helm/software/matita/nlibrary/logic/cprop.ma

   1 (**************************************************************************)
   2 (*       ___                                                              *)
   3 (*      ||M||                                                             *)
   4 (*      ||A||       A project by Andrea Asperti                           *)
   5 (*      ||T||                                                             *)
   6 (*      ||I||       Developers:                                           *)
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   9 (*      \   /                                                             *)
  10 (*       \ /        This file is distributed under the terms of the       *)
  11 (*        v         GNU General Public License Version 2                  *)
  12 (*                                                                        *)
  13 (**************************************************************************)
  14
  15 include "hints_declaration.ma".
  16 include "sets/setoids1.ma".
  17
  18 ndefinition CPROP: setoid1.
  19  napply mk_setoid1
  20   [ napply CProp[0]
  21   | napply (mk_equivalence_relation1 CProp[0])
  22      [ napply iff
  23      | #x; napply mk_iff; #H; nassumption
  24      | #x; #y; *; #H1; #H2; napply mk_iff; nassumption
  25      | #x; #y; #z; *; #H1; #H2; *; #H3; #H4; napply mk_iff; #w
  26         [ napply (H3 (H1 w)) | napply (H2 (H4 w))]##]##]
  27 nqed.
  28
  29 alias symbol "hint_decl" = "hint_decl_CProp2".
  30 unification hint 0 ≔ ⊢ CProp[0] ≡ carr1 CPROP.
  31
  32 (*ndefinition CProp0_of_CPROP: carr1 CPROP → CProp[0] ≝ λx.x.
  33 ncoercion CProp0_of_CPROP : ∀x: carr1 CPROP. CProp[0] ≝ CProp0_of_CPROP
  34  on _x: carr1 CPROP to CProp[0].*)
  35
  36 alias symbol "eq" = "setoid1 eq".
  37
  38 ndefinition fi': ∀A,B:CPROP. A = B → B → A.
  39  #A; #B; #H; napply (fi … H); nassumption.
  40 nqed.
  41
  42 notation ". r" with precedence 50 for @{'fi $r}.
  43 interpretation "fi" 'fi r = (fi' ?? r).
  44
  45 ndefinition and_morphism: binary_morphism1 CPROP CPROP CPROP.
  46  napply mk_binary_morphism1
  47   [ napply And
  48   | #a; #a'; #b; #b'; *; #H1; #H2; *; #H3; #H4; napply mk_iff; *; #K1; #K2; napply conj
  49      [ napply (H1 K1)
  50      | napply (H3 K2)
  51      | napply (H2 K1)
  52      | napply (H4 K2)]##]
  53 nqed.
  54
  55 unification hint 0 ≔ A,B ⊢ fun21 … (mk_binary_morphism1 … And (prop21 … and_morphism)) A B ≡ And A B.
  56
  57 (*nlemma test: ∀A,A',B: CProp[0]. A=A' → (B ∨ A) = B → (B ∧ A) ∧ B.
  58  #A; #A'; #B; #H1; #H2;
  59  napply (. ((#‡H1)‡H2^-1)); nnormalize;
  60 nqed.*)
  61
  62 ndefinition or_morphism: binary_morphism1 CPROP CPROP CPROP.
  63  napply mk_binary_morphism1
  64   [ napply Or
  65   | #a; #a'; #b; #b'; *; #H1; #H2; *; #H3; #H4; napply mk_iff; *; #H;
  66      ##[##1,3: napply or_introl |##*: napply or_intror ]
  67    ##[ napply (H1 H)
  68      | napply (H2 H)
  69      | napply (H3 H)
  70      | napply (H4 H)]##]
  71 nqed.
  72
  73 unification hint 0 ≔ A,B ⊢ fun21 … (mk_binary_morphism1 … Or (prop21 … or_morphism)) A B ≡ Or A B.
  74
  75 ndefinition if_morphism: binary_morphism1 CPROP CPROP CPROP.
  76  napply mk_binary_morphism1
  77   [ napply (λA,B. A → B)
  78   | #a; #a'; #b; #b'; #H1; #H2; napply mk_iff; #H; #w
  79      [ napply (if … H2); napply H; napply (fi … H1); nassumption
  80      | napply (fi … H2); napply H; napply (if … H1); nassumption]##]
  81 nqed.