--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/thys2/PDerivs.thy Sun Oct 10 18:35:21 2021 +0100
@@ -0,0 +1,561 @@
+
+theory PDerivs
+ imports Spec
+begin
+
+
+
+abbreviation
+ "SEQs rs r \<equiv> (\<Union>r' \<in> rs. {SEQ r' r})"
+
+lemma SEQs_eq_image:
+ "SEQs rs r = (\<lambda>r'. SEQ r' r) ` rs"
+ by auto
+
+primrec
+ pder :: "char \<Rightarrow> rexp \<Rightarrow> rexp set"
+where
+ "pder c ZERO = {}"
+| "pder c ONE = {}"
+| "pder c (CHAR d) = (if c = d then {ONE} else {})"
+| "pder c (ALT r1 r2) = (pder c r1) \<union> (pder c r2)"
+| "pder c (SEQ r1 r2) =
+ (if nullable r1 then SEQs (pder c r1) r2 \<union> pder c r2 else SEQs (pder c r1) r2)"
+| "pder c (STAR r) = SEQs (pder c r) (STAR r)"
+
+primrec
+ pders :: "char list \<Rightarrow> rexp \<Rightarrow> rexp set"
+where
+ "pders [] r = {r}"
+| "pders (c # s) r = \<Union> (pders s ` pder c r)"
+
+abbreviation
+ pder_set :: "char \<Rightarrow> rexp set \<Rightarrow> rexp set"
+where
+ "pder_set c rs \<equiv> \<Union> (pder c ` rs)"
+
+abbreviation
+ pders_set :: "char list \<Rightarrow> rexp set \<Rightarrow> rexp set"
+where
+ "pders_set s rs \<equiv> \<Union> (pders s ` rs)"
+
+lemma pders_append:
+ "pders (s1 @ s2) r = \<Union> (pders s2 ` pders s1 r)"
+by (induct s1 arbitrary: r) (simp_all)
+
+lemma pders_snoc:
+ shows "pders (s @ [c]) r = pder_set c (pders s r)"
+by (simp add: pders_append)
+
+lemma pders_simps [simp]:
+ shows "pders s ZERO = (if s = [] then {ZERO} else {})"
+ and "pders s ONE = (if s = [] then {ONE} else {})"
+ and "pders s (ALT r1 r2) = (if s = [] then {ALT r1 r2} else (pders s r1) \<union> (pders s r2))"
+by (induct s) (simp_all)
+
+lemma pders_CHAR:
+ shows "pders s (CHAR c) \<subseteq> {CHAR c, ONE}"
+by (induct s) (simp_all)
+
+subsection \<open>Relating left-quotients and partial derivatives\<close>
+
+lemma Sequ_UNION_distrib:
+shows "A ;; \<Union>(M ` I) = \<Union>((\<lambda>i. A ;; M i) ` I)"
+and "\<Union>(M ` I) ;; A = \<Union>((\<lambda>i. M i ;; A) ` I)"
+by (auto simp add: Sequ_def)
+
+
+lemma Der_pder:
+ shows "Der c (L r) = \<Union> (L ` pder c r)"
+by (induct r) (simp_all add: nullable_correctness Sequ_UNION_distrib)
+
+lemma Ders_pders:
+ shows "Ders s (L r) = \<Union> (L ` pders s r)"
+proof (induct s arbitrary: r)
+ case (Cons c s)
+ have ih: "\<And>r. Ders s (L r) = \<Union> (L ` pders s r)" by fact
+ have "Ders (c # s) (L r) = Ders s (Der c (L r))" by (simp add: Ders_def Der_def)
+ also have "\<dots> = Ders s (\<Union> (L ` pder c r))" by (simp add: Der_pder)
+ also have "\<dots> = (\<Union>A\<in>(L ` (pder c r)). (Ders s A))"
+ by (auto simp add: Ders_def)
+ also have "\<dots> = \<Union> (L ` (pders_set s (pder c r)))"
+ using ih by auto
+ also have "\<dots> = \<Union> (L ` (pders (c # s) r))" by simp
+ finally show "Ders (c # s) (L r) = \<Union> (L ` pders (c # s) r)" .
+qed (simp add: Ders_def)
+
+subsection \<open>Relating derivatives and partial derivatives\<close>
+
+lemma der_pder:
+ shows "\<Union> (L ` (pder c r)) = L (der c r)"
+unfolding der_correctness Der_pder by simp
+
+lemma ders_pders:
+ shows "\<Union> (L ` (pders s r)) = L (ders s r)"
+unfolding der_correctness ders_correctness Ders_pders by simp
+
+
+subsection \<open>Finiteness property of partial derivatives\<close>
+
+definition
+ pders_Set :: "string set \<Rightarrow> rexp \<Rightarrow> rexp set"
+where
+ "pders_Set A r \<equiv> \<Union>x \<in> A. pders x r"
+
+lemma pders_Set_subsetI:
+ assumes "\<And>s. s \<in> A \<Longrightarrow> pders s r \<subseteq> C"
+ shows "pders_Set A r \<subseteq> C"
+using assms unfolding pders_Set_def by (rule UN_least)
+
+lemma pders_Set_union:
+ shows "pders_Set (A \<union> B) r = (pders_Set A r \<union> pders_Set B r)"
+by (simp add: pders_Set_def)
+
+lemma pders_Set_subset:
+ shows "A \<subseteq> B \<Longrightarrow> pders_Set A r \<subseteq> pders_Set B r"
+by (auto simp add: pders_Set_def)
+
+definition
+ "UNIV1 \<equiv> UNIV - {[]}"
+
+lemma pders_Set_ZERO [simp]:
+ shows "pders_Set UNIV1 ZERO = {}"
+unfolding UNIV1_def pders_Set_def by auto
+
+lemma pders_Set_ONE [simp]:
+ shows "pders_Set UNIV1 ONE = {}"
+unfolding UNIV1_def pders_Set_def by (auto split: if_splits)
+
+lemma pders_Set_CHAR [simp]:
+ shows "pders_Set UNIV1 (CHAR c) = {ONE}"
+unfolding UNIV1_def pders_Set_def
+apply(auto)
+apply(frule rev_subsetD)
+apply(rule pders_CHAR)
+apply(simp)
+apply(case_tac xa)
+apply(auto split: if_splits)
+done
+
+lemma pders_Set_ALT [simp]:
+ shows "pders_Set UNIV1 (ALT r1 r2) = pders_Set UNIV1 r1 \<union> pders_Set UNIV1 r2"
+unfolding UNIV1_def pders_Set_def by auto
+
+
+text \<open>Non-empty suffixes of a string (needed for the cases of @{const SEQ} and @{const STAR} below)\<close>
+
+definition
+ "PSuf s \<equiv> {v. v \<noteq> [] \<and> (\<exists>u. u @ v = s)}"
+
+lemma PSuf_snoc:
+ shows "PSuf (s @ [c]) = (PSuf s) ;; {[c]} \<union> {[c]}"
+unfolding PSuf_def Sequ_def
+by (auto simp add: append_eq_append_conv2 append_eq_Cons_conv)
+
+lemma PSuf_Union:
+ shows "(\<Union>v \<in> PSuf s ;; {[c]}. f v) = (\<Union>v \<in> PSuf s. f (v @ [c]))"
+by (auto simp add: Sequ_def)
+
+lemma pders_Set_snoc:
+ shows "pders_Set (PSuf s ;; {[c]}) r = (pder_set c (pders_Set (PSuf s) r))"
+unfolding pders_Set_def
+by (simp add: PSuf_Union pders_snoc)
+
+lemma pders_SEQ:
+ shows "pders s (SEQ r1 r2) \<subseteq> SEQs (pders s r1) r2 \<union> (pders_Set (PSuf s) r2)"
+proof (induct s rule: rev_induct)
+ case (snoc c s)
+ have ih: "pders s (SEQ r1 r2) \<subseteq> SEQs (pders s r1) r2 \<union> (pders_Set (PSuf s) r2)"
+ by fact
+ have "pders (s @ [c]) (SEQ r1 r2) = pder_set c (pders s (SEQ r1 r2))"
+ by (simp add: pders_snoc)
+ also have "\<dots> \<subseteq> pder_set c (SEQs (pders s r1) r2 \<union> (pders_Set (PSuf s) r2))"
+ using ih by fastforce
+ also have "\<dots> = pder_set c (SEQs (pders s r1) r2) \<union> pder_set c (pders_Set (PSuf s) r2)"
+ by (simp)
+ also have "\<dots> = pder_set c (SEQs (pders s r1) r2) \<union> pders_Set (PSuf s ;; {[c]}) r2"
+ by (simp add: pders_Set_snoc)
+ also
+ have "\<dots> \<subseteq> pder_set c (SEQs (pders s r1) r2) \<union> pder c r2 \<union> pders_Set (PSuf s ;; {[c]}) r2"
+ by auto
+ also
+ have "\<dots> \<subseteq> SEQs (pder_set c (pders s r1)) r2 \<union> pder c r2 \<union> pders_Set (PSuf s ;; {[c]}) r2"
+ by (auto simp add: if_splits)
+ also have "\<dots> = SEQs (pders (s @ [c]) r1) r2 \<union> pder c r2 \<union> pders_Set (PSuf s ;; {[c]}) r2"
+ by (simp add: pders_snoc)
+ also have "\<dots> \<subseteq> SEQs (pders (s @ [c]) r1) r2 \<union> pders_Set (PSuf (s @ [c])) r2"
+ unfolding pders_Set_def by (auto simp add: PSuf_snoc)
+ finally show ?case .
+qed (simp)
+
+lemma pders_Set_SEQ_aux1:
+ assumes a: "s \<in> UNIV1"
+ shows "pders_Set (PSuf s) r \<subseteq> pders_Set UNIV1 r"
+using a unfolding UNIV1_def PSuf_def pders_Set_def by auto
+
+lemma pders_Set_SEQ_aux2:
+ assumes a: "s \<in> UNIV1"
+ shows "SEQs (pders s r1) r2 \<subseteq> SEQs (pders_Set UNIV1 r1) r2"
+using a unfolding pders_Set_def by auto
+
+lemma pders_Set_SEQ:
+ shows "pders_Set UNIV1 (SEQ r1 r2) \<subseteq> SEQs (pders_Set UNIV1 r1) r2 \<union> pders_Set UNIV1 r2"
+apply(rule pders_Set_subsetI)
+apply(rule subset_trans)
+apply(rule pders_SEQ)
+using pders_Set_SEQ_aux1 pders_Set_SEQ_aux2
+apply auto
+apply blast
+done
+
+lemma pders_STAR:
+ assumes a: "s \<noteq> []"
+ shows "pders s (STAR r) \<subseteq> SEQs (pders_Set (PSuf s) r) (STAR r)"
+using a
+proof (induct s rule: rev_induct)
+ case (snoc c s)
+ have ih: "s \<noteq> [] \<Longrightarrow> pders s (STAR r) \<subseteq> SEQs (pders_Set (PSuf s) r) (STAR r)" by fact
+ { assume asm: "s \<noteq> []"
+ have "pders (s @ [c]) (STAR r) = pder_set c (pders s (STAR r))" by (simp add: pders_snoc)
+ also have "\<dots> \<subseteq> pder_set c (SEQs (pders_Set (PSuf s) r) (STAR r))"
+ using ih[OF asm] by fast
+ also have "\<dots> \<subseteq> SEQs (pder_set c (pders_Set (PSuf s) r)) (STAR r) \<union> pder c (STAR r)"
+ by (auto split: if_splits)
+ also have "\<dots> \<subseteq> SEQs (pders_Set (PSuf (s @ [c])) r) (STAR r) \<union> (SEQs (pder c r) (STAR r))"
+ by (simp only: PSuf_snoc pders_Set_snoc pders_Set_union)
+ (auto simp add: pders_Set_def)
+ also have "\<dots> = SEQs (pders_Set (PSuf (s @ [c])) r) (STAR r)"
+ by (auto simp add: PSuf_snoc PSuf_Union pders_snoc pders_Set_def)
+ finally have ?case .
+ }
+ moreover
+ { assume asm: "s = []"
+ then have ?case by (auto simp add: pders_Set_def pders_snoc PSuf_def)
+ }
+ ultimately show ?case by blast
+qed (simp)
+
+lemma pders_Set_STAR:
+ shows "pders_Set UNIV1 (STAR r) \<subseteq> SEQs (pders_Set UNIV1 r) (STAR r)"
+apply(rule pders_Set_subsetI)
+apply(rule subset_trans)
+apply(rule pders_STAR)
+apply(simp add: UNIV1_def)
+apply(simp add: UNIV1_def PSuf_def)
+apply(auto simp add: pders_Set_def)
+done
+
+lemma finite_SEQs [simp]:
+ assumes a: "finite A"
+ shows "finite (SEQs A r)"
+using a by auto
+
+thm finite.intros
+
+lemma finite_pders_Set_UNIV1:
+ shows "finite (pders_Set UNIV1 r)"
+apply(induct r)
+apply(simp_all add:
+ finite_subset[OF pders_Set_SEQ]
+ finite_subset[OF pders_Set_STAR])
+done
+
+lemma pders_Set_UNIV:
+ shows "pders_Set UNIV r = pders [] r \<union> pders_Set UNIV1 r"
+unfolding UNIV1_def pders_Set_def
+by blast
+
+lemma finite_pders_Set_UNIV:
+ shows "finite (pders_Set UNIV r)"
+unfolding pders_Set_UNIV
+by (simp add: finite_pders_Set_UNIV1)
+
+lemma finite_pders_set:
+ shows "finite (pders_Set A r)"
+by (metis finite_pders_Set_UNIV pders_Set_subset rev_finite_subset subset_UNIV)
+
+
+text \<open>The following relationship between the alphabetic width of regular expressions
+(called \<open>awidth\<close> below) and the number of partial derivatives was proved
+by Antimirov~\cite{Antimirov95} and formalized by Max Haslbeck.\<close>
+
+fun awidth :: "rexp \<Rightarrow> nat" where
+"awidth ZERO = 0" |
+"awidth ONE = 0" |
+"awidth (CHAR a) = 1" |
+"awidth (ALT r1 r2) = awidth r1 + awidth r2" |
+"awidth (SEQ r1 r2) = awidth r1 + awidth r2" |
+"awidth (STAR r1) = awidth r1"
+
+lemma card_SEQs_pders_Set_le:
+ shows "card (SEQs (pders_Set A r) s) \<le> card (pders_Set A r)"
+ using finite_pders_set
+ unfolding SEQs_eq_image
+ by (rule card_image_le)
+
+lemma card_pders_set_UNIV1_le_awidth:
+ shows "card (pders_Set UNIV1 r) \<le> awidth r"
+proof (induction r)
+ case (ALT r1 r2)
+ have "card (pders_Set UNIV1 (ALT r1 r2)) = card (pders_Set UNIV1 r1 \<union> pders_Set UNIV1 r2)" by simp
+ also have "\<dots> \<le> card (pders_Set UNIV1 r1) + card (pders_Set UNIV1 r2)"
+ by(simp add: card_Un_le)
+ also have "\<dots> \<le> awidth (ALT r1 r2)" using ALT.IH by simp
+ finally show ?case .
+next
+ case (SEQ r1 r2)
+ have "card (pders_Set UNIV1 (SEQ r1 r2)) \<le> card (SEQs (pders_Set UNIV1 r1) r2 \<union> pders_Set UNIV1 r2)"
+ by (simp add: card_mono finite_pders_set pders_Set_SEQ)
+ also have "\<dots> \<le> card (SEQs (pders_Set UNIV1 r1) r2) + card (pders_Set UNIV1 r2)"
+ by (simp add: card_Un_le)
+ also have "\<dots> \<le> card (pders_Set UNIV1 r1) + card (pders_Set UNIV1 r2)"
+ by (simp add: card_SEQs_pders_Set_le)
+ also have "\<dots> \<le> awidth (SEQ r1 r2)" using SEQ.IH by simp
+ finally show ?case .
+next
+ case (STAR r)
+ have "card (pders_Set UNIV1 (STAR r)) \<le> card (SEQs (pders_Set UNIV1 r) (STAR r))"
+ by (simp add: card_mono finite_pders_set pders_Set_STAR)
+ also have "\<dots> \<le> card (pders_Set UNIV1 r)" by (rule card_SEQs_pders_Set_le)
+ also have "\<dots> \<le> awidth (STAR r)" by (simp add: STAR.IH)
+ finally show ?case .
+qed (auto)
+
+text\<open>Antimirov's Theorem 3.4:\<close>
+
+theorem card_pders_set_UNIV_le_awidth:
+ shows "card (pders_Set UNIV r) \<le> awidth r + 1"
+proof -
+ have "card (insert r (pders_Set UNIV1 r)) \<le> Suc (card (pders_Set UNIV1 r))"
+ by(auto simp: card_insert_if[OF finite_pders_Set_UNIV1])
+ also have "\<dots> \<le> Suc (awidth r)" by(simp add: card_pders_set_UNIV1_le_awidth)
+ finally show ?thesis by(simp add: pders_Set_UNIV)
+qed
+
+text\<open>Antimirov's Corollary 3.5:\<close>
+
+corollary card_pders_set_le_awidth:
+ shows "card (pders_Set A r) \<le> awidth r + 1"
+proof -
+ have "card (pders_Set A r) \<le> card (pders_Set UNIV r)"
+ by (simp add: card_mono finite_pders_set pders_Set_subset)
+ also have "... \<le> awidth r + 1"
+ by (rule card_pders_set_UNIV_le_awidth)
+ finally show "card (pders_Set A r) \<le> awidth r + 1" by simp
+qed
+
+(* other result by antimirov *)
+
+lemma card_pders_awidth:
+ shows "card (pders s r) \<le> awidth r + 1"
+proof -
+ have "pders s r \<subseteq> pders_Set UNIV r"
+ using pders_Set_def by auto
+ then have "card (pders s r) \<le> card (pders_Set UNIV r)"
+ by (simp add: card_mono finite_pders_set)
+ then show "card (pders s r) \<le> awidth r + 1"
+ using card_pders_set_le_awidth order_trans by blast
+qed
+
+
+
+
+
+fun subs :: "rexp \<Rightarrow> rexp set" where
+"subs ZERO = {ZERO}" |
+"subs ONE = {ONE}" |
+"subs (CHAR a) = {CHAR a, ONE}" |
+"subs (ALT r1 r2) = (subs r1 \<union> subs r2 \<union> {ALT r1 r2})" |
+"subs (SEQ r1 r2) = (subs r1 \<union> subs r2 \<union> {SEQ r1 r2} \<union> SEQs (subs r1) r2)" |
+"subs (STAR r1) = (subs r1 \<union> {STAR r1} \<union> SEQs (subs r1) (STAR r1))"
+
+lemma subs_finite:
+ shows "finite (subs r)"
+ apply(induct r)
+ apply(simp_all)
+ done
+
+
+
+lemma pders_subs:
+ shows "pders s r \<subseteq> subs r"
+ apply(induct r arbitrary: s)
+ apply(simp)
+ apply(simp)
+ apply(simp add: pders_CHAR)
+(* SEQ case *)
+ apply(simp)
+ apply(rule subset_trans)
+ apply(rule pders_SEQ)
+ defer
+(* ALT case *)
+ apply(simp)
+ apply(rule impI)
+ apply(rule conjI)
+ apply blast
+ apply blast
+(* STAR case *)
+ apply(case_tac s)
+ apply(simp)
+ apply(rule subset_trans)
+ thm pders_STAR
+ apply(rule pders_STAR)
+ apply(simp)
+ apply(auto simp add: pders_Set_def)[1]
+ apply(simp)
+ apply(rule conjI)
+ apply blast
+apply(auto simp add: pders_Set_def)[1]
+ done
+
+fun size2 :: "rexp \<Rightarrow> nat" where
+ "size2 ZERO = 1" |
+ "size2 ONE = 1" |
+ "size2 (CHAR c) = 1" |
+ "size2 (ALT r1 r2) = Suc (size2 r1 + size2 r2)" |
+ "size2 (SEQ r1 r2) = Suc (size2 r1 + size2 r2)" |
+ "size2 (STAR r1) = Suc (size2 r1)"
+
+
+lemma size_rexp:
+ fixes r :: rexp
+ shows "1 \<le> size2 r"
+ apply(induct r)
+ apply(simp)
+ apply(simp_all)
+ done
+
+lemma subs_card:
+ shows "card (subs r) \<le> Suc (size2 r + size2 r)"
+ apply(induct r)
+ apply(auto)
+ apply(subst card_insert)
+ apply(simp add: subs_finite)
+ apply(simp add: subs_finite)
+ oops
+
+lemma subs_size2:
+ shows "\<forall>r1 \<in> subs r. size2 r1 \<le> Suc (size2 r * size2 r)"
+ apply(induct r)
+ apply(simp)
+ apply(simp)
+ apply(simp)
+(* SEQ case *)
+ apply(simp)
+ apply(auto)[1]
+ apply (smt Suc_n_not_le_n add.commute distrib_left le_Suc_eq left_add_mult_distrib nat_le_linear trans_le_add1)
+ apply (smt Suc_le_mono Suc_n_not_le_n le_trans nat_le_linear power2_eq_square power2_sum semiring_normalization_rules(23) trans_le_add2)
+ apply (smt Groups.add_ac(3) Suc_n_not_le_n distrib_left le_Suc_eq left_add_mult_distrib nat_le_linear trans_le_add1)
+(* ALT case *)
+ apply(simp)
+ apply(auto)[1]
+ apply (smt Groups.add_ac(2) Suc_le_mono Suc_n_not_le_n le_add2 linear order_trans power2_eq_square power2_sum)
+ apply (smt Groups.add_ac(2) Suc_le_mono Suc_n_not_le_n left_add_mult_distrib linear mult.commute order.trans trans_le_add1)
+(* STAR case *)
+ apply(auto)[1]
+ apply(drule_tac x="r'" in bspec)
+ apply(simp)
+ apply(rule le_trans)
+ apply(assumption)
+ apply(simp)
+ using size_rexp
+ apply(simp)
+ done
+
+lemma awidth_size:
+ shows "awidth r \<le> size2 r"
+ apply(induct r)
+ apply(simp_all)
+ done
+
+lemma Sum1:
+ fixes A B :: "nat set"
+ assumes "A \<subseteq> B" "finite A" "finite B"
+ shows "\<Sum>A \<le> \<Sum>B"
+ using assms
+ by (simp add: sum_mono2)
+
+lemma Sum2:
+ fixes A :: "rexp set"
+ and f g :: "rexp \<Rightarrow> nat"
+ assumes "finite A" "\<forall>x \<in> A. f x \<le> g x"
+ shows "sum f A \<le> sum g A"
+ using assms
+ apply(induct A)
+ apply(auto)
+ done
+
+
+
+
+
+lemma pders_max_size:
+ shows "(sum size2 (pders s r)) \<le> (Suc (size2 r)) ^ 3"
+proof -
+ have "(sum size2 (pders s r)) \<le> sum (\<lambda>_. Suc (size2 r * size2 r)) (pders s r)"
+ apply(rule_tac Sum2)
+ apply (meson pders_subs rev_finite_subset subs_finite)
+ using pders_subs subs_size2 by blast
+ also have "... \<le> (Suc (size2 r * size2 r)) * (sum (\<lambda>_. 1) (pders s r))"
+ by simp
+ also have "... \<le> (Suc (size2 r * size2 r)) * card (pders s r)"
+ by simp
+ also have "... \<le> (Suc (size2 r * size2 r)) * (Suc (awidth r))"
+ using Suc_eq_plus1 card_pders_awidth mult_le_mono2 by presburger
+ also have "... \<le> (Suc (size2 r * size2 r)) * (Suc (size2 r))"
+ using Suc_le_mono awidth_size mult_le_mono2 by presburger
+ also have "... \<le> (Suc (size2 r)) ^ 3"
+ by (smt One_nat_def Suc_1 Suc_mult_le_cancel1 Suc_n_not_le_n antisym_conv le_Suc_eq mult.commute nat_le_linear numeral_3_eq_3 power2_eq_square power2_le_imp_le power_Suc size_rexp)
+ finally show ?thesis .
+qed
+
+lemma pders_Set_max_size:
+ shows "(sum size2 (pders_Set A r)) \<le> (Suc (size2 r)) ^ 3"
+proof -
+ have "(sum size2 (pders_Set A r)) \<le> sum (\<lambda>_. Suc (size2 r * size2 r)) (pders_Set A r)"
+ apply(rule_tac Sum2)
+ apply (simp add: finite_pders_set)
+ by (meson pders_Set_subsetI pders_subs subs_size2 subsetD)
+ also have "... \<le> (Suc (size2 r * size2 r)) * (sum (\<lambda>_. 1) (pders_Set A r))"
+ by simp
+ also have "... \<le> (Suc (size2 r * size2 r)) * card (pders_Set A r)"
+ by simp
+ also have "... \<le> (Suc (size2 r * size2 r)) * (Suc (awidth r))"
+ using Suc_eq_plus1 card_pders_set_le_awidth mult_le_mono2 by presburger
+ also have "... \<le> (Suc (size2 r * size2 r)) * (Suc (size2 r))"
+ using Suc_le_mono awidth_size mult_le_mono2 by presburger
+ also have "... \<le> (Suc (size2 r)) ^ 3"
+ by (smt One_nat_def Suc_1 Suc_mult_le_cancel1 Suc_n_not_le_n antisym_conv le_Suc_eq mult.commute nat_le_linear numeral_3_eq_3 power2_eq_square power2_le_imp_le power_Suc size_rexp)
+ finally show ?thesis .
+qed
+
+fun height :: "rexp \<Rightarrow> nat" where
+ "height ZERO = 1" |
+ "height ONE = 1" |
+ "height (CHAR c) = 1" |
+ "height (ALT r1 r2) = Suc (max (height r1) (height r2))" |
+ "height (SEQ r1 r2) = Suc (max (height r1) (height r2))" |
+ "height (STAR r1) = Suc (height r1)"
+
+lemma height_size2:
+ shows "height r \<le> size2 r"
+ apply(induct r)
+ apply(simp_all)
+ done
+
+lemma height_rexp:
+ fixes r :: rexp
+ shows "1 \<le> height r"
+ apply(induct r)
+ apply(simp_all)
+ done
+
+lemma subs_height:
+ shows "\<forall>r1 \<in> subs r. height r1 \<le> Suc (height r)"
+ apply(induct r)
+ apply(auto)+
+ done
+
+
+
+end
\ No newline at end of file