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