Require Export ZBase.

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

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

Hint Rewrite opp_0 : nz.

Theorem add_pred_l : forall n m, P n + m == P (n + m).

Theorem add_pred_r : forall n m, n + P m == P (n + m).

Theorem add_opp_r : forall n m, n + (- m) == n - m.

Theorem sub_0_l : forall n, 0 - n == - n.

Theorem sub_succ_l : forall n m, S n - m == S (n - m).

Theorem sub_pred_l : forall n m, P n - m == P (n - m).

Theorem sub_pred_r : forall n m, n - (P m) == S (n - m).

Theorem opp_pred : forall n, - (P n) == S (- n).

Theorem sub_diag : forall n, n - n == 0.

Theorem add_opp_diag_l : forall n, - n + n == 0.

Theorem add_opp_diag_r : forall n, n + (- n) == 0.

Theorem add_opp_l : forall n m, - m + n == n - m.

Theorem add_sub_assoc : forall n m p, n + (m - p) == (n + m) - p.

Theorem opp_involutive : forall n, - (- n) == n.

Theorem opp_add_distr : forall n m, - (n + m) == - n + (- m).

Theorem opp_sub_distr : forall n m, - (n - m) == - n + m.

Theorem opp_inj : forall n m, - n == - m -> n == m.

Theorem opp_inj_wd : forall n m, - n == - m <-> n == m.

Theorem eq_opp_l : forall n m, - n == m <-> n == - m.

Theorem eq_opp_r : forall n m, n == - m <-> - n == m.

Theorem sub_add_distr : forall n m p, n - (m + p) == (n - m) - p.

Theorem sub_sub_distr : forall n m p, n - (m - p) == (n - m) + p.

Theorem sub_opp_l : forall n m, - n - m == - m - n.

Theorem sub_opp_r : forall n m, n - (- m) == n + m.

Theorem add_sub_swap : forall n m p, n + m - p == n - p + m.

Theorem sub_cancel_l : forall n m p, n - m == n - p <-> m == p.

Theorem sub_cancel_r : forall n m p, n - p == m - p <-> n == m.

The next several theorems are devoted to moving terms from one side of an equation to the other. The name contains the operation in the original equation (add or sub) and the indication whether the left or right term is moved.

Theorem add_move_l : forall n m p, n + m == p <-> m == p - n.

Theorem add_move_r : forall n m p, n + m == p <-> n == p - m.

The two theorems above do not allow rewriting subformulas of the form n - m == p to n == p + m since subtraction is in the right-hand side of the equation. Hence the following two theorems.

Theorem sub_move_l : forall n m p, n - m == p <-> - m == p - n.

Theorem sub_move_r : forall n m p, n - m == p <-> n == p + m.

Theorem add_move_0_l : forall n m, n + m == 0 <-> m == - n.

Theorem add_move_0_r : forall n m, n + m == 0 <-> n == - m.

Theorem sub_move_0_l : forall n m, n - m == 0 <-> - m == - n.

Theorem sub_move_0_r : forall n m, n - m == 0 <-> n == m.

The following section is devoted to cancellation of like terms. The name includes the first operator and the position of the term being canceled.

Theorem add_simpl_l : forall n m, n + m - n == m.

Theorem add_simpl_r : forall n m, n + m - m == n.

Theorem sub_simpl_l : forall n m, - n - m + n == - m.

Theorem sub_simpl_r : forall n m, n - m + m == n.

Theorem sub_add : forall n m, m - n + n == m.

Now we have two sums or differences; the name includes the two operators and the position of the terms being canceled

Theorem add_add_simpl_l_l : forall n m p, (n + m) - (n + p) == m - p.

Theorem add_add_simpl_l_r : forall n m p, (n + m) - (p + n) == m - p.

Theorem add_add_simpl_r_l : forall n m p, (n + m) - (m + p) == n - p.

Theorem add_add_simpl_r_r : forall n m p, (n + m) - (p + m) == n - p.

Theorem sub_add_simpl_r_l : forall n m p, (n - m) + (m + p) == n + p.

Theorem sub_add_simpl_r_r : forall n m p, (n - m) + (p + m) == n + p.

Of course, there are many other variants