(* Title: HOL/Decision_Procs/ferrack_tac.ML Author: Amine Chaieb, TU Muenchen *) structure Ferrack_Tac = struct val trace = ref false; fun trace_msg s = if !trace then tracing s else (); val ferrack_ss = let val ths = [@{thm real_of_int_inject}, @{thm real_of_int_less_iff}, @{thm real_of_int_le_iff}] in @{simpset} delsimps ths addsimps (map (fn th => th RS sym) ths) end; val binarith = @{thms normalize_bin_simps} @ @{thms pred_bin_simps} @ @{thms succ_bin_simps} @ @{thms add_bin_simps} @ @{thms minus_bin_simps} @ @{thms mult_bin_simps}; val comp_arith = binarith @ simp_thms val zdvd_int = @{thm zdvd_int}; val zdiff_int_split = @{thm zdiff_int_split}; val all_nat = @{thm all_nat}; val ex_nat = @{thm ex_nat}; val number_of1 = @{thm number_of1}; val number_of2 = @{thm number_of2}; val split_zdiv = @{thm split_zdiv}; val split_zmod = @{thm split_zmod}; val mod_div_equality' = @{thm mod_div_equality'}; val split_div' = @{thm split_div'}; val Suc_plus1 = @{thm Suc_plus1}; val imp_le_cong = @{thm imp_le_cong}; val conj_le_cong = @{thm conj_le_cong}; val mod_add_left_eq = @{thm mod_add_left_eq} RS sym; val mod_add_right_eq = @{thm mod_add_right_eq} RS sym; val nat_div_add_eq = @{thm div_add1_eq} RS sym; val int_div_add_eq = @{thm zdiv_zadd1_eq} RS sym; val ZDIVISION_BY_ZERO_MOD = @{thm DIVISION_BY_ZERO} RS conjunct2; val ZDIVISION_BY_ZERO_DIV = @{thm DIVISION_BY_ZERO} RS conjunct1; fun prepare_for_linr sg q fm = let val ps = Logic.strip_params fm val hs = map HOLogic.dest_Trueprop (Logic.strip_assums_hyp fm) val c = HOLogic.dest_Trueprop (Logic.strip_assums_concl fm) fun mk_all ((s, T), (P,n)) = if 0 mem loose_bnos P then (HOLogic.all_const T $ Abs (s, T, P), n) else (incr_boundvars ~1 P, n-1) fun mk_all2 (v, t) = HOLogic.all_const (fastype_of v) $ lambda v t; val rhs = hs (* val (rhs,irhs) = List.partition (relevant (rev ps)) hs *) val np = length ps val (fm',np) = foldr (fn ((x, T), (fm,n)) => mk_all ((x, T), (fm,n))) (foldr HOLogic.mk_imp c rhs, np) ps val (vs, _) = List.partition (fn t => q orelse (type_of t) = HOLogic.natT) (OldTerm.term_frees fm' @ OldTerm.term_vars fm'); val fm2 = foldr mk_all2 fm' vs in (fm2, np + length vs, length rhs) end; (*Object quantifier to meta --*) fun spec_step n th = if (n=0) then th else (spec_step (n-1) th) RS spec ; (* object implication to meta---*) fun mp_step n th = if (n=0) then th else (mp_step (n-1) th) RS mp; fun linr_tac ctxt q i = (ObjectLogic.atomize_prems_tac i) THEN (REPEAT_DETERM (split_tac [@{thm split_min}, @{thm split_max}, @{thm abs_split}] i)) THEN (fn st => let val g = List.nth (prems_of st, i - 1) val thy = ProofContext.theory_of ctxt (* Transform the term*) val (t,np,nh) = prepare_for_linr thy q g (* Some simpsets for dealing with mod div abs and nat*) val simpset0 = Simplifier.theory_context thy HOL_basic_ss addsimps comp_arith val ct = cterm_of thy (HOLogic.mk_Trueprop t) (* Theorem for the nat --> int transformation *) val pre_thm = Seq.hd (EVERY [simp_tac simpset0 1, TRY (simp_tac (Simplifier.theory_context thy ferrack_ss) 1)] (trivial ct)) fun assm_tac i = REPEAT_DETERM_N nh (assume_tac i) (* The result of the quantifier elimination *) val (th, tac) = case (prop_of pre_thm) of Const ("==>", _) $ (Const ("Trueprop", _) $ t1) $ _ => let val pth = linr_oracle (cterm_of thy (Pattern.eta_long [] t1)) in (trace_msg ("calling procedure with term:\n" ^ Syntax.string_of_term ctxt t1); ((pth RS iffD2) RS pre_thm, assm_tac (i + 1) THEN (if q then I else TRY) (rtac TrueI i))) end | _ => (pre_thm, assm_tac i) in (rtac (((mp_step nh) o (spec_step np)) th) i THEN tac) st end handle Subscript => no_tac st); fun linr_meth src = Method.syntax (Args.mode "no_quantify") src #> (fn (q, ctxt) => SIMPLE_METHOD' (linr_tac ctxt (not q))); val setup = Method.add_method ("rferrack", linr_meth, "decision procedure for linear real arithmetic"); end