section‹General uiity lemms\close
java.lang.StringIndexOutOfBoundsException: Range [14, 12) out of bounds for length 25
‹
is a potpourri of various lemmas not specific to our project. Some of them could very well be included in the default Isabelle library. ›
‹
about the ‹single_valued› predicate. ›
single_valued_empty[simp]:"single_valued {}"
(rule single_valuedI) auto
single_valued_insert:
assumes "single_valued rel"
and "∧ x y . [(x,y) ∈ rel; x=a]==> y = b"
shows "single_valued (insert (a,b) rel)"
assms
(auto intro:single_valuedI dest:single_valuedD)
‹
about ‹ran›, the range of a finite map. ›
ran_upd: "ran (m (k ↦ v)) ⊆ ran m ∪ {v}"
ran_def by auto
ran_map_of: "ran (map_of xs) ⊆ snd ` set xs"
by (induct xs)(auto simp del:fun_upd_apply dest: ran_upd[THEN subsetD])
ran_concat: "ran (m1 ++ m2) ⊆ ran m1 ∪ ran m2"
ran_def
auto
ran_upds:
assumes eq_length: "length ks = length vs"
shows "ran (map_upds m ks vs) ⊆ ran m ∪ set vs"
-
have "ran (map_upds m ks vs) ⊆ ran (m++map_of (rev (zip ks vs)))"
unfolding map_upds_def by simp
also have "…⊆ ran m ∪ ran (map_of (rev (zip ks vs)))" by (rule ran_concat)
also have "…⊆ ran m ∪ snd ` set (rev (zip ks vs))"
by (intro Un_mono[of "ran m" "ran m"] subset_refl ran_map_of)
also have "…⊆ ran m ∪ set vs"
by (auto intro:Un_mono[of "ran m" "ran m"] subset_refl simp del:set_map simp add:set_map[THEN sym] map_snd_zip[OF eq_length])
finally show ?thesis .
ran_upd_mem[simp]: "v ∈ ran (m (k ↦ v))"
ran_def by auto
‹
about ‹map›, ‹zip› and ‹fst›/‹snd› ›
map_fst_zip: "length xs = length ys ==> map fst (zip xs ys) = xs"
(induct xs ys rule:list_induct2) by auto
map_snd_zip: "length xs = length ys ==> map snd (zip xs ys) = ys"
(induct xs ys rule:list_induct2) by auto
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