lean2/tests/lean/run/univ_problem.lean

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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
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
print "====================="
definition append {A : Type} {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 : below vnil), v)
(λ (n₁ : nat) (a₁ : A) (v₁ : vector A n₁) (B : below (vcons a₁ v₁)),
vcons a₁ (pr₁ B)))
exit
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
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