author  wenzelm 
Fri, 10 Oct 1997 15:52:12 +0200  
changeset 3835  9a5a4e123859 
parent 3610  7e5300420b03 
child 3910  1cc9b8ab161c 
permissions  rwrr 
1459  1 
(* Title: FOL/simpdata 
0  2 
ID: $Id$ 
1459  3 
Author: Lawrence C Paulson, Cambridge University Computer Laboratory 
282  4 
Copyright 1994 University of Cambridge 
0  5 

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Simplification data for FOL 

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*) 

8 

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(*** Rewrite rules ***) 

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fun int_prove_fun s = 

282  12 
(writeln s; 
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prove_goal IFOL.thy s 

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(fn prems => [ (cut_facts_tac prems 1), 

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(IntPr.fast_tac 1) ])); 
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val conj_simps = map int_prove_fun 
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["P & True <> P", "True & P <> P", 
0  19 
"P & False <> False", "False & P <> False", 
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"P & P <> P", "P & P & Q <> P & Q", 
1459  21 
"P & ~P <> False", "~P & P <> False", 
0  22 
"(P & Q) & R <> P & (Q & R)"]; 
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1953  24 
val disj_simps = map int_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", 
0  28 
"(P  Q)  R <> P  (Q  R)"]; 
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1953  30 
val not_simps = map int_prove_fun 
282  31 
["~(PQ) <> ~P & ~Q", 
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"~ False <> True", "~ True <> False"]; 
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1953  34 
val imp_simps = map int_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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1953  39 
val iff_simps = map int_prove_fun 
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["(True <> P) <> P", "(P <> True) <> P", 
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"(P <> P) <> True", 
1459  42 
"(False <> P) <> ~P", "(P <> False) <> ~P"]; 
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1953  44 
val quant_simps = map int_prove_fun 
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["(ALL x. P) <> P", "(EX x. P) <> P"]; 
0  46 

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(*These are NOT supplied by default!*) 

1953  48 
val distrib_simps = map int_prove_fun 
282  49 
["P & (Q  R) <> P&Q  P&R", 
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"(Q  R) & P <> Q&P  R&P", 

0  51 
"(P  Q > R) <> (P > R) & (Q > R)"]; 
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282  53 
(** Conversion into rewrite rules **) 
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53  55 
fun gen_all th = forall_elim_vars (#maxidx(rep_thm th)+1) th; 
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282  57 
(*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",_) $ p => 
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(case p of 
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Const("op >",_)$_$_ => atomize(r RS mp) 
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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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 _ => [r]; 
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282  71 

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val P_iff_F = int_prove_fun "~P ==> (P <> False)"; 

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val iff_reflection_F = P_iff_F RS iff_reflection; 

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val P_iff_T = int_prove_fun "P ==> (P <> True)"; 

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val iff_reflection_T = P_iff_T RS iff_reflection; 

77 

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

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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("op =",_)$_$_) => th RS eq_reflection 
282  82 
 _ $ (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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(*** Classical laws ***) 
282  88 

0  89 
fun prove_fun s = 
282  90 
(writeln s; 
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prove_goal FOL.thy s 

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(fn prems => [ (cut_facts_tac prems 1), 

1459  93 
(Cla.fast_tac FOL_cs 1) ])); 
745  94 

1953  95 
(*Avoids duplication of subgoals after expand_if, when the true and false 
96 
cases boil down to the same thing.*) 

97 
val cases_simp = prove_fun "(P > Q) & (~P > Q) <> Q"; 

98 

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(*At present, miniscoping is for classical logic only. We do NOT include 

100 
distribution of ALL over &, or dually that of EX over .*) 

0  101 

1953  102 
(*Miniscoping: pushing in existential quantifiers*) 
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val ex_simps = map prove_fun 

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

108 
"(EX x. P(x)  Q) <> (EX x. P(x))  Q", 

109 
"(EX x. P  Q(x)) <> P  (EX x. Q(x))", 

110 
"(EX x. P(x) > Q) <> (ALL x. P(x)) > Q", 

111 
"(EX x. P > Q(x)) <> P > (EX x. Q(x))"]; 

1953  112 

113 
(*Miniscoping: pushing in universal quantifiers*) 

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val all_simps = map prove_fun 

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["(ALL x. x=t > P(x)) <> P(t)", 
116 
"(ALL x. t=x > P(x)) <> P(t)", 

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"(ALL x. P(x) & Q) <> (ALL x. P(x)) & Q", 
118 
"(ALL x. P & Q(x)) <> P & (ALL x. Q(x))", 

119 
"(ALL x. P(x)  Q) <> (ALL x. P(x))  Q", 

120 
"(ALL x. P  Q(x)) <> P  (ALL x. Q(x))", 

121 
"(ALL x. P(x) > Q) <> (EX x. P(x)) > Q", 

122 
"(ALL x. P > Q(x)) <> P > (ALL x. Q(x))"]; 

1953  123 

1914  124 
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  127 

128 
fun prove nm thm = qed_goal nm FOL.thy thm (fn _ => [fast_tac FOL_cs 1]); 

129 

130 
int_prove "conj_commute" "P&Q <> Q&P"; 

131 
int_prove "conj_left_commute" "P&(Q&R) <> Q&(P&R)"; 

132 
val conj_comms = [conj_commute, conj_left_commute]; 

133 

134 
int_prove "disj_commute" "PQ <> QP"; 

135 
int_prove "disj_left_commute" "P(QR) <> Q(PR)"; 

136 
val disj_comms = [disj_commute, disj_left_commute]; 

137 

138 
int_prove "conj_disj_distribL" "P&(QR) <> (P&Q  P&R)"; 

139 
int_prove "conj_disj_distribR" "(PQ)&R <> (P&R  Q&R)"; 

140 

141 
int_prove "disj_conj_distribL" "P(Q&R) <> (PQ) & (PR)"; 

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

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144 
int_prove "imp_conj_distrib" "(P > (Q&R)) <> (P>Q) & (P>R)"; 

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

146 
int_prove "imp_disj" "(PQ > R) <> (P>R) & (Q>R)"; 

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148 
int_prove "de_Morgan_disj" "(~(P  Q)) <> (~P & ~Q)"; 

149 
prove "de_Morgan_conj" "(~(P & Q)) <> (~P  ~Q)"; 

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151 
prove "not_iff" "~(P <> Q) <> (P <> ~Q)"; 

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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)"; 

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

1914  156 
int_prove "imp_ex" "((EX x. P(x)) > Q) <> (ALL x. P(x) > Q)"; 
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158 
int_prove "ex_disj_distrib" 

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

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int_prove "all_conj_distrib" 

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

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(*Used in ZF, perhaps elsewhere?*) 
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val meta_eq_to_obj_eq = prove_goal IFOL.thy "x==y ==> x=y" 
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(fn [prem] => [rewtac prem, rtac refl 1]); 
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(*** case splitting ***) 
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qed_goal "meta_iffD" IFOL.thy "[ P==Q; Q ] ==> P" 
756  171 
(fn [prem1,prem2] => [rewtac prem1, rtac prem2 1]); 
282  172 

942  173 
local val mktac = mk_case_split_tac meta_iffD 
174 
in 

175 
fun split_tac splits = mktac (map mk_meta_eq splits) 

176 
end; 

1722  177 

178 
local val mktac = mk_case_split_inside_tac meta_iffD 

179 
in 

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fun split_inside_tac splits = mktac (map mk_meta_eq splits) 

181 
end; 

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(*** Standard simpsets ***) 
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structure Induction = InductionFun(struct val spec=IFOL.spec end); 
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open Simplifier Induction; 
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(*Add congruence rules for = or <> (instead of ==) *) 
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infix 4 addcongs delcongs; 
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fun ss addcongs congs = 
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ss addeqcongs (map standard (congs RL [eq_reflection,iff_reflection])); 
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fun ss delcongs congs = 
3566  195 
ss deleqcongs (map standard (congs RL [eq_reflection,iff_reflection])); 
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fun Addcongs congs = (simpset := !simpset addcongs congs); 

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fun Delcongs congs = (simpset := !simpset delcongs congs); 
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val 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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val notFalseI = int_prove_fun "~False"; 
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val triv_rls = [TrueI,refl,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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val FOL_basic_ss = empty_ss setsubgoaler asm_simp_tac 
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setSSolver safe_solver 
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setSolver unsafe_solver 
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setmksimps (map mk_meta_eq o atomize o gen_all); 
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val IFOL_ss = FOL_basic_ss addsimps IFOL_simps 
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addcongs [imp_cong]; 
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val cla_simps = 
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[de_Morgan_conj, de_Morgan_disj, not_all, not_ex, cases_simp] @ 
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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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val FOL_ss = IFOL_ss addsimps (cla_simps @ ex_simps @ all_simps); 
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2469  231 

232 

233 
(*** Install simpsets and datatypes in theory structure ***) 

234 

235 
simpset := FOL_ss; 

236 

237 
exception SS_DATA of simpset; 

238 

239 
let fun merge [] = SS_DATA empty_ss 

240 
 merge ss = let val ss = map (fn SS_DATA x => x) ss; 

241 
in SS_DATA (foldl merge_ss (hd ss, tl ss)) end; 

242 

243 
fun put (SS_DATA ss) = simpset := ss; 

244 

245 
fun get () = SS_DATA (!simpset); 

246 
in add_thydata "FOL" 

247 
("simpset", ThyMethods {merge = merge, put = put, get = get}) 

248 
end; 

249 

3610  250 
fun simpset_of tname = 
251 
case get_thydata tname "simpset" of 

252 
None => empty_ss 

253 
 Some (SS_DATA ss) => ss; 

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(*** Integration of simplifier with classical reasoner ***) 
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(* rot_eq_tac rotates the first equality premise of subgoal i to the front, 
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fails if there is no equaliy or if an equality is already at the front *) 
3537  261 
local 
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fun is_eq (Const ("Trueprop", _) $ (Const("op =" ,_) $ _ $ _)) = true 
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 is_eq (Const ("Trueprop", _) $ (Const("op <>",_) $ _ $ _)) = true 
264 
 is_eq _ = false; 

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fun find_eq n [] = None 
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 find_eq n (t :: ts) = if (is_eq t) then Some n 
267 
else find_eq (n + 1) ts; 

268 
in 

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val rot_eq_tac = 

270 
SUBGOAL (fn (Bi,i) => 

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case find_eq 0 (Logic.strip_assums_hyp Bi) of 

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None => no_tac 

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 Some 0 => no_tac 

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 Some n => rotate_tac n i) 

275 
end; 

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fun safe_asm_more_full_simp_tac ss = TRY o rot_eq_tac THEN' 
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safe_asm_full_simp_tac ss; 
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(*an unsatisfactory fix for the incomplete asm_full_simp_tac! 
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better: asm_really_full_simp_tac, a yet to be implemented version of 
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asm_full_simp_tac that applies all equalities in the 
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premises to all the premises *) 
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(*Add a simpset to a classical set!*) 
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infix 4 addSss addss; 
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fun cs addSss ss = cs addSaltern (CHANGED o (safe_asm_more_full_simp_tac ss)); 
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fun cs addss ss = cs addbefore asm_full_simp_tac ss; 
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fun Addss ss = (claset := !claset addss ss); 
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291 

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(*Designed to be idempotent, except if best_tac instantiates variables 
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in some of the subgoals*) 
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294 

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type clasimpset = (claset * simpset); 
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val FOL_css = (FOL_cs, FOL_ss); 
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fun pair_upd1 f ((a,b),x) = (f(a,x), b); 
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fun pair_upd2 f ((a,b),x) = (a, f(b,x)); 
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infix 4 addSIs2 addSEs2 addSDs2 addIs2 addEs2 addDs2 
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addsimps2 delsimps2 addcongs2 delcongs2; 
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fun op addSIs2 arg = pair_upd1 (op addSIs) arg; 
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fun op addSEs2 arg = pair_upd1 (op addSEs) arg; 
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fun op addSDs2 arg = pair_upd1 (op addSDs) arg; 
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fun op addIs2 arg = pair_upd1 (op addIs ) arg; 
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fun op addEs2 arg = pair_upd1 (op addEs ) arg; 
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fun op addDs2 arg = pair_upd1 (op addDs ) arg; 
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fun op addsimps2 arg = pair_upd2 (op addsimps) arg; 
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fun op delsimps2 arg = pair_upd2 (op delsimps) arg; 
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fun op addcongs2 arg = pair_upd2 (op addcongs) arg; 
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fun op delcongs2 arg = pair_upd2 (op delcongs) arg; 
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fun auto_tac (cs,ss) = 
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let val cs' = cs addss ss 
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in EVERY [TRY (safe_tac cs'), 
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REPEAT (FIRSTGOAL (fast_tac cs')), 
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TRY (safe_tac (cs addSss ss)), 
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320 
prune_params_tac] 
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321 
end; 
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322 

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fun Auto_tac () = auto_tac (!claset, !simpset); 
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324 

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fun auto () = by (Auto_tac ()); 