author | paulson |
Tue, 08 Oct 1996 10:28:02 +0200 | |
changeset 2072 | 6ac12b9478d5 |
parent 2064 | 5a5e508e2a2b |
child 2078 | b198b3d46fb4 |
permissions | -rw-r--r-- |
1934 | 1 |
(* Title: HOL/Auth/Message |
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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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Theory of Shared Keys (common to all symmetric-key protocols) |
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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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2072 | 12 |
Addsimps [de_Morgan_conj, de_Morgan_disj, not_all, not_ex]; |
13 |
(**Addsimps [all_conj_distrib, ex_disj_distrib];**) |
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14 |
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15 |
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2064 | 16 |
val prems = goal HOL.thy "[| P ==> Q(True); ~P ==> Q(False) |] ==> Q(P)"; |
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by (excluded_middle_tac "P" 1); |
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18 |
by (asm_simp_tac (!simpset addsimps prems) 1); |
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by (asm_simp_tac (!simpset addsimps prems) 1); |
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20 |
val expand_case = result(); |
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21 |
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22 |
fun expand_case_tac P i = |
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res_inst_tac [("P",P)] expand_case i THEN |
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Simp_tac (i+1) THEN |
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25 |
Simp_tac i; |
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26 |
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27 |
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1934 | 28 |
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2000 | 29 |
(*GOALS.ML??*) |
30 |
fun prlim n = (goals_limit:=n; pr()); |
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31 |
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1934 | 32 |
(*FUN.ML?? WE NEED A NOTION OF INVERSE IMAGE, OR GRAPH!!*) |
33 |
goal Set.thy "!!f. B <= range f = (B = f`` {x. f x: B})"; |
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34 |
by (fast_tac (!claset addEs [equalityE]) 1); |
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35 |
val subset_range_iff = result(); |
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36 |
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37 |
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open Shared; |
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1964 | 40 |
(*Holds because Friend is injective: thus cannot prove for all f*) |
41 |
goal thy "(Friend x : Friend``A) = (x:A)"; |
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42 |
by (Auto_tac()); |
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43 |
qed "Friend_image_eq"; |
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44 |
Addsimps [Friend_image_eq]; |
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45 |
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1934 | 46 |
Addsimps [Un_insert_left, Un_insert_right]; |
47 |
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48 |
(*By default only o_apply is built-in. But in the presence of eta-expansion |
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49 |
this means that some terms displayed as (f o g) will be rewritten, and others |
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50 |
will not!*) |
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51 |
Addsimps [o_def]; |
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52 |
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1943 | 53 |
(*** Basic properties of shrK and newK ***) |
1934 | 54 |
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1943 | 55 |
(* invKey (shrK A) = shrK A *) |
56 |
bind_thm ("invKey_shrK", rewrite_rule [isSymKey_def] isSym_shrK); |
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1934 | 57 |
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58 |
(* invKey (newK evs) = newK evs *) |
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59 |
bind_thm ("invKey_newK", rewrite_rule [isSymKey_def] isSym_newK); |
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1943 | 60 |
Addsimps [invKey_shrK, invKey_newK]; |
1934 | 61 |
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62 |
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1993 | 63 |
(*Injectiveness and freshness of new keys and nonces*) |
2032 | 64 |
AddIffs [inj_shrK RS inj_eq, inj_newN RS inj_eq, inj_newK RS inj_eq]; |
1934 | 65 |
|
66 |
(** Rewrites should not refer to initState(Friend i) |
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67 |
-- not in normal form! **) |
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68 |
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1943 | 69 |
Addsimps [newK_neq_shrK, newK_neq_shrK RS not_sym]; |
1934 | 70 |
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2032 | 71 |
goal thy "Key (newK evs) ~: parts (initState lost B)"; |
72 |
by (agent.induct_tac "B" 1); |
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1934 | 73 |
by (Auto_tac ()); |
2032 | 74 |
qed "newK_notin_initState"; |
1934 | 75 |
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2032 | 76 |
goal thy "Nonce (newN evs) ~: parts (initState lost B)"; |
77 |
by (agent.induct_tac "B" 1); |
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1934 | 78 |
by (Auto_tac ()); |
2032 | 79 |
qed "newN_notin_initState"; |
1934 | 80 |
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2032 | 81 |
AddIffs [newK_notin_initState, newN_notin_initState]; |
1934 | 82 |
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2032 | 83 |
goalw thy [keysFor_def] "keysFor (parts (initState lost C)) = {}"; |
1934 | 84 |
by (agent.induct_tac "C" 1); |
1993 | 85 |
by (auto_tac (!claset addIs [range_eqI], !simpset)); |
1934 | 86 |
qed "keysFor_parts_initState"; |
87 |
Addsimps [keysFor_parts_initState]; |
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88 |
||
89 |
goalw thy [keysFor_def] "keysFor (Key``E) = {}"; |
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by (Auto_tac ()); |
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91 |
qed "keysFor_image_Key"; |
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92 |
Addsimps [keysFor_image_Key]; |
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93 |
||
1943 | 94 |
goal thy "shrK A ~: newK``E"; |
1934 | 95 |
by (agent.induct_tac "A" 1); |
96 |
by (Auto_tac ()); |
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1943 | 97 |
qed "shrK_notin_image_newK"; |
98 |
Addsimps [shrK_notin_image_newK]; |
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1934 | 99 |
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2032 | 100 |
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101 |
(*** Function "sees" ***) |
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102 |
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103 |
goal thy |
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104 |
"!!evs. lost' <= lost ==> sees lost' A evs <= sees lost A evs"; |
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by (list.induct_tac "evs" 1); |
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by (agent.induct_tac "A" 1); |
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107 |
by (event.induct_tac "a" 2); |
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108 |
by (Auto_tac ()); |
|
109 |
qed "sees_mono"; |
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110 |
||
1964 | 111 |
(*Agents see their own shared keys!*) |
2032 | 112 |
goal thy "A ~= Spy --> Key (shrK A) : sees lost A evs"; |
1934 | 113 |
by (list.induct_tac "evs" 1); |
114 |
by (agent.induct_tac "A" 1); |
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2032 | 115 |
by (Auto_tac ()); |
116 |
qed_spec_mp "sees_own_shrK"; |
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1934 | 117 |
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2032 | 118 |
(*Spy sees shared keys of lost agents!*) |
2045
ae1030e66745
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2032
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119 |
goal thy "!!A. A: lost ==> Key (shrK A) : sees lost Spy evs"; |
2032 | 120 |
by (list.induct_tac "evs" 1); |
121 |
by (Auto_tac()); |
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2045
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
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2032
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|
122 |
qed "Spy_sees_lost"; |
1934 | 123 |
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2045
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
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124 |
AddSIs [sees_own_shrK, Spy_sees_lost]; |
2032 | 125 |
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126 |
(** Specialized rewrite rules for (sees lost A (Says...#evs)) **) |
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2026
0df5a96bf77e
Last working version prior to introduction of "lost"
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127 |
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2032 | 128 |
goal thy "sees lost A (Says A B X # evs) = insert X (sees lost A evs)"; |
1934 | 129 |
by (Simp_tac 1); |
130 |
qed "sees_own"; |
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131 |
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132 |
goal thy "!!A. Server ~= A ==> \ |
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2032 | 133 |
\ sees lost Server (Says A B X # evs) = sees lost Server evs"; |
1934 | 134 |
by (Asm_simp_tac 1); |
135 |
qed "sees_Server"; |
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136 |
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137 |
goal thy "!!A. Friend i ~= A ==> \ |
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2032 | 138 |
\ sees lost (Friend i) (Says A B X # evs) = sees lost (Friend i) evs"; |
1934 | 139 |
by (Asm_simp_tac 1); |
140 |
qed "sees_Friend"; |
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141 |
||
2032 | 142 |
goal thy "sees lost Spy (Says A B X # evs) = insert X (sees lost Spy evs)"; |
1934 | 143 |
by (Simp_tac 1); |
2032 | 144 |
qed "sees_Spy"; |
1934 | 145 |
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2032 | 146 |
goal thy "sees lost A (Says A' B X # evs) <= insert X (sees lost A evs)"; |
1934 | 147 |
by (simp_tac (!simpset setloop split_tac [expand_if]) 1); |
148 |
by (Fast_tac 1); |
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149 |
qed "sees_Says_subset_insert"; |
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150 |
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2032 | 151 |
goal thy "sees lost A evs <= sees lost A (Says A' B X # evs)"; |
1934 | 152 |
by (simp_tac (!simpset setloop split_tac [expand_if]) 1); |
153 |
by (Fast_tac 1); |
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154 |
qed "sees_subset_sees_Says"; |
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155 |
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2032 | 156 |
(*Pushing Unions into parts; one of the A's equals B, and thus sees lost Y*) |
157 |
goal thy "(UN A. parts (sees lost A (Says B C Y # evs))) = \ |
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158 |
\ parts {Y} Un (UN A. parts (sees lost A evs))"; |
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1934 | 159 |
by (Step_tac 1); |
2032 | 160 |
by (etac rev_mp 1); (*for some reason, split_tac does not work on assumptions*) |
1934 | 161 |
val ss = (!simpset addsimps [parts_Un, sees_Cons] |
2032 | 162 |
setloop split_tac [expand_if]); |
1934 | 163 |
by (ALLGOALS (fast_tac (!claset addss ss))); |
164 |
qed "UN_parts_sees_Says"; |
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165 |
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2032 | 166 |
goal thy "Says A B X : set_of_list evs --> X : sees lost Spy evs"; |
1934 | 167 |
by (list.induct_tac "evs" 1); |
168 |
by (Auto_tac ()); |
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2032 | 169 |
qed_spec_mp "Says_imp_sees_Spy"; |
1934 | 170 |
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2064 | 171 |
goal thy |
172 |
"!!evs. [| Says A B (Crypt X (shrK C)) : set_of_list evs; \ |
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173 |
\ C : lost |] \ |
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174 |
\ ==> X : analz (sees lost Spy evs)"; |
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175 |
by (fast_tac (!claset addSDs [Says_imp_sees_Spy RS analz.Inj] |
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176 |
addss (!simpset)) 1); |
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177 |
qed "Says_Crypt_lost"; |
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178 |
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2072 | 179 |
goal thy |
180 |
"!!evs. [| Says A B (Crypt X (shrK C)) : set_of_list evs; \ |
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181 |
\ X ~: analz (sees lost Spy evs) |] \ |
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182 |
\ ==> C ~: lost"; |
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183 |
by (fast_tac (!claset addSDs [Says_imp_sees_Spy RS analz.Inj] |
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184 |
addss (!simpset)) 1); |
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185 |
qed "Says_Crypt_not_lost"; |
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186 |
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2032 | 187 |
goal thy "initState lost C <= Key `` range shrK"; |
1934 | 188 |
by (agent.induct_tac "C" 1); |
189 |
by (Auto_tac ()); |
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190 |
qed "initState_subset"; |
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191 |
||
2032 | 192 |
goal thy "X : sees lost C evs --> \ |
1934 | 193 |
\ (EX A B. Says A B X : set_of_list evs) | \ |
1943 | 194 |
\ (EX A. X = Key (shrK A))"; |
1934 | 195 |
by (list.induct_tac "evs" 1); |
196 |
by (ALLGOALS Asm_simp_tac); |
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197 |
by (fast_tac (!claset addDs [impOfSubs initState_subset]) 1); |
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2032 | 198 |
by (rtac conjI 1); |
1934 | 199 |
by (Fast_tac 2); |
200 |
by (event.induct_tac "a" 1); |
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201 |
by (ALLGOALS (asm_simp_tac (!simpset addsimps [mem_if]))); |
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202 |
by (ALLGOALS Fast_tac); |
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203 |
qed_spec_mp "seesD"; |
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204 |
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2032 | 205 |
Addsimps [UN_parts_sees_Says, sees_own, sees_Server, sees_Friend, sees_Spy]; |
206 |
Delsimps [sees_Cons]; (**** NOTE REMOVAL -- laws above are cleaner ****) |
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1934 | 207 |
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208 |
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2049
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
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2045
diff
changeset
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209 |
(** Power of the Spy **) |
1934 | 210 |
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2049
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
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|
211 |
(*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
|
212 |
goal thy "sees lost A evs <= initState lost A Un sees lost Spy evs"; |
d3b93e1765bc
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paulson
parents:
2045
diff
changeset
|
213 |
by (list.induct_tac "evs" 1); |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
214 |
by (event.induct_tac "a" 2); |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
215 |
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
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|
216 |
addss (!simpset)))); |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
217 |
qed "sees_agent_subset_sees_Spy"; |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
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2045
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|
218 |
|
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
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|
219 |
(*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
|
220 |
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
|
221 |
by (list.induct_tac "evs" 1); |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
222 |
by (event.induct_tac "a" 2); |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
223 |
by (agent.induct_tac "A" 1); |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
224 |
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
|
225 |
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
|
226 |
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1934 | 227 |
|
228 |
(** Rewrites to push in Key and Crypt messages, so that other messages can |
|
229 |
be pulled out using the analz_insert rules **) |
|
230 |
||
231 |
fun insComm x y = read_instantiate_sg (sign_of thy) [("x",x), ("y",y)] |
|
232 |
insert_commute; |
|
233 |
||
234 |
val pushKeys = map (insComm "Key ?K") |
|
235 |
["Agent ?C", "Nonce ?N", "MPair ?X ?Y", "Crypt ?X ?K'"]; |
|
236 |
||
237 |
val pushCrypts = map (insComm "Crypt ?X ?K") |
|
238 |
["Agent ?C", "Nonce ?N", "MPair ?X' ?Y"]; |
|
239 |
||
240 |
(*Cannot be added with Addsimps -- we don't always want to re-order messages*) |
|
241 |
val pushes = pushKeys@pushCrypts; |
|
242 |
||
1943 | 243 |
val pushKey_newK = insComm "Key (newK ?evs)" "Key (shrK ?C)"; |
1934 | 244 |
|
245 |
||
1993 | 246 |
(*No premature instantiation of variables. For proving weak liveness.*) |
247 |
fun safe_solver prems = |
|
248 |
match_tac (TrueI::refl::prems) ORELSE' eq_assume_tac |
|
249 |
ORELSE' etac FalseE; |
|
250 |
||
2000 | 251 |
(*Analysis of Fake cases and of messages that forward unknown parts |
252 |
Abstraction over i is ESSENTIAL: it delays the dereferencing of claset*) |
|
2032 | 253 |
fun spy_analz_tac i = |
1993 | 254 |
SELECT_GOAL |
255 |
(EVERY [ (*push in occurrences of X...*) |
|
2032 | 256 |
(REPEAT o CHANGED) |
257 |
(res_inst_tac [("x1","X")] (insert_commute RS ssubst) 1), |
|
258 |
(*...allowing further simplifications*) |
|
259 |
simp_tac (!simpset setloop split_tac [expand_if]) 1, |
|
260 |
REPEAT (resolve_tac [impI,notI] 1), |
|
261 |
dtac (impOfSubs Fake_analz_insert) 1, |
|
262 |
eresolve_tac [asm_rl, synth.Inj] 1, |
|
263 |
Fast_tac 1, |
|
264 |
Asm_full_simp_tac 1, |
|
265 |
IF_UNSOLVED (deepen_tac (!claset addIs [impOfSubs analz_mono]) 0 1) |
|
266 |
]) i; |
|
1993 | 267 |
|
268 |
||
2032 | 269 |
(** 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
|
270 |
= parts {X} Un parts (sees lost A evs) -- since general case loops*) |
2012 | 271 |
|
272 |
val parts_insert_sees = |
|
2045
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
273 |
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
|
274 |
[("H", "sees lost A evs")] |
2012 | 275 |
|> standard; |
2045
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
276 |
|
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
277 |
|
2049
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
278 |
(*** Specialized rewriting for analz_insert_Key_newK ***) |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
279 |
|
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
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|
280 |
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
|
281 |
by (rtac (invKey_eq RS iffD1) 1); |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
282 |
by (Simp_tac 1); |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
283 |
val newK_invKey = result(); |
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
diff
changeset
|
284 |
|
d3b93e1765bc
Moved sees_lost_agent_subset_sees_Spy to common file
paulson
parents:
2045
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changeset
|
285 |
AddSDs [newK_invKey]; |
2045
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
286 |
|
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
287 |
Delsimps [image_insert]; |
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
288 |
Addsimps [image_insert RS sym]; |
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 |
Delsimps [image_Un]; |
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
291 |
Addsimps [image_Un RS sym]; |
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
292 |
|
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
293 |
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
|
294 |
by (Fast_tac 1); |
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
295 |
qed "insert_Key_singleton"; |
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
296 |
|
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
297 |
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
|
298 |
\ 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
|
299 |
by (Fast_tac 1); |
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
300 |
qed "insert_Key_image"; |
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
301 |
|
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 |
(*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
|
304 |
goal thy |
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
305 |
"!!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
|
306 |
\ (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
|
307 |
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
|
308 |
qed "analz_image_newK_lemma"; |
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
309 |
|
ae1030e66745
Removed some dead wood. Transferred lemmas used to prove analz_image_newK
paulson
parents:
2032
diff
changeset
|
310 |