1132 lines
46 KiB
Text
1132 lines
46 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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Module: data.list.basic
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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.basic algebra.function
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open eq.ops helper_tactics nat prod function
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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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/- 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 : add.succ_left
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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 := nat.no_confusion H
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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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-- 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 (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
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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 rewrite [aeqb]; 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, absurd xins nxins)
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(λ xint, absurd xint nxint)
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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 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))
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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, absurd Heq Hne)
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(assume Hp, absurd Hp Hn),
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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 not_eq_of_not_mem {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 {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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definition sublist (l₁ l₂ : list T) := ∀ ⦃a : T⦄, a ∈ l₁ → a ∈ l₂
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infix `⊆`:50 := sublist
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lemma nil_sub (l : list T) : [] ⊆ l :=
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λ b i, false.elim (iff.mp (mem_nil b) i)
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lemma sub.refl (l : list T) : l ⊆ l :=
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λ b i, i
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lemma 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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lemma sub_cons (a : T) (l : list T) : l ⊆ a::l :=
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λ b i, or.inr i
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lemma 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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lemma 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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lemma 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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lemma 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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lemma 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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lemma 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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lemma 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, eq.rec_on (eq.symm xeqa) ainm)
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(assume xinl : x ∈ l, lsubm xinl)
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lemma 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),
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or.elim (mem_or_mem_of_mem_append xinl₁l₂)
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(λ xinl₁ : x ∈ l₁, l₁subl xinl₁)
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(λ xinl₂ : x ∈ l₂, l₂subl xinl₂)
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/- find -/
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section
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variable [H : decidable_eq T]
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include H
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definition find : T → list T → nat
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| a [] := 0
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| a (b :: l) := if a = b then 0 else succ (find a l)
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theorem find_nil (x : T) : find x [] = 0
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theorem find_cons (x y : T) (l : list T) : find x (y::l) = if x = y then 0 else succ (find x l)
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theorem find.not_mem {l : list T} {x : T} : ¬x ∈ l → find x l = length l :=
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list.rec_on l
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(assume P₁ : ¬x ∈ [], _)
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(take y l,
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assume iH : ¬x ∈ l → find x l = length l,
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assume P₁ : ¬x ∈ y::l,
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have P₂ : ¬(x = y ∨ x ∈ l), from iff.elim_right (not_iff_not_of_iff !mem_cons_iff) P₁,
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have P₃ : ¬x = y ∧ ¬x ∈ l, from (iff.elim_left not_or_iff_not_and_not P₂),
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calc
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find x (y::l) = if x = y then 0 else succ (find x l) : !find_cons
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... = succ (find x l) : if_neg (and.elim_left P₃)
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... = succ (length l) : {iH (and.elim_right P₃)}
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... = length (y::l) : !length_cons⁻¹)
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end
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/- nth element -/
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definition nth [h : inhabited T] : list T → nat → T
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| [] n := arbitrary T
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| (a :: l) 0 := a
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| (a :: l) (n+1) := nth l n
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theorem nth_zero [h : inhabited T] (a : T) (l : list T) : nth (a :: l) 0 = a
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theorem nth_succ [h : inhabited T] (a : T) (l : list T) (n : nat) : nth (a::l) (n+1) = nth l n
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open decidable
|
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definition has_decidable_eq {A : Type} [H : decidable_eq A] : ∀ l₁ l₂ : list A, decidable (l₁ = l₂)
|
||
| [] [] := inl rfl
|
||
| [] (b::l₂) := inr (λ H, list.no_confusion H)
|
||
| (a::l₁) [] := inr (λ H, list.no_confusion H)
|
||
| (a::l₁) (b::l₂) :=
|
||
match H a b with
|
||
| inl Hab :=
|
||
match has_decidable_eq l₁ l₂ with
|
||
| inl He := inl (eq.rec_on Hab (eq.rec_on He rfl))
|
||
| inr Hn := inr (λ H, list.no_confusion H (λ Hab Ht, absurd Ht Hn))
|
||
end
|
||
| inr Hnab := inr (λ H, list.no_confusion H (λ Hab Ht, absurd Hab Hnab))
|
||
end
|
||
|
||
section combinators
|
||
variables {A B C : Type}
|
||
|
||
definition map (f : A → B) : list A → list B
|
||
| [] := []
|
||
| (a :: l) := f a :: map l
|
||
|
||
theorem map_nil (f : A → B) : map f [] = []
|
||
|
||
theorem map_cons (f : A → B) (a : A) (l : list A) : map f (a :: l) = f a :: map f l
|
||
|
||
theorem map_id : ∀ l : list A, map id l = l
|
||
| [] := rfl
|
||
| (x::xs) := begin rewrite [map_cons, map_id] end
|
||
|
||
theorem map_map (g : B → C) (f : A → B) : ∀ l, map g (map f l) = map (g ∘ f) l
|
||
| [] := rfl
|
||
| (a :: l) :=
|
||
show (g ∘ f) a :: map g (map f l) = map (g ∘ f) (a :: l),
|
||
by rewrite (map_map l)
|
||
|
||
theorem len_map (f : A → B) : ∀ l : list A, length (map f l) = length l
|
||
| [] := rfl
|
||
| (a :: l) :=
|
||
show length (map f l) + 1 = length l + 1,
|
||
by rewrite (len_map l)
|
||
|
||
theorem mem_map {A B : Type} (f : A → B) : ∀ {a l}, a ∈ l → f a ∈ map f l
|
||
| a [] i := absurd i !not_mem_nil
|
||
| a (x::xs) i := or.elim (eq_or_mem_of_mem_cons i)
|
||
(λ aeqx : a = x, by rewrite [aeqx, map_cons]; apply mem_cons)
|
||
(λ ainxs : a ∈ xs, or.inr (mem_map ainxs))
|
||
|
||
definition map₂ (f : A → B → C) : list A → list B → list C
|
||
| [] _ := []
|
||
| _ [] := []
|
||
| (x::xs) (y::ys) := f x y :: map₂ xs ys
|
||
|
||
definition foldl (f : A → B → A) : A → list B → A
|
||
| a [] := a
|
||
| a (b :: l) := foldl (f a b) l
|
||
|
||
theorem foldl_nil (f : A → B → A) (a : A) : foldl f a [] = a
|
||
|
||
theorem foldl_cons (f : A → B → A) (a : A) (b : B) (l : list B) : foldl f a (b::l) = foldl f (f a b) l
|
||
|
||
definition foldr (f : A → B → B) : B → list A → B
|
||
| b [] := b
|
||
| b (a :: l) := f a (foldr b l)
|
||
|
||
theorem foldr_nil (f : A → B → B) (b : B) : foldr f b [] = b
|
||
|
||
theorem foldr_cons (f : A → B → B) (b : B) (a : A) (l : list A) : foldr f b (a::l) = f a (foldr f b l)
|
||
|
||
section foldl_eq_foldr
|
||
-- foldl and foldr coincide when f is commutative and associative
|
||
parameters {α : Type} {f : α → α → α}
|
||
hypothesis (Hcomm : ∀ a b, f a b = f b a)
|
||
hypothesis (Hassoc : ∀ a b c, f (f a b) c = f a (f b c))
|
||
include Hcomm Hassoc
|
||
|
||
theorem foldl_eq_of_comm_of_assoc : ∀ a b l, foldl f a (b::l) = f b (foldl f a l)
|
||
| a b nil := Hcomm a b
|
||
| a b (c::l) :=
|
||
begin
|
||
change (foldl f (f (f a b) c) l = f b (foldl f (f a c) l)),
|
||
rewrite -foldl_eq_of_comm_of_assoc,
|
||
change (foldl f (f (f a b) c) l = foldl f (f (f a c) b) l),
|
||
have H₁ : f (f a b) c = f (f a c) b, by rewrite [Hassoc, Hassoc, Hcomm b c],
|
||
rewrite H₁
|
||
end
|
||
|
||
theorem foldl_eq_foldr : ∀ a l, foldl f a l = foldr f a l
|
||
| a nil := rfl
|
||
| a (b :: l) :=
|
||
begin
|
||
rewrite foldl_eq_of_comm_of_assoc,
|
||
esimp,
|
||
change (f b (foldl f a l) = f b (foldr f a l)),
|
||
rewrite foldl_eq_foldr
|
||
end
|
||
end foldl_eq_foldr
|
||
|
||
theorem foldl_append (f : B → A → B) : ∀ (b : B) (l₁ l₂ : list A), foldl f b (l₁++l₂) = foldl f (foldl f b l₁) l₂
|
||
| b [] l₂ := rfl
|
||
| b (a::l₁) l₂ := by rewrite [append_cons, *foldl_cons, foldl_append]
|
||
|
||
theorem foldr_append (f : A → B → B) : ∀ (b : B) (l₁ l₂ : list A), foldr f b (l₁++l₂) = foldr f (foldr f b l₂) l₁
|
||
| b [] l₂ := rfl
|
||
| b (a::l₁) l₂ := by rewrite [append_cons, *foldr_cons, foldr_append]
|
||
|
||
definition all (p : A → Prop) (l : list A) : Prop :=
|
||
foldr (λ a r, p a ∧ r) true l
|
||
|
||
definition any (p : A → Prop) (l : list A) : Prop :=
|
||
foldr (λ a r, p a ∨ r) false l
|
||
|
||
definition decidable_all (p : A → Prop) [H : decidable_pred p] : ∀ l, decidable (all p l)
|
||
| [] := decidable_true
|
||
| (a :: l) :=
|
||
match H a with
|
||
| inl Hp₁ :=
|
||
match decidable_all l with
|
||
| inl Hp₂ := inl (and.intro Hp₁ Hp₂)
|
||
| inr Hn₂ := inr (not_and_of_not_right (p a) Hn₂)
|
||
end
|
||
| inr Hn := inr (not_and_of_not_left (all p l) Hn)
|
||
end
|
||
|
||
definition decidable_any (p : A → Prop) [H : decidable_pred p] : ∀ l, decidable (any p l)
|
||
| [] := decidable_false
|
||
| (a :: l) :=
|
||
match H a with
|
||
| inl Hp := inl (or.inl Hp)
|
||
| inr Hn₁ :=
|
||
match decidable_any l with
|
||
| inl Hp₂ := inl (or.inr Hp₂)
|
||
| inr Hn₂ := inr (not_or Hn₁ Hn₂)
|
||
end
|
||
end
|
||
|
||
definition zip (l₁ : list A) (l₂ : list B) : list (A × B) :=
|
||
map₂ (λ a b, (a, b)) l₁ l₂
|
||
|
||
definition unzip : list (A × B) → list A × list B
|
||
| [] := ([], [])
|
||
| ((a, b) :: l) :=
|
||
match unzip l with
|
||
| (la, lb) := (a :: la, b :: lb)
|
||
end
|
||
|
||
theorem unzip_nil : unzip (@nil (A × B)) = ([], [])
|
||
|
||
theorem unzip_cons (a : A) (b : B) (l : list (A × B)) :
|
||
unzip ((a, b) :: l) = match unzip l with (la, lb) := (a :: la, b :: lb) end :=
|
||
rfl
|
||
|
||
theorem zip_unzip : ∀ (l : list (A × B)), zip (pr₁ (unzip l)) (pr₂ (unzip l)) = l
|
||
| [] := rfl
|
||
| ((a, b) :: l) :=
|
||
begin
|
||
rewrite unzip_cons,
|
||
have r : zip (pr₁ (unzip l)) (pr₂ (unzip l)) = l, from zip_unzip l,
|
||
revert r,
|
||
apply (prod.cases_on (unzip l)),
|
||
intros [la, lb, r],
|
||
rewrite -r
|
||
end
|
||
|
||
end combinators
|
||
|
||
/- flat -/
|
||
section
|
||
variable {A : Type}
|
||
|
||
definition flat (l : list (list A)) : list A :=
|
||
foldl append nil l
|
||
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'
|
||
|
||
lemma 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)
|
||
|
||
lemma 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)
|
||
|
||
lemma 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)))
|
||
|
||
lemma 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))))
|
||
|
||
lemma 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])
|
||
|
||
lemma 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))
|
||
|
||
lemma 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],
|
||
exact (and.intro rfl rfl)
|
||
end,
|
||
exists.intro (b::l₁) (exists.intro l₂ aux))
|
||
|
||
lemma 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, absurd (xeqa ▸ xinl) nainl)
|
||
(λ xinu : x ∈ u, xinu)
|
||
end qeq
|
||
|
||
section erase
|
||
variable {A : Type}
|
||
variable [H : decidable_eq A]
|
||
include H
|
||
|
||
definition erase (a : A) : list A → list A
|
||
| [] := []
|
||
| (b::l) :=
|
||
match H a b with
|
||
| inl e := l
|
||
| inr n := b :: erase l
|
||
end
|
||
|
||
lemma erase_nil (a : A) : erase a [] = [] :=
|
||
rfl
|
||
|
||
lemma erase_cons_head (a : A) (l : list A) : erase a (a :: l) = l :=
|
||
show match H a a with | inl e := l | inr n := a :: erase a l end = l,
|
||
by rewrite decidable_eq_inl_refl
|
||
|
||
lemma erase_cons_tail {a b : A} (l : list A) : a ≠ b → erase a (b::l) = b :: erase a l :=
|
||
assume h : a ≠ b,
|
||
show match H a b with | inl e := l | inr n₁ := b :: erase a l end = b :: erase a l,
|
||
by rewrite (decidable_eq_inr_neg h)
|
||
|
||
lemma length_erase_of_mem {a : A} : ∀ {l}, a ∈ l → length (erase a l) = pred (length l)
|
||
| [] h := rfl
|
||
| [x] h := by rewrite [mem_singleton h, erase_cons_head]
|
||
| (x::y::xs) h :=
|
||
by_cases
|
||
(λ aeqx : a = x, by rewrite [aeqx, erase_cons_head])
|
||
(λ anex : a ≠ x,
|
||
assert ainyxs : a ∈ y::xs, from or_resolve_right h anex,
|
||
by rewrite [erase_cons_tail _ anex, *length_cons, length_erase_of_mem ainyxs])
|
||
|
||
lemma length_erase_of_not_mem {a : A} : ∀ {l}, a ∉ l → length (erase a l) = length l
|
||
| [] h := rfl
|
||
| (x::xs) h :=
|
||
assert anex : a ≠ x, from λ aeqx : a = x, absurd (or.inl aeqx) h,
|
||
assert aninxs : a ∉ xs, from λ ainxs : a ∈ xs, absurd (or.inr ainxs) h,
|
||
by rewrite [erase_cons_tail _ anex, length_cons, length_erase_of_not_mem aninxs]
|
||
|
||
lemma erase_append_left {a : A} : ∀ {l₁} (l₂), a ∈ l₁ → erase a (l₁++l₂) = erase a l₁ ++ l₂
|
||
| [] l₂ h := absurd h !not_mem_nil
|
||
| (x::xs) l₂ h :=
|
||
by_cases
|
||
(λ aeqx : a = x, by rewrite [aeqx, append_cons, *erase_cons_head])
|
||
(λ anex : a ≠ x,
|
||
assert ainxs : a ∈ xs, from mem_of_ne_of_mem anex h,
|
||
by rewrite [append_cons, *erase_cons_tail _ anex, erase_append_left l₂ ainxs])
|
||
|
||
lemma erase_append_right {a : A} : ∀ {l₁} (l₂), a ∉ l₁ → erase a (l₁++l₂) = l₁ ++ erase a l₂
|
||
| [] l₂ h := _
|
||
| (x::xs) l₂ h :=
|
||
by_cases
|
||
(λ aeqx : a = x, by rewrite aeqx at h; exact (absurd !mem_cons h))
|
||
(λ anex : a ≠ x,
|
||
assert nainxs : a ∉ xs, from not_mem_of_not_mem h,
|
||
by rewrite [append_cons, *erase_cons_tail _ anex, erase_append_right l₂ nainxs])
|
||
|
||
lemma erase_sub (a : A) : ∀ l, erase a l ⊆ l
|
||
| [] := λ x xine, xine
|
||
| (x::xs) := λ y xine,
|
||
by_cases
|
||
(λ aeqx : a = x, by rewrite [aeqx at xine, erase_cons_head at xine]; exact (or.inr xine))
|
||
(λ anex : a ≠ x,
|
||
assert yinxe : y ∈ x :: erase a xs, by rewrite [erase_cons_tail _ anex at xine]; exact xine,
|
||
assert subxs : erase a xs ⊆ xs, from erase_sub xs,
|
||
by_cases
|
||
(λ yeqx : y = x, by rewrite yeqx; apply mem_cons)
|
||
(λ ynex : y ≠ x,
|
||
assert yine : y ∈ erase a xs, from mem_of_ne_of_mem ynex yinxe,
|
||
assert yinxs : y ∈ xs, from subxs yine,
|
||
or.inr yinxs))
|
||
|
||
theorem mem_erase_of_ne_of_mem {a b : A} : ∀ {l : list A}, a ≠ b → a ∈ l → a ∈ erase b l
|
||
| [] n i := absurd i !not_mem_nil
|
||
| (c::l) n i := by_cases
|
||
(λ beqc : b = c,
|
||
assert ainl : a ∈ l, from or.elim (eq_or_mem_of_mem_cons i)
|
||
(λ aeqc : a = c, absurd aeqc (beqc ▸ n))
|
||
(λ ainl : a ∈ l, ainl),
|
||
by rewrite [beqc, erase_cons_head]; exact ainl)
|
||
(λ bnec : b ≠ c, by_cases
|
||
(λ aeqc : a = c,
|
||
assert aux : a ∈ c :: erase b l, by rewrite [aeqc]; exact !mem_cons,
|
||
by rewrite [erase_cons_tail _ bnec]; exact aux)
|
||
(λ anec : a ≠ c,
|
||
have ainl : a ∈ l, from mem_of_ne_of_mem anec i,
|
||
have ainel : a ∈ erase b l, from mem_erase_of_ne_of_mem n ainl,
|
||
assert aux : a ∈ c :: erase b l, from mem_cons_of_mem _ ainel,
|
||
by rewrite [erase_cons_tail _ bnec]; exact aux)) --
|
||
|
||
theorem mem_of_mem_erase {a b : A} : ∀ {l}, a ∈ erase b l → a ∈ l
|
||
| [] i := absurd i !not_mem_nil
|
||
| (c::l) i := by_cases
|
||
(λ beqc : b = c, by rewrite [beqc at i, erase_cons_head at i]; exact (mem_cons_of_mem _ i))
|
||
(λ bnec : b ≠ c,
|
||
have i₁ : a ∈ c :: erase b l, by rewrite [erase_cons_tail _ bnec at i]; exact i,
|
||
or.elim (eq_or_mem_of_mem_cons i₁)
|
||
(λ aeqc : a = c, by rewrite [aeqc]; exact !mem_cons)
|
||
(λ ainel : a ∈ erase b l,
|
||
have ainl : a ∈ l, from mem_of_mem_erase ainel,
|
||
mem_cons_of_mem _ ainl))
|
||
end erase
|
||
|
||
/- disjoint -/
|
||
section disjoint
|
||
variable {A : Type}
|
||
|
||
definition disjoint (l₁ l₂ : list A) : Prop := ∀ a, (a ∈ l₁ → a ∉ l₂) ∧ (a ∈ l₂ → a ∉ l₁)
|
||
|
||
lemma disjoint_left {l₁ l₂ : list A} : disjoint l₁ l₂ → ∀ {a}, a ∈ l₁ → a ∉ l₂ :=
|
||
λ d a, and.elim_left (d a)
|
||
|
||
lemma disjoint_right {l₁ l₂ : list A} : disjoint l₁ l₂ → ∀ {a}, a ∈ l₂ → a ∉ l₁ :=
|
||
λ d a, and.elim_right (d a)
|
||
|
||
lemma disjoint.comm {l₁ l₂ : list A} : disjoint l₁ l₂ → disjoint l₂ l₁ :=
|
||
λ d a, and.intro
|
||
(λ ainl₂ : a ∈ l₂, disjoint_right d ainl₂)
|
||
(λ ainl₁ : a ∈ l₁, disjoint_left d ainl₁)
|
||
|
||
lemma disjoint_of_disjoint_cons_left {a : A} {l₁ l₂} : disjoint (a::l₁) l₂ → disjoint l₁ l₂ :=
|
||
λ d x, and.intro
|
||
(λ xinl₁ : x ∈ l₁, disjoint_left d (or.inr xinl₁))
|
||
(λ xinl₂ : x ∈ l₂,
|
||
have nxinal₁ : x ∉ a::l₁, from disjoint_right d xinl₂,
|
||
not_mem_of_not_mem nxinal₁)
|
||
|
||
lemma disjoint_of_disjoint_cons_right {a : A} {l₁ l₂} : disjoint l₁ (a::l₂) → disjoint l₁ l₂ :=
|
||
λ d, disjoint.comm (disjoint_of_disjoint_cons_left (disjoint.comm d))
|
||
|
||
lemma disjoint_nil_left (l : list A) : disjoint [] l :=
|
||
λ a, and.intro
|
||
(λ ab : a ∈ nil, absurd ab !not_mem_nil)
|
||
(λ ainl : a ∈ l, !not_mem_nil)
|
||
|
||
lemma disjoint_nil_right (l : list A) : disjoint l [] :=
|
||
disjoint.comm (disjoint_nil_left l)
|
||
|
||
lemma disjoint_cons_of_not_mem_of_disjoint {a : A} {l₁ l₂} : a ∉ l₂ → disjoint l₁ l₂ → disjoint (a::l₁) l₂ :=
|
||
λ nainl₂ d x, and.intro
|
||
(λ xinal₁ : x ∈ a::l₁, or.elim (eq_or_mem_of_mem_cons xinal₁)
|
||
(λ xeqa : x = a, xeqa⁻¹ ▸ nainl₂)
|
||
(λ xinl₁ : x ∈ l₁, disjoint_left d xinl₁))
|
||
(λ (xinl₂ : x ∈ l₂) (xinal₁ : x ∈ a::l₁), or.elim (eq_or_mem_of_mem_cons xinal₁)
|
||
(λ xeqa : x = a, absurd (xeqa ▸ xinl₂) nainl₂)
|
||
(λ xinl₁ : x ∈ l₁, absurd xinl₁ (disjoint_right d xinl₂)))
|
||
|
||
lemma disjoint_of_disjoint_append_left_left : ∀ {l₁ l₂ l : list A}, disjoint (l₁++l₂) l → disjoint l₁ l
|
||
| [] l₂ l d := disjoint_nil_left l
|
||
| (x::xs) l₂ l d :=
|
||
have nxinl : x ∉ l, from disjoint_left d !mem_cons,
|
||
have d₁ : disjoint (xs++l₂) l, from disjoint_of_disjoint_cons_left d,
|
||
have d₂ : disjoint xs l, from disjoint_of_disjoint_append_left_left d₁,
|
||
disjoint_cons_of_not_mem_of_disjoint nxinl d₂
|
||
|
||
lemma disjoint_of_disjoint_append_left_right : ∀ {l₁ l₂ l : list A}, disjoint (l₁++l₂) l → disjoint l₂ l
|
||
| [] l₂ l d := d
|
||
| (x::xs) l₂ l d :=
|
||
have d₁ : disjoint (xs++l₂) l, from disjoint_of_disjoint_cons_left d,
|
||
disjoint_of_disjoint_append_left_right d₁
|
||
|
||
lemma disjoint_of_disjoint_append_right_left : ∀ {l₁ l₂ l : list A}, disjoint l (l₁++l₂) → disjoint l l₁ :=
|
||
λ l₁ l₂ l d, disjoint.comm (disjoint_of_disjoint_append_left_left (disjoint.comm d))
|
||
|
||
lemma disjoint_of_disjoint_append_right_right : ∀ {l₁ l₂ l : list A}, disjoint l (l₁++l₂) → disjoint l l₂ :=
|
||
λ l₁ l₂ l d, disjoint.comm (disjoint_of_disjoint_append_left_right (disjoint.comm d))
|
||
|
||
end disjoint
|
||
|
||
/- no duplicates predicate -/
|
||
|
||
inductive nodup {A : Type} : list A → Prop :=
|
||
| ndnil : nodup []
|
||
| ndcons : ∀ {a l}, a ∉ l → nodup l → nodup (a::l)
|
||
|
||
section nodup
|
||
open nodup
|
||
variables {A B : Type}
|
||
|
||
theorem nodup_nil : @nodup A [] :=
|
||
ndnil
|
||
|
||
theorem nodup_cons {a : A} {l : list A} : a ∉ l → nodup l → nodup (a::l) :=
|
||
λ i n, ndcons i n
|
||
|
||
theorem nodup_of_nodup_cons : ∀ {a : A} {l : list A}, nodup (a::l) → nodup l
|
||
| a xs (ndcons i n) := n
|
||
|
||
theorem not_mem_of_nodup_cons : ∀ {a : A} {l : list A}, nodup (a::l) → a ∉ l
|
||
| a xs (ndcons i n) := i
|
||
|
||
theorem nodup_of_nodup_append_left : ∀ {l₁ l₂ : list A}, nodup (l₁++l₂) → nodup l₁
|
||
| [] l₂ n := nodup_nil
|
||
| (x::xs) l₂ n :=
|
||
have ndxs : nodup xs, from nodup_of_nodup_append_left (nodup_of_nodup_cons n),
|
||
have nxinxsl₂ : x ∉ xs++l₂, from not_mem_of_nodup_cons n,
|
||
have nxinxs : x ∉ xs, from not_mem_of_not_mem_append_left nxinxsl₂,
|
||
nodup_cons nxinxs ndxs
|
||
|
||
theorem nodup_of_nodup_append_right : ∀ {l₁ l₂ : list A}, nodup (l₁++l₂) → nodup l₂
|
||
| [] l₂ n := n
|
||
| (x::xs) l₂ n := nodup_of_nodup_append_right (nodup_of_nodup_cons n)
|
||
|
||
theorem disjoint_of_nodup_append : ∀ {l₁ l₂ : list A}, nodup (l₁++l₂) → disjoint l₁ l₂
|
||
| [] l₂ d := disjoint_nil_left l₂
|
||
| (x::xs) l₂ d :=
|
||
have d₁ : nodup (x::(xs++l₂)), from d,
|
||
have d₂ : nodup (xs++l₂), from nodup_of_nodup_cons d₁,
|
||
have nxin : x ∉ xs++l₂, from not_mem_of_nodup_cons d₁,
|
||
have nxinl₂ : x ∉ l₂, from not_mem_of_not_mem_append_right nxin,
|
||
have dsj : disjoint xs l₂, from disjoint_of_nodup_append d₂,
|
||
(λ a, and.intro
|
||
(λ ainxxs : a ∈ x::xs,
|
||
or.elim (eq_or_mem_of_mem_cons ainxxs)
|
||
(λ aeqx : a = x, aeqx⁻¹ ▸ nxinl₂)
|
||
(λ ainxs : a ∈ xs, disjoint_left dsj ainxs))
|
||
(λ ainl₂ : a ∈ l₂,
|
||
have nainxs : a ∉ xs, from disjoint_right dsj ainl₂,
|
||
assume ain : a ∈ x::xs, or.elim (eq_or_mem_of_mem_cons ain)
|
||
(λ aeqx : a = x, absurd (aeqx ▸ ainl₂) nxinl₂)
|
||
(λ ainxs : a ∈ xs, absurd ainxs nainxs)))
|
||
|
||
theorem nodup_append_of_nodup_of_nodup_of_disjoint : ∀ {l₁ l₂ : list A}, nodup l₁ → nodup l₂ → disjoint l₁ l₂ → nodup (l₁++l₂)
|
||
| [] l₂ d₁ d₂ dsj := by rewrite [append_nil_left]; exact d₂
|
||
| (x::xs) l₂ d₁ d₂ dsj :=
|
||
have dsj₁ : disjoint xs l₂, from disjoint_of_disjoint_cons_left dsj,
|
||
have ndxs : nodup xs, from nodup_of_nodup_cons d₁,
|
||
have ndxsl₂ : nodup (xs++l₂), from nodup_append_of_nodup_of_nodup_of_disjoint ndxs d₂ dsj₁,
|
||
have nxinxs : x ∉ xs, from not_mem_of_nodup_cons d₁,
|
||
have nxinl₂ : x ∉ l₂, from disjoint_left dsj !mem_cons,
|
||
have nxinxsl₂ : x ∉ xs++l₂, from not_mem_append nxinxs nxinl₂,
|
||
nodup_cons nxinxsl₂ ndxsl₂
|
||
|
||
theorem nodup_app_comm {l₁ l₂ : list A} (d : nodup (l₁++l₂)) : nodup (l₂++l₁) :=
|
||
have d₁ : nodup l₁, from nodup_of_nodup_append_left d,
|
||
have d₂ : nodup l₂, from nodup_of_nodup_append_right d,
|
||
have dsj : disjoint l₁ l₂, from disjoint_of_nodup_append d,
|
||
nodup_append_of_nodup_of_nodup_of_disjoint d₂ d₁ (disjoint.comm dsj)
|
||
|
||
theorem nodup_head {a : A} {l₁ l₂ : list A} (d : nodup (l₁++(a::l₂))) : nodup (a::(l₁++l₂)) :=
|
||
have d₁ : nodup (a::(l₂++l₁)), from nodup_app_comm d,
|
||
have d₂ : nodup (l₂++l₁), from nodup_of_nodup_cons d₁,
|
||
have d₃ : nodup (l₁++l₂), from nodup_app_comm d₂,
|
||
have nain : a ∉ l₂++l₁, from not_mem_of_nodup_cons d₁,
|
||
have nain₂ : a ∉ l₂, from not_mem_of_not_mem_append_left nain,
|
||
have nain₁ : a ∉ l₁, from not_mem_of_not_mem_append_right nain,
|
||
nodup_cons (not_mem_append nain₁ nain₂) d₃
|
||
|
||
theorem nodup_middle {a : A} {l₁ l₂ : list A} (d : nodup (a::(l₁++l₂))) : nodup (l₁++(a::l₂)) :=
|
||
have d₁ : nodup (l₁++l₂), from nodup_of_nodup_cons d,
|
||
have nain : a ∉ l₁++l₂, from not_mem_of_nodup_cons d,
|
||
have disj : disjoint l₁ l₂, from disjoint_of_nodup_append d₁,
|
||
have d₂ : nodup l₁, from nodup_of_nodup_append_left d₁,
|
||
have d₃ : nodup l₂, from nodup_of_nodup_append_right d₁,
|
||
have nain₂ : a ∉ l₂, from not_mem_of_not_mem_append_right nain,
|
||
have nain₁ : a ∉ l₁, from not_mem_of_not_mem_append_left nain,
|
||
have d₄ : nodup (a::l₂), from nodup_cons nain₂ d₃,
|
||
have disj₂ : disjoint l₁ (a::l₂), from disjoint.comm (disjoint_cons_of_not_mem_of_disjoint nain₁ (disjoint.comm disj)),
|
||
nodup_append_of_nodup_of_nodup_of_disjoint d₂ d₄ disj₂
|
||
|
||
theorem nodup_map {f : A → B} (inj : injective f) : ∀ {l : list A}, nodup l → nodup (map f l)
|
||
| [] n := begin rewrite [map_nil], apply nodup_nil end
|
||
| (x::xs) n :=
|
||
assert nxinxs : x ∉ xs, from not_mem_of_nodup_cons n,
|
||
assert ndxs : nodup xs, from nodup_of_nodup_cons n,
|
||
assert ndmfxs : nodup (map f xs), from nodup_map ndxs,
|
||
assert nfxinm : f x ∉ map f xs, from
|
||
λ ab : f x ∈ map f xs,
|
||
obtain (finv : B → A) (isinv : finv ∘ f = id), from inj,
|
||
assert finvfxin : finv (f x) ∈ map finv (map f xs), from mem_map finv ab,
|
||
assert xinxs : x ∈ xs,
|
||
begin
|
||
rewrite [map_map at finvfxin, isinv at finvfxin, left_inv_eq isinv at finvfxin],
|
||
rewrite [map_id at finvfxin],
|
||
exact finvfxin
|
||
end,
|
||
absurd xinxs nxinxs,
|
||
nodup_cons nfxinm ndmfxs
|
||
|
||
theorem nodup_erase_of_nodup [h : decidable_eq A] (a : A) : ∀ {l}, nodup l → nodup (erase a l)
|
||
| [] n := nodup_nil
|
||
| (b::l) n := by_cases
|
||
(λ aeqb : a = b, by rewrite [aeqb, erase_cons_head]; exact (nodup_of_nodup_cons n))
|
||
(λ aneb : a ≠ b,
|
||
have nbinl : b ∉ l, from not_mem_of_nodup_cons n,
|
||
have ndl : nodup l, from nodup_of_nodup_cons n,
|
||
have ndeal : nodup (erase a l), from nodup_erase_of_nodup ndl,
|
||
have nbineal : b ∉ erase a l, from λ i, absurd (erase_sub _ _ i) nbinl,
|
||
assert aux : nodup (b :: erase a l), from nodup_cons nbineal ndeal,
|
||
by rewrite [erase_cons_tail _ aneb]; exact aux)
|
||
|
||
theorem mem_erase_of_nodup [h : decidable_eq A] (a : A) : ∀ {l}, nodup l → a ∉ erase a l
|
||
| [] n := !not_mem_nil
|
||
| (b::l) n :=
|
||
have ndl : nodup l, from nodup_of_nodup_cons n,
|
||
have naineal : a ∉ erase a l, from mem_erase_of_nodup ndl,
|
||
assert nbinl : b ∉ l, from not_mem_of_nodup_cons n,
|
||
by_cases
|
||
(λ aeqb : a = b, by rewrite [aeqb, erase_cons_head]; exact nbinl)
|
||
(λ aneb : a ≠ b,
|
||
assert aux : a ∉ b :: erase a l, from
|
||
assume ainbeal : a ∈ b :: erase a l, or.elim (eq_or_mem_of_mem_cons ainbeal)
|
||
(λ aeqb : a = b, absurd aeqb aneb)
|
||
(λ aineal : a ∈ erase a l, absurd aineal naineal),
|
||
by rewrite [erase_cons_tail _ aneb]; exact aux)
|
||
|
||
definition erase_dup [H : decidable_eq A] : list A → list A
|
||
| [] := []
|
||
| (x :: xs) := if x ∈ xs then erase_dup xs else x :: erase_dup xs
|
||
|
||
theorem erase_dup_nil [H : decidable_eq A] : erase_dup [] = []
|
||
|
||
theorem erase_dup_cons_of_mem [H : decidable_eq A] {a : A} {l : list A} : a ∈ l → erase_dup (a::l) = erase_dup l :=
|
||
assume ainl, calc
|
||
erase_dup (a::l) = if a ∈ l then erase_dup l else a :: erase_dup l : rfl
|
||
... = erase_dup l : if_pos ainl
|
||
|
||
theorem erase_dup_cons_of_not_mem [H : decidable_eq A] {a : A} {l : list A} : a ∉ l → erase_dup (a::l) = a :: erase_dup l :=
|
||
assume nainl, calc
|
||
erase_dup (a::l) = if a ∈ l then erase_dup l else a :: erase_dup l : rfl
|
||
... = a :: erase_dup l : if_neg nainl
|
||
|
||
theorem mem_erase_dup [H : decidable_eq A] {a : A} : ∀ {l}, a ∈ l → a ∈ erase_dup l
|
||
| [] h := absurd h !not_mem_nil
|
||
| (b::l) h := by_cases
|
||
(λ binl : b ∈ l, or.elim (eq_or_mem_of_mem_cons h)
|
||
(λ aeqb : a = b, by rewrite [erase_dup_cons_of_mem binl, -aeqb at binl]; exact (mem_erase_dup binl))
|
||
(λ ainl : a ∈ l, by rewrite [erase_dup_cons_of_mem binl]; exact (mem_erase_dup ainl)))
|
||
(λ nbinl : b ∉ l, or.elim (eq_or_mem_of_mem_cons h)
|
||
(λ aeqb : a = b, by rewrite [erase_dup_cons_of_not_mem nbinl, aeqb]; exact !mem_cons)
|
||
(λ ainl : a ∈ l, by rewrite [erase_dup_cons_of_not_mem nbinl]; exact (or.inr (mem_erase_dup ainl))))
|
||
|
||
theorem mem_of_mem_erase_dup [H : decidable_eq A] {a : A} : ∀ {l}, a ∈ erase_dup l → a ∈ l
|
||
| [] h := by rewrite [erase_dup_nil at h]; exact h
|
||
| (b::l) h := by_cases
|
||
(λ binl : b ∈ l,
|
||
have h₁ : a ∈ erase_dup l, by rewrite [erase_dup_cons_of_mem binl at h]; exact h,
|
||
or.inr (mem_of_mem_erase_dup h₁))
|
||
(λ nbinl : b ∉ l,
|
||
have h₁ : a ∈ b :: erase_dup l, by rewrite [erase_dup_cons_of_not_mem nbinl at h]; exact h,
|
||
or.elim (eq_or_mem_of_mem_cons h₁)
|
||
(λ aeqb : a = b, by rewrite aeqb; exact !mem_cons)
|
||
(λ ainel : a ∈ erase_dup l, or.inr (mem_of_mem_erase_dup ainel)))
|
||
|
||
theorem nodup_erase_dup [H : decidable_eq A] : ∀ l : list A, nodup (erase_dup l)
|
||
| [] := by rewrite erase_dup_nil; exact nodup_nil
|
||
| (a::l) := by_cases
|
||
(λ ainl : a ∈ l, by rewrite [erase_dup_cons_of_mem ainl]; exact (nodup_erase_dup l))
|
||
(λ nainl : a ∉ l,
|
||
assert r : nodup (erase_dup l), from nodup_erase_dup l,
|
||
assert nin : a ∉ erase_dup l, from
|
||
assume ab : a ∈ erase_dup l, absurd (mem_of_mem_erase_dup ab) nainl,
|
||
by rewrite [erase_dup_cons_of_not_mem nainl]; exact (nodup_cons nin r))
|
||
|
||
theorem erase_dup_eq_of_nodup [H : decidable_eq A] : ∀ {l : list A}, nodup l → erase_dup l = l
|
||
| [] d := rfl
|
||
| (a::l) d :=
|
||
assert nainl : a ∉ l, from not_mem_of_nodup_cons d,
|
||
assert dl : nodup l, from nodup_of_nodup_cons d,
|
||
by rewrite [erase_dup_cons_of_not_mem nainl, erase_dup_eq_of_nodup dl]
|
||
end nodup
|
||
|
||
/- union -/
|
||
section union
|
||
variable {A : Type}
|
||
variable [H : decidable_eq A]
|
||
include H
|
||
|
||
definition union : list A → list A → list A
|
||
| [] l₂ := l₂
|
||
| (a::l₁) l₂ := if a ∈ l₂ then union l₁ l₂ else a :: union l₁ l₂
|
||
|
||
theorem union_nil (l : list A) : union [] l = l
|
||
|
||
theorem union_cons_of_mem {a : A} {l₂} : ∀ (l₁), a ∈ l₂ → union (a::l₁) l₂ = union l₁ l₂ :=
|
||
take l₁, assume ainl₂, calc
|
||
union (a::l₁) l₂ = if a ∈ l₂ then union l₁ l₂ else a :: union l₁ l₂ : rfl
|
||
... = union l₁ l₂ : if_pos ainl₂
|
||
|
||
theorem union_cons_of_not_mem {a : A} {l₂} : ∀ (l₁), a ∉ l₂ → union (a::l₁) l₂ = a :: union l₁ l₂ :=
|
||
take l₁, assume nainl₂, calc
|
||
union (a::l₁) l₂ = if a ∈ l₂ then union l₁ l₂ else a :: union l₁ l₂ : rfl
|
||
... = a :: union l₁ l₂ : if_neg nainl₂
|
||
|
||
theorem mem_or_mem_of_mem_union : ∀ {l₁ l₂} {a : A}, a ∈ union l₁ l₂ → a ∈ l₁ ∨ a ∈ l₂
|
||
| [] l₂ a ainl₂ := by rewrite union_nil at ainl₂; exact (or.inr (ainl₂))
|
||
| (b::l₁) l₂ a ainbl₁l₂ := by_cases
|
||
(λ binl₂ : b ∈ l₂,
|
||
have ainl₁l₂ : a ∈ union l₁ l₂, by rewrite [union_cons_of_mem l₁ binl₂ at ainbl₁l₂]; exact ainbl₁l₂,
|
||
or.elim (mem_or_mem_of_mem_union ainl₁l₂)
|
||
(λ ainl₁, or.inl (mem_cons_of_mem _ ainl₁))
|
||
(λ ainl₂, or.inr ainl₂))
|
||
(λ nbinl₂ : b ∉ l₂,
|
||
have ainb_l₁l₂ : a ∈ b :: union l₁ l₂, by rewrite [union_cons_of_not_mem l₁ nbinl₂ at ainbl₁l₂]; exact ainbl₁l₂,
|
||
or.elim (eq_or_mem_of_mem_cons ainb_l₁l₂)
|
||
(λ aeqb, by rewrite aeqb; exact (or.inl !mem_cons))
|
||
(λ ainl₁l₂,
|
||
or.elim (mem_or_mem_of_mem_union ainl₁l₂)
|
||
(λ ainl₁, or.inl (mem_cons_of_mem _ ainl₁))
|
||
(λ ainl₂, or.inr ainl₂)))
|
||
|
||
theorem mem_union_right {a : A} : ∀ (l₁) {l₂}, a ∈ l₂ → a ∈ union l₁ l₂
|
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| [] l₂ h := by rewrite union_nil; exact h
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| (b::l₁) l₂ h := by_cases
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(λ binl₂ : b ∈ l₂, by rewrite [union_cons_of_mem _ binl₂]; exact (mem_union_right _ h))
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(λ nbinl₂ : b ∉ l₂, by rewrite [union_cons_of_not_mem _ nbinl₂]; exact (mem_cons_of_mem _ (mem_union_right _ h)))
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theorem mem_union_left {a : A} : ∀ {l₁} (l₂), a ∈ l₁ → a ∈ union l₁ l₂
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| [] l₂ h := absurd h !not_mem_nil
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| (b::l₁) l₂ h := by_cases
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(λ binl₂ : b ∈ l₂, or.elim (eq_or_mem_of_mem_cons h)
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(λ aeqb : a = b,
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by rewrite [union_cons_of_mem l₁ binl₂, -aeqb at binl₂]; exact (mem_union_right _ binl₂))
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(λ ainl₁ : a ∈ l₁,
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by rewrite [union_cons_of_mem l₁ binl₂]; exact (mem_union_left _ ainl₁)))
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(λ nbinl₂ : b ∉ l₂, or.elim (eq_or_mem_of_mem_cons h)
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(λ aeqb : a = b,
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by rewrite [union_cons_of_not_mem l₁ nbinl₂, aeqb]; exact !mem_cons)
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(λ ainl₁ : a ∈ l₁,
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by rewrite [union_cons_of_not_mem l₁ nbinl₂]; exact (mem_cons_of_mem _ (mem_union_left _ ainl₁))))
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theorem nodup_union_of_nodup_of_nodup : ∀ {l₁ l₂ : list A}, nodup l₁ → nodup l₂ → nodup (union l₁ l₂)
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| [] l₂ n₁ nl₂ := by rewrite union_nil; exact nl₂
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| (a::l₁) l₂ nal₁ nl₂ :=
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assert nl₁ : nodup l₁, from nodup_of_nodup_cons nal₁,
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assert nl₁l₂ : nodup (union l₁ l₂), from nodup_union_of_nodup_of_nodup nl₁ nl₂,
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by_cases
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(λ ainl₂ : a ∈ l₂,
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by rewrite [union_cons_of_mem l₁ ainl₂]; exact nl₁l₂)
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(λ nainl₂ : a ∉ l₂,
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have nainl₁ : a ∉ l₁, from not_mem_of_nodup_cons nal₁,
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assert nainl₁l₂ : a ∉ union l₁ l₂, from
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assume ainl₁l₂ : a ∈ union l₁ l₂, or.elim (mem_or_mem_of_mem_union ainl₁l₂)
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(λ ainl₁, absurd ainl₁ nainl₁)
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(λ ainl₂, absurd ainl₂ nainl₂),
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by rewrite [union_cons_of_not_mem l₁ nainl₂]; exact (nodup_cons nainl₁l₂ nl₁l₂))
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theorem union_eq_append : ∀ {l₁ l₂ : list A}, disjoint l₁ l₂ → union l₁ l₂ = append l₁ l₂
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| [] l₂ d := rfl
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| (a::l₁) l₂ d :=
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assert nainl₂ : a ∉ l₂, from disjoint_left d !mem_cons,
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assert d₁ : disjoint l₁ l₂, from disjoint_of_disjoint_cons_left d,
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by rewrite [union_cons_of_not_mem _ nainl₂, append_cons, union_eq_append d₁]
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variable {B : Type}
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theorem foldl_union_of_disjoint (f : B → A → B) (b : B) {l₁ l₂ : list A} (d : disjoint l₁ l₂)
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: foldl f b (union l₁ l₂) = foldl f (foldl f b l₁) l₂ :=
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by rewrite [union_eq_append d, foldl_append]
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theorem foldr_union_of_dijoint (f : A → B → B) (b : B) {l₁ l₂ : list A} (d : disjoint l₁ l₂)
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: foldr f b (union l₁ l₂) = foldr f (foldr f b l₂) l₁ :=
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by rewrite [union_eq_append d, foldr_append]
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end union
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/- insert -/
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section insert
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variable {A : Type}
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variable [H : decidable_eq A]
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include H
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|
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definition insert (a : A) (l : list A) : list A :=
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if a ∈ l then l else a::l
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theorem insert_eq_of_mem {a : A} {l : list A} : a ∈ l → insert a l = l :=
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assume ainl, if_pos ainl
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theorem insert_eq_of_not_mem {a : A} {l : list A} : a ∉ l → insert a l = a::l :=
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assume nainl, if_neg nainl
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|
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theorem mem_insert (a : A) (l : list A) : a ∈ insert a l :=
|
||
by_cases
|
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(λ ainl : a ∈ l, by rewrite [insert_eq_of_mem ainl]; exact ainl)
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||
(λ nainl : a ∉ l, by rewrite [insert_eq_of_not_mem nainl]; exact !mem_cons)
|
||
|
||
theorem mem_insert_of_mem {a : A} (b : A) {l : list A} : a ∈ l → a ∈ insert b l :=
|
||
assume ainl, by_cases
|
||
(λ binl : b ∈ l, by rewrite [insert_eq_of_mem binl]; exact ainl)
|
||
(λ nbinl : b ∉ l, by rewrite [insert_eq_of_not_mem nbinl]; exact (mem_cons_of_mem _ ainl))
|
||
|
||
theorem nodup_insert (a : A) {l : list A} : nodup l → nodup (insert a l) :=
|
||
assume n, by_cases
|
||
(λ ainl : a ∈ l, by rewrite [insert_eq_of_mem ainl]; exact n)
|
||
(λ nainl : a ∉ l, by rewrite [insert_eq_of_not_mem nainl]; exact (nodup_cons nainl n))
|
||
|
||
theorem length_insert_of_mem {a : A} {l : list A} : a ∈ l → length (insert a l) = length l :=
|
||
assume ainl, by rewrite [insert_eq_of_mem ainl]
|
||
|
||
theorem length_insert_of_not_mem {a : A} {l : list A} : a ∉ l → length (insert a l) = length l + 1 :=
|
||
assume nainl, by rewrite [insert_eq_of_not_mem nainl]
|
||
end insert
|
||
end list
|
||
|
||
attribute list.has_decidable_eq [instance]
|
||
attribute list.decidable_mem [instance]
|
||
attribute list.decidable_any [instance]
|
||
attribute list.decidable_all [instance]
|