Adding: Field instance for Q.

      : Power function from Q -> Z -> Q.
      : Absolute value function.


git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@9901 85f007b7-540e-0410-9357-904b9bb8a0f7

1 files changed, 131 insertions, 0 deletions

diff --git a/theories/QArith/Qfield.v b/theories/QArith/Qfield.v
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+(************************************************************************)
+(*  v      *   The Coq Proof Assistant  /  The Coq Development Team     *)
+(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
+(*   \VV/  **************************************************************)
+(*    //   *      This file is distributed under the terms of the       *)
+(*         *       GNU Lesser General Public License Version 2.1        *)
+(************************************************************************)
+
+(*i $Id: Qring.v 9427 2006-12-11 18:46:35Z bgregoir $ i*)
+
+Require Export Field.
+Require Export QArith_base.
+Require Import NArithRing.
+
+(** * A ring tactic for rational numbers *)
+
+Definition Qeq_bool (x y : Q) :=
+  if Qeq_dec x y then true else false.
+
+Lemma Qeq_bool_correct : forall x y : Q, Qeq_bool x y = true -> x==y.
+Proof.
+  intros x y; unfold Qeq_bool in |- *; case (Qeq_dec x y); simpl in |- *; auto.
+  intros _ H; inversion H.
+Qed.
+
+Definition Qsft : field_theory 0 1 Qplus Qmult Qminus Qopp Qdiv Qinv Qeq.
+Proof.
+  constructor.
+  constructor.
+  exact Qplus_0_l.
+  exact Qplus_comm.
+  exact Qplus_assoc.
+  exact Qmult_1_l.
+  exact Qmult_comm.
+  exact Qmult_assoc.
+  exact Qmult_plus_distr_l.
+  reflexivity.
+  exact Qplus_opp_r.
+  discriminate.
+  reflexivity.
+  intros p Hp.
+  rewrite Qmult_comm.
+  apply Qmult_inv_r.
+  exact Hp.
+Qed.
+
+Lemma Qpower_theory : power_theory 1 Qmult Qeq Z_of_N Qpower.
+Proof.
+constructor.
+intros r [|n];
+reflexivity.
+Qed.
+
+Ltac isQcst t :=
+  match t with
+  | inject_Z ?z => isZcst z
+  | Qmake ?n ?d =>
+    match isZcst n with
+      true => isPcst d
+    | _ => false
+    end
+  | _ => false
+  end.
+
+Ltac Qcst t :=
+  match isQcst t with
+    true => t
+    | _ => NotConstant
+  end.
+
+Ltac Qpow_tac t :=
+  match t with
+  | Z_of_N ?n => Ncst n
+  | NtoZ ?n => Ncst n
+  | _ => NotConstant
+  end.
+
+Add Field Qfield : Qsft(decidable Qeq_bool_correct, constants [Qcst], power_tac Qpower_theory [Qpow_tac]).
+(** Exemple of use: *)
+
+Section Examples. 
+
+Let ex1 : forall x y z : Q, (x+y)*z ==  (x*z)+(y*z).
+  intros.
+  ring.
+Qed.
+
+Let ex2 : forall x y : Q, x+y == y+x.
+  intros. 
+  ring.
+Qed.
+
+Let ex3 : forall x y z : Q, (x+y)+z == x+(y+z).
+  intros.
+  ring.
+Qed.
+
+Let ex4 : (inject_Z 1)+(inject_Z 1)==(inject_Z 2).
+  ring.
+Qed.
+
+Let ex5 : 1+1 == 2#1.
+  ring.
+Qed.
+
+Let ex6 : (1#1)+(1#1) == 2#1.
+  ring.
+Qed.
+
+Let ex7 : forall x : Q, x-x== 0#1.
+  intro.
+  ring.
+Qed.
+
+Example test_field : forall x y : Q, ~(y==0) -> (x/y)*y == x.
+intros.
+field.
+auto.
+Qed.
+
+End Examples.
+
+Lemma Qopp_plus : forall a b,  -(a+b) == -a + -b.
+Proof.
+  intros; ring.
+Qed.
+
+Lemma Qopp_opp : forall q, - -q==q.
+Proof.
+  intros; ring.
+Qed.