lean2/tests/lean/run/662.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
definition count_vars : Π {t : type}, @term (λ x, unit) t -> nat
| count_vars (Var _) := 1
| count_vars (Const _) := 0
| count_vars (Plus e1 e2) := count_vars e1 + count_vars e2
| count_vars (Abs e1) := count_vars (e1 star)
| count_vars (App e1 e2) := count_vars e1 + count_vars e2
| count_vars (Let e1 e2) := count_vars e1 + count_vars (e2 star)
definition var (t : type) : @term (λ x, unit) t :=
Var star
example : count_vars (App (App (var (Func Nat (Func Nat Nat))) (var Nat)) (var Nat)) = 3 :=
rfl
definition count_vars2 : Π {t : type}, @term (λ x, unit) t -> nat
| _ (Var _) := 1
| _ (Const _) := 0
| _ (Plus e1 e2) := count_vars2 e1 + count_vars2 e2
| _ (Abs e1) := count_vars2 (e1 star)
| _ (App e1 e2) := count_vars2 e1 + count_vars2 e2
| _ (Let e1 e2) := count_vars2 e1 + count_vars2 (e2 star)