matita/contribs/LAMBDA-TYPES/Level-1/Base/ext/preamble.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 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 "bool" = "cic:/Coq/Init/Datatypes/bool.ind#xpointer(1/1)".
  49 alias id "conj" = "cic:/Coq/Init/Logic/and.ind#xpointer(1/1/1)".
  50 alias id "eq_add_S" = "cic:/Coq/Init/Peano/eq_add_S.con".
  51 alias id "eq" = "cic:/Coq/Init/Logic/eq.ind#xpointer(1/1)".
  52 alias id "eq_ind" = "cic:/Coq/Init/Logic/eq_ind.con".
  53 alias id "eq_ind_r" = "cic:/Coq/Init/Logic/eq_ind_r.con".
  54 alias id "ex2" = "cic:/Coq/Init/Logic/ex2.ind#xpointer(1/1)".
  55 alias id "ex2_ind" = "cic:/Coq/Init/Logic/ex2_ind.con".
  56 alias id "ex" = "cic:/Coq/Init/Logic/ex.ind#xpointer(1/1)".
  57 alias id "ex_intro2" = "cic:/Coq/Init/Logic/ex2.ind#xpointer(1/1/1)".
  58 alias id "false" = "cic:/Coq/Init/Datatypes/bool.ind#xpointer(1/1/2)".
  59 alias id "False" = "cic:/Coq/Init/Logic/False.ind#xpointer(1/1)".
  60 alias id "False_ind" = "cic:/Coq/Init/Logic/False_ind.con".
  61 alias id "I" = "cic:/Coq/Init/Logic/True.ind#xpointer(1/1/1)".
  62 alias id "le_antisym" = "cic:/Coq/Arith/Le/le_antisym.con".
  63 alias id "le" = "cic:/Coq/Init/Peano/le.ind#xpointer(1/1)".
  64 alias id "le_lt_n_Sm" = "cic:/Coq/Arith/Lt/le_lt_n_Sm.con".
  65 alias id "le_lt_or_eq" = "cic:/Coq/Arith/Lt/le_lt_or_eq.con".
  66 alias id "le_n" = "cic:/Coq/Init/Peano/le.ind#xpointer(1/1/1)".
  67 alias id "le_n_O_eq" = "cic:/Coq/Arith/Le/le_n_O_eq.con".
  68 alias id "le_not_lt" = "cic:/Coq/Arith/Lt/le_not_lt.con".
  69 alias id "le_n_S" = "cic:/Coq/Arith/Le/le_n_S.con".
  70 alias id "le_O_n" = "cic:/Coq/Arith/Le/le_O_n.con".
  71 alias id "le_or_lt" = "cic:/Coq/Arith/Lt/le_or_lt.con".
  72 alias id "le_plus_l" = "cic:/Coq/Arith/Plus/le_plus_l.con".
  73 alias id "le_plus_minus" = "cic:/Coq/Arith/Minus/le_plus_minus.con".
  74 alias id "le_plus_minus_r" = "cic:/Coq/Arith/Minus/le_plus_minus_r.con".
  75 alias id "le_plus_r" = "cic:/Coq/Arith/Plus/le_plus_r.con".
  76 alias id "le_S" = "cic:/Coq/Init/Peano/le.ind#xpointer(1/1/2)".
  77 alias id "le_S_n" = "cic:/Coq/Arith/Le/le_S_n.con".
  78 alias id "le_Sn_n" = "cic:/Coq/Arith/Le/le_Sn_n.con".
  79 alias id "le_trans" = "cic:/Coq/Arith/Le/le_trans.con".
  80 alias id "lt" = "cic:/Coq/Init/Peano/lt.con".
  81 alias id "lt_irrefl" = "cic:/Coq/Arith/Lt/lt_irrefl.con".
  82 alias id "lt_le_S" = "cic:/Coq/Arith/Lt/lt_le_S.con".
  83 alias id "lt_n_S" = "cic:/Coq/Arith/Lt/lt_n_S.con".
  84 alias id "minus" = "cic:/Coq/Init/Peano/minus.con".
  85 alias id "minus_n_O" = "cic:/Coq/Arith/Minus/minus_n_O.con".
  86 alias id "minus_plus" = "cic:/Coq/Arith/Minus/minus_plus.con".
  87 alias id "nat" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1)".
  88 alias id "nat_ind" = "cic:/Coq/Init/Datatypes/nat_ind.con".
  89 alias id "not" = "cic:/Coq/Init/Logic/not.con".
  90 alias id "not_eq_S" = "cic:/Coq/Init/Peano/not_eq_S.con".
  91 alias id "O" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1/1)".
  92 alias id "or" = "cic:/Coq/Init/Logic/or.ind#xpointer(1/1)".
  93 alias id "or_ind" = "cic:/Coq/Init/Logic/or_ind.con".
  94 alias id "or_introl" = "cic:/Coq/Init/Logic/or.ind#xpointer(1/1/1)".
  95 alias id "or_intror" = "cic:/Coq/Init/Logic/or.ind#xpointer(1/1/2)".
  96 alias id "O_S" = "cic:/Coq/Init/Peano/O_S.con".
  97 alias id "plus_assoc" = "cic:/Coq/Arith/Plus/plus_assoc.con".
  98 alias id "plus_assoc_reverse" = "cic:/Coq/Arith/Plus/plus_assoc_reverse.con".
  99 alias id "plus" = "cic:/Coq/Init/Peano/plus.con".
 100 alias id "plus_comm" = "cic:/Coq/Arith/Plus/plus_comm.con".
 101 alias id "plus_le_compat" = "cic:/Coq/Arith/Plus/plus_le_compat.con".
 102 alias id "plus_le_reg_l" = "cic:/Coq/Arith/Plus/plus_le_reg_l.con".
 103 alias id "plus_lt_reg_l" = "cic:/Coq/Arith/Plus/plus_lt_reg_l.con".
 104 alias id "plus_n_Sm" = "cic:/Coq/Init/Peano/plus_n_Sm.con".
 105 alias id "plus_reg_l" = "cic:/Coq/Arith/Plus/plus_reg_l.con".
 106 alias id "pred" = "cic:/Coq/Init/Peano/pred.con".
 107 alias id "refl_equal" = "cic:/Coq/Init/Logic/eq.ind#xpointer(1/1/1)".
 108 alias id "S" = "cic:/Coq/Init/Datatypes/nat.ind#xpointer(1/1/2)".
 109 alias id "true" = "cic:/Coq/Init/Datatypes/bool.ind#xpointer(1/1/1)".
 110 alias id "True" = "cic:/Coq/Init/Logic/True.ind#xpointer(1/1)".
 111 alias id "plus_lt_compat_r" = "cic:/Coq/Arith/Plus/plus_lt_compat_r.con".
 112 alias id "plus_n_O" = "cic:/Coq/Init/Peano/plus_n_O.con".
 113 alias id "plus_le_lt_compat" = "cic:/Coq/Arith/Plus/plus_le_lt_compat.con".
 114 alias id "lt_wf_ind" = "cic:/Coq/Arith/Wf_nat/lt_wf_ind.con".
 115 alias id "minus_Sn_m" = "cic:/Coq/Arith/Minus/minus_Sn_m.con".
 116 alias id "and_ind" = "cic:/Coq/Init/Logic/and_ind.con".
 117 alias id "le_lt_trans" = "cic:/Coq/Arith/Lt/le_lt_trans.con".
 118 alias id "lt_le_trans" = "cic:/Coq/Arith/Lt/lt_le_trans.con".
 119 alias id "le_lt_trans" = "cic:/Coq/Arith/Lt/le_lt_trans.con".
 120 alias id "plus_n_O" = "cic:/Coq/Init/Peano/plus_n_O.con".
 121 alias id "f_equal3" = "cic:/Coq/Init/Logic/f_equal3.con".
 122 alias id "S_pred" = "cic:/Coq/Arith/Lt/S_pred.con".
 123 alias id "lt_le_trans" = "cic:/Coq/Arith/Lt/lt_le_trans.con".
 124 alias id "plus_lt_compat_r" = "cic:/Coq/Arith/Plus/plus_lt_compat_r.con".
 125 alias id "le_plus_trans" = "cic:/Coq/Arith/Plus/le_plus_trans.con".
 126 alias id "f_equal2" = "cic:/Coq/Init/Logic/f_equal2.con".
 127 alias id "le_plus_trans" = "cic:/Coq/Arith/Plus/le_plus_trans.con".
 128 alias id "f_equal2" = "cic:/Coq/Init/Logic/f_equal2.con".
 129 alias id "plus_n_O" = "cic:/Coq/Init/Peano/plus_n_O.con".
 130 alias id "plus_n_O" = "cic:/Coq/Init/Peano/plus_n_O.con".
 131 alias id "lt_trans" = "cic:/Coq/Arith/Lt/lt_trans.con".
 132 alias id "minus_Sn_m" = "cic:/Coq/Arith/Minus/minus_Sn_m.con".
 133 alias id "ex_intro" = "cic:/Coq/Init/Logic/ex.ind#xpointer(1/1/1)".
 134 alias id "lt_trans" = "cic:/Coq/Arith/Lt/lt_trans.con".
 135 alias id "lt_n_Sn" = "cic:/Coq/Arith/Lt/lt_n_Sn.con".
 136 alias id "lt_le_trans" = "cic:/Coq/Arith/Lt/lt_le_trans.con".
 137 alias id "lt_wf_ind" = "cic:/Coq/Arith/Wf_nat/lt_wf_ind.con".
 138 alias id "bool_ind" = "cic:/Coq/Init/Datatypes/bool_ind.con".
 139 alias id "ex_ind" = "cic:/Coq/Init/Logic/ex_ind.con".
 140 alias id "plus_Snm_nSm" = "cic:/Coq/Arith/Plus/plus_Snm_nSm.con".
 141 alias id "plus_lt_le_compat" = "cic:/Coq/Arith/Plus/plus_lt_le_compat.con".
 142 alias id "plus_lt_compat" = "cic:/Coq/Arith/Plus/plus_lt_compat.con".
 143
 144 theorem f_equal: \forall A,B:Type. \forall f:A \to B.
 145                  \forall x,y:A. x = y \to f x = f y.
 146  intros. elim H. reflexivity.
 147 qed.
 148
 149 theorem sym_eq: \forall A:Type. \forall x,y:A. x = y \to y = x.
 150  intros. rewrite > H. reflexivity.
 151 qed.
 152
 153 theorem sym_not_eq: \forall A:Type. \forall x,y:A. x \neq y \to y \neq x.
 154  unfold not. intros. apply H. symmetry. assumption.
 155 qed.
 156
 157 theorem plus_reg_l: \forall (n,m,p:nat). n + m = n + p \to m = p.
 158  intros. apply plus_reg_l; auto.
 159 qed.
 160
 161 theorem plus_le_reg_l: \forall p,n,m. p + n <= p + m \to n <= m.
 162  intros. apply plus_le_reg_l; auto.
 163 qed.
 164
 165 definition sym_equal \def sym_eq.