# Theory JBasis

theory JBasis
imports Transitive_Closure_Table Eisbach
(*  Title:      HOL/MicroJava/J/JBasis.thy
Author:     David von Oheimb, TU Muenchen
*)

chapter ‹Java Source Language \label{cha:j}›

section ‹Some Auxiliary Definitions›

theory JBasis
imports
Main
"HOL-Library.Transitive_Closure_Table"
"HOL-Eisbach.Eisbach"
begin

lemmas [simp] = Let_def

subsection "unique"

definition unique :: "('a × 'b) list => bool" where
"unique == distinct ∘ map fst"

lemma fst_in_set_lemma: "(x, y) : set xys ==> x : fst  set xys"
by (induct xys) auto

lemma unique_Nil [simp]: "unique []"

lemma unique_Cons [simp]: "unique ((x,y)#l) = (unique l & (!y. (x,y) ~: set l))"
by (auto simp: unique_def dest: fst_in_set_lemma)

lemma unique_append: "unique l' ==> unique l ==>
(!(x,y):set l. !(x',y'):set l'. x' ~= x) ==> unique (l @ l')"
by (induct l) (auto dest: fst_in_set_lemma)

lemma unique_map_inj: "unique l ==> inj f ==> unique (map (%(k,x). (f k, g k x)) l)"
by (induct l) (auto dest: fst_in_set_lemma simp add: inj_eq)

lemma map_of_SomeI: "unique l ==> (k, x) : set l ==> map_of l k = Some x"
by (induct l) auto

lemma Ball_set_table: "(∀(x,y)∈set l. P x y) ==> (∀x. ∀y. map_of l x = Some y --> P x y)"
by (induct l) auto

lemma table_of_remap_SomeD:
"map_of (map (λ((k,k'),x). (k,(k',x))) t) k = Some (k',x) ==>
map_of t (k, k') = Some x"
by (atomize (full), induct t) auto

end
`