paulson@13437: (* Title: ZF/Constructible/Separation.thy paulson@13437: ID: \$Id\$ paulson@13437: Author: Lawrence C Paulson, Cambridge University Computer Laboratory paulson@13437: *) paulson@13437: paulson@13339: header{*Early Instances of Separation and Strong Replacement*} paulson@13323: paulson@13324: theory Separation = L_axioms + WF_absolute: paulson@13306: paulson@13564: text{*This theory proves all instances needed for locale @{text "M_basic"}*} paulson@13339: paulson@13306: text{*Helps us solve for de Bruijn indices!*} paulson@13306: lemma nth_ConsI: "[|nth(n,l) = x; n \ nat|] ==> nth(succ(n), Cons(a,l)) = x" paulson@13306: by simp paulson@13306: paulson@13316: lemmas nth_rules = nth_0 nth_ConsI nat_0I nat_succI wenzelm@13428: lemmas sep_rules = nth_0 nth_ConsI FOL_iff_sats function_iff_sats paulson@13323: fun_plus_iff_sats paulson@13306: paulson@13306: lemma Collect_conj_in_DPow: wenzelm@13428: "[| {x\A. P(x)} \ DPow(A); {x\A. Q(x)} \ DPow(A) |] paulson@13306: ==> {x\A. P(x) & Q(x)} \ DPow(A)" wenzelm@13428: by (simp add: Int_in_DPow Collect_Int_Collect_eq [symmetric]) paulson@13306: paulson@13306: lemma Collect_conj_in_DPow_Lset: paulson@13306: "[|z \ Lset(j); {x \ Lset(j). P(x)} \ DPow(Lset(j))|] paulson@13306: ==> {x \ Lset(j). x \ z & P(x)} \ DPow(Lset(j))" paulson@13306: apply (frule mem_Lset_imp_subset_Lset) wenzelm@13428: apply (simp add: Collect_conj_in_DPow Collect_mem_eq paulson@13306: subset_Int_iff2 elem_subset_in_DPow) paulson@13306: done paulson@13306: paulson@13306: lemma separation_CollectI: paulson@13306: "(\z. L(z) ==> L({x \ z . P(x)})) ==> separation(L, \x. P(x))" wenzelm@13428: apply (unfold separation_def, clarify) wenzelm@13428: apply (rule_tac x="{x\z. P(x)}" in rexI) paulson@13306: apply simp_all paulson@13306: done paulson@13306: paulson@13306: text{*Reduces the original comprehension to the reflected one*} paulson@13306: lemma reflection_imp_L_separation: paulson@13306: "[| \x\Lset(j). P(x) <-> Q(x); wenzelm@13428: {x \ Lset(j) . Q(x)} \ DPow(Lset(j)); paulson@13306: Ord(j); z \ Lset(j)|] ==> L({x \ z . P(x)})" paulson@13306: apply (rule_tac i = "succ(j)" in L_I) paulson@13306: prefer 2 apply simp paulson@13306: apply (subgoal_tac "{x \ z. P(x)} = {x \ Lset(j). x \ z & (Q(x))}") paulson@13306: prefer 2 wenzelm@13428: apply (blast dest: mem_Lset_imp_subset_Lset) paulson@13306: apply (simp add: Lset_succ Collect_conj_in_DPow_Lset) paulson@13306: done paulson@13306: paulson@13566: text{*Encapsulates the standard proof script for proving instances of paulson@13687: Separation.*} paulson@13566: lemma gen_separation: paulson@13566: assumes reflection: "REFLECTS [P,Q]" paulson@13566: and Lu: "L(u)" paulson@13566: and collI: "!!j. u \ Lset(j) paulson@13566: \ Collect(Lset(j), Q(j)) \ DPow(Lset(j))" paulson@13566: shows "separation(L,P)" paulson@13566: apply (rule separation_CollectI) paulson@13566: apply (rule_tac A="{u,z}" in subset_LsetE, blast intro: Lu) paulson@13566: apply (rule ReflectsE [OF reflection], assumption) paulson@13566: apply (drule subset_Lset_ltD, assumption) paulson@13566: apply (erule reflection_imp_L_separation) paulson@13566: apply (simp_all add: lt_Ord2, clarify) paulson@13691: apply (rule collI, assumption) paulson@13687: done paulson@13687: paulson@13687: text{*As above, but typically @{term u} is a finite enumeration such as paulson@13687: @{term "{a,b}"}; thus the new subgoal gets the assumption paulson@13687: @{term "{a,b} \ Lset(i)"}, which is logically equivalent to paulson@13687: @{term "a \ Lset(i)"} and @{term "b \ Lset(i)"}.*} paulson@13687: lemma gen_separation_multi: paulson@13687: assumes reflection: "REFLECTS [P,Q]" paulson@13687: and Lu: "L(u)" paulson@13687: and collI: "!!j. u \ Lset(j) paulson@13687: \ Collect(Lset(j), Q(j)) \ DPow(Lset(j))" paulson@13687: shows "separation(L,P)" paulson@13687: apply (rule gen_separation [OF reflection Lu]) paulson@13687: apply (drule mem_Lset_imp_subset_Lset) paulson@13687: apply (erule collI) paulson@13566: done paulson@13566: paulson@13306: paulson@13316: subsection{*Separation for Intersection*} paulson@13306: paulson@13306: lemma Inter_Reflects: wenzelm@13428: "REFLECTS[\x. \y[L]. y\A --> x \ y, paulson@13314: \i x. \y\Lset(i). y\A --> x \ y]" wenzelm@13428: by (intro FOL_reflections) paulson@13306: paulson@13306: lemma Inter_separation: paulson@13306: "L(A) ==> separation(L, \x. \y[L]. y\A --> x\y)" paulson@13566: apply (rule gen_separation [OF Inter_Reflects], simp) wenzelm@13428: apply (rule DPow_LsetI) paulson@13687: txt{*I leave this one example of a manual proof. The tedium of manually paulson@13687: instantiating @{term i}, @{term j} and @{term env} is obvious. *} wenzelm@13428: apply (rule ball_iff_sats) paulson@13306: apply (rule imp_iff_sats) paulson@13306: apply (rule_tac [2] i=1 and j=0 and env="[y,x,A]" in mem_iff_sats) paulson@13306: apply (rule_tac i=0 and j=2 in mem_iff_sats) paulson@13306: apply (simp_all add: succ_Un_distrib [symmetric]) paulson@13306: done paulson@13306: paulson@13437: subsection{*Separation for Set Difference*} paulson@13437: paulson@13437: lemma Diff_Reflects: paulson@13437: "REFLECTS[\x. x \ B, \i x. x \ B]" paulson@13437: by (intro FOL_reflections) paulson@13437: paulson@13437: lemma Diff_separation: paulson@13437: "L(B) ==> separation(L, \x. x \ B)" paulson@13566: apply (rule gen_separation [OF Diff_Reflects], simp) paulson@13687: apply (rule_tac env="[B]" in DPow_LsetI) paulson@13437: apply (rule sep_rules | simp)+ paulson@13437: done paulson@13437: paulson@13316: subsection{*Separation for Cartesian Product*} paulson@13306: paulson@13323: lemma cartprod_Reflects: paulson@13314: "REFLECTS[\z. \x[L]. x\A & (\y[L]. y\B & pair(L,x,y,z)), wenzelm@13428: \i z. \x\Lset(i). x\A & (\y\Lset(i). y\B & paulson@13807: pair(##Lset(i),x,y,z))]" paulson@13323: by (intro FOL_reflections function_reflections) paulson@13306: paulson@13306: lemma cartprod_separation: wenzelm@13428: "[| L(A); L(B) |] paulson@13306: ==> separation(L, \z. \x[L]. x\A & (\y[L]. y\B & pair(L,x,y,z)))" paulson@13687: apply (rule gen_separation_multi [OF cartprod_Reflects, of "{A,B}"], auto) paulson@13687: apply (rule_tac env="[A,B]" in DPow_LsetI) paulson@13316: apply (rule sep_rules | simp)+ paulson@13306: done paulson@13306: paulson@13316: subsection{*Separation for Image*} paulson@13306: paulson@13306: lemma image_Reflects: paulson@13314: "REFLECTS[\y. \p[L]. p\r & (\x[L]. x\A & pair(L,x,y,p)), paulson@13807: \i y. \p\Lset(i). p\r & (\x\Lset(i). x\A & pair(##Lset(i),x,y,p))]" paulson@13323: by (intro FOL_reflections function_reflections) paulson@13306: paulson@13306: lemma image_separation: wenzelm@13428: "[| L(A); L(r) |] paulson@13306: ==> separation(L, \y. \p[L]. p\r & (\x[L]. x\A & pair(L,x,y,p)))" paulson@13687: apply (rule gen_separation_multi [OF image_Reflects, of "{A,r}"], auto) paulson@13687: apply (rule_tac env="[A,r]" in DPow_LsetI) paulson@13316: apply (rule sep_rules | simp)+ paulson@13306: done paulson@13306: paulson@13306: paulson@13316: subsection{*Separation for Converse*} paulson@13306: paulson@13306: lemma converse_Reflects: paulson@13314: "REFLECTS[\z. \p[L]. p\r & (\x[L]. \y[L]. pair(L,x,y,p) & pair(L,y,x,z)), wenzelm@13428: \i z. \p\Lset(i). p\r & (\x\Lset(i). \y\Lset(i). paulson@13807: pair(##Lset(i),x,y,p) & pair(##Lset(i),y,x,z))]" paulson@13323: by (intro FOL_reflections function_reflections) paulson@13306: paulson@13306: lemma converse_separation: wenzelm@13428: "L(r) ==> separation(L, paulson@13306: \z. \p[L]. p\r & (\x[L]. \y[L]. pair(L,x,y,p) & pair(L,y,x,z)))" paulson@13566: apply (rule gen_separation [OF converse_Reflects], simp) paulson@13687: apply (rule_tac env="[r]" in DPow_LsetI) paulson@13316: apply (rule sep_rules | simp)+ paulson@13306: done paulson@13306: paulson@13306: paulson@13316: subsection{*Separation for Restriction*} paulson@13306: paulson@13306: lemma restrict_Reflects: paulson@13314: "REFLECTS[\z. \x[L]. x\A & (\y[L]. pair(L,x,y,z)), paulson@13807: \i z. \x\Lset(i). x\A & (\y\Lset(i). pair(##Lset(i),x,y,z))]" paulson@13323: by (intro FOL_reflections function_reflections) paulson@13306: paulson@13306: lemma restrict_separation: paulson@13306: "L(A) ==> separation(L, \z. \x[L]. x\A & (\y[L]. pair(L,x,y,z)))" paulson@13566: apply (rule gen_separation [OF restrict_Reflects], simp) paulson@13687: apply (rule_tac env="[A]" in DPow_LsetI) paulson@13316: apply (rule sep_rules | simp)+ paulson@13306: done paulson@13306: paulson@13306: paulson@13316: subsection{*Separation for Composition*} paulson@13306: paulson@13306: lemma comp_Reflects: wenzelm@13428: "REFLECTS[\xz. \x[L]. \y[L]. \z[L]. \xy[L]. \yz[L]. wenzelm@13428: pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) & paulson@13306: xy\s & yz\r, wenzelm@13428: \i xz. \x\Lset(i). \y\Lset(i). \z\Lset(i). \xy\Lset(i). \yz\Lset(i). paulson@13807: pair(##Lset(i),x,z,xz) & pair(##Lset(i),x,y,xy) & paulson@13807: pair(##Lset(i),y,z,yz) & xy\s & yz\r]" paulson@13323: by (intro FOL_reflections function_reflections) paulson@13306: paulson@13306: lemma comp_separation: paulson@13306: "[| L(r); L(s) |] wenzelm@13428: ==> separation(L, \xz. \x[L]. \y[L]. \z[L]. \xy[L]. \yz[L]. wenzelm@13428: pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) & paulson@13306: xy\s & yz\r)" paulson@13687: apply (rule gen_separation_multi [OF comp_Reflects, of "{r,s}"], auto) paulson@13687: txt{*Subgoals after applying general ``separation'' rule: paulson@13687: @{subgoals[display,indent=0,margin=65]}*} paulson@13687: apply (rule_tac env="[r,s]" in DPow_LsetI) paulson@13687: txt{*Subgoals ready for automatic synthesis of a formula: paulson@13687: @{subgoals[display,indent=0,margin=65]}*} paulson@13316: apply (rule sep_rules | simp)+ paulson@13306: done paulson@13306: paulson@13687: paulson@13316: subsection{*Separation for Predecessors in an Order*} paulson@13306: paulson@13306: lemma pred_Reflects: paulson@13314: "REFLECTS[\y. \p[L]. p\r & pair(L,y,x,p), paulson@13807: \i y. \p \ Lset(i). p\r & pair(##Lset(i),y,x,p)]" paulson@13323: by (intro FOL_reflections function_reflections) paulson@13306: paulson@13306: lemma pred_separation: paulson@13306: "[| L(r); L(x) |] ==> separation(L, \y. \p[L]. p\r & pair(L,y,x,p))" paulson@13687: apply (rule gen_separation_multi [OF pred_Reflects, of "{r,x}"], auto) paulson@13687: apply (rule_tac env="[r,x]" in DPow_LsetI) paulson@13316: apply (rule sep_rules | simp)+ paulson@13306: done paulson@13306: paulson@13306: paulson@13316: subsection{*Separation for the Membership Relation*} paulson@13306: paulson@13306: lemma Memrel_Reflects: paulson@13314: "REFLECTS[\z. \x[L]. \y[L]. pair(L,x,y,z) & x \ y, paulson@13807: \i z. \x \ Lset(i). \y \ Lset(i). pair(##Lset(i),x,y,z) & x \ y]" paulson@13323: by (intro FOL_reflections function_reflections) paulson@13306: paulson@13306: lemma Memrel_separation: paulson@13306: "separation(L, \z. \x[L]. \y[L]. pair(L,x,y,z) & x \ y)" paulson@13566: apply (rule gen_separation [OF Memrel_Reflects nonempty]) paulson@13687: apply (rule_tac env="[]" in DPow_LsetI) paulson@13316: apply (rule sep_rules | simp)+ paulson@13306: done paulson@13306: paulson@13306: paulson@13316: subsection{*Replacement for FunSpace*} wenzelm@13428: paulson@13306: lemma funspace_succ_Reflects: wenzelm@13428: "REFLECTS[\z. \p[L]. p\A & (\f[L]. \b[L]. \nb[L]. \cnbf[L]. wenzelm@13428: pair(L,f,b,p) & pair(L,n,b,nb) & is_cons(L,nb,f,cnbf) & wenzelm@13428: upair(L,cnbf,cnbf,z)), wenzelm@13428: \i z. \p \ Lset(i). p\A & (\f \ Lset(i). \b \ Lset(i). wenzelm@13428: \nb \ Lset(i). \cnbf \ Lset(i). paulson@13807: pair(##Lset(i),f,b,p) & pair(##Lset(i),n,b,nb) & paulson@13807: is_cons(##Lset(i),nb,f,cnbf) & upair(##Lset(i),cnbf,cnbf,z))]" paulson@13323: by (intro FOL_reflections function_reflections) paulson@13306: paulson@13306: lemma funspace_succ_replacement: wenzelm@13428: "L(n) ==> wenzelm@13428: strong_replacement(L, \p z. \f[L]. \b[L]. \nb[L]. \cnbf[L]. paulson@13306: pair(L,f,b,p) & pair(L,n,b,nb) & is_cons(L,nb,f,cnbf) & paulson@13306: upair(L,cnbf,cnbf,z))" wenzelm@13428: apply (rule strong_replacementI) paulson@13687: apply (rule_tac u="{n,B}" in gen_separation_multi [OF funspace_succ_Reflects], paulson@13687: auto) paulson@13687: apply (rule_tac env="[n,B]" in DPow_LsetI) paulson@13316: apply (rule sep_rules | simp)+ paulson@13306: done paulson@13306: paulson@13306: paulson@13634: subsection{*Separation for a Theorem about @{term "is_recfun"}*} paulson@13323: paulson@13323: lemma is_recfun_reflects: wenzelm@13428: "REFLECTS[\x. \xa[L]. \xb[L]. wenzelm@13428: pair(L,x,a,xa) & xa \ r & pair(L,x,b,xb) & xb \ r & wenzelm@13428: (\fx[L]. \gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) & paulson@13323: fx \ gx), wenzelm@13428: \i x. \xa \ Lset(i). \xb \ Lset(i). paulson@13807: pair(##Lset(i),x,a,xa) & xa \ r & pair(##Lset(i),x,b,xb) & xb \ r & paulson@13807: (\fx \ Lset(i). \gx \ Lset(i). fun_apply(##Lset(i),f,x,fx) & paulson@13807: fun_apply(##Lset(i),g,x,gx) & fx \ gx)]" paulson@13323: by (intro FOL_reflections function_reflections fun_plus_reflections) paulson@13323: paulson@13323: lemma is_recfun_separation: paulson@13323: --{*for well-founded recursion*} wenzelm@13428: "[| L(r); L(f); L(g); L(a); L(b) |] wenzelm@13428: ==> separation(L, wenzelm@13428: \x. \xa[L]. \xb[L]. wenzelm@13428: pair(L,x,a,xa) & xa \ r & pair(L,x,b,xb) & xb \ r & wenzelm@13428: (\fx[L]. \gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) & paulson@13323: fx \ gx))" paulson@13687: apply (rule gen_separation_multi [OF is_recfun_reflects, of "{r,f,g,a,b}"], paulson@13687: auto) paulson@13687: apply (rule_tac env="[r,f,g,a,b]" in DPow_LsetI) paulson@13323: apply (rule sep_rules | simp)+ paulson@13323: done paulson@13323: paulson@13323: paulson@13564: subsection{*Instantiating the locale @{text M_basic}*} paulson@13363: text{*Separation (and Strong Replacement) for basic set-theoretic constructions paulson@13363: such as intersection, Cartesian Product and image.*} paulson@13363: paulson@13564: lemma M_basic_axioms_L: "M_basic_axioms(L)" paulson@13564: apply (rule M_basic_axioms.intro) paulson@13437: apply (assumption | rule paulson@13437: Inter_separation Diff_separation cartprod_separation image_separation paulson@13437: converse_separation restrict_separation paulson@13437: comp_separation pred_separation Memrel_separation paulson@13634: funspace_succ_replacement is_recfun_separation)+ wenzelm@13428: done paulson@13323: paulson@13564: theorem M_basic_L: "PROP M_basic(L)" paulson@13564: by (rule M_basic.intro [OF M_trivial_L M_basic_axioms_L]) paulson@13437: paulson@13437: paulson@13564: lemmas cartprod_iff = M_basic.cartprod_iff [OF M_basic_L] paulson@13564: and cartprod_closed = M_basic.cartprod_closed [OF M_basic_L] paulson@13564: and sum_closed = M_basic.sum_closed [OF M_basic_L] paulson@13564: and M_converse_iff = M_basic.M_converse_iff [OF M_basic_L] paulson@13564: and converse_closed = M_basic.converse_closed [OF M_basic_L] paulson@13564: and converse_abs = M_basic.converse_abs [OF M_basic_L] paulson@13564: and image_closed = M_basic.image_closed [OF M_basic_L] paulson@13564: and vimage_abs = M_basic.vimage_abs [OF M_basic_L] paulson@13564: and vimage_closed = M_basic.vimage_closed [OF M_basic_L] paulson@13564: and domain_abs = M_basic.domain_abs [OF M_basic_L] paulson@13564: and domain_closed = M_basic.domain_closed [OF M_basic_L] paulson@13564: and range_abs = M_basic.range_abs [OF M_basic_L] paulson@13564: and range_closed = M_basic.range_closed [OF M_basic_L] paulson@13564: and field_abs = M_basic.field_abs [OF M_basic_L] paulson@13564: and field_closed = M_basic.field_closed [OF M_basic_L] paulson@13564: and relation_abs = M_basic.relation_abs [OF M_basic_L] paulson@13564: and function_abs = M_basic.function_abs [OF M_basic_L] paulson@13564: and apply_closed = M_basic.apply_closed [OF M_basic_L] paulson@13564: and apply_abs = M_basic.apply_abs [OF M_basic_L] paulson@13564: and typed_function_abs = M_basic.typed_function_abs [OF M_basic_L] paulson@13564: and injection_abs = M_basic.injection_abs [OF M_basic_L] paulson@13564: and surjection_abs = M_basic.surjection_abs [OF M_basic_L] paulson@13564: and bijection_abs = M_basic.bijection_abs [OF M_basic_L] paulson@13564: and M_comp_iff = M_basic.M_comp_iff [OF M_basic_L] paulson@13564: and comp_closed = M_basic.comp_closed [OF M_basic_L] paulson@13564: and composition_abs = M_basic.composition_abs [OF M_basic_L] paulson@13564: and restriction_is_function = M_basic.restriction_is_function [OF M_basic_L] paulson@13564: and restriction_abs = M_basic.restriction_abs [OF M_basic_L] paulson@13564: and M_restrict_iff = M_basic.M_restrict_iff [OF M_basic_L] paulson@13564: and restrict_closed = M_basic.restrict_closed [OF M_basic_L] paulson@13564: and Inter_abs = M_basic.Inter_abs [OF M_basic_L] paulson@13564: and Inter_closed = M_basic.Inter_closed [OF M_basic_L] paulson@13564: and Int_closed = M_basic.Int_closed [OF M_basic_L] paulson@13564: and is_funspace_abs = M_basic.is_funspace_abs [OF M_basic_L] paulson@13564: and succ_fun_eq2 = M_basic.succ_fun_eq2 [OF M_basic_L] paulson@13564: and funspace_succ = M_basic.funspace_succ [OF M_basic_L] paulson@13564: and finite_funspace_closed = M_basic.finite_funspace_closed [OF M_basic_L] paulson@13323: paulson@13564: lemmas is_recfun_equal = M_basic.is_recfun_equal [OF M_basic_L] paulson@13564: and is_recfun_cut = M_basic.is_recfun_cut [OF M_basic_L] paulson@13564: and is_recfun_functional = M_basic.is_recfun_functional [OF M_basic_L] paulson@13564: and is_recfun_relativize = M_basic.is_recfun_relativize [OF M_basic_L] paulson@13564: and is_recfun_restrict = M_basic.is_recfun_restrict [OF M_basic_L] paulson@13564: and univalent_is_recfun = M_basic.univalent_is_recfun [OF M_basic_L] paulson@13564: and wellfounded_exists_is_recfun = M_basic.wellfounded_exists_is_recfun [OF M_basic_L] paulson@13564: and wf_exists_is_recfun = M_basic.wf_exists_is_recfun [OF M_basic_L] paulson@13564: and is_recfun_abs = M_basic.is_recfun_abs [OF M_basic_L] paulson@13564: and irreflexive_abs = M_basic.irreflexive_abs [OF M_basic_L] paulson@13564: and transitive_rel_abs = M_basic.transitive_rel_abs [OF M_basic_L] paulson@13564: and linear_rel_abs = M_basic.linear_rel_abs [OF M_basic_L] paulson@13564: and wellordered_is_trans_on = M_basic.wellordered_is_trans_on [OF M_basic_L] paulson@13564: and wellordered_is_linear = M_basic.wellordered_is_linear [OF M_basic_L] paulson@13564: and wellordered_is_wellfounded_on = M_basic.wellordered_is_wellfounded_on [OF M_basic_L] paulson@13564: and wellfounded_imp_wellfounded_on = M_basic.wellfounded_imp_wellfounded_on [OF M_basic_L] paulson@13564: and wellfounded_on_subset_A = M_basic.wellfounded_on_subset_A [OF M_basic_L] paulson@13564: and wellfounded_on_iff_wellfounded = M_basic.wellfounded_on_iff_wellfounded [OF M_basic_L] paulson@13564: and wellfounded_on_imp_wellfounded = M_basic.wellfounded_on_imp_wellfounded [OF M_basic_L] paulson@13564: and wellfounded_on_field_imp_wellfounded = M_basic.wellfounded_on_field_imp_wellfounded [OF M_basic_L] paulson@13564: and wellfounded_iff_wellfounded_on_field = M_basic.wellfounded_iff_wellfounded_on_field [OF M_basic_L] paulson@13564: and wellfounded_induct = M_basic.wellfounded_induct [OF M_basic_L] paulson@13564: and wellfounded_on_induct = M_basic.wellfounded_on_induct [OF M_basic_L] paulson@13564: and linear_imp_relativized = M_basic.linear_imp_relativized [OF M_basic_L] paulson@13564: and trans_on_imp_relativized = M_basic.trans_on_imp_relativized [OF M_basic_L] paulson@13564: and wf_on_imp_relativized = M_basic.wf_on_imp_relativized [OF M_basic_L] paulson@13564: and wf_imp_relativized = M_basic.wf_imp_relativized [OF M_basic_L] paulson@13564: and well_ord_imp_relativized = M_basic.well_ord_imp_relativized [OF M_basic_L] paulson@13564: and order_isomorphism_abs = M_basic.order_isomorphism_abs [OF M_basic_L] paulson@13564: and pred_set_abs = M_basic.pred_set_abs [OF M_basic_L] paulson@13323: paulson@13564: lemmas pred_closed = M_basic.pred_closed [OF M_basic_L] paulson@13564: and membership_abs = M_basic.membership_abs [OF M_basic_L] paulson@13564: and M_Memrel_iff = M_basic.M_Memrel_iff [OF M_basic_L] paulson@13564: and Memrel_closed = M_basic.Memrel_closed [OF M_basic_L] paulson@13564: and wellfounded_on_asym = M_basic.wellfounded_on_asym [OF M_basic_L] paulson@13564: and wellordered_asym = M_basic.wellordered_asym [OF M_basic_L] wenzelm@13428: wenzelm@13429: declare cartprod_closed [intro, simp] wenzelm@13429: declare sum_closed [intro, simp] wenzelm@13429: declare converse_closed [intro, simp] paulson@13323: declare converse_abs [simp] wenzelm@13429: declare image_closed [intro, simp] paulson@13323: declare vimage_abs [simp] wenzelm@13429: declare vimage_closed [intro, simp] paulson@13323: declare domain_abs [simp] wenzelm@13429: declare domain_closed [intro, simp] paulson@13323: declare range_abs [simp] wenzelm@13429: declare range_closed [intro, simp] paulson@13323: declare field_abs [simp] wenzelm@13429: declare field_closed [intro, simp] paulson@13323: declare relation_abs [simp] paulson@13323: declare function_abs [simp] wenzelm@13429: declare apply_closed [intro, simp] paulson@13323: declare typed_function_abs [simp] paulson@13323: declare injection_abs [simp] paulson@13323: declare surjection_abs [simp] paulson@13323: declare bijection_abs [simp] wenzelm@13429: declare comp_closed [intro, simp] paulson@13323: declare composition_abs [simp] paulson@13323: declare restriction_abs [simp] wenzelm@13429: declare restrict_closed [intro, simp] paulson@13323: declare Inter_abs [simp] wenzelm@13429: declare Inter_closed [intro, simp] wenzelm@13429: declare Int_closed [intro, simp] paulson@13323: declare is_funspace_abs [simp] wenzelm@13429: declare finite_funspace_closed [intro, simp] paulson@13440: declare membership_abs [simp] paulson@13440: declare Memrel_closed [intro,simp] paulson@13323: paulson@13306: end