include "NPlus/fwd.ma".
theorem nplus_zero_1: \forall q. zero + q == q.
- intros. elim q; clear q; auto.
+ intros. elim q; clear q; auto new timeout=100.
qed.
theorem nplus_succ_1: \forall p,q,r. NPlus p q r \to
- (succ p) + q == (succ r).
+ (succ p) + q == (succ r).
intros 2. elim q; clear q;
[ lapply linear nplus_gen_zero_2 to H as H0.
- rewrite > H0. clear H0 p
+ subst
| lapply linear nplus_gen_succ_2 to H1 as H0.
decompose.
- rewrite > H2. clear H2 r
- ]; auto.
+ subst
+ ]; auto new timeout=100.
qed.
theorem nplus_sym: \forall p,q,r. (p + q == r) \to q + p == r.
intros 2. elim q; clear q;
[ lapply linear nplus_gen_zero_2 to H as H0.
- rewrite > H0. clear H0 p
+ subst
| lapply linear nplus_gen_succ_2 to H1 as H0.
decompose.
- rewrite > H2. clear H2 r
- ]; auto.
+ subst
+ ]; auto new timeout=100.
qed.
theorem nplus_shift_succ_sx: \forall p,q,r.
- (p + (succ q) == r) \to (succ p) + q == r.
+ (p + (succ q) == r) \to (succ p) + q == r.
intros.
lapply linear nplus_gen_succ_2 to H as H0.
- decompose.
- rewrite > H1. clear H1 r.
- auto.
+ 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.
+ ((succ p) + q == r) \to p + (succ q) == r.
intros.
lapply linear nplus_gen_succ_1 to H as H0.
- decompose.
- rewrite > H1. clear H1 r.
- auto.
+ decompose. subst. auto new timeout=100.
qed.
theorem nplus_trans_1: \forall p,q1,r1. (p + q1 == r1) \to
- \forall q2,r2. (r1 + q2 == r2) \to
- \exists q. (q1 + q2 == q) \land p + q == r2.
+ \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.
- rewrite > H0. clear H0 p
+ subst.
| lapply linear nplus_gen_succ_2 to H1 as H0.
- decompose.
- rewrite > H3. rewrite > H3 in H2. clear H3 r1.
+ decompose. subst.
lapply linear nplus_gen_succ_1 to H2 as H0.
- decompose.
- rewrite > H2. clear H2 r2.
+ decompose. subst.
lapply linear H to H4, H3 as H0.
decompose.
- ]; apply ex_intro; [| auto || auto ]. (**)
+ ]; apply ex_intro; [| auto new timeout=100 || auto new timeout=100 ]. (**)
qed.
theorem nplus_trans_2: \forall p1,q,r1. (p1 + q == r1) \to
- \forall p2,r2. (p2 + r1 == r2) \to
- \exists p. (p1 + p2 == p) \land p + q == r2.
+ \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.
- rewrite > H0. clear H0 p1
+ subst
| lapply linear nplus_gen_succ_2 to H1 as H0.
- decompose.
- rewrite > H3. rewrite > H3 in H2. clear H3 r1.
+ decompose. subst.
lapply linear nplus_gen_succ_2 to H2 as H0.
- decompose.
- rewrite > H2. clear H2 r2.
+ decompose. subst.
lapply linear H to H4, H3 as H0.
decompose.
- ]; apply ex_intro; [| auto || auto ]. (**)
+ ]; apply ex_intro; [| auto new timeout=100 || auto new timeout=100 ]. (**)
qed.
theorem nplus_conf: \forall p,q,r1. (p + q == r1) \to
- \forall r2. (p + q == r2) \to r1 = r2.
+ \forall r2. (p + q == r2) \to r1 = r2.
intros 2. elim q; clear q; intros;
[ lapply linear nplus_gen_zero_2 to H as H0.
- rewrite > H0 in H1. clear H0 p
+ subst
| lapply linear nplus_gen_succ_2 to H1 as H0.
- decompose.
- rewrite > H3. clear H3 r1.
+ decompose. subst.
lapply linear nplus_gen_succ_2 to H2 as H0.
- decompose.
- rewrite > H2. clear H2 r2.
- ]; auto.
+ decompose. subst.
+ ]; auto new timeout=100.
qed.