matitaB/matita/contribs/RELATIONAL/NPlus/monoid.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
   7 (*      ||T||         The HELM team.                                      *)
   8 (*      ||A||         http://helm.cs.unibo.it                             *)
   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15
  16
  17 include "NPlus/fun.ma".
  18
  19 (* Monoidal properties ******************************************************)
  20
  21 theorem nplus_zero_1: ∀q. zero ⊕ q ≍ q.
  22  intros; elim q; clear q; autobatch.
  23 qed.
  24
  25 theorem nplus_succ_1: ∀p,q,r. p ⊕ q ≍ r → succ p ⊕ q ≍ succ r.
  26  intros; elim H; clear H q r; autobatch.
  27 qed.
  28
  29 theorem nplus_comm: ∀p, q, x. p ⊕ q ≍ x → ∀y. q ⊕ p ≍ y → x = y.
  30  intros 4; elim H; clear H q x;
  31  [ lapply linear nplus_inv_zero_1 to H1
  32  | lapply linear nplus_inv_succ_1 to H3. decompose
  33  ]; destruct; autobatch.
  34 qed.
  35
  36 theorem nplus_comm_rew: ∀p,q,r. p ⊕ q ≍ r → q ⊕ p ≍ r.
  37  intros; elim H; clear H q r; autobatch.
  38 qed.
  39
  40 theorem nplus_ass: ∀p1, p2, r1. p1 ⊕ p2 ≍ r1 → ∀p3, s1. r1 ⊕ p3 ≍ s1 →
  41                    ∀r3. p2 ⊕ p3 ≍ r3 → ∀s3. p1 ⊕ r3 ≍ s3 → s1 = s3.
  42  intros 4; elim H; clear H p2 r1;
  43  [ lapply linear nplus_inv_zero_1 to H2. destruct.
  44    lapply nplus_mono to H1, H3. destruct. autobatch
  45  | lapply linear nplus_inv_succ_1 to H3. decompose. destruct.
  46    lapply linear nplus_inv_succ_1 to H4. decompose. destruct.
  47    lapply linear nplus_inv_succ_2 to H5. decompose. destruct. autobatch
  48  ].
  49 qed.
  50
  51 (* Corollaries of functional properties **************************************)
  52
  53 theorem nplus_inj_2: ∀p, q1, r. p ⊕ q1 ≍ r → ∀q2. p ⊕ q2 ≍ r → q1 = q2.
  54  intros. autobatch.
  55 qed.
  56
  57 (* Corollaries of nonoidal properties ***************************************)
  58
  59 theorem nplus_comm_1: ∀p1, q, r1. p1 ⊕ q ≍ r1 → ∀p2, r2. p2 ⊕ q ≍ r2 →
  60                       ∀x. p2 ⊕ r1 ≍ x → ∀y. p1 ⊕ r2 ≍ y → x = y.
  61  intros 4; elim H; clear H q r1;
  62  [ lapply linear nplus_inv_zero_2 to H1
  63  | lapply linear nplus_inv_succ_2 to H3.
  64    lapply linear nplus_inv_succ_2 to H4. decompose. destruct.
  65    lapply linear nplus_inv_succ_2 to H5. decompose
  66  ]; destruct; autobatch.
  67 qed.
  68
  69 theorem nplus_comm_1_rew: ∀p1,q,r1. p1 ⊕ q ≍ r1 → ∀p2,r2. p2 ⊕ q ≍ r2 →
  70                           ∀s. p1 ⊕ r2 ≍ s → p2 ⊕ r1 ≍ s.
  71  intros 4; elim H; clear H q r1;
  72  [ lapply linear nplus_inv_zero_2 to H1. destruct
  73  | lapply linear nplus_inv_succ_2 to H3. decompose. destruct.
  74    lapply linear nplus_inv_succ_2 to H4. decompose. destruct
  75  ]; autobatch.
  76 qed.
  77
  78 (*
  79 theorem nplus_shift_succ_sx: \forall p,q,r.
  80                              (p \oplus (succ q) \asymp r) \to (succ p) \oplus q \asymp r.
  81  intros.
  82  lapply linear nplus_inv_succ_2 to H as H0.
  83  decompose. destruct. auto new timeout=100.
  84 qed.
  85
  86 theorem nplus_shift_succ_dx: \forall p,q,r.
  87                              ((succ p) \oplus q \asymp r) \to p \oplus (succ q) \asymp r.
  88  intros.
  89  lapply linear nplus_inv_succ_1 to H as H0.
  90  decompose. destruct. auto new timeout=100.
  91 qed.
  92
  93 theorem nplus_trans_1: \forall p,q1,r1. (p \oplus q1 \asymp r1) \to
  94                        \forall q2,r2. (r1 \oplus q2 \asymp r2) \to
  95                        \exists q. (q1 \oplus q2 \asymp q) \land p \oplus q \asymp r2.
  96  intros 2; elim q1; clear q1; intros;
  97  [ lapply linear nplus_inv_zero_2 to H as H0.
  98    destruct.
  99  | lapply linear nplus_inv_succ_2 to H1 as H0.
 100    decompose. destruct.
 101    lapply linear nplus_inv_succ_1 to H2 as H0.
 102    decompose. destruct.
 103    lapply linear H to H4, H3 as H0.
 104    decompose.
 105  ]; apply ex_intro; [| auto new timeout=100 || auto new timeout=100 ]. (**)
 106 qed.
 107
 108 theorem nplus_trans_2: ∀p1,q,r1. p1 ⊕ q ≍ r1 → ∀p2,r2. p2 ⊕ r1 ≍ r2 →
 109                        ∃p. p1 ⊕ p2 ≍ p ∧ p ⊕ q ≍ r2.
 110  intros 2; elim q; clear q; intros;
 111  [ lapply linear nplus_inv_zero_2 to H as H0.
 112    destruct
 113  | lapply linear nplus_inv_succ_2 to H1 as H0.
 114    decompose. destruct.
 115    lapply linear nplus_inv_succ_2 to H2 as H0.
 116    decompose. destruct.
 117    lapply linear H to H4, H3 as H0.
 118    decompose.
 119  ]; autobatch. apply ex_intro; [| auto new timeout=100 || auto new timeout=100 ]. (**)
 120 qed.
 121 *)