author  nipkow 
Fri, 10 Sep 2004 00:19:15 +0200  
changeset 15195  197e00ce3f20 
parent 15185  8c43ffe2bb32 
child 15197  19e735596e51 
permissions  rwrr 
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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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13517  12 
structure NatArithUtils = 
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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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13517  54 
fun prove_conv expand_tac norm_tac sg tu = 
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mk_meta_eq (prove_goalw_cterm_nocheck [] (cterm_of sg (mk_eqv tu)) 

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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 tu)))); 
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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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13517  71 
fun prep_simproc (name, pats, proc) = 
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Simplifier.simproc (Theory.sign_of (the_context ())) name pats proc; 

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end; 

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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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end; 

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structure ArithData: ARITH_DATA = 

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struct 

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open NatArithUtils; 

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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 nat_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 nat_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 nat_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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(** prepare nat_cancel simprocs **) 
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val nat_cancel_sums_add = map prep_simproc 
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[("nateq_cancel_sums", 
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["(l::nat) + m = n", "(l::nat) = m + n", "Suc m = n", "m = Suc n"], EqCancelSums.proc), 

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("natless_cancel_sums", 

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["(l::nat) + m < n", "(l::nat) < m + n", "Suc m < n", "m < Suc n"], LessCancelSums.proc), 

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("natle_cancel_sums", 

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["(l::nat) + m <= n", "(l::nat) <= m + n", "Suc m <= n", "m <= Suc n"], LeCancelSums.proc)]; 

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val nat_cancel_sums = nat_cancel_sums_add @ 
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[prep_simproc ("natdiff_cancel_sums", 
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["((l::nat) + m)  n", "(l::nat)  (m + n)", "Suc m  n", "m  Suc n"], DiffCancelSums.proc)]; 

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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  205 
structure ArithTheoryDataArgs = 
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struct 
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val name = "HOL/arith"; 
15185  208 
type T = {splits: thm list, inj_consts: (string * typ)list, discrete: string list, presburger: (int > tactic) option}; 
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val empty = {splits = [], inj_consts = [], discrete = [], presburger = None}; 
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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, presburger= presburger1}, 
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{splits= splits2, inj_consts= inj_consts2, discrete= discrete2, presburger= presburger2}) = 
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{splits = Drule.merge_rules (splits1, splits2), 
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inj_consts = merge_lists inj_consts1 inj_consts2, 
15185  217 
discrete = merge_lists discrete1 discrete2, 
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presburger = (case presburger1 of None => presburger2  p => p)}; 
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fun print _ _ = (); 
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end; 
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9593  222 
structure ArithTheoryData = TheoryDataFun(ArithTheoryDataArgs); 
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fun arith_split_add (thy, thm) = (ArithTheoryData.map (fn {splits,inj_consts,discrete,presburger} => 
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{splits= thm::splits, inj_consts= inj_consts, discrete= discrete, presburger= presburger}) thy, thm); 
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fun arith_discrete d = ArithTheoryData.map (fn {splits,inj_consts,discrete,presburger} => 
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{splits = splits, inj_consts = inj_consts, discrete = d :: discrete, presburger= presburger}); 
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fun arith_inj_const c = ArithTheoryData.map (fn {splits,inj_consts,discrete,presburger} => 
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{splits = splits, inj_consts = c :: inj_consts, discrete = discrete, presburger = presburger}); 
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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  243 
 Some n => (overwrite(p,(t,ratadd(n,m))), i)); 
244 

245 
exception Zero; 

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10693  247 
fun rat_of_term(numt,dent) = 
248 
let val num = HOLogic.dest_binum numt and den = HOLogic.dest_binum dent 

249 
in if den = 0 then raise Zero else int_ratdiv(num,den) end; 

10718  250 

251 
(* Warning: in rare cases number_of encloses a nonnumeral, 

252 
in which case dest_binum raises TERM; hence all the handles below. 

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Same for Sucterms that turn out not to be numerals  
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although the simplifier should eliminate those anyway... 
10718  255 
*) 
256 

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fun number_of_Sucs (Const("Suc",_) $ n) = number_of_Sucs n + 1 
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 number_of_Sucs t = if HOLogic.is_zero t then 0 
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else raise TERM("number_of_Sucs",[]) 
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10718  261 
(* decompose nested multiplications, bracketing them to the right and combining all 
262 
their coefficients 

263 
*) 

264 

13499  265 
fun demult inj_consts = 
266 
let 

10718  267 
fun demult((mC as Const("op *",_)) $ s $ t,m) = ((case s of 
268 
Const("Numeral.number_of",_)$n 

269 
=> demult(t,ratmul(m,rat_of_int(HOLogic.dest_binum n))) 

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 Const("uminus",_)$(Const("Numeral.number_of",_)$n) 
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=> demult(t,ratmul(m,rat_of_int(~(HOLogic.dest_binum n)))) 
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 Const("Suc",_) $ _ 
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=> demult(t,ratmul(m,rat_of_int(number_of_Sucs s))) 
10718  274 
 Const("op *",_) $ s1 $ s2 => demult(mC $ s1 $ (mC $ s2 $ t),m) 
275 
 Const("HOL.divide",_) $ numt $ (Const("Numeral.number_of",_)$dent) => 

276 
let val den = HOLogic.dest_binum dent 

277 
in if den = 0 then raise Zero 

278 
else demult(mC $ numt $ t,ratmul(m, ratinv(rat_of_int den))) 

279 
end 

280 
 _ => atomult(mC,s,t,m) 

281 
) handle TERM _ => atomult(mC,s,t,m)) 

282 
 demult(atom as Const("HOL.divide",_) $ t $ (Const("Numeral.number_of",_)$dent), m) = 

283 
(let val den = HOLogic.dest_binum dent 

284 
in if den = 0 then raise Zero else demult(t,ratmul(m, ratinv(rat_of_int den))) end 

285 
handle TERM _ => (Some atom,m)) 

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 demult(Const("0",_),m) = (None, rat_of_int 0) 
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 demult(Const("1",_),m) = (None, m) 
10718  288 
 demult(t as Const("Numeral.number_of",_)$n,m) = 
289 
((None,ratmul(m,rat_of_int(HOLogic.dest_binum n))) 

290 
handle TERM _ => (Some t,m)) 

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 demult(Const("uminus",_)$t, m) = demult(t,ratmul(m,rat_of_int(~1))) 
13499  292 
 demult(t as Const f $ x, m) = 
293 
(if f mem inj_consts then Some x else Some t,m) 

10718  294 
 demult(atom,m) = (Some atom,m) 
295 

296 
and atomult(mC,atom,t,m) = (case demult(t,m) of (None,m') => (Some atom,m') 

297 
 (Some t',m') => (Some(mC $ atom $ t'),m')) 

13499  298 
in demult end; 
10718  299 

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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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if nT T then add_atom(all,m,pi) 
10693  306 
else poly(s,m,poly(t,ratneg m,pi)) 
307 
 poly(Const("uminus",_) $ t, m, pi) = poly(t,ratneg m,pi) 

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 poly(Const("0",_), _, pi) = pi 
11464  309 
 poly(Const("1",_), m, (p,i)) = (p,ratadd(i,m)) 
10693  310 
 poly(Const("Suc",_)$t, m, (p,i)) = poly(t, m, (p,ratadd(i,m))) 
10718  311 
 poly(t as Const("op *",_) $ _ $ _, m, pi as (p,i)) = 
13499  312 
(case demult inj_consts (t,m) of 
10718  313 
(None,m') => (p,ratadd(i,m)) 
314 
 (Some u,m') => add_atom(u,m',pi)) 

315 
 poly(t as Const("HOL.divide",_) $ _ $ _, m, pi as (p,i)) = 

13499  316 
(case demult inj_consts (t,m) of 
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(None,m') => (p,ratadd(i,m')) 
10718  318 
 (Some u,m') => add_atom(u,m',pi)) 
319 
 poly(all as (Const("Numeral.number_of",_)$t,m,(p,i))) = 

320 
((p,ratadd(i,ratmul(m,rat_of_int(HOLogic.dest_binum t)))) 

321 
handle TERM _ => add_atom all) 

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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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 poly x = add_atom x; 
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325 

10718  326 
val (p,i) = poly(lhs,rat_of_int 1,([],rat_of_int 0)) 
327 
and (q,j) = poly(rhs,rat_of_int 1,([],rat_of_int 0)) 

10693  328 

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in case rel of 
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"op <" => Some(p,i,"<",q,j) 
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 "op <=" => Some(p,i,"<=",q,j) 
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 "op =" => Some(p,i,"=",q,j) 
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 _ => None 
10693  334 
end handle Zero => None; 
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14509  336 
fun negate(Some(x,i,rel,y,j,d)) = Some(x,i,"~"^rel,y,j,d) 
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 negate None = None; 
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fun of_lin_arith_sort sg U = 
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Type.of_sort (Sign.tsig_of sg) (U,["Ring_and_Field.ordered_idom"]) 
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fun allows_lin_arith sg discrete (U as Type(D,[])) = 
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if of_lin_arith_sort sg U 
15185  344 
then (true, D mem discrete) 
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else (* special cases *) 
15185  346 
if D mem discrete then (true,true) else (false,false) 
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 allows_lin_arith sg discrete U = (of_lin_arith_sort sg U, false); 
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fun decomp1 (sg,discrete,inj_consts) (T,xxx) = 
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(case T of 
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Type("fun",[U,_]) => 
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(case allows_lin_arith sg discrete U of 
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(true,d) => (case decomp2 inj_consts xxx of None => None 
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 Some(p,i,rel,q,j) => Some(p,i,rel,q,j,d)) 
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355 
 (false,_) => None) 
9436
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356 
 _ => None); 
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357 

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358 
fun decomp2 data (_$(Const(rel,T)$lhs$rhs)) = decomp1 data (T,(rel,lhs,rhs)) 
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Linear arithmetic now copes with mixed nat/int formulae.
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359 
 decomp2 data (_$(Const("Not",_)$(Const(rel,T)$lhs$rhs))) = 
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Linear arithmetic now copes with mixed nat/int formulae.
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360 
negate(decomp1 data (T,(rel,lhs,rhs))) 
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Linear arithmetic now copes with mixed nat/int formulae.
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361 
 decomp2 data _ = None 
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362 

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363 
fun decomp sg = 
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364 
let val {discrete, inj_consts, ...} = ArithTheoryData.get_sg sg 
15121
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Initial changes to extend arithmetic from individual types to type classes.
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365 
in decomp2 (sg,discrete,inj_consts) end 
9436
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366 

10693  367 
fun number_of(n,T) = HOLogic.number_of_const T $ (HOLogic.mk_bin n) 
368 

9436
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369 
end; 
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370 

62bb04ab4b01
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371 

62bb04ab4b01
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372 
structure Fast_Arith = 
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373 
Fast_Lin_Arith(structure LA_Logic=LA_Logic and LA_Data=LA_Data_Ref); 
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374 

13499  375 
val fast_arith_tac = Fast_Arith.lin_arith_tac false 
376 
and fast_ex_arith_tac = Fast_Arith.lin_arith_tac 

14517  377 
and trace_arith = Fast_Arith.trace 
378 
and fast_arith_neq_limit = Fast_Arith.fast_arith_neq_limit; 

9436
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379 

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380 
local 
62bb04ab4b01
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381 

13902
b41fc9a31975
added isolate_Suc to counteract deficiencies in one of Larry's simprocs.
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382 
val isolateSuc = 
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383 
let val thy = theory "Nat" 
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384 
in prove_goal thy "Suc(i+j) = i+j + Suc 0" 
b41fc9a31975
added isolate_Suc to counteract deficiencies in one of Larry's simprocs.
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385 
(fn _ => [simp_tac (simpset_of thy) 1]) 
b41fc9a31975
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386 
end; 
b41fc9a31975
added isolate_Suc to counteract deficiencies in one of Larry's simprocs.
nipkow
parents:
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changeset

387 

9436
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388 
(* reduce contradictory <= to False. 
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389 
Most of the work is done by the cancel tactics. 
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390 
*) 
12931
2c0251fada94
solved the problem that Larry's simproce cancle_numerals(?) returns
nipkow
parents:
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diff
changeset

391 
val add_rules = 
14368
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converted Real/Lubs to Isar script. Converting arithmetic setup
paulson
parents:
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392 
[add_zero_left,add_zero_right,Zero_not_Suc,Suc_not_Zero,le_0_eq, 
15184  393 
One_nat_def,isolateSuc, 
394 
order_less_irrefl]; 

9436
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395 

14368
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converted Real/Lubs to Isar script. Converting arithmetic setup
paulson
parents:
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diff
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396 
val add_mono_thms_ordered_semiring = map (fn s => prove_goal (the_context ()) s 
9436
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397 
(fn prems => [cut_facts_tac prems 1, 
14368
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paulson
parents:
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diff
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398 
blast_tac (claset() addIs [add_mono]) 1])) 
15121
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Initial changes to extend arithmetic from individual types to type classes.
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parents:
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399 
["(i <= j) & (k <= l) ==> i + k <= j + (l::'a::pordered_ab_semigroup_add)", 
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Initial changes to extend arithmetic from individual types to type classes.
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400 
"(i = j) & (k <= l) ==> i + k <= j + (l::'a::pordered_ab_semigroup_add)", 
1198032bad25
Initial changes to extend arithmetic from individual types to type classes.
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parents:
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changeset

401 
"(i <= j) & (k = l) ==> i + k <= j + (l::'a::pordered_ab_semigroup_add)", 
1198032bad25
Initial changes to extend arithmetic from individual types to type classes.
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parents:
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changeset

402 
"(i = j) & (k = l) ==> i + k = j + (l::'a::pordered_ab_semigroup_add)" 
9436
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403 
]; 
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wenzelm
parents:
diff
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404 

15121
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Initial changes to extend arithmetic from individual types to type classes.
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parents:
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diff
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405 
val mono_ss = simpset() addsimps 
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Initial changes to extend arithmetic from individual types to type classes.
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parents:
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406 
[add_mono,add_strict_mono,add_less_le_mono,add_le_less_mono]; 
1198032bad25
Initial changes to extend arithmetic from individual types to type classes.
nipkow
parents:
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diff
changeset

407 

1198032bad25
Initial changes to extend arithmetic from individual types to type classes.
nipkow
parents:
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diff
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408 
val add_mono_thms_ordered_field = 
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diff
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409 
map (fn s => prove_goal (the_context ()) s 
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Initial changes to extend arithmetic from individual types to type classes.
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parents:
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410 
(fn prems => [cut_facts_tac prems 1, asm_simp_tac mono_ss 1])) 
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parents:
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411 
["(i<j) & (k=l) ==> i+k < j+(l::'a::pordered_cancel_ab_semigroup_add)", 
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Initial changes to extend arithmetic from individual types to type classes.
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parents:
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412 
"(i=j) & (k<l) ==> i+k < j+(l::'a::pordered_cancel_ab_semigroup_add)", 
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parents:
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413 
"(i<j) & (k<=l) ==> i+k < j+(l::'a::pordered_cancel_ab_semigroup_add)", 
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parents:
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414 
"(i<=j) & (k<l) ==> i+k < j+(l::'a::pordered_cancel_ab_semigroup_add)", 
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Initial changes to extend arithmetic from individual types to type classes.
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parents:
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415 
"(i<j) & (k<l) ==> i+k < j+(l::'a::pordered_cancel_ab_semigroup_add)"]; 
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Initial changes to extend arithmetic from individual types to type classes.
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parents:
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416 

9436
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wenzelm
parents:
diff
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417 
in 
62bb04ab4b01
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wenzelm
parents:
diff
changeset

418 

62bb04ab4b01
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diff
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419 
val init_lin_arith_data = 
62bb04ab4b01
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wenzelm
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420 
Fast_Arith.setup @ 
10693  421 
[Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, simpset = _} => 
15121
1198032bad25
Initial changes to extend arithmetic from individual types to type classes.
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parents:
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422 
{add_mono_thms = add_mono_thms @ 
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Initial changes to extend arithmetic from individual types to type classes.
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parents:
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diff
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423 
add_mono_thms_ordered_semiring @ add_mono_thms_ordered_field, 
10693  424 
mult_mono_thms = mult_mono_thms, 
10574
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diff
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425 
inj_thms = inj_thms, 
9436
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426 
lessD = lessD @ [Suc_leI], 
62bb04ab4b01
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diff
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427 
simpset = HOL_basic_ss addsimps add_rules addsimprocs nat_cancel_sums_add}), 
15185  428 
ArithTheoryData.init, arith_discrete "nat"]; 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
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diff
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429 

62bb04ab4b01
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wenzelm
parents:
diff
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430 
end; 
62bb04ab4b01
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wenzelm
parents:
diff
changeset

431 

13462  432 
val fast_nat_arith_simproc = 
433 
Simplifier.simproc (Theory.sign_of (the_context ())) "fast_nat_arith" 

434 
["(m::nat) < n","(m::nat) <= n", "(m::nat) = n"] Fast_Arith.lin_arith_prover; 

9436
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parents:
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435 

62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

436 

62bb04ab4b01
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wenzelm
parents:
diff
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437 
(* Because of fast_nat_arith_simproc, the arithmetic solver is really only 
62bb04ab4b01
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wenzelm
parents:
diff
changeset

438 
useful to detect inconsistencies among the premises for subgoals which are 
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

439 
*not* themselves (in)equalities, because the latter activate 
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

440 
fast_nat_arith_simproc anyway. However, it seems cheaper to activate the 
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

441 
solver all the time rather than add the additional check. *) 
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

442 

62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

443 

62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

444 
(* arith proof method *) 
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

445 

62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

446 
(* FIXME: K true should be replaced by a sensible test to speed things up 
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

447 
in case there are lots of irrelevant terms involved; 
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

448 
elimination of min/max can be optimized: 
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

449 
(max m n + k <= r) = (m+k <= r & n+k <= r) 
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

450 
(l <= min m n + k) = (l <= m+k & l <= n+k) 
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wenzelm
parents:
diff
changeset

451 
*) 
10516  452 
local 
453 

13499  454 
fun raw_arith_tac ex i st = 
455 
refute_tac (K true) 

456 
(REPEAT o split_tac (#splits (ArithTheoryData.get_sg (Thm.sign_of_thm st)))) 

14509  457 
((REPEAT_DETERM o etac linorder_neqE) THEN' fast_ex_arith_tac ex) 
458 
i st; 

9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

459 

13877
a6b825ee48d9
Added hook for presburger arithmetic decision procedure.
berghofe
parents:
13517
diff
changeset

460 
fun presburger_tac i st = 
a6b825ee48d9
Added hook for presburger arithmetic decision procedure.
berghofe
parents:
13517
diff
changeset

461 
(case ArithTheoryData.get_sg (sign_of_thm st) of 
a6b825ee48d9
Added hook for presburger arithmetic decision procedure.
berghofe
parents:
13517
diff
changeset

462 
{presburger = Some tac, ...} => 
a6b825ee48d9
Added hook for presburger arithmetic decision procedure.
berghofe
parents:
13517
diff
changeset

463 
(tracing "Simple arithmetic decision procedure failed.\nNow trying full Presburger arithmetic..."; tac i st) 
a6b825ee48d9
Added hook for presburger arithmetic decision procedure.
berghofe
parents:
13517
diff
changeset

464 
 _ => no_tac st); 
a6b825ee48d9
Added hook for presburger arithmetic decision procedure.
berghofe
parents:
13517
diff
changeset

465 

10516  466 
in 
467 

13877
a6b825ee48d9
Added hook for presburger arithmetic decision procedure.
berghofe
parents:
13517
diff
changeset

468 
val simple_arith_tac = FIRST' [fast_arith_tac, 
a6b825ee48d9
Added hook for presburger arithmetic decision procedure.
berghofe
parents:
13517
diff
changeset

469 
ObjectLogic.atomize_tac THEN' raw_arith_tac true]; 
a6b825ee48d9
Added hook for presburger arithmetic decision procedure.
berghofe
parents:
13517
diff
changeset

470 

a6b825ee48d9
Added hook for presburger arithmetic decision procedure.
berghofe
parents:
13517
diff
changeset

471 
val arith_tac = FIRST' [fast_arith_tac, 
a6b825ee48d9
Added hook for presburger arithmetic decision procedure.
berghofe
parents:
13517
diff
changeset

472 
ObjectLogic.atomize_tac THEN' raw_arith_tac true, 
a6b825ee48d9
Added hook for presburger arithmetic decision procedure.
berghofe
parents:
13517
diff
changeset

473 
presburger_tac]; 
a6b825ee48d9
Added hook for presburger arithmetic decision procedure.
berghofe
parents:
13517
diff
changeset

474 

a6b825ee48d9
Added hook for presburger arithmetic decision procedure.
berghofe
parents:
13517
diff
changeset

475 
val silent_arith_tac = FIRST' [fast_arith_tac, 
a6b825ee48d9
Added hook for presburger arithmetic decision procedure.
berghofe
parents:
13517
diff
changeset

476 
ObjectLogic.atomize_tac THEN' raw_arith_tac false, 
a6b825ee48d9
Added hook for presburger arithmetic decision procedure.
berghofe
parents:
13517
diff
changeset

477 
presburger_tac]; 
10516  478 

9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

479 
fun arith_method prems = 
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

480 
Method.METHOD (fn facts => HEADGOAL (Method.insert_tac (prems @ facts) THEN' arith_tac)); 
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

481 

10516  482 
end; 
483 

15195  484 
(* antisymmetry: 
485 
combines x <= y (or ~(y < x)) and y <= x (or ~(x < y)) into x = y *) 

486 

487 
local 

488 
val antisym = mk_meta_eq order_antisym 

489 
val not_lessD = linorder_not_less RS iffD1 

490 
fun prp t thm = (#prop(rep_thm thm) = t) 

491 
in 

492 
fun antisym_eq prems thm = 

493 
let 

494 
val r = #prop(rep_thm thm); 

495 
in 

496 
case r of 

497 
Tr $ ((c as Const("op <=",T)) $ s $ t) => 

498 
let val r' = Tr $ (c $ t $ s) 

499 
in 

500 
case Library.find_first (prp r') prems of 

501 
None => 

502 
let val r' = Tr $ (HOLogic.not_const $ (Const("op <",T) $ s $ t)) 

503 
in case Library.find_first (prp r') prems of 

504 
None => [] 

505 
 Some thm' => [(thm' RS not_lessD) RS (thm RS antisym)] 

506 
end 

507 
 Some thm' => [thm' RS (thm RS antisym)] 

508 
end 

509 
 Tr $ (Const("Not",_) $ (Const("op <",T) $ s $ t)) => 

510 
let val r' = Tr $ (Const("op <=",T) $ s $ t) 

511 
in 

512 
case Library.find_first (prp r') prems of 

513 
None => 

514 
let val r' = Tr $ (HOLogic.not_const $ (Const("op <",T) $ t $ s)) 

515 
in case Library.find_first (prp r') prems of 

516 
None => [] 

517 
 Some thm' => 

518 
[(thm' RS not_lessD) RS ((thm RS not_lessD) RS antisym)] 

519 
end 

520 
 Some thm' => [thm' RS ((thm RS not_lessD) RS antisym)] 

521 
end 

522 
 _ => [] 

523 
end 

524 
handle THM _ => [] 

525 
end; 

526 

9436
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rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

527 

62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

528 
(* theory setup *) 
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

529 

62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

530 
val arith_setup = 
15195  531 
[Simplifier.change_simpset_of (op setmksimps2) antisym_eq, 
532 
Simplifier.change_simpset_of (op addsimprocs) nat_cancel_sums] @ 

9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

533 
init_lin_arith_data @ 
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

534 
[Simplifier.change_simpset_of (op addSolver) 
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

535 
(mk_solver "lin. arith." Fast_Arith.cut_lin_arith_tac), 
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

536 
Simplifier.change_simpset_of (op addsimprocs) [fast_nat_arith_simproc], 
11704
3c50a2cd6f00
* sane numerals (stage 2): plain "num" syntax (removed "#");
wenzelm
parents:
11701
diff
changeset

537 
Method.add_methods [("arith", (arith_method o #2) oo Method.syntax Args.bang_facts, 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

538 
"decide linear arithmethic")], 
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

539 
Attrib.add_attributes [("arith_split", 
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

540 
(Attrib.no_args arith_split_add, Attrib.no_args Attrib.undef_local_attribute), 
9893  541 
"declaration of split rules for arithmetic procedure")]]; 