581 lines
22 KiB
Text
581 lines
22 KiB
Text
/-
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Copyright (c) 2014 Parikshit Khanna. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Parikshit Khanna, Jeremy Avigad, Leonardo de Moura
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Basic properties of lists.
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-/
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import logic tools.helper_tactics data.nat.order algebra.function
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open eq.ops helper_tactics nat prod function option
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inductive list (T : Type) : Type :=
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| nil {} : list T
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| cons : T → list T → list T
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namespace list
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notation h :: t := cons h t
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notation `[` l:(foldr `,` (h t, cons h t) nil `]`) := l
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variable {T : Type}
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lemma cons_ne_nil (a : T) (l : list T) : a::l ≠ [] :=
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by contradiction
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/- append -/
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definition append : list T → list T → list T
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| [] l := l
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| (h :: s) t := h :: (append s t)
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notation l₁ ++ l₂ := append l₁ l₂
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theorem append_nil_left (t : list T) : [] ++ t = t
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theorem append_cons (x : T) (s t : list T) : (x::s) ++ t = x::(s ++ t)
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theorem append_nil_right : ∀ (t : list T), t ++ [] = t
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| [] := rfl
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| (a :: l) := calc
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(a :: l) ++ [] = a :: (l ++ []) : rfl
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... = a :: l : append_nil_right l
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theorem append.assoc : ∀ (s t u : list T), s ++ t ++ u = s ++ (t ++ u)
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| [] t u := rfl
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| (a :: l) t u :=
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show a :: (l ++ t ++ u) = (a :: l) ++ (t ++ u),
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by rewrite (append.assoc l t u)
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/- length -/
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definition length : list T → nat
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| [] := 0
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| (a :: l) := length l + 1
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theorem length_nil : length (@nil T) = 0
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theorem length_cons (x : T) (t : list T) : length (x::t) = length t + 1
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theorem length_append : ∀ (s t : list T), length (s ++ t) = length s + length t
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| [] t := calc
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length ([] ++ t) = length t : rfl
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... = length [] + length t : zero_add
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| (a :: s) t := calc
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length (a :: s ++ t) = length (s ++ t) + 1 : rfl
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... = length s + length t + 1 : length_append
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... = (length s + 1) + length t : succ_add
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... = length (a :: s) + length t : rfl
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theorem eq_nil_of_length_eq_zero : ∀ {l : list T}, length l = 0 → l = []
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| [] H := rfl
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| (a::s) H := by contradiction
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theorem ne_nil_of_length_eq_succ : ∀ {l : list T} {n : nat}, length l = succ n → l ≠ []
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| [] n h := by contradiction
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| (a::l) n h := by contradiction
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-- add_rewrite length_nil length_cons
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/- concat -/
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definition concat : Π (x : T), list T → list T
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| a [] := [a]
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| a (b :: l) := b :: concat a l
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theorem concat_nil (x : T) : concat x [] = [x]
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theorem concat_cons (x y : T) (l : list T) : concat x (y::l) = y::(concat x l)
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theorem concat_eq_append (a : T) : ∀ (l : list T), concat a l = l ++ [a]
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| [] := rfl
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| (b :: l) :=
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show b :: (concat a l) = (b :: l) ++ (a :: []),
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by rewrite concat_eq_append
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theorem concat_ne_nil (a : T) : ∀ (l : list T), concat a l ≠ [] :=
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by intro l; induction l; repeat contradiction
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/- last -/
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definition last : Π l : list T, l ≠ [] → T
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| [] h := absurd rfl h
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| [a] h := a
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| (a₁::a₂::l) h := last (a₂::l) !cons_ne_nil
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lemma last_singleton (a : T) (h : [a] ≠ []) : last [a] h = a :=
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rfl
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lemma last_cons_cons (a₁ a₂ : T) (l : list T) (h : a₁::a₂::l ≠ []) : last (a₁::a₂::l) h = last (a₂::l) !cons_ne_nil :=
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rfl
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theorem last_congr {l₁ l₂ : list T} (h₁ : l₁ ≠ []) (h₂ : l₂ ≠ []) (h₃ : l₁ = l₂) : last l₁ h₁ = last l₂ h₂ :=
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by subst l₁
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theorem last_concat {x : T} : ∀ {l : list T} (h : concat x l ≠ []), last (concat x l) h = x
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| [] h := rfl
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| [a] h := rfl
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| (a₁::a₂::l) h :=
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begin
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change last (a₁::a₂::concat x l) !cons_ne_nil = x,
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rewrite last_cons_cons,
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change last (concat x (a₂::l)) !concat_ne_nil = x,
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apply last_concat
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end
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-- add_rewrite append_nil append_cons
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/- reverse -/
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definition reverse : list T → list T
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| [] := []
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| (a :: l) := concat a (reverse l)
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theorem reverse_nil : reverse (@nil T) = []
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theorem reverse_cons (x : T) (l : list T) : reverse (x::l) = concat x (reverse l)
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theorem reverse_singleton (x : T) : reverse [x] = [x]
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theorem reverse_append : ∀ (s t : list T), reverse (s ++ t) = (reverse t) ++ (reverse s)
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| [] t2 := calc
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reverse ([] ++ t2) = reverse t2 : rfl
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... = (reverse t2) ++ [] : append_nil_right
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... = (reverse t2) ++ (reverse []) : by rewrite reverse_nil
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| (a2 :: s2) t2 := calc
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reverse ((a2 :: s2) ++ t2) = concat a2 (reverse (s2 ++ t2)) : rfl
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... = concat a2 (reverse t2 ++ reverse s2) : reverse_append
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... = (reverse t2 ++ reverse s2) ++ [a2] : concat_eq_append
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... = reverse t2 ++ (reverse s2 ++ [a2]) : append.assoc
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... = reverse t2 ++ concat a2 (reverse s2) : concat_eq_append
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... = reverse t2 ++ reverse (a2 :: s2) : rfl
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theorem reverse_reverse : ∀ (l : list T), reverse (reverse l) = l
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| [] := rfl
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| (a :: l) := calc
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reverse (reverse (a :: l)) = reverse (concat a (reverse l)) : rfl
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... = reverse (reverse l ++ [a]) : concat_eq_append
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... = reverse [a] ++ reverse (reverse l) : reverse_append
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... = reverse [a] ++ l : reverse_reverse
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... = a :: l : rfl
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theorem concat_eq_reverse_cons (x : T) (l : list T) : concat x l = reverse (x :: reverse l) :=
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calc
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concat x l = concat x (reverse (reverse l)) : reverse_reverse
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... = reverse (x :: reverse l) : rfl
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/- head and tail -/
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definition head [h : inhabited T] : list T → T
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| [] := arbitrary T
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| (a :: l) := a
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theorem head_cons [h : inhabited T] (a : T) (l : list T) : head (a::l) = a
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theorem head_append [h : inhabited T] (t : list T) : ∀ {s : list T}, s ≠ [] → head (s ++ t) = head s
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| [] H := absurd rfl H
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| (a :: s) H :=
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show head (a :: (s ++ t)) = head (a :: s),
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by rewrite head_cons
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definition tail : list T → list T
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| [] := []
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| (a :: l) := l
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theorem tail_nil : tail (@nil T) = []
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theorem tail_cons (a : T) (l : list T) : tail (a::l) = l
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theorem cons_head_tail [h : inhabited T] {l : list T} : l ≠ [] → (head l)::(tail l) = l :=
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list.cases_on l
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(assume H : [] ≠ [], absurd rfl H)
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(take x l, assume H : x::l ≠ [], rfl)
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/- list membership -/
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definition mem : T → list T → Prop
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| a [] := false
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| a (b :: l) := a = b ∨ mem a l
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notation e ∈ s := mem e s
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notation e ∉ s := ¬ e ∈ s
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theorem mem_nil_iff (x : T) : x ∈ [] ↔ false :=
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iff.rfl
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theorem not_mem_nil (x : T) : x ∉ [] :=
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iff.mp !mem_nil_iff
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theorem mem_cons (x : T) (l : list T) : x ∈ x :: l :=
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or.inl rfl
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theorem mem_cons_of_mem (y : T) {x : T} {l : list T} : x ∈ l → x ∈ y :: l :=
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assume H, or.inr H
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theorem mem_cons_iff (x y : T) (l : list T) : x ∈ y::l ↔ (x = y ∨ x ∈ l) :=
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iff.rfl
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theorem eq_or_mem_of_mem_cons {x y : T} {l : list T} : x ∈ y::l → x = y ∨ x ∈ l :=
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assume h, h
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theorem mem_singleton {x a : T} : x ∈ [a] → x = a :=
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assume h : x ∈ [a], or.elim (eq_or_mem_of_mem_cons h)
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(λ xeqa : x = a, xeqa)
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(λ xinn : x ∈ [], absurd xinn !not_mem_nil)
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theorem mem_of_mem_cons_of_mem {a b : T} {l : list T} : a ∈ b::l → b ∈ l → a ∈ l :=
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assume ainbl binl, or.elim (eq_or_mem_of_mem_cons ainbl)
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(λ aeqb : a = b, by substvars; exact binl)
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(λ ainl : a ∈ l, ainl)
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theorem mem_or_mem_of_mem_append {x : T} {s t : list T} : x ∈ s ++ t → x ∈ s ∨ x ∈ t :=
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list.induction_on s or.inr
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(take y s,
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assume IH : x ∈ s ++ t → x ∈ s ∨ x ∈ t,
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assume H1 : x ∈ y::s ++ t,
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have H2 : x = y ∨ x ∈ s ++ t, from H1,
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have H3 : x = y ∨ x ∈ s ∨ x ∈ t, from or_of_or_of_imp_right H2 IH,
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iff.elim_right or.assoc H3)
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theorem mem_append_of_mem_or_mem {x : T} {s t : list T} : x ∈ s ∨ x ∈ t → x ∈ s ++ t :=
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list.induction_on s
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(take H, or.elim H false.elim (assume H, H))
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(take y s,
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assume IH : x ∈ s ∨ x ∈ t → x ∈ s ++ t,
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assume H : x ∈ y::s ∨ x ∈ t,
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or.elim H
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(assume H1,
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or.elim (eq_or_mem_of_mem_cons H1)
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(take H2 : x = y, or.inl H2)
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(take H2 : x ∈ s, or.inr (IH (or.inl H2))))
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(assume H1 : x ∈ t, or.inr (IH (or.inr H1))))
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theorem mem_append_iff (x : T) (s t : list T) : x ∈ s ++ t ↔ x ∈ s ∨ x ∈ t :=
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iff.intro mem_or_mem_of_mem_append mem_append_of_mem_or_mem
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theorem not_mem_of_not_mem_append_left {x : T} {s t : list T} : x ∉ s++t → x ∉ s :=
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λ nxinst xins, absurd (mem_append_of_mem_or_mem (or.inl xins)) nxinst
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theorem not_mem_of_not_mem_append_right {x : T} {s t : list T} : x ∉ s++t → x ∉ t :=
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λ nxinst xint, absurd (mem_append_of_mem_or_mem (or.inr xint)) nxinst
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theorem not_mem_append {x : T} {s t : list T} : x ∉ s → x ∉ t → x ∉ s++t :=
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λ nxins nxint xinst, or.elim (mem_or_mem_of_mem_append xinst)
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(λ xins, by contradiction)
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(λ xint, by contradiction)
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local attribute mem [reducible]
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local attribute append [reducible]
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theorem mem_split {x : T} {l : list T} : x ∈ l → ∃s t : list T, l = s ++ (x::t) :=
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list.induction_on l
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(take H : x ∈ [], false.elim (iff.elim_left !mem_nil_iff H))
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(take y l,
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assume IH : x ∈ l → ∃s t : list T, l = s ++ (x::t),
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assume H : x ∈ y::l,
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or.elim (eq_or_mem_of_mem_cons H)
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(assume H1 : x = y,
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exists.intro [] (!exists.intro (H1 ▸ rfl)))
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(assume H1 : x ∈ l,
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obtain s (H2 : ∃t : list T, l = s ++ (x::t)), from IH H1,
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obtain t (H3 : l = s ++ (x::t)), from H2,
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have H4 : y :: l = (y::s) ++ (x::t),
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from H3 ▸ rfl,
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!exists.intro (!exists.intro H4)))
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theorem mem_append_left {a : T} {l₁ : list T} (l₂ : list T) : a ∈ l₁ → a ∈ l₁ ++ l₂ :=
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assume ainl₁, mem_append_of_mem_or_mem (or.inl ainl₁)
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theorem mem_append_right {a : T} (l₁ : list T) {l₂ : list T} : a ∈ l₂ → a ∈ l₁ ++ l₂ :=
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assume ainl₂, mem_append_of_mem_or_mem (or.inr ainl₂)
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definition decidable_mem [instance] [H : decidable_eq T] (x : T) (l : list T) : decidable (x ∈ l) :=
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list.rec_on l
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(decidable.inr (not_of_iff_false !mem_nil_iff))
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(take (h : T) (l : list T) (iH : decidable (x ∈ l)),
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show decidable (x ∈ h::l), from
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decidable.rec_on iH
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(assume Hp : x ∈ l,
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decidable.rec_on (H x h)
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(assume Heq : x = h,
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decidable.inl (or.inl Heq))
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(assume Hne : x ≠ h,
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decidable.inl (or.inr Hp)))
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(assume Hn : ¬x ∈ l,
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decidable.rec_on (H x h)
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(assume Heq : x = h,
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decidable.inl (or.inl Heq))
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(assume Hne : x ≠ h,
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have H1 : ¬(x = h ∨ x ∈ l), from
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assume H2 : x = h ∨ x ∈ l, or.elim H2
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(assume Heq, by contradiction)
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(assume Hp, by contradiction),
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have H2 : ¬x ∈ h::l, from
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iff.elim_right (not_iff_not_of_iff !mem_cons_iff) H1,
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decidable.inr H2)))
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theorem mem_of_ne_of_mem {x y : T} {l : list T} (H₁ : x ≠ y) (H₂ : x ∈ y :: l) : x ∈ l :=
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or.elim (eq_or_mem_of_mem_cons H₂) (λe, absurd e H₁) (λr, r)
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theorem ne_of_not_mem_cons {a b : T} {l : list T} : a ∉ b::l → a ≠ b :=
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assume nin aeqb, absurd (or.inl aeqb) nin
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theorem not_mem_of_not_mem_cons {a b : T} {l : list T} : a ∉ b::l → a ∉ l :=
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assume nin nainl, absurd (or.inr nainl) nin
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lemma not_mem_cons_of_ne_of_not_mem {x y : T} {l : list T} : x ≠ y → x ∉ l → x ∉ y::l :=
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assume P1 P2, not.intro (assume Pxin, absurd (eq_or_mem_of_mem_cons Pxin) (not_or P1 P2))
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lemma ne_and_not_mem_of_not_mem_cons {x y : T} {l : list T} : x ∉ y::l → x ≠ y ∧ x ∉ l :=
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assume P, and.intro (ne_of_not_mem_cons P) (not_mem_of_not_mem_cons P)
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definition sublist (l₁ l₂ : list T) := ∀ ⦃a : T⦄, a ∈ l₁ → a ∈ l₂
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infix `⊆` := sublist
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theorem nil_sub (l : list T) : [] ⊆ l :=
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λ b i, false.elim (iff.mp (mem_nil_iff b) i)
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theorem sub.refl (l : list T) : l ⊆ l :=
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λ b i, i
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theorem sub.trans {l₁ l₂ l₃ : list T} (H₁ : l₁ ⊆ l₂) (H₂ : l₂ ⊆ l₃) : l₁ ⊆ l₃ :=
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λ b i, H₂ (H₁ i)
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theorem sub_cons (a : T) (l : list T) : l ⊆ a::l :=
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λ b i, or.inr i
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theorem sub_of_cons_sub {a : T} {l₁ l₂ : list T} : a::l₁ ⊆ l₂ → l₁ ⊆ l₂ :=
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λ s b i, s b (mem_cons_of_mem _ i)
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theorem cons_sub_cons {l₁ l₂ : list T} (a : T) (s : l₁ ⊆ l₂) : (a::l₁) ⊆ (a::l₂) :=
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λ b Hin, or.elim (eq_or_mem_of_mem_cons Hin)
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(λ e : b = a, or.inl e)
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(λ i : b ∈ l₁, or.inr (s i))
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theorem sub_append_left (l₁ l₂ : list T) : l₁ ⊆ l₁++l₂ :=
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λ b i, iff.mp' (mem_append_iff b l₁ l₂) (or.inl i)
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theorem sub_append_right (l₁ l₂ : list T) : l₂ ⊆ l₁++l₂ :=
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λ b i, iff.mp' (mem_append_iff b l₁ l₂) (or.inr i)
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theorem sub_cons_of_sub (a : T) {l₁ l₂ : list T} : l₁ ⊆ l₂ → l₁ ⊆ (a::l₂) :=
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λ (s : l₁ ⊆ l₂) (x : T) (i : x ∈ l₁), or.inr (s i)
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theorem sub_app_of_sub_left (l l₁ l₂ : list T) : l ⊆ l₁ → l ⊆ l₁++l₂ :=
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λ (s : l ⊆ l₁) (x : T) (xinl : x ∈ l),
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have xinl₁ : x ∈ l₁, from s xinl,
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mem_append_of_mem_or_mem (or.inl xinl₁)
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theorem sub_app_of_sub_right (l l₁ l₂ : list T) : l ⊆ l₂ → l ⊆ l₁++l₂ :=
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λ (s : l ⊆ l₂) (x : T) (xinl : x ∈ l),
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have xinl₁ : x ∈ l₂, from s xinl,
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mem_append_of_mem_or_mem (or.inr xinl₁)
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theorem cons_sub_of_sub_of_mem {a : T} {l m : list T} : a ∈ m → l ⊆ m → a::l ⊆ m :=
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λ (ainm : a ∈ m) (lsubm : l ⊆ m) (x : T) (xinal : x ∈ a::l), or.elim (eq_or_mem_of_mem_cons xinal)
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(assume xeqa : x = a, by substvars; exact ainm)
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(assume xinl : x ∈ l, lsubm xinl)
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theorem app_sub_of_sub_of_sub {l₁ l₂ l : list T} : l₁ ⊆ l → l₂ ⊆ l → l₁++l₂ ⊆ l :=
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λ (l₁subl : l₁ ⊆ l) (l₂subl : l₂ ⊆ l) (x : T) (xinl₁l₂ : x ∈ l₁++l₂),
|
||
or.elim (mem_or_mem_of_mem_append xinl₁l₂)
|
||
(λ xinl₁ : x ∈ l₁, l₁subl xinl₁)
|
||
(λ xinl₂ : x ∈ l₂, l₂subl xinl₂)
|
||
|
||
/- find -/
|
||
section
|
||
variable [H : decidable_eq T]
|
||
include H
|
||
|
||
definition find : T → list T → nat
|
||
| a [] := 0
|
||
| a (b :: l) := if a = b then 0 else succ (find a l)
|
||
|
||
theorem find_nil (x : T) : find x [] = 0
|
||
|
||
theorem find_cons (x y : T) (l : list T) : find x (y::l) = if x = y then 0 else succ (find x l)
|
||
|
||
theorem find_cons_of_eq {x y : T} (l : list T) : x = y → find x (y::l) = 0 :=
|
||
assume e, if_pos e
|
||
|
||
theorem find_cons_of_ne {x y : T} (l : list T) : x ≠ y → find x (y::l) = succ (find x l) :=
|
||
assume n, if_neg n
|
||
|
||
theorem find_of_not_mem {l : list T} {x : T} : ¬x ∈ l → find x l = length l :=
|
||
list.rec_on l
|
||
(assume P₁ : ¬x ∈ [], _)
|
||
(take y l,
|
||
assume iH : ¬x ∈ l → find x l = length l,
|
||
assume P₁ : ¬x ∈ y::l,
|
||
have P₂ : ¬(x = y ∨ x ∈ l), from iff.elim_right (not_iff_not_of_iff !mem_cons_iff) P₁,
|
||
have P₃ : ¬x = y ∧ ¬x ∈ l, from (iff.elim_left not_or_iff_not_and_not P₂),
|
||
calc
|
||
find x (y::l) = if x = y then 0 else succ (find x l) : !find_cons
|
||
... = succ (find x l) : if_neg (and.elim_left P₃)
|
||
... = succ (length l) : {iH (and.elim_right P₃)}
|
||
... = length (y::l) : !length_cons⁻¹)
|
||
|
||
lemma find_le_length : ∀ {a} {l : list T}, find a l ≤ length l
|
||
| a [] := !le.refl
|
||
| a (b::l) := decidable.rec_on (H a b)
|
||
(assume Peq, by rewrite [find_cons_of_eq l Peq]; exact !zero_le)
|
||
(assume Pne,
|
||
begin
|
||
rewrite [find_cons_of_ne l Pne, length_cons],
|
||
apply succ_le_succ, apply find_le_length
|
||
end)
|
||
|
||
lemma not_mem_of_find_eq_length : ∀ {a} {l : list T}, find a l = length l → a ∉ l
|
||
| a [] := assume Peq, !not_mem_nil
|
||
| a (b::l) := decidable.rec_on (H a b)
|
||
(assume Peq, by rewrite [find_cons_of_eq l Peq, length_cons]; contradiction)
|
||
(assume Pne,
|
||
begin
|
||
rewrite [find_cons_of_ne l Pne, length_cons, mem_cons_iff],
|
||
intro Plen, apply (not_or Pne),
|
||
exact not_mem_of_find_eq_length (succ_inj Plen)
|
||
end)
|
||
|
||
lemma find_lt_length {a} {l : list T} (Pin : a ∈ l) : find a l < length l :=
|
||
begin
|
||
apply nat.lt_of_le_and_ne,
|
||
apply find_le_length,
|
||
apply not.intro, intro Peq,
|
||
exact absurd Pin (not_mem_of_find_eq_length Peq)
|
||
end
|
||
|
||
end
|
||
|
||
/- nth element -/
|
||
section nth
|
||
definition nth : list T → nat → option T
|
||
| [] n := none
|
||
| (a :: l) 0 := some a
|
||
| (a :: l) (n+1) := nth l n
|
||
|
||
theorem nth_zero (a : T) (l : list T) : nth (a :: l) 0 = some a
|
||
|
||
theorem nth_succ (a : T) (l : list T) (n : nat) : nth (a::l) (succ n) = nth l n
|
||
|
||
theorem nth_eq_some : ∀ {l : list T} {n : nat}, n < length l → Σ a : T, nth l n = some a
|
||
| [] n h := absurd h !not_lt_zero
|
||
| (a::l) 0 h := ⟨a, rfl⟩
|
||
| (a::l) (succ n) h :=
|
||
have aux : n < length l, from lt_of_succ_lt_succ h,
|
||
obtain (r : T) (req : nth l n = some r), from nth_eq_some aux,
|
||
⟨r, by rewrite [nth_succ, req]⟩
|
||
|
||
open decidable
|
||
theorem find_nth [h : decidable_eq T] {a : T} : ∀ {l}, a ∈ l → nth l (find a l) = some a
|
||
| [] ain := absurd ain !not_mem_nil
|
||
| (b::l) ainbl := by_cases
|
||
(λ aeqb : a = b, by rewrite [find_cons_of_eq _ aeqb, nth_zero, aeqb])
|
||
(λ aneb : a ≠ b, or.elim (eq_or_mem_of_mem_cons ainbl)
|
||
(λ aeqb : a = b, absurd aeqb aneb)
|
||
(λ ainl : a ∈ l, by rewrite [find_cons_of_ne _ aneb, nth_succ, find_nth ainl]))
|
||
|
||
definition inth [h : inhabited T] (l : list T) (n : nat) : T :=
|
||
match nth l n with
|
||
| some a := a
|
||
| none := arbitrary T
|
||
end
|
||
|
||
theorem inth_zero [h : inhabited T] (a : T) (l : list T) : inth (a :: l) 0 = a
|
||
|
||
theorem inth_succ [h : inhabited T] (a : T) (l : list T) (n : nat) : inth (a::l) (n+1) = inth l n
|
||
end nth
|
||
|
||
open decidable
|
||
definition has_decidable_eq {A : Type} [H : decidable_eq A] : ∀ l₁ l₂ : list A, decidable (l₁ = l₂)
|
||
| [] [] := inl rfl
|
||
| [] (b::l₂) := inr (by contradiction)
|
||
| (a::l₁) [] := inr (by contradiction)
|
||
| (a::l₁) (b::l₂) :=
|
||
match H a b with
|
||
| inl Hab :=
|
||
match has_decidable_eq l₁ l₂ with
|
||
| inl He := inl (by congruence; repeat assumption)
|
||
| inr Hn := inr (by intro H; injection H; contradiction)
|
||
end
|
||
| inr Hnab := inr (by intro H; injection H; contradiction)
|
||
end
|
||
|
||
/- quasiequal a l l' means that l' is exactly l, with a added
|
||
once somewhere -/
|
||
section qeq
|
||
variable {A : Type}
|
||
inductive qeq (a : A) : list A → list A → Prop :=
|
||
| qhead : ∀ l, qeq a l (a::l)
|
||
| qcons : ∀ (b : A) {l l' : list A}, qeq a l l' → qeq a (b::l) (b::l')
|
||
|
||
open qeq
|
||
|
||
notation l' `≈`:50 a `|` l:50 := qeq a l l'
|
||
|
||
theorem qeq_app : ∀ (l₁ : list A) (a : A) (l₂ : list A), l₁++(a::l₂) ≈ a|l₁++l₂
|
||
| [] a l₂ := qhead a l₂
|
||
| (x::xs) a l₂ := qcons x (qeq_app xs a l₂)
|
||
|
||
theorem mem_head_of_qeq {a : A} {l₁ l₂ : list A} : l₁≈a|l₂ → a ∈ l₁ :=
|
||
take q, qeq.induction_on q
|
||
(λ l, !mem_cons)
|
||
(λ b l l' q r, or.inr r)
|
||
|
||
theorem mem_tail_of_qeq {a : A} {l₁ l₂ : list A} : l₁≈a|l₂ → ∀ x, x ∈ l₂ → x ∈ l₁ :=
|
||
take q, qeq.induction_on q
|
||
(λ l x i, or.inr i)
|
||
(λ b l l' q r x xinbl, or.elim (eq_or_mem_of_mem_cons xinbl)
|
||
(λ xeqb : x = b, xeqb ▸ mem_cons x l')
|
||
(λ xinl : x ∈ l, or.inr (r x xinl)))
|
||
|
||
theorem mem_cons_of_qeq {a : A} {l₁ l₂ : list A} : l₁≈a|l₂ → ∀ x, x ∈ l₁ → x ∈ a::l₂ :=
|
||
take q, qeq.induction_on q
|
||
(λ l x i, i)
|
||
(λ b l l' q r x xinbl', or.elim (eq_or_mem_of_mem_cons xinbl')
|
||
(λ xeqb : x = b, xeqb ▸ or.inr (mem_cons x l))
|
||
(λ xinl' : x ∈ l', or.elim (eq_or_mem_of_mem_cons (r x xinl'))
|
||
(λ xeqa : x = a, xeqa ▸ mem_cons x (b::l))
|
||
(λ xinl : x ∈ l, or.inr (or.inr xinl))))
|
||
|
||
theorem length_eq_of_qeq {a : A} {l₁ l₂ : list A} : l₁≈a|l₂ → length l₁ = succ (length l₂) :=
|
||
take q, qeq.induction_on q
|
||
(λ l, rfl)
|
||
(λ b l l' q r, by rewrite [*length_cons, r])
|
||
|
||
theorem qeq_of_mem {a : A} {l : list A} : a ∈ l → (∃l', l≈a|l') :=
|
||
list.induction_on l
|
||
(λ h : a ∈ nil, absurd h (not_mem_nil a))
|
||
(λ x xs r ainxxs, or.elim (eq_or_mem_of_mem_cons ainxxs)
|
||
(λ aeqx : a = x,
|
||
assert aux : ∃ l, x::xs≈x|l, from
|
||
exists.intro xs (qhead x xs),
|
||
by rewrite aeqx; exact aux)
|
||
(λ ainxs : a ∈ xs,
|
||
have ex : ∃l', xs ≈ a|l', from r ainxs,
|
||
obtain (l' : list A) (q : xs ≈ a|l'), from ex,
|
||
have q₂ : x::xs ≈ a | x::l', from qcons x q,
|
||
exists.intro (x::l') q₂))
|
||
|
||
theorem qeq_split {a : A} {l l' : list A} : l'≈a|l → ∃l₁ l₂, l = l₁++l₂ ∧ l' = l₁++(a::l₂) :=
|
||
take q, qeq.induction_on q
|
||
(λ t,
|
||
have aux : t = []++t ∧ a::t = []++(a::t), from and.intro rfl rfl,
|
||
exists.intro [] (exists.intro t aux))
|
||
(λ b t t' q r,
|
||
obtain (l₁ l₂ : list A) (h : t = l₁++l₂ ∧ t' = l₁++(a::l₂)), from r,
|
||
have aux : b::t = (b::l₁)++l₂ ∧ b::t' = (b::l₁)++(a::l₂),
|
||
begin
|
||
rewrite [and.elim_right h, and.elim_left h],
|
||
constructor, repeat reflexivity
|
||
end,
|
||
exists.intro (b::l₁) (exists.intro l₂ aux))
|
||
|
||
theorem sub_of_mem_of_sub_of_qeq {a : A} {l : list A} {u v : list A} : a ∉ l → a::l ⊆ v → v≈a|u → l ⊆ u :=
|
||
λ (nainl : a ∉ l) (s : a::l ⊆ v) (q : v≈a|u) (x : A) (xinl : x ∈ l),
|
||
have xinv : x ∈ v, from s (or.inr xinl),
|
||
have xinau : x ∈ a::u, from mem_cons_of_qeq q x xinv,
|
||
or.elim (eq_or_mem_of_mem_cons xinau)
|
||
(λ xeqa : x = a, by substvars; contradiction)
|
||
(λ xinu : x ∈ u, xinu)
|
||
end qeq
|
||
end list
|
||
|
||
attribute list.has_decidable_eq [instance]
|
||
attribute list.decidable_mem [instance]
|