int_mul_succ_r@{} (x y : Int) : x * y.+1 = x + x * y. Proof. induction x as [|x IHx|x IHx] in y |- *. - reflexivity. - rewrite 2 int_mul_succ_l. rewrite 2 int_add_succ_l. f_ap. rewrite IHx. rewrite !int_add_assoc; f_ap. by rewrite int_add_comm. - rewrite 2 int_mul_pred_l. rewrite int_neg_succ. rewrite int_mul_neg_l. rewrite 2 int_add_pred_l. f_ap. rewrite <- int_mul_neg_l. rewrite IHx. rewrite !int_add_assoc; f_ap. by rewrite int_add_comm. Defined.
Require Import Basics.Overture Basics.Nat Basics.Tactics Basics.Decidable Basics.Equivalences Basics.PathGroupoids Types.Paths Types.Universe. Require Import Spaces.Nat.Core.