feat(library/data/category): add vector
Signed-off-by: Leonardo de Moura <leonardo@microsoft.com>
This commit is contained in:
Floris van Doorn
Leonardo de Moura
parent
e9fc4f14a0
commit
02d72e4c40
1 changed files with 317 additions and 0 deletions
317
library/data/vector.lean
Normal file
317
library/data/vector.lean
Normal file
|
|
@ -0,0 +1,317 @@
|
|||
-- Copyright (c) 2014 Floris van Doorn. All rights reserved.
|
||||
-- Released under Apache 2.0 license as described in the file LICENSE.
|
||||
-- Author: Floris van Doorn
|
||||
import data.nat.basic data.empty
|
||||
open nat eq_ops
|
||||
|
||||
inductive vec (T : Type) : ℕ → Type :=
|
||||
nil {} : vec T 0,
|
||||
cons : T → ∀{n}, vec T n → vec T (succ n)
|
||||
|
||||
namespace vec
|
||||
infix `::` := cons --at what level?
|
||||
notation `[` l:(foldr `,` (h t, cons h t) nil) `]` := l
|
||||
|
||||
section sc_vec
|
||||
variable {T : Type}
|
||||
|
||||
theorem rec_on [protected] {C : ∀ (n : ℕ), vec T n → Type} {n : ℕ} (v : vec T n) (Hnil : C 0 nil)
|
||||
(Hcons : ∀(x : T) {n : ℕ} (w : vec T n), C n w → C (succ n) (cons x w)) : C n v :=
|
||||
rec Hnil Hcons v
|
||||
|
||||
theorem induction_on [protected] {C : ∀ (n : ℕ), vec T n → Prop} {n : ℕ} (v : vec T n) (Hnil : C 0 nil)
|
||||
(Hcons : ∀(x : T) {n : ℕ} (w : vec T n), C n w → C (succ n) (cons x w)) : C n v :=
|
||||
rec_on v Hnil Hcons
|
||||
|
||||
theorem case_on [protected] {C : ∀ (n : ℕ), vec T n → Type} {n : ℕ} (v : vec T n) (Hnil : C 0 nil)
|
||||
(Hcons : ∀(x : T) {n : ℕ} (w : vec T n), C (succ n) (cons x w)) : C n v :=
|
||||
rec_on v Hnil (take x n v IH, Hcons x v)
|
||||
|
||||
theorem case_zero_lem [private] {C : vec T 0 → Type} {n : ℕ} (v : vec T n) (Hnil : C nil) :
|
||||
∀ H : n = 0, C (cast (congr_arg (vec T) H) v) :=
|
||||
rec_on v (take H : 0 = 0, (eq.rec Hnil (cast_eq _ nil⁻¹)))
|
||||
(take (x : T) (n : ℕ) (w : vec T n) IH (H : succ n = 0),
|
||||
false.rec_type _ (absurd H succ_ne_zero))
|
||||
|
||||
theorem case_zero {C : vec T 0 → Type} (v : vec T 0) (Hnil : C nil) : C v :=
|
||||
eq.rec (case_zero_lem v Hnil (eq.refl 0)) (cast_eq _ v)
|
||||
|
||||
theorem rec_nonempty_lem [private] {C : Π{n}, vec T (succ n) → Type} {n : ℕ} (v : vec T n)
|
||||
(Hone : Πa, C [a]) (Hcons : Πa {n} (v : vec T (succ n)), C v → C (a :: v))
|
||||
: ∀{m} (H : n = succ m), C (cast (congr_arg (vec T) H) v) :=
|
||||
case_on v (take m (H : 0 = succ m), false.rec_type _ (absurd (H⁻¹) succ_ne_zero))
|
||||
(take x n v m H,
|
||||
have H2 : C (x::v), from
|
||||
sorry,
|
||||
-- rec_on v
|
||||
-- (Hone x)
|
||||
-- (take y n w IH, Hcons x (y::w)),
|
||||
show C (cast (congr_arg (vec T) H) (x::v)), from
|
||||
sorry
|
||||
)
|
||||
|
||||
theorem rec_nonempty {C : Π{n}, vec T (succ n) → Type} {n : ℕ} (v : vec T (succ n))
|
||||
(Hone : Πa, C [a]) (Hcons : Πa {n} (v : vec T (succ n)), C v → C (a :: v)) : C v :=
|
||||
sorry
|
||||
|
||||
theorem case_succ_lem [private] {C : Π{n}, vec T (succ n) → Type} {n : ℕ} (v : vec T n)
|
||||
(H : Πa {n} (v : vec T n), C (a :: v))
|
||||
: ∀{m} (H : n = succ m), C (cast (congr_arg (vec T) H) v) :=
|
||||
sorry
|
||||
|
||||
theorem case_succ {C : Π{n}, vec T (succ n) → Type} {n : ℕ} (v : vec T (succ n))
|
||||
(H : Πa {n} (v : vec T n), C (a :: v)) : C v :=
|
||||
sorry
|
||||
|
||||
theorem vec0_eq_nil (v : vec T 0) : v = nil :=
|
||||
case_zero v rfl
|
||||
|
||||
-- Concat
|
||||
-- ------
|
||||
|
||||
abbreviation cast_subst {A : Type} {P : A → Type} {a a' : A} (H : a = a') (B : P a) : P a' :=
|
||||
cast (congr_arg P H) B
|
||||
|
||||
definition concat {n m : ℕ} (v : vec T n) (w : vec T m) : vec T (n + m) :=
|
||||
vec.rec (cast_subst (add_zero_left⁻¹) w) (λx n w' u, cast_subst (add_succ_left⁻¹) (x::u)) v
|
||||
|
||||
infixl `++`:65 := concat
|
||||
|
||||
theorem nil_concat {n : ℕ} (v : vec T n) : nil ++ v = cast_subst (add_zero_left⁻¹) v := rfl
|
||||
|
||||
theorem cons_concat {n m : ℕ} (x : T) (v : vec T n) (w : vec T m)
|
||||
: (x :: v) ++ w = cast_subst (add_succ_left⁻¹) (x::(v++w)) := rfl
|
||||
|
||||
/-
|
||||
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),
|
||||
show concat (cons x l) nil = cons x l, from H ▸ refl _)
|
||||
|
||||
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),
|
||||
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 {n : ℕ} (v : vec T n) := n
|
||||
|
||||
theorem length_nil : length (@nil T) = 0 := rfl
|
||||
|
||||
-- theorem length_cons (x : T) (t : list T) : length (x :: t) = succ (length t) := rfl
|
||||
|
||||
-- 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 : rfl
|
||||
-- ... = zero + length t : {add_zero_left⁻¹}
|
||||
-- ... = length (@nil T) + length t : rfl)
|
||||
-- (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)) : rfl
|
||||
-- ... = succ (length s + length t) : { H }
|
||||
-- ... = succ (length s) + length t : {add_succ_left⁻¹}
|
||||
-- ... = length (cons x s) + length t : rfl)
|
||||
|
||||
-- -- add_rewrite length_nil length_cons
|
||||
|
||||
|
||||
-- -- Append
|
||||
-- -- ------
|
||||
|
||||
-- definition append (x : T) : list T → list T := list_rec [x] (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] := refl _
|
||||
|
||||
-- -- add_rewrite append_nil append_cons
|
||||
|
||||
|
||||
-- -- 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) = append x (reverse l) := refl _
|
||||
|
||||
-- theorem reverse_singleton (x : T) : reverse [x] = [x] := refl _
|
||||
|
||||
-- theorem reverse_concat (s t : list T) : reverse (s ++ t) = (reverse t) ++ (reverse s) :=
|
||||
-- list_induction_on s (symm (concat_nil _))
|
||||
-- (take x s,
|
||||
-- assume IH : reverse (s ++ t) = concat (reverse t) (reverse s),
|
||||
-- calc
|
||||
-- reverse ((x :: s) ++ t) = reverse (s ++ t) ++ [x] : refl _
|
||||
-- ... = reverse t ++ reverse s ++ [x] : {IH}
|
||||
-- ... = reverse t ++ (reverse s ++ [x]) : concat_assoc _ _ _
|
||||
-- ... = reverse t ++ (reverse (x :: s)) : refl _)
|
||||
|
||||
-- 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 (x :: l')) = x :: l', from
|
||||
-- calc
|
||||
-- reverse (reverse (x :: l')) = reverse (reverse l' ++ [x]) : refl _
|
||||
-- ... = reverse [x] ++ reverse (reverse l') : reverse_concat _ _
|
||||
-- ... = [x] ++ l' : { H }
|
||||
-- ... = x :: l' : refl _)
|
||||
|
||||
-- theorem append_eq_reverse_cons (x : T) (l : list T) : append x l = reverse (x :: reverse l) :=
|
||||
-- list_induction_on l (refl _)
|
||||
-- (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 _)
|
||||
|
||||
|
||||
-- -- 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 (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 (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 `∈` := mem
|
||||
|
||||
-- -- TODO: constructively, equality is stronger. Use that?
|
||||
-- theorem mem_nil (x : T) : 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 _
|
||||
|
||||
-- theorem mem_concat_imp_or (x : T) (s t : list T) : x ∈ s ++ t → x ∈ s ∨ x ∈ t :=
|
||||
-- list_induction_on s or_inr
|
||||
-- (take y s,
|
||||
-- assume IH : x ∈ s ++ t → x ∈ s ∨ x ∈ t,
|
||||
-- assume H1 : x ∈ (y :: s) ++ t,
|
||||
-- have H2 : x = y ∨ x ∈ s ++ t, from H1,
|
||||
-- have H3 : x = y ∨ x ∈ s ∨ x ∈ t, from or_imp_or_right H2 IH,
|
||||
-- iff_elim_right or_assoc H3)
|
||||
|
||||
-- theorem mem_or_imp_concat (x : T) (s t : list T) : x ∈ s ∨ x ∈ t → x ∈ s ++ t :=
|
||||
-- list_induction_on s
|
||||
-- (take H, or_elim H (false_elim _) (assume H, H))
|
||||
-- (take y s,
|
||||
-- assume IH : x ∈ s ∨ x ∈ t → x ∈ s ++ t,
|
||||
-- assume H : x ∈ y :: s ∨ x ∈ t,
|
||||
-- or_elim H
|
||||
-- (assume H1,
|
||||
-- or_elim H1
|
||||
-- (take H2 : x = y, or_inl H2)
|
||||
-- (take H2 : x ∈ s, or_inr (IH (or_inl H2))))
|
||||
-- (assume H1 : x ∈ t, or_inr (IH (or_inr H1))))
|
||||
|
||||
-- theorem mem_concat (x : T) (s t : list T) : x ∈ s ++ t ↔ x ∈ s ∨ x ∈ t
|
||||
-- := iff_intro (mem_concat_imp_or _ _ _) (mem_or_imp_concat _ _ _)
|
||||
|
||||
-- theorem mem_split (x : T) (l : list T) : x ∈ l → ∃s t : list T, l = s ++ (x :: t) :=
|
||||
-- list_induction_on l
|
||||
-- (take H : x ∈ nil, false_elim _ (iff_elim_left (mem_nil x) H))
|
||||
-- (take y l,
|
||||
-- assume IH : x ∈ l → ∃s t : list T, l = s ++ (x :: t),
|
||||
-- assume H : x ∈ y :: l,
|
||||
-- or_elim H
|
||||
-- (assume H1 : x = y,
|
||||
-- exists_intro nil
|
||||
-- (exists_intro l (subst H1 (refl _))))
|
||||
-- (assume H1 : x ∈ l,
|
||||
-- obtain s (H2 : ∃t : list T, l = s ++ (x :: t)), from IH H1,
|
||||
-- obtain t (H3 : l = s ++ (x :: t)), from H2,
|
||||
-- have H4 : y :: l = (y :: s) ++ (x :: t),
|
||||
-- from subst H3 (refl (y :: l)),
|
||||
-- exists_intro _ (exists_intro _ H4)))
|
||||
|
||||
-- -- Find
|
||||
-- -- ----
|
||||
|
||||
-- -- to do this: need decidability of = for nat
|
||||
-- -- definition find (x : T) : list T → nat
|
||||
-- -- := list_rec 0 (fun y l b, if x = y then 0 else succ b)
|
||||
|
||||
-- -- theorem find_nil (f : T) : find f nil = 0
|
||||
-- -- :=refl _
|
||||
|
||||
-- -- theorem find_cons (x y : T) (l : list T) : find x (cons y l) =
|
||||
-- -- if x = y then 0 else succ (find x l)
|
||||
-- -- := refl _
|
||||
|
||||
-- -- theorem not_mem_find (l : list T) (x : T) : ¬ mem x l → find x l = length l
|
||||
-- -- :=
|
||||
-- -- @list_induction_on T (λl, ¬ mem x l → find x l = length l) l
|
||||
-- -- -- list_induction_on l
|
||||
-- -- (assume P1 : ¬ mem x nil,
|
||||
-- -- show find x nil = length nil, from
|
||||
-- -- calc
|
||||
-- -- find x nil = 0 : find_nil _
|
||||
-- -- ... = length nil : by simp)
|
||||
-- -- (take y l,
|
||||
-- -- assume IH : ¬ (mem x l) → find x l = length l,
|
||||
-- -- assume P1 : ¬ (mem x (cons y l)),
|
||||
-- -- have P2 : ¬ (mem x l ∨ (y = x)), from subst P1 (mem_cons _ _ _),
|
||||
-- -- have P3 : ¬ (mem x l) ∧ (y ≠ x),from subst P2 (not_or _ _),
|
||||
-- -- have P4 : x ≠ y, from ne_symm (and_elim_right P3),
|
||||
-- -- calc
|
||||
-- -- find x (cons y l) = succ (find x l) :
|
||||
-- -- trans (find_cons _ _ _) (not_imp_if_eq P4 _ _)
|
||||
-- -- ... = succ (length l) : {IH (and_elim_left P3)}
|
||||
-- -- ... = length (cons y l) : symm (length_cons _ _))
|
||||
|
||||
-- -- nth element
|
||||
-- -- -----------
|
||||
|
||||
-- definition nth (x0 : T) (l : list T) (n : ℕ) : T :=
|
||||
-- nat_rec (λl, head x0 l) (λm f l, f (tail l)) n l
|
||||
|
||||
-- theorem nth_zero (x0 : T) (l : list T) : nth x0 l 0 = head x0 l := refl _
|
||||
|
||||
-- theorem nth_succ (x0 : T) (l : list T) (n : ℕ) : nth x0 l (succ n) = nth x0 (tail l) n := refl _
|
||||
|
||||
end sc_vec
|
||||
infixl `++`:65 := concat
|
||||
end vec
|
||||
Loading…
Reference in a new issue