src/HOL/Integ/cooper_proof.ML
author haftmann
Mon Jan 30 08:20:56 2006 +0100 (2006-01-30)
changeset 18851 9502ce541f01
parent 17985 d5d576b72371
child 19233 77ca20b0ed77
permissions -rw-r--r--
adaptions to codegen_package
     1 (*  Title:      HOL/Integ/cooper_proof.ML
     2     ID:         $Id$
     3     Author:     Amine Chaieb and Tobias Nipkow, TU Muenchen
     4 
     5 File containing the implementation of the proof
     6 generation for Cooper Algorithm
     7 *)
     8 
     9 
    10 signature COOPER_PROOF =
    11 sig
    12   val qe_Not : thm
    13   val qe_conjI : thm
    14   val qe_disjI : thm
    15   val qe_impI : thm
    16   val qe_eqI : thm
    17   val qe_exI : thm
    18   val list_to_set : typ -> term list -> term
    19   val qe_get_terms : thm -> term * term
    20   val cooper_prv  : Sign.sg -> term -> term -> thm
    21   val proof_of_evalc : Sign.sg -> term -> thm
    22   val proof_of_cnnf : Sign.sg -> term -> (term -> thm) -> thm
    23   val proof_of_linform : Sign.sg -> string list -> term -> thm
    24   val proof_of_adjustcoeffeq : Sign.sg -> term -> IntInf.int -> term -> thm
    25   val prove_elementar : Sign.sg -> string -> term -> thm
    26   val thm_of : Sign.sg -> (term -> (term list * (thm list -> thm))) -> term -> thm
    27 end;
    28 
    29 structure CooperProof : COOPER_PROOF =
    30 struct
    31 open CooperDec;
    32 
    33 (*
    34 val presburger_ss = simpset_of (theory "Presburger")
    35   addsimps [zdiff_def] delsimps [symmetric zdiff_def];
    36 *)
    37 
    38 val presburger_ss = simpset_of (theory "Presburger")
    39   addsimps[diff_int_def] delsimps [thm"diff_int_def_symmetric"];
    40 val cboolT = ctyp_of (sign_of HOL.thy) HOLogic.boolT;
    41 
    42 (*Theorems that will be used later for the proofgeneration*)
    43 
    44 val zdvd_iff_zmod_eq_0 = thm "zdvd_iff_zmod_eq_0";
    45 val unity_coeff_ex = thm "unity_coeff_ex";
    46 
    47 (* Thorems for proving the adjustment of the coeffitients*)
    48 
    49 val ac_lt_eq =  thm "ac_lt_eq";
    50 val ac_eq_eq = thm "ac_eq_eq";
    51 val ac_dvd_eq = thm "ac_dvd_eq";
    52 val ac_pi_eq = thm "ac_pi_eq";
    53 
    54 (* The logical compination of the sythetised properties*)
    55 val qe_Not = thm "qe_Not";
    56 val qe_conjI = thm "qe_conjI";
    57 val qe_disjI = thm "qe_disjI";
    58 val qe_impI = thm "qe_impI";
    59 val qe_eqI = thm "qe_eqI";
    60 val qe_exI = thm "qe_exI";
    61 val qe_ALLI = thm "qe_ALLI";
    62 
    63 (*Modulo D property for Pminusinf an Plusinf *)
    64 val fm_modd_minf = thm "fm_modd_minf";
    65 val not_dvd_modd_minf = thm "not_dvd_modd_minf";
    66 val dvd_modd_minf = thm "dvd_modd_minf";
    67 
    68 val fm_modd_pinf = thm "fm_modd_pinf";
    69 val not_dvd_modd_pinf = thm "not_dvd_modd_pinf";
    70 val dvd_modd_pinf = thm "dvd_modd_pinf";
    71 
    72 (* the minusinfinity proprty*)
    73 
    74 val fm_eq_minf = thm "fm_eq_minf";
    75 val neq_eq_minf = thm "neq_eq_minf";
    76 val eq_eq_minf = thm "eq_eq_minf";
    77 val le_eq_minf = thm "le_eq_minf";
    78 val len_eq_minf = thm "len_eq_minf";
    79 val not_dvd_eq_minf = thm "not_dvd_eq_minf";
    80 val dvd_eq_minf = thm "dvd_eq_minf";
    81 
    82 (* the Plusinfinity proprty*)
    83 
    84 val fm_eq_pinf = thm "fm_eq_pinf";
    85 val neq_eq_pinf = thm "neq_eq_pinf";
    86 val eq_eq_pinf = thm "eq_eq_pinf";
    87 val le_eq_pinf = thm "le_eq_pinf";
    88 val len_eq_pinf = thm "len_eq_pinf";
    89 val not_dvd_eq_pinf = thm "not_dvd_eq_pinf";
    90 val dvd_eq_pinf = thm "dvd_eq_pinf";
    91 
    92 (*Logical construction of the Property*)
    93 val eq_minf_conjI = thm "eq_minf_conjI";
    94 val eq_minf_disjI = thm "eq_minf_disjI";
    95 val modd_minf_disjI = thm "modd_minf_disjI";
    96 val modd_minf_conjI = thm "modd_minf_conjI";
    97 
    98 val eq_pinf_conjI = thm "eq_pinf_conjI";
    99 val eq_pinf_disjI = thm "eq_pinf_disjI";
   100 val modd_pinf_disjI = thm "modd_pinf_disjI";
   101 val modd_pinf_conjI = thm "modd_pinf_conjI";
   102 
   103 (*Cooper Backwards...*)
   104 (*Bset*)
   105 val not_bst_p_fm = thm "not_bst_p_fm";
   106 val not_bst_p_ne = thm "not_bst_p_ne";
   107 val not_bst_p_eq = thm "not_bst_p_eq";
   108 val not_bst_p_gt = thm "not_bst_p_gt";
   109 val not_bst_p_lt = thm "not_bst_p_lt";
   110 val not_bst_p_ndvd = thm "not_bst_p_ndvd";
   111 val not_bst_p_dvd = thm "not_bst_p_dvd";
   112 
   113 (*Aset*)
   114 val not_ast_p_fm = thm "not_ast_p_fm";
   115 val not_ast_p_ne = thm "not_ast_p_ne";
   116 val not_ast_p_eq = thm "not_ast_p_eq";
   117 val not_ast_p_gt = thm "not_ast_p_gt";
   118 val not_ast_p_lt = thm "not_ast_p_lt";
   119 val not_ast_p_ndvd = thm "not_ast_p_ndvd";
   120 val not_ast_p_dvd = thm "not_ast_p_dvd";
   121 
   122 (*Logical construction of the prop*)
   123 (*Bset*)
   124 val not_bst_p_conjI = thm "not_bst_p_conjI";
   125 val not_bst_p_disjI = thm "not_bst_p_disjI";
   126 val not_bst_p_Q_elim = thm "not_bst_p_Q_elim";
   127 
   128 (*Aset*)
   129 val not_ast_p_conjI = thm "not_ast_p_conjI";
   130 val not_ast_p_disjI = thm "not_ast_p_disjI";
   131 val not_ast_p_Q_elim = thm "not_ast_p_Q_elim";
   132 
   133 (*Cooper*)
   134 val cppi_eq = thm "cppi_eq";
   135 val cpmi_eq = thm "cpmi_eq";
   136 
   137 (*Others*)
   138 val simp_from_to = thm "simp_from_to";
   139 val P_eqtrue = thm "P_eqtrue";
   140 val P_eqfalse = thm "P_eqfalse";
   141 
   142 (*For Proving NNF*)
   143 
   144 val nnf_nn = thm "nnf_nn";
   145 val nnf_im = thm "nnf_im";
   146 val nnf_eq = thm "nnf_eq";
   147 val nnf_sdj = thm "nnf_sdj";
   148 val nnf_ncj = thm "nnf_ncj";
   149 val nnf_nim = thm "nnf_nim";
   150 val nnf_neq = thm "nnf_neq";
   151 val nnf_ndj = thm "nnf_ndj";
   152 
   153 (*For Proving term linearizition*)
   154 val linearize_dvd = thm "linearize_dvd";
   155 val lf_lt = thm "lf_lt";
   156 val lf_eq = thm "lf_eq";
   157 val lf_dvd = thm "lf_dvd";
   158 
   159 
   160 (* ------------------------------------------------------------------------- *)
   161 (*This function norm_zero_one  replaces the occurences of Numeral1 and Numeral0*)
   162 (*Respectively by their abstract representation Const("1",..) and COnst("0",..)*)
   163 (*this is necessary because the theorems use this representation.*)
   164 (* This function should be elminated in next versions...*)
   165 (* ------------------------------------------------------------------------- *)
   166 
   167 fun norm_zero_one fm = case fm of
   168   (Const ("op *",_) $ c $ t) => 
   169     if c = one then (norm_zero_one t)
   170     else if (dest_numeral c = ~1) 
   171          then (Const("uminus",HOLogic.intT --> HOLogic.intT) $ (norm_zero_one t))
   172          else (HOLogic.mk_binop "op *" (norm_zero_one c,norm_zero_one t))
   173   |(node $ rest) => ((norm_zero_one node)$(norm_zero_one rest))
   174   |(Abs(x,T,p)) => (Abs(x,T,(norm_zero_one p)))
   175   |_ => fm;
   176 
   177 (* ------------------------------------------------------------------------- *)
   178 (*function list to Set, constructs a set containing all elements of a given list.*)
   179 (* ------------------------------------------------------------------------- *)
   180 fun list_to_set T1 l = let val T = (HOLogic.mk_setT T1) in 
   181 	case l of 
   182 		[] => Const ("{}",T)
   183 		|(h::t) => Const("insert", T1 --> (T --> T)) $ h $(list_to_set T1 t)
   184 		end;
   185 		
   186 (* ------------------------------------------------------------------------- *)
   187 (* Returns both sides of an equvalence in the theorem*)
   188 (* ------------------------------------------------------------------------- *)
   189 fun qe_get_terms th = let val (_$(Const("op =",Type ("fun",[Type ("bool", []),_])) $ A $ B )) = prop_of th in (A,B) end;
   190 
   191 (* ------------------------------------------------------------------------- *)
   192 (*This function proove elementar will be used to generate proofs at
   193   runtime*) (*It is thought to prove properties such as a dvd b
   194   (essentially) that are only to make at runtime.*)
   195 (* ------------------------------------------------------------------------- *)
   196 fun prove_elementar thy s fm2 =
   197   Goal.prove thy [] [] (HOLogic.mk_Trueprop fm2) (fn _ => EVERY
   198   (case s of
   199   (*"ss" like simplification with simpset*)
   200   "ss" =>
   201     let val ss = presburger_ss addsimps [zdvd_iff_zmod_eq_0,unity_coeff_ex]
   202     in [simp_tac ss 1, TRY (simple_arith_tac 1)] end
   203 
   204   (*"bl" like blast tactic*)
   205   (* Is only used in the harrisons like proof procedure *)
   206   | "bl" => [blast_tac HOL_cs 1]
   207 
   208   (*"ed" like Existence disjunctions ...*)
   209   (* Is only used in the harrisons like proof procedure *)
   210   | "ed" =>
   211     let
   212       val ex_disj_tacs =
   213         let
   214           val tac1 = EVERY[REPEAT(resolve_tac [disjI1,disjI2] 1), etac exI 1]
   215           val tac2 = EVERY[etac exE 1, rtac exI 1,
   216             REPEAT(resolve_tac [disjI1,disjI2] 1), assumption 1]
   217 	in [rtac iffI 1,
   218           etac exE 1, REPEAT(EVERY[etac disjE 1, tac1]), tac1,
   219           REPEAT(EVERY[etac disjE 1, tac2]), tac2]
   220         end
   221     in ex_disj_tacs end
   222 
   223   | "fa" => [simple_arith_tac 1]
   224 
   225   | "sa" =>
   226     let val ss = presburger_ss addsimps zadd_ac
   227     in [simp_tac ss 1, TRY (simple_arith_tac 1)] end
   228 
   229   (* like Existance Conjunction *)
   230   | "ec" =>
   231     let val ss = presburger_ss addsimps zadd_ac
   232     in [simp_tac ss 1, TRY (blast_tac HOL_cs 1)] end
   233 
   234   | "ac" =>
   235     let val ss = HOL_basic_ss addsimps zadd_ac
   236     in [simp_tac ss 1] end
   237 
   238   | "lf" =>
   239     let val ss = presburger_ss addsimps zadd_ac
   240     in [simp_tac ss 1, TRY (simple_arith_tac 1)] end));
   241 
   242 (*=============================================================*)
   243 (*-------------------------------------------------------------*)
   244 (*              The new compact model                          *)
   245 (*-------------------------------------------------------------*)
   246 (*=============================================================*)
   247 
   248 fun thm_of sg decomp t = 
   249     let val (ts,recomb) = decomp t 
   250     in recomb (map (thm_of sg decomp) ts) 
   251     end;
   252 
   253 (*==================================================*)
   254 (*     Compact Version for adjustcoeffeq            *)
   255 (*==================================================*)
   256 
   257 fun decomp_adjustcoeffeq sg x l fm = case fm of
   258     (Const("Not",_)$(Const("op <",_) $(Const("0",_)) $(rt as (Const ("op +", _)$(Const ("op *",_) $    c $ y ) $z )))) => 
   259      let  
   260         val m = l div (dest_numeral c) 
   261         val n = if (x = y) then abs (m) else 1
   262         val xtm = (HOLogic.mk_binop "op *" ((mk_numeral ((m div n)*l) ), x)) 
   263         val rs = if (x = y) 
   264                  then (HOLogic.mk_binrel "op <" (zero,linear_sub [] (mk_numeral n) (HOLogic.mk_binop "op +" ( xtm ,( linear_cmul n z) )))) 
   265                  else HOLogic.mk_binrel "op <" (zero,linear_sub [] one rt )
   266         val ck = cterm_of sg (mk_numeral n)
   267         val cc = cterm_of sg c
   268         val ct = cterm_of sg z
   269         val cx = cterm_of sg y
   270         val pre = prove_elementar sg "lf" 
   271             (HOLogic.mk_binrel "op <" (Const("0",HOLogic.intT),(mk_numeral n)))
   272         val th1 = (pre RS (instantiate' [] [SOME ck,SOME cc, SOME cx, SOME ct] (ac_pi_eq)))
   273         in ([], fn [] => [th1,(prove_elementar sg "sa" (HOLogic.mk_eq (snd (qe_get_terms th1) ,rs)))] MRS trans)
   274         end
   275 
   276   |(Const(p,_) $a $( Const ("op +", _)$(Const ("op *",_) $ 
   277       c $ y ) $t )) => 
   278    if (is_arith_rel fm) andalso (x = y) 
   279    then  
   280         let val m = l div (dest_numeral c) 
   281            val k = (if p = "op <" then abs(m) else m)  
   282            val xtm = (HOLogic.mk_binop "op *" ((mk_numeral ((m div k)*l) ), x))
   283            val rs = (HOLogic.mk_binrel p ((linear_cmul k a),(HOLogic.mk_binop "op +" ( xtm ,( linear_cmul k t) )))) 
   284 
   285            val ck = cterm_of sg (mk_numeral k)
   286            val cc = cterm_of sg c
   287            val ct = cterm_of sg t
   288            val cx = cterm_of sg x
   289            val ca = cterm_of sg a
   290 
   291 	   in 
   292 	case p of
   293 	  "op <" => 
   294 	let val pre = prove_elementar sg "lf" 
   295 	    (HOLogic.mk_binrel "op <" (Const("0",HOLogic.intT),(mk_numeral k)))
   296             val th1 = (pre RS (instantiate' [] [SOME ck,SOME ca,SOME cc, SOME cx, SOME ct] (ac_lt_eq)))
   297 	in ([], fn [] => [th1,(prove_elementar sg "lf" (HOLogic.mk_eq (snd (qe_get_terms th1) ,rs)))] MRS trans)
   298          end
   299 
   300            |"op =" =>
   301 	     let val pre = prove_elementar sg "lf" 
   302 	    (HOLogic.Not $ (HOLogic.mk_binrel "op =" (Const("0",HOLogic.intT),(mk_numeral k))))
   303 	         val th1 = (pre RS(instantiate' [] [SOME ck,SOME ca,SOME cc, SOME cx, SOME ct] (ac_eq_eq)))
   304 	     in ([], fn [] => [th1,(prove_elementar sg "lf" (HOLogic.mk_eq (snd (qe_get_terms th1) ,rs)))] MRS trans)
   305              end
   306 
   307              |"Divides.op dvd" =>
   308 	       let val pre = prove_elementar sg "lf" 
   309 	   (HOLogic.Not $ (HOLogic.mk_binrel "op =" (Const("0",HOLogic.intT),(mk_numeral k))))
   310                    val th1 = (pre RS (instantiate' [] [SOME ck,SOME ca,SOME cc, SOME cx, SOME ct]) (ac_dvd_eq))
   311                in ([], fn [] => [th1,(prove_elementar sg "lf" (HOLogic.mk_eq (snd (qe_get_terms th1) ,rs)))] MRS trans)
   312                         
   313                end
   314               end
   315   else ([], fn [] => instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] refl)
   316 
   317  |( Const ("Not", _) $ p) => ([p], fn [th] => th RS qe_Not)
   318   |( Const ("op &",_) $ p $ q) => ([p,q], fn [th1,th2] => [th1,th2] MRS qe_conjI)
   319   |( Const ("op |",_) $ p $ q) =>([p,q], fn [th1,th2] => [th1,th2] MRS qe_disjI)
   320 
   321   |_ => ([], fn [] => instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] refl);
   322 
   323 fun proof_of_adjustcoeffeq sg x l = thm_of sg (decomp_adjustcoeffeq sg x l);
   324 
   325 
   326 
   327 (*==================================================*)
   328 (*   Finding rho for modd_minusinfinity             *)
   329 (*==================================================*)
   330 fun rho_for_modd_minf x dlcm sg fm1 =
   331 let
   332     (*Some certified Terms*)
   333     
   334    val ctrue = cterm_of sg HOLogic.true_const
   335    val cfalse = cterm_of sg HOLogic.false_const
   336    val fm = norm_zero_one fm1
   337   in  case fm1 of 
   338       (Const ("Not", _) $ (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z))) => 
   339          if (x=y) andalso (c1= zero) andalso (c2= one) then (instantiate' [SOME cboolT] [SOME ctrue] (fm_modd_minf))
   340            else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_minf))
   341 
   342       |(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z)) =>
   343   	   if (is_arith_rel fm) andalso (x=y) andalso (c1= zero) andalso (c2= one) 
   344 	   then (instantiate' [SOME cboolT] [SOME cfalse] (fm_modd_minf))
   345 	 	 else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_minf)) 
   346 
   347       |(Const("op <",_) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ pm1 $ y ) $ z )) =>
   348            if (y=x) andalso (c1 = zero) then 
   349             if (pm1 = one) then (instantiate' [SOME cboolT] [SOME cfalse] (fm_modd_minf)) else
   350 	     (instantiate' [SOME cboolT] [SOME ctrue] (fm_modd_minf))
   351 	    else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_minf))
   352   
   353       |Const ("Not",_) $ (Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) => 
   354          if y=x then  let val cz = cterm_of sg (norm_zero_one z)
   355 			  val fm2 = HOLogic.mk_binrel "op =" (HOLogic.mk_binop "Divides.op mod" (dlcm,d),norm_zero_one zero)
   356 	 	      in(instantiate' [] [SOME cz ] ((((prove_elementar sg "ss" fm2)) RS(((zdvd_iff_zmod_eq_0)RS sym) RS iffD1) ) RS (not_dvd_modd_minf)))
   357 		      end
   358 		else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_minf))
   359       |(Const("Divides.op dvd",_)$ d $ (db as (Const ("op +",_) $ (Const ("op *",_) $
   360       c $ y ) $ z))) => 
   361          if y=x then  let val cz = cterm_of sg (norm_zero_one z)
   362 			  val fm2 = HOLogic.mk_binrel "op =" (HOLogic.mk_binop "Divides.op mod" (dlcm,d),norm_zero_one zero)
   363 	 	      in(instantiate' [] [SOME cz ] ((((prove_elementar sg "ss" fm2)) RS (((zdvd_iff_zmod_eq_0)RS sym) RS iffD1) ) RS (dvd_modd_minf)))
   364 		      end
   365 		else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_minf))
   366 		
   367     
   368    |_ => instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_minf)
   369    end;	 
   370 (*=========================================================================*)
   371 (*=========================================================================*)
   372 fun rho_for_eq_minf x dlcm  sg fm1 =  
   373    let
   374    val fm = norm_zero_one fm1
   375     in  case fm1 of 
   376       (Const ("Not", _) $ (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z))) => 
   377          if  (x=y) andalso (c1=zero) andalso (c2=one) 
   378 	   then (instantiate' [] [SOME (cterm_of sg (norm_zero_one z))] (neq_eq_minf))
   379            else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_minf))
   380 
   381       |(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z)) =>
   382   	   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))
   383 	     then (instantiate' [] [SOME (cterm_of sg (norm_zero_one z))] (eq_eq_minf))
   384 	     else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_minf)) 
   385 
   386       |(Const("op <",_) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ pm1 $ y ) $ z )) =>
   387            if (y=x) andalso (c1 =zero) then 
   388             if pm1 = one then (instantiate' [] [SOME (cterm_of sg (norm_zero_one z))] (le_eq_minf)) else
   389 	     (instantiate' [] [SOME (cterm_of sg (norm_zero_one z))] (len_eq_minf))
   390 	    else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_minf))
   391       |Const ("Not",_) $ (Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) => 
   392          if y=x then  let val cd = cterm_of sg (norm_zero_one d)
   393 	 		  val cz = cterm_of sg (norm_zero_one z)
   394 	 	      in(instantiate' [] [SOME cd,  SOME cz] (not_dvd_eq_minf)) 
   395 		      end
   396 
   397 		else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_minf))
   398 		
   399       |(Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) => 
   400          if y=x then  let val cd = cterm_of sg (norm_zero_one d)
   401 	 		  val cz = cterm_of sg (norm_zero_one z)
   402 	 	      in(instantiate' [] [SOME cd, SOME cz ] (dvd_eq_minf))
   403 		      end
   404 		else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_minf))
   405 
   406       		
   407     |_ => (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_minf))
   408  end;
   409 
   410 (*=====================================================*)
   411 (*=====================================================*)
   412 (*=========== minf proofs with the compact version==========*)
   413 fun decomp_minf_eq x dlcm sg t =  case t of
   414    Const ("op &",_) $ p $q => ([p,q],fn [th1,th2] => [th1,th2] MRS eq_minf_conjI)
   415    |Const ("op |",_) $ p $q => ([p,q],fn [th1,th2] => [th1,th2] MRS eq_minf_disjI)
   416    |_ => ([],fn [] => rho_for_eq_minf x dlcm sg t);
   417 
   418 fun decomp_minf_modd x dlcm sg t = case t of
   419    Const ("op &",_) $ p $q => ([p,q],fn [th1,th2] => [th1,th2] MRS modd_minf_conjI)
   420    |Const ("op |",_) $ p $q => ([p,q],fn [th1,th2] => [th1,th2] MRS modd_minf_disjI)
   421    |_ => ([],fn [] => rho_for_modd_minf x dlcm sg t);
   422 
   423 (* -------------------------------------------------------------*)
   424 (*                    Finding rho for pinf_modd                 *)
   425 (* -------------------------------------------------------------*)
   426 fun rho_for_modd_pinf x dlcm sg fm1 = 
   427 let
   428     (*Some certified Terms*)
   429     
   430   val ctrue = cterm_of sg HOLogic.true_const
   431   val cfalse = cterm_of sg HOLogic.false_const
   432   val fm = norm_zero_one fm1
   433  in  case fm1 of 
   434       (Const ("Not", _) $ (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z))) => 
   435          if ((x=y) andalso (c1= zero) andalso (c2= one))
   436 	 then (instantiate' [SOME cboolT] [SOME ctrue] (fm_modd_pinf))
   437          else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_pinf))
   438 
   439       |(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z)) =>
   440   	if ((is_arith_rel fm) andalso (x = y) andalso (c1 = zero)  andalso (c2 = one)) 
   441 	then (instantiate' [SOME cboolT] [SOME cfalse] (fm_modd_pinf))
   442 	else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_pinf))
   443 
   444       |(Const("op <",_) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ pm1 $ y ) $ z )) =>
   445         if ((y=x) andalso (c1 = zero)) then 
   446           if (pm1 = one) 
   447 	  then (instantiate' [SOME cboolT] [SOME ctrue] (fm_modd_pinf)) 
   448 	  else (instantiate' [SOME cboolT] [SOME cfalse] (fm_modd_pinf))
   449 	else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_pinf))
   450   
   451       |Const ("Not",_) $ (Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) => 
   452          if y=x then  let val cz = cterm_of sg (norm_zero_one z)
   453 			  val fm2 = HOLogic.mk_binrel "op =" (HOLogic.mk_binop "Divides.op mod" (dlcm,d),norm_zero_one zero)
   454 	 	      in(instantiate' [] [SOME cz ] ((((prove_elementar sg "ss" fm2)) RS(((zdvd_iff_zmod_eq_0)RS sym) RS iffD1) ) RS (not_dvd_modd_pinf)))
   455 		      end
   456 		else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_pinf))
   457       |(Const("Divides.op dvd",_)$ d $ (db as (Const ("op +",_) $ (Const ("op *",_) $
   458       c $ y ) $ z))) => 
   459          if y=x then  let val cz = cterm_of sg (norm_zero_one z)
   460 			  val fm2 = HOLogic.mk_binrel "op =" (HOLogic.mk_binop "Divides.op mod" (dlcm,d),norm_zero_one zero)
   461 	 	      in(instantiate' [] [SOME cz ] ((((prove_elementar sg "ss" fm2)) RS (((zdvd_iff_zmod_eq_0)RS sym) RS iffD1) ) RS (dvd_modd_pinf)))
   462 		      end
   463 		else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_pinf))
   464 		
   465     
   466    |_ => instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_modd_pinf)
   467    end;	
   468 (* -------------------------------------------------------------*)
   469 (*                    Finding rho for pinf_eq                 *)
   470 (* -------------------------------------------------------------*)
   471 fun rho_for_eq_pinf x dlcm sg fm1 = 
   472   let
   473 					val fm = norm_zero_one fm1
   474     in  case fm1 of 
   475       (Const ("Not", _) $ (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z))) => 
   476          if  (x=y) andalso (c1=zero) andalso (c2=one) 
   477 	   then (instantiate' [] [SOME (cterm_of sg (norm_zero_one z))] (neq_eq_pinf))
   478            else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_pinf))
   479 
   480       |(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z)) =>
   481   	   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))
   482 	     then (instantiate' [] [SOME (cterm_of sg (norm_zero_one z))] (eq_eq_pinf))
   483 	     else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_pinf)) 
   484 
   485       |(Const("op <",_) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ pm1 $ y ) $ z )) =>
   486            if (y=x) andalso (c1 =zero) then 
   487             if pm1 = one then (instantiate' [] [SOME (cterm_of sg (norm_zero_one z))] (le_eq_pinf)) else
   488 	     (instantiate' [] [SOME (cterm_of sg (norm_zero_one z))] (len_eq_pinf))
   489 	    else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_pinf))
   490       |Const ("Not",_) $ (Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) => 
   491          if y=x then  let val cd = cterm_of sg (norm_zero_one d)
   492 	 		  val cz = cterm_of sg (norm_zero_one z)
   493 	 	      in(instantiate' [] [SOME cd,  SOME cz] (not_dvd_eq_pinf)) 
   494 		      end
   495 
   496 		else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_pinf))
   497 		
   498       |(Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) => 
   499          if y=x then  let val cd = cterm_of sg (norm_zero_one d)
   500 	 		  val cz = cterm_of sg (norm_zero_one z)
   501 	 	      in(instantiate' [] [SOME cd, SOME cz ] (dvd_eq_pinf))
   502 		      end
   503 		else (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_pinf))
   504 
   505       		
   506     |_ => (instantiate' [SOME cboolT] [SOME (cterm_of sg fm)] (fm_eq_pinf))
   507  end;
   508 
   509 
   510 
   511 fun  minf_proof_of_c sg x dlcm t =
   512   let val minf_eqth   = thm_of sg (decomp_minf_eq x dlcm sg) t
   513       val minf_moddth = thm_of sg (decomp_minf_modd x dlcm sg) t
   514   in (minf_eqth, minf_moddth)
   515 end;
   516 
   517 (*=========== pinf proofs with the compact version==========*)
   518 fun decomp_pinf_eq x dlcm sg t = case t of
   519    Const ("op &",_) $ p $q => ([p,q],fn [th1,th2] => [th1,th2] MRS eq_pinf_conjI)
   520    |Const ("op |",_) $ p $q => ([p,q],fn [th1,th2] => [th1,th2] MRS eq_pinf_disjI)
   521    |_ =>([],fn [] => rho_for_eq_pinf x dlcm sg t) ;
   522 
   523 fun decomp_pinf_modd x dlcm sg t =  case t of
   524    Const ("op &",_) $ p $q => ([p,q],fn [th1,th2] => [th1,th2] MRS modd_pinf_conjI)
   525    |Const ("op |",_) $ p $q => ([p,q],fn [th1,th2] => [th1,th2] MRS modd_pinf_disjI)
   526    |_ => ([],fn [] => rho_for_modd_pinf x dlcm sg t);
   527 
   528 fun  pinf_proof_of_c sg x dlcm t =
   529   let val pinf_eqth   = thm_of sg (decomp_pinf_eq x dlcm sg) t
   530       val pinf_moddth = thm_of sg (decomp_pinf_modd x dlcm sg) t
   531   in (pinf_eqth,pinf_moddth)
   532 end;
   533 
   534 
   535 (* ------------------------------------------------------------------------- *)
   536 (* Here we generate the theorem for the Bset Property in the simple direction*)
   537 (* It is just an instantiation*)
   538 (* ------------------------------------------------------------------------- *)
   539 (*
   540 fun bsetproof_of sg (x as Free(xn,xT)) fm bs dlcm   = 
   541   let
   542     val cp = cterm_of sg (absfree (xn,xT,(norm_zero_one fm)))
   543     val cdlcm = cterm_of sg dlcm
   544     val cB = cterm_of sg (list_to_set HOLogic.intT (map norm_zero_one bs))
   545   in instantiate' [] [SOME cdlcm,SOME cB, SOME cp] (bst_thm)
   546 end;
   547 
   548 fun asetproof_of sg (x as Free(xn,xT)) fm ast dlcm = 
   549   let
   550     val cp = cterm_of sg (absfree (xn,xT,(norm_zero_one fm)))
   551     val cdlcm = cterm_of sg dlcm
   552     val cA = cterm_of sg (list_to_set HOLogic.intT (map norm_zero_one ast))
   553   in instantiate' [] [SOME cdlcm,SOME cA, SOME cp] (ast_thm)
   554 end;
   555 *)
   556 
   557 (* For the generation of atomic Theorems*)
   558 (* Prove the premisses on runtime and then make RS*)
   559 (* ------------------------------------------------------------------------- *)
   560 
   561 (*========= this is rho ============*)
   562 fun generate_atomic_not_bst_p sg (x as Free(xn,xT)) fm dlcm B at = 
   563   let
   564     val cdlcm = cterm_of sg dlcm
   565     val cB = cterm_of sg B
   566     val cfma = cterm_of sg (absfree (xn,xT,(norm_zero_one fm)))
   567     val cat = cterm_of sg (norm_zero_one at)
   568   in
   569   case at of 
   570    (Const ("Not", _) $ (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z))) => 
   571       if  (x=y) andalso (c1=zero) andalso (c2=one) 
   572 	 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)
   573 	          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)))
   574 		  val th3 = prove_elementar sg "ss" (HOLogic.mk_binrel "op <" (Const("0",HOLogic.intT),dlcm))
   575 	 in  (instantiate' [] [SOME cfma]([th3,th1,th2] MRS (not_bst_p_ne)))
   576 	 end
   577          else (instantiate' [] [SOME cfma,  SOME cdlcm, SOME cB,SOME cat] (not_bst_p_fm))
   578 
   579    |(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $(Const ("op +", T) $(Const ("op *",_) $ c2 $ y) $z)) =>
   580      if (is_arith_rel at) andalso (x=y)
   581 	then let val bst_z = norm_zero_one (linear_neg (linear_add [] z (mk_numeral 1)))
   582 	         in let val th1 = prove_elementar sg "ss" (Const ("op :",HOLogic.intT --> (HOLogic.mk_setT HOLogic.intT) --> HOLogic.boolT) $ bst_z $ B)
   583 	          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))))
   584 		  val th3 = prove_elementar sg "ss" (HOLogic.mk_binrel "op <" (Const("0",HOLogic.intT),dlcm))
   585 	 in  (instantiate' [] [SOME cfma] ([th3,th1,th2] MRS (not_bst_p_eq)))
   586 	 end
   587        end
   588          else (instantiate' [] [SOME cfma,  SOME cdlcm, SOME cB,SOME cat] (not_bst_p_fm))
   589 
   590    |(Const("op <",_) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ pm1 $ y ) $ z )) =>
   591         if (y=x) andalso (c1 =zero) then 
   592         if pm1 = one then 
   593 	  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)
   594               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)))
   595 	  in  (instantiate' [] [SOME cfma,  SOME cdlcm]([th1,th2] MRS (not_bst_p_gt)))
   596 	    end
   597 	 else let val th1 = prove_elementar sg "ss" (HOLogic.mk_binrel "op <" (Const("0",HOLogic.intT),dlcm))
   598 	      in (instantiate' [] [SOME cfma, SOME cB,SOME (cterm_of sg (norm_zero_one z))] (th1 RS (not_bst_p_lt)))
   599 	      end
   600       else (instantiate' [] [SOME cfma,  SOME cdlcm, SOME cB,SOME cat] (not_bst_p_fm))
   601 
   602    |Const ("Not",_) $ (Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) => 
   603       if y=x then  
   604            let val cz = cterm_of sg (norm_zero_one z)
   605 	       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)
   606  	     in (instantiate' []  [SOME cfma, SOME cB,SOME cz] (th1 RS (not_bst_p_ndvd)))
   607 	     end
   608       else (instantiate' [] [SOME cfma,  SOME cdlcm, SOME cB,SOME cat] (not_bst_p_fm))
   609 
   610    |(Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) => 
   611        if y=x then  
   612 	 let val cz = cterm_of sg (norm_zero_one z)
   613 	     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)
   614  	    in (instantiate' []  [SOME cfma,SOME cB,SOME cz] (th1 RS (not_bst_p_dvd)))
   615 	  end
   616       else (instantiate' [] [SOME cfma,  SOME cdlcm, SOME cB,SOME cat] (not_bst_p_fm))
   617       		
   618    |_ => (instantiate' [] [SOME cfma,  SOME cdlcm, SOME cB,SOME cat] (not_bst_p_fm))
   619       		
   620     end;
   621     
   622 
   623 (* ------------------------------------------------------------------------- *)    
   624 (* Main interpretation function for this backwards dirction*)
   625 (* if atomic do generate atomis formulae else Construct theorems and then make RS with the construction theorems*)
   626 (*Help Function*)
   627 (* ------------------------------------------------------------------------- *)
   628 
   629 (*==================== Proof with the compact version   *)
   630 
   631 fun decomp_nbstp sg x dlcm B fm t = case t of 
   632    Const("op &",_) $ ls $ rs => ([ls,rs],fn [th1,th2] => [th1,th2] MRS not_bst_p_conjI )
   633   |Const("op |",_) $ ls $ rs => ([ls,rs],fn [th1,th2] => [th1,th2] MRS not_bst_p_disjI)
   634   |_ => ([], fn [] => generate_atomic_not_bst_p sg x fm dlcm B t);
   635 
   636 fun not_bst_p_proof_of_c sg (x as Free(xn,xT)) fm dlcm B t =
   637   let 
   638        val th =  thm_of sg (decomp_nbstp sg x dlcm (list_to_set xT (map norm_zero_one B)) fm) t
   639       val fma = absfree (xn,xT, norm_zero_one fm)
   640   in let val th1 =  prove_elementar sg "ss"  (HOLogic.mk_eq (fma,fma))
   641      in [th,th1] MRS (not_bst_p_Q_elim)
   642      end
   643   end;
   644 
   645 
   646 (* ------------------------------------------------------------------------- *)    
   647 (* Protokol interpretation function for the backwards direction for cooper's Theorem*)
   648 
   649 (* For the generation of atomic Theorems*)
   650 (* Prove the premisses on runtime and then make RS*)
   651 (* ------------------------------------------------------------------------- *)
   652 (*========= this is rho ============*)
   653 fun generate_atomic_not_ast_p sg (x as Free(xn,xT)) fm dlcm A at = 
   654   let
   655     val cdlcm = cterm_of sg dlcm
   656     val cA = cterm_of sg A
   657     val cfma = cterm_of sg (absfree (xn,xT,(norm_zero_one fm)))
   658     val cat = cterm_of sg (norm_zero_one at)
   659   in
   660   case at of 
   661    (Const ("Not", _) $ (Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $ (Const ("op +", _) $(Const ("op *",_) $ c2 $ y) $z))) => 
   662       if  (x=y) andalso (c1=zero) andalso (c2=one) 
   663 	 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)
   664 	          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)))
   665 		  val th3 = prove_elementar sg "ss" (HOLogic.mk_binrel "op <" (Const("0",HOLogic.intT),dlcm))
   666 	 in  (instantiate' [] [SOME cfma]([th3,th1,th2] MRS (not_ast_p_ne)))
   667 	 end
   668          else (instantiate' [] [SOME cfma,  SOME cdlcm, SOME cA,SOME cat] (not_ast_p_fm))
   669 
   670    |(Const("op =",Type ("fun",[Type ("IntDef.int", []),_])) $ c1 $(Const ("op +", T) $(Const ("op *",_) $ c2 $ y) $z)) =>
   671      if (is_arith_rel at) andalso (x=y)
   672 	then let val ast_z = norm_zero_one (linear_sub [] one z )
   673 	         val th1 = prove_elementar sg "ss" (Const ("op :",HOLogic.intT --> (HOLogic.mk_setT HOLogic.intT) --> HOLogic.boolT) $ ast_z $ A)
   674 	         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))))
   675 		 val th3 = prove_elementar sg "ss" (HOLogic.mk_binrel "op <" (Const("0",HOLogic.intT),dlcm))
   676 	 in  (instantiate' [] [SOME cfma] ([th3,th1,th2] MRS (not_ast_p_eq)))
   677        end
   678          else (instantiate' [] [SOME cfma,  SOME cdlcm, SOME cA,SOME cat] (not_ast_p_fm))
   679 
   680    |(Const("op <",_) $ c1 $(Const ("op +", _) $(Const ("op *",_) $ pm1 $ y ) $ z )) =>
   681         if (y=x) andalso (c1 =zero) then 
   682         if pm1 = (mk_numeral ~1) then 
   683 	  let val th1 = prove_elementar sg "ss" (Const ("op :",HOLogic.intT --> (HOLogic.mk_setT HOLogic.intT) --> HOLogic.boolT) $ (norm_zero_one z) $ A)
   684               val th2 =  prove_elementar sg "ss" (HOLogic.mk_binrel "op <" (zero,dlcm))
   685 	  in  (instantiate' [] [SOME cfma]([th2,th1] MRS (not_ast_p_lt)))
   686 	    end
   687 	 else let val th1 = prove_elementar sg "ss" (HOLogic.mk_binrel "op <" (Const("0",HOLogic.intT),dlcm))
   688 	      in (instantiate' [] [SOME cfma, SOME cA,SOME (cterm_of sg (norm_zero_one z))] (th1 RS (not_ast_p_gt)))
   689 	      end
   690       else (instantiate' [] [SOME cfma,  SOME cdlcm, SOME cA,SOME cat] (not_ast_p_fm))
   691 
   692    |Const ("Not",_) $ (Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) => 
   693       if y=x then  
   694            let val cz = cterm_of sg (norm_zero_one z)
   695 	       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)
   696  	     in (instantiate' []  [SOME cfma, SOME cA,SOME cz] (th1 RS (not_ast_p_ndvd)))
   697 	     end
   698       else (instantiate' [] [SOME cfma,  SOME cdlcm, SOME cA,SOME cat] (not_ast_p_fm))
   699 
   700    |(Const("Divides.op dvd",_)$ d $ (Const ("op +",_) $ (Const ("op *",_) $ c $ y ) $ z)) => 
   701        if y=x then  
   702 	 let val cz = cterm_of sg (norm_zero_one z)
   703 	     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)
   704  	    in (instantiate' []  [SOME cfma,SOME cA,SOME cz] (th1 RS (not_ast_p_dvd)))
   705 	  end
   706       else (instantiate' [] [SOME cfma,  SOME cdlcm, SOME cA,SOME cat] (not_ast_p_fm))
   707       		
   708    |_ => (instantiate' [] [SOME cfma,  SOME cdlcm, SOME cA,SOME cat] (not_ast_p_fm))
   709       		
   710     end;
   711 
   712 (* ------------------------------------------------------------------------ *)
   713 (* Main interpretation function for this backwards dirction*)
   714 (* if atomic do generate atomis formulae else Construct theorems and then make RS with the construction theorems*)
   715 (*Help Function*)
   716 (* ------------------------------------------------------------------------- *)
   717 
   718 fun decomp_nastp sg x dlcm A fm t = case t of 
   719    Const("op &",_) $ ls $ rs => ([ls,rs],fn [th1,th2] => [th1,th2] MRS not_ast_p_conjI )
   720   |Const("op |",_) $ ls $ rs => ([ls,rs],fn [th1,th2] => [th1,th2] MRS not_ast_p_disjI)
   721   |_ => ([], fn [] => generate_atomic_not_ast_p sg x fm dlcm A t);
   722 
   723 fun not_ast_p_proof_of_c sg (x as Free(xn,xT)) fm dlcm A t =
   724   let 
   725        val th =  thm_of sg (decomp_nastp sg x dlcm (list_to_set xT (map norm_zero_one A)) fm) t
   726       val fma = absfree (xn,xT, norm_zero_one fm)
   727   in let val th1 =  prove_elementar sg "ss"  (HOLogic.mk_eq (fma,fma))
   728      in [th,th1] MRS (not_ast_p_Q_elim)
   729      end
   730   end;
   731 
   732 
   733 (* -------------------------------*)
   734 (* Finding rho and beta for evalc *)
   735 (* -------------------------------*)
   736 
   737 fun rho_for_evalc sg at = case at of  
   738     (Const (p,_) $ s $ t) =>(  
   739     case AList.lookup (op =) operations p of 
   740         SOME f => 
   741            ((if (f ((dest_numeral s),(dest_numeral t))) 
   742              then prove_elementar sg "ss" (HOLogic.mk_eq(at,HOLogic.true_const)) 
   743              else prove_elementar sg "ss" (HOLogic.mk_eq(at, HOLogic.false_const)))  
   744 		   handle _ => instantiate' [SOME cboolT] [SOME (cterm_of sg at)] refl)
   745         | _ => instantiate' [SOME cboolT] [SOME (cterm_of sg at)] refl )
   746      |Const("Not",_)$(Const (p,_) $ s $ t) =>(  
   747        case AList.lookup (op =) operations p of 
   748          SOME f => 
   749            ((if (f ((dest_numeral s),(dest_numeral t))) 
   750              then prove_elementar sg "ss" (HOLogic.mk_eq(at, HOLogic.false_const))  
   751              else prove_elementar sg "ss" (HOLogic.mk_eq(at,HOLogic.true_const)))  
   752 		      handle _ => instantiate' [SOME cboolT] [SOME (cterm_of sg at)] refl) 
   753          | _ => instantiate' [SOME cboolT] [SOME (cterm_of sg at)] refl ) 
   754      | _ =>   instantiate' [SOME cboolT] [SOME (cterm_of sg at)] refl;
   755 
   756 
   757 (*=========================================================*)
   758 fun decomp_evalc sg t = case t of
   759    (Const("op &",_)$A$B) => ([A,B], fn [th1,th2] => [th1,th2] MRS qe_conjI)
   760    |(Const("op |",_)$A$B) => ([A,B], fn [th1,th2] => [th1,th2] MRS qe_disjI)
   761    |(Const("op -->",_)$A$B) => ([A,B], fn [th1,th2] => [th1,th2] MRS qe_impI)
   762    |(Const("op =", Type ("fun",[Type ("bool", []),_]))$A$B) => ([A,B], fn [th1,th2] => [th1,th2] MRS qe_eqI)
   763    |_ => ([], fn [] => rho_for_evalc sg t);
   764 
   765 
   766 fun proof_of_evalc sg fm = thm_of sg (decomp_evalc sg) fm;
   767 
   768 (*==================================================*)
   769 (*     Proof of linform with the compact model      *)
   770 (*==================================================*)
   771 
   772 
   773 fun decomp_linform sg vars t = case t of
   774    (Const("op &",_)$A$B) => ([A,B], fn [th1,th2] => [th1,th2] MRS qe_conjI)
   775    |(Const("op |",_)$A$B) => ([A,B], fn [th1,th2] => [th1,th2] MRS qe_disjI)
   776    |(Const("op -->",_)$A$B) => ([A,B], fn [th1,th2] => [th1,th2] MRS qe_impI)
   777    |(Const("op =", Type ("fun",[Type ("bool", []),_]))$A$B) => ([A,B], fn [th1,th2] => [th1,th2] MRS qe_eqI)
   778    |(Const("Not",_)$p) => ([p],fn [th] => th RS qe_Not)
   779    |(Const("Divides.op dvd",_)$d$r) => 
   780      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)))
   781      else (warning "Nonlinear Term --- Non numeral leftside at dvd";
   782        raise COOPER)
   783    |_ => ([], fn [] => prove_elementar sg "lf" (HOLogic.mk_eq (t, linform vars t)));
   784 
   785 fun proof_of_linform sg vars f = thm_of sg (decomp_linform sg vars) f;
   786 
   787 (* ------------------------------------------------------------------------- *)
   788 (* Interpretaion of Protocols of the cooper procedure : minusinfinity version*)
   789 (* ------------------------------------------------------------------------- *)
   790 fun coopermi_proof_of sg (x as Free(xn,xT)) fm B dlcm =
   791   (* Get the Bset thm*)
   792   let val (minf_eqth, minf_moddth) = minf_proof_of_c sg x dlcm fm 
   793       val dpos = prove_elementar sg "ss" (HOLogic.mk_binrel "op <" (zero,dlcm));
   794       val nbstpthm = not_bst_p_proof_of_c sg x fm dlcm B fm
   795   in (dpos,minf_eqth,nbstpthm,minf_moddth)
   796 end;
   797 
   798 (* ------------------------------------------------------------------------- *)
   799 (* Interpretaion of Protocols of the cooper procedure : plusinfinity version *)
   800 (* ------------------------------------------------------------------------- *)
   801 fun cooperpi_proof_of sg (x as Free(xn,xT)) fm A dlcm =
   802   let val (pinf_eqth,pinf_moddth) = pinf_proof_of_c sg x dlcm fm
   803       val dpos = prove_elementar sg "ss" (HOLogic.mk_binrel "op <" (zero,dlcm));
   804       val nastpthm = not_ast_p_proof_of_c sg x fm dlcm A fm
   805   in (dpos,pinf_eqth,nastpthm,pinf_moddth)
   806 end;
   807 
   808 (* ------------------------------------------------------------------------- *)
   809 (* Interpretaion of Protocols of the cooper procedure : full version*)
   810 (* ------------------------------------------------------------------------- *)
   811 fun cooper_thm sg s (x as Free(xn,xT)) cfm dlcm ast bst= case s of
   812   "pi" => let val (dpsthm,pinf_eqth,nbpth,pinf_moddth) = cooperpi_proof_of sg x cfm ast dlcm 
   813 	      in [dpsthm,pinf_eqth,nbpth,pinf_moddth] MRS (cppi_eq)
   814            end
   815   |"mi" => let val (dpsthm,minf_eqth,nbpth,minf_moddth) = coopermi_proof_of sg x cfm bst dlcm
   816 	       in [dpsthm,minf_eqth,nbpth,minf_moddth] MRS (cpmi_eq)
   817                 end
   818  |_ => error "parameter error";
   819 
   820 (* ------------------------------------------------------------------------- *)
   821 (* This function should evoluate to the end prove Procedure for one quantifier elimination for Presburger arithmetic*)
   822 (* It shoud be plugged in the qfnp argument of the quantifier elimination proof function*)
   823 (* ------------------------------------------------------------------------- *)
   824 
   825 (* val (timef:(unit->thm) -> thm,prtime,time_reset) = gen_timer();*)
   826 (* val (timef2:(unit->thm) -> thm,prtime2,time_reset2) = gen_timer(); *)
   827 
   828 fun cooper_prv sg (x as Free(xn,xT)) efm = let 
   829    (* lfm_thm : efm = linearized form of efm*)
   830    val lfm_thm = proof_of_linform sg [xn] efm
   831    (*efm2 is the linearized form of efm *) 
   832    val efm2 = snd(qe_get_terms lfm_thm)
   833    (* l is the lcm of all coefficients of x *)
   834    val l = formlcm x efm2
   835    (*ac_thm: efm = efm2 with adjusted coefficients of x *)
   836    val ac_thm = [lfm_thm , (proof_of_adjustcoeffeq sg x l efm2)] MRS trans
   837    (* fm is efm2 with adjusted coefficients of x *)
   838    val fm = snd (qe_get_terms ac_thm)
   839   (* cfm is l dvd x & fm' where fm' is fm where l*x is replaced by x*)
   840    val  cfm = unitycoeff x fm
   841    (*afm is fm where c*x is replaced by 1*x or -1*x *)
   842    val afm = adjustcoeff x l fm
   843    (* P = %x.afm*)
   844    val P = absfree(xn,xT,afm)
   845    (* This simpset allows the elimination of the sets in bex {1..d} *)
   846    val ss = presburger_ss addsimps
   847      [simp_from_to] delsimps [P_eqtrue, P_eqfalse, bex_triv, insert_iff]
   848    (* uth : EX x.P(l*x) = EX x. l dvd x & P x*)
   849    val uth = instantiate' [] [SOME (cterm_of sg P) , SOME (cterm_of sg (mk_numeral l))] (unity_coeff_ex)
   850    (* e_ac_thm : Ex x. efm = EX x. fm*)
   851    val e_ac_thm = (forall_intr (cterm_of sg x) ac_thm) COMP (qe_exI)
   852    (* A and B set of the formula*)
   853    val A = aset x cfm
   854    val B = bset x cfm
   855    (* the divlcm (delta) of the formula*)
   856    val dlcm = mk_numeral (divlcm x cfm)
   857    (* Which set is smaller to generate the (hoepfully) shorter proof*)
   858    val cms = if ((length A) < (length B )) then "pi" else "mi"
   859 (*   val _ = if cms = "pi" then writeln "Plusinfinity" else writeln "Minusinfinity"*)
   860    (* synthesize the proof of cooper's theorem*)
   861     (* cp_thm: EX x. cfm = Q*)
   862    val cp_thm =  cooper_thm sg cms x cfm dlcm A B
   863    (* Exxpand the right hand side to get rid of EX j : {1..d} to get a huge disjunction*)
   864    (* exp_cp_thm: EX x.cfm = Q' , where Q' is a simplified version of Q*)
   865 (*
   866    val _ = prth cp_thm
   867    val _ = writeln "Expanding the bounded EX..."
   868 *)
   869    val exp_cp_thm = refl RS (simplify ss (cp_thm RSN (2,trans)))
   870 (*
   871    val _ = writeln "Expanded" *)
   872    (* lsuth = EX.P(l*x) ; rsuth = EX x. l dvd x & P x*)
   873    val (lsuth,rsuth) = qe_get_terms (uth)
   874    (* lseacth = EX x. efm; rseacth = EX x. fm*)
   875    val (lseacth,rseacth) = qe_get_terms(e_ac_thm)
   876    (* lscth = EX x. cfm; rscth = Q' *)
   877    val (lscth,rscth) = qe_get_terms (exp_cp_thm)
   878    (* u_c_thm: EX x. P(l*x) = Q'*)
   879    val  u_c_thm = [([uth,prove_elementar sg "ss" (HOLogic.mk_eq (rsuth,lscth))] MRS trans),exp_cp_thm] MRS trans
   880    (* result: EX x. efm = Q'*)
   881  in  ([e_ac_thm,[(prove_elementar sg "ss" (HOLogic.mk_eq (rseacth,lsuth))),u_c_thm] MRS trans] MRS trans)
   882    end
   883 |cooper_prv _ _ _ =  error "Parameters format";
   884 
   885 (* **************************************** *)
   886 (*    An Other Version of cooper proving    *)
   887 (*     by giving a withness for EX          *)
   888 (* **************************************** *)
   889 
   890 
   891 
   892 fun cooper_prv_w sg (x as Free(xn,xT)) efm = let 
   893    (* lfm_thm : efm = linearized form of efm*)
   894    val lfm_thm = proof_of_linform sg [xn] efm
   895    (*efm2 is the linearized form of efm *) 
   896    val efm2 = snd(qe_get_terms lfm_thm)
   897    (* l is the lcm of all coefficients of x *)
   898    val l = formlcm x efm2
   899    (*ac_thm: efm = efm2 with adjusted coefficients of x *)
   900    val ac_thm = [lfm_thm , (proof_of_adjustcoeffeq sg x l efm2)] MRS trans
   901    (* fm is efm2 with adjusted coefficients of x *)
   902    val fm = snd (qe_get_terms ac_thm)
   903   (* cfm is l dvd x & fm' where fm' is fm where l*x is replaced by x*)
   904    val  cfm = unitycoeff x fm
   905    (*afm is fm where c*x is replaced by 1*x or -1*x *)
   906    val afm = adjustcoeff x l fm
   907    (* P = %x.afm*)
   908    val P = absfree(xn,xT,afm)
   909    (* This simpset allows the elimination of the sets in bex {1..d} *)
   910    val ss = presburger_ss addsimps
   911      [simp_from_to] delsimps [P_eqtrue, P_eqfalse, bex_triv, insert_iff]
   912    (* uth : EX x.P(l*x) = EX x. l dvd x & P x*)
   913    val uth = instantiate' [] [SOME (cterm_of sg P) , SOME (cterm_of sg (mk_numeral l))] (unity_coeff_ex)
   914    (* e_ac_thm : Ex x. efm = EX x. fm*)
   915    val e_ac_thm = (forall_intr (cterm_of sg x) ac_thm) COMP (qe_exI)
   916    (* lsuth = EX.P(l*x) ; rsuth = EX x. l dvd x & P x*)
   917    val (lsuth,rsuth) = qe_get_terms (uth)
   918    (* lseacth = EX x. efm; rseacth = EX x. fm*)
   919    val (lseacth,rseacth) = qe_get_terms(e_ac_thm)
   920 
   921    val (w,rs) = cooper_w [] cfm
   922    val exp_cp_thm =  case w of 
   923      (* FIXME - e_ac_thm just tipped to test syntactical correctness of the program!!!!*)
   924     SOME n =>  e_ac_thm (* Prove cfm (n) and use exI and then Eq_TrueI*)
   925    |_ => let 
   926     (* A and B set of the formula*)
   927     val A = aset x cfm
   928     val B = bset x cfm
   929     (* the divlcm (delta) of the formula*)
   930     val dlcm = mk_numeral (divlcm x cfm)
   931     (* Which set is smaller to generate the (hoepfully) shorter proof*)
   932     val cms = if ((length A) < (length B )) then "pi" else "mi"
   933     (* synthesize the proof of cooper's theorem*)
   934      (* cp_thm: EX x. cfm = Q*)
   935     val cp_thm = cooper_thm sg cms x cfm dlcm A B
   936      (* Exxpand the right hand side to get rid of EX j : {1..d} to get a huge disjunction*)
   937     (* exp_cp_thm: EX x.cfm = Q' , where Q' is a simplified version of Q*)
   938     in refl RS (simplify ss (cp_thm RSN (2,trans)))
   939     end
   940    (* lscth = EX x. cfm; rscth = Q' *)
   941    val (lscth,rscth) = qe_get_terms (exp_cp_thm)
   942    (* u_c_thm: EX x. P(l*x) = Q'*)
   943    val  u_c_thm = [([uth,prove_elementar sg "ss" (HOLogic.mk_eq (rsuth,lscth))] MRS trans),exp_cp_thm] MRS trans
   944    (* result: EX x. efm = Q'*)
   945  in  ([e_ac_thm,[(prove_elementar sg "ss" (HOLogic.mk_eq (rseacth,lsuth))),u_c_thm] MRS trans] MRS trans)
   946    end
   947 |cooper_prv_w _ _ _ =  error "Parameters format";
   948 
   949 
   950 
   951 fun decomp_cnnf sg lfnp P = case P of 
   952      Const ("op &",_) $ p $q => ([p,q] , fn [th1,th2] => [th1,th2] MRS qe_conjI )
   953    |Const ("op |",_) $ p $q => ([p,q] , fn [th1,th2] => [th1,th2] MRS  qe_disjI)
   954    |Const ("Not",_) $ (Const("Not",_) $ p) => ([p], fn [th] => th RS nnf_nn)
   955    |Const("Not",_) $ (Const(opn,T) $ p $ q) => 
   956      if opn = "op |" 
   957       then case (p,q) of 
   958          (A as (Const ("op &",_) $ r $ s),B as (Const ("op &",_) $ r1 $ t)) =>
   959           if r1 = negate r 
   960           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)
   961 
   962           else ([HOLogic.Not $ p,HOLogic.Not $ q ], fn [th1,th2] => [th1,th2] MRS nnf_ndj)
   963         |(_,_) => ([HOLogic.Not $ p,HOLogic.Not $ q ], fn [th1,th2] => [th1,th2] MRS nnf_ndj)
   964       else (
   965          case (opn,T) of 
   966            ("op &",_) => ([HOLogic.Not $ p,HOLogic.Not $ q ], fn [th1,th2] =>[th1,th2] MRS nnf_ncj )
   967            |("op -->",_) => ([p,HOLogic.Not $ q ], fn [th1,th2] =>[th1,th2] MRS nnf_nim )
   968            |("op =",Type ("fun",[Type ("bool", []),_])) => 
   969            ([HOLogic.conj $ p $ (HOLogic.Not $ q),HOLogic.conj $ (HOLogic.Not $ p) $ q], fn [th1,th2] => [th1,th2] MRS nnf_neq)
   970             |(_,_) => ([], fn [] => lfnp P)
   971 )
   972 
   973    |(Const ("op -->",_) $ p $ q) => ([HOLogic.Not$p,q], fn [th1,th2] => [th1,th2] MRS nnf_im)
   974 
   975    |(Const ("op =", Type ("fun",[Type ("bool", []),_])) $ p $ q) =>
   976      ([HOLogic.conj $ p $ q,HOLogic.conj $ (HOLogic.Not $ p) $ (HOLogic.Not $ q) ], fn [th1,th2] =>[th1,th2] MRS nnf_eq )
   977    |_ => ([], fn [] => lfnp P);
   978 
   979 
   980 
   981 
   982 fun proof_of_cnnf sg p lfnp = 
   983  let val th1 = thm_of sg (decomp_cnnf sg lfnp) p
   984      val rs = snd(qe_get_terms th1)
   985      val th2 = prove_elementar sg "ss" (HOLogic.mk_eq(rs,simpl rs))
   986   in [th1,th2] MRS trans
   987   end;
   988 
   989 end;
   990