- bug fix in destruct

[helm.git] / matita / contribs / RELATIONAL / NPlus / monoid.ma

diff --git a/matita/contribs/RELATIONAL/NPlus/monoid.ma b/matita/contribs/RELATIONAL/NPlus/monoid.ma
index 804485e5436680aecf63066c67d5baf3c739a5ac..a521c9a1dfef29b49f0176f484f0bf4e9eac3383 100644 (file)
--- a/matita/contribs/RELATIONAL/NPlus/monoid.ma
+++ b/matita/contribs/RELATIONAL/NPlus/monoid.ma
@@ -19,62 +19,96 @@ 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. 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
+                    \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
+ ]; 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 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. 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.
+ 
 (* 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 ***************************************)
 
+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. destruct.
+   lapply linear nplus_inv_succ_2 to H5. decompose
+ ]; 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_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
- ]; 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.
 
 (*                      
 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.
- decompose. subst. auto new timeout=100.
+ lapply linear nplus_inv_succ_2 to H as H0.
+ 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_gen_succ_1 to H as H0.
- decompose. subst. auto new timeout=100.
+ lapply linear nplus_inv_succ_1 to H as H0.
+ decompose. destruct. 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.
  intros 2; elim q1; clear q1; intros;
- [ lapply linear nplus_gen_zero_2 to H as H0.
-   subst.
- | lapply linear nplus_gen_succ_2 to H1 as H0.
-   decompose. subst.
-   lapply linear nplus_gen_succ_1 to H2 as H0.
-   decompose. subst.
+ [ lapply linear nplus_inv_zero_2 to H as H0.
+   destruct.
+ | lapply linear nplus_inv_succ_2 to H1 as H0.
+   decompose. destruct.
+   lapply linear nplus_inv_succ_1 to H2 as H0.
+   decompose. destruct.
    lapply linear H to H4, H3 as H0.
    decompose.
  ]; apply ex_intro; [| auto new timeout=100 || auto new timeout=100 ]. (**)
@@ -84,15 +118,14 @@ 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.
  intros 2; elim q; clear q; intros;
- [ lapply linear nplus_gen_zero_2 to H as H0.
-   subst
- | lapply linear nplus_gen_succ_2 to H1 as H0.
-   decompose. subst.
-   lapply linear nplus_gen_succ_2 to H2 as H0.
-   decompose. subst.
+ [ lapply linear nplus_inv_zero_2 to H as H0.
+   destruct
+ | lapply linear nplus_inv_succ_2 to H1 as H0.
+   decompose. destruct.
+   lapply linear nplus_inv_succ_2 to H2 as H0.
+   decompose. destruct.
    lapply linear H to H4, H3 as H0.
    decompose.
  ]; apply ex_intro; [| auto new timeout=100 || auto new timeout=100 ]. (**)
 qed.
 *)
-