(* Title: ZF/Constructible/Separation.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 2002 University of Cambridge
*)
header{*Early Instances of Separation and Strong Replacement*}
theory Separation = L_axioms + WF_absolute:
text{*This theory proves all instances needed for locale @{text "M_basic"}*}
text{*Helps us solve for de Bruijn indices!*}
lemma nth_ConsI: "[|nth(n,l) = x; n \<in> nat|] ==> nth(succ(n), Cons(a,l)) = x"
by simp
lemmas nth_rules = nth_0 nth_ConsI nat_0I nat_succI
lemmas sep_rules = nth_0 nth_ConsI FOL_iff_sats function_iff_sats
fun_plus_iff_sats
lemma Collect_conj_in_DPow:
"[| {x\<in>A. P(x)} \<in> DPow(A); {x\<in>A. Q(x)} \<in> DPow(A) |]
==> {x\<in>A. P(x) & Q(x)} \<in> DPow(A)"
by (simp add: Int_in_DPow Collect_Int_Collect_eq [symmetric])
lemma Collect_conj_in_DPow_Lset:
"[|z \<in> Lset(j); {x \<in> Lset(j). P(x)} \<in> DPow(Lset(j))|]
==> {x \<in> Lset(j). x \<in> z & P(x)} \<in> DPow(Lset(j))"
apply (frule mem_Lset_imp_subset_Lset)
apply (simp add: Collect_conj_in_DPow Collect_mem_eq
subset_Int_iff2 elem_subset_in_DPow)
done
lemma separation_CollectI:
"(\<And>z. L(z) ==> L({x \<in> z . P(x)})) ==> separation(L, \<lambda>x. P(x))"
apply (unfold separation_def, clarify)
apply (rule_tac x="{x\<in>z. P(x)}" in rexI)
apply simp_all
done
text{*Reduces the original comprehension to the reflected one*}
lemma reflection_imp_L_separation:
"[| \<forall>x\<in>Lset(j). P(x) <-> Q(x);
{x \<in> Lset(j) . Q(x)} \<in> DPow(Lset(j));
Ord(j); z \<in> Lset(j)|] ==> L({x \<in> z . P(x)})"
apply (rule_tac i = "succ(j)" in L_I)
prefer 2 apply simp
apply (subgoal_tac "{x \<in> z. P(x)} = {x \<in> Lset(j). x \<in> z & (Q(x))}")
prefer 2
apply (blast dest: mem_Lset_imp_subset_Lset)
apply (simp add: Lset_succ Collect_conj_in_DPow_Lset)
done
subsection{*Separation for Intersection*}
lemma Inter_Reflects:
"REFLECTS[\<lambda>x. \<forall>y[L]. y\<in>A --> x \<in> y,
\<lambda>i x. \<forall>y\<in>Lset(i). y\<in>A --> x \<in> y]"
by (intro FOL_reflections)
lemma Inter_separation:
"L(A) ==> separation(L, \<lambda>x. \<forall>y[L]. y\<in>A --> x\<in>y)"
apply (rule separation_CollectI)
apply (rule_tac A="{A,z}" in subset_LsetE, blast)
apply (rule ReflectsE [OF Inter_Reflects], assumption)
apply (drule subset_Lset_ltD, assumption)
apply (erule reflection_imp_L_separation)
apply (simp_all add: lt_Ord2, clarify)
apply (rule DPow_LsetI)
apply (rule ball_iff_sats)
apply (rule imp_iff_sats)
apply (rule_tac [2] i=1 and j=0 and env="[y,x,A]" in mem_iff_sats)
apply (rule_tac i=0 and j=2 in mem_iff_sats)
apply (simp_all add: succ_Un_distrib [symmetric])
done
subsection{*Separation for Set Difference*}
lemma Diff_Reflects:
"REFLECTS[\<lambda>x. x \<notin> B, \<lambda>i x. x \<notin> B]"
by (intro FOL_reflections)
lemma Diff_separation:
"L(B) ==> separation(L, \<lambda>x. x \<notin> B)"
apply (rule separation_CollectI)
apply (rule_tac A="{B,z}" in subset_LsetE, blast)
apply (rule ReflectsE [OF Diff_Reflects], assumption)
apply (drule subset_Lset_ltD, assumption)
apply (erule reflection_imp_L_separation)
apply (simp_all add: lt_Ord2, clarify)
apply (rule DPow_LsetI)
apply (rule not_iff_sats)
apply (rule_tac env="[x,B]" in mem_iff_sats)
apply (rule sep_rules | simp)+
done
subsection{*Separation for Cartesian Product*}
lemma cartprod_Reflects:
"REFLECTS[\<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. y\<in>B & pair(L,x,y,z)),
\<lambda>i z. \<exists>x\<in>Lset(i). x\<in>A & (\<exists>y\<in>Lset(i). y\<in>B &
pair(**Lset(i),x,y,z))]"
by (intro FOL_reflections function_reflections)
lemma cartprod_separation:
"[| L(A); L(B) |]
==> separation(L, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. y\<in>B & pair(L,x,y,z)))"
apply (rule separation_CollectI)
apply (rule_tac A="{A,B,z}" in subset_LsetE, blast)
apply (rule ReflectsE [OF cartprod_Reflects], assumption)
apply (drule subset_Lset_ltD, assumption)
apply (erule reflection_imp_L_separation)
apply (simp_all add: lt_Ord2, clarify)
apply (rule DPow_LsetI)
apply (rename_tac u)
apply (rule bex_iff_sats)
apply (rule conj_iff_sats)
apply (rule_tac i=0 and j=2 and env="[x,u,A,B]" in mem_iff_sats, simp_all)
apply (rule sep_rules | simp)+
done
subsection{*Separation for Image*}
lemma image_Reflects:
"REFLECTS[\<lambda>y. \<exists>p[L]. p\<in>r & (\<exists>x[L]. x\<in>A & pair(L,x,y,p)),
\<lambda>i y. \<exists>p\<in>Lset(i). p\<in>r & (\<exists>x\<in>Lset(i). x\<in>A & pair(**Lset(i),x,y,p))]"
by (intro FOL_reflections function_reflections)
lemma image_separation:
"[| L(A); L(r) |]
==> separation(L, \<lambda>y. \<exists>p[L]. p\<in>r & (\<exists>x[L]. x\<in>A & pair(L,x,y,p)))"
apply (rule separation_CollectI)
apply (rule_tac A="{A,r,z}" in subset_LsetE, blast)
apply (rule ReflectsE [OF image_Reflects], assumption)
apply (drule subset_Lset_ltD, assumption)
apply (erule reflection_imp_L_separation)
apply (simp_all add: lt_Ord2, clarify)
apply (rule DPow_LsetI)
apply (rule bex_iff_sats)
apply (rule conj_iff_sats)
apply (rule_tac env="[p,y,A,r]" in mem_iff_sats)
apply (rule sep_rules | simp)+
done
subsection{*Separation for Converse*}
lemma converse_Reflects:
"REFLECTS[\<lambda>z. \<exists>p[L]. p\<in>r & (\<exists>x[L]. \<exists>y[L]. pair(L,x,y,p) & pair(L,y,x,z)),
\<lambda>i z. \<exists>p\<in>Lset(i). p\<in>r & (\<exists>x\<in>Lset(i). \<exists>y\<in>Lset(i).
pair(**Lset(i),x,y,p) & pair(**Lset(i),y,x,z))]"
by (intro FOL_reflections function_reflections)
lemma converse_separation:
"L(r) ==> separation(L,
\<lambda>z. \<exists>p[L]. p\<in>r & (\<exists>x[L]. \<exists>y[L]. pair(L,x,y,p) & pair(L,y,x,z)))"
apply (rule separation_CollectI)
apply (rule_tac A="{r,z}" in subset_LsetE, blast)
apply (rule ReflectsE [OF converse_Reflects], assumption)
apply (drule subset_Lset_ltD, assumption)
apply (erule reflection_imp_L_separation)
apply (simp_all add: lt_Ord2, clarify)
apply (rule DPow_LsetI)
apply (rename_tac u)
apply (rule bex_iff_sats)
apply (rule conj_iff_sats)
apply (rule_tac i=0 and j=2 and env="[p,u,r]" in mem_iff_sats, simp_all)
apply (rule sep_rules | simp)+
done
subsection{*Separation for Restriction*}
lemma restrict_Reflects:
"REFLECTS[\<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. pair(L,x,y,z)),
\<lambda>i z. \<exists>x\<in>Lset(i). x\<in>A & (\<exists>y\<in>Lset(i). pair(**Lset(i),x,y,z))]"
by (intro FOL_reflections function_reflections)
lemma restrict_separation:
"L(A) ==> separation(L, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. pair(L,x,y,z)))"
apply (rule separation_CollectI)
apply (rule_tac A="{A,z}" in subset_LsetE, blast)
apply (rule ReflectsE [OF restrict_Reflects], assumption)
apply (drule subset_Lset_ltD, assumption)
apply (erule reflection_imp_L_separation)
apply (simp_all add: lt_Ord2, clarify)
apply (rule DPow_LsetI)
apply (rename_tac u)
apply (rule bex_iff_sats)
apply (rule conj_iff_sats)
apply (rule_tac i=0 and j=2 and env="[x,u,A]" in mem_iff_sats, simp_all)
apply (rule sep_rules | simp)+
done
subsection{*Separation for Composition*}
lemma comp_Reflects:
"REFLECTS[\<lambda>xz. \<exists>x[L]. \<exists>y[L]. \<exists>z[L]. \<exists>xy[L]. \<exists>yz[L].
pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) &
xy\<in>s & yz\<in>r,
\<lambda>i xz. \<exists>x\<in>Lset(i). \<exists>y\<in>Lset(i). \<exists>z\<in>Lset(i). \<exists>xy\<in>Lset(i). \<exists>yz\<in>Lset(i).
pair(**Lset(i),x,z,xz) & pair(**Lset(i),x,y,xy) &
pair(**Lset(i),y,z,yz) & xy\<in>s & yz\<in>r]"
by (intro FOL_reflections function_reflections)
lemma comp_separation:
"[| L(r); L(s) |]
==> separation(L, \<lambda>xz. \<exists>x[L]. \<exists>y[L]. \<exists>z[L]. \<exists>xy[L]. \<exists>yz[L].
pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) &
xy\<in>s & yz\<in>r)"
apply (rule separation_CollectI)
apply (rule_tac A="{r,s,z}" in subset_LsetE, blast)
apply (rule ReflectsE [OF comp_Reflects], assumption)
apply (drule subset_Lset_ltD, assumption)
apply (erule reflection_imp_L_separation)
apply (simp_all add: lt_Ord2, clarify)
apply (rule DPow_LsetI)
apply (rename_tac u)
apply (rule bex_iff_sats)+
apply (rename_tac x y z)
apply (rule conj_iff_sats)
apply (rule_tac env="[z,y,x,u,r,s]" in pair_iff_sats)
apply (rule sep_rules | simp)+
done
subsection{*Separation for Predecessors in an Order*}
lemma pred_Reflects:
"REFLECTS[\<lambda>y. \<exists>p[L]. p\<in>r & pair(L,y,x,p),
\<lambda>i y. \<exists>p \<in> Lset(i). p\<in>r & pair(**Lset(i),y,x,p)]"
by (intro FOL_reflections function_reflections)
lemma pred_separation:
"[| L(r); L(x) |] ==> separation(L, \<lambda>y. \<exists>p[L]. p\<in>r & pair(L,y,x,p))"
apply (rule separation_CollectI)
apply (rule_tac A="{r,x,z}" in subset_LsetE, blast)
apply (rule ReflectsE [OF pred_Reflects], assumption)
apply (drule subset_Lset_ltD, assumption)
apply (erule reflection_imp_L_separation)
apply (simp_all add: lt_Ord2, clarify)
apply (rule DPow_LsetI)
apply (rename_tac u)
apply (rule bex_iff_sats)
apply (rule conj_iff_sats)
apply (rule_tac env = "[p,u,r,x]" in mem_iff_sats)
apply (rule sep_rules | simp)+
done
subsection{*Separation for the Membership Relation*}
lemma Memrel_Reflects:
"REFLECTS[\<lambda>z. \<exists>x[L]. \<exists>y[L]. pair(L,x,y,z) & x \<in> y,
\<lambda>i z. \<exists>x \<in> Lset(i). \<exists>y \<in> Lset(i). pair(**Lset(i),x,y,z) & x \<in> y]"
by (intro FOL_reflections function_reflections)
lemma Memrel_separation:
"separation(L, \<lambda>z. \<exists>x[L]. \<exists>y[L]. pair(L,x,y,z) & x \<in> y)"
apply (rule separation_CollectI)
apply (rule_tac A="{z}" in subset_LsetE, blast)
apply (rule ReflectsE [OF Memrel_Reflects], assumption)
apply (drule subset_Lset_ltD, assumption)
apply (erule reflection_imp_L_separation)
apply (simp_all add: lt_Ord2)
apply (rule DPow_LsetI)
apply (rename_tac u)
apply (rule bex_iff_sats conj_iff_sats)+
apply (rule_tac env = "[y,x,u]" in pair_iff_sats)
apply (rule sep_rules | simp)+
done
subsection{*Replacement for FunSpace*}
lemma funspace_succ_Reflects:
"REFLECTS[\<lambda>z. \<exists>p[L]. p\<in>A & (\<exists>f[L]. \<exists>b[L]. \<exists>nb[L]. \<exists>cnbf[L].
pair(L,f,b,p) & pair(L,n,b,nb) & is_cons(L,nb,f,cnbf) &
upair(L,cnbf,cnbf,z)),
\<lambda>i z. \<exists>p \<in> Lset(i). p\<in>A & (\<exists>f \<in> Lset(i). \<exists>b \<in> Lset(i).
\<exists>nb \<in> Lset(i). \<exists>cnbf \<in> Lset(i).
pair(**Lset(i),f,b,p) & pair(**Lset(i),n,b,nb) &
is_cons(**Lset(i),nb,f,cnbf) & upair(**Lset(i),cnbf,cnbf,z))]"
by (intro FOL_reflections function_reflections)
lemma funspace_succ_replacement:
"L(n) ==>
strong_replacement(L, \<lambda>p z. \<exists>f[L]. \<exists>b[L]. \<exists>nb[L]. \<exists>cnbf[L].
pair(L,f,b,p) & pair(L,n,b,nb) & is_cons(L,nb,f,cnbf) &
upair(L,cnbf,cnbf,z))"
apply (rule strong_replacementI)
apply (rule rallI)
apply (rule separation_CollectI)
apply (rule_tac A="{n,A,z}" in subset_LsetE, blast)
apply (rule ReflectsE [OF funspace_succ_Reflects], assumption)
apply (drule subset_Lset_ltD, assumption)
apply (erule reflection_imp_L_separation)
apply (simp_all add: lt_Ord2)
apply (rule DPow_LsetI)
apply (rename_tac u)
apply (rule bex_iff_sats)
apply (rule conj_iff_sats)
apply (rule_tac env = "[p,u,n,A]" in mem_iff_sats)
apply (rule sep_rules | simp)+
done
subsection{*Separation for Order-Isomorphisms*}
lemma well_ord_iso_Reflects:
"REFLECTS[\<lambda>x. x\<in>A -->
(\<exists>y[L]. \<exists>p[L]. fun_apply(L,f,x,y) & pair(L,y,x,p) & p \<in> r),
\<lambda>i x. x\<in>A --> (\<exists>y \<in> Lset(i). \<exists>p \<in> Lset(i).
fun_apply(**Lset(i),f,x,y) & pair(**Lset(i),y,x,p) & p \<in> r)]"
by (intro FOL_reflections function_reflections)
lemma well_ord_iso_separation:
"[| L(A); L(f); L(r) |]
==> separation (L, \<lambda>x. x\<in>A --> (\<exists>y[L]. (\<exists>p[L].
fun_apply(L,f,x,y) & pair(L,y,x,p) & p \<in> r)))"
apply (rule separation_CollectI)
apply (rule_tac A="{A,f,r,z}" in subset_LsetE, blast)
apply (rule ReflectsE [OF well_ord_iso_Reflects], assumption)
apply (drule subset_Lset_ltD, assumption)
apply (erule reflection_imp_L_separation)
apply (simp_all add: lt_Ord2)
apply (rule DPow_LsetI)
apply (rename_tac u)
apply (rule imp_iff_sats)
apply (rule_tac env = "[u,A,f,r]" in mem_iff_sats)
apply (rule sep_rules | simp)+
done
subsection{*Separation for @{term "obase"}*}
lemma obase_reflects:
"REFLECTS[\<lambda>a. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L].
ordinal(L,x) & membership(L,x,mx) & pred_set(L,A,a,r,par) &
order_isomorphism(L,par,r,x,mx,g),
\<lambda>i a. \<exists>x \<in> Lset(i). \<exists>g \<in> Lset(i). \<exists>mx \<in> Lset(i). \<exists>par \<in> Lset(i).
ordinal(**Lset(i),x) & membership(**Lset(i),x,mx) & pred_set(**Lset(i),A,a,r,par) &
order_isomorphism(**Lset(i),par,r,x,mx,g)]"
by (intro FOL_reflections function_reflections fun_plus_reflections)
lemma obase_separation:
--{*part of the order type formalization*}
"[| L(A); L(r) |]
==> separation(L, \<lambda>a. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L].
ordinal(L,x) & membership(L,x,mx) & pred_set(L,A,a,r,par) &
order_isomorphism(L,par,r,x,mx,g))"
apply (rule separation_CollectI)
apply (rule_tac A="{A,r,z}" in subset_LsetE, blast)
apply (rule ReflectsE [OF obase_reflects], assumption)
apply (drule subset_Lset_ltD, assumption)
apply (erule reflection_imp_L_separation)
apply (simp_all add: lt_Ord2)
apply (rule DPow_LsetI)
apply (rename_tac u)
apply (rule bex_iff_sats)
apply (rule conj_iff_sats)
apply (rule_tac env = "[x,u,A,r]" in ordinal_iff_sats)
apply (rule sep_rules | simp)+
done
subsection{*Separation for a Theorem about @{term "obase"}*}
lemma obase_equals_reflects:
"REFLECTS[\<lambda>x. x\<in>A --> ~(\<exists>y[L]. \<exists>g[L].
ordinal(L,y) & (\<exists>my[L]. \<exists>pxr[L].
membership(L,y,my) & pred_set(L,A,x,r,pxr) &
order_isomorphism(L,pxr,r,y,my,g))),
\<lambda>i x. x\<in>A --> ~(\<exists>y \<in> Lset(i). \<exists>g \<in> Lset(i).
ordinal(**Lset(i),y) & (\<exists>my \<in> Lset(i). \<exists>pxr \<in> Lset(i).
membership(**Lset(i),y,my) & pred_set(**Lset(i),A,x,r,pxr) &
order_isomorphism(**Lset(i),pxr,r,y,my,g)))]"
by (intro FOL_reflections function_reflections fun_plus_reflections)
lemma obase_equals_separation:
"[| L(A); L(r) |]
==> separation (L, \<lambda>x. x\<in>A --> ~(\<exists>y[L]. \<exists>g[L].
ordinal(L,y) & (\<exists>my[L]. \<exists>pxr[L].
membership(L,y,my) & pred_set(L,A,x,r,pxr) &
order_isomorphism(L,pxr,r,y,my,g))))"
apply (rule separation_CollectI)
apply (rule_tac A="{A,r,z}" in subset_LsetE, blast)
apply (rule ReflectsE [OF obase_equals_reflects], assumption)
apply (drule subset_Lset_ltD, assumption)
apply (erule reflection_imp_L_separation)
apply (simp_all add: lt_Ord2)
apply (rule DPow_LsetI)
apply (rename_tac u)
apply (rule imp_iff_sats ball_iff_sats disj_iff_sats not_iff_sats)+
apply (rule_tac env = "[u,A,r]" in mem_iff_sats)
apply (rule sep_rules | simp)+
done
subsection{*Replacement for @{term "omap"}*}
lemma omap_reflects:
"REFLECTS[\<lambda>z. \<exists>a[L]. a\<in>B & (\<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L].
ordinal(L,x) & pair(L,a,x,z) & membership(L,x,mx) &
pred_set(L,A,a,r,par) & order_isomorphism(L,par,r,x,mx,g)),
\<lambda>i z. \<exists>a \<in> Lset(i). a\<in>B & (\<exists>x \<in> Lset(i). \<exists>g \<in> Lset(i). \<exists>mx \<in> Lset(i).
\<exists>par \<in> Lset(i).
ordinal(**Lset(i),x) & pair(**Lset(i),a,x,z) &
membership(**Lset(i),x,mx) & pred_set(**Lset(i),A,a,r,par) &
order_isomorphism(**Lset(i),par,r,x,mx,g))]"
by (intro FOL_reflections function_reflections fun_plus_reflections)
lemma omap_replacement:
"[| L(A); L(r) |]
==> strong_replacement(L,
\<lambda>a z. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L].
ordinal(L,x) & pair(L,a,x,z) & membership(L,x,mx) &
pred_set(L,A,a,r,par) & order_isomorphism(L,par,r,x,mx,g))"
apply (rule strong_replacementI)
apply (rule rallI)
apply (rename_tac B)
apply (rule separation_CollectI)
apply (rule_tac A="{A,B,r,z}" in subset_LsetE, blast)
apply (rule ReflectsE [OF omap_reflects], assumption)
apply (drule subset_Lset_ltD, assumption)
apply (erule reflection_imp_L_separation)
apply (simp_all add: lt_Ord2)
apply (rule DPow_LsetI)
apply (rename_tac u)
apply (rule bex_iff_sats conj_iff_sats)+
apply (rule_tac env = "[a,u,A,B,r]" in mem_iff_sats)
apply (rule sep_rules | simp)+
done
subsection{*Separation for a Theorem about @{term "obase"}*}
lemma is_recfun_reflects:
"REFLECTS[\<lambda>x. \<exists>xa[L]. \<exists>xb[L].
pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r &
(\<exists>fx[L]. \<exists>gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) &
fx \<noteq> gx),
\<lambda>i x. \<exists>xa \<in> Lset(i). \<exists>xb \<in> Lset(i).
pair(**Lset(i),x,a,xa) & xa \<in> r & pair(**Lset(i),x,b,xb) & xb \<in> r &
(\<exists>fx \<in> Lset(i). \<exists>gx \<in> Lset(i). fun_apply(**Lset(i),f,x,fx) &
fun_apply(**Lset(i),g,x,gx) & fx \<noteq> gx)]"
by (intro FOL_reflections function_reflections fun_plus_reflections)
lemma is_recfun_separation:
--{*for well-founded recursion*}
"[| L(r); L(f); L(g); L(a); L(b) |]
==> separation(L,
\<lambda>x. \<exists>xa[L]. \<exists>xb[L].
pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r &
(\<exists>fx[L]. \<exists>gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) &
fx \<noteq> gx))"
apply (rule separation_CollectI)
apply (rule_tac A="{r,f,g,a,b,z}" in subset_LsetE, blast)
apply (rule ReflectsE [OF is_recfun_reflects], assumption)
apply (drule subset_Lset_ltD, assumption)
apply (erule reflection_imp_L_separation)
apply (simp_all add: lt_Ord2)
apply (rule DPow_LsetI)
apply (rename_tac u)
apply (rule bex_iff_sats conj_iff_sats)+
apply (rule_tac env = "[xa,u,r,f,g,a,b]" in pair_iff_sats)
apply (rule sep_rules | simp)+
done
subsection{*Instantiating the locale @{text M_basic}*}
text{*Separation (and Strong Replacement) for basic set-theoretic constructions
such as intersection, Cartesian Product and image.*}
lemma M_basic_axioms_L: "M_basic_axioms(L)"
apply (rule M_basic_axioms.intro)
apply (assumption | rule
Inter_separation Diff_separation cartprod_separation image_separation
converse_separation restrict_separation
comp_separation pred_separation Memrel_separation
funspace_succ_replacement well_ord_iso_separation
obase_separation obase_equals_separation
omap_replacement is_recfun_separation)+
done
theorem M_basic_L: "PROP M_basic(L)"
by (rule M_basic.intro [OF M_trivial_L M_basic_axioms_L])
lemmas cartprod_iff = M_basic.cartprod_iff [OF M_basic_L]
and cartprod_closed = M_basic.cartprod_closed [OF M_basic_L]
and sum_closed = M_basic.sum_closed [OF M_basic_L]
and M_converse_iff = M_basic.M_converse_iff [OF M_basic_L]
and converse_closed = M_basic.converse_closed [OF M_basic_L]
and converse_abs = M_basic.converse_abs [OF M_basic_L]
and image_closed = M_basic.image_closed [OF M_basic_L]
and vimage_abs = M_basic.vimage_abs [OF M_basic_L]
and vimage_closed = M_basic.vimage_closed [OF M_basic_L]
and domain_abs = M_basic.domain_abs [OF M_basic_L]
and domain_closed = M_basic.domain_closed [OF M_basic_L]
and range_abs = M_basic.range_abs [OF M_basic_L]
and range_closed = M_basic.range_closed [OF M_basic_L]
and field_abs = M_basic.field_abs [OF M_basic_L]
and field_closed = M_basic.field_closed [OF M_basic_L]
and relation_abs = M_basic.relation_abs [OF M_basic_L]
and function_abs = M_basic.function_abs [OF M_basic_L]
and apply_closed = M_basic.apply_closed [OF M_basic_L]
and apply_abs = M_basic.apply_abs [OF M_basic_L]
and typed_function_abs = M_basic.typed_function_abs [OF M_basic_L]
and injection_abs = M_basic.injection_abs [OF M_basic_L]
and surjection_abs = M_basic.surjection_abs [OF M_basic_L]
and bijection_abs = M_basic.bijection_abs [OF M_basic_L]
and M_comp_iff = M_basic.M_comp_iff [OF M_basic_L]
and comp_closed = M_basic.comp_closed [OF M_basic_L]
and composition_abs = M_basic.composition_abs [OF M_basic_L]
and restriction_is_function = M_basic.restriction_is_function [OF M_basic_L]
and restriction_abs = M_basic.restriction_abs [OF M_basic_L]
and M_restrict_iff = M_basic.M_restrict_iff [OF M_basic_L]
and restrict_closed = M_basic.restrict_closed [OF M_basic_L]
and Inter_abs = M_basic.Inter_abs [OF M_basic_L]
and Inter_closed = M_basic.Inter_closed [OF M_basic_L]
and Int_closed = M_basic.Int_closed [OF M_basic_L]
and finite_fun_closed = M_basic.finite_fun_closed [OF M_basic_L]
and is_funspace_abs = M_basic.is_funspace_abs [OF M_basic_L]
and succ_fun_eq2 = M_basic.succ_fun_eq2 [OF M_basic_L]
and funspace_succ = M_basic.funspace_succ [OF M_basic_L]
and finite_funspace_closed = M_basic.finite_funspace_closed [OF M_basic_L]
lemmas is_recfun_equal = M_basic.is_recfun_equal [OF M_basic_L]
and is_recfun_cut = M_basic.is_recfun_cut [OF M_basic_L]
and is_recfun_functional = M_basic.is_recfun_functional [OF M_basic_L]
and is_recfun_relativize = M_basic.is_recfun_relativize [OF M_basic_L]
and is_recfun_restrict = M_basic.is_recfun_restrict [OF M_basic_L]
and univalent_is_recfun = M_basic.univalent_is_recfun [OF M_basic_L]
and exists_is_recfun_indstep = M_basic.exists_is_recfun_indstep [OF M_basic_L]
and wellfounded_exists_is_recfun = M_basic.wellfounded_exists_is_recfun [OF M_basic_L]
and wf_exists_is_recfun = M_basic.wf_exists_is_recfun [OF M_basic_L]
and is_recfun_abs = M_basic.is_recfun_abs [OF M_basic_L]
and irreflexive_abs = M_basic.irreflexive_abs [OF M_basic_L]
and transitive_rel_abs = M_basic.transitive_rel_abs [OF M_basic_L]
and linear_rel_abs = M_basic.linear_rel_abs [OF M_basic_L]
and wellordered_is_trans_on = M_basic.wellordered_is_trans_on [OF M_basic_L]
and wellordered_is_linear = M_basic.wellordered_is_linear [OF M_basic_L]
and wellordered_is_wellfounded_on = M_basic.wellordered_is_wellfounded_on [OF M_basic_L]
and wellfounded_imp_wellfounded_on = M_basic.wellfounded_imp_wellfounded_on [OF M_basic_L]
and wellfounded_on_subset_A = M_basic.wellfounded_on_subset_A [OF M_basic_L]
and wellfounded_on_iff_wellfounded = M_basic.wellfounded_on_iff_wellfounded [OF M_basic_L]
and wellfounded_on_imp_wellfounded = M_basic.wellfounded_on_imp_wellfounded [OF M_basic_L]
and wellfounded_on_field_imp_wellfounded = M_basic.wellfounded_on_field_imp_wellfounded [OF M_basic_L]
and wellfounded_iff_wellfounded_on_field = M_basic.wellfounded_iff_wellfounded_on_field [OF M_basic_L]
and wellfounded_induct = M_basic.wellfounded_induct [OF M_basic_L]
and wellfounded_on_induct = M_basic.wellfounded_on_induct [OF M_basic_L]
and wellfounded_on_induct2 = M_basic.wellfounded_on_induct2 [OF M_basic_L]
and linear_imp_relativized = M_basic.linear_imp_relativized [OF M_basic_L]
and trans_on_imp_relativized = M_basic.trans_on_imp_relativized [OF M_basic_L]
and wf_on_imp_relativized = M_basic.wf_on_imp_relativized [OF M_basic_L]
and wf_imp_relativized = M_basic.wf_imp_relativized [OF M_basic_L]
and well_ord_imp_relativized = M_basic.well_ord_imp_relativized [OF M_basic_L]
and order_isomorphism_abs = M_basic.order_isomorphism_abs [OF M_basic_L]
and pred_set_abs = M_basic.pred_set_abs [OF M_basic_L]
lemmas pred_closed = M_basic.pred_closed [OF M_basic_L]
and membership_abs = M_basic.membership_abs [OF M_basic_L]
and M_Memrel_iff = M_basic.M_Memrel_iff [OF M_basic_L]
and Memrel_closed = M_basic.Memrel_closed [OF M_basic_L]
and wellordered_iso_predD = M_basic.wellordered_iso_predD [OF M_basic_L]
and wellordered_iso_pred_eq = M_basic.wellordered_iso_pred_eq [OF M_basic_L]
and wellfounded_on_asym = M_basic.wellfounded_on_asym [OF M_basic_L]
and wellordered_asym = M_basic.wellordered_asym [OF M_basic_L]
and ord_iso_pred_imp_lt = M_basic.ord_iso_pred_imp_lt [OF M_basic_L]
and obase_iff = M_basic.obase_iff [OF M_basic_L]
and omap_iff = M_basic.omap_iff [OF M_basic_L]
and omap_unique = M_basic.omap_unique [OF M_basic_L]
and omap_yields_Ord = M_basic.omap_yields_Ord [OF M_basic_L]
and otype_iff = M_basic.otype_iff [OF M_basic_L]
and otype_eq_range = M_basic.otype_eq_range [OF M_basic_L]
and Ord_otype = M_basic.Ord_otype [OF M_basic_L]
and domain_omap = M_basic.domain_omap [OF M_basic_L]
and omap_subset = M_basic.omap_subset [OF M_basic_L]
and omap_funtype = M_basic.omap_funtype [OF M_basic_L]
and wellordered_omap_bij = M_basic.wellordered_omap_bij [OF M_basic_L]
and omap_ord_iso = M_basic.omap_ord_iso [OF M_basic_L]
and Ord_omap_image_pred = M_basic.Ord_omap_image_pred [OF M_basic_L]
and restrict_omap_ord_iso = M_basic.restrict_omap_ord_iso [OF M_basic_L]
and obase_equals = M_basic.obase_equals [OF M_basic_L]
and omap_ord_iso_otype = M_basic.omap_ord_iso_otype [OF M_basic_L]
and obase_exists = M_basic.obase_exists [OF M_basic_L]
and omap_exists = M_basic.omap_exists [OF M_basic_L]
and otype_exists = M_basic.otype_exists [OF M_basic_L]
and omap_ord_iso_otype' = M_basic.omap_ord_iso_otype' [OF M_basic_L]
and ordertype_exists = M_basic.ordertype_exists [OF M_basic_L]
and relativized_imp_well_ord = M_basic.relativized_imp_well_ord [OF M_basic_L]
and well_ord_abs = M_basic.well_ord_abs [OF M_basic_L]
declare cartprod_closed [intro, simp]
declare sum_closed [intro, simp]
declare converse_closed [intro, simp]
declare converse_abs [simp]
declare image_closed [intro, simp]
declare vimage_abs [simp]
declare vimage_closed [intro, simp]
declare domain_abs [simp]
declare domain_closed [intro, simp]
declare range_abs [simp]
declare range_closed [intro, simp]
declare field_abs [simp]
declare field_closed [intro, simp]
declare relation_abs [simp]
declare function_abs [simp]
declare apply_closed [intro, simp]
declare typed_function_abs [simp]
declare injection_abs [simp]
declare surjection_abs [simp]
declare bijection_abs [simp]
declare comp_closed [intro, simp]
declare composition_abs [simp]
declare restriction_abs [simp]
declare restrict_closed [intro, simp]
declare Inter_abs [simp]
declare Inter_closed [intro, simp]
declare Int_closed [intro, simp]
declare is_funspace_abs [simp]
declare finite_funspace_closed [intro, simp]
declare membership_abs [simp]
declare Memrel_closed [intro,simp]
end