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[helm.git] / matita / legacy / coq.ma

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+(**************************************************************************)
+(*       ___                                                             *)
+(*      ||M||                                                             *)
+(*      ||A||       A project by Andrea Asperti                           *)
+(*      ||T||                                                             *)
+(*      ||I||       Developers:                                           *)
+(*      ||T||       A.Asperti, C.Sacerdoti Coen,                          *)
+(*      ||A||       E.Tassi, S.Zacchiroli                                 *)
+(*      \   /                                                             *)
+(*       \ /        This file is distributed under the terms of the       *)
+(*        v         GNU Lesser General Public License Version 2.1         *)
+(*                                                                        *)
+(**************************************************************************)
+
+default "equality"
+ cic:/Coq/Init/Logic/eq.ind
+ cic:/Coq/Init/Logic/sym_eq.con
+ cic:/Coq/Init/Logic/trans_eq.con
+ cic:/Coq/Init/Logic/eq_ind.con
+ cic:/Coq/Init/Logic/eq_ind_r.con
+ cic:/Coq/Init/Logic/eq_rec.con
+ cic:/Coq/Init/Logic/eq_rec_r.con
+ cic:/Coq/Init/Logic/eq_rect.con
+ cic:/Coq/Init/Logic/eq_rect_r.con
+ cic:/Coq/Init/Logic/f_equal.con
+ cic:/matita/legacy/coq/f_equal1.con.
+
+default "true"
+ cic:/Coq/Init/Logic/True.ind. 
+default "false"
+ cic:/Coq/Init/Logic/False.ind. 
+default "absurd"
+ cic:/Coq/Init/Logic/absurd.con. 
+
+(* aritmetic operators *)
+
+interpretation "Coq's natural plus" 'plus x y = (cic:/Coq/Init/Peano/plus.con x y).
+interpretation "Coq's real plus" 'plus x y = (cic:/Coq/Reals/Rdefinitions/Rplus.con x y).
+interpretation "Coq's binary integer plus" 'plus x y = (cic:/Coq/ZArith/BinInt/Zplus.con x y).
+interpretation "Coq's binary positive plus" 'plus x y = (cic:/Coq/NArith/BinPos/Pplus.con x y).
+interpretation "Coq's natural minus" 'minus x y = (cic:/Coq/Init/Peano/minus.con x y).
+interpretation "Coq's real minus" 'minus x y = (cic:/Coq/Reals/Rdefinitions/Rminus.con x y).
+interpretation "Coq's binary integer minus" 'minus x y = (cic:/Coq/ZArith/BinInt/Zminus.con x y).
+interpretation "Coq's binary positive minus" 'minus x y = (cic:/Coq/NArith/BinPos/Pminus.con x y).
+interpretation "Coq's natural times" 'times x y = (cic:/Coq/Init/Peano/mult.con x y).
+interpretation "Coq's real times" 'times x y = (cic:/Coq/Reals/Rdefinitions/Rmult.con x y).
+interpretation "Coq's binary positive times" 'times x y = (cic:/Coq/NArith/BinPos/Pmult.con x y).
+interpretation "Coq's binary integer times" 'times x y = (cic:/Coq/ZArith/BinInt/Zmult.con x y).
+interpretation "Coq's real power" 'power x y = (cic:/Coq/Reals/Rfunctions/pow.con x y).
+interpretation "Coq's integer power" 'power x y = (cic:/Coq/ZArith/Zpower/Zpower.con x y).
+interpretation "Coq's real divide" 'divide x y = (cic:/Coq/Reals/Rdefinitions/Rdiv.con x y).
+interpretation "Coq's real unary minus" 'uminus x = (cic:/Coq/Reals/Rdefinitions/Ropp.con x).
+interpretation "Coq's binary integer negative sign" 'uminus x = (cic:/Coq/ZArith/BinInt/Z.ind#xpointer(1/1/3) x).
+interpretation "Coq's binary integer unary minus" 'uminus x = (cic:/Coq/ZArith/BinInt/Zopp.con x).
+
+(* logical operators *)
+
+interpretation "Coq's logical and" 'and x y = (cic:/Coq/Init/Logic/and.ind#xpointer(1/1) x y).
+interpretation "Coq's logical or" 'or x y = (cic:/Coq/Init/Logic/or.ind#xpointer(1/1) x y).
+interpretation "Coq's logical not" 'not x = (cic:/Coq/Init/Logic/not.con x).
+interpretation "Coq's exists" 'exists \eta.x = (cic:/Coq/Init/Logic/ex.ind#xpointer(1/1) _ x).
+
+(* relational operators *)
+
+interpretation "Coq's natural 'less or equal to'" 'leq x y = (cic:/Coq/Init/Peano/le.ind#xpointer(1/1) x y).
+interpretation "Coq's real 'less or equal to'" 'leq x y = (cic:/Coq/Reals/Rdefinitions/Rle.con x y).
+interpretation "Coq's natural 'greater or equal to'" 'geq x y = (cic:/Coq/Init/Peano/ge.con x y).
+interpretation "Coq's real 'greater or equal to'" 'geq x y = (cic:/Coq/Reals/Rdefinitions/Rge.con x y).
+interpretation "Coq's natural 'less than'" 'lt x y = (cic:/Coq/Init/Peano/lt.con x y).
+interpretation "Coq's real 'less than'" 'lt x y = (cic:/Coq/Reals/Rdefinitions/Rlt.con x y).
+interpretation "Coq's natural 'greater than'" 'gt x y = (cic:/Coq/Init/Peano/gt.con x y).
+interpretation "Coq's real 'greater than'" 'gt x y = (cic:/Coq/Reals/Rdefinitions/Rgt.con x y).
+
+interpretation "Coq's leibnitz's equality" 'eq x y = (cic:/Coq/Init/Logic/eq.ind#xpointer(1/1) _ x y).
+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)).
+
+interpretation "Coq's natural 'not less or equal than'"
+ 'nleq x y = (cic:/Coq/Init/Logic/not.con 
+               (cic:/Coq/Init/Peano/le.ind#xpointer(1/1) x y)).
+               
+theorem f_equal1 : \forall A,B:Type.\forall f:A\to B.\forall x,y:A.
+  x = y \to (f y) = (f x).
+  intros.
+  symmetry.
+  apply cic:/Coq/Init/Logic/f_equal.con.
+  assumption.
+qed.
+(* aliases *)
+
+(* FG: This is because "and" is a reserved keyword of the parser *)
+alias id "land" = "cic:/Coq/Init/Logic/and.ind#xpointer(1/1)".
+