src/HOL/Integ/int_arith1.ML
author paulson
Sun Feb 15 10:46:37 2004 +0100 (2004-02-15)
changeset 14387 e96d5c42c4b0
parent 14378 69c4d5997669
child 14390 55fe71faadda
permissions -rw-r--r--
Polymorphic treatment of binary arithmetic using axclasses
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(*  Title:      HOL/Integ/int_arith1.ML
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    ID:         $Id$
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    Authors:    Larry Paulson and Tobias Nipkow
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Simprocs and decision procedure for linear arithmetic.
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*)
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(** Misc ML bindings **)
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val NCons_Pls = thm"NCons_Pls";
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val NCons_Min = thm"NCons_Min";
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val NCons_BIT = thm"NCons_BIT";
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val number_of_Pls = thm"number_of_Pls";
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val number_of_Min = thm"number_of_Min";
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val number_of_BIT = thm"number_of_BIT";
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val bin_succ_Pls = thm"bin_succ_Pls";
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val bin_succ_Min = thm"bin_succ_Min";
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val bin_succ_BIT = thm"bin_succ_BIT";
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val bin_pred_Pls = thm"bin_pred_Pls";
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val bin_pred_Min = thm"bin_pred_Min";
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val bin_pred_BIT = thm"bin_pred_BIT";
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val bin_minus_Pls = thm"bin_minus_Pls";
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val bin_minus_Min = thm"bin_minus_Min";
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val bin_minus_BIT = thm"bin_minus_BIT";
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val bin_add_Pls = thm"bin_add_Pls";
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val bin_add_Min = thm"bin_add_Min";
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val bin_mult_Pls = thm"bin_mult_Pls";
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val bin_mult_Min = thm"bin_mult_Min";
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val bin_mult_BIT = thm"bin_mult_BIT";
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val neg_def = thm "neg_def";
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val iszero_def = thm "iszero_def";
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val NCons_Pls_0 = thm"NCons_Pls_0";
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val NCons_Pls_1 = thm"NCons_Pls_1";
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val NCons_Min_0 = thm"NCons_Min_0";
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val NCons_Min_1 = thm"NCons_Min_1";
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val bin_succ_1 = thm"bin_succ_1";
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val bin_succ_0 = thm"bin_succ_0";
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val bin_pred_1 = thm"bin_pred_1";
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val bin_pred_0 = thm"bin_pred_0";
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val bin_minus_1 = thm"bin_minus_1";
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val bin_minus_0 = thm"bin_minus_0";
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val bin_add_BIT_11 = thm"bin_add_BIT_11";
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val bin_add_BIT_10 = thm"bin_add_BIT_10";
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val bin_add_BIT_0 = thm"bin_add_BIT_0";
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val bin_add_Pls_right = thm"bin_add_Pls_right";
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val bin_add_Min_right = thm"bin_add_Min_right";
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val bin_add_BIT_BIT = thm"bin_add_BIT_BIT";
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val bin_mult_1 = thm"bin_mult_1";
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val bin_mult_0 = thm"bin_mult_0";
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val number_of_NCons = thm"number_of_NCons";
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val number_of_succ = thm"number_of_succ";
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val number_of_pred = thm"number_of_pred";
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val number_of_minus = thm"number_of_minus";
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val number_of_add = thm"number_of_add";
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val diff_number_of_eq = thm"diff_number_of_eq";
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val number_of_mult = thm"number_of_mult";
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val double_number_of_BIT = thm"double_number_of_BIT";
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val numeral_0_eq_0 = thm"numeral_0_eq_0";
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val numeral_1_eq_1 = thm"numeral_1_eq_1";
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val numeral_m1_eq_minus_1 = thm"numeral_m1_eq_minus_1";
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val mult_minus1 = thm"mult_minus1";
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val mult_minus1_right = thm"mult_minus1_right";
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val minus_number_of_mult = thm"minus_number_of_mult";
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val zero_less_nat_eq = thm"zero_less_nat_eq";
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val eq_number_of_eq = thm"eq_number_of_eq";
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val iszero_number_of_Pls = thm"iszero_number_of_Pls";
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val nonzero_number_of_Min = thm"nonzero_number_of_Min";
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val iszero_number_of_BIT = thm"iszero_number_of_BIT";
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val iszero_number_of_0 = thm"iszero_number_of_0";
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val iszero_number_of_1 = thm"iszero_number_of_1";
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val less_number_of_eq_neg = thm"less_number_of_eq_neg";
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val le_number_of_eq = thm"le_number_of_eq";
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val not_neg_number_of_Pls = thm"not_neg_number_of_Pls";
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val neg_number_of_Min = thm"neg_number_of_Min";
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val neg_number_of_BIT = thm"neg_number_of_BIT";
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val le_number_of_eq_not_less = thm"le_number_of_eq_not_less";
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val abs_number_of = thm"abs_number_of";
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val number_of_reorient = thm"number_of_reorient";
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val add_number_of_left = thm"add_number_of_left";
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val mult_number_of_left = thm"mult_number_of_left";
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val add_number_of_diff1 = thm"add_number_of_diff1";
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val add_number_of_diff2 = thm"add_number_of_diff2";
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val less_iff_diff_less_0 = thm"less_iff_diff_less_0";
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val eq_iff_diff_eq_0 = thm"eq_iff_diff_eq_0";
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val le_iff_diff_le_0 = thm"le_iff_diff_le_0";
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val NCons_simps = thms"NCons_simps";
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val bin_arith_extra_simps = thms"bin_arith_extra_simps";
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val bin_arith_simps = thms"bin_arith_simps";
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val bin_rel_simps = thms"bin_rel_simps";
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val zless_imp_add1_zle = thm "zless_imp_add1_zle";
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val combine_common_factor = thm"combine_common_factor";
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val eq_add_iff1 = thm"eq_add_iff1";
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val eq_add_iff2 = thm"eq_add_iff2";
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val less_add_iff1 = thm"less_add_iff1";
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val less_add_iff2 = thm"less_add_iff2";
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val le_add_iff1 = thm"le_add_iff1";
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val le_add_iff2 = thm"le_add_iff2";
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val arith_special = thms"arith_special";
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structure Bin_Simprocs =
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  struct
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  fun prove_conv tacs sg (hyps: thm list) xs (t, u) =
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    if t aconv u then None
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    else
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      let val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (t, u))
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      in Some (Tactic.prove sg xs [] eq (K (EVERY tacs))) end
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  fun prove_conv_nohyps tacs sg = prove_conv tacs sg [];
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  fun prove_conv_nohyps_novars tacs sg = prove_conv tacs sg [] [];
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  fun prep_simproc (name, pats, proc) =
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    Simplifier.simproc (Theory.sign_of (the_context())) name pats proc;
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  fun is_numeral (Const("Numeral.number_of", _) $ w) = true
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    | is_numeral _ = false
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  fun simplify_meta_eq f_number_of_eq f_eq =
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      mk_meta_eq ([f_eq, f_number_of_eq] MRS trans)
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  (*reorientation simprules using ==, for the following simproc*)
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  val meta_zero_reorient = zero_reorient RS eq_reflection
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  val meta_one_reorient = one_reorient RS eq_reflection
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  val meta_number_of_reorient = number_of_reorient RS eq_reflection
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  (*reorientation simplification procedure: reorients (polymorphic) 
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    0 = x, 1 = x, nnn = x provided x isn't 0, 1 or a numeral.*)
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  fun reorient_proc sg _ (_ $ t $ u) =
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    case u of
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	Const("0", _) => None
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      | Const("1", _) => None
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      | Const("Numeral.number_of", _) $ _ => None
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      | _ => Some (case t of
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		  Const("0", _) => meta_zero_reorient
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		| Const("1", _) => meta_one_reorient
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		| Const("Numeral.number_of", _) $ _ => meta_number_of_reorient)
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  val reorient_simproc = 
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      prep_simproc ("reorient_simproc", ["0=x", "1=x", "number_of w = x"], reorient_proc)
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  end;
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Addsimps arith_special;
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Addsimprocs [Bin_Simprocs.reorient_simproc];
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structure Int_Numeral_Simprocs =
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struct
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(*Maps 0 to Numeral0 and 1 to Numeral1 so that arithmetic in simprocs
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  isn't complicated by the abstract 0 and 1.*)
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val numeral_syms = [numeral_0_eq_0 RS sym, numeral_1_eq_1 RS sym];
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(*Utilities*)
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fun mk_numeral T n = HOLogic.number_of_const T $ HOLogic.mk_bin n;
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(*Decodes a binary INTEGER*)
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fun dest_numeral (Const("0", _)) = 0
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  | dest_numeral (Const("1", _)) = 1
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  | dest_numeral (Const("Numeral.number_of", _) $ w) =
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     (HOLogic.dest_binum w
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      handle TERM _ => raise TERM("Int_Numeral_Simprocs.dest_numeral:1", [w]))
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  | dest_numeral t = raise TERM("Int_Numeral_Simprocs.dest_numeral:2", [t]);
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fun find_first_numeral past (t::terms) =
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        ((dest_numeral t, rev past @ terms)
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         handle TERM _ => find_first_numeral (t::past) terms)
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  | find_first_numeral past [] = raise TERM("find_first_numeral", []);
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val mk_plus = HOLogic.mk_binop "op +";
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fun mk_minus t = 
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  let val T = Term.fastype_of t
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  in Const ("uminus", T --> T) $ t
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  end;
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(*Thus mk_sum[t] yields t+0; longer sums don't have a trailing zero*)
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fun mk_sum T []        = mk_numeral T 0
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  | mk_sum T [t,u]     = mk_plus (t, u)
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  | mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
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(*this version ALWAYS includes a trailing zero*)
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fun long_mk_sum T []        = mk_numeral T 0
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  | long_mk_sum T (t :: ts) = mk_plus (t, mk_sum T ts);
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val dest_plus = HOLogic.dest_bin "op +" Term.dummyT;
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(*decompose additions AND subtractions as a sum*)
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fun dest_summing (pos, Const ("op +", _) $ t $ u, ts) =
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        dest_summing (pos, t, dest_summing (pos, u, ts))
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  | dest_summing (pos, Const ("op -", _) $ t $ u, ts) =
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        dest_summing (pos, t, dest_summing (not pos, u, ts))
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  | dest_summing (pos, t, ts) =
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        if pos then t::ts else mk_minus t :: ts;
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fun dest_sum t = dest_summing (true, t, []);
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val mk_diff = HOLogic.mk_binop "op -";
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val dest_diff = HOLogic.dest_bin "op -" Term.dummyT;
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val mk_times = HOLogic.mk_binop "op *";
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fun mk_prod T = 
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  let val one = mk_numeral T 1
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  fun mk [] = one
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    | mk [t] = t
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    | mk (t :: ts) = if t = one then mk ts else mk_times (t, mk ts)
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  in mk end;
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(*This version ALWAYS includes a trailing one*)
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fun long_mk_prod T []        = mk_numeral T 1
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  | long_mk_prod T (t :: ts) = mk_times (t, mk_prod T ts);
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val dest_times = HOLogic.dest_bin "op *" Term.dummyT;
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fun dest_prod t =
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      let val (t,u) = dest_times t
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      in  dest_prod t @ dest_prod u  end
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      handle TERM _ => [t];
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(*DON'T do the obvious simplifications; that would create special cases*)
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fun mk_coeff (k, t) = mk_times (mk_numeral (Term.fastype_of t) k, t);
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(*Express t as a product of (possibly) a numeral with other sorted terms*)
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fun dest_coeff sign (Const ("uminus", _) $ t) = dest_coeff (~sign) t
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  | dest_coeff sign t =
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    let val ts = sort Term.term_ord (dest_prod t)
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        val (n, ts') = find_first_numeral [] ts
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                          handle TERM _ => (1, ts)
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    in (sign*n, mk_prod (Term.fastype_of t) ts') end;
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(*Find first coefficient-term THAT MATCHES u*)
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fun find_first_coeff past u [] = raise TERM("find_first_coeff", [])
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  | find_first_coeff past u (t::terms) =
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        let val (n,u') = dest_coeff 1 t
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        in  if u aconv u' then (n, rev past @ terms)
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                          else find_first_coeff (t::past) u terms
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        end
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        handle TERM _ => find_first_coeff (t::past) u terms;
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(*Simplify Numeral0+n, n+Numeral0, Numeral1*n, n*Numeral1*)
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val add_0s =  thms "add_0s";
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val mult_1s = thms "mult_1s";
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(*To perform binary arithmetic.  The "left" rewriting handles patterns
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  created by the simprocs, such as 3 * (5 * x). *)
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val bin_simps = [numeral_0_eq_0 RS sym, numeral_1_eq_1 RS sym,
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                 add_number_of_left, mult_number_of_left] @
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                bin_arith_simps @ bin_rel_simps;
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(*Binary arithmetic BUT NOT ADDITION since it may collapse adjacent terms
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  during re-arrangement*)
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val non_add_bin_simps = 
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    bin_simps \\ [add_number_of_left, number_of_add RS sym];
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(*To evaluate binary negations of coefficients*)
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val minus_simps = NCons_simps @
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                   [numeral_m1_eq_minus_1 RS sym, number_of_minus RS sym,
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                    bin_minus_1, bin_minus_0, bin_minus_Pls, bin_minus_Min,
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                    bin_pred_1, bin_pred_0, bin_pred_Pls, bin_pred_Min];
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(*To let us treat subtraction as addition*)
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val diff_simps = [diff_minus, minus_add_distrib, minus_minus];
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(*push the unary minus down: - x * y = x * - y *)
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val minus_mult_eq_1_to_2 =
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    [minus_mult_left RS sym, minus_mult_right] MRS trans |> standard;
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(*to extract again any uncancelled minuses*)
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val minus_from_mult_simps =
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    [minus_minus, minus_mult_left RS sym, minus_mult_right RS sym];
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(*combine unary minus with numeric literals, however nested within a product*)
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val mult_minus_simps =
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    [mult_assoc, minus_mult_left, minus_mult_eq_1_to_2];
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(*Apply the given rewrite (if present) just once*)
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fun trans_tac None      = all_tac
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  | trans_tac (Some th) = ALLGOALS (rtac (th RS trans));
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fun simplify_meta_eq rules =
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    simplify (HOL_basic_ss addeqcongs[eq_cong2] addsimps rules)
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    o mk_meta_eq;
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structure CancelNumeralsCommon =
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  struct
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  val mk_sum            = mk_sum
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   296
  val dest_sum          = dest_sum
wenzelm@13462
   297
  val mk_coeff          = mk_coeff
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   298
  val dest_coeff        = dest_coeff 1
wenzelm@13462
   299
  val find_first_coeff  = find_first_coeff []
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   300
  val trans_tac         = trans_tac
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   301
  val norm_tac =
paulson@11868
   302
     ALLGOALS (simp_tac (HOL_ss addsimps numeral_syms@add_0s@mult_1s@
paulson@14387
   303
                                         diff_simps@minus_simps@add_ac))
paulson@14387
   304
     THEN ALLGOALS (simp_tac (HOL_ss addsimps non_add_bin_simps@mult_minus_simps))
paulson@14387
   305
     THEN ALLGOALS (simp_tac (HOL_ss addsimps minus_from_mult_simps@
paulson@14387
   306
                                              add_ac@mult_ac))
wenzelm@13462
   307
  val numeral_simp_tac  = ALLGOALS (simp_tac (HOL_ss addsimps add_0s@bin_simps))
wenzelm@9436
   308
  val simplify_meta_eq  = simplify_meta_eq (add_0s@mult_1s)
wenzelm@9436
   309
  end;
wenzelm@9436
   310
wenzelm@9436
   311
wenzelm@9436
   312
structure EqCancelNumerals = CancelNumeralsFun
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   313
 (open CancelNumeralsCommon
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   314
  val prove_conv = Bin_Simprocs.prove_conv
wenzelm@9436
   315
  val mk_bal   = HOLogic.mk_eq
paulson@14387
   316
  val dest_bal = HOLogic.dest_bin "op =" Term.dummyT
wenzelm@9436
   317
  val bal_add1 = eq_add_iff1 RS trans
wenzelm@9436
   318
  val bal_add2 = eq_add_iff2 RS trans
wenzelm@9436
   319
);
wenzelm@9436
   320
wenzelm@9436
   321
structure LessCancelNumerals = CancelNumeralsFun
wenzelm@9436
   322
 (open CancelNumeralsCommon
wenzelm@13485
   323
  val prove_conv = Bin_Simprocs.prove_conv
wenzelm@9436
   324
  val mk_bal   = HOLogic.mk_binrel "op <"
paulson@14387
   325
  val dest_bal = HOLogic.dest_bin "op <" Term.dummyT
wenzelm@9436
   326
  val bal_add1 = less_add_iff1 RS trans
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   327
  val bal_add2 = less_add_iff2 RS trans
wenzelm@9436
   328
);
wenzelm@9436
   329
wenzelm@9436
   330
structure LeCancelNumerals = CancelNumeralsFun
wenzelm@9436
   331
 (open CancelNumeralsCommon
wenzelm@13485
   332
  val prove_conv = Bin_Simprocs.prove_conv
wenzelm@9436
   333
  val mk_bal   = HOLogic.mk_binrel "op <="
paulson@14387
   334
  val dest_bal = HOLogic.dest_bin "op <=" Term.dummyT
wenzelm@9436
   335
  val bal_add1 = le_add_iff1 RS trans
wenzelm@9436
   336
  val bal_add2 = le_add_iff2 RS trans
wenzelm@9436
   337
);
wenzelm@9436
   338
wenzelm@13462
   339
val cancel_numerals =
paulson@11868
   340
  map Bin_Simprocs.prep_simproc
wenzelm@9436
   341
   [("inteq_cancel_numerals",
paulson@14387
   342
     ["(l::'a::number_ring) + m = n",
paulson@14387
   343
      "(l::'a::number_ring) = m + n",
paulson@14387
   344
      "(l::'a::number_ring) - m = n",
paulson@14387
   345
      "(l::'a::number_ring) = m - n",
paulson@14387
   346
      "(l::'a::number_ring) * m = n",
paulson@14387
   347
      "(l::'a::number_ring) = m * n"],
wenzelm@9436
   348
     EqCancelNumerals.proc),
wenzelm@13462
   349
    ("intless_cancel_numerals",
paulson@14387
   350
     ["(l::'a::{ordered_ring,number_ring}) + m < n",
paulson@14387
   351
      "(l::'a::{ordered_ring,number_ring}) < m + n",
paulson@14387
   352
      "(l::'a::{ordered_ring,number_ring}) - m < n",
paulson@14387
   353
      "(l::'a::{ordered_ring,number_ring}) < m - n",
paulson@14387
   354
      "(l::'a::{ordered_ring,number_ring}) * m < n",
paulson@14387
   355
      "(l::'a::{ordered_ring,number_ring}) < m * n"],
wenzelm@9436
   356
     LessCancelNumerals.proc),
wenzelm@13462
   357
    ("intle_cancel_numerals",
paulson@14387
   358
     ["(l::'a::{ordered_ring,number_ring}) + m <= n",
paulson@14387
   359
      "(l::'a::{ordered_ring,number_ring}) <= m + n",
paulson@14387
   360
      "(l::'a::{ordered_ring,number_ring}) - m <= n",
paulson@14387
   361
      "(l::'a::{ordered_ring,number_ring}) <= m - n",
paulson@14387
   362
      "(l::'a::{ordered_ring,number_ring}) * m <= n",
paulson@14387
   363
      "(l::'a::{ordered_ring,number_ring}) <= m * n"],
wenzelm@9436
   364
     LeCancelNumerals.proc)];
wenzelm@9436
   365
wenzelm@9436
   366
wenzelm@9436
   367
structure CombineNumeralsData =
wenzelm@9436
   368
  struct
wenzelm@13462
   369
  val add               = op + : int*int -> int
wenzelm@13462
   370
  val mk_sum            = long_mk_sum    (*to work for e.g. 2*x + 3*x *)
wenzelm@13462
   371
  val dest_sum          = dest_sum
wenzelm@13462
   372
  val mk_coeff          = mk_coeff
wenzelm@13462
   373
  val dest_coeff        = dest_coeff 1
paulson@14272
   374
  val left_distrib      = combine_common_factor RS trans
wenzelm@13485
   375
  val prove_conv        = Bin_Simprocs.prove_conv_nohyps
wenzelm@9436
   376
  val trans_tac          = trans_tac
wenzelm@13462
   377
  val norm_tac =
paulson@11868
   378
     ALLGOALS (simp_tac (HOL_ss addsimps numeral_syms@add_0s@mult_1s@
paulson@14387
   379
                                         diff_simps@minus_simps@add_ac))
paulson@14387
   380
     THEN ALLGOALS (simp_tac (HOL_ss addsimps non_add_bin_simps@mult_minus_simps))
paulson@14387
   381
     THEN ALLGOALS (simp_tac (HOL_ss addsimps minus_from_mult_simps@
paulson@14387
   382
                                              add_ac@mult_ac))
wenzelm@13462
   383
  val numeral_simp_tac  = ALLGOALS
wenzelm@9436
   384
                    (simp_tac (HOL_ss addsimps add_0s@bin_simps))
wenzelm@9436
   385
  val simplify_meta_eq  = simplify_meta_eq (add_0s@mult_1s)
wenzelm@9436
   386
  end;
wenzelm@9436
   387
wenzelm@9436
   388
structure CombineNumerals = CombineNumeralsFun(CombineNumeralsData);
wenzelm@13462
   389
wenzelm@13462
   390
val combine_numerals =
wenzelm@13462
   391
  Bin_Simprocs.prep_simproc
paulson@14387
   392
    ("int_combine_numerals", 
paulson@14387
   393
     ["(i::'a::number_ring) + j", "(i::'a::number_ring) - j"], 
paulson@14387
   394
     CombineNumerals.proc);
wenzelm@9436
   395
wenzelm@9436
   396
end;
wenzelm@9436
   397
wenzelm@9436
   398
Addsimprocs Int_Numeral_Simprocs.cancel_numerals;
wenzelm@9436
   399
Addsimprocs [Int_Numeral_Simprocs.combine_numerals];
wenzelm@9436
   400
wenzelm@9436
   401
(*examples:
wenzelm@9436
   402
print_depth 22;
wenzelm@9436
   403
set timing;
wenzelm@9436
   404
set trace_simp;
wenzelm@13462
   405
fun test s = (Goal s, by (Simp_tac 1));
wenzelm@9436
   406
wenzelm@11704
   407
test "l + 2 + 2 + 2 + (l + 2) + (oo + 2) = (uu::int)";
wenzelm@9436
   408
wenzelm@11704
   409
test "2*u = (u::int)";
wenzelm@11704
   410
test "(i + j + 12 + (k::int)) - 15 = y";
wenzelm@11704
   411
test "(i + j + 12 + (k::int)) - 5 = y";
wenzelm@9436
   412
wenzelm@9436
   413
test "y - b < (b::int)";
wenzelm@11704
   414
test "y - (3*b + c) < (b::int) - 2*c";
wenzelm@9436
   415
wenzelm@11704
   416
test "(2*x - (u*v) + y) - v*3*u = (w::int)";
wenzelm@11704
   417
test "(2*x*u*v + (u*v)*4 + y) - v*u*4 = (w::int)";
wenzelm@11704
   418
test "(2*x*u*v + (u*v)*4 + y) - v*u = (w::int)";
wenzelm@11704
   419
test "u*v - (x*u*v + (u*v)*4 + y) = (w::int)";
wenzelm@9436
   420
wenzelm@11704
   421
test "(i + j + 12 + (k::int)) = u + 15 + y";
wenzelm@11704
   422
test "(i + j*2 + 12 + (k::int)) = j + 5 + y";
wenzelm@9436
   423
wenzelm@11704
   424
test "2*y + 3*z + 6*w + 2*y + 3*z + 2*u = 2*y' + 3*z' + 6*w' + 2*y' + 3*z' + u + (vv::int)";
wenzelm@9436
   425
wenzelm@9436
   426
test "a + -(b+c) + b = (d::int)";
wenzelm@9436
   427
test "a + -(b+c) - b = (d::int)";
wenzelm@9436
   428
wenzelm@9436
   429
(*negative numerals*)
wenzelm@11704
   430
test "(i + j + -2 + (k::int)) - (u + 5 + y) = zz";
wenzelm@11704
   431
test "(i + j + -3 + (k::int)) < u + 5 + y";
wenzelm@11704
   432
test "(i + j + 3 + (k::int)) < u + -6 + y";
wenzelm@11704
   433
test "(i + j + -12 + (k::int)) - 15 = y";
wenzelm@11704
   434
test "(i + j + 12 + (k::int)) - -15 = y";
wenzelm@11704
   435
test "(i + j + -12 + (k::int)) - -15 = y";
wenzelm@9436
   436
*)
wenzelm@9436
   437
wenzelm@9436
   438
paulson@14387
   439
(** Constant folding for multiplication in semirings **)
wenzelm@9436
   440
paulson@14387
   441
(*We do not need folding for addition: combine_numerals does the same thing*)
wenzelm@9436
   442
paulson@14387
   443
structure Semiring_Times_Assoc_Data : ASSOC_FOLD_DATA =
wenzelm@9436
   444
struct
wenzelm@13462
   445
  val ss                = HOL_ss
wenzelm@13462
   446
  val eq_reflection     = eq_reflection
wenzelm@9436
   447
  val sg_ref = Sign.self_ref (Theory.sign_of (the_context ()))
wenzelm@9436
   448
  val add_ac = mult_ac
wenzelm@9436
   449
end;
wenzelm@9436
   450
paulson@14387
   451
structure Semiring_Times_Assoc = Assoc_Fold (Semiring_Times_Assoc_Data);
wenzelm@9436
   452
paulson@14387
   453
val assoc_fold_simproc =
paulson@14387
   454
  Bin_Simprocs.prep_simproc
paulson@14387
   455
   ("semiring_assoc_fold", ["(a::'a::semiring) * b"],
paulson@14387
   456
    Semiring_Times_Assoc.proc);
paulson@14387
   457
paulson@14387
   458
Addsimprocs [assoc_fold_simproc];
paulson@14387
   459
paulson@14387
   460
wenzelm@9436
   461
wenzelm@9436
   462
wenzelm@9436
   463
(*** decision procedure for linear arithmetic ***)
wenzelm@9436
   464
wenzelm@9436
   465
(*---------------------------------------------------------------------------*)
wenzelm@9436
   466
(* Linear arithmetic                                                         *)
wenzelm@9436
   467
(*---------------------------------------------------------------------------*)
wenzelm@9436
   468
wenzelm@9436
   469
(*
wenzelm@9436
   470
Instantiation of the generic linear arithmetic package for int.
wenzelm@9436
   471
*)
wenzelm@9436
   472
wenzelm@9436
   473
(* Update parameters of arithmetic prover *)
wenzelm@9436
   474
local
wenzelm@9436
   475
wenzelm@9436
   476
(* reduce contradictory <= to False *)
wenzelm@13462
   477
val add_rules =
paulson@14387
   478
    simp_thms @ bin_arith_simps @ bin_rel_simps @ arith_special @
paulson@14387
   479
    [numeral_0_eq_0, numeral_1_eq_1,
paulson@14369
   480
     minus_zero, diff_minus, left_minus, right_minus,
paulson@14369
   481
     mult_zero_left, mult_zero_right, mult_1, mult_1_right,
paulson@14369
   482
     minus_mult_left RS sym, minus_mult_right RS sym,
paulson@14369
   483
     minus_add_distrib, minus_minus, mult_assoc,
paulson@14387
   484
     of_nat_0, of_nat_1, of_nat_Suc, of_nat_add, of_nat_mult,
paulson@14387
   485
     of_int_0, of_int_1, of_int_add, of_int_mult, int_eq_of_nat,
paulson@14387
   486
     zero_neq_one, zero_less_one, zero_le_one, 
paulson@14387
   487
     zero_neq_one RS not_sym, not_one_le_zero, not_one_less_zero];
wenzelm@9436
   488
paulson@14387
   489
val simprocs = [assoc_fold_simproc, Int_Numeral_Simprocs.combine_numerals]@
paulson@14387
   490
               Int_Numeral_Simprocs.cancel_numerals;
wenzelm@9436
   491
wenzelm@9436
   492
in
wenzelm@9436
   493
wenzelm@9436
   494
val int_arith_setup =
nipkow@10693
   495
 [Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, simpset} =>
paulson@14368
   496
   {add_mono_thms = add_mono_thms,
nipkow@10693
   497
    mult_mono_thms = mult_mono_thms,
nipkow@10574
   498
    inj_thms = [zle_int RS iffD2,int_int_eq RS iffD2] @ inj_thms,
paulson@14272
   499
    lessD = lessD @ [zless_imp_add1_zle],
wenzelm@9436
   500
    simpset = simpset addsimps add_rules
wenzelm@9436
   501
                      addsimprocs simprocs
wenzelm@9436
   502
                      addcongs [if_weak_cong]}),
paulson@14387
   503
  arith_inj_const ("IntDef.of_nat", HOLogic.natT --> HOLogic.intT),
nipkow@10834
   504
  arith_inj_const ("IntDef.int", HOLogic.natT --> HOLogic.intT),
wenzelm@9436
   505
  arith_discrete ("IntDef.int", true)];
wenzelm@9436
   506
wenzelm@9436
   507
end;
wenzelm@9436
   508
wenzelm@13462
   509
val fast_int_arith_simproc =
wenzelm@13462
   510
  Simplifier.simproc (Theory.sign_of (the_context()))
paulson@14387
   511
  "fast_int_arith" 
paulson@14387
   512
     ["(m::'a::{ordered_ring,number_ring}) < n",
paulson@14387
   513
      "(m::'a::{ordered_ring,number_ring}) <= n",
paulson@14387
   514
      "(m::'a::{ordered_ring,number_ring}) = n"] Fast_Arith.lin_arith_prover;
wenzelm@9436
   515
wenzelm@9436
   516
Addsimprocs [fast_int_arith_simproc]
wenzelm@13462
   517
wenzelm@9436
   518
wenzelm@9436
   519
(* Some test data
wenzelm@9436
   520
Goal "!!a::int. [| a <= b; c <= d; x+y<z |] ==> a+c <= b+d";
wenzelm@9436
   521
by (fast_arith_tac 1);
wenzelm@11704
   522
Goal "!!a::int. [| a < b; c < d |] ==> a-d+ 2 <= b+(-c)";
wenzelm@9436
   523
by (fast_arith_tac 1);
paulson@11868
   524
Goal "!!a::int. [| a < b; c < d |] ==> a+c+ 1 < b+d";
wenzelm@9436
   525
by (fast_arith_tac 1);
wenzelm@9436
   526
Goal "!!a::int. [| a <= b; b+b <= c |] ==> a+a <= c";
wenzelm@9436
   527
by (fast_arith_tac 1);
wenzelm@9436
   528
Goal "!!a::int. [| a+b <= i+j; a<=b; i<=j |] \
wenzelm@9436
   529
\     ==> a+a <= j+j";
wenzelm@9436
   530
by (fast_arith_tac 1);
wenzelm@9436
   531
Goal "!!a::int. [| a+b < i+j; a<b; i<j |] \
wenzelm@11704
   532
\     ==> a+a - - -1 < j+j - 3";
wenzelm@9436
   533
by (fast_arith_tac 1);
wenzelm@9436
   534
Goal "!!a::int. a+b+c <= i+j+k & a<=b & b<=c & i<=j & j<=k --> a+a+a <= k+k+k";
wenzelm@9436
   535
by (arith_tac 1);
wenzelm@9436
   536
Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
wenzelm@9436
   537
\     ==> a <= l";
wenzelm@9436
   538
by (fast_arith_tac 1);
wenzelm@9436
   539
Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
wenzelm@9436
   540
\     ==> a+a+a+a <= l+l+l+l";
wenzelm@9436
   541
by (fast_arith_tac 1);
wenzelm@9436
   542
Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
wenzelm@9436
   543
\     ==> a+a+a+a+a <= l+l+l+l+i";
wenzelm@9436
   544
by (fast_arith_tac 1);
wenzelm@9436
   545
Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
wenzelm@9436
   546
\     ==> a+a+a+a+a+a <= l+l+l+l+i+l";
wenzelm@9436
   547
by (fast_arith_tac 1);
wenzelm@9436
   548
Goal "!!a::int. [| a+b+c+d <= i+j+k+l; a<=b; b<=c; c<=d; i<=j; j<=k; k<=l |] \
wenzelm@11704
   549
\     ==> 6*a <= 5*l+i";
wenzelm@9436
   550
by (fast_arith_tac 1);
wenzelm@9436
   551
*)