author | haftmann |
Tue, 10 Jul 2007 17:30:54 +0200 | |
changeset 23713 | db10cf19f1f8 |
parent 23689 | 0410269099dc |
child 23881 | 851c74f1bb69 |
permissions | -rw-r--r-- |
23466 | 1 |
(* Title: HOL/Tools/Presburger/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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structure Integertab = TableFun(type key = Integer.int val ord = Integer.ord); |
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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("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 ("Orderings.ord_class.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 ("Orderings.ord_class.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("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 1)]); |
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fun linear_cmul 0 tm = zero |
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| linear_cmul n tm = |
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case tm of |
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Const("HOL.plus_class.plus",_)$a$b => addC$(linear_cmul n a)$(linear_cmul n b) |
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| Const ("HOL.times_class.times",_)$c$x => mulC$(numeral1 (Integer.mult n) c)$x |
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| Const("HOL.minus_class.minus",_)$a$b => subC$(linear_cmul n a)$(linear_cmul n b) |
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| (m as Const("HOL.minus_class.uminus",_))$a => m$(linear_cmul n a) |
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| _ => numeral1 (Integer.mult n) 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 = |
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case (tm1,tm2) of |
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(Const("HOL.plus_class.plus",_)$(Const("HOL.times_class.times",_)$c1$x1)$r1, |
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Const("HOL.plus_class.plus",_)$(Const("HOL.times_class.times",_)$c2$x2)$r2) => |
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if x1 = x2 then |
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let val c = numeral2 Integer.add 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("HOL.plus_class.plus",_) $ (Const("HOL.times_class.times",_)$c1$x1)$r1 ,_) => |
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addC$(mulC$c1$x1)$(linear_add vars r1 tm2) |
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| (_, Const("HOL.plus_class.plus",_)$(Const("HOL.times_class.times",_)$c2$x2)$r2) => |
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addC$(mulC$c2$x2)$(linear_add vars tm1 r2) |
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| (_,_) => numeral2 Integer.add tm1 tm2; |
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fun linear_neg tm = linear_cmul ~1 tm; |
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fun linear_sub vars tm1 tm2 = linear_add vars tm1 (linear_neg tm2); |
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fun lint vars tm = |
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if is_numeral tm then tm |
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else case tm of |
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Const("HOL.minus_class.uminus",_)$t => linear_neg (lint vars t) |
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| Const("HOL.plus_class.plus",_) $ s $ t => linear_add vars (lint vars s) (lint vars t) |
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| Const("HOL.minus_class.minus",_) $ s $ t => linear_sub vars (lint vars s) (lint vars t) |
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| Const ("HOL.times_class.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("Not",_)$(Const("Orderings.ord_class.less",T)$s$t)) = |
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lin vs (Const("Orderings.ord_class.less_eq",T)$t$s) |
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| lin (vs as x::_) (Const("Not",_)$(Const("Orderings.ord_class.less_eq",T)$s$t)) = |
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lin vs (Const("Orderings.ord_class.less",T)$t$s) |
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| lin vs (Const ("Not",T)$t) = Const ("Not",T)$ (lin vs t) |
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changeset
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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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RS eq_reflection |
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end; |
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fun is_intrel (b$_$_) = domain_type (fastype_of b) = HOLogic.intT |
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| is_intrel (@{term "Not"}$(b$_$_)) = domain_type (fastype_of b) = HOLogic.intT |
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| is_intrel _ = false; |
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fun linearize_conv ctxt vs ct = |
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case (term_of ct) of |
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23582
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changeset
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Const(@{const_name Divides.dvd},_)$d$t => |
23466 | 261 |
let |
262 |
val th = binop_conv (lint_conv ctxt vs) ct |
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val (d',t') = Thm.dest_binop (Thm.rhs_of th) |
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val (dt',tt') = (term_of d', term_of t') |
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in if is_numeral dt' andalso is_numeral tt' |
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then Conv.fconv_rule (arg_conv (Simplifier.rewrite presburger_ss)) th |
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else |
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268 |
let |
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val dth = |
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((if dest_numeral (term_of d') < 0 then |
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Conv.fconv_rule (arg_conv (arg1_conv (lint_conv ctxt vs))) |
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272 |
(Thm.transitive th (inst' [d',t'] dvd_uminus)) |
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else th) handle TERM _ => th) |
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val d'' = Thm.rhs_of dth |> Thm.dest_arg1 |
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in |
|
276 |
case tt' of |
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277 |
Const("HOL.plus_class.plus",_)$(Const("HOL.times_class.times",_)$c$_)$_ => |
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let val x = dest_numeral c |
|
279 |
in if x < 0 then Conv.fconv_rule (arg_conv (arg_conv (lint_conv ctxt vs))) |
|
280 |
(Thm.transitive dth (inst' [d'',t'] dvd_uminus')) |
|
281 |
else dth end |
|
282 |
| _ => dth |
|
283 |
end |
|
284 |
end |
|
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replaced code generator framework for reflected cooper
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diff
changeset
|
285 |
| Const("Not",_)$(Const(@{const_name Divides.dvd},_)$_$_) => arg_conv (linearize_conv ctxt vs) ct |
23466 | 286 |
| t => if is_intrel t |
287 |
then (provelin ctxt ((HOLogic.eq_const bT)$t$(lin vs t) |> HOLogic.mk_Trueprop)) |
|
288 |
RS eq_reflection |
|
289 |
else reflexive ct; |
|
290 |
||
291 |
val dvdc = @{cterm "op dvd :: int => _"}; |
|
292 |
||
293 |
fun unify ctxt q = |
|
294 |
let |
|
295 |
val (e,(cx,p)) = q |> Thm.dest_comb ||> Thm.dest_abs NONE |
|
296 |
val x = term_of cx |
|
297 |
val ins = insert (op = : integer*integer -> bool) |
|
298 |
fun h (acc,dacc) t = |
|
299 |
case (term_of t) of |
|
300 |
Const(s,_)$(Const("HOL.times_class.times",_)$c$y)$ _ => |
|
301 |
if x aconv y |
|
302 |
andalso s mem ["op =", "Orderings.ord_class.less", "Orderings.ord_class.less_eq"] |
|
303 |
then (ins (dest_numeral c) acc,dacc) else (acc,dacc) |
|
304 |
| Const(s,_)$_$(Const("HOL.times_class.times",_)$c$y) => |
|
305 |
if x aconv y |
|
306 |
andalso s mem ["Orderings.ord_class.less", "Orderings.ord_class.less_eq"] |
|
307 |
then (ins (dest_numeral c) acc, dacc) else (acc,dacc) |
|
23689
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23582
diff
changeset
|
308 |
| Const(@{const_name Divides.dvd},_)$_$(Const("HOL.plus_class.plus",_)$(Const("HOL.times_class.times",_)$c$y)$_) => |
23466 | 309 |
if x aconv y then (acc,ins (dest_numeral c) dacc) else (acc,dacc) |
310 |
| Const("op &",_)$_$_ => h (h (acc,dacc) (Thm.dest_arg1 t)) (Thm.dest_arg t) |
|
311 |
| Const("op |",_)$_$_ => h (h (acc,dacc) (Thm.dest_arg1 t)) (Thm.dest_arg t) |
|
312 |
| Const("Not",_)$_ => h (acc,dacc) (Thm.dest_arg t) |
|
313 |
| _ => (acc, dacc) |
|
314 |
val (cs,ds) = h ([],[]) p |
|
23514 | 315 |
val l = fold Integer.lcm (cs union ds) 1 |
23466 | 316 |
fun cv k ct = |
317 |
let val (tm as b$s$t) = term_of ct |
|
318 |
in ((HOLogic.eq_const bT)$tm$(b$(linear_cmul k s)$(linear_cmul k t)) |
|
319 |
|> HOLogic.mk_Trueprop |> provelin ctxt) RS eq_reflection end |
|
320 |
fun nzprop x = |
|
321 |
let |
|
322 |
val th = |
|
323 |
Simplifier.rewrite lin_ss |
|
324 |
(Thm.capply @{cterm Trueprop} (Thm.capply @{cterm "Not"} |
|
23689
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23582
diff
changeset
|
325 |
(Thm.capply (Thm.capply @{cterm "op = :: int => _"} (Numeral.mk_cnumber @{ctyp "int"} x)) |
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23582
diff
changeset
|
326 |
@{cterm "0::int"}))) |
23466 | 327 |
in equal_elim (Thm.symmetric th) TrueI end; |
328 |
val notz = let val tab = fold Integertab.update |
|
329 |
(ds ~~ (map (fn x => nzprop (Integer.div l x)) ds)) Integertab.empty |
|
330 |
in |
|
331 |
(fn ct => (valOf (Integertab.lookup tab (ct |> term_of |> dest_numeral)) |
|
332 |
handle Option => (writeln "noz: Theorems-Table contains no entry for"; |
|
333 |
print_cterm ct ; raise Option))) |
|
334 |
end |
|
335 |
fun unit_conv t = |
|
336 |
case (term_of t) of |
|
337 |
Const("op &",_)$_$_ => binop_conv unit_conv t |
|
338 |
| Const("op |",_)$_$_ => binop_conv unit_conv t |
|
339 |
| Const("Not",_)$_ => arg_conv unit_conv t |
|
340 |
| Const(s,_)$(Const("HOL.times_class.times",_)$c$y)$ _ => |
|
341 |
if x=y andalso s mem ["op =", "Orderings.ord_class.less", "Orderings.ord_class.less_eq"] |
|
342 |
then cv (Integer.div l (dest_numeral c)) t else Thm.reflexive t |
|
343 |
| Const(s,_)$_$(Const("HOL.times_class.times",_)$c$y) => |
|
344 |
if x=y andalso s mem ["Orderings.ord_class.less", "Orderings.ord_class.less_eq"] |
|
345 |
then cv (Integer.div l (dest_numeral c)) t else Thm.reflexive t |
|
23689
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23582
diff
changeset
|
346 |
| Const(@{const_name Divides.dvd},_)$d$(r as (Const("HOL.plus_class.plus",_)$(Const("HOL.times_class.times",_)$c$y)$_)) => |
23466 | 347 |
if x=y then |
348 |
let |
|
349 |
val k = Integer.div l (dest_numeral c) |
|
350 |
val kt = HOLogic.mk_number iT k |
|
351 |
val th1 = inst' [Thm.dest_arg1 t, Thm.dest_arg t] |
|
352 |
((Thm.dest_arg t |> funpow 2 Thm.dest_arg1 |> notz) RS zdvd_mono) |
|
353 |
val (d',t') = (mulC$kt$d, mulC$kt$r) |
|
354 |
val thc = (provelin ctxt ((HOLogic.eq_const iT)$d'$(lint [] d') |> HOLogic.mk_Trueprop)) |
|
355 |
RS eq_reflection |
|
356 |
val tht = (provelin ctxt ((HOLogic.eq_const iT)$t'$(linear_cmul k r) |> HOLogic.mk_Trueprop)) |
|
357 |
RS eq_reflection |
|
358 |
in Thm.transitive th1 (Thm.combination (Drule.arg_cong_rule dvdc thc) tht) end |
|
359 |
else Thm.reflexive t |
|
360 |
| _ => Thm.reflexive t |
|
361 |
val uth = unit_conv p |
|
23689
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23582
diff
changeset
|
362 |
val clt = Numeral.mk_cnumber @{ctyp "int"} l |
23466 | 363 |
val ltx = Thm.capply (Thm.capply cmulC clt) cx |
364 |
val th = Drule.arg_cong_rule e (Thm.abstract_rule (fst (dest_Free x )) cx uth) |
|
365 |
val th' = inst' [Thm.cabs ltx (Thm.rhs_of uth), clt] unity_coeff_ex |
|
366 |
val thf = transitive th |
|
367 |
(transitive (symmetric (beta_conversion true (cprop_of th' |> Thm.dest_arg1))) th') |
|
368 |
val (lth,rth) = Thm.dest_comb (cprop_of thf) |>> Thm.dest_arg |>> Thm.beta_conversion true |
|
369 |
||> beta_conversion true |>> Thm.symmetric |
|
370 |
in transitive (transitive lth thf) rth end; |
|
371 |
||
372 |
||
373 |
val emptyIS = @{cterm "{}::int set"}; |
|
374 |
val insert_tm = @{cterm "insert :: int => _"}; |
|
375 |
val mem_tm = Const("op :",[iT , HOLogic.mk_setT iT] ---> bT); |
|
376 |
fun mkISet cts = fold_rev (Thm.capply insert_tm #> Thm.capply) cts emptyIS; |
|
377 |
val cTrp = @{cterm "Trueprop"}; |
|
378 |
val eqelem_imp_imp = (thm"eqelem_imp_iff") RS iffD1; |
|
379 |
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 |
|
380 |
|> Thm.dest_abs NONE |> snd |> Thm.dest_fun |> Thm.dest_arg) |
|
381 |
[asetP,bsetP]; |
|
382 |
||
383 |
val D_tm = @{cpat "?D::int"}; |
|
384 |
||
385 |
val int_eq = (op =):integer*integer -> bool; |
|
386 |
fun cooperex_conv ctxt vs q = |
|
387 |
let |
|
388 |
||
389 |
val uth = unify ctxt q |
|
390 |
val (x,p) = Thm.dest_abs NONE (Thm.dest_arg (Thm.rhs_of uth)) |
|
391 |
val ins = insert (op aconvc) |
|
392 |
fun h t (bacc,aacc,dacc) = |
|
393 |
case (whatis x t) of |
|
394 |
And (p,q) => h q (h p (bacc,aacc,dacc)) |
|
395 |
| Or (p,q) => h q (h p (bacc,aacc,dacc)) |
|
396 |
| Eq t => (ins (minus1 t) bacc, |
|
397 |
ins (plus1 t) aacc,dacc) |
|
398 |
| NEq t => (ins t bacc, |
|
399 |
ins t aacc, dacc) |
|
400 |
| Lt t => (bacc, ins t aacc, dacc) |
|
401 |
| Le t => (bacc, ins (plus1 t) aacc,dacc) |
|
402 |
| Gt t => (ins t bacc, aacc,dacc) |
|
403 |
| Ge t => (ins (minus1 t) bacc, aacc,dacc) |
|
404 |
| Dvd (d,s) => (bacc,aacc,insert int_eq (term_of d |> dest_numeral) dacc) |
|
405 |
| NDvd (d,s) => (bacc,aacc,insert int_eq (term_of d|> dest_numeral) dacc) |
|
406 |
| _ => (bacc, aacc, dacc) |
|
407 |
val (b0,a0,ds) = h p ([],[],[]) |
|
23514 | 408 |
val d = fold Integer.lcm ds 1 |
23582 | 409 |
val cd = Numeral.mk_cnumber @{ctyp "int"} d |
23466 | 410 |
val dt = term_of cd |
411 |
fun divprop x = |
|
412 |
let |
|
413 |
val th = |
|
414 |
Simplifier.rewrite lin_ss |
|
415 |
(Thm.capply @{cterm Trueprop} |
|
23582 | 416 |
(Thm.capply (Thm.capply dvdc (Numeral.mk_cnumber @{ctyp "int"} x)) cd)) |
23466 | 417 |
in equal_elim (Thm.symmetric th) TrueI end; |
418 |
val dvd = let val tab = fold Integertab.update |
|
419 |
(ds ~~ (map divprop ds)) Integertab.empty in |
|
420 |
(fn ct => (valOf (Integertab.lookup tab (term_of ct |> dest_numeral)) |
|
421 |
handle Option => (writeln "dvd: Theorems-Table contains no entry for"; |
|
422 |
print_cterm ct ; raise Option))) |
|
423 |
end |
|
424 |
val dp = |
|
425 |
let val th = Simplifier.rewrite lin_ss |
|
426 |
(Thm.capply @{cterm Trueprop} |
|
427 |
(Thm.capply (Thm.capply @{cterm "op < :: int => _"} @{cterm "0::int"}) cd)) |
|
428 |
in equal_elim (Thm.symmetric th) TrueI end; |
|
429 |
(* A and B set *) |
|
430 |
local |
|
431 |
val insI1 = instantiate' [SOME @{ctyp "int"}] [] @{thm "insertI1"} |
|
432 |
val insI2 = instantiate' [SOME @{ctyp "int"}] [] @{thm "insertI2"} |
|
433 |
in |
|
434 |
fun provein x S = |
|
435 |
case term_of S of |
|
436 |
Const("{}",_) => error "Unexpected error in Cooper please email Amine Chaieb" |
|
437 |
| Const("insert",_)$y$_ => |
|
438 |
let val (cy,S') = Thm.dest_binop S |
|
439 |
in if term_of x aconv y then instantiate' [] [SOME x, SOME S'] insI1 |
|
440 |
else implies_elim (instantiate' [] [SOME x, SOME S', SOME cy] insI2) |
|
441 |
(provein x S') |
|
442 |
end |
|
443 |
end |
|
444 |
||
445 |
val al = map (lint vs o term_of) a0 |
|
446 |
val bl = map (lint vs o term_of) b0 |
|
447 |
val (sl,s0,f,abths,cpth) = |
|
448 |
if length (distinct (op aconv) bl) <= length (distinct (op aconv) al) |
|
449 |
then |
|
450 |
(bl,b0,decomp_minf, |
|
451 |
fn B => (map (fn th => implies_elim (Thm.instantiate ([],[(B_tm,B), (D_tm,cd)]) th) dp) |
|
452 |
[bseteq,bsetneq,bsetlt, bsetle, bsetgt,bsetge])@ |
|
453 |
(map (Thm.instantiate ([],[(B_tm,B), (D_tm,cd)])) |
|
454 |
[bsetdvd,bsetndvd,bsetP,infDdvd, infDndvd,bsetconj, |
|
455 |
bsetdisj,infDconj, infDdisj]), |
|
456 |
cpmi) |
|
457 |
else (al,a0,decomp_pinf,fn A => |
|
458 |
(map (fn th => implies_elim (Thm.instantiate ([],[(A_tm,A), (D_tm,cd)]) th) dp) |
|
459 |
[aseteq,asetneq,asetlt, asetle, asetgt,asetge])@ |
|
460 |
(map (Thm.instantiate ([],[(A_tm,A), (D_tm,cd)])) |
|
461 |
[asetdvd,asetndvd, asetP, infDdvd, infDndvd,asetconj, |
|
462 |
asetdisj,infDconj, infDdisj]),cppi) |
|
463 |
val cpth = |
|
464 |
let |
|
465 |
val sths = map (fn (tl,t0) => |
|
466 |
if tl = term_of t0 |
|
467 |
then instantiate' [SOME @{ctyp "int"}] [SOME t0] refl |
|
468 |
else provelin ctxt ((HOLogic.eq_const iT)$tl$(term_of t0) |
|
469 |
|> HOLogic.mk_Trueprop)) |
|
470 |
(sl ~~ s0) |
|
471 |
val csl = distinct (op aconvc) (map (cprop_of #> Thm.dest_arg #> Thm.dest_arg1) sths) |
|
472 |
val S = mkISet csl |
|
473 |
val inStab = fold (fn ct => fn tab => Termtab.update (term_of ct, provein ct S) tab) |
|
474 |
csl Termtab.empty |
|
475 |
val eqelem_th = instantiate' [SOME @{ctyp "int"}] [NONE,NONE, SOME S] eqelem_imp_imp |
|
476 |
val inS = |
|
477 |
let |
|
478 |
fun transmem th0 th1 = |
|
479 |
Thm.equal_elim |
|
480 |
(Drule.arg_cong_rule cTrp (Drule.fun_cong_rule (Drule.arg_cong_rule |
|
481 |
((Thm.dest_fun o Thm.dest_fun o Thm.dest_arg o cprop_of) th1) th0) S)) th1 |
|
482 |
val tab = fold Termtab.update |
|
483 |
(map (fn eq => |
|
484 |
let val (s,t) = cprop_of eq |> Thm.dest_arg |> Thm.dest_binop |
|
485 |
val th = if term_of s = term_of t |
|
486 |
then valOf(Termtab.lookup inStab (term_of s)) |
|
487 |
else FWD (instantiate' [] [SOME s, SOME t] eqelem_th) |
|
488 |
[eq, valOf(Termtab.lookup inStab (term_of s))] |
|
489 |
in (term_of t, th) end) |
|
490 |
sths) Termtab.empty |
|
491 |
in fn ct => |
|
492 |
(valOf (Termtab.lookup tab (term_of ct)) |
|
493 |
handle Option => (writeln "inS: No theorem for " ; print_cterm ct ; raise Option)) |
|
494 |
end |
|
495 |
val (inf, nb, pd) = divide_and_conquer (f x dvd inS (abths S)) p |
|
496 |
in [dp, inf, nb, pd] MRS cpth |
|
497 |
end |
|
498 |
val cpth' = Thm.transitive uth (cpth RS eq_reflection) |
|
499 |
in Thm.transitive cpth' ((simp_thms_conv then_conv eval_conv) (Thm.rhs_of cpth')) |
|
500 |
end; |
|
501 |
||
502 |
fun literals_conv bops uops env cv = |
|
503 |
let fun h t = |
|
504 |
case (term_of t) of |
|
505 |
b$_$_ => if member (op aconv) bops b then binop_conv h t else cv env t |
|
506 |
| u$_ => if member (op aconv) uops u then arg_conv h t else cv env t |
|
507 |
| _ => cv env t |
|
508 |
in h end; |
|
509 |
||
510 |
fun integer_nnf_conv ctxt env = |
|
511 |
nnf_conv then_conv literals_conv [HOLogic.conj, HOLogic.disj] [] env (linearize_conv ctxt); |
|
512 |
||
23689
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23582
diff
changeset
|
513 |
(* val my_term = ref (@{cterm "NotaHING"}); *) |
23466 | 514 |
local |
515 |
val pcv = Simplifier.rewrite |
|
516 |
(HOL_basic_ss addsimps (simp_thms @ (List.take(ex_simps,4)) |
|
517 |
@ [not_all,all_not_ex, ex_disj_distrib])) |
|
518 |
val postcv = Simplifier.rewrite presburger_ss |
|
519 |
fun conv ctxt p = |
|
520 |
let val _ = () (* my_term := p *) |
|
521 |
in |
|
23523 | 522 |
Qelim.gen_qelim_conv pcv postcv pcv (cons o term_of) |
23466 | 523 |
(term_frees (term_of p)) (linearize_conv ctxt) (integer_nnf_conv ctxt) |
524 |
(cooperex_conv ctxt) p |
|
525 |
end |
|
526 |
handle CTERM s => raise COOPER ("Cooper Failed", CTERM s) |
|
527 |
| THM s => raise COOPER ("Cooper Failed", THM s) |
|
23523 | 528 |
| TYPE s => raise COOPER ("Cooper Failed", TYPE s) |
23466 | 529 |
in val cooper_conv = conv |
530 |
end; |
|
531 |
end; |
|
532 |
||
533 |
||
534 |
||
535 |
structure Coopereif = |
|
536 |
struct |
|
537 |
||
23713 | 538 |
open GeneratedCooper; |
539 |
||
540 |
fun cooper s = raise Cooper.COOPER ("Cooper oracle failed", ERROR s); |
|
541 |
fun i_of_term vs t = case t |
|
542 |
of Free (xn, xT) => (case AList.lookup (op aconv) vs t |
|
543 |
of NONE => cooper "Variable not found in the list!" |
|
544 |
| SOME n => Bound n) |
|
545 |
| @{term "0::int"} => C 0 |
|
546 |
| @{term "1::int"} => C 1 |
|
547 |
| Term.Bound i => Bound (Integer.int i) |
|
548 |
| Const(@{const_name HOL.uminus},_)$t' => Neg (i_of_term vs t') |
|
549 |
| Const(@{const_name HOL.plus},_)$t1$t2 => Add (i_of_term vs t1,i_of_term vs t2) |
|
550 |
| Const(@{const_name HOL.minus},_)$t1$t2 => Sub (i_of_term vs t1,i_of_term vs t2) |
|
551 |
| Const(@{const_name HOL.times},_)$t1$t2 => |
|
552 |
(Mul (HOLogic.dest_number t1 |> snd, i_of_term vs t2) |
|
553 |
handle TERM _ => |
|
554 |
(Mul (HOLogic.dest_number t2 |> snd, i_of_term vs t1) |
|
555 |
handle TERM _ => cooper "Reification: Unsupported kind of multiplication")) |
|
556 |
| _ => (C (HOLogic.dest_number t |> snd) |
|
557 |
handle TERM _ => cooper "Reification: unknown term"); |
|
23689
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23582
diff
changeset
|
558 |
|
23713 | 559 |
fun qf_of_term ps vs t = case t |
560 |
of Const("True",_) => T |
|
561 |
| Const("False",_) => F |
|
562 |
| Const(@{const_name Orderings.less},_)$t1$t2 => Lt (Sub (i_of_term vs t1,i_of_term vs t2)) |
|
563 |
| Const(@{const_name Orderings.less_eq},_)$t1$t2 => Le (Sub(i_of_term vs t1,i_of_term vs t2)) |
|
564 |
| Const(@{const_name Divides.dvd},_)$t1$t2 => |
|
565 |
(Dvd(HOLogic.dest_number t1 |> snd, i_of_term vs t2) handle _ => cooper "Reification: unsupported dvd") |
|
566 |
| @{term "op = :: int => _"}$t1$t2 => Eq (Sub (i_of_term vs t1,i_of_term vs t2)) |
|
567 |
| @{term "op = :: bool => _ "}$t1$t2 => Iffa(qf_of_term ps vs t1,qf_of_term ps vs t2) |
|
568 |
| Const("op &",_)$t1$t2 => And(qf_of_term ps vs t1,qf_of_term ps vs t2) |
|
569 |
| Const("op |",_)$t1$t2 => Or(qf_of_term ps vs t1,qf_of_term ps vs t2) |
|
570 |
| Const("op -->",_)$t1$t2 => Impa(qf_of_term ps vs t1,qf_of_term ps vs t2) |
|
571 |
| Const("Not",_)$t' => Nota(qf_of_term ps vs t') |
|
572 |
| Const("Ex",_)$Abs(xn,xT,p) => |
|
573 |
let val (xn',p') = variant_abs (xn,xT,p) |
|
574 |
val vs' = (Free (xn',xT), 0) :: (map (fn(v,n) => (v,1+ n)) vs) |
|
575 |
in E (qf_of_term ps vs' p') |
|
576 |
end |
|
577 |
| Const("All",_)$Abs(xn,xT,p) => |
|
578 |
let val (xn',p') = variant_abs (xn,xT,p) |
|
579 |
val vs' = (Free (xn',xT), 0) :: (map (fn(v,n) => (v,1+ n)) vs) |
|
580 |
in A (qf_of_term ps vs' p') |
|
581 |
end |
|
582 |
| _ =>(case AList.lookup (op aconv) ps t of |
|
583 |
NONE => cooper "Reification: unknown term!" |
|
584 |
| SOME n => Closed n); |
|
23466 | 585 |
|
586 |
local |
|
587 |
val ops = [@{term "op &"}, @{term "op |"}, @{term "op -->"}, @{term "op = :: bool => _"}, |
|
588 |
@{term "op = :: int => _"}, @{term "op < :: int => _"}, |
|
589 |
@{term "op <= :: int => _"}, @{term "Not"}, @{term "All:: (int => _) => _"}, |
|
590 |
@{term "Ex:: (int => _) => _"}, @{term "True"}, @{term "False"}] |
|
591 |
fun ty t = Bool.not (fastype_of t = HOLogic.boolT) |
|
592 |
in |
|
593 |
fun term_bools acc t = |
|
594 |
case t of |
|
595 |
(l as f $ a) $ b => if ty t orelse f mem ops then term_bools (term_bools acc l)b |
|
596 |
else insert (op aconv) t acc |
|
597 |
| f $ a => if ty t orelse f mem ops then term_bools (term_bools acc f) a |
|
598 |
else insert (op aconv) t acc |
|
599 |
| Abs p => term_bools acc (snd (variant_abs p)) |
|
600 |
| _ => if ty t orelse t mem ops then acc else insert (op aconv) t acc |
|
601 |
end; |
|
602 |
||
603 |
fun myassoc2 l v = |
|
604 |
case l of |
|
605 |
[] => NONE |
|
23689
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23582
diff
changeset
|
606 |
| (x,v')::xs => if v = v' then SOME x |
23466 | 607 |
else myassoc2 xs v; |
608 |
||
23713 | 609 |
fun term_of_i vs t = case t |
610 |
of C i => HOLogic.mk_number HOLogic.intT i |
|
611 |
| Bound n => the (myassoc2 vs n) |
|
612 |
| Neg t' => @{term "uminus :: int => _"} $ term_of_i vs t' |
|
613 |
| Add (t1, t2) => @{term "op + :: int => _"} $ term_of_i vs t1 $ term_of_i vs t2 |
|
614 |
| Sub (t1, t2) => @{term "op - :: int => _"} $ term_of_i vs t1 $ term_of_i vs t2 |
|
615 |
| Mul (i, t2) => @{term "op * :: int => _"} $ |
|
616 |
HOLogic.mk_number HOLogic.intT i $ term_of_i vs t2 |
|
617 |
| Cx (i, t') => term_of_i vs (Add (Mul (i, Bound 0), t')); |
|
23466 | 618 |
|
619 |
fun term_of_qf ps vs t = |
|
620 |
case t of |
|
621 |
T => HOLogic.true_const |
|
622 |
| F => HOLogic.false_const |
|
623 |
| Lt t' => @{term "op < :: int => _ "}$ term_of_i vs t'$ @{term "0::int"} |
|
624 |
| Le t' => @{term "op <= :: int => _ "}$ term_of_i vs t' $ @{term "0::int"} |
|
625 |
| Gt t' => @{term "op < :: int => _ "}$ @{term "0::int"}$ term_of_i vs t' |
|
626 |
| Ge t' => @{term "op <= :: int => _ "}$ @{term "0::int"}$ term_of_i vs t' |
|
627 |
| Eq t' => @{term "op = :: int => _ "}$ term_of_i vs t'$ @{term "0::int"} |
|
23713 | 628 |
| NEq t' => term_of_qf ps vs (Nota (Eq t')) |
629 |
| Dvd(i,t') => @{term "op dvd :: int => _ "} $ |
|
630 |
HOLogic.mk_number HOLogic.intT i $ term_of_i vs t' |
|
23689
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23582
diff
changeset
|
631 |
| NDvd(i,t')=> term_of_qf ps vs (Nota(Dvd(i,t'))) |
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23582
diff
changeset
|
632 |
| Nota t' => HOLogic.Not$(term_of_qf ps vs t') |
23466 | 633 |
| And(t1,t2) => HOLogic.conj$(term_of_qf ps vs t1)$(term_of_qf ps vs t2) |
634 |
| Or(t1,t2) => HOLogic.disj$(term_of_qf ps vs t1)$(term_of_qf ps vs t2) |
|
23689
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23582
diff
changeset
|
635 |
| Impa(t1,t2) => HOLogic.imp$(term_of_qf ps vs t1)$(term_of_qf ps vs t2) |
23713 | 636 |
| Iffa(t1,t2) => @{term "op = :: bool => _"} $ term_of_qf ps vs t1 $ term_of_qf ps vs t2 |
637 |
| Closed n => the (myassoc2 ps n) |
|
23689
0410269099dc
replaced code generator framework for reflected cooper
haftmann
parents:
23582
diff
changeset
|
638 |
| NClosed n => term_of_qf ps vs (Nota (Closed n)) |
23466 | 639 |
| _ => cooper "If this is raised, Isabelle/HOL or generate_code is inconsistent!"; |
640 |
||
641 |
fun cooper_oracle thy t = |
|
23713 | 642 |
let |
643 |
val (vs, ps) = pairself (map_index (swap o apfst Integer.int)) |
|
644 |
(term_frees t, term_bools [] t); |
|
645 |
in |
|
646 |
equals propT $ HOLogic.mk_Trueprop t $ |
|
647 |
HOLogic.mk_Trueprop (term_of_qf ps vs (pa (qf_of_term ps vs t))) |
|
648 |
end; |
|
23466 | 649 |
|
650 |
end; |