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 end vector