lean2/tests/lean/slow/list_elab2.lean

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--- Copyright (c) 2014 Parikshit Khanna. All rights reserved.
--- Released under Apache 2.0 license as described in the file LICENSE.
--- Authors: Parikshit Khanna, Jeremy Avigad
----------------------------------------------------------------------------------------------------
-- Theory list
-- ===========
--
-- Basic properties of lists.
import logic data.nat
-- import congr
set_option unifier.expensive true
using nat
-- using congr
using eq_proofs
namespace list
-- Type
-- ----
inductive list (T : Type) : Type :=
| nil {} : list T
| cons : T → list T → list T
infix `::` : 65 := cons
section
variable {T : Type}
theorem list_induction_on {P : list T → Prop} (l : list T) (Hnil : P nil)
(Hind : forall x : T, forall l : list T, forall H : P l, P (cons x l)) : P l :=
list_rec Hnil Hind l
theorem list_cases_on {P : list T → Prop} (l : list T) (Hnil : P nil)
(Hcons : forall x : T, forall l : list T, P (cons x l)) : P l :=
list_induction_on l Hnil (take x l IH, Hcons x l)
notation `[` l:(foldr `,` (h t, cons h t) nil) `]` := l
-- Concat
-- ------
definition concat (s t : list T) : list T :=
list_rec t (fun x : T, fun l : list T, fun u : list T, cons x u) s
infixl `++` : 65 := concat
theorem nil_concat (t : list T) : nil ++ t = t := refl _
theorem cons_concat (x : T) (s t : list T) : (x :: s) ++ t = x :: (s ++ t) := refl _
theorem concat_nil (t : list T) : t ++ nil = t :=
list_induction_on t (refl _)
(take (x : T) (l : list T) (H : concat l nil = l),
H ▸ (refl (cons x (concat l nil))))
theorem concat_nil2 (t : list T) : t ++ nil = t :=
list_induction_on t (refl _)
(take (x : T) (l : list T) (H : concat l nil = l),
-- H ▸ (refl (cons x (concat l nil))))
H ▸ (refl (concat (cons x l) nil)))
theorem concat_assoc (s t u : list T) : s ++ t ++ u = s ++ (t ++ u) :=
list_induction_on s (refl _)
(take x l,
assume H : concat (concat l t) u = concat l (concat t u),
H ▸ refl _)
theorem concat_assoc2 (s t u : list T) : s ++ t ++ u = s ++ (t ++ u) :=
list_induction_on s (refl _)
(take x l,
assume H : concat (concat l t) u = concat l (concat t u),
calc concat (concat (cons x l) t) u = cons x (concat (concat l t) u) : refl _
... = concat (cons x l) (concat t u) : { H })
theorem concat_assoc3 (s t u : list T) : s ++ t ++ u = s ++ (t ++ u) :=
list_induction_on s (refl _)
(take x l,
assume H : concat (concat l t) u = concat l (concat t u),
calc concat (concat (cons x l) t) u = cons x (concat l (concat t u)) : { H }
... = concat (cons x l) (concat t u) : refl _)
theorem concat_assoc4 (s t u : list T) : s ++ t ++ u = s ++ (t ++ u) :=
list_induction_on s (refl _)
(take x l,
assume H : concat (concat l t) u = concat l (concat t u),
calc
concat (concat (cons x l) t) u = cons x (concat (concat l t) u) : refl _
... = cons x (concat l (concat t u)) : { H }
... = concat (cons x l) (concat t u) : refl _)
-- Length
-- ------
definition length : list T → := list_rec 0 (fun x l m, succ m)
-- TODO: cannot replace zero by 0
theorem length_nil : length (@nil T) = zero := refl _
theorem length_cons (x : T) (t : list T) : length (x :: t) = succ (length t) := refl _
theorem length_concat (s t : list T) : length (s ++ t) = length s + length t :=
list_induction_on s
(calc
length (concat nil t) = length t : refl _
... = zero + length t : {symm (add_zero_left (length t))}
... = length (@nil T) + length t : refl _)
(take x s,
assume H : length (concat s t) = length s + length t,
calc
length (concat (cons x s) t ) = succ (length (concat s t)) : refl _
... = succ (length s + length t) : { H }
... = succ (length s) + length t : {symm (add_succ_left _ _)}
... = length (cons x s) + length t : refl _)
-- Reverse
-- -------
definition reverse : list T → list T := list_rec nil (fun x l r, r ++ [x])
theorem reverse_nil : reverse (@nil T) = nil := refl _
theorem reverse_cons (x : T) (l : list T) : reverse (x :: l) = (reverse l) ++ (cons x nil) := refl _
-- opaque_hint (hiding reverse)
theorem reverse_concat (s t : list T) : reverse (s ++ t) = (reverse t) ++ (reverse s) :=
list_induction_on s
(calc
reverse (concat nil t) = reverse t : { nil_concat _ }
... = concat (reverse t) nil : symm (concat_nil _)
... = concat (reverse t) (reverse nil) : {symm (reverse_nil)})
(take x l,
assume H : reverse (concat l t) = concat (reverse t) (reverse l),
calc
reverse (concat (cons x l) t) = concat (reverse (concat l t)) (cons x nil) : refl _
... = concat (concat (reverse t) (reverse l)) (cons x nil) : { H }
... = concat (reverse t) (concat (reverse l) (cons x nil)) : concat_assoc _ _ _
... = concat (reverse t) (reverse (cons x l)) : refl _)
-- -- add_rewrite length_nil length_cons
theorem reverse_reverse (l : list T) : reverse (reverse l) = l :=
list_induction_on l (refl _)
(take x l',
assume H: reverse (reverse l') = l',
show reverse (reverse (cons x l')) = cons x l', from
calc
reverse (reverse (cons x l')) =
concat (reverse (cons x nil)) (reverse (reverse l')) : {reverse_concat _ _}
... = cons x l' : {H})
-- Append
-- ------
-- TODO: define reverse from append
definition append (x : T) : list T → list T := list_rec (x :: nil) (fun y l l', y :: l')
theorem append_nil (x : T) : append x nil = [x] := refl _
theorem append_cons (x : T) (y : T) (l : list T) : append x (y :: l) = y :: (append x l) := refl _
theorem append_eq_concat (x : T) (l : list T) : append x l = l ++ [x] :=
list_induction_on l (refl _)
(take y l,
assume P : append x l = concat l [x],
P ▸ refl _)
set_option unifier.expensive false
theorem append_eq_reverse_cons (x : T) (l : list T) : append x l = reverse (x :: reverse l) :=
list_induction_on l
(calc
append x nil = [x] : (refl _)
... = concat nil [x] : {symm (nil_concat _)}
... = concat (reverse nil) [x] : {symm (reverse_nil)}
... = reverse [x] : {symm (reverse_cons _ _)}
... = reverse (x :: (reverse nil)) : {symm (reverse_nil)})
(take y l',
assume H : append x l' = reverse (x :: reverse l'),
calc
append x (y :: l') = (y :: l') ++ [ x ] : append_eq_concat _ _
... = concat (reverse (reverse (y :: l'))) [ x ] : {symm (reverse_reverse _)}
... = reverse (x :: (reverse (y :: l'))) : refl _)
exit
-- Head and tail
-- -------------
definition head (x0 : T) : list T → T := list_rec x0 (fun x l h, x)
theorem head_nil (x0 : T) : head x0 (@nil T) = x0 := refl _
theorem head_cons (x : T) (x0 : T) (t : list T) : head x0 (x :: t) = x := refl _
theorem head_concat (s t : list T) (x0 : T) : s ≠ nil → (head x0 (s ++ t) = head x0 s) :=
list_cases_on s
(take H : nil ≠ nil, absurd_elim (head x0 (concat nil t) = head x0 nil) (refl nil) H)
(take x s,
take H : cons x s ≠ nil,
calc
head x0 (concat (cons x s) t) = head x0 (cons x (concat s t)) : {cons_concat _ _ _}
... = x : {head_cons _ _ _}
... = head x0 (cons x s) : {symm ( head_cons x x0 s)})
definition tail : list T → list T := list_rec nil (fun x l b, l)
theorem tail_nil : tail (@nil T) = nil := refl _
theorem tail_cons (x : T) (l : list T) : tail (cons x l) = l := refl _
theorem cons_head_tail (x0 : T) (l : list T) : l ≠ nil → (head x0 l) :: (tail l) = l :=
list_cases_on l
(assume H : nil ≠ nil, absurd_elim _ (refl _) H)
(take x l, assume H : cons x l ≠ nil, refl _)
-- List membership
-- ---------------
definition mem (x : T) : list T → Prop := list_rec false (fun y l H, x = y H)
infix `∈` : 50 := mem
theorem mem_nil (x : T) : mem x nil ↔ false := iff_refl _
theorem mem_cons (x : T) (y : T) (l : list T) : mem x (cons y l) ↔ (x = y mem x l) := iff_refl _