theory Rec_def2
imports Main
begin

datatype recf =  z
              |  s
              |  id nat nat
              |  Cn nat recf "recf list"
              |  Pr nat recf recf
              |  Mn nat recf 

function rec_exec :: "recf \<Rightarrow> nat list \<Rightarrow> nat"
  where
  "rec_exec z xs = 0" |
  "rec_exec s xs = (Suc (xs ! 0))" |
  "rec_exec (id m n) xs = (xs ! n)" |
  "rec_exec (Cn n f gs) xs = 
             (let ys = (map (\<lambda> a. rec_exec a xs) gs) in 
                                  rec_exec f ys)" | 
  "rec_exec (Pr n f g) xs = 
     (if hd xs = 0 then 
                  rec_exec f (tl xs)
      else rec_exec g ((hd xs - 1) # tl xs @ [rec_exec (Pr n f g) ((hd xs - 1) # tl xs)]))" |
  "rec_exec (Mn n f) xs = (LEAST x. rec_exec f (x # xs) = 0)"
by pat_completeness auto

termination
apply(relation "measures [\<lambda> (r, xs). size r, (\<lambda> (r, xs). hd xs)]")
apply(auto simp add: less_Suc_eq_le intro: trans_le_add2 list_size_estimation')
done


inductive terminate :: "recf \<Rightarrow> nat list \<Rightarrow> bool"
  where
  termi_z: "terminate z [n]"
| termi_s: "terminate s [n]"
| termi_id: "\<lbrakk>n < m; length xs = m\<rbrakk> \<Longrightarrow> terminate (id m n) xs"
| termi_cn: "\<lbrakk>terminate f (map (\<lambda>g. rec_exec g xs) gs); 
              \<forall>g \<in> set gs. terminate g xs; length xs = n\<rbrakk> \<Longrightarrow> terminate (Cn n f gs) xs"
| termi_pr_0: "\<lbrakk>terminate f xs; length xs = n\<rbrakk> \<Longrightarrow> terminate (Pr n f g) (0 # xs)"
| termi_pr_suc: "\<lbrakk>terminate (Pr n f gs) (x # xs);
                  terminate g (x # rec_exec (Pr n f gs) (x # xs) # xs)\<rbrakk> 
                  \<Longrightarrow> terminate (Pr n f gs) (Suc x # xs)"
| termi_mn: "\<lbrakk>length xs = n; rec_exec f (r # xs) = 0;
              \<forall> i < r. terminate f (i # xs) \<and> rec_exec f (i # xs) > 0\<rbrakk> \<Longrightarrow> terminate (Mn n f) xs"

end