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(* Title: ZF/Constructible/Wellorderings.thy 
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory 

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*) 

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header {*Relativized Wellorderings*} 
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theory Wellorderings imports Relative begin 
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text{*We define functions analogous to @{term ordermap} @{term ordertype} 

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but without using recursion. Instead, there is a direct appeal 

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to Replacement. This will be the basis for a version relativized 

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to some class @{text M}. The main result is Theorem I 7.6 in Kunen, 

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page 17.*} 

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subsection{*Wellorderings*} 

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definition 
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irreflexive :: "[i=>o,i,i]=>o" where 
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"irreflexive(M,A,r) == \<forall>x[M]. x\<in>A \<longrightarrow> <x,x> \<notin> r" 
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definition 
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transitive_rel :: "[i=>o,i,i]=>o" where 
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"transitive_rel(M,A,r) == 
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\<forall>x[M]. x\<in>A \<longrightarrow> (\<forall>y[M]. y\<in>A \<longrightarrow> (\<forall>z[M]. z\<in>A \<longrightarrow> 
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<x,y>\<in>r \<longrightarrow> <y,z>\<in>r \<longrightarrow> <x,z>\<in>r))" 

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definition 
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linear_rel :: "[i=>o,i,i]=>o" where 
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"linear_rel(M,A,r) == 
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\<forall>x[M]. x\<in>A \<longrightarrow> (\<forall>y[M]. y\<in>A \<longrightarrow> <x,y>\<in>r  x=y  <y,x>\<in>r)" 
13223  32 

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definition 
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wellfounded :: "[i=>o,i]=>o" where 
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{*EVERY nonempty set has an @{text r}minimal element*} 
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"wellfounded(M,r) == 

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\<forall>x[M]. x\<noteq>0 \<longrightarrow> (\<exists>y[M]. y\<in>x & ~(\<exists>z[M]. z\<in>x & <z,y> \<in> r))" 
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definition 
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wellfounded_on :: "[i=>o,i,i]=>o" where 
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{*every nonempty SUBSET OF @{text A} has an @{text r}minimal element*} 
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"wellfounded_on(M,A,r) == 

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\<forall>x[M]. x\<noteq>0 \<longrightarrow> x\<subseteq>A \<longrightarrow> (\<exists>y[M]. y\<in>x & ~(\<exists>z[M]. z\<in>x & <z,y> \<in> r))" 
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definition 
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wellordered :: "[i=>o,i,i]=>o" where 
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{*linear and wellfounded on @{text A}*} 
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"wellordered(M,A,r) == 
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transitive_rel(M,A,r) & linear_rel(M,A,r) & wellfounded_on(M,A,r)" 
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subsubsection {*Trivial absoluteness proofs*} 

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lemma (in M_basic) irreflexive_abs [simp]: 
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"M(A) ==> irreflexive(M,A,r) \<longleftrightarrow> irrefl(A,r)" 
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by (simp add: irreflexive_def irrefl_def) 
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lemma (in M_basic) transitive_rel_abs [simp]: 
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"M(A) ==> transitive_rel(M,A,r) \<longleftrightarrow> trans[A](r)" 
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by (simp add: transitive_rel_def trans_on_def) 
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lemma (in M_basic) linear_rel_abs [simp]: 
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"M(A) ==> linear_rel(M,A,r) \<longleftrightarrow> linear(A,r)" 
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by (simp add: linear_rel_def linear_def) 
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lemma (in M_basic) wellordered_is_trans_on: 
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"[ wellordered(M,A,r); M(A) ] ==> trans[A](r)" 
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by (auto simp add: wellordered_def) 
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lemma (in M_basic) wellordered_is_linear: 
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"[ wellordered(M,A,r); M(A) ] ==> linear(A,r)" 
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by (auto simp add: wellordered_def) 
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lemma (in M_basic) wellordered_is_wellfounded_on: 
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"[ wellordered(M,A,r); M(A) ] ==> wellfounded_on(M,A,r)" 
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by (auto simp add: wellordered_def) 
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lemma (in M_basic) wellfounded_imp_wellfounded_on: 
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"[ wellfounded(M,r); M(A) ] ==> wellfounded_on(M,A,r)" 
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by (auto simp add: wellfounded_def wellfounded_on_def) 

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lemma (in M_basic) wellfounded_on_subset_A: 
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"[ wellfounded_on(M,A,r); B<=A ] ==> wellfounded_on(M,B,r)" 
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by (simp add: wellfounded_on_def, blast) 

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subsubsection {*Wellfounded relations*} 

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lemma (in M_basic) wellfounded_on_iff_wellfounded: 
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"wellfounded_on(M,A,r) \<longleftrightarrow> wellfounded(M, r \<inter> A*A)" 
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apply (simp add: wellfounded_on_def wellfounded_def, safe) 
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apply force 
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apply (drule_tac x=x in rspec, assumption, blast) 
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done 
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lemma (in M_basic) wellfounded_on_imp_wellfounded: 
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"[wellfounded_on(M,A,r); r \<subseteq> A*A] ==> wellfounded(M,r)" 
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by (simp add: wellfounded_on_iff_wellfounded subset_Int_iff) 

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lemma (in M_basic) wellfounded_on_field_imp_wellfounded: 
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"wellfounded_on(M, field(r), r) ==> wellfounded(M,r)" 
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by (simp add: wellfounded_def wellfounded_on_iff_wellfounded, fast) 

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lemma (in M_basic) wellfounded_iff_wellfounded_on_field: 
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"M(r) ==> wellfounded(M,r) \<longleftrightarrow> wellfounded_on(M, field(r), r)" 
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by (blast intro: wellfounded_imp_wellfounded_on 
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wellfounded_on_field_imp_wellfounded) 

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(*Consider the least z in domain(r) such that P(z) does not hold...*) 
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lemma (in M_basic) wellfounded_induct: 
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"[ wellfounded(M,r); M(a); M(r); separation(M, \<lambda>x. ~P(x)); 
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\<forall>x. M(x) & (\<forall>y. <y,x> \<in> r \<longrightarrow> P(y)) \<longrightarrow> P(x) ] 
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==> P(a)" 
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apply (simp (no_asm_use) add: wellfounded_def) 
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apply (drule_tac x="{z \<in> domain(r). ~P(z)}" in rspec) 
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apply (blast dest: transM)+ 

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done 
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lemma (in M_basic) wellfounded_on_induct: 
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"[ a\<in>A; wellfounded_on(M,A,r); M(A); 
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separation(M, \<lambda>x. x\<in>A \<longrightarrow> ~P(x)); 
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\<forall>x\<in>A. M(x) & (\<forall>y\<in>A. <y,x> \<in> r \<longrightarrow> P(y)) \<longrightarrow> P(x) ] 

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==> P(a)" 
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apply (simp (no_asm_use) add: wellfounded_on_def) 
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apply (drule_tac x="{z\<in>A. z\<in>A \<longrightarrow> ~P(z)}" in rspec) 
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apply (blast intro: transM)+ 
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done 
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subsubsection {*Kunen's lemma IV 3.14, page 123*} 

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lemma (in M_basic) linear_imp_relativized: 
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"linear(A,r) ==> linear_rel(M,A,r)" 
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by (simp add: linear_def linear_rel_def) 

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lemma (in M_basic) trans_on_imp_relativized: 
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"trans[A](r) ==> transitive_rel(M,A,r)" 
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by (unfold transitive_rel_def trans_on_def, blast) 

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lemma (in M_basic) wf_on_imp_relativized: 
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"wf[A](r) ==> wellfounded_on(M,A,r)" 
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apply (simp add: wellfounded_on_def wf_def wf_on_def, clarify) 

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apply (drule_tac x=x in spec, blast) 
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done 
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lemma (in M_basic) wf_imp_relativized: 
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"wf(r) ==> wellfounded(M,r)" 
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apply (simp add: wellfounded_def wf_def, clarify) 

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apply (drule_tac x=x in spec, blast) 
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done 
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lemma (in M_basic) well_ord_imp_relativized: 
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"well_ord(A,r) ==> wellordered(M,A,r)" 
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by (simp add: wellordered_def well_ord_def tot_ord_def part_ord_def 

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linear_imp_relativized trans_on_imp_relativized wf_on_imp_relativized) 

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text{*The property being well founded (and hence of being well ordered) is not absolute: 
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the set that doesn't contain a minimal element may not exist in the class M. 

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However, every set that is well founded in a transitive model M is well founded (page 124).*} 

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subsection{* Relativized versions of orderisomorphisms and order types *} 

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lemma (in M_basic) order_isomorphism_abs [simp]: 
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"[ M(A); M(B); M(f) ] 
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==> order_isomorphism(M,A,r,B,s,f) \<longleftrightarrow> f \<in> ord_iso(A,r,B,s)" 
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by (simp add: apply_closed order_isomorphism_def ord_iso_def) 
13223  166 

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lemma (in M_basic) pred_set_abs [simp]: 
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"[ M(r); M(B) ] ==> pred_set(M,A,x,r,B) \<longleftrightarrow> B = Order.pred(A,x,r)" 
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apply (simp add: pred_set_def Order.pred_def) 
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apply (blast dest: transM) 

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done 

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lemma (in M_basic) pred_closed [intro,simp]: 
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"[ M(A); M(r); M(x) ] ==> M(Order.pred(A,x,r))" 
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apply (simp add: Order.pred_def) 

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apply (insert pred_separation [of r x], simp) 
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done 
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lemma (in M_basic) membership_abs [simp]: 
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"[ M(r); M(A) ] ==> membership(M,A,r) \<longleftrightarrow> r = Memrel(A)" 
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apply (simp add: membership_def Memrel_def, safe) 
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apply (rule equalityI) 

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apply clarify 

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apply (frule transM, assumption) 

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apply blast 

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apply clarify 

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apply (subgoal_tac "M(<xb,ya>)", blast) 

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apply (blast dest: transM) 

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apply auto 

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done 

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lemma (in M_basic) M_Memrel_iff: 
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"M(A) ==> 
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Memrel(A) = {z \<in> A*A. \<exists>x[M]. \<exists>y[M]. z = \<langle>x,y\<rangle> & x \<in> y}" 
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apply (simp add: Memrel_def) 
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apply (blast dest: transM) 

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done 

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lemma (in M_basic) Memrel_closed [intro,simp]: 
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"M(A) ==> M(Memrel(A))" 
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apply (simp add: M_Memrel_iff) 

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apply (insert Memrel_separation, simp) 
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done 
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subsection {* Main results of Kunen, Chapter 1 section 6 *} 

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text{*Subset properties proved outside the locale*} 

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lemma linear_rel_subset: 

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"[ linear_rel(M,A,r); B<=A ] ==> linear_rel(M,B,r)" 

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by (unfold linear_rel_def, blast) 

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lemma transitive_rel_subset: 

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"[ transitive_rel(M,A,r); B<=A ] ==> transitive_rel(M,B,r)" 

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by (unfold transitive_rel_def, blast) 

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lemma wellfounded_on_subset: 

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"[ wellfounded_on(M,A,r); B<=A ] ==> wellfounded_on(M,B,r)" 

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by (unfold wellfounded_on_def subset_def, blast) 

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lemma wellordered_subset: 

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"[ wellordered(M,A,r); B<=A ] ==> wellordered(M,B,r)" 

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apply (unfold wellordered_def) 

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apply (blast intro: linear_rel_subset transitive_rel_subset 

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wellfounded_on_subset) 
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done 
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lemma (in M_basic) wellfounded_on_asym: 
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"[ wellfounded_on(M,A,r); <a,x>\<in>r; a\<in>A; x\<in>A; M(A) ] ==> <x,a>\<notin>r" 
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apply (simp add: wellfounded_on_def) 

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apply (drule_tac x="{x,a}" in rspec) 
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apply (blast dest: transM)+ 

13223  234 
done 
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lemma (in M_basic) wellordered_asym: 
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"[ wellordered(M,A,r); <a,x>\<in>r; a\<in>A; x\<in>A; M(A) ] ==> <x,a>\<notin>r" 
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by (simp add: wellordered_def, blast dest: wellfounded_on_asym) 

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end 