433
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theory SizeBound6CT
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imports "Lexer" "PDerivs"
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begin
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section \<open>Bit-Encodings\<close>
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fun orderedSufAux :: "nat \<Rightarrow> char list \<Rightarrow> char list list"
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where
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"orderedSufAux (Suc i) ss = (drop i ss) # (orderedSufAux i ss)"
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|"orderedSufAux 0 ss = Nil"
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fun
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orderedSuf :: "char list \<Rightarrow> char list list"
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where
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"orderedSuf s = orderedSufAux (length s) s"
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fun orderedPrefAux :: "nat \<Rightarrow> char list \<Rightarrow> char list list"
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where
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"orderedPrefAux (Suc i) ss = (take i ss) # (orderedPrefAux i ss)"
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|"orderedPrefAux 0 ss = Nil"
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438
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fun ordsuf :: "char list \<Rightarrow> char list list"
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where
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"ordsuf [] = []"
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| "ordsuf (x # xs) = (ordsuf xs) @ [(x # xs)]"
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433
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fun orderedPref :: "char list \<Rightarrow> char list list"
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where
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"orderedPref s = orderedPrefAux (length s) s"
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lemma shape_of_pref_1list:
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shows "orderedPref [c] = [[]]"
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apply auto
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done
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lemma shape_of_suf_1list:
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shows "orderedSuf [c] = [[c]]"
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by auto
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lemma shape_of_suf_2list:
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shows "orderedSuf [c2, c3] = [[c3], [c2,c3]]"
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by auto
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lemma shape_of_prf_2list:
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shows "orderedPref [c1, c2] = [[c1], []]"
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by auto
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lemma shape_of_suf_3list:
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shows "orderedSuf [c1, c2, c3] = [[c3], [c2, c3], [c1, c2, c3]]"
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by auto
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436
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lemma ordsuf_last:
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shows "ordsuf (xs @ [x]) = [x] # (map (\<lambda>s. s @ [x]) (ordsuf xs))"
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apply(induct xs)
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apply(auto)
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done
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lemma ordsuf_append:
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shows "ordsuf (s1 @ s) = (ordsuf s) @ (map (\<lambda>s11. s11 @ s) (ordsuf s1))"
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apply(induct s1 arbitrary: s rule: rev_induct)
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apply(simp)
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apply(drule_tac x="[x] @ s" in meta_spec)
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apply(simp)
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apply(subst ordsuf_last)
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apply(simp)
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done
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(*
433
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lemma throwing_elem_around:
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shows "orderedSuf (s1 @ [a] @ s) = (orderedSuf s) @ (map (\<lambda>s11. s11 @ s) (orderedSuf ( s1 @ [a]) ))"
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and "orderedSuf (s1 @ [a] @ s) = (orderedSuf ([a] @ s) @ (map (\<lambda>s11. s11 @ ([a] @ s))) (orderedSuf s1) )"
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sorry
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lemma suf_cons:
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shows "orderedSuf (s1 @ s) = (orderedSuf s) @ (map (\<lambda>s11. s11 @ s) (orderedSuf s1))"
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apply(induct s arbitrary: s1)
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apply simp
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apply(subgoal_tac "s1 @ a # s = (s1 @ [a]) @ s")
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prefer 2
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apply simp
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apply(subgoal_tac "orderedSuf (s1 @ a # s) = orderedSuf ((s1 @ [a]) @ s)")
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prefer 2
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apply presburger
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apply(drule_tac x="s1 @ [a]" in meta_spec)
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sorry
435
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*)
433
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lemma shape_of_prf_3list:
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shows "orderedPref [c1, c2, c3] = [[c1, c2], [c1], []]"
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by auto
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fun zip_concat :: "char list list \<Rightarrow> char list list \<Rightarrow> char list list"
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where
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"zip_concat (s1#ss1) (s2#ss2) = (s1@s2) # (zip_concat ss1 ss2)"
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| "zip_concat [] [] = []"
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| "zip_concat [] (s2#ss2) = s2 # (zip_concat [] ss2)"
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| "zip_concat (s1#ss1) [] = s1 # (zip_concat ss1 [])"
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434
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(*
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lemma compliment_pref_suf:
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shows "zip_concat (orderedPref s) (orderedSuf s) = replicate (length s) s "
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apply(induct s)
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apply auto[1]
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sorry
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434
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*)
433
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438
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433
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datatype rrexp =
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RZERO
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| RONE
441
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| RCHAR char
433
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| RSEQ rrexp rrexp
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| RALTS "rrexp list"
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| RSTAR rrexp
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abbreviation
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"RALT r1 r2 \<equiv> RALTS [r1, r2]"
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fun
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rnullable :: "rrexp \<Rightarrow> bool"
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where
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"rnullable (RZERO) = False"
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| "rnullable (RONE ) = True"
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| "rnullable (RCHAR c) = False"
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| "rnullable (RALTS rs) = (\<exists>r \<in> set rs. rnullable r)"
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| "rnullable (RSEQ r1 r2) = (rnullable r1 \<and> rnullable r2)"
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| "rnullable (RSTAR r) = True"
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fun
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rder :: "char \<Rightarrow> rrexp \<Rightarrow> rrexp"
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where
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"rder c (RZERO) = RZERO"
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| "rder c (RONE) = RZERO"
441
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| "rder c (RCHAR d) = (if c = d then RONE else RZERO)"
433
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| "rder c (RALTS rs) = RALTS (map (rder c) rs)"
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| "rder c (RSEQ r1 r2) =
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(if rnullable r1
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then RALT (RSEQ (rder c r1) r2) (rder c r2)
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else RSEQ (rder c r1) r2)"
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| "rder c (RSTAR r) = RSEQ (rder c r) (RSTAR r)"
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fun
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rders :: "rrexp \<Rightarrow> string \<Rightarrow> rrexp"
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where
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"rders r [] = r"
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| "rders r (c#s) = rders (rder c r) s"
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fun rdistinct :: "'a list \<Rightarrow>'a set \<Rightarrow> 'a list"
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where
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"rdistinct [] acc = []"
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| "rdistinct (x#xs) acc =
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(if x \<in> acc then rdistinct xs acc
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else x # (rdistinct xs ({x} \<union> acc)))"
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fun rflts :: "rrexp list \<Rightarrow> rrexp list"
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where
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"rflts [] = []"
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| "rflts (RZERO # rs) = rflts rs"
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| "rflts ((RALTS rs1) # rs) = rs1 @ rflts rs"
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| "rflts (r1 # rs) = r1 # rflts rs"
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fun rsimp_ALTs :: " rrexp list \<Rightarrow> rrexp"
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where
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"rsimp_ALTs [] = RZERO"
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| "rsimp_ALTs [r] = r"
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| "rsimp_ALTs rs = RALTS rs"
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fun rsimp_SEQ :: " rrexp \<Rightarrow> rrexp \<Rightarrow> rrexp"
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where
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"rsimp_SEQ RZERO _ = RZERO"
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| "rsimp_SEQ _ RZERO = RZERO"
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| "rsimp_SEQ RONE r2 = r2"
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| "rsimp_SEQ r1 r2 = RSEQ r1 r2"
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fun rsimp :: "rrexp \<Rightarrow> rrexp"
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where
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"rsimp (RSEQ r1 r2) = rsimp_SEQ (rsimp r1) (rsimp r2)"
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| "rsimp (RALTS rs) = rsimp_ALTs (rdistinct (rflts (map rsimp rs)) {}) "
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| "rsimp r = r"
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fun
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rders_simp :: "rrexp \<Rightarrow> string \<Rightarrow> rrexp"
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where
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"rders_simp r [] = r"
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| "rders_simp r (c#s) = rders_simp (rsimp (rder c r)) s"
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fun rsize :: "rrexp \<Rightarrow> nat" where
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"rsize RZERO = 1"
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| "rsize (RONE) = 1"
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| "rsize (RCHAR c) = 1"
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| "rsize (RALTS rs) = Suc (sum_list (map rsize rs))"
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| "rsize (RSEQ r1 r2) = Suc (rsize r1 + rsize r2)"
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| "rsize (RSTAR r) = Suc (rsize r)"
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fun rlist_size :: "rrexp list \<Rightarrow> nat" where
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"rlist_size (r # rs) = rsize r + rlist_size rs"
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| "rlist_size [] = 0"
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thm neq_Nil_conv
435
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lemma hand_made_def_rlist_size:
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shows "rlist_size rs = sum_list (map rsize rs)"
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proof (induct rs)
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case Nil show ?case by simp
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next
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case (Cons a rs) thus ?case
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by simp
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qed
433
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lemma rsimp_aalts_smaller:
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shows "rsize (rsimp_ALTs rs) \<le> rsize (RALTS rs)"
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apply(induct rs)
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apply simp
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apply simp
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apply(case_tac "rs = []")
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apply simp
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apply(subgoal_tac "\<exists>rsp ap. rs = ap # rsp")
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apply(erule exE)+
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apply simp
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apply simp
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by(meson neq_Nil_conv)
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(*
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lemma rders_shape:
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shows "s \<noteq> [] \<Longrightarrow> rders_simp (RSEQ r1 r2) s =
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rsimp (RALTS ((RSEQ (rders r1 s) r2) #
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(map (rders r2) (orderedSuf s))) )"
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apply(induct s arbitrary: r1 r2 rule: rev_induct)
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apply simp
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apply simp
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sorry
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*)
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fun rders_cond_list :: "rrexp \<Rightarrow> bool list \<Rightarrow> char list list \<Rightarrow> rrexp list"
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where
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"rders_cond_list r2 (True # bs) (s # strs) = (rders_simp r2 s) # (rders_cond_list r2 bs strs)"
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| "rders_cond_list r2 (False # bs) (s # strs) = rders_cond_list r2 bs strs"
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| "rders_cond_list r2 [] s = []"
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| "rders_cond_list r2 bs [] = []"
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fun nullable_bools :: "rrexp \<Rightarrow> char list list \<Rightarrow> bool list"
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where
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"nullable_bools r (s#strs) = (rnullable (rders r s)) # (nullable_bools r strs) "
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|"nullable_bools r [] = []"
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fun cond_list :: "rrexp \<Rightarrow> rrexp \<Rightarrow> char list \<Rightarrow> rrexp list"
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where
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"cond_list r1 r2 s = rders_cond_list r2 (nullable_bools r1 (orderedPref s) ) (orderedSuf s)"
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435
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438
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fun seq_update :: " char \<Rightarrow> rrexp \<Rightarrow> char list list \<Rightarrow> char list list"
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where
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"seq_update c r Ss = (if (rnullable r) then ([c] # (map (\<lambda>s1. s1 @ [c]) Ss)) else (map (\<lambda>s1. s1 @ [c]) Ss))"
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433
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lemma rSEQ_mono:
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shows "rsize (rsimp_SEQ r1 r2) \<le>rsize ( RSEQ r1 r2)"
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apply auto
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apply(induct r1)
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apply auto
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apply(case_tac "r2")
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apply simp_all
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apply(case_tac r2)
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apply simp_all
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apply(case_tac r2)
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apply simp_all
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apply(case_tac r2)
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apply simp_all
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apply(case_tac r2)
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apply simp_all
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done
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435
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lemma ralts_cap_mono:
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shows "rsize (RALTS rs) \<le> Suc ( sum_list (map rsize rs)) "
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by simp
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lemma rflts_def_idiot:
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shows "\<lbrakk> a \<noteq> RZERO; \<nexists>rs1. a = RALTS rs1\<rbrakk>
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\<Longrightarrow> rflts (a # rs) = a # rflts rs"
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apply(case_tac a)
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apply simp_all
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done
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lemma rflts_mono:
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shows "sum_list (map rsize (rflts rs))\<le> sum_list (map rsize rs)"
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apply(induct rs)
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apply simp
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apply(case_tac "a = RZERO")
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apply simp
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apply(case_tac "\<exists>rs1. a = RALTS rs1")
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apply(erule exE)
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apply simp
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apply(subgoal_tac "rflts (a # rs) = a # (rflts rs)")
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prefer 2
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using rflts_def_idiot apply blast
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apply simp
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done
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lemma rdistinct_smaller: shows "sum_list (map rsize (rdistinct rs ss)) \<le>
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sum_list (map rsize rs )"
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apply (induct rs arbitrary: ss)
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apply simp
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by (simp add: trans_le_add2)
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lemma rdistinct_phi_smaller: "sum_list (map rsize (rdistinct rs {})) \<le> sum_list (map rsize rs)"
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by (simp add: rdistinct_smaller)
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lemma rsimp_alts_mono :
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shows "\<And>x. (\<And>xa. xa \<in> set x \<Longrightarrow> rsize (rsimp xa) \<le> rsize xa) \<Longrightarrow>
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rsize (rsimp_ALTs (rdistinct (rflts (map rsimp x)) {})) \<le> Suc (sum_list (map rsize x))"
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apply(subgoal_tac "rsize (rsimp_ALTs (rdistinct (rflts (map rsimp x)) {} ))
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\<le> rsize (RALTS (rdistinct (rflts (map rsimp x)) {} ))")
+ − 347
prefer 2
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using rsimp_aalts_smaller apply auto[1]
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apply(subgoal_tac "rsize (RALTS (rdistinct (rflts (map rsimp x)) {})) \<le>Suc( sum_list (map rsize (rdistinct (rflts (map rsimp x)) {})))")
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prefer 2
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using ralts_cap_mono apply blast
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apply(subgoal_tac "sum_list (map rsize (rdistinct (rflts (map rsimp x)) {})) \<le>
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sum_list (map rsize ( (rflts (map rsimp x))))")
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prefer 2
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using rdistinct_smaller apply presburger
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apply(subgoal_tac "sum_list (map rsize (rflts (map rsimp x))) \<le>
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sum_list (map rsize (map rsimp x))")
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prefer 2
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using rflts_mono apply blast
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apply(subgoal_tac "sum_list (map rsize (map rsimp x)) \<le> sum_list (map rsize x)")
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prefer 2
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apply (simp add: sum_list_mono)
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by linarith
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433
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lemma rsimp_mono:
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shows "rsize (rsimp r) \<le> rsize r"
435
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433
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apply(induct r)
435
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apply simp_all
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433
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apply(subgoal_tac "rsize (rsimp_SEQ (rsimp r1) (rsimp r2)) \<le> rsize (RSEQ (rsimp r1) (rsimp r2))")
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apply force
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using rSEQ_mono
435
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apply presburger
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using rsimp_alts_mono by auto
433
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lemma idiot:
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shows "rsimp_SEQ RONE r = r"
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apply(case_tac r)
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apply simp_all
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done
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435
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lemma no_alt_short_list_after_simp:
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shows "RALTS rs = rsimp r \<Longrightarrow> rsimp_ALTs rs = RALTS rs"
433
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sorry
+ − 392
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lemma no_further_dB_after_simp:
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shows "RALTS rs = rsimp r \<Longrightarrow> rdistinct rs {} = rs"
435
+ − 395
433
+ − 396
sorry
+ − 397
+ − 398
+ − 399
lemma idiot2:
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shows " \<lbrakk>r1 \<noteq> RZERO; r1 \<noteq> RONE;r2 \<noteq> RZERO\<rbrakk>
+ − 401
\<Longrightarrow> rsimp_SEQ r1 r2 = RSEQ r1 r2"
+ − 402
apply(case_tac r1)
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apply(case_tac r2)
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apply simp_all
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apply(case_tac r2)
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apply simp_all
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apply(case_tac r2)
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apply simp_all
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apply(case_tac r2)
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apply simp_all
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apply(case_tac r2)
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apply simp_all
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done
438
+ − 414
(*
433
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lemma rsimp_aalts_another:
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shows "\<forall>r \<in> (set (map rsimp ((RSEQ (rders r1 s) r2) #
+ − 417
(rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) )) ). (rsize r) < N "
+ − 418
sorry
+ − 419
+ − 420
+ − 421
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lemma shape_derssimpseq_onechar:
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shows " (rders_simp r [c]) = (rsimp (rders r [c]))"
+ − 424
and "rders_simp (RSEQ r1 r2) [c] =
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rsimp (RALTS ((RSEQ (rders r1 [c]) r2) #
+ − 426
(rders_cond_list r2 (nullable_bools r1 (orderedPref [c])) (orderedSuf [c]))) )"
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apply simp
+ − 428
apply(simp add: rders.simps)
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apply(case_tac "rsimp (rder c r1) = RZERO")
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apply auto
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apply(case_tac "rsimp (rder c r1) = RONE")
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apply auto
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apply(subgoal_tac "rsimp_SEQ RONE (rsimp r2) = rsimp r2")
+ − 434
prefer 2
+ − 435
using idiot
+ − 436
apply simp
+ − 437
apply(subgoal_tac "rsimp_SEQ RONE (rsimp r2) = rsimp_ALTs (rdistinct (rflts [rsimp r2]) {})")
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prefer 2
+ − 439
apply auto
+ − 440
apply(case_tac "rsimp r2")
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apply auto
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apply(subgoal_tac "rdistinct x5 {} = x5")
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prefer 2
+ − 444
using no_further_dB_after_simp
+ − 445
apply metis
+ − 446
apply(subgoal_tac "rsimp_ALTs (rdistinct x5 {}) = rsimp_ALTs x5")
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prefer 2
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apply fastforce
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apply auto
+ − 450
apply (metis no_alt_short_list_after_simp)
+ − 451
apply (case_tac "rsimp r2 = RZERO")
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apply(subgoal_tac "rsimp_SEQ (rsimp (rder c r1)) (rsimp r2) = RZERO")
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prefer 2
+ − 454
apply(case_tac "rsimp ( rder c r1)")
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apply auto
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apply(subgoal_tac "rsimp_SEQ (rsimp (rder c r1)) (rsimp r2) = RSEQ (rsimp (rder c r1)) (rsimp r2)")
+ − 457
prefer 2
+ − 458
apply auto
+ − 459
sorry
438
+ − 460
*)
433
+ − 461
438
+ − 462
(*
+ − 463
lemma shape_derssimpseq_onechar2:
+ − 464
shows "rders_simp (RSEQ r1 r2) [c] =
+ − 465
rsimp (RALTS ((RSEQ (rders_simp r1 [c]) r2) #
+ − 466
(rders_cond_list r2 (nullable_bools r1 (orderedPref [c])) (orderedSuf [c]))) )"
+ − 467
sorry
433
+ − 468
438
+ − 469
*)
+ − 470
+ − 471
lemma shape_derssimpseq_onechar:
+ − 472
shows " (rders_simp r [c]) = (rsimp (rders r [c]))"
+ − 473
and "rders_simp (RSEQ r1 r2) [c] =
+ − 474
rsimp (RALTS ((RSEQ (rders r1 [c]) r2) #
+ − 475
(rders_cond_list r2 (nullable_bools r1 (orderedPref [c])) (orderedSuf [c]))) )"
+ − 476
apply simp
+ − 477
apply(simp add: rders.simps)
+ − 478
apply(case_tac "rsimp (rder c r1) = RZERO")
+ − 479
apply auto
+ − 480
apply(case_tac "rsimp (rder c r1) = RONE")
+ − 481
apply auto
+ − 482
apply(subgoal_tac "rsimp_SEQ RONE (rsimp r2) = rsimp r2")
+ − 483
prefer 2
+ − 484
using idiot
+ − 485
apply simp
+ − 486
apply(subgoal_tac "rsimp_SEQ RONE (rsimp r2) = rsimp_ALTs (rdistinct (rflts [rsimp r2]) {})")
+ − 487
prefer 2
+ − 488
apply auto
+ − 489
apply(case_tac "rsimp r2")
+ − 490
apply auto
+ − 491
apply(subgoal_tac "rdistinct x5 {} = x5")
+ − 492
prefer 2
+ − 493
using no_further_dB_after_simp
+ − 494
apply metis
+ − 495
apply(subgoal_tac "rsimp_ALTs (rdistinct x5 {}) = rsimp_ALTs x5")
+ − 496
prefer 2
+ − 497
apply fastforce
+ − 498
apply auto
+ − 499
apply (metis no_alt_short_list_after_simp)
+ − 500
apply (case_tac "rsimp r2 = RZERO")
+ − 501
apply(subgoal_tac "rsimp_SEQ (rsimp (rder c r1)) (rsimp r2) = RZERO")
+ − 502
prefer 2
+ − 503
apply(case_tac "rsimp ( rder c r1)")
+ − 504
apply auto
+ − 505
apply(subgoal_tac "rsimp_SEQ (rsimp (rder c r1)) (rsimp r2) = RSEQ (rsimp (rder c r1)) (rsimp r2)")
+ − 506
prefer 2
+ − 507
apply auto
+ − 508
sorry
433
+ − 509
+ − 510
lemma shape_derssimpseq_onechar2:
+ − 511
shows "rders_simp (RSEQ r1 r2) [c] =
+ − 512
rsimp (RALTS ((RSEQ (rders_simp r1 [c]) r2) #
+ − 513
(rders_cond_list r2 (nullable_bools r1 (orderedPref [c])) (orderedSuf [c]))) )"
+ − 514
sorry
+ − 515
438
+ − 516
(*
+ − 517
lemma simp_helps_der_pierce_dB:
+ − 518
shows " rsimp
+ − 519
(rsimp_ALTs
+ − 520
(map (rder x)
+ − 521
(rdistinct rs {}))) =
+ − 522
rsimp (rsimp_ALTs (rdistinct (map (rder x) rs) {}))"
+ − 523
+ − 524
sorry
+ − 525
*)
+ − 526
(*
+ − 527
lemma simp_helps_der_pierce_flts:
+ − 528
shows " rsimp
+ − 529
(rsimp_ALTs
+ − 530
(rdistinct
+ − 531
(map (rder x)
+ − 532
(rflts rs )
+ − 533
) {}
+ − 534
)
+ − 535
) =
+ − 536
rsimp (rsimp_ALTs (rdistinct (rflts (map (rder x) rs)) {}) )"
+ − 537
+ − 538
sorry
+ − 539
+ − 540
*)
433
+ − 541
+ − 542
lemma rders__onechar:
+ − 543
shows " (rders_simp r [c]) = (rsimp (rders r [c]))"
+ − 544
by simp
+ − 545
+ − 546
lemma rders_append:
+ − 547
"rders c (s1 @ s2) = rders (rders c s1) s2"
+ − 548
apply(induct s1 arbitrary: c s2)
+ − 549
apply(simp_all)
+ − 550
done
+ − 551
+ − 552
lemma rders_simp_append:
+ − 553
"rders_simp c (s1 @ s2) = rders_simp (rders_simp c s1) s2"
+ − 554
apply(induct s1 arbitrary: c s2)
+ − 555
apply(simp_all)
+ − 556
done
+ − 557
+ − 558
lemma inside_simp_removal:
+ − 559
shows " rsimp (rder x (rsimp r)) = rsimp (rder x r)"
+ − 560
+ − 561
sorry
+ − 562
+ − 563
lemma set_related_list:
+ − 564
shows "distinct rs \<Longrightarrow> length rs = card (set rs)"
+ − 565
by (simp add: distinct_card)
+ − 566
(*this section deals with the property of distinctBy: creates a list without duplicates*)
+ − 567
lemma rdistinct_never_added_twice:
+ − 568
shows "rdistinct (a # rs) {a} = rdistinct rs {a}"
+ − 569
by force
+ − 570
+ − 571
+ − 572
lemma rdistinct_does_the_job:
+ − 573
shows "distinct (rdistinct rs s)"
+ − 574
apply(induct rs arbitrary: s)
+ − 575
apply simp
+ − 576
apply simp
+ − 577
sorry
+ − 578
+ − 579
+ − 580
lemma simp_helps_der_pierce:
+ − 581
shows " rsimp
+ − 582
(rder x
+ − 583
(rsimp_ALTs rs)) =
+ − 584
rsimp
+ − 585
(rsimp_ALTs
+ − 586
(map (rder x )
+ − 587
rs
+ − 588
)
+ − 589
)"
+ − 590
sorry
+ − 591
+ − 592
lemma unfold_ders_simp_inside_only:
+ − 593
shows " (rders_simp (RSEQ r1 r2) xs =
+ − 594
rsimp (RALTS (RSEQ (rders_simp r1 xs) r2 # rders_cond_list r2 (nullable_bools r1 (orderedPref xs)) (orderedSuf xs))))
+ − 595
\<Longrightarrow> rsimp (rder x (rders_simp (RSEQ r1 r2) xs)) = rsimp (rder x (rsimp (RALTS(RSEQ (rders_simp r1 xs) r2 # rders_cond_list r2 (nullable_bools r1 (orderedPref xs)) (orderedSuf xs)))))"
+ − 596
by presburger
+ − 597
+ − 598
+ − 599
+ − 600
lemma unfold_ders_simp_inside_only_nosimp:
+ − 601
shows " (rders_simp (RSEQ r1 r2) xs =
+ − 602
rsimp (RALTS (RSEQ (rders_simp r1 xs) r2 # rders_cond_list r2 (nullable_bools r1 (orderedPref xs)) (orderedSuf xs))))
+ − 603
\<Longrightarrow> rsimp (rder x (rders_simp (RSEQ r1 r2) xs)) = rsimp (rder x (RALTS(RSEQ (rders_simp r1 xs) r2 # rders_cond_list r2 (nullable_bools r1 (orderedPref xs)) (orderedSuf xs))))"
+ − 604
using inside_simp_removal by presburger
+ − 605
+ − 606
+ − 607
+ − 608
+ − 609
lemma rders_simp_one_char:
+ − 610
shows "rders_simp r [c] = rsimp (rder c r)"
+ − 611
apply auto
+ − 612
done
+ − 613
+ − 614
lemma rsimp_idem:
+ − 615
shows "rsimp (rsimp r) = rsimp r"
+ − 616
sorry
+ − 617
434
+ − 618
corollary rsimp_inner_idem1:
+ − 619
shows "rsimp r = RSEQ r1 r2 \<Longrightarrow> rsimp r1 = r1 \<and> rsimp r2 = r2"
+ − 620
+ − 621
sorry
+ − 622
+ − 623
corollary rsimp_inner_idem2:
+ − 624
shows "rsimp r = RALTS rs \<Longrightarrow> \<forall>r' \<in> (set rs). rsimp r' = r'"
+ − 625
sorry
+ − 626
+ − 627
corollary rsimp_inner_idem3:
+ − 628
shows "rsimp r = RALTS rs \<Longrightarrow> map rsimp rs = rs"
+ − 629
by (meson map_idI rsimp_inner_idem2)
+ − 630
+ − 631
corollary rsimp_inner_idem4:
+ − 632
shows "rsimp r = RALTS rs \<Longrightarrow> flts rs = rs"
+ − 633
sorry
+ − 634
+ − 635
433
+ − 636
lemma head_one_more_simp:
+ − 637
shows "map rsimp (r # rs) = map rsimp (( rsimp r) # rs)"
+ − 638
by (simp add: rsimp_idem)
+ − 639
+ − 640
lemma head_one_more_dersimp:
+ − 641
shows "map rsimp ((rder x (rders_simp r s) # rs)) = map rsimp ((rders_simp r (s@[x]) ) # rs)"
+ − 642
using head_one_more_simp rders_simp_append rders_simp_one_char by presburger
+ − 643
+ − 644
thm cond_list.simps
+ − 645
+ − 646
lemma suffix_plus1char:
+ − 647
shows "\<not> (rnullable (rders r1 s)) \<Longrightarrow> cond_list r1 r2 (s@[c]) = map (rder c) (cond_list r1 r2 s)"
+ − 648
apply simp
+ − 649
sorry
+ − 650
+ − 651
lemma suffix_plus1charn:
+ − 652
shows "rnullable (rders r1 s) \<Longrightarrow> cond_list r1 r2 (s@[c]) = (rder c r2) # (map (rder c) (cond_list r1 r2 s))"
+ − 653
sorry
+ − 654
+ − 655
lemma ders_simp_nullability:
+ − 656
shows "rnullable (rders r s) = rnullable (rders_simp r s)"
+ − 657
sorry
+ − 658
+ − 659
lemma first_elem_seqder:
+ − 660
shows "\<not>rnullable r1p \<Longrightarrow> map rsimp (rder x (RSEQ r1p r2)
+ − 661
# rs) = map rsimp ((RSEQ (rder x r1p) r2) # rs) "
+ − 662
by auto
+ − 663
+ − 664
lemma first_elem_seqder1:
+ − 665
shows "\<not>rnullable (rders_simp r xs) \<Longrightarrow> map rsimp ( (rder x (RSEQ (rders_simp r xs) r2)) # rs) =
+ − 666
map rsimp ( (RSEQ (rsimp (rder x (rders_simp r xs))) r2) # rs)"
+ − 667
by (simp add: rsimp_idem)
+ − 668
+ − 669
lemma first_elem_seqdersimps:
+ − 670
shows "\<not>rnullable (rders_simp r xs) \<Longrightarrow> map rsimp ( (rder x (RSEQ (rders_simp r xs) r2)) # rs) =
+ − 671
map rsimp ( (RSEQ (rders_simp r (xs @ [x])) r2) # rs)"
+ − 672
using first_elem_seqder1 rders_simp_append by auto
+ − 673
+ − 674
lemma first_elem_seqder_nullable:
+ − 675
shows "rnullable (rders_simp r1 xs) \<Longrightarrow> cond_list r1 r2 (xs @ [x]) = rder x r2 # map (rder x) (cond_list r1 r2 xs)"
+ − 676
sorry
+ − 677
+ − 678
438
+ − 679
+ − 680
+ − 681
lemma seq_update_seq_ders:
+ − 682
shows "rsimp (rder c ( rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) #
+ − 683
(map (rders_simp r2) Ss))))) =
+ − 684
rsimp (RALTS ((RSEQ (rders_simp r1 (s @ [c])) r2) #
+ − 685
(map (rders_simp r2) (seq_update c (rders_simp r1 s) Ss)))) "
+ − 686
sorry
+ − 687
+ − 688
lemma seq_ders_closed_form1:
+ − 689
shows "\<exists>Ss. rders_simp (RSEQ r1 r2) [c] = rsimp (RALTS ((RSEQ (rders_simp r1 [c]) r2) #
+ − 690
(map ( rders_simp r2 ) Ss)))"
+ − 691
apply(case_tac "rnullable r1")
+ − 692
apply(subgoal_tac " rders_simp (RSEQ r1 r2) [c] =
+ − 693
rsimp (RALTS ((RSEQ (rders_simp r1 [c]) r2) # (map (rders_simp r2) [[c]])))")
+ − 694
prefer 2
+ − 695
apply (simp add: rsimp_idem)
+ − 696
apply(rule_tac x = "[[c]]" in exI)
+ − 697
apply simp
+ − 698
apply(subgoal_tac " rders_simp (RSEQ r1 r2) [c] =
+ − 699
rsimp (RALTS ((RSEQ (rders_simp r1 [c]) r2) # (map (rders_simp r2) [])))")
+ − 700
apply blast
+ − 701
apply(simp add: rsimp_idem)
+ − 702
sorry
+ − 703
+ − 704
+ − 705
+ − 706
lemma seq_ders_closed_form:
+ − 707
shows "s \<noteq> [] \<Longrightarrow> \<exists>Ss. rders_simp (RSEQ r1 r2) s = rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) #
+ − 708
(map ( rders_simp r2 ) Ss)))"
+ − 709
apply(induct s rule: rev_induct)
+ − 710
apply simp
+ − 711
apply(case_tac xs)
+ − 712
using seq_ders_closed_form1 apply auto[1]
+ − 713
apply(subgoal_tac "\<exists>Ss. rders_simp (RSEQ r1 r2) xs = rsimp (RALTS (RSEQ (rders_simp r1 xs) r2 # map (rders_simp r2) Ss))")
+ − 714
prefer 2
+ − 715
+ − 716
apply blast
+ − 717
apply(erule exE)
+ − 718
apply(rule_tac x = "seq_update x (rders_simp r1 xs) Ss" in exI)
+ − 719
apply(subst rders_simp_append)
+ − 720
apply(subgoal_tac "rders_simp (rders_simp (RSEQ r1 r2) xs) [x] = rsimp (rder x (rders_simp (RSEQ r1 r2) xs))")
+ − 721
apply(simp only:)
+ − 722
apply(subst seq_update_seq_ders)
+ − 723
apply blast
+ − 724
using rders_simp_one_char by presburger
+ − 725
+ − 726
+ − 727
433
+ − 728
(*nullable_seq_with_list1 related *)
+ − 729
lemma LHS0_def_der_alt:
+ − 730
shows "rsimp (rder x (RALTS ( (RSEQ (rders_simp r1 xs) r2) # (cond_list r1 r2 xs)))) =
+ − 731
rsimp (RALTS ((rder x (RSEQ (rders_simp r1 xs) r2)) # (map (rder x) (cond_list r1 r2 xs))))"
+ − 732
by fastforce
+ − 733
+ − 734
lemma LHS1_def_der_seq:
+ − 735
shows "rnullable (rders_simp r1 xs) \<Longrightarrow>
+ − 736
rsimp (rder x (RALTS ( (RSEQ (rders_simp r1 xs) r2) # (cond_list r1 r2 xs)))) =
+ − 737
rsimp(RALTS ((RALTS ((RSEQ (rders_simp r1 (xs @ [x])) r2) # [rder x r2]) ) # (map (rder x ) (cond_list r1 r2 xs))))"
+ − 738
by (simp add: rders_simp_append rsimp_idem)
+ − 739
+ − 740
+ − 741
+ − 742
+ − 743
+ − 744
lemma cond_list_head_last:
+ − 745
shows "rnullable (rders r1 s) \<Longrightarrow>
+ − 746
RALTS (r # (cond_list r1 r2 (s @ [c]))) = RALTS (r # ((rder c r2) # (map (rder c) (cond_list r1 r2 s))))"
+ − 747
using suffix_plus1charn by blast
+ − 748
435
+ − 749
lemma simp_flatten2:
+ − 750
shows "rsimp (RALTS (r # [RALTS rs])) = rsimp (RALTS (r # rs))"
+ − 751
sorry
+ − 752
433
+ − 753
+ − 754
lemma simp_flatten:
+ − 755
shows "rsimp (RALTS ((RALTS rsa) # rsb)) = rsimp (RALTS (rsa @ rsb))"
+ − 756
+ − 757
sorry
+ − 758
+ − 759
lemma RHS1:
+ − 760
shows "rnullable (rders_simp r1 xs) \<Longrightarrow>
+ − 761
rsimp (RALTS ((RSEQ (rders_simp r1 (xs @ [x])) r2) #
+ − 762
(cond_list r1 r2 (xs @[x])))) =
+ − 763
rsimp (RALTS ((RSEQ (rders_simp r1 (xs @ [x])) r2) #
+ − 764
( ((rder x r2) # (map (rder x) (cond_list r1 r2 xs)))))) "
+ − 765
using first_elem_seqder_nullable by presburger
+ − 766
+ − 767
+ − 768
lemma nullable_seq_with_list1:
+ − 769
shows " rnullable (rders_simp r1 s) \<Longrightarrow>
+ − 770
rsimp (rder c (RALTS ( (RSEQ (rders_simp r1 s) r2) # (cond_list r1 r2 s)) )) =
+ − 771
rsimp (RALTS ( (RSEQ (rders_simp r1 (s @ [c])) r2) # (cond_list r1 r2 (s @ [c])) ) )"
434
+ − 772
using RHS1 LHS1_def_der_seq cond_list_head_last simp_flatten
+ − 773
by (metis append.left_neutral append_Cons)
+ − 774
+ − 775
+ − 776
(*^^^^^^^^^nullable_seq_with_list1 related ^^^^^^^^^^^^^^^^*)
+ − 777
+ − 778
+ − 779
+ − 780
+ − 781
+ − 782
433
+ − 783
+ − 784
+ − 785
+ − 786
+ − 787
lemma nullable_seq_with_list:
+ − 788
shows "rnullable (rders_simp r1 xs) \<Longrightarrow> rsimp (rder x (RALTS ((RSEQ (rders_simp r1 xs) r2) #
+ − 789
(rders_cond_list r2 (nullable_bools r1 (orderedPref xs)) (orderedSuf xs)) ))) =
+ − 790
rsimp (RALTS ((RSEQ (rders_simp r1 (xs@[x])) r2) #
+ − 791
(rders_cond_list r2 (nullable_bools r1 (orderedPref (xs@[x]))) (orderedSuf (xs@[x]))) ) )
+ − 792
"
+ − 793
apply(subgoal_tac "rsimp (rder x (RALTS ((RSEQ (rders_simp r1 xs) r2) # (cond_list r1 r2 xs)))) =
+ − 794
rsimp (RALTS ((RSEQ (rders_simp r1 (xs@[x])) r2) # (cond_list r1 r2 (xs@[x]))))")
+ − 795
apply auto[1]
+ − 796
using nullable_seq_with_list1 by auto
+ − 797
+ − 798
+ − 799
+ − 800
+ − 801
lemma r1r2ders_whole:
+ − 802
"rsimp
+ − 803
(RALTS
+ − 804
(rder x (RSEQ (rders_simp r1 xs) r2) #
+ − 805
map (rder x) (rders_cond_list r2 (nullable_bools r1 (orderedPref xs)) (orderedSuf xs)))) =
+ − 806
rsimp( RALTS( ( (RSEQ (rders_simp r1 (xs@[x])) r2)
+ − 807
# (rders_cond_list r2 (nullable_bools r1 (orderedPref (xs @ [x]))) (orderedSuf (xs @ [x])))))) "
+ − 808
using ders_simp_nullability first_elem_seqdersimps nullable_seq_with_list1 suffix_plus1char by auto
+ − 809
+ − 810
lemma rders_simp_same_simpders:
+ − 811
shows "s \<noteq> [] \<Longrightarrow> rders_simp r s = rsimp (rders r s)"
+ − 812
apply(induct s rule: rev_induct)
+ − 813
apply simp
+ − 814
apply(case_tac "xs = []")
+ − 815
apply simp
+ − 816
apply(simp add: rders_append rders_simp_append)
+ − 817
using inside_simp_removal by blast
+ − 818
+ − 819
lemma shape_derssimp_seq:
+ − 820
shows "\<lbrakk>s \<noteq> []\<rbrakk> \<Longrightarrow> rders_simp (RSEQ r1 r2) s =
+ − 821
rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) #
+ − 822
(rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))) )"
+ − 823
+ − 824
apply(induct s arbitrary: r1 r2 rule: rev_induct)
+ − 825
apply simp
+ − 826
apply(case_tac "xs = []")
+ − 827
using shape_derssimpseq_onechar2 apply force
+ − 828
apply(simp only: rders_simp_append)
+ − 829
apply(simp only: rders_simp_one_char)
+ − 830
+ − 831
apply(subgoal_tac "rsimp (rder x (rders_simp (RSEQ r1 r2) xs))
+ − 832
= rsimp (rder x (RALTS(RSEQ (rders_simp r1 xs) r2 # rders_cond_list r2 (nullable_bools r1 (orderedPref xs)) (orderedSuf xs))))")
+ − 833
prefer 2
+ − 834
using unfold_ders_simp_inside_only_nosimp apply presburger
+ − 835
apply(subgoal_tac "rsimp (rder x (RALTS (RSEQ (rders_simp r1 xs) r2
+ − 836
# rders_cond_list r2 (nullable_bools r1 (orderedPref xs)) (orderedSuf xs)))) =
+ − 837
rsimp ( (RALTS (rder x (RSEQ (rders_simp r1 xs) r2)
+ − 838
# (map (rder x) (rders_cond_list r2 (nullable_bools r1 (orderedPref xs)) (orderedSuf xs))))))
+ − 839
")
+ − 840
prefer 2
+ − 841
apply simp
+ − 842
using r1r2ders_whole rders_simp_append rders_simp_one_char by presburger
+ − 843
+ − 844
+ − 845
lemma shape_derssimp_alts:
+ − 846
shows "s \<noteq> [] \<Longrightarrow> rders_simp (RALTS rs) s = rsimp (RALTS (map (\<lambda>r. rders r s) rs))"
+ − 847
apply(case_tac "s")
+ − 848
apply simp
+ − 849
apply simp
+ − 850
sorry
+ − 851
(*
+ − 852
fun rexp_encode :: "rrexp \<Rightarrow> nat"
+ − 853
where
+ − 854
"rexp_encode RZERO = 0"
+ − 855
|"rexp_encode RONE = 1"
+ − 856
|"rexp_encode (RCHAR c) = 2"
+ − 857
|"rexp_encode (RSEQ r1 r2) = ( 2 ^ (rexp_encode r1)) "
+ − 858
*)
438
+ − 859
433
+ − 860
438
+ − 861
+ − 862
433
+ − 863
(*
+ − 864
fun rexp_enum :: "nat \<Rightarrow> rrexp list"
+ − 865
where
+ − 866
"rexp_enum 0 = []"
+ − 867
|"rexp_enum (Suc 0) = RALTS [] # RZERO # (RONE # (map RCHAR (all_chars 255)))"
+ − 868
|"rexp_enum (Suc n) = [(RSEQ r1 r2). r1 \<in> set (rexp_enum i) \<and> r2 \<in> set (rexp_enum j) \<and> i + j = n]"
+ − 869
+ − 870
*)
438
+ − 871
+ − 872
lemma non_zero_size:
+ − 873
shows "rsize r \<ge> Suc 0"
+ − 874
apply(induct r)
+ − 875
apply auto done
+ − 876
+ − 877
corollary size_geq1:
+ − 878
shows "rsize r \<ge> 1"
+ − 879
by (simp add: non_zero_size)
+ − 880
+ − 881
+ − 882
lemma rexp_size_induct:
+ − 883
shows "\<And>N r x5 a list.
+ − 884
\<lbrakk> rsize r = Suc N; r = RALTS x5;
+ − 885
x5 = a # list\<rbrakk> \<Longrightarrow>\<exists>i j. rsize a = i \<and> rsize (RALTS list) = j \<and> i + j = Suc N \<and> i \<le> N \<and> j \<le> N"
+ − 886
apply(rule_tac x = "rsize a" in exI)
+ − 887
apply(rule_tac x = "rsize (RALTS list)" in exI)
+ − 888
apply(subgoal_tac "rsize a \<ge> 1")
+ − 889
prefer 2
+ − 890
using One_nat_def non_zero_size apply presburger
+ − 891
apply(subgoal_tac "rsize (RALTS list) \<ge> 1 ")
+ − 892
prefer 2
+ − 893
using size_geq1 apply blast
+ − 894
apply simp
+ − 895
done
433
+ − 896
442
+ − 897
definition SEQ_set where
+ − 898
"SEQ_set A n \<equiv> {RSEQ r1 r2 | r1 r2. r1 \<in> A \<and> r2 \<in> A \<and> rsize r1 + rsize r2 \<le> n}"
438
+ − 899
442
+ − 900
definition SEQ_set_cartesian where
+ − 901
"SEQ_set_cartesian A n = {RSEQ r1 r2 | r1 r2. r1 \<in> A \<and> r2 \<in> A}"
441
+ − 902
442
+ − 903
definition ALT_set where
+ − 904
"ALT_set A n \<equiv> {RALTS rs | rs. set rs \<subseteq> A \<and> sum_list (map rsize rs) \<le> n}"
441
+ − 905
+ − 906
+ − 907
definition
+ − 908
"sizeNregex N \<equiv> {r. rsize r \<le> N}"
+ − 909
442
+ − 910
lemma sizenregex_induct:
+ − 911
shows "sizeNregex (Suc n) = sizeNregex n \<union> {RZERO, RONE, RALTS []} \<union> {RCHAR c| c. True} \<union>
+ − 912
SEQ_set ( sizeNregex n) n \<union> ALT_set (sizeNregex n) n \<union> (RSTAR ` (sizeNregex n))"
+ − 913
sorry
441
+ − 914
438
+ − 915
442
+ − 916
lemma chars_finite:
+ − 917
shows "finite (RCHAR ` (UNIV::(char set)))"
+ − 918
apply(simp)
+ − 919
done
+ − 920
+ − 921
thm full_SetCompr_eq
441
+ − 922
442
+ − 923
lemma size1finite:
+ − 924
shows "finite (sizeNregex (Suc 0))"
+ − 925
apply(subst sizenregex_induct)
+ − 926
apply(subst finite_Un)+
+ − 927
apply(subgoal_tac "sizeNregex 0 = {}")
+ − 928
apply(rule conjI)+
+ − 929
apply (metis Collect_empty_eq finite.emptyI non_zero_size not_less_eq_eq sizeNregex_def)
+ − 930
apply simp
+ − 931
apply (simp add: full_SetCompr_eq)
+ − 932
apply (simp add: SEQ_set_def)
+ − 933
apply (simp add: ALT_set_def)
+ − 934
apply(simp add: full_SetCompr_eq)
+ − 935
using non_zero_size not_less_eq_eq sizeNregex_def by fastforce
+ − 936
+ − 937
lemma seq_included_in_cart:
+ − 938
shows "SEQ_set A n \<subseteq> SEQ_set_cartesian A n"
+ − 939
sledgehammer
+ − 940
+ − 941
lemma finite_seq:
+ − 942
shows " finite (sizeNregex n) \<Longrightarrow> finite (SEQ_set (sizeNregex n) n)"
+ − 943
apply(rule finite_subset)
+ − 944
sorry
441
+ − 945
+ − 946
442
+ − 947
lemma finite_size_n:
+ − 948
shows "finite (sizeNregex n)"
+ − 949
apply(induct n)
+ − 950
apply (metis Collect_empty_eq finite.emptyI non_zero_size not_less_eq_eq sizeNregex_def)
+ − 951
apply(subst sizenregex_induct)
+ − 952
apply(subst finite_Un)+
+ − 953
apply(rule conjI)+
+ − 954
apply simp
+ − 955
apply simp
+ − 956
apply (simp add: full_SetCompr_eq)
+ − 957
+ − 958
sorry
433
+ − 959
+ − 960
(*below probably needs proved concurrently*)
+ − 961
+ − 962
lemma finite_r1r2_ders_list:
+ − 963
shows "(\<forall>s. rsize (rders_simp r1 s) < N1 \<and> rsize (rders_simp r2 s) < N2)
+ − 964
\<Longrightarrow> \<exists>l. \<forall>s.
+ − 965
(length (rdistinct (map rsimp (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))) {}) ) < l "
+ − 966
sorry
+ − 967
+ − 968
(*
+ − 969
\<lbrakk>s \<noteq> []\<rbrakk> \<Longrightarrow> rders_simp (RSEQ r1 r2) s =
+ − 970
rsimp (RALTS ((RSEQ (rders r1 s) r2) #
+ − 971
(rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))) )
+ − 972
*)
+ − 973
+ − 974
+ − 975
lemma sum_list_size2:
+ − 976
shows "\<forall>z \<in>set rs. (rsize z ) \<le> Nr \<Longrightarrow> rlist_size rs \<le> (length rs) * Nr"
+ − 977
apply(induct rs)
+ − 978
apply simp
+ − 979
by simp
+ − 980
+ − 981
lemma sum_list_size:
+ − 982
fixes rs
+ − 983
shows " \<forall>r \<in> (set rs). (rsize r) \<le> Nr \<and> (length rs) \<le> l \<Longrightarrow> rlist_size rs \<le> l * Nr"
+ − 984
by (metis dual_order.trans mult.commute mult_le_mono2 mult_zero_right sum_list_size2 zero_le)
+ − 985
+ − 986
lemma seq_second_term_chain1:
+ − 987
shows " \<forall>s. rlist_size (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) \<le>
+ − 988
rlist_size (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))"
+ − 989
+ − 990
sorry
+ − 991
+ − 992
+ − 993
lemma seq_second_term_chain2:
+ − 994
shows "\<forall>s. rlist_size (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) =
+ − 995
rlist_size (map rsimp (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)))"
+ − 996
+ − 997
oops
+ − 998
+ − 999
lemma seq_second_term_bounded:
+ − 1000
fixes r2 r1
+ − 1001
assumes "\<forall>s. rsize (rders_simp r2 s) < N2"
+ − 1002
shows "\<exists>N3. \<forall>s. rlist_size (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) < N3"
+ − 1003
+ − 1004
sorry
+ − 1005
+ − 1006
+ − 1007
lemma seq_first_term_bounded:
+ − 1008
fixes r1 r2
+ − 1009
shows "\<exists>Nr. \<forall>s. rsize (rders_simp r1 s) \<le> Nr \<Longrightarrow> \<exists>Nr'. \<forall>s. rsize (RSEQ (rders_simp r1 s) r2) \<le> Nr'"
+ − 1010
apply(erule exE)
+ − 1011
apply(rule_tac x = "Nr + (rsize r2) + 1" in exI)
+ − 1012
by simp
+ − 1013
+ − 1014
+ − 1015
lemma alts_triangle_inequality:
+ − 1016
shows "rsize (RALTS (r # rs)) \<le> rsize r + rlist_size rs + 1 "
+ − 1017
apply(subgoal_tac "rsize (RALTS (r # rs) ) = rsize r + rlist_size rs + 1")
+ − 1018
apply auto[1]
+ − 1019
apply(induct rs)
+ − 1020
apply simp
+ − 1021
apply auto
+ − 1022
done
+ − 1023
+ − 1024
lemma seq_equal_term_nosimp_entire_bounded:
+ − 1025
shows "(\<forall>s. rsize (rders_simp r1 s) < N1 \<and> rsize (rders_simp r2 s) < N2)
+ − 1026
\<Longrightarrow> \<exists>N3. \<forall>s. rsize (RALTS ((RSEQ (rders_simp r1 s) r2) #
+ − 1027
(rdistinct ((rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) ) {}) ) ) \<le> N3"
+ − 1028
apply(subgoal_tac "\<exists>N3. \<forall>s. rsize (RALTS ((RSEQ (rders_simp r1 s) r2) #
+ − 1029
(rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))) ) \<le>
+ − 1030
rsize (RSEQ (rders_simp r1 s) r2) +
+ − 1031
rlist_size (map rsimp (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))) + 1")
+ − 1032
prefer 2
+ − 1033
using alts_triangle_inequality apply presburger
+ − 1034
using seq_first_term_bounded
+ − 1035
using seq_second_term_bounded
+ − 1036
apply(subgoal_tac "\<exists>N3. \<forall>s. rlist_size (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) < N3")
+ − 1037
prefer 2
+ − 1038
apply meson
+ − 1039
apply(subgoal_tac "\<exists>Nr'. \<forall>s. rsize (RSEQ (rders_simp r1 s) r2) \<le> Nr'")
+ − 1040
prefer 2
+ − 1041
apply (meson order_le_less)
+ − 1042
apply(erule exE)
+ − 1043
apply(erule exE)
+ − 1044
sorry
+ − 1045
+ − 1046
lemma alts_simp_bounded_by_sloppy1_version:
+ − 1047
shows "\<forall>s. rsize (rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) #
+ − 1048
(rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))) )) \<le>
+ − 1049
rsize (RALTS (rdistinct ((RSEQ (rders_simp r1 s) r2) #
+ − 1050
(rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))
+ − 1051
)
+ − 1052
{}
+ − 1053
)
+ − 1054
) "
+ − 1055
sorry
+ − 1056
+ − 1057
lemma alts_simp_bounded_by_sloppy1:
+ − 1058
shows "rsize (rsimp (RALTS (rdistinct ((RSEQ (rders_simp r1 s) r2) #
+ − 1059
(rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))
+ − 1060
)
+ − 1061
{}
+ − 1062
)
+ − 1063
)) \<le>
+ − 1064
rsize (RALTS ((RSEQ (rders_simp r1 s) r2) #
+ − 1065
(rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))
+ − 1066
)
+ − 1067
)"
+ − 1068
sorry
+ − 1069
435
+ − 1070
433
+ − 1071
+ − 1072
(*this section deals with the property of distinctBy: creates a list without duplicates*)
+ − 1073
lemma distinct_mono:
+ − 1074
shows "rlist_size (rdistinct (a # s) {}) \<le> rlist_size (a # (rdistinct s {}) )"
+ − 1075
sorry
+ − 1076
+ − 1077
lemma distinct_acc_mono:
+ − 1078
shows "A \<subseteq> B \<Longrightarrow> rlist_size (rdistinct s A) \<ge> rlist_size (rdistinct s B)"
+ − 1079
apply(induct s arbitrary: A B)
+ − 1080
apply simp
+ − 1081
apply(case_tac "a \<in> A")
+ − 1082
apply(subgoal_tac "a \<in> B")
+ − 1083
+ − 1084
apply simp
+ − 1085
+ − 1086
apply blast
+ − 1087
apply(subgoal_tac "rlist_size (rdistinct (a # s) A) = rlist_size (a # (rdistinct s (A \<union> {a})))")
+ − 1088
apply(case_tac "a \<in> B")
+ − 1089
apply(subgoal_tac "rlist_size (rdistinct (a # s) B) = rlist_size ( (rdistinct s B))")
+ − 1090
apply (metis Un_insert_right dual_order.trans insert_subset le_add_same_cancel2 rlist_size.simps(1) sup_bot_right zero_order(1))
+ − 1091
apply simp
+ − 1092
apply auto
+ − 1093
by (meson insert_mono)
+ − 1094
+ − 1095
+ − 1096
lemma distinct_mono2:
+ − 1097
shows " rlist_size (rdistinct s {a}) \<le> rlist_size (rdistinct s {})"
+ − 1098
apply(induct s)
+ − 1099
apply simp
+ − 1100
apply simp
+ − 1101
using distinct_acc_mono by auto
+ − 1102
+ − 1103
+ − 1104
+ − 1105
lemma distinct_mono_spares_first_elem:
+ − 1106
shows "rsize (RALTS (rdistinct (a # s) {})) \<le> rsize (RALTS (a # (rdistinct s {})))"
+ − 1107
apply simp
+ − 1108
apply (subgoal_tac "rlist_size ( (rdistinct s {a})) \<le> rlist_size ( (rdistinct s {})) ")
+ − 1109
using hand_made_def_rlist_size apply auto[1]
+ − 1110
using distinct_mono2 by auto
+ − 1111
+ − 1112
lemma sloppy1_bounded_by_sloppiest:
+ − 1113
shows "rsize (RALTS (rdistinct ((RSEQ (rders_simp r1 s) r2) #
+ − 1114
(rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))
+ − 1115
)
+ − 1116
{}
+ − 1117
)
+ − 1118
) \<le> rsize (RALTS ((RSEQ (rders_simp r1 s) r2) #
+ − 1119
(rdistinct (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) {})
+ − 1120
+ − 1121
+ − 1122
)
+ − 1123
)"
+ − 1124
+ − 1125
sorry
+ − 1126
+ − 1127
+ − 1128
lemma alts_simp_bounded_by_sloppiest_version:
+ − 1129
shows "\<forall>s. rsize (rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) #
+ − 1130
(rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))) )) \<le>
+ − 1131
rsize (RALTS ((RSEQ (rders_simp r1 s) r2) #
+ − 1132
(rdistinct (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) {}) ) ) "
+ − 1133
by (meson alts_simp_bounded_by_sloppy1_version order_trans sloppy1_bounded_by_sloppiest)
+ − 1134
+ − 1135
+ − 1136
lemma seq_equal_term_entire_bounded:
+ − 1137
shows "(\<forall>s. rsize (rders_simp r1 s) < N1 \<and> rsize (rders_simp r2 s) < N2)
+ − 1138
\<Longrightarrow> \<exists>N3. \<forall>s. rsize (rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) #
+ − 1139
(rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))) )) \<le> N3"
+ − 1140
using seq_equal_term_nosimp_entire_bounded
+ − 1141
apply(subgoal_tac " \<exists>N3. \<forall>s. rsize (RALTS ((RSEQ (rders_simp r1 s) r2) #
+ − 1142
(rdistinct (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) {}) ) ) \<le> N3")
+ − 1143
apply(erule exE)
+ − 1144
prefer 2
+ − 1145
apply blast
+ − 1146
apply(subgoal_tac "\<forall>s. rsize (rsimp (RALTS ((RSEQ (rders_simp r1 s) r2) #
+ − 1147
(rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))) )) \<le>
+ − 1148
rsize (RALTS ((RSEQ (rders_simp r1 s) r2) #
+ − 1149
(rdistinct (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) {}) ) ) ")
+ − 1150
prefer 2
+ − 1151
using alts_simp_bounded_by_sloppiest_version apply blast
+ − 1152
apply(rule_tac x = "Suc N3 " in exI)
+ − 1153
apply(rule allI)
+ − 1154
+ − 1155
apply(subgoal_tac " rsize
+ − 1156
(rsimp
+ − 1157
(RALTS
+ − 1158
(RSEQ (rders_simp r1 s) r2 # rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s))))
+ − 1159
\<le> rsize
+ − 1160
(RALTS
+ − 1161
(RSEQ (rders_simp r1 s) r2 #
+ − 1162
rdistinct (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) {}))")
+ − 1163
prefer 2
+ − 1164
apply presburger
+ − 1165
apply(subgoal_tac " rsize
+ − 1166
(RALTS
+ − 1167
(RSEQ (rders_simp r1 s) r2 #
+ − 1168
rdistinct (rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)) {})) \<le> N3")
+ − 1169
+ − 1170
apply linarith
+ − 1171
apply simp
+ − 1172
done
+ − 1173
+ − 1174
+ − 1175
+ − 1176
lemma M1seq:
+ − 1177
fixes r1 r2
+ − 1178
shows "(\<forall>s. rsize (rders_simp r1 s) < N1 \<and> rsize (rders_simp r2 s) < N2)
+ − 1179
\<Longrightarrow> \<exists>N3.\<forall>s.(rsize (rders_simp (RSEQ r1 r2) s)) < N3"
+ − 1180
apply(frule seq_equal_term_entire_bounded)
+ − 1181
apply(erule exE)
+ − 1182
apply(rule_tac x = "max (N3+2) (Suc (Suc (rsize r1) + (rsize r2)))" in exI)
+ − 1183
apply(rule allI)
+ − 1184
apply(case_tac "s = []")
+ − 1185
prefer 2
434
+ − 1186
apply (metis add_2_eq_Suc' le_imp_less_Suc less_SucI max.strict_coboundedI1 shape_derssimp_seq)
+ − 1187
by (metis add.assoc less_Suc_eq less_max_iff_disj plus_1_eq_Suc rders_simp.simps(1) rsize.simps(5))
+ − 1188
433
+ − 1189
(* apply (simp add: less_SucI shape_derssimp_seq(2))
+ − 1190
apply (meson less_SucI less_max_iff_disj)
+ − 1191
apply simp
+ − 1192
done*)
+ − 1193
+ − 1194
(*lemma empty_diff:
+ − 1195
shows "s = [] \<Longrightarrow>
+ − 1196
(rsize (rders_simp (RSEQ r1 r2) s)) \<le>
+ − 1197
(max
+ − 1198
(rsize (rsimp (RALTS (RSEQ (rders r1 s) r2 # rders_cond_list r2 (nullable_bools r1 (orderedPref s)) (orderedSuf s)))))
+ − 1199
(Suc (rsize r1 + rsize r2)) ) "
+ − 1200
apply simp
+ − 1201
done*)
+ − 1202
(*For star related bound*)
+ − 1203
+ − 1204
lemma star_is_a_singleton_list_derc:
434
+ − 1205
shows " \<exists>Ss. rders_simp (RSTAR r) [c] = rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)))"
433
+ − 1206
apply simp
+ − 1207
apply(rule_tac x = "[[c]]" in exI)
+ − 1208
apply auto
434
+ − 1209
apply(case_tac "rsimp (rder c r)")
+ − 1210
apply simp
+ − 1211
apply auto
+ − 1212
apply(subgoal_tac "rsimp (RSEQ x41 x42) = RSEQ x41 x42")
+ − 1213
prefer 2
+ − 1214
apply (metis rsimp_idem)
+ − 1215
apply(subgoal_tac "rsimp x41 = x41")
+ − 1216
prefer 2
+ − 1217
using rsimp_inner_idem1 apply blast
+ − 1218
apply(subgoal_tac "rsimp x42 = x42")
+ − 1219
prefer 2
+ − 1220
using rsimp_inner_idem1 apply blast
+ − 1221
apply simp
+ − 1222
apply(subgoal_tac "map rsimp x5 = x5")
+ − 1223
prefer 2
+ − 1224
using rsimp_inner_idem3 apply blast
+ − 1225
apply simp
+ − 1226
apply(subgoal_tac "rflts x5 = x5")
+ − 1227
prefer 2
+ − 1228
using rsimp_inner_idem4 apply blast
+ − 1229
apply simp
+ − 1230
using rsimp_inner_idem4 by auto
+ − 1231
+ − 1232
433
+ − 1233
+ − 1234
lemma rder_rsimp_ALTs_commute:
+ − 1235
shows " (rder x (rsimp_ALTs rs)) = rsimp_ALTs (map (rder x) rs)"
+ − 1236
apply(induct rs)
+ − 1237
apply simp
+ − 1238
apply(case_tac rs)
+ − 1239
apply simp
+ − 1240
apply auto
+ − 1241
done
+ − 1242
434
+ − 1243
433
+ − 1244
+ − 1245
fun star_update :: "char \<Rightarrow> rrexp \<Rightarrow> char list list => char list list" where
+ − 1246
"star_update c r [] = []"
+ − 1247
|"star_update c r (s # Ss) = (if (rnullable (rders_simp r s))
+ − 1248
then (s@[c]) # [c] # (star_update c r Ss)
435
+ − 1249
else (s@[c]) # (star_update c r Ss) )"
433
+ − 1250
434
+ − 1251
lemma star_update_case1:
+ − 1252
shows "rnullable (rders_simp r s) \<Longrightarrow> star_update c r (s # Ss) = (s @ [c]) # [c] # (star_update c r Ss)"
+ − 1253
+ − 1254
by force
+ − 1255
+ − 1256
lemma star_update_case2:
435
+ − 1257
shows "\<not>rnullable (rders_simp r s) \<Longrightarrow> star_update c r (s # Ss) = (s @ [c]) # (star_update c r Ss)"
+ − 1258
by simp
+ − 1259
+ − 1260
lemma bubble_break: shows "rflts [r, RZERO] = rflts [r]"
+ − 1261
apply(case_tac r)
+ − 1262
apply simp+
+ − 1263
done
+ − 1264
+ − 1265
lemma rsimp_alts_idem_aux1:
+ − 1266
shows "rsimp_ALTs (rdistinct (rflts [rsimp a]) {}) = rsimp (RALTS [a])"
+ − 1267
by force
+ − 1268
+ − 1269
+ − 1270
+ − 1271
lemma rsimp_alts_idem_aux2:
+ − 1272
shows "rsimp a = rsimp (RALTS [a])"
+ − 1273
apply(simp)
+ − 1274
apply(case_tac "rsimp a")
+ − 1275
apply simp+
+ − 1276
apply (metis no_alt_short_list_after_simp no_further_dB_after_simp)
434
+ − 1277
by simp
+ − 1278
+ − 1279
lemma rsimp_alts_idem:
+ − 1280
shows "rsimp (rsimp_ALTs (a # as)) = rsimp (rsimp_ALTs (a # [(rsimp (rsimp_ALTs as))] ))"
435
+ − 1281
apply(induct as)
+ − 1282
apply(subgoal_tac "rsimp (rsimp_ALTs [a, rsimp (rsimp_ALTs [])]) = rsimp (rsimp_ALTs [a, RZERO])")
+ − 1283
prefer 2
+ − 1284
apply simp
+ − 1285
using bubble_break rsimp_alts_idem_aux2 apply auto[1]
+ − 1286
apply(case_tac as)
+ − 1287
apply(subgoal_tac "rsimp_ALTs( aa # as) = aa")
+ − 1288
prefer 2
+ − 1289
apply simp
+ − 1290
using head_one_more_simp apply fastforce
+ − 1291
apply(subgoal_tac "rsimp_ALTs (aa # as) = RALTS (aa # as)")
+ − 1292
prefer 2
+ − 1293
+ − 1294
using rsimp_ALTs.simps(3) apply presburger
+ − 1295
+ − 1296
apply(simp only:)
+ − 1297
apply(subgoal_tac "rsimp_ALTs (a # aa # aaa # list) = RALTS (a # aa # aaa # list)")
+ − 1298
prefer 2
+ − 1299
using rsimp_ALTs.simps(3) apply presburger
+ − 1300
apply(simp only:)
+ − 1301
apply(subgoal_tac "rsimp_ALTs [a, rsimp (RALTS (aa # aaa # list))] = RALTS (a # [rsimp (RALTS (aa # aaa # list))])")
+ − 1302
prefer 2
+ − 1303
+ − 1304
using rsimp_ALTs.simps(3) apply presburger
+ − 1305
apply(simp only:)
+ − 1306
using simp_flatten2
+ − 1307
apply(subgoal_tac " rsimp (RALT a (rsimp (RALTS (aa # aaa # list)))) = rsimp (RALT a ((RALTS (aa # aaa # list)))) ")
+ − 1308
prefer 2
+ − 1309
+ − 1310
apply (metis head_one_more_simp list.simps(9) rsimp.simps(2))
+ − 1311
apply (simp only:)
+ − 1312
done
+ − 1313
434
+ − 1314
+ − 1315
lemma rsimp_alts_idem2:
+ − 1316
shows "rsimp (rsimp_ALTs (a # as)) = rsimp (rsimp_ALTs ((rsimp a) # [(rsimp (rsimp_ALTs as))] ))"
435
+ − 1317
using head_one_more_simp rsimp_alts_idem by auto
+ − 1318
434
+ − 1319
+ − 1320
lemma evolution_step1:
+ − 1321
shows "rsimp
+ − 1322
(rsimp_ALTs
+ − 1323
(rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)) # map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
+ − 1324
rsimp
+ − 1325
(rsimp_ALTs
+ − 1326
(rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)) # [(rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)))])) "
+ − 1327
using rsimp_alts_idem by auto
+ − 1328
+ − 1329
lemma evolution_step2:
+ − 1330
assumes " rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
+ − 1331
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))"
+ − 1332
shows "rsimp
+ − 1333
(rsimp_ALTs
+ − 1334
(rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)) # map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
+ − 1335
rsimp
+ − 1336
(rsimp_ALTs
+ − 1337
(rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)) # [ rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))])) "
+ − 1338
by (simp add: assms rsimp_alts_idem)
+ − 1339
435
+ − 1340
lemma rsimp_seq_aux1:
+ − 1341
shows "r = RONE \<and> r2 = RSTAR r0 \<Longrightarrow> rsimp_SEQ r r2 = r2"
+ − 1342
apply simp
+ − 1343
done
+ − 1344
+ − 1345
lemma multiple_alts_simp_flatten:
+ − 1346
shows "rsimp (RALT (RALT r1 r2) (rsimp_ALTs rs)) = rsimp (RALTS (r1 # r2 # rs))"
+ − 1347
by (metis Cons_eq_appendI append_self_conv2 rsimp_ALTs.simps(2) rsimp_ALTs.simps(3) rsimp_alts_idem simp_flatten)
+ − 1348
+ − 1349
+ − 1350
lemma evo3_main_aux1:
+ − 1351
shows "rsimp
+ − 1352
(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
+ − 1353
(rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))) =
+ − 1354
rsimp
+ − 1355
(RALTS
+ − 1356
(RSEQ (rders_simp r (a @ [x])) (RSTAR r) #
+ − 1357
RSEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))"
+ − 1358
apply(subgoal_tac "rsimp
+ − 1359
(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
+ − 1360
(rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))) =
+ − 1361
rsimp
+ − 1362
(RALT (RALT (RSEQ ( (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
+ − 1363
(rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))) ")
+ − 1364
prefer 2
+ − 1365
apply (simp add: rsimp_idem)
+ − 1366
apply (simp only:)
+ − 1367
apply(subst multiple_alts_simp_flatten)
+ − 1368
by simp
+ − 1369
+ − 1370
+ − 1371
lemma evo3_main_nullable:
+ − 1372
shows "
+ − 1373
\<And>a Ss.
+ − 1374
\<lbrakk>rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
+ − 1375
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)));
+ − 1376
rders_simp r a \<noteq> RONE; rders_simp r a \<noteq> RZERO; rnullable (rders_simp r a)\<rbrakk>
+ − 1377
\<Longrightarrow> rsimp
+ − 1378
(rsimp_ALTs
+ − 1379
[rder x (RSEQ (rders_simp r a) (RSTAR r)),
+ − 1380
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))]) =
+ − 1381
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r (a # Ss))))"
+ − 1382
apply(subgoal_tac "rder x (RSEQ (rders_simp r a) (RSTAR r))
+ − 1383
= RALT (RSEQ (rder x (rders_simp r a)) (RSTAR r)) (RSEQ (rder x r) (RSTAR r))")
+ − 1384
prefer 2
+ − 1385
apply simp
+ − 1386
apply(simp only:)
+ − 1387
apply(subgoal_tac "star_update x r (a # Ss) = (a @ [x]) # [x] # (star_update x r Ss)")
+ − 1388
prefer 2
+ − 1389
using star_update_case1 apply presburger
+ − 1390
apply(simp only:)
+ − 1391
apply(subst List.list.map(2))+
+ − 1392
apply(subgoal_tac "rsimp
+ − 1393
(rsimp_ALTs
+ − 1394
[RALT (RSEQ (rder x (rders_simp r a)) (RSTAR r)) (RSEQ (rder x r) (RSTAR r)),
+ − 1395
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))]) =
+ − 1396
rsimp
+ − 1397
(RALTS
+ − 1398
[RALT (RSEQ (rder x (rders_simp r a)) (RSTAR r)) (RSEQ (rder x r) (RSTAR r)),
+ − 1399
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))])")
+ − 1400
prefer 2
+ − 1401
using rsimp_ALTs.simps(3) apply presburger
+ − 1402
apply(simp only:)
+ − 1403
apply(subgoal_tac " rsimp
+ − 1404
(rsimp_ALTs
+ − 1405
(rsimp_SEQ (rders_simp r (a @ [x])) (RSTAR r) #
+ − 1406
rsimp_SEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))
+ − 1407
=
+ − 1408
rsimp
+ − 1409
(RALTS
+ − 1410
(rsimp_SEQ (rders_simp r (a @ [x])) (RSTAR r) #
+ − 1411
rsimp_SEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))")
+ − 1412
+ − 1413
prefer 2
+ − 1414
using rsimp_ALTs.simps(3) apply presburger
+ − 1415
apply (simp only:)
+ − 1416
apply(subgoal_tac " rsimp
+ − 1417
(RALT (RALT (RSEQ (rder x (rders_simp r a)) (RSTAR r)) (RSEQ ( (rder x r)) (RSTAR r)))
+ − 1418
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))))) =
+ − 1419
rsimp
+ − 1420
(RALT (RALT (RSEQ (rsimp (rder x (rders_simp r a))) (RSTAR r)) (RSEQ (rsimp (rder x r)) (RSTAR r)))
+ − 1421
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))))")
+ − 1422
prefer 2
+ − 1423
apply (simp add: rsimp_idem)
+ − 1424
apply(simp only:)
+ − 1425
apply(subgoal_tac " rsimp
+ − 1426
(RALT (RALT (RSEQ (rsimp (rder x (rders_simp r a))) (RSTAR r)) (RSEQ (rsimp (rder x r)) (RSTAR r)))
+ − 1427
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))))) =
+ − 1428
rsimp
+ − 1429
(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
+ − 1430
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))))")
+ − 1431
prefer 2
+ − 1432
using rders_simp_append rders_simp_one_char rsimp_idem apply presburger
+ − 1433
apply(simp only:)
+ − 1434
apply(subgoal_tac " rsimp
+ − 1435
(RALTS
+ − 1436
(rsimp_SEQ (rders_simp r (a @ [x])) (RSTAR r) #
+ − 1437
rsimp_SEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))) =
+ − 1438
rsimp
+ − 1439
(RALTS
+ − 1440
(RSEQ (rders_simp r (a @ [x])) (RSTAR r) #
+ − 1441
RSEQ (rders_simp r [x]) (RSTAR r) # map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))")
+ − 1442
prefer 2
+ − 1443
apply (smt (z3) idiot2 list.simps(9) rrexp.distinct(9) rsimp.simps(1) rsimp.simps(2) rsimp.simps(3) rsimp.simps(4) rsimp.simps(6) rsimp_idem)
+ − 1444
apply(simp only:)
+ − 1445
apply(subgoal_tac " rsimp
+ − 1446
(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
+ − 1447
(rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))))) =
+ − 1448
rsimp
+ − 1449
(RALT (RALT (RSEQ (rsimp (rders_simp r (a @ [x]))) (RSTAR r)) (RSEQ (rders_simp r [x]) (RSTAR r)))
+ − 1450
( (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))))) ")
+ − 1451
prefer 2
+ − 1452
using rsimp_idem apply force
+ − 1453
apply(simp only:)
+ − 1454
using evo3_main_aux1 by blast
+ − 1455
+ − 1456
+ − 1457
lemma evo3_main_not1:
+ − 1458
shows " \<not>rnullable (rders_simp r a) \<Longrightarrow> rder x (RSEQ (rders_simp r a) (RSTAR r)) = RSEQ (rder x (rders_simp r a)) (RSTAR r)"
+ − 1459
by fastforce
+ − 1460
+ − 1461
+ − 1462
lemma evo3_main_not2:
+ − 1463
shows "\<not>rnullable (rders_simp r a) \<Longrightarrow> rsimp
+ − 1464
(rsimp_ALTs
+ − 1465
(rder x (RSEQ (rders_simp r a) (RSTAR r)) # rs)) = rsimp
+ − 1466
(rsimp_ALTs
+ − 1467
((RSEQ (rders_simp r (a @ [x])) (RSTAR r)) # rs))"
+ − 1468
by (simp add: rders_simp_append rsimp_alts_idem2 rsimp_idem)
+ − 1469
+ − 1470
lemma evo3_main_not3:
+ − 1471
shows "rsimp
+ − 1472
(rsimp_ALTs
+ − 1473
(rsimp_SEQ r1 (RSTAR r) # rs)) =
+ − 1474
rsimp (rsimp_ALTs
+ − 1475
(RSEQ r1 (RSTAR r) # rs))"
+ − 1476
by (metis idiot2 rrexp.distinct(9) rsimp.simps(1) rsimp.simps(3) rsimp.simps(4) rsimp.simps(6) rsimp_alts_idem rsimp_alts_idem2)
+ − 1477
+ − 1478
+ − 1479
lemma evo3_main_notnullable:
+ − 1480
shows "\<And>a Ss.
+ − 1481
\<lbrakk>rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
+ − 1482
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)));
+ − 1483
rders_simp r a \<noteq> RONE; rders_simp r a \<noteq> RZERO; \<not>rnullable (rders_simp r a)\<rbrakk>
+ − 1484
\<Longrightarrow> rsimp
+ − 1485
(rsimp_ALTs
+ − 1486
[rder x (RSEQ (rders_simp r a) (RSTAR r)),
+ − 1487
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))]) =
+ − 1488
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r (a # Ss))))"
+ − 1489
apply(subst star_update_case2)
+ − 1490
apply simp
+ − 1491
apply(subst List.list.map(2))
+ − 1492
apply(subst evo3_main_not2)
+ − 1493
apply simp
+ − 1494
apply(subst evo3_main_not3)
+ − 1495
using rsimp_alts_idem by presburger
+ − 1496
+ − 1497
+ − 1498
lemma evo3_aux2:
+ − 1499
shows "rders_simp r a = RONE \<Longrightarrow> rsimp_SEQ (rders_simp (rders_simp r a) [x]) (RSTAR r) = RZERO"
+ − 1500
by simp
+ − 1501
lemma evo3_aux3:
+ − 1502
shows "rsimp (rsimp_ALTs (RZERO # rs)) = rsimp (rsimp_ALTs rs)"
+ − 1503
by (metis list.simps(8) list.simps(9) rdistinct.simps(1) rflts.simps(1) rflts.simps(2) rsimp.simps(2) rsimp_ALTs.simps(1) rsimp_ALTs.simps(2) rsimp_ALTs.simps(3) rsimp_alts_idem)
+ − 1504
+ − 1505
lemma evo3_aux4:
+ − 1506
shows " rsimp
+ − 1507
(rsimp_ALTs
+ − 1508
[RSEQ (rder x r) (RSTAR r),
+ − 1509
rsimp (rsimp_ALTs rs)]) =
+ − 1510
rsimp
+ − 1511
(rsimp_ALTs
+ − 1512
(rsimp_SEQ (rders_simp r [x]) (RSTAR r) # rs))"
+ − 1513
by (metis rders_simp_one_char rsimp.simps(1) rsimp.simps(6) rsimp_alts_idem rsimp_alts_idem2)
+ − 1514
+ − 1515
lemma evo3_aux5:
+ − 1516
shows "rders_simp r a \<noteq> RONE \<and> rders_simp r a \<noteq> RZERO \<Longrightarrow> rsimp_SEQ (rders_simp r a) (RSTAR r) = RSEQ (rders_simp r a) (RSTAR r)"
+ − 1517
using idiot2 by blast
+ − 1518
+ − 1519
+ − 1520
lemma evolution_step3:
+ − 1521
shows" \<And>a Ss.
+ − 1522
rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss)) =
+ − 1523
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss))) \<Longrightarrow>
+ − 1524
rsimp
+ − 1525
(rsimp_ALTs
+ − 1526
[rder x (rsimp_SEQ (rders_simp r a) (RSTAR r)),
+ − 1527
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)))]) =
+ − 1528
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r (a # Ss))))"
+ − 1529
apply(case_tac "rders_simp r a = RONE")
+ − 1530
apply(subst rsimp_seq_aux1)
+ − 1531
apply simp
+ − 1532
apply(subst rder.simps(6))
+ − 1533
apply(subgoal_tac "rnullable (rders_simp r a)")
+ − 1534
prefer 2
+ − 1535
using rnullable.simps(2) apply presburger
+ − 1536
apply(subst star_update_case1)
+ − 1537
apply simp
+ − 1538
+ − 1539
apply(subst List.list.map)+
+ − 1540
apply(subst rders_simp_append)
+ − 1541
apply(subst evo3_aux2)
+ − 1542
apply simp
+ − 1543
apply(subst evo3_aux3)
+ − 1544
apply(subst evo3_aux4)
+ − 1545
apply simp
+ − 1546
apply(case_tac "rders_simp r a = RZERO")
+ − 1547
+ − 1548
apply (simp add: rsimp_alts_idem2)
+ − 1549
apply(subgoal_tac "rders_simp r (a @ [x]) = RZERO")
+ − 1550
prefer 2
+ − 1551
using rder.simps(1) rders_simp_append rders_simp_one_char rsimp.simps(3) apply presburger
+ − 1552
using rflts.simps(2) rsimp.simps(3) rsimp_SEQ.simps(1) apply presburger
+ − 1553
apply(subst evo3_aux5)
+ − 1554
apply simp
+ − 1555
apply(case_tac "rnullable (rders_simp r a) ")
+ − 1556
using evo3_main_nullable apply blast
+ − 1557
using evo3_main_notnullable apply blast
+ − 1558
done
434
+ − 1559
+ − 1560
(*
+ − 1561
proof (prove)
+ − 1562
goal (1 subgoal):
+ − 1563
1. map f (a # s) = f a # map f s
+ − 1564
Auto solve_direct: the current goal can be solved directly with
+ − 1565
HOL.nitpick_simp(115): map ?f (?x21.0 # ?x22.0) = ?f ?x21.0 # map ?f ?x22.0
+ − 1566
List.list.map(2): map ?f (?x21.0 # ?x22.0) = ?f ?x21.0 # map ?f ?x22.0
+ − 1567
List.list.simps(9): map ?f (?x21.0 # ?x22.0) = ?f ?x21.0 # map ?f ?x22.0
+ − 1568
*)
433
+ − 1569
lemma starseq_list_evolution:
+ − 1570
fixes r :: rrexp and Ss :: "char list list" and x :: char
+ − 1571
shows "rsimp (rsimp_ALTs (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss) ) =
+ − 1572
rsimp (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) (star_update x r Ss)) )"
+ − 1573
apply(induct Ss)
434
+ − 1574
apply simp
+ − 1575
apply(subst List.list.map(2))
+ − 1576
apply(subst evolution_step2)
+ − 1577
apply simp
435
+ − 1578
+ − 1579
433
+ − 1580
sorry
+ − 1581
+ − 1582
+ − 1583
lemma star_seqs_produce_star_seqs:
+ − 1584
shows "rsimp (rsimp_ALTs (map (rder x \<circ> (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss))
+ − 1585
= rsimp (rsimp_ALTs (map ( (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r)))) Ss))"
+ − 1586
by (meson comp_apply)
+ − 1587
434
+ − 1588
lemma map_der_lambda_composition:
+ − 1589
shows "map (rder x) (map (\<lambda>s. f s) Ss) = map (\<lambda>s. (rder x (f s))) Ss"
+ − 1590
by force
+ − 1591
+ − 1592
lemma ralts_vs_rsimpalts:
+ − 1593
shows "rsimp (RALTS rs) = rsimp (rsimp_ALTs rs)"
435
+ − 1594
by (metis evo3_aux3 rsimp_ALTs.simps(2) rsimp_ALTs.simps(3) simp_flatten2)
+ − 1595
433
+ − 1596
+ − 1597
lemma linearity_of_list_of_star_or_starseqs:
+ − 1598
fixes r::rrexp and Ss::"char list list" and x::char
+ − 1599
shows "\<exists>Ssa. rsimp (rder x (rsimp_ALTs (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))) =
434
+ − 1600
rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ssa)))"
+ − 1601
apply(subst rder_rsimp_ALTs_commute)
+ − 1602
apply(subst map_der_lambda_composition)
+ − 1603
using starseq_list_evolution
+ − 1604
apply(rule_tac x = "star_update x r Ss" in exI)
+ − 1605
apply(subst ralts_vs_rsimpalts)
+ − 1606
by simp
+ − 1607
433
+ − 1608
+ − 1609
434
+ − 1610
(*certified correctness---does not depend on any previous sorry*)
+ − 1611
lemma star_list_push_der: shows " \<lbrakk>xs \<noteq> [] \<Longrightarrow> \<exists>Ss. rders_simp (RSTAR r) xs = rsimp (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss));
+ − 1612
xs @ [x] \<noteq> []; xs \<noteq> []\<rbrakk> \<Longrightarrow>
+ − 1613
\<exists>Ss. rders_simp (RSTAR r ) (xs @ [x]) =
+ − 1614
rsimp (RALTS (map (\<lambda>s1. (rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) ) Ss) )"
+ − 1615
apply(subgoal_tac "\<exists>Ss. rders_simp (RSTAR r) xs = rsimp (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))")
+ − 1616
prefer 2
+ − 1617
apply blast
+ − 1618
apply(erule exE)
+ − 1619
apply(subgoal_tac "rders_simp (RSTAR r) (xs @ [x]) = rsimp (rder x (rsimp (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))))")
+ − 1620
prefer 2
+ − 1621
using rders_simp_append
+ − 1622
using rders_simp_one_char apply presburger
+ − 1623
apply(rule_tac x= "Ss" in exI)
+ − 1624
apply(subgoal_tac " rsimp (rder x (rsimp (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)))) =
+ − 1625
rsimp (rsimp (rder x (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))))")
+ − 1626
prefer 2
+ − 1627
using inside_simp_removal rsimp_idem apply presburger
+ − 1628
apply(subgoal_tac "rsimp (rsimp (rder x (RALTS (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)))) =
+ − 1629
rsimp (rsimp (RALTS (map (rder x) (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))))")
+ − 1630
prefer 2
+ − 1631
using rder.simps(4) apply presburger
+ − 1632
apply(subgoal_tac "rsimp (rsimp (RALTS (map (rder x) (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)))) =
+ − 1633
rsimp (rsimp (RALTS (map (\<lambda>s1. (rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r)))) Ss)))")
+ − 1634
apply (metis rsimp_idem)
+ − 1635
by (metis map_der_lambda_composition)
+ − 1636
+ − 1637
lemma simp_in_lambdas :
+ − 1638
shows "
+ − 1639
rsimp (RALTS (map (\<lambda>s1. (rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) ) Ss) ) =
+ − 1640
rsimp (RALTS (map (\<lambda>s1. (rsimp (rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))))) Ss))"
+ − 1641
by (metis (no_types, lifting) comp_apply list.map_comp map_eq_conv rsimp.simps(2) rsimp_idem)
+ − 1642
433
+ − 1643
+ − 1644
lemma starder_is_a_list_of_stars_or_starseqs:
434
+ − 1645
shows "s \<noteq> [] \<Longrightarrow> \<exists>Ss. rders_simp (RSTAR r) s = rsimp (RALTS( (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss)))"
433
+ − 1646
apply(induct s rule: rev_induct)
+ − 1647
apply simp
+ − 1648
apply(case_tac "xs = []")
+ − 1649
using star_is_a_singleton_list_derc
434
+ − 1650
apply(simp)
+ − 1651
apply(subgoal_tac "\<exists>Ss. rders_simp (RSTAR r) (xs @ [x]) =
+ − 1652
rsimp (RALTS (map (\<lambda>s1. rder x (rsimp_SEQ (rders_simp r s1) (RSTAR r))) Ss))")
+ − 1653
prefer 2
+ − 1654
using star_list_push_der apply presburger
+ − 1655
+ − 1656
435
+ − 1657
by (metis ralts_vs_rsimpalts starseq_list_evolution)
+ − 1658
+ − 1659
+ − 1660
lemma starder_is_a_list:
+ − 1661
shows " \<exists>Ss. rders_simp (RSTAR r) s = rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r s1) (RSTAR r)) Ss))) \<or> rders_simp (RSTAR r) s = RSTAR r"
+ − 1662
apply(case_tac s)
+ − 1663
prefer 2
+ − 1664
apply (metis neq_Nil_conv starder_is_a_list_of_stars_or_starseqs)
+ − 1665
apply simp
+ − 1666
done
+ − 1667
+ − 1668
+ − 1669
(** start about bounds here**)
+ − 1670
+ − 1671
+ − 1672
lemma list_simp_size:
+ − 1673
shows "rlist_size (map rsimp rs) \<le> rlist_size rs"
+ − 1674
apply(induct rs)
+ − 1675
apply simp
+ − 1676
apply simp
+ − 1677
apply (subgoal_tac "rsize (rsimp a) \<le> rsize a")
+ − 1678
prefer 2
+ − 1679
using rsimp_mono apply fastforce
+ − 1680
using add_le_mono by presburger
+ − 1681
+ − 1682
lemma inside_list_simp_inside_list:
+ − 1683
shows "r \<in> set rs \<Longrightarrow> rsimp r \<in> set (map rsimp rs)"
+ − 1684
apply (induct rs)
+ − 1685
apply simp
+ − 1686
apply auto
+ − 1687
done
+ − 1688
+ − 1689
+ − 1690
lemma rsize_star_seq_list:
+ − 1691
shows "(\<forall>s. rsize (rders_simp r0 s) < N0 ) \<Longrightarrow> \<exists>N3.\<forall>Ss.
+ − 1692
rlist_size (rdistinct (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss) {}) < N3"
434
+ − 1693
sorry
433
+ − 1694
+ − 1695
435
+ − 1696
lemma rdistinct_bound_by_no_simp:
+ − 1697
shows "
+ − 1698
+ − 1699
rlist_size (rdistinct (map rsimp rs) (set (map rsimp ss)))
+ − 1700
\<le> (rlist_size (rdistinct rs (set ss)))
+ − 1701
"
+ − 1702
apply(induct rs arbitrary: ss)
+ − 1703
apply simp
+ − 1704
apply(case_tac "a \<in> set ss")
+ − 1705
apply(subgoal_tac "rsimp a \<in> set (map rsimp ss)")
+ − 1706
prefer 2
+ − 1707
using inside_list_simp_inside_list apply blast
+ − 1708
+ − 1709
apply simp
+ − 1710
apply simp
+ − 1711
by (metis List.set_insert add_le_mono image_insert insert_absorb rsimp_mono trans_le_add2)
+ − 1712
+ − 1713
+ − 1714
lemma starder_closed_form_bound_aux1:
+ − 1715
shows
+ − 1716
"\<forall>Ss. rsize (rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss)))) \<le>
+ − 1717
Suc (rlist_size ( (rdistinct ( ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss))) {}))) "
+ − 1718
+ − 1719
sorry
+ − 1720
+ − 1721
lemma starder_closed_form_bound:
+ − 1722
shows "(\<forall>s. rsize (rders_simp r0 s) < N0 ) \<Longrightarrow> \<exists>N3.\<forall>Ss.
+ − 1723
rsize(rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss)))) < N3"
+ − 1724
apply(subgoal_tac " \<exists>N3.\<forall>Ss.
+ − 1725
rlist_size (rdistinct (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss) {}) < N3")
+ − 1726
prefer 2
+ − 1727
+ − 1728
using rsize_star_seq_list apply auto[1]
+ − 1729
apply(erule exE)
+ − 1730
apply(rule_tac x = "Suc N3" in exI)
+ − 1731
apply(subgoal_tac "\<forall>Ss. rsize (rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss)))) \<le>
+ − 1732
Suc (rlist_size ( (rdistinct ( ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss))) {})))")
+ − 1733
prefer 2
+ − 1734
using starder_closed_form_bound_aux1 apply blast
+ − 1735
by (meson less_trans_Suc linorder_not_le not_less_eq)
+ − 1736
+ − 1737
+ − 1738
thm starder_closed_form_bound_aux1
+ − 1739
+ − 1740
(*
+ − 1741
"ralts_vs_rsimpalts", , and "starder_closed_form_bound_aux1", which could be due to a bug in Sledgehammer or to inconsistent axioms (including "sorry"s)
+ − 1742
*)
+ − 1743
+ − 1744
lemma starder_size_bound:
+ − 1745
shows "(\<forall>s. rsize (rders_simp r0 s) < N0 ) \<Longrightarrow> \<exists>N3.\<forall>Ss.
+ − 1746
rsize(rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss)))) < N3 \<and>
+ − 1747
rsize (RSTAR r0) < N3"
+ − 1748
apply(subgoal_tac " \<exists>N3.\<forall>Ss.
+ − 1749
rsize(rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss)))) < N3")
+ − 1750
prefer 2
+ − 1751
using starder_closed_form_bound apply blast
+ − 1752
apply(erule exE)
+ − 1753
apply(rule_tac x = "max N3 (Suc (rsize (RSTAR r0)))" in exI)
+ − 1754
using less_max_iff_disj by blast
+ − 1755
+ − 1756
+ − 1757
+ − 1758
433
+ − 1759
lemma finite_star:
+ − 1760
shows "(\<forall>s. rsize (rders_simp r0 s) < N0 )
+ − 1761
\<Longrightarrow> \<exists>N3. \<forall>s.(rsize (rders_simp (RSTAR r0) s)) < N3"
435
+ − 1762
apply(subgoal_tac " \<exists>N3. \<forall>Ss.
+ − 1763
rsize(rsimp (RALTS ( (map (\<lambda>s1. rsimp_SEQ (rders_simp r0 s1) (RSTAR r0)) Ss)))) < N3 \<and>
+ − 1764
rsize (RSTAR r0) < N3")
+ − 1765
prefer 2
+ − 1766
using starder_size_bound apply blast
+ − 1767
apply(erule exE)
+ − 1768
apply(rule_tac x = N3 in exI)
+ − 1769
by (metis starder_is_a_list)
433
+ − 1770
+ − 1771
+ − 1772
lemma rderssimp_zero:
+ − 1773
shows"rders_simp RZERO s = RZERO"
+ − 1774
apply(induction s)
+ − 1775
apply simp
+ − 1776
by simp
+ − 1777
+ − 1778
lemma rderssimp_one:
+ − 1779
shows"rders_simp RONE (a # s) = RZERO"
+ − 1780
apply(induction s)
+ − 1781
apply simp
+ − 1782
by simp
+ − 1783
+ − 1784
lemma rderssimp_char:
+ − 1785
shows "rders_simp (RCHAR c) s = RONE \<or> rders_simp (RCHAR c) s = RZERO \<or> rders_simp (RCHAR c) s = (RCHAR c)"
+ − 1786
apply auto
+ − 1787
by (metis rder.simps(2) rder.simps(3) rders_simp.elims rders_simp.simps(2) rderssimp_one rsimp.simps(4))
+ − 1788
+ − 1789
lemma finite_size_ders:
+ − 1790
fixes r
+ − 1791
shows " \<exists>Nr. \<forall>s. rsize (rders_simp r s) < Nr"
+ − 1792
apply(induct r rule: rrexp.induct)
+ − 1793
apply auto
+ − 1794
apply(rule_tac x = "2" in exI)
+ − 1795
using rderssimp_zero rsize.simps(1) apply presburger
+ − 1796
apply(rule_tac x = "2" in exI)
+ − 1797
apply (metis Suc_1 lessI rders_simp.elims rderssimp_one rsize.simps(1) rsize.simps(2))
+ − 1798
apply(rule_tac x = "2" in meta_spec)
+ − 1799
apply (metis lessI rderssimp_char rsize.simps(1) rsize.simps(2) rsize.simps(3))
+ − 1800
+ − 1801
using M1seq apply blast
+ − 1802
prefer 2
+ − 1803
+ − 1804
apply (simp add: finite_star)
+ − 1805
sorry
+ − 1806
434
+ − 1807
lemma finite_list_of_ders:
+ − 1808
fixes r
+ − 1809
shows"\<exists>dersset. ( (finite dersset) \<and> (\<forall>s. (rders_simp r s) \<in> dersset) )"
+ − 1810
sorry
+ − 1811
+ − 1812
433
+ − 1813
+ − 1814
unused_thms
+ − 1815
lemma seq_ders_shape:
+ − 1816
shows "E"
+ − 1817
+ − 1818
oops
+ − 1819
+ − 1820
(*rsimp (rders (RSEQ r1 r2) s) =
+ − 1821
rsimp RALT [RSEQ (rders r1 s) r2, rders r2 si, ...]
+ − 1822
where si is the i-th shortest suffix of s such that si \<in> L r2"
+ − 1823
*)
+ − 1824
+ − 1825
+ − 1826
end