author  haftmann 
Fri, 20 Jul 2007 14:28:25 +0200  
changeset 23881  851c74f1bb69 
parent 23530  438c5d2db482 
child 24076  ae946f751c44 
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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22997  19 
val mk_plus = HOLogic.mk_binop @{const_name HOL.plus}; 
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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) = List.partition (equal HOLogic.Suc_zero) 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 @{const_name HOL.plus} 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 => HOLogic.Suc_zero :: 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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fun prove_conv expand_tac norm_tac ss tu = (* FIXME avoid standard *) 
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mk_meta_eq (standard (Goal.prove (Simplifier.the_context ss) [] [] 
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(HOLogic.mk_Trueprop (HOLogic.mk_eq tu)) 
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(K (EVERY [expand_tac, norm_tac ss])))); 
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(* rewriting *) 
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fun simp_all_tac rules = 
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let val ss0 = HOL_ss addsimps rules 

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in fn ss => ALLGOALS (simp_tac (Simplifier.inherit_context ss ss0)) end; 

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13517  61 
fun prep_simproc (name, pats, proc) = 
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Simplifier.simproc (the_context ()) name pats proc; 
13517  63 

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end; (* NatArithUtils *) 
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13517  66 

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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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13517  74 
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_tac1 = simp_all_tac [@{thm "add_Suc"}, @{thm "add_Suc_right"}, 
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@{thm "add_0"}, @{thm "add_0_right"}]; 

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val norm_tac2 = simp_all_tac @{thms add_ac}; 
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fun norm_tac ss = norm_tac1 ss THEN norm_tac2 ss; 
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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 @{thm 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 @{thm "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 @{const_name HOL.less}; 
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val dest_bal = HOLogic.dest_bin @{const_name HOL.less} HOLogic.natT; 

22838  112 
val uncancel_tac = gen_uncancel_tac @{thm "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 @{const_name HOL.less_eq}; 
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val dest_bal = HOLogic.dest_bin @{const_name HOL.less_eq} HOLogic.natT; 

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val uncancel_tac = gen_uncancel_tac @{thm "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 @{const_name HOL.minus}; 
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val dest_bal = HOLogic.dest_bin @{const_name HOL.minus} HOLogic.natT; 

22838  132 
val uncancel_tac = gen_uncancel_tac @{thm "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", 
20268  139 
["(l::nat) + m = n", "(l::nat) = m + n", "Suc m = n", "m = Suc n"], 
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K EqCancelSums.proc), 

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("natless_cancel_sums", 
20268  142 
["(l::nat) + m < n", "(l::nat) < m + n", "Suc m < n", "m < Suc n"], 
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K 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"], 
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K 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"], 
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K DiffCancelSums.proc)]; 

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end; (* ArithData *) 
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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 notI = notI; 
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val sym = sym; 
22548  171 
val not_lessD = @{thm linorder_not_less} RS iffD1; 
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val not_leD = @{thm linorder_not_le} RS iffD1; 

21243  173 
val le0 = thm "le0"; 
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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 atomize thm = case #prop(rep_thm thm) of 
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Const("Trueprop",_) $ (Const("op &",_) $ _ $ _) => 
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atomize(thm RS conjunct1) @ atomize(thm RS conjunct2) 
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 _ => [thm]; 
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23190  184 
fun neg_prop ((TP as Const("Trueprop",_)) $ (Const("Not",_) $ t)) = TP $ t 
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 neg_prop ((TP as Const("Trueprop",_)) $ t) = TP $ (HOLogic.Not $t) 

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 neg_prop t = raise TERM ("neg_prop", [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; (* LA_Logic *) 
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(* arith theory data *) 
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202 

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datatype arithtactic = ArithTactic of {name: string, tactic: int > tactic, id: stamp}; 
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fun mk_arith_tactic name tactic = ArithTactic {name = name, tactic = tactic, id = stamp ()}; 
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fun eq_arith_tactic (ArithTactic {id = id1, ...}, ArithTactic {id = id2, ...}) = (id1 = id2); 
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16424  209 
structure ArithTheoryData = TheoryDataFun 
22846  210 
( 
20268  211 
type T = {splits: thm list, 
212 
inj_consts: (string * typ) list, 

213 
discrete: string list, 

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tactics: arithtactic list}; 
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val empty = {splits = [], inj_consts = [], discrete = [], tactics = []}; 
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val copy = I; 
16424  217 
val extend = I; 
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fun merge _ ({splits= splits1, inj_consts= inj_consts1, discrete= discrete1, tactics= tactics1}, 
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{splits= splits2, inj_consts= inj_consts2, discrete= discrete2, tactics= tactics2}) = 
22634  220 
{splits = Library.merge Thm.eq_thm_prop (splits1, splits2), 
221 
inj_consts = Library.merge (op =) (inj_consts1, inj_consts2), 

222 
discrete = Library.merge (op =) (discrete1, discrete2), 

223 
tactics = Library.merge eq_arith_tactic (tactics1, tactics2)}; 

22846  224 
); 
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18728  226 
val arith_split_add = Thm.declaration_attribute (fn thm => 
20897  227 
Context.mapping (ArithTheoryData.map (fn {splits,inj_consts,discrete,tactics} => 
22634  228 
{splits= insert Thm.eq_thm_prop thm splits, inj_consts= inj_consts, discrete= discrete, tactics= tactics})) I); 
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fun arith_discrete d = ArithTheoryData.map (fn {splits,inj_consts,discrete,tactics} => 
22634  231 
{splits = splits, inj_consts = inj_consts, discrete = insert (op =) d discrete, tactics= tactics}); 
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fun arith_inj_const c = ArithTheoryData.map (fn {splits,inj_consts,discrete,tactics} => 
22634  234 
{splits = splits, inj_consts = insert (op =) c inj_consts, discrete = discrete, tactics= tactics}); 
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fun arith_tactic_add tac = ArithTheoryData.map (fn {splits,inj_consts,discrete,tactics} => 
22634  237 
{splits= splits, inj_consts= inj_consts, discrete= discrete, tactics= insert eq_arith_tactic tac tactics}); 
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238 

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signature HOL_LIN_ARITH_DATA = 
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sig 
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include LIN_ARITH_DATA 
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val fast_arith_split_limit : int ref 
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end; 
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structure LA_Data_Ref: HOL_LIN_ARITH_DATA = 
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struct 
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(* internal representation of linear (in)equations *) 
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type decompT = ((term * Rat.rat) list * Rat.rat * string * (term * Rat.rat) list * Rat.rat * bool); 
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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 : term) (m : Rat.rat) (p : (term * Rat.rat) list, i : Rat.rat) : 
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(term * Rat.rat) list * Rat.rat = 
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case AList.lookup (op =) p t of NONE => ((t, m) :: p, i) 
22947  260 
 SOME n => (AList.update (op =) (t, Rat.add n m) p, i); 
10693  261 

262 
exception Zero; 

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fun rat_of_term (numt, dent) = 
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let 
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val num = HOLogic.dest_numeral numt 
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val den = HOLogic.dest_numeral dent 
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in 
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if den = 0 then raise Zero else Rat.rat_of_quotient (num, den) 
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270 
end; 
10718  271 

272 
(* Warning: in rare cases number_of encloses a nonnumeral, 

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in which case dest_numeral 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  276 
*) 
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fun number_of_Sucs (Const ("Suc", _) $ n) : int = 
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number_of_Sucs n + 1 
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 number_of_Sucs t = 
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if HOLogic.is_zero t then 0 else raise TERM ("number_of_Sucs", []); 
10718  281 

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(* decompose nested multiplications, bracketing them to the right and combining 
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all their coefficients 
10718  284 
*) 
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fun demult (inj_consts : (string * typ) list) : term * Rat.rat > term option * Rat.rat = 
13499  286 
let 
22997  287 
fun demult ((mC as Const (@{const_name HOL.times}, _)) $ s $ t, m) = ( 
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(case s of 
22997  289 
Const ("Numeral.number_class.number_of", _) $ n => 
22947  290 
demult (t, Rat.mult m (Rat.rat_of_int (HOLogic.dest_numeral n))) 
22997  291 
 Const (@{const_name HOL.uminus}, _) $ (Const ("Numeral.number_class.number_of", _) $ n) => 
22947  292 
demult (t, Rat.mult m (Rat.rat_of_int (~(HOLogic.dest_numeral n)))) 
22997  293 
 Const (@{const_name Suc}, _) $ _ => 
22947  294 
demult (t, Rat.mult m (Rat.rat_of_int (HOLogic.dest_nat s))) 
22997  295 
 Const (@{const_name HOL.times}, _) $ s1 $ s2 => 
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demult (mC $ s1 $ (mC $ s2 $ t), m) 
22997  297 
 Const (@{const_name HOL.divide}, _) $ numt $ (Const ("Numeral.number_class.number_of", _) $ dent) => 
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let 
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val den = HOLogic.dest_numeral dent 
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in 
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301 
if den = 0 then 
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raise Zero 
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else 
22947  304 
demult (mC $ numt $ t, Rat.mult m (Rat.inv (Rat.rat_of_int den))) 
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end 
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306 
 _ => 
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307 
atomult (mC, s, t, m) 
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) handle TERM _ => atomult (mC, s, t, m) 
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309 
) 
22997  310 
 demult (atom as Const(@{const_name HOL.divide}, _) $ t $ (Const ("Numeral.number_class.number_of", _) $ dent), m) = 
20268  311 
(let 
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val den = HOLogic.dest_numeral dent 
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313 
in 
20268  314 
if den = 0 then 
315 
raise Zero 

316 
else 

22947  317 
demult (t, Rat.mult m (Rat.inv (Rat.rat_of_int den))) 
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318 
end 
20268  319 
handle TERM _ => (SOME atom, m)) 
22997  320 
 demult (Const (@{const_name HOL.zero}, _), m) = (NONE, Rat.zero) 
321 
 demult (Const (@{const_name HOL.one}, _), m) = (NONE, m) 

322 
 demult (t as Const ("Numeral.number_class.number_of", _) $ n, m) = 

22947  323 
((NONE, Rat.mult m (Rat.rat_of_int (HOLogic.dest_numeral n))) 
324 
handle TERM _ => (SOME t, m)) 

22997  325 
 demult (Const (@{const_name HOL.uminus}, _) $ t, m) = demult (t, Rat.neg m) 
20268  326 
 demult (t as Const f $ x, m) = 
22997  327 
(if member (op =) inj_consts f then SOME x else SOME t, m) 
20268  328 
 demult (atom, m) = (SOME atom, m) 
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329 
and 
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330 
atomult (mC, atom, t, m) = ( 
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331 
case demult (t, m) of (NONE, m') => (SOME atom, m') 
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332 
 (SOME t', m') => (SOME (mC $ atom $ t'), m') 
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333 
) 
13499  334 
in demult end; 
10718  335 

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336 
fun decomp0 (inj_consts : (string * typ) list) (rel : string, lhs : term, rhs : term) : 
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337 
((term * Rat.rat) list * Rat.rat * string * (term * Rat.rat) list * Rat.rat) option = 
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338 
let 
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339 
(* Turn term into list of summand * multiplicity plus a constant *) 
22997  340 
fun poly (Const (@{const_name HOL.plus}, _) $ s $ t, m : Rat.rat, pi : (term * Rat.rat) list * Rat.rat) = 
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341 
poly (s, m, poly (t, m, pi)) 
22997  342 
 poly (all as Const (@{const_name HOL.minus}, T) $ s $ t, m, pi) = 
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343 
if nT T then add_atom all m pi else poly (s, m, poly (t, Rat.neg m, pi)) 
22997  344 
 poly (all as Const (@{const_name HOL.uminus}, T) $ t, m, pi) = 
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345 
if nT T then add_atom all m pi else poly (t, Rat.neg m, pi) 
22997  346 
 poly (Const (@{const_name HOL.zero}, _), _, pi) = 
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347 
pi 
22997  348 
 poly (Const (@{const_name HOL.one}, _), m, (p, i)) = 
22947  349 
(p, Rat.add i m) 
22997  350 
 poly (Const (@{const_name Suc}, _) $ t, m, (p, i)) = 
22947  351 
poly (t, m, (p, Rat.add i m)) 
22997  352 
 poly (all as Const (@{const_name HOL.times}, _) $ _ $ _, m, pi as (p, i)) = 
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353 
(case demult inj_consts (all, m) of 
22947  354 
(NONE, m') => (p, Rat.add i m') 
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355 
 (SOME u, m') => add_atom u m' pi) 
22997  356 
 poly (all as Const (@{const_name HOL.divide}, _) $ _ $ _, m, pi as (p, i)) = 
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357 
(case demult inj_consts (all, m) of 
22947  358 
(NONE, m') => (p, Rat.add i m') 
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359 
 (SOME u, m') => add_atom u m' pi) 
22997  360 
 poly (all as Const ("Numeral.number_class.number_of", Type(_,[_,T])) $ t, m, pi as (p, i)) = 
21820
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introduced mk/dest_numeral/number for mk/dest_binum etc.
haftmann
parents:
21621
diff
changeset

361 
(let val k = HOLogic.dest_numeral t 
20859  362 
val k2 = if k < 0 andalso T = HOLogic.natT then 0 else k 
22947  363 
in (p, Rat.add i (Rat.mult m (Rat.rat_of_int k2))) end 
20859  364 
handle TERM _ => add_atom all m pi) 
20271
e76e77e0d615
fixed a bug in function poly: decomposition of products
webertj
parents:
20268
diff
changeset

365 
 poly (all as Const f $ x, m, pi) = 
e76e77e0d615
fixed a bug in function poly: decomposition of products
webertj
parents:
20268
diff
changeset

366 
if f mem inj_consts then poly (x, m, pi) else add_atom all m pi 
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fixed a bug in function poly: decomposition of products
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parents:
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diff
changeset

367 
 poly (all, m, pi) = 
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fixed a bug in function poly: decomposition of products
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diff
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368 
add_atom all m pi 
22947  369 
val (p, i) = poly (lhs, Rat.one, ([], Rat.zero)) 
370 
val (q, j) = poly (rhs, Rat.one, ([], Rat.zero)) 

20254
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371 
in 
58b71535ed00
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372 
case rel of 
23881  373 
@{const_name HOL.less} => SOME (p, i, "<", q, j) 
374 
 @{const_name HOL.less_eq} => SOME (p, i, "<=", q, j) 

20254
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375 
 "op =" => SOME (p, i, "=", q, j) 
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376 
 _ => NONE 
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377 
end handle Zero => NONE; 
9436
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rearranged setup of arithmetic procedures, avoiding global reference values;
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parents:
diff
changeset

378 

20271
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fixed a bug in function poly: decomposition of products
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379 
fun of_lin_arith_sort sg (U : typ) : bool = 
20254
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380 
Type.of_sort (Sign.tsig_of sg) (U, ["Ring_and_Field.ordered_idom"]) 
15121
1198032bad25
Initial changes to extend arithmetic from individual types to type classes.
nipkow
parents:
14738
diff
changeset

381 

20271
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fixed a bug in function poly: decomposition of products
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382 
fun allows_lin_arith sg (discrete : string list) (U as Type (D, [])) : bool * bool = 
20254
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383 
if of_lin_arith_sort sg U then 
58b71535ed00
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384 
(true, D mem discrete) 
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385 
else (* special cases *) 
20271
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fixed a bug in function poly: decomposition of products
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386 
if D mem discrete then (true, true) else (false, false) 
20254
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387 
 allows_lin_arith sg discrete U = 
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388 
(of_lin_arith_sort sg U, false); 
15121
1198032bad25
Initial changes to extend arithmetic from individual types to type classes.
nipkow
parents:
14738
diff
changeset

389 

20271
e76e77e0d615
fixed a bug in function poly: decomposition of products
webertj
parents:
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changeset

390 
fun decomp_typecheck (sg, discrete, inj_consts) (T : typ, xxx) : decompT option = 
e76e77e0d615
fixed a bug in function poly: decomposition of products
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changeset

391 
case T of 
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fixed a bug in function poly: decomposition of products
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392 
Type ("fun", [U, _]) => 
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fixed a bug in function poly: decomposition of products
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changeset

393 
(case allows_lin_arith sg discrete U of 
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fixed a bug in function poly: decomposition of products
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changeset

394 
(true, d) => 
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395 
(case decomp0 inj_consts xxx of 
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fixed a bug in function poly: decomposition of products
webertj
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396 
NONE => NONE 
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fixed a bug in function poly: decomposition of products
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parents:
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changeset

397 
 SOME (p, i, rel, q, j) => SOME (p, i, rel, q, j, d)) 
e76e77e0d615
fixed a bug in function poly: decomposition of products
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parents:
20268
diff
changeset

398 
 (false, _) => 
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fixed a bug in function poly: decomposition of products
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parents:
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diff
changeset

399 
NONE) 
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fixed a bug in function poly: decomposition of products
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parents:
20268
diff
changeset

400 
 _ => NONE; 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

401 

20271
e76e77e0d615
fixed a bug in function poly: decomposition of products
webertj
parents:
20268
diff
changeset

402 
fun negate (SOME (x, i, rel, y, j, d)) = SOME (x, i, "~" ^ rel, y, j, d) 
e76e77e0d615
fixed a bug in function poly: decomposition of products
webertj
parents:
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diff
changeset

403 
 negate NONE = NONE; 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

404 

23190  405 
fun decomp_negation data 
406 
((Const ("Trueprop", _)) $ (Const (rel, T) $ lhs $ rhs)) : decompT option = 

20271
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fixed a bug in function poly: decomposition of products
webertj
parents:
20268
diff
changeset

407 
decomp_typecheck data (T, (rel, lhs, rhs)) 
23190  408 
 decomp_negation data ((Const ("Trueprop", _)) $ 
409 
(Const ("Not", _) $ (Const (rel, T) $ lhs $ rhs))) = 

20271
e76e77e0d615
fixed a bug in function poly: decomposition of products
webertj
parents:
20268
diff
changeset

410 
negate (decomp_typecheck data (T, (rel, lhs, rhs))) 
e76e77e0d615
fixed a bug in function poly: decomposition of products
webertj
parents:
20268
diff
changeset

411 
 decomp_negation data _ = 
e76e77e0d615
fixed a bug in function poly: decomposition of products
webertj
parents:
20268
diff
changeset

412 
NONE; 
e76e77e0d615
fixed a bug in function poly: decomposition of products
webertj
parents:
20268
diff
changeset

413 

e76e77e0d615
fixed a bug in function poly: decomposition of products
webertj
parents:
20268
diff
changeset

414 
fun decomp sg : term > decompT option = 
20254
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
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parents:
20217
diff
changeset

415 
let 
58b71535ed00
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diff
changeset

416 
val {discrete, inj_consts, ...} = ArithTheoryData.get sg 
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset

417 
in 
20271
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fixed a bug in function poly: decomposition of products
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parents:
20268
diff
changeset

418 
decomp_negation (sg, discrete, inj_consts) 
20254
58b71535ed00
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parents:
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changeset

419 
end; 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

420 

20276
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20271
diff
changeset

421 
fun domain_is_nat (_ $ (Const (_, T) $ _ $ _)) = nT T 
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20271
diff
changeset

422 
 domain_is_nat (_ $ (Const ("Not", _) $ (Const (_, T) $ _ $ _))) = nT T 
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20271
diff
changeset

423 
 domain_is_nat _ = false; 
d94dc40673b1
possible disagreement between proof search and proof reconstruction when eliminating inequalities over different types fixed
webertj
parents:
20271
diff
changeset

424 

21820
2f2b6a965ccc
introduced mk/dest_numeral/number for mk/dest_binum etc.
haftmann
parents:
21621
diff
changeset

425 
fun number_of (n, T) = HOLogic.mk_number T n; 
10693  426 

20217
25b068a99d2b
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diff
changeset

427 
(**) 
25b068a99d2b
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webertj
parents:
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diff
changeset

428 
(* code that performs certain goal transformations for linear arithmetic *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
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diff
changeset

429 
(**) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
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diff
changeset

430 

25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
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diff
changeset

431 
(* A "do nothing" variant of pre_decomp and pre_tac: 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
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parents:
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diff
changeset

432 

25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
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diff
changeset

433 
fun pre_decomp sg Ts termitems = [termitems]; 
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linear arithmetic splits certain operators (e.g. min, max, abs)
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parents:
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diff
changeset

434 
fun pre_tac i = all_tac; 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
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diff
changeset

435 
*) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

436 

25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

437 
(**) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
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diff
changeset

438 
(* the following code performs splitting of certain constants (e.g. min, *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

439 
(* max) in a linear arithmetic problem; similar to what split_tac later does *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

440 
(* to the proof state *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

441 
(**) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

442 

25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
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diff
changeset

443 
val fast_arith_split_limit = ref 9; 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

444 

20268  445 
(* checks if splitting with 'thm' is implemented *) 
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
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20044
diff
changeset

446 

20268  447 
fun is_split_thm (thm : thm) : bool = 
448 
case concl_of thm of _ $ (_ $ (_ $ lhs) $ _) => ( 

449 
(* Trueprop $ ((op =) $ (?P $ lhs) $ rhs) *) 

450 
case head_of lhs of 

22997  451 
Const (a, _) => member (op =) [@{const_name Orderings.max}, 
452 
@{const_name Orderings.min}, 

453 
@{const_name HOL.abs}, 

454 
@{const_name HOL.minus}, 

20268  455 
"IntDef.nat", 
22997  456 
"Divides.div_class.mod", 
457 
"Divides.div_class.div"] a 

20268  458 
 _ => (warning ("Lin. Arith.: wrong format for split rule " ^ 
459 
Display.string_of_thm thm); 

460 
false)) 

461 
 _ => (warning ("Lin. Arith.: wrong format for split rule " ^ 

462 
Display.string_of_thm thm); 

463 
false); 

20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

464 

25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

465 
(* substitute new for occurrences of old in a term, incrementing bound *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

466 
(* variables as needed when substituting inside an abstraction *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

467 

20268  468 
fun subst_term ([] : (term * term) list) (t : term) = t 
469 
 subst_term pairs t = 

20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

470 
(case AList.lookup (op aconv) pairs t of 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

471 
SOME new => 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

472 
new 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

473 
 NONE => 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
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diff
changeset

474 
(case t of Abs (a, T, body) => 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

475 
let val pairs' = map (pairself (incr_boundvars 1)) pairs 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

476 
in Abs (a, T, subst_term pairs' body) end 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

477 
 t1 $ t2 => 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

478 
subst_term pairs t1 $ subst_term pairs t2 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

479 
 _ => t)); 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

480 

25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

481 
(* approximates the effect of one application of split_tac (followed by NNF *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

482 
(* normalization) on the subgoal represented by '(Ts, terms)'; returns a *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

483 
(* list of new subgoals (each again represented by a typ list for bound *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

484 
(* variables and a term list for premises), or NONE if split_tac would fail *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

485 
(* on the subgoal *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

486 

25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

487 
(* FIXME: currently only the effect of certain split theorems is reproduced *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

488 
(* (which is why we need 'is_split_thm'). A more canonical *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

489 
(* implementation should analyze the righthand side of the split *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

490 
(* theorem that can be applied, and modify the subgoal accordingly. *) 
20268  491 
(* Or even better, the splitter should be extended to provide *) 
492 
(* splitting on terms as well as splitting on theorems (where the *) 

493 
(* former can have a faster implementation as it does not need to be *) 

494 
(* proofproducing). *) 

20217
25b068a99d2b
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webertj
parents:
20044
diff
changeset

495 

20268  496 
fun split_once_items (sg : theory) (Ts : typ list, terms : term list) : 
497 
(typ list * term list) list option = 

20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

498 
let 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

499 
(* takes a list [t1, ..., tn] to the term *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
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diff
changeset

500 
(* tn' > ... > t1' > False , *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

501 
(* where ti' = HOLogic.dest_Trueprop ti *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

502 
(* term list > term *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

503 
fun REPEAT_DETERM_etac_rev_mp terms' = 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

504 
fold (curry HOLogic.mk_imp) (map HOLogic.dest_Trueprop terms') HOLogic.false_const 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
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diff
changeset

505 
val split_thms = filter is_split_thm (#splits (ArithTheoryData.get sg)) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

506 
val cmap = Splitter.cmap_of_split_thms split_thms 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

507 
val splits = Splitter.split_posns cmap sg Ts (REPEAT_DETERM_etac_rev_mp terms) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

508 
in 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

509 
if length splits > !fast_arith_split_limit then ( 
20268  510 
tracing ("fast_arith_split_limit exceeded (current value is " ^ 
511 
string_of_int (!fast_arith_split_limit) ^ ")"); 

20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

512 
NONE 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

513 
) else ( 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
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diff
changeset

514 
case splits of [] => 
20268  515 
(* split_tac would fail: no possible split *) 
516 
NONE 

517 
 ((_, _, _, split_type, split_term) :: _) => ( 

518 
(* ignore all but the first possible split *) 

20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

519 
case strip_comb split_term of 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

520 
(* ?P (max ?i ?j) = ((?i <= ?j > ?P ?j) & (~ ?i <= ?j > ?P ?i)) *) 
22997  521 
(Const (@{const_name Orderings.max}, _), [t1, t2]) => 
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522 
let 
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523 
val rev_terms = rev terms 
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524 
val terms1 = map (subst_term [(split_term, t1)]) rev_terms 
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525 
val terms2 = map (subst_term [(split_term, t2)]) rev_terms 
23881  526 
val t1_leq_t2 = Const (@{const_name HOL.less_eq}, 
20268  527 
split_type > split_type > HOLogic.boolT) $ t1 $ t2 
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528 
val not_t1_leq_t2 = HOLogic.Not $ t1_leq_t2 
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529 
val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const) 
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530 
val subgoal1 = (HOLogic.mk_Trueprop t1_leq_t2) :: terms2 @ [not_false] 
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531 
val subgoal2 = (HOLogic.mk_Trueprop not_t1_leq_t2) :: terms1 @ [not_false] 
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532 
in 
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533 
SOME [(Ts, subgoal1), (Ts, subgoal2)] 
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534 
end 
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535 
(* ?P (min ?i ?j) = ((?i <= ?j > ?P ?i) & (~ ?i <= ?j > ?P ?j)) *) 
22997  536 
 (Const (@{const_name Orderings.min}, _), [t1, t2]) => 
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537 
let 
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538 
val rev_terms = rev terms 
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539 
val terms1 = map (subst_term [(split_term, t1)]) rev_terms 
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540 
val terms2 = map (subst_term [(split_term, t2)]) rev_terms 
23881  541 
val t1_leq_t2 = Const (@{const_name HOL.less_eq}, 
20268  542 
split_type > split_type > HOLogic.boolT) $ t1 $ t2 
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543 
val not_t1_leq_t2 = HOLogic.Not $ t1_leq_t2 
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544 
val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const) 
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545 
val subgoal1 = (HOLogic.mk_Trueprop t1_leq_t2) :: terms1 @ [not_false] 
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546 
val subgoal2 = (HOLogic.mk_Trueprop not_t1_leq_t2) :: terms2 @ [not_false] 
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547 
in 
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548 
SOME [(Ts, subgoal1), (Ts, subgoal2)] 
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549 
end 
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550 
(* ?P (abs ?a) = ((0 <= ?a > ?P ?a) & (?a < 0 > ?P ( ?a))) *) 
22997  551 
 (Const (@{const_name HOL.abs}, _), [t1]) => 
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552 
let 
20268  553 
val rev_terms = rev terms 
554 
val terms1 = map (subst_term [(split_term, t1)]) rev_terms 

22997  555 
val terms2 = map (subst_term [(split_term, Const (@{const_name HOL.uminus}, 
20268  556 
split_type > split_type) $ t1)]) rev_terms 
22997  557 
val zero = Const (@{const_name HOL.zero}, split_type) 
23881  558 
val zero_leq_t1 = Const (@{const_name HOL.less_eq}, 
20268  559 
split_type > split_type > HOLogic.boolT) $ zero $ t1 
23881  560 
val t1_lt_zero = Const (@{const_name HOL.less}, 
20268  561 
split_type > split_type > HOLogic.boolT) $ t1 $ zero 
562 
val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const) 

563 
val subgoal1 = (HOLogic.mk_Trueprop zero_leq_t1) :: terms1 @ [not_false] 

564 
val subgoal2 = (HOLogic.mk_Trueprop t1_lt_zero) :: terms2 @ [not_false] 

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565 
in 
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566 
SOME [(Ts, subgoal1), (Ts, subgoal2)] 
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567 
end 
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568 
(* ?P (?a  ?b) = ((?a < ?b > ?P 0) & (ALL d. ?a = ?b + d > ?P d)) *) 
22997  569 
 (Const (@{const_name HOL.minus}, _), [t1, t2]) => 
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570 
let 
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571 
(* "d" in the above theorem becomes a new bound variable after NNF *) 
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572 
(* transformation, therefore some adjustment of indices is necessary *) 
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573 
val rev_terms = rev terms 
22997  574 
val zero = Const (@{const_name HOL.zero}, split_type) 
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575 
val d = Bound 0 
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576 
val terms1 = map (subst_term [(split_term, zero)]) rev_terms 
20268  577 
val terms2 = map (subst_term [(incr_boundvars 1 split_term, d)]) 
578 
(map (incr_boundvars 1) rev_terms) 

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579 
val t1' = incr_boundvars 1 t1 
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580 
val t2' = incr_boundvars 1 t2 
23881  581 
val t1_lt_t2 = Const (@{const_name HOL.less}, 
20268  582 
split_type > split_type > HOLogic.boolT) $ t1 $ t2 
583 
val t1_eq_t2_plus_d = Const ("op =", split_type > split_type > HOLogic.boolT) $ t1' $ 

22997  584 
(Const (@{const_name HOL.plus}, 
20268  585 
split_type > split_type > split_type) $ t2' $ d) 
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586 
val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const) 
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587 
val subgoal1 = (HOLogic.mk_Trueprop t1_lt_t2) :: terms1 @ [not_false] 
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588 
val subgoal2 = (HOLogic.mk_Trueprop t1_eq_t2_plus_d) :: terms2 @ [not_false] 
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589 
in 
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590 
SOME [(Ts, subgoal1), (split_type :: Ts, subgoal2)] 
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591 
end 
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592 
(* ?P (nat ?i) = ((ALL n. ?i = int n > ?P n) & (?i < 0 > ?P 0)) *) 
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593 
 (Const ("IntDef.nat", _), [t1]) => 
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594 
let 
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595 
val rev_terms = rev terms 
22997  596 
val zero_int = Const (@{const_name HOL.zero}, HOLogic.intT) 
597 
val zero_nat = Const (@{const_name HOL.zero}, HOLogic.natT) 

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598 
val n = Bound 0 
20268  599 
val terms1 = map (subst_term [(incr_boundvars 1 split_term, n)]) 
600 
(map (incr_boundvars 1) rev_terms) 

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601 
val terms2 = map (subst_term [(split_term, zero_nat)]) rev_terms 
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602 
val t1' = incr_boundvars 1 t1 
20268  603 
val t1_eq_int_n = Const ("op =", HOLogic.intT > HOLogic.intT > HOLogic.boolT) $ t1' $ 
23365  604 
(Const ("Nat.of_nat", HOLogic.natT > HOLogic.intT) $ n) 
23881  605 
val t1_lt_zero = Const (@{const_name HOL.less}, 
20268  606 
HOLogic.intT > HOLogic.intT > HOLogic.boolT) $ t1 $ zero_int 
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607 
val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const) 
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608 
val subgoal1 = (HOLogic.mk_Trueprop t1_eq_int_n) :: terms1 @ [not_false] 
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609 
val subgoal2 = (HOLogic.mk_Trueprop t1_lt_zero) :: terms2 @ [not_false] 
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610 
in 
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611 
SOME [(HOLogic.natT :: Ts, subgoal1), (Ts, subgoal2)] 
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612 
end 
20268  613 
(* "?P ((?n::nat) mod (number_of ?k)) = 
614 
((number_of ?k = 0 > ?P ?n) & (~ (number_of ?k = 0) > 

615 
(ALL i j. j < number_of ?k > ?n = number_of ?k * i + j > ?P j))) *) 

22997  616 
 (Const ("Divides.div_class.mod", Type ("fun", [Type ("nat", []), _])), [t1, t2]) => 
20217
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617 
let 
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618 
val rev_terms = rev terms 
22997  619 
val zero = Const (@{const_name HOL.zero}, split_type) 
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620 
val i = Bound 1 
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621 
val j = Bound 0 
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622 
val terms1 = map (subst_term [(split_term, t1)]) rev_terms 
20268  623 
val terms2 = map (subst_term [(incr_boundvars 2 split_term, j)]) 
624 
(map (incr_boundvars 2) rev_terms) 

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625 
val t1' = incr_boundvars 2 t1 
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626 
val t2' = incr_boundvars 2 t2 
20268  627 
val t2_eq_zero = Const ("op =", 
628 
split_type > split_type > HOLogic.boolT) $ t2 $ zero 

629 
val t2_neq_zero = HOLogic.mk_not (Const ("op =", 

630 
split_type > split_type > HOLogic.boolT) $ t2' $ zero) 

23881  631 
val j_lt_t2 = Const (@{const_name HOL.less}, 
20268  632 
split_type > split_type> HOLogic.boolT) $ j $ t2' 
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633 
val t1_eq_t2_times_i_plus_j = Const ("op =", split_type > split_type > HOLogic.boolT) $ t1' $ 
22997  634 
(Const (@{const_name HOL.plus}, split_type > split_type > split_type) $ 
635 
(Const (@{const_name HOL.times}, 

20268  636 
split_type > split_type > split_type) $ t2' $ i) $ j) 
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637 
val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const) 
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638 
val subgoal1 = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false] 
20268  639 
val subgoal2 = (map HOLogic.mk_Trueprop 
640 
[t2_neq_zero, j_lt_t2, t1_eq_t2_times_i_plus_j]) 

641 
@ terms2 @ [not_false] 

20217
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642 
in 
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643 
SOME [(Ts, subgoal1), (split_type :: split_type :: Ts, subgoal2)] 
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644 
end 
20268  645 
(* "?P ((?n::nat) div (number_of ?k)) = 
646 
((number_of ?k = 0 > ?P 0) & (~ (number_of ?k = 0) > 

647 
(ALL i j. j < number_of ?k > ?n = number_of ?k * i + j > ?P i))) *) 

22997  648 
 (Const ("Divides.div_class.div", Type ("fun", [Type ("nat", []), _])), [t1, t2]) => 
20217
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649 
let 
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650 
val rev_terms = rev terms 
22997  651 
val zero = Const (@{const_name HOL.zero}, split_type) 
20217
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652 
val i = Bound 1 
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653 
val j = Bound 0 
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654 
val terms1 = map (subst_term [(split_term, zero)]) rev_terms 
20268  655 
val terms2 = map (subst_term [(incr_boundvars 2 split_term, i)]) 
656 
(map (incr_boundvars 2) rev_terms) 

20217
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657 
val t1' = incr_boundvars 2 t1 
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658 
val t2' = incr_boundvars 2 t2 
20268  659 
val t2_eq_zero = Const ("op =", 
660 
split_type > split_type > HOLogic.boolT) $ t2 $ zero 

661 
val t2_neq_zero = HOLogic.mk_not (Const ("op =", 

662 
split_type > split_type > HOLogic.boolT) $ t2' $ zero) 

23881  663 
val j_lt_t2 = Const (@{const_name HOL.less}, 
20268  664 
split_type > split_type> HOLogic.boolT) $ j $ t2' 
20217
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665 
val t1_eq_t2_times_i_plus_j = Const ("op =", split_type > split_type > HOLogic.boolT) $ t1' $ 
22997  666 
(Const (@{const_name HOL.plus}, split_type > split_type > split_type) $ 
667 
(Const (@{const_name HOL.times}, 

20268  668 
split_type > split_type > split_type) $ t2' $ i) $ j) 
20217
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669 
val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const) 
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670 
val subgoal1 = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false] 
20268  671 
val subgoal2 = (map HOLogic.mk_Trueprop 
672 
[t2_neq_zero, j_lt_t2, t1_eq_t2_times_i_plus_j]) 

673 
@ terms2 @ [not_false] 

20217
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674 
in 
25b068a99d2b
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parents:
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changeset

675 
SOME [(Ts, subgoal1), (split_type :: split_type :: Ts, subgoal2)] 
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676 
end 
20268  677 
(* "?P ((?n::int) mod (number_of ?k)) = 
678 
((iszero (number_of ?k) > ?P ?n) & 

20485  679 
(neg (number_of (uminus ?k)) > 
20268  680 
(ALL i j. 0 <= j & j < number_of ?k & ?n = number_of ?k * i + j > ?P j)) & 
681 
(neg (number_of ?k) > 

682 
(ALL i j. number_of ?k < j & j <= 0 & ?n = number_of ?k * i + j > ?P j))) *) 

22997  683 
 (Const ("Divides.div_class.mod", 
20268  684 
Type ("fun", [Type ("IntDef.int", []), _])), [t1, t2 as (number_of $ k)]) => 
20217
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685 
let 
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686 
val rev_terms = rev terms 
22997  687 
val zero = Const (@{const_name HOL.zero}, split_type) 
20217
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688 
val i = Bound 1 
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parents:
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689 
val j = Bound 0 
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690 
val terms1 = map (subst_term [(split_term, t1)]) rev_terms 
20268  691 
val terms2_3 = map (subst_term [(incr_boundvars 2 split_term, j)]) 
692 
(map (incr_boundvars 2) rev_terms) 

20217
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693 
val t1' = incr_boundvars 2 t1 
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694 
val (t2' as (_ $ k')) = incr_boundvars 2 t2 
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695 
val iszero_t2 = Const ("IntDef.iszero", split_type > HOLogic.boolT) $ t2 
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696 
val neg_minus_k = Const ("IntDef.neg", split_type > HOLogic.boolT) $ 
20268  697 
(number_of $ 
22997  698 
(Const (@{const_name HOL.uminus}, 
20485  699 
HOLogic.intT > HOLogic.intT) $ k')) 
23881  700 
val zero_leq_j = Const (@{const_name HOL.less_eq}, 
20268  701 
split_type > split_type > HOLogic.boolT) $ zero $ j 
23881  702 
val j_lt_t2 = Const (@{const_name HOL.less}, 
20268  703 
split_type > split_type> HOLogic.boolT) $ j $ t2' 
20217
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704 
val t1_eq_t2_times_i_plus_j = Const ("op =", split_type > split_type > HOLogic.boolT) $ t1' $ 
22997  705 
(Const (@{const_name HOL.plus}, split_type > split_type > split_type) $ 
706 
(Const (@{const_name HOL.times}, 

20268  707 
split_type > split_type > split_type) $ t2' $ i) $ j) 
20217
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708 
val neg_t2 = Const ("IntDef.neg", split_type > HOLogic.boolT) $ t2' 
23881  709 
val t2_lt_j = Const (@{const_name HOL.less}, 
20268  710 
split_type > split_type> HOLogic.boolT) $ t2' $ j 
23881  711 
val j_leq_zero = Const (@{const_name HOL.less_eq}, 
20268  712 
split_type > split_type > HOLogic.boolT) $ j $ zero 
20217
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713 
val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const) 
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714 
val subgoal1 = (HOLogic.mk_Trueprop iszero_t2) :: terms1 @ [not_false] 
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715 
val subgoal2 = (map HOLogic.mk_Trueprop [neg_minus_k, zero_leq_j]) 
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716 
@ hd terms2_3 
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717 
:: (if tl terms2_3 = [] then [not_false] else []) 
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718 
@ (map HOLogic.mk_Trueprop [j_lt_t2, t1_eq_t2_times_i_plus_j]) 
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719 
@ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false]) 
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720 
val subgoal3 = (map HOLogic.mk_Trueprop [neg_t2, t2_lt_j]) 
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721 
@ hd terms2_3 
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722 
:: (if tl terms2_3 = [] then [not_false] else []) 
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723 
@ (map HOLogic.mk_Trueprop [j_leq_zero, t1_eq_t2_times_i_plus_j]) 
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724 
@ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false]) 
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725 
val Ts' = split_type :: split_type :: Ts 
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726 
in 
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parents:
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727 
SOME [(Ts, subgoal1), (Ts', subgoal2), (Ts', subgoal3)] 
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728 
end 
20268  729 
(* "?P ((?n::int) div (number_of ?k)) = 
730 
((iszero (number_of ?k) > ?P 0) & 

20485  731 
(neg (number_of (uminus ?k)) > 
20268  732 
(ALL i. (EX j. 0 <= j & j < number_of ?k & ?n = number_of ?k * i + j) > ?P i)) & 
733 
(neg (number_of ?k) > 

734 
(ALL i. (EX j. number_of ?k < j & j <= 0 & ?n = number_of ?k * i + j) > ?P i))) *) 

22997  735 
 (Const ("Divides.div_class.div", 
20268  736 
Type ("fun", [Type ("IntDef.int", []), _])), [t1, t2 as (number_of $ k)]) => 
20217
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737 
let 
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738 
val rev_terms = rev terms 
22997  739 
val zero = Const (@{const_name HOL.zero}, split_type) 
20217
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parents:
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740 
val i = Bound 1 
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parents:
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diff
changeset

741 
val j = Bound 0 
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changeset

742 
val terms1 = map (subst_term [(split_term, zero)]) rev_terms 
20268  743 
val terms2_3 = map (subst_term [(incr_boundvars 2 split_term, i)]) 
744 
(map (incr_boundvars 2) rev_terms) 

20217
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745 
val t1' = incr_boundvars 2 t1 
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746 
val (t2' as (_ $ k')) = incr_boundvars 2 t2 
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747 
val iszero_t2 = Const ("IntDef.iszero", split_type > HOLogic.boolT) $ t2 
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748 
val neg_minus_k = Const ("IntDef.neg", split_type > HOLogic.boolT) $ 
20268  749 
(number_of $ 
22997  750 
(Const (@{const_name HOL.uminus}, 
20485  751 
HOLogic.intT > HOLogic.intT) $ k')) 
23881  752 
val zero_leq_j = Const (@{const_name HOL.less_eq}, 
20268  753 
split_type > split_type > HOLogic.boolT) $ zero $ j 
23881  754 
val j_lt_t2 = Const (@{const_name HOL.less}, 
20268  755 
split_type > split_type> HOLogic.boolT) $ j $ t2' 
756 
val t1_eq_t2_times_i_plus_j = Const ("op =", 

757 
split_type > split_type > HOLogic.boolT) $ t1' $ 

22997  758 
(Const (@{const_name HOL.plus}, split_type > split_type > split_type) $ 
759 
(Const (@{const_name HOL.times}, 

20268  760 
split_type > split_type > split_type) $ t2' $ i) $ j) 
20217
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761 
val neg_t2 = Const ("IntDef.neg", split_type > HOLogic.boolT) $ t2' 
23881  762 
val t2_lt_j = Const (@{const_name HOL.less}, 
20268  763 
split_type > split_type> HOLogic.boolT) $ t2' $ j 
23881  764 
val j_leq_zero = Const (@{const_name HOL.less_eq}, 
20268  765 
split_type > split_type > HOLogic.boolT) $ j $ zero 
20217
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changeset

766 
val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const) 
25b068a99d2b
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changeset

767 
val subgoal1 = (HOLogic.mk_Trueprop iszero_t2) :: terms1 @ [not_false] 
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changeset

768 
val subgoal2 = (HOLogic.mk_Trueprop neg_minus_k) 
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diff
changeset

769 
:: terms2_3 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
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diff
changeset

770 
@ not_false 
20268  771 
:: (map HOLogic.mk_Trueprop 
772 
[zero_leq_j, j_lt_t2, t1_eq_t2_times_i_plus_j]) 

20217
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changeset

773 
val subgoal3 = (HOLogic.mk_Trueprop neg_t2) 
25b068a99d2b
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diff
changeset

774 
:: terms2_3 
25b068a99d2b
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diff
changeset

775 
@ not_false 
20268  776 
:: (map HOLogic.mk_Trueprop 
777 
[t2_lt_j, j_leq_zero, t1_eq_t2_times_i_plus_j]) 

20217
25b068a99d2b
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changeset

778 
val Ts' = split_type :: split_type :: Ts 
25b068a99d2b
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diff
changeset

779 
in 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
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parents:
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diff
changeset

780 
SOME [(Ts, subgoal1), (Ts', subgoal2), (Ts', subgoal3)] 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
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changeset

781 
end 
20268  782 
(* this will only happen if a split theorem can be applied for which no *) 
783 
(* code exists above  in which case either the split theorem should be *) 

784 
(* implemented above, or 'is_split_thm' should be modified to filter it *) 

785 
(* out *) 

20217
25b068a99d2b
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parents:
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diff
changeset

786 
 (t, ts) => ( 
20268  787 
warning ("Lin. Arith.: split rule for " ^ Sign.string_of_term sg t ^ 
788 
" (with " ^ Int.toString (length ts) ^ 

20217
25b068a99d2b
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diff
changeset

789 
" argument(s)) not implemented; proof reconstruction is likely to fail"); 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
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diff
changeset

790 
NONE 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
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diff
changeset

791 
)) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

792 
) 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

793 
end; 
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

794 

20268  795 
(* remove terms that do not satisfy 'p'; change the order of the remaining *) 
20217
25b068a99d2b
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changeset

796 
(* terms in the same way as filter_prems_tac does *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

797 

20268  798 
fun filter_prems_tac_items (p : term > bool) (terms : term list) : term list = 
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
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diff
changeset

799 
let 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
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parents:
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diff
changeset

800 
fun filter_prems (t, (left, right)) = 
25b068a99d2b
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parents:
20044
diff
changeset

801 
if p t then (left, right @ [t]) else (left @ right, []) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
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20044
diff
changeset

802 
val (left, right) = foldl filter_prems ([], []) terms 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

803 
in 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

804 
right @ left 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

805 
end; 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

806 

25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

807 
(* return true iff TRY (etac notE) THEN eq_assume_tac would succeed on a *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
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diff
changeset

808 
(* subgoal that has 'terms' as premises *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

809 

20268  810 
fun negated_term_occurs_positively (terms : term list) : bool = 
811 
List.exists 

812 
(fn (Trueprop $ (Const ("Not", _) $ t)) => member (op aconv) terms (Trueprop $ t) 

813 
 _ => false) 

814 
terms; 

20217
25b068a99d2b
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diff
changeset

815 

20268  816 
fun pre_decomp sg (Ts : typ list, terms : term list) : (typ list * term list) list = 
20217
25b068a99d2b
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parents:
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diff
changeset

817 
let 
25b068a99d2b
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webertj
parents:
20044
diff
changeset

818 
(* repeatedly split (including newly emerging subgoals) until no further *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

819 
(* splitting is possible *) 
20271
e76e77e0d615
fixed a bug in function poly: decomposition of products
webertj
parents:
20268
diff
changeset

820 
fun split_loop ([] : (typ list * term list) list) = ([] : (typ list * term list) list) 
20268  821 
 split_loop (subgoal::subgoals) = ( 
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
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diff
changeset

822 
case split_once_items sg subgoal of 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

823 
SOME new_subgoals => split_loop (new_subgoals @ subgoals) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
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parents:
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diff
changeset

824 
 NONE => subgoal :: split_loop subgoals 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

825 
) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

826 
fun is_relevant t = isSome (decomp sg t) 
20268  827 
(* filter_prems_tac is_relevant: *) 
828 
val relevant_terms = filter_prems_tac_items is_relevant terms 

829 
(* split_tac, NNF normalization: *) 

830 
val split_goals = split_loop [(Ts, relevant_terms)] 

831 
(* necessary because split_once_tac may normalize terms: *) 

832 
val beta_eta_norm = map (apsnd (map (Envir.eta_contract o Envir.beta_norm))) split_goals 

833 
(* TRY (etac notE) THEN eq_assume_tac: *) 

834 
val result = List.filter (not o negated_term_occurs_positively o snd) beta_eta_norm 

20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

835 
in 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

836 
result 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

837 
end; 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

838 

25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

839 
(* takes the ith subgoal [ A1; ...; An ] ==> B to *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
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parents:
20044
diff
changeset

840 
(* An > ... > A1 > B, performs splitting with the given 'split_thms' *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
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parents:
20044
diff
changeset

841 
(* (resulting in a different subgoal P), takes P to ~P ==> False, *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

842 
(* performs NNFnormalization of ~P, and eliminates conjunctions, *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

843 
(* disjunctions and existential quantifiers from the premises, possibly (in *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

844 
(* the case of disjunctions) resulting in several new subgoals, each of the *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

845 
(* general form [ Q1; ...; Qm ] ==> False. Fails if more than *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

846 
(* !fast_arith_split_limit splits are possible. *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

847 

20850  848 
local 
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

849 
val nnf_simpset = 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

850 
empty_ss setmkeqTrue mk_eq_True 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

851 
setmksimps (mksimps mksimps_pairs) 
20850  852 
addsimps [imp_conv_disj, iff_conv_conj_imp, de_Morgan_disj, de_Morgan_conj, 
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

853 
not_all, not_ex, not_not] 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

854 
fun prem_nnf_tac i st = 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

855 
full_simp_tac (Simplifier.theory_context (Thm.theory_of_thm st) nnf_simpset) i st 
20850  856 
in 
857 

858 
fun split_once_tac (split_thms : thm list) (i : int) : tactic = 

859 
let 

20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

860 
fun cond_split_tac i st = 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

861 
let 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

862 
val subgoal = Logic.nth_prem (i, Thm.prop_of st) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

863 
val Ts = rev (map snd (Logic.strip_params subgoal)) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

864 
val concl = HOLogic.dest_Trueprop (Logic.strip_assums_concl subgoal) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

865 
val cmap = Splitter.cmap_of_split_thms split_thms 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

866 
val splits = Splitter.split_posns cmap (theory_of_thm st) Ts concl 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

867 
in 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

868 
if length splits > !fast_arith_split_limit then 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

869 
no_tac st 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

870 
else 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

871 
split_tac split_thms i st 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

872 
end 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

873 
in 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

874 
EVERY' [ 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

875 
REPEAT_DETERM o etac rev_mp, 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

876 
cond_split_tac, 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

877 
rtac ccontr, 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

878 
prem_nnf_tac, 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

879 
TRY o REPEAT_ALL_NEW (DETERM o (eresolve_tac [conjE, exE] ORELSE' etac disjE)) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

880 
] i 
20850  881 
end 
882 

883 
end; (* local *) 

20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

884 

25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

885 
(* remove irrelevant premises, then split the ith subgoal (and all new *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

886 
(* subgoals) by using 'split_once_tac' repeatedly. Betaetanormalize new *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

887 
(* subgoals and finally attempt to solve them by finding an immediate *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

888 
(* contradiction (i.e. a term and its negation) in their premises. *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

889 

25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

890 
fun pre_tac i st = 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

891 
let 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

892 
val sg = theory_of_thm st 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

893 
val split_thms = filter is_split_thm (#splits (ArithTheoryData.get sg)) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

894 
fun is_relevant t = isSome (decomp sg t) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

895 
in 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

896 
DETERM ( 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

897 
TRY (filter_prems_tac is_relevant i) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

898 
THEN ( 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

899 
(TRY o REPEAT_ALL_NEW (split_once_tac split_thms)) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

900 
THEN_ALL_NEW 
23530  901 
(CONVERSION Drule.beta_eta_conversion 
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

902 
THEN' 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

903 
(TRY o (etac notE THEN' eq_assume_tac))) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

904 
) i 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

905 
) st 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

906 
end; 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

907 

25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

908 
end; (* LA_Data_Ref *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

909 

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

910 

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

911 
structure Fast_Arith = 
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

912 
Fast_Lin_Arith(structure LA_Logic=LA_Logic and LA_Data=LA_Data_Ref); 
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

913 

20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

914 
val fast_arith_tac = Fast_Arith.lin_arith_tac false; 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

915 
val fast_ex_arith_tac = Fast_Arith.lin_arith_tac; 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

916 
val trace_arith = Fast_Arith.trace; 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

917 
val fast_arith_neq_limit = Fast_Arith.fast_arith_neq_limit; 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

918 
val fast_arith_split_limit = LA_Data_Ref.fast_arith_split_limit; 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

919 

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

920 
(* reduce contradictory <= to False. 
22838  921 
Most of the work is done by the cancel tactics. *) 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

922 

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

923 
val init_lin_arith_data = 
18708  924 
Fast_Arith.map_data (fn {add_mono_thms, mult_mono_thms, inj_thms, lessD, ...} => 
15121
1198032bad25
Initial changes to extend arithmetic from individual types to type classes.
nipkow
parents:
14738
diff
changeset

925 
{add_mono_thms = add_mono_thms @ 
22838  926 
@{thms add_mono_thms_ordered_semiring} @ @{thms add_mono_thms_ordered_field}, 
10693  927 
mult_mono_thms = mult_mono_thms, 
10574
8f98f0301d67
Linear arithmetic now copes with mixed nat/int formulae.
nipkow
parents:
10516
diff
changeset

928 
inj_thms = inj_thms, 
21243  929 
lessD = lessD @ [thm "Suc_leI"], 
22887  930 
neqE = [@{thm linorder_neqE_nat}, @{thm linorder_neqE_ordered_idom}], 
22838  931 
simpset = HOL_basic_ss 
23085  932 
addsimps [@{thm "monoid_add_class.zero_plus.add_0_left"}, @{thm "monoid_add_class.zero_plus.add_0_right"}, 
22838  933 
@{thm "Zero_not_Suc"}, @{thm "Suc_not_Zero"}, @{thm "le_0_eq"}, @{thm "One_nat_def"}, 
934 
@{thm "order_less_irrefl"}, @{thm "zero_neq_one"}, @{thm "zero_less_one"}, 

935 
@{thm "zero_le_one"}, @{thm "zero_neq_one"} RS not_sym, @{thm "not_one_le_zero"}, 

936 
@{thm "not_one_less_zero"}] 

937 
addsimprocs [ab_group_add_cancel.sum_conv, ab_group_add_cancel.rel_conv] 

938 
(*abel_cancel helps it work in abstract algebraic domains*) 

939 
addsimprocs nat_cancel_sums_add}) #> 

18708  940 
arith_discrete "nat"; 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

941 

13462  942 
val fast_nat_arith_simproc = 
16834  943 
Simplifier.simproc (the_context ()) "fast_nat_arith" 
13462  944 
["(m::nat) < n","(m::nat) <= n", "(m::nat) = n"] Fast_Arith.lin_arith_prover; 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

945 

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

946 
(* Because of fast_nat_arith_simproc, the arithmetic solver is really only 
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

947 
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

948 
*not* themselves (in)equalities, because the latter activate 
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

949 
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

950 
solver all the time rather than add the additional check. *) 
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

951 

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

952 

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

953 
(* arith proof method *) 
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

954 

10516  955 
local 
956 

13499  957 
fun raw_arith_tac ex i st = 
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

958 
(* FIXME: K true should be replaced by a sensible test (perhaps "isSome o 
23200  959 
decomp sg"?  but note that the test is applied to terms already before 
960 
they are split/normalized) to speed things up in case there are lots of 

961 
irrelevant terms involved; elimination of min/max can be optimized: 

20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

962 
(max m n + k <= r) = (m+k <= r & n+k <= r) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

963 
(l <= min m n + k) = (l <= m+k & l <= n+k) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

964 
*) 
13499  965 
refute_tac (K true) 
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

966 
(* Splitting is also done inside fast_arith_tac, but not completely  *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

967 
(* split_tac may use split theorems that have not been implemented in *) 
20268  968 
(* fast_arith_tac (cf. pre_decomp and split_once_items above), and *) 
969 
(* fast_arith_split_limit may trigger. *) 

20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

970 
(* Therefore splitting outside of fast_arith_tac may allow us to prove *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

971 
(* some goals that fast_arith_tac alone would fail on. *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

972 
(REPEAT_DETERM o split_tac (#splits (ArithTheoryData.get (Thm.theory_of_thm st)))) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

973 
(fast_ex_arith_tac ex) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

974 
i st; 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

975 

20412
40757f475eb0
additional list of tactics that can be added to arith
webertj
parents:
20280
diff
changeset

976 
fun arith_theory_tac i st = 
40757f475eb0
additional list of tactics that can be added to arith
webertj
parents:
20280
diff
changeset

977 
let 
40757f475eb0
additional list of tactics that can be added to arith
webertj
parents:
20280
diff
changeset

978 
val tactics = #tactics (ArithTheoryData.get (Thm.theory_of_thm st)) 
40757f475eb0
additional list of tactics that can be added to arith
webertj
parents:
20280
diff
changeset

979 
in 
40757f475eb0
additional list of tactics that can be added to arith
webertj
parents:
20280
diff
changeset

980 
FIRST' (map (fn ArithTactic {tactic, ...} => tactic) tactics) i st 
40757f475eb0
additional list of tactics that can be added to arith
webertj
parents:
20280
diff
changeset

981 
end; 
13877
a6b825ee48d9
Added hook for presburger arithmetic decision procedure.
berghofe
parents:
13517
diff
changeset

982 

10516  983 
in 
984 

20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

985 
val simple_arith_tac = FIRST' [fast_arith_tac, 
23190  986 
ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_arith_tac true]; 
13877
a6b825ee48d9
Added hook for presburger arithmetic decision procedure.
berghofe
parents:
13517
diff
changeset

987 

20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

988 
val arith_tac = FIRST' [fast_arith_tac, 
23190  989 
ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_arith_tac true, 
20412
40757f475eb0
additional list of tactics that can be added to arith
webertj
parents:
20280
diff
changeset

990 
arith_theory_tac]; 
13877
a6b825ee48d9
Added hook for presburger arithmetic decision procedure.
berghofe
parents:
13517
diff
changeset

991 

20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

992 
val silent_arith_tac = FIRST' [fast_arith_tac, 
23190  993 
ObjectLogic.full_atomize_tac THEN' (REPEAT_DETERM o rtac impI) THEN' raw_arith_tac false, 
20412
40757f475eb0
additional list of tactics that can be added to arith
webertj
parents:
20280
diff
changeset

994 
arith_theory_tac]; 
10516  995 

20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

996 
fun arith_method prems = 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

997 
Method.METHOD (fn facts => HEADGOAL (Method.insert_tac (prems @ facts) THEN' arith_tac)); 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

998 

10516  999 
end; 
1000 

15195  1001 
(* antisymmetry: 
15197  1002 
combines x <= y (or ~(y < x)) and y <= x (or ~(x < y)) into x = y 
15195  1003 

1004 
local 

1005 
val antisym = mk_meta_eq order_antisym 

22548  1006 
val not_lessD = @{thm linorder_not_less} RS iffD1 
15195  1007 
fun prp t thm = (#prop(rep_thm thm) = t) 
1008 
in 

1009 
fun antisym_eq prems thm = 

1010 
let 

1011 
val r = #prop(rep_thm thm); 

1012 
in 

1013 
case r of 

23881  1014 
Tr $ ((c as Const(@{const_name HOL.less_eq},T)) $ s $ t) => 
15195  1015 
let val r' = Tr $ (c $ t $ s) 
1016 
in 

1017 
case Library.find_first (prp r') prems of 

15531  1018 
NONE => 
23881  1019 
let val r' = Tr $ (HOLogic.Not $ (Const(@{const_name HOL.less},T) $ s $ t)) 
15195  1020 
in case Library.find_first (prp r') prems of 
15531  1021 
NONE => [] 
1022 
 SOME thm' => [(thm' RS not_lessD) RS (thm RS antisym)] 

15195  1023 
end 
15531  1024 
 SOME thm' => [thm' RS (thm RS antisym)] 
15195  1025 
end 
23881  1026 
 Tr $ (Const("Not",_) $ (Const(@{const_name HOL.less},T) $ s $ t)) => 
1027 
let val r' = Tr $ (Const(@{const_name HOL.less_eq},T) $ s $ t) 

15195  1028 
in 
1029 
case Library.find_first (prp r') prems of 

15531  1030 
NONE => 
23881  1031 
let val r' = Tr $ (HOLogic.Not $ (Const(@{const_name HOL.less},T) $ t $ s)) 
15195  1032 
in case Library.find_first (prp r') prems of 
15531  1033 
NONE => [] 
1034 
 SOME thm' => 

15195  1035 
[(thm' RS not_lessD) RS ((thm RS not_lessD) RS antisym)] 
1036 
end 

15531  1037 
 SOME thm' => [thm' RS ((thm RS not_lessD) RS antisym)] 
15195  1038 
end 
1039 
 _ => [] 

1040 
end 

1041 
handle THM _ => [] 

1042 
end; 

15197  1043 
*) 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

1044 

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

1045 
(* theory setup *) 
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

1046 

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

1047 
val arith_setup = 
18708  1048 
init_lin_arith_data #> 
1049 
(fn thy => (Simplifier.change_simpset_of thy (fn ss => ss 

17875  1050 
addsimprocs (nat_cancel_sums @ [fast_nat_arith_simproc]) 
18708  1051 
addSolver (mk_solver' "lin. arith." Fast_Arith.cut_lin_arith_tac)); thy)) #> 
15221  1052 
Method.add_methods 
21879  1053 
[("arith", (arith_method o fst) oo Method.syntax Args.bang_facts, 
18708  1054 
"decide linear arithmethic")] #> 
18728  1055 
Attrib.add_attributes [("arith_split", Attrib.no_args arith_split_add, 
18708  1056 
"declaration of split rules for arithmetic procedure")]; 