src/ZF/Constructible/Separation.thy
 author ballarin Tue Jun 20 15:53:44 2006 +0200 (2006-06-20) changeset 19931 fb32b43e7f80 parent 16417 9bc16273c2d4 child 29223 e09c53289830 permissions -rw-r--r--
Restructured locales with predicates: import is now an interpretation.
New method intro_locales.
 paulson@13437 ` 1` ```(* Title: ZF/Constructible/Separation.thy ``` paulson@13437 ` 2` ``` ID: \$Id\$ ``` paulson@13437 ` 3` ``` Author: Lawrence C Paulson, Cambridge University Computer Laboratory ``` paulson@13437 ` 4` ```*) ``` paulson@13437 ` 5` paulson@13339 ` 6` ```header{*Early Instances of Separation and Strong Replacement*} ``` paulson@13323 ` 7` haftmann@16417 ` 8` ```theory Separation imports L_axioms WF_absolute begin ``` paulson@13306 ` 9` paulson@13564 ` 10` ```text{*This theory proves all instances needed for locale @{text "M_basic"}*} ``` paulson@13339 ` 11` paulson@13306 ` 12` ```text{*Helps us solve for de Bruijn indices!*} ``` paulson@13306 ` 13` ```lemma nth_ConsI: "[|nth(n,l) = x; n \ nat|] ==> nth(succ(n), Cons(a,l)) = x" ``` paulson@13306 ` 14` ```by simp ``` paulson@13306 ` 15` paulson@13316 ` 16` ```lemmas nth_rules = nth_0 nth_ConsI nat_0I nat_succI ``` wenzelm@13428 ` 17` ```lemmas sep_rules = nth_0 nth_ConsI FOL_iff_sats function_iff_sats ``` paulson@13323 ` 18` ``` fun_plus_iff_sats ``` paulson@13306 ` 19` paulson@13306 ` 20` ```lemma Collect_conj_in_DPow: ``` wenzelm@13428 ` 21` ``` "[| {x\A. P(x)} \ DPow(A); {x\A. Q(x)} \ DPow(A) |] ``` paulson@13306 ` 22` ``` ==> {x\A. P(x) & Q(x)} \ DPow(A)" ``` wenzelm@13428 ` 23` ```by (simp add: Int_in_DPow Collect_Int_Collect_eq [symmetric]) ``` paulson@13306 ` 24` paulson@13306 ` 25` ```lemma Collect_conj_in_DPow_Lset: ``` paulson@13306 ` 26` ``` "[|z \ Lset(j); {x \ Lset(j). P(x)} \ DPow(Lset(j))|] ``` paulson@13306 ` 27` ``` ==> {x \ Lset(j). x \ z & P(x)} \ DPow(Lset(j))" ``` paulson@13306 ` 28` ```apply (frule mem_Lset_imp_subset_Lset) ``` wenzelm@13428 ` 29` ```apply (simp add: Collect_conj_in_DPow Collect_mem_eq ``` paulson@13306 ` 30` ``` subset_Int_iff2 elem_subset_in_DPow) ``` paulson@13306 ` 31` ```done ``` paulson@13306 ` 32` paulson@13306 ` 33` ```lemma separation_CollectI: ``` paulson@13306 ` 34` ``` "(\z. L(z) ==> L({x \ z . P(x)})) ==> separation(L, \x. P(x))" ``` wenzelm@13428 ` 35` ```apply (unfold separation_def, clarify) ``` wenzelm@13428 ` 36` ```apply (rule_tac x="{x\z. P(x)}" in rexI) ``` paulson@13306 ` 37` ```apply simp_all ``` paulson@13306 ` 38` ```done ``` paulson@13306 ` 39` paulson@13306 ` 40` ```text{*Reduces the original comprehension to the reflected one*} ``` paulson@13306 ` 41` ```lemma reflection_imp_L_separation: ``` paulson@13306 ` 42` ``` "[| \x\Lset(j). P(x) <-> Q(x); ``` wenzelm@13428 ` 43` ``` {x \ Lset(j) . Q(x)} \ DPow(Lset(j)); ``` paulson@13306 ` 44` ``` Ord(j); z \ Lset(j)|] ==> L({x \ z . P(x)})" ``` paulson@13306 ` 45` ```apply (rule_tac i = "succ(j)" in L_I) ``` paulson@13306 ` 46` ``` prefer 2 apply simp ``` paulson@13306 ` 47` ```apply (subgoal_tac "{x \ z. P(x)} = {x \ Lset(j). x \ z & (Q(x))}") ``` paulson@13306 ` 48` ``` prefer 2 ``` wenzelm@13428 ` 49` ``` apply (blast dest: mem_Lset_imp_subset_Lset) ``` paulson@13306 ` 50` ```apply (simp add: Lset_succ Collect_conj_in_DPow_Lset) ``` paulson@13306 ` 51` ```done ``` paulson@13306 ` 52` paulson@13566 ` 53` ```text{*Encapsulates the standard proof script for proving instances of ``` paulson@13687 ` 54` ``` Separation.*} ``` paulson@13566 ` 55` ```lemma gen_separation: ``` paulson@13566 ` 56` ``` assumes reflection: "REFLECTS [P,Q]" ``` paulson@13566 ` 57` ``` and Lu: "L(u)" ``` paulson@13566 ` 58` ``` and collI: "!!j. u \ Lset(j) ``` paulson@13566 ` 59` ``` \ Collect(Lset(j), Q(j)) \ DPow(Lset(j))" ``` paulson@13566 ` 60` ``` shows "separation(L,P)" ``` paulson@13566 ` 61` ```apply (rule separation_CollectI) ``` paulson@13566 ` 62` ```apply (rule_tac A="{u,z}" in subset_LsetE, blast intro: Lu) ``` paulson@13566 ` 63` ```apply (rule ReflectsE [OF reflection], assumption) ``` paulson@13566 ` 64` ```apply (drule subset_Lset_ltD, assumption) ``` paulson@13566 ` 65` ```apply (erule reflection_imp_L_separation) ``` paulson@13566 ` 66` ``` apply (simp_all add: lt_Ord2, clarify) ``` paulson@13691 ` 67` ```apply (rule collI, assumption) ``` paulson@13687 ` 68` ```done ``` paulson@13687 ` 69` paulson@13687 ` 70` ```text{*As above, but typically @{term u} is a finite enumeration such as ``` paulson@13687 ` 71` ``` @{term "{a,b}"}; thus the new subgoal gets the assumption ``` paulson@13687 ` 72` ``` @{term "{a,b} \ Lset(i)"}, which is logically equivalent to ``` paulson@13687 ` 73` ``` @{term "a \ Lset(i)"} and @{term "b \ Lset(i)"}.*} ``` paulson@13687 ` 74` ```lemma gen_separation_multi: ``` paulson@13687 ` 75` ``` assumes reflection: "REFLECTS [P,Q]" ``` paulson@13687 ` 76` ``` and Lu: "L(u)" ``` paulson@13687 ` 77` ``` and collI: "!!j. u \ Lset(j) ``` paulson@13687 ` 78` ``` \ Collect(Lset(j), Q(j)) \ DPow(Lset(j))" ``` paulson@13687 ` 79` ``` shows "separation(L,P)" ``` paulson@13687 ` 80` ```apply (rule gen_separation [OF reflection Lu]) ``` paulson@13687 ` 81` ```apply (drule mem_Lset_imp_subset_Lset) ``` paulson@13687 ` 82` ```apply (erule collI) ``` paulson@13566 ` 83` ```done ``` paulson@13566 ` 84` paulson@13306 ` 85` paulson@13316 ` 86` ```subsection{*Separation for Intersection*} ``` paulson@13306 ` 87` paulson@13306 ` 88` ```lemma Inter_Reflects: ``` wenzelm@13428 ` 89` ``` "REFLECTS[\x. \y[L]. y\A --> x \ y, ``` paulson@13314 ` 90` ``` \i x. \y\Lset(i). y\A --> x \ y]" ``` wenzelm@13428 ` 91` ```by (intro FOL_reflections) ``` paulson@13306 ` 92` paulson@13306 ` 93` ```lemma Inter_separation: ``` paulson@13306 ` 94` ``` "L(A) ==> separation(L, \x. \y[L]. y\A --> x\y)" ``` paulson@13566 ` 95` ```apply (rule gen_separation [OF Inter_Reflects], simp) ``` wenzelm@13428 ` 96` ```apply (rule DPow_LsetI) ``` paulson@13687 ` 97` ``` txt{*I leave this one example of a manual proof. The tedium of manually ``` paulson@13687 ` 98` ``` instantiating @{term i}, @{term j} and @{term env} is obvious. *} ``` wenzelm@13428 ` 99` ```apply (rule ball_iff_sats) ``` paulson@13306 ` 100` ```apply (rule imp_iff_sats) ``` paulson@13306 ` 101` ```apply (rule_tac [2] i=1 and j=0 and env="[y,x,A]" in mem_iff_sats) ``` paulson@13306 ` 102` ```apply (rule_tac i=0 and j=2 in mem_iff_sats) ``` paulson@13306 ` 103` ```apply (simp_all add: succ_Un_distrib [symmetric]) ``` paulson@13306 ` 104` ```done ``` paulson@13306 ` 105` paulson@13437 ` 106` ```subsection{*Separation for Set Difference*} ``` paulson@13437 ` 107` paulson@13437 ` 108` ```lemma Diff_Reflects: ``` paulson@13437 ` 109` ``` "REFLECTS[\x. x \ B, \i x. x \ B]" ``` paulson@13437 ` 110` ```by (intro FOL_reflections) ``` paulson@13437 ` 111` paulson@13437 ` 112` ```lemma Diff_separation: ``` paulson@13437 ` 113` ``` "L(B) ==> separation(L, \x. x \ B)" ``` paulson@13566 ` 114` ```apply (rule gen_separation [OF Diff_Reflects], simp) ``` paulson@13687 ` 115` ```apply (rule_tac env="[B]" in DPow_LsetI) ``` paulson@13437 ` 116` ```apply (rule sep_rules | simp)+ ``` paulson@13437 ` 117` ```done ``` paulson@13437 ` 118` paulson@13316 ` 119` ```subsection{*Separation for Cartesian Product*} ``` paulson@13306 ` 120` paulson@13323 ` 121` ```lemma cartprod_Reflects: ``` paulson@13314 ` 122` ``` "REFLECTS[\z. \x[L]. x\A & (\y[L]. y\B & pair(L,x,y,z)), ``` wenzelm@13428 ` 123` ``` \i z. \x\Lset(i). x\A & (\y\Lset(i). y\B & ``` paulson@13807 ` 124` ``` pair(##Lset(i),x,y,z))]" ``` paulson@13323 ` 125` ```by (intro FOL_reflections function_reflections) ``` paulson@13306 ` 126` paulson@13306 ` 127` ```lemma cartprod_separation: ``` wenzelm@13428 ` 128` ``` "[| L(A); L(B) |] ``` paulson@13306 ` 129` ``` ==> separation(L, \z. \x[L]. x\A & (\y[L]. y\B & pair(L,x,y,z)))" ``` paulson@13687 ` 130` ```apply (rule gen_separation_multi [OF cartprod_Reflects, of "{A,B}"], auto) ``` paulson@13687 ` 131` ```apply (rule_tac env="[A,B]" in DPow_LsetI) ``` paulson@13316 ` 132` ```apply (rule sep_rules | simp)+ ``` paulson@13306 ` 133` ```done ``` paulson@13306 ` 134` paulson@13316 ` 135` ```subsection{*Separation for Image*} ``` paulson@13306 ` 136` paulson@13306 ` 137` ```lemma image_Reflects: ``` paulson@13314 ` 138` ``` "REFLECTS[\y. \p[L]. p\r & (\x[L]. x\A & pair(L,x,y,p)), ``` paulson@13807 ` 139` ``` \i y. \p\Lset(i). p\r & (\x\Lset(i). x\A & pair(##Lset(i),x,y,p))]" ``` paulson@13323 ` 140` ```by (intro FOL_reflections function_reflections) ``` paulson@13306 ` 141` paulson@13306 ` 142` ```lemma image_separation: ``` wenzelm@13428 ` 143` ``` "[| L(A); L(r) |] ``` paulson@13306 ` 144` ``` ==> separation(L, \y. \p[L]. p\r & (\x[L]. x\A & pair(L,x,y,p)))" ``` paulson@13687 ` 145` ```apply (rule gen_separation_multi [OF image_Reflects, of "{A,r}"], auto) ``` paulson@13687 ` 146` ```apply (rule_tac env="[A,r]" in DPow_LsetI) ``` paulson@13316 ` 147` ```apply (rule sep_rules | simp)+ ``` paulson@13306 ` 148` ```done ``` paulson@13306 ` 149` paulson@13306 ` 150` paulson@13316 ` 151` ```subsection{*Separation for Converse*} ``` paulson@13306 ` 152` paulson@13306 ` 153` ```lemma converse_Reflects: ``` paulson@13314 ` 154` ``` "REFLECTS[\z. \p[L]. p\r & (\x[L]. \y[L]. pair(L,x,y,p) & pair(L,y,x,z)), ``` wenzelm@13428 ` 155` ``` \i z. \p\Lset(i). p\r & (\x\Lset(i). \y\Lset(i). ``` paulson@13807 ` 156` ``` pair(##Lset(i),x,y,p) & pair(##Lset(i),y,x,z))]" ``` paulson@13323 ` 157` ```by (intro FOL_reflections function_reflections) ``` paulson@13306 ` 158` paulson@13306 ` 159` ```lemma converse_separation: ``` wenzelm@13428 ` 160` ``` "L(r) ==> separation(L, ``` paulson@13306 ` 161` ``` \z. \p[L]. p\r & (\x[L]. \y[L]. pair(L,x,y,p) & pair(L,y,x,z)))" ``` paulson@13566 ` 162` ```apply (rule gen_separation [OF converse_Reflects], simp) ``` paulson@13687 ` 163` ```apply (rule_tac env="[r]" in DPow_LsetI) ``` paulson@13316 ` 164` ```apply (rule sep_rules | simp)+ ``` paulson@13306 ` 165` ```done ``` paulson@13306 ` 166` paulson@13306 ` 167` paulson@13316 ` 168` ```subsection{*Separation for Restriction*} ``` paulson@13306 ` 169` paulson@13306 ` 170` ```lemma restrict_Reflects: ``` paulson@13314 ` 171` ``` "REFLECTS[\z. \x[L]. x\A & (\y[L]. pair(L,x,y,z)), ``` paulson@13807 ` 172` ``` \i z. \x\Lset(i). x\A & (\y\Lset(i). pair(##Lset(i),x,y,z))]" ``` paulson@13323 ` 173` ```by (intro FOL_reflections function_reflections) ``` paulson@13306 ` 174` paulson@13306 ` 175` ```lemma restrict_separation: ``` paulson@13306 ` 176` ``` "L(A) ==> separation(L, \z. \x[L]. x\A & (\y[L]. pair(L,x,y,z)))" ``` paulson@13566 ` 177` ```apply (rule gen_separation [OF restrict_Reflects], simp) ``` paulson@13687 ` 178` ```apply (rule_tac env="[A]" in DPow_LsetI) ``` paulson@13316 ` 179` ```apply (rule sep_rules | simp)+ ``` paulson@13306 ` 180` ```done ``` paulson@13306 ` 181` paulson@13306 ` 182` paulson@13316 ` 183` ```subsection{*Separation for Composition*} ``` paulson@13306 ` 184` paulson@13306 ` 185` ```lemma comp_Reflects: ``` wenzelm@13428 ` 186` ``` "REFLECTS[\xz. \x[L]. \y[L]. \z[L]. \xy[L]. \yz[L]. ``` wenzelm@13428 ` 187` ``` pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) & ``` paulson@13306 ` 188` ``` xy\s & yz\r, ``` wenzelm@13428 ` 189` ``` \i xz. \x\Lset(i). \y\Lset(i). \z\Lset(i). \xy\Lset(i). \yz\Lset(i). ``` paulson@13807 ` 190` ``` pair(##Lset(i),x,z,xz) & pair(##Lset(i),x,y,xy) & ``` paulson@13807 ` 191` ``` pair(##Lset(i),y,z,yz) & xy\s & yz\r]" ``` paulson@13323 ` 192` ```by (intro FOL_reflections function_reflections) ``` paulson@13306 ` 193` paulson@13306 ` 194` ```lemma comp_separation: ``` paulson@13306 ` 195` ``` "[| L(r); L(s) |] ``` wenzelm@13428 ` 196` ``` ==> separation(L, \xz. \x[L]. \y[L]. \z[L]. \xy[L]. \yz[L]. ``` wenzelm@13428 ` 197` ``` pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) & ``` paulson@13306 ` 198` ``` xy\s & yz\r)" ``` paulson@13687 ` 199` ```apply (rule gen_separation_multi [OF comp_Reflects, of "{r,s}"], auto) ``` paulson@13687 ` 200` ```txt{*Subgoals after applying general ``separation'' rule: ``` paulson@13687 ` 201` ``` @{subgoals[display,indent=0,margin=65]}*} ``` paulson@13687 ` 202` ```apply (rule_tac env="[r,s]" in DPow_LsetI) ``` paulson@13687 ` 203` ```txt{*Subgoals ready for automatic synthesis of a formula: ``` paulson@13687 ` 204` ``` @{subgoals[display,indent=0,margin=65]}*} ``` paulson@13316 ` 205` ```apply (rule sep_rules | simp)+ ``` paulson@13306 ` 206` ```done ``` paulson@13306 ` 207` paulson@13687 ` 208` paulson@13316 ` 209` ```subsection{*Separation for Predecessors in an Order*} ``` paulson@13306 ` 210` paulson@13306 ` 211` ```lemma pred_Reflects: ``` paulson@13314 ` 212` ``` "REFLECTS[\y. \p[L]. p\r & pair(L,y,x,p), ``` paulson@13807 ` 213` ``` \i y. \p \ Lset(i). p\r & pair(##Lset(i),y,x,p)]" ``` paulson@13323 ` 214` ```by (intro FOL_reflections function_reflections) ``` paulson@13306 ` 215` paulson@13306 ` 216` ```lemma pred_separation: ``` paulson@13306 ` 217` ``` "[| L(r); L(x) |] ==> separation(L, \y. \p[L]. p\r & pair(L,y,x,p))" ``` paulson@13687 ` 218` ```apply (rule gen_separation_multi [OF pred_Reflects, of "{r,x}"], auto) ``` paulson@13687 ` 219` ```apply (rule_tac env="[r,x]" in DPow_LsetI) ``` paulson@13316 ` 220` ```apply (rule sep_rules | simp)+ ``` paulson@13306 ` 221` ```done ``` paulson@13306 ` 222` paulson@13306 ` 223` paulson@13316 ` 224` ```subsection{*Separation for the Membership Relation*} ``` paulson@13306 ` 225` paulson@13306 ` 226` ```lemma Memrel_Reflects: ``` paulson@13314 ` 227` ``` "REFLECTS[\z. \x[L]. \y[L]. pair(L,x,y,z) & x \ y, ``` paulson@13807 ` 228` ``` \i z. \x \ Lset(i). \y \ Lset(i). pair(##Lset(i),x,y,z) & x \ y]" ``` paulson@13323 ` 229` ```by (intro FOL_reflections function_reflections) ``` paulson@13306 ` 230` paulson@13306 ` 231` ```lemma Memrel_separation: ``` paulson@13306 ` 232` ``` "separation(L, \z. \x[L]. \y[L]. pair(L,x,y,z) & x \ y)" ``` paulson@13566 ` 233` ```apply (rule gen_separation [OF Memrel_Reflects nonempty]) ``` paulson@13687 ` 234` ```apply (rule_tac env="[]" in DPow_LsetI) ``` paulson@13316 ` 235` ```apply (rule sep_rules | simp)+ ``` paulson@13306 ` 236` ```done ``` paulson@13306 ` 237` paulson@13306 ` 238` paulson@13316 ` 239` ```subsection{*Replacement for FunSpace*} ``` wenzelm@13428 ` 240` paulson@13306 ` 241` ```lemma funspace_succ_Reflects: ``` wenzelm@13428 ` 242` ``` "REFLECTS[\z. \p[L]. p\A & (\f[L]. \b[L]. \nb[L]. \cnbf[L]. ``` wenzelm@13428 ` 243` ``` pair(L,f,b,p) & pair(L,n,b,nb) & is_cons(L,nb,f,cnbf) & ``` wenzelm@13428 ` 244` ``` upair(L,cnbf,cnbf,z)), ``` wenzelm@13428 ` 245` ``` \i z. \p \ Lset(i). p\A & (\f \ Lset(i). \b \ Lset(i). ``` wenzelm@13428 ` 246` ``` \nb \ Lset(i). \cnbf \ Lset(i). ``` paulson@13807 ` 247` ``` pair(##Lset(i),f,b,p) & pair(##Lset(i),n,b,nb) & ``` paulson@13807 ` 248` ``` is_cons(##Lset(i),nb,f,cnbf) & upair(##Lset(i),cnbf,cnbf,z))]" ``` paulson@13323 ` 249` ```by (intro FOL_reflections function_reflections) ``` paulson@13306 ` 250` paulson@13306 ` 251` ```lemma funspace_succ_replacement: ``` wenzelm@13428 ` 252` ``` "L(n) ==> ``` wenzelm@13428 ` 253` ``` strong_replacement(L, \p z. \f[L]. \b[L]. \nb[L]. \cnbf[L]. ``` paulson@13306 ` 254` ``` pair(L,f,b,p) & pair(L,n,b,nb) & is_cons(L,nb,f,cnbf) & ``` paulson@13306 ` 255` ``` upair(L,cnbf,cnbf,z))" ``` wenzelm@13428 ` 256` ```apply (rule strong_replacementI) ``` paulson@13687 ` 257` ```apply (rule_tac u="{n,B}" in gen_separation_multi [OF funspace_succ_Reflects], ``` paulson@13687 ` 258` ``` auto) ``` paulson@13687 ` 259` ```apply (rule_tac env="[n,B]" in DPow_LsetI) ``` paulson@13316 ` 260` ```apply (rule sep_rules | simp)+ ``` paulson@13306 ` 261` ```done ``` paulson@13306 ` 262` paulson@13306 ` 263` paulson@13634 ` 264` ```subsection{*Separation for a Theorem about @{term "is_recfun"}*} ``` paulson@13323 ` 265` paulson@13323 ` 266` ```lemma is_recfun_reflects: ``` wenzelm@13428 ` 267` ``` "REFLECTS[\x. \xa[L]. \xb[L]. ``` wenzelm@13428 ` 268` ``` pair(L,x,a,xa) & xa \ r & pair(L,x,b,xb) & xb \ r & ``` wenzelm@13428 ` 269` ``` (\fx[L]. \gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) & ``` paulson@13323 ` 270` ``` fx \ gx), ``` wenzelm@13428 ` 271` ``` \i x. \xa \ Lset(i). \xb \ Lset(i). ``` paulson@13807 ` 272` ``` pair(##Lset(i),x,a,xa) & xa \ r & pair(##Lset(i),x,b,xb) & xb \ r & ``` paulson@13807 ` 273` ``` (\fx \ Lset(i). \gx \ Lset(i). fun_apply(##Lset(i),f,x,fx) & ``` paulson@13807 ` 274` ``` fun_apply(##Lset(i),g,x,gx) & fx \ gx)]" ``` paulson@13323 ` 275` ```by (intro FOL_reflections function_reflections fun_plus_reflections) ``` paulson@13323 ` 276` paulson@13323 ` 277` ```lemma is_recfun_separation: ``` paulson@13323 ` 278` ``` --{*for well-founded recursion*} ``` wenzelm@13428 ` 279` ``` "[| L(r); L(f); L(g); L(a); L(b) |] ``` wenzelm@13428 ` 280` ``` ==> separation(L, ``` wenzelm@13428 ` 281` ``` \x. \xa[L]. \xb[L]. ``` wenzelm@13428 ` 282` ``` pair(L,x,a,xa) & xa \ r & pair(L,x,b,xb) & xb \ r & ``` wenzelm@13428 ` 283` ``` (\fx[L]. \gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) & ``` paulson@13323 ` 284` ``` fx \ gx))" ``` paulson@13687 ` 285` ```apply (rule gen_separation_multi [OF is_recfun_reflects, of "{r,f,g,a,b}"], ``` paulson@13687 ` 286` ``` auto) ``` paulson@13687 ` 287` ```apply (rule_tac env="[r,f,g,a,b]" in DPow_LsetI) ``` paulson@13323 ` 288` ```apply (rule sep_rules | simp)+ ``` paulson@13323 ` 289` ```done ``` paulson@13323 ` 290` paulson@13323 ` 291` paulson@13564 ` 292` ```subsection{*Instantiating the locale @{text M_basic}*} ``` paulson@13363 ` 293` ```text{*Separation (and Strong Replacement) for basic set-theoretic constructions ``` paulson@13363 ` 294` ```such as intersection, Cartesian Product and image.*} ``` paulson@13363 ` 295` paulson@13564 ` 296` ```lemma M_basic_axioms_L: "M_basic_axioms(L)" ``` paulson@13564 ` 297` ``` apply (rule M_basic_axioms.intro) ``` paulson@13437 ` 298` ``` apply (assumption | rule ``` paulson@13437 ` 299` ``` Inter_separation Diff_separation cartprod_separation image_separation ``` paulson@13437 ` 300` ``` converse_separation restrict_separation ``` paulson@13437 ` 301` ``` comp_separation pred_separation Memrel_separation ``` paulson@13634 ` 302` ``` funspace_succ_replacement is_recfun_separation)+ ``` wenzelm@13428 ` 303` ``` done ``` paulson@13323 ` 304` paulson@13564 ` 305` ```theorem M_basic_L: "PROP M_basic(L)" ``` paulson@13564 ` 306` ```by (rule M_basic.intro [OF M_trivial_L M_basic_axioms_L]) ``` paulson@13437 ` 307` ballarin@19931 ` 308` ```interpretation M_basic [L] by (rule M_basic_L) ``` paulson@13323 ` 309` paulson@13323 ` 310` paulson@13306 ` 311` ```end ```