src/ZF/Constructible/Rec_Separation.thy
 author wenzelm Mon Jul 29 00:57:16 2002 +0200 (2002-07-29) changeset 13428 99e52e78eb65 parent 13422 af9bc8d87a75 child 13429 2232810416fc permissions -rw-r--r--
eliminate open locales and special ML code;
 paulson@13363 ` 1` ```header{*Separation for Facts About Recursion*} ``` paulson@13348 ` 2` paulson@13363 ` 3` ```theory Rec_Separation = Separation + Datatype_absolute: ``` paulson@13348 ` 4` paulson@13348 ` 5` ```text{*This theory proves all instances needed for locales @{text ``` paulson@13348 ` 6` ```"M_trancl"}, @{text "M_wfrank"} and @{text "M_datatypes"}*} ``` paulson@13348 ` 7` paulson@13363 ` 8` ```lemma eq_succ_imp_lt: "[|i = succ(j); Ord(i)|] ==> jnnat[M]. \n[M]. \n'[M]. ``` paulson@13348 ` 19` ``` omega(M,nnat) & n\nnat & successor(M,n,n') & ``` paulson@13348 ` 20` ``` (\f[M]. typed_function(M,n',A,f) & ``` wenzelm@13428 ` 21` ``` (\x[M]. \y[M]. \zero[M]. pair(M,x,y,p) & empty(M,zero) & ``` wenzelm@13428 ` 22` ``` fun_apply(M,f,zero,x) & fun_apply(M,f,n,y)) & ``` wenzelm@13428 ` 23` ``` (\j[M]. j\n --> ``` wenzelm@13428 ` 24` ``` (\fj[M]. \sj[M]. \fsj[M]. \ffp[M]. ``` wenzelm@13428 ` 25` ``` fun_apply(M,f,j,fj) & successor(M,j,sj) & ``` wenzelm@13428 ` 26` ``` fun_apply(M,f,sj,fsj) & pair(M,fj,fsj,ffp) & ffp \ r)))"*) ``` paulson@13348 ` 27` ```constdefs rtran_closure_mem_fm :: "[i,i,i]=>i" ``` wenzelm@13428 ` 28` ``` "rtran_closure_mem_fm(A,r,p) == ``` paulson@13348 ` 29` ``` Exists(Exists(Exists( ``` paulson@13348 ` 30` ``` And(omega_fm(2), ``` paulson@13348 ` 31` ``` And(Member(1,2), ``` paulson@13348 ` 32` ``` And(succ_fm(1,0), ``` paulson@13348 ` 33` ``` Exists(And(typed_function_fm(1, A#+4, 0), ``` wenzelm@13428 ` 34` ``` And(Exists(Exists(Exists( ``` wenzelm@13428 ` 35` ``` And(pair_fm(2,1,p#+7), ``` wenzelm@13428 ` 36` ``` And(empty_fm(0), ``` wenzelm@13428 ` 37` ``` And(fun_apply_fm(3,0,2), fun_apply_fm(3,5,1))))))), ``` wenzelm@13428 ` 38` ``` Forall(Implies(Member(0,3), ``` wenzelm@13428 ` 39` ``` Exists(Exists(Exists(Exists( ``` wenzelm@13428 ` 40` ``` And(fun_apply_fm(5,4,3), ``` wenzelm@13428 ` 41` ``` And(succ_fm(4,2), ``` wenzelm@13428 ` 42` ``` And(fun_apply_fm(5,2,1), ``` wenzelm@13428 ` 43` ``` And(pair_fm(3,1,0), Member(0,r#+9))))))))))))))))))))" ``` paulson@13348 ` 44` paulson@13348 ` 45` paulson@13348 ` 46` ```lemma rtran_closure_mem_type [TC]: ``` paulson@13348 ` 47` ``` "[| x \ nat; y \ nat; z \ nat |] ==> rtran_closure_mem_fm(x,y,z) \ formula" ``` wenzelm@13428 ` 48` ```by (simp add: rtran_closure_mem_fm_def) ``` paulson@13348 ` 49` paulson@13348 ` 50` ```lemma arity_rtran_closure_mem_fm [simp]: ``` wenzelm@13428 ` 51` ``` "[| x \ nat; y \ nat; z \ nat |] ``` paulson@13348 ` 52` ``` ==> arity(rtran_closure_mem_fm(x,y,z)) = succ(x) \ succ(y) \ succ(z)" ``` wenzelm@13428 ` 53` ```by (simp add: rtran_closure_mem_fm_def succ_Un_distrib [symmetric] Un_ac) ``` paulson@13348 ` 54` paulson@13348 ` 55` ```lemma sats_rtran_closure_mem_fm [simp]: ``` paulson@13348 ` 56` ``` "[| x \ nat; y \ nat; z \ nat; env \ list(A)|] ``` wenzelm@13428 ` 57` ``` ==> sats(A, rtran_closure_mem_fm(x,y,z), env) <-> ``` paulson@13348 ` 58` ``` rtran_closure_mem(**A, nth(x,env), nth(y,env), nth(z,env))" ``` paulson@13348 ` 59` ```by (simp add: rtran_closure_mem_fm_def rtran_closure_mem_def) ``` paulson@13348 ` 60` paulson@13348 ` 61` ```lemma rtran_closure_mem_iff_sats: ``` wenzelm@13428 ` 62` ``` "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; ``` paulson@13348 ` 63` ``` i \ nat; j \ nat; k \ nat; env \ list(A)|] ``` paulson@13348 ` 64` ``` ==> rtran_closure_mem(**A, x, y, z) <-> sats(A, rtran_closure_mem_fm(i,j,k), env)" ``` paulson@13348 ` 65` ```by (simp add: sats_rtran_closure_mem_fm) ``` paulson@13348 ` 66` paulson@13348 ` 67` ```theorem rtran_closure_mem_reflection: ``` wenzelm@13428 ` 68` ``` "REFLECTS[\x. rtran_closure_mem(L,f(x),g(x),h(x)), ``` paulson@13348 ` 69` ``` \i x. rtran_closure_mem(**Lset(i),f(x),g(x),h(x))]" ``` paulson@13348 ` 70` ```apply (simp only: rtran_closure_mem_def setclass_simps) ``` wenzelm@13428 ` 71` ```apply (intro FOL_reflections function_reflections fun_plus_reflections) ``` paulson@13348 ` 72` ```done ``` paulson@13348 ` 73` paulson@13348 ` 74` ```text{*Separation for @{term "rtrancl(r)"}.*} ``` paulson@13348 ` 75` ```lemma rtrancl_separation: ``` paulson@13348 ` 76` ``` "[| L(r); L(A) |] ==> separation (L, rtran_closure_mem(L,A,r))" ``` wenzelm@13428 ` 77` ```apply (rule separation_CollectI) ``` wenzelm@13428 ` 78` ```apply (rule_tac A="{r,A,z}" in subset_LsetE, blast ) ``` paulson@13348 ` 79` ```apply (rule ReflectsE [OF rtran_closure_mem_reflection], assumption) ``` wenzelm@13428 ` 80` ```apply (drule subset_Lset_ltD, assumption) ``` paulson@13348 ` 81` ```apply (erule reflection_imp_L_separation) ``` paulson@13348 ` 82` ``` apply (simp_all add: lt_Ord2) ``` paulson@13385 ` 83` ```apply (rule DPow_LsetI) ``` paulson@13348 ` 84` ```apply (rename_tac u) ``` paulson@13348 ` 85` ```apply (rule_tac env = "[u,r,A]" in rtran_closure_mem_iff_sats) ``` paulson@13348 ` 86` ```apply (rule sep_rules | simp)+ ``` paulson@13348 ` 87` ```done ``` paulson@13348 ` 88` paulson@13348 ` 89` paulson@13348 ` 90` ```subsubsection{*Reflexive/Transitive Closure, Internalized*} ``` paulson@13348 ` 91` wenzelm@13428 ` 92` ```(* "rtran_closure(M,r,s) == ``` paulson@13348 ` 93` ``` \A[M]. is_field(M,r,A) --> ``` wenzelm@13428 ` 94` ``` (\p[M]. p \ s <-> rtran_closure_mem(M,A,r,p))" *) ``` paulson@13348 ` 95` ```constdefs rtran_closure_fm :: "[i,i]=>i" ``` wenzelm@13428 ` 96` ``` "rtran_closure_fm(r,s) == ``` paulson@13348 ` 97` ``` Forall(Implies(field_fm(succ(r),0), ``` paulson@13348 ` 98` ``` Forall(Iff(Member(0,succ(succ(s))), ``` paulson@13348 ` 99` ``` rtran_closure_mem_fm(1,succ(succ(r)),0)))))" ``` paulson@13348 ` 100` paulson@13348 ` 101` ```lemma rtran_closure_type [TC]: ``` paulson@13348 ` 102` ``` "[| x \ nat; y \ nat |] ==> rtran_closure_fm(x,y) \ formula" ``` wenzelm@13428 ` 103` ```by (simp add: rtran_closure_fm_def) ``` paulson@13348 ` 104` paulson@13348 ` 105` ```lemma arity_rtran_closure_fm [simp]: ``` wenzelm@13428 ` 106` ``` "[| x \ nat; y \ nat |] ``` paulson@13348 ` 107` ``` ==> arity(rtran_closure_fm(x,y)) = succ(x) \ succ(y)" ``` paulson@13348 ` 108` ```by (simp add: rtran_closure_fm_def succ_Un_distrib [symmetric] Un_ac) ``` paulson@13348 ` 109` paulson@13348 ` 110` ```lemma sats_rtran_closure_fm [simp]: ``` paulson@13348 ` 111` ``` "[| x \ nat; y \ nat; env \ list(A)|] ``` wenzelm@13428 ` 112` ``` ==> sats(A, rtran_closure_fm(x,y), env) <-> ``` paulson@13348 ` 113` ``` rtran_closure(**A, nth(x,env), nth(y,env))" ``` paulson@13348 ` 114` ```by (simp add: rtran_closure_fm_def rtran_closure_def) ``` paulson@13348 ` 115` paulson@13348 ` 116` ```lemma rtran_closure_iff_sats: ``` wenzelm@13428 ` 117` ``` "[| nth(i,env) = x; nth(j,env) = y; ``` paulson@13348 ` 118` ``` i \ nat; j \ nat; env \ list(A)|] ``` paulson@13348 ` 119` ``` ==> rtran_closure(**A, x, y) <-> sats(A, rtran_closure_fm(i,j), env)" ``` paulson@13348 ` 120` ```by simp ``` paulson@13348 ` 121` paulson@13348 ` 122` ```theorem rtran_closure_reflection: ``` wenzelm@13428 ` 123` ``` "REFLECTS[\x. rtran_closure(L,f(x),g(x)), ``` paulson@13348 ` 124` ``` \i x. rtran_closure(**Lset(i),f(x),g(x))]" ``` paulson@13348 ` 125` ```apply (simp only: rtran_closure_def setclass_simps) ``` paulson@13348 ` 126` ```apply (intro FOL_reflections function_reflections rtran_closure_mem_reflection) ``` paulson@13348 ` 127` ```done ``` paulson@13348 ` 128` paulson@13348 ` 129` paulson@13348 ` 130` ```subsubsection{*Transitive Closure of a Relation, Internalized*} ``` paulson@13348 ` 131` paulson@13348 ` 132` ```(* "tran_closure(M,r,t) == ``` paulson@13348 ` 133` ``` \s[M]. rtran_closure(M,r,s) & composition(M,r,s,t)" *) ``` paulson@13348 ` 134` ```constdefs tran_closure_fm :: "[i,i]=>i" ``` wenzelm@13428 ` 135` ``` "tran_closure_fm(r,s) == ``` paulson@13348 ` 136` ``` Exists(And(rtran_closure_fm(succ(r),0), composition_fm(succ(r),0,succ(s))))" ``` paulson@13348 ` 137` paulson@13348 ` 138` ```lemma tran_closure_type [TC]: ``` paulson@13348 ` 139` ``` "[| x \ nat; y \ nat |] ==> tran_closure_fm(x,y) \ formula" ``` wenzelm@13428 ` 140` ```by (simp add: tran_closure_fm_def) ``` paulson@13348 ` 141` paulson@13348 ` 142` ```lemma arity_tran_closure_fm [simp]: ``` wenzelm@13428 ` 143` ``` "[| x \ nat; y \ nat |] ``` paulson@13348 ` 144` ``` ==> arity(tran_closure_fm(x,y)) = succ(x) \ succ(y)" ``` paulson@13348 ` 145` ```by (simp add: tran_closure_fm_def succ_Un_distrib [symmetric] Un_ac) ``` paulson@13348 ` 146` paulson@13348 ` 147` ```lemma sats_tran_closure_fm [simp]: ``` paulson@13348 ` 148` ``` "[| x \ nat; y \ nat; env \ list(A)|] ``` wenzelm@13428 ` 149` ``` ==> sats(A, tran_closure_fm(x,y), env) <-> ``` paulson@13348 ` 150` ``` tran_closure(**A, nth(x,env), nth(y,env))" ``` paulson@13348 ` 151` ```by (simp add: tran_closure_fm_def tran_closure_def) ``` paulson@13348 ` 152` paulson@13348 ` 153` ```lemma tran_closure_iff_sats: ``` wenzelm@13428 ` 154` ``` "[| nth(i,env) = x; nth(j,env) = y; ``` paulson@13348 ` 155` ``` i \ nat; j \ nat; env \ list(A)|] ``` paulson@13348 ` 156` ``` ==> tran_closure(**A, x, y) <-> sats(A, tran_closure_fm(i,j), env)" ``` paulson@13348 ` 157` ```by simp ``` paulson@13348 ` 158` paulson@13348 ` 159` ```theorem tran_closure_reflection: ``` wenzelm@13428 ` 160` ``` "REFLECTS[\x. tran_closure(L,f(x),g(x)), ``` paulson@13348 ` 161` ``` \i x. tran_closure(**Lset(i),f(x),g(x))]" ``` paulson@13348 ` 162` ```apply (simp only: tran_closure_def setclass_simps) ``` wenzelm@13428 ` 163` ```apply (intro FOL_reflections function_reflections ``` paulson@13348 ` 164` ``` rtran_closure_reflection composition_reflection) ``` paulson@13348 ` 165` ```done ``` paulson@13348 ` 166` paulson@13348 ` 167` paulson@13348 ` 168` ```subsection{*Separation for the Proof of @{text "wellfounded_on_trancl"}*} ``` paulson@13348 ` 169` paulson@13348 ` 170` ```lemma wellfounded_trancl_reflects: ``` wenzelm@13428 ` 171` ``` "REFLECTS[\x. \w[L]. \wx[L]. \rp[L]. ``` wenzelm@13428 ` 172` ``` w \ Z & pair(L,w,x,wx) & tran_closure(L,r,rp) & wx \ rp, ``` wenzelm@13428 ` 173` ``` \i x. \w \ Lset(i). \wx \ Lset(i). \rp \ Lset(i). ``` paulson@13348 ` 174` ``` w \ Z & pair(**Lset(i),w,x,wx) & tran_closure(**Lset(i),r,rp) & ``` paulson@13348 ` 175` ``` wx \ rp]" ``` wenzelm@13428 ` 176` ```by (intro FOL_reflections function_reflections fun_plus_reflections ``` paulson@13348 ` 177` ``` tran_closure_reflection) ``` paulson@13348 ` 178` paulson@13348 ` 179` paulson@13348 ` 180` ```lemma wellfounded_trancl_separation: ``` wenzelm@13428 ` 181` ``` "[| L(r); L(Z) |] ==> ``` wenzelm@13428 ` 182` ``` separation (L, \x. ``` wenzelm@13428 ` 183` ``` \w[L]. \wx[L]. \rp[L]. ``` wenzelm@13428 ` 184` ``` w \ Z & pair(L,w,x,wx) & tran_closure(L,r,rp) & wx \ rp)" ``` wenzelm@13428 ` 185` ```apply (rule separation_CollectI) ``` wenzelm@13428 ` 186` ```apply (rule_tac A="{r,Z,z}" in subset_LsetE, blast ) ``` paulson@13348 ` 187` ```apply (rule ReflectsE [OF wellfounded_trancl_reflects], assumption) ``` wenzelm@13428 ` 188` ```apply (drule subset_Lset_ltD, assumption) ``` paulson@13348 ` 189` ```apply (erule reflection_imp_L_separation) ``` paulson@13348 ` 190` ``` apply (simp_all add: lt_Ord2) ``` paulson@13385 ` 191` ```apply (rule DPow_LsetI) ``` wenzelm@13428 ` 192` ```apply (rename_tac u) ``` paulson@13348 ` 193` ```apply (rule bex_iff_sats conj_iff_sats)+ ``` wenzelm@13428 ` 194` ```apply (rule_tac env = "[w,u,r,Z]" in mem_iff_sats) ``` paulson@13348 ` 195` ```apply (rule sep_rules tran_closure_iff_sats | simp)+ ``` paulson@13348 ` 196` ```done ``` paulson@13348 ` 197` paulson@13363 ` 198` paulson@13363 ` 199` ```subsubsection{*Instantiating the locale @{text M_trancl}*} ``` wenzelm@13428 ` 200` wenzelm@13428 ` 201` ```theorem M_trancl_axioms_L: "M_trancl_axioms(L)" ``` wenzelm@13428 ` 202` ``` apply (rule M_trancl_axioms.intro) ``` wenzelm@13428 ` 203` ``` apply (assumption | rule ``` wenzelm@13428 ` 204` ``` rtrancl_separation wellfounded_trancl_separation)+ ``` wenzelm@13428 ` 205` ``` done ``` paulson@13363 ` 206` wenzelm@13428 ` 207` ```theorem M_trancl_L: "PROP M_trancl(L)" ``` wenzelm@13428 ` 208` ``` apply (rule M_trancl.intro) ``` wenzelm@13428 ` 209` ``` apply (rule M_triv_axioms_L) ``` wenzelm@13428 ` 210` ``` apply (rule M_axioms_axioms_L) ``` wenzelm@13428 ` 211` ``` apply (rule M_trancl_axioms_L) ``` wenzelm@13428 ` 212` ``` done ``` paulson@13363 ` 213` wenzelm@13428 ` 214` ```lemmas iterates_abs = M_trancl.iterates_abs [OF M_trancl_L] ``` wenzelm@13428 ` 215` ``` and rtran_closure_rtrancl = M_trancl.rtran_closure_rtrancl [OF M_trancl_L] ``` wenzelm@13428 ` 216` ``` and rtrancl_closed = M_trancl.rtrancl_closed [OF M_trancl_L] ``` wenzelm@13428 ` 217` ``` and rtrancl_abs = M_trancl.rtrancl_abs [OF M_trancl_L] ``` wenzelm@13428 ` 218` ``` and trancl_closed = M_trancl.trancl_closed [OF M_trancl_L] ``` wenzelm@13428 ` 219` ``` and trancl_abs = M_trancl.trancl_abs [OF M_trancl_L] ``` wenzelm@13428 ` 220` ``` and wellfounded_on_trancl = M_trancl.wellfounded_on_trancl [OF M_trancl_L] ``` wenzelm@13428 ` 221` ``` and wellfounded_trancl = M_trancl.wellfounded_trancl [OF M_trancl_L] ``` wenzelm@13428 ` 222` ``` and wfrec_relativize = M_trancl.wfrec_relativize [OF M_trancl_L] ``` wenzelm@13428 ` 223` ``` and trans_wfrec_relativize = M_trancl.trans_wfrec_relativize [OF M_trancl_L] ``` wenzelm@13428 ` 224` ``` and trans_wfrec_abs = M_trancl.trans_wfrec_abs [OF M_trancl_L] ``` wenzelm@13428 ` 225` ``` and trans_eq_pair_wfrec_iff = M_trancl.trans_eq_pair_wfrec_iff [OF M_trancl_L] ``` wenzelm@13428 ` 226` ``` and eq_pair_wfrec_iff = M_trancl.eq_pair_wfrec_iff [OF M_trancl_L] ``` paulson@13363 ` 227` paulson@13363 ` 228` ```declare rtrancl_closed [intro,simp] ``` paulson@13363 ` 229` ```declare rtrancl_abs [simp] ``` paulson@13363 ` 230` ```declare trancl_closed [intro,simp] ``` paulson@13363 ` 231` ```declare trancl_abs [simp] ``` paulson@13363 ` 232` paulson@13363 ` 233` paulson@13348 ` 234` ```subsection{*Well-Founded Recursion!*} ``` paulson@13348 ` 235` paulson@13352 ` 236` ```(* M_is_recfun :: "[i=>o, [i,i,i]=>o, i, i, i] => o" ``` wenzelm@13428 ` 237` ``` "M_is_recfun(M,MH,r,a,f) == ``` wenzelm@13428 ` 238` ``` \z[M]. z \ f <-> ``` paulson@13348 ` 239` ``` 5 4 3 2 1 0 ``` wenzelm@13428 ` 240` ``` (\x[M]. \y[M]. \xa[M]. \sx[M]. \r_sx[M]. \f_r_sx[M]. ``` wenzelm@13428 ` 241` ``` pair(M,x,y,z) & pair(M,x,a,xa) & upair(M,x,x,sx) & ``` paulson@13348 ` 242` ``` pre_image(M,r,sx,r_sx) & restriction(M,f,r_sx,f_r_sx) & ``` paulson@13348 ` 243` ``` xa \ r & MH(x, f_r_sx, y))" ``` paulson@13348 ` 244` ```*) ``` paulson@13348 ` 245` paulson@13348 ` 246` ```constdefs is_recfun_fm :: "[[i,i,i]=>i, i, i, i]=>i" ``` wenzelm@13428 ` 247` ``` "is_recfun_fm(p,r,a,f) == ``` paulson@13348 ` 248` ``` Forall(Iff(Member(0,succ(f)), ``` paulson@13348 ` 249` ``` Exists(Exists(Exists(Exists(Exists(Exists( ``` paulson@13348 ` 250` ``` And(pair_fm(5,4,6), ``` paulson@13348 ` 251` ``` And(pair_fm(5,a#+7,3), ``` paulson@13348 ` 252` ``` And(upair_fm(5,5,2), ``` paulson@13348 ` 253` ``` And(pre_image_fm(r#+7,2,1), ``` paulson@13348 ` 254` ``` And(restriction_fm(f#+7,1,0), ``` paulson@13348 ` 255` ``` And(Member(3,r#+7), p(5,0,4)))))))))))))))" ``` paulson@13348 ` 256` paulson@13348 ` 257` paulson@13348 ` 258` ```lemma is_recfun_type_0: ``` wenzelm@13428 ` 259` ``` "[| !!x y z. [| x \ nat; y \ nat; z \ nat |] ==> p(x,y,z) \ formula; ``` wenzelm@13428 ` 260` ``` x \ nat; y \ nat; z \ nat |] ``` paulson@13348 ` 261` ``` ==> is_recfun_fm(p,x,y,z) \ formula" ``` paulson@13348 ` 262` ```apply (unfold is_recfun_fm_def) ``` paulson@13348 ` 263` ```(*FIXME: FIND OUT why simp loops!*) ``` paulson@13348 ` 264` ```apply typecheck ``` wenzelm@13428 ` 265` ```by simp ``` paulson@13348 ` 266` paulson@13348 ` 267` ```lemma is_recfun_type [TC]: ``` wenzelm@13428 ` 268` ``` "[| p(5,0,4) \ formula; ``` wenzelm@13428 ` 269` ``` x \ nat; y \ nat; z \ nat |] ``` paulson@13348 ` 270` ``` ==> is_recfun_fm(p,x,y,z) \ formula" ``` wenzelm@13428 ` 271` ```by (simp add: is_recfun_fm_def) ``` paulson@13348 ` 272` paulson@13348 ` 273` ```lemma arity_is_recfun_fm [simp]: ``` wenzelm@13428 ` 274` ``` "[| arity(p(5,0,4)) le 8; x \ nat; y \ nat; z \ nat |] ``` paulson@13348 ` 275` ``` ==> arity(is_recfun_fm(p,x,y,z)) = succ(x) \ succ(y) \ succ(z)" ``` wenzelm@13428 ` 276` ```apply (frule lt_nat_in_nat, simp) ``` wenzelm@13428 ` 277` ```apply (simp add: is_recfun_fm_def succ_Un_distrib [symmetric] ) ``` wenzelm@13428 ` 278` ```apply (subst subset_Un_iff2 [of "arity(p(5,0,4))", THEN iffD1]) ``` wenzelm@13428 ` 279` ```apply (rule le_imp_subset) ``` wenzelm@13428 ` 280` ```apply (erule le_trans, simp) ``` wenzelm@13428 ` 281` ```apply (simp add: succ_Un_distrib [symmetric] Un_ac) ``` paulson@13348 ` 282` ```done ``` paulson@13348 ` 283` paulson@13348 ` 284` ```lemma sats_is_recfun_fm: ``` wenzelm@13428 ` 285` ``` assumes MH_iff_sats: ``` wenzelm@13428 ` 286` ``` "!!x y z env. ``` wenzelm@13428 ` 287` ``` [| x \ nat; y \ nat; z \ nat; env \ list(A)|] ``` wenzelm@13428 ` 288` ``` ==> MH(nth(x,env), nth(y,env), nth(z,env)) <-> sats(A, p(x,y,z), env)" ``` wenzelm@13428 ` 289` ``` shows ``` paulson@13348 ` 290` ``` "[|x \ nat; y \ nat; z \ nat; env \ list(A)|] ``` wenzelm@13428 ` 291` ``` ==> sats(A, is_recfun_fm(p,x,y,z), env) <-> ``` paulson@13352 ` 292` ``` M_is_recfun(**A, MH, nth(x,env), nth(y,env), nth(z,env))" ``` paulson@13348 ` 293` ```by (simp add: is_recfun_fm_def M_is_recfun_def MH_iff_sats [THEN iff_sym]) ``` paulson@13348 ` 294` paulson@13348 ` 295` ```lemma is_recfun_iff_sats: ``` paulson@13348 ` 296` ``` "[| (!!x y z env. [| x \ nat; y \ nat; z \ nat; env \ list(A)|] ``` paulson@13348 ` 297` ``` ==> MH(nth(x,env), nth(y,env), nth(z,env)) <-> ``` paulson@13348 ` 298` ``` sats(A, p(x,y,z), env)); ``` wenzelm@13428 ` 299` ``` nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; ``` paulson@13348 ` 300` ``` i \ nat; j \ nat; k \ nat; env \ list(A)|] ``` wenzelm@13428 ` 301` ``` ==> M_is_recfun(**A, MH, x, y, z) <-> sats(A, is_recfun_fm(p,i,j,k), env)" ``` paulson@13348 ` 302` ```by (simp add: sats_is_recfun_fm [of A MH]) ``` paulson@13348 ` 303` paulson@13348 ` 304` ```theorem is_recfun_reflection: ``` paulson@13348 ` 305` ``` assumes MH_reflection: ``` wenzelm@13428 ` 306` ``` "!!f g h. REFLECTS[\x. MH(L, f(x), g(x), h(x)), ``` paulson@13348 ` 307` ``` \i x. MH(**Lset(i), f(x), g(x), h(x))]" ``` wenzelm@13428 ` 308` ``` shows "REFLECTS[\x. M_is_recfun(L, MH(L), f(x), g(x), h(x)), ``` paulson@13352 ` 309` ``` \i x. M_is_recfun(**Lset(i), MH(**Lset(i)), f(x), g(x), h(x))]" ``` paulson@13348 ` 310` ```apply (simp (no_asm_use) only: M_is_recfun_def setclass_simps) ``` wenzelm@13428 ` 311` ```apply (intro FOL_reflections function_reflections ``` wenzelm@13428 ` 312` ``` restriction_reflection MH_reflection) ``` paulson@13348 ` 313` ```done ``` paulson@13348 ` 314` paulson@13363 ` 315` ```text{*Currently, @{text sats}-theorems for higher-order operators don't seem ``` paulson@13363 ` 316` ```useful. Reflection theorems do work, though. This one avoids the repetition ``` paulson@13363 ` 317` ```of the @{text MH}-term.*} ``` paulson@13363 ` 318` ```theorem is_wfrec_reflection: ``` paulson@13363 ` 319` ``` assumes MH_reflection: ``` wenzelm@13428 ` 320` ``` "!!f g h. REFLECTS[\x. MH(L, f(x), g(x), h(x)), ``` paulson@13363 ` 321` ``` \i x. MH(**Lset(i), f(x), g(x), h(x))]" ``` wenzelm@13428 ` 322` ``` shows "REFLECTS[\x. is_wfrec(L, MH(L), f(x), g(x), h(x)), ``` paulson@13363 ` 323` ``` \i x. is_wfrec(**Lset(i), MH(**Lset(i)), f(x), g(x), h(x))]" ``` paulson@13363 ` 324` ```apply (simp (no_asm_use) only: is_wfrec_def setclass_simps) ``` wenzelm@13428 ` 325` ```apply (intro FOL_reflections MH_reflection is_recfun_reflection) ``` paulson@13363 ` 326` ```done ``` paulson@13363 ` 327` paulson@13363 ` 328` ```subsection{*The Locale @{text "M_wfrank"}*} ``` paulson@13363 ` 329` paulson@13363 ` 330` ```subsubsection{*Separation for @{term "wfrank"}*} ``` paulson@13348 ` 331` paulson@13348 ` 332` ```lemma wfrank_Reflects: ``` paulson@13348 ` 333` ``` "REFLECTS[\x. \rplus[L]. tran_closure(L,r,rplus) --> ``` paulson@13352 ` 334` ``` ~ (\f[L]. M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f)), ``` paulson@13348 ` 335` ``` \i x. \rplus \ Lset(i). tran_closure(**Lset(i),r,rplus) --> ``` wenzelm@13428 ` 336` ``` ~ (\f \ Lset(i). ``` wenzelm@13428 ` 337` ``` M_is_recfun(**Lset(i), %x f y. is_range(**Lset(i),f,y), ``` paulson@13352 ` 338` ``` rplus, x, f))]" ``` wenzelm@13428 ` 339` ```by (intro FOL_reflections function_reflections is_recfun_reflection tran_closure_reflection) ``` paulson@13348 ` 340` paulson@13348 ` 341` ```lemma wfrank_separation: ``` paulson@13348 ` 342` ``` "L(r) ==> ``` paulson@13348 ` 343` ``` separation (L, \x. \rplus[L]. tran_closure(L,r,rplus) --> ``` paulson@13352 ` 344` ``` ~ (\f[L]. M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f)))" ``` wenzelm@13428 ` 345` ```apply (rule separation_CollectI) ``` wenzelm@13428 ` 346` ```apply (rule_tac A="{r,z}" in subset_LsetE, blast ) ``` paulson@13348 ` 347` ```apply (rule ReflectsE [OF wfrank_Reflects], assumption) ``` wenzelm@13428 ` 348` ```apply (drule subset_Lset_ltD, assumption) ``` paulson@13348 ` 349` ```apply (erule reflection_imp_L_separation) ``` paulson@13348 ` 350` ``` apply (simp_all add: lt_Ord2, clarify) ``` paulson@13385 ` 351` ```apply (rule DPow_LsetI) ``` wenzelm@13428 ` 352` ```apply (rename_tac u) ``` paulson@13348 ` 353` ```apply (rule ball_iff_sats imp_iff_sats)+ ``` paulson@13348 ` 354` ```apply (rule_tac env="[rplus,u,r]" in tran_closure_iff_sats) ``` paulson@13348 ` 355` ```apply (rule sep_rules is_recfun_iff_sats | simp)+ ``` paulson@13348 ` 356` ```done ``` paulson@13348 ` 357` paulson@13348 ` 358` paulson@13363 ` 359` ```subsubsection{*Replacement for @{term "wfrank"}*} ``` paulson@13348 ` 360` paulson@13348 ` 361` ```lemma wfrank_replacement_Reflects: ``` wenzelm@13428 ` 362` ``` "REFLECTS[\z. \x[L]. x \ A & ``` paulson@13348 ` 363` ``` (\rplus[L]. tran_closure(L,r,rplus) --> ``` wenzelm@13428 ` 364` ``` (\y[L]. \f[L]. pair(L,x,y,z) & ``` paulson@13352 ` 365` ``` M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f) & ``` paulson@13348 ` 366` ``` is_range(L,f,y))), ``` wenzelm@13428 ` 367` ``` \i z. \x \ Lset(i). x \ A & ``` paulson@13348 ` 368` ``` (\rplus \ Lset(i). tran_closure(**Lset(i),r,rplus) --> ``` wenzelm@13428 ` 369` ``` (\y \ Lset(i). \f \ Lset(i). pair(**Lset(i),x,y,z) & ``` paulson@13352 ` 370` ``` M_is_recfun(**Lset(i), %x f y. is_range(**Lset(i),f,y), rplus, x, f) & ``` paulson@13348 ` 371` ``` is_range(**Lset(i),f,y)))]" ``` paulson@13348 ` 372` ```by (intro FOL_reflections function_reflections fun_plus_reflections ``` paulson@13348 ` 373` ``` is_recfun_reflection tran_closure_reflection) ``` paulson@13348 ` 374` paulson@13348 ` 375` paulson@13348 ` 376` ```lemma wfrank_strong_replacement: ``` paulson@13348 ` 377` ``` "L(r) ==> ``` wenzelm@13428 ` 378` ``` strong_replacement(L, \x z. ``` paulson@13348 ` 379` ``` \rplus[L]. tran_closure(L,r,rplus) --> ``` wenzelm@13428 ` 380` ``` (\y[L]. \f[L]. pair(L,x,y,z) & ``` paulson@13352 ` 381` ``` M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f) & ``` paulson@13348 ` 382` ``` is_range(L,f,y)))" ``` wenzelm@13428 ` 383` ```apply (rule strong_replacementI) ``` paulson@13348 ` 384` ```apply (rule rallI) ``` wenzelm@13428 ` 385` ```apply (rename_tac B) ``` wenzelm@13428 ` 386` ```apply (rule separation_CollectI) ``` wenzelm@13428 ` 387` ```apply (rule_tac A="{B,r,z}" in subset_LsetE, blast ) ``` paulson@13348 ` 388` ```apply (rule ReflectsE [OF wfrank_replacement_Reflects], assumption) ``` wenzelm@13428 ` 389` ```apply (drule subset_Lset_ltD, assumption) ``` paulson@13348 ` 390` ```apply (erule reflection_imp_L_separation) ``` paulson@13348 ` 391` ``` apply (simp_all add: lt_Ord2) ``` paulson@13385 ` 392` ```apply (rule DPow_LsetI) ``` wenzelm@13428 ` 393` ```apply (rename_tac u) ``` paulson@13348 ` 394` ```apply (rule bex_iff_sats ball_iff_sats conj_iff_sats)+ ``` wenzelm@13428 ` 395` ```apply (rule_tac env = "[x,u,B,r]" in mem_iff_sats) ``` paulson@13348 ` 396` ```apply (rule sep_rules tran_closure_iff_sats is_recfun_iff_sats | simp)+ ``` paulson@13348 ` 397` ```done ``` paulson@13348 ` 398` paulson@13348 ` 399` paulson@13363 ` 400` ```subsubsection{*Separation for Proving @{text Ord_wfrank_range}*} ``` paulson@13348 ` 401` paulson@13348 ` 402` ```lemma Ord_wfrank_Reflects: ``` wenzelm@13428 ` 403` ``` "REFLECTS[\x. \rplus[L]. tran_closure(L,r,rplus) --> ``` wenzelm@13428 ` 404` ``` ~ (\f[L]. \rangef[L]. ``` paulson@13348 ` 405` ``` is_range(L,f,rangef) --> ``` paulson@13352 ` 406` ``` M_is_recfun(L, \x f y. is_range(L,f,y), rplus, x, f) --> ``` paulson@13348 ` 407` ``` ordinal(L,rangef)), ``` wenzelm@13428 ` 408` ``` \i x. \rplus \ Lset(i). tran_closure(**Lset(i),r,rplus) --> ``` wenzelm@13428 ` 409` ``` ~ (\f \ Lset(i). \rangef \ Lset(i). ``` paulson@13348 ` 410` ``` is_range(**Lset(i),f,rangef) --> ``` wenzelm@13428 ` 411` ``` M_is_recfun(**Lset(i), \x f y. is_range(**Lset(i),f,y), ``` paulson@13352 ` 412` ``` rplus, x, f) --> ``` paulson@13348 ` 413` ``` ordinal(**Lset(i),rangef))]" ``` wenzelm@13428 ` 414` ```by (intro FOL_reflections function_reflections is_recfun_reflection ``` paulson@13348 ` 415` ``` tran_closure_reflection ordinal_reflection) ``` paulson@13348 ` 416` paulson@13348 ` 417` ```lemma Ord_wfrank_separation: ``` paulson@13348 ` 418` ``` "L(r) ==> ``` paulson@13348 ` 419` ``` separation (L, \x. ``` wenzelm@13428 ` 420` ``` \rplus[L]. tran_closure(L,r,rplus) --> ``` wenzelm@13428 ` 421` ``` ~ (\f[L]. \rangef[L]. ``` paulson@13348 ` 422` ``` is_range(L,f,rangef) --> ``` paulson@13352 ` 423` ``` M_is_recfun(L, \x f y. is_range(L,f,y), rplus, x, f) --> ``` wenzelm@13428 ` 424` ``` ordinal(L,rangef)))" ``` wenzelm@13428 ` 425` ```apply (rule separation_CollectI) ``` wenzelm@13428 ` 426` ```apply (rule_tac A="{r,z}" in subset_LsetE, blast ) ``` paulson@13348 ` 427` ```apply (rule ReflectsE [OF Ord_wfrank_Reflects], assumption) ``` wenzelm@13428 ` 428` ```apply (drule subset_Lset_ltD, assumption) ``` paulson@13348 ` 429` ```apply (erule reflection_imp_L_separation) ``` paulson@13348 ` 430` ``` apply (simp_all add: lt_Ord2, clarify) ``` paulson@13385 ` 431` ```apply (rule DPow_LsetI) ``` wenzelm@13428 ` 432` ```apply (rename_tac u) ``` paulson@13348 ` 433` ```apply (rule ball_iff_sats imp_iff_sats)+ ``` paulson@13348 ` 434` ```apply (rule_tac env="[rplus,u,r]" in tran_closure_iff_sats) ``` paulson@13348 ` 435` ```apply (rule sep_rules is_recfun_iff_sats | simp)+ ``` paulson@13348 ` 436` ```done ``` paulson@13348 ` 437` paulson@13348 ` 438` paulson@13363 ` 439` ```subsubsection{*Instantiating the locale @{text M_wfrank}*} ``` wenzelm@13428 ` 440` wenzelm@13428 ` 441` ```theorem M_wfrank_axioms_L: "M_wfrank_axioms(L)" ``` wenzelm@13428 ` 442` ``` apply (rule M_wfrank_axioms.intro) ``` wenzelm@13428 ` 443` ``` apply (assumption | rule ``` wenzelm@13428 ` 444` ``` wfrank_separation wfrank_strong_replacement Ord_wfrank_separation)+ ``` wenzelm@13428 ` 445` ``` done ``` paulson@13363 ` 446` wenzelm@13428 ` 447` ```theorem M_wfrank_L: "PROP M_wfrank(L)" ``` wenzelm@13428 ` 448` ``` apply (rule M_wfrank.intro) ``` wenzelm@13428 ` 449` ``` apply (rule M_triv_axioms_L) ``` wenzelm@13428 ` 450` ``` apply (rule M_axioms_axioms_L) ``` wenzelm@13428 ` 451` ``` apply (rule M_trancl_axioms_L) ``` wenzelm@13428 ` 452` ``` apply (rule M_wfrank_axioms_L) ``` wenzelm@13428 ` 453` ``` done ``` paulson@13363 ` 454` wenzelm@13428 ` 455` ```lemmas iterates_closed = M_wfrank.iterates_closed [OF M_wfrank_L] ``` wenzelm@13428 ` 456` ``` and exists_wfrank = M_wfrank.exists_wfrank [OF M_wfrank_L] ``` wenzelm@13428 ` 457` ``` and M_wellfoundedrank = M_wfrank.M_wellfoundedrank [OF M_wfrank_L] ``` wenzelm@13428 ` 458` ``` and Ord_wfrank_range = M_wfrank.Ord_wfrank_range [OF M_wfrank_L] ``` wenzelm@13428 ` 459` ``` and Ord_range_wellfoundedrank = M_wfrank.Ord_range_wellfoundedrank [OF M_wfrank_L] ``` wenzelm@13428 ` 460` ``` and function_wellfoundedrank = M_wfrank.function_wellfoundedrank [OF M_wfrank_L] ``` wenzelm@13428 ` 461` ``` and domain_wellfoundedrank = M_wfrank.domain_wellfoundedrank [OF M_wfrank_L] ``` wenzelm@13428 ` 462` ``` and wellfoundedrank_type = M_wfrank.wellfoundedrank_type [OF M_wfrank_L] ``` wenzelm@13428 ` 463` ``` and Ord_wellfoundedrank = M_wfrank.Ord_wellfoundedrank [OF M_wfrank_L] ``` wenzelm@13428 ` 464` ``` and wellfoundedrank_eq = M_wfrank.wellfoundedrank_eq [OF M_wfrank_L] ``` wenzelm@13428 ` 465` ``` and wellfoundedrank_lt = M_wfrank.wellfoundedrank_lt [OF M_wfrank_L] ``` wenzelm@13428 ` 466` ``` and wellfounded_imp_subset_rvimage = M_wfrank.wellfounded_imp_subset_rvimage [OF M_wfrank_L] ``` wenzelm@13428 ` 467` ``` and wellfounded_imp_wf = M_wfrank.wellfounded_imp_wf [OF M_wfrank_L] ``` wenzelm@13428 ` 468` ``` and wellfounded_on_imp_wf_on = M_wfrank.wellfounded_on_imp_wf_on [OF M_wfrank_L] ``` wenzelm@13428 ` 469` ``` and wf_abs = M_wfrank.wf_abs [OF M_wfrank_L] ``` wenzelm@13428 ` 470` ``` and wf_on_abs = M_wfrank.wf_on_abs [OF M_wfrank_L] ``` wenzelm@13428 ` 471` ``` and wfrec_replacement_iff = M_wfrank.wfrec_replacement_iff [OF M_wfrank_L] ``` wenzelm@13428 ` 472` ``` and trans_wfrec_closed = M_wfrank.trans_wfrec_closed [OF M_wfrank_L] ``` wenzelm@13428 ` 473` ``` and wfrec_closed = M_wfrank.wfrec_closed [OF M_wfrank_L] ``` paulson@13363 ` 474` paulson@13363 ` 475` ```declare iterates_closed [intro,simp] ``` paulson@13363 ` 476` ```declare Ord_wfrank_range [rule_format] ``` paulson@13363 ` 477` ```declare wf_abs [simp] ``` paulson@13363 ` 478` ```declare wf_on_abs [simp] ``` paulson@13363 ` 479` paulson@13363 ` 480` paulson@13363 ` 481` ```subsection{*For Datatypes*} ``` paulson@13363 ` 482` paulson@13363 ` 483` ```subsubsection{*Binary Products, Internalized*} ``` paulson@13363 ` 484` paulson@13363 ` 485` ```constdefs cartprod_fm :: "[i,i,i]=>i" ``` wenzelm@13428 ` 486` ```(* "cartprod(M,A,B,z) == ``` wenzelm@13428 ` 487` ``` \u[M]. u \ z <-> (\x[M]. x\A & (\y[M]. y\B & pair(M,x,y,u)))" *) ``` wenzelm@13428 ` 488` ``` "cartprod_fm(A,B,z) == ``` paulson@13363 ` 489` ``` Forall(Iff(Member(0,succ(z)), ``` paulson@13363 ` 490` ``` Exists(And(Member(0,succ(succ(A))), ``` paulson@13363 ` 491` ``` Exists(And(Member(0,succ(succ(succ(B)))), ``` paulson@13363 ` 492` ``` pair_fm(1,0,2)))))))" ``` paulson@13363 ` 493` paulson@13363 ` 494` ```lemma cartprod_type [TC]: ``` paulson@13363 ` 495` ``` "[| x \ nat; y \ nat; z \ nat |] ==> cartprod_fm(x,y,z) \ formula" ``` wenzelm@13428 ` 496` ```by (simp add: cartprod_fm_def) ``` paulson@13363 ` 497` paulson@13363 ` 498` ```lemma arity_cartprod_fm [simp]: ``` wenzelm@13428 ` 499` ``` "[| x \ nat; y \ nat; z \ nat |] ``` paulson@13363 ` 500` ``` ==> arity(cartprod_fm(x,y,z)) = succ(x) \ succ(y) \ succ(z)" ``` wenzelm@13428 ` 501` ```by (simp add: cartprod_fm_def succ_Un_distrib [symmetric] Un_ac) ``` paulson@13363 ` 502` paulson@13363 ` 503` ```lemma sats_cartprod_fm [simp]: ``` paulson@13363 ` 504` ``` "[| x \ nat; y \ nat; z \ nat; env \ list(A)|] ``` wenzelm@13428 ` 505` ``` ==> sats(A, cartprod_fm(x,y,z), env) <-> ``` paulson@13363 ` 506` ``` cartprod(**A, nth(x,env), nth(y,env), nth(z,env))" ``` paulson@13363 ` 507` ```by (simp add: cartprod_fm_def cartprod_def) ``` paulson@13363 ` 508` paulson@13363 ` 509` ```lemma cartprod_iff_sats: ``` wenzelm@13428 ` 510` ``` "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; ``` paulson@13363 ` 511` ``` i \ nat; j \ nat; k \ nat; env \ list(A)|] ``` paulson@13363 ` 512` ``` ==> cartprod(**A, x, y, z) <-> sats(A, cartprod_fm(i,j,k), env)" ``` paulson@13363 ` 513` ```by (simp add: sats_cartprod_fm) ``` paulson@13363 ` 514` paulson@13363 ` 515` ```theorem cartprod_reflection: ``` wenzelm@13428 ` 516` ``` "REFLECTS[\x. cartprod(L,f(x),g(x),h(x)), ``` paulson@13363 ` 517` ``` \i x. cartprod(**Lset(i),f(x),g(x),h(x))]" ``` paulson@13363 ` 518` ```apply (simp only: cartprod_def setclass_simps) ``` wenzelm@13428 ` 519` ```apply (intro FOL_reflections pair_reflection) ``` paulson@13363 ` 520` ```done ``` paulson@13363 ` 521` paulson@13363 ` 522` paulson@13363 ` 523` ```subsubsection{*Binary Sums, Internalized*} ``` paulson@13363 ` 524` wenzelm@13428 ` 525` ```(* "is_sum(M,A,B,Z) == ``` wenzelm@13428 ` 526` ``` \A0[M]. \n1[M]. \s1[M]. \B1[M]. ``` paulson@13363 ` 527` ``` 3 2 1 0 ``` paulson@13363 ` 528` ``` number1(M,n1) & cartprod(M,n1,A,A0) & upair(M,n1,n1,s1) & ``` paulson@13363 ` 529` ``` cartprod(M,s1,B,B1) & union(M,A0,B1,Z)" *) ``` paulson@13363 ` 530` ```constdefs sum_fm :: "[i,i,i]=>i" ``` wenzelm@13428 ` 531` ``` "sum_fm(A,B,Z) == ``` paulson@13363 ` 532` ``` Exists(Exists(Exists(Exists( ``` wenzelm@13428 ` 533` ``` And(number1_fm(2), ``` paulson@13363 ` 534` ``` And(cartprod_fm(2,A#+4,3), ``` paulson@13363 ` 535` ``` And(upair_fm(2,2,1), ``` paulson@13363 ` 536` ``` And(cartprod_fm(1,B#+4,0), union_fm(3,0,Z#+4)))))))))" ``` paulson@13363 ` 537` paulson@13363 ` 538` ```lemma sum_type [TC]: ``` paulson@13363 ` 539` ``` "[| x \ nat; y \ nat; z \ nat |] ==> sum_fm(x,y,z) \ formula" ``` wenzelm@13428 ` 540` ```by (simp add: sum_fm_def) ``` paulson@13363 ` 541` paulson@13363 ` 542` ```lemma arity_sum_fm [simp]: ``` wenzelm@13428 ` 543` ``` "[| x \ nat; y \ nat; z \ nat |] ``` paulson@13363 ` 544` ``` ==> arity(sum_fm(x,y,z)) = succ(x) \ succ(y) \ succ(z)" ``` wenzelm@13428 ` 545` ```by (simp add: sum_fm_def succ_Un_distrib [symmetric] Un_ac) ``` paulson@13363 ` 546` paulson@13363 ` 547` ```lemma sats_sum_fm [simp]: ``` paulson@13363 ` 548` ``` "[| x \ nat; y \ nat; z \ nat; env \ list(A)|] ``` wenzelm@13428 ` 549` ``` ==> sats(A, sum_fm(x,y,z), env) <-> ``` paulson@13363 ` 550` ``` is_sum(**A, nth(x,env), nth(y,env), nth(z,env))" ``` paulson@13363 ` 551` ```by (simp add: sum_fm_def is_sum_def) ``` paulson@13363 ` 552` paulson@13363 ` 553` ```lemma sum_iff_sats: ``` wenzelm@13428 ` 554` ``` "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; ``` paulson@13363 ` 555` ``` i \ nat; j \ nat; k \ nat; env \ list(A)|] ``` paulson@13363 ` 556` ``` ==> is_sum(**A, x, y, z) <-> sats(A, sum_fm(i,j,k), env)" ``` paulson@13363 ` 557` ```by simp ``` paulson@13363 ` 558` paulson@13363 ` 559` ```theorem sum_reflection: ``` wenzelm@13428 ` 560` ``` "REFLECTS[\x. is_sum(L,f(x),g(x),h(x)), ``` paulson@13363 ` 561` ``` \i x. is_sum(**Lset(i),f(x),g(x),h(x))]" ``` paulson@13363 ` 562` ```apply (simp only: is_sum_def setclass_simps) ``` wenzelm@13428 ` 563` ```apply (intro FOL_reflections function_reflections cartprod_reflection) ``` paulson@13363 ` 564` ```done ``` paulson@13363 ` 565` paulson@13363 ` 566` paulson@13363 ` 567` ```subsubsection{*The Operator @{term quasinat}*} ``` paulson@13363 ` 568` paulson@13363 ` 569` ```(* "is_quasinat(M,z) == empty(M,z) | (\m[M]. successor(M,m,z))" *) ``` paulson@13363 ` 570` ```constdefs quasinat_fm :: "i=>i" ``` paulson@13363 ` 571` ``` "quasinat_fm(z) == Or(empty_fm(z), Exists(succ_fm(0,succ(z))))" ``` paulson@13363 ` 572` paulson@13363 ` 573` ```lemma quasinat_type [TC]: ``` paulson@13363 ` 574` ``` "x \ nat ==> quasinat_fm(x) \ formula" ``` wenzelm@13428 ` 575` ```by (simp add: quasinat_fm_def) ``` paulson@13363 ` 576` paulson@13363 ` 577` ```lemma arity_quasinat_fm [simp]: ``` paulson@13363 ` 578` ``` "x \ nat ==> arity(quasinat_fm(x)) = succ(x)" ``` wenzelm@13428 ` 579` ```by (simp add: quasinat_fm_def succ_Un_distrib [symmetric] Un_ac) ``` paulson@13363 ` 580` paulson@13363 ` 581` ```lemma sats_quasinat_fm [simp]: ``` paulson@13363 ` 582` ``` "[| x \ nat; env \ list(A)|] ``` paulson@13363 ` 583` ``` ==> sats(A, quasinat_fm(x), env) <-> is_quasinat(**A, nth(x,env))" ``` paulson@13363 ` 584` ```by (simp add: quasinat_fm_def is_quasinat_def) ``` paulson@13363 ` 585` paulson@13363 ` 586` ```lemma quasinat_iff_sats: ``` wenzelm@13428 ` 587` ``` "[| nth(i,env) = x; nth(j,env) = y; ``` paulson@13363 ` 588` ``` i \ nat; env \ list(A)|] ``` paulson@13363 ` 589` ``` ==> is_quasinat(**A, x) <-> sats(A, quasinat_fm(i), env)" ``` paulson@13363 ` 590` ```by simp ``` paulson@13363 ` 591` paulson@13363 ` 592` ```theorem quasinat_reflection: ``` wenzelm@13428 ` 593` ``` "REFLECTS[\x. is_quasinat(L,f(x)), ``` paulson@13363 ` 594` ``` \i x. is_quasinat(**Lset(i),f(x))]" ``` paulson@13363 ` 595` ```apply (simp only: is_quasinat_def setclass_simps) ``` wenzelm@13428 ` 596` ```apply (intro FOL_reflections function_reflections) ``` paulson@13363 ` 597` ```done ``` paulson@13363 ` 598` paulson@13363 ` 599` paulson@13363 ` 600` ```subsubsection{*The Operator @{term is_nat_case}*} ``` paulson@13363 ` 601` paulson@13363 ` 602` ```(* is_nat_case :: "[i=>o, i, [i,i]=>o, i, i] => o" ``` wenzelm@13428 ` 603` ``` "is_nat_case(M, a, is_b, k, z) == ``` paulson@13363 ` 604` ``` (empty(M,k) --> z=a) & ``` paulson@13363 ` 605` ``` (\m[M]. successor(M,m,k) --> is_b(m,z)) & ``` paulson@13363 ` 606` ``` (is_quasinat(M,k) | empty(M,z))" *) ``` paulson@13363 ` 607` ```text{*The formula @{term is_b} has free variables 1 and 0.*} ``` paulson@13363 ` 608` ```constdefs is_nat_case_fm :: "[i, [i,i]=>i, i, i]=>i" ``` wenzelm@13428 ` 609` ``` "is_nat_case_fm(a,is_b,k,z) == ``` paulson@13363 ` 610` ``` And(Implies(empty_fm(k), Equal(z,a)), ``` wenzelm@13428 ` 611` ``` And(Forall(Implies(succ_fm(0,succ(k)), ``` paulson@13363 ` 612` ``` Forall(Implies(Equal(0,succ(succ(z))), is_b(1,0))))), ``` paulson@13363 ` 613` ``` Or(quasinat_fm(k), empty_fm(z))))" ``` paulson@13363 ` 614` paulson@13363 ` 615` ```lemma is_nat_case_type [TC]: ``` wenzelm@13428 ` 616` ``` "[| is_b(1,0) \ formula; ``` wenzelm@13428 ` 617` ``` x \ nat; y \ nat; z \ nat |] ``` paulson@13363 ` 618` ``` ==> is_nat_case_fm(x,is_b,y,z) \ formula" ``` wenzelm@13428 ` 619` ```by (simp add: is_nat_case_fm_def) ``` paulson@13363 ` 620` paulson@13363 ` 621` ```lemma arity_is_nat_case_fm [simp]: ``` wenzelm@13428 ` 622` ``` "[| is_b(1,0) \ formula; x \ nat; y \ nat; z \ nat |] ``` wenzelm@13428 ` 623` ``` ==> arity(is_nat_case_fm(x,is_b,y,z)) = ``` wenzelm@13428 ` 624` ``` succ(x) \ succ(y) \ succ(z) \ (arity(is_b(1,0)) #- 2)" ``` wenzelm@13428 ` 625` ```apply (subgoal_tac "arity(is_b(1,0)) \ nat") ``` paulson@13363 ` 626` ```apply typecheck ``` paulson@13363 ` 627` ```(*FIXME: could nat_diff_split work?*) ``` paulson@13363 ` 628` ```apply (auto simp add: diff_def raw_diff_succ is_nat_case_fm_def nat_imp_quasinat ``` paulson@13363 ` 629` ``` succ_Un_distrib [symmetric] Un_ac ``` wenzelm@13428 ` 630` ``` split: split_nat_case) ``` paulson@13363 ` 631` ```done ``` paulson@13363 ` 632` paulson@13363 ` 633` ```lemma sats_is_nat_case_fm: ``` wenzelm@13428 ` 634` ``` assumes is_b_iff_sats: ``` wenzelm@13428 ` 635` ``` "!!a b. [| a \ A; b \ A|] ``` paulson@13363 ` 636` ``` ==> is_b(a,b) <-> sats(A, p(1,0), Cons(b, Cons(a,env)))" ``` wenzelm@13428 ` 637` ``` shows ``` paulson@13363 ` 638` ``` "[|x \ nat; y \ nat; z < length(env); env \ list(A)|] ``` wenzelm@13428 ` 639` ``` ==> sats(A, is_nat_case_fm(x,p,y,z), env) <-> ``` paulson@13363 ` 640` ``` is_nat_case(**A, nth(x,env), is_b, nth(y,env), nth(z,env))" ``` wenzelm@13428 ` 641` ```apply (frule lt_length_in_nat, assumption) ``` paulson@13363 ` 642` ```apply (simp add: is_nat_case_fm_def is_nat_case_def is_b_iff_sats [THEN iff_sym]) ``` paulson@13363 ` 643` ```done ``` paulson@13363 ` 644` paulson@13363 ` 645` ```lemma is_nat_case_iff_sats: ``` wenzelm@13428 ` 646` ``` "[| (!!a b. [| a \ A; b \ A|] ``` paulson@13363 ` 647` ``` ==> is_b(a,b) <-> sats(A, p(1,0), Cons(b, Cons(a,env)))); ``` wenzelm@13428 ` 648` ``` nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; ``` paulson@13363 ` 649` ``` i \ nat; j \ nat; k < length(env); env \ list(A)|] ``` wenzelm@13428 ` 650` ``` ==> is_nat_case(**A, x, is_b, y, z) <-> sats(A, is_nat_case_fm(i,p,j,k), env)" ``` paulson@13363 ` 651` ```by (simp add: sats_is_nat_case_fm [of A is_b]) ``` paulson@13363 ` 652` paulson@13363 ` 653` paulson@13363 ` 654` ```text{*The second argument of @{term is_b} gives it direct access to @{term x}, ``` wenzelm@13428 ` 655` ``` which is essential for handling free variable references. Without this ``` paulson@13363 ` 656` ``` argument, we cannot prove reflection for @{term iterates_MH}.*} ``` paulson@13363 ` 657` ```theorem is_nat_case_reflection: ``` paulson@13363 ` 658` ``` assumes is_b_reflection: ``` wenzelm@13428 ` 659` ``` "!!h f g. REFLECTS[\x. is_b(L, h(x), f(x), g(x)), ``` paulson@13363 ` 660` ``` \i x. is_b(**Lset(i), h(x), f(x), g(x))]" ``` wenzelm@13428 ` 661` ``` shows "REFLECTS[\x. is_nat_case(L, f(x), is_b(L,x), g(x), h(x)), ``` paulson@13363 ` 662` ``` \i x. is_nat_case(**Lset(i), f(x), is_b(**Lset(i), x), g(x), h(x))]" ``` paulson@13363 ` 663` ```apply (simp (no_asm_use) only: is_nat_case_def setclass_simps) ``` wenzelm@13428 ` 664` ```apply (intro FOL_reflections function_reflections ``` wenzelm@13428 ` 665` ``` restriction_reflection is_b_reflection quasinat_reflection) ``` paulson@13363 ` 666` ```done ``` paulson@13363 ` 667` paulson@13363 ` 668` paulson@13363 ` 669` paulson@13363 ` 670` ```subsection{*The Operator @{term iterates_MH}, Needed for Iteration*} ``` paulson@13363 ` 671` paulson@13363 ` 672` ```(* iterates_MH :: "[i=>o, [i,i]=>o, i, i, i, i] => o" ``` paulson@13363 ` 673` ``` "iterates_MH(M,isF,v,n,g,z) == ``` paulson@13363 ` 674` ``` is_nat_case(M, v, \m u. \gm[M]. fun_apply(M,g,m,gm) & isF(gm,u), ``` paulson@13363 ` 675` ``` n, z)" *) ``` paulson@13363 ` 676` ```constdefs iterates_MH_fm :: "[[i,i]=>i, i, i, i, i]=>i" ``` wenzelm@13428 ` 677` ``` "iterates_MH_fm(isF,v,n,g,z) == ``` wenzelm@13428 ` 678` ``` is_nat_case_fm(v, ``` wenzelm@13428 ` 679` ``` \m u. Exists(And(fun_apply_fm(succ(succ(succ(g))),succ(m),0), ``` wenzelm@13428 ` 680` ``` Forall(Implies(Equal(0,succ(succ(u))), isF(1,0))))), ``` paulson@13363 ` 681` ``` n, z)" ``` paulson@13363 ` 682` paulson@13363 ` 683` ```lemma iterates_MH_type [TC]: ``` wenzelm@13428 ` 684` ``` "[| p(1,0) \ formula; ``` wenzelm@13428 ` 685` ``` v \ nat; x \ nat; y \ nat; z \ nat |] ``` paulson@13363 ` 686` ``` ==> iterates_MH_fm(p,v,x,y,z) \ formula" ``` wenzelm@13428 ` 687` ```by (simp add: iterates_MH_fm_def) ``` paulson@13363 ` 688` paulson@13363 ` 689` paulson@13363 ` 690` ```lemma arity_iterates_MH_fm [simp]: ``` wenzelm@13428 ` 691` ``` "[| p(1,0) \ formula; ``` wenzelm@13428 ` 692` ``` v \ nat; x \ nat; y \ nat; z \ nat |] ``` wenzelm@13428 ` 693` ``` ==> arity(iterates_MH_fm(p,v,x,y,z)) = ``` paulson@13363 ` 694` ``` succ(v) \ succ(x) \ succ(y) \ succ(z) \ (arity(p(1,0)) #- 4)" ``` paulson@13363 ` 695` ```apply (subgoal_tac "arity(p(1,0)) \ nat") ``` paulson@13363 ` 696` ```apply typecheck ``` paulson@13363 ` 697` ```apply (simp add: nat_imp_quasinat iterates_MH_fm_def Un_ac ``` paulson@13363 ` 698` ``` split: split_nat_case, clarify) ``` paulson@13363 ` 699` ```apply (rename_tac i j) ``` wenzelm@13428 ` 700` ```apply (drule eq_succ_imp_eq_m1, simp) ``` paulson@13363 ` 701` ```apply (drule eq_succ_imp_eq_m1, simp) ``` paulson@13363 ` 702` ```apply (simp add: diff_Un_distrib succ_Un_distrib Un_ac diff_diff_left) ``` paulson@13363 ` 703` ```done ``` paulson@13363 ` 704` paulson@13363 ` 705` ```lemma sats_iterates_MH_fm: ``` wenzelm@13428 ` 706` ``` assumes is_F_iff_sats: ``` wenzelm@13428 ` 707` ``` "!!a b c d. [| a \ A; b \ A; c \ A; d \ A|] ``` paulson@13363 ` 708` ``` ==> is_F(a,b) <-> ``` paulson@13363 ` 709` ``` sats(A, p(1,0), Cons(b, Cons(a, Cons(c, Cons(d,env)))))" ``` wenzelm@13428 ` 710` ``` shows ``` paulson@13363 ` 711` ``` "[|v \ nat; x \ nat; y \ nat; z < length(env); env \ list(A)|] ``` wenzelm@13428 ` 712` ``` ==> sats(A, iterates_MH_fm(p,v,x,y,z), env) <-> ``` paulson@13363 ` 713` ``` iterates_MH(**A, is_F, nth(v,env), nth(x,env), nth(y,env), nth(z,env))" ``` wenzelm@13428 ` 714` ```by (simp add: iterates_MH_fm_def iterates_MH_def sats_is_nat_case_fm ``` paulson@13363 ` 715` ``` is_F_iff_sats [symmetric]) ``` paulson@13363 ` 716` paulson@13363 ` 717` ```lemma iterates_MH_iff_sats: ``` wenzelm@13428 ` 718` ``` "[| (!!a b c d. [| a \ A; b \ A; c \ A; d \ A|] ``` paulson@13363 ` 719` ``` ==> is_F(a,b) <-> ``` paulson@13363 ` 720` ``` sats(A, p(1,0), Cons(b, Cons(a, Cons(c, Cons(d,env)))))); ``` wenzelm@13428 ` 721` ``` nth(i',env) = v; nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; ``` paulson@13363 ` 722` ``` i' \ nat; i \ nat; j \ nat; k < length(env); env \ list(A)|] ``` wenzelm@13428 ` 723` ``` ==> iterates_MH(**A, is_F, v, x, y, z) <-> ``` paulson@13363 ` 724` ``` sats(A, iterates_MH_fm(p,i',i,j,k), env)" ``` wenzelm@13428 ` 725` ```apply (rule iff_sym) ``` wenzelm@13428 ` 726` ```apply (rule iff_trans) ``` wenzelm@13428 ` 727` ```apply (rule sats_iterates_MH_fm [of A is_F], blast, simp_all) ``` paulson@13363 ` 728` ```done ``` paulson@13363 ` 729` paulson@13363 ` 730` ```theorem iterates_MH_reflection: ``` paulson@13363 ` 731` ``` assumes p_reflection: ``` wenzelm@13428 ` 732` ``` "!!f g h. REFLECTS[\x. p(L, f(x), g(x)), ``` paulson@13363 ` 733` ``` \i x. p(**Lset(i), f(x), g(x))]" ``` wenzelm@13428 ` 734` ``` shows "REFLECTS[\x. iterates_MH(L, p(L), e(x), f(x), g(x), h(x)), ``` paulson@13363 ` 735` ``` \i x. iterates_MH(**Lset(i), p(**Lset(i)), e(x), f(x), g(x), h(x))]" ``` paulson@13363 ` 736` ```apply (simp (no_asm_use) only: iterates_MH_def) ``` paulson@13363 ` 737` ```txt{*Must be careful: simplifying with @{text setclass_simps} above would ``` paulson@13363 ` 738` ``` change @{text "\gm[**Lset(i)]"} into @{text "\gm \ Lset(i)"}, when ``` paulson@13363 ` 739` ``` it would no longer match rule @{text is_nat_case_reflection}. *} ``` wenzelm@13428 ` 740` ```apply (rule is_nat_case_reflection) ``` paulson@13363 ` 741` ```apply (simp (no_asm_use) only: setclass_simps) ``` paulson@13363 ` 742` ```apply (intro FOL_reflections function_reflections is_nat_case_reflection ``` wenzelm@13428 ` 743` ``` restriction_reflection p_reflection) ``` paulson@13363 ` 744` ```done ``` paulson@13363 ` 745` paulson@13363 ` 746` paulson@13363 ` 747` wenzelm@13428 ` 748` ```subsection{*@{term L} is Closed Under the Operator @{term list}*} ``` paulson@13363 ` 749` paulson@13386 ` 750` ```subsubsection{*The List Functor, Internalized*} ``` paulson@13386 ` 751` paulson@13386 ` 752` ```constdefs list_functor_fm :: "[i,i,i]=>i" ``` wenzelm@13428 ` 753` ```(* "is_list_functor(M,A,X,Z) == ``` wenzelm@13428 ` 754` ``` \n1[M]. \AX[M]. ``` paulson@13386 ` 755` ``` number1(M,n1) & cartprod(M,A,X,AX) & is_sum(M,n1,AX,Z)" *) ``` wenzelm@13428 ` 756` ``` "list_functor_fm(A,X,Z) == ``` paulson@13386 ` 757` ``` Exists(Exists( ``` wenzelm@13428 ` 758` ``` And(number1_fm(1), ``` paulson@13386 ` 759` ``` And(cartprod_fm(A#+2,X#+2,0), sum_fm(1,0,Z#+2)))))" ``` paulson@13386 ` 760` paulson@13386 ` 761` ```lemma list_functor_type [TC]: ``` paulson@13386 ` 762` ``` "[| x \ nat; y \ nat; z \ nat |] ==> list_functor_fm(x,y,z) \ formula" ``` wenzelm@13428 ` 763` ```by (simp add: list_functor_fm_def) ``` paulson@13386 ` 764` paulson@13386 ` 765` ```lemma arity_list_functor_fm [simp]: ``` wenzelm@13428 ` 766` ``` "[| x \ nat; y \ nat; z \ nat |] ``` paulson@13386 ` 767` ``` ==> arity(list_functor_fm(x,y,z)) = succ(x) \ succ(y) \ succ(z)" ``` wenzelm@13428 ` 768` ```by (simp add: list_functor_fm_def succ_Un_distrib [symmetric] Un_ac) ``` paulson@13386 ` 769` paulson@13386 ` 770` ```lemma sats_list_functor_fm [simp]: ``` paulson@13386 ` 771` ``` "[| x \ nat; y \ nat; z \ nat; env \ list(A)|] ``` wenzelm@13428 ` 772` ``` ==> sats(A, list_functor_fm(x,y,z), env) <-> ``` paulson@13386 ` 773` ``` is_list_functor(**A, nth(x,env), nth(y,env), nth(z,env))" ``` paulson@13386 ` 774` ```by (simp add: list_functor_fm_def is_list_functor_def) ``` paulson@13386 ` 775` paulson@13386 ` 776` ```lemma list_functor_iff_sats: ``` wenzelm@13428 ` 777` ``` "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; ``` paulson@13386 ` 778` ``` i \ nat; j \ nat; k \ nat; env \ list(A)|] ``` paulson@13386 ` 779` ``` ==> is_list_functor(**A, x, y, z) <-> sats(A, list_functor_fm(i,j,k), env)" ``` paulson@13386 ` 780` ```by simp ``` paulson@13386 ` 781` paulson@13386 ` 782` ```theorem list_functor_reflection: ``` wenzelm@13428 ` 783` ``` "REFLECTS[\x. is_list_functor(L,f(x),g(x),h(x)), ``` paulson@13386 ` 784` ``` \i x. is_list_functor(**Lset(i),f(x),g(x),h(x))]" ``` paulson@13386 ` 785` ```apply (simp only: is_list_functor_def setclass_simps) ``` paulson@13386 ` 786` ```apply (intro FOL_reflections number1_reflection ``` wenzelm@13428 ` 787` ``` cartprod_reflection sum_reflection) ``` paulson@13386 ` 788` ```done ``` paulson@13386 ` 789` paulson@13386 ` 790` paulson@13386 ` 791` ```subsubsection{*Instances of Replacement for Lists*} ``` paulson@13386 ` 792` paulson@13363 ` 793` ```lemma list_replacement1_Reflects: ``` paulson@13363 ` 794` ``` "REFLECTS ``` paulson@13363 ` 795` ``` [\x. \u[L]. u \ B \ (\y[L]. pair(L,u,y,x) \ ``` paulson@13363 ` 796` ``` is_wfrec(L, iterates_MH(L, is_list_functor(L,A), 0), memsn, u, y)), ``` paulson@13363 ` 797` ``` \i x. \u \ Lset(i). u \ B \ (\y \ Lset(i). pair(**Lset(i), u, y, x) \ ``` wenzelm@13428 ` 798` ``` is_wfrec(**Lset(i), ``` wenzelm@13428 ` 799` ``` iterates_MH(**Lset(i), ``` paulson@13363 ` 800` ``` is_list_functor(**Lset(i), A), 0), memsn, u, y))]" ``` wenzelm@13428 ` 801` ```by (intro FOL_reflections function_reflections is_wfrec_reflection ``` wenzelm@13428 ` 802` ``` iterates_MH_reflection list_functor_reflection) ``` paulson@13363 ` 803` wenzelm@13428 ` 804` ```lemma list_replacement1: ``` paulson@13363 ` 805` ``` "L(A) ==> iterates_replacement(L, is_list_functor(L,A), 0)" ``` paulson@13363 ` 806` ```apply (unfold iterates_replacement_def wfrec_replacement_def, clarify) ``` wenzelm@13428 ` 807` ```apply (rule strong_replacementI) ``` paulson@13363 ` 808` ```apply (rule rallI) ``` wenzelm@13428 ` 809` ```apply (rename_tac B) ``` wenzelm@13428 ` 810` ```apply (rule separation_CollectI) ``` wenzelm@13428 ` 811` ```apply (insert nonempty) ``` wenzelm@13428 ` 812` ```apply (subgoal_tac "L(Memrel(succ(n)))") ``` wenzelm@13428 ` 813` ```apply (rule_tac A="{B,A,n,z,0,Memrel(succ(n))}" in subset_LsetE, blast ) ``` paulson@13363 ` 814` ```apply (rule ReflectsE [OF list_replacement1_Reflects], assumption) ``` wenzelm@13428 ` 815` ```apply (drule subset_Lset_ltD, assumption) ``` paulson@13363 ` 816` ```apply (erule reflection_imp_L_separation) ``` paulson@13386 ` 817` ``` apply (simp_all add: lt_Ord2 Memrel_closed) ``` wenzelm@13428 ` 818` ```apply (elim conjE) ``` paulson@13385 ` 819` ```apply (rule DPow_LsetI) ``` wenzelm@13428 ` 820` ```apply (rename_tac v) ``` paulson@13363 ` 821` ```apply (rule bex_iff_sats conj_iff_sats)+ ``` paulson@13363 ` 822` ```apply (rule_tac env = "[u,v,A,n,B,0,Memrel(succ(n))]" in mem_iff_sats) ``` paulson@13363 ` 823` ```apply (rule sep_rules | simp)+ ``` paulson@13363 ` 824` ```txt{*Can't get sat rules to work for higher-order operators, so just expand them!*} ``` paulson@13363 ` 825` ```apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def) ``` paulson@13363 ` 826` ```apply (rule sep_rules list_functor_iff_sats quasinat_iff_sats | simp)+ ``` paulson@13363 ` 827` ```done ``` paulson@13363 ` 828` paulson@13363 ` 829` paulson@13363 ` 830` ```lemma list_replacement2_Reflects: ``` paulson@13363 ` 831` ``` "REFLECTS ``` paulson@13363 ` 832` ``` [\x. \u[L]. u \ B \ u \ nat \ ``` paulson@13363 ` 833` ``` (\sn[L]. \msn[L]. successor(L, u, sn) \ membership(L, sn, msn) \ ``` paulson@13363 ` 834` ``` is_wfrec (L, iterates_MH (L, is_list_functor(L, A), 0), ``` paulson@13363 ` 835` ``` msn, u, x)), ``` paulson@13363 ` 836` ``` \i x. \u \ Lset(i). u \ B \ u \ nat \ ``` wenzelm@13428 ` 837` ``` (\sn \ Lset(i). \msn \ Lset(i). ``` paulson@13363 ` 838` ``` successor(**Lset(i), u, sn) \ membership(**Lset(i), sn, msn) \ ``` wenzelm@13428 ` 839` ``` is_wfrec (**Lset(i), ``` paulson@13363 ` 840` ``` iterates_MH (**Lset(i), is_list_functor(**Lset(i), A), 0), ``` paulson@13363 ` 841` ``` msn, u, x))]" ``` wenzelm@13428 ` 842` ```by (intro FOL_reflections function_reflections is_wfrec_reflection ``` wenzelm@13428 ` 843` ``` iterates_MH_reflection list_functor_reflection) ``` paulson@13363 ` 844` paulson@13363 ` 845` wenzelm@13428 ` 846` ```lemma list_replacement2: ``` wenzelm@13428 ` 847` ``` "L(A) ==> strong_replacement(L, ``` wenzelm@13428 ` 848` ``` \n y. n\nat & ``` paulson@13363 ` 849` ``` (\sn[L]. \msn[L]. successor(L,n,sn) & membership(L,sn,msn) & ``` wenzelm@13428 ` 850` ``` is_wfrec(L, iterates_MH(L,is_list_functor(L,A), 0), ``` paulson@13363 ` 851` ``` msn, n, y)))" ``` wenzelm@13428 ` 852` ```apply (rule strong_replacementI) ``` paulson@13363 ` 853` ```apply (rule rallI) ``` wenzelm@13428 ` 854` ```apply (rename_tac B) ``` wenzelm@13428 ` 855` ```apply (rule separation_CollectI) ``` wenzelm@13428 ` 856` ```apply (insert nonempty) ``` wenzelm@13428 ` 857` ```apply (rule_tac A="{A,B,z,0,nat}" in subset_LsetE) ``` wenzelm@13428 ` 858` ```apply (blast intro: L_nat) ``` paulson@13363 ` 859` ```apply (rule ReflectsE [OF list_replacement2_Reflects], assumption) ``` wenzelm@13428 ` 860` ```apply (drule subset_Lset_ltD, assumption) ``` paulson@13363 ` 861` ```apply (erule reflection_imp_L_separation) ``` paulson@13363 ` 862` ``` apply (simp_all add: lt_Ord2) ``` paulson@13385 ` 863` ```apply (rule DPow_LsetI) ``` wenzelm@13428 ` 864` ```apply (rename_tac v) ``` paulson@13363 ` 865` ```apply (rule bex_iff_sats conj_iff_sats)+ ``` paulson@13363 ` 866` ```apply (rule_tac env = "[u,v,A,B,0,nat]" in mem_iff_sats) ``` paulson@13363 ` 867` ```apply (rule sep_rules | simp)+ ``` paulson@13363 ` 868` ```apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def) ``` paulson@13363 ` 869` ```apply (rule sep_rules list_functor_iff_sats quasinat_iff_sats | simp)+ ``` paulson@13363 ` 870` ```done ``` paulson@13363 ` 871` paulson@13386 ` 872` wenzelm@13428 ` 873` ```subsection{*@{term L} is Closed Under the Operator @{term formula}*} ``` paulson@13386 ` 874` paulson@13386 ` 875` ```subsubsection{*The Formula Functor, Internalized*} ``` paulson@13386 ` 876` paulson@13386 ` 877` ```constdefs formula_functor_fm :: "[i,i]=>i" ``` wenzelm@13428 ` 878` ```(* "is_formula_functor(M,X,Z) == ``` wenzelm@13428 ` 879` ``` \nat'[M]. \natnat[M]. \natnatsum[M]. \XX[M]. \X3[M]. ``` paulson@13398 ` 880` ``` 4 3 2 1 0 ``` wenzelm@13428 ` 881` ``` omega(M,nat') & cartprod(M,nat',nat',natnat) & ``` paulson@13386 ` 882` ``` is_sum(M,natnat,natnat,natnatsum) & ``` wenzelm@13428 ` 883` ``` cartprod(M,X,X,XX) & is_sum(M,XX,X,X3) & ``` wenzelm@13428 ` 884` ``` is_sum(M,natnatsum,X3,Z)" *) ``` wenzelm@13428 ` 885` ``` "formula_functor_fm(X,Z) == ``` paulson@13398 ` 886` ``` Exists(Exists(Exists(Exists(Exists( ``` wenzelm@13428 ` 887` ``` And(omega_fm(4), ``` paulson@13398 ` 888` ``` And(cartprod_fm(4,4,3), ``` paulson@13398 ` 889` ``` And(sum_fm(3,3,2), ``` paulson@13398 ` 890` ``` And(cartprod_fm(X#+5,X#+5,1), ``` paulson@13398 ` 891` ``` And(sum_fm(1,X#+5,0), sum_fm(2,0,Z#+5)))))))))))" ``` paulson@13386 ` 892` paulson@13386 ` 893` ```lemma formula_functor_type [TC]: ``` paulson@13386 ` 894` ``` "[| x \ nat; y \ nat |] ==> formula_functor_fm(x,y) \ formula" ``` wenzelm@13428 ` 895` ```by (simp add: formula_functor_fm_def) ``` paulson@13386 ` 896` paulson@13386 ` 897` ```lemma sats_formula_functor_fm [simp]: ``` paulson@13386 ` 898` ``` "[| x \ nat; y \ nat; env \ list(A)|] ``` wenzelm@13428 ` 899` ``` ==> sats(A, formula_functor_fm(x,y), env) <-> ``` paulson@13386 ` 900` ``` is_formula_functor(**A, nth(x,env), nth(y,env))" ``` paulson@13386 ` 901` ```by (simp add: formula_functor_fm_def is_formula_functor_def) ``` paulson@13386 ` 902` paulson@13386 ` 903` ```lemma formula_functor_iff_sats: ``` wenzelm@13428 ` 904` ``` "[| nth(i,env) = x; nth(j,env) = y; ``` paulson@13386 ` 905` ``` i \ nat; j \ nat; env \ list(A)|] ``` paulson@13386 ` 906` ``` ==> is_formula_functor(**A, x, y) <-> sats(A, formula_functor_fm(i,j), env)" ``` paulson@13386 ` 907` ```by simp ``` paulson@13386 ` 908` paulson@13386 ` 909` ```theorem formula_functor_reflection: ``` wenzelm@13428 ` 910` ``` "REFLECTS[\x. is_formula_functor(L,f(x),g(x)), ``` paulson@13386 ` 911` ``` \i x. is_formula_functor(**Lset(i),f(x),g(x))]" ``` paulson@13386 ` 912` ```apply (simp only: is_formula_functor_def setclass_simps) ``` paulson@13386 ` 913` ```apply (intro FOL_reflections omega_reflection ``` wenzelm@13428 ` 914` ``` cartprod_reflection sum_reflection) ``` paulson@13386 ` 915` ```done ``` paulson@13386 ` 916` paulson@13386 ` 917` ```subsubsection{*Instances of Replacement for Formulas*} ``` paulson@13386 ` 918` paulson@13386 ` 919` ```lemma formula_replacement1_Reflects: ``` paulson@13386 ` 920` ``` "REFLECTS ``` paulson@13386 ` 921` ``` [\x. \u[L]. u \ B \ (\y[L]. pair(L,u,y,x) \ ``` paulson@13386 ` 922` ``` is_wfrec(L, iterates_MH(L, is_formula_functor(L), 0), memsn, u, y)), ``` paulson@13386 ` 923` ``` \i x. \u \ Lset(i). u \ B \ (\y \ Lset(i). pair(**Lset(i), u, y, x) \ ``` wenzelm@13428 ` 924` ``` is_wfrec(**Lset(i), ``` wenzelm@13428 ` 925` ``` iterates_MH(**Lset(i), ``` paulson@13386 ` 926` ``` is_formula_functor(**Lset(i)), 0), memsn, u, y))]" ``` wenzelm@13428 ` 927` ```by (intro FOL_reflections function_reflections is_wfrec_reflection ``` wenzelm@13428 ` 928` ``` iterates_MH_reflection formula_functor_reflection) ``` paulson@13386 ` 929` wenzelm@13428 ` 930` ```lemma formula_replacement1: ``` paulson@13386 ` 931` ``` "iterates_replacement(L, is_formula_functor(L), 0)" ``` paulson@13386 ` 932` ```apply (unfold iterates_replacement_def wfrec_replacement_def, clarify) ``` wenzelm@13428 ` 933` ```apply (rule strong_replacementI) ``` paulson@13386 ` 934` ```apply (rule rallI) ``` wenzelm@13428 ` 935` ```apply (rename_tac B) ``` wenzelm@13428 ` 936` ```apply (rule separation_CollectI) ``` wenzelm@13428 ` 937` ```apply (insert nonempty) ``` wenzelm@13428 ` 938` ```apply (subgoal_tac "L(Memrel(succ(n)))") ``` wenzelm@13428 ` 939` ```apply (rule_tac A="{B,n,z,0,Memrel(succ(n))}" in subset_LsetE, blast ) ``` paulson@13386 ` 940` ```apply (rule ReflectsE [OF formula_replacement1_Reflects], assumption) ``` wenzelm@13428 ` 941` ```apply (drule subset_Lset_ltD, assumption) ``` paulson@13386 ` 942` ```apply (erule reflection_imp_L_separation) ``` paulson@13386 ` 943` ``` apply (simp_all add: lt_Ord2 Memrel_closed) ``` paulson@13386 ` 944` ```apply (rule DPow_LsetI) ``` wenzelm@13428 ` 945` ```apply (rename_tac v) ``` paulson@13386 ` 946` ```apply (rule bex_iff_sats conj_iff_sats)+ ``` paulson@13386 ` 947` ```apply (rule_tac env = "[u,v,n,B,0,Memrel(succ(n))]" in mem_iff_sats) ``` paulson@13386 ` 948` ```apply (rule sep_rules | simp)+ ``` paulson@13386 ` 949` ```txt{*Can't get sat rules to work for higher-order operators, so just expand them!*} ``` paulson@13386 ` 950` ```apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def) ``` paulson@13386 ` 951` ```apply (rule sep_rules formula_functor_iff_sats quasinat_iff_sats | simp)+ ``` paulson@13386 ` 952` ```txt{*SLOW: like 40 seconds!*} ``` paulson@13386 ` 953` ```done ``` paulson@13386 ` 954` paulson@13386 ` 955` ```lemma formula_replacement2_Reflects: ``` paulson@13386 ` 956` ``` "REFLECTS ``` paulson@13386 ` 957` ``` [\x. \u[L]. u \ B \ u \ nat \ ``` paulson@13386 ` 958` ``` (\sn[L]. \msn[L]. successor(L, u, sn) \ membership(L, sn, msn) \ ``` paulson@13386 ` 959` ``` is_wfrec (L, iterates_MH (L, is_formula_functor(L), 0), ``` paulson@13386 ` 960` ``` msn, u, x)), ``` paulson@13386 ` 961` ``` \i x. \u \ Lset(i). u \ B \ u \ nat \ ``` wenzelm@13428 ` 962` ``` (\sn \ Lset(i). \msn \ Lset(i). ``` paulson@13386 ` 963` ``` successor(**Lset(i), u, sn) \ membership(**Lset(i), sn, msn) \ ``` wenzelm@13428 ` 964` ``` is_wfrec (**Lset(i), ``` paulson@13386 ` 965` ``` iterates_MH (**Lset(i), is_formula_functor(**Lset(i)), 0), ``` paulson@13386 ` 966` ``` msn, u, x))]" ``` wenzelm@13428 ` 967` ```by (intro FOL_reflections function_reflections is_wfrec_reflection ``` wenzelm@13428 ` 968` ``` iterates_MH_reflection formula_functor_reflection) ``` paulson@13386 ` 969` paulson@13386 ` 970` wenzelm@13428 ` 971` ```lemma formula_replacement2: ``` wenzelm@13428 ` 972` ``` "strong_replacement(L, ``` wenzelm@13428 ` 973` ``` \n y. n\nat & ``` paulson@13386 ` 974` ``` (\sn[L]. \msn[L]. successor(L,n,sn) & membership(L,sn,msn) & ``` wenzelm@13428 ` 975` ``` is_wfrec(L, iterates_MH(L,is_formula_functor(L), 0), ``` paulson@13386 ` 976` ``` msn, n, y)))" ``` wenzelm@13428 ` 977` ```apply (rule strong_replacementI) ``` paulson@13386 ` 978` ```apply (rule rallI) ``` wenzelm@13428 ` 979` ```apply (rename_tac B) ``` wenzelm@13428 ` 980` ```apply (rule separation_CollectI) ``` wenzelm@13428 ` 981` ```apply (insert nonempty) ``` wenzelm@13428 ` 982` ```apply (rule_tac A="{B,z,0,nat}" in subset_LsetE) ``` wenzelm@13428 ` 983` ```apply (blast intro: L_nat) ``` paulson@13386 ` 984` ```apply (rule ReflectsE [OF formula_replacement2_Reflects], assumption) ``` wenzelm@13428 ` 985` ```apply (drule subset_Lset_ltD, assumption) ``` paulson@13386 ` 986` ```apply (erule reflection_imp_L_separation) ``` paulson@13386 ` 987` ``` apply (simp_all add: lt_Ord2) ``` paulson@13386 ` 988` ```apply (rule DPow_LsetI) ``` wenzelm@13428 ` 989` ```apply (rename_tac v) ``` paulson@13386 ` 990` ```apply (rule bex_iff_sats conj_iff_sats)+ ``` paulson@13386 ` 991` ```apply (rule_tac env = "[u,v,B,0,nat]" in mem_iff_sats) ``` paulson@13386 ` 992` ```apply (rule sep_rules | simp)+ ``` paulson@13386 ` 993` ```apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def) ``` paulson@13386 ` 994` ```apply (rule sep_rules formula_functor_iff_sats quasinat_iff_sats | simp)+ ``` paulson@13386 ` 995` ```done ``` paulson@13386 ` 996` paulson@13386 ` 997` ```text{*NB The proofs for type @{term formula} are virtually identical to those ``` paulson@13386 ` 998` ```for @{term "list(A)"}. It was a cut-and-paste job! *} ``` paulson@13386 ` 999` paulson@13387 ` 1000` paulson@13409 ` 1001` ```subsection{*Internalized Forms of Data Structuring Operators*} ``` paulson@13409 ` 1002` paulson@13409 ` 1003` ```subsubsection{*The Formula @{term is_Inl}, Internalized*} ``` paulson@13409 ` 1004` paulson@13409 ` 1005` ```(* is_Inl(M,a,z) == \zero[M]. empty(M,zero) & pair(M,zero,a,z) *) ``` paulson@13409 ` 1006` ```constdefs Inl_fm :: "[i,i]=>i" ``` paulson@13409 ` 1007` ``` "Inl_fm(a,z) == Exists(And(empty_fm(0), pair_fm(0,succ(a),succ(z))))" ``` paulson@13409 ` 1008` paulson@13409 ` 1009` ```lemma Inl_type [TC]: ``` paulson@13409 ` 1010` ``` "[| x \ nat; z \ nat |] ==> Inl_fm(x,z) \ formula" ``` wenzelm@13428 ` 1011` ```by (simp add: Inl_fm_def) ``` paulson@13409 ` 1012` paulson@13409 ` 1013` ```lemma sats_Inl_fm [simp]: ``` paulson@13409 ` 1014` ``` "[| x \ nat; z \ nat; env \ list(A)|] ``` paulson@13409 ` 1015` ``` ==> sats(A, Inl_fm(x,z), env) <-> is_Inl(**A, nth(x,env), nth(z,env))" ``` paulson@13409 ` 1016` ```by (simp add: Inl_fm_def is_Inl_def) ``` paulson@13409 ` 1017` paulson@13409 ` 1018` ```lemma Inl_iff_sats: ``` wenzelm@13428 ` 1019` ``` "[| nth(i,env) = x; nth(k,env) = z; ``` paulson@13409 ` 1020` ``` i \ nat; k \ nat; env \ list(A)|] ``` paulson@13409 ` 1021` ``` ==> is_Inl(**A, x, z) <-> sats(A, Inl_fm(i,k), env)" ``` paulson@13409 ` 1022` ```by simp ``` paulson@13409 ` 1023` paulson@13409 ` 1024` ```theorem Inl_reflection: ``` wenzelm@13428 ` 1025` ``` "REFLECTS[\x. is_Inl(L,f(x),h(x)), ``` paulson@13409 ` 1026` ``` \i x. is_Inl(**Lset(i),f(x),h(x))]" ``` paulson@13409 ` 1027` ```apply (simp only: is_Inl_def setclass_simps) ``` wenzelm@13428 ` 1028` ```apply (intro FOL_reflections function_reflections) ``` paulson@13409 ` 1029` ```done ``` paulson@13409 ` 1030` paulson@13409 ` 1031` paulson@13409 ` 1032` ```subsubsection{*The Formula @{term is_Inr}, Internalized*} ``` paulson@13409 ` 1033` paulson@13409 ` 1034` ```(* is_Inr(M,a,z) == \n1[M]. number1(M,n1) & pair(M,n1,a,z) *) ``` paulson@13409 ` 1035` ```constdefs Inr_fm :: "[i,i]=>i" ``` paulson@13409 ` 1036` ``` "Inr_fm(a,z) == Exists(And(number1_fm(0), pair_fm(0,succ(a),succ(z))))" ``` paulson@13409 ` 1037` paulson@13409 ` 1038` ```lemma Inr_type [TC]: ``` paulson@13409 ` 1039` ``` "[| x \ nat; z \ nat |] ==> Inr_fm(x,z) \ formula" ``` wenzelm@13428 ` 1040` ```by (simp add: Inr_fm_def) ``` paulson@13409 ` 1041` paulson@13409 ` 1042` ```lemma sats_Inr_fm [simp]: ``` paulson@13409 ` 1043` ``` "[| x \ nat; z \ nat; env \ list(A)|] ``` paulson@13409 ` 1044` ``` ==> sats(A, Inr_fm(x,z), env) <-> is_Inr(**A, nth(x,env), nth(z,env))" ``` paulson@13409 ` 1045` ```by (simp add: Inr_fm_def is_Inr_def) ``` paulson@13409 ` 1046` paulson@13409 ` 1047` ```lemma Inr_iff_sats: ``` wenzelm@13428 ` 1048` ``` "[| nth(i,env) = x; nth(k,env) = z; ``` paulson@13409 ` 1049` ``` i \ nat; k \ nat; env \ list(A)|] ``` paulson@13409 ` 1050` ``` ==> is_Inr(**A, x, z) <-> sats(A, Inr_fm(i,k), env)" ``` paulson@13409 ` 1051` ```by simp ``` paulson@13409 ` 1052` paulson@13409 ` 1053` ```theorem Inr_reflection: ``` wenzelm@13428 ` 1054` ``` "REFLECTS[\x. is_Inr(L,f(x),h(x)), ``` paulson@13409 ` 1055` ``` \i x. is_Inr(**Lset(i),f(x),h(x))]" ``` paulson@13409 ` 1056` ```apply (simp only: is_Inr_def setclass_simps) ``` wenzelm@13428 ` 1057` ```apply (intro FOL_reflections function_reflections) ``` paulson@13409 ` 1058` ```done ``` paulson@13409 ` 1059` paulson@13409 ` 1060` paulson@13409 ` 1061` ```subsubsection{*The Formula @{term is_Nil}, Internalized*} ``` paulson@13409 ` 1062` paulson@13409 ` 1063` ```(* is_Nil(M,xs) == \zero[M]. empty(M,zero) & is_Inl(M,zero,xs) *) ``` paulson@13409 ` 1064` paulson@13409 ` 1065` ```constdefs Nil_fm :: "i=>i" ``` paulson@13409 ` 1066` ``` "Nil_fm(x) == Exists(And(empty_fm(0), Inl_fm(0,succ(x))))" ``` wenzelm@13428 ` 1067` paulson@13409 ` 1068` ```lemma Nil_type [TC]: "x \ nat ==> Nil_fm(x) \ formula" ``` wenzelm@13428 ` 1069` ```by (simp add: Nil_fm_def) ``` paulson@13409 ` 1070` paulson@13409 ` 1071` ```lemma sats_Nil_fm [simp]: ``` paulson@13409 ` 1072` ``` "[| x \ nat; env \ list(A)|] ``` paulson@13409 ` 1073` ``` ==> sats(A, Nil_fm(x), env) <-> is_Nil(**A, nth(x,env))" ``` paulson@13409 ` 1074` ```by (simp add: Nil_fm_def is_Nil_def) ``` paulson@13409 ` 1075` paulson@13409 ` 1076` ```lemma Nil_iff_sats: ``` paulson@13409 ` 1077` ``` "[| nth(i,env) = x; i \ nat; env \ list(A)|] ``` paulson@13409 ` 1078` ``` ==> is_Nil(**A, x) <-> sats(A, Nil_fm(i), env)" ``` paulson@13409 ` 1079` ```by simp ``` paulson@13409 ` 1080` paulson@13409 ` 1081` ```theorem Nil_reflection: ``` wenzelm@13428 ` 1082` ``` "REFLECTS[\x. is_Nil(L,f(x)), ``` paulson@13409 ` 1083` ``` \i x. is_Nil(**Lset(i),f(x))]" ``` paulson@13409 ` 1084` ```apply (simp only: is_Nil_def setclass_simps) ``` wenzelm@13428 ` 1085` ```apply (intro FOL_reflections function_reflections Inl_reflection) ``` paulson@13409 ` 1086` ```done ``` paulson@13409 ` 1087` paulson@13409 ` 1088` paulson@13422 ` 1089` ```subsubsection{*The Formula @{term is_Cons}, Internalized*} ``` paulson@13395 ` 1090` paulson@13387 ` 1091` paulson@13409 ` 1092` ```(* "is_Cons(M,a,l,Z) == \p[M]. pair(M,a,l,p) & is_Inr(M,p,Z)" *) ``` paulson@13409 ` 1093` ```constdefs Cons_fm :: "[i,i,i]=>i" ``` wenzelm@13428 ` 1094` ``` "Cons_fm(a,l,Z) == ``` paulson@13409 ` 1095` ``` Exists(And(pair_fm(succ(a),succ(l),0), Inr_fm(0,succ(Z))))" ``` paulson@13409 ` 1096` paulson@13409 ` 1097` ```lemma Cons_type [TC]: ``` paulson@13409 ` 1098` ``` "[| x \ nat; y \ nat; z \ nat |] ==> Cons_fm(x,y,z) \ formula" ``` wenzelm@13428 ` 1099` ```by (simp add: Cons_fm_def) ``` paulson@13409 ` 1100` paulson@13409 ` 1101` ```lemma sats_Cons_fm [simp]: ``` paulson@13409 ` 1102` ``` "[| x \ nat; y \ nat; z \ nat; env \ list(A)|] ``` wenzelm@13428 ` 1103` ``` ==> sats(A, Cons_fm(x,y,z), env) <-> ``` paulson@13409 ` 1104` ``` is_Cons(**A, nth(x,env), nth(y,env), nth(z,env))" ``` paulson@13409 ` 1105` ```by (simp add: Cons_fm_def is_Cons_def) ``` paulson@13409 ` 1106` paulson@13409 ` 1107` ```lemma Cons_iff_sats: ``` wenzelm@13428 ` 1108` ``` "[| nth(i,env) = x; nth(j,env) = y; nth(k,env) = z; ``` paulson@13409 ` 1109` ``` i \ nat; j \ nat; k \ nat; env \ list(A)|] ``` paulson@13409 ` 1110` ``` ==>is_Cons(**A, x, y, z) <-> sats(A, Cons_fm(i,j,k), env)" ``` paulson@13409 ` 1111` ```by simp ``` paulson@13409 ` 1112` paulson@13409 ` 1113` ```theorem Cons_reflection: ``` wenzelm@13428 ` 1114` ``` "REFLECTS[\x. is_Cons(L,f(x),g(x),h(x)), ``` paulson@13409 ` 1115` ``` \i x. is_Cons(**Lset(i),f(x),g(x),h(x))]" ``` paulson@13409 ` 1116` ```apply (simp only: is_Cons_def setclass_simps) ``` wenzelm@13428 ` 1117` ```apply (intro FOL_reflections pair_reflection Inr_reflection) ``` paulson@13409 ` 1118` ```done ``` paulson@13409 ` 1119` paulson@13409 ` 1120` ```subsubsection{*The Formula @{term is_quasilist}, Internalized*} ``` paulson@13409 ` 1121` paulson@13409 ` 1122` ```(* is_quasilist(M,xs) == is_Nil(M,z) | (\x[M]. \l[M]. is_Cons(M,x,l,z))" *) ``` paulson@13409 ` 1123` paulson@13409 ` 1124` ```constdefs quasilist_fm :: "i=>i" ``` wenzelm@13428 ` 1125` ``` "quasilist_fm(x) == ``` paulson@13409 ` 1126` ``` Or(Nil_fm(x), Exists(Exists(Cons_fm(1,0,succ(succ(x))))))" ``` wenzelm@13428 ` 1127` paulson@13409 ` 1128` ```lemma quasilist_type [TC]: "x \ nat ==> quasilist_fm(x) \ formula" ``` wenzelm@13428 ` 1129` ```by (simp add: quasilist_fm_def) ``` paulson@13409 ` 1130` paulson@13409 ` 1131` ```lemma sats_quasilist_fm [simp]: ``` paulson@13409 ` 1132` ``` "[| x \ nat; env \ list(A)|] ``` paulson@13409 ` 1133` ``` ==> sats(A, quasilist_fm(x), env) <-> is_quasilist(**A, nth(x,env))" ``` paulson@13409 ` 1134` ```by (simp add: quasilist_fm_def is_quasilist_def) ``` paulson@13409 ` 1135` paulson@13409 ` 1136` ```lemma quasilist_iff_sats: ``` paulson@13409 ` 1137` ``` "[| nth(i,env) = x; i \ nat; env \ list(A)|] ``` paulson@13409 ` 1138` ``` ==> is_quasilist(**A, x) <-> sats(A, quasilist_fm(i), env)" ``` paulson@13409 ` 1139` ```by simp ``` paulson@13409 ` 1140` paulson@13409 ` 1141` ```theorem quasilist_reflection: ``` wenzelm@13428 ` 1142` ``` "REFLECTS[\x. is_quasilist(L,f(x)), ``` paulson@13409 ` 1143` ``` \i x. is_quasilist(**Lset(i),f(x))]" ``` paulson@13409 ` 1144` ```apply (simp only: is_quasilist_def setclass_simps) ``` wenzelm@13428 ` 1145` ```apply (intro FOL_reflections Nil_reflection Cons_reflection) ``` paulson@13409 ` 1146` ```done ``` paulson@13409 ` 1147` paulson@13409 ` 1148` paulson@13409 ` 1149` ```subsection{*Absoluteness for the Function @{term nth}*} ``` paulson@13409 ` 1150` paulson@13409 ` 1151` paulson@13409 ` 1152` ```subsubsection{*The Formula @{term is_tl}, Internalized*} ``` paulson@13409 ` 1153` wenzelm@13428 ` 1154` ```(* "is_tl(M,xs,T) == ``` paulson@13409 ` 1155` ``` (is_Nil(M,xs) --> T=xs) & ``` paulson@13409 ` 1156` ``` (\x[M]. \l[M]. ~ is_Cons(M,x,l,xs) | T=l) & ``` paulson@13409 ` 1157` ``` (is_quasilist(M,xs) | empty(M,T))" *) ``` paulson@13409 ` 1158` ```constdefs tl_fm :: "[i,i]=>i" ``` wenzelm@13428 ` 1159` ``` "tl_fm(xs,T) == ``` paulson@13409 ` 1160` ``` And(Implies(Nil_fm(xs), Equal(T,xs)), ``` paulson@13409 ` 1161` ``` And(Forall(Forall(Or(Neg(Cons_fm(1,0,xs#+2)), Equal(T#+2,0)))), ``` paulson@13409 ` 1162` ``` Or(quasilist_fm(xs), empty_fm(T))))" ``` paulson@13409 ` 1163` paulson@13409 ` 1164` ```lemma tl_type [TC]: ``` paulson@13409 ` 1165` ``` "[| x \ nat; y \ nat |] ==> tl_fm(x,y) \ formula" ``` wenzelm@13428 ` 1166` ```by (simp add: tl_fm_def) ``` paulson@13409 ` 1167` paulson@13409 ` 1168` ```lemma sats_tl_fm [simp]: ``` paulson@13409 ` 1169` ``` "[| x \ nat; y \ nat; env \ list(A)|] ``` paulson@13409 ` 1170` ``` ==> sats(A, tl_fm(x,y), env) <-> is_tl(**A, nth(x,env), nth(y,env))" ``` paulson@13409 ` 1171` ```by (simp add: tl_fm_def is_tl_def) ``` paulson@13409 ` 1172` paulson@13409 ` 1173` ```lemma tl_iff_sats: ``` paulson@13409 ` 1174` ``` "[| nth(i,env) = x; nth(j,env) = y; ``` paulson@13409 ` 1175` ``` i \ nat; j \ nat; env \ list(A)|] ``` paulson@13409 ` 1176` ``` ==> is_tl(**A, x, y) <-> sats(A, tl_fm(i,j), env)" ``` paulson@13409 ` 1177` ```by simp ``` paulson@13409 ` 1178` paulson@13409 ` 1179` ```theorem tl_reflection: ``` wenzelm@13428 ` 1180` ``` "REFLECTS[\x. is_tl(L,f(x),g(x)), ``` paulson@13409 ` 1181` ``` \i x. is_tl(**Lset(i),f(x),g(x))]" ``` paulson@13409 ` 1182` ```apply (simp only: is_tl_def setclass_simps) ``` paulson@13409 ` 1183` ```apply (intro FOL_reflections Nil_reflection Cons_reflection ``` wenzelm@13428 ` 1184` ``` quasilist_reflection empty_reflection) ``` paulson@13409 ` 1185` ```done ``` paulson@13409 ` 1186` paulson@13409 ` 1187` paulson@13409 ` 1188` ```subsubsection{*An Instance of Replacement for @{term nth}*} ``` paulson@13409 ` 1189` paulson@13409 ` 1190` ```lemma nth_replacement_Reflects: ``` paulson@13409 ` 1191` ``` "REFLECTS ``` paulson@13409 ` 1192` ``` [\x. \u[L]. u \ B \ (\y[L]. pair(L,u,y,x) \ ``` paulson@13409 ` 1193` ``` is_wfrec(L, iterates_MH(L, is_tl(L), z), memsn, u, y)), ``` paulson@13409 ` 1194` ``` \i x. \u \ Lset(i). u \ B \ (\y \ Lset(i). pair(**Lset(i), u, y, x) \ ``` wenzelm@13428 ` 1195` ``` is_wfrec(**Lset(i), ``` wenzelm@13428 ` 1196` ``` iterates_MH(**Lset(i), ``` paulson@13409 ` 1197` ``` is_tl(**Lset(i)), z), memsn, u, y))]" ``` wenzelm@13428 ` 1198` ```by (intro FOL_reflections function_reflections is_wfrec_reflection ``` wenzelm@13428 ` 1199` ``` iterates_MH_reflection list_functor_reflection tl_reflection) ``` paulson@13409 ` 1200` wenzelm@13428 ` 1201` ```lemma nth_replacement: ``` paulson@13409 ` 1202` ``` "L(w) ==> iterates_replacement(L, %l t. is_tl(L,l,t), w)" ``` paulson@13409 ` 1203` ```apply (unfold iterates_replacement_def wfrec_replacement_def, clarify) ``` wenzelm@13428 ` 1204` ```apply (rule strong_replacementI) ``` wenzelm@13428 ` 1205` ```apply (rule rallI) ``` wenzelm@13428 ` 1206` ```apply (rule separation_CollectI) ``` wenzelm@13428 ` 1207` ```apply (subgoal_tac "L(Memrel(succ(n)))") ``` wenzelm@13428 ` 1208` ```apply (rule_tac A="{A,n,w,z,Memrel(succ(n))}" in subset_LsetE, blast ) ``` paulson@13409 ` 1209` ```apply (rule ReflectsE [OF nth_replacement_Reflects], assumption) ``` wenzelm@13428 ` 1210` ```apply (drule subset_Lset_ltD, assumption) ``` paulson@13409 ` 1211` ```apply (erule reflection_imp_L_separation) ``` paulson@13409 ` 1212` ``` apply (simp_all add: lt_Ord2 Memrel_closed) ``` wenzelm@13428 ` 1213` ```apply (elim conjE) ``` paulson@13409 ` 1214` ```apply (rule DPow_LsetI) ``` wenzelm@13428 ` 1215` ```apply (rename_tac v) ``` paulson@13409 ` 1216` ```apply (rule bex_iff_sats conj_iff_sats)+ ``` paulson@13409 ` 1217` ```apply (rule_tac env = "[u,v,A,z,w,Memrel(succ(n))]" in mem_iff_sats) ``` paulson@13409 ` 1218` ```apply (rule sep_rules | simp)+ ``` paulson@13409 ` 1219` ```apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def) ``` paulson@13409 ` 1220` ```apply (rule sep_rules quasinat_iff_sats tl_iff_sats | simp)+ ``` paulson@13409 ` 1221` ```done ``` paulson@13409 ` 1222` paulson@13422 ` 1223` paulson@13422 ` 1224` paulson@13422 ` 1225` ```subsubsection{*Instantiating the locale @{text M_datatypes}*} ``` wenzelm@13428 ` 1226` wenzelm@13428 ` 1227` ```theorem M_datatypes_axioms_L: "M_datatypes_axioms(L)" ``` wenzelm@13428 ` 1228` ``` apply (rule M_datatypes_axioms.intro) ``` wenzelm@13428 ` 1229` ``` apply (assumption | rule ``` wenzelm@13428 ` 1230` ``` list_replacement1 list_replacement2 ``` wenzelm@13428 ` 1231` ``` formula_replacement1 formula_replacement2 ``` wenzelm@13428 ` 1232` ``` nth_replacement)+ ``` wenzelm@13428 ` 1233` ``` done ``` paulson@13422 ` 1234` wenzelm@13428 ` 1235` ```theorem M_datatypes_L: "PROP M_datatypes(L)" ``` wenzelm@13428 ` 1236` ``` apply (rule M_datatypes.intro) ``` wenzelm@13428 ` 1237` ``` apply (rule M_triv_axioms_L) ``` wenzelm@13428 ` 1238` ``` apply (rule M_axioms_axioms_L) ``` wenzelm@13428 ` 1239` ``` apply (rule M_trancl_axioms_L) ``` wenzelm@13428 ` 1240` ``` apply (rule M_wfrank_axioms_L) ``` wenzelm@13428 ` 1241` ``` apply (rule M_datatypes_axioms_L) ``` wenzelm@13428 ` 1242` ``` done ``` paulson@13422 ` 1243` wenzelm@13428 ` 1244` ```lemmas list_closed = M_datatypes.list_closed [OF M_datatypes_L] ``` wenzelm@13428 ` 1245` ``` and formula_closed = M_datatypes.formula_closed [OF M_datatypes_L] ``` wenzelm@13428 ` 1246` ``` and list_abs = M_datatypes.list_abs [OF M_datatypes_L] ``` wenzelm@13428 ` 1247` ``` and formula_abs = M_datatypes.formula_abs [OF M_datatypes_L] ``` wenzelm@13428 ` 1248` ``` and nth_abs = M_datatypes.nth_abs [OF M_datatypes_L] ``` paulson@13409 ` 1249` paulson@13422 ` 1250` ```declare list_closed [intro,simp] ``` paulson@13422 ` 1251` ```declare formula_closed [intro,simp] ``` paulson@13422 ` 1252` ```declare list_abs [simp] ``` paulson@13422 ` 1253` ```declare formula_abs [simp] ``` paulson@13422 ` 1254` ```declare nth_abs [simp] ``` paulson@13422 ` 1255` paulson@13422 ` 1256` wenzelm@13428 ` 1257` ```subsection{*@{term L} is Closed Under the Operator @{term eclose}*} ``` paulson@13422 ` 1258` paulson@13422 ` 1259` ```subsubsection{*Instances of Replacement for @{term eclose}*} ``` paulson@13422 ` 1260` paulson@13422 ` 1261` ```lemma eclose_replacement1_Reflects: ``` paulson@13422 ` 1262` ``` "REFLECTS ``` paulson@13422 ` 1263` ``` [\x. \u[L]. u \ B \ (\y[L]. pair(L,u,y,x) \ ``` paulson@13422 ` 1264` ``` is_wfrec(L, iterates_MH(L, big_union(L), A), memsn, u, y)), ``` paulson@13422 ` 1265` ``` \i x. \u \ Lset(i). u \ B \ (\y \ Lset(i). pair(**Lset(i), u, y, x) \ ``` wenzelm@13428 ` 1266` ``` is_wfrec(**Lset(i), ``` wenzelm@13428 ` 1267` ``` iterates_MH(**Lset(i), big_union(**Lset(i)), A), ``` paulson@13422 ` 1268` ``` memsn, u, y))]" ``` wenzelm@13428 ` 1269` ```by (intro FOL_reflections function_reflections is_wfrec_reflection ``` wenzelm@13428 ` 1270` ``` iterates_MH_reflection) ``` paulson@13422 ` 1271` wenzelm@13428 ` 1272` ```lemma eclose_replacement1: ``` paulson@13422 ` 1273` ``` "L(A) ==> iterates_replacement(L, big_union(L), A)" ``` paulson@13422 ` 1274` ```apply (unfold iterates_replacement_def wfrec_replacement_def, clarify) ``` wenzelm@13428 ` 1275` ```apply (rule strong_replacementI) ``` paulson@13422 ` 1276` ```apply (rule rallI) ``` wenzelm@13428 ` 1277` ```apply (rename_tac B) ``` wenzelm@13428 ` 1278` ```apply (rule separation_CollectI) ``` wenzelm@13428 ` 1279` ```apply (subgoal_tac "L(Memrel(succ(n)))") ``` wenzelm@13428 ` 1280` ```apply (rule_tac A="{B,A,n,z,Memrel(succ(n))}" in subset_LsetE, blast ) ``` paulson@13422 ` 1281` ```apply (rule ReflectsE [OF eclose_replacement1_Reflects], assumption) ``` wenzelm@13428 ` 1282` ```apply (drule subset_Lset_ltD, assumption) ``` paulson@13422 ` 1283` ```apply (erule reflection_imp_L_separation) ``` paulson@13422 ` 1284` ``` apply (simp_all add: lt_Ord2 Memrel_closed) ``` wenzelm@13428 ` 1285` ```apply (elim conjE) ``` paulson@13422 ` 1286` ```apply (rule DPow_LsetI) ``` wenzelm@13428 ` 1287` ```apply (rename_tac v) ``` paulson@13422 ` 1288` ```apply (rule bex_iff_sats conj_iff_sats)+ ``` paulson@13422 ` 1289` ```apply (rule_tac env = "[u,v,A,n,B,Memrel(succ(n))]" in mem_iff_sats) ``` paulson@13422 ` 1290` ```apply (rule sep_rules | simp)+ ``` paulson@13422 ` 1291` ```txt{*Can't get sat rules to work for higher-order operators, so just expand them!*} ``` paulson@13422 ` 1292` ```apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def) ``` paulson@13422 ` 1293` ```apply (rule sep_rules big_union_iff_sats quasinat_iff_sats | simp)+ ``` paulson@13409 ` 1294` ```done ``` paulson@13409 ` 1295` paulson@13422 ` 1296` paulson@13422 ` 1297` ```lemma eclose_replacement2_Reflects: ``` paulson@13422 ` 1298` ``` "REFLECTS ``` paulson@13422 ` 1299` ``` [\x. \u[L]. u \ B \ u \ nat \ ``` paulson@13422 ` 1300` ``` (\sn[L]. \msn[L]. successor(L, u, sn) \ membership(L, sn, msn) \ ``` paulson@13422 ` 1301` ``` is_wfrec (L, iterates_MH (L, big_union(L), A), ``` paulson@13422 ` 1302` ``` msn, u, x)), ``` paulson@13422 ` 1303` ``` \i x. \u \ Lset(i). u \ B \ u \ nat \ ``` wenzelm@13428 ` 1304` ``` (\sn \ Lset(i). \msn \ Lset(i). ``` paulson@13422 ` 1305` ``` successor(**Lset(i), u, sn) \ membership(**Lset(i), sn, msn) \ ``` wenzelm@13428 ` 1306` ``` is_wfrec (**Lset(i), ``` paulson@13422 ` 1307` ``` iterates_MH (**Lset(i), big_union(**Lset(i)), A), ``` paulson@13422 ` 1308` ``` msn, u, x))]" ``` wenzelm@13428 ` 1309` ```by (intro FOL_reflections function_reflections is_wfrec_reflection ``` wenzelm@13428 ` 1310` ``` iterates_MH_reflection) ``` paulson@13422 ` 1311` paulson@13422 ` 1312` wenzelm@13428 ` 1313` ```lemma eclose_replacement2: ``` wenzelm@13428 ` 1314` ``` "L(A) ==> strong_replacement(L, ``` wenzelm@13428 ` 1315` ``` \n y. n\nat & ``` paulson@13422 ` 1316` ``` (\sn[L]. \msn[L]. successor(L,n,sn) & membership(L,sn,msn) & ``` wenzelm@13428 ` 1317` ``` is_wfrec(L, iterates_MH(L,big_union(L), A), ``` paulson@13422 ` 1318` ``` msn, n, y)))" ``` wenzelm@13428 ` 1319` ```apply (rule strong_replacementI) ``` paulson@13422 ` 1320` ```apply (rule rallI) ``` wenzelm@13428 ` 1321` ```apply (rename_tac B) ``` wenzelm@13428 ` 1322` ```apply (rule separation_CollectI) ``` wenzelm@13428 ` 1323` ```apply (rule_tac A="{A,B,z,nat}" in subset_LsetE) ``` wenzelm@13428 ` 1324` ```apply (blast intro: L_nat) ``` paulson@13422 ` 1325` ```apply (rule ReflectsE [OF eclose_replacement2_Reflects], assumption) ``` wenzelm@13428 ` 1326` ```apply (drule subset_Lset_ltD, assumption) ``` paulson@13422 ` 1327` ```apply (erule reflection_imp_L_separation) ``` paulson@13422 ` 1328` ``` apply (simp_all add: lt_Ord2) ``` paulson@13422 ` 1329` ```apply (rule DPow_LsetI) ``` wenzelm@13428 ` 1330` ```apply (rename_tac v) ``` paulson@13422 ` 1331` ```apply (rule bex_iff_sats conj_iff_sats)+ ``` paulson@13422 ` 1332` ```apply (rule_tac env = "[u,v,A,B,nat]" in mem_iff_sats) ``` paulson@13422 ` 1333` ```apply (rule sep_rules | simp)+ ``` paulson@13422 ` 1334` ```apply (simp add: is_wfrec_def M_is_recfun_def iterates_MH_def is_nat_case_def) ``` paulson@13422 ` 1335` ```apply (rule sep_rules big_union_iff_sats quasinat_iff_sats | simp)+ ``` paulson@13422 ` 1336` ```done ``` paulson@13422 ` 1337` paulson@13422 ` 1338` paulson@13422 ` 1339` ```subsubsection{*Instantiating the locale @{text M_eclose}*} ``` paulson@13422 ` 1340` wenzelm@13428 ` 1341` ```theorem M_eclose_axioms_L: "M_eclose_axioms(L)" ``` wenzelm@13428 ` 1342` ``` apply (rule M_eclose_axioms.intro) ``` wenzelm@13428 ` 1343` ``` apply (assumption | rule eclose_replacement1 eclose_replacement2)+ ``` wenzelm@13428 ` 1344` ``` done ``` paulson@13422 ` 1345` wenzelm@13428 ` 1346` ```theorem M_eclose_L: "PROP M_eclose(L)" ``` wenzelm@13428 ` 1347` ``` apply (rule M_eclose.intro) ``` wenzelm@13428 ` 1348` ``` apply (rule M_triv_axioms_L) ``` wenzelm@13428 ` 1349` ``` apply (rule M_axioms_axioms_L) ``` wenzelm@13428 ` 1350` ``` apply (rule M_trancl_axioms_L) ``` wenzelm@13428 ` 1351` ``` apply (rule M_wfrank_axioms_L) ``` wenzelm@13428 ` 1352` ``` apply (rule M_datatypes_axioms_L) ``` wenzelm@13428 ` 1353` ``` apply (rule M_eclose_axioms_L) ``` wenzelm@13428 ` 1354` ``` done ``` paulson@13422 ` 1355` wenzelm@13428 ` 1356` ```lemmas eclose_closed [intro, simp] = M_eclose.eclose_closed [OF M_eclose_L] ``` wenzelm@13428 ` 1357` ``` and eclose_abs [intro, simp] = M_eclose.eclose_abs [OF M_eclose_L] ``` paulson@13422 ` 1358` paulson@13348 ` 1359` ```end ```