Theory Auxiliary

(*  Title:       CoreC++
    Author:      David von Oheimb, Tobias Nipkow, Daniel Wasserrab  
    Maintainer:  Daniel Wasserrab <wasserra at fmi.uni-passau.de>
*)

section ‹Auxiliary Definitions›

theory Auxiliary
imports Complex_Main "HOL-Library.While_Combinator"
begin

declare
 option.splits[split]
 Let_def[simp]
 subset_insertI2 [simp]
 Cons_eq_map_conv [iff]

(* 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  (‹(⌊_⌋)›)

lemma butlast_tail:
  "butlast (Xs@[X,Y]) = Xs@[X]"
by (induct Xs) auto


lemma butlast_noteq:"Cs ≠ [] ⟹ butlast Cs ≠ Cs"
by(induct Cs)simp_all


lemma app_hd_tl:"⟦Cs ≠ []; Cs = Cs' @ tl Cs⟧ ⟹ Cs' = [hd Cs]"

apply (subgoal_tac "[hd Cs] @ tl Cs = Cs' @ tl Cs")
 apply fast
apply simp
done



lemma only_one_append:"⟦C' ∉ set Cs; C' ∉ set Cs'; Ds@ C'#Ds' = Cs@ C'#Cs'⟧ 
⟹ Cs = Ds ∧ Cs' = Ds'"

  apply -
  apply (simp add:append_eq_append_conv2)
  apply (auto simp:in_set_conv_decomp)
     apply (subgoal_tac "hd (us @ C'#Ds') = C'")
      apply (case_tac us)
       apply simp
      apply fastforce
     apply simp
    apply (subgoal_tac "hd (us @ C'#Ds') = C'")
     apply (case_tac us)
      apply simp
     apply fastforce
    apply simp
   apply (subgoal_tac "hd (us @ C'#Cs') = C'")
    apply (case_tac us)
     apply simp
    apply fastforce
   apply (subgoal_tac "hd(C'#Ds') = C'")
    apply simp
   apply (simp (no_asm))
  apply (subgoal_tac "hd (us @ C'#Cs') = C'")
   apply (case_tac us)
    apply simp
   apply fastforce
  apply (subgoal_tac "hd(C'#Ds') = C'")
   apply simp
  apply (simp (no_asm))
  done


definition pick :: "'a set ⇒ 'a" where
  "pick A ≡ SOME x. x ∈ A"


lemma pick_is_element:"x ∈ A ⟹ pick A ∈ A"
by (unfold pick_def,rule_tac x="x" in someI)


definition set2list :: "'a set ⇒ 'a list" where
  "set2list A ≡ fst (while (λ(Es,S). S ≠ {})
                       (λ(Es,S). let x = pick S in (x#Es,S-{x}))
                       ([],A) )"

lemma card_pick:"⟦finite A; A ≠ {}⟧ ⟹ Suc(card(A-{pick(A)})) = card A"
by (drule card_Suc_Diff1,auto dest!:pick_is_element simp:ex_in_conv)


lemma set2list_prop:"⟦finite A; A ≠ {}⟧ ⟹ 
  ∃xs. while (λ(Es,S). S ≠ {})
             (λ(Es,S). let x = pick S in (x#Es,S-{x}))
             ([],A) = (xs,{}) ∧ (set xs ∪ {} = A)"

apply(rule_tac P="(λxs. (set(fst xs) ∪ snd xs = A))" and 
               r="measure (card o snd)"  in while_rule)
apply(auto dest:pick_is_element)
apply(auto dest:card_pick simp:ex_in_conv measure_def inv_image_def)
done


lemma set2list_correct:"⟦finite A; A ≠ {}; set2list A = xs⟧ ⟹ set xs = A"
by (auto dest:set2list_prop simp:set2list_def)



subsection ‹‹distinct_fst››
 
definition distinct_fst :: "('a × 'b) list ⇒ bool" where
  "distinct_fst  ≡  distinct ∘ 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 map_of_SomeI:
  "⟦ distinct_fst kxs; (k,x) ∈ set kxs ⟧ ⟹ map_of kxs k = Some x"
by (induct kxs) (auto simp:fun_upd_apply)


subsection ‹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