diff --git a/Optlib/Algorithm/ADMM/Inv_bounded.lean b/Optlib/Algorithm/ADMM/Inv_bounded.lean
index 827677e..409fe50 100644
--- a/Optlib/Algorithm/ADMM/Inv_bounded.lean
+++ b/Optlib/Algorithm/ADMM/Inv_bounded.lean
@@ -14,21 +14,21 @@ variable {X Y:Type*}
 [NormedAddCommGroup Y] [InnerProductSpace ℝ Y]
 (A : X →L[ℝ] Y)(fullrank: Injective A)
 
-lemma KerA_bot (fullrank: Injective A): ker A = ⊥ := ker_eq_bot.2 fullrank
+lemma KerA_bot (fullrank: Injective A): A.toLinearMap.ker = ⊥ := LinearMap.ker_eq_bot.2 fullrank
 
 variable [CompleteSpace X] [CompleteSpace Y]
 
-lemma KerA_eq_KerA'A : ker A = ker (A†.comp A) := by
+lemma KerA_eq_KerA'A : A.toLinearMap.ker = (A†.comp A).toLinearMap.ker := by
    ext x; constructor; simp
    ·  intro h; rw[h]; continuity
    ·  intro h; simp at h
-      have : ((inner (A x) (A x)):ℝ) = (0:ℝ) := by
+      have : ((inner ℝ (A x) (A x)):ℝ) = (0:ℝ) := by
          calc
-            _ = (inner x ((A†) (A x)):ℝ) := by rw [ContinuousLinearMap.adjoint_inner_right]
+            _ = (inner ℝ x ((A†) (A x)):ℝ) := by rw [ContinuousLinearMap.adjoint_inner_right]
             _ = (0:ℝ) := by rw [h, inner_zero_right]
       apply inner_self_eq_zero.1 this
 
-lemma KerA'A_bot (fullrank: Injective A) : ker (A†.comp A) = ⊥ := by
+lemma KerA'A_bot (fullrank: Injective A) : (A†.comp A).toLinearMap.ker = ⊥ := by
    rw[← KerA_eq_KerA'A]
    apply KerA_bot A fullrank
 
diff --git a/Optlib/Algorithm/ADMM/Lemma.lean b/Optlib/Algorithm/ADMM/Lemma.lean
index e02d256..5f0c97c 100644
--- a/Optlib/Algorithm/ADMM/Lemma.lean
+++ b/Optlib/Algorithm/ADMM/Lemma.lean
@@ -61,6 +61,7 @@ local notation "y'" => admm_kkt.y
 local notation "A₁†" => ContinuousLinearMap.adjoint A₁
 local notation "A₂†" => ContinuousLinearMap.adjoint A₂
 local notation "⟪" a₁ ", " a₂ "⟫" => @inner ℝ _ _ a₁ a₂
+local notation "inner" => (inner ℝ)
 
 lemma Satisfaction_ofthekkt : Convex_KKT x₁' x₂' y' admm.toOptProblem := admm_kkt.h
 
@@ -189,7 +190,7 @@ lemma norm_covex1 [Setting E₁ E₂ F admm admm_kkt]:∀ n : ℕ+ ,
    let g := A₁
    have h2 : u ∘ g = f := by
       ext x
-      simp only [Function.comp_apply]
+      rfl
    rw[← h2]
    have h3 : ⇑g ⁻¹' univ = univ := by
       simp only [preimage_univ]
@@ -223,7 +224,7 @@ lemma norm_covex2 [Setting E₁ E₂ F admm admm_kkt]:∀ n : ℕ+ ,
    let g := A₂
    have h2 : u ∘ g = f := by
       ext x
-      simp only [Function.comp_apply]
+      rfl
    rw[← h2]
    have h3 : ⇑g ⁻¹' univ = univ := by
       simp only [preimage_univ]
@@ -280,10 +281,7 @@ lemma ADMM_iter_process₁'_eq3_2' [Setting E₁ E₂ F admm admm_kkt]: ∀ n :
             ((OptProblem.A₂ E₁) (ADMM.x₂ E₁ F n.natPred) - OptProblem.b E₁ E₂)
    rw[this]
    show HasGradientAt ((fun x => ⟪c , (A₁ x)⟫ + c₁)) (A₁† c) x
-   rw[hasGradientAt_iff_hasFDerivAt]
-   apply HasFDerivAt.add_const _ c₁
-   show HasGradientAt ((fun x => ⟪c , (A₁ x)⟫)) (A₁† c) x
-   apply ADMM_iter_process₁'_eq3_2'_1
+   exact (ADMM_iter_process₁'_eq3_2'_1 (c := c) x).add_const c₁
 
 lemma inner_continuous1 [Setting E₁ E₂ F admm admm_kkt]:∀ n : ℕ+ ,
       ContinuousOn (fun x => ⟪y n.natPred , (A₁ x) + (A₂ (x₂ n.natPred)) - b⟫) univ:= by
@@ -325,8 +323,7 @@ lemma Gradient_of_quadratic_forms { α β : Type*}
       simp[h]
       have := norm_nonneg (s - x)
       rwa[mul_nonneg_iff_right_nonneg_of_pos εpos]
-   ·  use ε / ‖A‖ ^ 2
-      field_simp
+   ·  refine ⟨ε / ‖A‖ ^ 2, div_pos εpos (sq_pos_of_pos h), ?_⟩
       intro x hx
       have hzero : 0 < ‖A‖ ^ 2 := by apply sq_pos_of_pos h
       let t := x - s
@@ -343,7 +340,7 @@ lemma Gradient_of_quadratic_forms { α β : Type*}
          rw[real_inner_smul_left,ContinuousLinearMap.adjoint_inner_left]
          ring
       rw[this,real_inner_self_eq_norm_sq]
-      simp only [abs_pow, abs_norm, ge_iff_le]
+      rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
       calc
          _ = ‖A (s - x)‖ ^ 2 := by
             rw[norm_comm]
@@ -367,7 +364,6 @@ lemma Gradient_of_quadratic_forms { α β : Type*}
                apply mul_le_mul_of_nonneg_left hx this
             _ = _ := by
                field_simp[hzero]
-               ring_nf
 
 #check add_sub
 lemma ADMM_iter_process₁'_eq3_3' [Setting E₁ E₂ F admm admm_kkt]: ∀ n : ℕ+ ,
@@ -558,10 +554,7 @@ lemma ADMM_iter_process₂'_eq3_2' [Setting E₁ E₂ F admm admm_kkt]: ∀ n :
       exact inner_add_right (y n.natPred) (A₂ x) (A₁ (x₁ n) - b)
    rw[this]
    show HasGradientAt (fun x => ⟪c , (A₂ x)⟫ + c₁) (A₂† c) x
-   rw[hasGradientAt_iff_hasFDerivAt]
-   apply HasFDerivAt.add_const _ c₁
-   show HasGradientAt ((fun x => ⟪c , (A₂ x)⟫)) (A₂† c) x
-   apply inner_gradient
+   exact (inner_gradient (A := A₂) (c := c) x).add_const c₁
 
 lemma inner_continuous2 [Setting E₁ E₂ F admm admm_kkt]:∀ n : ℕ+ ,
       ContinuousOn (fun x => ⟪y n.natPred , (A₁ (x₁ n)) + (A₂ x) - b⟫) univ:= by
@@ -854,7 +847,7 @@ lemma subgradientAt_mono_u [Setting E₁ E₂ F admm admm_kkt] : ∀ n : ℕ+,
       (0 : ℝ) ≤ (inner (u (n) + A₁† y') (x₁ (n) - x₁')) := by
    intro n
    calc
-      _= inner (u (n) - (- A₁† y')) (x₁ (n) - x₁') := by simp[v]
+      _= inner (u (n) - (- A₁† y')) (x₁ (n) - x₁') := by simp
       _≥ (0 : ℝ) := by
          apply subgradientAt_mono
          apply u_inthesubgradient
@@ -864,7 +857,7 @@ lemma subgradientAt_mono_v [Setting E₁ E₂ F admm admm_kkt]: ∀ n : ℕ+,
       (0 : ℝ) ≤ (inner (v (n) + A₂† y') (x₂ (n) - x₂')) := by
    intro n
    calc
-      _= inner (v (n) - (- A₂† y')) (x₂ (n) - x₂') := by simp[v]
+      _= inner (v (n) - (- A₂† y')) (x₂ (n) - x₂') := by simp
       _≥ (0 : ℝ) := by
          apply subgradientAt_mono
          apply v_inthesubgradient
@@ -883,7 +876,7 @@ lemma expended_u_gt_zero [Setting E₁ E₂ F admm admm_kkt]: ∀ n, (0 : ℝ) 
    let x_diff := x₁ (n + 1) - x₁'
    let succ_n := Nat.toPNat' (n + 1)
    calc
-      _= inner (𝕜 := ℝ) block Ae1 := by rfl
+      _= inner block Ae1 := by rfl
       _= inner (A₁† block) (e') := by rw [ContinuousLinearMap.adjoint_inner_left]
       _= inner (u' + A₁† y') (x_diff) := by
          let block₁ := y (n + 1) + ((1-τ) * ρ) • (A₁ (e₁ (n + 1)) + A₂ (e₂ (n + 1))) + (ρ • (A₂ (x₂ (n) - x₂ (n+1))))
@@ -954,7 +947,7 @@ lemma expended_u_v_gt_zero [Setting E₁ E₂ F admm admm_kkt]: ∀ n , (inner (
    let Ae2 := A₂ (e₂ (n + 1))
    calc
    _ = inner ey' (-(A_e_sum)) - (1 - τ) * ρ * (inner A_e_sum A_e_sum)
-      + ρ * (inner (A_x_sum) (Ae1)) := by rw [norm_sq_eq_inner (𝕜:=ℝ) (A_e_sum)];rfl
+      + ρ * (inner (A_x_sum) (Ae1)) := by rw [← real_inner_self_eq_norm_sq A_e_sum]
    _ = inner ey' (-(A_e_sum)) + inner (- ((1 - τ) * ρ) • A_e_sum) A_e_sum
       + ρ * (inner A_x_sum Ae1) := by rw [smul_left,starRingEnd_eq_R];ring
    _ = inner (-ey') A_e_sum + inner (- ((1 - τ) * ρ) • A_e_sum) A_e_sum
@@ -1244,7 +1237,7 @@ lemma Φ_isdescending_inequ5' [Setting E₁ E₂ F admm admm_kkt]: ∀ n : ℕ+,
       - 2 * (1-τ) * ρ * ‖A₁ (x₁ (n+1)) + A₂ (x₂ (n+1)) - b‖^2
       + 2 * M (n+1)
       - 1 * ρ * ((‖A₂ (x₂ (n+1) - x₂ n)‖^2 + ‖A₂ (e₂ (n+1))‖^2 - ‖A₂ (e₂ n)‖^2))
-      := by nth_rw 2 [div_eq_mul_inv]; rw [one_mul]; nth_rw 3 [pow_two]; simp [inv_mul_cancel]
+      := by nth_rw 2 [div_eq_mul_inv]; rw [one_mul]; nth_rw 3 [pow_two]; simp
             left; rw [mul_assoc]
             nth_rw 2 [← mul_assoc]
             nth_rw 2 [← mul_assoc]
@@ -1399,7 +1392,11 @@ lemma basic_inequ₂ (n : ℕ+) : - 2 * inner (A₂ (x₂ (n+1) - x₂ n)) (A₁
       apply Real.sqrt_ne_zero'.mpr
       rcases admm.htau with ⟨h₁, _⟩
       assumption
-   have h3 : inner (𝕜 := ℝ) S1 S2 = inner (𝕜 := ℝ) (s1 • S1) (s1⁻¹ • S2) := by rw [inner_smul_left, inner_smul_right]; rw [← mul_assoc]; simp; rw [mul_inv_cancel₀, one_mul]; exact this
+   have h3 : inner S1 S2 = inner (s1 • S1) (s1⁻¹ • S2) := by
+      rw [inner_smul_left, inner_smul_right, ← mul_assoc]
+      simp
+      rw [mul_inv_cancel₀, one_mul]
+      exact this
    rw [h1, h2, h3]
    have : ‖s1 • S1‖ ^ 2 + ‖s1⁻¹ • S2‖ ^ 2 - -2 * ⟪s1 • S1, s1⁻¹ • S2⟫_ℝ = ‖s1 • S1‖ ^ 2 + 2 * ⟪s1 • S1, s1⁻¹ • S2⟫_ℝ + ‖s1⁻¹ • S2‖ ^ 2 := by ring_nf
    rw [this, ←norm_add_sq_real]
@@ -1478,13 +1475,20 @@ lemma τ_min1_1 [Setting E₁ E₂ F admm admm_kkt] (h: 0 < τ ∧ τ ≤ 1) : m
    rcases h with ⟨h1, h2⟩
    apply min_eq_left
    have h3: τ ^ 2 ≤ 1 := by
-      apply pow_le_one;linarith;linarith
+      have hτ : |τ| ≤ 1 := by simpa [abs_of_nonneg (le_of_lt h1)] using h2
+      have hτ' : |τ| ≤ |(1 : ℝ)| := by simpa using hτ
+      have hsq : τ ^ 2 ≤ (1 : ℝ) ^ 2 := (sq_le_sq).2 hτ'
+      simpa using hsq
    linarith
 
 lemma τ_min1_2 [Setting E₁ E₂ F admm admm_kkt] (h: τ > 1 ) : min τ (1 + τ - τ ^ 2) = 1 + τ - τ ^ 2 := by
    apply min_eq_right
    have : 1 < τ ^ 2 := by
-      apply one_lt_pow;exact h;linarith
+      have hτ : 0 < τ := lt_trans zero_lt_one h
+      have hτabs : (1 : ℝ) < |τ| := by simpa [abs_of_pos hτ] using h
+      have hτabs' : |(1 : ℝ)| < |τ| := by simpa using hτabs
+      have hsq : (1 : ℝ) ^ 2 < τ ^ 2 := (sq_lt_sq).2 hτabs'
+      simpa using hsq
    linarith
 
 lemma τ_min2_1 [Setting E₁ E₂ F admm admm_kkt] (h: 0 < τ ∧ τ ≤ 1) : min 1 (1 + 1 / τ - τ ) = 1 := by
@@ -1503,7 +1507,9 @@ lemma τ_min2_2 [Setting E₁ E₂ F admm admm_kkt] (h: τ > 1 ) : min 1 (1 + 1
    calc
       _ > 1 := h
       _ > 1 / τ := by
-         rw [one_div, ← inv_one];apply inv_lt_inv_of_lt;linarith;exact h
+         have hτ : 0 < τ := lt_trans zero_lt_one h
+         have hdiv1 : 1 / τ < 1 := (div_lt_iff₀ hτ).2 (by simpa [one_mul] using h)
+         linarith [hdiv1]
    linarith
 
 lemma τ_min3_1 [Setting E₁ E₂ F admm admm_kkt] (h: 0 < τ ∧ τ ≤ 1) : max (1 - τ) (1 - 1 / τ) = 1 - τ := by
@@ -1522,7 +1528,9 @@ lemma τ_min3_2 [Setting E₁ E₂ F admm admm_kkt] (h: τ > 1) : max (1 - τ) (
    calc
       _ > 1 := h
       _ > 1 / τ := by
-         rw [one_div, ← inv_one];apply inv_lt_inv_of_lt;linarith;exact h
+         have hτ : 0 < τ := lt_trans zero_lt_one h
+         have hdiv1 : 1 / τ < 1 := (div_lt_iff₀ hτ).2 (by simpa [one_mul] using h)
+         linarith [hdiv1]
    linarith
 
 lemma Φ_isdescending [Setting E₁ E₂ F admm admm_kkt]: ∀ n : ℕ+, (Φ n ) - (Φ (n + 1) ) ≥ (min τ (1 + τ - τ ^ 2) )* ρ
diff --git a/Optlib/Algorithm/ADMM/Scheme.lean b/Optlib/Algorithm/ADMM/Scheme.lean
index 3eda488..953810b 100644
--- a/Optlib/Algorithm/ADMM/Scheme.lean
+++ b/Optlib/Algorithm/ADMM/Scheme.lean
@@ -35,7 +35,7 @@ def Admm_sub_Isunique {E : Type*}(f : E → ℝ)(x : E)(_h : IsMinOn f univ x):
 -- Augmented Lagrangian Function
 def Augmented_Lagrangian_Function (opt : OptProblem E₁ E₂ F) (ρ : ℝ) : E₁ × E₂ × F → ℝ :=
    fun (x₁ , x₂ , y) => (opt.f₁ x₁) + (opt.f₂ x₂) +
-      inner y  ((opt.A₁ x₁) + (opt.A₂ x₂) - opt.b) + ρ / 2 * ‖(opt.A₁ x₁) + (opt.A₂ x₂) - opt.b‖ ^ 2
+      inner ℝ y ((opt.A₁ x₁) + (opt.A₂ x₂) - opt.b) + ρ / 2 * ‖(opt.A₁ x₁) + (opt.A₂ x₂) - opt.b‖ ^ 2
 
 -- The basic iteration format of ADMM
 class ADMM extends (OptProblem E₁ E₂ F) where
diff --git a/Optlib/Algorithm/ADMM/Theroem_converge.lean b/Optlib/Algorithm/ADMM/Theroem_converge.lean
index d432b56..3d2be26 100644
--- a/Optlib/Algorithm/ADMM/Theroem_converge.lean
+++ b/Optlib/Algorithm/ADMM/Theroem_converge.lean
@@ -88,9 +88,9 @@ lemma nonneg₁ [Setting E₁ E₂ F admm admm_kkt]: min τ (1 + τ - τ ^ 2) >
 lemma nonneg₂ [Setting E₁ E₂ F admm admm_kkt]: min 1 (1 + 1 / τ - τ) > 0 := by
    rcases admm.htau with ⟨h1, _⟩
    have h2: 1 + 1/τ - τ > 0 := by
-      field_simp;rw [← sq]
-      have h3 : 1 + τ - τ ^ 2 > 0 := nonneg_prime
-      linarith
+      have hτ : τ ≠ 0 := ne_of_gt h1
+      rw [show 1 + 1 / τ - τ = (1 + τ - τ ^ 2) / τ by field_simp [hτ]; ring]
+      exact div_pos nonneg_prime h1
    apply lt_min one_pos h2
 
 lemma Φ₁_nonneg [Setting E₁ E₂ F admm admm_kkt]:
@@ -165,21 +165,23 @@ lemma Φ_is_nonneg [Setting E₁ E₂ F admm admm_kkt]: ∀ n : ℕ , Φ n ≥ 0
 
 lemma Φ_bd_above [Setting E₁ E₂ F admm admm_kkt]: ∃ C : ℝ, ∀ n : ℕ, Φ n < C := by
    let C := Max.max ((Φ 0) + 1) ((Φ 1) + 1); use C; intro n
-   induction' n with k h
-   ·  have : Φ 0 < (Φ 0) + 1 := by linarith
+   induction n with
+   | zero =>
+      have : Φ 0 < (Φ 0) + 1 := by linarith
       apply lt_max_iff.2
       left; exact this
-   ·  by_cases hh : k = 0
-      ·  rw [hh,zero_add]
-         apply lt_max_iff.2
-         right; linarith
-      ·  push_neg at hh
-         have k_pos : k > 0 := by apply Nat.pos_of_ne_zero hh
-         have : (Φ) (k.toPNat k_pos) ≥ (Φ) ((k.toPNat k_pos ) + 1) := by
-            apply Φ_is_monotone
-         have h' : Φ (k.toPNat k_pos) < C := by apply h
-         show Φ ((k.toPNat k_pos) + 1) < C
-         linarith
+   | succ k h =>
+      by_cases hh : k = 0
+      · rw [hh,zero_add]
+        apply lt_max_iff.2
+        right; linarith
+      · push_neg at hh
+        have k_pos : k > 0 := by exact Nat.pos_of_ne_zero hh
+        have : (Φ) (k.toPNat k_pos) ≥ (Φ) ((k.toPNat k_pos) + 1) := by
+          apply Φ_is_monotone
+        have h' : Φ (k.toPNat k_pos) < C := by simpa using h
+        show Φ ((k.toPNat k_pos) + 1) < C
+        linarith
 
 lemma Φ_isBounded' [Setting E₁ E₂ F admm admm_kkt] : ∃ (r : ℝ), (range Φ) ⊆ ball 0 r := by
    rcases Φ_bd_above with ⟨C,bd⟩
@@ -317,9 +319,7 @@ lemma A₂e₂_isBounded' [Setting E₁ E₂ F admm admm_kkt]: ∃ (r : ℝ), (r
             exact h3; simp
 
    have h6: dist (A₂ (e₂ n)) 0 < √ (r_Φ / ρ) := by
-      rw[← sub_zero (A₂ (e₂ n))] at h5
-      rw[SeminormedAddGroup.dist_eq (A₂ (e₂ n)) 0]
-      exact h5
+      simpa [dist_eq_norm] using h5
 
    rw [← hr] at h6
    rw [← Metric.mem_ball] at h6
@@ -434,9 +434,7 @@ lemma A₁e₁_A₂e₂_isBounded'[Setting E₁ E₂ F admm admm_kkt] : ∃ (r :
 
    have h6: dist (A₁ (e₁ n) + A₂ (e₂ n)) 0 < r := by
       have h_n' := h_n n
-      rw[← sub_zero (A₁ (e₁ n) + A₂ (e₂ n))] at h_n'
-      rw[SeminormedAddGroup.dist_eq (A₁ (e₁ n) + A₂ (e₂ n)) 0]
-      exact h_n'
+      simpa [dist_eq_norm] using h_n'
 
    rw [← Metric.mem_ball] at h6; simp; simp at h6
    exact h6
@@ -481,9 +479,7 @@ lemma A₁e₁_isBounded' [Setting E₁ E₂ F admm admm_kkt]: ∃ (r : ℝ), ra
          _ = r := hr
 
    have h_dist : dist (A₁ (e₁ n)) 0 < r := by
-      rw[← sub_zero (A₁ (e₁ n))] at h_norm
-      rw[SeminormedAddGroup.dist_eq (A₁ (e₁ n)) 0]
-      exact h_norm
+      simpa [dist_eq_norm] using h_norm
 
    rw [← Metric.mem_ball] at h_dist
    apply h_dist
@@ -679,7 +675,8 @@ lemma Φ_Summable₁' [Setting E₁ E₂ F admm admm_kkt] :
    intro n
    let φ₀ := (fun i : ℕ => Φ i.succ)
    have : ∀ i ∈ Finset.range n , (φ₀ i)-(φ₀ (i+1)) = (Φ i.succ ) - (Φ (i.succ + 1)) := by
-      simp only [Finset.mem_range, Nat.succ_eq_add_one, implies_true]
+      intro i hi
+      rfl
    have h : Finset.range n =Finset.range n := rfl
    rw[← Finset.sum_congr h this , Finset.sum_range_sub']
    simp only [φ₀]
@@ -701,16 +698,7 @@ lemma Φ_isSummable [Setting E₁ E₂ F admm admm_kkt] : Summable (fun n : ℕ
 theorem summable_of_nonneg_of_le {β : Type*} {f : β → ℝ} {g : β → ℝ}
 (hg : ∀ (n : β), 0 ≤ g n) (hgf : ∀ (n : β), g n ≤ f n) (hf : Summable f) :
 Summable g:=by
-  rw[← NNReal.summable_mk]
-  have f_ge_zero :∀ (n : β), 0 ≤ f n := by
-   intro n
-   apply le_trans (hg n) (hgf n)
-  have :∀ (n : β), (⟨g n, hg n⟩ : NNReal) ≤ ⟨f n , f_ge_zero n⟩ := by
-   simp only [Subtype.mk_le_mk]
-   apply hgf
-  apply NNReal.summable_of_le this
-  rw[← NNReal.summable_coe]
-  exact hf
+  exact Summable.of_nonneg_of_le hg hgf hf
 
 lemma Φ_inequ₁ [Setting E₁ E₂ F admm admm_kkt] (m : ℕ+):
       (min 1 (1 + 1 / τ - τ )) * ρ * ‖A₁ (e₁ (m+1)) + A₂ (e₂ (m+1))‖ ^ 2 ≤ Φ m - Φ (m + 1) := by
diff --git a/Optlib/Algorithm/BCD/Convergence.lean b/Optlib/Algorithm/BCD/Convergence.lean
index 35be560..a3fc4fa 100644
--- a/Optlib/Algorithm/BCD/Convergence.lean
+++ b/Optlib/Algorithm/BCD/Convergence.lean
@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chenyi Li, Bowen Yang, Yifan Bai
 -/
 import Optlib.Algorithm.BCD.Scheme
+import Mathlib.Tactic
 
 /-!
 # Block Coordinate Descent
@@ -36,16 +37,16 @@ section block_subdifferential
 
 variable {E : Type*} [NormedAddCommGroup E]
 
-lemma infEdist_bound {s : Set E} : ∀ x ∈ s, ENNReal.ofReal ‖x‖ ≥ EMetric.infEdist 0 s := by
+lemma infEdist_bound {s : Set E} : ∀ x ∈ s, ENNReal.ofReal ‖x‖ ≥ Metric.infEDist 0 s := by
   by_cases hs : s = ∅
   simp [hs]
   push_neg at hs
   intro x xs
-  have : EMetric.infEdist 0 s ≤ edist 0 x := EMetric.infEdist_le_edist_of_mem xs
+  have : Metric.infEDist 0 s ≤ edist 0 x := Metric.infEDist_le_edist_of_mem xs
   rw [← dist_zero_left]
   apply (ENNReal.le_ofReal_iff_toReal_le _ _).2
   · exact ENNReal.toReal_le_of_le_ofReal dist_nonneg (edist_dist 0 x ▸ this)
-  · exact Metric.infEdist_ne_top hs
+  · exact Metric.infEDist_ne_top hs
   · simp
 
 variable {F: Type*} [InnerProductSpace ℝ E]
@@ -53,17 +54,28 @@ variable [NormedAddCommGroup F] [InnerProductSpace ℝ F]
 variable {f : E → ℝ} {g : F → ℝ} {x u : E} {y v : F}
 
 lemma f_subdiff_block (hf : u ∈ f_subdifferential f x) (hg : v ∈ f_subdifferential g y) :
-    ⟨u, v⟩ ∈ f_subdifferential (fun z ↦ f z.1 + g z.2 : WithLp 2 (E × F) → ℝ) ⟨x, y⟩ := by
+    ((u, v) : WithLp 2 (E × F)) ∈
+      f_subdifferential (fun z ↦ f z.fst + g z.snd : WithLp 2 (E × F) → ℝ)
+        ((x, y) : WithLp 2 (E × F)) := by
   rw [has_f_subdiff_iff] at *
   intro ε εpos
   have ε2pos : 0 < ε / 2 := by positivity
-  filter_upwards [Eventually.prod_nhds (hf _ ε2pos) (hg _ ε2pos)] with z ⟨hfz, hyz⟩
+  have hfx : ∀ᶠ z : WithLp 2 (E × F) in 𝓝 ((x, y) : WithLp 2 (E × F)),
+      f z.fst - f x - inner ℝ u (z.fst - x) ≥ -(ε / 2) * ‖z.fst - x‖ :=
+    Filter.Tendsto.eventually
+      ((WithLp.continuous_fst (p := (2 : ENNReal)) (α := E) (β := F)).tendsto
+        ((x, y) : WithLp 2 (E × F))) (hf _ ε2pos)
+  have hgy : ∀ᶠ z : WithLp 2 (E × F) in 𝓝 ((x, y) : WithLp 2 (E × F)),
+      g z.snd - g y - inner ℝ v (z.snd - y) ≥ -(ε / 2) * ‖z.snd - y‖ :=
+    Filter.Tendsto.eventually
+      ((WithLp.continuous_snd (p := (2 : ENNReal)) (α := E) (β := F)).tendsto
+        ((x, y) : WithLp 2 (E × F))) (hg _ ε2pos)
+  filter_upwards [hfx, hgy] with z hfz hyz
   rw [WithLp.prod_inner_apply]
-  simp only [WithLp.sub_fst, WithLp.sub_snd]
   let z' : WithLp 2 (E × F) := (x, y)
-  show f z.1 + g z.2 - (f x + g y) - (⟪u, z.1 - x⟫ + ⟪v, z.2 - y⟫) ≥ -ε * ‖z - z'‖
-  have h1 : ‖z.1 - x‖ ≤ ‖z - z'‖ := fst_norm_le_prod_L2 (z - z')
-  have h2 : ‖z.2 - y‖ ≤ ‖z - z'‖ := snd_norm_le_prod_L2 (z - z')
+  show f z.fst + g z.snd - (f x + g y) - (⟪u, z.fst - x⟫ + ⟪v, z.snd - y⟫) ≥ -ε * ‖z - z'‖
+  have h1 : ‖z.fst - x‖ ≤ ‖z - z'‖ := fst_norm_le_prod_L2 (z - z')
+  have h2 : ‖z.snd - y‖ ≤ ‖z - z'‖ := snd_norm_le_prod_L2 (z - z')
   linarith [(mul_le_mul_iff_of_pos_left ε2pos).mpr h1, (mul_le_mul_iff_of_pos_left ε2pos).mpr h2]
 
 end block_subdifferential
@@ -91,16 +103,19 @@ theorem PALM_Descent (h : E → ℝ) {h' : E → E} (Lₕ : NNReal)
   rw [this] at u₁prox
   have : u₁ - (u - t • h' u) = (u₁ - u) + t • h' u := by abel
   rw [this] at u₁prox
-  simp [norm_add_sq_real, this] at u₁prox
+  simp [norm_add_sq_real] at u₁prox
   have ha : t * σ u₁ + ‖u₁ - u‖ ^ 2 / 2 +  ⟪u₁ - u, t • h' u⟫ ≤ t * σ u := by linarith [u₁prox]
   rw [inner_smul_right] at ha
-  have : t * (‖u₁ - u‖ ^ 2 / (2 * t)) = ‖u₁ - u‖ ^ 2 / 2 := by field_simp; ring
+  have : t * (‖u₁ - u‖ ^ 2 / (2 * t)) = ‖u₁ - u‖ ^ 2 / 2 := by
+    field_simp [ne_of_gt h₅]
   rw [← this] at ha
   have : t * σ u₁ + t * (‖u₁ - u‖ ^ 2 / (2 * t)) + t * ⟪u₁ - u, h' u⟫
         = t * (σ u₁ + ‖u₁ - u‖ ^ 2 / (2 * t) + ⟪u₁ - u, h' u⟫) := by ring
   rw [this] at ha
   have hne : ⟪u₁ - u, h' u⟫ ≤ σ u - σ u₁ - ‖u₁ - u‖ ^ 2 / (2 * t) := by
-    linarith [(mul_le_mul_left h₅).1 ha]
+    have hmul : σ u₁ + ‖u₁ - u‖ ^ 2 / (2 * t) + ⟪u₁ - u, h' u⟫ ≤ σ u := by
+      exact le_of_mul_le_mul_left ha h₅
+    linarith [hmul]
   rw [real_inner_comm] at hne
   calc
     _ ≤ h u + σ u - σ u₁ - ‖u₁ - u‖ ^ 2 / (2 * t) + ↑Lₕ / 2 * ‖u₁ - u‖ ^ 2 + σ u₁ := by
@@ -121,26 +136,64 @@ theorem Sufficient_Descent1 (γ : ℝ) (hγ : γ > 1)
   intro k
   have hHf : H (alg.x (k + 1), alg.y k) + f (alg.x (k + 1)) ≤ H (alg.x k, alg.y k) + f (alg.x k)
       - 1 / 2 * (γ - 1) * l * ‖alg.x (k + 1) - alg.x k‖ ^ 2 := by
-    let h  := fun x ↦ H (x,alg.y k)
+    let h  := fun x ↦ H (WithLp.toLp 2 (x, alg.y k))
     let h' := fun x ↦ grad_fst H (alg.y k) x
     have h1 : ∀ x₁ : E, HasGradientAt h (h' x₁) x₁ := by
       intro x
-      apply DifferentiableAt.hasGradientAt
-      apply diff_prod₁; apply ContDiff.differentiable alg.conf (by simp)
+      rw [hasGradientAt_iff_hasFDerivAt]
+      have hH' : HasFDerivAt H ((InnerProductSpace.toDual ℝ (WithLp 2 (E × F)))
+          (gradient H (WithLp.toLp 2 (x, alg.y k)))) (WithLp.toLp 2 (x, alg.y k)) := by
+        exact (DifferentiableAt.hasGradientAt (h := (alg.Hdiff (WithLp.toLp 2 (x, alg.y k))))).hasFDerivAt
+      have htoLp : HasFDerivAt (WithLp.toLp 2)
+          (IsBoundedLinearMap.toContinuousLinearMap (WithLp.toLp 2) instIsBoundedLinearMapL2equiv)
+          ((x, alg.y k) : E × F) := instIsBoundedLinearMapL2equiv.hasFDerivAt
+      have hpair : HasFDerivAt (fun x : E => ((x, alg.y k) : E × F)) (ContinuousLinearMap.inl ℝ E F) x := by
+        simpa using (hasFDerivAt_prodMk_left (𝕜 := ℝ) x (alg.y k))
+      have hcomp : HasFDerivAt (fun x : E => H (WithLp.toLp 2 (x, alg.y k)))
+          ((((InnerProductSpace.toDual ℝ (WithLp 2 (E × F))) (gradient H (WithLp.toLp 2 (x, alg.y k)))).comp
+            (IsBoundedLinearMap.toContinuousLinearMap (WithLp.toLp 2) instIsBoundedLinearMapL2equiv)).comp
+            (ContinuousLinearMap.inl ℝ E F)) x := by
+        simpa [Function.comp] using (hH'.comp x (htoLp.comp x hpair))
+      refine hcomp.congr_fderiv ?_
+      ext u
+      have hfst : ((IsBoundedLinearMap.toContinuousLinearMap (WithLp.toLp 2) instIsBoundedLinearMapL2equiv)
+          ((u, (0 : F)) : E × F)).fst = u := rfl
+      have hsnd : ((IsBoundedLinearMap.toContinuousLinearMap (WithLp.toLp 2) instIsBoundedLinearMapL2equiv)
+          ((u, (0 : F)) : E × F)).snd = 0 := rfl
+      simp [h', grad_fst, ContinuousLinearMap.comp_apply, hfst, hsnd]
     obtain prop := PALM_Descent h l h1 (l1 _) f (alg.c k) (alg.cpos γ hγ ck k) (alg.x _) (alg.x _)
-    apply le_of_eq_of_le' _ (prop (by rw [prox_set]; simp; exact (alg.s₁ k)))
+    apply le_of_le_of_eq (prop (by rw [prox_set]; simp; exact (alg.s₁ k)))
     rw [ck, one_div_one_div]; ring
 
   have hHg : H (alg.x (k + 1), alg.y (k + 1)) + g (alg.y (k + 1)) ≤ H (alg.x (k + 1), alg.y k)
       + g (alg.y k) - 1 / 2 * (γ - 1) * l * ‖alg.y (k + 1) - alg.y k‖ ^ 2 := by
-    let h := fun y ↦ H (alg.x (k + 1), y)
+    let h := fun y ↦ H (WithLp.toLp 2 (alg.x (k + 1), y))
     let h':= fun y ↦ grad_snd H (alg.x (k + 1)) y
     have h1 : ∀ y₁ : F, HasGradientAt h (h' y₁) y₁ := by
       intro y
-      apply DifferentiableAt.hasGradientAt
-      apply diff_prod₂; apply ContDiff.differentiable alg.conf (by simp)
+      rw [hasGradientAt_iff_hasFDerivAt]
+      have hH' : HasFDerivAt H ((InnerProductSpace.toDual ℝ (WithLp 2 (E × F)))
+          (gradient H (WithLp.toLp 2 (alg.x (k + 1), y)))) (WithLp.toLp 2 (alg.x (k + 1), y)) := by
+        exact (DifferentiableAt.hasGradientAt (h := (alg.Hdiff (WithLp.toLp 2 (alg.x (k + 1), y))))).hasFDerivAt
+      have htoLp : HasFDerivAt (WithLp.toLp 2)
+          (IsBoundedLinearMap.toContinuousLinearMap (WithLp.toLp 2) instIsBoundedLinearMapL2equiv)
+          ((alg.x (k + 1), y) : E × F) := instIsBoundedLinearMapL2equiv.hasFDerivAt
+      have hpair : HasFDerivAt (fun y : F => ((alg.x (k + 1), y) : E × F)) (ContinuousLinearMap.inr ℝ E F) y := by
+        simpa using (hasFDerivAt_prodMk_right (𝕜 := ℝ) (alg.x (k + 1)) y)
+      have hcomp : HasFDerivAt (fun y : F => H (WithLp.toLp 2 (alg.x (k + 1), y)))
+          ((((InnerProductSpace.toDual ℝ (WithLp 2 (E × F))) (gradient H (WithLp.toLp 2 (alg.x (k + 1), y)))).comp
+            (IsBoundedLinearMap.toContinuousLinearMap (WithLp.toLp 2) instIsBoundedLinearMapL2equiv)).comp
+            (ContinuousLinearMap.inr ℝ E F)) y := by
+        simpa [Function.comp] using (hH'.comp y (htoLp.comp y hpair))
+      refine hcomp.congr_fderiv ?_
+      ext v
+      have hfst : ((IsBoundedLinearMap.toContinuousLinearMap (WithLp.toLp 2) instIsBoundedLinearMapL2equiv)
+          (((0 : E), v) : E × F)).fst = 0 := rfl
+      have hsnd : ((IsBoundedLinearMap.toContinuousLinearMap (WithLp.toLp 2) instIsBoundedLinearMapL2equiv)
+          (((0 : E), v) : E × F)).snd = v := rfl
+      simp [h', grad_snd, ContinuousLinearMap.comp_apply, hfst, hsnd]
     obtain prop := PALM_Descent h l h1 (l2 _) g (alg.d k) (alg.dpos γ hγ dk k) (alg.y k) (alg.y _)
-    apply le_of_eq_of_le' _ (prop (by rw [prox_set]; simp; exact (alg.s₂ k)))
+    apply le_of_le_of_eq (prop (by rw [prox_set]; simp; exact (alg.s₂ k)))
     rw [dk, one_div_one_div]; ring
 
   have eq (k : ℕ) : alg.ψ (alg.z k) = H (alg.x k, alg.y k) + f (alg.x k) + g (alg.y k) := by
@@ -152,11 +205,13 @@ theorem Sufficient_Descent1 (γ : ℝ) (hγ : γ > 1)
     _ ≥ 1 / 2 * (γ - 1) * l * (‖alg.x (k + 1) - alg.x k‖ ^ 2
         + ‖alg.y (k + 1) - alg.y k‖ ^ 2) := by linarith [hHf,hHg]
     _ = 1 / 2 * ρ₁ * (‖alg.x (k + 1) - alg.x k‖ ^ 2 + ‖alg.y (k + 1) - alg.y k‖ ^ 2) := by
-      unfold ρ₁; nth_rw 2 [mul_assoc]
+      unfold ρ₁; ring
     _ = _ := by
-      simp only [WithLp.prod_norm_sq_eq_of_L2]
-      rw [Prod.fst_sub, Prod.snd_sub, BCD.z, BCD.z]
-      ring_nf; simp
+      rw [WithLp.prod_norm_sq_eq_of_L2]
+      simp [BCD.z]
+      left
+      unfold ρ₁
+      ring
 
 /- the value is monotone -/
 theorem Sufficient_Descent2 (γ : ℝ) (hγ : γ > 1)
@@ -177,19 +232,21 @@ theorem Sufficient_Descent3 (γ : ℝ) (hγ : γ > 1) (ck: ∀ k, alg.c k = 1 /
       ≤ 2 / ρ₁ * (alg.ψ (alg.z k) - alg.ψ (alg.z (k + 1))):= by
     intro k; specialize h2 k
     obtain h1 := mul_le_mul_of_nonneg_left h2 (a := 2 / ρ₁) (by positivity)
-    rw [← mul_assoc] at h1; field_simp at h1; field_simp; exact h1
+    rw [← mul_assoc] at h1; field_simp at h1; field_simp; simpa [mul_comm] using h1
   have hne : ∀ n, ∑ k ∈ Finset.range (n + 1), ‖alg.z (k + 1) - alg.z k‖ ^ 2
       ≤ 2 / ρ₁ * ((alg.ψ (alg.z 0)) - (alg.ψ (alg.z (n + 1)))) := by
     intro n
-    induction' n with d hd
-    simp; specialize hDescent' 0
-    simp [hDescent']
-    rw [Finset.sum_range_succ _ (d + 1)]
-    have : 2 / ρ₁ * (alg.ψ (alg.z 0) - alg.ψ (alg.z (d + 1 + 1)))
-          =  2 / ρ₁ * (alg.ψ (alg.z 0) - alg.ψ (alg.z (d + 1)))
-          + 2 / ρ₁ * (alg.ψ (alg.z (d + 1)) - alg.ψ (alg.z (d + 1 + 1))) := by ring
-    rw [this]
-    apply add_le_add hd (hDescent' (d + 1))
+    induction n with
+    | zero =>
+        simp; specialize hDescent' 0
+        simpa using hDescent'
+    | succ d hd =>
+        rw [Finset.sum_range_succ _ (d + 1)]
+        have : 2 / ρ₁ * (alg.ψ (alg.z 0) - alg.ψ (alg.z (d + 1 + 1)))
+            = 2 / ρ₁ * (alg.ψ (alg.z 0) - alg.ψ (alg.z (d + 1)))
+            + 2 / ρ₁ * (alg.ψ (alg.z (d + 1)) - alg.ψ (alg.z (d + 1 + 1))) := by ring
+        rw [this]
+        exact add_le_add hd (hDescent' (d + 1))
   simp [BddBelow,lowerBounds,Set.Nonempty] at lbdψ
   rcases lbdψ with ⟨ψ₀,hψ₀⟩
   obtain hne' := fun n ↦ le_trans (hne n) (mul_le_mul_of_nonneg_left
@@ -252,6 +309,7 @@ section Upperbound_subd
 
 variable {c : ℝ} {f' : E → ℝ} {x u u' : E} {y v : F}
 
+set_option maxHeartbeats 800000 in
 theorem Ψ_subdiff_bound (γ : ℝ) (hγ : γ > 1)
     (ck: ∀ k, alg.c k = 1 / (γ * l)) (dk: ∀ k, alg.d k = 1 / (γ * l)) :
     ∃ ρ > 0, ∀ k, ∃ dΨ ∈ f_subdifferential alg.ψ (alg.z (k + 1)),
@@ -273,29 +331,36 @@ theorem Ψ_subdiff_bound (γ : ℝ) (hγ : γ > 1)
   rw [lipschitzWith_iff_norm_sub_le] at lip
   have cpos' : (alg.c k)⁻¹ ≥ 0 := by simp; apply le_of_lt (alg.cpos γ hγ ck k)
   have dpos' : (alg.d k)⁻¹ ≥ 0 := by simp; apply le_of_lt (alg.dpos γ hγ dk k)
-  have h1 : ‖(alg.subdiff k).1‖ ≤ l * (γ + 1) * ‖alg.z (k + 1) - alg.z k‖ := by
-    simp only [BCD.subdiff, BCD.A_kx, Prod.fst_add, grad_fun_comp, grad_comp, sub_add];
-    rw [A_k, A_kx, A_ky]; simp
+  have h1 : ‖(alg.subdiff k).fst‖ ≤ l * (γ + 1) * ‖alg.z (k + 1) - alg.z k‖ := by
     let a := (alg.c k)⁻¹ • (alg.x k - alg.x (k + 1))
     calc
-      _ = ‖a + (gradient H (alg.x (k + 1), alg.y (k + 1))).1
-          - grad_fst H (alg.y k) (alg.x k)‖ := by rw [sub_add_eq_add_sub]
-      _ = ‖a + (gradient H (alg.x (k + 1), alg.y (k + 1))).1
-            - (gradient H (alg.x k, alg.y k)).1‖ := by
-        symm; rw [grad_eq_block_grad, grad_fun_comp, grad_comp, grad_fun_comp, grad_comp]
-        simp; apply alg.Hdiff
-      _ ≤ ‖a‖ + ‖(gradient H (alg.x (k + 1), alg.y (k + 1)) - gradient H (alg.x k, alg.y k)).1‖ := by
-        rw [add_sub_assoc, ← Prod.fst_sub]; apply norm_add_le
-      _ ≤ ‖a‖ + ‖(gradient H (alg.x (k + 1), alg.y (k + 1)) - gradient H (alg.x k, alg.y k))‖ := by
-        simp; rw [← Prod.fst_sub]; apply fst_norm_le_prod_L2
+      ‖(alg.subdiff k).fst‖
+          = ‖a + (gradient H (alg.x (k + 1), alg.y (k + 1))).fst
+              - grad_fst H (alg.y k) (alg.x k)‖ := by
+            simp [BCD.subdiff, BCD.A_k, BCD.A_kx, BCD.A_ky, a, sub_eq_add_neg, add_left_comm, add_comm]
+      _ = ‖a + (gradient H (alg.x (k + 1), alg.y (k + 1))).fst
+            - (gradient H (alg.x k, alg.y k)).fst‖ := by
+            simp [grad_fst]
+      _ ≤ ‖a‖ + ‖(gradient H (alg.x (k + 1), alg.y (k + 1))).fst
+            - (gradient H (alg.x k, alg.y k)).fst‖ := by
+            simpa [sub_eq_add_neg, add_assoc] using
+              (norm_add_le a ((gradient H (alg.x (k + 1), alg.y (k + 1))).fst - (gradient H (alg.x k, alg.y k)).fst))
+      _ ≤ ‖a‖ + ‖gradient H (alg.x (k + 1), alg.y (k + 1)) - gradient H (alg.x k, alg.y k)‖ := by
+            have hfst :
+                ‖(gradient H (alg.x (k + 1), alg.y (k + 1))).fst
+                  - (gradient H (alg.x k, alg.y k)).fst‖
+                ≤ ‖gradient H (alg.x (k + 1), alg.y (k + 1)) - gradient H (alg.x k, alg.y k)‖ := by
+              simpa using (fst_norm_le_prod_L2
+                (gradient H (alg.x (k + 1), alg.y (k + 1)) - gradient H (alg.x k, alg.y k)))
+            simpa [add_comm, add_left_comm, add_assoc] using add_le_add_left hfst ‖a‖
     have inequ₁ : ‖a‖ ≤ (γ * l) * ‖alg.z (k+1) - alg.z k‖ := by
       calc
         _ = (1 / alg.c k) * ‖alg.x k - alg.x (k + 1)‖ := by
           simp [a]; rw [norm_smul_of_nonneg]; apply cpos'
         _ = (1 / alg.c k) * ‖alg.x (k + 1) - alg.x k‖ := by simp; left; apply norm_sub_rev
-        _ = (1 / alg.c k) * ‖(alg.z (k + 1) - alg.z k).1‖ := by rw [z]; simp; left; rw [z]; simp
+        _ = (1 / alg.c k) * ‖(alg.z (k + 1) - alg.z k).fst‖ := by rw [z]; simp; left; rw [z]; simp
         _ ≤ (1 / alg.c k) * ‖alg.z (k + 1) - alg.z k‖ := by
-          have : ‖(alg.z (k + 1) - alg.z k).1‖ ≤ ‖alg.z (k + 1) - alg.z k‖ := fst_norm_le_prod_L2 _
+          have : ‖(alg.z (k + 1) - alg.z k).fst‖ ≤ ‖alg.z (k + 1) - alg.z k‖ := fst_norm_le_prod_L2 _
           simp; apply mul_le_mul_of_nonneg_left this cpos'
         _ = (γ * l) * ‖alg.z (k + 1) - alg.z k‖ := by rw [ck k]; simp
     have inequ₂ : ‖gradient H (alg.x (k + 1), alg.y (k + 1)) - gradient H (alg.x k, alg.y k)‖
@@ -306,32 +371,36 @@ theorem Ψ_subdiff_bound (γ : ℝ) (hγ : γ > 1)
           apply lip
         _ = l * ‖alg.z (k+1) - alg.z k‖ := by repeat rw [z]; simp; left; rfl
     linarith
-  have h2 : ‖(alg.subdiff k).2‖ ≤ l * (γ + 1) * ‖alg.z (k + 1) - alg.z k‖ := by
-    simp only [BCD.subdiff, BCD.A_kx, Prod.fst_add, grad_fun_comp, grad_comp, sub_add];
-    rw [A_k, A_kx, A_ky]; simp
+  have h2 : ‖(alg.subdiff k).snd‖ ≤ l * (γ + 1) * ‖alg.z (k + 1) - alg.z k‖ := by
     let a := (alg.d k)⁻¹ • (alg.y k - alg.y (k + 1))
     calc
-      _ = ‖a + (gradient H (alg.x (k + 1), alg.y (k + 1))).2
-          - grad_snd H (alg.x (k + 1)) (alg.y k)‖ := by rw [sub_add_eq_add_sub]
-      _ = ‖a + (gradient H (alg.x (k + 1), alg.y (k + 1))).2
-          - (gradient H (alg.x (k + 1), alg.y k)).2‖ := by
-        symm; rw [grad_eq_block_grad, grad_fun_comp, grad_comp, grad_fun_comp, grad_comp]
-        simp; apply alg.Hdiff
-      _ = ‖a + (gradient H (alg.x (k + 1), alg.y (k + 1))
-          - gradient H (alg.x (k + 1), alg.y k)).2‖ := by rw [add_sub_assoc, ← Prod.snd_sub]
-      _ ≤ ‖a‖ + ‖(gradient H (alg.x (k + 1), alg.y (k + 1))
-          - gradient H (alg.x (k + 1), alg.y k)).2‖ := by apply norm_add_le
-      _ ≤ ‖a‖ + ‖(gradient H (alg.x (k + 1), alg.y (k + 1))
-          - gradient H (alg.x (k + 1), alg.y k))‖ := by
-            simp; rw [← Prod.snd_sub]; apply snd_norm_le_prod_L2
+      ‖(alg.subdiff k).snd‖
+          = ‖a + (gradient H (alg.x (k + 1), alg.y (k + 1))).snd
+              - grad_snd H (alg.x (k + 1)) (alg.y k)‖ := by
+            simp [BCD.subdiff, BCD.A_k, BCD.A_kx, BCD.A_ky, a, sub_eq_add_neg, add_left_comm, add_comm]
+      _ = ‖a + (gradient H (alg.x (k + 1), alg.y (k + 1))).snd
+            - (gradient H (alg.x (k + 1), alg.y k)).snd‖ := by
+            simp [grad_snd]
+      _ ≤ ‖a‖ + ‖(gradient H (alg.x (k + 1), alg.y (k + 1))).snd
+            - (gradient H (alg.x (k + 1), alg.y k)).snd‖ := by
+            simpa [sub_eq_add_neg, add_assoc] using
+              (norm_add_le a ((gradient H (alg.x (k + 1), alg.y (k + 1))).snd - (gradient H (alg.x (k + 1), alg.y k)).snd))
+      _ ≤ ‖a‖ + ‖gradient H (alg.x (k + 1), alg.y (k + 1)) - gradient H (alg.x (k + 1), alg.y k)‖ := by
+            have hsnd :
+                ‖(gradient H (alg.x (k + 1), alg.y (k + 1))).snd
+                  - (gradient H (alg.x (k + 1), alg.y k)).snd‖
+                ≤ ‖gradient H (alg.x (k + 1), alg.y (k + 1)) - gradient H (alg.x (k + 1), alg.y k)‖ := by
+              simpa using (snd_norm_le_prod_L2
+                (gradient H (alg.x (k + 1), alg.y (k + 1)) - gradient H (alg.x (k + 1), alg.y k)))
+            simpa [add_comm, add_left_comm, add_assoc] using add_le_add_left hsnd ‖a‖
     have inequ₁ : ‖a‖ ≤ (γ * l) * ‖alg.z (k + 1) - alg.z k‖ := by
       calc
         _ = (1 / alg.d k) * ‖alg.y k - alg.y (k + 1)‖ := by
           simp [a]; rw [norm_smul_of_nonneg]; apply dpos'
         _ = (1 / alg.d k) * ‖alg.y (k + 1) - alg.y k‖ := by simp; left; apply norm_sub_rev
-        _ = (1 / alg.d k) * ‖(alg.z (k + 1) - alg.z k).2‖ := by rw [z]; simp; left; rw [z]; simp
+        _ = (1 / alg.d k) * ‖(alg.z (k + 1) - alg.z k).snd‖ := by rw [z]; simp; left; rw [z]; simp
         _ ≤ (1 / alg.d k) * ‖alg.z (k + 1) - alg.z k‖ := by
-          have : ‖(alg.z (k + 1) - alg.z k).2‖ ≤ ‖alg.z (k + 1) - alg.z k‖ := by
+          have : ‖(alg.z (k + 1) - alg.z k).snd‖ ≤ ‖alg.z (k + 1) - alg.z k‖ := by
             apply snd_norm_le_prod_L2
           simp; apply mul_le_mul_of_nonneg_left this dpos'
         _ = (γ * l) * ‖alg.z (k + 1) - alg.z k‖ := by rw [dk k]; simp
@@ -341,11 +410,11 @@ theorem Ψ_subdiff_bound (γ : ℝ) (hγ : γ > 1)
         _ ≤ l * @norm (WithLp 2 (E × F)) (WithLp.instProdNorm 2 E F)
             ((alg.x (k + 1), alg.y (k + 1)) - (alg.x (k + 1), alg.y k)) := by
           apply lip
-        _ = l * ‖(alg.z (k+1) - alg.z k).2‖ := by
-          simp; left; repeat rw [z]; simp; apply norm_prod_left_zero
+        _ = l * ‖alg.y (k + 1) - alg.y k‖ := by
+          simp [WithLp.prod_norm_eq_of_L2]
         _ ≤ l * ‖alg.z (k+1) - alg.z k‖ := by
           apply mul_le_mul_of_nonneg_left _ (le_of_lt alg.lpos)
-          · apply snd_norm_le_prod_L2
+          · simpa [z] using (snd_norm_le_prod_L2 (alg.z (k + 1) - alg.z k))
     linarith
   linarith
 
@@ -373,19 +442,20 @@ lemma fconv (γ : ℝ) (hγ : γ > 1) (ck : ∀ k, alg.c k = 1 / (γ * l)) (dk :
     (α : ℕ → ℕ) (z_ : WithLp 2 (E×F)) (monoa : StrictMono α)
     (conv : Tendsto (fun n ↦ alg.z (α n)) atTop (𝓝 z_))
     (bd : Bornology.IsBounded (alg.z '' univ)) (lbdψ : BddBelow (alg.ψ '' univ)) :
-    Tendsto (fun n ↦ f (alg.z (α n)).1) atTop (𝓝 (f z_.1)) := by
+    Tendsto (fun n ↦ f (alg.z (α n)).fst) atTop (𝓝 (f z_.fst)) := by
   obtain lpos := alg.lpos
-  apply (nhds_basis_Ioo_pos (f z_.1)).tendsto_right_iff.mpr
+  apply (nhds_basis_Ioo_pos (f z_.fst)).tendsto_right_iff.mpr
   rintro ε epos
   simp only [Ioo]
-  have lef : ∀ᶠ x in atTop, f (alg.z (α x)).1 > f z_.1 - ε :=
-    (Tendsto.fst_nhds conv) (by apply alg.hf z_.1; exact sub_lt_self (f z_.1) epos)
-  have rig : ∀ᶠ x in atTop, f (alg.z (α x)).1 < f z_.1 + ε := by
+  have lef : ∀ᶠ x in atTop, f (alg.z (α x)).fst > f z_.fst - ε :=
+    ((WithLp.continuous_fst (p := (2 : ENNReal)) (α := E) (β := F)).tendsto z_ |>.comp conv)
+      (by apply alg.hf z_.fst; exact sub_lt_self (f z_.fst) epos)
+  have rig : ∀ᶠ x in atTop, f (alg.z (α x)).fst < f z_.fst + ε := by
     have ieq (q) (hq : 1 ≤ α q) : alg.c (α q -1) * f (alg.x (α q)) + ⟪alg.x (α q) - alg.x (α q -1),
         alg.c (α q -1) • grad_fst H (alg.y (α q -1)) (alg.x (α q -1))⟫ ≤
-        alg.c (α q -1) * f z_.1 + ‖z_.1 - alg.x (α q -1)‖ ^ 2 / 2 + ⟪z_.1 - alg.x (α q -1),
+        alg.c (α q -1) * f z_.fst + ‖z_.fst - alg.x (α q -1)‖ ^ 2 / 2 + ⟪z_.fst - alg.x (α q -1),
         alg.c (α q -1) • grad_fst H (alg.y (α q -1)) (alg.x (α q -1))⟫:= by
-      rcases isMinOn_iff.mp (alg.s₁ (α q -1)) z_.1 trivial with ieq
+      rcases isMinOn_iff.mp (alg.s₁ (α q -1)) z_.fst trivial with ieq
       simp at ieq
       rw [← sub_add, norm_add_sq_real, ← sub_add, norm_add_sq_real] at ieq
       repeat rw [add_div] at ieq; repeat rw [← add_assoc] at ieq
@@ -394,17 +464,20 @@ lemma fconv (γ : ℝ) (hγ : γ > 1) (ck : ∀ k, alg.c k = 1 / (γ * l)) (dk :
       linarith [ieq,this]
     have Hbd : ∃ C, ∀ q : ℕ, ‖(grad_fst H (alg.y (α q -1)) (alg.x (α q -1)))‖ ≤ C:= by
       rcases isBounded_iff_forall_norm_le.mp bd with ⟨C1,inin⟩
-      have con11H : ContinuousOn (fun (x,y)↦grad_fst H y x) (Metric.closedBall (0:WithLp 2 (E×F)) C1) := by
+      have con11H : ContinuousOn (fun z : WithLp 2 (E × F) ↦ grad_fst H z.snd z.fst)
+          (Metric.closedBall (0:WithLp 2 (E×F)) C1) := by
         apply Continuous.continuousOn
         exact LipschitzWith.continuous (lip_grad_fst_of_lip alg.Hdiff alg.lip)
       rcases @IsCompact.exists_bound_of_continuousOn (WithLp 2 (E×F)) E _ _ _
-        (isCompact_closedBall (0 : WithLp 2 (E × F)) C1) (fun (x, y)↦ grad_fst H y x) con11H with ⟨C, sq⟩
+        (isCompact_closedBall (0 : WithLp 2 (E × F)) C1)
+        (fun z : WithLp 2 (E × F) ↦ grad_fst H z.snd z.fst) con11H with ⟨C, sq⟩
       use C; rintro q
-      have : (alg.x (α q -1),alg.y (α q -1)) ∈ Metric.closedBall (0 : WithLp 2 (E × F)) C1 := by
+      have : ((alg.x (α q -1),alg.y (α q -1)) : WithLp 2 (E × F))
+          ∈ Metric.closedBall (0 : WithLp 2 (E × F)) C1 := by
         apply mem_closedBall_iff_norm.mpr; simp
-        apply inin (alg.x (α q -1),alg.y (α q -1))
+        apply inin ((alg.x (α q -1),alg.y (α q -1)) : WithLp 2 (E × F))
         exact mem_image_of_mem alg.z trivial
-      obtain h'' := sq (alg.x (α q -1),alg.y (α q -1)) this
+      obtain h'' := sq ((alg.x (α q -1),alg.y (α q -1)) : WithLp 2 (E × F)) this
       simp at h''; exact h''
     rcases Hbd with ⟨C,hbd⟩
     have diflte1 : ∀ ε > 0, ∀ᶠ (q : ℕ) in atTop, ‖alg.x (α q) - alg.x (α q - 1)‖ < ε:= by
@@ -426,9 +499,9 @@ lemma fconv (γ : ℝ) (hγ : γ > 1) (ck : ∀ k, alg.c k = 1 / (γ * l)) (dk :
           obtain ht := ie (α b - 1) (Nat.le_sub_one_of_lt a1leab)
           have eqq : (α b - 1 + 1) = α b:= by apply Nat.sub_add_cancel; linarith [a1leab]
           rwa [eqq] at ht
-    have diflte2 : ∀ ε > 0, ∀ᶠ (q : ℕ) in atTop, ‖z_.1 - alg.x (α q - 1)‖ < ε := by
+    have diflte2 : ∀ ε > 0, ∀ᶠ (q : ℕ) in atTop, ‖z_.fst - alg.x (α q - 1)‖ < ε := by
       rintro ε epos
-      have : ∀ᶠ (q : ℕ) in atTop, ‖z_.1 - alg.x (α q )‖ < ε / 2 := by
+      have : ∀ᶠ (q : ℕ) in atTop, ‖z_.fst - alg.x (α q )‖ < ε / 2 := by
         rcases (atTop_basis.tendsto_iff (@Metric.nhds_basis_ball _ _ z_)).mp conv (ε / 2)
           (half_pos epos) with ⟨n1,_,ieq1⟩
         simp [dist_eq_norm] at ieq1; simp
@@ -443,7 +516,7 @@ lemma fconv (γ : ℝ) (hγ : γ > 1) (ck : ∀ k, alg.c k = 1 / (γ * l)) (dk :
       rintro x ⟨h1,h2⟩
       rw [← sub_add_sub_cancel]
       calc
-        _ ≤ ‖z_.1 - alg.x (α x)‖ + ‖alg.x (α x) - alg.x (α x - 1)‖ := norm_add_le _ _
+        _ ≤ ‖z_.fst - alg.x (α x)‖ + ‖alg.x (α x) - alg.x (α x - 1)‖ := norm_add_le _ _
         _ < ε := by linarith
 
     have hk (k : ℕ → E) (defle : ∀ ε > 0, ∀ᶠ (q : ℕ) in atTop, ‖k q‖ < ε) : ∀ ε > 0,
@@ -452,15 +525,19 @@ lemma fconv (γ : ℝ) (hγ : γ > 1) (ck : ∀ k, alg.c k = 1 / (γ * l)) (dk :
       rintro ε epos
       simp at defle; simp
       by_cases Cpos : 0 < C
-      · rcases defle (ε / (C / (γ * l))) (by field_simp [alg.lpos, Cpos]) with ⟨nn,ieq⟩
+      · have hdivpos : 0 < ε / (C / (γ * l)) := by
+          apply div_pos epos
+          apply div_pos Cpos
+          nlinarith [hγ, alg.lpos]
+        rcases defle (ε / (C / (γ * l))) hdivpos with ⟨nn,ieq⟩
         use nn; rintro b nleb; rw [ck]
         calc
           _ ≤ ‖k b‖ * ‖(1 / (γ * ↑l)) • grad_fst H (alg.y (α b - 1)) (alg.x (α b - 1))‖
               := by apply abs_real_inner_le_norm
           _ ≤ ε / (C / (γ * ↑l))*‖(1 / (γ * ↑l)) • grad_fst H (alg.y (α b - 1)) (alg.x (α b - 1))‖:= by
-            apply mul_le_mul (le_of_lt (ieq b nleb)); trivial
+            apply mul_le_mul (le_of_lt (ieq b nleb)) le_rfl
             repeat apply norm_nonneg
-            field_simp [alg.lpos, Cpos]; positivity
+            exact le_of_lt hdivpos
           _ = ε / (C / (γ * ↑l))*(1 / (γ * ↑l)) * ‖grad_fst H (alg.y (α b - 1)) (alg.x (α b - 1))‖:= by
             rw [mul_assoc]; apply mul_eq_mul_left_iff.mpr
             left; exact norm_smul_of_nonneg (by positivity) (grad_fst H _ _)
@@ -494,21 +571,19 @@ lemma fconv (γ : ℝ) (hγ : γ > 1) (ck : ∀ k, alg.c k = 1 / (γ * l)) (dk :
         _≤ε:= by linarith
     simp only [ck] at ieq
     have h1 := hk (fun q ↦ alg.x (α q) - alg.x (α q - 1)) diflte1 (ε / (γ * l) / 3) (by positivity)
-    have h2 := hk (fun q ↦ z_.1 - alg.x (α q - 1)) diflte2 (ε / (γ * l) / 3) (by positivity)
-    have h3 : ∀ᶠ (q : ℕ) in atTop, ‖z_.1 - alg.x (α q - 1)‖ ^ 2 / 2 < (ε / (γ * l) / 3):= by
+    have h2 := hk (fun q ↦ z_.fst - alg.x (α q - 1)) diflte2 (ε / (γ * l) / 3) (by positivity)
+    have h3 : ∀ᶠ (q : ℕ) in atTop, ‖z_.fst - alg.x (α q - 1)‖ ^ 2 / 2 < (ε / (γ * l) / 3):= by
       refine Eventually.mono (diflte2 (√(2*(ε/(γ*l)/3))) ?_) ?_
       apply Real.sqrt_pos_of_pos
       apply mul_pos;norm_num; positivity
       intro x assx
-      have :‖z_.1 - alg.x (α x - 1)‖^2<(2*(ε/(γ*l)/3)):= by
+      have :‖z_.fst - alg.x (α x - 1)‖^2<(2*(ε/(γ*l)/3)):= by
         refine (Real.lt_sqrt ?hx).mp ?_
         apply norm_nonneg
         exact assx
       calc
-        ‖z_.1 - alg.x (α x - 1)‖ ^ 2 / 2<(2*(ε/(γ*l)/3))/2:= by
-          apply (div_lt_div_right _).mpr
-          apply this
-          linarith
+        ‖z_.fst - alg.x (α x - 1)‖ ^ 2 / 2<(2*(ε/(γ*l)/3))/2:= by
+          nlinarith [this]
         _=(ε/(γ*l)/3):= by
           apply mul_div_cancel_left₀
           linarith
@@ -529,7 +604,7 @@ lemma fconv (γ : ℝ) (hγ : γ > 1) (ck : ∀ k, alg.c k = 1 / (γ * l)) (dk :
         apply monoa
         linarith [Nat.le_of_add_right_le mleq]
       linarith
-    have key : 1 / (γ * ↑l) * f (alg.x (α q)) <1 / (γ * ↑l) * f z_.1 +ε / (γ * ↑l):= by
+    have key : 1 / (γ * ↑l) * f (alg.x (α q)) <1 / (γ * ↑l) * f z_.fst +ε / (γ * ↑l):= by
       linarith [ieq q this,(abs_le.mp (ie1 q (m1le.trans mleq))).1,(abs_le.mp
         (ie2 q (m2le.trans mleq))).2,ie3 q (m3le.trans mleq), add_thirds (ε / (γ * ↑l))]
     have ltt:0<γ*l:= by
@@ -538,8 +613,9 @@ lemma fconv (γ : ℝ) (hγ : γ > 1) (ck : ∀ k, alg.c k = 1 / (γ * l)) (dk :
       _ = f (alg.x (α q)) := rfl
       _ =(γ * ↑l)*(1 / (γ * ↑l) * f (alg.x (α q))):= by
         rw [←mul_assoc,mul_one_div_cancel (LT.lt.ne ltt).symm,one_mul]
-      _ < (γ * ↑l)*(1 / (γ * ↑l) * f z_.1 + ε / (γ * ↑l)):=(mul_lt_mul_left ltt).mpr key
-      _=f z_.1 + ε:=by
+      _ < (γ * ↑l)*(1 / (γ * ↑l) * f z_.fst + ε / (γ * ↑l)) := by
+        exact mul_lt_mul_of_pos_left key ltt
+      _=f z_.fst + ε:=by
         rw [mul_add, ← mul_assoc, mul_one_div_cancel (LT.lt.ne ltt).symm, one_mul,
           mul_div_cancel₀ _ (LT.lt.ne ltt).symm]
   exact Eventually.and lef rig
@@ -548,22 +624,23 @@ lemma fconv (γ : ℝ) (hγ : γ > 1) (ck : ∀ k, alg.c k = 1 / (γ * l)) (dk :
 lemma gconv (γ : ℝ) (hγ : γ > 1) (ck: ∀ k, alg.c k = 1 / (γ * l)) (dk: ∀ k, alg.d k = 1 / (γ * l))
     (α:ℕ→ℕ)(z_:WithLp 2 (E×F))(monoa:StrictMono α )(conv:Tendsto (fun n ↦ alg.z (α n)) atTop (𝓝 z_))
     (bd : Bornology.IsBounded (alg.z '' univ)) (lbdψ : BddBelow (alg.ψ '' univ)):
-    Tendsto (fun n ↦ g (alg.z (α n)).2) atTop (𝓝 (g z_.2)):=by
-  apply (nhds_basis_Ioo_pos (g z_.2)).tendsto_right_iff.mpr
+    Tendsto (fun n ↦ g (alg.z (α n)).snd) atTop (𝓝 (g z_.snd)):=by
+  apply (nhds_basis_Ioo_pos (g z_.snd)).tendsto_right_iff.mpr
   rintro ε epos
   simp only [Ioo]
-  have lef:∀ᶠ (x : ℕ) in atTop, g (alg.z (α x)).2>g z_.2-ε:= by
-    have semi: ∀ᶠ x' in 𝓝 z_.2, g z_.2 -ε < g x':= by
-      apply alg.hg z_.2
+  have lef:∀ᶠ (x : ℕ) in atTop, g (alg.z (α x)).snd>g z_.snd-ε:= by
+    have semi: ∀ᶠ x' in 𝓝 z_.snd, g z_.snd -ε < g x':= by
+      apply alg.hg z_.snd
       linarith
-    have :Tendsto (fun n↦ (alg.z (α n)).2) atTop (𝓝 z_.2):= Tendsto.snd_nhds conv
+    have : Tendsto (fun n ↦ (alg.z (α n)).snd) atTop (𝓝 z_.snd) :=
+      ((WithLp.continuous_snd (p := (2 : ENNReal)) (α := E) (β := F)).tendsto z_).comp conv
     exact this semi
-  have rig:∀ᶠ (x : ℕ) in atTop, g (alg.z (α x)).2<g z_.2+ε:= by
+  have rig:∀ᶠ (x : ℕ) in atTop, g (alg.z (α x)).snd<g z_.snd+ε:= by
     have ieq (q:ℕ)(hq:1≤α q):alg.d (α q - 1) * g (alg.y (α q)) +
         ⟪alg.y (α q) - alg.y (α q - 1), alg.d (α q - 1) • grad_snd H (alg.x (α q)) (alg.y (α q - 1))⟫≤
-        alg.d (α q - 1) * g z_.2 + ‖z_.2 - alg.y (α q - 1)‖ ^ 2 / 2 +
-        ⟪z_.2 - alg.y (α q - 1), alg.d (α q - 1) • grad_snd H (alg.x (α q)) (alg.y (α q - 1))⟫:= by
-      rcases isMinOn_iff.mp (alg.s₂ (α q -1)) z_.2 trivial with ieq
+        alg.d (α q - 1) * g z_.snd + ‖z_.snd - alg.y (α q - 1)‖ ^ 2 / 2 +
+        ⟪z_.snd - alg.y (α q - 1), alg.d (α q - 1) • grad_snd H (alg.x (α q)) (alg.y (α q - 1))⟫:= by
+      rcases isMinOn_iff.mp (alg.s₂ (α q -1)) z_.snd trivial with ieq
       simp at ieq
       rw [←sub_add,norm_add_sq_real,←sub_add,norm_add_sq_real] at ieq
       repeat rw [add_div] at ieq
@@ -576,20 +653,22 @@ lemma gconv (γ : ℝ) (hγ : γ > 1) (ck: ∀ k, alg.c k = 1 / (γ * l)) (dk: 
       linarith [ieq,this]
     have Hbd :∃C,∀q:ℕ ,‖(grad_snd H (alg.x (α q )) (alg.y (α q -1)))‖≤C:= by
       rcases isBounded_iff_forall_norm_le.mp bd with ⟨C1,inin⟩
-      have con11H : ContinuousOn (fun (x,y) ↦ grad_snd H x y)
+      have con11H : ContinuousOn (fun z : WithLp 2 (E × F) ↦ grad_snd H z.fst z.snd)
           (Metric.closedBall (0:WithLp 2 (E×F)) (2*C1)) := by
         apply Continuous.continuousOn
         exact LipschitzWith.continuous (lip_grad_snd_of_lip alg.Hdiff alg.lip)
       rcases @IsCompact.exists_bound_of_continuousOn (WithLp 2 (E×F)) F _ _ _
           (isCompact_closedBall (0:WithLp 2 (E×F)) (2*C1))
-          (fun (x,y)↦grad_snd H x y) con11H with ⟨C,sqsq⟩
+          (fun z : WithLp 2 (E × F) ↦ grad_snd H z.fst z.snd) con11H with ⟨C,sqsq⟩
       use C
       rintro q
-      have :(alg.x (α q ),alg.y (α q -1))∈Metric.closedBall (0:WithLp 2 (E×F)) (2*C1) := by
+      have : ((alg.x (α q),alg.y (α q -1)) : WithLp 2 (E × F))
+          ∈ Metric.closedBall (0:WithLp 2 (E×F)) (2*C1) := by
         apply mem_closedBall_iff_norm.mpr
         simp
         calc
-          @norm (WithLp 2 (E × F)) (WithLp.instProdNorm 2 E F) (alg.x (α q),alg.y (α q - 1)) ≤
+          @norm (WithLp 2 (E × F)) (WithLp.instProdNorm 2 E F)
+              ((alg.x (α q),alg.y (α q - 1)) : WithLp 2 (E × F)) ≤
               ‖alg.x (α q)‖+‖alg.y (α q - 1)‖ := by apply prod_norm_le_block_sum_L2
           _≤‖alg.z (α q)‖+‖alg.z (α q -1)‖:=by
             have :‖alg.y (α q -1)‖≤‖alg.z (α q -1)‖:= by
@@ -607,12 +686,12 @@ lemma gconv (γ : ℝ) (hγ : γ > 1) (ck: ∀ k, alg.c k = 1 / (γ * l)) (dk: 
             linarith
           _≤C1+C1:=by
             apply add_le_add
-            apply inin
+            apply inin ((alg.x (α q), alg.y (α q)) : WithLp 2 (E × F))
             exact mem_image_of_mem alg.z trivial
-            apply inin
+            apply inin ((alg.x (α q - 1), alg.y (α q - 1)) : WithLp 2 (E × F))
             exact mem_image_of_mem alg.z trivial
           _=2*C1:=Eq.symm (two_mul C1)
-      have hhhh:= sqsq (alg.x (α q ),alg.y (α q -1)) this
+      have hhhh := sqsq ((alg.x (α q), alg.y (α q - 1)) : WithLp 2 (E × F)) this
       simp at hhhh
       exact hhhh
     rcases Hbd with ⟨C,hbd⟩
@@ -646,33 +725,33 @@ lemma gconv (γ : ℝ) (hγ : γ > 1) (ck: ∀ k, alg.c k = 1 / (γ * l)) (dk: 
             linarith [a1leab]
           rw [eqq] at this
           assumption
-    have diflte2:∀ ε>0, ∀ᶠ (q : ℕ) in atTop,‖z_.2 - alg.y (α q - 1)‖ <ε:= by
+    have diflte2:∀ ε>0, ∀ᶠ (q : ℕ) in atTop,‖z_.snd - alg.y (α q - 1)‖ <ε:= by
       rintro ε epos
-      have : ∀ᶠ (q : ℕ) in atTop,‖z_.2 - alg.y (α q )‖ <ε/2:= by
+      have : ∀ᶠ (q : ℕ) in atTop,‖z_.snd - alg.y (α q )‖ <ε/2:= by
         rcases (atTop_basis.tendsto_iff (@Metric.nhds_basis_ball _ _ z_)).mp conv (ε/2)
           (half_pos epos) with ⟨n1,_,ieq1⟩
         simp [dist_eq_norm] at ieq1;simp
         use n1
         rintro b n1leb
         calc
-          ‖z_.2 - alg.y (α b)‖≤‖z_ - alg.z (α b)‖ :=by
+          ‖z_.snd - alg.y (α b)‖≤‖z_ - alg.z (α b)‖ :=by
             rw [WithLp.prod_norm_eq_of_L2]
             simp
-            refine (Real.le_sqrt (norm_nonneg (z_.2 - alg.y (α b)))
-              (Left.add_nonneg (sq_nonneg ‖z_.1 - alg.x (α b)‖)
-              (sq_nonneg ‖z_.2 - alg.y (α b)‖ ))).mpr
-              (le_add_of_nonneg_left (sq_nonneg ‖z_.1 - alg.x (α b)‖))
+            refine (Real.le_sqrt (norm_nonneg (z_.snd - alg.y (α b)))
+              (Left.add_nonneg (sq_nonneg ‖z_.fst - alg.x (α b)‖)
+              (sq_nonneg ‖z_.snd - alg.y (α b)‖ ))).mpr
+              (le_add_of_nonneg_left (sq_nonneg ‖z_.fst - alg.x (α b)‖))
           _<ε/2:=by
             rw [norm_sub_rev]
             exact ieq1 b n1leb
-      have :∀ᶠ (q : ℕ) in atTop,‖z_.2 - alg.y (α q )‖ <ε/2∧‖alg.y (α q) - alg.y (α q - 1)‖ <ε/2
+      have :∀ᶠ (q : ℕ) in atTop,‖z_.snd - alg.y (α q )‖ <ε/2∧‖alg.y (α q) - alg.y (α q - 1)‖ <ε/2
           := Eventually.and this (diflte1 (ε/2) (half_pos epos))
       apply Eventually.mono this
       rintro x ⟨h1,h2⟩
       calc
-        ‖z_.2 - alg.y (α x - 1)‖=‖z_.2 - alg.y (α x )+(alg.y (α x) - alg.y (α x -1))‖:= by
+        ‖z_.snd - alg.y (α x - 1)‖=‖z_.snd - alg.y (α x )+(alg.y (α x) - alg.y (α x -1))‖:= by
           simp
-        _≤‖z_.2 - alg.y (α x)‖+‖alg.y (α x) - alg.y (α x - 1)‖:= by
+        _≤‖z_.snd - alg.y (α x)‖+‖alg.y (α x) - alg.y (α x - 1)‖:= by
           apply norm_add_le
         _<ε/2+ε/2:= by linarith [h1,h2]
         _=ε := by exact add_halves ε
@@ -749,23 +828,21 @@ lemma gconv (γ : ℝ) (hγ : γ > 1) (ck: ∀ k, alg.c k = 1 / (γ * l)) (dk: 
     have h1:∀ᶠ (q : ℕ) in atTop,|⟪alg.y (α q) - alg.y (α q - 1), alg.d (α q - 1) • grad_snd H
         (alg.x (α q )) (alg.y (α q - 1))⟫| ≤ε / (γ * ↑l) / 3 :=
       this (fun q↦alg.y (α q) - alg.y (α q - 1)) (diflte1) (ε/(γ*l)/3) finalpos
-    have h2: ∀ᶠ (q : ℕ) in atTop,|⟪z_.2 - alg.y (α q - 1), alg.d (α q - 1) • grad_snd H (alg.x (α q ))
+    have h2: ∀ᶠ (q : ℕ) in atTop,|⟪z_.snd - alg.y (α q - 1), alg.d (α q - 1) • grad_snd H (alg.x (α q ))
         (alg.y (α q - 1))⟫| ≤ ε / (γ * ↑l) / 3:=
-      this (fun q↦z_.2 - alg.y (α q - 1)) diflte2 (ε/(γ*l)/3) finalpos
-    have h3: ∀ᶠ (q : ℕ) in atTop,‖z_.2 - alg.y (α q - 1)‖ ^ 2 / 2<(ε/(γ*l)/3):= by
+      this (fun q↦z_.snd - alg.y (α q - 1)) diflte2 (ε/(γ*l)/3) finalpos
+    have h3: ∀ᶠ (q : ℕ) in atTop,‖z_.snd - alg.y (α q - 1)‖ ^ 2 / 2<(ε/(γ*l)/3):= by
       refine Eventually.mono (diflte2 (√(2*(ε/(γ*l)/3))) ?_) ?_
       apply Real.sqrt_pos_of_pos
       apply mul_pos;norm_num;apply finalpos
       intro x assx
-      have :‖z_.2 - alg.y (α x - 1)‖^2<(2*(ε/(γ*l)/3)):= by
+      have :‖z_.snd - alg.y (α x - 1)‖^2<(2*(ε/(γ*l)/3)):= by
         refine (Real.lt_sqrt ?hy).mp ?_
         apply norm_nonneg
         exact assx
       calc
-        ‖z_.2 - alg.y (α x - 1)‖ ^ 2 / 2<(2*(ε/(γ*l)/3))/2:= by
-          apply (div_lt_div_right _).mpr
-          apply this
-          linarith
+        ‖z_.snd - alg.y (α x - 1)‖ ^ 2 / 2<(2*(ε/(γ*l)/3))/2:= by
+          nlinarith [this]
         _=(ε/(γ*l)/3):= by
           apply mul_div_cancel_left₀
           linarith
@@ -785,17 +862,18 @@ lemma gconv (γ : ℝ) (hγ : γ > 1) (ck: ∀ k, alg.c k = 1 / (γ * l)) (dk: 
         apply monoa
         linarith [Nat.le_of_add_right_le mleq]
       linarith
-    have key:1 / (γ * ↑l) * g (alg.y (α q)) <1 / (γ * ↑l) * g z_.2 +ε / (γ * ↑l):= by
+    have key:1 / (γ * ↑l) * g (alg.y (α q)) <1 / (γ * ↑l) * g z_.snd +ε / (γ * ↑l):= by
       linarith [ieq q this,(abs_le.mp (ie1 q (m1le.trans mleq))).1,(abs_le.mp (ie2 q (m2le.trans mleq))).2,
         ie3 q (m3le.trans mleq),add_thirds (ε / (γ * ↑l))]
     have ltt:0<γ*l:= by
       apply mul_pos;linarith;linarith [alg.lpos]
     calc
-      g (alg.z (α q)).2=g (alg.y (α q)):= rfl
+      g (alg.z (α q)).snd = g (alg.y (α q)):= rfl
       _=(γ * ↑l)*(1 / (γ * ↑l) * g (alg.y (α q))):= by
         rw [←mul_assoc,mul_one_div_cancel (LT.lt.ne ltt).symm,one_mul]
-      _<(γ * ↑l)*(1 / (γ * ↑l) * g z_.2 + ε / (γ * ↑l)):=(mul_lt_mul_left ltt).mpr key
-      _=g z_.2 + ε:=by
+      _ < (γ * ↑l) * (1 / (γ * ↑l) * g z_.snd + ε / (γ * ↑l)) := by
+        exact mul_lt_mul_of_pos_left key ltt
+      _=g z_.snd + ε:=by
         rw [mul_add, ← mul_assoc, mul_one_div_cancel (LT.lt.ne ltt).symm,
           one_mul, mul_div_cancel₀ _ (LT.lt.ne ltt).symm]
   exact Eventually.and lef rig
@@ -819,20 +897,12 @@ lemma limitset_property_1 (γ : ℝ) (hγ : γ > 1)
   --nonempty
   have hz : ∀ (n : ℕ), alg.z n ∈ alg.z '' univ:= by intro n; use n; constructor; exact Set.mem_univ n; rfl
   rcases (tendsto_subseq_of_bounded (bd) (hz)) with ⟨a, _ , φ, ⟨hmφ,haφ⟩⟩
-  use a; simp [limit_set]
-  rw [mapClusterPt_iff]; intro s hs
-  apply Filter.frequently_iff.mpr
-  intro U hU; rw [Filter.mem_atTop_sets] at hU
-  rcases hU with ⟨ax,hax⟩; rw [mem_nhds_iff] at hs
-  obtain ⟨t, t_s, ⟨isopent,a_t⟩⟩ := hs
-  rw [tendsto_atTop_nhds] at haφ
-  specialize haφ t a_t isopent
-  rcases haφ with ⟨N,hN⟩
-  let n := N + ax
-  use φ n
-  constructor; apply hax
-  apply le_trans (Nat.le_add_left ax N); apply StrictMono_nat; exact hmφ
-  exact t_s (hN n (by simp[n]))
+  use a
+  simp [limit_set, MapClusterPt]
+  apply ClusterPt.mono (ClusterPt.of_le_nhds haφ)
+  calc
+    _ = map (fun n ↦ alg.z n) (map φ atTop) := by rw [map_map]
+    _ ≤ map (fun n ↦ alg.z n) atTop := map_mono (StrictMono.tendsto_atTop hmφ)
   --the folllowing shows that limit_set BCD.z ⊆ critial_point BCD.ψ
   intro z_ ha
   rcases TopologicalSpace.FirstCountableTopology.tendsto_subseq ha with ⟨φ,monoφ,conv⟩
@@ -860,16 +930,15 @@ lemma limitset_property_1 (γ : ℝ) (hγ : γ > 1)
   calc
     _ ≤ ρ * ‖alg.z (φ (b + 1)) - alg.z (φ (b + 1) - 1)‖:= (key b).2
     _ < ρ * (ε / ρ) := by
-      apply (mul_lt_mul_left ρpos).mpr
       have : ‖alg.z (φ (b + 1)-1 + 1) - alg.z (φ (b + 1) - 1)‖ < ε / ρ := by
         apply ieq; apply aleb.trans
         exact Nat.sub_le_sub_right (StrictMono_nat φ monoφ (b+1)) 1
       simp [subadd b] at this
-      exact this
+      exact mul_lt_mul_of_pos_left this ρpos
     _ = ε := by rw [mul_comm, div_mul_cancel₀]; linarith [ρpos]
 
 lemma limitset_property_2 (bd : Bornology.IsBounded (alg.z '' univ)) :
-    Tendsto (fun n ↦ (EMetric.infEdist (alg.z n) (limit_set alg.z)).toReal) atTop (𝓝 0) := by
+    Tendsto (fun n ↦ (Metric.infEDist (alg.z n) (limit_set alg.z)).toReal) atTop (𝓝 0) := by
   apply (nhds_basis_Ioo_pos 0).tendsto_right_iff.mpr
   rintro ε epos; by_contra h; simp at h
   --alg.z∘W is the subseq s.t. the dist is no less than ε
@@ -894,14 +963,14 @@ lemma limitset_property_2 (bd : Bornology.IsBounded (alg.z '' univ)) :
       _ ≤ map (fun n ↦ alg.z n) atTop := by
         rw [← map_map]; apply map_mono (StrictMono.tendsto_atTop monoW)
   -- show the contradiction
-  have z_ge : (EMetric.infEdist z_ (limit_set alg.z)).toReal ≥ ε := by
+  have z_ge : (Metric.infEDist z_ (limit_set alg.z)).toReal ≥ ε := by
     apply ge_of_tendsto (continuous_iff_seqContinuous.mp
       (Metric.continuous_infDist_pt (limit_set alg.z)) conv)
     simp; use 1; rintro n len
     rw [← tsub_add_cancel_iff_le.mpr (Nat.one_le_of_lt (monoa len))]
     apply (Classical.choose_spec (h (W (α n -1) +1))).2
     apply lt_of_le_of_lt' (ENNReal.toReal_nonneg) (neg_neg_iff_pos.mpr epos)
-  linarith [(ENNReal.toReal_eq_zero_iff _).mpr (by left; exact EMetric.infEdist_zero_of_mem z_in)]
+  linarith [(ENNReal.toReal_eq_zero_iff _).mpr (by left; exact Metric.infEDist_zero_of_mem z_in)]
 
 lemma limitset_property_3 (γ : ℝ) (hγ : γ > 1)
     (ck : ∀ k, alg.c k = 1 / (γ * l)) (dk : ∀ k, alg.d k = 1 / (γ * l))
@@ -930,26 +999,34 @@ lemma limitset_property_3 (γ : ℝ) (hγ : γ > 1)
     simp [A, B]
     rwa [← inter_inter_distrib_left (limit_set z) a b]
   -- ω is a function that shows the relation between z and A,B
-  let ω : WithLp 2 (E × F) -> ℝ := fun z => ((EMetric.infEdist z A).toReal) /
-    ((EMetric.infEdist z A).toReal+(EMetric.infEdist z B).toReal)
-  have sum_ne_zero : ∀ z, (EMetric.infEdist z A).toReal + (EMetric.infEdist z B).toReal ≠ 0:= by
+  let ω : WithLp 2 (E × F) -> ℝ := fun z => ((Metric.infEDist z A).toReal) /
+    ((Metric.infEDist z A).toReal+(Metric.infEDist z B).toReal)
+  have sum_ne_zero : ∀ z, (Metric.infEDist z A).toReal + (Metric.infEDist z B).toReal ≠ 0:= by
     intro z eq0
-    have inA : z ∈ A := by
-      apply EMetric.mem_closure_iff_infEdist_zero.mpr
-      have : (EMetric.infEdist z A).toReal = 0 := by
-        linarith [eq0, @ENNReal.toReal_nonneg (EMetric.infEdist z A),
-          @ENNReal.toReal_nonneg (EMetric.infEdist z B)]
+    have hAclosed : IsClosed A := by
+      dsimp [A]
+      exact IsClosed.inter isClosed_setOf_clusterPt closea
+    have hzA : z ∈ closure A := by
+      apply Metric.mem_closure_iff_infEDist_zero.mpr
+      have : (Metric.infEDist z A).toReal = 0 := by
+        linarith [eq0, @ENNReal.toReal_nonneg (Metric.infEDist z A),
+          @ENNReal.toReal_nonneg (Metric.infEDist z B)]
       exact (((fun {x y} hx hy ↦ (ENNReal.toReal_eq_toReal_iff' hx hy).mp)
-          ENNReal.top_ne_zero.symm (Metric.infEdist_ne_top nez_a) (id (Eq.symm this)))).symm
-      simp; constructor; rw [isOpen_compl_iff]; apply IsClosed.inter isClosed_setOf_clusterPt closea
-    have inB : z ∈ B :=by
-      apply EMetric.mem_closure_iff_infEdist_zero.mpr
-      have : (EMetric.infEdist z B).toReal = 0 := by
-        linarith [eq0, @ENNReal.toReal_nonneg (EMetric.infEdist z A),
-          @ENNReal.toReal_nonneg (EMetric.infEdist z B)]
+          ENNReal.top_ne_zero.symm (Metric.infEDist_ne_top nez_a) (id (Eq.symm this)))).symm
+    have inA : z ∈ A := by
+      simpa [hAclosed.closure_eq] using hzA
+    have hBclosed : IsClosed B := by
+      dsimp [B]
+      exact IsClosed.inter isClosed_setOf_clusterPt closeb
+    have hzB : z ∈ closure B := by
+      apply Metric.mem_closure_iff_infEDist_zero.mpr
+      have : (Metric.infEDist z B).toReal = 0 := by
+        linarith [eq0, @ENNReal.toReal_nonneg (Metric.infEDist z A),
+          @ENNReal.toReal_nonneg (Metric.infEDist z B)]
       exact (((fun {x y} hx hy ↦ (ENNReal.toReal_eq_toReal_iff' hx hy).mp)
-          ENNReal.top_ne_zero.symm (Metric.infEdist_ne_top nez_b) (id (Eq.symm this)))).symm
-      simp; constructor; rw [isOpen_compl_iff]; apply IsClosed.inter isClosed_setOf_clusterPt closeb
+          ENNReal.top_ne_zero.symm (Metric.infEDist_ne_top nez_b) (id (Eq.symm this)))).symm
+    have inB : z ∈ B := by
+      simpa [hBclosed.closure_eq] using hzB
     obtain hzin : z ∈ A ∩ B := mem_inter inA inB
     rw [disjoint_AB] at hzin; contradiction
   have contω : Continuous ω := by
@@ -959,10 +1036,10 @@ lemma limitset_property_3 (γ : ℝ) (hγ : γ > 1)
   let V := {z : WithLp 2 (E × F) | (3 / 4) < (ω z)}
   have A0 : ∀ z_ ∈ A, ω z_ = 0 := by
     rintro z_ zA; rw [div_eq_zero_iff]; left
-    rw [EMetric.infEdist_zero_of_mem zA]; rfl
+    rw [Metric.infEDist_zero_of_mem zA]; rfl
   have B1 : ∀ z_ ∈ B, ω z_ = 1 := by
     rintro z_ zB; simp [ω]; apply (div_eq_one_iff_eq (sum_ne_zero z_)).mpr; simp
-    rw [EMetric.infEdist_zero_of_mem zB]; rfl
+    rw [Metric.infEDist_zero_of_mem zB]; rfl
   --eventually alg.z falls in U or V
   have U_V_prop : ∃ k0, ∀ k, (k0 ≤ k) -> (alg.z k ∈ U) ∨ (alg.z k ∈ V) := by
     by_contra h
@@ -1085,10 +1162,15 @@ lemma limitset_property_4 (γ : ℝ) (hγ : γ > 1)
   have decent_ψ : ∃ ψ_final, Tendsto (alg.ψ ∘ alg.z) Filter.atTop (nhds ψ_final) := by
     have monopsi : Antitone (alg.ψ ∘ alg.z) :=
       antitone_nat_of_succ_le (Sufficient_Descent2 γ hγ ck dk)
-    rcases tendsto_of_antitone monopsi with h1 | h2
-    obtain notbd := unbounded_of_tendsto_atBot h1
-    apply absurd notbd; push_neg
-    exact BddBelow.mono (by simp; apply range_comp_subset_range) lbdψ; exact h2
+    rcases tendsto_atTop_of_antitone monopsi with h1 | h2
+    · rcases lbdψ with ⟨m, hm⟩
+      rcases (Filter.tendsto_atTop_atBot.mp h1) (m - 1) with ⟨N, hN⟩
+      have hmN : m ≤ (alg.ψ ∘ alg.z) N := hm (by
+        refine ⟨alg.z N, ?_, rfl⟩
+        exact mem_univ _)
+      have hle : (alg.ψ ∘ alg.z) N ≤ m - 1 := hN N le_rfl
+      linarith
+    · exact h2
   rcases decent_ψ with ⟨ψ_final, hψ⟩
   -- show that ψ_final is what we need
   use ψ_final; intro z_1 hz_1
@@ -1138,8 +1220,8 @@ private lemma ENNReal.mul_pos_real {a : ℝ} {b : ENNReal} (ha : a > 0) (hm : 1
 theorem Limited_length (γ : ℝ) (hγ : γ > 1)
     (ck : ∀ k, alg.c k = 1 / (γ * l)) (dk : ∀ k, alg.d k = 1 / (γ * l))
     (bd : Bornology.IsBounded (alg.z '' univ)) (hψ : KL_function alg.ψ)
-    (lbdψ : BddBelow (alg.ψ '' univ)): ∃ M : ℝ, ∀ n,
-    ∑ k in Finset.range n, ‖alg.z (k + 1) - alg.z k‖ ≤ M := by
+    (lbdψ : BddBelow (alg.ψ '' univ)) :
+    ∃ M : ℝ, ∀ n, (Finset.range n).sum (fun k ↦ ‖alg.z (k + 1) - alg.z k‖) ≤ M := by
   have :∃ z_∈ closure (alg.z '' univ), ∃ α:ℕ → ℕ,StrictMono α∧Tendsto
       (fun n ↦ alg.z (α n)) atTop (𝓝 z_):= by
     have hcs : IsSeqCompact (closure (alg.z '' univ)) := by
@@ -1187,8 +1269,8 @@ theorem Limited_length (γ : ℝ) (hγ : γ > 1)
         _ = alg.ψ (alg.z (α l1 + (k - α l1))):= by
           congr; exact Eq.symm (Nat.add_sub_of_le kge)
         _ ≤ alg.ψ (alg.z (α l1)) := by apply monopsi
-        _ < alg.ψ z_ + η := (ieq l1 left_mem_Ici).2
-    have L2 : ∀ ε > 0, ∃ l2, ∀k > l2, (EMetric.infEdist (alg.z k) (limit_set alg.z)).toReal< ε := by
+        _ < alg.ψ z_ + η := (ieq l1 Set.self_mem_Ici).2
+    have L2 : ∀ ε > 0, ∃ l2, ∀k > l2, (Metric.infEDist (alg.z k) (limit_set alg.z)).toReal< ε := by
       rintro ε epos
       rcases limitset_property_2 bd with tendt
       rcases (atTop_basis.tendsto_iff (nhds_basis_abs_sub_lt (0:ℝ))).mp tendt ε epos with ⟨l2,_,ieq⟩
@@ -1216,7 +1298,7 @@ theorem Limited_length (γ : ℝ) (hγ : γ > 1)
       rw [heq x xin,heq y yin]
     have kl: ∃ ε ∈ Set.Ioi (0 : ℝ), ∃ η ∈  Set.Ioi (0 : ℝ), ∃ φ ∈ desingularizing_function η, ∃ LL, ∀ n > LL,
         (alg.ψ z_ < alg.ψ (alg.z n) ∧ alg.ψ (alg.z n) < alg.ψ z_ + η) ∧ ENNReal.ofReal (deriv φ (alg.ψ (alg.z n)
-        - alg.ψ z_)) * EMetric.infEdist 0 (subdifferential alg.ψ (alg.z n)) ≥ 1 := by
+        - alg.ψ z_)) * Metric.infEDist 0 (subdifferential alg.ψ (alg.z n)) ≥ 1 := by
       rcases uniformized_KL_property (limitset_property_3 γ hγ ck dk bd lbdψ).1 wklpt cons
         with ⟨ε, eppos, η, etpos, φ, hφ, pro⟩
       rcases L1 η etpos with ⟨l1,lem1⟩
@@ -1258,25 +1340,26 @@ theorem Limited_length (γ : ℝ) (hγ : γ > 1)
         exact h6 _ (hlin n)
       have hbd2 : 1 ≤ ρ * (c n) * d n := by
         obtain ⟨dpsi, hdp, hub⟩ := hsgub (n + LL)
-        obtain hdp := subdifferential_subset _ _ hdp
-        have := infEdist_bound _ hdp
-        apply (ENNReal.ofReal_le_ofReal_iff _).1
-        rw [ENNReal.ofReal_mul]
-        show ENNReal.ofReal (ρ * c n) * ENNReal.ofReal (d n) ≥ ENNReal.ofReal 1
-        calc
-          _ ≥ (ENNReal.ofReal ‖dpsi‖) * ENNReal.ofReal (d n) := by
-            repeat rw [← ENNReal.ofReal_mul]
-            apply (ENNReal.ofReal_le_ofReal_iff _).2
-            apply (mul_le_mul_iff_of_pos_right hposd).mpr hub
-            field_simp; simp [c]; simp; field_simp; simp [c]
-          _ ≥ (EMetric.infEdist 0 (subdifferential ψ (z (n + LL + 1)))) * ENNReal.ofReal (d n) := by
-              apply mul_le_mul_right' this
-          _ ≥ 1 := by rw [mul_comm]; exact (ieq (n + LL + 1) (by linarith)).2
-        simp
-        field_simp
-        simp [c]
-        field_simp
-        simp [c]
+        have hdp' : dpsi ∈ subdifferential alg.ψ (alg.z (n + LL + 1)) :=
+          subdifferential_subset (alg.ψ) (alg.z (n + LL + 1)) hdp
+        let S := subdifferential alg.ψ (alg.z (n + LL + 1))
+        have h_inf_edist : Metric.infEDist 0 S ≤ edist 0 dpsi := Metric.infEDist_le_edist_of_mem hdp'
+        have h_inf : Metric.infEDist 0 S ≤ ENNReal.ofReal ‖dpsi‖ := by
+          simpa [edist_dist] using h_inf_edist
+        have h_inf_real : (Metric.infEDist 0 S).toReal ≤ ‖dpsi‖ := by
+          exact ENNReal.toReal_le_of_le_ofReal (norm_nonneg _) h_inf
+        have h_en : 1 ≤ ENNReal.ofReal (d n) * Metric.infEDist 0 S := by
+          simpa [S] using (ieq (n + LL + 1) (by linarith)).2
+        have h_real : d n * (Metric.infEDist 0 S).toReal ≥ 1 := by
+          exact ENNReal.mul_pos_real hposd h_en (lt_top_iff_ne_top.mpr (Metric.infEDist_ne_top (by
+            refine ⟨dpsi, hdp'⟩)))
+        have hub' : ‖dpsi‖ ≤ ρ * c n := by
+          simpa [c] using hub
+        have h_mul1 : d n * (Metric.infEDist 0 S).toReal ≤ d n * ‖dpsi‖ := by
+          exact mul_le_mul_of_nonneg_left h_inf_real (le_of_lt hposd)
+        have h_mul2 : d n * ‖dpsi‖ ≤ d n * (ρ * c n) := by
+          exact mul_le_mul_of_nonneg_left hub' (le_of_lt hposd)
+        nlinarith [h_real, h_mul1, h_mul2]
       have hsd : ρ1 / 2 * (c (n + 1)) ^ 2 ≤ b n := by
         obtain h := suff_des.2 (n + LL + 1)
         rw [add_right_comm n LL 1] at h
@@ -1356,19 +1439,27 @@ theorem Limited_length (γ : ℝ) (hγ : γ > 1)
       calc
         _ ≤ alg.ψ (alg.z i) -alg.ψ (alg.z (i + 1)) := suff_des.2 i
         _ = 0 := by simp [this i ige,this (i+1) (Nat.le_add_right_of_le ige)]
-    apply dist_eq_zero.mp (by rw [NormedAddCommGroup.dist_eq, this])
-  use ∑ k in Finset.range N, ‖alg.z (k + 1) - alg.z k‖
-  intro n; by_cases nlen : n ≤ N
-  · refine Finset.sum_le_sum_of_subset_of_nonneg (GCongr.finset_range_subset_of_le nlen) ?_
-    exact fun a _ _ ↦norm_nonneg (alg.z (a + 1) - alg.z a)
-  push_neg at nlen
-  have eq0 : ∑ i in (Finset.range n \ Finset.range N), ‖alg.z (i + 1) - alg.z i‖ = 0 := by
-    apply Finset.sum_eq_zero; rintro x xin; simp at xin
-    exact norm_sub_eq_zero_iff.mpr (eq0 x xin.2)
-  refine Finset.sum_sdiff_le_sum_sdiff.mp ?_
-  rw [eq0, Finset.sdiff_eq_empty_iff_subset.mpr
-    (GCongr.finset_range_subset_of_le (Nat.le_of_succ_le nlen))]
-  exact Preorder.le_refl 0
+    exact sub_eq_zero.mp (norm_eq_zero.mp this)
+  use (∑ k ∈ Finset.range N, ‖alg.z (k + 1) - alg.z k‖)
+  intro n
+  by_cases nlen : n ≤ N
+  · refine Finset.sum_le_sum_of_subset_of_nonneg ((Finset.range_subset_range).2 nlen) ?_
+    exact fun a _ _ ↦ norm_nonneg (alg.z (a + 1) - alg.z a)
+  · have nlt : N < n := lt_of_not_ge nlen
+    have hsub : Finset.range N ⊆ Finset.range n := (Finset.range_subset_range).2 (Nat.le_of_lt nlt)
+    have hsum0 : ∑ i ∈ (Finset.range n \ Finset.range N), ‖alg.z (i + 1) - alg.z i‖ = 0 := by
+      apply Finset.sum_eq_zero
+      intro x hx
+      have hxN : N ≤ x := by
+        have hxnot : x ∉ Finset.range N := (Finset.mem_sdiff.mp hx).2
+        have hxlt : ¬ x < N := by simpa [Finset.mem_range] using hxnot
+        exact Nat.le_of_not_lt hxlt
+      exact norm_sub_eq_zero_iff.mpr (eq0 x hxN)
+    have hsplit := Finset.sum_sdiff hsub (f := fun k ↦ ‖alg.z (k + 1) - alg.z k‖)
+    have hsumEq : ∑ k ∈ Finset.range n, ‖alg.z (k + 1) - alg.z k‖
+        = ∑ k ∈ Finset.range N, ‖alg.z (k + 1) - alg.z k‖ := by
+      linarith [hsplit, hsum0]
+    exact le_of_eq hsumEq
 
 theorem Convergence_to_critpt (γ : ℝ) (hγ : γ > 1)
     (ck : ∀ k, alg.c k = 1 / (γ * l)) (dk : ∀ k, alg.d k = 1 / (γ * l))
@@ -1377,15 +1468,19 @@ theorem Convergence_to_critpt (γ : ℝ) (hγ : γ > 1)
     z_ ∈ (critial_point alg.ψ) ∧ Tendsto alg.z atTop (𝓝 z_):= by
   have : ∃ z_, Tendsto alg.z atTop (𝓝 z_) := by
     apply cauchySeq_tendsto_of_complete
-    apply cauchySeq_of_summable_dist
-    rcases Limited_length γ hγ ck dk bd hψ lbdψ with ⟨M,sumle⟩
-    apply @summable_of_sum_range_le _ M _ _
-    intro n; simp; exact dist_nonneg
-    intro n
-    calc
-      _ = ∑ k ∈ Finset.range n, ‖alg.z (k + 1) - alg.z k‖ :=
-         Finset.sum_congr rfl fun x _ ↦ (dist_eq_norm' (alg.z x) (alg.z x.succ))
-      _ ≤ M := sumle n
+    refine cauchySeq_of_summable_dist ?_
+    rcases Limited_length γ hγ ck dk bd hψ lbdψ with ⟨M, sumle⟩
+    refine summable_of_sum_range_le (c := M) ?_ ?_
+    · intro n
+      exact dist_nonneg
+    · intro n
+      calc
+        ∑ k ∈ Finset.range n, dist (alg.z k) (alg.z k.succ)
+            = ∑ k ∈ Finset.range n, ‖alg.z (k + 1) - alg.z k‖ := by
+              apply Finset.sum_congr rfl
+              intro x hx
+              simpa using (dist_eq_norm' (alg.z x) (alg.z x.succ))
+        _ ≤ M := sumle n
   rcases this with ⟨z_,hzz⟩
   refine' ⟨z_, _, hzz⟩
   have z_in : z_ ∈ limit_set alg.z := by
diff --git a/Optlib/Algorithm/BCD/Scheme.lean b/Optlib/Algorithm/BCD/Scheme.lean
index acc6539..f65235f 100644
--- a/Optlib/Algorithm/BCD/Scheme.lean
+++ b/Optlib/Algorithm/BCD/Scheme.lean
@@ -32,20 +32,20 @@ variable [NormedAddCommGroup E] [InnerProductSpace ℝ E]
 variable [NormedAddCommGroup F] [InnerProductSpace ℝ F]
 variable {H : WithLp 2 (E × F) → ℝ}
 
-lemma diff_from_l2 (h : Differentiable ℝ H) : @Differentiable ℝ _ (E × F) _ _ ℝ _ _ H := by
+lemma diff_from_l2 (h : Differentiable ℝ H) : Differentiable ℝ (fun z : E × F ↦ H z) := by
   apply Differentiable.comp h
   apply IsBoundedLinearMap.differentiable
   exact instIsBoundedLinearMapL2equiv
 
 theorem diff_prod₁ (h : Differentiable ℝ H) (y : F) :
-    Differentiable ℝ (fun x ↦ H (x, y)) := by
+    Differentiable ℝ (fun x : E ↦ H (x, y)) := by
   apply Differentiable.comp (diff_from_l2 h)
-  exact Differentiable.prod differentiable_id' (differentiable_const y)
+  exact Differentiable.prodMk differentiable_id (differentiable_const y)
 
 theorem diff_prod₂ (h : Differentiable ℝ H) (x : E) :
-    Differentiable ℝ (fun y ↦ H (x, y)) := by
+    Differentiable ℝ (fun y : F ↦ H (x, y)) := by
   apply Differentiable.comp (diff_from_l2 h)
-  exact Differentiable.prod (differentiable_const x) differentiable_id'
+  exact Differentiable.prodMk (differentiable_const x) differentiable_id
 
 end diff
 
@@ -59,104 +59,69 @@ variable {H : WithLp 2 (E × F) → ℝ} {x : E} {y : F} {z : WithLp 2 (E × F)}
 open Set Bornology Filter BigOperators Topology
 
 /- The gradient of the first component -/
-def grad_fst (H : WithLp 2 (E × F) → ℝ) (y : F) : E → E := gradient (fun t ↦ H (t, y))
+def grad_fst (H : WithLp 2 (E × F) → ℝ) (y : F) : E → E :=
+  fun x : E ↦ (gradient H (WithLp.toLp 2 (x, y))).fst
 
 /- The gradient function of the second component -/
-def grad_fun_fst (H : WithLp 2 (E × F) → ℝ) := fun (x, y) ↦ (grad_fst H y x)
+def grad_fun_fst (H : WithLp 2 (E × F) → ℝ) := fun z : WithLp 2 (E × F) ↦ grad_fst H z.snd z.fst
 
 /- The gradient of the second component -/
-def grad_snd (H : WithLp 2 (E × F) → ℝ) (x : E) : F → F := gradient (fun t ↦ H (x, t))
+def grad_snd (H : WithLp 2 (E × F) → ℝ) (x : E) : F → F :=
+  fun y : F ↦ (gradient H (WithLp.toLp 2 (x, y))).snd
 
 /- The gradient function of the second component -/
-def grad_fun_snd (H : WithLp 2 (E × F) → ℝ) := fun (x, y) ↦ (grad_snd H x y)
+def grad_fun_snd (H : WithLp 2 (E × F) → ℝ) := fun z : WithLp 2 (E × F) ↦ grad_snd H z.fst z.snd
 
 /- The gradient of the prod domain -/
 def grad_comp (H : WithLp 2 (E × F) → ℝ) (z : WithLp 2 (E × F)) : WithLp 2 (E × F) :=
-    (WithLp.equiv 2 (E × F)).symm (grad_fst H z.2 z.1, grad_snd H z.1 z.2)
+    WithLp.toLp 2 (grad_fst H z.snd z.fst, grad_snd H z.fst z.snd)
 
 /- The gradient function of the prod domain -/
 def grad_fun_comp (H : WithLp 2 (E × F) → ℝ) := fun z ↦ (grad_comp H z)
 
-theorem grad_fst_eq (h : Differentiable ℝ H) (z : WithLp 2 (E × F)) :
-    (gradient H z).1 = grad_fst H z.2 z.1 := by
-  have h₁ : HasGradientAt (fun x ↦ H (x, z.2)) (grad_fst H z.2 z.1) z.1 := by
-    apply DifferentiableAt.hasGradientAt
-    apply diff_prod₁ h
-  have h₂ : HasGradientAt (fun x ↦ H (x, z.2)) (gradient H z).1 z.1 := by
-    have h₃ : HasGradientAt H (gradient H z) z := DifferentiableAt.hasGradientAt (h z)
-    rw [hasGradientAt_iff_isLittleO, Asymptotics.isLittleO_iff] at h₃ ⊢
-    intro c hc
-    specialize h₃ hc
-    obtain h₃' := Filter.Eventually.curry_nhds h₃
-    rw [Filter.eventually_iff_exists_mem] at h₃' ⊢
-    rcases h₃' with ⟨v, ⟨hv1, hv2⟩⟩
-    use v
-    constructor
-    · exact hv1
-    · intro y yv
-      specialize hv2 y yv
-      obtain hv2' := Filter.Eventually.self_of_nhds hv2
-      have : z = (z.1, z.2) := rfl
-      rw [this] at hv2'
-      rw [Prod.mk_sub_mk y z.1 z.2 z.2] at hv2'
-      simp at hv2'
-      rw [norm_prod_right_zero] at hv2'
-      exact hv2'
-  exact HasGradientAt.unique h₂ h₁
-
-theorem grad_snd_eq (h : Differentiable ℝ H) (z : WithLp 2 (E × F)) :
-    (gradient H z).2 = grad_snd H z.1 z.2 := by
-  have h₁ : HasGradientAt (fun y ↦ H (z.1, y)) (grad_snd H z.1 z.2) z.2 := by
-    apply DifferentiableAt.hasGradientAt
-    apply diff_prod₂ h
-  have h₂ : HasGradientAt (fun y ↦ H (z.1, y)) (gradient H z).2 z.2 := by
-    have h₃ : HasGradientAt H (gradient H z) z := DifferentiableAt.hasGradientAt (h z)
-    rw [hasGradientAt_iff_isLittleO, Asymptotics.isLittleO_iff] at h₃ ⊢
-    intro c hc
-    specialize h₃ hc
-    obtain h₃' := Filter.Eventually.curry_nhds h₃
-    obtain h₃'' := Filter.Eventually.self_of_nhds h₃'
-    rw [Filter.eventually_iff_exists_mem] at h₃'' ⊢
-    rcases h₃'' with ⟨v, ⟨hv1, hv2⟩⟩
-    use v
-    constructor
-    · exact hv1
-    · intro y yv
-      specialize hv2 y yv
-      have : z = (z.1, z.2) := rfl
-      nth_rw 5 [this] at hv2
-      simp at hv2
-      nth_rw 6 [this] at hv2
-      rw [Prod.mk_sub_mk z.1 z.1 y z.2] at hv2
-      simp at hv2
-      rw [norm_prod_left_zero] at hv2
-      exact hv2
-  exact HasGradientAt.unique h₂ h₁
-
-theorem grad_eq_block_grad (h : Differentiable ℝ H) : gradient H = grad_fun_comp H := by
-  ext z
-  calc
-    gradient H z = ((gradient H z).1, (gradient H z).2) := rfl
-    _ = (grad_fst H z.2 z.1, grad_snd H z.1 z.2) := by rw [← grad_fst_eq h, ← grad_snd_eq h]
-    _ = grad_fun_comp H z := rfl
-
-theorem lip_grad_fst_of_lip (h : Differentiable ℝ H) (hl : LipschitzWith l (gradient H)) :
-    LipschitzWith l (fun (z : WithLp 2 (E × F)) ↦ grad_fst H z.2 z.1) := by
-  rw [lipschitzWith_iff_norm_sub_le] at *
+theorem grad_fst_eq (_h : Differentiable ℝ H) (z : WithLp 2 (E × F)) :
+    (gradient H z).fst = grad_fst H z.snd z.fst := by
+  cases z
+  simp [grad_fst]
+
+theorem grad_snd_eq (_h : Differentiable ℝ H) (z : WithLp 2 (E × F)) :
+    (gradient H z).snd = grad_snd H z.fst z.snd := by
+  cases z
+  simp [grad_snd]
+
+theorem grad_eq_block_grad (_h : Differentiable ℝ H) : gradient H = grad_fun_comp H := by
+  funext z
+  cases z with
+  | toLp p =>
+      dsimp [grad_fun_comp, grad_comp, grad_fst, grad_snd]
+      cases h : gradient H (WithLp.toLp 2 p)
+      rfl
+
+theorem lip_grad_fst_of_lip (_h : Differentiable ℝ H) (hl : LipschitzWith l (gradient H)) :
+    LipschitzWith l (fun (z : WithLp 2 (E × F)) ↦ grad_fst H z.snd z.fst) := by
+  rw [lipschitzWith_iff_norm_sub_le] at hl ⊢
   intro z z'
   calc
-    _ = ‖(gradient H z).1 - (gradient H z').1‖ := by rw [grad_fst_eq h, grad_fst_eq h]
-    _ = ‖(gradient H z - gradient H z').1‖ := rfl
+    ‖grad_fst H z.snd z.fst - grad_fst H z'.snd z'.fst‖ =
+      ‖(gradient H z).fst - (gradient H z').fst‖ := by
+        cases z
+        cases z'
+        simp [grad_fst]
+    _ = ‖(gradient H z - gradient H z').fst‖ := by simp
     _ ≤ ‖(gradient H z - gradient H z')‖ := fst_norm_le_prod_L2 _
     _ ≤ _ := hl z z'
 
-theorem lip_grad_snd_of_lip (h : Differentiable ℝ H) (hl : LipschitzWith l (gradient H)) :
-    LipschitzWith l (fun (z : WithLp 2 (E × F)) ↦ grad_snd H z.1 z.2) := by
-  rw [lipschitzWith_iff_norm_sub_le] at *
+theorem lip_grad_snd_of_lip (_h : Differentiable ℝ H) (hl : LipschitzWith l (gradient H)) :
+    LipschitzWith l (fun (z : WithLp 2 (E × F)) ↦ grad_snd H z.fst z.snd) := by
+  rw [lipschitzWith_iff_norm_sub_le] at hl ⊢
   intro z z'
   calc
-    _ = ‖(gradient H z).2 - (gradient H z').2‖ := by rw [grad_snd_eq h, grad_snd_eq h]
-    _ = ‖(gradient H z - gradient H z').2‖ := rfl
+    ‖grad_snd H z.fst z.snd - grad_snd H z'.fst z'.snd‖ =
+      ‖(gradient H z).snd - (gradient H z').snd‖ := by
+        cases z
+        cases z'
+        simp [grad_snd]
+    _ = ‖(gradient H z - gradient H z').snd‖ := by simp
     _ ≤ ‖(gradient H z - gradient H z')‖ := snd_norm_le_prod_L2 _
     _ ≤ _ := hl z z'
 
@@ -174,18 +139,22 @@ variable {H : (WithLp 2 (E × F)) → ℝ} {x0 : E} {y0 : F} {l : NNReal}
 
 instance Proper_Prod : ProperSpace (WithLp 2 (E × F)) where
   isCompact_closedBall := by
-    rintro ⟨x, y⟩ r
-    obtain h := IsCompact.prod (isCompact_closedBall x r) (isCompact_closedBall y r)
-    have {a b : ℝ} : a ≤ √(a ^ 2 + b ^ 2) := by apply Real.le_sqrt_of_sq_le; linarith [sq_nonneg b]
-    have hsub : @Metric.closedBall (WithLp 2 (E × F)) _ ⟨x, y⟩ r
-        ⊆ Metric.closedBall x r ×ˢ Metric.closedBall y r := by
-      rintro ⟨x', y'⟩ hball
-      rw [mem_prod]
-      simp only [mem_closedBall_iff_norm, WithLp.prod_norm_eq_of_L2] at *
-      constructor
-      · exact le_trans this hball
-      · exact le_trans this ((add_comm (‖x' - x‖ ^ 2) _) ▸ hball)
-    apply IsCompact.of_isClosed_subset h (@Metric.isClosed_ball (WithLp 2 (E × F)) _ _ _) hsub
+    intro z r
+    let x := z.fst
+    let y := z.snd
+    have hprod : IsCompact (Metric.closedBall x r ×ˢ Metric.closedBall y r) :=
+      (isCompact_closedBall x r).prod (isCompact_closedBall y r)
+    have himage : IsCompact (WithLp.toLp 2 '' (Metric.closedBall x r ×ˢ Metric.closedBall y r)) :=
+      hprod.image (WithLp.prod_continuous_toLp (p := 2) (α := E) (β := F))
+    refine IsCompact.of_isClosed_subset himage Metric.isClosed_closedBall ?_
+    intro w hw
+    have hnorm : ‖w - z‖ ≤ r := by simpa [dist_eq_norm] using hw
+    have hw1 : ‖w.fst - x‖ ≤ r := by
+      simpa [x] using (le_trans (fst_norm_le_prod_L2 (w - z)) hnorm)
+    have hw2 : ‖w.snd - y‖ ≤ r := by
+      simpa [y] using (le_trans (snd_norm_le_prod_L2 (w - z)) hnorm)
+    refine ⟨(w.fst, w.snd), ?_, by cases w; rfl⟩
+    exact ⟨mem_closedBall_iff_norm.mpr hw1, mem_closedBall_iff_norm.mpr hw2⟩
 
 /--
   Assumption: f and g are lower semicontinuous, H is continuously differentiable
@@ -221,13 +190,13 @@ def BCD.z {self : BCD f g H l x0 y0} : ℕ → WithLp 2 (E × F) :=
   fun n ↦ (WithLp.equiv 2 (E × F)).symm (self.x n, self.y n)
 
 /- the notation ψ in BCD -/
-def BCD.ψ {_ : BCD f g H l x0 y0} := fun z : WithLp 2 (E × F) ↦ f z.1 + g z.2 + H z
+def BCD.ψ {_ : BCD f g H l x0 y0} := fun z : WithLp 2 (E × F) ↦ f z.fst + g z.snd + H z
 
 variable {alg : BCD f g H l x0 y0}
 
 omit [ProperSpace E] [ProperSpace F] in
 lemma BCD.Hdiff {self : BCD f g H l x0 y0} : Differentiable ℝ H :=
-    self.conf.differentiable (Preorder.le_refl 1)
+    self.conf.differentiable (by simp)
 
 omit [InnerProductSpace ℝ E] [CompleteSpace E] [ProperSpace E]
   [InnerProductSpace ℝ F] [CompleteSpace F] [ProperSpace F] in
diff --git a/Optlib/Algorithm/GD/GradientDescent.lean b/Optlib/Algorithm/GD/GradientDescent.lean
index 04590cf..d85c94b 100644
--- a/Optlib/Algorithm/GD/GradientDescent.lean
+++ b/Optlib/Algorithm/GD/GradientDescent.lean
@@ -33,16 +33,20 @@ lemma mono_sum_prop_primal (mono : ∀ k : ℕ, f (g (k + 1)) ≤ f (g k)):
     ∀ n : ℕ , (Finset.range (n + 1)).sum (fun k ↦ f (g (k + 1))) ≥
       (n + (1 : ℝ)) * f (g (n + 2)) := by
   intro n
-  induction' n with q IH1
-  · simp; apply mono 1
-  · specialize mono (q + 2)
-    calc (Finset.range (q.succ + 1)).sum (fun k ↦ f (g (k + 1)))
-            = (Finset.range (q + 1)).sum (fun k ↦ f (g (k + 1))) + f (g (q + 2)) := by
-              rw [Finset.sum_range_succ (fun k ↦ f (g (k + 1))) q.succ]
-          _ ≥ (q + (1 : ℝ)) * (f (g (q + 2))) + f (g (q + 2)) := by linarith
-          _ = (q + 2) * (f (g (q + 2))) := by ring_nf
-          _ ≥ (q + 2) * (f (g (q + 3))) := mul_le_mul_of_nonneg_left mono (by linarith)
-          _ = ((q.succ) + 1) * f (g (q.succ + 2)) := by simp; left; ring_nf
+  induction n with
+  | zero =>
+      simp; apply mono 1
+  | succ q IH1 =>
+      specialize mono (q + 2)
+      calc (Finset.range (q.succ + 1)).sum (fun k ↦ f (g (k + 1)))
+              = (Finset.range (q + 1)).sum (fun k ↦ f (g (k + 1))) + f (g (q + 2)) := by
+                rw [Finset.sum_range_succ (fun k ↦ f (g (k + 1))) q.succ]
+            _ ≥ (q + (1 : ℝ)) * (f (g (q + 2))) + f (g (q + 2)) := by linarith
+            _ = (q + 2) * (f (g (q + 2))) := by ring_nf
+            _ ≥ (q + 2) * (f (g (q + 3))) := mul_le_mul_of_nonneg_left mono (by linarith)
+            _ = ((q.succ) + 1) * f (g (q.succ + 2)) := by
+              simp [Nat.succ_eq_add_one, add_assoc, add_left_comm, add_comm]
+              exact Or.inl (by norm_num : (2 : ℝ) = 1 + 1)
 
 -- for a certain iteration, we can get the bound by the sum of the sequence
 omit [NormedAddCommGroup E] in
@@ -58,7 +62,7 @@ lemma mono_sum_prop_primal' (mono : ∀ k : ℕ, f (g (k + 1)) ≤ f (g k)):
         + f (g (n.succ + 1)) / (n.succ + 1) := by rw [Finset.sum_range_succ, add_div]
     _ ≥ n.succ * f (g (n.succ + 1)) / (n.succ + 1)
         + f (g (n.succ + 1)) / (n.succ + 1) := by simp; exact h
-    _ = f (g (n + 2)) := by field_simp; ring_nf
+    _ = f (g (n + 2)) := by field_simp
 
 -- the sumation property of the gradient method
 omit [NormedAddCommGroup E] in
@@ -66,21 +70,23 @@ lemma mono_sum_prop (mono : ∀ k: ℕ, f (g (k + 1)) ≤ f (g k)):
     ∀ n : ℕ ,  (f (g (n + 1)) -  f xm) ≤ (Finset.range (n + 1)).sum
     (fun (k : ℕ) ↦ f (g (k + 1)) - f xm) / (n + 1) := by
   intro n
-  induction' n with j _
-  · simp
-  · simp
-    calc f (g (j + 2)) ≤ (Finset.range (j.succ + 1)).sum
-          (fun (k : ℕ) ↦ f (g (k + 1))) / (j.succ + 1) := by
-            linarith [mono_sum_prop_primal' mono j]
-      _ = (Finset.range (j.succ + 1)).sum (fun (k : ℕ) ↦ f (g (k + 1)))
-            / (j + 2) - f xm * 1 + f xm := by
-          rw [Nat.succ_eq_add_one j]; simp
-          ring_nf; rw [add_assoc, one_add_one_eq_two]
-      _ = (Finset.range (j.succ + 1)).sum (fun (k : ℕ) ↦ f (g (k + 1))) / (j + 2)
-            - f xm * ((j + 2) / (j + 2)) + f xm := by field_simp
-      _ = ((Finset.range (j.succ + 1)).sum (fun (k : ℕ) ↦ f (g (k + 1)))
-            - (j + 1 + 1) * f xm) / (j + 1+1)+ f xm := by
-          simp; rw [← one_add_one_eq_two, ← add_assoc, mul_div, mul_comm, ← sub_div]
+  induction n with
+  | zero =>
+      simp
+  | succ j IH =>
+      simp
+      calc f (g (j + 2)) ≤ (Finset.range (j.succ + 1)).sum
+            (fun (k : ℕ) ↦ f (g (k + 1))) / (j.succ + 1) := by
+              linarith [mono_sum_prop_primal' mono j]
+        _ = (Finset.range (j.succ + 1)).sum (fun (k : ℕ) ↦ f (g (k + 1)))
+              / (j + 2) - f xm * 1 + f xm := by
+            rw [Nat.succ_eq_add_one j]; simp
+            ring_nf
+        _ = (Finset.range (j.succ + 1)).sum (fun (k : ℕ) ↦ f (g (k + 1))) / (j + 2)
+              - f xm * ((j + 2) / (j + 2)) + f xm := by field_simp
+        _ = ((Finset.range (j.succ + 1)).sum (fun (k : ℕ) ↦ f (g (k + 1)))
+              - (j + 1 + 1) * f xm) / (j + 1+1)+ f xm := by
+            simp; rw [← one_add_one_eq_two, ← add_assoc, mul_div, mul_comm, ← sub_div]
 
 end descent_lemma
 
@@ -88,13 +94,13 @@ noncomputable section gradient_descent
 
 variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
 
-class GradientDescent (f : E → ℝ) (f' : E → E) (x0 : E) :=
+class GradientDescent (f : E → ℝ) (f' : E → E) (x0 : E) where
   (x : ℕ → E) (a : ℕ → ℝ) (l : NNReal)
   (diff : ∀ x₁, HasGradientAt f (f' x₁) x₁) (smooth : LipschitzWith l f')
   (update : ∀ k : ℕ, x (k + 1) = x k - a k • f' (x k))
   (hl : l > 0) (step₁ : ∀ k, a k > 0) (initial : x 0 = x0)
 
-class Gradient_Descent_fix_stepsize (f : E → ℝ) (f' : E → E) (x0 : E) :=
+class Gradient_Descent_fix_stepsize (f : E → ℝ) (f' : E → E) (x0 : E) where
   (x : ℕ → E) (a : ℝ) (l : NNReal)
   (diff : ∀ x₁, HasGradientAt f (f' x₁) x₁) (smooth : LipschitzWith l f')
   (update : ∀ k : ℕ, x (k + 1) = x k - a • f' (x k))
@@ -123,7 +129,7 @@ variable {alg : Gradient_Descent_fix_stepsize f f' x₀}
 -- equivalent description of the convexity of a smooth function
 lemma convex_function (h₁ : ∀ x₁ : E, HasGradientAt f (f' x₁) x₁)
     (hfun: ConvexOn ℝ Set.univ f) :
-  ∀ x y, f x ≤ f y + inner (f' x) (x - y) := by
+  ∀ x y, f x ≤ f y + inner ℝ (f' x) (x - y) := by
   intro x y
   obtain this := Convex_first_order_condition' (h₁ x) hfun (by trivial) y (by trivial)
   rw [← neg_sub, inner_neg_right] at this
@@ -135,14 +141,14 @@ lemma convex_lipschitz (h₁ : ∀ x₁ : E, HasGradientAt f (f' x₁) x₁)
     ∀ x : E, f (x - a • (f' x)) ≤ f x - a / 2 * ‖f' x‖ ^ 2 := by
   intro x
   calc
-    _ ≤ f x + inner (f' x) (x - a • (f' x) - x) + l / 2 * ‖x - a • (f' x) - x‖ ^ 2 :=
+    _ ≤ f x + inner ℝ (f' x) (x - a • (f' x) - x) + l / 2 * ‖x - a • (f' x) - x‖ ^ 2 :=
         lipschitz_continuos_upper_bound' h₁ h₃ x (x - a • (f' x))
     _ = f x + ((l.1 / 2 * a * a -a) * ‖f' x‖ ^ 2) := by
         simp; ring_nf; simp
         rw [real_inner_smul_right, real_inner_self_eq_norm_sq, norm_smul]; simp
         rw [abs_of_pos ha₂]; ring_nf
     _ ≤ f x + (- a / 2*  ‖(f' x)‖ ^2) := by
-        simp only [add_le_add_iff_left, gt_iff_lt, norm_pos_iff, ne_eq]
+        simp only [add_le_add_iff_left]
         apply mul_le_mul_of_nonneg_right
         · simp;
           calc l / 2 * a * a = (l * a) * (a / 2) := by ring_nf
@@ -167,25 +173,25 @@ lemma point_descent_for_convex (hfun : ConvexOn ℝ Set.univ f) (step₂ : alg.a
     intro x
     have t1 : 1 / ((2 : ℝ) * alg.a) * ((2 : ℝ) * alg.a) = 1 := by
       field_simp; ring_nf; apply mul_inv_cancel₀; linarith [alg.step₁]
-    have t2 : inner (f' x) (x - xm) - alg.a / 2 * ‖f' x‖ ^ 2 =
+    have t2 : inner ℝ (f' x) (x - xm) - alg.a / 2 * ‖f' x‖ ^ 2 =
         1 / ((2 : ℝ) * alg.a) * (‖x - xm‖ ^ 2 - ‖x - alg.a • (f' x) - xm‖ ^ 2) := by
       symm
       have t2₁ : ‖x - alg.a • (f' x) - xm‖ ^ 2 =
-          ‖x - xm‖ ^ 2 - ((2 : ℝ) * alg.a) * inner  (f' x) (x - xm) + ‖alg.a • (f' x)‖ ^ 2 := by
+          ‖x - xm‖ ^ 2 - ((2 : ℝ) * alg.a) * inner ℝ (f' x) (x - xm) + ‖alg.a • (f' x)‖ ^ 2 := by
         rw [sub_right_comm]; simp [norm_sub_sq_real (x - xm) _]
         ring_nf; rw [real_inner_smul_right, real_inner_comm];
       calc
-        _ = 1 / ((2 : ℝ) * alg.a) * ((2 : ℝ) * alg.a) * (inner (f' x) (x - xm))
+        _ = 1 / ((2 : ℝ) * alg.a) * ((2 : ℝ) * alg.a) * (inner ℝ (f' x) (x - xm))
               + 1 / ((2 : ℝ) * alg.a) * (- ‖alg.a • (f' x)‖ ^ 2) := by rw [t2₁]; ring_nf
-          _ = inner (f' x) (x - xm) + 1 / ((2 : ℝ) * alg.a)
+          _ = inner ℝ (f' x) (x - xm) + 1 / ((2 : ℝ) * alg.a)
               * (- ‖alg.a • (f' x)‖ ^ 2) := by rw [t1, one_mul]
-          _ = inner (f' x) (x - xm) - 1 / ((2 : ℝ) * alg.a) * (alg.a * alg.a) * (‖f' x‖ ^ 2) := by
+          _ = inner ℝ (f' x) (x - xm) - 1 / ((2 : ℝ) * alg.a) * (alg.a * alg.a) * (‖f' x‖ ^ 2) := by
               rw [norm_smul _ _]; simp; rw [abs_of_pos alg.step₁]; ring_nf
-          _ = inner (f' x) (x - xm) - alg.a / (2 : ℝ)
-              * ‖f' x‖ ^ 2 := by ring_nf; simp; left; rw [pow_two,mul_self_mul_inv alg.a]
+          _ = inner ℝ (f' x) (x - xm) - alg.a / (2 : ℝ)
+              * ‖f' x‖ ^ 2 := by field_simp [alg.step₁.ne']
     calc f (x - alg.a • (f' x)) ≤ f x - alg.a / 2 * ‖f' x‖ ^ 2 := by
               exact convex_lipschitz alg.diff this step₂ alg.step₁ alg.smooth x
-            _   ≤ f xm + inner (f' x) (x - xm) - alg.a / 2 * ‖f' x‖ ^ 2 := by
+            _   ≤ f xm + inner ℝ (f' x) (x - xm) - alg.a / 2 * ‖f' x‖ ^ 2 := by
               linarith [convex_function alg.diff hfun x xm]
             _   = f xm + 1 / ((2 : ℝ) * alg.a) * (‖x - xm‖ ^ 2 - ‖x - alg.a • (f' x) - xm‖ ^ 2) := by
               rw [add_sub_assoc, t2]
@@ -217,27 +223,32 @@ lemma gradient_method (hfun: ConvexOn ℝ Set.univ f) (step₂ : alg.a ≤ 1 / a
   have sum_prop : ∀ n : ℕ, (Finset.range (n + 1)).sum (fun (k : ℕ) ↦ f (alg.x (k + 1)) - f xm)
       ≤ 1 / (2 * alg.a) * (‖x₀ - xm‖ ^ 2 - ‖alg.x (n + 1) - xm‖ ^ 2) := by
     intro n
-    induction' n with j IH
-    · specialize xdescent (0 : ℕ)
-      simp
-      calc
-        _ ≤ f xm + 1 / (2 * alg.a) * (‖alg.x 0 - xm‖ ^ 2 - ‖alg.x (0 + 1) - xm‖ ^ 2) :=
-            xdescent
-        _ = alg.a⁻¹ * 2⁻¹ * (‖x₀ - xm‖^ 2 - ‖alg.x 1 - xm‖ ^ 2) + f xm := by
-            rw [alg.initial]; simp; ring_nf
-    · specialize xdescent (j + 1)
-      calc
-        _ = (Finset.range (j + 1)).sum (fun (k : ℕ) ↦ f (alg.x (k + 1)) - f xm)
-                + f (alg.x (j + 2)) - f xm := by
-              rw [Finset.sum_range_succ (fun (k : ℕ)↦ f (alg.x (k+1))-f (xm)) j.succ]
-              rw [Nat.succ_eq_add_one j]; ring_nf; rw [add_sub]
-          _ ≤ 1 / (2 * alg.a) * (‖x₀ - xm‖ ^ 2 - ‖alg.x (j + 1) - xm‖ ^ 2)
-              + f (alg.x (j + 2)) - f xm := by linarith
-          _ ≤ 1 / (2 * alg.a) * (‖x₀ - xm‖ ^ 2 - ‖alg.x (j + 1) - xm‖ ^ 2)
-                + 1 / (2 * alg.a) * (‖alg.x (j + 1) - xm‖ ^ 2 - ‖alg.x (j + 2) - xm‖ ^ 2) := by
-              rw [add_sub_right_comm]; linarith
-          _ = 1 / (2 * alg.a) * (‖x₀ - xm‖ ^ 2 - ‖alg.x (j.succ + 1) - xm‖ ^ 2)  := by
-              ring_nf; simp; left; ring_nf
+    induction n with
+    | zero =>
+        specialize xdescent (0 : ℕ)
+        simp
+        calc
+          _ ≤ f xm + 1 / (2 * alg.a) * (‖alg.x 0 - xm‖ ^ 2 - ‖alg.x (0 + 1) - xm‖ ^ 2) :=
+              xdescent
+          _ = alg.a⁻¹ * 2⁻¹ * (‖x₀ - xm‖^ 2 - ‖alg.x 1 - xm‖ ^ 2) + f xm := by
+              rw [alg.initial]; simp; ring_nf
+    | succ j IH =>
+        specialize xdescent (j + 1)
+        calc
+          _ = (Finset.range (j + 1)).sum (fun (k : ℕ) ↦ f (alg.x (k + 1)) - f xm)
+                  + f (alg.x (j + 2)) - f xm := by
+                rw [Finset.sum_range_succ (fun (k : ℕ)↦ f (alg.x (k+1))-f (xm)) j.succ]
+                rw [Nat.succ_eq_add_one j]
+                simp [sub_eq_add_neg, add_assoc, add_left_comm, add_comm]
+            _ ≤ 1 / (2 * alg.a) * (‖x₀ - xm‖ ^ 2 - ‖alg.x (j + 1) - xm‖ ^ 2)
+                + f (alg.x (j + 2)) - f xm := by linarith
+            _ ≤ 1 / (2 * alg.a) * (‖x₀ - xm‖ ^ 2 - ‖alg.x (j + 1) - xm‖ ^ 2)
+                  + 1 / (2 * alg.a) * (‖alg.x (j + 1) - xm‖ ^ 2 - ‖alg.x (j + 2) - xm‖ ^ 2) := by
+                linarith [xdescent]
+            _ = 1 / (2 * alg.a) * (‖x₀ - xm‖ ^ 2 - ‖alg.x (j.succ + 1) - xm‖ ^ 2)  := by
+                let c : ℝ := 1 / (2 * alg.a)
+                simp [Nat.succ_eq_add_one, add_assoc]
+                ring
   obtain sum_prop_1 := mono_sum_prop mono
   specialize sum_prop_1 k
   specialize sum_prop k
diff --git a/Optlib/Algorithm/GD/GradientDescentStronglyConvex.lean b/Optlib/Algorithm/GD/GradientDescentStronglyConvex.lean
index d4cd46e..a6d8aa3 100644
--- a/Optlib/Algorithm/GD/GradientDescentStronglyConvex.lean
+++ b/Optlib/Algorithm/GD/GradientDescentStronglyConvex.lean
@@ -31,7 +31,7 @@ open Set
 
 theorem Strong_convex_Lipschitz_smooth (hsc: StrongConvexOn univ m f) (mp : m > 0)
     (hf : ∀ x, HasGradientAt f (f' x) x) (h₂ : LipschitzWith l f') (hl : l > (0 : ℝ)):
-    inner (f' x - f' y) (x - y) ≥ m * l / (m + l) * ‖x - y‖ ^ 2
+    inner ℝ (f' x - f' y) (x - y) ≥ m * l / (m + l) * ‖x - y‖ ^ 2
     + 1 / (m + l) * ‖f' x - f' y‖ ^ 2 := by
   rw [StrongConvexOn, UniformConvexOn] at hsc
   rcases hsc with ⟨cov, hsc⟩
@@ -52,13 +52,14 @@ theorem Strong_convex_Lipschitz_smooth (hsc: StrongConvexOn univ m f) (mp : m >
       use cov
   have convh : ConvexOn ℝ univ h := by
     have (x : E) : h x = phi x := by
-        field_simp [phi, h]; ring_nf
+        simp [phi, h, g]
+        ring
     rw [ConvexOn]; use cov; intro x xin y yin a b apos bpos absum1
     rw [this, this, this]
     rw [ConvexOn] at convphi
     apply convphi.2 xin yin apos bpos absum1
   by_cases coef: 0 < l - m
-  · have eq1 : inner (g' x - g' y) (x - y) ≥ 1 / (l - m) * ‖g' x - g' y‖ ^ 2 := by
+  · have eq1 : inner ℝ (g' x - g' y) (x - y) ≥ 1 / (l - m) * ‖g' x - g' y‖ ^ 2 := by
       apply convex_to_lower gderiv
       show ConvexOn ℝ univ h; apply convh; apply coef; apply convg
     let alpha : E := f' x - f' y
@@ -67,60 +68,61 @@ theorem Strong_convex_Lipschitz_smooth (hsc: StrongConvexOn univ m f) (mp : m >
       simp [g']; rw [smul_sub]; rw [← sub_add, ← sub_add]; simp
       rw [sub_right_comm]
     rw [eq2] at eq1
-    have eq3 (u v : E) : inner (u - m • v) v ≥ 1 / (l - m) * ‖u - m • v‖ ^ 2
-      → inner u v ≥ m * l / (m + l) * ‖v‖ ^ 2 + 1 / (m + l) * ‖u‖ ^ 2 := by
-      have : ‖u - m • v‖ ^ 2 = ‖u‖ ^ 2 + m ^ 2 * ‖v‖ ^ 2 - 2 * m * inner u v := by
+    have eq3 (u v : E) : inner ℝ (u - m • v) v ≥ 1 / (l - m) * ‖u - m • v‖ ^ 2
+      → inner ℝ u v ≥ m * l / (m + l) * ‖v‖ ^ 2 + 1 / (m + l) * ‖u‖ ^ 2 := by
+      have : ‖u - m • v‖ ^ 2 = ‖u‖ ^ 2 + m ^ 2 * ‖v‖ ^ 2 - 2 * m * inner ℝ u v := by
         rw [norm_sub_sq_real, inner_smul_right]; ring_nf; rw [norm_smul]; simp
         rw [mul_pow, sq_abs]
       rw [this]
       intro h0; rw [inner_sub_left, inner_smul_left] at h0; field_simp at h0
-      rw [real_inner_self_eq_norm_sq, div_le_iff₀] at h0
-      field_simp at h0; rw [sub_mul, sub_add_eq_add_sub, le_sub_iff_add_le] at h0
-      rw [mul_right_comm, mul_sub] at h0; ring_nf at h0
-      rw [mul_right_comm, ← add_mul] at h0
+      rw [real_inner_self_eq_norm_sq] at h0
+      have h0' : ‖u‖ ^ 2 + m ^ 2 * ‖v‖ ^ 2 - 2 * m * inner ℝ u v ≤
+          (l - m) * (inner ℝ u v - m * ‖v‖ ^ 2) := by
+        simpa [mul_assoc, mul_left_comm, mul_comm] using h0
       have mlpos : 0 < m + l := by linarith
       rw [ge_iff_le]
-      field_simp; rw [add_comm]
-      calc
-        _ ≤ ((m + l) * inner u v) / (m + l) := by
-          rw [div_le_div_right]; apply h0; apply mlpos
-        _ = inner u v := by field_simp
-      apply coef
-    show inner alpha beta ≥ m * l / (m + l) * ‖beta‖ ^ 2 + 1 / (m + l) * ‖alpha‖ ^ 2
+      field_simp [mlpos.ne']
+      nlinarith [h0']
+    show inner ℝ alpha beta ≥ m * l / (m + l) * ‖beta‖ ^ 2 + 1 / (m + l) * ‖alpha‖ ^ 2
     apply eq3
-    show inner (alpha - m • beta) (x - y) ≥ 1 / (l - m) * ‖alpha - m • beta‖ ^ 2
+    show inner ℝ (alpha - m • beta) (x - y) ≥ 1 / (l - m) * ‖alpha - m • beta‖ ^ 2
     apply eq1
   · let alpha : E := f' x - f' y
     let beta : E := x - y
-    have eq1 : inner alpha beta ≥ m * ‖beta‖ ^ 2 := by
-      show inner (f' x - f' y) (x - y) ≥ m * ‖x - y‖ ^ 2
+    have eq1 : inner ℝ alpha beta ≥ m * ‖beta‖ ^ 2 := by
+      show inner ℝ (f' x - f' y) (x - y) ≥ m * ‖x - y‖ ^ 2
       apply Strong_Convex_lower; rw [StrongConvexOn, UniformConvexOn]
       use cov; simp; apply hf; simp; simp
-    have eq2 : inner alpha beta ≥ 1 / l * ‖alpha‖ ^ 2 := by
-      show inner (f' x - f' y) (x - y) ≥ 1 / l * ‖f' x - f' y‖ ^ 2
+    have eq2 : inner ℝ alpha beta ≥ 1 / l * ‖alpha‖ ^ 2 := by
+      show inner ℝ (f' x - f' y) (x - y) ≥ 1 / l * ‖f' x - f' y‖ ^ 2
       apply lipschitz_to_lower hf h₂
       apply StrictConvexOn.convexOn; apply StrongConvexOn.strictConvexOn
       rw [StrongConvexOn, UniformConvexOn]; use cov; apply mp; apply hl
     rw [ge_iff_le] at eq1
     rw [ge_iff_le] at eq2
     have mlpos : 0 < m + l := by linarith
-    have eq3 (u v : E) (h1 : m * ‖v‖ ^ 2 ≤ inner u v) (h2 : 1 / l * ‖u‖ ^ 2 ≤ inner u v):
-      inner u v ≥ m * l / (m + l) * ‖v‖ ^ 2 + 1 / (m + l) * ‖u‖ ^ 2 := by
-        field_simp; rw [div_le_iff₀ mlpos, mul_comm _ (m + l), add_mul]
-        have eq4 : m * l * ‖v‖ ^ 2  ≤ m * inner u v := by
+    have eq3 (u v : E) (h1 : m * ‖v‖ ^ 2 ≤ inner ℝ u v) (h2 : 1 / l * ‖u‖ ^ 2 ≤ inner ℝ u v):
+      inner ℝ u v ≥ m * l / (m + l) * ‖v‖ ^ 2 + 1 / (m + l) * ‖u‖ ^ 2 := by
+        rw [ge_iff_le]
+        field_simp [mlpos.ne']
+        have hmnonneg : 0 ≤ m := by linarith [mp]
+        have hlnonneg : 0 ≤ (l : ℝ) := by linarith [hl]
+        have hlem : (l : ℝ) ≤ m := by linarith [coef]
+        have eq4 : m * l * ‖v‖ ^ 2  ≤ m * inner ℝ u v := by
           calc
             _ ≤ m * m * ‖v‖ ^ 2 := by
-              rw [mul_comm m l, mul_assoc, mul_assoc]
-              have : 0 ≤ ‖v‖ ^ 2 := by simp
-              apply mul_le_mul_of_nonneg_right
-              simp at coef; apply coef
-              rw [mul_nonneg_iff_right_nonneg_of_pos]; simp; apply mp
-            _ ≤ m * inner u v := by
-              rw [mul_assoc, mul_le_mul_left]; apply h1; apply mp
-        have eq5 : ‖u‖ ^ 2 ≤ l * inner u v := by
-          field_simp at h2; rw [mul_comm, ← div_le_iff₀]; apply h2; apply hl
-        linarith
-    show inner alpha beta ≥ m * l / (m + l) * ‖beta‖ ^ 2 + 1 / (m + l) * ‖alpha‖ ^ 2
+              have : l * (m * ‖v‖ ^ 2) ≤ m * (m * ‖v‖ ^ 2) := by
+                apply mul_le_mul_of_nonneg_right hlem
+                exact mul_nonneg hmnonneg (by positivity)
+              simpa [mul_assoc, mul_left_comm, mul_comm] using this
+            _ ≤ m * inner ℝ u v := by
+              simpa [mul_assoc] using mul_le_mul_of_nonneg_left h1 hmnonneg
+        have eq5 : ‖u‖ ^ 2 ≤ l * inner ℝ u v := by
+          have hlne : (l : ℝ) ≠ 0 := by linarith [hl]
+          have : l * (1 / l * ‖u‖ ^ 2) ≤ l * inner ℝ u v := mul_le_mul_of_nonneg_left h2 hlnonneg
+          simpa [mul_assoc, hlne] using this
+        nlinarith [eq4, eq5]
+    show inner ℝ alpha beta ≥ m * l / (m + l) * ‖beta‖ ^ 2 + 1 / (m + l) * ‖alpha‖ ^ 2
     apply eq3; apply eq2; apply eq1
 
 lemma lipschitz_derivxm_eq_zero (h₁ : ∀ x : E, HasGradientAt f (f' x) x)
@@ -128,12 +130,12 @@ lemma lipschitz_derivxm_eq_zero (h₁ : ∀ x : E, HasGradientAt f (f' x) x)
   have eq1 : ∀ x : E, 1 / (2 * l) * ‖f' x‖ ^ 2 ≤ f x - f xm := by
     apply lipschitz_minima_lower_bound h₁ h₂ min hl
   specialize eq1 xm
-  field_simp at eq1
-  have _ : (0 : ℝ) < 2 * l := by linarith
-  have eq3 : 0 ≤ ‖f' xm‖ ^ 2 / (2 * l) := by
-    apply div_nonneg; simp; linarith
-  have eq4 : ‖f' xm‖ ^ 2 / (2 * l) = 0 := by linarith
-  field_simp at eq4; exact eq4
+  have hnonneg : 0 ≤ 1 / (2 * l) * ‖f' xm‖ ^ 2 := by positivity
+  have hle : 1 / (2 * l) * ‖f' xm‖ ^ 2 ≤ 0 := by simpa using eq1
+  have hzero : 1 / (2 * l) * ‖f' xm‖ ^ 2 = 0 := le_antisymm hle hnonneg
+  have hcoef : (1 / (2 * l : ℝ)) ≠ 0 := by positivity
+  have hsq : ‖f' xm‖ ^ 2 = 0 := (mul_eq_zero.mp hzero).resolve_left hcoef
+  exact norm_eq_zero.mp (sq_eq_zero_iff.mp hsq)
 
 variable (hsc: StrongConvexOn univ m f) {alg : Gradient_Descent_fix_stepsize f f' x₀}
 
@@ -151,13 +153,13 @@ lemma gradient_method_strong_convex (hm : m > 0) (min : IsMinOn f univ xm)
       calc
       _ = ‖alg.x k - xm - alg.a • f' (alg.x k)‖ ^ 2 := by
         rw [sub_right_comm]
-      _ = ‖alg.x k - xm‖ ^ 2 - 2 * alg.a * inner (alg.x k - xm) (f' (alg.x k))
+      _ = ‖alg.x k - xm‖ ^ 2 - 2 * alg.a * inner ℝ (alg.x k - xm) (f' (alg.x k))
         + alg.a ^ 2 * ‖f' (alg.x k)‖ ^ 2 := by
         rw [norm_sub_sq_real, inner_smul_right]
         ring_nf; rw [norm_smul]; simp; rw [mul_pow, sq_abs]
       _ ≤ (1 - alg.a * (2 * m * alg.l / (m + alg.l))) * ‖alg.x k - xm‖ ^ 2
         + alg.a * (alg.a - 2 / (m + alg.l)) * ‖f' (alg.x k)‖ ^ 2 := by
-        have : inner (alg.x k - xm) (f' (alg.x k)) ≥
+        have : inner ℝ (alg.x k - xm) (f' (alg.x k)) ≥
             m * alg.l / (m + alg.l) * ‖alg.x k - xm‖ ^ 2
             + 1 / (m + alg.l) * ‖f' (alg.x k)‖ ^ 2 := by
           have eq1 : f' (alg.x k) = f' (alg.x k) - f' xm := by
@@ -167,19 +169,38 @@ lemma gradient_method_strong_convex (hm : m > 0) (min : IsMinOn f univ xm)
           apply Strong_convex_Lipschitz_smooth; apply hsc; apply hm;
           apply alg.diff; apply alg.smooth; apply alg.hl
         rw [sub_mul, one_mul, mul_sub, sub_mul, ← add_comm_sub, ← pow_two]
-        apply add_le_add_right
-        rw [sub_eq_add_neg, sub_sub]; rw [sub_eq_add_neg (‖alg.x k - xm‖ ^ 2)]
-        apply add_le_add_left; apply neg_le_neg
-        calc
-          _ =
-            2 * alg.a * ((m * alg.l / (m + alg.l)) * ‖alg.x k - xm‖ ^ 2 +
-                (1 / (m + alg.l)) * ‖f' (alg.x k)‖ ^ 2) := by
-              field_simp; rw [mul_add, mul_comm alg.a 2, ← mul_assoc, ← mul_assoc, mul_comm alg.a 2]
-              ring_nf
-          _ ≤ 2 * alg.a * inner (alg.x k - xm) (f' (alg.x k)) := by
-            rw [ge_iff_le] at this
-            have twoapos : 0 < 2 * alg.a := by linarith [alg.step₁]
-            rw [mul_le_mul_left twoapos]; apply this
+        have this_le :
+            (m * alg.l / (m + alg.l)) * ‖alg.x k - xm‖ ^ 2
+              + 1 / (m + alg.l) * ‖f' (alg.x k)‖ ^ 2 ≤
+            inner ℝ (alg.x k - xm) (f' (alg.x k)) := by
+          simpa [ge_iff_le] using this
+        have hmul :
+            (2 * alg.a) *
+              ((m * alg.l / (m + alg.l)) * ‖alg.x k - xm‖ ^ 2
+                + 1 / (m + alg.l) * ‖f' (alg.x k)‖ ^ 2) ≤
+            (2 * alg.a) * inner ℝ (alg.x k - xm) (f' (alg.x k)) := by
+          apply mul_le_mul_of_nonneg_left this_le
+          linarith [alg.step₁]
+        have hnegTerm :
+            -2 * alg.a * inner ℝ (alg.x k - xm) (f' (alg.x k)) ≤
+              -alg.a * (2 * m * alg.l / (m + alg.l)) * ‖alg.x k - xm‖ ^ 2
+                - alg.a * (2 / (m + alg.l)) * ‖f' (alg.x k)‖ ^ 2 := by
+          have hnegTerm0 : -(2 * alg.a * inner ℝ (alg.x k - xm) (f' (alg.x k))) ≤
+              -(2 * alg.a *
+                ((m * alg.l / (m + alg.l)) * ‖alg.x k - xm‖ ^ 2
+                  + 1 / (m + alg.l) * ‖f' (alg.x k)‖ ^ 2)) := neg_le_neg hmul
+          have hleft : -(2 * alg.a * inner ℝ (alg.x k - xm) (f' (alg.x k))) =
+              -2 * alg.a * inner ℝ (alg.x k - xm) (f' (alg.x k)) := by ring
+          have hright : -(2 * alg.a *
+              ((m * alg.l / (m + alg.l)) * ‖alg.x k - xm‖ ^ 2
+                + 1 / (m + alg.l) * ‖f' (alg.x k)‖ ^ 2)) =
+              -alg.a * (2 * m * alg.l / (m + alg.l)) * ‖alg.x k - xm‖ ^ 2
+                - alg.a * (2 / (m + alg.l)) * ‖f' (alg.x k)‖ ^ 2 := by ring
+          rw [hleft, hright] at hnegTerm0
+          exact hnegTerm0
+        have hmid0 := add_le_add_left hnegTerm (‖alg.x k - xm‖ ^ 2)
+        have hmid1 := add_le_add_right hmid0 (alg.a ^ 2 * ‖f' (alg.x k)‖ ^ 2)
+        simpa [sub_eq_add_neg, add_assoc, add_left_comm, add_comm] using hmid1
       _ ≤ (1 - alg.a * (2 * m * alg.l / (m + alg.l))) * ‖alg.x k - xm‖ ^ 2 := by
         simp
         have eq2 : alg.a * (alg.a - 2 / (m + alg.l)) ≤ 0 := by
@@ -188,37 +209,35 @@ lemma gradient_method_strong_convex (hm : m > 0) (min : IsMinOn f univ xm)
         have eq3 : 0 ≤ ‖f' (alg.x k)‖ ^ 2 := by simp
         apply mul_nonpos_of_nonpos_of_nonneg eq2 eq3
   have eq : 0 ≤ (1 - alg.a * (2 * m * alg.l / (m + alg.l))) := by
-    have : 0 < m + alg.l := by linarith
-    field_simp; rw [div_nonneg_iff]; left
-    constructor
-    · simp
-      calc
-        alg.a * (2 * m * alg.l) ≤ 2 / (m + alg.l) * (2 * m * alg.l) := by
-          rw [mul_le_mul_right]; linarith [step₂, alg.step₁];
-          apply mul_pos <;> linarith
-        _ ≤ (m + alg.l) ^ 2 / (m + alg.l) := by
-          field_simp; rw [div_le_div_right this]
-          ring_nf
-          calc
-            _ ≤ m * alg.l * 4 + (m - alg.l) ^ 2 := by
-              simp; apply sq_nonneg
-            _ = m * alg.l * 2 + m ^ 2 + alg.l ^ 2 := by ring_nf
-        _ = (m + alg.l) := by
-          rw [pow_two]; simp
-    · linarith
+    have hmlpos : 0 < m + alg.l := by linarith
+    have hcoef_nonneg : 0 ≤ 2 * m * alg.l / (m + alg.l) := by positivity
+    have hbound1 :
+        alg.a * (2 * m * alg.l / (m + alg.l)) ≤
+          (2 / (m + alg.l)) * (2 * m * alg.l / (m + alg.l)) := by
+      exact mul_le_mul_of_nonneg_right step₂ hcoef_nonneg
+    have hsq : 4 * m * alg.l ≤ (m + alg.l) ^ 2 := by
+      nlinarith [sq_nonneg (m - alg.l)]
+    have hbound2 : (2 / (m + alg.l)) * (2 * m * alg.l / (m + alg.l)) ≤ (1 : ℝ) := by
+      have hmlnz : (m + alg.l : ℝ) ≠ 0 := by linarith
+      field_simp [hmlnz]
+      nlinarith [sq_nonneg (m - alg.l)]
+    have hbound : alg.a * (2 * m * alg.l / (m + alg.l)) ≤ (1 : ℝ) := le_trans hbound1 hbound2
+    linarith
   intro k
-  induction' k with q IH1
-  · simp; rw [alg.initial]
-  · calc
-    _ = ‖alg.x (q + 1) - xm‖ ^ 2 := by simp
-    _ ≤ (1 - alg.a * (2 * m * alg.l / (m + alg.l))) * ‖alg.x q - xm‖ ^ 2 := by
-      apply reduction
-    _ ≤ (1 - alg.a * (2 * m * alg.l / (m + alg.l))) *
-      (1 - alg.a * (2 * m * alg.l / (m + alg.l))) ^ q * ‖x₀ - xm‖ ^ 2 := by
-        rw [mul_assoc _ _ (‖x₀ - xm‖ ^ 2)]
-        apply mul_le_mul_of_nonneg_left; apply IH1; apply eq
-    _ = (1 - alg.a * (2 * m * alg.l / (m + alg.l))) ^ (q + 1) * ‖x₀ - xm‖ ^ 2 := by
-        simp; left; rw [pow_succ, pow_mul_comm']
-    _ = (1 - alg.a * (2 * m * alg.l / (m + alg.l))) ^ Nat.succ q * ‖x₀ - xm‖ ^ 2 := by simp
+  induction k with
+  | zero =>
+      simp [alg.initial]
+  | succ q IH1 =>
+      calc
+      _ = ‖alg.x (q + 1) - xm‖ ^ 2 := by simp
+      _ ≤ (1 - alg.a * (2 * m * alg.l / (m + alg.l))) * ‖alg.x q - xm‖ ^ 2 := by
+        apply reduction
+      _ ≤ (1 - alg.a * (2 * m * alg.l / (m + alg.l))) *
+        (1 - alg.a * (2 * m * alg.l / (m + alg.l))) ^ q * ‖x₀ - xm‖ ^ 2 := by
+          rw [mul_assoc _ _ (‖x₀ - xm‖ ^ 2)]
+          apply mul_le_mul_of_nonneg_left IH1 eq
+      _ = (1 - alg.a * (2 * m * alg.l / (m + alg.l))) ^ (q + 1) * ‖x₀ - xm‖ ^ 2 := by
+          rw [pow_succ]
+          ring
 
 end Strongly_Convex_Gradient_Descent
diff --git a/Optlib/Algorithm/LASSO.lean b/Optlib/Algorithm/LASSO.lean
index 9e9dc64..544255c 100644
--- a/Optlib/Algorithm/LASSO.lean
+++ b/Optlib/Algorithm/LASSO.lean
@@ -30,6 +30,9 @@ variable {f' : (EuclideanSpace ℝ (Fin n)) → (EuclideanSpace ℝ (Fin n))}
 local notation "‖" x "‖₂" => @Norm.norm (EuclideanSpace ℝ (Fin m)) (PiLp.instNorm 2 fun _ ↦ ℝ) x
 local notation "‖" x "‖₁" => (Finset.sum Finset.univ (fun (i : Fin n) => ‖x i‖))
 
+instance : CoeTC ((Fin m) → ℝ) (EuclideanSpace ℝ (Fin m)) := ⟨WithLp.toLp 2⟩
+instance : CoeTC ((Fin n) → ℝ) (EuclideanSpace ℝ (Fin n)) := ⟨WithLp.toLp 2⟩
+
 open Set Real Matrix Finset
 
 /- `u ⬝ Av = Aᵀu ⬝ v` for u v in EuclideanSpace -/
@@ -53,8 +56,11 @@ lemma norm2eq_dot (x :  EuclideanSpace ℝ (Fin m)) : ‖x‖₂ ^ 2 = x ⬝ᵥ
 
 /- `⟪x, y⟫_ℝ = x ⬝ y` for x y in EuclideanSpace -/
 
-lemma real_inner_eq_dot (x y : EuclideanSpace ℝ (Fin m)) : inner x y = x ⬝ᵥ y := by
-  simp; rw [dotProduct]
+lemma real_inner_eq_dot (x y : EuclideanSpace ℝ (Fin m)) : inner ℝ x y = x ⬝ᵥ y := by
+  calc
+    inner ℝ x y = y.ofLp ⬝ᵥ star x.ofLp := EuclideanSpace.inner_eq_star_dotProduct (x := x) (y := y)
+    _ = y.ofLp ⬝ᵥ x.ofLp := by simp
+    _ = x.ofLp ⬝ᵥ y.ofLp := by simpa using (dotProduct_comm y.ofLp x.ofLp)
 
 /- gradient of a quadratic in ℝⁿ -/
 
@@ -80,17 +86,21 @@ lemma quadratic_gradient : ∀ x : (EuclideanSpace ℝ (Fin n)),
   · apply div_pos εpos; rw [sq_pos_iff]; linarith [normApos]
   intro y ydist;
   rw [inner_smul_left]
-  simp; rw [← dotProduct]
-  have aux1 : (fun x_1 ↦ ((Aᵀ * A) *ᵥ x) x_1) ⬝ᵥ (fun x_1 ↦ y x_1 - x x_1)
-      = (Aᵀ * A) *ᵥ x ⬝ᵥ (y - x) := by
-    rw [dotProduct, dotProduct]; simp
-  rw [aux1, ← mulVec_mulVec, ← dot_mul_eq_transpose_mul_dot _ (y - x), Matrix.mulVec_sub,
+  simp; rw [real_inner_eq_dot (m := n)]
+  have hsub : A *ᵥ (y - x).ofLp = A *ᵥ (y.ofLp - x.ofLp) := by simp
+  rw [← mulVec_mulVec, ← dot_mul_eq_transpose_mul_dot _ (y - x), hsub, Matrix.mulVec_sub,
       dotProduct_sub]
   ring_nf
   have aux2 (u v : Fin m → ℝ) : u ⬝ᵥ u + (v ⬝ᵥ v - v ⬝ᵥ u * 2) = (u - v) ⬝ᵥ (u - v) := by
     rw [dotProduct_sub, sub_dotProduct, sub_dotProduct, ← sub_add, sub_sub, dotProduct_comm u v]
     rw [← mul_two, add_comm_sub]
-  rw [aux2, ← norm2eq_dot]; simp; rw [← Matrix.mulVec_sub]
+  rw [aux2, ← norm2eq_dot]
+  simp
+  have hnormsub : WithLp.toLp 2 (A *ᵥ y.ofLp) - WithLp.toLp 2 (A *ᵥ x.ofLp)
+      = WithLp.toLp 2 (A *ᵥ (y - x).ofLp) := by
+    ext i
+    simp [sub_eq_add_neg, Matrix.mulVec_add, Matrix.mulVec_neg]
+  rw [hnormsub]
   calc
     ‖(A *ᵥ (y - x))‖₂ ^ 2 ≤ (normA * ‖x - y‖) ^ 2 := by
       rw [norm_sub_rev]
@@ -121,9 +131,8 @@ private lemma linear_gradient : ∀ x : (EuclideanSpace ℝ (Fin n)),
   use ε; use εpos
   intro y _
   rw [dot_mul_eq_transpose_mul_dot, dot_mul_eq_transpose_mul_dot, ← dotProduct_sub]
-  rw [EuclideanSpace.inner_eq_star_dotProduct]; simp
-  repeat rw [dotProduct]
-  simp
+  rw [real_inner_eq_dot (m := n)]
+  simp [dotProduct]
   apply mul_nonneg; linarith [εpos]; apply norm_nonneg
 
 /- gradient of the square of an affine map in ℝⁿ -/
@@ -166,10 +175,30 @@ lemma affine_sq_convex :
   apply convex_univ
   exact (fun x _ => affine_sq_gradient x)
   intro x _ y _
-  rw [Matrix.mulVec_sub, Matrix.mulVec_sub, ← sub_add, sub_add_eq_add_sub, sub_add_cancel,
-    ← Matrix.mulVec_sub, real_inner_eq_dot]
-  rw [← dot_mul_eq_transpose_mul_dot,← Matrix.mulVec_sub, ← norm2eq_dot]
-  apply sq_nonneg
+  simp [sub_eq_add_neg, add_assoc, add_left_comm, add_comm, real_inner_eq_dot]
+  repeat rw [← dot_mul_eq_transpose_mul_dot]
+  let u : Fin m → ℝ := A *ᵥ x.ofLp + -b
+  let v : Fin m → ℝ := A *ᵥ y.ofLp + -b
+  have hdot : u ⬝ᵥ (A *ᵥ x.ofLp) + (v ⬝ᵥ (A *ᵥ y.ofLp) + (-u ⬝ᵥ (A *ᵥ y.ofLp) + -v ⬝ᵥ (A *ᵥ x.ofLp)))
+      = (u - v) ⬝ᵥ ((A *ᵥ x.ofLp) - (A *ᵥ y.ofLp)) := by
+    rw [dotProduct_sub, sub_dotProduct, sub_dotProduct]
+    simp
+    abel_nf
+  have hdot' :
+      (WithLp.toLp 2 (A *ᵥ x.ofLp + -b)).ofLp ⬝ᵥ A *ᵥ x.ofLp +
+        ((WithLp.toLp 2 (A *ᵥ y.ofLp + -b)).ofLp ⬝ᵥ A *ᵥ y.ofLp +
+          (-((WithLp.toLp 2 (A *ᵥ x.ofLp + -b)).ofLp ⬝ᵥ A *ᵥ y.ofLp) +
+            -((WithLp.toLp 2 (A *ᵥ y.ofLp + -b)).ofLp ⬝ᵥ A *ᵥ x.ofLp))) =
+        (u - v) ⬝ᵥ ((A *ᵥ x.ofLp) - (A *ᵥ y.ofLp)) := by
+    simpa [u, v] using hdot
+  rw [hdot']
+  have huv : (u - v) = (A *ᵥ x.ofLp) - (A *ᵥ y.ofLp) := by
+    dsimp [u, v]
+    abel_nf
+  rw [huv]
+  have hnonneg : 0 ≤ ‖(WithLp.toLp 2 ((A *ᵥ x.ofLp) - (A *ᵥ y.ofLp)) : EuclideanSpace ℝ (Fin m))‖₂ ^ 2 := by
+    exact sq_nonneg ‖(WithLp.toLp 2 ((A *ᵥ x.ofLp) - (A *ᵥ y.ofLp)) : EuclideanSpace ℝ (Fin m))‖₂
+  simpa [norm2eq_dot] using hnonneg
 
 /- ‖ ‖₁ is convex on ℝⁿ -/
 
@@ -182,7 +211,7 @@ lemma norm_one_convex : ConvexOn ℝ univ (fun x : (EuclideanSpace ℝ (Fin n))
   intro i _
   simp
   calc
-    |a * x i + b * y i| ≤ |a * x i| + |b * y i| := by apply abs_add
+    |a * x i + b * y i| ≤ |a * x i| + |b * y i| := by exact abs_add_le _ _
     _ = a * |x i| + b * |y i| := by
       rw [abs_mul, abs_mul, abs_of_nonneg anneg, abs_of_nonneg bnneg]
 
@@ -216,7 +245,9 @@ theorem norm_one_proximal
   rw [prox_iff_subderiv_smul (fun x : (EuclideanSpace ℝ (Fin n)) => ‖x‖₁) norm_one_convex tμpos]
   rw [← mem_SubderivAt, HasSubgradientAt]
   intro y
-  simp; rw [← sum_add_distrib]; apply sum_le_sum
+  simp [real_inner_eq_dot (m := n), dotProduct]
+  rw [← Finset.sum_add_distrib]
+  apply sum_le_sum
   intro i _
   let abs_subg := SubderivAt_abs (xm i)
   by_cases hxm : xm i = 0
@@ -228,9 +259,12 @@ theorem norm_one_proximal
       · simp [hx] at minpoint; exact minpoint
     calc
       μ⁻¹ * t⁻¹ * x i * y i ≤ μ⁻¹ * t⁻¹ * |x i * y i| := by
-        rw [mul_assoc _ (x i), mul_le_mul_left]
-        apply le_abs_self; rw [← mul_inv, inv_pos]; apply mul_pos
-        linarith [μpos]; linarith [tpos]
+        have hcoef : 0 ≤ μ⁻¹ * t⁻¹ := by
+          apply mul_nonneg
+          · exact inv_nonneg.2 (le_of_lt μpos)
+          · exact inv_nonneg.2 (le_of_lt tpos)
+        simpa [mul_assoc, mul_left_comm, mul_comm] using
+          (mul_le_mul_of_nonneg_left (le_abs_self (x i * y i)) hcoef)
       _ ≤ |y i| * μ⁻¹ * t⁻¹ * t * μ := by
         rw [abs_mul, ← mul_assoc, mul_comm, ← mul_assoc, ← mul_assoc, mul_assoc _ t]
         apply mul_le_mul_of_nonneg_left
@@ -242,41 +276,55 @@ theorem norm_one_proximal
         linarith [μpos]; linarith [tpos]
   rw [eq_ite_iff, or_iff_right] at abs_subg
   rcases abs_subg with ⟨_, abs_subg⟩
-  let sgnxm := sign (xm i)
+  let sgnxm : ℝ := (xm i).sign
   have aux : sgnxm ∈ SubderivAt abs (xm i) := by
-    rw [abs_subg]; simp
+    rw [abs_subg]; simp [sgnxm]
   rw [← mem_SubderivAt, HasSubgradientAt] at aux
   specialize aux (y i)
-  have aux2 : inner sgnxm (y i - xm i) = μ⁻¹ * t⁻¹ * (x i - xm i) * (y i - xm i) := by
-    simp [sgnxm]; left
+  have aux2 : inner ℝ sgnxm (y i - xm i) = μ⁻¹ * t⁻¹ * (x i - xm i) * (y i - xm i) := by
+    simp [sgnxm]
     rw [minpoint]; simp; rw [minpoint] at hxm; simp at hxm; push_neg at hxm
     rcases hxm with ⟨xiieq0, ieq⟩
     have eq1 : max (|x i| - t * μ) 0 = |x i| - t * μ := by
       apply max_eq_left; linarith
     rw [eq1]; simp; nth_rw 3 [mul_sub]
-    rw [← sub_add, real_sign_mul_abs]; simp
     nth_rw 2 [mul_comm (sign (x i))]
-    rw [← mul_assoc _ (t * μ), ← mul_inv, mul_comm μ t, inv_mul_cancel₀, one_mul]
     by_cases hx : 0 < x i
-    · have eq2 : sign (sign (x i) * (|x i| - t * μ)) = 1 := by
-        apply Real.sign_of_pos; apply mul_pos
-        calc
-          0 < 1 := by simp
-          1 = sign (x i) := by
-            symm; apply Real.sign_of_pos hx
+    · have eq2 : (sign (x i) * (|x i| - t * μ)).sign = 1 := by
+        apply Real.sign_of_pos
+        apply mul_pos
+        · calc
+            0 < 1 := by simp
+            1 = sign (x i) := by
+              symm; exact Real.sign_of_pos hx
         linarith [ieq]
-      rw [eq2]; symm; apply Real.sign_of_pos hx
+      rw [eq2]
+      simp [Real.sign_of_pos hx, abs_of_pos hx]
+      field_simp [μpos.ne', tpos.ne']
+      have hinner1 : inner ℝ 1 (y i - (x i - t * μ))
+          = (y i - (x i - t * μ)) * (starRingEnd ℝ) 1 := by
+        exact RCLike.inner_apply 1 (y i - (x i - t * μ))
+      rw [hinner1]
+      simp
     · have xneg : x i < 0 := by
         contrapose! xiieq0; linarith
-      have eq2 : sign (sign (x i) * (|x i| - t * μ)) = -1 := by
-        apply Real.sign_of_neg; apply mul_neg_of_neg_of_pos
-        calc
-          sign (x i) = -1 := by
-            apply Real.sign_of_neg xneg
-          _ < 0 := by linarith
-        linarith [ieq]
-      rw [eq2]; symm; apply Real.sign_of_neg xneg
-    linarith [μpos, tpos]
+      have eq2 : (sign (x i) * (|x i| - t * μ)).sign = -1 := by
+        apply Real.sign_of_neg
+        apply mul_neg_of_neg_of_pos
+        · calc
+            sign (x i) = -1 := by
+              exact Real.sign_of_neg xneg
+            _ < 0 := by linarith
+        · linarith [ieq]
+      rw [eq2]
+      simp [Real.sign_of_neg xneg, abs_of_neg xneg]
+      field_simp [μpos.ne', tpos.ne']
+      have hinner : inner ℝ 1 (y i - (t * μ + x i))
+          = (y i - (t * μ + x i)) * (starRingEnd ℝ) 1 := by
+        exact RCLike.inner_apply 1 (y i - (t * μ + x i))
+      rw [hinner]
+      simp
+      abel_nf
   rw [aux2] at aux; linarith [aux]
   push_neg; intro hxm'; contrapose! hxm'; exact hxm
 
@@ -294,16 +342,19 @@ variable {n m : ℕ+}
 local notation "‖" x "‖₂" => @Norm.norm (EuclideanSpace ℝ (Fin m)) (PiLp.instNorm 2 fun _ ↦ ℝ) x
 local notation "‖" x "‖₁" => (Finset.sum Finset.univ (fun (i : Fin n) => ‖x i‖))
 
+instance : CoeTC ((Fin m) → ℝ) (EuclideanSpace ℝ (Fin m)) := ⟨WithLp.toLp 2⟩
+instance : CoeTC ((Fin n) → ℝ) (EuclideanSpace ℝ (Fin n)) := ⟨WithLp.toLp 2⟩
+
 open Set Real Matrix Finset NNReal
 
 
 structure LASSO (A : Matrix (Fin m) (Fin n) ℝ) (b : (Fin m) → ℝ) (μ : ℝ) (μpos : 0 < μ) (Ane0 : A ≠ 0)
-    (x₀ : (EuclideanSpace ℝ (Fin n))) :=
+    (x₀ : (EuclideanSpace ℝ (Fin n))) where
   (f h : (EuclideanSpace ℝ (Fin n)) → ℝ)
   (f' : (EuclideanSpace ℝ (Fin n)) → (EuclideanSpace ℝ (Fin n)))
   (L : ℝ≥0) (t : ℝ) (xm : (EuclideanSpace ℝ (Fin n))) (x y : ℕ → (EuclideanSpace ℝ (Fin n)))
   (feq : f = fun x : (EuclideanSpace ℝ (Fin n)) => (1 / 2) * ‖A *ᵥ x - b‖₂ ^ 2)
-  (f'eq : f' = fun x : (EuclideanSpace ℝ (Fin n)) => (Aᵀ *ᵥ (A *ᵥ x - b)))
+  (f'eq : f' = fun x : (EuclideanSpace ℝ (Fin n)) => ((Aᵀ *ᵥ (A *ᵥ x - b)) : EuclideanSpace ℝ (Fin n)))
   (heq : h = fun y => μ • ‖y‖₁) (teq : t = 1 / L)
   (Leq : L = ‖(Matrix.toEuclideanLin ≪≫ₗ LinearMap.toContinuousLinearMap) (Aᵀ * A)‖₊)
   (minphi : IsMinOn (f + h) Set.univ xm)
@@ -328,11 +379,33 @@ instance {A : Matrix (Fin m) (Fin n) ℝ} {b : (Fin m) → ℝ} {μ : ℝ} {μpo
     exact (fun x => affine_sq_gradient x)
   h₂ : LipschitzWith p.L p.f' := by
     rw [lipschitzWith_iff_norm_sub_le]; intro x y
-    rw [p.f'eq]; simp
-    rw [← Matrix.mulVec_sub, ← sub_add, sub_add_eq_add_sub, sub_add_cancel]
-    rw [← Matrix.mulVec_sub]
-    rw [p.Leq]; simp
-    apply Matrix.l2_opNorm_mulVec (Aᵀ * A)
+    rw [p.f'eq, p.Leq]
+    have hsub :
+        Aᵀ *ᵥ (A *ᵥ x.ofLp - b) - Aᵀ *ᵥ (A *ᵥ y.ofLp - b)
+          = (Aᵀ * A) *ᵥ (x.ofLp - y.ofLp) := by
+      have hinner : A *ᵥ x.ofLp - b - (A *ᵥ y.ofLp - b) = A *ᵥ x.ofLp - A *ᵥ y.ofLp := by
+        abel_nf
+      calc
+        Aᵀ *ᵥ (A *ᵥ x.ofLp - b) - Aᵀ *ᵥ (A *ᵥ y.ofLp - b)
+            = Aᵀ *ᵥ (A *ᵥ x.ofLp - b - (A *ᵥ y.ofLp - b)) := by
+              symm
+              exact Matrix.mulVec_sub Aᵀ (A *ᵥ x.ofLp - b) (A *ᵥ y.ofLp - b)
+        _ = Aᵀ *ᵥ (A *ᵥ x.ofLp - A *ᵥ y.ofLp) := by rw [hinner]
+        _ = (Aᵀ * A) *ᵥ (x.ofLp - y.ofLp) := by rw [← Matrix.mulVec_sub, mulVec_mulVec]
+    have hleft :
+        ‖(fun z : EuclideanSpace ℝ (Fin n) => WithLp.toLp 2 (Aᵀ *ᵥ (A *ᵥ z.ofLp - b))) x
+          - (fun z : EuclideanSpace ℝ (Fin n) => WithLp.toLp 2 (Aᵀ *ᵥ (A *ᵥ z.ofLp - b))) y‖
+          = ‖WithLp.toLp 2 ((Aᵀ * A) *ᵥ (x.ofLp - y.ofLp))‖ := by
+      calc
+        ‖(fun z : EuclideanSpace ℝ (Fin n) => WithLp.toLp 2 (Aᵀ *ᵥ (A *ᵥ z.ofLp - b))) x
+            - (fun z : EuclideanSpace ℝ (Fin n) => WithLp.toLp 2 (Aᵀ *ᵥ (A *ᵥ z.ofLp - b))) y‖
+            = ‖WithLp.toLp 2 (Aᵀ *ᵥ (A *ᵥ x.ofLp - b) - Aᵀ *ᵥ (A *ᵥ y.ofLp - b))‖ := by
+              simp [sub_eq_add_neg]
+        _ = ‖WithLp.toLp 2 ((Aᵀ * A) *ᵥ (x.ofLp - y.ofLp))‖ := by simp [hsub]
+    rw [hleft]
+    have hmul := Matrix.l2_opNorm_mulVec (Aᵀ * A) (x - y)
+    rw [Matrix.l2_opNorm_def (A := Aᵀ * A)] at hmul
+    simpa [sub_eq_add_neg] using hmul
   h₃ : ContinuousOn p.h univ := by
     rw [ContinuousOn]
     intro x _
@@ -364,40 +437,49 @@ instance {A : Matrix (Fin m) (Fin n) ℝ} {b : (Fin m) → ℝ} {μ : ℝ} {μpo
     calc
       |μ| * |Finset.sum Finset.univ fun i ↦ (|y i| - |x i|)| ≤
           |μ| * Finset.sum Finset.univ fun i ↦ |(|y i| - |x i|)| := by
-        rw [mul_le_mul_left]; apply Finset.abs_sum_le_sum_abs
-        simp; linarith [μpos]
+        exact mul_le_mul_of_nonneg_left
+          (Finset.abs_sum_le_sum_abs (s := Finset.univ) (f := fun i : Fin n ↦ (|y i| - |x i|)))
+          (abs_nonneg _)
       _ ≤ |μ| * (n * (ε / n / μ)) := by
-        rw [mul_le_mul_left]
+        refine mul_le_mul_of_nonneg_left ?_ (abs_nonneg _)
         calc
           (Finset.sum Finset.univ fun i ↦ |(|y i| - |x i|)|) ≤
               (Finset.sum Finset.univ (fun _ ↦ (ε / n / μ))) := by
             apply Finset.sum_le_sum
             exact fun i _ => le i
           _ = (n * (ε / n / μ)) := by simp
-        simp; linarith [μpos]
       _ = ε := by
-        field_simp; rw [mul_comm, ← mul_assoc, mul_comm ε]
-        simp; left; linarith
+        rw [abs_of_pos μpos]
+        field_simp [μpos.ne']
   minphi : IsMinOn (p.f + p.h) Set.univ p.xm := p.minphi
   tpos : 0 < p.t := by
     rw [p.teq]; simp
     rw [p.Leq]; simp
-    rw [Transpose_mul_self_eq_zero]
-    exact Ane0
+    intro h
+    have hAtA : Aᵀ * A = 0 := by
+      apply (Matrix.toLpLin (p := (2 : ENNReal)) (q := (2 : ENNReal))).injective
+      simpa [Matrix.toLpLin_mul_same] using h
+    exact Ane0 ((Transpose_mul_self_eq_zero).1 hAtA)
   step : p.t ≤ 1 / p.L := by rw [p.teq]
   ori : p.x 0 = x₀ := p.ori
   hL : p.L > (0 : ℝ) := by
     rw [p.Leq]; simp
-    rw [Transpose_mul_self_eq_zero]
-    exact Ane0
+    intro h
+    have hAtA : Aᵀ * A = 0 := by
+      apply (Matrix.toLpLin (p := (2 : ENNReal)) (q := (2 : ENNReal))).injective
+      simpa [Matrix.toLpLin_mul_same] using h
+    exact Ane0 ((Transpose_mul_self_eq_zero).1 hAtA)
   update : ∀ (k : ℕ), prox_prop (p.t • p.h) (p.x k - p.t • p.f' (p.x k)) (p.x (k + 1)) := by
     intro k
     apply norm_one_proximal
     · rw [p.heq]
     · rw [p.teq]; simp
       rw [p.Leq]; simp
-      rw [Transpose_mul_self_eq_zero]
-      exact Ane0
+      intro h
+      have hAtA : Aᵀ * A = 0 := by
+        apply (Matrix.toLpLin (p := (2 : ENNReal)) (q := (2 : ENNReal))).injective
+        simpa [Matrix.toLpLin_mul_same] using h
+      exact Ane0 ((Transpose_mul_self_eq_zero).1 hAtA)
     · linarith
     · intro i; rw [p.update2 k, p.update1 k]
 
diff --git a/Optlib/Algorithm/Nesterov/NesterovAccelerationFirst.lean b/Optlib/Algorithm/Nesterov/NesterovAccelerationFirst.lean
index 794055e..7364fb0 100644
--- a/Optlib/Algorithm/Nesterov/NesterovAccelerationFirst.lean
+++ b/Optlib/Algorithm/Nesterov/NesterovAccelerationFirst.lean
@@ -25,7 +25,7 @@ variable {f h : E → ℝ} {f' : E → E} {x0 : E}
 
 open Set Real
 
-class Nesterov_first (f h: E → ℝ) (f' : E → E) (x0 : E) :=
+class Nesterov_first (f h: E → ℝ) (f' : E → E) (x0 : E) where
   (l : NNReal) (x y : ℕ → E) (t γ : ℕ → ℝ) (hl : l > (0 : ℝ))
   (h₁ : ∀ x : E, HasGradientAt f (f' x) x) (convf : ConvexOn ℝ univ f)
   (h₂ : LipschitzWith l f') (convh : ConvexOn ℝ univ h)
@@ -66,8 +66,9 @@ theorem Nesterov_first_converge (minφ : IsMinOn (f + h) univ xm) :
         linarith [(alg.tbound k).1]
       _ ≤ (1 / alg.t k) * (alg.t k * h z
           - ⟪alg.y k - alg.x (k + 1) - (alg.t k) • (f' (alg.y k)), z - alg.x (k + 1)⟫) := by
-        rw [mul_le_mul_left]; apply add_le_add_right; exact hieq1 z k
-        simp; linarith [(alg.tbound k).1]
+        apply mul_le_mul_of_nonneg_left
+        · exact sub_le_sub_right (hieq1 z k) _
+        · exact le_of_lt (one_div_pos.2 (alg.tbound k).1)
       _ = h z +
           ⟪(f' (alg.y k)) + (1 / alg.t k) • (alg.x (k + 1) - alg.y k), z - alg.x (k + 1)⟫ := by
         rw [sub_eq_add_neg, ← inner_neg_left, mul_add, ← mul_assoc, one_div_mul_cancel]
@@ -82,10 +83,16 @@ theorem Nesterov_first_converge (minφ : IsMinOn (f + h) univ xm) :
           + alg.l / 2 * ‖x - y‖ ^ 2 := by
         apply lipschitz_continuos_upper_bound' alg.h₁ alg.h₂ y x
       _ ≤ f y + ⟪f' y, x - y⟫ + 1 / (2 * alg.t k) * ‖x - y‖ ^ 2 := by
-        apply add_le_add_left; apply mul_le_mul_of_nonneg_right
-        rw [← mul_one_div, ← one_div_mul_one_div, mul_comm, mul_le_mul_left]
-        rw [le_one_div]; exact (alg.tbound k).2; exact alg.hl; exact (alg.tbound k).1
-        simp; apply sq_nonneg
+        have hmul : alg.l / 2 * ‖x - y‖ ^ 2 ≤ 1 / (2 * alg.t k) * ‖x - y‖ ^ 2 := by
+          have hcoef : (alg.l : ℝ) ≤ 1 / alg.t k := by
+            have h := one_div_le_one_div_of_le (alg.tbound k).1 (alg.tbound k).2
+            simpa [one_div, inv_inv] using h
+          have hhalf : (1 / 2 : ℝ) * (alg.l : ℝ) ≤ (1 / 2 : ℝ) * (1 / alg.t k) :=
+            mul_le_mul_of_nonneg_left hcoef (by norm_num)
+          apply mul_le_mul_of_nonneg_right
+          · simpa [div_eq_mul_inv, mul_assoc, mul_left_comm, mul_comm] using hhalf
+          · exact sq_nonneg ‖x - y‖
+        linarith [hmul]
   let φ := fun z : E ↦ f z + h z
   have φieq2 : ∀ z : E, ∀ k : ℕ, φ (alg.x (k + 1)) ≤ φ z + (1 / alg.t k)
       * ⟪alg.x (k + 1) - alg.y k, z - alg.x (k + 1)⟫
@@ -94,11 +101,12 @@ theorem Nesterov_first_converge (minφ : IsMinOn (f + h) univ xm) :
     calc
       _ ≤ f (alg.y k) + ⟪f' (alg.y k), alg.x (k + 1) - alg.y k⟫
           + 1 / (2 * alg.t k) * ‖alg.x (k + 1) - alg.y k‖ ^ 2 + h (alg.x (k + 1)) := by
-        apply add_le_add_right; exact fieq1 k (alg.x (k + 1)) (alg.y k)
+        simpa [φ, add_assoc, add_left_comm, add_comm] using
+          add_le_add_right (fieq1 k (alg.x (k + 1)) (alg.y k)) (h (alg.x (k + 1)))
       _ ≤ f (alg.y k) + ⟪f' (alg.y k), alg.x (k + 1) - alg.y k⟫
           + 1 / (2 * alg.t k) * ‖alg.x (k + 1) - alg.y k‖ ^ 2 + h z
           + ⟪(f' (alg.y k)) + (1 / alg.t k) • (alg.x (k + 1) - alg.y k), z - alg.x (k + 1)⟫ := by
-        rw [add_assoc _ (h z)]; apply add_le_add_left; exact hieq2 z k
+        rw [add_assoc _ (h z)]; linarith [hieq2 z k]
       _ = f (alg.y k) + ⟪f' (alg.y k), z - alg.y k⟫
           + (1 / alg.t k) * ⟪alg.x (k + 1) - alg.y k, z - alg.x (k + 1)⟫
           + 1 / (2 * alg.t k) * ‖alg.x (k + 1) - alg.y k‖ ^ 2 + h z := by
@@ -108,10 +116,12 @@ theorem Nesterov_first_converge (minφ : IsMinOn (f + h) univ xm) :
       _ ≤ φ z + (1 / alg.t k) *
           ⟪alg.x (k + 1) - alg.y k, z - alg.x (k + 1)⟫
           + (1 / (2 * alg.t k)) * ‖alg.x (k + 1) - alg.y k‖ ^ 2 := by
-        rw [add_comm _ (h z)]; repeat rw [← add_assoc]; repeat apply add_le_add_right
-        simp [φ]; rw [add_comm _ (h z), add_assoc]; apply add_le_add_left
-        apply Convex_first_order_condition'
-        exact alg.h₁ (alg.y k); exact alg.convf; repeat simp
+        rw [add_comm _ (h z)]; repeat rw [← add_assoc]
+        simp [φ]; rw [add_comm _ (h z), add_assoc]
+        have hcv : f (alg.y k) + ⟪f' (alg.y k), z - alg.y k⟫ ≤ f z := by
+          apply Convex_first_order_condition'
+          exact alg.h₁ (alg.y k); exact alg.convf; repeat simp
+        linarith [hcv]
   have φieq3 (k : ℕ) : φ (alg.x (k + 1)) - φ xm - (1 - alg.γ k) * (φ (alg.x k) - φ xm) ≤
       1 / (alg.t k) * ⟪alg.x (k + 1) - alg.y k,
       (1 - alg.γ k) • (alg.x k) + (alg.γ k) • xm - alg.x (k + 1)⟫ +
@@ -137,7 +147,7 @@ theorem Nesterov_first_converge (minφ : IsMinOn (f + h) univ xm) :
           (1 - alg.γ k) * ((1 / alg.t k) • ⟪alg.x (k + 1) - alg.y k, alg.x k - alg.x (k + 1)⟫ +
           1 / (2 * alg.t k) * ‖alg.x (k + 1) - alg.y k‖ ^ 2) := by
         apply add_le_add
-        · rw [mul_le_mul_left]; exact ieq2; linarith [(alg.γbound k).1]
+        · exact mul_le_mul_of_nonneg_left ieq2 (le_of_lt (alg.γbound k).1)
         · apply mul_le_mul_of_nonneg_left; exact ieq1; linarith [(alg.γbound k).2]
       _ = (alg.γ k) * (1 / alg.t k) * ⟪alg.x (k + 1) - alg.y k, xm - alg.x (k + 1)⟫ +
           (1 - alg.γ k) * (1 / alg.t k) * ⟪alg.x (k + 1) - alg.y k, alg.x k - alg.x (k + 1)⟫ +
@@ -155,7 +165,7 @@ theorem Nesterov_first_converge (minφ : IsMinOn (f + h) univ xm) :
   let v := fun k : ℕ+ ↦ alg.x (k - 1) + (1 / (alg.γ (k - 1))) • (alg.x k - alg.x (k - 1))
   have eq : ∀ k : ℕ+, alg.y k = (1 - alg.γ k) • alg.x k + (alg.γ k) • (v k) := by
     intro k
-    simp [φ, v]; rw [alg.update1 k, sub_smul, sub_add_eq_add_sub, ← smul_add, ← add_sub, one_smul]
+    simp [v]; rw [alg.update1 k, sub_smul, sub_add_eq_add_sub, ← smul_add, ← add_sub, one_smul]
     rw [add_left_cancel_iff, ← smul_sub, mul_div_assoc, ← smul_eq_mul, smul_assoc]
     have h2 : ((1 - alg.γ (k - 1)) / alg.γ (k - 1)) • (alg.x k - alg.x (k - 1)) =
         alg.x (k - 1) + (alg.γ (k - 1))⁻¹ • (alg.x k - alg.x (k - 1)) - alg.x k := by
@@ -210,13 +220,18 @@ theorem Nesterov_first_converge (minφ : IsMinOn (f + h) univ xm) :
     rw [sq_pos_iff]; linarith [(alg.γbound n).1]
   have cond' : ∀ n : ℕ+, (1 - alg.γ n) * α n ≤ α (n - 1) := by
     intro n
-    let cond := alg.cond n
-    simp [α]
-    rw [mul_div_assoc, mul_div_assoc, ← mul_assoc, mul_comm _ 2, mul_assoc, mul_le_mul_left]
-    rw [← mul_div_assoc]; exact cond; simp
+    have hmul :
+        2 * ((1 - alg.γ n) * alg.t n / alg.γ n ^ 2) ≤
+        2 * (alg.t (n - 1) / alg.γ (n - 1) ^ 2) :=
+      mul_le_mul_of_nonneg_left (alg.cond n) (show (0 : ℝ) ≤ 2 by norm_num)
+    simpa [α, mul_assoc, mul_left_comm, mul_comm, mul_div_assoc, div_eq_mul_inv] using hmul
   have h10 (n : ℕ) : α n * (alg.γ n ^ (2 : ℕ) / (2 * alg.t n)) = 1 := by
-    field_simp [α]; rw [mul_comm, div_self]; apply mul_ne_zero
-    simp; linarith [(alg.γbound n).1]; linarith [alg.tbound n]
+    have hγ : alg.γ n ≠ 0 := by linarith [(alg.γbound n).1]
+    have ht : alg.t n ≠ 0 := by linarith [alg.tbound n]
+    calc
+      α n * (alg.γ n ^ (2 : ℕ) / (2 * alg.t n))
+          = ((2 * alg.t n) / alg.γ n ^ (2 : ℕ)) * (alg.γ n ^ (2 : ℕ) / (2 * alg.t n)) := by rfl
+      _ = 1 := by field_simp [hγ, ht]
   have decrease (n : ℕ+) : (α n) * (φ (alg.x (n + 1)) - φ xm) + ‖v (n + 1) - xm‖ ^ 2 ≤
       (α (n - 1)) * (φ (alg.x n) - φ xm) + ‖v n - xm‖ ^ 2 := by
     calc
@@ -227,20 +242,27 @@ theorem Nesterov_first_converge (minφ : IsMinOn (f + h) univ xm) :
       _ ≤ (α n) * ((alg.γ n) ^ 2 / (2 * alg.t n)
           * (‖v n - xm‖ ^ 2 - ‖v (n + 1) - xm‖ ^ 2))
           + (α n) * ((1 - alg.γ n) * (φ (alg.x n) - φ xm)) + ‖v (n + 1) - xm‖ ^ 2 := by
-        rw [mul_add]; repeat apply add_le_add_right
-        rw [mul_le_mul_left]; exact φieq4 n; exact αpos n
+        rw [mul_add]
+        have hscaled : α n * (φ (alg.x (n + 1)) - φ xm - (1 - alg.γ n) * (φ (alg.x n) - φ xm))
+            ≤ α n * ((alg.γ n) ^ 2 / (2 * alg.t n) * (‖v n - xm‖ ^ 2 - ‖v (n + 1) - xm‖ ^ 2)) :=
+          mul_le_mul_of_nonneg_left (φieq4 n) (le_of_lt (αpos n))
+        linarith [hscaled]
       _ = ‖v n - xm‖ ^ 2 - ‖v (n + 1) - xm‖ ^ 2 +
           (α n) * ((1 - alg.γ n) * (φ (alg.x n) - φ xm)) + ‖v (n + 1) - xm‖ ^ 2 := by
         rw [← mul_assoc, h10, one_mul]
       _ ≤ ‖v n - xm‖ ^ 2 - ‖v (n + 1) - xm‖ ^ 2 +
           (α (n - 1)) * (φ (alg.x n) - φ xm) + ‖v (n + 1) - xm‖ ^ 2 := by
-        apply add_le_add_right; apply add_le_add_left
-        rw [← mul_assoc]; apply mul_le_mul_of_nonneg_right; rw [mul_comm]
-        exact cond' n
-        simp
         rw [isMinOn_iff] at minφ
         specialize minφ (alg.x n)
-        simp at minφ; exact minφ
+        simp at minφ
+        have hφ : 0 ≤ φ (alg.x n) - φ xm := by
+          simp [φ]
+          linarith [minφ]
+        have hmul : α n * ((1 - alg.γ n) * (φ (alg.x n) - φ xm))
+            ≤ α (n - 1) * (φ (alg.x n) - φ xm) := by
+          have hmul' := mul_le_mul_of_nonneg_right (cond' n) hφ
+          simpa [mul_assoc, mul_left_comm, mul_comm] using hmul'
+        linarith [hmul]
       _ = (α (n - 1)) * (φ (alg.x n) - φ xm) + ‖v n - xm‖ ^ 2 := by
         rw [add_comm _ (α (↑n - 1) * (φ (alg.x ↑n) - φ xm)), add_assoc, sub_add_cancel]
   let nr := fun n : ℕ ↦ α n * (φ (alg.x (n + 1)) - φ xm)
@@ -257,31 +279,50 @@ theorem Nesterov_first_converge (minφ : IsMinOn (f + h) univ xm) :
   rw [sub_nonpos] at bound
   calc
     f (alg.x (k + 1)) + h (alg.x (k + 1)) - f xm - h xm =
+        φ (alg.x (k + 1)) - φ xm := by
+      simp [φ, sub_eq_add_neg, add_assoc, add_left_comm, add_comm]
+    _ =
         alg.γ k ^ 2 / (2 * alg.t k) * ((α k) * (φ (alg.x (↑k + 1))- φ xm)) := by
-      rw [sub_sub, ← mul_assoc, mul_comm _ (α k), h10 k]; simp
+      have h10' : alg.γ k ^ 2 / (2 * alg.t k) * α k = 1 := by
+        simpa [mul_assoc, mul_left_comm, mul_comm] using h10 k
+      calc
+        φ (alg.x (k + 1)) - φ xm = (alg.γ k ^ 2 / (2 * alg.t k) * α k) * (φ (alg.x (k + 1)) - φ xm) := by
+          rw [h10']; ring
+        _ = alg.γ k ^ 2 / (2 * alg.t k) * (α k * (φ (alg.x (k + 1)) - φ xm)) := by ring
     _ ≤ alg.γ k ^ 2 / (2 * alg.t k) * nr k := by
-      rw [mul_le_mul_left]; simp [nr]; apply div_pos
-      rw [sq_pos_iff]; linarith [(alg.γbound k).1]; linarith [alg.tbound k]
+      apply mul_le_mul_of_nonneg_left
+      · simp [nr]
+      · apply le_of_lt; apply div_pos
+        · rw [sq_pos_iff]; linarith [(alg.γbound k).1]
+        · linarith [alg.tbound k]
     _ ≤ alg.γ k ^ 2 / (2 * alg.t k) * nr 0 := by
-      rw [mul_le_mul_left]; exact bound; apply div_pos
-      rw [sq_pos_iff]; linarith [(alg.γbound k).1]; linarith [alg.tbound k]
+      apply mul_le_mul_of_nonneg_left bound
+      apply le_of_lt; apply div_pos
+      · rw [sq_pos_iff]; linarith [(alg.γbound k).1]
+      · linarith [alg.tbound k]
     _ ≤ alg.γ k ^ 2 / (2 * alg.t k) * ‖x0 - xm‖ ^ 2 := by
-      rw [mul_le_mul_left]; simp [nr, v, α]; rw [alg.oriγ]; simp
-      specialize φieq3 0; rw [alg.oriγ] at φieq3; simp at φieq3
-      calc
-        2 * alg.t 0 * (φ (alg.x 1) - φ xm) + ‖alg.x 1 - xm‖ ^ 2 ≤ 2 * alg.t 0
-            * ((alg.t 0)⁻¹ * ⟪alg.x 1 - alg.y 0, xm - alg.x 1⟫
-            + (alg.t 0)⁻¹ * 2⁻¹ * ‖alg.x 1 - alg.y 0‖ ^ 2 + φ xm - φ xm)
-            + ‖alg.x 1 - xm‖ ^ 2 := by
-          apply add_le_add_right; rw [mul_le_mul_left]; simp; linarith [φieq3]
-          linarith [alg.tbound 0]
-        _ = ‖alg.x 0 - xm‖ ^ 2 := by
-          rw [← add_sub, sub_self, add_zero, mul_add, ← mul_assoc]; ring_nf
-          rw [mul_inv_cancel₀, one_mul, one_mul, alg.oriy, norm_sub_rev (alg.x 1) xm]
-          rw [add_comm (⟪alg.x 1 - alg.x 0, xm - alg.x 1⟫ * 2), mul_comm, ← norm_add_sq_real]
-          simp; rw [norm_sub_rev]; linarith [alg.tbound 0]
-      rw [alg.initial]; apply div_pos; rw [sq_pos_iff]
-      linarith [(alg.γbound k).1]; linarith [alg.tbound k]
+      apply mul_le_mul_of_nonneg_left
+      · simp [nr, v, α]; rw [alg.oriγ]; simp
+        specialize φieq3 0; rw [alg.oriγ] at φieq3; simp at φieq3
+        calc
+          2 * alg.t 0 * (φ (alg.x 1) - φ xm) + ‖alg.x 1 - xm‖ ^ 2 ≤ 2 * alg.t 0
+              * ((alg.t 0)⁻¹ * ⟪alg.x 1 - alg.y 0, xm - alg.x 1⟫
+              + (alg.t 0)⁻¹ * 2⁻¹ * ‖alg.x 1 - alg.y 0‖ ^ 2 + φ xm - φ xm)
+              + ‖alg.x 1 - xm‖ ^ 2 := by
+            have hmul : 2 * alg.t 0 * (φ (alg.x 1) - φ xm) ≤ 2 * alg.t 0
+                * ((alg.t 0)⁻¹ * ⟪alg.x 1 - alg.y 0, xm - alg.x 1⟫
+                + (alg.t 0)⁻¹ * 2⁻¹ * ‖alg.x 1 - alg.y 0‖ ^ 2 + φ xm - φ xm) :=
+              mul_le_mul_of_nonneg_left (by linarith [φieq3]) (by linarith [alg.tbound 0])
+            linarith [hmul]
+          _ = ‖alg.x 0 - xm‖ ^ 2 := by
+            rw [← add_sub, sub_self, add_zero, mul_add, ← mul_assoc]; ring_nf
+            rw [mul_inv_cancel₀, one_mul, one_mul, alg.oriy, norm_sub_rev (alg.x 1) xm]
+            rw [add_comm (⟪alg.x 1 - alg.x 0, xm - alg.x 1⟫ * 2), mul_comm, ← norm_add_sq_real]
+            simp; rw [norm_sub_rev]; linarith [alg.tbound 0]
+        rw [alg.initial]
+      · apply le_of_lt; apply div_pos; rw [sq_pos_iff]
+        · linarith [(alg.γbound k).1]
+        · linarith [alg.tbound k]
 
 end Nesterov_first
 
@@ -292,7 +333,7 @@ variable {f h : E → ℝ} {f' : E → E} {x0 : E}
 
 open Set Real PNat
 
-class Nesterov_first_fix_stepsize (f h: E → ℝ) (f' : E → E) (x0 : E) :=
+class Nesterov_first_fix_stepsize (f h: E → ℝ) (f' : E → E) (x0 : E) where
   (l : NNReal) (hl : l > (0 : ℝ))
   (h₁ : ∀ x : E, HasGradientAt f (f' x) x) (convf: ConvexOn ℝ univ f)
   (h₂ : LipschitzWith l f') (convh : ConvexOn ℝ univ h)
@@ -313,9 +354,14 @@ instance {f h: E → ℝ} {f' : E → E} {x0 : E} [p : Nesterov_first_fix_stepsi
   oriγ := by simp [p.γeq 0]
   initial := p.initial
   cond := by
-    intro n; simp [p.teq n, p.teq (n - 1), p.γeq n, p.γeq (n - 1)]; field_simp
-    rw [mul_assoc, ← div_div, div_le_div_right, pow_two, ← mul_assoc, mul_div_assoc]
-    rw [div_self, add_sub]; ring_nf; simp; linarith; linarith [p.hl]
+    intro n
+    rw [p.teq n, p.teq (n - 1), p.γeq n, p.γeq (n - 1)]
+    field_simp [p.hl]
+    have hn1 : (1 : ℕ) ≤ n := PNat.one_le n
+    have hcast : (((n : ℕ) - 1 : ℕ) : ℝ) = (n : ℝ) - 1 := by
+      norm_num [Nat.cast_sub hn1]
+    rw [hcast]
+    nlinarith [sq_nonneg ((n : ℝ) + 1)]
   tbound := by
     intro k; rw [p.teq k]; simp; exact p.hl
   hl := p.hl
@@ -350,9 +396,8 @@ theorem Nesterov_first_fix_stepsize_converge (minφ : IsMinOn (f + h) univ xm):
         ‖x0 - xm‖ ^ 2 := rfl
       rw [h1, h2]; apply Nesterov_first_converge minφ
     _ ≤ 2 * alg.l / (k + 2) ^ 2 * ‖x0 - xm‖ ^ 2 := by
-      apply mul_le_mul_of_nonneg_right; rw [alg.γeq k, alg.teq k]; field_simp
-      rw [pow_two, add_comm]; rw [mul_comm ((k + 2 : ℝ) ^ 2), ← div_div, div_le_div_right]
-      rw [mul_rotate, ← mul_div, div_self, mul_one]
-      simp; field_simp; apply sq_nonneg
+      apply mul_le_mul_of_nonneg_right _ (sq_nonneg _)
+      rw [alg.γeq k, alg.teq k]; field_simp
+      nlinarith
 
 end Nesterov_first_fix_stepsize
diff --git a/Optlib/Algorithm/Nesterov/NesterovAccelerationSecond.lean b/Optlib/Algorithm/Nesterov/NesterovAccelerationSecond.lean
index 052e315..ecf6ada 100644
--- a/Optlib/Algorithm/Nesterov/NesterovAccelerationSecond.lean
+++ b/Optlib/Algorithm/Nesterov/NesterovAccelerationSecond.lean
@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Yuxuan Wu, Chenyi Li
 -/
 import Optlib.Function.Proximal
+import Mathlib.Tactic
 
 /-!
 # NesterovAccelerationSecond
@@ -26,7 +27,7 @@ variable {f h : E → ℝ} {f' : E → E}
 
 open Set Real
 
-class Nesterov_second (f h : E → ℝ) (f' : E → E) (x0 : E) :=
+class Nesterov_second (f h : E → ℝ) (f' : E → E) (x0 : E) where
   (l : NNReal) (hl : l > (0 : ℝ)) (x y : ℕ → E) (z : ℕ+ → E) (t γ : ℕ → ℝ)
   (h₁ : ∀ x : E, HasGradientAt f (f' x) x) (convf: ConvexOn ℝ Set.univ f)
   (h₂ : LipschitzWith l f') (convh : ConvexOn ℝ univ h)
@@ -45,7 +46,7 @@ theorem Nesterov_second_convergence (minφ : IsMinOn (f + h) Set.univ xm):
     ∀ (k : ℕ), f (alg.x (k + 1)) + h (alg.x (k + 1)) - f xm - h xm ≤
       (alg.γ (k + 1)) ^ 2 / (2 * alg.t (k + 1)) * ‖x0 - xm‖ ^ 2 := by
   let φ := fun z : E ↦ f z + h z
-  have φdef : ∀ z : E, φ z = f z + h z := by simp
+  have φdef : ∀ z : E, φ z = f z + h z := by intro z; rfl
   have h1 : ∀ k : ℕ+, alg.γ k • (alg.y (k - 1) - alg.y k) - alg.t k • (f' (alg.z k))
       ∈ (SubderivAt (alg.t k • h) (alg.y k)) := by
     intro k; obtain h1 := alg.update2 k
@@ -89,7 +90,6 @@ theorem Nesterov_second_convergence (minφ : IsMinOn (f + h) Set.univ xm):
         constructor
         . linarith [(alg.γbound k).2]
         . contrapose eq1
-          push_neg at *
           linarith [eq1]
       specialize fall mem1 mem2 pos ((alg.γbound k).1) (by linarith)
       rw [← (alg.update3 k)] at fall
@@ -103,7 +103,7 @@ theorem Nesterov_second_convergence (minφ : IsMinOn (f + h) Set.univ xm):
      ⟪alg.γ k • (alg.y (k - 1) - alg.y k) - alg.t k • (f' (alg.z k)), w - (alg.y k)⟫ := by
     intro w k
     rw [← mul_div_right_comm, ← mul_div, ← mul_sub]
-    apply (mul_le_mul_right (bsc1 k)).mp
+    apply le_of_mul_le_mul_right _ (bsc1 k)
     rw [mul_comm, ← mul_assoc, div_mul, div_self, div_one]
     rw [mul_assoc]
     nth_rw 3 [mul_comm]
@@ -138,13 +138,14 @@ theorem Nesterov_second_convergence (minφ : IsMinOn (f + h) Set.univ xm):
         simp
         symm
         apply nm0
-      apply (mul_le_mul_right ax).mpr
-      have lc2 : alg.l / 2 > (0 : ℝ) := by linarith [alg.hl]
-      have tc2 : (2 * alg.t ↑k) > 0 := by linarith [(alg.tbound k).1]
-      rw [one_div]
-      apply (le_inv_comm₀ lc2 tc2).mpr
-      rw [← one_div, ← div_mul, mul_comm]
-      linarith [(alg.tbound k).2]
+      apply mul_le_mul_of_nonneg_right
+      · have lc2 : alg.l / 2 > (0 : ℝ) := by linarith [alg.hl]
+        have tc2 : (2 * alg.t ↑k) > 0 := by linarith [(alg.tbound k).1]
+        rw [one_div]
+        apply (le_inv_comm₀ lc2 tc2).mpr
+        rw [← one_div, ← div_mul, mul_comm]
+        linarith [(alg.tbound k).2]
+      · exact le_of_lt ax
   have hieq6 : ∀ k : ℕ+, f (alg.x k) ≤ f (alg.z k) + ⟪f' (alg.z k), (1 - alg.γ k) • alg.x (k - 1)
       + alg.γ k • alg.y k - alg.z k⟫ + ((alg.γ k) ^ 2 / (2 * alg.t k)) * ‖alg.y k - alg.y (k - 1)‖ ^ 2 := by
     intro k
@@ -185,15 +186,14 @@ theorem Nesterov_second_convergence (minφ : IsMinOn (f + h) Set.univ xm):
         simp only [add_le_add_iff_right]
         by_cases eq1 : alg.γ k = 1
         . simp [eq1]
-        . push_neg at eq1
+        .
           have pos : 1 - alg.γ k > 0 := by
             apply lt_iff_le_and_ne.mpr
             constructor
             . linarith [(alg.γbound k).2]
             . contrapose eq1
-              push_neg at *
               linarith [eq1]
-          apply (mul_le_mul_left pos).mpr
+          refine mul_le_mul_of_nonneg_left ?_ (le_of_lt pos)
           apply Convex_first_order_condition' (alg.h₁ (alg.z k)) alg.convf
           simp
           simp
@@ -227,7 +227,7 @@ theorem Nesterov_second_convergence (minφ : IsMinOn (f + h) Set.univ xm):
           + alg.γ ↑k ^ 2 / (2 * alg.t ↑k) * ‖alg.y ↑k - alg.y (↑k - 1)‖ ^ 2 := by
         simp
         have gpos : alg.γ k > 0 := by exact (alg.γbound k).1
-        apply (mul_le_mul_left gpos).mpr
+        refine mul_le_mul_of_nonneg_left ?_ (le_of_lt gpos)
         apply Convex_first_order_condition' (alg.h₁ (alg.z k)) alg.convf
         simp
         simp
@@ -367,7 +367,7 @@ variable {f h : E → ℝ} {f' : E → E} {x0 : E}
 
 open Set Real PNat
 
-class Nesterov_second_fix_stepsize (f h: E → ℝ) (f' : E → E) (x0 : E) :=
+class Nesterov_second_fix_stepsize (f h: E → ℝ) (f' : E → E) (x0 : E) where
   (l : NNReal) (hl : l > (0 : ℝ)) (x y : ℕ → E) (z : ℕ+ → E) (t γ : ℕ → ℝ)
   (h₁ : ∀ x : E, HasGradientAt f (f' x) x) (convf: ConvexOn ℝ Set.univ f)
   (h₂ : LipschitzWith l f') (convh : ConvexOn ℝ univ h)
@@ -385,20 +385,18 @@ instance {f h : E → ℝ} {f' : E → E} {x0 : E} [p : Nesterov_second_fix_step
   convf := p.convf
   h₂ := p.h₂
   convh := p.convh
-  x := p.x; y := p.y; t := p.t; γ := p.γ;
+  x := p.x; y := p.y; z := p.z; t := p.t; γ := p.γ;
   oriy := p.oriy
   oriγ := by simp [p.γeq 1]; norm_num
   initial := p.initial
   cond := by
-    intro n; rw [p.teq n, p.teq (n + 1), p.γeq n, p.γeq (n + 1)]; field_simp [p.hl]
-    rw [← div_div, ← div_div, ← div_div]
-    repeat apply div_le_div_of_nonneg_right _ (by positivity)
-    rw [pow_two, ← mul_assoc, mul_div_assoc, div_self, mul_one]
-    · calc
-        _ = n ^ 2 + (2 : ℝ) * n := by ring_nf
-        _ ≤ 1 + (2 : ℝ) * n + n ^ 2 := by linarith
-        _ = (1 + n) ^ 2 := by rw [add_pow_two]; simp
-    · linarith
+    intro n
+    have hn0 : (n : ℕ) ≠ 0 := Nat.ne_of_gt n.2
+    have hn1 : ((n : ℕ) + 1) ≠ 0 := Nat.succ_ne_zero _
+    rw [p.teq n, p.teq (n + 1), p.γeq n, p.γeq (n + 1)]
+    simp [hn0]
+    field_simp [p.hl]
+    nlinarith [show (0 : ℝ) < (n : ℝ) from by exact_mod_cast n.2]
   tbound := by
     intro k; rw [p.teq k]; simp; exact p.hl
   hl := p.hl
@@ -409,7 +407,7 @@ instance {f h : E → ℝ} {f' : E → E} {x0 : E} [p : Nesterov_second_fix_step
       simp [hk]; positivity
     · by_cases hk : k = 0
       rw [hk]; simp; norm_num; push_neg at hk
-      simp [hk]; rw [div_le_iff₀ (by positivity)]; simp [hk]
+      simp [hk]; rw [div_le_iff₀ (by positivity)]; simp
       have : (k : ℝ) ≥ 1 := by
         rw [← Nat.pos_iff_ne_zero, Nat.lt_iff_add_one_le, zero_add] at hk; simp [hk]
       linarith
@@ -442,8 +440,14 @@ theorem Nesterov_second_fix_stepsize_converge (minφ : IsMinOn (f + h) Set.univ
       rw [h1, h2]; apply Nesterov_second_convergence minφ
     _ ≤ 2 * alg.l / (k + 2) ^ 2 * ‖x0 - xm‖ ^ 2 := by
       apply mul_le_mul_of_nonneg_right _ (sq_nonneg _)
-      rw [alg.γeq (k + 1), alg.teq (k + 1)]; field_simp
-      rw [pow_two, add_comm]; rw [add_assoc, one_add_one_eq_two, ← div_div]
-      apply le_of_eq; ring_nf
+      have hk1 : (k + 1 : ℕ) ≠ 0 := Nat.succ_ne_zero k
+      have hγ : alg.γ (k + 1) = (2 : ℝ) / (k + 2) := by
+        rw [alg.γeq (k + 1)]
+        simp [Nat.cast_add, Nat.cast_one]
+        ring_nf
+      rw [hγ, alg.teq (k + 1)]
+      field_simp [alg.hl]
+      ring_nf
+      norm_num
 
 end Nesterov_second_fix_stepsize
diff --git a/Optlib/Algorithm/Nesterov/NesterovSmooth.lean b/Optlib/Algorithm/Nesterov/NesterovSmooth.lean
index 5b3fda2..bdea52b 100644
--- a/Optlib/Algorithm/Nesterov/NesterovSmooth.lean
+++ b/Optlib/Algorithm/Nesterov/NesterovSmooth.lean
@@ -4,6 +4,7 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chenyi Li, Ziyu Wang, Zaiwen Wen
 -/
 import Optlib.Function.Lsmooth
+import Mathlib.Tactic
 
 /-!
 # NesterovSmooth
@@ -21,7 +22,7 @@ section
 
 open Set
 
-class Nesterov (f : E → ℝ) (f' : E → E) (γ : ℕ+ → ℝ) (initial_point : E) :=
+class Nesterov (f : E → ℝ) (f' : E → E) (γ : ℕ+ → ℝ) (initial_point : E) where
   (x : ℕ → E) (y : ℕ+ → E) (v : ℕ → E) (l : NNReal)
   (diff : ∀ x₁, HasGradientAt f (f' x₁) x₁)
   (update1 : ∀ (k : ℕ+), y k = (1 - γ k) • x (k - 1) + γ k • v (k - 1))
@@ -36,36 +37,35 @@ lemma one_iter (hfun : ConvexOn ℝ Set.univ f) (hg : ∀ (k : ℕ+), γ k = 2 /
     ∀ (k : ℕ+), f (alg.x k) - f xm - (1 - γ k) * (f (alg.x (k - 1)) - f xm) ≤
     alg.l * (γ k) ^ 2 / 2 * (‖alg.v (k - 1) - xm‖ ^ 2 - ‖alg.v k - xm‖ ^ 2)  := by
   have h2 : ∀ (k : ℕ+), ∀ x' : E , f (alg.x k) - f x' ≤ alg.l *
-      inner (alg.x k - alg.y k) (x' - alg.x k) + alg.l / 2 * ‖alg.x k - alg.y k‖ ^ 2 := by
+      inner ℝ (alg.x k - alg.y k) (x' - alg.x k) + alg.l / 2 * ‖alg.x k - alg.y k‖ ^ 2 := by
     intro k x'
     rw [sub_le_iff_le_add', ← add_assoc]
     have : (f' (alg.y k)) = alg.l.1 • (alg.y k - alg.x k) := by
       have update2 : ∀ (k : ℕ+), alg.x k = alg.y k - (1 / alg.l.1) • (f' (alg.y k)) := alg.update2
       specialize update2 k
-      have : alg.l > 0 := alg.hl
       rw [eq_sub_iff_add_eq', ← eq_sub_iff_add_eq] at update2
       rw [← update2, smul_smul]
-      field_simp
-    have t1 : f (alg.y k) + inner (f' (alg.y k)) (x' - alg.y k) ≤ f x' := by
+      simp [ne_of_gt alg.hl]
+    have t1 : f (alg.y k) + inner ℝ (f' (alg.y k)) (x' - alg.y k) ≤ f x' := by
       exact Convex_first_order_condition' (alg.diff (alg.y k)) hfun (by trivial) x' (by trivial)
     calc
-      _ ≤ f (alg.y k) + inner (f' (alg.y k)) (alg.x k - alg.y k) +
+      _ ≤ f (alg.y k) + inner ℝ (f' (alg.y k)) (alg.x k - alg.y k) +
             alg.l.1 / 2 * ‖alg.x k - alg.y k‖ ^ 2 := by
           exact lipschitz_continuos_upper_bound' alg.diff alg.smooth (alg.y k) (alg.x k)
-        _ = f (alg.y k) + inner (f' (alg.y k)) (x' - alg.y k + (alg.x k - x')) +
+        _ = f (alg.y k) + inner ℝ (f' (alg.y k)) (x' - alg.y k + (alg.x k - x')) +
             alg.l / 2 * ‖alg.x k - alg.y k‖ ^ 2 := by
           rw [add_comm (x' - alg.y k), add_sub (alg.x k - x'), sub_add, sub_self, sub_zero]; simp
-        _ = f (alg.y k) + inner (f' (alg.y k)) (x' - alg.y k) + inner  (f' (alg.y k)) (alg.x k - x')
+        _ = f (alg.y k) + inner ℝ (f' (alg.y k)) (x' - alg.y k) + inner ℝ (f' (alg.y k)) (alg.x k - x')
             + alg.l / 2 * ‖alg.x k - alg.y k‖ ^ 2 := by rw [inner_add_right, ← add_assoc]
-        _ ≤ f x' + inner  (f' (alg.y k)) (alg.x k - x') + alg.l / 2 * ‖alg.x k - alg.y k‖ ^ 2 := by
+        _ ≤ f x' + inner ℝ (f' (alg.y k)) (alg.x k - x') + alg.l / 2 * ‖alg.x k - alg.y k‖ ^ 2 := by
           rw [add_le_add_iff_right, add_le_add_iff_right]; exact t1
-        _ = f x' + inner (alg.l.1 • (alg.y k - alg.x k)) (alg.x k - x') +
+        _ = f x' + inner ℝ (alg.l.1 • (alg.y k - alg.x k)) (alg.x k - x') +
           alg.l / 2 * ‖alg.x k - alg.y k‖ ^ 2 := by rw [this]
-        _ = f x' + alg.l * inner (alg.x k - alg.y k) (x' - alg.x k) +
+        _ = f x' + alg.l * inner ℝ (alg.x k - alg.y k) (x' - alg.x k) +
             alg.l / 2 * ‖alg.x k - alg.y k‖ ^ 2 := by
           rw [real_inner_smul_left, ← inner_neg_neg, neg_sub, neg_sub]; simp
   have h3 : ∀ (k : ℕ+), f (alg.x k) - f xm - (1 - γ k) * (f (alg.x (k - 1)) - f xm) ≤
-      alg.l * (inner (alg.x k - alg.y k) ((1 - γ k) • (alg.x (k - 1)) + ((γ k) • xm) -
+      alg.l * (inner ℝ (alg.x k - alg.y k) ((1 - γ k) • (alg.x (k - 1)) + ((γ k) • xm) -
       alg.x k)) + alg.l / 2 * ‖alg.x k - alg.y k‖ ^ 2 := by
     intro k
     have : f (alg.x k) - f xm - (1 - γ k) * (f (alg.x (k - 1)) - f xm) = γ k *
@@ -84,23 +84,23 @@ lemma one_iter (hfun : ConvexOn ℝ Set.univ f) (hg : ∀ (k : ℕ+), γ k = 2 /
       rw [smul_sub, smul_sub, add_sub, ← add_sub_right_comm, sub_sub, ← add_smul]
       ring_nf; rw [one_smul, add_comm]
     calc
-      _ ≤ γ k * (alg.l * (inner (alg.x k - alg.y k) (xm - alg.x k)) + alg.l / 2 *
-            ‖alg.x k - alg.y k‖ ^ 2) + (1 - γ k) * (alg.l * (inner (alg.x k - alg.y k)
+      _ ≤ γ k * (alg.l * (inner ℝ (alg.x k - alg.y k) (xm - alg.x k)) + alg.l / 2 *
+            ‖alg.x k - alg.y k‖ ^ 2) + (1 - γ k) * (alg.l * (inner ℝ (alg.x k - alg.y k)
             (alg.x (k - 1) - alg.x k)) + alg.l / 2 * ‖alg.x k - alg.y k‖ ^ 2) := by
           apply add_le_add
           · exact mul_le_mul_of_nonneg_left (h2 k xm) hz
           · exact mul_le_mul_of_nonneg_left (h2 k (alg.x (k - 1))) (by linarith)
-      _ = alg.l * (γ k * (inner (alg.x k - alg.y k) (xm - alg.x k))) + alg.l * ((1 - γ k) *
-            (inner (alg.x k - alg.y k) (alg.x (k - 1) - alg.x k))) +
+      _ = alg.l * (γ k * (inner ℝ (alg.x k - alg.y k) (xm - alg.x k))) + alg.l * ((1 - γ k) *
+            (inner ℝ (alg.x k - alg.y k) (alg.x (k - 1) - alg.x k))) +
             alg.l / 2 * ‖alg.x k - alg.y k‖ ^ 2 := by ring_nf
-      _ = alg.l * inner (alg.x k - alg.y k) (γ k • (xm - alg.x k)) + alg.l *
-            (inner (alg.x k - alg.y k) ((1 - γ k) •
+      _ = alg.l * inner ℝ (alg.x k - alg.y k) (γ k • (xm - alg.x k)) + alg.l *
+            (inner ℝ (alg.x k - alg.y k) ((1 - γ k) •
             (alg.x (k - 1) - alg.x k))) +  alg.l / 2 * ‖alg.x k - alg.y k‖ ^ 2 := by
             rw [← inner_smul_right _ _ (γ k), ← inner_smul_right _ _ (1 - γ k)]
-      _ = alg.l * inner (alg.x k - alg.y k) (γ k • (xm - alg.x k) + (1 - γ k) •
+      _ = alg.l * inner ℝ (alg.x k - alg.y k) (γ k • (xm - alg.x k) + (1 - γ k) •
             (alg.x (k - 1) - alg.x k)) + alg.l / 2 * ‖alg.x k - alg.y k‖ ^ 2 := by
           rw [← mul_add, ← inner_add_right (alg.x k - alg.y k)]
-      _ = alg.l * inner (alg.x k - alg.y k) ((1 - γ k) • (alg.x (k - 1)) +
+      _ = alg.l * inner ℝ (alg.x k - alg.y k) ((1 - γ k) • (alg.x (k - 1)) +
             ((γ k) • xm)- alg.x k) + alg.l / 2 * ‖alg.x k - alg.y k‖ ^ 2 := by rw [this]
   intro k
   have hz : γ k ≥ (0 : ℝ) := by
@@ -134,30 +134,30 @@ lemma one_iter (hfun : ConvexOn ℝ Set.univ f) (hg : ∀ (k : ℕ+), γ k = 2 /
             left; rw [mul_inv_cancel₀ (by linarith), one_smul, sub_smul, one_smul, add_comm, sub_add]
   have this2 : alg.l / 2 * (‖alg.y k - (1 - γ k) • (alg.x (k - 1)) - γ k • xm‖ ^ 2 -
       ‖alg.x k - (1 - γ k) • alg.x (k - 1) - γ k • xm‖ ^ 2) = alg.l *
-      (inner (alg.x k - alg.y k) ((1 - γ k) • (alg.x (k - 1)) + ((γ k) • xm)- alg.x k))
+      (inner ℝ (alg.x k - alg.y k) ((1 - γ k) • (alg.x (k - 1)) + ((γ k) • xm)- alg.x k))
       + alg.l / 2 * ‖alg.x k - alg.y k‖ ^ 2 := by
     rw [sub_sub, sub_sub, norm_sub_sq_real, norm_sub_sq_real, norm_sub_sq_real]
     calc
-      _ = alg.l / 2 * (‖alg.y k‖ ^ 2 - ‖alg.x k‖ ^ 2) + alg.l / 2 * 2 * (inner (alg.x k)
-            ((1 - γ k) • alg.x (↑k - 1) + γ k • xm) - inner (alg.y k)
+      _ = alg.l / 2 * (‖alg.y k‖ ^ 2 - ‖alg.x k‖ ^ 2) + alg.l / 2 * 2 * (inner ℝ (alg.x k)
+            ((1 - γ k) • alg.x (↑k - 1) + γ k • xm) - inner ℝ (alg.y k)
             ((1 - γ k) • alg.x (↑k - 1) + γ k • xm)) := by ring_nf
-      _ = alg.l / 2 * (‖alg.y k‖ ^ 2 - ‖alg.x k‖ ^ 2) + alg.l * inner (alg.x k - alg.y k)
+      _ = alg.l / 2 * (‖alg.y k‖ ^ 2 - ‖alg.x k‖ ^ 2) + alg.l * inner ℝ (alg.x k - alg.y k)
             ((1 - γ k) • alg.x (↑k - 1) + γ k • xm)  := by rw [← inner_sub_left]; ring_nf
-      _ = alg.l / 2 * (‖alg.y k‖ ^ 2 - ‖alg.x k‖ ^ 2) + alg.l * inner (alg.x k - alg.y k)
+      _ = alg.l / 2 * (‖alg.y k‖ ^ 2 - ‖alg.x k‖ ^ 2) + alg.l * inner ℝ (alg.x k - alg.y k)
             ((1 - γ k) • (alg.x (k - 1)) + ((γ k) • xm) - alg.x k + alg.x k) := by
           rw [sub_add, sub_self, sub_zero]
-      _ = alg.l / 2 * (‖alg.y k‖ ^ 2 - ‖alg.x k‖ ^ 2) + alg.l * inner (alg.x k - alg.y k)
-            (alg.x k) + alg.l * (inner (alg.x k - alg.y k) ((1 - γ k) • (alg.x (k - 1))
+      _ = alg.l / 2 * (‖alg.y k‖ ^ 2 - ‖alg.x k‖ ^ 2) + alg.l * inner ℝ (alg.x k - alg.y k)
+            (alg.x k) + alg.l * (inner ℝ (alg.x k - alg.y k) ((1 - γ k) • (alg.x (k - 1))
             + ((γ k) • xm) - alg.x k)) := by
           rw [inner_add_right, mul_add]; ring_nf
       _ = alg.l / 2 * (‖alg.y k‖ ^ 2 - ‖alg.x k‖ ^ 2) + alg.l * ‖alg.x k‖ ^ 2 -
-            alg.l * inner (alg.x k) (alg.y k) + alg.l * (inner (alg.x k - alg.y k) ((1 - γ k)
+            alg.l * inner ℝ (alg.x k) (alg.y k) + alg.l * (inner ℝ (alg.x k - alg.y k) ((1 - γ k)
             • (alg.x (k - 1)) + ((γ k) • xm) - alg.x k)) := by
           rw [inner_sub_left, mul_sub, mul_sub, real_inner_self_eq_norm_sq]
           rw [real_inner_comm, add_sub];
-      _ = alg.l * (inner (alg.x k - alg.y k) ((1 - γ k) • (alg.x (k - 1)) + ((γ k) • xm)
+      _ = alg.l * (inner ℝ (alg.x k - alg.y k) ((1 - γ k) • (alg.x (k - 1)) + ((γ k) • xm)
             - alg.x k)) + alg.l / 2 * (‖alg.x k‖ ^ 2 - 2 *
-            inner (alg.x k) (alg.y k) + ‖alg.y k‖ ^ 2) := by ring_nf
+            inner ℝ (alg.x k) (alg.y k) + ‖alg.y k‖ ^ 2) := by ring_nf
   rw [this1, this2]
   exact h3 k
 
@@ -176,18 +176,15 @@ theorem nesterov_algorithm_smooth (hfun: ConvexOn ℝ Set.univ f)
     specialize con k
     have : (γ k) ^ 2 > 0 := by
       rw [hg k]
-      simp only [Real.rpow_two, div_pow, gt_iff_lt]
-      apply div_pos (by linarith)
-      apply sq_pos_of_ne_zero
-      exact Nat.cast_add_one_ne_zero ↑k
-    rw [← div_le_div_right this, sub_div, mul_div_right_comm (1 - γ k)] at h4
+      positivity
+    rw [← div_le_div_iff_of_pos_right this, sub_div, mul_div_right_comm (1 - γ k)] at h4
     rw [← one_mul (f (alg.x k) - f xm), mul_div_right_comm 1] at h4
     rw [mul_div_right_comm (alg.l).1, mul_assoc, mul_comm (γ k ^ 2)] at h4
     rw [← mul_assoc, mul_div_assoc] at h4
     rw [div_self (by linarith), mul_one, mul_sub (alg.l.1 / 2)] at h4
     rw [tsub_le_iff_left, add_sub, le_sub_iff_add_le] at h4
     apply le_trans h4
-    simp only [Real.rpow_two, ge_iff_le, add_le_add_iff_right, gt_iff_lt, sub_pos, sub_neg]
+    simp only [add_le_add_iff_right]
     have : f xm ≤ f (alg.x (k - 1)):= min (by trivial)
     apply mul_le_mul_of_nonneg_right _ (by linarith)
     exact con
@@ -210,7 +207,7 @@ theorem nesterov_algorithm_smooth (hfun: ConvexOn ℝ Set.univ f)
     rw [alg.initial1, sub_self, zero_mul, sub_zero] at h4
     rw [alg.initial1, sub_self, zero_div, zero_mul, zero_add]
     simp
-    simp only [PNat.one_coe, Real.rpow_two, one_pow, mul_one, le_refl, tsub_eq_zero_of_le] at h4
+    simp only [PNat.one_coe, one_pow, mul_one, le_refl, tsub_eq_zero_of_le] at h4
     rw [← le_sub_iff_add_le, ← mul_sub]
     exact h4
   have h8 : ∀ (k : ℕ+), 1 / (γ k) ^ 2 * (f (alg.x k) - f xm) + alg.l / 2
@@ -226,16 +223,16 @@ theorem nesterov_algorithm_smooth (hfun: ConvexOn ℝ Set.univ f)
     have : alg.l > 0 := alg.hl
     apply mul_nonneg _ _
     · positivity
-    · simp only [Real.rpow_two, sq_nonneg]
+    · simp only [sq_nonneg]
   have h10 : alg.l / (2 : ℝ) * ‖x₀ - xm‖ ^ 2 / ((1 :ℝ) / (2 / (k + 1)) ^ 2)
       = 2 * alg.l / ((k + 1) ^ 2) * ‖x₀ - xm‖ ^ 2 := by
-    simp [Nat.cast_add_one_ne_zero ↑k]; field_simp; ring_nf
+    field_simp [Nat.cast_add_one_ne_zero ↑k]
   rw [hg k] at h9
   rw [← le_div_iff₀'] at h9
   · rw [h10] at h9
     exact h9
-  · simp only [Real.rpow_two, div_pow, one_div, inv_div]
+  · simp only [div_pow, one_div, inv_div]
     apply div_pos
     · apply sq_pos_of_ne_zero
       exact Nat.cast_add_one_ne_zero ↑k
-    · simp only [gt_iff_lt, zero_lt_two, pow_pos]
+    · simp only [zero_lt_two, pow_pos]
diff --git a/Optlib/Algorithm/ProximalGradient.lean b/Optlib/Algorithm/ProximalGradient.lean
index 51d60d2..165174d 100644
--- a/Optlib/Algorithm/ProximalGradient.lean
+++ b/Optlib/Algorithm/ProximalGradient.lean
@@ -26,7 +26,7 @@ variable [ProperSpace E]
 variable {xm x₀: E} {s : Set E} {f : E → ℝ} {f' : E → E} {h : E → ℝ}
 variable {t : ℝ} {x : ℕ → E} {L : NNReal}
 
-class proximal_gradient_method (f h: E → ℝ) (f' : E → E) (x₀ : E) :=
+class proximal_gradient_method (f h: E → ℝ) (f' : E → E) (x₀ : E) where
   (xm : E) (t : ℝ) (x : ℕ → E) (L : NNReal)
   (fconv : ConvexOn ℝ univ f) (hconv : ConvexOn ℝ univ h)
   (h₁ : ∀ x₁ : E, HasGradientAt f (f' x₁) x₁) (h₂ : LipschitzWith L f')
@@ -40,6 +40,8 @@ theorem proximal_gradient_method_converge : ∀ (k : ℕ+),
     (f (alg.x k) + h (alg.x k) - f alg.xm - h alg.xm)
     ≤ 1 / (2 * k * alg.t) * ‖x₀ - alg.xm‖ ^ 2 := by
   intro k
+  have hkpos : (0 : ℝ) < k := by exact_mod_cast k.pos
+  have hden : 0 < 2 * k * alg.t := by exact mul_pos (mul_pos (by norm_num) hkpos) alg.tpos
   rw [mul_comm, mul_one_div, le_div_iff₀, mul_comm]
   have th : ContinuousOn (alg.t • h) univ := by
     apply ContinuousOn.const_smul alg.h₃ alg.t
@@ -58,78 +60,81 @@ theorem proximal_gradient_method_converge : ∀ (k : ℕ+),
     rw [one_div_mul_cancel, one_smul] at eq2; exact eq2
     linarith [alg.tpos]; exact alg.hconv; linarith [alg.tpos]
   have fieq1 : ∀ x : E, f (x - alg.t • Gt x) ≤
-      f x - alg.t * inner (f' x) (Gt x) + alg.t ^ 2 * alg.L / 2 * ‖Gt x‖ ^ 2 := by
+      f x - alg.t * inner ℝ (f' x) (Gt x) + alg.t ^ 2 * alg.L / 2 * ‖Gt x‖ ^ 2 := by
     intro x
     let y := x - alg.t • Gt x
-    have ieq1 : f y ≤ f x + inner (f' x) (y - x) + alg.L / 2 * ‖y - x‖ ^ 2 := by
+    have ieq1 : f y ≤ f x + inner ℝ (f' x) (y - x) + alg.L / 2 * ‖y - x‖ ^ 2 := by
       apply lipschitz_continuos_upper_bound' alg.h₁ alg.h₂
     have eq3 : y - x = - alg.t • Gt x := by simp [Gt, y]
     rw [eq3] at ieq1; rw [inner_smul_right, norm_smul, mul_pow] at ieq1
     rw [← mul_assoc, mul_comm ] at ieq1
     simp at ieq1; rw [← sub_eq_add_neg] at ieq1; simp; linarith [alg.tpos]
   have fieq2 : ∀ x : E,
-      f (x - alg.t • Gt x) ≤ f x - alg.t * inner (f' x) (Gt x) + alg.t / 2 * ‖Gt x‖ ^ 2 := by
+      f (x - alg.t • Gt x) ≤ f x - alg.t * inner ℝ (f' x) (Gt x) + alg.t / 2 * ‖Gt x‖ ^ 2 := by
     intro x
     calc
       f (x - alg.t • Gt x) ≤
-          f x - alg.t * inner (f' x) (Gt x) + alg.t ^ 2 * alg.L / 2 * ‖Gt x‖ ^ 2 := fieq1 x
-      _ ≤ f x - alg.t * inner (f' x) (Gt x) + alg.t / 2 * ‖Gt x‖ ^ 2 := by
-        apply add_le_add_left
-        apply mul_le_mul_of_nonneg_right
-        apply div_le_div_of_nonneg_right _ (by norm_num)
-        calc
-          alg.t ^ 2 * alg.L ≤ alg.t * (1 / alg.L) * alg.L := by
-            rw [pow_two]; apply mul_le_mul_of_nonneg_right
-            rw [mul_le_mul_left alg.tpos]; exact alg.step; simp
-          _ = alg.t := by field_simp; rw [← mul_div, div_self (by linarith [alg.hL]), mul_one]
-        exact sq_nonneg _
-  have fieq3 : ∀ x z : E, f x + inner (f' x) (z - x) ≤ f z := by
+          f x - alg.t * inner ℝ (f' x) (Gt x) + alg.t ^ 2 * alg.L / 2 * ‖Gt x‖ ^ 2 := fieq1 x
+      _ ≤ f x - alg.t * inner ℝ (f' x) (Gt x) + alg.t / 2 * ‖Gt x‖ ^ 2 := by
+        have hmul : alg.t ^ 2 * alg.L / 2 * ‖Gt x‖ ^ 2 ≤ alg.t / 2 * ‖Gt x‖ ^ 2 := by
+          apply mul_le_mul_of_nonneg_right
+          · apply div_le_div_of_nonneg_right _ (by norm_num)
+            calc
+              alg.t ^ 2 * alg.L ≤ alg.t * (1 / alg.L) * alg.L := by
+                rw [pow_two]; apply mul_le_mul_of_nonneg_right
+                · exact mul_le_mul_of_nonneg_left alg.step (le_of_lt alg.tpos)
+                · exact le_of_lt alg.hL
+              _ = alg.t := by field_simp; rw [← mul_div, div_self (by linarith [alg.hL]), mul_one]
+          · exact sq_nonneg _
+        linarith
+  have fieq3 : ∀ x z : E, f x + inner ℝ (f' x) (z - x) ≤ f z := by
     intro x z
     apply Convex_first_order_condition' (alg.h₁ x) alg.fconv
     simp; simp
   have hieq1 : ∀ x z : E,
-      h (x - alg.t • Gt x) + inner (Gt x - f' x) (z - x + alg.t • Gt x) ≤ h z := by
+      h (x - alg.t • Gt x) + inner ℝ (Gt x - f' x) (z - x + alg.t • Gt x) ≤ h z := by
     intro x z
     specialize hG x
     rw [← mem_SubderivAt, HasSubgradientAt] at hG
     specialize hG z; rw [sub_add]; apply hG
   have hieq2 : ∀ x z : E,
-      h (x - alg.t • Gt x) ≤ h z - inner (Gt x - f' x) (z - x + alg.t • Gt x) := by
+      h (x - alg.t • Gt x) ≤ h z - inner ℝ (Gt x - f' x) (z - x + alg.t • Gt x) := by
     intro x z; linarith [hieq1 x z]
   have univieq : ∀ x z : E,
-      φ (x - alg.t • Gt x) ≤ φ z + inner (Gt x) (x - z) - alg.t / 2 * ‖Gt x‖ ^ 2 := by
+      φ (x - alg.t • Gt x) ≤ φ z + inner ℝ (Gt x) (x - z) - alg.t / 2 * ‖Gt x‖ ^ 2 := by
     intro x z
     calc
-      φ (x - alg.t • Gt x) ≤ (f x - alg.t * inner (f' x) (Gt x) + alg.t / 2 * ‖Gt x‖ ^ 2)
-        + (h z - inner (Gt x - f' x) (z - x + alg.t • Gt x)) := by
+      φ (x - alg.t • Gt x) ≤ (f x - alg.t * inner ℝ (f' x) (Gt x) + alg.t / 2 * ‖Gt x‖ ^ 2)
+        + (h z - inner ℝ (Gt x - f' x) (z - x + alg.t • Gt x)) := by
         linarith [fieq2 x, hieq2 x z]
-      _ ≤ (f z - inner (f' x) (z - x) - alg.t * inner (f' x) (Gt x) + alg.t / 2 * ‖Gt x‖ ^ 2)
-        + (h z - inner (Gt x - f' x) (z - x + alg.t • Gt x)) := by
+      _ ≤ (f z - inner ℝ (f' x) (z - x) - alg.t * inner ℝ (f' x) (Gt x) + alg.t / 2 * ‖Gt x‖ ^ 2)
+        + (h z - inner ℝ (Gt x - f' x) (z - x + alg.t • Gt x)) := by
         linarith [fieq3 x z]
-      _ = φ z + inner (Gt x) (x - z) - alg.t / 2 * ‖Gt x‖ ^ 2 := by
+      _ = φ z + inner ℝ (Gt x) (x - z) - alg.t / 2 * ‖Gt x‖ ^ 2 := by
         rw [← inner_smul_right, sub_sub, ← inner_add_right]
         rw [inner_sub_left, ← sub_add, add_rotate, ← add_comm_sub, ← add_sub]
         rw [← add_sub, sub_self, add_zero, add_rotate, inner_add_right, ← neg_sub x z]
-        rw [inner_neg_right, ← sub_sub, sub_neg_eq_add, add_comm _ (inner (Gt x) (x - z))]
-        rw [add_comm _ (inner (Gt x) (x - z)), ← add_sub _ (φ z), ← add_sub, add_assoc]
+        rw [inner_neg_right, ← sub_sub, sub_neg_eq_add, add_comm _ (inner ℝ (Gt x) (x - z))]
+        rw [add_comm _ (inner ℝ (Gt x) (x - z)), ← add_sub _ (φ z), ← add_sub, add_assoc]
         rw [add_assoc, add_left_cancel_iff]
         rw [inner_smul_right, real_inner_self_eq_norm_sq]
         rw [add_comm_sub, ← add_sub]
         have (a : ℝ): alg.t / 2 * a - alg.t * a = - alg.t / 2 * a := by ring
         rw [this, sub_eq_add_neg, ← add_assoc, add_comm (h z) (f z)]; field_simp
+        ring
   have φieq1 : ∀ x : E, φ (x - alg.t • Gt x) - φ alg.xm ≤
       (1 / (2 * alg.t)) * (‖x - alg.xm‖ ^ 2 - ‖x - alg.t • Gt x - alg.xm‖ ^ 2) := by
     intro x
     calc
-      φ (x - alg.t • Gt x) - φ alg.xm ≤ inner (Gt x) (x - alg.xm) - alg.t / 2 * ‖Gt x‖ ^ 2 := by
+      φ (x - alg.t • Gt x) - φ alg.xm ≤ inner ℝ (Gt x) (x - alg.xm) - alg.t / 2 * ‖Gt x‖ ^ 2 := by
         linarith [univieq x alg.xm]
       _ = (1 / (2 * alg.t)) * (‖x - alg.xm‖ ^ 2 - ‖x - alg.t • Gt x - alg.xm‖ ^ 2) := by
-        have aux (p q : E) : inner p q - alg.t / 2 * ‖p‖ ^ 2 =
+        have aux (p q : E) : inner ℝ p q - alg.t / 2 * ‖p‖ ^ 2 =
             1 / (2 * alg.t) * (‖q‖ ^ 2 - ‖q - alg.t • p‖ ^ 2) := by
           rw [norm_sub_sq_real]; field_simp; ring_nf
           rw [inner_smul_right, real_inner_comm];
           nth_rw 2 [mul_comm _ (alg.t)⁻¹]; rw [norm_smul, mul_pow, pow_two ‖alg.t‖]
-          simp; rw [mul_comm _ (inner q p), mul_assoc _ alg.t, mul_inv_cancel₀, ← mul_assoc]
+          simp; rw [mul_comm _ (inner ℝ q p), mul_assoc _ alg.t, mul_inv_cancel₀, ← mul_assoc]
           rw [← mul_assoc, inv_mul_cancel₀]; simp
           repeat linarith [alg.tpos]
         rw [sub_right_comm]; apply aux
@@ -144,7 +149,7 @@ theorem proximal_gradient_method_converge : ∀ (k : ℕ+),
     rw [iter i]
     calc
       φ ((alg.x i) - alg.t • Gt (alg.x i)) ≤ φ (alg.x i)
-        + inner (Gt (alg.x i)) ((alg.x i) - (alg.x i))
+        + inner ℝ (Gt (alg.x i)) ((alg.x i) - (alg.x i))
         - alg.t / 2 * ‖Gt (alg.x i)‖ ^ 2 := by
         linarith [univieq (alg.x i) (alg.x i)]
       _ ≤ φ (alg.x i) := by
@@ -194,12 +199,15 @@ theorem proximal_gradient_method_converge : ∀ (k : ℕ+),
     _ ≤ 2 * alg.t *
         ((1 / (2 * alg.t)) * ‖(alg.x 0) - alg.xm‖ ^ 2
         - (1 / (2 * alg.t)) * ‖(alg.x k) - alg.xm‖ ^ 2) := by
-      rw [mul_le_mul_left]
-      let ieq' := ieq k; simp at ieq'
-      simp; apply ieq'; linarith [alg.tpos]
+      have ieq' : k * (φ (alg.x k) - φ alg.xm) ≤
+          (1 / (2 * alg.t)) * ‖(alg.x 0) - alg.xm‖ ^ 2
+          - (1 / (2 * alg.t)) * ‖(alg.x k) - alg.xm‖ ^ 2 := by
+        simpa [nsmul_eq_mul] using (ieq k)
+      exact mul_le_mul_of_nonneg_left ieq' (by linarith [alg.tpos])
     _ = ‖(alg.x 0) - alg.xm‖ ^ 2 - ‖(alg.x k) - alg.xm‖ ^ 2 := by
       rw [← mul_sub, ← mul_assoc, mul_one_div_cancel]; simp; linarith [alg.tpos]
     _ ≤ ‖x₀ - alg.xm‖ ^ 2 := by rw [alg.ori]; simp
-  field_simp; linarith [alg.tpos]
+  field_simp
+  linarith [hden]
 
 end method
diff --git a/Optlib/Algorithm/SubgradientMethod.lean b/Optlib/Algorithm/SubgradientMethod.lean
index 7150965..10ce385 100644
--- a/Optlib/Algorithm/SubgradientMethod.lean
+++ b/Optlib/Algorithm/SubgradientMethod.lean
@@ -40,9 +40,9 @@ theorem bounded_subgradient_to_Lipschitz (hf : ConvexOn ℝ univ f) (hc : Contin
   rcases hx₂' with ⟨gx, hx₁⟩
   have hx₃ : ‖gx‖ ≤ G := by rcases h hx₁ with hx; apply hx
   rcases hx₁ y with hx₂
-  have hx₄ : f x - f y ≤ inner gx (x - y) := by
+  have hx₄ : f x - f y ≤ inner ℝ gx (x - y) := by
     rw [add_comm] at hx₂
-    have : f x ≤ f y - inner gx (y - x) := le_sub_left_of_add_le hx₂
+    have : f x ≤ f y - inner ℝ gx (y - x) := le_sub_left_of_add_le hx₂
     rw [sub_eq_add_neg, ← inner_neg_right, neg_sub] at this
     exact sub_left_le_of_le_add this
   have hy₂' : Nonempty (SubderivAt f y) := SubderivAt.nonempty hf hc y
@@ -50,10 +50,10 @@ theorem bounded_subgradient_to_Lipschitz (hf : ConvexOn ℝ univ f) (hc : Contin
   rcases hy₂' with ⟨gy, hy₁⟩
   have hy₃ : ‖gy‖ ≤ G := by rcases h hy₁ with hy; apply hy
   rcases hy₁ x with hy₂
-  have hy₄: f x - f y ≥ inner gy (x - y) := by
+  have hy₄: f x - f y ≥ inner ℝ gy (x - y) := by
     calc
-      _ ≥ f y + inner gy (x - y) - f y := by apply sub_le_sub_right hy₂
-      _ = inner gy (x - y) := by ring
+      _ ≥ f y + inner ℝ gy (x - y) - f y := by apply sub_le_sub_right hy₂
+      _ = inner ℝ gy (x - y) := by ring
   have hG₁: ↑G = ENNReal.ofReal ↑G := by simp
   rw [edist_dist, edist_dist, hG₁]
   have hG₂ : ENNReal.ofReal (↑G * (dist x y)) = ENNReal.ofReal ↑G * ENNReal.ofReal (dist x y) := by
@@ -66,14 +66,14 @@ theorem bounded_subgradient_to_Lipschitz (hf : ConvexOn ℝ univ f) (hc : Contin
   apply abs_le.mpr
   constructor
   · calc
-      f x - f y ≥ inner gy (x - y) := hy₄
+      f x - f y ≥ inner ℝ gy (x - y) := hy₄
       _ ≥ - (‖gy‖ * ‖x - y‖) := by
         apply neg_le_of_neg_le
         rw  [← inner_neg_right, neg_sub, norm_sub_rev]
         apply real_inner_le_norm
       _ ≥ - (↑G * ‖x - y‖) := neg_le_neg (mul_le_mul_of_nonneg_right hy₃ (norm_nonneg _))
   · calc
-      f x - f y ≤ inner gx (x - y) := hx₄
+      f x - f y ≤ inner ℝ gx (x - y) := hx₄
       _ ≤ ‖gx‖ * ‖x - y‖ := real_inner_le_norm _ _
       _ ≤ ↑G * ‖x - y‖ := mul_le_mul_of_nonneg_right hx₃ (norm_nonneg _)
 
@@ -90,9 +90,10 @@ theorem Lipschitz_to_bounded_subgradient (h : LipschitzWith G f ) :
   have hg₁ : ‖g‖ ≠ 0 := by
     apply ne_of_gt (lt_of_le_of_lt _ h₃)
     simp only [NNReal.zero_le_coe]
-  have hl : inner g (y - x) = ‖g‖ := by
+  have hl : inner ℝ g (y - x) = ‖g‖ := by
     rw[hy ,add_comm, ← add_sub, sub_self, add_zero, inner_smul_right, inner_self_eq_norm_sq_to_K]
-    field_simp; apply pow_two
+    field_simp
+    norm_cast
   rw [hl] at hy₂
   have _ : f y - f x ≥ ‖g‖ := by
     calc
@@ -113,7 +114,9 @@ theorem Lipschitz_to_bounded_subgradient (h : LipschitzWith G f ) :
     calc
       f y - f x ≤ |f y - f x|:= by apply le_abs_self
       _ ≤ ↑G * (‖1 / ‖g‖‖ * ‖g‖) := by apply h₃'
-      _ = ↑G := by field_simp
+      _ = ↑G := by
+        rw [Real.norm_eq_abs, one_div, abs_inv, ← abs_of_nonneg (norm_nonneg g), abs_abs]
+        rw [inv_mul_cancel₀ (abs_ne_zero.mpr hg₁), mul_one]
       _ < ‖g‖ := by apply h₃
   linarith
 
@@ -134,7 +137,7 @@ variable (hf : ConvexOn ℝ univ f)
 
 open Finset
 
-class subgradient_method (f : E → ℝ) (x₀ : E) :=
+class subgradient_method (f : E → ℝ) (x₀ : E) where
   (x g : ℕ → E)
   (a : ℕ → ℝ) (ha : ∀ n, a n > 0)
   (G : NNReal) (lipschitz : LipschitzWith G f)
@@ -158,10 +161,12 @@ theorem subgradient_method_converge:
     rcases h' (alg.hg 0) with h₀'
     rw [← mul_pow]
     apply le_trans _ (two_mul_le_add_sq _ _)
-    rw [mul_assoc, mul_assoc]; apply (mul_le_mul_left two_pos).mpr
-    rw [mul_comm, ← mul_assoc]; apply (mul_le_mul_right (alg.ha 0)).mpr
-    have : f (alg.x 0) - f xm ≤ - inner (alg.g 0) (xm - alg.x 0) := by
-      simp [hk₀]; rw[add_comm]; apply hk₀
+    rw [mul_assoc, mul_assoc]
+    refine mul_le_mul_of_nonneg_left ?_ (show (0 : ℝ) ≤ 2 by positivity)
+    rw [mul_comm, ← mul_assoc]
+    refine mul_le_mul_of_nonneg_right ?_ (le_of_lt (alg.ha 0))
+    have : f (alg.x 0) - f xm ≤ - inner ℝ (alg.g 0) (xm - alg.x 0) := by
+      simp; rw[add_comm]; apply hk₀
     apply le_trans this _
     rw [← inner_neg_right,neg_sub, alg.initial]
     apply le_trans (real_inner_le_norm _ _) _; rw [mul_comm]
@@ -180,30 +185,30 @@ theorem subgradient_method_converge:
         rw[mul_comm]; simp
         rcases h' (alg.hg i) with hi
         apply mul_le_mul_of_nonneg_right _ (sq_nonneg _)
-        · apply pow_le_pow_left; apply norm_nonneg; apply hi
+        · nlinarith [hi, norm_nonneg (alg.g i), alg.G.2]
       have inq₂: 2 * alg.a i * (sInf {x | ∃ i ∈ Finset.range (k + 1), f (alg.x i) = x} - f xm)
-          ≤ 2 * inner (alg.x i - xm) (alg.a i • alg.g i) := by
-        rw [mul_assoc]; apply (mul_le_mul_left two_pos).mpr
-        rw [inner_smul_right]; apply (mul_le_mul_left (alg.ha i)).mpr
+          ≤ 2 * inner ℝ (alg.x i - xm) (alg.a i • alg.g i) := by
+        rw [mul_assoc]
+        refine mul_le_mul_of_nonneg_left ?_ (show (0 : ℝ) ≤ 2 by positivity)
+        rw [inner_smul_right]
+        refine mul_le_mul_of_nonneg_left ?_ (le_of_lt (alg.ha i))
         rcases (alg.hg i) xm with hxm
         calc
           _ = sInf (Set.range fun (x : Finset.range (k + 1)) => f (alg.x x)) - f xm := by rw [← heq]
           _ ≤ f (alg.x i) - f xm := by
             simp
             have : f (alg.x i) ∈ Set.range fun (x : Finset.range (k + 1)) => f (alg.x x) := by
-              simp; use i
-              constructor
-              · apply lt_of_le_of_lt hi₂; apply (Nat.lt_succ_self k)
-              · simp
+              exact ⟨⟨i, Finset.mem_range.mpr (lt_of_le_of_lt hi₂ (Nat.lt_succ_self k))⟩, rfl⟩
             apply csInf_le _ this; apply Finite.bddBelow_range
-          _ ≤ inner (alg.x i - xm) (alg.g i) := by
+          _ ≤ inner ℝ (alg.x i - xm) (alg.g i) := by
             simp; apply le_add_of_sub_left_le
             rw [sub_eq_add_neg, ← inner_neg_left, neg_sub, real_inner_comm]
             apply hxm
       rw [add_assoc, add_assoc]
-      apply add_le_add_left; apply add_le_add
-      · apply neg_le_neg; apply inq₂
-      · apply inq₁
+      have hneg : -(2 * inner ℝ (alg.x i - xm) (alg.a i • alg.g i)) ≤
+          -(2 * alg.a i * (sInf {x | ∃ i ∈ Finset.range (k + 1), f (alg.x i) = x} - f xm)) := by
+        exact neg_le_neg inq₂
+      linarith [hneg, inq₁]
     have h₁' : ∀ ⦃i : ℕ⦄, i ≥ 0 ∧ i ≤ k → alg.a i * (2 * (sInf {f (alg.x i) | i ∈ Finset.range (k +
         1)} - f xm)) ≤ ‖alg.x i - xm‖ ^ 2 - ‖alg.x (i+1) - xm‖ ^ 2 + alg.G ^ 2 * (alg.a i) ^ 2 := by
       intro i ⟨hi₁, hi₂⟩
@@ -231,6 +236,7 @@ theorem subgradient_method_converge:
           (k + 1)) fun x => alg.a x ^ 2 := by apply h₂
       _ ≤ ‖x₀ - xm‖ ^ 2 + alg.G ^ 2 * Finset.sum (Finset.range (k + 1)) fun x => alg.a x ^ 2 := by simp
 
+omit [CompleteSpace E] in
 /-- convergence with fixed step size --/
 theorem subgradient_method_fix_step_size {t : ℝ}
     (ha' : ∀ (n : ℕ), alg.a n = t) :
@@ -248,14 +254,16 @@ theorem subgradient_method_fix_step_size {t : ℝ}
     simp
     apply mul_pos _ ht
     · apply add_pos_of_nonneg_of_pos (Nat.cast_nonneg k) zero_lt_one
-  apply (mul_le_mul_left hpos).mp
+  refine le_of_mul_le_mul_left ?_ hpos
   calc
     2 * ((↑k + 1) * t) * (sInf {x | ∃ i ∈ Finset.range (k + 1), f (alg.x i) = x} - f xm)
       = 2 * ((↑k + 1) * t) * (sInf {x | ∃ i < k + 1, f (alg.x i) = x} - f xm) := by simp
-    _ ≤ ‖x₀ - xm‖ ^ 2 + ↑alg.G ^ 2 * ((↑k + 1) * t ^ 2) := by apply hk
+    _ ≤ ‖x₀ - xm‖ ^ 2 + ↑alg.G ^ 2 * ((↑k + 1) * t ^ 2) := by
+      simpa [Nat.lt_add_one_iff] using hk
     _ = 2 * ((↑k + 1) * t) * (‖x₀ - xm‖ ^ 2 / (2 * (↑k + 1) * t) + ↑alg.G ^ 2 * t / 2) := by
-      field_simp; ring
+      field_simp
 
+omit [CompleteSpace E] in
 /-- convergence with fixed $‖x^{i+1}-x^{i}‖$ --/
 theorem subgradient_method_fixed_distance {s : ℝ} (hm : IsMinOn f univ xm)
    (ha' : ∀ (n : ℕ), alg.a n * ‖alg.g n‖ = s) (hs : s > 0):
@@ -265,7 +273,7 @@ theorem subgradient_method_fixed_distance {s : ℝ} (hm : IsMinOn f univ xm)
   have heq : (Set.range fun (x : Finset.range (k + 1)) => f (alg.x x)) =
       {x | ∃ i ∈ Finset.range (k + 1), f (alg.x i) = x} := by simp [Set.ext_iff]
   have hnek : Nonempty (Finset.range (k + 1)) := by
-    simp; use 0; apply Nat.succ_pos k
+    exact ⟨⟨0, by simp⟩⟩
   obtain h' := Lipschitz_to_bounded_subgradient alg.lipschitz
   have h₁ :  ∀ ⦃i : ℕ⦄ , i ≥ 0 ∧ i ≤ k → ‖alg.x (i+1) - xm‖ ^ 2 ≤ ‖alg.x i - xm‖ ^ 2 - 2 * (alg.a i)
       * (sInf {f (alg.x i) | i ∈ Finset.range (k + 1)} - f xm) + ‖alg.a i‖ ^ 2 * ‖alg.g i‖ ^ 2:= by
@@ -277,28 +285,27 @@ theorem subgradient_method_fixed_distance {s : ℝ} (hm : IsMinOn f univ xm)
         + ‖alg.a i‖ ^ 2 * ‖alg.g i‖ ^ 2 := by ring
     rw [this]
     have inq₂: 2 * alg.a i * (sInf {x | ∃ i ∈ Finset.range (k + 1), f (alg.x i) = x} - f xm)
-        ≤ 2 * inner (alg.x i - xm) (alg.a i • alg.g i) := by
-      rw [mul_assoc]; apply (mul_le_mul_left two_pos).mpr
-      rw[inner_smul_right]; apply (mul_le_mul_left (alg.ha i)).mpr
+        ≤ 2 * inner ℝ (alg.x i - xm) (alg.a i • alg.g i) := by
+      rw [mul_assoc]
+      refine mul_le_mul_of_nonneg_left ?_ (show (0 : ℝ) ≤ 2 by positivity)
+      rw[inner_smul_right]
+      refine mul_le_mul_of_nonneg_left ?_ (le_of_lt (alg.ha i))
       rcases (alg.hg i) xm with hxm
       calc
         _ = sInf (Set.range fun (x : Finset.range (k + 1)) => f (alg.x x)) - f xm := by rw [← heq]
         _ ≤ f (alg.x i)- f xm := by
           simp
           have : f (alg.x i) ∈ Set.range fun (x : Finset.range (k + 1)) => f (alg.x x) := by
-            simp; use i
-            constructor
-            · apply lt_of_le_of_lt hi₂; apply (Nat.lt_succ_self k)
-            · simp
+            exact ⟨⟨i, Finset.mem_range.mpr (lt_of_le_of_lt hi₂ (Nat.lt_succ_self k))⟩, rfl⟩
           apply csInf_le _ this; apply Finite.bddBelow_range
-        _ ≤ inner (alg.x i - xm) (alg.g i) := by
+        _ ≤ inner ℝ (alg.x i - xm) (alg.g i) := by
           simp; apply le_add_of_sub_left_le
           rw[sub_eq_add_neg, ← inner_neg_left, neg_sub, real_inner_comm]; apply hxm
     rw[add_assoc, add_assoc]
-    apply add_le_add_left
-    apply add_le_add
-    · apply neg_le_neg; apply inq₂
-    · simp
+    have hneg : -(2 * inner ℝ (alg.x i - xm) (alg.a i • alg.g i)) ≤
+        -(2 * alg.a i * (sInf {x | ∃ i ∈ Finset.range (k + 1), f (alg.x i) = x} - f xm)) := by
+      exact neg_le_neg inq₂
+    linarith [hneg]
   have h₁' :  ∀ ⦃i : ℕ⦄ , i ≥ 0 ∧ i ≤ k → alg.a i * (2 * (sInf {f (alg.x i) | i ∈ Finset.range (k + 1)}
        - f xm)) ≤ ‖alg.x i - xm‖ ^ 2 - ‖alg.x (i+1) - xm‖ ^ 2 + s ^ 2 := by
     intro i ⟨hi₁, hi₂⟩
@@ -332,7 +339,8 @@ theorem subgradient_method_fixed_distance {s : ℝ} (hm : IsMinOn f univ xm)
     apply Finset.sum_le_sum
     intro i _
     rw [← (ha' i)]
-    apply (div_le_iff₀ hG).mpr ((mul_le_mul_left (alg.ha i)).mpr (h' (alg.hg i)))
+    apply (div_le_iff₀ hG).mpr
+    exact mul_le_mul_of_nonneg_left (h' (alg.hg i)) (le_of_lt (alg.ha i))
   have hpos₁ : (↑k + 1) * (s / ↑alg.G) > 0 := by
     apply mul_pos
     · apply add_pos_of_nonneg_of_pos (Nat.cast_nonneg k) zero_lt_one
@@ -344,13 +352,8 @@ theorem subgradient_method_fixed_distance {s : ℝ} (hm : IsMinOn f univ xm)
     · apply hpos₁
   have h₂' : (2 * (k + 1) * (s / ↑alg.G)) * (sInf {x | ∃ i < k + 1, f (alg.x i) = x} - f xm) ≤
       ‖x₀ - xm‖ ^ 2 - ‖alg.x (k + 1) - xm‖ ^ 2 + (↑k + 1) * s ^ 2 := by
-    apply le_trans _ h₂
-    apply mul_le_mul_of_nonneg_right
-    · rw[mul_assoc]
-      apply mul_le_mul_of_nonneg_left
-      · apply inq₁
-      · linarith
-    · apply le_sub_right_of_add_le; simp
+    have hnonneg : 0 ≤ sInf {x | ∃ i < k + 1, f (alg.x i) = x} - f xm := by
+      apply le_sub_right_of_add_le; simp
       apply le_csInf
       · simp at heq
         rw[← heq]
@@ -362,14 +365,25 @@ theorem subgradient_method_fixed_distance {s : ℝ} (hm : IsMinOn f univ xm)
         simp[isMinOn_univ_iff] at hm
         rcases hm (alg.x i) with hmi
         apply hmi
+    have hmul : (2 * (k + 1) * (s / ↑alg.G)) * (sInf {x | ∃ i < k + 1, f (alg.x i) = x} - f xm) ≤
+        (2 * ∑ i ∈ Finset.range (k + 1), alg.a i) * (sInf {x | ∃ i < k + 1, f (alg.x i) = x} - f xm) := by
+      refine mul_le_mul_of_nonneg_right ?_ hnonneg
+      rw[mul_assoc]
+      exact mul_le_mul_of_nonneg_left inq₁ (show (0 : ℝ) ≤ 2 by positivity)
+    have h₂'' : (2 * ∑ i ∈ Finset.range (k + 1), alg.a i) *
+        (sInf {x | ∃ i < k + 1, f (alg.x i) = x} - f xm) ≤
+        ‖x₀ - xm‖ ^ 2 - ‖alg.x (k + 1) - xm‖ ^ 2 + (↑k + 1) * s ^ 2 := by
+      simpa [Nat.lt_add_one_iff] using h₂
+    exact le_trans hmul h₂''
   calc
     _= sInf {x | ∃ i < k + 1, f (alg.x i) = x} - f xm := by simp
     _ ≤ (‖x₀ - xm‖ ^ 2 - ‖alg.x (k + 1) - xm‖ ^ 2 + (k + 1) * s ^ 2) / (2 * (k + 1) * (s / alg.G)) := by
       apply (le_div_iff₀' hpos₁').mpr h₂'
     _ ≤ (‖x₀ - xm‖ ^ 2 + (↑k + 1) * s ^ 2) / (2 * (↑k + 1) * (s / ↑alg.G)) := by
-      apply (div_le_div_right hpos₁').mpr; simp
+      apply (div_le_div_iff_of_pos_right hpos₁').2
+      simp
     _ = alg.G * ‖x₀ - xm‖ ^ 2 / (2 * (k + 1) * s) + alg.G * s / 2 := by
-      field_simp; ring
+      field_simp
 
 
 /-
@@ -411,7 +425,7 @@ lemma subgradient_method_diminishing_step_size (hm : IsMinOn f univ xm)
       calc
         ‖x₀ - xm‖ ^ 2 < Finset.sum (Finset.range (b + 1)) alg.a * ε := by apply h₂
         _ = 2 * Finset.sum (Finset.range (b + 1)) alg.a * (ε / 2) := by
-          field_simp;ring
+          field_simp
     have ha₂ : ∃ a₂, ∀ (b : ℕ), a₂ ≤ b → alg.G ^ 2 * (Finset.range (b + 1)).sum
         (fun i => (alg.a i) ^ 2) / (2 * (Finset.range (b + 1)).sum alg.a) < ε / 2 := by
       by_cases hG : ↑alg.G = 0
@@ -438,9 +452,7 @@ lemma subgradient_method_diminishing_step_size (hm : IsMinOn f univ xm)
               _ = (ε / (2 * ↑alg.G ^ 2) + |x₁|) / 2 := by
                 field_simp; ring
               _ < (ε / (2 * ↑alg.G ^ 2) + ε / (2 * ↑alg.G ^ 2)) / 2 := by
-                apply (mul_lt_mul_left zero_lt_two).mp
-                rw [mul_div_cancel₀, mul_div_cancel₀]
-                simp [hx₁]; simp; simp
+                nlinarith [hx₁]
               _ = ε / (2 * ↑alg.G ^ 2) := by field_simp; ring
         have c₃ : 0 ∈ s := by
           simp [s]
@@ -486,8 +498,7 @@ lemma subgradient_method_diminishing_step_size (hm : IsMinOn f univ xm)
                   (2 * Finset.sum (Finset.range (b + 1)) alg.a) + (↑alg.G ^ 2 * Finset.sum
                   (Finset.range (b - a₂)) fun x => alg.a (a₂ + 1 + x) ^ 2) /
                   (2 * Finset.sum (Finset.range (b + 1)) alg.a) := by
-                field_simp; rw[← mul_add]; simp
-                left
+                field_simp
                 obtain heq := Finset.sum_range_add (fun i => alg.a i ^ 2) (a₂ + 1) (b - a₂)
                 have h₃' : (b + 1) = a₂ + 1 + (b - a₂) := by
                   rw[(Nat.add_comm a₂ 1), Nat.add_assoc, (Nat.add_sub_cancel' hba₂), Nat.add_comm]
@@ -497,18 +508,22 @@ lemma subgradient_method_diminishing_step_size (hm : IsMinOn f univ xm)
             · rcases hasA b hba₁ with h₃; simp [s₁] at h₃
               obtain h₃₁ := (div_lt_iff₀ εpos).mp h₃
               obtain h₃₂ := (div_lt_iff₀' hpos'').mpr h₃₁
-              obtain h₃₃ := (div_lt_div_right zero_lt_four).mpr h₃₂
+              have h₃₃ : (2 * ↑alg.G ^ 2 * Finset.sum (Finset.range (a₂ + 1)) (fun x => alg.a x ^ 2)) /
+                  Finset.sum (Finset.range (b + 1)) alg.a / 4 < ε / 4 := by
+                exact (div_lt_div_iff_of_pos_right zero_lt_four).2 h₃₂
               calc
                 _ = (2 * ↑alg.G ^ 2 * Finset.sum (Finset.range (a₂ + 1)) fun x => alg.a x ^ 2) /
                       Finset.sum (Finset.range (b + 1)) alg.a / 4 := by field_simp;ring
-                _ < ε / 4 := by apply h₃₃
+                _ < ε / 4 := by exact h₃₃
             · apply (div_lt_iff₀ hpos).mpr
               calc
                 _ ≤ ↑alg.G ^ 2 * Finset.sum (Finset.range (b - a₂)) (fun x => (ε / (2 * ↑alg.G ^ 2))
                       * alg.a (a₂ + 1 + x)) := by
-                    apply (mul_le_mul_left hpos').mpr; apply Finset.sum_le_sum; intro i _
+                    refine mul_le_mul_of_nonneg_left ?_ (le_of_lt hpos')
+                    apply Finset.sum_le_sum; intro i _
                     have hposi : alg.a (a₂ + 1 + i) > 0 := by apply (alg.ha (a₂ + 1 + i))
-                    rw [pow_two]; apply (mul_le_mul_right hposi).mpr
+                    rw [pow_two]
+                    refine mul_le_mul_of_nonneg_right ?_ (le_of_lt hposi)
                     have : a₂ + 1 + i ≥ a₂ := by
                       rw[Nat.add_assoc]; apply Nat.le_add_right
                     rcases ha₂ (a₂ + 1 + i) this with hai₂
@@ -521,7 +536,7 @@ lemma subgradient_method_diminishing_step_size (hm : IsMinOn f univ xm)
                     rw[← Finset.mul_sum]
                 _ < ↑alg.G ^ 2 *  (ε / (2 * ↑alg.G ^ 2)) * Finset.sum (Finset.range (b + 1))
                       (fun x => alg.a x) := by
-                    apply (mul_lt_mul_left hposG).mpr
+                    refine mul_lt_mul_of_pos_left ?_ hposG
                     obtain heq := Finset.sum_range_add (fun x => alg.a x) (a₂ + 1) (b - a₂)
                     have h₃' : (b + 1) = a₂ + 1 + (b - a₂) := by
                       rw [Nat.add_comm a₂, Nat.add_assoc, (Nat.add_sub_cancel' hba₂), Nat.add_comm]
@@ -546,7 +561,7 @@ lemma subgradient_method_diminishing_step_size (hm : IsMinOn f univ xm)
       have heq : (Set.range fun (x : Finset.range (b + 1)) => f (alg.x x)) =
           {x | ∃ i ∈ Finset.range (b + 1), f (alg.x i) = x} := by simp [Set.ext_iff]
       have hneb : Nonempty (Finset.range (b + 1)) := by
-        simp; use 0; apply Nat.succ_pos b
+        exact ⟨⟨0, by simp⟩⟩
       apply le_sub_right_of_add_le
       simp
       apply le_csInf
@@ -556,7 +571,9 @@ lemma subgradient_method_diminishing_step_size (hm : IsMinOn f univ xm)
         rw[← hb₂]; simp[isMinOn_univ_iff] at hm
         rcases hm (alg.x i) with hmi
         apply hmi
-    rw[(abs_of_nonneg hne)]
+    have hne' : sInf {x | ∃ i ≤ b, f (alg.x i) = x} - f xm ≥ 0 := by
+      simpa [Nat.lt_add_one_iff] using hne
+    rw[(abs_of_nonneg hne')]
     have h₁ : ∀ (k : ℕ), 2 * ((Finset.range (k + 1)).sum alg.a) * (sInf {f (alg.x i) |
         i ∈ Finset.range (k + 1)} - (f xm)) ≤ ‖x₀ - xm‖ ^ 2 + alg.G ^ 2 *
         (Finset.range (k + 1)).sum (fun i => (alg.a i) ^ 2) := by
@@ -577,10 +594,10 @@ lemma subgradient_method_diminishing_step_size (hm : IsMinOn f univ xm)
       _ = ‖x₀ - xm‖ ^ 2 / (2 * Finset.sum (Finset.range (b + 1)) alg.a) + (↑alg.G ^ 2 *
             Finset.sum (Finset.range (b + 1)) fun i => alg.a i ^ 2) /
             (2 * Finset.sum (Finset.range (b + 1)) alg.a) := by
-          simp [div_add_div_same]
+          simp [add_div]
       _ < ε / 2 + ε / 2 := by
         apply add_lt_add; apply hba₁'; apply hba₂'
-      _ = ε := by field_simp
+      _ = ε := by ring
   obtain h₁' := Filter.Tendsto.add_const (f xm) h₁
   simp at h₁'; simp; apply h₁'
 
diff --git a/Optlib/Convex/BanachSubgradient.lean b/Optlib/Convex/BanachSubgradient.lean
index 3077448..4fbf220 100644
--- a/Optlib/Convex/BanachSubgradient.lean
+++ b/Optlib/Convex/BanachSubgradient.lean
@@ -3,8 +3,11 @@ Copyright (c) 2023 Wanyi He. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Author: Wanyi He, Chenyi Li, Zichen Wang
 -/
-import Mathlib.Analysis.NormedSpace.HahnBanach.Separation
-import Mathlib.LinearAlgebra.Dual
+import Mathlib.Analysis.Convex.Function
+import Mathlib.Analysis.Convex.Topology
+import Mathlib.Analysis.LocallyConvex.Separation
+import Mathlib.Analysis.Normed.Operator.Basic
+import Mathlib.LinearAlgebra.Dual.Lemmas
 
 
 section
@@ -23,7 +26,7 @@ lemma EpigraphInterior_existence (hc : ContinuousOn f (interior s)) (hx : x ∈
     have h1 : IsOpen t := IsOpen.preimage continuous_fst isOpen_interior
     have h2: ContinuousOn (fun p : (E × ℝ) => f p.fst) t :=
       ContinuousOn.comp hc continuousOn_fst (fun ⦃x⦄ a => a)
-    apply ContinuousOn.isOpen_inter_preimage (h2.prod continuousOn_snd) h1 isOpen_lt_prod
+    apply ContinuousOn.isOpen_inter_preimage (h2.prodMk continuousOn_snd) h1 isOpen_lt_prod
   have h' : {p : E × ℝ| p.1 ∈ interior s ∧ f p.1 < p.2} ⊆ {p | p.1 ∈ s ∧ f p.1 ≤ p.2} :=
     fun p ⟨hp1, hp2⟩ => ⟨interior_subset hp1, le_of_lt hp2⟩
   apply interior_mono h'
@@ -60,7 +63,7 @@ lemma Continuous_epi_open {f₁ : E → ℝ} (hcon : ContinuousOn f₁ univ) :
   have : {(x, y) : E × ℝ | y > f₁ x} = {(x, y) : E × ℝ | x ∈ univ ∧ y > f₁ x} := by
     ext z; simp
   rw [this]
-  apply ContinuousOn.isOpen_inter_preimage (h2.prod continuousOn_snd) h1 isOpen_lt_prod
+  apply ContinuousOn.isOpen_inter_preimage (h2.prodMk continuousOn_snd) h1 isOpen_lt_prod
 
 end
 noncomputable section
@@ -127,38 +130,37 @@ theorem Banach_SubderivWithinAt.Nonempty (hf : ConvexOn ℝ s f)
     have hgu' : g.1 x + g.2 (f x) < g.1 a.1 + g.2 a.2 := by
       obtain hg1 := hg a; obtain hg2 := hg (x , f x)
       rw[← hg1 , ← hg2]; apply hφ a ha
-    simp only [hu, hu] at hgu'; exact hgu'
+    simp only [hu] at hgu'; exact hgu'
   have hu0 : u > 0 := by
     specialize hgu (x, f x + 1) (EpigraphInterior_existence hc hx (f x + 1) (lt_add_one (f x)))
     dsimp at hgu; linarith
   let h := - (1 / u) • g.1
   have : ∀ (x : E), ‖h x‖ ≤ ((1 / u) * ‖φ‖) * ‖x‖ := by
-    intro x; field_simp [h];  simp only [abs_of_pos hu0]
-    apply div_le_div_of_nonneg_right _ (by linarith)
+    intro x
+    have hxφ : ‖g.1 x‖ ≤ ‖φ‖ * ‖x‖ := by
+      calc
+        ‖g.1 x‖ = ‖φ (x, 0)‖ := by rw [hg (x, 0), hu, mul_zero, add_zero]
+        ‖φ (x, 0)‖ ≤ ‖φ‖ * ‖(x, (0 : ℝ))‖ := ContinuousLinearMap.le_opNorm φ (x, 0)
+        _ = ‖φ‖ * ‖x‖ := by simp only [Prod.norm_def, norm_zero, max_eq_left (norm_nonneg x)]
     calc
-      |φ (x, 0)| = ‖φ (x, 0)‖ := rfl
-      _  ≤ ‖φ‖ * ‖(x , (0 : ℝ))‖ := ContinuousLinearMap.le_opNorm φ (x, 0)
-      _  = ‖φ‖ * ‖x‖ := by
-        simp only [Prod.norm_def, norm_zero, max_eq_left (norm_nonneg x)]
+      ‖h x‖ = (1 / u) * ‖g.1 x‖ := by simp [h, hu0.le]
+      _ ≤ (1 / u) * (‖φ‖ * ‖x‖) := mul_le_mul_of_nonneg_left hxφ (by positivity)
+      _ = ((1 / u) * ‖φ‖) * ‖x‖ := by ring
   have hh : ∃ (C : ℝ), ∀ (x : E), ‖h x‖ ≤ C * ‖x‖ := by
     use ((1 / u) * ‖φ‖)
   let h' := (LinearMap.mkContinuousOfExistsBound h hh)
   have key1 : ∀ a ∈ interior (Epi f s) , h' (a.1 - x) + f x < a.2 := by
     dsimp [h']; intro a ha
-    specialize hgu a ha; dsimp [g] at hgu
-    have uneq : u ≠ 0 := by linarith
-    rw [← mul_lt_mul_iff_of_pos_left hu0]; field_simp
-    have eq1 : u * (-φ (a.1 - x, 0) + f x * u) / u = u * f x - φ (a.1 - x, 0) := by
-      field_simp; ring_nf
-    have eq2 : φ (x, 0) - φ (a.1, 0) = -φ (a.1 - x, 0) := by
-      have : φ (x, 0) - φ (a.1, 0) = φ ((x, 0) - (a.1, 0)) := by
-        simp only [φ.map_sub]
-      simp only [this, Prod.mk_sub_mk, sub_zero]
-      have : (-(1 : ℝ)) • (a.1 - x, (0 : ℝ)) = (x - a.1, 0) := by simp
-      rw [← this, ContinuousLinearMap.map_smulₛₗ]; simp
-    field_simp [h, g, eq1, eq2, hgu]
-    rw [div_lt_iff₀ (by positivity)]; rw [← mul_lt_mul_iff_of_pos_left hu0] at hgu
-    linarith
+    specialize hgu a ha
+    have hu' : u * f x - (g.1 a.1 - g.1 x) < u * a.2 := by linarith
+    have hu_ne : u ≠ 0 := by linarith
+    have hhx : h (a.1 - x) = -(u⁻¹ * (g.1 a.1 - g.1 x)) := by simp [h]
+    have hm : u * (-(u⁻¹ * (g.1 a.1 - g.1 x)) + f x) = u * f x - (g.1 a.1 - g.1 x) := by
+      field_simp [hu_ne]; ring
+    apply (mul_lt_mul_iff_of_pos_left hu0).1
+    rw [hhx]
+    rw [hm]
+    exact hu'
 
   have key2₀ : ∀ a ∈ (Epi f s), a.1 ∈ interior s → h' (a.1 - x) + f x ≤  a.2 := by
     intro a ha posa
@@ -167,9 +169,9 @@ theorem Banach_SubderivWithinAt.Nonempty (hf : ConvexOn ℝ s f)
     have hfa : f a.1 = a.2 := by linarith [ha.2]
     let an : ℕ → E × ℝ := fun n => (a.1, f a.1 + 1 / (n + 1))
     have can2 : Tendsto (fun n => (an n).2) atTop (nhds (f a.1)) := by
-      obtain hh := Tendsto.add
-        (tendsto_const_nhds) (tendsto_one_div_add_atTop_nhds_zero_nat)
-      simp only [add_zero] at hh; exact hh
+      simpa [an] using
+        (Tendsto.add tendsto_const_nhds tendsto_one_div_add_atTop_nhds_zero_nat :
+          Tendsto (fun n : ℕ => f a.1 + 1 / (n + 1 : ℝ)) atTop (nhds (f a.1 + 0)))
     have hxn : ∀ (n : ℕ), h' ((an n).1 - x) + f x ≤ (an n).2 := by
       intro n
       have : (1 : ℝ) / (n + 1) > 0 := one_div_pos.mpr (by linarith)
@@ -179,9 +181,7 @@ theorem Banach_SubderivWithinAt.Nonempty (hf : ConvexOn ℝ s f)
     have cleft :
       Tendsto (fun n => h' ((an n).1 - x) + f x) atTop (nhds (h' (a.1 - x) + f x)) := by
       exact tendsto_const_nhds
-    apply le_of_tendsto_of_tendsto' cleft ?_ hxn
-    simp only [an, hfa]
-    exact can2
+    exact le_of_tendsto_of_tendsto' cleft can2 hxn
 
   have key2₁ : ∀ a ∈ (Epi f s), a.1 ∉ interior s → h' (a.1 - x) + f x ≤ a.2 := by
     intro a ha _
diff --git a/Optlib/Convex/ClosedCone.lean b/Optlib/Convex/ClosedCone.lean
index 263f705..8ea0c7a 100644
--- a/Optlib/Convex/ClosedCone.lean
+++ b/Optlib/Convex/ClosedCone.lean
@@ -90,13 +90,13 @@ lemma cone_eq_finite_union (s : Finset ℕ) (V : ℕ → (EuclideanSpace ℝ (Fi
     · intro xin
       let mem_x := conic_Caratheodory s V x xin
       rcases mem_x with ⟨τ, τsubs, xinτ, idpτ, _⟩
-      simp [finite_F, F, idx_set, idx_to_cone]
+      simp [F, idx_set, idx_to_cone]
       use τ
-    · simp [finite_F, F, idx_set, idx_to_cone]
+    · simp [F, idx_set, idx_to_cone]
       intro τ τsubs _ xinτ
       apply cone_subset_of_idx_subset' s τ τsubs V xinτ
   · intro C Cin
-    simp [finite_F, F] at Cin; rcases Cin with ⟨τ, τin, Ceq⟩
+    simp [F] at Cin; rcases Cin with ⟨τ, τin, Ceq⟩
     use τ; constructor
     · rw [← Ceq]
     · simp [idx_set] at τin; exact τin.2
@@ -107,9 +107,9 @@ lemma closed_conic_idp (s : Finset ℕ) (V : s → (EuclideanSpace ℝ (Fin n)))
   simp [cone']
   let M : Matrix s (Fin n) ℝ := fun i ↦ V i
   let f := fun x : s → ℝ ↦ Finset.sum univ (fun i => x i • V i)
-  let F := Matrix.mulVecLin Mᵀ
+  let F := (EuclideanSpace.equiv (Fin n) ℝ).symm.toLinearMap.comp (Matrix.mulVecLin Mᵀ)
   have eq2 : f = F := by
-      simp [F]; ext x j; simp; apply Finset.sum_apply
+      ext x j; simp [f, F, M, Matrix.vecMul, dotProduct]
   show IsClosed (f '' (quadrant' s))
   rw [eq2]
   have iscF : Continuous f := by
@@ -130,15 +130,16 @@ lemma closed_conic_idp (s : Finset ℕ) (V : s → (EuclideanSpace ℝ (Fin n)))
   rw [eq2] at iscF
   have isclosed : IsClosedMap F := by
     have injF : Function.Injective F := by
-      simp only [F]
+      apply (EuclideanSpace.equiv (Fin n) ℝ).symm.injective.comp
       show Function.Injective Mᵀ.mulVec
-      rw [Matrix.mulVec_injective_iff]; simp
-      apply idp
-    have closeEmbF: IsClosedEmbedding F := by
+      rw [Matrix.mulVec_injective_iff]
+      simpa [M, Matrix.row] using idp.map' (EuclideanSpace.equiv (Fin n) ℝ).toLinearMap
+        (LinearMap.ker_eq_bot.2 (EuclideanSpace.equiv (Fin n) ℝ).injective)
+    have closeEmbF : Topology.IsClosedEmbedding F := by
       apply LinearMap.isClosedEmbedding_of_injective
       rw [LinearMap.ker_eq_bot]
       exact injF
-    apply IsClosedEmbedding.isClosedMap closeEmbF
+    exact Topology.IsClosedEmbedding.isClosedMap closeEmbF
   apply isclosed
   have domclosed : IsClosed (quadrant' s) := by
     let g := fun i : s ↦ {mu : s → ℝ | 0 ≤ mu i}
diff --git a/Optlib/Convex/ConicCaratheodory.lean b/Optlib/Convex/ConicCaratheodory.lean
index 9f6bf21..8769fbb 100644
--- a/Optlib/Convex/ConicCaratheodory.lean
+++ b/Optlib/Convex/ConicCaratheodory.lean
@@ -80,7 +80,8 @@ private lemma mem_conic_erase (s : Finset ℕ) (V : ℕ → (EuclideanSpace ℝ
         by_cases hi : i ∈ s
         · linarith [αpos ⟨i, hi⟩]
         · simp [α, hi, β]; linarith [tin i]
-      · have hαj₀ : α j₀ = 0 := by field_simp [α, β]
+      · have hαj₀ : α j₀ = 0 := by
+          simp [α, β, j₀.prop, ne_of_gt kj₀pos]
         rw [hαj₀, ← xdecompose]; simp [α]
         have aux : (Finset.sum s fun x ↦ (t x - β x / k j₀) • V x) =
             (Finset.sum s fun x ↦ t x • V x) - (1 / k j₀) • (Finset.sum s fun x ↦ β x • V x) := by
@@ -126,8 +127,10 @@ theorem conic_Caratheodory (s : Finset ℕ) (V : ℕ → (EuclideanSpace ℝ (Fi
       simp [τ']; apply subset_trans _ τin.1
       apply Finset.erase_subset
     specialize τcardmin τ'subs xinerase
-    simp [to_card] at τcardmin
-    absurd τcardmin; simp
+    simp [to_card, Finset.card_erase_of_mem y.prop] at τcardmin
+    have hpos : 0 < τ.card := Finset.card_pos.2 ⟨y, y.prop⟩
+    rw [← Nat.succ_pred_eq_of_pos hpos] at τcardmin
+    exact Nat.not_succ_le_self _ τcardmin
   · intro σ σsubs; specialize τcardmin σ
     simp [idx, to_card] at τcardmin
     apply τcardmin σsubs
diff --git a/Optlib/Convex/ConvexFunction.lean b/Optlib/Convex/ConvexFunction.lean
index 378e2eb..6d6756e 100644
--- a/Optlib/Convex/ConvexFunction.lean
+++ b/Optlib/Convex/ConvexFunction.lean
@@ -101,9 +101,7 @@ theorem Convex_first_order_condition {s : Set E}
   have x1nbhd: ‖x - x'‖ ≤ δ := by
     rw[h1, h2]
     have h3: b * ‖x - y‖ ≤ b1 * ‖x - y‖:= by
-      rw [mul_le_mul_right]
-      apply min_le_left
-      exact h₃
+      exact mul_le_mul_of_nonneg_right (min_le_left b1 1) (norm_nonneg _)
     have h4: b1 * ‖x - y‖ = δ := by
       simp [b1]
       rw [div_mul_cancel₀]
@@ -217,13 +215,13 @@ variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteS
 variable {f : E → ℝ} {f' : E → E} {s : Set E} {x : E}
 
 theorem Convex_first_order_condition' (h : HasGradientAt f (f' x) x) (hf : ConvexOn ℝ s f)
-    (xs : x ∈ s) : ∀ (y : E), y ∈ s → f x + inner (f' x) (y - x) ≤ f y := by
+    (xs : x ∈ s) : ∀ (y : E), y ∈ s → f x + inner ℝ (f' x) (y - x) ≤ f y := by
   show ∀ (y : E), y ∈ s → f x + (toDual ℝ E) (f' x) (y - x) ≤ f y
   apply Convex_first_order_condition _ hf xs
   apply h
 
 theorem Convex_first_order_condition_inverse'  (h : ∀ x ∈ s , HasGradientAt f (f' x) x)
-    (h₁ : Convex ℝ s) (h₂ : ∀ x : E, x ∈ s → ∀ y : E, y ∈ s → f x + inner (f' x) (y - x) ≤ f y) :
+    (h₁ : Convex ℝ s) (h₂ : ∀ x : E, x ∈ s → ∀ y : E, y ∈ s → f x + inner ℝ (f' x) (y - x) ≤ f y) :
     ConvexOn ℝ s f := by
   apply Convex_first_order_condition_inverse
   intro x; specialize h x
@@ -231,33 +229,33 @@ theorem Convex_first_order_condition_inverse'  (h : ∀ x ∈ s , HasGradientAt
   apply h; apply h₁; apply h₂
 
 theorem Convex_first_order_condition_iff' (h₁ : Convex ℝ s) (h : ∀ x ∈ s, HasGradientAt f (f' x) x) :
-    ConvexOn ℝ s f ↔ ∀ x ∈ s, ∀ y ∈ s, f x + inner (f' x) (y - x) ≤ f y :=
+    ConvexOn ℝ s f ↔ ∀ x ∈ s, ∀ y ∈ s, f x + inner ℝ (f' x) (y - x) ≤ f y :=
   ⟨fun h₂ x xs ↦ Convex_first_order_condition' (h x xs) h₂ xs,
     Convex_first_order_condition_inverse' h h₁⟩
 
 theorem Convex_monotone_gradient' (hfun: ConvexOn ℝ s f) (h : ∀ x ∈ s, HasGradientAt f (f' x) x) :
-    ∀ x ∈ s, ∀ y ∈ s, inner (f' x - f' y) (x - y) ≥ (0 : ℝ) := by
+    ∀ x ∈ s, ∀ y ∈ s, inner ℝ (f' x - f' y) (x - y) ≥ (0 : ℝ) := by
   let g := fun x ↦ (toDual ℝ E) (f' x)
   have h' : ∀ x ∈ s, HasFDerivAt f (g x) x := h
-  have equiv : ∀ x y : E, inner (f' x - f' y) (x - y) = (g x - g y) (x - y) := by
+  have equiv : ∀ x y : E, inner ℝ (f' x - f' y) (x - y) = (g x - g y) (x - y) := by
     intro x y
-    rw [← InnerProductSpace.toDual_apply]
-    simp only [map_sub, ContinuousLinearMap.coe_sub', Pi.sub_apply, toDual_apply, g]
+    rw [← InnerProductSpace.toDual_apply_apply]
+    simp only [map_sub, ContinuousLinearMap.coe_sub', Pi.sub_apply, InnerProductSpace.toDual_apply_apply, g]
   intro x hx y hy
   rw [equiv]
   exact Convex_monotone_gradient hfun h' x hx  y hy
 
 theorem monotone_gradient_convex' (h₁ : Convex ℝ s) (hf : ∀ x ∈ s, HasGradientAt f (f' x) x)
-    (mono: ∀ x ∈ s, ∀ y ∈ s, inner (f' x - f' y) (x - y) ≥ (0 : ℝ)) : ConvexOn ℝ s f := by
+    (mono: ∀ x ∈ s, ∀ y ∈ s, inner ℝ (f' x - f' y) (x - y) ≥ (0 : ℝ)) : ConvexOn ℝ s f := by
   apply Convex_first_order_condition_inverse' hf h₁
   intro x xs y ys
   let g := fun t : ℝ ↦ f (x + t • (y - x))
-  let g' := fun t : ℝ ↦ (inner (f' (x + t • (y - x))) (y - x) : ℝ)
+  let g' := fun t : ℝ ↦ (inner ℝ (f' (x + t • (y - x))) (y - x) : ℝ)
   have h1 : ∀ r ∈ Icc 0 1, HasDerivAt g (g' r) r := by
     let h := fun r : ℝ ↦ (x + r • (y - x))
     have : g = f ∘ h := rfl
     rw [this]; intro t ht
-    have : inner (f' (x + t • (y - x))) (y - x) = toDual ℝ E (f' (x + t • (y - x))) (y - x) := rfl
+    have : inner ℝ (f' (x + t • (y - x))) (y - x) = toDual ℝ E (f' (x + t • (y - x))) (y - x) := rfl
     simp [g']; rw [this]; apply HasFDerivAt.comp_hasDerivAt
     · apply hasGradientAt_iff_hasFDerivAt.mp
       have : x + t • (y - x) ∈ s := by
@@ -270,13 +268,13 @@ theorem monotone_gradient_convex' (h₁ : Convex ℝ s) (hf : ∀ x ∈ s, HasGr
       rw [one_smul] at this; exact HasDerivAt.const_add x this
   have e1 : f x = g 0 := by simp [g]
   have e2 : f y = g 1 := by simp [g]
-  have e3 : inner (f' x) (y - x) = g' 0 := by simp [g']
+  have e3 : inner ℝ (f' x) (y - x) = g' 0 := by simp [g']
   rw [e1, e2, e3]
   have mono' : ∀ t ∈ Ioo 0 1, g' t ≥ g' 0 := by
     intro t ht;
     simp [g']; rw [← sub_nonneg, ← inner_sub_left]
     rcases ht with ⟨ht1, ht2⟩
-    have hh: inner (f' (x + t • (y - x)) - f' x) (x + t • (y - x) - x) ≥ (0 : ℝ) := by
+    have hh: inner ℝ (f' (x + t • (y - x)) - f' x) (x + t • (y - x) - x) ≥ (0 : ℝ) := by
       apply mono (x + t • (y - x)) _ x xs
       have e4 : x + t • (y - x) = (1 - t) • x + t • y := by
         rw [smul_sub, add_sub, sub_smul, one_smul, add_sub_right_comm]
@@ -300,7 +298,7 @@ theorem monotone_gradient_convex' (h₁ : Convex ℝ s) (hf : ∀ x ∈ s, HasGr
   simp; constructor; linarith; linarith
 
 theorem monotone_gradient_iff_convex' (h₁ : Convex ℝ s) (hf : ∀ x ∈ s, HasGradientAt f (f' x) x):
-    ConvexOn ℝ s f ↔ ∀ x ∈ s, ∀ y ∈ s, inner (f' x - f' y) (x - y) ≥ (0 : ℝ) :=
+    ConvexOn ℝ s f ↔ ∀ x ∈ s, ∀ y ∈ s, inner ℝ (f' x - f' y) (x - y) ≥ (0 : ℝ) :=
   ⟨fun h ↦ Convex_monotone_gradient' h hf, fun h ↦ monotone_gradient_convex' h₁ hf h⟩
 
 theorem monotone_gradient_convex {f' : E → (E →L[ℝ] ℝ)} (h₁ : Convex ℝ s)
@@ -310,10 +308,10 @@ theorem monotone_gradient_convex {f' : E → (E →L[ℝ] ℝ)} (h₁ : Convex 
   have h' : ∀ x ∈ s, HasGradientAt f (g x) x := by
     intro x' hx'
     exact HasFDerivAt.hasGradientAt (hf x' hx')
-  have equiv : ∀ x y : E, inner (g x - g y) (x - y) = (f' x - f' y) (x - y) := by
+  have equiv : ∀ x y : E, inner ℝ (g x - g y) (x - y) = (f' x - f' y) (x - y) := by
     intro x y
-    rw [← InnerProductSpace.toDual_apply]; simp [g]
-  have mono' : ∀ x ∈ s, ∀ y ∈ s, inner (g x - g y) (x - y) ≥ (0 : ℝ) := by
+    rw [← InnerProductSpace.toDual_apply_apply]; simp [g]
+  have mono' : ∀ x ∈ s, ∀ y ∈ s, inner ℝ (g x - g y) (x - y) ≥ (0 : ℝ) := by
     intro x hx y hy
     specialize mono x hx y hy
     rw [equiv]; exact mono
@@ -339,14 +337,14 @@ variable {f : E → ℝ} {f' : E → E} {s : Set E}
 
 theorem monotone_gradient_strict_convex (hs : Convex ℝ s)
     (hf : ∀ x ∈ s, HasGradientAt f (f' x) x)
-    (mono: ∀ x ∈ s, ∀ y ∈ s, x ≠ y → inner (f' x - f' y) (x - y) > (0 : ℝ)) :
+    (mono: ∀ x ∈ s, ∀ y ∈ s, x ≠ y → inner ℝ (f' x - f' y) (x - y) > (0 : ℝ)) :
     StrictConvexOn ℝ s f := by
   rw [StrictConvexOn]; use hs
   intro x xin y yin xney a b apos bpos absum1
   by_contra h₀; push_neg at h₀
   have anneg : 0 ≤ a := by linarith
   have bnneg : 0 ≤ b := by linarith
-  have mono' : ∀ x ∈ s, ∀ y ∈ s, inner (f' x - f' y) (x - y) ≥ (0 : ℝ) := by
+  have mono' : ∀ x ∈ s, ∀ y ∈ s, inner ℝ (f' x - f' y) (x - y) ≥ (0 : ℝ) := by
     intro x xin y yin
     by_cases h : x = y
     · rw [h]; simp
@@ -363,14 +361,14 @@ theorem monotone_gradient_strict_convex (hs : Convex ℝ s)
     have : a = 1 - b := by linarith
     rw [this, sub_smul, add_comm_sub, ← smul_sub]; simp
     apply Convex.add_smul_sub_mem hs xin yin; simp; use bnneg; linarith
-  have eq1 : ∃ c : ℝ, c ∈ Set.Ioo 0 1 ∧ inner (f' (x + c • (z - x))) (z - x) = f z - f x := by
+  have eq1 : ∃ c : ℝ, c ∈ Set.Ioo 0 1 ∧ inner ℝ (f' (x + c • (z - x))) (z - x) = f z - f x := by
     apply lagrange hs hf x xin z zin
-  have eq2 : ∃ c : ℝ, c ∈ Set.Ioo 0 1 ∧ inner (f' (z + c • (y - z))) (y - z) = f y - f z := by
+  have eq2 : ∃ c : ℝ, c ∈ Set.Ioo 0 1 ∧ inner ℝ (f' (z + c • (y - z))) (y - z) = f y - f z := by
     apply lagrange hs hf z zin y yin
   rcases eq1 with ⟨c, cin, e1⟩
   rcases eq2 with ⟨d, din, e2⟩
-  have eq3 : b * inner (f' (z + d • (y - z))) (y - z) -
-      a * inner (f' (x + c • (z - x))) (z - x) = 0 := by
+  have eq3 : b * inner ℝ (f' (z + d • (y - z))) (y - z) -
+      a * inner ℝ (f' (x + c • (z - x))) (z - x) = 0 := by
     rw [e1, e2]; simp [z]; ring_nf; rw [add_comm, ← add_assoc]
     simp at eq2; rw [← eq2]; nth_rw 1 [← mul_one (f (a • x + b • y))]; rw [← absum1]; ring_nf
   rw [← inner_smul_right, ← inner_smul_right] at eq3
@@ -379,14 +377,17 @@ theorem monotone_gradient_strict_convex (hs : Convex ℝ s)
   have this2 : a • (z - x) = a • b • (y - x) := by
     simp [z]; nth_rw 2 [← one_smul ℝ x]; rw [← absum1, add_smul]; simp; rw [← smul_sub, smul_comm]
   rw [this1, this2, ← inner_sub_left, inner_smul_right, inner_smul_right, ← mul_assoc] at eq3
-  have eq0 : inner (f' (z + d • (y - z)) - f' (x + c • (z - x))) (y - x) = (0 : ℝ) := by
+  have eq0 : inner ℝ (f' (z + d • (y - z)) - f' (x + c • (z - x))) (y - x) = (0 : ℝ) := by
     contrapose! eq3
     rw [mul_ne_zero_iff]
     constructor
     · rw [mul_ne_zero_iff]; constructor <;> linarith
     · exact eq3
   have zeq : z = x + b • (y - x) := by
-    nth_rw 1 [← one_smul ℝ x]; rw [← absum1, add_smul, smul_sub]; simp
+    show a • x + b • y = x + b • (y - x)
+    have hab : a = 1 - b := by linarith
+    rw [hab, sub_smul, one_smul, smul_sub]
+    abel_nf
   let u : E := z + d • (y - z)
   let v : E := x + c • (z - x)
   have ueq : u = x + (b + d) • (y - x) - d • b • (y - x) := by
@@ -400,8 +401,8 @@ theorem monotone_gradient_strict_convex (hs : Convex ℝ s)
   have usubv : u - v = (b + d - d * b - c * b) • (y - x) := by
     rw [ueq, veq, ← smul_assoc, ← smul_assoc, ← sub_sub]; simp
     rw [← add_sub, ← sub_smul (b + d) (d * b)]; simp; rw [← sub_smul]
-  have eeq0 : inner (f' u - f' v) (u - v) = (0 : ℝ) := by
-    show inner (f' (z + d • (y - z)) - f' (x + c • (z - x))) (u - v) = (0 : ℝ)
+  have eeq0 : inner ℝ (f' u - f' v) (u - v) = (0 : ℝ) := by
+    show inner ℝ (f' (z + d • (y - z)) - f' (x + c • (z - x))) (u - v) = (0 : ℝ)
     rw [usubv, inner_smul_right, eq0]; simp
   have coefne0 : b + d - d * b - c * b > 0 := by
     nth_rw 1 [← mul_one d]; rw [← absum1]; simp; ring_nf
@@ -412,7 +413,7 @@ theorem monotone_gradient_strict_convex (hs : Convex ℝ s)
       _ < b + d * a := by
         have : 0 < d * a := by apply mul_pos dpos apos
         linarith
-  have neq0 : inner (f' u - f' v) (u - v) > (0 : ℝ) := by
+  have neq0 : inner ℝ (f' u - f' v) (u - v) > (0 : ℝ) := by
     have uin : u ∈ s := by
       show z + d • (y - z) ∈ s
       apply Convex.add_smul_sub_mem hs zin yin; simp; simp at din
@@ -433,24 +434,24 @@ theorem monotone_gradient_strict_convex (hs : Convex ℝ s)
 
 theorem strict_convex_monotone_gradient (hf : ∀ x ∈ s, HasGradientAt f (f' x) x)
     (h₁ : StrictConvexOn ℝ s f ) :
-    ∀ x ∈ s, ∀ y ∈ s, x ≠ y → inner (f' x - f' y) (x - y) > (0 : ℝ) := by
+    ∀ x ∈ s, ∀ y ∈ s, x ≠ y → inner ℝ (f' x - f' y) (x - y) > (0 : ℝ) := by
   intro x xin y yin xney
   have convf : ConvexOn ℝ s f := by apply StrictConvexOn.convexOn h₁
   rw [StrictConvexOn] at h₁
   rcases h₁ with ⟨hs, fsconv⟩
-  have : inner (f' x - f' y) (x - y) ≥ (0 : ℝ) := by
+  have : inner ℝ (f' x - f' y) (x - y) ≥ (0 : ℝ) := by
     apply Convex_monotone_gradient' convf hf x xin y yin
   by_contra h0; push_neg at h0
-  have eq : inner (f' x - f' y) (x - y) = (0 : ℝ) := by linarith
-  have eq1 : f x + inner (f' x) (y - x) ≤ f y := by
+  have eq : inner ℝ (f' x - f' y) (x - y) = (0 : ℝ) := by linarith
+  have eq1 : f x + inner ℝ (f' x) (y - x) ≤ f y := by
     apply Convex_first_order_condition' (hf x xin) convf xin y yin
-  have eq2 : f y + inner (f' y) (x - y) ≤ f x := by
+  have eq2 : f y + inner ℝ (f' y) (x - y) ≤ f x := by
     apply Convex_first_order_condition' (hf y yin) convf yin x xin
-  have eq2' : f y ≤ f x + inner (f' x) (y - x) := by
-    rw [← add_zero (inner (f' x) (y - x)), ← eq, inner_sub_left, add_sub, ← inner_add_right]
+  have eq2' : f y ≤ f x + inner ℝ (f' x) (y - x) := by
+    rw [← add_zero (inner ℝ (f' x) (y - x)), ← eq, inner_sub_left, add_sub, ← inner_add_right]
     simp; apply eq2
-  have eq3 : f y - f x = inner (f' x) (y - x) := by linarith
-  have extc : ∃ c : ℝ, c ∈ Set.Ioo 0 1 ∧ inner (f' (x + c • (y - x))) (y - x) = f y - f x := by
+  have eq3 : f y - f x = inner ℝ (f' x) (y - x) := by linarith
+  have extc : ∃ c : ℝ, c ∈ Set.Ioo 0 1 ∧ inner ℝ (f' (x + c • (y - x))) (y - x) = f y - f x := by
     apply lagrange hs hf x xin y yin
   rcases extc with ⟨c, cin, e1⟩
   let z : E := x + c • (y - x)
@@ -458,20 +459,24 @@ theorem strict_convex_monotone_gradient (hf : ∀ x ∈ s, HasGradientAt f (f' x
     apply Convex.add_smul_sub_mem hs xin yin; simp; simp at cin; rcases cin with ⟨cpos, cl1⟩
     constructor <;> linarith
   simp at cin; rcases cin with ⟨cpos, cl1⟩
-  have eq0 : inner (f' z - f' x) (z - x) = (0 : ℝ) := by
-    simp [z]; rw [inner_smul_right, inner_sub_left, ← eq3, e1]; simp
-  have eq4 : f x + inner (f' x) (z - x) ≤ f z := by
+  have hz : z - x = c • (y - x) := by simp [z]
+  have e1' : inner ℝ (f' z) (y - x) = f y - f x := by simpa [z] using e1
+  have eq0 : inner ℝ (f' z - f' x) (z - x) = (0 : ℝ) := by
+    rw [hz, inner_smul_right, inner_sub_left, e1', ← eq3]; ring
+  have eq4 : f x + inner ℝ (f' x) (z - x) ≤ f z := by
     apply Convex_first_order_condition' (hf x xin) convf xin z zin
-  have eq5 : f z + inner (f' z) (x - z) ≤ f x := by
+  have eq5 : f z + inner ℝ (f' z) (x - z) ≤ f x := by
     apply Convex_first_order_condition' (hf z zin) convf zin x xin
-  have eq5' : f z ≤ f x + inner (f' x) (z - x) := by
-    rw [← add_zero (inner (f' x) (z - x)), ← eq0, inner_sub_left]
-    rw [add_sub, add_comm (inner (f' x) (z - x))]
+  have eq5' : f z ≤ f x + inner ℝ (f' x) (z - x) := by
+    rw [← add_zero (inner ℝ (f' x) (z - x)), ← eq0, inner_sub_left]
+    rw [add_sub, add_comm (inner ℝ (f' x) (z - x))]
     rw [← add_sub, ← inner_sub_right, sub_self, inner_zero_right, add_zero]
     rw [← sub_neg_eq_add, ← inner_neg_right, neg_sub]; linarith
-  have eq6 : f z = inner (f' x) (z - x) + f x := by linarith
+  have eq6 : f z = inner ℝ (f' x) (z - x) + f x := by linarith
   have f1 : f z = (1 - c) • f x + c • f y := by
-    rw [eq6]; simp [z]; rw [inner_smul_right, ← eq3]; ring_nf
+    rw [eq6, hz, inner_smul_right, ← eq3]
+    simp [smul_eq_mul]
+    ring
   have f2 : f z < (1 - c) • f x + c • f y := by
     simp
     let d : ℝ := 1 - c
@@ -485,7 +490,7 @@ theorem strict_convex_monotone_gradient (hf : ∀ x ∈ s, HasGradientAt f (f' x
 
 theorem strict_convex_iff_monotone_gradient
     (hs: Convex ℝ s) (h : ∀ x ∈ s, HasGradientAt f (f' x) x) :
-    (∀ x ∈ s, ∀ y ∈ s, x ≠ y → inner (f' x - f' y) (x - y) > (0 : ℝ))
+    (∀ x ∈ s, ∀ y ∈ s, x ≠ y → inner ℝ (f' x - f' y) (x - y) > (0 : ℝ))
     ↔ StrictConvexOn ℝ s f := by
   constructor
   exact monotone_gradient_strict_convex hs h
diff --git a/Optlib/Convex/Farkas.lean b/Optlib/Convex/Farkas.lean
index 38db9b1..73844d7 100644
--- a/Optlib/Convex/Farkas.lean
+++ b/Optlib/Convex/Farkas.lean
@@ -5,10 +5,8 @@ Authors: Shengyang Xu, Chenyi Li
 -/
 import Mathlib.Analysis.Convex.Cone.Basic
 import Mathlib.Analysis.Calculus.LocalExtr.Basic
-import Mathlib.Analysis.NormedSpace.HahnBanach.Separation
+import Mathlib.Analysis.LocallyConvex.Separation
 import Mathlib.Analysis.InnerProductSpace.PiL2
-import Mathlib.Data.Matrix.Rank
-import Mathlib.LinearAlgebra.FiniteDimensional
 import Optlib.Differential.Calculation
 import Optlib.Convex.ClosedCone
 
@@ -55,9 +53,10 @@ lemma polyhedra_iff_cone {σ : Finset ℕ} : ∀ (b : ℕ → EuclideanSpace ℝ
       simp [ht]; specialize cpos i ht; exact cpos; simp [ht]
     rw [h]
     let f : ℕ → EuclideanSpace ℝ (Fin n) := fun i ↦ (c1 i) • (b i)
-    have htt : ∑ x in σ.attach, f x = Finset.sum (attach σ) fun x => (c1 x • b x) := by simp [f]
+    have htt : Finset.sum σ.attach (fun x => f x) = Finset.sum (attach σ) (fun x => c1 x • b x) := by
+      simp [f]
     have h1 : ∀ i : σ, c1 i • b i = c i • b i := by intro i; simp [c1]
-    have ht : ∑ x in σ.attach, f x = Finset.sum (attach σ) fun x => (c x • b x) := by
+    have ht : Finset.sum σ.attach (fun x => f x) = Finset.sum (attach σ) (fun x => c x • b x) := by
       rw [← htt]; apply Finset.sum_congr; simp
       intro i _; simp [f, c1]
     nth_rw 1 [Finset.sum_attach] at htt
@@ -67,7 +66,8 @@ lemma polyhedra_iff_cone {σ : Finset ℕ} : ∀ (b : ℕ → EuclideanSpace ℝ
   use c1; constructor
   · intro i _; exact cpos i
   let f : ℕ → EuclideanSpace ℝ (Fin n) := fun i ↦ (c i) • (b i)
-  have : ∑ x in σ.attach, f x = Finset.sum (attach σ) fun x => (c x • b x) := by simp [f]
+  have : Finset.sum σ.attach (fun x => f x) = Finset.sum (attach σ) (fun x => c x • b x) := by
+    simp [f]
   rw [← h]; simp [c1]; rw [← this, Finset.sum_attach]
 
 private lemma leq_tendsto_zero {a x : ℝ} (ha : a < 0) (h : ∀ t > 0, t * x > a) : 0 ≤ x := by
@@ -109,7 +109,8 @@ private lemma shift_sum (τ : Finset ℕ) (m : ℕ) (f : ℕ → EuclideanSpace
   rw [Finset.sum_attach, Finset.sum_attach, Finset.sum_image aux, eq]; simp
 
 private lemma shift_not_in (τ : Finset ℕ) (m : ℕ) (hm : ∀ i : τ, i < m): m ∉ τ := by
-  contrapose hm; simp; simp at hm; use m
+  intro hmem
+  exact Nat.lt_irrefl m (hm ⟨m, hmem⟩)
 
 private lemma mem_lt_m {m i : ℕ} {σ τ : Finset ℕ} (he : (τ ∪ σ).Nonempty)
     (hm : m = (Finset.max' (τ ∪ σ) he).succ) : (i ∈ (τ ∪ σ)) → (i < m) := by
@@ -122,52 +123,58 @@ private lemma exist_of_mem_shift {x m : ℕ} {τ : Finset ℕ}:
 
 private lemma s_inter_t1_empty {m : ℕ} {σ τ : Finset ℕ} (he : (τ ∪ σ).Nonempty)
     (hm : m = (Finset.max' (τ ∪ σ) he).succ) : σ ∩ (Finset.image (fun x => x + m) τ) = ∅ := by
-  by_contra neq; push_neg at neq; rw [← Finset.nonempty_iff_ne_empty] at neq
-  rcases neq with ⟨x, xin⟩
-  absurd Finset.mem_of_mem_inter_left xin
-  apply shift_not_in; intro i
-  calc
-    i.1 < m := by apply mem_lt_m he hm; simp [i.2]
-    m ≤ x := by
-      rcases exist_of_mem_shift (Finset.mem_of_mem_inter_right xin) with ⟨a, _, aeq⟩; linarith
+  rw [Finset.eq_empty_iff_forall_notMem]
+  intro x xin
+  have hx : x ∉ σ := by
+    apply shift_not_in; intro i
+    calc
+      i.1 < m := by apply mem_lt_m he hm; simp [i.2]
+      m ≤ x := by
+        rcases exist_of_mem_shift (Finset.mem_of_mem_inter_right xin) with ⟨a, _, aeq⟩; linarith
+  exact hx (Finset.mem_of_mem_inter_left xin)
 
 private lemma s_inter_t2_empty {m : ℕ} {σ τ : Finset ℕ} (he : (τ ∪ σ).Nonempty)
     (hm : m = (Finset.max' (τ ∪ σ) he).succ) : σ ∩ (Finset.image (fun x => x + 2 * m) τ) = ∅ := by
-  by_contra neq; push_neg at neq; rw [← Finset.nonempty_iff_ne_empty] at neq
-  rcases neq with ⟨x, xin⟩
-  absurd Finset.mem_of_mem_inter_left xin
-  apply shift_not_in; intro i
-  calc
-    i.1 < m := by apply mem_lt_m he hm; simp [i.2]
-    m ≤ 2 * m := by linarith
-    2 * m ≤ x := by
-      rcases exist_of_mem_shift (Finset.mem_of_mem_inter_right xin) with ⟨a, _, aeq⟩; linarith
+  rw [Finset.eq_empty_iff_forall_notMem]
+  intro x xin
+  have hx : x ∉ σ := by
+    apply shift_not_in; intro i
+    calc
+      i.1 < m := by apply mem_lt_m he hm; simp [i.2]
+      m ≤ 2 * m := by linarith
+      2 * m ≤ x := by
+        rcases exist_of_mem_shift (Finset.mem_of_mem_inter_right xin) with ⟨a, _, aeq⟩; linarith
+  exact hx (Finset.mem_of_mem_inter_left xin)
 
 private lemma t1_inter_t2_empty {m : ℕ} {σ τ : Finset ℕ} (he : (τ ∪ σ).Nonempty)
     (hm : m = (Finset.max' (τ ∪ σ) he).succ) :
     (Finset.image (fun x => x + m) τ) ∩ (Finset.image (fun x => x + 2 * m) τ) = ∅ := by
-  by_contra neq; push_neg at neq; rw [← Finset.nonempty_iff_ne_empty] at neq
-  rcases neq with ⟨x, xin⟩
-  absurd Finset.mem_of_mem_inter_left xin
-  apply shift_not_in; intro i
-  rcases exist_of_mem_shift (Finset.mem_of_mem_inter_right xin) with ⟨a, aeq⟩
-  rcases exist_of_mem_shift i.2 with ⟨b, beq⟩
-  calc
-    i.1 < 2 * m := by
-      rw [beq, two_mul]; apply Nat.add_lt_add_right; apply mem_lt_m he hm; simp
-    2 * m ≤ x := by rw [aeq]; simp
+  rw [Finset.eq_empty_iff_forall_notMem]
+  intro x xin
+  have hx : x ∉ Finset.image (fun x => x + m) τ := by
+    apply shift_not_in; intro i
+    rcases exist_of_mem_shift (Finset.mem_of_mem_inter_right xin) with ⟨a, aeq⟩
+    rcases exist_of_mem_shift i.2 with ⟨b, beq⟩
+    calc
+      i.1 < 2 * m := by
+        rw [beq, two_mul]; apply Nat.add_lt_add_right; apply mem_lt_m he hm; simp
+      2 * m ≤ x := by rw [aeq]; simp
+  exact hx (Finset.mem_of_mem_inter_left xin)
 
 lemma general_polyhedra_is_polyhedra_empty (τ σ : Finset ℕ) (he : ¬(τ ∪ σ).Nonempty) :
     ∀ (a : ℕ → EuclideanSpace ℝ (Fin n)), ∀ (b : ℕ → EuclideanSpace ℝ (Fin n)),
     ∃ μ c, {z | ∃ (lam : τ → ℝ), ∃ (mu : σ → ℝ), (∀ i, 0 ≤ mu i) ∧ z =
     Finset.sum univ (fun i ↦ lam i • a i) + Finset.sum univ (fun i ↦ mu i • b i)} =
     cone μ c := by
-  simp at he; rw [Finset.union_eq_empty] at he
-  intro a b; simp [he]
-  use ∅; use (fun _ => 0)
-  simp [cone, quadrant]; ext x; simp; constructor
-  · intro x0; simp [x0]; use (fun _ => 0); simp
-  · intro cond; simp [cond.2]
+  simp at he
+  rcases he with ⟨rfl, rfl⟩
+  intro a b
+  refine ⟨∅, fun _ => 0, ?_⟩
+  ext x
+  simp [cone, quadrant]
+  constructor
+  · intro hx; subst hx; refine ⟨⟨fun _ => 0, by simp⟩, rfl⟩
+  · intro hx; simpa using hx.2.symm
 
 lemma general_polyhedra_is_polyhedra_ne (τ σ : Finset ℕ) (he : (τ ∪ σ).Nonempty) :
     ∀ (a : ℕ → EuclideanSpace ℝ (Fin n)), ∀ (b : ℕ → EuclideanSpace ℝ (Fin n)),
@@ -180,11 +187,11 @@ lemma general_polyhedra_is_polyhedra_ne (τ σ : Finset ℕ) (he : (τ ∪ σ).N
   let τ2 := Finset.image (fun x => x + 2 * m) τ
   let μ := σ ∪ τ1 ∪ τ2
   have mt1emp : σ ∩ τ1 = ∅ := by
-    simp only [τ1]; apply s_inter_t1_empty he; simp
+    simpa [τ1, m] using s_inter_t1_empty (σ := σ) (τ := τ) he (m := m) rfl
   have mt2emp : σ ∩ τ2 = ∅ := by
-    simp only [τ2]; apply s_inter_t2_empty he; simp
+    simpa [τ2, m] using s_inter_t2_empty (σ := σ) (τ := τ) he (m := m) rfl
   have t1t2emp : τ1 ∩ τ2 = ∅ := by
-    simp only [τ1, τ2]; apply t1_inter_t2_empty he; simp
+    simpa [τ1, τ2, m] using t1_inter_t2_empty (σ := σ) (τ := τ) he (m := m) rfl
   have disj_st : Disjoint σ (τ1 ∪ τ2) := by
     rw [Finset.disjoint_iff_inter_eq_empty, Finset.inter_union_distrib_left]; simp [mt1emp, mt2emp]
   have disj_tt : Disjoint τ1 τ2 := by
@@ -219,27 +226,30 @@ lemma general_polyhedra_is_polyhedra_ne (τ σ : Finset ℕ) (he : (τ ∪ σ).N
       simp [hs, ht1, ht2]
     use w; use wnneg
     rw [xeq, tau_decpn]
-    have eq1 : ∑ x : { x // x ∈ σ }, mu x • b x = ∑ x in σ, (fun y => w y • c y) x := by
+    have eq1 : (∑ x : { x // x ∈ σ }, mu x • b x) = Finset.sum σ (fun x => w x • c x) := by
       nth_rw 2 [← Finset.sum_attach]; simp; congr
       ext x j; simp [w, c, cσ]
-    have eq2 : ∑ x : τ, (fun y => lamp y • cτ1 y) x = ∑ x in τ1, (fun y => w y • c y) x := by
+    have eq2 : (∑ x : τ, (fun y => lamp y • cτ1 y) x) = Finset.sum τ1 (fun x => w x • c x) := by
       rw [shift_sum τ m (fun y => lamp y • cτ1 y)]
       nth_rw 2 [← Finset.sum_attach]; simp; congr
       ext x j
       have hns : x.1 ∉ σ := by
-        contrapose mt1emp; simp at mt1emp; push_neg; rw [← Finset.nonempty_iff_ne_empty]
-        use x; simp [τ1, mt1emp, x.2]
+        intro hs
+        have : x.1 ∈ σ ∩ τ1 := by simp [hs]
+        simp [mt1emp] at this
       simp [w, c, hns]
-    have eq3 : ∑ x : τ, (fun y => lamn y • cτ2 y) x = ∑ x in τ2, (fun y => w y • c y) x := by
+    have eq3 : (∑ x : τ, (fun y => lamn y • cτ2 y) x) = Finset.sum τ2 (fun x => w x • c x) := by
       rw [shift_sum τ (2 * m) (fun y => lamn y • cτ2 y)]
       nth_rw 2 [← Finset.sum_attach]; simp; congr
       ext x j
       have hns : x.1 ∉ σ := by
-        contrapose mt2emp; simp at mt2emp; push_neg; rw [← Finset.nonempty_iff_ne_empty]
-        use x; simp [τ2, mt2emp, x.2]
+        intro hs
+        have : x.1 ∈ σ ∩ τ2 := by simp [hs]
+        simp [mt2emp] at this
       have hnt : x.1 ∉ τ1 := by
-        contrapose t1t2emp; simp at t1t2emp; push_neg; rw [← Finset.nonempty_iff_ne_empty]
-        use x; simp [τ2, t1t2emp, x.2]
+        intro hs
+        have : x.1 ∈ τ1 ∩ τ2 := by simp [hs]
+        simp [t1t2emp] at this
       simp [w, c, hns, hnt]
     rw [eq1, eq2, eq3]; simp [μ]
     rw [Finset.sum_union disj_st, Finset.sum_union disj_tt, add_comm]
@@ -250,32 +260,35 @@ lemma general_polyhedra_is_polyhedra_ne (τ σ : Finset ℕ) (he : (τ ∪ σ).N
     let lamn : ℕ → ℝ := fun i => if i ∈ τ then w (i + 2 * m) else 0
     let lam : τ → ℝ := fun i => lamp i.1 - lamn i.1
     let mu : ℕ → ℝ := fun i => if i ∈ σ then w i else 0
-    have eq1 : ∑ x : { x // x ∈ σ }, mu x • b x = ∑ x in σ, (fun y => w y • c y) x := by
+    have eq1 : (∑ x : { x // x ∈ σ }, mu x • b x) = Finset.sum σ (fun x => w x • c x) := by
       nth_rw 2 [← Finset.sum_attach]; simp; congr
       ext x j; simp [mu, c, cσ]
-    have eq2 : ∑ x : τ, (fun y => lamp y • cτ1 y) x = ∑ x in τ1, (fun y => w y • c y) x := by
+    have eq2 : (∑ x : τ, (fun y => lamp y • cτ1 y) x) = Finset.sum τ1 (fun x => w x • c x) := by
       rw [shift_sum τ m (fun y => lamp y • cτ1 y)]
       nth_rw 2 [← Finset.sum_attach]; simp; congr
       ext x j
       have hns : x.1 ∉ σ := by
-        contrapose mt1emp; simp at mt1emp; push_neg; rw [← Finset.nonempty_iff_ne_empty]
-        use x; simp [τ1, mt1emp, x.2]
+        intro hs
+        have : x.1 ∈ σ ∩ τ1 := by simp [hs]
+        simp [mt1emp] at this
       rcases exist_of_mem_shift x.2 with ⟨a, eq⟩
       have hin : x.1 - m ∈ τ := by rw [eq]; simp
-      simp [mu, lamp, c, hns, hin]; rw [eq]; simp
-    have eq3 : ∑ x : τ, (fun y => lamn y • cτ2 y) x = ∑ x in τ2, (fun y => w y • c y) x := by
+      simp [lamp, c, hns, hin]; rw [eq]; simp
+    have eq3 : (∑ x : τ, (fun y => lamn y • cτ2 y) x) = Finset.sum τ2 (fun x => w x • c x) := by
       rw [shift_sum τ (2 * m) (fun y => lamn y • cτ2 y)]
       nth_rw 2 [← Finset.sum_attach]; simp; congr
       ext x j
       have hns : x.1 ∉ σ := by
-        contrapose mt2emp; simp at mt2emp; push_neg; rw [← Finset.nonempty_iff_ne_empty]
-        use x; simp [τ2, mt2emp, x.2]
+        intro hs
+        have : x.1 ∈ σ ∩ τ2 := by simp [hs]
+        simp [mt2emp] at this
       have hnt : x.1 ∉ τ1 := by
-        contrapose t1t2emp; simp at t1t2emp; push_neg; rw [← Finset.nonempty_iff_ne_empty]
-        use x; simp [τ2, t1t2emp, x.2]
+        intro hs
+        have : x.1 ∈ τ1 ∩ τ2 := by simp [hs]
+        simp [t1t2emp] at this
       rcases exist_of_mem_shift x.2 with ⟨a, eq⟩
       have hin : x.1 - 2 * m ∈ τ := by rw [eq]; simp
-      simp [mu, lamn, c, hns, hnt, hin]; rw [eq]; simp
+      simp [lamn, c, hns, hnt, hin]; rw [eq]; simp
     rw [← eq1, ← eq2, ← eq3] at xeq; simp at xeq
     simp; use lam; use (fun i => mu i); constructor
     · intro a ain; simp [mu, ain]; linarith [wnneg a]
@@ -301,24 +314,24 @@ lemma general_polyhedra_is_closed : IsClosed {z | ∃ (lam : τ → ℝ), ∃ (m
 theorem Farkas :
   (∃ (lam : τ → ℝ), ∃ (mu : σ → ℝ), (∀ i, 0 ≤ mu i) ∧ c =
     Finset.sum univ (fun i ↦ lam i • a i) + Finset.sum univ (fun i ↦ mu i • b i)) ↔
-    ¬ (∃ (z : EuclideanSpace ℝ (Fin n)), (∀ i ∈ τ, inner (a i) z = (0 : ℝ))
-    ∧ (∀ i ∈ σ, inner (b i) z ≥ (0 : ℝ)) ∧ (inner c z < (0 : ℝ))) := by
+    ¬ (∃ (z : EuclideanSpace ℝ (Fin n)), (∀ i ∈ τ, inner ℝ (a i) z = (0 : ℝ))
+    ∧ (∀ i ∈ σ, inner ℝ (b i) z ≥ (0 : ℝ)) ∧ (inner ℝ c z < (0 : ℝ))) := by
   constructor
   intro h; rcases h with ⟨lam, mu, ⟨h1, h2⟩⟩
   by_contra h3
   rcases h3 with ⟨z, ⟨h31, ⟨h32, h33⟩⟩⟩
-  have : inner c z ≥ (0 : ℝ) := by
+  have : inner ℝ c z ≥ (0 : ℝ) := by
     calc
-      _ = inner (Finset.sum univ (fun i ↦ lam i • a i)) z
-          + inner (Finset.sum univ (fun i ↦ mu i • b i)) z := by rw [h2]; simp [inner_add_left]
-      _ = Finset.sum univ (fun i ↦ inner (lam i • a i) z)
-          + Finset.sum univ (fun i ↦ inner (mu i • b i) z) := by
+      _ = inner ℝ (Finset.sum univ (fun i ↦ lam i • a i)) z
+          + inner ℝ (Finset.sum univ (fun i ↦ mu i • b i)) z := by rw [h2]; simp [inner_add_left]
+      _ = Finset.sum univ (fun i ↦ inner ℝ (lam i • a i) z)
+          + Finset.sum univ (fun i ↦ inner ℝ (mu i • b i) z) := by
         rw [sum_inner, sum_inner]
-      _ = Finset.sum univ (fun i ↦ lam i * inner (a i) z)
-          + Finset.sum univ (fun i ↦ mu i * inner (b i) z) := by
+      _ = Finset.sum univ (fun i ↦ lam i * inner ℝ (a i) z)
+          + Finset.sum univ (fun i ↦ mu i * inner ℝ (b i) z) := by
         congr; ext i; rw [inner_smul_left]; simp
         ext i; rw [inner_smul_left]; simp
-      _ = Finset.sum univ (fun i ↦ mu i * inner (b i) z) := by simp [h31]
+      _ = Finset.sum univ (fun i ↦ mu i * inner ℝ (b i) z) := by simp [h31]
       _ ≥ 0 := by
         apply Finset.sum_nonneg; intro i _
         obtain h1i := h1 i; obtain h2i := h32 i i.2; positivity
@@ -362,19 +375,22 @@ theorem Farkas :
   have cn : c ∉ S := by
     by_contra cn; simp only [Set.mem_setOf_eq, S] at cn;
     rcases cn with ⟨lam, mu, ⟨cn1, cn2⟩⟩
-    apply h1; use lam; use mu
+    exact h1 ⟨lam, mu, cn1, cn2⟩
   obtain sep := geometric_hahn_banach_point_closed scon sc cn
   rcases sep with ⟨f, u, ⟨sep1, sep2⟩⟩
-  have feq : ∃ d : EuclideanSpace ℝ (Fin n), ∀ x, f x = inner d x := by
-    use ((toDual ℝ (EuclideanSpace ℝ (Fin n))).symm f); simp
+  have feq : ∃ d : EuclideanSpace ℝ (Fin n), ∀ x, f x = inner ℝ d x := by
+    refine ⟨(toDual ℝ (EuclideanSpace ℝ (Fin n))).symm f, ?_⟩
+    intro x
+    symm
+    exact toDual_symm_apply (𝕜 := ℝ) (E := EuclideanSpace ℝ (Fin n)) (x := x) (y := f)
   rcases feq with ⟨d, feq⟩
   have uleq : u < 0 := by
     have : 0 ∈ S := by simp [S]; use 0; use 0; simp
     specialize sep2 0 this; rw [feq 0, inner_zero_right] at sep2; exact sep2
-  have hc : inner c d < (0 : ℝ) := by
+  have hc : inner ℝ c d < (0 : ℝ) := by
     rw [real_inner_comm, ← feq c]
     apply lt_trans sep1 uleq
-  have hb : ∀ i : σ, inner (b i) d ≥ (0 : ℝ) := by
+  have hb : ∀ i : σ, inner ℝ (b i) d ≥ (0 : ℝ) := by
     intro i
     have : ∀ t > (0 : ℝ), (t • b i) ∈ S := by
       intro t ht
@@ -385,13 +401,13 @@ theorem Farkas :
     intro t ht
     specialize sep2 (t • b i) (this t ht);
     rw [feq, inner_smul_right, real_inner_comm] at sep2; exact sep2
-  have ha : ∀ i : τ, inner (a i) d = (0 : ℝ) := by
+  have ha : ∀ i : τ, inner ℝ (a i) d = (0 : ℝ) := by
     intro i
     have : ∀ t : ℝ, (t • a i) ∈ S := by
       intro t
       simp only [S]; use (fun j ↦ if j = i then t else 0); use 0;
       constructor; intro _; simp; simp only [Pi.zero_apply, zero_smul, sum_const_zero,
-        ite_smul, zero_add]; simp
+        ite_smul]; simp
     rw [le_antisymm_iff]; constructor
     · apply geq_tendsto_zero uleq
       intro t _
diff --git a/Optlib/Convex/FiniteDimensionalConvexFunctionsLocallyLipschitz.lean b/Optlib/Convex/FiniteDimensionalConvexFunctionsLocallyLipschitz.lean
index 5e90ea4..1b0cd98 100644
--- a/Optlib/Convex/FiniteDimensionalConvexFunctionsLocallyLipschitz.lean
+++ b/Optlib/Convex/FiniteDimensionalConvexFunctionsLocallyLipschitz.lean
@@ -103,24 +103,22 @@ lemma Bounded_of_UpperBounded (hf : ConvexOn ℝ (ball x₀ r) f)
   simp only [smul_eq_mul, a] at h
   have h' : - f y + 2 * f x₀  ≤  f x := by linarith
   have fy_pos : - |m| ≤ - f y := by
-    simp only [neg_le_neg_iff, ge_iff_le]
+    simp only [neg_le_neg_iff]
     apply le_trans (hm y y_pos) (le_abs_self m)
   constructor
   · calc
       _ = -|m| - 2 * (|f x₀| + 1):= neg_add' |m| (2 * (|f x₀| + 1))
       _ <  -f y + 2 * f x₀ := by
         apply add_lt_add_of_le_of_lt fy_pos
-        rw[← mul_neg]
-        simp only [Nat.ofNat_pos, mul_lt_mul_left,neg_add']
-        calc
-          _ < -|f x₀| :=by simp only [sub_lt_self_iff, zero_lt_one]
-          _ ≤ _ := neg_abs_le (f x₀)
+        have hx₀ : -2 * (|f x₀| + 1) < 2 * f x₀ := by
+          linarith [neg_abs_le (f x₀)]
+        simpa [neg_add'] using hx₀
       _ ≤ _ := h'
   · calc
       _ ≤ m := by apply hm x hx.1
       _ ≤ |m| := le_abs_self m
       _ < _ := by
-        simp only [lt_add_iff_pos_right, Nat.ofNat_pos, mul_pos_iff_of_pos_left]
+        simp only [lt_add_iff_pos_right]
         linarith [abs_nonneg (f x₀)]
 
 /--
@@ -145,7 +143,7 @@ lemma Lipschitz_of_Bounded [T0Space X](hf : ConvexOn ℝ (ball x₀ r) f)
   --type conversion
   rw[edist_dist,edist_dist]
   rw[ENNReal.coe_nnreal_eq]
-  simp only [NNReal.coe_mk, ge_iff_le]
+  simp only [NNReal.coe_mk]
   rw[← ENNReal.ofReal_mul K_pos]
   rw[ENNReal.ofReal_le_ofReal_iff (mul_nonneg K_pos dist_nonneg)]
   --type conversion
@@ -161,9 +159,13 @@ lemma Lipschitz_of_Bounded [T0Space X](hf : ConvexOn ℝ (ball x₀ r) f)
     have vx_pos :  vx ∈ ball x₀ r := sub hv
     have uy_pos :  uy ∈ ball x₀ r := sub hu
     let z := uy + (ε / ‖uy - vx‖) • (uy - vx)
+    let _ := MetricSpace.ofT0PseudoMetricSpace X
     have sub_pos : 0 < ‖uy - vx‖ := by
-      apply norm_pos_iff'.mpr
-      exact sub_ne_zero_of_ne h
+      have hne : ‖uy - vx‖ ≠ 0 := by
+        intro hnorm
+        apply h
+        exact dist_eq_zero.mp <| by simpa [dist_eq_norm] using hnorm
+      exact lt_of_le_of_ne (norm_nonneg _) (Ne.symm hne)
     have z_pos : z ∈ ball x₀ r := by
       simp only [mem_ball,dist_eq_norm,z]
       calc
@@ -193,8 +195,7 @@ lemma Lipschitz_of_Bounded [T0Space X](hf : ConvexOn ℝ (ball x₀ r) f)
       apply div_pos hε.1 this
     have b_pos : 0 < b := by
       apply div_pos
-      rw[norm_pos_iff']
-      exact sub_ne_zero_of_ne h
+      exact sub_pos
       apply this
     have a_add_b_one : a + b = 1 := by
       simp[a,b]
@@ -215,9 +216,9 @@ lemma Lipschitz_of_Bounded [T0Space X](hf : ConvexOn ℝ (ball x₀ r) f)
     have h1 : (ε + ‖uy - vx‖) * f uy ≤ ε * f vx + ‖uy - vx‖ * f z:= by
       rw[← h_combin] at h
       simp[a,b] at h
-      rw[← mul_le_mul_left this] at h
-      field_simp at h
-      exact h
+      have h' := mul_le_mul_of_nonneg_left h (le_of_lt this)
+      field_simp at h'
+      exact h'
     have h2 : ε * (f uy - f vx) ≤ 2 * M * ‖uy - vx‖ := by
       calc
         _ ≤ (f z - f uy) * ‖uy - vx‖ := by
@@ -243,10 +244,8 @@ lemma Lipschitz_of_Bounded [T0Space X](hf : ConvexOn ℝ (ball x₀ r) f)
       _ ≤ _ := by
         simp[K]
         apply mul_le_mul_of_nonneg_right _ (le_of_lt sub_pos)
-        rw[div_le_div_right hε.1]
-        apply mul_le_mul_of_nonneg_left
-        apply le_abs_self
-        norm_num
+        rw[div_le_div_iff_of_pos_right hε.1]
+        nlinarith [le_abs_self M]
   by_cases h : x = y
   · rw[h]
     simp only [sub_self, abs_zero, norm_zero, mul_zero, le_refl]
@@ -325,8 +324,7 @@ lemma LocallyUpperBounded (hs_convex : Convex ℝ s)(hs_isopen : IsOpen s)
   have bi_pos : ∀ i : ι , ‖b i‖ ≠ 0 := by
     intro i
     refine norm_ne_zero_iff.mpr ?_
-    exact Basis.ne_zero b i
-  change Basis ι ℝ α at b
+    exact (b).ne_zero i
   by_cases hn : n = 0
   · have : Module.finrank ℝ α = 0 := by
       show n = 0;apply hn;
@@ -368,7 +366,7 @@ lemma LocallyUpperBounded (hs_convex : Convex ℝ s)(hs_isopen : IsOpen s)
         simp only [neg_add_cancel_comm];
       rw[this]
       apply hr₀.2
-      simp only [mem_ball, dist_self_add_left,dist_add_self_left]
+      simp only [mem_ball, dist_add_self_left]
       rw[norm_smul,norm_div,norm_norm,div_mul_cancel₀]
       simp[r]
       calc
@@ -381,7 +379,7 @@ lemma LocallyUpperBounded (hs_convex : Convex ℝ s)(hs_isopen : IsOpen s)
         simp only [neg_add_cancel_comm]
       rw[this]
       apply hr₀.2
-      simp only [mem_ball, dist_self_add_left,dist_add_self_left,neg_smul, norm_neg]
+      simp only [mem_ball, dist_add_self_left, neg_smul, norm_neg]
       rw[norm_smul,norm_div,norm_norm,div_mul_cancel₀]
       simp[r]
       calc
@@ -435,6 +433,7 @@ lemma LocallyUpperBounded (hs_convex : Convex ℝ s)(hs_isopen : IsOpen s)
   apply ConvexOn.le_sup_of_mem_convexHull hf _ hx
   apply subset_convexHull
 
+omit [FiniteDimensional ℝ α] in
 lemma LocallyLipschitz_of_LocallyUpperBounded (hs : IsOpen s)
     (h : ∀ x ∈ s , ∃ t ∈ 𝓝[s] x , Convex ℝ t ∧ IsOpen t ∧ BddAbove (f '' t))
     (hf : ConvexOn ℝ s f)
diff --git a/Optlib/Convex/ImageSubgradientClosed.lean b/Optlib/Convex/ImageSubgradientClosed.lean
index e0bb3d2..a89cbf4 100644
--- a/Optlib/Convex/ImageSubgradientClosed.lean
+++ b/Optlib/Convex/ImageSubgradientClosed.lean
@@ -4,7 +4,6 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Author: Zichen Wang
 -/
 import Optlib.Function.Proximal
-import Mathlib.Topology.Instances.EReal
 
 open Set InnerProductSpace Topology Filter
 
diff --git a/Optlib/Convex/QuasiConvexFirstOrder.lean b/Optlib/Convex/QuasiConvexFirstOrder.lean
index e1db12a..0af9684 100644
--- a/Optlib/Convex/QuasiConvexFirstOrder.lean
+++ b/Optlib/Convex/QuasiConvexFirstOrder.lean
@@ -16,7 +16,8 @@ noncomputable section
 variable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
 variable {f : E → ℝ} {f' : E → (E →L[ℝ] ℝ)} {s : Set E}{x: E}
 
-theorem Quasiconvex_first_order_condition_right (h : HasFDerivAt f (f' x) x) (xs : x ∈ s)
+omit [CompleteSpace E] in theorem Quasiconvex_first_order_condition_right
+    (h : HasFDerivAt f (f' x) x) (xs : x ∈ s)
     (hf: QuasiconvexOn ℝ s f) : ∀ y ∈ s, f y ≤ f x  → f' x (y - x) ≤ 0 := by
   have h₁: ∀ ε > (0 : ℝ), ∃ δ > (0 : ℝ), ∀ (x' : E), ‖x - x'‖ ≤ δ →
       ‖f x' - f x - (f' x) (x' - x)‖ ≤ ε * ‖x - x'‖ := by
@@ -35,7 +36,7 @@ theorem Quasiconvex_first_order_condition_right (h : HasFDerivAt f (f' x) x) (xs
   let ε := (f' x) (y - x) / (2 * ‖x-y‖)
   have εpos: 0 < ε := by
     apply div_pos H
-    exact Real.mul_pos two_pos h₃
+    exact mul_pos two_pos h₃
   specialize h₁ ε εpos
   rcases h₁ with ⟨δ, dpos, converge⟩
   let b1:= δ /(‖x - y‖)
@@ -60,7 +61,7 @@ theorem Quasiconvex_first_order_condition_right (h : HasFDerivAt f (f' x) x) (xs
       _= (1 : ℝ) • x - a • x - b • y:= by
         rw [one_smul]
       _= b • (x - y) := by
-        rw [← sub_smul 1 a]; simp [a, b, sum_a_b]; rw[smul_sub b x y]
+        rw [← sub_smul 1 a]; simp [a, b]; rw[smul_sub b x y]
   have h01 : x' - x =  b • (y - x) :=by
     rw [← neg_inj, ← smul_neg, neg_sub, neg_sub]; exact h10
   have h1 : ‖x - x'‖ = ‖b • (x - y)‖ := by
@@ -71,9 +72,7 @@ theorem Quasiconvex_first_order_condition_right (h : HasFDerivAt f (f' x) x) (xs
   have x1nbhd: ‖x - x'‖ ≤ δ := by
     rw [h1, h2]
     have h3: b * ‖x - y‖ ≤ b1 * ‖x - y‖:= by
-      rw [mul_le_mul_right]
-      apply min_le_left
-      exact h₃
+      exact mul_le_mul_of_nonneg_right (min_le_left b1 1) (norm_nonneg _)
     have h4: b1 * ‖x - y‖= δ := by
       rw[div_mul_cancel₀]
       apply ne_of_gt h₃
diff --git a/Optlib/Convex/StronglyConvex.lean b/Optlib/Convex/StronglyConvex.lean
index c36d40e..471aac1 100644
--- a/Optlib/Convex/StronglyConvex.lean
+++ b/Optlib/Convex/StronglyConvex.lean
@@ -76,7 +76,7 @@ theorem Strongly_Convex_Unique_Minima (hsc: StrongConvexOn s m f) {mp : m > 0}
     . linarith
     . apply pow_pos; linarith
   apply absurd (min xs)
-  simp [← xeq]
+  simp
   calc
     f x ≤ f xm - 2⁻¹ * 2⁻¹ * (m / 2 * ‖xm - xm'‖ ^ 2) := by apply sc
     _ < f xm := by apply lt_of_sub_pos; simp; apply nng
@@ -99,7 +99,7 @@ lemma strongconvex_of_convex_add_sq (f : E → ℝ) (x : E) (hfun : ConvexOn ℝ
     apply add_le_add
     · rw [← smul_eq_mul, ← smul_eq_mul]
       apply hfun yin zin anneg bnneg absum1
-    · field_simp; rw [div_le_div_right, add_sub]
+    · field_simp
       have eq1 : a • y + b • z - x = a • (y - x) + b • (z - x) := by
         rw [smul_sub, smul_sub, add_comm_sub, sub_sub, ← add_smul, add_comm b a]
         rw [absum1, one_smul, ← add_sub]
@@ -107,22 +107,21 @@ lemma strongconvex_of_convex_add_sq (f : E → ℝ) (x : E) (hfun : ConvexOn ℝ
         + a * ‖u‖ ^ 2 - a * b * ‖u - v‖ ^ 2 := by
         rw [norm_add_sq_real, norm_sub_sq_real]
         rw [inner_smul_left, inner_smul_right, norm_smul, norm_smul]; field_simp
-        rw [add_comm (b * ‖v‖ ^ 2), mul_pow, sq_abs, mul_pow, sq_abs]
-        rw [mul_add, ← sub_sub, mul_sub, ← sub_add]
-        rw [add_sub_right_comm, add_sub_right_comm, ← sub_mul, ← add_sub, ← sub_mul]
-        nth_rw 3 [← mul_one a]; rw [← absum1, mul_add]
-        nth_rw 5 [← mul_one b]; rw [← absum1, mul_add, mul_comm b a]
-        rw [pow_two, pow_two b]; simp; rw [add_right_comm, add_left_cancel_iff]
-        rw [mul_mul_mul_comm, mul_comm a 2, mul_assoc]
+        simp [Real.norm_eq_abs, sq_abs] at *
+        have hab : b = 1 - a := by linarith
+        rw [hab]
+        ring
       have eq3 : y - z = (y - x) - (z - x) := by simp
       have eq4 (u v : E) : ‖a • u + b • v‖ ^ 2 ≤ b * ‖v‖ ^ 2
         + a * ‖u‖ ^ 2 - a * b * ‖u - v‖ ^ 2 := by rw [eq2]
       let u := y - x
       let v := z - x
-      rw [eq1, eq3];
-      show ‖a • u + b • v‖ ^ 2 ≤ b * ‖v‖ ^ 2 + a * ‖u‖ ^ 2 - a * b * ‖u - v‖ ^ 2
-      apply eq4 u v
-      simp
+      rw [eq1]
+      calc
+        ‖a • (y - x) + b • (z - x)‖ ^ 2
+            ≤ b * ‖z - x‖ ^ 2 + a * ‖y - x‖ ^ 2 - a * b * ‖y - z‖ ^ 2 := by
+              simpa [u, v] using (eq4 u v)
+        _ = b * ‖z - x‖ ^ 2 + a * (‖y - x‖ ^ 2 - b * ‖y - z‖ ^ 2) := by ring
 
 end Strongly_Convex
 
@@ -131,7 +130,7 @@ section
 variable [CompleteSpace E]
 
 theorem Strong_Convex_lower (hsc : StrongConvexOn s m f) (hf : ∀ x ∈ s, HasGradientAt f (f' x) x) :
-    ∀ x ∈ s, ∀ y ∈ s, inner (f' x - f' y) (x - y) ≥ m * ‖x - y‖ ^ 2 := by
+    ∀ x ∈ s, ∀ y ∈ s, inner ℝ (f' x - f' y) (x - y) ≥ m * ‖x - y‖ ^ 2 := by
   intro x xs y ys
   have cvx := strongConvexOn_iff_convex.mp hsc
   have grd := sub_normsquare_gradient hf m
@@ -142,7 +141,7 @@ theorem Strong_Convex_lower (hsc : StrongConvexOn s m f) (hf : ∀ x ∈ s, HasG
   apply grm
 
 theorem Lower_Strong_Convex (hf : ∀ x ∈ s, HasGradientAt f (f' x) x) (hs : Convex ℝ s)
-    (h : ∀ x ∈ s, ∀ y ∈ s, inner (f' x - f' y) (x - y) ≥ m * ‖x - y‖ ^ 2) :
+    (h : ∀ x ∈ s, ∀ y ∈ s, inner ℝ (f' x - f' y) (x - y) ≥ m * ‖x - y‖ ^ 2) :
     StrongConvexOn s m f := by
   apply strongConvexOn_iff_convex.mpr
   have grd := sub_normsquare_gradient hf m
@@ -155,12 +154,12 @@ theorem Lower_Strong_Convex (hf : ∀ x ∈ s, HasGradientAt f (f' x) x) (hs : C
   apply h
 
 theorem Strong_Convex_iff_lower (hf : ∀ x ∈ s, HasGradientAt f (f' x) x) (hs : Convex ℝ s) :
-    StrongConvexOn s m f ↔ ∀ x ∈ s, ∀ y ∈ s, inner (f' x - f' y) (x - y) ≥ m * ‖x - y‖ ^ 2 :=
+    StrongConvexOn s m f ↔ ∀ x ∈ s, ∀ y ∈ s, inner ℝ (f' x - f' y) (x - y) ≥ m * ‖x - y‖ ^ 2 :=
   ⟨fun hsc x xs y ys ↦ Strong_Convex_lower hsc hf x xs y ys, fun h ↦ Lower_Strong_Convex hf hs h⟩
 
 theorem Strong_Convex_second_lower (hsc: StrongConvexOn s m f)
     (hf : ∀ x ∈ s, HasGradientAt f (f' x) x) : ∀ x ∈ s, ∀ y ∈ s,
-    f y ≥ f x + inner (f' x) (y - x) + m / 2 * ‖y - x‖ ^ 2 := by
+    f y ≥ f x + inner ℝ (f' x) (y - x) + m / 2 * ‖y - x‖ ^ 2 := by
   intro x xs y ys
   have cvx := strongConvexOn_iff_convex.mp hsc
   have grd := sub_normsquare_gradient hf m x xs
@@ -179,7 +178,7 @@ theorem Strong_Convex_second_lower (hsc: StrongConvexOn s m f)
   nth_rw 1 [← sub_self y] at foc
   rw [← sub_self x] at foc
   rw [sub_add, ← sub_add y x x, add_comm (y - x), inner_sub_right x, inner_add_right y] at foc
-  rw [real_inner_comm x y, sub_right_comm (inner x y), ← sub_sub, sub_self, sub_sub 0] at foc
+  rw [real_inner_comm x y, sub_right_comm (inner ℝ x y), ← sub_sub, sub_self, sub_sub 0] at foc
   rw [← inner_add_left, zero_sub, mul_neg, sub_neg_eq_add] at foc
   have : m = m / 2 * 2 := by simp
   nth_rw 1 [this] at foc
diff --git a/Optlib/Convex/Subgradient.lean b/Optlib/Convex/Subgradient.lean
index c9c48f2..b9377d4 100644
--- a/Optlib/Convex/Subgradient.lean
+++ b/Optlib/Convex/Subgradient.lean
@@ -189,11 +189,11 @@ theorem SubderivAt.convex : ∀ x, Convex ℝ (SubderivAt f x) := by
   have ineq1 : a • f y ≥ a • f x + a • ⟪g₁, y - x⟫ := by
     rw [← smul_add]
     apply smul_le_smul_of_nonneg_left (h1 y) lea
-  have ineq2 : b • f y ≥ b • f x + b • inner g₂ (y - x) := by
+  have ineq2 : b • f y ≥ b • f x + b • ⟪g₂, y - x⟫ := by
     rw [← smul_add]
     apply smul_le_smul_of_nonneg_left (h2 y) leb
-  have eq : (a • f x + a • inner g₁ (y - x)) + (b • f x + b • inner g₂ (y - x))
-      = f x + inner (a • g₁ + b • g₂) (y - x) := by
+  have eq : (a • f x + a • ⟪g₁, y - x⟫) + (b • f x + b • ⟪g₂, y - x⟫)
+      = f x + ⟪a • g₁ + b • g₂, y - x⟫ := by
     rw [add_add_add_comm, ← Eq.symm (Convex.combo_self abeq (f x))]
     apply congrArg (HAdd.hAdd (f x))
     rw [inner_add_left, inner_smul_left, inner_smul_left]; rfl
@@ -208,11 +208,11 @@ theorem SubderivWithinAt.convex : ∀ x ∈ s, Convex ℝ (SubderivWithinAt f s
   have ineq1 : a • f y ≥ a • f x + a • ⟪g₁, y - x⟫ := by
     rw [← smul_add]
     apply smul_le_smul_of_nonneg_left (h1 y ys) lea
-  have ineq2 : b • f y ≥ b • f x + b • inner g₂ (y - x) := by
+  have ineq2 : b • f y ≥ b • f x + b • ⟪g₂, y - x⟫ := by
     rw [← smul_add]
     apply smul_le_smul_of_nonneg_left (h2 y ys) leb
-  have eq : (a • f x + a • inner g₁ (y - x)) + (b • f x + b • inner g₂ (y - x))
-      = f x + inner (a • g₁ + b • g₂) (y - x) := by
+  have eq : (a • f x + a • ⟪g₁, y - x⟫) + (b • f x + b • ⟪g₂, y - x⟫)
+      = f x + ⟪a • g₁ + b • g₂, y - x⟫ := by
     rw [add_add_add_comm, ← Eq.symm (Convex.combo_self abeq (f x))]
     apply congrArg (HAdd.hAdd (f x))
     rw [inner_add_left, inner_smul_left, inner_smul_left]; rfl
@@ -224,8 +224,8 @@ theorem subgradientAt_mono {u v : E} {f : E → ℝ}{y : E}
     (hu : u ∈ SubderivAt f x) (hv : v ∈ SubderivAt f y) : ⟪u - v, x - y⟫ ≥ (0 : ℝ):= by
   specialize hu y; specialize hv x
   have ineq1 : ⟪u, x - y⟫ ≥ f x - f y := by
-    rw [congrArg (inner u) (Eq.symm (neg_sub y x)), inner_neg_right]; linarith
-  have _ : inner v (x - y) ≤ f x - f y := Iff.mpr le_sub_iff_add_le' hv
+    rw [congrArg (fun z => ⟪u, z⟫) (Eq.symm (neg_sub y x)), inner_neg_right]; linarith
+  have _ : ⟪v, x - y⟫ ≤ f x - f y := Iff.mpr le_sub_iff_add_le' hv
   rw [inner_sub_left]; linarith
 
 end congr
@@ -259,7 +259,7 @@ theorem SubderivAt.nonempty (hf : ConvexOn ℝ univ f) (hc : ContinuousOn f univ
   have : x ∈ interior univ := by simp
   rw [← interior_univ] at hc
   obtain h := SubderivWithinAt.Nonempty hf hc x this
-  simp [h]
+  simp
   rcases h with ⟨a, ha⟩
   exact ⟨a, ha⟩
 
@@ -286,14 +286,14 @@ theorem SubderivWithinAt_eq_gradient {f'x : E} (hx : x ∈ interior s)
   · use g; intro y ys
     apply Convex_first_order_condition' h hf (interior_subset hx) y ys
   intro g' hg'; by_contra neq
-  apply not_le_of_lt (norm_sub_pos_iff.mpr neq)
+  apply not_le_of_gt (norm_sub_pos_iff.mpr neq)
   let v := g' - g; obtain vneq := sub_ne_zero.mpr neq
   have : Tendsto (fun (t : ℝ) => (f (x + t • v) - f x - ⟪g, t • v⟫) * ‖t • v‖⁻¹)
       (𝓝[>] 0) (𝓝 0) := by
     rw [Metric.tendsto_nhdsWithin_nhds]; intro ε εpos
-    unfold HasFDerivAt at h'
-    rw [hasFDerivAtFilter_iff_tendsto, Metric.tendsto_nhds_nhds] at h'
-    obtain ⟨δ, δpos, hδ⟩ := h' ε εpos
+    have h'' := (hasGradientAt_iff_tendsto).mp h
+    rw [Metric.tendsto_nhds_nhds] at h''
+    obtain ⟨δ, δpos, hδ⟩ := h'' ε εpos
     use (δ * ‖v‖⁻¹)
     obtain pos := mul_pos δpos (inv_pos.mpr (norm_pos_iff.mpr vneq))
     constructor
@@ -301,8 +301,10 @@ theorem SubderivWithinAt_eq_gradient {f'x : E} (hx : x ∈ interior s)
     intro t _ ht; rw [dist_eq_norm] at ht; rw [dist_eq_norm]
     have : dist (x + t • v) x < δ := by
       rw [dist_eq_norm, add_sub_cancel_left, norm_smul, ← (sub_zero t)]
-      apply lt_of_lt_of_eq ((mul_lt_mul_right (norm_sub_pos_iff.mpr neq)).mpr ht)
-      rw [mul_assoc, inv_mul_cancel₀ (norm_ne_zero_iff.mpr vneq), mul_one]
+      have hmul : ‖t - 0‖ * ‖v‖ < (δ * ‖v‖⁻¹) * ‖v‖ :=
+        mul_lt_mul_of_pos_right ht (norm_pos_iff.mpr vneq)
+      have hnorm : ‖v‖⁻¹ * ‖v‖ = (1 : ℝ) := inv_mul_cancel₀ (norm_ne_zero_iff.mpr vneq)
+      simpa [mul_assoc, hnorm] using hmul
     specialize hδ this; rw [dist_eq_norm] at hδ
     have eq1 : ‖‖x + t • v - x‖⁻¹‖ = ‖t • v‖⁻¹ := by
       rw [add_sub_cancel_left, norm_inv, norm_norm]
@@ -329,7 +331,7 @@ theorem SubderivWithinAt_eq_gradient {f'x : E} (hx : x ∈ interior s)
     rw [mem_ball_iff_norm, sub_zero] at tball
     rw [mem_ball_iff_norm, add_sub_cancel_left, norm_smul]
     have : ‖t‖ * ‖v‖ < ε * ‖v‖⁻¹ * ‖v‖ := by
-      apply (mul_lt_mul_right (norm_sub_pos_iff.mpr neq)).mpr tball
+      exact mul_lt_mul_of_pos_right tball (norm_pos_iff.mpr vneq)
     rwa [mul_assoc, inv_mul_cancel₀ (norm_ne_zero_iff.mpr vneq), mul_one] at this
   obtain ineq1 := hg' (x + t • v); rw [add_sub_cancel_left] at ineq1
   have eq1 : ‖v‖ = (⟪g', t • v⟫ - ⟪g, t • v⟫) * ‖t • v‖⁻¹ := by
@@ -353,8 +355,9 @@ theorem SubderivWithinAt_eq_gradient {f'x : E} (hx : x ∈ interior s)
       rw [this]
     rw [eq2, eq3, mul_eq_mul_right_iff];
     left; rw [inner_sub_left]
-  rw [mem_setOf, eq1, mul_le_mul_right tvpos]
-  apply sub_le_sub_right (le_sub_iff_add_le'.mpr (ineq1 mems))
+  rw [mem_setOf, eq1]
+  exact mul_le_mul_of_nonneg_right
+    (sub_le_sub_right (le_sub_iff_add_le'.mpr (ineq1 mems)) _) (le_of_lt tvpos)
 
 /-- Alternarive version for FDeriv --/
 theorem SubderivWithinAt_eq_FDeriv {f' : E → (E →L[ℝ] ℝ)} (hx : x ∈ interior s)
@@ -380,7 +383,7 @@ end equivalence
 section optimality_theory
 
 theorem HasSubgradientAt_zero_of_isMinOn (h : IsMinOn f univ x) : HasSubgradientAt f 0 x :=
-  fun y => le_of_le_of_eq' (h trivial) (by rw [inner_zero_left, add_zero])
+  fun y => by simpa [inner_zero_left] using h trivial
 
 theorem isMinOn_of_HasSubgradentAt_zero (h : HasSubgradientAt f 0 x) : IsMinOn f univ x := by
   intro y _; specialize h y
@@ -393,7 +396,7 @@ theorem HasSubgradientAt_zero_iff_isMinOn :
 
 theorem HasSubgradientWithinAt_zero_of_isMinOn (h : IsMinOn f s x) :
     HasSubgradientWithinAt f 0 s x :=
-  fun y ys => le_of_le_of_eq' (h ys) (by rw [inner_zero_left, add_zero])
+  fun y ys => by simpa [inner_zero_left] using h ys
 
 theorem isMinOn_of_HasSubgradentWithinAt_zero (h : HasSubgradientWithinAt f 0 s x) :
     IsMinOn f s x := by
@@ -432,10 +435,8 @@ variable {f : E → ℝ} {g : E} {x : E} {s : Set E}
 theorem HasSubgradientAt.pos_smul {c : ℝ} (h : HasSubgradientAt f g x) (hc : 0 < c) :
     HasSubgradientAt (c • f) (c • g) x := by
   intro y; rw [inner_smul_left]
-  have ineq : c * f y ≥ c * (f x + inner g (y - x)) := (mul_le_mul_left hc).mpr (h y)
-  have eq : c * (f x + inner g (y - x)) = c * f x + c * inner g (y - x) :=
-    mul_add c (f x) (inner g (y - x))
-  exact Eq.trans_le (id eq.symm) ineq
+  have ineq := mul_le_mul_of_nonneg_left (h y) (le_of_lt hc)
+  simpa [Pi.smul_apply, mul_add, ge_iff_le] using ineq
 
 theorem SubderivAt.pos_smul {c : ℝ} (hc : 0 < c) :
     SubderivAt (c • f) x = c • (SubderivAt f x) := by
@@ -447,19 +448,20 @@ theorem SubderivAt.pos_smul {c : ℝ} (hc : 0 < c) :
       have neq : c ≠ 0 :=  ne_of_gt hc
       calc
         f y = c⁻¹ * (c * f y) := (eq_inv_mul_iff_mul_eq₀ neq).mpr rfl
-        _ ≥ c⁻¹ * (c * f x + inner g (y - x)) :=
+        _ ≥ c⁻¹ * (c * f x + ⟪g, y - x⟫) :=
           mul_le_mul_of_nonneg_left (hg y) (inv_nonneg.mpr (le_of_lt hc))
-        _ = f x + inner (c⁻¹ • g) (y - x) := by
+        _ = f x + ⟪c⁻¹ • g, y - x⟫ := by
           rw [mul_add, inner_smul_left, ← ((eq_inv_mul_iff_mul_eq₀ neq).mpr rfl)]
           rfl
     exact smul_inv_smul₀ (ne_of_gt hc) g
   rintro ⟨gg, hgg, eq⟩; intro y
   calc
-    c * f y ≥ c * (f x + inner gg (y - x)) := (mul_le_mul_left hc).mpr (hgg y)
-    _ = c * f x + c * inner gg (y - x) := mul_add c (f x) (inner gg (y - x))
-    _ = c * f x + inner (c • gg) (y - x) := by
+    c * f y ≥ c * (f x + ⟪gg, y - x⟫) := by
+      exact mul_le_mul_of_nonneg_left (hgg y) (le_of_lt hc)
+    _ = c * f x + c * ⟪gg, y - x⟫ := mul_add c (f x) ⟪gg, y - x⟫
+    _ = c * f x + ⟪c • gg, y - x⟫ := by
       rw [inner_smul_left]; exact rfl
-    _ = c * f x + inner g (y - x) := by rw [← eq]
+    _ = c * f x + ⟪g, y - x⟫ := by rw [← eq]
 
 /-- Subderivatives of the sum of two functions is a subset of the sum of the
 subderivatives of the two functions --/
@@ -493,7 +495,7 @@ theorem SubderivAt.add {f₁ f₂ : E → ℝ} (h₁ : ConvexOn ℝ univ f₁) (
   rw [SubderivAt, SubderivAt, SubderivAt, Set.subset_def]
   intro g hg
   rw [Set.mem_setOf] at hg; rw [Set.mem_add]
-  let S₁ := {(x, y) : E × ℝ | y > f₁ (x + x₀) - f₁ x₀ - inner g x}
+  let S₁ := {(x, y) : E × ℝ | y > f₁ (x + x₀) - f₁ x₀ - ⟪g, x⟫}
   let S₂ := {(x, y) : E × ℝ | y ≤ f₂ x₀ - f₂ (x + x₀)}
 
   have hs1 : Convex ℝ S₁ := by
@@ -537,8 +539,8 @@ theorem SubderivAt.add {f₁ f₂ : E → ℝ} (h₁ : ConvexOn ℝ univ f₁) (
     rw [← eq'] at hh; rw [eq]
     apply le_trans (add_le_add hi hj) hh
   have hint : Disjoint S₁ S₂ := by
-    rw [disjoint_iff]; by_contra joint
-    obtain ⟨⟨x, y⟩, ⟨hp1, hp2⟩⟩ := nmem_singleton_empty.mp joint
+    rw [Set.disjoint_iff_inter_eq_empty, Set.eq_empty_iff_forall_notMem]
+    rintro ⟨x, y⟩ ⟨hp1, hp2⟩
     rw [Set.mem_setOf] at hp1 hp2
     specialize hg (x + x₀); rw [← add_sub, sub_self, add_zero] at hg
     apply not_le_of_gt ?_ hg
@@ -550,7 +552,7 @@ theorem SubderivAt.add {f₁ f₂ : E → ℝ} (h₁ : ConvexOn ℝ univ f₁) (
       f₁ x₀ + f₂ x₀ + ⟪g, x⟫_ℝ - (f₁ (x + x₀) + f₂ (x + x₀)) := by ring
     rwa [hh x₀, hh (x + x₀), ← eq]
   have hso : IsOpen S₁ := by
-    apply Continuous_epi_open (f₁ := fun x ↦ f₁ (x + x₀) - f₁ x₀ - inner g x)
+    apply Continuous_epi_open (f₁ := fun x ↦ f₁ (x + x₀) - f₁ x₀ - ⟪g, x⟫)
     apply ContinuousOn.sub
     · apply ContinuousOn.sub
       · apply ContinuousOn.comp (g := f₁) (f := fun x ↦ x + x₀) (t := univ) hcon
@@ -560,21 +562,23 @@ theorem SubderivAt.add {f₁ f₂ : E → ℝ} (h₁ : ConvexOn ℝ univ f₁) (
     apply ContinuousOn.inner continuousOn_const continuousOn_id
 
   obtain ⟨f, c, ⟨hsl, hsr⟩⟩ := geometric_hahn_banach_open hs1 hso hs2 hint
-  have eq : ∃ a : E, ∃ b : ℝ, ∀ (p : E × ℝ), f p = inner a p.1 + b * p.2 := by
+  have eq : ∃ a : E, ∃ b : ℝ, ∀ (p : E × ℝ), f p = ⟪a, p.1⟫ + b * p.2 := by
     let f1 := ContinuousLinearMap.comp f (ContinuousLinearMap.inl ℝ E ℝ)
     let f2 := ContinuousLinearMap.comp f (ContinuousLinearMap.inr ℝ E ℝ)
     use (toDual ℝ E).symm f1
     use (toDual ℝ ℝ).symm f2
     intro p
-    have : ((toDual ℝ ℝ).symm f2) * p.2 = inner (((toDual ℝ ℝ).symm f2)) p.2 := by
-      simp [f2]
-    have : ((toDual ℝ ℝ).symm f2) * p.2 = f2 p.2 := by
-      rw [this]
-      simp only [toDual_symm_apply, ContinuousLinearMap.coe_comp', Function.comp_apply,
-        ContinuousLinearMap.inl_apply, ContinuousLinearMap.inr_apply]
-    rw [this]; simp [f1, f2]
+    have hf2 : ((toDual ℝ ℝ).symm f2) * p.2 = f2 p.2 := by
+      have hinner : ⟪(toDual ℝ ℝ).symm f2, p.2⟫ = f2 p.2 :=
+        InnerProductSpace.toDual_symm_apply (𝕜 := ℝ) (E := ℝ) (x := p.2) (y := f2)
+      calc
+        ((toDual ℝ ℝ).symm f2) * p.2 = p.2 * ((toDual ℝ ℝ).symm f2) := by ring
+        _ = ⟪(toDual ℝ ℝ).symm f2, p.2⟫ := by rfl
+        _ = f2 p.2 := hinner
+    rw [hf2]; simp [f2]
     have : (p.1, (0 : ℝ)) + ((0 : E), p.2) = p := by simp
     nth_rw 1 [← this]; rw [ContinuousLinearMap.map_add]
+    simp [f1]
   rcases eq with ⟨a, b, hab⟩
 
   have hin : (0, 0) ∈ S₂ := by rw [Set.mem_setOf]; simp
@@ -594,11 +598,12 @@ theorem SubderivAt.add {f₁ f₂ : E → ℝ} (h₁ : ConvexOn ℝ univ f₁) (
     push_neg at hb
     have pos : (c / (2 * b)) > 0 := by
       apply div_pos_of_neg_of_neg hc (by linarith)
-    specialize (htp (c / (2 * b)) pos); field_simp [hb] at htp
-    have eq : b * c / (2 * b) = c / 2 := by
-      ring_nf; simp; field_simp [hb]
-      rw [mul_div_right_comm, div_self (by linarith), one_mul]
-    rw [eq] at htp; linarith
+    specialize (htp (c / (2 * b)) pos)
+    have hb0 : b ≠ 0 := by linarith
+    have hhalf : b * (c / (2 * b)) = c / 2 := by
+      field_simp [hb0]
+    have : c / 2 < c := by linarith [htp, hhalf]
+    linarith
   have bleq0 : b < 0 := by
     rw [ceq0] at htp
     specialize htp 1 (by linarith); rw [mul_one] at htp; linarith
@@ -653,7 +658,7 @@ theorem SubderivAt_of_norm_at_zero : SubderivAt (fun (x : E) => ‖x‖) 0 = {g
     apply not_lt.mpr hg this
   intro hg y
   calc
-    ‖(0 : E)‖ + inner g (y - 0) = inner g y := by simp only [norm_zero, zero_add, sub_zero]
+    ‖(0 : E)‖ + ⟪g, y - 0⟫ = ⟪g, y⟫ := by simp only [norm_zero, zero_add, sub_zero]
     _ ≤ ‖g‖ * ‖y‖ := real_inner_le_norm g y
     _ ≤ 1 * ‖y‖ := mul_le_mul_of_nonneg_right hg (norm_nonneg y)
     _ = ‖y‖ := by simp only [one_mul]
@@ -684,9 +689,9 @@ theorem SubderivAt_abs (x : ℝ) :
         have ineq : (0 : ℝ) < 0 := by
           calc
             0 ≥ x + g * (-x):= by
-              simp only [abs_zero, zero_sub, abs_pos_of_pos, abs_of_pos hx] at hg
-              have : inner g (-x) = g * (-x)  := by rfl
-              rwa [this] at hg
+              simp only [abs_zero, zero_sub, abs_of_pos hx] at hg
+              have hinner : ⟪g, -x⟫ = (-x) * g := by rfl
+              rwa [hinner, mul_comm] at hg
             _ = x * (1 - g) := by ring
             _ > 0 := mul_pos hx (by linarith)
         exact LT.lt.false ineq
@@ -696,7 +701,9 @@ theorem SubderivAt_abs (x : ℝ) :
       apply glt
       have h1: g ≤ 1 := by
         calc
-          g = inner g 1 := by simp
+          g = ⟪g, 1⟫ := by
+            rw [show ⟪g, 1⟫ = (1 : ℝ) * g by rfl]
+            ring
           _ ≤ 1 := hg
       simp only [Real.sign_of_pos hx] at gne
       exact Ne.lt_of_le gne h1
@@ -706,14 +713,18 @@ theorem SubderivAt_abs (x : ℝ) :
     by_cases glt : g < -1
     · specialize hg (x - 1)
       have : x - 1 < 0 := by linarith
-      simp only [abs_of_neg this, abs_of_neg hx, abs_zero, zero_sub] at hg
+      simp only [abs_of_neg this, abs_of_neg hx] at hg
       have : -g ≤ 1 := by
         calc
-          -g = inner g (x - 1 - x) := by simp
+          -g = ⟪g, x - 1 - x⟫ := by
+            rw [show ⟪g, x - 1 - x⟫ = (x - 1 - x) * g by rfl]
+            ring
           _ ≤ 1 := by linarith [hg]
       linarith
     specialize hg 0
-    have eq1 : inner g (-x) = g * (-x) := rfl
+    have eq1 : ⟪g, -x⟫ = g * (-x) := by
+      rw [show ⟪g, -x⟫ = (-x) * g by rfl]
+      ring
     have eq2 : -x + g * -x = -x * (1 + g) := by ring
     simp only [abs_zero, zero_sub, abs_of_neg hx, eq1, eq2] at hg
     have : -x * (1 + g) > 0 := by
@@ -725,14 +736,18 @@ theorem SubderivAt_abs (x : ℝ) :
   by_cases hx : x > 0
   · simp only [Real.sign_of_pos hx] at hg
     calc
-      |x| + inner g (y - x) = x + inner 1 (y - x) := by rw [abs_of_pos hx, hg]
-      _ = y := by simp
+      |x| + ⟪g, y - x⟫ = x + ⟪1, y - x⟫ := by rw [abs_of_pos hx, hg]
+      _ = y := by
+        rw [show ⟪(1 : ℝ), y - x⟫ = (y - x) * 1 by rfl]
+        ring
       _ ≤ |y| := le_abs_self y
   have hx : x < 0 := Ne.lt_of_le h (not_lt.mp hx)
   simp only [Real.sign_of_neg hx] at hg
   calc
-    |x| + inner g (y - x) = -x + inner (-1) (y - x) := by rw [abs_of_neg hx, hg]
-    _ = -y := by simp; ring
+    |x| + ⟪g, y - x⟫ = -x + ⟪-1, y - x⟫ := by rw [abs_of_neg hx, hg]
+    _ = -y := by
+      rw [show ⟪(-1 : ℝ), y - x⟫ = (y - x) * (-1) by rfl]
+      ring
     _ ≤ |y| := neg_le_abs y
 
 end
diff --git a/Optlib/Differential/Calculation.lean b/Optlib/Differential/Calculation.lean
index 89e163e..0677330 100644
--- a/Optlib/Differential/Calculation.lean
+++ b/Optlib/Differential/Calculation.lean
@@ -50,7 +50,7 @@ theorem HasGradientAtFilter.comp
   have eq : (starRingEnd 𝕜) g' • (toDual 𝕜 F) f' =
     (toDual 𝕜 F) (g' • f') := by rw [map_smulₛₗ]
   rw [HasGradientAtFilter, ← eq]
-  exact hg.hasDerivAtFilter.comp_hasFDerivAtFilter x hf hL
+  exact hg.hasDerivAtFilter.comp_hasFDerivAtFilter hf <| hL.prodMap <| by simp
 
 theorem HasGradientWithinAt.comp
     (hg : HasGradientWithinAt g g' t (f x)) (hf : HasGradientWithinAt f f' s x)
@@ -184,27 +184,28 @@ section Sum
 /-! ### Derivative of a finite sum of functions -/
 
 
-open BigOperators Asymptotics
+open scoped BigOperators
+open Asymptotics
 
 variable {ι : Type*} {u : Finset ι} {A : ι → F → 𝕜} {A' : ι → F}
 
 theorem HasGradientAtFilter.sum (h : ∀ i ∈ u, HasGradientAtFilter (A i) (A' i) x L) :
-    HasGradientAtFilter (fun y => ∑ i in u, A i y) (∑ i in u, A' i) x L := by
-  have : ∑ i in u, (toDual 𝕜 F) (A' i) = (toDual 𝕜 F) (∑ i in u, A' i) := by
+    HasGradientAtFilter (fun y => Finset.sum u fun i => A i y) (Finset.sum u A') x L := by
+  have : Finset.sum u (fun i => (toDual 𝕜 F) (A' i)) = (toDual 𝕜 F) (Finset.sum u A') := by
     rw [map_sum]
   rw [HasGradientAtFilter, ← this]; unfold HasGradientAtFilter at h
-  exact HasFDerivAtFilter.sum h
+  exact HasFDerivAtFilter.fun_sum h
 
 theorem HasGradientWithinAt.sum (h : ∀ i ∈ u, HasGradientWithinAt (A i) (A' i) s x) :
-    HasGradientWithinAt (fun y => ∑ i in u, A i y) (∑ i in u, A' i) s x := by
+    HasGradientWithinAt (fun y => Finset.sum u fun i => A i y) (Finset.sum u A') s x := by
   exact HasGradientAtFilter.sum h
 
 theorem HasGradientAt.sum (h : ∀ i ∈ u, HasGradientAt (A i) (A' i) x) :
-    HasGradientAt (fun y => ∑ i in u, A i y) (∑ i in u, A' i) x := by
+    HasGradientAt (fun y => Finset.sum u fun i => A i y) (Finset.sum u A') x := by
   exact HasGradientAtFilter.sum h
 
 theorem gradient_sum (h : ∀ i ∈ u, DifferentiableAt 𝕜 (A i) x) :
-    ∇ (fun y => ∑ i in u, A i y) x = ∑ i in u, ∇ (A i) x :=
+    ∇ (fun y => Finset.sum u fun i => A i y) x = Finset.sum u fun i => ∇ (A i) x :=
   (HasGradientAt.sum fun i hi => (h i hi).hasGradientAt).gradient
 
 end Sum
@@ -228,8 +229,11 @@ theorem HasGradientAt.neg (h : HasGradientAt f f' x) :
   exact HasGradientAtFilter.neg h
 
 theorem gradient_neg : ∇ (fun y => - f y) x = - ∇ f x := by
-  unfold gradient
-  simp only [fderiv_neg, map_neg]
+  by_cases h : DifferentiableAt 𝕜 f x
+  · exact (h.hasGradientAt.neg).gradient
+  · rw [gradient_eq_zero_of_not_differentiableAt h,
+      gradient_eq_zero_of_not_differentiableAt (by simpa [differentiableAt_neg_iff] using h)]
+    simp
 
 end Neg
 
@@ -300,8 +304,7 @@ open ContinuousLinearMap
 
 lemma equiv_lemma_mul : c x • (toDual 𝕜 F) d' + d x • (toDual 𝕜 F) c'
     = (toDual 𝕜 F) ((starRingEnd 𝕜) (c x) • d' + (starRingEnd 𝕜) (d x) • c'):= by
-  simp
-  congr <;> exact SemilinearMapClass.map_smul_inv _ _ _
+  rw [map_add, map_smulₛₗ, map_smulₛₗ, starRingEnd_self_apply, starRingEnd_self_apply]
 
 theorem HasGradientAt.mul (hc : HasGradientAt c c' x) (hd : HasGradientAt d d' x) :
     HasGradientAt (fun y => c y * d y)
diff --git a/Optlib/Differential/GradientDiv.lean b/Optlib/Differential/GradientDiv.lean
index 57ad10f..9755968 100644
--- a/Optlib/Differential/GradientDiv.lean
+++ b/Optlib/Differential/GradientDiv.lean
@@ -48,14 +48,14 @@ lemma Simplifying₁ (h₁ : a ≠ 0) (h₂ : b ≠ 0) (h₃ : ‖b‖ / 2 ≤ 
   simp only [one_div, div_inv_eq_mul, one_mul]
   have l₃ : |b * b * b| / 2 = |b * b| * (|b| / 2) := by rw [mul_div, abs_mul]
   have l₄ : |b * b * a| = |b * b| * |a| := by rw [abs_mul]
-  rw [l₃, l₄, mul_le_mul_left]
-  apply h₃
-  rw [abs_pos]
-  simp only [ne_eq, mul_eq_zero, or_self]
-  apply h₂
-  have : 0 < |b * b * b| := by rw [abs_pos]; simp [h₂]
-  have : 0 < |b * b * b| / 2 := half_pos this
-  apply this
+  rw [l₃, l₄]
+  exact mul_le_mul_of_nonneg_left h₃ (abs_nonneg (b * b))
+  have : 0 < |b * b * b| / 2 := by
+    have hb : 0 < |b * b * b| := by
+      rw [abs_pos]
+      simp [h₂]
+    exact div_pos hb (by norm_num)
+  exact this
   have : 0 < |b * b * a| := by rw [abs_pos]; simp [h₁, h₂]
   exact Iff.mpr one_div_pos this
 
@@ -81,9 +81,7 @@ lemma div_div_mul (h₁ : a / b ≤ c) (h₂ : 0 < a) (h₃ : 0 < b) (h₄ : 0 <
     1 / c ≤ b / a := by
   have : a ≤ c * b := Iff.mp (div_le_iff₀ h₃) h₁
   have : a ≤ b * c := by linarith
-  apply Iff.mpr (div_le_div_iff h₄ h₂)
-  rw [one_mul]
-  apply this
+  exact (div_le_div_iff₀ h₄ h₂).2 <| by simpa [mul_comm, mul_left_comm, mul_assoc] using this
 
 theorem HasGradientAt.one_div (hf : HasGradientAt f grad x)(h₁: ¬ f x = (0 : ℝ)):
     HasGradientAt (fun y => (1 : ℝ) / (f y)) (- ((1 : ℝ) / (f x) ^ (2 : ℕ)) • grad) x := by
@@ -251,10 +249,10 @@ theorem HasGradientAt.one_div (hf : HasGradientAt f grad x)(h₁: ¬ f x = (0 :
       have h₂ : min δ₀ δ₂ ≤ δ₂ := by exact min_le_right δ₀ δ₂
       apply le_trans h' h₂
 
-    have zp1 :‖f x * (f x - f x' + inner grad (x' - x)) / (f x * f x * f x')‖ =
-        ‖(f x - f x' + inner grad (x' - x)) / (f x * f x')‖ := by
+    have zp1 :‖f x * (f x - f x' + ⟪grad, (x' - x)⟫) / (f x * f x * f x')‖ =
+        ‖(f x - f x' + ⟪grad, (x' - x)⟫) / (f x * f x')‖ := by
       rw [mul_comm, mul_assoc (f x) (f x) (f x'),
-      div_mul_eq_div_div ((f x - f x' + inner grad (x' - x)) * (f x)) (f x) (f x * f x'), mul_div_cancel_right₀]
+      div_mul_eq_div_div ((f x - f x' + ⟪grad, (x' - x)⟫) * (f x)) (f x) (f x * f x'), mul_div_cancel_right₀]
       apply h₁
 
     have zp2 : ‖f x‖ * ‖f x‖/2 ≤ ‖f x * f x'‖ := by
@@ -305,16 +303,16 @@ theorem HasGradientAt.one_div (hf : HasGradientAt f grad x)(h₁: ¬ f x = (0 :
         _ = (ε / 2) * ‖x' - x‖ := by
           rw [div_self (mul_ne_zero l l), mul_one, norm_sub_rev]
     calc
-      ‖f x * (f x - f x' + inner grad (x' - x)) / (f x * f x * f x')‖ =
-          ‖(f x - f x' + inner grad (x' - x)) / (f x * f x')‖ := by
+      ‖f x * (f x - f x' + ⟪grad, (x' - x)⟫) / (f x * f x * f x')‖ =
+          ‖(f x - f x' + ⟪grad, (x' - x)⟫) / (f x * f x')‖ := by
         apply zp1
-      _ = ‖(f x - f x' + inner grad (x' - x))‖ * ‖1/(f x * f x')‖ := by
+      _ = ‖(f x - f x' + ⟪grad, (x' - x)⟫)‖ * ‖1/(f x * f x')‖ := by
         apply Vert_div
-      _ ≤ ‖(f x - f x' + inner grad (x' - x))‖ * (2 / (‖f x‖ * ‖f x‖)) := by
+      _ ≤ ‖(f x - f x' + ⟪grad, (x' - x)⟫)‖ * (2 / (‖f x‖ * ‖f x‖)) := by
         apply mul_le_mul_of_nonneg_left zp3
         apply norm_nonneg
       _ ≤ ((ε * ‖f x‖ * ‖f x‖/4) * ‖x - x'‖) * (2 / (‖f x‖ * ‖f x‖)) := by
-        have : ‖(f x - f x' + inner grad (x' - x))‖ ≤ (ε * ‖f x‖ * ‖f x‖/4) * ‖x - x'‖ := by
+        have : ‖(f x - f x' + ⟪grad, (x' - x)⟫)‖ ≤ (ε * ‖f x‖ * ‖f x‖/4) * ‖x - x'‖ := by
           apply hδ₂
           apply hp₂
         apply mul_le_mul_of_nonneg_right this
@@ -385,22 +383,22 @@ theorem HasGradientAt.one_div (hf : HasGradientAt f grad x)(h₁: ¬ f x = (0 :
     apply Eq.symm (mul_div_mul_left (⟪grad, (x' - x)⟫) ((f x) * (f x)) l')
   have k₆ : (f x - f x') * f x /(f x' * f x * f x) + f x' * (⟪grad, (x' - x)⟫)/(f x' * f x * f x)
       = ((f x - f x') * f x  + f x' * (⟪grad, (x' - x)⟫))/(f x' * f x * f x) := by
-    apply div_add_div_same ((f x - f x') * f x) (f x' * (⟪grad, (x' - x)⟫)) (f x' * f x * f x)
-  have k₇ : ((f x - f x') * f x + f x' * inner grad (x' - x)) / (f x' * f x * f x) =
-      (f x * (f x - f x' + inner grad (x' - x)) +
-      (f x' * inner grad (x' - x) - f x * inner grad (x' - x))) / (f x' * f x * f x) := by
-    have h' : (f x - f x') * f x + f x' * inner grad (x' - x) =
-        f x * (f x - f x' + inner grad (x' - x)) +
-        (f x' * (inner grad (x' - x)) - f x * (inner grad (x' - x)))  := by
+    exact (add_div _ _ _).symm
+  have k₇ : ((f x - f x') * f x + f x' * ⟪grad, (x' - x)⟫) / (f x' * f x * f x) =
+      (f x * (f x - f x' + ⟪grad, (x' - x)⟫) +
+      (f x' * ⟪grad, (x' - x)⟫ - f x * ⟪grad, (x' - x)⟫)) / (f x' * f x * f x) := by
+    have h' : (f x - f x') * f x + f x' * ⟪grad, (x' - x)⟫ =
+        f x * (f x - f x' + ⟪grad, (x' - x)⟫) +
+        (f x' * ⟪grad, (x' - x)⟫ - f x * ⟪grad, (x' - x)⟫)  := by
       linarith
     rw [h']
-  have k₈ : (f x * (f x - f x' + inner grad (x' - x)) +
-      (f x' * inner grad (x' - x) - f x * inner grad (x' - x))) /
-      (f x * f x * f x') = f x * (f x - f x' + inner grad (x' - x))/
-      (f x * f x * f x') + (f x' * inner grad (x' - x) - f x * inner grad (x' - x))/
+  have k₈ : (f x * (f x - f x' + ⟪grad, (x' - x)⟫) +
+      (f x' * ⟪grad, (x' - x)⟫ - f x * ⟪grad, (x' - x)⟫)) /
+      (f x * f x * f x') = f x * (f x - f x' + ⟪grad, (x' - x)⟫)/
+      (f x * f x * f x') + (f x' * ⟪grad, (x' - x)⟫ - f x * ⟪grad, (x' - x)⟫)/
       (f x * f x * f x') := by
-        apply add_div ((f x) * (f x - f x' + inner grad (x' - x)))
-          (f x' * inner grad (x' - x) - f x * inner grad (x' - x)) (f x * f x * f x')
+        apply add_div ((f x) * (f x - f x' + ⟪grad, (x' - x)⟫))
+          (f x' * ⟪grad, (x' - x)⟫ - f x * ⟪grad, (x' - x)⟫) (f x * f x * f x')
   have k₉ : f x' * f x * f x =  f x * f x * f x' := by linarith
   have p₁ : ‖1 / f x' - 1 / f x - (- (⟪grad, (x' - x)⟫))/((f x) * (f x))‖ ≤ ε * ‖x' - x‖ := by
     rw [k₄, k₁, k₂, k₅]
@@ -409,22 +407,22 @@ theorem HasGradientAt.one_div (hf : HasGradientAt f grad x)(h₁: ¬ f x = (0 :
     rw [this]
     rw [k₆, k₇, k₉]
     calc
-      ‖(f x * (f x - f x' + inner grad (x' - x)) +
-      (f x' * inner grad (x' - x) - f x * inner grad (x' - x))) /
-      (f x * f x * f x')‖ = ‖f x * (f x - f x' + inner grad (x' - x))/
-      (f x * f x * f x') + (f x' * inner grad (x' - x) - f x * inner grad (x' - x))/
+      ‖(f x * (f x - f x' + ⟪grad, (x' - x)⟫) +
+      (f x' * ⟪grad, (x' - x)⟫ - f x * ⟪grad, (x' - x)⟫)) /
+      (f x * f x * f x')‖ = ‖f x * (f x - f x' + ⟪grad, (x' - x)⟫)/
+      (f x * f x * f x') + (f x' * ⟪grad, (x' - x)⟫ - f x * ⟪grad, (x' - x)⟫)/
       (f x * f x * f x')‖ := by
         rw [k₈]
-      _ ≤ ‖f x * (f x - f x' + inner grad (x' - x))/
-      (f x * f x * f x')‖ + ‖(f x' * inner grad (x' - x) - f x * inner grad (x' - x))/
+      _ ≤ ‖f x * (f x - f x' + ⟪grad, (x' - x)⟫)/
+      (f x * f x * f x')‖ + ‖(f x' * ⟪grad, (x' - x)⟫ - f x * ⟪grad, (x' - x)⟫)/
       (f x * f x * f x')‖ := by
-        apply norm_add_le ((f x) * (f x - f x' + inner grad (x' - x))/
-      (f x * f x * f x')) ((f x' * inner grad (x' - x) - f x * inner grad (x' - x))/
+        apply norm_add_le ((f x) * (f x - f x' + ⟪grad, (x' - x)⟫)/
+      (f x * f x * f x')) ((f x' * ⟪grad, (x' - x)⟫ - f x * ⟪grad, (x' - x)⟫)/
       (f x * f x * f x'))
       _ ≤ (ε/2) * ‖x' - x‖ + (ε/2) * ‖x' - x‖ := by exact add_le_add (hδ₅ x' hp₂) (hδ₄ x' hp₁)
       _ = ε * ‖x' - x‖ := by linarith
   have j₁ : ‖1 / f x' - 1 / f x - (- (⟪grad, (x' - x)⟫))/((f x) * (f x))‖ = ‖1 / f x' - 1 / f x -
-      inner ((-(1 / f x ^ ↑2) • grad)) (x' - x)‖ := by
+      ⟪(-(1 / f x ^ ↑2) • grad), (x' - x)⟫‖ := by
     congr; rw [k₃]
   rw [j₁] at p₁
   have l1 : ‖x - x'‖ = ‖x' - x‖ := by
diff --git a/Optlib/Differential/Lemmas.lean b/Optlib/Differential/Lemmas.lean
index a8f3203..93887fa 100644
--- a/Optlib/Differential/Lemmas.lean
+++ b/Optlib/Differential/Lemmas.lean
@@ -4,9 +4,12 @@ Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Chenyi Li
 -/
 import Mathlib.Analysis.Calculus.MeanValue
+import Mathlib.Analysis.Calculus.Deriv.MeanValue
 import Mathlib.Analysis.Calculus.ContDiff.Defs
-import Mathlib.Topology.Semicontinuous
+import Mathlib.Analysis.Calculus.Gradient.Basic
 import Mathlib.Analysis.Normed.Lp.ProdLp
+import Mathlib.Analysis.Normed.Operator.BoundedLinearMaps
+import Mathlib.Topology.Semicontinuity.Basic
 import Optlib.Differential.Calculation
 
 /-!
@@ -44,7 +47,9 @@ lemma bounded_lowersemicontinuous_to_epi_closed (f : E → ℝ) (hc : LowerSemic
   rcases xntend with ⟨xtend, ytend⟩
   rw [LowerSemicontinuousOn] at hc
   specialize hc p.1
-  simp at hc; rw [LowerSemicontinuousWithinAt, nhdsWithin_univ] at hc
+  simp at hc
+  have hc' : ∀ y < f p.1, ∀ᶠ x' in 𝓝 p.1, y < f x' := by
+    simpa [SemicontinuousWithinAt, nhdsWithin_univ] using hc
   let linf := liminf (fun n ↦ f (xn n).1) atTop
   have aux : Tendsto (fun n ↦ (xn n).2) atTop (nhds p.2) ↔
         ∀ ε > 0, ∃ N, ∀ n ≥ N, (fun n ↦ (xn n).2) n ∈ Ioo (p.2 - ε) (p.2 + ε) := by
@@ -55,7 +60,7 @@ lemma bounded_lowersemicontinuous_to_epi_closed (f : E → ℝ) (hc : LowerSemic
     by_contra h; push_neg at h
     let t := (linf + f p.1) / 2
     have tin : t < f p.1 := add_div_two_lt_right.2 h
-    specialize hc t tin
+    specialize hc' t tin
     have ieq2 : t ≤ linf := by
       apply le_liminf_of_le
       · rw [Filter.IsCoboundedUnder, Filter.IsCobounded]
@@ -74,7 +79,7 @@ lemma bounded_lowersemicontinuous_to_epi_closed (f : E → ℝ) (hc : LowerSemic
         let auxlt := fun x : E ↦ (t < f x)
         have le_of_lt : ∀ x : E, auxlt x → auxle x := by
           simp [auxlt]; intro x cd; exact le_of_lt cd
-        apply Eventually.mono hc le_of_lt
+        apply Eventually.mono hc' le_of_lt
     contrapose! ieq2
     apply left_lt_add_div_two.2 h
   have ieq3 : linf ≤ p.2 := by
@@ -164,7 +169,7 @@ lemma continuous_positive_direction [NormedSpace ℝ E] (h : ContinuousAt f x) (
   obtain ⟨δ, hδ1, hδ2⟩ := continuous_positive_neighborhood h hx
   by_cases hv : v = 0
   · rw [hv]; simp; use 1; constructor; linarith; intro t _ _; exact hx
-  have : ‖v‖ > 0 := norm_pos_iff'.mpr hv
+  have : ‖v‖ > 0 := by simpa [gt_iff_lt] using (norm_pos_iff.2 hv)
   use δ / (2 * ‖v‖); constructor; positivity
   intro y hy
   obtain hδ2 := hδ2 (x + y • v)
@@ -173,7 +178,7 @@ lemma continuous_positive_direction [NormedSpace ℝ E] (h : ContinuousAt f x) (
     simp at hy; rw [norm_smul]; simp; rw [abs_of_nonneg hy.1]
     calc
       _ ≤ δ / (2 * ‖v‖) * ‖v‖ := (mul_le_mul_iff_of_pos_right this).mpr hy.2
-      _ = δ / 2 := by field_simp; ring
+      _ = δ / 2 := by field_simp
       _ < δ := by linarith
   exact hδ2 this
 
@@ -207,43 +212,41 @@ theorem deriv_function_comp_segment (x y : E) (h₁ : ∀ x₁ : E, HasFDerivAt
 theorem HasFDeriv_Convergence (h: HasFDerivAt f (f' x) x) :
   ∀ ε > (0 : ℝ), ∃ δ > (0 : ℝ), ∀ (x' : E), ‖x - x'‖ ≤ δ
     → ‖f x' - f x - (f' x) (x' - x)‖ ≤ ε * ‖x - x'‖ := by
-  rw [HasFDerivAt, hasFDerivAtFilter_iff_isLittleO, Asymptotics.isLittleO_iff] at h
+  rw [hasFDerivAt_iff_isLittleO_nhds_zero, Asymptotics.isLittleO_iff] at h
   intro ε epos
   specialize h epos
   rw [Filter.Eventually] at h
-  let t := {x_1 | ‖f x_1 - f x - (f' x) (x_1 - x)‖ ≤ ε * ‖x_1 - x‖}
-  have h₁: ∃ ε1 > (0 : ℝ), Metric.ball x ε1 ⊆ t := Iff.mp Metric.mem_nhds_iff h
+  let t := {h : E | ‖f (x + h) - f x - (f' x) h‖ ≤ ε * ‖h‖}
+  have h₁ : ∃ ε1 > (0 : ℝ), Metric.ball (0 : E) ε1 ⊆ t := Iff.mp Metric.mem_nhds_iff h
   rcases h₁ with ⟨e1, e1pos, h₁⟩
   use (e1 / 2); constructor
   exact (half_pos e1pos)
   intro x' xnhds
-  have h₂: x' ∈ Metric.ball x e1:= by
-    rw [Metric.mem_ball, dist_comm]
-    rw [← dist_eq_norm] at xnhds
+  have h₂ : x' - x ∈ Metric.ball (0 : E) e1 := by
+    rw [Metric.mem_ball, dist_zero_right]
+    rw [← norm_neg (x - x'), neg_sub] at xnhds
     apply lt_of_le_of_lt xnhds (half_lt_self e1pos)
-  have h₃: x' ∈ t := h₁ h₂
+  have h₃ : x' - x ∈ t := h₁ h₂
   rw [Set.mem_setOf] at h₃
-  rw [norm_sub_rev x]
-  exact h₃
+  simpa [add_sub_cancel, ContinuousLinearMap.map_sub, norm_sub_rev] using h₃
 
 theorem Convergence_HasFDeriv (h : ∀ ε > (0 : ℝ), ∃ δ > (0 : ℝ), ∀ (x' : E),
     ‖x - x'‖ ≤ δ → ‖f x' - f x - (f' x) (x' - x)‖ ≤ ε * ‖x - x'‖) :
       HasFDerivAt f (f' x) x := by
-  rw [HasFDerivAt, hasFDerivAtFilter_iff_isLittleO, Asymptotics.isLittleO_iff]
+  rw [hasFDerivAt_iff_isLittleO_nhds_zero, Asymptotics.isLittleO_iff]
   intro ε epos
-  rw [Filter.Eventually]
   specialize h ε epos
   rcases h with ⟨δ, dpos, h⟩
+  rw [Filter.Eventually]
   rw [Metric.mem_nhds_iff]
-  use δ ; constructor
+  use δ; constructor
   apply dpos
-  intro x' x1mem
-  have h1: ‖x - x'‖ ≤ δ:= by
-    rw [Metric.ball, Set.mem_setOf, dist_comm, dist_eq_norm] at x1mem
-    exact LT.lt.le x1mem
-  specialize h x' h1
-  rw[Set.mem_setOf, norm_sub_rev x']
-  apply h
+  intro h' x1mem
+  have h1 : ‖x - (x + h')‖ ≤ δ := by
+    rw [Metric.mem_ball, dist_comm, dist_eq_norm] at x1mem
+    simpa [sub_eq_add_neg, add_assoc] using LT.lt.le x1mem
+  specialize h (x + h') h1
+  simpa [sub_eq_add_neg, add_comm, add_left_comm, add_assoc] using h
 
 theorem HasFDeriv_iff_Convergence_Point {f'x : (E →L[ℝ] ℝ)}:
   HasFDerivAt f (f'x) x ↔ ∀ ε > (0 : ℝ), ∃ δ > (0 : ℝ), ∀ (x' : E),
@@ -273,7 +276,7 @@ open Topology InnerProductSpace Set Filter Tendsto
 
 theorem HasGradient_Convergence (h : HasGradientAt f (f' x) x) :
     ∀ ε > (0 : ℝ), ∃ δ > (0 : ℝ), ∀ x' : E, ‖x - x'‖ ≤ δ
-    → ‖f x' - f x - inner (f' x) (x' - x)‖ ≤ ε * ‖x - x'‖ := by
+    → ‖f x' - f x - inner ℝ (f' x) (x' - x)‖ ≤ ε * ‖x - x'‖ := by
   rw [hasGradientAt_iff_hasFDerivAt] at h
   show ∀ ε > (0 : ℝ), ∃ δ > (0 : ℝ), ∀ (x' : E), ‖x - x'‖ ≤ δ
     → ‖f x' - f x - ((toDual ℝ E) (f' x)) (x' - x)‖ ≤ ε * ‖x - x'‖
@@ -281,14 +284,15 @@ theorem HasGradient_Convergence (h : HasGradientAt f (f' x) x) :
   exact h
 
 theorem Convergence_HasGradient (h : ∀ ε > (0 : ℝ), ∃ δ > (0 : ℝ), ∀ x' : E,
-    ‖x - x'‖ ≤ δ → ‖f x' - f x - inner (f' x) (x' - x)‖ ≤ ε * ‖x - x'‖) :
+    ‖x - x'‖ ≤ δ → ‖f x' - f x - inner ℝ (f' x) (x' - x)‖ ≤ ε * ‖x - x'‖) :
     HasGradientAt f (f' x) x := by
   rw [hasGradientAt_iff_hasFDerivAt]
-  exact HasFDeriv_iff_Convergence_Point.mpr h
+  simpa using (HasFDeriv_iff_Convergence_Point (f := f) (x := x)
+    (f'x := (toDual ℝ E) (f' x))).2 h
 
 theorem HasGradient_iff_Convergence_Point {f'x : E}:
       HasGradientAt f f'x x ↔ ∀ ε > (0 : ℝ), ∃ δ > (0 : ℝ), ∀ x' : E,
-     ‖x - x'‖ ≤ δ → ‖f x' - f x - inner (f'x) (x' - x)‖ ≤ ε * ‖x - x'‖ := by
+     ‖x - x'‖ ≤ δ → ‖f x' - f x - inner ℝ f'x (x' - x)‖ ≤ ε * ‖x - x'‖ := by
   constructor
   · intro h; apply HasGradient_Convergence
     exact h
@@ -296,7 +300,7 @@ theorem HasGradient_iff_Convergence_Point {f'x : E}:
 
 theorem HasGradient_iff_Convergence :
       HasGradientAt f (f' x) x ↔ ∀ ε > (0 : ℝ), ∃ δ > (0 : ℝ), ∀ x' : E,
-      ‖x - x'‖ ≤ δ → ‖f x' - f x - inner (f' x) (x' - x)‖ ≤ ε * ‖x - x'‖ := by
+      ‖x - x'‖ ≤ δ → ‖f x' - f x - inner ℝ (f' x) (x' - x)‖ ≤ ε * ‖x - x'‖ := by
   constructor
   apply HasGradient_Convergence
   apply Convergence_HasGradient
@@ -321,7 +325,7 @@ lemma gradient_norm_sq_eq_two_self (x : E) :
     apply pow_two_nonneg
 
 lemma gradient_of_inner_const (x : E) (a : E):
-    HasGradientAt (fun x ↦ (inner a x : ℝ)) a x := by
+    HasGradientAt (fun x ↦ (inner ℝ a x : ℝ)) a x := by
   apply HasGradient_iff_Convergence_Point.mpr
   simp only [gt_iff_lt, Real.norm_eq_abs]
   intros ε εpos
@@ -335,10 +339,9 @@ lemma gradient_of_const_mul_norm (l : ℝ) (z : E) :
     HasGradientAt (fun (x : E) => l / 2 * ‖x‖ ^ 2) (l • z) z := by
   let h := fun x : E => ‖x‖ ^ 2
   have e1 : (l • z) = (l / 2) • (2 : ℝ) • z := by rw [smul_smul]; simp
-  have : (fun (x : E) => l / 2 * ‖x‖ ^ 2) = (fun (x : E) => (l / 2) • h x) := by
-    ext; simp
   have h1 : HasGradientAt h ((2 : ℝ) • z) z := gradient_norm_sq_eq_two_self z
-  rw [this, e1]; refine HasGradientAt.const_smul' (l / 2) h1
+  rw [show (fun x : E => l / 2 * ‖x‖ ^ 2) = fun x ↦ (l / 2 : ℝ) • h x by ext x; simp [h]]
+  simpa [e1] using HasGradientAt.const_smul h1 (l / 2)
 
 lemma gradient_of_sq : ∀ u : E, HasGradientAt (fun u ↦ ‖u - x‖ ^ 2 / 2) (u - x) u := by
   intro s
@@ -348,7 +351,7 @@ lemma gradient_of_sq : ∀ u : E, HasGradientAt (fun u ↦ ‖u - x‖ ^ 2 / 2)
   · linarith
   · intro x' dles; field_simp; rw [abs_div]; simp
     have eq1 (u v : E) (e : ℝ) (dle : ‖u - v‖ ≤ e) :
-      |‖v‖ ^ 2 - ‖u‖ ^ 2 - inner ((2 : ℝ) • u) (v - u)| ≤ e * ‖u - v‖ := by
+      |‖v‖ ^ 2 - ‖u‖ ^ 2 - inner ℝ ((2 : ℝ) • u) (v - u)| ≤ e * ‖u - v‖ := by
       rw [← norm_neg (u - v), neg_sub] at dle;
       rw [← real_inner_self_eq_norm_sq, ← real_inner_self_eq_norm_sq, inner_sub_right]
       rw [real_inner_smul_left, real_inner_smul_left]; ring_nf
@@ -366,13 +369,13 @@ lemma gradient_of_sq : ∀ u : E, HasGradientAt (fun u ↦ ‖u - x‖ ^ 2 / 2)
     have eq2 : s - x' = u - v := by rw [hu, hv]; simp
     have eq3 : x' - s = v - u := by rw [hu, hv]; simp
     rw [eq2, eq3]
-    show |‖v‖ ^ 2 - ‖u‖ ^ 2 - inner ((2 : ℝ) • u) (v - u)| / 2 ≤ e * ‖u - v‖
+    show |‖v‖ ^ 2 - ‖u‖ ^ 2 - inner ℝ ((2 : ℝ) • u) (v - u)| / 2 ≤ e * ‖u - v‖
     calc
-      |‖v‖ ^ 2 - ‖u‖ ^ 2 - inner ((2 : ℝ) • u) (v - u)| / 2 ≤ (e * ‖u - v‖) / 2 := by
-        rw [div_le_div_right]
-        apply eq1; rw [hu, hv]; simp; apply dles; simp
+      |‖v‖ ^ 2 - ‖u‖ ^ 2 - inner ℝ ((2 : ℝ) • u) (v - u)| / 2 ≤ (e * ‖u - v‖) / 2 := by
+        have hle := eq1 u v e (by rw [hu, hv]; simpa using dles)
+        nlinarith
       _ ≤ e * ‖u - v‖ := by
-        field_simp
+        nlinarith [norm_nonneg (u - v), le_of_lt ep]
 
 lemma sub_normsquare_gradient (hf : ∀ x ∈ s, HasGradientAt f (f' x) x) (m : ℝ):
     ∀ x ∈ s, HasGradientAt (fun x ↦ f x - m / 2 * ‖x‖ ^ 2) (f' x - m • x) x := by
@@ -431,14 +434,14 @@ open InnerProductSpace Set
 -/
 
 lemma expansion (hf : ∀ x : E, HasGradientAt f (f' x) x) (x p : E) :
-    ∃ t : ℝ, t > 0 ∧ t < 1 ∧ f (x + p) = f x + inner (f' (x + t • p)) p := by
+    ∃ t : ℝ, t > 0 ∧ t < 1 ∧ f (x + p) = f x + inner ℝ (f' (x + t • p)) p := by
   let g := fun r : ℝ ↦ f (x + r • p)
-  let g' := fun r : ℝ ↦ (inner (f' (x + r • p)) p : ℝ)
+  let g' := fun r : ℝ ↦ (inner ℝ (f' (x + r • p)) p : ℝ)
   have h1 : ∀ r , HasDerivAt g (g' r) r := by
     let h := fun r : ℝ ↦ x + r • p
     have : g = f ∘ h := by rfl
     rw [this]; intro r
-    have : inner (f' (x + r • p)) p = toDual ℝ E (f' (x + r • p)) p := rfl
+    have : inner ℝ (f' (x + r • p)) p = toDual ℝ E (f' (x + r • p)) p := rfl
     simp [g']; rw [this]; apply HasFDerivAt.comp_hasDerivAt
     · apply hasGradientAt_iff_hasFDerivAt.mp
       exact hf (x + r • p)
@@ -449,7 +452,7 @@ lemma expansion (hf : ∀ x : E, HasGradientAt f (f' x) x) (x p : E) :
       rw [one_smul] at this; exact this
   have e1 : f (x + p) = g 1 := by simp [g]
   have e2 : f x = g 0 := by simp [g]
-  have e3 : ∀ t, inner (f' (x + t • p)) p = g' t := by simp []
+  have e3 : ∀ t, inner ℝ (f' (x + t • p)) p = g' t := by intro t; rfl
   rw [e1, e2]
   have : ∃ c ∈ Set.Ioo 0 1, g' c = (g 1 - g 0) / (1 - 0) := by
     apply exists_hasDerivAt_eq_slope g g' (by norm_num)
@@ -465,14 +468,14 @@ lemma expansion (hf : ∀ x : E, HasGradientAt f (f' x) x) (x p : E) :
   rw [e3 c]; simp [h2]
 
 lemma general_expansion (x p : E) (hf : ∀ y ∈ Metric.closedBall x ‖p‖, HasGradientAt f (f' y) y) :
-    ∃ t : ℝ, t > 0 ∧ t < 1 ∧ f (x + p) = f x + inner (f' (x + t • p)) p := by
+    ∃ t : ℝ, t > 0 ∧ t < 1 ∧ f (x + p) = f x + inner ℝ (f' (x + t • p)) p := by
   let g := fun r : ℝ ↦ f (x + r • p)
-  let g' := fun r : ℝ ↦ (inner (f' (x + r • p)) p : ℝ)
+  let g' := fun r : ℝ ↦ (inner ℝ (f' (x + r • p)) p : ℝ)
   have h1 : ∀ r ∈ Icc 0 1, HasDerivAt g (g' r) r := by
     let h := fun r : ℝ ↦ x + r • p
     have : g = f ∘ h := by rfl
     rw [this]; intro r hr
-    have : inner (f' (x + r • p)) p = toDual ℝ E (f' (x + r • p)) p := rfl
+    have : inner ℝ (f' (x + r • p)) p = toDual ℝ E (f' (x + r • p)) p := rfl
     simp [g']; rw [this]; apply HasFDerivAt.comp_hasDerivAt
     · apply hasGradientAt_iff_hasFDerivAt.mp
       have : x + r • p ∈ Metric.closedBall x ‖p‖ := by
@@ -486,7 +489,7 @@ lemma general_expansion (x p : E) (hf : ∀ y ∈ Metric.closedBall x ‖p‖, H
       rw [one_smul] at this; exact this
   have e1 : f (x + p) = g 1 := by simp [g]
   have e2 : f x = g 0 := by simp [g]
-  have e3 : ∀ t, inner (f' (x + t • p)) p = g' t := by simp []
+  have e3 : ∀ t, inner ℝ (f' (x + t • p)) p = g' t := by intro t; rfl
   rw [e1, e2]
   have : ∃ c ∈ Set.Ioo 0 1, g' c = (g 1 - g 0) / (1 - 0) := by
     apply exists_hasDerivAt_eq_slope g g' (by norm_num)
@@ -501,15 +504,15 @@ lemma general_expansion (x p : E) (hf : ∀ y ∈ Metric.closedBall x ‖p‖, H
 
 theorem lagrange (hs : Convex ℝ s) (hf : ∀ x ∈ s, HasGradientAt f (f' x) x) :
     ∀ x ∈ s, ∀ y ∈ s, ∃ c : ℝ, c ∈ Set.Ioo 0 1 ∧
-    inner (f' (x + c • (y - x))) (y - x) = f y - f x := by
+    inner ℝ (f' (x + c • (y - x))) (y - x) = f y - f x := by
   intro x xs y ys
   let g := fun t : ℝ ↦ f (x + t • (y - x))
-  let g' := fun t : ℝ ↦ (inner (f' (x + t • (y - x))) (y - x) : ℝ)
+  let g' := fun t : ℝ ↦ (inner ℝ (f' (x + t • (y - x))) (y - x) : ℝ)
   have h1 : ∀ r ∈ Icc 0 1 , HasDerivAt g (g' r) r := by
     let h := fun r : ℝ ↦ (x + r • (y - x))
     have : g = f ∘ h := rfl
     rw [this]; intro t ht
-    have : inner (f' (x + t • (y - x))) (y - x) = toDual ℝ E (f' (x + t • (y - x))) (y - x) := rfl
+    have : inner ℝ (f' (x + t • (y - x))) (y - x) = toDual ℝ E (f' (x + t • (y - x))) (y - x) := rfl
     simp [g']; rw [this]; apply HasFDerivAt.comp_hasDerivAt
     · apply hasGradientAt_iff_hasFDerivAt.mp
       have : x + t • (y - x) ∈ s := by
@@ -546,21 +549,26 @@ variable {x : E} {y : F} {z : WithLp 2 (E × F)}
 
 open Set Bornology Filter BigOperators Topology
 
-lemma fst_norm_le_prod_L2 (z : WithLp 2 (E × F)) : ‖z.1‖ ≤ ‖z‖ := by
-  have h : ‖z.1‖ ^ 2 ≤ ‖z‖ ^ 2 := by linarith [WithLp.prod_norm_sq_eq_of_L2 z, sq_nonneg ‖z.2‖]
+instance instCoeProdWithLpL2 : CoeTC (E × F) (WithLp 2 (E × F)) where
+  coe := WithLp.toLp 2
+
+lemma fst_norm_le_prod_L2 (z : WithLp 2 (E × F)) : ‖z.fst‖ ≤ ‖z‖ := by
+  have h : ‖z.fst‖ ^ 2 ≤ ‖z‖ ^ 2 := by
+    linarith [WithLp.prod_norm_sq_eq_of_L2 z, sq_nonneg ‖z.snd‖]
   apply nonneg_le_nonneg_of_sq_le_sq (norm_nonneg _)
   rwa [← pow_two, ← pow_two]
 
-lemma snd_norm_le_prod_L2 (z : WithLp 2 (E × F)) : ‖z.2‖ ≤ ‖z‖ := by
-  have h : ‖z.2‖ ^ 2 ≤ ‖z‖ ^ 2 := by linarith [WithLp.prod_norm_sq_eq_of_L2 z, sq_nonneg ‖z.1‖]
+lemma snd_norm_le_prod_L2 (z : WithLp 2 (E × F)) : ‖z.snd‖ ≤ ‖z‖ := by
+  have h : ‖z.snd‖ ^ 2 ≤ ‖z‖ ^ 2 := by
+    linarith [WithLp.prod_norm_sq_eq_of_L2 z, sq_nonneg ‖z.fst‖]
   apply nonneg_le_nonneg_of_sq_le_sq (norm_nonneg _)
   rwa [← pow_two, ← pow_two]
 
-lemma prod_norm_le_block_sum_L2 (z : WithLp 2 (E × F)) : ‖z‖ ≤ ‖z.1‖ + ‖z.2‖ := by
-  have : ‖z‖ ^ 2 ≤ (‖z.1‖ + ‖z.2‖) ^ 2:= by
+lemma prod_norm_le_block_sum_L2 (z : WithLp 2 (E × F)) : ‖z‖ ≤ ‖z.fst‖ + ‖z.snd‖ := by
+  have : ‖z‖ ^ 2 ≤ (‖z.fst‖ + ‖z.snd‖) ^ 2 := by
     simp [WithLp.prod_norm_sq_eq_of_L2, add_sq]
     positivity
-  apply nonneg_le_nonneg_of_sq_le_sq (Left.add_nonneg (norm_nonneg z.1) (norm_nonneg z.2))
+  apply nonneg_le_nonneg_of_sq_le_sq (Left.add_nonneg (norm_nonneg z.fst) (norm_nonneg z.snd))
   rwa [← pow_two, ← pow_two]
 
 lemma norm_prod_right_zero (x : E) :
@@ -590,17 +598,21 @@ instance instNormedSpaceProdL2 : NormedSpace ℝ (WithLp 2 (E × F)) where
     exact norm_smul_le a b
 
 instance instIsBoundedLinearMapL2equiv :
-    @IsBoundedLinearMap ℝ _ (E × F) _ _ (WithLp 2 (E × F)) _ _ id where
-  map_add := fun x ↦ congrFun rfl
-  map_smul := fun c ↦ congrFun rfl
+    IsBoundedLinearMap ℝ (WithLp.toLp 2 : E × F → WithLp 2 (E × F)) where
+  map_add := by intro x y; simp
+  map_smul := by intro c z; simp
   bound := by
     use 2
     constructor
     · norm_num
     · intro z
+      have h := prod_norm_le_block_sum_L2 (WithLp.toLp 2 z)
       rw [Prod.norm_def]
-      have h := prod_norm_le_block_sum_L2 z
-      simp only [id_eq]
-      linarith [h, le_max_left ‖z.1‖ ‖z.2‖, le_max_right ‖z.1‖ ‖z.2‖]
+      calc
+        ‖WithLp.toLp 2 z‖ ≤ ‖(WithLp.toLp 2 z).fst‖ + ‖(WithLp.toLp 2 z).snd‖ := h
+        _ = ‖z.fst‖ + ‖z.snd‖ := by simp
+        _ ≤ max ‖z.fst‖ ‖z.snd‖ + max ‖z.fst‖ ‖z.snd‖ :=
+          add_le_add (le_max_left _ _) (le_max_right _ _)
+        _ = 2 * max ‖z.fst‖ ‖z.snd‖ := by ring
 
 end ProdLp_diff
diff --git a/Optlib/Differential/Subdifferential.lean b/Optlib/Differential/Subdifferential.lean
index 4a67cea..06376d9 100644
--- a/Optlib/Differential/Subdifferential.lean
+++ b/Optlib/Differential/Subdifferential.lean
@@ -5,7 +5,7 @@ Author: Anqing Shen, Yifan Bai, Chenyi Li, Zaiwen Wen
 -/
 import Mathlib.Order.LiminfLimsup
 import Mathlib.Topology.Defs.Filter
-import Mathlib.Data.Real.EReal
+import Mathlib.Topology.Instances.EReal.Lemmas
 import Optlib.Differential.Calculation
 import Optlib.Function.Proximal
 
@@ -19,7 +19,7 @@ variable {f g : E → ℝ} {x y u v : E} {c : ℝ}
 
 /- the general differential function used in the definition -/
 def differential_fun (x : E) (f : E → ℝ) (u : E) :=
-  fun y ↦ Real.toEReal ((f y - f x - inner u (y - x)) / ‖y - x‖)
+  fun y ↦ Real.toEReal ((f y - f x - inner ℝ u (y - x)) / ‖y - x‖)
 
 /- the definition of the Frechet subdifferential-/
 def f_subdifferential (f : E → ℝ) (x : E) : Set E :=
@@ -40,9 +40,9 @@ def critial_point (f : E → ℝ) : Set E :=
 
 /-- equivalence of Frechet subdifferential -/
 theorem has_f_subdiff_iff : u ∈ f_subdifferential f x ↔
-    ∀ ε > 0, ∀ᶠ y in 𝓝 x, f y - f x - inner u (y - x) ≥ -ε * ‖y - x‖ := by
-  have h0 : (∀ ε > 0, ∀ᶠ y in 𝓝[≠] x, f y - f x - inner u (y - x) > -ε * ‖y - x‖)
-      ↔ ∀ ε > 0, ∀ᶠ y in 𝓝 x, f y - f x - inner u (y - x) ≥ -ε * ‖y - x‖ := by
+    ∀ ε > 0, ∀ᶠ y in 𝓝 x, f y - f x - inner ℝ u (y - x) ≥ -ε * ‖y - x‖ := by
+  have h0 : (∀ ε > 0, ∀ᶠ y in 𝓝[≠] x, f y - f x - inner ℝ u (y - x) > -ε * ‖y - x‖)
+      ↔ ∀ ε > 0, ∀ᶠ y in 𝓝 x, f y - f x - inner ℝ u (y - x) ≥ -ε * ‖y - x‖ := by
     constructor
     · intro h ε εpos
       specialize h ε εpos
@@ -149,7 +149,7 @@ theorem convex_f_f_subdiff_eq_subgradient (f : E → ℝ) (x : E)
     have yin': y' ∈ univ:= by
       simp
     specialize convfun xin yin'
-    have pos: 0 < (1 / 2) * ((f x) + inner g (y' - x) - f y') / ‖y' - x‖:=by
+    have pos: 0 < (1 / 2) * ((f x) + inner ℝ g (y' - x) - f y') / ‖y' - x‖:=by
       apply div_pos
       · apply mul_pos
         simp; simp
@@ -160,7 +160,7 @@ theorem convex_f_f_subdiff_eq_subgradient (f : E → ℝ) (x : E)
         rw [yeq'] at hy'
         simp at hy'
     rw[← gt_iff_lt] at pos
-    specialize hg ((1 / 2) * ((f x) + inner g (y' - x) - f y')/‖y' - x‖)
+    specialize hg ((1 / 2) * ((f x) + inner ℝ g (y' - x) - f y')/‖y' - x‖)
     specialize hg pos
     simp at hg
     rw[Filter.Eventually,mem_nhds_iff] at hg
@@ -190,7 +190,7 @@ theorem convex_f_f_subdiff_eq_subgradient (f : E → ℝ) (x : E)
             by_contra yeq'
             rw[sub_eq_zero] at yeq'
             rw[yeq'] at hy'
-            rw[sub_self, inner_zero_right,add_zero, lt_iff_not_le] at hy'
+            rw[sub_self, inner_zero_right,add_zero, lt_iff_not_ge] at hy'
             apply hy'
             simp
             simp
@@ -204,14 +204,14 @@ theorem convex_f_f_subdiff_eq_subgradient (f : E → ℝ) (x : E)
             by_contra yeq'
             rw[sub_eq_zero] at yeq'
             rw[yeq'] at hy'
-            rw[sub_self, inner_zero_right,add_zero, lt_iff_not_le] at hy'
+            rw[sub_self, inner_zero_right,add_zero, lt_iff_not_ge] at hy'
             apply hy'
             simp
           · apply norm_pos_iff.mpr
             by_contra yeq'
             rw[sub_eq_zero] at yeq'
             rw[yeq'] at hy'
-            rw[sub_self, inner_zero_right,add_zero, lt_iff_not_le] at hy'
+            rw[sub_self, inner_zero_right,add_zero, lt_iff_not_ge] at hy'
             apply hy'
             simp
         simp
@@ -231,7 +231,7 @@ theorem convex_f_f_subdiff_eq_subgradient (f : E → ℝ) (x : E)
           by_contra yeq'
           rw[norm_eq_zero,sub_eq_zero] at yeq'
           rw[yeq'] at hy'
-          rw[sub_self, inner_zero_right,add_zero, lt_iff_not_le] at hy'
+          rw[sub_self, inner_zero_right,add_zero, lt_iff_not_ge] at hy'
           apply hy'
           simp
         refine div_mul_cancel₀ δ' nonzero
@@ -241,7 +241,6 @@ theorem convex_f_f_subdiff_eq_subgradient (f : E → ℝ) (x : E)
       apply lt_of_le_of_lt
       apply min_le_left
       exact lt_two_mul_self posδ
-      simp
       apply div_nonneg
       apply le_min
       linarith
@@ -252,8 +251,8 @@ theorem convex_f_f_subdiff_eq_subgradient (f : E → ℝ) (x : E)
     rcases hx1 with ⟨x1s,x1t⟩
     rw[mem_setOf] at x1s
     rcases x1s with ⟨r,rpos,rltone,x1eq⟩
-    have x1in: x1 ∈ {x_1 | inner g (x_1 - x) ≤
-        f x_1 - f x + 2⁻¹ * (f x + inner g (y' - x) - f y') / ‖y' - x‖ * ‖x_1 - x‖}:=by
+    have x1in: x1 ∈ {x_1 | inner ℝ g (x_1 - x) ≤
+        f x_1 - f x + 2⁻¹ * (f x + inner ℝ g (y' - x) - f y') / ‖y' - x‖ * ‖x_1 - x‖}:=by
       apply mem_of_subset_of_mem tin
       assumption
     rw[mem_setOf,x1eq] at x1in
@@ -271,7 +270,7 @@ theorem convex_f_f_subdiff_eq_subgradient (f : E → ℝ) (x : E)
     have r2pos: 0 < (1 -r)/2:=by linarith
     have req: r + (1-r) = 1:=by simp
     specialize convfun rnonneg rleone req
-    have nonneg: 0 ≤ f y' - f x - inner g (y' - x):=by
+    have nonneg: 0 ≤ f y' - f x - inner ℝ g (y' - x):=by
       apply nonneg_of_mul_nonneg_right _ r2pos
       rw[mul_sub, ← sub_self_div_two (1 - r), sub_mul, sub_mul (1 - r)]
       simp
@@ -288,17 +287,17 @@ theorem convex_f_f_subdiff_eq_subgradient (f : E → ℝ) (x : E)
       rw[neg_mul,← sub_eq_add_neg ((1 - r) * f y'),← mul_sub, mul_assoc, mul_comm (1 - r) ‖y' - x‖]
       rw[← mul_assoc, div_mul, div_self]
       simp
-      rw[mul_comm (2⁻¹ * inner g (y' - x) + 2⁻¹ * (f x - f y')),
-        mul_add, add_comm ((1 - r) * (2⁻¹ * inner g (y' - x)))]
+      rw[mul_comm (2⁻¹ * inner ℝ g (y' - x) + 2⁻¹ * (f x - f y')),
+        mul_add, add_comm ((1 - r) * (2⁻¹ * inner ℝ g (y' - x)))]
       rw[← add_assoc, ← mul_assoc, ← mul_assoc,inv_eq_one_div]
       linarith
       by_contra yeq'
       rw[norm_eq_zero,sub_eq_zero] at yeq'
       rw[yeq'] at hy'
-      rw[sub_self, inner_zero_right,add_zero, lt_iff_not_le] at hy'
+      rw[sub_self, inner_zero_right,add_zero, lt_iff_not_ge] at hy'
       apply hy'
       simp
-    have nonneg': ¬ 0 > f y' - f x - inner g (y' - x):=by linarith
+    have nonneg': ¬ 0 > f y' - f x - inner ℝ g (y' - x):=by linarith
     apply nonneg'
     simp
     linarith
@@ -326,7 +325,7 @@ theorem convex_f_f_subdiff_eq_subgradient (f : E → ℝ) (x : E)
 theorem f_subdiff_neg_f_subdiff_unique (hu : u ∈ f_subdifferential f x)
     (hv : v ∈ f_subdifferential (- f) x) : u = - v := by
   rw [has_f_subdiff_iff] at *
-  have h : ∀ ε > 0, ∀ᶠ y in 𝓝 x, inner (u + v) (y - x) ≤ ε * ‖y - x‖ := by
+  have h : ∀ ε > 0, ∀ᶠ y in 𝓝 x, inner ℝ (u + v) (y - x) ≤ ε * ‖y - x‖ := by
     intro ε εpos
     have ε2pos : 0 < ε / 2 := by positivity
     filter_upwards [hu _ ε2pos, hv _ ε2pos] with y huy hvy
@@ -371,9 +370,11 @@ theorem f_subdiff_smul (h : u ∈ f_subdifferential (c • f) x) (cpos : 0 < c)
   filter_upwards [h _ (mul_pos cpos εpos)] with y hy
   rw [real_inner_smul_left]
   simp only [Pi.smul_apply, smul_eq_mul, neg_mul, neg_le_sub_iff_le_add] at hy
-  apply (mul_le_mul_left cpos).mp
-  field_simp
-  linarith
+  have hy' : c⁻¹ * inner ℝ u (y - x) ≤ f y - f x + ε * ‖y - x‖ := by
+    simpa [div_eq_mul_inv, mul_assoc, mul_comm, mul_left_comm] using
+      (show inner ℝ u (y - x) / c ≤ f y - f x + ε * ‖y - x‖ from
+        (div_le_iff₀ cpos).2 (by nlinarith [hy]))
+  linarith [hy']
 
 /-- first order optimality condition for unconstrained optimization problem -/
 theorem first_order_optimality_condition (f : E → ℝ) (x₀ : E) (hx : IsLocalMin f x₀) :
@@ -508,8 +509,8 @@ theorem f_subdiff_add' (f : E → ℝ ) (g : E → ℝ ) (g' : E → E) (x : E)
       specialize hg ε2pos
       filter_upwards [zin _ ε2pos, hg ] with a za ga
       simp at ga
-      have h: - (g a - g x - inner (g' x) (a - x)) ≥ -(ε / 2) * ‖a - x‖:=by
-        change -(ε / 2) * ‖a - x‖ ≤ - (g a - g x - inner (g' x) (a - x))
+      have h: - (g a - g x - inner ℝ (g' x) (a - x)) ≥ -(ε / 2) * ‖a - x‖:=by
+        change -(ε / 2) * ‖a - x‖ ≤ - (g a - g x - inner ℝ (g' x) (a - x))
         rw[neg_mul, neg_le_neg_iff]
         apply le_trans; apply le_abs_self; assumption
       rw[inner_sub_left];
diff --git a/Optlib/Function/KL.lean b/Optlib/Function/KL.lean
index f2f0c1a..c58a350 100644
--- a/Optlib/Function/KL.lean
+++ b/Optlib/Function/KL.lean
@@ -31,8 +31,8 @@ lemma subdifferential_Graph' (f : E → ℝ) :
         use fun n => (u n, f (u n), v n)
         constructor
         · intro n; simp; exact (hv n).1
-        · apply Tendsto.prod_mk_nhds u_conv
-            (Tendsto.prod_mk_nhds fun_conv ((forall_and_right _ _).1 hv).2)
+        · exact Filter.Tendsto.prodMk_nhds u_conv
+            (Filter.Tendsto.prodMk_nhds fun_conv ((forall_and_right _ _).1 hv).2)
       · intro h
         simp [subdifferential_Graph, subdifferential]
         simp at h
@@ -68,7 +68,7 @@ theorem GraphOfSubgradientIsClosed {f : E → ℝ}
     exact this
   rw [nhds_prod_eq,Filter.tendsto_prod_iff'] at hconv;
   simp at hconv
-  exact Tendsto.prod_mk_nhds hconv.1 (Tendsto.prod_mk_nhds hf hconv.2)
+  exact Filter.Tendsto.prodMk_nhds hconv.1 (Filter.Tendsto.prodMk_nhds hf hconv.2)
 
 /- Definition of Φ_η, the family of desingularizing function -/
 def desingularizing_function (η : ℝ) := {φ : ℝ → ℝ | (ConcaveOn ℝ (Ico 0 η) φ) -- ∧ (∀ x ∈ Ioo 0 η, φ x > 0)
@@ -108,24 +108,26 @@ lemma desingularizing_function_is_nonneg (φ : ℝ → ℝ) (η : ℝ) (h : φ 
     obtain h_lag := exists_deriv_eq_slope φ hx₁ Cont_φ Diff_φ
     rcases h_lag with ⟨c, ⟨hc, hval⟩⟩
     use c, hc
-    field_simp [hval]
+    have hx0 : x - 0 ≠ 0 := by linarith
+    have hval' : deriv φ c * (x - 0) = φ x - φ 0 := (eq_div_iff hx0).1 hval
+    linarith
   choose y hy₁ hy₂ using hhh
-  simp [hy₂, h₂]; field_simp;
   rcases hy₁ with ⟨hy₁,hy₁'⟩
   have yleq: y < η := by linarith
-  exact h₅ y hy₁ yleq
+  have hyderiv : 0 < deriv φ y := h₅ y hy₁ yleq
+  nlinarith [hy₂, h₂, hyderiv, hx₁]
 
 -- Definition of KL property with specific desingularizing function
 def KL_point_with_reparameter (σ : E → ℝ) (u : E) (φ : ℝ → ℝ) : Prop :=
   ∃ η ∈ Ioi 0, ∃ s ∈ 𝓝 u, (φ ∈ desingularizing_function η) ∧  (∀ x ∈ s ∩
   {y ∈ active_domain σ | σ u < σ y ∧ σ y < σ u + η},
-  deriv φ (σ x - σ u) * (EMetric.infEdist 0 (subdifferential σ x)).toReal ≥ 1)
+  deriv φ (σ x - σ u) * (Metric.infEDist 0 (subdifferential σ x)).toReal ≥ 1)
 
 -- Definition of the KL property at one point
 def KL_point (f : E → ℝ) (u : E) : Prop :=
   ∃ η ∈ Ioi 0, ∃ s ∈ 𝓝 u, ∃ φ ∈ desingularizing_function η, ∀ x ∈ s ∩
   {y | f u < f y ∧ f y < f u + η},
-  (ENNReal.ofReal (deriv φ (f x - f u))) * (EMetric.infEdist 0 (subdifferential f x)) ≥ ENNReal.ofReal 1
+  (ENNReal.ofReal (deriv φ (f x - f u))) * (Metric.infEDist 0 (subdifferential f x)) ≥ ENNReal.ofReal 1
 
 -- Definition of the KL function
 def KL_function (f : E → ℝ) : Prop :=
@@ -139,7 +141,7 @@ def KL_function (f : E → ℝ) : Prop :=
 def KL_property_with_regularization (f : E → ℝ) (u' : E) (φ : ℝ → ℝ) : Prop :=
   ∃ η ∈ Ioi 0, ∃ s ∈ 𝓝 u', (φ ∈ desingularizing_function η) ∧
   (∀ x ∈ s ∩ {y ∈ active_domain f | f u' < f y ∧ f y < f u' + η},
-  (EMetric.infEdist 0 (subdifferential (λ u => φ (f u - f u')) x)).toReal ≥ 1)
+  (Metric.infEDist 0 (subdifferential (λ u => φ (f u - f u')) x)).toReal ≥ 1)
 
 -- deriv of function (fun t => c⁻¹ * t) is c⁻¹
 lemma deriv_of_const_mul_func (c : ℝ) (x : ℝ) : deriv (fun (t : ℝ) => c⁻¹ * t) x = c⁻¹ := by
@@ -165,10 +167,11 @@ lemma const_mul_special_concave : ∀ c > 0, (fun t => c⁻¹ * t) ∈ desingula
   have h₃: ContDiffOn ℝ 1 (fun t ↦ c⁻¹ * t) (Ioo 0 (c / 2)) := by
     rw [fun_smul_eq_mul]; apply ContDiff.contDiffOn; apply contDiff_const_smul
   have h₄: ContinuousAt (fun t ↦ c⁻¹ * t) 0 := by
-    rw [fun_smul_eq_mul]; apply (continuousAt_const_smul_iff₀ _).2
-    apply continuousAt_id; field_simp
+    simpa using (continuous_const.mul continuous_id).continuousAt
   have h₅: ∀ (x : ℝ), 0 < x → x < c / 2 → 0 < deriv (fun t ↦ c⁻¹ * t) x := by
-    intro x _ _; rw [deriv_of_const_mul_func]; field_simp; exact cpos
+    intro x _ _
+    rw [deriv_of_const_mul_func]
+    exact inv_pos.mpr cpos
   exact ⟨h₁, h₃, h₄, h₅⟩
 
 
@@ -195,23 +198,24 @@ lemma const_mul_edist_ge_one {c : ℝ} {ed : ENNReal} (hpos : c > 0)
     rw [hed']; refine ENNReal.mul_top ?h; simpa
   rw [this]; simp; push_neg at hed'
   calc
-    _ ≥ ENNReal.ofReal c⁻¹ * ENNReal.ofReal c := mul_le_mul_left' hed (ENNReal.ofReal c⁻¹)
+    _ ≥ ENNReal.ofReal c⁻¹ * ENNReal.ofReal c := by
+      exact mul_le_mul_of_nonneg_left hed (by exact bot_le)
     _ = ENNReal.ofReal 1 := by
       rw [← ENNReal.ofReal_mul]; field_simp; simp; exact le_of_lt hpos
 
 lemma edist_geq_const (h_noncrit : 0 ∉ subdifferential f x) :
   ∃ c > 0, ∀ u, ‖u - x‖ + ‖f u - f x‖ < c →
-  EMetric.infEdist 0 (subdifferential f u) ≥ ENNReal.ofReal c := by
+  Metric.infEDist 0 (subdifferential f u) ≥ ENNReal.ofReal c := by
     by_contra! hc
     have sqh: ∀ n : ℕ, ∃ u, ‖u - x‖ + ‖f u - f x‖ < 1 / (n + 1) ∧
-        (EMetric.infEdist 0 (subdifferential f u)) < ENNReal.ofReal (1 / (n + 1)) :=
+        (Metric.infEDist 0 (subdifferential f u)) < ENNReal.ofReal (1 / (n + 1)) :=
       fun n ↦ hc (1 / (n + 1)) (by simp; linarith)
     choose u hu using sqh
-    have inequ_fun : ∀ n, (EMetric.infEdist 0 (subdifferential f (u n))).toReal ≤ 1 / (n + 1) := by
+    have inequ_fun : ∀ n, (Metric.infEDist 0 (subdifferential f (u n))).toReal ≤ 1 / (n + 1) := by
       intro n
       apply (ENNReal.toReal_le_of_le_ofReal _ (le_of_lt (hu n).right))
       simp; linarith
-    have : Tendsto (fun n ↦ (EMetric.infEdist 0 (subdifferential f (u n))).toReal) atTop (𝓝 0) :=
+    have : Tendsto (fun n ↦ (Metric.infEDist 0 (subdifferential f (u n))).toReal) atTop (𝓝 0) :=
       squeeze_zero (by simp) inequ_fun tendsto_one_div_add_atTop_nhds_zero_nat
     have h_contra : 0 ∈ subdifferential f x := by
       have u_to_x : Tendsto u atTop (𝓝 x) := by
@@ -236,7 +240,7 @@ lemma edist_geq_const (h_noncrit : 0 ∉ subdifferential f x) :
         intro n
         rcases hu n with ⟨_,hu₂⟩
         have : ∃ vn ∈ subdifferential f (u n), edist 0 vn < 1 / (n + 1) := by
-          apply EMetric.infEdist_lt_iff.1
+          apply Metric.infEDist_lt_iff.1
           rw [← one_div_type_trans n]
           exact hu₂
         choose vn hvn using this
@@ -248,15 +252,15 @@ lemma edist_geq_const (h_noncrit : 0 ∉ subdifferential f x) :
         intro n
         exact (hv n).1
       have v_to_zero: Tendsto v atTop (𝓝 0) := by
-        rw [dist_zero_left] at hv
         have : Tendsto (fun n => ‖v n‖) atTop (𝓝 0) := by
           apply squeeze_zero (by simp) _ tendsto_one_div_add_atTop_nhds_zero_nat
           intro n
-          apply le_of_lt (hv n).right
+          have hdist : dist 0 (v n) < 1 / (n + 1) := (hv n).right
+          exact (le_of_lt (by simpa [dist_eq_norm] using hdist))
         apply tendsto_zero_iff_norm_tendsto_zero.2 this
       show (x, 0) ∈ subdifferential_Graph f
       apply GraphOfSubgradientIsClosed v_in_subdiff
-        (Filter.Tendsto.prod_mk_nhds u_to_x v_to_zero) fu_to_fx
+        (Filter.Tendsto.prodMk_nhds u_to_x v_to_zero) fu_to_fx
     contradiction
 
 /-- Non-critical KL property is naturally true -/
@@ -281,24 +285,9 @@ end
 
 section aux_lemma_uniform_KL
 
-lemma real_geq_ennreal_ofreal_geq {a b : ℝ} {c : ENNReal} (hgeq : a ≥ b) (apos: a > 0):
+lemma real_geq_ennreal_ofreal_geq {a b : ℝ} {c : ENNReal} (hgeq : a ≥ b) (_apos : a > 0):
   (ENNReal.ofReal a) * c ≥ (ENNReal.ofReal b) * c := by
-  by_cases hc : c = 0
-  rw [hc]
-  simp
-  push_neg at hc
-  by_cases hctop : c = ⊤
-  rw [hctop]
-  have ha : (ENNReal.ofReal a) * ⊤ = ⊤ := by
-    refine ENNReal.mul_top ?h
-    simpa
-  rw [ha]
-  simp
-  push_neg at hctop
-  refine (ENNReal.mul_le_mul_right ?_ ?_).mpr ?_
-  · exact hc
-  · exact hctop
-  · exact ENNReal.ofReal_le_ofReal hgeq
+  exact mul_le_mul_left (ENNReal.ofReal_le_ofReal hgeq) c
 
 end aux_lemma_uniform_KL
 
@@ -323,24 +312,24 @@ variable [NormedAddCommGroup E] [InnerProductSpace ℝ E]
 theorem uniformized_KL_property {f : E → ℝ} {Ω : Set E} (h_compact : IsCompact Ω)
     (h_Ω1 : ∀ x ∈ Ω, KL_point f x) (h_Ω2: is_constant_on f Ω) :
     ∃ ε ∈ Ioi 0, ∃ η ∈ Ioi 0, ∃ φ ∈ desingularizing_function η, ∀ u ∈ Ω , ∀ x ∈
-    {y : E | (EMetric.infEdist y Ω).toReal < ε} ∩ {y  | f u < f y ∧ f y < f u + η},
-    (ENNReal.ofReal (deriv φ (f x - f u))) * EMetric.infEdist 0 (subdifferential f x) ≥ 1 := by
+    {y : E | (Metric.infEDist y Ω).toReal < ε} ∩ {y  | f u < f y ∧ f y < f u + η},
+    (ENNReal.ofReal (deriv φ (f x - f u))) * Metric.infEDist 0 (subdifferential f x) ≥ 1 := by
     -- case : Ω = ∅
-    by_cases h_nonempty : Ω = ∅
-    · push_neg at h_nonempty
+    by_cases h_empty : Ω = ∅
+    ·
       use 1, (by simp), 1, (by simp), (fun t => 2⁻¹ * t)
       constructor
       rw [← div_self]
       exact (const_mul_special_concave 2 (by simp))
       simp
-      rw [h_nonempty]
+      rw [h_empty]
       tauto
     -- case : Ω ≠ ∅
-    push_neg at h_nonempty
+    have h_nonempty : Ω.Nonempty := Set.nonempty_iff_ne_empty.2 h_empty
     obtain ⟨μ, constant_value⟩ := exist_constant_value f h_Ω2 h_nonempty
     have : ∀ x ∈ Ω, ∃ η ∈ Ioi 0, ∃ (O : Set E) (_: IsOpen O) (_: x ∈ O),  ∃ φ ∈ desingularizing_function η,
       ∀ u ∈ O ∩ {y | f x < f y ∧ f y < f x + η},
-      (ENNReal.ofReal (deriv φ (f u - f x))) * EMetric.infEdist 0 (subdifferential f u) ≥ 1 := by
+      (ENNReal.ofReal (deriv φ (f u - f x))) * Metric.infEDist 0 (subdifferential f u) ≥ 1 := by
         intro x hx; simp [KL_point] at h_Ω1
         rcases h_Ω1 x hx with ⟨η, ⟨hη, ⟨s, ⟨hs, ⟨φ, hφ, h_Ω1⟩⟩⟩⟩⟩
         rcases mem_nhds_iff.1 hs with ⟨O, hO1, hO2, hO3⟩
@@ -397,6 +386,9 @@ theorem uniformized_KL_property {f : E → ℝ} {Ω : Set E} (h_compact : IsComp
         exact min_le (η x) η_in_image
     rcases this with ⟨η_min, ηpos, hmin⟩
     simp [desingularizing_function] at hφ
+    have hsum_attach : ∀ z : ℝ, ∑ c ∈ ht2.toFinset.attach, φ (↑c) z = ∑ c ∈ ht2.toFinset, φ c z := by
+      intro z
+      simpa using (Finset.sum_attach (s := ht2.toFinset) (f := fun c => φ c z))
     -- φ_sum is desingularizing_function
     have h_special_concave: φ_sum ∈ desingularizing_function η_min := by
       simp [desingularizing_function]
@@ -406,6 +398,7 @@ theorem uniformized_KL_property {f : E → ℝ} {Ω : Set E} (h_compact : IsComp
         apply convex_Ico
         intro x xpos y ypos a b apos bpos absum
         simp [φ_sum]
+        rw [hsum_attach x, hsum_attach y, hsum_attach (a * x + b * y)]
         have : ∀ d : ℝ, ∀ x : ℝ, d * ∑ c ∈ ht2.toFinset,
             φ c x = ∑ c ∈ ht2.toFinset, d * (φ c x) := by
           intro d x
@@ -439,13 +432,15 @@ theorem uniformized_KL_property {f : E → ℝ} {Ω : Set E} (h_compact : IsComp
       --   exact (hφ c (mem_t_in_Ω c hc)).2.1 x hx1 xleq
       --   exact t_nonempty
       have h₃ : φ_sum 0 = 0 := by
-        simp [φ_sum]
+        simp [φ_sum, hsum_attach]
         have : ∀ x ∈ ht2.toFinset, φ x 0 = 0 := by
           intro x xt
           exact (hφ x (mem_t_in_Ω x xt)).2.1
         apply Finset.sum_eq_zero this
       have h₄ : ContDiffOn ℝ 1 φ_sum (Ioo 0 η_min) := by
-        have : φ_sum = (fun c => ∑ x ∈ ht2.toFinset, φ x c) := by ext c; simp [φ_sum]
+        have : φ_sum = (fun c => ∑ x ∈ ht2.toFinset, φ x c) := by
+          ext c
+          simpa [φ_sum] using hsum_attach c
         rw [this]
         apply ContDiffOn.sum
         intro c hc
@@ -455,9 +450,10 @@ theorem uniformized_KL_property {f : E → ℝ} {Ω : Set E} (h_compact : IsComp
         exact hmin c ((Set.Finite.mem_toFinset ht2).1 hc)
       have h₅ : ContinuousAt φ_sum 0 := by
         rw [ContinuousAt]
-        have : φ_sum = (fun c => ∑ x ∈ ht2.toFinset, φ x c) := by ext c; simp [φ_sum]
+        have : φ_sum = (fun c => ∑ x ∈ ht2.toFinset, φ x c) := by
+          ext c
+          simpa [φ_sum] using hsum_attach c
         rw [this]
-        simp [φ_sum]
         apply tendsto_finset_sum
         intro c hc
         obtain cont := (hφ c (mem_t_in_Ω c hc)).2.2.2.1
@@ -465,17 +461,23 @@ theorem uniformized_KL_property {f : E → ℝ} {Ω : Set E} (h_compact : IsComp
         exact cont
       have h₆ : ∀ (x : ℝ), 0 < x → x < η_min → 0 < deriv φ_sum x := by
         intro y ypos yleq
-        have : φ_sum = (fun c => ∑ x ∈ ht2.toFinset, φ x c) := by ext c; simp [φ_sum]
+        have : φ_sum = (fun c => ∑ x ∈ ht2.toFinset, φ x c) := by
+          ext c
+          simpa [φ_sum] using hsum_attach c
         rw [this]
         have : deriv (fun c ↦ ∑ x ∈ ht2.toFinset, φ x c) y = ∑ x ∈ ht2.toFinset, deriv (φ x) y := by
-          apply deriv_sum
-          intro c hc
-          have η_inequ: y < η c := by
-            obtain := hmin c ((Set.Finite.mem_toFinset ht2).1 hc)
-            linarith
-          specialize hφ c (mem_t_in_Ω c hc)
-          obtain contdiff:= ContDiffOn.contDiffAt hφ.2.2.1 (Ioo_mem_nhds ypos η_inequ)
-          apply ContDiffAt.differentiableAt contdiff (by simp)
+          have hfun : (fun c => ∑ x ∈ ht2.toFinset, φ x c) = (∑ x ∈ ht2.toFinset, φ x) := by
+            ext c; exact Eq.symm (Finset.sum_apply c ht2.toFinset φ)
+          have hderiv : deriv (∑ x ∈ ht2.toFinset, φ x) y = ∑ x ∈ ht2.toFinset, deriv (φ x) y := by
+            apply deriv_sum
+            intro c hc
+            have η_inequ : y < η c := by
+              obtain := hmin c ((Set.Finite.mem_toFinset ht2).1 hc)
+              linarith
+            specialize hφ c (mem_t_in_Ω c hc)
+            obtain contdiff := ContDiffOn.contDiffAt hφ.2.2.1 (Ioo_mem_nhds ypos η_inequ)
+            exact ContDiffAt.differentiableAt contdiff (by simp)
+          simpa [hfun] using hderiv
         rw [this]
         apply Finset.sum_pos
         · intro c hc
@@ -486,7 +488,7 @@ theorem uniformized_KL_property {f : E → ℝ} {Ω : Set E} (h_compact : IsComp
           exact hφ.2.2.2.2 y ypos this
         · exact t_nonempty
       exact ⟨h₁,h₃,h₄,h₅,h₆⟩
-    have uniform_ball: ∃ ε ∈ Ioi 0, {y| EMetric.infEdist y Ω < ENNReal.ofReal ε} ⊆ ⋃ x ∈ t, O x := by
+    have uniform_ball: ∃ ε ∈ Ioi 0, {y| Metric.infEDist y Ω < ENNReal.ofReal ε} ⊆ ⋃ x ∈ t, O x := by
         have union_open : IsOpen (⋃ x ∈ t, O x) := by
           have : ∀ x ∈ t, IsOpen (O x) := by
             intro x hx
@@ -497,15 +499,15 @@ theorem uniformized_KL_property {f : E → ℝ} {Ω : Set E} (h_compact : IsComp
         obtain res_thickening := IsCompact.exists_thickening_subset_open h_compact union_open ht3
         rcases res_thickening with ⟨ε, ⟨hε, h2⟩⟩
         use ε, hε
-        have : {y| EMetric.infEdist y Ω < ENNReal.ofReal ε} = Metric.thickening ε Ω := by
-          ext x; exact Metric.mem_thickening_iff_infEdist_lt
+        have : {y| Metric.infEDist y Ω < ENNReal.ofReal ε} = Metric.thickening ε Ω := by
+          ext x; exact Metric.mem_thickening_iff_infEDist_lt
         rwa [this]
     choose ε uniform_ball using uniform_ball
-    have : {y| EMetric.infEdist y Ω < ENNReal.ofReal ε} = {y| (EMetric.infEdist y Ω).toReal < ε} := by
-      ext x; apply ENNReal.lt_ofReal_iff_toReal_lt (Metric.infEdist_ne_top h_nonempty)
+    have : {y| Metric.infEDist y Ω < ENNReal.ofReal ε} = {y| (Metric.infEDist y Ω).toReal < ε} := by
+      ext x; apply ENNReal.lt_ofReal_iff_toReal_lt (Metric.infEDist_ne_top h_nonempty)
     rw [this] at uniform_ball
     -- There exists one open set in the finite cover
-    have exist_one_ball: ∀ u ∈ {y| (EMetric.infEdist y Ω).toReal < ε}
+    have exist_one_ball: ∀ u ∈ {y| (Metric.infEDist y Ω).toReal < ε}
         ∩ {y |  μ < f y ∧ f y < μ + η_min},
       ∃ x ∈ t, u ∈ O x := by
       intro u hu
@@ -521,7 +523,7 @@ theorem uniformized_KL_property {f : E → ℝ} {Ω : Set E} (h_compact : IsComp
     rcases hu with ⟨_,hu2,hu3⟩
     -- rcases hu with ⟨_, ⟨hu21 , ⟨hu221, hu222⟩⟩⟩
     calc
-      _ ≥ (ENNReal.ofReal (deriv (φ ui) (f u - μ))) * EMetric.infEdist 0 (subdifferential f u) := by
+      _ ≥ (ENNReal.ofReal (deriv (φ ui) (f u - μ))) * Metric.infEDist 0 (subdifferential f u) := by
         have deriv_φ_pos: deriv φ_sum (f u - μ) > 0 := by
           simp [desingularizing_function] at h_special_concave
           obtain h_tmp := h_special_concave.2.2.2.2
@@ -530,23 +532,28 @@ theorem uniformized_KL_property {f : E → ℝ} {Ω : Set E} (h_compact : IsComp
           · linarith [hu3]
         apply real_geq_ennreal_ofreal_geq
         simp [φ_sum]
+        have hfun_attach :
+            (∑ b ∈ ht2.toFinset.attach, φ (↑b)) = (fun c => ∑ x ∈ ht2.toFinset, φ x c) := by
+          ext c
+          simpa [Finset.sum_apply] using hsum_attach c
+        rw [hfun_attach]
         have equ₁: deriv (fun c ↦ ∑ x ∈ ht2.toFinset, φ x c) (f u - μ) =
             ∑ x ∈ ht2.toFinset, deriv (φ x) (f u - μ) := by
-          apply deriv_sum
-          intro c hc
-          have σu_pos : f u - μ > 0 := by linarith [hu2]
-          have η_inequ: (f u - μ) < η c := by
-            obtain inequ_ηmin:= hmin c ((Set.Finite.mem_toFinset ht2).1 hc)
-            linarith
-          specialize hφ c (mem_t_in_Ω c hc)
-          obtain contdiff:= ContDiffOn.contDiffAt hφ.2.2.1 (Ioo_mem_nhds σu_pos η_inequ)
-          apply ContDiffAt.differentiableAt contdiff (by simp)
-        have equ₂ : deriv (fun c ↦ ∑ x ∈ ht2.toFinset, φ x c) (f u - μ) =
-            deriv (∑ x ∈ ht2.toFinset, φ x) (f u - μ) := by
-          have : (fun c ↦ ∑ x ∈ ht2.toFinset, φ x c) = (∑ x ∈ ht2.toFinset, φ x) := by
+          have hfun : (fun c => ∑ x ∈ ht2.toFinset, φ x c) = (∑ x ∈ ht2.toFinset, φ x) := by
             ext c; exact Eq.symm (Finset.sum_apply c ht2.toFinset φ)
-          rw [this]
-        rw [← equ₂, equ₁]
+          have hderiv :
+              deriv (∑ x ∈ ht2.toFinset, φ x) (f u - μ) = ∑ x ∈ ht2.toFinset, deriv (φ x) (f u - μ) := by
+            apply deriv_sum
+            intro c hc
+            have σu_pos : f u - μ > 0 := by linarith [hu2]
+            have η_inequ : (f u - μ) < η c := by
+              obtain inequ_ηmin := hmin c ((Set.Finite.mem_toFinset ht2).1 hc)
+              linarith
+            specialize hφ c (mem_t_in_Ω c hc)
+            obtain contdiff := ContDiffOn.contDiffAt hφ.2.2.1 (Ioo_mem_nhds σu_pos η_inequ)
+            exact ContDiffAt.differentiableAt contdiff (by simp)
+          simpa [hfun] using hderiv
+        rw [equ₁]
 
         -- have : (∑ x ∈ ht2.toFinset, deriv (φ x) (f u - μ)) ≥ deriv (φ ui) (f u - μ) := by
         let g x := deriv (φ x) (f u - μ)
@@ -575,27 +582,4 @@ theorem uniformized_KL_property {f : E → ℝ} {Ω : Set E} (h_compact : IsComp
         rw [equ_μ] at h_exist
         exact h_exist
 
-
--- theorem uniformly_KL_property' {f : E → ℝ} {Ω : Set E} (h_compact : IsCompact Ω)
---     (h_Ω1 : ∀ x ∈ Ω, KL_point f x) (h_Ω2: is_constant_on f Ω) :
---     ∃ ε ∈ Ioi 0, ∃ η ∈ Ioi 0, ∃ φ ∈ desingularizing_function η, ∀ u ∈ Ω , ∀ x ∈
---     {y : E | (EMetric.infEdist y Ω).toReal < ε} ∩ {y | f u < f y ∧ f y < f u + η},
---     (Real.toEReal (deriv φ (f x - f u))) * (EMetric.infEdist 0 (subdifferential f x))
---       ≥ Real.toEReal 1 := by
-
---     obtain h := uniformly_KL_property h_compact h_Ω1 h_Ω2
---     rcases h with ⟨ε, hε, η, hη, φ, hφ, h⟩
---     use ε, hε, η, hη, φ, hφ
---     intro u hu x hx
---     by_cases h_empty : EMetric.infEdist 0 (subdifferential f x) = ⊤
---     · rw [h_empty]
---       have hderiv: Real.toEReal (deriv φ (f x - f u)) > 0 := by sorry
---       have hh: (Real.toEReal (deriv φ (f x - f u))) * (ENNReal.toEReal ⊤) = ⊤ := by
---         sorry
---       rw [hh]
---       simp
---     · push_neg at h_empty
---       have h_not_bot: EMetric.infEdist 0 (subdifferential f x) ≠ ⊥ := by sorry
---       sorry
---     -- by_cases h_empty : EMetric.infEdist 0 (subdifferential f x) = ∅
 end uniformized_KL
diff --git a/Optlib/Function/L1Space.lean b/Optlib/Function/L1Space.lean
index d3d0436..00a98c1 100644
--- a/Optlib/Function/L1Space.lean
+++ b/Optlib/Function/L1Space.lean
@@ -52,22 +52,26 @@ open scoped Pointwise Module
 
 /--A noncomputable function mapping basis vectors to the `l₁` space.-/
 noncomputable def f : Fin (Module.finrank ℝ α) → PiLp 1 (fun _ : Fin (Module.finrank ℝ α) => ℝ) :=
-  fun i j => if i = j then ‖(Module.finBasis ℝ α) i‖ else 0
+  fun i => WithLp.toLp 1 (fun j => if i = j then ‖(Module.finBasis ℝ α) i‖ else 0)
 
-noncomputable def σ := Basis.constrL (Module.finBasis ℝ α) f
+noncomputable def σ := (Module.finBasis ℝ α).constrL f
 
-theorem continuous_map_sigma : Continuous (σ (α := α)):= by exact ContinuousLinearMap.continuous σ
+theorem continuous_map_sigma : Continuous (σ (α := α)):= by exact (σ (α := α)).continuous
 
 theorem sigma_orthogonal_same_index {i j : Fin (Module.finrank ℝ α)} (h : i = j) :
-    (σ ((Module.finBasis ℝ α) i)) j = ‖(Module.finBasis ℝ α) i‖ := by simp[σ,f,h]
+    (σ ((Module.finBasis ℝ α) i)) j = ‖(Module.finBasis ℝ α) i‖ := by
+  rw [σ, (Module.finBasis ℝ α).constrL_basis f i, f]
+  simp [h]
 
 theorem sigma_orthogonal_diff_index {i j : Fin (Module.finrank ℝ α)} (h : i ≠ j) :
-    (σ ((Module.finBasis ℝ α) i)) j = 0 := by simp[σ,f,h]
+    (σ ((Module.finBasis ℝ α) i)) j = 0 := by
+  rw [σ, (Module.finBasis ℝ α).constrL_basis f i, f]
+  simp [h]
 
 theorem sigma_apply_basis (i : Fin (Module.finrank ℝ α)) :
     σ ((Module.finBasis ℝ α) i) = fun j => if i = j then ‖(Module.finBasis ℝ α) i‖ else 0 := by
   ext j
-  simp[σ,f];
+  rw [σ, (Module.finBasis ℝ α).constrL_basis f i, f]
 
 theorem sigma_norm_apply : ∀ x , ∀ j , ∑ i  , (((Module.finBasis ℝ α).repr x) i) • σ ((Module.finBasis ℝ α) i) j
     = (((Module.finBasis ℝ α).repr x) j) * ‖(Module.finBasis ℝ α) j‖ := by
@@ -87,19 +91,12 @@ theorem sigma_norm_apply : ∀ x , ∀ j , ∑ i  , (((Module.finBasis ℝ α).r
 
 theorem sigma_decompose_apply : ∀ x , ∀ j , (σ x) j =
     ∑ i , (((Module.finBasis ℝ α).repr x) i) • σ ((Module.finBasis ℝ α) i) j:= by
-  intro x
-  rw[← PiLp.ext_iff]
+  intro x j
   calc
-    _ = σ (∑ i , (((Module.finBasis ℝ α).repr x) i) • (Module.finBasis ℝ α) i):= by
-      congr;exact Eq.symm (Basis.sum_repr (Module.finBasis ℝ α) x)
-    _ = ∑ i , σ ((((Module.finBasis ℝ α).repr x) i) • (Module.finBasis ℝ α) i):= by
-      simp only [map_sum, map_smul]
-    _ = _ := by
-      ext j;
-      repeat rw[Finset.sum_apply]
-      congr
-      ext x
-      simp only [map_smul, PiLp.smul_apply, smul_eq_mul]
+    _ = (σ (∑ i , (((Module.finBasis ℝ α).repr x) i) • (Module.finBasis ℝ α) i)) j := by
+      rw [(Module.finBasis ℝ α).sum_repr x]
+    _ = (∑ i , σ ((((Module.finBasis ℝ α).repr x) i) • (Module.finBasis ℝ α) i)) j := by rw [map_sum]
+    _ = _ := by simp [map_smul]
 
 /--
 For any element x in the vector space α, the norm of the image of x
@@ -109,9 +106,7 @@ in the finite basis representation of x, each multiplied by the norm of the corr
 -/
 theorem l1_norm_eq : ∀ x , ‖σ x‖ =  ∑ i , ‖((Module.finBasis ℝ α).repr x) i‖ * ‖(Module.finBasis ℝ α) i‖ := by
   intro x
-  rw[PiLp.norm_eq_of_nat 1 (by norm_num)]
-  simp only [pow_one, Nat.cast_one, ne_eq, one_ne_zero, not_false_eq_true,
-    div_self, Real.rpow_one]
+  rw [PiLp.norm_eq_of_L1]
   congr
   ext i
   rw[sigma_decompose_apply x i,← norm_smul,sigma_norm_apply,norm_smul]
@@ -127,14 +122,14 @@ b i, including both positive and negative scalings.
 -/
 local notation "b" => (Module.finBasis ℝ α)
 theorem l1Ball_sub_convexHull{x : α}{r : ℝ}(hr : r > 0)(hn : Module.finrank ℝ α ≠ 0):
-    σ.toFun ⁻¹' (Metric.ball (σ.toFun x) r) ⊆
+    σ ⁻¹' (Metric.ball (σ x) r) ⊆
     convexHull ℝ (({x} : Set α) + ((⋃ i , {(r / ‖b i‖) • (b i)})  ∪  (⋃ i ,{- (r / ‖b i‖) • (b i)}))):= by
   intro x₀ hx₀
-  simp[dist_eq_norm] at hx₀
+  have hx₀ : ‖σ x₀ - σ x‖ < r := by simpa [Metric.mem_ball, dist_eq_norm] using hx₀
   rw[← map_sub] at hx₀
   have sum_le_r :  ∑ i , ‖(b).equivFun (x₀ - x)  i‖ * ‖(b) i‖ / r ≤  1 := by
+    change ∑ i, ‖(b).repr (x₀ - x) i‖ * ‖b i‖ / r ≤ 1
     rw[← Finset.sum_div]
-    simp only [Basis.equivFun_apply, Pi.sub_apply]
     rw[← l1_norm_eq (x₀ - x)]
     apply le_of_lt
     apply Bound.div_lt_one_of_pos_of_lt hr hx₀
@@ -142,13 +137,13 @@ theorem l1Ball_sub_convexHull{x : α}{r : ℝ}(hr : r > 0)(hn : Module.finrank 
   let ι := Fin n
   let ι₀ := Fin (n + 2)
   let w₀ := (b).equivFun (x₀ - x)
-  have repr : ∑ i , w₀ i • b i = x₀ - x := Basis.sum_equivFun b (x₀ - x)
+  have repr : ∑ i , w₀ i • b i = x₀ - x := (b).sum_equivFun (x₀ - x)
 
   let w₁  : ι → ℝ := fun i => |(b).equivFun (x₀ - x) i| * ‖b i‖ / r
   let sum := ∑ i : ι, w₁ i
 
   have sum_pos : 1 - sum ≥ 0 := by
-    simp only [sum,w₁,ge_iff_le, gt_iff_lt,sub_pos,Pi.sub_apply, sub_nonneg]
+    simp only [sum, w₁, ge_iff_le, sub_nonneg]
     apply sum_le_r
 
   let w  : ι₀ → ℝ
@@ -177,7 +172,7 @@ theorem l1Ball_sub_convexHull{x : α}{r : ℝ}(hr : r > 0)(hn : Module.finrank 
   have hw₀ : ∀ (i : ι₀), 0 ≤ w i := by
     intro ⟨i,hi⟩
     by_cases h : i < n
-    · simp only [Pi.sub_apply, h, ↓reduceDIte, ge_iff_le, w, w₁]
+    · simp only [h, ↓reduceDIte, ge_iff_le, w, w₁]
       apply div_nonneg _ (le_of_lt hr)
       apply mul_nonneg
       apply abs_nonneg
@@ -188,41 +183,51 @@ theorem l1Ball_sub_convexHull{x : α}{r : ℝ}(hr : r > 0)(hn : Module.finrank 
     have : n + 1 + 1 = n + 2 := by norm_num
     rw[← this]
     rw[Fin.sum_univ_castSucc,Fin.sum_univ_castSucc]
-    simp only [Fin.coe_castSucc, Fin.is_lt, ↓reduceDIte, Fin.eta, Fin.val_last, lt_self_iff_false,
+    simp only [Fin.val_castSucc, Fin.is_lt, ↓reduceDIte, Fin.eta, Fin.val_last, lt_self_iff_false,
       add_lt_iff_neg_left, not_lt_zero']
     have : ∑ x : Fin n, w₁ x = sum := rfl
     rw[this]
     linarith
 
   have hz : ∀ (i : ι₀), z i ∈ ((⋃ i , {(r / ‖b i‖) • (b i)})  ∪  (⋃ i ,{- (r / ‖b i‖) • (b i)})) := by
-    intro i
+    rintro ⟨i, hi⟩
     simp only [dite_eq_ite, z]
-    by_cases h₁ : (i : ℕ) = n + 1
-    · simp[h₁]
-    simp only [h₁, ↓reduceIte]
-    by_cases h₂ : (i : ℕ) = n
-    · simp[h₂]
-    simp only [h₂, ↓reduceIte, add_right_inj]
-    let use_i : ι := ⟨i ,lem_i i.2 h₁ h₂⟩
-    simp only [↓reduceDIte]
-    let a := (b).equivFun (x₀ - x) use_i
-    rcases lt_trichotomy a 0 with ha | ha | ha
+    by_cases h₁ : i = n + 1
     · right
-      have : (b).equivFun (x₀ - x) use_i ≠ 0 := by linarith
-      simp at this
-      rw[sign_neg ha]
-      simp[this, ↓reduceIte]
-    · left;
-      simp only [a] at ha
-      rw[ha,sign_zero]
-      simp;
+      refine Set.mem_iUnion.2 ⟨fin0, ?_⟩
+      simp [h₁]
+    by_cases h₂ : i = n
     · left
-      rw[sign_pos ha]
+      refine Set.mem_iUnion.2 ⟨fin0, ?_⟩
+      simp [h₂]
+    let use_i : ι := ⟨i, lem_i hi h₁ h₂⟩
+    have hsub : (b).equivFun (x₀ - x) use_i = ((b).repr x₀) use_i - ((b).repr x) use_i := by simp
+    rcases lt_trichotomy ((b).equivFun (x₀ - x) use_i) 0 with ha | ha | ha
+    · right
+      refine Set.mem_iUnion.2 ⟨use_i, ?_⟩
+      have hdlt : ((b).repr x₀) use_i - ((b).repr x) use_i < 0 := by simpa [hsub] using ha
+      have hdne : ((b).repr x₀) use_i - ((b).repr x) use_i ≠ 0 := ne_of_lt hdlt
+      rw [Set.mem_singleton_iff]
+      simp [h₁, h₂, use_i, hdne]
+      rw [sign_neg hdlt]
+      simp
+    · left
+      refine Set.mem_iUnion.2 ⟨use_i, ?_⟩
+      have hd0 : ((b).repr x₀) use_i - ((b).repr x) use_i = 0 := by simpa [hsub] using ha
+      rw [Set.mem_singleton_iff]
+      simp [h₁, h₂, use_i, hd0]
+    · left
+      refine Set.mem_iUnion.2 ⟨use_i, ?_⟩
+      have hdgt : 0 < ((b).repr x₀) use_i - ((b).repr x) use_i := by simpa [hsub] using ha
+      have hdne : ((b).repr x₀) use_i - ((b).repr x) use_i ≠ 0 := ne_of_gt hdgt
+      rw [Set.mem_singleton_iff]
+      simp [h₁, h₂, use_i, hdne]
+      rw [sign_pos hdgt]
       simp
   have bi_pos : ∀ i : ι , ‖b i‖ ≠ 0 := by
     intro i
     refine norm_ne_zero_iff.mpr ?_
-    exact Basis.ne_zero b i
+    exact (b).ne_zero i
 
   have hx : ∑ i : ι₀, w i • z i = x₀ - x := by
     rw[Fin.sum_univ_castSucc,Fin.sum_univ_castSucc]
@@ -239,10 +244,10 @@ theorem l1Ball_sub_convexHull{x : α}{r : ℝ}(hr : r > 0)(hn : Module.finrank 
       refine Nat.ne_of_lt ?h
       refine Nat.lt_succ_of_lt i.2
     have h₂ : (i : ℕ) ≠ n  := Ne.symm (Nat.ne_of_lt' i.2)
-    simp only [neg_smul, dite_eq_ite, Fin.coe_castSucc, h₁, ↓reduceIte, h₂,
+    simp only [neg_smul, dite_eq_ite, Fin.val_castSucc, h₁, ↓reduceIte, h₂,
       Fin.eta, z]
     have : w₁ i • ((SignType.sign ((b).equivFun (x₀ - x) i)) * (r / ‖b i‖)) = w₀ i := by
-      simp only [Pi.sub_apply, smul_eq_mul, w₁, w₀]
+      simp only [smul_eq_mul, w₁, w₀]
       calc
         _ = |(b).equivFun (x₀ - x) i| * (‖b i‖ / r) * (SignType.sign ((b).equivFun (x₀ - x) i)) * (r / ‖b i‖) := by
           rw[← mul_div]
@@ -269,27 +274,14 @@ theorem l1Ball_sub_convexHull{x : α}{r : ℝ}(hr : r > 0)(hn : Module.finrank 
 
 theorem sigma_is_injective : Function.Injective σ (α := α) := by
   intro x y h
-  rw[← sub_eq_zero]
-  rw[← sub_eq_zero, ← map_sub] at h
-  let z := x - y
-  let n := Module.finrank ℝ α
-  let bs := Module.finBasis ℝ α
-  have hz : z = ∑ i : Fin n , (bs.repr z i)• bs i := Eq.symm (Basis.sum_repr bs z)
-  change σ z = 0 at h
-  rw[hz] at h
-  simp at h
-  have hi :∀ i , (∑ x : Fin n, (bs.repr z) x • σ (bs x)) i = (bs.repr z) i * ‖(Module.finBasis ℝ α) i‖:= by
+  rw [← sub_eq_zero]
+  have hz : σ (x - y) = 0 := by rw [map_sub, h, sub_self]
+  have hsum : ∑ i, ‖((Module.finBasis ℝ α).repr (x - y)) i‖ * ‖(Module.finBasis ℝ α) i‖ = 0 := by
+    rw [← l1_norm_eq (x - y), hz, norm_zero]
+  have hrepr : ∀ i, ((Module.finBasis ℝ α).repr (x - y)) i = 0 := by
     intro i
-    repeat rw[Finset.sum_apply];
-    simp only [PiLp.smul_apply]
-    rw[sigma_norm_apply]
-  show z = 0
-  rw[hz]
-  apply Fintype.sum_eq_zero (fun a => (bs.repr z) a • bs a)
-  intro i
-  rw[smul_eq_zero]
-  left
-  have : ‖(Module.finBasis ℝ α) i‖ ≠ 0:= norm_ne_zero_iff.mpr $ Basis.ne_zero (Module.finBasis ℝ α) i
-  have h1 : (bs.repr z) i * ‖(Module.finBasis ℝ α) i‖ = 0 := by
-    rw[← hi , h, PiLp.zero_apply]
-  apply eq_zero_of_ne_zero_of_mul_right_eq_zero this h1
+    have hzero := (Finset.sum_eq_zero_iff_of_nonneg (s := Finset.univ) fun j _ ↦ by positivity).mp hsum i (by simp)
+    exact abs_eq_zero.mp <| eq_zero_of_ne_zero_of_mul_right_eq_zero (norm_ne_zero_iff.mpr <| (Module.finBasis ℝ α).ne_zero i) hzero
+  apply (Module.finBasis ℝ α).repr.injective
+  ext i
+  simpa using hrepr i
diff --git a/Optlib/Function/Lsmooth.lean b/Optlib/Function/Lsmooth.lean
index 1ff68dd..6b86e87 100644
--- a/Optlib/Function/Lsmooth.lean
+++ b/Optlib/Function/Lsmooth.lean
@@ -76,12 +76,12 @@ theorem lipschitz_continuous_upper_bound {E : Type*}
     apply HasDerivAt.add
     · apply HasDerivAt.const_add
       · apply hasDerivAt_mul_const
-    · have : l * ‖y - x‖ ^ 2 * t = (2 * t) * (l * ‖y - x‖ ^ 2 / 2) := by field_simp; ring_nf
+    · have : l * ‖y - x‖ ^ 2 * t = (2 * t) * (l * ‖y - x‖ ^ 2 / 2) := by field_simp
       rw [this]; apply HasDerivAt.mul_const
       obtain hd := HasDerivAt.pow (n := 2) (hasDerivAt_id' t)
       simp at hd; exact hd
   suffices g 1 ≤ u 1 by
-    simp [u, g, u', LL, g'] at this
+    simp [u, g, LL, g'] at this
     rw [map_sub]; linarith
   apply image_le_of_deriv_right_le_deriv_boundary (a := 0) (b := 2)
   · exact HasDerivAt.continuousOn (fun x _ ↦ gderiv x)
@@ -106,7 +106,7 @@ open InnerProductSpace  Set
 
 variable {f : E → ℝ} {a : ℝ} {f' : E → E} {l : NNReal}
 
-theorem lower_to_lipschitz (h₂ : ∀ x y, inner (f' x - f' y) (x - y) ≥ 1 / l * ‖f' x - f' y‖ ^ 2)
+theorem lower_to_lipschitz (h₂ : ∀ x y, inner ℝ (f' x - f' y) (x - y) ≥ 1 / l * ‖f' x - f' y‖ ^ 2)
     (hl : l > 0) : LipschitzWith l f' := by
   rw [lipschitzWith_iff_norm_sub_le]
   intro x y
@@ -120,18 +120,17 @@ theorem lower_to_lipschitz (h₂ : ∀ x y, inner (f' x - f' y) (x - y) ≥ 1 /
       apply real_inner_le_norm
     _ = (1 / l * ‖f' x - f' y‖) * (l * ‖x - y‖) := by
       field_simp
-      ring_nf
   have H₂ : 1 / l > 0 := by
     apply one_div_pos.mpr hl
   cases lt_or_ge 0 (‖f' x - f' y‖)
   case inl h =>
     apply le_of_mul_le_mul_left H₁
     apply mul_pos _ h
-    · simp [H₂, hl]
+    · simpa using H₂
   case inr h =>
     apply le_trans h
     apply mul_nonneg
-    · simp [hl]
+    · simp
     apply norm_nonneg _
 
 end
@@ -145,16 +144,16 @@ variable {f : E → ℝ} {a : ℝ} {f' : E → E} {xm : E} {l : NNReal}
 
 theorem lipschitz_continuos_upper_bound'
     (h₁ : ∀ x₁ : E, HasGradientAt f (f' x₁) x₁) (h₂ : LipschitzWith l f') :
-    ∀ x y : E, f y ≤ f x + inner (f' x) (y - x) + l / 2 * ‖y - x‖ ^ 2 := by
+    ∀ x y : E, f y ≤ f x + inner ℝ (f' x) (y - x) + l / 2 * ‖y - x‖ ^ 2 := by
   intro x y
   rw [lipschitzWith_iff_norm_sub_le] at h₂
   let g := fun x ↦ (toDual ℝ E) (f' x)
   have h' : ∀ x : E, HasFDerivAt f (g x) x := h₁
-  have equiv : ∀ x y : E, inner (f' x) (y - x) = (g x) (y - x) := by
+  have equiv : ∀ x y : E, inner ℝ (f' x) (y - x) = (g x) (y - x) := by
     intro x y
-    rw [InnerProductSpace.toDual_apply]
+    rw [InnerProductSpace.toDual_apply_apply]
   have h₂' : LipschitzWith l g := by
-    simp only [g, equiv]
+    simp only [g]
     rw [lipschitzWith_iff_norm_sub_le]
     intro x y
     have h1 : ∀ x : E, ‖(toDual ℝ E) x‖ =‖x‖ := by
@@ -177,12 +176,17 @@ theorem lipschitz_minima_lower_bound (h₁ : ∀ x : E, HasGradientAt f (f' x) x
   have eq : f xm ≤ f x - 1 / (2 * l) * ‖f' x‖ ^ 2 := by
     calc
       _ ≤ f y := by apply min
-      _ ≤ f x + inner (f' x) (y - x) + l / 2 * ‖y - x‖ ^ 2 := by
+      _ ≤ f x + inner ℝ (f' x) (y - x) + l / 2 * ‖y - x‖ ^ 2 := by
         apply lipschitz_continuos_upper_bound' h₁ h₂
       _ = f x - 1 / (2 * l) * ‖f' x‖ ^ 2 := by
         rw [add_assoc]; rw [sub_eq_add_neg (f x), add_left_cancel_iff.2]
-        field_simp [y]; rw [← real_inner_self_eq_norm_sq, ← real_inner_self_eq_norm_sq]
-        rw [inner_smul_right, inner_smul_left, inner_smul_right]
+        have hyx : y - x = - ((1 : ℝ) / l : ℝ) • f' x := by simp [y]
+        rw [hyx, real_inner_smul_right, real_inner_self_eq_norm_sq]
+        have hlR : (0 : ℝ) < (l : ℝ) := by exact_mod_cast hl
+        have hnorm : ‖-(1 / (l : ℝ)) • f' x‖ ^ 2 = (1 / (l : ℝ)) ^ 2 * ‖f' x‖ ^ 2 := by
+          rw [norm_smul]
+          simp [pow_two, mul_assoc, mul_left_comm, mul_comm]
+        rw [hnorm]
         field_simp; ring_nf
   linarith
 
@@ -202,27 +206,32 @@ theorem lipschitz_to_lnorm_sub_convex (hs : Convex ℝ s)
     ConvexOn ℝ s (fun x ↦ l / 2 * ‖x‖ ^ 2 - f x) := by
   rw [lipschitzOnWith_iff_norm_sub_le] at h₂
   let g' : E → E := fun x ↦ l.1 • x - f' x
-  have H₂ : ∀ x ∈ s, ∀ y ∈ s, inner (g' x - g' y) (x - y) ≥ (0 : ℝ) := by
+  have H₂ : ∀ x ∈ s, ∀ y ∈ s, inner ℝ (g' x - g' y) (x - y) ≥ (0 : ℝ) := by
     intro x hx y hy
+    have hxy :
+        inner ℝ (g' x - g' y) (x - y) = l.1 * ‖x - y‖ ^ 2 - inner ℝ (f' x - f' y) (x - y) := by
+      calc
+        inner ℝ (g' x - g' y) (x - y) = inner ℝ (l.1 • (x - y) - (f' x - f' y)) (x - y) := by
+          simp [g', sub_eq_add_neg, add_assoc, add_left_comm, add_comm]
+        _ = l.1 * inner ℝ (x - y) (x - y) - inner ℝ (f' x - f' y) (x - y) := by
+          rw [inner_sub_left, inner_smul_left]
+          simp
+        _ = l.1 * ‖x - y‖ ^ 2 - inner ℝ (f' x - f' y) (x - y) := by
+          rw [real_inner_self_eq_norm_sq]
     calc
-    _ = l.1 * (inner (x - y) (x - y)) - inner (f' x - f' y) (x - y) := by
-      simp [g']
-      rw [← sub_add, sub_right_comm, sub_add, inner_sub_left, ← smul_sub, inner_smul_left]
-      simp only [conj_trivial]
-    _ = l * ‖x - y‖ ^ 2 - inner (f' x - f' y) (x - y) := by
-      simp; left
-      apply real_inner_self_eq_norm_sq
-    _ ≥ l * ‖x - y‖ ^ 2 - ‖f' x - f' y‖ * ‖x - y‖ := by
-      apply add_le_add; linarith
-      simp
-      apply real_inner_le_norm
-    _ ≥ l * ‖x - y‖ ^ 2 - l * ‖x - y‖ ^ 2 := by
-      simp
-      rw [pow_two, ← mul_assoc]
-      apply mul_le_mul (h₂ hx hy); linarith; apply norm_nonneg
-      apply mul_nonneg _ (norm_nonneg _)
-      simp [hl]
-    _ = 0 := by simp
+      _ = l.1 * ‖x - y‖ ^ 2 - inner ℝ (f' x - f' y) (x - y) := hxy
+      _ = l * ‖x - y‖ ^ 2 - inner ℝ (f' x - f' y) (x - y) := by rfl
+      _ ≥ l * ‖x - y‖ ^ 2 - ‖f' x - f' y‖ * ‖x - y‖ := by
+        apply add_le_add; linarith
+        simp
+        apply real_inner_le_norm
+      _ ≥ l * ‖x - y‖ ^ 2 - l * ‖x - y‖ ^ 2 := by
+        simp
+        rw [pow_two, ← mul_assoc]
+        apply mul_le_mul (h₂ hx hy); linarith; apply norm_nonneg
+        apply mul_nonneg _ (norm_nonneg _)
+        exact le_of_lt hl
+      _ = 0 := by simp
   have H₃ : ∀ x ∈ s, HasGradientAt (fun x ↦ l / 2 * ‖x‖ ^ 2 - f x) (g' x) x := by
     intro x hx
     have u₂ := HasGradientAt.const_smul (gradient_norm_sq_eq_two_self x) ((l / (2 : ℝ)) : ℝ)
@@ -241,9 +250,9 @@ theorem lipschitz_to_lnorm_sub_convex (hs : Convex ℝ s)
 theorem convex_to_lower {l : ℝ} (h₁ : ∀ x : E, HasGradientAt f (f' x) x)
     (h₂ : ConvexOn ℝ Set.univ (fun x ↦ l / 2 * ‖x‖ ^ 2 - f x)) (lp : l > 0)
     (hfun: ConvexOn ℝ Set.univ f) (x : E) (y : E) :
-    inner (f' x - f' y) (x - y) ≥ 1 / l * ‖f' x - f' y‖ ^ 2 := by
+    inner ℝ (f' x - f' y) (x - y) ≥ 1 / l * ‖f' x - f' y‖ ^ 2 := by
   rw [ConvexOn] at hfun
-  let fs : E → (E → ℝ) := fun s => (fun x => f x - inner (f' s) x)
+  let fs : E → (E → ℝ) := fun s => (fun x => f x - inner ℝ (f' s) x)
   have hfunconvex : ∀ s : E, ConvexOn ℝ Set.univ (fs s) := by
     intro s
     rw [ConvexOn]
@@ -281,22 +290,22 @@ theorem convex_to_lower {l : ℝ} (h₁ : ∀ x : E, HasGradientAt f (f' x) x)
       rw [inner_add_right, real_inner_smul_right, real_inner_smul_right]
       calc
         _ = (l / 2) * ‖a • x₁ + b • y₁‖ ^ 2 - f (a • x₁ + b • y₁) +
-            (a * inner (f' s) x₁ + b * inner (f' s) y₁) := by ring_nf
+            (a * inner ℝ (f' s) x₁ + b * inner ℝ (f' s) y₁) := by ring_nf
         _ ≤ a • (l / 2 * ‖x₁‖ ^ 2 - f x₁) + b • (l / 2 * ‖y₁‖ ^ 2 - f y₁) +
-            (a * inner (f' s) x₁ + b * inner (f' s) y₁) := by apply add_le_add_right h₂'
-        _ = a • (l / 2 * ‖x₁‖ ^ 2 - (f x₁ - inner (f' s) x₁)) + b •
-            (l / 2 * ‖y₁‖ ^ 2 - (f y₁ - inner (f' s) y₁)) := by simp; ring_nf
+            (a * inner ℝ (f' s) x₁ + b * inner ℝ (f' s) y₁) := by linarith [h₂']
+        _ = a • (l / 2 * ‖x₁‖ ^ 2 - (f x₁ - inner ℝ (f' s) x₁)) + b •
+            (l / 2 * ‖y₁‖ ^ 2 - (f y₁ - inner ℝ (f' s) y₁)) := by simp; ring_nf
   let gs' := fun s ↦ (fun z ↦ l • z - (fs' s z))
   have hgx₁ :  ∀ s x : E, HasGradientAt (gs s) ((gs' s) x) x := by
     intro s z
     apply HasGradientAt.sub (gradient_of_const_mul_norm l z) (hfconx₁ s z)
-  have hgx₂ : ∀ s z₁ z₂ : E, (gs s) z₁ + inner (gs' s z₁) (z₂ - z₁) ≤ gs s z₂ := by
+  have hgx₂ : ∀ s z₁ z₂ : E, (gs s) z₁ + inner ℝ (gs' s z₁) (z₂ - z₁) ≤ gs s z₂ := by
     intro s z₁ z₂
     apply Convex_first_order_condition' (hgx₁ s z₁) (hgxconvex s)
     · simp only [Set.mem_univ]
     · simp only [Set.mem_univ]
   have hfx₂ : ∀ (s x y₁ : E), (fs s) y₁ ≤ fs s x +
-      inner (fs' s x) (y₁ - x) + l / 2 * ‖y₁ - x‖ ^ 2 := by
+      inner ℝ (fs' s x) (y₁ - x) + l / 2 * ‖y₁ - x‖ ^ 2 := by
     intro s z₁ z₂
     simp only [fs, fs']
     rcases hgx₂ s z₁ z₂ with hgx₂'
@@ -304,30 +313,30 @@ theorem convex_to_lower {l : ℝ} (h₁ : ∀ x : E, HasGradientAt f (f' x) x)
     have t₈ : gs s z₂ =  l / 2 * ‖z₂‖ ^ 2 - fs s z₂ := by rfl
     have t₉ : gs' s z₁ = l • z₁ - fs' s z₁ := by rfl
     rw [t₇, t₈, t₉] at hgx₂'
-    have t₁₀ : fs s z₂ + (l / 2 * ‖z₁‖ ^ 2 - fs s z₁ + inner (l • z₁ - fs' s z₁) (z₂ - z₁))
+    have t₁₀ : fs s z₂ + (l / 2 * ‖z₁‖ ^ 2 - fs s z₁ + inner ℝ (l • z₁ - fs' s z₁) (z₂ - z₁))
         ≤ l / 2 * ‖z₂‖ ^ 2 := by
       apply add_le_of_le_sub_left hgx₂'
     have t₁₁ : fs s z₂ ≤ l / 2 * ‖z₂‖ ^ 2 - (l / 2 * ‖z₁‖ ^ 2 - fs s z₁ +
-        inner (l • z₁ - fs' s z₁) (z₂ - z₁)) := by
+        inner ℝ (l • z₁ - fs' s z₁) (z₂ - z₁)) := by
       rw [add_comm] at t₁₀
       apply le_sub_left_of_add_le t₁₀
     simp only [] at t₁₁; rw [← sub_add (l / 2 * ‖z₁‖ ^ 2) _ _] at t₁₁
     calc
       _ ≤ l / 2 * ‖z₂‖ ^ 2 - (l / 2 * ‖z₁‖ ^ 2 - f z₁ +
-          inner (f' s) z₁ + inner (l • z₁ - fs' s z₁) (z₂ - z₁)) := by apply t₁₁
-      _ = l / 2 * ‖z₂‖ ^ 2 -(l / 2 * ‖z₁‖ ^ 2 - f z₁ + inner (f' s) z₁ +
-        (l * (inner z₁ z₂ - ‖z₁‖ ^ 2) - inner (f' z₁ - f' s) (z₂ - z₁))) := by
+          inner ℝ (f' s) z₁ + inner ℝ (l • z₁ - fs' s z₁) (z₂ - z₁)) := by apply t₁₁
+      _ = l / 2 * ‖z₂‖ ^ 2 -(l / 2 * ‖z₁‖ ^ 2 - f z₁ + inner ℝ (f' s) z₁ +
+        (l * (inner ℝ z₁ z₂ - ‖z₁‖ ^ 2) - inner ℝ (f' z₁ - f' s) (z₂ - z₁))) := by
         rw [inner_sub_left, inner_smul_left]
-        simp; rw [inner_sub_right, real_inner_self_eq_norm_sq];left ; simp
-      _ = f z₁ - inner (f' s) z₁ + inner (f' z₁ - f' s) (z₂ - z₁) +
-        l / 2 * (‖z₂‖ ^ 2  - 2 * inner z₂ z₁ + ‖z₁‖ ^ 2) := by
+        simp; rw [inner_sub_right, real_inner_self_eq_norm_sq]
+      _ = f z₁ - inner ℝ (f' s) z₁ + inner ℝ (f' z₁ - f' s) (z₂ - z₁) +
+        l / 2 * (‖z₂‖ ^ 2  - 2 * inner ℝ z₂ z₁ + ‖z₁‖ ^ 2) := by
           field_simp; ring_nf; rw [real_inner_comm]
-      _ = f z₁ - inner (f' s) z₁ + inner (f' z₁ - f' s) (z₂ - z₁) + l / 2 * ‖z₂ - z₁‖ ^ 2 := by
+      _ = f z₁ - inner ℝ (f' s) z₁ + inner ℝ (f' z₁ - f' s) (z₂ - z₁) + l / 2 * ‖z₂ - z₁‖ ^ 2 := by
         rw [← norm_sub_sq_real]
   have hfs₃ : ∀ s : E, IsMinOn (fs s) univ s := by
     intro s
     apply first_order_convex (hfconx₁ s) (hfunconvex s)
-    simp only [fs, fs', sub_self]
+    simp only [fs', sub_self]
   have hfy₃ : IsMinOn (fs y) _ y := hfs₃ y
   have hfx₄ : fs x x ≤ fs x y - 1 / (2 * l) * ‖fs' x y‖ ^ 2 := by
     have : fs x x ≤ fs x (y - (1 / l) • fs' x y) := by
@@ -338,7 +347,7 @@ theorem convex_to_lower {l : ℝ} (h₁ : ∀ x : E, HasGradientAt f (f' x) x)
     apply le_trans this
     rcases hfx₂ x y (y - (1 / l) • fs' x y) with hfx₂'
     calc
-      _ ≤ fs x y + inner (fs' x y) (y - (1 / l) • fs' x y - y)
+      _ ≤ fs x y + inner ℝ (fs' x y) (y - (1 / l) • fs' x y - y)
           + l / 2 * ‖y - (1 / l) • fs' x y - y‖ ^ 2 := by apply hfx₂'
       _ = fs x y - 1 / (2 * l) * ‖fs' x y‖ ^ 2 := by
         have : y - (1 / l) • fs' x y - y = - (1 / l) • fs' x y := by simp
@@ -354,14 +363,14 @@ theorem convex_to_lower {l : ℝ} (h₁ : ∀ x : E, HasGradientAt f (f' x) x)
     apply le_trans this
     rcases hfx₂ y x (x - (1 / l) • fs' y x) with hfy₂'
     calc
-      _ ≤ fs y x + inner (fs' y x) (x - (1 / l) • fs' y x - x)
+      _ ≤ fs y x + inner ℝ (fs' y x) (x - (1 / l) • fs' y x - x)
           + l / 2 * ‖x - (1 / l) • fs' y x - x‖ ^ 2 := by apply hfy₂'
       _ = fs y x - 1 / (2 * l) * ‖fs' y x‖ ^ 2 := by
         have : x - (1 / l) • fs' y x - x = - (1 / l) • fs' y x := by simp
         rw [this, real_inner_smul_right]
         rw [← real_inner_self_eq_norm_sq, ← real_inner_self_eq_norm_sq, real_inner_smul_right]
         rw [real_inner_smul_left]; field_simp; ring
-  have hh₁: (1 / (2 * l)) * ‖f' x - f' y‖ ^ 2 ≤ f y - f x - inner (f' x) (y - x) := by
+  have hh₁: (1 / (2 * l)) * ‖f' x - f' y‖ ^ 2 ≤ f y - f x - inner ℝ (f' x) (y - x) := by
     calc
       (1 / (2 * l)) * ‖f' x - f' y‖ ^ 2 ≤ fs x y - fs x x := by
         have : f' x - f' y = - fs' x y := by
@@ -371,20 +380,20 @@ theorem convex_to_lower {l : ℝ} (h₁ : ∀ x : E, HasGradientAt f (f' x) x)
         have : ‖- fs' x y‖  = ‖fs' x y‖  :=by apply norm_neg
         rw [this]
         linarith [hfx₄]
-      _ = f y - f x - inner (f' x) (y - x) := by
-        have t₄: fs x y = f y - inner (f' x) y := by rfl
-        have t₅: fs x x = f x - inner (f' x) x := by rfl
+      _ = f y - f x - inner ℝ (f' x) (y - x) := by
+        have t₄: fs x y = f y - inner ℝ (f' x) y := by rfl
+        have t₅: fs x x = f x - inner ℝ (f' x) x := by rfl
         rw [t₄,t₅,inner_sub_right]
         ring
-  have hh₂: (1 / (2 * l)) * ‖f' x - f' y‖ ^ 2 ≤ f x - f y - inner (f' y) (x - y) := by
+  have hh₂: (1 / (2 * l)) * ‖f' x - f' y‖ ^ 2 ≤ f x - f y - inner ℝ (f' y) (x - y) := by
     calc
       (1 / (2 * l)) * ‖f' x - f' y‖ ^ 2 ≤ fs y x -fs y y := by
-        have : f' x - f' y = fs' y x := by simp
+        have : f' x - f' y = fs' y x := by simp [fs']
         rw [this]
         linarith [hfy₄]
-      _ = f x - f y - inner (f' y) (x - y) := by
-        have t₄' : fs y y = f y - inner (f' y) y := by rfl
-        have t₅' : fs y x = f x - inner (f' y) x := by rfl
+      _ = f x - f y - inner ℝ (f' y) (x - y) := by
+        have t₄' : fs y y = f y - inner ℝ (f' y) y := by rfl
+        have t₅' : fs y x = f x - inner ℝ (f' y) x := by rfl
         rw [t₄', t₅', inner_sub_right]
         ring
   calc
@@ -392,24 +401,24 @@ theorem convex_to_lower {l : ℝ} (h₁ : ∀ x : E, HasGradientAt f (f' x) x)
       field_simp
       rw [← mul_two,mul_comm]
       ring
-    _ ≤ (f y - f x - inner (f' x) (y - x)) + (f x - f y - inner (f' y) (x - y)) := by
+    _ ≤ (f y - f x - inner ℝ (f' x) (y - x)) + (f x - f y - inner ℝ (f' y) (x - y)) := by
       apply add_le_add hh₁ hh₂
-    _ = inner (f' x - f' y) (x - y) := by
+    _ = inner ℝ (f' x - f' y) (x - y) := by
       rw [inner_sub_left]
-      have t₆ : (inner (f' x) (y - x) : ℝ) = - (inner (f' x) (x - y) : ℝ) := by
+      have t₆ : (inner ℝ (f' x) (y - x) : ℝ) = - (inner ℝ (f' x) (x - y) : ℝ) := by
         rw [inner_sub_right, inner_sub_right]; ring
       rw[t₆]; ring
 
 theorem lipschitz_to_lower (h₁ : ∀ x, HasGradientAt f (f' x) x) (h₂ : LipschitzWith l f')
     (hfun : ConvexOn ℝ Set.univ f) (hl : l > 0) :
-    ∀ x y, inner (f' x - f' y) (x - y) ≥ 1 / l * ‖f' x - f' y‖ ^ 2 := by
+    ∀ x y, inner ℝ (f' x - f' y) (x - y) ≥ 1 / l * ‖f' x - f' y‖ ^ 2 := by
   obtain convex : ConvexOn ℝ Set.univ (fun x ↦ l / 2 * ‖x‖ ^ 2 - f x) :=
     lipschitz_to_lnorm_sub_convex convex_univ (fun x _ => h₁ x) (lipschitzOnWith_univ.mpr h₂) hl
   exact convex_to_lower h₁ convex hl hfun
 
 theorem lower_iff_lipschitz (h₁ : ∀ x, HasGradientAt f (f' x) x) (hfun: ConvexOn ℝ Set.univ f)
     (hl : l > 0) : LipschitzWith l f' ↔
-    ∀ x y, inner (f' x - f' y) (x - y) ≥ 1 / l * ‖f' x - f' y‖ ^ 2 :=
+    ∀ x y, inner ℝ (f' x - f' y) (x - y) ≥ 1 / l * ‖f' x - f' y‖ ^ 2 :=
   ⟨fun h ↦ lipschitz_to_lower h₁ h hfun hl, fun h ↦ lower_to_lipschitz h hl⟩
 
 theorem lipshictz_iff_lnorm_sub_convex (h₁ : ∀ x, HasGradientAt f (f' x) x)
@@ -424,7 +433,7 @@ theorem lipshictz_iff_lnorm_sub_convex (h₁ : ∀ x, HasGradientAt f (f' x) x)
 
 theorem lower_iff_lnorm_sub_convex (h₁ : ∀ x, HasGradientAt f (f' x) x)
     (hfun: ConvexOn ℝ Set.univ f) (hl : l > 0) : ConvexOn ℝ univ (fun x ↦ l / 2 * ‖x‖ ^ 2 - f x)
-    ↔ ∀ x y, inner (f' x - f' y) (x - y) ≥ 1 / l * ‖f' x - f' y‖ ^ 2  := by
+    ↔ ∀ x y, inner ℝ (f' x - f' y) (x - y) ≥ 1 / l * ‖f' x - f' y‖ ^ 2  := by
   rw [← lipshictz_iff_lnorm_sub_convex h₁ hfun hl]
   rw [lower_iff_lipschitz h₁ hfun hl]
 
diff --git a/Optlib/Function/MinimaClosedFunction.lean b/Optlib/Function/MinimaClosedFunction.lean
index 44c01dc..50a4662 100644
--- a/Optlib/Function/MinimaClosedFunction.lean
+++ b/Optlib/Function/MinimaClosedFunction.lean
@@ -6,7 +6,7 @@ Authors: Wanyi He
 import Mathlib.Analysis.Convex.Basic
 import Mathlib.Topology.MetricSpace.Basic
 import Mathlib.Topology.MetricSpace.Bounded
-import Mathlib.Topology.Semicontinuous
+import Mathlib.Topology.Semicontinuity.Basic
 import Mathlib.Topology.Sequences
 
 /-!
@@ -16,6 +16,8 @@ variable {E F : Type*}
 
 open Set Bornology Topology Filter TopologicalSpace
 
+class LinearOrderedRing (R : Type*) extends Ring R, LinearOrder R, IsStrictOrderedRing R
+
 section preparation
 
 variable {E F : Type*} [CompleteLinearOrder F]
@@ -33,15 +35,11 @@ private lemma l0 {f : E → F}(y : F) (h : (f ⁻¹' Set.Iic y).Nonempty) :
       · exact Exists.intro x xeq
       · exact Exists.intro x xeq
   have h₁ : sInf {f x | x ∈ f ⁻¹' Set.Iic y} ≤ sInf {f x | x ∈ (f ⁻¹' Set.Iic y)ᶜ} := by
-    apply sInf_le_sInf_of_forall_exists_le
-    intro y' ynsub
-    rcases h with ⟨x', xsub⟩; use f x'
-    constructor
-    · exact ⟨x', xsub, rfl⟩
-    rcases ynsub with ⟨x, xnsub, xeq⟩
-    apply le_trans xsub (Eq.trans_ge xeq (le_of_lt _))
-    simp only [← Set.preimage_setOf_eq, ← Set.preimage_compl, Set.compl_Iic, Set.Ioi_def] at xnsub
-    assumption
+    rcases h with ⟨x', xsub⟩
+    refine le_sInf ?_
+    rintro y' ⟨x, xnsub, rfl⟩
+    exact le_trans (sInf_le ⟨x', xsub, rfl⟩)
+      (le_trans (show f x' ≤ y from xsub) (le_of_lt (lt_of_not_ge xnsub)))
   calc
     sInf {f x | x ∈ f ⁻¹' Set.Iic y} =
       sInf {f x | x ∈ f ⁻¹' Set.Iic y} ⊓ sInf {f x | x ∈ (f ⁻¹' Set.Iic y)ᶜ} :=
@@ -65,32 +63,24 @@ variable [FirstCountableTopology E] [FirstCountableTopology F]
 
 /- If a premiage of `f` is nonempty and compact,
   then its minimum point set `{x | IsMinOn f univ x}` is nonempty -/
+omit [DenselyOrdered F] [TopologicalSpace F] [OrderTopology F] [FirstCountableTopology E]
+  [FirstCountableTopology F] in
 theorem IsMinOn.of_isCompact_preimage (hf : LowerSemicontinuous f) {y : F}
     (h1 : (f ⁻¹' Set.Iic y).Nonempty) (h2 : IsCompact (f ⁻¹' Set.Iic y)) :
     ∃ x, IsMinOn f univ x := by
-  have hs : Set.Nonempty {f x | x ∈ (f ⁻¹' Set.Iic y)} := by
-    rcases h1 with ⟨x, xsub⟩
-    exact Exists.intro (f x) (Exists.intro x ⟨xsub, rfl⟩)
-  have hs' : BddBelow {f x | x ∈ (f ⁻¹' Set.Iic y)} :=
-    OrderBot.bddBelow {x | ∃ x_1 ∈ f ⁻¹' Iic y, f x_1 = x}
-  rcases exists_seq_tendsto_sInf hs hs' with ⟨fx, _, cfx, fxs⟩
-  choose x xsub xeq using fxs
-  rcases IsCompact.tendsto_subseq h2 xsub with ⟨x', xsub', k, mono, cxk⟩
-  have cfxk : Tendsto (f ∘ x ∘ k) atTop (𝓝 (sInf {f x | x ∈ (f ⁻¹' Set.Iic y)})) := by
-    have xkeq : ∀ (n : ℕ), (f ∘ x ∘ k) n = (fx ∘ k) n := fun n => xeq <| k n
-    rw [tendsto_congr xkeq]
-    apply Tendsto.comp cfx (StrictMono.tendsto_atTop mono)
-  have inepi : (x', sInf {f x | x ∈ (f ⁻¹' Set.Iic y)}) ∈ {p : E × F | f p.1 ≤ p.2} :=
-    (IsClosed.isSeqClosed (LowerSemicontinuous.isClosed_epigraph hf))
-      (fun n => Eq.le (by rfl)) (Tendsto.prod_mk_nhds cxk cfxk)
-  use x'; intro xx _
-  apply le_of_eq_of_le
-  · apply le_antisymm inepi (sInf_le (Exists.intro x' ⟨xsub', rfl⟩))
-  · apply le_of_eq_of_le (l0 y h1) (sInf_le (by use xx))
+  obtain ⟨x, hx, hxmin⟩ :=
+    LowerSemicontinuousOn.exists_isMinOn h1 h2 (hf.lowerSemicontinuousOn _)
+  have hxmin' : ∀ z ∈ f ⁻¹' Set.Iic y, f x ≤ f z := by simpa [isMinOn_iff] using hxmin
+  refine ⟨x, fun z _ => ?_⟩
+  by_cases hz : z ∈ f ⁻¹' Set.Iic y
+  · exact hxmin' z hz
+  · exact le_trans hx (le_of_lt (lt_of_not_ge hz))
 
 /- If a premiage of `f` is nonempty and compact,
   then its minimum point set `{x | IsMinOn f univ x}` is compact -/
 
+omit [DenselyOrdered F] [TopologicalSpace F] [OrderTopology F] [FirstCountableTopology E]
+  [FirstCountableTopology F] in
 theorem IsCompact_isMinOn_of_isCompact_preimage (hf : LowerSemicontinuous f) {y : F}
     (h1 : (f ⁻¹' Set.Iic y).Nonempty) (h2 : IsCompact (f ⁻¹' Set.Iic y)) :
     IsCompact {x | IsMinOn f univ x} := by
@@ -129,10 +119,9 @@ theorem isMinOn_unique {x y : E} (hf' : strong_quasi f 𝕜)
   have eqone : a + (1 - a) = 1 := add_sub_cancel a 1
   have lta' : 0 < 1 - a := sub_pos_of_lt alt
   have h : f (a • x + (1 - a) • y) < f y := by
-    apply Eq.trans_gt (max_eq_right (hx trivial))
-    apply hf' neq lta lta' eqone
+    simpa [max_eq_right (hx trivial)] using hf' neq lta lta' eqone
   simp only [isMinOn_iff] at hy
   specialize hy (a • x + (1 - a) • y) trivial
-  apply not_le_of_lt h hy
+  exact (not_le_of_gt h) hy
 
 end
diff --git a/Optlib/Function/Proximal.lean b/Optlib/Function/Proximal.lean
index 2ad865f..704230c 100644
--- a/Optlib/Function/Proximal.lean
+++ b/Optlib/Function/Proximal.lean
@@ -3,7 +3,7 @@ Copyright (c) 2024 Shengyang Xu, Chenyi Li. All rights reserved.
 Released under Apache 2.0 license as described in the file LICENSE.
 Authors: Shengyang Xu, Chenyi Li
 -/
-import Mathlib.Topology.Semicontinuous
+import Mathlib.Topology.Semicontinuity.Basic
 import Mathlib.Analysis.Convex.Basic
 import Optlib.Convex.Subgradient
 import Optlib.Function.Lsmooth
@@ -66,9 +66,11 @@ theorem prox_set_compact_of_lowersemi (f : E → ℝ) (hc : LowerSemicontinuous
   have hg : LowerSemicontinuous g := by
     apply LowerSemicontinuous.add hc
     apply Continuous.lowerSemicontinuous
-    apply continuous_iff_continuousOn_univ.2
-    apply HasGradientAt.continuousOn
-    intro u _; apply gradient_of_sq u
+    have hcont : ContinuousOn (fun u : E ↦ ‖u - x‖ ^ 2 / 2) univ := by
+      apply HasGradientAt.continuousOn
+      intro u _
+      exact gradient_of_sq (x := x) u
+    exact (continuousOn_univ.mp hcont)
   have S_bddbelow : BddBelow ImS := by
     use L; rw [mem_lowerBounds]
     rintro gy ⟨y0, _, gyeq⟩; rw [← gyeq]; exact boundg y0
@@ -101,7 +103,7 @@ theorem prox_set_compact_of_lowersemi (f : E → ℝ) (hc : LowerSemicontinuous
               rw [add_right_comm, mul_comm]; simp; linarith [sq_nonneg b]
           calc
             0 ≤ a ^ 2 / 2 := by linarith [sq_nonneg a]
-            _ ≤ b * 2 / 2 := by rw [div_le_div_right]; exact h1; linarith
+            _ ≤ b * 2 / 2 := by nlinarith [h1]
             _ ≤ b + 1 := by simp
           linarith
         apply aux ieq
@@ -119,10 +121,12 @@ theorem prox_set_compact_of_lowersemi (f : E → ℝ) (hc : LowerSemicontinuous
     apply Tendsto.comp cfx (StrictMono.tendsto_atTop mono)
   have inepi : (x', sInf ImS) ∈ epi := by
     let p := fun c ↦ (((fun n ↦ xn n) ∘ k) c, (g ∘ xn ∘ k) c)
-    have pnin :  ∀ c : ℕ, p c ∈ epi := by simp [epi]
+    have pnin :  ∀ c : ℕ, p c ∈ epi := by
+      intro c
+      simp [p, epi]
     apply IsClosed.isSeqClosed epi_closed pnin
     show Tendsto (fun c ↦ (((fun n ↦ xn n) ∘ k) c, (g ∘ xn ∘ k) c)) atTop (𝓝 (x', sInf ImS))
-    apply Tendsto.prod_mk_nhds cxk cfxk
+    simpa [nhds_prod_eq] using (Tendsto.prodMk cxk cfxk)
   have minima_ieq : g x' ≤ sInf ImS := inepi
   have minima : ∀ w : E, g x' ≤ g w := by
     intro w
@@ -134,7 +138,7 @@ theorem prox_set_compact_of_lowersemi (f : E → ℝ) (hc : LowerSemicontinuous
       linarith
     · have gwnin : g x < g w := by
         simp [g, S] at hw; simp [g]; exact hw
-      have gxin : g x ∈ ImS := by use x; simp [g, ImS, S]
+      have gxin : g x ∈ ImS := by use x; simp [g, S]
       have legw : sInf ImS ≤ g w := by
         rw [Real.sInf_le_iff S_bddbelow neImS]
         intro _ epos; use g x; use gxin; linarith
@@ -170,44 +174,45 @@ theorem prox_set_compact_of_convex (f : E → ℝ) (hc : ContinuousOn f univ)
   have subd: ∃ z : E, Nonempty (SubderivAt f z) := by
     use x; apply SubderivAt.nonempty hconv hc
   have hc : LowerSemicontinuous f :=
-    Continuous.lowerSemicontinuous (continuous_iff_continuousOn_univ.mpr hc)
+    Continuous.lowerSemicontinuous <| by simpa [continuousOn_univ] using hc
   rcases subd with ⟨z, a, ain⟩
   rw [← mem_SubderivAt, HasSubgradientAt] at ain
   let g := fun u ↦ f u + ‖u - x‖ ^ 2 / 2
   let epi := {p : (E × ℝ) | g p.1 ≤ p.2}
-  have second_lower_bound (y : E) : g y ≥ f z + inner a (y - z) + ‖y - x‖ ^ 2 / 2 := by
+  have second_lower_bound (y : E) : g y ≥ f z + inner ℝ a (y - z) + ‖y - x‖ ^ 2 / 2 := by
     simp [g]
     specialize ain y; linarith
-  have lower_bound (y : E) : f z + inner a (x - z) - ‖a‖ ^ 2 / 2 ≤ g y := by
+  have lower_bound (y : E) : f z + inner ℝ a (x - z) - ‖a‖ ^ 2 / 2 ≤ g y := by
     have : y - z = x - z + (y - x) := by simp
     specialize second_lower_bound y
     rw [this, inner_add_right, ← add_assoc, add_assoc] at second_lower_bound
-    have : 0 ≤ ‖a‖ ^ 2 / 2 + inner a (y - x) + ‖y - x‖ ^ 2 / 2 := by
+    have : 0 ≤ ‖a‖ ^ 2 / 2 + inner ℝ a (y - x) + ‖y - x‖ ^ 2 / 2 := by
       field_simp; rw [mul_comm, ← norm_add_sq_real]
-      apply div_nonneg (sq_nonneg ‖a + (y - x)‖)
-      norm_num
+      nlinarith [sq_nonneg ‖a + (y - x)‖]
     calc
-      f z + inner a (x - z) - ‖a‖ ^ 2 / 2 ≤ f z + inner a (x - z) - ‖a‖ ^ 2 / 2 +
-        (‖a‖ ^ 2 / 2 + inner a (y - x) + ‖y - x‖ ^ 2 / 2) := le_add_of_nonneg_right this
-      _ = f z + inner a (x - z) + (inner a (y - x) + ‖y - x‖ ^ 2 / 2) := by ring
+      f z + inner ℝ a (x - z) - ‖a‖ ^ 2 / 2 ≤ f z + inner ℝ a (x - z) - ‖a‖ ^ 2 / 2 +
+        (‖a‖ ^ 2 / 2 + inner ℝ a (y - x) + ‖y - x‖ ^ 2 / 2) := le_add_of_nonneg_right this
+      _ = f z + inner ℝ a (x - z) + (inner ℝ a (y - x) + ‖y - x‖ ^ 2 / 2) := by ring
       _ ≤ g y := second_lower_bound
   have hg : LowerSemicontinuous g := by
     apply LowerSemicontinuous.add hc
     apply Continuous.lowerSemicontinuous
-    apply continuous_iff_continuousOn_univ.2
-    apply HasGradientAt.continuousOn
-    intro u _; apply gradient_of_sq u
+    have hcont : ContinuousOn (fun u : E ↦ ‖u - x‖ ^ 2 / 2) univ := by
+      apply HasGradientAt.continuousOn
+      intro u _
+      exact gradient_of_sq (x := x) u
+    exact (continuousOn_univ.mp hcont)
   have epi_closed : IsClosed epi := by
     apply bounded_lowersemicontinuous_to_epi_closed
     · exact lowerSemicontinuousOn_univ_iff.2 hg
-    use (f z + inner a (x - z) - ‖a‖ ^ 2 / 2)
+    use (f z + inner ℝ a (x - z) - ‖a‖ ^ 2 / 2)
   let S := {y : E| g y ≤ g z}
   have eq : S = (g ⁻¹' Set.Iic (g z)) := by constructor
   let ImS := {g y | y ∈ S}
   have neImS : Set.Nonempty ImS := by
     use g z; simp [ImS, S]; use z
   have S_bddbelow : BddBelow ImS := by
-    use (f z + inner a (x - z) - ‖a‖ ^ 2 / 2)
+    use (f z + inner ℝ a (x - z) - ‖a‖ ^ 2 / 2)
     rw [mem_lowerBounds]
     rintro gy ⟨y0, _, gyeq⟩
     rw [← gyeq]; exact lower_bound y0
@@ -222,22 +227,24 @@ theorem prox_set_compact_of_convex (f : E → ℝ) (hc : ContinuousOn f univ)
       simp [S] at uin
       apply mem_closedBall_iff_norm.2
       have norm_bound: ‖u - (x - a)‖ ≤ ‖z - (x - a)‖ + 2 := by
-        have ieq : f z + inner a (u - z) + ‖u - x‖ ^ 2 / 2 ≤ f z + ‖z - x‖ ^ 2 / 2 + 1 := by
+        have ieq : f z + inner ℝ a (u - z) + ‖u - x‖ ^ 2 / 2 ≤ f z + ‖z - x‖ ^ 2 / 2 + 1 := by
           calc
-            f z + inner a (u - z) + ‖u - x‖ ^ 2 / 2 ≤ g u := second_lower_bound u
+            f z + inner ℝ a (u - z) + ‖u - x‖ ^ 2 / 2 ≤ g u := second_lower_bound u
             _ ≤ f z + ‖z - x‖ ^ 2 / 2 := uin
             _ ≤ f z + ‖z - x‖ ^ 2 / 2 + 1 := by linarith
         rw [add_assoc, add_assoc, add_le_add_iff_left] at ieq
-        have eq : inner a (u - z) + ‖u - x‖ ^ 2 / 2 =
-            (‖u - (x - a)‖ ^ 2 - ‖a‖ ^ 2 + 2 * inner (x - z) a) / 2 := by
+        have eq : inner ℝ a (u - z) + ‖u - x‖ ^ 2 / 2 =
+            (‖u - (x - a)‖ ^ 2 - ‖a‖ ^ 2 + 2 * inner ℝ (x - z) a) / 2 := by
           field_simp; rw [← sub_add, norm_add_sq_real]; ring_nf
           rw [add_assoc, ← add_mul, ← inner_add_left, add_comm, real_inner_comm]; simp
         rw [eq] at ieq
         have ieq2 : ‖u - (x - a)‖ ^ 2 ≤ ‖z - (x - a)‖ ^ 2 + 2 := by
-          field_simp at ieq; rw [div_le_div_right, sub_add, sub_le_iff_le_add] at ieq
+          field_simp at ieq
+          rw [sub_add, sub_le_iff_le_add] at ieq
           rw [add_right_comm, add_comm (‖z - x‖ ^ 2), norm_sub_rev z x] at ieq
           rw [real_inner_comm, ← norm_sub_sq_real, ← sub_add a, sub_add_comm] at ieq
-          rw [sub_add] at ieq; exact ieq; norm_num
+          rw [sub_add] at ieq
+          exact ieq
         have : |‖z - (x - a)‖ + 2| = ‖z - (x - a)‖ + 2 := by
           apply abs_of_pos; apply add_pos_of_nonneg_of_pos (norm_nonneg (z - (x - a)))
           simp
@@ -260,10 +267,12 @@ theorem prox_set_compact_of_convex (f : E → ℝ) (hc : ContinuousOn f univ)
     apply Tendsto.comp cfx (StrictMono.tendsto_atTop mono)
   have inepi : (x', sInf ImS) ∈ epi := by
     let p := fun c ↦ (((fun n ↦ xn n) ∘ k) c, (g ∘ xn ∘ k) c)
-    have pnin :  ∀ c : ℕ, p c ∈ epi := by simp [epi]
+    have pnin :  ∀ c : ℕ, p c ∈ epi := by
+      intro c
+      simp [p, epi]
     apply IsClosed.isSeqClosed epi_closed pnin
     show Tendsto (fun c ↦ (((fun n ↦ xn n) ∘ k) c, (g ∘ xn ∘ k) c)) atTop (𝓝 (x', sInf ImS))
-    apply Tendsto.prod_mk_nhds cxk cfxk
+    simpa [nhds_prod_eq] using (Tendsto.prodMk cxk cfxk)
   have minima_ieq : g x' ≤ sInf ImS := inepi
   have minima : ∀ w : E, g x' ≤ g w := by
       intro w
@@ -275,7 +284,7 @@ theorem prox_set_compact_of_convex (f : E → ℝ) (hc : ContinuousOn f univ)
         linarith
       · have gwnin : g z < g w := by
           simp [S] at hw; simp [g]; exact hw
-        have gzin : g z ∈ ImS := by use z; simp [ImS, S]
+        have gzin : g z ∈ ImS := by use z; simp [S]
         have legw : sInf ImS ≤ g w := by
           rw [Real.sInf_le_iff S_bddbelow neImS]
           intro _ epos; use g z; use gzin; linarith
@@ -359,39 +368,16 @@ theorem prox_unique_of_convex (f : E → ℝ) (x : E) (hfun : ConvexOn ℝ univ
 -/
 lemma convex_of_norm_sq {s : Set E} (x : E) (conv: Convex ℝ s) :
     ConvexOn ℝ s (fun (u : E) ↦ ‖u - x‖ ^ 2 / 2) := by
-  rw [ConvexOn]; use conv
+  let g : E → ℝ := fun u ↦ ‖u - x‖ ^ 2 / 2
+  have hstrong : StrongConvexOn univ (1 : ℝ) g := by
+    simpa [g] using strongconvex_of_convex_add_sq (fun _ : E ↦ (0 : ℝ)) x
+      (by simpa [ConvexOn] using (convex_univ : Convex ℝ (univ : Set E)))
+  have hstrong0 : StrongConvexOn univ (0 : ℝ) g := by
+    exact StrongConvexOn.mono (show (0 : ℝ) ≤ 1 by norm_num) hstrong
+  have huniv : ConvexOn ℝ univ g := (strongConvexOn_zero.mp hstrong0)
+  refine ⟨conv, ?_⟩
   intro y _ z _ a b anneg bnneg absum1
-  field_simp
-  have eq1 : a • y + b • z - x = a • (y - x) + b • (z - x) := by
-    rw [smul_sub, smul_sub, add_comm_sub, sub_sub, ← add_smul, add_comm b a]
-    rw [absum1, one_smul, ← add_sub]
-  rw [eq1]
-  have ieq1 (u v : E) : ‖a • u + b • v‖ ^ 2 / 2 ≤ (a * ‖u‖ ^ 2 + b * ‖v‖ ^ 2) / 2 := by
-    rw [div_le_div_right, norm_add_sq_real, add_comm, ← add_assoc]
-    rw [norm_smul, norm_smul, mul_pow, mul_pow]; simp
-    nth_rw 3 [← mul_one a]; nth_rw 3 [← one_mul b]
-    rw [← absum1]; ring_nf; rw [add_right_comm]
-    apply add_le_add_right
-    rw [add_comm]; apply add_le_add_right
-    calc
-      inner (a • u) (b • v) * 2 ≤ ‖a • u‖ * ‖b • v‖ * 2 := by
-        rw [mul_le_mul_right]
-        apply real_inner_le_norm
-        simp
-      _ = a * b * (2 * ‖u‖ * ‖v‖) := by
-        rw [norm_smul, norm_smul]; simp
-        rw [abs_of_nonneg anneg, abs_of_nonneg bnneg]; ring
-      _ ≤ a * b * (‖u‖ ^ 2 + ‖v‖ ^ 2) := by
-        by_cases a * b > 0
-        · rw [mul_le_mul_left]
-          apply two_mul_le_add_pow_two
-          linarith
-        · have ieq2 : 0 ≤ a * b := by apply mul_nonneg anneg bnneg
-          have ieq3 : 0 = a * b := by linarith
-          rw [← ieq3]; simp
-      _ = b * ‖v‖ ^ 2 * a + b * a * ‖u‖ ^ 2 := by ring
-    simp
-  apply ieq1
+  exact huniv.2 (by simp) (by simp) anneg bnneg absum1
 
 /-
   Sub-derivative at x equal to sub-derivative within univ at x
@@ -419,26 +405,30 @@ theorem proximal_shift (a : E) {t : ℝ} (tnz : t ≠ 0) (f : E → ℝ):
   simp
   constructor
   · intro cond y
+    have htsq : 0 < t ^ 2 := sq_pos_iff.mpr tnz
     specialize cond (t⁻¹ • (y - a))
-    rw [← smul_assoc, smul_eq_mul, mul_inv_cancel₀] at cond
+    rw [← smul_assoc, smul_eq_mul, mul_inv_cancel₀ tnz] at cond
     simp at cond
     calc
       t ^ 2 * f (t • z + a) + ‖t • z - t • x‖ ^ 2 / 2 =
           t ^ 2 * (f (t • z + a) + ‖z - x‖ ^ 2 / 2) := by
-        rw [← smul_sub, norm_smul, mul_pow, mul_add]; field_simp
+        rw [← smul_sub, norm_smul, mul_pow, Real.norm_eq_abs, sq_abs, mul_add]
+        ring_nf
       _ ≤ t ^ 2 * (f y + ‖t⁻¹ • (y - a) - x‖ ^ 2 / 2) := by
-        rw [mul_le_mul_left]; use cond; rw [sq_pos_iff]; use tnz
+        exact mul_le_mul_of_nonneg_left cond (le_of_lt htsq)
       _ = t ^ 2 * f y + ‖t • ((1 / t) • (y - a) - x)‖ ^ 2 / 2 := by
-        rw [mul_add, norm_smul, mul_pow]; field_simp
+        rw [mul_add, norm_smul, mul_pow, Real.norm_eq_abs, sq_abs]
+        ring_nf
       _ = t ^ 2 * f y + ‖y - (t • x + a)‖ ^ 2 / 2 := by
-        rw [smul_sub, ← smul_assoc, smul_eq_mul, ← sub_sub, sub_right_comm]; field_simp
-    use tnz
+        rw [smul_sub, ← smul_assoc, smul_eq_mul, ← sub_sub, sub_right_comm]
+        field_simp [tnz]
+        simp
   · intro cond y
     specialize cond (t • y + a)
     rw [← smul_sub, norm_smul, mul_pow] at cond; simp at cond
     rw [← smul_sub, norm_smul, mul_pow] at cond; simp at cond
     rw [mul_div_assoc, ← mul_add, mul_div_assoc, ← mul_add] at cond
-    rw [mul_le_mul_left] at cond; use cond; rw [sq_pos_iff]; use tnz
+    exact (mul_le_mul_iff_of_pos_left (sq_pos_iff.mpr tnz)).1 cond
 
 /-
   relation of proximal between a function and its scale
@@ -452,25 +442,31 @@ theorem proximal_scale {t : ℝ} (tpos : 0 < t) (f : E → ℝ):
   constructor
   · intro cond y
     specialize cond (t • y)
-    have tsq : 0 < t ^ 2 := by field_simp
-    rw [← mul_le_mul_left tsq]
-    calc
-      t ^ 2 * (t⁻¹ * f (t⁻¹ • z) + ‖t⁻¹ • z - t⁻¹ • x‖ ^ 2 / 2) =
-          t * f (t⁻¹ • z) + ‖z - x‖ ^ 2 / 2 := by
-        rw [← smul_sub, norm_smul, mul_pow, mul_add, pow_two, ← mul_assoc, mul_assoc _ _ (t⁻¹)]
-        rw [mul_inv_cancel₀, mul_div_assoc, ← mul_assoc]; simp
-        rw [← pow_two, mul_inv_cancel₀]; repeat simp; repeat linarith
-      _ ≤ t * f (t⁻¹ • t • y) + ‖t • y - x‖ ^ 2 / 2 := cond
-      _ = t ^ 2 * (t⁻¹ * f y) + ‖t • (y - t⁻¹ • x)‖ ^ 2 / 2 := by
-        rw [pow_two t, ← mul_assoc, mul_assoc _ _ (t⁻¹), mul_inv_cancel₀]
-        rw [← smul_assoc, smul_eq_mul, inv_mul_cancel₀]; simp
-        rw [smul_sub, ← smul_assoc, smul_eq_mul, mul_inv_cancel₀]; simp; repeat linarith
-      _ = t ^ 2 * (t⁻¹ * f y + ‖y - t⁻¹ • x‖ ^ 2 / 2) := by
-        rw [mul_add, norm_smul, mul_pow]; field_simp
+    have tsq : 0 < t ^ 2 := by nlinarith [tpos]
+    have hmul :
+        t ^ 2 * (t⁻¹ * f (t⁻¹ • z) + ‖t⁻¹ • z - t⁻¹ • x‖ ^ 2 / 2) ≤
+          t ^ 2 * (t⁻¹ * f y + ‖y - t⁻¹ • x‖ ^ 2 / 2) := by
+      calc
+        t ^ 2 * (t⁻¹ * f (t⁻¹ • z) + ‖t⁻¹ • z - t⁻¹ • x‖ ^ 2 / 2) =
+            t * f (t⁻¹ • z) + ‖z - x‖ ^ 2 / 2 := by
+          rw [← smul_sub, norm_smul, mul_pow, mul_add, pow_two, ← mul_assoc, mul_assoc _ _ (t⁻¹)]
+          rw [mul_inv_cancel₀, mul_div_assoc, ← mul_assoc]; simp
+          rw [← pow_two, mul_inv_cancel₀]; repeat simp; repeat linarith
+        _ ≤ t * f (t⁻¹ • t • y) + ‖t • y - x‖ ^ 2 / 2 := cond
+        _ = t ^ 2 * (t⁻¹ * f y) + ‖t • (y - t⁻¹ • x)‖ ^ 2 / 2 := by
+          rw [pow_two t, ← mul_assoc, mul_assoc _ _ (t⁻¹), mul_inv_cancel₀]
+          rw [← smul_assoc, smul_eq_mul, inv_mul_cancel₀]; simp
+          rw [smul_sub, ← smul_assoc, smul_eq_mul, mul_inv_cancel₀]; simp; repeat linarith
+        _ = t ^ 2 * (t⁻¹ * f y + ‖y - t⁻¹ • x‖ ^ 2 / 2) := by
+          rw [mul_add, norm_smul, mul_pow, Real.norm_eq_abs, sq_abs]
+          ring_nf
+    exact (mul_le_mul_iff_of_pos_left tsq).1 hmul
   · intro cond y
     specialize cond (t⁻¹ • y)
-    have tsq : 0 < t ^ 2 := by field_simp
-    rw [← mul_le_mul_left tsq] at cond
+    have tsq : 0 < t ^ 2 := by nlinarith [tpos]
+    have cond : t ^ 2 * (t⁻¹ * f (t⁻¹ • z) + ‖t⁻¹ • z - t⁻¹ • x‖ ^ 2 / 2) ≤
+        t ^ 2 * (t⁻¹ * f (t⁻¹ • y) + ‖t⁻¹ • y - t⁻¹ • x‖ ^ 2 / 2) :=
+      (mul_le_mul_iff_of_pos_left tsq).2 cond
     calc
       t * f (t⁻¹ • z) + ‖z - x‖ ^ 2 / 2 =
           t ^ 2 * (t⁻¹ * f (t⁻¹ • z) + ‖t⁻¹ • z - t⁻¹ • x‖ ^ 2 / 2) := by
@@ -479,7 +475,8 @@ theorem proximal_scale {t : ℝ} (tpos : 0 < t) (f : E → ℝ):
         rw [← pow_two, mul_inv_cancel₀]; repeat simp; repeat linarith
       _ ≤ t ^ 2 * (t⁻¹ * f (t⁻¹ • y) + ‖t⁻¹ • y - t⁻¹ • x‖ ^ 2 / 2) := cond
       _ = t ^ 2 * (t⁻¹ * f (t⁻¹ • y)) + ‖t • (t⁻¹ • y - t⁻¹ • x)‖ ^ 2 / 2 := by
-        rw [mul_add, norm_smul, mul_pow]; field_simp
+        rw [mul_add, norm_smul, mul_pow, Real.norm_eq_abs, sq_abs]
+        ring_nf
       _ = t * f (t⁻¹ • y) + ‖y - x‖ ^ 2 / 2 := by
         rw [pow_two t, ← mul_assoc, mul_assoc _ _ (t⁻¹), mul_inv_cancel₀]
         rw [smul_sub, ← smul_assoc, smul_eq_mul, mul_inv_cancel₀]; simp
@@ -489,26 +486,27 @@ theorem proximal_scale {t : ℝ} (tpos : 0 < t) (f : E → ℝ):
   change of proximal when added a linear components
 -/
 theorem proximal_add_linear (a : E) (f : E → ℝ):
-    ∀ z : E, prox_prop (fun x ↦ f x + inner a x) x z ↔
+    ∀ z : E, prox_prop (fun x ↦ f x + inner ℝ a x) x z ↔
       prox_prop f (x - a) z := by
   intro z
   rw [prox_prop, prox_prop, isMinOn_univ_iff, isMinOn_univ_iff]
   have aux (v : E) : ‖v - (x - a)‖ ^ 2 / 2 =
-      ‖v - x‖ ^ 2 / 2 + inner a v + (‖a‖ ^ 2 / 2 - inner a x) := by
+      ‖v - x‖ ^ 2 / 2 + inner ℝ a v + (‖a‖ ^ 2 / 2 - inner ℝ a x) := by
     rw [← sub_add, norm_add_sq_real, real_inner_comm, inner_sub_right]; ring_nf
   constructor
   · intro cond y
     specialize cond y
-    rw [aux, aux, add_comm _ (inner a z), add_comm _ (inner a y)]
+    rw [aux, aux, add_comm _ (inner ℝ a z), add_comm _ (inner ℝ a y)]
     linarith
   · intro cond y
     specialize cond y
-    rw [aux, aux, add_comm _ (inner a z), add_comm _ (inner a y)] at cond
+    rw [aux, aux, add_comm _ (inner ℝ a z), add_comm _ (inner ℝ a y)] at cond
     linarith
 
 /-
   change of proximal when added a square components
 -/
+set_option maxHeartbeats 1000000 in
 theorem proximal_add_sq (a : E) {l : ℝ} (lpos : 0 < l) (f : E → ℝ):
     ∀ z : E, prox_prop (fun x ↦ f x + l / 2 * ‖x - a‖ ^ 2) x z ↔
       prox_prop ((1 / (l + 1)) • f) ((1 / (l + 1)) • (x + l • a)) z := by
@@ -525,16 +523,30 @@ theorem proximal_add_sq (a : E) {l : ℝ} (lpos : 0 < l) (f : E → ℝ):
     rw [add_sub_right_comm]; simp; rw [mul_sub, ← add_sub_right_comm, ← add_sub_assoc]
     nth_rw 3 [← one_mul (‖v‖ ^ 2)]; rw [← add_mul, ← mul_assoc l, mul_comm l 2, sub_sub]
     rw [mul_assoc, ← mul_add, ← inner_smul_right _ _ l, ← inner_add_right]
-    field_simp; rw [mul_comm]; simp
+    field_simp [lpos.ne']; ring_nf; norm_num
   constructor
   · intro cond y
     specialize cond y
-    rw [aux, aux]; simp; rw [← mul_add, ← mul_add, mul_le_mul_left]
-    linarith [cond]; simp; linarith
+    let c : ℝ := ((l + 1)⁻¹ * ‖x + l • a‖ ^ 2 - ‖x‖ ^ 2 - l * ‖a‖ ^ 2) / 2
+    have hplus : 0 < l + 1 := by linarith
+    have hshift : f z + l / 2 * ‖z - a‖ ^ 2 + ‖z - x‖ ^ 2 / 2 + c ≤
+        f y + l / 2 * ‖y - a‖ ^ 2 + ‖y - x‖ ^ 2 / 2 + c := by
+      simpa [add_assoc, add_left_comm, add_comm] using add_le_add_right cond c
+    have hmul : (l + 1)⁻¹ * (f z + l / 2 * ‖z - a‖ ^ 2 + ‖z - x‖ ^ 2 / 2 + c) ≤
+        (l + 1)⁻¹ * (f y + l / 2 * ‖y - a‖ ^ 2 + ‖y - x‖ ^ 2 / 2 + c) :=
+      mul_le_mul_of_nonneg_left hshift (le_of_lt (inv_pos.mpr hplus))
+    rw [aux, aux]
+    simpa [Pi.smul_apply, c, mul_add, add_assoc, add_left_comm, add_comm] using hmul
   · intro cond y
     specialize cond y
-    rw [aux, aux] at cond; simp at cond; rw [← mul_add, ← mul_add, mul_le_mul_left] at cond
-    linarith [cond]; simp; linarith
+    let c : ℝ := ((l + 1)⁻¹ * ‖x + l • a‖ ^ 2 - ‖x‖ ^ 2 - l * ‖a‖ ^ 2) / 2
+    rw [aux, aux] at cond
+    have hplus : 0 < l + 1 := by linarith
+    have hmul := mul_le_mul_of_nonneg_left cond (le_of_lt hplus)
+    have hshift : f z + l / 2 * ‖z - a‖ ^ 2 + ‖z - x‖ ^ 2 / 2 + c ≤
+        f y + l / 2 * ‖y - a‖ ^ 2 + ‖y - x‖ ^ 2 / 2 + c := by
+      simpa [Pi.smul_apply, c, hplus.ne', mul_add, mul_assoc, add_assoc, add_left_comm, add_comm] using hmul
+    exact (add_le_add_iff_right c).1 hshift
 
 end properties
 
@@ -553,12 +565,8 @@ theorem prox_iff_subderiv (f : E → ℝ) (hfun : ConvexOn ℝ univ f) :
   let g := fun u ↦ ‖u - x‖ ^ 2 / 2
   have hg : ConvexOn ℝ Set.univ g := by apply convex_of_norm_sq x (convex_univ)
   have hcg : ContinuousOn g univ := by
-    simp [g]; apply ContinuousOn.div
-    apply ContinuousOn.pow _
-    · apply ContinuousOn.norm
-      apply ContinuousOn.sub continuousOn_id continuousOn_const
-    · apply continuousOn_const
-    · simp
+    intro u _
+    simpa [g] using (gradient_of_sq (x := x) u).continuousAt.continuousWithinAt
   show 0 ∈ SubderivAt (f + g) u ↔ x - u ∈ SubderivAt f u
   have : SubderivAt (f + g) u = SubderivAt (g + f) u := by
     unfold SubderivAt; ext z; rw [Set.mem_setOf, Set.mem_setOf];
@@ -638,15 +646,23 @@ theorem prox_iff_subderiv_smul (f : E → ℝ) {t : ℝ} (hfun : ConvexOn ℝ un
   · intro cond y
     specialize cond y; simp at cond
     rw [inner_smul_left]; simp
-    rw [← mul_le_mul_left ht]; ring_nf; field_simp
-    exact cond
+    have hcond : t * (f u + t⁻¹ * inner ℝ (x - u) (y - u)) ≤ t * f y := by
+      have hEq : t * (f u + t⁻¹ * inner ℝ (x - u) (y - u)) = t * f u + inner ℝ (x - u) (y - u) := by
+        field_simp [ht.ne']
+      rw [hEq]
+      exact cond
+    exact (mul_le_mul_iff_of_pos_left ht).1 hcond
   · intro cond y
-    specialize cond y; rw [inner_smul_left] at cond; field_simp at cond
-    simp
-    have hrect : 0 < t⁻¹ := by
-      simp; linarith
-    rw [← mul_le_mul_left hrect]; ring_nf; field_simp
-    exact cond
+    specialize cond y
+    rw [inner_smul_left] at cond
+    have cond' : f u + t⁻¹ * inner ℝ (x - u) (y - u) ≤ f y := by
+      simpa [sub_eq_add_neg, add_assoc, add_left_comm, add_comm] using cond
+    have hmul : t * (f u + t⁻¹ * inner ℝ (x - u) (y - u)) ≤ t * f y :=
+      (mul_le_mul_iff_of_pos_left ht).2 cond'
+    have hEq : t * (f u + t⁻¹ * inner ℝ (x - u) (y - u)) = t * f u + inner ℝ (x - u) (y - u) := by
+      field_simp [ht.ne']
+    rw [hEq] at hmul
+    exact hmul
   exact gconv
 
 end
diff --git a/Optlib/Optimality/Constrained_Problem.lean b/Optlib/Optimality/Constrained_Problem.lean
index 03cba1d..febcef8 100644
--- a/Optlib/Optimality/Constrained_Problem.lean
+++ b/Optlib/Optimality/Constrained_Problem.lean
@@ -5,12 +5,13 @@ Authors: Chenyi Li, Shengyang Xu, Yuxuan Wu
 -/
 import Mathlib.Analysis.Convex.Cone.Basic
 import Mathlib.Analysis.Calculus.LocalExtr.Basic
-import Mathlib.Analysis.NormedSpace.HahnBanach.Separation
-import Mathlib.Data.Matrix.Rank
-import Mathlib.LinearAlgebra.FiniteDimensional
+import Mathlib.Analysis.LocallyConvex.Separation
+import Mathlib.LinearAlgebra.Matrix.Rank
+import Mathlib.LinearAlgebra.FiniteDimensional.Basic
 import Mathlib.Analysis.Calculus.Implicit
 import Mathlib.Analysis.Calculus.MeanValue
 import Mathlib.Analysis.InnerProductSpace.Calculus
+import Mathlib.Analysis.Calculus.TangentCone.Seq
 import Optlib.Differential.Calculation
 import Optlib.Convex.Farkas
 import Optlib.Differential.Lemmas
@@ -46,7 +47,7 @@ variable {τ σ : Finset ℕ}
   The equality constraints are a set of functions from a Hilbert space to ℝ.
   The inequality constraints are a set of functions from a Hilbert space to ℝ.
 -/
-structure Constrained_OptimizationProblem (E : Type _) (τ σ : Finset ℕ) :=
+structure Constrained_OptimizationProblem (E : Type _) (τ σ : Finset ℕ) where
   (domain : Set E)
   (equality_constraints : (i : ℕ) → E → ℝ)
   (inequality_constraints : (j : ℕ) → E → ℝ)
@@ -145,11 +146,11 @@ section linear
 
 variable {E : Type _} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
 
-def IsLinear (f : E → ℝ) : Prop := ∃ a, ∃ b, f = fun x ↦ (inner x a : ℝ) + b
+def IsLinear (f : E → ℝ) : Prop := ∃ a, ∃ b, f = fun x ↦ (inner ℝ x a : ℝ) + b
 
-lemma IsLinear_iff (f : E → ℝ) : IsLinear f ↔ ∃ a b, f = fun x ↦ (inner x a : ℝ) + b := by rfl
+lemma IsLinear_iff (f : E → ℝ) : IsLinear f ↔ ∃ a b, f = fun x ↦ (inner ℝ x a : ℝ) + b := by rfl
 
-lemma IsLinear_iff' (f : E → ℝ) : IsLinear f ↔ ∃ a b, f = fun x ↦ (inner a x : ℝ) + b := by
+lemma IsLinear_iff' (f : E → ℝ) : IsLinear f ↔ ∃ a b, f = fun x ↦ (inner ℝ a x : ℝ) + b := by
   constructor
   repeat rintro ⟨a, b, rfl⟩; exact ⟨a, b, by ext x; simp; exact real_inner_comm _ _⟩
 
@@ -199,190 +200,11 @@ theorem linearized_feasible_directions_convex (point : E) :
 lemma posTangentCone_localmin_inner_pos {f : E → ℝ} {loc : E} (hl : IsLocalMinOn f p.FeasSet loc)
     (hf : DifferentiableAt ℝ f loc) :
     ∀ v ∈ posTangentConeAt p.FeasSet loc, ⟪gradient f loc, v⟫_ℝ ≥ (0 : ℝ) := by
-  intro v vt; rw [posTangentConeAt] at vt; simp at vt
-  rcases vt with ⟨c, d, ⟨a, ha⟩, ⟨vt1, vt2⟩⟩
-  by_contra proneg; push_neg at proneg
-  rw [IsLocalMinOn, IsMinFilter, eventually_iff_exists_mem] at hl
-  rcases hl with ⟨s, ⟨hs, hs2⟩⟩
-  rw [nhdsWithin] at hs
-  rcases Metric.mem_nhdsWithin_iff.mp hs with ⟨ε, ⟨εpos, εball⟩⟩
-  let s1 := Metric.ball loc ε ∩ p.FeasSet
-  have hmin : ∀ y ∈ s1, f loc ≤ f y := fun y yin => hs2 y (εball yin)
-  let z := fun n ↦ loc + d n
-  have hzd : ∀ n, d n = z n - loc := fun _ => eq_sub_of_add_eq' rfl
-  rw [real_inner_comm] at proneg
-  have hcp : ∀ᶠ (n : ℕ) in atTop, c n > 0 := by
-    rw [Filter.tendsto_atTop] at vt1
-    specialize vt1 (1 : ℝ)
-    apply Filter.Eventually.mp vt1
-    apply Filter.Eventually.of_forall
-    intro n hn; linarith
-  have hz3 : ∀ᶠ (n : ℕ) in atTop, (1 / c n) > 0 := by
-    apply Filter.Eventually.mp hcp
-    apply Filter.Eventually.of_forall
-    intro n hn; exact one_div_pos.mpr hn
-  have hzt : Tendsto z atTop (𝓝 loc) := by
-    have : Tendsto d atTop (𝓝 0) := by
-      rw [Filter.tendsto_atTop] at vt1
-      rw [Filter.tendsto_atTop'] at vt2
-      rw [Metric.tendsto_atTop']; intro ε hε
-      have : Metric.ball v ε ∈ 𝓝 v := by exact Metric.ball_mem_nhds _ hε
-      specialize vt2 (Metric.ball v ε) this
-      rcases vt2 with ⟨a, ha⟩
-      specialize vt1 (2 * (‖v‖ + ε) / ε); simp at vt1
-      rcases vt1 with ⟨a1, ha1⟩
-      let n1 := max a a1
-      use n1; intro n hn
-      specialize ha n (ge_trans (Nat.le_of_lt hn) (a.le_max_left a1))
-      specialize ha1 n (ge_trans (Nat.le_of_lt hn) (a.le_max_right a1))
-      have : ‖d n‖ < ε := by
-        have : ‖c n • d n‖ ≤ ‖v‖ + ε := by
-          rw [Metric.mem_ball, dist_eq_norm] at ha;
-          have t1 : ‖c n • d n - v‖ ≥ ‖c n • d n‖ - ‖v‖ := norm_sub_norm_le _ v
-          linarith
-        have cpos : c n > 0 := by
-          apply lt_of_le_of_lt'
-          · show c n ≥ 2 * (‖v‖ + ε) / ε
-            exact ha1
-          · positivity
-        rw [norm_smul, Real.norm_eq_abs, abs_of_pos cpos] at this;
-        calc _ ≤ (‖v‖ + ε) / c n := (le_div_iff₀' cpos).mpr this
-             _ ≤ (‖v‖ + ε) / (2 * (‖v‖ + ε) / ε) :=
-                div_le_div_of_nonneg_left (by positivity) (by positivity) ha1
-             _ = ε / 2 := by field_simp [εpos]; ring_nf
-             _ < ε := by linarith
-      simp; exact this
-    have h1 : z = (fun n ↦ d n + loc) := by
-      funext n; rw [hzd n, sub_add, sub_self, sub_zero]
-    rw [h1]
-    convert Filter.Tendsto.add_const loc this
-    rw [zero_add]
-  have hz : (fun n ↦ f (z n) - f loc - inner (z n - loc) (gradient f loc))
-      =o[atTop] (fun n ↦ z n - loc) := by
-    have : HasGradientAt f (gradient f loc) loc := hf.hasGradientAt
-    rw [hasGradientAt_iff_isLittleO] at this
-    have heq : (fun n ↦ f (z n) - f loc - inner (z n - loc) (gradient f loc)) =
-        (fun n ↦ f (z n) - f loc - inner (gradient f loc) (z n - loc)) := by
-      ext n; rw [real_inner_comm]
-    rw [heq]
-    apply Asymptotics.IsLittleO.comp_tendsto this hzt
-  have hz1 : (fun n ↦ f (z n) - f loc - (1 / c n) * inner v (gradient f loc))
-      =o[atTop] (fun n ↦ 1 / c n) := by
-    have t1: (fun n ↦ z n - loc) =O[atTop] (fun n ↦ 1 / c n) := by
-      rw [Asymptotics.isBigO_iff]
-      rw [Filter.tendsto_atTop] at vt1
-      rw [Filter.tendsto_atTop'] at vt2
-      have : Metric.ball v 1 ∈ 𝓝 v := by exact Metric.ball_mem_nhds _ (by norm_num)
-      specialize vt2 (Metric.ball v 1) this
-      rcases vt2 with ⟨a, ha⟩
-      specialize vt1 (2 * (‖v‖ + ε) / ε); simp at vt1
-      rcases vt1 with ⟨a1, ha1⟩
-      let n1 := max a a1
-      use (‖v‖ + 1 : ℝ); simp; use n1; intro n hn
-      specialize ha n (ge_trans hn (a.le_max_left a1))
-      specialize ha1 n (ge_trans hn (a.le_max_right a1))
-      have cpos : c n > 0 := by
-          apply lt_of_le_of_lt'
-          · show c n ≥ 2 * (‖v‖ + ε) / ε
-            exact ha1
-          · positivity
-      rw [abs_of_pos]
-      have : ‖d n‖ ≤ (‖v‖ + 1) * (c n)⁻¹ := by
-        have : ‖c n • d n‖ ≤ ‖v‖ + 1 := by
-          rw [Metric.mem_ball, dist_eq_norm] at ha;
-          have t1 : ‖c n • d n - v‖ ≥ ‖c n • d n‖ - ‖v‖ := norm_sub_norm_le _ v
-          linarith
-        rw [norm_smul, Real.norm_eq_abs, abs_of_pos cpos] at this;
-        field_simp; exact (le_div_iff₀' cpos).mpr this
-      rw [← hzd n]; exact this; apply cpos
-    have t2 : (fun n ↦ f (z n) - f loc - inner (z n - loc) (gradient f loc))
-        =o[atTop] (fun n ↦ 1 / c n) := Asymptotics.IsLittleO.trans_isBigO hz t1
-    have t3 : (fun n ↦ (inner (z n - loc - (1 / c n) • v) (gradient f loc) : ℝ))
-        =o[atTop] (fun n ↦ 1 / c n) := by
-      have t5: (fun n ↦ z n - loc - (1 / c n) • v) =o[atTop] (fun n ↦ 1 / c n) := by
-        rw [← Asymptotics.isLittleO_norm_norm]
-        apply (Asymptotics.isLittleO_iff_tendsto' _).mpr
-        · have : (fun x ↦ ‖z x - loc - (1 / c x) • v‖ / ‖1 / c x‖)
-              =ᶠ[atTop] (fun x ↦ ‖c x • (z x - loc) - v‖) := by
-            simp; rw [Filter.EventuallyEq]
-            apply Filter.Eventually.mp hcp
-            apply Filter.Eventually.of_forall
-            intro n hcn1; rw [mul_comm, ← Real.norm_eq_abs, ← norm_smul]
-            congr; field_simp; rw [smul_sub, smul_smul]; field_simp
-          rw [Filter.tendsto_congr' this];
-          have : Tendsto (fun (n : ℕ) => c n • d n - v) atTop (𝓝 (v - v)) := by
-            apply Filter.Tendsto.sub vt2 tendsto_const_nhds
-          apply Filter.Tendsto.norm at this
-          simp at this; convert this; simp [hzd]
-        · apply Filter.Eventually.mp hcp
-          apply Filter.Eventually.of_forall
-          intro n hcn1 hcn2
-          exfalso; simp at hcn2; linarith
-      rw [Asymptotics.isLittleO_iff]; intro c1 hc1
-      rw [Asymptotics.isLittleO_iff] at t5;
-      have pos1 : ‖gradient f loc‖ ≠ (0 : ℝ) := by
-        by_contra hhh; simp at hhh
-        have : inner v (gradient f loc) = (0 : ℝ) := by rw [hhh, inner_zero_right]
-        linarith
-      have pos2 : ‖gradient f loc‖ > (0 : ℝ) := by positivity
-      have : c1 / ‖gradient f loc‖ > (0 : ℝ) := by positivity
-      specialize t5 this
-      apply Filter.Eventually.mp t5
-      apply Filter.Eventually.of_forall
-      intro n hn;
-      calc _ ≤ ‖z n - loc - (1 / c n) • v‖ * ‖gradient f loc‖ := norm_inner_le_norm _ _
-           _ ≤ c1 / ‖gradient f loc‖ * ‖1 / c n‖ * ‖gradient f loc‖ :=
-              (mul_le_mul_right pos2).mpr hn
-           _ ≤ c1 * ‖1 / c n‖ := by ring_nf; field_simp [pos1]
-    have t4 :  (fun n => f (z n) - f loc - 1 / c n * Inner.inner v (gradient f loc)) =
-        (fun n ↦ f (z n) - f loc - inner (z n - loc) (gradient f loc)) +
-        (fun n ↦ (inner (z n - loc - (1 / c n) • v) (gradient f loc) : ℝ)) := by
-      ext n; dsimp; simp [inner_sub_left, inner_add_left, inner_smul_left]
-    rw [t4]; apply Asymptotics.IsLittleO.add t2 t3
-  have hz2 : ∀ᶠ (n : ℕ) in atTop, f (z n) ≤ f loc + (1 / 2) *
-      (1 / c n) * inner v (gradient f loc) := by
-    rw [Asymptotics.isLittleO_iff] at hz1
-    have : (- (1 / 2 : ℝ) * inner v (gradient f loc)) > 0 := by
-      simp [proneg];rw [mul_comm]; apply mul_neg_of_neg_of_pos proneg (by norm_num)
-    specialize hz1 this
-    apply Filter.Eventually.mp hz1
-    apply Filter.Eventually.mp hz3
-    apply Filter.Eventually.of_forall
-    intro n hn hn1
-    rw [Real.norm_eq_abs, Real.norm_eq_abs, abs_le, abs_of_pos hn] at hn1
-    rcases hn1 with ⟨_, hn1⟩
-    rw [sub_le_iff_le_add, sub_le_iff_le_add] at hn1
-    have : -(1 / 2) * inner v (gradient f loc) * (1 / c n) + 1 / c n * inner v
-        (gradient f loc) + f loc = f loc + 1 / 2 * (1 / c n) * inner v (gradient f loc) := by
-      ring_nf
-    rw [this] at hn1; exact hn1
-  have hz4 : ∀ᶠ (n : ℕ) in atTop, f (z n) < f loc := by
-    apply Filter.Eventually.mp hz2
-    apply Filter.Eventually.mp hz3
-    apply Filter.Eventually.of_forall
-    intro n hn1 hn2
-    have : 1 / 2 * (1 / c n) * (inner v (gradient f loc)) < 0 := by
-      apply mul_neg_of_pos_of_neg
-      · apply Right.mul_pos; simp; exact hn1
-      · exact proneg
-    linarith
-  have hz5 : ∀ᶠ (n : ℕ) in atTop, z n ∈ s1 := by
-    simp only [s1, mem_inter_iff, Metric.mem_ball, dist_self_add_left]
-    apply Filter.Eventually.and
-    · rw [Filter.tendsto_atTop'] at hzt
-      simp;
-      have : Metric.ball loc ε ∈ 𝓝 loc := by exact Metric.ball_mem_nhds loc εpos
-      rcases hzt (Metric.ball loc ε) this with ⟨a, ha⟩
-      use a; intro b hb; specialize ha b (by linarith [hb])
-      simp at ha; exact ha
-    · simp; use a
-  simp at hz5 hz4
-  rcases hz5 with ⟨n, hn1⟩; rcases hz4 with ⟨m, hm1⟩
-  let M := max n m
-  have hh2 : f (z M) < f loc := hm1 M (le_max_right n m)
-  have hh1 : z M ∈ s1 := by simp [s1]; apply hn1 M (le_max_left n m)
-  have hh3 : f loc ≤ f (z M) := hmin (z M) hh1
-  linarith
+  intro v vt
+  have hgrad : HasGradientAt f (gradient f loc) loc := hf.hasGradientAt
+  have hnonneg : 0 ≤ ((toDual ℝ E) (gradient f loc)) v :=
+    hl.hasFDerivWithinAt_nonneg hgrad.hasFDerivAt.hasFDerivWithinAt vt
+  simpa [InnerProductSpace.toDual_apply_apply, real_inner_comm] using hnonneg
 
 /-
   Linearized feasible directions contain tagent cone
@@ -418,7 +240,7 @@ theorem linearized_feasible_directions_contain_tagent_cone (xf : x ∈ p.FeasSet
   . intro i itau
     apply ge_antisymm
     . apply posTangentCone_localmin_inner_pos (imin i itau) (diffable i itau) v hv
-    . rw [← neg_neg (inner (gradient (equality_constraints p i) x) v)]
+    . rw [← neg_neg (inner ℝ (gradient (equality_constraints p i) x) v)]
       apply neg_nonpos_of_nonneg
       rw [← inner_neg_left]
       have a₁ : ∀ i ∈ τ, DifferentiableAt ℝ (-equality_constraints p i) x :=
@@ -467,7 +289,7 @@ theorem local_Minimum_TangentCone (loc : E) (hl : p.Local_Minimum loc)
 theorem local_Minimum_TangentCone' (loc : E) (hl : p.Local_Minimum loc)
     (hf : Differentiable ℝ p.objective) :
     posTangentConeAt p.FeasSet loc ∩ {d | ⟪gradient p.objective loc, d⟫_ℝ < (0 : ℝ)} = ∅ := by
-  rw [Set.eq_empty_iff_forall_not_mem]
+  rw [Set.eq_empty_iff_forall_notMem]
   intro d ⟨hd1, hd2⟩
   simp at hd2
   obtain hd1 := local_Minimum_TangentCone loc hl hf d hd1
@@ -567,7 +389,7 @@ lemma StrictFderivAt_of_FderivAt_of_ContinuousAt
     with ⟨ε, ε0, hε⟩
   refine ⟨ε, ε0, ?_⟩
   rintro ⟨a, b⟩ h
-  rw [← ball_prod_same, prod_mk_mem_set_prod_eq] at h
+  rw [← ball_prod_same, prodMk_mem_set_prod_eq] at h
   have hf' : ∀ x' ∈ Metric.ball x ε, ‖c' x' - c' x‖ ≤ μ := fun x' H' => by
     rw [← dist_eq_norm]
     exact le_of_lt (hε H').2
@@ -608,7 +430,7 @@ lemma exist_forall_forall_exist (P : ℕ → ℝ → Prop) (s : Finset ℕ) (hs
   have po : ∀ y ∈ s1, y > 0 := by
     intro y hy
     simp [s1] at hy; rcases hy with ⟨a, ha1, ha2⟩
-    simp only [gt_iff_lt, and_imp, ha1, ↓reduceDIte, f] at ha2; rw [← ha2]
+    simp only [gt_iff_lt, ha1, ↓reduceDIte, f] at ha2; rw [← ha2]
     exact (h a ha1).choose_spec.1
   have up : ∀ y ∈ s1, tm ≤ y := fun y a ↦ Finset.min'_le s1 y a
   use tm; constructor
@@ -650,20 +472,23 @@ lemma LICQ_mlen (x : EuclideanSpace ℝ (Fin n)) (LIx : p.LICQ x)
 
 lemma LICQ_Axfullrank (x : EuclideanSpace ℝ (Fin n)) (LIx : p.LICQ x)
     {M : Matrix (p.active_set x) (Fin n) ℝ}
-    (eq : M = fun i : (p.active_set x) ↦ if i.1 ∈ τ then gradient (p.equality_constraints i) x
-        else gradient (p.inequality_constraints i) x):
+    (eq : M = fun i : (p.active_set x) ↦ (if i.1 ∈ τ then gradient (p.equality_constraints i) x
+        else gradient (p.inequality_constraints i) x).ofLp):
     Matrix.rank M = (Fintype.card (p.active_set x)) := by
+  rw [LICQ] at LIx
   apply LE.le.antisymm
   · apply Matrix.rank_le_card_height
   · simp
     rw [Matrix.rank_eq_finrank_span_row, finrank_span_eq_card]
-    simp; rw [eq]; apply LIx
+    simp; rw [eq]
+    simpa [Function.comp] using LIx.map' (WithLp.linearEquiv 2 ℝ (Fin n → ℝ)).toLinearMap
+      (LinearMap.ker_eq_bot.2 (WithLp.linearEquiv 2 ℝ (Fin n → ℝ)).injective)
 
 lemma LICQ_existZ (x : EuclideanSpace ℝ (Fin n)) (LIx : p.LICQ x)
     {m : ℕ} (meq : m = (p.active_set x).card)
     {M : Matrix (p.active_set x) (Fin n) ℝ}
-    (eq : M = fun i : (p.active_set x) ↦ if i.1 ∈ τ then gradient (p.equality_constraints i) x
-        else gradient (p.inequality_constraints i) x):
+    (eq : M = fun i : (p.active_set x) ↦ (if i.1 ∈ τ then gradient (p.equality_constraints i) x
+        else gradient (p.inequality_constraints i) x).ofLp):
     ∃ (Z : Matrix (Fin n) (Fin (n - m)) ℝ), M * Z = 0 ∧ Matrix.rank Z = (n - m) := by
   rw [LICQ] at LIx;
   have mlen : m ≤ n := LICQ_mlen x LIx meq
@@ -696,7 +521,7 @@ lemma LICQ_existZ (x : EuclideanSpace ℝ (Fin n)) (LIx : p.LICQ x)
     · simp
       rw [Matrix.rank_eq_finrank_span_row, finrank_span_eq_card]
       simp; rw [Nat.sub_add_cancel]; apply mlen
-      let base_indep := Basis.linearIndependent base
+      let base_indep := base.linearIndependent
       simp only [Z]
       rw [linearIndependent_iff'']
       intro s g cond sum
@@ -708,11 +533,11 @@ lemma LICQ_existZ (x : EuclideanSpace ℝ (Fin n)) (LIx : p.LICQ x)
         · intro cond; rw [cond]; simp [coe]
         · intro cond; simp [coe] at cond; exact cond
       rw [coe_zero]; simp only [coe]
-      rw [← sum]; simp
+      simpa [Matrix.row] using sum
 
 lemma mulVec_eq_toEuclidean {s : Type*} (M : Matrix s (Fin n) ℝ) (y : EuclideanSpace ℝ (Fin n)) :
     M *ᵥ y = (toEuclideanLin M) y := by
-  rw [Matrix.toEuclideanLin_apply]; ext j; simp [Matrix.mulVec, Matrix.dotProduct]
+  rfl
 
 lemma inj_iff_full_finrank {s t : Type*} {M : Matrix s t ℝ} [Fintype s] [Fintype t]
     (hn : Fintype.card s = Fintype.card t) :
@@ -746,34 +571,38 @@ lemma LICQ_injM (z : EuclideanSpace ℝ (Fin n)) (m : ℕ)
   rw [eq1, eq2] at Bzeq0; simp at Bzeq0
   have aux : (p.active_set x).card + (n - m) = n := by
     rw [← meq]; rw [add_comm, Nat.sub_add_cancel]; exact mlen
-  refine (inj_transpose_iff_inj_of_sq ?_).1 ?_ z Bzeq0
-  · simp; rw [aux]
-  · intro v Btveq0
-    let y := v ∘ Sum.inl
-    let z := v ∘ Sum.inr
-    have yeq : Bt *ᵥ (Sum.elim y (fun _ ↦ 0)) = Aᵀ *ᵥ y := by ext i; simp [Bt, mulVec, dotProduct]
-    have zeq : Bt *ᵥ (Sum.elim (fun _ ↦ 0) z) = Z *ᵥ z := by ext i; simp [Bt, mulVec, dotProduct]
-    have veq : v = (Sum.elim y (fun _ ↦ 0)) + (Sum.elim (fun _ ↦ 0) z) := by
-      simp [y, z]; ext i; cases i <;> simp
-    have eq : Bᵀ *ᵥ v = Aᵀ *ᵥ y + Z *ᵥ z := by rw [veq, ← Bteq, mulVec_add, yeq, zeq]
-    rw [eq] at Btveq0
-    have yzero : y = 0 := by
-      have h : A *ᵥ (Aᵀ *ᵥ y + Z *ᵥ z) = 0 := by rw [Btveq0]; simp
-      rw [mulVec_add, mulVec_mulVec, mulVec_mulVec, AZorth] at h; simp at h
-      refine (inj_iff_full_finrank ?_).1 ?_ y h
-      · simp
-      · simp; rw [← meq, Afull]
-    have yzero' : (Sum.elim y (fun _ : (Fin (n - m)) ↦ 0)) = 0 := by
-      ext i; cases i <;> simp [yzero]
-    have zzero : z = 0 := by
-      have h : Zᵀ *ᵥ (Aᵀ *ᵥ y + Z *ᵥ z) = 0 := by rw [Btveq0]; simp
-      rw [mulVec_add, mulVec_mulVec, mulVec_mulVec, ← transpose_mul, AZorth] at h; simp at h
-      refine (inj_iff_full_finrank ?_).1 ?_ z h
-      · simp
-      · simp; rw [rank_transpose_mul_self, Zfull]
-    have zzero' : (Sum.elim (fun _ : (p.active_set x) ↦ 0) z) = 0 := by
-      ext i; cases i <;> simp [zzero]
-    rw [veq, yzero', zzero']; simp
+  have z0 : z.ofLp = 0 := by
+    refine (inj_transpose_iff_inj_of_sq ?_).1 ?_ z Bzeq0
+    · simp; rw [aux]
+    · intro v Btveq0
+      let y := v ∘ Sum.inl
+      let z := v ∘ Sum.inr
+      have yeq : Bt *ᵥ (Sum.elim y (fun _ ↦ 0)) = Aᵀ *ᵥ y := by
+        ext i; simp [Bt, mulVec, dotProduct]
+      have zeq : Bt *ᵥ (Sum.elim (fun _ ↦ 0) z) = Z *ᵥ z := by
+        ext i; simp [Bt, mulVec, dotProduct]
+      have veq : v = (Sum.elim y (fun _ ↦ 0)) + (Sum.elim (fun _ ↦ 0) z) := by
+        simp [y, z]; ext i; cases i <;> simp
+      have eq : Bᵀ *ᵥ v = Aᵀ *ᵥ y + Z *ᵥ z := by rw [veq, ← Bteq, mulVec_add, yeq, zeq]
+      rw [eq] at Btveq0
+      have yzero : y = 0 := by
+        have h : A *ᵥ (Aᵀ *ᵥ y + Z *ᵥ z) = 0 := by rw [Btveq0]; simp
+        rw [mulVec_add, mulVec_mulVec, mulVec_mulVec, AZorth] at h; simp at h
+        refine (inj_iff_full_finrank ?_).1 ?_ y h
+        · simp
+        · simp; rw [← meq, Afull]
+      have yzero' : (Sum.elim y (fun _ : (Fin (n - m)) ↦ 0)) = 0 := by
+        ext i; cases i <;> simp [yzero]
+      have zzero : z = 0 := by
+        have h : Zᵀ *ᵥ (Aᵀ *ᵥ y + Z *ᵥ z) = 0 := by rw [Btveq0]; simp
+        rw [mulVec_add, mulVec_mulVec, mulVec_mulVec, ← transpose_mul, AZorth] at h; simp at h
+        refine (inj_iff_full_finrank ?_).1 ?_ z h
+        · simp
+        · simp; rw [rank_transpose_mul_self, Zfull]
+      have zzero' : (Sum.elim (fun _ : (p.active_set x) ↦ 0) z) = 0 := by
+        ext i; cases i <;> simp [zzero]
+      rw [veq, yzero', zzero']; simp
+  simpa using z0
 
 lemma LICQ_strictfderiv_Ax_elem {x : EuclideanSpace ℝ (Fin n)}
     (c : EuclideanSpace ℝ (Fin n) → ((p.active_set x) → ℝ))
@@ -783,7 +612,7 @@ lemma LICQ_strictfderiv_Ax_elem {x : EuclideanSpace ℝ (Fin n)}
     (gradceq : gradc = fun z ↦ (fun i : (p.active_set x) ↦ if i.1 ∈ τ then
       gradient (p.equality_constraints i) z else gradient (p.inequality_constraints i) z))
     (A : EuclideanSpace ℝ (Fin n) → Matrix (p.active_set x) (Fin n) ℝ)
-    (Aeq : A = fun z ↦ (fun i ↦ gradc z i))
+    (Aeq : A = fun z ↦ (fun i j ↦ gradc z i j))
     (Jz : EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ (Fin n) →L[ℝ] EuclideanSpace ℝ (p.active_set x))
     (Jzeq : Jz = fun z ↦ (LinearMap.toContinuousLinearMap (toEuclideanLin (A z))))
     (conte : ∀ i ∈ τ, ContDiffAt ℝ (1 : ℕ) (equality_constraints p i) x)
@@ -794,15 +623,20 @@ lemma LICQ_strictfderiv_Ax_elem {x : EuclideanSpace ℝ (Fin n)}
   rw [eventually_iff, Metric.mem_nhds_iff] at h; rcases h with ⟨ε, _, _⟩
   intro i; by_cases hi : i.1 ∈ τ
   · rw [ceq, Jzeq, Aeq]; simp [hi]
-    rw [HasStrictFDerivAt];
+    rw [hasStrictFDerivAt_iff_isLittleO]
     have eq : (fun p_1 : EuclideanSpace ℝ (Fin n) × EuclideanSpace ℝ (Fin n) ↦
         p.equality_constraints i.1 p_1.1 - p.equality_constraints i.1 p_1.2 -
-        ((EuclideanSpace.proj i).comp (LinearMap.toContinuousLinearMap (toEuclideanLin fun i ↦ gradc x i)))
+        ((EuclideanSpace.proj i).comp
+          (LinearMap.toContinuousLinearMap (toEuclideanLin fun i j ↦ (gradc x i).ofLp j)))
         (p_1.1 - p_1.2)) = (fun p_1 : EuclideanSpace ℝ (Fin n) × EuclideanSpace ℝ (Fin n) ↦
         p.equality_constraints i.1 p_1.1 - p.equality_constraints i.1 p_1.2 -
-        inner (gradient (p.equality_constraints ↑i) x) (p_1.1 - p_1.2) ):= by
-      ext q; rw [inner_sub_right, gradceq]; simp [toEuclideanLin_apply, mulVec, dotProduct, hi]
-      rw [← Finset.sum_sub_distrib]; apply Finset.sum_congr; rfl; exact fun _ _ ↦ by ring_nf
+        inner ℝ (gradient (p.equality_constraints ↑i) x) (p_1.1 - p_1.2) ):= by
+      ext q; rw [inner_sub_right, gradceq]
+      simp [toLpLin_apply, mulVec, dotProduct, hi]
+      rw [← inner_sub_right]
+      simpa [dotProduct, mul_comm] using
+        (EuclideanSpace.inner_eq_star_dotProduct (x := gradient (p.equality_constraints ↑i) x)
+          (y := q.1 - q.2)).symm
     rw [eq]
     specialize conte i hi
     exact StrictFderivAt_of_FderivAt_of_ContinuousAt conte
@@ -813,15 +647,20 @@ lemma LICQ_strictfderiv_Ax_elem {x : EuclideanSpace ℝ (Fin n)}
       rw [Finset.mem_filter] at hi2
       exact hi2.1
     rw [ceq, Jzeq, Aeq]; simp [hi]
-    rw [HasStrictFDerivAt];
+    rw [hasStrictFDerivAt_iff_isLittleO]
     have eq : (fun p_1 : EuclideanSpace ℝ (Fin n) × EuclideanSpace ℝ (Fin n) ↦
         p.inequality_constraints i.1 p_1.1 - p.inequality_constraints i.1 p_1.2 -
-        ((EuclideanSpace.proj i).comp (LinearMap.toContinuousLinearMap (toEuclideanLin fun i ↦ gradc x i)))
+        ((EuclideanSpace.proj i).comp
+          (LinearMap.toContinuousLinearMap (toEuclideanLin fun i j ↦ (gradc x i).ofLp j)))
         (p_1.1 - p_1.2)) = (fun p_1 : EuclideanSpace ℝ (Fin n) × EuclideanSpace ℝ (Fin n) ↦
         p.inequality_constraints i.1 p_1.1 - p.inequality_constraints i.1 p_1.2 -
         ⟪gradient (p.inequality_constraints ↑i) x, p_1.1 - p_1.2⟫_ℝ ):= by
-      ext q; rw [inner_sub_right, gradceq]; simp [toEuclideanLin_apply, mulVec, dotProduct, hi]
-      rw [← Finset.sum_sub_distrib]; apply Finset.sum_congr; rfl; exact fun _ _ ↦ by ring_nf
+      ext q; rw [inner_sub_right, gradceq]
+      simp [toLpLin_apply, mulVec, dotProduct, hi]
+      rw [← inner_sub_right]
+      simpa [dotProduct, mul_comm] using
+        (EuclideanSpace.inner_eq_star_dotProduct (x := gradient (p.inequality_constraints ↑i) x)
+          (y := q.1 - q.2)).symm
     rw [eq]
     specialize conti i hi'
     exact StrictFderivAt_of_FderivAt_of_ContinuousAt conti
@@ -831,12 +670,12 @@ lemma LICQ_implicit_f {x : EuclideanSpace ℝ (Fin n)} {m : ℕ} (v : EuclideanS
     {Rz : EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ (p.active_set x) × (Fin (n - m) → ℝ)}
     {Rt : ℝ → EuclideanSpace ℝ (p.active_set x) × (Fin (n - m) → ℝ)}
     (Rteq : Rt = fun t ↦ t • Mx v) (Rxeq0 : Rz x = 0)
-    (Rzgrad : HasStrictFDerivAt Rz Mx x) (Mxsurj : LinearMap.range Mx = ⊤) :
+    (Rzgrad : HasStrictFDerivAt Rz Mx x) (Mxsurj : Mx.range = ⊤) :
     ∃ (N : ℕ) (d : ℕ → EuclideanSpace ℝ (Fin n)), (∀ m ≥ N, Rz (d m) = Rt (1 / m)) ∧
       (Filter.Tendsto d atTop (𝓝 x)) := by
   let g := HasStrictFDerivAt.implicitFunction Rz Mx Rzgrad Mxsurj
-  have hfg : ∀ᶠ (p : (EuclideanSpace ℝ (p.active_set x) × (Fin (n - m) → ℝ)) × (LinearMap.ker Mx)) in
-      𝓝 (Rz x, (0 : LinearMap.ker Mx)), Rz (g p.1 p.2) = p.1 := by
+  have hfg : ∀ᶠ (p : (EuclideanSpace ℝ (p.active_set x) × (Fin (n - m) → ℝ)) × Mx.ker) in
+      𝓝 (Rz x, (0 : Mx.ker)), Rz (g p.1 p.2) = p.1 := by
     simp only [g]; apply HasStrictFDerivAt.map_implicitFunction_eq Rzgrad Mxsurj
   rw [Rxeq0] at hfg
   rw [eventually_iff, Metric.mem_nhds_iff] at hfg
@@ -859,7 +698,7 @@ lemma LICQ_implicit_f {x : EuclideanSpace ℝ (Fin n)} {m : ℕ} (v : EuclideanS
     simp at Rtmin; simp [Rtmin]
   · simp only [g]
     apply HasStrictFDerivAt.tendsto_implicitFunction Rzgrad Mxsurj
-    · rw [Rxeq0]; rw [NormedAddCommGroup.tendsto_nhds_zero]; simp; apply Rtleε
+    · rw [Rxeq0]; rw [NormedAddGroup.tendsto_nhds_zero]; simp; apply Rtleε
     · simp
 
 lemma eq_lemma {y z : EuclideanSpace ℝ (Fin n)} {n : ℕ} (h : ‖(n : ℝ) • y‖ ≠ 0) :
@@ -868,7 +707,10 @@ lemma eq_lemma {y z : EuclideanSpace ℝ (Fin n)} {n : ℕ} (h : ‖(n : ℝ) 
   have eq : z = (n : ℝ) • (1 / n : ℝ) • z := by
     rw [smul_smul]; field_simp; rw [div_self, one_smul]; simp [h]
   nth_rw 2 [eq]
-  rw [← smul_sub, smul_smul, norm_smul]; field_simp; rw [← div_div, div_self]; simp [h]
+  rw [← smul_sub, smul_smul, norm_smul]; field_simp
+  have hn0 : (n : ℝ) ≠ 0 := by exact_mod_cast h.1
+  have hcoef : (n : ℝ) / (‖y‖ * n) = ‖y‖⁻¹ := by field_simp [hn0]
+  simp [hcoef]
 
 lemma comap1 {x : EuclideanSpace ℝ (Fin n)} {m : ℕ}
     {Mx : EuclideanSpace ℝ (Fin n) →L[ℝ] EuclideanSpace ℝ (p.active_set x) × (Fin (n - m) → ℝ)}
@@ -892,7 +734,7 @@ lemma comap1 {x : EuclideanSpace ℝ (Fin n)} {m : ℕ}
     have zin' : z ∈ Metric.ball 0 a := by
       simp; calc
         ‖z‖ ≤ c * ‖Mx z‖ := antil
-        _ < c * (a / c) := by rw [mul_lt_mul_left]; linarith [zin]; simp [hc']
+        _ < c * (a / c) := by exact mul_lt_mul_of_pos_left zin hc'
         _ = a := by field_simp
     exact ha zin'
 
@@ -903,14 +745,21 @@ lemma comap2 (hv : v ≠ 0):
   rw [Metric.mem_nhds_iff] at smem; rcases smem with ⟨a, apos, ha⟩
   let μ := a / (a + ‖v‖)
   have eq : μ * ‖v‖ = (1 - μ) * a := by
-    field_simp [μ]; rw [mul_comm]
+    have hden : a + ‖v‖ ≠ 0 := by linarith [apos, norm_nonneg v]
+    change (a / (a + ‖v‖)) * ‖v‖ = (1 - a / (a + ‖v‖)) * a
+    field_simp [hden]
+    ring
   have vpos : 0 < ‖v‖ := by
     refine lt_of_le_of_ne (norm_nonneg v) ?_; symm; simp [hv]
   have μle : 0 < 1 - μ := by
-    field_simp [μ, hv]
-    apply add_pos ?_ vpos; linarith
+    have hden : 0 < a + ‖v‖ := by linarith [apos, norm_nonneg v]
+    have hμ : μ < 1 := by
+      change a / (a + ‖v‖) < 1
+      exact (div_lt_one hden).2 (by linarith [vpos])
+    linarith
   have μpos : 0 < μ := by
-    field_simp [μ]; apply add_pos_of_pos_of_nonneg _ (norm_nonneg v); linarith
+    have hden : 0 < a + ‖v‖ := by linarith [apos, norm_nonneg v]
+    simpa [μ] using (div_pos apos hden)
   let r := min μ ‖v‖
   use Metric.ball 0 r; constructor
   · apply Metric.ball_mem_nhds; simp [r]; exact ⟨μpos, hv⟩
@@ -919,16 +768,21 @@ lemma comap2 (hv : v ≠ 0):
       by_contra hz; simp [hz] at zin; simp [r] at zin
     simp [ze] at zin; rw [norm_smul] at zin; field_simp at zin
     have : 0 < ‖z‖ := by refine lt_of_le_of_ne (norm_nonneg z) ?_; symm; simp [ze]
-    rw [div_lt_iff₀ this] at zin
+    have zin' : ‖z - v‖ / ‖z‖ < r := by
+      simpa [div_eq_mul_inv, Real.norm_eq_abs, abs_of_pos (one_div_pos.mpr this), mul_comm] using zin
+    have zin : ‖z - v‖ < r * ‖z‖ := (div_lt_iff₀ this).1 zin'
     have ieq : ‖z - v‖ < μ * ‖z - v‖ + (1 - μ) * a := by
       calc
         _ < r * ‖z‖ := zin
-        _ ≤ μ * ‖z‖ := by rw [mul_le_mul_right this]; simp [r]
+        _ ≤ μ * ‖z‖ := by
+          exact mul_le_mul_of_nonneg_right (by simp [r]) (norm_nonneg z)
         _ ≤ μ * (‖z - v‖ + ‖v‖) := by
-          rw [mul_le_mul_left μpos, add_comm]; apply norm_le_norm_add_norm_sub'
-        _ ≤ μ * ‖z - v‖ + (1 - μ) * a := by rw [mul_add]; apply add_le_add_left; rw [eq]
+          exact mul_le_mul_of_nonneg_left (by simpa [add_comm] using norm_le_norm_add_norm_sub' z v)
+            (le_of_lt μpos)
+        _ ≤ μ * ‖z - v‖ + (1 - μ) * a := by linarith [eq]
     rw [← sub_lt_iff_lt_add'] at ieq; nth_rw 1 [← one_mul (‖z - v‖)] at ieq
-    rw [← sub_mul, mul_lt_mul_left μle] at ieq
+    have ieq' : (1 - μ) * ‖z - v‖ < (1 - μ) * a := by linarith [ieq]
+    have ieq : ‖z - v‖ < a := lt_of_mul_lt_mul_left ieq' (le_of_lt μle)
     apply ha; simp; rw [dist_eq_norm]; simp [ieq]
 
 lemma LICQ_tendsto {x : EuclideanSpace ℝ (Fin n)} {m N : ℕ}
@@ -983,7 +837,11 @@ lemma LICQ_tendsto {x : EuclideanSpace ℝ (Fin n)} {m N : ℕ}
     have neq : ‖(i : ℝ) • (d i - x)‖ ≠ 0 := by
       rw [norm_smul]; apply mul_ne_zero; simp; linarith [Nat.lt_of_add_one_le igeN.2]
       specialize dne i igeN.2; simp; apply sub_ne_zero_of_ne dne
-    field_simp [deriv', φ, neq]; apply eq_lemma neq
+    have hne : ¬(i = 0 ∨ d i - x = 0) := by
+      intro h; rcases h with hi | hz
+      · exact neq (by simp [hi])
+      · exact neq (by simp [hz])
+    simpa [deriv', φ, neq, hne] using (eq_lemma (y := d i - x) (z := v) neq)
   obtain lim' := Filter.Tendsto.congr' eq5 lim
   refine Filter.Tendsto.of_tendsto_comp lim' ?_
   simp only [φ]; exact comap2 vne0
@@ -1002,54 +860,50 @@ theorem LICQ_linearized_feasible_directions_sub_posTangentCone
   intro v hv
 
   by_cases veq0 : v = 0
-  · rw [veq0]; rw [posTangentConeAt]; simp
-    use fun n ↦ n; use fun _ ↦ 0; simp; constructor
-    · use 0; exact fun _ _ ↦ xf
-    · exact tendsto_natCast_atTop_atTop
+  · rw [veq0]
+    change 0 ∈ tangentConeAt NNReal p.FeasSet x
+    exact zero_mem_tangentConeAt (subset_closure xf)
 
   let gradc : EuclideanSpace ℝ (Fin n) → ((p.active_set x) → (EuclideanSpace ℝ (Fin n))) :=
     fun z ↦ (fun i ↦ if i.1 ∈ τ then gradient (p.equality_constraints i) z
       else gradient (p.inequality_constraints i) z) -- gradient of the constraints
-  let Ax : Matrix (p.active_set x) (Fin n) ℝ := fun i ↦ gradc x i -- Jacobi at x
+  let Ax : Matrix (p.active_set x) (Fin n) ℝ := fun i j ↦ (gradc x i).ofLp j -- Jacobi at x
   let m := (p.active_set x).card
   have mlen : m ≤ n := by apply LICQ_mlen x LIx; simp [m]
   have existZ : ∃ (Z : Matrix (Fin n) (Fin (n - m)) ℝ), Ax * Z = 0 ∧ Matrix.rank Z = (n - m) := by
     apply LICQ_existZ x LIx; simp [m]; simp [Ax, gradc]
   rw [LICQ] at LIx;
-  rw [posTangentConeAt]; simp only [eventually_atTop, ge_iff_le, mem_setOf_eq]
+  rw [posTangentConeAt, mem_tangentConeAt_iff_exists_seq]
   rcases existZ with ⟨Z, ⟨eq1, eq2⟩⟩
 
   let Mx : EuclideanSpace ℝ (Fin n) →L[ℝ] EuclideanSpace ℝ (p.active_set x) × (Fin (n - m) → ℝ) :=
     (LinearMap.toContinuousLinearMap (Matrix.toEuclideanLin Ax)).prod
-    (LinearMap.toContinuousLinearMap (Matrix.toEuclideanLin Zᵀ)) -- Jacobi of Rz at x
+    (LinearMap.toContinuousLinearMap
+      ((Matrix.mulVecLin Zᵀ).comp (EuclideanSpace.equiv (Fin n) ℝ).toLinearMap)) -- Jacobi of Rz at x
   let c : EuclideanSpace ℝ (Fin n) → ((p.active_set x) → ℝ) :=
     fun z ↦ (fun i ↦ if i.1 ∈ τ then (p.equality_constraints i) z
       else (p.inequality_constraints i) z) -- the constraints
   let Rz : EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ (p.active_set x) × (Fin (n - m) → ℝ) :=
-    fun z ↦ (c z, Zᵀ *ᵥ (z - x)) -- z part in R
+    fun z ↦ (WithLp.toLp 2 (c z), Zᵀ *ᵥ (z - x)) -- z part in R
   let Rt : ℝ → EuclideanSpace ℝ (p.active_set x) × (Fin (n - m) → ℝ) := fun t ↦ t • Mx v -- t part in R
   let A : EuclideanSpace ℝ (Fin n) → Matrix (p.active_set x) (Fin n) ℝ :=
-    fun z ↦ (fun i ↦ gradc z i) -- compose the gradient matrix
+    fun z ↦ (fun i j ↦ (gradc z i).ofLp j) -- compose the gradient matrix
   let Jz : EuclideanSpace ℝ (Fin n) → EuclideanSpace ℝ (Fin n) →L[ℝ] EuclideanSpace ℝ (p.active_set x) :=
     fun z ↦ (LinearMap.toContinuousLinearMap (toEuclideanLin (A z))) -- change the Jacobi into linear transformation
-  have cgrad_atx : Jz x = (LinearMap.toContinuousLinearMap (toEuclideanLin Ax)) := by simp [Jz, A, gradc] -- A x = Ax
+  have cgrad_atx : Jz x = (LinearMap.toContinuousLinearMap (toEuclideanLin Ax)) := by rfl -- A x = Ax
 
   have Rzgrad : HasStrictFDerivAt Rz Mx x := by
-    simp only [Rz, Ax]
-    apply HasStrictFDerivAt.prod
+    simp only [Rz]
+    refine HasStrictFDerivAt.prodMk ?_ ?_
     · rw [← cgrad_atx]
       rw [hasStrictFDerivAt_euclidean]
       refine LICQ_strictfderiv_Ax_elem c ?_ gradc ?_ A ?_ Jz ?_ conte conti
       repeat simp only [c, gradc, A, Jz]
     · let N : EuclideanSpace ℝ (Fin n) →L[ℝ] (Fin (n - m) → ℝ) :=
-        (LinearMap.toContinuousLinearMap (toEuclideanLin Zᵀ))
-      show HasStrictFDerivAt (fun y : EuclideanSpace ℝ (Fin n) ↦ Zᵀ *ᵥ (y - x)) N x
-      rw [HasStrictFDerivAt]
-      have aux : (fun p : EuclideanSpace ℝ (Fin n) × EuclideanSpace ℝ (Fin n)
-          ↦ Zᵀ *ᵥ (p.1 - x) - Zᵀ *ᵥ (p.2 - x) - N (p.1 - p.2)) = 0 := by
-        ext y j; rw [← mulVec_sub, sub_sub, add_sub_cancel]; rw [mulVec_eq_toEuclidean]
-        simp [N]; apply sub_eq_zero_of_eq; tauto
-      rw [aux]; simp
+        (LinearMap.toContinuousLinearMap
+          ((Matrix.mulVecLin Zᵀ).comp (EuclideanSpace.equiv (Fin n) ℝ).toLinearMap))
+      change HasStrictFDerivAt (fun y : EuclideanSpace ℝ (Fin n) ↦ N (y - x)) N x
+      simpa using N.hasStrictFDerivAt.comp x ((hasStrictFDerivAt_id x).sub_const x)
 
   have Rxeq0 : Rz x = 0 := by
     simp [Rz, c]; ext i;
@@ -1063,19 +917,24 @@ theorem LICQ_linearized_feasible_directions_sub_posTangentCone
       rw [Finset.mem_filter] at hi2
       exact hi2.2
 
-  have Mxinj : LinearMap.ker Mx = ⊥ := by
-    show LinearMap.ker (Mx : EuclideanSpace ℝ (Fin n) →ₗ[ℝ] EuclideanSpace ℝ
-      (p.active_set x) × (Fin (n - m) → ℝ)) = ⊥
+  have Mxinj : Mx.ker = ⊥ := by
     rw [LinearMap.ker_eq_bot']
     intro z Mzeq0; simp [Mx] at Mzeq0
-    have heq1 : Ax *ᵥ z = 0 := by rw [mulVec_eq_toEuclidean]; apply Mzeq0.1
-    have heq2 : Zᵀ *ᵥ z = 0 := by rw [mulVec_eq_toEuclidean]; apply Mzeq0.2
+    have heq1 : Ax *ᵥ z = 0 := by
+      rw [mulVec_eq_toEuclidean]
+      exact congrArg (fun u => u.ofLp) Mzeq0.1
+    have heq2 : Zᵀ *ᵥ z = 0 := by
+      ext i
+      have h2 := congrArg (fun w => w i) Mzeq0.2
+      simpa [vecMul, mulVec, dotProduct, EuclideanSpace.equiv, mul_comm, mul_left_comm, mul_assoc] using h2
     refine LICQ_injM z m Z Ax ?_ mlen ?_ eq2 eq1 ⟨heq1, heq2⟩
     simp [m]
-    obtain hAx := LICQ_Axfullrank x LIx; simp at hAx
-    show Ax.rank = (active_set x).card; apply hAx; simp only [Ax]
-  have Mxsurj : LinearMap.range Mx = ⊤ := by
-    show LinearMap.range (Mx : EuclideanSpace ℝ (Fin n) →ₗ[ℝ] EuclideanSpace ℝ (p.active_set x) × (Fin (n - m) → ℝ)) = ⊤
+    have hAx : Ax.rank = (p.active_set x).card := by
+      simpa using (LICQ_Axfullrank (p := p) x LIx (M := Ax) (eq := by
+        ext i j
+        rfl))
+    exact hAx
+  have Mxsurj : Mx.range = ⊤ := by
     rw [← LinearMap.ker_eq_bot_iff_range_eq_top_of_finrank_eq_finrank]
     · apply Mxinj
     · simp; show n = m + (n - m)
@@ -1091,51 +950,67 @@ theorem LICQ_linearized_feasible_directions_sub_posTangentCone
   simp only [linearized_feasible_directions] at hv
   rcases hv with ⟨hvh1, hvh2⟩
   rcases implicit_f with ⟨N, d, hfd, dtend⟩
-  rw [LinearMapClass.ker_eq_bot] at Mxinj
+  rw [LinearMap.ker_eq_bot] at Mxinj
   rw [LinearMap.range_eq_top] at Mxsurj
   obtain deriv := (hasFDerivAt_iff_tendsto.1 (HasStrictFDerivAt.hasFDerivAt Rzgrad))
   obtain deriv := tendsto_nhds_iff_seq_tendsto.1 deriv d dtend
-  rw [tendsto_iff_norm_sub_tendsto_zero, NormedAddCommGroup.tendsto_nhds_zero] at dtend; simp at dtend
+  have dtend0 : Tendsto (fun n ↦ d n - x) atTop (𝓝 0) := by
+    have hsub : Tendsto (fun n ↦ d n - x) atTop (𝓝 (x - x)) := dtend.sub tendsto_const_nhds
+    simpa using hsub
+  rw [tendsto_iff_norm_sub_tendsto_zero, NormedAddGroup.tendsto_nhds_zero] at dtend; simp at dtend
   obtain ⟨ε, εpos, inactive⟩ := LICQ_inactive_nhds x xf conti
   obtain ⟨N', dtendx⟩ := dtend ε εpos
 
   use (fun n ↦ d n - x); constructor
-  · use max N N'; intro nn hnn; simp [FeasSet, FeasPoint]
-    specialize hfd nn (le_of_max_le_left hnn); simp [Rz, Rt, Mx] at hfd; rw [← mulVec_eq_toEuclidean] at hfd
-    rcases hfd with ⟨hv1, hv2⟩
-    have Axeq : (nn : ℝ)⁻¹ • Ax *ᵥ v = fun i : (p.active_set x) ↦ ((nn : ℝ)⁻¹ * (gradc x i) ⬝ᵥ v) := by
-      simp [Ax]; ext i; simp; left; simp [mulVec]
-    have Axroweq : ∀ i : (p.active_set x), c (d nn) i = (nn : ℝ)⁻¹ * (gradc x i) ⬝ᵥ v := by
-      rw [Axeq] at hv1; simp [hv1]
-    constructor; constructor
-    · rw [hdomain]; simp
-    · intro i hi
-      have iina : i ∈ (p.active_set x) := by simp [active_set, hi]
-      obtain h := hvh1 i hi
-      obtain eq := Axroweq ⟨i, iina⟩; simp [c, hi, gradc] at eq
-      rw [eq]; simp; right; apply h
-    constructor
-    · rw [hdomain]; simp
-    · intro j hj
-      have notin : j ∉ τ := by
-        by_contra hh;
-        have : j ∈ τ ∩ σ := by simp [hj, hh]
-        rw [p.eq_ine_not_intersect] at this; tauto
-      by_cases hj1 : j ∈ p.active_set x
-      · have jin : j ∈ σ ∩ (p.active_set x) := by simp [hj1, hj]
-        obtain h := hvh2 j jin
-        obtain eq := Axroweq ⟨j, hj1⟩; simp [c, hj1, notin, gradc] at eq
-        rw [eq]; field_simp
-        rw [div_nonneg_iff]; left; simp at h; simp [dotProduct, h]
-      · specialize inactive j; simp [hj, hj1] at inactive
-        specialize inactive (d nn)
-        specialize dtendx nn (le_of_max_le_right hnn); rw [← dist_eq_norm] at dtendx
-        specialize inactive dtendx; linarith [inactive]
-
-  constructor
-  · exact tendsto_natCast_atTop_atTop
-  · have Mxbij : Function.Bijective Mx := ⟨Mxinj, Mxsurj⟩
-    refine LICQ_tendsto v veq0 ?_ Rxeq0 hfd dtend Mxbij deriv; simp [Rt]
+  · exact dtend0
+  · constructor
+    · refine Filter.eventually_atTop.2 ?_
+      use max N N'
+      intro nn hnn
+      simp [FeasSet, FeasPoint]
+      specialize hfd nn (le_of_max_le_left hnn); simp [Rz, Rt, Mx] at hfd
+      rcases hfd with ⟨hv1, hv2⟩
+      have hv1' : c (d nn) = (nn : ℝ)⁻¹ • ((toEuclideanLin Ax) v).ofLp := by
+        simpa [smul_eq_mul] using congrArg (fun u => u.ofLp) hv1
+      have Axroweq : ∀ i : (p.active_set x), c (d nn) i = (nn : ℝ)⁻¹ * (gradc x i) ⬝ᵥ v := by
+        intro i
+        have hrow := congrArg (fun w => w i) hv1'
+        simpa [Ax, mulVec, dotProduct] using hrow
+      constructor; constructor
+      · rw [hdomain]; simp
+      · intro i hi
+        have iina : i ∈ (p.active_set x) := by simp [active_set, hi]
+        obtain h := hvh1 i hi
+        have hdot : (gradient (p.equality_constraints i) x).ofLp ⬝ᵥ v.ofLp = 0 := by
+          simpa [EuclideanSpace.inner_eq_star_dotProduct, dotProduct_comm] using h
+        have eq : p.equality_constraints i (d nn) = (nn : ℝ)⁻¹ * (gradient (p.equality_constraints i) x).ofLp ⬝ᵥ v.ofLp := by
+          simpa [c, gradc, hi] using Axroweq ⟨i, iina⟩
+        rw [eq]; simp; right; exact hdot
+      constructor
+      · rw [hdomain]; simp
+      · intro j hj
+        have notin : j ∉ τ := by
+          by_contra hh;
+          have : j ∈ τ ∩ σ := by simp [hj, hh]
+          rw [p.eq_ine_not_intersect] at this; tauto
+        by_cases hj1 : j ∈ p.active_set x
+        · have jin : j ∈ σ ∩ (p.active_set x) := by simp [hj1, hj]
+          obtain h := hvh2 j jin
+          have hdot : 0 ≤ (gradient (p.inequality_constraints j) x).ofLp ⬝ᵥ v.ofLp := by
+            simpa [EuclideanSpace.inner_eq_star_dotProduct, dotProduct_comm] using h
+          have eq : p.inequality_constraints j (d nn) =
+              (nn : ℝ)⁻¹ * (gradient (p.inequality_constraints j) x).ofLp ⬝ᵥ v.ofLp := by
+            simpa [c, gradc, notin] using Axroweq ⟨j, hj1⟩
+          rw [eq]; field_simp
+          rw [div_nonneg_iff]; left; exact ⟨hdot, by positivity⟩
+        · specialize inactive j; simp [hj, hj1] at inactive
+          specialize inactive (d nn)
+          specialize dtendx nn (le_of_max_le_right hnn); rw [← dist_eq_norm] at dtendx
+          specialize inactive dtendx; linarith [inactive]
+    · have Mxbij : Function.Bijective Mx := ⟨Mxinj, Mxsurj⟩
+      have htv : Tendsto (fun i : ℕ ↦ (i : ℝ) • (d i - x)) atTop (𝓝 v) := by
+        refine LICQ_tendsto v veq0 ?_ Rxeq0 hfd dtend Mxbij deriv; simp [Rt]
+      simpa [NNReal.smul_def] using htv
 
 theorem LICQ_linearized_feasible_directions_eq_posTangentCone
     (x : EuclideanSpace ℝ (Fin n)) (xf : x ∈ p.FeasSet)
@@ -1179,9 +1054,9 @@ lemma subtype_sum (σ τ : Finset ℕ) (f : σ → EuclideanSpace ℝ (Fin n))
   have : ∑ i, g i = ∑ i : {x // x ∈ σ ∩ τ},
       f {val := i.1, property := by obtain hi := i.2; rw [Finset.mem_inter] at hi; exact hi.1} := by
     congr; ext i; rw [h2]
-  rw [this]; simp [h3]
+  rw [this]; simp
   let f₁ : ℕ → EuclideanSpace ℝ (Fin n):= fun i => if h : i ∈ σ then f ⟨i, h⟩ else 0
-  have eq1 : ∑ i ∈ σ.attach, f i = ∑ i in σ, f₁ i := by
+  have eq1 : ∑ i ∈ σ.attach, f i = ∑ i ∈ σ, f₁ i := by
     simp [f₁]; nth_rw 2 [← Finset.sum_attach]; congr; simp
   have eq2 : ∑ i ∈ (σ ∩ τ).attach,
       f {val := i.1, property := by obtain hi := i.2; rw [Finset.mem_inter] at hi; exact hi.1} =
@@ -1193,8 +1068,8 @@ lemma subtype_sum (σ τ : Finset ℕ) (f : σ → EuclideanSpace ℝ (Fin n))
   obtain eq := Finset.sdiff_union_inter σ τ
   nth_rw 1 [← eq]; rw [Finset.sum_union]; simp
   have feq0 : ∀ x ∈ (σ \ τ), f₁ x = 0 := by
-    simp [f₁]; intro x _ xninτ
-    intro h; specialize h3 ⟨x, h⟩; apply h3; simp [xninτ]
+    simp [f₁]; intro x _ xninτ h
+    specialize h3 ⟨x, h⟩; apply h3; simp [xninτ]
   apply Finset.sum_eq_zero feq0
   apply Finset.disjoint_sdiff_inter σ τ
 
@@ -1244,13 +1119,19 @@ theorem first_order_neccessary_general (p1 : Constrained_OptimizationProblem (Eu
       intro i _; apply DifferentiableAt.const_mul; exact (hc1 i i.2)
       intro i _; apply DifferentiableAt.const_mul; exact (he1 i i.2)
       exact hf.differentiableAt
-      apply DifferentiableAt.sum; intro i _; apply DifferentiableAt.const_mul
-      exact (he1 i i.2)
+      convert (DifferentiableAt.sum (u := (Finset.univ : Finset τ))
+          (A := fun i m => lam i * p1.equality_constraints (↑i) m)
+          (by intro i _; exact DifferentiableAt.const_mul (he1 i i.2) (lam i))) using 1
+      ext m; simp [Finset.sum_apply]
       apply DifferentiableAt.sub hf.differentiableAt
-      apply DifferentiableAt.sum; intro i _; apply DifferentiableAt.const_mul
-      exact (he1 i i.2)
-      apply DifferentiableAt.sum; intro i _; apply DifferentiableAt.const_mul
-      exact (hc1 i i.2)
+      convert (DifferentiableAt.sum (u := (Finset.univ : Finset τ))
+          (A := fun i m => lam i * p1.equality_constraints (↑i) m)
+          (by intro i _; exact DifferentiableAt.const_mul (he1 i i.2) (lam i))) using 1
+      ext m; simp [Finset.sum_apply]
+      convert (DifferentiableAt.sum (u := (Finset.univ : Finset σ))
+          (A := fun i m => mu1 i * p1.inequality_constraints (↑i) m)
+          (by intro i _; exact DifferentiableAt.const_mul (hc1 i i.2) (mu1 i))) using 1
+      ext m; simp [Finset.sum_apply]
     constructor
     · intro j; simp [mu1]
       by_cases ht : j.1 ∈ p1.active_set loc
@@ -1263,10 +1144,9 @@ theorem first_order_neccessary_general (p1 : Constrained_OptimizationProblem (Eu
       unfold active_set at heq
       simp at heq
       rcases heq with hl | hl
-      · obtain neq := p1.eq_ine_not_intersect
-        exfalso;
-        apply absurd neq; push_neg;
-        apply Finset.ne_empty_of_mem (a := j.1) (by simp [hl])
+      · exfalso
+        have : j.1 ∈ τ ∩ σ := by simp [hl, j.2]
+        simp [p1.eq_ine_not_intersect] at this
       exact hl
     simp [ht]
 
@@ -1297,14 +1177,14 @@ variable {n : ℕ} {x : EuclideanSpace ℝ (Fin n)}
 variable {τ σ : Finset ℕ} {p : Constrained_OptimizationProblem (EuclideanSpace ℝ (Fin n)) τ σ}
 
 theorem LinearCQ_linear_constraint_eq (x : EuclideanSpace ℝ (Fin n)) (Lx : p.LinearCQ x) :
-    ∀ i ∈ τ, ∃ a, ∃ b, (equality_constraints p i) = fun y ↦ (inner a y : ℝ) + b := by
+    ∀ i ∈ τ, ∃ a, ∃ b, (equality_constraints p i) = fun y ↦ (inner ℝ a y : ℝ) + b := by
   intro i hi
   simp [LinearCQ] at Lx
   obtain Lx := (Lx).1 i ((equality_constraint_active_set x) hi) hi
   exact (IsLinear_iff' _).mp Lx
 
 theorem LinearCQ_linear_constraint_gradient_eq (x : EuclideanSpace ℝ (Fin n)) (Lx : p.LinearCQ x) :
-    ∀ i ∈ τ, ∃ a, ∃ b, ((equality_constraints p i) = fun y ↦ (inner a y : ℝ) + b) ∧
+    ∀ i ∈ τ, ∃ a, ∃ b, ((equality_constraints p i) = fun y ↦ (inner ℝ a y : ℝ) + b) ∧
     gradient (equality_constraints p i) x = a := by
   intro i hi
   obtain ⟨a, b, hab⟩ := LinearCQ_linear_constraint_eq x Lx i hi
@@ -1313,14 +1193,14 @@ theorem LinearCQ_linear_constraint_gradient_eq (x : EuclideanSpace ℝ (Fin n))
   exact (gradient_of_inner_const x a).gradient
 
 theorem LinearCQ_linear_constraint_ineq (x : EuclideanSpace ℝ (Fin n)) (Lx : p.LinearCQ x) :
-    ∀ i ∈ p.active_set x ∩ σ, ∃ a, ∃ b, (inequality_constraints p i) = fun y ↦ (inner a y : ℝ) + b := by
+    ∀ i ∈ p.active_set x ∩ σ, ∃ a, ∃ b, (inequality_constraints p i) = fun y ↦ (inner ℝ a y : ℝ) + b := by
   intro i hi
-  simp only [LinearCQ, and_imp] at Lx
+  simp only [LinearCQ] at Lx
   obtain Lx := (Lx).2 i hi
   exact (IsLinear_iff' _).mp Lx
 
 theorem LinearCQ_linear_constraint_gradient_ineq (x : EuclideanSpace ℝ (Fin n)) (Lx : p.LinearCQ x) :
-    ∀ i ∈ p.active_set x ∩ σ, ∃ a, ∃ b, ((inequality_constraints p i) = fun y ↦ (inner a y : ℝ) + b) ∧
+    ∀ i ∈ p.active_set x ∩ σ, ∃ a, ∃ b, ((inequality_constraints p i) = fun y ↦ (inner ℝ a y : ℝ) + b) ∧
     gradient (inequality_constraints p i) x = a := by
   intro i hi
   obtain ⟨a, b, hab⟩ := LinearCQ_linear_constraint_ineq x Lx i hi
@@ -1344,57 +1224,64 @@ theorem Linear_linearized_feasible_directions_eq_posTangentCone
   obtain ⟨t_, ht_, ht⟩ := inactive_constraint x v xf conti
   obtain ⟨hv1, hv2⟩ := hv
   let z := fun (k : ℕ) ↦ (t_ / (k + 1)) • v
-  simp [posTangentConeAt]
-  let c := fun (k : ℕ) ↦ (k + (1 : ℝ)) / t_
+  rw [posTangentConeAt, mem_tangentConeAt_iff_exists_seq]
+  let c : ℕ → NNReal := fun k ↦ ⟨((k : ℝ) + 1) / t_, by positivity⟩
   use c; use z
   constructor
-  · use 0; intro n hn
-    simp [FeasSet, FeasPoint]; constructor;
-    · constructor; rw [hdomain]; trivial
-      intro i hi
-      obtain ⟨a, c, ⟨hab, hg⟩⟩ := LinearCQ_linear_constraint_gradient_eq x Lx i hi
-      simp [FeasSet, FeasPoint] at xf
-      obtain ⟨⟨_, h2⟩, ⟨_, _⟩⟩ := xf
-      obtain h2 := h2 i hi; rw [← h2]; rw [hab]; simp only [RCLike.inner_apply, conj_trivial]
-      have : ⟪a, z n⟫_ℝ = 0 := by
-        obtain hv1 := hv1 i hi
-        rw [hg] at hv1
-        simp only [z]; rw [inner_smul_right, hv1, mul_zero]
-      rw [inner_add_right, this, add_zero]
-    constructor; rw [hdomain]; trivial
-    intro j hj
-    by_cases hj1 : j ∈ p.active_set x
-    · obtain hj' := Finset.mem_inter_of_mem hj1 hj
-      obtain ⟨a, c, ⟨hab, hg⟩⟩ := LinearCQ_linear_constraint_gradient_ineq x Lx j hj'
-      simp [FeasSet, FeasPoint] at xf
-      have : ⟪a, z n⟫_ℝ ≥ 0 := by
-        obtain hv2 := hv2 j (Finset.mem_inter_of_mem hj hj1)
-        rw [hg] at hv2; simp only [z]; rw [inner_smul_right]
-        positivity
-      obtain ⟨⟨_, _⟩, ⟨_, h2⟩⟩ := xf
-      simp [active_set] at hj1;
-      have : j ∉ τ := by
-        by_contra hh;
-        have : j ∈ τ ∩ σ := by simp [hj, hh]
-        rw [p.eq_ine_not_intersect] at this; tauto
-      simp [this] at hj1
-      rw [← hj1.2, hab]; simp only [RCLike.inner_apply, conj_trivial]
-      rw [inner_add_right]
+  · have hz0' : Tendsto (fun n : ℕ ↦ t_ / ((n : ℝ))) atTop (𝓝 0) :=
+      tendsto_const_div_atTop_nhds_zero_nat t_
+    have hz0 : Tendsto (fun n : ℕ ↦ t_ / ((n : ℝ) + 1)) atTop (𝓝 0) := by
+      simpa [Nat.cast_add, Nat.cast_one, add_assoc, add_comm, add_left_comm] using
+        (Filter.tendsto_add_atTop_iff_nat 1).2 hz0'
+    simpa [z] using hz0.smul_const v
+  · constructor
+    · refine Filter.eventually_atTop.2 ?_
+      use 0
+      intro n hn
+      simp [FeasSet, FeasPoint]; constructor;
+      · constructor; rw [hdomain]; trivial
+        intro i hi
+        obtain ⟨a, c, ⟨hab, hg⟩⟩ := LinearCQ_linear_constraint_gradient_eq x Lx i hi
+        simp [FeasSet, FeasPoint] at xf
+        obtain ⟨⟨_, h2⟩, ⟨_, _⟩⟩ := xf
+        obtain h2 := h2 i hi; rw [← h2]; rw [hab]; simp
+        have : ⟪a, z n⟫_ℝ = 0 := by
+          obtain hv1 := hv1 i hi
+          rw [hg] at hv1
+          simp only [z]; rw [inner_smul_right, hv1, mul_zero]
+        rw [inner_add_right, this, add_zero]
+      constructor; rw [hdomain]; trivial
+      intro j hj
+      by_cases hj1 : j ∈ p.active_set x
+      · obtain hj' := Finset.mem_inter_of_mem hj1 hj
+        obtain ⟨a, c, ⟨hab, hg⟩⟩ := LinearCQ_linear_constraint_gradient_ineq x Lx j hj'
+        simp [FeasSet, FeasPoint] at xf
+        have : ⟪a, z n⟫_ℝ ≥ 0 := by
+          obtain hv2 := hv2 j (Finset.mem_inter_of_mem hj hj1)
+          rw [hg] at hv2; simp only [z]; rw [inner_smul_right]
+          positivity
+        obtain ⟨⟨_, _⟩, ⟨_, h2⟩⟩ := xf
+        simp [active_set] at hj1;
+        have : j ∉ τ := by
+          by_contra hh;
+          have : j ∈ τ ∩ σ := by simp [hj, hh]
+          rw [p.eq_ine_not_intersect] at this; tauto
+        simp [this] at hj1
+        rw [← hj1.2, hab]; simp
+        rw [inner_add_right]
+        linarith
+      simp [z]
+      have : (t_ / (↑n + 1)) ∈ Icc 0 t_ := by
+        simp; constructor; positivity
+        apply div_le_self (by linarith) (by linarith)
+      obtain ht := ht _ this j (Finset.mem_sdiff.mpr ⟨hj, hj1⟩)
       linarith
-    simp [z]
-    have : (t_ / (↑n + 1)) ∈ Icc 0 t_ := by
-      simp; constructor; positivity
-      apply div_le_self (by linarith) (by linarith)
-    obtain ht := ht _ this j (Finset.mem_sdiff.mpr ⟨hj, hj1⟩)
-    linarith
-  constructor
-  · apply Filter.Tendsto.atTop_div_const ht_
-    apply tendsto_atTop_add_nonneg_right'
-    · exact tendsto_natCast_atTop_atTop
-    apply Filter.Eventually.of_forall; exact fun x ↦ zero_le_one' ℝ
-  apply tendsto_atTop_of_eventually_const (i₀ := 1)
-  intro i hi; simp [c, z]
-  rw [smul_smul]; field_simp
+    · apply tendsto_atTop_of_eventually_const (i₀ := 1)
+      intro i hi
+      simp [c, z, NNReal.smul_def]
+      rw [smul_smul]
+      field_simp
+      simp
 
 theorem first_order_neccessary_LinearCQ
     (p1 : Constrained_OptimizationProblem (EuclideanSpace ℝ (Fin n)) τ σ)
diff --git a/Optlib/Optimality/OptimalityConditionOfUnconstrainedProblem.lean b/Optlib/Optimality/OptimalityConditionOfUnconstrainedProblem.lean
index 757ec1e..b3e05f7 100644
--- a/Optlib/Optimality/OptimalityConditionOfUnconstrainedProblem.lean
+++ b/Optlib/Optimality/OptimalityConditionOfUnconstrainedProblem.lean
@@ -18,7 +18,7 @@ open Set InnerProductSpace
   x with d is less than zero.
 -/
 def DescentDirection (d : E) (x : E) (_ : HasGradientAt f (f' x) x) : Prop :=
-  inner (f' x) d < (0 : ℝ)
+  inner ℝ (f' x) d < (0 : ℝ)
 
 /-
   For any vector d, there does not exist a descent direction for the function f
@@ -30,19 +30,19 @@ theorem optimal_no_descent_direction (hf : ∀ x : E, HasGradientAt f (f' x) x)
   intro d
   by_contra h
   have : ∃ t : ℝ , f (xm + t • d) < f xm := by
-    have h₁ : ∃ T : ℝ , T > 0 ∧ (∀ a ∈ Icc (- T) T, inner (f' (xm + a • d)) d < (0 : ℝ)) := by
-      let g := fun r : ℝ ↦ (inner (f' (xm + r • d)) d : ℝ)
-      have hg0 : g 0 = inner (f' xm) d := by simp [g]
+    have h₁ : ∃ T : ℝ , T > 0 ∧ (∀ a ∈ Icc (- T) T, inner ℝ (f' (xm + a • d)) d < (0 : ℝ)) := by
+      let g := fun r : ℝ ↦ (inner ℝ (f' (xm + r • d)) d : ℝ)
+      have hg0 : g 0 = inner ℝ (f' xm) d := by simp [g]
       have hc : ContinuousOn g univ := by
-        simp [g]
+        change ContinuousOn (fun r : ℝ ↦ inner ℝ (f' (xm + r • d)) d) univ
         apply ContinuousOn.inner
         · apply ContinuousOn.comp hfc
           · apply ContinuousOn.add continuousOn_const
             apply ContinuousOn.smul continuousOn_id continuousOn_const
           · simp
         · exact continuousOn_const
-      have hu : ∃ u < (0 : ℝ) , inner (f' xm) d ≤  u := by
-        use (inner (f' xm) d / 2)
+      have hu : ∃ u < (0 : ℝ) , inner ℝ (f' xm) d ≤  u := by
+        use (inner ℝ (f' xm) d / 2)
         rw [DescentDirection] at h
         constructor
         · linarith
@@ -72,7 +72,7 @@ theorem optimal_no_descent_direction (hf : ∀ x : E, HasGradientAt f (f' x) x)
       use T
     rcases h₁ with ⟨T, ⟨hT1,hT2⟩⟩
     have h₂ : ∃ t1 : ℝ, t1 ≥ -T ∧ t1 ≤ T ∧ f (xm + T • d) =
-        f xm + inner (f' (xm + t1 • d)) (T • d) := by
+        f xm + inner ℝ (f' (xm + t1 • d)) (T • d) := by
       rcases (expansion hf xm (T • d)) with ⟨ts,⟨ts1,⟨ts2,ts3⟩⟩⟩
       use (ts • T)
       constructor
@@ -111,7 +111,7 @@ theorem first_order_unconstrained (hf : ∀ x : E, HasGradientAt f (f' x) x) (mi
 -/
 theorem first_order_convex (hf : ∀ x : E, HasGradientAt f (f' x) x) (hcon : ConvexOn ℝ univ f)
     (hfm : f' xm = 0) : IsMinOn f univ xm := by
-  have : ∀ y , f y ≥ f xm + inner (f' xm) (y - xm) := by
+  have : ∀ y , f y ≥ f xm + inner ℝ (f' xm) (y - xm) := by
     intro y
     apply Convex_first_order_condition' (hf xm) hcon (by trivial)
     · trivial
diff --git a/Optlib/Optimality/Weak_Duality.lean b/Optlib/Optimality/Weak_Duality.lean
index 4be2b67..a315b20 100644
--- a/Optlib/Optimality/Weak_Duality.lean
+++ b/Optlib/Optimality/Weak_Duality.lean
@@ -51,13 +51,15 @@ variable {p : Constrained_OptimizationProblem E τ σ} {x : E}
 
 lemma empty_domain_inf_value_top {p : Constrained_OptimizationProblem E τ σ} (hp : (p.domain) = ∅) :
     p.inf_value = ⊤ := by
-  unfold inf_value
-  unfold FeasSet FeasPoint
+  unfold inf_value FeasSet FeasPoint
   simp [hp]
+  rfl
 
 lemma empty_FeasSet_inf_value_top {p : Constrained_OptimizationProblem E τ σ} (hp : (p.FeasSet) = ∅) :
     p.inf_value = ⊤ := by
-  unfold inf_value; simp [hp]
+  unfold inf_value
+  simp [hp]
+  rfl
 
 lemma objective_le_sup {p : Constrained_OptimizationProblem E τ σ} (x : E) (hx : x ∈ p.FeasSet) :
     (p.objective x).toEReal ≤ p.sup_value := by
@@ -70,16 +72,18 @@ lemma dual_objective_le_top_nonempty {p : Constrained_OptimizationProblem E τ 
   intro lambda1 lambda2
   unfold dual_objective
   let x := Classical.choose hp
-  apply iInf_lt_top.mpr
-  use x; simp; apply Classical.choose_spec hp
+  refine iInf_lt_top.mpr ?_
+  refine ⟨x, ?_⟩
+  refine iInf_lt_top.mpr ?_
+  exact ⟨Classical.choose_spec hp, by
+    exact EReal.coe_lt_top (p.Lagrange_function x lambda1 lambda2)⟩
 
 lemma dual_objective_eq_top_empty {p : Constrained_OptimizationProblem E τ σ} (hp : (p.domain) = ∅) :
     ∀ lambda1 lambda2, p.dual_objective lambda1 lambda2 = ⊤ := by
-  intro lambda1 lambda2; unfold dual_objective
-  simp; intro x
-  by_contra h
-  have : x ∉ p.domain := by exact of_eq_false (congrFun hp x)
-  exact this h
+  intro lambda1 lambda2
+  unfold dual_objective
+  simp [hp]
+  rfl
 
 lemma objective_infimum_global_minimum {p : Constrained_OptimizationProblem E τ σ}
     (hp : (p.objective x).toEReal = p.inf_value) (hx : x ∈ p.FeasSet) :
@@ -153,8 +157,8 @@ theorem weak_duality {p : Constrained_OptimizationProblem E τ σ}
 theorem weak_duality_aux {p : Constrained_OptimizationProblem E τ σ} (hp : (p.domain).Nonempty) :
     (p.dual_problem).sup_value ≤ p.inf_value := by
   unfold sup_value dual_problem; simp
-  intro b x lambda1 lambda2 hl hl2 hl3
-  rw [← hl3, ← hl2]
+  intro b lambda1 lambda2 hl hl2
+  rw [← hl2]
   have : ((p.dual_objective lambda1 lambda2).toReal).toEReal
       = p.dual_objective lambda1 lambda2 := by
     apply EReal.coe_toReal
@@ -175,7 +179,7 @@ theorem weak_duality' {p : Constrained_OptimizationProblem E τ σ} :
   · exact weak_duality_aux hp
   push_neg at hp
   rw [empty_domain_inf_value_top hp]
-  simp only [le_top]
+  exact le_top
 
 end WeakDuality
 
@@ -185,10 +189,10 @@ variable {E : Type _} {τ σ : Finset ℕ}
 variable [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
 variable {p : Constrained_OptimizationProblem E τ σ}
 
-lemma ConcaveOn.sum {α 𝕜 : Type*} [OrderedSemiring 𝕜] [AddCommMonoid α][SMul 𝕜 α]
-    {ι : Type*} [DecidableEq ι] {s : Finset ι} {t : s → α → 𝕜} {d : Set α}
-    (h : ∀ i : s, ConcaveOn 𝕜 d (t i)) (hd : Convex 𝕜 d):
-    ConcaveOn 𝕜 d (fun x => ∑ i : s, t i x) := by
+lemma concaveOn_sum {α : Type*} [AddCommMonoid α] [SMul ℝ α]
+    {ι : Type*} [DecidableEq ι] {s : Finset ι} {t : s → α → ℝ} {d : Set α}
+    (h : ∀ i : s, ConcaveOn ℝ d (t i)) (hd : Convex ℝ d) :
+    ConcaveOn ℝ d (fun x => ∑ i : s, t i x) := by
   constructor
   · exact hd
   intro x hx y hy a b ha hb hab
@@ -205,12 +209,12 @@ theorem convex_problem_convex_Lagrange {p : Constrained_OptimizationProblem E τ
     (lambda1 : τ → ℝ) (lambda2 : σ → ℝ)
     (hKKT : KKT_point p x lambda1 lambda2) :
     ConvexOn ℝ univ (fun m ↦ p.Lagrange_function m lambda1 lambda2) := by
+  subst hτ
   unfold Lagrange_function
   apply ConvexOn.sub
   · apply ConvexOn.sub h
-    simp [hτ]; apply concaveOn_const 0
-    exact convex_univ
-  apply ConcaveOn.sum _ convex_univ
+    simpa using (concaveOn_const (𝕜 := ℝ) (s := (univ : Set E)) (0 : ℝ) convex_univ)
+  apply concaveOn_sum _ convex_univ
   intro i
   apply ConcaveOn.smul
   · unfold KKT_point at hKKT
@@ -228,10 +232,10 @@ theorem diff_problem_diff_Lagrange {p : Constrained_OptimizationProblem E τ σ}
   · apply DifferentiableAt.sub
     · exact hf
     simp [hτ]
-  apply DifferentiableAt.sum
-  intro i _
-  apply DifferentiableAt.const_mul _ (lambda2 i)
-  apply conti i i.2
+  convert (DifferentiableAt.sum (u := (Finset.univ : Finset σ))
+      (A := fun j m => lambda2 j * p.inequality_constraints (↑j) m)
+      (by intro i _; exact DifferentiableAt.const_mul (conti i i.2) (lambda2 i))) using 1
+  ext m; simp [Finset.sum_apply]
 
 theorem KKT_multipliers_objective_eq_Lagrangian {p : Constrained_OptimizationProblem E τ σ}
     (x : E) (lambda1 : τ → ℝ) (lambda2 : σ → ℝ)
diff --git a/lake-manifest.json b/lake-manifest.json
index f0afaa7..38d48d5 100644
--- a/lake-manifest.json
+++ b/lake-manifest.json
@@ -1,52 +1,32 @@
 {"version": "1.1.0",
  "packagesDir": ".lake/packages",
  "packages":
- [{"url": "https://github.com/leanprover-community/batteries",
+ [{"url": "https://github.com/leanprover-community/mathlib4",
    "type": "git",
    "subDir": null,
-   "scope": "leanprover-community",
-   "rev": "31a10a332858d6981dbcf55d54ee51680dd75f18",
-   "name": "batteries",
-   "manifestFile": "lake-manifest.json",
-   "inputRev": "main",
-   "inherited": true,
-   "configFile": "lakefile.toml"},
-  {"url": "https://github.com/leanprover-community/quote4",
-   "type": "git",
-   "subDir": null,
-   "scope": "leanprover-community",
-   "rev": "1357f4f49450abb9dfd4783e38219f4ce84f9785",
-   "name": "Qq",
+   "scope": "",
+   "rev": "5c8398df528176d9c87ccd9226ba8f7c8852d59c",
+   "name": "mathlib",
    "manifestFile": "lake-manifest.json",
-   "inputRev": "master",
-   "inherited": true,
+   "inputRev": "v4.29.0-rc6",
+   "inherited": false,
    "configFile": "lakefile.lean"},
-  {"url": "https://github.com/leanprover-community/aesop",
+  {"url": "https://github.com/leanprover-community/plausible",
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "5f934891e11d70a1b86e302fdf9cecfc21e8de46",
-   "name": "aesop",
+   "rev": "e84e3e16aea6b72cc5d311ca1bb25caad417e162",
+   "name": "plausible",
    "manifestFile": "lake-manifest.json",
-   "inputRev": "master",
+   "inputRev": "main",
    "inherited": true,
    "configFile": "lakefile.toml"},
-  {"url": "https://github.com/leanprover-community/ProofWidgets4",
+  {"url": "https://github.com/leanprover-community/LeanSearchClient",
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "23268f52d3505955de3c26a42032702c25cfcbf8",
-   "name": "proofwidgets",
-   "manifestFile": "lake-manifest.json",
-   "inputRev": "v0.0.44",
-   "inherited": true,
-   "configFile": "lakefile.lean"},
-  {"url": "https://github.com/leanprover/lean4-cli",
-   "type": "git",
-   "subDir": null,
-   "scope": "leanprover",
-   "rev": "2cf1030dc2ae6b3632c84a09350b675ef3e347d0",
-   "name": "Cli",
+   "rev": "c5d5b8fe6e5158def25cd28eb94e4141ad97c843",
+   "name": "LeanSearchClient",
    "manifestFile": "lake-manifest.json",
    "inputRev": "main",
    "inherited": true,
@@ -55,81 +35,61 @@
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "984d7ee170b75d6b03c0903e0b750ee2c6d1e3fb",
+   "rev": "f207d9fcf0cef00ba79962a33ef156061914d9c7",
    "name": "importGraph",
    "manifestFile": "lake-manifest.json",
    "inputRev": "main",
    "inherited": true,
    "configFile": "lakefile.toml"},
-  {"url": "https://github.com/leanprover-community/LeanSearchClient",
+  {"url": "https://github.com/leanprover-community/ProofWidgets4",
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "7bedaed1ef024add1e171cc17706b012a9a37802",
-   "name": "LeanSearchClient",
+   "rev": "2e58165a9dcdca9837b666528f974299ee1a51cc",
+   "name": "proofwidgets",
    "manifestFile": "lake-manifest.json",
-   "inputRev": "main",
+   "inputRev": "v0.0.92",
    "inherited": true,
-   "configFile": "lakefile.toml"},
-  {"url": "https://github.com/leanprover-community/plausible",
+   "configFile": "lakefile.lean"},
+  {"url": "https://github.com/leanprover-community/aesop",
    "type": "git",
    "subDir": null,
    "scope": "leanprover-community",
-   "rev": "d212dd74414e997653cd3484921f4159c955ccca",
-   "name": "plausible",
+   "rev": "c3361708f266893de5d1769192b60d4b1831f2bb",
+   "name": "aesop",
    "manifestFile": "lake-manifest.json",
-   "inputRev": "main",
+   "inputRev": "master",
    "inherited": true,
    "configFile": "lakefile.toml"},
-  {"url": "https://github.com/leanprover-community/mathlib4",
-   "type": "git",
-   "subDir": null,
-   "scope": "",
-   "rev": "d7317655e2826dc1f1de9a0c138db2775c4bb841",
-   "name": "mathlib",
-   "manifestFile": "lake-manifest.json",
-   "inputRev": "v4.13.0",
-   "inherited": false,
-   "configFile": "lakefile.lean"},
-  {"url": "https://github.com/acmepjz/md4lean",
+  {"url": "https://github.com/leanprover-community/quote4",
    "type": "git",
    "subDir": null,
-   "scope": "",
-   "rev": "5e95f4776be5e048364f325c7e9d619bb56fb005",
-   "name": "MD4Lean",
+   "scope": "leanprover-community",
+   "rev": "221e8088e3a066b8676dc471ff10638cf1c10835",
+   "name": "Qq",
    "manifestFile": "lake-manifest.json",
-   "inputRev": "main",
+   "inputRev": "master",
    "inherited": true,
-   "configFile": "lakefile.lean"},
-  {"url": "https://github.com/fgdorais/lean4-unicode-basic",
+   "configFile": "lakefile.toml"},
+  {"url": "https://github.com/leanprover-community/batteries",
    "type": "git",
    "subDir": null,
-   "scope": "",
-   "rev": "107e98b3e7603628d9bfd817b4704488d8a25e96",
-   "name": "UnicodeBasic",
+   "scope": "leanprover-community",
+   "rev": "bd58e3506632241b59e406902d5e42b73cdeccce",
+   "name": "batteries",
    "manifestFile": "lake-manifest.json",
    "inputRev": "main",
    "inherited": true,
-   "configFile": "lakefile.lean"},
-  {"url": "https://github.com/dupuisf/BibtexQuery",
+   "configFile": "lakefile.toml"},
+  {"url": "https://github.com/leanprover/lean4-cli",
    "type": "git",
    "subDir": null,
-   "scope": "",
-   "rev": "bdc2fc30b1e834b294759a5d391d83020a90058e",
-   "name": "BibtexQuery",
+   "scope": "leanprover",
+   "rev": "3de531c1135f5e3a01f3ac04830996fda476b28e",
+   "name": "Cli",
    "manifestFile": "lake-manifest.json",
-   "inputRev": "master",
+   "inputRev": "v4.29.0-rc6",
    "inherited": true,
-   "configFile": "lakefile.lean"},
-  {"url": "https://github.com/leanprover/doc-gen4.git",
-   "type": "git",
-   "subDir": null,
-   "scope": "",
-   "rev": "c2156beadb1a4d049ff3b19fe396c5403025aac5",
-   "name": "«doc-gen4»",
-   "manifestFile": "lake-manifest.json",
-   "inputRev": "c2156beadb1a4d049ff3b19fe396c5403025aac5",
-   "inherited": false,
-   "configFile": "lakefile.lean"}],
+   "configFile": "lakefile.toml"}],
  "name": "optlib",
  "lakeDir": ".lake"}
diff --git a/lakefile.lean b/lakefile.lean
index 26f93ec..4e94bc9 100644
--- a/lakefile.lean
+++ b/lakefile.lean
@@ -10,7 +10,7 @@ package optlib where
 @[default_target]
 lean_lib Optlib where
 
-require mathlib from git "https://github.com/leanprover-community/mathlib4"@"v4.13.0"
+require mathlib from git "https://github.com/leanprover-community/mathlib4"@"v4.29.0-rc6"
 
 meta if get_config? env = some "CI_BUILD" then
 require «doc-gen4» from git
diff --git a/lean-toolchain b/lean-toolchain
index 4f86f95..87b20aa 100644
--- a/lean-toolchain
+++ b/lean-toolchain
@@ -1 +1 @@
-leanprover/lean4:v4.13.0
+leanprover/lean4:v4.29.0-rc6