src/HOL/Tools/Presburger/cooper_proof.ML
author chaieb
Tue Jun 14 09:46:35 2005 +0200 (2005-06-14)
changeset 16389 48832eba5b1e
parent 15661 9ef583b08647
child 17485 c39871c52977
permissions -rw-r--r--
int --> IntInt.int
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(*  Title:      HOL/Integ/cooper_proof.ML
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    ID:         $Id$
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    Author:     Amine Chaieb and Tobias Nipkow, TU Muenchen
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File containing the implementation of the proof
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generation for Cooper Algorithm
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*)
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signature COOPER_PROOF =
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sig
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  val qe_Not : thm
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  val qe_conjI : thm
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  val qe_disjI : thm
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  val qe_impI : thm
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  val qe_eqI : thm
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  val qe_exI : thm
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  val list_to_set : typ -> term list -> term
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  val qe_get_terms : thm -> term * term
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  val cooper_prv  : Sign.sg -> term -> term -> thm
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  val proof_of_evalc : Sign.sg -> term -> thm
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  val proof_of_cnnf : Sign.sg -> term -> (term -> thm) -> thm
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  val proof_of_linform : Sign.sg -> string list -> term -> thm
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  val proof_of_adjustcoeffeq : Sign.sg -> term -> IntInf.int -> term -> thm
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  val prove_elementar : Sign.sg -> string -> term -> thm
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  val thm_of : Sign.sg -> (term -> (term list * (thm list -> thm))) -> term -> thm
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end;
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structure CooperProof : COOPER_PROOF =
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struct
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open CooperDec;
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(*
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val presburger_ss = simpset_of (theory "Presburger")
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  addsimps [zdiff_def] delsimps [symmetric zdiff_def];
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*)
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val presburger_ss = simpset_of (theory "Presburger")
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  addsimps[diff_int_def] delsimps [thm"diff_int_def_symmetric"];
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val cboolT = ctyp_of (sign_of HOL.thy) HOLogic.boolT;
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(*Theorems that will be used later for the proofgeneration*)
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val zdvd_iff_zmod_eq_0 = thm "zdvd_iff_zmod_eq_0";
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val unity_coeff_ex = thm "unity_coeff_ex";
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(* Thorems for proving the adjustment of the coeffitients*)
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val ac_lt_eq =  thm "ac_lt_eq";
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val ac_eq_eq = thm "ac_eq_eq";
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val ac_dvd_eq = thm "ac_dvd_eq";
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val ac_pi_eq = thm "ac_pi_eq";
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(* The logical compination of the sythetised properties*)
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val qe_Not = thm "qe_Not";
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val qe_conjI = thm "qe_conjI";
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val qe_disjI = thm "qe_disjI";
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val qe_impI = thm "qe_impI";
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val qe_eqI = thm "qe_eqI";
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val qe_exI = thm "qe_exI";
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val qe_ALLI = thm "qe_ALLI";
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(*Modulo D property for Pminusinf an Plusinf *)
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val fm_modd_minf = thm "fm_modd_minf";
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val not_dvd_modd_minf = thm "not_dvd_modd_minf";
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val dvd_modd_minf = thm "dvd_modd_minf";
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val fm_modd_pinf = thm "fm_modd_pinf";
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val not_dvd_modd_pinf = thm "not_dvd_modd_pinf";
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val dvd_modd_pinf = thm "dvd_modd_pinf";
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(* the minusinfinity proprty*)
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val fm_eq_minf = thm "fm_eq_minf";
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val neq_eq_minf = thm "neq_eq_minf";
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val eq_eq_minf = thm "eq_eq_minf";
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val le_eq_minf = thm "le_eq_minf";
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val len_eq_minf = thm "len_eq_minf";
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val not_dvd_eq_minf = thm "not_dvd_eq_minf";
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val dvd_eq_minf = thm "dvd_eq_minf";
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(* the Plusinfinity proprty*)
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val fm_eq_pinf = thm "fm_eq_pinf";
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val neq_eq_pinf = thm "neq_eq_pinf";
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val eq_eq_pinf = thm "eq_eq_pinf";
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val le_eq_pinf = thm "le_eq_pinf";
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val len_eq_pinf = thm "len_eq_pinf";
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val not_dvd_eq_pinf = thm "not_dvd_eq_pinf";
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val dvd_eq_pinf = thm "dvd_eq_pinf";
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(*Logical construction of the Property*)
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val eq_minf_conjI = thm "eq_minf_conjI";
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val eq_minf_disjI = thm "eq_minf_disjI";
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val modd_minf_disjI = thm "modd_minf_disjI";
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val modd_minf_conjI = thm "modd_minf_conjI";
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val eq_pinf_conjI = thm "eq_pinf_conjI";
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val eq_pinf_disjI = thm "eq_pinf_disjI";
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val modd_pinf_disjI = thm "modd_pinf_disjI";
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val modd_pinf_conjI = thm "modd_pinf_conjI";
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(*Cooper Backwards...*)
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(*Bset*)
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val not_bst_p_fm = thm "not_bst_p_fm";
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val not_bst_p_ne = thm "not_bst_p_ne";
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val not_bst_p_eq = thm "not_bst_p_eq";
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val not_bst_p_gt = thm "not_bst_p_gt";
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val not_bst_p_lt = thm "not_bst_p_lt";
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val not_bst_p_ndvd = thm "not_bst_p_ndvd";
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val not_bst_p_dvd = thm "not_bst_p_dvd";
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(*Aset*)
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val not_ast_p_fm = thm "not_ast_p_fm";
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val not_ast_p_ne = thm "not_ast_p_ne";
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val not_ast_p_eq = thm "not_ast_p_eq";
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val not_ast_p_gt = thm "not_ast_p_gt";
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val not_ast_p_lt = thm "not_ast_p_lt";
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val not_ast_p_ndvd = thm "not_ast_p_ndvd";
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val not_ast_p_dvd = thm "not_ast_p_dvd";
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(*Logical construction of the prop*)
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(*Bset*)
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val not_bst_p_conjI = thm "not_bst_p_conjI";
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val not_bst_p_disjI = thm "not_bst_p_disjI";
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val not_bst_p_Q_elim = thm "not_bst_p_Q_elim";
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(*Aset*)
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val not_ast_p_conjI = thm "not_ast_p_conjI";
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val not_ast_p_disjI = thm "not_ast_p_disjI";
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val not_ast_p_Q_elim = thm "not_ast_p_Q_elim";
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(*Cooper*)
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val cppi_eq = thm "cppi_eq";
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val cpmi_eq = thm "cpmi_eq";
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(*Others*)
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val simp_from_to = thm "simp_from_to";
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val P_eqtrue = thm "P_eqtrue";
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val P_eqfalse = thm "P_eqfalse";
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(*For Proving NNF*)
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val nnf_nn = thm "nnf_nn";
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val nnf_im = thm "nnf_im";
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val nnf_eq = thm "nnf_eq";
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val nnf_sdj = thm "nnf_sdj";
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val nnf_ncj = thm "nnf_ncj";
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val nnf_nim = thm "nnf_nim";
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val nnf_neq = thm "nnf_neq";
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val nnf_ndj = thm "nnf_ndj";
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(*For Proving term linearizition*)
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val linearize_dvd = thm "linearize_dvd";
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val lf_lt = thm "lf_lt";
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val lf_eq = thm "lf_eq";
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val lf_dvd = thm "lf_dvd";
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(* ------------------------------------------------------------------------- *)
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(*This function norm_zero_one  replaces the occurences of Numeral1 and Numeral0*)
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(*Respectively by their abstract representation Const("1",..) and COnst("0",..)*)
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(*this is necessary because the theorems use this representation.*)
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(* This function should be elminated in next versions...*)
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(* ------------------------------------------------------------------------- *)
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fun norm_zero_one fm = case fm of
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  (Const ("op *",_) $ c $ t) => 
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    if c = one then (norm_zero_one t)
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    else if (dest_numeral c = ~1) 
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         then (Const("uminus",HOLogic.intT --> HOLogic.intT) $ (norm_zero_one t))
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         else (HOLogic.mk_binop "op *" (norm_zero_one c,norm_zero_one t))
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  |(node $ rest) => ((norm_zero_one node)$(norm_zero_one rest))
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  |(Abs(x,T,p)) => (Abs(x,T,(norm_zero_one p)))
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  |_ => fm;
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(* ------------------------------------------------------------------------- *)
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(*function list to Set, constructs a set containing all elements of a given list.*)
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(* ------------------------------------------------------------------------- *)
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fun list_to_set T1 l = let val T = (HOLogic.mk_setT T1) in 
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	case l of 
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		[] => Const ("{}",T)
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		|(h::t) => Const("insert", T1 --> (T --> T)) $ h $(list_to_set T1 t)
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		end;
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(* ------------------------------------------------------------------------- *)
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(* Returns both sides of an equvalence in the theorem*)
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(* ------------------------------------------------------------------------- *)
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fun qe_get_terms th = let val (_$(Const("op =",Type ("fun",[Type ("bool", []),_])) $ A $ B )) = prop_of th in (A,B) end;
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(* ------------------------------------------------------------------------- *)
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(* Modified version of the simple version with minimal amount of checking and postprocessing*)
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(* ------------------------------------------------------------------------- *)
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fun simple_prove_goal_cterm2 G tacs =
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  let
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    fun check NONE = error "prove_goal: tactic failed"
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      | check (SOME (thm, _)) = (case nprems_of thm of
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            0 => thm
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          | i => !result_error_fn thm (string_of_int i ^ " unsolved goals!"))
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  in check (Seq.pull (EVERY tacs (trivial G))) end;
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(*-------------------------------------------------------------*)
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(*-------------------------------------------------------------*)
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fun cert_Trueprop sg t = cterm_of sg (HOLogic.mk_Trueprop t);
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(* ------------------------------------------------------------------------- *)
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(*This function proove elementar will be used to generate proofs at runtime*)
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(*It is is based on the isabelle function proove_goalw_cterm and is thought to *)
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(*prove properties such as a dvd b (essentially) that are only to make at
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runtime.*)
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(* ------------------------------------------------------------------------- *)
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fun prove_elementar sg s fm2 = case s of 
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  (*"ss" like simplification with simpset*)
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  "ss" =>
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    let
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      val ss = presburger_ss addsimps
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        [zdvd_iff_zmod_eq_0,unity_coeff_ex]
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      val ct =  cert_Trueprop sg fm2
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    in 
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      simple_prove_goal_cterm2 ct [simp_tac ss 1, TRY (simple_arith_tac 1)] 
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    end
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  (*"bl" like blast tactic*)
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  (* Is only used in the harrisons like proof procedure *)
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  | "bl" =>
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     let val ct = cert_Trueprop sg fm2
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     in
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       simple_prove_goal_cterm2 ct [blast_tac HOL_cs 1]
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     end
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  (*"ed" like Existence disjunctions ...*)
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  (* Is only used in the harrisons like proof procedure *)
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  | "ed" =>
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    let
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      val ex_disj_tacs =
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        let
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          val tac1 = EVERY[REPEAT(resolve_tac [disjI1,disjI2] 1), etac exI 1]
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          val tac2 = EVERY[etac exE 1, rtac exI 1,
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            REPEAT(resolve_tac [disjI1,disjI2] 1), assumption 1]
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	in [rtac iffI 1,
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          etac exE 1, REPEAT(EVERY[etac disjE 1, tac1]), tac1,
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          REPEAT(EVERY[etac disjE 1, tac2]), tac2]
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        end
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      val ct = cert_Trueprop sg fm2
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    in 
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      simple_prove_goal_cterm2 ct ex_disj_tacs
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    end
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  | "fa" =>
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    let val ct = cert_Trueprop sg fm2
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    in simple_prove_goal_cterm2 ct [simple_arith_tac 1]  
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    end
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  | "sa" =>
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    let
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      val ss = presburger_ss addsimps zadd_ac
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      val ct = cert_Trueprop sg fm2
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    in 
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      simple_prove_goal_cterm2 ct [simp_tac ss 1, TRY (simple_arith_tac 1)]  
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    end
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  (* like Existance Conjunction *)
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  | "ec" =>
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    let
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      val ss = presburger_ss addsimps zadd_ac
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      val ct = cert_Trueprop sg fm2
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    in 
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      simple_prove_goal_cterm2 ct [simp_tac ss 1, TRY (blast_tac HOL_cs 1)]
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    end
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  | "ac" =>
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    let
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      val ss = HOL_basic_ss addsimps zadd_ac
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      val ct = cert_Trueprop sg fm2
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    in 
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      simple_prove_goal_cterm2 ct [simp_tac ss 1]
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    end
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  | "lf" =>
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    let
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      val ss = presburger_ss addsimps zadd_ac
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      val ct = cert_Trueprop sg fm2
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    in 
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      simple_prove_goal_cterm2 ct [simp_tac ss 1, TRY (simple_arith_tac 1)]  
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    end;
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(*=============================================================*)
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(*-------------------------------------------------------------*)
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(*              The new compact model                          *)
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(*-------------------------------------------------------------*)
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(*=============================================================*)
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fun thm_of sg decomp t = 
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    let val (ts,recomb) = decomp t 
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    in recomb (map (thm_of sg decomp) ts) 
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    end;
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(*==================================================*)
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(*     Compact Version for adjustcoeffeq            *)
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(*==================================================*)
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fun decomp_adjustcoeffeq sg x l fm = case fm of
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    (Const("Not",_)$(Const("op <",_) $(Const("0",_)) $(rt as (Const ("op +", _)$(Const ("op *",_) $    c $ y ) $z )))) => 
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     let  
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        val m = l div (dest_numeral c) 
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        val n = if (x = y) then abs (m) else 1
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        val xtm = (HOLogic.mk_binop "op *" ((mk_numeral ((m div n)*l) ), x)) 
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        val rs = if (x = y) 
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                 then (HOLogic.mk_binrel "op <" (zero,linear_sub [] (mk_numeral n) (HOLogic.mk_binop "op +" ( xtm ,( linear_cmul n z) )))) 
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   316
                 else HOLogic.mk_binrel "op <" (zero,linear_sub [] one rt )
chaieb@14758
   317
        val ck = cterm_of sg (mk_numeral n)
chaieb@14758
   318
        val cc = cterm_of sg c
chaieb@14758
   319
        val ct = cterm_of sg z
chaieb@14758
   320
        val cx = cterm_of sg y
chaieb@14758
   321
        val pre = prove_elementar sg "lf" 
chaieb@14758
   322
            (HOLogic.mk_binrel "op <" (Const("0",HOLogic.intT),(mk_numeral n)))
skalberg@15531
   323
        val th1 = (pre RS (instantiate' [] [SOME ck,SOME cc, SOME cx, SOME ct] (ac_pi_eq)))
chaieb@14758
   324
        in ([], fn [] => [th1,(prove_elementar sg "sa" (HOLogic.mk_eq (snd (qe_get_terms th1) ,rs)))] MRS trans)
chaieb@14758
   325
        end
berghofe@13876
   326
chaieb@14758
   327
  |(Const(p,_) $a $( Const ("op +", _)$(Const ("op *",_) $ 
chaieb@14758
   328
      c $ y ) $t )) => 
chaieb@14758
   329
   if (is_arith_rel fm) andalso (x = y) 
chaieb@14758
   330
   then  
chaieb@14758
   331
        let val m = l div (dest_numeral c) 
chaieb@14758
   332
           val k = (if p = "op <" then abs(m) else m)  
chaieb@14758
   333
           val xtm = (HOLogic.mk_binop "op *" ((mk_numeral ((m div k)*l) ), x))
chaieb@14758
   334
           val rs = (HOLogic.mk_binrel p ((linear_cmul k a),(HOLogic.mk_binop "op +" ( xtm ,( linear_cmul k t) )))) 
chaieb@14758
   335
chaieb@14758
   336
           val ck = cterm_of sg (mk_numeral k)
chaieb@14758
   337
           val cc = cterm_of sg c
chaieb@14758
   338
           val ct = cterm_of sg t
chaieb@14758
   339
           val cx = cterm_of sg x
chaieb@14758
   340
           val ca = cterm_of sg a
chaieb@14758
   341
chaieb@14758
   342
	   in 
chaieb@14758
   343
	case p of
chaieb@14758
   344
	  "op <" => 
chaieb@14758
   345
	let val pre = prove_elementar sg "lf" 
chaieb@14758
   346
	    (HOLogic.mk_binrel "op <" (Const("0",HOLogic.intT),(mk_numeral k)))
skalberg@15531
   347
            val th1 = (pre RS (instantiate' [] [SOME ck,SOME ca,SOME cc, SOME cx, SOME ct] (ac_lt_eq)))
chaieb@14758
   348
	in ([], fn [] => [th1,(prove_elementar sg "lf" (HOLogic.mk_eq (snd (qe_get_terms th1) ,rs)))] MRS trans)
chaieb@14758
   349
         end
chaieb@14758
   350
chaieb@14758
   351
           |"op =" =>
chaieb@14758
   352
	     let val pre = prove_elementar sg "lf" 
berghofe@13876
   353
	    (HOLogic.Not $ (HOLogic.mk_binrel "op =" (Const("0",HOLogic.intT),(mk_numeral k))))
skalberg@15531
   354
	         val th1 = (pre RS(instantiate' [] [SOME ck,SOME ca,SOME cc, SOME cx, SOME ct] (ac_eq_eq)))
chaieb@14758
   355
	     in ([], fn [] => [th1,(prove_elementar sg "lf" (HOLogic.mk_eq (snd (qe_get_terms th1) ,rs)))] MRS trans)
chaieb@14758
   356
             end
chaieb@14758
   357
chaieb@14758
   358
             |"Divides.op dvd" =>
chaieb@14758
   359
	       let val pre = prove_elementar sg "lf" 
berghofe@13876
   360
	   (HOLogic.Not $ (HOLogic.mk_binrel "op =" (Const("0",HOLogic.intT),(mk_numeral k))))
skalberg@15531
   361
                   val th1 = (pre RS (instantiate' [] [SOME ck,SOME ca,SOME cc, SOME cx, SOME ct]) (ac_dvd_eq))
chaieb@14758
   362
               in ([], fn [] => [th1,(prove_elementar sg "lf" (HOLogic.mk_eq (snd (qe_get_terms th1) ,rs)))] MRS trans)
berghofe@13876
   363
                        
chaieb@14758
   364
               end
chaieb@14758
   365
              end
skalberg@15531
   366
  else ([], fn [] => instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] refl)
chaieb@14758
   367
chaieb@14758
   368
 |( Const ("Not", _) $ p) => ([p], fn [th] => th RS qe_Not)
chaieb@14758
   369
  |( Const ("op &",_) $ p $ q) => ([p,q], fn [th1,th2] => [th1,th2] MRS qe_conjI)
chaieb@14758
   370
  |( Const ("op |",_) $ p $ q) =>([p,q], fn [th1,th2] => [th1,th2] MRS qe_disjI)
berghofe@13876
   371
skalberg@15531
   372
  |_ => ([], fn [] => instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] refl);
berghofe@13876
   373
chaieb@14877
   374
fun proof_of_adjustcoeffeq sg x l = thm_of sg (decomp_adjustcoeffeq sg x l);
chaieb@14877
   375
chaieb@14877
   376
chaieb@14877
   377
chaieb@14758
   378
(*==================================================*)
chaieb@14758
   379
(*   Finding rho for modd_minusinfinity             *)
chaieb@14758
   380
(*==================================================*)
chaieb@14758
   381
fun rho_for_modd_minf x dlcm sg fm1 =
chaieb@14758
   382
let
wenzelm@15661
   383
    (*Some certified Terms*)
berghofe@13876
   384
    
berghofe@13876
   385
   val ctrue = cterm_of sg HOLogic.true_const
berghofe@13876
   386
   val cfalse = cterm_of sg HOLogic.false_const
berghofe@13876
   387
   val fm = norm_zero_one fm1
berghofe@13876
   388
  in  case fm1 of 
berghofe@13876
   389
      (Const ("Not", _) $ (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z))) => 
skalberg@15531
   390
         if (x=y) andalso (c1= zero) andalso (c2= one) then (instantiate' [SOME cboolT] [SOME ctrue] (fm_modd_minf))
skalberg@15531
   391
           else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_minf))
berghofe@13876
   392
berghofe@13876
   393
      |(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z)) =>
berghofe@13876
   394
  	   if (is_arith_rel fm) andalso (x=y) andalso (c1= zero) andalso (c2= one) 
skalberg@15531
   395
	   then (instantiate' [SOME cboolT] [SOME cfalse] (fm_modd_minf))
skalberg@15531
   396
	 	 else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_minf)) 
berghofe@13876
   397
berghofe@13876
   398
      |(Const("op <",_) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ pm1 $ y ) $ z )) =>
berghofe@13876
   399
           if (y=x) andalso (c1 = zero) then 
skalberg@15531
   400
            if (pm1 = one) then (instantiate' [SOME cboolT] [SOME cfalse] (fm_modd_minf)) else
skalberg@15531
   401
	     (instantiate' [SOME cboolT] [SOME ctrue] (fm_modd_minf))
skalberg@15531
   402
	    else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_minf))
berghofe@13876
   403
  
berghofe@13876
   404
      |Const ("Not",_) $ (Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) => 
berghofe@13876
   405
         if y=x then  let val cz = cterm_of sg (norm_zero_one z)
berghofe@13876
   406
			  val fm2 = HOLogic.mk_binrel "op =" (HOLogic.mk_binop "Divides.op mod" (dlcm,d),norm_zero_one zero)
skalberg@15531
   407
	 	      in(instantiate' [] [SOME cz ] ((((prove_elementar sg "ss" fm2)) RS(((zdvd_iff_zmod_eq_0)RS sym) RS iffD1) ) RS (not_dvd_modd_minf)))
berghofe@13876
   408
		      end
skalberg@15531
   409
		else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_minf))
berghofe@13876
   410
      |(Const("Divides.op dvd",_)$ d $ (db as (Const ("op +",_) $ (Const ("op *",_) $
berghofe@13876
   411
      c $ y ) $ z))) => 
berghofe@13876
   412
         if y=x then  let val cz = cterm_of sg (norm_zero_one z)
berghofe@13876
   413
			  val fm2 = HOLogic.mk_binrel "op =" (HOLogic.mk_binop "Divides.op mod" (dlcm,d),norm_zero_one zero)
skalberg@15531
   414
	 	      in(instantiate' [] [SOME cz ] ((((prove_elementar sg "ss" fm2)) RS (((zdvd_iff_zmod_eq_0)RS sym) RS iffD1) ) RS (dvd_modd_minf)))
berghofe@13876
   415
		      end
skalberg@15531
   416
		else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_minf))
berghofe@13876
   417
		
berghofe@13876
   418
    
skalberg@15531
   419
   |_ => instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_minf)
chaieb@14758
   420
   end;	 
chaieb@14758
   421
(*=========================================================================*)
chaieb@14758
   422
(*=========================================================================*)
chaieb@14758
   423
fun rho_for_eq_minf x dlcm  sg fm1 =  
chaieb@14758
   424
   let
berghofe@13876
   425
   val fm = norm_zero_one fm1
berghofe@13876
   426
    in  case fm1 of 
berghofe@13876
   427
      (Const ("Not", _) $ (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z))) => 
berghofe@13876
   428
         if  (x=y) andalso (c1=zero) andalso (c2=one) 
skalberg@15531
   429
	   then (instantiate' [] [SOME (cterm_of sg (norm_zero_one z))] (neq_eq_minf))
skalberg@15531
   430
           else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_minf))
berghofe@13876
   431
berghofe@13876
   432
      |(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z)) =>
berghofe@13876
   433
  	   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))
skalberg@15531
   434
	     then (instantiate' [] [SOME (cterm_of sg (norm_zero_one z))] (eq_eq_minf))
skalberg@15531
   435
	     else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_minf)) 
berghofe@13876
   436
berghofe@13876
   437
      |(Const("op <",_) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ pm1 $ y ) $ z )) =>
berghofe@13876
   438
           if (y=x) andalso (c1 =zero) then 
skalberg@15531
   439
            if pm1 = one then (instantiate' [] [SOME (cterm_of sg (norm_zero_one z))] (le_eq_minf)) else
skalberg@15531
   440
	     (instantiate' [] [SOME (cterm_of sg (norm_zero_one z))] (len_eq_minf))
skalberg@15531
   441
	    else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_minf))
berghofe@13876
   442
      |Const ("Not",_) $ (Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) => 
berghofe@13876
   443
         if y=x then  let val cd = cterm_of sg (norm_zero_one d)
berghofe@13876
   444
	 		  val cz = cterm_of sg (norm_zero_one z)
skalberg@15531
   445
	 	      in(instantiate' [] [SOME cd,  SOME cz] (not_dvd_eq_minf)) 
berghofe@13876
   446
		      end
berghofe@13876
   447
skalberg@15531
   448
		else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_minf))
berghofe@13876
   449
		
berghofe@13876
   450
      |(Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) => 
berghofe@13876
   451
         if y=x then  let val cd = cterm_of sg (norm_zero_one d)
berghofe@13876
   452
	 		  val cz = cterm_of sg (norm_zero_one z)
skalberg@15531
   453
	 	      in(instantiate' [] [SOME cd, SOME cz ] (dvd_eq_minf))
berghofe@13876
   454
		      end
skalberg@15531
   455
		else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_minf))
berghofe@13876
   456
berghofe@13876
   457
      		
skalberg@15531
   458
    |_ => (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_minf))
berghofe@13876
   459
 end;
berghofe@13876
   460
chaieb@14758
   461
(*=====================================================*)
chaieb@14758
   462
(*=====================================================*)
chaieb@14758
   463
(*=========== minf proofs with the compact version==========*)
chaieb@14758
   464
fun decomp_minf_eq x dlcm sg t =  case t of
chaieb@14758
   465
   Const ("op &",_) $ p $q => ([p,q],fn [th1,th2] => [th1,th2] MRS eq_minf_conjI)
chaieb@14758
   466
   |Const ("op |",_) $ p $q => ([p,q],fn [th1,th2] => [th1,th2] MRS eq_minf_disjI)
chaieb@14758
   467
   |_ => ([],fn [] => rho_for_eq_minf x dlcm sg t);
berghofe@13876
   468
chaieb@14758
   469
fun decomp_minf_modd x dlcm sg t = case t of
chaieb@14758
   470
   Const ("op &",_) $ p $q => ([p,q],fn [th1,th2] => [th1,th2] MRS modd_minf_conjI)
chaieb@14758
   471
   |Const ("op |",_) $ p $q => ([p,q],fn [th1,th2] => [th1,th2] MRS modd_minf_disjI)
chaieb@14758
   472
   |_ => ([],fn [] => rho_for_modd_minf x dlcm sg t);
berghofe@13876
   473
chaieb@14758
   474
(* -------------------------------------------------------------*)
chaieb@14758
   475
(*                    Finding rho for pinf_modd                 *)
chaieb@14758
   476
(* -------------------------------------------------------------*)
chaieb@14758
   477
fun rho_for_modd_pinf x dlcm sg fm1 = 
chaieb@14758
   478
let
wenzelm@15661
   479
    (*Some certified Terms*)
berghofe@13876
   480
    
berghofe@13876
   481
  val ctrue = cterm_of sg HOLogic.true_const
berghofe@13876
   482
  val cfalse = cterm_of sg HOLogic.false_const
berghofe@13876
   483
  val fm = norm_zero_one fm1
berghofe@13876
   484
 in  case fm1 of 
berghofe@13876
   485
      (Const ("Not", _) $ (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z))) => 
berghofe@13876
   486
         if ((x=y) andalso (c1= zero) andalso (c2= one))
skalberg@15531
   487
	 then (instantiate' [SOME cboolT] [SOME ctrue] (fm_modd_pinf))
skalberg@15531
   488
         else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_pinf))
berghofe@13876
   489
berghofe@13876
   490
      |(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z)) =>
berghofe@13876
   491
  	if ((is_arith_rel fm) andalso (x = y) andalso (c1 = zero)  andalso (c2 = one)) 
skalberg@15531
   492
	then (instantiate' [SOME cboolT] [SOME cfalse] (fm_modd_pinf))
skalberg@15531
   493
	else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_pinf))
berghofe@13876
   494
berghofe@13876
   495
      |(Const("op <",_) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ pm1 $ y ) $ z )) =>
berghofe@13876
   496
        if ((y=x) andalso (c1 = zero)) then 
berghofe@13876
   497
          if (pm1 = one) 
skalberg@15531
   498
	  then (instantiate' [SOME cboolT] [SOME ctrue] (fm_modd_pinf)) 
skalberg@15531
   499
	  else (instantiate' [SOME cboolT] [SOME cfalse] (fm_modd_pinf))
skalberg@15531
   500
	else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_pinf))
berghofe@13876
   501
  
berghofe@13876
   502
      |Const ("Not",_) $ (Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) => 
berghofe@13876
   503
         if y=x then  let val cz = cterm_of sg (norm_zero_one z)
berghofe@13876
   504
			  val fm2 = HOLogic.mk_binrel "op =" (HOLogic.mk_binop "Divides.op mod" (dlcm,d),norm_zero_one zero)
skalberg@15531
   505
	 	      in(instantiate' [] [SOME cz ] ((((prove_elementar sg "ss" fm2)) RS(((zdvd_iff_zmod_eq_0)RS sym) RS iffD1) ) RS (not_dvd_modd_pinf)))
berghofe@13876
   506
		      end
skalberg@15531
   507
		else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_pinf))
berghofe@13876
   508
      |(Const("Divides.op dvd",_)$ d $ (db as (Const ("op +",_) $ (Const ("op *",_) $
berghofe@13876
   509
      c $ y ) $ z))) => 
berghofe@13876
   510
         if y=x then  let val cz = cterm_of sg (norm_zero_one z)
berghofe@13876
   511
			  val fm2 = HOLogic.mk_binrel "op =" (HOLogic.mk_binop "Divides.op mod" (dlcm,d),norm_zero_one zero)
skalberg@15531
   512
	 	      in(instantiate' [] [SOME cz ] ((((prove_elementar sg "ss" fm2)) RS (((zdvd_iff_zmod_eq_0)RS sym) RS iffD1) ) RS (dvd_modd_pinf)))
berghofe@13876
   513
		      end
skalberg@15531
   514
		else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_pinf))
berghofe@13876
   515
		
berghofe@13876
   516
    
skalberg@15531
   517
   |_ => instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_pinf)
chaieb@14758
   518
   end;	
chaieb@14758
   519
(* -------------------------------------------------------------*)
chaieb@14758
   520
(*                    Finding rho for pinf_eq                 *)
chaieb@14758
   521
(* -------------------------------------------------------------*)
chaieb@14758
   522
fun rho_for_eq_pinf x dlcm sg fm1 = 
chaieb@14758
   523
  let
berghofe@13876
   524
					val fm = norm_zero_one fm1
berghofe@13876
   525
    in  case fm1 of 
berghofe@13876
   526
      (Const ("Not", _) $ (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z))) => 
berghofe@13876
   527
         if  (x=y) andalso (c1=zero) andalso (c2=one) 
skalberg@15531
   528
	   then (instantiate' [] [SOME (cterm_of sg (norm_zero_one z))] (neq_eq_pinf))
skalberg@15531
   529
           else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_pinf))
berghofe@13876
   530
berghofe@13876
   531
      |(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z)) =>
berghofe@13876
   532
  	   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))
skalberg@15531
   533
	     then (instantiate' [] [SOME (cterm_of sg (norm_zero_one z))] (eq_eq_pinf))
skalberg@15531
   534
	     else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_pinf)) 
berghofe@13876
   535
berghofe@13876
   536
      |(Const("op <",_) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ pm1 $ y ) $ z )) =>
berghofe@13876
   537
           if (y=x) andalso (c1 =zero) then 
skalberg@15531
   538
            if pm1 = one then (instantiate' [] [SOME (cterm_of sg (norm_zero_one z))] (le_eq_pinf)) else
skalberg@15531
   539
	     (instantiate' [] [SOME (cterm_of sg (norm_zero_one z))] (len_eq_pinf))
skalberg@15531
   540
	    else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_pinf))
berghofe@13876
   541
      |Const ("Not",_) $ (Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) => 
berghofe@13876
   542
         if y=x then  let val cd = cterm_of sg (norm_zero_one d)
berghofe@13876
   543
	 		  val cz = cterm_of sg (norm_zero_one z)
skalberg@15531
   544
	 	      in(instantiate' [] [SOME cd,  SOME cz] (not_dvd_eq_pinf)) 
berghofe@13876
   545
		      end
berghofe@13876
   546
skalberg@15531
   547
		else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_pinf))
berghofe@13876
   548
		
berghofe@13876
   549
      |(Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) => 
berghofe@13876
   550
         if y=x then  let val cd = cterm_of sg (norm_zero_one d)
berghofe@13876
   551
	 		  val cz = cterm_of sg (norm_zero_one z)
skalberg@15531
   552
	 	      in(instantiate' [] [SOME cd, SOME cz ] (dvd_eq_pinf))
berghofe@13876
   553
		      end
skalberg@15531
   554
		else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_pinf))
berghofe@13876
   555
berghofe@13876
   556
      		
skalberg@15531
   557
    |_ => (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_pinf))
berghofe@13876
   558
 end;
berghofe@13876
   559
berghofe@13876
   560
chaieb@14758
   561
chaieb@14758
   562
fun  minf_proof_of_c sg x dlcm t =
chaieb@14758
   563
  let val minf_eqth   = thm_of sg (decomp_minf_eq x dlcm sg) t
chaieb@14758
   564
      val minf_moddth = thm_of sg (decomp_minf_modd x dlcm sg) t
chaieb@14758
   565
  in (minf_eqth, minf_moddth)
chaieb@14758
   566
end;
berghofe@13876
   567
chaieb@14758
   568
(*=========== pinf proofs with the compact version==========*)
chaieb@14758
   569
fun decomp_pinf_eq x dlcm sg t = case t of
chaieb@14758
   570
   Const ("op &",_) $ p $q => ([p,q],fn [th1,th2] => [th1,th2] MRS eq_pinf_conjI)
chaieb@14758
   571
   |Const ("op |",_) $ p $q => ([p,q],fn [th1,th2] => [th1,th2] MRS eq_pinf_disjI)
chaieb@14758
   572
   |_ =>([],fn [] => rho_for_eq_pinf x dlcm sg t) ;
berghofe@13876
   573
chaieb@14758
   574
fun decomp_pinf_modd x dlcm sg t =  case t of
chaieb@14758
   575
   Const ("op &",_) $ p $q => ([p,q],fn [th1,th2] => [th1,th2] MRS modd_pinf_conjI)
chaieb@14758
   576
   |Const ("op |",_) $ p $q => ([p,q],fn [th1,th2] => [th1,th2] MRS modd_pinf_disjI)
chaieb@14758
   577
   |_ => ([],fn [] => rho_for_modd_pinf x dlcm sg t);
chaieb@14758
   578
chaieb@14758
   579
fun  pinf_proof_of_c sg x dlcm t =
chaieb@14758
   580
  let val pinf_eqth   = thm_of sg (decomp_pinf_eq x dlcm sg) t
chaieb@14758
   581
      val pinf_moddth = thm_of sg (decomp_pinf_modd x dlcm sg) t
chaieb@14758
   582
  in (pinf_eqth,pinf_moddth)
chaieb@14758
   583
end;
chaieb@14758
   584
berghofe@13876
   585
berghofe@13876
   586
(* ------------------------------------------------------------------------- *)
chaieb@14758
   587
(* Here we generate the theorem for the Bset Property in the simple direction*)
chaieb@14758
   588
(* It is just an instantiation*)
berghofe@13876
   589
(* ------------------------------------------------------------------------- *)
chaieb@14758
   590
(*
chaieb@14758
   591
fun bsetproof_of sg (x as Free(xn,xT)) fm bs dlcm   = 
chaieb@14758
   592
  let
chaieb@14758
   593
    val cp = cterm_of sg (absfree (xn,xT,(norm_zero_one fm)))
chaieb@14758
   594
    val cdlcm = cterm_of sg dlcm
chaieb@14758
   595
    val cB = cterm_of sg (list_to_set HOLogic.intT (map norm_zero_one bs))
skalberg@15531
   596
  in instantiate' [] [SOME cdlcm,SOME cB, SOME cp] (bst_thm)
chaieb@14758
   597
end;
berghofe@13876
   598
chaieb@14758
   599
fun asetproof_of sg (x as Free(xn,xT)) fm ast dlcm = 
chaieb@14758
   600
  let
chaieb@14758
   601
    val cp = cterm_of sg (absfree (xn,xT,(norm_zero_one fm)))
chaieb@14758
   602
    val cdlcm = cterm_of sg dlcm
chaieb@14758
   603
    val cA = cterm_of sg (list_to_set HOLogic.intT (map norm_zero_one ast))
skalberg@15531
   604
  in instantiate' [] [SOME cdlcm,SOME cA, SOME cp] (ast_thm)
berghofe@13876
   605
end;
chaieb@14758
   606
*)
berghofe@13876
   607
berghofe@13876
   608
(* For the generation of atomic Theorems*)
berghofe@13876
   609
(* Prove the premisses on runtime and then make RS*)
berghofe@13876
   610
(* ------------------------------------------------------------------------- *)
chaieb@14758
   611
chaieb@14758
   612
(*========= this is rho ============*)
berghofe@13876
   613
fun generate_atomic_not_bst_p sg (x as Free(xn,xT)) fm dlcm B at = 
berghofe@13876
   614
  let
berghofe@13876
   615
    val cdlcm = cterm_of sg dlcm
berghofe@13876
   616
    val cB = cterm_of sg B
berghofe@13876
   617
    val cfma = cterm_of sg (absfree (xn,xT,(norm_zero_one fm)))
berghofe@13876
   618
    val cat = cterm_of sg (norm_zero_one at)
berghofe@13876
   619
  in
berghofe@13876
   620
  case at of 
berghofe@13876
   621
   (Const ("Not", _) $ (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z))) => 
berghofe@13876
   622
      if  (x=y) andalso (c1=zero) andalso (c2=one) 
berghofe@13876
   623
	 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)
berghofe@13876
   624
	          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)))
berghofe@13876
   625
		  val th3 = prove_elementar sg "ss" (HOLogic.mk_binrel "op <" (Const("0",HOLogic.intT),dlcm))
skalberg@15531
   626
	 in  (instantiate' [] [SOME cfma]([th3,th1,th2] MRS (not_bst_p_ne)))
berghofe@13876
   627
	 end
skalberg@15531
   628
         else (instantiate' [] [SOME cfma,  SOME cdlcm, SOME cB,SOME cat] (not_bst_p_fm))
berghofe@13876
   629
berghofe@13876
   630
   |(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $(Const ("op +", T) $(Const ("op *",_) $ c2 $ y) $z)) =>
berghofe@13876
   631
     if (is_arith_rel at) andalso (x=y)
berghofe@13876
   632
	then let val bst_z = norm_zero_one (linear_neg (linear_add [] z (mk_numeral 1)))
berghofe@13876
   633
	         in let val th1 = prove_elementar sg "ss" (Const ("op :",HOLogic.intT --> (HOLogic.mk_setT HOLogic.intT) --> HOLogic.boolT) $ bst_z $ B)
berghofe@13876
   634
	          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))))
berghofe@13876
   635
		  val th3 = prove_elementar sg "ss" (HOLogic.mk_binrel "op <" (Const("0",HOLogic.intT),dlcm))
skalberg@15531
   636
	 in  (instantiate' [] [SOME cfma] ([th3,th1,th2] MRS (not_bst_p_eq)))
berghofe@13876
   637
	 end
berghofe@13876
   638
       end
skalberg@15531
   639
         else (instantiate' [] [SOME cfma,  SOME cdlcm, SOME cB,SOME cat] (not_bst_p_fm))
berghofe@13876
   640
berghofe@13876
   641
   |(Const("op <",_) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ pm1 $ y ) $ z )) =>
berghofe@13876
   642
        if (y=x) andalso (c1 =zero) then 
berghofe@13876
   643
        if pm1 = one then 
berghofe@13876
   644
	  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)
berghofe@13876
   645
              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)))
skalberg@15531
   646
	  in  (instantiate' [] [SOME cfma,  SOME cdlcm]([th1,th2] MRS (not_bst_p_gt)))
berghofe@13876
   647
	    end
berghofe@13876
   648
	 else let val th1 = prove_elementar sg "ss" (HOLogic.mk_binrel "op <" (Const("0",HOLogic.intT),dlcm))
skalberg@15531
   649
	      in (instantiate' [] [SOME cfma, SOME cB,SOME (cterm_of sg (norm_zero_one z))] (th1 RS (not_bst_p_lt)))
berghofe@13876
   650
	      end
skalberg@15531
   651
      else (instantiate' [] [SOME cfma,  SOME cdlcm, SOME cB,SOME cat] (not_bst_p_fm))
berghofe@13876
   652
berghofe@13876
   653
   |Const ("Not",_) $ (Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) => 
berghofe@13876
   654
      if y=x then  
berghofe@13876
   655
           let val cz = cterm_of sg (norm_zero_one z)
berghofe@13876
   656
	       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)
skalberg@15531
   657
 	     in (instantiate' []  [SOME cfma, SOME cB,SOME cz] (th1 RS (not_bst_p_ndvd)))
berghofe@13876
   658
	     end
skalberg@15531
   659
      else (instantiate' [] [SOME cfma,  SOME cdlcm, SOME cB,SOME cat] (not_bst_p_fm))
berghofe@13876
   660
berghofe@13876
   661
   |(Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) => 
berghofe@13876
   662
       if y=x then  
berghofe@13876
   663
	 let val cz = cterm_of sg (norm_zero_one z)
berghofe@13876
   664
	     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)
skalberg@15531
   665
 	    in (instantiate' []  [SOME cfma,SOME cB,SOME cz] (th1 RS (not_bst_p_dvd)))
berghofe@13876
   666
	  end
skalberg@15531
   667
      else (instantiate' [] [SOME cfma,  SOME cdlcm, SOME cB,SOME cat] (not_bst_p_fm))
berghofe@13876
   668
      		
skalberg@15531
   669
   |_ => (instantiate' [] [SOME cfma,  SOME cdlcm, SOME cB,SOME cat] (not_bst_p_fm))
berghofe@13876
   670
      		
berghofe@13876
   671
    end;
berghofe@13876
   672
    
chaieb@14758
   673
berghofe@13876
   674
(* ------------------------------------------------------------------------- *)    
berghofe@13876
   675
(* Main interpretation function for this backwards dirction*)
berghofe@13876
   676
(* if atomic do generate atomis formulae else Construct theorems and then make RS with the construction theorems*)
berghofe@13876
   677
(*Help Function*)
berghofe@13876
   678
(* ------------------------------------------------------------------------- *)
chaieb@14758
   679
chaieb@14758
   680
(*==================== Proof with the compact version   *)
berghofe@13876
   681
chaieb@14758
   682
fun decomp_nbstp sg x dlcm B fm t = case t of 
chaieb@14758
   683
   Const("op &",_) $ ls $ rs => ([ls,rs],fn [th1,th2] => [th1,th2] MRS not_bst_p_conjI )
chaieb@14758
   684
  |Const("op |",_) $ ls $ rs => ([ls,rs],fn [th1,th2] => [th1,th2] MRS not_bst_p_disjI)
chaieb@14758
   685
  |_ => ([], fn [] => generate_atomic_not_bst_p sg x fm dlcm B t);
chaieb@14758
   686
chaieb@14758
   687
fun not_bst_p_proof_of_c sg (x as Free(xn,xT)) fm dlcm B t =
chaieb@14758
   688
  let 
chaieb@14758
   689
       val th =  thm_of sg (decomp_nbstp sg x dlcm (list_to_set xT (map norm_zero_one B)) fm) t
berghofe@13876
   690
      val fma = absfree (xn,xT, norm_zero_one fm)
berghofe@13876
   691
  in let val th1 =  prove_elementar sg "ss"  (HOLogic.mk_eq (fma,fma))
berghofe@13876
   692
     in [th,th1] MRS (not_bst_p_Q_elim)
berghofe@13876
   693
     end
berghofe@13876
   694
  end;
berghofe@13876
   695
berghofe@13876
   696
berghofe@13876
   697
(* ------------------------------------------------------------------------- *)    
berghofe@13876
   698
(* Protokol interpretation function for the backwards direction for cooper's Theorem*)
berghofe@13876
   699
berghofe@13876
   700
(* For the generation of atomic Theorems*)
berghofe@13876
   701
(* Prove the premisses on runtime and then make RS*)
berghofe@13876
   702
(* ------------------------------------------------------------------------- *)
chaieb@14758
   703
(*========= this is rho ============*)
berghofe@13876
   704
fun generate_atomic_not_ast_p sg (x as Free(xn,xT)) fm dlcm A at = 
berghofe@13876
   705
  let
berghofe@13876
   706
    val cdlcm = cterm_of sg dlcm
berghofe@13876
   707
    val cA = cterm_of sg A
berghofe@13876
   708
    val cfma = cterm_of sg (absfree (xn,xT,(norm_zero_one fm)))
berghofe@13876
   709
    val cat = cterm_of sg (norm_zero_one at)
berghofe@13876
   710
  in
berghofe@13876
   711
  case at of 
berghofe@13876
   712
   (Const ("Not", _) $ (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z))) => 
berghofe@13876
   713
      if  (x=y) andalso (c1=zero) andalso (c2=one) 
berghofe@13876
   714
	 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)
berghofe@13876
   715
	          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)))
berghofe@13876
   716
		  val th3 = prove_elementar sg "ss" (HOLogic.mk_binrel "op <" (Const("0",HOLogic.intT),dlcm))
skalberg@15531
   717
	 in  (instantiate' [] [SOME cfma]([th3,th1,th2] MRS (not_ast_p_ne)))
berghofe@13876
   718
	 end
skalberg@15531
   719
         else (instantiate' [] [SOME cfma,  SOME cdlcm, SOME cA,SOME cat] (not_ast_p_fm))
berghofe@13876
   720
berghofe@13876
   721
   |(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $(Const ("op +", T) $(Const ("op *",_) $ c2 $ y) $z)) =>
berghofe@13876
   722
     if (is_arith_rel at) andalso (x=y)
berghofe@13876
   723
	then let val ast_z = norm_zero_one (linear_sub [] one z )
berghofe@13876
   724
	         val th1 = prove_elementar sg "ss" (Const ("op :",HOLogic.intT --> (HOLogic.mk_setT HOLogic.intT) --> HOLogic.boolT) $ ast_z $ A)
berghofe@13876
   725
	         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))))
berghofe@13876
   726
		 val th3 = prove_elementar sg "ss" (HOLogic.mk_binrel "op <" (Const("0",HOLogic.intT),dlcm))
skalberg@15531
   727
	 in  (instantiate' [] [SOME cfma] ([th3,th1,th2] MRS (not_ast_p_eq)))
berghofe@13876
   728
       end
skalberg@15531
   729
         else (instantiate' [] [SOME cfma,  SOME cdlcm, SOME cA,SOME cat] (not_ast_p_fm))
berghofe@13876
   730
berghofe@13876
   731
   |(Const("op <",_) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ pm1 $ y ) $ z )) =>
berghofe@13876
   732
        if (y=x) andalso (c1 =zero) then 
berghofe@13876
   733
        if pm1 = (mk_numeral ~1) then 
berghofe@13876
   734
	  let val th1 = prove_elementar sg "ss" (Const ("op :",HOLogic.intT --> (HOLogic.mk_setT HOLogic.intT) --> HOLogic.boolT) $ (norm_zero_one z) $ A)
berghofe@13876
   735
              val th2 =  prove_elementar sg "ss" (HOLogic.mk_binrel "op <" (zero,dlcm))
skalberg@15531
   736
	  in  (instantiate' [] [SOME cfma]([th2,th1] MRS (not_ast_p_lt)))
berghofe@13876
   737
	    end
berghofe@13876
   738
	 else let val th1 = prove_elementar sg "ss" (HOLogic.mk_binrel "op <" (Const("0",HOLogic.intT),dlcm))
skalberg@15531
   739
	      in (instantiate' [] [SOME cfma, SOME cA,SOME (cterm_of sg (norm_zero_one z))] (th1 RS (not_ast_p_gt)))
berghofe@13876
   740
	      end
skalberg@15531
   741
      else (instantiate' [] [SOME cfma,  SOME cdlcm, SOME cA,SOME cat] (not_ast_p_fm))
berghofe@13876
   742
berghofe@13876
   743
   |Const ("Not",_) $ (Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) => 
berghofe@13876
   744
      if y=x then  
berghofe@13876
   745
           let val cz = cterm_of sg (norm_zero_one z)
berghofe@13876
   746
	       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)
skalberg@15531
   747
 	     in (instantiate' []  [SOME cfma, SOME cA,SOME cz] (th1 RS (not_ast_p_ndvd)))
berghofe@13876
   748
	     end
skalberg@15531
   749
      else (instantiate' [] [SOME cfma,  SOME cdlcm, SOME cA,SOME cat] (not_ast_p_fm))
berghofe@13876
   750
berghofe@13876
   751
   |(Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) => 
berghofe@13876
   752
       if y=x then  
berghofe@13876
   753
	 let val cz = cterm_of sg (norm_zero_one z)
berghofe@13876
   754
	     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)
skalberg@15531
   755
 	    in (instantiate' []  [SOME cfma,SOME cA,SOME cz] (th1 RS (not_ast_p_dvd)))
berghofe@13876
   756
	  end
skalberg@15531
   757
      else (instantiate' [] [SOME cfma,  SOME cdlcm, SOME cA,SOME cat] (not_ast_p_fm))
berghofe@13876
   758
      		
skalberg@15531
   759
   |_ => (instantiate' [] [SOME cfma,  SOME cdlcm, SOME cA,SOME cat] (not_ast_p_fm))
berghofe@13876
   760
      		
berghofe@13876
   761
    end;
chaieb@14758
   762
chaieb@14758
   763
(* ------------------------------------------------------------------------ *)
berghofe@13876
   764
(* Main interpretation function for this backwards dirction*)
berghofe@13876
   765
(* if atomic do generate atomis formulae else Construct theorems and then make RS with the construction theorems*)
berghofe@13876
   766
(*Help Function*)
berghofe@13876
   767
(* ------------------------------------------------------------------------- *)
chaieb@14758
   768
chaieb@14758
   769
fun decomp_nastp sg x dlcm A fm t = case t of 
chaieb@14758
   770
   Const("op &",_) $ ls $ rs => ([ls,rs],fn [th1,th2] => [th1,th2] MRS not_ast_p_conjI )
chaieb@14758
   771
  |Const("op |",_) $ ls $ rs => ([ls,rs],fn [th1,th2] => [th1,th2] MRS not_ast_p_disjI)
chaieb@14758
   772
  |_ => ([], fn [] => generate_atomic_not_ast_p sg x fm dlcm A t);
berghofe@13876
   773
chaieb@14758
   774
fun not_ast_p_proof_of_c sg (x as Free(xn,xT)) fm dlcm A t =
chaieb@14758
   775
  let 
chaieb@14758
   776
       val th =  thm_of sg (decomp_nastp sg x dlcm (list_to_set xT (map norm_zero_one A)) fm) t
berghofe@13876
   777
      val fma = absfree (xn,xT, norm_zero_one fm)
chaieb@14758
   778
  in let val th1 =  prove_elementar sg "ss"  (HOLogic.mk_eq (fma,fma))
chaieb@14758
   779
     in [th,th1] MRS (not_ast_p_Q_elim)
chaieb@14758
   780
     end
chaieb@14758
   781
  end;
berghofe@13876
   782
berghofe@13876
   783
chaieb@14758
   784
(* -------------------------------*)
chaieb@14758
   785
(* Finding rho and beta for evalc *)
chaieb@14758
   786
(* -------------------------------*)
berghofe@13876
   787
chaieb@14758
   788
fun rho_for_evalc sg at = case at of  
chaieb@14758
   789
    (Const (p,_) $ s $ t) =>(  
chaieb@14758
   790
    case assoc (operations,p) of 
skalberg@15531
   791
        SOME f => 
chaieb@14758
   792
           ((if (f ((dest_numeral s),(dest_numeral t))) 
chaieb@14758
   793
             then prove_elementar sg "ss" (HOLogic.mk_eq(at,HOLogic.true_const)) 
chaieb@14758
   794
             else prove_elementar sg "ss" (HOLogic.mk_eq(at, HOLogic.false_const)))  
skalberg@15531
   795
		   handle _ => instantiate' [SOME cboolT] [SOME (cterm_of sg at)] refl)
skalberg@15531
   796
        | _ => instantiate' [SOME cboolT] [SOME (cterm_of sg at)] refl )
chaieb@14758
   797
     |Const("Not",_)$(Const (p,_) $ s $ t) =>(  
chaieb@14758
   798
       case assoc (operations,p) of 
skalberg@15531
   799
         SOME f => 
chaieb@14758
   800
           ((if (f ((dest_numeral s),(dest_numeral t))) 
chaieb@14758
   801
             then prove_elementar sg "ss" (HOLogic.mk_eq(at, HOLogic.false_const))  
chaieb@14758
   802
             else prove_elementar sg "ss" (HOLogic.mk_eq(at,HOLogic.true_const)))  
skalberg@15531
   803
		      handle _ => instantiate' [SOME cboolT] [SOME (cterm_of sg at)] refl) 
skalberg@15531
   804
         | _ => instantiate' [SOME cboolT] [SOME (cterm_of sg at)] refl ) 
skalberg@15531
   805
     | _ =>   instantiate' [SOME cboolT] [SOME (cterm_of sg at)] refl;
chaieb@14758
   806
chaieb@14758
   807
chaieb@14758
   808
(*=========================================================*)
chaieb@14758
   809
fun decomp_evalc sg t = case t of
chaieb@14758
   810
   (Const("op &",_)$A$B) => ([A,B], fn [th1,th2] => [th1,th2] MRS qe_conjI)
chaieb@14758
   811
   |(Const("op |",_)$A$B) => ([A,B], fn [th1,th2] => [th1,th2] MRS qe_disjI)
chaieb@14758
   812
   |(Const("op -->",_)$A$B) => ([A,B], fn [th1,th2] => [th1,th2] MRS qe_impI)
chaieb@14758
   813
   |(Const("op =", Type ("fun",[Type ("bool", []),_]))$A$B) => ([A,B], fn [th1,th2] => [th1,th2] MRS qe_eqI)
chaieb@14758
   814
   |_ => ([], fn [] => rho_for_evalc sg t);
chaieb@14758
   815
chaieb@14758
   816
chaieb@14758
   817
fun proof_of_evalc sg fm = thm_of sg (decomp_evalc sg) fm;
chaieb@14758
   818
chaieb@14758
   819
(*==================================================*)
chaieb@14758
   820
(*     Proof of linform with the compact model      *)
chaieb@14758
   821
(*==================================================*)
chaieb@14758
   822
chaieb@14758
   823
chaieb@14758
   824
fun decomp_linform sg vars t = case t of
chaieb@14758
   825
   (Const("op &",_)$A$B) => ([A,B], fn [th1,th2] => [th1,th2] MRS qe_conjI)
chaieb@14758
   826
   |(Const("op |",_)$A$B) => ([A,B], fn [th1,th2] => [th1,th2] MRS qe_disjI)
chaieb@14758
   827
   |(Const("op -->",_)$A$B) => ([A,B], fn [th1,th2] => [th1,th2] MRS qe_impI)
chaieb@14758
   828
   |(Const("op =", Type ("fun",[Type ("bool", []),_]))$A$B) => ([A,B], fn [th1,th2] => [th1,th2] MRS qe_eqI)
chaieb@14758
   829
   |(Const("Not",_)$p) => ([p],fn [th] => th RS qe_Not)
chaieb@15164
   830
   |(Const("Divides.op dvd",_)$d$r) => 
skalberg@15531
   831
     if is_numeral d then ([], fn [] => (prove_elementar sg "lf" (HOLogic.mk_eq (r, lint vars r))) RS (instantiate' [] [NONE , NONE, SOME (cterm_of sg d)](linearize_dvd)))
chaieb@15164
   832
     else (warning "Nonlinear Term --- Non numeral leftside at dvd";
chaieb@15164
   833
       raise COOPER)
chaieb@14758
   834
   |_ => ([], fn [] => prove_elementar sg "lf" (HOLogic.mk_eq (t, linform vars t)));
chaieb@14758
   835
chaieb@14758
   836
fun proof_of_linform sg vars f = thm_of sg (decomp_linform sg vars) f;
berghofe@13876
   837
berghofe@13876
   838
(* ------------------------------------------------------------------------- *)
berghofe@13876
   839
(* Interpretaion of Protocols of the cooper procedure : minusinfinity version*)
berghofe@13876
   840
(* ------------------------------------------------------------------------- *)
chaieb@14758
   841
fun coopermi_proof_of sg (x as Free(xn,xT)) fm B dlcm =
berghofe@13876
   842
  (* Get the Bset thm*)
chaieb@14758
   843
  let val (minf_eqth, minf_moddth) = minf_proof_of_c sg x dlcm fm 
berghofe@13876
   844
      val dpos = prove_elementar sg "ss" (HOLogic.mk_binrel "op <" (zero,dlcm));
chaieb@14758
   845
      val nbstpthm = not_bst_p_proof_of_c sg x fm dlcm B fm
chaieb@14758
   846
  in (dpos,minf_eqth,nbstpthm,minf_moddth)
berghofe@13876
   847
end;
berghofe@13876
   848
berghofe@13876
   849
(* ------------------------------------------------------------------------- *)
berghofe@13876
   850
(* Interpretaion of Protocols of the cooper procedure : plusinfinity version *)
berghofe@13876
   851
(* ------------------------------------------------------------------------- *)
chaieb@14758
   852
fun cooperpi_proof_of sg (x as Free(xn,xT)) fm A dlcm =
chaieb@14758
   853
  let val (pinf_eqth,pinf_moddth) = pinf_proof_of_c sg x dlcm fm
berghofe@13876
   854
      val dpos = prove_elementar sg "ss" (HOLogic.mk_binrel "op <" (zero,dlcm));
chaieb@14758
   855
      val nastpthm = not_ast_p_proof_of_c sg x fm dlcm A fm
chaieb@14758
   856
  in (dpos,pinf_eqth,nastpthm,pinf_moddth)
berghofe@13876
   857
end;
berghofe@13876
   858
berghofe@13876
   859
(* ------------------------------------------------------------------------- *)
berghofe@13876
   860
(* Interpretaion of Protocols of the cooper procedure : full version*)
berghofe@13876
   861
(* ------------------------------------------------------------------------- *)
chaieb@14758
   862
fun cooper_thm sg s (x as Free(xn,xT)) cfm dlcm ast bst= case s of
chaieb@14758
   863
  "pi" => let val (dpsthm,pinf_eqth,nbpth,pinf_moddth) = cooperpi_proof_of sg x cfm ast dlcm 
chaieb@14758
   864
	      in [dpsthm,pinf_eqth,nbpth,pinf_moddth] MRS (cppi_eq)
berghofe@13876
   865
           end
chaieb@14758
   866
  |"mi" => let val (dpsthm,minf_eqth,nbpth,minf_moddth) = coopermi_proof_of sg x cfm bst dlcm
chaieb@14758
   867
	       in [dpsthm,minf_eqth,nbpth,minf_moddth] MRS (cpmi_eq)
berghofe@13876
   868
                end
berghofe@13876
   869
 |_ => error "parameter error";
berghofe@13876
   870
berghofe@13876
   871
(* ------------------------------------------------------------------------- *)
berghofe@13876
   872
(* This function should evoluate to the end prove Procedure for one quantifier elimination for Presburger arithmetic*)
berghofe@13876
   873
(* It shoud be plugged in the qfnp argument of the quantifier elimination proof function*)
berghofe@13876
   874
(* ------------------------------------------------------------------------- *)
berghofe@13876
   875
chaieb@15165
   876
(* val (timef:(unit->thm) -> thm,prtime,time_reset) = gen_timer();*)
chaieb@15165
   877
(* val (timef2:(unit->thm) -> thm,prtime2,time_reset2) = gen_timer(); *)
chaieb@15164
   878
chaieb@14758
   879
fun cooper_prv sg (x as Free(xn,xT)) efm = let 
chaieb@14877
   880
   (* lfm_thm : efm = linearized form of efm*)
chaieb@14758
   881
   val lfm_thm = proof_of_linform sg [xn] efm
chaieb@14877
   882
   (*efm2 is the linearized form of efm *) 
chaieb@14758
   883
   val efm2 = snd(qe_get_terms lfm_thm)
chaieb@14877
   884
   (* l is the lcm of all coefficients of x *)
chaieb@14758
   885
   val l = formlcm x efm2
chaieb@14877
   886
   (*ac_thm: efm = efm2 with adjusted coefficients of x *)
chaieb@14877
   887
   val ac_thm = [lfm_thm , (proof_of_adjustcoeffeq sg x l efm2)] MRS trans
chaieb@14877
   888
   (* fm is efm2 with adjusted coefficients of x *)
berghofe@13876
   889
   val fm = snd (qe_get_terms ac_thm)
chaieb@14877
   890
  (* cfm is l dvd x & fm' where fm' is fm where l*x is replaced by x*)
berghofe@13876
   891
   val  cfm = unitycoeff x fm
chaieb@14877
   892
   (*afm is fm where c*x is replaced by 1*x or -1*x *)
berghofe@13876
   893
   val afm = adjustcoeff x l fm
chaieb@14877
   894
   (* P = %x.afm*)
berghofe@13876
   895
   val P = absfree(xn,xT,afm)
chaieb@14877
   896
   (* This simpset allows the elimination of the sets in bex {1..d} *)
berghofe@13876
   897
   val ss = presburger_ss addsimps
berghofe@13876
   898
     [simp_from_to] delsimps [P_eqtrue, P_eqfalse, bex_triv, insert_iff]
chaieb@14877
   899
   (* uth : EX x.P(l*x) = EX x. l dvd x & P x*)
skalberg@15531
   900
   val uth = instantiate' [] [SOME (cterm_of sg P) , SOME (cterm_of sg (mk_numeral l))] (unity_coeff_ex)
chaieb@14877
   901
   (* e_ac_thm : Ex x. efm = EX x. fm*)
berghofe@13876
   902
   val e_ac_thm = (forall_intr (cterm_of sg x) ac_thm) COMP (qe_exI)
chaieb@14877
   903
   (* A and B set of the formula*)
chaieb@14758
   904
   val A = aset x cfm
chaieb@14758
   905
   val B = bset x cfm
chaieb@14877
   906
   (* the divlcm (delta) of the formula*)
chaieb@14758
   907
   val dlcm = mk_numeral (divlcm x cfm)
chaieb@14877
   908
   (* Which set is smaller to generate the (hoepfully) shorter proof*)
chaieb@14758
   909
   val cms = if ((length A) < (length B )) then "pi" else "mi"
chaieb@15165
   910
(*   val _ = if cms = "pi" then writeln "Plusinfinity" else writeln "Minusinfinity"*)
chaieb@14877
   911
   (* synthesize the proof of cooper's theorem*)
chaieb@14877
   912
    (* cp_thm: EX x. cfm = Q*)
chaieb@15165
   913
   val cp_thm =  cooper_thm sg cms x cfm dlcm A B
chaieb@14877
   914
   (* Exxpand the right hand side to get rid of EX j : {1..d} to get a huge disjunction*)
chaieb@14877
   915
   (* exp_cp_thm: EX x.cfm = Q' , where Q' is a simplified version of Q*)
chaieb@15165
   916
(*
chaieb@15164
   917
   val _ = prth cp_thm
chaieb@15164
   918
   val _ = writeln "Expanding the bounded EX..."
chaieb@15165
   919
*)
chaieb@15165
   920
   val exp_cp_thm = refl RS (simplify ss (cp_thm RSN (2,trans)))
chaieb@15165
   921
(*
chaieb@15165
   922
   val _ = writeln "Expanded" *)
chaieb@14877
   923
   (* lsuth = EX.P(l*x) ; rsuth = EX x. l dvd x & P x*)
berghofe@13876
   924
   val (lsuth,rsuth) = qe_get_terms (uth)
chaieb@14877
   925
   (* lseacth = EX x. efm; rseacth = EX x. fm*)
berghofe@13876
   926
   val (lseacth,rseacth) = qe_get_terms(e_ac_thm)
chaieb@14877
   927
   (* lscth = EX x. cfm; rscth = Q' *)
berghofe@13876
   928
   val (lscth,rscth) = qe_get_terms (exp_cp_thm)
chaieb@14877
   929
   (* u_c_thm: EX x. P(l*x) = Q'*)
berghofe@13876
   930
   val  u_c_thm = [([uth,prove_elementar sg "ss" (HOLogic.mk_eq (rsuth,lscth))] MRS trans),exp_cp_thm] MRS trans
chaieb@14877
   931
   (* result: EX x. efm = Q'*)
berghofe@13876
   932
 in  ([e_ac_thm,[(prove_elementar sg "ss" (HOLogic.mk_eq (rseacth,lsuth))),u_c_thm] MRS trans] MRS trans)
berghofe@13876
   933
   end
chaieb@14758
   934
|cooper_prv _ _ _ =  error "Parameters format";
chaieb@14758
   935
chaieb@15107
   936
(* **************************************** *)
chaieb@15107
   937
(*    An Other Version of cooper proving    *)
chaieb@15107
   938
(*     by giving a withness for EX          *)
chaieb@15107
   939
(* **************************************** *)
chaieb@15107
   940
chaieb@15107
   941
chaieb@15107
   942
chaieb@15107
   943
fun cooper_prv_w sg (x as Free(xn,xT)) efm = let 
chaieb@15107
   944
   (* lfm_thm : efm = linearized form of efm*)
chaieb@15107
   945
   val lfm_thm = proof_of_linform sg [xn] efm
chaieb@15107
   946
   (*efm2 is the linearized form of efm *) 
chaieb@15107
   947
   val efm2 = snd(qe_get_terms lfm_thm)
chaieb@15107
   948
   (* l is the lcm of all coefficients of x *)
chaieb@15107
   949
   val l = formlcm x efm2
chaieb@15107
   950
   (*ac_thm: efm = efm2 with adjusted coefficients of x *)
chaieb@15107
   951
   val ac_thm = [lfm_thm , (proof_of_adjustcoeffeq sg x l efm2)] MRS trans
chaieb@15107
   952
   (* fm is efm2 with adjusted coefficients of x *)
chaieb@15107
   953
   val fm = snd (qe_get_terms ac_thm)
chaieb@15107
   954
  (* cfm is l dvd x & fm' where fm' is fm where l*x is replaced by x*)
chaieb@15107
   955
   val  cfm = unitycoeff x fm
chaieb@15107
   956
   (*afm is fm where c*x is replaced by 1*x or -1*x *)
chaieb@15107
   957
   val afm = adjustcoeff x l fm
chaieb@15107
   958
   (* P = %x.afm*)
chaieb@15107
   959
   val P = absfree(xn,xT,afm)
chaieb@15107
   960
   (* This simpset allows the elimination of the sets in bex {1..d} *)
chaieb@15107
   961
   val ss = presburger_ss addsimps
chaieb@15107
   962
     [simp_from_to] delsimps [P_eqtrue, P_eqfalse, bex_triv, insert_iff]
chaieb@15107
   963
   (* uth : EX x.P(l*x) = EX x. l dvd x & P x*)
skalberg@15531
   964
   val uth = instantiate' [] [SOME (cterm_of sg P) , SOME (cterm_of sg (mk_numeral l))] (unity_coeff_ex)
chaieb@15107
   965
   (* e_ac_thm : Ex x. efm = EX x. fm*)
chaieb@15107
   966
   val e_ac_thm = (forall_intr (cterm_of sg x) ac_thm) COMP (qe_exI)
chaieb@15107
   967
   (* lsuth = EX.P(l*x) ; rsuth = EX x. l dvd x & P x*)
chaieb@15107
   968
   val (lsuth,rsuth) = qe_get_terms (uth)
chaieb@15107
   969
   (* lseacth = EX x. efm; rseacth = EX x. fm*)
chaieb@15107
   970
   val (lseacth,rseacth) = qe_get_terms(e_ac_thm)
chaieb@15107
   971
chaieb@15107
   972
   val (w,rs) = cooper_w [] cfm
chaieb@15107
   973
   val exp_cp_thm =  case w of 
chaieb@15107
   974
     (* FIXME - e_ac_thm just tipped to test syntactical correctness of the program!!!!*)
skalberg@15531
   975
    SOME n =>  e_ac_thm (* Prove cfm (n) and use exI and then Eq_TrueI*)
chaieb@15107
   976
   |_ => let 
chaieb@15107
   977
    (* A and B set of the formula*)
chaieb@15107
   978
    val A = aset x cfm
chaieb@15107
   979
    val B = bset x cfm
chaieb@15107
   980
    (* the divlcm (delta) of the formula*)
chaieb@15107
   981
    val dlcm = mk_numeral (divlcm x cfm)
chaieb@15107
   982
    (* Which set is smaller to generate the (hoepfully) shorter proof*)
chaieb@15107
   983
    val cms = if ((length A) < (length B )) then "pi" else "mi"
chaieb@15107
   984
    (* synthesize the proof of cooper's theorem*)
chaieb@15107
   985
     (* cp_thm: EX x. cfm = Q*)
chaieb@15107
   986
    val cp_thm = cooper_thm sg cms x cfm dlcm A B
chaieb@15107
   987
     (* Exxpand the right hand side to get rid of EX j : {1..d} to get a huge disjunction*)
chaieb@15107
   988
    (* exp_cp_thm: EX x.cfm = Q' , where Q' is a simplified version of Q*)
chaieb@15107
   989
    in refl RS (simplify ss (cp_thm RSN (2,trans)))
chaieb@15107
   990
    end
chaieb@15107
   991
   (* lscth = EX x. cfm; rscth = Q' *)
chaieb@15107
   992
   val (lscth,rscth) = qe_get_terms (exp_cp_thm)
chaieb@15107
   993
   (* u_c_thm: EX x. P(l*x) = Q'*)
chaieb@15107
   994
   val  u_c_thm = [([uth,prove_elementar sg "ss" (HOLogic.mk_eq (rsuth,lscth))] MRS trans),exp_cp_thm] MRS trans
chaieb@15107
   995
   (* result: EX x. efm = Q'*)
chaieb@15107
   996
 in  ([e_ac_thm,[(prove_elementar sg "ss" (HOLogic.mk_eq (rseacth,lsuth))),u_c_thm] MRS trans] MRS trans)
chaieb@15107
   997
   end
chaieb@15107
   998
|cooper_prv_w _ _ _ =  error "Parameters format";
chaieb@15107
   999
chaieb@15107
  1000
berghofe@13876
  1001
chaieb@14758
  1002
fun decomp_cnnf sg lfnp P = case P of 
chaieb@14758
  1003
     Const ("op &",_) $ p $q => ([p,q] , fn [th1,th2] => [th1,th2] MRS qe_conjI )
chaieb@14758
  1004
   |Const ("op |",_) $ p $q => ([p,q] , fn [th1,th2] => [th1,th2] MRS  qe_disjI)
chaieb@14758
  1005
   |Const ("Not",_) $ (Const("Not",_) $ p) => ([p], fn [th] => th RS nnf_nn)
chaieb@14758
  1006
   |Const("Not",_) $ (Const(opn,T) $ p $ q) => 
chaieb@14758
  1007
     if opn = "op |" 
chaieb@14758
  1008
      then case (p,q) of 
chaieb@14758
  1009
         (A as (Const ("op &",_) $ r $ s),B as (Const ("op &",_) $ r1 $ t)) =>
chaieb@14758
  1010
          if r1 = negate r 
chaieb@14758
  1011
          then  ([r,HOLogic.Not$s,r1,HOLogic.Not$t],fn [th1_1,th1_2,th2_1,th2_2] => [[th1_1,th1_1] MRS qe_conjI,[th2_1,th2_2] MRS qe_conjI] MRS nnf_sdj)
berghofe@13876
  1012
chaieb@14758
  1013
          else ([HOLogic.Not $ p,HOLogic.Not $ q ], fn [th1,th2] => [th1,th2] MRS nnf_ndj)
chaieb@14758
  1014
        |(_,_) => ([HOLogic.Not $ p,HOLogic.Not $ q ], fn [th1,th2] => [th1,th2] MRS nnf_ndj)
chaieb@14758
  1015
      else (
chaieb@14758
  1016
         case (opn,T) of 
chaieb@14758
  1017
           ("op &",_) => ([HOLogic.Not $ p,HOLogic.Not $ q ], fn [th1,th2] =>[th1,th2] MRS nnf_ncj )
chaieb@14758
  1018
           |("op -->",_) => ([p,HOLogic.Not $ q ], fn [th1,th2] =>[th1,th2] MRS nnf_nim )
chaieb@14758
  1019
           |("op =",Type ("fun",[Type ("bool", []),_])) => 
chaieb@14758
  1020
           ([HOLogic.conj $ p $ (HOLogic.Not $ q),HOLogic.conj $ (HOLogic.Not $ p) $ q], fn [th1,th2] => [th1,th2] MRS nnf_neq)
chaieb@14758
  1021
            |(_,_) => ([], fn [] => lfnp P)
chaieb@14758
  1022
)
chaieb@14758
  1023
chaieb@14758
  1024
   |(Const ("op -->",_) $ p $ q) => ([HOLogic.Not$p,q], fn [th1,th2] => [th1,th2] MRS nnf_im)
chaieb@14758
  1025
chaieb@14758
  1026
   |(Const ("op =", Type ("fun",[Type ("bool", []),_])) $ p $ q) =>
chaieb@14758
  1027
     ([HOLogic.conj $ p $ q,HOLogic.conj $ (HOLogic.Not $ p) $ (HOLogic.Not $ q) ], fn [th1,th2] =>[th1,th2] MRS nnf_eq )
chaieb@14758
  1028
   |_ => ([], fn [] => lfnp P);
berghofe@13876
  1029
berghofe@13876
  1030
berghofe@13876
  1031
berghofe@13876
  1032
chaieb@14758
  1033
fun proof_of_cnnf sg p lfnp = 
chaieb@14758
  1034
 let val th1 = thm_of sg (decomp_cnnf sg lfnp) p
chaieb@14758
  1035
     val rs = snd(qe_get_terms th1)
chaieb@14758
  1036
     val th2 = prove_elementar sg "ss" (HOLogic.mk_eq(rs,simpl rs))
chaieb@14758
  1037
  in [th1,th2] MRS trans
chaieb@14758
  1038
  end;
berghofe@13876
  1039
berghofe@13876
  1040
end;
chaieb@14920
  1041