12 | Lam "\<guillemotleft>var\<guillemotright>trm" ("Lam [_]._" [100,100] 100) |
12 | Lam "\<guillemotleft>var\<guillemotright>trm" ("Lam [_]._" [100,100] 100) |
13 | App "trm" "trm" |
13 | App "trm" "trm" |
14 | Pss "mvar" "trm" |
14 | Pss "mvar" "trm" |
15 | Act "\<guillemotleft>mvar\<guillemotright>trm" ("Act [_]._" [100,100] 100) |
15 | Act "\<guillemotleft>mvar\<guillemotright>trm" ("Act [_]._" [100,100] 100) |
16 |
16 |
17 section {* strong induction principle *} |
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18 |
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19 (* FIXME: this induction rule needs some slight change to conform *) |
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20 (* with the convention from lam_substs *) |
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21 |
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22 lemma trm_induct_aux: |
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23 fixes P :: "trm \<Rightarrow> 'a \<Rightarrow> bool" |
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24 and f1 :: "'a \<Rightarrow> var set" |
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25 and f2 :: "'a \<Rightarrow> mvar set" |
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26 assumes fs1: "\<And>x. finite (f1 x)" |
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27 and fs2: "\<And>x. finite (f2 x)" |
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28 and h1: "\<And>k x. P (Var x) k" |
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29 and h2: "\<And>k x t. x\<notin>f1 k \<Longrightarrow> (\<forall>l. P t l) \<Longrightarrow> P (Lam [x].t) k" |
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30 and h3: "\<And>k t1 t2. (\<forall>l. P t1 l) \<Longrightarrow> (\<forall>l. P t2 l) \<Longrightarrow> P (App t1 t2) k" |
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31 and h4: "\<And>k a t1. (\<forall>l. P t1 l) \<Longrightarrow> P (Pss a t1) k" |
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32 and h5: "\<And>k a t1. a\<notin>f2 k \<Longrightarrow> (\<forall>l. P t1 l) \<Longrightarrow> P (Act [a].t1) k" |
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33 shows "\<forall>(pi1::var prm) (pi2::mvar prm) k. P (pi1\<bullet>(pi2\<bullet>t)) k" |
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34 proof (induct rule: trm.induct_weak) |
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35 case (goal1 a) |
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36 show ?case using h1 by simp |
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37 next |
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38 case (goal2 x t) |
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39 assume i1: "\<forall>(pi1::var prm)(pi2::mvar prm) k. P (pi1\<bullet>(pi2\<bullet>t)) k" |
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40 show ?case |
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41 proof (intro strip, simp add: abs_perm) |
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42 fix pi1::"var prm" and pi2::"mvar prm" and k::"'a" |
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43 have f: "\<exists>c::var. c\<sharp>(f1 k,pi1\<bullet>(pi2\<bullet>x),pi1\<bullet>(pi2\<bullet>t))" |
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44 by (rule at_exists_fresh[OF at_var_inst], simp add: supp_prod fs_var1 |
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45 at_fin_set_supp[OF at_var_inst, OF fs1] fs1) |
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46 then obtain c::"var" |
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47 where f1: "c\<noteq>(pi1\<bullet>(pi2\<bullet>x))" and f2: "c\<sharp>(f1 k)" and f3: "c\<sharp>(pi1\<bullet>(pi2\<bullet>t))" |
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48 by (force simp add: fresh_prod at_fresh[OF at_var_inst]) |
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49 have g: "Lam [c].([(c,pi1\<bullet>(pi2\<bullet>x))]\<bullet>(pi1\<bullet>(pi2\<bullet>t))) = Lam [(pi1\<bullet>(pi2\<bullet>x))].(pi1\<bullet>(pi2\<bullet>t))" using f1 f3 |
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50 by (simp add: trm.inject alpha) |
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51 from i1 have "\<forall>k. P (([(c,pi1\<bullet>(pi2\<bullet>x))]@pi1)\<bullet>(pi2\<bullet>t)) k" by force |
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52 hence i1b: "\<forall>k. P ([(c,pi1\<bullet>(pi2\<bullet>x))]\<bullet>(pi1\<bullet>(pi2\<bullet>t))) k" by (simp add: pt_var2[symmetric]) |
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53 with h3 f2 have "P (Lam [c].([(c,pi1\<bullet>(pi2\<bullet>x))]\<bullet>(pi1\<bullet>(pi2\<bullet>t)))) k" |
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54 by (auto simp add: fresh_def at_fin_set_supp[OF at_var_inst, OF fs1]) |
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55 with g show "P (Lam [(pi1\<bullet>(pi2\<bullet>x))].(pi1\<bullet>(pi2\<bullet>t))) k" by simp |
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56 qed |
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57 next |
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58 case (goal3 t1 t2) |
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59 assume i1: "\<forall>(pi1::var prm)(pi2::mvar prm) k. P (pi1\<bullet>(pi2\<bullet>t1)) k" |
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60 assume i2: "\<forall>(pi1::var prm)(pi2::mvar prm) k. P (pi1\<bullet>(pi2\<bullet>t2)) k" |
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61 show ?case |
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62 proof (intro strip) |
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63 fix pi1::"var prm" and pi2::"mvar prm" and k::"'a" |
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64 from h3 i1 i2 have "P (App (pi1\<bullet>(pi2\<bullet>t1)) (pi1\<bullet>(pi2\<bullet>t2))) k" by force |
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65 thus "P (pi1\<bullet>(pi2\<bullet>(App t1 t2))) k" by simp |
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66 qed |
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67 next |
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68 case (goal4 b t) |
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69 assume i1: "\<forall>(pi1::var prm)(pi2::mvar prm) k. P (pi1\<bullet>(pi2\<bullet>t)) k" |
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70 show ?case |
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71 proof (intro strip) |
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72 fix pi1::"var prm" and pi2::"mvar prm" and k::"'a" |
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73 from h4 i1 have "P (Pss (pi1\<bullet>(pi2\<bullet>b)) (pi1\<bullet>(pi2\<bullet>t))) k" by force |
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74 thus "P (pi1\<bullet>(pi2\<bullet>(Pss b t))) k" by simp |
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75 qed |
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76 next |
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77 case (goal5 b t) |
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78 assume i1: "\<forall>(pi1::var prm)(pi2::mvar prm) k. P (pi1\<bullet>(pi2\<bullet>t)) k" |
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79 show ?case |
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80 proof (intro strip, simp add: abs_perm) |
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81 fix pi1::"var prm" and pi2::"mvar prm" and k::"'a" |
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82 have f: "\<exists>c::mvar. c\<sharp>(f2 k,pi1\<bullet>(pi2\<bullet>b),pi1\<bullet>(pi2\<bullet>t))" |
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83 by (rule at_exists_fresh[OF at_mvar_inst], simp add: supp_prod fs_mvar1 |
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84 at_fin_set_supp[OF at_mvar_inst, OF fs2] fs2) |
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85 then obtain c::"mvar" |
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86 where f1: "c\<noteq>(pi1\<bullet>(pi2\<bullet>b))" and f2: "c\<sharp>(f2 k)" and f3: "c\<sharp>(pi1\<bullet>(pi2\<bullet>t))" |
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87 by (force simp add: fresh_prod at_fresh[OF at_mvar_inst]) |
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88 have g: "Act [c].(pi1\<bullet>([(c,pi1\<bullet>(pi2\<bullet>b))]\<bullet>(pi2\<bullet>t))) = Act [(pi1\<bullet>(pi2\<bullet>b))].(pi1\<bullet>(pi2\<bullet>t))" using f1 f3 |
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89 by (simp add: trm.inject alpha, simp add: dj_cp[OF cp_mvar_var_inst, OF dj_var_mvar]) |
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90 from i1 have "\<forall>k. P (pi1\<bullet>(([(c,pi1\<bullet>(pi2\<bullet>b))]@pi2)\<bullet>t)) k" by force |
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91 hence i1b: "\<forall>k. P (pi1\<bullet>([(c,pi1\<bullet>(pi2\<bullet>b))]\<bullet>(pi2\<bullet>t))) k" by (simp add: pt_mvar2[symmetric]) |
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92 with h5 f2 have "P (Act [c].(pi1\<bullet>([(c,pi1\<bullet>(pi2\<bullet>b))]\<bullet>(pi2\<bullet>t)))) k" |
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93 by (auto simp add: fresh_def at_fin_set_supp[OF at_mvar_inst, OF fs2]) |
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94 with g show "P (Act [(pi1\<bullet>(pi2\<bullet>b))].(pi1\<bullet>(pi2\<bullet>t))) k" by simp |
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95 qed |
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96 qed |
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97 |
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98 lemma trm_induct'[case_names Var Lam App Pss Act]: |
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99 fixes P :: "trm \<Rightarrow> 'a \<Rightarrow> bool" |
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100 and f1 :: "'a \<Rightarrow> var set" |
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101 and f2 :: "'a \<Rightarrow> mvar set" |
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102 assumes fs1: "\<And>x. finite (f1 x)" |
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103 and fs2: "\<And>x. finite (f2 x)" |
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104 and h1: "\<And>k x. P (Var x) k" |
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105 and h2: "\<And>k x t. x\<notin>f1 k \<Longrightarrow> (\<forall>l. P t l) \<Longrightarrow> P (Lam [x].t) k" |
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106 and h3: "\<And>k t1 t2. (\<forall>l. P t1 l) \<Longrightarrow> (\<forall>l. P t2 l) \<Longrightarrow> P (App t1 t2) k" |
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107 and h4: "\<And>k a t1. (\<forall>l. P t1 l) \<Longrightarrow> P (Pss a t1) k" |
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108 and h5: "\<And>k a t1. a\<notin>f2 k \<Longrightarrow> (\<forall>l. P t1 l) \<Longrightarrow> P (Act [a].t1) k" |
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109 shows "P t k" |
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110 proof - |
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111 have "\<forall>(pi1::var prm)(pi2::mvar prm) k. P (pi1\<bullet>(pi2\<bullet>t)) k" |
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112 using fs1 fs2 h1 h2 h3 h4 h5 by (rule trm_induct_aux, auto) |
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113 hence "P (([]::var prm)\<bullet>(([]::mvar prm)\<bullet>t)) k" by blast |
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114 thus "P t k" by simp |
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115 qed |
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116 |
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117 lemma trm_induct[case_names Var Lam App Pss Act]: |
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118 fixes P :: "trm \<Rightarrow> ('a::{fs_var,fs_mvar}) \<Rightarrow> bool" |
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119 assumes h1: "\<And>k x. P (Var x) k" |
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120 and h2: "\<And>k x t. x\<sharp>k \<Longrightarrow> (\<forall>l. P t l) \<Longrightarrow> P (Lam [x].t) k" |
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121 and h3: "\<And>k t1 t2. (\<forall>l. P t1 l) \<Longrightarrow> (\<forall>l. P t2 l) \<Longrightarrow> P (App t1 t2) k" |
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122 and h4: "\<And>k a t1. (\<forall>l. P t1 l) \<Longrightarrow> P (Pss a t1) k" |
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123 and h5: "\<And>k a t1. a\<sharp>k \<Longrightarrow> (\<forall>l. P t1 l) \<Longrightarrow> P (Act [a].t1) k" |
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124 shows "P t k" |
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125 by (rule trm_induct'[of "\<lambda>x. ((supp x)::var set)" "\<lambda>x. ((supp x)::mvar set)" "P"], |
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126 simp_all add: fs_var1 fs_mvar1 fresh_def[symmetric], auto intro: h1 h2 h3 h4 h5) |
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