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-rw-r--r--Makefile.build42
-rw-r--r--Makefile.common47
-rw-r--r--contrib/micromega/CheckerMaker.v20
-rw-r--r--contrib/micromega/Env.v182
-rw-r--r--contrib/micromega/EnvRing.v (renamed from contrib/micromega/NRing.v)224
-rw-r--r--contrib/micromega/Examples.v117
-rw-r--r--contrib/micromega/LICENSE.sos29
-rw-r--r--contrib/micromega/MExtraction.v23
-rw-r--r--contrib/micromega/Micromegatac.v79
-rw-r--r--contrib/micromega/OrderedRing.v25
-rw-r--r--contrib/micromega/QMicromega.v259
-rw-r--r--contrib/micromega/RMicromega.v148
-rw-r--r--contrib/micromega/Refl.v69
-rw-r--r--contrib/micromega/RingMicromega.v390
-rw-r--r--contrib/micromega/Tauto.v324
-rw-r--r--contrib/micromega/VarMap.v36
-rw-r--r--contrib/micromega/ZCoeff.v29
-rw-r--r--contrib/micromega/ZMicromega.v714
-rw-r--r--contrib/micromega/certificate.ml618
-rw-r--r--contrib/micromega/coq_micromega.ml1289
-rw-r--r--contrib/micromega/csdpcert.ml333
-rw-r--r--contrib/micromega/g_micromega.ml459
-rw-r--r--contrib/micromega/mfourier.ml667
-rw-r--r--contrib/micromega/micromega.ml1512
-rw-r--r--contrib/micromega/micromega.mli398
-rw-r--r--contrib/micromega/mutils.ml305
-rw-r--r--contrib/micromega/sos.ml1919
-rw-r--r--contrib/micromega/sos.mli66
-rw-r--r--contrib/micromega/vector.ml674
-rw-r--r--scripts/coqmktop.ml2
-rwxr-xr-xtest-suite/check2
-rw-r--r--test-suite/csdp.cache13100
-rw-r--r--test-suite/micromega/bertot.v22
-rw-r--r--test-suite/micromega/example.v347
-rw-r--r--test-suite/micromega/heap3_vcgen_25.v38
-rw-r--r--test-suite/micromega/qexample.v62
-rw-r--r--test-suite/micromega/rexample.v62
-rw-r--r--test-suite/micromega/square.v61
-rw-r--r--test-suite/micromega/zomicron.v27
-rw-r--r--test-suite/success/unification.v8
40 files changed, 23960 insertions, 368 deletions
diff --git a/Makefile.build b/Makefile.build
index 4b4ba73e9..598047dec 100644
--- a/Makefile.build
+++ b/Makefile.build
@@ -59,7 +59,7 @@ LOCALINCLUDES=-I config -I tools -I tools/coqdoc \
-I scripts -I lib -I kernel -I kernel/byterun -I library \
-I proofs -I tactics -I pretyping \
-I interp -I toplevel -I parsing -I ide/utils -I ide \
- -I contrib/omega -I contrib/romega \
+ -I contrib/omega -I contrib/romega -I contrib/micromega \
-I contrib/ring -I contrib/dp -I contrib/setoid_ring \
-I contrib/xml -I contrib/extraction \
-I contrib/interface -I contrib/fourier \
@@ -163,12 +163,12 @@ coqlight: theories-light tools coqbinaries
states:: states/initial.coq
-$(COQTOPOPT): $(COQMKTOP) $(LINKCMX) $(LIBCOQRUN) $(USERTACCMX)
+$(COQTOPOPT): $(COQMKTOP) $(LINKCMX) $(LIBCOQRUN)
$(SHOW)'COQMKTOP -o $@'
$(HIDE)$(COQMKTOP) -opt $(OPTFLAGS) -o $@
$(STRIP) $@
-$(COQTOPBYTE): $(COQMKTOP) $(LINKCMO) $(LIBCOQRUN) $(USERTACCMO)
+$(COQTOPBYTE): $(COQMKTOP) $(LINKCMO) $(LIBCOQRUN)
$(SHOW)'COQMKTOP -o $@'
$(HIDE)$(COQMKTOP) -top $(BYTEFLAGS) -o $@
@@ -307,14 +307,13 @@ parsing/highparsing.cmxa: $(HIGHPARSING:.cmo=.cmx)
$(SHOW)'OCAMLOPT -a -o $@'
$(HIDE)$(OCAMLOPT) $(OPTFLAGS) -a -o $@ $(HIGHPARSING:.cmo=.cmx)
-tactics/hightactics.cma: $(HIGHTACTICS) $(USERTACCMO)
+tactics/hightactics.cma: $(HIGHTACTICS)
$(SHOW)'OCAMLC -a -o $@'
- $(HIDE)$(OCAMLC) $(BYTEFLAGS) -a -o $@ $(HIGHTACTICS) $(USERTACCMO)
+ $(HIDE)$(OCAMLC) $(BYTEFLAGS) -a -o $@ $(HIGHTACTICS)
-tactics/hightactics.cmxa: $(HIGHTACTICS:.cmo=.cmx) $(USERTACCMO:.cmo=.cmx)
+tactics/hightactics.cmxa: $(HIGHTACTICS:.cmo=.cmx)
$(SHOW)'OCAMLOPT -a -o $@'
- $(HIDE)$(OCAMLOPT) $(OPTFLAGS) -a -o $@ $(HIGHTACTICS:.cmo=.cmx) \
- $(USERTACCMO:.cmo=.cmx)
+ $(HIDE)$(OCAMLOPT) $(OPTFLAGS) -a -o $@ $(HIGHTACTICS:.cmo=.cmx)
contrib/contrib.cma: $(CONTRIB)
$(SHOW)'OCAMLC -a -o $@'
@@ -325,6 +324,20 @@ contrib/contrib.cmxa: $(CONTRIB:.cmo=.cmx)
$(HIDE)$(OCAMLOPT) $(OPTFLAGS) -a -o $@ $(CONTRIB:.cmo=.cmx)
###########################################################################
+# Csdp to micromega special targets
+###########################################################################
+
+ifeq ($(BEST),opt)
+bin/csdpcert$(EXE): $(CSDPCERTCMX)
+ $(SHOW)'OCAMLOPT -o $@'
+ $(HIDE)$(OCAMLOPT) $(OPTFLAGS) nums.cmxa -o $@ $(CSDPCERTCMX)
+else
+bin/csdpcert$(EXE): $(CSDPCERTCMO)
+ $(SHOW)'OCAMLC -o $@'
+ $(HIDE)$(OCAMLC) -custom $(BYTEFLAGS) nums.cma -o $@ $(CSDPCERTCMO)
+endif
+
+###########################################################################
# CoqIde special targets
###########################################################################
@@ -343,12 +356,12 @@ coqide-byte: $(COQIDEBYTE) $(COQIDE)
coqide-opt: $(COQIDEBYTE) $(COQIDEOPT) $(COQIDE)
coqide-files: $(IDEFILES)
-$(COQIDEOPT): $(COQMKTOP) $(LINKCMX) $(LIBCOQRUN) $(USERTACCMX) ide/ide.cmxa
+$(COQIDEOPT): $(COQMKTOP) $(LINKCMX) $(LIBCOQRUN) ide/ide.cmxa
$(SHOW)'COQMKTOP -o $@'
$(HIDE)$(COQMKTOP) -ide -opt $(OPTFLAGS) -o $@
$(STRIP) $@
-$(COQIDEBYTE): $(COQMKTOP) $(LINKCMO) $(LIBCOQRUN) $(USERTACCMO) ide/ide.cma
+$(COQIDEBYTE): $(COQMKTOP) $(LINKCMO) $(LIBCOQRUN) ide/ide.cma
$(SHOW)'COQMKTOP -o $@'
$(HIDE)$(COQMKTOP) -g -ide -top $(BYTEFLAGS) -o $@
@@ -410,23 +423,23 @@ pcoq: pcoq-binaries pcoq-files
pcoq-binaries:: $(COQINTERFACE)
-bin/coq-interface$(EXE): $(COQMKTOP) $(LINKCMO) $(LIBCOQRUN) $(USERTACCMO) $(INTERFACE)
+bin/coq-interface$(EXE): $(COQMKTOP) $(LINKCMO) $(LIBCOQRUN) $(INTERFACE)
$(SHOW)'COQMKTOP -o $@'
$(HIDE)$(COQMKTOP) -top $(BYTEFLAGS) -o $@ $(INTERFACE)
-bin/coq-interface.opt$(EXE): $(COQMKTOP) $(LINKCMX) $(LIBCOQRUN) $(USERTACCMX) $(INTERFACECMX)
+bin/coq-interface.opt$(EXE): $(COQMKTOP) $(LINKCMX) $(LIBCOQRUN) $(INTERFACECMX)
$(SHOW)'COQMKTOP -o $@'
$(HIDE)$(COQMKTOP) -opt $(OPTFLAGS) -o $@ $(INTERFACECMX)
bin/parser$(EXE):$(LIBCOQRUN) $(PARSERCMO)
$(SHOW)'OCAMLC -o $@'
$(HIDE)$(OCAMLC) -custom -linkall $(BYTEFLAGS) -o $@ \
- dynlink.cma $(LIBCOQRUN) $(CMA) $(PARSERCMO)
+ dynlink.cma nums.cma $(LIBCOQRUN) $(CMA) $(PARSERCMO)
bin/parser.opt$(EXE): $(LIBCOQRUN) $(PARSERCMX)
$(SHOW)'OCAMLOPT -o $@'
$(HIDE)$(OCAMLOPT) -linkall $(OPTFLAGS) -o $@ \
- $(LIBCOQRUN) $(CMXA) $(PARSERCMX)
+ $(LIBCOQRUN) nums.cmxa $(CMXA) $(PARSERCMX)
pcoq-files:: $(INTERFACEVO) $(INTERFACERC)
@@ -497,6 +510,7 @@ noreal: logic arith bool zarith qarith lists sets fsets relations \
contrib: $(CONTRIBVO) $(CONTRIBCMO)
omega: $(OMEGAVO) $(OMEGACMO) $(ROMEGAVO) $(ROMEGACMO)
+micromega: $(MICROMEGAVO) $(MICROMEGACMO) $(CSDPCERT)
ring: $(RINGVO) $(RINGCMO)
setoid_ring: $(NEWRINGVO) $(NEWRINGCMO)
dp: $(DPCMO)
diff --git a/Makefile.common b/Makefile.common
index 4c86d6716..354a14724 100644
--- a/Makefile.common
+++ b/Makefile.common
@@ -38,6 +38,8 @@ OTHERBINARIES:=$(COQMKTOPBYTE) $(COQCBYTE)
MINICOQ:=bin/minicoq$(EXE)
+CSDPCERT:=bin/csdpcert
+
###########################################################################
# tools
###########################################################################
@@ -212,13 +214,8 @@ TOPLEVEL:=\
HIGHTACTICS:=\
tactics/refine.cmo tactics/extraargs.cmo \
- tactics/extratactics.cmo tactics/eauto.cmo tactics/class_tactics.cmo
-
-SPECTAC:= tactics/tauto.ml4 tactics/eqdecide.ml4
-USERTAC:= $(SPECTAC)
-
-USERTACCMO:=$(USERTAC:.ml4=.cmo)
-USERTACCMX:=$(USERTAC:.ml4=.cmx)
+ tactics/extratactics.cmo tactics/eauto.cmo tactics/class_tactics.cmo \
+ tactics/tauto.cmo tactics/eqdecide.cmo
OMEGACMO:=\
contrib/omega/omega.cmo contrib/omega/coq_omega.cmo \
@@ -228,6 +225,12 @@ ROMEGACMO:=\
contrib/romega/const_omega.cmo \
contrib/romega/refl_omega.cmo contrib/romega/g_romega.cmo
+MICROMEGACMO:=\
+ contrib/micromega/mutils.cmo contrib/micromega/vector.cmo \
+ contrib/micromega/micromega.cmo contrib/micromega/mfourier.cmo \
+ contrib/micromega/certificate.cmo \
+ contrib/micromega/coq_micromega.cmo contrib/micromega/g_micromega.cmo
+
RINGCMO:=\
contrib/ring/quote.cmo contrib/ring/g_quote.cmo \
contrib/ring/ring.cmo contrib/ring/g_ring.cmo
@@ -302,7 +305,8 @@ SUBTACCMO:=contrib/subtac/subtac_utils.cmo contrib/subtac/eterm.cmo \
RTAUTOCMO:=contrib/rtauto/proof_search.cmo contrib/rtauto/refl_tauto.cmo \
contrib/rtauto/g_rtauto.cmo
-CONTRIB:=$(OMEGACMO) $(ROMEGACMO) $(RINGCMO) $(NEWRINGCMO) $(DPCMO) $(FIELDCMO) \
+CONTRIB:=$(OMEGACMO) $(ROMEGACMO) $(MICROMEGACMO) \
+ $(RINGCMO) $(NEWRINGCMO) $(DPCMO) $(FIELDCMO) \
$(FOURIERCMO) $(EXTRACTIONCMO) $(JPROVERCMO) $(XMLCMO) \
$(CCCMO) $(FOCMO) $(SUBTACCMO) $(RTAUTOCMO) \
$(FUNINDCMO)
@@ -325,7 +329,7 @@ LINKCMX:=$(LINKCMOCMXA:.cmo=.cmx)
# objects known by the toplevel of Coq
OBJSCMO:=$(CONFIG) $(LIBREP) $(KERNEL) $(LIBRARY) $(PRETYPING) $(INTERP) \
$(PROOFS) $(PARSING) $(TACTICS) $(TOPLEVEL) $(HIGHPARSING) \
- $(HIGHTACTICS) $(USERTACMO) $(CONTRIB)
+ $(HIGHTACTICS) $(CONTRIB)
COQIDECMO:=ide/utils/okey.cmo ide/utils/config_file.cmo \
ide/utils/configwin_keys.cmo ide/utils/configwin_types.cmo \
@@ -375,6 +379,12 @@ PARSERCMX:= $(PARSERREQUIRESCMX) $(PARSERCODE:.cmo=.cmx)
INTERFACERC:= contrib/interface/vernacrc
+CSDPCERTCMO:= contrib/micromega/mutils.cmo contrib/micromega/micromega.cmo \
+ contrib/micromega/vector.cmo contrib/micromega/mfourier.cmo \
+ contrib/micromega/certificate.cmo \
+ contrib/micromega/sos.cmo contrib/micromega/csdpcert.cmo
+CSDPCERTCMX:= $(CSDPCERTCMO:.cmo=.cmx)
+
DEBUGPRINTERS:=dev/top_printers.cmo dev/vm_printers.cmo dev/printers.cma
COQDEPCMO:=config/coq_config.cmo tools/coqdep_lexer.cmo tools/coqdep.cmo
@@ -704,6 +714,15 @@ OMEGAVO:=$(addprefix contrib/omega/, \
ROMEGAVO:=$(addprefix contrib/romega/, \
ReflOmegaCore.vo ROmega.vo )
+MICROMEGAVO:=$(addprefix contrib/micromega/, \
+ CheckerMaker.vo Refl.vo \
+ Env.vo RingMicromega.vo \
+ EnvRing.vo VarMap.vo \
+ OrderedRing.vo ZCoeff.vo \
+ Micromegatac.vo ZMicromega.vo \
+ QMicromega.vo RMicromega.vo \
+ Tauto.vo )
+
RINGVO:=$(addprefix contrib/ring/, \
LegacyArithRing.vo Ring_normalize.vo \
LegacyRing_theory.vo LegacyRing.vo \
@@ -751,9 +770,9 @@ SUBTACVO:=$(addprefix theories/Program/, \
RTAUTOVO:=$(addprefix contrib/rtauto/, \
Bintree.vo Rtauto.vo )
-CONTRIBVO:= $(OMEGAVO) $(ROMEGAVO) $(RINGVO) $(FIELDVO) $(XMLVO) \
- $(FOURIERVO) $(JPROVERVO) $(CCVO) $(FUNINDVO) $(SUBTACVO) \
- $(RTAUTOVO) $(RECDEFVO) $(NEWRINGVO) $(DPVO)
+CONTRIBVO:= $(OMEGAVO) $(ROMEGAVO) $(MICROMEGAVO) $(RINGVO) $(FIELDVO) \
+ $(XMLVO) $(FOURIERVO) $(JPROVERVO) $(CCVO) $(FUNINDVO) \
+ $(SUBTACVO) $(RTAUTOVO) $(RECDEFVO) $(NEWRINGVO) $(DPVO)
ALLVO:= $(INITVO) $(THEORIESVO) $(CONTRIBVO)
@@ -808,12 +827,12 @@ STAGE1_TARGETS:= $(STAGE1) \
STAGE2_TARGETS:=$(COQBINARIES) lib kernel byterun library proofs tactics \
interp parsing pretyping highparsing toplevel hightactics \
coqide-binaries coqide-byte coqide-opt $(COQIDEOPT) $(COQIDEBYTE) $(COQIDE) \
- pcoq-binaries $(COQINTERFACE) coqbinaries pcoq $(TOOLS) tools \
+ pcoq-binaries $(COQINTERFACE) $(CSDPCERT) coqbinaries pcoq $(TOOLS) tools \
printers $(MINICOQ) debug
VO_TARGETS:=logic arith bool narith zarith qarith lists strings sets \
fsets allfsets relations wellfounded ints reals allreals \
setoids sorting natural integer rational numbers noreal \
- omega ring setoid_ring dp xml extraction field fourier jprover \
+ omega micromega ring setoid_ring dp xml extraction field fourier jprover \
funind cc programs subtac rtauto
DOC_TARGETS:=doc doc-html doc-ps doc-pdf stdlib refman tutorial faq rectutorial
STAGE3_TARGETS:=world install coqide coqide-files coq coqlib \
diff --git a/contrib/micromega/CheckerMaker.v b/contrib/micromega/CheckerMaker.v
index 8c491a55e..93b4d213f 100644
--- a/contrib/micromega/CheckerMaker.v
+++ b/contrib/micromega/CheckerMaker.v
@@ -1,10 +1,16 @@
-(********************************************************************)
-(* *)
-(* Micromega: A reflexive tactics using the Positivstellensatz *)
-(* *)
-(* Frédéric Besson (Irisa/Inria) 2006 *)
-(* *)
-(********************************************************************)
+(************************************************************************)
+(* 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 *)
+(************************************************************************)
+(* *)
+(* Micromega: A reflexive tactic using the Positivstellensatz *)
+(* *)
+(* Frédéric Besson (Irisa/Inria) 2006-2008 *)
+(* *)
+(************************************************************************)
Require Import Setoid.
Require Import Decidable.
diff --git a/contrib/micromega/Env.v b/contrib/micromega/Env.v
new file mode 100644
index 000000000..40db9e464
--- /dev/null
+++ b/contrib/micromega/Env.v
@@ -0,0 +1,182 @@
+(************************************************************************)
+(* 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 *)
+(************************************************************************)
+(* *)
+(* Micromega: A reflexive tactic using the Positivstellensatz *)
+(* *)
+(* Frédéric Besson (Irisa/Inria) 2006-2008 *)
+(* *)
+(************************************************************************)
+
+Require Import ZArith.
+Require Import Coq.Arith.Max.
+Require Import List.
+Set Implicit Arguments.
+
+(* I have addded a Leaf constructor to the varmap data structure (/contrib/ring/Quote.v)
+ -- this is harmless and spares a lot of Empty.
+ This means smaller proof-terms.
+ BTW, by dropping the polymorphism, I get small (yet noticeable) speed-up.
+*)
+
+Section S.
+
+ Variable D :Type.
+
+ Definition Env := positive -> D.
+
+ Definition jump (j:positive) (e:Env) := fun x => e (Pplus x j).
+
+ Definition nth (n:positive) (e : Env ) := e n.
+
+ Definition hd (x:D) (e: Env) := nth xH e.
+
+ Definition tail (e: Env) := jump xH e.
+
+ Lemma psucc : forall p, (match p with
+ | xI y' => xO (Psucc y')
+ | xO y' => xI y'
+ | 1%positive => 2%positive
+ end) = (p+1)%positive.
+ Proof.
+ destruct p.
+ auto with zarith.
+ rewrite xI_succ_xO.
+ auto with zarith.
+ reflexivity.
+ Qed.
+
+ Lemma jump_Pplus : forall i j l,
+ forall x, jump (i + j) l x = jump i (jump j l) x.
+ Proof.
+ unfold jump.
+ intros.
+ rewrite Pplus_assoc.
+ reflexivity.
+ Qed.
+
+ Lemma jump_simpl : forall p l,
+ forall x, jump p l x =
+ match p with
+ | xH => tail l x
+ | xO p => jump p (jump p l) x
+ | xI p => jump p (jump p (tail l)) x
+ end.
+ Proof.
+ destruct p ; unfold tail ; intros ; repeat rewrite <- jump_Pplus.
+ (* xI p = p + p + 1 *)
+ rewrite xI_succ_xO.
+ rewrite Pplus_diag.
+ rewrite <- Pplus_one_succ_r.
+ reflexivity.
+ (* xO p = p + p *)
+ rewrite Pplus_diag.
+ reflexivity.
+ reflexivity.
+ Qed.
+
+ Ltac jump_s :=
+ repeat
+ match goal with
+ | |- context [jump xH ?e] => rewrite (jump_simpl xH)
+ | |- context [jump (xO ?p) ?e] => rewrite (jump_simpl (xO p))
+ | |- context [jump (xI ?p) ?e] => rewrite (jump_simpl (xI p))
+ end.
+
+ Lemma jump_tl : forall j l, forall x, tail (jump j l) x = jump j (tail l) x.
+ Proof.
+ unfold tail.
+ intros.
+ repeat rewrite <- jump_Pplus.
+ rewrite Pplus_comm.
+ reflexivity.
+ Qed.
+
+ Lemma jump_Psucc : forall j l,
+ forall x, (jump (Psucc j) l x) = (jump 1 (jump j l) x).
+ Proof.
+ intros.
+ rewrite <- jump_Pplus.
+ rewrite Pplus_one_succ_r.
+ rewrite Pplus_comm.
+ reflexivity.
+ Qed.
+
+ Lemma jump_Pdouble_minus_one : forall i l,
+ forall x, (jump (Pdouble_minus_one i) (tail l)) x = (jump i (jump i l)) x.
+ Proof.
+ unfold tail.
+ intros.
+ repeat rewrite <- jump_Pplus.
+ rewrite <- Pplus_one_succ_r.
+ rewrite Psucc_o_double_minus_one_eq_xO.
+ rewrite Pplus_diag.
+ reflexivity.
+ Qed.
+
+ Lemma jump_x0_tail : forall p l, forall x, jump (xO p) (tail l) x = jump (xI p) l x.
+ Proof.
+ intros.
+ unfold jump.
+ unfold tail.
+ unfold jump.
+ rewrite <- Pplus_assoc.
+ simpl.
+ reflexivity.
+ Qed.
+
+ Lemma nth_spec : forall p l x,
+ nth p l =
+ match p with
+ | xH => hd x l
+ | xO p => nth p (jump p l)
+ | xI p => nth p (jump p (tail l))
+ end.
+ Proof.
+ unfold nth.
+ destruct p.
+ intros.
+ unfold jump, tail.
+ unfold jump.
+ rewrite Pplus_diag.
+ rewrite xI_succ_xO.
+ simpl.
+ reflexivity.
+ unfold jump.
+ rewrite Pplus_diag.
+ reflexivity.
+ unfold hd.
+ unfold nth.
+ reflexivity.
+ Qed.
+
+
+ Lemma nth_jump : forall p l x, nth p (tail l) = hd x (jump p l).
+ Proof.
+ unfold tail.
+ unfold hd.
+ unfold jump.
+ unfold nth.
+ intros.
+ rewrite Pplus_comm.
+ reflexivity.
+ Qed.
+
+ Lemma nth_Pdouble_minus_one :
+ forall p l, nth (Pdouble_minus_one p) (tail l) = nth p (jump p l).
+ Proof.
+ intros.
+ unfold tail.
+ unfold nth, jump.
+ rewrite Pplus_diag.
+ rewrite <- Psucc_o_double_minus_one_eq_xO.
+ rewrite Pplus_one_succ_r.
+ reflexivity.
+ Qed.
+
+End S.
+
diff --git a/contrib/micromega/NRing.v b/contrib/micromega/EnvRing.v
index b4c9cffe6..04e68272e 100644
--- a/contrib/micromega/NRing.v
+++ b/contrib/micromega/EnvRing.v
@@ -13,7 +13,7 @@
Set Implicit Arguments.
Require Import Setoid.
Require Import BinList.
-Require Import VarMap.
+Require Import Env.
Require Import BinPos.
Require Import BinNat.
Require Import BinInt.
@@ -393,7 +393,7 @@ Section MakeRingPol.
| zmon: positive -> Mon -> Mon
| vmon: positive -> Mon -> Mon.
- Fixpoint Mphi(l:off_map R) (M: Mon) {struct M} : R :=
+ Fixpoint Mphi(l:Env R) (M: Mon) {struct M} : R :=
match M with
mon0 => rI
| zmon j M1 => Mphi (jump j l) M1
@@ -490,7 +490,7 @@ Section MakeRingPol.
(** Evaluation of a polynomial towards R *)
- Fixpoint Pphi(l:off_map R) (P:Pol) {struct P} : R :=
+ Fixpoint Pphi(l:Env R) (P:Pol) {struct P} : R :=
match P with
| Pc c => [c]
| Pinj j Q => Pphi (jump j l) Q
@@ -554,6 +554,54 @@ Section MakeRingPol.
intros;simpl;apply (morph0 CRmorph).
Qed.
+Lemma env_morph : forall p e1 e2, (forall x, e1 x = e2 x) ->
+ p @ e1 = p @ e2.
+Proof.
+ induction p ; simpl.
+ reflexivity.
+ intros.
+ apply IHp.
+ intros.
+ unfold jump.
+ apply H.
+ intros.
+ rewrite (IHp1 e1 e2) ; auto.
+ rewrite (IHp2 (tail e1) (tail e2)) ; auto.
+ unfold hd. unfold nth. rewrite H. reflexivity.
+ unfold tail. unfold jump. intros ; apply H.
+Qed.
+
+Lemma Pjump_Pplus : forall P i j l, P @ (jump (i + j) l ) = P @ (jump j (jump i l)).
+Proof.
+ intros. apply env_morph. intros. rewrite <- jump_Pplus.
+ rewrite Pplus_comm.
+ reflexivity.
+Qed.
+
+Lemma Pjump_xO_tail : forall P p l,
+ P @ (jump (xO p) (tail l)) = P @ (jump (xI p) l).
+Proof.
+ intros.
+ apply env_morph.
+ intros.
+ rewrite (@jump_simpl R (xI p)).
+ rewrite (@jump_simpl R (xO p)).
+ reflexivity.
+Qed.
+
+Lemma Pjump_Pdouble_minus_one : forall P p l,
+ P @ (jump (Pdouble_minus_one p) (tail l)) = P @ (jump (xO p) l).
+Proof.
+ intros.
+ apply env_morph.
+ intros.
+ rewrite jump_Pdouble_minus_one.
+ rewrite (@jump_simpl R (xO p)).
+ reflexivity.
+Qed.
+
+
+
Lemma Pphi1 : forall l, P1@l == 1.
Proof.
intros;simpl;apply (morph1 CRmorph).
@@ -562,7 +610,8 @@ Section MakeRingPol.
Lemma mkPinj_ok : forall j l P, (mkPinj j P)@l == P@(jump j l).
Proof.
intros j l p;destruct p;simpl;rsimpl.
- rewrite <-jump_Pplus;rewrite Pplus_comm;rrefl.
+ rewrite Pjump_Pplus.
+ reflexivity.
Qed.
Let pow_pos_Pplus :=
@@ -581,13 +630,6 @@ Section MakeRingPol.
rewrite Pphi0. rewrite pow_pos_Pplus;rsimpl.
Qed.
- Ltac jump_simpl :=
- repeat (progress (
- match goal with
- | |- context [jump xH ?e] => rewrite (@jump_simpl R xH)
- | |- context [jump (xO ?p) ?e] => rewrite (@jump_simpl R (xO p))
- | |- context [jump (xI ?p) ?e] => rewrite (@jump_simpl R (xI p))
- end)).
Ltac Esimpl :=
repeat (progress (
@@ -666,15 +708,14 @@ Section MakeRingPol.
assert (H := ZPminus_spec p p0);destruct (ZPminus p p0).
rewrite H;Esimpl. rewrite IHP';rrefl.
rewrite H;Esimpl. rewrite IHP';Esimpl.
- rewrite <- jump_Pplus;rewrite Pplus_comm;rrefl.
+ rewrite Pjump_Pplus. rrefl.
rewrite H;Esimpl. rewrite IHP.
- rewrite <- jump_Pplus;rewrite Pplus_comm;rrefl.
+ rewrite Pjump_Pplus. rrefl.
destruct p0;simpl.
- rewrite IHP2;simpl; jump_simpl ;rsimpl.
- Esimpl.
+ rewrite IHP2;simpl. rsimpl.
+ rewrite Pjump_xO_tail. Esimpl.
rewrite IHP2;simpl.
- rewrite jump_Pdouble_minus_one.
- jump_simpl.
+ rewrite Pjump_Pdouble_minus_one.
rsimpl.
rewrite IHP'.
rsimpl.
@@ -682,10 +723,10 @@ Section MakeRingPol.
Esimpl2;add_push [c];rrefl.
destruct p0;simpl;Esimpl2.
rewrite IHP'2;simpl.
- jump_simpl.
+ rewrite Pjump_xO_tail.
rsimpl;add_push (P'1@l * (pow_pos rmul (hd 0 l) p));rrefl.
rewrite IHP'2;simpl.
- rewrite jump_Pdouble_minus_one;jump_simpl ; rsimpl.
+ rewrite Pjump_Pdouble_minus_one. rsimpl.
add_push (P'1@l * (pow_pos rmul (hd 0 l) p));rrefl.
rewrite IHP'2;rsimpl.
unfold tail.
@@ -700,9 +741,11 @@ Section MakeRingPol.
assert (forall P k l,
(PaddX Padd P'1 k P) @ l == P@l + P'1@l * pow_pos rmul (hd 0 l) k).
induction P;simpl;intros;try apply (ARadd_comm ARth).
- destruct p2; simpl; jump_simpl;try apply (ARadd_comm ARth).
- rewrite jump_Pdouble_minus_one.
- apply (ARadd_comm ARth).
+ destruct p2; simpl; try apply (ARadd_comm ARth).
+ rewrite Pjump_xO_tail.
+ apply (ARadd_comm ARth).
+ rewrite Pjump_Pdouble_minus_one.
+ apply (ARadd_comm ARth).
assert (H1 := ZPminus_spec p2 k);destruct (ZPminus p2 k);Esimpl2.
rewrite IHP'1;rsimpl; rewrite H1;add_push (P5 @ (tail l0));rrefl.
rewrite IHP'1;simpl;Esimpl.
@@ -728,25 +771,22 @@ Section MakeRingPol.
assert (H := ZPminus_spec p p0);destruct (ZPminus p p0).
rewrite H;Esimpl. rewrite IHP';rsimpl.
rewrite H;Esimpl. rewrite IHP';Esimpl.
- rewrite <- jump_Pplus;rewrite Pplus_comm;rrefl.
+ rewrite <- Pjump_Pplus;rewrite Pplus_comm;rrefl.
rewrite H;Esimpl. rewrite IHP.
- rewrite <- jump_Pplus;rewrite Pplus_comm;rrefl.
+ rewrite <- Pjump_Pplus;rewrite Pplus_comm;rrefl.
destruct p0;simpl.
- rewrite IHP2;simpl; jump_simpl ; rsimpl.
+ rewrite IHP2;simpl; try rewrite Pjump_xO_tail ; rsimpl.
rewrite IHP2;simpl.
- rewrite jump_Pdouble_minus_one;rsimpl.
+ rewrite Pjump_Pdouble_minus_one;rsimpl.
unfold tail ; rsimpl.
- jump_simpl.
- rsimpl.
rewrite IHP';rsimpl.
destruct P;simpl.
repeat rewrite Popp_ok;Esimpl2;rsimpl;add_push [c];try rrefl.
destruct p0;simpl;Esimpl2.
rewrite IHP'2;simpl;rsimpl;add_push (P'1@l * (pow_pos rmul (hd 0 l) p));trivial.
- jump_simpl.
- add_push (P @ ((jump p0 (jump p0 (tail l)))));rrefl.
- rewrite IHP'2;simpl;rewrite jump_Pdouble_minus_one;rsimpl.
- jump_simpl.
+ rewrite Pjump_xO_tail.
+ add_push (P @ ((jump (xI p0) l)));rrefl.
+ rewrite IHP'2;simpl;rewrite Pjump_Pdouble_minus_one;rsimpl.
add_push (- (P'1 @ l * pow_pos rmul (hd 0 l) p));rrefl.
unfold tail.
rewrite IHP'2;rsimpl;add_push (P @ (jump 1 l));rrefl.
@@ -761,9 +801,10 @@ Section MakeRingPol.
(PsubX Psub P'1 k P) @ l == P@l + - P'1@l * pow_pos rmul (hd 0 l) k).
induction P;simpl;intros.
rewrite Popp_ok;rsimpl;apply (ARadd_comm ARth);trivial.
- destruct p2;simpl; jump_simpl ; rewrite Popp_ok;rsimpl.
+ destruct p2;simpl; rewrite Popp_ok;rsimpl.
+ rewrite Pjump_xO_tail.
apply (ARadd_comm ARth);trivial.
- rewrite jump_Pdouble_minus_one.
+ rewrite Pjump_Pdouble_minus_one.
apply (ARadd_comm ARth);trivial.
apply (ARadd_comm ARth);trivial.
assert (H1 := ZPminus_spec p2 k);destruct (ZPminus p2 k);Esimpl2;rsimpl.
@@ -783,8 +824,8 @@ Section MakeRingPol.
Lemma PmulI_ok :
forall P',
- (forall (P : Pol) (l : off_map R), (Pmul P P') @ l == P @ l * P' @ l) ->
- forall (P : Pol) (p : positive) (l : off_map R),
+ (forall (P : Pol) (l : Env R), (Pmul P P') @ l == P @ l * P' @ l) ->
+ forall (P : Pol) (p : positive) (l : Env R),
(PmulI Pmul P' p P) @ l == P @ l * P' @ (jump p l).
Proof.
induction P;simpl;intros.
@@ -792,16 +833,15 @@ Section MakeRingPol.
assert (H1 := ZPminus_spec p p0);destruct (ZPminus p p0);Esimpl2.
rewrite H1; rewrite H;rrefl.
rewrite H1; rewrite H.
- rewrite Pplus_comm.
- rewrite jump_Pplus;simpl;rrefl.
- rewrite H1;rewrite Pplus_comm.
- rewrite jump_Pplus;rewrite IHP;rrefl.
+ rewrite Pjump_Pplus;simpl;rrefl.
+ rewrite H1.
+ rewrite Pjump_Pplus;rewrite IHP;rrefl.
destruct p0;Esimpl2.
- rewrite IHP1;rewrite IHP2;jump_simpl;rsimpl.
+ rewrite IHP1;rewrite IHP2;rsimpl.
+ rewrite Pjump_xO_tail.
mul_push (pow_pos rmul (hd 0 l) p);rrefl.
rewrite IHP1;rewrite IHP2;simpl;rsimpl.
- mul_push (pow_pos rmul (hd 0 l) p); rewrite jump_Pdouble_minus_one.
- jump_simpl.
+ mul_push (pow_pos rmul (hd 0 l) p); rewrite Pjump_Pdouble_minus_one.
rrefl.
rewrite IHP1;simpl;rsimpl.
mul_push (pow_pos rmul (hd 0 l) p).
@@ -857,9 +897,9 @@ Section MakeRingPol.
| xO j => Pinj (Pdouble_minus_one j) P ** P'2
| 1 => P ** P'2
end @ (tail l) == P @ (jump p0 l) * P'2 @ (tail l)).
- destruct p0;jump_simpl;rewrite IHP'2;Esimpl.
- jump_simpl ; reflexivity.
- rewrite jump_Pdouble_minus_one;Esimpl.
+ destruct p0;rewrite IHP'2;Esimpl.
+ rewrite Pjump_xO_tail. reflexivity.
+ rewrite Pjump_Pdouble_minus_one;Esimpl.
rewrite H;Esimpl.
rewrite Padd_ok; Esimpl2. rewrite Padd_ok; Esimpl2.
repeat (rewrite IHP'1 || rewrite IHP'2);simpl.
@@ -891,6 +931,57 @@ Lemma Pmul_ok : forall P P' l, (P**P')@l == P@l * P'@l.
rrefl.
Qed.
+ Lemma Mphi_morph : forall P env env', (forall x, env x = env' x ) ->
+ Mphi env P = Mphi env' P.
+ Proof.
+ induction P ; simpl.
+ reflexivity.
+ intros.
+ apply IHP.
+ intros.
+ unfold jump.
+ apply H.
+ (**)
+ intros.
+ replace (Mphi (tail env) P) with (Mphi (tail env') P).
+ unfold hd. unfold nth.
+ rewrite H.
+ reflexivity.
+ apply IHP.
+ unfold tail,jump.
+ intros. symmetry. apply H.
+ Qed.
+
+Lemma Mjump_xO_tail : forall M p l,
+ Mphi (jump (xO p) (tail l)) M = Mphi (jump (xI p) l) M.
+Proof.
+ intros.
+ apply Mphi_morph.
+ intros.
+ rewrite (@jump_simpl R (xI p)).
+ rewrite (@jump_simpl R (xO p)).
+ reflexivity.
+Qed.
+
+Lemma Mjump_Pdouble_minus_one : forall M p l,
+ Mphi (jump (Pdouble_minus_one p) (tail l)) M = Mphi (jump (xO p) l) M.
+Proof.
+ intros.
+ apply Mphi_morph.
+ intros.
+ rewrite jump_Pdouble_minus_one.
+ rewrite (@jump_simpl R (xO p)).
+ reflexivity.
+Qed.
+
+Lemma Mjump_Pplus : forall M i j l, Mphi (jump (i + j) l ) M = Mphi (jump j (jump i l)) M.
+Proof.
+ intros. apply Mphi_morph. intros. rewrite <- jump_Pplus.
+ rewrite Pplus_comm.
+ reflexivity.
+Qed.
+
+
Lemma mkZmon_ok: forall M j l,
Mphi l (mkZmon j M) == Mphi l (zmon j M).
@@ -900,14 +991,13 @@ Lemma Pmul_ok : forall P P' l, (P**P')@l == P@l * P'@l.
Lemma zmon_pred_ok : forall M j l,
Mphi (tail l) (zmon_pred j M) == Mphi l (zmon j M).
Proof.
- destruct j; simpl;intros auto; rsimpl.
+ destruct j; simpl;intros l; rsimpl.
rewrite mkZmon_ok;rsimpl.
simpl.
- jump_simpl.
+ rewrite Mjump_xO_tail.
reflexivity.
rewrite mkZmon_ok;simpl.
- rewrite jump_Pdouble_minus_one; rsimpl.
- jump_simpl ; reflexivity.
+ rewrite Mjump_Pdouble_minus_one; rsimpl.
Qed.
Lemma mkVmon_ok : forall M i l, Mphi l (mkVmon i M) == Mphi l M*pow_pos rmul (hd 0 l) i.
@@ -937,7 +1027,7 @@ Lemma Pmul_ok : forall P P' l, (P**P')@l == P@l * P'@l.
case (MFactor P (zmon (j -i) M)); simpl.
intros P2 Q2 H; repeat rewrite mkPinj_ok; auto.
rewrite <- (Pplus_minus _ _ (ZC2 _ _ He)).
- rewrite Pplus_comm; rewrite jump_Pplus; auto.
+ rewrite Mjump_Pplus; auto.
rewrite (morph0 CRmorph); rsimpl.
intros P2 m; rewrite (morph0 CRmorph); rsimpl.
@@ -1027,12 +1117,12 @@ Lemma Pmul_ok : forall P P' l, (P**P')@l == P@l * P'@l.
rewrite Padd_ok; rewrite PmulC_ok; rsimpl.
intros i P5 H; rewrite H.
intros HH H1; injection HH; intros; subst; rsimpl.
- rewrite Padd_ok; rewrite PmulI_ok by (intros; apply Pmul_ok). rewrite H1; rsimpl.
+ rewrite Padd_ok; rewrite PmulI_ok by (intros;apply Pmul_ok). rewrite H1; rsimpl.
intros i P5 P6 H1 H2 H3; rewrite H1; rewrite H3.
assert (P4 = Q1 ++ P3 ** PX i P5 P6).
injection H2; intros; subst;trivial.
rewrite H;rewrite Padd_ok;rewrite Pmul_ok;rsimpl.
- Qed.
+Qed.
(*
Lemma POneSubst_ok: forall P1 M1 P2 P3 l,
POneSubst P1 M1 P2 = Some P3 -> Mphi l M1 == P2@l -> P1@l == P3@l.
@@ -1082,7 +1172,7 @@ Proof.
rewrite <- PNSubst1_ok; auto.
Qed.
- Fixpoint MPcond (LM1: list (Mon * Pol)) (l: off_map R) {struct LM1} : Prop :=
+ Fixpoint MPcond (LM1: list (Mon * Pol)) (l: Env R) {struct LM1} : Prop :=
match LM1 with
cons (M1,P2) LM2 => (Mphi l M1 == P2@l) /\ (MPcond LM2 l)
| _ => True
@@ -1138,10 +1228,10 @@ Proof.
(** evaluation of polynomial expressions towards R *)
- Fixpoint PEeval (l:off_map R) (pe:PExpr) {struct pe} : R :=
+ Fixpoint PEeval (l:Env R) (pe:PExpr) {struct pe} : R :=
match pe with
| PEc c => phi c
- | PEX j => nth 0 j l
+ | PEX j => nth j l
| PEadd pe1 pe2 => (PEeval l pe1) + (PEeval l pe2)
| PEsub pe1 pe2 => (PEeval l pe1) - (PEeval l pe2)
| PEmul pe1 pe2 => (PEeval l pe1) * (PEeval l pe2)
@@ -1151,17 +1241,14 @@ Proof.
(** Correctness proofs *)
- Lemma mkX_ok : forall p l, nth 0 p l == (mk_X p) @ l.
+ Lemma mkX_ok : forall p l, nth p l == (mk_X p) @ l.
Proof.
destruct p;simpl;intros;Esimpl;trivial.
- rewrite nth_spec.
- jump_simpl.
- rewrite <- jump_tl.
- rewrite nth_jump.
+ rewrite nth_spec ; auto.
+ unfold hd.
+ rewrite <- nth_Pdouble_minus_one.
+ rewrite (nth_jump (Pdouble_minus_one p) l 1).
reflexivity.
- rewrite nth_spec.
- rewrite <- nth_jump.
- rewrite nth_Pdouble_minus_one;rrefl.
Qed.
Ltac Esimpl3 :=
@@ -1226,7 +1313,7 @@ Section POWER.
Lemma Ppow_N_ok : forall l, (forall P, subst_l P@l == P@l) ->
forall P n, (Ppow_N P n)@l == (pow_N P1 Pmul P n)@l.
- Proof. destruct n;simpl. rrefl. rewrite Ppow_pos_ok by trivial. Esimpl. Qed.
+ Proof. destruct n;simpl. rrefl. rewrite Ppow_pos_ok. trivial. Esimpl. auto. Qed.
End POWER.
@@ -1298,16 +1385,17 @@ Section POWER.
intros.
induction pe;simpl;Esimpl3.
apply mkX_ok.
- rewrite IHpe1;rewrite IHpe2;destruct pe1;destruct pe2;Esimpl3.
+ rewrite IHpe1;rewrite IHpe2;destruct pe1;destruct pe2;Esimpl3.
rewrite IHpe1;rewrite IHpe2;rrefl.
rewrite IHpe1;rewrite IHpe2. rewrite Pmul_ok. rrefl.
rewrite IHpe;rrefl.
- rewrite Ppow_N_ok by (intros; rrefl).
+ rewrite Ppow_N_ok by reflexivity.
rewrite pow_th.(rpow_pow_N). destruct n0;Esimpl3.
- induction p;simpl;try rewrite IHp;try rewrite IHpe;repeat rewrite Pms_ok;
+ induction p;simpl;try rewrite IHp;try rewrite IHpe;repeat rewrite Pms_ok;
repeat rewrite Pmul_ok;rrefl.
Qed.
+
End NORM_SUBST_REC.
diff --git a/contrib/micromega/Examples.v b/contrib/micromega/Examples.v
deleted file mode 100644
index 6bb9311a5..000000000
--- a/contrib/micromega/Examples.v
+++ /dev/null
@@ -1,117 +0,0 @@
-Require Import OrderedRing.
-Require Import RingMicromega.
-Require Import ZCoeff.
-Require Import Refl.
-Require Import ZArith.
-Require Import List.
-(*****)
-Require Import NRing.
-Require Import VarMap.
-(*****)
-(*Require Import Ring_polynom.*)
-(*****)
-
-Import OrderedRingSyntax.
-
-Section Examples.
-
-Variable R : Type.
-Variables rO rI : R.
-Variables rplus rtimes rminus: R -> R -> R.
-Variable ropp : R -> R.
-Variables req rle rlt : R -> R -> Prop.
-
-Variable sor : SOR rO rI rplus rtimes rminus ropp req rle rlt.
-
-Notation "0" := rO.
-Notation "1" := rI.
-Notation "x + y" := (rplus x y).
-Notation "x * y " := (rtimes x y).
-Notation "x - y " := (rminus x y).
-Notation "- x" := (ropp x).
-Notation "x == y" := (req x y).
-Notation "x ~= y" := (~ req x y).
-Notation "x <= y" := (rle x y).
-Notation "x < y" := (rlt x y).
-
-Definition phi : Z -> R := gen_order_phi_Z 0 1 rplus rtimes ropp.
-
-Lemma ZSORaddon :
- SORaddon 0 1 rplus rtimes rminus ropp req rle (* ring elements *)
- 0%Z 1%Z Zplus Zmult Zminus Zopp (* coefficients *)
- Zeq_bool Zle_bool
- phi (fun x => x) (pow_N 1 rtimes).
-Proof.
-constructor.
-exact (Zring_morph sor).
-exact (pow_N_th 1 rtimes sor.(SORsetoid)).
-apply (Zcneqb_morph sor).
-apply (Zcleb_morph sor).
-Qed.
-
-Definition Zeval_formula :=
- eval_formula 0 rplus rtimes rminus ropp req rle rlt phi (fun x => x) (pow_N 1 rtimes).
-Definition Z_In := S_In 0%Z Zeq_bool Zle_bool.
-Definition Z_Square := S_Square 0%Z Zeq_bool Zle_bool.
-
-(* Example: forall x y : Z, x + y = 0 -> x - y = 0 -> x < 0 -> False *)
-
-Lemma plus_minus : forall x y : R, x + y == 0 -> x - y == 0 -> x < 0 -> False.
-Proof.
-intros x y.
-Open Scope Z_scope.
-(*****)
-set (env := fun x y : R => Node (Leaf y) x (Empty _)).
-(*****)
-(*set (env := fun (x y : R) => x :: y :: nil).*)
-(*****)
-set (expr :=
- Build_Formula (PEadd (PEX Z 1) (PEX Z 2)) OpEq (PEc 0)
- :: Build_Formula (PEsub (PEX Z 1) (PEX Z 2)) OpEq (PEc 0)
- :: Build_Formula (PEX Z 1) OpLt (PEc 0) :: nil).
-set (cert :=
- S_Add (S_Mult (S_Pos 0 Zeq_bool Zle_bool 2 (refl_equal true)) (Z_In 2))
- (S_Add (S_Ideal (PEc 1) (Z_In 1)) (S_Ideal (PEc 1) (Z_In 0)))).
-change (make_impl (Zeval_formula (env x y)) expr False).
-apply (check_formulas_sound sor ZSORaddon expr cert).
-reflexivity.
-Close Scope Z_scope.
-Qed.
-
-(* Example *)
-
-Let four : R := ((1 + 1) * (1 + 1)).
-Lemma Zdiscr :
- forall a b c x : R,
- a * (x * x) + b * x + c == 0 -> 0 <= b * b - four * a * c.
-Proof.
-Open Scope Z_scope.
-(*****)
-set (env := fun (a b c x : R) => Node (Node (Leaf x) b (Empty _)) a (Leaf c)).
-(*****)
-(*set (env := fun (a b c x : R) => a :: b :: c :: x:: nil).*)
-(*****)
-set (poly1 :=
- (Build_Formula
- (PEadd
- (PEadd (PEmul (PEX Z 1) (PEmul (PEX Z 4) (PEX Z 4)))
- (PEmul (PEX Z 2) (PEX Z 4))) (PEX Z 3)) OpEq (PEc 0)) :: nil).
-set (poly2 :=
- (Build_Formula
- (PEsub (PEmul (PEX Z 2) (PEX Z 2))
- (PEmul (PEmul (PEc 4) (PEX Z 1)) (PEX Z 3))) OpGe (PEc 0)) :: nil).
-set (wit :=
- (S_Add (Z_In 0)
- (S_Add (S_Ideal (PEmul (PEc (-4)) (PEX Z 1)) (Z_In 1))
- (Z_Square
- (PEadd (PEmul (PEc 2) (PEmul (PEX Z 1) (PEX Z 4))) (PEX Z 2))))) :: nil).
-intros a b c x.
-change (make_impl (Zeval_formula (env a b c x)) poly1
- (make_conj (Zeval_formula (env a b c x)) poly2)).
-apply (check_conj_formulas_sound sor ZSORaddon poly1 poly2 wit).
-reflexivity.
-Close Scope Z_scope.
-Qed.
-
-End Examples.
-
diff --git a/contrib/micromega/LICENSE.sos b/contrib/micromega/LICENSE.sos
new file mode 100644
index 000000000..5aadfa2a6
--- /dev/null
+++ b/contrib/micromega/LICENSE.sos
@@ -0,0 +1,29 @@
+ HOL Light copyright notice, licence and disclaimer
+
+ (c) University of Cambridge 1998
+ (c) Copyright, John Harrison 1998-2006
+
+HOL Light version 2.20, hereinafter referred to as "the software", is a
+computer theorem proving system written by John Harrison. Much of the
+software was developed at the University of Cambridge Computer Laboratory,
+New Museums Site, Pembroke Street, Cambridge, CB2 3QG, England. The
+software is copyright, University of Cambridge 1998 and John Harrison
+1998-2006.
+
+Permission to use, copy, modify, and distribute the software and its
+documentation for any purpose and without fee is hereby granted. In the
+case of further distribution of the software the present text, including
+copyright notice, licence and disclaimer of warranty, must be included in
+full and unmodified form in any release. Distribution of derivative
+software obtained by modifying the software, or incorporating it into
+other software, is permitted, provided the inclusion of the software is
+acknowledged and that any changes made to the software are clearly
+documented.
+
+John Harrison and the University of Cambridge disclaim all warranties
+with regard to the software, including all implied warranties of
+merchantability and fitness. In no event shall John Harrison or the
+University of Cambridge be liable for any special, indirect,
+incidental or consequential damages or any damages whatsoever,
+including, but not limited to, those arising from computer failure or
+malfunction, work stoppage, loss of profit or loss of contracts.
diff --git a/contrib/micromega/MExtraction.v b/contrib/micromega/MExtraction.v
new file mode 100644
index 000000000..a5ac92db8
--- /dev/null
+++ b/contrib/micromega/MExtraction.v
@@ -0,0 +1,23 @@
+(************************************************************************)
+(* 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 *)
+(************************************************************************)
+(* *)
+(* Micromega: A reflexive tactic using the Positivstellensatz *)
+(* *)
+(* Frédéric Besson (Irisa/Inria) 2006-2008 *)
+(* *)
+(************************************************************************)
+
+(* Used to generate micromega.ml *)
+
+Require Import ZMicromega.
+Require Import QMicromega.
+Require Import VarMap.
+Require Import RingMicromega.
+Require Import NArith.
+
+Extraction "micromega.ml" List.map simpl_cone map_cone indexes n_of_Z Nnat.N_of_nat ZTautoChecker QTautoChecker find.
diff --git a/contrib/micromega/Micromegatac.v b/contrib/micromega/Micromegatac.v
new file mode 100644
index 000000000..5459634c9
--- /dev/null
+++ b/contrib/micromega/Micromegatac.v
@@ -0,0 +1,79 @@
+(************************************************************************)
+(* 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 *)
+(************************************************************************)
+(* *)
+(* Micromega: A reflexive tactic using the Positivstellensatz *)
+(* *)
+(* Frédéric Besson (Irisa/Inria) 2006-2008 *)
+(* *)
+(************************************************************************)
+
+Require Import ZMicromega.
+Require Import QMicromega.
+Require Import RMicromega.
+Require Import QArith.
+Require Export Ring_normalize.
+Require Import ZArith.
+Require Import Reals.
+Require Export RingMicromega.
+Require Import VarMap.
+Require Tauto.
+
+Ltac micromegac dom d :=
+ let tac := lazymatch dom with
+ | Z =>
+ micromegap d ;
+ intros __wit __varmap __ff ;
+ change (Tauto.eval_f (Zeval_formula (@find Z Z0 __varmap)) __ff) ;
+ apply (ZTautoChecker_sound __ff __wit); vm_compute ; reflexivity
+ | R =>
+ rmicromegap d ;
+ intros __wit __varmap __ff ;
+ change (Tauto.eval_f (Reval_formula (@find R 0%R __varmap)) __ff) ;
+ apply (RTautoChecker_sound __ff __wit); vm_compute ; reflexivity
+ | _ => fail "Unsupported domain"
+ end in tac.
+
+Tactic Notation "micromega" constr(dom) int_or_var(n) := micromegac dom n.
+Tactic Notation "micromega" constr(dom) := micromegac dom ltac:-1.
+
+Ltac zfarkas := omicronp ;
+ intros __wit __varmap __ff ;
+ change (Tauto.eval_f (Zeval_formula (@find Z Z0 __varmap)) __ff) ;
+ apply (ZTautoChecker_sound __ff __wit); vm_compute ; reflexivity.
+
+Ltac omicron dom :=
+ let tac := lazymatch dom with
+ | Z =>
+ zomicronp ;
+ intros __wit __varmap __ff ;
+ change (Tauto.eval_f (Zeval_formula (@find Z Z0 __varmap)) __ff) ;
+ apply (ZTautoChecker_sound __ff __wit); vm_compute ; reflexivity
+ | Q =>
+ qomicronp ;
+ intros __wit __varmap __ff ;
+ change (Tauto.eval_f (Qeval_formula (@find Q 0%Q __varmap)) __ff) ;
+ apply (QTautoChecker_sound __ff __wit); vm_compute ; reflexivity
+ | R =>
+ romicronp ;
+ intros __wit __varmap __ff ;
+ change (Tauto.eval_f (Reval_formula (@find R 0%R __varmap)) __ff) ;
+ apply (RTautoChecker_sound __ff __wit); vm_compute ; reflexivity
+ | _ => fail "Unsupported domain"
+ end in tac.
+
+Ltac sos dom :=
+ let tac := lazymatch dom with
+ | Z =>
+ sosp ;
+ intros __wit __varmap __ff ;
+ change (Tauto.eval_f (Zeval_formula (@find Z Z0 __varmap)) __ff) ;
+ apply (ZTautoChecker_sound __ff __wit); vm_compute ; reflexivity
+ | _ => fail "Unsupported domain"
+ end in tac.
+
+
diff --git a/contrib/micromega/OrderedRing.v b/contrib/micromega/OrderedRing.v
index c6dd5ccf0..149b77316 100644
--- a/contrib/micromega/OrderedRing.v
+++ b/contrib/micromega/OrderedRing.v
@@ -1,6 +1,18 @@
+(************************************************************************)
+(* 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 *)
+(************************************************************************)
+(* Evgeny Makarov, INRIA, 2007 *)
+(************************************************************************)
+
Require Import Setoid.
Require Import Ring.
+(** Generic properties of ordered rings on a setoid equality *)
+
Set Implicit Arguments.
Module Import OrderedRingSyntax.
@@ -29,7 +41,7 @@ Notation "x ~= y" := (~ req x y).
Notation "x <= y" := (rle x y).
Notation "x < y" := (rlt x y).
-Record SOR : Prop := mk_SOR_theory {
+Record SOR : Type := mk_SOR_theory {
SORsetoid : Setoid_Theory R req;
SORplus_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 + y1 == x2 + y2;
SORtimes_wd : forall x1 x2, x1 == x2 -> forall y1 y2, y1 == y2 -> x1 * y1 == x2 * y2;
@@ -71,12 +83,14 @@ Notation "x ~= y" := (~ req x y).
Notation "x <= y" := (rle x y).
Notation "x < y" := (rlt x y).
+
Add Relation R req
- reflexivity proved by sor.(SORsetoid).(@Equivalence_Reflexive _ _)
- symmetry proved by sor.(SORsetoid).(@Equivalence_Symmetric _ _)
- transitivity proved by sor.(SORsetoid).(@Equivalence_Transitive _ _)
+ reflexivity proved by sor.(SORsetoid).(@Equivalence_Reflexive _ _ )
+ symmetry proved by sor.(SORsetoid).(@Equivalence_Symmetric _ _ )
+ transitivity proved by sor.(SORsetoid).(@Equivalence_Transitive _ _ )
as sor_setoid.
+
Add Morphism rplus with signature req ==> req ==> req as rplus_morph.
Proof.
exact sor.(SORplus_wd).
@@ -148,7 +162,8 @@ Qed.
Theorem Rminus_eq_0 : forall n m : R, n - m == 0 <-> n == m.
Proof.
intros n m.
-split; intro H. setoid_replace n with ((n - m) + m) by ring. rewrite H. now rewrite Rplus_0_l.
+split; intro H. setoid_replace n with ((n - m) + m) by ring. rewrite H.
+now rewrite Rplus_0_l.
rewrite H; ring.
Qed.
diff --git a/contrib/micromega/QMicromega.v b/contrib/micromega/QMicromega.v
new file mode 100644
index 000000000..9e95f6c49
--- /dev/null
+++ b/contrib/micromega/QMicromega.v
@@ -0,0 +1,259 @@
+(************************************************************************)
+(* 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 *)
+(************************************************************************)
+(* *)
+(* Micromega: A reflexive tactic using the Positivstellensatz *)
+(* *)
+(* Frédéric Besson (Irisa/Inria) 2006-2008 *)
+(* *)
+(************************************************************************)
+
+Require Import OrderedRing.
+Require Import RingMicromega.
+Require Import Refl.
+Require Import QArith.
+Require Import Qring.
+
+(* Qsrt has been removed from the library ? *)
+Definition Qsrt : ring_theory 0 1 Qplus Qmult Qminus Qopp Qeq.
+Proof.
+ 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.
+Qed.
+
+
+Add Ring Qring : Qsrt.
+
+Lemma Qmult_neutral : forall x , 0 * x == 0.
+Proof.
+ intros.
+ compute.
+ reflexivity.
+Qed.
+
+(* Is there any qarith database ? *)
+
+Lemma Qsor : SOR 0 1 Qplus Qmult Qminus Qopp Qeq Qle Qlt.
+Proof.
+ constructor; intros ; subst ; try (intuition (subst; auto with qarith)).
+ apply Q_Setoid.
+ rewrite H ; rewrite H0 ; reflexivity.
+ rewrite H ; rewrite H0 ; reflexivity.
+ rewrite H ; auto ; reflexivity.
+ rewrite <- H ; rewrite <- H0 ; auto.
+ rewrite H ; rewrite H0 ; auto.
+ rewrite <- H ; rewrite <- H0 ; auto.
+ rewrite H ; rewrite H0 ; auto.
+ apply Qsrt.
+ apply Qle_refl.
+ apply Qle_antisym ; auto.
+ eapply Qle_trans ; eauto.
+ apply Qlt_le_weak ; auto.
+ apply (Qlt_not_eq n m H H0) ; auto.
+ destruct (Qle_lt_or_eq _ _ H0) ; auto.
+ tauto.
+ destruct(Q_dec n m) as [[H1 |H1] | H1 ] ; tauto.
+ apply (Qplus_le_compat p p n m (Qle_refl p) H).
+ generalize (Qmult_lt_compat_r 0 n m H0 H).
+ rewrite Qmult_neutral.
+ auto.
+ compute in H.
+ discriminate.
+Qed.
+
+Definition Qeq_bool (p q : Q) : bool := Zeq_bool (Qnum p * ' Qden q)%Z (Qnum q * ' Qden p)%Z.
+
+Definition Qle_bool (x y : Q) : bool := Zle_bool (Qnum x * ' Qden y)%Z (Qnum y * ' Qden x)%Z.
+
+Require ZMicromega.
+
+Lemma Qeq_bool_ok : forall x y, Qeq_bool x y = true -> x == y.
+Proof.
+ intros.
+ unfold Qeq_bool in H.
+ unfold Qeq.
+ apply (Zeqb_ok _ _ H).
+Qed.
+
+
+Lemma Qeq_bool_neq : forall x y, Qeq_bool x y = false -> ~ x == y.
+Proof.
+ unfold Qeq_bool,Qeq.
+ red ; intros ; subst.
+ rewrite H0 in H.
+ apply (ZMicromega.Zeq_bool_neq _ _ H).
+ reflexivity.
+Qed.
+
+Lemma Qle_bool_imp_le : forall x y : Q, Qle_bool x y = true -> x <= y.
+Proof.
+ unfold Qle_bool, Qle.
+ intros.
+ apply Zle_bool_imp_le ; auto.
+Qed.
+
+
+
+
+Lemma QSORaddon :
+ SORaddon 0 1 Qplus Qmult Qminus Qopp Qeq Qle (* ring elements *)
+ 0 1 Qplus Qmult Qminus Qopp (* coefficients *)
+ Qeq_bool Qle_bool
+ (fun x => x) (fun x => x) (pow_N 1 Qmult).
+Proof.
+ constructor.
+ constructor ; intros ; try reflexivity.
+ apply Qeq_bool_ok ; auto.
+ constructor.
+ reflexivity.
+ intros x y.
+ apply Qeq_bool_neq ; auto.
+ apply Qle_bool_imp_le.
+Qed.
+
+
+(*Definition Zeval_expr := eval_pexpr 0 Zplus Zmult Zminus Zopp (fun x => x) (fun x => Z_of_N x) (Zpower).*)
+Require Import EnvRing.
+
+Fixpoint Qeval_expr (env: PolEnv Q) (e: PExpr Q) : Q :=
+ match e with
+ | PEc c => c
+ | PEX j => env j
+ | PEadd pe1 pe2 => (Qeval_expr env pe1) + (Qeval_expr env pe2)
+ | PEsub pe1 pe2 => (Qeval_expr env pe1) - (Qeval_expr env pe2)
+ | PEmul pe1 pe2 => (Qeval_expr env pe1) * (Qeval_expr env pe2)
+ | PEopp pe1 => - (Qeval_expr env pe1)
+ | PEpow pe1 n => Qpower (Qeval_expr env pe1) (Z_of_N n)
+ end.
+
+Lemma Qeval_expr_simpl : forall env e,
+ Qeval_expr env e =
+ match e with
+ | PEc c => c
+ | PEX j => env j
+ | PEadd pe1 pe2 => (Qeval_expr env pe1) + (Qeval_expr env pe2)
+ | PEsub pe1 pe2 => (Qeval_expr env pe1) - (Qeval_expr env pe2)
+ | PEmul pe1 pe2 => (Qeval_expr env pe1) * (Qeval_expr env pe2)
+ | PEopp pe1 => - (Qeval_expr env pe1)
+ | PEpow pe1 n => Qpower (Qeval_expr env pe1) (Z_of_N n)
+ end.
+Proof.
+ destruct e ; reflexivity.
+Qed.
+
+Definition Qeval_expr' := eval_pexpr Qplus Qmult Qminus Qopp (fun x => x) (fun x => x) (pow_N 1 Qmult).
+
+Lemma QNpower : forall r n, r ^ Z_of_N n = pow_N 1 Qmult r n.
+Proof.
+ destruct n ; reflexivity.
+Qed.
+
+
+Lemma Qeval_expr_compat : forall env e, Qeval_expr env e = Qeval_expr' env e.
+Proof.
+ induction e ; simpl ; subst ; try congruence.
+ rewrite IHe.
+ apply QNpower.
+Qed.
+
+Definition Qeval_op2 (o : Op2) : Q -> Q -> Prop :=
+match o with
+| OpEq => Qeq
+| OpNEq => fun x y => ~ x == y
+| OpLe => Qle
+| OpGe => Qge
+| OpLt => Qlt
+| OpGt => Qgt
+end.
+
+Definition Qeval_formula (e:PolEnv Q) (ff : Formula Q) :=
+ let (lhs,o,rhs) := ff in Qeval_op2 o (Qeval_expr e lhs) (Qeval_expr e rhs).
+
+Definition Qeval_formula' :=
+ eval_formula Qplus Qmult Qminus Qopp Qeq Qle Qlt (fun x => x) (fun x => x) (pow_N 1 Qmult).
+
+Lemma Qeval_formula_compat : forall env f, Qeval_formula env f <-> Qeval_formula' env f.
+Proof.
+ intros.
+ unfold Qeval_formula.
+ destruct f.
+ repeat rewrite Qeval_expr_compat.
+ unfold Qeval_formula'.
+ unfold Qeval_expr'.
+ split ; destruct Fop ; simpl; auto.
+Qed.
+
+
+
+Definition Qeval_nformula :=
+ eval_nformula 0 Qplus Qmult Qminus Qopp Qeq Qle Qlt (fun x => x) (fun x => x) (pow_N 1 Qmult).
+
+Definition Qeval_op1 (o : Op1) : Q -> Prop :=
+match o with
+| Equal => fun x : Q => x == 0
+| NonEqual => fun x : Q => ~ x == 0
+| Strict => fun x : Q => 0 < x
+| NonStrict => fun x : Q => 0 <= x
+end.
+
+Lemma Qeval_nformula_simpl : forall env f, Qeval_nformula env f = (let (p, op) := f in Qeval_op1 op (Qeval_expr env p)).
+Proof.
+ intros.
+ destruct f.
+ rewrite Qeval_expr_compat.
+ reflexivity.
+Qed.
+
+Lemma Qeval_nformula_dec : forall env d, (Qeval_nformula env d) \/ ~ (Qeval_nformula env d).
+Proof.
+ exact (fun env d =>eval_nformula_dec Qsor (fun x => x) (fun x => x) (pow_N 1 Qmult) env d).
+Qed.
+
+Definition QWitness := ConeMember Q.
+
+Definition QWeakChecker := check_normalised_formulas 0 1 Qplus Qmult Qminus Qopp Qeq_bool Qle_bool.
+
+Require Import List.
+
+Lemma QWeakChecker_sound : forall (l : list (NFormula Q)) (cm : QWitness),
+ QWeakChecker l cm = true ->
+ forall env, make_impl (Qeval_nformula env) l False.
+Proof.
+ intros l cm H.
+ intro.
+ unfold Qeval_nformula.
+ apply (checker_nf_sound Qsor QSORaddon l cm).
+ unfold QWeakChecker in H.
+ exact H.
+Qed.
+
+Require Import Tauto.
+
+Definition QTautoChecker (f : BFormula (Formula Q)) (w: list QWitness) : bool :=
+ @tauto_checker (Formula Q) (NFormula Q) (@cnf_normalise Q) (@cnf_negate Q) QWitness QWeakChecker f w.
+
+Lemma QTautoChecker_sound : forall f w, QTautoChecker f w = true -> forall env, eval_f (Qeval_formula env) f.
+Proof.
+ intros f w.
+ unfold QTautoChecker.
+ apply (tauto_checker_sound Qeval_formula Qeval_nformula).
+ apply Qeval_nformula_dec.
+ intros. rewrite Qeval_formula_compat. unfold Qeval_formula'. now apply (cnf_normalise_correct Qsor).
+ intros. rewrite Qeval_formula_compat. unfold Qeval_formula'. now apply (cnf_negate_correct Qsor).
+ intros t w0.
+ apply QWeakChecker_sound.
+Qed.
+
+
diff --git a/contrib/micromega/RMicromega.v b/contrib/micromega/RMicromega.v
new file mode 100644
index 000000000..a662744bc
--- /dev/null
+++ b/contrib/micromega/RMicromega.v
@@ -0,0 +1,148 @@
+(************************************************************************)
+(* 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 *)
+(************************************************************************)
+(* *)
+(* Micromega: A reflexive tactic using the Positivstellensatz *)
+(* *)
+(* Frédéric Besson (Irisa/Inria) 2006-2008 *)
+(* *)
+(************************************************************************)
+
+Require Import OrderedRing.
+Require Import RingMicromega.
+Require Import Refl.
+Require Import Reals.
+Require Setoid.
+
+Definition Rsrt : ring_theory R0 R1 Rplus Rmult Rminus Ropp (@eq R).
+Proof.
+ constructor.
+ exact Rplus_0_l.
+ exact Rplus_comm.
+ intros. rewrite Rplus_assoc. auto.
+ exact Rmult_1_l.
+ exact Rmult_comm.
+ intros ; rewrite Rmult_assoc ; auto.
+ intros. rewrite Rmult_comm. rewrite Rmult_plus_distr_l.
+ rewrite (Rmult_comm z). rewrite (Rmult_comm z). auto.
+ reflexivity.
+ exact Rplus_opp_r.
+Qed.
+
+Add Ring Rring : Rsrt.
+Open Scope R_scope.
+
+Lemma Rmult_neutral : forall x:R , 0 * x = 0.
+Proof.
+ intro ; ring.
+Qed.
+
+
+Lemma Rsor : SOR R0 R1 Rplus Rmult Rminus Ropp (@eq R) Rle Rlt.
+Proof.
+ constructor; intros ; subst ; try (intuition (subst; try ring ; auto with real)).
+ constructor.
+ constructor.
+ unfold RelationClasses.Symmetric. auto.
+ unfold RelationClasses.Transitive. intros. subst. reflexivity.
+ apply Rsrt.
+ eapply Rle_trans ; eauto.
+ apply (Rlt_irrefl m) ; auto.
+ apply Rnot_le_lt. auto with real.
+ destruct (total_order_T n m) as [ [H1 | H1] | H1] ; auto.
+ intros.
+ rewrite <- (Rmult_neutral m).
+ apply (Rmult_lt_compat_r) ; auto.
+Qed.
+
+Require ZMicromega.
+
+(* R with coeffs in Z *)
+
+Lemma RZSORaddon :
+ SORaddon R0 R1 Rplus Rmult Rminus Ropp (@eq R) Rle (* ring elements *)
+ 0%Z 1%Z Zplus Zmult Zminus Zopp (* coefficients *)
+ Zeq_bool Zle_bool
+ IZR Nnat.nat_of_N pow.
+Proof.
+ constructor.
+ constructor ; intros ; try reflexivity.
+ apply plus_IZR.
+ symmetry. apply Z_R_minus.
+ apply mult_IZR.
+ apply Ropp_Ropp_IZR.
+ apply IZR_eq.
+ apply Zeqb_ok ; auto.
+ apply R_power_theory.
+ intros x y.
+ intro.
+ apply IZR_neq.
+ apply ZMicromega.Zeq_bool_neq ; auto.
+ intros. apply IZR_le. apply Zle_bool_imp_le. auto.
+Qed.
+
+
+Require Import EnvRing.
+
+Definition INZ (n:N) : R :=
+ match n with
+ | N0 => IZR 0%Z
+ | Npos p => IZR (Zpos p)
+ end.
+
+Definition Reval_expr := eval_pexpr Rplus Rmult Rminus Ropp IZR Nnat.nat_of_N pow.
+
+
+Definition Reval_formula :=
+ eval_formula Rplus Rmult Rminus Ropp (@eq R) Rle Rlt IZR Nnat.nat_of_N pow.
+
+
+Definition Reval_nformula :=
+ eval_nformula 0 Rplus Rmult Rminus Ropp (@eq R) Rle Rlt IZR Nnat.nat_of_N pow.
+
+
+Lemma Reval_nformula_dec : forall env d, (Reval_nformula env d) \/ ~ (Reval_nformula env d).
+Proof.
+ exact (fun env d =>eval_nformula_dec Rsor IZR Nnat.nat_of_N pow env d).
+Qed.
+
+Definition RWitness := ConeMember Z.
+
+Definition RWeakChecker := check_normalised_formulas 0%Z 1%Z Zplus Zmult Zminus Zopp Zeq_bool Zle_bool.
+
+Require Import List.
+
+Lemma RWeakChecker_sound : forall (l : list (NFormula Z)) (cm : RWitness),
+ RWeakChecker l cm = true ->
+ forall env, make_impl (Reval_nformula env) l False.
+Proof.
+ intros l cm H.
+ intro.
+ unfold Reval_nformula.
+ apply (checker_nf_sound Rsor RZSORaddon l cm).
+ unfold RWeakChecker in H.
+ exact H.
+Qed.
+
+Require Import Tauto.
+
+Definition RTautoChecker (f : BFormula (Formula Z)) (w: list RWitness) : bool :=
+ @tauto_checker (Formula Z) (NFormula Z) (@cnf_normalise Z) (@cnf_negate Z) RWitness RWeakChecker f w.
+
+Lemma RTautoChecker_sound : forall f w, RTautoChecker f w = true -> forall env, eval_f (Reval_formula env) f.
+Proof.
+ intros f w.
+ unfold RTautoChecker.
+ apply (tauto_checker_sound Reval_formula Reval_nformula).
+ apply Reval_nformula_dec.
+ intros. unfold Reval_formula. now apply (cnf_normalise_correct Rsor).
+ intros. unfold Reval_formula. now apply (cnf_negate_correct Rsor).
+ intros t w0.
+ apply RWeakChecker_sound.
+Qed.
+
+
diff --git a/contrib/micromega/Refl.v b/contrib/micromega/Refl.v
index 8dced223b..801d8b212 100644
--- a/contrib/micromega/Refl.v
+++ b/contrib/micromega/Refl.v
@@ -1,11 +1,19 @@
-(********************************************************************)
-(* *)
-(* Micromega: A reflexive tactics using the Positivstellensatz *)
-(* *)
-(* Frédéric Besson (Irisa/Inria) 2006 *)
-(* *)
-(********************************************************************)
+(************************************************************************)
+(* 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 *)
+(************************************************************************)
+(* *)
+(* Micromega: A reflexive tactic using the Positivstellensatz *)
+(* *)
+(* Frédéric Besson (Irisa/Inria) 2006-2008 *)
+(* *)
+(************************************************************************)
+
Require Import List.
+Require Setoid.
Set Implicit Arguments.
@@ -38,6 +46,7 @@ Proof.
intros; destruct l; simpl; tauto.
Qed.
+
Lemma make_conj_impl : forall (A : Type) (eval : A -> Prop) (l : list A) (g : Prop),
(make_conj eval l -> g) <-> make_impl eval l g.
Proof.
@@ -72,3 +81,49 @@ Proof.
apply IHl; auto.
Qed.
+
+
+Lemma make_conj_app : forall A eval l1 l2, @make_conj A eval (l1 ++ l2) <-> @make_conj A eval l1 /\ @make_conj A eval l2.
+Proof.
+ induction l1.
+ simpl.
+ tauto.
+ intros.
+ change ((a::l1) ++ l2) with (a :: (l1 ++ l2)).
+ rewrite make_conj_cons.
+ rewrite IHl1.
+ rewrite make_conj_cons.
+ tauto.
+Qed.
+
+Lemma not_make_conj_cons : forall (A:Type) (t:A) a eval (no_middle_eval : (eval t) \/ ~ (eval t)),
+ ~ make_conj eval (t ::a) -> ~ (eval t) \/ (~ make_conj eval a).
+Proof.
+ intros.
+ simpl in H.
+ destruct a.
+ tauto.
+ tauto.
+Qed.
+
+Lemma not_make_conj_app : forall (A:Type) (t:list A) a eval
+ (no_middle_eval : forall d, eval d \/ ~ eval d) ,
+ ~ make_conj eval (t ++ a) -> (~ make_conj eval t) \/ (~ make_conj eval a).
+Proof.
+ induction t.
+ simpl.
+ tauto.
+ intros.
+ simpl ((a::t)++a0)in H.
+ destruct (@not_make_conj_cons _ _ _ _ (no_middle_eval a) H).
+ left ; red ; intros.
+ apply H0.
+ rewrite make_conj_cons in H1.
+ tauto.
+ destruct (IHt _ _ no_middle_eval H0).
+ left ; red ; intros.
+ apply H1.
+ rewrite make_conj_cons in H2.
+ tauto.
+ right ; auto.
+Qed.
diff --git a/contrib/micromega/RingMicromega.v b/contrib/micromega/RingMicromega.v
index 5aca6e697..6885b82cd 100644
--- a/contrib/micromega/RingMicromega.v
+++ b/contrib/micromega/RingMicromega.v
@@ -1,17 +1,25 @@
+(************************************************************************)
+(* 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 *)
+(************************************************************************)
+(* Evgeny Makarov, INRIA, 2007 *)
+(************************************************************************)
+
Require Import NArith.
Require Import Relation_Definitions.
Require Import Setoid.
(*****)
-Require Import NRing.
-(*****)
-(*Require Import Ring_polynom.*)
+Require Import Env.
+Require Import EnvRing.
(*****)
Require Import List.
Require Import Bool.
Require Import OrderedRing.
Require Import Refl.
-Require Import CheckerMaker.
-Require VarMap.
+
Set Implicit Arguments.
@@ -71,9 +79,9 @@ Record SORaddon := mk_SOR_addon {
Variable addon : SORaddon.
Add Relation R req
- reflexivity proved by sor.(SORsetoid).(@Equivalence_Reflexive _ _)
- symmetry proved by sor.(SORsetoid).(@Equivalence_Symmetric _ _)
- transitivity proved by sor.(SORsetoid).(@Equivalence_Transitive _ _)
+ reflexivity proved by sor.(SORsetoid).(@Equivalence_Reflexive _ _ )
+ symmetry proved by sor.(SORsetoid).(@Equivalence_Symmetric _ _ )
+ transitivity proved by sor.(SORsetoid).(@Equivalence_Transitive _ _ )
as micomega_sor_setoid.
Add Morphism rplus with signature req ==> req ==> req as rplus_morph.
@@ -132,26 +140,26 @@ Qed.
(* Begin Micromega *)
Definition PExprC := PExpr C. (* arbitrary expressions built from +, *, - *)
-Definition PolC := Pol C. (* polynomials in generalized Horner form, defined in Ring_polynom or NRing *)
+Definition PolC := Pol C. (* polynomials in generalized Horner form, defined in Ring_polynom or EnvRing *)
(*****)
-Definition Env := VarMap.t R. (* For interpreting PExprC *)
-Definition PolEnv := VarMap.off_map R. (* For interpreting PolC *)
+(*Definition Env := Env R. (* For interpreting PExprC *)*)
+Definition PolEnv := Env R. (* For interpreting PolC *)
(*****)
(*Definition Env := list R.
Definition PolEnv := list R.*)
(*****)
-(* What benefit do we get, in the case of NRing, from defining eval_pexpr
+(* What benefit do we get, in the case of EnvRing, from defining eval_pexpr
explicitely below and not through PEeval, as the following lemma says? The
function eval_pexpr seems to be a straightforward special case of PEeval
when the environment (i.e., the second last argument of PEeval) of type
off_map (which is (option positive * t)) is (None, env). *)
(*****)
-Fixpoint eval_pexpr (l : Env) (pe : PExprC) {struct pe} : R :=
+Fixpoint eval_pexpr (l : PolEnv) (pe : PExprC) {struct pe} : R :=
match pe with
| PEc c => phi c
-| PEX j => VarMap.find 0 j l
+| PEX j => l j
| PEadd pe1 pe2 => (eval_pexpr l pe1) + (eval_pexpr l pe2)
| PEsub pe1 pe2 => (eval_pexpr l pe1) - (eval_pexpr l pe2)
| PEmul pe1 pe2 => (eval_pexpr l pe1) * (eval_pexpr l pe2)
@@ -159,9 +167,27 @@ match pe with
| PEpow pe1 n => rpow (eval_pexpr l pe1) (pow_phi n)
end.
-Lemma eval_pexpr_PEeval : forall (env : Env) (pe : PExprC),
+
+Lemma eval_pexpr_simpl : forall (l : PolEnv) (pe : PExprC),
+ eval_pexpr l pe =
+ match pe with
+ | PEc c => phi c
+ | PEX j => l j
+ | PEadd pe1 pe2 => (eval_pexpr l pe1) + (eval_pexpr l pe2)
+ | PEsub pe1 pe2 => (eval_pexpr l pe1) - (eval_pexpr l pe2)
+ | PEmul pe1 pe2 => (eval_pexpr l pe1) * (eval_pexpr l pe2)
+ | PEopp pe1 => - (eval_pexpr l pe1)
+ | PEpow pe1 n => rpow (eval_pexpr l pe1) (pow_phi n)
+ end.
+Proof.
+ intros ; destruct pe ; reflexivity.
+Qed.
+
+
+
+Lemma eval_pexpr_PEeval : forall (env : PolEnv) (pe : PExprC),
eval_pexpr env pe =
- PEeval 0 rplus rtimes rminus ropp phi pow_phi rpow (None, env) pe.
+ PEeval rplus rtimes rminus ropp phi pow_phi rpow env pe.
Proof.
induction pe; simpl; intros.
reflexivity.
@@ -177,36 +203,6 @@ Qed.
PEeval 0 rplus rtimes rminus ropp phi pow_phi rpow.*)
(*****)
-Inductive Op2 : Set := (* binary relations *)
-| OpEq
-| OpNEq
-| OpLe
-| OpGe
-| OpLt
-| OpGt.
-
-Definition eval_op2 (o : Op2) : R -> R -> Prop :=
-match o with
-| OpEq => req
-| OpNEq => fun x y : R => x ~= y
-| OpLe => rle
-| OpGe => fun x y : R => y <= x
-| OpLt => fun x y : R => x < y
-| OpGt => fun x y : R => y < x
-end.
-
-Record Formula : Type := {
- Flhs : PExprC;
- Fop : Op2;
- Frhs : PExprC
-}.
-
-Definition eval_formula (env : Env) (f : Formula) : Prop :=
- let (lhs, op, rhs) := f in
- (eval_op2 op) (eval_pexpr env lhs) (eval_pexpr env rhs).
-
-(* We normalize Formulas by moving terms to one side *)
-
Inductive Op1 : Set := (* relations with 0 *)
| Equal (* == 0 *)
| NonEqual (* ~= 0 *)
@@ -223,59 +219,9 @@ match o with
| NonStrict => fun x : R => 0 <= x
end.
-Definition eval_nformula (env : Env) (f : NFormula) : Prop :=
+Definition eval_nformula (env : PolEnv) (f : NFormula) : Prop :=
let (p, op) := f in eval_op1 op (eval_pexpr env p).
-Definition normalise (f : Formula) : NFormula :=
-let (lhs, op, rhs) := f in
- match op with
- | OpEq => (PEsub lhs rhs, Equal)
- | OpNEq => (PEsub lhs rhs, NonEqual)
- | OpLe => (PEsub rhs lhs, NonStrict)
- | OpGe => (PEsub lhs rhs, NonStrict)
- | OpGt => (PEsub lhs rhs, Strict)
- | OpLt => (PEsub rhs lhs, Strict)
- end.
-
-Definition negate (f : Formula) : NFormula :=
-let (lhs, op, rhs) := f in
- match op with
- | OpEq => (PEsub rhs lhs, NonEqual)
- | OpNEq => (PEsub rhs lhs, Equal)
- | OpLe => (PEsub lhs rhs, Strict) (* e <= e' == ~ e > e' *)
- | OpGe => (PEsub rhs lhs, Strict)
- | OpGt => (PEsub rhs lhs, NonStrict)
- | OpLt => (PEsub lhs rhs, NonStrict)
-end.
-
-Theorem normalise_sound :
- forall (env : Env) (f : Formula),
- eval_formula env f -> eval_nformula env (normalise f).
-Proof.
-intros env f H; destruct f as [lhs op rhs]; simpl in *.
-destruct op; simpl in *.
-now apply <- (Rminus_eq_0 sor).
-intros H1. apply -> (Rminus_eq_0 sor) in H1. now apply H.
-now apply -> (Rle_le_minus sor).
-now apply -> (Rle_le_minus sor).
-now apply -> (Rlt_lt_minus sor).
-now apply -> (Rlt_lt_minus sor).
-Qed.
-
-Theorem negate_correct :
- forall (env : Env) (f : Formula),
- eval_formula env f <-> ~ (eval_nformula env (negate f)).
-Proof.
-intros env f; destruct f as [lhs op rhs]; simpl.
-destruct op; simpl.
-symmetry. rewrite (Rminus_eq_0 sor).
-split; intro H; [symmetry; now apply -> (Req_dne sor) | symmetry in H; now apply <- (Req_dne sor)].
-rewrite (Rminus_eq_0 sor). split; intro; now apply (Rneq_symm sor).
-rewrite <- (Rlt_lt_minus sor). now rewrite <- (Rle_ngt sor).
-rewrite <- (Rlt_lt_minus sor). now rewrite <- (Rle_ngt sor).
-rewrite <- (Rle_le_minus sor). now rewrite <- (Rlt_nge sor).
-rewrite <- (Rle_le_minus sor). now rewrite <- (Rlt_nge sor).
-Qed.
Definition OpMult (o o' : Op1) : Op1 :=
match o with
@@ -365,6 +311,12 @@ Inductive Monoid (l : list NFormula) : PExprC -> Prop :=
| M_In : forall p : PExprC, In (p, NonEqual) l -> Monoid l p
| M_Mult : forall (e1 e2 : PExprC), Monoid l e1 -> Monoid l e2 -> Monoid l (PEmul e1 e2).
+(* Do we really need to rely on the intermediate definition of monoid ?
+ InC why the restriction NonEqual ?
+ Could not we consider the IsIdeal as a IsMult ?
+ The same for IsSquare ?
+*)
+
Inductive Cone (l : list (NFormula)) : PExprC -> Op1 -> Prop :=
| InC : forall p op, In (p, op) l -> op <> NonEqual -> Cone l p op
| IsIdeal : forall p, Cone l p Equal -> forall p', Cone l (PEmul p p') Equal
@@ -378,7 +330,7 @@ Inductive Cone (l : list (NFormula)) : PExprC -> Op1 -> Prop :=
(* As promised, if all hypotheses are true in some environment, then every
member of the monoid is nonzero in this environment *)
-Lemma monoid_nonzero : forall (l : list NFormula) (env : Env),
+Lemma monoid_nonzero : forall (l : list NFormula) (env : PolEnv),
(forall f : NFormula, In f l -> eval_nformula env f) ->
forall p : PExprC, Monoid l p -> eval_pexpr env p ~= 0.
Proof.
@@ -392,7 +344,7 @@ Qed.
member of the cone is true as well *)
Lemma cone_true :
- forall (l : list NFormula) (env : Env),
+ forall (l : list NFormula) (env : PolEnv),
(forall (f : NFormula), In f l -> eval_nformula env f) ->
forall (p : PExprC) (op : Op1), Cone l p op ->
op <> NonEqual /\ eval_nformula env (p, op).
@@ -423,7 +375,7 @@ Inductive ConeMember : Type :=
| S_Monoid : MonoidMember -> ConeMember
| S_Mult : ConeMember -> ConeMember -> ConeMember
| S_Add : ConeMember -> ConeMember -> ConeMember
-| S_Pos : forall c : C, cltb cO c = true -> ConeMember (* the proof of cltb 0 c = true should be (refl_equal true) *)
+| S_Pos : C -> ConeMember
| S_Z : ConeMember.
Definition nformula_times (f f' : NFormula) : NFormula :=
@@ -477,7 +429,7 @@ match cm with
| S_Monoid m => let p := eval_monoid l m in (PEmul p p, Strict)
| S_Mult p q => nformula_times (eval_cone l p) (eval_cone l q)
| S_Add p q => nformula_plus (eval_cone l p) (eval_cone l q)
-| S_Pos c _ => (PEc c, Strict)
+| S_Pos c => if cltb cO c then (PEc c, Strict) else (PEc cO, Equal)
| S_Z => (PEc cO, Equal)
end.
@@ -494,7 +446,8 @@ apply IsSquare.
apply IsMonoid. apply eval_monoid_in_monoid.
destruct (eval_cone l cm1). destruct (eval_cone l cm2). unfold nformula_times. now apply IsMult.
destruct (eval_cone l cm1). destruct (eval_cone l cm2). unfold nformula_plus. now apply IsAdd.
-now apply IsPos. apply IsZ.
+case_eq (cO [<] c) ; intros ; [apply IsPos ; auto| apply IsZ].
+apply IsZ.
Qed.
(* (inconsistent_cone_member l p) means (p, op) is in the cone for some op
@@ -507,7 +460,7 @@ verified if we have a certificate. *)
Definition inconsistent_cone_member (l : list NFormula) (p : PExprC) :=
exists op : Op1, Cone l p op /\
- forall env : Env, ~ eval_op1 op (eval_pexpr env p).
+ forall env : PolEnv, ~ eval_op1 op (eval_pexpr env p).
(* If some element of a cone is inconsistent, then the base of the cone
is also inconsistent *)
@@ -540,7 +493,7 @@ let Rops_wd := mk_reqe rplus rtimes ropp req
sor.(SORplus_wd)
sor.(SORtimes_wd)
sor.(SORopp_wd) in
- norm_aux_spec sor.(SORsetoid) Rops_wd (Rth_ARth sor.(SORsetoid) Rops_wd sor.(SORrt))
+ norm_aux_spec sor.(SORsetoid) Rops_wd (Rth_ARth (SORsetoid sor) Rops_wd sor.(SORrt))
addon.(SORrm) addon.(SORpower).
(*****)
(*Definition normalise_pexpr_correct :=
@@ -593,7 +546,7 @@ Definition check_normalised_formulas : list NFormula -> ConeMember -> bool :=
Lemma checker_nf_sound :
forall (l : list NFormula) (cm : ConeMember),
check_normalised_formulas l cm = true ->
- forall env : Env, make_impl (eval_nformula env) l False.
+ forall env : PolEnv, make_impl (eval_nformula env) l False.
Proof.
intros l cm H env.
unfold check_normalised_formulas in H.
@@ -603,29 +556,224 @@ pose proof (eval_cone_in_cone l cm) as H2. now rewrite H1 in H2.
apply check_inconsistent_sound. now rewrite <- H1.
Qed.
-Definition check_formulas :=
- CheckerMaker.check_formulas normalise check_normalised_formulas.
+(** Normalisation of formulae **)
-Theorem check_formulas_sound :
- forall (l : list Formula) (w : ConeMember),
- check_formulas l w = true ->
- forall env : Env, make_impl (eval_formula env) l False.
+Inductive Op2 : Set := (* binary relations *)
+| OpEq
+| OpNEq
+| OpLe
+| OpGe
+| OpLt
+| OpGt.
+
+Definition eval_op2 (o : Op2) : R -> R -> Prop :=
+match o with
+| OpEq => req
+| OpNEq => fun x y : R => x ~= y
+| OpLe => rle
+| OpGe => fun x y : R => y <= x
+| OpLt => fun x y : R => x < y
+| OpGt => fun x y : R => y < x
+end.
+
+Record Formula : Type := {
+ Flhs : PExprC;
+ Fop : Op2;
+ Frhs : PExprC
+}.
+
+Definition eval_formula (env : PolEnv) (f : Formula) : Prop :=
+ let (lhs, op, rhs) := f in
+ (eval_op2 op) (eval_pexpr env lhs) (eval_pexpr env rhs).
+
+(* We normalize Formulas by moving terms to one side *)
+
+Definition normalise (f : Formula) : NFormula :=
+let (lhs, op, rhs) := f in
+ match op with
+ | OpEq => (PEsub lhs rhs, Equal)
+ | OpNEq => (PEsub lhs rhs, NonEqual)
+ | OpLe => (PEsub rhs lhs, NonStrict)
+ | OpGe => (PEsub lhs rhs, NonStrict)
+ | OpGt => (PEsub lhs rhs, Strict)
+ | OpLt => (PEsub rhs lhs, Strict)
+ end.
+
+Definition negate (f : Formula) : NFormula :=
+let (lhs, op, rhs) := f in
+ match op with
+ | OpEq => (PEsub rhs lhs, NonEqual)
+ | OpNEq => (PEsub rhs lhs, Equal)
+ | OpLe => (PEsub lhs rhs, Strict) (* e <= e' == ~ e > e' *)
+ | OpGe => (PEsub rhs lhs, Strict)
+ | OpGt => (PEsub rhs lhs, NonStrict)
+ | OpLt => (PEsub lhs rhs, NonStrict)
+end.
+
+Theorem normalise_sound :
+ forall (env : PolEnv) (f : Formula),
+ eval_formula env f -> eval_nformula env (normalise f).
+Proof.
+intros env f H; destruct f as [lhs op rhs]; simpl in *.
+destruct op; simpl in *.
+now apply <- (Rminus_eq_0 sor).
+intros H1. apply -> (Rminus_eq_0 sor) in H1. now apply H.
+now apply -> (Rle_le_minus sor).
+now apply -> (Rle_le_minus sor).
+now apply -> (Rlt_lt_minus sor).
+now apply -> (Rlt_lt_minus sor).
+Qed.
+
+Theorem negate_correct :
+ forall (env : PolEnv) (f : Formula),
+ eval_formula env f <-> ~ (eval_nformula env (negate f)).
+Proof.
+intros env f; destruct f as [lhs op rhs]; simpl.
+destruct op; simpl.
+symmetry. rewrite (Rminus_eq_0 sor).
+split; intro H; [symmetry; now apply -> (Req_dne sor) | symmetry in H; now apply <- (Req_dne sor)].
+rewrite (Rminus_eq_0 sor). split; intro; now apply (Rneq_symm sor).
+rewrite <- (Rlt_lt_minus sor). now rewrite <- (Rle_ngt sor).
+rewrite <- (Rlt_lt_minus sor). now rewrite <- (Rle_ngt sor).
+rewrite <- (Rle_le_minus sor). now rewrite <- (Rlt_nge sor).
+rewrite <- (Rle_le_minus sor). now rewrite <- (Rlt_nge sor).
+Qed.
+
+(** Another normalistion - this is used for cnf conversion **)
+
+Definition xnormalise (t:Formula) : list (NFormula) :=
+ let (lhs,o,rhs) := t in
+ match o with
+ | OpEq =>
+ (PEsub lhs rhs, Strict)::(PEsub rhs lhs , Strict)::nil
+ | OpNEq => (PEsub lhs rhs,Equal) :: nil
+ | OpGt => (PEsub rhs lhs,NonStrict) :: nil
+ | OpLt => (PEsub lhs rhs,NonStrict) :: nil
+ | OpGe => (PEsub rhs lhs , Strict) :: nil
+ | OpLe => (PEsub lhs rhs ,Strict) :: nil
+ end.
+
+Require Import Tauto.
+
+Definition cnf_normalise (t:Formula) : cnf (NFormula) :=
+ List.map (fun x => x::nil) (xnormalise t).
+
+
+Add Ring SORRing : sor.(SORrt).
+
+Lemma cnf_normalise_correct : forall env t, eval_cnf (eval_nformula env) (cnf_normalise t) -> eval_formula env t.
Proof.
-exact (CheckerMaker.check_formulas_sound eval_formula eval_nformula normalise
- normalise_sound check_normalised_formulas checker_nf_sound).
+ unfold cnf_normalise, xnormalise ; simpl ; intros env t.
+ unfold eval_cnf.
+ destruct t as [lhs o rhs]; case_eq o ; simpl;
+ generalize (eval_pexpr env lhs);
+ generalize (eval_pexpr env rhs) ; intros z1 z2 ; intros.
+ (**)
+ apply sor.(SORle_antisymm).
+ rewrite (Rle_ngt sor). rewrite (Rlt_lt_minus sor). tauto.
+ rewrite (Rle_ngt sor). rewrite (Rlt_lt_minus sor). tauto.
+ now rewrite <- (Rminus_eq_0 sor).
+ rewrite (Rle_ngt sor). rewrite (Rlt_lt_minus sor). auto.
+ rewrite (Rle_ngt sor). rewrite (Rlt_lt_minus sor). auto.
+ rewrite (Rlt_nge sor). rewrite (Rle_le_minus sor). auto.
+ rewrite (Rlt_nge sor). rewrite (Rle_le_minus sor). auto.
+Qed.
+
+Definition xnegate (t:Formula) : list (NFormula) :=
+ let (lhs,o,rhs) := t in
+ match o with
+ | OpEq => (PEsub lhs rhs,Equal) :: nil
+ | OpNEq => (PEsub lhs rhs ,Strict)::(PEsub rhs lhs,Strict)::nil
+ | OpGt => (PEsub lhs rhs,Strict) :: nil
+ | OpLt => (PEsub rhs lhs,Strict) :: nil
+ | OpGe => (PEsub lhs rhs,NonStrict) :: nil
+ | OpLe => (PEsub rhs lhs,NonStrict) :: nil
+ end.
+
+Definition cnf_negate (t:Formula) : cnf (NFormula) :=
+ List.map (fun x => x::nil) (xnegate t).
+
+Lemma cnf_negate_correct : forall env t, eval_cnf (eval_nformula env) (cnf_negate t) -> ~ eval_formula env t.
+Proof.
+ unfold cnf_negate, xnegate ; simpl ; intros env t.
+ unfold eval_cnf.
+ destruct t as [lhs o rhs]; case_eq o ; simpl ;
+ generalize (eval_pexpr env lhs);
+ generalize (eval_pexpr env rhs) ; intros z1 z2 ; intros ;
+ intuition.
+ (**)
+ apply H0.
+ rewrite H1 ; ring.
+ (**)
+ apply H1.
+ apply sor.(SORle_antisymm).
+ rewrite (Rle_ngt sor). rewrite (Rlt_lt_minus sor). tauto.
+ rewrite (Rle_ngt sor). rewrite (Rlt_lt_minus sor). tauto.
+ (**)
+ apply H0. now rewrite (Rle_le_minus sor) in H1.
+ apply H0. now rewrite (Rle_le_minus sor) in H1.
+ apply H0. now rewrite (Rlt_lt_minus sor) in H1.
+ apply H0. now rewrite (Rlt_lt_minus sor) in H1.
Qed.
-Definition check_conj_formulas :=
- CheckerMaker.check_conj_formulas normalise negate check_normalised_formulas.
-Theorem check_conj_formulas_sound :
- forall (l1 : list Formula) (l2 : list Formula) (ws : list ConeMember),
- check_conj_formulas l1 ws l2 = true ->
- forall env : Env, make_impl (eval_formula env) l1 (make_conj (eval_formula env) l2).
+Lemma eval_nformula_dec : forall env d, (eval_nformula env d) \/ ~ (eval_nformula env d).
Proof.
-exact (check_conj_formulas_sound eval_formula eval_nformula normalise negate
- normalise_sound negate_correct check_normalised_formulas checker_nf_sound).
+ intros.
+ destruct d ; simpl.
+ generalize (eval_pexpr env p); intros.
+ destruct o ; simpl.
+ apply (Req_em sor r 0).
+ destruct (Req_em sor r 0) ; tauto.
+ rewrite <- (Rle_ngt sor r 0). generalize (Rle_gt_cases sor r 0). tauto.
+ rewrite <- (Rlt_nge sor r 0). generalize (Rle_gt_cases sor 0 r). tauto.
Qed.
+(** Some syntactic simplifications of expressions and cone elements *)
+
+
+Fixpoint simpl_expr (e:PExprC) : PExprC :=
+ match e with
+ | PEmul y z => let y' := simpl_expr y in let z' := simpl_expr z in
+ match y' , z' with
+ | PEc c , z' => if ceqb c cI then z' else PEmul y' z'
+ | _ , _ => PEmul y' z'
+ end
+ | PEadd x y => PEadd (simpl_expr x) (simpl_expr y)
+ | _ => e
+ end.
+
+
+Definition simpl_cone (e:ConeMember) : ConeMember :=
+ match e with
+ | S_Square t => match simpl_expr t with
+ | PEc c => if ceqb cO c then S_Z else S_Pos (ctimes c c)
+ | x => S_Square x
+ end
+ | S_Mult t1 t2 =>
+ match t1 , t2 with
+ | S_Z , x => S_Z
+ | x , S_Z => S_Z
+ | S_Pos c , S_Pos c' => S_Pos (ctimes c c')
+ | S_Pos p1 , S_Mult (S_Pos p2) x => S_Mult (S_Pos (ctimes p1 p2)) x
+ | S_Pos p1 , S_Mult x (S_Pos p2) => S_Mult (S_Pos (ctimes p1 p2)) x
+ | S_Mult (S_Pos p2) x , S_Pos p1 => S_Mult (S_Pos (ctimes p1 p2)) x
+ | S_Mult x (S_Pos p2) , S_Pos p1 => S_Mult (S_Pos (ctimes p1 p2)) x
+ | S_Pos x , S_Add y z => S_Add (S_Mult (S_Pos x) y) (S_Mult (S_Pos x) z)
+ | S_Pos c , _ => if ceqb cI c then t2 else S_Mult t1 t2
+ | _ , S_Pos c => if ceqb cI c then t1 else S_Mult t1 t2
+ | _ , _ => e
+ end
+ | S_Add t1 t2 =>
+ match t1 , t2 with
+ | S_Z , x => x
+ | x , S_Z => x
+ | x , y => S_Add x y
+ end
+ | _ => e
+ end.
+
+
+
End Micromega.
diff --git a/contrib/micromega/Tauto.v b/contrib/micromega/Tauto.v
new file mode 100644
index 000000000..ef48efa6d
--- /dev/null
+++ b/contrib/micromega/Tauto.v
@@ -0,0 +1,324 @@
+(************************************************************************)
+(* 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 *)
+(************************************************************************)
+(* *)
+(* Micromega: A reflexive tactic using the Positivstellensatz *)
+(* *)
+(* Frédéric Besson (Irisa/Inria) 2006-2008 *)
+(* *)
+(************************************************************************)
+
+Require Import List.
+Require Import Refl.
+Require Import Bool.
+
+Set Implicit Arguments.
+
+
+ Inductive BFormula (A:Type) : Type :=
+ | TT : BFormula A
+ | FF : BFormula A
+ | X : Prop -> BFormula A
+ | A : A -> BFormula A
+ | Cj : BFormula A -> BFormula A -> BFormula A
+ | D : BFormula A-> BFormula A -> BFormula A
+ | N : BFormula A -> BFormula A
+ | I : BFormula A-> BFormula A-> BFormula A.
+
+ Fixpoint eval_f (A:Type) (ev:A -> Prop ) (f:BFormula A) {struct f}: Prop :=
+ match f with
+ | TT => True
+ | FF => False
+ | A a => ev a
+ | X p => p
+ | Cj e1 e2 => (eval_f ev e1) /\ (eval_f ev e2)
+ | D e1 e2 => (eval_f ev e1) \/ (eval_f ev e2)
+ | N e => ~ (eval_f ev e)
+ | I f1 f2 => (eval_f ev f1) -> (eval_f ev f2)
+ end.
+
+
+ Lemma map_simpl : forall A B f l, @map A B f l = match l with
+ | nil => nil
+ | a :: l=> (f a) :: (@map A B f l)
+ end.
+ Proof.
+ destruct l ; reflexivity.
+ Qed.
+
+
+
+ Section S.
+
+ Variable Env : Type.
+ Variable Term : Type.
+ Variable eval : Env -> Term -> Prop.
+ Variable Term' : Type.
+ Variable eval' : Env -> Term' -> Prop.
+
+
+
+ Variable no_middle_eval' : forall env d, (eval' env d) \/ ~ (eval' env d).
+
+
+ Definition clause := list Term'.
+ Definition cnf := list clause.
+
+ Variable normalise : Term -> cnf.
+ Variable negate : Term -> cnf.
+
+
+ Definition tt : cnf := @nil clause.
+ Definition ff : cnf := cons (@nil Term') nil.
+
+
+ Definition or_clause_cnf (t:clause) (f:cnf) : cnf :=
+ List.map (fun x => (t++x)) f.
+
+ Fixpoint or_cnf (f : cnf) (f' : cnf) {struct f}: cnf :=
+ match f with
+ | nil => tt
+ | e :: rst => (or_cnf rst f') ++ (or_clause_cnf e f')
+ end.
+
+
+ Definition and_cnf (f1 : cnf) (f2 : cnf) : cnf :=
+ f1 ++ f2.
+
+ Fixpoint xcnf (pol : bool) (f : BFormula Term) {struct f}: cnf :=
+ match f with
+ | TT => if pol then tt else ff
+ | FF => if pol then ff else tt
+ | X p => if pol then ff else ff (* This is not complete - cannot negate any proposition *)
+ | A x => if pol then normalise x else negate x
+ | N e => xcnf (negb pol) e
+ | Cj e1 e2 =>
+ (if pol then and_cnf else or_cnf) (xcnf pol e1) (xcnf pol e2)
+ | D e1 e2 => (if pol then or_cnf else and_cnf) (xcnf pol e1) (xcnf pol e2)
+ | I e1 e2 => (if pol then or_cnf else and_cnf) (xcnf (negb pol) e1) (xcnf pol e2)
+ end.
+
+ Definition eval_cnf (env : Term' -> Prop) (f:cnf) := make_conj (fun cl => ~ make_conj env cl) f.
+
+
+ Lemma eval_cnf_app : forall env x y, eval_cnf (eval' env) (x++y) -> eval_cnf (eval' env) x /\ eval_cnf (eval' env) y.
+ Proof.
+ unfold eval_cnf.
+ intros.
+ rewrite make_conj_app in H ; auto.
+ Qed.
+
+
+ Lemma or_clause_correct : forall env t f, eval_cnf (eval' env) (or_clause_cnf t f) -> (~ make_conj (eval' env) t) \/ (eval_cnf (eval' env) f).
+ Proof.
+ unfold eval_cnf.
+ unfold or_clause_cnf.
+ induction f.
+ simpl.
+ intros ; right;auto.
+ (**)
+ rewrite map_simpl.
+ intros.
+ rewrite make_conj_cons in H.
+ destruct H as [HH1 HH2].
+ generalize (IHf HH2) ; clear IHf ; intro.
+ destruct H.
+ left ; auto.
+ rewrite make_conj_cons.
+ destruct (not_make_conj_app _ _ _ (no_middle_eval' env) HH1).
+ tauto.
+ tauto.
+ Qed.
+
+ Lemma eval_cnf_cons : forall env a f, (~ make_conj (eval' env) a) -> eval_cnf (eval' env) f -> eval_cnf (eval' env) (a::f).
+ Proof.
+ intros.
+ unfold eval_cnf in *.
+ rewrite make_conj_cons ; eauto.
+ Qed.
+
+ Lemma or_cnf_correct : forall env f f', eval_cnf (eval' env) (or_cnf f f') -> (eval_cnf (eval' env) f) \/ (eval_cnf (eval' env) f').
+ Proof.
+ induction f.
+ unfold eval_cnf.
+ simpl.
+ tauto.
+ (**)
+ intros.
+ simpl in H.
+ destruct (eval_cnf_app _ _ _ H).
+ clear H.
+ destruct (IHf _ H0).
+ destruct (or_clause_correct _ _ _ H1).
+ left.
+ apply eval_cnf_cons ; auto.
+ right ; auto.
+ right ; auto.
+ Qed.
+
+ Variable normalise_correct : forall env t, eval_cnf (eval' env) (normalise t) -> eval env t.
+
+ Variable negate_correct : forall env t, eval_cnf (eval' env) (negate t) -> ~ eval env t.
+
+
+ Lemma xcnf_correct : forall f pol env, eval_cnf (eval' env) (xcnf pol f) -> eval_f (eval env) (if pol then f else N f).
+ Proof.
+ induction f.
+ (* TT *)
+ unfold eval_cnf.
+ simpl.
+ destruct pol ; simpl ; auto.
+ (* FF *)
+ unfold eval_cnf.
+ destruct pol; simpl ; auto.
+ (* P *)
+ simpl.
+ destruct pol ; intros ;simpl.
+ unfold eval_cnf in H.
+ (* Here I have to drop the proposition *)
+ simpl in H.
+ tauto.
+ (* Here, I could store P in the clause *)
+ unfold eval_cnf in H;simpl in H.
+ tauto.
+ (* A *)
+ simpl.
+ destruct pol ; simpl.
+ intros.
+ apply normalise_correct ; auto.
+ (* A 2 *)
+ intros.
+ apply negate_correct ; auto.
+ auto.
+ (* Cj *)
+ destruct pol ; simpl.
+ (* pol = true *)
+ intros.
+ unfold and_cnf in H.
+ destruct (eval_cnf_app _ _ _ H).
+ clear H.
+ split.
+ apply (IHf1 _ _ H0).
+ apply (IHf2 _ _ H1).
+ (* pol = false *)
+ intros.
+ destruct (or_cnf_correct _ _ _ H).
+ generalize (IHf1 false env H0).
+ simpl.
+ tauto.
+ generalize (IHf2 false env H0).
+ simpl.
+ tauto.
+ (* D *)
+ simpl.
+ destruct pol.
+ (* pol = true *)
+ intros.
+ destruct (or_cnf_correct _ _ _ H).
+ generalize (IHf1 _ env H0).
+ simpl.
+ tauto.
+ generalize (IHf2 _ env H0).
+ simpl.
+ tauto.
+ (* pol = true *)
+ unfold and_cnf.
+ intros.
+ destruct (eval_cnf_app _ _ _ H).
+ clear H.
+ simpl.
+ generalize (IHf1 _ _ H0).
+ generalize (IHf2 _ _ H1).
+ simpl.
+ tauto.
+ (**)
+ simpl.
+ destruct pol ; simpl.
+ intros.
+ apply (IHf false) ; auto.
+ intros.
+ generalize (IHf _ _ H).
+ tauto.
+ (* I *)
+ simpl; intros.
+ destruct pol.
+ simpl.
+ intro.
+ destruct (or_cnf_correct _ _ _ H).
+ generalize (IHf1 _ _ H1).
+ simpl in *.
+ tauto.
+ generalize (IHf2 _ _ H1).
+ auto.
+ (* pol = false *)
+ unfold and_cnf in H.
+ simpl in H.
+ destruct (eval_cnf_app _ _ _ H).
+ generalize (IHf1 _ _ H0).
+ generalize (IHf2 _ _ H1).
+ simpl.
+ tauto.
+ Qed.
+
+
+ Variable Witness : Type.
+ Variable checker : list Term' -> Witness -> bool.
+
+ Variable checker_sound : forall t w, checker t w = true -> forall env, make_impl (eval' env) t False.
+
+ Fixpoint cnf_checker (f : cnf) (l : list Witness) {struct f}: bool :=
+ match f with
+ | nil => true
+ | e::f => match l with
+ | nil => false
+ | c::l => match checker e c with
+ | true => cnf_checker f l
+ | _ => false
+ end
+ end
+ end.
+
+ Lemma cnf_checker_sound : forall t w, cnf_checker t w = true -> forall env, eval_cnf (eval' env) t.
+ Proof.
+ unfold eval_cnf.
+ induction t.
+ (* bc *)
+ simpl.
+ auto.
+ (* ic *)
+ simpl.
+ destruct w.
+ intros ; discriminate.
+ case_eq (checker a w) ; intros ; try discriminate.
+ generalize (@checker_sound _ _ H env).
+ generalize (IHt _ H0 env) ; intros.
+ destruct t.
+ red ; intro.
+ rewrite <- make_conj_impl in H2.
+ tauto.
+ rewrite <- make_conj_impl in H2.
+ tauto.
+ Qed.
+
+
+ Definition tauto_checker (f:BFormula Term) (w:list Witness) : bool :=
+ cnf_checker (xcnf true f) w.
+
+ Lemma tauto_checker_sound : forall t w, tauto_checker t w = true -> forall env, eval_f (eval env) t.
+ Proof.
+ unfold tauto_checker.
+ intros.
+ change (eval_f (eval env) t) with (eval_f (eval env) (if true then t else TT Term)).
+ apply (xcnf_correct t true).
+ eapply cnf_checker_sound ; eauto.
+ Qed.
+
+
+
+
+End S.
+
diff --git a/contrib/micromega/VarMap.v b/contrib/micromega/VarMap.v
index 327f6d2d4..240c0fb7c 100644
--- a/contrib/micromega/VarMap.v
+++ b/contrib/micromega/VarMap.v
@@ -1,14 +1,20 @@
-(********************************************************************)
-(* *)
-(* Micromega:A reflexive tactics using the Positivstellensatz *)
-(* *)
-(* Frédéric Besson (Irisa/Inria) 2006 *)
-(* *)
-(********************************************************************)
+(************************************************************************)
+(* 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 *)
+(************************************************************************)
+(* *)
+(* Micromega: A reflexive tactic using the Positivstellensatz *)
+(* *)
+(* Frédéric Besson (Irisa/Inria) 2006-2008 *)
+(* *)
+(************************************************************************)
+
Require Import ZArith.
Require Import Coq.Arith.Max.
Require Import List.
-(*Require Import BinList.*)
Set Implicit Arguments.
(* I have addded a Leaf constructor to the varmap data structure (/contrib/ring/Quote.v)
@@ -26,19 +32,19 @@ Section MakeVarMap.
| Leaf : A -> t
| Node : t -> A -> t -> t .
- Fixpoint find (p:positive) (vm : t ) {struct vm} : A :=
+ Fixpoint find (vm : t ) (p:positive) {struct vm} : A :=
match vm with
| Empty => default
| Leaf i => i
| Node l e r => match p with
| xH => e
- | xO p => find p l
- | xI p => find p r
+ | xO p => find l p
+ | xI p => find r p
end
end.
- (* an off_map (a map with offset) offers the same functionalites as /contrib/setoid_ring/BinList.v - it is used in NRing.v *)
-
+ (* an off_map (a map with offset) offers the same functionalites as /contrib/setoid_ring/BinList.v - it is used in EnvRing.v *)
+(*
Definition off_map := (option positive *t )%type.
@@ -58,8 +64,10 @@ Section MakeVarMap.
end%positive in
find idx m.
+
Definition hd (l:off_map) := nth xH l.
+
Definition tail (l:off_map ) := jump xH l.
@@ -244,7 +252,7 @@ Section MakeVarMap.
reflexivity.
Qed.
-
+*)
End MakeVarMap.
diff --git a/contrib/micromega/ZCoeff.v b/contrib/micromega/ZCoeff.v
index 791e00a90..ced67e39d 100644
--- a/contrib/micromega/ZCoeff.v
+++ b/contrib/micromega/ZCoeff.v
@@ -1,3 +1,13 @@
+(************************************************************************)
+(* 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 *)
+(************************************************************************)
+(* Evgeny Makarov, INRIA, 2007 *)
+(************************************************************************)
+
Require Import OrderedRing.
Require Import RingMicromega.
Require Import ZArith.
@@ -29,6 +39,25 @@ Notation "x ~= y" := (~ req x y).
Notation "x <= y" := (rle x y).
Notation "x < y" := (rlt x y).
+Lemma req_refl : forall x, req x x.
+Proof.
+ destruct sor.(SORsetoid).
+ apply Equivalence_Reflexive.
+Qed.
+
+Lemma req_sym : forall x y, req x y -> req y x.
+Proof.
+ destruct sor.(SORsetoid).
+ apply Equivalence_Symmetric.
+Qed.
+
+Lemma req_trans : forall x y z, req x y -> req y z -> req x z.
+Proof.
+ destruct sor.(SORsetoid).
+ apply Equivalence_Transitive.
+Qed.
+
+
Add Relation R req
reflexivity proved by sor.(SORsetoid).(@Equivalence_Reflexive _ _)
symmetry proved by sor.(SORsetoid).(@Equivalence_Symmetric _ _)
diff --git a/contrib/micromega/ZMicromega.v b/contrib/micromega/ZMicromega.v
new file mode 100644
index 000000000..94c83f73d
--- /dev/null
+++ b/contrib/micromega/ZMicromega.v
@@ -0,0 +1,714 @@
+(************************************************************************)
+(* 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 *)
+(************************************************************************)
+(* *)
+(* Micromega: A reflexive tactic using the Positivstellensatz *)
+(* *)
+(* Frédéric Besson (Irisa/Inria) 2006-2008 *)
+(* *)
+(************************************************************************)
+
+Require Import OrderedRing.
+Require Import RingMicromega.
+Require Import ZCoeff.
+Require Import Refl.
+Require Import ZArith.
+Require Import List.
+Require Import Bool.
+
+Ltac flatten_bool :=
+ repeat match goal with
+ [ id : (_ && _)%bool = true |- _ ] => destruct (andb_prop _ _ id); clear id
+ | [ id : (_ || _)%bool = true |- _ ] => destruct (orb_prop _ _ id); clear id
+ end.
+
+Require Import EnvRing.
+
+Open Scope Z_scope.
+
+Lemma Zsor : SOR 0 1 Zplus Zmult Zminus Zopp (@eq Z) Zle Zlt.
+Proof.
+ constructor ; intros ; subst ; try (intuition (auto with zarith)).
+ apply Zsth.
+ apply Zth.
+ destruct (Ztrichotomy n m) ; intuition (auto with zarith).
+ apply Zmult_lt_0_compat ; auto.
+Qed.
+
+Lemma Zeq_bool_neq : forall x y, Zeq_bool x y = false -> x <> y.
+Proof.
+ red ; intros.
+ subst.
+ unfold Zeq_bool in H.
+ rewrite Zcompare_refl in H.
+ discriminate.
+Qed.
+
+Lemma ZSORaddon :
+ SORaddon 0 1 Zplus Zmult Zminus Zopp (@eq Z) Zle (* ring elements *)
+ 0%Z 1%Z Zplus Zmult Zminus Zopp (* coefficients *)
+ Zeq_bool Zle_bool
+ (fun x => x) (fun x => x) (pow_N 1 Zmult).
+Proof.
+ constructor.
+ constructor ; intros ; try reflexivity.
+ apply Zeqb_ok ; auto.
+ constructor.
+ reflexivity.
+ intros x y.
+ apply Zeq_bool_neq ; auto.
+ apply Zle_bool_imp_le.
+Qed.
+
+
+(*Definition Zeval_expr := eval_pexpr 0 Zplus Zmult Zminus Zopp (fun x => x) (fun x => Z_of_N x) (Zpower).*)
+
+Fixpoint Zeval_expr (env: PolEnv Z) (e: PExpr Z) : Z :=
+ match e with
+ | PEc c => c
+ | PEX j => env j
+ | PEadd pe1 pe2 => (Zeval_expr env pe1) + (Zeval_expr env pe2)
+ | PEsub pe1 pe2 => (Zeval_expr env pe1) - (Zeval_expr env pe2)
+ | PEmul pe1 pe2 => (Zeval_expr env pe1) * (Zeval_expr env pe2)
+ | PEopp pe1 => - (Zeval_expr env pe1)
+ | PEpow pe1 n => Zpower (Zeval_expr env pe1) (Z_of_N n)
+ end.
+
+Lemma Zeval_expr_simpl : forall env e,
+ Zeval_expr env e =
+ match e with
+ | PEc c => c
+ | PEX j => env j
+ | PEadd pe1 pe2 => (Zeval_expr env pe1) + (Zeval_expr env pe2)
+ | PEsub pe1 pe2 => (Zeval_expr env pe1) - (Zeval_expr env pe2)
+ | PEmul pe1 pe2 => (Zeval_expr env pe1) * (Zeval_expr env pe2)
+ | PEopp pe1 => - (Zeval_expr env pe1)
+ | PEpow pe1 n => Zpower (Zeval_expr env pe1) (Z_of_N n)
+ end.
+Proof.
+ destruct e ; reflexivity.
+Qed.
+
+
+Definition Zeval_expr' := eval_pexpr Zplus Zmult Zminus Zopp (fun x => x) (fun x => x) (pow_N 1 Zmult).
+
+Lemma ZNpower : forall r n, r ^ Z_of_N n = pow_N 1 Zmult r n.
+Proof.
+ destruct n.
+ reflexivity.
+ simpl.
+ unfold Zpower_pos.
+ replace (pow_pos Zmult r p) with (1 * (pow_pos Zmult r p)) by ring.
+ generalize 1.
+ induction p; simpl ; intros ; repeat rewrite IHp ; ring.
+Qed.
+
+
+
+Lemma Zeval_expr_compat : forall env e, Zeval_expr env e = Zeval_expr' env e.
+Proof.
+ induction e ; simpl ; subst ; try congruence.
+ rewrite IHe.
+ apply ZNpower.
+Qed.
+
+Definition Zeval_op2 (o : Op2) : Z -> Z -> Prop :=
+match o with
+| OpEq => @eq Z
+| OpNEq => fun x y => ~ x = y
+| OpLe => Zle
+| OpGe => Zge
+| OpLt => Zlt
+| OpGt => Zgt
+end.
+
+Definition Zeval_formula (e: PolEnv Z) (ff : Formula Z) :=
+ let (lhs,o,rhs) := ff in Zeval_op2 o (Zeval_expr e lhs) (Zeval_expr e rhs).
+
+Definition Zeval_formula' :=
+ eval_formula Zplus Zmult Zminus Zopp (@eq Z) Zle Zlt (fun x => x) (fun x => x) (pow_N 1 Zmult).
+
+Lemma Zeval_formula_compat : forall env f, Zeval_formula env f <-> Zeval_formula' env f.
+Proof.
+ intros.
+ unfold Zeval_formula.
+ destruct f.
+ repeat rewrite Zeval_expr_compat.
+ unfold Zeval_formula'.
+ unfold Zeval_expr'.
+ split ; destruct Fop ; simpl; auto with zarith.
+Qed.
+
+
+
+Definition Zeval_nformula :=
+ eval_nformula 0 Zplus Zmult Zminus Zopp (@eq Z) Zle Zlt (fun x => x) (fun x => x) (pow_N 1 Zmult).
+
+Definition Zeval_op1 (o : Op1) : Z -> Prop :=
+match o with
+| Equal => fun x : Z => x = 0
+| NonEqual => fun x : Z => x <> 0
+| Strict => fun x : Z => 0 < x
+| NonStrict => fun x : Z => 0 <= x
+end.
+
+Lemma Zeval_nformula_simpl : forall env f, Zeval_nformula env f = (let (p, op) := f in Zeval_op1 op (Zeval_expr env p)).
+Proof.
+ intros.
+ destruct f.
+ rewrite Zeval_expr_compat.
+ reflexivity.
+Qed.
+
+Lemma Zeval_nformula_dec : forall env d, (Zeval_nformula env d) \/ ~ (Zeval_nformula env d).
+Proof.
+ exact (fun env d =>eval_nformula_dec Zsor (fun x => x) (fun x => x) (pow_N 1%Z Zmult) env d).
+Qed.
+
+Definition ZWitness := ConeMember Z.
+
+Definition ZWeakChecker := check_normalised_formulas 0 1 Zplus Zmult Zminus Zopp Zeq_bool Zle_bool.
+
+Lemma ZWeakChecker_sound : forall (l : list (NFormula Z)) (cm : ZWitness),
+ ZWeakChecker l cm = true ->
+ forall env, make_impl (Zeval_nformula env) l False.
+Proof.
+ intros l cm H.
+ intro.
+ unfold Zeval_nformula.
+ apply (checker_nf_sound Zsor ZSORaddon l cm).
+ unfold ZWeakChecker in H.
+ exact H.
+Qed.
+
+Definition xnormalise (t:Formula Z) : list (NFormula Z) :=
+ let (lhs,o,rhs) := t in
+ match o with
+ | OpEq =>
+ ((PEsub lhs (PEadd rhs (PEc 1))),NonStrict)::((PEsub rhs (PEadd lhs (PEc 1))),NonStrict)::nil
+ | OpNEq => (PEsub lhs rhs,Equal) :: nil
+ | OpGt => (PEsub rhs lhs,NonStrict) :: nil
+ | OpLt => (PEsub lhs rhs,NonStrict) :: nil
+ | OpGe => (PEsub rhs (PEadd lhs (PEc 1)),NonStrict) :: nil
+ | OpLe => (PEsub lhs (PEadd rhs (PEc 1)),NonStrict) :: nil
+ end.
+
+Require Import Tauto.
+
+Definition normalise (t:Formula Z) : cnf (NFormula Z) :=
+ List.map (fun x => x::nil) (xnormalise t).
+
+
+Lemma normalise_correct : forall env t, eval_cnf (Zeval_nformula env) (normalise t) <-> Zeval_formula env t.
+Proof.
+ unfold normalise, xnormalise ; simpl ; intros env t.
+ rewrite Zeval_formula_compat.
+ unfold eval_cnf.
+ destruct t as [lhs o rhs]; case_eq o ; simpl;
+ generalize ( eval_pexpr Zplus Zmult Zminus Zopp (fun x : Z => x)
+ (fun x : BinNat.N => x) (pow_N 1 Zmult) env lhs);
+ generalize (eval_pexpr Zplus Zmult Zminus Zopp (fun x : Z => x)
+ (fun x : BinNat.N => x) (pow_N 1 Zmult) env rhs) ; intros z1 z2 ; intros ; subst;
+ intuition (auto with zarith).
+Qed.
+
+Definition xnegate (t:RingMicromega.Formula Z) : list (NFormula Z) :=
+ let (lhs,o,rhs) := t in
+ match o with
+ | OpEq => (PEsub lhs rhs,Equal) :: nil
+ | OpNEq => ((PEsub lhs (PEadd rhs (PEc 1))),NonStrict)::((PEsub rhs (PEadd lhs (PEc 1))),NonStrict)::nil
+ | OpGt => (PEsub lhs (PEadd rhs (PEc 1)),NonStrict) :: nil
+ | OpLt => (PEsub rhs (PEadd lhs (PEc 1)),NonStrict) :: nil
+ | OpGe => (PEsub lhs rhs,NonStrict) :: nil
+ | OpLe => (PEsub rhs lhs,NonStrict) :: nil
+ end.
+
+Definition negate (t:RingMicromega.Formula Z) : cnf (NFormula Z) :=
+ List.map (fun x => x::nil) (xnegate t).
+
+Lemma negate_correct : forall env t, eval_cnf (Zeval_nformula env) (negate t) <-> ~ Zeval_formula env t.
+Proof.
+ unfold negate, xnegate ; simpl ; intros env t.
+ rewrite Zeval_formula_compat.
+ unfold eval_cnf.
+ destruct t as [lhs o rhs]; case_eq o ; simpl ;
+ generalize ( eval_pexpr Zplus Zmult Zminus Zopp (fun x : Z => x)
+ (fun x : BinNat.N => x) (pow_N 1 Zmult) env lhs);
+ generalize (eval_pexpr Zplus Zmult Zminus Zopp (fun x : Z => x)
+ (fun x : BinNat.N => x) (pow_N 1 Zmult) env rhs) ; intros z1 z2 ; intros ;
+ intuition (auto with zarith).
+Qed.
+
+
+Definition ZweakTautoChecker (w: list ZWitness) (f : BFormula (Formula Z)) : bool :=
+ @tauto_checker (Formula Z) (NFormula Z) normalise negate ZWitness ZWeakChecker f w.
+
+(* To get a complete checker, the proof format has to be enriched *)
+
+Require Import Zdiv.
+Open Scope Z_scope.
+
+Definition ceiling (a b:Z) : Z :=
+ let (q,r) := Zdiv_eucl a b in
+ match r with
+ | Z0 => q
+ | _ => q + 1
+ end.
+
+Lemma narrow_interval_lower_bound : forall a b x, a > 0 -> a * x >= b -> x >= ceiling b a.
+Proof.
+ unfold ceiling.
+ intros.
+ generalize (Z_div_mod b a H).
+ destruct (Zdiv_eucl b a).
+ intros.
+ destruct H1.
+ destruct H2.
+ subst.
+ destruct (Ztrichotomy z0 0) as [ HH1 | [HH2 | HH3]]; destruct z0 ; try auto with zarith ; try discriminate.
+ assert (HH :x >= z \/ x < z) by (destruct (Ztrichotomy x z) ; auto with zarith).
+ destruct HH ;auto.
+ generalize (Zmult_lt_compat_l _ _ _ H3 H1).
+ auto with zarith.
+ clear H2.
+ assert (HH :x >= z +1 \/ x <= z) by (destruct (Ztrichotomy x z) ; intuition (auto with zarith)).
+ destruct HH ;auto.
+ assert (0 < a) by auto with zarith.
+ generalize (Zmult_lt_0_le_compat_r _ _ _ H2 H1).
+ intros.
+ rewrite Zmult_comm in H4.
+ rewrite (Zmult_comm z) in H4.
+ auto with zarith.
+Qed.
+
+Lemma narrow_interval_upper_bound : forall a b x, a > 0 -> a * x <= b -> x <= Zdiv b a.
+Proof.
+ unfold Zdiv.
+ intros.
+ generalize (Z_div_mod b a H).
+ destruct (Zdiv_eucl b a).
+ intros.
+ destruct H1.
+ destruct H2.
+ subst.
+ assert (HH :x <= z \/ z <= x -1) by (destruct (Ztrichotomy x z) ; intuition (auto with zarith)).
+ destruct HH ;auto.
+ assert (0 < a) by auto with zarith.
+ generalize (Zmult_lt_0_le_compat_r _ _ _ H4 H1).
+ intros.
+ ring_simplify in H5.
+ rewrite Zmult_comm in H5.
+ auto with zarith.
+Qed.
+
+
+(* In this case, a certificate is made of a pair of inequations, in 1 variable,
+ that do not have an integer solution.
+ => modify the fourier elimination
+ *)
+Require Import QArith.
+
+
+Inductive ProofTerm : Type :=
+| RatProof : ZWitness -> ProofTerm
+| CutProof : PExprC Z -> Q -> ZWitness -> ProofTerm -> ProofTerm
+| EnumProof : Q -> PExprC Z -> Q -> ZWitness -> ZWitness -> list ProofTerm -> ProofTerm.
+
+(* n/d <= x -> d*x - n >= 0 *)
+
+Definition makeLb (v:PExpr Z) (q:Q) : NFormula Z :=
+ let (n,d) := q in (PEsub (PEmul (PEc (Zpos d)) v) (PEc n),NonStrict).
+
+(* x <= n/d -> d * x <= d *)
+Definition makeUb (v:PExpr Z) (q:Q) : NFormula Z :=
+ let (n,d) := q in
+ (PEsub (PEc n) (PEmul (PEc (Zpos d)) v), NonStrict).
+
+Definition qceiling (q:Q) : Z :=
+ let (n,d) := q in ceiling n (Zpos d).
+
+Definition qfloor (q:Q) : Z :=
+ let (n,d) := q in Zdiv n (Zpos d).
+
+Definition makeLbCut (v:PExprC Z) (q:Q) : NFormula Z :=
+ (PEsub v (PEc (qceiling q)), NonStrict).
+
+Definition neg_nformula (f : NFormula Z) :=
+ let (e,o) := f in
+ (PEopp (PEadd e (PEc 1%Z)), o).
+
+Lemma neg_nformula_sound : forall env f, snd f = NonStrict ->( ~ (Zeval_nformula env (neg_nformula f)) <-> Zeval_nformula env f).
+Proof.
+ unfold neg_nformula.
+ destruct f.
+ simpl.
+ intros ; subst ; simpl in *.
+ split; auto with zarith.
+Qed.
+
+
+Definition cutChecker (l:list (NFormula Z)) (e: PExpr Z) (lb:Q) (pf : ZWitness) : option (NFormula Z) :=
+ let (lb,lc) := (makeLb e lb,makeLbCut e lb) in
+ if ZWeakChecker (neg_nformula lb::l) pf then Some lc else None.
+
+
+Fixpoint ZChecker (l:list (NFormula Z)) (pf : ProofTerm) {struct pf} : bool :=
+ match pf with
+ | RatProof pf => ZWeakChecker l pf
+ | CutProof e q pf rst =>
+ match cutChecker l e q pf with
+ | None => false
+ | Some c => ZChecker (c::l) rst
+ end
+ | EnumProof lb e ub pf1 pf2 rst =>
+ match cutChecker l e lb pf1 , cutChecker l (PEopp e) (Qopp ub) pf2 with
+ | None , _ | _ , None => false
+ | Some _ , Some _ => let (lb',ub') := (qceiling lb, Zopp (qceiling (- ub))) in
+ (fix label (pfs:list ProofTerm) :=
+ fun lb ub =>
+ match pfs with
+ | nil => if Z_gt_dec lb ub then true else false
+ | pf::rsr => andb (ZChecker ((PEsub e (PEc lb), Equal) :: l) pf) (label rsr (Zplus lb 1%Z) ub)
+ end)
+ rst lb' ub'
+ end
+ end.
+
+
+Lemma ZChecker_simpl : forall (pf : ProofTerm) (l:list (NFormula Z)),
+ ZChecker l pf =
+ match pf with
+ | RatProof pf => ZWeakChecker l pf
+ | CutProof e q pf rst =>
+ match cutChecker l e q pf with
+ | None => false
+ | Some c => ZChecker (c::l) rst
+ end
+ | EnumProof lb e ub pf1 pf2 rst =>
+ match cutChecker l e lb pf1 , cutChecker l (PEopp e) (Qopp ub) pf2 with
+ | None , _ | _ , None => false
+ | Some _ , Some _ => let (lb',ub') := (qceiling lb, Zopp (qceiling (- ub))) in
+ (fix label (pfs:list ProofTerm) :=
+ fun lb ub =>
+ match pfs with
+ | nil => if Z_gt_dec lb ub then true else false
+ | pf::rsr => andb (ZChecker ((PEsub e (PEc lb), Equal) :: l) pf) (label rsr (Zplus lb 1%Z) ub)
+ end)
+ rst lb' ub'
+ end
+ end.
+Proof.
+ destruct pf ; reflexivity.
+Qed.
+
+(*
+Fixpoint depth (n:nat) : ProofTerm -> option nat :=
+ match n with
+ | O => fun pf => None
+ | S n =>
+ fun pf =>
+ match pf with
+ | RatProof _ => Some O
+ | CutProof _ _ _ p => option_map S (depth n p)
+ | EnumProof _ _ _ _ _ l =>
+ let f := fun pf x =>
+ match x , depth n pf with
+ | None , _ | _ , None => None
+ | Some n1 , Some n2 => Some (Max.max n1 n2)
+ end in
+ List.fold_right f (Some O) l
+ end
+ end.
+*)
+Fixpoint bdepth (pf : ProofTerm) : nat :=
+ match pf with
+ | RatProof _ => O
+ | CutProof _ _ _ p => S (bdepth p)
+ | EnumProof _ _ _ _ _ l => S (List.fold_right (fun pf x => Max.max (bdepth pf) x) O l)
+ end.
+
+Require Import Wf_nat.
+
+Lemma in_bdepth : forall l a b p c c0 y, In y l -> ltof ProofTerm bdepth y (EnumProof a b p c c0 l).
+Proof.
+ induction l.
+ simpl.
+ tauto.
+ simpl.
+ intros.
+ destruct H.
+ subst.
+ unfold ltof.
+ simpl.
+ generalize ( (fold_right
+ (fun (pf : ProofTerm) (x : nat) => Max.max (bdepth pf) x) 0%nat l)).
+ intros.
+ generalize (bdepth y) ; intros.
+ generalize (Max.max_l n0 n) (Max.max_r n0 n).
+ omega.
+ generalize (IHl a0 b p c c0 y H).
+ unfold ltof.
+ simpl.
+ generalize ( (fold_right (fun (pf : ProofTerm) (x : nat) => Max.max (bdepth pf) x) 0%nat
+ l)).
+ intros.
+ generalize (Max.max_l (bdepth a) n) (Max.max_r (bdepth a) n).
+ omega.
+Qed.
+
+Lemma lb_lbcut : forall env e q, Zeval_nformula env (makeLb e q) -> Zeval_nformula env (makeLbCut e q).
+Proof.
+ unfold makeLb, makeLbCut.
+ destruct q.
+ rewrite Zeval_nformula_simpl.
+ rewrite Zeval_nformula_simpl.
+ unfold Zeval_op1.
+ rewrite Zeval_expr_simpl.
+ rewrite Zeval_expr_simpl.
+ rewrite Zeval_expr_simpl.
+ intro.
+ rewrite Zeval_expr_simpl.
+ revert H.
+ generalize (Zeval_expr env e).
+ rewrite Zeval_expr_simpl.
+ rewrite Zeval_expr_simpl.
+ unfold qceiling.
+ intros.
+ assert ( z >= ceiling Qnum (' Qden))%Z.
+ apply narrow_interval_lower_bound.
+ compute.
+ reflexivity.
+ destruct z ; auto with zarith.
+ auto with zarith.
+Qed.
+
+Lemma cutChecker_sound : forall e lb pf l res, cutChecker l e lb pf = Some res ->
+ forall env, make_impl (Zeval_nformula env) l (Zeval_nformula env res).
+Proof.
+ unfold cutChecker.
+ intros.
+ revert H.
+ case_eq (ZWeakChecker (neg_nformula (makeLb e lb) :: l) pf); intros ; [idtac | discriminate].
+ generalize (ZWeakChecker_sound _ _ H env).
+ intros.
+ inversion H0 ; subst ; clear H0.
+ apply -> make_conj_impl.
+ simpl in H1.
+ rewrite <- make_conj_impl in H1.
+ intros.
+ apply -> neg_nformula_sound ; auto.
+ red ; intros.
+ apply H1 ; auto.
+ clear H H1 H0.
+ generalize (lb_lbcut env e lb).
+ intros.
+ destruct (Zeval_nformula_dec env ((neg_nformula (makeLb e lb)))).
+ auto.
+ rewrite -> neg_nformula_sound in H0.
+ assert (HH := H H0).
+ rewrite <- neg_nformula_sound in HH.
+ tauto.
+ reflexivity.
+ unfold makeLb.
+ destruct lb.
+ reflexivity.
+Qed.
+
+
+Lemma cutChecker_sound_bound : forall e lb pf l res, cutChecker l e lb pf = Some res ->
+ forall env, make_conj (Zeval_nformula env) l -> (Zeval_expr env e >= qceiling lb)%Z.
+Proof.
+ intros.
+ generalize (cutChecker_sound _ _ _ _ _ H env).
+ intros.
+ rewrite <- (make_conj_impl) in H1.
+ generalize (H1 H0).
+ unfold cutChecker in H.
+ destruct (ZWeakChecker (neg_nformula (makeLb e lb) :: l) pf).
+ unfold makeLbCut in H.
+ inversion H ; subst.
+ clear H.
+ simpl.
+ rewrite Zeval_expr_compat.
+ unfold Zeval_expr'.
+ auto with zarith.
+ discriminate.
+Qed.
+
+
+Lemma ZChecker_sound : forall w l, ZChecker l w = true -> forall env, make_impl (Zeval_nformula env) l False.
+Proof.
+ induction w using (well_founded_ind (well_founded_ltof _ bdepth)).
+ destruct w.
+ (* RatProof *)
+ simpl.
+ intros.
+ eapply ZWeakChecker_sound.
+ apply H0.
+ (* CutProof *)
+ simpl.
+ intro.
+ case_eq (cutChecker l p q z) ; intros.
+ generalize (cutChecker_sound _ _ _ _ _ H0 env).
+ intro.
+ assert (make_impl (Zeval_nformula env) (n::l) False).
+ eapply (H w) ; auto.
+ unfold ltof.
+ simpl.
+ auto with arith.
+ simpl in H3.
+ rewrite <- make_conj_impl in H2.
+ rewrite <- make_conj_impl in H3.
+ rewrite <- make_conj_impl.
+ tauto.
+ discriminate.
+ (* EnumProof *)
+ intro.
+ rewrite ZChecker_simpl.
+ case_eq (cutChecker l0 p q z).
+ rename q into llb.
+ case_eq (cutChecker l0 (PEopp p) (- q0) z0).
+ intros.
+ rename q0 into uub.
+ (* get the bounds of the enum *)
+ rewrite <- make_conj_impl.
+ intro.
+ assert (qceiling llb <= Zeval_expr env p <= - qceiling ( - uub))%Z.
+ generalize (cutChecker_sound_bound _ _ _ _ _ H0 env H3).
+ generalize (cutChecker_sound_bound _ _ _ _ _ H1 env H3).
+ intros.
+ rewrite Zeval_expr_simpl in H5.
+ auto with zarith.
+ clear H0 H1.
+ revert H2 H3 H4.
+ generalize (qceiling llb) (- qceiling (- uub))%Z.
+ set (FF := (fix label (pfs : list ProofTerm) (lb ub : Z) {struct pfs} : bool :=
+ match pfs with
+ | nil => if Z_gt_dec lb ub then true else false
+ | pf :: rsr =>
+ (ZChecker ((PEsub p (PEc lb), Equal) :: l0) pf &&
+ label rsr (lb + 1)%Z ub)%bool
+ end)).
+ intros z1 z2.
+ intros.
+ assert (forall x, z1 <= x <= z2 -> exists pr,
+ (In pr l /\
+ ZChecker ((PEsub p (PEc x),Equal) :: l0) pr = true))%Z.
+ clear H.
+ revert H2.
+ clear H4.
+ revert z1 z2.
+ induction l;simpl ;intros.
+ destruct (Z_gt_dec z1 z2).
+ intros.
+ apply False_ind ; omega.
+ discriminate.
+ intros.
+ simpl in H2.
+ flatten_bool.
+ assert (HH:(x = z1 \/ z1 +1 <=x)%Z) by omega.
+ destruct HH.
+ subst.
+ exists a ; auto.
+ assert (z1 + 1 <= x <= z2)%Z by omega.
+ destruct (IHl _ _ H1 _ H4).
+ destruct H5.
+ exists x0 ; split;auto.
+ (*/asser *)
+ destruct (H0 _ H4) as [pr [Hin Hcheker]].
+ assert (make_impl (Zeval_nformula env) ((PEsub p (PEc (Zeval_expr env p)),Equal) :: l0) False).
+ apply (H pr);auto.
+ apply in_bdepth ; auto.
+ rewrite <- make_conj_impl in H1.
+ apply H1.
+ rewrite make_conj_cons.
+ split ;auto.
+ rewrite Zeval_nformula_simpl;
+ unfold Zeval_op1;
+ rewrite Zeval_expr_simpl.
+ generalize (Zeval_expr env p).
+ intros.
+ rewrite Zeval_expr_simpl.
+ auto with zarith.
+ intros ; discriminate.
+ intros ; discriminate.
+Qed.
+
+Definition ZTautoChecker (f : BFormula (Formula Z)) (w: list ProofTerm): bool :=
+ @tauto_checker (Formula Z) (NFormula Z) normalise negate ProofTerm ZChecker f w.
+
+Lemma ZTautoChecker_sound : forall f w, ZTautoChecker f w = true -> forall env, eval_f (Zeval_formula env) f.
+Proof.
+ intros f w.
+ unfold ZTautoChecker.
+ apply (tauto_checker_sound Zeval_formula Zeval_nformula).
+ apply Zeval_nformula_dec.
+ intros env t.
+ rewrite normalise_correct ; auto.
+ intros env t.
+ rewrite negate_correct ; auto.
+ intros t w0.
+ apply ZChecker_sound.
+Qed.
+
+
+Open Scope Z_scope.
+
+
+Fixpoint map_cone (f: nat -> nat) (e:ZWitness) : ZWitness :=
+ match e with
+ | S_In n => S_In _ (f n)
+ | S_Ideal e cm => S_Ideal e (map_cone f cm)
+ | S_Square _ => e
+ | S_Monoid l => S_Monoid _ (List.map f l)
+ | S_Mult cm1 cm2 => S_Mult (map_cone f cm1) (map_cone f cm2)
+ | S_Add cm1 cm2 => S_Add (map_cone f cm1) (map_cone f cm2)
+ | _ => e
+ end.
+
+Fixpoint indexes (e:ZWitness) : list nat :=
+ match e with
+ | S_In n => n::nil
+ | S_Ideal e cm => indexes cm
+ | S_Square e => nil
+ | S_Monoid l => l
+ | S_Mult cm1 cm2 => (indexes cm1)++ (indexes cm2)
+ | S_Add cm1 cm2 => (indexes cm1)++ (indexes cm2)
+ | _ => nil
+ end.
+
+(** To ease bindings from ml code **)
+(*Definition varmap := Quote.varmap.*)
+Definition make_impl := Refl.make_impl.
+Definition make_conj := Refl.make_conj.
+
+Require VarMap.
+
+(*Definition varmap_type := VarMap.t Z. *)
+Definition env := PolEnv Z.
+Definition node := @VarMap.Node Z.
+Definition empty := @VarMap.Empty Z.
+Definition leaf := @VarMap.Leaf Z.
+
+Definition coneMember := ZWitness.
+
+Definition eval := Zeval_formula.
+
+Definition prod_pos_nat := prod positive nat.
+
+Require Import Int.
+
+
+Definition n_of_Z (z:Z) : BinNat.N :=
+ match z with
+ | Z0 => N0
+ | Zpos p => Npos p
+ | Zneg p => N0
+ end.
+
+
+
diff --git a/contrib/micromega/certificate.ml b/contrib/micromega/certificate.ml
new file mode 100644
index 000000000..88e882e61
--- /dev/null
+++ b/contrib/micromega/certificate.ml
@@ -0,0 +1,618 @@
+(************************************************************************)
+(* 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 *)
+(************************************************************************)
+(* *)
+(* Micromega: A reflexive tactic using the Positivstellensatz *)
+(* *)
+(* Frédéric Besson (Irisa/Inria) 2006-2008 *)
+(* *)
+(************************************************************************)
+
+(* We take as input a list of polynomials [p1...pn] and return an unfeasibility
+ certificate polynomial. *)
+
+(*open Micromega.Polynomial*)
+open Big_int
+open Num
+
+module Mc = Micromega
+module Ml2C = Mutils.CamlToCoq
+module C2Ml = Mutils.CoqToCaml
+
+let (<+>) = add_num
+let (<->) = minus_num
+let (<*>) = mult_num
+
+type var = Mc.positive
+
+module Monomial :
+sig
+ type t
+ val const : t
+ val var : var -> t
+ val find : var -> t -> int
+ val mult : var -> t -> t
+ val prod : t -> t -> t
+ val compare : t -> t -> int
+ val pp : out_channel -> t -> unit
+ val fold : (var -> int -> 'a -> 'a) -> t -> 'a -> 'a
+end
+ =
+struct
+ (* A monomial is represented by a multiset of variables *)
+ module Map = Map.Make(struct type t = var let compare = Pervasives.compare end)
+ open Map
+
+ type t = int Map.t
+
+ (* The monomial that corresponds to a constant *)
+ let const = Map.empty
+
+ (* The monomial 'x' *)
+ let var x = Map.add x 1 Map.empty
+
+ (* Get the degre of a variable in a monomial *)
+ let find x m = try find x m with Not_found -> 0
+
+ (* Multiply a monomial by a variable *)
+ let mult x m = add x ( (find x m) + 1) m
+
+ (* Product of monomials *)
+ let prod m1 m2 = Map.fold (fun k d m -> add k ((find k m) + d) m) m1 m2
+
+ (* Total ordering of monomials *)
+ let compare m1 m2 = Map.compare Pervasives.compare m1 m2
+
+ let pp o m = Map.iter (fun k v ->
+ if v = 1 then Printf.fprintf o "x%i." (C2Ml.index k)
+ else Printf.fprintf o "x%i^%i." (C2Ml.index k) v) m
+
+ let fold = fold
+
+end
+
+
+module Poly :
+ (* A polynomial is a map of monomials *)
+ (*
+ This is probably a naive implementation
+ (expected to be fast enough - Coq is probably the bottleneck)
+ *The new ring contribution is using a sparse Horner representation.
+ *)
+sig
+ type t
+ val get : Monomial.t -> t -> num
+ val variable : var -> t
+ val add : Monomial.t -> num -> t -> t
+ val constant : num -> t
+ val mult : Monomial.t -> num -> t -> t
+ val product : t -> t -> t
+ val addition : t -> t -> t
+ val uminus : t -> t
+ val fold : (Monomial.t -> num -> 'a -> 'a) -> t -> 'a -> 'a
+ val pp : out_channel -> t -> unit
+ val compare : t -> t -> int
+end =
+struct
+ (*normalisation bug : 0*x ... *)
+ module P = Map.Make(Monomial)
+ open P
+
+ type t = num P.t
+
+ let pp o p = P.iter (fun k v ->
+ if compare_num v (Int 0) <> 0
+ then
+ if Monomial.compare Monomial.const k = 0
+ then Printf.fprintf o "%s " (string_of_num v)
+ else Printf.fprintf o "%s*%a " (string_of_num v) Monomial.pp k) p
+
+ (* Get the coefficient of monomial mn *)
+ let get : Monomial.t -> t -> num =
+ fun mn p -> try find mn p with Not_found -> (Int 0)
+
+
+ (* The polynomial 1.x *)
+ let variable : var -> t =
+ fun x -> add (Monomial.var x) (Int 1) empty
+
+ (*The constant polynomial *)
+ let constant : num -> t =
+ fun c -> add (Monomial.const) c empty
+
+ (* The addition of a monomial *)
+
+ let add : Monomial.t -> num -> t -> t =
+ fun mn v p ->
+ let vl = (get mn p) <+> v in
+ add mn vl p
+
+
+ (** Design choice: empty is not a polynomial
+ I do not remember why ....
+ **)
+
+ (* The product by a monomial *)
+ let mult : Monomial.t -> num -> t -> t =
+ fun mn v p ->
+ fold (fun mn' v' res -> P.add (Monomial.prod mn mn') (v<*>v') res) p empty
+
+
+ let addition : t -> t -> t =
+ fun p1 p2 -> fold (fun mn v p -> add mn v p) p1 p2
+
+
+ let product : t -> t -> t =
+ fun p1 p2 ->
+ fold (fun mn v res -> addition (mult mn v p2) res ) p1 empty
+
+
+ let uminus : t -> t =
+ fun p -> map (fun v -> minus_num v) p
+
+ let fold = P.fold
+
+ let compare = compare compare_num
+end
+
+open Mutils
+type 'a number_spec = {
+ bigint_to_number : big_int -> 'a;
+ number_to_num : 'a -> num;
+ zero : 'a;
+ unit : 'a;
+ mult : 'a -> 'a -> 'a;
+ eqb : 'a -> 'a -> Mc.bool
+}
+
+let z_spec = {
+ bigint_to_number = Ml2C.bigint ;
+ number_to_num = (fun x -> Big_int (C2Ml.z_big_int x));
+ zero = Mc.Z0;
+ unit = Mc.Zpos Mc.XH;
+ mult = Mc.zmult;
+ eqb = Mc.zeq_bool
+}
+
+
+let q_spec = {
+ bigint_to_number = (fun x -> {Mc.qnum = Ml2C.bigint x; Mc.qden = Mc.XH});
+ number_to_num = C2Ml.q_to_num;
+ zero = {Mc.qnum = Mc.Z0;Mc.qden = Mc.XH};
+ unit = {Mc.qnum = (Mc.Zpos Mc.XH) ; Mc.qden = Mc.XH};
+ mult = Mc.qmult;
+ eqb = Mc.qeq_bool
+}
+
+let r_spec = z_spec
+
+
+
+
+let dev_form n_spec p =
+ let rec dev_form p =
+ match p with
+ | Mc.PEc z -> Poly.constant (n_spec.number_to_num z)
+ | Mc.PEX v -> Poly.variable v
+ | Mc.PEmul(p1,p2) ->
+ let p1 = dev_form p1 in
+ let p2 = dev_form p2 in
+ Poly.product p1 p2
+ | Mc.PEadd(p1,p2) -> Poly.addition (dev_form p1) (dev_form p2)
+ | Mc.PEopp p -> Poly.uminus (dev_form p)
+ | Mc.PEsub(p1,p2) -> Poly.addition (dev_form p1) (Poly.uminus (dev_form p2))
+ | Mc.PEpow(p,n) ->
+ let p = dev_form p in
+ let n = C2Ml.n n in
+ let rec pow n =
+ if n = 0
+ then Poly.constant (n_spec.number_to_num n_spec.unit)
+ else Poly.product p (pow (n-1)) in
+ pow n in
+ dev_form p
+
+
+let monomial_to_polynomial mn =
+ Monomial.fold
+ (fun v i acc ->
+ let mn = if i = 1 then Mc.PEX v else Mc.PEpow (Mc.PEX v ,Ml2C.n i) in
+ if acc = Mc.PEc (Mc.Zpos Mc.XH)
+ then mn
+ else Mc.PEmul(mn,acc))
+ mn
+ (Mc.PEc (Mc.Zpos Mc.XH))
+
+let list_to_polynomial vars l =
+ assert (List.for_all (fun x -> ceiling_num x =/ x) l);
+ let var x = monomial_to_polynomial (List.nth vars x) in
+ let rec xtopoly p i = function
+ | [] -> p
+ | c::l -> if c =/ (Int 0) then xtopoly p (i+1) l
+ else let c = Mc.PEc (Ml2C.bigint (numerator c)) in
+ let mn =
+ if c = Mc.PEc (Mc.Zpos Mc.XH)
+ then var i
+ else Mc.PEmul (c,var i) in
+ let p' = if p = Mc.PEc Mc.Z0 then mn else
+ Mc.PEadd (mn, p) in
+ xtopoly p' (i+1) l in
+
+ xtopoly (Mc.PEc Mc.Z0) 0 l
+
+let rec fixpoint f x =
+ let y' = f x in
+ if y' = x then y'
+ else fixpoint f y'
+
+
+
+
+
+
+
+
+let rec_simpl_cone n_spec e =
+ let simpl_cone =
+ Mc.simpl_cone n_spec.zero n_spec.unit n_spec.mult n_spec.eqb in
+
+ let rec rec_simpl_cone = function
+ | Mc.S_Mult(t1, t2) ->
+ simpl_cone (Mc.S_Mult (rec_simpl_cone t1, rec_simpl_cone t2))
+ | Mc.S_Add(t1,t2) ->
+ simpl_cone (Mc.S_Add (rec_simpl_cone t1, rec_simpl_cone t2))
+ | x -> simpl_cone x in
+ rec_simpl_cone e
+
+
+let simplify_cone n_spec c = fixpoint (rec_simpl_cone n_spec) c
+
+type cone_prod =
+ Const of cone
+ | Ideal of cone *cone
+ | Mult of cone * cone
+ | Other of cone
+and cone = Mc.zWitness
+
+
+
+let factorise_linear_cone c =
+
+ let rec cone_list c l =
+ match c with
+ | Mc.S_Add (x,r) -> cone_list r (x::l)
+ | _ -> c :: l in
+
+ let factorise c1 c2 =
+ match c1 , c2 with
+ | Mc.S_Ideal(x,y) , Mc.S_Ideal(x',y') ->
+ if x = x' then Some (Mc.S_Ideal(x, Mc.S_Add(y,y'))) else None
+ | Mc.S_Mult(x,y) , Mc.S_Mult(x',y') ->
+ if x = x' then Some (Mc.S_Mult(x, Mc.S_Add(y,y'))) else None
+ | _ -> None in
+
+ let rec rebuild_cone l pending =
+ match l with
+ | [] -> (match pending with
+ | None -> Mc.S_Z
+ | Some p -> p
+ )
+ | e::l ->
+ (match pending with
+ | None -> rebuild_cone l (Some e)
+ | Some p -> (match factorise p e with
+ | None -> Mc.S_Add(p, rebuild_cone l (Some e))
+ | Some f -> rebuild_cone l (Some f) )
+ ) in
+
+ (rebuild_cone (List.sort Pervasives.compare (cone_list c [])) None)
+
+
+
+(* The binding with Fourier might be a bit obsolete
+ -- how does it handle equalities ? *)
+
+(* Certificates are elements of the cone such that P = 0 *)
+
+(* To begin with, we search for certificates of the form:
+ a1.p1 + ... an.pn + b1.q1 +... + bn.qn + c = 0
+ where pi >= 0 qi > 0
+ ai >= 0
+ bi >= 0
+ Sum bi + c >= 1
+ This is a linear problem: each monomial is considered as a variable.
+ Hence, we can use fourier.
+
+ The variable c is at index 0
+*)
+
+open Mfourier
+ (*module Fourier = Fourier(Vector.VList)(SysSet(Vector.VList))*)
+ (*module Fourier = Fourier(Vector.VSparse)(SysSetAlt(Vector.VSparse))*)
+module Fourier = Mfourier.Fourier(Vector.VSparse)(*(SysSetAlt(Vector.VMap))*)
+
+module Vect = Fourier.Vect
+open Fourier.Cstr
+
+(* fold_left followed by a rev ! *)
+
+let constrain_monomial mn l =
+ let coeffs = List.fold_left (fun acc p -> (Poly.get mn p)::acc) [] l in
+ if mn = Monomial.const
+ then
+ { coeffs = Vect.from_list ((Big_int unit_big_int):: (List.rev coeffs)) ;
+ op = Eq ;
+ cst = Big_int zero_big_int }
+ else
+ { coeffs = Vect.from_list ((Big_int zero_big_int):: (List.rev coeffs)) ;
+ op = Eq ;
+ cst = Big_int zero_big_int }
+
+
+let positivity l =
+ let rec xpositivity i l =
+ match l with
+ | [] -> []
+ | (_,Mc.Equal)::l -> xpositivity (i+1) l
+ | (_,_)::l ->
+ {coeffs = Vect.update (i+1) (fun _ -> Int 1) Vect.null ;
+ op = Ge ;
+ cst = Int 0 } :: (xpositivity (i+1) l)
+ in
+ xpositivity 0 l
+
+
+let string_of_op = function
+ | Mc.Strict -> "> 0"
+ | Mc.NonStrict -> ">= 0"
+ | Mc.Equal -> "= 0"
+ | Mc.NonEqual -> "<> 0"
+
+
+
+(* If the certificate includes at least one strict inequality,
+ the obtained polynomial can also be 0 *)
+let build_linear_system l =
+
+ (* Gather the monomials: HINT add up of the polynomials *)
+ let l' = List.map fst l in
+ let monomials =
+ List.fold_left (fun acc p -> Poly.addition p acc) (Poly.constant (Int 0)) l'
+ in (* For each monomial, compute a constraint *)
+ let s0 =
+ Poly.fold (fun mn _ res -> (constrain_monomial mn l')::res) monomials [] in
+ (* I need at least something strictly positive *)
+ let strict = {
+ coeffs = Vect.from_list ((Big_int unit_big_int)::
+ (List.map (fun (x,y) ->
+ match y with Mc.Strict ->
+ Big_int unit_big_int
+ | _ -> Big_int zero_big_int) l));
+ op = Ge ; cst = Big_int unit_big_int } in
+ (* Add the positivity constraint *)
+ {coeffs = Vect.from_list ([Big_int unit_big_int]) ;
+ op = Ge ;
+ cst = Big_int zero_big_int}::(strict::(positivity l)@s0)
+
+
+let big_int_to_z = Ml2C.bigint
+
+(* For Q, this is a pity that the certificate has been scaled
+ -- at a lower layer, certificates are using nums... *)
+let make_certificate n_spec cert li =
+ let bint_to_cst = n_spec.bigint_to_number in
+ match cert with
+ | [] -> None
+ | e::cert' ->
+ let cst = match compare_big_int e zero_big_int with
+ | 0 -> Mc.S_Z
+ | 1 -> Mc.S_Pos (bint_to_cst e)
+ | _ -> failwith "positivity error"
+ in
+ let rec scalar_product cert l =
+ match cert with
+ | [] -> Mc.S_Z
+ | c::cert -> match l with
+ | [] -> failwith "make_certificate(1)"
+ | i::l ->
+ let r = scalar_product cert l in
+ match compare_big_int c zero_big_int with
+ | -1 -> Mc.S_Add (
+ Mc.S_Ideal (Mc.PEc ( bint_to_cst c), Mc.S_In (Ml2C.nat i)),
+ r)
+ | 0 -> r
+ | _ -> Mc.S_Add (
+ Mc.S_Mult (Mc.S_Pos (bint_to_cst c), Mc.S_In (Ml2C.nat i)),
+ r) in
+
+ Some ((factorise_linear_cone
+ (simplify_cone n_spec (Mc.S_Add (cst, scalar_product cert' li)))))
+
+
+exception Found of Monomial.t
+
+let raw_certificate l =
+ let sys = build_linear_system l in
+ try
+ match Fourier.find_point sys with
+ | None -> None
+ | Some cert -> Some (rats_to_ints (Vect.to_list cert))
+ (* should not use rats_to_ints *)
+ with x ->
+ if debug
+ then (Printf.printf "raw certificate %s" (Printexc.to_string x);
+ flush stdout) ;
+ None
+
+
+let simple_linear_prover to_constant l =
+ let (lc,li) = List.split l in
+ match raw_certificate lc with
+ | None -> None (* No certificate *)
+ | Some cert -> make_certificate to_constant cert li
+
+
+
+let linear_prover n_spec l =
+ let li = List.combine l (interval 0 (List.length l -1)) in
+ let (l1,l') = List.partition
+ (fun (x,_) -> if snd' x = Mc.NonEqual then true else false) li in
+ let l' = List.map
+ (fun (c,i) -> let (Mc.Pair(x,y)) = c in
+ match y with
+ Mc.NonEqual -> failwith "cannot happen"
+ | y -> ((dev_form n_spec x, y),i)) l' in
+
+ simple_linear_prover n_spec l'
+
+
+let linear_prover n_spec l =
+ try linear_prover n_spec l with
+ x -> (print_string (Printexc.to_string x); None)
+
+(* zprover.... *)
+
+(* I need to gather the set of variables --->
+ Then go for fold
+ Once I have an interval, I need a certificate : 2 other fourier elims.
+ (I could probably get the certificate directly
+ as it is done in the fourier contrib.)
+*)
+
+let make_linear_system l =
+ let l' = List.map fst l in
+ let monomials = List.fold_left (fun acc p -> Poly.addition p acc)
+ (Poly.constant (Int 0)) l' in
+ let monomials = Poly.fold
+ (fun mn _ l -> if mn = Monomial.const then l else mn::l) monomials [] in
+ (List.map (fun (c,op) ->
+ {coeffs = Vect.from_list (List.map (fun mn -> (Poly.get mn c)) monomials) ;
+ op = op ;
+ cst = minus_num ( (Poly.get Monomial.const c))}) l
+ ,monomials)
+
+
+open Interval
+let pplus x y = Mc.PEadd(x,y)
+let pmult x y = Mc.PEmul(x,y)
+let pconst x = Mc.PEc x
+let popp x = Mc.PEopp x
+
+let debug = false
+
+(* keep track of enumerated vectors *)
+let rec mem p x l =
+ match l with [] -> false | e::l -> if p x e then true else mem p x l
+
+let rec remove_assoc p x l =
+ match l with [] -> [] | e::l -> if p x (fst e) then
+ remove_assoc p x l else e::(remove_assoc p x l)
+
+let eq x y = Vect.compare x y = 0
+
+(* Beurk... this code is a shame *)
+
+let rec zlinear_prover sys = xzlinear_prover [] sys
+
+and xzlinear_prover enum l : (Mc.proofTerm option) =
+ match linear_prover z_spec l with
+ | Some prf -> Some (Mc.RatProof prf)
+ | None ->
+ let ll = List.fold_right (fun (Mc.Pair(e,k)) r -> match k with
+ Mc.NonEqual -> r
+ | k -> (dev_form z_spec e ,
+ match k with
+ | Mc.Strict | Mc.NonStrict -> Ge
+ (* Loss of precision -- weakness of fourier*)
+ | Mc.Equal -> Eq
+ | Mc.NonEqual -> failwith "Cannot happen") :: r) l [] in
+
+ let (sys,var) = make_linear_system ll in
+ let res =
+ match Fourier.find_Q_interval sys with
+ | Some(i,x,j) -> if i =/ j
+ then Some(i,Vect.set x (Int 1) Vect.null,i) else None
+ | None -> None in
+ let res = match res with
+ | None ->
+ begin
+ let candidates = List.fold_right
+ (fun cstr acc ->
+ let gcd = Big_int (Vect.gcd cstr.coeffs) in
+ let vect = Vect.mul (Int 1 // gcd) cstr.coeffs in
+ if mem eq vect enum then acc
+ else ((vect,Fourier.optimise vect sys)::acc)) sys [] in
+ let candidates = List.fold_left (fun l (x,i) ->
+ match i with
+ None -> (x,Empty)::l
+ | Some i -> (x,i)::l) [] (candidates) in
+ match List.fold_left (fun (x1,i1) (x2,i2) ->
+ if smaller_itv i1 i2
+ then (x1,i1) else (x2,i2)) (Vect.null,Itv(None,None)) candidates
+ with
+ | (i,Empty) -> None
+ | (x,Itv(Some i, Some j)) -> Some(i,x,j)
+ | (x,Point n) -> Some(n,x,n)
+ | x -> match Fourier.find_Q_interval sys with
+ | None -> None
+ | Some(i,x,j) ->
+ if i =/ j
+ then Some(i,Vect.set x (Int 1) Vect.null,i)
+ else None
+ end
+ | _ -> res in
+
+ match res with
+ | Some (lb,e,ub) ->
+ let (lbn,lbd) =
+ (Ml2C.bigint (sub_big_int (numerator lb) unit_big_int),
+ Ml2C.bigint (denominator lb)) in
+ let (ubn,ubd) =
+ (Ml2C.bigint (add_big_int unit_big_int (numerator ub)) ,
+ Ml2C.bigint (denominator ub)) in
+ let expr = list_to_polynomial var (Vect.to_list e) in
+ (match
+ (*x <= ub -> x > ub *)
+ linear_prover z_spec
+ (Mc.Pair(pplus (pmult (pconst ubd) expr) (popp (pconst ubn)),
+ Mc.NonStrict) :: l),
+ (* lb <= x -> lb > x *)
+ linear_prover z_spec
+ (Mc.Pair( pplus (popp (pmult (pconst lbd) expr)) (pconst lbn) ,
+ Mc.NonStrict)::l)
+ with
+ | Some cub , Some clb ->
+ (match zlinear_enum (e::enum) expr
+ (ceiling_num lb) (floor_num ub) l
+ with
+ | None -> None
+ | Some prf ->
+ Some (Mc.EnumProof(Ml2C.q lb,expr,Ml2C.q ub,clb,cub,prf)))
+ | _ -> None
+ )
+ | _ -> None
+and xzlinear_enum enum expr clb cub l =
+ if clb >/ cub
+ then Some Mc.Nil
+ else
+ let pexpr = pplus (popp (pconst (Ml2C.bigint (numerator clb)))) expr in
+ let sys' = (Mc.Pair(pexpr, Mc.Equal))::l in
+ match xzlinear_prover enum sys' with
+ | None -> if debug then print_string "zlp?"; None
+ | Some prf -> if debug then print_string "zlp!";
+ match zlinear_enum enum expr (clb +/ (Int 1)) cub l with
+ | None -> None
+ | Some prfl -> Some (Mc.Cons(prf,prfl))
+
+and zlinear_enum enum expr clb cub l =
+ let res = xzlinear_enum enum expr clb cub l in
+ if debug then Printf.printf "zlinear_enum %s %s -> %s\n"
+ (string_of_num clb)
+ (string_of_num cub)
+ (match res with
+ | None -> "None"
+ | Some r -> "Some") ; res
+
diff --git a/contrib/micromega/coq_micromega.ml b/contrib/micromega/coq_micromega.ml
new file mode 100644
index 000000000..9fcc69173
--- /dev/null
+++ b/contrib/micromega/coq_micromega.ml
@@ -0,0 +1,1289 @@
+(************************************************************************)
+(* 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 *)
+(************************************************************************)
+(* *)
+(* Micromega: A reflexive tactic using the Positivstellensatz *)
+(* *)
+(* Frédéric Besson (Irisa/Inria) 2006-2008 *)
+(* *)
+(************************************************************************)
+
+open Mutils
+let debug = false
+
+let time str f x =
+ let t0 = (Unix.times()).Unix.tms_utime in
+ let res = f x in
+ let t1 = (Unix.times()).Unix.tms_utime in
+ (*if debug then*) (Printf.printf "time %s %f\n" str (t1 -. t0) ;
+ flush stdout);
+ res
+
+type ('a,'b) formula =
+ | TT
+ | FF
+ | X of 'b
+ | A of 'a * Names.name
+ | C of ('a,'b) formula * ('a,'b) formula * Names.name
+ | D of ('a,'b) formula * ('a,'b) formula * Names.name
+ | N of ('a,'b) formula * Names.name
+ | I of ('a,'b) formula * ('a,'b) formula * Names.name
+
+let none = Names.Anonymous
+
+let tag_formula t f =
+ match f with
+ | A(x,_) -> A(x,t)
+ | C(x,y,_) -> C(x,y,t)
+ | D(x,y,_) -> D(x,y,t)
+ | N(x,_) -> N(x,t)
+ | I(x,y,_) -> I(x,y,t)
+ | _ -> f
+
+let tt = []
+let ff = [ [] ]
+
+
+type ('constant,'contr) sentence =
+ ('constant Micromega.formula, 'contr) formula
+
+let cnf negate normalise f =
+ let negate a =
+ CoqToCaml.list (fun cl -> CoqToCaml.list (fun x -> x) cl) (negate a) in
+
+ let normalise a =
+ CoqToCaml.list (fun cl -> CoqToCaml.list (fun x -> x) cl) (normalise a) in
+
+ let and_cnf x y = x @ y in
+ let or_clause_cnf t f = List.map (fun x -> t@x ) f in
+
+ let rec or_cnf f f' =
+ match f with
+ | [] -> tt
+ | e :: rst -> (or_cnf rst f') @ (or_clause_cnf e f') in
+
+ let rec xcnf (pol : bool) f =
+ match f with
+ | TT -> if pol then tt else ff (* ?? *)
+ | FF -> if pol then ff else tt (* ?? *)
+ | X p -> if pol then ff else ff (* ?? *)
+ | A(x,t) -> if pol then normalise x else negate x
+ | N(e,t) -> xcnf (not pol) e
+ | C(e1,e2,t) ->
+ (if pol then and_cnf else or_cnf) (xcnf pol e1) (xcnf pol e2)
+ | D(e1,e2,t) ->
+ (if pol then or_cnf else and_cnf) (xcnf pol e1) (xcnf pol e2)
+ | I(e1,e2,t) ->
+ (if pol then or_cnf else and_cnf) (xcnf (not pol) e1) (xcnf pol e2) in
+
+ xcnf true f
+
+
+
+module M =
+struct
+ open Coqlib
+ open Term
+ (* let constant = gen_constant_in_modules "Omicron" coq_modules*)
+
+
+ let logic_dir = ["Coq";"Logic";"Decidable"]
+ let coq_modules =
+ init_modules @
+ [logic_dir] @ arith_modules @ zarith_base_modules @
+ [ ["Coq";"Lists";"List"];
+ ["ZMicromega"];
+ ["Tauto"];
+ ["RingMicromega"];
+ ["EnvRing"];
+ ["Coq"; "micromega"; "ZMicromega"];
+ ["Coq" ; "micromega" ; "Tauto"];
+ ["Coq" ; "micromega" ; "RingMicromega"];
+ ["Coq" ; "micromega" ; "EnvRing"];
+ ["Coq";"QArith"; "QArith_base"];
+ ["Coq";"Reals" ; "Rdefinitions"];
+ ["LRing_normalise"]]
+
+ let constant = gen_constant_in_modules "ZMicromega" coq_modules
+
+ let coq_and = lazy (constant "and")
+ let coq_or = lazy (constant "or")
+ let coq_not = lazy (constant "not")
+ let coq_iff = lazy (constant "iff")
+ let coq_True = lazy (constant "True")
+ let coq_False = lazy (constant "False")
+
+ let coq_cons = lazy (constant "cons")
+ let coq_nil = lazy (constant "nil")
+ let coq_list = lazy (constant "list")
+
+ let coq_O = lazy (constant "O")
+ let coq_S = lazy (constant "S")
+ let coq_nat = lazy (constant "nat")
+
+ let coq_NO = lazy
+ (gen_constant_in_modules "N" [ ["Coq";"NArith";"BinNat" ]] "N0")
+ let coq_Npos = lazy
+ (gen_constant_in_modules "N" [ ["Coq";"NArith"; "BinNat"]] "Npos")
+ (* let coq_n = lazy (constant "N")*)
+
+ let coq_pair = lazy (constant "pair")
+ let coq_None = lazy (constant "None")
+ let coq_option = lazy (constant "option")
+ let coq_positive = lazy (constant "positive")
+ let coq_xH = lazy (constant "xH")
+ let coq_xO = lazy (constant "xO")
+ let coq_xI = lazy (constant "xI")
+
+ let coq_N0 = lazy (constant "N0")
+ let coq_N0 = lazy (constant "Npos")
+
+
+ let coq_Z = lazy (constant "Z")
+ let coq_Q = lazy (constant "Q")
+ let coq_R = lazy (constant "R")
+
+ let coq_ZERO = lazy (constant "Z0")
+ let coq_POS = lazy (constant "Zpos")
+ let coq_NEG = lazy (constant "Zneg")
+
+ let coq_QWitness = lazy
+ (gen_constant_in_modules "QMicromega"
+ [["Coq"; "micromega"; "QMicromega"]] "QWitness")
+ let coq_ZWitness = lazy
+ (gen_constant_in_modules "QMicromega"
+ [["Coq"; "micromega"; "ZMicromega"]] "ZWitness")
+
+
+ let coq_Build_Witness = lazy (constant "Build_Witness")
+
+
+ let coq_Qmake = lazy (constant "Qmake")
+
+ let coq_proofTerm = lazy (constant "ProofTerm")
+ let coq_ratProof = lazy (constant "RatProof")
+ let coq_cutProof = lazy (constant "CutProof")
+ let coq_enumProof = lazy (constant "EnumProof")
+
+ let coq_Zgt = lazy (constant "Zgt")
+ let coq_Zge = lazy (constant "Zge")
+ let coq_Zle = lazy (constant "Zle")
+ let coq_Zlt = lazy (constant "Zlt")
+ let coq_Eq = lazy (constant "eq")
+
+ let coq_Zplus = lazy (constant "Zplus")
+ let coq_Zminus = lazy (constant "Zminus")
+ let coq_Zopp = lazy (constant "Zopp")
+ let coq_Zmult = lazy (constant "Zmult")
+ let coq_N_of_Z = lazy
+ (gen_constant_in_modules "ZArithRing"
+ [["Coq";"setoid_ring";"ZArithRing"]] "N_of_Z")
+
+
+ let coq_PEX = lazy (constant "PEX" )
+ let coq_PEc = lazy (constant"PEc")
+ let coq_PEadd = lazy (constant "PEadd")
+ let coq_PEopp = lazy (constant "PEopp")
+ let coq_PEmul = lazy (constant "PEmul")
+ let coq_PEsub = lazy (constant "PEsub")
+ let coq_PEpow = lazy (constant "PEpow")
+
+
+ let coq_OpEq = lazy (constant "OpEq")
+ let coq_OpNEq = lazy (constant "OpNEq")
+ let coq_OpLe = lazy (constant "OpLe")
+ let coq_OpLt = lazy (constant "OpLt")
+ let coq_OpGe = lazy (constant "OpGe")
+ let coq_OpGt = lazy (constant "OpGt")
+
+
+ let coq_S_In = lazy (constant "S_In")
+ let coq_S_Square = lazy (constant "S_Square")
+ let coq_S_Monoid = lazy (constant "S_Monoid")
+ let coq_S_Ideal = lazy (constant "S_Ideal")
+ let coq_S_Mult = lazy (constant "S_Mult")
+ let coq_S_Add = lazy (constant "S_Add")
+ let coq_S_Pos = lazy (constant "S_Pos")
+ let coq_S_Z = lazy (constant "S_Z")
+ let coq_coneMember = lazy (constant "coneMember")
+
+
+ let coq_make_impl = lazy
+ (gen_constant_in_modules "Zmicromega" [["Refl"]] "make_impl")
+ let coq_make_conj = lazy
+ (gen_constant_in_modules "Zmicromega" [["Refl"]] "make_conj")
+
+ let coq_Build = lazy
+ (gen_constant_in_modules "RingMicromega"
+ [["Coq" ; "micromega" ; "RingMicromega"] ; ["RingMicromega"] ]
+ "Build_Formula")
+ let coq_Cstr = lazy
+ (gen_constant_in_modules "RingMicromega"
+ [["Coq" ; "micromega" ; "RingMicromega"] ; ["RingMicromega"] ] "Formula")
+
+ type parse_error =
+ | Ukn
+ | BadStr of string
+ | BadNum of int
+ | BadTerm of Term.constr
+ | Msg of string
+ | Goal of (Term.constr list ) * Term.constr * parse_error
+
+ let string_of_error = function
+ | Ukn -> "ukn"
+ | BadStr s -> s
+ | BadNum i -> string_of_int i
+ | BadTerm _ -> "BadTerm"
+ | Msg s -> s
+ | Goal _ -> "Goal"
+
+
+ exception ParseError
+
+
+
+
+ let get_left_construct term =
+ match Term.kind_of_term term with
+ | Term.Construct(_,i) -> (i,[| |])
+ | Term.App(l,rst) ->
+ (match Term.kind_of_term l with
+ | Term.Construct(_,i) -> (i,rst)
+ | _ -> raise ParseError
+ )
+ | _ -> raise ParseError
+
+ module Mc = Micromega
+
+ let rec parse_nat term =
+ let (i,c) = get_left_construct term in
+ match i with
+ | 1 -> Mc.O
+ | 2 -> Mc.S (parse_nat (c.(0)))
+ | i -> raise ParseError
+
+
+ let pp_nat o n = Printf.fprintf o "%i" (CoqToCaml.nat n)
+
+
+ let rec dump_nat x =
+ match x with
+ | Mc.O -> Lazy.force coq_O
+ | Mc.S p -> Term.mkApp(Lazy.force coq_S,[| dump_nat p |])
+
+
+ let rec parse_positive term =
+ let (i,c) = get_left_construct term in
+ match i with
+ | 1 -> Mc.XI (parse_positive c.(0))
+ | 2 -> Mc.XO (parse_positive c.(0))
+ | 3 -> Mc.XH
+ | i -> raise ParseError
+
+
+ let rec dump_positive x =
+ match x with
+ | Mc.XH -> Lazy.force coq_xH
+ | Mc.XO p -> Term.mkApp(Lazy.force coq_xO,[| dump_positive p |])
+ | Mc.XI p -> Term.mkApp(Lazy.force coq_xI,[| dump_positive p |])
+
+ let pp_positive o x = Printf.fprintf o "%i" (CoqToCaml.positive x)
+
+
+ let rec dump_n x =
+ match x with
+ | Mc.N0 -> Lazy.force coq_N0
+ | Mc.Npos p -> Term.mkApp(Lazy.force coq_Npos,[| dump_positive p|])
+
+ let rec dump_index x =
+ match x with
+ | Mc.XH -> Lazy.force coq_xH
+ | Mc.XO p -> Term.mkApp(Lazy.force coq_xO,[| dump_index p |])
+ | Mc.XI p -> Term.mkApp(Lazy.force coq_xI,[| dump_index p |])
+
+
+ let pp_index o x = Printf.fprintf o "%i" (CoqToCaml.index x)
+
+ let rec dump_n x =
+ match x with
+ | Mc.N0 -> Lazy.force coq_NO
+ | Mc.Npos p -> Term.mkApp(Lazy.force coq_Npos,[| dump_positive p |])
+
+ let rec pp_n o x = output_string o (string_of_int (CoqToCaml.n x))
+
+ let dump_pair t1 t2 dump_t1 dump_t2 (Mc.Pair (x,y)) =
+ Term.mkApp(Lazy.force coq_pair,[| t1 ; t2 ; dump_t1 x ; dump_t2 y|])
+
+
+ let rec parse_z term =
+ let (i,c) = get_left_construct term in
+ match i with
+ | 1 -> Mc.Z0
+ | 2 -> Mc.Zpos (parse_positive c.(0))
+ | 3 -> Mc.Zneg (parse_positive c.(0))
+ | i -> raise ParseError
+
+ let dump_z x =
+ match x with
+ | Mc.Z0 ->Lazy.force coq_ZERO
+ | Mc.Zpos p -> Term.mkApp(Lazy.force coq_POS,[| dump_positive p|])
+ | Mc.Zneg p -> Term.mkApp(Lazy.force coq_NEG,[| dump_positive p|])
+
+ let pp_z o x = Printf.fprintf o "%i" (CoqToCaml.z x)
+
+let dump_num bd1 =
+ Term.mkApp(Lazy.force coq_Qmake,
+ [|dump_z (CamlToCoq.bigint (numerator bd1)) ;
+ dump_positive (CamlToCoq.positive_big_int (denominator bd1)) |])
+
+
+let dump_q q =
+ Term.mkApp(Lazy.force coq_Qmake,
+ [| dump_z q.Micromega.qnum ; dump_positive q.Micromega.qden|])
+
+let parse_q term =
+ match Term.kind_of_term term with
+ | Term.App(c, args) ->
+ (
+ match Term.kind_of_term c with
+ Term.Construct((n,j),i) ->
+ if Names.string_of_kn n = "Coq.QArith.QArith_base#<>#Q"
+ then {Mc.qnum = parse_z args.(0) ; Mc.qden = parse_positive args.(1) }
+ else raise ParseError
+ | _ -> raise ParseError
+ )
+ | _ -> raise ParseError
+
+ let rec parse_list parse_elt term =
+ let (i,c) = get_left_construct term in
+ match i with
+ | 1 -> Mc.Nil
+ | 2 -> Mc.Cons(parse_elt c.(1), parse_list parse_elt c.(2))
+ | i -> raise ParseError
+
+
+ let rec dump_list typ dump_elt l =
+ match l with
+ | Mc.Nil -> Term.mkApp(Lazy.force coq_nil,[| typ |])
+ | Mc.Cons(e,l) -> Term.mkApp(Lazy.force coq_cons,
+ [| typ; dump_elt e;dump_list typ dump_elt l|])
+
+ let rec dump_ml_list typ dump_elt l =
+ match l with
+ | [] -> Term.mkApp(Lazy.force coq_nil,[| typ |])
+ | e::l -> Term.mkApp(Lazy.force coq_cons,
+ [| typ; dump_elt e;dump_ml_list typ dump_elt l|])
+
+
+
+ let pp_list op cl elt o l =
+ let rec _pp o l =
+ match l with
+ | Mc.Nil -> ()
+ | Mc.Cons(e,Mc.Nil) -> Printf.fprintf o "%a" elt e
+ | Mc.Cons(e,l) -> Printf.fprintf o "%a ,%a" elt e _pp l in
+ Printf.fprintf o "%s%a%s" op _pp l cl
+
+
+
+ let pp_var = pp_positive
+ let dump_var = dump_positive
+
+ let rec pp_expr o e =
+ match e with
+ | Mc.PEX n -> Printf.fprintf o "V %a" pp_var n
+ | Mc.PEc z -> pp_z o z
+ | Mc.PEadd(e1,e2) -> Printf.fprintf o "(%a)+(%a)" pp_expr e1 pp_expr e2
+ | Mc.PEmul(e1,e2) -> Printf.fprintf o "%a*(%a)" pp_expr e1 pp_expr e2
+ | Mc.PEopp e -> Printf.fprintf o "-(%a)" pp_expr e
+ | Mc.PEsub(e1,e2) -> Printf.fprintf o "(%a)-(%a)" pp_expr e1 pp_expr e2
+ | Mc.PEpow(e,n) -> Printf.fprintf o "(%a)^(%a)" pp_expr e pp_n n
+
+
+ let dump_expr typ dump_z e =
+ let rec dump_expr e =
+ match e with
+ | Mc.PEX n -> mkApp(Lazy.force coq_PEX,[| typ; dump_var n |])
+ | Mc.PEc z -> mkApp(Lazy.force coq_PEc,[| typ ; dump_z z |])
+ | Mc.PEadd(e1,e2) -> mkApp(Lazy.force coq_PEadd,
+ [| typ; dump_expr e1;dump_expr e2|])
+ | Mc.PEsub(e1,e2) -> mkApp(Lazy.force coq_PEsub,
+ [| typ; dump_expr e1;dump_expr e2|])
+ | Mc.PEopp e -> mkApp(Lazy.force coq_PEopp,
+ [| typ; dump_expr e|])
+ | Mc.PEmul(e1,e2) -> mkApp(Lazy.force coq_PEmul,
+ [| typ; dump_expr e1;dump_expr e2|])
+ | Mc.PEpow(e,n) -> mkApp(Lazy.force coq_PEpow,
+ [| typ; dump_expr e; dump_n n|])
+ in
+ dump_expr e
+
+ let rec dump_monoid l = dump_list (Lazy.force coq_nat) dump_nat l
+
+ let rec dump_cone typ dump_z e =
+ let z = Lazy.force typ in
+ let rec dump_cone e =
+ match e with
+ | Mc.S_In n -> mkApp(Lazy.force coq_S_In,[| z; dump_nat n |])
+ | Mc.S_Ideal(e,c) -> mkApp(Lazy.force coq_S_Ideal,
+ [| z; dump_expr z dump_z e ; dump_cone c |])
+ | Mc.S_Square e -> mkApp(Lazy.force coq_S_Square,
+ [| z;dump_expr z dump_z e|])
+ | Mc.S_Monoid l -> mkApp (Lazy.force coq_S_Monoid,
+ [|z; dump_monoid l|])
+ | Mc.S_Add(e1,e2) -> mkApp(Lazy.force coq_S_Add,
+ [| z; dump_cone e1; dump_cone e2|])
+ | Mc.S_Mult(e1,e2) -> mkApp(Lazy.force coq_S_Mult,
+ [| z; dump_cone e1; dump_cone e2|])
+ | Mc.S_Pos p -> mkApp(Lazy.force coq_S_Pos,[| z; dump_z p|])
+ | Mc.S_Z -> mkApp( Lazy.force coq_S_Z,[| z|]) in
+ dump_cone e
+
+
+ let pp_cone pp_z o e =
+ let rec pp_cone o e =
+ match e with
+ | Mc.S_In n ->
+ Printf.fprintf o "(S_In %a)%%nat" pp_nat n
+ | Mc.S_Ideal(e,c) ->
+ Printf.fprintf o "(S_Ideal %a %a)" pp_expr e pp_cone c
+ | Mc.S_Square e ->
+ Printf.fprintf o "(S_Square %a)" pp_expr e
+ | Mc.S_Monoid l ->
+ Printf.fprintf o "(S_Monoid %a)" (pp_list "[" "]" pp_nat) l
+ | Mc.S_Add(e1,e2) ->
+ Printf.fprintf o "(S_Add %a %a)" pp_cone e1 pp_cone e2
+ | Mc.S_Mult(e1,e2) ->
+ Printf.fprintf o "(S_Mult %a %a)" pp_cone e1 pp_cone e2
+ | Mc.S_Pos p ->
+ Printf.fprintf o "(S_Pos %a)%%positive" pp_z p
+ | Mc.S_Z ->
+ Printf.fprintf o "S_Z" in
+ pp_cone o e
+
+
+
+
+ let rec parse_op term =
+ let (i,c) = get_left_construct term in
+ match i with
+ | 1 -> Mc.OpEq
+ | 2 -> Mc.OpLe
+ | 3 -> Mc.OpGe
+ | 4 -> Mc.OpGt
+ | 5 -> Mc.OpLt
+ | i -> raise ParseError
+
+
+ let rec dump_op = function
+ | Mc.OpEq-> Lazy.force coq_OpEq
+ | Mc.OpNEq-> Lazy.force coq_OpNEq
+ | Mc.OpLe -> Lazy.force coq_OpLe
+ | Mc.OpGe -> Lazy.force coq_OpGe
+ | Mc.OpGt-> Lazy.force coq_OpGt
+ | Mc.OpLt-> Lazy.force coq_OpLt
+
+
+
+ let pp_op o e=
+ match e with
+ | Mc.OpEq-> Printf.fprintf o "="
+ | Mc.OpNEq-> Printf.fprintf o "<>"
+ | Mc.OpLe -> Printf.fprintf o "=<"
+ | Mc.OpGe -> Printf.fprintf o ">="
+ | Mc.OpGt-> Printf.fprintf o ">"
+ | Mc.OpLt-> Printf.fprintf o "<"
+
+
+
+
+ let pp_cstr o {Mc.flhs = l ; Mc.fop = op ; Mc.frhs = r } =
+ Printf.fprintf o"(%a %a %a)" pp_expr l pp_op op pp_expr r
+
+ let dump_cstr typ dump_constant {Mc.flhs = e1 ; Mc.fop = o ; Mc.frhs = e2} =
+ Term.mkApp(Lazy.force coq_Build,
+ [| typ; dump_expr typ dump_constant e1 ;
+ dump_op o ;
+ dump_expr typ dump_constant e2|])
+
+
+
+ let parse_zop (op,args) =
+ match kind_of_term op with
+ | Const x ->
+ (match Names.string_of_con x with
+ | "Coq.ZArith.BinInt#<>#Zgt" -> (Mc.OpGt, args.(0), args.(1))
+ | "Coq.ZArith.BinInt#<>#Zge" -> (Mc.OpGe, args.(0), args.(1))
+ | "Coq.ZArith.BinInt#<>#Zlt" -> (Mc.OpLt, args.(0), args.(1))
+ | "Coq.ZArith.BinInt#<>#Zle" -> (Mc.OpLe, args.(0), args.(1))
+ (*| "Coq.Init.Logic#<>#not" -> Mc.OpNEq (* for backward compat *)*)
+ | s -> raise ParseError
+ )
+ | Ind(n,0) ->
+ (match Names.string_of_kn n with
+ | "Coq.Init.Logic#<>#eq" ->
+ if args.(0) <> Lazy.force coq_Z
+ then raise ParseError
+ else (Mc.OpEq, args.(1), args.(2))
+ | _ -> raise ParseError)
+ | _ -> failwith "parse_zop"
+
+
+ let parse_rop (op,args) =
+ try
+ match kind_of_term op with
+ | Const x ->
+ (match Names.string_of_con x with
+ | "Coq.Reals.Rdefinitions#<>#Rgt" -> (Mc.OpGt, args.(0), args.(1))
+ | "Coq.Reals.Rdefinitions#<>#Rge" -> (Mc.OpGe, args.(0), args.(1))
+ | "Coq.Reals.Rdefinitions#<>#Rlt" -> (Mc.OpLt, args.(0), args.(1))
+ | "Coq.Reals.Rdefinitions#<>#Rle" -> (Mc.OpLe, args.(0), args.(1))
+ (*| "Coq.Init.Logic#<>#not"-> Mc.OpNEq (* for backward compat *)*)
+ | s -> raise ParseError
+ )
+ | Ind(n,0) ->
+ (match Names.string_of_kn n with
+ | "Coq.Init.Logic#<>#eq" ->
+ (* if args.(0) <> Lazy.force coq_R
+ then raise ParseError
+ else*) (Mc.OpEq, args.(1), args.(2))
+ | _ -> raise ParseError)
+ | _ -> failwith "parse_rop"
+ with x ->
+ (Pp.pp (Pp.str "parse_rop failure ") ;
+ Pp.pp (Printer.prterm op) ; Pp.pp_flush ())
+ ; raise x
+
+
+ let parse_qop (op,args) =
+ (
+ (match kind_of_term op with
+ | Const x ->
+ (match Names.string_of_con x with
+ | "Coq.QArith.QArith_base#<>#Qgt" -> Mc.OpGt
+ | "Coq.QArith.QArith_base#<>#Qge" -> Mc.OpGe
+ | "Coq.QArith.QArith_base#<>#Qlt" -> Mc.OpLt
+ | "Coq.QArith.QArith_base#<>#Qle" -> Mc.OpLe
+ | "Coq.QArith.QArith_base#<>#Qeq" -> Mc.OpEq
+ | s -> raise ParseError
+ )
+ | _ -> failwith "parse_zop") , args.(0) , args.(1))
+
+
+ module Env =
+ struct
+ type t = constr list
+
+ let compute_rank_add env v =
+ let rec _add env n v =
+ match env with
+ | [] -> ([v],n)
+ | e::l ->
+ if eq_constr e v
+ then (env,n)
+ else
+ let (env,n) = _add l ( n+1) v in
+ (e::env,n) in
+ let (env, n) = _add env 1 v in
+ (env, CamlToCoq.idx n)
+
+
+ let empty = []
+
+ let elements env = env
+
+ end
+
+
+ let is_constant t = (* This is an approx *)
+ match kind_of_term t with
+ | Construct(i,_) -> true
+ | _ -> false
+
+
+ type 'a op =
+ | Binop of ('a Mc.pExpr -> 'a Mc.pExpr -> 'a Mc.pExpr)
+ | Opp
+ | Power
+ | Ukn of string
+
+
+ let parse_expr parse_constant parse_exp ops_spec env term =
+ if debug
+ then (Pp.pp (Pp.str "parse_expr: ");
+ Pp.pp_flush ();Pp.pp (Printer.prterm term); Pp.pp_flush ());
+
+ let constant_or_variable env term =
+ try
+ ( Mc.PEc (parse_constant term) , env)
+ with ParseError ->
+ let (env,n) = Env.compute_rank_add env term in
+ (Mc.PEX n , env) in
+
+ let rec parse_expr env term =
+ let combine env op (t1,t2) =
+ let (expr1,env) = parse_expr env t1 in
+ let (expr2,env) = parse_expr env t2 in
+ (op expr1 expr2,env) in
+ match kind_of_term term with
+ | App(t,args) ->
+ (
+ match kind_of_term t with
+ | Const c ->
+ ( match ops_spec (Names.string_of_con c) with
+ | Binop f -> combine env f (args.(0),args.(1))
+ | Opp -> let (expr,env) = parse_expr env args.(0) in
+ (Mc.PEopp expr, env)
+ | Power ->
+ let (expr,env) = parse_expr env args.(0) in
+ let exp = (parse_exp args.(1)) in
+ (Mc.PEpow(expr, exp) , env)
+ | Ukn s ->
+ if debug
+ then (Printf.printf "unknown op: %s\n" s; flush stdout;);
+ let (env,n) = Env.compute_rank_add env term in (Mc.PEX n, env)
+ )
+ | _ -> constant_or_variable env term
+ )
+ | _ -> constant_or_variable env term in
+ parse_expr env term
+
+
+let zop_spec = function
+ | "Coq.ZArith.BinInt#<>#Zplus" -> Binop (fun x y -> Mc.PEadd(x,y))
+ | "Coq.ZArith.BinInt#<>#Zminus" -> Binop (fun x y -> Mc.PEsub(x,y))
+ | "Coq.ZArith.BinInt#<>#Zmult" -> Binop (fun x y -> Mc.PEmul (x,y))
+ | "Coq.ZArith.BinInt#<>#Zopp" -> Opp
+ | "Coq.ZArith.Zpow_def#<>#Zpower" -> Power
+ | s -> Ukn s
+
+let qop_spec = function
+ | "Coq.QArith.QArith_base#<>#Qplus" -> Binop (fun x y -> Mc.PEadd(x,y))
+ | "Coq.QArith.QArith_base#<>#Qminus" -> Binop (fun x y -> Mc.PEsub(x,y))
+ | "Coq.QArith.QArith_base#<>#Qmult" -> Binop (fun x y -> Mc.PEmul (x,y))
+ | "Coq.QArith.QArith_base#<>#Qopp" -> Opp
+ | "Coq.QArith.QArith_base#<>#Qpower" -> Power
+ | s -> Ukn s
+
+let rop_spec = function
+ | "Coq.Reals.Rdefinitions#<>#Rplus" -> Binop (fun x y -> Mc.PEadd(x,y))
+ | "Coq.Reals.Rdefinitions#<>#Rminus" -> Binop (fun x y -> Mc.PEsub(x,y))
+ | "Coq.Reals.Rdefinitions#<>#Rmult" -> Binop (fun x y -> Mc.PEmul (x,y))
+ | "Coq.Reals.Rdefinitions#<>#Ropp" -> Opp
+ | "Coq.Reals.Rpow_def#<>#pow" -> Power
+ | s -> Ukn s
+
+
+
+
+
+let zconstant = parse_z
+let qconstant = parse_q
+
+
+let rconstant term =
+ if debug
+ then (Pp.pp_flush ();
+ Pp.pp (Pp.str "rconstant: ");
+ Pp.pp (Printer.prterm term); Pp.pp_flush ());
+ match Term.kind_of_term term with
+ | Const x ->
+ (match Names.string_of_con x with
+ | "Coq.Reals.Rdefinitions#<>#R0" -> Mc.Z0
+ | "Coq.Reals.Rdefinitions#<>#R1" -> Mc.Zpos Mc.XH
+ | _ -> raise ParseError
+ )
+ | _ -> raise ParseError
+
+
+let parse_zexpr =
+ parse_expr zconstant (fun x -> Mc.n_of_Z (parse_z x)) zop_spec
+let parse_qexpr =
+ parse_expr qconstant (fun x -> Mc.n_of_Z (parse_z x)) qop_spec
+let parse_rexpr =
+ parse_expr rconstant (fun x -> Mc.n_of_nat (parse_nat x)) rop_spec
+
+
+ let parse_arith parse_op parse_expr env cstr =
+ if debug
+ then (Pp.pp_flush ();
+ Pp.pp (Pp.str "parse_arith: ");
+ Pp.pp (Printer.prterm cstr);
+ Pp.pp_flush ());
+ match kind_of_term cstr with
+ | App(op,args) ->
+ let (op,lhs,rhs) = parse_op (op,args) in
+ let (e1,env) = parse_expr env lhs in
+ let (e2,env) = parse_expr env rhs in
+ ({Mc.flhs = e1; Mc.fop = op;Mc.frhs = e2},env)
+ | _ -> failwith "error : parse_arith(2)"
+
+ let parse_zarith = parse_arith parse_zop parse_zexpr
+
+ let parse_qarith = parse_arith parse_qop parse_qexpr
+
+ let parse_rarith = parse_arith parse_rop parse_rexpr
+
+
+ (* generic parsing of arithmetic expressions *)
+
+ let rec parse_conj parse_arith env term =
+ match kind_of_term term with
+ | App(l,rst) ->
+ (match kind_of_term l with
+ | Ind (n,_) ->
+ ( match Names.string_of_kn n with
+ | "Coq.Init.Logic#<>#and" ->
+ let (e1,env) = parse_arith env rst.(0) in
+ let (e2,env) = parse_conj parse_arith env rst.(1) in
+ (Mc.Cons(e1,e2),env)
+ | _ -> (* This might be an equality *)
+ let (e,env) = parse_arith env term in
+ (Mc.Cons(e,Mc.Nil),env))
+ | _ -> (* This is an arithmetic expression *)
+ let (e,env) = parse_arith env term in
+ (Mc.Cons(e,Mc.Nil),env))
+ | _ -> failwith "parse_conj(2)"
+
+
+
+ let rec f2f = function
+ | TT -> Mc.TT
+ | FF -> Mc.FF
+ | X _ -> Mc.X
+ | A (x,_) -> Mc.A x
+ | C (a,b,_) -> Mc.Cj(f2f a,f2f b)
+ | D (a,b,_) -> Mc.D(f2f a,f2f b)
+ | N (a,_) -> Mc.N(f2f a)
+ | I(a,b,_) -> Mc.I(f2f a,f2f b)
+
+ let is_prop t =
+ match t with
+ | Names.Anonymous -> true (* Not quite right *)
+ | Names.Name x -> false
+
+ let mkC f1 f2 = C(f1,f2,none)
+ let mkD f1 f2 = D(f1,f2,none)
+ let mkIff f1 f2 = C(I(f1,f2,none),I(f2,f2,none),none)
+ let mkI f1 f2 = I(f1,f2,none)
+
+ let mkformula_binary g term f1 f2 =
+ match f1 , f2 with
+ | X _ , X _ -> X(term)
+ | _ -> g f1 f2
+
+ let parse_formula parse_atom env term =
+ let parse_atom env t = try let (at,env) = parse_atom env t in (A(at,none), env) with _ -> (X(t),env) in
+
+ let rec xparse_formula env term =
+ match kind_of_term term with
+ | App(l,rst) ->
+ (match rst with
+ | [|a;b|] when l = Lazy.force coq_and ->
+ let f,env = xparse_formula env a in
+ let g,env = xparse_formula env b in
+ mkformula_binary mkC term f g,env
+ | [|a;b|] when l = Lazy.force coq_or ->
+ let f,env = xparse_formula env a in
+ let g,env = xparse_formula env b in
+ mkformula_binary mkD term f g,env
+ | [|a|] when l = Lazy.force coq_not ->
+ let (f,env) = xparse_formula env a in (N(f,none), env)
+ | [|a;b|] when l = Lazy.force coq_iff ->
+ let f,env = xparse_formula env a in
+ let g,env = xparse_formula env b in
+ mkformula_binary mkIff term f g,env
+ | _ -> parse_atom env term)
+ | Prod(typ,a,b) when not (Termops.dependent (mkRel 1) b) ->
+ let f,env = xparse_formula env a in
+ let g,env = xparse_formula env b in
+ mkformula_binary mkI term f g,env
+ | _ when term = Lazy.force coq_True -> (TT,env)
+ | _ -> X(term),env in
+ xparse_formula env term
+
+ let coq_TT = lazy
+ (gen_constant_in_modules "ZMicromega"
+ [["Coq" ; "micromega" ; "Tauto"];["Tauto"]] "TT")
+ let coq_FF = lazy
+ (gen_constant_in_modules "ZMicromega"
+ [["Coq" ; "micromega" ; "Tauto"];["Tauto"]] "FF")
+ let coq_And = lazy
+ (gen_constant_in_modules "ZMicromega"
+ [["Coq" ; "micromega" ; "Tauto"];["Tauto"]] "Cj")
+ let coq_Or = lazy
+ (gen_constant_in_modules "ZMicromega"
+ [["Coq" ; "micromega" ; "Tauto"];["Tauto"]] "D")
+ let coq_Neg = lazy
+ (gen_constant_in_modules "ZMicromega"
+ [["Coq" ; "micromega" ; "Tauto"];["Tauto"]] "N")
+ let coq_Atom = lazy
+ (gen_constant_in_modules "ZMicromega"
+ [["Coq" ; "micromega" ; "Tauto"];["Tauto"]] "A")
+ let coq_X = lazy
+ (gen_constant_in_modules "ZMicromega"
+ [["Coq" ; "micromega" ; "Tauto"];["Tauto"]] "X")
+ let coq_Impl = lazy
+ (gen_constant_in_modules "ZMicromega"
+ [["Coq" ; "micromega" ; "Tauto"];["Tauto"]] "I")
+ let coq_Formula = lazy
+ (gen_constant_in_modules "ZMicromega"
+ [["Coq" ; "micromega" ; "Tauto"];["Tauto"]] "BFormula")
+
+ let dump_formula typ dump_atom f =
+ let rec xdump f =
+ match f with
+ | TT -> mkApp(Lazy.force coq_TT,[| typ|])
+ | FF -> mkApp(Lazy.force coq_FF,[| typ|])
+ | C(x,y,_) -> mkApp(Lazy.force coq_And,[| typ ; xdump x ; xdump y|])
+ | D(x,y,_) -> mkApp(Lazy.force coq_Or,[| typ ; xdump x ; xdump y|])
+ | I(x,y,_) -> mkApp(Lazy.force coq_Impl,[| typ ; xdump x ; xdump y|])
+ | N(x,_) -> mkApp(Lazy.force coq_Neg,[| typ ; xdump x|])
+ | A(x,_) -> mkApp(Lazy.force coq_Atom,[| typ ; dump_atom x|])
+ | X(t) -> mkApp(Lazy.force coq_X,[| typ ; t|]) in
+
+ xdump f
+
+
+ (* Backward compat *)
+
+ let rec parse_concl parse_arith env term =
+ match kind_of_term term with
+ | Prod(_,expr,rst) -> (* a -> b *)
+ let (lhs,rhs,env) = parse_concl parse_arith env rst in
+ let (e,env) = parse_arith env expr in
+ (Mc.Cons(e,lhs),rhs,env)
+ | App(_,_) ->
+ let (conj, env) = parse_conj parse_arith env term in
+ (Mc.Nil,conj,env)
+ | Ind(n,_) ->
+ (match (Names.string_of_kn n) with
+ | "Coq.Init.Logic#<>#False" -> (Mc.Nil,Mc.Nil,env)
+ | s ->
+ print_string s ; flush stdout;
+ failwith "parse_concl")
+ | _ -> failwith "parse_concl"
+
+
+ let rec parse_hyps parse_arith env goal_hyps hyps =
+ match hyps with
+ | [] -> ([],goal_hyps,env)
+ | (i,t)::l ->
+ let (li,lt,env) = parse_hyps parse_arith env goal_hyps l in
+ try
+ let (c,env) = parse_arith env t in
+ (i::li, Mc.Cons(c,lt), env)
+ with x ->
+ (*(if debug then Printf.printf "parse_arith : %s\n" x);*)
+ (li,lt,env)
+
+
+ let parse_goal parse_arith env hyps term =
+ try
+ let (lhs,rhs,env) = parse_concl parse_arith env term in
+ let (li,lt,env) = parse_hyps parse_arith env lhs hyps in
+ (li,lt,rhs,env)
+ with Failure x -> raise ParseError
+ (* backward compat *)
+
+
+ (* ! reverse the list of bindings *)
+ let set l concl =
+ let rec _set acc = function
+ | [] -> acc
+ | (e::l) ->
+ let (name,expr,typ) = e in
+ _set (Term.mkNamedLetIn
+ (Names.id_of_string name)
+ expr typ acc) l in
+ _set concl l
+
+
+end
+
+open M
+
+
+let rec sig_of_cone = function
+ | Mc.S_In n -> [CoqToCaml.nat n]
+ | Mc.S_Ideal(e,w) -> sig_of_cone w
+ | Mc.S_Mult(w1,w2) ->
+ (sig_of_cone w1)@(sig_of_cone w2)
+ | Mc.S_Add(w1,w2) -> (sig_of_cone w1)@(sig_of_cone w2)
+ | _ -> []
+
+let same_proof sg cl1 cl2 =
+ let cl1 = CoqToCaml.list (fun x -> x) cl1 in
+ let cl2 = CoqToCaml.list (fun x -> x) cl2 in
+ let rec xsame_proof sg =
+ match sg with
+ | [] -> true
+ | n::sg -> (try List.nth cl1 n = List.nth cl2 n with _ -> false)
+ && (xsame_proof sg ) in
+ xsame_proof sg
+
+
+
+
+let tags_of_clause tgs wit clause =
+ let rec xtags tgs = function
+ | Mc.S_In n -> Names.Idset.union tgs
+ (snd (List.nth clause (CoqToCaml.nat n) ))
+ | Mc.S_Ideal(e,w) -> xtags tgs w
+ | Mc.S_Mult (w1,w2) | Mc.S_Add(w1,w2) -> xtags (xtags tgs w1) w2
+ | _ -> tgs in
+ xtags tgs wit
+
+let tags_of_cnf wits cnf =
+ List.fold_left2 (fun acc w cl -> tags_of_clause acc w cl)
+ Names.Idset.empty wits cnf
+
+
+let find_witness prover polys1 =
+ let l = CoqToCaml.list (fun x -> x) polys1 in
+ try_any prover l
+
+let rec witness prover l1 l2 =
+ match l2 with
+ | Micromega.Nil -> Some (Micromega.Nil)
+ | Micromega.Cons(e,l2) ->
+ match find_witness prover (Micromega.Cons( e,l1)) with
+ | None -> None
+ | Some w ->
+ (match witness prover l1 l2 with
+ | None -> None
+ | Some l -> Some (Micromega.Cons (w,l))
+ )
+
+
+let rec apply_ids t ids =
+ match ids with
+ | [] -> t
+ | i::ids -> apply_ids (Term.mkApp(t,[| Term.mkVar i |])) ids
+
+
+let coq_Node = lazy
+ (Coqlib.gen_constant_in_modules "VarMap"
+ [["Coq" ; "micromega" ; "VarMap"];["VarMap"]] "Node")
+let coq_Leaf = lazy
+ (Coqlib.gen_constant_in_modules "VarMap"
+ [["Coq" ; "micromega" ; "VarMap"];["VarMap"]] "Leaf")
+let coq_Empty = lazy
+ (Coqlib.gen_constant_in_modules "VarMap"
+ [["Coq" ; "micromega" ;"VarMap"];["VarMap"]] "Empty")
+
+
+let btree_of_array typ a =
+ let size_of_a = Array.length a in
+ let semi_size_of_a = size_of_a lsr 1 in
+ let node = Lazy.force coq_Node
+ and leaf = Lazy.force coq_Leaf
+ and empty = Term.mkApp (Lazy.force coq_Empty, [| typ |]) in
+ let rec aux n =
+ if n > size_of_a
+ then empty
+ else if n > semi_size_of_a
+ then Term.mkApp (leaf, [| typ; a.(n-1) |])
+ else Term.mkApp (node, [| typ; aux (2*n); a.(n-1); aux (2*n+1) |])
+ in
+ aux 1
+
+let btree_of_array typ a =
+ try
+ btree_of_array typ a
+ with x ->
+ failwith (Printf.sprintf "btree of array : %s" (Printexc.to_string x))
+
+let dump_varmap typ env =
+ btree_of_array typ (Array.of_list env)
+
+
+let rec pp_varmap o vm =
+ match vm with
+ | Mc.Empty -> output_string o "[]"
+ | Mc.Leaf z -> Printf.fprintf o "[%a]" pp_z z
+ | Mc.Node(l,z,r) -> Printf.fprintf o "[%a, %a, %a]" pp_varmap l pp_z z pp_varmap r
+
+
+
+let rec dump_proof_term = function
+ | Micromega.RatProof cone ->
+ Term.mkApp(Lazy.force coq_ratProof, [|dump_cone coq_Z dump_z cone|])
+ | Micromega.CutProof(e,q,cone,prf) ->
+ Term.mkApp(Lazy.force coq_cutProof,
+ [| dump_expr (Lazy.force coq_Z) dump_z e ;
+ dump_q q ;
+ dump_cone coq_Z dump_z cone ;
+ dump_proof_term prf|])
+ | Micromega.EnumProof( q1,e1,q2,c1,c2,prfs) ->
+ Term.mkApp (Lazy.force coq_enumProof,
+ [| dump_q q1 ; dump_expr (Lazy.force coq_Z) dump_z e1 ; dump_q q2;
+ dump_cone coq_Z dump_z c1 ; dump_cone coq_Z dump_z c2 ;
+ dump_list (Lazy.force coq_proofTerm) dump_proof_term prfs |])
+
+let pp_q o q = Printf.fprintf o "%a/%a" pp_z q.Micromega.qnum pp_positive q.Micromega.qden
+
+
+let rec pp_proof_term o = function
+ | Micromega.RatProof cone -> Printf.fprintf o "R[%a]" (pp_cone pp_z) cone
+ | Micromega.CutProof(e,q,_,p) -> failwith "not implemented"
+ | Micromega.EnumProof(q1,e1,q2,c1,c2,rst) ->
+ Printf.fprintf o "EP[%a,%a,%a,%a,%a,%a]"
+ pp_q q1 pp_expr e1 pp_q q2 (pp_cone pp_z) c1 (pp_cone pp_z) c2
+ (pp_list "[" "]" pp_proof_term) rst
+
+let rec parse_hyps parse_arith env hyps =
+ match hyps with
+ | [] -> ([],env)
+ | (i,t)::l ->
+ let (lhyps,env) = parse_hyps parse_arith env l in
+ try
+ let (c,env) = parse_formula parse_arith env t in
+ ((i,c)::lhyps, env)
+ with _ -> (lhyps,env)
+ (*(if debug then Printf.printf "parse_arith : %s\n" x);*)
+
+
+exception ParseError
+
+let parse_goal parse_arith env hyps term =
+ (* try*)
+ let (f,env) = parse_formula parse_arith env term in
+ let (lhyps,env) = parse_hyps parse_arith env hyps in
+ (lhyps,f,env)
+ (* with Failure x -> raise ParseError*)
+
+
+type ('a, 'b) domain_spec = {
+ typ : Term.constr; (* Z, Q , R *)
+ coeff : Term.constr ; (* Z, Q *)
+ dump_coeff : 'a -> Term.constr ;
+ proof_typ : Term.constr ;
+ dump_proof : 'b -> Term.constr
+}
+
+let zz_domain_spec = lazy {
+ typ = Lazy.force coq_Z;
+ coeff = Lazy.force coq_Z;
+ dump_coeff = dump_z ;
+ proof_typ = Lazy.force coq_proofTerm ;
+ dump_proof = dump_proof_term
+}
+
+let qq_domain_spec = lazy {
+ typ = Lazy.force coq_Q;
+ coeff = Lazy.force coq_Q;
+ dump_coeff = dump_q ;
+ proof_typ = Lazy.force coq_QWitness ;
+ dump_proof = dump_cone coq_Q dump_q
+}
+
+let rz_domain_spec = lazy {
+ typ = Lazy.force coq_R;
+ coeff = Lazy.force coq_Z;
+ dump_coeff = dump_z;
+ proof_typ = Lazy.force coq_ZWitness ;
+ dump_proof = dump_cone coq_Z dump_z
+}
+
+
+
+
+let micromega_order_change spec cert cert_typ env ff gl =
+ let formula_typ = (Term.mkApp( Lazy.force coq_Cstr,[| spec.coeff|])) in
+
+ let ff = dump_formula formula_typ (dump_cstr spec.coeff spec.dump_coeff) ff in
+ let vm = dump_varmap ( spec.typ) env in
+ Tactics.change_in_concl None
+ (set
+ [
+ ("__ff", ff, Term.mkApp(Lazy.force coq_Formula ,[| formula_typ |]));
+ ("__varmap", vm , Term.mkApp
+ (Coqlib.gen_constant_in_modules "VarMap"
+ [["Coq" ; "micromega" ; "VarMap"];["VarMap"]] "t", [| spec.typ|]));
+ ("__wit", cert,cert_typ)
+ ]
+ (Tacmach.pf_concl gl )
+
+ )
+ gl
+
+
+let detect_duplicates cnf wit =
+ let cnf = CoqToCaml.list (fun x -> x) cnf in
+ let wit = CoqToCaml.list (fun x -> x) wit in
+
+ let rec xdup cnf wit =
+ match wit with
+ | [] -> []
+ | w :: wit ->
+ let sg = sig_of_cone w in
+ match cnf with
+ | [] -> []
+ | e::cnf ->
+ let (dups,cnf) = (List.partition (fun x -> same_proof sg e x) cnf) in
+ dups@(xdup cnf wit) in
+ xdup cnf wit
+
+let find_witness prover polys1 =
+ try_any prover polys1
+
+
+let witness_list_with_tags prover l =
+
+ let rec xwitness_list l =
+ match l with
+ | [] -> Some([])
+ | e::l ->
+ match find_witness prover (List.map fst e) with
+ | None -> None
+ | Some w ->
+ (match xwitness_list l with
+ | None -> None
+ | Some l -> Some (w::l)
+ ) in
+ xwitness_list l
+
+let witness_list_without_tags prover l =
+
+ let rec xwitness_list l =
+ match l with
+ | [] -> Some([])
+ | e::l ->
+ match find_witness prover e with
+ | None -> None
+ | Some w ->
+ (match xwitness_list l with
+ | None -> None
+ | Some l -> Some (w::l)
+ ) in
+ xwitness_list l
+
+let witness_list prover l =
+ let rec xwitness_list l =
+ match l with
+ | Micromega.Nil -> Some(Micromega.Nil)
+ | Micromega.Cons(e,l) ->
+ match find_witness prover e with
+ | None -> None
+ | Some w ->
+ (match xwitness_list l with
+ | None -> None
+ | Some l -> Some (Micromega.Cons(w,l))
+ ) in
+ xwitness_list l
+
+
+
+
+let is_singleton = function [] -> true | [e] -> true | _ -> false
+
+
+let micromega_tauto negate normalise spec prover env polys1 polys2 gl =
+ let spec = Lazy.force spec in
+ let (ff,ids) =
+ List.fold_right
+ (fun (id,f) (cc,ids) ->
+ match f with
+ X _ -> (cc,ids)
+ | _ -> (I(tag_formula (Names.Name id) f,cc,none), id::ids))
+ polys1 (polys2,[]) in
+
+ let cnf_ff = cnf negate normalise ff in
+
+ if debug then
+ (Pp.pp (Pp.str "Formula....\n") ;
+ let formula_typ = (Term.mkApp( Lazy.force coq_Cstr,[| spec.coeff|])) in
+ let ff = dump_formula formula_typ
+ (dump_cstr spec.typ spec.dump_coeff) ff in
+ Pp.pp (Printer.prterm ff) ; Pp.pp_flush ()) ;
+
+ match witness_list_without_tags prover cnf_ff with
+ | None -> Tacticals.tclFAIL 0 (Pp.str "Cannot find witness") gl
+ | Some res -> (*Printf.printf "\nList %i" (List.length res); *)
+ let (ff,res,ids) = (ff,res,List.map Term.mkVar ids) in
+ let res' = dump_ml_list (spec.proof_typ) spec.dump_proof res in
+ (Tacticals.tclTHENSEQ
+ [
+ Tactics.generalize ids;
+ micromega_order_change spec res'
+ (Term.mkApp(Lazy.force coq_list,[| spec.proof_typ|])) env ff ;
+ ]) gl
+
+
+let micromega_gen parse_arith negate normalise spec prover gl =
+ let concl = Tacmach.pf_concl gl in
+ let hyps = Tacmach.pf_hyps_types gl in
+ try
+ let (hyps,concl,env) = parse_goal parse_arith Env.empty hyps concl in
+ let env = Env.elements env in
+ micromega_tauto negate normalise spec prover env hyps concl gl
+ with
+ | Failure x -> flush stdout ; Pp.pp_flush () ;
+ Tacticals.tclFAIL 0 (Pp.str x) gl
+ | ParseError -> Tacticals.tclFAIL 0 (Pp.str "Bad logical fragment") gl
+
+
+let lift_ratproof prover l =
+ match prover l with
+ | None -> None
+ | Some c -> Some (Mc.RatProof c)
+
+
+type csdpcert = Certificate.Mc.z Certificate.Mc.coneMember option
+type micromega_polys = (Micromega.z Mc.pExpr, Mc.op1) Micromega.prod list
+type provername = string * int option
+
+let call_csdpcert provername poly =
+ let tmp_to,ch_to = Filename.open_temp_file "csdpcert" ".in" in
+ let tmp_from = Filename.temp_file "csdpcert" ".out" in
+ output_value ch_to (provername,poly : provername * micromega_polys);
+ close_out ch_to;
+ let cmdname =
+ Filename.concat Coq_config.bindir
+ ("csdpcert" ^ Coq_config.exec_extension) in
+ let c = Sys.command (cmdname ^" "^ tmp_to ^" "^ tmp_from) in
+ (try Sys.remove tmp_to with _ -> ());
+ if c <> 0 then Util.error ("Failed to call csdp certificate generator");
+ let ch_from = open_in tmp_from in
+ let cert = (input_value ch_from : csdpcert) in
+ close_in ch_from; Sys.remove tmp_from;
+ cert
+
+let omicron gl =
+ micromega_gen parse_zarith Mc.negate Mc.normalise zz_domain_spec
+ [lift_ratproof
+ (Certificate.linear_prover Certificate.z_spec), "fourier refutation" ] gl
+
+
+let qomicron gl =
+ micromega_gen parse_qarith Mc.cnf_negate Mc.cnf_normalise qq_domain_spec
+ [ Certificate.linear_prover Certificate.q_spec, "fourier refutation" ] gl
+
+let romicron gl =
+ micromega_gen parse_rarith Mc.cnf_negate Mc.cnf_normalise rz_domain_spec
+ [ Certificate.linear_prover Certificate.z_spec, "fourier refutation" ] gl
+
+
+let rmicromega i gl =
+ micromega_gen parse_rarith Mc.negate Mc.normalise rz_domain_spec
+ [ call_csdpcert ("real_nonlinear_prover", Some i), "fourier refutation" ] gl
+
+
+let micromega i gl =
+ micromega_gen parse_zarith Mc.negate Mc.normalise zz_domain_spec
+ [lift_ratproof (call_csdpcert ("real_nonlinear_prover",Some i)),
+ "fourier refutation" ] gl
+
+
+let sos gl =
+ micromega_gen parse_zarith Mc.negate Mc.normalise zz_domain_spec
+ [lift_ratproof (call_csdpcert ("pure_sos", None)), "pure sos refutation"] gl
+
+let zomicron gl =
+ micromega_gen parse_zarith Mc.negate Mc.normalise zz_domain_spec
+ [Certificate.zlinear_prover, "zprover"] gl
diff --git a/contrib/micromega/csdpcert.ml b/contrib/micromega/csdpcert.ml
new file mode 100644
index 000000000..cfaf6ae1a
--- /dev/null
+++ b/contrib/micromega/csdpcert.ml
@@ -0,0 +1,333 @@
+(************************************************************************)
+(* 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 *)
+(************************************************************************)
+(* *)
+(* Micromega: A reflexive tactic using the Positivstellensatz *)
+(* *)
+(* Frédéric Besson (Irisa/Inria) 2006-2008 *)
+(* *)
+(************************************************************************)
+
+open Big_int
+open Num
+open Sos
+
+module Mc = Micromega
+module Ml2C = Mutils.CamlToCoq
+module C2Ml = Mutils.CoqToCaml
+
+let debug = false
+
+module M =
+struct
+ open Mc
+
+ let rec expr_to_term = function
+ | PEc z -> Const (Big_int (C2Ml.z_big_int z))
+ | PEX v -> Var ("x"^(string_of_int (C2Ml.index v)))
+ | PEmul(p1,p2) ->
+ let p1 = expr_to_term p1 in
+ let p2 = expr_to_term p2 in
+ let res = Mul(p1,p2) in res
+
+ | PEadd(p1,p2) -> Add(expr_to_term p1, expr_to_term p2)
+ | PEsub(p1,p2) -> Sub(expr_to_term p1, expr_to_term p2)
+ | PEpow(p,n) -> Pow(expr_to_term p , C2Ml.n n)
+ | PEopp p -> Opp (expr_to_term p)
+
+
+ let rec term_to_expr = function
+ | Const n -> PEc (Ml2C.bigint (big_int_of_num n))
+ | Zero -> PEc ( Z0)
+ | Var s -> PEX (Ml2C.index
+ (int_of_string (String.sub s 1 (String.length s - 1))))
+ | Mul(p1,p2) -> PEmul(term_to_expr p1, term_to_expr p2)
+ | Add(p1,p2) -> PEadd(term_to_expr p1, term_to_expr p2)
+ | Opp p -> PEopp (term_to_expr p)
+ | Pow(t,n) -> PEpow (term_to_expr t,Ml2C.n n)
+ | Sub(t1,t2) -> PEsub (term_to_expr t1, term_to_expr t2)
+ | _ -> failwith "term_to_expr: not implemented"
+
+ let term_to_expr e =
+ let e' = term_to_expr e in
+ if debug
+ then Printf.printf "term_to_expr : %s - %s\n"
+ (string_of_poly (poly_of_term e))
+ (string_of_poly (poly_of_term (expr_to_term e')));
+ e'
+
+end
+open M
+
+open List
+open Mutils
+
+let rec scale_term t =
+ match t with
+ | Zero -> unit_big_int , Zero
+ | Const n -> (denominator n) , Const (Big_int (numerator n))
+ | Var n -> unit_big_int , Var n
+ | Inv _ -> failwith "scale_term : not implemented"
+ | Opp t -> let s, t = scale_term t in s, Opp t
+ | Add(t1,t2) -> let s1,y1 = scale_term t1 and s2,y2 = scale_term t2 in
+ let g = gcd_big_int s1 s2 in
+ let s1' = div_big_int s1 g in
+ let s2' = div_big_int s2 g in
+ let e = mult_big_int g (mult_big_int s1' s2') in
+ if (compare_big_int e unit_big_int) = 0
+ then (unit_big_int, Add (y1,y2))
+ else e, Add (Mul(Const (Big_int s2'), y1),
+ Mul (Const (Big_int s1'), y2))
+ | Sub _ -> failwith "scale term: not implemented"
+ | Mul(y,z) -> let s1,y1 = scale_term y and s2,y2 = scale_term z in
+ mult_big_int s1 s2 , Mul (y1, y2)
+ | Pow(t,n) -> let s,t = scale_term t in
+ power_big_int_positive_int s n , Pow(t,n)
+ | _ -> failwith "scale_term : not implemented"
+
+let scale_term t =
+ let (s,t') = scale_term t in
+ s,t'
+
+
+
+
+let rec scale_certificate pos = match pos with
+ | Axiom_eq i -> unit_big_int , Axiom_eq i
+ | Axiom_le i -> unit_big_int , Axiom_le i
+ | Axiom_lt i -> unit_big_int , Axiom_lt i
+ | Monoid l -> unit_big_int , Monoid l
+ | Rational_eq n -> (denominator n) , Rational_eq (Big_int (numerator n))
+ | Rational_le n -> (denominator n) , Rational_le (Big_int (numerator n))
+ | Rational_lt n -> (denominator n) , Rational_lt (Big_int (numerator n))
+ | Square t -> let s,t' = scale_term t in
+ mult_big_int s s , Square t'
+ | Eqmul (t, y) -> let s1,y1 = scale_term t and s2,y2 = scale_certificate y in
+ mult_big_int s1 s2 , Eqmul (y1,y2)
+ | Sum (y, z) -> let s1,y1 = scale_certificate y
+ and s2,y2 = scale_certificate z in
+ let g = gcd_big_int s1 s2 in
+ let s1' = div_big_int s1 g in
+ let s2' = div_big_int s2 g in
+ mult_big_int g (mult_big_int s1' s2'),
+ Sum (Product(Rational_le (Big_int s2'), y1),
+ Product (Rational_le (Big_int s1'), y2))
+ | Product (y, z) ->
+ let s1,y1 = scale_certificate y and s2,y2 = scale_certificate z in
+ mult_big_int s1 s2 , Product (y1,y2)
+
+
+let is_eq = function Mc.Equal -> true | _ -> false
+let is_le = function Mc.NonStrict -> true | _ -> false
+let is_lt = function Mc.Strict -> true | _ -> false
+
+let get_index_of_ith_match f i l =
+ let rec get j res l =
+ match l with
+ | [] -> failwith "bad index"
+ | e::l -> if f e
+ then
+ (if j = i then res else get (j+1) (res+1) l )
+ else get j (res+1) l in
+ get 0 0 l
+
+
+let cert_of_pos eq le lt ll l pos =
+ let s,pos = (scale_certificate pos) in
+ let rec _cert_of_pos = function
+ Axiom_eq i -> let idx = get_index_of_ith_match is_eq i l in
+ Mc.S_In (Ml2C.nat idx)
+ | Axiom_le i -> let idx = get_index_of_ith_match is_le i l in
+ Mc.S_In (Ml2C.nat idx)
+ | Axiom_lt i -> let idx = get_index_of_ith_match is_lt i l in
+ Mc.S_In (Ml2C.nat idx)
+ | Monoid l -> Mc.S_Monoid (Ml2C.list Ml2C.nat l)
+ | Rational_eq n | Rational_le n | Rational_lt n ->
+ if compare_num n (Int 0) = 0 then Mc.S_Z else
+ Mc.S_Pos (Ml2C.bigint (big_int_of_num n))
+ | Square t -> Mc.S_Square (term_to_expr t)
+ | Eqmul (t, y) -> Mc.S_Ideal(term_to_expr t, _cert_of_pos y)
+ | Sum (y, z) -> Mc.S_Add (_cert_of_pos y, _cert_of_pos z)
+ | Product (y, z) -> Mc.S_Mult (_cert_of_pos y, _cert_of_pos z) in
+ s, Certificate.simplify_cone Certificate.z_spec (_cert_of_pos pos)
+
+
+let term_of_cert l pos =
+ let l = List.map fst' l in
+ let rec _cert_of_pos = function
+ | Mc.S_In i -> expr_to_term (List.nth l (C2Ml.nat i))
+ | Mc.S_Pos p -> Const (C2Ml.num p)
+ | Mc.S_Z -> Const (Int 0)
+ | Mc.S_Square t -> Mul(expr_to_term t, expr_to_term t)
+ | Mc.S_Monoid m -> List.fold_right
+ (fun x m -> Mul (expr_to_term (List.nth l (C2Ml.nat x)),m))
+ (C2Ml.list (fun x -> x) m) (Const (Int 1))
+ | Mc.S_Ideal (t, y) -> Mul(expr_to_term t, _cert_of_pos y)
+ | Mc.S_Add (y, z) -> Add (_cert_of_pos y, _cert_of_pos z)
+ | Mc.S_Mult (y, z) -> Mul (_cert_of_pos y, _cert_of_pos z) in
+ (_cert_of_pos pos)
+
+let rec canonical_sum_to_string = function s -> failwith "not implemented"
+
+let print_canonical_sum m = Format.print_string (canonical_sum_to_string m)
+
+let print_list_term l =
+ print_string "print_list_term\n";
+ List.iter (fun (Mc.Pair(e,k)) -> Printf.printf "q: %s %s ;"
+ (string_of_poly (poly_of_term (expr_to_term e)))
+ (match k with
+ Mc.Equal -> "= "
+ | Mc.Strict -> "> "
+ | Mc.NonStrict -> ">= "
+ | _ -> failwith "not_implemented")) l ;
+ print_string "\n"
+
+
+let partition_expr l =
+ let rec f i = function
+ | [] -> ([],[],[])
+ | Mc.Pair(e,k)::l ->
+ let (eq,ge,neq) = f (i+1) l in
+ match k with
+ | Mc.Equal -> ((e,i)::eq,ge,neq)
+ | Mc.NonStrict -> (eq,(e,Axiom_le i)::ge,neq)
+ | Mc.Strict -> (* e > 0 == e >= 0 /\ e <> 0 *)
+ (eq, (e,Axiom_lt i)::ge,(e,Axiom_lt i)::neq)
+ | Mc.NonEqual -> (eq,ge,(e,Axiom_eq i)::neq)
+ (* Not quite sure -- Coq interface has changed *)
+ in f 0 l
+
+
+let rec sets_of_list l =
+ match l with
+ | [] -> [[]]
+ | e::l -> let s = sets_of_list l in
+ s@(List.map (fun s0 -> e::s0) s)
+
+let cert_of_pos pos =
+ let s,pos = (scale_certificate pos) in
+ let rec _cert_of_pos = function
+ Axiom_eq i -> Mc.S_In (Ml2C.nat i)
+ | Axiom_le i -> Mc.S_In (Ml2C.nat i)
+ | Axiom_lt i -> Mc.S_In (Ml2C.nat i)
+ | Monoid l -> Mc.S_Monoid (Ml2C.list Ml2C.nat l)
+ | Rational_eq n | Rational_le n | Rational_lt n ->
+ if compare_num n (Int 0) = 0 then Mc.S_Z else
+ Mc.S_Pos (Ml2C.bigint (big_int_of_num n))
+ | Square t -> Mc.S_Square (term_to_expr t)
+ | Eqmul (t, y) -> Mc.S_Ideal(term_to_expr t, _cert_of_pos y)
+ | Sum (y, z) -> Mc.S_Add (_cert_of_pos y, _cert_of_pos z)
+ | Product (y, z) -> Mc.S_Mult (_cert_of_pos y, _cert_of_pos z) in
+ s, Certificate.simplify_cone Certificate.z_spec (_cert_of_pos pos)
+
+(* The exploration is probably not complete - for simple cases, it works... *)
+let real_nonlinear_prover d l =
+ try
+ let (eq,ge,neq) = partition_expr l in
+
+ let rec elim_const = function
+ [] -> []
+ | (x,y)::l -> let p = poly_of_term (expr_to_term x) in
+ if poly_isconst p
+ then elim_const l
+ else (p,y)::(elim_const l) in
+
+ let eq = elim_const eq in
+ let peq = List.map fst eq in
+
+ let pge = List.map
+ (fun (e,psatz) -> poly_of_term (expr_to_term e),psatz) ge in
+
+ let monoids = List.map (fun m -> (List.fold_right (fun (p,kd) y ->
+ let p = poly_of_term (expr_to_term p) in
+ match kd with
+ | Axiom_lt i -> poly_mul p y
+ | Axiom_eq i -> poly_mul (poly_pow p 2) y
+ | _ -> failwith "monoids") m (poly_const (Int 1)) , map snd m))
+ (sets_of_list neq) in
+
+ let (cert_ideal, cert_cone,monoid) = deepen_until d (fun d ->
+ list_try_find (fun m -> let (ci,cc) =
+ real_positivnullstellensatz_general false d peq pge (poly_neg (fst m) ) in
+ (ci,cc,snd m)) monoids) 0 in
+
+ let proofs_ideal = map2 (fun q i -> Eqmul(term_of_poly q,Axiom_eq i))
+ cert_ideal (List.map snd eq) in
+
+ let proofs_cone = map term_of_sos cert_cone in
+
+ let proof_ne =
+ let (neq , lt) = List.partition
+ (function Axiom_eq _ -> true | _ -> false ) monoid in
+ let sq = match
+ (List.map (function Axiom_eq i -> i | _ -> failwith "error") neq)
+ with
+ | [] -> Rational_lt (Int 1)
+ | l -> Monoid l in
+ List.fold_right (fun x y -> Product(x,y)) lt sq in
+
+ let proof = list_fold_right_elements
+ (fun s t -> Sum(s,t)) (proof_ne :: proofs_ideal @ proofs_cone) in
+
+ let s,proof' = scale_certificate proof in
+ let cert = snd (cert_of_pos proof') in
+ if debug
+ then Printf.printf "cert poly : %s\n"
+ (string_of_poly (poly_of_term (term_of_cert l cert)));
+ match Mc.zWeakChecker (Ml2C.list (fun x -> x) l) cert with
+ | Mc.True -> Some cert
+ | Mc.False -> (print_string "buggy certificate" ; flush stdout) ;None
+ with
+ | Sos.TooDeep -> None
+
+
+(* This is somewhat buggy, over Z, strict inequality vanish... *)
+let pure_sos l =
+ (* If there is no strict inequality,
+ I should nonetheless be able to try something - over Z > is equivalent to -1 >= *)
+ try
+ let l = List.combine l (interval 0 (length l -1)) in
+ let (lt,i) = try (List.find (fun (x,_) -> snd' x = Mc.Strict) l)
+ with Not_found -> List.hd l in
+ let plt = poly_neg (poly_of_term (expr_to_term (fst' lt))) in
+ let (n,polys) = sumofsquares plt in (* n * (ci * pi^2) *)
+ let pos = Product (Rational_lt n,
+ List.fold_right (fun (c,p) rst -> Sum (Product (Rational_lt c, Square
+ (term_of_poly p)), rst))
+ polys (Rational_lt (Int 0))) in
+ let proof = Sum(Axiom_lt i, pos) in
+ let s,proof' = scale_certificate proof in
+ let cert = snd (cert_of_pos proof') in
+ Some cert
+ with
+ | Not_found -> (* This is no strict inequality *) None
+ | x -> None
+
+
+type micromega_polys = (Micromega.z Mc.pExpr, Mc.op1) Micromega.prod list
+type csdp_certificate = Certificate.Mc.z Certificate.Mc.coneMember option
+type provername = string * int option
+
+let main () =
+ if Array.length Sys.argv <> 3 then
+ (Printf.printf "Usage: csdpcert inputfile outputfile\n"; exit 1);
+ let input_file = Sys.argv.(1) in
+ let output_file = Sys.argv.(2) in
+ let inch = open_in input_file in
+ let (prover,poly) = (input_value inch : provername * micromega_polys) in
+ close_in inch;
+ let cert =
+ match prover with
+ | "real_nonlinear_prover", Some d -> real_nonlinear_prover d poly
+ | "pure_sos", None -> pure_sos poly
+ | prover, _ -> (Printf.printf "unknown prover: %s\n" prover; exit 1) in
+ let outch = open_out output_file in
+ output_value outch (cert:csdp_certificate);
+ close_out outch;
+ exit 0;;
+
+let _ = main () in ()
diff --git a/contrib/micromega/g_micromega.ml4 b/contrib/micromega/g_micromega.ml4
new file mode 100644
index 000000000..3ea49dada
--- /dev/null
+++ b/contrib/micromega/g_micromega.ml4
@@ -0,0 +1,59 @@
+(************************************************************************)
+(* 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 *)
+(************************************************************************)
+(* *)
+(* Micromega: A reflexive tactic using the Positivstellensatz *)
+(* *)
+(* Frédéric Besson (Irisa/Inria) 2006-2008 *)
+(* *)
+(************************************************************************)
+
+(*i camlp4deps: "parsing/grammar.cma" i*)
+
+(* $Id:$ *)
+
+open Quote
+open Ring
+open Mutils
+open Rawterm
+open Util
+
+let out_arg = function
+ | ArgVar _ -> anomaly "Unevaluated or_var variable"
+ | ArgArg x -> x
+
+TACTIC EXTEND Micromega
+| [ "micromegap" int_or_var(i) ] -> [ Coq_micromega.micromega (out_arg i) ]
+| [ "micromegap" ] -> [ Coq_micromega.micromega (-1) ]
+END
+
+TACTIC EXTEND Sos
+[ "sosp" ] -> [ Coq_micromega.sos]
+END
+
+
+TACTIC EXTEND Omicron
+[ "omicronp" ] -> [ Coq_micromega.omicron]
+END
+
+TACTIC EXTEND QOmicron
+[ "qomicronp" ] -> [ Coq_micromega.qomicron]
+END
+
+
+TACTIC EXTEND ZOmicron
+[ "zomicronp" ] -> [ Coq_micromega.zomicron]
+END
+
+TACTIC EXTEND ROmicron
+[ "romicronp" ] -> [ Coq_micromega.romicron]
+END
+
+TACTIC EXTEND RMicromega
+| [ "rmicromegap" int_or_var(i) ] -> [ Coq_micromega.rmicromega (out_arg i) ]
+| [ "rmicromegap" ] -> [ Coq_micromega.rmicromega (-1) ]
+END
diff --git a/contrib/micromega/mfourier.ml b/contrib/micromega/mfourier.ml
new file mode 100644
index 000000000..415d3a3e2
--- /dev/null
+++ b/contrib/micromega/mfourier.ml
@@ -0,0 +1,667 @@
+(************************************************************************)
+(* 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 *)
+(************************************************************************)
+(* *)
+(* Micromega: A reflexive tactic using the Positivstellensatz *)
+(* *)
+(* Frédéric Besson (Irisa/Inria) 2006-2008 *)
+(* *)
+(************************************************************************)
+
+(* Yet another implementation of Fourier *)
+open Num
+
+module Cmp =
+ (* How to compare pairs, lists ... *)
+struct
+ let rec compare_lexical l =
+ match l with
+ | [] -> 0 (* Equal *)
+ | f::l ->
+ let cmp = f () in
+ if cmp = 0 then compare_lexical l else cmp
+
+ let rec compare_list cmp l1 l2 =
+ match l1 , l2 with
+ | [] , [] -> 0
+ | [] , _ -> -1
+ | _ , [] -> 1
+ | e1::l1 , e2::l2 ->
+ let c = cmp e1 e2 in
+ if c = 0 then compare_list cmp l1 l2 else c
+
+ let hash_list hash l =
+ let rec xhash res l =
+ match l with
+ | [] -> res
+ | e::l -> xhash ((hash e) lxor res) l in
+ xhash (Hashtbl.hash []) l
+
+end
+
+module Interval =
+struct
+ (** The type of intervals. **)
+ type intrvl = Empty | Point of num | Itv of num option * num option
+
+ (**
+ Different intervals can denote the same set of variables e.g.,
+ Point n && Itv (Some n, Some n)
+ Itv (Some x) (Some y) && Empty if x > y
+ see the 'belongs_to' function.
+ **)
+
+ (* The set of numerics that belong to an interval *)
+ let belongs_to n = function
+ | Empty -> false
+ | Point x -> n =/ x
+ | Itv(Some x, Some y) -> x <=/ n && n <=/ y
+ | Itv(None,Some y) -> n <=/ y
+ | Itv(Some x,None) -> x <=/ n
+ | Itv(None,None) -> true
+
+ let string_of_bound = function
+ | None -> "oo"
+ | Some n -> Printf.sprintf "Bd(%s)" (string_of_num n)
+
+ let string_of_intrvl = function
+ | Empty -> "[]"
+ | Point n -> Printf.sprintf "[%s]" (string_of_num n)
+ | Itv(bd1,bd2) ->
+ Printf.sprintf "[%s,%s]" (string_of_bound bd1) (string_of_bound bd2)
+
+ let pick_closed_to_zero = function
+ | Empty -> None
+ | Point n -> Some n
+ | Itv(None,None) -> Some (Int 0)
+ | Itv(None,Some i) ->
+ Some (if (Int 0) <=/ (floor_num i) then Int 0 else floor_num i)
+ | Itv(Some i,None) ->
+ Some (if i <=/ (Int 0) then Int 0 else ceiling_num i)
+ | Itv(Some i,Some j) ->
+ Some (
+ if i <=/ Int 0 && Int 0 <=/ j
+ then Int 0
+ else if ceiling_num i <=/ floor_num j
+ then ceiling_num i (* why not *) else i)
+
+ type status =
+ | O | Qonly | Z | Q
+
+ let interval_kind = function
+ | Empty -> O
+ | Point n -> if ceiling_num n =/ n then Z else Qonly
+ | Itv(None,None) -> Z
+ | Itv(None,Some i) -> if ceiling_num i <>/ i then Q else Z
+ | Itv(Some i,None) -> if ceiling_num i <>/ i then Q else Z
+ | Itv(Some i,Some j) ->
+ if ceiling_num i <>/ i or floor_num j <>/ j then Q else Z
+
+ let empty_z = function
+ | Empty -> true
+ | Point n -> ceiling_num n <>/ n
+ | Itv(None,None) | Itv(None,Some _) | Itv(Some _,None) -> false
+ | Itv(Some i,Some j) -> ceiling_num i >/ floor_num j
+
+
+ let normalise b1 b2 =
+ match b1 , b2 with
+ | Some i , Some j ->
+ (match compare_num i j with
+ | 1 -> Empty
+ | 0 -> Point i
+ | _ -> Itv(b1,b2)
+ )
+ | _ -> Itv(b1,b2)
+
+
+
+ let min x y =
+ match x , y with
+ | None , x | x , None -> x
+ | Some i , Some j -> Some (min_num i j)
+
+ let max x y =
+ match x , y with
+ | None , x | x , None -> x
+ | Some i , Some j -> Some (max_num i j)
+
+ let inter i1 i2 =
+ match i1,i2 with
+ | Empty , _ -> Empty
+ | _ , Empty -> Empty
+ | Point n , Point m -> if n =/ m then i1 else Empty
+ | Point n , Itv (mn,mx) | Itv (mn,mx) , Point n->
+ if (match mn with
+ | None -> true
+ | Some mn -> mn <=/ n) &&
+ (match mx with
+ | None -> true
+ | Some mx -> n <=/ mx) then Point n else Empty
+ | Itv (min1,max1) , Itv (min2,max2) ->
+ let bmin = max min1 min2
+ and bmax = min max1 max2 in
+ normalise bmin bmax
+
+ (* a.x >= b*)
+ let bound_of_constraint (a,b) =
+ match compare_num a (Int 0) with
+ | 0 ->
+ if compare_num b (Int 0) = 1
+ then Empty
+ (*actually this is a contradiction failwith "bound_of_constraint" *)
+ else Itv (None,None)
+ | 1 -> Itv (Some (div_num b a),None)
+ | -1 -> Itv (None, Some (div_num b a))
+ | x -> failwith "bound_of_constraint(2)"
+
+
+ let bounded x =
+ match x with
+ | Itv(None,_) | Itv(_,None) -> false
+ | _ -> true
+
+
+ let range = function
+ | Empty -> Some (Int 0)
+ | Point n -> Some (Int (if ceiling_num n =/ n then 1 else 0))
+ | Itv(None,_) | Itv(_,None)-> None
+ | Itv(Some i,Some j) -> Some (floor_num j -/ceiling_num i +/ (Int 1))
+
+ (* Returns the interval of smallest range *)
+ let smaller_itv i1 i2 =
+ match range i1 , range i2 with
+ | None , _ -> false
+ | _ , None -> true
+ | Some i , Some j -> i <=/ j
+
+end
+open Interval
+
+(* A set of constraints *)
+module Sys(V:Vector.S) (* : Vector.SystemS with module Vect = V*) =
+struct
+
+ module Vect = V
+
+ module Cstr = Vector.Cstr(V)
+ open Cstr
+
+
+ module CMap = Map.Make(
+ struct
+ type t = Vect.t
+ let compare = Vect.compare
+ end)
+
+ module CstrBag =
+ struct
+
+ type mut_itv = { mutable itv : intrvl}
+
+ type t = mut_itv CMap.t
+
+ exception Contradiction
+
+ let cstr_to_itv cstr =
+ let (n,l) = V.normalise cstr.coeffs in
+ if n =/ (Int 0)
+ then (Vect.null, bound_of_constraint (Int 0,cstr.cst)) (* Might be empty *)
+ else
+ match cstr.op with
+ | Eq -> let n = cstr.cst // n in (l, Point n)
+ | Ge ->
+ match compare_num n (Int 0) with
+ | 0 -> failwith "intrvl_of_constraint"
+ | 1 -> (l,Itv (Some (cstr.cst // n), None))
+ | -1 -> (l, Itv(None,Some (cstr.cst // n)))
+ | _ -> failwith "cstr_to_itv"
+
+
+ let empty = CMap.empty
+
+
+
+
+ let is_empty = CMap.is_empty
+
+ let find_vect v bag =
+ try
+ (bag,CMap.find v bag)
+ with Not_found -> let x = { itv = Itv(None,None)} in (CMap.add v x bag ,x)
+
+
+ let add (v,b) bag =
+ match b with
+ | Empty -> raise Contradiction
+ | Itv(None,None) -> bag
+ | _ ->
+ let (bag,intrl) = find_vect v bag in
+ match inter b intrl.itv with
+ | Empty -> raise Contradiction
+ | itv -> intrl.itv <- itv ; bag
+
+ exception Found of cstr
+
+ let find_equation bag =
+ try
+ CMap.fold (fun v i () ->
+ match i.itv with
+ | Point n -> let e = {coeffs = v ; op = Eq ; cst = n}
+ in raise (Found e)
+ | _ -> () ) bag () ; None
+ with Found c -> Some c
+
+
+ let fold f bag acc =
+ CMap.fold (fun v itv acc ->
+ match itv.itv with
+ | Empty | Itv(None,None) -> failwith "fold Empty"
+ | Itv(None ,Some i) ->
+ f {coeffs = V.mul (Int (-1)) v ; op = Ge ; cst = minus_num i} acc
+ | Point n -> f {coeffs = v ; op = Eq ; cst = n} acc
+ | Itv(x,y) ->
+ (match x with
+ | None -> (fun x -> x)
+ | Some i -> f {coeffs = v ; op = Ge ; cst = i})
+ (match y with
+ | None -> acc
+ | Some i ->
+ f {coeffs = V.mul (Int (-1)) v ; op = Ge ; cst = minus_num i} acc
+ ) ) bag acc
+
+
+ let remove l _ = failwith "remove:Not implemented"
+
+ module Map =
+ Map.Make(
+ struct
+ type t = int
+ let compare : int -> int -> int = Pervasives.compare
+ end)
+
+ let split f (t:t) =
+ let res =
+ fold (fun e m -> let i = f e in
+ Map.add i (add (cstr_to_itv e)
+ (try Map.find i m with
+ Not_found -> empty)) m) t Map.empty in
+ (fun i -> try Map.find i res with Not_found -> empty)
+
+ type map = (int list * int list) Map.t
+
+
+ let status (b:t) =
+ let _ , map = fold (fun c ( (idx:int),(res: map)) ->
+ ( idx + 1,
+ List.fold_left (fun (res:map) (pos,s) ->
+ let (lp,ln) = try Map.find pos res with Not_found -> ([],[]) in
+ match s with
+ | Vect.Pos -> Map.add pos (idx::lp,ln) res
+ | Vect.Neg ->
+ Map.add pos (lp, idx::ln) res) res
+ (Vect.status c.coeffs))) b (0,Map.empty) in
+ Map.fold (fun k e res -> (k,e)::res) map []
+
+
+ type it = num CMap.t
+
+ let iterator x = x
+
+ let element it = failwith "element:Not implemented"
+
+ end
+end
+
+module Fourier(Vect : Vector.S) =
+struct
+ module Vect = Vect
+ module Sys = Sys( Vect)
+ module Cstr = Sys.Cstr
+ module Bag = Sys.CstrBag
+
+ open Cstr
+ open Sys
+
+ let debug = false
+
+ let print_bag msg b =
+ print_endline msg;
+ CstrBag.fold (fun e () -> print_endline (Cstr.string_of_cstr e)) b ()
+
+ let print_bag_file file msg b =
+ let f = open_out file in
+ output_string f msg;
+ CstrBag.fold (fun e () ->
+ Printf.fprintf f "%s\n" (Cstr.string_of_cstr e)) b ()
+
+
+ (* A system with only inequations --
+ *)
+ let partition i m =
+ let splitter cstr = compare_num (Vect.get i cstr.coeffs ) (Int 0) in
+ let split = CstrBag.split splitter m in
+ (split (-1) , split 0, split 1)
+
+
+ (* op of the result is arbitrary Ge *)
+ let lin_comb n1 c1 n2 c2 =
+ { coeffs = Vect.lin_comb n1 c1.coeffs n2 c2.coeffs ;
+ op = Ge ;
+ cst = (n1 */ c1.cst) +/ (n2 */ c2.cst)}
+
+ (* BUG? : operator of the result ? *)
+
+ let combine_project i c1 c2 =
+ let p = Vect.get i c1.coeffs
+ and n = Vect.get i c2.coeffs in
+ assert (n </ Int 0 && p >/ Int 0) ;
+ let nopp = minus_num n in
+ let c =lin_comb nopp c1 p c2 in
+ let op = if c1.op = Ge || c2.op = Ge then Ge else Eq in
+ CstrBag.cstr_to_itv {coeffs = c.coeffs ; op = op ; cst= c.cst }
+
+
+ let project i m =
+ let (neg,zero,pos) = partition i m in
+ let project1 cpos acc =
+ CstrBag.fold (fun cneg res ->
+ CstrBag.add (combine_project i cpos cneg) res) neg acc in
+ (CstrBag.fold project1 pos zero)
+
+ (* Given a vector [x1 -> v1; ... ; xn -> vn]
+ and a constraint {x1 ; .... xn >= c }
+ *)
+ let evaluate_constraint i map cstr =
+ let {coeffs = _coeffs ; op = _op ; cst = _cst} = cstr in
+ let vi = Vect.get i _coeffs in
+ let v = Vect.set i (Int 0) _coeffs in
+ (vi, _cst -/ Vect.dotp map v)
+
+
+ let rec bounds m itv =
+ match m with
+ | [] -> itv
+ | e::m -> bounds m (inter itv (bound_of_constraint e))
+
+
+
+ let compare_status (i,(lp,ln)) (i',(lp',ln')) =
+ let cmp = Pervasives.compare
+ ((List.length lp) * (List.length ln))
+ ((List.length lp') * (List.length ln')) in
+ if cmp = 0
+ then Pervasives.compare i i'
+ else cmp
+
+ let cardinal m = CstrBag.fold (fun _ x -> x + 1) m 0
+
+ let lightest_projection l c m =
+ let bound = c in
+ if debug then (Printf.printf "l%i" bound; flush stdout) ;
+ let rec xlight best l =
+ match l with
+ | [] -> best
+ | i::l ->
+ let proj = (project i m) in
+ let cproj = cardinal proj in
+ (*Printf.printf " p %i " cproj; flush stdout;*)
+ match best with
+ | None ->
+ if cproj < bound
+ then Some(cproj,proj,i)
+ else xlight (Some(cproj,proj,i)) l
+ | Some (cbest,_,_) ->
+ if cproj < cbest
+ then
+ if cproj < bound then Some(cproj,proj,i)
+ else xlight (Some(cproj,proj,i)) l
+ else xlight best l in
+ match xlight None l with
+ | None -> None
+ | Some(_,p,i) -> Some (p,i)
+
+
+
+ exception Equality of cstr
+
+ let find_equality m = Bag.find_equation m
+
+
+
+ let pivot (n,v) eq ge =
+ assert (eq.op = Eq) ;
+ let res =
+ match
+ compare_num v (Int 0),
+ compare_num (Vect.get n ge.coeffs) (Int 0)
+ with
+ | 0 , _ -> failwith "Buggy"
+ | _ ,0 -> (CstrBag.cstr_to_itv ge)
+ | 1 , -1 -> combine_project n eq ge
+ | -1 , 1 -> combine_project n ge eq
+ | 1 , 1 ->
+ combine_project n ge
+ {coeffs = Vect.mul (Int (-1)) eq.coeffs;
+ op = eq.op ;
+ cst = minus_num eq.cst}
+ | -1 , -1 ->
+ combine_project n
+ {coeffs = Vect.mul (Int (-1)) eq.coeffs;
+ op = eq.op ; cst = minus_num eq.cst} ge
+ | _ -> failwith "pivot" in
+ res
+
+ let check_cstr v c =
+ let {coeffs = _coeffs ; op = _op ; cst = _cst} = c in
+ let vl = Vect.dotp v _coeffs in
+ match _op with
+ | Eq -> vl =/ _cst
+ | Ge -> vl >= _cst
+
+
+ let forall p sys =
+ try
+ CstrBag.fold (fun c () -> if p c then () else raise Not_found) sys (); true
+ with Not_found -> false
+
+
+ let check_sys v sys = forall (check_cstr v) sys
+
+ let check_null_cstr c =
+ let {coeffs = _coeffs ; op = _op ; cst = _cst} = c in
+ match _op with
+ | Eq -> (Int 0) =/ _cst
+ | Ge -> (Int 0) >= _cst
+
+ let check_null sys = forall check_null_cstr sys
+
+
+ let optimise_ge
+ quick_check choose choose_idx return_empty return_ge return_eq m =
+ let c = cardinal m in
+ let bound = 2 * c in
+ if debug then (Printf.printf "optimise_ge: %i\n" c; flush stdout);
+
+ let rec xoptimise m =
+ if debug then (Printf.printf "x%i" (cardinal m) ; flush stdout);
+ if debug then (print_bag "xoptimise" m ; flush stdout);
+ if quick_check m
+ then return_empty m
+ else
+ match find_equality m with
+ | None -> xoptimise_ge m
+ | Some eq -> xoptimise_eq eq m
+
+ and xoptimise_ge m =
+ begin
+ let c = cardinal m in
+ let l = List.map fst (List.sort compare_status (CstrBag.status m)) in
+ let idx = choose bound l c m in
+ match idx with
+ | None -> return_empty m
+ | Some (proj,i) ->
+ match xoptimise proj with
+ | None -> None
+ | Some mapping -> return_ge m i mapping
+ end
+ and xoptimise_eq eq m =
+ let l = List.map fst (Vect.status eq.coeffs) in
+ match choose_idx l with
+ | None -> (*if l = [] then None else*) return_empty m
+ | Some i ->
+ let p = (i,Vect.get i eq.coeffs) in
+ let m' = CstrBag.fold
+ (fun ge res -> CstrBag.add (pivot p eq ge) res) m CstrBag.empty in
+ match xoptimise ( m') with
+ | None -> None
+ | Some mapp -> return_eq m eq i mapp in
+ try
+ let res = xoptimise m in res
+ with CstrBag.Contradiction -> (*print_string "contradiction" ;*) None
+
+
+
+ let minimise m =
+ let opt_zero_choose bound l c m =
+ if c > bound
+ then lightest_projection l c m
+ else match l with
+ | [] -> None
+ | i::_ -> Some (project i m, i) in
+
+ let choose_idx = function [] -> None | x::l -> Some x in
+
+ let opt_zero_return_empty m = Some Vect.null in
+
+
+ let opt_zero_return_ge m i mapping =
+ let (it:intrvl) = CstrBag.fold (fun cstr itv -> Interval.inter
+ (bound_of_constraint (evaluate_constraint i mapping cstr)) itv) m
+ (Itv (None, None)) in
+ match pick_closed_to_zero it with
+ | None -> print_endline "Cannot pick" ; None
+ | Some v ->
+ let res = (Vect.set i v mapping) in
+ if debug
+ then Printf.printf "xoptimise res %i [%s]" i (Vect.string res) ;
+ Some res in
+
+ let opt_zero_return_eq m eq i mapp =
+ let (a,b) = evaluate_constraint i mapp eq in
+ Some (Vect.set i (div_num b a) mapp) in
+
+ optimise_ge check_null opt_zero_choose
+ choose_idx opt_zero_return_empty opt_zero_return_ge opt_zero_return_eq m
+
+ let normalise cstr = [CstrBag.cstr_to_itv cstr]
+
+ let find_point l =
+ (* List.iter (fun e -> print_endline (Cstr.string_of_cstr e)) l;*)
+ try
+ let m = List.fold_left (fun sys e -> CstrBag.add (CstrBag.cstr_to_itv e) sys)
+ CstrBag.empty l in
+ match minimise m with
+ | None -> None
+ | Some res ->
+ if debug then Printf.printf "[%s]" (Vect.string res);
+ Some res
+ with CstrBag.Contradiction -> None
+
+
+ let find_q_interval_for x m =
+ if debug then Printf.printf "find_q_interval_for %i\n" x ;
+
+ let choose bound l c m =
+ let rec xchoose l =
+ match l with
+ | [] -> None
+ | i::l -> if i = x then xchoose l else Some (project i m,i) in
+ xchoose l in
+
+ let rec choose_idx = function
+ [] -> None
+ | e::l -> if e = x then choose_idx l else Some e in
+
+ let return_empty m = (* Beurk *)
+ (* returns the interval of x *)
+ Some (CstrBag.fold (fun cstr itv ->
+ let i = if cstr.op = Eq
+ then Point (cstr.cst // Vect.get x cstr.coeffs)
+ else if Vect.is_null (Vect.set x (Int 0) cstr.coeffs)
+ then bound_of_constraint (Vect.get x cstr.coeffs , cstr.cst)
+ else itv
+ in
+ Interval.inter i itv) m (Itv (None, None))) in
+
+ let return_ge m i res = Some res in
+
+ let return_eq m eq i res = Some res in
+
+ try
+ optimise_ge
+ (fun x -> false) choose choose_idx return_empty return_ge return_eq m
+ with CstrBag.Contradiction -> None
+
+
+ let find_q_intervals sys =
+ let variables =
+ List.map fst (List.sort compare_status (CstrBag.status sys)) in
+ List.map (fun x -> (x,find_q_interval_for x sys)) variables
+
+ let pp_option f o = function
+ None -> Printf.fprintf o "None"
+ | Some x -> Printf.fprintf o "Some %a" f x
+
+ let optimise vect sys =
+ (* we have to modify the system with a dummy variable *)
+ let fresh =
+ List.fold_left (fun fr c -> Pervasives.max fr (Vect.fresh c.coeffs)) 0 sys in
+ assert (List.for_all (fun x -> Vect.get fresh x.coeffs =/ Int 0) sys);
+ let cstr = {
+ coeffs = Vect.set fresh (Int (-1)) vect ;
+ op = Eq ;
+ cst = (Int 0)} in
+ try
+ find_q_interval_for fresh
+ (List.fold_left
+ (fun bg c -> CstrBag.add (CstrBag.cstr_to_itv c) bg)
+ CstrBag.empty (cstr::sys))
+ with CstrBag.Contradiction -> None
+
+
+ let optimise vect sys =
+ let res = optimise vect sys in
+ if debug
+ then Printf.printf "optimise %s -> %a\n"
+ (Vect.string vect) (pp_option (fun o x -> Printf.printf "%s" (string_of_intrvl x))) res
+ ; res
+
+ let find_Q_interval sys =
+ try
+ let sys =
+ (List.fold_left
+ (fun bg c -> CstrBag.add (CstrBag.cstr_to_itv c) bg) CstrBag.empty sys) in
+ let candidates =
+ List.fold_left
+ (fun l (x,i) -> match i with
+ None -> (x,Empty)::l
+ | Some i -> (x,i)::l) [] (find_q_intervals sys) in
+ match List.fold_left
+ (fun (x1,i1) (x2,i2) ->
+ if smaller_itv i1 i2
+ then (x1,i1) else (x2,i2)) (-1,Itv(None,None)) candidates
+ with
+ | (i,Empty) -> None
+ | (x,Itv(Some i, Some j)) -> Some(i,x,j)
+ | (x,Point n) -> Some(n,x,n)
+ | _ -> None
+ with CstrBag.Contradiction -> None
+
+
+end
+
diff --git a/contrib/micromega/micromega.ml b/contrib/micromega/micromega.ml
new file mode 100644
index 000000000..e151e4e1d
--- /dev/null
+++ b/contrib/micromega/micromega.ml
@@ -0,0 +1,1512 @@
+type __ = Obj.t
+let __ = let rec f _ = Obj.repr f in Obj.repr f
+
+type bool =
+ | True
+ | False
+
+(** val negb : bool -> bool **)
+
+let negb = function
+ | True -> False
+ | False -> True
+
+type nat =
+ | O
+ | S of nat
+
+type 'a option =
+ | Some of 'a
+ | None
+
+type ('a, 'b) prod =
+ | Pair of 'a * 'b
+
+type comparison =
+ | Eq
+ | Lt
+ | Gt
+
+(** val compOpp : comparison -> comparison **)
+
+let compOpp = function
+ | Eq -> Eq
+ | Lt -> Gt
+ | Gt -> Lt
+
+type sumbool =
+ | Left
+ | Right
+
+type 'a sumor =
+ | Inleft of 'a
+ | Inright
+
+type 'a list =
+ | Nil
+ | Cons of 'a * 'a list
+
+(** val app : 'a1 list -> 'a1 list -> 'a1 list **)
+
+let rec app l m =
+ match l with
+ | Nil -> m
+ | Cons (a, l1) -> Cons (a, (app l1 m))
+
+(** val nth : nat -> 'a1 list -> 'a1 -> 'a1 **)
+
+let rec nth n0 l default =
+ match n0 with
+ | O -> (match l with
+ | Nil -> default
+ | Cons (x, l') -> x)
+ | S m ->
+ (match l with
+ | Nil -> default
+ | Cons (x, t0) -> nth m t0 default)
+
+(** val map : ('a1 -> 'a2) -> 'a1 list -> 'a2 list **)
+
+let rec map f = function
+ | Nil -> Nil
+ | Cons (a, t0) -> Cons ((f a), (map f t0))
+
+type positive =
+ | XI of positive
+ | XO of positive
+ | XH
+
+(** val psucc : positive -> positive **)
+
+let rec psucc = function
+ | XI p -> XO (psucc p)
+ | XO p -> XI p
+ | XH -> XO XH
+
+(** val pplus : positive -> positive -> positive **)
+
+let rec pplus x y =
+ match x with
+ | XI p ->
+ (match y with
+ | XI q0 -> XO (pplus_carry p q0)
+ | XO q0 -> XI (pplus p q0)
+ | XH -> XO (psucc p))
+ | XO p ->
+ (match y with
+ | XI q0 -> XI (pplus p q0)
+ | XO q0 -> XO (pplus p q0)
+ | XH -> XI p)
+ | XH ->
+ (match y with
+ | XI q0 -> XO (psucc q0)
+ | XO q0 -> XI q0
+ | XH -> XO XH)
+
+(** val pplus_carry : positive -> positive -> positive **)
+
+and pplus_carry x y =
+ match x with
+ | XI p ->
+ (match y with
+ | XI q0 -> XI (pplus_carry p q0)
+ | XO q0 -> XO (pplus_carry p q0)
+ | XH -> XI (psucc p))
+ | XO p ->
+ (match y with
+ | XI q0 -> XO (pplus_carry p q0)
+ | XO q0 -> XI (pplus p q0)
+ | XH -> XO (psucc p))
+ | XH ->
+ (match y with
+ | XI q0 -> XI (psucc q0)
+ | XO q0 -> XO (psucc q0)
+ | XH -> XI XH)
+
+(** val p_of_succ_nat : nat -> positive **)
+
+let rec p_of_succ_nat = function
+ | O -> XH
+ | S x -> psucc (p_of_succ_nat x)
+
+(** val pdouble_minus_one : positive -> positive **)
+
+let rec pdouble_minus_one = function
+ | XI p -> XI (XO p)
+ | XO p -> XI (pdouble_minus_one p)
+ | XH -> XH
+
+type positive_mask =
+ | IsNul
+ | IsPos of positive
+ | IsNeg
+
+(** val pdouble_plus_one_mask : positive_mask -> positive_mask **)
+
+let pdouble_plus_one_mask = function
+ | IsNul -> IsPos XH
+ | IsPos p -> IsPos (XI p)
+ | IsNeg -> IsNeg
+
+(** val pdouble_mask : positive_mask -> positive_mask **)
+
+let pdouble_mask = function
+ | IsNul -> IsNul
+ | IsPos p -> IsPos (XO p)
+ | IsNeg -> IsNeg
+
+(** val pdouble_minus_two : positive -> positive_mask **)
+
+let pdouble_minus_two = function
+ | XI p -> IsPos (XO (XO p))
+ | XO p -> IsPos (XO (pdouble_minus_one p))
+ | XH -> IsNul
+
+(** val pminus_mask : positive -> positive -> positive_mask **)
+
+let rec pminus_mask x y =
+ match x with
+ | XI p ->
+ (match y with
+ | XI q0 -> pdouble_mask (pminus_mask p q0)
+ | XO q0 -> pdouble_plus_one_mask (pminus_mask p q0)
+ | XH -> IsPos (XO p))
+ | XO p ->
+ (match y with
+ | XI q0 -> pdouble_plus_one_mask (pminus_mask_carry p q0)
+ | XO q0 -> pdouble_mask (pminus_mask p q0)
+ | XH -> IsPos (pdouble_minus_one p))
+ | XH -> (match y with
+ | XH -> IsNul
+ | _ -> IsNeg)
+
+(** val pminus_mask_carry : positive -> positive -> positive_mask **)
+
+and pminus_mask_carry x y =
+ match x with
+ | XI p ->
+ (match y with
+ | XI q0 -> pdouble_plus_one_mask (pminus_mask_carry p q0)
+ | XO q0 -> pdouble_mask (pminus_mask p q0)
+ | XH -> IsPos (pdouble_minus_one p))
+ | XO p ->
+ (match y with
+ | XI q0 -> pdouble_mask (pminus_mask_carry p q0)
+ | XO q0 -> pdouble_plus_one_mask (pminus_mask_carry p q0)
+ | XH -> pdouble_minus_two p)
+ | XH -> IsNeg
+
+(** val pminus : positive -> positive -> positive **)
+
+let pminus x y =
+ match pminus_mask x y with
+ | IsPos z0 -> z0
+ | _ -> XH
+
+(** val pmult : positive -> positive -> positive **)
+
+let rec pmult x y =
+ match x with
+ | XI p -> pplus y (XO (pmult p y))
+ | XO p -> XO (pmult p y)
+ | XH -> y
+
+(** val pcompare : positive -> positive -> comparison -> comparison **)
+
+let rec pcompare x y r =
+ match x with
+ | XI p ->
+ (match y with
+ | XI q0 -> pcompare p q0 r
+ | XO q0 -> pcompare p q0 Gt
+ | XH -> Gt)
+ | XO p ->
+ (match y with
+ | XI q0 -> pcompare p q0 Lt
+ | XO q0 -> pcompare p q0 r
+ | XH -> Gt)
+ | XH -> (match y with
+ | XH -> r
+ | _ -> Lt)
+
+type n =
+ | N0
+ | Npos of positive
+
+type z =
+ | Z0
+ | Zpos of positive
+ | Zneg of positive
+
+(** val zdouble_plus_one : z -> z **)
+
+let zdouble_plus_one = function
+ | Z0 -> Zpos XH
+ | Zpos p -> Zpos (XI p)
+ | Zneg p -> Zneg (pdouble_minus_one p)
+
+(** val zdouble_minus_one : z -> z **)
+
+let zdouble_minus_one = function
+ | Z0 -> Zneg XH
+ | Zpos p -> Zpos (pdouble_minus_one p)
+ | Zneg p -> Zneg (XI p)
+
+(** val zdouble : z -> z **)
+
+let zdouble = function
+ | Z0 -> Z0
+ | Zpos p -> Zpos (XO p)
+ | Zneg p -> Zneg (XO p)
+
+(** val zPminus : positive -> positive -> z **)
+
+let rec zPminus x y =
+ match x with
+ | XI p ->
+ (match y with
+ | XI q0 -> zdouble (zPminus p q0)
+ | XO q0 -> zdouble_plus_one (zPminus p q0)
+ | XH -> Zpos (XO p))
+ | XO p ->
+ (match y with
+ | XI q0 -> zdouble_minus_one (zPminus p q0)
+ | XO q0 -> zdouble (zPminus p q0)
+ | XH -> Zpos (pdouble_minus_one p))
+ | XH ->
+ (match y with
+ | XI q0 -> Zneg (XO q0)
+ | XO q0 -> Zneg (pdouble_minus_one q0)
+ | XH -> Z0)
+
+(** val zplus : z -> z -> z **)
+
+let zplus x y =
+ match x with
+ | Z0 -> y
+ | Zpos x' ->
+ (match y with
+ | Z0 -> Zpos x'
+ | Zpos y' -> Zpos (pplus x' y')
+ | Zneg y' ->
+ (match pcompare x' y' Eq with
+ | Eq -> Z0
+ | Lt -> Zneg (pminus y' x')
+ | Gt -> Zpos (pminus x' y')))
+ | Zneg x' ->
+ (match y with
+ | Z0 -> Zneg x'
+ | Zpos y' ->
+ (match pcompare x' y' Eq with
+ | Eq -> Z0
+ | Lt -> Zpos (pminus y' x')
+ | Gt -> Zneg (pminus x' y'))
+ | Zneg y' -> Zneg (pplus x' y'))
+
+(** val zopp : z -> z **)
+
+let zopp = function
+ | Z0 -> Z0
+ | Zpos x0 -> Zneg x0
+ | Zneg x0 -> Zpos x0
+
+(** val zminus : z -> z -> z **)
+
+let zminus m n0 =
+ zplus m (zopp n0)
+
+(** val zmult : z -> z -> z **)
+
+let zmult x y =
+ match x with
+ | Z0 -> Z0
+ | Zpos x' ->
+ (match y with
+ | Z0 -> Z0
+ | Zpos y' -> Zpos (pmult x' y')
+ | Zneg y' -> Zneg (pmult x' y'))
+ | Zneg x' ->
+ (match y with
+ | Z0 -> Z0
+ | Zpos y' -> Zneg (pmult x' y')
+ | Zneg y' -> Zpos (pmult x' y'))
+
+(** val zcompare : z -> z -> comparison **)
+
+let zcompare x y =
+ match x with
+ | Z0 -> (match y with
+ | Z0 -> Eq
+ | Zpos y' -> Lt
+ | Zneg y' -> Gt)
+ | Zpos x' -> (match y with
+ | Zpos y' -> pcompare x' y' Eq
+ | _ -> Gt)
+ | Zneg x' ->
+ (match y with
+ | Zneg y' -> compOpp (pcompare x' y' Eq)
+ | _ -> Lt)
+
+(** val dcompare_inf : comparison -> sumbool sumor **)
+
+let dcompare_inf = function
+ | Eq -> Inleft Left
+ | Lt -> Inleft Right
+ | Gt -> Inright
+
+(** val zcompare_rec :
+ z -> z -> (__ -> 'a1) -> (__ -> 'a1) -> (__ -> 'a1) -> 'a1 **)
+
+let zcompare_rec x y h1 h2 h3 =
+ match dcompare_inf (zcompare x y) with
+ | Inleft x0 -> (match x0 with
+ | Left -> h1 __
+ | Right -> h2 __)
+ | Inright -> h3 __
+
+(** val z_gt_dec : z -> z -> sumbool **)
+
+let z_gt_dec x y =
+ zcompare_rec x y (fun _ -> Right) (fun _ -> Right) (fun _ -> Left)
+
+(** val zle_bool : z -> z -> bool **)
+
+let zle_bool x y =
+ match zcompare x y with
+ | Gt -> False
+ | _ -> True
+
+(** val zge_bool : z -> z -> bool **)
+
+let zge_bool x y =
+ match zcompare x y with
+ | Lt -> False
+ | _ -> True
+
+(** val zgt_bool : z -> z -> bool **)
+
+let zgt_bool x y =
+ match zcompare x y with
+ | Gt -> True
+ | _ -> False
+
+(** val zeq_bool : z -> z -> bool **)
+
+let zeq_bool x y =
+ match zcompare x y with
+ | Eq -> True
+ | _ -> False
+
+(** val n_of_nat : nat -> n **)
+
+let n_of_nat = function
+ | O -> N0
+ | S n' -> Npos (p_of_succ_nat n')
+
+(** val zdiv_eucl_POS : positive -> z -> (z, z) prod **)
+
+let rec zdiv_eucl_POS a b =
+ match a with
+ | XI a' ->
+ let Pair (q0, r) = zdiv_eucl_POS a' b in
+ let r' = zplus (zmult (Zpos (XO XH)) r) (Zpos XH) in
+ (match zgt_bool b r' with
+ | True -> Pair ((zmult (Zpos (XO XH)) q0), r')
+ | False -> Pair ((zplus (zmult (Zpos (XO XH)) q0) (Zpos XH)),
+ (zminus r' b)))
+ | XO a' ->
+ let Pair (q0, r) = zdiv_eucl_POS a' b in
+ let r' = zmult (Zpos (XO XH)) r in
+ (match zgt_bool b r' with
+ | True -> Pair ((zmult (Zpos (XO XH)) q0), r')
+ | False -> Pair ((zplus (zmult (Zpos (XO XH)) q0) (Zpos XH)),
+ (zminus r' b)))
+ | XH ->
+ (match zge_bool b (Zpos (XO XH)) with
+ | True -> Pair (Z0, (Zpos XH))
+ | False -> Pair ((Zpos XH), Z0))
+
+(** val zdiv_eucl : z -> z -> (z, z) prod **)
+
+let zdiv_eucl a b =
+ match a with
+ | Z0 -> Pair (Z0, Z0)
+ | Zpos a' ->
+ (match b with
+ | Z0 -> Pair (Z0, Z0)
+ | Zpos p -> zdiv_eucl_POS a' b
+ | Zneg b' ->
+ let Pair (q0, r) = zdiv_eucl_POS a' (Zpos b') in
+ (match r with
+ | Z0 -> Pair ((zopp q0), Z0)
+ | _ -> Pair ((zopp (zplus q0 (Zpos XH))), (zplus b r))))
+ | Zneg a' ->
+ (match b with
+ | Z0 -> Pair (Z0, Z0)
+ | Zpos p ->
+ let Pair (q0, r) = zdiv_eucl_POS a' b in
+ (match r with
+ | Z0 -> Pair ((zopp q0), Z0)
+ | _ -> Pair ((zopp (zplus q0 (Zpos XH))), (zminus b r)))
+ | Zneg b' ->
+ let Pair (q0, r) = zdiv_eucl_POS a' (Zpos b') in
+ Pair (q0, (zopp r)))
+
+type 'c pol =
+ | Pc of 'c
+ | Pinj of positive * 'c pol
+ | PX of 'c pol * positive * 'c pol
+
+(** val p0 : 'a1 -> 'a1 pol **)
+
+let p0 cO =
+ Pc cO
+
+(** val p1 : 'a1 -> 'a1 pol **)
+
+let p1 cI =
+ Pc cI
+
+(** val peq : ('a1 -> 'a1 -> bool) -> 'a1 pol -> 'a1 pol -> bool **)
+
+let rec peq ceqb p p' =
+ match p with
+ | Pc c -> (match p' with
+ | Pc c' -> ceqb c c'
+ | _ -> False)
+ | Pinj (j, q0) ->
+ (match p' with
+ | Pinj (j', q') ->
+ (match pcompare j j' Eq with
+ | Eq -> peq ceqb q0 q'
+ | _ -> False)
+ | _ -> False)
+ | PX (p2, i, q0) ->
+ (match p' with
+ | PX (p'0, i', q') ->
+ (match pcompare i i' Eq with
+ | Eq ->
+ (match peq ceqb p2 p'0 with
+ | True -> peq ceqb q0 q'
+ | False -> False)
+ | _ -> False)
+ | _ -> False)
+
+(** val mkPinj_pred : positive -> 'a1 pol -> 'a1 pol **)
+
+let mkPinj_pred j p =
+ match j with
+ | XI j0 -> Pinj ((XO j0), p)
+ | XO j0 -> Pinj ((pdouble_minus_one j0), p)
+ | XH -> p
+
+(** val mkPX :
+ 'a1 -> ('a1 -> 'a1 -> bool) -> 'a1 pol -> positive -> 'a1 pol -> 'a1 pol **)
+
+let mkPX cO ceqb p i q0 =
+ match p with
+ | Pc c ->
+ (match ceqb c cO with
+ | True ->
+ (match q0 with
+ | Pc c0 -> q0
+ | Pinj (j', q1) -> Pinj ((pplus XH j'), q1)
+ | PX (p2, p3, p4) -> Pinj (XH, q0))
+ | False -> PX (p, i, q0))
+ | Pinj (p2, p3) -> PX (p, i, q0)
+ | PX (p', i', q') ->
+ (match peq ceqb q' (p0 cO) with
+ | True -> PX (p', (pplus i' i), q0)
+ | False -> PX (p, i, q0))
+
+(** val mkXi : 'a1 -> 'a1 -> positive -> 'a1 pol **)
+
+let mkXi cO cI i =
+ PX ((p1 cI), i, (p0 cO))
+
+(** val mkX : 'a1 -> 'a1 -> 'a1 pol **)
+
+let mkX cO cI =
+ mkXi cO cI XH
+
+(** val popp : ('a1 -> 'a1) -> 'a1 pol -> 'a1 pol **)
+
+let rec popp copp = function
+ | Pc c -> Pc (copp c)
+ | Pinj (j, q0) -> Pinj (j, (popp copp q0))
+ | PX (p2, i, q0) -> PX ((popp copp p2), i, (popp copp q0))
+
+(** val paddC : ('a1 -> 'a1 -> 'a1) -> 'a1 pol -> 'a1 -> 'a1 pol **)
+
+let rec paddC cadd p c =
+ match p with
+ | Pc c1 -> Pc (cadd c1 c)
+ | Pinj (j, q0) -> Pinj (j, (paddC cadd q0 c))
+ | PX (p2, i, q0) -> PX (p2, i, (paddC cadd q0 c))
+
+(** val psubC : ('a1 -> 'a1 -> 'a1) -> 'a1 pol -> 'a1 -> 'a1 pol **)
+
+let rec psubC csub p c =
+ match p with
+ | Pc c1 -> Pc (csub c1 c)
+ | Pinj (j, q0) -> Pinj (j, (psubC csub q0 c))
+ | PX (p2, i, q0) -> PX (p2, i, (psubC csub q0 c))
+
+(** val paddI :
+ ('a1 -> 'a1 -> 'a1) -> ('a1 pol -> 'a1 pol -> 'a1 pol) -> 'a1 pol ->
+ positive -> 'a1 pol -> 'a1 pol **)
+
+let rec paddI cadd pop q0 j = function
+ | Pc c ->
+ let p2 = paddC cadd q0 c in
+ (match p2 with
+ | Pc c0 -> p2
+ | Pinj (j', q1) -> Pinj ((pplus j j'), q1)
+ | PX (p3, p4, p5) -> Pinj (j, p2))
+ | Pinj (j', q') ->
+ (match zPminus j' j with
+ | Z0 ->
+ let p2 = pop q' q0 in
+ (match p2 with
+ | Pc c -> p2
+ | Pinj (j'0, q1) -> Pinj ((pplus j j'0), q1)
+ | PX (p3, p4, p5) -> Pinj (j, p2))
+ | Zpos k ->
+ let p2 = pop (Pinj (k, q')) q0 in
+ (match p2 with
+ | Pc c -> p2
+ | Pinj (j'0, q1) -> Pinj ((pplus j j'0), q1)
+ | PX (p3, p4, p5) -> Pinj (j, p2))
+ | Zneg k ->
+ let p2 = paddI cadd pop q0 k q' in
+ (match p2 with
+ | Pc c -> p2
+ | Pinj (j'0, q1) -> Pinj ((pplus j' j'0), q1)
+ | PX (p3, p4, p5) -> Pinj (j', p2)))
+ | PX (p2, i, q') ->
+ (match j with
+ | XI j0 -> PX (p2, i, (paddI cadd pop q0 (XO j0) q'))
+ | XO j0 -> PX (p2, i, (paddI cadd pop q0 (pdouble_minus_one j0) q'))
+ | XH -> PX (p2, i, (pop q' q0)))
+
+(** val psubI :
+ ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1) -> ('a1 pol -> 'a1 pol -> 'a1 pol) ->
+ 'a1 pol -> positive -> 'a1 pol -> 'a1 pol **)
+
+let rec psubI cadd copp pop q0 j = function
+ | Pc c ->
+ let p2 = paddC cadd (popp copp q0) c in
+ (match p2 with
+ | Pc c0 -> p2
+ | Pinj (j', q1) -> Pinj ((pplus j j'), q1)
+ | PX (p3, p4, p5) -> Pinj (j, p2))
+ | Pinj (j', q') ->
+ (match zPminus j' j with
+ | Z0 ->
+ let p2 = pop q' q0 in
+ (match p2 with
+ | Pc c -> p2
+ | Pinj (j'0, q1) -> Pinj ((pplus j j'0), q1)
+ | PX (p3, p4, p5) -> Pinj (j, p2))
+ | Zpos k ->
+ let p2 = pop (Pinj (k, q')) q0 in
+ (match p2 with
+ | Pc c -> p2
+ | Pinj (j'0, q1) -> Pinj ((pplus j j'0), q1)
+ | PX (p3, p4, p5) -> Pinj (j, p2))
+ | Zneg k ->
+ let p2 = psubI cadd copp pop q0 k q' in
+ (match p2 with
+ | Pc c -> p2
+ | Pinj (j'0, q1) -> Pinj ((pplus j' j'0), q1)
+ | PX (p3, p4, p5) -> Pinj (j', p2)))
+ | PX (p2, i, q') ->
+ (match j with
+ | XI j0 -> PX (p2, i, (psubI cadd copp pop q0 (XO j0) q'))
+ | XO j0 -> PX (p2, i,
+ (psubI cadd copp pop q0 (pdouble_minus_one j0) q'))
+ | XH -> PX (p2, i, (pop q' q0)))
+
+(** val paddX :
+ 'a1 -> ('a1 -> 'a1 -> bool) -> ('a1 pol -> 'a1 pol -> 'a1 pol) -> 'a1 pol
+ -> positive -> 'a1 pol -> 'a1 pol **)
+
+let rec paddX cO ceqb pop p' i' p = match p with
+ | Pc c -> PX (p', i', p)
+ | Pinj (j, q') ->
+ (match j with
+ | XI j0 -> PX (p', i', (Pinj ((XO j0), q')))
+ | XO j0 -> PX (p', i', (Pinj ((pdouble_minus_one j0), q')))
+ | XH -> PX (p', i', q'))
+ | PX (p2, i, q') ->
+ (match zPminus i i' with
+ | Z0 -> mkPX cO ceqb (pop p2 p') i q'
+ | Zpos k -> mkPX cO ceqb (pop (PX (p2, k, (p0 cO))) p') i' q'
+ | Zneg k -> mkPX cO ceqb (paddX cO ceqb pop p' k p2) i q')
+
+(** val psubX :
+ 'a1 -> ('a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> ('a1 pol -> 'a1 pol -> 'a1
+ pol) -> 'a1 pol -> positive -> 'a1 pol -> 'a1 pol **)
+
+let rec psubX cO copp ceqb pop p' i' p = match p with
+ | Pc c -> PX ((popp copp p'), i', p)
+ | Pinj (j, q') ->
+ (match j with
+ | XI j0 -> PX ((popp copp p'), i', (Pinj ((XO j0), q')))
+ | XO j0 -> PX ((popp copp p'), i', (Pinj (
+ (pdouble_minus_one j0), q')))
+ | XH -> PX ((popp copp p'), i', q'))
+ | PX (p2, i, q') ->
+ (match zPminus i i' with
+ | Z0 -> mkPX cO ceqb (pop p2 p') i q'
+ | Zpos k -> mkPX cO ceqb (pop (PX (p2, k, (p0 cO))) p') i' q'
+ | Zneg k -> mkPX cO ceqb (psubX cO copp ceqb pop p' k p2) i q')
+
+(** val padd :
+ 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 pol -> 'a1 pol
+ -> 'a1 pol **)
+
+let rec padd cO cadd ceqb p = function
+ | Pc c' -> paddC cadd p c'
+ | Pinj (j', q') -> paddI cadd (fun x x0 -> padd cO cadd ceqb x x0) q' j' p
+ | PX (p'0, i', q') ->
+ (match p with
+ | Pc c -> PX (p'0, i', (paddC cadd q' c))
+ | Pinj (j, q0) ->
+ (match j with
+ | XI j0 -> PX (p'0, i',
+ (padd cO cadd ceqb (Pinj ((XO j0), q0)) q'))
+ | XO j0 -> PX (p'0, i',
+ (padd cO cadd ceqb (Pinj ((pdouble_minus_one j0), q0))
+ q'))
+ | XH -> PX (p'0, i', (padd cO cadd ceqb q0 q')))
+ | PX (p2, i, q0) ->
+ (match zPminus i i' with
+ | Z0 ->
+ mkPX cO ceqb (padd cO cadd ceqb p2 p'0) i
+ (padd cO cadd ceqb q0 q')
+ | Zpos k ->
+ mkPX cO ceqb
+ (padd cO cadd ceqb (PX (p2, k, (p0 cO))) p'0) i'
+ (padd cO cadd ceqb q0 q')
+ | Zneg k ->
+ mkPX cO ceqb
+ (paddX cO ceqb (fun x x0 -> padd cO cadd ceqb x x0) p'0
+ k p2) i (padd cO cadd ceqb q0 q')))
+
+(** val psub :
+ 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1) -> ('a1
+ -> 'a1 -> bool) -> 'a1 pol -> 'a1 pol -> 'a1 pol **)
+
+let rec psub cO cadd csub copp ceqb p = function
+ | Pc c' -> psubC csub p c'
+ | Pinj (j', q') ->
+ psubI cadd copp (fun x x0 -> psub cO cadd csub copp ceqb x x0) q' j' p
+ | PX (p'0, i', q') ->
+ (match p with
+ | Pc c -> PX ((popp copp p'0), i', (paddC cadd (popp copp q') c))
+ | Pinj (j, q0) ->
+ (match j with
+ | XI j0 -> PX ((popp copp p'0), i',
+ (psub cO cadd csub copp ceqb (Pinj ((XO j0), q0)) q'))
+ | XO j0 -> PX ((popp copp p'0), i',
+ (psub cO cadd csub copp ceqb (Pinj
+ ((pdouble_minus_one j0), q0)) q'))
+ | XH -> PX ((popp copp p'0), i',
+ (psub cO cadd csub copp ceqb q0 q')))
+ | PX (p2, i, q0) ->
+ (match zPminus i i' with
+ | Z0 ->
+ mkPX cO ceqb (psub cO cadd csub copp ceqb p2 p'0) i
+ (psub cO cadd csub copp ceqb q0 q')
+ | Zpos k ->
+ mkPX cO ceqb
+ (psub cO cadd csub copp ceqb (PX (p2, k, (p0 cO))) p'0)
+ i' (psub cO cadd csub copp ceqb q0 q')
+ | Zneg k ->
+ mkPX cO ceqb
+ (psubX cO copp ceqb (fun x x0 ->
+ psub cO cadd csub copp ceqb x x0) p'0 k p2) i
+ (psub cO cadd csub copp ceqb q0 q')))
+
+(** val pmulC_aux :
+ 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 pol -> 'a1 ->
+ 'a1 pol **)
+
+let rec pmulC_aux cO cmul ceqb p c =
+ match p with
+ | Pc c' -> Pc (cmul c' c)
+ | Pinj (j, q0) ->
+ let p2 = pmulC_aux cO cmul ceqb q0 c in
+ (match p2 with
+ | Pc c0 -> p2
+ | Pinj (j', q1) -> Pinj ((pplus j j'), q1)
+ | PX (p3, p4, p5) -> Pinj (j, p2))
+ | PX (p2, i, q0) ->
+ mkPX cO ceqb (pmulC_aux cO cmul ceqb p2 c) i
+ (pmulC_aux cO cmul ceqb q0 c)
+
+(** val pmulC :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 pol ->
+ 'a1 -> 'a1 pol **)
+
+let pmulC cO cI cmul ceqb p c =
+ match ceqb c cO with
+ | True -> p0 cO
+ | False ->
+ (match ceqb c cI with
+ | True -> p
+ | False -> pmulC_aux cO cmul ceqb p c)
+
+(** val pmulI :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> ('a1 pol ->
+ 'a1 pol -> 'a1 pol) -> 'a1 pol -> positive -> 'a1 pol -> 'a1 pol **)
+
+let rec pmulI cO cI cmul ceqb pmul0 q0 j = function
+ | Pc c ->
+ let p2 = pmulC cO cI cmul ceqb q0 c in
+ (match p2 with
+ | Pc c0 -> p2
+ | Pinj (j', q1) -> Pinj ((pplus j j'), q1)
+ | PX (p3, p4, p5) -> Pinj (j, p2))
+ | Pinj (j', q') ->
+ (match zPminus j' j with
+ | Z0 ->
+ let p2 = pmul0 q' q0 in
+ (match p2 with
+ | Pc c -> p2
+ | Pinj (j'0, q1) -> Pinj ((pplus j j'0), q1)
+ | PX (p3, p4, p5) -> Pinj (j, p2))
+ | Zpos k ->
+ let p2 = pmul0 (Pinj (k, q')) q0 in
+ (match p2 with
+ | Pc c -> p2
+ | Pinj (j'0, q1) -> Pinj ((pplus j j'0), q1)
+ | PX (p3, p4, p5) -> Pinj (j, p2))
+ | Zneg k ->
+ let p2 = pmulI cO cI cmul ceqb pmul0 q0 k q' in
+ (match p2 with
+ | Pc c -> p2
+ | Pinj (j'0, q1) -> Pinj ((pplus j' j'0), q1)
+ | PX (p3, p4, p5) -> Pinj (j', p2)))
+ | PX (p', i', q') ->
+ (match j with
+ | XI j' ->
+ mkPX cO ceqb (pmulI cO cI cmul ceqb pmul0 q0 j p') i'
+ (pmulI cO cI cmul ceqb pmul0 q0 (XO j') q')
+ | XO j' ->
+ mkPX cO ceqb (pmulI cO cI cmul ceqb pmul0 q0 j p') i'
+ (pmulI cO cI cmul ceqb pmul0 q0 (pdouble_minus_one j') q')
+ | XH ->
+ mkPX cO ceqb (pmulI cO cI cmul ceqb pmul0 q0 XH p') i'
+ (pmul0 q' q0))
+
+(** val pmul :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1
+ -> bool) -> 'a1 pol -> 'a1 pol -> 'a1 pol **)
+
+let rec pmul cO cI cadd cmul ceqb p p'' = match p'' with
+ | Pc c -> pmulC cO cI cmul ceqb p c
+ | Pinj (j', q') ->
+ pmulI cO cI cmul ceqb (fun x x0 -> pmul cO cI cadd cmul ceqb x x0) q'
+ j' p
+ | PX (p', i', q') ->
+ (match p with
+ | Pc c -> pmulC cO cI cmul ceqb p'' c
+ | Pinj (j, q0) ->
+ mkPX cO ceqb (pmul cO cI cadd cmul ceqb p p') i'
+ (match j with
+ | XI j0 ->
+ pmul cO cI cadd cmul ceqb (Pinj ((XO j0), q0)) q'
+ | XO j0 ->
+ pmul cO cI cadd cmul ceqb (Pinj
+ ((pdouble_minus_one j0), q0)) q'
+ | XH -> pmul cO cI cadd cmul ceqb q0 q')
+ | PX (p2, i, q0) ->
+ padd cO cadd ceqb
+ (mkPX cO ceqb
+ (padd cO cadd ceqb
+ (mkPX cO ceqb (pmul cO cI cadd cmul ceqb p2 p') i (p0 cO))
+ (pmul cO cI cadd cmul ceqb
+ (match q0 with
+ | Pc c -> q0
+ | Pinj (j', q1) -> Pinj ((pplus XH j'), q1)
+ | PX (p3, p4, p5) -> Pinj (XH, q0)) p')) i'
+ (p0 cO))
+ (mkPX cO ceqb
+ (pmulI cO cI cmul ceqb (fun x x0 ->
+ pmul cO cI cadd cmul ceqb x x0) q' XH p2) i
+ (pmul cO cI cadd cmul ceqb q0 q')))
+
+type 'c pExpr =
+ | PEc of 'c
+ | PEX of positive
+ | PEadd of 'c pExpr * 'c pExpr
+ | PEsub of 'c pExpr * 'c pExpr
+ | PEmul of 'c pExpr * 'c pExpr
+ | PEopp of 'c pExpr
+ | PEpow of 'c pExpr * n
+
+(** val mk_X : 'a1 -> 'a1 -> positive -> 'a1 pol **)
+
+let mk_X cO cI j =
+ mkPinj_pred j (mkX cO cI)
+
+(** val ppow_pos :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1
+ -> bool) -> ('a1 pol -> 'a1 pol) -> 'a1 pol -> 'a1 pol -> positive -> 'a1
+ pol **)
+
+let rec ppow_pos cO cI cadd cmul ceqb subst_l res p = function
+ | XI p3 ->
+ subst_l
+ (pmul cO cI cadd cmul ceqb
+ (ppow_pos cO cI cadd cmul ceqb subst_l
+ (ppow_pos cO cI cadd cmul ceqb subst_l res p p3) p p3) p)
+ | XO p3 ->
+ ppow_pos cO cI cadd cmul ceqb subst_l
+ (ppow_pos cO cI cadd cmul ceqb subst_l res p p3) p p3
+ | XH -> subst_l (pmul cO cI cadd cmul ceqb res p)
+
+(** val ppow_N :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1
+ -> bool) -> ('a1 pol -> 'a1 pol) -> 'a1 pol -> n -> 'a1 pol **)
+
+let ppow_N cO cI cadd cmul ceqb subst_l p = function
+ | N0 -> p1 cI
+ | Npos p2 -> ppow_pos cO cI cadd cmul ceqb subst_l (p1 cI) p p2
+
+(** val norm_aux :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1
+ -> 'a1) -> ('a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 pExpr -> 'a1 pol **)
+
+let rec norm_aux cO cI cadd cmul csub copp ceqb = function
+ | PEc c -> Pc c
+ | PEX j -> mk_X cO cI j
+ | PEadd (pe1, pe2) ->
+ (match pe1 with
+ | PEopp pe3 ->
+ psub cO cadd csub copp ceqb
+ (norm_aux cO cI cadd cmul csub copp ceqb pe2)
+ (norm_aux cO cI cadd cmul csub copp ceqb pe3)
+ | _ ->
+ (match pe2 with
+ | PEopp pe3 ->
+ psub cO cadd csub copp ceqb
+ (norm_aux cO cI cadd cmul csub copp ceqb pe1)
+ (norm_aux cO cI cadd cmul csub copp ceqb pe3)
+ | _ ->
+ padd cO cadd ceqb
+ (norm_aux cO cI cadd cmul csub copp ceqb pe1)
+ (norm_aux cO cI cadd cmul csub copp ceqb pe2)))
+ | PEsub (pe1, pe2) ->
+ psub cO cadd csub copp ceqb
+ (norm_aux cO cI cadd cmul csub copp ceqb pe1)
+ (norm_aux cO cI cadd cmul csub copp ceqb pe2)
+ | PEmul (pe1, pe2) ->
+ pmul cO cI cadd cmul ceqb (norm_aux cO cI cadd cmul csub copp ceqb pe1)
+ (norm_aux cO cI cadd cmul csub copp ceqb pe2)
+ | PEopp pe1 -> popp copp (norm_aux cO cI cadd cmul csub copp ceqb pe1)
+ | PEpow (pe1, n0) ->
+ ppow_N cO cI cadd cmul ceqb (fun p -> p)
+ (norm_aux cO cI cadd cmul csub copp ceqb pe1) n0
+
+type 'a bFormula =
+ | TT
+ | FF
+ | X
+ | A of 'a
+ | Cj of 'a bFormula * 'a bFormula
+ | D of 'a bFormula * 'a bFormula
+ | N of 'a bFormula
+ | I of 'a bFormula * 'a bFormula
+
+type 'term' clause = 'term' list
+
+type 'term' cnf = 'term' clause list
+
+(** val tt : 'a1 cnf **)
+
+let tt =
+ Nil
+
+(** val ff : 'a1 cnf **)
+
+let ff =
+ Cons (Nil, Nil)
+
+(** val or_clause_cnf : 'a1 clause -> 'a1 cnf -> 'a1 cnf **)
+
+let or_clause_cnf t0 f =
+ map (fun x -> app t0 x) f
+
+(** val or_cnf : 'a1 cnf -> 'a1 cnf -> 'a1 cnf **)
+
+let rec or_cnf f f' =
+ match f with
+ | Nil -> tt
+ | Cons (e, rst) -> app (or_cnf rst f') (or_clause_cnf e f')
+
+(** val and_cnf : 'a1 cnf -> 'a1 cnf -> 'a1 cnf **)
+
+let and_cnf f1 f2 =
+ app f1 f2
+
+(** val xcnf :
+ ('a1 -> 'a2 cnf) -> ('a1 -> 'a2 cnf) -> bool -> 'a1 bFormula -> 'a2 cnf **)
+
+let rec xcnf normalise0 negate0 pol0 = function
+ | TT -> (match pol0 with
+ | True -> tt
+ | False -> ff)
+ | FF -> (match pol0 with
+ | True -> ff
+ | False -> tt)
+ | X -> ff
+ | A x -> (match pol0 with
+ | True -> normalise0 x
+ | False -> negate0 x)
+ | Cj (e1, e2) ->
+ (match pol0 with
+ | True ->
+ and_cnf (xcnf normalise0 negate0 pol0 e1)
+ (xcnf normalise0 negate0 pol0 e2)
+ | False ->
+ or_cnf (xcnf normalise0 negate0 pol0 e1)
+ (xcnf normalise0 negate0 pol0 e2))
+ | D (e1, e2) ->
+ (match pol0 with
+ | True ->
+ or_cnf (xcnf normalise0 negate0 pol0 e1)
+ (xcnf normalise0 negate0 pol0 e2)
+ | False ->
+ and_cnf (xcnf normalise0 negate0 pol0 e1)
+ (xcnf normalise0 negate0 pol0 e2))
+ | N e -> xcnf normalise0 negate0 (negb pol0) e
+ | I (e1, e2) ->
+ (match pol0 with
+ | True ->
+ or_cnf (xcnf normalise0 negate0 (negb pol0) e1)
+ (xcnf normalise0 negate0 pol0 e2)
+ | False ->
+ and_cnf (xcnf normalise0 negate0 (negb pol0) e1)
+ (xcnf normalise0 negate0 pol0 e2))
+
+(** val cnf_checker :
+ ('a1 list -> 'a2 -> bool) -> 'a1 cnf -> 'a2 list -> bool **)
+
+let rec cnf_checker checker f l =
+ match f with
+ | Nil -> True
+ | Cons (e, f0) ->
+ (match l with
+ | Nil -> False
+ | Cons (c, l0) ->
+ (match checker e c with
+ | True -> cnf_checker checker f0 l0
+ | False -> False))
+
+(** val tauto_checker :
+ ('a1 -> 'a2 cnf) -> ('a1 -> 'a2 cnf) -> ('a2 list -> 'a3 -> bool) -> 'a1
+ bFormula -> 'a3 list -> bool **)
+
+let tauto_checker normalise0 negate0 checker f w =
+ cnf_checker checker (xcnf normalise0 negate0 True f) w
+
+type 'c pExprC = 'c pExpr
+
+type 'c polC = 'c pol
+
+type op1 =
+ | Equal
+ | NonEqual
+ | Strict
+ | NonStrict
+
+type 'c nFormula = ('c pExprC, op1) prod
+
+type monoidMember = nat list
+
+type 'c coneMember =
+ | S_In of nat
+ | S_Ideal of 'c pExprC * 'c coneMember
+ | S_Square of 'c pExprC
+ | S_Monoid of monoidMember
+ | S_Mult of 'c coneMember * 'c coneMember
+ | S_Add of 'c coneMember * 'c coneMember
+ | S_Pos of 'c
+ | S_Z
+
+(** val nformula_times : 'a1 nFormula -> 'a1 nFormula -> 'a1 nFormula **)
+
+let nformula_times f f' =
+ let Pair (p, op) = f in
+ let Pair (p', op') = f' in
+ Pair ((PEmul (p, p')),
+ (match op with
+ | Equal -> Equal
+ | NonEqual -> NonEqual
+ | Strict -> op'
+ | NonStrict -> NonStrict))
+
+(** val nformula_plus : 'a1 nFormula -> 'a1 nFormula -> 'a1 nFormula **)
+
+let nformula_plus f f' =
+ let Pair (p, op) = f in
+ let Pair (p', op') = f' in
+ Pair ((PEadd (p, p')),
+ (match op with
+ | Equal -> op'
+ | NonEqual -> NonEqual
+ | Strict -> Strict
+ | NonStrict -> (match op' with
+ | Strict -> Strict
+ | _ -> NonStrict)))
+
+(** val eval_monoid :
+ 'a1 -> 'a1 nFormula list -> monoidMember -> 'a1 pExprC **)
+
+let rec eval_monoid cI l = function
+ | Nil -> PEc cI
+ | Cons (n0, ns0) -> PEmul
+ ((let Pair (q0, o) = nth n0 l (Pair ((PEc cI), NonEqual)) in
+ (match o with
+ | NonEqual -> q0
+ | _ -> PEc cI)), (eval_monoid cI l ns0))
+
+(** val eval_cone :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> bool) -> ('a1 -> 'a1 -> bool) -> 'a1
+ nFormula list -> 'a1 coneMember -> 'a1 nFormula **)
+
+let rec eval_cone cO cI ceqb cleb l = function
+ | S_In n0 ->
+ let Pair (p, o) = nth n0 l (Pair ((PEc cO), Equal)) in
+ (match o with
+ | NonEqual -> Pair ((PEc cO), Equal)
+ | _ -> nth n0 l (Pair ((PEc cO), Equal)))
+ | S_Ideal (p, cm') ->
+ let f = eval_cone cO cI ceqb cleb l cm' in
+ let Pair (q0, op) = f in
+ (match op with
+ | Equal -> Pair ((PEmul (q0, p)), Equal)
+ | _ -> f)
+ | S_Square p -> Pair ((PEmul (p, p)), NonStrict)
+ | S_Monoid m -> let p = eval_monoid cI l m in Pair ((PEmul (p, p)), Strict)
+ | S_Mult (p, q0) ->
+ nformula_times (eval_cone cO cI ceqb cleb l p)
+ (eval_cone cO cI ceqb cleb l q0)
+ | S_Add (p, q0) ->
+ nformula_plus (eval_cone cO cI ceqb cleb l p)
+ (eval_cone cO cI ceqb cleb l q0)
+ | S_Pos c ->
+ (match match cleb cO c with
+ | True -> negb (ceqb cO c)
+ | False -> False with
+ | True -> Pair ((PEc c), Strict)
+ | False -> Pair ((PEc cO), Equal))
+ | S_Z -> Pair ((PEc cO), Equal)
+
+(** val normalise_pexpr :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1
+ -> 'a1) -> ('a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 pExprC -> 'a1 polC **)
+
+let normalise_pexpr cO cI cplus ctimes cminus copp ceqb x =
+ norm_aux cO cI cplus ctimes cminus copp ceqb x
+
+(** val check_inconsistent :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1
+ -> 'a1) -> ('a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> ('a1 -> 'a1 -> bool)
+ -> 'a1 nFormula -> bool **)
+
+let check_inconsistent cO cI cplus ctimes cminus copp ceqb cleb = function
+ | Pair (e, op) ->
+ (match normalise_pexpr cO cI cplus ctimes cminus copp ceqb e with
+ | Pc c ->
+ (match op with
+ | Equal -> negb (ceqb c cO)
+ | NonEqual -> False
+ | Strict -> cleb c cO
+ | NonStrict ->
+ (match cleb c cO with
+ | True -> negb (ceqb c cO)
+ | False -> False))
+ | _ -> False)
+
+(** val check_normalised_formulas :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1
+ -> 'a1) -> ('a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> ('a1 -> 'a1 -> bool)
+ -> 'a1 nFormula list -> 'a1 coneMember -> bool **)
+
+let check_normalised_formulas cO cI cplus ctimes cminus copp ceqb cleb l cm =
+ check_inconsistent cO cI cplus ctimes cminus copp ceqb cleb
+ (eval_cone cO cI ceqb cleb l cm)
+
+type op2 =
+ | OpEq
+ | OpNEq
+ | OpLe
+ | OpGe
+ | OpLt
+ | OpGt
+
+type 'c formula = { flhs : 'c pExprC; fop : op2; frhs : 'c pExprC }
+
+(** val flhs : 'a1 formula -> 'a1 pExprC **)
+
+let flhs x = x.flhs
+
+(** val fop : 'a1 formula -> op2 **)
+
+let fop x = x.fop
+
+(** val frhs : 'a1 formula -> 'a1 pExprC **)
+
+let frhs x = x.frhs
+
+(** val xnormalise : 'a1 formula -> 'a1 nFormula list **)
+
+let xnormalise t0 =
+ let { flhs = lhs; fop = o; frhs = rhs } = t0 in
+ (match o with
+ | OpEq -> Cons ((Pair ((PEsub (lhs, rhs)), Strict)), (Cons ((Pair
+ ((PEsub (rhs, lhs)), Strict)), Nil)))
+ | OpNEq -> Cons ((Pair ((PEsub (lhs, rhs)), Equal)), Nil)
+ | OpLe -> Cons ((Pair ((PEsub (lhs, rhs)), Strict)), Nil)
+ | OpGe -> Cons ((Pair ((PEsub (rhs, lhs)), Strict)), Nil)
+ | OpLt -> Cons ((Pair ((PEsub (lhs, rhs)), NonStrict)), Nil)
+ | OpGt -> Cons ((Pair ((PEsub (rhs, lhs)), NonStrict)), Nil))
+
+(** val cnf_normalise : 'a1 formula -> 'a1 nFormula cnf **)
+
+let cnf_normalise t0 =
+ map (fun x -> Cons (x, Nil)) (xnormalise t0)
+
+(** val xnegate : 'a1 formula -> 'a1 nFormula list **)
+
+let xnegate t0 =
+ let { flhs = lhs; fop = o; frhs = rhs } = t0 in
+ (match o with
+ | OpEq -> Cons ((Pair ((PEsub (lhs, rhs)), Equal)), Nil)
+ | OpNEq -> Cons ((Pair ((PEsub (lhs, rhs)), Strict)), (Cons ((Pair
+ ((PEsub (rhs, lhs)), Strict)), Nil)))
+ | OpLe -> Cons ((Pair ((PEsub (rhs, lhs)), NonStrict)), Nil)
+ | OpGe -> Cons ((Pair ((PEsub (lhs, rhs)), NonStrict)), Nil)
+ | OpLt -> Cons ((Pair ((PEsub (rhs, lhs)), Strict)), Nil)
+ | OpGt -> Cons ((Pair ((PEsub (lhs, rhs)), Strict)), Nil))
+
+(** val cnf_negate : 'a1 formula -> 'a1 nFormula cnf **)
+
+let cnf_negate t0 =
+ map (fun x -> Cons (x, Nil)) (xnegate t0)
+
+(** val simpl_expr :
+ 'a1 -> ('a1 -> 'a1 -> bool) -> 'a1 pExprC -> 'a1 pExprC **)
+
+let rec simpl_expr cI ceqb e = match e with
+ | PEadd (x, y) -> PEadd ((simpl_expr cI ceqb x), (simpl_expr cI ceqb y))
+ | PEmul (y, z0) ->
+ let y' = simpl_expr cI ceqb y in
+ (match y' with
+ | PEc c ->
+ (match ceqb c cI with
+ | True -> simpl_expr cI ceqb z0
+ | False -> PEmul (y', (simpl_expr cI ceqb z0)))
+ | _ -> PEmul (y', (simpl_expr cI ceqb z0)))
+ | _ -> e
+
+(** val simpl_cone :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1
+ coneMember -> 'a1 coneMember **)
+
+let simpl_cone cO cI ctimes ceqb e = match e with
+ | S_Square t0 ->
+ (match simpl_expr cI ceqb t0 with
+ | PEc c ->
+ (match ceqb cO c with
+ | True -> S_Z
+ | False -> S_Pos (ctimes c c))
+ | _ -> S_Square (simpl_expr cI ceqb t0))
+ | S_Mult (t1, t2) ->
+ (match t1 with
+ | S_Mult (x, x0) ->
+ (match x with
+ | S_Pos p2 ->
+ (match t2 with
+ | S_Pos c -> S_Mult ((S_Pos (ctimes c p2)), x0)
+ | S_Z -> S_Z
+ | _ -> e)
+ | _ ->
+ (match x0 with
+ | S_Pos p2 ->
+ (match t2 with
+ | S_Pos c -> S_Mult ((S_Pos (ctimes c p2)), x)
+ | S_Z -> S_Z
+ | _ -> e)
+ | _ ->
+ (match t2 with
+ | S_Pos c ->
+ (match ceqb cI c with
+ | True -> t1
+ | False -> S_Mult (t1, t2))
+ | S_Z -> S_Z
+ | _ -> e)))
+ | S_Pos c ->
+ (match t2 with
+ | S_Mult (x, x0) ->
+ (match x with
+ | S_Pos p2 -> S_Mult ((S_Pos (ctimes c p2)), x0)
+ | _ ->
+ (match x0 with
+ | S_Pos p2 -> S_Mult ((S_Pos (ctimes c p2)), x)
+ | _ ->
+ (match ceqb cI c with
+ | True -> t2
+ | False -> S_Mult (t1, t2))))
+ | S_Add (y, z0) -> S_Add ((S_Mult ((S_Pos c), y)), (S_Mult
+ ((S_Pos c), z0)))
+ | S_Pos c0 -> S_Pos (ctimes c c0)
+ | S_Z -> S_Z
+ | _ ->
+ (match ceqb cI c with
+ | True -> t2
+ | False -> S_Mult (t1, t2)))
+ | S_Z -> S_Z
+ | _ ->
+ (match t2 with
+ | S_Pos c ->
+ (match ceqb cI c with
+ | True -> t1
+ | False -> S_Mult (t1, t2))
+ | S_Z -> S_Z
+ | _ -> e))
+ | S_Add (t1, t2) ->
+ (match t1 with
+ | S_Z -> t2
+ | _ -> (match t2 with
+ | S_Z -> t1
+ | _ -> S_Add (t1, t2)))
+ | _ -> e
+
+type q = { qnum : z; qden : positive }
+
+(** val qnum : q -> z **)
+
+let qnum x = x.qnum
+
+(** val qden : q -> positive **)
+
+let qden x = x.qden
+
+(** val qplus : q -> q -> q **)
+
+let qplus x y =
+ { qnum = (zplus (zmult x.qnum (Zpos y.qden)) (zmult y.qnum (Zpos x.qden)));
+ qden = (pmult x.qden y.qden) }
+
+(** val qmult : q -> q -> q **)
+
+let qmult x y =
+ { qnum = (zmult x.qnum y.qnum); qden = (pmult x.qden y.qden) }
+
+(** val qopp : q -> q **)
+
+let qopp x =
+ { qnum = (zopp x.qnum); qden = x.qden }
+
+(** val qminus : q -> q -> q **)
+
+let qminus x y =
+ qplus x (qopp y)
+
+type 'a t =
+ | Empty
+ | Leaf of 'a
+ | Node of 'a t * 'a * 'a t
+
+(** val find : 'a1 -> 'a1 t -> positive -> 'a1 **)
+
+let rec find default vm p =
+ match vm with
+ | Empty -> default
+ | Leaf i -> i
+ | Node (l, e, r) ->
+ (match p with
+ | XI p2 -> find default r p2
+ | XO p2 -> find default l p2
+ | XH -> e)
+
+type zWitness = z coneMember
+
+(** val zWeakChecker : z nFormula list -> z coneMember -> bool **)
+
+let zWeakChecker x x0 =
+ check_normalised_formulas Z0 (Zpos XH) zplus zmult zminus zopp zeq_bool
+ zle_bool x x0
+
+(** val xnormalise0 : z formula -> z nFormula list **)
+
+let xnormalise0 t0 =
+ let { flhs = lhs; fop = o; frhs = rhs } = t0 in
+ (match o with
+ | OpEq -> Cons ((Pair ((PEsub (lhs, (PEadd (rhs, (PEc (Zpos XH)))))),
+ NonStrict)), (Cons ((Pair ((PEsub (rhs, (PEadd (lhs, (PEc (Zpos
+ XH)))))), NonStrict)), Nil)))
+ | OpNEq -> Cons ((Pair ((PEsub (lhs, rhs)), Equal)), Nil)
+ | OpLe -> Cons ((Pair ((PEsub (lhs, (PEadd (rhs, (PEc (Zpos XH)))))),
+ NonStrict)), Nil)
+ | OpGe -> Cons ((Pair ((PEsub (rhs, (PEadd (lhs, (PEc (Zpos XH)))))),
+ NonStrict)), Nil)
+ | OpLt -> Cons ((Pair ((PEsub (lhs, rhs)), NonStrict)), Nil)
+ | OpGt -> Cons ((Pair ((PEsub (rhs, lhs)), NonStrict)), Nil))
+
+(** val normalise : z formula -> z nFormula cnf **)
+
+let normalise t0 =
+ map (fun x -> Cons (x, Nil)) (xnormalise0 t0)
+
+(** val xnegate0 : z formula -> z nFormula list **)
+
+let xnegate0 t0 =
+ let { flhs = lhs; fop = o; frhs = rhs } = t0 in
+ (match o with
+ | OpEq -> Cons ((Pair ((PEsub (lhs, rhs)), Equal)), Nil)
+ | OpNEq -> Cons ((Pair ((PEsub (lhs, (PEadd (rhs, (PEc (Zpos XH)))))),
+ NonStrict)), (Cons ((Pair ((PEsub (rhs, (PEadd (lhs, (PEc (Zpos
+ XH)))))), NonStrict)), Nil)))
+ | OpLe -> Cons ((Pair ((PEsub (rhs, lhs)), NonStrict)), Nil)
+ | OpGe -> Cons ((Pair ((PEsub (lhs, rhs)), NonStrict)), Nil)
+ | OpLt -> Cons ((Pair ((PEsub (rhs, (PEadd (lhs, (PEc (Zpos XH)))))),
+ NonStrict)), Nil)
+ | OpGt -> Cons ((Pair ((PEsub (lhs, (PEadd (rhs, (PEc (Zpos XH)))))),
+ NonStrict)), Nil))
+
+(** val negate : z formula -> z nFormula cnf **)
+
+let negate t0 =
+ map (fun x -> Cons (x, Nil)) (xnegate0 t0)
+
+(** val ceiling : z -> z -> z **)
+
+let ceiling a b =
+ let Pair (q0, r) = zdiv_eucl a b in
+ (match r with
+ | Z0 -> q0
+ | _ -> zplus q0 (Zpos XH))
+
+type proofTerm =
+ | RatProof of zWitness
+ | CutProof of z pExprC * q * zWitness * proofTerm
+ | EnumProof of q * z pExprC * q * zWitness * zWitness * proofTerm list
+
+(** val makeLb : z pExpr -> q -> z nFormula **)
+
+let makeLb v q0 =
+ let { qnum = n0; qden = d } = q0 in
+ Pair ((PEsub ((PEmul ((PEc (Zpos d)), v)), (PEc n0))), NonStrict)
+
+(** val qceiling : q -> z **)
+
+let qceiling q0 =
+ let { qnum = n0; qden = d } = q0 in ceiling n0 (Zpos d)
+
+(** val makeLbCut : z pExprC -> q -> z nFormula **)
+
+let makeLbCut v q0 =
+ Pair ((PEsub (v, (PEc (qceiling q0)))), NonStrict)
+
+(** val neg_nformula : z nFormula -> (z pExpr, op1) prod **)
+
+let neg_nformula = function
+ | Pair (e, o) -> Pair ((PEopp (PEadd (e, (PEc (Zpos XH))))), o)
+
+(** val cutChecker :
+ z nFormula list -> z pExpr -> q -> zWitness -> z nFormula option **)
+
+let cutChecker l e lb pf =
+ match zWeakChecker (Cons ((neg_nformula (makeLb e lb)), l)) pf with
+ | True -> Some (makeLbCut e lb)
+ | False -> None
+
+(** val zChecker : z nFormula list -> proofTerm -> bool **)
+
+let rec zChecker l = function
+ | RatProof pf0 -> zWeakChecker l pf0
+ | CutProof (e, q0, pf0, rst) ->
+ (match cutChecker l e q0 pf0 with
+ | Some c -> zChecker (Cons (c, l)) rst
+ | None -> False)
+ | EnumProof (lb, e, ub, pf1, pf2, rst) ->
+ (match cutChecker l e lb pf1 with
+ | Some n0 ->
+ (match cutChecker l (PEopp e) (qopp ub) pf2 with
+ | Some n1 ->
+ let rec label pfs lb0 ub0 =
+ match pfs with
+ | Nil ->
+ (match z_gt_dec lb0 ub0 with
+ | Left -> True
+ | Right -> False)
+ | Cons (pf0, rsr) ->
+ (match zChecker (Cons ((Pair ((PEsub (e, (PEc
+ lb0))), Equal)), l)) pf0 with
+ | True -> label rsr (zplus lb0 (Zpos XH)) ub0
+ | False -> False)
+ in label rst (qceiling lb) (zopp (qceiling (qopp ub)))
+ | None -> False)
+ | None -> False)
+
+(** val zTautoChecker : z formula bFormula -> proofTerm list -> bool **)
+
+let zTautoChecker f w =
+ tauto_checker normalise negate zChecker f w
+
+(** val map_cone : (nat -> nat) -> zWitness -> zWitness **)
+
+let rec map_cone f e = match e with
+ | S_In n0 -> S_In (f n0)
+ | S_Ideal (e0, cm) -> S_Ideal (e0, (map_cone f cm))
+ | S_Monoid l -> S_Monoid (map f l)
+ | S_Mult (cm1, cm2) -> S_Mult ((map_cone f cm1), (map_cone f cm2))
+ | S_Add (cm1, cm2) -> S_Add ((map_cone f cm1), (map_cone f cm2))
+ | _ -> e
+
+(** val indexes : zWitness -> nat list **)
+
+let rec indexes = function
+ | S_In n0 -> Cons (n0, Nil)
+ | S_Ideal (e0, cm) -> indexes cm
+ | S_Monoid l -> l
+ | S_Mult (cm1, cm2) -> app (indexes cm1) (indexes cm2)
+ | S_Add (cm1, cm2) -> app (indexes cm1) (indexes cm2)
+ | _ -> Nil
+
+(** val n_of_Z : z -> n **)
+
+let n_of_Z = function
+ | Zpos p -> Npos p
+ | _ -> N0
+
+(** val qeq_bool : q -> q -> bool **)
+
+let qeq_bool p q0 =
+ zeq_bool (zmult p.qnum (Zpos q0.qden)) (zmult q0.qnum (Zpos p.qden))
+
+(** val qle_bool : q -> q -> bool **)
+
+let qle_bool x y =
+ zle_bool (zmult x.qnum (Zpos y.qden)) (zmult y.qnum (Zpos x.qden))
+
+type qWitness = q coneMember
+
+(** val qWeakChecker : q nFormula list -> q coneMember -> bool **)
+
+let qWeakChecker x x0 =
+ check_normalised_formulas { qnum = Z0; qden = XH } { qnum = (Zpos XH);
+ qden = XH } qplus qmult qminus qopp qeq_bool qle_bool x x0
+
+(** val qTautoChecker : q formula bFormula -> qWitness list -> bool **)
+
+let qTautoChecker f w =
+ tauto_checker (fun x -> cnf_normalise x) (fun x ->
+ cnf_negate x) qWeakChecker f w
+
diff --git a/contrib/micromega/micromega.mli b/contrib/micromega/micromega.mli
new file mode 100644
index 000000000..f94f091e8
--- /dev/null
+++ b/contrib/micromega/micromega.mli
@@ -0,0 +1,398 @@
+type __ = Obj.t
+
+type bool =
+ | True
+ | False
+
+val negb : bool -> bool
+
+type nat =
+ | O
+ | S of nat
+
+type 'a option =
+ | Some of 'a
+ | None
+
+type ('a, 'b) prod =
+ | Pair of 'a * 'b
+
+type comparison =
+ | Eq
+ | Lt
+ | Gt
+
+val compOpp : comparison -> comparison
+
+type sumbool =
+ | Left
+ | Right
+
+type 'a sumor =
+ | Inleft of 'a
+ | Inright
+
+type 'a list =
+ | Nil
+ | Cons of 'a * 'a list
+
+val app : 'a1 list -> 'a1 list -> 'a1 list
+
+val nth : nat -> 'a1 list -> 'a1 -> 'a1
+
+val map : ('a1 -> 'a2) -> 'a1 list -> 'a2 list
+
+type positive =
+ | XI of positive
+ | XO of positive
+ | XH
+
+val psucc : positive -> positive
+
+val pplus : positive -> positive -> positive
+
+val pplus_carry : positive -> positive -> positive
+
+val p_of_succ_nat : nat -> positive
+
+val pdouble_minus_one : positive -> positive
+
+type positive_mask =
+ | IsNul
+ | IsPos of positive
+ | IsNeg
+
+val pdouble_plus_one_mask : positive_mask -> positive_mask
+
+val pdouble_mask : positive_mask -> positive_mask
+
+val pdouble_minus_two : positive -> positive_mask
+
+val pminus_mask : positive -> positive -> positive_mask
+
+val pminus_mask_carry : positive -> positive -> positive_mask
+
+val pminus : positive -> positive -> positive
+
+val pmult : positive -> positive -> positive
+
+val pcompare : positive -> positive -> comparison -> comparison
+
+type n =
+ | N0
+ | Npos of positive
+
+type z =
+ | Z0
+ | Zpos of positive
+ | Zneg of positive
+
+val zdouble_plus_one : z -> z
+
+val zdouble_minus_one : z -> z
+
+val zdouble : z -> z
+
+val zPminus : positive -> positive -> z
+
+val zplus : z -> z -> z
+
+val zopp : z -> z
+
+val zminus : z -> z -> z
+
+val zmult : z -> z -> z
+
+val zcompare : z -> z -> comparison
+
+val dcompare_inf : comparison -> sumbool sumor
+
+val zcompare_rec : z -> z -> (__ -> 'a1) -> (__ -> 'a1) -> (__ -> 'a1) -> 'a1
+
+val z_gt_dec : z -> z -> sumbool
+
+val zle_bool : z -> z -> bool
+
+val zge_bool : z -> z -> bool
+
+val zgt_bool : z -> z -> bool
+
+val zeq_bool : z -> z -> bool
+
+val n_of_nat : nat -> n
+
+val zdiv_eucl_POS : positive -> z -> (z, z) prod
+
+val zdiv_eucl : z -> z -> (z, z) prod
+
+type 'c pol =
+ | Pc of 'c
+ | Pinj of positive * 'c pol
+ | PX of 'c pol * positive * 'c pol
+
+val p0 : 'a1 -> 'a1 pol
+
+val p1 : 'a1 -> 'a1 pol
+
+val peq : ('a1 -> 'a1 -> bool) -> 'a1 pol -> 'a1 pol -> bool
+
+val mkPinj_pred : positive -> 'a1 pol -> 'a1 pol
+
+val mkPX :
+ 'a1 -> ('a1 -> 'a1 -> bool) -> 'a1 pol -> positive -> 'a1 pol -> 'a1 pol
+
+val mkXi : 'a1 -> 'a1 -> positive -> 'a1 pol
+
+val mkX : 'a1 -> 'a1 -> 'a1 pol
+
+val popp : ('a1 -> 'a1) -> 'a1 pol -> 'a1 pol
+
+val paddC : ('a1 -> 'a1 -> 'a1) -> 'a1 pol -> 'a1 -> 'a1 pol
+
+val psubC : ('a1 -> 'a1 -> 'a1) -> 'a1 pol -> 'a1 -> 'a1 pol
+
+val paddI :
+ ('a1 -> 'a1 -> 'a1) -> ('a1 pol -> 'a1 pol -> 'a1 pol) -> 'a1 pol ->
+ positive -> 'a1 pol -> 'a1 pol
+
+val psubI :
+ ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1) -> ('a1 pol -> 'a1 pol -> 'a1 pol) ->
+ 'a1 pol -> positive -> 'a1 pol -> 'a1 pol
+
+val paddX :
+ 'a1 -> ('a1 -> 'a1 -> bool) -> ('a1 pol -> 'a1 pol -> 'a1 pol) -> 'a1 pol
+ -> positive -> 'a1 pol -> 'a1 pol
+
+val psubX :
+ 'a1 -> ('a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> ('a1 pol -> 'a1 pol -> 'a1
+ pol) -> 'a1 pol -> positive -> 'a1 pol -> 'a1 pol
+
+val padd :
+ 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 pol -> 'a1 pol ->
+ 'a1 pol
+
+val psub :
+ 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1) -> ('a1
+ -> 'a1 -> bool) -> 'a1 pol -> 'a1 pol -> 'a1 pol
+
+val pmulC_aux :
+ 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 pol -> 'a1 -> 'a1
+ pol
+
+val pmulC :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 pol -> 'a1
+ -> 'a1 pol
+
+val pmulI :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> ('a1 pol ->
+ 'a1 pol -> 'a1 pol) -> 'a1 pol -> positive -> 'a1 pol -> 'a1 pol
+
+val pmul :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 ->
+ bool) -> 'a1 pol -> 'a1 pol -> 'a1 pol
+
+type 'c pExpr =
+ | PEc of 'c
+ | PEX of positive
+ | PEadd of 'c pExpr * 'c pExpr
+ | PEsub of 'c pExpr * 'c pExpr
+ | PEmul of 'c pExpr * 'c pExpr
+ | PEopp of 'c pExpr
+ | PEpow of 'c pExpr * n
+
+val mk_X : 'a1 -> 'a1 -> positive -> 'a1 pol
+
+val ppow_pos :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 ->
+ bool) -> ('a1 pol -> 'a1 pol) -> 'a1 pol -> 'a1 pol -> positive -> 'a1 pol
+
+val ppow_N :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 ->
+ bool) -> ('a1 pol -> 'a1 pol) -> 'a1 pol -> n -> 'a1 pol
+
+val norm_aux :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 ->
+ 'a1) -> ('a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 pExpr -> 'a1 pol
+
+type 'a bFormula =
+ | TT
+ | FF
+ | X
+ | A of 'a
+ | Cj of 'a bFormula * 'a bFormula
+ | D of 'a bFormula * 'a bFormula
+ | N of 'a bFormula
+ | I of 'a bFormula * 'a bFormula
+
+type 'term' clause = 'term' list
+
+type 'term' cnf = 'term' clause list
+
+val tt : 'a1 cnf
+
+val ff : 'a1 cnf
+
+val or_clause_cnf : 'a1 clause -> 'a1 cnf -> 'a1 cnf
+
+val or_cnf : 'a1 cnf -> 'a1 cnf -> 'a1 cnf
+
+val and_cnf : 'a1 cnf -> 'a1 cnf -> 'a1 cnf
+
+val xcnf :
+ ('a1 -> 'a2 cnf) -> ('a1 -> 'a2 cnf) -> bool -> 'a1 bFormula -> 'a2 cnf
+
+val cnf_checker : ('a1 list -> 'a2 -> bool) -> 'a1 cnf -> 'a2 list -> bool
+
+val tauto_checker :
+ ('a1 -> 'a2 cnf) -> ('a1 -> 'a2 cnf) -> ('a2 list -> 'a3 -> bool) -> 'a1
+ bFormula -> 'a3 list -> bool
+
+type 'c pExprC = 'c pExpr
+
+type 'c polC = 'c pol
+
+type op1 =
+ | Equal
+ | NonEqual
+ | Strict
+ | NonStrict
+
+type 'c nFormula = ('c pExprC, op1) prod
+
+type monoidMember = nat list
+
+type 'c coneMember =
+ | S_In of nat
+ | S_Ideal of 'c pExprC * 'c coneMember
+ | S_Square of 'c pExprC
+ | S_Monoid of monoidMember
+ | S_Mult of 'c coneMember * 'c coneMember
+ | S_Add of 'c coneMember * 'c coneMember
+ | S_Pos of 'c
+ | S_Z
+
+val nformula_times : 'a1 nFormula -> 'a1 nFormula -> 'a1 nFormula
+
+val nformula_plus : 'a1 nFormula -> 'a1 nFormula -> 'a1 nFormula
+
+val eval_monoid : 'a1 -> 'a1 nFormula list -> monoidMember -> 'a1 pExprC
+
+val eval_cone :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> bool) -> ('a1 -> 'a1 -> bool) -> 'a1 nFormula
+ list -> 'a1 coneMember -> 'a1 nFormula
+
+val normalise_pexpr :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 ->
+ 'a1) -> ('a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 pExprC -> 'a1 polC
+
+val check_inconsistent :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 ->
+ 'a1) -> ('a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> ('a1 -> 'a1 -> bool) -> 'a1
+ nFormula -> bool
+
+val check_normalised_formulas :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 ->
+ 'a1) -> ('a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> ('a1 -> 'a1 -> bool) -> 'a1
+ nFormula list -> 'a1 coneMember -> bool
+
+type op2 =
+ | OpEq
+ | OpNEq
+ | OpLe
+ | OpGe
+ | OpLt
+ | OpGt
+
+type 'c formula = { flhs : 'c pExprC; fop : op2; frhs : 'c pExprC }
+
+val flhs : 'a1 formula -> 'a1 pExprC
+
+val fop : 'a1 formula -> op2
+
+val frhs : 'a1 formula -> 'a1 pExprC
+
+val xnormalise : 'a1 formula -> 'a1 nFormula list
+
+val cnf_normalise : 'a1 formula -> 'a1 nFormula cnf
+
+val xnegate : 'a1 formula -> 'a1 nFormula list
+
+val cnf_negate : 'a1 formula -> 'a1 nFormula cnf
+
+val simpl_expr : 'a1 -> ('a1 -> 'a1 -> bool) -> 'a1 pExprC -> 'a1 pExprC
+
+val simpl_cone :
+ 'a1 -> 'a1 -> ('a1 -> 'a1 -> 'a1) -> ('a1 -> 'a1 -> bool) -> 'a1 coneMember
+ -> 'a1 coneMember
+
+type q = { qnum : z; qden : positive }
+
+val qnum : q -> z
+
+val qden : q -> positive
+
+val qplus : q -> q -> q
+
+val qmult : q -> q -> q
+
+val qopp : q -> q
+
+val qminus : q -> q -> q
+
+type 'a t =
+ | Empty
+ | Leaf of 'a
+ | Node of 'a t * 'a * 'a t
+
+val find : 'a1 -> 'a1 t -> positive -> 'a1
+
+type zWitness = z coneMember
+
+val zWeakChecker : z nFormula list -> z coneMember -> bool
+
+val xnormalise0 : z formula -> z nFormula list
+
+val normalise : z formula -> z nFormula cnf
+
+val xnegate0 : z formula -> z nFormula list
+
+val negate : z formula -> z nFormula cnf
+
+val ceiling : z -> z -> z
+
+type proofTerm =
+ | RatProof of zWitness
+ | CutProof of z pExprC * q * zWitness * proofTerm
+ | EnumProof of q * z pExprC * q * zWitness * zWitness * proofTerm list
+
+val makeLb : z pExpr -> q -> z nFormula
+
+val qceiling : q -> z
+
+val makeLbCut : z pExprC -> q -> z nFormula
+
+val neg_nformula : z nFormula -> (z pExpr, op1) prod
+
+val cutChecker :
+ z nFormula list -> z pExpr -> q -> zWitness -> z nFormula option
+
+val zChecker : z nFormula list -> proofTerm -> bool
+
+val zTautoChecker : z formula bFormula -> proofTerm list -> bool
+
+val map_cone : (nat -> nat) -> zWitness -> zWitness
+
+val indexes : zWitness -> nat list
+
+val n_of_Z : z -> n
+
+val qeq_bool : q -> q -> bool
+
+val qle_bool : q -> q -> bool
+
+type qWitness = q coneMember
+
+val qWeakChecker : q nFormula list -> q coneMember -> bool
+
+val qTautoChecker : q formula bFormula -> qWitness list -> bool
+
diff --git a/contrib/micromega/mutils.ml b/contrib/micromega/mutils.ml
new file mode 100644
index 000000000..2473608f2
--- /dev/null
+++ b/contrib/micromega/mutils.ml
@@ -0,0 +1,305 @@
+(************************************************************************)
+(* 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 *)
+(************************************************************************)
+(* *)
+(* Micromega: A reflexive tactic using the Positivstellensatz *)
+(* *)
+(* Frédéric Besson (Irisa/Inria) 2006-2008 *)
+(* *)
+(************************************************************************)
+
+let debug = false
+
+let fst' (Micromega.Pair(x,y)) = x
+let snd' (Micromega.Pair(x,y)) = y
+
+let rec try_any l x =
+ match l with
+ | [] -> None
+ | (f,s)::l -> match f x with
+ | None -> try_any l x
+ | x -> x
+
+let list_try_find f =
+ let rec try_find_f = function
+ | [] -> failwith "try_find"
+ | h::t -> try f h with Failure _ -> try_find_f t
+ in
+ try_find_f
+
+let rec list_fold_right_elements f l =
+ let rec aux = function
+ | [] -> invalid_arg "list_fold_right_elements"
+ | [x] -> x
+ | x::l -> f x (aux l) in
+ aux l
+
+let interval n m =
+ let rec interval_n (l,m) =
+ if n > m then l else interval_n (m::l,pred m)
+ in
+ interval_n ([],m)
+
+open Num
+open Big_int
+
+let ppcm x y =
+ let g = gcd_big_int x y in
+ let x' = div_big_int x g in
+ let y' = div_big_int y g in
+ mult_big_int g (mult_big_int x' y')
+
+
+let denominator = function
+ | Int _ | Big_int _ -> unit_big_int
+ | Ratio r -> Ratio.denominator_ratio r
+
+let numerator = function
+ | Ratio r -> Ratio.numerator_ratio r
+ | Int i -> Big_int.big_int_of_int i
+ | Big_int i -> i
+
+let rec ppcm_list c l =
+ match l with
+ | [] -> c
+ | e::l -> ppcm_list (ppcm c (denominator e)) l
+
+let rec rec_gcd_list c l =
+ match l with
+ | [] -> c
+ | e::l -> rec_gcd_list (gcd_big_int c (numerator e)) l
+
+let rec gcd_list l =
+ let res = rec_gcd_list zero_big_int l in
+ if compare_big_int res zero_big_int = 0
+ then unit_big_int else res
+
+
+
+let rats_to_ints l =
+ let c = ppcm_list unit_big_int l in
+ List.map (fun x -> (div_big_int (mult_big_int (numerator x) c)
+ (denominator x))) l
+
+(* Nasty reordering of lists - useful to trim certificate down *)
+let mapi f l =
+ let rec xmapi i l =
+ match l with
+ | [] -> []
+ | e::l -> (f e i)::(xmapi (i+1) l) in
+ xmapi 0 l
+
+
+let concatMapi f l = List.rev (mapi (fun e i -> (i,f e)) l)
+
+(* assoc_pos j [a0...an] = [j,a0....an,j+n],j+n+1 *)
+let assoc_pos j l = (mapi (fun e i -> e,i+j) l, j + (List.length l))
+
+let assoc_pos_assoc l =
+ let rec xpos i l =
+ match l with
+ | [] -> []
+ | (x,l) ::rst -> let (l',j) = assoc_pos i l in
+ (x,l')::(xpos j rst) in
+ xpos 0 l
+
+let filter_pos f l =
+ (* Could sort ... take care of duplicates... *)
+ let rec xfilter l =
+ match l with
+ | [] -> []
+ | (x,e)::l ->
+ if List.exists (fun ee -> List.mem ee f) (List.map snd e)
+ then (x,e)::(xfilter l)
+ else xfilter l in
+ xfilter l
+
+let select_pos lpos l =
+ let rec xselect i lpos l =
+ match lpos with
+ | [] -> []
+ | j::rpos ->
+ match l with
+ | [] -> failwith "select_pos"
+ | e::l ->
+ if i = j
+ then e:: (xselect (i+1) rpos l)
+ else xselect (i+1) lpos l in
+ xselect 0 lpos l
+
+
+module CoqToCaml =
+struct
+ open Micromega
+
+ let rec nat = function
+ | O -> 0
+ | S n -> (nat n) + 1
+
+
+ let rec positive p =
+ match p with
+ | XH -> 1
+ | XI p -> 1+ 2*(positive p)
+ | XO p -> 2*(positive p)
+
+
+ let n nt =
+ match nt with
+ | N0 -> 0
+ | Npos p -> positive p
+
+
+ let rec index i = (* Swap left-right ? *)
+ match i with
+ | XH -> 1
+ | XI i -> 1+(2*(index i))
+ | XO i -> 2*(index i)
+
+
+ let z x =
+ match x with
+ | Z0 -> 0
+ | Zpos p -> (positive p)
+ | Zneg p -> - (positive p)
+
+ open Big_int
+
+ let rec positive_big_int p =
+ match p with
+ | XH -> unit_big_int
+ | XI p -> add_int_big_int 1 (mult_int_big_int 2 (positive_big_int p))
+ | XO p -> (mult_int_big_int 2 (positive_big_int p))
+
+
+ let z_big_int x =
+ match x with
+ | Z0 -> zero_big_int
+ | Zpos p -> (positive_big_int p)
+ | Zneg p -> minus_big_int (positive_big_int p)
+
+
+ let num x = Num.Big_int (z_big_int x)
+
+ let rec list elt l =
+ match l with
+ | Nil -> []
+ | Cons(e,l) -> (elt e)::(list elt l)
+
+ let q_to_num {qnum = x ; qden = y} =
+ Big_int (z_big_int x) // (Big_int (z_big_int (Zpos y)))
+
+end
+
+
+module CamlToCoq =
+struct
+ open Micromega
+
+ let rec nat = function
+ | 0 -> O
+ | n -> S (nat (n-1))
+
+
+ let rec positive n =
+ if n=1 then XH
+ else if n land 1 = 1 then XI (positive (n lsr 1))
+ else XO (positive (n lsr 1))
+
+ let n nt =
+ if nt < 0
+ then assert false
+ else if nt = 0 then N0
+ else Npos (positive nt)
+
+
+
+
+
+ let rec index n =
+ if n=1 then XH
+ else if n land 1 = 1 then XI (index (n lsr 1))
+ else XO (index (n lsr 1))
+
+
+ let idx n =
+ (*a.k.a path_of_int *)
+ (* returns the list of digits of n in reverse order with
+ initial 1 removed *)
+ let rec digits_of_int n =
+ if n=1 then []
+ else (n mod 2 = 1)::(digits_of_int (n lsr 1))
+ in
+ List.fold_right
+ (fun b c -> (if b then XI c else XO c))
+ (List.rev (digits_of_int n))
+ (XH)
+
+
+
+ let z x =
+ match compare x 0 with
+ | 0 -> Z0
+ | 1 -> Zpos (positive x)
+ | _ -> (* this should be -1 *)
+ Zneg (positive (-x))
+
+ open Big_int
+
+ let positive_big_int n =
+ let two = big_int_of_int 2 in
+ let rec _pos n =
+ if eq_big_int n unit_big_int then XH
+ else
+ let (q,m) = quomod_big_int n two in
+ if eq_big_int unit_big_int m
+ then XI (_pos q)
+ else XO (_pos q) in
+ _pos n
+
+ let bigint x =
+ match sign_big_int x with
+ | 0 -> Z0
+ | 1 -> Zpos (positive_big_int x)
+ | _ -> Zneg (positive_big_int (minus_big_int x))
+
+ let q n =
+ {Micromega.qnum = bigint (numerator n) ;
+ Micromega.qden = positive_big_int (denominator n)}
+
+
+ let list elt l = List.fold_right (fun x l -> Cons(elt x, l)) l Nil
+
+end
+
+module Cmp =
+struct
+
+ let rec compare_lexical l =
+ match l with
+ | [] -> 0 (* Equal *)
+ | f::l ->
+ let cmp = f () in
+ if cmp = 0 then compare_lexical l else cmp
+
+ let rec compare_list cmp l1 l2 =
+ match l1 , l2 with
+ | [] , [] -> 0
+ | [] , _ -> -1
+ | _ , [] -> 1
+ | e1::l1 , e2::l2 ->
+ let c = cmp e1 e2 in
+ if c = 0 then compare_list cmp l1 l2 else c
+
+ let hash_list hash l =
+ let rec _hash_list l h =
+ match l with
+ | [] -> h lxor (Hashtbl.hash [])
+ | e::l -> _hash_list l ((hash e) lxor h) in
+
+ _hash_list l 0
+end
diff --git a/contrib/micromega/sos.ml b/contrib/micromega/sos.ml
new file mode 100644
index 000000000..e3d72ed9a
--- /dev/null
+++ b/contrib/micromega/sos.ml
@@ -0,0 +1,1919 @@
+(* ========================================================================= *)
+(* - This code originates from John Harrison's HOL LIGHT 2.20 *)
+(* (see file LICENSE.sos for license, copyright and disclaimer) *)
+(* - Laurent Théry (thery@sophia.inria.fr) has isolated the HOL *)
+(* independent bits *)
+(* - Frédéric Besson (fbesson@irisa.fr) is using it to feed micromega *)
+(* - Addition of a csdp cache by the Coq development team *)
+(* ========================================================================= *)
+
+(* ========================================================================= *)
+(* Nonlinear universal reals procedure using SOS decomposition. *)
+(* ========================================================================= *)
+
+open Num;;
+open List;;
+
+let debugging = ref false;;
+
+exception Sanity;;
+
+exception Unsolvable;;
+
+(* ------------------------------------------------------------------------- *)
+(* Comparisons that are reflexive on NaN and also short-circuiting. *)
+(* ------------------------------------------------------------------------- *)
+
+let (=?) = fun x y -> Pervasives.compare x y = 0;;
+let (<?) = fun x y -> Pervasives.compare x y < 0;;
+let (<=?) = fun x y -> Pervasives.compare x y <= 0;;
+let (>?) = fun x y -> Pervasives.compare x y > 0;;
+let (>=?) = fun x y -> Pervasives.compare x y >= 0;;
+
+(* ------------------------------------------------------------------------- *)
+(* Combinators. *)
+(* ------------------------------------------------------------------------- *)
+
+let (o) = fun f g x -> f(g x);;
+
+(* ------------------------------------------------------------------------- *)
+(* Some useful functions on "num" type. *)
+(* ------------------------------------------------------------------------- *)
+
+
+let num_0 = Int 0
+and num_1 = Int 1
+and num_2 = Int 2
+and num_10 = Int 10;;
+
+let pow2 n = power_num num_2 (Int n);;
+let pow10 n = power_num num_10 (Int n);;
+
+let numdom r =
+ let r' = Ratio.normalize_ratio (ratio_of_num r) in
+ num_of_big_int(Ratio.numerator_ratio r'),
+ num_of_big_int(Ratio.denominator_ratio r');;
+
+let numerator = (o) fst numdom
+and denominator = (o) snd numdom;;
+
+let gcd_num n1 n2 =
+ num_of_big_int(Big_int.gcd_big_int (big_int_of_num n1) (big_int_of_num n2));;
+
+let lcm_num x y =
+ if x =/ num_0 & y =/ num_0 then num_0
+ else abs_num((x */ y) // gcd_num x y);;
+
+
+(* ------------------------------------------------------------------------- *)
+(* List basics. *)
+(* ------------------------------------------------------------------------- *)
+
+let rec el n l =
+ if n = 0 then hd l else el (n - 1) (tl l);;
+
+
+(* ------------------------------------------------------------------------- *)
+(* Various versions of list iteration. *)
+(* ------------------------------------------------------------------------- *)
+
+let rec itlist f l b =
+ match l with
+ [] -> b
+ | (h::t) -> f h (itlist f t b);;
+
+let rec end_itlist f l =
+ match l with
+ [] -> failwith "end_itlist"
+ | [x] -> x
+ | (h::t) -> f h (end_itlist f t);;
+
+let rec itlist2 f l1 l2 b =
+ match (l1,l2) with
+ ([],[]) -> b
+ | (h1::t1,h2::t2) -> f h1 h2 (itlist2 f t1 t2 b)
+ | _ -> failwith "itlist2";;
+
+(* ------------------------------------------------------------------------- *)
+(* All pairs arising from applying a function over two lists. *)
+(* ------------------------------------------------------------------------- *)
+
+let rec allpairs f l1 l2 =
+ match l1 with
+ h1::t1 -> itlist (fun x a -> f h1 x :: a) l2 (allpairs f t1 l2)
+ | [] -> [];;
+
+(* ------------------------------------------------------------------------- *)
+(* String operations (surely there is a better way...) *)
+(* ------------------------------------------------------------------------- *)
+
+let implode l = itlist (^) l "";;
+
+let explode s =
+ let rec exap n l =
+ if n < 0 then l else
+ exap (n - 1) ((String.sub s n 1)::l) in
+ exap (String.length s - 1) [];;
+
+
+(* ------------------------------------------------------------------------- *)
+(* Attempting function or predicate applications. *)
+(* ------------------------------------------------------------------------- *)
+
+let can f x = try (f x; true) with Failure _ -> false;;
+
+
+(* ------------------------------------------------------------------------- *)
+(* Repetition of a function. *)
+(* ------------------------------------------------------------------------- *)
+
+let rec funpow n f x =
+ if n < 1 then x else funpow (n-1) f (f x);;
+
+
+(* ------------------------------------------------------------------------- *)
+(* term?? *)
+(* ------------------------------------------------------------------------- *)
+
+type vname = string;;
+
+type term =
+| Zero
+| Const of Num.num
+| Var of vname
+| Inv of term
+| Opp of term
+| Add of (term * term)
+| Sub of (term * term)
+| Mul of (term * term)
+| Div of (term * term)
+| Pow of (term * int);;
+
+
+(* ------------------------------------------------------------------------- *)
+(* Data structure for Positivstellensatz refutations. *)
+(* ------------------------------------------------------------------------- *)
+
+type positivstellensatz =
+ Axiom_eq of int
+ | Axiom_le of int
+ | Axiom_lt of int
+ | Rational_eq of num
+ | Rational_le of num
+ | Rational_lt of num
+ | Square of term
+ | Monoid of int list
+ | Eqmul of term * positivstellensatz
+ | Sum of positivstellensatz * positivstellensatz
+ | Product of positivstellensatz * positivstellensatz;;
+
+
+
+(* ------------------------------------------------------------------------- *)
+(* Replication and sequences. *)
+(* ------------------------------------------------------------------------- *)
+
+let rec replicate x n =
+ if n < 1 then []
+ else x::(replicate x (n - 1));;
+
+let rec (--) = fun m n -> if m > n then [] else m::((m + 1) -- n);;
+
+(* ------------------------------------------------------------------------- *)
+(* Various useful list operations. *)
+(* ------------------------------------------------------------------------- *)
+
+let rec forall p l =
+ match l with
+ [] -> true
+ | h::t -> p(h) & forall p t;;
+
+let rec tryfind f l =
+ match l with
+ [] -> failwith "tryfind"
+ | (h::t) -> try f h with Failure _ -> tryfind f t;;
+
+let index x =
+ let rec ind n l =
+ match l with
+ [] -> failwith "index"
+ | (h::t) -> if x =? h then n else ind (n + 1) t in
+ ind 0;;
+
+(* ------------------------------------------------------------------------- *)
+(* "Set" operations on lists. *)
+(* ------------------------------------------------------------------------- *)
+
+let rec mem x lis =
+ match lis with
+ [] -> false
+ | (h::t) -> x =? h or mem x t;;
+
+let insert x l =
+ if mem x l then l else x::l;;
+
+let union l1 l2 = itlist insert l1 l2;;
+
+let subtract l1 l2 = filter (fun x -> not (mem x l2)) l1;;
+
+(* ------------------------------------------------------------------------- *)
+(* Merging and bottom-up mergesort. *)
+(* ------------------------------------------------------------------------- *)
+
+let rec merge ord l1 l2 =
+ match l1 with
+ [] -> l2
+ | h1::t1 -> match l2 with
+ [] -> l1
+ | h2::t2 -> if ord h1 h2 then h1::(merge ord t1 l2)
+ else h2::(merge ord l1 t2);;
+
+
+(* ------------------------------------------------------------------------- *)
+(* Common measure predicates to use with "sort". *)
+(* ------------------------------------------------------------------------- *)
+
+let increasing f x y = f x <? f y;;
+
+let decreasing f x y = f x >? f y;;
+
+(* ------------------------------------------------------------------------- *)
+(* Zipping, unzipping etc. *)
+(* ------------------------------------------------------------------------- *)
+
+let rec zip l1 l2 =
+ match (l1,l2) with
+ ([],[]) -> []
+ | (h1::t1,h2::t2) -> (h1,h2)::(zip t1 t2)
+ | _ -> failwith "zip";;
+
+let rec unzip =
+ function [] -> [],[]
+ | ((a,b)::rest) -> let alist,blist = unzip rest in
+ (a::alist,b::blist);;
+
+(* ------------------------------------------------------------------------- *)
+(* Iterating functions over lists. *)
+(* ------------------------------------------------------------------------- *)
+
+let rec do_list f l =
+ match l with
+ [] -> ()
+ | (h::t) -> (f h; do_list f t);;
+
+(* ------------------------------------------------------------------------- *)
+(* Sorting. *)
+(* ------------------------------------------------------------------------- *)
+
+let rec sort cmp lis =
+ match lis with
+ [] -> []
+ | piv::rest ->
+ let r,l = partition (cmp piv) rest in
+ (sort cmp l) @ (piv::(sort cmp r));;
+
+(* ------------------------------------------------------------------------- *)
+(* Removing adjacent (NB!) equal elements from list. *)
+(* ------------------------------------------------------------------------- *)
+
+let rec uniq l =
+ match l with
+ x::(y::_ as t) -> let t' = uniq t in
+ if x =? y then t' else
+ if t'==t then l else x::t'
+ | _ -> l;;
+
+(* ------------------------------------------------------------------------- *)
+(* Convert list into set by eliminating duplicates. *)
+(* ------------------------------------------------------------------------- *)
+
+let setify s = uniq (sort (<=?) s);;
+
+(* ------------------------------------------------------------------------- *)
+(* Polymorphic finite partial functions via Patricia trees. *)
+(* *)
+(* The point of this strange representation is that it is canonical (equal *)
+(* functions have the same encoding) yet reasonably efficient on average. *)
+(* *)
+(* Idea due to Diego Olivier Fernandez Pons (OCaml list, 2003/11/10). *)
+(* ------------------------------------------------------------------------- *)
+
+type ('a,'b)func =
+ Empty
+ | Leaf of int * ('a*'b)list
+ | Branch of int * int * ('a,'b)func * ('a,'b)func;;
+
+(* ------------------------------------------------------------------------- *)
+(* Undefined function. *)
+(* ------------------------------------------------------------------------- *)
+
+let undefined = Empty;;
+
+(* ------------------------------------------------------------------------- *)
+(* In case of equality comparison worries, better use this. *)
+(* ------------------------------------------------------------------------- *)
+
+let is_undefined f =
+ match f with
+ Empty -> true
+ | _ -> false;;
+
+(* ------------------------------------------------------------------------- *)
+(* Operation analagous to "map" for lists. *)
+(* ------------------------------------------------------------------------- *)
+
+let mapf =
+ let rec map_list f l =
+ match l with
+ [] -> []
+ | (x,y)::t -> (x,f(y))::(map_list f t) in
+ let rec mapf f t =
+ match t with
+ Empty -> Empty
+ | Leaf(h,l) -> Leaf(h,map_list f l)
+ | Branch(p,b,l,r) -> Branch(p,b,mapf f l,mapf f r) in
+ mapf;;
+
+(* ------------------------------------------------------------------------- *)
+(* Operations analogous to "fold" for lists. *)
+(* ------------------------------------------------------------------------- *)
+
+let foldl =
+ let rec foldl_list f a l =
+ match l with
+ [] -> a
+ | (x,y)::t -> foldl_list f (f a x y) t in
+ let rec foldl f a t =
+ match t with
+ Empty -> a
+ | Leaf(h,l) -> foldl_list f a l
+ | Branch(p,b,l,r) -> foldl f (foldl f a l) r in
+ foldl;;
+
+let foldr =
+ let rec foldr_list f l a =
+ match l with
+ [] -> a
+ | (x,y)::t -> f x y (foldr_list f t a) in
+ let rec foldr f t a =
+ match t with
+ Empty -> a
+ | Leaf(h,l) -> foldr_list f l a
+ | Branch(p,b,l,r) -> foldr f l (foldr f r a) in
+ foldr;;
+
+(* ------------------------------------------------------------------------- *)
+(* Redefinition and combination. *)
+(* ------------------------------------------------------------------------- *)
+
+let (|->),combine =
+ let ldb x y = let z = x lxor y in z land (-z) in
+ let newbranch p1 t1 p2 t2 =
+ let b = ldb p1 p2 in
+ let p = p1 land (b - 1) in
+ if p1 land b = 0 then Branch(p,b,t1,t2)
+ else Branch(p,b,t2,t1) in
+ let rec define_list (x,y as xy) l =
+ match l with
+ (a,b as ab)::t ->
+ if x =? a then xy::t
+ else if x <? a then xy::l
+ else ab::(define_list xy t)
+ | [] -> [xy]
+ and combine_list op z l1 l2 =
+ match (l1,l2) with
+ [],_ -> l2
+ | _,[] -> l1
+ | ((x1,y1 as xy1)::t1,(x2,y2 as xy2)::t2) ->
+ if x1 <? x2 then xy1::(combine_list op z t1 l2)
+ else if x2 <? x1 then xy2::(combine_list op z l1 t2) else
+ let y = op y1 y2 and l = combine_list op z t1 t2 in
+ if z(y) then l else (x1,y)::l in
+ let (|->) x y =
+ let k = Hashtbl.hash x in
+ let rec upd t =
+ match t with
+ Empty -> Leaf (k,[x,y])
+ | Leaf(h,l) ->
+ if h = k then Leaf(h,define_list (x,y) l)
+ else newbranch h t k (Leaf(k,[x,y]))
+ | Branch(p,b,l,r) ->
+ if k land (b - 1) <> p then newbranch p t k (Leaf(k,[x,y]))
+ else if k land b = 0 then Branch(p,b,upd l,r)
+ else Branch(p,b,l,upd r) in
+ upd in
+ let rec combine op z t1 t2 =
+ match (t1,t2) with
+ Empty,_ -> t2
+ | _,Empty -> t1
+ | Leaf(h1,l1),Leaf(h2,l2) ->
+ if h1 = h2 then
+ let l = combine_list op z l1 l2 in
+ if l = [] then Empty else Leaf(h1,l)
+ else newbranch h1 t1 h2 t2
+ | (Leaf(k,lis) as lf),(Branch(p,b,l,r) as br) |
+ (Branch(p,b,l,r) as br),(Leaf(k,lis) as lf) ->
+ if k land (b - 1) = p then
+ if k land b = 0 then
+ let l' = combine op z lf l in
+ if is_undefined l' then r else Branch(p,b,l',r)
+ else
+ let r' = combine op z lf r in
+ if is_undefined r' then l else Branch(p,b,l,r')
+ else
+ newbranch k lf p br
+ | Branch(p1,b1,l1,r1),Branch(p2,b2,l2,r2) ->
+ if b1 < b2 then
+ if p2 land (b1 - 1) <> p1 then newbranch p1 t1 p2 t2
+ else if p2 land b1 = 0 then
+ let l = combine op z l1 t2 in
+ if is_undefined l then r1 else Branch(p1,b1,l,r1)
+ else
+ let r = combine op z r1 t2 in
+ if is_undefined r then l1 else Branch(p1,b1,l1,r)
+ else if b2 < b1 then
+ if p1 land (b2 - 1) <> p2 then newbranch p1 t1 p2 t2
+ else if p1 land b2 = 0 then
+ let l = combine op z t1 l2 in
+ if is_undefined l then r2 else Branch(p2,b2,l,r2)
+ else
+ let r = combine op z t1 r2 in
+ if is_undefined r then l2 else Branch(p2,b2,l2,r)
+ else if p1 = p2 then
+ let l = combine op z l1 l2 and r = combine op z r1 r2 in
+ if is_undefined l then r
+ else if is_undefined r then l else Branch(p1,b1,l,r)
+ else
+ newbranch p1 t1 p2 t2 in
+ (|->),combine;;
+
+(* ------------------------------------------------------------------------- *)
+(* Special case of point function. *)
+(* ------------------------------------------------------------------------- *)
+
+let (|=>) = fun x y -> (x |-> y) undefined;;
+
+
+(* ------------------------------------------------------------------------- *)
+(* Grab an arbitrary element. *)
+(* ------------------------------------------------------------------------- *)
+
+let rec choose t =
+ match t with
+ Empty -> failwith "choose: completely undefined function"
+ | Leaf(h,l) -> hd l
+ | Branch(b,p,t1,t2) -> choose t1;;
+
+(* ------------------------------------------------------------------------- *)
+(* Application. *)
+(* ------------------------------------------------------------------------- *)
+
+let applyd =
+ let rec apply_listd l d x =
+ match l with
+ (a,b)::t -> if x =? a then b
+ else if x >? a then apply_listd t d x else d x
+ | [] -> d x in
+ fun f d x ->
+ let k = Hashtbl.hash x in
+ let rec look t =
+ match t with
+ Leaf(h,l) when h = k -> apply_listd l d x
+ | Branch(p,b,l,r) -> look (if k land b = 0 then l else r)
+ | _ -> d x in
+ look f;;
+
+let apply f = applyd f (fun x -> failwith "apply");;
+
+let tryapplyd f a d = applyd f (fun x -> d) a;;
+
+let defined f x = try apply f x; true with Failure _ -> false;;
+
+(* ------------------------------------------------------------------------- *)
+(* Undefinition. *)
+(* ------------------------------------------------------------------------- *)
+
+let undefine =
+ let rec undefine_list x l =
+ match l with
+ (a,b as ab)::t ->
+ if x =? a then t
+ else if x <? a then l else
+ let t' = undefine_list x t in
+ if t' == t then l else ab::t'
+ | [] -> [] in
+ fun x ->
+ let k = Hashtbl.hash x in
+ let rec und t =
+ match t with
+ Leaf(h,l) when h = k ->
+ let l' = undefine_list x l in
+ if l' == l then t
+ else if l' = [] then Empty
+ else Leaf(h,l')
+ | Branch(p,b,l,r) when k land (b - 1) = p ->
+ if k land b = 0 then
+ let l' = und l in
+ if l' == l then t
+ else if is_undefined l' then r
+ else Branch(p,b,l',r)
+ else
+ let r' = und r in
+ if r' == r then t
+ else if is_undefined r' then l
+ else Branch(p,b,l,r')
+ | _ -> t in
+ und;;
+
+
+(* ------------------------------------------------------------------------- *)
+(* Mapping to sorted-list representation of the graph, domain and range. *)
+(* ------------------------------------------------------------------------- *)
+
+let graph f = setify (foldl (fun a x y -> (x,y)::a) [] f);;
+
+let dom f = setify(foldl (fun a x y -> x::a) [] f);;
+
+let ran f = setify(foldl (fun a x y -> y::a) [] f);;
+
+(* ------------------------------------------------------------------------- *)
+(* Turn a rational into a decimal string with d sig digits. *)
+(* ------------------------------------------------------------------------- *)
+
+let decimalize =
+ let rec normalize y =
+ if abs_num y </ Int 1 // Int 10 then normalize (Int 10 */ y) - 1
+ else if abs_num y >=/ Int 1 then normalize (y // Int 10) + 1
+ else 0 in
+ fun d x ->
+ if x =/ Int 0 then "0.0" else
+ let y = abs_num x in
+ let e = normalize y in
+ let z = pow10(-e) */ y +/ Int 1 in
+ let k = round_num(pow10 d */ z) in
+ (if x </ Int 0 then "-0." else "0.") ^
+ implode(tl(explode(string_of_num k))) ^
+ (if e = 0 then "" else "e"^string_of_int e);;
+
+
+(* ------------------------------------------------------------------------- *)
+(* Iterations over numbers, and lists indexed by numbers. *)
+(* ------------------------------------------------------------------------- *)
+
+let rec itern k l f a =
+ match l with
+ [] -> a
+ | h::t -> itern (k + 1) t f (f h k a);;
+
+let rec iter (m,n) f a =
+ if n < m then a
+ else iter (m+1,n) f (f m a);;
+
+(* ------------------------------------------------------------------------- *)
+(* The main types. *)
+(* ------------------------------------------------------------------------- *)
+
+type vector = int*(int,num)func;;
+
+type matrix = (int*int)*(int*int,num)func;;
+
+type monomial = (vname,int)func;;
+
+type poly = (monomial,num)func;;
+
+(* ------------------------------------------------------------------------- *)
+(* Assignment avoiding zeros. *)
+(* ------------------------------------------------------------------------- *)
+
+let (|-->) x y a = if y =/ Int 0 then a else (x |-> y) a;;
+
+(* ------------------------------------------------------------------------- *)
+(* This can be generic. *)
+(* ------------------------------------------------------------------------- *)
+
+let element (d,v) i = tryapplyd v i (Int 0);;
+
+let mapa f (d,v) =
+ d,foldl (fun a i c -> (i |--> f(c)) a) undefined v;;
+
+let is_zero (d,v) =
+ match v with
+ Empty -> true
+ | _ -> false;;
+
+(* ------------------------------------------------------------------------- *)
+(* Vectors. Conventionally indexed 1..n. *)
+(* ------------------------------------------------------------------------- *)
+
+let vector_0 n = (n,undefined:vector);;
+
+let dim (v:vector) = fst v;;
+
+let vector_const c n =
+ if c =/ Int 0 then vector_0 n
+ else (n,itlist (fun k -> k |-> c) (1--n) undefined :vector);;
+
+let vector_1 = vector_const (Int 1);;
+
+let vector_cmul c (v:vector) =
+ let n = dim v in
+ if c =/ Int 0 then vector_0 n
+ else n,mapf (fun x -> c */ x) (snd v);;
+
+let vector_neg (v:vector) = (fst v,mapf minus_num (snd v) :vector);;
+
+let vector_add (v1:vector) (v2:vector) =
+ let m = dim v1 and n = dim v2 in
+ if m <> n then failwith "vector_add: incompatible dimensions" else
+ (n,combine (+/) (fun x -> x =/ Int 0) (snd v1) (snd v2) :vector);;
+
+let vector_sub v1 v2 = vector_add v1 (vector_neg v2);;
+
+let vector_of_list l =
+ let n = length l in
+ (n,itlist2 (|->) (1--n) l undefined :vector);;
+
+(* ------------------------------------------------------------------------- *)
+(* Matrices; again rows and columns indexed from 1. *)
+(* ------------------------------------------------------------------------- *)
+
+let matrix_0 (m,n) = ((m,n),undefined:matrix);;
+
+let dimensions (m:matrix) = fst m;;
+
+let matrix_const c (m,n as mn) =
+ if m <> n then failwith "matrix_const: needs to be square"
+ else if c =/ Int 0 then matrix_0 mn
+ else (mn,itlist (fun k -> (k,k) |-> c) (1--n) undefined :matrix);;
+
+let matrix_1 = matrix_const (Int 1);;
+
+let matrix_cmul c (m:matrix) =
+ let (i,j) = dimensions m in
+ if c =/ Int 0 then matrix_0 (i,j)
+ else (i,j),mapf (fun x -> c */ x) (snd m);;
+
+let matrix_neg (m:matrix) = (dimensions m,mapf minus_num (snd m) :matrix);;
+
+let matrix_add (m1:matrix) (m2:matrix) =
+ let d1 = dimensions m1 and d2 = dimensions m2 in
+ if d1 <> d2 then failwith "matrix_add: incompatible dimensions"
+ else (d1,combine (+/) (fun x -> x =/ Int 0) (snd m1) (snd m2) :matrix);;
+
+let matrix_sub m1 m2 = matrix_add m1 (matrix_neg m2);;
+
+let row k (m:matrix) =
+ let i,j = dimensions m in
+ (j,
+ foldl (fun a (i,j) c -> if i = k then (j |-> c) a else a) undefined (snd m)
+ : vector);;
+
+let column k (m:matrix) =
+ let i,j = dimensions m in
+ (i,
+ foldl (fun a (i,j) c -> if j = k then (i |-> c) a else a) undefined (snd m)
+ : vector);;
+
+let transp (m:matrix) =
+ let i,j = dimensions m in
+ ((j,i),foldl (fun a (i,j) c -> ((j,i) |-> c) a) undefined (snd m) :matrix);;
+
+let diagonal (v:vector) =
+ let n = dim v in
+ ((n,n),foldl (fun a i c -> ((i,i) |-> c) a) undefined (snd v) : matrix);;
+
+let matrix_of_list l =
+ let m = length l in
+ if m = 0 then matrix_0 (0,0) else
+ let n = length (hd l) in
+ (m,n),itern 1 l (fun v i -> itern 1 v (fun c j -> (i,j) |-> c)) undefined;;
+
+(* ------------------------------------------------------------------------- *)
+(* Monomials. *)
+(* ------------------------------------------------------------------------- *)
+
+let monomial_eval assig (m:monomial) =
+ foldl (fun a x k -> a */ power_num (apply assig x) (Int k))
+ (Int 1) m;;
+
+let monomial_1 = (undefined:monomial);;
+
+let monomial_var x = (x |=> 1 :monomial);;
+
+let (monomial_mul:monomial->monomial->monomial) =
+ combine (+) (fun x -> false);;
+
+let monomial_pow (m:monomial) k =
+ if k = 0 then monomial_1
+ else mapf (fun x -> k * x) m;;
+
+let monomial_divides (m1:monomial) (m2:monomial) =
+ foldl (fun a x k -> tryapplyd m2 x 0 >= k & a) true m1;;
+
+let monomial_div (m1:monomial) (m2:monomial) =
+ let m = combine (+) (fun x -> x = 0) m1 (mapf (fun x -> -x) m2) in
+ if foldl (fun a x k -> k >= 0 & a) true m then m
+ else failwith "monomial_div: non-divisible";;
+
+let monomial_degree x (m:monomial) = tryapplyd m x 0;;
+
+let monomial_lcm (m1:monomial) (m2:monomial) =
+ (itlist (fun x -> x |-> max (monomial_degree x m1) (monomial_degree x m2))
+ (union (dom m1) (dom m2)) undefined :monomial);;
+
+let monomial_multidegree (m:monomial) = foldl (fun a x k -> k + a) 0 m;;
+
+let monomial_variables m = dom m;;
+
+(* ------------------------------------------------------------------------- *)
+(* Polynomials. *)
+(* ------------------------------------------------------------------------- *)
+
+let eval assig (p:poly) =
+ foldl (fun a m c -> a +/ c */ monomial_eval assig m) (Int 0) p;;
+
+let poly_0 = (undefined:poly);;
+
+let poly_isconst (p:poly) = foldl (fun a m c -> m = monomial_1 & a) true p;;
+
+let poly_var x = ((monomial_var x) |=> Int 1 :poly);;
+
+let poly_const c =
+ if c =/ Int 0 then poly_0 else (monomial_1 |=> c);;
+
+let poly_cmul c (p:poly) =
+ if c =/ Int 0 then poly_0
+ else mapf (fun x -> c */ x) p;;
+
+let poly_neg (p:poly) = (mapf minus_num p :poly);;
+
+let poly_add (p1:poly) (p2:poly) =
+ (combine (+/) (fun x -> x =/ Int 0) p1 p2 :poly);;
+
+let poly_sub p1 p2 = poly_add p1 (poly_neg p2);;
+
+let poly_cmmul (c,m) (p:poly) =
+ if c =/ Int 0 then poly_0
+ else if m = monomial_1 then mapf (fun d -> c */ d) p
+ else foldl (fun a m' d -> (monomial_mul m m' |-> c */ d) a) poly_0 p;;
+
+let poly_mul (p1:poly) (p2:poly) =
+ foldl (fun a m c -> poly_add (poly_cmmul (c,m) p2) a) poly_0 p1;;
+
+let poly_div (p1:poly) (p2:poly) =
+ if not(poly_isconst p2) then failwith "poly_div: non-constant" else
+ let c = eval undefined p2 in
+ if c =/ Int 0 then failwith "poly_div: division by zero"
+ else poly_cmul (Int 1 // c) p1;;
+
+let poly_square p = poly_mul p p;;
+
+let rec poly_pow p k =
+ if k = 0 then poly_const (Int 1)
+ else if k = 1 then p
+ else let q = poly_square(poly_pow p (k / 2)) in
+ if k mod 2 = 1 then poly_mul p q else q;;
+
+let poly_exp p1 p2 =
+ if not(poly_isconst p2) then failwith "poly_exp: not a constant" else
+ poly_pow p1 (Num.int_of_num (eval undefined p2));;
+
+let degree x (p:poly) = foldl (fun a m c -> max (monomial_degree x m) a) 0 p;;
+
+let multidegree (p:poly) =
+ foldl (fun a m c -> max (monomial_multidegree m) a) 0 p;;
+
+let poly_variables (p:poly) =
+ foldr (fun m c -> union (monomial_variables m)) p [];;
+
+(* ------------------------------------------------------------------------- *)
+(* Order monomials for human presentation. *)
+(* ------------------------------------------------------------------------- *)
+
+let humanorder_varpow (x1,k1) (x2,k2) = x1 < x2 or (x1 = x2 & k1 > k2);;
+
+let humanorder_monomial =
+ let rec ord l1 l2 = match (l1,l2) with
+ _,[] -> true
+ | [],_ -> false
+ | h1::t1,h2::t2 -> humanorder_varpow h1 h2 or (h1 = h2 & ord t1 t2) in
+ fun m1 m2 -> m1 = m2 or
+ ord (sort humanorder_varpow (graph m1))
+ (sort humanorder_varpow (graph m2));;
+
+(* ------------------------------------------------------------------------- *)
+(* Conversions to strings. *)
+(* ------------------------------------------------------------------------- *)
+
+let string_of_vector min_size max_size (v:vector) =
+ let n_raw = dim v in
+ if n_raw = 0 then "[]" else
+ let n = max min_size (min n_raw max_size) in
+ let xs = map ((o) string_of_num (element v)) (1--n) in
+ "[" ^ end_itlist (fun s t -> s ^ ", " ^ t) xs ^
+ (if n_raw > max_size then ", ...]" else "]");;
+
+let string_of_matrix max_size (m:matrix) =
+ let i_raw,j_raw = dimensions m in
+ let i = min max_size i_raw and j = min max_size j_raw in
+ let rstr = map (fun k -> string_of_vector j j (row k m)) (1--i) in
+ "["^end_itlist(fun s t -> s^";\n "^t) rstr ^
+ (if j > max_size then "\n ...]" else "]");;
+
+let string_of_vname (v:vname): string = (v: string);;
+
+let rec string_of_term t =
+ match t with
+ Opp t1 -> "(- " ^ string_of_term t1 ^ ")"
+| Add (t1, t2) ->
+ "(" ^ (string_of_term t1) ^ " + " ^ (string_of_term t2) ^ ")"
+| Sub (t1, t2) ->
+ "(" ^ (string_of_term t1) ^ " - " ^ (string_of_term t2) ^ ")"
+| Mul (t1, t2) ->
+ "(" ^ (string_of_term t1) ^ " * " ^ (string_of_term t2) ^ ")"
+| Inv t1 -> "(/ " ^ string_of_term t1 ^ ")"
+| Div (t1, t2) ->
+ "(" ^ (string_of_term t1) ^ " / " ^ (string_of_term t2) ^ ")"
+| Pow (t1, n1) ->
+ "(" ^ (string_of_term t1) ^ " ^ " ^ (string_of_int n1) ^ ")"
+| Zero -> "0"
+| Var v -> "x" ^ (string_of_vname v)
+| Const x -> string_of_num x;;
+
+
+let string_of_varpow x k =
+ if k = 1 then string_of_vname x else string_of_vname x^"^"^string_of_int k;;
+
+let string_of_monomial m =
+ if m = monomial_1 then "1" else
+ let vps = List.fold_right (fun (x,k) a -> string_of_varpow x k :: a)
+ (sort humanorder_varpow (graph m)) [] in
+ end_itlist (fun s t -> s^"*"^t) vps;;
+
+let string_of_cmonomial (c,m) =
+ if m = monomial_1 then string_of_num c
+ else if c =/ Int 1 then string_of_monomial m
+ else string_of_num c ^ "*" ^ string_of_monomial m;;
+
+let string_of_poly (p:poly) =
+ if p = poly_0 then "<<0>>" else
+ let cms = sort (fun (m1,_) (m2,_) -> humanorder_monomial m1 m2) (graph p) in
+ let s =
+ List.fold_left (fun a (m,c) ->
+ if c </ Int 0 then a ^ " - " ^ string_of_cmonomial(minus_num c,m)
+ else a ^ " + " ^ string_of_cmonomial(c,m))
+ "" cms in
+ let s1 = String.sub s 0 3
+ and s2 = String.sub s 3 (String.length s - 3) in
+ "<<" ^(if s1 = " + " then s2 else "-"^s2)^">>";;
+
+(* ------------------------------------------------------------------------- *)
+(* Printers. *)
+(* ------------------------------------------------------------------------- *)
+
+let print_vector v = Format.print_string(string_of_vector 0 20 v);;
+
+let print_matrix m = Format.print_string(string_of_matrix 20 m);;
+
+let print_monomial m = Format.print_string(string_of_monomial m);;
+
+let print_poly m = Format.print_string(string_of_poly m);;
+
+(*
+#install_printer print_vector;;
+#install_printer print_matrix;;
+#install_printer print_monomial;;
+#install_printer print_poly;;
+*)
+
+(* ------------------------------------------------------------------------- *)
+(* Conversion from term. *)
+(* ------------------------------------------------------------------------- *)
+
+let rec poly_of_term t = match t with
+ Zero -> poly_0
+| Const n -> poly_const n
+| Var x -> poly_var x
+| Opp t1 -> poly_neg (poly_of_term t1)
+| Inv t1 ->
+ let p = poly_of_term t1 in
+ if poly_isconst p then poly_const(Int 1 // eval undefined p)
+ else failwith "poly_of_term: inverse of non-constant polyomial"
+| Add (l, r) -> poly_add (poly_of_term l) (poly_of_term r)
+| Sub (l, r) -> poly_sub (poly_of_term l) (poly_of_term r)
+| Mul (l, r) -> poly_mul (poly_of_term l) (poly_of_term r)
+| Div (l, r) ->
+ let p = poly_of_term l and q = poly_of_term r in
+ if poly_isconst q then poly_cmul (Int 1 // eval undefined q) p
+ else failwith "poly_of_term: division by non-constant polynomial"
+| Pow (t, n) ->
+ poly_pow (poly_of_term t) n;;
+
+(* ------------------------------------------------------------------------- *)
+(* String of vector (just a list of space-separated numbers). *)
+(* ------------------------------------------------------------------------- *)
+
+let sdpa_of_vector (v:vector) =
+ let n = dim v in
+ let strs = map (o (decimalize 20) (element v)) (1--n) in
+ end_itlist (fun x y -> x ^ " " ^ y) strs ^ "\n";;
+
+(* ------------------------------------------------------------------------- *)
+(* String for block diagonal matrix numbered k. *)
+(* ------------------------------------------------------------------------- *)
+
+let sdpa_of_blockdiagonal k m =
+ let pfx = string_of_int k ^" " in
+ let ents =
+ foldl (fun a (b,i,j) c -> if i > j then a else ((b,i,j),c)::a) [] m in
+ let entss = sort (increasing fst) ents in
+ itlist (fun ((b,i,j),c) a ->
+ pfx ^ string_of_int b ^ " " ^ string_of_int i ^ " " ^ string_of_int j ^
+ " " ^ decimalize 20 c ^ "\n" ^ a) entss "";;
+
+(* ------------------------------------------------------------------------- *)
+(* String for a matrix numbered k, in SDPA sparse format. *)
+(* ------------------------------------------------------------------------- *)
+
+let sdpa_of_matrix k (m:matrix) =
+ let pfx = string_of_int k ^ " 1 " in
+ let ms = foldr (fun (i,j) c a -> if i > j then a else ((i,j),c)::a)
+ (snd m) [] in
+ let mss = sort (increasing fst) ms in
+ itlist (fun ((i,j),c) a ->
+ pfx ^ string_of_int i ^ " " ^ string_of_int j ^
+ " " ^ decimalize 20 c ^ "\n" ^ a) mss "";;
+
+(* ------------------------------------------------------------------------- *)
+(* String in SDPA sparse format for standard SDP problem: *)
+(* *)
+(* X = v_1 * [M_1] + ... + v_m * [M_m] - [M_0] must be PSD *)
+(* Minimize obj_1 * v_1 + ... obj_m * v_m *)
+(* ------------------------------------------------------------------------- *)
+
+let sdpa_of_problem comment obj mats =
+ let m = length mats - 1
+ and n,_ = dimensions (hd mats) in
+ "\"" ^ comment ^ "\"\n" ^
+ string_of_int m ^ "\n" ^
+ "1\n" ^
+ string_of_int n ^ "\n" ^
+ sdpa_of_vector obj ^
+ itlist2 (fun k m a -> sdpa_of_matrix (k - 1) m ^ a)
+ (1--length mats) mats "";;
+
+(* ------------------------------------------------------------------------- *)
+(* More parser basics. *)
+(* ------------------------------------------------------------------------- *)
+
+exception Noparse;;
+
+
+let isspace,issep,isbra,issymb,isalpha,isnum,isalnum =
+ let charcode s = Char.code(String.get s 0) in
+ let spaces = " \t\n\r"
+ and separators = ",;"
+ and brackets = "()[]{}"
+ and symbs = "\\!@#$%^&*-+|\\<=>/?~.:"
+ and alphas = "'abcdefghijklmnopqrstuvwxyz_ABCDEFGHIJKLMNOPQRSTUVWXYZ"
+ and nums = "0123456789" in
+ let allchars = spaces^separators^brackets^symbs^alphas^nums in
+ let csetsize = itlist ((o) max charcode) (explode allchars) 256 in
+ let ctable = Array.make csetsize 0 in
+ do_list (fun c -> Array.set ctable (charcode c) 1) (explode spaces);
+ do_list (fun c -> Array.set ctable (charcode c) 2) (explode separators);
+ do_list (fun c -> Array.set ctable (charcode c) 4) (explode brackets);
+ do_list (fun c -> Array.set ctable (charcode c) 8) (explode symbs);
+ do_list (fun c -> Array.set ctable (charcode c) 16) (explode alphas);
+ do_list (fun c -> Array.set ctable (charcode c) 32) (explode nums);
+ let isspace c = Array.get ctable (charcode c) = 1
+ and issep c = Array.get ctable (charcode c) = 2
+ and isbra c = Array.get ctable (charcode c) = 4
+ and issymb c = Array.get ctable (charcode c) = 8
+ and isalpha c = Array.get ctable (charcode c) = 16
+ and isnum c = Array.get ctable (charcode c) = 32
+ and isalnum c = Array.get ctable (charcode c) >= 16 in
+ isspace,issep,isbra,issymb,isalpha,isnum,isalnum;;
+
+let (||) parser1 parser2 input =
+ try parser1 input
+ with Noparse -> parser2 input;;
+
+let (++) parser1 parser2 input =
+ let result1,rest1 = parser1 input in
+ let result2,rest2 = parser2 rest1 in
+ (result1,result2),rest2;;
+
+let rec many prs input =
+ try let result,next = prs input in
+ let results,rest = many prs next in
+ (result::results),rest
+ with Noparse -> [],input;;
+
+let (>>) prs treatment input =
+ let result,rest = prs input in
+ treatment(result),rest;;
+
+let fix err prs input =
+ try prs input
+ with Noparse -> failwith (err ^ " expected");;
+
+let rec listof prs sep err =
+ prs ++ many (sep ++ fix err prs >> snd) >> (fun (h,t) -> h::t);;
+
+let possibly prs input =
+ try let x,rest = prs input in [x],rest
+ with Noparse -> [],input;;
+
+let some p =
+ function
+ [] -> raise Noparse
+ | (h::t) -> if p h then (h,t) else raise Noparse;;
+
+let a tok = some (fun item -> item = tok);;
+
+let rec atleast n prs i =
+ (if n <= 0 then many prs
+ else prs ++ atleast (n - 1) prs >> (fun (h,t) -> h::t)) i;;
+
+let finished input =
+ if input = [] then 0,input else failwith "Unparsed input";;
+
+let word s =
+ end_itlist (fun p1 p2 -> (p1 ++ p2) >> (fun (s,t) -> s^t))
+ (map a (explode s));;
+
+let token s =
+ many (some isspace) ++ word s ++ many (some isspace)
+ >> (fun ((_,t),_) -> t);;
+
+let decimal =
+ let numeral = some isnum in
+ let decimalint = atleast 1 numeral >> ((o) Num.num_of_string implode) in
+ let decimalfrac = atleast 1 numeral
+ >> (fun s -> Num.num_of_string(implode s) // pow10 (length s)) in
+ let decimalsig =
+ decimalint ++ possibly (a "." ++ decimalfrac >> snd)
+ >> (function (h,[]) -> h | (h,[x]) -> h +/ x | _ -> failwith "decimalsig") in
+ let signed prs =
+ a "-" ++ prs >> ((o) minus_num snd)
+ || a "+" ++ prs >> snd
+ || prs in
+ let exponent = (a "e" || a "E") ++ signed decimalint >> snd in
+ signed decimalsig ++ possibly exponent
+ >> (function (h,[]) -> h | (h,[x]) -> h */ power_num (Int 10) x | _ ->
+ failwith "exponent");;
+
+let mkparser p s =
+ let x,rst = p(explode s) in
+ if rst = [] then x else failwith "mkparser: unparsed input";;
+
+let parse_decimal = mkparser decimal;;
+
+(* ------------------------------------------------------------------------- *)
+(* Parse back a vector. *)
+(* ------------------------------------------------------------------------- *)
+
+let parse_csdpoutput =
+ let rec skipupto dscr prs inp =
+ (dscr ++ prs >> snd
+ || some (fun c -> true) ++ skipupto dscr prs >> snd) inp in
+ let ignore inp = (),[] in
+ let csdpoutput =
+ (decimal ++ many(a " " ++ decimal >> snd) >> (fun (h,t) -> h::t)) ++
+ (a " " ++ a "\n" ++ ignore) >> ((o) vector_of_list fst) in
+ mkparser csdpoutput;;
+
+(* ------------------------------------------------------------------------- *)
+(* CSDP parameters; so far I'm sticking with the defaults. *)
+(* ------------------------------------------------------------------------- *)
+
+let csdp_default_parameters =
+"axtol=1.0e-8
+atytol=1.0e-8
+objtol=1.0e-8
+pinftol=1.0e8
+dinftol=1.0e8
+maxiter=100
+minstepfrac=0.9
+maxstepfrac=0.97
+minstepp=1.0e-8
+minstepd=1.0e-8
+usexzgap=1
+tweakgap=0
+affine=0
+printlevel=1
+";;
+
+let csdp_params = csdp_default_parameters;;
+
+(* ------------------------------------------------------------------------- *)
+(* The same thing with CSDP. *)
+(* Modified by the Coq development team to use a cache *)
+(* ------------------------------------------------------------------------- *)
+
+let buffer_add_line buff line =
+ Buffer.add_string buff line; Buffer.add_char buff '\n'
+
+let string_of_file filename =
+ let fd = open_in filename in
+ let buff = Buffer.create 16 in
+ try while true do buffer_add_line buff (input_line fd) done; failwith ""
+ with End_of_file -> (close_in fd; Buffer.contents buff)
+
+let file_of_string filename s =
+ let fd = Pervasives.open_out filename in
+ output_string fd s; close_out fd
+
+let request_mark = "*** REQUEST ***"
+let answer_mark = "*** ANSWER ***"
+let end_mark = "*** END ***"
+let infeasible_mark = "Infeasible\n"
+let failure_mark = "Failure\n"
+
+let cache_name = "csdp.cache"
+
+let look_in_cache string_problem =
+ let n = String.length string_problem in
+ try
+ let inch = open_in cache_name in
+ let rec search () =
+ while input_line inch <> request_mark do () done;
+ let i = ref 0 in
+ while !i < n & string_problem.[!i] = input_char inch do incr i done;
+ if !i < n or input_line inch <> answer_mark then
+ search ()
+ else begin
+ let buff = Buffer.create 16 in
+ let line = ref (input_line inch) in
+ while (!line <> end_mark) do
+ buffer_add_line buff !line; line := input_line inch
+ done;
+ close_in inch;
+ Buffer.contents buff
+ end in
+ try search () with End_of_file -> close_in inch; raise Not_found
+ with Sys_error _ -> raise Not_found
+
+let flush_to_cache string_problem string_result =
+ try
+ let flags = [Open_append;Open_text;Open_creat] in
+ let outch = open_out_gen flags 0o666 cache_name in
+ begin
+ try
+ Printf.fprintf outch "%s\n" request_mark;
+ Printf.fprintf outch "%s" string_problem;
+ Printf.fprintf outch "%s\n" answer_mark;
+ Printf.fprintf outch "%s" string_result;
+ Printf.fprintf outch "%s\n" end_mark;
+ with Sys_error _ as e -> close_out outch; raise e
+ end;
+ close_out outch
+ with Sys_error _ ->
+ print_endline "Warning: Could not open or write to csdp cache"
+
+exception CsdpInfeasible
+
+let run_csdp dbg string_problem =
+ try
+ let res = look_in_cache string_problem in
+ if res = infeasible_mark then raise CsdpInfeasible;
+ if res = failure_mark then failwith "csdp error";
+ res
+ with Not_found ->
+ let input_file = Filename.temp_file "sos" ".dat-s" in
+ let output_file = Filename.temp_file "sos" ".dat-s" in
+ let temp_path = Filename.dirname input_file in
+ let params_file = Filename.concat temp_path "param.csdp" in
+ file_of_string input_file string_problem;
+ file_of_string params_file csdp_params;
+ let rv = Sys.command("cd "^temp_path^"; csdp "^input_file^" "^output_file^
+ (if dbg then "" else "> /dev/null")) in
+ if rv = 1 or rv = 2 then
+ (flush_to_cache string_problem infeasible_mark; raise CsdpInfeasible);
+ if rv = 127 then
+ (print_string "csdp not found, exiting..."; exit 1);
+ if rv <> 0 & rv <> 3 (* reduced accuracy *) then
+ (flush_to_cache string_problem failure_mark;
+ failwith("csdp: error "^string_of_int rv));
+ let string_result = string_of_file output_file in
+ flush_to_cache string_problem string_result;
+ if not dbg then
+ (Sys.remove input_file; Sys.remove output_file; Sys.remove params_file);
+ string_result
+
+let csdp obj mats =
+ try parse_csdpoutput (run_csdp !debugging (sdpa_of_problem "" obj mats))
+ with CsdpInfeasible -> failwith "csdp: Problem is infeasible"
+
+(* ------------------------------------------------------------------------- *)
+(* Try some apparently sensible scaling first. Note that this is purely to *)
+(* get a cleaner translation to floating-point, and doesn't affect any of *)
+(* the results, in principle. In practice it seems a lot better when there *)
+(* are extreme numbers in the original problem. *)
+(* ------------------------------------------------------------------------- *)
+
+let scale_then =
+ let common_denominator amat acc =
+ foldl (fun a m c -> lcm_num (denominator c) a) acc amat
+ and maximal_element amat acc =
+ foldl (fun maxa m c -> max_num maxa (abs_num c)) acc amat in
+ fun solver obj mats ->
+ let cd1 = itlist common_denominator mats (Int 1)
+ and cd2 = common_denominator (snd obj) (Int 1) in
+ let mats' = map (mapf (fun x -> cd1 */ x)) mats
+ and obj' = vector_cmul cd2 obj in
+ let max1 = itlist maximal_element mats' (Int 0)
+ and max2 = maximal_element (snd obj') (Int 0) in
+ let scal1 = pow2 (20-int_of_float(log(float_of_num max1) /. log 2.0))
+ and scal2 = pow2 (20-int_of_float(log(float_of_num max2) /. log 2.0)) in
+ let mats'' = map (mapf (fun x -> x */ scal1)) mats'
+ and obj'' = vector_cmul scal2 obj' in
+ solver obj'' mats'';;
+
+(* ------------------------------------------------------------------------- *)
+(* Round a vector to "nice" rationals. *)
+(* ------------------------------------------------------------------------- *)
+
+let nice_rational n x = round_num (n */ x) // n;;
+
+let nice_vector n = mapa (nice_rational n);;
+
+(* ------------------------------------------------------------------------- *)
+(* Reduce linear program to SDP (diagonal matrices) and test with CSDP. This *)
+(* one tests A [-1;x1;..;xn] >= 0 (i.e. left column is negated constants). *)
+(* ------------------------------------------------------------------------- *)
+
+let linear_program_basic a =
+ let m,n = dimensions a in
+ let mats = map (fun j -> diagonal (column j a)) (1--n)
+ and obj = vector_const (Int 1) m in
+ try ignore (run_csdp false (sdpa_of_problem "" obj mats)); true
+ with CsdpInfeasible -> false
+
+(* ------------------------------------------------------------------------- *)
+(* Test whether a point is in the convex hull of others. Rather than use *)
+(* computational geometry, express as linear inequalities and call CSDP. *)
+(* This is a bit lazy of me, but it's easy and not such a bottleneck so far. *)
+(* ------------------------------------------------------------------------- *)
+
+let in_convex_hull pts pt =
+ let pts1 = (1::pt) :: map (fun x -> 1::x) pts in
+ let pts2 = map (fun p -> map (fun x -> -x) p @ p) pts1 in
+ let n = length pts + 1
+ and v = 2 * (length pt + 1) in
+ let m = v + n - 1 in
+ let mat =
+ (m,n),
+ itern 1 pts2 (fun pts j -> itern 1 pts (fun x i -> (i,j) |-> Int x))
+ (iter (1,n) (fun i -> (v + i,i+1) |-> Int 1) undefined) in
+ linear_program_basic mat;;
+
+(* ------------------------------------------------------------------------- *)
+(* Filter down a set of points to a minimal set with the same convex hull. *)
+(* ------------------------------------------------------------------------- *)
+
+let minimal_convex_hull =
+ let augment1 = function (m::ms) -> if in_convex_hull ms m then ms else ms@[m]
+ | _ -> failwith "augment1"
+ in
+ let augment m ms = funpow 3 augment1 (m::ms) in
+ fun mons ->
+ let mons' = itlist augment (tl mons) [hd mons] in
+ funpow (length mons') augment1 mons';;
+
+(* ------------------------------------------------------------------------- *)
+(* Stuff for "equations" (generic A->num functions). *)
+(* ------------------------------------------------------------------------- *)
+
+let equation_cmul c eq =
+ if c =/ Int 0 then Empty else mapf (fun d -> c */ d) eq;;
+
+let equation_add eq1 eq2 = combine (+/) (fun x -> x =/ Int 0) eq1 eq2;;
+
+let equation_eval assig eq =
+ let value v = apply assig v in
+ foldl (fun a v c -> a +/ value(v) */ c) (Int 0) eq;;
+
+(* ------------------------------------------------------------------------- *)
+(* Eliminate among linear equations: return unconstrained variables and *)
+(* assignments for the others in terms of them. We give one pseudo-variable *)
+(* "one" that's used for a constant term. *)
+(* ------------------------------------------------------------------------- *)
+
+
+let eliminate_equations =
+ let rec extract_first p l =
+ match l with
+ [] -> failwith "extract_first"
+ | h::t -> if p(h) then h,t else
+ let k,s = extract_first p t in
+ k,h::s in
+ let rec eliminate vars dun eqs =
+ match vars with
+ [] -> if forall is_undefined eqs then dun
+ else (raise Unsolvable)
+ | v::vs ->
+ try let eq,oeqs = extract_first (fun e -> defined e v) eqs in
+ let a = apply eq v in
+ let eq' = equation_cmul (Int(-1) // a) (undefine v eq) in
+ let elim e =
+ let b = tryapplyd e v (Int 0) in
+ if b =/ Int 0 then e else
+ equation_add e (equation_cmul (minus_num b // a) eq) in
+ eliminate vs ((v |-> eq') (mapf elim dun)) (map elim oeqs)
+ with Failure _ -> eliminate vs dun eqs in
+ fun one vars eqs ->
+ let assig = eliminate vars undefined eqs in
+ let vs = foldl (fun a x f -> subtract (dom f) [one] @ a) [] assig in
+ setify vs,assig;;
+
+(* ------------------------------------------------------------------------- *)
+(* Eliminate all variables, in an essentially arbitrary order. *)
+(* ------------------------------------------------------------------------- *)
+
+let eliminate_all_equations one =
+ let choose_variable eq =
+ let (v,_) = choose eq in
+ if v = one then
+ let eq' = undefine v eq in
+ if is_undefined eq' then failwith "choose_variable" else
+ let (w,_) = choose eq' in w
+ else v in
+ let rec eliminate dun eqs =
+ match eqs with
+ [] -> dun
+ | eq::oeqs ->
+ if is_undefined eq then eliminate dun oeqs else
+ let v = choose_variable eq in
+ let a = apply eq v in
+ let eq' = equation_cmul (Int(-1) // a) (undefine v eq) in
+ let elim e =
+ let b = tryapplyd e v (Int 0) in
+ if b =/ Int 0 then e else
+ equation_add e (equation_cmul (minus_num b // a) eq) in
+ eliminate ((v |-> eq') (mapf elim dun)) (map elim oeqs) in
+ fun eqs ->
+ let assig = eliminate undefined eqs in
+ let vs = foldl (fun a x f -> subtract (dom f) [one] @ a) [] assig in
+ setify vs,assig;;
+
+(* ------------------------------------------------------------------------- *)
+(* Solve equations by assigning arbitrary numbers. *)
+(* ------------------------------------------------------------------------- *)
+
+let solve_equations one eqs =
+ let vars,assigs = eliminate_all_equations one eqs in
+ let vfn = itlist (fun v -> (v |-> Int 0)) vars (one |=> Int(-1)) in
+ let ass =
+ combine (+/) (fun c -> false) (mapf (equation_eval vfn) assigs) vfn in
+ if forall (fun e -> equation_eval ass e =/ Int 0) eqs
+ then undefine one ass else raise Sanity;;
+
+(* ------------------------------------------------------------------------- *)
+(* Hence produce the "relevant" monomials: those whose squares lie in the *)
+(* Newton polytope of the monomials in the input. (This is enough according *)
+(* to Reznik: "Extremal PSD forms with few terms", Duke Math. Journal, *)
+(* vol 45, pp. 363--374, 1978. *)
+(* *)
+(* These are ordered in sort of decreasing degree. In particular the *)
+(* constant monomial is last; this gives an order in diagonalization of the *)
+(* quadratic form that will tend to display constants. *)
+(* ------------------------------------------------------------------------- *)
+
+let newton_polytope pol =
+ let vars = poly_variables pol in
+ let mons = map (fun m -> map (fun x -> monomial_degree x m) vars) (dom pol)
+ and ds = map (fun x -> (degree x pol + 1) / 2) vars in
+ let all = itlist (fun n -> allpairs (fun h t -> h::t) (0--n)) ds [[]]
+ and mons' = minimal_convex_hull mons in
+ let all' =
+ filter (fun m -> in_convex_hull mons' (map (fun x -> 2 * x) m)) all in
+ map (fun m -> itlist2 (fun v i a -> if i = 0 then a else (v |-> i) a)
+ vars m monomial_1) (rev all');;
+
+(* ------------------------------------------------------------------------- *)
+(* Diagonalize (Cholesky/LDU) the matrix corresponding to a quadratic form. *)
+(* ------------------------------------------------------------------------- *)
+
+let diag m =
+ let nn = dimensions m in
+ let n = fst nn in
+ if snd nn <> n then failwith "diagonalize: non-square matrix" else
+ let rec diagonalize i m =
+ if is_zero m then [] else
+ let a11 = element m (i,i) in
+ if a11 </ Int 0 then failwith "diagonalize: not PSD"
+ else if a11 =/ Int 0 then
+ if is_zero(row i m) then diagonalize (i + 1) m
+ else failwith "diagonalize: not PSD"
+ else
+ let v = row i m in
+ let v' = mapa (fun a1k -> a1k // a11) v in
+ let m' =
+ (n,n),
+ iter (i+1,n) (fun j ->
+ iter (i+1,n) (fun k ->
+ ((j,k) |--> (element m (j,k) -/ element v j */ element v' k))))
+ undefined in
+ (a11,v')::diagonalize (i + 1) m' in
+ diagonalize 1 m;;
+
+(* ------------------------------------------------------------------------- *)
+(* Adjust a diagonalization to collect rationals at the start. *)
+(* ------------------------------------------------------------------------- *)
+
+let deration d =
+ if d = [] then Int 0,d else
+ let adj(c,l) =
+ let a = foldl (fun a i c -> lcm_num a (denominator c)) (Int 1) (snd l) //
+ foldl (fun a i c -> gcd_num a (numerator c)) (Int 0) (snd l) in
+ (c // (a */ a)),mapa (fun x -> a */ x) l in
+ let d' = map adj d in
+ let a = itlist ((o) lcm_num ((o) denominator fst)) d' (Int 1) //
+ itlist ((o) gcd_num ((o) numerator fst)) d' (Int 0) in
+ (Int 1 // a),map (fun (c,l) -> (a */ c,l)) d';;
+
+(* ------------------------------------------------------------------------- *)
+(* Enumeration of monomials with given multidegree bound. *)
+(* ------------------------------------------------------------------------- *)
+
+let rec enumerate_monomials d vars =
+ if d < 0 then []
+ else if d = 0 then [undefined]
+ else if vars = [] then [monomial_1] else
+ let alts =
+ map (fun k -> let oths = enumerate_monomials (d - k) (tl vars) in
+ map (fun ks -> if k = 0 then ks else (hd vars |-> k) ks) oths)
+ (0--d) in
+ end_itlist (@) alts;;
+
+(* ------------------------------------------------------------------------- *)
+(* Enumerate products of distinct input polys with degree <= d. *)
+(* We ignore any constant input polynomials. *)
+(* Give the output polynomial and a record of how it was derived. *)
+(* ------------------------------------------------------------------------- *)
+
+let rec enumerate_products d pols =
+ if d = 0 then [poly_const num_1,Rational_lt num_1] else if d < 0 then [] else
+ match pols with
+ [] -> [poly_const num_1,Rational_lt num_1]
+ | (p,b)::ps -> let e = multidegree p in
+ if e = 0 then enumerate_products d ps else
+ enumerate_products d ps @
+ map (fun (q,c) -> poly_mul p q,Product(b,c))
+ (enumerate_products (d - e) ps);;
+
+(* ------------------------------------------------------------------------- *)
+(* Multiply equation-parametrized poly by regular poly and add accumulator. *)
+(* ------------------------------------------------------------------------- *)
+
+let epoly_pmul p q acc =
+ foldl (fun a m1 c ->
+ foldl (fun b m2 e ->
+ let m = monomial_mul m1 m2 in
+ let es = tryapplyd b m undefined in
+ (m |-> equation_add (equation_cmul c e) es) b)
+ a q) acc p;;
+
+(* ------------------------------------------------------------------------- *)
+(* Usual operations on equation-parametrized poly. *)
+(* ------------------------------------------------------------------------- *)
+
+let epoly_cmul c l =
+ if c =/ Int 0 then undefined else mapf (equation_cmul c) l;;
+
+let epoly_neg x = epoly_cmul (Int(-1)) x;;
+
+let epoly_add x = combine equation_add is_undefined x;;
+
+let epoly_sub p q = epoly_add p (epoly_neg q);;
+
+(* ------------------------------------------------------------------------- *)
+(* Convert regular polynomial. Note that we treat (0,0,0) as -1. *)
+(* ------------------------------------------------------------------------- *)
+
+let epoly_of_poly p =
+ foldl (fun a m c -> (m |-> ((0,0,0) |=> minus_num c)) a) undefined p;;
+
+(* ------------------------------------------------------------------------- *)
+(* String for block diagonal matrix numbered k. *)
+(* ------------------------------------------------------------------------- *)
+
+let sdpa_of_blockdiagonal k m =
+ let pfx = string_of_int k ^" " in
+ let ents =
+ foldl (fun a (b,i,j) c -> if i > j then a else ((b,i,j),c)::a) [] m in
+ let entss = sort (increasing fst) ents in
+ itlist (fun ((b,i,j),c) a ->
+ pfx ^ string_of_int b ^ " " ^ string_of_int i ^ " " ^ string_of_int j ^
+ " " ^ decimalize 20 c ^ "\n" ^ a) entss "";;
+
+(* ------------------------------------------------------------------------- *)
+(* SDPA for problem using block diagonal (i.e. multiple SDPs) *)
+(* ------------------------------------------------------------------------- *)
+
+let sdpa_of_blockproblem comment nblocks blocksizes obj mats =
+ let m = length mats - 1 in
+ "\"" ^ comment ^ "\"\n" ^
+ string_of_int m ^ "\n" ^
+ string_of_int nblocks ^ "\n" ^
+ (end_itlist (fun s t -> s^" "^t) (map string_of_int blocksizes)) ^
+ "\n" ^
+ sdpa_of_vector obj ^
+ itlist2 (fun k m a -> sdpa_of_blockdiagonal (k - 1) m ^ a)
+ (1--length mats) mats "";;
+
+(* ------------------------------------------------------------------------- *)
+(* Hence run CSDP on a problem in block diagonal form. *)
+(* ------------------------------------------------------------------------- *)
+
+let csdp_blocks nblocks blocksizes obj mats =
+ let string_problem = sdpa_of_blockproblem "" nblocks blocksizes obj mats in
+ try parse_csdpoutput (run_csdp !debugging string_problem)
+ with CsdpInfeasible -> failwith "csdp: Problem is infeasible"
+
+(* ------------------------------------------------------------------------- *)
+(* 3D versions of matrix operations to consider blocks separately. *)
+(* ------------------------------------------------------------------------- *)
+
+let bmatrix_add = combine (+/) (fun x -> x =/ Int 0);;
+
+let bmatrix_cmul c bm =
+ if c =/ Int 0 then undefined
+ else mapf (fun x -> c */ x) bm;;
+
+let bmatrix_neg = bmatrix_cmul (Int(-1));;
+
+let bmatrix_sub m1 m2 = bmatrix_add m1 (bmatrix_neg m2);;
+
+(* ------------------------------------------------------------------------- *)
+(* Smash a block matrix into components. *)
+(* ------------------------------------------------------------------------- *)
+
+let blocks blocksizes bm =
+ map (fun (bs,b0) ->
+ let m = foldl
+ (fun a (b,i,j) c -> if b = b0 then ((i,j) |-> c) a else a)
+ undefined bm in
+ (*let d = foldl (fun a (i,j) c -> max a (max i j)) 0 m in*)
+ (((bs,bs),m):matrix))
+ (zip blocksizes (1--length blocksizes));;
+
+(* ------------------------------------------------------------------------- *)
+(* Positiv- and Nullstellensatz. Flag "linf" forces a linear representation. *)
+(* ------------------------------------------------------------------------- *)
+
+let real_positivnullstellensatz_general linf d eqs leqs pol
+ : poly list * (positivstellensatz * (num * poly) list) list =
+
+ let vars = itlist ((o) union poly_variables) (pol::eqs @ map fst leqs) [] in
+ let monoid =
+ if linf then
+ (poly_const num_1,Rational_lt num_1)::
+ (filter (fun (p,c) -> multidegree p <= d) leqs)
+ else enumerate_products d leqs in
+ let nblocks = length monoid in
+ let mk_idmultiplier k p =
+ let e = d - multidegree p in
+ let mons = enumerate_monomials e vars in
+ let nons = zip mons (1--length mons) in
+ mons,
+ itlist (fun (m,n) -> (m |-> ((-k,-n,n) |=> Int 1))) nons undefined in
+ let mk_sqmultiplier k (p,c) =
+ let e = (d - multidegree p) / 2 in
+ let mons = enumerate_monomials e vars in
+ let nons = zip mons (1--length mons) in
+ mons,
+ itlist (fun (m1,n1) ->
+ itlist (fun (m2,n2) a ->
+ let m = monomial_mul m1 m2 in
+ if n1 > n2 then a else
+ let c = if n1 = n2 then Int 1 else Int 2 in
+ let e = tryapplyd a m undefined in
+ (m |-> equation_add ((k,n1,n2) |=> c) e) a)
+ nons)
+ nons undefined in
+ let sqmonlist,sqs = unzip(map2 mk_sqmultiplier (1--length monoid) monoid)
+ and idmonlist,ids = unzip(map2 mk_idmultiplier (1--length eqs) eqs) in
+ let blocksizes = map length sqmonlist in
+ let bigsum =
+ itlist2 (fun p q a -> epoly_pmul p q a) eqs ids
+ (itlist2 (fun (p,c) s a -> epoly_pmul p s a) monoid sqs
+ (epoly_of_poly(poly_neg pol))) in
+ let eqns = foldl (fun a m e -> e::a) [] bigsum in
+ let pvs,assig = eliminate_all_equations (0,0,0) eqns in
+ let qvars = (0,0,0)::pvs in
+ let allassig = itlist (fun v -> (v |-> (v |=> Int 1))) pvs assig in
+ let mk_matrix v =
+ foldl (fun m (b,i,j) ass -> if b < 0 then m else
+ let c = tryapplyd ass v (Int 0) in
+ if c =/ Int 0 then m else
+ ((b,j,i) |-> c) (((b,i,j) |-> c) m))
+ undefined allassig in
+ let diagents = foldl
+ (fun a (b,i,j) e -> if b > 0 & i = j then equation_add e a else a)
+ undefined allassig in
+ let mats = map mk_matrix qvars
+ and obj = length pvs,
+ itern 1 pvs (fun v i -> (i |--> tryapplyd diagents v (Int 0)))
+ undefined in
+ let raw_vec = if pvs = [] then vector_0 0
+ else scale_then (csdp_blocks nblocks blocksizes) obj mats in
+ let find_rounding d =
+ (if !debugging then
+ (Format.print_string("Trying rounding with limit "^string_of_num d);
+ Format.print_newline())
+ else ());
+ let vec = nice_vector d raw_vec in
+ let blockmat = iter (1,dim vec)
+ (fun i a -> bmatrix_add (bmatrix_cmul (element vec i) (el i mats)) a)
+ (bmatrix_neg (el 0 mats)) in
+ let allmats = blocks blocksizes blockmat in
+ vec,map diag allmats in
+ let vec,ratdias =
+ if pvs = [] then find_rounding num_1
+ else tryfind find_rounding (map Num.num_of_int (1--31) @
+ map pow2 (5--66)) in
+ let newassigs =
+ itlist (fun k -> el (k - 1) pvs |-> element vec k)
+ (1--dim vec) ((0,0,0) |=> Int(-1)) in
+ let finalassigs =
+ foldl (fun a v e -> (v |-> equation_eval newassigs e) a) newassigs
+ allassig in
+ let poly_of_epoly p =
+ foldl (fun a v e -> (v |--> equation_eval finalassigs e) a)
+ undefined p in
+ let mk_sos mons =
+ let mk_sq (c,m) =
+ c,itlist (fun k a -> (el (k - 1) mons |--> element m k) a)
+ (1--length mons) undefined in
+ map mk_sq in
+ let sqs = map2 mk_sos sqmonlist ratdias
+ and cfs = map poly_of_epoly ids in
+ let msq = filter (fun (a,b) -> b <> []) (map2 (fun a b -> a,b) monoid sqs) in
+ let eval_sq sqs = itlist
+ (fun (c,q) -> poly_add (poly_cmul c (poly_mul q q))) sqs poly_0 in
+ let sanity =
+ itlist (fun ((p,c),s) -> poly_add (poly_mul p (eval_sq s))) msq
+ (itlist2 (fun p q -> poly_add (poly_mul p q)) cfs eqs
+ (poly_neg pol)) in
+ if not(is_undefined sanity) then raise Sanity else
+ cfs,map (fun (a,b) -> snd a,b) msq;;
+
+
+let term_of_monoid l1 m = itlist (fun i m -> Mul (nth l1 i,m)) m (Const num_1)
+
+let rec term_of_pos l1 x = match x with
+ Axiom_eq i -> failwith "term_of_pos"
+ | Axiom_le i -> nth l1 i
+ | Axiom_lt i -> nth l1 i
+ | Monoid m -> term_of_monoid l1 m
+ | Rational_eq n -> Const n
+ | Rational_le n -> Const n
+ | Rational_lt n -> Const n
+ | Square t -> Pow (t, 2)
+ | Eqmul (t, y) -> Mul (t, term_of_pos l1 y)
+ | Sum (y, z) -> Add (term_of_pos l1 y, term_of_pos l1 z)
+ | Product (y, z) -> Mul (term_of_pos l1 y, term_of_pos l1 z);;
+
+
+let dest_monomial mon = sort (increasing fst) (graph mon);;
+
+let monomial_order =
+ let rec lexorder l1 l2 =
+ match (l1,l2) with
+ [],[] -> true
+ | vps,[] -> false
+ | [],vps -> true
+ | ((x1,n1)::vs1),((x2,n2)::vs2) ->
+ if x1 < x2 then true
+ else if x2 < x1 then false
+ else if n1 < n2 then false
+ else if n2 < n1 then true
+ else lexorder vs1 vs2 in
+ fun m1 m2 ->
+ if m2 = monomial_1 then true else if m1 = monomial_1 then false else
+ let mon1 = dest_monomial m1 and mon2 = dest_monomial m2 in
+ let deg1 = itlist ((o) (+) snd) mon1 0
+ and deg2 = itlist ((o) (+) snd) mon2 0 in
+ if deg1 < deg2 then false else if deg1 > deg2 then true
+ else lexorder mon1 mon2;;
+
+let dest_poly p =
+ map (fun (m,c) -> c,dest_monomial m)
+ (sort (fun (m1,_) (m2,_) -> monomial_order m1 m2) (graph p));;
+
+(* ------------------------------------------------------------------------- *)
+(* Map back polynomials and their composites to term. *)
+(* ------------------------------------------------------------------------- *)
+
+let term_of_varpow =
+ fun x k ->
+ if k = 1 then Var x else Pow (Var x, k);;
+
+let term_of_monomial =
+ fun m -> if m = monomial_1 then Const num_1 else
+ let m' = dest_monomial m in
+ let vps = itlist (fun (x,k) a -> term_of_varpow x k :: a) m' [] in
+ end_itlist (fun s t -> Mul (s,t)) vps;;
+
+let term_of_cmonomial =
+ fun (m,c) ->
+ if m = monomial_1 then Const c
+ else if c =/ num_1 then term_of_monomial m
+ else Mul (Const c,term_of_monomial m);;
+
+let term_of_poly =
+ fun p ->
+ if p = poly_0 then Zero else
+ let cms = map term_of_cmonomial
+ (sort (fun (m1,_) (m2,_) -> monomial_order m1 m2) (graph p)) in
+ end_itlist (fun t1 t2 -> Add (t1,t2)) cms;;
+
+let term_of_sqterm (c,p) =
+ Product(Rational_lt c,Square(term_of_poly p));;
+
+let term_of_sos (pr,sqs) =
+ if sqs = [] then pr
+ else Product(pr,end_itlist (fun a b -> Sum(a,b)) (map term_of_sqterm sqs));;
+
+let rec deepen f n =
+ try (*print_string "Searching with depth limit ";
+ print_int n; print_newline();*) f n
+ with Failure _ -> deepen f (n + 1);;
+
+
+
+
+
+exception TooDeep
+
+let deepen_until limit f n =
+ match compare limit 0 with
+ | 0 -> raise TooDeep
+ | -1 -> deepen f n
+ | _ ->
+ let rec d_until f n =
+ try if !debugging
+ then (print_string "Searching with depth limit ";
+ print_int n; print_newline()) ; f n
+ with Failure x ->
+ if !debugging then (Printf.printf "solver error : %s\n" x) ;
+ if n = limit then raise TooDeep else d_until f (n + 1) in
+ d_until f n
+
+
+(* patch to remove zero polynomials from equalities.
+ In this case, hol light loops *)
+
+let real_nonlinear_prover depthmax eqs les lts =
+ let eq = map poly_of_term eqs
+ and le = map poly_of_term les
+ and lt = map poly_of_term lts in
+ let pol = itlist poly_mul lt (poly_const num_1)
+ and lep = map (fun (t,i) -> t,Axiom_le i) (zip le (0--(length le - 1)))
+ and ltp = map (fun (t,i) -> t,Axiom_lt i) (zip lt (0--(length lt - 1)))
+ and eqp = itlist2 (fun t i res ->
+ if t = undefined then res else (t,Axiom_eq i)::res) eq (0--(length eq - 1)) []
+ in
+
+ let proof =
+ let leq = lep @ ltp in
+ let eq = List.map fst eqp in
+ let tryall d =
+ let e = multidegree pol (*and pol' = poly_neg pol*) in
+ let k = if e = 0 then 1 else d / e in
+ tryfind (fun i -> d,i,
+ real_positivnullstellensatz_general false d eq leq
+ (poly_neg(poly_pow pol i)))
+ (0--k) in
+ let d,i,(cert_ideal,cert_cone) = deepen_until depthmax tryall 0 in
+ let proofs_ideal =
+ map2 (fun q i -> Eqmul(term_of_poly q,i))
+ cert_ideal (List.map snd eqp)
+ and proofs_cone = map term_of_sos cert_cone
+ and proof_ne =
+ if lt = [] then Rational_lt num_1 else
+ let p = end_itlist (fun s t -> Product(s,t)) (map snd ltp) in
+ funpow i (fun q -> Product(p,q)) (Rational_lt num_1) in
+ end_itlist (fun s t -> Sum(s,t)) (proof_ne :: proofs_ideal @ proofs_cone) in
+ if !debugging then (print_string("Translating proof certificate to Coq"); print_newline());
+ proof;;
+
+
+(* ------------------------------------------------------------------------- *)
+(* Now pure SOS stuff. *)
+(* ------------------------------------------------------------------------- *)
+
+(* ------------------------------------------------------------------------- *)
+(* Some combinatorial helper functions. *)
+(* ------------------------------------------------------------------------- *)
+
+let rec allpermutations l =
+ if l = [] then [[]] else
+ itlist (fun h acc -> map (fun t -> h::t)
+ (allpermutations (subtract l [h])) @ acc) l [];;
+
+let allvarorders l =
+ map (fun vlis x -> index x vlis) (allpermutations l);;
+
+let changevariables_monomial zoln (m:monomial) =
+ foldl (fun a x k -> (assoc x zoln |-> k) a) monomial_1 m;;
+
+let changevariables zoln pol =
+ foldl (fun a m c -> (changevariables_monomial zoln m |-> c) a)
+ poly_0 pol;;
+
+(* ------------------------------------------------------------------------- *)
+(* Sum-of-squares function with some lowbrow symmetry reductions. *)
+(* ------------------------------------------------------------------------- *)
+
+let sumofsquares_general_symmetry tool pol =
+ let vars = poly_variables pol
+ and lpps = newton_polytope pol in
+ let n = length lpps in
+ let sym_eqs =
+ let invariants = filter
+ (fun vars' ->
+ is_undefined(poly_sub pol (changevariables (zip vars vars') pol)))
+ (allpermutations vars) in
+(* let lpps2 = allpairs monomial_mul lpps lpps in*)
+(* let lpp2_classes =
+ setify(map (fun m ->
+ setify(map (fun vars' -> changevariables_monomial (zip vars vars') m)
+ invariants)) lpps2) in *)
+ let lpns = zip lpps (1--length lpps) in
+ let lppcs =
+ filter (fun (m,(n1,n2)) -> n1 <= n2)
+ (allpairs
+ (fun (m1,n1) (m2,n2) -> (m1,m2),(n1,n2)) lpns lpns) in
+ let clppcs = end_itlist (@)
+ (map (fun ((m1,m2),(n1,n2)) ->
+ map (fun vars' ->
+ (changevariables_monomial (zip vars vars') m1,
+ changevariables_monomial (zip vars vars') m2),(n1,n2))
+ invariants)
+ lppcs) in
+ let clppcs_dom = setify(map fst clppcs) in
+ let clppcs_cls = map (fun d -> filter (fun (e,_) -> e = d) clppcs)
+ clppcs_dom in
+ let eqvcls = map (o setify (map snd)) clppcs_cls in
+ let mk_eq cls acc =
+ match cls with
+ [] -> raise Sanity
+ | [h] -> acc
+ | h::t -> map (fun k -> (k |-> Int(-1)) (h |=> Int 1)) t @ acc in
+ itlist mk_eq eqvcls [] in
+ let eqs = foldl (fun a x y -> y::a) []
+ (itern 1 lpps (fun m1 n1 ->
+ itern 1 lpps (fun m2 n2 f ->
+ let m = monomial_mul m1 m2 in
+ if n1 > n2 then f else
+ let c = if n1 = n2 then Int 1 else Int 2 in
+ (m |-> ((n1,n2) |-> c) (tryapplyd f m undefined)) f))
+ (foldl (fun a m c -> (m |-> ((0,0)|=>c)) a)
+ undefined pol)) @
+ sym_eqs in
+ let pvs,assig = eliminate_all_equations (0,0) eqs in
+ let allassig = itlist (fun v -> (v |-> (v |=> Int 1))) pvs assig in
+ let qvars = (0,0)::pvs in
+ let diagents =
+ end_itlist equation_add (map (fun i -> apply allassig (i,i)) (1--n)) in
+ let mk_matrix v =
+ ((n,n),
+ foldl (fun m (i,j) ass -> let c = tryapplyd ass v (Int 0) in
+ if c =/ Int 0 then m else
+ ((j,i) |-> c) (((i,j) |-> c) m))
+ undefined allassig :matrix) in
+ let mats = map mk_matrix qvars
+ and obj = length pvs,
+ itern 1 pvs (fun v i -> (i |--> tryapplyd diagents v (Int 0)))
+ undefined in
+ let raw_vec = if pvs = [] then vector_0 0 else tool obj mats in
+ let find_rounding d =
+ (if !debugging then
+ (Format.print_string("Trying rounding with limit "^string_of_num d);
+ Format.print_newline())
+ else ());
+ let vec = nice_vector d raw_vec in
+ let mat = iter (1,dim vec)
+ (fun i a -> matrix_add (matrix_cmul (element vec i) (el i mats)) a)
+ (matrix_neg (el 0 mats)) in
+ deration(diag mat) in
+ let rat,dia =
+ if pvs = [] then
+ let mat = matrix_neg (el 0 mats) in
+ deration(diag mat)
+ else
+ tryfind find_rounding (map Num.num_of_int (1--31) @
+ map pow2 (5--66)) in
+ let poly_of_lin(d,v) =
+ d,foldl(fun a i c -> (el (i - 1) lpps |-> c) a) undefined (snd v) in
+ let lins = map poly_of_lin dia in
+ let sqs = map (fun (d,l) -> poly_mul (poly_const d) (poly_pow l 2)) lins in
+ let sos = poly_cmul rat (end_itlist poly_add sqs) in
+ if is_undefined(poly_sub sos pol) then rat,lins else raise Sanity;;
+
+let (sumofsquares: poly -> Num.num * (( Num.num * poly) list)) =
+sumofsquares_general_symmetry csdp;;
diff --git a/contrib/micromega/sos.mli b/contrib/micromega/sos.mli
new file mode 100644
index 000000000..31c9518c8
--- /dev/null
+++ b/contrib/micromega/sos.mli
@@ -0,0 +1,66 @@
+(************************************************************************)
+(* 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 *)
+(************************************************************************)
+
+
+type vname = string;;
+
+type term =
+| Zero
+| Const of Num.num
+| Var of vname
+| Inv of term
+| Opp of term
+| Add of (term * term)
+| Sub of (term * term)
+| Mul of (term * term)
+| Div of (term * term)
+| Pow of (term * int)
+
+type positivstellensatz =
+ Axiom_eq of int
+ | Axiom_le of int
+ | Axiom_lt of int
+ | Rational_eq of Num.num
+ | Rational_le of Num.num
+ | Rational_lt of Num.num
+ | Square of term
+ | Monoid of int list
+ | Eqmul of term * positivstellensatz
+ | Sum of positivstellensatz * positivstellensatz
+ | Product of positivstellensatz * positivstellensatz
+
+type poly
+
+val poly_isconst : poly -> bool
+
+val poly_neg : poly -> poly
+
+val poly_mul : poly -> poly -> poly
+
+val poly_pow : poly -> int -> poly
+
+val poly_const : Num.num -> poly
+
+val poly_of_term : term -> poly
+
+val term_of_poly : poly -> term
+
+val term_of_sos : positivstellensatz * (Num.num * poly) list ->
+ positivstellensatz
+
+val string_of_poly : poly -> string
+
+exception TooDeep
+
+val deepen_until : int -> (int -> 'a) -> int -> 'a
+
+val real_positivnullstellensatz_general : bool -> int -> poly list ->
+ (poly * positivstellensatz) list ->
+ poly -> poly list * (positivstellensatz * (Num.num * poly) list) list
+
+val sumofsquares : poly -> Num.num * ( Num.num * poly) list
diff --git a/contrib/micromega/vector.ml b/contrib/micromega/vector.ml
new file mode 100644
index 000000000..fee4ebfc1
--- /dev/null
+++ b/contrib/micromega/vector.ml
@@ -0,0 +1,674 @@
+(************************************************************************)
+(* 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 *)
+(************************************************************************)
+(* *)
+(* Micromega: A reflexive tactic using the Positivstellensatz *)
+(* *)
+(* Frédéric Besson (Irisa/Inria) 2006-2008 *)
+(* *)
+(************************************************************************)
+
+open Num
+
+module type S =
+sig
+ type t
+
+ val fresh : t -> int
+
+ val null : t
+
+ val is_null : t -> bool
+
+ val get : int -> t -> num
+
+ val update : int -> (num -> num) -> t -> t
+ (* behaviour is undef if index < 0 -- might loop*)
+
+ val set : int -> num -> t -> t
+
+ (*
+ For efficiency...
+
+ val get_update : int -> (num -> num) -> t -> num * t
+ *)
+
+ val mul : num -> t -> t
+
+ val uminus : t -> t
+
+ val add : t -> t -> t
+
+ val dotp : t -> t -> num
+
+ val lin_comb : num -> t -> num -> t -> t
+ (* lin_comb n1 t1 n2 t2 = (n1 * t1) + (n2 * t2) *)
+
+ val gcd : t -> Big_int.big_int
+
+ val normalise : t -> num * t
+
+ val hash : t -> int
+
+ val compare : t -> t -> int
+
+ type it
+
+ val iterator : t -> it
+ val element : it -> (num*it) option
+
+ val string : t -> string
+
+ type status = Pos | Neg
+
+ (* the result list is ordered by fst *)
+ val status : t -> (int * status) list
+
+ val from_list : num list -> t
+ val to_list : t -> num list
+
+end
+
+
+module type SystemS =
+sig
+
+ module Vect : S
+
+ module Cstr :
+ sig
+ type kind = Eq | Ge
+ val string_of_kind : kind -> string
+ type cstr = {coeffs : Vect.t ; op : kind ; cst : num}
+ val string_of_cstr : cstr -> string
+ val compare : cstr -> cstr -> int
+ end
+ open Cstr
+
+
+ module CstrBag :
+ sig
+ type t
+ exception Contradiction
+
+ val empty : t
+
+ val is_empty : t -> bool
+
+ val add : cstr -> t -> t
+ (* c can be deduced from add c t *)
+
+ val find : (cstr -> bool) -> t -> cstr option
+
+ val fold : (cstr -> 'a -> 'a) -> t -> 'a -> 'a
+
+ val status : t -> (int * (int list * int list)) list
+ (* aggregate of vector statuses *)
+
+ val remove : cstr -> t -> t
+
+ (* remove_list the ith element -- it is the ith element visited by 'fold' *)
+
+ val split : (cstr -> int) -> t -> (int -> t)
+
+ type it
+ val iterator : t -> it
+ val element : it -> (cstr*it) option
+
+ end
+
+end
+
+let zero_num = Int 0
+let unit_num = Int 1
+
+
+
+
+module Cstr(V:S) =
+struct
+ type kind = Eq | Ge
+ let string_of_kind = function Eq -> "Eq" | Ge -> "Ge"
+
+ type cstr = {coeffs : V.t ; op : kind ; cst : num}
+
+ let string_of_cstr {coeffs =a ; op = b ; cst =c} =
+ Printf.sprintf "{coeffs = %s;op=%s;cst=%s}" (V.string a) (string_of_kind b) (string_of_num c)
+
+ type t = cstr
+ let compare
+ {coeffs = v1 ; op = op1 ; cst = c1}
+ {coeffs = v2 ; op = op2 ; cst = c2} =
+ Mutils.Cmp.compare_lexical [
+ (fun () -> V.compare v1 v2);
+ (fun () -> Pervasives.compare op1 op2);
+ (fun () -> compare_num c1 c2)
+ ]
+
+
+end
+
+
+
+module VList : S with type t = num list =
+struct
+ type t = num list
+
+ let fresh l = failwith "not implemented"
+
+ let null = []
+
+ let is_null = List.for_all ((=/) zero_num)
+
+ let normalise l = failwith "Not implemented"
+ (*match l with (* Buggy : What if the first num is zero! *)
+ | [] -> (Int 0,[])
+ | [n] -> (n,[Int 1])
+ | n::l -> (n, (Int 1) :: List.map (fun x -> x // n) l)
+ *)
+
+
+ let get i l = try List.nth l i with _ -> zero_num
+
+ (* This is not tail-recursive *)
+ let rec update i f t =
+ match t with
+ | [] -> if i = 0 then [f zero_num] else (zero_num)::(update (i-1) f [])
+ | e::t -> if i = 0 then (f e)::t else e::(update (i-1) f t)
+
+ let rec set i n t =
+ match t with
+ | [] -> if i = 0 then [n] else (zero_num)::(set (i-1) n [])
+ | e::t -> if i = 0 then (n)::t else e::(set (i-1) n t)
+
+
+
+
+ let rec mul z t =
+ match z with
+ | Int 0 -> null
+ | Int 1 -> t
+ | _ -> List.map (mult_num z) t
+
+ let uminus t = mul (Int (-1)) t
+
+ let rec add t1 t2 =
+ match t1,t2 with
+ | [], _ -> t2
+ | _ , [] -> t1
+ | e1::t1,e2::t2 -> (e1 +/ e2 )::(add t1 t2)
+
+ let dotp t1 t2 =
+ let rec _dotp t1 t2 acc =
+ match t1, t2 with
+ | [] , _ -> acc
+ | _ , [] -> acc
+ | e1::t1,e2::t2 -> _dotp t1 t2 (acc +/ (e1 */ e2)) in
+ _dotp t1 t2 zero_num
+
+ let add_mul n t1 t2 =
+ match n with
+ | Int 0 -> t2
+ | Int 1 -> add t1 t2
+ | _ ->
+ let rec _add_mul t1 t2 =
+ match t1,t2 with
+ | [], _ -> t2
+ | _ , [] -> mul n t1
+ | e1::t1,e2::t2 -> ( (n */e1) +/ e2 )::(_add_mul t1 t2) in
+ _add_mul t1 t2
+
+ let lin_comb n1 t1 n2 t2 =
+ match n1,n2 with
+ | Int 0 , _ -> mul n2 t2
+ | Int 1 , _ -> add_mul n2 t2 t1
+ | _ , Int 0 -> mul n1 t1
+ | _ , Int 1 -> add_mul n1 t1 t2
+ | _ ->
+ let rec _lin_comb t1 t2 =
+ match t1,t2 with
+ | [], _ -> mul n2 t2
+ | _ , [] -> mul n1 t1
+ | e1::t1,e2::t2 -> ( (n1 */e1) +/ (n2 */ e2 ))::(_lin_comb t1 t2) in
+ _lin_comb t1 t2
+
+ (* could be computed on the fly *)
+ let gcd t =Mutils.gcd_list t
+
+
+
+
+ let hash = Mutils.Cmp.hash_list int_of_num
+
+ let compare = Mutils.Cmp.compare_list compare_num
+
+ type it = t
+ let iterator (x:t) : it = x
+ let element it =
+ match it with
+ | [] -> None
+ | e::l -> Some (e,l)
+
+ (* TODO: Buffer! *)
+ let string l = List.fold_right (fun n s -> (string_of_num n)^";"^s) l ""
+
+ type status = Pos | Neg
+
+ let status l =
+ let rec xstatus i l =
+ match l with
+ | [] -> []
+ | e::l ->
+ begin
+ match compare_num e (Int 0) with
+ | 1 -> (i,Pos):: (xstatus (i+1) l)
+ | 0 -> xstatus (i+1) l
+ | -1 -> (i,Neg) :: (xstatus (i+1) l)
+ | _ -> assert false
+ end in
+ xstatus 0 l
+
+ let from_list l = l
+ let to_list l = l
+
+end
+
+module VMap : S =
+struct
+ module Map = Map.Make(struct type t = int let compare (x:int) (y:int) = Pervasives.compare x y end)
+
+ type t = num Map.t
+
+ let null = Map.empty
+
+ let fresh m = failwith "not implemented"
+
+ let is_null = Map.is_empty
+
+ let normalise m = failwith "Not implemented"
+
+
+
+ let get i l = try Map.find i l with _ -> zero_num
+
+ let update i f t =
+ try
+ let res = f (Map.find i t) in
+ if res =/ zero_num
+ then Map.remove i t
+ else Map.add i res t
+ with
+ Not_found ->
+ let res = f zero_num in
+ if res =/ zero_num then t else Map.add i res t
+
+ let set i n t =
+ if n =/ zero_num then Map.remove i t
+ else Map.add i n t
+
+
+ let rec mul z t =
+ match z with
+ | Int 0 -> null
+ | Int 1 -> t
+ | _ -> Map.map (mult_num z) t
+
+ let uminus t = mul (Int (-1)) t
+
+
+ let map2 f m1 m2 =
+ let res,m2' =
+ Map.fold (fun k e (res,m2) ->
+ let v = f e (get k m2) in
+ if v =/ zero_num
+ then (res,Map.remove k m2)
+ else (Map.add k v res,Map.remove k m2)) m1 (Map.empty,m2) in
+ Map.fold (fun k e res ->
+ let v = f zero_num e in
+ if v =/ zero_num
+ then res else Map.add k v res) m2' res
+
+ let add t1 t2 = map2 (+/) t1 t2
+
+
+ let dotp t1 t2 =
+ Map.fold (fun k e res ->
+ res +/ (e */ get k t2)) t1 zero_num
+
+
+
+ let add_mul n t1 t2 =
+ match n with
+ | Int 0 -> t2
+ | Int 1 -> add t1 t2
+ | _ -> map2 (fun x y -> (n */ x) +/ y) t1 t2
+
+ let lin_comb n1 t1 n2 t2 =
+ match n1,n2 with
+ | Int 0 , _ -> mul n2 t2
+ | Int 1 , _ -> add_mul n2 t2 t1
+ | _ , Int 0 -> mul n1 t1
+ | _ , Int 1 -> add_mul n1 t1 t2
+ | _ -> map2 (fun x y -> (n1 */ x) +/ (n2 */ y)) t1 t2
+
+
+ let hash map = Map.fold (fun k e res -> k lxor (int_of_num e) lxor res) map 0
+
+ let compare = Map.compare compare_num
+
+ type it = t * int
+
+ let iterator (x:t) : it = (x,0)
+
+ let element (mp,id) =
+ try
+ Some (Map.find id mp, (mp, id+1))
+ with
+ Not_found -> None
+
+ (* TODO: Buffer! *)
+ type status = Pos | Neg
+
+ let status l = Map.fold (fun k e l ->
+ match compare_num e (Int 0) with
+ | 1 -> (k,Pos)::l
+ | 0 -> l
+ | -1 -> (k,Neg) :: l
+ | _ -> assert false) l []
+ let from_list l =
+ let rec from_list i l map =
+ match l with
+ | [] -> map
+ | e::l -> from_list (i+1) l (if e <>/ Int 0 then Map.add i e map else map) in
+ from_list 0 l Map.empty
+
+ let gcd m =
+ let res = Map.fold (fun _ e x -> Big_int.gcd_big_int x (Mutils.numerator e)) m Big_int.zero_big_int in
+ if Big_int.compare_big_int res Big_int.zero_big_int = 0
+ then Big_int.unit_big_int else res
+
+
+ let to_list m =
+ let l = List.rev (Map.fold (fun k e l -> (k,e)::l) m []) in
+ let rec xto_list i l =
+ match l with
+ | [] -> []
+ | (x,v)::l' -> if i = x then v::(xto_list (i+1) l') else zero_num ::(xto_list (i+1) l) in
+ xto_list 0 l
+
+ let string l = VList.string (to_list l)
+
+
+end
+
+
+module VSparse : S =
+struct
+
+ type t = (int*num) list
+
+ let null = []
+
+ let fresh l = List.fold_left (fun acc (i,_) -> max (i+1) acc) 0 l
+
+ let is_null l = l = []
+
+ let rec is_sorted l =
+ match l with
+ | [] -> true
+ | [e] -> true
+ | (i,_)::(j,x)::l -> i < j && is_sorted ((j,x)::l)
+
+
+ let check l = (List.for_all (fun (_,n) -> compare_num n (Int 0) <> 0) l) && (is_sorted l)
+
+ (* let get i t =
+ assert (check t);
+ try List.assoc i t with Not_found -> zero_num *)
+
+ let rec get (i:int) t =
+ match t with
+ | [] -> zero_num
+ | (j,n)::t ->
+ match compare i j with
+ | 0 -> n
+ | 1 -> get i t
+ | _ -> zero_num
+
+ let cons i v rst = if v =/ Int 0 then rst else (i,v)::rst
+
+ let rec update i f t =
+ match t with
+ | [] -> cons i (f zero_num) []
+ | (k,v)::l ->
+ match Pervasives.compare i k with
+ | 0 -> cons k (f v) l
+ | -1 -> cons i (f zero_num) t
+ | 1 -> (k,v) ::(update i f l)
+ | _ -> failwith "compare_num"
+
+ let update i f t =
+ assert (check t);
+ let res = update i f t in
+ assert (check t) ; res
+
+
+ let rec set i n t =
+ match t with
+ | [] -> cons i n []
+ | (k,v)::l ->
+ match Pervasives.compare i k with
+ | 0 -> cons k n l
+ | -1 -> cons i n t
+ | 1 -> (k,v) :: (set i n l)
+ | _ -> failwith "compare_num"
+
+
+ let rec map f l =
+ match l with
+ | [] -> []
+ | (i,e)::l -> cons i (f e) (map f l)
+
+ let rec mul z t =
+ match z with
+ | Int 0 -> null
+ | Int 1 -> t
+ | _ -> List.map (fun (i,n) -> (i, mult_num z n)) t
+
+ let mul z t =
+ assert (check t) ;
+ let res = mul z t in
+ assert (check res) ;
+ res
+
+ let uminus t = mul (Int (-1)) t
+
+
+ let normalise l =
+ match l with
+ | [] -> (Int 0,[])
+ | (i,n)::_ -> (n, mul ((Int 1) // n) l)
+
+
+ let rec map2 f m1 m2 =
+ match m1, m2 with
+ | [] , [] -> []
+ | l , [] -> map (fun x -> f x zero_num) l
+ | [] ,l -> map (f zero_num) l
+ | (i,e)::l1,(i',e')::l2 ->
+ match Pervasives.compare i i' with
+ | 0 -> cons i (f e e') (map2 f l1 l2)
+ | -1 -> cons i (f e zero_num) (map2 f l1 m2)
+ | 1 -> cons i' (f zero_num e') (map2 f m1 l2)
+ | _ -> assert false
+
+ (* let add t1 t2 = map2 (+/) t1 t2*)
+
+ let rec add (m1:t) (m2:t) =
+ match m1, m2 with
+ | [] , [] -> []
+ | l , [] -> l
+ | [] ,l -> l
+ | (i,e)::l1,(i',e')::l2 ->
+ match Pervasives.compare i i' with
+ | 0 -> cons i ( e +/ e') (add l1 l2)
+ | -1 -> (i,e) :: (add l1 m2)
+ | 1 -> (i', e') :: (add m1 l2)
+ | _ -> assert false
+
+
+
+
+ let add t1 t2 =
+ assert (check t1 && check t2);
+ let res = add t1 t2 in
+ assert (check res);
+ res
+
+
+ let rec dotp (t1:t) (t2:t) =
+ match t1, t2 with
+ | [] , _ -> zero_num
+ | _ , [] -> zero_num
+ | (i,e)::l1 , (i',e')::l2 ->
+ match Pervasives.compare i i' with
+ | 0 -> (e */ e') +/ (dotp l1 l2)
+ | -1 -> dotp l1 t2
+ | 1 -> dotp t1 l2
+ | _ -> assert false
+
+ let dotp t1 t2 =
+ assert (check t1 && check t2) ; dotp t1 t2
+
+ let add_mul n t1 t2 =
+ match n with
+ | Int 0 -> t2
+ | Int 1 -> add t1 t2
+ | _ -> map2 (fun x y -> (n */ x) +/ y) t1 t2
+
+ let add_mul n (t1:t) (t2:t) =
+ match n with
+ | Int 0 -> t2
+ | Int 1 -> add t1 t2
+ | _ ->
+ let rec xadd_mul m1 m2 =
+ match m1, m2 with
+ | [] , [] -> []
+ | _ , [] -> mul n m1
+ | [] , _ -> m2
+ | (i,e)::l1,(i',e')::l2 ->
+ match Pervasives.compare i i' with
+ | 0 -> cons i ( n */ e +/ e') (xadd_mul l1 l2)
+ | -1 -> (i,n */ e) :: (xadd_mul l1 m2)
+ | 1 -> (i', e') :: (xadd_mul m1 l2)
+ | _ -> assert false in
+ xadd_mul t1 t2
+
+
+
+
+ let lin_comb n1 t1 n2 t2 =
+ match n1,n2 with
+ | Int 0 , _ -> mul n2 t2
+ | Int 1 , _ -> add_mul n2 t2 t1
+ | _ , Int 0 -> mul n1 t1
+ | _ , Int 1 -> add_mul n1 t1 t2
+ | _ -> (*map2 (fun x y -> (n1 */ x) +/ (n2 */ y)) t1 t2*)
+ let rec xlin_comb m1 m2 =
+ match m1, m2 with
+ | [] , [] -> []
+ | _ , [] -> mul n1 m1
+ | [] , _ -> mul n2 m2
+ | (i,e)::l1,(i',e')::l2 ->
+ match Pervasives.compare i i' with
+ | 0 -> cons i ( n1 */ e +/ n2 */ e') (xlin_comb l1 l2)
+ | -1 -> (i,n1 */ e) :: (xlin_comb l1 m2)
+ | 1 -> (i', n2 */ e') :: (xlin_comb m1 l2)
+ | _ -> assert false in
+ xlin_comb t1 t2
+
+
+
+
+
+ let lin_comb n1 t1 n2 t2 =
+ assert (check t1 && check t2);
+ let res = lin_comb n1 t1 n2 t2 in
+ assert (check res); res
+
+ let hash = Mutils.Cmp.hash_list (fun (x,y) -> (Hashtbl.hash x) lxor (int_of_num y))
+
+
+ let compare = Mutils.Cmp.compare_list (fun x y -> Mutils.Cmp.compare_lexical
+ [
+ (fun () -> Pervasives.compare (fst x) (fst y));
+ (fun () -> compare_num (snd x) (snd y))])
+
+ (*
+ let compare (x:t) (y:t) =
+ let rec xcompare acc1 acc2 x y =
+ match x , y with
+ | [] , [] -> xcomp acc1 acc2
+ | [] , _ -> -1
+ | _ , [] -> 1
+ | (i,n1)::l1 , (j,n2)::l2 ->
+ match Pervasives.compare i j with
+ | 0 -> xcompare (n1::acc1) (n2::acc2) l1 l2
+ | c -> c
+ and xcomp acc1 acc2 = Mutils.Cmp.compare_list compare_num acc1 acc2 in
+ xcompare [] [] x y
+ *)
+
+ type it = t
+
+ let iterator (x:t) : it = x
+
+ let element l = failwith "Not_implemented"
+
+ (* TODO: Buffer! *)
+ type status = Pos | Neg
+
+ let status l = List.map (fun (i,e) ->
+ match compare_num e (Int 0) with
+ | 1 -> i,Pos
+ | -1 -> i,Neg
+ | _ -> assert false) l
+
+ let from_list (l: num list) =
+ let rec xfrom_list i l =
+ match l with
+ | [] -> []
+ | e::l ->
+ if e <>/ Int 0
+ then (i,e)::(xfrom_list (i+1) l)
+ else xfrom_list (i+1) l in
+
+ let res = xfrom_list 0 l in
+ assert (check res) ; res
+
+
+ let gcd m =
+ let res = List.fold_left (fun x (i,e) -> Big_int.gcd_big_int x (Mutils.numerator e)) Big_int.zero_big_int m in
+ if Big_int.compare_big_int res Big_int.zero_big_int = 0
+ then Big_int.unit_big_int else res
+
+ let to_list m =
+ let rec xto_list i l =
+ match l with
+ | [] -> []
+ | (x,v)::l' ->
+ if i = x then v::(xto_list (i+1) l') else zero_num ::(xto_list (i+1) l) in
+ xto_list 0 m
+
+ let to_list l =
+ assert (check l);
+ to_list l
+
+
+ let string l = VList.string (to_list l)
+
+end
diff --git a/scripts/coqmktop.ml b/scripts/coqmktop.ml
index bd155abfe..f6d4709fe 100644
--- a/scripts/coqmktop.ml
+++ b/scripts/coqmktop.ml
@@ -17,7 +17,7 @@ open Unix
(* Objects to link *)
(* 1. Core objects *)
-let ocamlobjs = ["unix.cma"]
+let ocamlobjs = ["unix.cma";"nums.cma"]
let dynobjs = ["dynlink.cma"]
let camlp4objs = ["gramlib.cma"]
let libobjs = ocamlobjs @ camlp4objs
diff --git a/test-suite/check b/test-suite/check
index 4c98f8ac2..f2307d405 100755
--- a/test-suite/check
+++ b/test-suite/check
@@ -248,6 +248,8 @@ echo "Parser tests"
test_parser parser
echo "Interactive tests"
test_interactive interactive
+echo "Micromega tests"
+test_success micromega
echo "Complexity tests"
test_complexity complexity
echo "Module tests"
diff --git a/test-suite/csdp.cache b/test-suite/csdp.cache
new file mode 100644
index 000000000..10196f5c4
--- /dev/null
+++ b/test-suite/csdp.cache
@@ -0,0 +1,13100 @@
+*** REQUEST ***
+""
+1
+3
+1 1 1
+0.10485760000000000000e7
+0 1 1 1 0.10485760000000000000e7
+1 2 1 1 0.10485760000000000000e7
+1 3 1 1 0.10485760000000000000e7
+*** ANSWER ***
+Infeasible
+*** END ***
+*** REQUEST ***
+""
+5
+5
+4 1 1 1 1
+0.52428800000000000000e6 0.10485760000000000000e7 -0.52428800000000000000e6 -0.15728640000000000000e7 0.10485760000000000000e7
+0 1 2 2 -0.52428800000000000000e6
+0 1 4 4 0.52428800000000000000e6
+0 2 1 1 -0.52428800000000000000e6
+0 4 1 1 0.52428800000000000000e6
+1 1 1 1 0.52428800000000000000e6
+1 1 2 2 0.52428800000000000000e6
+1 1 4 4 -0.52428800000000000000e6
+1 2 1 1 0.52428800000000000000e6
+1 4 1 1 -0.52428800000000000000e6
+2 1 1 2 0.52428800000000000000e6
+2 1 1 4 -0.52428800000000000000e6
+2 4 1 1 0.10485760000000000000e7
+3 1 1 3 0.52428800000000000000e6
+3 1 1 4 -0.26214400000000000000e6
+3 1 4 4 -0.52428800000000000000e6
+4 1 2 2 -0.52428800000000000000e6
+4 1 2 4 0.52428800000000000000e6
+4 1 4 4 -0.52428800000000000000e6
+4 5 1 1 -0.52428800000000000000e6
+5 3 1 1 0.52428800000000000000e6
+5 4 1 1 0.52428800000000000000e6
+*** ANSWER ***
+Failure
+*** END ***
+*** REQUEST ***
+""
+20
+7
+4 1 4 1 4 1 1
+0.52428800000000000000e6 0.10485760000000000000e7 -0.52428800000000000000e6 0.0 -0.10485760000000000000e7 0.52428800000000000000e6 0.0 0.0 0.10485760000000000000e7 0.10485760000000000000e7 0.10485760000000000000e7 0.52428800000000000000e6 0.0 0.0 0.0 0.0 0.0 0.0 0.10485760000000000000e7 0.52428800000000000000e6
+0 1 1 2 -0.26214400000000000000e6
+0 1 1 4 0.26214400000000000000e6
+0 2 1 1 -0.52428800000000000000e6
+0 3 1 2 -0.26214400000000000000e6
+0 5 1 4 0.26214400000000000000e6
+1 1 2 3 0.52428800000000000000e6
+1 1 3 4 -0.52428800000000000000e6
+1 5 1 4 0.26214400000000000000e6
+1 5 4 4 0.52428800000000000000e6
+2 1 1 1 0.52428800000000000000e6
+2 1 1 2 0.26214400000000000000e6
+2 1 1 4 -0.26214400000000000000e6
+2 2 1 1 0.52428800000000000000e6
+2 3 1 2 0.26214400000000000000e6
+2 5 1 4 -0.26214400000000000000e6
+3 1 1 3 0.52428800000000000000e6
+3 1 1 4 -0.26214400000000000000e6
+3 1 4 4 -0.52428800000000000000e6
+4 1 2 2 0.52428800000000000000e6
+4 1 4 4 -0.52428800000000000000e6
+4 3 1 2 -0.26214400000000000000e6
+4 5 1 4 0.26214400000000000000e6
+5 1 2 4 0.52428800000000000000e6
+5 1 4 4 -0.10485760000000000000e7
+5 5 1 4 0.52428800000000000000e6
+6 1 1 2 -0.26214400000000000000e6
+6 1 1 4 0.26214400000000000000e6
+6 3 1 1 0.52428800000000000000e6
+7 1 2 3 -0.52428800000000000000e6
+7 1 3 4 0.52428800000000000000e6
+7 3 1 3 0.52428800000000000000e6
+8 3 1 4 0.52428800000000000000e6
+8 5 1 4 0.52428800000000000000e6
+9 1 1 2 0.26214400000000000000e6
+9 1 1 4 -0.26214400000000000000e6
+9 1 2 3 -0.52428800000000000000e6
+9 1 3 4 0.52428800000000000000e6
+9 3 1 2 0.26214400000000000000e6
+9 3 2 2 0.52428800000000000000e6
+9 4 1 1 0.52428800000000000000e6
+10 3 3 3 0.52428800000000000000e6
+10 5 3 3 0.52428800000000000000e6
+11 3 4 4 0.52428800000000000000e6
+11 5 4 4 0.52428800000000000000e6
+12 1 1 2 0.26214400000000000000e6
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+19 6 1 1 0.52428800000000000000e6
+20 3 1 2 0.26214400000000000000e6
+20 5 1 4 0.26214400000000000000e6
+20 7 1 1 0.52428800000000000000e6
+*** ANSWER ***
+3.216324555313973399e-09 3.115755179085740881e-10 -3.324145964397584252e-13 5.224535444587893355e-10 -5.221424158161548355e-10 1.077350271294886586e+00 3.852719564190705860e-09 3.943375637517087529e-01 1.443375660692466089e-01 8.791397631563578409e-10 1.443375650484708950e-01 7.735027285744282277e-02 -1.056624343830985385e-01 -1.053741115158292411e-09 1.443375677592817763e-01 1.439267380244287369e-09 1.443375649633401314e-01 1.426488358509380043e-09 2.331632031590821230e-09 9.881736230754907151e-10
+1 1 1 1 1.633559363251345159e-04
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+2 5 4 4 4.226776984244937641e-01
+2 6 1 1 4.226367928629287118e-01
+2 7 1 1 1.000000471613313069e+00
+*** END ***
+*** REQUEST ***
+""
+5
+5
+4 1 1 1 1
+0.15728640000000000000e7 0.10485760000000000000e7 -0.52428800000000000000e6 -0.15728640000000000000e7 0.10485760000000000000e7
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+4 5 1 1 -0.52428800000000000000e6
+5 3 1 1 0.52428800000000000000e6
+5 4 1 1 0.52428800000000000000e6
+*** ANSWER ***
+Failure
+*** END ***
+*** REQUEST ***
+""
+20
+7
+4 1 4 1 4 1 1
+0.52428800000000000000e6 0.10485760000000000000e7 -0.52428800000000000000e6 0.0 -0.10485760000000000000e7 0.52428800000000000000e6 0.0 0.0 0.0 0.10485760000000000000e7 0.10485760000000000000e7 0.52428800000000000000e6 0.0 0.0 0.10485760000000000000e7 0.0 0.0 0.0 0.10485760000000000000e7 0.52428800000000000000e6
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+19 6 1 1 0.52428800000000000000e6
+20 3 1 2 0.26214400000000000000e6
+20 5 1 4 0.26214400000000000000e6
+20 7 1 1 0.52428800000000000000e6
+*** ANSWER ***
+-2.383868949298717141e-09 1.339230337975028206e-10 -1.554096679939493489e-13 2.248394946691001431e-10 -2.248297835432341409e-10 7.735027041131968539e-02 -3.465380614782425474e-10 -1.056624335885569255e-01 1.443375668010194179e-01 4.146098741655366720e-10 1.443375684724064745e-01 1.077350269708958752e+00 3.943375673759518252e-01 1.282342148333882059e-09 1.443375675560557292e-01 4.652808533053771065e-10 1.443375665686819631e-01 4.783976336776815147e-10 2.716615800832226185e-10 4.224315968550843865e-10
+1 1 1 1 7.021443402755951407e-05
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+2 6 1 1 1.577342158716849241e+00
+2 7 1 1 9.999952624970135373e-01
+*** END ***
+*** REQUEST ***
+""
+2
+4
+1 1 1 1
+0.15728640000000000000e7 0.10485760000000000000e7
+0 2 1 1 -0.10485760000000000000e7
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+1 4 1 1 0.10485760000000000000e7
+2 2 1 1 0.10485760000000000000e7
+2 3 1 1 0.10485760000000000000e7
+*** ANSWER ***
+1.481747076424031107e-35 9.158740256512039830e-35
+1 1 1 1 1.553724422408404842e-29
+1 2 1 1 1.048576000000000000e+06
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+2 2 1 1 2.162052517325988483e-12
+2 3 1 1 9.999999979853768650e-01
+2 4 1 1 2.162052536023870663e-12
+*** END ***
+*** REQUEST ***
+""
+1
+3
+1 1 1
+0.10485760000000000000e7
+0 1 1 1 0.10485760000000000000e7
+1 1 1 1 0.10485760000000000000e7
+1 2 1 1 -0.10485760000000000000e7
+1 3 1 1 0.10485760000000000000e7
+*** ANSWER ***
+Infeasible
+*** END ***
+*** REQUEST ***
+""
+5
+5
+4 1 1 1 1
+0.10485760000000000000e7 0.0 0.52428800000000000000e6 0.52428800000000000000e6 0.0
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+5 1 3 4 0.52428800000000000000e6
+5 1 4 4 -0.52428800000000000000e6
+5 5 1 1 0.52428800000000000000e6
+*** ANSWER ***
+3.195988761043818697e-15 2.434891739362865648e-17 1.131563150190301381e-17 1.866162834975372311e-17 -4.329478161627238877e-20
+1 1 1 1 1.675706186858708813e-09
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+1 1 2 2 8.763130856697872339e-14
+1 1 3 3 6.020281117436706900e-12
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+1 1 4 4 1.103302430310100170e-13
+1 2 1 1 5.242880000000016298e+05
+1 3 1 1 9.871699112802661190e-12
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+1 5 1 1 6.493237410294658411e-14
+2 1 1 1 2.000000000000000444e+00
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+2 2 1 1 5.422933505993135752e-16
+2 3 1 1 5.440902829587604117e+01
+2 4 1 1 5.340902829587604828e+01
+2 5 1 1 2.858830790951610652e+03
+*** END ***
+*** REQUEST ***
+""
+4
+2
+3 1
+-0.52428800000000000000e6 0.52428800000000000000e6 0.10485760000000000000e7 0.0
+0 1 2 2 0.10485760000000000000e7
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+3 2 1 1 0.10485760000000000000e7
+4 1 1 2 -0.10485760000000000000e7
+4 1 1 3 0.10485760000000000000e7
+*** ANSWER ***
+Failure
+*** END ***
+*** REQUEST ***
+""
+22
+2
+6 3
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+21 1 1 2 0.52428800000000000000e6
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+22 1 1 6 0.52428800000000000000e6
+22 1 3 6 -0.52428800000000000000e6
+22 1 5 6 0.52428800000000000000e6
+22 2 2 3 0.52428800000000000000e6
+*** ANSWER ***
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+*** REQUEST ***
+""
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+*** REQUEST ***
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+*** REQUEST ***
+""
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+*** ANSWER ***
+Failure
+*** END ***
+*** REQUEST ***
+""
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+*** REQUEST ***
+""
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+*** ANSWER ***
+Failure
+*** END ***
+*** REQUEST ***
+""
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+*** END ***
+*** REQUEST ***
+""
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+*** ANSWER ***
+Infeasible
+*** END ***
+*** REQUEST ***
+""
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+12 10 1 1 0.52428800000000000000e6
+13 9 1 1 0.52428800000000000000e6
+13 11 1 1 0.52428800000000000000e6
+*** ANSWER ***
+1.402333343794533736e-09 -1.552307416644435207e-23 1.604266199290855281e-09 1.343624328966796008e-09 -6.063371200683696295e-24 1.170820376268658425e+00 1.856810394853954153e-09 3.064822680874087664e-09 1.447213615551183663e+00 3.064822680874067398e-09 3.064822680874108757e-09 5.527780141197477439e-09 3.064822680874105862e-09
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+2 10 1 1 7.917802692350457416e-01
+2 11 1 1 1.000000000000000444e+00
+*** END ***
+*** REQUEST ***
+""
+1
+5
+1 1 1 1 1
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+1 1 1 1 0.10485760000000000000e7
+1 2 1 1 0.10485760000000000000e7
+1 5 1 1 0.10485760000000000000e7
+*** ANSWER ***
+1.000000005357981392e+00
+1 1 1 1 5.618250780273614797e-03
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+1 3 1 1 8.122239242601498837e-11
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+2 4 1 1 1.146570973783460818e+07
+2 5 1 1 1.236459417730440554e-16
+*** END ***
+*** REQUEST ***
+""
+10
+8
+5 1 1 1 1 1 1 1
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+10 1 3 4 0.26214400000000000000e6
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+10 7 1 1 0.52428800000000000000e6
+*** ANSWER ***
+-1.000000001260698657e+00 1.272997840951505179e-09 -6.071624507107061208e-12 -8.373302285347945428e-12 -2.976157420808034464e-14 -2.694196303457895409e-14 5.028154957630146827e-15 -1.020526200179788434e-14 1.222103204680506058e-11 5.571813919773229819e-14
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+2 7 1 1 3.310306905083069751e+02
+2 8 1 1 3.356160917931259746e+02
+*** END ***
+*** REQUEST ***
+""
+10
+8
+5 1 1 1 1 1 1 1
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+10 1 3 4 0.26214400000000000000e6
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+10 7 1 1 0.52428800000000000000e6
+*** ANSWER ***
+1.000000001261144300e+00 1.273327614775283784e-09 -6.072847097869460078e-12 -8.374957741020938299e-12 -2.982374356006742620e-14 -2.692657132842880265e-14 5.016088016331483008e-15 -1.025952210899453636e-14 1.222344619435513675e-11 5.568582828826051022e-14
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+2 7 1 1 3.310334702982946737e+02
+2 8 1 1 3.356188109513807376e+02
+*** END ***
+*** REQUEST ***
+""
+13
+11
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+12 10 1 1 0.52428800000000000000e6
+13 9 1 1 0.52428800000000000000e6
+13 11 1 1 0.52428800000000000000e6
+*** ANSWER ***
+1.402060991199688664e-09 -2.483883181138067524e-24 1.604502441709942906e-09 1.343505189877832168e-09 -4.530249279552842563e-24 1.170820376345740321e+00 1.856673489285015473e-09 3.064714009346125858e-09 1.447213615473736281e+00 3.064714009346132476e-09 3.064714009346106420e-09 5.528147447133072850e-09 3.064714009346072092e-09
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+2 10 1 1 7.917777905066630817e-01
+2 11 1 1 1.000000000000000444e+00
+*** END ***
+*** REQUEST ***
+""
+10
+8
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+10 1 3 4 -0.26214400000000000000e6
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+10 7 1 1 0.52428800000000000000e6
+*** ANSWER ***
+-1.000000001255223259e+00 1.267468020168873510e-09 -6.044810283915911358e-12 -8.336279969166351627e-12 -2.966149009571424797e-14 -2.678696495272935225e-14 -4.981786850069985191e-15 -1.021178361537661619e-14 1.216694793804843487e-11 5.539710179627446917e-14
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+2 7 1 1 3.309312209905145323e+02
+2 8 1 1 3.355487258131824433e+02
+*** END ***
+*** REQUEST ***
+""
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+7 1 2 4 0.52428800000000000000e6
+7 4 1 1 -0.10485760000000000000e7
+7 5 1 1 -0.10485760000000000000e7
+8 1 3 3 -0.52428800000000000000e6
+8 1 5 5 0.52428800000000000000e6
+9 1 1 4 -0.26214400000000000000e6
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+9 1 4 4 -0.52428800000000000000e6
+9 2 1 1 -0.52428800000000000000e6
+9 3 1 1 0.52428800000000000000e6
+10 1 3 4 -0.26214400000000000000e6
+10 6 1 1 0.52428800000000000000e6
+10 7 1 1 0.52428800000000000000e6
+*** ANSWER ***
+1.000000001255168192e+00 1.267293763581419573e-09 -6.044105779378328542e-12 -8.335319418073156169e-12 -2.963781190224006830e-14 -2.679132962591727316e-14 -4.985908279021202008e-15 -1.019181010909363138e-14 1.216554667291012868e-11 5.540629928084288642e-14
+1 1 1 1 6.645169428966527696e-04
+1 1 1 2 -3.168852130858705115e-06
+1 1 1 3 -4.370107947062738902e-06
+1 1 1 4 -3.189125067023352774e-06
+1 1 2 2 7.449142695091282828e-08
+1 1 2 3 -1.404637262691291531e-08
+1 1 2 4 -2.614051879791467958e-09
+1 1 3 3 1.570682662591977723e-07
+1 1 3 4 -1.452442891867727762e-08
+1 1 4 4 7.473300759368737514e-08
+1 1 5 5 8.468673233903780460e-08
+1 2 1 1 5.242880006581288762e+05
+1 3 1 1 6.468280310124216867e-06
+1 4 1 1 6.432962541554478388e-06
+1 5 1 1 9.525827983709759290e-08
+1 6 1 1 8.859294928040332590e-06
+1 7 1 1 1.190790339148690542e-07
+1 8 1 1 1.181229213313399491e-07
+2 1 1 1 9.999999982950379396e-01
+2 1 1 2 2.000547863553214967e+01
+2 1 1 3 1.446887684143138486e+01
+2 1 1 4 2.032657820528472570e+01
+2 1 2 2 9.554827697322816675e+02
+2 1 2 3 3.350478457791923574e+02
+2 1 2 4 4.304603230330404244e+02
+2 1 3 3 4.779913848659312521e+02
+2 1 3 4 3.448991980958476233e+02
+2 1 4 4 9.554827697345083379e+02
+2 1 5 5 4.779913848655970696e+02
+2 2 1 1 1.173653297509985344e-10
+2 3 1 1 2.032657820476820731e+01
+2 4 1 1 2.050547863553248362e+01
+2 5 1 1 4.109548443971845586e+02
+2 6 1 1 1.496887684146408048e+01
+2 7 1 1 3.309303212543836707e+02
+2 8 1 1 3.355478457795697977e+02
+*** END ***
+*** REQUEST ***
+""
+1
+4
+1 1 1 1
+0.10485760000000000000e7
+0 1 1 1 0.52428800000000000000e6
+1 1 1 1 0.10485760000000000000e7
+1 2 1 1 0.52428800000000000000e6
+1 3 1 1 0.52428800000000000000e6
+*** ANSWER ***
+5.000000014551048144e-01
+1 1 1 1 1.544610222759798968e-03
+1 2 1 1 2.621440007802954642e+05
+1 3 1 1 2.621440007802954642e+05
+1 4 1 1 1.740143951048130479e-05
+2 1 1 1 9.999999999999995559e-01
+2 2 1 1 5.202230325434514910e-16
+2 3 1 1 5.202230325434514910e-16
+2 4 1 1 4.287886701374642229e+01
+*** END ***
+*** REQUEST ***
+""
+1
+4
+1 1 1 1
+0.15728640000000000000e7
+0 1 1 1 0.10485760000000000000e7
+1 1 1 1 0.10485760000000000000e7
+1 2 1 1 0.52428800000000000000e6
+1 3 1 1 0.10485760000000000000e7
+1 4 1 1 -0.10485760000000000000e7
+*** ANSWER ***
+Infeasible
+*** END ***
+*** REQUEST ***
+""
+7
+7
+3 1 1 1 1 1 1
+0.10485760000000000000e7 0.52428800000000000000e6 0.26214400000000000000e6 0.10485760000000000000e7 0.13107200000000000000e7 0.52428800000000000000e6 0.52428800000000000000e6
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+7 1 1 2 0.52428800000000000000e6
+7 1 2 3 -0.52428800000000000000e6
+7 7 1 1 0.10485760000000000000e7
+*** ANSWER ***
+1.000000000705873582e+00 -2.045960686369595755e-10 7.924832100231817025e-10 3.022200793306051867e-10 1.000000000541151124e+00 2.587418279244411078e-10 9.999999995005985909e-01
+1 1 1 1 5.674380923163807470e-04
+1 1 1 2 -4.666699145296535718e-04
+1 1 1 3 -2.145345272670685238e-04
+1 1 2 2 8.309788744332677769e-04
+1 1 2 3 5.797144830422352796e-05
+1 1 3 3 3.169007219041686642e-04
+1 2 1 1 1.048576000597716076e+06
+1 3 1 1 1.048576000567438081e+06
+1 4 1 1 5.242880005285314983e+05
+1 5 1 1 2.713104709576987590e-04
+1 6 1 1 1.048576000324672787e+06
+1 7 1 1 1.048575999476339668e+06
+2 1 1 1 2.749999998446686966e+00
+2 1 1 2 1.499999998968473358e+00
+2 1 1 3 1.749999998806539114e+00
+2 1 2 2 1.249999999370027703e+00
+2 1 2 3 9.999999992261674464e-01
+2 1 3 3 1.999999998740054519e+00
+2 2 1 1 3.238682098779669515e-10
+2 3 1 1 3.598521421425696570e-10
+2 4 1 1 6.483565585491693585e-10
+2 5 1 1 1.999999998968473802e+00
+2 6 1 1 2.877200339276443111e-10
+2 7 1 1 2.576940442501448964e-10
+*** END ***
+*** REQUEST ***
+""
+1
+4
+1 1 1 1
+-0.10485760000000000000e7
+0 1 1 1 0.52428800000000000000e6
+1 1 1 1 0.52428800000000000000e6
+1 2 1 1 -0.52428800000000000000e6
+1 3 1 1 0.52428800000000000000e6
+1 4 1 1 -0.10485760000000000000e7
+*** ANSWER ***
+Infeasible
+*** END ***
+*** REQUEST ***
+""
+7
+7
+3 1 1 1 1 1 1
+-0.15728640000000000000e7 0.52428800000000000000e6 -0.52428800000000000000e6 0.78643200000000000000e6 0.10485760000000000000e7 0.78643200000000000000e6 0.26214400000000000000e6
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+7 1 2 3 -0.52428800000000000000e6
+7 2 1 1 -0.52428800000000000000e6
+7 7 1 1 0.10485760000000000000e7
+*** ANSWER ***
+-2.122682931605306855e+00 7.752799720276692952e-02 1.165860635762879349e+00 9.815257583734828678e-03 1.612372277731500825e+00 9.994238642109199466e-10 8.079226242726201418e-09
+1 1 1 1 6.421188734945860924e+05
+1 1 1 2 -8.859944334888096200e+05
+1 1 1 3 8.129399719488853589e+04
+1 1 2 2 1.222493482005696977e+06
+1 1 2 3 -1.121693082800859265e+05
+1 1 3 3 1.029204353612233172e+04
+1 2 1 1 1.084594889168302296e+06
+1 3 1 1 1.690694873494586209e+06
+1 4 1 1 1.003300895685269032e+06
+1 5 1 1 1.047971877838829594e-03
+1 6 1 1 2.230940399459027220e+06
+1 7 1 1 8.471682736692869378e-03
+2 1 1 1 4.259591788484874186e+00
+2 1 1 2 3.259591789052942890e+00
+2 1 1 3 1.879795894064171691e+00
+2 1 2 2 2.499999996290436233e+00
+2 1 2 3 1.499999996792355184e+00
+2 1 3 3 1.499999998145202795e+00
+2 2 1 1 9.245035287456098323e-10
+2 3 1 1 5.680374703195812657e-10
+2 4 1 1 1.781200139772412850e-09
+2 5 1 1 2.379795894064223649e+00
+2 6 1 1 5.019469081845273553e-10
+2 7 1 1 1.202041027281383762e-01
+*** END ***
+*** REQUEST ***
+""
+1
+4
+1 1 1 1
+0.15728640000000000000e7
+0 1 1 1 0.52428800000000000000e6
+1 1 1 1 0.10485760000000000000e7
+1 2 1 1 0.52428800000000000000e6
+1 3 1 1 0.52428800000000000000e6
+1 4 1 1 -0.52428800000000000000e6
+*** ANSWER ***
+Infeasible
+*** END ***
+*** REQUEST ***
+""
+7
+7
+3 1 1 1 1 1 1
+0.10485760000000000000e7 0.10485760000000000000e7 0.78643200000000000000e6 0.52428800000000000000e6 -0.52428800000000000000e6 0.78643200000000000000e6 -0.26214400000000000000e6
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+7 4 1 1 -0.26214400000000000000e6
+7 6 1 1 -0.52428800000000000000e6
+7 7 1 1 0.52428800000000000000e6
+*** ANSWER ***
+5.903058981601223648e-10 -2.184572009349524755e-10 1.973300667196124216e-10 5.096774878147903724e-10 5.714285719996095025e-01 2.322382768664171276e-10 1.428571425918901405e-01
+1 1 1 1 3.094902987345742344e-04
+1 1 1 2 -1.564451036551416774e-05
+1 1 1 3 -1.145344889637843635e-04
+1 1 2 2 1.034577860202921573e-04
+1 1 2 3 -3.853356493497662549e-05
+1 1 3 3 2.672177907314408148e-04
+1 2 1 1 2.995931431252422626e+05
+1 3 1 1 2.995931431565312669e+05
+1 4 1 1 7.489828597228313447e+04
+1 5 1 1 1.217597417017401030e-04
+1 6 1 1 1.497965714704329439e+05
+1 7 1 1 7.489828557521689800e+04
+2 1 1 1 1.357142856178165102e+00
+2 1 1 2 3.571428564608609690e-01
+2 1 1 3 7.142857129259727600e-01
+2 1 2 2 1.357142854846152158e+00
+2 1 2 3 2.142857139726029037e-01
+2 1 3 3 9.285714274230765231e-01
+2 2 1 1 2.869472882751067396e-10
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+2 4 1 1 2.378641678314248517e-09
+2 5 1 1 1.214285712925973204e+00
+2 6 1 1 5.245009816462727640e-10
+2 7 1 1 1.046630460345233254e-09
+*** END ***
+*** REQUEST ***
+""
+5
+5
+3 1 1 1 1
+0.10485760000000000000e7 -0.10485760000000000000e7 -0.52428800000000000000e6 0.0 0.10485760000000000000e7
+0 2 1 1 -0.52428800000000000000e6
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+5 1 2 3 -0.26214400000000000000e6
+5 4 1 1 0.52428800000000000000e6
+5 5 1 1 0.52428800000000000000e6
+*** ANSWER ***
+1.506710087973945430e-15 -1.502730211901189099e-17 -9.706111672358592842e-20 2.145065614622134004e-17 2.059684481769621228e-19
+1 1 1 1 7.901533611063818646e-10
+1 1 1 2 -7.878634173372506303e-12
+1 1 1 3 -5.623160804795046962e-12
+1 1 2 2 3.051184602274949299e-13
+1 1 2 3 -5.399339287890155872e-14
+1 1 3 3 1.524545239331687038e-13
+1 2 1 1 5.242880000000008149e+05
+1 3 1 1 1.144966411228804734e-11
+1 4 1 1 1.606859763520075239e-11
+1 5 1 1 3.113292884557458962e-13
+2 1 1 1 1.999999999999999334e+00
+2 1 1 2 1.920412171343920349e+01
+2 1 1 3 2.772488039363634726e+01
+2 1 2 2 4.974516755779263235e+02
+2 1 2 3 3.423236123876088755e+02
+2 1 3 3 9.939033511558531018e+02
+2 2 1 1 2.414464203176402792e-16
+2 3 1 1 2.772488039363634726e+01
+2 4 1 1 2.020412171343918573e+01
+2 5 1 1 3.241194906741699242e+02
+*** END ***
+*** REQUEST ***
+""
+1
+2
+2 1
+0.10485760000000000000e7
+0 1 1 2 -0.39321600000000000000e6
+0 1 2 2 -0.10485760000000000000e7
+0 2 1 1 -0.26214400000000000000e6
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+1 1 1 2 0.39321600000000000000e6
+1 1 2 2 0.10485760000000000000e7
+1 2 1 1 0.26214400000000000000e6
+*** ANSWER ***
+2.307692307692307543e-01
+1 1 1 1 1.814843076923076878e+05
+1 1 1 2 4.839581538461538148e+05
+1 1 2 2 1.290555076923076995e+06
+1 2 1 1 3.226387692307692487e+05
+2 1 1 1 1.641025640075068459e+00
+2 1 1 2 -6.154038509302057758e-01
+2 1 2 2 2.307836578517253046e-01
+2 2 1 1 1.158509909295606053e-09
+*** END ***
+*** REQUEST ***
+""
+1
+2
+5 1
+0.15728640000000000000e7
+0 1 2 2 -0.52428800000000000000e6
+0 1 3 4 0.10485760000000000000e7
+0 2 1 1 -0.52428800000000000000e6
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+1 1 2 2 0.52428800000000000000e6
+1 1 3 4 -0.10485760000000000000e7
+1 2 1 1 0.52428800000000000000e6
+*** ANSWER ***
+Infeasible
+*** END ***
+*** REQUEST ***
+""
+70
+2
+15 5
+0.0 0.0 0.13107200000000000000e6 0.0 0.0 0.39321600000000000000e6 -0.26214400000000000000e6 0.26214400000000000000e6 0.0 0.0 0.26214400000000000000e6 0.0 0.0 0.52428800000000000000e6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.39321600000000000000e6 0.0 0.0 0.0 0.0 0.0 -0.26214400000000000000e6 0.26214400000000000000e6 0.0 0.0 0.0 0.0 0.0 0.13107200000000000000e6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 -0.26214400000000000000e6 0.0 0.0 0.0 0.26214400000000000000e6 0.0 0.0 0.0 0.0 0.13107200000000000000e6 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.39321600000000000000e6 0.0 0.0 0.13107200000000000000e6 0.13107200000000000000e6 0.0 0.0 -0.10485760000000000000e7 0.0
+0 1 2 2 -0.13107200000000000000e6
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+*** REQUEST ***
+""
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+*** ANSWER ***
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+*** REQUEST ***
+""
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+*** ANSWER ***
+Infeasible
+*** END ***
+*** REQUEST ***
+""
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+*** REQUEST ***
+""
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+*** ANSWER ***
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+*** END ***
+*** REQUEST ***
+""
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+*** ANSWER ***
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+*** END ***
+*** REQUEST ***
+""
+57
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+*** ANSWER ***
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+*** END ***
+*** REQUEST ***
+""
+1
+5
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+1 1 1 1 0.10485760000000000000e7
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+1 4 1 1 0.52428800000000000000e6
+*** ANSWER ***
+5.000000015457306546e-01
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+2 5 1 1 5.258797495207707584e+01
+*** END ***
+*** REQUEST ***
+""
+57
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+*** ANSWER ***
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+*** END ***
+*** REQUEST ***
+""
+47
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+*** ANSWER ***
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+*** END ***
+*** REQUEST ***
+""
+47
+20
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+*** ANSWER ***
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+*** END ***
+*** REQUEST ***
+""
+1
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+*** ANSWER ***
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+2 6 1 1 6.282786184129422935e+01
+*** END ***
+*** REQUEST ***
+""
+57
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+*** ANSWER ***
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+*** END ***
+*** REQUEST ***
+""
+47
+20
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+*** ANSWER ***
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+*** END ***
+*** REQUEST ***
+""
+47
+20
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+*** ANSWER ***
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+*** REQUEST ***
+""
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+*** ANSWER ***
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+*** END ***
+*** REQUEST ***
+""
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+*** END ***
+*** REQUEST ***
+""
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+*** END ***
+*** REQUEST ***
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+*** ANSWER ***
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+*** END ***
+*** REQUEST ***
+""
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+*** ANSWER ***
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+*** END ***
+*** REQUEST ***
+""
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+*** ANSWER ***
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+*** END ***
+*** REQUEST ***
+""
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+*** ANSWER ***
+-4.334665259043807452e-15 -1.000000000000002442e+00 2.614572177519653489e-15 1.621893095060768740e-28 1.118406671931714504e-17
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+*** END ***
+*** REQUEST ***
+""
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+*** ANSWER ***
+-4.334665259043807452e-15 1.000000000000002442e+00 2.614572177519653489e-15 -1.621893095060768740e-28 1.118406671931714504e-17
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+*** END ***
+*** REQUEST ***
+""
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+*** END ***
+*** REQUEST ***
+""
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+*** ANSWER ***
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+*** END ***
+*** REQUEST ***
+""
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+*** END ***
+*** REQUEST ***
+""
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+*** ANSWER ***
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+*** END ***
+*** REQUEST ***
+""
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+*** END ***
+*** REQUEST ***
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+*** ANSWER ***
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+*** END ***
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+*** END ***
+*** REQUEST ***
+""
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+*** END ***
+*** REQUEST ***
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+*** ANSWER ***
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+*** END ***
+*** REQUEST ***
+""
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+*** ANSWER ***
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+*** END ***
+*** REQUEST ***
+""
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+*** ANSWER ***
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+*** END ***
+*** REQUEST ***
+""
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+*** ANSWER ***
+Infeasible
+*** END ***
+*** REQUEST ***
+""
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+*** ANSWER ***
+Failure
+*** END ***
+*** REQUEST ***
+""
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+*** ANSWER ***
+Failure
+*** END ***
+*** REQUEST ***
+""
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+*** ANSWER ***
+Failure
+*** END ***
+*** REQUEST ***
+""
+1085
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+1085 7 12 13 0.13107200000000000000e6
+1085 7 14 15 -0.13107200000000000000e6
+1085 8 5 5 0.26214400000000000000e6
+*** ANSWER ***
+Failure
+*** END ***
diff --git a/test-suite/micromega/bertot.v b/test-suite/micromega/bertot.v
new file mode 100644
index 000000000..6951fcd33
--- /dev/null
+++ b/test-suite/micromega/bertot.v
@@ -0,0 +1,22 @@
+(************************************************************************)
+(* *)
+(* Micromega: A reflexive tactic using the Positivstellensatz *)
+(* *)
+(* Frédéric Besson (Irisa/Inria) 2006-2008 *)
+(* *)
+(************************************************************************)
+
+Require Import ZArith.
+Require Import Micromegatac.
+
+Open Scope Z_scope.
+
+Goal (forall x y n,
+ ( ~ x < n /\ x <= n /\ 2 * y = x*(x+1) -> 2 * y = n*(n+1))
+ /\
+ (x < n /\ x <= n /\ 2 * y = x * (x+1) -> x + 1 <= n /\ 2 *(x+1+y) = (x+1)*(x+2))).
+Proof.
+ intros.
+ micromega Z.
+Qed.
+
diff --git a/test-suite/micromega/example.v b/test-suite/micromega/example.v
new file mode 100644
index 000000000..dc78ace5d
--- /dev/null
+++ b/test-suite/micromega/example.v
@@ -0,0 +1,347 @@
+(************************************************************************)
+(* *)
+(* Micromega: A reflexive tactic using the Positivstellensatz *)
+(* *)
+(* Frédéric Besson (Irisa/Inria) 2006-2008 *)
+(* *)
+(************************************************************************)
+
+Require Import ZArith.
+Require Import Micromegatac.
+Require Import Ring_normalize.
+Open Scope Z_scope.
+Require Import ZMicromega.
+Require Import VarMap.
+
+(* false in Q : x=1/2 and n=1 *)
+
+Lemma not_so_easy : forall x n : Z,
+ 2*x + 1 <= 2 *n -> x <= n-1.
+Proof.
+ intros.
+ zfarkas.
+Qed.
+
+
+(* From Laurent Théry *)
+
+Lemma some_pol : forall x, 4 * x ^ 2 + 3 * x + 2 >= 0.
+Proof.
+ intros.
+ micromega Z.
+Qed.
+
+
+Lemma Zdiscr: forall a b c x,
+ a * x ^ 2 + b * x + c = 0 -> b ^ 2 - 4 * a * c >= 0.
+Proof.
+ intros ; micromega Z.
+Qed.
+
+
+Lemma plus_minus : forall x y,
+ 0 = x + y -> 0 = x -y -> 0 = x /\ 0 = y.
+Proof.
+ intros.
+ zfarkas.
+Qed.
+
+
+
+Lemma mplus_minus : forall x y,
+ x + y >= 0 -> x -y >= 0 -> x^2 - y^2 >= 0.
+Proof.
+ intros; micromega Z.
+Qed.
+
+Lemma pol3: forall x y, 0 <= x + y ->
+ x^3 + 3*x^2*y + 3*x* y^2 + y^3 >= 0.
+Proof.
+ intros; micromega Z.
+Qed.
+
+
+(* Motivating example from: Expressiveness + Automation + Soundness:
+ Towards COmbining SMT Solvers and Interactive Proof Assistants *)
+Parameter rho : Z.
+Parameter rho_ge : rho >= 0.
+Parameter correct : Z -> Z -> Prop.
+
+
+Definition rbound1 (C:Z -> Z -> Z) : Prop :=
+ forall p s t, correct p t /\ s <= t -> C p t - C p s <= (1-rho)*(t-s).
+
+Definition rbound2 (C:Z -> Z -> Z) : Prop :=
+ forall p s t, correct p t /\ s <= t -> (1-rho)*(t-s) <= C p t - C p s.
+
+
+Lemma bounded_drift : forall s t p q C D, s <= t /\ correct p t /\ correct q t /\
+ rbound1 C /\ rbound2 C /\ rbound1 D /\ rbound2 D ->
+ Zabs (C p t - D q t) <= Zabs (C p s - D q s) + 2 * rho * (t- s).
+Proof.
+ intros.
+ generalize (Zabs_eq (C p t - D q t)).
+ generalize (Zabs_non_eq (C p t - D q t)).
+ generalize (Zabs_eq (C p s -D q s)).
+ generalize (Zabs_non_eq (C p s - D q s)).
+ unfold rbound2 in H.
+ unfold rbound1 in H.
+ intuition.
+ generalize (H6 _ _ _ (conj H H4)).
+ generalize (H7 _ _ _ (conj H H4)).
+ generalize (H8 _ _ _ (conj H H4)).
+ generalize (H10 _ _ _ (conj H H4)).
+ generalize (H6 _ _ _ (conj H5 H4)).
+ generalize (H7 _ _ _ (conj H5 H4)).
+ generalize (H8 _ _ _ (conj H5 H4)).
+ generalize (H10 _ _ _ (conj H5 H4)).
+ generalize rho_ge.
+ micromega Z.
+Qed.
+
+(* Rule of signs *)
+
+Lemma sign_pos_pos: forall x y,
+ x > 0 -> y > 0 -> x*y > 0.
+Proof.
+ intros; micromega Z.
+Qed.
+
+Lemma sign_pos_zero: forall x y,
+ x > 0 -> y = 0 -> x*y = 0.
+Proof.
+ intros; micromega Z.
+Qed.
+
+Lemma sign_pos_neg: forall x y,
+ x > 0 -> y < 0 -> x*y < 0.
+Proof.
+ intros; micromega Z.
+Qed.
+
+Lemma sign_zer_pos: forall x y,
+ x = 0 -> y > 0 -> x*y = 0.
+Proof.
+ intros; micromega Z.
+Qed.
+
+Lemma sign_zero_zero: forall x y,
+ x = 0 -> y = 0 -> x*y = 0.
+Proof.
+ intros; micromega Z.
+Qed.
+
+Lemma sign_zero_neg: forall x y,
+ x = 0 -> y < 0 -> x*y = 0.
+Proof.
+ intros; micromega Z.
+Qed.
+
+Lemma sign_neg_pos: forall x y,
+ x < 0 -> y > 0 -> x*y < 0.
+Proof.
+ intros; micromega Z.
+Qed.
+
+Lemma sign_neg_zero: forall x y,
+ x < 0 -> y = 0 -> x*y = 0.
+Proof.
+ intros; micromega Z.
+Qed.
+
+Lemma sign_neg_neg: forall x y,
+ x < 0 -> y < 0 -> x*y > 0.
+Proof.
+ intros; micromega Z.
+Qed.
+
+
+(* Other (simple) examples *)
+
+Lemma binomial : forall x y, (x+y)^2 = x^2 + 2*x*y + y^2.
+Proof.
+ intros.
+ zfarkas.
+Qed.
+
+Lemma product : forall x y, x >= 0 -> y >= 0 -> x * y >= 0.
+Proof.
+ intros.
+ micromega Z.
+Qed.
+
+
+Lemma product_strict : forall x y, x > 0 -> y > 0 -> x * y > 0.
+Proof.
+ intros.
+ micromega Z.
+Qed.
+
+
+Lemma pow_2_pos : forall x, x ^ 2 + 1 = 0 -> False.
+Proof.
+ intros ; micromega Z.
+Qed.
+
+(* Found in Parrilo's talk *)
+(* BUG?: certificate with **very** big coefficients *)
+Lemma parrilo_ex : forall x y, x - y^2 + 3 >= 0 -> y + x^2 + 2 = 0 -> False.
+Proof.
+ intros.
+ micromega Z.
+Qed.
+
+
+(* from hol_light/Examples/sos.ml *)
+
+Lemma hol_light1 : forall a1 a2 b1 b2,
+ a1 >= 0 -> a2 >= 0 ->
+ (a1 * a1 + a2 * a2 = b1 * b1 + b2 * b2 + 2) ->
+ (a1 * b1 + a2 * b2 = 0) -> a1 * a2 - b1 * b2 >= 0.
+Proof.
+ intros ; micromega Z.
+Qed.
+
+
+Lemma hol_light2 : forall x a,
+ 3 * x + 7 * a < 4 -> 3 < 2 * x -> a < 0.
+Proof.
+ intros ; micromega Z.
+Qed.
+
+Lemma hol_light3 : forall b a c x,
+ b ^ 2 < 4 * a * c -> (a * x ^2 + b * x + c = 0) -> False.
+Proof.
+intros ; micromega Z.
+Qed.
+
+Lemma hol_light4 : forall a c b x,
+ a * x ^ 2 + b * x + c = 0 -> b ^ 2 >= 4 * a * c.
+Proof.
+intros ; micromega Z.
+Qed.
+
+Lemma hol_light5 : forall x y,
+ 0 <= x /\ x <= 1 /\ 0 <= y /\ y <= 1
+ -> x ^ 2 + y ^ 2 < 1 \/
+ (x - 1) ^ 2 + y ^ 2 < 1 \/
+ x ^ 2 + (y - 1) ^ 2 < 1 \/
+ (x - 1) ^ 2 + (y - 1) ^ 2 < 1.
+Proof.
+intros; micromega Z.
+Qed.
+
+
+
+Lemma hol_light7 : forall x y z,
+ 0<= x /\ 0 <= y /\ 0 <= z /\ x + y + z <= 3
+ -> x * y + x * z + y * z >= 3 * x * y * z.
+Proof.
+intros ; micromega Z.
+Qed.
+
+Lemma hol_light8 : forall x y z,
+ x ^ 2 + y ^ 2 + z ^ 2 = 1 -> (x + y + z) ^ 2 <= 3.
+Proof.
+ intros ; micromega Z.
+Qed.
+
+Lemma hol_light9 : forall w x y z,
+ w ^ 2 + x ^ 2 + y ^ 2 + z ^ 2 = 1
+ -> (w + x + y + z) ^ 2 <= 4.
+Proof.
+ intros;micromega Z.
+Qed.
+
+Lemma hol_light10 : forall x y,
+ x >= 1 /\ y >= 1 -> x * y >= x + y - 1.
+Proof.
+ intros ; micromega Z.
+Qed.
+
+Lemma hol_light11 : forall x y,
+ x > 1 /\ y > 1 -> x * y > x + y - 1.
+Proof.
+ intros ; micromega Z.
+Qed.
+
+
+Lemma hol_light12: forall x y z,
+ 2 <= x /\ x <= 125841 / 50000 /\
+ 2 <= y /\ y <= 125841 / 50000 /\
+ 2 <= z /\ z <= 125841 / 50000
+ -> 2 * (x * z + x * y + y * z) - (x * x + y * y + z * z) >= 0.
+Proof.
+ intros x y z ; set (e:= (125841 / 50000)).
+ compute in e.
+ unfold e ; intros ; micromega Z.
+Qed.
+
+Lemma hol_light14 : forall x y z,
+ 2 <= x /\ x <= 4 /\ 2 <= y /\ y <= 4 /\ 2 <= z /\ z <= 4
+ -> 12 <= 2 * (x * z + x * y + y * z) - (x * x + y * y + z * z).
+Proof.
+ intros ;micromega Z.
+Qed.
+
+(* ------------------------------------------------------------------------- *)
+(* Inequality from sci.math (see "Leon-Sotelo, por favor"). *)
+(* ------------------------------------------------------------------------- *)
+
+Lemma hol_light16 : forall x y,
+ 0 <= x /\ 0 <= y /\ (x * y = 1)
+ -> x + y <= x ^ 2 + y ^ 2.
+Proof.
+ intros ; micromega Z.
+Qed.
+
+Lemma hol_light17 : forall x y,
+ 0 <= x /\ 0 <= y /\ (x * y = 1)
+ -> x * y * (x + y) <= x ^ 2 + y ^ 2.
+Proof.
+ intros ; micromega Z.
+Qed.
+
+Lemma hol_light18 : forall x y,
+ 0 <= x /\ 0 <= y -> x * y * (x + y) ^ 2 <= (x ^ 2 + y ^ 2) ^ 2.
+Proof.
+ intros ; micromega Z.
+Qed.
+
+(* ------------------------------------------------------------------------- *)
+(* Some examples over integers and natural numbers. *)
+(* ------------------------------------------------------------------------- *)
+
+Lemma hol_light19 : forall m n, 2 * m + n = (n + m) + m.
+Proof.
+ intros ; zfarkas.
+Qed.
+
+Lemma hol_light22 : forall n, n >= 0 -> n <= n * n.
+Proof.
+ intros.
+ micromega Z.
+Qed.
+
+
+Lemma hol_light24 : forall x1 y1 x2 y2, x1 >= 0 -> x2 >= 0 -> y1 >= 0 -> y2 >= 0 ->
+ ((x1 + y1) ^2 + x1 + 1 = (x2 + y2) ^ 2 + x2 + 1)
+ -> (x1 + y1 = x2 + y2).
+Proof.
+ intros.
+ micromega Z.
+Qed.
+
+Lemma motzkin' : forall x y, (x^2+y^2+1)*(x^2*y^4 + x^4*y^2 + 1 - 3*x^2*y^2) >= 0.
+Proof.
+ intros ; sos Z.
+Qed.
+
+
+
+Lemma motzkin : forall x y, (x^2*y^4 + x^4*y^2 + 1 - 3*x^2*y^2) >= 0.
+Proof.
+ intros.
+ generalize (motzkin' x y).
+ micromega Z.
+Qed.
diff --git a/test-suite/micromega/heap3_vcgen_25.v b/test-suite/micromega/heap3_vcgen_25.v
new file mode 100644
index 000000000..ec88b68dd
--- /dev/null
+++ b/test-suite/micromega/heap3_vcgen_25.v
@@ -0,0 +1,38 @@
+(************************************************************************)
+(* *)
+(* Micromega: A reflexive tactic using the Positivstellensatz *)
+(* *)
+(* Frédéric Besson (Irisa/Inria) 2006-2008 *)
+(* *)
+(************************************************************************)
+
+Require Import ZArith.
+Require Import Micromegatac.
+
+Open Scope Z_scope.
+
+Lemma vcgen_25 : forall
+ (n : Z)
+ (m : Z)
+ (jt : Z)
+ (j : Z)
+ (it : Z)
+ (i : Z)
+ (H0 : 1 * it + -2 * i + -1 = 0)
+ (H : 1 * jt + -2 * j + -1 = 0)
+ (H1 : 1 * n + -10 = 0)
+ (H2 : 0 <= -4028 * i + 6222 * j + 705 * m + -16674)
+ (H3 : 0 <= -418 * i + 651 * j + 94 * m + -1866)
+ (H4 : 0 <= -209 * i + 302 * j + 47 * m + -839)
+ (H5 : 0 <= -1 * i + 1 * j + -1)
+ (H6 : 0 <= -1 * j + 1 * m + 0)
+ (H7 : 0 <= 1 * j + 5 * m + -27)
+ (H8 : 0 <= 2 * j + -1 * m + 2)
+ (H9 : 0 <= 7 * j + 10 * m + -74)
+ (H10 : 0 <= 18 * j + -139 * m + 1188)
+ (H11 : 0 <= 1 * i + 0)
+ (H13 : 0 <= 121 * i + 810 * j + -7465 * m + 64350),
+ (1 = -2 * i + it).
+Proof.
+ intros ; zfarkas.
+Qed.
diff --git a/test-suite/micromega/qexample.v b/test-suite/micromega/qexample.v
new file mode 100644
index 000000000..42aff5a47
--- /dev/null
+++ b/test-suite/micromega/qexample.v
@@ -0,0 +1,62 @@
+(************************************************************************)
+(* *)
+(* Micromega: A reflexive tactic using the Positivstellensatz *)
+(* *)
+(* Frédéric Besson (Irisa/Inria) 2006-2008 *)
+(* *)
+(************************************************************************)
+
+Require Import Micromegatac.
+Require Import QArith.
+Require Import Ring_normalize.
+
+Lemma plus_minus : forall x y,
+ 0 == x + y -> 0 == x -y -> 0 == x /\ 0 == y.
+Proof.
+ intros.
+ omicron Q.
+Qed.
+
+(* Other (simple) examples *)
+Open Scope Q_scope.
+
+Lemma binomial : forall x y:Q, ((x+y)^2 == x^2 + (2 # 1) *x*y + y^2).
+Proof.
+ intros.
+ omicron Q.
+Qed.
+
+
+Lemma hol_light19 : forall m n, (2 # 1) * m + n == (n + m) + m.
+Proof.
+ intros ; omicron Q.
+Qed.
+Open Scope Z_scope.
+Open Scope Q_scope.
+
+Lemma vcgen_25 : forall
+ (n : Q)
+ (m : Q)
+ (jt : Q)
+ (j : Q)
+ (it : Q)
+ (i : Q)
+ (H0 : 1 * it + (-2 # 1) * i + (-1 # 1) == 0)
+ (H : 1 * jt + (-2 # 1) * j + (-1 # 1) == 0)
+ (H1 : 1 * n + (-10 # 1) = 0)
+ (H2 : 0 <= (-4028 # 1) * i + (6222 # 1) * j + (705 # 1) * m + (-16674 # 1))
+ (H3 : 0 <= (-418 # 1) * i + (651 # 1) * j + (94 # 1) * m + (-1866 # 1))
+ (H4 : 0 <= (-209 # 1) * i + (302 # 1) * j + (47 # 1) * m + (-839 # 1))
+ (H5 : 0 <= (-1 # 1) * i + 1 * j + (-1 # 1))
+ (H6 : 0 <= (-1 # 1) * j + 1 * m + (0 # 1))
+ (H7 : 0 <= (1 # 1) * j + (5 # 1) * m + (-27 # 1))
+ (H8 : 0 <= (2 # 1) * j + (-1 # 1) * m + (2 # 1))
+ (H9 : 0 <= (7 # 1) * j + (10 # 1) * m + (-74 # 1))
+ (H10 : 0 <= (18 # 1) * j + (-139 # 1) * m + (1188 # 1))
+ (H11 : 0 <= 1 * i + (0 # 1))
+ (H13 : 0 <= (121 # 1) * i + (810 # 1) * j + (-7465 # 1) * m + (64350 # 1)),
+ (( 1# 1) == (-2 # 1) * i + it).
+Proof.
+ intros.
+ omicron Q.
+Qed.
diff --git a/test-suite/micromega/rexample.v b/test-suite/micromega/rexample.v
new file mode 100644
index 000000000..5132dc4ef
--- /dev/null
+++ b/test-suite/micromega/rexample.v
@@ -0,0 +1,62 @@
+(************************************************************************)
+(* *)
+(* Micromega: A reflexive tactic using the Positivstellensatz *)
+(* *)
+(* Frédéric Besson (Irisa/Inria) 2006-2008 *)
+(* *)
+(************************************************************************)
+
+Require Import Micromegatac.
+Require Import Reals.
+Require Import Ring_normalize.
+
+Open Scope R_scope.
+
+Lemma plus_minus : forall x y,
+ 0 = x + y -> 0 = x -y -> 0 = x /\ 0 = y.
+Proof.
+ intros.
+ omicron R.
+Qed.
+
+(* Other (simple) examples *)
+
+Lemma binomial : forall x y, ((x+y)^2 = x^2 + 2 *x*y + y^2).
+Proof.
+ intros.
+ omicron R.
+Qed.
+
+
+Lemma hol_light19 : forall m n, 2 * m + n = (n + m) + m.
+Proof.
+ intros ; omicron R.
+Qed.
+
+
+Lemma vcgen_25 : forall
+ (n : R)
+ (m : R)
+ (jt : R)
+ (j : R)
+ (it : R)
+ (i : R)
+ (H0 : 1 * it + (-2%R ) * i + (-1%R ) = 0)
+ (H : 1 * jt + (-2 ) * j + (-1 ) = 0)
+ (H1 : 1 * n + (-10 ) = 0)
+ (H2 : 0 <= (-4028 ) * i + (6222 ) * j + (705 ) * m + (-16674 ))
+ (H3 : 0 <= (-418 ) * i + (651 ) * j + (94 ) * m + (-1866 ))
+ (H4 : 0 <= (-209 ) * i + (302 ) * j + (47 ) * m + (-839 ))
+ (H5 : 0 <= (-1 ) * i + 1 * j + (-1 ))
+ (H6 : 0 <= (-1 ) * j + 1 * m + (0 ))
+ (H7 : 0 <= (1 ) * j + (5 ) * m + (-27 ))
+ (H8 : 0 <= (2 ) * j + (-1 ) * m + (2 ))
+ (H9 : 0 <= (7 ) * j + (10 ) * m + (-74 ))
+ (H10 : 0 <= (18 ) * j + (-139 ) * m + (1188 ))
+ (H11 : 0 <= 1 * i + (0 ))
+ (H13 : 0 <= (121 ) * i + (810 ) * j + (-7465 ) * m + (64350 )),
+ (( 1 ) = (-2 ) * i + it).
+Proof.
+ intros.
+ omicron R.
+Qed.
diff --git a/test-suite/micromega/square.v b/test-suite/micromega/square.v
new file mode 100644
index 000000000..30c72e8c1
--- /dev/null
+++ b/test-suite/micromega/square.v
@@ -0,0 +1,61 @@
+(************************************************************************)
+(* *)
+(* Micromega: A reflexive tactic using the Positivstellensatz *)
+(* *)
+(* Frédéric Besson (Irisa/Inria) 2006-2008 *)
+(* *)
+(************************************************************************)
+
+Require Import ZArith Zwf Micromegatac QArith.
+Open Scope Z_scope.
+
+Lemma Zabs_square : forall x, (Zabs x)^2 = x^2.
+Proof.
+ intros ; case (Zabs_dec x) ; intros ; micromega Z.
+Qed.
+Hint Resolve Zabs_pos Zabs_square.
+
+Lemma integer_statement : ~exists n, exists p, n^2 = 2*p^2 /\ n <> 0.
+Proof.
+intros [n [p [Heq Hnz]]]; pose (n' := Zabs n); pose (p':=Zabs p).
+assert (facts : 0 <= Zabs n /\ 0 <= Zabs p /\ Zabs n^2=n^2
+ /\ Zabs p^2 = p^2) by auto.
+assert (H : (0 < n' /\ 0 <= p' /\ n' ^2 = 2* p' ^2)) by
+ (destruct facts as [Hf1 [Hf2 [Hf3 Hf4]]]; unfold n', p' ; micromega Z).
+generalize p' H; elim n' using (well_founded_ind (Zwf_well_founded 0)); clear.
+intros n IHn p [Hn [Hp Heq]].
+assert (Hzwf : Zwf 0 (2*p-n) n) by (unfold Zwf; micromega Z).
+assert (Hdecr : 0 < 2*p-n /\ 0 <= n-p /\ (2*p-n)^2=2*(n-p)^2) by micromega Z.
+apply (IHn (2*p-n) Hzwf (n-p) Hdecr).
+Qed.
+
+Open Scope Q_scope.
+
+Lemma QnumZpower : forall x : Q, Qnum (x ^ 2)%Q = ((Qnum x) ^ 2) %Z.
+Proof.
+ intros.
+ destruct x.
+ cbv beta iota zeta delta - [Zmult].
+ ring.
+Qed.
+
+
+Lemma QdenZpower : forall x : Q, ' Qden (x ^ 2)%Q = ('(Qden x) ^ 2) %Z.
+Proof.
+ intros.
+ destruct x.
+ cbv beta iota zeta delta - [Pmult].
+ rewrite Pmult_1_r.
+ reflexivity.
+Qed.
+
+Theorem sqrt2_not_rational : ~exists x:Q, x^2==2#1.
+Proof.
+ unfold Qeq; intros [x]; simpl (Qden (2#1)); rewrite Zmult_1_r.
+ intros HQeq.
+ assert (Heq : (Qnum x ^ 2 = 2 * ' Qden x ^ 2%Q)%Z) by
+ (rewrite QnumZpower in HQeq ; rewrite QdenZpower in HQeq ; auto).
+ assert (Hnx : (Qnum x <> 0)%Z)
+ by (intros Hx; simpl in HQeq; rewrite Hx in HQeq; discriminate HQeq).
+ apply integer_statement; exists (Qnum x); exists (' Qden x); auto.
+Qed.
diff --git a/test-suite/micromega/zomicron.v b/test-suite/micromega/zomicron.v
new file mode 100644
index 000000000..b13654d69
--- /dev/null
+++ b/test-suite/micromega/zomicron.v
@@ -0,0 +1,27 @@
+Require Import ZArith.
+Require Import Micromegatac.
+Open Scope Z_scope.
+
+Lemma two_x_eq_1 : forall x, 2 * x = 1 -> False.
+Proof.
+ intros.
+ omicron Z.
+Qed.
+
+Lemma two_x_y_eq_1 : forall x y, 2 * x + 2 * y = 1 -> False.
+Proof.
+ intros.
+ omicron Z.
+Qed.
+
+Lemma two_x_y_z_eq_1 : forall x y z, 2 * x + 2 * y + 2 * z= 1 -> False.
+Proof.
+ intros.
+ omicron Z.
+Qed.
+
+Lemma omega_nightmare : forall x y, 27 <= 11 * x + 13 * y <= 45 -> 7 * x - 9 * y = 4 -> -10 <= 7 * x - 9 * y <= 4 -> False.
+Proof.
+ intros ; intuition auto.
+ omicron Z.
+Qed.
diff --git a/test-suite/success/unification.v b/test-suite/success/unification.v
index 91ee18ea4..3d935cd1d 100644
--- a/test-suite/success/unification.v
+++ b/test-suite/success/unification.v
@@ -126,3 +126,11 @@ intros.
exists (fun n => match n with O => a | S n' => f' n' end).
constructor.
Qed.
+
+(* Why this not work, even, without the "fix" to #1851 *)
+
+Goal forall X (a:X 0) (f':forall n:nat, X n), (exists f : forall n:nat, X n, True).
+intros.
+exists (fun n => match n with O => a | S n' => f' n' end).
+constructor.
+Qed.