1 (* Title: HOL/Old_Datatype.thy |
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2 Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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3 Author: Stefan Berghofer and Markus Wenzel, TU Muenchen |
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4 *) |
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5 |
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6 header {* Old Datatype package: constructing datatypes from Cartesian Products and Disjoint Sums *} |
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7 |
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8 theory Old_Datatype |
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9 imports Power |
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10 keywords "old_datatype" :: thy_decl |
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11 begin |
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12 |
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13 subsection {* The datatype universe *} |
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14 |
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15 definition "Node = {p. EX f x k. p = (f :: nat => 'b + nat, x ::'a + nat) & f k = Inr 0}" |
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16 |
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17 typedef ('a, 'b) node = "Node :: ((nat => 'b + nat) * ('a + nat)) set" |
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18 morphisms Rep_Node Abs_Node |
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19 unfolding Node_def by auto |
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20 |
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21 text{*Datatypes will be represented by sets of type @{text node}*} |
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22 |
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23 type_synonym 'a item = "('a, unit) node set" |
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24 type_synonym ('a, 'b) dtree = "('a, 'b) node set" |
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25 |
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26 consts |
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27 Push :: "[('b + nat), nat => ('b + nat)] => (nat => ('b + nat))" |
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28 |
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29 Push_Node :: "[('b + nat), ('a, 'b) node] => ('a, 'b) node" |
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30 ndepth :: "('a, 'b) node => nat" |
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31 |
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32 Atom :: "('a + nat) => ('a, 'b) dtree" |
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33 Leaf :: "'a => ('a, 'b) dtree" |
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34 Numb :: "nat => ('a, 'b) dtree" |
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35 Scons :: "[('a, 'b) dtree, ('a, 'b) dtree] => ('a, 'b) dtree" |
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36 In0 :: "('a, 'b) dtree => ('a, 'b) dtree" |
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37 In1 :: "('a, 'b) dtree => ('a, 'b) dtree" |
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38 Lim :: "('b => ('a, 'b) dtree) => ('a, 'b) dtree" |
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39 |
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40 ntrunc :: "[nat, ('a, 'b) dtree] => ('a, 'b) dtree" |
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41 |
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42 uprod :: "[('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set" |
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43 usum :: "[('a, 'b) dtree set, ('a, 'b) dtree set]=> ('a, 'b) dtree set" |
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44 |
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45 Split :: "[[('a, 'b) dtree, ('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c" |
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46 Case :: "[[('a, 'b) dtree]=>'c, [('a, 'b) dtree]=>'c, ('a, 'b) dtree] => 'c" |
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47 |
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48 dprod :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set] |
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49 => (('a, 'b) dtree * ('a, 'b) dtree)set" |
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50 dsum :: "[(('a, 'b) dtree * ('a, 'b) dtree)set, (('a, 'b) dtree * ('a, 'b) dtree)set] |
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51 => (('a, 'b) dtree * ('a, 'b) dtree)set" |
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52 |
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53 |
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54 defs |
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55 |
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56 Push_Node_def: "Push_Node == (%n x. Abs_Node (apfst (Push n) (Rep_Node x)))" |
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57 |
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58 (*crude "lists" of nats -- needed for the constructions*) |
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59 Push_def: "Push == (%b h. case_nat b h)" |
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60 |
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61 (** operations on S-expressions -- sets of nodes **) |
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62 |
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63 (*S-expression constructors*) |
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64 Atom_def: "Atom == (%x. {Abs_Node((%k. Inr 0, x))})" |
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65 Scons_def: "Scons M N == (Push_Node (Inr 1) ` M) Un (Push_Node (Inr (Suc 1)) ` N)" |
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66 |
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67 (*Leaf nodes, with arbitrary or nat labels*) |
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68 Leaf_def: "Leaf == Atom o Inl" |
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69 Numb_def: "Numb == Atom o Inr" |
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70 |
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71 (*Injections of the "disjoint sum"*) |
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72 In0_def: "In0(M) == Scons (Numb 0) M" |
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73 In1_def: "In1(M) == Scons (Numb 1) M" |
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74 |
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75 (*Function spaces*) |
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76 Lim_def: "Lim f == Union {z. ? x. z = Push_Node (Inl x) ` (f x)}" |
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77 |
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78 (*the set of nodes with depth less than k*) |
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79 ndepth_def: "ndepth(n) == (%(f,x). LEAST k. f k = Inr 0) (Rep_Node n)" |
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80 ntrunc_def: "ntrunc k N == {n. n:N & ndepth(n)<k}" |
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81 |
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82 (*products and sums for the "universe"*) |
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83 uprod_def: "uprod A B == UN x:A. UN y:B. { Scons x y }" |
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84 usum_def: "usum A B == In0`A Un In1`B" |
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85 |
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86 (*the corresponding eliminators*) |
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87 Split_def: "Split c M == THE u. EX x y. M = Scons x y & u = c x y" |
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88 |
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89 Case_def: "Case c d M == THE u. (EX x . M = In0(x) & u = c(x)) |
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90 | (EX y . M = In1(y) & u = d(y))" |
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91 |
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92 |
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93 (** equality for the "universe" **) |
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94 |
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95 dprod_def: "dprod r s == UN (x,x'):r. UN (y,y'):s. {(Scons x y, Scons x' y')}" |
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96 |
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97 dsum_def: "dsum r s == (UN (x,x'):r. {(In0(x),In0(x'))}) Un |
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98 (UN (y,y'):s. {(In1(y),In1(y'))})" |
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99 |
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100 |
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101 |
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102 lemma apfst_convE: |
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103 "[| q = apfst f p; !!x y. [| p = (x,y); q = (f(x),y) |] ==> R |
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104 |] ==> R" |
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105 by (force simp add: apfst_def) |
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106 |
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107 (** Push -- an injection, analogous to Cons on lists **) |
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108 |
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109 lemma Push_inject1: "Push i f = Push j g ==> i=j" |
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110 apply (simp add: Push_def fun_eq_iff) |
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111 apply (drule_tac x=0 in spec, simp) |
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112 done |
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113 |
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114 lemma Push_inject2: "Push i f = Push j g ==> f=g" |
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115 apply (auto simp add: Push_def fun_eq_iff) |
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116 apply (drule_tac x="Suc x" in spec, simp) |
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117 done |
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118 |
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119 lemma Push_inject: |
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120 "[| Push i f =Push j g; [| i=j; f=g |] ==> P |] ==> P" |
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121 by (blast dest: Push_inject1 Push_inject2) |
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122 |
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123 lemma Push_neq_K0: "Push (Inr (Suc k)) f = (%z. Inr 0) ==> P" |
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124 by (auto simp add: Push_def fun_eq_iff split: nat.split_asm) |
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125 |
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126 lemmas Abs_Node_inj = Abs_Node_inject [THEN [2] rev_iffD1] |
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127 |
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128 |
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129 (*** Introduction rules for Node ***) |
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130 |
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131 lemma Node_K0_I: "(%k. Inr 0, a) : Node" |
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132 by (simp add: Node_def) |
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133 |
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134 lemma Node_Push_I: "p: Node ==> apfst (Push i) p : Node" |
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135 apply (simp add: Node_def Push_def) |
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136 apply (fast intro!: apfst_conv nat.case(2)[THEN trans]) |
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137 done |
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138 |
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139 |
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140 subsection{*Freeness: Distinctness of Constructors*} |
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141 |
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142 (** Scons vs Atom **) |
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143 |
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144 lemma Scons_not_Atom [iff]: "Scons M N \<noteq> Atom(a)" |
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145 unfolding Atom_def Scons_def Push_Node_def One_nat_def |
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146 by (blast intro: Node_K0_I Rep_Node [THEN Node_Push_I] |
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147 dest!: Abs_Node_inj |
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148 elim!: apfst_convE sym [THEN Push_neq_K0]) |
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149 |
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150 lemmas Atom_not_Scons [iff] = Scons_not_Atom [THEN not_sym] |
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151 |
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152 |
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153 (*** Injectiveness ***) |
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154 |
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155 (** Atomic nodes **) |
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156 |
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157 lemma inj_Atom: "inj(Atom)" |
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158 apply (simp add: Atom_def) |
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159 apply (blast intro!: inj_onI Node_K0_I dest!: Abs_Node_inj) |
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160 done |
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161 lemmas Atom_inject = inj_Atom [THEN injD] |
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162 |
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163 lemma Atom_Atom_eq [iff]: "(Atom(a)=Atom(b)) = (a=b)" |
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164 by (blast dest!: Atom_inject) |
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165 |
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166 lemma inj_Leaf: "inj(Leaf)" |
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167 apply (simp add: Leaf_def o_def) |
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168 apply (rule inj_onI) |
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169 apply (erule Atom_inject [THEN Inl_inject]) |
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170 done |
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171 |
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172 lemmas Leaf_inject [dest!] = inj_Leaf [THEN injD] |
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173 |
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174 lemma inj_Numb: "inj(Numb)" |
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175 apply (simp add: Numb_def o_def) |
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176 apply (rule inj_onI) |
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177 apply (erule Atom_inject [THEN Inr_inject]) |
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178 done |
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179 |
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180 lemmas Numb_inject [dest!] = inj_Numb [THEN injD] |
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181 |
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182 |
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183 (** Injectiveness of Push_Node **) |
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184 |
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185 lemma Push_Node_inject: |
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186 "[| Push_Node i m =Push_Node j n; [| i=j; m=n |] ==> P |
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187 |] ==> P" |
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188 apply (simp add: Push_Node_def) |
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189 apply (erule Abs_Node_inj [THEN apfst_convE]) |
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190 apply (rule Rep_Node [THEN Node_Push_I])+ |
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191 apply (erule sym [THEN apfst_convE]) |
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192 apply (blast intro: Rep_Node_inject [THEN iffD1] trans sym elim!: Push_inject) |
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193 done |
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194 |
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195 |
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196 (** Injectiveness of Scons **) |
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197 |
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198 lemma Scons_inject_lemma1: "Scons M N <= Scons M' N' ==> M<=M'" |
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199 unfolding Scons_def One_nat_def |
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200 by (blast dest!: Push_Node_inject) |
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201 |
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202 lemma Scons_inject_lemma2: "Scons M N <= Scons M' N' ==> N<=N'" |
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203 unfolding Scons_def One_nat_def |
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204 by (blast dest!: Push_Node_inject) |
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205 |
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206 lemma Scons_inject1: "Scons M N = Scons M' N' ==> M=M'" |
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207 apply (erule equalityE) |
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208 apply (iprover intro: equalityI Scons_inject_lemma1) |
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209 done |
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210 |
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211 lemma Scons_inject2: "Scons M N = Scons M' N' ==> N=N'" |
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212 apply (erule equalityE) |
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213 apply (iprover intro: equalityI Scons_inject_lemma2) |
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214 done |
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215 |
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216 lemma Scons_inject: |
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217 "[| Scons M N = Scons M' N'; [| M=M'; N=N' |] ==> P |] ==> P" |
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218 by (iprover dest: Scons_inject1 Scons_inject2) |
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219 |
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220 lemma Scons_Scons_eq [iff]: "(Scons M N = Scons M' N') = (M=M' & N=N')" |
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221 by (blast elim!: Scons_inject) |
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222 |
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223 (*** Distinctness involving Leaf and Numb ***) |
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224 |
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225 (** Scons vs Leaf **) |
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226 |
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227 lemma Scons_not_Leaf [iff]: "Scons M N \<noteq> Leaf(a)" |
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228 unfolding Leaf_def o_def by (rule Scons_not_Atom) |
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229 |
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230 lemmas Leaf_not_Scons [iff] = Scons_not_Leaf [THEN not_sym] |
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231 |
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232 (** Scons vs Numb **) |
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233 |
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234 lemma Scons_not_Numb [iff]: "Scons M N \<noteq> Numb(k)" |
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235 unfolding Numb_def o_def by (rule Scons_not_Atom) |
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236 |
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237 lemmas Numb_not_Scons [iff] = Scons_not_Numb [THEN not_sym] |
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238 |
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239 |
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240 (** Leaf vs Numb **) |
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241 |
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242 lemma Leaf_not_Numb [iff]: "Leaf(a) \<noteq> Numb(k)" |
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243 by (simp add: Leaf_def Numb_def) |
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244 |
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245 lemmas Numb_not_Leaf [iff] = Leaf_not_Numb [THEN not_sym] |
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246 |
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247 |
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248 (*** ndepth -- the depth of a node ***) |
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249 |
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250 lemma ndepth_K0: "ndepth (Abs_Node(%k. Inr 0, x)) = 0" |
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251 by (simp add: ndepth_def Node_K0_I [THEN Abs_Node_inverse] Least_equality) |
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252 |
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253 lemma ndepth_Push_Node_aux: |
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254 "case_nat (Inr (Suc i)) f k = Inr 0 --> Suc(LEAST x. f x = Inr 0) <= k" |
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255 apply (induct_tac "k", auto) |
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256 apply (erule Least_le) |
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257 done |
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258 |
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259 lemma ndepth_Push_Node: |
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260 "ndepth (Push_Node (Inr (Suc i)) n) = Suc(ndepth(n))" |
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261 apply (insert Rep_Node [of n, unfolded Node_def]) |
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262 apply (auto simp add: ndepth_def Push_Node_def |
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263 Rep_Node [THEN Node_Push_I, THEN Abs_Node_inverse]) |
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264 apply (rule Least_equality) |
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265 apply (auto simp add: Push_def ndepth_Push_Node_aux) |
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266 apply (erule LeastI) |
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267 done |
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268 |
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269 |
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270 (*** ntrunc applied to the various node sets ***) |
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271 |
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272 lemma ntrunc_0 [simp]: "ntrunc 0 M = {}" |
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273 by (simp add: ntrunc_def) |
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274 |
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275 lemma ntrunc_Atom [simp]: "ntrunc (Suc k) (Atom a) = Atom(a)" |
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276 by (auto simp add: Atom_def ntrunc_def ndepth_K0) |
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277 |
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278 lemma ntrunc_Leaf [simp]: "ntrunc (Suc k) (Leaf a) = Leaf(a)" |
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279 unfolding Leaf_def o_def by (rule ntrunc_Atom) |
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280 |
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281 lemma ntrunc_Numb [simp]: "ntrunc (Suc k) (Numb i) = Numb(i)" |
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282 unfolding Numb_def o_def by (rule ntrunc_Atom) |
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283 |
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284 lemma ntrunc_Scons [simp]: |
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285 "ntrunc (Suc k) (Scons M N) = Scons (ntrunc k M) (ntrunc k N)" |
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286 unfolding Scons_def ntrunc_def One_nat_def |
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287 by (auto simp add: ndepth_Push_Node) |
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288 |
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289 |
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290 |
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291 (** Injection nodes **) |
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292 |
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293 lemma ntrunc_one_In0 [simp]: "ntrunc (Suc 0) (In0 M) = {}" |
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294 apply (simp add: In0_def) |
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295 apply (simp add: Scons_def) |
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296 done |
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297 |
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298 lemma ntrunc_In0 [simp]: "ntrunc (Suc(Suc k)) (In0 M) = In0 (ntrunc (Suc k) M)" |
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299 by (simp add: In0_def) |
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300 |
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301 lemma ntrunc_one_In1 [simp]: "ntrunc (Suc 0) (In1 M) = {}" |
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302 apply (simp add: In1_def) |
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303 apply (simp add: Scons_def) |
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304 done |
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305 |
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306 lemma ntrunc_In1 [simp]: "ntrunc (Suc(Suc k)) (In1 M) = In1 (ntrunc (Suc k) M)" |
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307 by (simp add: In1_def) |
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308 |
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309 |
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310 subsection{*Set Constructions*} |
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311 |
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312 |
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313 (*** Cartesian Product ***) |
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314 |
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315 lemma uprodI [intro!]: "[| M:A; N:B |] ==> Scons M N : uprod A B" |
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316 by (simp add: uprod_def) |
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317 |
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318 (*The general elimination rule*) |
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319 lemma uprodE [elim!]: |
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320 "[| c : uprod A B; |
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321 !!x y. [| x:A; y:B; c = Scons x y |] ==> P |
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322 |] ==> P" |
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323 by (auto simp add: uprod_def) |
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324 |
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325 |
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326 (*Elimination of a pair -- introduces no eigenvariables*) |
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327 lemma uprodE2: "[| Scons M N : uprod A B; [| M:A; N:B |] ==> P |] ==> P" |
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328 by (auto simp add: uprod_def) |
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329 |
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330 |
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331 (*** Disjoint Sum ***) |
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332 |
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333 lemma usum_In0I [intro]: "M:A ==> In0(M) : usum A B" |
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334 by (simp add: usum_def) |
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335 |
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336 lemma usum_In1I [intro]: "N:B ==> In1(N) : usum A B" |
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337 by (simp add: usum_def) |
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338 |
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339 lemma usumE [elim!]: |
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340 "[| u : usum A B; |
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341 !!x. [| x:A; u=In0(x) |] ==> P; |
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342 !!y. [| y:B; u=In1(y) |] ==> P |
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343 |] ==> P" |
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344 by (auto simp add: usum_def) |
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345 |
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346 |
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347 (** Injection **) |
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348 |
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349 lemma In0_not_In1 [iff]: "In0(M) \<noteq> In1(N)" |
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350 unfolding In0_def In1_def One_nat_def by auto |
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351 |
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352 lemmas In1_not_In0 [iff] = In0_not_In1 [THEN not_sym] |
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353 |
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354 lemma In0_inject: "In0(M) = In0(N) ==> M=N" |
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355 by (simp add: In0_def) |
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356 |
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357 lemma In1_inject: "In1(M) = In1(N) ==> M=N" |
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358 by (simp add: In1_def) |
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359 |
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360 lemma In0_eq [iff]: "(In0 M = In0 N) = (M=N)" |
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361 by (blast dest!: In0_inject) |
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362 |
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363 lemma In1_eq [iff]: "(In1 M = In1 N) = (M=N)" |
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364 by (blast dest!: In1_inject) |
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365 |
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366 lemma inj_In0: "inj In0" |
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367 by (blast intro!: inj_onI) |
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368 |
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369 lemma inj_In1: "inj In1" |
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370 by (blast intro!: inj_onI) |
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371 |
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372 |
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373 (*** Function spaces ***) |
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374 |
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375 lemma Lim_inject: "Lim f = Lim g ==> f = g" |
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376 apply (simp add: Lim_def) |
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377 apply (rule ext) |
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378 apply (blast elim!: Push_Node_inject) |
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379 done |
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380 |
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381 |
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382 (*** proving equality of sets and functions using ntrunc ***) |
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383 |
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384 lemma ntrunc_subsetI: "ntrunc k M <= M" |
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385 by (auto simp add: ntrunc_def) |
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386 |
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387 lemma ntrunc_subsetD: "(!!k. ntrunc k M <= N) ==> M<=N" |
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388 by (auto simp add: ntrunc_def) |
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389 |
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390 (*A generalized form of the take-lemma*) |
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391 lemma ntrunc_equality: "(!!k. ntrunc k M = ntrunc k N) ==> M=N" |
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392 apply (rule equalityI) |
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393 apply (rule_tac [!] ntrunc_subsetD) |
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394 apply (rule_tac [!] ntrunc_subsetI [THEN [2] subset_trans], auto) |
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395 done |
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396 |
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397 lemma ntrunc_o_equality: |
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398 "[| !!k. (ntrunc(k) o h1) = (ntrunc(k) o h2) |] ==> h1=h2" |
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399 apply (rule ntrunc_equality [THEN ext]) |
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400 apply (simp add: fun_eq_iff) |
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401 done |
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402 |
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403 |
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404 (*** Monotonicity ***) |
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405 |
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406 lemma uprod_mono: "[| A<=A'; B<=B' |] ==> uprod A B <= uprod A' B'" |
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407 by (simp add: uprod_def, blast) |
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408 |
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409 lemma usum_mono: "[| A<=A'; B<=B' |] ==> usum A B <= usum A' B'" |
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410 by (simp add: usum_def, blast) |
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411 |
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412 lemma Scons_mono: "[| M<=M'; N<=N' |] ==> Scons M N <= Scons M' N'" |
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413 by (simp add: Scons_def, blast) |
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414 |
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415 lemma In0_mono: "M<=N ==> In0(M) <= In0(N)" |
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416 by (simp add: In0_def Scons_mono) |
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417 |
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418 lemma In1_mono: "M<=N ==> In1(M) <= In1(N)" |
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419 by (simp add: In1_def Scons_mono) |
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420 |
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421 |
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422 (*** Split and Case ***) |
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423 |
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424 lemma Split [simp]: "Split c (Scons M N) = c M N" |
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425 by (simp add: Split_def) |
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426 |
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427 lemma Case_In0 [simp]: "Case c d (In0 M) = c(M)" |
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428 by (simp add: Case_def) |
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429 |
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430 lemma Case_In1 [simp]: "Case c d (In1 N) = d(N)" |
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431 by (simp add: Case_def) |
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432 |
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433 |
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434 |
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435 (**** UN x. B(x) rules ****) |
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436 |
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437 lemma ntrunc_UN1: "ntrunc k (UN x. f(x)) = (UN x. ntrunc k (f x))" |
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438 by (simp add: ntrunc_def, blast) |
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439 |
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440 lemma Scons_UN1_x: "Scons (UN x. f x) M = (UN x. Scons (f x) M)" |
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441 by (simp add: Scons_def, blast) |
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442 |
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443 lemma Scons_UN1_y: "Scons M (UN x. f x) = (UN x. Scons M (f x))" |
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444 by (simp add: Scons_def, blast) |
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445 |
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446 lemma In0_UN1: "In0(UN x. f(x)) = (UN x. In0(f(x)))" |
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447 by (simp add: In0_def Scons_UN1_y) |
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448 |
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449 lemma In1_UN1: "In1(UN x. f(x)) = (UN x. In1(f(x)))" |
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450 by (simp add: In1_def Scons_UN1_y) |
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451 |
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452 |
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453 (*** Equality for Cartesian Product ***) |
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454 |
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455 lemma dprodI [intro!]: |
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456 "[| (M,M'):r; (N,N'):s |] ==> (Scons M N, Scons M' N') : dprod r s" |
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457 by (auto simp add: dprod_def) |
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458 |
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459 (*The general elimination rule*) |
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460 lemma dprodE [elim!]: |
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461 "[| c : dprod r s; |
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462 !!x y x' y'. [| (x,x') : r; (y,y') : s; |
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463 c = (Scons x y, Scons x' y') |] ==> P |
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464 |] ==> P" |
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465 by (auto simp add: dprod_def) |
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466 |
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467 |
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468 (*** Equality for Disjoint Sum ***) |
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469 |
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470 lemma dsum_In0I [intro]: "(M,M'):r ==> (In0(M), In0(M')) : dsum r s" |
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471 by (auto simp add: dsum_def) |
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472 |
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473 lemma dsum_In1I [intro]: "(N,N'):s ==> (In1(N), In1(N')) : dsum r s" |
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474 by (auto simp add: dsum_def) |
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475 |
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476 lemma dsumE [elim!]: |
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477 "[| w : dsum r s; |
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478 !!x x'. [| (x,x') : r; w = (In0(x), In0(x')) |] ==> P; |
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479 !!y y'. [| (y,y') : s; w = (In1(y), In1(y')) |] ==> P |
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480 |] ==> P" |
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481 by (auto simp add: dsum_def) |
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482 |
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483 |
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484 (*** Monotonicity ***) |
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485 |
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486 lemma dprod_mono: "[| r<=r'; s<=s' |] ==> dprod r s <= dprod r' s'" |
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487 by blast |
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488 |
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489 lemma dsum_mono: "[| r<=r'; s<=s' |] ==> dsum r s <= dsum r' s'" |
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490 by blast |
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491 |
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492 |
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493 (*** Bounding theorems ***) |
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494 |
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495 lemma dprod_Sigma: "(dprod (A <*> B) (C <*> D)) <= (uprod A C) <*> (uprod B D)" |
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496 by blast |
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497 |
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498 lemmas dprod_subset_Sigma = subset_trans [OF dprod_mono dprod_Sigma] |
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499 |
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500 (*Dependent version*) |
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501 lemma dprod_subset_Sigma2: |
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502 "(dprod (Sigma A B) (Sigma C D)) <= Sigma (uprod A C) (Split (%x y. uprod (B x) (D y)))" |
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503 by auto |
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504 |
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505 lemma dsum_Sigma: "(dsum (A <*> B) (C <*> D)) <= (usum A C) <*> (usum B D)" |
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506 by blast |
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507 |
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508 lemmas dsum_subset_Sigma = subset_trans [OF dsum_mono dsum_Sigma] |
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509 |
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510 |
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511 (*** Domain theorems ***) |
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512 |
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513 lemma Domain_dprod [simp]: "Domain (dprod r s) = uprod (Domain r) (Domain s)" |
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514 by auto |
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515 |
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516 lemma Domain_dsum [simp]: "Domain (dsum r s) = usum (Domain r) (Domain s)" |
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517 by auto |
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518 |
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519 |
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520 text {* hides popular names *} |
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521 hide_type (open) node item |
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522 hide_const (open) Push Node Atom Leaf Numb Lim Split Case |
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523 |
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524 ML_file "Tools/Old_Datatype/old_datatype.ML" |
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525 |
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526 ML_file "Tools/inductive_realizer.ML" |
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527 setup InductiveRealizer.setup |
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528 |
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529 end |
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