theory FBound
  imports "BlexerSimp" "ClosedFormsBounds"
begin

fun distinctBy :: "'a list \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b set \<Rightarrow> 'a list"
  where
  "distinctBy [] f acc = []"
| "distinctBy (x#xs) f acc = 
     (if (f x) \<in> acc then distinctBy xs f acc 
      else x # (distinctBy xs f ({f x} \<union> acc)))"

fun rerase :: "arexp \<Rightarrow> rrexp"
where
  "rerase AZERO = RZERO"
| "rerase (AONE _) = RONE"
| "rerase (ACHAR _ c) = RCHAR c"
| "rerase (AALTs bs rs) = RALTS (map rerase rs)"
| "rerase (ASEQ _ r1 r2) = RSEQ (rerase r1) (rerase r2)"
| "rerase (ASTAR _ r) = RSTAR (rerase r)"
| "rerase (ANTIMES _ r n) = RNTIMES (rerase r) n"

lemma eq1_rerase:
  shows "x ~1 y \<longleftrightarrow> (rerase x) = (rerase y)"
  apply(induct x y rule: eq1.induct)
  apply(auto)
  done


lemma distinctBy_distinctWith:
  shows "distinctBy xs f (f ` acc) = distinctWith xs (\<lambda>x y. f x = f y) acc"
  apply(induct xs arbitrary: acc)
  apply(auto)
  by (metis image_insert)

lemma distinctBy_distinctWith2:
  shows "distinctBy xs rerase {} = distinctWith xs eq1 {}"
  apply(subst distinctBy_distinctWith[of _ _ "{}", simplified])
  using eq1_rerase by presburger
  
lemma asize_rsize:
  shows "rsize (rerase r) = asize r"
  apply(induct r rule: rerase.induct)
  apply(auto)
  apply (metis (mono_tags, lifting) comp_apply map_eq_conv)
  done

lemma rerase_fuse:
  shows "rerase (fuse bs r) = rerase r"
  apply(induct r)
       apply simp+
  done

lemma rerase_bsimp_ASEQ:
  shows "rerase (bsimp_ASEQ x1 a1 a2) = rsimp_SEQ (rerase a1) (rerase a2)"
  apply(induct x1 a1 a2 rule: bsimp_ASEQ.induct)
  apply(auto)
  done

lemma rerase_bsimp_AALTs:
  shows "rerase (bsimp_AALTs bs rs) = rsimp_ALTs (map rerase rs)"
  apply(induct bs rs rule: bsimp_AALTs.induct)
  apply(auto simp add: rerase_fuse)
  done

fun anonalt :: "arexp \<Rightarrow> bool"
  where
  "anonalt (AALTs bs2 rs) = False"
| "anonalt r = True"


fun agood :: "arexp \<Rightarrow> bool" where
  "agood AZERO = False"
| "agood (AONE cs) = True" 
| "agood (ACHAR cs c) = True"
| "agood (AALTs cs []) = False"
| "agood (AALTs cs [r]) = False"
| "agood (AALTs cs (r1#r2#rs)) = (distinct (map rerase (r1 # r2 # rs)) \<and>(\<forall>r' \<in> set (r1#r2#rs). agood r' \<and> anonalt r'))"
| "agood (ASEQ _ AZERO _) = False"
| "agood (ASEQ _ (AONE _) _) = False"
| "agood (ASEQ _ _ AZERO) = False"
| "agood (ASEQ cs r1 r2) = (agood r1 \<and> agood r2)"
| "agood (ASTAR cs r) = True"


fun anonnested :: "arexp \<Rightarrow> bool"
  where
  "anonnested (AALTs bs2 []) = True"
| "anonnested (AALTs bs2 ((AALTs bs1 rs1) # rs2)) = False"
| "anonnested (AALTs bs2 (r # rs2)) = anonnested (AALTs bs2 rs2)"
| "anonnested r = True"


lemma asize0:
  shows "0 < asize r"
  apply(induct  r)
  apply(auto)
  done

lemma rnullable:
  shows "rnullable (rerase r) = bnullable r"
  apply(induct r rule: rerase.induct)
  apply(auto)
  done

lemma rder_bder_rerase:
  shows "rder c (rerase r ) = rerase (bder c r)"
  apply (induct r)
  apply (auto)
  using rerase_fuse apply presburger
  using rnullable apply blast
  using rnullable by blast

lemma rerase_map_bsimp:
  assumes "\<And> r. r \<in> set rs \<Longrightarrow> rerase (bsimp r) = (rsimp \<circ> rerase) r"
  shows "map rerase (map bsimp rs) =  map (rsimp \<circ> rerase) rs"
  using assms
  apply(induct rs)
  by simp_all


lemma rerase_flts:
  shows "map rerase (flts rs) = rflts (map rerase rs)"
  apply(induct rs rule: flts.induct)
  apply(auto simp add: rerase_fuse)
  done

lemma rerase_dB:
  shows "map rerase (distinctBy rs rerase acc) = rdistinct (map rerase rs) acc"
  apply(induct rs arbitrary: acc)
  apply simp+
  done
  
lemma rerase_earlier_later_same:
  assumes " \<And>r. r \<in> set rs \<Longrightarrow> rerase (bsimp r) = rsimp (rerase r)"
  shows " (map rerase (distinctBy (flts (map bsimp rs)) rerase {})) =
          (rdistinct (rflts (map (rsimp \<circ> rerase) rs)) {})"
  apply(subst rerase_dB)
  apply(subst rerase_flts)
  apply(subst rerase_map_bsimp)
  apply auto
  using assms
  apply simp
  done

lemma bsimp_rerase:
  shows "rerase (bsimp a) = rsimp (rerase a)"
  apply(induct a rule: bsimp.induct)
  apply(auto)
  using rerase_bsimp_ASEQ apply presburger
  using distinctBy_distinctWith2 rerase_bsimp_AALTs rerase_earlier_later_same by fastforce

lemma rders_simp_size:
  shows "rders_simp (rerase r) s  = rerase (bders_simp r s)"
  apply(induct s rule: rev_induct)
  apply simp
  by (simp add: bders_simp_append rder_bder_rerase rders_simp_append bsimp_rerase)


corollary aders_simp_finiteness:
  assumes "\<exists>N. \<forall>s. rsize (rders_simp (rerase r) s) \<le> N"
  shows " \<exists>N. \<forall>s. asize (bders_simp r s) \<le> N"
proof - 
  from assms obtain N where "\<forall>s. rsize (rders_simp (rerase r) s) \<le> N"
    by blast
  then have "\<forall>s. rsize (rerase (bders_simp r s)) \<le> N"
    by (simp add: rders_simp_size) 
  then have "\<forall>s. asize (bders_simp r s) \<le> N"
    by (simp add: asize_rsize) 
  then show "\<exists>N. \<forall>s. asize (bders_simp r s) \<le> N" by blast
qed
  
theorem annotated_size_bound:
  shows "\<exists>N. \<forall>s. asize (bders_simp r s) \<le> N"
  apply(insert aders_simp_finiteness)
  by (simp add: rders_simp_bounded)


unused_thms

end