760 lines
38 KiB
Text
760 lines
38 KiB
Text
/-
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Copyright (c) 2015 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Module: data.list.perm
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Author: Leonardo de Moura
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List permutations
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-/
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import data.list.basic data.list.set
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open list setoid nat binary
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variables {A B : Type}
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inductive perm : list A → list A → Prop :=
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| nil : perm [] []
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| skip : Π (x : A) {l₁ l₂ : list A}, perm l₁ l₂ → perm (x::l₁) (x::l₂)
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| swap : Π (x y : A) (l : list A), perm (y::x::l) (x::y::l)
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| trans : Π {l₁ l₂ l₃ : list A}, perm l₁ l₂ → perm l₂ l₃ → perm l₁ l₃
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namespace perm
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infix ~:50 := perm
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theorem eq_nil_of_perm_nil {l₁ : list A} (p : [] ~ l₁) : l₁ = [] :=
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have gen : ∀ (l₂ : list A) (p : l₂ ~ l₁), l₂ = [] → l₁ = [], from
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take l₂ p, perm.induction_on p
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(λ h, h)
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(by contradiction)
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(by contradiction)
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(λ l₁ l₂ l₃ p₁ p₂ r₁ r₂ e, r₂ (r₁ e)),
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gen [] p rfl
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theorem not_perm_nil_cons (x : A) (l : list A) : ¬ [] ~ (x::l) :=
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have gen : ∀ (l₁ l₂ : list A) (p : l₁ ~ l₂), l₁ = [] → l₂ = (x::l) → false, from
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take l₁ l₂ p, perm.induction_on p
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(by contradiction)
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(by contradiction)
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(by contradiction)
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(λ l₁ l₂ l₃ p₁ p₂ r₁ r₂ e₁ e₂,
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begin
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rewrite [e₂ at *, e₁ at *],
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have e₃ : l₂ = [], from eq_nil_of_perm_nil p₁,
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exact (r₂ e₃ rfl)
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end),
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assume p, gen [] (x::l) p rfl rfl
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protected theorem refl : ∀ (l : list A), l ~ l
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| [] := nil
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| (x::xs) := skip x (refl xs)
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protected theorem symm : ∀ {l₁ l₂ : list A}, l₁ ~ l₂ → l₂ ~ l₁ :=
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take l₁ l₂ p, perm.induction_on p
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nil
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(λ x l₁ l₂ p₁ r₁, skip x r₁)
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(λ x y l, swap y x l)
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(λ l₁ l₂ l₃ p₁ p₂ r₁ r₂, trans r₂ r₁)
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theorem eqv (A : Type) : equivalence (@perm A) :=
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mk_equivalence (@perm A) (@perm.refl A) (@perm.symm A) (@perm.trans A)
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protected definition is_setoid [instance] (A : Type) : setoid (list A) :=
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setoid.mk (@perm A) (perm.eqv A)
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calc_refl perm.refl
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calc_symm perm.symm
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calc_trans perm.trans
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theorem mem_perm {a : A} {l₁ l₂ : list A} : l₁ ~ l₂ → a ∈ l₁ → a ∈ l₂ :=
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assume p, perm.induction_on p
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(λ h, h)
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(λ x l₁ l₂ p₁ r₁ i, or.elim (eq_or_mem_of_mem_cons i)
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(assume aeqx : a = x, by rewrite aeqx; apply !mem_cons)
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(assume ainl₁ : a ∈ l₁, or.inr (r₁ ainl₁)))
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(λ x y l ainyxl, or.elim (eq_or_mem_of_mem_cons ainyxl)
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(assume aeqy : a = y, by rewrite aeqy; exact (or.inr !mem_cons))
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(assume ainxl : a ∈ x::l, or.elim (eq_or_mem_of_mem_cons ainxl)
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(assume aeqx : a = x, or.inl aeqx)
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(assume ainl : a ∈ l, or.inr (or.inr ainl))))
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(λ l₁ l₂ l₃ p₁ p₂ r₁ r₂ ainl₁, r₂ (r₁ ainl₁))
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theorem not_mem_perm {a : A} {l₁ l₂ : list A} : l₁ ~ l₂ → a ∉ l₁ → a ∉ l₂ :=
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assume p nainl₁ ainl₂, absurd (mem_perm (symm p) ainl₂) nainl₁
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theorem perm_app_left {l₁ l₂ : list A} (t₁ : list A) : l₁ ~ l₂ → (l₁++t₁) ~ (l₂++t₁) :=
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assume p, perm.induction_on p
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!refl
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(λ x l₁ l₂ p₁ r₁, skip x r₁)
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(λ x y l, !swap)
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(λ l₁ l₂ l₃ p₁ p₂ r₁ r₂, trans r₁ r₂)
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theorem perm_app_right (l : list A) {t₁ t₂ : list A} : t₁ ~ t₂ → (l++t₁) ~ (l++t₂) :=
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list.induction_on l
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(λ p, p)
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(λ x xs r p, skip x (r p))
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theorem perm_app {l₁ l₂ t₁ t₂ : list A} : l₁ ~ l₂ → t₁ ~ t₂ → (l₁++t₁) ~ (l₂++t₂) :=
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assume p₁ p₂, trans (perm_app_left t₁ p₁) (perm_app_right l₂ p₂)
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theorem perm_app_cons (a : A) {h₁ h₂ t₁ t₂ : list A} : h₁ ~ h₂ → t₁ ~ t₂ → (h₁ ++ (a::t₁)) ~ (h₂ ++ (a::t₂)) :=
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assume p₁ p₂, perm_app p₁ (skip a p₂)
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theorem perm_cons_app (a : A) : ∀ (l : list A), (a::l) ~ (l ++ [a])
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| [] := !refl
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| (x::xs) := calc
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a::x::xs ~ x::a::xs : swap x a xs
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... ~ x::(xs++[a]) : skip x (perm_cons_app xs)
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theorem perm_app_comm {l₁ l₂ : list A} : (l₁++l₂) ~ (l₂++l₁) :=
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list.induction_on l₁
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(by rewrite [append_nil_right, append_nil_left]; apply refl)
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(λ a t r, calc
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a::(t++l₂) ~ a::(l₂++t) : skip a r
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... ~ l₂++t++[a] : perm_cons_app
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... = l₂++(t++[a]) : append.assoc
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... ~ l₂++(a::t) : perm_app_right l₂ (symm (perm_cons_app a t)))
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theorem length_eq_length_of_perm {l₁ l₂ : list A} : l₁ ~ l₂ → length l₁ = length l₂ :=
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assume p, perm.induction_on p
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rfl
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(λ x l₁ l₂ p r, by rewrite [*length_cons, r])
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(λ x y l, by rewrite *length_cons)
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(λ l₁ l₂ l₃ p₁ p₂ r₁ r₂, eq.trans r₁ r₂)
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theorem eq_singlenton_of_perm_inv (a : A) {l : list A} : [a] ~ l → l = [a] :=
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have gen : ∀ l₂, perm l₂ l → l₂ = [a] → l = [a], from
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take l₂, assume p, perm.induction_on p
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(λ e, e)
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(λ x l₁ l₂ p r e,
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begin
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injection e with e₁ e₂,
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rewrite [e₁, e₂ at p],
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have h₁ : l₂ = [], from eq_nil_of_perm_nil p,
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rewrite h₁
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end)
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(λ x y l e, by injection e; contradiction)
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(λ l₁ l₂ l₃ p₁ p₂ r₁ r₂ e, r₂ (r₁ e)),
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assume p, gen [a] p rfl
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theorem eq_singlenton_of_perm (a b : A) : [a] ~ [b] → a = b :=
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assume p,
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begin
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injection eq_singlenton_of_perm_inv a p with e₁,
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rewrite e₁
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end
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theorem perm_rev : ∀ (l : list A), l ~ (reverse l)
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| [] := nil
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| (x::xs) := calc
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x::xs ~ xs++[x] : perm_cons_app x xs
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... ~ reverse xs ++ [x] : perm_app_left [x] (perm_rev xs)
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... = reverse (x::xs) : by rewrite [reverse_cons, concat_eq_append]
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theorem perm_middle (a : A) (l₁ l₂ : list A) : (a::l₁)++l₂ ~ l₁++(a::l₂) :=
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calc
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(a::l₁) ++ l₂ = a::(l₁++l₂) : rfl
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... ~ l₁++l₂++[a] : perm_cons_app
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... = l₁++(l₂++[a]) : append.assoc
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... ~ l₁++(a::l₂) : perm_app_right l₁ (symm (perm_cons_app a l₂))
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theorem perm_cons_app_cons {l l₁ l₂ : list A} (a : A) : l ~ l₁++l₂ → a::l ~ l₁++(a::l₂) :=
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assume p, calc
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a::l ~ l++[a] : perm_cons_app
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... ~ l₁++l₂++[a] : perm_app_left [a] p
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... = l₁++(l₂++[a]) : append.assoc
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... ~ l₁++(a::l₂) : perm_app_right l₁ (symm (perm_cons_app a l₂))
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open decidable
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theorem perm_erase [H : decidable_eq A] {a : A} : ∀ {l : list A}, a ∈ l → l ~ a::(erase a l)
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| [] h := absurd h !not_mem_nil
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| (x::t) h :=
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by_cases
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(assume aeqx : a = x, by rewrite [aeqx, erase_cons_head]; exact !perm.refl)
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(assume naeqx : a ≠ x,
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have aint : a ∈ t, from mem_of_ne_of_mem naeqx h,
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have aux : t ~ a :: erase a t, from perm_erase aint,
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calc x::t ~ x::a::(erase a t) : skip x aux
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... ~ a::x::(erase a t) : swap
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... = a::(erase a (x::t)) : by rewrite [!erase_cons_tail naeqx])
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theorem erase_perm_erase_of_perm [H : decidable_eq A] (a : A) {l₁ l₂ : list A} : l₁ ~ l₂ → erase a l₁ ~ erase a l₂ :=
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assume p, perm.induction_on p
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nil
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(λ x t₁ t₂ p r,
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by_cases
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(assume aeqx : a = x, by rewrite [aeqx, *erase_cons_head]; exact p)
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(assume naeqx : a ≠ x, by rewrite [*erase_cons_tail _ naeqx]; exact (skip x r)))
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(λ x y l,
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by_cases
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(assume aeqx : a = x,
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by_cases
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(assume aeqy : a = y, by rewrite [-aeqx, -aeqy]; exact !perm.refl)
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(assume naeqy : a ≠ y, by rewrite [-aeqx, erase_cons_tail _ naeqy, *erase_cons_head]; exact !perm.refl))
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(assume naeqx : a ≠ x,
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by_cases
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(assume aeqy : a = y, by rewrite [-aeqy, erase_cons_tail _ naeqx, *erase_cons_head]; exact !perm.refl)
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(assume naeqy : a ≠ y, by rewrite[erase_cons_tail _ naeqx, *erase_cons_tail _ naeqy, erase_cons_tail _ naeqx];
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exact !swap)))
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(λ l₁ l₂ l₃ p₁ p₂ r₁ r₂, trans r₁ r₂)
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theorem perm_induction_on {P : list A → list A → Prop} {l₁ l₂ : list A} (p : l₁ ~ l₂)
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(h₁ : P [] [])
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(h₂ : ∀ x l₁ l₂, l₁ ~ l₂ → P l₁ l₂ → P (x::l₁) (x::l₂))
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(h₃ : ∀ x y l₁ l₂, l₁ ~ l₂ → P l₁ l₂ → P (y::x::l₁) (x::y::l₂))
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(h₄ : ∀ l₁ l₂ l₃, l₁ ~ l₂ → l₂ ~ l₃ → P l₁ l₂ → P l₂ l₃ → P l₁ l₃)
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: P l₁ l₂ :=
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have P_refl : ∀ l, P l l
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| [] := h₁
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| (x::xs) := h₂ x xs xs !refl (P_refl xs),
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perm.induction_on p h₁ h₂ (λ x y l, h₃ x y l l !refl !P_refl) h₄
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theorem xswap {l₁ l₂ : list A} (x y : A) : l₁ ~ l₂ → x::y::l₁ ~ y::x::l₂ :=
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assume p, calc
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x::y::l₁ ~ y::x::l₁ : swap
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... ~ y::x::l₂ : skip y (skip x p)
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theorem perm_map (f : A → B) {l₁ l₂ : list A} : l₁ ~ l₂ → map f l₁ ~ map f l₂ :=
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assume p, perm_induction_on p
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nil
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(λ x l₁ l₂ p r, skip (f x) r)
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(λ x y l₁ l₂ p r, xswap (f y) (f x) r)
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(λ l₁ l₂ l₃ p₁ p₂ r₁ r₂, trans r₁ r₂)
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lemma perm_of_qeq {a : A} {l₁ l₂ : list A} : l₁≈a|l₂ → l₁~a::l₂ :=
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assume q, qeq.induction_on q
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(λ h, !refl)
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(λ b t₁ t₂ q₁ r₁, calc
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b::t₂ ~ b::a::t₁ : skip b r₁
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... ~ a::b::t₁ : swap)
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/- permutation is decidable if A has decidable equality -/
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section dec
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open decidable
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variable [Ha : decidable_eq A]
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include Ha
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definition decidable_perm_aux : ∀ (n : nat) (l₁ l₂ : list A), length l₁ = n → length l₂ = n → decidable (l₁ ~ l₂)
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| 0 l₁ l₂ H₁ H₂ :=
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assert l₁n : l₁ = [], from eq_nil_of_length_eq_zero H₁,
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assert l₂n : l₂ = [], from eq_nil_of_length_eq_zero H₂,
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by rewrite [l₁n, l₂n]; exact (inl perm.nil)
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| (n+1) (x::t₁) l₂ H₁ H₂ :=
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by_cases
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(assume xinl₂ : x ∈ l₂,
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let t₂ : list A := erase x l₂ in
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have len_t₁ : length t₁ = n, begin injection H₁ with e, exact e end,
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assert len_t₂_aux : length t₂ = pred (length l₂), from length_erase_of_mem xinl₂,
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assert len_t₂ : length t₂ = n, by rewrite [len_t₂_aux, H₂],
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match decidable_perm_aux n t₁ t₂ len_t₁ len_t₂ with
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| inl p := inl (calc
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x::t₁ ~ x::(erase x l₂) : skip x p
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... ~ l₂ : perm_erase xinl₂)
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| inr np := inr (λ p : x::t₁ ~ l₂,
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assert p₁ : erase x (x::t₁) ~ erase x l₂, from erase_perm_erase_of_perm x p,
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have p₂ : t₁ ~ erase x l₂, by rewrite [erase_cons_head at p₁]; exact p₁,
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absurd p₂ np)
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end)
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(assume nxinl₂ : x ∉ l₂,
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inr (λ p : x::t₁ ~ l₂, absurd (mem_perm p !mem_cons) nxinl₂))
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definition decidable_perm [instance] : ∀ (l₁ l₂ : list A), decidable (l₁ ~ l₂) :=
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λ l₁ l₂,
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by_cases
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(assume eql : length l₁ = length l₂,
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decidable_perm_aux (length l₂) l₁ l₂ eql rfl)
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(assume neql : length l₁ ≠ length l₂,
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inr (λ p : l₁ ~ l₂, absurd (length_eq_length_of_perm p) neql))
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end dec
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-- Auxiliary theorem for performing cases-analysis on l₂.
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-- We use it to prove perm_inv_core.
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private theorem discr {P : Prop} {a b : A} {l₁ l₂ l₃ : list A} :
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a::l₁ = l₂++(b::l₃) →
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(l₂ = [] → a = b → l₁ = l₃ → P) →
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(∀ t, l₂ = a::t → l₁ = t++(b::l₃) → P) → P :=
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match l₂ with
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| [] := λ e h₁ h₂, by injection e with e₁ e₂; exact h₁ rfl e₁ e₂
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| h::t := λ e h₁ h₂,
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begin
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injection e with e₁ e₂,
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rewrite e₁ at h₂,
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exact h₂ t rfl e₂
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end
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end
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-- Auxiliary theorem for performing cases-analysis on l₂.
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-- We use it to prove perm_inv_core.
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private theorem discr₂ {P : Prop} {a b c : A} {l₁ l₂ l₃ : list A} :
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a::b::l₁ = l₂++(c::l₃) →
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(l₂ = [] → l₃ = b::l₁ → a = c → P) →
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(l₂ = [a] → b = c → l₁ = l₃ → P) →
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(∀ t, l₂ = a::b::t → l₁ = t++(c::l₃) → P) → P :=
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match l₂ with
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| [] := λ e H₁ H₂ H₃,
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begin
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injection e with a_eq_c b_l₁_eq_l₃,
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exact H₁ rfl (eq.symm b_l₁_eq_l₃) a_eq_c
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end
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| [h₁] := λ e H₁ H₂ H₃,
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begin
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rewrite [append_cons at e, append_nil_left at e],
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injection e with a_eq_h₁ b_eq_c l₁_eq_l₃,
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rewrite [a_eq_h₁ at H₂, b_eq_c at H₂, l₁_eq_l₃ at H₂],
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exact H₂ rfl rfl rfl
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end
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| h₁::h₂::t₂ := λ e H₁ H₂ H₃,
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begin
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injection e with a_eq_h₁ b_eq_h₂ l₁_eq,
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rewrite [a_eq_h₁ at H₃, b_eq_h₂ at H₃],
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exact H₃ t₂ rfl l₁_eq
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end
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end
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/- permutation inversion -/
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theorem perm_inv_core {l₁ l₂ : list A} (p' : l₁ ~ l₂) : ∀ {a s₁ s₂}, l₁≈a|s₁ → l₂≈a|s₂ → s₁ ~ s₂ :=
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perm_induction_on p'
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(λ a s₁ s₂ e₁ e₂,
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have innil : a ∈ [], from mem_head_of_qeq e₁,
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absurd innil !not_mem_nil)
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(λ x t₁ t₂ p (r : ∀{a s₁ s₂}, t₁≈a|s₁ → t₂≈a|s₂ → s₁ ~ s₂) a s₁ s₂ e₁ e₂,
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obtain (s₁₁ s₁₂ : list A) (C₁ : s₁ = s₁₁ ++ s₁₂ ∧ x::t₁ = s₁₁++(a::s₁₂)), from qeq_split e₁,
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obtain (s₂₁ s₂₂ : list A) (C₂ : s₂ = s₂₁ ++ s₂₂ ∧ x::t₂ = s₂₁++(a::s₂₂)), from qeq_split e₂,
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discr (and.elim_right C₁)
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(λ (s₁₁_eq : s₁₁ = []) (x_eq_a : x = a) (t₁_eq : t₁ = s₁₂),
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assert s₁_p : s₁ ~ t₂, from calc
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s₁ = s₁₁ ++ s₁₂ : and.elim_left C₁
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... = t₁ : by rewrite [-t₁_eq, s₁₁_eq, append_nil_left]
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... ~ t₂ : p,
|
|
discr (and.elim_right C₂)
|
|
(λ (s₂₁_eq : s₂₁ = []) (x_eq_a : x = a) (t₂_eq: t₂ = s₂₂),
|
|
proof calc
|
|
s₁ ~ t₂ : s₁_p
|
|
... = s₂₁ ++ s₂₂ : by rewrite [-t₂_eq, s₂₁_eq, append_nil_left]
|
|
... = s₂ : by rewrite [and.elim_left C₂]
|
|
qed)
|
|
(λ (ts₂₁ : list A) (s₂₁_eq : s₂₁ = x::ts₂₁) (t₂_eq : t₂ = ts₂₁++(a::s₂₂)),
|
|
proof calc
|
|
s₁ ~ t₂ : s₁_p
|
|
... = ts₂₁++(a::s₂₂) : t₂_eq
|
|
... ~ (a::ts₂₁)++s₂₂ : !perm_middle
|
|
... = s₂₁ ++ s₂₂ : by rewrite [-x_eq_a, -s₂₁_eq]
|
|
... = s₂ : by rewrite [and.elim_left C₂]
|
|
qed))
|
|
(λ (ts₁₁ : list A) (s₁₁_eq : s₁₁ = x::ts₁₁) (t₁_eq : t₁ = ts₁₁++(a::s₁₂)),
|
|
assert t₁_qeq : t₁ ≈ a|(ts₁₁++s₁₂), by rewrite t₁_eq; exact !qeq_app,
|
|
assert s₁_eq : s₁ = x::(ts₁₁++s₁₂), from calc
|
|
s₁ = s₁₁ ++ s₁₂ : and.elim_left C₁
|
|
... = x::(ts₁₁++ s₁₂) : by rewrite s₁₁_eq,
|
|
discr (and.elim_right C₂)
|
|
(λ (s₂₁_eq : s₂₁ = []) (x_eq_a : x = a) (t₂_eq: t₂ = s₂₂),
|
|
proof calc
|
|
s₁ = a::(ts₁₁++s₁₂) : by rewrite [s₁_eq, x_eq_a]
|
|
... ~ ts₁₁++(a::s₁₂) : !perm_middle
|
|
... = t₁ : t₁_eq
|
|
... ~ t₂ : p
|
|
... = s₂ : by rewrite [t₂_eq, and.elim_left C₂, s₂₁_eq, append_nil_left]
|
|
qed)
|
|
(λ (ts₂₁ : list A) (s₂₁_eq : s₂₁ = x::ts₂₁) (t₂_eq : t₂ = ts₂₁++(a::s₂₂)),
|
|
assert t₂_qeq : t₂ ≈ a|(ts₂₁++s₂₂), by rewrite t₂_eq; exact !qeq_app,
|
|
proof calc
|
|
s₁ = x::(ts₁₁++s₁₂) : s₁_eq
|
|
... ~ x::(ts₂₁++s₂₂) : skip x (r t₁_qeq t₂_qeq)
|
|
... = s₂ : by rewrite [-append_cons, -s₂₁_eq, and.elim_left C₂]
|
|
qed)))
|
|
(λ x y t₁ t₂ p (r : ∀{a s₁ s₂}, t₁≈a|s₁ → t₂≈a|s₂ → s₁ ~ s₂) a s₁ s₂ e₁ e₂,
|
|
obtain (s₁₁ s₁₂ : list A) (C₁ : s₁ = s₁₁ ++ s₁₂ ∧ y::x::t₁ = s₁₁++(a::s₁₂)), from qeq_split e₁,
|
|
obtain (s₂₁ s₂₂ : list A) (C₂ : s₂ = s₂₁ ++ s₂₂ ∧ x::y::t₂ = s₂₁++(a::s₂₂)), from qeq_split e₂,
|
|
discr₂ (and.elim_right C₁)
|
|
(λ (s₁₁_eq : s₁₁ = []) (s₁₂_eq : s₁₂ = x::t₁) (y_eq_a : y = a),
|
|
assert s₁_p : s₁ ~ x::t₂, from calc
|
|
s₁ = s₁₁ ++ s₁₂ : and.elim_left C₁
|
|
... = x::t₁ : by rewrite [s₁₂_eq, s₁₁_eq, append_nil_left]
|
|
... ~ x::t₂ : skip x p,
|
|
discr₂ (and.elim_right C₂)
|
|
(λ (s₂₁_eq : s₂₁ = []) (s₂₂_eq : s₂₂ = y::t₂) (x_eq_a : x = a),
|
|
proof calc
|
|
s₁ ~ x::t₂ : s₁_p
|
|
... = s₂₁ ++ s₂₂ : by rewrite [x_eq_a, -y_eq_a, -s₂₂_eq, s₂₁_eq, append_nil_left]
|
|
... = s₂ : by rewrite [and.elim_left C₂]
|
|
qed)
|
|
(λ (s₂₁_eq : s₂₁ = [x]) (y_eq_a : y = a) (t₂_eq : t₂ = s₂₂),
|
|
proof calc
|
|
s₁ ~ x::t₂ : s₁_p
|
|
... = s₂₁ ++ s₂₂ : by rewrite [t₂_eq, s₂₁_eq, append_cons]
|
|
... = s₂ : by rewrite [and.elim_left C₂]
|
|
qed)
|
|
(λ (ts₂₁ : list A) (s₂₁_eq : s₂₁ = x::y::ts₂₁) (t₂_eq : t₂ = ts₂₁++(a::s₂₂)),
|
|
proof calc
|
|
s₁ ~ x::t₂ : s₁_p
|
|
... = x::(ts₂₁++(y::s₂₂)) : by rewrite [t₂_eq, -y_eq_a]
|
|
... ~ x::y::(ts₂₁++s₂₂) : skip x !perm_middle
|
|
... = s₂₁ ++ s₂₂ : by rewrite [s₂₁_eq, append_cons]
|
|
... = s₂ : by rewrite [and.elim_left C₂]
|
|
qed))
|
|
(λ (s₁₁_eq : s₁₁ = [y]) (x_eq_a : x = a) (t₁_eq : t₁ = s₁₂),
|
|
assert s₁_p : s₁ ~ y::t₂, from calc
|
|
s₁ = y::t₁ : by rewrite [and.elim_left C₁, s₁₁_eq, t₁_eq]
|
|
... ~ y::t₂ : skip y p,
|
|
discr₂ (and.elim_right C₂)
|
|
(λ (s₂₁_eq : s₂₁ = []) (s₂₂_eq : s₂₂ = y::t₂) (x_eq_a : x = a),
|
|
proof calc
|
|
s₁ ~ y::t₂ : s₁_p
|
|
... = s₂₁ ++ s₂₂ : by rewrite [s₂₁_eq, s₂₂_eq]
|
|
... = s₂ : by rewrite [and.elim_left C₂]
|
|
qed)
|
|
(λ (s₂₁_eq : s₂₁ = [x]) (y_eq_a : y = a) (t₂_eq : t₂ = s₂₂),
|
|
proof calc
|
|
s₁ ~ y::t₂ : s₁_p
|
|
... = s₂₁ ++ s₂₂ : by rewrite [s₂₁_eq, t₂_eq, y_eq_a, -x_eq_a]
|
|
... = s₂ : by rewrite [and.elim_left C₂]
|
|
qed)
|
|
(λ (ts₂₁ : list A) (s₂₁_eq : s₂₁ = x::y::ts₂₁) (t₂_eq : t₂ = ts₂₁++(a::s₂₂)),
|
|
proof calc
|
|
s₁ ~ y::t₂ : s₁_p
|
|
... = y::(ts₂₁++(x::s₂₂)) : by rewrite [t₂_eq, -x_eq_a]
|
|
... ~ y::x::(ts₂₁++s₂₂) : skip y !perm_middle
|
|
... ~ x::y::(ts₂₁++s₂₂) : swap
|
|
... = s₂₁ ++ s₂₂ : by rewrite [s₂₁_eq]
|
|
... = s₂ : by rewrite [and.elim_left C₂]
|
|
qed))
|
|
(λ (ts₁₁ : list A) (s₁₁_eq : s₁₁ = y::x::ts₁₁) (t₁_eq : t₁ = ts₁₁++(a::s₁₂)),
|
|
assert s₁_eq : s₁ = y::x::(ts₁₁++s₁₂), by rewrite [and.elim_left C₁, s₁₁_eq],
|
|
discr₂ (and.elim_right C₂)
|
|
(λ (s₂₁_eq : s₂₁ = []) (s₂₂_eq : s₂₂ = y::t₂) (x_eq_a : x = a),
|
|
proof calc
|
|
s₁ = y::a::(ts₁₁++s₁₂) : by rewrite [s₁_eq, x_eq_a]
|
|
... ~ y::(ts₁₁++(a::s₁₂)) : skip y !perm_middle
|
|
... = y::t₁ : by rewrite t₁_eq
|
|
... ~ y::t₂ : skip y p
|
|
... = s₂₁ ++ s₂₂ : by rewrite [s₂₁_eq, s₂₂_eq]
|
|
... = s₂ : by rewrite [and.elim_left C₂]
|
|
qed)
|
|
(λ (s₂₁_eq : s₂₁ = [x]) (y_eq_a : y = a) (t₂_eq : t₂ = s₂₂),
|
|
proof calc
|
|
s₁ = y::x::(ts₁₁++s₁₂) : by rewrite s₁_eq
|
|
... ~ x::y::(ts₁₁++s₁₂) : swap
|
|
... = x::a::(ts₁₁++s₁₂) : by rewrite y_eq_a
|
|
... ~ x::(ts₁₁++(a::s₁₂)) : skip x !perm_middle
|
|
... = x::t₁ : by rewrite t₁_eq
|
|
... ~ x::t₂ : skip x p
|
|
... = s₂₁ ++ s₂₂ : by rewrite [t₂_eq, s₂₁_eq]
|
|
... = s₂ : by rewrite [and.elim_left C₂]
|
|
qed)
|
|
(λ (ts₂₁ : list A) (s₂₁_eq : s₂₁ = x::y::ts₂₁) (t₂_eq : t₂ = ts₂₁++(a::s₂₂)),
|
|
assert t₁_qeq : t₁ ≈ a|(ts₁₁++s₁₂), by rewrite t₁_eq; exact !qeq_app,
|
|
assert t₂_qeq : t₂ ≈ a|(ts₂₁++s₂₂), by rewrite t₂_eq; exact !qeq_app,
|
|
assert p_aux : ts₁₁++s₁₂ ~ ts₂₁++s₂₂, from r t₁_qeq t₂_qeq,
|
|
proof calc
|
|
s₁ = y::x::(ts₁₁++s₁₂) : by rewrite s₁_eq
|
|
... ~ y::x::(ts₂₁++s₂₂) : skip y (skip x p_aux)
|
|
... ~ x::y::(ts₂₁++s₂₂) : swap
|
|
... = s₂₁ ++ s₂₂ : by rewrite s₂₁_eq
|
|
... = s₂ : by rewrite [and.elim_left C₂]
|
|
qed)))
|
|
(λ t₁ t₂ t₃ p₁ p₂
|
|
(r₁ : ∀{a s₁ s₂}, t₁ ≈ a|s₁ → t₂≈a|s₂ → s₁ ~ s₂)
|
|
(r₂ : ∀{a s₁ s₂}, t₂ ≈ a|s₁ → t₃≈a|s₂ → s₁ ~ s₂)
|
|
a s₁ s₂ e₁ e₂,
|
|
have aint₁ : a ∈ t₁, from mem_head_of_qeq e₁,
|
|
have aint₂ : a ∈ t₂, from mem_perm p₁ aint₁,
|
|
obtain (t₂' : list A) (e₂' : t₂≈a|t₂'), from qeq_of_mem aint₂,
|
|
calc s₁ ~ t₂' : r₁ e₁ e₂'
|
|
... ~ s₂ : r₂ e₂' e₂)
|
|
|
|
theorem perm_cons_inv {a : A} {l₁ l₂ : list A} : a::l₁ ~ a::l₂ → l₁ ~ l₂ :=
|
|
assume p, perm_inv_core p (qeq.qhead a l₁) (qeq.qhead a l₂)
|
|
|
|
theorem perm_app_inv {a : A} {l₁ l₂ l₃ l₄ : list A} : l₁++(a::l₂) ~ l₃++(a::l₄) → l₁++l₂ ~ l₃++l₄ :=
|
|
assume p : l₁++(a::l₂) ~ l₃++(a::l₄),
|
|
have p' : a::(l₁++l₂) ~ a::(l₃++l₄), from calc
|
|
a::(l₁++l₂) ~ l₁++(a::l₂) : perm_middle
|
|
... ~ l₃++(a::l₄) : p
|
|
... ~ a::(l₃++l₄) : symm (!perm_middle),
|
|
perm_cons_inv p'
|
|
|
|
section foldl
|
|
variables {f : B → A → B} {l₁ l₂ : list A}
|
|
variable rcomm : right_commutative f
|
|
include rcomm
|
|
|
|
theorem foldl_eq_of_perm : l₁ ~ l₂ → ∀ b, foldl f b l₁ = foldl f b l₂ :=
|
|
assume p, perm_induction_on p
|
|
(λ b, by rewrite *foldl_nil)
|
|
(λ x t₁ t₂ p r b, calc
|
|
foldl f b (x::t₁) = foldl f (f b x) t₁ : foldl_cons
|
|
... = foldl f (f b x) t₂ : r (f b x)
|
|
... = foldl f b (x::t₂) : foldl_cons)
|
|
(λ x y t₁ t₂ p r b, calc
|
|
foldl f b (y :: x :: t₁) = foldl f (f (f b y) x) t₁ : by rewrite foldl_cons
|
|
... = foldl f (f (f b x) y) t₁ : by rewrite rcomm
|
|
... = foldl f (f (f b x) y) t₂ : r (f (f b x) y)
|
|
... = foldl f b (x :: y :: t₂) : by rewrite foldl_cons)
|
|
(λ t₁ t₂ t₃ p₁ p₂ r₁ r₂ b, eq.trans (r₁ b) (r₂ b))
|
|
end foldl
|
|
|
|
section foldr
|
|
variables {f : A → B → B} {l₁ l₂ : list A}
|
|
variable lcomm : left_commutative f
|
|
include lcomm
|
|
|
|
theorem foldr_eq_of_perm : l₁ ~ l₂ → ∀ b, foldr f b l₁ = foldr f b l₂ :=
|
|
assume p, perm_induction_on p
|
|
(λ b, by rewrite *foldl_nil)
|
|
(λ x t₁ t₂ p r b, calc
|
|
foldr f b (x::t₁) = f x (foldr f b t₁) : foldr_cons
|
|
... = f x (foldr f b t₂) : by rewrite [r b]
|
|
... = foldr f b (x::t₂) : foldr_cons)
|
|
(λ x y t₁ t₂ p r b, calc
|
|
foldr f b (y :: x :: t₁) = f y (f x (foldr f b t₁)) : by rewrite foldr_cons
|
|
... = f x (f y (foldr f b t₁)) : by rewrite lcomm
|
|
... = f x (f y (foldr f b t₂)) : by rewrite [r b]
|
|
... = foldr f b (x :: y :: t₂) : by rewrite foldr_cons)
|
|
(λ t₁ t₂ t₃ p₁ p₂ r₁ r₂ a, eq.trans (r₁ a) (r₂ a))
|
|
end foldr
|
|
|
|
theorem perm_erase_dup_of_perm [H : decidable_eq A] {l₁ l₂ : list A} : l₁ ~ l₂ → erase_dup l₁ ~ erase_dup l₂ :=
|
|
assume p, perm_induction_on p
|
|
nil
|
|
(λ x t₁ t₂ p r, by_cases
|
|
(λ xint₁ : x ∈ t₁,
|
|
assert xint₂ : x ∈ t₂, from mem_of_mem_erase_dup (mem_perm r (mem_erase_dup xint₁)),
|
|
by rewrite [erase_dup_cons_of_mem xint₁, erase_dup_cons_of_mem xint₂]; exact r)
|
|
(λ nxint₁ : x ∉ t₁,
|
|
assert nxint₂ : x ∉ t₂, from
|
|
assume xint₂ : x ∈ t₂, absurd (mem_of_mem_erase_dup (mem_perm (symm r) (mem_erase_dup xint₂))) nxint₁,
|
|
by rewrite [erase_dup_cons_of_not_mem nxint₂, erase_dup_cons_of_not_mem nxint₁]; exact (skip x r)))
|
|
(λ y x t₁ t₂ p r, by_cases
|
|
(λ xinyt₁ : x ∈ y::t₁, by_cases
|
|
(λ yint₁ : y ∈ t₁,
|
|
assert yint₂ : y ∈ t₂, from mem_of_mem_erase_dup (mem_perm r (mem_erase_dup yint₁)),
|
|
assert yinxt₂ : y ∈ x::t₂, from or.inr (yint₂),
|
|
or.elim (eq_or_mem_of_mem_cons xinyt₁)
|
|
(λ xeqy : x = y,
|
|
assert xint₂ : x ∈ t₂, by rewrite [-xeqy at yint₂]; exact yint₂,
|
|
begin
|
|
rewrite [erase_dup_cons_of_mem xinyt₁, erase_dup_cons_of_mem yinxt₂,
|
|
erase_dup_cons_of_mem yint₁, erase_dup_cons_of_mem xint₂],
|
|
exact r
|
|
end)
|
|
(λ xint₁ : x ∈ t₁,
|
|
assert xint₂ : x ∈ t₂, from mem_of_mem_erase_dup (mem_perm r (mem_erase_dup xint₁)),
|
|
begin
|
|
rewrite [erase_dup_cons_of_mem xinyt₁, erase_dup_cons_of_mem yinxt₂,
|
|
erase_dup_cons_of_mem yint₁, erase_dup_cons_of_mem xint₂],
|
|
exact r
|
|
end))
|
|
(λ nyint₁ : y ∉ t₁,
|
|
assert nyint₂ : y ∉ t₂, from
|
|
assume yint₂ : y ∈ t₂, absurd (mem_of_mem_erase_dup (mem_perm (symm r) (mem_erase_dup yint₂))) nyint₁,
|
|
by_cases
|
|
(λ xeqy : x = y,
|
|
assert nxint₂ : x ∉ t₂, by rewrite [-xeqy at nyint₂]; exact nyint₂,
|
|
assert yinxt₂ : y ∈ x::t₂, by rewrite [xeqy]; exact !mem_cons,
|
|
begin
|
|
rewrite [erase_dup_cons_of_mem xinyt₁, erase_dup_cons_of_mem yinxt₂,
|
|
erase_dup_cons_of_not_mem nyint₁, erase_dup_cons_of_not_mem nxint₂, xeqy],
|
|
exact skip y r
|
|
end)
|
|
(λ xney : x ≠ y,
|
|
have xint₁ : x ∈ t₁, from or_resolve_right xinyt₁ xney,
|
|
assert xint₂ : x ∈ t₂, from mem_of_mem_erase_dup (mem_perm r (mem_erase_dup xint₁)),
|
|
assert nyinxt₂ : y ∉ x::t₂, from
|
|
assume yinxt₂ : y ∈ x::t₂, or.elim (eq_or_mem_of_mem_cons yinxt₂)
|
|
(λ h, absurd h (ne.symm xney))
|
|
(λ h, absurd h nyint₂),
|
|
begin
|
|
rewrite [erase_dup_cons_of_mem xinyt₁, erase_dup_cons_of_not_mem nyinxt₂,
|
|
erase_dup_cons_of_not_mem nyint₁, erase_dup_cons_of_mem xint₂],
|
|
exact skip y r
|
|
end)))
|
|
(λ nxinyt₁ : x ∉ y::t₁,
|
|
have xney : x ≠ y, from not_eq_of_not_mem nxinyt₁,
|
|
have nxint₁ : x ∉ t₁, from not_mem_of_not_mem nxinyt₁,
|
|
assert nxint₂ : x ∉ t₂, from
|
|
assume xint₂ : x ∈ t₂, absurd (mem_of_mem_erase_dup (mem_perm (symm r) (mem_erase_dup xint₂))) nxint₁,
|
|
by_cases
|
|
(λ yint₁ : y ∈ t₁,
|
|
assert yinxt₂ : y ∈ x::t₂, from or.inr (mem_of_mem_erase_dup (mem_perm r (mem_erase_dup yint₁))),
|
|
begin
|
|
rewrite [erase_dup_cons_of_not_mem nxinyt₁, erase_dup_cons_of_mem yinxt₂,
|
|
erase_dup_cons_of_mem yint₁, erase_dup_cons_of_not_mem nxint₂],
|
|
exact skip x r
|
|
end)
|
|
(λ nyint₁ : y ∉ t₁,
|
|
assert nyinxt₂ : y ∉ x::t₂, from
|
|
assume yinxt₂ : y ∈ x::t₂, or.elim (eq_or_mem_of_mem_cons yinxt₂)
|
|
(λ h, absurd h (ne.symm xney))
|
|
(λ h, absurd (mem_of_mem_erase_dup (mem_perm (symm r) (mem_erase_dup h))) nyint₁),
|
|
begin
|
|
rewrite [erase_dup_cons_of_not_mem nxinyt₁, erase_dup_cons_of_not_mem nyinxt₂,
|
|
erase_dup_cons_of_not_mem nyint₁, erase_dup_cons_of_not_mem nxint₂],
|
|
exact xswap x y r
|
|
end)))
|
|
(λ t₁ t₂ t₃ p₁ p₂ r₁ r₂, trans r₁ r₂)
|
|
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section perm_union
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variable [H : decidable_eq A]
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include H
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theorem perm_union_left {l₁ l₂ : list A} (t₁ : list A) : l₁ ~ l₂ → (union l₁ t₁) ~ (union l₂ t₁) :=
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assume p, perm.induction_on p
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(by rewrite [nil_union]; exact !refl)
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(λ x l₁ l₂ p₁ r₁, by_cases
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(λ xint₁ : x ∈ t₁, by rewrite [*union_cons_of_mem _ xint₁]; exact r₁)
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(λ nxint₁ : x ∉ t₁, by rewrite [*union_cons_of_not_mem _ nxint₁]; exact (skip _ r₁)))
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(λ x y l, by_cases
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(λ yint : y ∈ t₁, by_cases
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(λ xint : x ∈ t₁,
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by rewrite [*union_cons_of_mem _ xint, *union_cons_of_mem _ yint, *union_cons_of_mem _ xint]; exact !refl)
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(λ nxint : x ∉ t₁,
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by rewrite [*union_cons_of_mem _ yint, *union_cons_of_not_mem _ nxint, union_cons_of_mem _ yint]; exact !refl))
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(λ nyint : y ∉ t₁, by_cases
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(λ xint : x ∈ t₁,
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by rewrite [*union_cons_of_mem _ xint, *union_cons_of_not_mem _ nyint, union_cons_of_mem _ xint]; exact !refl)
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(λ nxint : x ∉ t₁,
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by rewrite [*union_cons_of_not_mem _ nxint, *union_cons_of_not_mem _ nyint, union_cons_of_not_mem _ nxint]; exact !swap)))
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(λ l₁ l₂ l₃ p₁ p₂ r₁ r₂, trans r₁ r₂)
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theorem perm_union_right (l : list A) {t₁ t₂ : list A} : t₁ ~ t₂ → (union l t₁) ~ (union l t₂) :=
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list.induction_on l
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(λ p, by rewrite [*union_nil]; exact p)
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(λ x xs r p, by_cases
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(λ xint₁ : x ∈ t₁,
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assert xint₂ : x ∈ t₂, from mem_perm p xint₁,
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by rewrite [union_cons_of_mem _ xint₁, union_cons_of_mem _ xint₂]; exact (r p))
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(λ nxint₁ : x ∉ t₁,
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assert nxint₂ : x ∉ t₂, from not_mem_perm p nxint₁,
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by rewrite [union_cons_of_not_mem _ nxint₁, union_cons_of_not_mem _ nxint₂]; exact (skip _ (r p))))
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theorem perm_union {l₁ l₂ t₁ t₂ : list A} : l₁ ~ l₂ → t₁ ~ t₂ → (union l₁ t₁) ~ (union l₂ t₂) :=
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assume p₁ p₂, trans (perm_union_left t₁ p₁) (perm_union_right l₂ p₂)
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end perm_union
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section perm_insert
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variable [H : decidable_eq A]
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include H
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theorem perm_insert (a : A) {l₁ l₂ : list A} : l₁ ~ l₂ → (insert a l₁) ~ (insert a l₂) :=
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assume p, by_cases
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(λ ainl₁ : a ∈ l₁,
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assert ainl₂ : a ∈ l₂, from mem_perm p ainl₁,
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by rewrite [insert_eq_of_mem ainl₁, insert_eq_of_mem ainl₂]; exact p)
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(λ nainl₁ : a ∉ l₁,
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assert nainl₂ : a ∉ l₂, from not_mem_perm p nainl₁,
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by rewrite [insert_eq_of_not_mem nainl₁, insert_eq_of_not_mem nainl₂]; exact (skip _ p))
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end perm_insert
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section perm_intersection
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variable [H : decidable_eq A]
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include H
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theorem perm_intersection_left {l₁ l₂ : list A} (t₁ : list A) : l₁ ~ l₂ → (intersection l₁ t₁) ~ (intersection l₂ t₁) :=
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assume p, perm.induction_on p
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!refl
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(λ x l₁ l₂ p₁ r₁, by_cases
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(λ xint₁ : x ∈ t₁, by rewrite [*intersection_cons_of_mem _ xint₁]; exact (skip x r₁))
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(λ nxint₁ : x ∉ t₁, by rewrite [*intersection_cons_of_not_mem _ nxint₁]; exact r₁))
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(λ x y l, by_cases
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(λ yint : y ∈ t₁, by_cases
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(λ xint : x ∈ t₁,
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by rewrite [*intersection_cons_of_mem _ xint, *intersection_cons_of_mem _ yint, *intersection_cons_of_mem _ xint];
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exact !swap)
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(λ nxint : x ∉ t₁,
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by rewrite [*intersection_cons_of_mem _ yint, *intersection_cons_of_not_mem _ nxint, intersection_cons_of_mem _ yint];
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exact !refl))
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(λ nyint : y ∉ t₁, by_cases
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(λ xint : x ∈ t₁,
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by rewrite [*intersection_cons_of_mem _ xint, *intersection_cons_of_not_mem _ nyint, intersection_cons_of_mem _ xint];
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exact !refl)
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(λ nxint : x ∉ t₁,
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by rewrite [*intersection_cons_of_not_mem _ nxint, *intersection_cons_of_not_mem _ nyint,
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intersection_cons_of_not_mem _ nxint];
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exact !refl)))
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(λ l₁ l₂ l₃ p₁ p₂ r₁ r₂, trans r₁ r₂)
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theorem perm_intersection_right (l : list A) {t₁ t₂ : list A} : t₁ ~ t₂ → (intersection l t₁) ~ (intersection l t₂) :=
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list.induction_on l
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(λ p, by rewrite [*intersection_nil]; exact !refl)
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(λ x xs r p, by_cases
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(λ xint₁ : x ∈ t₁,
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assert xint₂ : x ∈ t₂, from mem_perm p xint₁,
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by rewrite [intersection_cons_of_mem _ xint₁, intersection_cons_of_mem _ xint₂]; exact (skip _ (r p)))
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(λ nxint₁ : x ∉ t₁,
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assert nxint₂ : x ∉ t₂, from not_mem_perm p nxint₁,
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by rewrite [intersection_cons_of_not_mem _ nxint₁, intersection_cons_of_not_mem _ nxint₂]; exact (r p)))
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theorem perm_intersection {l₁ l₂ t₁ t₂ : list A} : l₁ ~ l₂ → t₁ ~ t₂ → (intersection l₁ t₁) ~ (intersection l₂ t₂) :=
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assume p₁ p₂, trans (perm_intersection_left t₁ p₁) (perm_intersection_right l₂ p₂)
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end perm_intersection
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/- extensionality -/
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section ext
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open eq.ops
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theorem perm_ext : ∀ {l₁ l₂ : list A}, nodup l₁ → nodup l₂ → (∀a, a ∈ l₁ ↔ a ∈ l₂) → l₁ ~ l₂
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| [] [] d₁ d₂ e := !perm.nil
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| [] (a₂::t₂) d₁ d₂ e := absurd (iff.mp' (e a₂) !mem_cons) (not_mem_nil a₂)
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| (a₁::t₁) [] d₁ d₂ e := absurd (iff.mp (e a₁) !mem_cons) (not_mem_nil a₁)
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| (a₁::t₁) (a₂::t₂) d₁ d₂ e :=
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have a₁inl₂ : a₁ ∈ a₂::t₂, from iff.mp (e a₁) !mem_cons,
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have dt₁ : nodup t₁, from nodup_of_nodup_cons d₁,
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have na₁int₁ : a₁ ∉ t₁, from not_mem_of_nodup_cons d₁,
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have ex : ∃s₁ s₂, a₂::t₂ = s₁++(a₁::s₂), from mem_split a₁inl₂,
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obtain (s₁ s₂ : list A) (t₂_eq : a₂::t₂ = s₁++(a₁::s₂)), from ex,
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have dt₂' : nodup (a₁::(s₁++s₂)), from nodup_head (by rewrite [t₂_eq at d₂]; exact d₂),
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have na₁s₁s₂ : a₁ ∉ s₁++s₂, from not_mem_of_nodup_cons dt₂',
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have na₁s₁ : a₁ ∉ s₁, from not_mem_of_not_mem_append_left na₁s₁s₂,
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have na₁s₂ : a₁ ∉ s₂, from not_mem_of_not_mem_append_right na₁s₁s₂,
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have ds₁s₂ : nodup (s₁++s₂), from nodup_of_nodup_cons dt₂',
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have eqv : ∀a, a ∈ t₁ ↔ a ∈ s₁++s₂, from
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take a, iff.intro
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(λ aint₁ : a ∈ t₁,
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assert aina₂t₂ : a ∈ a₂::t₂, from iff.mp (e a) (mem_cons_of_mem _ aint₁),
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have ains₁a₁s₂ : a ∈ s₁++(a₁::s₂), by rewrite [t₂_eq at aina₂t₂]; exact aina₂t₂,
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or.elim (mem_or_mem_of_mem_append ains₁a₁s₂)
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(λ ains₁ : a ∈ s₁, mem_append_left s₂ ains₁)
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(λ aina₁s₂ : a ∈ a₁::s₂, or.elim (eq_or_mem_of_mem_cons aina₁s₂)
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(λ aeqa₁ : a = a₁, absurd (aeqa₁ ▸ aint₁) na₁int₁)
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(λ ains₂ : a ∈ s₂, mem_append_right s₁ ains₂)))
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(λ ains₁s₂ : a ∈ s₁ ++ s₂, or.elim (mem_or_mem_of_mem_append ains₁s₂)
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(λ ains₁ : a ∈ s₁,
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have aina₂t₂ : a ∈ a₂::t₂, from by rewrite [t₂_eq]; exact (mem_append_left _ ains₁),
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have aina₁t₁ : a ∈ a₁::t₁, from iff.mp' (e a) aina₂t₂,
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or.elim (eq_or_mem_of_mem_cons aina₁t₁)
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(λ aeqa₁ : a = a₁, absurd (aeqa₁ ▸ ains₁) na₁s₁)
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(λ aint₁ : a ∈ t₁, aint₁))
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(λ ains₂ : a ∈ s₂,
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have aina₂t₂ : a ∈ a₂::t₂, from by rewrite [t₂_eq]; exact (mem_append_right _ (mem_cons_of_mem _ ains₂)),
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have aina₁t₁ : a ∈ a₁::t₁, from iff.mp' (e a) aina₂t₂,
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or.elim (eq_or_mem_of_mem_cons aina₁t₁)
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(λ aeqa₁ : a = a₁, absurd (aeqa₁ ▸ ains₂) na₁s₂)
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(λ aint₁ : a ∈ t₁, aint₁))),
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calc a₁::t₁ ~ a₁::(s₁++s₂) : skip a₁ (perm_ext dt₁ ds₁s₂ eqv)
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... ~ s₁++(a₁::s₂) : !perm_middle
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... = a₂::t₂ : by rewrite t₂_eq
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end ext
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/- cross_product -/
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section cross_product
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theorem perm_cross_product_left {l₁ l₂ : list A} (t₁ : list B) : l₁ ~ l₂ → (cross_product l₁ t₁) ~ (cross_product l₂ t₁) :=
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assume p : l₁ ~ l₂, perm.induction_on p
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!perm.refl
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(λ x l₁ l₂ p r, perm_app !perm.refl r)
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(λ x y l,
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let m₁ := map (λ b, (x, b)) t₁ in
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let m₂ := map (λ b, (y, b)) t₁ in
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let c := cross_product l t₁ in
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calc m₂ ++ (m₁ ++ c) = (m₂ ++ m₁) ++ c : by rewrite append.assoc
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... ~ (m₁ ++ m₂) ++ c : perm_app !perm_app_comm !perm.refl
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... = m₁ ++ (m₂ ++ c) : by rewrite append.assoc)
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(λ l₁ l₂ l₃ p₁ p₂ r₁ r₂, trans r₁ r₂)
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theorem perm_cross_product_right (l : list A) {t₁ t₂ : list B} : t₁ ~ t₂ → (cross_product l t₁) ~ (cross_product l t₂) :=
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list.induction_on l
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(λ p, by rewrite [*nil_cross_product]; exact !perm.refl)
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(λ a t r p,
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perm_app (perm_map _ p) (r p))
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theorem perm_cross_product {l₁ l₂ : list A} {t₁ t₂ : list B} : l₁ ~ l₂ → t₁ ~ t₂ → (cross_product l₁ t₁) ~ (cross_product l₂ t₂) :=
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assume p₁ p₂, trans (perm_cross_product_left t₁ p₁) (perm_cross_product_right l₂ p₂)
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end cross_product
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end perm
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