structure LinZTac = struct val trace = ref false; fun trace_msg s = if !trace then tracing s else (); val cooper_ss = @{simpset}; val nT = HOLogic.natT; val binarith = map thm ["Pls_0_eq", "Min_1_eq"]; 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 nat_mod_add_eq = @{thm mod_add1_eq} RS sym; val nat_mod_add_left_eq = @{thm mod_add_left_eq} RS sym; val nat_mod_add_right_eq = @{thm mod_add_right_eq} RS sym; val int_mod_add_eq = @{thm "zmod_zadd1_eq"} RS sym; val int_mod_add_left_eq = @{thm "zmod_zadd_left_eq"} RS sym; val int_mod_add_right_eq = @{thm "zmod_zadd_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; (* val fn_rews = List.concat (map thms ["allpairs.simps","iupt.simps","decr.simps", "decrnum.simps","disjuncts.simps","simpnum.simps", "simpfm.simps","numadd.simps","nummul.simps","numneg_def","numsub","simp_num_pair_def","not.simps","prep.simps","qelim.simps","minusinf.simps","plusinf.simps","rsplit0.simps","rlfm.simps","Υ.simps","υ.simps","linrqe_def", "ferrack_def", "Let_def", "numsub_def", "numneg_def","DJ_def", "imp_def", "evaldjf_def", "djf_def", "split_def", "eq_def", "disj_def", "simp_num_pair_def", "conj_def", "lt_def", "neq_def","gt_def"]); *) fun prepare_for_linz 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) = nT) (term_frees fm' @ 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 linz_tac ctxt q i = ObjectLogic.atomize_prems_tac 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_linz q g (* Some simpsets for dealing with mod div abs and nat*) val mod_div_simpset = HOL_basic_ss addsimps [refl,nat_mod_add_eq, nat_mod_add_left_eq, nat_mod_add_right_eq, int_mod_add_eq, int_mod_add_right_eq, int_mod_add_left_eq, nat_div_add_eq, int_div_add_eq, @{thm mod_self}, @{thm "zmod_self"}, @{thm DIVISION_BY_ZERO_MOD}, @{thm DIVISION_BY_ZERO_DIV}, ZDIVISION_BY_ZERO_MOD,ZDIVISION_BY_ZERO_DIV, @{thm "zdiv_zero"}, @{thm "zmod_zero"}, @{thm "div_0"}, @{thm "mod_0"}, @{thm "zdiv_1"}, @{thm "zmod_1"}, @{thm "div_1"}, @{thm "mod_1"}, Suc_plus1] addsimps @{thms add_ac} addsimprocs [cancel_div_mod_proc] val simpset0 = HOL_basic_ss addsimps [mod_div_equality', Suc_plus1] addsimps comp_arith addsplits [split_zdiv, split_zmod, split_div', @{thm "split_min"}, @{thm "split_max"}] (* Simp rules for changing (n::int) to int n *) val simpset1 = HOL_basic_ss addsimps [nat_number_of_def, zdvd_int] @ map (fn r => r RS sym) [@{thm int_int_eq}, @{thm zle_int}, @{thm zless_int}, @{thm zadd_int}, @{thm zmult_int}] addsplits [zdiff_int_split] (*simp rules for elimination of int n*) val simpset2 = HOL_basic_ss addsimps [@{thm nat_0_le}, @{thm all_nat}, @{thm ex_nat}, @{thm number_of1}, @{thm number_of2}, @{thm int_0}, @{thm int_1}] addcongs [@{thm conj_le_cong}, @{thm imp_le_cong}] (* simp rules for elimination of abs *) val simpset3 = HOL_basic_ss addsplits [@{thm abs_split}] val ct = cterm_of thy (HOLogic.mk_Trueprop t) (* Theorem for the nat --> int transformation *) val pre_thm = Seq.hd (EVERY [simp_tac mod_div_simpset 1, simp_tac simpset0 1, TRY (simp_tac simpset1 1), TRY (simp_tac simpset2 1), TRY (simp_tac simpset3 1), TRY (simp_tac cooper_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 = linzqe_oracle thy (Pattern.eta_long [] t1) in ((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 linz_args meth = let val parse_flag = Args.$$$ "no_quantify" >> (K (K false)); in Method.simple_args (Scan.optional (Args.$$$ "(" |-- Scan.repeat1 parse_flag --| Args.$$$ ")") [] >> curry (Library.foldl op |>) true) (fn q => fn ctxt => meth ctxt q 1) end; fun linz_method ctxt q i = Method.METHOD (fn facts => Method.insert_tac facts 1 THEN linz_tac ctxt q i); val setup = Method.add_method ("cooper", linz_args linz_method, "decision procedure for linear integer arithmetic"); end