author | wenzelm |
Wed, 14 Jun 2000 17:59:53 +0200 | |
changeset 9066 | b1e874e38dab |
parent 9018 | b16bc0b5ad21 |
child 9342 | d66f25a206b4 |
permissions | -rw-r--r-- |
3981 | 1 |
(* Title: HOL/Map.ML |
2 |
ID: $Id$ |
|
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Author: Tobias Nipkow |
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4 |
Copyright 1997 TU Muenchen |
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Map lemmas |
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*) |
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5300 | 9 |
section "empty"; |
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||
5069 | 11 |
Goalw [empty_def] "empty k = None"; |
4423 | 12 |
by (Simp_tac 1); |
3981 | 13 |
qed "empty_def2"; |
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Addsimps [empty_def2]; |
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15 |
||
5300 | 16 |
|
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section "map_upd"; |
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18 |
||
8529 | 19 |
Goal "t k = Some x ==> t(k|->x) = t"; |
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by (rtac ext 1); |
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by (Asm_simp_tac 1); |
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22 |
qed "map_upd_triv"; |
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5300 | 23 |
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24 |
Goalw [image_def] "finite (range f) ==> finite (range (f(a|->b)))"; |
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by (full_simp_tac (simpset() addsimps [full_SetCompr_eq]) 1); |
8254 | 26 |
by (rtac finite_subset 1); |
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by (assume_tac 2); |
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7961
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by Auto_tac; |
837a6b515005
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qed "finite_range_updI"; |
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30 |
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5300 | 31 |
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section "map_upds"; |
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3981 | 33 |
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5300 | 34 |
Goal "a ~: set as --> (!m bs. (m(a|->b)(as[|->]bs)) = (m(as[|->]bs)(a|->b)))"; |
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by (induct_tac "as" 1); |
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by (auto_tac (claset(), simpset() delsimps[fun_upd_apply])); |
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by (REPEAT(dtac spec 1)); |
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by (rotate_tac ~1 1); |
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by (etac subst 1); |
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by (etac (fun_upd_twist RS subst) 1); |
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by (rtac refl 1); |
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qed_spec_mp "map_upds_twist"; |
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Addsimps [map_upds_twist]; |
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45 |
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section "chg_map"; |
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8529 | 48 |
Goalw [chg_map_def] "m a = None ==> chg_map f a m = m"; |
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by Auto_tac; |
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qed "chg_map_new"; |
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51 |
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Goalw [chg_map_def] "m a = Some b ==> chg_map f a m = m(a|->f b)"; |
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by Auto_tac; |
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qed "chg_map_upd"; |
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5300 | 56 |
Addsimps[chg_map_new, chg_map_upd]; |
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7958 | 59 |
section "map_of"; |
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Goal "map_of xs k = Some y --> (k,y):set xs"; |
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by (induct_tac "xs" 1); |
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9018 | 63 |
by Auto_tac; |
7958 | 64 |
qed_spec_mp "map_of_SomeD"; |
65 |
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9018 | 66 |
Goal "inj f ==> map_of t k = Some x --> \ |
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\ map_of (map (split (%k. Pair (f k))) t) (f k) = Some x"; |
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by (induct_tac "t" 1); |
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by (auto_tac (claset(),simpset()addsimps[inj_eq])); |
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qed_spec_mp "map_of_mapk_SomeI"; |
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Goal "(k, x) : set l --> (? x. map_of l k = Some x)"; |
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by (induct_tac "l" 1); |
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by Auto_tac; |
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qed_spec_mp "weak_map_of_SomeI"; |
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7958 | 77 |
Goal "[| map_of xs k = Some z; P z |] ==> map_of [(k,z):xs . P z] k = Some z"; |
7961 | 78 |
by (rtac mp 1); |
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by (assume_tac 2); |
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7958 | 80 |
by (etac thin_rl 1); |
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by (induct_tac "xs" 1); |
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by Auto_tac; |
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qed "map_of_filter_in"; |
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84 |
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7961
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changeset
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85 |
Goal "finite (range (map_of l))"; |
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86 |
by (induct_tac "l" 1); |
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by (ALLGOALS (simp_tac (simpset() addsimps [image_constant]))); |
8254 | 88 |
by (rtac finite_subset 1); |
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by (assume_tac 2); |
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90 |
by Auto_tac; |
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qed "finite_range_map_of"; |
7958 | 92 |
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9018 | 93 |
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7958 | 94 |
section "option_map related"; |
4990 | 95 |
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8529 | 96 |
Goal "option_map f o empty = empty"; |
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by (rtac ext 1); |
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by (Simp_tac 1); |
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qed "option_map_o_empty"; |
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4990 | 100 |
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8529 | 101 |
Goal "option_map f o m(a|->b) = (option_map f o m)(a|->f b)"; |
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by (rtac ext 1); |
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by (Simp_tac 1); |
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qed "option_map_o_map_upd"; |
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5195 | 106 |
Addsimps[option_map_o_empty, option_map_o_map_upd]; |
4883 | 107 |
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3981 | 108 |
section "++"; |
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5069 | 110 |
Goalw [override_def] "m ++ empty = m"; |
4423 | 111 |
by (Simp_tac 1); |
3981 | 112 |
qed "override_empty"; |
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Addsimps [override_empty]; |
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5069 | 115 |
Goalw [override_def] "empty ++ m = m"; |
4423 | 116 |
by (Simp_tac 1); |
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by (rtac ext 1); |
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5183 | 118 |
by (split_tac [option.split] 1); |
4423 | 119 |
by (Simp_tac 1); |
3981 | 120 |
qed "empty_override"; |
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Addsimps [empty_override]; |
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5069 | 123 |
Goalw [override_def] |
3981 | 124 |
"((m ++ n) k = Some x) = (n k = Some x | n k = None & m k = Some x)"; |
5183 | 125 |
by (simp_tac (simpset() addsplits [option.split]) 1); |
3981 | 126 |
qed_spec_mp "override_Some_iff"; |
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4926 | 128 |
bind_thm ("override_SomeD", standard(override_Some_iff RS iffD1)); |
5444 | 129 |
AddSDs[override_SomeD]; |
3981 | 130 |
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7958 | 131 |
Goal "!!xx. n k = Some xx ==> (m ++ n) k = Some xx"; |
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by (stac override_Some_iff 1); |
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by (Fast_tac 1); |
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qed "override_find_right"; |
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Addsimps[override_find_right]; |
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5444 | 137 |
Goalw [override_def] "((m ++ n) k = None) = (n k = None & m k = None)"; |
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by (simp_tac (simpset() addsplits [option.split]) 1); |
3981 | 139 |
qed "override_None"; |
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AddIffs [override_None]; |
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141 |
||
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parents:
7961
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Goalw [override_def] "f ++ g(x|->y) = (f ++ g)(x|->y)"; |
8254 | 143 |
by (rtac ext 1); |
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by (auto_tac (claset_of Map.thy, simpset_of Map.thy)); |
837a6b515005
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qed "override_upd"; |
837a6b515005
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Addsimps[override_upd]; |
837a6b515005
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7961
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5444 | 148 |
Goalw [override_def] "map_of ys ++ map_of xs = map_of (xs@ys)"; |
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by (rtac sym 1); |
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4423 | 150 |
by (induct_tac "xs" 1); |
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by (Simp_tac 1); |
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by (rtac ext 1); |
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5183 | 153 |
by (asm_simp_tac (simpset() addsplits [option.split]) 1); |
5444 | 154 |
qed "map_of_override"; |
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Addsimps [map_of_override]; |
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3981 | 156 |
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Delsimps[fun_upd_apply]; |
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Goal "finite (range f) ==> finite (range (f ++ map_of l))"; |
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by (induct_tac "l" 1); |
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160 |
by Auto_tac; |
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161 |
by (fold_goals_tac [empty_def]); |
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162 |
by (Asm_simp_tac 1); |
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163 |
by (etac finite_range_updI 1); |
837a6b515005
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164 |
qed "finite_range_map_of_override"; |
837a6b515005
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165 |
Addsimps [fun_upd_apply]; |
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166 |
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5300 | 167 |
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3981 | 168 |
section "dom"; |
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8259 | 170 |
Goalw [dom_def] "m a = Some b ==> a : dom m"; |
171 |
by Auto_tac; |
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172 |
qed "domI"; |
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173 |
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174 |
Goalw [dom_def] "a : dom m ==> ? b. m a = Some b"; |
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by Auto_tac; |
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qed "domD"; |
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AddSDs [domD]; |
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178 |
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5069 | 179 |
Goalw [dom_def] "dom empty = {}"; |
4423 | 180 |
by (Simp_tac 1); |
3981 | 181 |
qed "dom_empty"; |
182 |
Addsimps [dom_empty]; |
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183 |
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5300 | 184 |
Goalw [dom_def] "dom(m(a|->b)) = insert a (dom m)"; |
4686 | 185 |
by (Simp_tac 1); |
4423 | 186 |
by (Blast_tac 1); |
5195 | 187 |
qed "dom_map_upd"; |
188 |
Addsimps [dom_map_upd]; |
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3981 | 189 |
|
8529 | 190 |
Goalw [dom_def] "finite (dom (map_of l))"; |
191 |
by (induct_tac "l" 1); |
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192 |
by (auto_tac (claset(), |
|
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simpset() addsimps [insert_Collect RS sym])); |
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qed "finite_dom_map_of"; |
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4883 | 195 |
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5069 | 196 |
Goalw [dom_def] "dom(m++n) = dom n Un dom m"; |
8529 | 197 |
by Auto_tac; |
3981 | 198 |
qed "dom_override"; |
199 |
Addsimps [dom_override]; |
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200 |
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201 |
section "ran"; |
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202 |
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5069 | 203 |
Goalw [ran_def] "ran empty = {}"; |
4423 | 204 |
by (Simp_tac 1); |
3981 | 205 |
qed "ran_empty"; |
206 |
Addsimps [ran_empty]; |
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4883 | 207 |
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7961
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208 |
Goalw [ran_def] "ran (%u. None) = {}"; |
837a6b515005
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209 |
by Auto_tac; |
837a6b515005
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7961
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|
210 |
qed "ran_empty'"; |
837a6b515005
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parents:
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changeset
|
211 |
Addsimps[ran_empty']; |
837a6b515005
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oheimb
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7961
diff
changeset
|
212 |
|
5300 | 213 |
Goalw [ran_def] "m a = None ==> ran(m(a|->b)) = insert b (ran m)"; |
4883 | 214 |
by Auto_tac; |
215 |
by (subgoal_tac "~(aa = a)" 1); |
|
216 |
by Auto_tac; |
|
5195 | 217 |
qed "ran_map_upd"; |
218 |
Addsimps [ran_map_upd]; |