author  paulson 
Fri, 05 Oct 2007 09:59:03 +0200  
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parent 22896  1c2abcabea61 
child 26928  ca87aff1ad2d 
permissions  rwrr 
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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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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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(fast_tac (LK_pack add_safes @{thms 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)"]; 
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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) 
7123  57 
show that this step can increase proof length! 
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***) 

59 

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(*existential miniscoping*) 

9713  61 
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", 

9713  72 
" (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("imp",_)$_$_ => atomize(r RS @{thm mp_R}) 
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 Const("conj",_)$_$_ => atomize(r RS @{thm conjunct1}) @ 

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atomize(r RS @{thm conjunct2}) 

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 Const("All",_)$_ => atomize(r RS @{thm spec}) 

9713  97 
 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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Goal " ~P ==>  (P <> False)"; 
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by (etac (@{thm thinR} RS @{thm cut}) 1); 
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by (fast_tac LK_pack 1); 
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qed "P_iff_F"; 
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22896  108 
bind_thm ("iff_reflection_F", P_iff_F RS @{thm iff_reflection}); 
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Goal " P ==>  (P <> True)"; 
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by (etac (@{thm thinR} RS @{thm cut}) 1); 
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by (fast_tac LK_pack 1); 
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qed "P_iff_T"; 
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22896  115 
bind_thm ("iff_reflection_T", P_iff_T RS @{thm 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("equal",_)$_$_) => th RS @{thm eq_reflection} 
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 (Const("iff",_)$_$_) => th RS @{thm 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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7123  131 
(*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 [@{thm meta_eq_to_obj_eq}, @{thm 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 (the_context ()) 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  176 
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); 
21428  179 
by (safe_tac LK_pack 1); 
22896  180 
by (REPEAT (rtac @{thm cut} 1 
9713  181 
THEN 
22896  182 
DEPTH_SOLVE_1 (resolve_tac [@{thm thinL}, @{thm thinR}, p2 COMP @{thm monotonic}] 1) 
9713  183 
THEN 
21428  184 
safe_tac LK_pack 1)); 
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qed "imp_cong"; 
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9713  187 
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); 
21428  190 
by (safe_tac LK_pack 1); 
22896  191 
by (REPEAT (rtac @{thm cut} 1 
9713  192 
THEN 
22896  193 
DEPTH_SOLVE_1 (resolve_tac [@{thm thinL}, @{thm thinR}, p2 COMP @{thm monotonic}] 1) 
9713  194 
THEN 
21428  195 
safe_tac LK_pack 1)); 
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qed "conj_cong"; 
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7123  198 
Goal " (x=y) <> (y=x)"; 
22896  199 
by (fast_tac (LK_pack add_safes @{thms subst}) 1); 
7123  200 
qed "eq_sym_conv"; 
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202 

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(*** Standard simpsets ***) 
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21426  206 
val triv_rls = [thm "FalseL", thm "TrueR", thm "basic", thm "refl", 
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thm "iff_refl", reflexive_thm]; 

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fun unsafe_solver prems = FIRST'[resolve_tac (triv_rls@prems), 
9713  210 
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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Simplifier.theory_context (the_context ()) empty_ss 
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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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@{thm 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 [thm "left_cong"] 
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addcongs [imp_cong]; 
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change_simpset (fn _ => LK_ss); 
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(* To create substition rules *) 
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qed_goal "eq_imp_subst" (the_context ()) " 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") ] (thm "cut") 1); 
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by (simp_tac (simpset() addsimps [thm "if_P"]) 2); 

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

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

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by (fast_tac LK_pack 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 LK_pack 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 LK_pack 1); 
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by (rtac (thm "symL") 1); 
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by (rtac (thm "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.*) 