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[Merged by Bors] - chore(Logic/Equiv/Fin/Rotate): generalize from Fin (n + 1) to Fin n #37338

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[Merged by Bors] - chore(Logic/Equiv/Fin/Rotate): generalize from Fin (n + 1) to Fin n #37338

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be09f6d
generalize
vihdzp Mar 29, 2026
202eaaf
fix
vihdzp Mar 29, 2026
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1 change: 0 additions & 1 deletion Mathlib/Geometry/Polygon/Basic.lean
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Expand Up @@ -84,7 +84,6 @@ theorem HasNondegenerateVertices.hasNondegenerateEdges [NeZero n] [Nontrivial R]
(h : poly.HasNondegenerateVertices R) : poly.HasNondegenerateEdges := by
obtain ⟨m, rfl⟩ := Nat.exists_eq_succ_of_ne_zero (NeZero.ne n)
intro i
simp only [finRotate_succ_apply]
simpa using (h i).injective.ne (by decide : (0 : Fin 3) ≠ 1)

theorem HasNondegenerateVertices.three_le [NeZero n] [Nontrivial R] {poly : Polygon P n}
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42 changes: 27 additions & 15 deletions Mathlib/Logic/Equiv/Fin/Rotate.lean
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Expand Up @@ -64,27 +64,34 @@ theorem Fin.snoc_eq_cons_rotate {α : Type*} (v : Fin n → α) (a : α) :
theorem finRotate_one : finRotate 1 = Equiv.refl _ :=
Subsingleton.elim _ _

@[simp] theorem finRotate_succ_apply (i : Fin (n + 1)) : finRotate (n + 1) i = i + 1 := by
cases n
· exact @Subsingleton.elim (Fin 1) _ _ _
obtain rfl | h := Fin.eq_or_lt_of_le i.le_last
· simp [finRotate_last]
· cases i
simp only [Fin.lt_def, Fin.val_last] at h
simp [finRotate_of_lt h, Fin.add_def, Nat.mod_eq_of_lt (Nat.succ_lt_succ h)]
@[simp]
theorem finRotate_apply (i : Fin n) : haveI := i.neZero; finRotate n i = i + 1 := by
match n with
| 0 => exact i.elim0
| 1 => exact @Subsingleton.elim (Fin 1) _ _ _
| n + 2 =>
obtain rfl | h := Fin.eq_or_lt_of_le i.le_last
· simp [finRotate_last]
· cases i
simp only [Fin.lt_def, Fin.val_last] at h
simp [finRotate_of_lt h, Fin.add_def, Nat.mod_eq_of_lt (Nat.succ_lt_succ h)]

@[deprecated finRotate_apply (since := "2026-03-29")]
theorem finRotate_succ_apply (i : Fin (n + 1)) : finRotate (n + 1) i = i + 1 := by
simp

theorem finRotate_apply_zero : finRotate n.succ 0 = 1 := by
simp

theorem coe_finRotate_of_ne_last {i : Fin n.succ} (h : i ≠ Fin.last n) :
(finRotate (n + 1) i : ℕ) = i + 1 := by
rw [finRotate_succ_apply]
rw [finRotate_apply]
have : (i : ℕ) < n := Fin.val_lt_last h
exact Fin.val_add_one_of_lt this

theorem coe_finRotate (i : Fin n.succ) :
(finRotate n.succ i : ℕ) = if i = Fin.last n then (0 : ℕ) else i + 1 := by
rw [finRotate_succ_apply, Fin.val_add_one i]
rw [finRotate_apply, Fin.val_add_one i]

theorem lt_finRotate_iff_ne_last (i : Fin (n + 1)) :
i < finRotate _ i ↔ i ≠ Fin.last n := by
Expand All @@ -95,14 +102,19 @@ theorem lt_finRotate_iff_ne_neg_one [NeZero n] (i : Fin n) :
obtain ⟨n, rfl⟩ := exists_eq_succ_of_ne_zero (NeZero.ne n)
rw [lt_finRotate_iff_ne_last, ne_eq, not_iff_not, ← Fin.neg_last, neg_neg]

@[simp] lemma finRotate_succ_symm_apply [NeZero n] (i : Fin n) : (finRotate _).symm i = i - 1 := by
obtain ⟨n, rfl⟩ := exists_eq_succ_of_ne_zero (NeZero.ne n)
@[simp]
lemma finRotate_symm_apply (i : Fin n) : haveI := i.neZero; (finRotate _).symm i = i - 1 := by
obtain ⟨n, rfl⟩ := exists_eq_succ_of_ne_zero i.pos.ne'
apply (finRotate n.succ).symm_apply_eq.mpr
rw [finRotate_succ_apply, sub_add_cancel]
rw [finRotate_apply, sub_add_cancel]

@[deprecated finRotate_symm_apply (since := "2026-03-29")]
lemma finRotate_succ_symm_apply [NeZero n] (i : Fin n) : (finRotate _).symm i = i - 1 := by
simp

lemma coe_finRotate_symm_of_ne_zero [NeZero n] {i : Fin n} (hi : i ≠ 0) :
((finRotate _).symm i : ℕ) = i - 1 := by
rwa [finRotate_succ_symm_apply, Fin.val_sub_one_of_ne_zero]
rwa [finRotate_symm_apply, Fin.val_sub_one_of_ne_zero]

theorem finRotate_symm_lt_iff_ne_zero [NeZero n] (i : Fin n) :
(finRotate _).symm i < i ↔ i ≠ 0 := by
Expand All @@ -128,5 +140,5 @@ lemma finCycle_eq_finRotate_iterate {k : Fin n} : finCycle k = (finRotate n)^[k.
| zero => simp
| succ k ih =>
rw [Fin.val_eq_val, Fin.val_castSucc] at ih
rw [Fin.val_succ, Function.iterate_succ', Function.comp_apply, ← ih, finRotate_succ_apply,
rw [Fin.val_succ, Function.iterate_succ', Function.comp_apply, ← ih, finRotate_apply,
finCycle_apply, finCycle_apply, add_assoc, Fin.coeSucc_eq_succ]
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