author | wenzelm |
Thu, 04 Jul 2002 15:03:03 +0200 | |
changeset 13295 | ca2e9b273472 |
parent 13293 | 09276ee04361 |
child 13298 | b4f370679c65 |
permissions | -rw-r--r-- |
13223 | 1 |
header {*Relativized Wellorderings*} |
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theory Wellorderings = Relative: |
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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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constdefs |
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irreflexive :: "[i=>o,i,i]=>o" |
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"irreflexive(M,A,r) == \<forall>x\<in>A. M(x) --> <x,x> \<notin> r" |
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transitive_rel :: "[i=>o,i,i]=>o" |
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"transitive_rel(M,A,r) == |
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\<forall>x\<in>A. M(x) --> (\<forall>y\<in>A. M(y) --> (\<forall>z\<in>A. M(z) --> |
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<x,y>\<in>r --> <y,z>\<in>r --> <x,z>\<in>r))" |
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linear_rel :: "[i=>o,i,i]=>o" |
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"linear_rel(M,A,r) == |
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\<forall>x\<in>A. M(x) --> (\<forall>y\<in>A. M(y) --> <x,y>\<in>r | x=y | <y,x>\<in>r)" |
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wellfounded :: "[i=>o,i]=>o" |
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--{*EVERY non-empty set has an @{text r}-minimal element*} |
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"wellfounded(M,r) == |
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\<forall>x. M(x) --> ~ empty(M,x) |
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--> (\<exists>y\<in>x. M(y) & ~(\<exists>z\<in>x. M(z) & <z,y> \<in> r))" |
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wellfounded_on :: "[i=>o,i,i]=>o" |
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--{*every non-empty 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) --> ~ empty(M,x) --> subset(M,x,A) |
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--> (\<exists>y\<in>x. M(y) & ~(\<exists>z\<in>x. M(z) & <z,y> \<in> r))" |
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wellordered :: "[i=>o,i,i]=>o" |
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--{*every non-empty subset of @{text A} has an @{text r}-minimal element*} |
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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_axioms) irreflexive_abs [simp]: |
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"M(A) ==> irreflexive(M,A,r) <-> irrefl(A,r)" |
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by (simp add: irreflexive_def irrefl_def) |
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lemma (in M_axioms) transitive_rel_abs [simp]: |
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"M(A) ==> transitive_rel(M,A,r) <-> trans[A](r)" |
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by (simp add: transitive_rel_def trans_on_def) |
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lemma (in M_axioms) linear_rel_abs [simp]: |
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"M(A) ==> linear_rel(M,A,r) <-> linear(A,r)" |
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by (simp add: linear_rel_def linear_def) |
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lemma (in M_axioms) 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_axioms) 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_axioms) 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_axioms) 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_axioms) 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 {*Well-founded relations*} |
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lemma (in M_axioms) wellfounded_on_iff_wellfounded: |
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"wellfounded_on(M,A,r) <-> wellfounded(M, r \<inter> A*A)" |
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apply (simp add: wellfounded_on_def wellfounded_def, safe) |
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apply blast |
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apply (drule_tac x=x in spec, blast) |
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done |
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lemma (in M_axioms) 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_axioms) 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_axioms) wellfounded_iff_wellfounded_on_field: |
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"M(r) ==> wellfounded(M,r) <-> 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_axioms) 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 --> P(y)) --> 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 spec) |
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apply (blast dest: transM) |
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done |
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lemma (in M_axioms) 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 --> ~P(x)); |
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\<forall>x\<in>A. M(x) & (\<forall>y\<in>A. <y,x> \<in> r --> P(y)) --> 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 --> ~P(z)}" in spec) |
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apply (blast intro: transM) |
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done |
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text{*The assumption @{term "r \<subseteq> A*A"} justifies strengthening the induction |
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hypothesis by removing the restriction to @{term A}.*} |
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lemma (in M_axioms) wellfounded_on_induct2: |
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"[| a\<in>A; wellfounded_on(M,A,r); M(A); r \<subseteq> A*A; |
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separation(M, \<lambda>x. x\<in>A --> ~P(x)); |
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\<forall>x\<in>A. M(x) & (\<forall>y. <y,x> \<in> r --> P(y)) --> P(x) |] |
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==> P(a)"; |
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by (rule wellfounded_on_induct, assumption+, blast) |
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subsubsection {*Kunen's lemma IV 3.14, page 123*} |
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lemma (in M_axioms) 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_axioms) 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_axioms) 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_axioms) 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_axioms) 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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subsection{* Relativized versions of order-isomorphisms and order types *} |
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lemma (in M_axioms) 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) <-> f \<in> ord_iso(A,r,B,s)" |
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by (simp add: typed_apply_abs [OF bij_is_fun] apply_closed |
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order_isomorphism_def ord_iso_def) |
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lemma (in M_axioms) pred_set_abs [simp]: |
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"[| M(r); M(B) |] ==> pred_set(M,A,x,r,B) <-> 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_axioms) 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_axioms) membership_abs [simp]: |
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"[| M(r); M(A) |] ==> membership(M,A,r) <-> 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_axioms) M_Memrel_iff: |
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"M(A) ==> |
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Memrel(A) = {z \<in> A*A. \<exists>x. M(x) \<and> (\<exists>y. M(y) \<and> z = \<langle>x,y\<rangle> \<and> 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_axioms) 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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text{*Inductive argument for Kunen's Lemma 6.1, etc. |
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Simple proof from Halmos, page 72*} |
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lemma (in M_axioms) wellordered_iso_subset_lemma: |
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"[| wellordered(M,A,r); f \<in> ord_iso(A,r, A',r); A'<= A; y \<in> A; |
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M(A); M(f); M(r) |] ==> ~ <f`y, y> \<in> r" |
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apply (unfold wellordered_def ord_iso_def) |
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apply (elim conjE CollectE) |
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apply (erule wellfounded_on_induct, assumption+) |
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apply (insert well_ord_iso_separation [of A f r]) |
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apply (simp add: typed_apply_abs [OF bij_is_fun] apply_closed, clarify) |
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apply (drule_tac a = x in bij_is_fun [THEN apply_type], assumption, blast) |
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done |
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text{*Kunen's Lemma 6.1: there's no order-isomorphism to an initial segment |
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of a well-ordering*} |
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lemma (in M_axioms) wellordered_iso_predD: |
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"[| wellordered(M,A,r); f \<in> ord_iso(A, r, Order.pred(A,x,r), r); |
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M(A); M(f); M(r) |] ==> x \<notin> A" |
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apply (rule notI) |
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apply (frule wellordered_iso_subset_lemma, assumption) |
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apply (auto elim: predE) |
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(*Now we know ~ (f`x < x) *) |
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apply (drule ord_iso_is_bij [THEN bij_is_fun, THEN apply_type], assumption) |
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(*Now we also know f`x \<in> pred(A,x,r); contradiction! *) |
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apply (simp add: Order.pred_def) |
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done |
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lemma (in M_axioms) wellordered_iso_pred_eq_lemma: |
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"[| f \<in> \<langle>Order.pred(A,y,r), r\<rangle> \<cong> \<langle>Order.pred(A,x,r), r\<rangle>; |
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wellordered(M,A,r); x\<in>A; y\<in>A; M(A); M(f); M(r) |] ==> <x,y> \<notin> r" |
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apply (frule wellordered_is_trans_on, assumption) |
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apply (rule notI) |
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apply (drule_tac x2=y and x=x and r2=r in |
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wellordered_subset [OF _ pred_subset, THEN wellordered_iso_predD]) |
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apply (simp add: trans_pred_pred_eq) |
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apply (blast intro: predI dest: transM)+ |
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done |
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text{*Simple consequence of Lemma 6.1*} |
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lemma (in M_axioms) wellordered_iso_pred_eq: |
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"[| wellordered(M,A,r); |
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f \<in> ord_iso(Order.pred(A,a,r), r, Order.pred(A,c,r), r); |
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M(A); M(f); M(r); a\<in>A; c\<in>A |] ==> a=c" |
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apply (frule wellordered_is_trans_on, assumption) |
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apply (frule wellordered_is_linear, assumption) |
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apply (erule_tac x=a and y=c in linearE, auto) |
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apply (drule ord_iso_sym) |
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(*two symmetric cases*) |
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apply (blast dest: wellordered_iso_pred_eq_lemma)+ |
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done |
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lemma (in M_axioms) 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 spec) |
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apply (simp add: cons_closed) |
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apply (blast dest: transM) |
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done |
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lemma (in M_axioms) 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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text{*Surely a shorter proof using lemmas in @{text Order}? |
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Like @{text well_ord_iso_preserving}?*} |
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lemma (in M_axioms) ord_iso_pred_imp_lt: |
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"[| f \<in> ord_iso(Order.pred(A,x,r), r, i, Memrel(i)); |
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g \<in> ord_iso(Order.pred(A,y,r), r, j, Memrel(j)); |
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wellordered(M,A,r); x \<in> A; y \<in> A; M(A); M(r); M(f); M(g); M(j); |
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Ord(i); Ord(j); \<langle>x,y\<rangle> \<in> r |] |
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==> i < j" |
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apply (frule wellordered_is_trans_on, assumption) |
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apply (frule_tac y=y in transM, assumption) |
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apply (rule_tac i=i and j=j in Ord_linear_lt, auto) |
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txt{*case @{term "i=j"} yields a contradiction*} |
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apply (rule_tac x1=x and A1="Order.pred(A,y,r)" in |
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wellordered_iso_predD [THEN notE]) |
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apply (blast intro: wellordered_subset [OF _ pred_subset]) |
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apply (simp add: trans_pred_pred_eq) |
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apply (blast intro: Ord_iso_implies_eq ord_iso_sym ord_iso_trans) |
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apply (simp_all add: pred_iff pred_closed converse_closed comp_closed) |
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txt{*case @{term "j<i"} also yields a contradiction*} |
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apply (frule restrict_ord_iso2, assumption+) |
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apply (frule ord_iso_sym [THEN ord_iso_is_bij, THEN bij_is_fun]) |
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apply (frule apply_type, blast intro: ltD) |
|
319 |
--{*thus @{term "converse(f)`j \<in> Order.pred(A,x,r)"}*} |
|
320 |
apply (simp add: pred_iff) |
|
321 |
apply (subgoal_tac |
|
322 |
"\<exists>h. M(h) & h \<in> ord_iso(Order.pred(A,y,r), r, |
|
323 |
Order.pred(A, converse(f)`j, r), r)") |
|
324 |
apply (clarify, frule wellordered_iso_pred_eq, assumption+) |
|
325 |
apply (blast dest: wellordered_asym) |
|
326 |
apply (intro exI conjI) |
|
327 |
prefer 2 apply (blast intro: Ord_iso_implies_eq ord_iso_sym ord_iso_trans)+ |
|
328 |
done |
|
329 |
||
330 |
||
331 |
lemma ord_iso_converse1: |
|
332 |
"[| f: ord_iso(A,r,B,s); <b, f`a>: s; a:A; b:B |] |
|
333 |
==> <converse(f) ` b, a> : r" |
|
334 |
apply (frule ord_iso_converse, assumption+) |
|
335 |
apply (blast intro: ord_iso_is_bij [THEN bij_is_fun, THEN apply_funtype]) |
|
336 |
apply (simp add: left_inverse_bij [OF ord_iso_is_bij]) |
|
337 |
done |
|
338 |
||
339 |
||
340 |
subsection {* Order Types: A Direct Construction by Replacement*} |
|
341 |
||
342 |
text{*This follows Kunen's Theorem I 7.6, page 17.*} |
|
343 |
||
344 |
constdefs |
|
345 |
||
346 |
obase :: "[i=>o,i,i,i] => o" |
|
347 |
--{*the domain of @{text om}, eventually shown to equal @{text A}*} |
|
348 |
"obase(M,A,r,z) == |
|
13293 | 349 |
\<forall>a[M]. |
350 |
a \<in> z <-> |
|
13223 | 351 |
(a\<in>A & (\<exists>x g mx par. M(x) & M(g) & M(mx) & M(par) & ordinal(M,x) & |
352 |
membership(M,x,mx) & pred_set(M,A,a,r,par) & |
|
13293 | 353 |
order_isomorphism(M,par,r,x,mx,g)))" |
13223 | 354 |
|
355 |
||
356 |
omap :: "[i=>o,i,i,i] => o" |
|
357 |
--{*the function that maps wosets to order types*} |
|
358 |
"omap(M,A,r,f) == |
|
13293 | 359 |
\<forall>z[M]. |
360 |
z \<in> f <-> |
|
13223 | 361 |
(\<exists>a\<in>A. M(a) & |
362 |
(\<exists>x g mx par. M(x) & M(g) & M(mx) & M(par) & ordinal(M,x) & |
|
363 |
pair(M,a,x,z) & membership(M,x,mx) & |
|
364 |
pred_set(M,A,a,r,par) & |
|
13293 | 365 |
order_isomorphism(M,par,r,x,mx,g)))" |
13223 | 366 |
|
367 |
||
368 |
otype :: "[i=>o,i,i,i] => o" --{*the order types themselves*} |
|
369 |
"otype(M,A,r,i) == \<exists>f. M(f) & omap(M,A,r,f) & is_range(M,f,i)" |
|
370 |
||
371 |
||
372 |
||
373 |
lemma (in M_axioms) obase_iff: |
|
374 |
"[| M(A); M(r); M(z) |] |
|
375 |
==> obase(M,A,r,z) <-> |
|
376 |
z = {a\<in>A. \<exists>x g. M(x) & M(g) & Ord(x) & |
|
377 |
g \<in> ord_iso(Order.pred(A,a,r),r,x,Memrel(x))}" |
|
378 |
apply (simp add: obase_def Memrel_closed pred_closed) |
|
379 |
apply (rule iffI) |
|
380 |
prefer 2 apply blast |
|
381 |
apply (rule equalityI) |
|
382 |
apply (clarify, frule transM, assumption, rotate_tac -1, simp) |
|
383 |
apply (clarify, frule transM, assumption, force) |
|
384 |
done |
|
385 |
||
386 |
text{*Can also be proved with the premise @{term "M(z)"} instead of |
|
387 |
@{term "M(f)"}, but that version is less useful.*} |
|
388 |
lemma (in M_axioms) omap_iff: |
|
389 |
"[| omap(M,A,r,f); M(A); M(r); M(f) |] |
|
390 |
==> z \<in> f <-> |
|
391 |
(\<exists>a\<in>A. \<exists>x g. M(x) & M(g) & z = <a,x> & Ord(x) & |
|
392 |
g \<in> ord_iso(Order.pred(A,a,r),r,x,Memrel(x)))" |
|
393 |
apply (rotate_tac 1) |
|
394 |
apply (simp add: omap_def Memrel_closed pred_closed) |
|
13293 | 395 |
apply (rule iffI) |
396 |
apply (drule_tac [2] x=z in rspec) |
|
397 |
apply (drule_tac x=z in rspec) |
|
398 |
apply (blast dest: transM)+ |
|
13223 | 399 |
done |
400 |
||
401 |
lemma (in M_axioms) omap_unique: |
|
402 |
"[| omap(M,A,r,f); omap(M,A,r,f'); M(A); M(r); M(f); M(f') |] ==> f' = f" |
|
403 |
apply (rule equality_iffI) |
|
404 |
apply (simp add: omap_iff) |
|
405 |
done |
|
406 |
||
407 |
lemma (in M_axioms) omap_yields_Ord: |
|
408 |
"[| omap(M,A,r,f); \<langle>a,x\<rangle> \<in> f; M(a); M(x) |] ==> Ord(x)" |
|
409 |
apply (simp add: omap_def, blast) |
|
410 |
done |
|
411 |
||
412 |
lemma (in M_axioms) otype_iff: |
|
413 |
"[| otype(M,A,r,i); M(A); M(r); M(i) |] |
|
414 |
==> x \<in> i <-> |
|
415 |
(\<exists>a\<in>A. \<exists>g. M(x) & M(g) & Ord(x) & |
|
416 |
g \<in> ord_iso(Order.pred(A,a,r),r,x,Memrel(x)))" |
|
417 |
apply (simp add: otype_def, auto) |
|
418 |
apply (blast dest: transM) |
|
419 |
apply (blast dest!: omap_iff intro: transM) |
|
420 |
apply (rename_tac a g) |
|
421 |
apply (rule_tac a=a in rangeI) |
|
422 |
apply (frule transM, assumption) |
|
423 |
apply (simp add: omap_iff, blast) |
|
424 |
done |
|
425 |
||
426 |
lemma (in M_axioms) otype_eq_range: |
|
427 |
"[| omap(M,A,r,f); otype(M,A,r,i); M(A); M(r); M(f); M(i) |] ==> i = range(f)" |
|
428 |
apply (auto simp add: otype_def omap_iff) |
|
429 |
apply (blast dest: omap_unique) |
|
430 |
done |
|
431 |
||
432 |
||
433 |
lemma (in M_axioms) Ord_otype: |
|
434 |
"[| otype(M,A,r,i); trans[A](r); M(A); M(r); M(i) |] ==> Ord(i)" |
|
435 |
apply (rotate_tac 1) |
|
436 |
apply (rule OrdI) |
|
437 |
prefer 2 |
|
438 |
apply (simp add: Ord_def otype_def omap_def) |
|
439 |
apply clarify |
|
440 |
apply (frule pair_components_in_M, assumption) |
|
441 |
apply blast |
|
442 |
apply (auto simp add: Transset_def otype_iff) |
|
443 |
apply (blast intro: transM) |
|
444 |
apply (rename_tac y a g) |
|
445 |
apply (frule ord_iso_sym [THEN ord_iso_is_bij, THEN bij_is_fun, |
|
446 |
THEN apply_funtype], assumption) |
|
447 |
apply (rule_tac x="converse(g)`y" in bexI) |
|
448 |
apply (frule_tac a="converse(g) ` y" in ord_iso_restrict_pred, assumption) |
|
449 |
apply (safe elim!: predE) |
|
450 |
apply (intro conjI exI) |
|
451 |
prefer 3 |
|
452 |
apply (blast intro: restrict_ord_iso ord_iso_sym ltI) |
|
453 |
apply (blast intro: transM) |
|
454 |
apply (blast intro: Ord_in_Ord) |
|
455 |
done |
|
456 |
||
457 |
lemma (in M_axioms) domain_omap: |
|
458 |
"[| omap(M,A,r,f); obase(M,A,r,B); M(A); M(r); M(B); M(f) |] |
|
459 |
==> domain(f) = B" |
|
460 |
apply (rotate_tac 2) |
|
461 |
apply (simp add: domain_closed obase_iff) |
|
462 |
apply (rule equality_iffI) |
|
463 |
apply (simp add: domain_iff omap_iff, blast) |
|
464 |
done |
|
465 |
||
466 |
lemma (in M_axioms) omap_subset: |
|
467 |
"[| omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i); |
|
468 |
M(A); M(r); M(f); M(B); M(i) |] ==> f \<subseteq> B * i" |
|
469 |
apply (rotate_tac 3, clarify) |
|
470 |
apply (simp add: omap_iff obase_iff) |
|
471 |
apply (force simp add: otype_iff) |
|
472 |
done |
|
473 |
||
474 |
lemma (in M_axioms) omap_funtype: |
|
475 |
"[| omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i); |
|
476 |
M(A); M(r); M(f); M(B); M(i) |] ==> f \<in> B -> i" |
|
477 |
apply (rotate_tac 3) |
|
478 |
apply (simp add: domain_omap omap_subset Pi_iff function_def omap_iff) |
|
479 |
apply (blast intro: Ord_iso_implies_eq ord_iso_sym ord_iso_trans) |
|
480 |
done |
|
481 |
||
482 |
||
483 |
lemma (in M_axioms) wellordered_omap_bij: |
|
484 |
"[| wellordered(M,A,r); omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i); |
|
485 |
M(A); M(r); M(f); M(B); M(i) |] ==> f \<in> bij(B,i)" |
|
486 |
apply (insert omap_funtype [of A r f B i]) |
|
487 |
apply (auto simp add: bij_def inj_def) |
|
488 |
prefer 2 apply (blast intro: fun_is_surj dest: otype_eq_range) |
|
489 |
apply (frule_tac a="w" in apply_Pair, assumption) |
|
490 |
apply (frule_tac a="x" in apply_Pair, assumption) |
|
491 |
apply (simp add: omap_iff) |
|
492 |
apply (blast intro: wellordered_iso_pred_eq ord_iso_sym ord_iso_trans) |
|
493 |
done |
|
494 |
||
495 |
||
496 |
text{*This is not the final result: we must show @{term "oB(A,r) = A"}*} |
|
497 |
lemma (in M_axioms) omap_ord_iso: |
|
498 |
"[| wellordered(M,A,r); omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i); |
|
499 |
M(A); M(r); M(f); M(B); M(i) |] ==> f \<in> ord_iso(B,r,i,Memrel(i))" |
|
500 |
apply (rule ord_isoI) |
|
501 |
apply (erule wellordered_omap_bij, assumption+) |
|
502 |
apply (insert omap_funtype [of A r f B i], simp) |
|
503 |
apply (frule_tac a="x" in apply_Pair, assumption) |
|
504 |
apply (frule_tac a="y" in apply_Pair, assumption) |
|
505 |
apply (auto simp add: omap_iff) |
|
506 |
txt{*direction 1: assuming @{term "\<langle>x,y\<rangle> \<in> r"}*} |
|
507 |
apply (blast intro: ltD ord_iso_pred_imp_lt) |
|
508 |
txt{*direction 2: proving @{term "\<langle>x,y\<rangle> \<in> r"} using linearity of @{term r}*} |
|
509 |
apply (rename_tac x y g ga) |
|
510 |
apply (frule wellordered_is_linear, assumption, |
|
511 |
erule_tac x=x and y=y in linearE, assumption+) |
|
512 |
txt{*the case @{term "x=y"} leads to immediate contradiction*} |
|
513 |
apply (blast elim: mem_irrefl) |
|
514 |
txt{*the case @{term "\<langle>y,x\<rangle> \<in> r"}: handle like the opposite direction*} |
|
515 |
apply (blast dest: ord_iso_pred_imp_lt ltD elim: mem_asym) |
|
516 |
done |
|
517 |
||
518 |
lemma (in M_axioms) Ord_omap_image_pred: |
|
519 |
"[| wellordered(M,A,r); omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i); |
|
520 |
M(A); M(r); M(f); M(B); M(i); b \<in> A |] ==> Ord(f `` Order.pred(A,b,r))" |
|
521 |
apply (frule wellordered_is_trans_on, assumption) |
|
522 |
apply (rule OrdI) |
|
523 |
prefer 2 apply (simp add: image_iff omap_iff Ord_def, blast) |
|
524 |
txt{*Hard part is to show that the image is a transitive set.*} |
|
525 |
apply (rotate_tac 3) |
|
526 |
apply (simp add: Transset_def, clarify) |
|
527 |
apply (simp add: image_iff pred_iff apply_iff [OF omap_funtype [of A r f B i]]) |
|
528 |
apply (rename_tac c j, clarify) |
|
529 |
apply (frule omap_funtype [of A r f B, THEN apply_funtype], assumption+) |
|
530 |
apply (subgoal_tac "j : i") |
|
531 |
prefer 2 apply (blast intro: Ord_trans Ord_otype) |
|
532 |
apply (subgoal_tac "converse(f) ` j : B") |
|
533 |
prefer 2 |
|
534 |
apply (blast dest: wellordered_omap_bij [THEN bij_converse_bij, |
|
535 |
THEN bij_is_fun, THEN apply_funtype]) |
|
536 |
apply (rule_tac x="converse(f) ` j" in bexI) |
|
537 |
apply (simp add: right_inverse_bij [OF wellordered_omap_bij]) |
|
538 |
apply (intro predI conjI) |
|
539 |
apply (erule_tac b=c in trans_onD) |
|
540 |
apply (rule ord_iso_converse1 [OF omap_ord_iso [of A r f B i]]) |
|
541 |
apply (auto simp add: obase_iff) |
|
542 |
done |
|
543 |
||
544 |
lemma (in M_axioms) restrict_omap_ord_iso: |
|
545 |
"[| wellordered(M,A,r); omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i); |
|
546 |
D \<subseteq> B; M(A); M(r); M(f); M(B); M(i) |] |
|
547 |
==> restrict(f,D) \<in> (\<langle>D,r\<rangle> \<cong> \<langle>f``D, Memrel(f``D)\<rangle>)" |
|
548 |
apply (frule ord_iso_restrict_image [OF omap_ord_iso [of A r f B i]], |
|
549 |
assumption+) |
|
550 |
apply (drule ord_iso_sym [THEN subset_ord_iso_Memrel]) |
|
551 |
apply (blast dest: subsetD [OF omap_subset]) |
|
552 |
apply (drule ord_iso_sym, simp) |
|
553 |
done |
|
554 |
||
555 |
lemma (in M_axioms) obase_equals: |
|
556 |
"[| wellordered(M,A,r); omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i); |
|
557 |
M(A); M(r); M(f); M(B); M(i) |] ==> B = A" |
|
558 |
apply (rotate_tac 4) |
|
559 |
apply (rule equalityI, force simp add: obase_iff, clarify) |
|
560 |
apply (subst obase_iff [of A r B, THEN iffD1], assumption+, simp) |
|
561 |
apply (frule wellordered_is_wellfounded_on, assumption) |
|
562 |
apply (erule wellfounded_on_induct, assumption+) |
|
563 |
apply (insert obase_equals_separation, simp add: Memrel_closed pred_closed, clarify) |
|
564 |
apply (rename_tac b) |
|
565 |
apply (subgoal_tac "Order.pred(A,b,r) <= B") |
|
566 |
prefer 2 apply (force simp add: pred_iff obase_iff) |
|
567 |
apply (intro conjI exI) |
|
568 |
prefer 4 apply (blast intro: restrict_omap_ord_iso) |
|
569 |
apply (blast intro: Ord_omap_image_pred)+ |
|
570 |
done |
|
571 |
||
572 |
||
573 |
||
574 |
text{*Main result: @{term om} gives the order-isomorphism |
|
575 |
@{term "\<langle>A,r\<rangle> \<cong> \<langle>i, Memrel(i)\<rangle>"} *} |
|
576 |
theorem (in M_axioms) omap_ord_iso_otype: |
|
577 |
"[| wellordered(M,A,r); omap(M,A,r,f); obase(M,A,r,B); otype(M,A,r,i); |
|
578 |
M(A); M(r); M(f); M(B); M(i) |] ==> f \<in> ord_iso(A, r, i, Memrel(i))" |
|
579 |
apply (frule omap_ord_iso, assumption+) |
|
580 |
apply (frule obase_equals, assumption+, blast) |
|
13293 | 581 |
done |
13223 | 582 |
|
583 |
lemma (in M_axioms) obase_exists: |
|
13293 | 584 |
"[| M(A); M(r) |] ==> \<exists>z[M]. obase(M,A,r,z)" |
13223 | 585 |
apply (simp add: obase_def) |
586 |
apply (insert obase_separation [of A r]) |
|
587 |
apply (simp add: separation_def) |
|
588 |
done |
|
589 |
||
590 |
lemma (in M_axioms) omap_exists: |
|
13293 | 591 |
"[| M(A); M(r) |] ==> \<exists>z[M]. omap(M,A,r,z)" |
13223 | 592 |
apply (insert obase_exists [of A r]) |
593 |
apply (simp add: omap_def) |
|
594 |
apply (insert omap_replacement [of A r]) |
|
595 |
apply (simp add: strong_replacement_def, clarify) |
|
13293 | 596 |
apply (drule_tac x=x in spec, clarify) |
13223 | 597 |
apply (simp add: Memrel_closed pred_closed obase_iff) |
598 |
apply (erule impE) |
|
599 |
apply (clarsimp simp add: univalent_def) |
|
600 |
apply (blast intro: Ord_iso_implies_eq ord_iso_sym ord_iso_trans, clarify) |
|
13293 | 601 |
apply (rule_tac x=Y in rexI) |
602 |
apply (simp add: Memrel_closed pred_closed obase_iff, blast, assumption) |
|
13223 | 603 |
done |
604 |
||
13293 | 605 |
declare rall_simps [simp] rex_simps [simp] |
606 |
||
13223 | 607 |
lemma (in M_axioms) otype_exists: |
608 |
"[| wellordered(M,A,r); M(A); M(r) |] ==> \<exists>i. M(i) & otype(M,A,r,i)" |
|
13293 | 609 |
apply (insert omap_exists [of A r]) |
610 |
apply (simp add: otype_def, safe) |
|
611 |
apply (rule_tac x="range(x)" in exI) |
|
13223 | 612 |
apply blast |
613 |
done |
|
614 |
||
615 |
theorem (in M_axioms) omap_ord_iso_otype: |
|
616 |
"[| wellordered(M,A,r); M(A); M(r) |] |
|
617 |
==> \<exists>f. M(f) & (\<exists>i. M(i) & Ord(i) & f \<in> ord_iso(A, r, i, Memrel(i)))" |
|
618 |
apply (insert obase_exists [of A r] omap_exists [of A r] otype_exists [of A r], simp, clarify) |
|
619 |
apply (subgoal_tac "Ord(i)", blast intro: omap_ord_iso_otype) |
|
620 |
apply (rule Ord_otype) |
|
621 |
apply (force simp add: otype_def range_closed) |
|
622 |
apply (simp_all add: wellordered_is_trans_on) |
|
623 |
done |
|
624 |
||
625 |
lemma (in M_axioms) ordertype_exists: |
|
626 |
"[| wellordered(M,A,r); M(A); M(r) |] |
|
627 |
==> \<exists>f. M(f) & (\<exists>i. M(i) & Ord(i) & f \<in> ord_iso(A, r, i, Memrel(i)))" |
|
628 |
apply (insert obase_exists [of A r] omap_exists [of A r] otype_exists [of A r], simp, clarify) |
|
629 |
apply (subgoal_tac "Ord(i)", blast intro: omap_ord_iso_otype) |
|
630 |
apply (rule Ord_otype) |
|
631 |
apply (force simp add: otype_def range_closed) |
|
632 |
apply (simp_all add: wellordered_is_trans_on) |
|
633 |
done |
|
634 |
||
635 |
||
636 |
lemma (in M_axioms) relativized_imp_well_ord: |
|
637 |
"[| wellordered(M,A,r); M(A); M(r) |] ==> well_ord(A,r)" |
|
638 |
apply (insert ordertype_exists [of A r], simp) |
|
639 |
apply (blast intro: well_ord_ord_iso well_ord_Memrel ) |
|
640 |
done |
|
641 |
||
642 |
subsection {*Kunen's theorem 5.4, poage 127*} |
|
643 |
||
644 |
text{*(a) The notion of Wellordering is absolute*} |
|
645 |
theorem (in M_axioms) well_ord_abs [simp]: |
|
646 |
"[| M(A); M(r) |] ==> wellordered(M,A,r) <-> well_ord(A,r)" |
|
647 |
by (blast intro: well_ord_imp_relativized relativized_imp_well_ord) |
|
648 |
||
649 |
||
650 |
text{*(b) Order types are absolute*} |
|
651 |
lemma (in M_axioms) |
|
652 |
"[| wellordered(M,A,r); f \<in> ord_iso(A, r, i, Memrel(i)); |
|
653 |
M(A); M(r); M(f); M(i); Ord(i) |] ==> i = ordertype(A,r)" |
|
654 |
by (blast intro: Ord_ordertype relativized_imp_well_ord ordertype_ord_iso |
|
655 |
Ord_iso_implies_eq ord_iso_sym ord_iso_trans) |
|
656 |
||
657 |
end |