author | paulson |
Fri, 18 Oct 1996 11:42:17 +0200 | |
changeset 2109 | fabc35243cea |
parent 2078 | b198b3d46fb4 |
child 2124 | 9677fdf5fc23 |
permissions | -rw-r--r-- |
1934 | 1 |
(* Title: HOL/Auth/Message |
2 |
ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Copyright 1996 University of Cambridge |
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5 |
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Theory of Shared Keys (common to all symmetric-key protocols) |
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7 |
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Server keys; initial states of agents; new nonces and keys; function "sees" |
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2032 | 9 |
*) |
10 |
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1934 | 11 |
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2064 | 12 |
val prems = goal HOL.thy "[| P ==> Q(True); ~P ==> Q(False) |] ==> Q(P)"; |
2109
fabc35243cea
Deleted obsolete rewrites (they are now in HOL/simpdata)
paulson
parents:
2078
diff
changeset
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13 |
by (case_tac "P" 1); |
fabc35243cea
Deleted obsolete rewrites (they are now in HOL/simpdata)
paulson
parents:
2078
diff
changeset
|
14 |
by (ALLGOALS (asm_simp_tac (!simpset addsimps prems))); |
2064 | 15 |
val expand_case = result(); |
16 |
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17 |
fun expand_case_tac P i = |
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18 |
res_inst_tac [("P",P)] expand_case i THEN |
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19 |
Simp_tac (i+1) THEN |
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20 |
Simp_tac i; |
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21 |
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22 |
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1934 | 23 |
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2000 | 24 |
(*GOALS.ML??*) |
25 |
fun prlim n = (goals_limit:=n; pr()); |
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26 |
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1934 | 27 |
(*FUN.ML?? WE NEED A NOTION OF INVERSE IMAGE, OR GRAPH!!*) |
28 |
goal Set.thy "!!f. B <= range f = (B = f`` {x. f x: B})"; |
|
29 |
by (fast_tac (!claset addEs [equalityE]) 1); |
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30 |
val subset_range_iff = result(); |
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31 |
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32 |
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33 |
open Shared; |
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34 |
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1964 | 35 |
(*Holds because Friend is injective: thus cannot prove for all f*) |
36 |
goal thy "(Friend x : Friend``A) = (x:A)"; |
|
37 |
by (Auto_tac()); |
|
38 |
qed "Friend_image_eq"; |
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39 |
Addsimps [Friend_image_eq]; |
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40 |
||
1934 | 41 |
Addsimps [Un_insert_left, Un_insert_right]; |
42 |
||
43 |
(*By default only o_apply is built-in. But in the presence of eta-expansion |
|
44 |
this means that some terms displayed as (f o g) will be rewritten, and others |
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45 |
will not!*) |
|
46 |
Addsimps [o_def]; |
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47 |
||
1943 | 48 |
(*** Basic properties of shrK and newK ***) |
1934 | 49 |
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1943 | 50 |
(* invKey (shrK A) = shrK A *) |
51 |
bind_thm ("invKey_shrK", rewrite_rule [isSymKey_def] isSym_shrK); |
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1934 | 52 |
|
53 |
(* invKey (newK evs) = newK evs *) |
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54 |
bind_thm ("invKey_newK", rewrite_rule [isSymKey_def] isSym_newK); |
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1943 | 55 |
Addsimps [invKey_shrK, invKey_newK]; |
1934 | 56 |
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57 |
||
1993 | 58 |
(*Injectiveness and freshness of new keys and nonces*) |
2032 | 59 |
AddIffs [inj_shrK RS inj_eq, inj_newN RS inj_eq, inj_newK RS inj_eq]; |
1934 | 60 |
|
61 |
(** Rewrites should not refer to initState(Friend i) |
|
62 |
-- not in normal form! **) |
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63 |
||
1943 | 64 |
Addsimps [newK_neq_shrK, newK_neq_shrK RS not_sym]; |
1934 | 65 |
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2032 | 66 |
goal thy "Key (newK evs) ~: parts (initState lost B)"; |
67 |
by (agent.induct_tac "B" 1); |
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1934 | 68 |
by (Auto_tac ()); |
2032 | 69 |
qed "newK_notin_initState"; |
1934 | 70 |
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2032 | 71 |
goal thy "Nonce (newN evs) ~: parts (initState lost B)"; |
72 |
by (agent.induct_tac "B" 1); |
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1934 | 73 |
by (Auto_tac ()); |
2032 | 74 |
qed "newN_notin_initState"; |
1934 | 75 |
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2032 | 76 |
AddIffs [newK_notin_initState, newN_notin_initState]; |
1934 | 77 |
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2032 | 78 |
goalw thy [keysFor_def] "keysFor (parts (initState lost C)) = {}"; |
1934 | 79 |
by (agent.induct_tac "C" 1); |
1993 | 80 |
by (auto_tac (!claset addIs [range_eqI], !simpset)); |
1934 | 81 |
qed "keysFor_parts_initState"; |
82 |
Addsimps [keysFor_parts_initState]; |
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83 |
||
84 |
goalw thy [keysFor_def] "keysFor (Key``E) = {}"; |
|
85 |
by (Auto_tac ()); |
|
86 |
qed "keysFor_image_Key"; |
|
87 |
Addsimps [keysFor_image_Key]; |
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88 |
||
1943 | 89 |
goal thy "shrK A ~: newK``E"; |
1934 | 90 |
by (agent.induct_tac "A" 1); |
91 |
by (Auto_tac ()); |
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1943 | 92 |
qed "shrK_notin_image_newK"; |
93 |
Addsimps [shrK_notin_image_newK]; |
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1934 | 94 |
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2032 | 95 |
|
96 |
(*** Function "sees" ***) |
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97 |
||
98 |
goal thy |
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99 |
"!!evs. lost' <= lost ==> sees lost' A evs <= sees lost A evs"; |
|
100 |
by (list.induct_tac "evs" 1); |
|
101 |
by (agent.induct_tac "A" 1); |
|
102 |
by (event.induct_tac "a" 2); |
|
103 |
by (Auto_tac ()); |
|
104 |
qed "sees_mono"; |
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105 |
||
1964 | 106 |
(*Agents see their own shared keys!*) |
2032 | 107 |
goal thy "A ~= Spy --> Key (shrK A) : sees lost A evs"; |
1934 | 108 |
by (list.induct_tac "evs" 1); |
109 |
by (agent.induct_tac "A" 1); |
|
2032 | 110 |
by (Auto_tac ()); |
111 |
qed_spec_mp "sees_own_shrK"; |
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1934 | 112 |
|
2032 | 113 |
(*Spy sees shared keys of lost agents!*) |
2045
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parents:
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|
114 |
goal thy "!!A. A: lost ==> Key (shrK A) : sees lost Spy evs"; |
2032 | 115 |
by (list.induct_tac "evs" 1); |
116 |
by (Auto_tac()); |
|
2045
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
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parents:
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|
117 |
qed "Spy_sees_lost"; |
1934 | 118 |
|
2045
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
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parents:
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|
119 |
AddSIs [sees_own_shrK, Spy_sees_lost]; |
2032 | 120 |
|
121 |
(** Specialized rewrite rules for (sees lost A (Says...#evs)) **) |
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2026
0df5a96bf77e
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|
122 |
|
2078 | 123 |
goal thy "sees lost B (Says A B X # evs) = insert X (sees lost B evs)"; |
1934 | 124 |
by (Simp_tac 1); |
125 |
qed "sees_own"; |
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126 |
||
2078 | 127 |
goal thy "!!A. Server ~= B ==> \ |
128 |
\ sees lost Server (Says A B X # evs) = sees lost Server evs"; |
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1934 | 129 |
by (Asm_simp_tac 1); |
130 |
qed "sees_Server"; |
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131 |
||
2078 | 132 |
goal thy "!!A. Friend i ~= B ==> \ |
133 |
\ sees lost (Friend i) (Says A B X # evs) = sees lost (Friend i) evs"; |
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1934 | 134 |
by (Asm_simp_tac 1); |
135 |
qed "sees_Friend"; |
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136 |
||
2032 | 137 |
goal thy "sees lost Spy (Says A B X # evs) = insert X (sees lost Spy evs)"; |
1934 | 138 |
by (Simp_tac 1); |
2032 | 139 |
qed "sees_Spy"; |
1934 | 140 |
|
2032 | 141 |
goal thy "sees lost A (Says A' B X # evs) <= insert X (sees lost A evs)"; |
1934 | 142 |
by (simp_tac (!simpset setloop split_tac [expand_if]) 1); |
143 |
by (Fast_tac 1); |
|
144 |
qed "sees_Says_subset_insert"; |
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145 |
||
2032 | 146 |
goal thy "sees lost A evs <= sees lost A (Says A' B X # evs)"; |
1934 | 147 |
by (simp_tac (!simpset setloop split_tac [expand_if]) 1); |
148 |
by (Fast_tac 1); |
|
149 |
qed "sees_subset_sees_Says"; |
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150 |
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2032 | 151 |
(*Pushing Unions into parts; one of the A's equals B, and thus sees lost Y*) |
152 |
goal thy "(UN A. parts (sees lost A (Says B C Y # evs))) = \ |
|
153 |
\ parts {Y} Un (UN A. parts (sees lost A evs))"; |
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1934 | 154 |
by (Step_tac 1); |
2032 | 155 |
by (etac rev_mp 1); (*for some reason, split_tac does not work on assumptions*) |
1934 | 156 |
val ss = (!simpset addsimps [parts_Un, sees_Cons] |
2032 | 157 |
setloop split_tac [expand_if]); |
1934 | 158 |
by (ALLGOALS (fast_tac (!claset addss ss))); |
159 |
qed "UN_parts_sees_Says"; |
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160 |
||
2032 | 161 |
goal thy "Says A B X : set_of_list evs --> X : sees lost Spy evs"; |
1934 | 162 |
by (list.induct_tac "evs" 1); |
163 |
by (Auto_tac ()); |
|
2032 | 164 |
qed_spec_mp "Says_imp_sees_Spy"; |
1934 | 165 |
|
2064 | 166 |
goal thy |
2078 | 167 |
"!!evs. [| Says A B (Crypt X (shrK C)) : set_of_list evs; C : lost |] \ |
2064 | 168 |
\ ==> X : analz (sees lost Spy evs)"; |
169 |
by (fast_tac (!claset addSDs [Says_imp_sees_Spy RS analz.Inj] |
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170 |
addss (!simpset)) 1); |
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171 |
qed "Says_Crypt_lost"; |
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172 |
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2072 | 173 |
goal thy |
174 |
"!!evs. [| Says A B (Crypt X (shrK C)) : set_of_list evs; \ |
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175 |
\ X ~: analz (sees lost Spy evs) |] \ |
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176 |
\ ==> C ~: lost"; |
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177 |
by (fast_tac (!claset addSDs [Says_imp_sees_Spy RS analz.Inj] |
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178 |
addss (!simpset)) 1); |
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179 |
qed "Says_Crypt_not_lost"; |
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180 |
||
2032 | 181 |
goal thy "initState lost C <= Key `` range shrK"; |
1934 | 182 |
by (agent.induct_tac "C" 1); |
183 |
by (Auto_tac ()); |
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184 |
qed "initState_subset"; |
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185 |
||
2032 | 186 |
goal thy "X : sees lost C evs --> \ |
2078 | 187 |
\ (EX A B. Says A B X : set_of_list evs) | (EX A. X = Key (shrK A))"; |
1934 | 188 |
by (list.induct_tac "evs" 1); |
189 |
by (ALLGOALS Asm_simp_tac); |
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190 |
by (fast_tac (!claset addDs [impOfSubs initState_subset]) 1); |
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2032 | 191 |
by (rtac conjI 1); |
1934 | 192 |
by (Fast_tac 2); |
193 |
by (event.induct_tac "a" 1); |
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194 |
by (ALLGOALS (asm_simp_tac (!simpset addsimps [mem_if]))); |
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195 |
by (ALLGOALS Fast_tac); |
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196 |
qed_spec_mp "seesD"; |
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197 |
||
2032 | 198 |
Addsimps [UN_parts_sees_Says, sees_own, sees_Server, sees_Friend, sees_Spy]; |
199 |
Delsimps [sees_Cons]; (**** NOTE REMOVAL -- laws above are cleaner ****) |
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1934 | 200 |
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201 |
||
2049
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
202 |
(** Power of the Spy **) |
1934 | 203 |
|
2049
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
204 |
(*The Spy can see more than anybody else, except for their initial state*) |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
205 |
goal thy "sees lost A evs <= initState lost A Un sees lost Spy evs"; |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
206 |
by (list.induct_tac "evs" 1); |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
207 |
by (event.induct_tac "a" 2); |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
208 |
by (ALLGOALS (fast_tac (!claset addDs [sees_Says_subset_insert RS subsetD] |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
209 |
addss (!simpset)))); |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
210 |
qed "sees_agent_subset_sees_Spy"; |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
211 |
|
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
212 |
(*The Spy can see more than anybody else who's lost their key!*) |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
213 |
goal thy "A: lost --> A ~= Server --> sees lost A evs <= sees lost Spy evs"; |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
214 |
by (list.induct_tac "evs" 1); |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
215 |
by (event.induct_tac "a" 2); |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
216 |
by (agent.induct_tac "A" 1); |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
217 |
by (auto_tac (!claset addDs [sees_Says_subset_insert RS subsetD], (!simpset))); |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
218 |
qed_spec_mp "sees_lost_agent_subset_sees_Spy"; |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
219 |
|
1934 | 220 |
|
221 |
(** Rewrites to push in Key and Crypt messages, so that other messages can |
|
222 |
be pulled out using the analz_insert rules **) |
|
223 |
||
224 |
fun insComm x y = read_instantiate_sg (sign_of thy) [("x",x), ("y",y)] |
|
225 |
insert_commute; |
|
226 |
||
227 |
val pushKeys = map (insComm "Key ?K") |
|
228 |
["Agent ?C", "Nonce ?N", "MPair ?X ?Y", "Crypt ?X ?K'"]; |
|
229 |
||
230 |
val pushCrypts = map (insComm "Crypt ?X ?K") |
|
231 |
["Agent ?C", "Nonce ?N", "MPair ?X' ?Y"]; |
|
232 |
||
233 |
(*Cannot be added with Addsimps -- we don't always want to re-order messages*) |
|
234 |
val pushes = pushKeys@pushCrypts; |
|
235 |
||
1943 | 236 |
val pushKey_newK = insComm "Key (newK ?evs)" "Key (shrK ?C)"; |
1934 | 237 |
|
238 |
||
1993 | 239 |
(*No premature instantiation of variables. For proving weak liveness.*) |
240 |
fun safe_solver prems = |
|
241 |
match_tac (TrueI::refl::prems) ORELSE' eq_assume_tac |
|
242 |
ORELSE' etac FalseE; |
|
243 |
||
2000 | 244 |
(*Analysis of Fake cases and of messages that forward unknown parts |
245 |
Abstraction over i is ESSENTIAL: it delays the dereferencing of claset*) |
|
2032 | 246 |
fun spy_analz_tac i = |
1993 | 247 |
SELECT_GOAL |
248 |
(EVERY [ (*push in occurrences of X...*) |
|
2032 | 249 |
(REPEAT o CHANGED) |
250 |
(res_inst_tac [("x1","X")] (insert_commute RS ssubst) 1), |
|
251 |
(*...allowing further simplifications*) |
|
252 |
simp_tac (!simpset setloop split_tac [expand_if]) 1, |
|
253 |
REPEAT (resolve_tac [impI,notI] 1), |
|
254 |
dtac (impOfSubs Fake_analz_insert) 1, |
|
255 |
eresolve_tac [asm_rl, synth.Inj] 1, |
|
256 |
Fast_tac 1, |
|
257 |
Asm_full_simp_tac 1, |
|
258 |
IF_UNSOLVED (deepen_tac (!claset addIs [impOfSubs analz_mono]) 0 1) |
|
259 |
]) i; |
|
1993 | 260 |
|
261 |
||
2032 | 262 |
(** Simplifying parts (insert X (sees lost A evs)) |
2045
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
263 |
= parts {X} Un parts (sees lost A evs) -- since general case loops*) |
2012 | 264 |
|
265 |
val parts_insert_sees = |
|
2045
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
266 |
parts_insert |> read_instantiate_sg (sign_of thy) |
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
267 |
[("H", "sees lost A evs")] |
2012 | 268 |
|> standard; |
2045
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
269 |
|
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
270 |
|
2049
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
271 |
(*** Specialized rewriting for analz_insert_Key_newK ***) |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
272 |
|
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
273 |
goal thy "!!K. newK evs = invKey K ==> newK evs = K"; |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
274 |
by (rtac (invKey_eq RS iffD1) 1); |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
275 |
by (Simp_tac 1); |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
276 |
val newK_invKey = result(); |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
277 |
|
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
278 |
AddSDs [newK_invKey]; |
2045
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
279 |
|
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
280 |
Delsimps [image_insert]; |
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
281 |
Addsimps [image_insert RS sym]; |
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
282 |
|
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
283 |
Delsimps [image_Un]; |
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
284 |
Addsimps [image_Un RS sym]; |
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
285 |
|
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
286 |
goal thy "insert (Key (newK x)) H = Key `` (newK``{x}) Un H"; |
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
287 |
by (Fast_tac 1); |
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
288 |
qed "insert_Key_singleton"; |
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
289 |
|
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
290 |
goal thy "insert (Key (f x)) (Key``(f``E) Un C) = \ |
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
291 |
\ Key `` (f `` (insert x E)) Un C"; |
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
292 |
by (Fast_tac 1); |
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
293 |
qed "insert_Key_image"; |
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
294 |
|
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
295 |
|
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
296 |
(*Lemma for the trivial direction of the if-and-only-if*) |
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
297 |
goal thy |
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
298 |
"!!evs. (Key K : analz (Key``nE Un H)) --> (K : nE | Key K : analz H) ==> \ |
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
299 |
\ (Key K : analz (Key``nE Un H)) = (K : nE | Key K : analz H)"; |
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
300 |
by (fast_tac (!claset addSEs [impOfSubs analz_mono]) 1); |
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
301 |
qed "analz_image_newK_lemma"; |
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
302 |
|
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
303 |