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1 |
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2 |
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3 theory Unification = Main + Terms + Fresh + Equ + Substs + Mgu: |
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4 |
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5 (* problems to which no reduction applies *) |
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6 |
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7 consts stuck :: "problem_type set" |
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8 defs |
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9 stuck_def: "stuck \<equiv> { P1. \<not>(\<exists>P2 nabla s. P1 \<Turnstile>(nabla,s)\<Rightarrow>P2)}" |
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10 |
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11 (* all problems which are stuck and have no unifier *) |
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12 |
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13 consts fail :: "problem_type set" |
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14 inductive fail |
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15 intros |
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16 [intro!]: "\<lbrakk>occurs X t\<rbrakk>\<Longrightarrow>(Susp pi X\<approx>?Abst a t#xs,ys)\<in>fail" |
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17 [intro!]: "\<lbrakk>occurs X t\<rbrakk>\<Longrightarrow>(Susp pi X\<approx>?Func F t#xs,ys)\<in>fail" |
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18 [intro!]: "\<lbrakk>occurs X t1\<or>occurs X t2\<rbrakk>\<Longrightarrow>(Susp pi X\<approx>?Paar t1 t2#xs,ys)\<in>fail" |
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19 [intro!]: "\<lbrakk>occurs X t\<rbrakk>\<Longrightarrow>(Abst a t\<approx>?Susp pi X#xs,ys)\<in>fail" |
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20 [intro!]: "\<lbrakk>occurs X t\<rbrakk>\<Longrightarrow>(Func F t\<approx>?Susp pi X#xs,ys)\<in>fail" |
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21 [intro!]: "\<lbrakk>occurs X t1\<or>occurs X t2\<rbrakk>\<Longrightarrow>(Paar t1 t2\<approx>?Susp pi X#xs,ys)\<in>fail" |
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22 [intro!,dest!]: "([],a\<sharp>? Atom a#ys)\<in>fail" |
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23 [intro!]: "a\<noteq>b\<Longrightarrow>(Atom a\<approx>? Atom b#xs,ys)\<in>fail" |
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24 [intro!,dest!]: "(Abst a t\<approx>?Unit#xs,ys)\<in>fail" |
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25 [intro!,dest!]: "(Unit\<approx>?Abst a t#xs,ys)\<in>fail" |
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26 [intro!,dest!]: "(Abst a t\<approx>?Atom b#xs,ys)\<in>fail" |
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27 [intro!,dest!]: "(Atom b\<approx>?Abst a t#xs,ys)\<in>fail" |
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28 [intro!,dest!]: "(Abst a t\<approx>?Paar t1 t2#xs,ys)\<in>fail" |
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29 [intro!,dest!]: "(Paar t1 t2\<approx>?Abst a t#xs,ys)\<in>fail" |
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30 [intro!,dest!]: "(Abst a t\<approx>?Func F t1#xs,ys)\<in>fail" |
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31 [intro!,dest!]: "(Func F t1\<approx>?Abst a t#xs,ys)\<in>fail" |
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32 [intro!,dest!]: "(Unit\<approx>?Atom b#xs,ys)\<in>fail" |
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33 [intro!,dest!]: "(Atom b\<approx>?Unit#xs,ys)\<in>fail" |
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34 [intro!,dest!]: "(Unit\<approx>?Paar t1 t2#xs,ys)\<in>fail" |
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35 [intro!,dest!]: "(Paar t1 t2\<approx>?Unit#xs,ys)\<in>fail" |
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36 [intro!,dest!]: "(Unit\<approx>?Func F t1#xs,ys)\<in>fail" |
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37 [intro!,dest!]: "(Func F t1\<approx>?Unit#xs,ys)\<in>fail" |
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38 [intro!,dest!]: "(Atom a\<approx>?Paar t1 t2#xs,ys)\<in>fail" |
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39 [intro!,dest!]: "(Paar t1 t2\<approx>?Atom a#xs,ys)\<in>fail" |
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40 [intro!,dest!]: "(Atom a\<approx>?Func F t1#xs,ys)\<in>fail" |
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41 [intro!,dest!]: "(Func F t1\<approx>?Atom a#xs,ys)\<in>fail" |
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42 [intro!,dest!]: "(Func F t\<approx>?Paar t1 t2#xs,ys)\<in>fail" |
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43 [intro!,dest!]: "(Paar t1 t2\<approx>?Func F t#xs,ys)\<in>fail" |
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44 [intro!]: "\<lbrakk>F1\<noteq>F2\<rbrakk>\<Longrightarrow>(Func F1 t1\<approx>?Func F2 t2#xs,ys)\<in>fail" |
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45 |
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46 (* the results that are interesting are the stuck ones *) |
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47 |
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48 consts |
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49 results :: "problem_type \<Rightarrow> problem_type set" |
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50 defs |
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51 results_def: |
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52 "results P1 \<equiv> if P1\<in>stuck then {P1} else {P2. \<exists>nabla s. P1\<Turnstile>(nabla,s)\<Rightarrow>P2 \<and> P2\<in>stuck}" |
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53 |
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54 (* a "failed" problem has no unifier *) |
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55 |
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56 lemma fail_then_empty: |
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57 "(P1\<in>fail) \<Longrightarrow> (U P1={})" |
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58 apply(erule fail.cases) |
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59 apply(simp add: all_solutions_def) |
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60 apply(rule allI)+ |
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61 apply(rule impI) |
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62 apply(drule_tac nabla1="aa" and s1="b" and "pi1.1"="pi" in occurs_sub_trm_equ[THEN mp]) |
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63 apply(subgoal_tac "\<forall>t1\<in>psub_trms (Abst a (subst b t)).\<not>(\<exists>pi2. aa\<turnstile>Abst a (subst b t)\<approx>swap pi2 t1)") |
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64 apply(simp) |
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65 apply(drule equ_sym) |
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66 apply(clarify) |
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67 apply(drule_tac "t1.0"="Abst a (subst b t)" and |
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68 "t2.0"="subst b (Susp pi X)" and |
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69 "t3.0"="swap pia t2" in equ_trans) |
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70 apply(simp) |
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71 apply(best) |
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72 apply(rule psub_trm_not_equ) |
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73 apply(simp add: all_solutions_def) |
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74 apply(rule allI)+ |
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75 apply(rule impI) |
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76 apply(drule_tac nabla1="a" and s1="b" and "pi1.1"="pi" in occurs_sub_trm_equ[THEN mp]) |
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77 apply(subgoal_tac "\<forall>t1\<in>psub_trms (Func F (subst b t)).\<not>(\<exists>pi2. a\<turnstile>Func F (subst b t)\<approx>swap pi2 t1)") |
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78 apply(simp) |
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79 apply(drule equ_sym) |
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80 apply(clarify) |
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81 apply(drule_tac "t1.0"="Func F (subst b t)" and |
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82 "t2.0"="subst b (Susp pi X)" and |
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83 "t3.0"="swap pia t2" in equ_trans) |
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84 apply(simp) |
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85 apply(best) |
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86 apply(rule psub_trm_not_equ) |
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87 apply(simp add: all_solutions_def) |
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88 apply(rule allI)+ |
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89 apply(rule impI) |
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90 apply(erule disjE) |
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91 apply(drule_tac nabla1="a" and s1="b" and "pi1.1"="pi" in occurs_sub_trm_equ[THEN mp]) |
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92 apply(subgoal_tac "\<forall>t3\<in>psub_trms (Paar (subst b t1) (subst b t2)). |
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93 \<not>(\<exists>pi2. a\<turnstile>Paar (subst b t1) (subst b t2)\<approx>swap pi2 t3)") |
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94 apply(simp) |
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95 apply(drule equ_sym) |
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96 apply(clarify) |
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97 apply(drule_tac "t1.0"="Paar (subst b t1) (subst b t2)" and |
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98 "t2.0"="subst b (Susp pi X)" and |
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99 "t3.0"="swap pia t2a" in equ_trans) |
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100 apply(simp) |
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101 apply(best) |
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102 apply(rule psub_trm_not_equ) |
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103 apply(drule_tac nabla1="a" and s1="b" and "pi1.1"="pi" in occurs_sub_trm_equ[THEN mp]) |
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104 apply(subgoal_tac "\<forall>t3\<in>psub_trms (Paar (subst b t1) (subst b t2)). |
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105 \<not>(\<exists>pi2. a\<turnstile>Paar (subst b t1) (subst b t2)\<approx>swap pi2 t3)") |
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106 apply(simp) |
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107 apply(drule equ_sym) |
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108 apply(clarify) |
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109 apply(drule_tac "t1.0"="Paar (subst b t1) (subst b t2)" and |
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110 "t2.0"="subst b (Susp pi X)" and |
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111 "t3.0"="swap pia t2a" in equ_trans) |
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112 apply(simp) |
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113 apply(best) |
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114 apply(rule psub_trm_not_equ) |
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115 apply(simp add: all_solutions_def) |
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116 apply(rule allI)+ |
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117 apply(rule impI) |
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118 apply(drule_tac nabla1="aa" and s1="b" and "pi1.1"="pi" in occurs_sub_trm_equ[THEN mp]) |
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119 apply(subgoal_tac "\<forall>t3\<in>psub_trms (Abst a (subst b t)). |
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120 \<not>(\<exists>pi2. aa\<turnstile>Abst a (subst b t)\<approx>swap pi2 t3)") |
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121 apply(simp) |
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122 apply(clarify) |
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123 apply(drule_tac "t1.0"="Abst a (subst b t)" and |
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124 "t2.0"="subst b (Susp pi X)" and |
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125 "t3.0"="swap pia t2" in equ_trans) |
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126 apply(simp) |
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127 apply(best) |
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128 apply(rule psub_trm_not_equ) |
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129 apply(simp add: all_solutions_def) |
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130 apply(rule allI)+ |
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131 apply(rule impI) |
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132 apply(drule_tac nabla1="a" and s1="b" and "pi1.1"="pi" in occurs_sub_trm_equ[THEN mp]) |
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133 apply(subgoal_tac "\<forall>t1\<in>psub_trms (Func F (subst b t)).\<not>(\<exists>pi2. a\<turnstile>Func F (subst b t)\<approx>swap pi2 t1)") |
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134 apply(simp) |
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135 apply(clarify) |
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136 apply(drule_tac "t1.0"="Func F (subst b t)" and |
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137 "t2.0"="subst b (Susp pi X)" and |
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138 "t3.0"="swap pia t2" in equ_trans) |
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139 apply(simp) |
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140 apply(best) |
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141 apply(rule psub_trm_not_equ) |
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142 apply(simp add: all_solutions_def) |
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143 apply(rule allI)+ |
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144 apply(rule impI) |
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145 apply(erule disjE) |
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146 apply(drule_tac nabla1="a" and s1="b" and "pi1.1"="pi" in occurs_sub_trm_equ[THEN mp]) |
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147 apply(subgoal_tac "\<forall>t3\<in>psub_trms (Paar (subst b t1) (subst b t2)). |
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148 \<not>(\<exists>pi2. a\<turnstile>Paar (subst b t1) (subst b t2)\<approx>swap pi2 t3)") |
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149 apply(simp) |
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150 apply(clarify) |
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151 apply(drule_tac "t1.0"="Paar (subst b t1) (subst b t2)" and |
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152 "t2.0"="subst b (Susp pi X)" and |
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153 "t3.0"="swap pia t2a" in equ_trans) |
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154 apply(simp) |
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155 apply(best) |
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156 apply(rule psub_trm_not_equ) |
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157 apply(drule_tac nabla1="a" and s1="b" and "pi1.1"="pi" in occurs_sub_trm_equ[THEN mp]) |
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158 apply(subgoal_tac "\<forall>t3\<in>psub_trms (Paar (subst b t1) (subst b t2)). |
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159 \<not>(\<exists>pi2. a\<turnstile>Paar (subst b t1) (subst b t2)\<approx>swap pi2 t3)") |
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160 apply(simp) |
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161 apply(clarify) |
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162 apply(drule_tac "t1.0"="Paar (subst b t1) (subst b t2)" and |
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163 "t2.0"="subst b (Susp pi X)" and |
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164 "t3.0"="swap pia t2a" in equ_trans) |
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165 apply(simp) |
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166 apply(best) |
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167 apply(rule psub_trm_not_equ) |
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168 apply(simp add: all_solutions_def, fast dest!: fresh.cases) |
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169 apply(simp add: all_solutions_def, fast dest!: equ.cases)+ |
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170 done |
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171 |
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172 (* the only stuck problems are the "failed" problems and the empty problem *) |
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173 |
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174 lemma stuck_equiv: "stuck = {([],[])}\<union>fail" |
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175 apply(subgoal_tac "([],[])\<in>stuck") |
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176 apply(subgoal_tac "\<forall>P\<in>fail. P\<in>stuck") |
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177 apply(subgoal_tac "\<forall>P\<in>stuck. P=([],[]) \<or> P\<in>fail") |
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178 apply(force) |
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179 apply(rule ballI) |
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180 apply(thin_tac "([], []) \<in> stuck") |
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181 apply(thin_tac "\<forall>P\<in>fail. P \<in> stuck") |
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182 apply(simp add: stuck_def) |
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183 apply(clarify) |
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184 apply(case_tac a) |
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185 apply(simp) |
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186 apply(case_tac b) |
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187 apply(simp) |
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188 apply(simp) |
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189 apply(case_tac aa) |
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190 apply(simp) |
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191 apply(case_tac ba) |
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192 apply(simp_all) |
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193 apply(case_tac "ab=lista") |
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194 apply(force) |
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195 apply(force) |
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196 apply(force) |
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197 apply(force) |
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198 apply(force) |
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199 apply(force) |
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200 apply(force) |
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201 apply(case_tac aa) |
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202 apply(simp) |
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203 apply(case_tac ab) |
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204 apply(simp_all) |
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205 apply(case_tac ba) |
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206 apply(simp_all) |
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207 apply(case_tac "lista=listb") |
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208 apply(force) |
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209 apply(force) |
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210 apply(case_tac "occurs list2 (Abst lista trm)") |
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211 apply(force) |
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212 apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) (Abst lista trm))] (list,b))" in spec) |
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213 apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) (Abst lista trm))] (list,b))" in spec) |
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214 apply(drule_tac x="{}" in spec) |
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215 apply(drule_tac x="[(list2, swap (rev list1) (Abst lista trm))]" in spec) |
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216 apply(simp only: surjective_pairing[THEN sym]) |
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217 apply(force) |
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218 apply(force) |
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219 apply(force) |
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220 apply(force) |
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221 apply(force) |
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222 apply(case_tac ba) |
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223 apply(simp_all) |
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224 apply(case_tac "occurs list2 (Abst lista trm)") |
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225 apply(force) |
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226 apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) (Abst lista trm))] (list,b))" in spec) |
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227 apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) (Abst lista trm))] (list,b))" in spec) |
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228 apply(drule_tac x="{}" in spec) |
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229 apply(drule_tac x="[(list2, swap (rev list1) (Abst lista trm))]" in spec) |
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230 apply(simp only: surjective_pairing[THEN sym]) |
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231 apply(force) |
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232 apply(case_tac "list2=list2a") |
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233 apply(force) |
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234 apply(case_tac "occurs list2 (Susp list1a list2a)") |
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235 apply(simp) |
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236 apply(drule_tac |
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237 x="fst (apply_subst [(list2,swap (rev list1) (Susp list1a list2a))] (list,b))" in spec) |
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238 apply(drule_tac |
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239 x="snd (apply_subst [(list2,swap (rev list1) (Susp list1a list2a))] (list,b))" in spec) |
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240 apply(drule_tac x="{}" in spec) |
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241 apply(drule_tac x="[(list2, swap (rev list1) (Susp list1a list2a))]" in spec) |
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242 apply(simp only: surjective_pairing[THEN sym]) |
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243 apply(force) |
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244 apply(case_tac "occurs list2 Unit") |
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245 apply(simp) |
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246 apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) Unit)] (list,b))" in spec) |
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247 apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) Unit)] (list,b))" in spec) |
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248 apply(drule_tac x="{}" in spec) |
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249 apply(drule_tac x="[(list2, swap (rev list1) Unit)]" in spec) |
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250 apply(simp only: surjective_pairing[THEN sym]) |
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251 apply(force) |
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252 apply(case_tac "occurs list2 (Atom lista)") |
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253 apply(simp) |
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254 apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) (Atom lista))] (list,b))" in spec) |
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255 apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) (Atom lista))] (list,b))" in spec) |
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256 apply(drule_tac x="{}" in spec) |
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257 apply(drule_tac x="[(list2, swap (rev list1) (Atom lista))]" in spec) |
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258 apply(simp only: surjective_pairing[THEN sym]) |
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259 apply(force) |
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260 apply(case_tac "occurs list2 (Paar trm1 trm2)") |
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261 apply(force) |
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262 apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) (Paar trm1 trm2))] (list,b))" in spec) |
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263 apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) (Paar trm1 trm2))] (list,b))" in spec) |
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264 apply(drule_tac x="{}" in spec) |
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265 apply(drule_tac x="[(list2, swap (rev list1) (Paar trm1 trm2))]" in spec) |
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266 apply(simp only: surjective_pairing[THEN sym]) |
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267 apply(force) |
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268 apply(case_tac "occurs list2 (Func lista trm)") |
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269 apply(force) |
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270 apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) (Func lista trm))] (list,b))" in spec) |
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271 apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) (Func lista trm))] (list,b))" in spec) |
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272 apply(drule_tac x="{}" in spec) |
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273 apply(drule_tac x="[(list2, swap (rev list1) (Func lista trm))]" in spec) |
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274 apply(simp only: surjective_pairing[THEN sym]) |
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275 apply(force) |
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276 apply(case_tac ba) |
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277 apply(simp_all) |
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278 apply(force) |
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279 apply(case_tac "occurs list2 Unit") |
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280 apply(simp) |
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281 apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) Unit)] (list,b))" in spec) |
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282 apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) Unit)] (list,b))" in spec) |
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283 apply(drule_tac x="{}" in spec) |
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284 apply(drule_tac x="[(list2, swap (rev list1) Unit)]" in spec) |
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285 apply(simp only: surjective_pairing[THEN sym]) |
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286 apply(force) |
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287 apply(force) |
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288 apply(force) |
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289 apply(force) |
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290 apply(force) |
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291 apply(case_tac ba) |
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292 apply(simp_all) |
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293 apply(force) |
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294 apply(case_tac "occurs list2 (Atom lista)") |
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295 apply(simp) |
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296 apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) (Atom lista))] (list,b))" in spec) |
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297 apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) (Atom lista))] (list,b))" in spec) |
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298 apply(drule_tac x="{}" in spec) |
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299 apply(drule_tac x="[(list2, swap (rev list1) (Atom lista))]" in spec) |
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300 apply(simp only: surjective_pairing[THEN sym]) |
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301 apply(force) |
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302 apply(force) |
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303 apply(force) |
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304 apply(force) |
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305 apply(force) |
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306 apply(case_tac ba) |
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307 apply(simp_all) |
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308 apply(force) |
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309 apply(case_tac "occurs list2 (Paar trm1 trm2)") |
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310 apply(force) |
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311 apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) (Paar trm1 trm2))] (list,b))" in spec) |
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312 apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) (Paar trm1 trm2))] (list,b))" in spec) |
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313 apply(drule_tac x="{}" in spec) |
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314 apply(drule_tac x="[(list2, swap (rev list1) (Paar trm1 trm2))]" in spec) |
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315 apply(simp only: surjective_pairing[THEN sym]) |
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316 apply(force) |
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317 apply(force) |
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318 apply(force) |
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319 apply(force) |
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320 apply(force) |
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321 apply(case_tac ba) |
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322 apply(simp_all) |
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323 apply(force) |
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324 apply(case_tac "occurs list2 (Func lista trm)") |
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325 apply(force) |
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326 apply(drule_tac x="fst (apply_subst [(list2,swap (rev list1) (Func lista trm))] (list,b))" in spec) |
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327 apply(drule_tac x="snd (apply_subst [(list2,swap (rev list1) (Func lista trm))] (list,b))" in spec) |
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328 apply(drule_tac x="{}" in spec) |
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329 apply(drule_tac x="[(list2, swap (rev list1) (Func lista trm))]" in spec) |
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330 apply(simp only: surjective_pairing[THEN sym]) |
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331 apply(force) |
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332 apply(force) |
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333 apply(force) |
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334 apply(force) |
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335 apply(force) |
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336 apply(rule ballI) |
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337 apply(thin_tac "([], []) \<in> stuck") |
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338 apply(simp add: stuck_def) |
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339 apply(clarify) |
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340 apply(ind_cases "((a, b), (nabla, s), aa, ba) \<in> red_plus") |
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341 apply(ind_cases "((a, b), s, aa, ba) \<in> s_red") |
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342 apply(simp_all) |
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343 apply(ind_cases "((Unit, Unit) # aa, b) \<in> fail") |
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344 apply(ind_cases "((Paar t1 t2, Paar s1 s2) # xs, b) \<in> fail") |
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345 apply(ind_cases "((Func F t1, Func F t2) # xs, b) \<in> fail") |
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346 apply(simp) |
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347 apply(ind_cases "((Abst ab t1, Abst ab t2) # xs, b) \<in> fail") |
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348 apply(ind_cases "((Abst ab t1, Abst bb t2) # xs, b) \<in> fail") |
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349 apply(ind_cases "((Atom ab, Atom ab) # aa, b) \<in> fail") |
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350 apply(simp) |
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351 apply(ind_cases "((Susp pi1 X, Susp pi2 X) # aa, b) \<in> fail") |
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352 apply(ind_cases "((Susp pi X, t) # xs, b) \<in> fail") |
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353 apply(simp add: apply_subst_def)+ |
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354 apply(case_tac "occurs X t1") |
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355 apply(simp) |
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356 apply(simp) |
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357 apply(ind_cases "((t, Susp pi X) # xs, b) \<in> fail") |
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358 apply(simp add: apply_subst_def) |
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359 apply(case_tac "occurs X t1") |
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360 apply(simp) |
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361 apply(simp) |
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362 apply(case_tac "occurs X t1") |
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363 apply(simp) |
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364 apply(simp) |
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365 apply(ind_cases "((a, b), nabla, aa, ba) \<in> c_red") |
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366 apply(simp_all) |
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367 apply(ind_cases "([], (ab, Unit) # ba) \<in> fail") |
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368 apply(ind_cases "([], (ab, Paar t1 t2) # xs) \<in> fail") |
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369 apply(ind_cases "([], (ab, Func F t) # xs) \<in> fail") |
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370 apply(ind_cases "([], (ab, Abst ab t) # ba) \<in> fail") |
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371 apply(ind_cases "([], (ab, Abst bb t) # xs) \<in> fail") |
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372 apply(ind_cases "([], (ab, Atom bb) # ba) \<in> fail") |
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373 apply(simp) |
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374 apply(ind_cases "([], (ab, Susp pi X) # ba) \<in> fail") |
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375 apply(ind_cases "(a, b) \<turnstile> s1 \<leadsto> P2") |
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376 apply(simp_all) |
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377 apply(ind_cases "((Unit, Unit) # xs, b) \<in> fail") |
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378 apply(ind_cases "((Paar t1 t2, Paar s1 s2) # xs, b) \<in> fail") |
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379 apply(ind_cases "((Func F t1, Func F t2) # xs, b) \<in> fail") |
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380 apply(simp) |
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381 apply(ind_cases "((Abst ab t1, Abst ab t2) # xs, b) \<in> fail") |
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382 apply(ind_cases "((Abst ab t1, Abst bb t2) # xs, b) \<in> fail") |
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383 apply(ind_cases "((Atom ab, Atom ab) # aa, b) \<in> fail") |
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384 apply(simp) |
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385 apply(ind_cases "((Susp pi1 X, Susp pi2 X) # aa, b) \<in> fail") |
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386 apply(ind_cases "((Susp pi X, t) # xs, b) \<in> fail") |
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387 apply(simp add: apply_subst_def)+ |
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388 apply(case_tac "occurs X t1") |
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389 apply(simp) |
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390 apply(simp) |
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391 apply(ind_cases "((t, Susp pi X) # xs, b) \<in> fail") |
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392 apply(simp add: apply_subst_def) |
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393 apply(case_tac "occurs X t1") |
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394 apply(simp) |
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395 apply(simp) |
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396 apply(case_tac "occurs X t1") |
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397 apply(simp) |
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398 apply(simp) |
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399 apply(ind_cases "(a, b) \<turnstile> nabla1 \<rightarrow> P2") |
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400 apply(simp_all) |
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401 apply(ind_cases "([], (ab, Unit) # ba) \<in> fail") |
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402 apply(ind_cases "([], (ab, Paar t1 t2) # xs) \<in> fail") |
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403 apply(ind_cases "([], (ab, Func F t) # xs) \<in> fail") |
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404 apply(ind_cases "([], (ab, Abst ab t) # ba) \<in> fail") |
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405 apply(ind_cases "([], (ab, Abst bb t) # xs) \<in> fail") |
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406 apply(ind_cases "([], (ab, Atom bb) # ba) \<in> fail") |
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407 apply(simp) |
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408 apply(ind_cases "([], (ab, Susp pi X) # ba) \<in> fail") |
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409 apply(simp add: stuck_def) |
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410 apply(rule allI)+ |
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411 apply(clarify) |
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412 apply(ind_cases "(([], []), (nabla, s), a, b) \<in> red_plus") |
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413 apply(ind_cases "(([], []), s, a, b) \<in> s_red") |
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414 apply(ind_cases "(([], []), nabla, a, b) \<in> c_red") |
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415 apply(ind_cases "([], []) \<turnstile> s1 \<leadsto> P2") |
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416 apply(ind_cases "([], []) \<turnstile> nabla1 \<rightarrow> P2") |
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417 done |
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418 |
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419 lemma u_empty_sred: |
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420 "P1\<turnstile>s\<leadsto>P2 \<longrightarrow> U P2 ={} \<longrightarrow> U P1={}" |
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421 apply(rule impI) |
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422 apply(ind_cases "P1 \<turnstile> s \<leadsto> P2") |
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423 apply(rule impI, simp add: all_solutions_def) |
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424 apply(rule impI, simp add: all_solutions_def) |
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425 apply(fast dest!: equ_paar_elim) |
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426 apply(rule impI, simp add: all_solutions_def) |
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427 apply(fast dest!: equ_func_elim) |
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428 apply(rule impI, simp add: all_solutions_def) |
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429 apply(fast dest!: equ_abst_aa_elim) |
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430 apply(rule impI, simp add: all_solutions_def) |
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431 apply(force dest!: equ_abst_ab_elim simp add: subst_swap_comm[THEN sym]) |
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432 apply(rule impI, simp add: all_solutions_def) |
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433 apply(rule impI, simp add: all_solutions_def) |
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434 apply(simp add: ds_list_equ_ds) |
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435 apply(rule allI)+ |
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436 apply(rule impI) |
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437 apply(drule_tac x="a" in spec) |
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438 apply(drule_tac x="b" in spec) |
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439 apply(erule disjE) |
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440 apply(force) |
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441 apply(simp add: subst_susp) |
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442 apply(drule equ_pi1_pi2_dec[THEN mp]) |
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443 apply(force simp add: subst_susp) |
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444 apply(auto) |
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445 apply(simp add: all_solutions_def) |
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446 apply(simp_all add: apply_subst_def) |
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447 apply(auto) |
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448 apply(drule_tac x="a" in spec) |
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449 apply(drule_tac x="b" in spec) |
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450 apply(drule unif_1) |
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451 apply(auto) |
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452 apply(drule_tac x="(aa,ba)" in bspec) |
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453 apply(assumption) |
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454 apply(simp) |
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455 apply(drule_tac "t1.0"="aa" and "t2.0"="ba" in unif_2a) |
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456 apply(simp add: subst_comp_expand) |
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457 apply(drule_tac x="(aa,ba)" in bspec) |
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458 apply(assumption) |
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459 apply(simp) |
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460 apply(drule_tac a="aa" and "t"="ba" in unif_2b) |
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461 apply(simp add: subst_comp_expand) |
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462 apply(simp add: all_solutions_def) |
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463 apply(auto) |
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464 apply(drule_tac x="a" in spec) |
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465 apply(drule_tac x="b" in spec) |
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466 apply(drule equ_sym) |
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467 apply(drule unif_1) |
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468 apply(auto) |
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469 apply(drule_tac x="(aa,ba)" in bspec) |
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470 apply(assumption) |
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471 apply(simp) |
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472 apply(drule_tac "t1.0"="aa" and "t2.0"="ba" in unif_2a) |
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473 apply(simp add: subst_comp_expand) |
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474 apply(drule_tac x="(aa,ba)" in bspec) |
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475 apply(assumption) |
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476 apply(simp) |
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477 apply(drule_tac a="aa" and "t"="ba" in unif_2b) |
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478 apply(simp add: subst_comp_expand) |
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479 done |
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480 |
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481 lemma u_empty_cred: |
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482 "P1\<turnstile>nabla\<rightarrow>P2 \<longrightarrow> U P2 ={} \<longrightarrow> U P1={}" |
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483 apply(rule impI) |
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484 apply(ind_cases "P1 \<turnstile>nabla\<rightarrow>P2") |
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485 apply(rule impI, simp add: all_solutions_def) |
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486 apply(rule impI, simp add: all_solutions_def) |
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487 apply(fast dest!: fresh_paar_elim) |
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488 apply(rule impI, simp add: all_solutions_def) |
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489 apply(fast dest!: fresh_func_elim) |
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490 apply(rule impI, simp add: all_solutions_def) |
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491 apply(rule impI, simp add: all_solutions_def) |
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492 apply(force dest!: fresh_abst_ab_elim) |
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493 apply(rule impI, simp add: all_solutions_def) |
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494 apply(rule impI, simp add: all_solutions_def) |
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495 done |
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496 |
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497 lemma u_empty_red_plus: |
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498 "P1\<Turnstile>(nabla,s)\<Rightarrow>P2 \<longrightarrow> U P2 ={} \<longrightarrow> U P1={}" |
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499 apply(rule impI) |
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500 apply(erule red_plus.induct) |
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501 apply(drule u_empty_sred[THEN mp], assumption) |
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502 apply(drule u_empty_cred[THEN mp], assumption) |
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503 apply(drule u_empty_sred[THEN mp], force) |
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504 apply(drule u_empty_cred[THEN mp], force) |
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505 done |
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506 |
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507 (* all problems that cannot be solved produce "failed" problems only *) |
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508 |
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509 lemma empty_then_fail: "U P1={} \<longrightarrow> (\<forall>P\<in>results P1. P\<in>fail)" |
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510 apply(simp add: results_def) |
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511 apply(rule conjI) |
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512 apply(rule impI) |
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513 apply(rule impI) |
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514 apply(simp add: stuck_equiv) |
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515 apply(erule disjE) |
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516 apply(subgoal_tac "({},[])\<in>U ([],[])") |
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517 apply(simp) |
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518 apply(simp add: all_solutions_def) |
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519 apply(assumption) |
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520 apply(rule impI)+ |
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521 apply(rule allI)+ |
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522 apply(rule impI) |
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523 apply(erule conjE) |
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524 apply(simp add: stuck_equiv) |
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525 apply(auto) |
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526 apply(subgoal_tac "({},[])\<in>U ([],[])") |
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527 apply(drule_tac "nabla3.0"="nabla" and "nabla1.0"="{}" and "s1.0"="[]" in P1_from_P2_red_plus) |
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528 apply(simp add: ext_subst_def) |
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529 apply(auto) |
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530 apply(simp add: all_solutions_def) |
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531 done |
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532 |
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533 (* if a problem can be solved then no "failed" problem is produced *) |
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534 |
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535 lemma not_empty_then_not_fail: "U P1\<noteq>{} \<longrightarrow> \<not>(\<exists>P\<in>results P1. P\<in>fail)" |
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536 apply(rule impI) |
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537 apply(simp) |
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538 apply(rule ballI) |
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539 apply(clarify) |
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540 apply(simp add: results_def) |
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541 apply(case_tac "P1\<in>stuck") |
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542 apply(simp_all) |
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543 apply(drule fail_then_empty) |
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544 apply(simp) |
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545 apply(drule fail_then_empty) |
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546 apply(erule conjE) |
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547 apply(clarify) |
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548 apply(drule u_empty_red_plus[THEN mp]) |
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549 apply(simp) |
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550 done |
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551 |
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552 end |
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553 |
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554 |
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555 |
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619 |
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621 |
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623 |
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