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|
(************************************************************************)
(* v * The Coq Proof Assistant / The Coq Development Team *)
(* <O___,, * CNRS-Ecole Polytechnique-INRIA Futurs-Universite Paris Sud *)
(* \VV/ **************************************************************)
(* // * This file is distributed under the terms of the *)
(* * GNU Lesser General Public License Version 2.1 *)
(************************************************************************)
(* *)
(* Micromega: A reflexive tactic using the Positivstellensatz *)
(* *)
(* ** Toplevel definition of tactics ** *)
(* *)
(* - Modules ISet, M, Mc, Env, Cache, CacheZ *)
(* *)
(* Frédéric Besson (Irisa/Inria) 2006-2009 *)
(* *)
(************************************************************************)
open Mutils
(**
* Debug flag
*)
let debug = false
(**
* Time function
*)
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
(**
* Initialize a tag type to the Tag module declaration (see Mutils).
*)
type tag = Tag.t
(**
* An atom is of the form:
* pExpr1 {<,>,=,<>,<=,>=} pExpr2
* where pExpr1, pExpr2 are polynomial expressions (see Micromega). pExprs are
* parametrized by 'cst, which is used as the type of constants.
*)
type 'cst atom = 'cst Micromega.formula
(**
* Micromega's encoding of formulas.
* By order of appearance: boolean constants, variables, atoms, conjunctions,
* disjunctions, negation, implication.
*)
type 'cst formula =
| TT
| FF
| X of Term.constr
| A of 'cst atom * tag * Term.constr
| C of 'cst formula * 'cst formula
| D of 'cst formula * 'cst formula
| N of 'cst formula
| I of 'cst formula * Names.identifier option * 'cst formula
(**
* Formula pretty-printer.
*)
let rec pp_formula o f =
match f with
| TT -> output_string o "tt"
| FF -> output_string o "ff"
| X c -> output_string o "X "
| A(_,t,_) -> Printf.fprintf o "A(%a)" Tag.pp t
| C(f1,f2) -> Printf.fprintf o "C(%a,%a)" pp_formula f1 pp_formula f2
| D(f1,f2) -> Printf.fprintf o "D(%a,%a)" pp_formula f1 pp_formula f2
| I(f1,n,f2) -> Printf.fprintf o "I(%a%s,%a)"
pp_formula f1
(match n with
| Some id -> Names.string_of_id id
| None -> "") pp_formula f2
| N(f) -> Printf.fprintf o "N(%a)" pp_formula f
(**
* Collect the identifiers of a (string of) implications. Implication labels
* are inherited from Coq/CoC's higher order dependent type constructor (Pi).
*)
let rec ids_of_formula f =
match f with
| I(f1,Some id,f2) -> id::(ids_of_formula f2)
| _ -> []
(**
* A clause is a list of (tagged) nFormulas.
* nFormulas are normalized formulas, i.e., of the form:
* cPol {=,<>,>,>=} 0
* with cPol compact polynomials (see the Pol inductive type in EnvRing.v).
*)
type 'cst clause = ('cst Micromega.nFormula * tag) list
(**
* A CNF is a list of clauses.
*)
type 'cst cnf = ('cst clause) list
(**
* True and False are empty cnfs and clauses.
*)
let tt : 'cst cnf = []
let ff : 'cst cnf = [ [] ]
(**
* A refinement of cnf with tags left out. This is an intermediary form
* between the cnf tagged list representation ('cst cnf) used to solve psatz,
* and the freeform formulas ('cst formula) that is retrieved from Coq.
*)
type 'cst mc_cnf = ('cst Micromega.nFormula) list list
(**
* From a freeform formula, build a cnf.
* The parametric functions negate and normalize are theory-dependent, and
* originate in micromega.ml (extracted, e.g. for rnegate, from RMicromega.v
* and RingMicromega.v).
*)
let cnf (negate: 'cst atom -> 'cst mc_cnf) (normalise:'cst atom -> 'cst mc_cnf) (f:'cst formula) =
let negate a t =
List.map (fun cl -> List.map (fun x -> (x,t)) cl) (negate a) in
let normalise a t =
List.map (fun cl -> List.map (fun x -> (x,t)) 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 (polarity : bool) f =
match f with
| TT -> if polarity then tt else ff
| FF -> if polarity then ff else tt
| X p -> if polarity then ff else ff
| A(x,t,_) -> if polarity then normalise x t else negate x t
| N(e) -> xcnf (not polarity) e
| C(e1,e2) ->
(if polarity then and_cnf else or_cnf) (xcnf polarity e1) (xcnf polarity e2)
| D(e1,e2) ->
(if polarity then or_cnf else and_cnf) (xcnf polarity e1) (xcnf polarity e2)
| I(e1,_,e2) ->
(if polarity then or_cnf else and_cnf) (xcnf (not polarity) e1) (xcnf polarity e2) in
xcnf true f
(**
* MODULE: Ordered set of integers.
*)
module ISet = Set.Make(struct type t = int let compare : int -> int -> int = Pervasives.compare end)
(**
* Given a set of integers s={i0,...,iN} and a list m, return the list of
* elements of m that are at position i0,...,iN.
*)
let selecti s m =
let rec xselecti i m =
match m with
| [] -> []
| e::m -> if ISet.mem i s then e::(xselecti (i+1) m) else xselecti (i+1) m in
xselecti 0 m
(**
* MODULE: Mapping of the Coq data-strustures into Caml and Caml extracted
* code. This includes initializing Caml variables based on Coq terms, parsing
* various Coq expressions into Caml, and dumping Caml expressions into Coq.
*
* Opened here and in csdpcert.ml.
*)
module M =
struct
open Coqlib
open Term
(**
* Location of the Coq libraries.
*)
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"];
["Coq";"Reals" ; "Rpow_def"];
["LRing_normalise"]]
(**
* Initialization : a large amount of Caml symbols are derived from
* ZMicromega.v
*)
let init_constant = gen_constant_in_modules "ZMicromega" init_modules
let constant = gen_constant_in_modules "ZMicromega" coq_modules
(* let constant = gen_constant_in_modules "Omicron" coq_modules *)
let coq_and = lazy (init_constant "and")
let coq_or = lazy (init_constant "or")
let coq_not = lazy (init_constant "not")
let coq_iff = lazy (init_constant "iff")
let coq_True = lazy (init_constant "True")
let coq_False = lazy (init_constant "False")
let coq_cons = lazy (constant "cons")
let coq_nil = lazy (constant "nil")
let coq_list = lazy (constant "list")
let coq_O = lazy (init_constant "O")
let coq_S = lazy (init_constant "S")
let coq_nat = lazy (init_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_Build_Witness = lazy (constant "Build_Witness")
let coq_Qmake = lazy (constant "Qmake")
let coq_R0 = lazy (constant "R0")
let coq_R1 = lazy (constant "R1")
let coq_proofTerm = lazy (constant "ZArithProof")
let coq_doneProof = lazy (constant "DoneProof")
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 (init_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_Zpower = lazy (constant "Zpower")
let coq_Qgt = lazy (constant "Qgt")
let coq_Qge = lazy (constant "Qge")
let coq_Qle = lazy (constant "Qle")
let coq_Qlt = lazy (constant "Qlt")
let coq_Qeq = lazy (constant "Qeq")
let coq_Qplus = lazy (constant "Qplus")
let coq_Qminus = lazy (constant "Qminus")
let coq_Qopp = lazy (constant "Qopp")
let coq_Qmult = lazy (constant "Qmult")
let coq_Qpower = lazy (constant "Qpower")
let coq_Rgt = lazy (constant "Rgt")
let coq_Rge = lazy (constant "Rge")
let coq_Rle = lazy (constant "Rle")
let coq_Rlt = lazy (constant "Rlt")
let coq_Rplus = lazy (constant "Rplus")
let coq_Rminus = lazy (constant "Rminus")
let coq_Ropp = lazy (constant "Ropp")
let coq_Rmult = lazy (constant "Rmult")
let coq_Rpower = lazy (constant "pow")
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_PX = lazy (constant "PX" )
let coq_Pc = lazy (constant"Pc")
let coq_Pinj = lazy (constant "Pinj")
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_PsatzIn = lazy (constant "PsatzIn")
let coq_PsatzSquare = lazy (constant "PsatzSquare")
let coq_PsatzMulE = lazy (constant "PsatzMulE")
let coq_PsatzMultC = lazy (constant "PsatzMulC")
let coq_PsatzAdd = lazy (constant "PsatzAdd")
let coq_PsatzC = lazy (constant "PsatzC")
let coq_PsatzZ = lazy (constant "PsatzZ")
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_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")
(**
* Initialization : a few Caml symbols are derived from other libraries;
* QMicromega, ZArithRing, RingMicromega.
*)
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_N_of_Z = lazy
(gen_constant_in_modules "ZArithRing"
[["Coq";"setoid_ring";"ZArithRing"]] "N_of_Z")
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")
(**
* Parsing and dumping : transformation functions between Caml and Coq
* data-structures.
*
* dump_* functions go from Micromega to Coq terms
* parse_* functions go from Coq to Micromega terms
* pp_* functions pretty-print Coq terms.
*)
(* Error datastructures *)
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
(* A simple but useful getter function *)
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
(* Access the Micromega module *)
module Mc = Micromega
(* parse/dump/print from numbers up to expressions and formulas *)
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 (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) -> if c = Lazy.force coq_Qmake then
{Mc.qnum = parse_z args.(0) ; Mc.qden = parse_positive args.(1) }
else raise ParseError
| _ -> raise ParseError
let rec parse_list parse_elt term =
let (i,c) = get_left_construct term in
match i with
| 1 -> []
| 2 -> parse_elt c.(1) :: parse_list parse_elt c.(2)
| i -> raise ParseError
let rec dump_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_list typ dump_elt l|])
let pp_list op cl elt o l =
let rec _pp o l =
match l with
| [] -> ()
| [e] -> Printf.fprintf o "%a" elt e
| 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 pp_expr pp_z o e =
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 in
pp_expr o e
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 dump_pol typ dump_c e =
let rec dump_pol e =
match e with
| Mc.Pc n -> mkApp(Lazy.force coq_Pc, [|typ ; dump_c n|])
| Mc.Pinj(p,pol) -> mkApp(Lazy.force coq_Pinj , [| typ ; dump_positive p ; dump_pol pol|])
| Mc.PX(pol1,p,pol2) -> mkApp(Lazy.force coq_PX, [| typ ; dump_pol pol1 ; dump_positive p ; dump_pol pol2|]) in
dump_pol e
let pp_pol pp_c o e =
let rec pp_pol o e =
match e with
| Mc.Pc n -> Printf.fprintf o "Pc %a" pp_c n
| Mc.Pinj(p,pol) -> Printf.fprintf o "Pinj(%a,%a)" pp_positive p pp_pol pol
| Mc.PX(pol1,p,pol2) -> Printf.fprintf o "PX(%a,%a,%a)" pp_pol pol1 pp_positive p pp_pol pol2 in
pp_pol o e
let pp_cnf pp_c o f =
let pp_clause o l = List.iter (fun ((p,_),t) -> Printf.fprintf o "(%a @%a)" (pp_pol pp_c) p Tag.pp t) l in
List.iter (fun l -> Printf.fprintf o "[%a]" pp_clause l) f
let dump_psatz typ dump_z e =
let z = Lazy.force typ in
let rec dump_cone e =
match e with
| Mc.PsatzIn n -> mkApp(Lazy.force coq_PsatzIn,[| z; dump_nat n |])
| Mc.PsatzMulC(e,c) -> mkApp(Lazy.force coq_PsatzMultC,
[| z; dump_pol z dump_z e ; dump_cone c |])
| Mc.PsatzSquare e -> mkApp(Lazy.force coq_PsatzSquare,
[| z;dump_pol z dump_z e|])
| Mc.PsatzAdd(e1,e2) -> mkApp(Lazy.force coq_PsatzAdd,
[| z; dump_cone e1; dump_cone e2|])
| Mc.PsatzMulE(e1,e2) -> mkApp(Lazy.force coq_PsatzMulE,
[| z; dump_cone e1; dump_cone e2|])
| Mc.PsatzC p -> mkApp(Lazy.force coq_PsatzC,[| z; dump_z p|])
| Mc.PsatzZ -> mkApp( Lazy.force coq_PsatzZ,[| z|]) in
dump_cone e
let pp_psatz pp_z o e =
let rec pp_cone o e =
match e with
| Mc.PsatzIn n ->
Printf.fprintf o "(In %a)%%nat" pp_nat n
| Mc.PsatzMulC(e,c) ->
Printf.fprintf o "( %a [*] %a)" (pp_pol pp_z) e pp_cone c
| Mc.PsatzSquare e ->
Printf.fprintf o "(%a^2)" (pp_pol pp_z) e
| Mc.PsatzAdd(e1,e2) ->
Printf.fprintf o "(%a [+] %a)" pp_cone e1 pp_cone e2
| Mc.PsatzMulE(e1,e2) ->
Printf.fprintf o "(%a [*] %a)" pp_cone e1 pp_cone e2
| Mc.PsatzC p ->
Printf.fprintf o "(%a)%%positive" pp_z p
| Mc.PsatzZ ->
Printf.fprintf o "0" in
pp_cone o e
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 pp_z o {Mc.flhs = l ; Mc.fop = op ; Mc.frhs = r } =
Printf.fprintf o"(%a %a %a)" (pp_expr pp_z) l pp_op op (pp_expr pp_z) 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 assoc_const x l =
try
snd (List.find (fun (x',y) -> x = Lazy.force x') l)
with
Not_found -> raise ParseError
let zop_table = [
coq_Zgt, Mc.OpGt ;
coq_Zge, Mc.OpGe ;
coq_Zlt, Mc.OpLt ;
coq_Zle, Mc.OpLe ]
let rop_table = [
coq_Rgt, Mc.OpGt ;
coq_Rge, Mc.OpGe ;
coq_Rlt, Mc.OpLt ;
coq_Rle, Mc.OpLe ]
let qop_table = [
coq_Qlt, Mc.OpLt ;
coq_Qle, Mc.OpLe ;
coq_Qeq, Mc.OpEq
]
let parse_zop (op,args) =
match kind_of_term op with
| Const x -> (assoc_const op zop_table, args.(0) , args.(1))
| Ind(n,0) ->
if op = Lazy.force coq_Eq && args.(0) = Lazy.force coq_Z
then (Mc.OpEq, args.(1), args.(2))
else raise ParseError
| _ -> failwith "parse_zop"
let parse_rop (op,args) =
match kind_of_term op with
| Const x -> (assoc_const op rop_table, args.(0) , args.(1))
| Ind(n,0) ->
if op = Lazy.force coq_Eq && args.(0) = Lazy.force coq_R
then (Mc.OpEq, args.(1), args.(2))
else raise ParseError
| _ -> failwith "parse_zop"
let parse_qop (op,args) =
(assoc_const op qop_table, args.(0) , args.(1))
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 assoc_ops x l =
try
snd (List.find (fun (x',y) -> x = Lazy.force x') l)
with
Not_found -> Ukn "Oups"
(**
* MODULE: Env is for environment.
*)
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 (* MODULE END: Env *)
(**
* This is the big generic function for expression parsers.
*)
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 assoc_ops t ops_spec 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 ->
begin
try
let (expr,env) = parse_expr env args.(0) in
let power = (parse_exp expr args.(1)) in
(power , env)
with _ -> (* if the exponent is a variable *)
let (env,n) = Env.compute_rank_add env term in (Mc.PEX n, env)
end
| 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 =
[
coq_Zplus , Binop (fun x y -> Mc.PEadd(x,y)) ;
coq_Zminus , Binop (fun x y -> Mc.PEsub(x,y)) ;
coq_Zmult , Binop (fun x y -> Mc.PEmul (x,y)) ;
coq_Zopp , Opp ;
coq_Zpower , Power]
let qop_spec =
[
coq_Qplus , Binop (fun x y -> Mc.PEadd(x,y)) ;
coq_Qminus , Binop (fun x y -> Mc.PEsub(x,y)) ;
coq_Qmult , Binop (fun x y -> Mc.PEmul (x,y)) ;
coq_Qopp , Opp ;
coq_Qpower , Power]
let rop_spec =
[
coq_Rplus , Binop (fun x y -> Mc.PEadd(x,y)) ;
coq_Rminus , Binop (fun x y -> Mc.PEsub(x,y)) ;
coq_Rmult , Binop (fun x y -> Mc.PEmul (x,y)) ;
coq_Ropp , Opp ;
coq_Rpower , Power]
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 ->
if term = Lazy.force coq_R0
then Mc.Z0
else if term = Lazy.force coq_R1
then Mc.Zpos Mc.XH
else raise ParseError
| _ -> raise ParseError
let parse_zexpr = parse_expr
zconstant
(fun expr x ->
let exp = (parse_z x) in
match exp with
| Mc.Zneg _ -> Mc.PEc Mc.Z0
| _ -> Mc.PEpow(expr, Mc.n_of_Z exp))
zop_spec
let parse_qexpr = parse_expr
qconstant
(fun expr x ->
let exp = parse_z x in
match exp with
| Mc.Zneg _ ->
begin
match expr with
| Mc.PEc q -> Mc.PEc (Mc.qpower q exp)
| _ -> print_string "parse_qexpr parse error" ; flush stdout ; raise ParseError
end
| _ -> let exp = Mc.n_of_Z exp in
Mc.PEpow(expr,exp))
qop_spec
let parse_rexpr = parse_expr
rconstant
(fun expr x ->
let exp = Mc.n_of_nat (parse_nat x) in
Mc.PEpow(expr,exp))
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 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)
let mkD f1 f2 = D(f1,f2)
let mkIff f1 f2 = C(I(f1,None,f2),I(f2,None,f1))
let mkI f1 f2 = I(f1,None,f2)
let mkformula_binary g term f1 f2 =
match f1 , f2 with
| X _ , X _ -> X(term)
| _ -> g f1 f2
(**
* This is the big generic function for formula parsers.
*)
let parse_formula parse_atom env term =
let parse_atom env tg t = try let (at,env) = parse_atom env t in
(A(at,tg,t), env,Tag.next tg) with _ -> (X(t),env,tg) in
let rec xparse_formula env tg term =
match kind_of_term term with
| App(l,rst) ->
(match rst with
| [|a;b|] when l = Lazy.force coq_and ->
let f,env,tg = xparse_formula env tg a in
let g,env, tg = xparse_formula env tg b in
mkformula_binary mkC term f g,env,tg
| [|a;b|] when l = Lazy.force coq_or ->
let f,env,tg = xparse_formula env tg a in
let g,env,tg = xparse_formula env tg b in
mkformula_binary mkD term f g,env,tg
| [|a|] when l = Lazy.force coq_not ->
let (f,env,tg) = xparse_formula env tg a in (N(f), env,tg)
| [|a;b|] when l = Lazy.force coq_iff ->
let f,env,tg = xparse_formula env tg a in
let g,env,tg = xparse_formula env tg b in
mkformula_binary mkIff term f g,env,tg
| _ -> parse_atom env tg term)
| Prod(typ,a,b) when not (Termops.dependent (mkRel 1) b) ->
let f,env,tg = xparse_formula env tg a in
let g,env,tg = xparse_formula env tg b in
mkformula_binary mkI term f g,env,tg
| _ when term = Lazy.force coq_True -> (TT,env,tg)
| _ when term = Lazy.force coq_False -> (FF,env,tg)
| _ -> X(term),env,tg in
xparse_formula env term
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
(**
* Given a conclusion and a list of affectations, rebuild a term prefixed by
* the appropriate letins.
* TODO: reverse the list of bindings!
*)
let set l concl =
let rec xset acc = function
| [] -> acc
| (e::l) ->
let (name,expr,typ) = e in
xset (Term.mkNamedLetIn
(Names.id_of_string name)
expr typ acc) l in
xset concl l
end (**
* MODULE END: M
*)
open M
let rec sig_of_cone = function
| Mc.PsatzIn n -> [CoqToCaml.nat n]
| Mc.PsatzMulE(w1,w2) -> (sig_of_cone w1)@(sig_of_cone w2)
| Mc.PsatzMulC(w1,w2) -> (sig_of_cone w2)
| Mc.PsatzAdd(w1,w2) -> (sig_of_cone w1)@(sig_of_cone w2)
| _ -> []
let same_proof sg cl1 cl2 =
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.PsatzIn n -> Names.Idset.union tgs
(snd (List.nth clause (CoqToCaml.nat n) ))
| Mc.PsatzMulC(e,w) -> xtags tgs w
| Mc.PsatzMulE (w1,w2) | Mc.PsatzAdd(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 = try_any prover polys1
let rec witness prover l1 l2 =
match l2 with
| [] -> Some []
| e :: l2 ->
match find_witness prover (e::l1) with
| None -> None
| Some w ->
(match witness prover l1 l2 with
| None -> None
| Some l -> Some (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.DoneProof -> Lazy.force coq_doneProof
| Micromega.RatProof(cone,rst) ->
Term.mkApp(Lazy.force coq_ratProof, [| dump_psatz coq_Z dump_z cone; dump_proof_term rst|])
| Micromega.CutProof(cone,prf) ->
Term.mkApp(Lazy.force coq_cutProof,
[| dump_psatz coq_Z dump_z cone ;
dump_proof_term prf|])
| Micromega.EnumProof(c1,c2,prfs) ->
Term.mkApp (Lazy.force coq_enumProof,
[| dump_psatz coq_Z dump_z c1 ; dump_psatz 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.DoneProof -> Printf.fprintf o "D"
| Micromega.RatProof(cone,rst) -> Printf.fprintf o "R[%a,%a]" (pp_psatz pp_z) cone pp_proof_term rst
| Micromega.CutProof(cone,rst) -> Printf.fprintf o "C[%a,%a]" (pp_psatz pp_z) cone pp_proof_term rst
| Micromega.EnumProof(c1,c2,rst) ->
Printf.fprintf o "EP[%a,%a,%a]"
(pp_psatz pp_z) c1 (pp_psatz pp_z) c2
(pp_list "[" "]" pp_proof_term) rst
let rec parse_hyps parse_arith env tg hyps =
match hyps with
| [] -> ([],env,tg)
| (i,t)::l ->
let (lhyps,env,tg) = parse_hyps parse_arith env tg l in
try
let (c,env,tg) = parse_formula parse_arith env tg t in
((i,c)::lhyps, env,tg)
with _ -> (lhyps,env,tg)
(*(if debug then Printf.printf "parse_arith : %s\n" x);*)
(*exception ParseError*)
let parse_goal parse_arith env hyps term =
(* try*)
let (f,env,tg) = parse_formula parse_arith env (Tag.from 0) term in
let (lhyps,env,tg) = parse_hyps parse_arith env tg hyps in
(lhyps,f,env)
(* with Failure x -> raise ParseError*)
(**
* The datastructures that aggregate theory-dependent proof values.
*)
type ('d, 'prf) domain_spec = {
typ : Term.constr; (* Z, Q , R *)
coeff : Term.constr ; (* Z, Q *)
dump_coeff : 'd -> Term.constr ;
proof_typ : Term.constr ;
dump_proof : 'prf -> 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_psatz 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_psatz coq_Z dump_z
}
(**
* Instanciate the current Coq goal with a Micromega formula, a varmap, and a
* witness.
*)
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
(**
* The datastructures that aggregate prover attributes.
*)
type ('a,'prf) prover = {
name : string ; (* name of the prover *)
prover : 'a list -> 'prf option ; (* the prover itself *)
hyps : 'prf -> ISet.t ; (* extract the indexes of the hypotheses really used in the proof *)
compact : 'prf -> (int -> int) -> 'prf ; (* remap the hyp indexes according to function *)
pp_prf : out_channel -> 'prf -> unit ;(* pretting printing of proof *)
pp_f : out_channel -> 'a -> unit (* pretty printing of the formulas (polynomials)*)
}
(**
* Given a list of provers and a disjunction of atoms, find a proof of any of
* the atoms. Returns an (optional) pair of a proof and a prover
* datastructure.
*)
let find_witness provers polys1 =
let provers = List.map (fun p ->
(fun l ->
match p.prover l with
| None -> None
| Some prf -> Some(prf,p)) , p.name) provers in
try_any provers (List.map fst polys1)
(**
* Given a list of provers and a CNF, find a proof for each of the clauses.
* Return the proofs as a list.
*)
let witness_list 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_tags = witness_list
(* *Deprecated* let is_singleton = function [] -> true | [e] -> true | _ -> false *)
let pp_ml_list pp_elt o l =
output_string o "[" ;
List.iter (fun x -> Printf.fprintf o "%a ;" pp_elt x) l ;
output_string o "]"
(**
* Prune the proof object, according to the 'diff' between two cnf formulas.
*)
let compact_proofs (cnf_ff: 'cst cnf) res (cnf_ff': 'cst cnf) =
let compact_proof (old_cl:'cst clause) (prf,prover) (new_cl:'cst clause) =
let new_cl = Mutils.mapi (fun (f,_) i -> (f,i)) new_cl in
let remap i =
let formula = try fst (List.nth old_cl i) with Failure _ -> failwith "bad old index" in
List.assoc formula new_cl in
if debug then
begin
Printf.printf "\ncompact_proof : %a %a %a"
(pp_ml_list prover.pp_f) (List.map fst old_cl)
prover.pp_prf prf
(pp_ml_list prover.pp_f) (List.map fst new_cl) ;
flush stdout
end ;
let res = try prover.compact prf remap with x ->
if debug then Printf.fprintf stdout "Proof compaction %s" (Printexc.to_string x) ;
(* This should not happen -- this is the recovery plan... *)
match prover.prover (List.map fst new_cl) with
| None -> failwith "proof compaction error"
| Some p -> p
in
if debug then
begin
Printf.printf " -> %a\n"
prover.pp_prf res ;
flush stdout
end ;
res in
let is_proof_compatible (old_cl:'cst clause) (prf,prover) (new_cl:'cst clause) =
let hyps_idx = prover.hyps prf in
let hyps = selecti hyps_idx old_cl in
is_sublist hyps new_cl in
let cnf_res = List.combine cnf_ff res in (* we get pairs clause * proof *)
List.map (fun x ->
let (o,p) = List.find (fun (l,p) -> is_proof_compatible l p x) cnf_res
in compact_proof o p x) cnf_ff'
(**
* "Hide out" tagged atoms of a formula by transforming them into generic
* variables. See the Tag module in mutils.ml for more.
*)
let abstract_formula hyps f =
let rec xabs f =
match f with
| X c -> X c
| A(a,t,term) -> if TagSet.mem t hyps then A(a,t,term) else X(term)
| C(f1,f2) ->
(match xabs f1 , xabs f2 with
| X a1 , X a2 -> X (Term.mkApp(Lazy.force coq_and, [|a1;a2|]))
| f1 , f2 -> C(f1,f2) )
| D(f1,f2) ->
(match xabs f1 , xabs f2 with
| X a1 , X a2 -> X (Term.mkApp(Lazy.force coq_or, [|a1;a2|]))
| f1 , f2 -> D(f1,f2) )
| N(f) ->
(match xabs f with
| X a -> X (Term.mkApp(Lazy.force coq_not, [|a|]))
| f -> N f)
| I(f1,hyp,f2) ->
(match xabs f1 , hyp, xabs f2 with
| X a1 , Some _ , af2 -> af2
| X a1 , None , X a2 -> X (Term.mkArrow a1 a2)
| af1 , _ , af2 -> I(af1,hyp,af2)
)
| FF -> FF
| TT -> TT
in xabs f
(**
* This exception is raised by really_call_csdpcert if Coq's configure didn't
* find a CSDP executable.
*)
exception CsdpNotFound
(**
* This is the core of Micromega: apply the prover, analyze the result and
* prune unused fomulas, and finally modify the proof state.
*)
let micromega_tauto negate normalise spec prover env polys1 polys2 gl =
let spec = Lazy.force spec in
(* Express the goal as one big implication *)
let (ff,ids) =
List.fold_right
(fun (id,f) (cc,ids) ->
match f with
X _ -> (cc,ids)
| _ -> (I(f,Some id,cc), id::ids))
polys1 (polys2,[]) in
(* Convert the aplpication into a (mc_)cnf (a list of lists of formulas) *)
let cnf_ff = cnf negate normalise ff in
if debug then
begin
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 ();
Printf.fprintf stdout "cnf : %a\n" (pp_cnf (fun o _ -> ())) cnf_ff
end;
match witness_list_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 hyps = List.fold_left (fun s (cl,(prf,p)) ->
let tags = ISet.fold (fun i s -> let t = snd (List.nth cl i) in
if debug then (Printf.fprintf stdout "T : %i -> %a" i Tag.pp t) ;
(*try*) TagSet.add t s (* with Invalid_argument _ -> s*)) (p.hyps prf) TagSet.empty in
TagSet.union s tags) TagSet.empty (List.combine cnf_ff res) in
if debug then (Printf.printf "TForm : %a\n" pp_formula ff ; flush stdout;
Printf.printf "Hyps : %a\n" (fun o s -> TagSet.fold (fun i _ -> Printf.fprintf o "%a " Tag.pp i) s ()) hyps) ;
let ff' = abstract_formula hyps ff in
let cnf_ff' = cnf negate normalise ff' in
if debug then
begin
Pp.pp (Pp.str "\nAFormula\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 ();
Printf.fprintf stdout "cnf : %a\n" (pp_cnf (fun o _ -> ())) cnf_ff'
end;
(* Even if it does not work, this does not mean it is not provable
-- the prover is REALLY incomplete *)
(* if debug then
begin
(* recompute the proofs *)
match witness_list_tags prover cnf_ff' with
| None -> failwith "abstraction is wrong"
| Some res -> ()
end ; *)
let res' = compact_proofs cnf_ff res cnf_ff' in
let (ff',res',ids) = (ff',res',List.map Term.mkVar (ids_of_formula ff')) in
let res' = dump_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
(**
* Parse the proof environment, and call micromega_tauto
*)
let micromega_gen
parse_arith
(negate:'cst atom -> 'cst mc_cnf)
(normalise:'cst atom -> 'cst mc_cnf)
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
| CsdpNotFound -> flush stdout ; Pp.pp_flush () ;
Tacticals.tclFAIL 0 (Pp.str
(" Skipping what remains of this tactic: the complexity of the goal requires "
^ "the use of a specialized external tool called csdp. \n\n"
^ "Unfortunately Coq isn't aware of the presence of any \"csdp\" executable in the path. \n\n"
^ "Csdp packages are provided by some OS distributions; binaries and source code can be downloaded from https://projects.coin-or.org/Csdp")) gl
let lift_ratproof prover l =
match prover l with
| None -> None
| Some c -> Some (Mc.RatProof( c,Mc.DoneProof))
type micromega_polys = (Micromega.q Mc.pol * Mc.op1) list
type csdp_certificate = S of Sos_types.positivstellensatz option | F of string
type provername = string * int option
(**
* The caching mechanism.
*)
open Persistent_cache
module Cache = PHashtable(struct
type t = (provername * micromega_polys)
let equal = (=)
let hash = Hashtbl.hash
end)
let csdp_cache = "csdp.cache"
(**
* Build the command to call csdpcert, and launch it. This in turn will call
* the sos driver to the csdp executable.
* Throw CsdpNotFound if Coq isn't aware of any csdp executable.
*)
let require_csdp =
if System.is_in_system_path "csdp"
then lazy ()
else lazy (raise CsdpNotFound)
let really_call_csdpcert : provername -> micromega_polys -> Sos_types.positivstellensatz option =
fun provername poly ->
Lazy.force require_csdp;
let cmdname =
List.fold_left Filename.concat (Envars.coqlib ())
["plugins"; "micromega"; "csdpcert" ^ Coq_config.exec_extension] in
match ((command cmdname [|cmdname|] (provername,poly)) : csdp_certificate) with
| F str -> failwith str
| S res -> res
(**
* Check the cache before calling the prover.
*)
let xcall_csdpcert =
Cache.memo csdp_cache (fun (prover,pb) -> really_call_csdpcert prover pb)
(**
* Prover callback functions.
*)
let call_csdpcert prover pb = xcall_csdpcert (prover,pb)
let rec z_to_q_pol e =
match e with
| Mc.Pc z -> Mc.Pc {Mc.qnum = z ; Mc.qden = Mc.XH}
| Mc.Pinj(p,pol) -> Mc.Pinj(p,z_to_q_pol pol)
| Mc.PX(pol1,p,pol2) -> Mc.PX(z_to_q_pol pol1, p, z_to_q_pol pol2)
let call_csdpcert_q provername poly =
match call_csdpcert provername poly with
| None -> None
| Some cert ->
let cert = Certificate.q_cert_of_pos cert in
if Mc.qWeakChecker poly cert
then Some cert
else ((print_string "buggy certificate" ; flush stdout) ;None)
let call_csdpcert_z provername poly =
let l = List.map (fun (e,o) -> (z_to_q_pol e,o)) poly in
match call_csdpcert provername l with
| None -> None
| Some cert ->
let cert = Certificate.z_cert_of_pos cert in
if Mc.zWeakChecker poly cert
then Some cert
else ((print_string "buggy certificate" ; flush stdout) ;None)
let xhyps_of_cone base acc prf =
let rec xtract e acc =
match e with
| Mc.PsatzC _ | Mc.PsatzZ | Mc.PsatzSquare _ -> acc
| Mc.PsatzIn n -> let n = (CoqToCaml.nat n) in
if n >= base
then ISet.add (n-base) acc
else acc
| Mc.PsatzMulC(_,c) -> xtract c acc
| Mc.PsatzAdd(e1,e2) | Mc.PsatzMulE(e1,e2) -> xtract e1 (xtract e2 acc) in
xtract prf acc
let hyps_of_cone prf = xhyps_of_cone 0 ISet.empty prf
let compact_cone prf f =
let np n = CamlToCoq.nat (f (CoqToCaml.nat n)) in
let rec xinterp prf =
match prf with
| Mc.PsatzC _ | Mc.PsatzZ | Mc.PsatzSquare _ -> prf
| Mc.PsatzIn n -> Mc.PsatzIn (np n)
| Mc.PsatzMulC(e,c) -> Mc.PsatzMulC(e,xinterp c)
| Mc.PsatzAdd(e1,e2) -> Mc.PsatzAdd(xinterp e1,xinterp e2)
| Mc.PsatzMulE(e1,e2) -> Mc.PsatzMulE(xinterp e1,xinterp e2) in
xinterp prf
let hyps_of_pt pt =
let rec xhyps base pt acc =
match pt with
| Mc.DoneProof -> acc
| Mc.RatProof(c,pt) -> xhyps (base+1) pt (xhyps_of_cone base acc c)
| Mc.CutProof(c,pt) -> xhyps (base+1) pt (xhyps_of_cone base acc c)
| Mc.EnumProof(c1,c2,l) ->
let s = xhyps_of_cone base (xhyps_of_cone base acc c2) c1 in
List.fold_left (fun s x -> xhyps (base + 1) x s) s l in
xhyps 0 pt ISet.empty
let hyps_of_pt pt =
let res = hyps_of_pt pt in
if debug
then (Printf.fprintf stdout "\nhyps_of_pt : %a -> " pp_proof_term pt ; ISet.iter (fun i -> Printf.printf "%i " i) res);
res
let compact_pt pt f =
let translate ofset x =
if x < ofset then x
else (f (x-ofset) + ofset) in
let rec compact_pt ofset pt =
match pt with
| Mc.DoneProof -> Mc.DoneProof
| Mc.RatProof(c,pt) -> Mc.RatProof(compact_cone c (translate (ofset)), compact_pt (ofset+1) pt )
| Mc.CutProof(c,pt) -> Mc.CutProof(compact_cone c (translate (ofset)), compact_pt (ofset+1) pt )
| Mc.EnumProof(c1,c2,l) -> Mc.EnumProof(compact_cone c1 (translate (ofset)), compact_cone c2 (translate (ofset)),
Mc.map (fun x -> compact_pt (ofset+1) x) l) in
compact_pt 0 pt
(**
* Definition of provers.
* Instantiates the type ('a,'prf) prover defined above.
*)
let lift_pexpr_prover p l = p (List.map (fun (e,o) -> Mc.denorm e , o) l)
let linear_prover_Z = {
name = "linear prover" ;
prover = lift_ratproof (lift_pexpr_prover (Certificate.linear_prover_with_cert Certificate.z_spec)) ;
hyps = hyps_of_pt ;
compact = compact_pt ;
pp_prf = pp_proof_term;
pp_f = fun o x -> pp_pol pp_z o (fst x)
}
let linear_prover_Q = {
name = "linear prover";
prover = lift_pexpr_prover (Certificate.linear_prover_with_cert Certificate.q_spec) ;
hyps = hyps_of_cone ;
compact = compact_cone ;
pp_prf = pp_psatz pp_q ;
pp_f = fun o x -> pp_pol pp_q o (fst x)
}
let linear_prover_R = {
name = "linear prover";
prover = lift_pexpr_prover (Certificate.linear_prover_with_cert Certificate.z_spec) ;
hyps = hyps_of_cone ;
compact = compact_cone ;
pp_prf = pp_psatz pp_z ;
pp_f = fun o x -> pp_pol pp_z o (fst x)
}
let non_linear_prover_Q str o = {
name = "real nonlinear prover";
prover = call_csdpcert_q (str, o);
hyps = hyps_of_cone;
compact = compact_cone ;
pp_prf = pp_psatz pp_q ;
pp_f = fun o x -> pp_pol pp_q o (fst x)
}
let non_linear_prover_R str o = {
name = "real nonlinear prover";
prover = call_csdpcert_z (str, o);
hyps = hyps_of_cone;
compact = compact_cone;
pp_prf = pp_psatz pp_z;
pp_f = fun o x -> pp_pol pp_z o (fst x)
}
let non_linear_prover_Z str o = {
name = "real nonlinear prover";
prover = lift_ratproof (call_csdpcert_z (str, o));
hyps = hyps_of_pt;
compact = compact_pt;
pp_prf = pp_proof_term;
pp_f = fun o x -> pp_pol pp_z o (fst x)
}
module CacheZ = PHashtable(struct
type t = (Mc.z Mc.pol * Mc.op1) list
let equal = (=)
let hash = Hashtbl.hash
end)
let memo_zlinear_prover = CacheZ.memo "lia.cache" (lift_pexpr_prover Certificate.zlinear_prover)
let linear_Z = {
name = "lia";
prover = memo_zlinear_prover ;
hyps = hyps_of_pt;
compact = compact_pt;
pp_prf = pp_proof_term;
pp_f = fun o x -> pp_pol pp_z o (fst x)
}
(**
* Functions instantiating micromega_gen with the appropriate theories and
* solvers
*)
let psatzl_Z gl =
micromega_gen parse_zarith Mc.negate Mc.normalise zz_domain_spec
[ linear_prover_Z ] gl
let psatzl_Q gl =
micromega_gen parse_qarith Mc.qnegate Mc.qnormalise qq_domain_spec
[ linear_prover_Q ] gl
let psatz_Q i gl =
micromega_gen parse_qarith Mc.qnegate Mc.qnormalise qq_domain_spec
[ non_linear_prover_Q "real_nonlinear_prover" (Some i) ] gl
let psatzl_R gl =
micromega_gen parse_rarith Mc.rnegate Mc.rnormalise rz_domain_spec
[ linear_prover_R ] gl
let psatz_R i gl =
micromega_gen parse_rarith Mc.rnegate Mc.rnormalise rz_domain_spec
[ non_linear_prover_R "real_nonlinear_prover" (Some i) ] gl
let psatz_Z i gl =
micromega_gen parse_zarith Mc.negate Mc.normalise zz_domain_spec
[ non_linear_prover_Z "real_nonlinear_prover" (Some i) ] gl
let sos_Z gl =
micromega_gen parse_zarith Mc.negate Mc.normalise zz_domain_spec
[ non_linear_prover_Z "pure_sos" None ] gl
let sos_Q gl =
micromega_gen parse_qarith Mc.qnegate Mc.qnormalise qq_domain_spec
[ non_linear_prover_Q "pure_sos" None ] gl
let sos_R gl =
micromega_gen parse_rarith Mc.rnegate Mc.rnormalise rz_domain_spec
[ non_linear_prover_R "pure_sos" None ] gl
let xlia gl =
micromega_gen parse_zarith Mc.negate Mc.normalise zz_domain_spec
[ linear_Z ] gl
(* Local Variables: *)
(* coding: utf-8 *)
(* End: *)
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