lean2/tests/lean/run/662b.lean

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open nat
inductive type : Type :=
| Nat : type
| Func : type → type → type
open type
section var
variable {var : type → Type}
inductive term : type → Type :=
| Var : ∀ {t}, var t → term t
| Const : nat → term Nat
| Plus : term Nat → term Nat → term Nat
| Abs : ∀ {dom ran}, (var dom → term ran) → term (Func dom ran)
| App : ∀ {dom ran}, term (Func dom ran) → term dom → term ran
| Let : ∀ {t1 t2}, term t1 → (var t1 → term t2) → term t2
end var
open term
definition Term t := Π (var : type → Type), @term var t
open unit
-- Define count_vars using tactics
definition count_vars1 {t : type} (T : @term (λ x, unit) t) : nat :=
begin
induction T,
{exact 1},
{exact 0},
{exact v_0 + v_1},
{exact v_0 star},
{exact v_0 + v_1},
{exact v_0 + v_1 star},
end
-- Define count_vars using recursor
definition count_vars2 {t : type} (T : @term (λ x, unit) t) : nat :=
term.rec_on T
(λ t u, 1)
(λ n, 0)
(λ T₁ T₂ n₁ n₂, n₁ + n₂)
(λ d r f n, n star)
(λ d r T₁ T₂ n₁ n₂, n₁ + n₂)
(λ t₁ t₂ T₁ T₂ n₁ n₂, n₁ + n₂ star)
definition var (t : type) : @term (λ x, unit) t :=
Var star
example : count_vars1 (App (App (var (Func Nat (Func Nat Nat))) (var Nat)) (var Nat)) = 3 :=
rfl
example : count_vars2 (App (App (var (Func Nat (Func Nat Nat))) (var Nat)) (var Nat)) = 3 :=
rfl