(* Title: ZF/AC/AC1_AC17.thy ID: $Id: AC17_AC1.thy,v 1.8 2005/06/17 14:15:10 haftmann Exp $ Author: Krzysztof Grabczewski The equivalence of AC0, AC1 and AC17 Also, the proofs needed to show that each of AC2, AC3, ..., AC6 is equivalent to AC0 and AC1. *) theory AC17_AC1 imports HH begin (** AC0 is equivalent to AC1. AC0 comes from Suppes, AC1 from Rubin & Rubin **) lemma AC0_AC1_lemma: "[| f:(Π X ∈ A. X); D ⊆ A |] ==> ∃g. g:(Π X ∈ D. X)" by (fast intro!: lam_type apply_type) lemma AC0_AC1: "AC0 ==> AC1" apply (unfold AC0_def AC1_def) apply (blast intro: AC0_AC1_lemma) done lemma AC1_AC0: "AC1 ==> AC0" by (unfold AC0_def AC1_def, blast) (**** The proof of AC1 ==> AC17 ****) lemma AC1_AC17_lemma: "f ∈ (Π X ∈ Pow(A) - {0}. X) ==> f ∈ (Pow(A) - {0} -> A)" apply (rule Pi_type, assumption) apply (drule apply_type, assumption, fast) done lemma AC1_AC17: "AC1 ==> AC17" apply (unfold AC1_def AC17_def) apply (rule allI) apply (rule ballI) apply (erule_tac x = "Pow (A) -{0}" in allE) apply (erule impE, fast) apply (erule exE) apply (rule bexI) apply (erule_tac [2] AC1_AC17_lemma) apply (rule apply_type, assumption) apply (fast dest!: AC1_AC17_lemma elim!: apply_type) done (**** The proof of AC17 ==> AC1 ****) (* *********************************************************************** *) (* more properties of HH *) (* *********************************************************************** *) lemma UN_eq_imp_well_ord: "[| x - (\<Union>j ∈ LEAST i. HH(λX ∈ Pow(x)-{0}. {f`X}, x, i) = {x}. HH(λX ∈ Pow(x)-{0}. {f`X}, x, j)) = 0; f ∈ Pow(x)-{0} -> x |] ==> ∃r. well_ord(x,r)" apply (rule exI) apply (erule well_ord_rvimage [OF bij_Least_HH_x [THEN bij_converse_bij, THEN bij_is_inj] Ord_Least [THEN well_ord_Memrel]], assumption) done (* *********************************************************************** *) (* theorems closer to the proof *) (* *********************************************************************** *) lemma not_AC1_imp_ex: "~AC1 ==> ∃A. ∀f ∈ Pow(A)-{0} -> A. ∃u ∈ Pow(A)-{0}. f`u ∉ u" apply (unfold AC1_def) apply (erule swap) apply (rule allI) apply (erule swap) apply (rule_tac x = "Union (A)" in exI) apply (blast intro: lam_type) done lemma AC17_AC1_aux1: "[| ∀f ∈ Pow(x) - {0} -> x. ∃u ∈ Pow(x) - {0}. f`u∉u; ∃f ∈ Pow(x)-{0}->x. x - (\<Union>a ∈ (LEAST i. HH(λX ∈ Pow(x)-{0}. {f`X},x,i)={x}). HH(λX ∈ Pow(x)-{0}. {f`X},x,a)) = 0 |] ==> P" apply (erule bexE) apply (erule UN_eq_imp_well_ord [THEN exE], assumption) apply (erule ex_choice_fun_Pow [THEN exE]) apply (erule ballE) apply (fast intro: apply_type del: DiffE) apply (erule notE) apply (rule Pi_type, assumption) apply (blast dest: apply_type) done lemma AC17_AC1_aux2: "~ (∃f ∈ Pow(x)-{0}->x. x - F(f) = 0) ==> (λf ∈ Pow(x)-{0}->x . x - F(f)) ∈ (Pow(x) -{0} -> x) -> Pow(x) - {0}" by (fast intro!: lam_type dest!: Diff_eq_0_iff [THEN iffD1]) lemma AC17_AC1_aux3: "[| f`Z ∈ Z; Z ∈ Pow(x)-{0} |] ==> (λX ∈ Pow(x)-{0}. {f`X})`Z ∈ Pow(Z)-{0}" by auto lemma AC17_AC1_aux4: "∃f ∈ F. f`((λf ∈ F. Q(f))`f) ∈ (λf ∈ F. Q(f))`f ==> ∃f ∈ F. f`Q(f) ∈ Q(f)" by simp lemma AC17_AC1: "AC17 ==> AC1" apply (unfold AC17_def) apply (rule classical) apply (erule not_AC1_imp_ex [THEN exE]) apply (case_tac "∃f ∈ Pow(x)-{0} -> x. x - (\<Union>a ∈ (LEAST i. HH (λX ∈ Pow (x) -{0}. {f`X},x,i) ={x}) . HH (λX ∈ Pow (x) -{0}. {f`X},x,a)) = 0") apply (erule AC17_AC1_aux1, assumption) apply (drule AC17_AC1_aux2) apply (erule allE) apply (drule bspec, assumption) apply (drule AC17_AC1_aux4) apply (erule bexE) apply (drule apply_type, assumption) apply (simp add: HH_Least_eq_x del: Diff_iff ) apply (drule AC17_AC1_aux3, assumption) apply (fast dest!: subst_elem [OF _ HH_Least_eq_x [symmetric]] f_subset_imp_HH_subset elim!: mem_irrefl) done (* ********************************************************************** AC1 ==> AC2 ==> AC1 AC1 ==> AC4 ==> AC3 ==> AC1 AC4 ==> AC5 ==> AC4 AC1 <-> AC6 ************************************************************************* *) (* ********************************************************************** *) (* AC1 ==> AC2 *) (* ********************************************************************** *) lemma AC1_AC2_aux1: "[| f:(Π X ∈ A. X); B ∈ A; 0∉A |] ==> {f`B} ⊆ B Int {f`C. C ∈ A}" by (fast elim!: apply_type) lemma AC1_AC2_aux2: "[| pairwise_disjoint(A); B ∈ A; C ∈ A; D ∈ B; D ∈ C |] ==> f`B = f`C" by (unfold pairwise_disjoint_def, fast) lemma AC1_AC2: "AC1 ==> AC2" apply (unfold AC1_def AC2_def) apply (rule allI) apply (rule impI) apply (elim asm_rl conjE allE exE impE, assumption) apply (intro exI ballI equalityI) prefer 2 apply (rule AC1_AC2_aux1, assumption+) apply (fast elim!: AC1_AC2_aux2 elim: apply_type) done (* ********************************************************************** *) (* AC2 ==> AC1 *) (* ********************************************************************** *) lemma AC2_AC1_aux1: "0∉A ==> 0 ∉ {B*{B}. B ∈ A}" by (fast dest!: sym [THEN Sigma_empty_iff [THEN iffD1]]) lemma AC2_AC1_aux2: "[| X*{X} Int C = {y}; X ∈ A |] ==> (THE y. X*{X} Int C = {y}): X*A" apply (rule subst_elem [of y]) apply (blast elim!: equalityE) apply (auto simp add: singleton_eq_iff) done lemma AC2_AC1_aux3: "∀D ∈ {E*{E}. E ∈ A}. ∃y. D Int C = {y} ==> (λx ∈ A. fst(THE z. (x*{x} Int C = {z}))) ∈ (Π X ∈ A. X)" apply (rule lam_type) apply (drule bspec, blast) apply (blast intro: AC2_AC1_aux2 fst_type) done lemma AC2_AC1: "AC2 ==> AC1" apply (unfold AC1_def AC2_def pairwise_disjoint_def) apply (intro allI impI) apply (elim allE impE) prefer 2 apply (fast elim!: AC2_AC1_aux3) apply (blast intro!: AC2_AC1_aux1) done (* ********************************************************************** *) (* AC1 ==> AC4 *) (* ********************************************************************** *) lemma empty_notin_images: "0 ∉ {R``{x}. x ∈ domain(R)}" by blast lemma AC1_AC4: "AC1 ==> AC4" apply (unfold AC1_def AC4_def) apply (intro allI impI) apply (drule spec, drule mp [OF _ empty_notin_images]) apply (best intro!: lam_type elim!: apply_type) done (* ********************************************************************** *) (* AC4 ==> AC3 *) (* ********************************************************************** *) lemma AC4_AC3_aux1: "f ∈ A->B ==> (\<Union>z ∈ A. {z}*f`z) ⊆ A*Union(B)" by (fast dest!: apply_type) lemma AC4_AC3_aux2: "domain(\<Union>z ∈ A. {z}*f(z)) = {a ∈ A. f(a)≠0}" by blast lemma AC4_AC3_aux3: "x ∈ A ==> (\<Union>z ∈ A. {z}*f(z))``{x} = f(x)" by fast lemma AC4_AC3: "AC4 ==> AC3" apply (unfold AC3_def AC4_def) apply (intro allI ballI) apply (elim allE impE) apply (erule AC4_AC3_aux1) apply (simp add: AC4_AC3_aux2 AC4_AC3_aux3 cong add: Pi_cong) done (* ********************************************************************** *) (* AC3 ==> AC1 *) (* ********************************************************************** *) lemma AC3_AC1_lemma: "b∉A ==> (Π x ∈ {a ∈ A. id(A)`a≠b}. id(A)`x) = (Π x ∈ A. x)" apply (simp add: id_def cong add: Pi_cong) apply (rule_tac b = A in subst_context, fast) done lemma AC3_AC1: "AC3 ==> AC1" apply (unfold AC1_def AC3_def) apply (fast intro!: id_type elim: AC3_AC1_lemma [THEN subst]) done (* ********************************************************************** *) (* AC4 ==> AC5 *) (* ********************************************************************** *) lemma AC4_AC5: "AC4 ==> AC5" apply (unfold range_def AC4_def AC5_def) apply (intro allI ballI) apply (elim allE impE) apply (erule fun_is_rel [THEN converse_type]) apply (erule exE) apply (rename_tac g) apply (rule_tac x=g in bexI) apply (blast dest: apply_equality range_type) apply (blast intro: Pi_type dest: apply_type fun_is_rel) done (* ********************************************************************** *) (* AC5 ==> AC4, Rubin & Rubin, p. 11 *) (* ********************************************************************** *) lemma AC5_AC4_aux1: "R ⊆ A*B ==> (λx ∈ R. fst(x)) ∈ R -> A" by (fast intro!: lam_type fst_type) lemma AC5_AC4_aux2: "R ⊆ A*B ==> range(λx ∈ R. fst(x)) = domain(R)" by (unfold lam_def, force) lemma AC5_AC4_aux3: "[| ∃f ∈ A->C. P(f,domain(f)); A=B |] ==> ∃f ∈ B->C. P(f,B)" apply (erule bexE) apply (frule domain_of_fun, fast) done lemma AC5_AC4_aux4: "[| R ⊆ A*B; g ∈ C->R; ∀x ∈ C. (λz ∈ R. fst(z))` (g`x) = x |] ==> (λx ∈ C. snd(g`x)): (Π x ∈ C. R``{x})" apply (rule lam_type) apply (force dest: apply_type) done lemma AC5_AC4: "AC5 ==> AC4" apply (unfold AC4_def AC5_def, clarify) apply (elim allE ballE) apply (drule AC5_AC4_aux3 [OF _ AC5_AC4_aux2], assumption) apply (fast elim!: AC5_AC4_aux4) apply (blast intro: AC5_AC4_aux1) done (* ********************************************************************** *) (* AC1 <-> AC6 *) (* ********************************************************************** *) lemma AC1_iff_AC6: "AC1 <-> AC6" by (unfold AC1_def AC6_def, blast) end
lemma AC0_AC1_lemma:
[| f ∈ (ΠX∈A. X); D ⊆ A |] ==> ∃g. g ∈ (ΠX∈D. X)
lemma AC0_AC1:
AC0 ==> AC1
lemma AC1_AC0:
AC1 ==> AC0
lemma AC1_AC17_lemma:
f ∈ (ΠX∈Pow(A) - {0}. X) ==> f ∈ Pow(A) - {0} -> A
lemma AC1_AC17:
AC1 ==> AC17
lemma UN_eq_imp_well_ord:
[| x - (\<Union>j∈μi. HH(λX∈Pow(x) - {0}. {f ` X}, x, i) = {x}. HH(λX∈Pow(x) - {0}. {f ` X}, x, j)) = 0; f ∈ Pow(x) - {0} -> x |] ==> ∃r. well_ord(x, r)
lemma not_AC1_imp_ex:
¬ AC1 ==> ∃A. ∀f∈Pow(A) - {0} -> A. ∃u∈Pow(A) - {0}. f ` u ∉ u
lemma AC17_AC1_aux1:
[| ∀f∈Pow(x) - {0} -> x. ∃u∈Pow(x) - {0}. f ` u ∉ u; ∃f∈Pow(x) - {0} -> x. x - (\<Union>a∈μi. HH(λX∈Pow(x) - {0}. {f ` X}, x, i) = {x}. HH(λX∈Pow(x) - {0}. {f ` X}, x, a)) = 0 |] ==> P
lemma AC17_AC1_aux2:
¬ (∃f∈Pow(x) - {0} -> x. x - F(f) = 0) ==> (λf∈Pow(x) - {0} -> x. x - F(f)) ∈ (Pow(x) - {0} -> x) -> Pow(x) - {0}
lemma AC17_AC1_aux3:
[| f ` Z ∈ Z; Z ∈ Pow(x) - {0} |] ==> (λX∈Pow(x) - {0}. {f ` X}) ` Z ∈ Pow(Z) - {0}
lemma AC17_AC1_aux4:
∃f∈F. f ` ((λf∈F. Q(f)) ` f) ∈ (λf∈F. Q(f)) ` f ==> ∃f∈F. f ` Q(f) ∈ Q(f)
lemma AC17_AC1:
AC17 ==> AC1
lemma AC1_AC2_aux1:
[| f ∈ (ΠX∈A. X); B ∈ A; 0 ∉ A |] ==> {f ` B} ⊆ B ∩ {f ` C . C ∈ A}
lemma AC1_AC2_aux2:
[| pairwise_disjoint(A); B ∈ A; C ∈ A; D ∈ B; D ∈ C |] ==> f ` B = f ` C
lemma AC1_AC2:
AC1 ==> AC2
lemma AC2_AC1_aux1:
0 ∉ A ==> 0 ∉ {B × {B} . B ∈ A}
lemma AC2_AC1_aux2:
[| X × {X} ∩ C = {y}; X ∈ A |] ==> (THE y. X × {X} ∩ C = {y}) ∈ X × A
lemma AC2_AC1_aux3:
∀D∈{E × {E} . E ∈ A}. ∃y. D ∩ C = {y} ==> (λx∈A. fst(THE z. x × {x} ∩ C = {z})) ∈ (ΠX∈A. X)
lemma AC2_AC1:
AC2 ==> AC1
lemma empty_notin_images:
0 ∉ {R `` {x} . x ∈ domain(R)}
lemma AC1_AC4:
AC1 ==> AC4
lemma AC4_AC3_aux1:
f ∈ A -> B ==> (\<Union>z∈A. {z} × f ` z) ⊆ A × \<Union>B
lemma AC4_AC3_aux2:
domain(\<Union>z∈A. {z} × f(z)) = {a ∈ A . f(a) ≠ 0}
lemma AC4_AC3_aux3:
x ∈ A ==> (\<Union>z∈A. {z} × f(z)) `` {x} = f(x)
lemma AC4_AC3:
AC4 ==> AC3
lemma AC3_AC1_lemma:
b ∉ A ==> (Πx∈{a ∈ A . id(A) ` a ≠ b}. id(A) ` x) = (Πx∈A. x)
lemma AC3_AC1:
AC3 ==> AC1
lemma AC4_AC5:
AC4 ==> AC5
lemma AC5_AC4_aux1:
R ⊆ A × B ==> (λx∈R. fst(x)) ∈ R -> A
lemma AC5_AC4_aux2:
R ⊆ A × B ==> range(λx∈R. fst(x)) = domain(R)
lemma AC5_AC4_aux3:
[| ∃f∈A -> C. P(f, domain(f)); A = B |] ==> ∃f∈B -> C. P(f, B)
lemma AC5_AC4_aux4:
[| R ⊆ A × B; g ∈ C -> R; ∀x∈C. (λz∈R. fst(z)) ` (g ` x) = x |] ==> (λx∈C. snd(g ` x)) ∈ (Πx∈C. R `` {x})
lemma AC5_AC4:
AC5 ==> AC4
lemma AC1_iff_AC6:
AC1 <-> AC6