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Diffstat (limited to 'theories/QArith/QOrderedType.v')
-rw-r--r-- | theories/QArith/QOrderedType.v | 58 |
1 files changed, 58 insertions, 0 deletions
diff --git a/theories/QArith/QOrderedType.v b/theories/QArith/QOrderedType.v new file mode 100644 index 00000000..692bfd92 --- /dev/null +++ b/theories/QArith/QOrderedType.v @@ -0,0 +1,58 @@ +(************************************************************************) +(* 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 *) +(************************************************************************) + +Require Import QArith_base Equalities Orders OrdersTac. + +Local Open Scope Q_scope. + +(** * DecidableType structure for rational numbers *) + +Module Q_as_DT <: DecidableTypeFull. + Definition t := Q. + Definition eq := Qeq. + Definition eq_equiv := Q_Setoid. + Definition eqb := Qeq_bool. + Definition eqb_eq := Qeq_bool_iff. + + Include BackportEq. (** eq_refl, eq_sym, eq_trans *) + Include HasEqBool2Dec. (** eq_dec *) + +End Q_as_DT. + +(** Note that the last module fulfills by subtyping many other + interfaces, such as [DecidableType] or [EqualityType]. *) + + + +(** * OrderedType structure for rational numbers *) + +Module Q_as_OT <: OrderedTypeFull. + Include Q_as_DT. + Definition lt := Qlt. + Definition le := Qle. + Definition compare := Qcompare. + + Instance lt_strorder : StrictOrder Qlt. + Proof. split; [ exact Qlt_irrefl | exact Qlt_trans ]. Qed. + + Instance lt_compat : Proper (Qeq==>Qeq==>iff) Qlt. + Proof. auto with *. Qed. + + Definition le_lteq := Qle_lteq. + Definition compare_spec := Qcompare_spec. + +End Q_as_OT. + + +(** * An [order] tactic for [Q] numbers *) + +Module QOrder := OTF_to_OrderTac Q_as_OT. +Ltac q_order := QOrder.order. + +(** Note that [q_order] is domain-agnostic: it will not prove + [1<=2] or [x<=x+x], but rather things like [x<=y -> y<=x -> x==y]. *) |