src/ZF/Constructible/Rank_Separation.thy
 author ballarin Tue Jun 20 15:53:44 2006 +0200 (2006-06-20) changeset 19931 fb32b43e7f80 parent 16417 9bc16273c2d4 child 32960 69916a850301 permissions -rw-r--r--
Restructured locales with predicates: import is now an interpretation.
New method intro_locales.
```     1 (*  Title:      ZF/Constructible/Rank_Separation.thy
```
```     2     ID:   \$Id\$
```
```     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
```
```     4 *)
```
```     5
```
```     6 header {*Separation for Facts About Order Types, Rank Functions and
```
```     7       Well-Founded Relations*}
```
```     8
```
```     9 theory Rank_Separation imports Rank Rec_Separation begin
```
```    10
```
```    11
```
```    12 text{*This theory proves all instances needed for locales
```
```    13  @{text "M_ordertype"} and  @{text "M_wfrank"}.  But the material is not
```
```    14  needed for proving the relative consistency of AC.*}
```
```    15
```
```    16 subsection{*The Locale @{text "M_ordertype"}*}
```
```    17
```
```    18 subsubsection{*Separation for Order-Isomorphisms*}
```
```    19
```
```    20 lemma well_ord_iso_Reflects:
```
```    21   "REFLECTS[\<lambda>x. x\<in>A -->
```
```    22                 (\<exists>y[L]. \<exists>p[L]. fun_apply(L,f,x,y) & pair(L,y,x,p) & p \<in> r),
```
```    23         \<lambda>i x. x\<in>A --> (\<exists>y \<in> Lset(i). \<exists>p \<in> Lset(i).
```
```    24                 fun_apply(##Lset(i),f,x,y) & pair(##Lset(i),y,x,p) & p \<in> r)]"
```
```    25 by (intro FOL_reflections function_reflections)
```
```    26
```
```    27 lemma well_ord_iso_separation:
```
```    28      "[| L(A); L(f); L(r) |]
```
```    29       ==> separation (L, \<lambda>x. x\<in>A --> (\<exists>y[L]. (\<exists>p[L].
```
```    30                      fun_apply(L,f,x,y) & pair(L,y,x,p) & p \<in> r)))"
```
```    31 apply (rule gen_separation_multi [OF well_ord_iso_Reflects, of "{A,f,r}"],
```
```    32        auto)
```
```    33 apply (rule_tac env="[A,f,r]" in DPow_LsetI)
```
```    34 apply (rule sep_rules | simp)+
```
```    35 done
```
```    36
```
```    37
```
```    38 subsubsection{*Separation for @{term "obase"}*}
```
```    39
```
```    40 lemma obase_reflects:
```
```    41   "REFLECTS[\<lambda>a. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L].
```
```    42              ordinal(L,x) & membership(L,x,mx) & pred_set(L,A,a,r,par) &
```
```    43              order_isomorphism(L,par,r,x,mx,g),
```
```    44         \<lambda>i a. \<exists>x \<in> Lset(i). \<exists>g \<in> Lset(i). \<exists>mx \<in> Lset(i). \<exists>par \<in> Lset(i).
```
```    45              ordinal(##Lset(i),x) & membership(##Lset(i),x,mx) & pred_set(##Lset(i),A,a,r,par) &
```
```    46              order_isomorphism(##Lset(i),par,r,x,mx,g)]"
```
```    47 by (intro FOL_reflections function_reflections fun_plus_reflections)
```
```    48
```
```    49 lemma obase_separation:
```
```    50      --{*part of the order type formalization*}
```
```    51      "[| L(A); L(r) |]
```
```    52       ==> separation(L, \<lambda>a. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L].
```
```    53              ordinal(L,x) & membership(L,x,mx) & pred_set(L,A,a,r,par) &
```
```    54              order_isomorphism(L,par,r,x,mx,g))"
```
```    55 apply (rule gen_separation_multi [OF obase_reflects, of "{A,r}"], auto)
```
```    56 apply (rule_tac env="[A,r]" in DPow_LsetI)
```
```    57 apply (rule ordinal_iff_sats sep_rules | simp)+
```
```    58 done
```
```    59
```
```    60
```
```    61 subsubsection{*Separation for a Theorem about @{term "obase"}*}
```
```    62
```
```    63 lemma obase_equals_reflects:
```
```    64   "REFLECTS[\<lambda>x. x\<in>A --> ~(\<exists>y[L]. \<exists>g[L].
```
```    65                 ordinal(L,y) & (\<exists>my[L]. \<exists>pxr[L].
```
```    66                 membership(L,y,my) & pred_set(L,A,x,r,pxr) &
```
```    67                 order_isomorphism(L,pxr,r,y,my,g))),
```
```    68         \<lambda>i x. x\<in>A --> ~(\<exists>y \<in> Lset(i). \<exists>g \<in> Lset(i).
```
```    69                 ordinal(##Lset(i),y) & (\<exists>my \<in> Lset(i). \<exists>pxr \<in> Lset(i).
```
```    70                 membership(##Lset(i),y,my) & pred_set(##Lset(i),A,x,r,pxr) &
```
```    71                 order_isomorphism(##Lset(i),pxr,r,y,my,g)))]"
```
```    72 by (intro FOL_reflections function_reflections fun_plus_reflections)
```
```    73
```
```    74 lemma obase_equals_separation:
```
```    75      "[| L(A); L(r) |]
```
```    76       ==> separation (L, \<lambda>x. x\<in>A --> ~(\<exists>y[L]. \<exists>g[L].
```
```    77                               ordinal(L,y) & (\<exists>my[L]. \<exists>pxr[L].
```
```    78                               membership(L,y,my) & pred_set(L,A,x,r,pxr) &
```
```    79                               order_isomorphism(L,pxr,r,y,my,g))))"
```
```    80 apply (rule gen_separation_multi [OF obase_equals_reflects, of "{A,r}"], auto)
```
```    81 apply (rule_tac env="[A,r]" in DPow_LsetI)
```
```    82 apply (rule sep_rules | simp)+
```
```    83 done
```
```    84
```
```    85
```
```    86 subsubsection{*Replacement for @{term "omap"}*}
```
```    87
```
```    88 lemma omap_reflects:
```
```    89  "REFLECTS[\<lambda>z. \<exists>a[L]. a\<in>B & (\<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L].
```
```    90      ordinal(L,x) & pair(L,a,x,z) & membership(L,x,mx) &
```
```    91      pred_set(L,A,a,r,par) & order_isomorphism(L,par,r,x,mx,g)),
```
```    92  \<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).
```
```    93         \<exists>par \<in> Lset(i).
```
```    94          ordinal(##Lset(i),x) & pair(##Lset(i),a,x,z) &
```
```    95          membership(##Lset(i),x,mx) & pred_set(##Lset(i),A,a,r,par) &
```
```    96          order_isomorphism(##Lset(i),par,r,x,mx,g))]"
```
```    97 by (intro FOL_reflections function_reflections fun_plus_reflections)
```
```    98
```
```    99 lemma omap_replacement:
```
```   100      "[| L(A); L(r) |]
```
```   101       ==> strong_replacement(L,
```
```   102              \<lambda>a z. \<exists>x[L]. \<exists>g[L]. \<exists>mx[L]. \<exists>par[L].
```
```   103              ordinal(L,x) & pair(L,a,x,z) & membership(L,x,mx) &
```
```   104              pred_set(L,A,a,r,par) & order_isomorphism(L,par,r,x,mx,g))"
```
```   105 apply (rule strong_replacementI)
```
```   106 apply (rule_tac u="{A,r,B}" in gen_separation_multi [OF omap_reflects], auto)
```
```   107 apply (rule_tac env="[A,B,r]" in DPow_LsetI)
```
```   108 apply (rule sep_rules | simp)+
```
```   109 done
```
```   110
```
```   111
```
```   112
```
```   113 subsection{*Instantiating the locale @{text M_ordertype}*}
```
```   114 text{*Separation (and Strong Replacement) for basic set-theoretic constructions
```
```   115 such as intersection, Cartesian Product and image.*}
```
```   116
```
```   117 lemma M_ordertype_axioms_L: "M_ordertype_axioms(L)"
```
```   118   apply (rule M_ordertype_axioms.intro)
```
```   119        apply (assumption | rule well_ord_iso_separation
```
```   120 	 obase_separation obase_equals_separation
```
```   121 	 omap_replacement)+
```
```   122   done
```
```   123
```
```   124 theorem M_ordertype_L: "PROP M_ordertype(L)"
```
```   125   apply (rule M_ordertype.intro)
```
```   126    apply (rule M_basic_L)
```
```   127   apply (rule M_ordertype_axioms_L)
```
```   128   done
```
```   129
```
```   130
```
```   131 subsection{*The Locale @{text "M_wfrank"}*}
```
```   132
```
```   133 subsubsection{*Separation for @{term "wfrank"}*}
```
```   134
```
```   135 lemma wfrank_Reflects:
```
```   136  "REFLECTS[\<lambda>x. \<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   137               ~ (\<exists>f[L]. M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f)),
```
```   138       \<lambda>i x. \<forall>rplus \<in> Lset(i). tran_closure(##Lset(i),r,rplus) -->
```
```   139          ~ (\<exists>f \<in> Lset(i).
```
```   140             M_is_recfun(##Lset(i), %x f y. is_range(##Lset(i),f,y),
```
```   141                         rplus, x, f))]"
```
```   142 by (intro FOL_reflections function_reflections is_recfun_reflection tran_closure_reflection)
```
```   143
```
```   144 lemma wfrank_separation:
```
```   145      "L(r) ==>
```
```   146       separation (L, \<lambda>x. \<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   147          ~ (\<exists>f[L]. M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f)))"
```
```   148 apply (rule gen_separation [OF wfrank_Reflects], simp)
```
```   149 apply (rule_tac env="[r]" in DPow_LsetI)
```
```   150 apply (rule sep_rules tran_closure_iff_sats is_recfun_iff_sats | simp)+
```
```   151 done
```
```   152
```
```   153
```
```   154 subsubsection{*Replacement for @{term "wfrank"}*}
```
```   155
```
```   156 lemma wfrank_replacement_Reflects:
```
```   157  "REFLECTS[\<lambda>z. \<exists>x[L]. x \<in> A &
```
```   158         (\<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   159          (\<exists>y[L]. \<exists>f[L]. pair(L,x,y,z)  &
```
```   160                         M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f) &
```
```   161                         is_range(L,f,y))),
```
```   162  \<lambda>i z. \<exists>x \<in> Lset(i). x \<in> A &
```
```   163       (\<forall>rplus \<in> Lset(i). tran_closure(##Lset(i),r,rplus) -->
```
```   164        (\<exists>y \<in> Lset(i). \<exists>f \<in> Lset(i). pair(##Lset(i),x,y,z)  &
```
```   165          M_is_recfun(##Lset(i), %x f y. is_range(##Lset(i),f,y), rplus, x, f) &
```
```   166          is_range(##Lset(i),f,y)))]"
```
```   167 by (intro FOL_reflections function_reflections fun_plus_reflections
```
```   168              is_recfun_reflection tran_closure_reflection)
```
```   169
```
```   170 lemma wfrank_strong_replacement:
```
```   171      "L(r) ==>
```
```   172       strong_replacement(L, \<lambda>x z.
```
```   173          \<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   174          (\<exists>y[L]. \<exists>f[L]. pair(L,x,y,z)  &
```
```   175                         M_is_recfun(L, %x f y. is_range(L,f,y), rplus, x, f) &
```
```   176                         is_range(L,f,y)))"
```
```   177 apply (rule strong_replacementI)
```
```   178 apply (rule_tac u="{r,B}"
```
```   179          in gen_separation_multi [OF wfrank_replacement_Reflects],
```
```   180        auto)
```
```   181 apply (rule_tac env="[r,B]" in DPow_LsetI)
```
```   182 apply (rule sep_rules tran_closure_iff_sats is_recfun_iff_sats | simp)+
```
```   183 done
```
```   184
```
```   185
```
```   186 subsubsection{*Separation for Proving @{text Ord_wfrank_range}*}
```
```   187
```
```   188 lemma Ord_wfrank_Reflects:
```
```   189  "REFLECTS[\<lambda>x. \<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   190           ~ (\<forall>f[L]. \<forall>rangef[L].
```
```   191              is_range(L,f,rangef) -->
```
```   192              M_is_recfun(L, \<lambda>x f y. is_range(L,f,y), rplus, x, f) -->
```
```   193              ordinal(L,rangef)),
```
```   194       \<lambda>i x. \<forall>rplus \<in> Lset(i). tran_closure(##Lset(i),r,rplus) -->
```
```   195           ~ (\<forall>f \<in> Lset(i). \<forall>rangef \<in> Lset(i).
```
```   196              is_range(##Lset(i),f,rangef) -->
```
```   197              M_is_recfun(##Lset(i), \<lambda>x f y. is_range(##Lset(i),f,y),
```
```   198                          rplus, x, f) -->
```
```   199              ordinal(##Lset(i),rangef))]"
```
```   200 by (intro FOL_reflections function_reflections is_recfun_reflection
```
```   201           tran_closure_reflection ordinal_reflection)
```
```   202
```
```   203 lemma  Ord_wfrank_separation:
```
```   204      "L(r) ==>
```
```   205       separation (L, \<lambda>x.
```
```   206          \<forall>rplus[L]. tran_closure(L,r,rplus) -->
```
```   207           ~ (\<forall>f[L]. \<forall>rangef[L].
```
```   208              is_range(L,f,rangef) -->
```
```   209              M_is_recfun(L, \<lambda>x f y. is_range(L,f,y), rplus, x, f) -->
```
```   210              ordinal(L,rangef)))"
```
```   211 apply (rule gen_separation [OF Ord_wfrank_Reflects], simp)
```
```   212 apply (rule_tac env="[r]" in DPow_LsetI)
```
```   213 apply (rule sep_rules tran_closure_iff_sats is_recfun_iff_sats | simp)+
```
```   214 done
```
```   215
```
```   216
```
```   217 subsubsection{*Instantiating the locale @{text M_wfrank}*}
```
```   218
```
```   219 lemma M_wfrank_axioms_L: "M_wfrank_axioms(L)"
```
```   220   apply (rule M_wfrank_axioms.intro)
```
```   221    apply (assumption | rule
```
```   222      wfrank_separation wfrank_strong_replacement Ord_wfrank_separation)+
```
```   223   done
```
```   224
```
```   225 theorem M_wfrank_L: "PROP M_wfrank(L)"
```
```   226   apply (rule M_wfrank.intro)
```
```   227    apply (rule M_trancl_L)
```
```   228   apply (rule M_wfrank_axioms_L)
```
```   229   done
```
```   230
```
```   231 lemmas exists_wfrank = M_wfrank.exists_wfrank [OF M_wfrank_L]
```
```   232   and M_wellfoundedrank = M_wfrank.M_wellfoundedrank [OF M_wfrank_L]
```
```   233   and Ord_wfrank_range = M_wfrank.Ord_wfrank_range [OF M_wfrank_L]
```
```   234   and Ord_range_wellfoundedrank = M_wfrank.Ord_range_wellfoundedrank [OF M_wfrank_L]
```
```   235   and function_wellfoundedrank = M_wfrank.function_wellfoundedrank [OF M_wfrank_L]
```
```   236   and domain_wellfoundedrank = M_wfrank.domain_wellfoundedrank [OF M_wfrank_L]
```
```   237   and wellfoundedrank_type = M_wfrank.wellfoundedrank_type [OF M_wfrank_L]
```
```   238   and Ord_wellfoundedrank = M_wfrank.Ord_wellfoundedrank [OF M_wfrank_L]
```
```   239   and wellfoundedrank_eq = M_wfrank.wellfoundedrank_eq [OF M_wfrank_L]
```
```   240   and wellfoundedrank_lt = M_wfrank.wellfoundedrank_lt [OF M_wfrank_L]
```
```   241   and wellfounded_imp_subset_rvimage = M_wfrank.wellfounded_imp_subset_rvimage [OF M_wfrank_L]
```
```   242   and wellfounded_imp_wf = M_wfrank.wellfounded_imp_wf [OF M_wfrank_L]
```
```   243   and wellfounded_on_imp_wf_on = M_wfrank.wellfounded_on_imp_wf_on [OF M_wfrank_L]
```
```   244   and wf_abs = M_wfrank.wf_abs [OF M_wfrank_L]
```
```   245   and wf_on_abs = M_wfrank.wf_on_abs [OF M_wfrank_L]
```
```   246
```
`   247 end`