intros. elim H; clear H q r; auto.
qed.
+theorem nplus_comm: \forall p, q, x. (p + q == x) \to
+ \forall y. (q + p == y) \to x = y.
+ 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.
+qed.
+
theorem nplus_comm_rew: \forall p,q,r. (p + q == r) \to q + p == r.
intros. elim H; clear H q r; auto.
qed.
(* Corollaries of nonoidal properties ***************************************)
+theorem nplus_comm_1: \forall p1, q, r1. (p1 + q == r1) \to
+ \forall p2, r2. (p2 + q == r2) \to
+ \forall x. (p2 + r1 == x) \to
+ \forall y. (p1 + r2 == y) \to
+ x = y.
+ intros 4. elim H; clear H q r1;
+ [ lapply linear nplus_inv_zero_2 to H1
+ | 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.
+qed.
+
theorem nplus_comm_1_rew: \forall p1,q,r1. (p1 + q == r1) \to
\forall p2,r2. (p2 + q == r2) \to
\forall s. (p1 + r2 == s) \to (p2 + r1 == s).
intros 4. elim H; clear H q r1;
- [ lapply linear nplus_gen_zero_2 to H1. subst
- | lapply linear nplus_gen_succ_2 to H3. decompose. subst.
- lapply linear nplus_gen_succ_2 to H4. decompose. subst
+ [ 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.
qed.
theorem nplus_shift_succ_sx: \forall p,q,r.
(p + (succ q) == r) \to (succ p) + q == r.
intros.
- lapply linear nplus_gen_succ_2 to H as H0.
+ lapply linear nplus_inv_succ_2 to H as H0.
decompose. subst. auto new timeout=100.
qed.
theorem nplus_shift_succ_dx: \forall p,q,r.
((succ p) + q == r) \to p + (succ q) == r.
intros.
- lapply linear nplus_gen_succ_1 to H as H0.
+ lapply linear nplus_inv_succ_1 to H as H0.
decompose. subst. auto new timeout=100.
qed.
\forall q2,r2. (r1 + q2 == r2) \to
\exists q. (q1 + q2 == q) \land p + q == r2.
intros 2; elim q1; clear q1; intros;
- [ lapply linear nplus_gen_zero_2 to H as H0.
+ [ lapply linear nplus_inv_zero_2 to H as H0.
subst.
- | lapply linear nplus_gen_succ_2 to H1 as H0.
+ | lapply linear nplus_inv_succ_2 to H1 as H0.
decompose. subst.
- lapply linear nplus_gen_succ_1 to H2 as H0.
+ lapply linear nplus_inv_succ_1 to H2 as H0.
decompose. subst.
lapply linear H to H4, H3 as H0.
decompose.
\forall p2,r2. (p2 + r1 == r2) \to
\exists p. (p1 + p2 == p) \land p + q == r2.
intros 2; elim q; clear q; intros;
- [ lapply linear nplus_gen_zero_2 to H as H0.
+ [ lapply linear nplus_inv_zero_2 to H as H0.
subst
- | lapply linear nplus_gen_succ_2 to H1 as H0.
+ | lapply linear nplus_inv_succ_2 to H1 as H0.
decompose. subst.
- lapply linear nplus_gen_succ_2 to H2 as H0.
+ lapply linear nplus_inv_succ_2 to H2 as H0.
decompose. subst.
lapply linear H to H4, H3 as H0.
decompose.
]; apply ex_intro; [| auto new timeout=100 || auto new timeout=100 ]. (**)
qed.
*)
-