diff options
| -rw-r--r-- | .depend.coq | 77 | ||||
| -rw-r--r-- | Makefile | 9 | ||||
| -rw-r--r-- | theories/Logic/ClassicalEpsilon.v | 24 | ||||
| -rw-r--r-- | theories/Logic/DecidableType.v (renamed from theories/FSets/DecidableType.v) | 0 | ||||
| -rw-r--r-- | theories/Logic/DecidableTypeEx.v (renamed from theories/FSets/DecidableTypeEx.v) | 0 | ||||
| -rw-r--r-- | theories/ZArith/Int.v (renamed from theories/FSets/Int.v) | 0 |
6 files changed, 80 insertions, 30 deletions
diff --git a/.depend.coq b/.depend.coq index 658d38a6f..949cc501c 100644 --- a/.depend.coq +++ b/.depend.coq @@ -1,5 +1,3 @@ -theories/FSets/DecidableType.vo: theories/FSets/DecidableType.v theories/Lists/SetoidList.vo -theories/FSets/DecidableTypeEx.vo: theories/FSets/DecidableTypeEx.v theories/FSets/DecidableType.vo theories/FSets/OrderedType.vo theories/FSets/OrderedTypeEx.vo theories/FSets/OrderedType.vo: theories/FSets/OrderedType.v theories/Lists/SetoidList.vo theories/FSets/OrderedTypeEx.vo: theories/FSets/OrderedTypeEx.v theories/FSets/OrderedType.vo theories/ZArith/ZArith.vo contrib/omega/Omega.vo theories/NArith/NArith.vo theories/NArith/Ndec.vo theories/Arith/Compare_dec.vo theories/FSets/OrderedTypeAlt.vo: theories/FSets/OrderedTypeAlt.v theories/FSets/OrderedType.vo @@ -11,10 +9,10 @@ theories/FSets/FSetProperties.vo: theories/FSets/FSetProperties.v theories/FSets theories/FSets/FSetEqProperties.vo: theories/FSets/FSetEqProperties.v theories/FSets/FSetProperties.vo theories/Bool/Zerob.vo theories/Bool/Sumbool.vo contrib/omega/Omega.vo theories/FSets/FSets.vo: theories/FSets/FSets.v theories/FSets/OrderedType.vo theories/FSets/OrderedTypeEx.vo theories/FSets/OrderedTypeAlt.vo theories/FSets/FSetInterface.vo theories/FSets/FSetBridge.vo theories/FSets/FSetProperties.vo theories/FSets/FSetEqProperties.vo theories/FSets/FSetList.vo theories/FSets/FSetWeakProperties.vo: theories/FSets/FSetWeakProperties.v theories/FSets/FSetWeakInterface.vo theories/FSets/FSetWeakFacts.vo -theories/FSets/FSetWeakInterface.vo: theories/FSets/FSetWeakInterface.v theories/Bool/Bool.vo theories/FSets/DecidableType.vo +theories/FSets/FSetWeakInterface.vo: theories/FSets/FSetWeakInterface.v theories/Bool/Bool.vo theories/Logic/DecidableType.vo theories/FSets/FSetWeakList.vo: theories/FSets/FSetWeakList.v theories/FSets/FSetWeakInterface.vo theories/FSets/FSetWeakFacts.vo: theories/FSets/FSetWeakFacts.v theories/FSets/FSetWeakInterface.vo -theories/FSets/FSetWeak.vo: theories/FSets/FSetWeak.v theories/FSets/DecidableType.vo theories/FSets/DecidableTypeEx.vo theories/FSets/FSetWeakInterface.vo theories/FSets/FSetFacts.vo theories/FSets/FSetProperties.vo theories/FSets/FSetWeakList.vo +theories/FSets/FSetWeak.vo: theories/FSets/FSetWeak.v theories/Logic/DecidableType.vo theories/Logic/DecidableTypeEx.vo theories/FSets/FSetWeakInterface.vo theories/FSets/FSetFacts.vo theories/FSets/FSetProperties.vo theories/FSets/FSetWeakList.vo theories/FSets/FMapInterface.vo: theories/FSets/FMapInterface.v theories/FSets/FSetInterface.vo theories/FSets/FMapList.vo: theories/FSets/FMapList.v theories/FSets/FSetInterface.vo theories/FSets/FMapInterface.vo theories/FSets/FMaps.vo: theories/FSets/FMaps.v theories/FSets/OrderedType.vo theories/FSets/OrderedTypeEx.vo theories/FSets/OrderedTypeAlt.vo theories/FSets/FMapInterface.vo theories/FSets/FMapList.vo theories/FSets/FMapPositive.vo theories/FSets/FMapIntMap.vo theories/FSets/FMapFacts.vo @@ -22,13 +20,12 @@ theories/FSets/FMapFacts.vo: theories/FSets/FMapFacts.v theories/Bool/Bool.vo th theories/FSets/FMapWeakFacts.vo: theories/FSets/FMapWeakFacts.v theories/Bool/Bool.vo theories/FSets/OrderedType.vo theories/FSets/FMapWeakInterface.vo theories/FSets/FMapWeakInterface.vo: theories/FSets/FMapWeakInterface.v theories/FSets/FSetInterface.vo theories/FSets/FSetWeakInterface.vo theories/FSets/FMapWeakList.vo: theories/FSets/FMapWeakList.v theories/FSets/FSetInterface.vo theories/FSets/FSetWeakInterface.vo theories/FSets/FMapWeakInterface.vo -theories/FSets/FMapWeak.vo: theories/FSets/FMapWeak.v theories/FSets/DecidableType.vo theories/FSets/DecidableTypeEx.vo theories/FSets/FMapWeakInterface.vo theories/FSets/FMapWeakList.vo theories/FSets/FMapWeakFacts.vo +theories/FSets/FMapWeak.vo: theories/FSets/FMapWeak.v theories/Logic/DecidableType.vo theories/Logic/DecidableTypeEx.vo theories/FSets/FMapWeakInterface.vo theories/FSets/FMapWeakList.vo theories/FSets/FMapWeakFacts.vo theories/FSets/FMapPositive.vo: theories/FSets/FMapPositive.v theories/ZArith/ZArith.vo theories/FSets/OrderedType.vo theories/FSets/FMapInterface.vo theories/FSets/FMapIntMap.vo: theories/FSets/FMapIntMap.v theories/Bool/Bool.vo theories/NArith/NArith.vo theories/NArith/Ndigits.vo theories/NArith/Ndec.vo theories/NArith/Nnat.vo theories/IntMap/Allmaps.vo theories/FSets/OrderedType.vo theories/FSets/OrderedTypeEx.vo theories/FSets/FMapInterface.vo theories/FSets/FMapList.vo theories/FSets/FSetToFiniteSet.vo: theories/FSets/FSetToFiniteSet.v theories/Sets/Ensembles.vo theories/Sets/Finite_sets.vo theories/FSets/FSetInterface.vo theories/FSets/FSetProperties.vo theories/FSets/OrderedTypeEx.vo -theories/FSets/Int.vo: theories/FSets/Int.v theories/ZArith/ZArith.vo contrib/romega/ROmega.vo -theories/FSets/FMapAVL.vo: theories/FSets/FMapAVL.v theories/FSets/FSetInterface.vo theories/FSets/FMapInterface.vo theories/FSets/FMapList.vo theories/ZArith/ZArith.vo theories/FSets/Int.vo -theories/FSets/FSetAVL.vo: theories/FSets/FSetAVL.v theories/FSets/FSetInterface.vo theories/FSets/FSetList.vo theories/ZArith/ZArith.vo theories/FSets/Int.vo +theories/FSets/FMapAVL.vo: theories/FSets/FMapAVL.v theories/FSets/FSetInterface.vo theories/FSets/FMapInterface.vo theories/FSets/FMapList.vo theories/ZArith/ZArith.vo theories/ZArith/Int.vo +theories/FSets/FSetAVL.vo: theories/FSets/FSetAVL.v theories/FSets/FSetInterface.vo theories/FSets/FSetList.vo theories/ZArith/ZArith.vo theories/ZArith/Int.vo theories/Reals/Rdefinitions.vo: theories/Reals/Rdefinitions.v theories/ZArith/ZArith_base.vo theories/Reals/Raxioms.vo: theories/Reals/Raxioms.v theories/ZArith/ZArith_base.vo theories/Reals/Rdefinitions.vo theories/Reals/RIneq.vo: theories/Reals/RIneq.v theories/Reals/Raxioms.vo contrib/ring/ZArithRing.vo contrib/omega/Omega.vo contrib/field/Field.vo @@ -122,6 +119,8 @@ theories/Logic/EqdepFacts.vo: theories/Logic/EqdepFacts.v theories/Logic/ProofIrrelevanceFacts.vo: theories/Logic/ProofIrrelevanceFacts.v theories/Logic/EqdepFacts.vo theories/Logic/ClassicalEpsilon.vo: theories/Logic/ClassicalEpsilon.v theories/Logic/Classical.vo theories/Logic/ChoiceFacts.vo theories/Logic/ClassicalUniqueChoice.vo: theories/Logic/ClassicalUniqueChoice.v theories/Logic/Classical.vo theories/Setoids/Setoid.vo +theories/Logic/DecidableType.vo: theories/Logic/DecidableType.v theories/Lists/SetoidList.vo +theories/Logic/DecidableTypeEx.vo: theories/Logic/DecidableTypeEx.v theories/Logic/DecidableType.vo theories/FSets/OrderedType.vo theories/FSets/OrderedTypeEx.vo theories/Arith/Arith.vo: theories/Arith/Arith.v theories/Arith/Le.vo theories/Arith/Lt.vo theories/Arith/Plus.vo theories/Arith/Gt.vo theories/Arith/Minus.vo theories/Arith/Mult.vo theories/Arith/Between.vo theories/Arith/Peano_dec.vo theories/Arith/Compare_dec.vo theories/Arith/Factorial.vo theories/Arith/Gt.vo: theories/Arith/Gt.v theories/Arith/Le.vo theories/Arith/Lt.vo theories/Arith/Plus.vo theories/Arith/Between.vo: theories/Arith/Between.v theories/Arith/Le.vo theories/Arith/Lt.vo @@ -182,6 +181,7 @@ theories/ZArith/ZArith_base.vo: theories/ZArith/ZArith_base.v theories/NArith/Bi theories/ZArith/Zbool.vo: theories/ZArith/Zbool.v theories/ZArith/BinInt.vo theories/ZArith/Zeven.vo theories/ZArith/Zorder.vo theories/ZArith/Zcompare.vo theories/ZArith/ZArith_dec.vo theories/Bool/Sumbool.vo theories/ZArith/Zbinary.vo: theories/ZArith/Zbinary.v theories/Bool/Bvector.vo theories/ZArith/ZArith.vo theories/ZArith/Zpower.vo contrib/omega/Omega.vo theories/ZArith/Znumtheory.vo: theories/ZArith/Znumtheory.v theories/ZArith/ZArith_base.vo contrib/ring/ZArithRing.vo theories/ZArith/Zcomplements.vo theories/ZArith/Zdiv.vo +theories/ZArith/Int.vo: theories/ZArith/Int.v theories/ZArith/ZArith.vo contrib/romega/ROmega.vo theories/Setoids/Setoid.vo: theories/Setoids/Setoid.v theories/Relations/Relation_Definitions.vo theories/Lists/MonoList.vo: theories/Lists/MonoList.v theories/Arith/Le.vo theories/Lists/ListSet.vo: theories/Lists/ListSet.v theories/Lists/List.vo @@ -213,8 +213,6 @@ theories/Sets/Multiset.vo: theories/Sets/Multiset.v theories/Sets/Permut.vo theo theories/Sets/Relations_3_facts.vo: theories/Sets/Relations_3_facts.v theories/Sets/Relations_1.vo theories/Sets/Relations_1_facts.vo theories/Sets/Relations_2.vo theories/Sets/Relations_2_facts.vo theories/Sets/Relations_3.vo theories/Sets/Partial_Order.vo: theories/Sets/Partial_Order.v theories/Sets/Ensembles.vo theories/Sets/Relations_1.vo theories/Sets/Uniset.vo: theories/Sets/Uniset.v theories/Bool/Bool.vo theories/Sets/Permut.vo -theories/FSets/DecidableType.vo: theories/FSets/DecidableType.v theories/Lists/SetoidList.vo -theories/FSets/DecidableTypeEx.vo: theories/FSets/DecidableTypeEx.v theories/FSets/DecidableType.vo theories/FSets/OrderedType.vo theories/FSets/OrderedTypeEx.vo theories/FSets/OrderedType.vo: theories/FSets/OrderedType.v theories/Lists/SetoidList.vo theories/FSets/OrderedTypeEx.vo: theories/FSets/OrderedTypeEx.v theories/FSets/OrderedType.vo theories/ZArith/ZArith.vo contrib/omega/Omega.vo theories/NArith/NArith.vo theories/NArith/Ndec.vo theories/Arith/Compare_dec.vo theories/FSets/OrderedTypeAlt.vo: theories/FSets/OrderedTypeAlt.v theories/FSets/OrderedType.vo @@ -226,10 +224,10 @@ theories/FSets/FSetProperties.vo: theories/FSets/FSetProperties.v theories/FSets theories/FSets/FSetEqProperties.vo: theories/FSets/FSetEqProperties.v theories/FSets/FSetProperties.vo theories/Bool/Zerob.vo theories/Bool/Sumbool.vo contrib/omega/Omega.vo theories/FSets/FSets.vo: theories/FSets/FSets.v theories/FSets/OrderedType.vo theories/FSets/OrderedTypeEx.vo theories/FSets/OrderedTypeAlt.vo theories/FSets/FSetInterface.vo theories/FSets/FSetBridge.vo theories/FSets/FSetProperties.vo theories/FSets/FSetEqProperties.vo theories/FSets/FSetList.vo theories/FSets/FSetWeakProperties.vo: theories/FSets/FSetWeakProperties.v theories/FSets/FSetWeakInterface.vo theories/FSets/FSetWeakFacts.vo -theories/FSets/FSetWeakInterface.vo: theories/FSets/FSetWeakInterface.v theories/Bool/Bool.vo theories/FSets/DecidableType.vo +theories/FSets/FSetWeakInterface.vo: theories/FSets/FSetWeakInterface.v theories/Bool/Bool.vo theories/Logic/DecidableType.vo theories/FSets/FSetWeakList.vo: theories/FSets/FSetWeakList.v theories/FSets/FSetWeakInterface.vo theories/FSets/FSetWeakFacts.vo: theories/FSets/FSetWeakFacts.v theories/FSets/FSetWeakInterface.vo -theories/FSets/FSetWeak.vo: theories/FSets/FSetWeak.v theories/FSets/DecidableType.vo theories/FSets/DecidableTypeEx.vo theories/FSets/FSetWeakInterface.vo theories/FSets/FSetFacts.vo theories/FSets/FSetProperties.vo theories/FSets/FSetWeakList.vo +theories/FSets/FSetWeak.vo: theories/FSets/FSetWeak.v theories/Logic/DecidableType.vo theories/Logic/DecidableTypeEx.vo theories/FSets/FSetWeakInterface.vo theories/FSets/FSetFacts.vo theories/FSets/FSetProperties.vo theories/FSets/FSetWeakList.vo theories/FSets/FMapInterface.vo: theories/FSets/FMapInterface.v theories/FSets/FSetInterface.vo theories/FSets/FMapList.vo: theories/FSets/FMapList.v theories/FSets/FSetInterface.vo theories/FSets/FMapInterface.vo theories/FSets/FMaps.vo: theories/FSets/FMaps.v theories/FSets/OrderedType.vo theories/FSets/OrderedTypeEx.vo theories/FSets/OrderedTypeAlt.vo theories/FSets/FMapInterface.vo theories/FSets/FMapList.vo theories/FSets/FMapPositive.vo theories/FSets/FMapIntMap.vo theories/FSets/FMapFacts.vo @@ -237,13 +235,12 @@ theories/FSets/FMapFacts.vo: theories/FSets/FMapFacts.v theories/Bool/Bool.vo th theories/FSets/FMapWeakFacts.vo: theories/FSets/FMapWeakFacts.v theories/Bool/Bool.vo theories/FSets/OrderedType.vo theories/FSets/FMapWeakInterface.vo theories/FSets/FMapWeakInterface.vo: theories/FSets/FMapWeakInterface.v theories/FSets/FSetInterface.vo theories/FSets/FSetWeakInterface.vo theories/FSets/FMapWeakList.vo: theories/FSets/FMapWeakList.v 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theories/IntMap/Mapiter.vo theories/IntMap/Lsort.vo theories/IntMap/Mapsubset.vo theories/Lists/List.vo @@ -277,6 +274,54 @@ theories/Reals/Raxioms.vo: theories/Reals/Raxioms.v theories/ZArith/ZArith_base. theories/Reals/RIneq.vo: theories/Reals/RIneq.v theories/Reals/Raxioms.vo contrib/ring/ZArithRing.vo contrib/omega/Omega.vo contrib/field/Field.vo theories/Reals/DiscrR.vo: theories/Reals/DiscrR.v theories/Reals/RIneq.vo contrib/omega/Omega.vo theories/Reals/Rbase.vo: theories/Reals/Rbase.v theories/Reals/Rdefinitions.vo theories/Reals/Raxioms.vo theories/Reals/RIneq.vo theories/Reals/DiscrR.vo +theories/Reals/R_Ifp.vo: theories/Reals/R_Ifp.v theories/Reals/Rbase.vo contrib/omega/Omega.vo +theories/Reals/Rbasic_fun.vo: theories/Reals/Rbasic_fun.v theories/Reals/Rbase.vo theories/Reals/R_Ifp.vo contrib/fourier/Fourier.vo +theories/Reals/R_sqr.vo: theories/Reals/R_sqr.v theories/Reals/Rbase.vo theories/Reals/Rbasic_fun.vo +theories/Reals/SplitAbsolu.vo: theories/Reals/SplitAbsolu.v 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theories/Reals/RiemannInt.vo +theories/Reals/Reals.vo: theories/Reals/Reals.v theories/Reals/Rbase.vo theories/Reals/Rfunctions.vo theories/Reals/SeqSeries.vo theories/Reals/Rtrigo.vo theories/Reals/Ranalysis.vo theories/Reals/Integration.vo theories/Sorting/Heap.vo: theories/Sorting/Heap.v theories/Lists/List.vo theories/Sets/Multiset.vo theories/Sorting/Permutation.vo theories/Relations/Relations.vo theories/Sorting/Sorting.vo theories/Sorting/Permutation.vo: theories/Sorting/Permutation.v theories/Relations/Relations.vo theories/Lists/List.vo theories/Sets/Multiset.vo theories/Arith/Arith.vo theories/Sorting/Sorting.vo: theories/Sorting/Sorting.v theories/Lists/List.vo theories/Sets/Multiset.vo theories/Sorting/Permutation.vo theories/Relations/Relations.vo @@ -823,7 +823,8 @@ LOGICVO=\ theories/Logic/ClassicalChoice.vo theories/Logic/ClassicalDescription.vo \ theories/Logic/RelationalChoice.vo theories/Logic/Diaconescu.vo \ theories/Logic/EqdepFacts.vo theories/Logic/ProofIrrelevanceFacts.vo \ - theories/Logic/ClassicalEpsilon.vo theories/Logic/ClassicalUniqueChoice.vo + theories/Logic/ClassicalEpsilon.vo theories/Logic/ClassicalUniqueChoice.vo \ + theories/Logic/DecidableType.vo theories/Logic/DecidableTypeEx.vo ARITHVO=\ theories/Arith/Arith.vo theories/Arith/Gt.vo \ @@ -868,7 +869,7 @@ ZARITHVO=\ theories/ZArith/Zdiv.vo theories/ZArith/Zsqrt.vo \ theories/ZArith/Zwf.vo theories/ZArith/ZArith_base.vo \ theories/ZArith/Zbool.vo theories/ZArith/Zbinary.vo \ - theories/ZArith/Znumtheory.vo + theories/ZArith/Znumtheory.vo theories/ZArith/Int.vo QARITHVO=\ theories/QArith/QArith_base.vo theories/QArith/Qreduction.vo \ @@ -898,7 +899,6 @@ SETSVO=\ theories/Sets/Partial_Order.vo theories/Sets/Uniset.vo FSETSBASEVO=\ - theories/FSets/DecidableType.vo theories/FSets/DecidableTypeEx.vo \ theories/FSets/OrderedType.vo \ theories/FSets/OrderedTypeEx.vo theories/FSets/OrderedTypeAlt.vo \ theories/FSets/FSetInterface.vo theories/FSets/FSetList.vo \ @@ -912,8 +912,7 @@ FSETSBASEVO=\ theories/FSets/FMapWeakFacts.vo \ theories/FSets/FMapWeakInterface.vo theories/FSets/FMapWeakList.vo \ theories/FSets/FMapWeak.vo theories/FSets/FMapPositive.vo \ - theories/FSets/FMapIntMap.vo theories/FSets/FSetToFiniteSet.vo \ - theories/FSets/Int.vo \ + theories/FSets/FMapIntMap.vo theories/FSets/FSetToFiniteSet.vo FSETS_basic= diff --git a/theories/Logic/ClassicalEpsilon.v b/theories/Logic/ClassicalEpsilon.v index b3efa5fad..79c1e56d2 100644 --- a/theories/Logic/ClassicalEpsilon.v +++ b/theories/Logic/ClassicalEpsilon.v @@ -6,9 +6,9 @@ (* * GNU Lesser General Public License Version 2.1 *) (************************************************************************) -(*i $Id:$ i*) +(*i $Id$ i*) -(** This file provides classical logic and indefinite description +(** *** This file provides classical logic and indefinite description (Hilbert's epsilon operator) *) (** Classical epsilon's operator (i.e. indefinite description) implies @@ -25,7 +25,7 @@ Notation Local "'inhabited' A" := A (at level 200, only parsing). Axiom constructive_indefinite_description : forall (A : Type) (P : A->Prop), - (exists x : A, P x) -> { x : A | P x }. + (ex P) -> { x : A | P x }. Lemma constructive_definite_description : forall (A : Type) (P : A->Prop), @@ -43,11 +43,11 @@ Qed. Theorem classical_indefinite_description : forall (A : Type) (P : A->Prop), inhabited A -> - { x : A | (exists x : A, P x) -> P x }. + { x : A | ex P -> P x }. Proof. intros A P i. destruct (excluded_middle_informative (exists x, P x)) as [Hex|HnonP]. - apply constructive_indefinite_description with (P:= fun x => (exists x, P x) -> P x). + apply constructive_indefinite_description with (P:= fun x => ex P -> P x). destruct Hex as (x,Hx). exists x; intros _; exact Hx. firstorder. @@ -59,9 +59,11 @@ Definition epsilon (A : Type) (i:inhabited A) (P : A->Prop) : A := proj1_sig (classical_indefinite_description P i). Definition epsilon_spec (A : Type) (i:inhabited A) (P : A->Prop) : - (exists x:A, P x) -> P (epsilon i P) + (ex P) -> P (epsilon i P) := proj2_sig (classical_indefinite_description P i). +Opaque epsilon. + (** Open question: is classical_indefinite_description constructively provable from [relational_choice] and [constructive_definite_description] (at least, using the fact that @@ -70,7 +72,10 @@ Definition epsilon_spec (A : Type) (i:inhabited A) (P : A->Prop) : [classical_indefinite_description] is provable (see [relative_non_contradiction_of_indefinite_desc]). *) -(** Weaker lemmas (compatibility lemmas) *) +(** Remark: we use [ex P] rather than [exists x, P x] (which is [ex + (fun x => P x)] to ease unification *) + +(** *** Weaker lemmas (compatibility lemmas) *) Theorem choice : forall (A B : Type) (R : A->B->Prop), @@ -78,7 +83,8 @@ Theorem choice : (exists f : A->B, forall x : A, R x (f x)). Proof. intros A B R H. -exists (fun x => proj1_sig (constructive_indefinite_description (R x) (H x))). +exists (fun x => proj1_sig (constructive_indefinite_description (H x))). intro x. -apply (proj2_sig (constructive_indefinite_description (R x) (H x))). +apply (proj2_sig (constructive_indefinite_description (H x))). Qed. + diff --git a/theories/FSets/DecidableType.v b/theories/Logic/DecidableType.v index a4de6ca7f..a4de6ca7f 100644 --- a/theories/FSets/DecidableType.v +++ b/theories/Logic/DecidableType.v diff --git a/theories/FSets/DecidableTypeEx.v b/theories/Logic/DecidableTypeEx.v index dcca37095..dcca37095 100644 --- a/theories/FSets/DecidableTypeEx.v +++ b/theories/Logic/DecidableTypeEx.v diff --git a/theories/FSets/Int.v b/theories/ZArith/Int.v index ee8b24561..ee8b24561 100644 --- a/theories/FSets/Int.v +++ b/theories/ZArith/Int.v |