lean2/tests/lean/hott/rw_eta.hlean

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open eq is_equiv funext
constant f : nat → nat → nat
example : (λ x y : nat, f x y) = f :=
rfl
namespace hide
variables {A : Type} {B : A → Type} {C : Πa, B a → Type}
definition homotopy2 [reducible] (f g : Πa b, C a b) : Type :=
Πa b, f a b = g a b
notation f `2`:50 g := homotopy2 f g
definition eq_of_homotopy2 {f g : Πa b, C a b} (H : f 2 g) : f = g :=
eq_of_homotopy (λa, eq_of_homotopy (H a))
definition apD100 {f g : Πa b, C a b} (p : f = g) : f 2 g :=
λa b, apd10 (apd10 p a) b
local attribute eq_of_homotopy [reducible]
definition foo (f g : Πa b, C a b) (H : f 2 g) (a : A)
: apD100 (eq_of_homotopy2 H) a = H a :=
begin
esimp [apD100, eq_of_homotopy2, eq_of_homotopy],
rewrite (right_inv apd10 (λ(a : A), eq_of_homotopy (H a))),
apply (right_inv apd10)
end
end hide