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Require Import Coq.ZArith.ZArith.
Require Import Coq.micromega.Lia.
Require Import Crypto.Util.ZUtil.Hints.Core.
Local Open Scope Z_scope.
Module Z.
Definition opp_distr_if (b : bool) x y : -(if b then x else y) = if b then -x else -y.
Proof. destruct b; reflexivity. Qed.
Hint Rewrite opp_distr_if : push_Zopp.
Hint Rewrite <- opp_distr_if : pull_Zopp.
Lemma mul_r_distr_if (b : bool) x y z : z * (if b then x else y) = if b then z * x else z * y.
Proof. destruct b; reflexivity. Qed.
Hint Rewrite mul_r_distr_if : push_Zmul.
Hint Rewrite <- mul_r_distr_if : pull_Zmul.
Lemma mul_l_distr_if (b : bool) x y z : (if b then x else y) * z = if b then x * z else y * z.
Proof. destruct b; reflexivity. Qed.
Hint Rewrite mul_l_distr_if : push_Zmul.
Hint Rewrite <- mul_l_distr_if : pull_Zmul.
Lemma add_r_distr_if (b : bool) x y z : z + (if b then x else y) = if b then z + x else z + y.
Proof. destruct b; reflexivity. Qed.
Hint Rewrite add_r_distr_if : push_Zadd.
Hint Rewrite <- add_r_distr_if : pull_Zadd.
Lemma add_l_distr_if (b : bool) x y z : (if b then x else y) + z = if b then x + z else y + z.
Proof. destruct b; reflexivity. Qed.
Hint Rewrite add_l_distr_if : push_Zadd.
Hint Rewrite <- add_l_distr_if : pull_Zadd.
Lemma sub_r_distr_if (b : bool) x y z : z - (if b then x else y) = if b then z - x else z - y.
Proof. destruct b; reflexivity. Qed.
Hint Rewrite sub_r_distr_if : push_Zsub.
Hint Rewrite <- sub_r_distr_if : pull_Zsub.
Lemma sub_l_distr_if (b : bool) x y z : (if b then x else y) - z = if b then x - z else y - z.
Proof. destruct b; reflexivity. Qed.
Hint Rewrite sub_l_distr_if : push_Zsub.
Hint Rewrite <- sub_l_distr_if : pull_Zsub.
Lemma div_r_distr_if (b : bool) x y z : z / (if b then x else y) = if b then z / x else z / y.
Proof. destruct b; reflexivity. Qed.
Hint Rewrite div_r_distr_if : push_Zdiv.
Hint Rewrite <- div_r_distr_if : pull_Zdiv.
Lemma div_l_distr_if (b : bool) x y z : (if b then x else y) / z = if b then x / z else y / z.
Proof. destruct b; reflexivity. Qed.
Hint Rewrite div_l_distr_if : push_Zdiv.
Hint Rewrite <- div_l_distr_if : pull_Zdiv.
End Z.
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