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refactor(library/data/list/perm): use by_cases instead of dependent-if

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Leonardo de Moura 2015-04-07 09:26:24 -07:00
parent 9306830d8c
commit c6a35e718d
1 changed files with 44 additions and 39 deletions
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83
library/data/list/perm.lean
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@ -155,33 +155,37 @@ assume p, calc
... = l₁++(l₂++[a]) : append.assoc ... = l₁++(l₂++[a]) : append.assoc
... ~ l₁++(a::l₂) : perm_app_right l₁ (symm (perm_cons_app a l₂)) ... ~ l₁++(a::l₂) : perm_app_right l₁ (symm (perm_cons_app a l₂))
open decidable
theorem perm_erase [H : decidable_eq A] {a : A} : ∀ {l : list A}, a ∈ l → l ~ a::(erase a l) theorem perm_erase [H : decidable_eq A] {a : A} : ∀ {l : list A}, a ∈ l → l ~ a::(erase a l)
| [] h := absurd h !not_mem_nil | [] h := absurd h !not_mem_nil
| (x::t) h := | (x::t) h :=
if Heq : a = x then by_cases
by rewrite [Heq, erase_cons_head]; exact !perm.refl (assume aeqx : a = x, by rewrite [aeqx, erase_cons_head]; exact !perm.refl)
else (assume naeqx : a ≠ x,
have aint : a ∈ t, from mem_of_ne_of_mem Heq h, have aint : a ∈ t, from mem_of_ne_of_mem naeqx h,
have aux : t ~ a :: erase a t, from perm_erase aint, have aux : t ~ a :: erase a t, from perm_erase aint,
calc x::t ~ x::a::(erase a t) : skip x aux calc x::t ~ x::a::(erase a t) : skip x aux
... ~ a::x::(erase a t) : swap ... ~ a::x::(erase a t) : swap
... = a::(erase a (x::t)) : by rewrite [!erase_cons_tail Heq] ... = a::(erase a (x::t)) : by rewrite [!erase_cons_tail naeqx])
theorem erase_perm_erase_of_perm [H : decidable_eq A] (a : A) {l₁ l₂ : list A} : l₁ ~ l₂ → erase a l₁ ~ erase a l₂ := theorem erase_perm_erase_of_perm [H : decidable_eq A] (a : A) {l₁ l₂ : list A} : l₁ ~ l₂ → erase a l₁ ~ erase a l₂ :=
assume p, perm.induction_on p assume p, perm.induction_on p
nil nil
(λ x t₁ t₂ p r, (λ x t₁ t₂ p r,
if Hax : a = x by_cases
then by rewrite [Hax, *erase_cons_head]; exact p (assume aeqx : a = x, by rewrite [aeqx, *erase_cons_head]; exact p)
else by rewrite [*erase_cons_tail _ Hax]; exact (skip x r)) (assume naeqx : a ≠ x, by rewrite [*erase_cons_tail _ naeqx]; exact (skip x r)))
(λ x y l, (λ x y l,
if Hax : a = x by_cases
then (if Hay : a = y (assume aeqx : a = x,
then by rewrite [-Hax, -Hay]; exact !perm.refl by_cases
else by rewrite [-Hax, erase_cons_tail _ Hay, *erase_cons_head]; exact !perm.refl) (assume aeqy : a = y, by rewrite [-aeqx, -aeqy]; exact !perm.refl)
else (if Hay : a = y (assume naeqy : a ≠ y, by rewrite [-aeqx, erase_cons_tail _ naeqy, *erase_cons_head]; exact !perm.refl))
then by rewrite [-Hay, erase_cons_tail _ Hax, *erase_cons_head]; exact !perm.refl (assume naeqx : a ≠ x,
else by rewrite[erase_cons_tail _ Hax, *erase_cons_tail _ Hay, erase_cons_tail _ Hax]; exact !swap)) by_cases
(assume aeqy : a = y, by rewrite [-aeqy, erase_cons_tail _ naeqx, *erase_cons_head]; exact !perm.refl)
(assume naeqy : a ≠ y, by rewrite[erase_cons_tail _ naeqx, *erase_cons_tail _ naeqy, erase_cons_tail _ naeqx];
exact !swap)))
(λ l₁ l₂ l₃ p₁ p₂ r₁ r₂, trans r₁ r₂) (λ l₁ l₂ l₃ p₁ p₂ r₁ r₂, trans r₁ r₂)
theorem perm_induction_on {P : list A → list A → Prop} {l₁ l₂ : list A} (p : l₁ ~ l₂) theorem perm_induction_on {P : list A → list A → Prop} {l₁ l₂ : list A} (p : l₁ ~ l₂)
@ -226,29 +230,30 @@ definition decidable_perm_aux : ∀ (n : nat) (l₁ l₂ : list A), length l₁
assert l₂n : l₂ = [], from eq_nil_of_length_eq_zero H₂, assert l₂n : l₂ = [], from eq_nil_of_length_eq_zero H₂,
by rewrite [l₁n, l₂n]; exact (inl perm.nil) by rewrite [l₁n, l₂n]; exact (inl perm.nil)
| (n+1) (x::t₁) l₂ H₁ H₂ := | (n+1) (x::t₁) l₂ H₁ H₂ :=
if xinl₂ : x ∈ l₂ then by_cases
let t₂ : list A := erase x l₂ in (assume xinl₂ : x ∈ l₂,
have len_t₁ : length t₁ = n, from nat.no_confusion H₁ (λ e, e), let t₂ : list A := erase x l₂ in
assert len_t₂_aux : length t₂ = pred (length l₂), from length_erase_of_mem x l₂ xinl₂, have len_t₁ : length t₁ = n, from nat.no_confusion H₁ (λ e, e),
assert len_t₂ : length t₂ = n, by rewrite [len_t₂_aux, H₂], assert len_t₂_aux : length t₂ = pred (length l₂), from length_erase_of_mem x l₂ xinl₂,
match decidable_perm_aux n t₁ t₂ len_t₁ len_t₂ with assert len_t₂ : length t₂ = n, by rewrite [len_t₂_aux, H₂],
| inl p := inl (calc match decidable_perm_aux n t₁ t₂ len_t₁ len_t₂ with
x::t₁ ~ x::(erase x l₂) : skip x p | inl p := inl (calc
... ~ l₂ : perm_erase xinl₂) x::t₁ ~ x::(erase x l₂) : skip x p
| inr np := inr (λ p : x::t₁ ~ l₂, ... ~ l₂ : perm_erase xinl₂)
assert p₁ : erase x (x::t₁) ~ erase x l₂, from erase_perm_erase_of_perm x p, | inr np := inr (λ p : x::t₁ ~ l₂,
have p₂ : t₁ ~ erase x l₂, by rewrite [erase_cons_head at p₁]; exact p₁, assert p₁ : erase x (x::t₁) ~ erase x l₂, from erase_perm_erase_of_perm x p,
absurd p₂ np) have p₂ : t₁ ~ erase x l₂, by rewrite [erase_cons_head at p₁]; exact p₁,
end absurd p₂ np)
else end)
inr (λ p : x::t₁ ~ l₂, absurd (mem_perm x (x::t₁) l₂ p !mem_cons) xinl₂) (assume nxinl₂ : x ∉ l₂,
inr (λ p : x::t₁ ~ l₂, absurd (mem_perm x (x::t₁) l₂ p !mem_cons) nxinl₂))
definition decidable_perm [instance] : ∀ (l₁ l₂ : list A), decidable (l₁ ~ l₂) := definition decidable_perm [instance] : ∀ (l₁ l₂ : list A), decidable (l₁ ~ l₂) :=
λ l₁ l₂, λ l₁ l₂,
if Hl : length l₁ = length l₂ then by_cases
decidable_perm_aux (length l₂) l₁ l₂ Hl rfl (assume eql : length l₁ = length l₂,
else decidable_perm_aux (length l₂) l₁ l₂ eql rfl)
inr (λ p : l₁ ~ l₂, absurd (length_eq_length_of_perm p) Hl) (assume neql : length l₁ ≠ length l₂,
inr (λ p : l₁ ~ l₂, absurd (length_eq_length_of_perm p) neql))
end dec end dec
end perm end perm