author  wenzelm 
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(* Title: Sequents/simpdata.ML 
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ID: $Id$ 
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Author: Lawrence C Paulson 
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Copyright 1999 University of Cambridge 
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Instantiation of the generic simplifier for LK 
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Borrows from the DC simplifier of Soren Heilmann. 
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*) 
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(*** Rewrite rules ***) 
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9713  13 
fun prove_fun s = 
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(writeln s; 

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prove_goal LK.thy s 
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(fn prems => [ (cut_facts_tac prems 1), 
7123  17 
(fast_tac (pack() add_safes [subst]) 1) ])); 
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val conj_simps = map prove_fun 
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[" P & True <> P", " True & P <> P", 
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" P & False <> False", " False & P <> False", 
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" P & P <> P", "  P & P & Q <> P & Q", 
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" P & ~P <> False", " ~P & P <> False", 
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" (P & Q) & R <> P & (Q & R)"]; 
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val disj_simps = map prove_fun 
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[" P  True <> True", " True  P <> True", 
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" P  False <> P", " False  P <> P", 
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" P  P <> P", " P  P  Q <> P  Q", 
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" (P  Q)  R <> P  (Q  R)"]; 
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val not_simps = map prove_fun 
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[" ~ False <> True", " ~ True <> False"]; 
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val imp_simps = map prove_fun 
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[" (P > False) <> ~P", " (P > True) <> True", 
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" (False > P) <> True", " (True > P) <> P", 
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" (P > P) <> True", " (P > ~P) <> ~P"]; 
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val iff_simps = map prove_fun 
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[" (True <> P) <> P", " (P <> True) <> P", 
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" (P <> P) <> True", 
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" (False <> P) <> ~P", " (P <> False) <> ~P"]; 
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7123  45 

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val quant_simps = map 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", 

9713  51 
" (EX x. x=t & P(x)) <> P(t)", 
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" (EX x. t=x & P(x)) <> P(t)"]; 
53 

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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  

9713  56 
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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***) 

59 

60 
(*existential miniscoping*) 

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val ex_simps = map prove_fun 
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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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" (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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" (EX x. P(x) > Q) <> (ALL x. P(x)) > Q", 

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

7123  68 

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(*universal miniscoping*) 

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val 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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" (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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" (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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(*These are NOT supplied by default!*) 
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val distrib_simps = map prove_fun 
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[" P & (Q  R) <> P&Q  P&R", 
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" (Q  R) & P <> Q&P  R&P", 
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" (P  Q > R) <> (P > R) & (Q > R)"]; 
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(** Conversion into rewrite rules **) 
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(*Make atomic rewrite rules*) 
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fun atomize r = 
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case concl_of r of 
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Const("Trueprop",_) $ Abs(_,_,a) $ Abs(_,_,c) => 
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(case (forms_of_seq a, forms_of_seq c) of 
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([], [p]) => 
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(case p of 

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Const("op >",_)$_$_ => atomize(r RS mp_R) 

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 Const("op &",_)$_$_ => atomize(r RS conjunct1) @ 

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atomize(r RS conjunct2) 

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 Const("All",_)$_ => atomize(r RS spec) 

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 Const("True",_) => [] (*True is DELETED*) 

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 Const("False",_) => [] (*should False do something?*) 

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 _ => [r]) 

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 _ => []) (*ignore theorem unless it has precisely one conclusion*) 
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 _ => [r]; 
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9259  104 
Goal " ~P ==>  (P <> False)"; 
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by (etac (thinR RS cut) 1); 

9713  106 
by (Fast_tac 1); 
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qed "P_iff_F"; 
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val iff_reflection_F = P_iff_F RS iff_reflection; 
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9259  111 
Goal " P ==>  (P <> True)"; 
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by (etac (thinR RS cut) 1); 

9713  113 
by (Fast_tac 1); 
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qed "P_iff_T"; 
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val iff_reflection_T = P_iff_T RS iff_reflection; 
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(*Make metaequalities.*) 
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fun mk_meta_eq th = case concl_of th of 
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Const("==",_)$_$_ => th 
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 Const("Trueprop",_) $ Abs(_,_,a) $ Abs(_,_,c) => 
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(case (forms_of_seq a, forms_of_seq c) of 
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([], [p]) => 

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(case p of 

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(Const("op =",_)$_$_) => th RS eq_reflection 

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

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 (Const("Not",_)$_) => th RS iff_reflection_F 

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 _ => th RS iff_reflection_T) 

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 _ => error ("addsimps: unable to use theorem\n" ^ 

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string_of_thm th)); 

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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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(*** Named rewrite rules ***) 

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fun prove nm thm = qed_goal nm LK.thy thm 
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(fn prems => [ (cut_facts_tac prems 1), 
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(fast_tac LK_pack 1) ]); 
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prove "conj_commute" " P&Q <> Q&P"; 
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prove "conj_left_commute" " P&(Q&R) <> Q&(P&R)"; 
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val conj_comms = [conj_commute, conj_left_commute]; 
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prove "disj_commute" " PQ <> QP"; 
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prove "disj_left_commute" " P(QR) <> Q(PR)"; 
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val disj_comms = [disj_commute, disj_left_commute]; 
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prove "conj_disj_distribL" " P&(QR) <> (P&Q  P&R)"; 
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prove "conj_disj_distribR" " (PQ)&R <> (P&R  Q&R)"; 
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prove "disj_conj_distribL" " P(Q&R) <> (PQ) & (PR)"; 
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prove "disj_conj_distribR" " (P&Q)R <> (PR) & (QR)"; 
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prove "imp_conj_distrib" " (P > (Q&R)) <> (P>Q) & (P>R)"; 
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prove "imp_conj" " ((P&Q)>R) <> (P > (Q > R))"; 
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prove "imp_disj" " (PQ > R) <> (P>R) & (Q>R)"; 
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prove "imp_disj1" " (P>Q)  R <> (P>Q  R)"; 
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prove "imp_disj2" " Q  (P>R) <> (P>Q  R)"; 
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prove "de_Morgan_disj" " (~(P  Q)) <> (~P & ~Q)"; 
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prove "de_Morgan_conj" " (~(P & Q)) <> (~P  ~Q)"; 
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prove "not_iff" " ~(P <> Q) <> (P <> ~Q)"; 
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9713  178 
val [p1,p2] = Goal 
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"[  P <> P';  P' ==>  Q <> Q' ] ==>  (P>Q) <> (P'>Q')"; 
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by (lemma_tac p1 1); 
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by (Safe_tac 1); 
9713  182 
by (REPEAT (rtac cut 1 
183 
THEN 

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DEPTH_SOLVE_1 (resolve_tac [thinL, thinR, p2 COMP monotonic] 1) 

185 
THEN 

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Safe_tac 1)); 

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qed "imp_cong"; 
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9713  189 
val [p1,p2] = Goal 
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"[  P <> P';  P' ==>  Q <> Q' ] ==>  (P&Q) <> (P'&Q')"; 
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by (lemma_tac p1 1); 
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by (Safe_tac 1); 
9713  193 
by (REPEAT (rtac cut 1 
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THEN 

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DEPTH_SOLVE_1 (resolve_tac [thinL, thinR, p2 COMP monotonic] 1) 

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THEN 

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Safe_tac 1)); 

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qed "conj_cong"; 
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7123  200 
Goal " (x=y) <> (y=x)"; 
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by (fast_tac (pack() add_safes [subst]) 1); 

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qed "eq_sym_conv"; 

203 

204 

205 
(** ifthenelse rules **) 

206 

207 
Goalw [If_def] " (if True then x else y) = x"; 

208 
by (Fast_tac 1); 

209 
qed "if_True"; 

210 

211 
Goalw [If_def] " (if False then x else y) = y"; 

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by (Fast_tac 1); 

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qed "if_False"; 

214 

215 
Goalw [If_def] " P ==>  (if P then x else y) = x"; 

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by (etac (thinR RS cut) 1); 

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by (Fast_tac 1); 

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qed "if_P"; 

219 

220 
Goalw [If_def] " ~P ==>  (if P then x else y) = y"; 

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by (etac (thinR RS cut) 1); 

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by (Fast_tac 1); 

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qed "if_not_P"; 

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open Simplifier; 
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(*** Standard simpsets ***) 
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val triv_rls = [FalseL, TrueR, basic, refl, iff_refl, reflexive_thm]; 
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fun unsafe_solver prems = FIRST'[resolve_tac (triv_rls@prems), 
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assume_tac]; 
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(*No premature instantiation of variables during simplification*) 
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fun safe_solver prems = FIRST'[fn i => DETERM (match_tac (triv_rls@prems) i), 
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eq_assume_tac]; 
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(*No simprules, but basic infrastructure for simplification*) 
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val LK_basic_ss = 
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empty_ss setsubgoaler asm_simp_tac 

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setSSolver (mk_solver "safe" safe_solver) 

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setSolver (mk_solver "unsafe" unsafe_solver) 

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setmksimps (map mk_meta_eq o atomize o gen_all) 
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setmkcong mk_meta_cong; 
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val LK_simps = 
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[triv_forall_equality, (* prunes params *) 
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refl RS P_iff_T] @ 
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conj_simps @ disj_simps @ not_simps @ 

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imp_simps @ iff_simps @quant_simps @ all_simps @ ex_simps @ 
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[de_Morgan_conj, de_Morgan_disj, imp_disj1, imp_disj2] @ 
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map prove_fun 
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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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val LK_ss = 
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LK_basic_ss addsimps LK_simps 

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addeqcongs [left_cong] 

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addcongs [imp_cong]; 

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simpset_ref() := LK_ss; 
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(* To create substition rules *) 
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qed_goal "eq_imp_subst" LK.thy " a=b ==> $H, A(a), $G  $E, A(b), $F" 
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(fn prems => 
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[cut_facts_tac prems 1, 
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asm_simp_tac LK_basic_ss 1]); 
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Goal " P(if Q then x else y) <> ((Q > P(x)) & (~Q > P(y)))"; 
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by (res_inst_tac [ ("P","Q") ] cut 1); 

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by (simp_tac (simpset() addsimps [if_P]) 2); 

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by (res_inst_tac [ ("P","~Q") ] cut 1); 

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by (simp_tac (simpset() addsimps [if_not_P]) 2); 

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by (Fast_tac 1); 

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qed "split_if"; 

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Goal " (if P then x else x) = x"; 
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by (lemma_tac split_if 1); 

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by (Fast_tac 1); 

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qed "if_cancel"; 

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Goal " (if x=y then y else x) = x"; 

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by (lemma_tac split_if 1); 

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by (Safe_tac 1); 

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by (rtac symL 1); 

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by (rtac basic 1); 

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qed "if_eq_cancel"; 

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(*Putting in automatic case splits seems to require a lot of work.*) 