refactor(library/data/list/perm): use by_cases instead of dependent-if
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Leonardo de Moura
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9306830d8c
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c6a35e718d
1 changed files with 44 additions and 39 deletions
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@ -155,33 +155,37 @@ assume p, calc
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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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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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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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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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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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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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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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@ -226,29 +230,30 @@ definition decidable_perm_aux : ∀ (n : nat) (l₁ l₂ : list A), length l₁
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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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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, 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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(assume nxinl₂ : x ∉ l₂,
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inr (λ p : x::t₁ ~ l₂, absurd (mem_perm x (x::t₁) l₂ 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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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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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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end perm
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