(* Title: HOL/Decision_Procs/mir_tac.ML Author: Amine Chaieb, TU Muenchen *) structure Mir_Tac = struct val trace = ref false; fun trace_msg s = if !trace then tracing s else (); val mir_ss = let val ths = map thm ["real_of_int_inject", "real_of_int_less_iff", "real_of_int_le_iff"] in @{simpset} delsimps ths addsimps (map (fn th => th RS sym) ths) end; val nT = HOLogic.natT; val nat_arith = map thm ["add_nat_number_of", "diff_nat_number_of", "mult_nat_number_of", "eq_nat_number_of", "less_nat_number_of"]; val comp_arith = (map thm ["Let_def", "if_False", "if_True", "add_0", "add_Suc", "add_number_of_left", "mult_number_of_left", "Suc_eq_add_numeral_1"])@ (map (fn s => thm s RS sym) ["numeral_1_eq_1", "numeral_0_eq_0"]) @ @{thms arith_simps} @ nat_arith @ @{thms rel_simps} val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"}, @{thm "real_of_nat_number_of"}, @{thm "real_of_nat_Suc"}, @{thm "real_of_nat_one"}, @{thm "real_of_one"}, @{thm "real_of_int_zero"}, @{thm "real_of_nat_zero"}, @{thm "Ring_and_Field.divide_zero"}, @{thm "divide_divide_eq_left"}, @{thm "times_divide_eq_right"}, @{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"}, @{thm "diff_def"}, @{thm "minus_divide_left"}] val comp_ths = ths @ comp_arith @ 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_eq = @{thm "mod_add_eq"} RS sym; 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_mir thy 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) (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 mir_tac ctxt q i = (ObjectLogic.atomize_prems_tac i) THEN (simp_tac (HOL_basic_ss addsimps [@{thm "abs_ge_zero"}] addsimps simp_thms) 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_mir thy q g (* Some simpsets for dealing with mod div abs and nat*) val mod_div_simpset = HOL_basic_ss addsimps [refl, mod_add_eq, @{thm "mod_self"}, @{thm "zmod_self"}, @{thm "zdiv_zero"},@{thm "zmod_zero"},@{thm "div_0"}, @{thm "mod_0"}, @{thm "div_by_1"}, @{thm "mod_by_1"}, @{thm "div_1"}, @{thm "mod_1"}, @{thm "Suc_plus1"}] addsimps @{thms add_ac} addsimprocs [cancel_div_mod_proc] val simpset0 = HOL_basic_ss addsimps [mod_div_equality', Suc_plus1] addsimps comp_ths addsplits [@{thm "split_zdiv"}, @{thm "split_zmod"}, @{thm "split_div'"}, @{thm "split_min"}, @{thm "split_max"}] (* Simp rules for changing (n::int) to int n *) val simpset1 = HOL_basic_ss addsimps [@{thm "nat_number_of_def"}, @{thm "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 [@{thm "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 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 mir_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 = (* If quick_and_dirty then run without proof generation as oracle*) if !quick_and_dirty then mirfr_oracle (false, cterm_of thy (Pattern.eta_long [] t1)) else mirfr_oracle (true, 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 mir_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) end; fun mir_method ctxt q = SIMPLE_METHOD' (mir_tac ctxt q); val setup = Method.add_method ("mir", mir_args mir_method, "decision procedure for MIR arithmetic"); end