(* Title: HOL/simpdata.ML
ID: $Id$
Author: Tobias Nipkow
Copyright 1991 University of Cambridge
Instantiation of the generic simplifier for HOL.
*)
section "Simplifier";
(*** Addition of rules to simpsets and clasets simultaneously ***) (* FIXME move to Provers/clasimp.ML? *)
infix 4 addIffs delIffs;
(*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 ((cla, simp), th) =
(case HOLogic.dest_Trueprop (#prop (rep_thm th)) of
(Const("Not", _) $ A) =>
cla addSEs [zero_var_indexes (th RS notE)]
| (con $ _ $ _) =>
if con = iff_const
then cla addSIs [zero_var_indexes (th RS iffD2)]
addSDs [zero_var_indexes (th RS iffD1)]
else cla addSIs [th]
| _ => cla addSIs [th],
simp addsimps [th])
handle _ => error ("AddIffs: theorem must be unconditional\n" ^
string_of_thm th);
fun delIff ((cla, simp), th) =
(case HOLogic.dest_Trueprop (#prop (rep_thm th)) of
(Const ("Not", _) $ A) =>
cla delrules [zero_var_indexes (th RS notE)]
| (con $ _ $ _) =>
if con = iff_const
then cla delrules [zero_var_indexes (th RS iffD2),
make_elim (zero_var_indexes (th RS iffD1))]
else cla delrules [th]
| _ => cla delrules [th],
simp delsimps [th])
handle _ => (warning("DelIffs: ignoring conditional theorem\n" ^
string_of_thm th); (cla, simp));
fun store_clasimp (cla, simp) = (claset_ref () := cla; simpset_ref () := simp)
in
val op addIffs = foldl addIff;
val op delIffs = foldl delIff;
fun AddIffs thms = store_clasimp ((claset (), simpset ()) addIffs thms);
fun DelIffs thms = store_clasimp ((claset (), simpset ()) delIffs thms);
end;
(* "iff" attribute *)
local
fun change_global_css f (thy, th) =
let
val cs_ref = Classical.claset_ref_of thy;
val ss_ref = Simplifier.simpset_ref_of thy;
val (cs', ss') = f ((! cs_ref, ! ss_ref), [th]);
in cs_ref := cs'; ss_ref := ss'; (thy, th) end;
fun change_local_css f (ctxt, th) =
let
val cs = Classical.get_local_claset ctxt;
val ss = Simplifier.get_local_simpset ctxt;
val (cs', ss') = f ((cs, ss), [th]);
val ctxt' =
ctxt
|> Classical.put_local_claset cs'
|> Simplifier.put_local_simpset ss';
in (ctxt', th) end;
in
val iff_add_global = change_global_css (op addIffs);
val iff_add_local = change_local_css (op addIffs);
val simpdata_setup =
[Attrib.add_attributes [("iff", (Attrib.no_args iff_add_global, Attrib.no_args iff_add_local),
"add rules to simpset and claset simultaneously")]];
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 1]);
in
(*Make meta-equalities. The operator below is Trueprop*)
fun mk_meta_eq r = r RS eq_reflection;
val Eq_TrueI = mk_meta_eq(prover "P --> (P = True)" RS mp);
val Eq_FalseI = mk_meta_eq(prover "~P --> (P = False)" RS mp);
fun mk_eq th = case concl_of th of
Const("==",_)$_$_ => th
| _$(Const("op =",_)$_$_) => mk_meta_eq th
| _$(Const("Not",_)$_) => th RS Eq_FalseI
| _ => th RS Eq_TrueI;
(* last 2 lines requires all formulae to be of the from Trueprop(.) *)
fun mk_eq_True r = Some(r RS meta_eq_to_obj_eq RS Eq_TrueI);
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");
val not_not = prover "(~ ~ P) = P";
val simp_thms = [not_not] @ map prover
[ "(x=x) = True",
"(~True) = False", "(~False) = True",
"(~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 ==) *)
(* ###FIXME: Move to simplifier,
taking mk_meta_cong as input, eliminating addeqcongs and deleqcongs *)
infix 4 addcongs delcongs;
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);
val imp_cong = impI RSN
(2, prove_goal HOL.thy "(P=P')--> (P'--> (Q=Q'))--> ((P-->Q) = (P'-->Q'))"
(fn _=> [Blast_tac 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 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 :-( *)
prove "imp_conv_disj" "(P --> Q) = ((~P) | Q)";
prove "iff_conv_conj_imp" "(P = Q) = ((P --> Q) & (Q --> P))";
(*Avoids duplication of subgoals after split_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 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 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 1])
in bind_thm("disj_cong",standard (impI RSN (2, th RS mp RS mp))) end;
prove "eq_sym_conv" "(x=y) = (y=x)";
(** if-then-else rules **)
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_goalw "if_P" HOL.thy [if_def] "!!P. P ==> (if P then x else y) = x"
(K [Blast_tac 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 "split_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)]);
(* for backwards compatibility: *)
val expand_if = split_if;
qed_goal "split_if_asm" HOL.thy
"P(if Q then x else y) = (~((Q & ~P x) | (~Q & ~P y)))"
(K [stac split_if 1,
Blast_tac 1]);
qed_goal "if_cancel" HOL.thy "(if c then x else x) = x"
(K [stac split_if 1, Blast_tac 1]);
qed_goal "if_eq_cancel" HOL.thy "(if x = y then y else x) = x"
(K [stac split_if 1, Blast_tac 1]);
(*This form is useful for expanding IFs on the RIGHT of the ==> symbol*)
qed_goal "if_bool_eq_conj" HOL.thy
"(if P then Q else R) = ((P-->Q) & (~P-->R))"
(K [rtac split_if 1]);
(*And this form is useful for expanding IFs on the LEFT*)
qed_goal "if_bool_eq_disj" HOL.thy
"(if P then Q else R) = ((P&Q) | (~P&R))"
(K [stac split_if 1,
Blast_tac 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 =
Thm.read_cterm (Theory.sign_of HOL.thy) ("EX x. P(x) & Q(x)",HOLogic.boolT)
val all_pattern =
Thm.read_cterm (Theory.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 ***)
structure SplitterData =
struct
structure Simplifier = Simplifier
val mk_eq = mk_eq
val meta_eq_to_iff = meta_eq_to_obj_eq
val iffD = iffD2
val disjE = disjE
val conjE = conjE
val exE = exE
val contrapos = contrapos
val contrapos2 = contrapos2
val notnotD = notnotD
end;
structure Splitter = SplitterFun(SplitterData);
val split_tac = Splitter.split_tac;
val split_inside_tac = Splitter.split_inside_tac;
val split_asm_tac = Splitter.split_asm_tac;
val op addsplits = Splitter.addsplits;
val op delsplits = Splitter.delsplits;
val Addsplits = Splitter.Addsplits;
val Delsplits = Splitter.Delsplits;
(** 'if' congruence rules: neither included by default! *)
(*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.*)
fun gen_all th = forall_elim_vars (#maxidx(rep_thm th)+1) th;
val mksimps_pairs =
[("op -->", [mp]), ("op &", [conjunct1,conjunct2]),
("All", [spec]), ("True", []), ("False", []),
("If", [if_bool_eq_conj RS iffD1])];
(* ###FIXME: move to Provers/simplifier.ML
val mk_atomize: (string * thm list) list -> thm -> thm list
*)
(* ###FIXME: move to Provers/simplifier.ML *)
fun mk_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 mksimps pairs = (map mk_eq o mk_atomize pairs o gen_all);
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_eq_True;
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,
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, Eps_eq, Eps_sym_eq]
@ ex_simps @ all_simps @ simp_thms)
addsimprocs [defALL_regroup,defEX_regroup]
addcongs [imp_cong]
addsplits [split_if];
(*Simplifies x assuming c and y assuming ~c*)
val prems = Goalw [if_def]
"[| b=c; c ==> x=u; ~c ==> y=v |] ==> \
\ (if b then x else y) = (if c then u else v)";
by (asm_simp_tac (HOL_ss addsimps prems) 1);
qed "if_cong";
(*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]);
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 [split_if])) 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 addcongs [if_weak_cong];
(*** integration of simplifier with classical reasoner ***)
structure Clasimp = ClasimpFun
(structure Simplifier = Simplifier
and Classical = Classical
and Blast = Blast);
open Clasimp;
val HOL_css = (HOL_cs, HOL_ss);
(*** A general 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 ?
*)
fun refute_tac test prep_tac ref_tac =
let val nnf_simps =
[imp_conv_disj,iff_conv_conj_imp,de_Morgan_disj,de_Morgan_conj,
not_all,not_ex,not_not];
val nnf_simpset =
empty_ss setmkeqTrue mk_eq_True
setmksimps (mksimps mksimps_pairs)
addsimps nnf_simps;
val prem_nnf_tac = full_simp_tac nnf_simpset;
val refute_prems_tac =
REPEAT(eresolve_tac [conjE, exE] 1 ORELSE
filter_prems_tac test 1 ORELSE
etac disjE 1) THEN
ref_tac 1;
in EVERY'[TRY o filter_prems_tac test,
DETERM o REPEAT o etac rev_mp, prep_tac, rtac ccontr, prem_nnf_tac,
SELECT_GOAL (DEPTH_SOLVE refute_prems_tac)]
end;