(* *)
(**************************************************************************)
-set "baseuri" "cic:/matita/RELATIONAL/NPlus/monoid".
+
include "NPlus/fun.ma".
(* 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. (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.
+ ]; destruct; 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 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. auto
- | 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. auto
+ [ lapply linear nplus_inv_zero_1 to H2. destruct.
+ lapply nplus_mono to H1, H3. destruct. autobatch
+ | lapply linear nplus_inv_succ_1 to H3. decompose. destruct.
+ lapply linear nplus_inv_succ_1 to H4. decompose. destruct.
+ lapply linear nplus_inv_succ_2 to H5. decompose. destruct. autobatch
].
qed.
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 ***************************************)
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 H4. decompose. destruct.
lapply linear nplus_inv_succ_2 to H5. decompose
- ]; subst; auto.
+ ]; destruct; autobatch.
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_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.
+ [ lapply linear nplus_inv_zero_2 to H1. destruct
+ | lapply linear nplus_inv_succ_2 to H3. decompose. destruct.
+ lapply linear nplus_inv_succ_2 to H4. decompose. destruct
+ ]; autobatch.
qed.
(*
(p + (succ q) == r) \to (succ p) + q == r.
intros.
lapply linear nplus_inv_succ_2 to H as H0.
- decompose. subst. auto new timeout=100.
+ decompose. destruct. 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_inv_succ_1 to H as H0.
- decompose. subst. auto new timeout=100.
+ decompose. destruct. auto new timeout=100.
qed.
theorem nplus_trans_1: \forall p,q1,r1. (p + q1 == r1) \to
\exists q. (q1 + q2 == q) \land p + q == r2.
intros 2; elim q1; clear q1; intros;
[ lapply linear nplus_inv_zero_2 to H as H0.
- subst.
+ destruct.
| lapply linear nplus_inv_succ_2 to H1 as H0.
- decompose. subst.
+ decompose. destruct.
lapply linear nplus_inv_succ_1 to H2 as H0.
- decompose. subst.
+ decompose. destruct.
lapply linear H to H4, H3 as H0.
decompose.
]; apply ex_intro; [| auto new timeout=100 || auto new timeout=100 ]. (**)
\exists p. (p1 + p2 == p) \land p + q == r2.
intros 2; elim q; clear q; intros;
[ lapply linear nplus_inv_zero_2 to H as H0.
- subst
+ destruct
| lapply linear nplus_inv_succ_2 to H1 as H0.
- decompose. subst.
+ decompose. destruct.
lapply linear nplus_inv_succ_2 to H2 as H0.
- decompose. subst.
+ decompose. destruct.
lapply linear H to H4, H3 as H0.
decompose.
]; apply ex_intro; [| auto new timeout=100 || auto new timeout=100 ]. (**)