author  nipkow 
Mon, 18 Dec 2000 14:59:05 +0100  
changeset 10693  9e4a0e84d0d6 
parent 10574  8f98f0301d67 
child 10718  c058f78c3544 
permissions  rwrr 
9436
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(* Title: HOL/arith_data.ML 
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ID: $Id$ 
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Author: Markus Wenzel, Stefan Berghofer and Tobias Nipkow 
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Various arithmetic proof procedures. 
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*) 
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(**) 
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(* 1. Cancellation of common terms *) 
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(**) 
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signature ARITH_DATA = 
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sig 
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val nat_cancel_sums_add: simproc list 
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val nat_cancel_sums: simproc list 
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val nat_cancel_factor: simproc list 
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val nat_cancel: simproc list 
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end; 
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structure ArithData: ARITH_DATA = 
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struct 
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(** abstract syntax of structure nat: 0, Suc, + **) 
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(* mk_sum, mk_norm_sum *) 
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val one = HOLogic.mk_nat 1; 
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val mk_plus = HOLogic.mk_binop "op +"; 
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fun mk_sum [] = HOLogic.zero 
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 mk_sum [t] = t 
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 mk_sum (t :: ts) = mk_plus (t, mk_sum ts); 
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(*normal form of sums: Suc (... (Suc (a + (b + ...))))*) 
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fun mk_norm_sum ts = 
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let val (ones, sums) = partition (equal one) ts in 
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funpow (length ones) HOLogic.mk_Suc (mk_sum sums) 
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end; 
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(* dest_sum *) 
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val dest_plus = HOLogic.dest_bin "op +" HOLogic.natT; 
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fun dest_sum tm = 
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if HOLogic.is_zero tm then [] 
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else 
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(case try HOLogic.dest_Suc tm of 
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Some t => one :: dest_sum t 
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 None => 
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(case try dest_plus tm of 
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Some (t, u) => dest_sum t @ dest_sum u 
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 None => [tm])); 
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(** generic proof tools **) 
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(* prove conversions *) 
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val mk_eqv = HOLogic.mk_Trueprop o HOLogic.mk_eq; 
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fun prove_conv expand_tac norm_tac sg (t, u) = 
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mk_meta_eq (prove_goalw_cterm_nocheck [] (cterm_of sg (mk_eqv (t, u))) 
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(K [expand_tac, norm_tac])) 
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handle ERROR => error ("The error(s) above occurred while trying to prove " ^ 
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(string_of_cterm (cterm_of sg (mk_eqv (t, u))))); 
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val subst_equals = prove_goal HOL.thy "[ t = s; u = t ] ==> u = s" 
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(fn prems => [cut_facts_tac prems 1, SIMPSET' asm_simp_tac 1]); 
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(* rewriting *) 
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fun simp_all rules = ALLGOALS (simp_tac (HOL_ss addsimps rules)); 
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val add_rules = [add_Suc, add_Suc_right, add_0, add_0_right]; 
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val mult_rules = [mult_Suc, mult_Suc_right, mult_0, mult_0_right]; 
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(** cancel common summands **) 
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structure Sum = 
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struct 
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val mk_sum = mk_norm_sum; 
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val dest_sum = dest_sum; 
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val prove_conv = prove_conv; 
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val norm_tac = simp_all add_rules THEN simp_all add_ac; 
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end; 
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fun gen_uncancel_tac rule ct = 
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rtac (instantiate' [] [None, Some ct] (rule RS subst_equals)) 1; 
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(* nat eq *) 
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structure EqCancelSums = CancelSumsFun 
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(struct 
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open Sum; 
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val mk_bal = HOLogic.mk_eq; 
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val dest_bal = HOLogic.dest_bin "op =" HOLogic.natT; 
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val uncancel_tac = gen_uncancel_tac add_left_cancel; 
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end); 
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(* nat less *) 
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structure LessCancelSums = CancelSumsFun 
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(struct 
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open Sum; 
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val mk_bal = HOLogic.mk_binrel "op <"; 
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val dest_bal = HOLogic.dest_bin "op <" HOLogic.natT; 
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val uncancel_tac = gen_uncancel_tac add_left_cancel_less; 
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end); 
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(* nat le *) 
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structure LeCancelSums = CancelSumsFun 
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(struct 
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open Sum; 
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val mk_bal = HOLogic.mk_binrel "op <="; 
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val dest_bal = HOLogic.dest_bin "op <=" HOLogic.natT; 
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val uncancel_tac = gen_uncancel_tac add_left_cancel_le; 
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end); 
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(* nat diff *) 
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structure DiffCancelSums = CancelSumsFun 
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(struct 
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open Sum; 
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val mk_bal = HOLogic.mk_binop "op "; 
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val dest_bal = HOLogic.dest_bin "op " HOLogic.natT; 
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val uncancel_tac = gen_uncancel_tac diff_cancel; 
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end); 
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(** cancel common factor **) 
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structure Factor = 
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struct 
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val mk_sum = mk_norm_sum; 
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val dest_sum = dest_sum; 
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val prove_conv = prove_conv; 
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val norm_tac = simp_all (add_rules @ mult_rules) THEN simp_all add_ac; 
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end; 
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fun mk_cnat n = cterm_of (Theory.sign_of (the_context ())) (HOLogic.mk_nat n); 
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fun gen_multiply_tac rule k = 
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if k > 0 then 
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rtac (instantiate' [] [None, Some (mk_cnat (k  1))] (rule RS subst_equals)) 1 
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else no_tac; 
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(* nat eq *) 
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structure EqCancelFactor = CancelFactorFun 
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(struct 
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open Factor; 
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val mk_bal = HOLogic.mk_eq; 
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val dest_bal = HOLogic.dest_bin "op =" HOLogic.natT; 
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val multiply_tac = gen_multiply_tac Suc_mult_cancel1; 
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end); 
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(* nat less *) 
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structure LessCancelFactor = CancelFactorFun 
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(struct 
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open Factor; 
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val mk_bal = HOLogic.mk_binrel "op <"; 
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val dest_bal = HOLogic.dest_bin "op <" HOLogic.natT; 
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val multiply_tac = gen_multiply_tac Suc_mult_less_cancel1; 
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end); 
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(* nat le *) 
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structure LeCancelFactor = CancelFactorFun 
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(struct 
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open Factor; 
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val mk_bal = HOLogic.mk_binrel "op <="; 
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val dest_bal = HOLogic.dest_bin "op <=" HOLogic.natT; 
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val multiply_tac = gen_multiply_tac Suc_mult_le_cancel1; 
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end); 
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(** prepare nat_cancel simprocs **) 
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fun prep_pat s = Thm.read_cterm (Theory.sign_of (the_context ())) (s, HOLogic.termT); 
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val prep_pats = map prep_pat; 
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fun prep_simproc (name, pats, proc) = Simplifier.mk_simproc name pats proc; 
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val eq_pats = prep_pats ["(l::nat) + m = n", "(l::nat) = m + n", "Suc m = n", "m = Suc n"]; 
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val less_pats = prep_pats ["(l::nat) + m < n", "(l::nat) < m + n", "Suc m < n", "m < Suc n"]; 
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val le_pats = prep_pats ["(l::nat) + m <= n", "(l::nat) <= m + n", "Suc m <= n", "m <= Suc n"]; 
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val diff_pats = prep_pats ["((l::nat) + m)  n", "(l::nat)  (m + n)", "Suc m  n", "m  Suc n"]; 
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val nat_cancel_sums_add = map prep_simproc 
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[("nateq_cancel_sums", eq_pats, EqCancelSums.proc), 
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("natless_cancel_sums", less_pats, LessCancelSums.proc), 
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("natle_cancel_sums", le_pats, LeCancelSums.proc)]; 
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val nat_cancel_sums = nat_cancel_sums_add @ 
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[prep_simproc("natdiff_cancel_sums", diff_pats, DiffCancelSums.proc)]; 
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val nat_cancel_factor = map prep_simproc 
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[("nateq_cancel_factor", eq_pats, EqCancelFactor.proc), 
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("natless_cancel_factor", less_pats, LessCancelFactor.proc), 
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("natle_cancel_factor", le_pats, LeCancelFactor.proc)]; 
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val nat_cancel = nat_cancel_factor @ nat_cancel_sums; 
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end; 
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open ArithData; 
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(**) 
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(* 2. Linear arithmetic *) 
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(**) 
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(* Parameters data for general linear arithmetic functor *) 
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structure LA_Logic: LIN_ARITH_LOGIC = 
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struct 
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val ccontr = ccontr; 
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val conjI = conjI; 
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val neqE = linorder_neqE; 
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val notI = notI; 
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val sym = sym; 
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val not_lessD = linorder_not_less RS iffD1; 
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val not_leD = linorder_not_le RS iffD1; 
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fun mk_Eq thm = (thm RS Eq_FalseI) handle THM _ => (thm RS Eq_TrueI); 
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val mk_Trueprop = HOLogic.mk_Trueprop; 
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fun neg_prop(TP$(Const("Not",_)$t)) = TP$t 
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 neg_prop(TP$t) = TP $ (Const("Not",HOLogic.boolT>HOLogic.boolT)$t); 
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fun is_False thm = 
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let val _ $ t = #prop(rep_thm thm) 
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in t = Const("False",HOLogic.boolT) end; 
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fun is_nat(t) = fastype_of1 t = HOLogic.natT; 
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fun mk_nat_thm sg t = 
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let val ct = cterm_of sg t and cn = cterm_of sg (Var(("n",0),HOLogic.natT)) 
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in instantiate ([],[(cn,ct)]) le0 end; 
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end; 
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(* arith theory data *) 
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9593  265 
structure ArithTheoryDataArgs = 
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struct 
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val name = "HOL/arith"; 
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type T = {splits: thm list, inj_consts: (string * typ)list, discrete: (string * bool) list}; 
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val empty = {splits = [], inj_consts = [], discrete = []}; 
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val copy = I; 
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val prep_ext = I; 
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fun merge ({splits= splits1, inj_consts= inj_consts1, discrete= discrete1}, 
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{splits= splits2, inj_consts= inj_consts2, discrete= discrete2}) = 
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{splits = Drule.merge_rules (splits1, splits2), 
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inj_consts = merge_lists inj_consts1 inj_consts2, 
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discrete = merge_alists discrete1 discrete2}; 
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fun print _ _ = (); 
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end; 
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9593  281 
structure ArithTheoryData = TheoryDataFun(ArithTheoryDataArgs); 
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fun arith_split_add (thy, thm) = (ArithTheoryData.map (fn {splits,inj_consts,discrete} => 
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{splits= thm::splits, inj_consts= inj_consts, discrete= discrete}) thy, thm); 
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fun arith_discrete d = ArithTheoryData.map (fn {splits,inj_consts,discrete} => 
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{splits = splits, inj_consts = inj_consts, discrete = d :: discrete}); 
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fun arith_inj_const c = ArithTheoryData.map (fn {splits,inj_consts,discrete} => 
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{splits = splits, inj_consts = c :: inj_consts, discrete = discrete}); 
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structure LA_Data_Ref: LIN_ARITH_DATA = 
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struct 
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(* Decomposition of terms *) 
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fun nT (Type("fun",[N,_])) = N = HOLogic.natT 
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 nT _ = false; 
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fun add_atom(t,m,(p,i)) = (case assoc(p,t) of None => ((t,m)::p,i) 
10693  302 
 Some n => (overwrite(p,(t,ratadd(n,m))), i)); 
303 

304 
exception Zero; 

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10693  306 
fun rat_of_term(numt,dent) = 
307 
let val num = HOLogic.dest_binum numt and den = HOLogic.dest_binum dent 

308 
in if den = 0 then raise Zero else int_ratdiv(num,den) end; 

309 

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fun decomp2 inj_consts (rel,lhs,rhs) = 
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let 
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(* Turn term into list of summand * multiplicity plus a constant *) 
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fun poly(Const("op +",_) $ s $ t, m, pi) = poly(s,m,poly(t,m,pi)) 
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 poly(all as Const("op ",T) $ s $ t, m, pi) = 
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316 
if nT T then add_atom(all,m,pi) 
10693  317 
else poly(s,m,poly(t,ratneg m,pi)) 
318 
 poly(Const("uminus",_) $ t, m, pi) = poly(t,ratneg m,pi) 

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319 
 poly(Const("0",_), _, pi) = pi 
10693  320 
 poly(Const("Suc",_)$t, m, (p,i)) = poly(t, m, (p,ratadd(i,m))) 
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321 
 poly(all as Const("op *",_) $ (Const("Numeral.number_of",_)$c) $ t, m, pi)= 
10693  322 
(poly(t,ratmul(m, rat_of_int(HOLogic.dest_binum c)),pi) 
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323 
handle TERM _ => add_atom(all,m,pi)) 
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 poly(all as Const("op *",_) $ t $ (Const("Numeral.number_of",_)$c), m, pi)= 
10693  325 
(poly(t,ratmul(m, rat_of_int(HOLogic.dest_binum c)),pi) 
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326 
handle TERM _ => add_atom(all,m,pi)) 
10693  327 
 poly(Const("op *",_) $ (Const("HOL.divide",_) $ (Const("Numeral.number_of",_)$numt) $ 
328 
(Const("Numeral.number_of",_)$dent)) $ t, m, pi) = 

329 
poly(t,ratmul(m,rat_of_term(numt,dent)),pi) 

330 
 poly(Const("op *",_) $ t $ (Const("HOL.divide",_) $ (Const("Numeral.number_of",_)$numt) $ 

331 
(Const("Numeral.number_of",_)$dent)), m, pi) = 

332 
poly(t,ratmul(m,rat_of_term(numt,dent)),pi) 

333 
 poly(all as Const("HOL.divide",_) $ t $ (Const("Numeral.number_of",_)$dent), m, pi) = 

334 
let val den = HOLogic.dest_binum dent 

335 
in if den = 0 then raise Zero else poly(t,ratmul(m, ratinv(rat_of_int den)),pi) end 

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336 
 poly(all as Const("Numeral.number_of",_)$t,m,(p,i)) = 
10693  337 
((p,ratadd(i,ratmul(m,rat_of_int(HOLogic.dest_binum t)))) 
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338 
handle TERM _ => add_atom(all,m,(p,i))) 
10693  339 
 poly(Const("HOL.divide",_) $ (Const("Numeral.number_of",_)$numt) $ 
340 
(Const("Numeral.number_of",_)$dent), m, (p,i)) = 

341 
(p,ratadd(i,ratmul(m,rat_of_term(numt,dent)))) 

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 poly(all as Const f $ x, m, pi) = 
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if f mem inj_consts then poly(x,m,pi) else add_atom(all,m,pi) 
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344 
 poly x = add_atom x; 
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345 

10693  346 
val (p,i) = poly(lhs,rat_of_int 1,([],rat_of_int 0)) 
347 
and (q,j) = poly(rhs,rat_of_int 1,([],rat_of_int 0)) 

348 

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349 
in case rel of 
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350 
"op <" => Some(p,i,"<",q,j) 
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351 
 "op <=" => Some(p,i,"<=",q,j) 
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352 
 "op =" => Some(p,i,"=",q,j) 
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353 
 _ => None 
10693  354 
end handle Zero => None; 
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355 

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356 
fun negate(Some(x,i,rel,y,j,d)) = Some(x,i,"~"^rel,y,j,d) 
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357 
 negate None = None; 
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358 

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359 
fun decomp1 (discrete,inj_consts) (T,xxx) = 
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360 
(case T of 
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361 
Type("fun",[Type(D,[]),_]) => 
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362 
(case assoc(discrete,D) of 
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363 
None => None 
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364 
 Some d => (case decomp2 inj_consts xxx of 
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365 
None => None 
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366 
 Some(p,i,rel,q,j) => Some(p,i,rel,q,j,d))) 
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367 
 _ => None); 
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368 

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fun decomp2 data (_$(Const(rel,T)$lhs$rhs)) = decomp1 data (T,(rel,lhs,rhs)) 
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370 
 decomp2 data (_$(Const("Not",_)$(Const(rel,T)$lhs$rhs))) = 
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371 
negate(decomp1 data (T,(rel,lhs,rhs))) 
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372 
 decomp2 data _ = None 
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373 

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374 
fun decomp sg = 
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375 
let val {discrete, inj_consts, ...} = ArithTheoryData.get_sg sg 
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376 
in decomp2 (discrete,inj_consts) end 
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377 

10693  378 
fun number_of(n,T) = HOLogic.number_of_const T $ (HOLogic.mk_bin n) 
379 

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380 
end; 
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381 

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382 

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383 
structure Fast_Arith = 
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384 
Fast_Lin_Arith(structure LA_Logic=LA_Logic and LA_Data=LA_Data_Ref); 
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385 

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386 
val fast_arith_tac = Fast_Arith.lin_arith_tac 
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387 
and trace_arith = Fast_Arith.trace; 
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388 

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389 
local 
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390 

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391 
(* reduce contradictory <= to False. 
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392 
Most of the work is done by the cancel tactics. 
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393 
*) 
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394 
val add_rules = [add_0,add_0_right,Zero_not_Suc,Suc_not_Zero,le_0_eq]; 
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395 

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396 
val add_mono_thms_nat = map (fn s => prove_goal (the_context ()) s 
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397 
(fn prems => [cut_facts_tac prems 1, 
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398 
blast_tac (claset() addIs [add_le_mono]) 1])) 
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399 
["(i <= j) & (k <= l) ==> i + k <= j + (l::nat)", 
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400 
"(i = j) & (k <= l) ==> i + k <= j + (l::nat)", 
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401 
"(i <= j) & (k = l) ==> i + k <= j + (l::nat)", 
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402 
"(i = j) & (k = l) ==> i + k = j + (l::nat)" 
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403 
]; 
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404 

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405 
in 
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406 

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407 
val init_lin_arith_data = 
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408 
Fast_Arith.setup @ 
10693  409 
[Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, simpset = _} => 
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410 
{add_mono_thms = add_mono_thms @ add_mono_thms_nat, 
10693  411 
mult_mono_thms = mult_mono_thms, 
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412 
inj_thms = inj_thms, 
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413 
lessD = lessD @ [Suc_leI], 
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414 
simpset = HOL_basic_ss addsimps add_rules addsimprocs nat_cancel_sums_add}), 
9593  415 
ArithTheoryData.init, arith_discrete ("nat", true)]; 
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416 

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417 
end; 
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418 

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419 

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420 
local 
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421 
val nat_arith_simproc_pats = 
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422 
map (fn s => Thm.read_cterm (Theory.sign_of (the_context ())) (s, HOLogic.boolT)) 
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423 
["(m::nat) < n","(m::nat) <= n", "(m::nat) = n"]; 
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424 
in 
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425 
val fast_nat_arith_simproc = mk_simproc 
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426 
"fast_nat_arith" nat_arith_simproc_pats Fast_Arith.lin_arith_prover; 
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427 
end; 
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428 

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429 
(* Because of fast_nat_arith_simproc, the arithmetic solver is really only 
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430 
useful to detect inconsistencies among the premises for subgoals which are 
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431 
*not* themselves (in)equalities, because the latter activate 
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432 
fast_nat_arith_simproc anyway. However, it seems cheaper to activate the 
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433 
solver all the time rather than add the additional check. *) 
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434 

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435 

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436 
(* arith proof method *) 
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437 

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438 
(* FIXME: K true should be replaced by a sensible test to speed things up 
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439 
in case there are lots of irrelevant terms involved; 
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440 
elimination of min/max can be optimized: 
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441 
(max m n + k <= r) = (m+k <= r & n+k <= r) 
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442 
(l <= min m n + k) = (l <= m+k & l <= n+k) 
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443 
*) 
10516  444 
local 
445 

446 
val atomize_tac = Method.atomize_tac (thms "atomize'"); 

447 

448 
fun raw_arith_tac i st = 

9593  449 
refute_tac (K true) (REPEAT o split_tac (#splits (ArithTheoryData.get_sg (Thm.sign_of_thm st)))) 
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450 
((REPEAT_DETERM o etac linorder_neqE) THEN' fast_arith_tac) i st; 
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451 

10516  452 
in 
453 

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454 
val arith_tac = fast_arith_tac ORELSE' (atomize_tac THEN' raw_arith_tac); 
10516  455 

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456 
fun arith_method prems = 
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457 
Method.METHOD (fn facts => HEADGOAL (Method.insert_tac (prems @ facts) THEN' arith_tac)); 
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458 

10516  459 
end; 
460 

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461 

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462 
(* theory setup *) 
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463 

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464 
val arith_setup = 
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465 
[Simplifier.change_simpset_of (op addsimprocs) nat_cancel] @ 
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466 
init_lin_arith_data @ 
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467 
[Simplifier.change_simpset_of (op addSolver) 
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468 
(mk_solver "lin. arith." Fast_Arith.cut_lin_arith_tac), 
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469 
Simplifier.change_simpset_of (op addsimprocs) [fast_nat_arith_simproc], 
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470 
Method.add_methods [("arith", (arith_method o #2) oo Method.syntax Args.bang_facts, 
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471 
"decide linear arithmethic")], 
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rearranged setup of arithmetic procedures, avoiding global reference values;
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472 
Attrib.add_attributes [("arith_split", 
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rearranged setup of arithmetic procedures, avoiding global reference values;
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473 
(Attrib.no_args arith_split_add, Attrib.no_args Attrib.undef_local_attribute), 
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"declaration of split rules for arithmetic procedure")]]; 