feat(library/data/perm): cleanup and add calculational proof support to perm

This commit is contained in:
Leonardo de Moura 2015-04-02 08:50:05 -07:00
parent 54136c1ec0
commit 92d6e93971

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@ -19,8 +19,9 @@ inductive perm : list A → list A → Prop :=
| trans : Π {l₁ l₂ l₃ : list A}, perm l₁ l₂ → perm l₂ l₃ → perm l₁ l₃
namespace perm
theorem eq_nil_of_perm_nil {l₁ : list A} (p : perm [] l₁) : l₁ = [] :=
have gen : ∀ (l₂ : list A) (p : perm l₂ l₁), l₂ = [] → l₁ = [], from
infix ~:50 := perm
theorem eq_nil_of_perm_nil {l₁ : list A} (p : [] ~ l₁) : l₁ = [] :=
have gen : ∀ (l₂ : list A) (p : l₂ ~ l₁), l₂ = [] → l₁ = [], from
take l₂ p, perm.induction_on p
(λ h, h)
(λ x y l₁ l₂ p₁ r₁, list.no_confusion r₁)
@ -28,8 +29,8 @@ namespace perm
(λ l₁ l₂ l₃ p₁ p₂ r₁ r₂ e, r₂ (r₁ e)),
gen [] p rfl
theorem not_perm_nil_cons (x : A) (l : list A) : ¬ perm [] (x::l) :=
have gen : ∀ (l₁ l₂ : list A) (p : perm l₁ l₂), l₁ = [] → l₂ = (x::l) → false, from
theorem not_perm_nil_cons (x : A) (l : list A) : ¬ [] ~ (x::l) :=
have gen : ∀ (l₁ l₂ : list A) (p : l₁ ~ l₂), l₁ = [] → l₂ = (x::l) → false, from
take l₁ l₂ p, perm.induction_on p
(λ e₁ e₂, list.no_confusion e₂)
(λ x l₁ l₂ p₁ r₁ e₁ e₂, list.no_confusion e₁)
@ -42,85 +43,89 @@ namespace perm
end),
assume p, gen [] (x::l) p rfl rfl
protected theorem refl : ∀ (l : list A), perm l l
protected theorem refl : ∀ (l : list A), l ~ l
| [] := nil
| (x::xs) := skip x (refl xs)
protected theorem symm : ∀ {l₁ l₂ : list A}, perm l₁ l₂ → perm l₂ l₁ :=
protected theorem symm : ∀ {l₁ l₂ : list A}, l₁ ~ l₂ → l₂ ~ l₁ :=
take l₁ l₂ p, perm.induction_on p
nil
(λ x l₁ l₂ p₁ r₁, skip x r₁)
(λ x y l, swap y x l)
(λ l₁ l₂ l₃ p₁ p₂ r₁ r₂, trans r₂ r₁)
theorem is_eqv (A : Type) : equivalence (@perm A) :=
mk_equivalence (@perm A) (@refl A) (@symm A) (@trans A)
theorem eqv (A : Type) : equivalence (@perm A) :=
mk_equivalence (@perm A) (@perm.refl A) (@perm.symm A) (@perm.trans A)
protected definition is_setoid [instance] (A : Type) : setoid (list A) :=
setoid.mk (@perm A) (is_eqv A)
setoid.mk (@perm A) (perm.eqv A)
theorem mem_perm (a : A) (l₁ l₂ : list A) : perm l₁ l₂ → a ∈ l₁ → a ∈ l₂ :=
calc_refl perm.refl
calc_symm perm.symm
calc_trans perm.trans
-- TODO: remove this theorems after we improve calc
theorem perm_of_eq_of_perm (l₁ l₂ l₃ : list A) : l₁ = l₂ → l₂ ~ l₃ → l₁ ~ l₃ :=
assume e p, eq.rec_on (eq.symm e) p
theorem perm_of_perm_of_eq (l₁ l₂ l₃ : list A) : l₁ ~ l₂ → l₂ = l₃ → l₁ ~ l₃ :=
assume p e, eq.rec_on e p
calc_trans perm_of_perm_of_eq
calc_trans perm_of_eq_of_perm
theorem mem_perm (a : A) (l₁ l₂ : list A) : l₁ ~ l₂ → a ∈ l₁ → a ∈ l₂ :=
assume p, perm.induction_on p
(λ h, h)
(λ x l₁ l₂ p₁ r₁ i, or.elim i
(λ aeqx, by rewrite aeqx; apply !mem_cons)
(λ ainl₁ : a ∈ l₁, or.inr (r₁ ainl₁)))
(assume aeqx : a = x, by rewrite aeqx; apply !mem_cons)
(assume ainl₁ : a ∈ l₁, or.inr (r₁ ainl₁)))
(λ x y l ainyxl, or.elim ainyxl
(λ aeqy : a = y, by rewrite aeqy; exact (or.inr !mem_cons))
(λ ainxl : a ∈ x::l, or.elim ainxl
(λ aeqx : a = x, or.inl aeqx)
(λ ainl : a ∈ l, or.inr (or.inr ainl))))
(assume aeqy : a = y, by rewrite aeqy; exact (or.inr !mem_cons))
(assume ainxl : a ∈ x::l, or.elim ainxl
(assume aeqx : a = x, or.inl aeqx)
(assume ainl : a ∈ l, or.inr (or.inr ainl))))
(λ l₁ l₂ l₃ p₁ p₂ r₁ r₂ ainl₁, r₂ (r₁ ainl₁))
theorem perm_app_left {l₁ l₂ : list A} (t₁ : list A) : perm l₁ l₂ → perm (l₁++t₁) (l₂++t₁) :=
theorem perm_app_left {l₁ l₂ : list A} (t₁ : list A) : l₁ ~ l₂ → (l₁++t₁) ~ (l₂++t₁) :=
assume p, perm.induction_on p
!refl
(λ x l₁ l₂ p₁ r₁, skip x r₁)
(λ x y l, !swap)
(λ l₁ l₂ l₃ p₁ p₂ r₁ r₂, trans r₁ r₂)
theorem perm_app_right (l : list A) {t₁ t₂ : list A} : perm t₁ t₂ → perm (l++t₁) (l++t₂) :=
theorem perm_app_right (l : list A) {t₁ t₂ : list A} : t₁ ~ t₂ → (l++t₁) ~ (l++t₂) :=
list.induction_on l
(λ p, p)
(λ x xs r p, skip x (r p))
theorem perm_app {l₁ l₂ t₁ t₂ : list A} : perm l₁ l₂ → perm t₁ t₂ → perm (l₁++t₁) (l₂++t₂) :=
theorem perm_app {l₁ l₂ t₁ t₂ : list A} : l₁ ~ l₂ → t₁ ~ t₂ → (l₁++t₁) ~ (l₂++t₂) :=
assume p₁ p₂, trans (perm_app_left t₁ p₁) (perm_app_right l₂ p₂)
theorem perm_app_cons (a : A) {h₁ h₂ t₁ t₂ : list A} : perm h₁ h₂ → perm t₁ t₂ → perm (h₁ ++ (a::t₁)) (h₂ ++ (a::t₂)) :=
theorem perm_app_cons (a : A) {h₁ h₂ t₁ t₂ : list A} : h₁ ~ h₂ → t₁ ~ t₂ → (h₁ ++ (a::t₁)) ~ (h₂ ++ (a::t₂)) :=
assume p₁ p₂, perm_app p₁ (skip a p₂)
theorem perm_cons_app (a : A) : ∀ (l : list A), perm (a::l) (l ++ [a])
theorem perm_cons_app (a : A) : ∀ (l : list A), (a::l) ~ (l ++ [a])
| [] := !refl
| (x::xs) :=
show perm (a::x::xs) (x::(xs ++ [a])), from
have p₁ : perm (a::xs) (xs++[a]), from perm_cons_app xs,
have p₂ : perm (x::a::xs) (x::(xs++[a])), from skip x p₁,
have p₃ : perm (a::x::xs) (x::a::xs), from swap x a xs,
trans p₃ p₂
| (x::xs) := calc
a::x::xs ~ x::a::xs : swap x a xs
... ~ x::(xs++[a]) : skip x (perm_cons_app xs)
theorem perm_app_comm {l₁ l₂ : list A} : perm (l₁++l₂) (l₂++l₁) :=
theorem perm_app_comm {l₁ l₂ : list A} : (l₁++l₂) ~ (l₂++l₁) :=
list.induction_on l₁
(by rewrite [append_nil_right, append_nil_left]; apply refl)
(λ a t r,
show perm (a::(t++l₂)) (l₂++(a::t)), from
begin
have p₀ : perm (a::(t++l₂)) (a::(l₂++t)), from skip a r,
have p₁ : perm (a::(l₂++t)) (l₂++t++[a]), from !perm_cons_app,
have p₂ : perm (t++[a]) (a::t), from symm (perm_cons_app a t),
have p₃ : perm (l₂++(t++[a])) (l₂++(a::t)), from perm_app_right l₂ p₂,
rewrite [append.assoc at p₁],
exact (trans p₀ (trans p₁ p₃))
end)
(λ a t r, calc
a::(t++l₂) ~ a::(l₂++t) : skip a r
... ~ l₂++t++[a] : perm_cons_app
... = l₂++(t++[a]) : append.assoc
... ~ l₂++(a::t) : perm_app_right l₂ (symm (perm_cons_app a t)))
theorem length_eq_lenght_of_perm {l₁ l₂ : list A} : perm l₁ l₂ → length l₁ = length l₂ :=
theorem length_eq_length_of_perm {l₁ l₂ : list A} : l₁ ~ l₂ → length l₁ = length l₂ :=
assume p, perm.induction_on p
rfl
(λ x l₁ l₂ p r, by rewrite [*length_cons, r])
(λ x y l, by rewrite *length_cons)
(λ l₁ l₂ l₃ p₁ p₂ r₁ r₂, eq.trans r₁ r₂)
theorem eq_singlenton_of_perm_inv (a : A) {l : list A} : perm [a] l → l = [a] :=
theorem eq_singlenton_of_perm_inv (a : A) {l : list A} : [a] ~ l → l = [a] :=
have gen : ∀ l₂, perm l₂ l → l₂ = [a] → l = [a], from
take l₂, assume p, perm.induction_on p
(λ e, e)
@ -134,15 +139,13 @@ namespace perm
(λ l₁ l₂ l₃ p₁ p₂ r₁ r₂ e, r₂ (r₁ e)),
assume p, gen [a] p rfl
theorem eq_singlenton_of_perm (a b : A) : perm [a] [b] → a = b :=
theorem eq_singlenton_of_perm (a b : A) : [a] ~ [b] → a = b :=
assume p, list.no_confusion (eq_singlenton_of_perm_inv a p) (λ e₁ e₂, by rewrite e₁)
theorem perm_rev : ∀ (l : list A), perm l (reverse l)
theorem perm_rev : ∀ (l : list A), l ~ (reverse l)
| [] := nil
| (x::xs) :=
begin
rewrite [reverse_cons, concat_eq_append],
apply (trans (perm_cons_app x xs)),
exact (perm_app_left [x] (perm_rev xs))
end
| (x::xs) := calc
x::xs ~ xs++[x] : perm_cons_app x xs
... ~ reverse xs ++ [x] : perm_app_left [x] (perm_rev xs)
... = reverse (x::xs) : by rewrite [reverse_cons, concat_eq_append]
end perm