author  oheimb 
Wed, 12 Aug 1998 16:21:18 +0200  
changeset 5304  c133f16febc7 
parent 5278  a903b66822e2 
child 5307  6a699d5cdef4 
permissions  rwrr 
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(* Title: HOL/simpdata.ML 
923  2 
ID: $Id$ 
1465  3 
Author: Tobias Nipkow 
923  4 
Copyright 1991 University of Cambridge 
5 

5304  6 
Instantiation of the generic simplifier for HOL. 
923  7 
*) 
8 

1984  9 
section "Simplifier"; 
10 

11 
(*** Addition of rules to simpsets and clasets simultaneously ***) 

12 

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infix 4 addIffs delIffs; 
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1984  15 
(*Takes UNCONDITIONAL theorems of the form A<>B to 
2031  16 
the Safe Intr rule B==>A and 
17 
the Safe Destruct rule A==>B. 

1984  18 
Also ~A goes to the Safe Elim rule A ==> ?R 
19 
Failing other cases, A is added as a Safe Intr rule*) 

20 
local 

21 
val iff_const = HOLogic.eq_const HOLogic.boolT; 

22 

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fun addIff ((cla, simp), th) = 
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(case HOLogic.dest_Trueprop (#prop (rep_thm th)) of 
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(Const("Not", _) $ A) => 
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cla addSEs [zero_var_indexes (th RS notE)] 
2031  27 
 (con $ _ $ _) => 
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if con = iff_const 
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then cla addSIs [zero_var_indexes (th RS iffD2)] 
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addSDs [zero_var_indexes (th RS iffD1)] 
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else cla addSIs [th] 
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 _ => cla addSIs [th], 
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simp addsimps [th]) 
1984  34 
handle _ => error ("AddIffs: theorem must be unconditional\n" ^ 
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string_of_thm th); 
1984  36 

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fun delIff ((cla, simp), th) = 
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(case HOLogic.dest_Trueprop (#prop (rep_thm th)) of 
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(Const ("Not", _) $ A) => 
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cla delrules [zero_var_indexes (th RS notE)] 
2031  41 
 (con $ _ $ _) => 
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if con = iff_const 
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then cla delrules [zero_var_indexes (th RS iffD2), 
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make_elim (zero_var_indexes (th RS iffD1))] 
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else cla delrules [th] 
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 _ => cla delrules [th], 
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simp delsimps [th]) 
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handle _ => (warning("DelIffs: ignoring conditional theorem\n" ^ 
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string_of_thm th); (cla, simp)); 
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fun store_clasimp (cla, simp) = (claset_ref () := cla; simpset_ref () := simp) 
1984  52 
in 
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val op addIffs = foldl addIff; 
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val op delIffs = foldl delIff; 
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fun AddIffs thms = store_clasimp ((claset (), simpset ()) addIffs thms); 
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fun DelIffs thms = store_clasimp ((claset (), simpset ()) delIffs thms); 
1984  57 
end; 
58 

5304  59 

4640  60 
qed_goal "meta_eq_to_obj_eq" HOL.thy "x==y ==> x=y" 
61 
(fn [prem] => [rewtac prem, rtac refl 1]); 

62 

923  63 
local 
64 

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fun prover s = prove_goal HOL.thy s (K [Blast_tac 1]); 
923  66 

1922  67 
val P_imp_P_iff_True = prover "P > (P = True)" RS mp; 
68 
val P_imp_P_eq_True = P_imp_P_iff_True RS eq_reflection; 

923  69 

1922  70 
val not_P_imp_P_iff_F = prover "~P > (P = False)" RS mp; 
71 
val not_P_imp_P_eq_False = not_P_imp_P_iff_F RS eq_reflection; 

923  72 

2134  73 
in 
74 

5304  75 
fun meta_eq r = r RS eq_reflection; 
76 

77 
fun mk_meta_eq th = case concl_of th of 

78 
Const("==",_)$_$_ => th 

79 
 _$(Const("op =",_)$_$_) => meta_eq th 

80 
 _$(Const("Not",_)$_) => th RS not_P_imp_P_eq_False 

81 
 _ => th RS P_imp_P_eq_True; 

82 
(* last 2 lines requires all formulae to be of the from Trueprop(.) *) 

83 

4677  84 
fun mk_meta_eq_True r = Some(r RS meta_eq_to_obj_eq RS P_imp_P_eq_True); 
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923  86 

2082  87 
val simp_thms = map prover 
88 
[ "(x=x) = True", 

89 
"(~True) = False", "(~False) = True", "(~ ~ P) = P", 

90 
"(~P) ~= P", "P ~= (~P)", "(P ~= Q) = (P = (~Q))", 

4640  91 
"(True=P) = P", "(P=True) = P", "(False=P) = (~P)", "(P=False) = (~P)", 
2082  92 
"(True > P) = P", "(False > P) = True", 
93 
"(P > True) = True", "(P > P) = True", 

94 
"(P > False) = (~P)", "(P > ~P) = (~P)", 

95 
"(P & True) = P", "(True & P) = P", 

2800  96 
"(P & False) = False", "(False & P) = False", 
97 
"(P & P) = P", "(P & (P & Q)) = (P & Q)", 

3913  98 
"(P & ~P) = False", "(~P & P) = False", 
2082  99 
"(P  True) = True", "(True  P) = True", 
2800  100 
"(P  False) = P", "(False  P) = P", 
101 
"(P  P) = P", "(P  (P  Q)) = (P  Q)", 

3913  102 
"(P  ~P) = True", "(~P  P) = True", 
2082  103 
"((~P) = (~Q)) = (P=Q)", 
3842  104 
"(!x. P) = P", "(? x. P) = P", "? x. x=t", "? x. t=x", 
4351  105 
(*two needed for the onepointrule quantifier simplification procs*) 
106 
"(? x. x=t & P(x)) = P(t)", (*essential for termination!!*) 

107 
"(! x. t=x > P(x)) = P(t)" ]; (*covers a stray case*) 

923  108 

988  109 
(*Add congruence rules for = (instead of ==) *) 
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infix 4 addcongs delcongs; 
4351  111 

4640  112 
fun mk_meta_cong rl = 
5304  113 
standard(meta_eq(replicate (nprems_of rl) meta_eq_to_obj_eq MRS rl)) 
4640  114 
handle THM _ => 
115 
error("Premises and conclusion of congruence rules must be =equalities"); 

116 

117 
fun ss addcongs congs = ss addeqcongs (map mk_meta_cong congs); 

118 

119 
fun ss delcongs congs = ss deleqcongs (map mk_meta_cong congs); 

923  120 

4086  121 
fun Addcongs congs = (simpset_ref() := simpset() addcongs congs); 
122 
fun Delcongs congs = (simpset_ref() := simpset() delcongs congs); 

1264  123 

1922  124 
val imp_cong = impI RSN 
125 
(2, prove_goal HOL.thy "(P=P')> (P'> (Q=Q'))> ((P>Q) = (P'>Q'))" 

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(fn _=> [Blast_tac 1]) RS mp RS mp); 
1922  127 

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(*Miniscoping: pushing in existential quantifiers*) 
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val ex_simps = map prover 
3842  130 
["(EX x. P x & Q) = ((EX x. P x) & Q)", 
131 
"(EX x. P & Q x) = (P & (EX x. Q x))", 

132 
"(EX x. P x  Q) = ((EX x. P x)  Q)", 

133 
"(EX x. P  Q x) = (P  (EX x. Q x))", 

134 
"(EX x. P x > Q) = ((ALL x. P x) > Q)", 

135 
"(EX x. P > Q x) = (P > (EX x. Q x))"]; 

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(*Miniscoping: pushing in universal quantifiers*) 
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val all_simps = map prover 
3842  139 
["(ALL x. P x & Q) = ((ALL x. P x) & Q)", 
140 
"(ALL x. P & Q x) = (P & (ALL x. Q x))", 

141 
"(ALL x. P x  Q) = ((ALL x. P x)  Q)", 

142 
"(ALL x. P  Q x) = (P  (ALL x. Q x))", 

143 
"(ALL x. P x > Q) = ((EX x. P x) > Q)", 

144 
"(ALL x. P > Q x) = (P > (ALL x. Q x))"]; 

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923  146 

2022  147 
(* elimination of existential quantifiers in assumptions *) 
923  148 

149 
val ex_all_equiv = 

150 
let val lemma1 = prove_goal HOL.thy 

151 
"(? x. P(x) ==> PROP Q) ==> (!!x. P(x) ==> PROP Q)" 

152 
(fn prems => [resolve_tac prems 1, etac exI 1]); 

153 
val lemma2 = prove_goalw HOL.thy [Ex_def] 

154 
"(!!x. P(x) ==> PROP Q) ==> (? x. P(x) ==> PROP Q)" 

155 
(fn prems => [REPEAT(resolve_tac prems 1)]) 

156 
in equal_intr lemma1 lemma2 end; 

157 

158 
end; 

159 

3654  160 
(* Elimination of True from asumptions: *) 
161 

162 
val True_implies_equals = prove_goal HOL.thy 

163 
"(True ==> PROP P) == PROP P" 

4525  164 
(K [rtac equal_intr_rule 1, atac 2, 
3654  165 
METAHYPS (fn prems => resolve_tac prems 1) 1, 
166 
rtac TrueI 1]); 

167 

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fun prove nm thm = qed_goal nm HOL.thy thm (K [Blast_tac 1]); 
923  169 

170 
prove "conj_commute" "(P&Q) = (Q&P)"; 

171 
prove "conj_left_commute" "(P&(Q&R)) = (Q&(P&R))"; 

172 
val conj_comms = [conj_commute, conj_left_commute]; 

2134  173 
prove "conj_assoc" "((P&Q)&R) = (P&(Q&R))"; 
923  174 

1922  175 
prove "disj_commute" "(PQ) = (QP)"; 
176 
prove "disj_left_commute" "(P(QR)) = (Q(PR))"; 

177 
val disj_comms = [disj_commute, disj_left_commute]; 

2134  178 
prove "disj_assoc" "((PQ)R) = (P(QR))"; 
1922  179 

923  180 
prove "conj_disj_distribL" "(P&(QR)) = (P&Q  P&R)"; 
181 
prove "conj_disj_distribR" "((PQ)&R) = (P&R  Q&R)"; 

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1892  183 
prove "disj_conj_distribL" "(P(Q&R)) = ((PQ) & (PR))"; 
184 
prove "disj_conj_distribR" "((P&Q)R) = ((PR) & (QR))"; 

185 

2134  186 
prove "imp_conjR" "(P > (Q&R)) = ((P>Q) & (P>R))"; 
187 
prove "imp_conjL" "((P&Q) >R) = (P > (Q > R))"; 

188 
prove "imp_disjL" "((PQ) > R) = ((P>R)&(Q>R))"; 

1892  189 

3448  190 
(*These two are specialized, but imp_disj_not1 is useful in Auth/Yahalom.ML*) 
191 
prove "imp_disj_not1" "((P > Q  R)) = (~Q > P > R)"; 

192 
prove "imp_disj_not2" "((P > Q  R)) = (~R > P > Q)"; 

193 

3904  194 
prove "imp_disj1" "((P>Q)R) = (P> QR)"; 
195 
prove "imp_disj2" "(Q(P>R)) = (P> QR)"; 

196 

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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_imp" "(~(P > Q)) = (P & ~Q)"; 
1922  200 
prove "not_iff" "(P~=Q) = (P = (~Q))"; 
4743  201 
prove "disj_not1" "(~P  Q) = (P > Q)"; 
202 
prove "disj_not2" "(P  ~Q) = (Q > P)"; (* changes orientation :( *) 

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203 

4830  204 
(*Avoids duplication of subgoals after split_if, when the true and false 
2134  205 
cases boil down to the same thing.*) 
206 
prove "cases_simp" "((P > Q) & (~P > Q)) = Q"; 

207 

3842  208 
prove "not_all" "(~ (! x. P(x))) = (? x.~P(x))"; 
1922  209 
prove "imp_all" "((! x. P x) > Q) = (? x. P x > Q)"; 
3842  210 
prove "not_ex" "(~ (? x. P(x))) = (! x.~P(x))"; 
1922  211 
prove "imp_ex" "((? x. P x) > Q) = (! x. P x > Q)"; 
1660  212 

1655  213 
prove "ex_disj_distrib" "(? x. P(x)  Q(x)) = ((? x. P(x))  (? x. Q(x)))"; 
214 
prove "all_conj_distrib" "(!x. P(x) & Q(x)) = ((! x. P(x)) & (! x. Q(x)))"; 

215 

2134  216 
(* '&' congruence rule: not included by default! 
217 
May slow rewrite proofs down by as much as 50% *) 

218 

219 
let val th = prove_goal HOL.thy 

220 
"(P=P')> (P'> (Q=Q'))> ((P&Q) = (P'&Q'))" 

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221 
(fn _=> [Blast_tac 1]) 
2134  222 
in bind_thm("conj_cong",standard (impI RSN (2, th RS mp RS mp))) end; 
223 

224 
let val th = prove_goal HOL.thy 

225 
"(Q=Q')> (Q'> (P=P'))> ((P&Q) = (P'&Q'))" 

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(fn _=> [Blast_tac 1]) 
2134  227 
in bind_thm("rev_conj_cong",standard (impI RSN (2, th RS mp RS mp))) end; 
228 

229 
(* '' congruence rule: not included by default! *) 

230 

231 
let val th = prove_goal HOL.thy 

232 
"(P=P')> (~P'> (Q=Q'))> ((PQ) = (P'Q'))" 

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(fn _=> [Blast_tac 1]) 
2134  234 
in bind_thm("disj_cong",standard (impI RSN (2, th RS mp RS mp))) end; 
235 

236 
prove "eq_sym_conv" "(x=y) = (y=x)"; 

237 

5278  238 

239 
(** ifthenelse rules **) 

240 

2134  241 
qed_goalw "if_True" HOL.thy [if_def] "(if True then x else y) = x" 
4525  242 
(K [Blast_tac 1]); 
2134  243 

244 
qed_goalw "if_False" HOL.thy [if_def] "(if False then x else y) = y" 

4525  245 
(K [Blast_tac 1]); 
2134  246 

5304  247 
qed_goalw "if_P" HOL.thy [if_def] "!!P. P ==> (if P then x else y) = x" 
248 
(K [Blast_tac 1]); 

249 

2134  250 
qed_goalw "if_not_P" HOL.thy [if_def] "!!P. ~P ==> (if P then x else y) = y" 
4525  251 
(K [Blast_tac 1]); 
2134  252 

4830  253 
qed_goal "split_if" HOL.thy 
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"P(if Q then x else y) = ((Q > P(x)) & (~Q > P(y)))" (K [ 
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255 
res_inst_tac [("Q","Q")] (excluded_middle RS disjE) 1, 
2134  256 
stac if_P 2, 
257 
stac if_not_P 1, 

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258 
ALLGOALS (Blast_tac)]); 
4830  259 
(* for backwards compatibility: *) 
260 
val expand_if = split_if; 

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261 

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262 
qed_goal "split_if_asm" HOL.thy 
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263 
"P(if Q then x else y) = (~((Q & ~P x)  (~Q & ~P y)))" 
4830  264 
(K [stac split_if 1, 
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265 
Blast_tac 1]); 
2134  266 

5304  267 
qed_goal "if_cancel" HOL.thy "(if c then x else x) = x" 
268 
(K [stac split_if 1, Blast_tac 1]); 

269 

270 
qed_goal "if_eq_cancel" HOL.thy "(if x = y then y else x) = x" 

271 
(K [stac split_if 1, Blast_tac 1]); 

272 

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273 
(*This form is useful for expanding IFs on the RIGHT of the ==> symbol*) 
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274 
qed_goal "if_bool_eq_conj" HOL.thy 
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275 
"(if P then Q else R) = ((P>Q) & (~P>R))" 
4830  276 
(K [rtac split_if 1]); 
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277 

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278 
(*And this form is useful for expanding IFs on the LEFT*) 
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279 
qed_goal "if_bool_eq_disj" HOL.thy 
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280 
"(if P then Q else R) = ((P&Q)  (~P&R))" 
4830  281 
(K [stac split_if 1, 
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282 
Blast_tac 1]); 
2134  283 

4351  284 

285 
(*** make simplification procedures for quantifier elimination ***) 

286 

287 
structure Quantifier1 = Quantifier1Fun( 

288 
struct 

289 
(*abstract syntax*) 

290 
fun dest_eq((c as Const("op =",_)) $ s $ t) = Some(c,s,t) 

291 
 dest_eq _ = None; 

292 
fun dest_conj((c as Const("op &",_)) $ s $ t) = Some(c,s,t) 

293 
 dest_conj _ = None; 

294 
val conj = HOLogic.conj 

295 
val imp = HOLogic.imp 

296 
(*rules*) 

297 
val iff_reflection = eq_reflection 

298 
val iffI = iffI 

299 
val sym = sym 

300 
val conjI= conjI 

301 
val conjE= conjE 

302 
val impI = impI 

303 
val impE = impE 

304 
val mp = mp 

305 
val exI = exI 

306 
val exE = exE 

307 
val allI = allI 

308 
val allE = allE 

309 
end); 

310 

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311 
local 
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312 
val ex_pattern = 
4351  313 
read_cterm (sign_of HOL.thy) ("EX x. P(x) & Q(x)",HOLogic.boolT) 
3913  314 

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315 
val all_pattern = 
4351  316 
read_cterm (sign_of HOL.thy) ("ALL x. P(x) & P'(x) > Q(x)",HOLogic.boolT) 
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317 

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318 
in 
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319 
val defEX_regroup = 
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320 
mk_simproc "defined EX" [ex_pattern] Quantifier1.rearrange_ex; 
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321 
val defALL_regroup = 
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322 
mk_simproc "defined ALL" [all_pattern] Quantifier1.rearrange_all; 
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323 
end; 
3913  324 

4351  325 

326 
(*** Case splitting ***) 

3913  327 

5304  328 
structure SplitterData = 
329 
struct 

330 
structure Simplifier = Simplifier 

331 
val mk_meta_eq = mk_meta_eq 

332 
val meta_eq_to_iff = meta_eq_to_obj_eq 

333 
val iffD = iffD2 

334 
val disjE = disjE 

335 
val conjE = conjE 

336 
val exE = exE 

337 
val contrapos = contrapos 

338 
val contrapos2 = contrapos2 

339 
val notnotD = notnotD 

340 
end; 

4681  341 

5304  342 
structure Splitter = SplitterFun(SplitterData); 
2263  343 

5304  344 
val split_tac = Splitter.split_tac; 
345 
val split_inside_tac = Splitter.split_inside_tac; 

346 
val split_asm_tac = Splitter.split_asm_tac; 

347 
val addsplits = Splitter.addsplits; 

348 
val delsplits = Splitter.delsplits; 

349 
val Addsplits = Splitter.Addsplits; 

350 
val Delsplits = Splitter.Delsplits; 

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351 

2134  352 
(** 'if' congruence rules: neither included by default! *) 
353 

354 
(*Simplifies x assuming c and y assuming ~c*) 

355 
qed_goal "if_cong" HOL.thy 

356 
"[ b=c; c ==> x=u; ~c ==> y=v ] ==>\ 

357 
\ (if b then x else y) = (if c then u else v)" 

358 
(fn rew::prems => 

4830  359 
[stac rew 1, stac split_if 1, stac split_if 1, 
2935  360 
blast_tac (HOL_cs addDs prems) 1]); 
2134  361 

362 
(*Prevents simplification of x and y: much faster*) 

363 
qed_goal "if_weak_cong" HOL.thy 

364 
"b=c ==> (if b then x else y) = (if c then x else y)" 

365 
(fn [prem] => [rtac (prem RS arg_cong) 1]); 

366 

367 
(*Prevents simplification of t: much faster*) 

368 
qed_goal "let_weak_cong" HOL.thy 

369 
"a = b ==> (let x=a in t(x)) = (let x=b in t(x))" 

370 
(fn [prem] => [rtac (prem RS arg_cong) 1]); 

371 

372 
(*In general it seems wrong to add distributive laws by default: they 

373 
might cause exponential blowup. But imp_disjL has been in for a while 

374 
and cannot be removed without affecting existing proofs. Moreover, 

375 
rewriting by "(PQ > R) = ((P>R)&(Q>R))" might be justified on the 

376 
grounds that it allows simplification of R in the two cases.*) 

377 

5304  378 
fun gen_all th = forall_elim_vars (#maxidx(rep_thm th)+1) th; 
379 

2134  380 
val mksimps_pairs = 
381 
[("op >", [mp]), ("op &", [conjunct1,conjunct2]), 

382 
("All", [spec]), ("True", []), ("False", []), 

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383 
("If", [if_bool_eq_conj RS iffD1])]; 
1758  384 

5304  385 
(* FIXME: move to Provers/simplifier.ML 
386 
val mk_atomize: (string * thm list) list > thm > thm list 

387 
*) 

388 
(* FIXME: move to Provers/simplifier.ML*) 

389 
fun mk_atomize pairs = 

390 
let fun atoms th = 

391 
(case concl_of th of 

392 
Const("Trueprop",_) $ p => 

393 
(case head_of p of 

394 
Const(a,_) => 

395 
(case assoc(pairs,a) of 

396 
Some(rls) => flat (map atoms ([th] RL rls)) 

397 
 None => [th]) 

398 
 _ => [th]) 

399 
 _ => [th]) 

400 
in atoms end; 

401 

402 
fun mksimps pairs = (map mk_meta_eq o mk_atomize pairs o gen_all); 

403 

4640  404 
fun unsafe_solver prems = FIRST'[resolve_tac (reflexive_thm::TrueI::refl::prems), 
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405 
atac, etac FalseE]; 
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406 
(*No premature instantiation of variables during simplification*) 
4640  407 
fun safe_solver prems = FIRST'[match_tac (reflexive_thm::TrueI::prems), 
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408 
eq_assume_tac, ematch_tac [FalseE]]; 
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409 

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410 
val HOL_basic_ss = empty_ss setsubgoaler asm_simp_tac 
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411 
setSSolver safe_solver 
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412 
setSolver unsafe_solver 
4677  413 
setmksimps (mksimps mksimps_pairs) 
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414 
setmkeqTrue mk_meta_eq_True; 
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415 

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416 
val HOL_ss = 
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417 
HOL_basic_ss addsimps 
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418 
([triv_forall_equality, (* prunes params *) 
3654  419 
True_implies_equals, (* prune asms `True' *) 
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420 
if_True, if_False, if_cancel, if_eq_cancel, 
5304  421 
imp_disjL, conj_assoc, disj_assoc, 
3904  422 
de_Morgan_conj, de_Morgan_disj, imp_disj1, imp_disj2, not_imp, 
5304  423 
disj_not1, not_all, not_ex, cases_simp, Eps_eq] 
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424 
@ ex_simps @ all_simps @ simp_thms) 
4032
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425 
addsimprocs [defALL_regroup,defEX_regroup] 
4744
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426 
addcongs [imp_cong] 
4830  427 
addsplits [split_if]; 
2082  428 

1655  429 
qed_goal "if_distrib" HOL.thy 
430 
"f(if c then x else y) = (if c then f x else f y)" 

4830  431 
(K [simp_tac (HOL_ss setloop (split_tac [split_if])) 1]); 
1655  432 

1984  433 

4327  434 
(*For expand_case_tac*) 
2948
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435 
val prems = goal HOL.thy "[ P ==> Q(True); ~P ==> Q(False) ] ==> Q(P)"; 
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436 
by (case_tac "P" 1); 
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437 
by (ALLGOALS (asm_simp_tac (HOL_ss addsimps prems))); 
f18035b1d531
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438 
val expand_case = result(); 
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439 

4327  440 
(*Used in Auth proofs. Typically P contains Vars that become instantiated 
441 
during unification.*) 

2948
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442 
fun expand_case_tac P i = 
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443 
res_inst_tac [("P",P)] expand_case i THEN 
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444 
Simp_tac (i+1) THEN 
f18035b1d531
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445 
Simp_tac i; 
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446 

f18035b1d531
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447 

4119  448 
(* install implicit simpset *) 
1984  449 

4086  450 
simpset_ref() := HOL_ss; 
1984  451 

3615
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Moved some functions which used to be part of thy_data.ML
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452 

4652
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453 

5219  454 
(*** integration of simplifier with classical reasoner ***) 
2636
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455 

5219  456 
structure Clasimp = ClasimpFun 
457 
(structure Simplifier = Simplifier and Classical = Classical and Blast = Blast 

5220  458 
val op addcongs = op addcongs and op delcongs = op delcongs 
459 
and op addSaltern = op addSaltern and op addbefore = op addbefore); 

5219  460 

4652
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461 
open Clasimp; 
2636
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462 

4b30dbe4a020
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463 
val HOL_css = (HOL_cs, HOL_ss); 