X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=blobdiff_plain;f=helm%2Fsoftware%2Fmatita%2Fcontribs%2FRELATIONAL%2FNPlus%2Fmonoid.ma;h=7a9bb8da047c4ee1443eef83ae70691cc6f5c504;hb=d9824956d9132109ed5f23380a0a1f9c5181d18a;hp=a521c9a1dfef29b49f0176f484f0bf4e9eac3383;hpb=eefb7b4c9f5c4c531199c95e4bb72d8b8c88bc2e;p=helm.git

diff --git a/helm/software/matita/contribs/RELATIONAL/NPlus/monoid.ma b/helm/software/matita/contribs/RELATIONAL/NPlus/monoid.ma
index a521c9a1d..7a9bb8da0 100644
--- a/helm/software/matita/contribs/RELATIONAL/NPlus/monoid.ma
+++ b/helm/software/matita/contribs/RELATIONAL/NPlus/monoid.ma
@@ -12,38 +12,34 @@
 (*                                                                        *)
 (**************************************************************************)
 
-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; autobatch.
+theorem nplus_zero_1: ∀q. zero ⊕ q ≍ q.
+ 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; autobatch.
+theorem nplus_succ_1: ∀p,q,r. p ⊕ q ≍ r → succ p ⊕ q ≍ succ r.
+ 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.
+theorem nplus_comm: ∀p, q, x. p ⊕ q ≍ x → ∀y. q ⊕ p ≍ y → 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; autobatch.
+theorem nplus_comm_rew: ∀p,q,r. p ⊕ q ≍ r → q ⊕ p ≍ r.
+ 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;
+theorem nplus_ass: ∀p1, p2, r1. p1 ⊕ p2 ≍ r1 → ∀p3, s1. r1 ⊕ p3 ≍ s1 →
+                   ∀r3. p2 ⊕ p3 ≍ r3 → ∀s3. p1 ⊕ r3 ≍ s3 → 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.
@@ -54,19 +50,15 @@ 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.
+theorem nplus_inj_2: ∀p, q1, r. p ⊕ q1 ≍ r → ∀q2. p ⊕ q2 ≍ r → q1 = q2.
  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;
+theorem nplus_comm_1: ∀p1, q, r1. p1 ⊕ q ≍ r1 → ∀p2, r2. p2 ⊕ q ≍ r2 →
+                      ∀x. p2 ⊕ r1 ≍ x → ∀y. p1 ⊕ r2 ≍ y → 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.
@@ -74,10 +66,9 @@ theorem nplus_comm_1: \forall p1, q, r1. (p1 + q == r1) \to
  ]; 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;
+theorem nplus_comm_1_rew: ∀p1,q,r1. p1 ⊕ q ≍ r1 → ∀p2,r2. p2 ⊕ q ≍ r2 →
+                          ∀s. p1 ⊕ r2 ≍ s → p2 ⊕ r1 ≍ s.
+ intros 4; elim H; clear H q r1;
  [ 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
@@ -86,22 +77,22 @@ qed.
 
 (*                      
 theorem nplus_shift_succ_sx: \forall p,q,r. 
-                             (p + (succ q) == r) \to (succ p) + q == r.
+                             (p \oplus (succ q) \asymp r) \to (succ p) \oplus q \asymp r.
  intros.
  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.
+                             ((succ p) \oplus q \asymp r) \to p \oplus (succ q) \asymp r.
  intros.
  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.
+theorem nplus_trans_1: \forall p,q1,r1. (p \oplus q1 \asymp r1) \to 
+                       \forall q2,r2. (r1 \oplus q2 \asymp r2) \to
+                       \exists q. (q1 \oplus q2 \asymp q) \land p \oplus q \asymp r2.
  intros 2; elim q1; clear q1; intros;
  [ lapply linear nplus_inv_zero_2 to H as H0.
    destruct.
@@ -114,9 +105,8 @@ theorem nplus_trans_1: \forall p,q1,r1. (p + q1 == r1) \to
  ]; 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.
+theorem nplus_trans_2: ∀p1,q,r1. p1 ⊕ q ≍ r1 → ∀p2,r2. p2 ⊕ r1 ≍ r2 →
+                       ∃p. p1 ⊕ p2 ≍ p ∧ p ⊕ q ≍ r2.
  intros 2; elim q; clear q; intros;
  [ lapply linear nplus_inv_zero_2 to H as H0.
    destruct
@@ -126,6 +116,6 @@ theorem nplus_trans_2: \forall p1,q,r1. (p1 + q == r1) \to
    decompose. destruct.
    lapply linear H to H4, H3 as H0.
    decompose.
- ]; apply ex_intro; [| auto new timeout=100 || auto new timeout=100 ]. (**)
+ ]; autobatch. apply ex_intro; [| auto new timeout=100 || auto new timeout=100 ]. (**)
 qed.
 *)