refactor: Simp to simp only ...
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jstoobysmith
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b2ac704d80
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855dc5146d
12 changed files with 257 additions and 192 deletions
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@ -31,44 +31,44 @@ def predAboveI (i x : Fin n.succ.succ) : Fin n.succ :=
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· omega
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· omega⟩
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lemma predAboveI_self (i : Fin n.succ.succ) : predAboveI i i = ⟨i.val - 1, by omega⟩ := by
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lemma predAboveI_self (i : Fin n.succ.succ) : predAboveI i i = ⟨i.val - 1, by omega⟩ := by
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simp [predAboveI]
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@[simp]
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lemma predAboveI_succAbove (i : Fin n.succ.succ) (x : Fin n.succ) :
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predAboveI i (Fin.succAbove i x) = x := by
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simp [predAboveI, Fin.succAbove, Fin.ext_iff]
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simp only [Nat.succ_eq_add_one, predAboveI, Fin.succAbove, Fin.val_fin_lt, Fin.ext_iff]
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split_ifs
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· rfl
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· rename_i h1 h2
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simp [Fin.lt_def] at h1 h2
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simp only [Fin.lt_def, Fin.coe_castSucc, not_lt, Nat.succ_eq_add_one, Fin.val_succ] at h1 h2
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omega
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· rfl
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lemma succsAbove_predAboveI {i x : Fin n.succ.succ} (h : i ≠ x) :
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Fin.succAbove i (predAboveI i x) = x := by
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simp [Fin.succAbove, predAboveI, Fin.ext_iff]
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simp only [Fin.succAbove, predAboveI, Nat.succ_eq_add_one, Fin.val_fin_lt, Fin.ext_iff]
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split_ifs
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· rfl
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· rename_i h1 h2
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rw [Fin.lt_def] at h1 h2
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simp
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simp at h2
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simp only [Fin.succ_mk, Nat.succ_eq_add_one, add_right_eq_self, one_ne_zero]
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simp only [Fin.castSucc_mk, Fin.eta, Fin.val_fin_lt, not_lt] at h2
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rw [Fin.le_def] at h2
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omega
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· rename_i h1 h2
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simp at h1
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simp only [not_lt] at h1
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rw [Fin.le_def] at h1
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rw [Fin.lt_def] at h2
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simp at h2
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simp only [Fin.castSucc_mk] at h2
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omega
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· rename_i h1 h2
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simp
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simp at h1
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simp only [Fin.succ_mk, Nat.succ_eq_add_one]
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simp only [not_lt] at h1
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rw [Fin.le_def] at h1
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omega
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lemma predAboveI_eq_iff {i x : Fin n.succ.succ} (h : i ≠ x) (y : Fin n.succ) :
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y = predAboveI i x ↔ i.succAbove y = x := by
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y = predAboveI i x ↔ i.succAbove y = x := by
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apply Iff.intro
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· intro h
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subst h
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@ -83,19 +83,19 @@ lemma predAboveI_lt {i x : Fin n.succ.succ} (h : x.val < i.val) :
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lemma predAboveI_ge {i x : Fin n.succ.succ} (h : i.val < x.val) :
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predAboveI i x = ⟨x.val - 1, by omega⟩ := by
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simp [predAboveI, h]
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simp only [Nat.succ_eq_add_one, predAboveI, Fin.val_fin_lt, dite_eq_else, Fin.mk.injEq]
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omega
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lemma succAbove_succAbove_predAboveI (i : Fin n.succ.succ) (j : Fin n.succ) (x : Fin n) :
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i.succAbove (j.succAbove x) =
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(i.succAbove j).succAbove ((predAboveI (i.succAbove j) i).succAbove x) := by
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by_cases h1 : j.castSucc < i
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· have hx := Fin.succAbove_of_castSucc_lt _ _ h1
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· have hx := Fin.succAbove_of_castSucc_lt _ _ h1
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rw [hx]
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rw [predAboveI_ge h1]
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by_cases hx1 : x.castSucc < j
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· rw [Fin.succAbove_of_castSucc_lt _ _ hx1]
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rw [Fin.succAbove_of_castSucc_lt _ _]
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· rw [Fin.succAbove_of_castSucc_lt _ _ hx1]
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rw [Fin.succAbove_of_castSucc_lt _ _]
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· nth_rewrite 2 [Fin.succAbove_of_castSucc_lt _ _]
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· rw [Fin.succAbove_of_castSucc_lt _ _]
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exact hx1
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@ -103,7 +103,7 @@ lemma succAbove_succAbove_predAboveI (i : Fin n.succ.succ) (j : Fin n.succ) (x :
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simp_all
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omega
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· exact Nat.lt_trans hx1 h1
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· simp at hx1
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· simp only [not_lt] at hx1
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rw [Fin.le_def] at hx1
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rw [Fin.lt_def] at h1
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rw [Fin.succAbove_of_le_castSucc _ _ hx1]
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@ -116,7 +116,7 @@ lemma succAbove_succAbove_predAboveI (i : Fin n.succ.succ) (j : Fin n.succ) (x :
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· rw [Fin.lt_def] at hx2 ⊢
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simp_all
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omega
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· simp at hx2
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· simp only [Nat.succ_eq_add_one, not_lt] at hx2
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rw [Fin.succAbove_of_le_castSucc _ _ hx2]
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nth_rewrite 2 [Fin.succAbove_of_le_castSucc]
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· rw [Fin.succAbove_of_le_castSucc]
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@ -124,8 +124,8 @@ lemma succAbove_succAbove_predAboveI (i : Fin n.succ.succ) (j : Fin n.succ) (x :
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exact Nat.le_succ_of_le hx1
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· rw [Fin.le_def] at hx2 ⊢
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simp_all
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· simp at h1
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have hx := Fin.succAbove_of_le_castSucc _ _ h1
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· simp only [Nat.succ_eq_add_one, not_lt] at h1
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have hx := Fin.succAbove_of_le_castSucc _ _ h1
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rw [hx]
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rw [predAboveI_lt (Nat.lt_add_one_of_le h1)]
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by_cases hx1 : j ≤ x.castSucc
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@ -141,7 +141,7 @@ lemma succAbove_succAbove_predAboveI (i : Fin n.succ.succ) (j : Fin n.succ) (x :
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· rw [Fin.le_def] at hx1 h1 ⊢
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simp_all
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omega
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· simp at hx1
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· simp only [Nat.succ_eq_add_one, not_le] at hx1
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rw [Fin.lt_def] at hx1
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rw [Fin.le_def] at h1
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rw [Fin.succAbove_of_castSucc_lt _ _ hx1]
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@ -149,12 +149,12 @@ lemma succAbove_succAbove_predAboveI (i : Fin n.succ.succ) (j : Fin n.succ) (x :
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· rw [Fin.succAbove_of_castSucc_lt _ _ hx2]
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nth_rewrite 2 [Fin.succAbove_of_castSucc_lt _ _]
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· rw [Fin.succAbove_of_castSucc_lt _ _]
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rw [Fin.lt_def] at hx2 ⊢
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rw [Fin.lt_def] at hx2 ⊢
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simp_all
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omega
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· rw [Fin.lt_def] at hx2 ⊢
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· rw [Fin.lt_def] at hx2 ⊢
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simp_all
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· simp at hx2
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· simp only [not_lt] at hx2
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rw [Fin.succAbove_of_le_castSucc _ _ hx2]
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nth_rewrite 2 [Fin.succAbove_of_le_castSucc]
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· rw [Fin.succAbove_of_castSucc_lt]
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@ -175,11 +175,12 @@ def finExtractOne {n : ℕ} (i : Fin n.succ) : Fin n.succ ≃ Fin 1 ⊕ Fin n :=
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lemma finExtractOne_apply_eq {n : ℕ} (i : Fin n.succ) :
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finExtractOne i i = Sum.inl 0 := by
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simp [finExtractOne]
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simp only [Nat.succ_eq_add_one, finExtractOne, Equiv.trans_apply, finCongr_apply,
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Equiv.sumCongr_apply, Equiv.coe_trans, Equiv.sumComm_apply, Equiv.coe_refl, Fin.isValue]
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have h1 : Fin.cast (finExtractOne.proof_1 i) i = Fin.castAdd ((n - ↑i)) ⟨i.1, lt_add_one i.1⟩ := by
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simp [Fin.ext_iff]
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rw [h1, finSumFinEquiv_symm_apply_castAdd]
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simp
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simp only [Nat.succ_eq_add_one, Sum.map_inl, Function.comp_apply, Fin.isValue]
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have h2 : @Fin.mk (↑i + 1) ↑i (lt_add_one i.1) = Fin.natAdd i.val 1 := by
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simp [Fin.ext_iff]
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rw [h2, finSumFinEquiv_symm_apply_natAdd]
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@ -245,12 +246,13 @@ lemma finExtractOne_symm_inl_apply {n : ℕ} (i : Fin n.succ) :
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rfl
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def finExtractOnPermHom (i : Fin n.succ.succ) (σ : Fin n.succ.succ ≃ Fin n.succ.succ) :
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Fin n.succ → Fin n.succ := fun x => predAboveI (σ i) (σ ((finExtractOne i).symm (Sum.inr x)))
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Fin n.succ → Fin n.succ := fun x => predAboveI (σ i) (σ ((finExtractOne i).symm (Sum.inr x)))
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lemma finExtractOnPermHom_inv (i : Fin n.succ.succ) (σ : Fin n.succ.succ ≃ Fin n.succ.succ) :
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(finExtractOnPermHom (σ i) σ.symm) ∘ (finExtractOnPermHom i σ) = id := by
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(finExtractOnPermHom (σ i) σ.symm) ∘ (finExtractOnPermHom i σ) = id := by
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funext x
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simp [finExtractOnPermHom]
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simp only [Nat.succ_eq_add_one, Function.comp_apply, finExtractOnPermHom, Equiv.symm_apply_apply,
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finExtractOne_symm_inr_apply, id_eq]
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by_cases h : σ (i.succAbove x) < σ i
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· rw [predAboveI_lt h, Fin.succAbove_of_castSucc_lt]
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· simp
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@ -261,18 +263,18 @@ lemma finExtractOnPermHom_inv (i : Fin n.succ.succ) (σ : Fin n.succ.succ ≃ Fi
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have hn : σ i < σ (i.succAbove x) := by omega
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rw [predAboveI_ge hn]
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rw [Fin.succAbove_of_le_castSucc]
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· simp
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trans predAboveI i (σ.symm (σ (i.succAbove x)))
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· simp only [Nat.succ_eq_add_one, Fin.succ_mk]
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trans predAboveI i (σ.symm (σ (i.succAbove x)))
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· congr
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exact Nat.sub_add_cancel (Fin.lt_of_le_of_lt (Fin.zero_le (σ i)) hn)
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simp
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rw [Fin.le_def]
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simp
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simp only [Nat.succ_eq_add_one, Fin.castSucc_mk]
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omega
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def finExtractOnePerm (i : Fin n.succ.succ) (σ : Fin n.succ.succ ≃ Fin n.succ.succ) :
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Fin n.succ ≃ Fin n.succ where
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toFun x := finExtractOnPermHom i σ x
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toFun x := finExtractOnPermHom i σ x
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invFun x := finExtractOnPermHom (σ i) σ.symm x
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left_inv x := by
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simpa using congrFun (finExtractOnPermHom_inv i σ) x
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@ -290,7 +292,8 @@ def finExtractTwo {n : ℕ} (i : Fin n.succ.succ) (j : Fin n.succ) :
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@[simp]
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lemma finExtractTwo_apply_fst {n : ℕ} (i : Fin n.succ.succ) (j : Fin n.succ) :
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finExtractTwo i j i = Sum.inl (Sum.inl 0) := by
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simp [finExtractTwo]
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simp only [Nat.succ_eq_add_one, finExtractTwo, Equiv.trans_apply, Equiv.sumCongr_apply,
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Equiv.coe_refl, Fin.isValue]
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simp [finExtractOne_apply_eq]
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lemma finExtractTwo_symm_inr {n : ℕ} (i : Fin n.succ.succ) (j : Fin n.succ) :
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