set_option pp.notation false
constant A    : Type
constants a b : A
constant P    : A → Type
constant H₁   : a = a
constant H₂   : P a
constant H₃   : a = b
constant f {A : Type} (a : A) : a = a
#reduce (eq.rec H₂ (@f A a) : P a)
#reduce (eq.rec H₂ H₁ : P a)
#reduce (eq.rec H₂ H₃ : P b)
#reduce (eq.rec H₂ (@eq.refl A a) : P a)
-- eval λ (A : Type) (a b : A) (H₁ : a = a) (P : A → Prop) (H₂ : P a) (H₃ : a = a) (c : A), eq.rec (eq.rec H₂ H₁) H₃
#check @eq.rec A a P H₂ a
#check λ H : a = a, H₂
inductive to_type {B : Type} : B → Type
| mk : Π (b : B), to_type b

noncomputable definition tst1 : to_type (λ H : a = a, H₂) := to_type.mk (@eq.rec A a P H₂ a)
#check to_type.mk(λ H : a = a, H₂)
#check to_type.mk(@eq.rec A a P H₂ a)
#check to_type.mk(λ H : a = a, H₂) = to_type.mk(@eq.rec A a P H₂ a)
-- #check to_type.mk(eq.rec H₂ H₁) = to_type.mk(H₂)
-- #check to_type.mk(eq.rec H₂ (f a)) = to_type.mk(H₂)