(* Title: HOL/simpdata.ML ID: $Id$ Author: Tobias Nipkow Copyright 1991 University of Cambridge Instantiation of the generic simplifier *) section "Simplifier"; open Simplifier; (*** Addition of rules to simpsets and clasets simultaneously ***) (*Takes UNCONDITIONAL theorems of the form A<->B to the Safe Intr rule B==>A and the Safe Destruct rule A==>B. Also ~A goes to the Safe Elim rule A ==> ?R Failing other cases, A is added as a Safe Intr rule*) local val iff_const = HOLogic.eq_const HOLogic.boolT; fun addIff th = (case HOLogic.dest_Trueprop (#prop(rep_thm th)) of (Const("Not",_) $ A) => AddSEs [zero_var_indexes (th RS notE)] | (con $ _ $ _) => if con=iff_const then (AddSIs [zero_var_indexes (th RS iffD2)]; AddSDs [zero_var_indexes (th RS iffD1)]) else AddSIs [th] | _ => AddSIs [th]; Addsimps [th]) handle _ => error ("AddIffs: theorem must be unconditional\n" ^ string_of_thm th) fun delIff th = (case HOLogic.dest_Trueprop (#prop(rep_thm th)) of (Const("Not",_) $ A) => Delrules [zero_var_indexes (th RS notE)] | (con $ _ $ _) => if con=iff_const then Delrules [zero_var_indexes (th RS iffD2), make_elim (zero_var_indexes (th RS iffD1))] else Delrules [th] | _ => Delrules [th]; Delsimps [th]) handle _ => warning("DelIffs: ignoring conditional theorem\n" ^ string_of_thm th) in val AddIffs = seq addIff val DelIffs = seq delIff end; qed_goal "meta_eq_to_obj_eq" HOL.thy "x==y ==> x=y" (fn [prem] => [rewtac prem, rtac refl 1]); local fun prover s = prove_goal HOL.thy s (K [blast_tac HOL_cs 1]); val P_imp_P_iff_True = prover "P --> (P = True)" RS mp; val P_imp_P_eq_True = P_imp_P_iff_True RS eq_reflection; val not_P_imp_P_iff_F = prover "~P --> (P = False)" RS mp; val not_P_imp_P_eq_False = not_P_imp_P_iff_F RS eq_reflection; fun atomize pairs = let fun atoms th = (case concl_of th of Const("Trueprop",_) $ p => (case head_of p of Const(a,_) => (case assoc(pairs,a) of Some(rls) => flat (map atoms ([th] RL rls)) | None => [th]) | _ => [th]) | _ => [th]) in atoms end; fun gen_all th = forall_elim_vars (#maxidx(rep_thm th)+1) th; in fun mk_meta_eq r = r RS eq_reflection; fun mk_meta_eq_True r = Some(r RS meta_eq_to_obj_eq RS P_imp_P_eq_True); fun mk_meta_eq_simp r = case concl_of r of Const("==",_)$_$_ => r | _$(Const("op =",_)$lhs$rhs) => mk_meta_eq r | _$(Const("Not",_)$_) => r RS not_P_imp_P_eq_False | _ => r RS P_imp_P_eq_True; (* last 2 lines requires all formulae to be of the from Trueprop(.) *) val simp_thms = map prover [ "(x=x) = True", "(~True) = False", "(~False) = True", "(~ ~ P) = P", "(~P) ~= P", "P ~= (~P)", "(P ~= Q) = (P = (~Q))", "(True=P) = P", "(P=True) = P", "(False=P) = (~P)", "(P=False) = (~P)", "(True --> P) = P", "(False --> P) = True", "(P --> True) = True", "(P --> P) = True", "(P --> False) = (~P)", "(P --> ~P) = (~P)", "(P & True) = P", "(True & P) = P", "(P & False) = False", "(False & P) = False", "(P & P) = P", "(P & (P & Q)) = (P & Q)", "(P & ~P) = False", "(~P & P) = False", "(P | True) = True", "(True | P) = True", "(P | False) = P", "(False | P) = P", "(P | P) = P", "(P | (P | Q)) = (P | Q)", "(P | ~P) = True", "(~P | P) = True", "((~P) = (~Q)) = (P=Q)", "(!x. P) = P", "(? x. P) = P", "? x. x=t", "? x. t=x", (*two needed for the one-point-rule quantifier simplification procs*) "(? x. x=t & P(x)) = P(t)", (*essential for termination!!*) "(! x. t=x --> P(x)) = P(t)" ]; (*covers a stray case*) (*Add congruence rules for = (instead of ==) *) infix 4 addcongs delcongs; fun mk_meta_cong rl = standard(mk_meta_eq(replicate (nprems_of rl) meta_eq_to_obj_eq MRS rl)) handle THM _ => error("Premises and conclusion of congruence rules must be =-equalities"); fun ss addcongs congs = ss addeqcongs (map mk_meta_cong congs); fun ss delcongs congs = ss deleqcongs (map mk_meta_cong congs); fun Addcongs congs = (simpset_ref() := simpset() addcongs congs); fun Delcongs congs = (simpset_ref() := simpset() delcongs congs); fun mksimps pairs = map mk_meta_eq_simp o atomize pairs o gen_all; val imp_cong = impI RSN (2, prove_goal HOL.thy "(P=P')--> (P'--> (Q=Q'))--> ((P-->Q) = (P'-->Q'))" (fn _=> [blast_tac HOL_cs 1]) RS mp RS mp); (*Miniscoping: pushing in existential quantifiers*) val ex_simps = map prover ["(EX x. P x & Q) = ((EX x. P x) & Q)", "(EX x. P & Q x) = (P & (EX x. Q x))", "(EX x. P x | Q) = ((EX x. P x) | Q)", "(EX x. P | Q x) = (P | (EX x. Q x))", "(EX x. P x --> Q) = ((ALL x. P x) --> Q)", "(EX x. P --> Q x) = (P --> (EX x. Q x))"]; (*Miniscoping: pushing in universal quantifiers*) val all_simps = map prover ["(ALL x. P x & Q) = ((ALL x. P x) & Q)", "(ALL x. P & Q x) = (P & (ALL x. Q x))", "(ALL x. P x | Q) = ((ALL x. P x) | Q)", "(ALL x. P | Q x) = (P | (ALL x. Q x))", "(ALL x. P x --> Q) = ((EX x. P x) --> Q)", "(ALL x. P --> Q x) = (P --> (ALL x. Q x))"]; (* elimination of existential quantifiers in assumptions *) val ex_all_equiv = let val lemma1 = prove_goal HOL.thy "(? x. P(x) ==> PROP Q) ==> (!!x. P(x) ==> PROP Q)" (fn prems => [resolve_tac prems 1, etac exI 1]); val lemma2 = prove_goalw HOL.thy [Ex_def] "(!!x. P(x) ==> PROP Q) ==> (? x. P(x) ==> PROP Q)" (fn prems => [REPEAT(resolve_tac prems 1)]) in equal_intr lemma1 lemma2 end; end; (* Elimination of True from asumptions: *) val True_implies_equals = prove_goal HOL.thy "(True ==> PROP P) == PROP P" (K [rtac equal_intr_rule 1, atac 2, METAHYPS (fn prems => resolve_tac prems 1) 1, rtac TrueI 1]); fun prove nm thm = qed_goal nm HOL.thy thm (K [blast_tac HOL_cs 1]); prove "conj_commute" "(P&Q) = (Q&P)"; prove "conj_left_commute" "(P&(Q&R)) = (Q&(P&R))"; val conj_comms = [conj_commute, conj_left_commute]; prove "conj_assoc" "((P&Q)&R) = (P&(Q&R))"; prove "disj_commute" "(P|Q) = (Q|P)"; prove "disj_left_commute" "(P|(Q|R)) = (Q|(P|R))"; val disj_comms = [disj_commute, disj_left_commute]; prove "disj_assoc" "((P|Q)|R) = (P|(Q|R))"; prove "conj_disj_distribL" "(P&(Q|R)) = (P&Q | P&R)"; prove "conj_disj_distribR" "((P|Q)&R) = (P&R | Q&R)"; prove "disj_conj_distribL" "(P|(Q&R)) = ((P|Q) & (P|R))"; prove "disj_conj_distribR" "((P&Q)|R) = ((P|R) & (Q|R))"; prove "imp_conjR" "(P --> (Q&R)) = ((P-->Q) & (P-->R))"; prove "imp_conjL" "((P&Q) -->R) = (P --> (Q --> R))"; prove "imp_disjL" "((P|Q) --> R) = ((P-->R)&(Q-->R))"; (*These two are specialized, but imp_disj_not1 is useful in Auth/Yahalom.ML*) prove "imp_disj_not1" "((P --> Q | R)) = (~Q --> P --> R)"; prove "imp_disj_not2" "((P --> Q | R)) = (~R --> P --> Q)"; prove "imp_disj1" "((P-->Q)|R) = (P--> Q|R)"; prove "imp_disj2" "(Q|(P-->R)) = (P--> Q|R)"; prove "de_Morgan_disj" "(~(P | Q)) = (~P & ~Q)"; prove "de_Morgan_conj" "(~(P & Q)) = (~P | ~Q)"; prove "not_imp" "(~(P --> Q)) = (P & ~Q)"; prove "not_iff" "(P~=Q) = (P = (~Q))"; prove "disj_not1" "(~P | Q) = (P --> Q)"; prove "disj_not2" "(P | ~Q) = (Q --> P)"; (* changes orientation :-( *) (*Avoids duplication of subgoals after expand_if, when the true and false cases boil down to the same thing.*) prove "cases_simp" "((P --> Q) & (~P --> Q)) = Q"; prove "not_all" "(~ (! x. P(x))) = (? x.~P(x))"; prove "imp_all" "((! x. P x) --> Q) = (? x. P x --> Q)"; prove "not_ex" "(~ (? x. P(x))) = (! x.~P(x))"; prove "imp_ex" "((? x. P x) --> Q) = (! x. P x --> Q)"; prove "ex_disj_distrib" "(? x. P(x) | Q(x)) = ((? x. P(x)) | (? x. Q(x)))"; prove "all_conj_distrib" "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))"; (* '&' congruence rule: not included by default! May slow rewrite proofs down by as much as 50% *) let val th = prove_goal HOL.thy "(P=P')--> (P'--> (Q=Q'))--> ((P&Q) = (P'&Q'))" (fn _=> [blast_tac HOL_cs 1]) in bind_thm("conj_cong",standard (impI RSN (2, th RS mp RS mp))) end; let val th = prove_goal HOL.thy "(Q=Q')--> (Q'--> (P=P'))--> ((P&Q) = (P'&Q'))" (fn _=> [blast_tac HOL_cs 1]) in bind_thm("rev_conj_cong",standard (impI RSN (2, th RS mp RS mp))) end; (* '|' congruence rule: not included by default! *) let val th = prove_goal HOL.thy "(P=P')--> (~P'--> (Q=Q'))--> ((P|Q) = (P'|Q'))" (fn _=> [blast_tac HOL_cs 1]) in bind_thm("disj_cong",standard (impI RSN (2, th RS mp RS mp))) end; prove "eq_sym_conv" "(x=y) = (y=x)"; qed_goalw "o_apply" HOL.thy [o_def] "(f o g) x = f (g x)" (K [rtac refl 1]); qed_goalw "if_True" HOL.thy [if_def] "(if True then x else y) = x" (K [Blast_tac 1]); qed_goalw "if_False" HOL.thy [if_def] "(if False then x else y) = y" (K [Blast_tac 1]); qed_goal "if_P" HOL.thy "P ==> (if P then x else y) = x" (fn [prem] => [ stac (prem RS eqTrueI) 1, rtac if_True 1 ]); (* qed_goal "if_not_P" HOL.thy "~P ==> (if P then x else y) = y" (fn [prem] => [ stac (prem RS not_P_imp_P_iff_F) 1, rtac if_False 1 ]); *) qed_goalw "if_not_P" HOL.thy [if_def] "!!P. ~P ==> (if P then x else y) = y" (K [Blast_tac 1]); qed_goal "expand_if" HOL.thy "P(if Q then x else y) = ((Q --> P(x)) & (~Q --> P(y)))" (K [ res_inst_tac [("Q","Q")] (excluded_middle RS disjE) 1, stac if_P 2, stac if_not_P 1, ALLGOALS (blast_tac HOL_cs)]); qed_goal "split_if_asm" HOL.thy "P(if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))" (K [ stac expand_if 1, blast_tac HOL_cs 1]); qed_goal "if_bool_eq" HOL.thy "(if P then Q else R) = ((P-->Q) & (~P-->R))" (K [rtac expand_if 1]); (*** make simplification procedures for quantifier elimination ***) 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; val conj = HOLogic.conj val imp = HOLogic.imp (*rules*) val iff_reflection = eq_reflection val iffI = iffI val sym = sym val conjI= conjI val conjE= conjE val impI = impI val impE = impE val mp = mp val exI = exI val exE = exE val allI = allI val allE = allE end); local val ex_pattern = read_cterm (sign_of HOL.thy) ("EX x. P(x) & Q(x)",HOLogic.boolT) val all_pattern = read_cterm (sign_of HOL.thy) ("ALL x. P(x) & P'(x) --> Q(x)",HOLogic.boolT) in val defEX_regroup = mk_simproc "defined EX" [ex_pattern] Quantifier1.rearrange_ex; val defALL_regroup = mk_simproc "defined ALL" [all_pattern] Quantifier1.rearrange_all; end; (*** Case splitting ***) local val mktac = mk_case_split_tac (meta_eq_to_obj_eq RS iffD2) in fun split_tac splits = mktac (map mk_meta_eq splits) end; local val mktac = mk_case_split_inside_tac (meta_eq_to_obj_eq RS iffD2) in fun split_inside_tac splits = mktac (map mk_meta_eq splits) end; val split_asm_tac = mk_case_split_asm_tac split_tac (disjE,conjE,exE,contrapos,contrapos2,notnotD); infix 4 addsplits delsplits; fun ss addsplits splits = let fun addsplit(ss,split) = let val name = "split " ^ const_of_split_thm split in ss addloop (name,split_tac [split]) end in foldl addsplit (ss,splits) end; fun ss delsplits splits = let fun delsplit(ss,split) = let val name = "split " ^ const_of_split_thm split in ss delloop name end in foldl delsplit (ss,splits) end; fun Addsplits splits = (simpset_ref() := simpset() addsplits splits); fun Delsplits splits = (simpset_ref() := simpset() delsplits splits); qed_goal "if_cancel" HOL.thy "(if c then x else x) = x" (K [split_tac [expand_if] 1, blast_tac HOL_cs 1]); qed_goal "if_eq_cancel" HOL.thy "(if x = y then y else x) = x" (K [split_tac [expand_if] 1, blast_tac HOL_cs 1]); (** 'if' congruence rules: neither included by default! *) (*Simplifies x assuming c and y assuming ~c*) qed_goal "if_cong" HOL.thy "[| b=c; c ==> x=u; ~c ==> y=v |] ==>\ \ (if b then x else y) = (if c then u else v)" (fn rew::prems => [stac rew 1, stac expand_if 1, stac expand_if 1, blast_tac (HOL_cs addDs prems) 1]); (*Prevents simplification of x and y: much faster*) qed_goal "if_weak_cong" HOL.thy "b=c ==> (if b then x else y) = (if c then x else y)" (fn [prem] => [rtac (prem RS arg_cong) 1]); (*Prevents simplification of t: much faster*) qed_goal "let_weak_cong" HOL.thy "a = b ==> (let x=a in t(x)) = (let x=b in t(x))" (fn [prem] => [rtac (prem RS arg_cong) 1]); (*In general it seems wrong to add distributive laws by default: they might cause exponential blow-up. But imp_disjL has been in for a while and cannot be removed without affecting existing proofs. Moreover, rewriting by "(P|Q --> R) = ((P-->R)&(Q-->R))" might be justified on the grounds that it allows simplification of R in the two cases.*) val mksimps_pairs = [("op -->", [mp]), ("op &", [conjunct1,conjunct2]), ("All", [spec]), ("True", []), ("False", []), ("If", [if_bool_eq RS iffD1])]; fun unsafe_solver prems = FIRST'[resolve_tac (reflexive_thm::TrueI::refl::prems), atac, etac FalseE]; (*No premature instantiation of variables during simplification*) fun safe_solver prems = FIRST'[match_tac (reflexive_thm::TrueI::prems), eq_assume_tac, ematch_tac [FalseE]]; val HOL_basic_ss = empty_ss setsubgoaler asm_simp_tac setSSolver safe_solver setSolver unsafe_solver setmksimps (mksimps mksimps_pairs) setmkeqTrue mk_meta_eq_True addsplits [expand_if]; val HOL_ss = HOL_basic_ss addsimps ([triv_forall_equality, (* prunes params *) True_implies_equals, (* prune asms `True' *) if_True, if_False, if_cancel, if_eq_cancel, o_apply, imp_disjL, conj_assoc, disj_assoc, de_Morgan_conj, de_Morgan_disj, imp_disj1, imp_disj2, not_imp, disj_not1, not_all, not_ex, cases_simp] @ ex_simps @ all_simps @ simp_thms) addsimprocs [defALL_regroup,defEX_regroup] addcongs [imp_cong]; qed_goal "if_distrib" HOL.thy "f(if c then x else y) = (if c then f x else f y)" (K [simp_tac (HOL_ss setloop (split_tac [expand_if])) 1]); qed_goalw "o_assoc" HOL.thy [o_def] "f o (g o h) = f o g o h" (K [rtac ext 1, rtac refl 1]); (*For expand_case_tac*) val prems = goal HOL.thy "[| P ==> Q(True); ~P ==> Q(False) |] ==> Q(P)"; by (case_tac "P" 1); by (ALLGOALS (asm_simp_tac (HOL_ss addsimps prems))); val expand_case = result(); (*Used in Auth proofs. Typically P contains Vars that become instantiated during unification.*) fun expand_case_tac P i = res_inst_tac [("P",P)] expand_case i THEN Simp_tac (i+1) THEN Simp_tac i; (* install implicit simpset *) simpset_ref() := HOL_ss; (*** Integration of simplifier with classical reasoner ***) (* rot_eq_tac rotates the first equality premise of subgoal i to the front, fails if there is no equaliy or if an equality is already at the front *) local fun is_eq (Const ("Trueprop", _) $ (Const("op =" ,_) $ _ $ _)) = true | is_eq _ = false; val find_eq = find_index is_eq; in val rot_eq_tac = SUBGOAL (fn (Bi,i) => let val n = find_eq (Logic.strip_assums_hyp Bi) in if n>0 then rotate_tac n i else no_tac end) end; use "$ISABELLE_HOME/src/Provers/clasimp.ML"; open Clasimp; val HOL_css = (HOL_cs, HOL_ss);