Library Coq.Numbers.Integer.Abstract.ZMul

Require Export ZAdd.

Module ZMulProp (Import Z : ZAxiomsMiniSig').
Include ZAddProp Z.

A note on naming: right (correspondingly, left) distributivity happens when the sum is multiplied by a number on the right (left), not when the sum itself is the right (left) factor in the product (see planetmath.org and mathworld.wolfram.com). In the old library BinInt, distributivity over subtraction was named correctly, but distributivity over addition was named incorrectly. The names in Isabelle/HOL library are also incorrect.

Theorems that are either not valid on N or have different proofs on N and Z

Theorem mul_pred_r : forall n m, n * (P m ) == n * m - n.
Proof.
intros n m.
rewrite <- (succ_pred m) at 2.
now rewrite mul_succ_r, <- add_sub_assoc, sub_diag, add_0_r.
Qed.

Theorem mul_pred_l : forall n m, (P n ) * m == n * m - m.
Proof.
intros n m; rewrite (mul_comm (P n) m), (mul_comm n m). apply mul_pred_r.
Qed.

Theorem mul_opp_l : forall n m, (- n ) * m == - (n * m ).
Proof.
intros n m. apply add_move_0_r.
now rewrite <- mul_add_distr_r, add_opp_diag_l, mul_0_l.
Qed.

Theorem mul_opp_r : forall n m, n * (- m ) == - (n * m ).
Proof.
intros n m; rewrite (mul_comm n (- m)), (mul_comm n m); apply mul_opp_l.
Qed.

Theorem mul_opp_opp : forall n m, (- n ) * (- m ) == n * m.
Proof.
intros n m; now rewrite mul_opp_l, mul_opp_r, opp_involutive.
Qed.

Theorem mul_opp_comm : forall n m, (- n ) * m == n * (- m ).
Proof.
intros n m. now rewrite mul_opp_l, <- mul_opp_r.
Qed.

Theorem mul_sub_distr_l : forall n m p, n * (m - p ) == n * m - n * p.
Proof.
intros n m p. do 2 rewrite <- add_opp_r. rewrite mul_add_distr_l.
now rewrite mul_opp_r.
Qed.

Theorem mul_sub_distr_r : forall n m p, (n - m ) * p == n * p - m * p.
Proof.
intros n m p; rewrite (mul_comm (n - m) p), (mul_comm n p), (mul_comm m p);
now apply mul_sub_distr_l.
Qed.

End ZMulProp.