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1 (* Title: HOL/Library/Wfrec.thy |
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2 Author: Tobias Nipkow |
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3 Author: Lawrence C Paulson |
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4 Author: Konrad Slind |
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5 *) |
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6 |
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7 header {* Well-Founded Recursion Combinator *} |
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8 |
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9 theory Wfrec |
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10 imports Main |
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11 begin |
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12 |
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13 inductive |
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14 wfrec_rel :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b => bool" |
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15 for R :: "('a * 'a) set" |
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16 and F :: "('a => 'b) => 'a => 'b" |
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17 where |
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18 wfrecI: "ALL z. (z, x) : R --> wfrec_rel R F z (g z) ==> |
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19 wfrec_rel R F x (F g x)" |
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20 |
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21 definition |
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22 cut :: "('a => 'b) => ('a * 'a)set => 'a => 'a => 'b" where |
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23 "cut f r x == (%y. if (y,x):r then f y else undefined)" |
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24 |
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25 definition |
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26 adm_wf :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => bool" where |
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27 "adm_wf R F == ALL f g x. |
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28 (ALL z. (z, x) : R --> f z = g z) --> F f x = F g x" |
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29 |
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30 definition |
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31 wfrec :: "('a * 'a) set => (('a => 'b) => 'a => 'b) => 'a => 'b" where |
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32 "wfrec R F == %x. THE y. wfrec_rel R (%f x. F (cut f R x) x) x y" |
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33 |
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34 lemma cuts_eq: "(cut f r x = cut g r x) = (ALL y. (y,x):r --> f(y)=g(y))" |
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35 by (simp add: fun_eq_iff cut_def) |
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36 |
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37 lemma cut_apply: "(x,a):r ==> (cut f r a)(x) = f(x)" |
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38 by (simp add: cut_def) |
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39 |
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40 text{*Inductive characterization of wfrec combinator; for details see: |
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41 John Harrison, "Inductive definitions: automation and application"*} |
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42 |
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43 lemma wfrec_unique: "[| adm_wf R F; wf R |] ==> EX! y. wfrec_rel R F x y" |
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44 apply (simp add: adm_wf_def) |
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45 apply (erule_tac a=x in wf_induct) |
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46 apply (rule ex1I) |
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47 apply (rule_tac g = "%x. THE y. wfrec_rel R F x y" in wfrec_rel.wfrecI) |
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48 apply (fast dest!: theI') |
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49 apply (erule wfrec_rel.cases, simp) |
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50 apply (erule allE, erule allE, erule allE, erule mp) |
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51 apply (fast intro: the_equality [symmetric]) |
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52 done |
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53 |
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54 lemma adm_lemma: "adm_wf R (%f x. F (cut f R x) x)" |
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55 apply (simp add: adm_wf_def) |
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56 apply (intro strip) |
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57 apply (rule cuts_eq [THEN iffD2, THEN subst], assumption) |
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58 apply (rule refl) |
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59 done |
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60 |
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61 lemma wfrec: "wf(r) ==> wfrec r H a = H (cut (wfrec r H) r a) a" |
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62 apply (simp add: wfrec_def) |
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63 apply (rule adm_lemma [THEN wfrec_unique, THEN the1_equality], assumption) |
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64 apply (rule wfrec_rel.wfrecI) |
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65 apply (intro strip) |
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66 apply (erule adm_lemma [THEN wfrec_unique, THEN theI']) |
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67 done |
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68 |
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69 |
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70 text{** This form avoids giant explosions in proofs. NOTE USE OF ==*} |
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71 lemma def_wfrec: "[| f==wfrec r H; wf(r) |] ==> f(a) = H (cut f r a) a" |
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72 apply auto |
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73 apply (blast intro: wfrec) |
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74 done |
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75 |
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76 |
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77 subsection {* Nitpick setup *} |
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78 |
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79 axiomatization wf_wfrec :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" |
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80 |
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81 definition wf_wfrec' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where |
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82 [nitpick_simp]: "wf_wfrec' R F x = F (cut (wf_wfrec R F) R x) x" |
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83 |
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84 definition wfrec' :: "('a \<times> 'a \<Rightarrow> bool) \<Rightarrow> (('a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b) \<Rightarrow> 'a \<Rightarrow> 'b" where |
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85 "wfrec' R F x \<equiv> if wf R then wf_wfrec' R F x |
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86 else THE y. wfrec_rel R (%f x. F (cut f R x) x) x y" |
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87 |
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88 setup {* |
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89 Nitpick_HOL.register_ersatz_global |
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90 [(@{const_name wf_wfrec}, @{const_name wf_wfrec'}), |
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91 (@{const_name wfrec}, @{const_name wfrec'})] |
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92 *} |
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93 |
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94 hide_const (open) wf_wfrec wf_wfrec' wfrec' |
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95 hide_fact (open) wf_wfrec'_def wfrec'_def |
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96 |
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97 subsection {* Wellfoundedness of @{text same_fst} *} |
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98 |
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99 definition |
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100 same_fst :: "('a => bool) => ('a => ('b * 'b)set) => (('a*'b)*('a*'b))set" |
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101 where |
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102 "same_fst P R == {((x',y'),(x,y)) . x'=x & P x & (y',y) : R x}" |
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103 --{*For @{text rec_def} declarations where the first n parameters |
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104 stay unchanged in the recursive call. *} |
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105 |
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106 lemma same_fstI [intro!]: |
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107 "[| P x; (y',y) : R x |] ==> ((x,y'),(x,y)) : same_fst P R" |
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108 by (simp add: same_fst_def) |
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109 |
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110 lemma wf_same_fst: |
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111 assumes prem: "(!!x. P x ==> wf(R x))" |
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112 shows "wf(same_fst P R)" |
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113 apply (simp cong del: imp_cong add: wf_def same_fst_def) |
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114 apply (intro strip) |
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115 apply (rename_tac a b) |
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116 apply (case_tac "wf (R a)") |
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117 apply (erule_tac a = b in wf_induct, blast) |
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118 apply (blast intro: prem) |
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119 done |
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120 |
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121 end |