author | paulson |
Mon, 08 Jul 2002 17:51:56 +0200 | |
changeset 13316 | d16629fd0f95 |
parent 13306 | 6eebcddee32b |
child 13319 | 23de7b3af453 |
permissions | -rw-r--r-- |
13223 | 1 |
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 \<notin> z" |
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subset :: "[i=>o,i,i] => o" |
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"subset(M,A,B) == \<forall>x[M]. x\<in>A --> 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[M]. x\<in>z --> x = a | x = b)" |
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pair :: "[i=>o,i,i,i] => o" |
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13254 | 18 |
"pair(M,a,b,z) == \<exists>x[M]. upair(M,a,a,x) & |
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(\<exists>y[M]. upair(M,a,b,y) & upair(M,x,y,z))" |
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13223 | 20 |
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13306 | 21 |
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union :: "[i=>o,i,i,i] => o" |
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"union(M,a,b,z) == \<forall>x[M]. x \<in> z <-> x \<in> a | x \<in> b" |
13245 | 24 |
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is_cons :: "[i=>o,i,i,i] => o" |
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"is_cons(M,a,b,z) == \<exists>x[M]. upair(M,a,a,x) & union(M,x,b,z)" |
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13223 | 28 |
successor :: "[i=>o,i,i] => o" |
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"successor(M,a,z) == is_cons(M,a,a,z)" |
13223 | 30 |
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powerset :: "[i=>o,i,i] => o" |
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13254 | 32 |
"powerset(M,A,z) == \<forall>x[M]. x \<in> z <-> subset(M,x,A)" |
13223 | 33 |
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inter :: "[i=>o,i,i,i] => o" |
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13254 | 35 |
"inter(M,a,b,z) == \<forall>x[M]. x \<in> z <-> x \<in> a & x \<in> b" |
13223 | 36 |
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setdiff :: "[i=>o,i,i,i] => o" |
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"setdiff(M,a,b,z) == \<forall>x[M]. x \<in> z <-> x \<in> a & x \<notin> b" |
13223 | 39 |
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big_union :: "[i=>o,i,i] => o" |
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"big_union(M,A,z) == \<forall>x[M]. x \<in> z <-> (\<exists>y[M]. y\<in>A & x \<in> y)" |
13223 | 42 |
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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 \<in> z <-> (\<forall>y[M]. y\<in>A --> x \<in> y)))" |
13223 | 47 |
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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 \<in> z <-> (\<exists>x[M]. x\<in>A & (\<exists>y[M]. y\<in>B & pair(M,x,y,u)))" |
13223 | 51 |
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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 \<in> z <-> |
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(\<exists>w[M]. w\<in>r & (\<exists>u[M]. \<exists>v[M]. 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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13299 | 59 |
\<forall>x[M]. x \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>y[M]. y\<in>A & pair(M,x,y,w)))" |
13223 | 60 |
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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 \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>y[M]. 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 \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>x[M]. x\<in>A & 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]. 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 \<in> z <-> (\<exists>w[M]. w\<in>r & (\<exists>x[M]. 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]. is_domain(M,r,dr) & |
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(\<exists>rr[M]. 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[M]. z\<in>r --> (\<exists>x[M]. \<exists>y[M]. 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[M]. \<forall>y[M]. \<forall>y'[M]. \<forall>p[M]. \<forall>p'[M]. |
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pair(M,x,y,p) --> pair(M,x,y',p') --> p\<in>r --> p'\<in>r --> 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]. (\<exists>u[M]. u\<in>f & 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[M]. u\<in>r --> (\<forall>x[M]. \<forall>y[M]. pair(M,x,y,u) --> y\<in>B))" |
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is_funspace :: "[i=>o,i,i,i] => o" |
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"is_funspace(M,A,B,F) == |
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\<forall>f[M]. f \<in> F <-> typed_function(M,A,B,f)" |
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13245 | 104 |
composition :: "[i=>o,i,i,i] => o" |
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"composition(M,r,s,t) == |
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\<forall>p[M]. p \<in> t <-> |
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(\<exists>x[M]. \<exists>y[M]. \<exists>z[M]. pair(M,x,z,p) & \<langle>x,y\<rangle> \<in> s & \<langle>y,z\<rangle> \<in> r)" |
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13245 | 108 |
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13223 | 110 |
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[M]. \<forall>x'[M]. \<forall>y[M]. \<forall>p[M]. \<forall>p'[M]. |
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pair(M,x,y,p) --> pair(M,x',y,p') --> p\<in>f --> p'\<in>f --> x=x')" |
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13223 | 115 |
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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[M]. y\<in>B --> (\<exists>x[M]. x\<in>A & fun_apply(M,f,x,y)))" |
13223 | 120 |
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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 \<in> z <-> (x \<in> r & (\<exists>u[M]. u\<in>A & (\<exists>v[M]. pair(M,u,v,x))))" |
13223 | 127 |
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transitive_set :: "[i=>o,i] => o" |
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13299 | 129 |
"transitive_set(M,a) == \<forall>x[M]. x\<in>a --> subset(M,x,a)" |
13223 | 130 |
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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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13299 | 133 |
"ordinal(M,a) == transitive_set(M,a) & (\<forall>x[M]. x\<in>a --> 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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13299 | 139 |
(\<forall>x[M]. x\<in>a --> (\<exists>y[M]. y\<in>a & successor(M,x,y)))" |
13223 | 140 |
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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[M]. x\<in>a --> ~ limit_ordinal(M,x))" |
13223 | 151 |
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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[M]. x\<in>a --> ~ limit_ordinal(M,x))" |
13223 | 155 |
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number1 :: "[i=>o,i] => o" |
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13299 | 157 |
"number1(M,a) == (\<exists>x[M]. empty(M,x) & successor(M,x,a))" |
13223 | 158 |
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number2 :: "[i=>o,i] => o" |
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"number2(M,a) == (\<exists>x[M]. number1(M,x) & successor(M,x,a))" |
13223 | 161 |
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number3 :: "[i=>o,i] => o" |
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"number3(M,a) == (\<exists>x[M]. 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[M]. \<forall>y[M]. (\<forall>z[M]. z \<in> x <-> z \<in> y) --> x=y" |
13223 | 172 |
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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]. \<exists>y[M]. \<forall>x[M]. x \<in> y <-> x \<in> z & P(x)" |
13223 | 178 |
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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]. upair(M,x,y,z))" |
13223 | 181 |
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Union_ax :: "(i=>o) => o" |
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"Union_ax(M) == \<forall>x[M]. (\<exists>z[M]. big_union(M,x,z))" |
13223 | 184 |
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power_ax :: "(i=>o) => o" |
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"power_ax(M) == \<forall>x[M]. (\<exists>z[M]. powerset(M,x,z))" |
13223 | 187 |
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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[M]. x\<in>A --> (\<forall>y z. M(y) --> M(z) --> P(x,y) & P(x,z) --> y=z))" |
13223 | 191 |
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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]. univalent(M,A,P) --> |
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(\<exists>Y[M]. (\<forall>b[M]. ((\<exists>x[M]. x\<in>A & P(x,b)) --> b \<in> Y)))" |
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13223 | 196 |
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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]. univalent(M,A,P) --> |
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(\<exists>Y[M]. (\<forall>b[M]. (b \<in> Y <-> (\<exists>x[M]. x\<in>A & P(x,b)))))" |
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13223 | 201 |
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foundation_ax :: "(i=>o) => o" |
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"foundation_ax(M) == |
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\<forall>x[M]. (\<exists>y\<in>x. M(y)) |
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--> (\<exists>y[M]. y\<in>x & ~(\<exists>z[M]. z\<in>x & z \<in> y))" |
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13223 | 206 |
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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 separation_cong [cong]: |
13223 | 229 |
"(!!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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13254 | 233 |
lemma univalent_cong [cong]: |
13223 | 234 |
"[| 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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13254 | 238 |
lemma strong_replacement_cong [cong]: |
13223 | 239 |
"[| !!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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247 |
done |
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248 |
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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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255 |
done |
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256 |
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257 |
lemma Collect_in_VLimit: |
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"[| X \<in> Vfrom(A,i); Limit(i); Transset(A) |] |
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259 |
==> Collect(X,P) \<in> Vfrom(A,i)" |
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260 |
apply (rule Limit_VfromE, assumption+) |
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261 |
apply (blast intro: Limit_has_succ VfromI Collect_in_Vfrom) |
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262 |
done |
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263 |
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264 |
lemma Collect_in_univ: |
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265 |
"[| X \<in> univ(A); Transset(A) |] ==> Collect(X,P) \<in> univ(A)" |
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266 |
by (simp add: univ_def Collect_in_VLimit Limit_nat) |
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268 |
lemma "separation(\<lambda>x. x \<in> univ(0), P)" |
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apply (simp add: separation_def, clarify) |
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apply (rule_tac x = "Collect(x,P)" in bexI) |
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apply (blast intro: Collect_in_univ Transset_0)+ |
13223 | 272 |
done |
273 |
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274 |
text{*Unordered pairing axiom*} |
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275 |
lemma "upair_ax(\<lambda>x. x \<in> univ(0))" |
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276 |
apply (simp add: upair_ax_def upair_def) |
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277 |
apply (blast intro: doubleton_in_univ) |
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278 |
done |
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279 |
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280 |
text{*Union axiom*} |
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281 |
lemma "Union_ax(\<lambda>x. x \<in> univ(0))" |
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13299 | 282 |
apply (simp add: Union_ax_def big_union_def, clarify) |
283 |
apply (rule_tac x="\<Union>x" in bexI) |
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284 |
apply (blast intro: univ0_downwards_mem) |
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285 |
apply (blast intro: Union_in_univ Transset_0) |
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13223 | 286 |
done |
287 |
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288 |
text{*Powerset axiom*} |
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289 |
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290 |
lemma Pow_in_univ: |
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291 |
"[| X \<in> univ(A); Transset(A) |] ==> Pow(X) \<in> univ(A)" |
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292 |
apply (simp add: univ_def Pow_in_VLimit Limit_nat) |
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293 |
done |
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294 |
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295 |
lemma "power_ax(\<lambda>x. x \<in> univ(0))" |
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13299 | 296 |
apply (simp add: power_ax_def powerset_def subset_def, clarify) |
297 |
apply (rule_tac x="Pow(x)" in bexI) |
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298 |
apply (blast intro: univ0_downwards_mem) |
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299 |
apply (blast intro: Pow_in_univ Transset_0) |
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13223 | 300 |
done |
301 |
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302 |
text{*Foundation axiom*} |
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303 |
lemma "foundation_ax(\<lambda>x. x \<in> univ(0))" |
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304 |
apply (simp add: foundation_ax_def, clarify) |
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13299 | 305 |
apply (cut_tac A=x in foundation) |
306 |
apply (blast intro: univ0_downwards_mem) |
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13223 | 307 |
done |
308 |
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309 |
lemma "replacement(\<lambda>x. x \<in> univ(0), P)" |
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310 |
apply (simp add: replacement_def, clarify) |
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311 |
oops |
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312 |
text{*no idea: maybe prove by induction on the rank of A?*} |
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313 |
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314 |
text{*Still missing: Replacement, Choice*} |
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||
316 |
subsection{*lemmas needed to reduce some set constructions to instances |
|
317 |
of Separation*} |
|
318 |
||
319 |
lemma image_iff_Collect: "r `` A = {y \<in> Union(Union(r)). \<exists>p\<in>r. \<exists>x\<in>A. p=<x,y>}" |
|
320 |
apply (rule equalityI, auto) |
|
321 |
apply (simp add: Pair_def, blast) |
|
322 |
done |
|
323 |
||
324 |
lemma vimage_iff_Collect: |
|
325 |
"r -`` A = {x \<in> Union(Union(r)). \<exists>p\<in>r. \<exists>y\<in>A. p=<x,y>}" |
|
326 |
apply (rule equalityI, auto) |
|
327 |
apply (simp add: Pair_def, blast) |
|
328 |
done |
|
329 |
||
330 |
text{*These two lemmas lets us prove @{text domain_closed} and |
|
331 |
@{text range_closed} without new instances of separation*} |
|
332 |
||
333 |
lemma domain_eq_vimage: "domain(r) = r -`` Union(Union(r))" |
|
334 |
apply (rule equalityI, auto) |
|
335 |
apply (rule vimageI, assumption) |
|
336 |
apply (simp add: Pair_def, blast) |
|
337 |
done |
|
338 |
||
339 |
lemma range_eq_image: "range(r) = r `` Union(Union(r))" |
|
340 |
apply (rule equalityI, auto) |
|
341 |
apply (rule imageI, assumption) |
|
342 |
apply (simp add: Pair_def, blast) |
|
343 |
done |
|
344 |
||
345 |
lemma replacementD: |
|
346 |
"[| replacement(M,P); M(A); univalent(M,A,P) |] |
|
13299 | 347 |
==> \<exists>Y[M]. (\<forall>b[M]. ((\<exists>x[M]. x\<in>A & P(x,b)) --> b \<in> Y))" |
13223 | 348 |
by (simp add: replacement_def) |
349 |
||
350 |
lemma strong_replacementD: |
|
351 |
"[| strong_replacement(M,P); M(A); univalent(M,A,P) |] |
|
13299 | 352 |
==> \<exists>Y[M]. (\<forall>b[M]. (b \<in> Y <-> (\<exists>x[M]. x\<in>A & P(x,b))))" |
13223 | 353 |
by (simp add: strong_replacement_def) |
354 |
||
355 |
lemma separationD: |
|
13290
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
356 |
"[| separation(M,P); M(z) |] ==> \<exists>y[M]. \<forall>x[M]. x \<in> y <-> x \<in> z & P(x)" |
13223 | 357 |
by (simp add: separation_def) |
358 |
||
359 |
||
360 |
text{*More constants, for order types*} |
|
361 |
constdefs |
|
362 |
||
363 |
order_isomorphism :: "[i=>o,i,i,i,i,i] => o" |
|
364 |
"order_isomorphism(M,A,r,B,s,f) == |
|
365 |
bijection(M,A,B,f) & |
|
13306 | 366 |
(\<forall>x[M]. x\<in>A --> (\<forall>y[M]. y\<in>A --> |
367 |
(\<forall>p[M]. \<forall>fx[M]. \<forall>fy[M]. \<forall>q[M]. |
|
13223 | 368 |
pair(M,x,y,p) --> fun_apply(M,f,x,fx) --> fun_apply(M,f,y,fy) --> |
13306 | 369 |
pair(M,fx,fy,q) --> (p\<in>r <-> q\<in>s))))" |
13223 | 370 |
|
371 |
pred_set :: "[i=>o,i,i,i,i] => o" |
|
372 |
"pred_set(M,A,x,r,B) == |
|
13299 | 373 |
\<forall>y[M]. y \<in> B <-> (\<exists>p[M]. p\<in>r & y \<in> A & pair(M,y,x,p))" |
13223 | 374 |
|
375 |
membership :: "[i=>o,i,i] => o" --{*membership relation*} |
|
376 |
"membership(M,A,r) == |
|
13306 | 377 |
\<forall>p[M]. p \<in> r <-> (\<exists>x[M]. x\<in>A & (\<exists>y[M]. y\<in>A & x\<in>y & pair(M,x,y,p)))" |
13223 | 378 |
|
379 |
||
380 |
subsection{*Absoluteness for a transitive class model*} |
|
381 |
||
382 |
text{*The class M is assumed to be transitive and to satisfy some |
|
383 |
relativized ZF axioms*} |
|
13290
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
384 |
locale M_triv_axioms = |
13223 | 385 |
fixes M |
386 |
assumes transM: "[| y\<in>x; M(x) |] ==> M(y)" |
|
387 |
and nonempty [simp]: "M(0)" |
|
388 |
and upair_ax: "upair_ax(M)" |
|
389 |
and Union_ax: "Union_ax(M)" |
|
390 |
and power_ax: "power_ax(M)" |
|
391 |
and replacement: "replacement(M,P)" |
|
13268 | 392 |
and M_nat [iff]: "M(nat)" (*i.e. the axiom of infinity*) |
13290
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
393 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
394 |
lemma (in M_triv_axioms) ball_abs [simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
395 |
"M(A) ==> (\<forall>x\<in>A. M(x) --> P(x)) <-> (\<forall>x\<in>A. P(x))" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
396 |
by (blast intro: transM) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
397 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
398 |
lemma (in M_triv_axioms) rall_abs [simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
399 |
"M(A) ==> (\<forall>x[M]. x\<in>A --> P(x)) <-> (\<forall>x\<in>A. P(x))" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
400 |
by (blast intro: transM) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
401 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
402 |
lemma (in M_triv_axioms) bex_abs [simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
403 |
"M(A) ==> (\<exists>x\<in>A. M(x) & P(x)) <-> (\<exists>x\<in>A. P(x))" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
404 |
by (blast intro: transM) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
405 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
406 |
lemma (in M_triv_axioms) rex_abs [simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
407 |
"M(A) ==> (\<exists>x[M]. x\<in>A & P(x)) <-> (\<exists>x\<in>A. P(x))" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
408 |
by (blast intro: transM) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
409 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
410 |
lemma (in M_triv_axioms) ball_iff_equiv: |
13299 | 411 |
"M(A) ==> (\<forall>x[M]. (x\<in>A <-> P(x))) <-> |
13290
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
412 |
(\<forall>x\<in>A. P(x)) & (\<forall>x. P(x) --> M(x) --> x\<in>A)" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
413 |
by (blast intro: transM) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
414 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
415 |
text{*Simplifies proofs of equalities when there's an iff-equality |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
416 |
available for rewriting, universally quantified over M. *} |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
417 |
lemma (in M_triv_axioms) M_equalityI: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
418 |
"[| !!x. M(x) ==> x\<in>A <-> x\<in>B; M(A); M(B) |] ==> A=B" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
419 |
by (blast intro!: equalityI dest: transM) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
420 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
421 |
lemma (in M_triv_axioms) empty_abs [simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
422 |
"M(z) ==> empty(M,z) <-> z=0" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
423 |
apply (simp add: empty_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
424 |
apply (blast intro: transM) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
425 |
done |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
426 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
427 |
lemma (in M_triv_axioms) subset_abs [simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
428 |
"M(A) ==> subset(M,A,B) <-> A \<subseteq> B" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
429 |
apply (simp add: subset_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
430 |
apply (blast intro: transM) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
431 |
done |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
432 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
433 |
lemma (in M_triv_axioms) upair_abs [simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
434 |
"M(z) ==> upair(M,a,b,z) <-> z={a,b}" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
435 |
apply (simp add: upair_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
436 |
apply (blast intro: transM) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
437 |
done |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
438 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
439 |
lemma (in M_triv_axioms) upair_in_M_iff [iff]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
440 |
"M({a,b}) <-> M(a) & M(b)" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
441 |
apply (insert upair_ax, simp add: upair_ax_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
442 |
apply (blast intro: transM) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
443 |
done |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
444 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
445 |
lemma (in M_triv_axioms) singleton_in_M_iff [iff]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
446 |
"M({a}) <-> M(a)" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
447 |
by (insert upair_in_M_iff [of a a], simp) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
448 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
449 |
lemma (in M_triv_axioms) pair_abs [simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
450 |
"M(z) ==> pair(M,a,b,z) <-> z=<a,b>" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
451 |
apply (simp add: pair_def ZF.Pair_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
452 |
apply (blast intro: transM) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
453 |
done |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
454 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
455 |
lemma (in M_triv_axioms) pair_in_M_iff [iff]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
456 |
"M(<a,b>) <-> M(a) & M(b)" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
457 |
by (simp add: ZF.Pair_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
458 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
459 |
lemma (in M_triv_axioms) pair_components_in_M: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
460 |
"[| <x,y> \<in> A; M(A) |] ==> M(x) & M(y)" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
461 |
apply (simp add: Pair_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
462 |
apply (blast dest: transM) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
463 |
done |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
464 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
465 |
lemma (in M_triv_axioms) cartprod_abs [simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
466 |
"[| M(A); M(B); M(z) |] ==> cartprod(M,A,B,z) <-> z = A*B" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
467 |
apply (simp add: cartprod_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
468 |
apply (rule iffI) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
469 |
apply (blast intro!: equalityI intro: transM dest!: rspec) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
470 |
apply (blast dest: transM) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
471 |
done |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
472 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
473 |
lemma (in M_triv_axioms) union_abs [simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
474 |
"[| M(a); M(b); M(z) |] ==> union(M,a,b,z) <-> z = a Un b" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
475 |
apply (simp add: union_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
476 |
apply (blast intro: transM) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
477 |
done |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
478 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
479 |
lemma (in M_triv_axioms) inter_abs [simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
480 |
"[| M(a); M(b); M(z) |] ==> inter(M,a,b,z) <-> z = a Int b" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
481 |
apply (simp add: inter_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
482 |
apply (blast intro: transM) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
483 |
done |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
484 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
485 |
lemma (in M_triv_axioms) setdiff_abs [simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
486 |
"[| M(a); M(b); M(z) |] ==> setdiff(M,a,b,z) <-> z = a-b" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
487 |
apply (simp add: setdiff_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
488 |
apply (blast intro: transM) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
489 |
done |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
490 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
491 |
lemma (in M_triv_axioms) Union_abs [simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
492 |
"[| M(A); M(z) |] ==> big_union(M,A,z) <-> z = Union(A)" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
493 |
apply (simp add: big_union_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
494 |
apply (blast intro!: equalityI dest: transM) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
495 |
done |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
496 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
497 |
lemma (in M_triv_axioms) Union_closed [intro,simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
498 |
"M(A) ==> M(Union(A))" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
499 |
by (insert Union_ax, simp add: Union_ax_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
500 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
501 |
lemma (in M_triv_axioms) Un_closed [intro,simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
502 |
"[| M(A); M(B) |] ==> M(A Un B)" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
503 |
by (simp only: Un_eq_Union, blast) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
504 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
505 |
lemma (in M_triv_axioms) cons_closed [intro,simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
506 |
"[| M(a); M(A) |] ==> M(cons(a,A))" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
507 |
by (subst cons_eq [symmetric], blast) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
508 |
|
13306 | 509 |
lemma (in M_triv_axioms) cons_abs [simp]: |
510 |
"[| M(b); M(z) |] ==> is_cons(M,a,b,z) <-> z = cons(a,b)" |
|
511 |
by (simp add: is_cons_def, blast intro: transM) |
|
512 |
||
13290
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
513 |
lemma (in M_triv_axioms) successor_abs [simp]: |
13306 | 514 |
"[| M(a); M(z) |] ==> successor(M,a,z) <-> z = succ(a)" |
13290
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
515 |
by (simp add: successor_def, blast) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
516 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
517 |
lemma (in M_triv_axioms) succ_in_M_iff [iff]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
518 |
"M(succ(a)) <-> M(a)" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
519 |
apply (simp add: succ_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
520 |
apply (blast intro: transM) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
521 |
done |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
522 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
523 |
lemma (in M_triv_axioms) separation_closed [intro,simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
524 |
"[| separation(M,P); M(A) |] ==> M(Collect(A,P))" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
525 |
apply (insert separation, simp add: separation_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
526 |
apply (drule rspec, assumption, clarify) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
527 |
apply (subgoal_tac "y = Collect(A,P)", blast) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
528 |
apply (blast dest: transM) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
529 |
done |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
530 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
531 |
text{*Probably the premise and conclusion are equivalent*} |
13306 | 532 |
lemma (in M_triv_axioms) strong_replacementI [OF rallI]: |
533 |
"[| \<forall>A[M]. separation(M, %u. \<exists>x[M]. x\<in>A & P(x,u)) |] |
|
13290
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
534 |
==> strong_replacement(M,P)" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
535 |
apply (simp add: strong_replacement_def, clarify) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
536 |
apply (frule replacementD [OF replacement], assumption, clarify) |
13299 | 537 |
apply (drule_tac x=A in rspec, clarify) |
13290
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
538 |
apply (drule_tac z=Y in separationD, assumption, clarify) |
13299 | 539 |
apply (rule_tac x=y in rexI) |
540 |
apply (blast dest: transM)+ |
|
13290
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
541 |
done |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
542 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
543 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
544 |
(*The last premise expresses that P takes M to M*) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
545 |
lemma (in M_triv_axioms) strong_replacement_closed [intro,simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
546 |
"[| strong_replacement(M,P); M(A); univalent(M,A,P); |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
547 |
!!x y. [| x\<in>A; P(x,y); M(x) |] ==> M(y) |] ==> M(Replace(A,P))" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
548 |
apply (simp add: strong_replacement_def) |
13299 | 549 |
apply (drule rspec, auto) |
13290
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
550 |
apply (subgoal_tac "Replace(A,P) = Y") |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
551 |
apply simp |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
552 |
apply (rule equality_iffI) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
553 |
apply (simp add: Replace_iff, safe) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
554 |
apply (blast dest: transM) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
555 |
apply (frule transM, assumption) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
556 |
apply (simp add: univalent_def) |
13299 | 557 |
apply (drule rspec [THEN iffD1], assumption, assumption) |
13290
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
558 |
apply (blast dest: transM) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
559 |
done |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
560 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
561 |
(*The first premise can't simply be assumed as a schema. |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
562 |
It is essential to take care when asserting instances of Replacement. |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
563 |
Let K be a nonconstructible subset of nat and define |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
564 |
f(x) = x if x:K and f(x)=0 otherwise. Then RepFun(nat,f) = cons(0,K), a |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
565 |
nonconstructible set. So we cannot assume that M(X) implies M(RepFun(X,f)) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
566 |
even for f : M -> M. |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
567 |
*) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
568 |
lemma (in M_triv_axioms) RepFun_closed [intro,simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
569 |
"[| strong_replacement(M, \<lambda>x y. y = f(x)); M(A); \<forall>x\<in>A. M(f(x)) |] |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
570 |
==> M(RepFun(A,f))" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
571 |
apply (simp add: RepFun_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
572 |
apply (rule strong_replacement_closed) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
573 |
apply (auto dest: transM simp add: univalent_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
574 |
done |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
575 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
576 |
lemma (in M_triv_axioms) lam_closed [intro,simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
577 |
"[| strong_replacement(M, \<lambda>x y. y = <x,b(x)>); M(A); \<forall>x\<in>A. M(b(x)) |] |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
578 |
==> M(\<lambda>x\<in>A. b(x))" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
579 |
by (simp add: lam_def, blast dest: transM) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
580 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
581 |
lemma (in M_triv_axioms) image_abs [simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
582 |
"[| M(r); M(A); M(z) |] ==> image(M,r,A,z) <-> z = r``A" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
583 |
apply (simp add: image_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
584 |
apply (rule iffI) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
585 |
apply (blast intro!: equalityI dest: transM, blast) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
586 |
done |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
587 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
588 |
text{*What about @{text Pow_abs}? Powerset is NOT absolute! |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
589 |
This result is one direction of absoluteness.*} |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
590 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
591 |
lemma (in M_triv_axioms) powerset_Pow: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
592 |
"powerset(M, x, Pow(x))" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
593 |
by (simp add: powerset_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
594 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
595 |
text{*But we can't prove that the powerset in @{text M} includes the |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
596 |
real powerset.*} |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
597 |
lemma (in M_triv_axioms) powerset_imp_subset_Pow: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
598 |
"[| powerset(M,x,y); M(y) |] ==> y <= Pow(x)" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
599 |
apply (simp add: powerset_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
600 |
apply (blast dest: transM) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
601 |
done |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
602 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
603 |
lemma (in M_triv_axioms) nat_into_M [intro]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
604 |
"n \<in> nat ==> M(n)" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
605 |
by (induct n rule: nat_induct, simp_all) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
606 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
607 |
lemma (in M_triv_axioms) nat_case_closed: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
608 |
"[|M(k); M(a); \<forall>m[M]. M(b(m))|] ==> M(nat_case(a,b,k))" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
609 |
apply (case_tac "k=0", simp) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
610 |
apply (case_tac "\<exists>m. k = succ(m)", force) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
611 |
apply (simp add: nat_case_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
612 |
done |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
613 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
614 |
lemma (in M_triv_axioms) Inl_in_M_iff [iff]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
615 |
"M(Inl(a)) <-> M(a)" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
616 |
by (simp add: Inl_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
617 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
618 |
lemma (in M_triv_axioms) Inr_in_M_iff [iff]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
619 |
"M(Inr(a)) <-> M(a)" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
620 |
by (simp add: Inr_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
621 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
622 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
623 |
subsection{*Absoluteness for ordinals*} |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
624 |
text{*These results constitute Theorem IV 5.1 of Kunen (page 126).*} |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
625 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
626 |
lemma (in M_triv_axioms) lt_closed: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
627 |
"[| j<i; M(i) |] ==> M(j)" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
628 |
by (blast dest: ltD intro: transM) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
629 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
630 |
lemma (in M_triv_axioms) transitive_set_abs [simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
631 |
"M(a) ==> transitive_set(M,a) <-> Transset(a)" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
632 |
by (simp add: transitive_set_def Transset_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
633 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
634 |
lemma (in M_triv_axioms) ordinal_abs [simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
635 |
"M(a) ==> ordinal(M,a) <-> Ord(a)" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
636 |
by (simp add: ordinal_def Ord_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
637 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
638 |
lemma (in M_triv_axioms) limit_ordinal_abs [simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
639 |
"M(a) ==> limit_ordinal(M,a) <-> Limit(a)" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
640 |
apply (simp add: limit_ordinal_def Ord_0_lt_iff Limit_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
641 |
apply (simp add: lt_def, blast) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
642 |
done |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
643 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
644 |
lemma (in M_triv_axioms) successor_ordinal_abs [simp]: |
13299 | 645 |
"M(a) ==> successor_ordinal(M,a) <-> Ord(a) & (\<exists>b[M]. a = succ(b))" |
13290
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
646 |
apply (simp add: successor_ordinal_def, safe) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
647 |
apply (drule Ord_cases_disj, auto) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
648 |
done |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
649 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
650 |
lemma finite_Ord_is_nat: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
651 |
"[| Ord(a); ~ Limit(a); \<forall>x\<in>a. ~ Limit(x) |] ==> a \<in> nat" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
652 |
by (induct a rule: trans_induct3, simp_all) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
653 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
654 |
lemma naturals_not_limit: "a \<in> nat ==> ~ Limit(a)" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
655 |
by (induct a rule: nat_induct, auto) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
656 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
657 |
lemma (in M_triv_axioms) finite_ordinal_abs [simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
658 |
"M(a) ==> finite_ordinal(M,a) <-> a \<in> nat" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
659 |
apply (simp add: finite_ordinal_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
660 |
apply (blast intro: finite_Ord_is_nat intro: nat_into_Ord |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
661 |
dest: Ord_trans naturals_not_limit) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
662 |
done |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
663 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
664 |
lemma Limit_non_Limit_implies_nat: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
665 |
"[| Limit(a); \<forall>x\<in>a. ~ Limit(x) |] ==> a = nat" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
666 |
apply (rule le_anti_sym) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
667 |
apply (rule all_lt_imp_le, blast, blast intro: Limit_is_Ord) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
668 |
apply (simp add: lt_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
669 |
apply (blast intro: Ord_in_Ord Ord_trans finite_Ord_is_nat) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
670 |
apply (erule nat_le_Limit) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
671 |
done |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
672 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
673 |
lemma (in M_triv_axioms) omega_abs [simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
674 |
"M(a) ==> omega(M,a) <-> a = nat" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
675 |
apply (simp add: omega_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
676 |
apply (blast intro: Limit_non_Limit_implies_nat dest: naturals_not_limit) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
677 |
done |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
678 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
679 |
lemma (in M_triv_axioms) number1_abs [simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
680 |
"M(a) ==> number1(M,a) <-> a = 1" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
681 |
by (simp add: number1_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
682 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
683 |
lemma (in M_triv_axioms) number1_abs [simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
684 |
"M(a) ==> number2(M,a) <-> a = succ(1)" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
685 |
by (simp add: number2_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
686 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
687 |
lemma (in M_triv_axioms) number3_abs [simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
688 |
"M(a) ==> number3(M,a) <-> a = succ(succ(1))" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
689 |
by (simp add: number3_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
690 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
691 |
text{*Kunen continued to 20...*} |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
692 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
693 |
(*Could not get this to work. The \<lambda>x\<in>nat is essential because everything |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
694 |
but the recursion variable must stay unchanged. But then the recursion |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
695 |
equations only hold for x\<in>nat (or in some other set) and not for the |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
696 |
whole of the class M. |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
697 |
consts |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
698 |
natnumber_aux :: "[i=>o,i] => i" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
699 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
700 |
primrec |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
701 |
"natnumber_aux(M,0) = (\<lambda>x\<in>nat. if empty(M,x) then 1 else 0)" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
702 |
"natnumber_aux(M,succ(n)) = |
13299 | 703 |
(\<lambda>x\<in>nat. if (\<exists>y[M]. natnumber_aux(M,n)`y=1 & successor(M,y,x)) |
13290
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
704 |
then 1 else 0)" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
705 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
706 |
constdefs |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
707 |
natnumber :: "[i=>o,i,i] => o" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
708 |
"natnumber(M,n,x) == natnumber_aux(M,n)`x = 1" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
709 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
710 |
lemma (in M_triv_axioms) [simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
711 |
"natnumber(M,0,x) == x=0" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
712 |
*) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
713 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
714 |
subsection{*Some instances of separation and strong replacement*} |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
715 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
716 |
locale M_axioms = M_triv_axioms + |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
717 |
assumes Inter_separation: |
13268 | 718 |
"M(A) ==> separation(M, \<lambda>x. \<forall>y[M]. y\<in>A --> x\<in>y)" |
13223 | 719 |
and cartprod_separation: |
720 |
"[| M(A); M(B) |] |
|
13298 | 721 |
==> separation(M, \<lambda>z. \<exists>x[M]. x\<in>A & (\<exists>y[M]. y\<in>B & pair(M,x,y,z)))" |
13223 | 722 |
and image_separation: |
723 |
"[| M(A); M(r) |] |
|
13268 | 724 |
==> separation(M, \<lambda>y. \<exists>p[M]. p\<in>r & (\<exists>x[M]. x\<in>A & pair(M,x,y,p)))" |
13223 | 725 |
and converse_separation: |
13298 | 726 |
"M(r) ==> separation(M, |
727 |
\<lambda>z. \<exists>p[M]. p\<in>r & (\<exists>x[M]. \<exists>y[M]. pair(M,x,y,p) & pair(M,y,x,z)))" |
|
13223 | 728 |
and restrict_separation: |
13268 | 729 |
"M(A) ==> separation(M, \<lambda>z. \<exists>x[M]. x\<in>A & (\<exists>y[M]. pair(M,x,y,z)))" |
13223 | 730 |
and comp_separation: |
731 |
"[| M(r); M(s) |] |
|
13268 | 732 |
==> separation(M, \<lambda>xz. \<exists>x[M]. \<exists>y[M]. \<exists>z[M]. \<exists>xy[M]. \<exists>yz[M]. |
733 |
pair(M,x,z,xz) & pair(M,x,y,xy) & pair(M,y,z,yz) & |
|
734 |
xy\<in>s & yz\<in>r)" |
|
13223 | 735 |
and pred_separation: |
13298 | 736 |
"[| M(r); M(x) |] ==> separation(M, \<lambda>y. \<exists>p[M]. p\<in>r & pair(M,y,x,p))" |
13223 | 737 |
and Memrel_separation: |
13298 | 738 |
"separation(M, \<lambda>z. \<exists>x[M]. \<exists>y[M]. pair(M,x,y,z) & x \<in> y)" |
13268 | 739 |
and funspace_succ_replacement: |
740 |
"M(n) ==> |
|
13306 | 741 |
strong_replacement(M, \<lambda>p z. \<exists>f[M]. \<exists>b[M]. \<exists>nb[M]. \<exists>cnbf[M]. |
742 |
pair(M,f,b,p) & pair(M,n,b,nb) & is_cons(M,nb,f,cnbf) & |
|
743 |
upair(M,cnbf,cnbf,z))" |
|
13223 | 744 |
and well_ord_iso_separation: |
745 |
"[| M(A); M(f); M(r) |] |
|
13299 | 746 |
==> separation (M, \<lambda>x. x\<in>A --> (\<exists>y[M]. (\<exists>p[M]. |
13245 | 747 |
fun_apply(M,f,x,y) & pair(M,y,x,p) & p \<in> r)))" |
13306 | 748 |
and obase_separation: |
749 |
--{*part of the order type formalization*} |
|
750 |
"[| M(A); M(r) |] |
|
751 |
==> separation(M, \<lambda>a. \<exists>x[M]. \<exists>g[M]. \<exists>mx[M]. \<exists>par[M]. |
|
752 |
ordinal(M,x) & membership(M,x,mx) & pred_set(M,A,a,r,par) & |
|
753 |
order_isomorphism(M,par,r,x,mx,g))" |
|
13223 | 754 |
and obase_equals_separation: |
755 |
"[| M(A); M(r) |] |
|
13316 | 756 |
==> separation (M, \<lambda>x. x\<in>A --> ~(\<exists>y[M]. \<exists>g[M]. |
757 |
ordinal(M,y) & (\<exists>my[M]. \<exists>pxr[M]. |
|
758 |
membership(M,y,my) & pred_set(M,A,x,r,pxr) & |
|
759 |
order_isomorphism(M,pxr,r,y,my,g))))" |
|
13306 | 760 |
and omap_replacement: |
761 |
"[| M(A); M(r) |] |
|
762 |
==> strong_replacement(M, |
|
763 |
\<lambda>a z. \<exists>x[M]. \<exists>g[M]. \<exists>mx[M]. \<exists>par[M]. |
|
764 |
ordinal(M,x) & pair(M,a,x,z) & membership(M,x,mx) & |
|
765 |
pred_set(M,A,a,r,par) & order_isomorphism(M,par,r,x,mx,g))" |
|
13223 | 766 |
and is_recfun_separation: |
767 |
--{*for well-founded recursion. NEEDS RELATIVIZATION*} |
|
768 |
"[| M(A); M(f); M(g); M(a); M(b) |] |
|
13251
74cb2af8811e
new treatment of wfrec, replacing wf[A](r) by wf(r)
paulson
parents:
13247
diff
changeset
|
769 |
==> separation(M, \<lambda>x. \<langle>x,a\<rangle> \<in> r & \<langle>x,b\<rangle> \<in> r & f`x \<noteq> g`x)" |
13223 | 770 |
|
771 |
lemma (in M_axioms) cartprod_iff_lemma: |
|
13254 | 772 |
"[| M(C); \<forall>u[M]. u \<in> C <-> (\<exists>x\<in>A. \<exists>y\<in>B. u = {{x}, {x,y}}); |
773 |
powerset(M, A \<union> B, p1); powerset(M, p1, p2); M(p2) |] |
|
13223 | 774 |
==> C = {u \<in> p2 . \<exists>x\<in>A. \<exists>y\<in>B. u = {{x}, {x,y}}}" |
775 |
apply (simp add: powerset_def) |
|
13254 | 776 |
apply (rule equalityI, clarify, simp) |
777 |
apply (frule transM, assumption) |
|
13223 | 778 |
apply (frule transM, assumption, simp) |
779 |
apply blast |
|
780 |
apply clarify |
|
781 |
apply (frule transM, assumption, force) |
|
782 |
done |
|
783 |
||
784 |
lemma (in M_axioms) cartprod_iff: |
|
785 |
"[| M(A); M(B); M(C) |] |
|
786 |
==> cartprod(M,A,B,C) <-> |
|
787 |
(\<exists>p1 p2. M(p1) & M(p2) & powerset(M,A Un B,p1) & powerset(M,p1,p2) & |
|
788 |
C = {z \<in> p2. \<exists>x\<in>A. \<exists>y\<in>B. z = <x,y>})" |
|
789 |
apply (simp add: Pair_def cartprod_def, safe) |
|
790 |
defer 1 |
|
791 |
apply (simp add: powerset_def) |
|
792 |
apply blast |
|
793 |
txt{*Final, difficult case: the left-to-right direction of the theorem.*} |
|
794 |
apply (insert power_ax, simp add: power_ax_def) |
|
13299 | 795 |
apply (frule_tac x="A Un B" and P="\<lambda>x. rex(M,?Q(x))" in rspec) |
796 |
apply (blast, clarify) |
|
797 |
apply (drule_tac x=z and P="\<lambda>x. rex(M,?Q(x))" in rspec) |
|
798 |
apply assumption |
|
13223 | 799 |
apply (blast intro: cartprod_iff_lemma) |
800 |
done |
|
801 |
||
802 |
lemma (in M_axioms) cartprod_closed_lemma: |
|
13299 | 803 |
"[| M(A); M(B) |] ==> \<exists>C[M]. cartprod(M,A,B,C)" |
13223 | 804 |
apply (simp del: cartprod_abs add: cartprod_iff) |
805 |
apply (insert power_ax, simp add: power_ax_def) |
|
13299 | 806 |
apply (frule_tac x="A Un B" and P="\<lambda>x. rex(M,?Q(x))" in rspec) |
807 |
apply (blast, clarify) |
|
808 |
apply (drule_tac x=z and P="\<lambda>x. rex(M,?Q(x))" in rspec) |
|
809 |
apply (blast, clarify) |
|
810 |
apply (intro rexI exI conjI) |
|
811 |
prefer 5 apply (rule refl) |
|
812 |
prefer 3 apply assumption |
|
813 |
prefer 3 apply assumption |
|
13245 | 814 |
apply (insert cartprod_separation [of A B], auto) |
13223 | 815 |
done |
816 |
||
817 |
text{*All the lemmas above are necessary because Powerset is not absolute. |
|
818 |
I should have used Replacement instead!*} |
|
13245 | 819 |
lemma (in M_axioms) cartprod_closed [intro,simp]: |
13223 | 820 |
"[| M(A); M(B) |] ==> M(A*B)" |
821 |
by (frule cartprod_closed_lemma, assumption, force) |
|
822 |
||
13268 | 823 |
lemma (in M_axioms) sum_closed [intro,simp]: |
824 |
"[| M(A); M(B) |] ==> M(A+B)" |
|
825 |
by (simp add: sum_def) |
|
826 |
||
13290
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
827 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
828 |
subsubsection {*converse of a relation*} |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
829 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
830 |
lemma (in M_axioms) M_converse_iff: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
831 |
"M(r) ==> |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
832 |
converse(r) = |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
833 |
{z \<in> Union(Union(r)) * Union(Union(r)). |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
834 |
\<exists>p\<in>r. \<exists>x[M]. \<exists>y[M]. p = \<langle>x,y\<rangle> & z = \<langle>y,x\<rangle>}" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
835 |
apply (rule equalityI) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
836 |
prefer 2 apply (blast dest: transM, clarify, simp) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
837 |
apply (simp add: Pair_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
838 |
apply (blast dest: transM) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
839 |
done |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
840 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
841 |
lemma (in M_axioms) converse_closed [intro,simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
842 |
"M(r) ==> M(converse(r))" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
843 |
apply (simp add: M_converse_iff) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
844 |
apply (insert converse_separation [of r], simp) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
845 |
done |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
846 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
847 |
lemma (in M_axioms) converse_abs [simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
848 |
"[| M(r); M(z) |] ==> is_converse(M,r,z) <-> z = converse(r)" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
849 |
apply (simp add: is_converse_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
850 |
apply (rule iffI) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
851 |
prefer 2 apply blast |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
852 |
apply (rule M_equalityI) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
853 |
apply simp |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
854 |
apply (blast dest: transM)+ |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
855 |
done |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
856 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
857 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
858 |
subsubsection {*image, preimage, domain, range*} |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
859 |
|
13245 | 860 |
lemma (in M_axioms) image_closed [intro,simp]: |
13223 | 861 |
"[| M(A); M(r) |] ==> M(r``A)" |
862 |
apply (simp add: image_iff_Collect) |
|
13245 | 863 |
apply (insert image_separation [of A r], simp) |
13223 | 864 |
done |
865 |
||
866 |
lemma (in M_axioms) vimage_abs [simp]: |
|
867 |
"[| M(r); M(A); M(z) |] ==> pre_image(M,r,A,z) <-> z = r-``A" |
|
868 |
apply (simp add: pre_image_def) |
|
869 |
apply (rule iffI) |
|
870 |
apply (blast intro!: equalityI dest: transM, blast) |
|
871 |
done |
|
872 |
||
13245 | 873 |
lemma (in M_axioms) vimage_closed [intro,simp]: |
13223 | 874 |
"[| M(A); M(r) |] ==> M(r-``A)" |
13290
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
875 |
by (simp add: vimage_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
876 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
877 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
878 |
subsubsection{*Domain, range and field*} |
13223 | 879 |
|
880 |
lemma (in M_axioms) domain_abs [simp]: |
|
881 |
"[| M(r); M(z) |] ==> is_domain(M,r,z) <-> z = domain(r)" |
|
882 |
apply (simp add: is_domain_def) |
|
883 |
apply (blast intro!: equalityI dest: transM) |
|
884 |
done |
|
885 |
||
13245 | 886 |
lemma (in M_axioms) domain_closed [intro,simp]: |
13223 | 887 |
"M(r) ==> M(domain(r))" |
888 |
apply (simp add: domain_eq_vimage) |
|
889 |
done |
|
890 |
||
891 |
lemma (in M_axioms) range_abs [simp]: |
|
892 |
"[| M(r); M(z) |] ==> is_range(M,r,z) <-> z = range(r)" |
|
893 |
apply (simp add: is_range_def) |
|
894 |
apply (blast intro!: equalityI dest: transM) |
|
895 |
done |
|
896 |
||
13245 | 897 |
lemma (in M_axioms) range_closed [intro,simp]: |
13223 | 898 |
"M(r) ==> M(range(r))" |
899 |
apply (simp add: range_eq_image) |
|
900 |
done |
|
901 |
||
13245 | 902 |
lemma (in M_axioms) field_abs [simp]: |
903 |
"[| M(r); M(z) |] ==> is_field(M,r,z) <-> z = field(r)" |
|
904 |
by (simp add: domain_closed range_closed is_field_def field_def) |
|
905 |
||
906 |
lemma (in M_axioms) field_closed [intro,simp]: |
|
907 |
"M(r) ==> M(field(r))" |
|
908 |
by (simp add: domain_closed range_closed Un_closed field_def) |
|
909 |
||
910 |
||
13290
28ce81eff3de
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diff
changeset
|
911 |
subsubsection{*Relations, functions and application*} |
13254 | 912 |
|
13223 | 913 |
lemma (in M_axioms) relation_abs [simp]: |
914 |
"M(r) ==> is_relation(M,r) <-> relation(r)" |
|
915 |
apply (simp add: is_relation_def relation_def) |
|
916 |
apply (blast dest!: bspec dest: pair_components_in_M)+ |
|
917 |
done |
|
918 |
||
919 |
lemma (in M_axioms) function_abs [simp]: |
|
920 |
"M(r) ==> is_function(M,r) <-> function(r)" |
|
921 |
apply (simp add: is_function_def function_def, safe) |
|
922 |
apply (frule transM, assumption) |
|
923 |
apply (blast dest: pair_components_in_M)+ |
|
924 |
done |
|
925 |
||
13245 | 926 |
lemma (in M_axioms) apply_closed [intro,simp]: |
13223 | 927 |
"[|M(f); M(a)|] ==> M(f`a)" |
13290
28ce81eff3de
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paulson
parents:
13269
diff
changeset
|
928 |
by (simp add: apply_def) |
13223 | 929 |
|
930 |
lemma (in M_axioms) apply_abs: |
|
931 |
"[| function(f); M(f); M(y) |] |
|
932 |
==> fun_apply(M,f,x,y) <-> x \<in> domain(f) & f`x = y" |
|
933 |
apply (simp add: fun_apply_def) |
|
934 |
apply (blast intro: function_apply_equality function_apply_Pair) |
|
935 |
done |
|
936 |
||
937 |
lemma (in M_axioms) typed_apply_abs: |
|
938 |
"[| f \<in> A -> B; M(f); M(y) |] |
|
939 |
==> fun_apply(M,f,x,y) <-> x \<in> A & f`x = y" |
|
940 |
by (simp add: apply_abs fun_is_function domain_of_fun) |
|
941 |
||
942 |
lemma (in M_axioms) typed_function_abs [simp]: |
|
943 |
"[| M(A); M(f) |] ==> typed_function(M,A,B,f) <-> f \<in> A -> B" |
|
944 |
apply (auto simp add: typed_function_def relation_def Pi_iff) |
|
945 |
apply (blast dest: pair_components_in_M)+ |
|
946 |
done |
|
947 |
||
948 |
lemma (in M_axioms) injection_abs [simp]: |
|
949 |
"[| M(A); M(f) |] ==> injection(M,A,B,f) <-> f \<in> inj(A,B)" |
|
950 |
apply (simp add: injection_def apply_iff inj_def apply_closed) |
|
13247 | 951 |
apply (blast dest: transM [of _ A]) |
13223 | 952 |
done |
953 |
||
954 |
lemma (in M_axioms) surjection_abs [simp]: |
|
955 |
"[| M(A); M(B); M(f) |] ==> surjection(M,A,B,f) <-> f \<in> surj(A,B)" |
|
956 |
by (simp add: typed_apply_abs surjection_def surj_def) |
|
957 |
||
958 |
lemma (in M_axioms) bijection_abs [simp]: |
|
959 |
"[| M(A); M(B); M(f) |] ==> bijection(M,A,B,f) <-> f \<in> bij(A,B)" |
|
960 |
by (simp add: bijection_def bij_def) |
|
961 |
||
962 |
||
13290
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separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
963 |
subsubsection{*Composition of relations*} |
13223 | 964 |
|
965 |
lemma (in M_axioms) M_comp_iff: |
|
966 |
"[| M(r); M(s) |] |
|
967 |
==> r O s = |
|
968 |
{xz \<in> domain(s) * range(r). |
|
13268 | 969 |
\<exists>x[M]. \<exists>y[M]. \<exists>z[M]. xz = \<langle>x,z\<rangle> & \<langle>x,y\<rangle> \<in> s & \<langle>y,z\<rangle> \<in> r}" |
13223 | 970 |
apply (simp add: comp_def) |
971 |
apply (rule equalityI) |
|
13247 | 972 |
apply clarify |
973 |
apply simp |
|
13223 | 974 |
apply (blast dest: transM)+ |
975 |
done |
|
976 |
||
13245 | 977 |
lemma (in M_axioms) comp_closed [intro,simp]: |
13223 | 978 |
"[| M(r); M(s) |] ==> M(r O s)" |
979 |
apply (simp add: M_comp_iff) |
|
13245 | 980 |
apply (insert comp_separation [of r s], simp) |
981 |
done |
|
982 |
||
983 |
lemma (in M_axioms) composition_abs [simp]: |
|
984 |
"[| M(r); M(s); M(t) |] |
|
985 |
==> composition(M,r,s,t) <-> t = r O s" |
|
13247 | 986 |
apply safe |
13245 | 987 |
txt{*Proving @{term "composition(M, r, s, r O s)"}*} |
988 |
prefer 2 |
|
989 |
apply (simp add: composition_def comp_def) |
|
990 |
apply (blast dest: transM) |
|
991 |
txt{*Opposite implication*} |
|
992 |
apply (rule M_equalityI) |
|
993 |
apply (simp add: composition_def comp_def) |
|
994 |
apply (blast del: allE dest: transM)+ |
|
13223 | 995 |
done |
996 |
||
13290
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
997 |
text{*no longer needed*} |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
998 |
lemma (in M_axioms) restriction_is_function: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
999 |
"[| restriction(M,f,A,z); function(f); M(f); M(A); M(z) |] |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
1000 |
==> function(z)" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
1001 |
apply (rotate_tac 1) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
1002 |
apply (simp add: restriction_def ball_iff_equiv) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
1003 |
apply (unfold function_def, blast) |
13269 | 1004 |
done |
1005 |
||
13290
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
1006 |
lemma (in M_axioms) restriction_abs [simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
1007 |
"[| M(f); M(A); M(z) |] |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
1008 |
==> restriction(M,f,A,z) <-> z = restrict(f,A)" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
1009 |
apply (simp add: ball_iff_equiv restriction_def restrict_def) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
1010 |
apply (blast intro!: equalityI dest: transM) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
1011 |
done |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
1012 |
|
13223 | 1013 |
|
13290
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
1014 |
lemma (in M_axioms) M_restrict_iff: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
1015 |
"M(r) ==> restrict(r,A) = {z \<in> r . \<exists>x\<in>A. \<exists>y[M]. z = \<langle>x, y\<rangle>}" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
1016 |
by (simp add: restrict_def, blast dest: transM) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
1017 |
|
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
1018 |
lemma (in M_axioms) restrict_closed [intro,simp]: |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
1019 |
"[| M(A); M(r) |] ==> M(restrict(r,A))" |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
1020 |
apply (simp add: M_restrict_iff) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
1021 |
apply (insert restrict_separation [of A], simp) |
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
1022 |
done |
13223 | 1023 |
|
1024 |
lemma (in M_axioms) Inter_abs [simp]: |
|
1025 |
"[| M(A); M(z) |] ==> big_inter(M,A,z) <-> z = Inter(A)" |
|
1026 |
apply (simp add: big_inter_def Inter_def) |
|
1027 |
apply (blast intro!: equalityI dest: transM) |
|
1028 |
done |
|
1029 |
||
13245 | 1030 |
lemma (in M_axioms) Inter_closed [intro,simp]: |
13223 | 1031 |
"M(A) ==> M(Inter(A))" |
13245 | 1032 |
by (insert Inter_separation, simp add: Inter_def) |
13223 | 1033 |
|
13245 | 1034 |
lemma (in M_axioms) Int_closed [intro,simp]: |
13223 | 1035 |
"[| M(A); M(B) |] ==> M(A Int B)" |
1036 |
apply (subgoal_tac "M({A,B})") |
|
13247 | 1037 |
apply (frule Inter_closed, force+) |
13223 | 1038 |
done |
1039 |
||
13290
28ce81eff3de
separation of M_axioms into M_triv_axioms and M_axioms
paulson
parents:
13269
diff
changeset
|
1040 |
subsubsection{*Functions and function space*} |
13268 | 1041 |
|
13245 | 1042 |
text{*M contains all finite functions*} |
1043 |
lemma (in M_axioms) finite_fun_closed_lemma [rule_format]: |
|
1044 |
"[| n \<in> nat; M(A) |] ==> \<forall>f \<in> n -> A. M(f)" |
|
1045 |
apply (induct_tac n, simp) |
|
1046 |
apply (rule ballI) |
|
1047 |
apply (simp add: succ_def) |
|
1048 |
apply (frule fun_cons_restrict_eq) |
|
1049 |
apply (erule ssubst) |
|
1050 |
apply (subgoal_tac "M(f`x) & restrict(f,x) \<in> x -> A") |
|
1051 |
apply (simp add: cons_closed nat_into_M apply_closed) |
|
1052 |
apply (blast intro: apply_funtype transM restrict_type2) |
|
1053 |
done |
|
1054 |
||
1055 |
lemma (in M_axioms) finite_fun_closed [rule_format]: |
|
1056 |
"[| f \<in> n -> A; n \<in> nat; M(A) |] ==> M(f)" |
|
1057 |
by (blast intro: finite_fun_closed_lemma) |
|
1058 |
||
13268 | 1059 |
text{*The assumption @{term "M(A->B)"} is unusual, but essential: in |
1060 |
all but trivial cases, A->B cannot be expected to belong to @{term M}.*} |
|
1061 |
lemma (in M_axioms) is_funspace_abs [simp]: |
|
1062 |
"[|M(A); M(B); M(F); M(A->B)|] ==> is_funspace(M,A,B,F) <-> F = A->B"; |
|
1063 |
apply (simp add: is_funspace_def) |
|
1064 |
apply (rule iffI) |
|
1065 |
prefer 2 apply blast |
|
1066 |
apply (rule M_equalityI) |
|
1067 |
apply simp_all |
|
1068 |
done |
|
1069 |
||
1070 |
lemma (in M_axioms) succ_fun_eq2: |
|
1071 |
"[|M(B); M(n->B)|] ==> |
|
1072 |
succ(n) -> B = |
|
1073 |
\<Union>{z. p \<in> (n->B)*B, \<exists>f[M]. \<exists>b[M]. p = <f,b> & z = {cons(<n,b>, f)}}" |
|
1074 |
apply (simp add: succ_fun_eq) |
|
1075 |
apply (blast dest: transM) |
|
1076 |
done |
|
1077 |
||
1078 |
lemma (in M_axioms) funspace_succ: |
|
1079 |
"[|M(n); M(B); M(n->B) |] ==> M(succ(n) -> B)" |
|
13306 | 1080 |
apply (insert funspace_succ_replacement [of n], simp) |
13268 | 1081 |
apply (force simp add: succ_fun_eq2 univalent_def) |
1082 |
done |
|
1083 |
||
1084 |
text{*@{term M} contains all finite function spaces. Needed to prove the |
|
1085 |
absoluteness of transitive closure.*} |
|
1086 |
lemma (in M_axioms) finite_funspace_closed [intro,simp]: |
|
1087 |
"[|n\<in>nat; M(B)|] ==> M(n->B)" |
|
1088 |
apply (induct_tac n, simp) |
|
1089 |
apply (simp add: funspace_succ nat_into_M) |
|
1090 |
done |
|
1091 |
||
13223 | 1092 |
end |