author | lcp |
Fri, 23 Dec 1994 16:33:37 +0100 | |
changeset 833 | ba386650df2c |
parent 803 | 4c8333ab3eae |
child 845 | 825e96b87ef7 |
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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||
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(** Lemma: Banach's Decomposition Theorem **) |
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||
17 |
goal Cardinal.thy "bnd_mono(X, %W. X - g``(Y - f``W))"; |
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18 |
by (rtac bnd_monoI 1); |
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19 |
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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23 |
"g: Y->X ==> \ |
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\ g``(Y - f`` lfp(X, %W. X - g``(Y - f``W))) = \ |
|
25 |
\ X - lfp(X, %W. X - g``(Y - f``W)) "; |
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26 |
by (res_inst_tac [("P", "%u. ?v = X-u")] |
|
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(decomp_bnd_mono RS lfp_Tarski RS ssubst) 1); |
|
28 |
by (simp_tac (ZF_ss addsimps [subset_refl, double_complement, |
|
29 |
gfun RS fun_is_rel RS image_subset]) 1); |
|
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"; |
|
37 |
by (REPEAT |
|
38 |
(FIRSTGOAL |
|
39 |
(resolve_tac [refl, exI, conjI, Diff_disjoint, Diff_partition]))); |
|
40 |
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 |
|
44 |
val prems = goal Cardinal.thy |
|
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"[| f: inj(X,Y); g: inj(Y,X) |] ==> EX h. h: bij(X,Y)"; |
|
46 |
by (cut_facts_tac prems 1); |
|
47 |
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 |
||
57 |
goalw Cardinal.thy [eqpoll_def] "X eqpoll X"; |
|
437 | 58 |
by (rtac exI 1); |
59 |
by (rtac id_bij 1); |
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760 | 60 |
qed "eqpoll_refl"; |
435 | 61 |
|
62 |
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); |
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760 | 64 |
qed "eqpoll_sym"; |
435 | 65 |
|
66 |
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 | 69 |
qed "eqpoll_trans"; |
435 | 70 |
|
71 |
(** Le-pollence is a partial ordering **) |
|
72 |
||
73 |
goalw Cardinal.thy [lepoll_def] "!!X Y. X<=Y ==> X lepoll Y"; |
|
437 | 74 |
by (rtac exI 1); |
75 |
by (etac id_subset_inj 1); |
|
760 | 76 |
qed "subset_imp_lepoll"; |
435 | 77 |
|
78 |
val lepoll_refl = subset_refl RS subset_imp_lepoll; |
|
79 |
||
80 |
goalw Cardinal.thy [eqpoll_def, bij_def, lepoll_def] |
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81 |
"!!X Y. X eqpoll Y ==> X lepoll Y"; |
|
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by (fast_tac ZF_cs 1); |
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760 | 83 |
qed "eqpoll_imp_lepoll"; |
435 | 84 |
|
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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); |
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760 | 88 |
qed "lepoll_trans"; |
435 | 89 |
|
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(*Asymmetry law*) |
|
91 |
goalw Cardinal.thy [lepoll_def,eqpoll_def] |
|
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"!!X Y. [| X lepoll Y; Y lepoll X |] ==> X eqpoll Y"; |
|
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by (REPEAT (etac exE 1)); |
|
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by (rtac schroeder_bernstein 1); |
|
95 |
by (REPEAT (assume_tac 1)); |
|
760 | 96 |
qed "eqpollI"; |
435 | 97 |
|
98 |
val [major,minor] = goal Cardinal.thy |
|
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"[| X eqpoll Y; [| X lepoll Y; Y lepoll X |] ==> P |] ==> P"; |
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437 | 100 |
by (rtac minor 1); |
435 | 101 |
by (REPEAT (resolve_tac [major, eqpoll_imp_lepoll, eqpoll_sym] 1)); |
760 | 102 |
qed "eqpollE"; |
435 | 103 |
|
104 |
goal Cardinal.thy "X eqpoll Y <-> X lepoll Y & Y lepoll X"; |
|
105 |
by (fast_tac (ZF_cs addIs [eqpollI] addSEs [eqpollE]) 1); |
|
760 | 106 |
qed "eqpoll_iff"; |
435 | 107 |
|
833
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
108 |
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
|
109 |
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
|
110 |
qed "lepoll_0_is_0"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
111 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
112 |
(*0 lepoll Y*) |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
113 |
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
|
114 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
115 |
(*A eqpoll 0 ==> A=0*) |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
116 |
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
|
117 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
118 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
119 |
(*** lesspoll: contributions by Krzysztof Grabczewski ***) |
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 |
goalw Cardinal.thy [inj_def, surj_def] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
122 |
"!!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
|
123 |
by (safe_tac lemmas_cs); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
124 |
by (swap_res_tac [exI] 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
125 |
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
|
126 |
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
|
127 |
addEs [apply_funtype RS succE]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
128 |
(*Proving it's injective*) |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
129 |
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
|
130 |
by (fast_tac ZF_cs 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
131 |
qed "inj_not_surj_succ"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
132 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
133 |
(** Variations on transitivity **) |
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 |
goalw Cardinal.thy [lesspoll_def] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
136 |
"!!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
|
137 |
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
|
138 |
qed "lesspoll_trans"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
139 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
140 |
goalw Cardinal.thy [lesspoll_def] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
141 |
"!!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
|
142 |
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
|
143 |
qed "lesspoll_lepoll_lesspoll"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
144 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
145 |
goalw Cardinal.thy [lesspoll_def] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
146 |
"!!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
|
147 |
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
|
148 |
qed "lepoll_lesspoll_lesspoll"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
149 |
|
435 | 150 |
|
151 |
(** LEAST -- the least number operator [from HOL/Univ.ML] **) |
|
152 |
||
153 |
val [premP,premOrd,premNot] = goalw Cardinal.thy [Least_def] |
|
154 |
"[| P(i); Ord(i); !!x. x<i ==> ~P(x) |] ==> (LEAST x.P(x)) = i"; |
|
155 |
by (rtac the_equality 1); |
|
156 |
by (fast_tac (ZF_cs addSIs [premP,premOrd,premNot]) 1); |
|
157 |
by (REPEAT (etac conjE 1)); |
|
437 | 158 |
by (etac (premOrd RS Ord_linear_lt) 1); |
435 | 159 |
by (ALLGOALS (fast_tac (ZF_cs addSIs [premP] addSDs [premNot]))); |
760 | 160 |
qed "Least_equality"; |
435 | 161 |
|
162 |
goal Cardinal.thy "!!i. [| P(i); Ord(i) |] ==> P(LEAST x.P(x))"; |
|
163 |
by (etac rev_mp 1); |
|
164 |
by (trans_ind_tac "i" [] 1); |
|
165 |
by (rtac impI 1); |
|
166 |
by (rtac classical 1); |
|
167 |
by (EVERY1 [rtac (Least_equality RS ssubst), assume_tac, assume_tac]); |
|
168 |
by (assume_tac 2); |
|
169 |
by (fast_tac (ZF_cs addSEs [ltE]) 1); |
|
760 | 170 |
qed "LeastI"; |
435 | 171 |
|
172 |
(*Proof is almost identical to the one above!*) |
|
173 |
goal Cardinal.thy "!!i. [| P(i); Ord(i) |] ==> (LEAST x.P(x)) le i"; |
|
174 |
by (etac rev_mp 1); |
|
175 |
by (trans_ind_tac "i" [] 1); |
|
176 |
by (rtac impI 1); |
|
177 |
by (rtac classical 1); |
|
178 |
by (EVERY1 [rtac (Least_equality RS ssubst), assume_tac, assume_tac]); |
|
179 |
by (etac le_refl 2); |
|
180 |
by (fast_tac (ZF_cs addEs [ltE, lt_trans1] addIs [leI, ltI]) 1); |
|
760 | 181 |
qed "Least_le"; |
435 | 182 |
|
183 |
(*LEAST really is the smallest*) |
|
184 |
goal Cardinal.thy "!!i. [| P(i); i < (LEAST x.P(x)) |] ==> Q"; |
|
437 | 185 |
by (rtac (Least_le RSN (2,lt_trans2) RS lt_irrefl) 1); |
435 | 186 |
by (REPEAT (eresolve_tac [asm_rl, ltE] 1)); |
760 | 187 |
qed "less_LeastE"; |
435 | 188 |
|
437 | 189 |
(*If there is no such P then LEAST is vacuously 0*) |
190 |
goalw Cardinal.thy [Least_def] |
|
191 |
"!!P. [| ~ (EX i. Ord(i) & P(i)) |] ==> (LEAST x.P(x)) = 0"; |
|
192 |
by (rtac the_0 1); |
|
193 |
by (fast_tac ZF_cs 1); |
|
760 | 194 |
qed "Least_0"; |
437 | 195 |
|
435 | 196 |
goal Cardinal.thy "Ord(LEAST x.P(x))"; |
437 | 197 |
by (excluded_middle_tac "EX i. Ord(i) & P(i)" 1); |
435 | 198 |
by (safe_tac ZF_cs); |
437 | 199 |
by (rtac (Least_le RS ltE) 2); |
435 | 200 |
by (REPEAT_SOME assume_tac); |
437 | 201 |
by (etac (Least_0 RS ssubst) 1); |
202 |
by (rtac Ord_0 1); |
|
760 | 203 |
qed "Ord_Least"; |
435 | 204 |
|
205 |
||
206 |
(** Basic properties of cardinals **) |
|
207 |
||
208 |
(*Not needed for simplification, but helpful below*) |
|
209 |
val prems = goal Cardinal.thy |
|
210 |
"[| !!y. P(y) <-> Q(y) |] ==> (LEAST x.P(x)) = (LEAST x.Q(x))"; |
|
211 |
by (simp_tac (FOL_ss addsimps prems) 1); |
|
760 | 212 |
qed "Least_cong"; |
435 | 213 |
|
765 | 214 |
(*Need AC to prove X lepoll Y ==> |X| le |Y| ; |
215 |
see well_ord_lepoll_imp_Card_le *) |
|
435 | 216 |
goalw Cardinal.thy [eqpoll_def,cardinal_def] "!!X Y. X eqpoll Y ==> |X| = |Y|"; |
437 | 217 |
by (rtac Least_cong 1); |
435 | 218 |
by (fast_tac (ZF_cs addEs [comp_bij,bij_converse_bij]) 1); |
760 | 219 |
qed "cardinal_cong"; |
435 | 220 |
|
221 |
(*Under AC, the premise becomes trivial; one consequence is ||A|| = |A|*) |
|
222 |
goalw Cardinal.thy [eqpoll_def, cardinal_def] |
|
223 |
"!!A. well_ord(A,r) ==> |A| eqpoll A"; |
|
437 | 224 |
by (rtac LeastI 1); |
225 |
by (etac Ord_ordertype 2); |
|
226 |
by (rtac exI 1); |
|
467 | 227 |
by (etac (ordermap_bij RS bij_converse_bij) 1); |
760 | 228 |
qed "well_ord_cardinal_eqpoll"; |
435 | 229 |
|
803
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
792
diff
changeset
|
230 |
bind_thm ("Ord_cardinal_eqpoll", well_ord_Memrel RS well_ord_cardinal_eqpoll); |
435 | 231 |
|
232 |
goal Cardinal.thy |
|
233 |
"!!X Y. [| well_ord(X,r); well_ord(Y,s); |X| = |Y| |] ==> X eqpoll Y"; |
|
437 | 234 |
by (rtac (eqpoll_sym RS eqpoll_trans) 1); |
235 |
by (etac well_ord_cardinal_eqpoll 1); |
|
435 | 236 |
by (asm_simp_tac (ZF_ss addsimps [well_ord_cardinal_eqpoll]) 1); |
760 | 237 |
qed "well_ord_cardinal_eqE"; |
435 | 238 |
|
239 |
||
240 |
(** Observations from Kunen, page 28 **) |
|
241 |
||
242 |
goalw Cardinal.thy [cardinal_def] "!!i. Ord(i) ==> |i| le i"; |
|
437 | 243 |
by (etac (eqpoll_refl RS Least_le) 1); |
760 | 244 |
qed "Ord_cardinal_le"; |
435 | 245 |
|
484 | 246 |
goalw Cardinal.thy [Card_def] "!!K. Card(K) ==> |K| = K"; |
437 | 247 |
by (etac sym 1); |
760 | 248 |
qed "Card_cardinal_eq"; |
435 | 249 |
|
250 |
val prems = goalw Cardinal.thy [Card_def,cardinal_def] |
|
251 |
"[| Ord(i); !!j. j<i ==> ~(j eqpoll i) |] ==> Card(i)"; |
|
437 | 252 |
by (rtac (Least_equality RS ssubst) 1); |
435 | 253 |
by (REPEAT (ares_tac ([refl,eqpoll_refl]@prems) 1)); |
760 | 254 |
qed "CardI"; |
435 | 255 |
|
256 |
goalw Cardinal.thy [Card_def, cardinal_def] "!!i. Card(i) ==> Ord(i)"; |
|
437 | 257 |
by (etac ssubst 1); |
258 |
by (rtac Ord_Least 1); |
|
760 | 259 |
qed "Card_is_Ord"; |
435 | 260 |
|
765 | 261 |
goal Cardinal.thy "!!K. Card(K) ==> K le |K|"; |
262 |
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
|
263 |
qed "Card_cardinal_le"; |
765 | 264 |
|
484 | 265 |
goalw Cardinal.thy [cardinal_def] "Ord(|A|)"; |
437 | 266 |
by (rtac Ord_Least 1); |
760 | 267 |
qed "Ord_cardinal"; |
435 | 268 |
|
437 | 269 |
goal Cardinal.thy "Card(0)"; |
270 |
by (rtac (Ord_0 RS CardI) 1); |
|
271 |
by (fast_tac (ZF_cs addSEs [ltE]) 1); |
|
760 | 272 |
qed "Card_0"; |
437 | 273 |
|
522 | 274 |
val [premK,premL] = goal Cardinal.thy |
275 |
"[| Card(K); Card(L) |] ==> Card(K Un L)"; |
|
276 |
by (rtac ([premK RS Card_is_Ord, premL RS Card_is_Ord] MRS Ord_linear_le) 1); |
|
277 |
by (asm_simp_tac |
|
278 |
(ZF_ss addsimps [premL, le_imp_subset, subset_Un_iff RS iffD1]) 1); |
|
279 |
by (asm_simp_tac |
|
280 |
(ZF_ss addsimps [premK, le_imp_subset, subset_Un_iff2 RS iffD1]) 1); |
|
760 | 281 |
qed "Card_Un"; |
522 | 282 |
|
283 |
(*Infinite unions of cardinals? See Devlin, Lemma 6.7, page 98*) |
|
284 |
||
484 | 285 |
goalw Cardinal.thy [cardinal_def] "Card(|A|)"; |
437 | 286 |
by (excluded_middle_tac "EX i. Ord(i) & i eqpoll A" 1); |
287 |
by (etac (Least_0 RS ssubst) 1 THEN rtac Card_0 1); |
|
288 |
by (rtac (Ord_Least RS CardI) 1); |
|
289 |
by (safe_tac ZF_cs); |
|
290 |
by (rtac less_LeastE 1); |
|
291 |
by (assume_tac 2); |
|
292 |
by (etac eqpoll_trans 1); |
|
293 |
by (REPEAT (ares_tac [LeastI] 1)); |
|
760 | 294 |
qed "Card_cardinal"; |
437 | 295 |
|
435 | 296 |
(*Kunen's Lemma 10.5*) |
297 |
goal Cardinal.thy "!!i j. [| |i| le j; j le i |] ==> |j| = |i|"; |
|
437 | 298 |
by (rtac (eqpollI RS cardinal_cong) 1); |
299 |
by (etac (le_imp_subset RS subset_imp_lepoll) 1); |
|
300 |
by (rtac lepoll_trans 1); |
|
301 |
by (etac (le_imp_subset RS subset_imp_lepoll) 2); |
|
302 |
by (rtac (eqpoll_sym RS eqpoll_imp_lepoll) 1); |
|
303 |
by (rtac Ord_cardinal_eqpoll 1); |
|
435 | 304 |
by (REPEAT (eresolve_tac [ltE, Ord_succD] 1)); |
760 | 305 |
qed "cardinal_eq_lemma"; |
435 | 306 |
|
307 |
goal Cardinal.thy "!!i j. i le j ==> |i| le |j|"; |
|
308 |
by (res_inst_tac [("i","|i|"),("j","|j|")] Ord_linear_le 1); |
|
309 |
by (REPEAT_FIRST (ares_tac [Ord_cardinal, le_eqI])); |
|
437 | 310 |
by (rtac cardinal_eq_lemma 1); |
311 |
by (assume_tac 2); |
|
312 |
by (etac le_trans 1); |
|
313 |
by (etac ltE 1); |
|
314 |
by (etac Ord_cardinal_le 1); |
|
760 | 315 |
qed "cardinal_mono"; |
435 | 316 |
|
317 |
(*Since we have |succ(nat)| le |nat|, the converse of cardinal_mono fails!*) |
|
318 |
goal Cardinal.thy "!!i j. [| |i| < |j|; Ord(i); Ord(j) |] ==> i < j"; |
|
437 | 319 |
by (rtac Ord_linear2 1); |
435 | 320 |
by (REPEAT_SOME assume_tac); |
437 | 321 |
by (etac (lt_trans2 RS lt_irrefl) 1); |
322 |
by (etac cardinal_mono 1); |
|
760 | 323 |
qed "cardinal_lt_imp_lt"; |
435 | 324 |
|
484 | 325 |
goal Cardinal.thy "!!i j. [| |i| < K; Ord(i); Card(K) |] ==> i < K"; |
435 | 326 |
by (asm_simp_tac (ZF_ss addsimps |
327 |
[cardinal_lt_imp_lt, Card_is_Ord, Card_cardinal_eq]) 1); |
|
760 | 328 |
qed "Card_lt_imp_lt"; |
435 | 329 |
|
484 | 330 |
goal Cardinal.thy "!!i j. [| Ord(i); Card(K) |] ==> (|i| < K) <-> (i < K)"; |
331 |
by (fast_tac (ZF_cs addEs [Card_lt_imp_lt, Ord_cardinal_le RS lt_trans1]) 1); |
|
760 | 332 |
qed "Card_lt_iff"; |
484 | 333 |
|
334 |
goal Cardinal.thy "!!i j. [| Ord(i); Card(K) |] ==> (K le |i|) <-> (K le i)"; |
|
335 |
by (asm_simp_tac (ZF_ss addsimps |
|
336 |
[Card_lt_iff, Card_is_Ord, Ord_cardinal, |
|
337 |
not_lt_iff_le RS iff_sym]) 1); |
|
760 | 338 |
qed "Card_le_iff"; |
484 | 339 |
|
435 | 340 |
|
341 |
(*** The finite cardinals ***) |
|
342 |
||
833
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
343 |
goalw Cardinal.thy [lepoll_def, inj_def] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
344 |
"!!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
|
345 |
by (safe_tac ZF_cs); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
346 |
by (res_inst_tac [("x", "lam x:A. if(f`x=v, f`u, f`x)")] exI 1); |
437 | 347 |
by (rtac CollectI 1); |
833
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
348 |
(*Proving it's in the function space A->B*) |
437 | 349 |
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
|
350 |
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
|
351 |
by (fast_tac (ZF_cs addSEs [mem_irrefl] addEs [apply_funtype RS consE]) 1); |
435 | 352 |
(*Proving it's injective*) |
353 |
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
|
354 |
by (fast_tac ZF_cs 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
355 |
qed "cons_lepoll_consD"; |
435 | 356 |
|
833
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
357 |
goal Cardinal.thy |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
358 |
"!!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
|
359 |
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
|
360 |
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
|
361 |
qed "cons_eqpoll_consD"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
362 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
363 |
(*Lemma suggested by Mike Fourman*) |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
364 |
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
|
365 |
by (etac cons_lepoll_consD 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
366 |
by (REPEAT (rtac mem_not_refl 1)); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
367 |
qed "succ_lepoll_succD"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
368 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
369 |
val [prem] = goal Cardinal.thy |
435 | 370 |
"m:nat ==> ALL n: nat. m lepoll n --> m le n"; |
371 |
by (nat_ind_tac "m" [prem] 1); |
|
372 |
by (fast_tac (ZF_cs addSIs [nat_0_le]) 1); |
|
437 | 373 |
by (rtac ballI 1); |
435 | 374 |
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
|
375 |
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
|
376 |
succI1 RS Pi_empty2]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
377 |
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
|
378 |
val nat_lepoll_imp_le_lemma = result(); |
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
792
diff
changeset
|
379 |
|
4c8333ab3eae
changed useless "qed" calls for lemmas back to uses of "result",
lcp
parents:
792
diff
changeset
|
380 |
bind_thm ("nat_lepoll_imp_le", nat_lepoll_imp_le_lemma RS bspec RS mp); |
435 | 381 |
|
382 |
goal Cardinal.thy |
|
383 |
"!!m n. [| m:nat; n: nat |] ==> m eqpoll n <-> m = n"; |
|
437 | 384 |
by (rtac iffI 1); |
435 | 385 |
by (asm_simp_tac (ZF_ss addsimps [eqpoll_refl]) 2); |
437 | 386 |
by (fast_tac (ZF_cs addIs [nat_lepoll_imp_le, le_anti_sym] |
387 |
addSEs [eqpollE]) 1); |
|
760 | 388 |
qed "nat_eqpoll_iff"; |
435 | 389 |
|
390 |
goalw Cardinal.thy [Card_def,cardinal_def] |
|
391 |
"!!n. n: nat ==> Card(n)"; |
|
437 | 392 |
by (rtac (Least_equality RS ssubst) 1); |
435 | 393 |
by (REPEAT_FIRST (ares_tac [eqpoll_refl, nat_into_Ord, refl])); |
394 |
by (asm_simp_tac (ZF_ss addsimps [lt_nat_in_nat RS nat_eqpoll_iff]) 1); |
|
437 | 395 |
by (fast_tac (ZF_cs addSEs [lt_irrefl]) 1); |
760 | 396 |
qed "nat_into_Card"; |
435 | 397 |
|
398 |
(*Part of Kunen's Lemma 10.6*) |
|
399 |
goal Cardinal.thy "!!n. [| succ(n) lepoll n; n:nat |] ==> P"; |
|
437 | 400 |
by (rtac (nat_lepoll_imp_le RS lt_irrefl) 1); |
435 | 401 |
by (REPEAT (ares_tac [nat_succI] 1)); |
760 | 402 |
qed "succ_lepoll_natE"; |
435 | 403 |
|
404 |
||
833
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
405 |
(** lepoll, lesspoll and natural numbers **) |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
406 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
407 |
goalw Cardinal.thy [lesspoll_def] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
408 |
"!!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
|
409 |
by (rtac conjI 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
410 |
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
|
411 |
by (rtac notI 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
412 |
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
|
413 |
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
|
414 |
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
|
415 |
qed "lepoll_imp_lesspoll_succ"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
416 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
417 |
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
|
418 |
"!!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
|
419 |
by (step_tac ZF_cs 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
420 |
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
|
421 |
qed "lesspoll_succ_imp_lepoll"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
422 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
423 |
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
|
424 |
\ A lepoll m | A eqpoll succ(m)"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
425 |
by (rtac disjCI 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
426 |
by (rtac lesspoll_succ_imp_lepoll 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
427 |
by (assume_tac 2); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
428 |
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
|
429 |
qed "lepoll_succ_disj"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
430 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
431 |
|
435 | 432 |
(*** The first infinite cardinal: Omega, or nat ***) |
433 |
||
434 |
(*This implies Kunen's Lemma 10.6*) |
|
435 |
goal Cardinal.thy "!!n. [| n<i; n:nat |] ==> ~ i lepoll n"; |
|
437 | 436 |
by (rtac notI 1); |
435 | 437 |
by (rtac succ_lepoll_natE 1 THEN assume_tac 2); |
438 |
by (rtac lepoll_trans 1 THEN assume_tac 2); |
|
437 | 439 |
by (etac ltE 1); |
435 | 440 |
by (REPEAT (ares_tac [Ord_succ_subsetI RS subset_imp_lepoll] 1)); |
760 | 441 |
qed "lt_not_lepoll"; |
435 | 442 |
|
443 |
goal Cardinal.thy "!!i n. [| Ord(i); n:nat |] ==> i eqpoll n <-> i=n"; |
|
437 | 444 |
by (rtac iffI 1); |
435 | 445 |
by (asm_simp_tac (ZF_ss addsimps [eqpoll_refl]) 2); |
446 |
by (rtac Ord_linear_lt 1); |
|
447 |
by (REPEAT_SOME (eresolve_tac [asm_rl, nat_into_Ord])); |
|
448 |
by (etac (lt_nat_in_nat RS nat_eqpoll_iff RS iffD1) 1 THEN |
|
449 |
REPEAT (assume_tac 1)); |
|
450 |
by (rtac (lt_not_lepoll RS notE) 1 THEN (REPEAT (assume_tac 1))); |
|
437 | 451 |
by (etac eqpoll_imp_lepoll 1); |
760 | 452 |
qed "Ord_nat_eqpoll_iff"; |
435 | 453 |
|
437 | 454 |
goalw Cardinal.thy [Card_def,cardinal_def] "Card(nat)"; |
455 |
by (rtac (Least_equality RS ssubst) 1); |
|
456 |
by (REPEAT_FIRST (ares_tac [eqpoll_refl, Ord_nat, refl])); |
|
457 |
by (etac ltE 1); |
|
458 |
by (asm_simp_tac (ZF_ss addsimps [eqpoll_iff, lt_not_lepoll, ltI]) 1); |
|
760 | 459 |
qed "Card_nat"; |
435 | 460 |
|
437 | 461 |
(*Allows showing that |i| is a limit cardinal*) |
462 |
goal Cardinal.thy "!!i. nat le i ==> nat le |i|"; |
|
463 |
by (rtac (Card_nat RS Card_cardinal_eq RS subst) 1); |
|
464 |
by (etac cardinal_mono 1); |
|
760 | 465 |
qed "nat_le_cardinal"; |
437 | 466 |
|
571
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
467 |
|
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
468 |
(*** Towards Cardinal Arithmetic ***) |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
469 |
(** Congruence laws for successor, cardinal addition and multiplication **) |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
470 |
|
792
5d2a7634da46
case_ss: now built upon ZF/Order/bij_inverse_ss. Deleted
lcp
parents:
782
diff
changeset
|
471 |
val case_ss = |
5d2a7634da46
case_ss: now built upon ZF/Order/bij_inverse_ss. Deleted
lcp
parents:
782
diff
changeset
|
472 |
bij_inverse_ss addsimps [Inl_iff, Inl_Inr_iff, Inr_iff, Inr_Inl_iff, |
5d2a7634da46
case_ss: now built upon ZF/Order/bij_inverse_ss. Deleted
lcp
parents:
782
diff
changeset
|
473 |
case_Inl, case_Inr, InlI, InrI]; |
571
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
474 |
|
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
475 |
(*Congruence law for cons under equipollence*) |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
476 |
goalw Cardinal.thy [lepoll_def] |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
477 |
"!!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
|
478 |
by (safe_tac ZF_cs); |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
479 |
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
|
480 |
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
|
481 |
lam_injective 1); |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
482 |
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
|
483 |
setloop etac consE') 1); |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
484 |
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
|
485 |
setloop etac consE') 1); |
760 | 486 |
qed "cons_lepoll_cong"; |
571
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
487 |
|
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
488 |
goal Cardinal.thy |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
489 |
"!!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
|
490 |
by (asm_full_simp_tac (ZF_ss addsimps [eqpoll_iff, cons_lepoll_cong]) 1); |
760 | 491 |
qed "cons_eqpoll_cong"; |
571
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
492 |
|
833
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
493 |
goal Cardinal.thy |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
494 |
"!!A B. [| a ~: A; b ~: B |] ==> \ |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
495 |
\ 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
|
496 |
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
|
497 |
qed "cons_lepoll_cons_iff"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
498 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
499 |
goal Cardinal.thy |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
500 |
"!!A B. [| a ~: A; b ~: B |] ==> \ |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
501 |
\ 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
|
502 |
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
|
503 |
qed "cons_eqpoll_cons_iff"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
504 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
505 |
goalw Cardinal.thy [succ_def] "{a} eqpoll 1"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
506 |
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
|
507 |
qed "singleton_eqpoll_1"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
508 |
|
571
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
509 |
(*Congruence law for succ under equipollence*) |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
510 |
goalw Cardinal.thy [succ_def] |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
511 |
"!!A B. A eqpoll B ==> succ(A) eqpoll succ(B)"; |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
512 |
by (REPEAT (ares_tac [cons_eqpoll_cong, mem_not_refl] 1)); |
760 | 513 |
qed "succ_eqpoll_cong"; |
571
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
514 |
|
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
515 |
(*Congruence law for + under equipollence*) |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
516 |
goalw Cardinal.thy [eqpoll_def] |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
517 |
"!!A B C D. [| A eqpoll C; B eqpoll D |] ==> A+B eqpoll C+D"; |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
518 |
by (safe_tac ZF_cs); |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
519 |
by (rtac exI 1); |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
520 |
by (res_inst_tac [("c", "case(%x. Inl(f`x), %y. Inr(fa`y))"), |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
521 |
("d", "case(%x. Inl(converse(f)`x), %y. Inr(converse(fa)`y))")] |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
522 |
lam_bijective 1); |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
523 |
by (safe_tac (ZF_cs addSEs [sumE])); |
792
5d2a7634da46
case_ss: now built upon ZF/Order/bij_inverse_ss. Deleted
lcp
parents:
782
diff
changeset
|
524 |
by (ALLGOALS (asm_simp_tac case_ss)); |
760 | 525 |
qed "sum_eqpoll_cong"; |
571
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
526 |
|
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
527 |
(*Congruence law for * under equipollence*) |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
528 |
goalw Cardinal.thy [eqpoll_def] |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
529 |
"!!A B C D. [| A eqpoll C; B eqpoll D |] ==> A*B eqpoll C*D"; |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
530 |
by (safe_tac ZF_cs); |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
531 |
by (rtac exI 1); |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
532 |
by (res_inst_tac [("c", "split(%x y. <f`x, fa`y>)"), |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
533 |
("d", "split(%x y. <converse(f)`x, converse(fa)`y>)")] |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
534 |
lam_bijective 1); |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
535 |
by (safe_tac ZF_cs); |
792
5d2a7634da46
case_ss: now built upon ZF/Order/bij_inverse_ss. Deleted
lcp
parents:
782
diff
changeset
|
536 |
by (ALLGOALS (asm_simp_tac case_ss)); |
760 | 537 |
qed "prod_eqpoll_cong"; |
571
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
538 |
|
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
539 |
goalw Cardinal.thy [eqpoll_def] |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
540 |
"!!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
|
541 |
by (rtac exI 1); |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
542 |
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
|
543 |
("d", "%y. if(y: range(f), converse(f)`y, y)")] |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
544 |
lam_bijective 1); |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
545 |
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
|
546 |
by (asm_simp_tac |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
547 |
(ZF_ss addsimps [inj_converse_fun RS apply_funtype] |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
548 |
setloop split_tac [expand_if]) 1); |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
549 |
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
|
550 |
setloop etac UnE') 1); |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
551 |
by (asm_simp_tac |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
552 |
(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
|
553 |
setloop split_tac [expand_if]) 1); |
0b03ce5b62f7
ZF/Cardinal: some results moved here from CardinalArith
lcp
parents:
522
diff
changeset
|
554 |
by (fast_tac (ZF_cs addEs [equals0D]) 1); |
760 | 555 |
qed "inj_disjoint_eqpoll"; |
833
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
556 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
557 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
558 |
(*** 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
|
559 |
Could easily generalise from succ to cons. ***) |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
560 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
561 |
goalw Cardinal.thy [succ_def] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
562 |
"!!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
|
563 |
by (rtac cons_lepoll_consD 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
564 |
by (rtac mem_not_refl 3); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
565 |
by (eresolve_tac [cons_Diff RS ssubst] 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
566 |
by (safe_tac ZF_cs); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
567 |
qed "diff_sing_lepoll"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
568 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
569 |
goalw Cardinal.thy [succ_def] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
570 |
"!!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
|
571 |
by (rtac cons_lepoll_consD 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
572 |
by (rtac mem_not_refl 2); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
573 |
by (eresolve_tac [cons_Diff RS ssubst] 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
574 |
by (safe_tac ZF_cs); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
575 |
qed "lepoll_diff_sing"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
576 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
577 |
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
|
578 |
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
|
579 |
addIs [diff_sing_lepoll,lepoll_diff_sing]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
580 |
qed "diff_sing_eqpoll"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
581 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
582 |
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
|
583 |
by (forward_tac [diff_sing_lepoll] 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
584 |
by (assume_tac 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
585 |
by (dtac lepoll_0_is_0 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
586 |
by (fast_tac (eq_cs addEs [equalityE]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
587 |
qed "lepoll_1_is_sing"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
588 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
589 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
590 |
(*** Finite and infinite sets ***) |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
591 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
592 |
goalw Cardinal.thy [Finite_def] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
593 |
"!!A. [| A lepoll n; n:nat |] ==> Finite(A)"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
594 |
by (etac rev_mp 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
595 |
by (etac nat_induct 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
596 |
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
|
597 |
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
|
598 |
qed "lepoll_nat_imp_Finite"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
599 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
600 |
goalw Cardinal.thy [Finite_def] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
601 |
"!!X. [| Y lepoll X; Finite(X) |] ==> Finite(Y)"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
602 |
by (fast_tac (ZF_cs addSEs [eqpollE] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
603 |
addEs [lepoll_trans RS |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
604 |
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
|
605 |
qed "lepoll_Finite"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
606 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
607 |
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
|
608 |
by (excluded_middle_tac "y:x" 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
609 |
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
|
610 |
by (etac bexE 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
611 |
by (rtac bexI 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
612 |
by (etac nat_succI 2); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
613 |
by (asm_simp_tac |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
614 |
(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
|
615 |
qed "Finite_imp_cons_Finite"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
616 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
617 |
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
|
618 |
by (etac Finite_imp_cons_Finite 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
619 |
qed "Finite_imp_succ_Finite"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
620 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
621 |
goalw Cardinal.thy [Finite_def] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
622 |
"!!i. [| Ord(i); ~ Finite(i) |] ==> nat le i"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
623 |
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
|
624 |
by (assume_tac 2); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
625 |
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
|
626 |
qed "nat_le_infinite_Ord"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
627 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
628 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
629 |
(*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
|
630 |
set is well-ordered. Proofs simplified by lcp. *) |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
631 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
632 |
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
|
633 |
by (etac nat_induct 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
634 |
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
|
635 |
by (rtac wf_onI 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
636 |
by (asm_full_simp_tac |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
637 |
(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
|
638 |
by (excluded_middle_tac "x:Z" 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
639 |
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
|
640 |
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
|
641 |
by (dres_inst_tac [("x", "Z")] spec 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
642 |
by (safe_tac ZF_cs); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
643 |
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
|
644 |
by (fast_tac ZF_cs 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
645 |
qed "nat_wf_on_converse_Memrel"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
646 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
647 |
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
|
648 |
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
|
649 |
by (rewtac well_ord_def); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
650 |
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
|
651 |
qed "nat_well_ord_converse_Memrel"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
652 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
653 |
goal Cardinal.thy |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
654 |
"!!A. [| well_ord(A,r); \ |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
655 |
\ 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
|
656 |
\ |] ==> well_ord(A,converse(r))"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
657 |
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
|
658 |
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
|
659 |
by (assume_tac 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
660 |
by (asm_full_simp_tac |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
661 |
(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
|
662 |
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
|
663 |
qed "well_ord_converse"; |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
664 |
|
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
665 |
goal Cardinal.thy |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
666 |
"!!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
|
667 |
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
|
668 |
REPEAT (assume_tac 1)); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
669 |
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
|
670 |
by (rewtac eqpoll_def); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
671 |
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
|
672 |
qed "ordertype_eq_n"; |
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 |
goalw Cardinal.thy [Finite_def] |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
675 |
"!!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
|
676 |
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
|
677 |
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
|
678 |
addSIs [nat_well_ord_converse_Memrel]) 1); |
ba386650df2c
Proved cons_lepoll_consD, succ_lepoll_succD, cons_eqpoll_consD,
lcp
parents:
803
diff
changeset
|
679 |
qed "Finite_well_ord_converse"; |