author  haftmann 
Thu, 17 May 2007 19:49:40 +0200  
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parent 22947  f53486e661a7 
child 23085  fd30d75a6614 
permissions  rwrr 
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(* Title: HOL/arith_data.ML 
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ID: $Id$ 
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Author: Markus Wenzel, Stefan Berghofer and Tobias Nipkow 
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Various arithmetic proof procedures. 
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*) 
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(**) 
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(* 1. Cancellation of common terms *) 
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(**) 
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13517  12 
structure NatArithUtils = 
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struct 
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(** abstract syntax of structure nat: 0, Suc, + **) 
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(* mk_sum, mk_norm_sum *) 
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val 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; 
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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 Orderings.less}; 
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val dest_bal = HOLogic.dest_bin @{const_name Orderings.less} HOLogic.natT; 

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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 Orderings.less_eq}; 
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val dest_bal = HOLogic.dest_bin @{const_name Orderings.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; 

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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", 
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["(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", 
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["(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; 
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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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fun neg_prop(TP$(Const("Not",_)$t)) = TP$t 
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 neg_prop(TP$t) = TP $ (Const("Not",HOLogic.boolT>HOLogic.boolT)$t); 
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fun is_False thm = 
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let val _ $ t = #prop(rep_thm thm) 
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in t = Const("False",HOLogic.boolT) end; 
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fun is_nat(t) = fastype_of1 t = HOLogic.natT; 
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fun mk_nat_thm sg t = 
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let val ct = cterm_of sg t and cn = cterm_of sg (Var(("n",0),HOLogic.natT)) 
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in instantiate ([],[(cn,ct)]) le0 end; 
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end; (* LA_Logic *) 
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(* arith theory data *) 
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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  208 
structure ArithTheoryData = TheoryDataFun 
22846  209 
( 
20268  210 
type T = {splits: thm list, 
211 
inj_consts: (string * typ) list, 

212 
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  216 
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  219 
{splits = Library.merge Thm.eq_thm_prop (splits1, splits2), 
220 
inj_consts = Library.merge (op =) (inj_consts1, inj_consts2), 

221 
discrete = Library.merge (op =) (discrete1, discrete2), 

222 
tactics = Library.merge eq_arith_tactic (tactics1, tactics2)}; 

22846  223 
); 
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18728  225 
val arith_split_add = Thm.declaration_attribute (fn thm => 
20897  226 
Context.mapping (ArithTheoryData.map (fn {splits,inj_consts,discrete,tactics} => 
22634  227 
{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  230 
{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  233 
{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  236 
{splits= splits, inj_consts= inj_consts, discrete= discrete, tactics= insert eq_arith_tactic tac tactics}); 
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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  259 
 SOME n => (AList.update (op =) (t, Rat.add n m) p, i); 
10693  260 

261 
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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end; 
10718  270 

271 
(* 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  275 
*) 
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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  280 

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(* decompose nested multiplications, bracketing them to the right and combining 
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all their coefficients 
10718  283 
*) 
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fun demult (inj_consts : (string * typ) list) : term * Rat.rat > term option * Rat.rat = 
13499  285 
let 
22997  286 
fun demult ((mC as Const (@{const_name HOL.times}, _)) $ s $ t, m) = ( 
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(case s of 
22997  288 
Const ("Numeral.number_class.number_of", _) $ n => 
22947  289 
demult (t, Rat.mult m (Rat.rat_of_int (HOLogic.dest_numeral n))) 
22997  290 
 Const (@{const_name HOL.uminus}, _) $ (Const ("Numeral.number_class.number_of", _) $ n) => 
22947  291 
demult (t, Rat.mult m (Rat.rat_of_int (~(HOLogic.dest_numeral n)))) 
22997  292 
 Const (@{const_name Suc}, _) $ _ => 
22947  293 
demult (t, Rat.mult m (Rat.rat_of_int (HOLogic.dest_nat s))) 
22997  294 
 Const (@{const_name HOL.times}, _) $ s1 $ s2 => 
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demult (mC $ s1 $ (mC $ s2 $ t), m) 
22997  296 
 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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300 
if den = 0 then 
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raise Zero 
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else 
22947  303 
demult (mC $ numt $ t, Rat.mult m (Rat.inv (Rat.rat_of_int den))) 
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end 
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305 
 _ => 
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306 
atomult (mC, s, t, m) 
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) handle TERM _ => atomult (mC, s, t, m) 
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308 
) 
22997  309 
 demult (atom as Const(@{const_name HOL.divide}, _) $ t $ (Const ("Numeral.number_class.number_of", _) $ dent), m) = 
20268  310 
(let 
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val den = HOLogic.dest_numeral dent 
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312 
in 
20268  313 
if den = 0 then 
314 
raise Zero 

315 
else 

22947  316 
demult (t, Rat.mult m (Rat.inv (Rat.rat_of_int den))) 
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317 
end 
20268  318 
handle TERM _ => (SOME atom, m)) 
22997  319 
 demult (Const (@{const_name HOL.zero}, _), m) = (NONE, Rat.zero) 
320 
 demult (Const (@{const_name HOL.one}, _), m) = (NONE, m) 

321 
 demult (t as Const ("Numeral.number_class.number_of", _) $ n, m) = 

22947  322 
((NONE, Rat.mult m (Rat.rat_of_int (HOLogic.dest_numeral n))) 
323 
handle TERM _ => (SOME t, m)) 

22997  324 
 demult (Const (@{const_name HOL.uminus}, _) $ t, m) = demult (t, Rat.neg m) 
20268  325 
 demult (t as Const f $ x, m) = 
22997  326 
(if member (op =) inj_consts f then SOME x else SOME t, m) 
20268  327 
 demult (atom, m) = (SOME atom, m) 
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328 
and 
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atomult (mC, atom, t, m) = ( 
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330 
case demult (t, m) of (NONE, m') => (SOME atom, m') 
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331 
 (SOME t', m') => (SOME (mC $ atom $ t'), m') 
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332 
) 
13499  333 
in demult end; 
10718  334 

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335 
fun decomp0 (inj_consts : (string * typ) list) (rel : string, lhs : term, rhs : term) : 
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336 
((term * Rat.rat) list * Rat.rat * string * (term * Rat.rat) list * Rat.rat) option = 
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337 
let 
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338 
(* Turn term into list of summand * multiplicity plus a constant *) 
22997  339 
fun poly (Const (@{const_name HOL.plus}, _) $ s $ t, m : Rat.rat, pi : (term * Rat.rat) list * Rat.rat) = 
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340 
poly (s, m, poly (t, m, pi)) 
22997  341 
 poly (all as Const (@{const_name HOL.minus}, T) $ s $ t, m, pi) = 
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342 
if nT T then add_atom all m pi else poly (s, m, poly (t, Rat.neg m, pi)) 
22997  343 
 poly (all as Const (@{const_name HOL.uminus}, T) $ t, m, pi) = 
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344 
if nT T then add_atom all m pi else poly (t, Rat.neg m, pi) 
22997  345 
 poly (Const (@{const_name HOL.zero}, _), _, pi) = 
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346 
pi 
22997  347 
 poly (Const (@{const_name HOL.one}, _), m, (p, i)) = 
22947  348 
(p, Rat.add i m) 
22997  349 
 poly (Const (@{const_name Suc}, _) $ t, m, (p, i)) = 
22947  350 
poly (t, m, (p, Rat.add i m)) 
22997  351 
 poly (all as Const (@{const_name HOL.times}, _) $ _ $ _, m, pi as (p, i)) = 
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352 
(case demult inj_consts (all, m) of 
22947  353 
(NONE, m') => (p, Rat.add i m') 
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354 
 (SOME u, m') => add_atom u m' pi) 
22997  355 
 poly (all as Const (@{const_name HOL.divide}, _) $ _ $ _, m, pi as (p, i)) = 
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356 
(case demult inj_consts (all, m) of 
22947  357 
(NONE, m') => (p, Rat.add i m') 
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358 
 (SOME u, m') => add_atom u m' pi) 
22997  359 
 poly (all as Const ("Numeral.number_class.number_of", Type(_,[_,T])) $ t, m, pi as (p, i)) = 
21820
2f2b6a965ccc
introduced mk/dest_numeral/number for mk/dest_binum etc.
haftmann
parents:
21621
diff
changeset

360 
(let val k = HOLogic.dest_numeral t 
20859  361 
val k2 = if k < 0 andalso T = HOLogic.natT then 0 else k 
22947  362 
in (p, Rat.add i (Rat.mult m (Rat.rat_of_int k2))) end 
20859  363 
handle TERM _ => add_atom all m pi) 
20271
e76e77e0d615
fixed a bug in function poly: decomposition of products
webertj
parents:
20268
diff
changeset

364 
 poly (all as Const f $ x, m, pi) = 
e76e77e0d615
fixed a bug in function poly: decomposition of products
webertj
parents:
20268
diff
changeset

365 
if f mem inj_consts then poly (x, m, pi) else add_atom all m pi 
e76e77e0d615
fixed a bug in function poly: decomposition of products
webertj
parents:
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diff
changeset

366 
 poly (all, m, pi) = 
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fixed a bug in function poly: decomposition of products
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20268
diff
changeset

367 
add_atom all m pi 
22947  368 
val (p, i) = poly (lhs, Rat.one, ([], Rat.zero)) 
369 
val (q, j) = poly (rhs, Rat.one, ([], Rat.zero)) 

20254
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370 
in 
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diff
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371 
case rel of 
22997  372 
@{const_name Orderings.less} => SOME (p, i, "<", q, j) 
373 
 @{const_name Orderings.less_eq} => SOME (p, i, "<=", q, j) 

20254
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374 
 "op =" => SOME (p, i, "=", q, j) 
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375 
 _ => NONE 
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376 
end handle Zero => NONE; 
9436
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rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

377 

20271
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fixed a bug in function poly: decomposition of products
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20268
diff
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378 
fun of_lin_arith_sort sg (U : typ) : bool = 
20254
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lin_arith_prover splits certain operators (e.g. min, max, abs)
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diff
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379 
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

380 

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

381 
fun allows_lin_arith sg (discrete : string list) (U as Type (D, [])) : bool * bool = 
20254
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382 
if of_lin_arith_sort sg U then 
58b71535ed00
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diff
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383 
(true, D mem discrete) 
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
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384 
else (* special cases *) 
20271
e76e77e0d615
fixed a bug in function poly: decomposition of products
webertj
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diff
changeset

385 
if D mem discrete then (true, true) else (false, false) 
20254
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lin_arith_prover splits certain operators (e.g. min, max, abs)
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parents:
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diff
changeset

386 
 allows_lin_arith sg discrete U = 
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387 
(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

388 

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

389 
fun decomp_typecheck (sg, discrete, inj_consts) (T : typ, xxx) : decompT option = 
e76e77e0d615
fixed a bug in function poly: decomposition of products
webertj
parents:
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diff
changeset

390 
case T of 
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fixed a bug in function poly: decomposition of products
webertj
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diff
changeset

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

392 
(case allows_lin_arith sg discrete U of 
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fixed a bug in function poly: decomposition of products
webertj
parents:
20268
diff
changeset

393 
(true, d) => 
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fixed a bug in function poly: decomposition of products
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parents:
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diff
changeset

394 
(case decomp0 inj_consts xxx of 
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fixed a bug in function poly: decomposition of products
webertj
parents:
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diff
changeset

395 
NONE => NONE 
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fixed a bug in function poly: decomposition of products
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parents:
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diff
changeset

396 
 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:
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diff
changeset

397 
 (false, _) => 
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fixed a bug in function poly: decomposition of products
webertj
parents:
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diff
changeset

398 
NONE) 
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fixed a bug in function poly: decomposition of products
webertj
parents:
20268
diff
changeset

399 
 _ => NONE; 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

400 

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

401 
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:
20268
diff
changeset

402 
 negate NONE = NONE; 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

403 

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

404 
fun decomp_negation data (_ $ (Const (rel, T) $ lhs $ rhs)) : decompT option = 
e76e77e0d615
fixed a bug in function poly: decomposition of products
webertj
parents:
20268
diff
changeset

405 
decomp_typecheck data (T, (rel, lhs, rhs)) 
e76e77e0d615
fixed a bug in function poly: decomposition of products
webertj
parents:
20268
diff
changeset

406 
 decomp_negation data (_ $ (Const ("Not", _) $ (Const (rel, T) $ lhs $ rhs))) = 
e76e77e0d615
fixed a bug in function poly: decomposition of products
webertj
parents:
20268
diff
changeset

407 
negate (decomp_typecheck data (T, (rel, lhs, rhs))) 
e76e77e0d615
fixed a bug in function poly: decomposition of products
webertj
parents:
20268
diff
changeset

408 
 decomp_negation data _ = 
e76e77e0d615
fixed a bug in function poly: decomposition of products
webertj
parents:
20268
diff
changeset

409 
NONE; 
e76e77e0d615
fixed a bug in function poly: decomposition of products
webertj
parents:
20268
diff
changeset

410 

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

411 
fun decomp sg : term > decompT option = 
20254
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset

412 
let 
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
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diff
changeset

413 
val {discrete, inj_consts, ...} = ArithTheoryData.get sg 
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
20217
diff
changeset

414 
in 
20271
e76e77e0d615
fixed a bug in function poly: decomposition of products
webertj
parents:
20268
diff
changeset

415 
decomp_negation (sg, discrete, inj_consts) 
20254
58b71535ed00
lin_arith_prover splits certain operators (e.g. min, max, abs)
webertj
parents:
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changeset

416 
end; 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

417 

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

418 
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

419 
 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

420 
 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

421 

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

422 
fun number_of (n, T) = HOLogic.mk_number T n; 
10693  423 

20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
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diff
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424 
(**) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

425 
(* code that performs certain goal transformations for linear arithmetic *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

426 
(**) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

427 

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

428 
(* A "do nothing" variant of pre_decomp and pre_tac: 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

429 

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

430 
fun pre_decomp sg Ts termitems = [termitems]; 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

431 
fun pre_tac i = all_tac; 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

432 
*) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

433 

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

434 
(**) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

435 
(* 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

436 
(* 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

437 
(* to the proof state *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

438 
(**) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

439 

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

440 
val fast_arith_split_limit = ref 9; 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

441 

20268  442 
(* checks if splitting with 'thm' is implemented *) 
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

443 

20268  444 
fun is_split_thm (thm : thm) : bool = 
445 
case concl_of thm of _ $ (_ $ (_ $ lhs) $ _) => ( 

446 
(* Trueprop $ ((op =) $ (?P $ lhs) $ rhs) *) 

447 
case head_of lhs of 

22997  448 
Const (a, _) => member (op =) [@{const_name Orderings.max}, 
449 
@{const_name Orderings.min}, 

450 
@{const_name HOL.abs}, 

451 
@{const_name HOL.minus}, 

20268  452 
"IntDef.nat", 
22997  453 
"Divides.div_class.mod", 
454 
"Divides.div_class.div"] a 

20268  455 
 _ => (warning ("Lin. Arith.: wrong format for split rule " ^ 
456 
Display.string_of_thm thm); 

457 
false)) 

458 
 _ => (warning ("Lin. Arith.: wrong format for split rule " ^ 

459 
Display.string_of_thm thm); 

460 
false); 

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

461 

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

462 
(* 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

463 
(* variables as needed when substituting inside an abstraction *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

464 

20268  465 
fun subst_term ([] : (term * term) list) (t : term) = t 
466 
 subst_term pairs t = 

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

467 
(case AList.lookup (op aconv) pairs t of 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

468 
SOME new => 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

469 
new 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

470 
 NONE => 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

471 
(case t of Abs (a, T, body) => 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

472 
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

473 
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

474 
 t1 $ t2 => 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

475 
subst_term pairs t1 $ subst_term pairs t2 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

476 
 _ => t)); 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

477 

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

478 
(* 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

479 
(* 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

480 
(* 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

481 
(* 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

482 
(* on the subgoal *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

483 

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

484 
(* 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

485 
(* (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

486 
(* 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

487 
(* theorem that can be applied, and modify the subgoal accordingly. *) 
20268  488 
(* Or even better, the splitter should be extended to provide *) 
489 
(* splitting on terms as well as splitting on theorems (where the *) 

490 
(* former can have a faster implementation as it does not need to be *) 

491 
(* proofproducing). *) 

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

492 

20268  493 
fun split_once_items (sg : theory) (Ts : typ list, terms : term list) : 
494 
(typ list * term list) list option = 

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

495 
let 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

496 
(* takes a list [t1, ..., tn] to the term *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

497 
(* tn' > ... > t1' > False , *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
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diff
changeset

498 
(* where ti' = HOLogic.dest_Trueprop ti *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

499 
(* term list > term *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

500 
fun REPEAT_DETERM_etac_rev_mp terms' = 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

501 
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:
20044
diff
changeset

502 
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

503 
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

504 
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

505 
in 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

506 
if length splits > !fast_arith_split_limit then ( 
20268  507 
tracing ("fast_arith_split_limit exceeded (current value is " ^ 
508 
string_of_int (!fast_arith_split_limit) ^ ")"); 

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

509 
NONE 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

510 
) else ( 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

511 
case splits of [] => 
20268  512 
(* split_tac would fail: no possible split *) 
513 
NONE 

514 
 ((_, _, _, split_type, split_term) :: _) => ( 

515 
(* ignore all but the first possible split *) 

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

516 
case strip_comb split_term of 
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517 
(* ?P (max ?i ?j) = ((?i <= ?j > ?P ?j) & (~ ?i <= ?j > ?P ?i)) *) 
22997  518 
(Const (@{const_name Orderings.max}, _), [t1, t2]) => 
20217
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519 
let 
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520 
val rev_terms = rev terms 
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521 
val terms1 = map (subst_term [(split_term, t1)]) rev_terms 
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522 
val terms2 = map (subst_term [(split_term, t2)]) rev_terms 
22997  523 
val t1_leq_t2 = Const (@{const_name Orderings.less_eq}, 
20268  524 
split_type > split_type > HOLogic.boolT) $ t1 $ t2 
20217
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525 
val not_t1_leq_t2 = HOLogic.Not $ t1_leq_t2 
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526 
val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const) 
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527 
val subgoal1 = (HOLogic.mk_Trueprop t1_leq_t2) :: terms2 @ [not_false] 
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528 
val subgoal2 = (HOLogic.mk_Trueprop not_t1_leq_t2) :: terms1 @ [not_false] 
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529 
in 
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530 
SOME [(Ts, subgoal1), (Ts, subgoal2)] 
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531 
end 
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532 
(* ?P (min ?i ?j) = ((?i <= ?j > ?P ?i) & (~ ?i <= ?j > ?P ?j)) *) 
22997  533 
 (Const (@{const_name Orderings.min}, _), [t1, t2]) => 
20217
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534 
let 
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535 
val rev_terms = rev terms 
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536 
val terms1 = map (subst_term [(split_term, t1)]) rev_terms 
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537 
val terms2 = map (subst_term [(split_term, t2)]) rev_terms 
22997  538 
val t1_leq_t2 = Const (@{const_name Orderings.less_eq}, 
20268  539 
split_type > split_type > HOLogic.boolT) $ t1 $ t2 
20217
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540 
val not_t1_leq_t2 = HOLogic.Not $ t1_leq_t2 
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541 
val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const) 
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542 
val subgoal1 = (HOLogic.mk_Trueprop t1_leq_t2) :: terms1 @ [not_false] 
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543 
val subgoal2 = (HOLogic.mk_Trueprop not_t1_leq_t2) :: terms2 @ [not_false] 
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544 
in 
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545 
SOME [(Ts, subgoal1), (Ts, subgoal2)] 
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546 
end 
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547 
(* ?P (abs ?a) = ((0 <= ?a > ?P ?a) & (?a < 0 > ?P ( ?a))) *) 
22997  548 
 (Const (@{const_name HOL.abs}, _), [t1]) => 
20217
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549 
let 
20268  550 
val rev_terms = rev terms 
551 
val terms1 = map (subst_term [(split_term, t1)]) rev_terms 

22997  552 
val terms2 = map (subst_term [(split_term, Const (@{const_name HOL.uminus}, 
20268  553 
split_type > split_type) $ t1)]) rev_terms 
22997  554 
val zero = Const (@{const_name HOL.zero}, split_type) 
555 
val zero_leq_t1 = Const (@{const_name Orderings.less_eq}, 

20268  556 
split_type > split_type > HOLogic.boolT) $ zero $ t1 
22997  557 
val t1_lt_zero = Const (@{const_name Orderings.less}, 
20268  558 
split_type > split_type > HOLogic.boolT) $ t1 $ zero 
559 
val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const) 

560 
val subgoal1 = (HOLogic.mk_Trueprop zero_leq_t1) :: terms1 @ [not_false] 

561 
val subgoal2 = (HOLogic.mk_Trueprop t1_lt_zero) :: terms2 @ [not_false] 

20217
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562 
in 
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563 
SOME [(Ts, subgoal1), (Ts, subgoal2)] 
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564 
end 
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565 
(* ?P (?a  ?b) = ((?a < ?b > ?P 0) & (ALL d. ?a = ?b + d > ?P d)) *) 
22997  566 
 (Const (@{const_name HOL.minus}, _), [t1, t2]) => 
20217
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567 
let 
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568 
(* "d" in the above theorem becomes a new bound variable after NNF *) 
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569 
(* transformation, therefore some adjustment of indices is necessary *) 
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570 
val rev_terms = rev terms 
22997  571 
val zero = Const (@{const_name HOL.zero}, split_type) 
20217
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572 
val d = Bound 0 
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573 
val terms1 = map (subst_term [(split_term, zero)]) rev_terms 
20268  574 
val terms2 = map (subst_term [(incr_boundvars 1 split_term, d)]) 
575 
(map (incr_boundvars 1) rev_terms) 

20217
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576 
val t1' = incr_boundvars 1 t1 
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577 
val t2' = incr_boundvars 1 t2 
22997  578 
val t1_lt_t2 = Const (@{const_name Orderings.less}, 
20268  579 
split_type > split_type > HOLogic.boolT) $ t1 $ t2 
580 
val t1_eq_t2_plus_d = Const ("op =", split_type > split_type > HOLogic.boolT) $ t1' $ 

22997  581 
(Const (@{const_name HOL.plus}, 
20268  582 
split_type > split_type > split_type) $ t2' $ d) 
20217
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583 
val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const) 
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584 
val subgoal1 = (HOLogic.mk_Trueprop t1_lt_t2) :: terms1 @ [not_false] 
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585 
val subgoal2 = (HOLogic.mk_Trueprop t1_eq_t2_plus_d) :: terms2 @ [not_false] 
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586 
in 
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587 
SOME [(Ts, subgoal1), (split_type :: Ts, subgoal2)] 
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588 
end 
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589 
(* ?P (nat ?i) = ((ALL n. ?i = int n > ?P n) & (?i < 0 > ?P 0)) *) 
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590 
 (Const ("IntDef.nat", _), [t1]) => 
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591 
let 
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592 
val rev_terms = rev terms 
22997  593 
val zero_int = Const (@{const_name HOL.zero}, HOLogic.intT) 
594 
val zero_nat = Const (@{const_name HOL.zero}, HOLogic.natT) 

20217
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595 
val n = Bound 0 
20268  596 
val terms1 = map (subst_term [(incr_boundvars 1 split_term, n)]) 
597 
(map (incr_boundvars 1) rev_terms) 

20217
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598 
val terms2 = map (subst_term [(split_term, zero_nat)]) rev_terms 
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599 
val t1' = incr_boundvars 1 t1 
20268  600 
val t1_eq_int_n = Const ("op =", HOLogic.intT > HOLogic.intT > HOLogic.boolT) $ t1' $ 
601 
(Const ("IntDef.int", HOLogic.natT > HOLogic.intT) $ n) 

22997  602 
val t1_lt_zero = Const (@{const_name Orderings.less}, 
20268  603 
HOLogic.intT > HOLogic.intT > HOLogic.boolT) $ t1 $ zero_int 
20217
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604 
val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const) 
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605 
val subgoal1 = (HOLogic.mk_Trueprop t1_eq_int_n) :: terms1 @ [not_false] 
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606 
val subgoal2 = (HOLogic.mk_Trueprop t1_lt_zero) :: terms2 @ [not_false] 
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changeset

607 
in 
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608 
SOME [(HOLogic.natT :: Ts, subgoal1), (Ts, subgoal2)] 
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609 
end 
20268  610 
(* "?P ((?n::nat) mod (number_of ?k)) = 
611 
((number_of ?k = 0 > ?P ?n) & (~ (number_of ?k = 0) > 

612 
(ALL i j. j < number_of ?k > ?n = number_of ?k * i + j > ?P j))) *) 

22997  613 
 (Const ("Divides.div_class.mod", Type ("fun", [Type ("nat", []), _])), [t1, t2]) => 
20217
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614 
let 
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changeset

615 
val rev_terms = rev terms 
22997  616 
val zero = Const (@{const_name HOL.zero}, split_type) 
20217
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617 
val i = Bound 1 
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618 
val j = Bound 0 
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619 
val terms1 = map (subst_term [(split_term, t1)]) rev_terms 
20268  620 
val terms2 = map (subst_term [(incr_boundvars 2 split_term, j)]) 
621 
(map (incr_boundvars 2) rev_terms) 

20217
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changeset

622 
val t1' = incr_boundvars 2 t1 
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changeset

623 
val t2' = incr_boundvars 2 t2 
20268  624 
val t2_eq_zero = Const ("op =", 
625 
split_type > split_type > HOLogic.boolT) $ t2 $ zero 

626 
val t2_neq_zero = HOLogic.mk_not (Const ("op =", 

627 
split_type > split_type > HOLogic.boolT) $ t2' $ zero) 

22997  628 
val j_lt_t2 = Const (@{const_name Orderings.less}, 
20268  629 
split_type > split_type> HOLogic.boolT) $ j $ t2' 
20217
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630 
val t1_eq_t2_times_i_plus_j = Const ("op =", split_type > split_type > HOLogic.boolT) $ t1' $ 
22997  631 
(Const (@{const_name HOL.plus}, split_type > split_type > split_type) $ 
632 
(Const (@{const_name HOL.times}, 

20268  633 
split_type > split_type > split_type) $ t2' $ i) $ j) 
20217
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634 
val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const) 
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635 
val subgoal1 = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false] 
20268  636 
val subgoal2 = (map HOLogic.mk_Trueprop 
637 
[t2_neq_zero, j_lt_t2, t1_eq_t2_times_i_plus_j]) 

638 
@ terms2 @ [not_false] 

20217
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changeset

639 
in 
25b068a99d2b
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parents:
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changeset

640 
SOME [(Ts, subgoal1), (split_type :: split_type :: Ts, subgoal2)] 
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changeset

641 
end 
20268  642 
(* "?P ((?n::nat) div (number_of ?k)) = 
643 
((number_of ?k = 0 > ?P 0) & (~ (number_of ?k = 0) > 

644 
(ALL i j. j < number_of ?k > ?n = number_of ?k * i + j > ?P i))) *) 

22997  645 
 (Const ("Divides.div_class.div", Type ("fun", [Type ("nat", []), _])), [t1, t2]) => 
20217
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646 
let 
25b068a99d2b
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changeset

647 
val rev_terms = rev terms 
22997  648 
val zero = Const (@{const_name HOL.zero}, split_type) 
20217
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parents:
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changeset

649 
val i = Bound 1 
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parents:
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changeset

650 
val j = Bound 0 
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changeset

651 
val terms1 = map (subst_term [(split_term, zero)]) rev_terms 
20268  652 
val terms2 = map (subst_term [(incr_boundvars 2 split_term, i)]) 
653 
(map (incr_boundvars 2) rev_terms) 

20217
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changeset

654 
val t1' = incr_boundvars 2 t1 
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changeset

655 
val t2' = incr_boundvars 2 t2 
20268  656 
val t2_eq_zero = Const ("op =", 
657 
split_type > split_type > HOLogic.boolT) $ t2 $ zero 

658 
val t2_neq_zero = HOLogic.mk_not (Const ("op =", 

659 
split_type > split_type > HOLogic.boolT) $ t2' $ zero) 

22997  660 
val j_lt_t2 = Const (@{const_name Orderings.less}, 
20268  661 
split_type > split_type> HOLogic.boolT) $ j $ t2' 
20217
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changeset

662 
val t1_eq_t2_times_i_plus_j = Const ("op =", split_type > split_type > HOLogic.boolT) $ t1' $ 
22997  663 
(Const (@{const_name HOL.plus}, split_type > split_type > split_type) $ 
664 
(Const (@{const_name HOL.times}, 

20268  665 
split_type > split_type > split_type) $ t2' $ i) $ j) 
20217
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changeset

666 
val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const) 
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changeset

667 
val subgoal1 = (HOLogic.mk_Trueprop t2_eq_zero) :: terms1 @ [not_false] 
20268  668 
val subgoal2 = (map HOLogic.mk_Trueprop 
669 
[t2_neq_zero, j_lt_t2, t1_eq_t2_times_i_plus_j]) 

670 
@ terms2 @ [not_false] 

20217
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changeset

671 
in 
25b068a99d2b
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parents:
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changeset

672 
SOME [(Ts, subgoal1), (split_type :: split_type :: Ts, subgoal2)] 
25b068a99d2b
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changeset

673 
end 
20268  674 
(* "?P ((?n::int) mod (number_of ?k)) = 
675 
((iszero (number_of ?k) > ?P ?n) & 

20485  676 
(neg (number_of (uminus ?k)) > 
20268  677 
(ALL i j. 0 <= j & j < number_of ?k & ?n = number_of ?k * i + j > ?P j)) & 
678 
(neg (number_of ?k) > 

679 
(ALL i j. number_of ?k < j & j <= 0 & ?n = number_of ?k * i + j > ?P j))) *) 

22997  680 
 (Const ("Divides.div_class.mod", 
20268  681 
Type ("fun", [Type ("IntDef.int", []), _])), [t1, t2 as (number_of $ k)]) => 
20217
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parents:
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diff
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682 
let 
25b068a99d2b
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683 
val rev_terms = rev terms 
22997  684 
val zero = Const (@{const_name HOL.zero}, split_type) 
20217
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parents:
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685 
val i = Bound 1 
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parents:
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diff
changeset

686 
val j = Bound 0 
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687 
val terms1 = map (subst_term [(split_term, t1)]) rev_terms 
20268  688 
val terms2_3 = map (subst_term [(incr_boundvars 2 split_term, j)]) 
689 
(map (incr_boundvars 2) rev_terms) 

20217
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690 
val t1' = incr_boundvars 2 t1 
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691 
val (t2' as (_ $ k')) = incr_boundvars 2 t2 
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692 
val iszero_t2 = Const ("IntDef.iszero", split_type > HOLogic.boolT) $ t2 
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693 
val neg_minus_k = Const ("IntDef.neg", split_type > HOLogic.boolT) $ 
20268  694 
(number_of $ 
22997  695 
(Const (@{const_name HOL.uminus}, 
20485  696 
HOLogic.intT > HOLogic.intT) $ k')) 
22997  697 
val zero_leq_j = Const (@{const_name Orderings.less_eq}, 
20268  698 
split_type > split_type > HOLogic.boolT) $ zero $ j 
22997  699 
val j_lt_t2 = Const (@{const_name Orderings.less}, 
20268  700 
split_type > split_type> HOLogic.boolT) $ j $ t2' 
20217
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701 
val t1_eq_t2_times_i_plus_j = Const ("op =", split_type > split_type > HOLogic.boolT) $ t1' $ 
22997  702 
(Const (@{const_name HOL.plus}, split_type > split_type > split_type) $ 
703 
(Const (@{const_name HOL.times}, 

20268  704 
split_type > split_type > split_type) $ t2' $ i) $ j) 
20217
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705 
val neg_t2 = Const ("IntDef.neg", split_type > HOLogic.boolT) $ t2' 
22997  706 
val t2_lt_j = Const (@{const_name Orderings.less}, 
20268  707 
split_type > split_type> HOLogic.boolT) $ t2' $ j 
22997  708 
val j_leq_zero = Const (@{const_name Orderings.less_eq}, 
20268  709 
split_type > split_type > HOLogic.boolT) $ j $ zero 
20217
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710 
val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const) 
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711 
val subgoal1 = (HOLogic.mk_Trueprop iszero_t2) :: terms1 @ [not_false] 
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712 
val subgoal2 = (map HOLogic.mk_Trueprop [neg_minus_k, zero_leq_j]) 
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713 
@ hd terms2_3 
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714 
:: (if tl terms2_3 = [] then [not_false] else []) 
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715 
@ (map HOLogic.mk_Trueprop [j_lt_t2, t1_eq_t2_times_i_plus_j]) 
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716 
@ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false]) 
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717 
val subgoal3 = (map HOLogic.mk_Trueprop [neg_t2, t2_lt_j]) 
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718 
@ hd terms2_3 
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719 
:: (if tl terms2_3 = [] then [not_false] else []) 
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720 
@ (map HOLogic.mk_Trueprop [j_leq_zero, t1_eq_t2_times_i_plus_j]) 
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721 
@ (if tl terms2_3 = [] then [] else tl terms2_3 @ [not_false]) 
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722 
val Ts' = split_type :: split_type :: Ts 
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diff
changeset

723 
in 
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diff
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724 
SOME [(Ts, subgoal1), (Ts', subgoal2), (Ts', subgoal3)] 
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725 
end 
20268  726 
(* "?P ((?n::int) div (number_of ?k)) = 
727 
((iszero (number_of ?k) > ?P 0) & 

20485  728 
(neg (number_of (uminus ?k)) > 
20268  729 
(ALL i. (EX j. 0 <= j & j < number_of ?k & ?n = number_of ?k * i + j) > ?P i)) & 
730 
(neg (number_of ?k) > 

731 
(ALL i. (EX j. number_of ?k < j & j <= 0 & ?n = number_of ?k * i + j) > ?P i))) *) 

22997  732 
 (Const ("Divides.div_class.div", 
20268  733 
Type ("fun", [Type ("IntDef.int", []), _])), [t1, t2 as (number_of $ k)]) => 
20217
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734 
let 
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735 
val rev_terms = rev terms 
22997  736 
val zero = Const (@{const_name HOL.zero}, split_type) 
20217
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737 
val i = Bound 1 
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parents:
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diff
changeset

738 
val j = Bound 0 
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diff
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739 
val terms1 = map (subst_term [(split_term, zero)]) rev_terms 
20268  740 
val terms2_3 = map (subst_term [(incr_boundvars 2 split_term, i)]) 
741 
(map (incr_boundvars 2) rev_terms) 

20217
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742 
val t1' = incr_boundvars 2 t1 
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743 
val (t2' as (_ $ k')) = incr_boundvars 2 t2 
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744 
val iszero_t2 = Const ("IntDef.iszero", split_type > HOLogic.boolT) $ t2 
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745 
val neg_minus_k = Const ("IntDef.neg", split_type > HOLogic.boolT) $ 
20268  746 
(number_of $ 
22997  747 
(Const (@{const_name HOL.uminus}, 
20485  748 
HOLogic.intT > HOLogic.intT) $ k')) 
22997  749 
val zero_leq_j = Const (@{const_name Orderings.less_eq}, 
20268  750 
split_type > split_type > HOLogic.boolT) $ zero $ j 
22997  751 
val j_lt_t2 = Const (@{const_name Orderings.less}, 
20268  752 
split_type > split_type> HOLogic.boolT) $ j $ t2' 
753 
val t1_eq_t2_times_i_plus_j = Const ("op =", 

754 
split_type > split_type > HOLogic.boolT) $ t1' $ 

22997  755 
(Const (@{const_name HOL.plus}, split_type > split_type > split_type) $ 
756 
(Const (@{const_name HOL.times}, 

20268  757 
split_type > split_type > split_type) $ t2' $ i) $ j) 
20217
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changeset

758 
val neg_t2 = Const ("IntDef.neg", split_type > HOLogic.boolT) $ t2' 
22997  759 
val t2_lt_j = Const (@{const_name Orderings.less}, 
20268  760 
split_type > split_type> HOLogic.boolT) $ t2' $ j 
22997  761 
val j_leq_zero = Const (@{const_name Orderings.less_eq}, 
20268  762 
split_type > split_type > HOLogic.boolT) $ j $ zero 
20217
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changeset

763 
val not_false = HOLogic.mk_Trueprop (HOLogic.Not $ HOLogic.false_const) 
25b068a99d2b
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diff
changeset

764 
val subgoal1 = (HOLogic.mk_Trueprop iszero_t2) :: terms1 @ [not_false] 
25b068a99d2b
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changeset

765 
val subgoal2 = (HOLogic.mk_Trueprop neg_minus_k) 
25b068a99d2b
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diff
changeset

766 
:: terms2_3 
25b068a99d2b
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diff
changeset

767 
@ not_false 
20268  768 
:: (map HOLogic.mk_Trueprop 
769 
[zero_leq_j, j_lt_t2, t1_eq_t2_times_i_plus_j]) 

20217
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changeset

770 
val subgoal3 = (HOLogic.mk_Trueprop neg_t2) 
25b068a99d2b
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parents:
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diff
changeset

771 
:: terms2_3 
25b068a99d2b
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webertj
parents:
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diff
changeset

772 
@ not_false 
20268  773 
:: (map HOLogic.mk_Trueprop 
774 
[t2_lt_j, j_leq_zero, t1_eq_t2_times_i_plus_j]) 

20217
25b068a99d2b
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diff
changeset

775 
val Ts' = split_type :: split_type :: Ts 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
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parents:
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diff
changeset

776 
in 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

777 
SOME [(Ts, subgoal1), (Ts', subgoal2), (Ts', subgoal3)] 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
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diff
changeset

778 
end 
20268  779 
(* this will only happen if a split theorem can be applied for which no *) 
780 
(* code exists above  in which case either the split theorem should be *) 

781 
(* implemented above, or 'is_split_thm' should be modified to filter it *) 

782 
(* out *) 

20217
25b068a99d2b
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parents:
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diff
changeset

783 
 (t, ts) => ( 
20268  784 
warning ("Lin. Arith.: split rule for " ^ Sign.string_of_term sg t ^ 
785 
" (with " ^ Int.toString (length ts) ^ 

20217
25b068a99d2b
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webertj
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diff
changeset

786 
" argument(s)) not implemented; proof reconstruction is likely to fail"); 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

787 
NONE 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

788 
)) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

789 
) 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

790 
end; 
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

791 

20268  792 
(* remove terms that do not satisfy 'p'; change the order of the remaining *) 
20217
25b068a99d2b
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parents:
20044
diff
changeset

793 
(* terms in the same way as filter_prems_tac does *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
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diff
changeset

794 

20268  795 
fun filter_prems_tac_items (p : term > bool) (terms : term list) : term list = 
20217
25b068a99d2b
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parents:
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diff
changeset

796 
let 
25b068a99d2b
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parents:
20044
diff
changeset

797 
fun filter_prems (t, (left, right)) = 
25b068a99d2b
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webertj
parents:
20044
diff
changeset

798 
if p t then (left, right @ [t]) else (left @ right, []) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

799 
val (left, right) = foldl filter_prems ([], []) terms 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

800 
in 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

801 
right @ left 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

802 
end; 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

803 

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

804 
(* 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:
20044
diff
changeset

805 
(* subgoal that has 'terms' as premises *) 
25b068a99d2b
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webertj
parents:
20044
diff
changeset

806 

20268  807 
fun negated_term_occurs_positively (terms : term list) : bool = 
808 
List.exists 

809 
(fn (Trueprop $ (Const ("Not", _) $ t)) => member (op aconv) terms (Trueprop $ t) 

810 
 _ => false) 

811 
terms; 

20217
25b068a99d2b
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diff
changeset

812 

20268  813 
fun pre_decomp sg (Ts : typ list, terms : term list) : (typ list * term list) list = 
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

814 
let 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

815 
(* 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

816 
(* splitting is possible *) 
20271
e76e77e0d615
fixed a bug in function poly: decomposition of products
webertj
parents:
20268
diff
changeset

817 
fun split_loop ([] : (typ list * term list) list) = ([] : (typ list * term list) list) 
20268  818 
 split_loop (subgoal::subgoals) = ( 
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

819 
case split_once_items sg subgoal of 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

820 
SOME new_subgoals => split_loop (new_subgoals @ subgoals) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

821 
 NONE => subgoal :: split_loop subgoals 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

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

823 
fun is_relevant t = isSome (decomp sg t) 
20268  824 
(* filter_prems_tac is_relevant: *) 
825 
val relevant_terms = filter_prems_tac_items is_relevant terms 

826 
(* split_tac, NNF normalization: *) 

827 
val split_goals = split_loop [(Ts, relevant_terms)] 

828 
(* necessary because split_once_tac may normalize terms: *) 

829 
val beta_eta_norm = map (apsnd (map (Envir.eta_contract o Envir.beta_norm))) split_goals 

830 
(* TRY (etac notE) THEN eq_assume_tac: *) 

831 
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

832 
in 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

833 
result 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

834 
end; 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

835 

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

836 
(* takes the ith subgoal [ A1; ...; An ] ==> B to *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

837 
(* An > ... > A1 > B, performs splitting with the given 'split_thms' *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

838 
(* (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

839 
(* performs NNFnormalization of ~P, and eliminates conjunctions, *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

840 
(* 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

841 
(* 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

842 
(* 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

843 
(* !fast_arith_split_limit splits are possible. *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

844 

20850  845 
local 
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

846 
val nnf_simpset = 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

847 
empty_ss setmkeqTrue mk_eq_True 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

848 
setmksimps (mksimps mksimps_pairs) 
20850  849 
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

850 
not_all, not_ex, not_not] 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

851 
fun prem_nnf_tac i st = 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

852 
full_simp_tac (Simplifier.theory_context (Thm.theory_of_thm st) nnf_simpset) i st 
20850  853 
in 
854 

855 
fun split_once_tac (split_thms : thm list) (i : int) : tactic = 

856 
let 

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

857 
fun cond_split_tac i st = 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

858 
let 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

859 
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

860 
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

861 
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

862 
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

863 
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

864 
in 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

865 
if length splits > !fast_arith_split_limit then 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

866 
no_tac st 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

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

868 
split_tac split_thms i st 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

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

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

871 
EVERY' [ 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

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

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

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

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

876 
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

877 
] i 
20850  878 
end 
879 

880 
end; (* local *) 

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

881 

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

882 
(* 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

883 
(* 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

884 
(* 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

885 
(* 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

886 

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

887 
fun pre_tac i st = 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

888 
let 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

889 
val sg = theory_of_thm st 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

890 
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

891 
fun is_relevant t = isSome (decomp sg t) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

892 
in 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

893 
DETERM ( 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

894 
TRY (filter_prems_tac is_relevant i) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

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

896 
(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

897 
THEN_ALL_NEW 
20268  898 
((fn j => PRIMITIVE 
22900  899 
(Conv.fconv_rule 
900 
(Conv.goals_conv (equal j) (Drule.beta_eta_conversion)))) 

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

901 
THEN' 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

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

903 
) i 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

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

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

906 

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

907 
end; (* LA_Data_Ref *) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

908 

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

909 

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

910 
structure Fast_Arith = 
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

911 
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

912 

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

913 
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

914 
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

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

916 
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

917 
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

918 

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

919 
(* reduce contradictory <= to False. 
22838  920 
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

921 

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

922 
val init_lin_arith_data = 
18708  923 
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

924 
{add_mono_thms = add_mono_thms @ 
22838  925 
@{thms add_mono_thms_ordered_semiring} @ @{thms add_mono_thms_ordered_field}, 
10693  926 
mult_mono_thms = mult_mono_thms, 
10574
8f98f0301d67
Linear arithmetic now copes with mixed nat/int formulae.
nipkow
parents:
10516
diff
changeset

927 
inj_thms = inj_thms, 
21243  928 
lessD = lessD @ [thm "Suc_leI"], 
22887  929 
neqE = [@{thm linorder_neqE_nat}, @{thm linorder_neqE_ordered_idom}], 
22838  930 
simpset = HOL_basic_ss 
931 
addsimps [@{thm "add_zero_left"}, @{thm "add_zero_right"}, 

932 
@{thm "Zero_not_Suc"}, @{thm "Suc_not_Zero"}, @{thm "le_0_eq"}, @{thm "One_nat_def"}, 

933 
@{thm "order_less_irrefl"}, @{thm "zero_neq_one"}, @{thm "zero_less_one"}, 

934 
@{thm "zero_le_one"}, @{thm "zero_neq_one"} RS not_sym, @{thm "not_one_le_zero"}, 

935 
@{thm "not_one_less_zero"}] 

936 
addsimprocs [ab_group_add_cancel.sum_conv, ab_group_add_cancel.rel_conv] 

937 
(*abel_cancel helps it work in abstract algebraic domains*) 

938 
addsimprocs nat_cancel_sums_add}) #> 

18708  939 
arith_discrete "nat"; 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

940 

13462  941 
val fast_nat_arith_simproc = 
16834  942 
Simplifier.simproc (the_context ()) "fast_nat_arith" 
13462  943 
["(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

944 

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

945 
(* 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

946 
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

947 
*not* themselves (in)equalities, because the latter activate 
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

948 
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

949 
solver all the time rather than add the additional check. *) 
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

950 

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

953 

10516  954 
local 
955 

13499  956 
fun raw_arith_tac ex i st = 
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

957 
(* FIXME: K true should be replaced by a sensible test (perhaps "isSome o 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

958 
decomp sg"?) to speed things up in case there are lots of irrelevant 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

959 
terms involved; elimination of min/max can be optimized: 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

960 
(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

961 
(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

962 
*) 
13499  963 
refute_tac (K true) 
20217
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

964 
(* 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

965 
(* split_tac may use split theorems that have not been implemented in *) 
20268  966 
(* fast_arith_tac (cf. pre_decomp and split_once_items above), and *) 
967 
(* fast_arith_split_limit may trigger. *) 

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

968 
(* 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

969 
(* 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

970 
(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

971 
(fast_ex_arith_tac ex) 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

972 
i st; 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

973 

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

974 
fun arith_theory_tac i st = 
40757f475eb0
additional list of tactics that can be added to arith
webertj
parents:
20280
diff
changeset

975 
let 
40757f475eb0
additional list of tactics that can be added to arith
webertj
parents:
20280
diff
changeset

976 
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

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

978 
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

979 
end; 
13877
a6b825ee48d9
Added hook for presburger arithmetic decision procedure.
berghofe
parents:
13517
diff
changeset

980 

10516  981 
in 
982 

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

983 
val simple_arith_tac = FIRST' [fast_arith_tac, 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

984 
ObjectLogic.atomize_tac THEN' raw_arith_tac true]; 
13877
a6b825ee48d9
Added hook for presburger arithmetic decision procedure.
berghofe
parents:
13517
diff
changeset

985 

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

986 
val arith_tac = FIRST' [fast_arith_tac, 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

987 
ObjectLogic.atomize_tac THEN' raw_arith_tac true, 
20412
40757f475eb0
additional list of tactics that can be added to arith
webertj
parents:
20280
diff
changeset

988 
arith_theory_tac]; 
13877
a6b825ee48d9
Added hook for presburger arithmetic decision procedure.
berghofe
parents:
13517
diff
changeset

989 

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

990 
val silent_arith_tac = FIRST' [fast_arith_tac, 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

991 
ObjectLogic.atomize_tac THEN' raw_arith_tac false, 
20412
40757f475eb0
additional list of tactics that can be added to arith
webertj
parents:
20280
diff
changeset

992 
arith_theory_tac]; 
10516  993 

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

994 
fun arith_method prems = 
25b068a99d2b
linear arithmetic splits certain operators (e.g. min, max, abs)
webertj
parents:
20044
diff
changeset

995 
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

996 

10516  997 
end; 
998 

15195  999 
(* antisymmetry: 
15197  1000 
combines x <= y (or ~(y < x)) and y <= x (or ~(x < y)) into x = y 
15195  1001 

1002 
local 

1003 
val antisym = mk_meta_eq order_antisym 

22548  1004 
val not_lessD = @{thm linorder_not_less} RS iffD1 
15195  1005 
fun prp t thm = (#prop(rep_thm thm) = t) 
1006 
in 

1007 
fun antisym_eq prems thm = 

1008 
let 

1009 
val r = #prop(rep_thm thm); 

1010 
in 

1011 
case r of 

22997  1012 
Tr $ ((c as Const(@{const_name Orderings.less_eq},T)) $ s $ t) => 
15195  1013 
let val r' = Tr $ (c $ t $ s) 
1014 
in 

1015 
case Library.find_first (prp r') prems of 

15531  1016 
NONE => 
22997  1017 
let val r' = Tr $ (HOLogic.Not $ (Const(@{const_name Orderings.less},T) $ s $ t)) 
15195  1018 
in case Library.find_first (prp r') prems of 
15531  1019 
NONE => [] 
1020 
 SOME thm' => [(thm' RS not_lessD) RS (thm RS antisym)] 

15195  1021 
end 
15531  1022 
 SOME thm' => [thm' RS (thm RS antisym)] 
15195  1023 
end 
22997  1024 
 Tr $ (Const("Not",_) $ (Const(@{const_name Orderings.less},T) $ s $ t)) => 
1025 
let val r' = Tr $ (Const(@{const_name Orderings.less_eq},T) $ s $ t) 

15195  1026 
in 
1027 
case Library.find_first (prp r') prems of 

15531  1028 
NONE => 
22997  1029 
let val r' = Tr $ (HOLogic.Not $ (Const(@{const_name Orderings.less},T) $ t $ s)) 
15195  1030 
in case Library.find_first (prp r') prems of 
15531  1031 
NONE => [] 
1032 
 SOME thm' => 

15195  1033 
[(thm' RS not_lessD) RS ((thm RS not_lessD) RS antisym)] 
1034 
end 

15531  1035 
 SOME thm' => [thm' RS ((thm RS not_lessD) RS antisym)] 
15195  1036 
end 
1037 
 _ => [] 

1038 
end 

1039 
handle THM _ => [] 

1040 
end; 

15197  1041 
*) 
9436
62bb04ab4b01
rearranged setup of arithmetic procedures, avoiding global reference values;
wenzelm
parents:
diff
changeset

1042 

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

1043 
(* theory setup *) 
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 
val arith_setup = 
18708  1046 
init_lin_arith_data #> 
1047 
(fn thy => (Simplifier.change_simpset_of thy (fn ss => ss 

17875  1048 
addsimprocs (nat_cancel_sums @ [fast_nat_arith_simproc]) 
18708  1049 
addSolver (mk_solver' "lin. arith." Fast_Arith.cut_lin_arith_tac)); thy)) #> 
15221  1050 
Method.add_methods 
21879  1051 
[("arith", (arith_method o fst) oo Method.syntax Args.bang_facts, 
18708  1052 
"decide linear arithmethic")] #> 
18728  1053 
Attrib.add_attributes [("arith_split", Attrib.no_args arith_split_add, 
18708  1054 
"declaration of split rules for arithmetic procedure")]; 