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

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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||         The HELM team.                                      *)
+(*      ||A||         http://helm.cs.unibo.it                             *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU General Public License Version 2                  *)
+(*                                                                        *)
+(**************************************************************************)
+
+
+
+include "NPlus/fun.ma".
+
+(* Monoidal properties ******************************************************)
+
+theorem nplus_zero_1: \forall 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.
+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; 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. 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_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_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_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_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 ]. (**)
+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.
+ intros 2; elim q; clear q; intros;
+ [ 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.
+*)