author | lcp |
Thu, 13 Apr 1995 11:30:57 +0200 | |
changeset 1031 | a53cbb4b06c5 |
parent 845 | 825e96b87ef7 |
child 1055 | 67f5344605b7 |
permissions | -rw-r--r-- |
435 | 1 |
(* Title: ZF/Cardinal.ML |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1994 University of Cambridge |
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Cardinals in Zermelo-Fraenkel Set Theory |
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This theory does NOT assume the Axiom of Choice |
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*) |
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open Cardinal; |
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(*** The Schroeder-Bernstein Theorem -- see Davey & Priestly, page 106 ***) |
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(** Lemma: Banach's Decomposition Theorem **) |
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goal Cardinal.thy "bnd_mono(X, %W. X - g``(Y - f``W))"; |
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by (rtac bnd_monoI 1); |
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by (REPEAT (ares_tac [Diff_subset, subset_refl, Diff_mono, image_mono] 1)); |
|
760 | 20 |
qed "decomp_bnd_mono"; |
435 | 21 |
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val [gfun] = goal Cardinal.thy |
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"g: Y->X ==> \ |
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\ g``(Y - f`` lfp(X, %W. X - g``(Y - f``W))) = \ |
|
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\ X - lfp(X, %W. X - g``(Y - f``W)) "; |
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by (res_inst_tac [("P", "%u. ?v = X-u")] |
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(decomp_bnd_mono RS lfp_Tarski RS ssubst) 1); |
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by (simp_tac (ZF_ss addsimps [subset_refl, double_complement, |
|
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gfun RS fun_is_rel RS image_subset]) 1); |
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760 | 30 |
qed "Banach_last_equation"; |
435 | 31 |
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val prems = goal Cardinal.thy |
|
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"[| f: X->Y; g: Y->X |] ==> \ |
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\ EX XA XB YA YB. (XA Int XB = 0) & (XA Un XB = X) & \ |
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\ (YA Int YB = 0) & (YA Un YB = Y) & \ |
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\ f``XA=YA & g``YB=XB"; |
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37 |
by (REPEAT |
|
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(FIRSTGOAL |
|
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(resolve_tac [refl, exI, conjI, Diff_disjoint, Diff_partition]))); |
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by (rtac Banach_last_equation 3); |
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by (REPEAT (resolve_tac (prems@[fun_is_rel, image_subset, lfp_subset]) 1)); |
|
760 | 42 |
qed "decomposition"; |
435 | 43 |
|
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val prems = goal Cardinal.thy |
|
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"[| f: inj(X,Y); g: inj(Y,X) |] ==> EX h. h: bij(X,Y)"; |
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by (cut_facts_tac prems 1); |
|
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by (cut_facts_tac [(prems RL [inj_is_fun]) MRS decomposition] 1); |
|
48 |
by (fast_tac (ZF_cs addSIs [restrict_bij,bij_disjoint_Un] |
|
49 |
addIs [bij_converse_bij]) 1); |
|
50 |
(* The instantiation of exI to "restrict(f,XA) Un converse(restrict(g,YB))" |
|
51 |
is forced by the context!! *) |
|
760 | 52 |
qed "schroeder_bernstein"; |
435 | 53 |
|
54 |
||
55 |
(** Equipollence is an equivalence relation **) |
|
56 |
||
845
825e96b87ef7
Added Krzysztof's theorem LeastI2. Proof of sum_eqpoll_cong
lcp
parents:
833
diff
changeset
|
57 |
goalw Cardinal.thy [eqpoll_def] "!!f A B. f: bij(A,B) ==> A eqpoll B"; |
825e96b87ef7
Added Krzysztof's theorem LeastI2. Proof of sum_eqpoll_cong
lcp
parents:
833
diff
changeset
|
58 |
by (etac exI 1); |
825e96b87ef7
Added Krzysztof's theorem LeastI2. Proof of sum_eqpoll_cong
lcp
parents:
833
diff
changeset
|
59 |
qed "bij_imp_eqpoll"; |
825e96b87ef7
Added Krzysztof's theorem LeastI2. Proof of sum_eqpoll_cong
lcp
parents:
833
diff
changeset
|
60 |
|
825e96b87ef7
Added Krzysztof's theorem LeastI2. Proof of sum_eqpoll_cong
lcp
parents:
833
diff
changeset
|
61 |
(*A eqpoll A*) |
825e96b87ef7
Added Krzysztof's theorem LeastI2. Proof of sum_eqpoll_cong
lcp
parents:
833
diff
changeset
|
62 |
bind_thm ("eqpoll_refl", id_bij RS bij_imp_eqpoll); |
435 | 63 |
|
64 |
goalw Cardinal.thy [eqpoll_def] "!!X Y. X eqpoll Y ==> Y eqpoll X"; |
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by (fast_tac (ZF_cs addIs [bij_converse_bij]) 1); |
|
760 | 66 |
qed "eqpoll_sym"; |
435 | 67 |
|
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goalw Cardinal.thy [eqpoll_def] |
|
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"!!X Y. [| X eqpoll Y; Y eqpoll Z |] ==> X eqpoll Z"; |
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by (fast_tac (ZF_cs addIs [comp_bij]) 1); |
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760 | 71 |
qed "eqpoll_trans"; |
435 | 72 |
|
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(** Le-pollence is a partial ordering **) |
|
74 |
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75 |
goalw Cardinal.thy [lepoll_def] "!!X Y. X<=Y ==> X lepoll Y"; |
|
437 | 76 |
by (rtac exI 1); |
77 |
by (etac id_subset_inj 1); |
|
760 | 78 |
qed "subset_imp_lepoll"; |
435 | 79 |
|
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val lepoll_refl = subset_refl RS subset_imp_lepoll; |
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81 |
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82 |
goalw Cardinal.thy [eqpoll_def, bij_def, lepoll_def] |
|
83 |
"!!X Y. X eqpoll Y ==> X lepoll Y"; |
|
84 |
by (fast_tac ZF_cs 1); |
|
760 | 85 |
qed "eqpoll_imp_lepoll"; |
435 | 86 |
|
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goalw Cardinal.thy [lepoll_def] |
|
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"!!X Y. [| X lepoll Y; Y lepoll Z |] ==> X lepoll Z"; |
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by (fast_tac (ZF_cs addIs [comp_inj]) 1); |
|
760 | 90 |
qed "lepoll_trans"; |
435 | 91 |
|
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(*Asymmetry law*) |
|
93 |
goalw Cardinal.thy [lepoll_def,eqpoll_def] |
|
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"!!X Y. [| X lepoll Y; Y lepoll X |] ==> X eqpoll Y"; |
|
95 |
by (REPEAT (etac exE 1)); |
|
96 |
by (rtac schroeder_bernstein 1); |
|
97 |
by (REPEAT (assume_tac 1)); |
|
760 | 98 |
qed "eqpollI"; |
435 | 99 |
|
100 |
val [major,minor] = goal Cardinal.thy |
|
101 |
"[| X eqpoll Y; [| X lepoll Y; Y lepoll X |] ==> P |] ==> P"; |
|
437 | 102 |
by (rtac minor 1); |
435 | 103 |
by (REPEAT (resolve_tac [major, eqpoll_imp_lepoll, eqpoll_sym] 1)); |
760 | 104 |
qed "eqpollE"; |
435 | 105 |
|
106 |
goal Cardinal.thy "X eqpoll Y <-> X lepoll Y & Y lepoll X"; |
|
107 |
by (fast_tac (ZF_cs addIs [eqpollI] addSEs [eqpollE]) 1); |
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760 | 108 |
qed "eqpoll_iff"; |
435 | 109 |
|
833
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
110 |
goalw Cardinal.thy [lepoll_def, inj_def] "!!A. A lepoll 0 ==> A = 0"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
111 |
by (fast_tac (eq_cs addDs [apply_type]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
112 |
qed "lepoll_0_is_0"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
113 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
114 |
(*0 lepoll Y*) |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
115 |
bind_thm ("empty_lepollI", empty_subsetI RS subset_imp_lepoll); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
116 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
117 |
(*A eqpoll 0 ==> A=0*) |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
118 |
bind_thm ("eqpoll_0_is_0", eqpoll_imp_lepoll RS lepoll_0_is_0); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
119 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
120 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
121 |
(*** lesspoll: contributions by Krzysztof Grabczewski ***) |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
122 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
123 |
goalw Cardinal.thy [inj_def, surj_def] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
124 |
"!!f. [| f : inj(A, succ(m)); f ~: surj(A, succ(m)) |] ==> EX f. f:inj(A,m)"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
125 |
by (safe_tac lemmas_cs); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
126 |
by (swap_res_tac [exI] 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
127 |
by (res_inst_tac [("a", "lam z:A. if(f`z=m, y, f`z)")] CollectI 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
128 |
by (fast_tac (ZF_cs addSIs [if_type RS lam_type] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
129 |
addEs [apply_funtype RS succE]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
130 |
(*Proving it's injective*) |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
131 |
by (asm_simp_tac (ZF_ss setloop split_tac [expand_if]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
132 |
by (fast_tac ZF_cs 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
133 |
qed "inj_not_surj_succ"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
134 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
135 |
(** Variations on transitivity **) |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
136 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
137 |
goalw Cardinal.thy [lesspoll_def] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
138 |
"!!X. [| X lesspoll Y; Y lesspoll Z |] ==> X lesspoll Z"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
139 |
by (fast_tac (ZF_cs addSIs [eqpollI] addSEs [eqpollE] addIs [lepoll_trans]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
140 |
qed "lesspoll_trans"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
141 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
142 |
goalw Cardinal.thy [lesspoll_def] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
143 |
"!!X. [| X lesspoll Y; Y lepoll Z |] ==> X lesspoll Z"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
144 |
by (fast_tac (ZF_cs addSIs [eqpollI] addSEs [eqpollE] addIs [lepoll_trans]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
145 |
qed "lesspoll_lepoll_lesspoll"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
146 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
147 |
goalw Cardinal.thy [lesspoll_def] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
148 |
"!!X. [| X lesspoll Y; Z lepoll X |] ==> Z lesspoll Y"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
149 |
by (fast_tac (ZF_cs addSIs [eqpollI] addSEs [eqpollE] addIs [lepoll_trans]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
150 |
qed "lepoll_lesspoll_lesspoll"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
151 |
|
435 | 152 |
|
153 |
(** LEAST -- the least number operator [from HOL/Univ.ML] **) |
|
154 |
||
155 |
val [premP,premOrd,premNot] = goalw Cardinal.thy [Least_def] |
|
156 |
"[| P(i); Ord(i); !!x. x<i ==> ~P(x) |] ==> (LEAST x.P(x)) = i"; |
|
157 |
by (rtac the_equality 1); |
|
158 |
by (fast_tac (ZF_cs addSIs [premP,premOrd,premNot]) 1); |
|
159 |
by (REPEAT (etac conjE 1)); |
|
437 | 160 |
by (etac (premOrd RS Ord_linear_lt) 1); |
435 | 161 |
by (ALLGOALS (fast_tac (ZF_cs addSIs [premP] addSDs [premNot]))); |
760 | 162 |
qed "Least_equality"; |
435 | 163 |
|
164 |
goal Cardinal.thy "!!i. [| P(i); Ord(i) |] ==> P(LEAST x.P(x))"; |
|
165 |
by (etac rev_mp 1); |
|
166 |
by (trans_ind_tac "i" [] 1); |
|
167 |
by (rtac impI 1); |
|
168 |
by (rtac classical 1); |
|
169 |
by (EVERY1 [rtac (Least_equality RS ssubst), assume_tac, assume_tac]); |
|
170 |
by (assume_tac 2); |
|
171 |
by (fast_tac (ZF_cs addSEs [ltE]) 1); |
|
760 | 172 |
qed "LeastI"; |
435 | 173 |
|
174 |
(*Proof is almost identical to the one above!*) |
|
175 |
goal Cardinal.thy "!!i. [| P(i); Ord(i) |] ==> (LEAST x.P(x)) le i"; |
|
176 |
by (etac rev_mp 1); |
|
177 |
by (trans_ind_tac "i" [] 1); |
|
178 |
by (rtac impI 1); |
|
179 |
by (rtac classical 1); |
|
180 |
by (EVERY1 [rtac (Least_equality RS ssubst), assume_tac, assume_tac]); |
|
181 |
by (etac le_refl 2); |
|
182 |
by (fast_tac (ZF_cs addEs [ltE, lt_trans1] addIs [leI, ltI]) 1); |
|
760 | 183 |
qed "Least_le"; |
435 | 184 |
|
185 |
(*LEAST really is the smallest*) |
|
186 |
goal Cardinal.thy "!!i. [| P(i); i < (LEAST x.P(x)) |] ==> Q"; |
|
437 | 187 |
by (rtac (Least_le RSN (2,lt_trans2) RS lt_irrefl) 1); |
435 | 188 |
by (REPEAT (eresolve_tac [asm_rl, ltE] 1)); |
760 | 189 |
qed "less_LeastE"; |
435 | 190 |
|
1031 | 191 |
(*Easier to apply than LeastI: conclusion has only one occurrence of P*) |
845
825e96b87ef7
Added Krzysztof's theorem LeastI2. Proof of sum_eqpoll_cong
lcp
parents:
833
diff
changeset
|
192 |
qed_goal "LeastI2" Cardinal.thy |
825e96b87ef7
Added Krzysztof's theorem LeastI2. Proof of sum_eqpoll_cong
lcp
parents:
833
diff
changeset
|
193 |
"[| P(i); Ord(i); !!j. P(j) ==> Q(j) |] ==> Q(LEAST j. P(j))" |
825e96b87ef7
Added Krzysztof's theorem LeastI2. Proof of sum_eqpoll_cong
lcp
parents:
833
diff
changeset
|
194 |
(fn prems => [ resolve_tac prems 1, |
825e96b87ef7
Added Krzysztof's theorem LeastI2. Proof of sum_eqpoll_cong
lcp
parents:
833
diff
changeset
|
195 |
rtac LeastI 1, |
825e96b87ef7
Added Krzysztof's theorem LeastI2. Proof of sum_eqpoll_cong
lcp
parents:
833
diff
changeset
|
196 |
resolve_tac prems 1, |
825e96b87ef7
Added Krzysztof's theorem LeastI2. Proof of sum_eqpoll_cong
lcp
parents:
833
diff
changeset
|
197 |
resolve_tac prems 1 ]); |
825e96b87ef7
Added Krzysztof's theorem LeastI2. Proof of sum_eqpoll_cong
lcp
parents:
833
diff
changeset
|
198 |
|
437 | 199 |
(*If there is no such P then LEAST is vacuously 0*) |
200 |
goalw Cardinal.thy [Least_def] |
|
201 |
"!!P. [| ~ (EX i. Ord(i) & P(i)) |] ==> (LEAST x.P(x)) = 0"; |
|
202 |
by (rtac the_0 1); |
|
203 |
by (fast_tac ZF_cs 1); |
|
760 | 204 |
qed "Least_0"; |
437 | 205 |
|
435 | 206 |
goal Cardinal.thy "Ord(LEAST x.P(x))"; |
437 | 207 |
by (excluded_middle_tac "EX i. Ord(i) & P(i)" 1); |
435 | 208 |
by (safe_tac ZF_cs); |
437 | 209 |
by (rtac (Least_le RS ltE) 2); |
435 | 210 |
by (REPEAT_SOME assume_tac); |
437 | 211 |
by (etac (Least_0 RS ssubst) 1); |
212 |
by (rtac Ord_0 1); |
|
760 | 213 |
qed "Ord_Least"; |
435 | 214 |
|
215 |
||
216 |
(** Basic properties of cardinals **) |
|
217 |
||
218 |
(*Not needed for simplification, but helpful below*) |
|
219 |
val prems = goal Cardinal.thy |
|
220 |
"[| !!y. P(y) <-> Q(y) |] ==> (LEAST x.P(x)) = (LEAST x.Q(x))"; |
|
221 |
by (simp_tac (FOL_ss addsimps prems) 1); |
|
760 | 222 |
qed "Least_cong"; |
435 | 223 |
|
765 | 224 |
(*Need AC to prove X lepoll Y ==> |X| le |Y| ; |
225 |
see well_ord_lepoll_imp_Card_le *) |
|
435 | 226 |
goalw Cardinal.thy [eqpoll_def,cardinal_def] "!!X Y. X eqpoll Y ==> |X| = |Y|"; |
437 | 227 |
by (rtac Least_cong 1); |
435 | 228 |
by (fast_tac (ZF_cs addEs [comp_bij,bij_converse_bij]) 1); |
760 | 229 |
qed "cardinal_cong"; |
435 | 230 |
|
231 |
(*Under AC, the premise becomes trivial; one consequence is ||A|| = |A|*) |
|
845
825e96b87ef7
Added Krzysztof's theorem LeastI2. Proof of sum_eqpoll_cong
lcp
parents:
833
diff
changeset
|
232 |
goalw Cardinal.thy [cardinal_def] |
435 | 233 |
"!!A. well_ord(A,r) ==> |A| eqpoll A"; |
437 | 234 |
by (rtac LeastI 1); |
235 |
by (etac Ord_ordertype 2); |
|
845
825e96b87ef7
Added Krzysztof's theorem LeastI2. Proof of sum_eqpoll_cong
lcp
parents:
833
diff
changeset
|
236 |
by (etac (ordermap_bij RS bij_converse_bij RS bij_imp_eqpoll) 1); |
760 | 237 |
qed "well_ord_cardinal_eqpoll"; |
435 | 238 |
|
803
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
792
diff
changeset
|
239 |
bind_thm ("Ord_cardinal_eqpoll", well_ord_Memrel RS well_ord_cardinal_eqpoll); |
435 | 240 |
|
241 |
goal Cardinal.thy |
|
242 |
"!!X Y. [| well_ord(X,r); well_ord(Y,s); |X| = |Y| |] ==> X eqpoll Y"; |
|
437 | 243 |
by (rtac (eqpoll_sym RS eqpoll_trans) 1); |
244 |
by (etac well_ord_cardinal_eqpoll 1); |
|
435 | 245 |
by (asm_simp_tac (ZF_ss addsimps [well_ord_cardinal_eqpoll]) 1); |
760 | 246 |
qed "well_ord_cardinal_eqE"; |
435 | 247 |
|
248 |
||
249 |
(** Observations from Kunen, page 28 **) |
|
250 |
||
251 |
goalw Cardinal.thy [cardinal_def] "!!i. Ord(i) ==> |i| le i"; |
|
437 | 252 |
by (etac (eqpoll_refl RS Least_le) 1); |
760 | 253 |
qed "Ord_cardinal_le"; |
435 | 254 |
|
484 | 255 |
goalw Cardinal.thy [Card_def] "!!K. Card(K) ==> |K| = K"; |
437 | 256 |
by (etac sym 1); |
760 | 257 |
qed "Card_cardinal_eq"; |
435 | 258 |
|
845
825e96b87ef7
Added Krzysztof's theorem LeastI2. Proof of sum_eqpoll_cong
lcp
parents:
833
diff
changeset
|
259 |
(* Could replace the ~(j eqpoll i) by ~(i lepoll j) *) |
435 | 260 |
val prems = goalw Cardinal.thy [Card_def,cardinal_def] |
261 |
"[| Ord(i); !!j. j<i ==> ~(j eqpoll i) |] ==> Card(i)"; |
|
437 | 262 |
by (rtac (Least_equality RS ssubst) 1); |
435 | 263 |
by (REPEAT (ares_tac ([refl,eqpoll_refl]@prems) 1)); |
760 | 264 |
qed "CardI"; |
435 | 265 |
|
266 |
goalw Cardinal.thy [Card_def, cardinal_def] "!!i. Card(i) ==> Ord(i)"; |
|
437 | 267 |
by (etac ssubst 1); |
268 |
by (rtac Ord_Least 1); |
|
760 | 269 |
qed "Card_is_Ord"; |
435 | 270 |
|
765 | 271 |
goal Cardinal.thy "!!K. Card(K) ==> K le |K|"; |
272 |
by (asm_simp_tac (ZF_ss addsimps [le_refl, Card_is_Ord, Card_cardinal_eq]) 1); |
|
782
200a16083201
added bind_thm for theorems defined by "standard ..."
clasohm
parents:
765
diff
changeset
|
273 |
qed "Card_cardinal_le"; |
765 | 274 |
|
484 | 275 |
goalw Cardinal.thy [cardinal_def] "Ord(|A|)"; |
437 | 276 |
by (rtac Ord_Least 1); |
760 | 277 |
qed "Ord_cardinal"; |
435 | 278 |
|
845
825e96b87ef7
Added Krzysztof's theorem LeastI2. Proof of sum_eqpoll_cong
lcp
parents:
833
diff
changeset
|
279 |
(*The cardinals are the initial ordinals*) |
825e96b87ef7
Added Krzysztof's theorem LeastI2. Proof of sum_eqpoll_cong
lcp
parents:
833
diff
changeset
|
280 |
goal Cardinal.thy "Card(K) <-> Ord(K) & (ALL j. j<K --> ~ j eqpoll K)"; |
825e96b87ef7
Added Krzysztof's theorem LeastI2. Proof of sum_eqpoll_cong
lcp
parents:
833
diff
changeset
|
281 |
by (safe_tac (ZF_cs addSIs [CardI, Card_is_Ord])); |
825e96b87ef7
Added Krzysztof's theorem LeastI2. Proof of sum_eqpoll_cong
lcp
parents:
833
diff
changeset
|
282 |
by (fast_tac ZF_cs 2); |
825e96b87ef7
Added Krzysztof's theorem LeastI2. Proof of sum_eqpoll_cong
lcp
parents:
833
diff
changeset
|
283 |
by (rewrite_goals_tac [Card_def, cardinal_def]); |
825e96b87ef7
Added Krzysztof's theorem LeastI2. Proof of sum_eqpoll_cong
lcp
parents:
833
diff
changeset
|
284 |
by (resolve_tac [less_LeastE] 1); |
825e96b87ef7
Added Krzysztof's theorem LeastI2. Proof of sum_eqpoll_cong
lcp
parents:
833
diff
changeset
|
285 |
by (eresolve_tac [subst] 2); |
825e96b87ef7
Added Krzysztof's theorem LeastI2. Proof of sum_eqpoll_cong
lcp
parents:
833
diff
changeset
|
286 |
by (ALLGOALS assume_tac); |
825e96b87ef7
Added Krzysztof's theorem LeastI2. Proof of sum_eqpoll_cong
lcp
parents:
833
diff
changeset
|
287 |
qed "Card_iff_initial"; |
825e96b87ef7
Added Krzysztof's theorem LeastI2. Proof of sum_eqpoll_cong
lcp
parents:
833
diff
changeset
|
288 |
|
437 | 289 |
goal Cardinal.thy "Card(0)"; |
290 |
by (rtac (Ord_0 RS CardI) 1); |
|
291 |
by (fast_tac (ZF_cs addSEs [ltE]) 1); |
|
760 | 292 |
qed "Card_0"; |
437 | 293 |
|
522 | 294 |
val [premK,premL] = goal Cardinal.thy |
295 |
"[| Card(K); Card(L) |] ==> Card(K Un L)"; |
|
296 |
by (rtac ([premK RS Card_is_Ord, premL RS Card_is_Ord] MRS Ord_linear_le) 1); |
|
297 |
by (asm_simp_tac |
|
298 |
(ZF_ss addsimps [premL, le_imp_subset, subset_Un_iff RS iffD1]) 1); |
|
299 |
by (asm_simp_tac |
|
300 |
(ZF_ss addsimps [premK, le_imp_subset, subset_Un_iff2 RS iffD1]) 1); |
|
760 | 301 |
qed "Card_Un"; |
522 | 302 |
|
303 |
(*Infinite unions of cardinals? See Devlin, Lemma 6.7, page 98*) |
|
304 |
||
484 | 305 |
goalw Cardinal.thy [cardinal_def] "Card(|A|)"; |
437 | 306 |
by (excluded_middle_tac "EX i. Ord(i) & i eqpoll A" 1); |
307 |
by (etac (Least_0 RS ssubst) 1 THEN rtac Card_0 1); |
|
308 |
by (rtac (Ord_Least RS CardI) 1); |
|
309 |
by (safe_tac ZF_cs); |
|
310 |
by (rtac less_LeastE 1); |
|
311 |
by (assume_tac 2); |
|
312 |
by (etac eqpoll_trans 1); |
|
313 |
by (REPEAT (ares_tac [LeastI] 1)); |
|
760 | 314 |
qed "Card_cardinal"; |
437 | 315 |
|
435 | 316 |
(*Kunen's Lemma 10.5*) |
317 |
goal Cardinal.thy "!!i j. [| |i| le j; j le i |] ==> |j| = |i|"; |
|
437 | 318 |
by (rtac (eqpollI RS cardinal_cong) 1); |
319 |
by (etac (le_imp_subset RS subset_imp_lepoll) 1); |
|
320 |
by (rtac lepoll_trans 1); |
|
321 |
by (etac (le_imp_subset RS subset_imp_lepoll) 2); |
|
322 |
by (rtac (eqpoll_sym RS eqpoll_imp_lepoll) 1); |
|
323 |
by (rtac Ord_cardinal_eqpoll 1); |
|
435 | 324 |
by (REPEAT (eresolve_tac [ltE, Ord_succD] 1)); |
760 | 325 |
qed "cardinal_eq_lemma"; |
435 | 326 |
|
327 |
goal Cardinal.thy "!!i j. i le j ==> |i| le |j|"; |
|
328 |
by (res_inst_tac [("i","|i|"),("j","|j|")] Ord_linear_le 1); |
|
329 |
by (REPEAT_FIRST (ares_tac [Ord_cardinal, le_eqI])); |
|
437 | 330 |
by (rtac cardinal_eq_lemma 1); |
331 |
by (assume_tac 2); |
|
332 |
by (etac le_trans 1); |
|
333 |
by (etac ltE 1); |
|
334 |
by (etac Ord_cardinal_le 1); |
|
760 | 335 |
qed "cardinal_mono"; |
435 | 336 |
|
337 |
(*Since we have |succ(nat)| le |nat|, the converse of cardinal_mono fails!*) |
|
338 |
goal Cardinal.thy "!!i j. [| |i| < |j|; Ord(i); Ord(j) |] ==> i < j"; |
|
437 | 339 |
by (rtac Ord_linear2 1); |
435 | 340 |
by (REPEAT_SOME assume_tac); |
437 | 341 |
by (etac (lt_trans2 RS lt_irrefl) 1); |
342 |
by (etac cardinal_mono 1); |
|
760 | 343 |
qed "cardinal_lt_imp_lt"; |
435 | 344 |
|
484 | 345 |
goal Cardinal.thy "!!i j. [| |i| < K; Ord(i); Card(K) |] ==> i < K"; |
435 | 346 |
by (asm_simp_tac (ZF_ss addsimps |
347 |
[cardinal_lt_imp_lt, Card_is_Ord, Card_cardinal_eq]) 1); |
|
760 | 348 |
qed "Card_lt_imp_lt"; |
435 | 349 |
|
484 | 350 |
goal Cardinal.thy "!!i j. [| Ord(i); Card(K) |] ==> (|i| < K) <-> (i < K)"; |
351 |
by (fast_tac (ZF_cs addEs [Card_lt_imp_lt, Ord_cardinal_le RS lt_trans1]) 1); |
|
760 | 352 |
qed "Card_lt_iff"; |
484 | 353 |
|
354 |
goal Cardinal.thy "!!i j. [| Ord(i); Card(K) |] ==> (K le |i|) <-> (K le i)"; |
|
355 |
by (asm_simp_tac (ZF_ss addsimps |
|
356 |
[Card_lt_iff, Card_is_Ord, Ord_cardinal, |
|
357 |
not_lt_iff_le RS iff_sym]) 1); |
|
760 | 358 |
qed "Card_le_iff"; |
484 | 359 |
|
435 | 360 |
|
361 |
(*** The finite cardinals ***) |
|
362 |
||
833
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
363 |
goalw Cardinal.thy [lepoll_def, inj_def] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
364 |
"!!A B. [| cons(u,A) lepoll cons(v,B); u~:A; v~:B |] ==> A lepoll B"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
365 |
by (safe_tac ZF_cs); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
366 |
by (res_inst_tac [("x", "lam x:A. if(f`x=v, f`u, f`x)")] exI 1); |
437 | 367 |
by (rtac CollectI 1); |
833
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
368 |
(*Proving it's in the function space A->B*) |
437 | 369 |
by (rtac (if_type RS lam_type) 1); |
833
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
370 |
by (fast_tac (ZF_cs addEs [apply_funtype RS consE]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
371 |
by (fast_tac (ZF_cs addSEs [mem_irrefl] addEs [apply_funtype RS consE]) 1); |
435 | 372 |
(*Proving it's injective*) |
373 |
by (asm_simp_tac (ZF_ss setloop split_tac [expand_if]) 1); |
|
833
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
374 |
by (fast_tac ZF_cs 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
375 |
qed "cons_lepoll_consD"; |
435 | 376 |
|
833
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
377 |
goal Cardinal.thy |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
378 |
"!!A B. [| cons(u,A) eqpoll cons(v,B); u~:A; v~:B |] ==> A eqpoll B"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
379 |
by (asm_full_simp_tac (ZF_ss addsimps [eqpoll_iff]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
380 |
by (fast_tac (ZF_cs addIs [cons_lepoll_consD]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
381 |
qed "cons_eqpoll_consD"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
382 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
383 |
(*Lemma suggested by Mike Fourman*) |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
384 |
goalw Cardinal.thy [succ_def] "!!m n. succ(m) lepoll succ(n) ==> m lepoll n"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
385 |
by (etac cons_lepoll_consD 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
386 |
by (REPEAT (rtac mem_not_refl 1)); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
387 |
qed "succ_lepoll_succD"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
388 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
389 |
val [prem] = goal Cardinal.thy |
435 | 390 |
"m:nat ==> ALL n: nat. m lepoll n --> m le n"; |
391 |
by (nat_ind_tac "m" [prem] 1); |
|
392 |
by (fast_tac (ZF_cs addSIs [nat_0_le]) 1); |
|
437 | 393 |
by (rtac ballI 1); |
435 | 394 |
by (eres_inst_tac [("n","n")] natE 1); |
833
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
395 |
by (asm_simp_tac (ZF_ss addsimps [lepoll_def, inj_def, |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
396 |
succI1 RS Pi_empty2]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
397 |
by (fast_tac (ZF_cs addSIs [succ_leI] addSDs [succ_lepoll_succD]) 1); |
803
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
792
diff
changeset
|
398 |
val nat_lepoll_imp_le_lemma = result(); |
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
792
diff
changeset
|
399 |
|
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
792
diff
changeset
|
400 |
bind_thm ("nat_lepoll_imp_le", nat_lepoll_imp_le_lemma RS bspec RS mp); |
435 | 401 |
|
402 |
goal Cardinal.thy |
|
403 |
"!!m n. [| m:nat; n: nat |] ==> m eqpoll n <-> m = n"; |
|
437 | 404 |
by (rtac iffI 1); |
435 | 405 |
by (asm_simp_tac (ZF_ss addsimps [eqpoll_refl]) 2); |
437 | 406 |
by (fast_tac (ZF_cs addIs [nat_lepoll_imp_le, le_anti_sym] |
407 |
addSEs [eqpollE]) 1); |
|
760 | 408 |
qed "nat_eqpoll_iff"; |
435 | 409 |
|
410 |
goalw Cardinal.thy [Card_def,cardinal_def] |
|
411 |
"!!n. n: nat ==> Card(n)"; |
|
437 | 412 |
by (rtac (Least_equality RS ssubst) 1); |
435 | 413 |
by (REPEAT_FIRST (ares_tac [eqpoll_refl, nat_into_Ord, refl])); |
414 |
by (asm_simp_tac (ZF_ss addsimps [lt_nat_in_nat RS nat_eqpoll_iff]) 1); |
|
437 | 415 |
by (fast_tac (ZF_cs addSEs [lt_irrefl]) 1); |
760 | 416 |
qed "nat_into_Card"; |
435 | 417 |
|
418 |
(*Part of Kunen's Lemma 10.6*) |
|
419 |
goal Cardinal.thy "!!n. [| succ(n) lepoll n; n:nat |] ==> P"; |
|
437 | 420 |
by (rtac (nat_lepoll_imp_le RS lt_irrefl) 1); |
435 | 421 |
by (REPEAT (ares_tac [nat_succI] 1)); |
760 | 422 |
qed "succ_lepoll_natE"; |
435 | 423 |
|
424 |
||
833
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
425 |
(** lepoll, lesspoll and natural numbers **) |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
426 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
427 |
goalw Cardinal.thy [lesspoll_def] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
428 |
"!!m. [| A lepoll m; m:nat |] ==> A lesspoll succ(m)"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
429 |
by (rtac conjI 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
430 |
by (fast_tac (ZF_cs addIs [subset_imp_lepoll RSN (2,lepoll_trans)]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
431 |
by (rtac notI 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
432 |
by (dresolve_tac [eqpoll_sym RS eqpoll_imp_lepoll] 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
433 |
by (dtac lepoll_trans 1 THEN assume_tac 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
434 |
by (etac succ_lepoll_natE 1 THEN assume_tac 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
435 |
qed "lepoll_imp_lesspoll_succ"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
436 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
437 |
goalw Cardinal.thy [lesspoll_def, lepoll_def, eqpoll_def, bij_def] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
438 |
"!!m. [| A lesspoll succ(m); m:nat |] ==> A lepoll m"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
439 |
by (step_tac ZF_cs 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
440 |
by (fast_tac (ZF_cs addSIs [inj_not_surj_succ]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
441 |
qed "lesspoll_succ_imp_lepoll"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
442 |
|
1031 | 443 |
goal Cardinal.thy "!!m. m:nat ==> A lesspoll succ(m) <-> A lepoll m"; |
444 |
by (fast_tac (ZF_cs addSIs [lepoll_imp_lesspoll_succ, |
|
445 |
lesspoll_succ_imp_lepoll]) 1); |
|
446 |
qed "lesspoll_succ_iff"; |
|
447 |
||
833
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
448 |
goal Cardinal.thy "!!A m. [| A lepoll succ(m); m:nat |] ==> \ |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
449 |
\ A lepoll m | A eqpoll succ(m)"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
450 |
by (rtac disjCI 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
451 |
by (rtac lesspoll_succ_imp_lepoll 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
452 |
by (assume_tac 2); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
453 |
by (asm_simp_tac (ZF_ss addsimps [lesspoll_def]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
454 |
qed "lepoll_succ_disj"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
455 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
456 |
|
435 | 457 |
(*** The first infinite cardinal: Omega, or nat ***) |
458 |
||
459 |
(*This implies Kunen's Lemma 10.6*) |
|
460 |
goal Cardinal.thy "!!n. [| n<i; n:nat |] ==> ~ i lepoll n"; |
|
437 | 461 |
by (rtac notI 1); |
435 | 462 |
by (rtac succ_lepoll_natE 1 THEN assume_tac 2); |
463 |
by (rtac lepoll_trans 1 THEN assume_tac 2); |
|
437 | 464 |
by (etac ltE 1); |
435 | 465 |
by (REPEAT (ares_tac [Ord_succ_subsetI RS subset_imp_lepoll] 1)); |
760 | 466 |
qed "lt_not_lepoll"; |
435 | 467 |
|
468 |
goal Cardinal.thy "!!i n. [| Ord(i); n:nat |] ==> i eqpoll n <-> i=n"; |
|
437 | 469 |
by (rtac iffI 1); |
435 | 470 |
by (asm_simp_tac (ZF_ss addsimps [eqpoll_refl]) 2); |
471 |
by (rtac Ord_linear_lt 1); |
|
472 |
by (REPEAT_SOME (eresolve_tac [asm_rl, nat_into_Ord])); |
|
473 |
by (etac (lt_nat_in_nat RS nat_eqpoll_iff RS iffD1) 1 THEN |
|
474 |
REPEAT (assume_tac 1)); |
|
475 |
by (rtac (lt_not_lepoll RS notE) 1 THEN (REPEAT (assume_tac 1))); |
|
437 | 476 |
by (etac eqpoll_imp_lepoll 1); |
760 | 477 |
qed "Ord_nat_eqpoll_iff"; |
435 | 478 |
|
437 | 479 |
goalw Cardinal.thy [Card_def,cardinal_def] "Card(nat)"; |
480 |
by (rtac (Least_equality RS ssubst) 1); |
|
481 |
by (REPEAT_FIRST (ares_tac [eqpoll_refl, Ord_nat, refl])); |
|
482 |
by (etac ltE 1); |
|
483 |
by (asm_simp_tac (ZF_ss addsimps [eqpoll_iff, lt_not_lepoll, ltI]) 1); |
|
760 | 484 |
qed "Card_nat"; |
435 | 485 |
|
437 | 486 |
(*Allows showing that |i| is a limit cardinal*) |
487 |
goal Cardinal.thy "!!i. nat le i ==> nat le |i|"; |
|
488 |
by (rtac (Card_nat RS Card_cardinal_eq RS subst) 1); |
|
489 |
by (etac cardinal_mono 1); |
|
760 | 490 |
qed "nat_le_cardinal"; |
437 | 491 |
|
571
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
492 |
|
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
493 |
(*** Towards Cardinal Arithmetic ***) |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
494 |
(** Congruence laws for successor, cardinal addition and multiplication **) |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
495 |
|
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
496 |
(*Congruence law for cons under equipollence*) |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
497 |
goalw Cardinal.thy [lepoll_def] |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
498 |
"!!A B. [| A lepoll B; b ~: B |] ==> cons(a,A) lepoll cons(b,B)"; |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
499 |
by (safe_tac ZF_cs); |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
500 |
by (res_inst_tac [("x", "lam y: cons(a,A).if(y=a, b, f`y)")] exI 1); |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
501 |
by (res_inst_tac [("d","%z.if(z:B, converse(f)`z, a)")] |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
502 |
lam_injective 1); |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
503 |
by (asm_simp_tac (ZF_ss addsimps [inj_is_fun RS apply_type, cons_iff] |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
504 |
setloop etac consE') 1); |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
505 |
by (asm_simp_tac (ZF_ss addsimps [inj_is_fun RS apply_type, left_inverse] |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
506 |
setloop etac consE') 1); |
760 | 507 |
qed "cons_lepoll_cong"; |
571
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
508 |
|
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
509 |
goal Cardinal.thy |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
510 |
"!!A B. [| A eqpoll B; a ~: A; b ~: B |] ==> cons(a,A) eqpoll cons(b,B)"; |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
511 |
by (asm_full_simp_tac (ZF_ss addsimps [eqpoll_iff, cons_lepoll_cong]) 1); |
760 | 512 |
qed "cons_eqpoll_cong"; |
571
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
513 |
|
833
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
514 |
goal Cardinal.thy |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
515 |
"!!A B. [| a ~: A; b ~: B |] ==> \ |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
516 |
\ cons(a,A) lepoll cons(b,B) <-> A lepoll B"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
517 |
by (fast_tac (ZF_cs addIs [cons_lepoll_cong, cons_lepoll_consD]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
518 |
qed "cons_lepoll_cons_iff"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
519 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
520 |
goal Cardinal.thy |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
521 |
"!!A B. [| a ~: A; b ~: B |] ==> \ |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
522 |
\ cons(a,A) eqpoll cons(b,B) <-> A eqpoll B"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
523 |
by (fast_tac (ZF_cs addIs [cons_eqpoll_cong, cons_eqpoll_consD]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
524 |
qed "cons_eqpoll_cons_iff"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
525 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
526 |
goalw Cardinal.thy [succ_def] "{a} eqpoll 1"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
527 |
by (fast_tac (ZF_cs addSIs [eqpoll_refl RS cons_eqpoll_cong]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
528 |
qed "singleton_eqpoll_1"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
529 |
|
571
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
530 |
(*Congruence law for succ under equipollence*) |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
531 |
goalw Cardinal.thy [succ_def] |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
532 |
"!!A B. A eqpoll B ==> succ(A) eqpoll succ(B)"; |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
533 |
by (REPEAT (ares_tac [cons_eqpoll_cong, mem_not_refl] 1)); |
760 | 534 |
qed "succ_eqpoll_cong"; |
571
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
535 |
|
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
536 |
(*Congruence law for + under equipollence*) |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
537 |
goalw Cardinal.thy [eqpoll_def] |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
538 |
"!!A B C D. [| A eqpoll C; B eqpoll D |] ==> A+B eqpoll C+D"; |
845
825e96b87ef7
Added Krzysztof's theorem LeastI2. Proof of sum_eqpoll_cong
lcp
parents:
833
diff
changeset
|
539 |
by (fast_tac (ZF_cs addSIs [sum_bij]) 1); |
760 | 540 |
qed "sum_eqpoll_cong"; |
571
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
541 |
|
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
542 |
(*Congruence law for * under equipollence*) |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
543 |
goalw Cardinal.thy [eqpoll_def] |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
544 |
"!!A B C D. [| A eqpoll C; B eqpoll D |] ==> A*B eqpoll C*D"; |
845
825e96b87ef7
Added Krzysztof's theorem LeastI2. Proof of sum_eqpoll_cong
lcp
parents:
833
diff
changeset
|
545 |
by (fast_tac (ZF_cs addSIs [prod_bij]) 1); |
760 | 546 |
qed "prod_eqpoll_cong"; |
571
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
547 |
|
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
548 |
goalw Cardinal.thy [eqpoll_def] |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
549 |
"!!f. [| f: inj(A,B); A Int B = 0 |] ==> A Un (B - range(f)) eqpoll B"; |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
550 |
by (rtac exI 1); |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
551 |
by (res_inst_tac [("c", "%x. if(x:A, f`x, x)"), |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
552 |
("d", "%y. if(y: range(f), converse(f)`y, y)")] |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
553 |
lam_bijective 1); |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
554 |
by (fast_tac (ZF_cs addSIs [if_type, apply_type] addIs [inj_is_fun]) 1); |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
555 |
by (asm_simp_tac |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
556 |
(ZF_ss addsimps [inj_converse_fun RS apply_funtype] |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
557 |
setloop split_tac [expand_if]) 1); |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
558 |
by (asm_simp_tac (ZF_ss addsimps [inj_is_fun RS apply_rangeI, left_inverse] |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
559 |
setloop etac UnE') 1); |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
560 |
by (asm_simp_tac |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
561 |
(ZF_ss addsimps [inj_converse_fun RS apply_funtype, right_inverse] |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
562 |
setloop split_tac [expand_if]) 1); |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
563 |
by (fast_tac (ZF_cs addEs [equals0D]) 1); |
760 | 564 |
qed "inj_disjoint_eqpoll"; |
833
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
565 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
566 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
567 |
(*** Lemmas by Krzysztof Grabczewski. New proofs using cons_lepoll_cons. |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
568 |
Could easily generalise from succ to cons. ***) |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
569 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
570 |
goalw Cardinal.thy [succ_def] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
571 |
"!!A a n. [| a:A; A lepoll succ(n) |] ==> A - {a} lepoll n"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
572 |
by (rtac cons_lepoll_consD 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
573 |
by (rtac mem_not_refl 3); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
574 |
by (eresolve_tac [cons_Diff RS ssubst] 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
575 |
by (safe_tac ZF_cs); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
576 |
qed "diff_sing_lepoll"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
577 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
578 |
goalw Cardinal.thy [succ_def] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
579 |
"!!A a n. [| a:A; succ(n) lepoll A |] ==> n lepoll A - {a}"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
580 |
by (rtac cons_lepoll_consD 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
581 |
by (rtac mem_not_refl 2); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
582 |
by (eresolve_tac [cons_Diff RS ssubst] 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
583 |
by (safe_tac ZF_cs); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
584 |
qed "lepoll_diff_sing"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
585 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
586 |
goal Cardinal.thy "!!A a n. [| a:A; A eqpoll succ(n) |] ==> A - {a} eqpoll n"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
587 |
by (fast_tac (ZF_cs addSIs [eqpollI] addSEs [eqpollE] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
588 |
addIs [diff_sing_lepoll,lepoll_diff_sing]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
589 |
qed "diff_sing_eqpoll"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
590 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
591 |
goal Cardinal.thy "!!A. [| A lepoll 1; a:A |] ==> A = {a}"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
592 |
by (forward_tac [diff_sing_lepoll] 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
593 |
by (assume_tac 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
594 |
by (dtac lepoll_0_is_0 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
595 |
by (fast_tac (eq_cs addEs [equalityE]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
596 |
qed "lepoll_1_is_sing"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
597 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
598 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
599 |
(*** Finite and infinite sets ***) |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
600 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
601 |
goalw Cardinal.thy [Finite_def] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
602 |
"!!A. [| A lepoll n; n:nat |] ==> Finite(A)"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
603 |
by (etac rev_mp 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
604 |
by (etac nat_induct 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
605 |
by (fast_tac (ZF_cs addSDs [lepoll_0_is_0] addSIs [eqpoll_refl,nat_0I]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
606 |
by (fast_tac (ZF_cs addSDs [lepoll_succ_disj] addSIs [nat_succI]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
607 |
qed "lepoll_nat_imp_Finite"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
608 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
609 |
goalw Cardinal.thy [Finite_def] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
610 |
"!!X. [| Y lepoll X; Finite(X) |] ==> Finite(Y)"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
611 |
by (fast_tac (ZF_cs addSEs [eqpollE] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
612 |
addEs [lepoll_trans RS |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
613 |
rewrite_rule [Finite_def] lepoll_nat_imp_Finite]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
614 |
qed "lepoll_Finite"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
615 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
616 |
goalw Cardinal.thy [Finite_def] "!!x. Finite(x) ==> Finite(cons(y,x))"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
617 |
by (excluded_middle_tac "y:x" 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
618 |
by (asm_simp_tac (ZF_ss addsimps [cons_absorb]) 2); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
619 |
by (etac bexE 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
620 |
by (rtac bexI 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
621 |
by (etac nat_succI 2); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
622 |
by (asm_simp_tac |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
623 |
(ZF_ss addsimps [succ_def, cons_eqpoll_cong, mem_not_refl]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
624 |
qed "Finite_imp_cons_Finite"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
625 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
626 |
goalw Cardinal.thy [succ_def] "!!x. Finite(x) ==> Finite(succ(x))"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
627 |
by (etac Finite_imp_cons_Finite 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
628 |
qed "Finite_imp_succ_Finite"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
629 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
630 |
goalw Cardinal.thy [Finite_def] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
631 |
"!!i. [| Ord(i); ~ Finite(i) |] ==> nat le i"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
632 |
by (eresolve_tac [Ord_nat RSN (2,Ord_linear2)] 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
633 |
by (assume_tac 2); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
634 |
by (fast_tac (ZF_cs addSIs [eqpoll_refl] addSEs [ltE]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
635 |
qed "nat_le_infinite_Ord"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
636 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
637 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
638 |
(*Krzysztof Grabczewski's proof that the converse of a finite, well-ordered |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
639 |
set is well-ordered. Proofs simplified by lcp. *) |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
640 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
641 |
goal Nat.thy "!!n. n:nat ==> wf[n](converse(Memrel(n)))"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
642 |
by (etac nat_induct 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
643 |
by (fast_tac (ZF_cs addIs [wf_onI]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
644 |
by (rtac wf_onI 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
645 |
by (asm_full_simp_tac |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
646 |
(ZF_ss addsimps [wf_on_def, wf_def, converse_iff, Memrel_iff]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
647 |
by (excluded_middle_tac "x:Z" 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
648 |
by (dres_inst_tac [("x", "x")] bspec 2 THEN assume_tac 2); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
649 |
by (fast_tac (ZF_cs addSEs [mem_irrefl] addEs [mem_asym]) 2); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
650 |
by (dres_inst_tac [("x", "Z")] spec 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
651 |
by (safe_tac ZF_cs); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
652 |
by (dres_inst_tac [("x", "xa")] bspec 1 THEN assume_tac 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
653 |
by (fast_tac ZF_cs 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
654 |
qed "nat_wf_on_converse_Memrel"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
655 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
656 |
goal Cardinal.thy "!!n. n:nat ==> well_ord(n,converse(Memrel(n)))"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
657 |
by (forward_tac [Ord_nat RS Ord_in_Ord RS well_ord_Memrel] 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
658 |
by (rewtac well_ord_def); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
659 |
by (fast_tac (ZF_cs addSIs [tot_ord_converse, nat_wf_on_converse_Memrel]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
660 |
qed "nat_well_ord_converse_Memrel"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
661 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
662 |
goal Cardinal.thy |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
663 |
"!!A. [| well_ord(A,r); \ |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
664 |
\ well_ord(ordertype(A,r), converse(Memrel(ordertype(A, r)))) \ |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
665 |
\ |] ==> well_ord(A,converse(r))"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
666 |
by (resolve_tac [well_ord_Int_iff RS iffD1] 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
667 |
by (forward_tac [ordermap_bij RS bij_is_inj RS well_ord_rvimage] 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
668 |
by (assume_tac 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
669 |
by (asm_full_simp_tac |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
670 |
(ZF_ss addsimps [rvimage_converse, converse_Int, converse_prod, |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
671 |
ordertype_ord_iso RS ord_iso_rvimage_eq]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
672 |
qed "well_ord_converse"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
673 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
674 |
goal Cardinal.thy |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
675 |
"!!A. [| well_ord(A,r); A eqpoll n; n:nat |] ==> ordertype(A,r)=n"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
676 |
by (rtac (Ord_ordertype RS Ord_nat_eqpoll_iff RS iffD1) 1 THEN |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
677 |
REPEAT (assume_tac 1)); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
678 |
by (rtac eqpoll_trans 1 THEN assume_tac 2); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
679 |
by (rewtac eqpoll_def); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
680 |
by (fast_tac (ZF_cs addSIs [ordermap_bij RS bij_converse_bij]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
681 |
qed "ordertype_eq_n"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
682 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
683 |
goalw Cardinal.thy [Finite_def] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
684 |
"!!A. [| Finite(A); well_ord(A,r) |] ==> well_ord(A,converse(r))"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
685 |
by (rtac well_ord_converse 1 THEN assume_tac 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
686 |
by (fast_tac (ZF_cs addDs [ordertype_eq_n] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
687 |
addSIs [nat_well_ord_converse_Memrel]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
688 |
qed "Finite_well_ord_converse"; |