234 lines
7.8 KiB
Text
234 lines
7.8 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
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----------------------------------------------------------------------------------------------------
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-- Theory list
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-- ===========
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--
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-- Basic properties of lists.
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import tactic
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import nat
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-- import congr
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set_option unifier.expensive true
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using nat
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-- using congr
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using eq_proofs
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namespace list
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-- Type
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-- ----
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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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infix `::` : 65 := cons
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section
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variable {T : Type}
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theorem list_induction_on {P : list T → Prop} (l : list T) (Hnil : P nil)
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(Hind : forall x : T, forall l : list T, forall H : P l, P (cons x l)) : P l :=
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list_rec Hnil Hind l
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theorem list_cases_on {P : list T → Prop} (l : list T) (Hnil : P nil)
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(Hcons : forall x : T, forall l : list T, P (cons x l)) : P l :=
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list_induction_on l Hnil (take x l IH, Hcons x l)
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notation `[` l:(foldr `,` (h t, cons h t) nil) `]` := l
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-- Concat
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-- ------
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definition concat (s t : list T) : list T :=
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list_rec t (fun x : T, fun l : list T, fun u : list T, cons x u) s
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infixl `++` : 65 := concat
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theorem nil_concat (t : list T) : nil ++ t = t := refl _
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theorem cons_concat (x : T) (s t : list T) : (x :: s) ++ t = x :: (s ++ t) := refl _
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theorem concat_nil (t : list T) : t ++ nil = t :=
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list_induction_on t (refl _)
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(take (x : T) (l : list T) (H : concat l nil = l),
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H ▸ (refl (cons x (concat l nil))))
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theorem concat_nil2 (t : list T) : t ++ nil = t :=
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list_induction_on t (refl _)
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(take (x : T) (l : list T) (H : concat l nil = l),
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-- H ▸ (refl (cons x (concat l nil))))
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H ▸ (refl (concat (cons x l) nil)))
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theorem concat_assoc (s t u : list T) : s ++ t ++ u = s ++ (t ++ u) :=
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list_induction_on s (refl _)
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(take x l,
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assume H : concat (concat l t) u = concat l (concat t u),
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H ▸ refl _)
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theorem concat_assoc2 (s t u : list T) : s ++ t ++ u = s ++ (t ++ u) :=
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list_induction_on s (refl _)
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(take x l,
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assume H : concat (concat l t) u = concat l (concat t u),
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calc concat (concat (cons x l) t) u = cons x (concat (concat l t) u) : refl _
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... = concat (cons x l) (concat t u) : { H })
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theorem concat_assoc3 (s t u : list T) : s ++ t ++ u = s ++ (t ++ u) :=
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list_induction_on s (refl _)
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(take x l,
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assume H : concat (concat l t) u = concat l (concat t u),
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calc concat (concat (cons x l) t) u = cons x (concat l (concat t u)) : { H }
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... = concat (cons x l) (concat t u) : refl _)
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theorem concat_assoc4 (s t u : list T) : s ++ t ++ u = s ++ (t ++ u) :=
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list_induction_on s (refl _)
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(take x l,
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assume H : concat (concat l t) u = concat l (concat t u),
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calc
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concat (concat (cons x l) t) u = cons x (concat (concat l t) u) : refl _
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... = cons x (concat l (concat t u)) : { H }
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... = concat (cons x l) (concat t u) : refl _)
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-- Length
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-- ------
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definition length : list T → ℕ := list_rec 0 (fun x l m, succ m)
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-- TODO: cannot replace zero by 0
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theorem length_nil : length (@nil T) = zero := refl _
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theorem length_cons (x : T) (t : list T) : length (x :: t) = succ (length t) := refl _
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theorem length_concat (s t : list T) : length (s ++ t) = length s + length t :=
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list_induction_on s
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(calc
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length (concat nil t) = length t : refl _
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... = zero + length t : {symm (add_zero_left (length t))}
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... = length (@nil T) + length t : refl _)
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(take x s,
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assume H : length (concat s t) = length s + length t,
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calc
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length (concat (cons x s) t ) = succ (length (concat s t)) : refl _
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... = succ (length s + length t) : { H }
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... = succ (length s) + length t : {symm (add_succ_left _ _)}
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... = length (cons x s) + length t : refl _)
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-- Reverse
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-- -------
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definition reverse : list T → list T := list_rec nil (fun x l r, r ++ [x])
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theorem reverse_nil : reverse (@nil T) = nil := refl _
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theorem reverse_cons (x : T) (l : list T) : reverse (x :: l) = (reverse l) ++ (cons x nil) := refl _
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-- opaque_hint (hiding reverse)
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theorem reverse_concat (s t : list T) : reverse (s ++ t) = (reverse t) ++ (reverse s) :=
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list_induction_on s
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(calc
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reverse (concat nil t) = reverse t : { nil_concat _ }
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... = concat (reverse t) nil : symm (concat_nil _)
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... = concat (reverse t) (reverse nil) : {symm (reverse_nil)})
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(take x l,
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assume H : reverse (concat l t) = concat (reverse t) (reverse l),
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calc
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reverse (concat (cons x l) t) = concat (reverse (concat l t)) (cons x nil) : refl _
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... = concat (concat (reverse t) (reverse l)) (cons x nil) : { H }
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... = concat (reverse t) (concat (reverse l) (cons x nil)) : concat_assoc _ _ _
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... = concat (reverse t) (reverse (cons x l)) : refl _)
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-- -- add_rewrite length_nil length_cons
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theorem reverse_reverse (l : list T) : reverse (reverse l) = l :=
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list_induction_on l (refl _)
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(take x l',
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assume H: reverse (reverse l') = l',
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show reverse (reverse (cons x l')) = cons x l', from
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calc
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reverse (reverse (cons x l')) =
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concat (reverse (cons x nil)) (reverse (reverse l')) : {reverse_concat _ _}
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... = cons x l' : {H})
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-- Append
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-- ------
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-- TODO: define reverse from append
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definition append (x : T) : list T → list T := list_rec (x :: nil) (fun y l l', y :: l')
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theorem append_nil (x : T) : append x nil = [x] := refl _
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theorem append_cons (x : T) (y : T) (l : list T) : append x (y :: l) = y :: (append x l) := refl _
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theorem append_eq_concat (x : T) (l : list T) : append x l = l ++ [x] :=
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list_induction_on l (refl _)
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(take y l,
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assume P : append x l = concat l [x],
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P ▸ refl _)
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set_option unifier.expensive false
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theorem append_eq_reverse_cons (x : T) (l : list T) : append x l = reverse (x :: reverse l) :=
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list_induction_on l
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(calc
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append x nil = [x] : (refl _)
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... = concat nil [x] : {symm (nil_concat _)}
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... = concat (reverse nil) [x] : {symm (reverse_nil)}
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... = reverse [x] : {symm (reverse_cons _ _)}
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... = reverse (x :: (reverse nil)) : {symm (reverse_nil)})
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(take y l',
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assume H : append x l' = reverse (x :: reverse l'),
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calc
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append x (y :: l') = (y :: l') ++ [ x ] : append_eq_concat _ _
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... = concat (reverse (reverse (y :: l'))) [ x ] : {symm (reverse_reverse _)}
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... = reverse (x :: (reverse (y :: l'))) : refl _)
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exit
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-- Head and tail
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-- -------------
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definition head (x0 : T) : list T → T := list_rec x0 (fun x l h, x)
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theorem head_nil (x0 : T) : head x0 (@nil T) = x0 := refl _
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theorem head_cons (x : T) (x0 : T) (t : list T) : head x0 (x :: t) = x := refl _
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theorem head_concat (s t : list T) (x0 : T) : s ≠ nil → (head x0 (s ++ t) = head x0 s) :=
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list_cases_on s
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(take H : nil ≠ nil, absurd_elim (head x0 (concat nil t) = head x0 nil) (refl nil) H)
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(take x s,
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take H : cons x s ≠ nil,
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calc
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head x0 (concat (cons x s) t) = head x0 (cons x (concat s t)) : {cons_concat _ _ _}
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... = x : {head_cons _ _ _}
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... = head x0 (cons x s) : {symm ( head_cons x x0 s)})
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definition tail : list T → list T := list_rec nil (fun x l b, l)
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theorem tail_nil : tail (@nil T) = nil := refl _
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theorem tail_cons (x : T) (l : list T) : tail (cons x l) = l := refl _
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theorem cons_head_tail (x0 : T) (l : list T) : l ≠ nil → (head x0 l) :: (tail l) = l :=
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list_cases_on l
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(assume H : nil ≠ nil, absurd_elim _ (refl _) H)
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(take x l, assume H : cons x l ≠ nil, refl _)
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-- List membership
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-- ---------------
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definition mem (x : T) : list T → Prop := list_rec false (fun y l H, x = y ∨ H)
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infix `∈` : 50 := mem
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theorem mem_nil (x : T) : mem x nil ↔ false := iff_refl _
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theorem mem_cons (x : T) (y : T) (l : list T) : mem x (cons y l) ↔ (x = y ∨ mem x l) := iff_refl _
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