Theory Auxiliary

(*  Title:      Jinja/Common/Basis.thy

    Author:     David von Oheimb, Tobias Nipkow
    Copyright   1999 TU Muenchen
*)

header {* 
  \chapter{Jinja Source Language}\label{cha:j}
  \isaheader{Auxiliary Definitions}
*}

theory Auxiliary imports Main begin
(* FIXME move and possibly turn into a general simproc *)
lemma nat_add_max_le[simp]:
  "((n::nat) + max i j ≤ m) = (n + i ≤ m ∧ n + j ≤ m)"
 (*<*)by arith(*>*)

lemma Suc_add_max_le[simp]:
  "(Suc(n + max i j) ≤ m) = (Suc(n + i) ≤ m ∧ Suc(n + j) ≤ m)"
(*<*)by arith(*>*)


notation Some  ("(⌊_⌋)")

(*<*)
declare
 option.splits[split]
 Let_def[simp]
 subset_insertI2 [simp]
 Cons_eq_map_conv [iff]
(*>*)


section {*@{text distinct_fst}*}
 
definition distinct_fst  :: "('a × 'b) list => bool"
where
  "distinct_fst  ≡  distinct o map fst"

lemma distinct_fst_Nil [simp]:
  "distinct_fst []"
 (*<*)
apply (unfold distinct_fst_def)
apply (simp (no_asm))
done
(*>*)

lemma distinct_fst_Cons [simp]:
  "distinct_fst ((k,x)#kxs) = (distinct_fst kxs ∧ (∀y. (k,y) ∉ set kxs))"
(*<*)
apply (unfold distinct_fst_def)
apply (auto simp:image_def)
done
(*>*)
(*
lemma distinct_fst_append:
 "[| distinct_fst kxs'; distinct_fst kxs; ∀(k,x) ∈ set kxs. ∀(k',x') ∈ set kxs'. k' ≠ k |]
  ==> distinct_fst(kxs @ kxs')"
by (induct kxs) (auto dest: fst_in_set_lemma)

lemma distinct_fst_map_inj:
  "[| distinct_fst kxs; inj f |] ==> distinct_fst (map (λ(k,x). (f k, g k x)) kxs)"
by (induct kxs) (auto dest: fst_in_set_lemma simp: inj_eq)
*)

lemma map_of_SomeI:
  "[| distinct_fst kxs; (k,x) ∈ set kxs |] ==> map_of kxs k = Some x"
(*<*)by (induct kxs) (auto simp:fun_upd_apply)(*>*)


section {* Using @{term list_all2} for relations *}

definition fun_of :: "('a × 'b) set => 'a => 'b => bool"
where
  "fun_of S ≡ λx y. (x,y) ∈ S"

text {* Convenience lemmas *}
(*<*)
declare fun_of_def [simp]
(*>*)
lemma rel_list_all2_Cons [iff]:
  "list_all2 (fun_of S) (x#xs) (y#ys) = 
   ((x,y) ∈ S ∧ list_all2 (fun_of S) xs ys)"
  (*<*)by simp(*>*)

lemma rel_list_all2_Cons1:
  "list_all2 (fun_of S) (x#xs) ys = 
  (∃z zs. ys = z#zs ∧ (x,z) ∈ S ∧ list_all2 (fun_of S) xs zs)"
  (*<*)by (cases ys) auto(*>*)

lemma rel_list_all2_Cons2:
  "list_all2 (fun_of S) xs (y#ys) = 
  (∃z zs. xs = z#zs ∧ (z,y) ∈ S ∧ list_all2 (fun_of S) zs ys)"
  (*<*)by (cases xs) auto(*>*)

lemma rel_list_all2_refl:
  "(!!x. (x,x) ∈ S) ==> list_all2 (fun_of S) xs xs"
  (*<*)by (simp add: list_all2_refl)(*>*)

lemma rel_list_all2_antisym:
  "[| (!!x y. [|(x,y) ∈ S; (y,x) ∈ T|] ==> x = y); 
     list_all2 (fun_of S) xs ys; list_all2 (fun_of T) ys xs |] ==> xs = ys"
  (*<*)by (rule list_all2_antisym) auto(*>*)

lemma rel_list_all2_trans: 
  "[| !!a b c. [|(a,b) ∈ R; (b,c) ∈ S|] ==> (a,c) ∈ T;
    list_all2 (fun_of R) as bs; list_all2 (fun_of S) bs cs|] 
  ==> list_all2 (fun_of T) as cs"
  (*<*)by (rule list_all2_trans) auto(*>*)

lemma rel_list_all2_update_cong:
  "[| i<size xs; list_all2 (fun_of S) xs ys; (x,y) ∈ S |] 
  ==> list_all2 (fun_of S) (xs[i:=x]) (ys[i:=y])"
  (*<*)by (simp add: list_all2_update_cong)(*>*)

lemma rel_list_all2_nthD:
  "[| list_all2 (fun_of S) xs ys; p < size xs |] ==> (xs!p,ys!p) ∈ S"
  (*<*)by (drule list_all2_nthD) auto(*>*)

lemma rel_list_all2I:
  "[| length a = length b; !!n. n < length a ==> (a!n,b!n) ∈ S |] ==> list_all2 (fun_of S) a b"
  (*<*)by (erule list_all2_all_nthI) simp(*>*)

(*<*)declare fun_of_def [simp del](*>*)

end