Definition eq_sind_r :
forall (A:Type) (x:A) (P:A -> SProp), P x -> forall y:A, y = x -> P y.
Register f_equal2 as core.eq.congr2.

(* Specifications: *)
Theorem f_equal3 :
forall (A1 A2 A3 B:Type) (f:A1 -> A2 -> A3 -> B) (x1 y1:A1)
(x2 y2:A2) (x3 y3:A3),
x1 = y1 -> x2 = y2 -> x3 = y3 -> f x1 x2 x3 = f y1 y2 y3.
Theorem f_equal4 :
forall (A1 A2 A3 A4 B:Type) (f:A1 -> A2 -> A3 -> A4 -> B)
(x1 y1:A1) (x2 y2:A2) (x3 y3:A3) (x4 y4:A4),
x1 = y1 -> x2 = y2 -> x3 = y3 -> x4 = y4 -> f x1 x2 x3 x4 = f y1 y2 y3 y4.
Theorem f_equal5 :
forall (A1 A2 A3 A4 A5 B:Type) (f:A1 -> A2 -> A3 -> A4 -> A5 -> B)
(x1 y1:A1) (x2 y2:A2) (x3 y3:A3) (x4 y4:A4) (x5 y5:A5),
x1 = y1 ->
x2 = y2 ->
x3 = y3 -> x4 = y4 -> x5 = y5 -> f x1 x2 x3 x4 x5 = f y1 y2 y3 y4 y5.
Theorem f_equal_compose A B C (a b:A) (f:A->B) (g:B->C) (e:a=b) :
f_equal g (f_equal f e) = f_equal (fun a => g (f a)) e.
Lemma eq_trans_map_distr A B x y z (f:A->B) (e:x=y) (e':y=z) : f_equal f (eq_trans e e') = eq_trans (f_equal f e) (f_equal f e').
Lemma rew_const A P (x y:A) (e:x=y) (k:P) :
rew [fun _ => P] e in k = k.