author | wenzelm |
Mon, 29 Jul 2002 00:57:16 +0200 | |
changeset 13428 | 99e52e78eb65 |
parent 13385 | 31df66ca0780 |
child 13429 | 2232810416fc |
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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|
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text{*Helps us solve for de Bruijn indices!*} |
8 |
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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10 |
||
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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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|
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lemma Collect_conj_in_DPow: |
|
13428 | 16 |
"[| {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)" |
13428 | 18 |
by (simp add: Int_in_DPow Collect_Int_Collect_eq [symmetric]) |
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|
20 |
lemma Collect_conj_in_DPow_Lset: |
|
21 |
"[|z \<in> Lset(j); {x \<in> Lset(j). P(x)} \<in> DPow(Lset(j))|] |
|
22 |
==> {x \<in> Lset(j). x \<in> z & P(x)} \<in> DPow(Lset(j))" |
|
23 |
apply (frule mem_Lset_imp_subset_Lset) |
|
13428 | 24 |
apply (simp add: Collect_conj_in_DPow Collect_mem_eq |
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subset_Int_iff2 elem_subset_in_DPow) |
26 |
done |
|
27 |
||
28 |
lemma separation_CollectI: |
|
29 |
"(\<And>z. L(z) ==> L({x \<in> z . P(x)})) ==> separation(L, \<lambda>x. P(x))" |
|
13428 | 30 |
apply (unfold separation_def, clarify) |
31 |
apply (rule_tac x="{x\<in>z. P(x)}" in rexI) |
|
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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); |
|
13428 | 38 |
{x \<in> Lset(j) . Q(x)} \<in> DPow(Lset(j)); |
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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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apply (blast dest: mem_Lset_imp_subset_Lset) |
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apply (simp add: Lset_succ Collect_conj_in_DPow_Lset) |
46 |
done |
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47 |
||
48 |
||
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subsection{*Separation for Intersection*} |
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|
51 |
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]" |
13428 | 54 |
by (intro FOL_reflections) |
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|
56 |
lemma Inter_separation: |
|
57 |
"L(A) ==> separation(L, \<lambda>x. \<forall>y[L]. y\<in>A --> x\<in>y)" |
|
13428 | 58 |
apply (rule separation_CollectI) |
59 |
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) |
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apply (erule reflection_imp_L_separation) |
63 |
apply (simp_all add: lt_Ord2, clarify) |
|
13428 | 64 |
apply (rule DPow_LsetI) |
65 |
apply (rule ball_iff_sats) |
|
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apply (rule imp_iff_sats) |
67 |
apply (rule_tac [2] i=1 and j=0 and env="[y,x,A]" in mem_iff_sats) |
|
68 |
apply (rule_tac i=0 and j=2 in mem_iff_sats) |
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69 |
apply (simp_all add: succ_Un_distrib [symmetric]) |
|
70 |
done |
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71 |
||
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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)))" |
13428 | 83 |
apply (rule separation_CollectI) |
84 |
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) |
13428 | 88 |
apply (simp_all add: lt_Ord2, clarify) |
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apply (rule DPow_LsetI) |
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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) |
93 |
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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done |
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subsection{*Separation for Image*} |
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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)))" |
13428 | 107 |
apply (rule separation_CollectI) |
108 |
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 DPow_LsetI) |
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apply (rule bex_iff_sats) |
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apply (rule conj_iff_sats) |
116 |
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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done |
119 |
||
120 |
||
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subsection{*Separation for Converse*} |
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|
123 |
lemma converse_Reflects: |
|
13314 | 124 |
"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)), |
13428 | 125 |
\<lambda>i z. \<exists>p\<in>Lset(i). p\<in>r & (\<exists>x\<in>Lset(i). \<exists>y\<in>Lset(i). |
13314 | 126 |
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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13428 | 130 |
"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)))" |
13428 | 132 |
apply (rule separation_CollectI) |
133 |
apply (rule_tac A="{r,z}" in subset_LsetE, blast ) |
|
13306 | 134 |
apply (rule ReflectsE [OF converse_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 DPow_LsetI) |
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apply (rename_tac u) |
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apply (rule bex_iff_sats) |
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13306 | 141 |
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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done |
145 |
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146 |
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subsection{*Separation for Restriction*} |
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|
149 |
lemma restrict_Reflects: |
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"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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|
154 |
lemma restrict_separation: |
|
155 |
"L(A) ==> separation(L, \<lambda>z. \<exists>x[L]. x\<in>A & (\<exists>y[L]. pair(L,x,y,z)))" |
|
13428 | 156 |
apply (rule separation_CollectI) |
157 |
apply (rule_tac A="{A,z}" in subset_LsetE, blast ) |
|
13306 | 158 |
apply (rule ReflectsE [OF restrict_Reflects], assumption) |
13428 | 159 |
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 DPow_LsetI) |
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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]" in mem_iff_sats, simp_all) |
13316 | 167 |
apply (rule sep_rules | simp)+ |
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done |
169 |
||
170 |
||
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subsection{*Separation for Composition*} |
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173 |
lemma comp_Reflects: |
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13428 | 174 |
"REFLECTS[\<lambda>xz. \<exists>x[L]. \<exists>y[L]. \<exists>z[L]. \<exists>xy[L]. \<exists>yz[L]. |
175 |
pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) & |
|
13306 | 176 |
xy\<in>s & yz\<in>r, |
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\<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). |
178 |
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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|
182 |
lemma comp_separation: |
|
183 |
"[| L(r); L(s) |] |
|
13428 | 184 |
==> separation(L, \<lambda>xz. \<exists>x[L]. \<exists>y[L]. \<exists>z[L]. \<exists>xy[L]. \<exists>yz[L]. |
185 |
pair(L,x,z,xz) & pair(L,x,y,xy) & pair(L,y,z,yz) & |
|
13306 | 186 |
xy\<in>s & yz\<in>r)" |
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apply (rule separation_CollectI) |
188 |
apply (rule_tac A="{r,s,z}" in subset_LsetE, blast ) |
|
13306 | 189 |
apply (rule ReflectsE [OF comp_Reflects], assumption) |
13428 | 190 |
apply (drule subset_Lset_ltD, assumption) |
13306 | 191 |
apply (erule reflection_imp_L_separation) |
192 |
apply (simp_all add: lt_Ord2, clarify) |
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apply (rule DPow_LsetI) |
13428 | 194 |
apply (rename_tac u) |
13306 | 195 |
apply (rule bex_iff_sats)+ |
13428 | 196 |
apply (rename_tac x y z) |
13306 | 197 |
apply (rule conj_iff_sats) |
198 |
apply (rule_tac env="[z,y,x,u,r,s]" in pair_iff_sats) |
|
13316 | 199 |
apply (rule sep_rules | simp)+ |
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done |
201 |
||
13316 | 202 |
subsection{*Separation for Predecessors in an Order*} |
13306 | 203 |
|
204 |
lemma pred_Reflects: |
|
13314 | 205 |
"REFLECTS[\<lambda>y. \<exists>p[L]. p\<in>r & pair(L,y,x,p), |
206 |
\<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) |
13306 | 208 |
|
209 |
lemma pred_separation: |
|
210 |
"[| L(r); L(x) |] ==> separation(L, \<lambda>y. \<exists>p[L]. p\<in>r & pair(L,y,x,p))" |
|
13428 | 211 |
apply (rule separation_CollectI) |
212 |
apply (rule_tac A="{r,x,z}" in subset_LsetE, blast ) |
|
13306 | 213 |
apply (rule ReflectsE [OF pred_Reflects], assumption) |
13428 | 214 |
apply (drule subset_Lset_ltD, assumption) |
13306 | 215 |
apply (erule reflection_imp_L_separation) |
216 |
apply (simp_all add: lt_Ord2, clarify) |
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apply (rule DPow_LsetI) |
13428 | 218 |
apply (rename_tac u) |
13306 | 219 |
apply (rule bex_iff_sats) |
220 |
apply (rule conj_iff_sats) |
|
13428 | 221 |
apply (rule_tac env = "[p,u,r,x]" in mem_iff_sats) |
13316 | 222 |
apply (rule sep_rules | simp)+ |
13306 | 223 |
done |
224 |
||
225 |
||
13316 | 226 |
subsection{*Separation for the Membership Relation*} |
13306 | 227 |
|
228 |
lemma Memrel_Reflects: |
|
13314 | 229 |
"REFLECTS[\<lambda>z. \<exists>x[L]. \<exists>y[L]. pair(L,x,y,z) & x \<in> y, |
230 |
\<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 | 232 |
|
233 |
lemma Memrel_separation: |
|
234 |
"separation(L, \<lambda>z. \<exists>x[L]. \<exists>y[L]. pair(L,x,y,z) & x \<in> y)" |
|
13428 | 235 |
apply (rule separation_CollectI) |
236 |
apply (rule_tac A="{z}" in subset_LsetE, blast ) |
|
13306 | 237 |
apply (rule ReflectsE [OF Memrel_Reflects], assumption) |
13428 | 238 |
apply (drule subset_Lset_ltD, assumption) |
13306 | 239 |
apply (erule reflection_imp_L_separation) |
240 |
apply (simp_all add: lt_Ord2) |
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241 |
apply (rule DPow_LsetI) |
13428 | 242 |
apply (rename_tac u) |
13316 | 243 |
apply (rule bex_iff_sats conj_iff_sats)+ |
13428 | 244 |
apply (rule_tac env = "[y,x,u]" in pair_iff_sats) |
13316 | 245 |
apply (rule sep_rules | simp)+ |
13306 | 246 |
done |
247 |
||
248 |
||
13316 | 249 |
subsection{*Replacement for FunSpace*} |
13428 | 250 |
|
13306 | 251 |
lemma funspace_succ_Reflects: |
13428 | 252 |
"REFLECTS[\<lambda>z. \<exists>p[L]. p\<in>A & (\<exists>f[L]. \<exists>b[L]. \<exists>nb[L]. \<exists>cnbf[L]. |
253 |
pair(L,f,b,p) & pair(L,n,b,nb) & is_cons(L,nb,f,cnbf) & |
|
254 |
upair(L,cnbf,cnbf,z)), |
|
255 |
\<lambda>i z. \<exists>p \<in> Lset(i). p\<in>A & (\<exists>f \<in> Lset(i). \<exists>b \<in> Lset(i). |
|
256 |
\<exists>nb \<in> Lset(i). \<exists>cnbf \<in> Lset(i). |
|
257 |
pair(**Lset(i),f,b,p) & pair(**Lset(i),n,b,nb) & |
|
258 |
is_cons(**Lset(i),nb,f,cnbf) & upair(**Lset(i),cnbf,cnbf,z))]" |
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by (intro FOL_reflections function_reflections) |
13306 | 260 |
|
261 |
lemma funspace_succ_replacement: |
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13428 | 262 |
"L(n) ==> |
263 |
strong_replacement(L, \<lambda>p z. \<exists>f[L]. \<exists>b[L]. \<exists>nb[L]. \<exists>cnbf[L]. |
|
13306 | 264 |
pair(L,f,b,p) & pair(L,n,b,nb) & is_cons(L,nb,f,cnbf) & |
265 |
upair(L,cnbf,cnbf,z))" |
|
13428 | 266 |
apply (rule strong_replacementI) |
267 |
apply (rule rallI) |
|
268 |
apply (rule separation_CollectI) |
|
269 |
apply (rule_tac A="{n,A,z}" in subset_LsetE, blast ) |
|
13306 | 270 |
apply (rule ReflectsE [OF funspace_succ_Reflects], assumption) |
13428 | 271 |
apply (drule subset_Lset_ltD, assumption) |
13306 | 272 |
apply (erule reflection_imp_L_separation) |
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apply (simp_all add: lt_Ord2) |
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apply (rule DPow_LsetI) |
13428 | 275 |
apply (rename_tac u) |
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apply (rule bex_iff_sats) |
277 |
apply (rule conj_iff_sats) |
|
13428 | 278 |
apply (rule_tac env = "[p,u,n,A]" in mem_iff_sats) |
13316 | 279 |
apply (rule sep_rules | simp)+ |
13306 | 280 |
done |
281 |
||
282 |
||
13316 | 283 |
subsection{*Separation for Order-Isomorphisms*} |
13306 | 284 |
|
285 |
lemma well_ord_iso_Reflects: |
|
13428 | 286 |
"REFLECTS[\<lambda>x. x\<in>A --> |
13314 | 287 |
(\<exists>y[L]. \<exists>p[L]. fun_apply(L,f,x,y) & pair(L,y,x,p) & p \<in> r), |
13428 | 288 |
\<lambda>i x. x\<in>A --> (\<exists>y \<in> Lset(i). \<exists>p \<in> Lset(i). |
13314 | 289 |
fun_apply(**Lset(i),f,x,y) & pair(**Lset(i),y,x,p) & p \<in> r)]" |
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290 |
by (intro FOL_reflections function_reflections) |
13306 | 291 |
|
292 |
lemma well_ord_iso_separation: |
|
13428 | 293 |
"[| L(A); L(f); L(r) |] |
294 |
==> separation (L, \<lambda>x. x\<in>A --> (\<exists>y[L]. (\<exists>p[L]. |
|
295 |
fun_apply(L,f,x,y) & pair(L,y,x,p) & p \<in> r)))" |
|
296 |
apply (rule separation_CollectI) |
|
297 |
apply (rule_tac A="{A,f,r,z}" in subset_LsetE, blast ) |
|
13306 | 298 |
apply (rule ReflectsE [OF well_ord_iso_Reflects], assumption) |
13428 | 299 |
apply (drule subset_Lset_ltD, assumption) |
13306 | 300 |
apply (erule reflection_imp_L_separation) |
301 |
apply (simp_all add: lt_Ord2) |
|
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|
302 |
apply (rule DPow_LsetI) |
13428 | 303 |
apply (rename_tac u) |
13306 | 304 |
apply (rule imp_iff_sats) |
13428 | 305 |
apply (rule_tac env = "[u,A,f,r]" in mem_iff_sats) |
13316 | 306 |
apply (rule sep_rules | simp)+ |
307 |
done |
|
308 |
||
309 |
||
310 |
subsection{*Separation for @{term "obase"}*} |
|
311 |
||
312 |
lemma obase_reflects: |
|
13428 | 313 |
"REFLECTS[\<lambda>a. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L]. |
314 |
ordinal(L,x) & membership(L,x,mx) & pred_set(L,A,a,r,par) & |
|
315 |
order_isomorphism(L,par,r,x,mx,g), |
|
316 |
\<lambda>i a. \<exists>x \<in> Lset(i). \<exists>g \<in> Lset(i). \<exists>mx \<in> Lset(i). \<exists>par \<in> Lset(i). |
|
317 |
ordinal(**Lset(i),x) & membership(**Lset(i),x,mx) & pred_set(**Lset(i),A,a,r,par) & |
|
318 |
order_isomorphism(**Lset(i),par,r,x,mx,g)]" |
|
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|
319 |
by (intro FOL_reflections function_reflections fun_plus_reflections) |
13316 | 320 |
|
321 |
lemma obase_separation: |
|
322 |
--{*part of the order type formalization*} |
|
13428 | 323 |
"[| L(A); L(r) |] |
324 |
==> separation(L, \<lambda>a. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L]. |
|
325 |
ordinal(L,x) & membership(L,x,mx) & pred_set(L,A,a,r,par) & |
|
326 |
order_isomorphism(L,par,r,x,mx,g))" |
|
327 |
apply (rule separation_CollectI) |
|
328 |
apply (rule_tac A="{A,r,z}" in subset_LsetE, blast ) |
|
13316 | 329 |
apply (rule ReflectsE [OF obase_reflects], assumption) |
13428 | 330 |
apply (drule subset_Lset_ltD, assumption) |
13316 | 331 |
apply (erule reflection_imp_L_separation) |
332 |
apply (simp_all add: lt_Ord2) |
|
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|
333 |
apply (rule DPow_LsetI) |
13428 | 334 |
apply (rename_tac u) |
13306 | 335 |
apply (rule bex_iff_sats) |
336 |
apply (rule conj_iff_sats) |
|
13428 | 337 |
apply (rule_tac env = "[x,u,A,r]" in ordinal_iff_sats) |
13316 | 338 |
apply (rule sep_rules | simp)+ |
339 |
done |
|
340 |
||
341 |
||
13319 | 342 |
subsection{*Separation for a Theorem about @{term "obase"}*} |
13316 | 343 |
|
344 |
lemma obase_equals_reflects: |
|
13428 | 345 |
"REFLECTS[\<lambda>x. x\<in>A --> ~(\<exists>y[L]. \<exists>g[L]. |
346 |
ordinal(L,y) & (\<exists>my[L]. \<exists>pxr[L]. |
|
347 |
membership(L,y,my) & pred_set(L,A,x,r,pxr) & |
|
348 |
order_isomorphism(L,pxr,r,y,my,g))), |
|
349 |
\<lambda>i x. x\<in>A --> ~(\<exists>y \<in> Lset(i). \<exists>g \<in> Lset(i). |
|
350 |
ordinal(**Lset(i),y) & (\<exists>my \<in> Lset(i). \<exists>pxr \<in> Lset(i). |
|
351 |
membership(**Lset(i),y,my) & pred_set(**Lset(i),A,x,r,pxr) & |
|
352 |
order_isomorphism(**Lset(i),pxr,r,y,my,g)))]" |
|
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|
353 |
by (intro FOL_reflections function_reflections fun_plus_reflections) |
13316 | 354 |
|
355 |
||
356 |
lemma obase_equals_separation: |
|
13428 | 357 |
"[| L(A); L(r) |] |
358 |
==> separation (L, \<lambda>x. x\<in>A --> ~(\<exists>y[L]. \<exists>g[L]. |
|
359 |
ordinal(L,y) & (\<exists>my[L]. \<exists>pxr[L]. |
|
360 |
membership(L,y,my) & pred_set(L,A,x,r,pxr) & |
|
361 |
order_isomorphism(L,pxr,r,y,my,g))))" |
|
362 |
apply (rule separation_CollectI) |
|
363 |
apply (rule_tac A="{A,r,z}" in subset_LsetE, blast ) |
|
13316 | 364 |
apply (rule ReflectsE [OF obase_equals_reflects], assumption) |
13428 | 365 |
apply (drule subset_Lset_ltD, assumption) |
13316 | 366 |
apply (erule reflection_imp_L_separation) |
367 |
apply (simp_all add: lt_Ord2) |
|
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|
368 |
apply (rule DPow_LsetI) |
13428 | 369 |
apply (rename_tac u) |
13316 | 370 |
apply (rule imp_iff_sats ball_iff_sats disj_iff_sats not_iff_sats)+ |
13428 | 371 |
apply (rule_tac env = "[u,A,r]" in mem_iff_sats) |
13316 | 372 |
apply (rule sep_rules | simp)+ |
373 |
done |
|
374 |
||
375 |
||
376 |
subsection{*Replacement for @{term "omap"}*} |
|
377 |
||
378 |
lemma omap_reflects: |
|
13428 | 379 |
"REFLECTS[\<lambda>z. \<exists>a[L]. a\<in>B & (\<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L]. |
380 |
ordinal(L,x) & pair(L,a,x,z) & membership(L,x,mx) & |
|
13316 | 381 |
pred_set(L,A,a,r,par) & order_isomorphism(L,par,r,x,mx,g)), |
13428 | 382 |
\<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). |
383 |
\<exists>par \<in> Lset(i). |
|
384 |
ordinal(**Lset(i),x) & pair(**Lset(i),a,x,z) & |
|
385 |
membership(**Lset(i),x,mx) & pred_set(**Lset(i),A,a,r,par) & |
|
13316 | 386 |
order_isomorphism(**Lset(i),par,r,x,mx,g))]" |
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387 |
by (intro FOL_reflections function_reflections fun_plus_reflections) |
13316 | 388 |
|
389 |
lemma omap_replacement: |
|
13428 | 390 |
"[| L(A); L(r) |] |
13316 | 391 |
==> strong_replacement(L, |
13428 | 392 |
\<lambda>a z. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L]. |
393 |
ordinal(L,x) & pair(L,a,x,z) & membership(L,x,mx) & |
|
394 |
pred_set(L,A,a,r,par) & order_isomorphism(L,par,r,x,mx,g))" |
|
395 |
apply (rule strong_replacementI) |
|
13316 | 396 |
apply (rule rallI) |
13428 | 397 |
apply (rename_tac B) |
398 |
apply (rule separation_CollectI) |
|
399 |
apply (rule_tac A="{A,B,r,z}" in subset_LsetE, blast ) |
|
13316 | 400 |
apply (rule ReflectsE [OF omap_reflects], assumption) |
13428 | 401 |
apply (drule subset_Lset_ltD, assumption) |
13316 | 402 |
apply (erule reflection_imp_L_separation) |
403 |
apply (simp_all add: lt_Ord2) |
|
13385
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|
404 |
apply (rule DPow_LsetI) |
13428 | 405 |
apply (rename_tac u) |
13316 | 406 |
apply (rule bex_iff_sats conj_iff_sats)+ |
13428 | 407 |
apply (rule_tac env = "[a,u,A,B,r]" in mem_iff_sats) |
13316 | 408 |
apply (rule sep_rules | simp)+ |
13306 | 409 |
done |
410 |
||
13323
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|
411 |
|
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|
412 |
subsection{*Separation for a Theorem about @{term "obase"}*} |
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|
413 |
|
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|
414 |
lemma is_recfun_reflects: |
13428 | 415 |
"REFLECTS[\<lambda>x. \<exists>xa[L]. \<exists>xb[L]. |
416 |
pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r & |
|
417 |
(\<exists>fx[L]. \<exists>gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) & |
|
13323
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|
418 |
fx \<noteq> gx), |
13428 | 419 |
\<lambda>i x. \<exists>xa \<in> Lset(i). \<exists>xb \<in> Lset(i). |
13323
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|
420 |
pair(**Lset(i),x,a,xa) & xa \<in> r & pair(**Lset(i),x,b,xb) & xb \<in> r & |
13428 | 421 |
(\<exists>fx \<in> Lset(i). \<exists>gx \<in> Lset(i). fun_apply(**Lset(i),f,x,fx) & |
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|
422 |
fun_apply(**Lset(i),g,x,gx) & fx \<noteq> gx)]" |
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|
423 |
by (intro FOL_reflections function_reflections fun_plus_reflections) |
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|
424 |
|
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|
425 |
lemma is_recfun_separation: |
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|
426 |
--{*for well-founded recursion*} |
13428 | 427 |
"[| L(r); L(f); L(g); L(a); L(b) |] |
428 |
==> separation(L, |
|
429 |
\<lambda>x. \<exists>xa[L]. \<exists>xb[L]. |
|
430 |
pair(L,x,a,xa) & xa \<in> r & pair(L,x,b,xb) & xb \<in> r & |
|
431 |
(\<exists>fx[L]. \<exists>gx[L]. fun_apply(L,f,x,fx) & fun_apply(L,g,x,gx) & |
|
13323
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|
432 |
fx \<noteq> gx))" |
13428 | 433 |
apply (rule separation_CollectI) |
434 |
apply (rule_tac A="{r,f,g,a,b,z}" in subset_LsetE, blast ) |
|
13323
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More relativization, reflection and proofs of separation
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|
435 |
apply (rule ReflectsE [OF is_recfun_reflects], assumption) |
13428 | 436 |
apply (drule subset_Lset_ltD, assumption) |
13323
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|
437 |
apply (erule reflection_imp_L_separation) |
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|
438 |
apply (simp_all add: lt_Ord2) |
13385
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Expressing Lset and L without using length and arity; simplifies Separation
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parents:
13363
diff
changeset
|
439 |
apply (rule DPow_LsetI) |
13428 | 440 |
apply (rename_tac u) |
13323
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|
441 |
apply (rule bex_iff_sats conj_iff_sats)+ |
13428 | 442 |
apply (rule_tac env = "[xa,u,r,f,g,a,b]" in pair_iff_sats) |
13323
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|
443 |
apply (rule sep_rules | simp)+ |
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|
444 |
done |
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|
445 |
|
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|
446 |
|
13363 | 447 |
subsection{*Instantiating the locale @{text M_axioms}*} |
448 |
text{*Separation (and Strong Replacement) for basic set-theoretic constructions |
|
449 |
such as intersection, Cartesian Product and image.*} |
|
450 |
||
13428 | 451 |
theorem M_axioms_axioms_L: "M_axioms_axioms(L)" |
452 |
apply (rule M_axioms_axioms.intro) |
|
453 |
apply (assumption | rule |
|
454 |
Inter_separation cartprod_separation image_separation |
|
455 |
converse_separation restrict_separation |
|
456 |
comp_separation pred_separation Memrel_separation |
|
457 |
funspace_succ_replacement well_ord_iso_separation |
|
458 |
obase_separation obase_equals_separation |
|
459 |
omap_replacement is_recfun_separation)+ |
|
460 |
done |
|
461 |
||
462 |
theorem M_axioms_L: "PROP M_axioms(L)" |
|
463 |
apply (rule M_axioms.intro) |
|
464 |
apply (rule M_triv_axioms_L) |
|
465 |
apply (rule M_axioms_axioms_L) |
|
466 |
done |
|
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|
467 |
|
13428 | 468 |
lemmas cartprod_iff = M_axioms.cartprod_iff [OF M_axioms_L] |
469 |
and cartprod_closed = M_axioms.cartprod_closed [OF M_axioms_L] |
|
470 |
and sum_closed = M_axioms.sum_closed [OF M_axioms_L] |
|
471 |
and M_converse_iff = M_axioms.M_converse_iff [OF M_axioms_L] |
|
472 |
and converse_closed = M_axioms.converse_closed [OF M_axioms_L] |
|
473 |
and converse_abs = M_axioms.converse_abs [OF M_axioms_L] |
|
474 |
and image_closed = M_axioms.image_closed [OF M_axioms_L] |
|
475 |
and vimage_abs = M_axioms.vimage_abs [OF M_axioms_L] |
|
476 |
and vimage_closed = M_axioms.vimage_closed [OF M_axioms_L] |
|
477 |
and domain_abs = M_axioms.domain_abs [OF M_axioms_L] |
|
478 |
and domain_closed = M_axioms.domain_closed [OF M_axioms_L] |
|
479 |
and range_abs = M_axioms.range_abs [OF M_axioms_L] |
|
480 |
and range_closed = M_axioms.range_closed [OF M_axioms_L] |
|
481 |
and field_abs = M_axioms.field_abs [OF M_axioms_L] |
|
482 |
and field_closed = M_axioms.field_closed [OF M_axioms_L] |
|
483 |
and relation_abs = M_axioms.relation_abs [OF M_axioms_L] |
|
484 |
and function_abs = M_axioms.function_abs [OF M_axioms_L] |
|
485 |
and apply_closed = M_axioms.apply_closed [OF M_axioms_L] |
|
486 |
and apply_abs = M_axioms.apply_abs [OF M_axioms_L] |
|
487 |
and typed_function_abs = M_axioms.typed_function_abs [OF M_axioms_L] |
|
488 |
and injection_abs = M_axioms.injection_abs [OF M_axioms_L] |
|
489 |
and surjection_abs = M_axioms.surjection_abs [OF M_axioms_L] |
|
490 |
and bijection_abs = M_axioms.bijection_abs [OF M_axioms_L] |
|
491 |
and M_comp_iff = M_axioms.M_comp_iff [OF M_axioms_L] |
|
492 |
and comp_closed = M_axioms.comp_closed [OF M_axioms_L] |
|
493 |
and composition_abs = M_axioms.composition_abs [OF M_axioms_L] |
|
494 |
and restriction_is_function = M_axioms.restriction_is_function [OF M_axioms_L] |
|
495 |
and restriction_abs = M_axioms.restriction_abs [OF M_axioms_L] |
|
496 |
and M_restrict_iff = M_axioms.M_restrict_iff [OF M_axioms_L] |
|
497 |
and restrict_closed = M_axioms.restrict_closed [OF M_axioms_L] |
|
498 |
and Inter_abs = M_axioms.Inter_abs [OF M_axioms_L] |
|
499 |
and Inter_closed = M_axioms.Inter_closed [OF M_axioms_L] |
|
500 |
and Int_closed = M_axioms.Int_closed [OF M_axioms_L] |
|
501 |
and finite_fun_closed = M_axioms.finite_fun_closed [OF M_axioms_L] |
|
502 |
and is_funspace_abs = M_axioms.is_funspace_abs [OF M_axioms_L] |
|
503 |
and succ_fun_eq2 = M_axioms.succ_fun_eq2 [OF M_axioms_L] |
|
504 |
and funspace_succ = M_axioms.funspace_succ [OF M_axioms_L] |
|
505 |
and finite_funspace_closed = M_axioms.finite_funspace_closed [OF M_axioms_L] |
|
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|
506 |
|
13428 | 507 |
lemmas is_recfun_equal = M_axioms.is_recfun_equal [OF M_axioms_L] |
508 |
and is_recfun_cut = M_axioms.is_recfun_cut [OF M_axioms_L] |
|
509 |
and is_recfun_functional = M_axioms.is_recfun_functional [OF M_axioms_L] |
|
510 |
and is_recfun_relativize = M_axioms.is_recfun_relativize [OF M_axioms_L] |
|
511 |
and is_recfun_restrict = M_axioms.is_recfun_restrict [OF M_axioms_L] |
|
512 |
and univalent_is_recfun = M_axioms.univalent_is_recfun [OF M_axioms_L] |
|
513 |
and exists_is_recfun_indstep = M_axioms.exists_is_recfun_indstep [OF M_axioms_L] |
|
514 |
and wellfounded_exists_is_recfun = M_axioms.wellfounded_exists_is_recfun [OF M_axioms_L] |
|
515 |
and wf_exists_is_recfun = M_axioms.wf_exists_is_recfun [OF M_axioms_L] |
|
516 |
and is_recfun_abs = M_axioms.is_recfun_abs [OF M_axioms_L] |
|
517 |
and irreflexive_abs = M_axioms.irreflexive_abs [OF M_axioms_L] |
|
518 |
and transitive_rel_abs = M_axioms.transitive_rel_abs [OF M_axioms_L] |
|
519 |
and linear_rel_abs = M_axioms.linear_rel_abs [OF M_axioms_L] |
|
520 |
and wellordered_is_trans_on = M_axioms.wellordered_is_trans_on [OF M_axioms_L] |
|
521 |
and wellordered_is_linear = M_axioms.wellordered_is_linear [OF M_axioms_L] |
|
522 |
and wellordered_is_wellfounded_on = M_axioms.wellordered_is_wellfounded_on [OF M_axioms_L] |
|
523 |
and wellfounded_imp_wellfounded_on = M_axioms.wellfounded_imp_wellfounded_on [OF M_axioms_L] |
|
524 |
and wellfounded_on_subset_A = M_axioms.wellfounded_on_subset_A [OF M_axioms_L] |
|
525 |
and wellfounded_on_iff_wellfounded = M_axioms.wellfounded_on_iff_wellfounded [OF M_axioms_L] |
|
526 |
and wellfounded_on_imp_wellfounded = M_axioms.wellfounded_on_imp_wellfounded [OF M_axioms_L] |
|
527 |
and wellfounded_on_field_imp_wellfounded = M_axioms.wellfounded_on_field_imp_wellfounded [OF M_axioms_L] |
|
528 |
and wellfounded_iff_wellfounded_on_field = M_axioms.wellfounded_iff_wellfounded_on_field [OF M_axioms_L] |
|
529 |
and wellfounded_induct = M_axioms.wellfounded_induct [OF M_axioms_L] |
|
530 |
and wellfounded_on_induct = M_axioms.wellfounded_on_induct [OF M_axioms_L] |
|
531 |
and wellfounded_on_induct2 = M_axioms.wellfounded_on_induct2 [OF M_axioms_L] |
|
532 |
and linear_imp_relativized = M_axioms.linear_imp_relativized [OF M_axioms_L] |
|
533 |
and trans_on_imp_relativized = M_axioms.trans_on_imp_relativized [OF M_axioms_L] |
|
534 |
and wf_on_imp_relativized = M_axioms.wf_on_imp_relativized [OF M_axioms_L] |
|
535 |
and wf_imp_relativized = M_axioms.wf_imp_relativized [OF M_axioms_L] |
|
536 |
and well_ord_imp_relativized = M_axioms.well_ord_imp_relativized [OF M_axioms_L] |
|
537 |
and order_isomorphism_abs = M_axioms.order_isomorphism_abs [OF M_axioms_L] |
|
538 |
and pred_set_abs = M_axioms.pred_set_abs [OF M_axioms_L] |
|
13323
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|
539 |
|
13428 | 540 |
lemmas pred_closed = M_axioms.pred_closed [OF M_axioms_L] |
541 |
and membership_abs = M_axioms.membership_abs [OF M_axioms_L] |
|
542 |
and M_Memrel_iff = M_axioms.M_Memrel_iff [OF M_axioms_L] |
|
543 |
and Memrel_closed = M_axioms.Memrel_closed [OF M_axioms_L] |
|
544 |
and wellordered_iso_predD = M_axioms.wellordered_iso_predD [OF M_axioms_L] |
|
545 |
and wellordered_iso_pred_eq = M_axioms.wellordered_iso_pred_eq [OF M_axioms_L] |
|
546 |
and wellfounded_on_asym = M_axioms.wellfounded_on_asym [OF M_axioms_L] |
|
547 |
and wellordered_asym = M_axioms.wellordered_asym [OF M_axioms_L] |
|
548 |
and ord_iso_pred_imp_lt = M_axioms.ord_iso_pred_imp_lt [OF M_axioms_L] |
|
549 |
and obase_iff = M_axioms.obase_iff [OF M_axioms_L] |
|
550 |
and omap_iff = M_axioms.omap_iff [OF M_axioms_L] |
|
551 |
and omap_unique = M_axioms.omap_unique [OF M_axioms_L] |
|
552 |
and omap_yields_Ord = M_axioms.omap_yields_Ord [OF M_axioms_L] |
|
553 |
and otype_iff = M_axioms.otype_iff [OF M_axioms_L] |
|
554 |
and otype_eq_range = M_axioms.otype_eq_range [OF M_axioms_L] |
|
555 |
and Ord_otype = M_axioms.Ord_otype [OF M_axioms_L] |
|
556 |
and domain_omap = M_axioms.domain_omap [OF M_axioms_L] |
|
557 |
and omap_subset = M_axioms.omap_subset [OF M_axioms_L] |
|
558 |
and omap_funtype = M_axioms.omap_funtype [OF M_axioms_L] |
|
559 |
and wellordered_omap_bij = M_axioms.wellordered_omap_bij [OF M_axioms_L] |
|
560 |
and omap_ord_iso = M_axioms.omap_ord_iso [OF M_axioms_L] |
|
561 |
and Ord_omap_image_pred = M_axioms.Ord_omap_image_pred [OF M_axioms_L] |
|
562 |
and restrict_omap_ord_iso = M_axioms.restrict_omap_ord_iso [OF M_axioms_L] |
|
563 |
and obase_equals = M_axioms.obase_equals [OF M_axioms_L] |
|
564 |
and omap_ord_iso_otype = M_axioms.omap_ord_iso_otype [OF M_axioms_L] |
|
565 |
and obase_exists = M_axioms.obase_exists [OF M_axioms_L] |
|
566 |
and omap_exists = M_axioms.omap_exists [OF M_axioms_L] |
|
567 |
and otype_exists = M_axioms.otype_exists [OF M_axioms_L] |
|
568 |
and omap_ord_iso_otype' = M_axioms.omap_ord_iso_otype' [OF M_axioms_L] |
|
569 |
and ordertype_exists = M_axioms.ordertype_exists [OF M_axioms_L] |
|
570 |
and relativized_imp_well_ord = M_axioms.relativized_imp_well_ord [OF M_axioms_L] |
|
571 |
and well_ord_abs = M_axioms.well_ord_abs [OF M_axioms_L] |
|
572 |
||
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|
573 |
|
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|
574 |
declare cartprod_closed [intro,simp] |
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|
575 |
declare sum_closed [intro,simp] |
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|
576 |
declare converse_closed [intro,simp] |
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|
577 |
declare converse_abs [simp] |
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|
578 |
declare image_closed [intro,simp] |
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|
579 |
declare vimage_abs [simp] |
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|
580 |
declare vimage_closed [intro,simp] |
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|
581 |
declare domain_abs [simp] |
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|
582 |
declare domain_closed [intro,simp] |
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|
583 |
declare range_abs [simp] |
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|
584 |
declare range_closed [intro,simp] |
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|
585 |
declare field_abs [simp] |
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|
586 |
declare field_closed [intro,simp] |
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|
587 |
declare relation_abs [simp] |
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|
588 |
declare function_abs [simp] |
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|
589 |
declare apply_closed [intro,simp] |
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|
590 |
declare typed_function_abs [simp] |
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|
591 |
declare injection_abs [simp] |
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|
592 |
declare surjection_abs [simp] |
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|
593 |
declare bijection_abs [simp] |
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|
594 |
declare comp_closed [intro,simp] |
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|
595 |
declare composition_abs [simp] |
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|
596 |
declare restriction_abs [simp] |
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|
597 |
declare restrict_closed [intro,simp] |
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|
598 |
declare Inter_abs [simp] |
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|
599 |
declare Inter_closed [intro,simp] |
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|
600 |
declare Int_closed [intro,simp] |
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|
601 |
declare is_funspace_abs [simp] |
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|
602 |
declare finite_funspace_closed [intro,simp] |
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|
603 |
|
13306 | 604 |
end |