(* Title: HOL/simpdata.ML ID: $Id: simpdata.ML,v 1.197 2007/07/29 12:29:49 wenzelm Exp $ Author: Tobias Nipkow Copyright 1991 University of Cambridge Instantiation of the generic simplifier for HOL. *) (** tools setup **) structure Quantifier1 = Quantifier1Fun (struct (*abstract syntax*) fun dest_eq ((c as Const("op =",_)) $ s $ t) = SOME (c, s, t) | dest_eq _ = NONE; fun dest_conj ((c as Const("op &",_)) $ s $ t) = SOME (c, s, t) | dest_conj _ = NONE; fun dest_imp ((c as Const("op -->",_)) $ s $ t) = SOME (c, s, t) | dest_imp _ = NONE; val conj = HOLogic.conj val imp = HOLogic.imp (*rules*) val iff_reflection = @{thm eq_reflection} val iffI = @{thm iffI} val iff_trans = @{thm trans} val conjI= @{thm conjI} val conjE= @{thm conjE} val impI = @{thm impI} val mp = @{thm mp} val uncurry = @{thm uncurry} val exI = @{thm exI} val exE = @{thm exE} val iff_allI = @{thm iff_allI} val iff_exI = @{thm iff_exI} val all_comm = @{thm all_comm} val ex_comm = @{thm ex_comm} end); structure Simpdata = struct fun mk_meta_eq r = r RS @{thm eq_reflection}; fun safe_mk_meta_eq r = mk_meta_eq r handle Thm.THM _ => r; fun mk_eq th = case concl_of th (*expects Trueprop if not == *) of Const ("==",_) $ _ $ _ => th | _ $ (Const ("op =", _) $ _ $ _) => mk_meta_eq th | _ $ (Const ("Not", _) $ _) => th RS @{thm Eq_FalseI} | _ => th RS @{thm Eq_TrueI} fun mk_eq_True r = SOME (r RS @{thm meta_eq_to_obj_eq} RS @{thm Eq_TrueI}) handle Thm.THM _ => NONE; (* Produce theorems of the form (P1 =simp=> ... =simp=> Pn => x == y) ==> (P1 =simp=> ... =simp=> Pn => x = y) *) fun lift_meta_eq_to_obj_eq i st = let fun count_imp (Const ("HOL.simp_implies", _) $ P $ Q) = 1 + count_imp Q | count_imp _ = 0; val j = count_imp (Logic.strip_assums_concl (List.nth (prems_of st, i - 1))) in if j = 0 then @{thm meta_eq_to_obj_eq} else let val Ps = map (fn k => Free ("P" ^ string_of_int k, propT)) (1 upto j); fun mk_simp_implies Q = foldr (fn (R, S) => Const ("HOL.simp_implies", propT --> propT --> propT) $ R $ S) Q Ps val aT = TFree ("'a", HOLogic.typeS); val x = Free ("x", aT); val y = Free ("y", aT) in Goal.prove_global (Thm.theory_of_thm st) [] [mk_simp_implies (Logic.mk_equals (x, y))] (mk_simp_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (x, y)))) (fn prems => EVERY [rewrite_goals_tac @{thms simp_implies_def}, REPEAT (ares_tac (@{thm meta_eq_to_obj_eq} :: map (rewrite_rule @{thms simp_implies_def}) prems) 1)]) end end; (*Congruence rules for = (instead of ==)*) fun mk_meta_cong rl = zero_var_indexes (let val rl' = Seq.hd (TRYALL (fn i => fn st => rtac (lift_meta_eq_to_obj_eq i st) i st) rl) in mk_meta_eq rl' handle THM _ => if can Logic.dest_equals (concl_of rl') then rl' else error "Conclusion of congruence rules must be =-equality" end); fun mk_atomize pairs = let fun atoms thm = let fun res th = map (fn rl => th RS rl); (*exception THM*) fun res_fixed rls = if Thm.maxidx_of (Thm.adjust_maxidx_thm ~1 thm) = ~1 then res thm rls else Variable.trade (K (fn [thm'] => res thm' rls)) (Variable.thm_context thm) [thm]; in case concl_of thm of Const ("Trueprop", _) $ p => (case head_of p of Const (a, _) => (case AList.lookup (op =) pairs a of SOME rls => (maps atoms (res_fixed rls) handle THM _ => [thm]) | NONE => [thm]) | _ => [thm]) | _ => [thm] end; in atoms end; fun mksimps pairs = map_filter (try mk_eq) o mk_atomize pairs o gen_all; fun unsafe_solver_tac prems = (fn i => REPEAT_DETERM (match_tac @{thms simp_impliesI} i)) THEN' FIRST' [resolve_tac (reflexive_thm :: @{thm TrueI} :: @{thm refl} :: prems), atac, etac @{thm FalseE}]; val unsafe_solver = mk_solver "HOL unsafe" unsafe_solver_tac; (*No premature instantiation of variables during simplification*) fun safe_solver_tac prems = (fn i => REPEAT_DETERM (match_tac @{thms simp_impliesI} i)) THEN' FIRST' [match_tac (reflexive_thm :: @{thm TrueI} :: @{thm refl} :: prems), eq_assume_tac, ematch_tac @{thms FalseE}]; val safe_solver = mk_solver "HOL safe" safe_solver_tac; structure SplitterData = struct structure Simplifier = Simplifier val mk_eq = mk_eq val meta_eq_to_iff = @{thm meta_eq_to_obj_eq} val iffD = @{thm iffD2} val disjE = @{thm disjE} val conjE = @{thm conjE} val exE = @{thm exE} val contrapos = @{thm contrapos_nn} val contrapos2 = @{thm contrapos_pp} val notnotD = @{thm notnotD} end; structure Splitter = SplitterFun(SplitterData); val split_tac = Splitter.split_tac; val split_inside_tac = Splitter.split_inside_tac; val op addsplits = Splitter.addsplits; val op delsplits = Splitter.delsplits; val Addsplits = Splitter.Addsplits; val Delsplits = Splitter.Delsplits; (* integration of simplifier with classical reasoner *) structure Clasimp = ClasimpFun (structure Simplifier = Simplifier and Splitter = Splitter and Classical = Classical and Blast = Blast val iffD1 = @{thm iffD1} val iffD2 = @{thm iffD2} val notE = @{thm notE}); open Clasimp; val _ = ML_Context.value_antiq "clasimpset" (Scan.succeed ("clasimpset", "Clasimp.local_clasimpset_of (ML_Context.the_local_context ())")); val mksimps_pairs = [("op -->", [@{thm mp}]), ("op &", [@{thm conjunct1}, @{thm conjunct2}]), ("All", [@{thm spec}]), ("True", []), ("False", []), ("HOL.If", [@{thm if_bool_eq_conj} RS @{thm iffD1}])]; val HOL_basic_ss = Simplifier.theory_context @{theory} empty_ss setsubgoaler asm_simp_tac setSSolver safe_solver setSolver unsafe_solver setmksimps (mksimps mksimps_pairs) setmkeqTrue mk_eq_True setmkcong mk_meta_cong; fun hol_simplify rews = Simplifier.full_simplify (HOL_basic_ss addsimps rews); fun unfold_tac ths = let val ss0 = Simplifier.clear_ss HOL_basic_ss addsimps ths in fn ss => ALLGOALS (full_simp_tac (Simplifier.inherit_context ss ss0)) end; (** generic refutation procedure **) (* parameters: test: term -> bool tests if a term is at all relevant to the refutation proof; if not, then it can be discarded. Can improve performance, esp. if disjunctions can be discarded (no case distinction needed!). prep_tac: int -> tactic A preparation tactic to be applied to the goal once all relevant premises have been moved to the conclusion. ref_tac: int -> tactic the actual refutation tactic. Should be able to deal with goals [| A1; ...; An |] ==> False where the Ai are atomic, i.e. no top-level &, | or EX *) local val nnf_simpset = empty_ss setmkeqTrue mk_eq_True setmksimps (mksimps mksimps_pairs) addsimps [@{thm imp_conv_disj}, @{thm iff_conv_conj_imp}, @{thm de_Morgan_disj}, @{thm de_Morgan_conj}, @{thm not_all}, @{thm not_ex}, @{thm not_not}]; fun prem_nnf_tac i st = full_simp_tac (Simplifier.theory_context (Thm.theory_of_thm st) nnf_simpset) i st; in fun refute_tac test prep_tac ref_tac = let val refute_prems_tac = REPEAT_DETERM (eresolve_tac [@{thm conjE}, @{thm exE}] 1 ORELSE filter_prems_tac test 1 ORELSE etac @{thm disjE} 1) THEN (DETERM (etac @{thm notE} 1 THEN eq_assume_tac 1) ORELSE ref_tac 1); in EVERY'[TRY o filter_prems_tac test, REPEAT_DETERM o etac @{thm rev_mp}, prep_tac, rtac @{thm ccontr}, prem_nnf_tac, SELECT_GOAL (DEPTH_SOLVE refute_prems_tac)] end; end; val defALL_regroup = Simplifier.simproc @{theory} "defined ALL" ["ALL x. P x"] Quantifier1.rearrange_all; val defEX_regroup = Simplifier.simproc @{theory} "defined EX" ["EX x. P x"] Quantifier1.rearrange_ex; val simpset_simprocs = HOL_basic_ss addsimprocs [defALL_regroup, defEX_regroup] end; structure Splitter = Simpdata.Splitter; structure Clasimp = Simpdata.Clasimp;