254 lines
10 KiB
Text
254 lines
10 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
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open list setoid nat
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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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(λ x y l₁ l₂ p₁ r₁, list.no_confusion r₁)
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(λ x y l e, list.no_confusion e)
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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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(λ e₁ e₂, list.no_confusion e₂)
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(λ x l₁ l₂ p₁ r₁ e₁ e₂, list.no_confusion e₁)
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(λ x y l e₁ e₂, list.no_confusion e₁)
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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 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 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 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 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, list.no_confusion e (λ (e₁ : x = a) (e₂ : l₁ = []),
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begin
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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, list.no_confusion e (λ e₁ e₂, list.no_confusion e₂))
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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, list.no_confusion (eq_singlenton_of_perm_inv a p) (λ e₁ e₂, by rewrite e₁)
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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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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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if Heq : a = x then
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by rewrite [Heq, erase_cons_head]; exact !perm.refl
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else
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have aint : a ∈ t, from mem_of_ne_of_mem Heq 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 Heq]
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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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if Hax : a = x
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then by rewrite [Hax, *erase_cons_head]; exact p
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else by rewrite [*erase_cons_tail _ Hax]; exact (skip x r))
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(λ x y l,
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if Hax : a = x
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then (if Hay : a = y
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then by rewrite [-Hax, -Hay]; exact !perm.refl
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else by rewrite [-Hax, erase_cons_tail _ Hay, *erase_cons_head]; exact !perm.refl)
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else (if Hay : a = y
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then by rewrite [-Hay, erase_cons_tail _ Hax, *erase_cons_head]; exact !perm.refl
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else by rewrite[erase_cons_tail _ Hax, *erase_cons_tail _ Hay, erase_cons_tail _ Hax]; 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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if xinl₂ : x ∈ l₂ then
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let t₂ : list A := erase x l₂ in
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have len_t₁ : length t₁ = n, from nat.no_confusion H₁ (λ e, e),
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assert len_t₂_aux : length t₂ = pred (length l₂), from length_erase_of_mem x l₂ 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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else
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inr (λ p : x::t₁ ~ l₂, absurd (mem_perm x (x::t₁) l₂ p !mem_cons) xinl₂)
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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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if Hl : length l₁ = length l₂ then
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decidable_perm_aux (length l₂) l₁ l₂ Hl rfl
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else
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inr (λ p : l₁ ~ l₂, absurd (length_eq_length_of_perm p) Hl)
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end dec
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end perm
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