Stuff moved from old Matita.

[helm.git] / matita / 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: ∀q. zero ⊕ q ≍ q.
+ intros; elim q; clear q; autobatch.
+qed.
+
+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: ∀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: ∀p,q,r. p ⊕ q ≍ r → q ⊕ p ≍ r.
+ intros; elim H; clear H q r; autobatch.
+qed.
+
+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.
+   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: ∀p, q1, r. p ⊕ q1 ≍ r → ∀q2. p ⊕ q2 ≍ r → q1 = q2.
+ intros. autobatch.
+qed.
+
+(* Corollaries of nonoidal properties ***************************************)
+
+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.
+   lapply linear nplus_inv_succ_2 to H5. decompose
+ ]; destruct; autobatch.
+qed.
+
+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
+ ]; autobatch.
+qed.
+
+(*                      
+theorem nplus_shift_succ_sx: \forall 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) \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 \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.
+ | 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: ∀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
+ | 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.
+ ]; autobatch. apply ex_intro; [| auto new timeout=100 || auto new timeout=100 ]. (**)
+qed.
+*)