feat(library/standard/list.lean): add facts about lists
This commit is contained in:
Jeremy Avigad
Leonardo de Moura
parent
e846c8c76b
commit
c89c96b913
1 changed files with 102 additions and 194 deletions
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@ -11,13 +11,26 @@
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import tactic
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import nat
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import congr
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import if -- for find
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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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-- TODO: move this
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theorem or_right_comm (a b c : Prop) : (a ∨ b) ∨ c ↔ (a ∨ c) ∨ b :=
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calc
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(a ∨ b) ∨ c ↔ a ∨ (b ∨ c) : or_assoc _ _ _
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... ↔ a ∨ (c ∨ b) : congr.infer iff iff _ (or_comm b c)
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... ↔ (a ∨ c) ∨ b : iff_symm (or_assoc _ _ _)
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-- TODO: add or_left_comm, and_right_comm, and_left_comm
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-- Type
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-- ----
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@ -40,9 +53,6 @@ theorem list_cases_on {P : list T → Prop} (l : list T) (Hnil : P nil)
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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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notation `[` `]` := nil
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-- TODO: should this be needed?
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notation `[` x `]` := cons x nil
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-- Concat
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@ -62,16 +72,6 @@ list_induction_on t (refl _)
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(take (x : T) (l : list T) (H : concat l nil = l),
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show concat (cons x l) nil = cons x l, from H ▸ refl _)
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-- TODO: these work:
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-- calc
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-- concat (cons x l) nil = cons x (concat l nil) : refl (concat (cons x l) nil)
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-- ... = cons x l : {H})
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-- H ▸ (refl (cons x (concat l nil))))
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-- doesn't work:
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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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@ -81,29 +81,13 @@ list_induction_on s (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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-- TODO: deleting refl doesn't work, nor does
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-- H ▸ refl _)
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-- 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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-- 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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-- 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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-- add_rewrite nil_concat cons_concat concat_nil concat_assoc
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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_nil : length (@nil T) = 0 := refl _
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theorem length_cons (x : T) (t : list T) : length (x :: t) = succ (length t) := refl _
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@ -121,7 +105,21 @@ list_induction_on s
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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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-- -- add_rewrite length_nil length_cons
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-- add_rewrite length_nil length_cons
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-- Append
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-- ------
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definition append (x : T) : list T → list T := list_rec [x] (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] := refl _
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-- add_rewrite append_nil append_cons
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-- Reverse
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@ -131,104 +129,39 @@ 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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theorem reverse_cons (x : T) (l : list T) : reverse (x :: l) = append x (reverse l) := refl _
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-- opaque_hint (hiding reverse)
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theorem reverse_singleton (x : T) : reverse [x] = [x] := refl _
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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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list_induction_on s (symm (concat_nil _))
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(take x s,
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assume IH : reverse (s ++ t) = concat (reverse t) (reverse s),
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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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reverse ((x :: s) ++ t) = reverse (s ++ t) ++ [x] : refl _
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... = reverse t ++ reverse s ++ [x] : {IH}
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... = reverse t ++ (reverse s ++ [x]) : concat_assoc _ _ _
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... = reverse t ++ (reverse (x :: s)) : refl _)
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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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show reverse (reverse (x :: l')) = 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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-- longer versions:
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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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-- ... = concat (reverse (cons x nil)) l : {H}
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-- ... = cons x l : refl _)
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-- calc
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-- reverse (reverse (cons x l)) = reverse (concat (reverse l) (cons x nil))
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-- : refl _
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-- ... = concat (reverse (cons x nil)) (reverse (reverse l)) : {reverse_concat _ _}
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-- ... = concat (reverse (cons x nil)) l : {H}
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-- ... = cons x l : refl _)
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-- before:
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-- calc
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-- reverse (reverse (cons x l)) = reverse (concat (reverse l) (cons x nil))
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-- : {reverse_cons x l}
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-- ... = concat (reverse (cons x nil)) (reverse (reverse l)) : {reverse_concat _ _}
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-- ... = concat (reverse (cons x nil)) l : {H}
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-- ... = concat (concat (reverse nil) (cons x nil)) l : {reverse_cons _ _}
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-- ... = concat (concat nil (cons x nil)) l : {reverse_nil}
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-- ... = concat (cons x nil) l : {nil_concat _}
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-- ... = cons x (concat nil l) : cons_concat _ _ _
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-- ... = cons x l : {nil_concat _})
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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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-- calc
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-- append x (cons y l) = concat (cons y l) (cons x nil) : { P })
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-- calc
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-- append x (cons y l) = cons y (concat l (cons x nil)) : { P }
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-- ... = concat (cons y l) (cons x nil) : refl _)
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-- here it works!
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-- append x (cons y l) = cons y (append x l) : refl _
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-- ... = cons y (concat l (cons x nil)) : { P }
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-- ... = concat (cons y l) (cons x nil) : refl _)
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reverse (reverse (x :: l')) = reverse (reverse l' ++ [x]) : refl _
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... = reverse [x] ++ reverse (reverse l') : reverse_concat _ _
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... = [x] ++ l' : { H }
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... = x :: l' : refl _)
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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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list_induction_on l (refl _)
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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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... = reverse (x :: (reverse (y :: l'))) : refl _)
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-- Head and tail
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@ -265,89 +198,64 @@ list_cases_on l
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-- List membership
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-- ---------------
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definition mem (f : T) : list T → Prop := list_rec false (fun x l H, (H ∨ (x = f)))
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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 (f : T) : mem f nil ↔ false := iff_refl _
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-- TODO: constructively, equality is stronger. Use that?
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theorem mem_nil (x : T) : x ∈ nil ↔ false := iff_refl _
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theorem mem_cons (x : T) (f : T) (l : list T) : mem f (cons x l) ↔ (mem f l ∨ x = f) := 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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-- TODO: fix this!
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-- theorem or_right_comm : ∀a b c, (a ∨ b) ∨ c ↔ (a ∨ c) ∨ b :=
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-- take a b c, calc
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-- (a ∨ b) ∨ c ↔ a ∨ (b ∨ c) : or_assoc _ _ _
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-- ... ↔ a ∨ (c ∨ b) : {or_comm _ _}
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-- ... ↔ (a ∨ c) ∨ b : (or_assoc _ _ _)⁻¹
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theorem mem_concat_imp_or (x : T) (s t : list T) : x ∈ s ++ t → x ∈ s ∨ x ∈ t :=
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list_induction_on s (or_intro_right _)
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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 imp_or_right H2 IH,
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iff_elim_right (or_assoc _ _ _) H3)
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-- theorem mem_concat_imp_or (f : T) (s t : list T) : mem f (concat s t) → mem f s ∨ mem f t :=
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-- list_induction_on s
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-- (assume H : mem f (concat nil t),
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-- have H1 : mem f t, from subst H (nil_concat t),
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-- show mem f nil ∨ mem f t, from or_intro_right _ H1)
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-- (take x l,
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-- assume IH : mem f (concat l t) → mem f l ∨ mem f t,
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-- assume H : mem f (concat (cons x l) t),
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-- have H1 : mem f (cons x (concat l t)), from subst H (cons_concat _ _ _),
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-- have H2 : mem f (concat l t) ∨ x = f, from (mem_cons _ _ _) ◂ H1,
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-- have H3 : (mem f l ∨ mem f t) ∨ x = f, from imp_or_left H2 IH,
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-- have H4 : (mem f l ∨ x = f) ∨ mem f t, from or_right_comm _ _ _ ◂ H3,
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-- show mem f (cons x l) ∨ mem f t, from subst H4 (symm (mem_cons _ _ _)))
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theorem mem_or_imp_concat (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 H1
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(take H2 : x = y, or_intro_left _ H2)
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(take H2 : x ∈ s, or_intro_right _ (IH (or_intro_left _ H2))))
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(assume H1 : x ∈ t, or_intro_right _ (IH (or_intro_right _ H1))))
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-- theorem mem_or_imp_concat (f : T) (s t : list T) :
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-- mem f s ∨ mem f t → mem f (concat s t)
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-- :=
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-- list_induction_on s
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-- (assume H : mem f nil ∨ mem f t,
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-- have H1 : false ∨ mem f t, from subst H (mem_nil f),
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-- have H2 : mem f t, from subst H1 (or_false_right _),
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-- show mem f (concat nil t), from subst H2 (symm (nil_concat _)))
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-- (take x l,
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-- assume IH : mem f l ∨ mem f t → mem f (concat l t),
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-- assume H : (mem f (cons x l)) ∨ (mem f t),
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-- have H1 : ((mem f l) ∨ (x = f)) ∨ (mem f t), from subst H (mem_cons _ _ _),
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-- have H2 : (mem f t) ∨ ((mem f l) ∨ (x = f)), from subst H1 (or_comm _ _),
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-- have H3 : ((mem f t) ∨ (mem f l)) ∨ (x = f), from subst H2 (symm (or_assoc _ _ _)),
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-- have H4 : ((mem f l) ∨ (mem f t)) ∨ (x = f), from subst H3 (or_comm _ _),
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-- have H5 : (mem f (concat l t)) ∨ (x = f), from (or_imp_or_left H4 IH),
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-- have H6 : (mem f (cons x (concat l t))), from subst H5 (symm (mem_cons _ _ _)),
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-- show (mem f (concat (cons x l) t)), from subst H6 (symm (cons_concat _ _ _)))
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theorem mem_concat (x : T) (s t : list T) : x ∈ s ++ t ↔ x ∈ s ∨ x ∈ t
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:= iff_intro (mem_concat_imp_or _ _ _) (mem_or_imp_concat _ _ _)
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-- theorem mem_concat (f : T) (s t : list T) : mem f (concat s t) ↔ mem f s ∨ mem f t
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-- := iff_intro (mem_concat_imp_or _ _ _) (mem_or_imp_concat _ _ _)
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axiom sorry {P : Prop} : P
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-- theorem mem_split (f : T) (s : list T) :
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-- mem f s → ∃ a b : list T, s = concat a (cons f b)
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-- :=
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-- list_induction_on s
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-- (assume H : mem f nil,
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-- have H1 : mem f nil ↔ false, from (mem_nil f),
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-- show ∃ a b : list T, nil = concat a (cons f b), from absurd_elim _ H (eqf_elim H1))
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-- (take x l,
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-- assume P1 : mem f l → ∃ a b : list T, l = concat a (cons f b),
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-- assume P2 : mem f (cons x l),
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-- have P3 : mem f l ∨ x = f, from subst P2 (mem_cons _ _ _),
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-- show ∃ a b : list T, cons x l = concat a (cons f b), from
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-- or_elim P3
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-- (assume Q1 : mem f l,
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-- obtain (a : list T) (PQ : ∃ b, l = concat a (cons f b)), from P1 Q1,
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-- obtain (b : list T) (RS : l = concat a (cons f b)), from PQ,
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-- exists_intro (cons x a)
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-- (exists_intro b
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-- (calc
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-- cons x l = cons x (concat a (cons f b)) : { RS }
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-- ... = concat (cons x a) (cons f b) : (symm (cons_concat _ _ _)))))
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-- (assume Q2 : x = f,
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-- exists_intro nil
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-- (exists_intro l
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-- (calc
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-- cons x l = concat nil (cons x l) : (symm (nil_concat _))
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-- ... = concat nil (cons f l) : {Q2}))))
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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 ∈ nil, false_elim _ (iff_elim_left (mem_nil x) 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 H
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(assume H1 : x = y,
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exists_intro nil
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(exists_intro l (subst H1 (refl _))))
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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 trans (subst H3 (refl (y :: l))) (cons_concat _ _ _),
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exists_intro _ (exists_intro _ H4)))
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-- -- Find
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-- -- ----
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-- Find
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-- ----
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-- definition find (x : T) : list T → ℕ
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-- to do this: need decidability of = for nat
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-- definition find (x : T) : list T → nat
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-- := list_rec 0 (fun y l b, if x = y then 0 else succ b)
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-- theorem find_nil (f : T) : find f nil = 0
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@ -378,17 +286,17 @@ theorem mem_cons (x : T) (f : T) (l : list T) : mem f (cons x l) ↔ (mem f l
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-- ... = succ (length l) : {IH (and_elim_left P3)}
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-- ... = length (cons y l) : symm (length_cons _ _))
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-- -- nth element
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-- -- -----------
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-- nth element
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-- -----------
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-- definition nth (x0 : T) (l : list T) (n : ℕ) : T
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-- := nat.rec (λl, head x0 l) (λm f l, f (tail l)) n l
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definition nth (x0 : T) (l : list T) (n : ℕ) : T :=
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nat_rec (λl, head x0 l) (λm f l, f (tail l)) n l
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-- theorem nth (x0 : T) (l : list T) : nth_element x0 l 0 = head x0 l
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-- := hcongr1 (nat::rec_zero _ _) l
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theorem nth_zero (x0 : T) (l : list T) : nth x0 l 0 = head x0 l := refl _
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-- theorem nth_element_succ (x0 : T) (l : list T) (n : ℕ) :
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-- nth_element x0 l (succ n) = nth_element x0 (tail l) n
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-- := hcongr1 (nat::rec_succ _ _ _) l
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theorem nth_succ (x0 : T) (l : list T) (n : ℕ) : nth x0 l (succ n) = nth x0 (tail l) n := refl _
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-- end
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||||
end
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||||
|
||||
-- declare global notation outside the section
|
||||
infixl `++` : 65 := concat
|
||||
|
|
|
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Loading…
Reference in a new issue