author | paulson |
Fri, 12 Jul 2002 11:24:40 +0200 | |
changeset 13352 | 3cd767f8d78b |
parent 13350 | 626b79677dfa |
child 13363 | c26eeb000470 |
permissions | -rw-r--r-- |
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header{*Early Instances of Separation and Strong Replacement*} |
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theory Separation = L_axioms + WF_absolute: |
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text{*This theory proves all instances needed for locale @{text "M_axioms"}*} |
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text{*Helps us solve for de Bruijn indices!*} |
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lemma nth_ConsI: "[|nth(n,l) = x; n \<in> nat|] ==> nth(succ(n), Cons(a,l)) = x" |
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9 |
by simp |
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lemmas nth_rules = nth_0 nth_ConsI nat_0I nat_succI |
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lemmas sep_rules = nth_0 nth_ConsI FOL_iff_sats function_iff_sats |
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fun_plus_iff_sats |
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lemma Collect_conj_in_DPow: |
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"[| {x\<in>A. P(x)} \<in> DPow(A); {x\<in>A. Q(x)} \<in> DPow(A) |] |
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==> {x\<in>A. P(x) & Q(x)} \<in> DPow(A)" |
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by (simp add: Int_in_DPow Collect_Int_Collect_eq [symmetric]) |
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||
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lemma Collect_conj_in_DPow_Lset: |
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"[|z \<in> Lset(j); {x \<in> Lset(j). P(x)} \<in> DPow(Lset(j))|] |
|
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==> {x \<in> Lset(j). x \<in> z & P(x)} \<in> DPow(Lset(j))" |
|
23 |
apply (frule mem_Lset_imp_subset_Lset) |
|
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apply (simp add: Collect_conj_in_DPow Collect_mem_eq |
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subset_Int_iff2 elem_subset_in_DPow) |
|
26 |
done |
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||
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lemma separation_CollectI: |
|
29 |
"(\<And>z. L(z) ==> L({x \<in> z . P(x)})) ==> separation(L, \<lambda>x. P(x))" |
|
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apply (unfold separation_def, clarify) |
|
31 |
apply (rule_tac x="{x\<in>z. P(x)}" in rexI) |
|
32 |
apply simp_all |
|
33 |
done |
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34 |
||
35 |
text{*Reduces the original comprehension to the reflected one*} |
|
36 |
lemma reflection_imp_L_separation: |
|
37 |
"[| \<forall>x\<in>Lset(j). P(x) <-> Q(x); |
|
38 |
{x \<in> Lset(j) . Q(x)} \<in> DPow(Lset(j)); |
|
39 |
Ord(j); z \<in> Lset(j)|] ==> L({x \<in> z . P(x)})" |
|
40 |
apply (rule_tac i = "succ(j)" in L_I) |
|
41 |
prefer 2 apply simp |
|
42 |
apply (subgoal_tac "{x \<in> z. P(x)} = {x \<in> Lset(j). x \<in> z & (Q(x))}") |
|
43 |
prefer 2 |
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44 |
apply (blast dest: mem_Lset_imp_subset_Lset) |
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apply (simp add: Lset_succ Collect_conj_in_DPow_Lset) |
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46 |
done |
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subsection{*Separation for Intersection*} |
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lemma Inter_Reflects: |
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"REFLECTS[\<lambda>x. \<forall>y[L]. y\<in>A --> x \<in> y, |
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\<lambda>i x. \<forall>y\<in>Lset(i). y\<in>A --> x \<in> y]" |
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by (intro FOL_reflections) |
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lemma Inter_separation: |
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"L(A) ==> separation(L, \<lambda>x. \<forall>y[L]. y\<in>A --> x\<in>y)" |
|
58 |
apply (rule separation_CollectI) |
|
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apply (rule_tac A="{A,z}" in subset_LsetE, blast ) |
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apply (rule ReflectsE [OF Inter_Reflects], assumption) |
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apply (drule subset_Lset_ltD, assumption) |
|
62 |
apply (erule reflection_imp_L_separation) |
|
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apply (simp_all add: lt_Ord2, clarify) |
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apply (rule DPowI2) |
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apply (rule ball_iff_sats) |
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apply (rule imp_iff_sats) |
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apply (rule_tac [2] i=1 and j=0 and env="[y,x,A]" in mem_iff_sats) |
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apply (rule_tac i=0 and j=2 in mem_iff_sats) |
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apply (simp_all add: succ_Un_distrib [symmetric]) |
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done |
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||
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subsection{*Separation for Cartesian Product*} |
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lemma cartprod_Reflects: |
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"REFLECTS[\<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. y\<in>B & pair(L,x,y,z)), |
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\<lambda>i z. \<exists>x\<in>Lset(i). x\<in>A & (\<exists>y\<in>Lset(i). y\<in>B & |
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pair(**Lset(i),x,y,z))]" |
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by (intro FOL_reflections function_reflections) |
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lemma cartprod_separation: |
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"[| L(A); L(B) |] |
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==> separation(L, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. y\<in>B & pair(L,x,y,z)))" |
|
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apply (rule separation_CollectI) |
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apply (rule_tac A="{A,B,z}" in subset_LsetE, blast ) |
|
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apply (rule ReflectsE [OF cartprod_Reflects], assumption) |
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apply (drule subset_Lset_ltD, assumption) |
|
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apply (erule reflection_imp_L_separation) |
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apply (simp_all add: lt_Ord2, clarify) |
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apply (rule DPowI2) |
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apply (rename_tac u) |
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apply (rule bex_iff_sats) |
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apply (rule conj_iff_sats) |
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apply (rule_tac i=0 and j=2 and env="[x,u,A,B]" in mem_iff_sats, simp_all) |
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apply (rule sep_rules | simp)+ |
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apply (simp_all add: succ_Un_distrib [symmetric]) |
96 |
done |
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||
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subsection{*Separation for Image*} |
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100 |
lemma image_Reflects: |
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"REFLECTS[\<lambda>y. \<exists>p[L]. p\<in>r & (\<exists>x[L]. x\<in>A & pair(L,x,y,p)), |
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\<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))]" |
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by (intro FOL_reflections function_reflections) |
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lemma image_separation: |
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"[| L(A); L(r) |] |
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==> separation(L, \<lambda>y. \<exists>p[L]. p\<in>r & (\<exists>x[L]. x\<in>A & pair(L,x,y,p)))" |
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apply (rule separation_CollectI) |
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apply (rule_tac A="{A,r,z}" in subset_LsetE, blast ) |
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apply (rule ReflectsE [OF image_Reflects], assumption) |
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apply (drule subset_Lset_ltD, assumption) |
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apply (erule reflection_imp_L_separation) |
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apply (simp_all add: lt_Ord2, clarify) |
|
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apply (rule DPowI2) |
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apply (rule bex_iff_sats) |
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apply (rule conj_iff_sats) |
|
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apply (rule_tac env="[p,y,A,r]" in mem_iff_sats) |
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apply (rule sep_rules | simp)+ |
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apply (simp_all add: succ_Un_distrib [symmetric]) |
120 |
done |
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||
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subsection{*Separation for Converse*} |
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lemma converse_Reflects: |
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"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)), |
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\<lambda>i z. \<exists>p\<in>Lset(i). p\<in>r & (\<exists>x\<in>Lset(i). \<exists>y\<in>Lset(i). |
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pair(**Lset(i),x,y,p) & pair(**Lset(i),y,x,z))]" |
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by (intro FOL_reflections function_reflections) |
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lemma converse_separation: |
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132 |
"L(r) ==> separation(L, |
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\<lambda>z. \<exists>p[L]. p\<in>r & (\<exists>x[L]. \<exists>y[L]. pair(L,x,y,p) & pair(L,y,x,z)))" |
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134 |
apply (rule separation_CollectI) |
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apply (rule_tac A="{r,z}" in subset_LsetE, blast ) |
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apply (rule ReflectsE [OF converse_Reflects], assumption) |
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137 |
apply (drule subset_Lset_ltD, assumption) |
|
138 |
apply (erule reflection_imp_L_separation) |
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139 |
apply (simp_all add: lt_Ord2, clarify) |
|
140 |
apply (rule DPowI2) |
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141 |
apply (rename_tac u) |
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apply (rule bex_iff_sats) |
|
143 |
apply (rule conj_iff_sats) |
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apply (rule_tac i=0 and j=2 and env="[p,u,r]" in mem_iff_sats, simp_all) |
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apply (rule sep_rules | simp)+ |
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apply (simp_all add: succ_Un_distrib [symmetric]) |
147 |
done |
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subsection{*Separation for Restriction*} |
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lemma restrict_Reflects: |
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13314 | 153 |
"REFLECTS[\<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. pair(L,x,y,z)), |
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\<lambda>i z. \<exists>x\<in>Lset(i). x\<in>A & (\<exists>y\<in>Lset(i). pair(**Lset(i),x,y,z))]" |
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by (intro FOL_reflections function_reflections) |
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lemma restrict_separation: |
|
158 |
"L(A) ==> separation(L, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. pair(L,x,y,z)))" |
|
159 |
apply (rule separation_CollectI) |
|
160 |
apply (rule_tac A="{A,z}" in subset_LsetE, blast ) |
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161 |
apply (rule ReflectsE [OF restrict_Reflects], assumption) |
|
162 |
apply (drule subset_Lset_ltD, assumption) |
|
163 |
apply (erule reflection_imp_L_separation) |
|
164 |
apply (simp_all add: lt_Ord2, clarify) |
|
165 |
apply (rule DPowI2) |
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166 |
apply (rename_tac u) |
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167 |
apply (rule bex_iff_sats) |
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apply (rule conj_iff_sats) |
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apply (rule_tac i=0 and j=2 and env="[x,u,A]" in mem_iff_sats, simp_all) |
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apply (rule sep_rules | simp)+ |
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apply (simp_all add: succ_Un_distrib [symmetric]) |
172 |
done |
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subsection{*Separation for Composition*} |
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177 |
lemma comp_Reflects: |
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"REFLECTS[\<lambda>xz. \<exists>x[L]. \<exists>y[L]. \<exists>z[L]. \<exists>xy[L]. \<exists>yz[L]. |
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pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) & |
180 |
xy\<in>s & yz\<in>r, |
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181 |
\<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). |
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pair(**Lset(i),x,z,xz) & pair(**Lset(i),x,y,xy) & |
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pair(**Lset(i),y,z,yz) & xy\<in>s & yz\<in>r]" |
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by (intro FOL_reflections function_reflections) |
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186 |
lemma comp_separation: |
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"[| L(r); L(s) |] |
|
188 |
==> separation(L, \<lambda>xz. \<exists>x[L]. \<exists>y[L]. \<exists>z[L]. \<exists>xy[L]. \<exists>yz[L]. |
|
189 |
pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) & |
|
190 |
xy\<in>s & yz\<in>r)" |
|
191 |
apply (rule separation_CollectI) |
|
192 |
apply (rule_tac A="{r,s,z}" in subset_LsetE, blast ) |
|
193 |
apply (rule ReflectsE [OF comp_Reflects], assumption) |
|
194 |
apply (drule subset_Lset_ltD, assumption) |
|
195 |
apply (erule reflection_imp_L_separation) |
|
196 |
apply (simp_all add: lt_Ord2, clarify) |
|
197 |
apply (rule DPowI2) |
|
198 |
apply (rename_tac u) |
|
199 |
apply (rule bex_iff_sats)+ |
|
200 |
apply (rename_tac x y z) |
|
201 |
apply (rule conj_iff_sats) |
|
202 |
apply (rule_tac env="[z,y,x,u,r,s]" in pair_iff_sats) |
|
13316 | 203 |
apply (rule sep_rules | simp)+ |
13306 | 204 |
apply (simp_all add: succ_Un_distrib [symmetric]) |
205 |
done |
|
206 |
||
13316 | 207 |
subsection{*Separation for Predecessors in an Order*} |
13306 | 208 |
|
209 |
lemma pred_Reflects: |
|
13314 | 210 |
"REFLECTS[\<lambda>y. \<exists>p[L]. p\<in>r & pair(L,y,x,p), |
211 |
\<lambda>i y. \<exists>p \<in> Lset(i). p\<in>r & pair(**Lset(i),y,x,p)]" |
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by (intro FOL_reflections function_reflections) |
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|
214 |
lemma pred_separation: |
|
215 |
"[| L(r); L(x) |] ==> separation(L, \<lambda>y. \<exists>p[L]. p\<in>r & pair(L,y,x,p))" |
|
216 |
apply (rule separation_CollectI) |
|
217 |
apply (rule_tac A="{r,x,z}" in subset_LsetE, blast ) |
|
218 |
apply (rule ReflectsE [OF pred_Reflects], assumption) |
|
219 |
apply (drule subset_Lset_ltD, assumption) |
|
220 |
apply (erule reflection_imp_L_separation) |
|
221 |
apply (simp_all add: lt_Ord2, clarify) |
|
222 |
apply (rule DPowI2) |
|
223 |
apply (rename_tac u) |
|
224 |
apply (rule bex_iff_sats) |
|
225 |
apply (rule conj_iff_sats) |
|
226 |
apply (rule_tac env = "[p,u,r,x]" in mem_iff_sats) |
|
13316 | 227 |
apply (rule sep_rules | simp)+ |
13306 | 228 |
apply (simp_all add: succ_Un_distrib [symmetric]) |
229 |
done |
|
230 |
||
231 |
||
13316 | 232 |
subsection{*Separation for the Membership Relation*} |
13306 | 233 |
|
234 |
lemma Memrel_Reflects: |
|
13314 | 235 |
"REFLECTS[\<lambda>z. \<exists>x[L]. \<exists>y[L]. pair(L,x,y,z) & x \<in> y, |
236 |
\<lambda>i z. \<exists>x \<in> Lset(i). \<exists>y \<in> Lset(i). pair(**Lset(i),x,y,z) & x \<in> y]" |
|
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by (intro FOL_reflections function_reflections) |
13306 | 238 |
|
239 |
lemma Memrel_separation: |
|
240 |
"separation(L, \<lambda>z. \<exists>x[L]. \<exists>y[L]. pair(L,x,y,z) & x \<in> y)" |
|
241 |
apply (rule separation_CollectI) |
|
242 |
apply (rule_tac A="{z}" in subset_LsetE, blast ) |
|
243 |
apply (rule ReflectsE [OF Memrel_Reflects], assumption) |
|
244 |
apply (drule subset_Lset_ltD, assumption) |
|
245 |
apply (erule reflection_imp_L_separation) |
|
246 |
apply (simp_all add: lt_Ord2) |
|
247 |
apply (rule DPowI2) |
|
248 |
apply (rename_tac u) |
|
13316 | 249 |
apply (rule bex_iff_sats conj_iff_sats)+ |
13306 | 250 |
apply (rule_tac env = "[y,x,u]" in pair_iff_sats) |
13316 | 251 |
apply (rule sep_rules | simp)+ |
13306 | 252 |
apply (simp_all add: succ_Un_distrib [symmetric]) |
253 |
done |
|
254 |
||
255 |
||
13316 | 256 |
subsection{*Replacement for FunSpace*} |
13306 | 257 |
|
258 |
lemma funspace_succ_Reflects: |
|
13314 | 259 |
"REFLECTS[\<lambda>z. \<exists>p[L]. p\<in>A & (\<exists>f[L]. \<exists>b[L]. \<exists>nb[L]. \<exists>cnbf[L]. |
13306 | 260 |
pair(L,f,b,p) & pair(L,n,b,nb) & is_cons(L,nb,f,cnbf) & |
261 |
upair(L,cnbf,cnbf,z)), |
|
262 |
\<lambda>i z. \<exists>p \<in> Lset(i). p\<in>A & (\<exists>f \<in> Lset(i). \<exists>b \<in> Lset(i). |
|
263 |
\<exists>nb \<in> Lset(i). \<exists>cnbf \<in> Lset(i). |
|
264 |
pair(**Lset(i),f,b,p) & pair(**Lset(i),n,b,nb) & |
|
13314 | 265 |
is_cons(**Lset(i),nb,f,cnbf) & upair(**Lset(i),cnbf,cnbf,z))]" |
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by (intro FOL_reflections function_reflections) |
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lemma funspace_succ_replacement: |
|
269 |
"L(n) ==> |
|
270 |
strong_replacement(L, \<lambda>p z. \<exists>f[L]. \<exists>b[L]. \<exists>nb[L]. \<exists>cnbf[L]. |
|
271 |
pair(L,f,b,p) & pair(L,n,b,nb) & is_cons(L,nb,f,cnbf) & |
|
272 |
upair(L,cnbf,cnbf,z))" |
|
273 |
apply (rule strong_replacementI) |
|
274 |
apply (rule rallI) |
|
275 |
apply (rule separation_CollectI) |
|
276 |
apply (rule_tac A="{n,A,z}" in subset_LsetE, blast ) |
|
277 |
apply (rule ReflectsE [OF funspace_succ_Reflects], assumption) |
|
278 |
apply (drule subset_Lset_ltD, assumption) |
|
279 |
apply (erule reflection_imp_L_separation) |
|
280 |
apply (simp_all add: lt_Ord2) |
|
281 |
apply (rule DPowI2) |
|
282 |
apply (rename_tac u) |
|
283 |
apply (rule bex_iff_sats) |
|
284 |
apply (rule conj_iff_sats) |
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apply (rule_tac env = "[p,u,n,A]" in mem_iff_sats) |
13316 | 286 |
apply (rule sep_rules | simp)+ |
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apply (simp_all add: succ_Un_distrib [symmetric]) |
288 |
done |
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290 |
||
13316 | 291 |
subsection{*Separation for Order-Isomorphisms*} |
13306 | 292 |
|
293 |
lemma well_ord_iso_Reflects: |
|
13314 | 294 |
"REFLECTS[\<lambda>x. x\<in>A --> |
295 |
(\<exists>y[L]. \<exists>p[L]. fun_apply(L,f,x,y) & pair(L,y,x,p) & p \<in> r), |
|
296 |
\<lambda>i x. x\<in>A --> (\<exists>y \<in> Lset(i). \<exists>p \<in> Lset(i). |
|
297 |
fun_apply(**Lset(i),f,x,y) & pair(**Lset(i),y,x,p) & p \<in> r)]" |
|
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298 |
by (intro FOL_reflections function_reflections) |
13306 | 299 |
|
300 |
lemma well_ord_iso_separation: |
|
301 |
"[| L(A); L(f); L(r) |] |
|
302 |
==> separation (L, \<lambda>x. x\<in>A --> (\<exists>y[L]. (\<exists>p[L]. |
|
303 |
fun_apply(L,f,x,y) & pair(L,y,x,p) & p \<in> r)))" |
|
304 |
apply (rule separation_CollectI) |
|
305 |
apply (rule_tac A="{A,f,r,z}" in subset_LsetE, blast ) |
|
306 |
apply (rule ReflectsE [OF well_ord_iso_Reflects], assumption) |
|
307 |
apply (drule subset_Lset_ltD, assumption) |
|
308 |
apply (erule reflection_imp_L_separation) |
|
309 |
apply (simp_all add: lt_Ord2) |
|
310 |
apply (rule DPowI2) |
|
311 |
apply (rename_tac u) |
|
312 |
apply (rule imp_iff_sats) |
|
313 |
apply (rule_tac env = "[u,A,f,r]" in mem_iff_sats) |
|
13316 | 314 |
apply (rule sep_rules | simp)+ |
315 |
apply (simp_all add: succ_Un_distrib [symmetric]) |
|
316 |
done |
|
317 |
||
318 |
||
319 |
subsection{*Separation for @{term "obase"}*} |
|
320 |
||
321 |
lemma obase_reflects: |
|
322 |
"REFLECTS[\<lambda>a. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L]. |
|
323 |
ordinal(L,x) & membership(L,x,mx) & pred_set(L,A,a,r,par) & |
|
324 |
order_isomorphism(L,par,r,x,mx,g), |
|
325 |
\<lambda>i a. \<exists>x \<in> Lset(i). \<exists>g \<in> Lset(i). \<exists>mx \<in> Lset(i). \<exists>par \<in> Lset(i). |
|
326 |
ordinal(**Lset(i),x) & membership(**Lset(i),x,mx) & pred_set(**Lset(i),A,a,r,par) & |
|
327 |
order_isomorphism(**Lset(i),par,r,x,mx,g)]" |
|
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328 |
by (intro FOL_reflections function_reflections fun_plus_reflections) |
13316 | 329 |
|
330 |
lemma obase_separation: |
|
331 |
--{*part of the order type formalization*} |
|
332 |
"[| L(A); L(r) |] |
|
333 |
==> separation(L, \<lambda>a. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L]. |
|
334 |
ordinal(L,x) & membership(L,x,mx) & pred_set(L,A,a,r,par) & |
|
335 |
order_isomorphism(L,par,r,x,mx,g))" |
|
336 |
apply (rule separation_CollectI) |
|
337 |
apply (rule_tac A="{A,r,z}" in subset_LsetE, blast ) |
|
338 |
apply (rule ReflectsE [OF obase_reflects], assumption) |
|
339 |
apply (drule subset_Lset_ltD, assumption) |
|
340 |
apply (erule reflection_imp_L_separation) |
|
341 |
apply (simp_all add: lt_Ord2) |
|
342 |
apply (rule DPowI2) |
|
343 |
apply (rename_tac u) |
|
13306 | 344 |
apply (rule bex_iff_sats) |
345 |
apply (rule conj_iff_sats) |
|
13316 | 346 |
apply (rule_tac env = "[x,u,A,r]" in ordinal_iff_sats) |
347 |
apply (rule sep_rules | simp)+ |
|
348 |
apply (simp_all add: succ_Un_distrib [symmetric]) |
|
349 |
done |
|
350 |
||
351 |
||
13319 | 352 |
subsection{*Separation for a Theorem about @{term "obase"}*} |
13316 | 353 |
|
354 |
lemma obase_equals_reflects: |
|
355 |
"REFLECTS[\<lambda>x. x\<in>A --> ~(\<exists>y[L]. \<exists>g[L]. |
|
356 |
ordinal(L,y) & (\<exists>my[L]. \<exists>pxr[L]. |
|
357 |
membership(L,y,my) & pred_set(L,A,x,r,pxr) & |
|
358 |
order_isomorphism(L,pxr,r,y,my,g))), |
|
359 |
\<lambda>i x. x\<in>A --> ~(\<exists>y \<in> Lset(i). \<exists>g \<in> Lset(i). |
|
360 |
ordinal(**Lset(i),y) & (\<exists>my \<in> Lset(i). \<exists>pxr \<in> Lset(i). |
|
361 |
membership(**Lset(i),y,my) & pred_set(**Lset(i),A,x,r,pxr) & |
|
362 |
order_isomorphism(**Lset(i),pxr,r,y,my,g)))]" |
|
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363 |
by (intro FOL_reflections function_reflections fun_plus_reflections) |
13316 | 364 |
|
365 |
||
366 |
lemma obase_equals_separation: |
|
367 |
"[| L(A); L(r) |] |
|
368 |
==> separation (L, \<lambda>x. x\<in>A --> ~(\<exists>y[L]. \<exists>g[L]. |
|
369 |
ordinal(L,y) & (\<exists>my[L]. \<exists>pxr[L]. |
|
370 |
membership(L,y,my) & pred_set(L,A,x,r,pxr) & |
|
371 |
order_isomorphism(L,pxr,r,y,my,g))))" |
|
372 |
apply (rule separation_CollectI) |
|
373 |
apply (rule_tac A="{A,r,z}" in subset_LsetE, blast ) |
|
374 |
apply (rule ReflectsE [OF obase_equals_reflects], assumption) |
|
375 |
apply (drule subset_Lset_ltD, assumption) |
|
376 |
apply (erule reflection_imp_L_separation) |
|
377 |
apply (simp_all add: lt_Ord2) |
|
378 |
apply (rule DPowI2) |
|
379 |
apply (rename_tac u) |
|
380 |
apply (rule imp_iff_sats ball_iff_sats disj_iff_sats not_iff_sats)+ |
|
381 |
apply (rule_tac env = "[u,A,r]" in mem_iff_sats) |
|
382 |
apply (rule sep_rules | simp)+ |
|
383 |
apply (simp_all add: succ_Un_distrib [symmetric]) |
|
384 |
done |
|
385 |
||
386 |
||
387 |
subsection{*Replacement for @{term "omap"}*} |
|
388 |
||
389 |
lemma omap_reflects: |
|
390 |
"REFLECTS[\<lambda>z. \<exists>a[L]. a\<in>B & (\<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L]. |
|
391 |
ordinal(L,x) & pair(L,a,x,z) & membership(L,x,mx) & |
|
392 |
pred_set(L,A,a,r,par) & order_isomorphism(L,par,r,x,mx,g)), |
|
393 |
\<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). |
|
394 |
\<exists>par \<in> Lset(i). |
|
395 |
ordinal(**Lset(i),x) & pair(**Lset(i),a,x,z) & |
|
396 |
membership(**Lset(i),x,mx) & pred_set(**Lset(i),A,a,r,par) & |
|
397 |
order_isomorphism(**Lset(i),par,r,x,mx,g))]" |
|
13323
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|
398 |
by (intro FOL_reflections function_reflections fun_plus_reflections) |
13316 | 399 |
|
400 |
lemma omap_replacement: |
|
401 |
"[| L(A); L(r) |] |
|
402 |
==> strong_replacement(L, |
|
403 |
\<lambda>a z. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L]. |
|
404 |
ordinal(L,x) & pair(L,a,x,z) & membership(L,x,mx) & |
|
405 |
pred_set(L,A,a,r,par) & order_isomorphism(L,par,r,x,mx,g))" |
|
406 |
apply (rule strong_replacementI) |
|
407 |
apply (rule rallI) |
|
408 |
apply (rename_tac B) |
|
409 |
apply (rule separation_CollectI) |
|
410 |
apply (rule_tac A="{A,B,r,z}" in subset_LsetE, blast ) |
|
411 |
apply (rule ReflectsE [OF omap_reflects], assumption) |
|
412 |
apply (drule subset_Lset_ltD, assumption) |
|
413 |
apply (erule reflection_imp_L_separation) |
|
414 |
apply (simp_all add: lt_Ord2) |
|
415 |
apply (rule DPowI2) |
|
416 |
apply (rename_tac u) |
|
417 |
apply (rule bex_iff_sats conj_iff_sats)+ |
|
13339
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Fixed quantified variable name preservation for ball and bex (bounded quants)
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13324
diff
changeset
|
418 |
apply (rule_tac env = "[a,u,A,B,r]" in mem_iff_sats) |
13316 | 419 |
apply (rule sep_rules | simp)+ |
13306 | 420 |
apply (simp_all add: succ_Un_distrib [symmetric]) |
421 |
done |
|
422 |
||
13323
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|
423 |
|
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|
424 |
subsection{*Separation for a Theorem about @{term "obase"}*} |
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|
425 |
|
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|
426 |
lemma is_recfun_reflects: |
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|
427 |
"REFLECTS[\<lambda>x. \<exists>xa[L]. \<exists>xb[L]. |
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|
428 |
pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r & |
2c287f50c9f3
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diff
changeset
|
429 |
(\<exists>fx[L]. \<exists>gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) & |
2c287f50c9f3
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parents:
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diff
changeset
|
430 |
fx \<noteq> gx), |
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changeset
|
431 |
\<lambda>i x. \<exists>xa \<in> Lset(i). \<exists>xb \<in> Lset(i). |
2c287f50c9f3
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parents:
13319
diff
changeset
|
432 |
pair(**Lset(i),x,a,xa) & xa \<in> r & pair(**Lset(i),x,b,xb) & xb \<in> r & |
2c287f50c9f3
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parents:
13319
diff
changeset
|
433 |
(\<exists>fx \<in> Lset(i). \<exists>gx \<in> Lset(i). fun_apply(**Lset(i),f,x,fx) & |
2c287f50c9f3
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changeset
|
434 |
fun_apply(**Lset(i),g,x,gx) & fx \<noteq> gx)]" |
2c287f50c9f3
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diff
changeset
|
435 |
by (intro FOL_reflections function_reflections fun_plus_reflections) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
436 |
|
2c287f50c9f3
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parents:
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diff
changeset
|
437 |
lemma is_recfun_separation: |
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parents:
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changeset
|
438 |
--{*for well-founded recursion*} |
2c287f50c9f3
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parents:
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changeset
|
439 |
"[| L(r); L(f); L(g); L(a); L(b) |] |
2c287f50c9f3
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parents:
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diff
changeset
|
440 |
==> separation(L, |
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changeset
|
441 |
\<lambda>x. \<exists>xa[L]. \<exists>xb[L]. |
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More relativization, reflection and proofs of separation
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parents:
13319
diff
changeset
|
442 |
pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r & |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
443 |
(\<exists>fx[L]. \<exists>gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) & |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
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diff
changeset
|
444 |
fx \<noteq> gx))" |
2c287f50c9f3
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paulson
parents:
13319
diff
changeset
|
445 |
apply (rule separation_CollectI) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
446 |
apply (rule_tac A="{r,f,g,a,b,z}" in subset_LsetE, blast ) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
447 |
apply (rule ReflectsE [OF is_recfun_reflects], assumption) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
448 |
apply (drule subset_Lset_ltD, assumption) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
449 |
apply (erule reflection_imp_L_separation) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
450 |
apply (simp_all add: lt_Ord2) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
451 |
apply (rule DPowI2) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
452 |
apply (rename_tac u) |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
453 |
apply (rule bex_iff_sats conj_iff_sats)+ |
13339
0f89104dd377
Fixed quantified variable name preservation for ball and bex (bounded quants)
paulson
parents:
13324
diff
changeset
|
454 |
apply (rule_tac env = "[xa,u,r,f,g,a,b]" in pair_iff_sats) |
13323
2c287f50c9f3
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paulson
parents:
13319
diff
changeset
|
455 |
apply (rule sep_rules | simp)+ |
2c287f50c9f3
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paulson
parents:
13319
diff
changeset
|
456 |
apply (simp_all add: succ_Un_distrib [symmetric]) |
2c287f50c9f3
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paulson
parents:
13319
diff
changeset
|
457 |
done |
2c287f50c9f3
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paulson
parents:
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diff
changeset
|
458 |
|
2c287f50c9f3
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paulson
parents:
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diff
changeset
|
459 |
|
2c287f50c9f3
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paulson
parents:
13319
diff
changeset
|
460 |
ML |
2c287f50c9f3
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parents:
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diff
changeset
|
461 |
{* |
2c287f50c9f3
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paulson
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changeset
|
462 |
val Inter_separation = thm "Inter_separation"; |
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|
463 |
val cartprod_separation = thm "cartprod_separation"; |
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diff
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|
464 |
val image_separation = thm "image_separation"; |
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changeset
|
465 |
val converse_separation = thm "converse_separation"; |
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parents:
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diff
changeset
|
466 |
val restrict_separation = thm "restrict_separation"; |
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paulson
parents:
13319
diff
changeset
|
467 |
val comp_separation = thm "comp_separation"; |
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More relativization, reflection and proofs of separation
paulson
parents:
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diff
changeset
|
468 |
val pred_separation = thm "pred_separation"; |
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parents:
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diff
changeset
|
469 |
val Memrel_separation = thm "Memrel_separation"; |
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More relativization, reflection and proofs of separation
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parents:
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diff
changeset
|
470 |
val funspace_succ_replacement = thm "funspace_succ_replacement"; |
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More relativization, reflection and proofs of separation
paulson
parents:
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diff
changeset
|
471 |
val well_ord_iso_separation = thm "well_ord_iso_separation"; |
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parents:
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changeset
|
472 |
val obase_separation = thm "obase_separation"; |
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More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
473 |
val obase_equals_separation = thm "obase_equals_separation"; |
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More relativization, reflection and proofs of separation
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parents:
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diff
changeset
|
474 |
val omap_replacement = thm "omap_replacement"; |
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More relativization, reflection and proofs of separation
paulson
parents:
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diff
changeset
|
475 |
val is_recfun_separation = thm "is_recfun_separation"; |
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paulson
parents:
13319
diff
changeset
|
476 |
|
2c287f50c9f3
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paulson
parents:
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diff
changeset
|
477 |
val m_axioms = |
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paulson
parents:
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diff
changeset
|
478 |
[Inter_separation, cartprod_separation, image_separation, |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
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changeset
|
479 |
converse_separation, restrict_separation, comp_separation, |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
480 |
pred_separation, Memrel_separation, funspace_succ_replacement, |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
481 |
well_ord_iso_separation, obase_separation, |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
482 |
obase_equals_separation, omap_replacement, is_recfun_separation] |
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More relativization, reflection and proofs of separation
paulson
parents:
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diff
changeset
|
483 |
|
2c287f50c9f3
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paulson
parents:
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diff
changeset
|
484 |
fun axiomsL th = |
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paulson
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diff
changeset
|
485 |
kill_flex_triv_prems (m_axioms MRS (trivaxL th)); |
2c287f50c9f3
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paulson
parents:
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diff
changeset
|
486 |
|
2c287f50c9f3
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changeset
|
487 |
bind_thm ("cartprod_iff", axiomsL (thm "M_axioms.cartprod_iff")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
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changeset
|
488 |
bind_thm ("cartprod_closed", axiomsL (thm "M_axioms.cartprod_closed")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
489 |
bind_thm ("sum_closed", axiomsL (thm "M_axioms.sum_closed")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
490 |
bind_thm ("M_converse_iff", axiomsL (thm "M_axioms.M_converse_iff")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
491 |
bind_thm ("converse_closed", axiomsL (thm "M_axioms.converse_closed")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
492 |
bind_thm ("converse_abs", axiomsL (thm "M_axioms.converse_abs")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
493 |
bind_thm ("image_closed", axiomsL (thm "M_axioms.image_closed")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
494 |
bind_thm ("vimage_abs", axiomsL (thm "M_axioms.vimage_abs")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
495 |
bind_thm ("vimage_closed", axiomsL (thm "M_axioms.vimage_closed")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
496 |
bind_thm ("domain_abs", axiomsL (thm "M_axioms.domain_abs")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
497 |
bind_thm ("domain_closed", axiomsL (thm "M_axioms.domain_closed")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
498 |
bind_thm ("range_abs", axiomsL (thm "M_axioms.range_abs")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
499 |
bind_thm ("range_closed", axiomsL (thm "M_axioms.range_closed")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
500 |
bind_thm ("field_abs", axiomsL (thm "M_axioms.field_abs")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
501 |
bind_thm ("field_closed", axiomsL (thm "M_axioms.field_closed")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
502 |
bind_thm ("relation_abs", axiomsL (thm "M_axioms.relation_abs")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
503 |
bind_thm ("function_abs", axiomsL (thm "M_axioms.function_abs")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
504 |
bind_thm ("apply_closed", axiomsL (thm "M_axioms.apply_closed")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
505 |
bind_thm ("apply_abs", axiomsL (thm "M_axioms.apply_abs")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
506 |
bind_thm ("typed_function_abs", axiomsL (thm "M_axioms.typed_function_abs")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
507 |
bind_thm ("injection_abs", axiomsL (thm "M_axioms.injection_abs")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
508 |
bind_thm ("surjection_abs", axiomsL (thm "M_axioms.surjection_abs")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
509 |
bind_thm ("bijection_abs", axiomsL (thm "M_axioms.bijection_abs")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
510 |
bind_thm ("M_comp_iff", axiomsL (thm "M_axioms.M_comp_iff")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
511 |
bind_thm ("comp_closed", axiomsL (thm "M_axioms.comp_closed")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
512 |
bind_thm ("composition_abs", axiomsL (thm "M_axioms.composition_abs")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
513 |
bind_thm ("restriction_is_function", axiomsL (thm "M_axioms.restriction_is_function")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
514 |
bind_thm ("restriction_abs", axiomsL (thm "M_axioms.restriction_abs")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
515 |
bind_thm ("M_restrict_iff", axiomsL (thm "M_axioms.M_restrict_iff")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
516 |
bind_thm ("restrict_closed", axiomsL (thm "M_axioms.restrict_closed")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
517 |
bind_thm ("Inter_abs", axiomsL (thm "M_axioms.Inter_abs")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
518 |
bind_thm ("Inter_closed", axiomsL (thm "M_axioms.Inter_closed")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
519 |
bind_thm ("Int_closed", axiomsL (thm "M_axioms.Int_closed")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
520 |
bind_thm ("finite_fun_closed", axiomsL (thm "M_axioms.finite_fun_closed")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
521 |
bind_thm ("is_funspace_abs", axiomsL (thm "M_axioms.is_funspace_abs")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
522 |
bind_thm ("succ_fun_eq2", axiomsL (thm "M_axioms.succ_fun_eq2")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
523 |
bind_thm ("funspace_succ", axiomsL (thm "M_axioms.funspace_succ")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
524 |
bind_thm ("finite_funspace_closed", axiomsL (thm "M_axioms.finite_funspace_closed")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
525 |
*} |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
526 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
527 |
ML |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
528 |
{* |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
529 |
bind_thm ("is_recfun_equal", axiomsL (thm "M_axioms.is_recfun_equal")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
530 |
bind_thm ("is_recfun_cut", axiomsL (thm "M_axioms.is_recfun_cut")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
531 |
bind_thm ("is_recfun_functional", axiomsL (thm "M_axioms.is_recfun_functional")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
532 |
bind_thm ("is_recfun_relativize", axiomsL (thm "M_axioms.is_recfun_relativize")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
533 |
bind_thm ("is_recfun_restrict", axiomsL (thm "M_axioms.is_recfun_restrict")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
534 |
bind_thm ("univalent_is_recfun", axiomsL (thm "M_axioms.univalent_is_recfun")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
535 |
bind_thm ("exists_is_recfun_indstep", axiomsL (thm "M_axioms.exists_is_recfun_indstep")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
536 |
bind_thm ("wellfounded_exists_is_recfun", axiomsL (thm "M_axioms.wellfounded_exists_is_recfun")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
537 |
bind_thm ("wf_exists_is_recfun", axiomsL (thm "M_axioms.wf_exists_is_recfun")); |
13350 | 538 |
bind_thm ("is_recfun_abs", axiomsL (thm "M_axioms.is_recfun_abs")); |
13323
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
539 |
bind_thm ("irreflexive_abs", axiomsL (thm "M_axioms.irreflexive_abs")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
540 |
bind_thm ("transitive_rel_abs", axiomsL (thm "M_axioms.transitive_rel_abs")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
541 |
bind_thm ("linear_rel_abs", axiomsL (thm "M_axioms.linear_rel_abs")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
542 |
bind_thm ("wellordered_is_trans_on", axiomsL (thm "M_axioms.wellordered_is_trans_on")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
543 |
bind_thm ("wellordered_is_linear", axiomsL (thm "M_axioms.wellordered_is_linear")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
544 |
bind_thm ("wellordered_is_wellfounded_on", axiomsL (thm "M_axioms.wellordered_is_wellfounded_on")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
545 |
bind_thm ("wellfounded_imp_wellfounded_on", axiomsL (thm "M_axioms.wellfounded_imp_wellfounded_on")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
546 |
bind_thm ("wellfounded_on_subset_A", axiomsL (thm "M_axioms.wellfounded_on_subset_A")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
547 |
bind_thm ("wellfounded_on_iff_wellfounded", axiomsL (thm "M_axioms.wellfounded_on_iff_wellfounded")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
548 |
bind_thm ("wellfounded_on_imp_wellfounded", axiomsL (thm "M_axioms.wellfounded_on_imp_wellfounded")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
549 |
bind_thm ("wellfounded_on_field_imp_wellfounded", axiomsL (thm "M_axioms.wellfounded_on_field_imp_wellfounded")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
550 |
bind_thm ("wellfounded_iff_wellfounded_on_field", axiomsL (thm "M_axioms.wellfounded_iff_wellfounded_on_field")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
551 |
bind_thm ("wellfounded_induct", axiomsL (thm "M_axioms.wellfounded_induct")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
552 |
bind_thm ("wellfounded_on_induct", axiomsL (thm "M_axioms.wellfounded_on_induct")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
553 |
bind_thm ("wellfounded_on_induct2", axiomsL (thm "M_axioms.wellfounded_on_induct2")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
554 |
bind_thm ("linear_imp_relativized", axiomsL (thm "M_axioms.linear_imp_relativized")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
555 |
bind_thm ("trans_on_imp_relativized", axiomsL (thm "M_axioms.trans_on_imp_relativized")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
556 |
bind_thm ("wf_on_imp_relativized", axiomsL (thm "M_axioms.wf_on_imp_relativized")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
557 |
bind_thm ("wf_imp_relativized", axiomsL (thm "M_axioms.wf_imp_relativized")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
558 |
bind_thm ("well_ord_imp_relativized", axiomsL (thm "M_axioms.well_ord_imp_relativized")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
559 |
bind_thm ("order_isomorphism_abs", axiomsL (thm "M_axioms.order_isomorphism_abs")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
560 |
bind_thm ("pred_set_abs", axiomsL (thm "M_axioms.pred_set_abs")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
561 |
*} |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
562 |
|
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
563 |
ML |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
564 |
{* |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
565 |
bind_thm ("pred_closed", axiomsL (thm "M_axioms.pred_closed")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
566 |
bind_thm ("membership_abs", axiomsL (thm "M_axioms.membership_abs")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
567 |
bind_thm ("M_Memrel_iff", axiomsL (thm "M_axioms.M_Memrel_iff")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
568 |
bind_thm ("Memrel_closed", axiomsL (thm "M_axioms.Memrel_closed")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
569 |
bind_thm ("wellordered_iso_predD", axiomsL (thm "M_axioms.wellordered_iso_predD")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
570 |
bind_thm ("wellordered_iso_pred_eq", axiomsL (thm "M_axioms.wellordered_iso_pred_eq")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
571 |
bind_thm ("wellfounded_on_asym", axiomsL (thm "M_axioms.wellfounded_on_asym")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
572 |
bind_thm ("wellordered_asym", axiomsL (thm "M_axioms.wellordered_asym")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
573 |
bind_thm ("ord_iso_pred_imp_lt", axiomsL (thm "M_axioms.ord_iso_pred_imp_lt")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
574 |
bind_thm ("obase_iff", axiomsL (thm "M_axioms.obase_iff")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
575 |
bind_thm ("omap_iff", axiomsL (thm "M_axioms.omap_iff")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
576 |
bind_thm ("omap_unique", axiomsL (thm "M_axioms.omap_unique")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
577 |
bind_thm ("omap_yields_Ord", axiomsL (thm "M_axioms.omap_yields_Ord")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
578 |
bind_thm ("otype_iff", axiomsL (thm "M_axioms.otype_iff")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
579 |
bind_thm ("otype_eq_range", axiomsL (thm "M_axioms.otype_eq_range")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
580 |
bind_thm ("Ord_otype", axiomsL (thm "M_axioms.Ord_otype")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
581 |
bind_thm ("domain_omap", axiomsL (thm "M_axioms.domain_omap")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
582 |
bind_thm ("omap_subset", axiomsL (thm "M_axioms.omap_subset")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
583 |
bind_thm ("omap_funtype", axiomsL (thm "M_axioms.omap_funtype")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
584 |
bind_thm ("wellordered_omap_bij", axiomsL (thm "M_axioms.wellordered_omap_bij")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
585 |
bind_thm ("omap_ord_iso", axiomsL (thm "M_axioms.omap_ord_iso")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
586 |
bind_thm ("Ord_omap_image_pred", axiomsL (thm "M_axioms.Ord_omap_image_pred")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
587 |
bind_thm ("restrict_omap_ord_iso", axiomsL (thm "M_axioms.restrict_omap_ord_iso")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
588 |
bind_thm ("obase_equals", axiomsL (thm "M_axioms.obase_equals")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
589 |
bind_thm ("omap_ord_iso_otype", axiomsL (thm "M_axioms.omap_ord_iso_otype")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
590 |
bind_thm ("obase_exists", axiomsL (thm "M_axioms.obase_exists")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
591 |
bind_thm ("omap_exists", axiomsL (thm "M_axioms.omap_exists")); |
2c287f50c9f3
More relativization, reflection and proofs of separation
paulson
parents:
13319
diff
changeset
|
592 |
bind_thm ("otype_exists", axiomsL (thm "M_axioms.otype_exists")); |
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593 |
bind_thm ("omap_ord_iso_otype", axiomsL (thm "M_axioms.omap_ord_iso_otype")); |
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594 |
bind_thm ("ordertype_exists", axiomsL (thm "M_axioms.ordertype_exists")); |
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595 |
bind_thm ("relativized_imp_well_ord", axiomsL (thm "M_axioms.relativized_imp_well_ord")); |
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596 |
bind_thm ("well_ord_abs", axiomsL (thm "M_axioms.well_ord_abs")); |
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|
597 |
*} |
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|
598 |
|
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|
599 |
declare cartprod_closed [intro,simp] |
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|
600 |
declare sum_closed [intro,simp] |
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|
601 |
declare converse_closed [intro,simp] |
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|
602 |
declare converse_abs [simp] |
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|
603 |
declare image_closed [intro,simp] |
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|
604 |
declare vimage_abs [simp] |
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|
605 |
declare vimage_closed [intro,simp] |
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|
606 |
declare domain_abs [simp] |
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|
607 |
declare domain_closed [intro,simp] |
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|
608 |
declare range_abs [simp] |
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|
609 |
declare range_closed [intro,simp] |
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|
610 |
declare field_abs [simp] |
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|
611 |
declare field_closed [intro,simp] |
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|
612 |
declare relation_abs [simp] |
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|
613 |
declare function_abs [simp] |
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|
614 |
declare apply_closed [intro,simp] |
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|
615 |
declare typed_function_abs [simp] |
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|
616 |
declare injection_abs [simp] |
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|
617 |
declare surjection_abs [simp] |
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|
618 |
declare bijection_abs [simp] |
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|
619 |
declare comp_closed [intro,simp] |
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|
620 |
declare composition_abs [simp] |
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|
621 |
declare restriction_abs [simp] |
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|
622 |
declare restrict_closed [intro,simp] |
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|
623 |
declare Inter_abs [simp] |
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|
624 |
declare Inter_closed [intro,simp] |
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|
625 |
declare Int_closed [intro,simp] |
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|
626 |
declare finite_fun_closed [rule_format] |
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|
627 |
declare is_funspace_abs [simp] |
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|
628 |
declare finite_funspace_closed [intro,simp] |
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629 |
|
13306 | 630 |
end |