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refactor: free simps

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jstoobysmith 2025-01-05 16:46:15 +00:00
parent 9184f6087c
commit 7026d8ce66
7 changed files with 150 additions and 146 deletions
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12
HepLean/Mathematics/Fin.lean
View file

@ -101,7 +101,7 @@ lemma succAbove_succAbove_predAboveI (i : Fin n.succ.succ) (j : Fin n.succ) (x :
· rw [Fin.succAbove_of_castSucc_lt _ _]
exact hx1
· rw [Fin.lt_def] at h1 hx1 ⊢
simp_all
simp_all only [Nat.succ_eq_add_one, Fin.coe_castSucc]
omega
· exact Nat.lt_trans hx1 h1
· simp only [not_lt] at hx1
@ -115,7 +115,7 @@ lemma succAbove_succAbove_predAboveI (i : Fin n.succ.succ) (j : Fin n.succ) (x :
· rfl
· exact hx1
· rw [Fin.lt_def] at hx2 ⊢
simp_all
simp_all only [Nat.succ_eq_add_one, Fin.coe_castSucc, Fin.val_succ]
omega
· simp only [Nat.succ_eq_add_one, not_lt] at hx2
rw [Fin.succAbove_of_le_castSucc _ _ hx2]
@ -135,12 +135,12 @@ lemma succAbove_succAbove_predAboveI (i : Fin n.succ.succ) (j : Fin n.succ) (x :
· nth_rewrite 2 [Fin.succAbove_of_le_castSucc _ _]
· rw [Fin.succAbove_of_le_castSucc _ _]
rw [Fin.le_def] at hx1 ⊢
simp_all
simp_all only [Nat.succ_eq_add_one, Fin.coe_castSucc, Fin.val_succ, add_le_add_iff_right]
· rw [Fin.le_def] at h1 hx1 ⊢
simp_all
simp_all only [Nat.succ_eq_add_one, Fin.coe_castSucc]
omega
· rw [Fin.le_def] at hx1 h1 ⊢
simp_all
simp_all only [Nat.succ_eq_add_one, Fin.coe_castSucc, Fin.val_succ]
omega
· simp only [Nat.succ_eq_add_one, not_le] at hx1
rw [Fin.lt_def] at hx1
@ -151,7 +151,7 @@ lemma succAbove_succAbove_predAboveI (i : Fin n.succ.succ) (j : Fin n.succ) (x :
nth_rewrite 2 [Fin.succAbove_of_castSucc_lt _ _]
· rw [Fin.succAbove_of_castSucc_lt _ _]
rw [Fin.lt_def] at hx2 ⊢
simp_all
simp_all only [Nat.succ_eq_add_one, Fin.coe_castSucc, Fin.val_succ]
omega
· rw [Fin.lt_def] at hx2 ⊢
simp_all

132
HepLean/Mathematics/Fin/Involutions.lean
View file

@ -29,14 +29,15 @@ def involutionCons (n : ℕ) : {f : Fin n.succ → Fin n.succ // Function.Involu
intro i
by_cases h : f.1 i.succ = 0
· simp [h]
· simp [h]
simp [f.2 i.succ]
· simp only [succ_eq_add_one, h, ↓reduceDIte, Fin.succ_pred]
simp only [f.2 i.succ, Fin.pred_succ, dite_eq_ite, ite_eq_right_iff]
intro h
exact False.elim (Fin.succ_ne_zero i h)⟩,
⟨if h : f.1 0 = 0 then none else Fin.pred (f.1 0) h , by
by_cases h0 : f.1 0 = 0
· simp [h0]
· simp [h0]
· simp only [succ_eq_add_one, h0, ↓reduceDIte, Option.isSome_some, Option.get_some,
Fin.succ_pred, dite_eq_left_iff, Fin.pred_inj, forall_const]
refine fun h => False.elim (h (f.2 0))⟩⟩
invFun f := ⟨
if h : (f.2.1).isSome then
@ -60,7 +61,7 @@ def involutionCons (n : ℕ) : {f : Fin n.succ → Fin n.succ // Function.Involu
simp
· rw [Function.update_apply ]
rw [if_neg hja]
simp
simp only [Function.comp_apply, Fin.cons_succ]
have hf2 := f.2.2 hs
change f.1.1 a = a at hf2
have hjf1 : f.1.1 j ≠ a := by
@ -71,40 +72,41 @@ def involutionCons (n : ℕ) : {f : Fin n.succ → Fin n.succ // Function.Involu
rw [hf2] at haj
exact hja haj
rw [Function.update_apply, if_neg hjf1]
simp
simp only [Function.comp_apply, Fin.succ_inj]
rw [f.1.2]
· simp [hs]
· simp only [succ_eq_add_one, hs, Bool.false_eq_true, ↓reduceDIte]
intro i
rcases Fin.eq_zero_or_eq_succ i with hi | ⟨j, hj⟩
· subst hi
simp
· subst hj
simp
simp only [Fin.cons_succ, Function.comp_apply, Fin.succ_inj]
rw [f.1.2]⟩
left_inv f := by
match f with
| ⟨f, hf⟩ =>
simp
simp only [succ_eq_add_one, Option.isSome_dite', Option.get_dite', Fin.succ_pred,
Fin.cons_update, dite_eq_ite, ite_not, Subtype.mk.injEq]
ext i
by_cases h0 : f 0 = 0
· simp [h0]
· simp only [h0, ↓reduceIte]
rcases Fin.eq_zero_or_eq_succ i with hi | ⟨j, hj⟩
· subst hi
simp [h0]
· subst hj
simp [h0]
simp only [Fin.cons_succ, Function.comp_apply, Fin.val_succ]
by_cases hj : f j.succ =0
· rw [← h0] at hj
have hn := Function.Involutive.injective hf hj
exact False.elim (Fin.succ_ne_zero j hn)
· simp [hj]
· simp only [hj, ↓reduceDIte, Fin.coe_pred]
rw [Fin.ext_iff] at hj
simp at hj
simp only [succ_eq_add_one, Fin.val_zero] at hj
omega
· rw [if_neg h0]
by_cases hf' : i = f 0
· subst hf'
simp
simp only [Function.update_same, Fin.val_zero]
rw [hf]
simp
· rw [Function.update_apply, if_neg hf']
@ -112,31 +114,33 @@ def involutionCons (n : ℕ) : {f : Fin n.succ → Fin n.succ // Function.Involu
· subst hi
simp
· subst hj
simp
simp only [Fin.cons_succ, Function.comp_apply, Fin.val_succ]
by_cases hj : f j.succ =0
· rw [← hj] at hf'
rw [hf] at hf'
simp at hf'
· simp [hj]
simp only [not_true_eq_false] at hf'
· simp only [hj, ↓reduceDIte, Fin.coe_pred]
rw [Fin.ext_iff] at hj
simp at hj
simp only [succ_eq_add_one, Fin.val_zero] at hj
omega
right_inv f := by
match f with
| ⟨⟨f, hf⟩, ⟨f0, hf0⟩⟩ =>
ext i
· simp
· simp only [succ_eq_add_one, Fin.cons_update]
by_cases hs : f0.isSome
· simp [hs]
· simp only [hs, ↓reduceDIte]
by_cases hi : i = f0.get hs
· simp [hi, Function.update_apply]
· simp only [Function.update_apply, hi, ↓reduceIte, ↓reduceDIte]
exact Eq.symm (Fin.val_eq_of_eq (hf0 hs))
· simp [hi]
· simp only [ne_eq, Fin.succ_inj, hi, not_false_eq_true, Function.update_noteq,
Fin.cons_succ, Function.comp_apply, Fin.pred_succ, dite_eq_ite]
split
· rename_i h
exact False.elim (Fin.succ_ne_zero (f i) h)
· rfl
· simp [hs]
· simp only [hs, Bool.false_eq_true, ↓reduceDIte, Fin.cons_succ, Function.comp_apply,
Fin.pred_succ, dite_eq_ite]
split
· rename_i h
exact False.elim (Fin.succ_ne_zero (f i) h)
@ -145,9 +149,9 @@ def involutionCons (n : ℕ) : {f : Fin n.succ → Fin n.succ // Function.Involu
Option.dite_none_left_eq_some, Option.some.injEq]
by_cases hs : f0.isSome
· simp only [hs, ↓reduceDIte]
simp [Fin.cons_zero]
simp only [Fin.cons_zero, Fin.pred_succ, exists_prop]
have hx : ¬ (f0.get hs).succ = 0 := (Fin.succ_ne_zero (f0.get hs))
simp [hx]
simp only [hx, not_false_eq_true, true_and]
refine Iff.intro (fun hi => ?_) (fun hi => ?_)
· rw [← hi]
exact
@ -156,8 +160,9 @@ def involutionCons (n : ℕ) : {f : Fin n.succ → Fin n.succ // Function.Involu
(of_eq_true (Eq.trans (congrArg (fun x => x = true) hs) (eq_self true))))
· subst hi
exact rfl
· simp [hs]
simp at hs
· simp only [hs, Bool.false_eq_true, ↓reduceDIte, Fin.cons_zero, not_true_eq_false,
IsEmpty.exists_iff, false_iff]
simp only [Bool.not_eq_true, Option.not_isSome, Option.isNone_iff_eq_none] at hs
subst hs
exact ne_of_beq_false rfl
@ -166,7 +171,7 @@ lemma involutionCons_ext {n : ℕ} {f1 f2 : (f : {f : Fin n → Fin n // Functi
(h1 : f1.1 = f2.1) (h2 : f1.2 = Equiv.subtypeEquivRight (by rw [h1]; simp) f2.2) : f1 = f2 := by
cases f1
cases f2
simp at h1 h2
simp only at h1 h2
subst h1
rename_i fst snd snd_1
simp_all only [Sigma.mk.inj_iff, heq_eq_eq, true_and]
@ -234,7 +239,7 @@ lemma involutionAddEquiv_cast {n : ℕ} {f1 f2 : {f : Fin n → Fin n // Functio
involutionAddEquiv f1 = (Equiv.subtypeEquivRight (by rw [hf]; simp)).trans
((involutionAddEquiv f2).trans (Equiv.optionCongr (finCongr (by rw [hf])))):= by
subst hf
simp
simp only [finCongr_refl, Equiv.optionCongr_refl, Equiv.trans_refl]
rfl
@ -249,7 +254,7 @@ lemma involutionAddEquiv_none_succ {n : ℕ}
{f : {f : Fin n.succ → Fin n.succ // Function.Involutive f}}
(h : involutionAddEquiv (involutionCons n f).1 (involutionCons n f).2 = none)
(x : Fin n) : f.1 x.succ = x.succ ↔ (involutionCons n f).1.1 x = x := by
simp [involutionCons]
simp only [succ_eq_add_one, involutionCons, Fin.cons_update, Equiv.coe_fn_mk, dite_eq_left_iff]
have hn' := involutionAddEquiv_none_image_zero h
have hx : ¬ f.1 x.succ = ⟨0, Nat.zero_lt_succ n⟩:= by
rw [← hn']
@ -268,7 +273,9 @@ lemma involutionAddEquiv_isSome_image_zero {n : ℕ} :
→ (involutionAddEquiv (involutionCons n f).1 (involutionCons n f).2).isSome
→ ¬ f.1 ⟨0, Nat.zero_lt_succ n⟩ = ⟨0, Nat.zero_lt_succ n⟩ := by
intro f hf
simp [involutionAddEquiv, involutionCons] at hf
simp only [succ_eq_add_one, involutionCons, Equiv.coe_fn_mk, involutionAddEquiv,
Option.isSome_some, Option.get_some, Option.isSome_none, Equiv.trans_apply,
Equiv.optionCongr_apply, Equiv.coe_trans, RelIso.coe_fn_toEquiv, Option.isSome_map'] at hf
simp_all only [List.length_cons, Fin.zero_eta]
obtain ⟨val, property⟩ := f
simp_all only [List.length_cons]
@ -286,9 +293,9 @@ def involutionNoFixedEquivSum {n : ℕ} :
left_inv f := by
rfl
right_inv f := by
simp
simp only [succ_eq_add_one, ne_eq, mul_eq]
ext
· simp
· simp only [Fin.coe_pred]
rw [f.2.2.2.2]
simp
· simp
@ -309,12 +316,12 @@ def involutionNoFixedZeroSetEquivEquiv {n : ℕ}
(∀ i, (e.symm ∘ f ∘ e) i ≠ i) ∧ (e.symm ∘ f ∘ e) 0 = k.succ} where
toFun f := ⟨e ∘ f.1 ∘ e.symm, by
intro i
simp
simp only [succ_eq_add_one, ne_eq, Function.comp_apply, Equiv.symm_apply_apply]
rw [f.2.1], by
simpa using f.2.2.1, by simpa using f.2.2.2⟩
invFun f := ⟨e.symm ∘ f.1 ∘ e, by
intro i
simp
simp only [succ_eq_add_one, Function.comp_apply, ne_eq, Equiv.apply_symm_apply]
have hf2 := f.2.1 i
simpa using hf2, by
simpa using f.2.2.1, by simpa using f.2.2.2⟩
@ -341,7 +348,7 @@ def involutionNoFixedZeroSetEquivSetEquiv {n : ℕ} (k : Fin (2 * n + 1)) (e : F
have hi := h (e i)
simp [hi]
rw [h1]
simp
simp only [succ_eq_add_one, Function.comp_apply, ne_eq, and_congr_right_iff, and_congr_left_iff]
intro h1 h2
apply Iff.intro
· intro h i
@ -360,7 +367,7 @@ def involutionNoFixedZeroSetEquivEquiv' {n : ℕ} (k : Fin (2 * n + 1)) (e : Fin
≃ {f : Fin (2 * n.succ) → Fin (2 * n.succ) // Function.Involutive f ∧
(∀ i, f i ≠ i) ∧ f (e 0) = e k.succ} := by
refine Equiv.subtypeEquivRight ?_
simp
simp only [succ_eq_add_one, ne_eq, Function.comp_apply, and_congr_right_iff]
intro f hi h1
exact Equiv.symm_apply_eq e
@ -373,7 +380,7 @@ def involutionNoFixedZeroSetEquivSetOne {n : ℕ} (k : Fin (2 * n + 1)) :
refine Equiv.trans (involutionNoFixedZeroSetEquivSetEquiv k (Equiv.swap k.succ 1)) ?_
refine Equiv.trans (involutionNoFixedZeroSetEquivEquiv' k (Equiv.swap k.succ 1)) ?_
refine Equiv.subtypeEquivRight ?_
simp
simp only [succ_eq_add_one, ne_eq, Equiv.swap_apply_left, and_congr_right_iff]
intro f hi h1
rw [Equiv.swap_apply_of_ne_of_ne]
· exact Ne.symm (Fin.succ_ne_zero k)
@ -387,33 +394,32 @@ def involutionNoFixedSetOne {n : ℕ} :
toFun f := by
have hf1 : f.1 1 = 0 := by
have hf := f.2.2.2
simp [← hf]
simp only [succ_eq_add_one, ne_eq, ← hf]
rw [f.2.1]
let f' := f.1 ∘ Fin.succ ∘ Fin.succ
have hf' (i : Fin (2 * n)) : f' i ≠ 0 := by
simp [f']
simp [← hf1]
simp only [succ_eq_add_one, mul_eq, ne_eq, Function.comp_apply, f']
simp only [← hf1, succ_eq_add_one, ne_eq]
by_contra hn
have hn' := Function.Involutive.injective f.2.1 hn
simp [Fin.ext_iff] at hn'
let f'' := fun i => (f' i).pred (hf' i)
have hf'' (i : Fin (2 * n)) : f'' i ≠ 0 := by
simp [f'']
simp only [mul_eq, ne_eq, f'']
rw [@Fin.pred_eq_iff_eq_succ]
simp [f']
simp [← f.2.2.2 ]
simp only [mul_eq, succ_eq_add_one, ne_eq, Function.comp_apply, Fin.succ_zero_eq_one, f']
simp only [← f.2.2.2, succ_eq_add_one, ne_eq]
by_contra hn
have hn' := Function.Involutive.injective f.2.1 hn
simp [Fin.ext_iff] at hn'
let f''' := fun i => (f'' i).pred (hf'' i)
refine ⟨f''', ?_, ?_⟩
· intro i
simp [f''', f'', f']
simp only [mul_eq, succ_eq_add_one, ne_eq, Function.comp_apply, Fin.succ_pred, f''', f'', f']
simp [f.2.1 i.succ.succ]
· intro i
simp [f''', f'', f']
rw [@Fin.pred_eq_iff_eq_succ]
rw [@Fin.pred_eq_iff_eq_succ]
simp only [mul_eq, succ_eq_add_one, ne_eq, Function.comp_apply, f''', f'', f']
rw [Fin.pred_eq_iff_eq_succ, Fin.pred_eq_iff_eq_succ]
exact f.2.2.1 i.succ.succ
invFun f := by
let f' := fun (i : Fin (2 * n.succ))=>
@ -427,16 +433,16 @@ def involutionNoFixedSetOne {n : ℕ} :
| ⟨0, h⟩ => rfl
| ⟨1, h⟩ => rfl
| ⟨(Nat.succ (Nat.succ m)), h⟩ =>
simp [f']
simp only [succ_eq_add_one, ne_eq, f']
split
· rename_i h
simp at h
simp only [succ_eq_add_one, Fin.zero_eta] at h
exact False.elim (Fin.succ_ne_zero (f.1 ⟨m, _⟩).succ h)
· rename_i h
simp [Fin.ext_iff] at h
· rename_i h
rename_i x r
simp_all [Fin.ext_iff]
simp_all only [succ_eq_add_one, Fin.ext_iff, Fin.val_succ, add_left_inj]
have hfn {a b : ℕ} {ha : a < 2 * n} {hb : b < 2 * n}
(hab : ↑(f.1 ⟨a, ha⟩) = b): ↑(f.1 ⟨b, hb⟩) = a := by
have ht : f.1 ⟨a, ha⟩ = ⟨b, hb⟩ := by
@ -446,7 +452,7 @@ def involutionNoFixedSetOne {n : ℕ} :
· intro i
match i with
| ⟨0, h⟩ =>
simp [f']
simp only [succ_eq_add_one, ne_eq, Fin.zero_eta, f']
split
· rename_i h
simp
@ -455,7 +461,7 @@ def involutionNoFixedSetOne {n : ℕ} :
· rename_i h
simp [Fin.ext_iff] at h
| ⟨1, h⟩ =>
simp [f']
simp only [succ_eq_add_one, ne_eq, Fin.mk_one, f']
split
· rename_i h
simp at h
@ -464,11 +470,11 @@ def involutionNoFixedSetOne {n : ℕ} :
· rename_i h
simp [Fin.ext_iff] at h
| ⟨(Nat.succ (Nat.succ m)), h⟩ =>
simp [f', Fin.ext_iff]
simp only [succ_eq_add_one, ne_eq, Fin.ext_iff, Fin.val_succ, add_left_inj, f']
have hf:= f.2.2 ⟨m, by exact Nat.add_lt_add_iff_right.mp h⟩
simp [Fin.ext_iff] at hf
simp only [ne_eq, Fin.ext_iff] at hf
omega
· simp [f']
· simp only [succ_eq_add_one, ne_eq, f']
split
· rename_i h
simp
@ -479,30 +485,30 @@ def involutionNoFixedSetOne {n : ℕ} :
left_inv f := by
have hf1 : f.1 1 = 0 := by
have hf := f.2.2.2
simp [← hf]
simp only [succ_eq_add_one, ne_eq, ← hf]
rw [f.2.1]
simp
simp only [succ_eq_add_one, ne_eq, mul_eq, Function.comp_apply, Fin.succ_mk, Fin.succ_pred]
ext i
simp
simp only
split
· simp
· simp only [Fin.val_one, succ_eq_add_one, Fin.zero_eta]
rw [f.2.2.2]
simp
· simp
· simp only [Fin.val_zero, succ_eq_add_one, Fin.mk_one]
rw [hf1]
simp
· rfl
right_inv f := by
simp
simp only [ne_eq, mul_eq, succ_eq_add_one, Function.comp_apply]
ext i
simp
simp only [Fin.coe_pred]
split
· rename_i h
simp [Fin.ext_iff] at h
· rename_i h
simp [Fin.ext_iff] at h
· rename_i h
simp
simp only [Fin.val_succ, add_tsub_cancel_right]
congr
apply congrArg
simp_all [Fin.ext_iff]

2
HepLean/Mathematics/List.lean
View file

@ -45,7 +45,7 @@ lemma takeWile_eraseIdx {I : Type} (P : I → Prop) [DecidablePred P] :
| a :: b :: l, Nat.succ n, h => by
simp only [Nat.succ_eq_add_one, List.eraseIdx_cons_succ]
by_cases hPa : P a
· dsimp [List.takeWhile]
· dsimp only [List.takeWhile]
simp only [hPa, decide_True, List.eraseIdx_cons_succ, List.cons.injEq, true_and]
rw [takeWile_eraseIdx]
rfl