119 lines
4.2 KiB
Text
119 lines
4.2 KiB
Text
import logic data.nat.basic data.prod data.unit
|
||
open nat prod
|
||
|
||
inductive vector (A : Type) : nat → Type :=
|
||
vnil {} : vector A zero,
|
||
vcons : Π {n : nat}, A → vector A n → vector A (succ n)
|
||
|
||
namespace vector
|
||
print definition no_confusion
|
||
infixr `::` := vcons
|
||
|
||
theorem vcons.inj₁ {A : Type} {n : nat} (a₁ a₂ : A) (v₁ v₂ : vector A n) : vcons a₁ v₁ = vcons a₂ v₂ → a₁ = a₂ :=
|
||
begin
|
||
intro h, apply (no_confusion h), intros, assumption
|
||
end
|
||
|
||
theorem vcons.inj₂ {A : Type} {n : nat} (a₁ a₂ : A) (v₁ v₂ : vector A n) : vcons a₁ v₁ = vcons a₂ v₂ → v₁ = v₂ :=
|
||
begin
|
||
intro h, apply heq.to_eq, apply (no_confusion h), intros, eassumption,
|
||
end
|
||
|
||
section
|
||
universe variables l₁ l₂
|
||
variable {A : Type.{l₁}}
|
||
variable {C : Π (n : nat), vector A n → Type.{l₂+1}}
|
||
definition brec_on {n : nat} (v : vector A n) (H : Π (n : nat) (v : vector A n), @below A C n v → C n v) : C n v :=
|
||
have general : C n v × @below A C n v, from
|
||
rec_on v
|
||
(pair (H zero vnil unit.star) unit.star)
|
||
(λ (n₁ : nat) (a₁ : A) (v₁ : vector A n₁) (r₁ : C n₁ v₁ × @below A C n₁ v₁),
|
||
have b : @below A C _ (vcons a₁ v₁), from
|
||
r₁,
|
||
have c : C (succ n₁) (vcons a₁ v₁), from
|
||
H (succ n₁) (vcons a₁ v₁) b,
|
||
pair c b),
|
||
pr₁ general
|
||
end
|
||
|
||
check brec_on
|
||
|
||
definition bw := @below
|
||
|
||
definition sum {n : nat} (v : vector nat n) : nat :=
|
||
brec_on v (λ (n : nat) (v : vector nat n),
|
||
cases_on v
|
||
(λ (B : bw vnil), zero)
|
||
(λ (n₁ : nat) (a : nat) (v₁ : vector nat n₁) (B : bw (vcons a v₁)),
|
||
a + pr₁ B))
|
||
|
||
example : sum (10 :: 20 :: vnil) = 30 :=
|
||
rfl
|
||
|
||
definition addk {n : nat} (v : vector nat n) (k : nat) : vector nat n :=
|
||
brec_on v (λ (n : nat) (v : vector nat n),
|
||
cases_on v
|
||
(λ (B : bw vnil), vnil)
|
||
(λ (n₁ : nat) (a₁ : nat) (v₁ : vector nat n₁) (B : bw (vcons a₁ v₁)),
|
||
vcons (a₁+k) (pr₁ B)))
|
||
|
||
example : addk (1 :: 2 :: vnil) 3 = 4 :: 5 :: vnil :=
|
||
rfl
|
||
|
||
definition append.{l} {A : Type.{l+1}} {n m : nat} (w : vector A m) (v : vector A n) : vector A (n + m) :=
|
||
brec_on w (λ (n : nat) (w : vector A n),
|
||
cases_on w
|
||
(λ (B : bw vnil), v)
|
||
(λ (n₁ : nat) (a₁ : A) (v₁ : vector A n₁) (B : bw (vcons a₁ v₁)),
|
||
vcons a₁ (pr₁ B)))
|
||
|
||
example : append (1 :: 2 :: vnil) (3 :: vnil) = 1 :: 2 :: 3 :: vnil :=
|
||
rfl
|
||
|
||
definition head {A : Type} {n : nat} (v : vector A (succ n)) : A :=
|
||
cases_on v
|
||
(λ H : succ n = 0, nat.no_confusion H)
|
||
(λn' h t (H : succ n = succ n'), h)
|
||
rfl
|
||
|
||
definition tail {A : Type} {n : nat} (v : vector A (succ n)) : vector A n :=
|
||
@cases_on A (λn' v, succ n = n' → vector A (pred n')) (succ n) v
|
||
(λ H : succ n = 0, nat.no_confusion H)
|
||
(λ (n' : nat) (h : A) (t : vector A n') (H : succ n = succ n'),
|
||
t)
|
||
rfl
|
||
|
||
definition add {n : nat} (w v : vector nat n) : vector nat n :=
|
||
@brec_on nat (λ (n : nat) (v : vector nat n), vector nat n → vector nat n) n w
|
||
(λ (n : nat) (w : vector nat n),
|
||
cases_on w
|
||
(λ (B : bw vnil) (w : vector nat zero), vnil)
|
||
(λ (n₁ : nat) (a₁ : nat) (v₁ : vector nat n₁) (B : bw (vcons a₁ v₁)) (v : vector nat (succ n₁)),
|
||
vcons (a₁ + head v) (pr₁ B (tail v)))) v
|
||
|
||
example : add (1 :: 2 :: vnil) (3 :: 5 :: vnil) = 4 :: 7 :: vnil :=
|
||
rfl
|
||
|
||
definition map {A B C : Type} {n : nat} (f : A → B → C) (w : vector A n) (v : vector B n) : vector C n :=
|
||
let P := λ (n : nat) (v : vector A n), vector B n → vector C n in
|
||
@brec_on A P n w
|
||
(λ (n : nat) (w : vector A n),
|
||
begin
|
||
cases w with (n₁, h₁, t₁),
|
||
show @below A P zero vnil → vector B zero → vector C zero, from
|
||
λ b v, vnil,
|
||
show @below A P (succ n₁) (h₁ :: t₁) → vector B (succ n₁) → vector C (succ n₁), from
|
||
λ b v,
|
||
begin
|
||
cases v with (n₂, h₂, t₂),
|
||
have r : vector B n₂ → vector C n₂, from pr₁ b,
|
||
(f h₁ h₂) :: r t₂,
|
||
end
|
||
end) v
|
||
|
||
example : map num.add (1 :: 2 :: vnil) (3 :: 5 :: vnil) = 4 :: 7 :: vnil :=
|
||
rfl
|
||
|
||
print definition map
|
||
|
||
end vector
|