theory Happen_withinimports Main Momentbegin(* lemma fixes P :: "('a list) \<Rightarrow> bool" and Q :: "('a list) \<Rightarrow> bool" and k :: nat and f :: "('a list) \<Rightarrow> nat" assumes "\<And> s t. \<lbrakk>P s; \<not> Q s; P (t@s); k < length t\<rbrakk> \<Longrightarrow> f (t@s) < f s" shows "\<And> s t. \<lbrakk> P s; P(t @ s); f(s) * k < length t\<rbrakk> \<Longrightarrow> Q (t@s)" sorry*)text {* The following two notions are introduced to improve the situation. *}definition all_future :: "(('a list) \<Rightarrow> bool) \<Rightarrow> (('a list) \<Rightarrow> bool) \<Rightarrow> ('a list) \<Rightarrow> bool"where "all_future G R s = (\<forall> t. G (t@s) \<longrightarrow> R t)"definition happen_within :: "(('a list) \<Rightarrow> bool) \<Rightarrow> (('a list) \<Rightarrow> bool) \<Rightarrow> nat \<Rightarrow> ('a list) \<Rightarrow> bool"where "happen_within G R k s = all_future G (\<lambda> t. k < length t \<longrightarrow> (\<exists> i \<le> k. R (moment i t @ s) \<and> G (moment i t @ s))) s"lemma happen_within_intro: fixes P :: "('a list) \<Rightarrow> bool" and Q :: "('a list) \<Rightarrow> bool" and k :: nat and f :: "('a list) \<Rightarrow> nat" assumes lt_k: "0 < k" and step: "\<And> s. \<lbrakk>P s; \<not> Q s\<rbrakk> \<Longrightarrow> happen_within P (\<lambda> s'. f s' < f s) k s" shows "\<And> s. P s \<Longrightarrow> happen_within P Q ((f s + 1) * k) s"proof - fix s assume "P s" thus "happen_within P Q ((f s + 1) * k) s" proof(induct n == "f s + 1" arbitrary:s rule:nat_less_induct) fix s assume ih [rule_format]: "\<forall>m<f s + 1. \<forall>x. m = f x + 1 \<longrightarrow> P x \<longrightarrow> happen_within P Q ((f x + 1) * k) x" and ps: "P s" show "happen_within P Q ((f s + 1) * k) s" proof(cases "Q s") case True show ?thesis proof - { fix t from True and ps have "0 \<le> ((f s + 1)*k) \<and> Q (moment 0 t @ s) \<and> P (moment 0 t @ s)" by auto hence "\<exists>i\<le>(f s + 1) * k. Q (moment i t @ s) \<and> P (moment i t @ s)" by auto } thus ?thesis by (auto simp: happen_within_def all_future_def) qed next case False from step [OF ps False] have kk: "happen_within P (\<lambda>s'. f s' < f s) k s" . show ?thesis proof - { fix t assume pts: "P (t @ s)" and ltk: "(f s + 1) * k < length t" from ltk have lt_k_lt: "k < length t" by auto with kk pts obtain i where le_ik: "i \<le> k" and lt_f: "f (moment i t @ s) < f s" and p_m: "P (moment i t @ s)" by (auto simp:happen_within_def all_future_def) from ih [of "f (moment i t @ s) + 1" "(moment i t @ s)", OF _ _ p_m] and lt_f have hw: "happen_within P Q ((f (moment i t @ s) + 1) * k) (moment i t @ s)" by auto have "(\<exists>j\<le>(f s + 1) * k. Q (moment j t @ s) \<and> P (moment j t @ s))" (is "\<exists> j. ?T j") proof - let ?t = "restm i t" have eq_t: "t = ?t @ moment i t" by (simp add:moment_restm_s) have h1: "P (restm i t @ moment i t @ s)" proof - from pts and eq_t have "P ((restm i t @ moment i t) @ s)" by simp thus ?thesis by simp qed moreover have h2: "(f (moment i t @ s) + 1) * k < length (restm i t)" proof - have h: "\<And> x y z. (x::nat) \<le> y \<Longrightarrow> x * z \<le> y * z" by simp from lt_f have "(f (moment i t @ s) + 1) \<le> f s " by simp from h [OF this, of k] have "(f (moment i t @ s) + 1) * k \<le> f s * k" . moreover from le_ik have "\<dots> \<le> ((f s) * k + k - i)" by simp moreover from le_ik lt_k_lt and ltk have "(f s) * k + k - i < length t - i" by simp moreover have "length (restm i t) = length t - i" using length_restm by metis ultimately show ?thesis by simp qed from hw [unfolded happen_within_def all_future_def, rule_format, OF h1 h2] obtain m where le_m: "m \<le> (f (moment i t @ s) + 1) * k" and q_m: "Q (moment m ?t @ moment i t @ s)" and p_m: "P (moment m ?t @ moment i t @ s)" by auto have eq_mm: "moment m ?t @ moment i t @ s = (moment (m+i) t)@s" proof - have "moment m (restm i t) @ moment i t = moment (m + i) t" using moment_plus_split by metis thus ?thesis by simp qed let ?j = "m + i" have "?T ?j" proof - have "m + i \<le> (f s + 1) * k" proof - have h: "\<And> x y z. (x::nat) \<le> y \<Longrightarrow> x * z \<le> y * z" by simp from lt_f have "(f (moment i t @ s) + 1) \<le> f s " by simp from h [OF this, of k] have "(f (moment i t @ s) + 1) * k \<le> f s * k" . with le_m have "m \<le> f s * k" by simp hence "m + i \<le> f s * k + i" by simp with le_ik show ?thesis by simp qed moreover from eq_mm q_m have " Q (moment (m + i) t @ s)" by metis moreover from eq_mm p_m have " P (moment (m + i) t @ s)" by metis ultimately show ?thesis by blast qed thus ?thesis by blast qed } thus ?thesis by (simp add:happen_within_def all_future_def firstn.simps) qed qed qedqedend