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)

definition bitcode_agnostic :: "(arexp \<Rightarrow> arexp ) \<Rightarrow> bool"
  where " bitcode_agnostic f = (\<forall>a1 a2. rerase a1 = rerase a2 \<longrightarrow> rerase (f a1) = rerase (f a2))  "

lemma bitcode_agnostic_bsimp:
  shows  "bitcode_agnostic bsimp"
  by (simp add: bitcode_agnostic_def bsimp_rerase)

thm bsimp_rerase

lemma cant1:
  shows "\<lbrakk> bsimp a = b; rerase a = rerase b; a = ASEQ bs r1 r2 \<rbrakk> \<Longrightarrow>
        \<exists>bs' r1' r2'. b = ASEQ bs' r1' r2' \<and> rerase r1' = rerase r1 \<and> rerase r2' = rerase r2"
  sorry



(*"part is less than whole" thm for rrexp, since rrexp is always finite*)
lemma rrexp_finite1:
  shows "\<lbrakk> bsimp_ASEQ bs1 (bsimp ra1) (bsimp ra2) = ASEQ bs2 rb1 rb2; ra1 ~1 rb1; ra2 ~1 rb2 \<rbrakk> \<Longrightarrow> rerase ra1 = rerase (bsimp ra1) "
  apply(case_tac ra1 )
        apply simp+
     apply(case_tac rb1)
           apply simp+

  sorry

lemma unsure_unchanging:
  assumes "bsimp a = bsimp b"
and "a ~1 b"
shows "a = b"
  using assms
  apply(induct rule: eq1.induct)
                      apply simp+
  oops

lemma eq1rerase:
  shows "rerase r1 = rerase r2 \<longleftrightarrow> r1 ~1 r2"
  using eq1_rerase by presburger

thm contrapos_pp

lemma r_part_neq_whole:
  shows "RSEQ r1 r2 \<noteq> r2"
  apply simp
  done

lemma r_part_neq_whole2:
  shows "RSEQ r1 r2 \<noteq> rsimp r2"
  by (metis good.simps(7) good.simps(8) good1 good_SEQ r_part_neq_whole rrexp.distinct(5) rsimp.simps(3) test)



lemma arexpfiniteaux1:
  shows "rerase (bsimp_ASEQ x41 (bsimp x42) (bsimp x43)) = RSEQ (rerase x42) (rerase x43) \<Longrightarrow> \<forall>bs. bsimp x42 \<noteq> AONE bs"
  apply(erule contrapos_pp)
  apply simp
  apply(erule exE)
  apply simp
  by (metis bsimp_rerase r_part_neq_whole2 rerase_fuse)

lemma arexpfiniteaux2:
  shows "rerase (bsimp_ASEQ x41 (bsimp x42) (bsimp x43)) = RSEQ (rerase x42) (rerase x43) \<Longrightarrow> bsimp x42 \<noteq> AZERO "
  apply(erule contrapos_pp)
  apply simp
  done

lemma arexpfiniteaux3:
  shows "rerase (bsimp_ASEQ x41 (bsimp x42) (bsimp x43)) = RSEQ (rerase x42) (rerase x43) \<Longrightarrow> bsimp x43 \<noteq> AZERO "
  apply(erule contrapos_pp)
  apply simp
  done


lemma arexp_finite1:
  shows "rerase (bsimp b) = rerase b \<Longrightarrow> bsimp b = b"
  apply(induct b)
        apply simp+
         apply(case_tac "bsimp b2 = AZERO")
          apply simp
     apply (case_tac "bsimp b1 = AZERO")
      apply simp
  apply(case_tac "\<exists>bs. bsimp b1 = AONE bs")
  using arexpfiniteaux1 apply blast
     apply simp
     apply(subgoal_tac "bsimp_ASEQ x1 (bsimp b1) (bsimp b2) = ASEQ x1 (bsimp b1) (bsimp b2)")
  apply simp
  using bsimp_ASEQ1 apply presburger
  apply simp

  sorry

lemma bitcodes_unchanging2:
  assumes "bsimp a = b"
and "a ~1 b"
shows "a = b"
  using assms
  apply(induct rule: eq1.induct)
                      apply simp
                      apply simp
                      apply simp

                      apply auto
  
  sorry



lemma bitcodes_unchanging:
  shows "\<lbrakk>bsimp a = b; rerase a = rerase b \<rbrakk> \<Longrightarrow> a = b"
  apply(induction a arbitrary: b)
        apply simp+
     apply(case_tac "\<exists>bs. bsimp a1 = AONE bs")
      apply(erule exE)
      apply simp
      prefer 2
      apply(case_tac "bsimp a1 = AZERO")
       apply simp
      apply simp
      apply (metis BlexerSimp.bsimp_ASEQ0 bsimp_ASEQ1 rerase.simps(1) rerase.simps(5) rrexp.distinct(5) rrexp.inject(2))
  
  sorry


lemma bagnostic_shows_bsimp_idem:
  assumes "bitcode_agnostic bsimp"
and "rerase (bsimp a) = rsimp (rerase a)"
and "rsimp r = rsimp (rsimp r)"
shows "bsimp a = bsimp (bsimp a)"
  
  oops

theorem bsimp_idem:
  shows "bsimp (bsimp a) = bsimp a"
  using bitcodes_unchanging bsimp_rerase rsimp_idem by auto

unused_thms

end