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child 16019  0e1405402d53 
permissions  rwrr 
9889  1 
(* Title: FOL/simpdata.ML 
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ID: $Id$ 
1459  3 
Author: Lawrence C Paulson, Cambridge University Computer Laboratory 
282  4 
Copyright 1994 University of Cambridge 
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9889  6 
Simplification data for FOL. 
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*) 
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9300  9 

5496  10 
(* Elimination of True from asumptions: *) 
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12038  12 
bind_thm ("True_implies_equals", prove_goal IFOL.thy 
5496  13 
"(True ==> PROP P) == PROP P" 
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(K [rtac equal_intr_rule 1, atac 2, 

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METAHYPS (fn prems => resolve_tac prems 1) 1, 

12038  16 
rtac TrueI 1])); 
5496  17 

18 

0  19 
(*** Rewrite rules ***) 
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10431  21 
fun int_prove_fun s = 
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(writeln s; 

282  23 
prove_goal IFOL.thy s 
10431  24 
(fn prems => [ (cut_facts_tac prems 1), 
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(IntPr.fast_tac 1) ])); 
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bind_thms ("conj_simps", map int_prove_fun 
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["P & True <> P", "True & P <> P", 
0  29 
"P & False <> False", "False & P <> False", 
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"P & P <> P", "P & P & Q <> P & Q", 
1459  31 
"P & ~P <> False", "~P & P <> False", 
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"(P & Q) & R <> P & (Q & R)"]); 
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12038  34 
bind_thms ("disj_simps", map int_prove_fun 
1459  35 
["P  True <> True", "True  P <> True", 
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"P  False <> P", "False  P <> P", 

2801  37 
"P  P <> P", "P  P  Q <> P  Q", 
12038  38 
"(P  Q)  R <> P  (Q  R)"]); 
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12038  40 
bind_thms ("not_simps", map int_prove_fun 
282  41 
["~(PQ) <> ~P & ~Q", 
12038  42 
"~ False <> True", "~ True <> False"]); 
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12038  44 
bind_thms ("imp_simps", map int_prove_fun 
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["(P > False) <> ~P", "(P > True) <> True", 
10431  46 
"(False > P) <> True", "(True > P) <> P", 
12038  47 
"(P > P) <> True", "(P > ~P) <> ~P"]); 
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12038  49 
bind_thms ("iff_simps", map int_prove_fun 
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["(True <> P) <> P", "(P <> True) <> P", 
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"(P <> P) <> True", 
12038  52 
"(False <> P) <> ~P", "(P <> False) <> ~P"]); 
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(*The x=t versions are needed for the simplification procedures*) 
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bind_thms ("quant_simps", map int_prove_fun 
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["(ALL x. P) <> P", 
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"(ALL x. x=t > P(x)) <> P(t)", 
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"(ALL x. t=x > P(x)) <> P(t)", 
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"(EX x. P) <> P", 
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"EX x. x=t", "EX x. t=x", 
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"(EX x. x=t & P(x)) <> P(t)", 
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"(EX x. t=x & P(x)) <> P(t)"]); 
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64 
(*These are NOT supplied by default!*) 

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bind_thms ("distrib_simps", map int_prove_fun 
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["P & (Q  R) <> P&Q  P&R", 
282  67 
"(Q  R) & P <> Q&P  R&P", 
12038  68 
"(P  Q > R) <> (P > R) & (Q > R)"]); 
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282  70 
(** Conversion into rewrite rules **) 
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12038  72 
bind_thm ("P_iff_F", int_prove_fun "~P ==> (P <> False)"); 
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bind_thm ("iff_reflection_F", P_iff_F RS iff_reflection); 

282  74 

12038  75 
bind_thm ("P_iff_T", int_prove_fun "P ==> (P <> True)"); 
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bind_thm ("iff_reflection_T", P_iff_T RS iff_reflection); 

282  77 

78 
(*Make metaequalities. The operator below is Trueprop*) 

5555  79 

282  80 
fun mk_meta_eq th = case concl_of th of 
5555  81 
_ $ (Const("op =",_)$_$_) => th RS eq_reflection 
82 
 _ $ (Const("op <>",_)$_$_) => th RS iff_reflection 

10431  83 
 _ => 
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error("conclusion must be a =equality or <>");; 
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86 
fun mk_eq th = case concl_of th of 

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Const("==",_)$_$_ => th 
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 _ $ (Const("op =",_)$_$_) => mk_meta_eq th 
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 _ $ (Const("op <>",_)$_$_) => mk_meta_eq th 

282  90 
 _ $ (Const("Not",_)$_) => th RS iff_reflection_F 
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 _ => th RS iff_reflection_T; 

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(*Replace premises x=y, X<>Y by X==Y*) 
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val mk_meta_prems = 
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rule_by_tactic 

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(REPEAT_FIRST (resolve_tac [meta_eq_to_obj_eq, def_imp_iff])); 
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(*Congruence rules for = or <> (instead of ==)*) 
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fun mk_meta_cong rl = 
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standard(mk_meta_eq (mk_meta_prems rl)) 
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handle THM _ => 
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error("Premises and conclusion of congruence rules must use =equality or <>"); 
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5304  104 
val mksimps_pairs = 
105 
[("op >", [mp]), ("op &", [conjunct1,conjunct2]), 

106 
("All", [spec]), ("True", []), ("False", [])]; 

107 

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(* ###FIXME: move to Provers/simplifier.ML 
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val mk_atomize: (string * thm list) list > thm > thm list 
110 
*) 

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(* ###FIXME: move to Provers/simplifier.ML *) 
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fun mk_atomize pairs = 
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let fun atoms th = 

114 
(case concl_of th of 

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Const("Trueprop",_) $ p => 

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(case head_of p of 

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Const(a,_) => 

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(case assoc(pairs,a) of 

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SOME(rls) => List.concat (map atoms ([th] RL rls)) 
15531  120 
 NONE => [th]) 
5304  121 
 _ => [th]) 
122 
 _ => [th]) 

123 
in atoms end; 

124 

12725  125 
fun mksimps pairs = (map mk_eq o mk_atomize pairs o gen_all); 
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(*** Classical laws ***) 
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10431  129 
fun prove_fun s = 
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(writeln s; 

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prove_goal (the_context ()) s 
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(fn prems => [ (cut_facts_tac prems 1), 
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(Cla.fast_tac FOL_cs 1) ])); 
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(*Avoids duplication of subgoals after expand_if, when the true and false 
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cases boil down to the same thing.*) 

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bind_thm ("cases_simp", prove_fun "(P > Q) & (~P > Q) <> Q"); 
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(*** Miniscoping: pushing quantifiers in 
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We do NOT distribute of ALL over &, or dually that of EX over  
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Baaz and Leitsch, On Skolemization and Proof Complexity (1994) 
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show that this step can increase proof length! 
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***) 
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(*existential miniscoping*) 
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bind_thms ("int_ex_simps", map int_prove_fun 
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["(EX x. P(x) & Q) <> (EX x. P(x)) & Q", 

149 
"(EX x. P & Q(x)) <> P & (EX x. Q(x))", 

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"(EX x. P(x)  Q) <> (EX x. P(x))  Q", 

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"(EX x. P  Q(x)) <> P  (EX x. Q(x))"]); 

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(*classical rules*) 
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bind_thms ("cla_ex_simps", map prove_fun 
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["(EX x. P(x) > Q) <> (ALL x. P(x)) > Q", 

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"(EX x. P > Q(x)) <> P > (EX x. Q(x))"]); 

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12038  158 
bind_thms ("ex_simps", int_ex_simps @ cla_ex_simps); 
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(*universal miniscoping*) 
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bind_thms ("int_all_simps", map int_prove_fun 
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["(ALL x. P(x) & Q) <> (ALL x. P(x)) & Q", 

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"(ALL x. P & Q(x)) <> P & (ALL x. Q(x))", 

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"(ALL x. P(x) > Q) <> (EX x. P(x)) > Q", 

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"(ALL x. P > Q(x)) <> P > (ALL x. Q(x))"]); 

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(*classical rules*) 
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bind_thms ("cla_all_simps", map prove_fun 
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["(ALL x. P(x)  Q) <> (ALL x. P(x))  Q", 

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"(ALL x. P  Q(x)) <> P  (ALL x. Q(x))"]); 

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bind_thms ("all_simps", int_all_simps @ cla_all_simps); 
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(*** Named rewrite rules proved for IFOL ***) 
1953  176 

1914  177 
fun int_prove nm thm = qed_goal nm IFOL.thy thm 
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(fn prems => [ (cut_facts_tac prems 1), 
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(IntPr.fast_tac 1) ]); 
1914  180 

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fun prove nm thm = qed_goal nm (the_context ()) thm (fn _ => [Blast_tac 1]); 
1914  182 

183 
int_prove "conj_commute" "P&Q <> Q&P"; 

184 
int_prove "conj_left_commute" "P&(Q&R) <> Q&(P&R)"; 

12038  185 
bind_thms ("conj_comms", [conj_commute, conj_left_commute]); 
1914  186 

187 
int_prove "disj_commute" "PQ <> QP"; 

188 
int_prove "disj_left_commute" "P(QR) <> Q(PR)"; 

12038  189 
bind_thms ("disj_comms", [disj_commute, disj_left_commute]); 
1914  190 

191 
int_prove "conj_disj_distribL" "P&(QR) <> (P&Q  P&R)"; 

192 
int_prove "conj_disj_distribR" "(PQ)&R <> (P&R  Q&R)"; 

193 

194 
int_prove "disj_conj_distribL" "P(Q&R) <> (PQ) & (PR)"; 

195 
int_prove "disj_conj_distribR" "(P&Q)R <> (PR) & (QR)"; 

196 

197 
int_prove "imp_conj_distrib" "(P > (Q&R)) <> (P>Q) & (P>R)"; 

198 
int_prove "imp_conj" "((P&Q)>R) <> (P > (Q > R))"; 

199 
int_prove "imp_disj" "(PQ > R) <> (P>R) & (Q>R)"; 

200 

3910  201 
prove "imp_disj1" "(P>Q)  R <> (P>Q  R)"; 
202 
prove "imp_disj2" "Q  (P>R) <> (P>Q  R)"; 

203 

1914  204 
int_prove "de_Morgan_disj" "(~(P  Q)) <> (~P & ~Q)"; 
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prove "de_Morgan_conj" "(~(P & Q)) <> (~P  ~Q)"; 

206 

12765  207 
prove "not_imp" "~(P > Q) <> (P & ~Q)"; 
1914  208 
prove "not_iff" "~(P <> Q) <> (P <> ~Q)"; 
209 

3835  210 
prove "not_all" "(~ (ALL x. P(x))) <> (EX x.~P(x))"; 
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prove "imp_all" "((ALL x. P(x)) > Q) <> (EX x. P(x) > Q)"; 

212 
int_prove "not_ex" "(~ (EX x. P(x))) <> (ALL x.~P(x))"; 

1914  213 
int_prove "imp_ex" "((EX x. P(x)) > Q) <> (ALL x. P(x) > Q)"; 
214 

215 
int_prove "ex_disj_distrib" 

216 
"(EX x. P(x)  Q(x)) <> ((EX x. P(x))  (EX x. Q(x)))"; 

217 
int_prove "all_conj_distrib" 

218 
"(ALL x. P(x) & Q(x)) <> ((ALL x. P(x)) & (ALL x. Q(x)))"; 

219 

220 

11232  221 
local 
222 
val uncurry = prove_goal (the_context()) "P > Q > R ==> P & Q > R" 

223 
(fn prems => [cut_facts_tac prems 1, Blast_tac 1]); 

224 

225 
val iff_allI = allI RS 

226 
prove_goal (the_context()) "ALL x. P(x) <> Q(x) ==> (ALL x. P(x)) <> (ALL x. Q(x))" 

227 
(fn prems => [cut_facts_tac prems 1, Blast_tac 1]) 

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val iff_exI = allI RS 
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prove_goal (the_context()) "ALL x. P(x) <> Q(x) ==> (EX x. P(x)) <> (EX x. Q(x))" 
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(fn prems => [cut_facts_tac prems 1, Blast_tac 1]) 
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val all_comm = prove_goal (the_context()) "(ALL x y. P(x,y)) <> (ALL y x. P(x,y))" 
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(fn _ => [Blast_tac 1]) 
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val ex_comm = prove_goal (the_context()) "(EX x y. P(x,y)) <> (EX y x. P(x,y))" 
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(fn _ => [Blast_tac 1]) 
11232  236 
in 
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(** make simplification procedures for quantifier elimination **) 
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structure Quantifier1 = Quantifier1Fun( 
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struct 
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(*abstract syntax*) 
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fun dest_eq((c as Const("op =",_)) $ s $ t) = SOME(c,s,t) 
243 
 dest_eq _ = NONE; 

244 
fun dest_conj((c as Const("op &",_)) $ s $ t) = SOME(c,s,t) 

245 
 dest_conj _ = NONE; 

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fun dest_imp((c as Const("op >",_)) $ s $ t) = SOME(c,s,t) 

247 
 dest_imp _ = NONE; 

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val conj = FOLogic.conj 
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val imp = FOLogic.imp 
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(*rules*) 
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val iff_reflection = iff_reflection 
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val iffI = iffI 
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val iff_trans = iff_trans 
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val conjI= conjI 
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val conjE= conjE 
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val impI = impI 
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val mp = mp 
11232  258 
val uncurry = uncurry 
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val exI = exI 
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val exE = exE 
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val iff_allI = iff_allI 
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val iff_exI = iff_exI 
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val all_comm = all_comm 
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val ex_comm = ex_comm 
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end); 
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11232  267 
end; 
268 

13462  269 
val defEX_regroup = 
270 
Simplifier.simproc (Theory.sign_of (the_context ())) 

271 
"defined EX" ["EX x. P(x)"] Quantifier1.rearrange_ex; 

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val defALL_regroup = 
13462  274 
Simplifier.simproc (Theory.sign_of (the_context ())) 
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"defined ALL" ["ALL x. P(x)"] Quantifier1.rearrange_all; 

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(*** Case splitting ***) 
0  279 

12038  280 
bind_thm ("meta_eq_to_iff", prove_goal IFOL.thy "x==y ==> x<>y" 
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(fn [prem] => [rewtac prem, rtac iffI 1, atac 1, atac 1])); 

1722  282 

5304  283 
structure SplitterData = 
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struct 

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structure Simplifier = Simplifier 

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val mk_eq = mk_eq 
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val meta_eq_to_iff = meta_eq_to_iff 
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val iffD = iffD2 

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val disjE = disjE 

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val conjE = conjE 

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val exE = exE 

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val contrapos = contrapos 

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val contrapos2 = contrapos2 

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val notnotD = notnotD 

295 
end; 

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5304  297 
structure Splitter = SplitterFun(SplitterData); 
1722  298 

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val split_tac = Splitter.split_tac; 
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val split_inside_tac = Splitter.split_inside_tac; 

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val split_asm_tac = Splitter.split_asm_tac; 

5307  302 
val op addsplits = Splitter.addsplits; 
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val op delsplits = Splitter.delsplits; 

5304  304 
val Addsplits = Splitter.Addsplits; 
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val Delsplits = Splitter.Delsplits; 

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307 

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(*** Standard simpsets ***) 
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structure Induction = InductionFun(struct val spec=IFOL.spec end); 
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open Induction; 
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5555  314 

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bind_thms ("meta_simps", 
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[triv_forall_equality, (* prunes params *) 

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True_implies_equals]); (* prune asms `True' *) 

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12038  319 
bind_thms ("IFOL_simps", 
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[refl RS P_iff_T] @ conj_simps @ disj_simps @ not_simps @ 

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imp_simps @ iff_simps @ quant_simps); 

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12038  323 
bind_thm ("notFalseI", int_prove_fun "~False"); 
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bind_thms ("triv_rls", [TrueI,refl,reflexive_thm,iff_refl,notFalseI]); 

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fun unsafe_solver prems = FIRST'[resolve_tac (triv_rls@prems), 
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atac, etac FalseE]; 
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(*No premature instantiation of variables during simplification*) 
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fun safe_solver prems = FIRST'[match_tac (triv_rls@prems), 
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eq_assume_tac, ematch_tac [FalseE]]; 
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3910  332 
(*No simprules, but basic infastructure for simplification*) 
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val FOL_basic_ss = empty_ss 
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setsubgoaler asm_simp_tac 

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setSSolver (mk_solver "FOL safe" safe_solver) 

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setSolver (mk_solver "FOL unsafe" unsafe_solver) 

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setmksimps (mksimps mksimps_pairs) 

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setmkcong mk_meta_cong; 

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3910  341 
(*intuitionistic simprules only*) 
10431  342 
val IFOL_ss = FOL_basic_ss 
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addsimps (meta_simps @ IFOL_simps @ int_ex_simps @ int_all_simps) 

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addsimprocs [defALL_regroup, defEX_regroup] 

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addcongs [imp_cong]; 

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12038  347 
bind_thms ("cla_simps", 
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[de_Morgan_conj, de_Morgan_disj, imp_disj1, imp_disj2, 

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not_imp, not_all, not_ex, cases_simp] @ 
12038  350 
map prove_fun 
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["~(P&Q) <> ~P  ~Q", 

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"P  ~P", "~P  P", 

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"~ ~ P <> P", "(~P > P) <> P", 

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"(~P <> ~Q) <> (P<>Q)"]); 

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3910  356 
(*classical simprules too*) 
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val FOL_ss = IFOL_ss addsimps (cla_simps @ cla_ex_simps @ cla_all_simps); 
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val simpsetup = [fn thy => (simpset_ref_of thy := FOL_ss; thy)]; 
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5219  362 
(*** integration of simplifier with classical reasoner ***) 
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5219  364 
structure Clasimp = ClasimpFun 
8472  365 
(structure Simplifier = Simplifier and Splitter = Splitter 
9851  366 
and Classical = Cla and Blast = Blast 
11344  367 
val iffD1 = iffD1 val iffD2 = iffD2 val notE = notE 
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val cla_make_elim = cla_make_elim); 
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open Clasimp; 
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val FOL_css = (FOL_cs, FOL_ss); 