theory Happen_within
imports Main Moment
begin

(* 
  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
  qed
qed

end