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theory Wellorderings(* Title: ZF/Constructible/Wellorderings.thy ID: $Id: Wellorderings.thy,v 1.22 2005/06/17 14:15:10 haftmann Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory *) header {*Relativized Wellorderings*} theory Wellorderings imports Relative begin text{*We define functions analogous to @{term ordermap} @{term ordertype} but without using recursion. Instead, there is a direct appeal to Replacement. This will be the basis for a version relativized to some class @{text M}. The main result is Theorem I 7.6 in Kunen, page 17.*} subsection{*Wellorderings*} constdefs irreflexive :: "[i=>o,i,i]=>o" "irreflexive(M,A,r) == ∀x[M]. x∈A --> <x,x> ∉ r" transitive_rel :: "[i=>o,i,i]=>o" "transitive_rel(M,A,r) == ∀x[M]. x∈A --> (∀y[M]. y∈A --> (∀z[M]. z∈A --> <x,y>∈r --> <y,z>∈r --> <x,z>∈r))" linear_rel :: "[i=>o,i,i]=>o" "linear_rel(M,A,r) == ∀x[M]. x∈A --> (∀y[M]. y∈A --> <x,y>∈r | x=y | <y,x>∈r)" wellfounded :: "[i=>o,i]=>o" --{*EVERY non-empty set has an @{text r}-minimal element*} "wellfounded(M,r) == ∀x[M]. x≠0 --> (∃y[M]. y∈x & ~(∃z[M]. z∈x & <z,y> ∈ r))" wellfounded_on :: "[i=>o,i,i]=>o" --{*every non-empty SUBSET OF @{text A} has an @{text r}-minimal element*} "wellfounded_on(M,A,r) == ∀x[M]. x≠0 --> x⊆A --> (∃y[M]. y∈x & ~(∃z[M]. z∈x & <z,y> ∈ r))" wellordered :: "[i=>o,i,i]=>o" --{*linear and wellfounded on @{text A}*} "wellordered(M,A,r) == transitive_rel(M,A,r) & linear_rel(M,A,r) & wellfounded_on(M,A,r)" subsubsection {*Trivial absoluteness proofs*} lemma (in M_basic) irreflexive_abs [simp]: "M(A) ==> irreflexive(M,A,r) <-> irrefl(A,r)" by (simp add: irreflexive_def irrefl_def) lemma (in M_basic) transitive_rel_abs [simp]: "M(A) ==> transitive_rel(M,A,r) <-> trans[A](r)" by (simp add: transitive_rel_def trans_on_def) lemma (in M_basic) linear_rel_abs [simp]: "M(A) ==> linear_rel(M,A,r) <-> linear(A,r)" by (simp add: linear_rel_def linear_def) lemma (in M_basic) wellordered_is_trans_on: "[| wellordered(M,A,r); M(A) |] ==> trans[A](r)" by (auto simp add: wellordered_def) lemma (in M_basic) wellordered_is_linear: "[| wellordered(M,A,r); M(A) |] ==> linear(A,r)" by (auto simp add: wellordered_def) lemma (in M_basic) wellordered_is_wellfounded_on: "[| wellordered(M,A,r); M(A) |] ==> wellfounded_on(M,A,r)" by (auto simp add: wellordered_def) lemma (in M_basic) wellfounded_imp_wellfounded_on: "[| wellfounded(M,r); M(A) |] ==> wellfounded_on(M,A,r)" by (auto simp add: wellfounded_def wellfounded_on_def) lemma (in M_basic) wellfounded_on_subset_A: "[| wellfounded_on(M,A,r); B<=A |] ==> wellfounded_on(M,B,r)" by (simp add: wellfounded_on_def, blast) subsubsection {*Well-founded relations*} lemma (in M_basic) wellfounded_on_iff_wellfounded: "wellfounded_on(M,A,r) <-> wellfounded(M, r ∩ A*A)" apply (simp add: wellfounded_on_def wellfounded_def, safe) apply force apply (drule_tac x=x in rspec, assumption, blast) done lemma (in M_basic) wellfounded_on_imp_wellfounded: "[|wellfounded_on(M,A,r); r ⊆ A*A|] ==> wellfounded(M,r)" by (simp add: wellfounded_on_iff_wellfounded subset_Int_iff) lemma (in M_basic) wellfounded_on_field_imp_wellfounded: "wellfounded_on(M, field(r), r) ==> wellfounded(M,r)" by (simp add: wellfounded_def wellfounded_on_iff_wellfounded, fast) lemma (in M_basic) wellfounded_iff_wellfounded_on_field: "M(r) ==> wellfounded(M,r) <-> wellfounded_on(M, field(r), r)" by (blast intro: wellfounded_imp_wellfounded_on wellfounded_on_field_imp_wellfounded) (*Consider the least z in domain(r) such that P(z) does not hold...*) lemma (in M_basic) wellfounded_induct: "[| wellfounded(M,r); M(a); M(r); separation(M, λx. ~P(x)); ∀x. M(x) & (∀y. <y,x> ∈ r --> P(y)) --> P(x) |] ==> P(a)"; apply (simp (no_asm_use) add: wellfounded_def) apply (drule_tac x="{z ∈ domain(r). ~P(z)}" in rspec) apply (blast dest: transM)+ done lemma (in M_basic) wellfounded_on_induct: "[| a∈A; wellfounded_on(M,A,r); M(A); separation(M, λx. x∈A --> ~P(x)); ∀x∈A. M(x) & (∀y∈A. <y,x> ∈ r --> P(y)) --> P(x) |] ==> P(a)"; apply (simp (no_asm_use) add: wellfounded_on_def) apply (drule_tac x="{z∈A. z∈A --> ~P(z)}" in rspec) apply (blast intro: transM)+ done subsubsection {*Kunen's lemma IV 3.14, page 123*} lemma (in M_basic) linear_imp_relativized: "linear(A,r) ==> linear_rel(M,A,r)" by (simp add: linear_def linear_rel_def) lemma (in M_basic) trans_on_imp_relativized: "trans[A](r) ==> transitive_rel(M,A,r)" by (unfold transitive_rel_def trans_on_def, blast) lemma (in M_basic) wf_on_imp_relativized: "wf[A](r) ==> wellfounded_on(M,A,r)" apply (simp add: wellfounded_on_def wf_def wf_on_def, clarify) apply (drule_tac x=x in spec, blast) done lemma (in M_basic) wf_imp_relativized: "wf(r) ==> wellfounded(M,r)" apply (simp add: wellfounded_def wf_def, clarify) apply (drule_tac x=x in spec, blast) done lemma (in M_basic) well_ord_imp_relativized: "well_ord(A,r) ==> wellordered(M,A,r)" by (simp add: wellordered_def well_ord_def tot_ord_def part_ord_def linear_imp_relativized trans_on_imp_relativized wf_on_imp_relativized) subsection{* Relativized versions of order-isomorphisms and order types *} lemma (in M_basic) order_isomorphism_abs [simp]: "[| M(A); M(B); M(f) |] ==> order_isomorphism(M,A,r,B,s,f) <-> f ∈ ord_iso(A,r,B,s)" by (simp add: apply_closed order_isomorphism_def ord_iso_def) lemma (in M_basic) pred_set_abs [simp]: "[| M(r); M(B) |] ==> pred_set(M,A,x,r,B) <-> B = Order.pred(A,x,r)" apply (simp add: pred_set_def Order.pred_def) apply (blast dest: transM) done lemma (in M_basic) pred_closed [intro,simp]: "[| M(A); M(r); M(x) |] ==> M(Order.pred(A,x,r))" apply (simp add: Order.pred_def) apply (insert pred_separation [of r x], simp) done lemma (in M_basic) membership_abs [simp]: "[| M(r); M(A) |] ==> membership(M,A,r) <-> r = Memrel(A)" apply (simp add: membership_def Memrel_def, safe) apply (rule equalityI) apply clarify apply (frule transM, assumption) apply blast apply clarify apply (subgoal_tac "M(<xb,ya>)", blast) apply (blast dest: transM) apply auto done lemma (in M_basic) M_Memrel_iff: "M(A) ==> Memrel(A) = {z ∈ A*A. ∃x[M]. ∃y[M]. z = 〈x,y〉 & x ∈ y}" apply (simp add: Memrel_def) apply (blast dest: transM) done lemma (in M_basic) Memrel_closed [intro,simp]: "M(A) ==> M(Memrel(A))" apply (simp add: M_Memrel_iff) apply (insert Memrel_separation, simp) done subsection {* Main results of Kunen, Chapter 1 section 6 *} text{*Subset properties-- proved outside the locale*} lemma linear_rel_subset: "[| linear_rel(M,A,r); B<=A |] ==> linear_rel(M,B,r)" by (unfold linear_rel_def, blast) lemma transitive_rel_subset: "[| transitive_rel(M,A,r); B<=A |] ==> transitive_rel(M,B,r)" by (unfold transitive_rel_def, blast) lemma wellfounded_on_subset: "[| wellfounded_on(M,A,r); B<=A |] ==> wellfounded_on(M,B,r)" by (unfold wellfounded_on_def subset_def, blast) lemma wellordered_subset: "[| wellordered(M,A,r); B<=A |] ==> wellordered(M,B,r)" apply (unfold wellordered_def) apply (blast intro: linear_rel_subset transitive_rel_subset wellfounded_on_subset) done lemma (in M_basic) wellfounded_on_asym: "[| wellfounded_on(M,A,r); <a,x>∈r; a∈A; x∈A; M(A) |] ==> <x,a>∉r" apply (simp add: wellfounded_on_def) apply (drule_tac x="{x,a}" in rspec) apply (blast dest: transM)+ done lemma (in M_basic) wellordered_asym: "[| wellordered(M,A,r); <a,x>∈r; a∈A; x∈A; M(A) |] ==> <x,a>∉r" by (simp add: wellordered_def, blast dest: wellfounded_on_asym) end
lemma irreflexive_abs:
[| PROP M_basic(M); M(A) |] ==> irreflexive(M, A, r) <-> irrefl(A, r)
lemma transitive_rel_abs:
[| PROP M_basic(M); M(A) |] ==> transitive_rel(M, A, r) <-> trans[A](r)
lemma linear_rel_abs:
[| PROP M_basic(M); M(A) |] ==> linear_rel(M, A, r) <-> linear(A, r)
lemma wellordered_is_trans_on:
[| PROP M_basic(M); wellordered(M, A, r); M(A) |] ==> trans[A](r)
lemma wellordered_is_linear:
[| PROP M_basic(M); wellordered(M, A, r); M(A) |] ==> linear(A, r)
lemma wellordered_is_wellfounded_on:
[| PROP M_basic(M); wellordered(M, A, r); M(A) |] ==> wellfounded_on(M, A, r)
lemma wellfounded_imp_wellfounded_on:
[| PROP M_basic(M); wellfounded(M, r); M(A) |] ==> wellfounded_on(M, A, r)
lemma wellfounded_on_subset_A:
[| PROP M_basic(M); wellfounded_on(M, A, r); B ⊆ A |] ==> wellfounded_on(M, B, r)
lemma wellfounded_on_iff_wellfounded:
PROP M_basic(M) ==> wellfounded_on(M, A, r) <-> wellfounded(M, r ∩ A × A)
lemma wellfounded_on_imp_wellfounded:
[| PROP M_basic(M); wellfounded_on(M, A, r); r ⊆ A × A |] ==> wellfounded(M, r)
lemma wellfounded_on_field_imp_wellfounded:
[| PROP M_basic(M); wellfounded_on(M, field(r), r) |] ==> wellfounded(M, r)
lemma wellfounded_iff_wellfounded_on_field:
[| PROP M_basic(M); M(r) |] ==> wellfounded(M, r) <-> wellfounded_on(M, field(r), r)
lemma wellfounded_induct:
[| PROP M_basic(M); wellfounded(M, r); M(a); M(r); separation(M, %x. ¬ P(x)); ∀x. M(x) ∧ (∀y. 〈y, x〉 ∈ r --> P(y)) --> P(x) |] ==> P(a)
lemma wellfounded_on_induct:
[| PROP M_basic(M); a ∈ A; wellfounded_on(M, A, r); M(A); separation(M, %x. x ∈ A --> ¬ P(x)); ∀x∈A. M(x) ∧ (∀y∈A. 〈y, x〉 ∈ r --> P(y)) --> P(x) |] ==> P(a)
lemma linear_imp_relativized:
[| PROP M_basic(M); linear(A, r) |] ==> linear_rel(M, A, r)
lemma trans_on_imp_relativized:
[| PROP M_basic(M); trans[A](r) |] ==> transitive_rel(M, A, r)
lemma wf_on_imp_relativized:
[| PROP M_basic(M); wf[A](r) |] ==> wellfounded_on(M, A, r)
lemma wf_imp_relativized:
[| PROP M_basic(M); wf(r) |] ==> wellfounded(M, r)
lemma well_ord_imp_relativized:
[| PROP M_basic(M); well_ord(A, r) |] ==> wellordered(M, A, r)
lemma order_isomorphism_abs:
[| PROP M_basic(M); M(A); M(B); M(f) |] ==> order_isomorphism(M, A, r, B, s, f) <-> f ∈ ord_iso(A, r, B, s)
lemma pred_set_abs:
[| PROP M_basic(M); M(r); M(B) |] ==> pred_set(M, A, x, r, B) <-> B = Order.pred(A, x, r)
lemma pred_closed:
[| PROP M_basic(M); M(A); M(r); M(x) |] ==> M(Order.pred(A, x, r))
lemma membership_abs:
[| PROP M_basic(M); M(r); M(A) |] ==> membership(M, A, r) <-> r = Memrel(A)
lemma M_Memrel_iff:
[| PROP M_basic(M); M(A) |] ==> Memrel(A) = {z ∈ A × A . ∃x[M]. ∃y[M]. z = 〈x, y〉 ∧ x ∈ y}
lemma Memrel_closed:
[| PROP M_basic(M); M(A) |] ==> M(Memrel(A))
lemma linear_rel_subset:
[| linear_rel(M, A, r); B ⊆ A |] ==> linear_rel(M, B, r)
lemma transitive_rel_subset:
[| transitive_rel(M, A, r); B ⊆ A |] ==> transitive_rel(M, B, r)
lemma wellfounded_on_subset:
[| wellfounded_on(M, A, r); B ⊆ A |] ==> wellfounded_on(M, B, r)
lemma wellordered_subset:
[| wellordered(M, A, r); B ⊆ A |] ==> wellordered(M, B, r)
lemma wellfounded_on_asym:
[| PROP M_basic(M); wellfounded_on(M, A, r); 〈a, x〉 ∈ r; a ∈ A; x ∈ A; M(A) |] ==> 〈x, a〉 ∉ r
lemma wellordered_asym:
[| PROP M_basic(M); wellordered(M, A, r); 〈a, x〉 ∈ r; a ∈ A; x ∈ A; M(A) |] ==> 〈x, a〉 ∉ r