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header {*Relativization and Absoluteness*}
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theory Relative = Main:
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subsection{* Relativized versions of standard set-theoretic concepts *}
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constdefs
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empty :: "[i=>o,i] => o"
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"empty(M,z) == \<forall>x. M(x) --> x \<notin> z"
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subset :: "[i=>o,i,i] => o"
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"subset(M,A,B) == \<forall>x\<in>A. M(x) --> x \<in> B"
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upair :: "[i=>o,i,i,i] => o"
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"upair(M,a,b,z) == a \<in> z & b \<in> z & (\<forall>x\<in>z. M(x) --> x = a | x = b)"
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pair :: "[i=>o,i,i,i] => o"
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"pair(M,a,b,z) == \<exists>x. M(x) & upair(M,a,a,x) &
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(\<exists>y. M(y) & upair(M,a,b,y) & upair(M,x,y,z))"
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union :: "[i=>o,i,i,i] => o"
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"union(M,a,b,z) == \<forall>x. M(x) --> (x \<in> z <-> x \<in> a | x \<in> b)"
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successor :: "[i=>o,i,i] => o"
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"successor(M,a,z) == \<exists>x. M(x) & upair(M,a,a,x) & union(M,x,a,z)"
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powerset :: "[i=>o,i,i] => o"
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"powerset(M,A,z) == \<forall>x. M(x) --> (x \<in> z <-> subset(M,x,A))"
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inter :: "[i=>o,i,i,i] => o"
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"inter(M,a,b,z) == \<forall>x. M(x) --> (x \<in> z <-> x \<in> a & x \<in> b)"
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setdiff :: "[i=>o,i,i,i] => o"
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"setdiff(M,a,b,z) == \<forall>x. M(x) --> (x \<in> z <-> x \<in> a & x \<notin> b)"
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big_union :: "[i=>o,i,i] => o"
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"big_union(M,A,z) == \<forall>x. M(x) --> (x \<in> z <-> (\<exists>y\<in>A. M(y) & x \<in> y))"
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big_inter :: "[i=>o,i,i] => o"
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"big_inter(M,A,z) ==
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(A=0 --> z=0) &
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(A\<noteq>0 --> (\<forall>x. M(x) --> (x \<in> z <-> (\<forall>y\<in>A. M(y) --> x \<in> y))))"
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cartprod :: "[i=>o,i,i,i] => o"
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"cartprod(M,A,B,z) ==
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\<forall>u. M(u) --> (u \<in> z <-> (\<exists>x\<in>A. M(x) & (\<exists>y\<in>B. M(y) & pair(M,x,y,u))))"
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is_converse :: "[i=>o,i,i] => o"
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"is_converse(M,r,z) ==
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\<forall>x. M(x) -->
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(x \<in> z <->
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(\<exists>w\<in>r. M(w) &
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(\<exists>u v. M(u) & M(v) & pair(M,u,v,w) & pair(M,v,u,x))))"
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pre_image :: "[i=>o,i,i,i] => o"
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"pre_image(M,r,A,z) ==
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\<forall>x. M(x) --> (x \<in> z <-> (\<exists>w\<in>r. M(w) & (\<exists>y\<in>A. M(y) & pair(M,x,y,w))))"
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is_domain :: "[i=>o,i,i] => o"
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"is_domain(M,r,z) ==
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\<forall>x. M(x) --> (x \<in> z <-> (\<exists>w\<in>r. M(w) & (\<exists>y. M(y) & pair(M,x,y,w))))"
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image :: "[i=>o,i,i,i] => o"
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"image(M,r,A,z) ==
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\<forall>y. M(y) --> (y \<in> z <-> (\<exists>w\<in>r. M(w) & (\<exists>x\<in>A. M(x) & pair(M,x,y,w))))"
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is_range :: "[i=>o,i,i] => o"
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--{*the cleaner
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@{term "\<exists>r'. M(r') & is_converse(M,r,r') & is_domain(M,r',z)"}
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unfortunately needs an instance of separation in order to prove
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@{term "M(converse(r))"}.*}
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"is_range(M,r,z) ==
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\<forall>y. M(y) --> (y \<in> z <-> (\<exists>w\<in>r. M(w) & (\<exists>x. M(x) & pair(M,x,y,w))))"
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is_field :: "[i=>o,i,i] => o"
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"is_field(M,r,z) ==
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\<exists>dr. M(dr) & is_domain(M,r,dr) &
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(\<exists>rr. M(rr) & is_range(M,r,rr) & union(M,dr,rr,z))"
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is_relation :: "[i=>o,i] => o"
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"is_relation(M,r) ==
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(\<forall>z\<in>r. M(z) --> (\<exists>x y. M(x) & M(y) & pair(M,x,y,z)))"
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is_function :: "[i=>o,i] => o"
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"is_function(M,r) ==
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(\<forall>x y y' p p'. M(x) --> M(y) --> M(y') --> M(p) --> M(p') -->
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pair(M,x,y,p) --> pair(M,x,y',p') --> p\<in>r --> p'\<in>r -->
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y=y')"
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fun_apply :: "[i=>o,i,i,i] => o"
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"fun_apply(M,f,x,y) ==
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(\<forall>y'. M(y') --> ((\<exists>u\<in>f. M(u) & pair(M,x,y',u)) <-> y=y'))"
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typed_function :: "[i=>o,i,i,i] => o"
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"typed_function(M,A,B,r) ==
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is_function(M,r) & is_relation(M,r) & is_domain(M,r,A) &
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(\<forall>u\<in>r. M(u) --> (\<forall>x y. M(x) & M(y) & pair(M,x,y,u) --> y\<in>B))"
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composition :: "[i=>o,i,i,i] => o"
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"composition(M,r,s,t) ==
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\<forall>p. M(p) --> (p \<in> t <->
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(\<exists>x. M(x) & (\<exists>y. M(y) & (\<exists>z. M(z) &
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p = \<langle>x,z\<rangle> & \<langle>x,y\<rangle> \<in> s & \<langle>y,z\<rangle> \<in> r))))"
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injection :: "[i=>o,i,i,i] => o"
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"injection(M,A,B,f) ==
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typed_function(M,A,B,f) &
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(\<forall>x x' y p p'. M(x) --> M(x') --> M(y) --> M(p) --> M(p') -->
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pair(M,x,y,p) --> pair(M,x',y,p') --> p\<in>f --> p'\<in>f -->
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x=x')"
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surjection :: "[i=>o,i,i,i] => o"
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"surjection(M,A,B,f) ==
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typed_function(M,A,B,f) &
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(\<forall>y\<in>B. M(y) --> (\<exists>x\<in>A. M(x) & fun_apply(M,f,x,y)))"
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bijection :: "[i=>o,i,i,i] => o"
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"bijection(M,A,B,f) == injection(M,A,B,f) & surjection(M,A,B,f)"
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restriction :: "[i=>o,i,i,i] => o"
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"restriction(M,r,A,z) ==
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\<forall>x. M(x) -->
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(x \<in> z <->
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(x \<in> r & (\<exists>u\<in>A. M(u) & (\<exists>v. M(v) & pair(M,u,v,x)))))"
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transitive_set :: "[i=>o,i] => o"
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"transitive_set(M,a) == \<forall>x\<in>a. M(x) --> subset(M,x,a)"
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ordinal :: "[i=>o,i] => o"
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--{*an ordinal is a transitive set of transitive sets*}
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"ordinal(M,a) == transitive_set(M,a) & (\<forall>x\<in>a. M(x) --> transitive_set(M,x))"
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limit_ordinal :: "[i=>o,i] => o"
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--{*a limit ordinal is a non-empty, successor-closed ordinal*}
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"limit_ordinal(M,a) ==
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ordinal(M,a) & ~ empty(M,a) &
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(\<forall>x\<in>a. M(x) --> (\<exists>y\<in>a. M(y) & successor(M,x,y)))"
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successor_ordinal :: "[i=>o,i] => o"
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--{*a successor ordinal is any ordinal that is neither empty nor limit*}
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"successor_ordinal(M,a) ==
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ordinal(M,a) & ~ empty(M,a) & ~ limit_ordinal(M,a)"
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finite_ordinal :: "[i=>o,i] => o"
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--{*an ordinal is finite if neither it nor any of its elements are limit*}
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"finite_ordinal(M,a) ==
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ordinal(M,a) & ~ limit_ordinal(M,a) &
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(\<forall>x\<in>a. M(x) --> ~ limit_ordinal(M,x))"
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omega :: "[i=>o,i] => o"
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--{*omega is a limit ordinal none of whose elements are limit*}
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"omega(M,a) == limit_ordinal(M,a) & (\<forall>x\<in>a. M(x) --> ~ limit_ordinal(M,x))"
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number1 :: "[i=>o,i] => o"
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"number1(M,a) == (\<exists>x. M(x) & empty(M,x) & successor(M,x,a))"
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number2 :: "[i=>o,i] => o"
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"number2(M,a) == (\<exists>x. M(x) & number1(M,x) & successor(M,x,a))"
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number3 :: "[i=>o,i] => o"
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"number3(M,a) == (\<exists>x. M(x) & number2(M,x) & successor(M,x,a))"
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subsection {*The relativized ZF axioms*}
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constdefs
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extensionality :: "(i=>o) => o"
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"extensionality(M) ==
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\<forall>x y. M(x) --> M(y) --> (\<forall>z. M(z) --> (z \<in> x <-> z \<in> y)) --> x=y"
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separation :: "[i=>o, i=>o] => o"
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--{*Big problem: the formula @{text P} should only involve parameters
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belonging to @{text M}. Don't see how to enforce that.*}
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"separation(M,P) ==
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\<forall>z. M(z) --> (\<exists>y. M(y) & (\<forall>x. M(x) --> (x \<in> y <-> x \<in> z & P(x))))"
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upair_ax :: "(i=>o) => o"
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"upair_ax(M) == \<forall>x y. M(x) --> M(y) --> (\<exists>z. M(z) & upair(M,x,y,z))"
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Union_ax :: "(i=>o) => o"
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"Union_ax(M) == \<forall>x. M(x) --> (\<exists>z. M(z) & big_union(M,x,z))"
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power_ax :: "(i=>o) => o"
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"power_ax(M) == \<forall>x. M(x) --> (\<exists>z. M(z) & powerset(M,x,z))"
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univalent :: "[i=>o, i, [i,i]=>o] => o"
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"univalent(M,A,P) ==
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(\<forall>x\<in>A. M(x) --> (\<forall>y z. M(y) --> M(z) --> P(x,y) & P(x,z) --> y=z))"
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replacement :: "[i=>o, [i,i]=>o] => o"
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"replacement(M,P) ==
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\<forall>A. M(A) --> univalent(M,A,P) -->
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(\<exists>Y. M(Y) & (\<forall>b. M(b) --> ((\<exists>x\<in>A. M(x) & P(x,b)) --> b \<in> Y)))"
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strong_replacement :: "[i=>o, [i,i]=>o] => o"
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"strong_replacement(M,P) ==
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\<forall>A. M(A) --> univalent(M,A,P) -->
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(\<exists>Y. M(Y) & (\<forall>b. M(b) --> (b \<in> Y <-> (\<exists>x\<in>A. M(x) & P(x,b)))))"
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foundation_ax :: "(i=>o) => o"
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"foundation_ax(M) ==
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\<forall>x. M(x) --> (\<exists>y\<in>x. M(y))
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--> (\<exists>y\<in>x. M(y) & ~(\<exists>z\<in>x. M(z) & z \<in> y))"
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subsection{*A trivial consistency proof for $V_\omega$ *}
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text{*We prove that $V_\omega$
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(or @{text univ} in Isabelle) satisfies some ZF axioms.
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Kunen, Theorem IV 3.13, page 123.*}
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lemma univ0_downwards_mem: "[| y \<in> x; x \<in> univ(0) |] ==> y \<in> univ(0)"
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apply (insert Transset_univ [OF Transset_0])
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apply (simp add: Transset_def, blast)
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done
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lemma univ0_Ball_abs [simp]:
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"A \<in> univ(0) ==> (\<forall>x\<in>A. x \<in> univ(0) --> P(x)) <-> (\<forall>x\<in>A. P(x))"
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by (blast intro: univ0_downwards_mem)
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lemma univ0_Bex_abs [simp]:
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"A \<in> univ(0) ==> (\<exists>x\<in>A. x \<in> univ(0) & P(x)) <-> (\<exists>x\<in>A. P(x))"
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by (blast intro: univ0_downwards_mem)
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text{*Congruence rule for separation: can assume the variable is in @{text M}*}
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lemma [cong]:
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"(!!x. M(x) ==> P(x) <-> P'(x)) ==> separation(M,P) <-> separation(M,P')"
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by (simp add: separation_def)
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text{*Congruence rules for replacement*}
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lemma [cong]:
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"[| A=A'; !!x y. [| x\<in>A; M(x); M(y) |] ==> P(x,y) <-> P'(x,y) |]
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==> univalent(M,A,P) <-> univalent(M,A',P')"
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by (simp add: univalent_def)
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lemma [cong]:
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"[| !!x y. [| M(x); M(y) |] ==> P(x,y) <-> P'(x,y) |]
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==> strong_replacement(M,P) <-> strong_replacement(M,P')"
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by (simp add: strong_replacement_def)
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text{*The extensionality axiom*}
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lemma "extensionality(\<lambda>x. x \<in> univ(0))"
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apply (simp add: extensionality_def)
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apply (blast intro: univ0_downwards_mem)
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done
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text{*The separation axiom requires some lemmas*}
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lemma Collect_in_Vfrom:
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"[| X \<in> Vfrom(A,j); Transset(A) |] ==> Collect(X,P) \<in> Vfrom(A, succ(j))"
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apply (drule Transset_Vfrom)
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apply (rule subset_mem_Vfrom)
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apply (unfold Transset_def, blast)
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done
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lemma Collect_in_VLimit:
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"[| X \<in> Vfrom(A,i); Limit(i); Transset(A) |]
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==> Collect(X,P) \<in> Vfrom(A,i)"
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apply (rule Limit_VfromE, assumption+)
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apply (blast intro: Limit_has_succ VfromI Collect_in_Vfrom)
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done
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lemma Collect_in_univ:
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"[| X \<in> univ(A); Transset(A) |] ==> Collect(X,P) \<in> univ(A)"
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by (simp add: univ_def Collect_in_VLimit Limit_nat)
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lemma "separation(\<lambda>x. x \<in> univ(0), P)"
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apply (simp add: separation_def)
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apply (blast intro: Collect_in_univ Transset_0)
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done
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text{*Unordered pairing axiom*}
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lemma "upair_ax(\<lambda>x. x \<in> univ(0))"
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apply (simp add: upair_ax_def upair_def)
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apply (blast intro: doubleton_in_univ)
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done
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text{*Union axiom*}
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lemma "Union_ax(\<lambda>x. x \<in> univ(0))"
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apply (simp add: Union_ax_def big_union_def)
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apply (blast intro: Union_in_univ Transset_0 univ0_downwards_mem)
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done
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text{*Powerset axiom*}
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lemma Pow_in_univ:
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"[| X \<in> univ(A); Transset(A) |] ==> Pow(X) \<in> univ(A)"
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apply (simp add: univ_def Pow_in_VLimit Limit_nat)
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done
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lemma "power_ax(\<lambda>x. x \<in> univ(0))"
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apply (simp add: power_ax_def powerset_def subset_def)
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apply (blast intro: Pow_in_univ Transset_0 univ0_downwards_mem)
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done
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text{*Foundation axiom*}
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lemma "foundation_ax(\<lambda>x. x \<in> univ(0))"
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apply (simp add: foundation_ax_def, clarify)
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apply (cut_tac A=x in foundation, blast)
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done
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|
|
302 |
lemma "replacement(\<lambda>x. x \<in> univ(0), P)"
|
|
303 |
apply (simp add: replacement_def, clarify)
|
|
304 |
oops
|
|
305 |
text{*no idea: maybe prove by induction on the rank of A?*}
|
|
306 |
|
|
307 |
text{*Still missing: Replacement, Choice*}
|
|
308 |
|
|
309 |
subsection{*lemmas needed to reduce some set constructions to instances
|
|
310 |
of Separation*}
|
|
311 |
|
|
312 |
lemma image_iff_Collect: "r `` A = {y \<in> Union(Union(r)). \<exists>p\<in>r. \<exists>x\<in>A. p=<x,y>}"
|
|
313 |
apply (rule equalityI, auto)
|
|
314 |
apply (simp add: Pair_def, blast)
|
|
315 |
done
|
|
316 |
|
|
317 |
lemma vimage_iff_Collect:
|
|
318 |
"r -`` A = {x \<in> Union(Union(r)). \<exists>p\<in>r. \<exists>y\<in>A. p=<x,y>}"
|
|
319 |
apply (rule equalityI, auto)
|
|
320 |
apply (simp add: Pair_def, blast)
|
|
321 |
done
|
|
322 |
|
|
323 |
text{*These two lemmas lets us prove @{text domain_closed} and
|
|
324 |
@{text range_closed} without new instances of separation*}
|
|
325 |
|
|
326 |
lemma domain_eq_vimage: "domain(r) = r -`` Union(Union(r))"
|
|
327 |
apply (rule equalityI, auto)
|
|
328 |
apply (rule vimageI, assumption)
|
|
329 |
apply (simp add: Pair_def, blast)
|
|
330 |
done
|
|
331 |
|
|
332 |
lemma range_eq_image: "range(r) = r `` Union(Union(r))"
|
|
333 |
apply (rule equalityI, auto)
|
|
334 |
apply (rule imageI, assumption)
|
|
335 |
apply (simp add: Pair_def, blast)
|
|
336 |
done
|
|
337 |
|
|
338 |
lemma replacementD:
|
|
339 |
"[| replacement(M,P); M(A); univalent(M,A,P) |]
|
|
340 |
==> \<exists>Y. M(Y) & (\<forall>b. M(b) --> ((\<exists>x\<in>A. M(x) & P(x,b)) --> b \<in> Y))"
|
|
341 |
by (simp add: replacement_def)
|
|
342 |
|
|
343 |
lemma strong_replacementD:
|
|
344 |
"[| strong_replacement(M,P); M(A); univalent(M,A,P) |]
|
|
345 |
==> \<exists>Y. M(Y) & (\<forall>b. M(b) --> (b \<in> Y <-> (\<exists>x\<in>A. M(x) & P(x,b))))"
|
|
346 |
by (simp add: strong_replacement_def)
|
|
347 |
|
|
348 |
lemma separationD:
|
|
349 |
"[| separation(M,P); M(z) |]
|
|
350 |
==> \<exists>y. M(y) & (\<forall>x. M(x) --> (x \<in> y <-> x \<in> z & P(x)))"
|
|
351 |
by (simp add: separation_def)
|
|
352 |
|
|
353 |
|
|
354 |
text{*More constants, for order types*}
|
|
355 |
constdefs
|
|
356 |
|
|
357 |
order_isomorphism :: "[i=>o,i,i,i,i,i] => o"
|
|
358 |
"order_isomorphism(M,A,r,B,s,f) ==
|
|
359 |
bijection(M,A,B,f) &
|
|
360 |
(\<forall>x\<in>A. \<forall>y\<in>A. \<forall>p fx fy q.
|
|
361 |
M(x) --> M(y) --> M(p) --> M(fx) --> M(fy) --> M(q) -->
|
|
362 |
pair(M,x,y,p) --> fun_apply(M,f,x,fx) --> fun_apply(M,f,y,fy) -->
|
|
363 |
pair(M,fx,fy,q) --> (p\<in>r <-> q\<in>s))"
|
|
364 |
|
|
365 |
|
|
366 |
pred_set :: "[i=>o,i,i,i,i] => o"
|
|
367 |
"pred_set(M,A,x,r,B) ==
|
|
368 |
\<forall>y. M(y) --> (y \<in> B <-> (\<exists>p\<in>r. M(p) & y \<in> A & pair(M,y,x,p)))"
|
|
369 |
|
|
370 |
membership :: "[i=>o,i,i] => o" --{*membership relation*}
|
|
371 |
"membership(M,A,r) ==
|
|
372 |
\<forall>p. M(p) -->
|
|
373 |
(p \<in> r <-> (\<exists>x\<in>A. \<exists>y\<in>A. M(x) & M(y) & x\<in>y & pair(M,x,y,p)))"
|
|
374 |
|
|
375 |
|
|
376 |
subsection{*Absoluteness for a transitive class model*}
|
|
377 |
|
|
378 |
text{*The class M is assumed to be transitive and to satisfy some
|
|
379 |
relativized ZF axioms*}
|
|
380 |
locale M_axioms =
|
|
381 |
fixes M
|
|
382 |
assumes transM: "[| y\<in>x; M(x) |] ==> M(y)"
|
|
383 |
and nonempty [simp]: "M(0)"
|
|
384 |
and upair_ax: "upair_ax(M)"
|
|
385 |
and Union_ax: "Union_ax(M)"
|
|
386 |
and power_ax: "power_ax(M)"
|
|
387 |
and replacement: "replacement(M,P)"
|
13245
|
388 |
and M_nat: "M(nat)" (*i.e. the axiom of infinity*)
|
13223
|
389 |
and Inter_separation:
|
|
390 |
"M(A) ==> separation(M, \<lambda>x. \<forall>y\<in>A. M(y) --> x\<in>y)"
|
|
391 |
and cartprod_separation:
|
|
392 |
"[| M(A); M(B) |]
|
|
393 |
==> separation(M, \<lambda>z. \<exists>x\<in>A. \<exists>y\<in>B. M(x) & M(y) & pair(M,x,y,z))"
|
|
394 |
and image_separation:
|
|
395 |
"[| M(A); M(r) |]
|
|
396 |
==> separation(M, \<lambda>y. \<exists>p\<in>r. M(p) & (\<exists>x\<in>A. M(x) & pair(M,x,y,p)))"
|
|
397 |
and vimage_separation:
|
|
398 |
"[| M(A); M(r) |]
|
|
399 |
==> separation(M, \<lambda>x. \<exists>p\<in>r. M(p) & (\<exists>y\<in>A. M(x) & pair(M,x,y,p)))"
|
|
400 |
and converse_separation:
|
|
401 |
"M(r) ==> separation(M, \<lambda>z. \<exists>p\<in>r. M(p) & (\<exists>x y. M(x) & M(y) &
|
|
402 |
pair(M,x,y,p) & pair(M,y,x,z)))"
|
|
403 |
and restrict_separation:
|
|
404 |
"M(A)
|
|
405 |
==> separation(M, \<lambda>z. \<exists>x\<in>A. M(x) & (\<exists>y. M(y) & pair(M,x,y,z)))"
|
|
406 |
and comp_separation:
|
|
407 |
"[| M(r); M(s) |]
|
|
408 |
==> separation(M, \<lambda>xz. \<exists>x y z. M(x) & M(y) & M(z) &
|
|
409 |
(\<exists>xy\<in>s. \<exists>yz\<in>r. M(xy) & M(yz) &
|
|
410 |
pair(M,x,z,xz) & pair(M,x,y,xy) & pair(M,y,z,yz)))"
|
|
411 |
and pred_separation:
|
|
412 |
"[| M(r); M(x) |] ==> separation(M, \<lambda>y. \<exists>p\<in>r. M(p) & pair(M,y,x,p))"
|
|
413 |
and Memrel_separation:
|
13245
|
414 |
"separation(M, \<lambda>z. \<exists>x y. M(x) & M(y) & pair(M,x,y,z) & x \<in> y)"
|
13223
|
415 |
and obase_separation:
|
|
416 |
--{*part of the order type formalization*}
|
|
417 |
"[| M(A); M(r) |]
|
|
418 |
==> separation(M, \<lambda>a. \<exists>x g mx par. M(x) & M(g) & M(mx) & M(par) &
|
|
419 |
ordinal(M,x) & membership(M,x,mx) & pred_set(M,A,a,r,par) &
|
|
420 |
order_isomorphism(M,par,r,x,mx,g))"
|
|
421 |
and well_ord_iso_separation:
|
|
422 |
"[| M(A); M(f); M(r) |]
|
13245
|
423 |
==> separation (M, \<lambda>x. x\<in>A --> (\<exists>y. M(y) & (\<exists>p. M(p) &
|
|
424 |
fun_apply(M,f,x,y) & pair(M,y,x,p) & p \<in> r)))"
|
13223
|
425 |
and obase_equals_separation:
|
|
426 |
"[| M(A); M(r) |]
|
|
427 |
==> separation
|
|
428 |
(M, \<lambda>x. x\<in>A --> ~(\<exists>y. M(y) & (\<exists>g. M(g) &
|
|
429 |
ordinal(M,y) & (\<exists>my pxr. M(my) & M(pxr) &
|
|
430 |
membership(M,y,my) & pred_set(M,A,x,r,pxr) &
|
|
431 |
order_isomorphism(M,pxr,r,y,my,g)))))"
|
|
432 |
and is_recfun_separation:
|
|
433 |
--{*for well-founded recursion. NEEDS RELATIVIZATION*}
|
|
434 |
"[| M(A); M(f); M(g); M(a); M(b) |]
|
13245
|
435 |
==> separation(M, \<lambda>x. x \<in> A --> \<langle>x,a\<rangle> \<in> r & \<langle>x,b\<rangle> \<in> r & f`x \<noteq> g`x)"
|
13223
|
436 |
and omap_replacement:
|
|
437 |
"[| M(A); M(r) |]
|
|
438 |
==> strong_replacement(M,
|
|
439 |
\<lambda>a z. \<exists>x g mx par. M(x) & M(g) & M(mx) & M(par) &
|
|
440 |
ordinal(M,x) & pair(M,a,x,z) & membership(M,x,mx) &
|
|
441 |
pred_set(M,A,a,r,par) & order_isomorphism(M,par,r,x,mx,g))"
|
|
442 |
|
|
443 |
lemma (in M_axioms) Ball_abs [simp]:
|
|
444 |
"M(A) ==> (\<forall>x\<in>A. M(x) --> P(x)) <-> (\<forall>x\<in>A. P(x))"
|
|
445 |
by (blast intro: transM)
|
|
446 |
|
|
447 |
lemma (in M_axioms) Bex_abs [simp]:
|
|
448 |
"M(A) ==> (\<exists>x\<in>A. M(x) & P(x)) <-> (\<exists>x\<in>A. P(x))"
|
|
449 |
by (blast intro: transM)
|
|
450 |
|
|
451 |
lemma (in M_axioms) Ball_iff_equiv:
|
|
452 |
"M(A) ==> (\<forall>x. M(x) --> (x\<in>A <-> P(x))) <->
|
|
453 |
(\<forall>x\<in>A. P(x)) & (\<forall>x. P(x) --> M(x) --> x\<in>A)"
|
|
454 |
by (blast intro: transM)
|
|
455 |
|
13245
|
456 |
text{*Simplifies proofs of equalities when there's an iff-equality
|
|
457 |
available for rewriting, universally quantified over M. *}
|
|
458 |
lemma (in M_axioms) M_equalityI:
|
|
459 |
"[| !!x. M(x) ==> x\<in>A <-> x\<in>B; M(A); M(B) |] ==> A=B"
|
|
460 |
by (blast intro!: equalityI dest: transM)
|
|
461 |
|
13223
|
462 |
lemma (in M_axioms) empty_abs [simp]:
|
|
463 |
"M(z) ==> empty(M,z) <-> z=0"
|
|
464 |
apply (simp add: empty_def)
|
|
465 |
apply (blast intro: transM)
|
|
466 |
done
|
|
467 |
|
|
468 |
lemma (in M_axioms) subset_abs [simp]:
|
|
469 |
"M(A) ==> subset(M,A,B) <-> A \<subseteq> B"
|
|
470 |
apply (simp add: subset_def)
|
|
471 |
apply (blast intro: transM)
|
|
472 |
done
|
|
473 |
|
|
474 |
lemma (in M_axioms) upair_abs [simp]:
|
|
475 |
"M(z) ==> upair(M,a,b,z) <-> z={a,b}"
|
|
476 |
apply (simp add: upair_def)
|
|
477 |
apply (blast intro: transM)
|
|
478 |
done
|
|
479 |
|
|
480 |
lemma (in M_axioms) upair_in_M_iff [iff]:
|
|
481 |
"M({a,b}) <-> M(a) & M(b)"
|
|
482 |
apply (insert upair_ax, simp add: upair_ax_def)
|
|
483 |
apply (blast intro: transM)
|
|
484 |
done
|
|
485 |
|
|
486 |
lemma (in M_axioms) singleton_in_M_iff [iff]:
|
|
487 |
"M({a}) <-> M(a)"
|
|
488 |
by (insert upair_in_M_iff [of a a], simp)
|
|
489 |
|
|
490 |
lemma (in M_axioms) pair_abs [simp]:
|
|
491 |
"M(z) ==> pair(M,a,b,z) <-> z=<a,b>"
|
|
492 |
apply (simp add: pair_def ZF.Pair_def)
|
|
493 |
apply (blast intro: transM)
|
|
494 |
done
|
|
495 |
|
|
496 |
lemma (in M_axioms) pair_in_M_iff [iff]:
|
|
497 |
"M(<a,b>) <-> M(a) & M(b)"
|
|
498 |
by (simp add: ZF.Pair_def)
|
|
499 |
|
|
500 |
lemma (in M_axioms) pair_components_in_M:
|
|
501 |
"[| <x,y> \<in> A; M(A) |] ==> M(x) & M(y)"
|
|
502 |
apply (simp add: Pair_def)
|
|
503 |
apply (blast dest: transM)
|
|
504 |
done
|
|
505 |
|
|
506 |
lemma (in M_axioms) cartprod_abs [simp]:
|
|
507 |
"[| M(A); M(B); M(z) |] ==> cartprod(M,A,B,z) <-> z = A*B"
|
|
508 |
apply (simp add: cartprod_def)
|
|
509 |
apply (rule iffI)
|
|
510 |
apply (blast intro!: equalityI intro: transM dest!: spec)
|
|
511 |
apply (blast dest: transM)
|
|
512 |
done
|
|
513 |
|
|
514 |
lemma (in M_axioms) union_abs [simp]:
|
|
515 |
"[| M(a); M(b); M(z) |] ==> union(M,a,b,z) <-> z = a Un b"
|
|
516 |
apply (simp add: union_def)
|
|
517 |
apply (blast intro: transM)
|
|
518 |
done
|
|
519 |
|
|
520 |
lemma (in M_axioms) inter_abs [simp]:
|
|
521 |
"[| M(a); M(b); M(z) |] ==> inter(M,a,b,z) <-> z = a Int b"
|
|
522 |
apply (simp add: inter_def)
|
|
523 |
apply (blast intro: transM)
|
|
524 |
done
|
|
525 |
|
|
526 |
lemma (in M_axioms) setdiff_abs [simp]:
|
|
527 |
"[| M(a); M(b); M(z) |] ==> setdiff(M,a,b,z) <-> z = a-b"
|
|
528 |
apply (simp add: setdiff_def)
|
|
529 |
apply (blast intro: transM)
|
|
530 |
done
|
|
531 |
|
|
532 |
lemma (in M_axioms) Union_abs [simp]:
|
|
533 |
"[| M(A); M(z) |] ==> big_union(M,A,z) <-> z = Union(A)"
|
|
534 |
apply (simp add: big_union_def)
|
|
535 |
apply (blast intro!: equalityI dest: transM)
|
|
536 |
done
|
|
537 |
|
13245
|
538 |
lemma (in M_axioms) Union_closed [intro,simp]:
|
13223
|
539 |
"M(A) ==> M(Union(A))"
|
|
540 |
by (insert Union_ax, simp add: Union_ax_def)
|
|
541 |
|
13245
|
542 |
lemma (in M_axioms) Un_closed [intro,simp]:
|
13223
|
543 |
"[| M(A); M(B) |] ==> M(A Un B)"
|
|
544 |
by (simp only: Un_eq_Union, blast)
|
|
545 |
|
13245
|
546 |
lemma (in M_axioms) cons_closed [intro,simp]:
|
13223
|
547 |
"[| M(a); M(A) |] ==> M(cons(a,A))"
|
|
548 |
by (subst cons_eq [symmetric], blast)
|
|
549 |
|
|
550 |
lemma (in M_axioms) successor_abs [simp]:
|
|
551 |
"[| M(a); M(z) |] ==> successor(M,a,z) <-> z=succ(a)"
|
|
552 |
by (simp add: successor_def, blast)
|
|
553 |
|
|
554 |
lemma (in M_axioms) succ_in_M_iff [iff]:
|
|
555 |
"M(succ(a)) <-> M(a)"
|
|
556 |
apply (simp add: succ_def)
|
|
557 |
apply (blast intro: transM)
|
|
558 |
done
|
|
559 |
|
13245
|
560 |
lemma (in M_axioms) separation_closed [intro,simp]:
|
13223
|
561 |
"[| separation(M,P); M(A) |] ==> M(Collect(A,P))"
|
|
562 |
apply (insert separation, simp add: separation_def)
|
|
563 |
apply (drule spec [THEN mp], assumption, clarify)
|
|
564 |
apply (subgoal_tac "y = Collect(A,P)", blast)
|
|
565 |
apply (blast dest: transM)
|
|
566 |
done
|
|
567 |
|
|
568 |
text{*Probably the premise and conclusion are equivalent*}
|
|
569 |
lemma (in M_axioms) strong_replacementI [rule_format]:
|
|
570 |
"[| \<forall>A. M(A) --> separation(M, %u. \<exists>x\<in>A. P(x,u)) |]
|
|
571 |
==> strong_replacement(M,P)"
|
13247
|
572 |
apply (simp add: strong_replacement_def, clarify)
|
|
573 |
apply (frule replacementD [OF replacement], assumption)
|
|
574 |
apply clarify
|
|
575 |
apply (drule_tac x=A in spec, clarify)
|
|
576 |
apply (drule_tac z=Y in separationD, assumption)
|
|
577 |
apply clarify
|
13223
|
578 |
apply (blast dest: transM)
|
|
579 |
done
|
|
580 |
|
|
581 |
|
|
582 |
(*The last premise expresses that P takes M to M*)
|
13245
|
583 |
lemma (in M_axioms) strong_replacement_closed [intro,simp]:
|
13223
|
584 |
"[| strong_replacement(M,P); M(A); univalent(M,A,P);
|
13247
|
585 |
!!x y. [| x\<in>A; P(x,y); M(x) |] ==> M(y) |] ==> M(Replace(A,P))"
|
13223
|
586 |
apply (simp add: strong_replacement_def)
|
|
587 |
apply (drule spec [THEN mp], auto)
|
|
588 |
apply (subgoal_tac "Replace(A,P) = Y")
|
13247
|
589 |
apply simp
|
13223
|
590 |
apply (rule equality_iffI)
|
13247
|
591 |
apply (simp add: Replace_iff, safe)
|
13223
|
592 |
apply (blast dest: transM)
|
|
593 |
apply (frule transM, assumption)
|
13247
|
594 |
apply (simp add: univalent_def)
|
13223
|
595 |
apply (drule spec [THEN mp, THEN iffD1], assumption, assumption)
|
|
596 |
apply (blast dest: transM)
|
|
597 |
done
|
|
598 |
|
|
599 |
(*The first premise can't simply be assumed as a schema.
|
|
600 |
It is essential to take care when asserting instances of Replacement.
|
|
601 |
Let K be a nonconstructible subset of nat and define
|
|
602 |
f(x) = x if x:K and f(x)=0 otherwise. Then RepFun(nat,f) = cons(0,K), a
|
|
603 |
nonconstructible set. So we cannot assume that M(X) implies M(RepFun(X,f))
|
|
604 |
even for f : M -> M.
|
|
605 |
*)
|
13245
|
606 |
lemma (in M_axioms) RepFun_closed [intro,simp]:
|
13247
|
607 |
"[| strong_replacement(M, \<lambda>x y. y = f(x)); M(A); \<forall>x\<in>A. M(f(x)) |]
|
13223
|
608 |
==> M(RepFun(A,f))"
|
|
609 |
apply (simp add: RepFun_def)
|
|
610 |
apply (rule strong_replacement_closed)
|
|
611 |
apply (auto dest: transM simp add: univalent_def)
|
|
612 |
done
|
|
613 |
|
13247
|
614 |
lemma (in M_axioms) lam_closed [intro,simp]:
|
|
615 |
"[| strong_replacement(M, \<lambda>x y. y = <x,b(x)>); M(A); \<forall>x\<in>A. M(b(x)) |]
|
|
616 |
==> M(\<lambda>x\<in>A. b(x))"
|
|
617 |
by (simp add: lam_def, blast dest: transM)
|
|
618 |
|
13223
|
619 |
lemma (in M_axioms) converse_abs [simp]:
|
|
620 |
"[| M(r); M(z) |] ==> is_converse(M,r,z) <-> z = converse(r)"
|
|
621 |
apply (simp add: is_converse_def)
|
|
622 |
apply (rule iffI)
|
|
623 |
apply (rule equalityI)
|
|
624 |
apply (blast dest: transM)
|
|
625 |
apply (clarify, frule transM, assumption, simp, blast)
|
|
626 |
done
|
|
627 |
|
|
628 |
lemma (in M_axioms) image_abs [simp]:
|
|
629 |
"[| M(r); M(A); M(z) |] ==> image(M,r,A,z) <-> z = r``A"
|
|
630 |
apply (simp add: image_def)
|
|
631 |
apply (rule iffI)
|
|
632 |
apply (blast intro!: equalityI dest: transM, blast)
|
|
633 |
done
|
|
634 |
|
|
635 |
text{*What about @{text Pow_abs}? Powerset is NOT absolute!
|
|
636 |
This result is one direction of absoluteness.*}
|
|
637 |
|
|
638 |
lemma (in M_axioms) powerset_Pow:
|
|
639 |
"powerset(M, x, Pow(x))"
|
|
640 |
by (simp add: powerset_def)
|
|
641 |
|
|
642 |
text{*But we can't prove that the powerset in @{text M} includes the
|
|
643 |
real powerset.*}
|
|
644 |
lemma (in M_axioms) powerset_imp_subset_Pow:
|
|
645 |
"[| powerset(M,x,y); M(y) |] ==> y <= Pow(x)"
|
|
646 |
apply (simp add: powerset_def)
|
|
647 |
apply (blast dest: transM)
|
|
648 |
done
|
|
649 |
|
|
650 |
lemma (in M_axioms) cartprod_iff_lemma:
|
|
651 |
"[| M(C); \<forall>u. M(u) --> u \<in> C <-> (\<exists>x\<in>A. \<exists>y\<in>B. u = {{x}, {x,y}});
|
|
652 |
powerset(M, A \<union> B, p1); powerset(M, p1, p2); M(p2) |]
|
|
653 |
==> C = {u \<in> p2 . \<exists>x\<in>A. \<exists>y\<in>B. u = {{x}, {x,y}}}"
|
|
654 |
apply (simp add: powerset_def)
|
|
655 |
apply (rule equalityI, clarify, simp)
|
|
656 |
apply (frule transM, assumption, simp)
|
|
657 |
apply blast
|
|
658 |
apply clarify
|
|
659 |
apply (frule transM, assumption, force)
|
|
660 |
done
|
|
661 |
|
|
662 |
lemma (in M_axioms) cartprod_iff:
|
|
663 |
"[| M(A); M(B); M(C) |]
|
|
664 |
==> cartprod(M,A,B,C) <->
|
|
665 |
(\<exists>p1 p2. M(p1) & M(p2) & powerset(M,A Un B,p1) & powerset(M,p1,p2) &
|
|
666 |
C = {z \<in> p2. \<exists>x\<in>A. \<exists>y\<in>B. z = <x,y>})"
|
|
667 |
apply (simp add: Pair_def cartprod_def, safe)
|
|
668 |
defer 1
|
|
669 |
apply (simp add: powerset_def)
|
|
670 |
apply blast
|
|
671 |
txt{*Final, difficult case: the left-to-right direction of the theorem.*}
|
|
672 |
apply (insert power_ax, simp add: power_ax_def)
|
|
673 |
apply (frule_tac x="A Un B" and P="\<lambda>x. M(x) --> Ex(?Q(x))" in spec)
|
|
674 |
apply (erule impE, blast, clarify)
|
|
675 |
apply (drule_tac x=z and P="\<lambda>x. M(x) --> Ex(?Q(x))" in spec)
|
|
676 |
apply (blast intro: cartprod_iff_lemma)
|
|
677 |
done
|
|
678 |
|
|
679 |
lemma (in M_axioms) cartprod_closed_lemma:
|
|
680 |
"[| M(A); M(B) |] ==> \<exists>C. M(C) & cartprod(M,A,B,C)"
|
|
681 |
apply (simp del: cartprod_abs add: cartprod_iff)
|
|
682 |
apply (insert power_ax, simp add: power_ax_def)
|
|
683 |
apply (frule_tac x="A Un B" and P="\<lambda>x. M(x) --> Ex(?Q(x))" in spec)
|
|
684 |
apply (erule impE, blast, clarify)
|
|
685 |
apply (drule_tac x=z and P="\<lambda>x. M(x) --> Ex(?Q(x))" in spec)
|
|
686 |
apply (erule impE, blast, clarify)
|
|
687 |
apply (intro exI conjI)
|
|
688 |
prefer 6 apply (rule refl)
|
|
689 |
prefer 4 apply assumption
|
|
690 |
prefer 4 apply assumption
|
13245
|
691 |
apply (insert cartprod_separation [of A B], auto)
|
13223
|
692 |
done
|
|
693 |
|
|
694 |
|
|
695 |
text{*All the lemmas above are necessary because Powerset is not absolute.
|
|
696 |
I should have used Replacement instead!*}
|
13245
|
697 |
lemma (in M_axioms) cartprod_closed [intro,simp]:
|
13223
|
698 |
"[| M(A); M(B) |] ==> M(A*B)"
|
|
699 |
by (frule cartprod_closed_lemma, assumption, force)
|
|
700 |
|
13245
|
701 |
lemma (in M_axioms) image_closed [intro,simp]:
|
13223
|
702 |
"[| M(A); M(r) |] ==> M(r``A)"
|
|
703 |
apply (simp add: image_iff_Collect)
|
13245
|
704 |
apply (insert image_separation [of A r], simp)
|
13223
|
705 |
done
|
|
706 |
|
|
707 |
lemma (in M_axioms) vimage_abs [simp]:
|
|
708 |
"[| M(r); M(A); M(z) |] ==> pre_image(M,r,A,z) <-> z = r-``A"
|
|
709 |
apply (simp add: pre_image_def)
|
|
710 |
apply (rule iffI)
|
|
711 |
apply (blast intro!: equalityI dest: transM, blast)
|
|
712 |
done
|
|
713 |
|
13245
|
714 |
lemma (in M_axioms) vimage_closed [intro,simp]:
|
13223
|
715 |
"[| M(A); M(r) |] ==> M(r-``A)"
|
|
716 |
apply (simp add: vimage_iff_Collect)
|
13245
|
717 |
apply (insert vimage_separation [of A r], simp)
|
13223
|
718 |
done
|
|
719 |
|
|
720 |
lemma (in M_axioms) domain_abs [simp]:
|
|
721 |
"[| M(r); M(z) |] ==> is_domain(M,r,z) <-> z = domain(r)"
|
|
722 |
apply (simp add: is_domain_def)
|
|
723 |
apply (blast intro!: equalityI dest: transM)
|
|
724 |
done
|
|
725 |
|
13245
|
726 |
lemma (in M_axioms) domain_closed [intro,simp]:
|
13223
|
727 |
"M(r) ==> M(domain(r))"
|
|
728 |
apply (simp add: domain_eq_vimage)
|
|
729 |
done
|
|
730 |
|
|
731 |
lemma (in M_axioms) range_abs [simp]:
|
|
732 |
"[| M(r); M(z) |] ==> is_range(M,r,z) <-> z = range(r)"
|
|
733 |
apply (simp add: is_range_def)
|
|
734 |
apply (blast intro!: equalityI dest: transM)
|
|
735 |
done
|
|
736 |
|
13245
|
737 |
lemma (in M_axioms) range_closed [intro,simp]:
|
13223
|
738 |
"M(r) ==> M(range(r))"
|
|
739 |
apply (simp add: range_eq_image)
|
|
740 |
done
|
|
741 |
|
13245
|
742 |
lemma (in M_axioms) field_abs [simp]:
|
|
743 |
"[| M(r); M(z) |] ==> is_field(M,r,z) <-> z = field(r)"
|
|
744 |
by (simp add: domain_closed range_closed is_field_def field_def)
|
|
745 |
|
|
746 |
lemma (in M_axioms) field_closed [intro,simp]:
|
|
747 |
"M(r) ==> M(field(r))"
|
|
748 |
by (simp add: domain_closed range_closed Un_closed field_def)
|
|
749 |
|
|
750 |
|
13223
|
751 |
lemma (in M_axioms) M_converse_iff:
|
|
752 |
"M(r) ==>
|
|
753 |
converse(r) =
|
|
754 |
{z \<in> range(r) * domain(r).
|
13245
|
755 |
\<exists>p\<in>r. \<exists>x. M(x) & (\<exists>y. M(y) & p = \<langle>x,y\<rangle> & z = \<langle>y,x\<rangle>)}"
|
13223
|
756 |
by (blast dest: transM)
|
|
757 |
|
13245
|
758 |
lemma (in M_axioms) converse_closed [intro,simp]:
|
13223
|
759 |
"M(r) ==> M(converse(r))"
|
|
760 |
apply (simp add: M_converse_iff)
|
13245
|
761 |
apply (insert converse_separation [of r], simp)
|
13223
|
762 |
done
|
|
763 |
|
|
764 |
lemma (in M_axioms) relation_abs [simp]:
|
|
765 |
"M(r) ==> is_relation(M,r) <-> relation(r)"
|
|
766 |
apply (simp add: is_relation_def relation_def)
|
|
767 |
apply (blast dest!: bspec dest: pair_components_in_M)+
|
|
768 |
done
|
|
769 |
|
|
770 |
lemma (in M_axioms) function_abs [simp]:
|
|
771 |
"M(r) ==> is_function(M,r) <-> function(r)"
|
|
772 |
apply (simp add: is_function_def function_def, safe)
|
|
773 |
apply (frule transM, assumption)
|
|
774 |
apply (blast dest: pair_components_in_M)+
|
|
775 |
done
|
|
776 |
|
13245
|
777 |
lemma (in M_axioms) apply_closed [intro,simp]:
|
13223
|
778 |
"[|M(f); M(a)|] ==> M(f`a)"
|
13245
|
779 |
apply (simp add: apply_def)
|
13223
|
780 |
done
|
|
781 |
|
|
782 |
lemma (in M_axioms) apply_abs:
|
|
783 |
"[| function(f); M(f); M(y) |]
|
|
784 |
==> fun_apply(M,f,x,y) <-> x \<in> domain(f) & f`x = y"
|
|
785 |
apply (simp add: fun_apply_def)
|
|
786 |
apply (blast intro: function_apply_equality function_apply_Pair)
|
|
787 |
done
|
|
788 |
|
|
789 |
lemma (in M_axioms) typed_apply_abs:
|
|
790 |
"[| f \<in> A -> B; M(f); M(y) |]
|
|
791 |
==> fun_apply(M,f,x,y) <-> x \<in> A & f`x = y"
|
|
792 |
by (simp add: apply_abs fun_is_function domain_of_fun)
|
|
793 |
|
|
794 |
lemma (in M_axioms) typed_function_abs [simp]:
|
|
795 |
"[| M(A); M(f) |] ==> typed_function(M,A,B,f) <-> f \<in> A -> B"
|
|
796 |
apply (auto simp add: typed_function_def relation_def Pi_iff)
|
|
797 |
apply (blast dest: pair_components_in_M)+
|
|
798 |
done
|
|
799 |
|
|
800 |
lemma (in M_axioms) injection_abs [simp]:
|
|
801 |
"[| M(A); M(f) |] ==> injection(M,A,B,f) <-> f \<in> inj(A,B)"
|
|
802 |
apply (simp add: injection_def apply_iff inj_def apply_closed)
|
13247
|
803 |
apply (blast dest: transM [of _ A])
|
13223
|
804 |
done
|
|
805 |
|
|
806 |
lemma (in M_axioms) surjection_abs [simp]:
|
|
807 |
"[| M(A); M(B); M(f) |] ==> surjection(M,A,B,f) <-> f \<in> surj(A,B)"
|
|
808 |
by (simp add: typed_apply_abs surjection_def surj_def)
|
|
809 |
|
|
810 |
lemma (in M_axioms) bijection_abs [simp]:
|
|
811 |
"[| M(A); M(B); M(f) |] ==> bijection(M,A,B,f) <-> f \<in> bij(A,B)"
|
|
812 |
by (simp add: bijection_def bij_def)
|
|
813 |
|
|
814 |
text{*no longer needed*}
|
|
815 |
lemma (in M_axioms) restriction_is_function:
|
|
816 |
"[| restriction(M,f,A,z); function(f); M(f); M(A); M(z) |]
|
|
817 |
==> function(z)"
|
|
818 |
apply (rotate_tac 1)
|
|
819 |
apply (simp add: restriction_def Ball_iff_equiv)
|
|
820 |
apply (unfold function_def, blast)
|
|
821 |
done
|
|
822 |
|
|
823 |
lemma (in M_axioms) restriction_abs [simp]:
|
|
824 |
"[| M(f); M(A); M(z) |]
|
|
825 |
==> restriction(M,f,A,z) <-> z = restrict(f,A)"
|
|
826 |
apply (simp add: Ball_iff_equiv restriction_def restrict_def)
|
|
827 |
apply (blast intro!: equalityI dest: transM)
|
|
828 |
done
|
|
829 |
|
|
830 |
|
|
831 |
lemma (in M_axioms) M_restrict_iff:
|
|
832 |
"M(r) ==> restrict(r,A) = {z \<in> r . \<exists>x\<in>A. \<exists>y. M(y) & z = \<langle>x, y\<rangle>}"
|
|
833 |
by (simp add: restrict_def, blast dest: transM)
|
|
834 |
|
13245
|
835 |
lemma (in M_axioms) restrict_closed [intro,simp]:
|
13223
|
836 |
"[| M(A); M(r) |] ==> M(restrict(r,A))"
|
|
837 |
apply (simp add: M_restrict_iff)
|
13245
|
838 |
apply (insert restrict_separation [of A], simp)
|
13223
|
839 |
done
|
|
840 |
|
|
841 |
lemma (in M_axioms) M_comp_iff:
|
|
842 |
"[| M(r); M(s) |]
|
|
843 |
==> r O s =
|
|
844 |
{xz \<in> domain(s) * range(r).
|
13245
|
845 |
\<exists>x. M(x) & (\<exists>y. M(y) & (\<exists>z. M(z) &
|
|
846 |
xz = \<langle>x,z\<rangle> & \<langle>x,y\<rangle> \<in> s & \<langle>y,z\<rangle> \<in> r))}"
|
13223
|
847 |
apply (simp add: comp_def)
|
|
848 |
apply (rule equalityI)
|
13247
|
849 |
apply clarify
|
|
850 |
apply simp
|
13223
|
851 |
apply (blast dest: transM)+
|
|
852 |
done
|
|
853 |
|
13245
|
854 |
lemma (in M_axioms) comp_closed [intro,simp]:
|
13223
|
855 |
"[| M(r); M(s) |] ==> M(r O s)"
|
|
856 |
apply (simp add: M_comp_iff)
|
13245
|
857 |
apply (insert comp_separation [of r s], simp)
|
|
858 |
done
|
|
859 |
|
|
860 |
lemma (in M_axioms) composition_abs [simp]:
|
|
861 |
"[| M(r); M(s); M(t) |]
|
|
862 |
==> composition(M,r,s,t) <-> t = r O s"
|
13247
|
863 |
apply safe
|
13245
|
864 |
txt{*Proving @{term "composition(M, r, s, r O s)"}*}
|
|
865 |
prefer 2
|
|
866 |
apply (simp add: composition_def comp_def)
|
|
867 |
apply (blast dest: transM)
|
|
868 |
txt{*Opposite implication*}
|
|
869 |
apply (rule M_equalityI)
|
|
870 |
apply (simp add: composition_def comp_def)
|
|
871 |
apply (blast del: allE dest: transM)+
|
13223
|
872 |
done
|
|
873 |
|
|
874 |
lemma (in M_axioms) nat_into_M [intro]:
|
|
875 |
"n \<in> nat ==> M(n)"
|
|
876 |
by (induct n rule: nat_induct, simp_all)
|
|
877 |
|
|
878 |
lemma (in M_axioms) Inl_in_M_iff [iff]:
|
|
879 |
"M(Inl(a)) <-> M(a)"
|
|
880 |
by (simp add: Inl_def)
|
|
881 |
|
|
882 |
lemma (in M_axioms) Inr_in_M_iff [iff]:
|
|
883 |
"M(Inr(a)) <-> M(a)"
|
|
884 |
by (simp add: Inr_def)
|
|
885 |
|
|
886 |
lemma (in M_axioms) Inter_abs [simp]:
|
|
887 |
"[| M(A); M(z) |] ==> big_inter(M,A,z) <-> z = Inter(A)"
|
|
888 |
apply (simp add: big_inter_def Inter_def)
|
|
889 |
apply (blast intro!: equalityI dest: transM)
|
|
890 |
done
|
|
891 |
|
13245
|
892 |
lemma (in M_axioms) Inter_closed [intro,simp]:
|
13223
|
893 |
"M(A) ==> M(Inter(A))"
|
13245
|
894 |
by (insert Inter_separation, simp add: Inter_def)
|
13223
|
895 |
|
13245
|
896 |
lemma (in M_axioms) Int_closed [intro,simp]:
|
13223
|
897 |
"[| M(A); M(B) |] ==> M(A Int B)"
|
|
898 |
apply (subgoal_tac "M({A,B})")
|
13247
|
899 |
apply (frule Inter_closed, force+)
|
13223
|
900 |
done
|
|
901 |
|
13245
|
902 |
text{*M contains all finite functions*}
|
|
903 |
lemma (in M_axioms) finite_fun_closed_lemma [rule_format]:
|
|
904 |
"[| n \<in> nat; M(A) |] ==> \<forall>f \<in> n -> A. M(f)"
|
|
905 |
apply (induct_tac n, simp)
|
|
906 |
apply (rule ballI)
|
|
907 |
apply (simp add: succ_def)
|
|
908 |
apply (frule fun_cons_restrict_eq)
|
|
909 |
apply (erule ssubst)
|
|
910 |
apply (subgoal_tac "M(f`x) & restrict(f,x) \<in> x -> A")
|
|
911 |
apply (simp add: cons_closed nat_into_M apply_closed)
|
|
912 |
apply (blast intro: apply_funtype transM restrict_type2)
|
|
913 |
done
|
|
914 |
|
|
915 |
lemma (in M_axioms) finite_fun_closed [rule_format]:
|
|
916 |
"[| f \<in> n -> A; n \<in> nat; M(A) |] ==> M(f)"
|
|
917 |
by (blast intro: finite_fun_closed_lemma)
|
|
918 |
|
|
919 |
|
13223
|
920 |
subsection{*Absoluteness for ordinals*}
|
|
921 |
text{*These results constitute Theorem IV 5.1 of Kunen (page 126).*}
|
|
922 |
|
|
923 |
lemma (in M_axioms) lt_closed:
|
|
924 |
"[| j<i; M(i) |] ==> M(j)"
|
|
925 |
by (blast dest: ltD intro: transM)
|
|
926 |
|
|
927 |
lemma (in M_axioms) transitive_set_abs [simp]:
|
|
928 |
"M(a) ==> transitive_set(M,a) <-> Transset(a)"
|
|
929 |
by (simp add: transitive_set_def Transset_def)
|
|
930 |
|
|
931 |
lemma (in M_axioms) ordinal_abs [simp]:
|
|
932 |
"M(a) ==> ordinal(M,a) <-> Ord(a)"
|
|
933 |
by (simp add: ordinal_def Ord_def)
|
|
934 |
|
|
935 |
lemma (in M_axioms) limit_ordinal_abs [simp]:
|
|
936 |
"M(a) ==> limit_ordinal(M,a) <-> Limit(a)"
|
|
937 |
apply (simp add: limit_ordinal_def Ord_0_lt_iff Limit_def)
|
|
938 |
apply (simp add: lt_def, blast)
|
|
939 |
done
|
|
940 |
|
|
941 |
lemma (in M_axioms) successor_ordinal_abs [simp]:
|
|
942 |
"M(a) ==> successor_ordinal(M,a) <-> Ord(a) & (\<exists>b. M(b) & a = succ(b))"
|
|
943 |
apply (simp add: successor_ordinal_def, safe)
|
|
944 |
apply (drule Ord_cases_disj, auto)
|
|
945 |
done
|
|
946 |
|
|
947 |
lemma finite_Ord_is_nat:
|
|
948 |
"[| Ord(a); ~ Limit(a); \<forall>x\<in>a. ~ Limit(x) |] ==> a \<in> nat"
|
|
949 |
by (induct a rule: trans_induct3, simp_all)
|
|
950 |
|
|
951 |
lemma naturals_not_limit: "a \<in> nat ==> ~ Limit(a)"
|
|
952 |
by (induct a rule: nat_induct, auto)
|
|
953 |
|
|
954 |
lemma (in M_axioms) finite_ordinal_abs [simp]:
|
|
955 |
"M(a) ==> finite_ordinal(M,a) <-> a \<in> nat"
|
|
956 |
apply (simp add: finite_ordinal_def)
|
|
957 |
apply (blast intro: finite_Ord_is_nat intro: nat_into_Ord
|
|
958 |
dest: Ord_trans naturals_not_limit)
|
|
959 |
done
|
|
960 |
|
|
961 |
lemma Limit_non_Limit_implies_nat: "[| Limit(a); \<forall>x\<in>a. ~ Limit(x) |] ==> a = nat"
|
|
962 |
apply (rule le_anti_sym)
|
|
963 |
apply (rule all_lt_imp_le, blast, blast intro: Limit_is_Ord)
|
|
964 |
apply (simp add: lt_def)
|
|
965 |
apply (blast intro: Ord_in_Ord Ord_trans finite_Ord_is_nat)
|
|
966 |
apply (erule nat_le_Limit)
|
|
967 |
done
|
|
968 |
|
|
969 |
lemma (in M_axioms) omega_abs [simp]:
|
|
970 |
"M(a) ==> omega(M,a) <-> a = nat"
|
|
971 |
apply (simp add: omega_def)
|
|
972 |
apply (blast intro: Limit_non_Limit_implies_nat dest: naturals_not_limit)
|
|
973 |
done
|
|
974 |
|
|
975 |
lemma (in M_axioms) number1_abs [simp]:
|
|
976 |
"M(a) ==> number1(M,a) <-> a = 1"
|
|
977 |
by (simp add: number1_def)
|
|
978 |
|
|
979 |
lemma (in M_axioms) number1_abs [simp]:
|
|
980 |
"M(a) ==> number2(M,a) <-> a = succ(1)"
|
|
981 |
by (simp add: number2_def)
|
|
982 |
|
|
983 |
lemma (in M_axioms) number3_abs [simp]:
|
|
984 |
"M(a) ==> number3(M,a) <-> a = succ(succ(1))"
|
|
985 |
by (simp add: number3_def)
|
|
986 |
|
|
987 |
text{*Kunen continued to 20...*}
|
|
988 |
|
|
989 |
(*Could not get this to work. The \<lambda>x\<in>nat is essential because everything
|
|
990 |
but the recursion variable must stay unchanged. But then the recursion
|
|
991 |
equations only hold for x\<in>nat (or in some other set) and not for the
|
|
992 |
whole of the class M.
|
|
993 |
consts
|
|
994 |
natnumber_aux :: "[i=>o,i] => i"
|
|
995 |
|
|
996 |
primrec
|
|
997 |
"natnumber_aux(M,0) = (\<lambda>x\<in>nat. if empty(M,x) then 1 else 0)"
|
|
998 |
"natnumber_aux(M,succ(n)) =
|
|
999 |
(\<lambda>x\<in>nat. if (\<exists>y. M(y) & natnumber_aux(M,n)`y=1 & successor(M,y,x))
|
|
1000 |
then 1 else 0)"
|
|
1001 |
|
|
1002 |
constdefs
|
|
1003 |
natnumber :: "[i=>o,i,i] => o"
|
|
1004 |
"natnumber(M,n,x) == natnumber_aux(M,n)`x = 1"
|
|
1005 |
|
|
1006 |
lemma (in M_axioms) [simp]:
|
|
1007 |
"natnumber(M,0,x) == x=0"
|
|
1008 |
*)
|
|
1009 |
|
|
1010 |
|
|
1011 |
end
|