src/HOL/Tools/Qelim/cooper.ML
author wenzelm
Sat May 17 13:54:30 2008 +0200 (2008-05-17)
changeset 26928 ca87aff1ad2d
parent 25768 1c1ca4b20ec6
child 27018 b3e63f39fc0f
permissions -rw-r--r--
structure Display: less pervasive operations;
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(*  Title:      HOL/Tools/Qelim/cooper.ML
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    ID:         $Id$
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    Author:     Amine Chaieb, TU Muenchen
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*)
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signature COOPER =
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 sig
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  val cooper_conv : Proof.context -> conv
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  exception COOPER of string * exn
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end;
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structure Cooper: COOPER =
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struct
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open Conv;
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open Normalizer;
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exception COOPER of string * exn;
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val simp_thms_conv = Simplifier.rewrite (HOL_basic_ss addsimps simp_thms);
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val FWD = Drule.implies_elim_list;
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val true_tm = @{cterm "True"};
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val false_tm = @{cterm "False"};
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val zdvd1_eq = @{thm "zdvd1_eq"};
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val presburger_ss = @{simpset} addsimps [zdvd1_eq];
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val lin_ss = presburger_ss addsimps (@{thm "dvd_eq_mod_eq_0"}::zdvd1_eq::@{thms zadd_ac});
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val iT = HOLogic.intT
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val bT = HOLogic.boolT;
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val dest_numeral = HOLogic.dest_number #> snd;
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val [miconj, midisj, mieq, mineq, milt, mile, migt, mige, midvd, mindvd, miP] = 
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    map(instantiate' [SOME @{ctyp "int"}] []) @{thms "minf"};
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val [infDconj, infDdisj, infDdvd,infDndvd,infDP] = 
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    map(instantiate' [SOME @{ctyp "int"}] []) @{thms "inf_period"};
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val [piconj, pidisj, pieq,pineq,pilt,pile,pigt,pige,pidvd,pindvd,piP] = 
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    map (instantiate' [SOME @{ctyp "int"}] []) @{thms "pinf"};
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val [miP, piP] = map (instantiate' [SOME @{ctyp "bool"}] []) [miP, piP];
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val infDP = instantiate' (map SOME [@{ctyp "int"}, @{ctyp "bool"}]) [] infDP;
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val [[asetconj, asetdisj, aseteq, asetneq, asetlt, asetle, 
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      asetgt, asetge, asetdvd, asetndvd,asetP],
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     [bsetconj, bsetdisj, bseteq, bsetneq, bsetlt, bsetle, 
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      bsetgt, bsetge, bsetdvd, bsetndvd,bsetP]]  = [@{thms "aset"}, @{thms "bset"}];
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val [miex, cpmi, piex, cppi] = [@{thm "minusinfinity"}, @{thm "cpmi"}, 
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                                @{thm "plusinfinity"}, @{thm "cppi"}];
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val unity_coeff_ex = instantiate' [SOME @{ctyp "int"}] [] @{thm "unity_coeff_ex"};
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val [zdvd_mono,simp_from_to,all_not_ex] = 
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     [@{thm "zdvd_mono"}, @{thm "simp_from_to"}, @{thm "all_not_ex"}];
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val [dvd_uminus, dvd_uminus'] = @{thms "uminus_dvd_conv"};
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val eval_ss = presburger_ss addsimps [simp_from_to] delsimps [insert_iff,bex_triv];
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val eval_conv = Simplifier.rewrite eval_ss;
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(* recognising cterm without moving to terms *)
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datatype fm = And of cterm*cterm| Or of cterm*cterm| Eq of cterm | NEq of cterm 
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            | Lt of cterm | Le of cterm | Gt of cterm | Ge of cterm
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            | Dvd of cterm*cterm | NDvd of cterm*cterm | Nox
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fun whatis x ct = 
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( case (term_of ct) of 
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  Const("op &",_)$_$_ => And (Thm.dest_binop ct)
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| Const ("op |",_)$_$_ => Or (Thm.dest_binop ct)
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| Const ("op =",ty)$y$_ => if term_of x aconv y then Eq (Thm.dest_arg ct) else Nox
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| Const (@{const_name Not},_) $ (Const ("op =",_)$y$_) => 
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  if term_of x aconv y then NEq (funpow 2 Thm.dest_arg ct) else Nox
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| Const (@{const_name HOL.less}, _) $ y$ z =>
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   if term_of x aconv y then Lt (Thm.dest_arg ct) 
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   else if term_of x aconv z then Gt (Thm.dest_arg1 ct) else Nox
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| Const (@{const_name HOL.less_eq}, _) $ y $ z => 
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   if term_of x aconv y then Le (Thm.dest_arg ct) 
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   else if term_of x aconv z then Ge (Thm.dest_arg1 ct) else Nox
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| Const (@{const_name Divides.dvd},_)$_$(Const(@{const_name "HOL.plus"},_)$y$_) =>
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   if term_of x aconv y then Dvd (Thm.dest_binop ct ||> Thm.dest_arg) else Nox 
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| Const (@{const_name Not},_) $ (Const (@{const_name Divides.dvd},_)$_$(Const(@{const_name "HOL.plus"},_)$y$_)) =>
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   if term_of x aconv y then 
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   NDvd (Thm.dest_binop (Thm.dest_arg ct) ||> Thm.dest_arg) else Nox 
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| _ => Nox)
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  handle CTERM _ => Nox; 
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fun get_pmi_term t = 
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  let val (x,eq) = 
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     (Thm.dest_abs NONE o Thm.dest_arg o snd o Thm.dest_abs NONE o Thm.dest_arg)
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        (Thm.dest_arg t)
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in (Thm.cabs x o Thm.dest_arg o Thm.dest_arg) eq end;
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val get_pmi = get_pmi_term o cprop_of;
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val p_v' = @{cpat "?P' :: int => bool"}; 
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val q_v' = @{cpat "?Q' :: int => bool"};
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val p_v = @{cpat "?P:: int => bool"};
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val q_v = @{cpat "?Q:: int => bool"};
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fun myfwd (th1, th2, th3) p q 
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      [(th_1,th_2,th_3), (th_1',th_2',th_3')] = 
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  let  
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   val (mp', mq') = (get_pmi th_1, get_pmi th_1')
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   val mi_th = FWD (instantiate ([],[(p_v,p),(q_v,q), (p_v',mp'),(q_v',mq')]) th1) 
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                   [th_1, th_1']
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   val infD_th = FWD (instantiate ([],[(p_v,mp'), (q_v, mq')]) th3) [th_3,th_3']
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   val set_th = FWD (instantiate ([],[(p_v,p), (q_v,q)]) th2) [th_2, th_2']
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  in (mi_th, set_th, infD_th)
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  end;
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val inst' = fn cts => instantiate' [] (map SOME cts);
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val infDTrue = instantiate' [] [SOME true_tm] infDP;
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val infDFalse = instantiate' [] [SOME false_tm] infDP;
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val cadd =  @{cterm "op + :: int => _"}
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val cmulC =  @{cterm "op * :: int => _"}
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val cminus =  @{cterm "op - :: int => _"}
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val cone =  @{cterm "1 :: int"}
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val cneg = @{cterm "uminus :: int => _"}
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val [addC, mulC, subC, negC] = map term_of [cadd, cmulC, cminus, cneg]
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val [zero, one] = [@{term "0 :: int"}, @{term "1 :: int"}];
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val is_numeral = can dest_numeral; 
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fun numeral1 f n = HOLogic.mk_number iT (f (dest_numeral n)); 
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fun numeral2 f m n = HOLogic.mk_number iT (f (dest_numeral m) (dest_numeral n));
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val [minus1,plus1] = 
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    map (fn c => fn t => Thm.capply (Thm.capply c t) cone) [cminus,cadd];
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fun decomp_pinf x dvd inS [aseteq, asetneq, asetlt, asetle, 
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                           asetgt, asetge,asetdvd,asetndvd,asetP,
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                           infDdvd, infDndvd, asetconj,
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                           asetdisj, infDconj, infDdisj] cp =
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 case (whatis x cp) of
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  And (p,q) => ([p,q], myfwd (piconj, asetconj, infDconj) (Thm.cabs x p) (Thm.cabs x q))
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| Or (p,q) => ([p,q], myfwd (pidisj, asetdisj, infDdisj) (Thm.cabs x p) (Thm.cabs x q))
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| Eq t => ([], K (inst' [t] pieq, FWD (inst' [t] aseteq) [inS (plus1 t)], infDFalse))
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| NEq t => ([], K (inst' [t] pineq, FWD (inst' [t] asetneq) [inS t], infDTrue))
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| Lt t => ([], K (inst' [t] pilt, FWD (inst' [t] asetlt) [inS t], infDFalse))
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| Le t => ([], K (inst' [t] pile, FWD (inst' [t] asetle) [inS (plus1 t)], infDFalse))
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| Gt t => ([], K (inst' [t] pigt, (inst' [t] asetgt), infDTrue))
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| Ge t => ([], K (inst' [t] pige, (inst' [t] asetge), infDTrue))
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| Dvd (d,s) => 
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   ([],let val dd = dvd d
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	     in K (inst' [d,s] pidvd, FWD (inst' [d,s] asetdvd) [dd],FWD (inst' [d,s] infDdvd) [dd]) end)
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| NDvd(d,s) => ([],let val dd = dvd d
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	      in K (inst' [d,s] pindvd, FWD (inst' [d,s] asetndvd) [dd], FWD (inst' [d,s] infDndvd) [dd]) end)
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| _ => ([], K (inst' [cp] piP, inst' [cp] asetP, inst' [cp] infDP));
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fun decomp_minf x dvd inS [bseteq,bsetneq,bsetlt, bsetle, bsetgt,
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                           bsetge,bsetdvd,bsetndvd,bsetP,
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                           infDdvd, infDndvd, bsetconj,
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                           bsetdisj, infDconj, infDdisj] cp =
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 case (whatis x cp) of
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  And (p,q) => ([p,q], myfwd (miconj, bsetconj, infDconj) (Thm.cabs x p) (Thm.cabs x q))
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| Or (p,q) => ([p,q], myfwd (midisj, bsetdisj, infDdisj) (Thm.cabs x p) (Thm.cabs x q))
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| Eq t => ([], K (inst' [t] mieq, FWD (inst' [t] bseteq) [inS (minus1 t)], infDFalse))
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| NEq t => ([], K (inst' [t] mineq, FWD (inst' [t] bsetneq) [inS t], infDTrue))
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| Lt t => ([], K (inst' [t] milt, (inst' [t] bsetlt), infDTrue))
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| Le t => ([], K (inst' [t] mile, (inst' [t] bsetle), infDTrue))
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| Gt t => ([], K (inst' [t] migt, FWD (inst' [t] bsetgt) [inS t], infDFalse))
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| Ge t => ([], K (inst' [t] mige,FWD (inst' [t] bsetge) [inS (minus1 t)], infDFalse))
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| Dvd (d,s) => ([],let val dd = dvd d
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	      in K (inst' [d,s] midvd, FWD (inst' [d,s] bsetdvd) [dd] , FWD (inst' [d,s] infDdvd) [dd]) end)
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| NDvd (d,s) => ([],let val dd = dvd d
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	      in K (inst' [d,s] mindvd, FWD (inst' [d,s] bsetndvd) [dd], FWD (inst' [d,s] infDndvd) [dd]) end)
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| _ => ([], K (inst' [cp] miP, inst' [cp] bsetP, inst' [cp] infDP))
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    (* Canonical linear form for terms, formulae etc.. *)
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fun provelin ctxt t = Goal.prove ctxt [] [] t 
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  (fn _ => EVERY [simp_tac lin_ss 1, TRY (simple_arith_tac ctxt 1)]);
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fun linear_cmul 0 tm = zero 
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  | linear_cmul n tm = case tm of  
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      Const (@{const_name HOL.plus}, _) $ a $ b => addC $ linear_cmul n a $ linear_cmul n b
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    | Const (@{const_name HOL.times}, _) $ c $ x => mulC $ numeral1 (fn m => n * m) c $ x
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    | Const (@{const_name HOL.minus}, _) $ a $ b => subC $ linear_cmul n a $ linear_cmul n b
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    | (m as Const (@{const_name HOL.uminus}, _)) $ a => m $ linear_cmul n a
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    | _ => numeral1 (fn m => n * m) tm;
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fun earlier [] x y = false 
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	| earlier (h::t) x y = 
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    if h aconv y then false else if h aconv x then true else earlier t x y; 
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fun linear_add vars tm1 tm2 = case (tm1, tm2) of 
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    (Const (@{const_name HOL.plus}, _) $ (Const (@{const_name HOL.times}, _) $ c1 $ x1) $ r1,
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    Const (@{const_name HOL.plus}, _) $ (Const (@{const_name HOL.times}, _) $ c2 $ x2) $ r2) =>
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   if x1 = x2 then 
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     let val c = numeral2 (curry op +) c1 c2
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      in if c = zero then linear_add vars r1 r2
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         else addC$(mulC$c$x1)$(linear_add vars r1 r2)
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     end 
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     else if earlier vars x1 x2 then addC $ (mulC $ c1 $ x1) $ linear_add vars r1 tm2
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   else addC $ (mulC $ c2 $ x2) $ linear_add vars tm1 r2
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 | (Const (@{const_name HOL.plus}, _) $ (Const (@{const_name HOL.times}, _) $ c1 $ x1) $ r1, _) =>
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      addC $ (mulC $ c1 $ x1) $ linear_add vars r1 tm2
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 | (_, Const (@{const_name HOL.plus}, _) $ (Const (@{const_name HOL.times}, _) $ c2 $ x2) $ r2) => 
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      addC $ (mulC $ c2 $ x2) $ linear_add vars tm1 r2
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 | (_, _) => numeral2 (curry op +) tm1 tm2;
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fun linear_neg tm = linear_cmul ~1 tm; 
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fun linear_sub vars tm1 tm2 = linear_add vars tm1 (linear_neg tm2); 
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fun lint vars tm =  if is_numeral tm then tm  else case tm of 
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  Const (@{const_name HOL.uminus}, _) $ t => linear_neg (lint vars t)
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| Const (@{const_name HOL.plus}, _) $ s $ t => linear_add vars (lint vars s) (lint vars t)
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| Const (@{const_name HOL.minus}, _) $ s $ t => linear_sub vars (lint vars s) (lint vars t)
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| Const (@{const_name HOL.times}, _) $ s $ t =>
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  let val s' = lint vars s  
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      val t' = lint vars t  
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  in if is_numeral s' then (linear_cmul (dest_numeral s') t') 
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     else if is_numeral t' then (linear_cmul (dest_numeral t') s') 
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     else raise COOPER ("Cooper Failed", TERM ("lint: not linear",[tm]))
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  end 
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 | _ => addC $ (mulC $ one $ tm) $ zero;
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fun lin (vs as x::_) (Const (@{const_name Not}, _) $ (Const (@{const_name HOL.less}, T) $ s $ t)) = 
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    lin vs (Const (@{const_name HOL.less_eq}, T) $ t $ s)
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  | lin (vs as x::_) (Const (@{const_name Not},_) $ (Const(@{const_name HOL.less_eq}, T) $ s $ t)) = 
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    lin vs (Const (@{const_name HOL.less}, T) $ t $ s)
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  | lin vs (Const (@{const_name Not},T)$t) = Const (@{const_name Not},T)$ (lin vs t)
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  | lin (vs as x::_) (Const(@{const_name Divides.dvd},_)$d$t) = 
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    HOLogic.mk_binrel @{const_name Divides.dvd} (numeral1 abs d, lint vs t)
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  | lin (vs as x::_) ((b as Const("op =",_))$s$t) = 
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     (case lint vs (subC$t$s) of 
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      (t as a$(m$c$y)$r) => 
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        if x <> y then b$zero$t
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        else if dest_numeral c < 0 then b$(m$(numeral1 ~ c)$y)$r
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        else b$(m$c$y)$(linear_neg r)
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      | t => b$zero$t)
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  | lin (vs as x::_) (b$s$t) = 
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     (case lint vs (subC$t$s) of 
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      (t as a$(m$c$y)$r) => 
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        if x <> y then b$zero$t
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        else if dest_numeral c < 0 then b$(m$(numeral1 ~ c)$y)$r
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        else b$(linear_neg r)$(m$c$y)
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      | t => b$zero$t)
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  | lin vs fm = fm;
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fun lint_conv ctxt vs ct = 
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let val t = term_of ct
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in (provelin ctxt ((HOLogic.eq_const iT)$t$(lint vs t) |> HOLogic.mk_Trueprop))
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   246
             RS eq_reflection
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   247
end;
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   248
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   249
fun is_intrel (b$_$_) = domain_type (fastype_of b) = HOLogic.intT
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   250
  | is_intrel (@{term "Not"}$(b$_$_)) = domain_type (fastype_of b) = HOLogic.intT
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   251
  | is_intrel _ = false;
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   252
 
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   253
fun linearize_conv ctxt vs ct = case term_of ct of
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   254
  Const(@{const_name Divides.dvd},_)$d$t => 
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   255
  let 
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   256
    val th = binop_conv (lint_conv ctxt vs) ct
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   257
    val (d',t') = Thm.dest_binop (Thm.rhs_of th)
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   258
    val (dt',tt') = (term_of d', term_of t')
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   259
  in if is_numeral dt' andalso is_numeral tt' 
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   260
     then Conv.fconv_rule (arg_conv (Simplifier.rewrite presburger_ss)) th
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   261
     else 
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   262
     let 
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   263
      val dth = 
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   264
      ((if dest_numeral (term_of d') < 0 then 
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   265
          Conv.fconv_rule (arg_conv (arg1_conv (lint_conv ctxt vs)))
wenzelm@23466
   266
                           (Thm.transitive th (inst' [d',t'] dvd_uminus))
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   267
        else th) handle TERM _ => th)
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   268
      val d'' = Thm.rhs_of dth |> Thm.dest_arg1
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   269
     in
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   270
      case tt' of 
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   271
        Const(@{const_name HOL.plus},_)$(Const(@{const_name HOL.times},_)$c$_)$_ => 
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   272
        let val x = dest_numeral c
wenzelm@23466
   273
        in if x < 0 then Conv.fconv_rule (arg_conv (arg_conv (lint_conv ctxt vs)))
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   274
                                       (Thm.transitive dth (inst' [d'',t'] dvd_uminus'))
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   275
        else dth end
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   276
      | _ => dth
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   277
     end
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   278
  end
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   279
| Const (@{const_name Not},_)$(Const(@{const_name Divides.dvd},_)$_$_) => arg_conv (linearize_conv ctxt vs) ct
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   280
| t => if is_intrel t 
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   281
      then (provelin ctxt ((HOLogic.eq_const bT)$t$(lin vs t) |> HOLogic.mk_Trueprop))
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   282
       RS eq_reflection
wenzelm@23466
   283
      else reflexive ct;
wenzelm@23466
   284
wenzelm@23466
   285
val dvdc = @{cterm "op dvd :: int => _"};
wenzelm@23466
   286
wenzelm@23466
   287
fun unify ctxt q = 
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   288
 let
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   289
  val (e,(cx,p)) = q |> Thm.dest_comb ||> Thm.dest_abs NONE
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   290
  val x = term_of cx 
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   291
  val ins = insert (op = : int * int -> bool)
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   292
  fun h (acc,dacc) t = 
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   293
   case (term_of t) of
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   294
    Const(s,_)$(Const(@{const_name HOL.times},_)$c$y)$ _ => 
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   295
    if x aconv y andalso member (op =)
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   296
      ["op =", @{const_name HOL.less}, @{const_name HOL.less_eq}] s
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   297
    then (ins (dest_numeral c) acc,dacc) else (acc,dacc)
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   298
  | Const(s,_)$_$(Const(@{const_name HOL.times},_)$c$y) => 
haftmann@23881
   299
    if x aconv y andalso member (op =)
haftmann@23881
   300
       [@{const_name HOL.less}, @{const_name HOL.less_eq}] s 
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   301
    then (ins (dest_numeral c) acc, dacc) else (acc,dacc)
haftmann@25768
   302
  | Const(@{const_name Divides.dvd},_)$_$(Const(@{const_name HOL.plus},_)$(Const(@{const_name HOL.times},_)$c$y)$_) => 
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   303
    if x aconv y then (acc,ins (dest_numeral c) dacc) else (acc,dacc)
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   304
  | Const("op &",_)$_$_ => h (h (acc,dacc) (Thm.dest_arg1 t)) (Thm.dest_arg t)
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   305
  | Const("op |",_)$_$_ => h (h (acc,dacc) (Thm.dest_arg1 t)) (Thm.dest_arg t)
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   306
  | Const (@{const_name Not},_)$_ => h (acc,dacc) (Thm.dest_arg t)
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   307
  | _ => (acc, dacc)
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   308
  val (cs,ds) = h ([],[]) p
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   309
  val l = Integer.lcms (cs union ds)
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   310
  fun cv k ct = 
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   311
    let val (tm as b$s$t) = term_of ct 
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   312
    in ((HOLogic.eq_const bT)$tm$(b$(linear_cmul k s)$(linear_cmul k t))
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   313
         |> HOLogic.mk_Trueprop |> provelin ctxt) RS eq_reflection end
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   314
  fun nzprop x = 
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   315
   let 
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   316
    val th = 
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   317
     Simplifier.rewrite lin_ss 
wenzelm@23466
   318
      (Thm.capply @{cterm Trueprop} (Thm.capply @{cterm "Not"} 
haftmann@23689
   319
           (Thm.capply (Thm.capply @{cterm "op = :: int => _"} (Numeral.mk_cnumber @{ctyp "int"} x)) 
haftmann@23689
   320
           @{cterm "0::int"})))
wenzelm@23466
   321
   in equal_elim (Thm.symmetric th) TrueI end;
wenzelm@24630
   322
  val notz = let val tab = fold Inttab.update 
wenzelm@24630
   323
                               (ds ~~ (map (fn x => nzprop (l div x)) ds)) Inttab.empty 
wenzelm@23466
   324
            in 
wenzelm@24630
   325
             (fn ct => (valOf (Inttab.lookup tab (ct |> term_of |> dest_numeral)) 
wenzelm@23466
   326
                handle Option => (writeln "noz: Theorems-Table contains no entry for"; 
wenzelm@26928
   327
                                    Display.print_cterm ct ; raise Option)))
wenzelm@23466
   328
           end
wenzelm@23466
   329
  fun unit_conv t = 
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   330
   case (term_of t) of
wenzelm@23466
   331
   Const("op &",_)$_$_ => binop_conv unit_conv t
wenzelm@23466
   332
  | Const("op |",_)$_$_ => binop_conv unit_conv t
haftmann@25768
   333
  | Const (@{const_name Not},_)$_ => arg_conv unit_conv t
haftmann@25768
   334
  | Const(s,_)$(Const(@{const_name HOL.times},_)$c$y)$ _ => 
haftmann@23881
   335
    if x=y andalso member (op =)
haftmann@23881
   336
      ["op =", @{const_name HOL.less}, @{const_name HOL.less_eq}] s
wenzelm@24630
   337
    then cv (l div dest_numeral c) t else Thm.reflexive t
haftmann@25768
   338
  | Const(s,_)$_$(Const(@{const_name HOL.times},_)$c$y) => 
haftmann@23881
   339
    if x=y andalso member (op =)
haftmann@23881
   340
      [@{const_name HOL.less}, @{const_name HOL.less_eq}] s
wenzelm@24630
   341
    then cv (l div dest_numeral c) t else Thm.reflexive t
haftmann@25768
   342
  | Const(@{const_name Divides.dvd},_)$d$(r as (Const(@{const_name HOL.plus},_)$(Const(@{const_name HOL.times},_)$c$y)$_)) => 
wenzelm@23466
   343
    if x=y then 
wenzelm@23466
   344
      let 
wenzelm@24630
   345
       val k = l div dest_numeral c
wenzelm@23466
   346
       val kt = HOLogic.mk_number iT k
wenzelm@23466
   347
       val th1 = inst' [Thm.dest_arg1 t, Thm.dest_arg t] 
wenzelm@23466
   348
             ((Thm.dest_arg t |> funpow 2 Thm.dest_arg1 |> notz) RS zdvd_mono)
wenzelm@23466
   349
       val (d',t') = (mulC$kt$d, mulC$kt$r)
wenzelm@23466
   350
       val thc = (provelin ctxt ((HOLogic.eq_const iT)$d'$(lint [] d') |> HOLogic.mk_Trueprop))
wenzelm@23466
   351
                   RS eq_reflection
wenzelm@23466
   352
       val tht = (provelin ctxt ((HOLogic.eq_const iT)$t'$(linear_cmul k r) |> HOLogic.mk_Trueprop))
wenzelm@23466
   353
                 RS eq_reflection
wenzelm@23466
   354
      in Thm.transitive th1 (Thm.combination (Drule.arg_cong_rule dvdc thc) tht) end                 
wenzelm@23466
   355
    else Thm.reflexive t
wenzelm@23466
   356
  | _ => Thm.reflexive t
wenzelm@23466
   357
  val uth = unit_conv p
haftmann@23689
   358
  val clt =  Numeral.mk_cnumber @{ctyp "int"} l
wenzelm@23466
   359
  val ltx = Thm.capply (Thm.capply cmulC clt) cx
wenzelm@23466
   360
  val th = Drule.arg_cong_rule e (Thm.abstract_rule (fst (dest_Free x )) cx uth)
wenzelm@23466
   361
  val th' = inst' [Thm.cabs ltx (Thm.rhs_of uth), clt] unity_coeff_ex
wenzelm@23466
   362
  val thf = transitive th 
wenzelm@23466
   363
      (transitive (symmetric (beta_conversion true (cprop_of th' |> Thm.dest_arg1))) th')
wenzelm@23466
   364
  val (lth,rth) = Thm.dest_comb (cprop_of thf) |>> Thm.dest_arg |>> Thm.beta_conversion true
wenzelm@23466
   365
                  ||> beta_conversion true |>> Thm.symmetric
wenzelm@23466
   366
 in transitive (transitive lth thf) rth end;
wenzelm@23466
   367
wenzelm@23466
   368
wenzelm@23466
   369
val emptyIS = @{cterm "{}::int set"};
wenzelm@23466
   370
val insert_tm = @{cterm "insert :: int => _"};
wenzelm@23466
   371
val mem_tm = Const("op :",[iT , HOLogic.mk_setT iT] ---> bT);
wenzelm@23466
   372
fun mkISet cts = fold_rev (Thm.capply insert_tm #> Thm.capply) cts emptyIS;
wenzelm@23466
   373
val cTrp = @{cterm "Trueprop"};
wenzelm@23466
   374
val eqelem_imp_imp = (thm"eqelem_imp_iff") RS iffD1;
wenzelm@23466
   375
val [A_tm,B_tm] = map (fn th => cprop_of th |> funpow 2 Thm.dest_arg |> Thm.dest_abs NONE |> snd |> Thm.dest_arg1 |> Thm.dest_arg 
wenzelm@23466
   376
                                      |> Thm.dest_abs NONE |> snd |> Thm.dest_fun |> Thm.dest_arg)
wenzelm@23466
   377
                      [asetP,bsetP];
wenzelm@23466
   378
wenzelm@23466
   379
val D_tm = @{cpat "?D::int"};
wenzelm@23466
   380
wenzelm@23466
   381
fun cooperex_conv ctxt vs q = 
wenzelm@23466
   382
let 
wenzelm@23466
   383
wenzelm@23466
   384
 val uth = unify ctxt q
wenzelm@23466
   385
 val (x,p) = Thm.dest_abs NONE (Thm.dest_arg (Thm.rhs_of uth))
wenzelm@23466
   386
 val ins = insert (op aconvc)
wenzelm@23466
   387
 fun h t (bacc,aacc,dacc) = 
wenzelm@23466
   388
  case (whatis x t) of
wenzelm@23466
   389
    And (p,q) => h q (h p (bacc,aacc,dacc))
wenzelm@23466
   390
  | Or (p,q) => h q  (h p (bacc,aacc,dacc))
wenzelm@23466
   391
  | Eq t => (ins (minus1 t) bacc, 
wenzelm@23466
   392
             ins (plus1 t) aacc,dacc)
wenzelm@23466
   393
  | NEq t => (ins t bacc, 
wenzelm@23466
   394
              ins t aacc, dacc)
wenzelm@23466
   395
  | Lt t => (bacc, ins t aacc, dacc)
wenzelm@23466
   396
  | Le t => (bacc, ins (plus1 t) aacc,dacc)
wenzelm@23466
   397
  | Gt t => (ins t bacc, aacc,dacc)
wenzelm@23466
   398
  | Ge t => (ins (minus1 t) bacc, aacc,dacc)
wenzelm@24630
   399
  | Dvd (d,s) => (bacc,aacc,insert (op =) (term_of d |> dest_numeral) dacc)
wenzelm@24630
   400
  | NDvd (d,s) => (bacc,aacc,insert (op =) (term_of d|> dest_numeral) dacc)
wenzelm@23466
   401
  | _ => (bacc, aacc, dacc)
wenzelm@23466
   402
 val (b0,a0,ds) = h p ([],[],[])
wenzelm@24630
   403
 val d = Integer.lcms ds
wenzelm@23582
   404
 val cd = Numeral.mk_cnumber @{ctyp "int"} d
wenzelm@23466
   405
 val dt = term_of cd
wenzelm@23466
   406
 fun divprop x = 
wenzelm@23466
   407
   let 
wenzelm@23466
   408
    val th = 
wenzelm@23466
   409
     Simplifier.rewrite lin_ss 
wenzelm@23466
   410
      (Thm.capply @{cterm Trueprop} 
wenzelm@23582
   411
           (Thm.capply (Thm.capply dvdc (Numeral.mk_cnumber @{ctyp "int"} x)) cd))
wenzelm@23466
   412
   in equal_elim (Thm.symmetric th) TrueI end;
wenzelm@24630
   413
 val dvd = let val tab = fold Inttab.update
wenzelm@24630
   414
                               (ds ~~ (map divprop ds)) Inttab.empty in 
wenzelm@24630
   415
           (fn ct => (valOf (Inttab.lookup tab (term_of ct |> dest_numeral)) 
wenzelm@23466
   416
                    handle Option => (writeln "dvd: Theorems-Table contains no entry for"; 
wenzelm@26928
   417
                                      Display.print_cterm ct ; raise Option)))
wenzelm@23466
   418
           end
wenzelm@23466
   419
 val dp = 
wenzelm@23466
   420
   let val th = Simplifier.rewrite lin_ss 
wenzelm@23466
   421
      (Thm.capply @{cterm Trueprop} 
wenzelm@23466
   422
           (Thm.capply (Thm.capply @{cterm "op < :: int => _"} @{cterm "0::int"}) cd))
wenzelm@23466
   423
   in equal_elim (Thm.symmetric th) TrueI end;
wenzelm@23466
   424
    (* A and B set *)
wenzelm@23466
   425
   local 
wenzelm@23466
   426
     val insI1 = instantiate' [SOME @{ctyp "int"}] [] @{thm "insertI1"}
wenzelm@23466
   427
     val insI2 = instantiate' [SOME @{ctyp "int"}] [] @{thm "insertI2"}
wenzelm@23466
   428
   in
wenzelm@23466
   429
    fun provein x S = 
wenzelm@23466
   430
     case term_of S of
wenzelm@23466
   431
        Const("{}",_) => error "Unexpected error in Cooper please email Amine Chaieb"
wenzelm@23466
   432
      | Const("insert",_)$y$_ => 
wenzelm@23466
   433
         let val (cy,S') = Thm.dest_binop S
wenzelm@23466
   434
         in if term_of x aconv y then instantiate' [] [SOME x, SOME S'] insI1
wenzelm@23466
   435
         else implies_elim (instantiate' [] [SOME x, SOME S', SOME cy] insI2) 
wenzelm@23466
   436
                           (provein x S')
wenzelm@23466
   437
         end
wenzelm@23466
   438
   end
wenzelm@23466
   439
 
wenzelm@23466
   440
 val al = map (lint vs o term_of) a0
wenzelm@23466
   441
 val bl = map (lint vs o term_of) b0
wenzelm@23466
   442
 val (sl,s0,f,abths,cpth) = 
wenzelm@23466
   443
   if length (distinct (op aconv) bl) <= length (distinct (op aconv) al) 
wenzelm@23466
   444
   then  
wenzelm@23466
   445
    (bl,b0,decomp_minf,
wenzelm@23466
   446
     fn B => (map (fn th => implies_elim (Thm.instantiate ([],[(B_tm,B), (D_tm,cd)]) th) dp) 
wenzelm@23466
   447
                     [bseteq,bsetneq,bsetlt, bsetle, bsetgt,bsetge])@
wenzelm@23466
   448
                   (map (Thm.instantiate ([],[(B_tm,B), (D_tm,cd)])) 
wenzelm@23466
   449
                        [bsetdvd,bsetndvd,bsetP,infDdvd, infDndvd,bsetconj,
wenzelm@23466
   450
                         bsetdisj,infDconj, infDdisj]),
wenzelm@23466
   451
                       cpmi) 
wenzelm@23466
   452
     else (al,a0,decomp_pinf,fn A => 
wenzelm@23466
   453
          (map (fn th => implies_elim (Thm.instantiate ([],[(A_tm,A), (D_tm,cd)]) th) dp)
wenzelm@23466
   454
                   [aseteq,asetneq,asetlt, asetle, asetgt,asetge])@
wenzelm@23466
   455
                   (map (Thm.instantiate ([],[(A_tm,A), (D_tm,cd)])) 
wenzelm@23466
   456
                   [asetdvd,asetndvd, asetP, infDdvd, infDndvd,asetconj,
wenzelm@23466
   457
                         asetdisj,infDconj, infDdisj]),cppi)
wenzelm@23466
   458
 val cpth = 
wenzelm@23466
   459
  let
wenzelm@23466
   460
   val sths = map (fn (tl,t0) => 
wenzelm@23466
   461
                      if tl = term_of t0 
wenzelm@23466
   462
                      then instantiate' [SOME @{ctyp "int"}] [SOME t0] refl
wenzelm@23466
   463
                      else provelin ctxt ((HOLogic.eq_const iT)$tl$(term_of t0) 
wenzelm@23466
   464
                                 |> HOLogic.mk_Trueprop)) 
wenzelm@23466
   465
                   (sl ~~ s0)
wenzelm@23466
   466
   val csl = distinct (op aconvc) (map (cprop_of #> Thm.dest_arg #> Thm.dest_arg1) sths)
wenzelm@23466
   467
   val S = mkISet csl
wenzelm@23466
   468
   val inStab = fold (fn ct => fn tab => Termtab.update (term_of ct, provein ct S) tab) 
wenzelm@23466
   469
                    csl Termtab.empty
wenzelm@23466
   470
   val eqelem_th = instantiate' [SOME @{ctyp "int"}] [NONE,NONE, SOME S] eqelem_imp_imp
wenzelm@23466
   471
   val inS = 
wenzelm@23466
   472
     let 
wenzelm@23466
   473
      fun transmem th0 th1 = 
wenzelm@23466
   474
       Thm.equal_elim 
wenzelm@23466
   475
        (Drule.arg_cong_rule cTrp (Drule.fun_cong_rule (Drule.arg_cong_rule 
wenzelm@23466
   476
               ((Thm.dest_fun o Thm.dest_fun o Thm.dest_arg o cprop_of) th1) th0) S)) th1
wenzelm@23466
   477
      val tab = fold Termtab.update
wenzelm@23466
   478
        (map (fn eq => 
wenzelm@23466
   479
                let val (s,t) = cprop_of eq |> Thm.dest_arg |> Thm.dest_binop 
wenzelm@23466
   480
                    val th = if term_of s = term_of t 
wenzelm@23466
   481
                             then valOf(Termtab.lookup inStab (term_of s))
wenzelm@23466
   482
                             else FWD (instantiate' [] [SOME s, SOME t] eqelem_th) 
wenzelm@23466
   483
                                [eq, valOf(Termtab.lookup inStab (term_of s))]
wenzelm@23466
   484
                 in (term_of t, th) end)
wenzelm@23466
   485
                  sths) Termtab.empty
wenzelm@23466
   486
        in fn ct => 
wenzelm@23466
   487
          (valOf (Termtab.lookup tab (term_of ct))
wenzelm@26928
   488
           handle Option => (writeln "inS: No theorem for " ; Display.print_cterm ct ; raise Option))
wenzelm@23466
   489
        end
wenzelm@23466
   490
       val (inf, nb, pd) = divide_and_conquer (f x dvd inS (abths S)) p
wenzelm@23466
   491
   in [dp, inf, nb, pd] MRS cpth
wenzelm@23466
   492
   end
wenzelm@23466
   493
 val cpth' = Thm.transitive uth (cpth RS eq_reflection)
wenzelm@23466
   494
in Thm.transitive cpth' ((simp_thms_conv then_conv eval_conv) (Thm.rhs_of cpth'))
wenzelm@23466
   495
end;
wenzelm@23466
   496
wenzelm@23466
   497
fun literals_conv bops uops env cv = 
wenzelm@23466
   498
 let fun h t =
wenzelm@23466
   499
  case (term_of t) of 
wenzelm@23466
   500
   b$_$_ => if member (op aconv) bops b then binop_conv h t else cv env t
wenzelm@23466
   501
 | u$_ => if member (op aconv) uops u then arg_conv h t else cv env t
wenzelm@23466
   502
 | _ => cv env t
wenzelm@23466
   503
 in h end;
wenzelm@23466
   504
wenzelm@23466
   505
fun integer_nnf_conv ctxt env =
wenzelm@23466
   506
 nnf_conv then_conv literals_conv [HOLogic.conj, HOLogic.disj] [] env (linearize_conv ctxt);
wenzelm@23466
   507
wenzelm@23466
   508
local
wenzelm@23466
   509
 val pcv = Simplifier.rewrite 
wenzelm@23466
   510
     (HOL_basic_ss addsimps (simp_thms @ (List.take(ex_simps,4)) 
wenzelm@23466
   511
                      @ [not_all,all_not_ex, ex_disj_distrib]))
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   512
 val postcv = Simplifier.rewrite presburger_ss
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   513
 fun conv ctxt p = 
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   514
  let val _ = ()
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   515
  in
chaieb@23523
   516
   Qelim.gen_qelim_conv pcv postcv pcv (cons o term_of) 
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   517
      (term_frees (term_of p)) (linearize_conv ctxt) (integer_nnf_conv ctxt) 
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   518
      (cooperex_conv ctxt) p 
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   519
  end
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   520
  handle  CTERM s => raise COOPER ("Cooper Failed", CTERM s)
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   521
        | THM s => raise COOPER ("Cooper Failed", THM s) 
chaieb@23523
   522
        | TYPE s => raise COOPER ("Cooper Failed", TYPE s) 
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   523
in val cooper_conv = conv 
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   524
end;
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   525
end;
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   526
wenzelm@23466
   527
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   528
wenzelm@23466
   529
structure Coopereif =
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   530
struct
wenzelm@23466
   531
haftmann@23713
   532
open GeneratedCooper;
haftmann@23713
   533
haftmann@23713
   534
fun cooper s = raise Cooper.COOPER ("Cooper oracle failed", ERROR s);
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   535
fun i_of_term vs t = case t
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   536
 of Free (xn, xT) => (case AList.lookup (op aconv) vs t
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   537
     of NONE   => cooper "Variable not found in the list!"
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   538
      | SOME n => Bound n)
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   539
  | @{term "0::int"} => C 0
haftmann@23713
   540
  | @{term "1::int"} => C 1
wenzelm@24630
   541
  | Term.Bound i => Bound i
haftmann@23713
   542
  | Const(@{const_name HOL.uminus},_)$t' => Neg (i_of_term vs t')
haftmann@23713
   543
  | Const(@{const_name HOL.plus},_)$t1$t2 => Add (i_of_term vs t1,i_of_term vs t2)
haftmann@23713
   544
  | Const(@{const_name HOL.minus},_)$t1$t2 => Sub (i_of_term vs t1,i_of_term vs t2)
haftmann@23713
   545
  | Const(@{const_name HOL.times},_)$t1$t2 => 
haftmann@23713
   546
     (Mul (HOLogic.dest_number t1 |> snd, i_of_term vs t2)
haftmann@23713
   547
    handle TERM _ => 
haftmann@23713
   548
       (Mul (HOLogic.dest_number t2 |> snd, i_of_term vs t1)
haftmann@23713
   549
        handle TERM _ => cooper "Reification: Unsupported kind of multiplication"))
haftmann@23713
   550
  | _ => (C (HOLogic.dest_number t |> snd) 
haftmann@23713
   551
           handle TERM _ => cooper "Reification: unknown term");
haftmann@23689
   552
haftmann@23713
   553
fun qf_of_term ps vs t =  case t
haftmann@23713
   554
 of Const("True",_) => T
haftmann@23713
   555
  | Const("False",_) => F
haftmann@23881
   556
  | Const(@{const_name HOL.less},_)$t1$t2 => Lt (Sub (i_of_term vs t1,i_of_term vs t2))
haftmann@23881
   557
  | Const(@{const_name HOL.less_eq},_)$t1$t2 => Le (Sub(i_of_term vs t1,i_of_term vs t2))
haftmann@23713
   558
  | Const(@{const_name Divides.dvd},_)$t1$t2 => 
haftmann@23713
   559
      (Dvd(HOLogic.dest_number t1 |> snd, i_of_term vs t2) handle _ => cooper "Reification: unsupported dvd")
haftmann@23713
   560
  | @{term "op = :: int => _"}$t1$t2 => Eq (Sub (i_of_term vs t1,i_of_term vs t2))
haftmann@23713
   561
  | @{term "op = :: bool => _ "}$t1$t2 => Iffa(qf_of_term ps vs t1,qf_of_term ps vs t2)
haftmann@23713
   562
  | Const("op &",_)$t1$t2 => And(qf_of_term ps vs t1,qf_of_term ps vs t2)
haftmann@23713
   563
  | Const("op |",_)$t1$t2 => Or(qf_of_term ps vs t1,qf_of_term ps vs t2)
haftmann@23713
   564
  | Const("op -->",_)$t1$t2 => Impa(qf_of_term ps vs t1,qf_of_term ps vs t2)
haftmann@25768
   565
  | Const (@{const_name Not},_)$t' => Nota(qf_of_term ps vs t')
haftmann@23713
   566
  | Const("Ex",_)$Abs(xn,xT,p) => 
haftmann@23713
   567
     let val (xn',p') = variant_abs (xn,xT,p)
haftmann@23713
   568
         val vs' = (Free (xn',xT), 0) :: (map (fn(v,n) => (v,1+ n)) vs)
haftmann@23713
   569
     in E (qf_of_term ps vs' p')
haftmann@23713
   570
     end
haftmann@23713
   571
  | Const("All",_)$Abs(xn,xT,p) => 
haftmann@23713
   572
     let val (xn',p') = variant_abs (xn,xT,p)
haftmann@23713
   573
         val vs' = (Free (xn',xT), 0) :: (map (fn(v,n) => (v,1+ n)) vs)
haftmann@23713
   574
     in A (qf_of_term ps vs' p')
haftmann@23713
   575
     end
haftmann@23713
   576
  | _ =>(case AList.lookup (op aconv) ps t of 
haftmann@23713
   577
           NONE => cooper "Reification: unknown term!"
haftmann@23713
   578
         | SOME n => Closed n);
wenzelm@23466
   579
wenzelm@23466
   580
local
wenzelm@23466
   581
 val ops = [@{term "op &"}, @{term "op |"}, @{term "op -->"}, @{term "op = :: bool => _"},
wenzelm@23466
   582
             @{term "op = :: int => _"}, @{term "op < :: int => _"}, 
wenzelm@23466
   583
             @{term "op <= :: int => _"}, @{term "Not"}, @{term "All:: (int => _) => _"}, 
wenzelm@23466
   584
             @{term "Ex:: (int => _) => _"}, @{term "True"}, @{term "False"}]
wenzelm@23466
   585
fun ty t = Bool.not (fastype_of t = HOLogic.boolT)
wenzelm@23466
   586
in
wenzelm@23466
   587
fun term_bools acc t =
wenzelm@23466
   588
case t of 
wenzelm@23466
   589
    (l as f $ a) $ b => if ty t orelse f mem ops then term_bools (term_bools acc l)b 
wenzelm@23466
   590
            else insert (op aconv) t acc
wenzelm@23466
   591
  | f $ a => if ty t orelse f mem ops then term_bools (term_bools acc f) a  
wenzelm@23466
   592
            else insert (op aconv) t acc
wenzelm@23466
   593
  | Abs p => term_bools acc (snd (variant_abs p))
wenzelm@23466
   594
  | _ => if ty t orelse t mem ops then acc else insert (op aconv) t acc
wenzelm@23466
   595
end;
wenzelm@23466
   596
 
wenzelm@23466
   597
fun myassoc2 l v =
wenzelm@23466
   598
    case l of
wenzelm@23466
   599
	[] => NONE
haftmann@23689
   600
      | (x,v')::xs => if v = v' then SOME x
wenzelm@23466
   601
		      else myassoc2 xs v;
wenzelm@23466
   602
haftmann@23713
   603
fun term_of_i vs t = case t
haftmann@23713
   604
 of C i => HOLogic.mk_number HOLogic.intT i
haftmann@23713
   605
  | Bound n => the (myassoc2 vs n)
haftmann@23713
   606
  | Neg t' => @{term "uminus :: int => _"} $ term_of_i vs t'
haftmann@23713
   607
  | Add (t1, t2) => @{term "op + :: int => _"} $ term_of_i vs t1 $ term_of_i vs t2
haftmann@23713
   608
  | Sub (t1, t2) => @{term "op - :: int => _"} $ term_of_i vs t1 $ term_of_i vs t2
haftmann@23713
   609
  | Mul (i, t2) => @{term "op * :: int => _"} $
haftmann@23713
   610
      HOLogic.mk_number HOLogic.intT i $ term_of_i vs t2
haftmann@23713
   611
  | Cx (i, t') => term_of_i vs (Add (Mul (i, Bound 0), t'));
wenzelm@23466
   612
wenzelm@23466
   613
fun term_of_qf ps vs t = 
wenzelm@23466
   614
 case t of 
wenzelm@23466
   615
   T => HOLogic.true_const 
wenzelm@23466
   616
 | F => HOLogic.false_const
wenzelm@23466
   617
 | Lt t' => @{term "op < :: int => _ "}$ term_of_i vs t'$ @{term "0::int"}
wenzelm@23466
   618
 | Le t' => @{term "op <= :: int => _ "}$ term_of_i vs t' $ @{term "0::int"}
wenzelm@23466
   619
 | Gt t' => @{term "op < :: int => _ "}$ @{term "0::int"}$ term_of_i vs t'
wenzelm@23466
   620
 | Ge t' => @{term "op <= :: int => _ "}$ @{term "0::int"}$ term_of_i vs t'
wenzelm@23466
   621
 | Eq t' => @{term "op = :: int => _ "}$ term_of_i vs t'$ @{term "0::int"}
haftmann@23713
   622
 | NEq t' => term_of_qf ps vs (Nota (Eq t'))
haftmann@23713
   623
 | Dvd(i,t') => @{term "op dvd :: int => _ "} $ 
haftmann@23713
   624
    HOLogic.mk_number HOLogic.intT i $ term_of_i vs t'
haftmann@23689
   625
 | NDvd(i,t')=> term_of_qf ps vs (Nota(Dvd(i,t')))
haftmann@23689
   626
 | Nota t' => HOLogic.Not$(term_of_qf ps vs t')
wenzelm@23466
   627
 | And(t1,t2) => HOLogic.conj$(term_of_qf ps vs t1)$(term_of_qf ps vs t2)
wenzelm@23466
   628
 | Or(t1,t2) => HOLogic.disj$(term_of_qf ps vs t1)$(term_of_qf ps vs t2)
haftmann@23689
   629
 | Impa(t1,t2) => HOLogic.imp$(term_of_qf ps vs t1)$(term_of_qf ps vs t2)
haftmann@23713
   630
 | Iffa(t1,t2) => @{term "op = :: bool => _"} $ term_of_qf ps vs t1 $ term_of_qf ps vs t2
haftmann@23713
   631
 | Closed n => the (myassoc2 ps n)
haftmann@23689
   632
 | NClosed n => term_of_qf ps vs (Nota (Closed n))
wenzelm@23466
   633
 | _ => cooper "If this is raised, Isabelle/HOL or generate_code is inconsistent!";
wenzelm@23466
   634
wenzelm@23466
   635
fun cooper_oracle thy t = 
haftmann@23713
   636
  let
wenzelm@24630
   637
    val (vs, ps) = pairself (map_index swap) (term_frees t, term_bools [] t);
haftmann@23713
   638
  in
haftmann@23713
   639
    equals propT $ HOLogic.mk_Trueprop t $
haftmann@23713
   640
      HOLogic.mk_Trueprop (term_of_qf ps vs (pa (qf_of_term ps vs t)))
haftmann@23713
   641
  end;
wenzelm@23466
   642
wenzelm@23466
   643
end;