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1 theory SCT_Examples |
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2 imports Size_Change_Termination |
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3 begin |
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4 |
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5 |
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6 |
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7 function f :: "nat \<Rightarrow> nat \<Rightarrow> nat" |
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8 where |
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9 "f n 0 = n" |
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10 | "f 0 (Suc m) = f (Suc m) m" |
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11 | "f (Suc n) (Suc m) = f m n" |
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12 by pat_completeness auto |
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13 |
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14 |
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15 termination |
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16 unfolding f_rel_def lfp_const |
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17 apply (rule SCT_on_relations) |
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18 apply (tactic "SCT.abs_rel_tac") (* Build call descriptors *) |
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19 apply (rule ext, rule ext, simp) (* Show that they are correct *) |
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20 apply (tactic "SCT.mk_call_graph") (* Build control graph *) |
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21 apply (rule LJA_apply) (* Apply main theorem *) |
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22 apply (simp add:finite_acg_ins finite_acg_empty) (* show finiteness *) |
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23 apply (rule SCT'_exec) |
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24 by eval (* Evaluate to true *) |
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25 |
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26 |
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27 function p :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat" |
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28 where |
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29 "p m n r = (if r>0 then p m (r - 1) n else |
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30 if n>0 then p r (n - 1) m |
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31 else m)" |
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32 by pat_completeness auto |
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33 |
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34 termination |
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35 unfolding p_rel_def lfp_const |
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36 apply (rule SCT_on_relations) |
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37 apply (tactic "SCT.abs_rel_tac") |
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38 apply (rule ext, rule ext, simp) |
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39 apply (tactic "SCT.mk_call_graph") |
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40 apply (rule LJA_apply) |
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41 apply (simp add:finite_acg_ins finite_acg_empty) |
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42 apply (rule SCT'_exec) |
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43 by eval |
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44 |
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45 function foo :: "bool \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat" |
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46 where |
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47 "foo True (Suc n) m = foo True n (Suc m)" |
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48 "foo True 0 m = foo False 0 m" |
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49 "foo False n (Suc m) = foo False (Suc n) m" |
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50 "foo False n 0 = n" |
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51 by pat_completeness auto |
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52 |
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53 termination |
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54 unfolding foo_rel_def lfp_const |
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55 apply (rule SCT_on_relations) |
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56 apply (tactic "SCT.abs_rel_tac") |
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57 apply (rule ext, rule ext, simp) |
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58 apply (tactic "SCT.mk_call_graph") |
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59 apply (rule LJA_apply) |
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60 apply (simp add:finite_acg_ins finite_acg_empty) |
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61 apply (rule SCT'_exec) |
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62 by eval |
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63 |
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64 |
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65 function (sequential) |
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66 bar :: "nat \<Rightarrow> nat \<Rightarrow> nat \<Rightarrow> nat" |
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67 where |
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68 "bar 0 (Suc n) m = bar m m m" |
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69 | "bar k n m = 0" |
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70 by pat_completeness auto |
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71 |
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72 termination |
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73 unfolding bar_rel_def lfp_const |
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74 apply (rule SCT_on_relations) |
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75 apply (tactic "SCT.abs_rel_tac") |
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76 apply (rule ext, rule ext, simp) |
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77 apply (tactic "SCT.mk_call_graph") |
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78 apply (rule LJA_apply) |
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79 apply (simp add:finite_acg_ins finite_acg_empty) |
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80 by (rule SCT'_empty) |
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81 |
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82 |
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83 end |