author  haftmann 
Tue, 17 Aug 2010 19:36:39 +0200  
changeset 38500  d5477ee35820 
parent 36319  8feb2c4bef1a 
child 41526  54b4686704af 
permissions  rwrr 
17456  1 
(* Title: CCL/Wfd.thy 
1474  2 
Author: Martin Coen, Cambridge University Computer Laboratory 
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Copyright 1993 University of Cambridge 
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*) 

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header {* Wellfounded relations in CCL *} 
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theory Wfd 

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imports Trancl Type Hered 

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begin 

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consts 

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(*** Predicates ***) 

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Wfd :: "[i set] => o" 

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(*** Relations ***) 

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wf :: "[i set] => i set" 

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wmap :: "[i=>i,i set] => i set" 

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lex :: "[i set,i set] => i set" (infixl "**" 70) 
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NatPR :: "i set" 
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ListPR :: "i set => i set" 

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24825  22 
defs 
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Wfd_def: 
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"Wfd(R) == ALL P.(ALL x.(ALL y.<y,x> : R > y:P) > x:P) > (ALL a. a:P)" 
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wf_def: "wf(R) == {x. x:R & Wfd(R)}" 
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wmap_def: "wmap(f,R) == {p. EX x y. p=<x,y> & <f(x),f(y)> : R}" 
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lex_def: 

3837  31 
"ra**rb == {p. EX a a' b b'. p = <<a,b>,<a',b'>> & (<a,a'> : ra  (a=a' & <b,b'> : rb))}" 
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NatPR_def: "NatPR == {p. EX x:Nat. p=<x,succ(x)>}" 
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ListPR_def: "ListPR(A) == {p. EX h:A. EX t:List(A). p=<t,h$t>}" 

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lemma wfd_induct: 

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assumes 1: "Wfd(R)" 

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and 2: "!!x.[ ALL y. <y,x>: R > P(y) ] ==> P(x)" 

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shows "P(a)" 

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apply (rule 1 [unfolded Wfd_def, rule_format, THEN CollectD]) 

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using 2 apply blast 

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done 

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lemma wfd_strengthen_lemma: 

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assumes 1: "!!x y.<x,y> : R ==> Q(x)" 

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and 2: "ALL x. (ALL y. <y,x> : R > y : P) > x : P" 

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and 3: "!!x. Q(x) ==> x:P" 

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shows "a:P" 

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apply (rule 2 [rule_format]) 

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using 1 3 

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apply blast 

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done 

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ML {* 

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fun wfd_strengthen_tac ctxt s i = 
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res_inst_tac ctxt [(("Q", 0), s)] @{thm wfd_strengthen_lemma} i THEN assume_tac (i+1) 
20140  58 
*} 
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lemma wf_anti_sym: "[ Wfd(r); <a,x>:r; <x,a>:r ] ==> P" 

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apply (subgoal_tac "ALL x. <a,x>:r > <x,a>:r > P") 

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apply blast 

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apply (erule wfd_induct) 

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apply blast 

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done 

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lemma wf_anti_refl: "[ Wfd(r); <a,a>: r ] ==> P" 

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apply (rule wf_anti_sym) 

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apply assumption+ 

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done 

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subsection {* Irreflexive transitive closure *} 

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lemma trancl_wf: 

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assumes 1: "Wfd(R)" 

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shows "Wfd(R^+)" 

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apply (unfold Wfd_def) 

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apply (rule allI ballI impI)+ 

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(*must retain the universal formula for later use!*) 

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apply (rule allE, assumption) 

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apply (erule mp) 

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apply (rule 1 [THEN wfd_induct]) 

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apply (rule impI [THEN allI]) 

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apply (erule tranclE) 

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apply blast 

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apply (erule spec [THEN mp, THEN spec, THEN mp]) 

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apply assumption+ 

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done 

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91 

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subsection {* Lexicographic Ordering *} 

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lemma lexXH: 

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"p : ra**rb <> (EX a a' b b'. p = <<a,b>,<a',b'>> & (<a,a'> : ra  a=a' & <b,b'> : rb))" 

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unfolding lex_def by blast 

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lemma lexI1: "<a,a'> : ra ==> <<a,b>,<a',b'>> : ra**rb" 

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by (blast intro!: lexXH [THEN iffD2]) 

100 

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lemma lexI2: "<b,b'> : rb ==> <<a,b>,<a,b'>> : ra**rb" 

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by (blast intro!: lexXH [THEN iffD2]) 

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lemma lexE: 

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assumes 1: "p : ra**rb" 

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and 2: "!!a a' b b'.[ <a,a'> : ra; p=<<a,b>,<a',b'>> ] ==> R" 

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and 3: "!!a b b'.[ <b,b'> : rb; p = <<a,b>,<a,b'>> ] ==> R" 

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shows R 

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apply (rule 1 [THEN lexXH [THEN iffD1], THEN exE]) 

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using 2 3 

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apply blast 

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done 

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lemma lex_pair: "[ p : r**s; !!a a' b b'. p = <<a,b>,<a',b'>> ==> P ] ==>P" 

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apply (erule lexE) 

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apply blast+ 

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done 

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lemma lex_wf: 

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assumes 1: "Wfd(R)" 

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and 2: "Wfd(S)" 

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shows "Wfd(R**S)" 

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apply (unfold Wfd_def) 

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apply safe 

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apply (tactic {* wfd_strengthen_tac @{context} "%x. EX a b. x=<a,b>" 1 *}) 
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apply (blast elim!: lex_pair) 
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apply (subgoal_tac "ALL a b.<a,b>:P") 

128 
apply blast 

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apply (rule 1 [THEN wfd_induct, THEN allI]) 

130 
apply (rule 2 [THEN wfd_induct, THEN allI]) back 

131 
apply (fast elim!: lexE) 

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done 

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134 

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subsection {* Mapping *} 

136 

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lemma wmapXH: "p : wmap(f,r) <> (EX x y. p=<x,y> & <f(x),f(y)> : r)" 

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unfolding wmap_def by blast 

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lemma wmapI: "<f(a),f(b)> : r ==> <a,b> : wmap(f,r)" 

141 
by (blast intro!: wmapXH [THEN iffD2]) 

142 

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lemma wmapE: "[ p : wmap(f,r); !!a b.[ <f(a),f(b)> : r; p=<a,b> ] ==> R ] ==> R" 

144 
by (blast dest!: wmapXH [THEN iffD1]) 

145 

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lemma wmap_wf: 

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assumes 1: "Wfd(r)" 

148 
shows "Wfd(wmap(f,r))" 

149 
apply (unfold Wfd_def) 

150 
apply clarify 

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apply (subgoal_tac "ALL b. ALL a. f (a) =b>a:P") 

152 
apply blast 

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apply (rule 1 [THEN wfd_induct, THEN allI]) 

154 
apply clarify 

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apply (erule spec [THEN mp]) 

156 
apply (safe elim!: wmapE) 

157 
apply (erule spec [THEN mp, THEN spec, THEN mp]) 

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apply assumption 

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apply (rule refl) 

160 
done 

161 

162 

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subsection {* Projections *} 

164 

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lemma wfstI: "<xa,ya> : r ==> <<xa,xb>,<ya,yb>> : wmap(fst,r)" 

166 
apply (rule wmapI) 

167 
apply simp 

168 
done 

169 

170 
lemma wsndI: "<xb,yb> : r ==> <<xa,xb>,<ya,yb>> : wmap(snd,r)" 

171 
apply (rule wmapI) 

172 
apply simp 

173 
done 

174 

175 
lemma wthdI: "<xc,yc> : r ==> <<xa,<xb,xc>>,<ya,<yb,yc>>> : wmap(thd,r)" 

176 
apply (rule wmapI) 

177 
apply simp 

178 
done 

179 

180 

181 
subsection {* Ground wellfounded relations *} 

182 

183 
lemma wfI: "[ Wfd(r); a : r ] ==> a : wf(r)" 

184 
unfolding wf_def by blast 

185 

186 
lemma Empty_wf: "Wfd({})" 

187 
unfolding Wfd_def by (blast elim: EmptyXH [THEN iffD1, THEN FalseE]) 

188 

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lemma wf_wf: "Wfd(wf(R))" 

190 
unfolding wf_def 

191 
apply (rule_tac Q = "Wfd(R)" in excluded_middle [THEN disjE]) 

192 
apply simp_all 

193 
apply (rule Empty_wf) 

194 
done 

195 

196 
lemma NatPRXH: "p : NatPR <> (EX x:Nat. p=<x,succ(x)>)" 

197 
unfolding NatPR_def by blast 

198 

199 
lemma ListPRXH: "p : ListPR(A) <> (EX h:A. EX t:List(A).p=<t,h$t>)" 

200 
unfolding ListPR_def by blast 

201 

202 
lemma NatPRI: "x : Nat ==> <x,succ(x)> : NatPR" 

203 
by (auto simp: NatPRXH) 

204 

205 
lemma ListPRI: "[ t : List(A); h : A ] ==> <t,h $ t> : ListPR(A)" 

206 
by (auto simp: ListPRXH) 

207 

208 
lemma NatPR_wf: "Wfd(NatPR)" 

209 
apply (unfold Wfd_def) 

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apply clarify 

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apply (tactic {* wfd_strengthen_tac @{context} "%x. x:Nat" 1 *}) 
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apply (fastsimp iff: NatPRXH) 
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apply (erule Nat_ind) 

214 
apply (fastsimp iff: NatPRXH)+ 

215 
done 

216 

217 
lemma ListPR_wf: "Wfd(ListPR(A))" 

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apply (unfold Wfd_def) 

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apply clarify 

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apply (tactic {* wfd_strengthen_tac @{context} "%x. x:List (A)" 1 *}) 
20140  221 
apply (fastsimp iff: ListPRXH) 
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apply (erule List_ind) 

223 
apply (fastsimp iff: ListPRXH)+ 

224 
done 

225 

226 

227 
subsection {* General Recursive Functions *} 

228 

229 
lemma letrecT: 

230 
assumes 1: "a : A" 

231 
and 2: "!!p g.[ p:A; ALL x:{x: A. <x,p>:wf(R)}. g(x) : D(x) ] ==> h(p,g) : D(p)" 

232 
shows "letrec g x be h(x,g) in g(a) : D(a)" 

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apply (rule 1 [THEN rev_mp]) 

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apply (rule wf_wf [THEN wfd_induct]) 

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apply (subst letrecB) 

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apply (rule impI) 

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apply (erule 2) 

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apply blast 

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done 

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241 
lemma SPLITB: "SPLIT(<a,b>,B) = B(a,b)" 

242 
unfolding SPLIT_def 

243 
apply (rule set_ext) 

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apply blast 

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done 

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20140  247 
lemma letrec2T: 
248 
assumes "a : A" 

249 
and "b : B" 

250 
and "!!p q g.[ p:A; q:B; 

251 
ALL x:A. ALL y:{y: B. <<x,y>,<p,q>>:wf(R)}. g(x,y) : D(x,y) ] ==> 

252 
h(p,q,g) : D(p,q)" 

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shows "letrec g x y be h(x,y,g) in g(a,b) : D(a,b)" 

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apply (unfold letrec2_def) 

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apply (rule SPLITB [THEN subst]) 

256 
apply (assumption  rule letrecT pairT splitT prems)+ 

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apply (subst SPLITB) 

258 
apply (assumption  rule ballI SubtypeI prems)+ 

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apply (rule SPLITB [THEN subst]) 

260 
apply (assumption  rule letrecT SubtypeI pairT splitT prems  

261 
erule bspec SubtypeE sym [THEN subst])+ 

262 
done 

263 

264 
lemma lem: "SPLIT(<a,<b,c>>,%x xs. SPLIT(xs,%y z. B(x,y,z))) = B(a,b,c)" 

265 
by (simp add: SPLITB) 

266 

267 
lemma letrec3T: 

268 
assumes "a : A" 

269 
and "b : B" 

270 
and "c : C" 

271 
and "!!p q r g.[ p:A; q:B; r:C; 

272 
ALL x:A. ALL y:B. ALL z:{z:C. <<x,<y,z>>,<p,<q,r>>> : wf(R)}. 

273 
g(x,y,z) : D(x,y,z) ] ==> 

274 
h(p,q,r,g) : D(p,q,r)" 

275 
shows "letrec g x y z be h(x,y,z,g) in g(a,b,c) : D(a,b,c)" 

276 
apply (unfold letrec3_def) 

277 
apply (rule lem [THEN subst]) 

278 
apply (assumption  rule letrecT pairT splitT prems)+ 

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apply (simp add: SPLITB) 

280 
apply (assumption  rule ballI SubtypeI prems)+ 

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apply (rule lem [THEN subst]) 

282 
apply (assumption  rule letrecT SubtypeI pairT splitT prems  

283 
erule bspec SubtypeE sym [THEN subst])+ 

284 
done 

285 

286 
lemmas letrecTs = letrecT letrec2T letrec3T 

287 

288 

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subsection {* Type Checking for Recursive Calls *} 

290 

291 
lemma rcallT: 

292 
"[ ALL x:{x:A.<x,p>:wf(R)}.g(x):D(x); 

293 
g(a) : D(a) ==> g(a) : E; a:A; <a,p>:wf(R) ] ==> 

294 
g(a) : E" 

295 
by blast 

296 

297 
lemma rcall2T: 

298 
"[ ALL x:A. ALL y:{y:B.<<x,y>,<p,q>>:wf(R)}.g(x,y):D(x,y); 

299 
g(a,b) : D(a,b) ==> g(a,b) : E; a:A; b:B; <<a,b>,<p,q>>:wf(R) ] ==> 

300 
g(a,b) : E" 

301 
by blast 

302 

303 
lemma rcall3T: 

304 
"[ ALL x:A. ALL y:B. ALL z:{z:C.<<x,<y,z>>,<p,<q,r>>>:wf(R)}. g(x,y,z):D(x,y,z); 

305 
g(a,b,c) : D(a,b,c) ==> g(a,b,c) : E; 

306 
a:A; b:B; c:C; <<a,<b,c>>,<p,<q,r>>> : wf(R) ] ==> 

307 
g(a,b,c) : E" 

308 
by blast 

309 

310 
lemmas rcallTs = rcallT rcall2T rcall3T 

311 

312 

313 
subsection {* Instantiating an induction hypothesis with an equality assumption *} 

314 

315 
lemma hyprcallT: 

316 
assumes 1: "g(a) = b" 

317 
and 2: "ALL x:{x:A.<x,p>:wf(R)}.g(x):D(x)" 

318 
and 3: "ALL x:{x:A.<x,p>:wf(R)}.g(x):D(x) ==> b=g(a) ==> g(a) : D(a) ==> P" 

319 
and 4: "ALL x:{x:A.<x,p>:wf(R)}.g(x):D(x) ==> a:A" 

320 
and 5: "ALL x:{x:A.<x,p>:wf(R)}.g(x):D(x) ==> <a,p>:wf(R)" 

321 
shows P 

322 
apply (rule 3 [OF 2, OF 1 [symmetric]]) 

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apply (rule rcallT [OF 2]) 

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apply assumption 

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apply (rule 4 [OF 2]) 

326 
apply (rule 5 [OF 2]) 

327 
done 

328 

329 
lemma hyprcall2T: 

330 
assumes 1: "g(a,b) = c" 

331 
and 2: "ALL x:A. ALL y:{y:B.<<x,y>,<p,q>>:wf(R)}.g(x,y):D(x,y)" 

332 
and 3: "[ c=g(a,b); g(a,b) : D(a,b) ] ==> P" 

333 
and 4: "a:A" 

334 
and 5: "b:B" 

335 
and 6: "<<a,b>,<p,q>>:wf(R)" 

336 
shows P 

337 
apply (rule 3) 

338 
apply (rule 1 [symmetric]) 

339 
apply (rule rcall2T) 

23467  340 
apply (rule 2) 
341 
apply assumption 

342 
apply (rule 4) 

343 
apply (rule 5) 

344 
apply (rule 6) 

20140  345 
done 
346 

347 
lemma hyprcall3T: 

348 
assumes 1: "g(a,b,c) = d" 

349 
and 2: "ALL x:A. ALL y:B. ALL z:{z:C.<<x,<y,z>>,<p,<q,r>>>:wf(R)}.g(x,y,z):D(x,y,z)" 

350 
and 3: "[ d=g(a,b,c); g(a,b,c) : D(a,b,c) ] ==> P" 

351 
and 4: "a:A" 

352 
and 5: "b:B" 

353 
and 6: "c:C" 

354 
and 7: "<<a,<b,c>>,<p,<q,r>>> : wf(R)" 

355 
shows P 

356 
apply (rule 3) 

357 
apply (rule 1 [symmetric]) 

358 
apply (rule rcall3T) 

23467  359 
apply (rule 2) 
360 
apply assumption 

361 
apply (rule 4) 

362 
apply (rule 5) 

363 
apply (rule 6) 

364 
apply (rule 7) 

20140  365 
done 
366 

367 
lemmas hyprcallTs = hyprcallT hyprcall2T hyprcall3T 

368 

369 

370 
subsection {* Rules to Remove Induction Hypotheses after Type Checking *} 

371 

372 
lemma rmIH1: "[ ALL x:{x:A.<x,p>:wf(R)}.g(x):D(x); P ] ==> P" . 

373 

374 
lemma rmIH2: "[ ALL x:A. ALL y:{y:B.<<x,y>,<p,q>>:wf(R)}.g(x,y):D(x,y); P ] ==> P" . 

375 

376 
lemma rmIH3: 

377 
"[ ALL x:A. ALL y:B. ALL z:{z:C.<<x,<y,z>>,<p,<q,r>>>:wf(R)}.g(x,y,z):D(x,y,z); 

378 
P ] ==> 

379 
P" . 

380 

381 
lemmas rmIHs = rmIH1 rmIH2 rmIH3 

382 

383 

384 
subsection {* Lemmas for constructors and subtypes *} 

385 

386 
(* 0ary constructors do not need additional rules as they are handled *) 

387 
(* correctly by applying SubtypeI *) 

388 

389 
lemma Subtype_canTs: 

390 
"!!a b A B P. a : {x:A. b:{y:B(a).P(<x,y>)}} ==> <a,b> : {x:Sigma(A,B).P(x)}" 

391 
"!!a A B P. a : {x:A. P(inl(x))} ==> inl(a) : {x:A+B. P(x)}" 

392 
"!!b A B P. b : {x:B. P(inr(x))} ==> inr(b) : {x:A+B. P(x)}" 

393 
"!!a P. a : {x:Nat. P(succ(x))} ==> succ(a) : {x:Nat. P(x)}" 

394 
"!!h t A P. h : {x:A. t : {y:List(A).P(x$y)}} ==> h$t : {x:List(A).P(x)}" 

395 
by (assumption  rule SubtypeI canTs icanTs  erule SubtypeE)+ 

396 

397 
lemma letT: "[ f(t):B; ~t=bot ] ==> let x be t in f(x) : B" 

398 
apply (erule letB [THEN ssubst]) 

399 
apply assumption 

400 
done 

401 

402 
lemma applyT2: "[ a:A; f : Pi(A,B) ] ==> f ` a : B(a)" 

403 
apply (erule applyT) 

404 
apply assumption 

405 
done 

406 

407 
lemma rcall_lemma1: "[ a:A; a:A ==> P(a) ] ==> a : {x:A. P(x)}" 

408 
by blast 

409 

410 
lemma rcall_lemma2: "[ a:{x:A. Q(x)}; [ a:A; Q(a) ] ==> P(a) ] ==> a : {x:A. P(x)}" 

411 
by blast 

412 

413 
lemmas rcall_lemmas = asm_rl rcall_lemma1 SubtypeD1 rcall_lemma2 

414 

415 

416 
subsection {* Typechecking *} 

417 

418 
ML {* 

419 

420 
local 

421 

422 
val type_rls = 

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@{thms canTs} @ @{thms icanTs} @ @{thms applyT2} @ @{thms ncanTs} @ @{thms incanTs} @ 
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@{thms precTs} @ @{thms letrecTs} @ @{thms letT} @ @{thms Subtype_canTs}; 
20140  425 

38500  426 
fun bvars (Const(@{const_name all},_) $ Abs(s,_,t)) l = bvars t (s::l) 
20140  427 
 bvars _ l = l 
428 

38500  429 
fun get_bno l n (Const(@{const_name all},_) $ Abs(s,_,t)) = get_bno (s::l) n t 
430 
 get_bno l n (Const(@{const_name Trueprop},_) $ t) = get_bno l n t 

431 
 get_bno l n (Const(@{const_name Ball},_) $ _ $ Abs(s,_,t)) = get_bno (s::l) (n+1) t 

432 
 get_bno l n (Const(@{const_name mem},_) $ t $ _) = get_bno l n t 

20140  433 
 get_bno l n (t $ s) = get_bno l n t 
434 
 get_bno l n (Bound m) = (mlength(l),n) 

435 

436 
(* Not a great way of identifying induction hypothesis! *) 

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fun could_IH x = Term.could_unify(x,hd (prems_of @{thm rcallT})) orelse 
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Term.could_unify(x,hd (prems_of @{thm rcall2T})) orelse 
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439 
Term.could_unify(x,hd (prems_of @{thm rcall3T})) 
20140  440 

441 
fun IHinst tac rls = SUBGOAL (fn (Bi,i) => 

442 
let val bvs = bvars Bi [] 

33317  443 
val ihs = filter could_IH (Logic.strip_assums_hyp Bi) 
20140  444 
val rnames = map (fn x=> 
445 
let val (a,b) = get_bno [] 0 x 

446 
in (List.nth(bvs,a),b) end) ihs 

447 
fun try_IHs [] = no_tac 

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 try_IHs ((x,y)::xs) = tac [(("g", 0), x)] (List.nth(rls,y1)) i ORELSE (try_IHs xs) 
20140  449 
in try_IHs rnames end) 
450 

451 
fun is_rigid_prog t = 

452 
case (Logic.strip_assums_concl t) of 

38500  453 
(Const(@{const_name Trueprop},_) $ (Const(@{const_name mem},_) $ a $ _)) => null (Term.add_vars a []) 
20140  454 
 _ => false 
455 
in 

456 

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fun rcall_tac ctxt i = 
27239  458 
let fun tac ps rl i = res_inst_tac ctxt ps rl i THEN atac i 
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in IHinst tac @{thms rcallTs} i end 
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THEN eresolve_tac @{thms rcall_lemmas} i 
20140  461 

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fun raw_step_tac ctxt prems i = ares_tac (prems@type_rls) i ORELSE 
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rcall_tac ctxt i ORELSE 
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464 
ematch_tac [@{thm SubtypeE}] i ORELSE 
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match_tac [@{thm SubtypeI}] i 
20140  466 

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fun tc_step_tac ctxt prems = SUBGOAL (fn (Bi,i) => 
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if is_rigid_prog Bi then raw_step_tac ctxt prems i else no_tac) 
20140  469 

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fun typechk_tac ctxt rls i = SELECT_GOAL (REPEAT_FIRST (tc_step_tac ctxt rls)) i 
20140  471 

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fun tac ctxt = typechk_tac ctxt [] 1 
20140  473 

474 
(*** Clean up Correctness Condictions ***) 

475 

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val clean_ccs_tac = REPEAT_FIRST (eresolve_tac ([@{thm SubtypeE}] @ @{thms rmIHs}) ORELSE' 
20140  477 
hyp_subst_tac) 
478 

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fun clean_ccs_tac ctxt = 
27239  480 
let fun tac ps rl i = eres_inst_tac ctxt ps rl i THEN atac i in 
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TRY (REPEAT_FIRST (IHinst tac @{thms hyprcallTs} ORELSE' 
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eresolve_tac ([asm_rl, @{thm SubtypeE}] @ @{thms rmIHs}) ORELSE' 
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483 
hyp_subst_tac)) 
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484 
end 
20140  485 

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fun gen_ccs_tac ctxt rls i = 
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SELECT_GOAL (REPEAT_FIRST (tc_step_tac ctxt rls) THEN clean_ccs_tac ctxt) i 
17456  488 

0  489 
end 
20140  490 
*} 
491 

492 

493 
subsection {* Evaluation *} 

494 

495 
ML {* 

496 

497 
local 

31902  498 
structure Data = Named_Thms(val name = "eval" val description = "evaluation rules"); 
20140  499 
in 
500 

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fun eval_tac ths = 
32283  502 
Subgoal.FOCUS_PREMS (fn {context, prems, ...} => 
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503 
DEPTH_SOLVE_1 (resolve_tac (ths @ prems @ Data.get context) 1)); 
20140  504 

505 
val eval_setup = 

24034  506 
Data.setup #> 
30515  507 
Method.setup @{binding eval} 
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508 
(Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD' (CHANGED o eval_tac ths ctxt))) 
30515  509 
"evaluation"; 
20140  510 

511 
end; 

512 

513 
*} 

514 

515 
setup eval_setup 

516 

517 
lemmas eval_rls [eval] = trueV falseV pairV lamV caseVtrue caseVfalse caseVpair caseVlam 

518 

519 
lemma applyV [eval]: 

520 
assumes "f > lam x. b(x)" 

521 
and "b(a) > c" 

522 
shows "f ` a > c" 

523 
unfolding apply_def by (eval prems) 

524 

525 
lemma letV: 

526 
assumes 1: "t > a" 

527 
and 2: "f(a) > c" 

528 
shows "let x be t in f(x) > c" 

529 
apply (unfold let_def) 

530 
apply (rule 1 [THEN canonical]) 

531 
apply (tactic {* 

26391  532 
REPEAT (DEPTH_SOLVE_1 (resolve_tac (@{thms assms} @ @{thms eval_rls}) 1 ORELSE 
533 
etac @{thm substitute} 1)) *}) 

20140  534 
done 
535 

536 
lemma fixV: "f(fix(f)) > c ==> fix(f) > c" 

537 
apply (unfold fix_def) 

538 
apply (rule applyV) 

539 
apply (rule lamV) 

540 
apply assumption 

541 
done 

542 

543 
lemma letrecV: 

544 
"h(t,%y. letrec g x be h(x,g) in g(y)) > c ==> 

545 
letrec g x be h(x,g) in g(t) > c" 

546 
apply (unfold letrec_def) 

547 
apply (assumption  rule fixV applyV lamV)+ 

548 
done 

549 

550 
lemmas [eval] = letV letrecV fixV 

551 

552 
lemma V_rls [eval]: 

553 
"true > true" 

554 
"false > false" 

555 
"!!b c t u. [ b>true; t>c ] ==> if b then t else u > c" 

556 
"!!b c t u. [ b>false; u>c ] ==> if b then t else u > c" 

557 
"!!a b. <a,b> > <a,b>" 

558 
"!!a b c t h. [ t > <a,b>; h(a,b) > c ] ==> split(t,h) > c" 

559 
"zero > zero" 

560 
"!!n. succ(n) > succ(n)" 

561 
"!!c n t u. [ n > zero; t > c ] ==> ncase(n,t,u) > c" 

562 
"!!c n t u x. [ n > succ(x); u(x) > c ] ==> ncase(n,t,u) > c" 

563 
"!!c n t u. [ n > zero; t > c ] ==> nrec(n,t,u) > c" 

564 
"!!c n t u x. [ n>succ(x); u(x,nrec(x,t,u))>c ] ==> nrec(n,t,u)>c" 

565 
"[] > []" 

566 
"!!h t. h$t > h$t" 

567 
"!!c l t u. [ l > []; t > c ] ==> lcase(l,t,u) > c" 

568 
"!!c l t u x xs. [ l > x$xs; u(x,xs) > c ] ==> lcase(l,t,u) > c" 

569 
"!!c l t u. [ l > []; t > c ] ==> lrec(l,t,u) > c" 

570 
"!!c l t u x xs. [ l>x$xs; u(x,xs,lrec(xs,t,u))>c ] ==> lrec(l,t,u)>c" 

571 
unfolding data_defs by eval+ 

572 

573 

574 
subsection {* Factorial *} 

575 

36319  576 
schematic_lemma 
20140  577 
"letrec f n be ncase(n,succ(zero),%x. nrec(n,zero,%y g. nrec(f(x),g,%z h. succ(h)))) 
578 
in f(succ(succ(zero))) > ?a" 

579 
by eval 

580 

36319  581 
schematic_lemma 
20140  582 
"letrec f n be ncase(n,succ(zero),%x. nrec(n,zero,%y g. nrec(f(x),g,%z h. succ(h)))) 
583 
in f(succ(succ(succ(zero)))) > ?a" 

584 
by eval 

585 

586 
subsection {* Less Than Or Equal *} 

587 

36319  588 
schematic_lemma 
20140  589 
"letrec f p be split(p,%m n. ncase(m,true,%x. ncase(n,false,%y. f(<x,y>)))) 
590 
in f(<succ(zero), succ(zero)>) > ?a" 

591 
by eval 

592 

36319  593 
schematic_lemma 
20140  594 
"letrec f p be split(p,%m n. ncase(m,true,%x. ncase(n,false,%y. f(<x,y>)))) 
595 
in f(<succ(zero), succ(succ(succ(succ(zero))))>) > ?a" 

596 
by eval 

597 

36319  598 
schematic_lemma 
20140  599 
"letrec f p be split(p,%m n. ncase(m,true,%x. ncase(n,false,%y. f(<x,y>)))) 
600 
in f(<succ(succ(succ(succ(succ(zero))))), succ(succ(succ(succ(zero))))>) > ?a" 

601 
by eval 

602 

603 

604 
subsection {* Reverse *} 

605 

36319  606 
schematic_lemma 
20140  607 
"letrec id l be lcase(l,[],%x xs. x$id(xs)) 
608 
in id(zero$succ(zero)$[]) > ?a" 

609 
by eval 

610 

36319  611 
schematic_lemma 
20140  612 
"letrec rev l be lcase(l,[],%x xs. lrec(rev(xs),x$[],%y ys g. y$g)) 
613 
in rev(zero$succ(zero)$(succ((lam x. x)`succ(zero)))$([])) > ?a" 

614 
by eval 

615 

616 
end 