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1 theory Rec_def2 |
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2 imports Main |
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3 begin |
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4 |
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5 datatype recf = z |
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6 | s |
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7 | id nat nat |
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8 | Cn nat recf "recf list" |
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9 | Pr nat recf recf |
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10 | Mn nat recf |
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11 |
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12 function rec_exec :: "recf \<Rightarrow> nat list \<Rightarrow> nat" |
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13 where |
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14 "rec_exec z xs = 0" | |
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15 "rec_exec s xs = (Suc (xs ! 0))" | |
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16 "rec_exec (id m n) xs = (xs ! n)" | |
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17 "rec_exec (Cn n f gs) xs = |
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18 (let ys = (map (\<lambda> a. rec_exec a xs) gs) in |
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19 rec_exec f ys)" | |
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20 "rec_exec (Pr n f g) xs = |
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21 (if hd xs = 0 then |
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22 rec_exec f (tl xs) |
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23 else rec_exec g ((hd xs - 1) # tl xs @ [rec_exec (Pr n f g) ((hd xs - 1) # tl xs)]))" | |
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24 "rec_exec (Mn n f) xs = (LEAST x. rec_exec f (x # xs) = 0)" |
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25 by pat_completeness auto |
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26 |
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27 termination |
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28 apply(relation "measures [\<lambda> (r, xs). size r, (\<lambda> (r, xs). hd xs)]") |
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29 apply(auto simp add: less_Suc_eq_le intro: trans_le_add2 list_size_estimation') |
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30 done |
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31 |
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32 |
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33 inductive terminate :: "recf \<Rightarrow> nat list \<Rightarrow> bool" |
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34 where |
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35 termi_z: "terminate z [n]" |
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36 | termi_s: "terminate s [n]" |
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37 | termi_id: "\<lbrakk>n < m; length xs = m\<rbrakk> \<Longrightarrow> terminate (id m n) xs" |
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38 | termi_cn: "\<lbrakk>terminate f (map (\<lambda>g. rec_exec g xs) gs); |
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39 \<forall>g \<in> set gs. terminate g xs; length xs = n\<rbrakk> \<Longrightarrow> terminate (Cn n f gs) xs" |
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40 | termi_pr_0: "\<lbrakk>terminate f xs; length xs = n\<rbrakk> \<Longrightarrow> terminate (Pr n f g) (0 # xs)" |
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41 | termi_pr_suc: "\<lbrakk>terminate (Pr n f gs) (x # xs); |
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42 terminate g (x # rec_exec (Pr n f gs) (x # xs) # xs)\<rbrakk> |
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43 \<Longrightarrow> terminate (Pr n f gs) (Suc x # xs)" |
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44 | termi_mn: "\<lbrakk>length xs = n; rec_exec f (r # xs) = 0; |
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45 \<forall> i < r. terminate f (i # xs) \<and> rec_exec f (i # xs) > 0\<rbrakk> \<Longrightarrow> terminate (Mn n f) xs" |
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46 |
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47 end |