helm/software/matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/preamble.ma

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   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 set "baseuri" "cic:/matita/LAMBDA-TYPES/Level-1/Base/ext/preamble".
  16
  17 (* FG: We should include legacy/coq.ma bit it is not working *)
  18 (* include "legacy/coq.ma".                                  *)
  19
  20 default "equality"
  21  cic:/Coq/Init/Logic/eq.ind
  22  cic:/Coq/Init/Logic/sym_eq.con
  23  cic:/Coq/Init/Logic/trans_eq.con
  24  cic:/Coq/Init/Logic/eq_ind.con
  25  cic:/Coq/Init/Logic/eq_ind_r.con
  26  cic:/Coq/Init/Logic/f_equal.con
  27  cic:/Coq/Init/Logic/f_equal1.con.
  28
  29 default "true"
  30  cic:/Coq/Init/Logic/True.ind.
  31
  32 default "false"
  33  cic:/Coq/Init/Logic/False.ind.
  34
  35 default "absurd"
  36  cic:/Coq/Init/Logic/absurd.con.
  37
  38 interpretation "Coq's leibnitz's equality" 'eq x y = (cic:/Coq/Init/Logic/eq.ind#xpointer(1/1) _ x y).
  39 interpretation "Coq's not equal to (leibnitz)" 'neq x y = (cic:/Coq/Init/Logic/not.con (cic:/Coq/Init/Logic/eq.ind#xpointer(1/1) _ x y)).
  40 interpretation "Coq's natural plus" 'plus x y = (cic:/Coq/Init/Peano/plus.con x y).
  41 interpretation "Coq's natural 'less or equal to'" 'leq x y = (cic:/Coq/Init/Peano/le.ind#xpointer(1/1) x y).
  42
  43 alias id "land" = "cic:/Coq/Init/Logic/and.ind#xpointer(1/1)".
  44
  45 (* FG/CSC: These aliases should disappear: we would like to write something
  46  *         like: "disambiguate in cic:/Coq/*"
  47  *)
  48 alias id "True" = "cic:/Coq/Init/Logic/True.ind#xpointer(1/1)".
  49 alias id "or" = "cic:/Coq/Init/Logic/or.ind#xpointer(1/1)".
  50 alias id "nat" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1)".
  51 alias id "eq" = "cic:/Coq/Init/Logic/eq.ind#xpointer(1/1)".
  52 alias id "plus" = "cic:/Coq/Init/Peano/plus.con".
  53 alias id "le_trans" = "cic:/Coq/Arith/Le/le_trans.con".
  54 alias id "le_plus_r" = "cic:/Coq/Arith/Plus/le_plus_r.con".
  55 alias id "le" = "cic:/Coq/Init/Peano/le.ind#xpointer(1/1)".
  56 alias id "ex" = "cic:/Coq/Init/Logic/ex.ind#xpointer(1/1)".
  57 alias id "ex2" = "cic:/Coq/Init/Logic/ex2.ind#xpointer(1/1)".
  58 alias id "true" = "cic:/Coq/Init/Datatypes/bool.ind#xpointer(1/1/1)".
  59 alias id "false" = "cic:/Coq/Init/Datatypes/bool.ind#xpointer(1/1/2)".
  60 alias id "bool" = "cic:/Coq/Init/Datatypes/bool.ind#xpointer(1/1)".
  61 alias id "O" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1/1)".
  62 alias id "S" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1/2)".
  63 alias id "eq_ind" = "cic:/Coq/Init/Logic/eq_ind.con".
  64 alias id "or_introl" = "cic:/Coq/Init/Logic/or.ind#xpointer(1/1/1)".
  65 alias id "or_intror" = "cic:/Coq/Init/Logic/or.ind#xpointer(1/1/2)".
  66 alias id "False_ind" = "cic:/Coq/Init/Logic/False_ind.con".
  67 alias id "False" = "cic:/Coq/Init/Logic/False.ind#xpointer(1/1)".
  68 alias id "I" = "cic:/Coq/Init/Logic/True.ind#xpointer(1/1/1)".
  69 alias id "minus" = "cic:/Coq/Init/Peano/minus.con".
  70 alias id "le_n" = "cic:/Coq/Init/Peano/le.ind#xpointer(1/1/1)".
  71 alias id "le_antisym" = "cic:/Coq/Arith/Le/le_antisym.con".
  72 alias id "eq_ind_r" = "cic:/Coq/Init/Logic/eq_ind_r.con".
  73
  74 theorem f_equal: \forall A,B:Type. \forall f:A \to B.
  75                  \forall x,y:A. x = y \to f x = f y.
  76  intros. elim H. reflexivity.
  77 qed.
  78
  79 theorem sym_eq: \forall A:Type. \forall x,y:A. x = y \to y = x.
  80  intros. rewrite > H. reflexivity.
  81 qed.
  82
  83 theorem sym_not_eq: \forall A:Type. \forall x,y:A. x \neq y \to y \neq x.
  84  unfold not. intros. apply H. symmetry. assumption.
  85 qed.
  86
  87 theorem plus_reg_l: \forall (n,m,p:nat). n + m = n + p \to m = p.
  88  intros. apply plus_reg_l; auto.
  89 qed.
  90
  91 theorem plus_le_reg_l: \forall p,n,m. p + n <= p + m \to n <= m.
  92  intros. apply plus_le_reg_l; auto.
  93 qed.
  94
  95 definition sym_equal \def sym_eq.