author | haftmann |
Wed, 23 Jul 2025 13:22:58 +0200 | |
changeset 82900 | bd3685e5f883 |
parent 76219 | cf7db6353322 |
permissions | -rw-r--r-- |
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(* Title: ZF/AC/AC17_AC1.thy |
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Author: Krzysztof Grabczewski |
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The equivalence of AC0, AC1 and AC17 |
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Also, the proofs needed to show that each of AC2, AC3, ..., AC6 is equivalent |
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to AC0 and AC1. |
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*) |
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theory AC17_AC1 |
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imports HH |
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begin |
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(** AC0 is equivalent to AC1. |
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AC0 comes from Suppes, AC1 from Rubin & Rubin **) |
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lemma AC0_AC1_lemma: "\<lbrakk>f:(\<Prod>X \<in> A. X); D \<subseteq> A\<rbrakk> \<Longrightarrow> \<exists>g. g:(\<Prod>X \<in> D. X)" |
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by (fast intro!: lam_type apply_type) |
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lemma AC0_AC1: "AC0 \<Longrightarrow> AC1" |
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unfolding AC0_def AC1_def |
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apply (blast intro: AC0_AC1_lemma) |
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done |
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lemma AC1_AC0: "AC1 \<Longrightarrow> AC0" |
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by (unfold AC0_def AC1_def, blast) |
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(**** The proof of AC1 \<Longrightarrow> AC17 ****) |
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lemma AC1_AC17_lemma: "f \<in> (\<Prod>X \<in> Pow(A) - {0}. X) \<Longrightarrow> f \<in> (Pow(A) - {0} -> A)" |
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apply (rule Pi_type, assumption) |
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apply (drule apply_type, assumption, fast) |
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done |
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lemma AC1_AC17: "AC1 \<Longrightarrow> AC17" |
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unfolding AC1_def AC17_def |
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apply (rule allI) |
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apply (rule ballI) |
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apply (erule_tac x = "Pow (A) -{0}" in allE) |
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apply (erule impE, fast) |
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apply (erule exE) |
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apply (rule bexI) |
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apply (erule_tac [2] AC1_AC17_lemma) |
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apply (rule apply_type, assumption) |
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apply (fast dest!: AC1_AC17_lemma elim!: apply_type) |
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done |
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(**** The proof of AC17 \<Longrightarrow> AC1 ****) |
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(* *********************************************************************** *) |
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(* more properties of HH *) |
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(* *********************************************************************** *) |
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lemma UN_eq_imp_well_ord: |
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"\<lbrakk>x - (\<Union>j \<in> \<mu> i. HH(\<lambda>X \<in> Pow(x)-{0}. {f`X}, x, i) = {x}. |
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HH(\<lambda>X \<in> Pow(x)-{0}. {f`X}, x, j)) = 0; |
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f \<in> Pow(x)-{0} -> x\<rbrakk> |
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\<Longrightarrow> \<exists>r. well_ord(x,r)" |
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apply (rule exI) |
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apply (erule well_ord_rvimage |
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[OF bij_Least_HH_x [THEN bij_converse_bij, THEN bij_is_inj] |
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Ord_Least [THEN well_ord_Memrel]], assumption) |
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done |
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(* *********************************************************************** *) |
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(* theorems closer to the proof *) |
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(* *********************************************************************** *) |
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lemma not_AC1_imp_ex: |
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"\<not>AC1 \<Longrightarrow> \<exists>A. \<forall>f \<in> Pow(A)-{0} -> A. \<exists>u \<in> Pow(A)-{0}. f`u \<notin> u" |
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unfolding AC1_def |
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apply (erule swap) |
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apply (rule allI) |
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apply (erule swap) |
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apply (rule_tac x = "\<Union>(A)" in exI) |
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apply (blast intro: lam_type) |
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done |
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lemma AC17_AC1_aux1: |
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"\<lbrakk>\<forall>f \<in> Pow(x) - {0} -> x. \<exists>u \<in> Pow(x) - {0}. f`u\<notin>u; |
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\<exists>f \<in> Pow(x)-{0}->x. |
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x - (\<Union>a \<in> (\<mu> i. HH(\<lambda>X \<in> Pow(x)-{0}. {f`X},x,i)={x}). |
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HH(\<lambda>X \<in> Pow(x)-{0}. {f`X},x,a)) = 0\<rbrakk> |
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\<Longrightarrow> P" |
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apply (erule bexE) |
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apply (erule UN_eq_imp_well_ord [THEN exE], assumption) |
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apply (erule ex_choice_fun_Pow [THEN exE]) |
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apply (erule ballE) |
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apply (fast intro: apply_type del: DiffE) |
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apply (erule notE) |
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apply (rule Pi_type, assumption) |
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apply (blast dest: apply_type) |
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done |
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lemma AC17_AC1_aux2: |
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"\<not> (\<exists>f \<in> Pow(x)-{0}->x. x - F(f) = 0) |
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\<Longrightarrow> (\<lambda>f \<in> Pow(x)-{0}->x . x - F(f)) |
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\<in> (Pow(x) -{0} -> x) -> Pow(x) - {0}" |
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by (fast intro!: lam_type dest!: Diff_eq_0_iff [THEN iffD1]) |
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lemma AC17_AC1_aux3: |
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"\<lbrakk>f`Z \<in> Z; Z \<in> Pow(x)-{0}\<rbrakk> |
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\<Longrightarrow> (\<lambda>X \<in> Pow(x)-{0}. {f`X})`Z \<in> Pow(Z)-{0}" |
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by auto |
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lemma AC17_AC1_aux4: |
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"\<exists>f \<in> F. f`((\<lambda>f \<in> F. Q(f))`f) \<in> (\<lambda>f \<in> F. Q(f))`f |
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\<Longrightarrow> \<exists>f \<in> F. f`Q(f) \<in> Q(f)" |
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by simp |
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lemma AC17_AC1: "AC17 \<Longrightarrow> AC1" |
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unfolding AC17_def |
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apply (rule classical) |
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apply (erule not_AC1_imp_ex [THEN exE]) |
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apply (case_tac |
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"\<exists>f \<in> Pow(x)-{0} -> x. |
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x - (\<Union>a \<in> (\<mu> i. HH (\<lambda>X \<in> Pow (x) -{0}. {f`X},x,i) ={x}) . HH (\<lambda>X \<in> Pow (x) -{0}. {f`X},x,a)) = 0") |
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apply (erule AC17_AC1_aux1, assumption) |
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apply (drule AC17_AC1_aux2) |
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apply (erule allE) |
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apply (drule bspec, assumption) |
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apply (drule AC17_AC1_aux4) |
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apply (erule bexE) |
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apply (drule apply_type, assumption) |
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apply (simp add: HH_Least_eq_x del: Diff_iff ) |
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apply (drule AC17_AC1_aux3, assumption) |
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apply (fast dest!: subst_elem [OF _ HH_Least_eq_x [symmetric]] |
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f_subset_imp_HH_subset elim!: mem_irrefl) |
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done |
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(* ********************************************************************** |
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AC1 \<Longrightarrow> AC2 \<Longrightarrow> AC1 |
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AC1 \<Longrightarrow> AC4 \<Longrightarrow> AC3 \<Longrightarrow> AC1 |
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AC4 \<Longrightarrow> AC5 \<Longrightarrow> AC4 |
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AC1 \<longleftrightarrow> AC6 |
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************************************************************************* *) |
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(* ********************************************************************** *) |
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(* AC1 \<Longrightarrow> AC2 *) |
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(* ********************************************************************** *) |
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lemma AC1_AC2_aux1: |
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"\<lbrakk>f:(\<Prod>X \<in> A. X); B \<in> A; 0\<notin>A\<rbrakk> \<Longrightarrow> {f`B} \<subseteq> B \<inter> {f`C. C \<in> A}" |
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by (fast elim!: apply_type) |
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lemma AC1_AC2_aux2: |
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"\<lbrakk>pairwise_disjoint(A); B \<in> A; C \<in> A; D \<in> B; D \<in> C\<rbrakk> \<Longrightarrow> f`B = f`C" |
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by (unfold pairwise_disjoint_def, fast) |
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lemma AC1_AC2: "AC1 \<Longrightarrow> AC2" |
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unfolding AC1_def AC2_def |
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apply (rule allI) |
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apply (rule impI) |
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apply (elim asm_rl conjE allE exE impE, assumption) |
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apply (intro exI ballI equalityI) |
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prefer 2 apply (rule AC1_AC2_aux1, assumption+) |
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apply (fast elim!: AC1_AC2_aux2 elim: apply_type) |
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done |
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(* ********************************************************************** *) |
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(* AC2 \<Longrightarrow> AC1 *) |
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(* ********************************************************************** *) |
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lemma AC2_AC1_aux1: "0\<notin>A \<Longrightarrow> 0 \<notin> {B*{B}. B \<in> A}" |
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by (fast dest!: sym [THEN Sigma_empty_iff [THEN iffD1]]) |
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lemma AC2_AC1_aux2: "\<lbrakk>X*{X} \<inter> C = {y}; X \<in> A\<rbrakk> |
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\<Longrightarrow> (THE y. X*{X} \<inter> C = {y}): X*A" |
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apply (rule subst_elem [of y]) |
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apply (blast elim!: equalityE) |
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apply (auto simp add: singleton_eq_iff) |
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done |
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lemma AC2_AC1_aux3: |
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"\<forall>D \<in> {E*{E}. E \<in> A}. \<exists>y. D \<inter> C = {y} |
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\<Longrightarrow> (\<lambda>x \<in> A. fst(THE z. (x*{x} \<inter> C = {z}))) \<in> (\<Prod>X \<in> A. X)" |
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apply (rule lam_type) |
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apply (drule bspec, blast) |
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apply (blast intro: AC2_AC1_aux2 fst_type) |
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done |
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lemma AC2_AC1: "AC2 \<Longrightarrow> AC1" |
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unfolding AC1_def AC2_def pairwise_disjoint_def |
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apply (intro allI impI) |
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apply (elim allE impE) |
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prefer 2 apply (fast elim!: AC2_AC1_aux3) |
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apply (blast intro!: AC2_AC1_aux1) |
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done |
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(* ********************************************************************** *) |
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(* AC1 \<Longrightarrow> AC4 *) |
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(* ********************************************************************** *) |
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lemma empty_notin_images: "0 \<notin> {R``{x}. x \<in> domain(R)}" |
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by blast |
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lemma AC1_AC4: "AC1 \<Longrightarrow> AC4" |
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unfolding AC1_def AC4_def |
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apply (intro allI impI) |
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apply (drule spec, drule mp [OF _ empty_notin_images]) |
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apply (best intro!: lam_type elim!: apply_type) |
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done |
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(* ********************************************************************** *) |
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(* AC4 \<Longrightarrow> AC3 *) |
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(* ********************************************************************** *) |
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lemma AC4_AC3_aux1: "f \<in> A->B \<Longrightarrow> (\<Union>z \<in> A. {z}*f`z) \<subseteq> A*\<Union>(B)" |
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by (fast dest!: apply_type) |
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lemma AC4_AC3_aux2: "domain(\<Union>z \<in> A. {z}*f(z)) = {a \<in> A. f(a)\<noteq>0}" |
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by blast |
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lemma AC4_AC3_aux3: "x \<in> A \<Longrightarrow> (\<Union>z \<in> A. {z}*f(z))``{x} = f(x)" |
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by fast |
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lemma AC4_AC3: "AC4 \<Longrightarrow> AC3" |
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unfolding AC3_def AC4_def |
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apply (intro allI ballI) |
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apply (elim allE impE) |
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apply (erule AC4_AC3_aux1) |
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apply (simp add: AC4_AC3_aux2 AC4_AC3_aux3 cong add: Pi_cong) |
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done |
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(* ********************************************************************** *) |
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(* AC3 \<Longrightarrow> AC1 *) |
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(* ********************************************************************** *) |
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lemma AC3_AC1_lemma: |
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"b\<notin>A \<Longrightarrow> (\<Prod>x \<in> {a \<in> A. id(A)`a\<noteq>b}. id(A)`x) = (\<Prod>x \<in> A. x)" |
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apply (simp add: id_def cong add: Pi_cong) |
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apply (rule_tac b = A in subst_context, fast) |
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done |
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lemma AC3_AC1: "AC3 \<Longrightarrow> AC1" |
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unfolding AC1_def AC3_def |
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apply (fast intro!: id_type elim: AC3_AC1_lemma [THEN subst]) |
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done |
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(* ********************************************************************** *) |
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(* AC4 \<Longrightarrow> AC5 *) |
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(* ********************************************************************** *) |
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lemma AC4_AC5: "AC4 \<Longrightarrow> AC5" |
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unfolding range_def AC4_def AC5_def |
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apply (intro allI ballI) |
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apply (elim allE impE) |
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apply (erule fun_is_rel [THEN converse_type]) |
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apply (erule exE) |
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apply (rename_tac g) |
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apply (rule_tac x=g in bexI) |
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apply (blast dest: apply_equality range_type) |
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apply (blast intro: Pi_type dest: apply_type fun_is_rel) |
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done |
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(* ********************************************************************** *) |
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(* AC5 \<Longrightarrow> AC4, Rubin & Rubin, p. 11 *) |
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(* ********************************************************************** *) |
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lemma AC5_AC4_aux1: "R \<subseteq> A*B \<Longrightarrow> (\<lambda>x \<in> R. fst(x)) \<in> R -> A" |
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by (fast intro!: lam_type fst_type) |
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lemma AC5_AC4_aux2: "R \<subseteq> A*B \<Longrightarrow> range(\<lambda>x \<in> R. fst(x)) = domain(R)" |
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by (unfold lam_def, force) |
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lemma AC5_AC4_aux3: "\<lbrakk>\<exists>f \<in> A->C. P(f,domain(f)); A=B\<rbrakk> \<Longrightarrow> \<exists>f \<in> B->C. P(f,B)" |
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apply (erule bexE) |
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apply (frule domain_of_fun, fast) |
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done |
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lemma AC5_AC4_aux4: "\<lbrakk>R \<subseteq> A*B; g \<in> C->R; \<forall>x \<in> C. (\<lambda>z \<in> R. fst(z))` (g`x) = x\<rbrakk> |
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\<Longrightarrow> (\<lambda>x \<in> C. snd(g`x)): (\<Prod>x \<in> C. R``{x})" |
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apply (rule lam_type) |
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apply (force dest: apply_type) |
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done |
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lemma AC5_AC4: "AC5 \<Longrightarrow> AC4" |
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apply (unfold AC4_def AC5_def, clarify) |
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apply (elim allE ballE) |
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apply (drule AC5_AC4_aux3 [OF _ AC5_AC4_aux2], assumption) |
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apply (fast elim!: AC5_AC4_aux4) |
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apply (blast intro: AC5_AC4_aux1) |
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done |
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(* ********************************************************************** *) |
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(* AC1 \<longleftrightarrow> AC6 *) |
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(* ********************************************************************** *) |
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lemma AC1_iff_AC6: "AC1 \<longleftrightarrow> AC6" |
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by (unfold AC1_def AC6_def, blast) |
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end |