Fixed bug in adjustcoeffeq_wp.
(* Title: HOL/Integ/cooper_proof.ML
ID: $Id$
Author: Amine Chaieb and Tobias Nipkow, TU Muenchen
License: GPL (GNU GENERAL PUBLIC LICENSE)
File containing the implementation of the proof
generation for Cooper Algorithm
*)
signature COOPER_PROOF =
sig
val qe_Not : thm
val qe_conjI : thm
val qe_disjI : thm
val qe_impI : thm
val qe_eqI : thm
val qe_exI : thm
val qe_get_terms : thm -> term * term
val cooper_prv : Sign.sg -> term -> term -> string list -> thm
val proof_of_evalc : Sign.sg -> term -> thm
val proof_of_cnnf : Sign.sg -> term -> (term -> thm) -> thm
val proof_of_linform : Sign.sg -> string list -> term -> thm
end;
structure CooperProof : COOPER_PROOF =
struct
open CooperDec;
(*-----------------------------------------------------------------*)
(*-----------------------------------------------------------------*)
(*-----------------------------------------------------------------*)
(*--- ---*)
(*--- ---*)
(*--- Protocoling part ---*)
(*--- ---*)
(*--- includes the protocolling datastructure ---*)
(*--- ---*)
(*--- and the protocolling fuctions ---*)
(*--- ---*)
(*--- ---*)
(*-----------------------------------------------------------------*)
(*-----------------------------------------------------------------*)
(*-----------------------------------------------------------------*)
val presburger_ss = simpset_of (theory "Presburger")
addsimps [zdiff_def] delsimps [symmetric zdiff_def];
val cboolT = ctyp_of (sign_of HOL.thy) HOLogic.boolT;
(*Theorems that will be used later for the proofgeneration*)
val zdvd_iff_zmod_eq_0 = thm "zdvd_iff_zmod_eq_0";
val unity_coeff_ex = thm "unity_coeff_ex";
(* Thorems for proving the adjustment of the coeffitients*)
val ac_lt_eq = thm "ac_lt_eq";
val ac_eq_eq = thm "ac_eq_eq";
val ac_dvd_eq = thm "ac_dvd_eq";
val ac_pi_eq = thm "ac_pi_eq";
(* The logical compination of the sythetised properties*)
val qe_Not = thm "qe_Not";
val qe_conjI = thm "qe_conjI";
val qe_disjI = thm "qe_disjI";
val qe_impI = thm "qe_impI";
val qe_eqI = thm "qe_eqI";
val qe_exI = thm "qe_exI";
val qe_ALLI = thm "qe_ALLI";
(*Modulo D property for Pminusinf an Plusinf *)
val fm_modd_minf = thm "fm_modd_minf";
val not_dvd_modd_minf = thm "not_dvd_modd_minf";
val dvd_modd_minf = thm "dvd_modd_minf";
val fm_modd_pinf = thm "fm_modd_pinf";
val not_dvd_modd_pinf = thm "not_dvd_modd_pinf";
val dvd_modd_pinf = thm "dvd_modd_pinf";
(* the minusinfinity proprty*)
val fm_eq_minf = thm "fm_eq_minf";
val neq_eq_minf = thm "neq_eq_minf";
val eq_eq_minf = thm "eq_eq_minf";
val le_eq_minf = thm "le_eq_minf";
val len_eq_minf = thm "len_eq_minf";
val not_dvd_eq_minf = thm "not_dvd_eq_minf";
val dvd_eq_minf = thm "dvd_eq_minf";
(* the Plusinfinity proprty*)
val fm_eq_pinf = thm "fm_eq_pinf";
val neq_eq_pinf = thm "neq_eq_pinf";
val eq_eq_pinf = thm "eq_eq_pinf";
val le_eq_pinf = thm "le_eq_pinf";
val len_eq_pinf = thm "len_eq_pinf";
val not_dvd_eq_pinf = thm "not_dvd_eq_pinf";
val dvd_eq_pinf = thm "dvd_eq_pinf";
(*Logical construction of the Property*)
val eq_minf_conjI = thm "eq_minf_conjI";
val eq_minf_disjI = thm "eq_minf_disjI";
val modd_minf_disjI = thm "modd_minf_disjI";
val modd_minf_conjI = thm "modd_minf_conjI";
val eq_pinf_conjI = thm "eq_pinf_conjI";
val eq_pinf_disjI = thm "eq_pinf_disjI";
val modd_pinf_disjI = thm "modd_pinf_disjI";
val modd_pinf_conjI = thm "modd_pinf_conjI";
(*A/B - set Theorem *)
val bst_thm = thm "bst_thm";
val ast_thm = thm "ast_thm";
(*Cooper Backwards...*)
(*Bset*)
val not_bst_p_fm = thm "not_bst_p_fm";
val not_bst_p_ne = thm "not_bst_p_ne";
val not_bst_p_eq = thm "not_bst_p_eq";
val not_bst_p_gt = thm "not_bst_p_gt";
val not_bst_p_lt = thm "not_bst_p_lt";
val not_bst_p_ndvd = thm "not_bst_p_ndvd";
val not_bst_p_dvd = thm "not_bst_p_dvd";
(*Aset*)
val not_ast_p_fm = thm "not_ast_p_fm";
val not_ast_p_ne = thm "not_ast_p_ne";
val not_ast_p_eq = thm "not_ast_p_eq";
val not_ast_p_gt = thm "not_ast_p_gt";
val not_ast_p_lt = thm "not_ast_p_lt";
val not_ast_p_ndvd = thm "not_ast_p_ndvd";
val not_ast_p_dvd = thm "not_ast_p_dvd";
(*Logical construction of the prop*)
(*Bset*)
val not_bst_p_conjI = thm "not_bst_p_conjI";
val not_bst_p_disjI = thm "not_bst_p_disjI";
val not_bst_p_Q_elim = thm "not_bst_p_Q_elim";
(*Aset*)
val not_ast_p_conjI = thm "not_ast_p_conjI";
val not_ast_p_disjI = thm "not_ast_p_disjI";
val not_ast_p_Q_elim = thm "not_ast_p_Q_elim";
(*Cooper*)
val cppi_eq = thm "cppi_eq";
val cpmi_eq = thm "cpmi_eq";
(*Others*)
val simp_from_to = thm "simp_from_to";
val P_eqtrue = thm "P_eqtrue";
val P_eqfalse = thm "P_eqfalse";
(*For Proving NNF*)
val nnf_nn = thm "nnf_nn";
val nnf_im = thm "nnf_im";
val nnf_eq = thm "nnf_eq";
val nnf_sdj = thm "nnf_sdj";
val nnf_ncj = thm "nnf_ncj";
val nnf_nim = thm "nnf_nim";
val nnf_neq = thm "nnf_neq";
val nnf_ndj = thm "nnf_ndj";
(*For Proving term linearizition*)
val linearize_dvd = thm "linearize_dvd";
val lf_lt = thm "lf_lt";
val lf_eq = thm "lf_eq";
val lf_dvd = thm "lf_dvd";
(* ------------------------------------------------------------------------- *)
(*Datatatype declarations for Proofprotocol for the cooperprocedure.*)
(* ------------------------------------------------------------------------- *)
(* ------------------------------------------------------------------------- *)
(*Datatatype declarations for Proofprotocol for the adjustcoeff step.*)
(* ------------------------------------------------------------------------- *)
datatype CpLog = No
|Simp of term*CpLog
|Blast of CpLog*CpLog
|Aset of (term*term*(term list)*term)
|Bset of (term*term*(term list)*term)
|Minusinf of CpLog*CpLog
|Cooper of term*CpLog*CpLog*CpLog
|Eq_minf of term*term
|Modd_minf of term*term
|Eq_minf_conjI of CpLog*CpLog
|Modd_minf_conjI of CpLog*CpLog
|Modd_minf_disjI of CpLog*CpLog
|Eq_minf_disjI of CpLog*CpLog
|Not_bst_p of term*term*term*term*CpLog
|Not_bst_p_atomic of term
|Not_bst_p_conjI of CpLog*CpLog
|Not_bst_p_disjI of CpLog*CpLog
|Not_ast_p of term*term*term*term*CpLog
|Not_ast_p_atomic of term
|Not_ast_p_conjI of CpLog*CpLog
|Not_ast_p_disjI of CpLog*CpLog
|CpLogError;
datatype ACLog = ACAt of int*term
|ACPI of int*term
|ACfm of term
|ACNeg of ACLog
|ACConst of string*ACLog*ACLog;
(* ------------------------------------------------------------------------- *)
(*Datatatype declarations for Proofprotocol for the CNNF step.*)
(* ------------------------------------------------------------------------- *)
datatype NNFLog = NNFAt of term
|NNFSimp of NNFLog
|NNFNN of NNFLog
|NNFConst of string*NNFLog*NNFLog;
(* ------------------------------------------------------------------------- *)
(*Datatatype declarations for Proofprotocol for the linform step.*)
(* ------------------------------------------------------------------------- *)
datatype LfLog = LfAt of term
|LfAtdvd of term
|Lffm of term
|LfConst of string*LfLog*LfLog
|LfNot of LfLog
|LfQ of string*string*typ*LfLog;
(* ------------------------------------------------------------------------- *)
(*Datatatype declarations for Proofprotocol for the evaluation- evalc- step.*)
(* ------------------------------------------------------------------------- *)
datatype EvalLog = EvalAt of term
|Evalfm of term
|EvalConst of string*EvalLog*EvalLog;
(* ------------------------------------------------------------------------- *)
(*This function norm_zero_one replaces the occurences of Numeral1 and Numeral0*)
(*Respectively by their abstract representation Const("1",..) and COnst("0",..)*)
(*this is necessary because the theorems use this representation.*)
(* This function should be elminated in next versions...*)
(* ------------------------------------------------------------------------- *)
fun norm_zero_one fm = case fm of
(Const ("op *",_) $ c $ t) =>
if c = one then (norm_zero_one t)
else if (dest_numeral c = ~1)
then (Const("uminus",HOLogic.intT --> HOLogic.intT) $ (norm_zero_one t))
else (HOLogic.mk_binop "op *" (norm_zero_one c,norm_zero_one t))
|(node $ rest) => ((norm_zero_one node)$(norm_zero_one rest))
|(Abs(x,T,p)) => (Abs(x,T,(norm_zero_one p)))
|_ => fm;
(* ------------------------------------------------------------------------- *)
(* Intended to tell that here we changed the structure of the formula with respect to the posineq theorem : ~(0 < t) = 0 < 1-t*)
(* ------------------------------------------------------------------------- *)
fun adjustcoeffeq_wp x l fm =
case fm of
(Const("Not",_)$(Const("op <",_) $(Const("0",_)) $(rt as (Const ("op +", _)$(Const ("op *",_) $ c $ y ) $z )))) =>
if (x = y)
then let
val m = l div (dest_numeral c)
val n = abs (m)
val xtm = (HOLogic.mk_binop "op *" ((mk_numeral ((m div n)*l) ), x))
val rs = (HOLogic.mk_binrel "op <" (zero,linear_sub [] (mk_numeral n) (HOLogic.mk_binop "op +" ( xtm ,( linear_cmul n z) ))))
in (ACPI(n,fm),rs)
end
else let val rs = (HOLogic.mk_binrel "op <" (zero,linear_sub [] one rt ))
in (ACPI(1,fm),rs)
end
|(Const(p,_) $d $( Const ("op +", _)$(Const ("op *",_) $
c $ y ) $z )) => if (is_arith_rel fm) andalso (x = y) then
let val m = l div (dest_numeral c)
val n = (if p = "op <" then abs(m) else m)
val xtm = (HOLogic.mk_binop "op *" ((mk_numeral ((m div n)*l) ), x))
val rs = (HOLogic.mk_binrel p ((linear_cmul n d),(HOLogic.mk_binop "op +" ( xtm ,( linear_cmul n z) ))))
in (ACAt(n,fm),rs)
end
else (ACfm(fm),fm)
|( Const ("Not", _) $ p) => let val (rsp,rsr) = adjustcoeffeq_wp x l p
in (ACNeg(rsp),HOLogic.Not $ rsr)
end
|( Const ("op &",_) $ p $ q) =>let val (rspp,rspr) = adjustcoeffeq_wp x l p
val (rsqp,rsqr) = adjustcoeffeq_wp x l q
in (ACConst ("CJ",rspp,rsqp), HOLogic.mk_conj (rspr,rsqr))
end
|( Const ("op |",_) $ p $ q) =>let val (rspp,rspr) = adjustcoeffeq_wp x l p
val (rsqp,rsqr) = adjustcoeffeq_wp x l q
in (ACConst ("DJ",rspp,rsqp), HOLogic.mk_disj (rspr,rsqr))
end
|_ => (ACfm(fm),fm);
(*_________________________________________*)
(*-----------------------------------------*)
(* Protocol generation for the liform step *)
(*_________________________________________*)
(*-----------------------------------------*)
fun linform_wp fm =
let fun at_linform_wp at =
case at of
(Const("op <=",_)$s$t) => LfAt(at)
|(Const("op <",_)$s$t) => LfAt(at)
|(Const("op =",_)$s$t) => LfAt(at)
|(Const("Divides.op dvd",_)$s$t) => LfAtdvd(at)
in
if is_arith_rel fm
then at_linform_wp fm
else case fm of
(Const("Not",_) $ A) => LfNot(linform_wp A)
|(Const("op &",_)$ A $ B) => LfConst("CJ",linform_wp A, linform_wp B)
|(Const("op |",_)$ A $ B) => LfConst("DJ",linform_wp A, linform_wp B)
|(Const("op -->",_)$ A $ B) => LfConst("IM",linform_wp A, linform_wp B)
|(Const("op =",Type ("fun",[Type ("bool", []),_]))$ A $ B) => LfConst("EQ",linform_wp A, linform_wp B)
|Const("Ex",_)$Abs(x,T,p) =>
let val (xn,p1) = variant_abs(x,T,p)
in LfQ("Ex",xn,T,linform_wp p1)
end
|Const("All",_)$Abs(x,T,p) =>
let val (xn,p1) = variant_abs(x,T,p)
in LfQ("All",xn,T,linform_wp p1)
end
end;
(* ------------------------------------------------------------------------- *)
(*For simlified formulas we just notice the original formula, for whitch we habe been
intendes to make the proof.*)
(* ------------------------------------------------------------------------- *)
fun simpl_wp (fm,pr) = let val fm2 = simpl fm
in (fm2,Simp(fm,pr))
end;
(* ------------------------------------------------------------------------- *)
(*Help function for the generation of the proof EX.P_{minus \infty} --> EX. P(x) *)
(* ------------------------------------------------------------------------- *)
fun minusinf_wph x fm = let fun mk_atomar_minusinf_proof x fm = (Modd_minf(x,fm),Eq_minf(x,fm))
fun combine_minusinf_proofs opr (ppr1,ppr2) (qpr1,qpr2) = case opr of
"CJ" => (Modd_minf_conjI(ppr1,qpr1),Eq_minf_conjI(ppr2,qpr2))
|"DJ" => (Modd_minf_disjI(ppr1,qpr1),Eq_minf_disjI(ppr2,qpr2))
in
case fm of
(Const ("Not", _) $ (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z))) =>
if (x=y) andalso (c1= zero) andalso (c2= one) then (HOLogic.true_const ,(mk_atomar_minusinf_proof x fm))
else (fm ,(mk_atomar_minusinf_proof x fm))
|(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z)) =>
if (is_arith_rel fm) andalso (x=y) andalso (c1= zero) andalso (c2= one)
then (HOLogic.false_const ,(mk_atomar_minusinf_proof x fm))
else (fm,(mk_atomar_minusinf_proof x fm))
|(Const("op <",_) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ c2 $ y ) $ z )) =>
if (y=x) andalso (c1 = zero) then
if c2 = one then (HOLogic.false_const,(mk_atomar_minusinf_proof x fm)) else
(HOLogic.true_const,(mk_atomar_minusinf_proof x fm))
else (fm,(mk_atomar_minusinf_proof x fm))
|(Const("Not",_)$(Const ("Divides.op dvd",_) $_ )) => (fm,mk_atomar_minusinf_proof x fm)
|(Const ("Divides.op dvd",_) $_ ) => (fm,mk_atomar_minusinf_proof x fm)
|(Const ("op &",_) $ p $ q) => let val (pfm,ppr) = minusinf_wph x p
val (qfm,qpr) = minusinf_wph x q
val pr = (combine_minusinf_proofs "CJ" ppr qpr)
in
(HOLogic.conj $ pfm $qfm , pr)
end
|(Const ("op |",_) $ p $ q) => let val (pfm,ppr) = minusinf_wph x p
val (qfm,qpr) = minusinf_wph x q
val pr = (combine_minusinf_proofs "DJ" ppr qpr)
in
(HOLogic.disj $ pfm $qfm , pr)
end
|_ => (fm,(mk_atomar_minusinf_proof x fm))
end;
(* ------------------------------------------------------------------------- *) (* Protokol for the Proof of the property of the minusinfinity formula*)
(* Just combines the to protokols *)
(* ------------------------------------------------------------------------- *)
fun minusinf_wp x fm = let val (fm2,pr) = (minusinf_wph x fm)
in (fm2,Minusinf(pr))
end;
(* ------------------------------------------------------------------------- *)
(*Help function for the generation of the proof EX.P_{plus \infty} --> EX. P(x) *)
(* ------------------------------------------------------------------------- *)
fun plusinf_wph x fm = let fun mk_atomar_plusinf_proof x fm = (Modd_minf(x,fm),Eq_minf(x,fm))
fun combine_plusinf_proofs opr (ppr1,ppr2) (qpr1,qpr2) = case opr of
"CJ" => (Modd_minf_conjI(ppr1,qpr1),Eq_minf_conjI(ppr2,qpr2))
|"DJ" => (Modd_minf_disjI(ppr1,qpr1),Eq_minf_disjI(ppr2,qpr2))
in
case fm of
(Const ("Not", _) $ (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z))) =>
if (x=y) andalso (c1= zero) andalso (c2= one) then (HOLogic.true_const ,(mk_atomar_plusinf_proof x fm))
else (fm ,(mk_atomar_plusinf_proof x fm))
|(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z)) =>
if (is_arith_rel fm) andalso (x=y) andalso (c1= zero) andalso (c2= one)
then (HOLogic.false_const ,(mk_atomar_plusinf_proof x fm))
else (fm,(mk_atomar_plusinf_proof x fm))
|(Const("op <",_) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ c2 $ y ) $ z )) =>
if (y=x) andalso (c1 = zero) then
if c2 = one then (HOLogic.true_const,(mk_atomar_plusinf_proof x fm)) else
(HOLogic.false_const,(mk_atomar_plusinf_proof x fm))
else (fm,(mk_atomar_plusinf_proof x fm))
|(Const("Not",_)$(Const ("Divides.op dvd",_) $_ )) => (fm,mk_atomar_plusinf_proof x fm)
|(Const ("Divides.op dvd",_) $_ ) => (fm,mk_atomar_plusinf_proof x fm)
|(Const ("op &",_) $ p $ q) => let val (pfm,ppr) = plusinf_wph x p
val (qfm,qpr) = plusinf_wph x q
val pr = (combine_plusinf_proofs "CJ" ppr qpr)
in
(HOLogic.conj $ pfm $qfm , pr)
end
|(Const ("op |",_) $ p $ q) => let val (pfm,ppr) = plusinf_wph x p
val (qfm,qpr) = plusinf_wph x q
val pr = (combine_plusinf_proofs "DJ" ppr qpr)
in
(HOLogic.disj $ pfm $qfm , pr)
end
|_ => (fm,(mk_atomar_plusinf_proof x fm))
end;
(* ------------------------------------------------------------------------- *) (* Protokol for the Proof of the property of the minusinfinity formula*)
(* Just combines the to protokols *)
(* ------------------------------------------------------------------------- *)
fun plusinf_wp x fm = let val (fm2,pr) = (plusinf_wph x fm)
in (fm2,Minusinf(pr))
end;
(* ------------------------------------------------------------------------- *)
(*Protocol that we here uses Bset.*)
(* ------------------------------------------------------------------------- *)
fun bset_wp x fm = let val bs = bset x fm in
(bs,Bset(x,fm,bs,mk_numeral (divlcm x fm)))
end;
(* ------------------------------------------------------------------------- *)
(*Protocol that we here uses Aset.*)
(* ------------------------------------------------------------------------- *)
fun aset_wp x fm = let val ast = aset x fm in
(ast,Aset(x,fm,ast,mk_numeral (divlcm x fm)))
end;
(* ------------------------------------------------------------------------- *)
(*function list to Set, constructs a set containing all elements of a given list.*)
(* ------------------------------------------------------------------------- *)
fun list_to_set T1 l = let val T = (HOLogic.mk_setT T1) in
case l of
[] => Const ("{}",T)
|(h::t) => Const("insert", T1 --> (T --> T)) $ h $(list_to_set T1 t)
end;
(*====================================================================*)
(* ------------------------------------------------------------------------- *)
(* ------------------------------------------------------------------------- *)
(*Protocol for the proof of the backward direction of the cooper theorem.*)
(* Helpfunction - Protokols evereything about the proof reconstruction*)
(* ------------------------------------------------------------------------- *)
fun not_bst_p_wph fm = case fm of
Const("Not",_) $ R => if (is_arith_rel R) then (Not_bst_p_atomic (fm)) else CpLogError
|Const("op &",_) $ ls $ rs => Not_bst_p_conjI((not_bst_p_wph ls),(not_bst_p_wph rs))
|Const("op |",_) $ ls $ rs => Not_bst_p_disjI((not_bst_p_wph ls),(not_bst_p_wph rs))
|_ => Not_bst_p_atomic (fm);
(* ------------------------------------------------------------------------- *)
(* Main protocoling function for the backward direction gives the Bset and the divlcm and the Formula herself. Needed as inherited attributes for the proof reconstruction*)
(* ------------------------------------------------------------------------- *)
fun not_bst_p_wp x fm = let val prt = not_bst_p_wph fm
val D = mk_numeral (divlcm x fm)
val B = map norm_zero_one (bset x fm)
in (Not_bst_p (x,fm,D,(list_to_set HOLogic.intT B) , prt))
end;
(*====================================================================*)
(* ------------------------------------------------------------------------- *)
(* ------------------------------------------------------------------------- *)
(*Protocol for the proof of the backward direction of the cooper theorem.*)
(* Helpfunction - Protokols evereything about the proof reconstruction*)
(* ------------------------------------------------------------------------- *)
fun not_ast_p_wph fm = case fm of
Const("Not",_) $ R => if (is_arith_rel R) then (Not_ast_p_atomic (fm)) else CpLogError
|Const("op &",_) $ ls $ rs => Not_ast_p_conjI((not_ast_p_wph ls),(not_ast_p_wph rs))
|Const("op |",_) $ ls $ rs => Not_ast_p_disjI((not_ast_p_wph ls),(not_ast_p_wph rs))
|_ => Not_ast_p_atomic (fm);
(* ------------------------------------------------------------------------- *)
(* Main protocoling function for the backward direction gives the Bset and the divlcm and the Formula herself. Needed as inherited attributes for the proof reconstruction*)
(* ------------------------------------------------------------------------- *)
fun not_ast_p_wp x fm = let val prt = not_ast_p_wph fm
val D = mk_numeral (divlcm x fm)
val B = map norm_zero_one (aset x fm)
in (Not_ast_p (x,fm,D,(list_to_set HOLogic.intT B) , prt))
end;
(*======================================================*)
(* Protokolgeneration for the formula evaluation process*)
(*======================================================*)
fun evalc_wp fm =
let fun evalc_atom_wp at =case at of
(Const (p,_) $ s $ t) =>(
case assoc (operations,p) of
Some f => ((if (f ((dest_numeral s),(dest_numeral t))) then EvalAt(HOLogic.mk_eq(at,HOLogic.true_const)) else EvalAt(HOLogic.mk_eq(at, HOLogic.false_const)))
handle _ => Evalfm(at))
| _ => Evalfm(at))
|Const("Not",_)$(Const (p,_) $ s $ t) =>(
case assoc (operations,p) of
Some f => ((if (f ((dest_numeral s),(dest_numeral t))) then
EvalAt(HOLogic.mk_eq(at, HOLogic.false_const)) else EvalAt(HOLogic.mk_eq(at,HOLogic.true_const)))
handle _ => Evalfm(at))
| _ => Evalfm(at))
| _ => Evalfm(at)
in
case fm of
(Const("op &",_)$A$B) => EvalConst("CJ",evalc_wp A,evalc_wp B)
|(Const("op |",_)$A$B) => EvalConst("DJ",evalc_wp A,evalc_wp B)
|(Const("op -->",_)$A$B) => EvalConst("IM",evalc_wp A,evalc_wp B)
|(Const("op =", Type ("fun",[Type ("bool", []),_]))$A$B) => EvalConst("EQ",evalc_wp A,evalc_wp B)
|_ => evalc_atom_wp fm
end;
(*======================================================*)
(* Protokolgeneration for the NNF Transformation *)
(*======================================================*)
fun cnnf_wp f =
let fun hcnnf_wp fm =
case fm of
(Const ("op &",_) $ p $ q) => NNFConst("CJ",hcnnf_wp p,hcnnf_wp q)
| (Const ("op |",_) $ p $ q) => NNFConst("DJ",hcnnf_wp p,hcnnf_wp q)
| (Const ("op -->",_) $ p $q) => NNFConst("IM",hcnnf_wp (HOLogic.Not $ p),hcnnf_wp q)
| (Const ("op =",Type ("fun",[Type ("bool", []),_])) $ p $ q) => NNFConst("EQ",hcnnf_wp (HOLogic.mk_conj(p,q)),hcnnf_wp (HOLogic.mk_conj((HOLogic.Not $ p), (HOLogic.Not $ q))))
| (Const ("Not",_) $ (Const("Not",_) $ p)) => NNFNN(hcnnf_wp p)
| (Const ("Not",_) $ (Const ("op &",_) $ p $ q)) => NNFConst ("NCJ",(hcnnf_wp(HOLogic.Not $ p)),(hcnnf_wp(HOLogic.Not $ q)))
| (Const ("Not",_) $(Const ("op |",_) $ (A as (Const ("op &",_) $ p $ q)) $
(B as (Const ("op &",_) $ p1 $ r)))) => if p1 = negate p then
NNFConst("SDJ",
NNFConst("CJ",hcnnf_wp p,hcnnf_wp(HOLogic.Not $ q)),
NNFConst("CJ",hcnnf_wp p1,hcnnf_wp(HOLogic.Not $ r)))
else NNFConst ("NDJ",(hcnnf_wp(HOLogic.Not $ A)),(hcnnf_wp(HOLogic.Not $ B)))
| (Const ("Not",_) $ (Const ("op |",_) $ p $ q)) => NNFConst ("NDJ",(hcnnf_wp(HOLogic.Not $ p)),(hcnnf_wp(HOLogic.Not $ q)))
| (Const ("Not",_) $ (Const ("op -->",_) $ p $q)) => NNFConst ("NIM",(hcnnf_wp(p)),(hcnnf_wp(HOLogic.Not $ q)))
| (Const ("Not",_) $ (Const ("op =",Type ("fun",[Type ("bool", []),_])) $ p $ q)) =>NNFConst ("NEQ",(NNFConst("CJ",hcnnf_wp p,hcnnf_wp(HOLogic.Not $ q))),(NNFConst("CJ",hcnnf_wp(HOLogic.Not $ p),hcnnf_wp q)))
| _ => NNFAt(fm)
in NNFSimp(hcnnf_wp f)
end;
(* ------------------------------------------------------------------------- *)
(*Cooper decision Procedure with proof protocoling*)
(* ------------------------------------------------------------------------- *)
fun coopermi_wp vars fm =
case fm of
Const ("Ex",_) $ Abs(xo,T,po) => let
val (xn,np) = variant_abs(xo,T,po)
val x = (Free(xn , T))
val p = np (* Is this a legal proof for the P=NP Problem??*)
val (p_inf,miprt) = simpl_wp (minusinf_wp x p)
val (bset,bsprt) = bset_wp x p
val nbst_p_prt = not_bst_p_wp x p
val dlcm = divlcm x p
val js = 1 upto dlcm
fun p_element j b = linrep vars x (linear_add vars b (mk_numeral j)) p
fun stage j = list_disj (linrep vars x (mk_numeral j) p_inf :: map (p_element j) bset)
in (list_disj (map stage js),Cooper(mk_numeral dlcm,miprt,bsprt,nbst_p_prt))
end
| _ => (error "cooper: not an existential formula",No);
fun cooperpi_wp vars fm =
case fm of
Const ("Ex",_) $ Abs(xo,T,po) => let
val (xn,np) = variant_abs(xo,T,po)
val x = (Free(xn , T))
val p = np (* Is this a legal proof for the P=NP Problem??*)
val (p_inf,piprt) = simpl_wp (plusinf_wp x p)
val (aset,asprt) = aset_wp x p
val nast_p_prt = not_ast_p_wp x p
val dlcm = divlcm x p
val js = 1 upto dlcm
fun p_element j a = linrep vars x (linear_sub vars a (mk_numeral j)) p
fun stage j = list_disj (linrep vars x (mk_numeral j) p_inf :: map (p_element j) aset)
in (list_disj (map stage js),Cooper(mk_numeral dlcm,piprt,asprt,nast_p_prt))
end
| _ => (error "cooper: not an existential formula",No);
(*-----------------------------------------------------------------*)
(*-----------------------------------------------------------------*)
(*-----------------------------------------------------------------*)
(*--- ---*)
(*--- ---*)
(*--- Interpretation and Proofgeneration Part ---*)
(*--- ---*)
(*--- Protocole interpretation functions ---*)
(*--- ---*)
(*--- and proofgeneration functions ---*)
(*--- ---*)
(*--- ---*)
(*--- ---*)
(*--- ---*)
(*-----------------------------------------------------------------*)
(*-----------------------------------------------------------------*)
(*-----------------------------------------------------------------*)
(* ------------------------------------------------------------------------- *)
(* Returns both sides of an equvalence in the theorem*)
(* ------------------------------------------------------------------------- *)
fun qe_get_terms th = let val (_$(Const("op =",Type ("fun",[Type ("bool", []),_])) $ A $ B )) = prop_of th in (A,B) end;
(*-------------------------------------------------------------*)
(*-------------------------------------------------------------*)
(*-------------------------------------------------------------*)
(*-------------------------------------------------------------*)
(* ------------------------------------------------------------------------- *)
(* Modified version of the simple version with minimal amount of checking and postprocessing*)
(* ------------------------------------------------------------------------- *)
fun simple_prove_goal_cterm2 G tacs =
let
fun check None = error "prove_goal: tactic failed"
| check (Some (thm, _)) = (case nprems_of thm of
0 => thm
| i => !result_error_fn thm (string_of_int i ^ " unsolved goals!"))
in check (Seq.pull (EVERY tacs (trivial G))) end;
(*-------------------------------------------------------------*)
(*-------------------------------------------------------------*)
(*-------------------------------------------------------------*)
(*-------------------------------------------------------------*)
(*-------------------------------------------------------------*)
fun cert_Trueprop sg t = cterm_of sg (HOLogic.mk_Trueprop t);
(* ------------------------------------------------------------------------- *)
(*This function proove elementar will be used to generate proofs at runtime*)
(*It is is based on the isabelle function proove_goalw_cterm and is thought to *)
(*prove properties such as a dvd b (essentially) that are only to make at
runtime.*)
(* ------------------------------------------------------------------------- *)
fun prove_elementar sg s fm2 = case s of
(*"ss" like simplification with simpset*)
"ss" =>
let
val ss = presburger_ss addsimps
[zdvd_iff_zmod_eq_0,unity_coeff_ex]
val ct = cert_Trueprop sg fm2
in
simple_prove_goal_cterm2 ct [simp_tac ss 1, TRY (simple_arith_tac 1)]
end
(*"bl" like blast tactic*)
(* Is only used in the harrisons like proof procedure *)
| "bl" =>
let val ct = cert_Trueprop sg fm2
in
simple_prove_goal_cterm2 ct [blast_tac HOL_cs 1]
end
(*"ed" like Existence disjunctions ...*)
(* Is only used in the harrisons like proof procedure *)
| "ed" =>
let
val ex_disj_tacs =
let
val tac1 = EVERY[REPEAT(resolve_tac [disjI1,disjI2] 1), etac exI 1]
val tac2 = EVERY[etac exE 1, rtac exI 1,
REPEAT(resolve_tac [disjI1,disjI2] 1), assumption 1]
in [rtac iffI 1,
etac exE 1, REPEAT(EVERY[etac disjE 1, tac1]), tac1,
REPEAT(EVERY[etac disjE 1, tac2]), tac2]
end
val ct = cert_Trueprop sg fm2
in
simple_prove_goal_cterm2 ct ex_disj_tacs
end
| "fa" =>
let val ct = cert_Trueprop sg fm2
in simple_prove_goal_cterm2 ct [simple_arith_tac 1]
end
| "sa" =>
let
val ss = presburger_ss addsimps zadd_ac
val ct = cert_Trueprop sg fm2
in
simple_prove_goal_cterm2 ct [simp_tac ss 1, TRY (simple_arith_tac 1)]
end
| "ac" =>
let
val ss = HOL_basic_ss addsimps zadd_ac
val ct = cert_Trueprop sg fm2
in
simple_prove_goal_cterm2 ct [simp_tac ss 1]
end
| "lf" =>
let
val ss = presburger_ss addsimps zadd_ac
val ct = cert_Trueprop sg fm2
in
simple_prove_goal_cterm2 ct [simp_tac ss 1, TRY (simple_arith_tac 1)]
end;
(* ------------------------------------------------------------------------- *)
(* This function return an Isabelle proof, of the adjustcoffeq result.*)
(* The proofs are in Presburger.thy and are generally based on the arithmetic *)
(* ------------------------------------------------------------------------- *)
fun proof_of_adjustcoeffeq sg (prt,rs) = case prt of
ACfm fm => instantiate' [Some cboolT]
[Some (cterm_of sg fm)] refl
| ACAt (k,at as (Const(p,_) $a $( Const ("op +", _)$(Const ("op *",_) $
c $ x ) $t ))) =>
let
val ck = cterm_of sg (mk_numeral k)
val cc = cterm_of sg c
val ct = cterm_of sg t
val cx = cterm_of sg x
val ca = cterm_of sg a
in case p of
"op <" => let val pre = prove_elementar sg "lf"
(HOLogic.mk_binrel "op <" (Const("0",HOLogic.intT),(mk_numeral k)))
val th1 = (pre RS (instantiate' [] [Some ck,Some ca,Some cc, Some cx, Some ct] (ac_lt_eq)))
in [th1,(prove_elementar sg "lf" (HOLogic.mk_eq (snd (qe_get_terms th1) ,rs)))] MRS trans
end
|"op =" =>let val pre = prove_elementar sg "lf"
(HOLogic.Not $ (HOLogic.mk_binrel "op =" (Const("0",HOLogic.intT),(mk_numeral k))))
in let val th1 = (pre RS(instantiate' [] [Some ck,Some ca,Some cc, Some cx, Some ct] (ac_eq_eq)))
in [th1,(prove_elementar sg "lf" (HOLogic.mk_eq (snd (qe_get_terms th1) ,rs)))] MRS trans
end
end
|"Divides.op dvd" =>let val pre = prove_elementar sg "lf"
(HOLogic.Not $ (HOLogic.mk_binrel "op =" (Const("0",HOLogic.intT),(mk_numeral k))))
val th1 = (pre RS (instantiate' [] [Some ck,Some ca,Some cc, Some cx, Some ct]) (ac_dvd_eq))
in [th1,(prove_elementar sg "lf" (HOLogic.mk_eq (snd (qe_get_terms th1) ,rs)))] MRS trans
end
end
|ACPI(k,at as (Const("Not",_)$(Const("op <",_) $a $( Const ("op +", _)$(Const ("op *",_) $ c $ x ) $t )))) =>
let
val ck = cterm_of sg (mk_numeral k)
val cc = cterm_of sg c
val ct = cterm_of sg t
val cx = cterm_of sg x
val pre = prove_elementar sg "lf"
(HOLogic.mk_binrel "op <" (Const("0",HOLogic.intT),(mk_numeral k)))
val th1 = (pre RS (instantiate' [] [Some ck,Some cc, Some cx, Some ct] (ac_pi_eq)))
in [th1,(prove_elementar sg "sa" (HOLogic.mk_eq (snd (qe_get_terms th1) ,rs)))] MRS trans
end
|ACNeg(pr) => let val (Const("Not",_)$nrs) = rs
in (proof_of_adjustcoeffeq sg (pr,nrs)) RS (qe_Not)
end
|ACConst(s,pr1,pr2) =>
let val (Const(_,_)$rs1$rs2) = rs
val th1 = proof_of_adjustcoeffeq sg (pr1,rs1)
val th2 = proof_of_adjustcoeffeq sg (pr2,rs2)
in case s of
"CJ" => [th1,th2] MRS (qe_conjI)
|"DJ" => [th1,th2] MRS (qe_disjI)
|"IM" => [th1,th2] MRS (qe_impI)
|"EQ" => [th1,th2] MRS (qe_eqI)
end;
(* ------------------------------------------------------------------------- *)
(* This function return an Isabelle proof, of some properties on the atoms*)
(* The proofs are in Presburger.thy and are generally based on the arithmetic *)
(* This function doese only instantiate the the theorems in the theory *)
(* ------------------------------------------------------------------------- *)
fun atomar_minf_proof_of sg dlcm (Modd_minf (x,fm1)) =
let
(*Some certified Terms*)
val ctrue = cterm_of sg HOLogic.true_const
val cfalse = cterm_of sg HOLogic.false_const
val fm = norm_zero_one fm1
in case fm1 of
(Const ("Not", _) $ (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z))) =>
if (x=y) andalso (c1= zero) andalso (c2= one) then (instantiate' [Some cboolT] [Some ctrue] (fm_modd_minf))
else (instantiate' [Some cboolT] [Some (cterm_of sg fm)] (fm_modd_minf))
|(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z)) =>
if (is_arith_rel fm) andalso (x=y) andalso (c1= zero) andalso (c2= one)
then (instantiate' [Some cboolT] [Some cfalse] (fm_modd_minf))
else (instantiate' [Some cboolT] [Some (cterm_of sg fm)] (fm_modd_minf))
|(Const("op <",_) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ pm1 $ y ) $ z )) =>
if (y=x) andalso (c1 = zero) then
if (pm1 = one) then (instantiate' [Some cboolT] [Some cfalse] (fm_modd_minf)) else
(instantiate' [Some cboolT] [Some ctrue] (fm_modd_minf))
else (instantiate' [Some cboolT] [Some (cterm_of sg fm)] (fm_modd_minf))
|Const ("Not",_) $ (Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) =>
if y=x then let val cz = cterm_of sg (norm_zero_one z)
val fm2 = HOLogic.mk_binrel "op =" (HOLogic.mk_binop "Divides.op mod" (dlcm,d),norm_zero_one zero)
in(instantiate' [] [Some cz ] ((((prove_elementar sg "ss" fm2)) RS(((zdvd_iff_zmod_eq_0)RS sym) RS iffD1) ) RS (not_dvd_modd_minf)))
end
else (instantiate' [Some cboolT] [Some (cterm_of sg fm)] (fm_modd_minf))
|(Const("Divides.op dvd",_)$ d $ (db as (Const ("op +",_) $ (Const ("op *",_) $
c $ y ) $ z))) =>
if y=x then let val cz = cterm_of sg (norm_zero_one z)
val fm2 = HOLogic.mk_binrel "op =" (HOLogic.mk_binop "Divides.op mod" (dlcm,d),norm_zero_one zero)
in(instantiate' [] [Some cz ] ((((prove_elementar sg "ss" fm2)) RS (((zdvd_iff_zmod_eq_0)RS sym) RS iffD1) ) RS (dvd_modd_minf)))
end
else (instantiate' [Some cboolT] [Some (cterm_of sg fm)] (fm_modd_minf))
|_ => instantiate' [Some cboolT] [Some (cterm_of sg fm)] (fm_modd_minf)
end
|atomar_minf_proof_of sg dlcm (Eq_minf (x,fm1)) = let
(*Some certified types*)
val fm = norm_zero_one fm1
in case fm1 of
(Const ("Not", _) $ (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z))) =>
if (x=y) andalso (c1=zero) andalso (c2=one)
then (instantiate' [] [Some (cterm_of sg (norm_zero_one z))] (neq_eq_minf))
else (instantiate' [Some cboolT] [Some (cterm_of sg fm)] (fm_eq_minf))
|(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z)) =>
if (is_arith_rel fm) andalso (x=y) andalso ((c1=zero) orelse (c1 = norm_zero_one zero)) andalso ((c2=one) orelse (c1 = norm_zero_one one))
then (instantiate' [] [Some (cterm_of sg (norm_zero_one z))] (eq_eq_minf))
else (instantiate' [Some cboolT] [Some (cterm_of sg fm)] (fm_eq_minf))
|(Const("op <",_) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ pm1 $ y ) $ z )) =>
if (y=x) andalso (c1 =zero) then
if pm1 = one then (instantiate' [] [Some (cterm_of sg (norm_zero_one z))] (le_eq_minf)) else
(instantiate' [] [Some (cterm_of sg (norm_zero_one z))] (len_eq_minf))
else (instantiate' [Some cboolT] [Some (cterm_of sg fm)] (fm_eq_minf))
|Const ("Not",_) $ (Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) =>
if y=x then let val cd = cterm_of sg (norm_zero_one d)
val cz = cterm_of sg (norm_zero_one z)
in(instantiate' [] [Some cd, Some cz] (not_dvd_eq_minf))
end
else (instantiate' [Some cboolT] [Some (cterm_of sg fm)] (fm_eq_minf))
|(Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) =>
if y=x then let val cd = cterm_of sg (norm_zero_one d)
val cz = cterm_of sg (norm_zero_one z)
in(instantiate' [] [Some cd, Some cz ] (dvd_eq_minf))
end
else (instantiate' [Some cboolT] [Some (cterm_of sg fm)] (fm_eq_minf))
|_ => (instantiate' [Some cboolT] [Some (cterm_of sg fm)] (fm_eq_minf))
end;
(* ------------------------------------------------------------------------- *)
(* This function combines proofs of some special form already synthetised from the subtrees to make*)
(* a new proof of the same form. The combination occures whith isabelle theorems which have been already prooved *)
(*these Theorems are in Presburger.thy and mostly do not relay on the arithmetic.*)
(* These are Theorems for the Property of P_{-infty}*)
(* ------------------------------------------------------------------------- *)
fun combine_minf_proof s pr1 pr2 = case s of
"ECJ" => [pr1 , pr2] MRS (eq_minf_conjI)
|"EDJ" => [pr1 , pr2] MRS (eq_minf_disjI)
|"MCJ" => [pr1 , pr2] MRS (modd_minf_conjI)
|"MDJ" => [pr1 , pr2] MRS (modd_minf_disjI);
(* ------------------------------------------------------------------------- *)
(*This function return an isabelle Proof for the minusinfinity theorem*)
(* It interpretates the protool and gives the protokoles property of P_{...} as a theorem*)
(* ------------------------------------------------------------------------- *)
fun minf_proof_ofh sg dlcm prl = case prl of
Eq_minf (_) => atomar_minf_proof_of sg dlcm prl
|Modd_minf (_) => atomar_minf_proof_of sg dlcm prl
|Eq_minf_conjI (prl1,prl2) => let val pr1 = minf_proof_ofh sg dlcm prl1
val pr2 = minf_proof_ofh sg dlcm prl2
in (combine_minf_proof "ECJ" pr1 pr2)
end
|Eq_minf_disjI (prl1,prl2) => let val pr1 = minf_proof_ofh sg dlcm prl1
val pr2 = minf_proof_ofh sg dlcm prl2
in (combine_minf_proof "EDJ" pr1 pr2)
end
|Modd_minf_conjI (prl1,prl2) => let val pr1 = minf_proof_ofh sg dlcm prl1
val pr2 = minf_proof_ofh sg dlcm prl2
in (combine_minf_proof "MCJ" pr1 pr2)
end
|Modd_minf_disjI (prl1,prl2) => let val pr1 = minf_proof_ofh sg dlcm prl1
val pr2 = minf_proof_ofh sg dlcm prl2
in (combine_minf_proof "MDJ" pr1 pr2)
end;
(* ------------------------------------------------------------------------- *)
(* Main function For the rest both properies of P_{..} are needed and here both theorems are returned.*)
(* ------------------------------------------------------------------------- *)
fun minf_proof_of sg dlcm (Minusinf (prl1,prl2)) =
let val pr1 = minf_proof_ofh sg dlcm prl1
val pr2 = minf_proof_ofh sg dlcm prl2
in (pr1, pr2)
end;
(* ------------------------------------------------------------------------- *)
(* This function return an Isabelle proof, of some properties on the atoms*)
(* The proofs are in Presburger.thy and are generally based on the arithmetic *)
(* This function doese only instantiate the the theorems in the theory *)
(* ------------------------------------------------------------------------- *)
fun atomar_pinf_proof_of sg dlcm (Modd_minf (x,fm1)) =
let
(*Some certified Terms*)
val ctrue = cterm_of sg HOLogic.true_const
val cfalse = cterm_of sg HOLogic.false_const
val fm = norm_zero_one fm1
in case fm1 of
(Const ("Not", _) $ (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z))) =>
if ((x=y) andalso (c1= zero) andalso (c2= one))
then (instantiate' [Some cboolT] [Some ctrue] (fm_modd_pinf))
else (instantiate' [Some cboolT] [Some (cterm_of sg fm)] (fm_modd_pinf))
|(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z)) =>
if ((is_arith_rel fm) andalso (x = y) andalso (c1 = zero) andalso (c2 = one))
then (instantiate' [Some cboolT] [Some cfalse] (fm_modd_pinf))
else (instantiate' [Some cboolT] [Some (cterm_of sg fm)] (fm_modd_pinf))
|(Const("op <",_) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ pm1 $ y ) $ z )) =>
if ((y=x) andalso (c1 = zero)) then
if (pm1 = one)
then (instantiate' [Some cboolT] [Some ctrue] (fm_modd_pinf))
else (instantiate' [Some cboolT] [Some cfalse] (fm_modd_pinf))
else (instantiate' [Some cboolT] [Some (cterm_of sg fm)] (fm_modd_pinf))
|Const ("Not",_) $ (Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) =>
if y=x then let val cz = cterm_of sg (norm_zero_one z)
val fm2 = HOLogic.mk_binrel "op =" (HOLogic.mk_binop "Divides.op mod" (dlcm,d),norm_zero_one zero)
in(instantiate' [] [Some cz ] ((((prove_elementar sg "ss" fm2)) RS(((zdvd_iff_zmod_eq_0)RS sym) RS iffD1) ) RS (not_dvd_modd_pinf)))
end
else (instantiate' [Some cboolT] [Some (cterm_of sg fm)] (fm_modd_pinf))
|(Const("Divides.op dvd",_)$ d $ (db as (Const ("op +",_) $ (Const ("op *",_) $
c $ y ) $ z))) =>
if y=x then let val cz = cterm_of sg (norm_zero_one z)
val fm2 = HOLogic.mk_binrel "op =" (HOLogic.mk_binop "Divides.op mod" (dlcm,d),norm_zero_one zero)
in(instantiate' [] [Some cz ] ((((prove_elementar sg "ss" fm2)) RS (((zdvd_iff_zmod_eq_0)RS sym) RS iffD1) ) RS (dvd_modd_pinf)))
end
else (instantiate' [Some cboolT] [Some (cterm_of sg fm)] (fm_modd_pinf))
|_ => instantiate' [Some cboolT] [Some (cterm_of sg fm)] (fm_modd_pinf)
end
|atomar_pinf_proof_of sg dlcm (Eq_minf (x,fm1)) = let
val fm = norm_zero_one fm1
in case fm1 of
(Const ("Not", _) $ (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z))) =>
if (x=y) andalso (c1=zero) andalso (c2=one)
then (instantiate' [] [Some (cterm_of sg (norm_zero_one z))] (neq_eq_pinf))
else (instantiate' [Some cboolT] [Some (cterm_of sg fm)] (fm_eq_pinf))
|(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z)) =>
if (is_arith_rel fm) andalso (x=y) andalso ((c1=zero) orelse (c1 = norm_zero_one zero)) andalso ((c2=one) orelse (c1 = norm_zero_one one))
then (instantiate' [] [Some (cterm_of sg (norm_zero_one z))] (eq_eq_pinf))
else (instantiate' [Some cboolT] [Some (cterm_of sg fm)] (fm_eq_pinf))
|(Const("op <",_) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ pm1 $ y ) $ z )) =>
if (y=x) andalso (c1 =zero) then
if pm1 = one then (instantiate' [] [Some (cterm_of sg (norm_zero_one z))] (le_eq_pinf)) else
(instantiate' [] [Some (cterm_of sg (norm_zero_one z))] (len_eq_pinf))
else (instantiate' [Some cboolT] [Some (cterm_of sg fm)] (fm_eq_pinf))
|Const ("Not",_) $ (Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) =>
if y=x then let val cd = cterm_of sg (norm_zero_one d)
val cz = cterm_of sg (norm_zero_one z)
in(instantiate' [] [Some cd, Some cz] (not_dvd_eq_pinf))
end
else (instantiate' [Some cboolT] [Some (cterm_of sg fm)] (fm_eq_pinf))
|(Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) =>
if y=x then let val cd = cterm_of sg (norm_zero_one d)
val cz = cterm_of sg (norm_zero_one z)
in(instantiate' [] [Some cd, Some cz ] (dvd_eq_pinf))
end
else (instantiate' [Some cboolT] [Some (cterm_of sg fm)] (fm_eq_pinf))
|_ => (instantiate' [Some cboolT] [Some (cterm_of sg fm)] (fm_eq_pinf))
end;
(* ------------------------------------------------------------------------- *)
(* This function combines proofs of some special form already synthetised from the subtrees to make*)
(* a new proof of the same form. The combination occures whith isabelle theorems which have been already prooved *)
(*these Theorems are in Presburger.thy and mostly do not relay on the arithmetic.*)
(* These are Theorems for the Property of P_{+infty}*)
(* ------------------------------------------------------------------------- *)
fun combine_pinf_proof s pr1 pr2 = case s of
"ECJ" => [pr1 , pr2] MRS (eq_pinf_conjI)
|"EDJ" => [pr1 , pr2] MRS (eq_pinf_disjI)
|"MCJ" => [pr1 , pr2] MRS (modd_pinf_conjI)
|"MDJ" => [pr1 , pr2] MRS (modd_pinf_disjI);
(* ------------------------------------------------------------------------- *)
(*This function return an isabelle Proof for the minusinfinity theorem*)
(* It interpretates the protool and gives the protokoles property of P_{...} as a theorem*)
(* ------------------------------------------------------------------------- *)
fun pinf_proof_ofh sg dlcm prl = case prl of
Eq_minf (_) => atomar_pinf_proof_of sg dlcm prl
|Modd_minf (_) => atomar_pinf_proof_of sg dlcm prl
|Eq_minf_conjI (prl1,prl2) => let val pr1 = pinf_proof_ofh sg dlcm prl1
val pr2 = pinf_proof_ofh sg dlcm prl2
in (combine_pinf_proof "ECJ" pr1 pr2)
end
|Eq_minf_disjI (prl1,prl2) => let val pr1 = pinf_proof_ofh sg dlcm prl1
val pr2 = pinf_proof_ofh sg dlcm prl2
in (combine_pinf_proof "EDJ" pr1 pr2)
end
|Modd_minf_conjI (prl1,prl2) => let val pr1 = pinf_proof_ofh sg dlcm prl1
val pr2 = pinf_proof_ofh sg dlcm prl2
in (combine_pinf_proof "MCJ" pr1 pr2)
end
|Modd_minf_disjI (prl1,prl2) => let val pr1 = pinf_proof_ofh sg dlcm prl1
val pr2 = pinf_proof_ofh sg dlcm prl2
in (combine_pinf_proof "MDJ" pr1 pr2)
end;
(* ------------------------------------------------------------------------- *)
(* Main function For the rest both properies of P_{..} are needed and here both theorems are returned.*)
(* ------------------------------------------------------------------------- *)
fun pinf_proof_of sg dlcm (Minusinf (prl1,prl2)) =
let val pr1 = pinf_proof_ofh sg dlcm prl1
val pr2 = pinf_proof_ofh sg dlcm prl2
in (pr1, pr2)
end;
(* ------------------------------------------------------------------------- *)
(* Here we generate the theorem for the Bset Property in the simple direction*)
(* It is just an instantiation*)
(* ------------------------------------------------------------------------- *)
fun bsetproof_of sg (Bset(x as Free(xn,xT),fm,bs,dlcm)) =
let
val cp = cterm_of sg (absfree (xn,xT,(norm_zero_one fm)))
val cdlcm = cterm_of sg dlcm
val cB = cterm_of sg (list_to_set HOLogic.intT (map norm_zero_one bs))
in instantiate' [] [Some cdlcm,Some cB, Some cp] (bst_thm)
end;
(* ------------------------------------------------------------------------- *)
(* Here we generate the theorem for the Bset Property in the simple direction*)
(* It is just an instantiation*)
(* ------------------------------------------------------------------------- *)
fun asetproof_of sg (Aset(x as Free(xn,xT),fm,ast,dlcm)) =
let
val cp = cterm_of sg (absfree (xn,xT,(norm_zero_one fm)))
val cdlcm = cterm_of sg dlcm
val cA = cterm_of sg (list_to_set HOLogic.intT (map norm_zero_one ast))
in instantiate' [] [Some cdlcm,Some cA, Some cp] (ast_thm)
end;
(* ------------------------------------------------------------------------- *)
(* Protokol interpretation function for the backwards direction for cooper's Theorem*)
(* For the generation of atomic Theorems*)
(* Prove the premisses on runtime and then make RS*)
(* ------------------------------------------------------------------------- *)
fun generate_atomic_not_bst_p sg (x as Free(xn,xT)) fm dlcm B at =
let
val cdlcm = cterm_of sg dlcm
val cB = cterm_of sg B
val cfma = cterm_of sg (absfree (xn,xT,(norm_zero_one fm)))
val cat = cterm_of sg (norm_zero_one at)
in
case at of
(Const ("Not", _) $ (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z))) =>
if (x=y) andalso (c1=zero) andalso (c2=one)
then let val th1 = prove_elementar sg "ss" (Const ("op :",HOLogic.intT --> (HOLogic.mk_setT HOLogic.intT) --> HOLogic.boolT) $ (norm_zero_one (linear_cmul ~1 z)) $ B)
val th2 = prove_elementar sg "ss" (HOLogic.mk_eq ((norm_zero_one (linear_cmul ~1 z)),Const("uminus",HOLogic.intT --> HOLogic.intT) $(norm_zero_one z)))
val th3 = prove_elementar sg "ss" (HOLogic.mk_binrel "op <" (Const("0",HOLogic.intT),dlcm))
in (instantiate' [] [Some cfma]([th3,th1,th2] MRS (not_bst_p_ne)))
end
else (instantiate' [] [Some cfma, Some cdlcm, Some cB,Some cat] (not_bst_p_fm))
|(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $(Const ("op +", T) $(Const ("op *",_) $ c2 $ y) $z)) =>
if (is_arith_rel at) andalso (x=y)
then let val bst_z = norm_zero_one (linear_neg (linear_add [] z (mk_numeral 1)))
in let val th1 = prove_elementar sg "ss" (Const ("op :",HOLogic.intT --> (HOLogic.mk_setT HOLogic.intT) --> HOLogic.boolT) $ bst_z $ B)
val th2 = prove_elementar sg "ss" (HOLogic.mk_eq (bst_z,Const("op -",T) $ (Const("uminus",HOLogic.intT --> HOLogic.intT) $(norm_zero_one z)) $ (Const("1",HOLogic.intT))))
val th3 = prove_elementar sg "ss" (HOLogic.mk_binrel "op <" (Const("0",HOLogic.intT),dlcm))
in (instantiate' [] [Some cfma] ([th3,th1,th2] MRS (not_bst_p_eq)))
end
end
else (instantiate' [] [Some cfma, Some cdlcm, Some cB,Some cat] (not_bst_p_fm))
|(Const("op <",_) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ pm1 $ y ) $ z )) =>
if (y=x) andalso (c1 =zero) then
if pm1 = one then
let val th1 = prove_elementar sg "ss" (Const ("op :",HOLogic.intT --> (HOLogic.mk_setT HOLogic.intT) --> HOLogic.boolT) $ (norm_zero_one (linear_cmul ~1 z)) $ B)
val th2 = prove_elementar sg "ss" (HOLogic.mk_eq ((norm_zero_one (linear_cmul ~1 z)),Const("uminus",HOLogic.intT --> HOLogic.intT) $(norm_zero_one z)))
in (instantiate' [] [Some cfma, Some cdlcm]([th1,th2] MRS (not_bst_p_gt)))
end
else let val th1 = prove_elementar sg "ss" (HOLogic.mk_binrel "op <" (Const("0",HOLogic.intT),dlcm))
in (instantiate' [] [Some cfma, Some cB,Some (cterm_of sg (norm_zero_one z))] (th1 RS (not_bst_p_lt)))
end
else (instantiate' [] [Some cfma, Some cdlcm, Some cB,Some cat] (not_bst_p_fm))
|Const ("Not",_) $ (Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) =>
if y=x then
let val cz = cterm_of sg (norm_zero_one z)
val th1 = (prove_elementar sg "ss" (HOLogic.mk_binrel "op =" (HOLogic.mk_binop "Divides.op mod" (dlcm,d),norm_zero_one zero))) RS (((zdvd_iff_zmod_eq_0)RS sym) RS iffD1)
in (instantiate' [] [Some cfma, Some cB,Some cz] (th1 RS (not_bst_p_ndvd)))
end
else (instantiate' [] [Some cfma, Some cdlcm, Some cB,Some cat] (not_bst_p_fm))
|(Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) =>
if y=x then
let val cz = cterm_of sg (norm_zero_one z)
val th1 = (prove_elementar sg "ss" (HOLogic.mk_binrel "op =" (HOLogic.mk_binop "Divides.op mod" (dlcm,d),norm_zero_one zero))) RS (((zdvd_iff_zmod_eq_0)RS sym) RS iffD1)
in (instantiate' [] [Some cfma,Some cB,Some cz] (th1 RS (not_bst_p_dvd)))
end
else (instantiate' [] [Some cfma, Some cdlcm, Some cB,Some cat] (not_bst_p_fm))
|_ => (instantiate' [] [Some cfma, Some cdlcm, Some cB,Some cat] (not_bst_p_fm))
end;
(* ------------------------------------------------------------------------- *)
(* Main interpretation function for this backwards dirction*)
(* if atomic do generate atomis formulae else Construct theorems and then make RS with the construction theorems*)
(*Help Function*)
(* ------------------------------------------------------------------------- *)
fun not_bst_p_proof_of_h sg x fm dlcm B prt = case prt of
(Not_bst_p_atomic(fm2)) => (generate_atomic_not_bst_p sg x fm dlcm B fm2)
|(Not_bst_p_conjI(pr1,pr2)) =>
let val th1 = (not_bst_p_proof_of_h sg x fm dlcm B pr1)
val th2 = (not_bst_p_proof_of_h sg x fm dlcm B pr2)
in ([th1,th2] MRS (not_bst_p_conjI))
end
|(Not_bst_p_disjI(pr1,pr2)) =>
let val th1 = (not_bst_p_proof_of_h sg x fm dlcm B pr1)
val th2 = (not_bst_p_proof_of_h sg x fm dlcm B pr2)
in ([th1,th2] MRS not_bst_p_disjI)
end;
(* Main function*)
fun not_bst_p_proof_of sg (Not_bst_p(x as Free(xn,xT),fm,dlcm,B,prl)) =
let val th = not_bst_p_proof_of_h sg x fm dlcm B prl
val fma = absfree (xn,xT, norm_zero_one fm)
in let val th1 = prove_elementar sg "ss" (HOLogic.mk_eq (fma,fma))
in [th,th1] MRS (not_bst_p_Q_elim)
end
end;
(* ------------------------------------------------------------------------- *)
(* Protokol interpretation function for the backwards direction for cooper's Theorem*)
(* For the generation of atomic Theorems*)
(* Prove the premisses on runtime and then make RS*)
(* ------------------------------------------------------------------------- *)
fun generate_atomic_not_ast_p sg (x as Free(xn,xT)) fm dlcm A at =
let
val cdlcm = cterm_of sg dlcm
val cA = cterm_of sg A
val cfma = cterm_of sg (absfree (xn,xT,(norm_zero_one fm)))
val cat = cterm_of sg (norm_zero_one at)
in
case at of
(Const ("Not", _) $ (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z))) =>
if (x=y) andalso (c1=zero) andalso (c2=one)
then let val th1 = prove_elementar sg "ss" (Const ("op :",HOLogic.intT --> (HOLogic.mk_setT HOLogic.intT) --> HOLogic.boolT) $ (norm_zero_one (linear_cmul ~1 z)) $ A)
val th2 = prove_elementar sg "ss" (HOLogic.mk_eq ((norm_zero_one (linear_cmul ~1 z)),Const("uminus",HOLogic.intT --> HOLogic.intT) $(norm_zero_one z)))
val th3 = prove_elementar sg "ss" (HOLogic.mk_binrel "op <" (Const("0",HOLogic.intT),dlcm))
in (instantiate' [] [Some cfma]([th3,th1,th2] MRS (not_ast_p_ne)))
end
else (instantiate' [] [Some cfma, Some cdlcm, Some cA,Some cat] (not_ast_p_fm))
|(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $(Const ("op +", T) $(Const ("op *",_) $ c2 $ y) $z)) =>
if (is_arith_rel at) andalso (x=y)
then let val ast_z = norm_zero_one (linear_sub [] one z )
val th1 = prove_elementar sg "ss" (Const ("op :",HOLogic.intT --> (HOLogic.mk_setT HOLogic.intT) --> HOLogic.boolT) $ ast_z $ A)
val th2 = prove_elementar sg "ss" (HOLogic.mk_eq (ast_z,Const("op +",T) $ (Const("uminus",HOLogic.intT --> HOLogic.intT) $(norm_zero_one z)) $ (Const("1",HOLogic.intT))))
val th3 = prove_elementar sg "ss" (HOLogic.mk_binrel "op <" (Const("0",HOLogic.intT),dlcm))
in (instantiate' [] [Some cfma] ([th3,th1,th2] MRS (not_ast_p_eq)))
end
else (instantiate' [] [Some cfma, Some cdlcm, Some cA,Some cat] (not_ast_p_fm))
|(Const("op <",_) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ pm1 $ y ) $ z )) =>
if (y=x) andalso (c1 =zero) then
if pm1 = (mk_numeral ~1) then
let val th1 = prove_elementar sg "ss" (Const ("op :",HOLogic.intT --> (HOLogic.mk_setT HOLogic.intT) --> HOLogic.boolT) $ (norm_zero_one z) $ A)
val th2 = prove_elementar sg "ss" (HOLogic.mk_binrel "op <" (zero,dlcm))
in (instantiate' [] [Some cfma]([th2,th1] MRS (not_ast_p_lt)))
end
else let val th1 = prove_elementar sg "ss" (HOLogic.mk_binrel "op <" (Const("0",HOLogic.intT),dlcm))
in (instantiate' [] [Some cfma, Some cA,Some (cterm_of sg (norm_zero_one z))] (th1 RS (not_ast_p_gt)))
end
else (instantiate' [] [Some cfma, Some cdlcm, Some cA,Some cat] (not_ast_p_fm))
|Const ("Not",_) $ (Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) =>
if y=x then
let val cz = cterm_of sg (norm_zero_one z)
val th1 = (prove_elementar sg "ss" (HOLogic.mk_binrel "op =" (HOLogic.mk_binop "Divides.op mod" (dlcm,d),norm_zero_one zero))) RS (((zdvd_iff_zmod_eq_0)RS sym) RS iffD1)
in (instantiate' [] [Some cfma, Some cA,Some cz] (th1 RS (not_ast_p_ndvd)))
end
else (instantiate' [] [Some cfma, Some cdlcm, Some cA,Some cat] (not_ast_p_fm))
|(Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) =>
if y=x then
let val cz = cterm_of sg (norm_zero_one z)
val th1 = (prove_elementar sg "ss" (HOLogic.mk_binrel "op =" (HOLogic.mk_binop "Divides.op mod" (dlcm,d),norm_zero_one zero))) RS (((zdvd_iff_zmod_eq_0)RS sym) RS iffD1)
in (instantiate' [] [Some cfma,Some cA,Some cz] (th1 RS (not_ast_p_dvd)))
end
else (instantiate' [] [Some cfma, Some cdlcm, Some cA,Some cat] (not_ast_p_fm))
|_ => (instantiate' [] [Some cfma, Some cdlcm, Some cA,Some cat] (not_ast_p_fm))
end;
(* ------------------------------------------------------------------------- *)
(* Main interpretation function for this backwards dirction*)
(* if atomic do generate atomis formulae else Construct theorems and then make RS with the construction theorems*)
(*Help Function*)
(* ------------------------------------------------------------------------- *)
fun not_ast_p_proof_of_h sg x fm dlcm A prt = case prt of
(Not_ast_p_atomic(fm2)) => (generate_atomic_not_ast_p sg x fm dlcm A fm2)
|(Not_ast_p_conjI(pr1,pr2)) =>
let val th1 = (not_ast_p_proof_of_h sg x fm dlcm A pr1)
val th2 = (not_ast_p_proof_of_h sg x fm dlcm A pr2)
in ([th1,th2] MRS (not_ast_p_conjI))
end
|(Not_ast_p_disjI(pr1,pr2)) =>
let val th1 = (not_ast_p_proof_of_h sg x fm dlcm A pr1)
val th2 = (not_ast_p_proof_of_h sg x fm dlcm A pr2)
in ([th1,th2] MRS (not_ast_p_disjI))
end;
(* Main function*)
fun not_ast_p_proof_of sg (Not_ast_p(x as Free(xn,xT),fm,dlcm,A,prl)) =
let val th = not_ast_p_proof_of_h sg x fm dlcm A prl
val fma = absfree (xn,xT, norm_zero_one fm)
val th1 = prove_elementar sg "ss" (HOLogic.mk_eq (fma,fma))
in [th,th1] MRS (not_ast_p_Q_elim)
end;
(* ------------------------------------------------------------------------- *)
(* Interpretaion of Protocols of the cooper procedure : minusinfinity version*)
(* ------------------------------------------------------------------------- *)
fun coopermi_proof_of sg x (Cooper (dlcm,Simp(fm,miprt),bsprt,nbst_p_prt)) =
(* Get the Bset thm*)
let val bst = bsetproof_of sg bsprt
val (mit1,mit2) = minf_proof_of sg dlcm miprt
val fm1 = norm_zero_one (simpl fm)
val dpos = prove_elementar sg "ss" (HOLogic.mk_binrel "op <" (zero,dlcm));
val nbstpthm = not_bst_p_proof_of sg nbst_p_prt
(* Return the four theorems needed to proove the whole Cooper Theorem*)
in (dpos,mit2,bst,nbstpthm,mit1)
end;
(* ------------------------------------------------------------------------- *)
(* Interpretaion of Protocols of the cooper procedure : plusinfinity version *)
(* ------------------------------------------------------------------------- *)
fun cooperpi_proof_of sg x (Cooper (dlcm,Simp(fm,miprt),bsprt,nast_p_prt)) =
let val ast = asetproof_of sg bsprt
val (mit1,mit2) = pinf_proof_of sg dlcm miprt
val fm1 = norm_zero_one (simpl fm)
val dpos = prove_elementar sg "ss" (HOLogic.mk_binrel "op <" (zero,dlcm));
val nastpthm = not_ast_p_proof_of sg nast_p_prt
in (dpos,mit2,ast,nastpthm,mit1)
end;
(* ------------------------------------------------------------------------- *)
(* Interpretaion of Protocols of the cooper procedure : full version*)
(* ------------------------------------------------------------------------- *)
fun cooper_thm sg s (x as Free(xn,xT)) vars cfm = case s of
"pi" => let val (rs,prt) = cooperpi_wp (xn::vars) (HOLogic.mk_exists(xn,xT,cfm))
val (dpsthm,th1,th2,nbpth,th3) = cooperpi_proof_of sg x prt
in [dpsthm,th1,th2,nbpth,th3] MRS (cppi_eq)
end
|"mi" => let val (rs,prt) = coopermi_wp (xn::vars) (HOLogic.mk_exists(xn,xT,cfm))
val (dpsthm,th1,th2,nbpth,th3) = coopermi_proof_of sg x prt
in [dpsthm,th1,th2,nbpth,th3] MRS (cpmi_eq)
end
|_ => error "parameter error";
(* ------------------------------------------------------------------------- *)
(* This function should evoluate to the end prove Procedure for one quantifier elimination for Presburger arithmetic*)
(* It shoud be plugged in the qfnp argument of the quantifier elimination proof function*)
(* ------------------------------------------------------------------------- *)
fun cooper_prv sg (x as Free(xn,xT)) efm vars = let
val l = formlcm x efm
val ac_thm = proof_of_adjustcoeffeq sg (adjustcoeffeq_wp x l efm)
val fm = snd (qe_get_terms ac_thm)
val cfm = unitycoeff x fm
val afm = adjustcoeff x l fm
val P = absfree(xn,xT,afm)
val ss = presburger_ss addsimps
[simp_from_to] delsimps [P_eqtrue, P_eqfalse, bex_triv, insert_iff]
val uth = instantiate' [] [Some (cterm_of sg P) , Some (cterm_of sg (mk_numeral l))] (unity_coeff_ex)
val e_ac_thm = (forall_intr (cterm_of sg x) ac_thm) COMP (qe_exI)
val cms = if ((length (aset x cfm)) < (length (bset x cfm))) then "pi" else "mi"
val cp_thm = cooper_thm sg cms x vars cfm
val exp_cp_thm = refl RS (simplify ss (cp_thm RSN (2,trans)))
val (lsuth,rsuth) = qe_get_terms (uth)
val (lseacth,rseacth) = qe_get_terms(e_ac_thm)
val (lscth,rscth) = qe_get_terms (exp_cp_thm)
val u_c_thm = [([uth,prove_elementar sg "ss" (HOLogic.mk_eq (rsuth,lscth))] MRS trans),exp_cp_thm] MRS trans
in ([e_ac_thm,[(prove_elementar sg "ss" (HOLogic.mk_eq (rseacth,lsuth))),u_c_thm] MRS trans] MRS trans)
end
|cooper_prv _ _ _ _ = error "Parameters format";
(*====================================================*)
(*Interpretation function for the evaluation protokol *)
(*====================================================*)
fun proof_of_evalc sg fm =
let
fun proof_of_evalch prt = case prt of
EvalAt(at) => prove_elementar sg "ss" at
|Evalfm(fm) => instantiate' [Some cboolT] [Some (cterm_of sg fm)] refl
|EvalConst(s,pr1,pr2) =>
let val th1 = proof_of_evalch pr1
val th2 = proof_of_evalch pr2
in case s of
"CJ" =>[th1,th2] MRS (qe_conjI)
|"DJ" =>[th1,th2] MRS (qe_disjI)
|"IM" =>[th1,th2] MRS (qe_impI)
|"EQ" =>[th1,th2] MRS (qe_eqI)
end
in proof_of_evalch (evalc_wp fm)
end;
(*============================================================*)
(*Interpretation function for the NNF-Transformation protokol *)
(*============================================================*)
fun proof_of_cnnf sg fm pf =
let fun proof_of_cnnfh prt pat = case prt of
NNFAt(at) => pat at
|NNFSimp (pr) => let val th1 = proof_of_cnnfh pr pat
in let val fm2 = snd (qe_get_terms th1)
in [th1,prove_elementar sg "ss" (HOLogic.mk_eq(fm2 ,simpl fm2))] MRS trans
end
end
|NNFNN (pr) => (proof_of_cnnfh pr pat) RS (nnf_nn)
|NNFConst (s,pr1,pr2) =>
let val th1 = proof_of_cnnfh pr1 pat
val th2 = proof_of_cnnfh pr2 pat
in case s of
"CJ" => [th1,th2] MRS (qe_conjI)
|"DJ" => [th1,th2] MRS (qe_disjI)
|"IM" => [th1,th2] MRS (nnf_im)
|"EQ" => [th1,th2] MRS (nnf_eq)
|"SDJ" => let val (Const("op &",_)$A$_) = fst (qe_get_terms th1)
val (Const("op &",_)$C$_) = fst (qe_get_terms th2)
in [th1,th2,prove_elementar sg "ss" (HOLogic.mk_eq (A,HOLogic.Not $ C))] MRS (nnf_sdj)
end
|"NCJ" => [th1,th2] MRS (nnf_ncj)
|"NIM" => [th1,th2] MRS (nnf_nim)
|"NEQ" => [th1,th2] MRS (nnf_neq)
|"NDJ" => [th1,th2] MRS (nnf_ndj)
end
in proof_of_cnnfh (cnnf_wp fm) pf
end;
(*====================================================*)
(* Interpretation function for the linform protokol *)
(*====================================================*)
fun proof_of_linform sg vars f =
let fun proof_of_linformh prt =
case prt of
(LfAt (at)) => prove_elementar sg "lf" (HOLogic.mk_eq (at, linform vars at))
|(LfAtdvd (Const("Divides.op dvd",_)$d$t)) => (prove_elementar sg "lf" (HOLogic.mk_eq (t, lint vars t))) RS (instantiate' [] [None , None, Some (cterm_of sg d)](linearize_dvd))
|(Lffm (fm)) => (instantiate' [Some cboolT] [Some (cterm_of sg fm)] refl)
|(LfConst (s,pr1,pr2)) =>
let val th1 = proof_of_linformh pr1
val th2 = proof_of_linformh pr2
in case s of
"CJ" => [th1,th2] MRS (qe_conjI)
|"DJ" =>[th1,th2] MRS (qe_disjI)
|"IM" =>[th1,th2] MRS (qe_impI)
|"EQ" =>[th1,th2] MRS (qe_eqI)
end
|(LfNot(pr)) =>
let val th = proof_of_linformh pr
in (th RS (qe_Not))
end
|(LfQ(s,xn,xT,pr)) =>
let val th = forall_intr (cterm_of sg (Free(xn,xT)))(proof_of_linformh pr)
in if s = "Ex"
then (th COMP(qe_exI) )
else (th COMP(qe_ALLI) )
end
in
proof_of_linformh (linform_wp f)
end;
end;