(* Monoidal properties ******************************************************)
theorem nplus_zero_1: \forall q. zero + q == q.
- intros. elim q; clear q; auto.
+ intros. elim q; clear q; autobatch.
qed.
-theorem nplus_succ_1: \forall p,q,r. NPlus p q r \to
+theorem nplus_succ_1: \forall p,q,r. (p + q == r) \to
(succ p) + q == (succ r).
- intros. elim H; clear H q r; auto.
+ intros. elim H; clear H q r; autobatch.
qed.
theorem nplus_comm: \forall p, q, x. (p + q == x) \to
intros 4; elim H; clear H q x;
[ lapply linear nplus_inv_zero_1 to H1
| lapply linear nplus_inv_succ_1 to H3. decompose
- ]; subst; auto.
+ ]; subst; autobatch.
qed.
theorem nplus_comm_rew: \forall p,q,r. (p + q == r) \to q + p == r.
- intros. elim H; clear H q r; auto.
+ intros. elim H; clear H q r; autobatch.
qed.
+theorem nplus_ass: \forall p1, p2, r1. (p1 + p2 == r1) \to
+ \forall p3, s1. (r1 + p3 == s1) \to
+ \forall r3. (p2 + p3 == r3) \to
+ \forall s3. (p1 + r3 == s3) \to s1 = s3.
+ intros 4. elim H; clear H p2 r1;
+ [ lapply linear nplus_inv_zero_1 to H2. subst.
+ lapply nplus_mono to H1, H3. subst. autobatch
+ | lapply linear nplus_inv_succ_1 to H3. decompose. subst.
+ lapply linear nplus_inv_succ_1 to H4. decompose. subst.
+ lapply linear nplus_inv_succ_2 to H5. decompose. subst. autobatch
+ ].
+qed.
+
(* Corollaries of functional properties **************************************)
theorem nplus_inj_2: \forall p, q1, r. (p + q1 == r) \to
\forall q2. (p + q2 == r) \to q1 = q2.
- intros. auto.
+ intros. autobatch.
qed.
(* Corollaries of nonoidal properties ***************************************)
| lapply linear nplus_inv_succ_2 to H3.
lapply linear nplus_inv_succ_2 to H4. decompose. subst.
lapply linear nplus_inv_succ_2 to H5. decompose
- ]; subst; auto.
+ ]; subst; autobatch.
qed.
theorem nplus_comm_1_rew: \forall p1,q,r1. (p1 + q == r1) \to
[ lapply linear nplus_inv_zero_2 to H1. subst
| lapply linear nplus_inv_succ_2 to H3. decompose. subst.
lapply linear nplus_inv_succ_2 to H4. decompose. subst
- ]; auto.
+ ]; autobatch.
qed.
(*