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1 (* Title: HOL/LList.thy |
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2 ID: $Id$ |
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3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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4 Copyright 1994 University of Cambridge |
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5 |
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6 Definition of type 'a llist by a greatest fixed point |
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7 |
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8 Shares NIL, CONS, List_case with List.thy |
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9 |
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10 Still needs filter and flatten functions -- hard because they need |
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11 bounds on the amount of lookahead required. |
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12 |
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13 Could try (but would it work for the gfp analogue of term?) |
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14 LListD_Fun_def "LListD_Fun(A) == (%Z.diag({Numb(0)}) <++> diag(A) <**> Z)" |
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15 |
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16 A nice but complex example would be [ML for the Working Programmer, page 176] |
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17 from(1) = enumerate (Lmap (Lmap(pack), makeqq(from(1),from(1)))) |
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18 |
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19 Previous definition of llistD_Fun was explicit: |
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20 llistD_Fun_def |
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21 "llistD_Fun(r) == |
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22 {(LNil,LNil)} Un |
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23 (UN x. (split(%l1 l2.(LCons(x,l1),LCons(x,l2))))``r)" |
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24 *) |
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25 |
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26 LList = Gfp + SList + |
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27 |
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28 types |
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29 'a llist |
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30 |
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31 arities |
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32 llist :: (term)term |
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33 |
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34 consts |
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35 list_Fun :: ['a item set, 'a item set] => 'a item set |
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36 LListD_Fun :: |
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37 "[('a item * 'a item)set, ('a item * 'a item)set] => |
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38 ('a item * 'a item)set" |
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39 |
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40 llist :: 'a item set => 'a item set |
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41 LListD :: "('a item * 'a item)set => ('a item * 'a item)set" |
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42 llistD_Fun :: "('a llist * 'a llist)set => ('a llist * 'a llist)set" |
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43 |
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44 Rep_llist :: 'a llist => 'a item |
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45 Abs_llist :: 'a item => 'a llist |
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46 LNil :: 'a llist |
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47 LCons :: ['a, 'a llist] => 'a llist |
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48 |
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49 llist_case :: ['b, ['a, 'a llist]=>'b, 'a llist] => 'b |
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50 |
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51 LList_corec_fun :: "[nat, 'a=>unit+('b item * 'a), 'a] => 'b item" |
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52 LList_corec :: "['a, 'a => unit + ('b item * 'a)] => 'b item" |
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53 llist_corec :: "['a, 'a => unit + ('b * 'a)] => 'b llist" |
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54 |
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55 Lmap :: ('a item => 'b item) => ('a item => 'b item) |
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56 lmap :: ('a=>'b) => ('a llist => 'b llist) |
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57 |
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58 iterates :: ['a => 'a, 'a] => 'a llist |
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59 |
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60 Lconst :: 'a item => 'a item |
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61 Lappend :: ['a item, 'a item] => 'a item |
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62 lappend :: ['a llist, 'a llist] => 'a llist |
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63 |
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64 |
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65 coinductive "llist(A)" |
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66 intrs |
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67 NIL_I "NIL: llist(A)" |
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68 CONS_I "[| a: A; M: llist(A) |] ==> CONS a M : llist(A)" |
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69 |
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70 coinductive "LListD(r)" |
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71 intrs |
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72 NIL_I "(NIL, NIL) : LListD(r)" |
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73 CONS_I "[| (a,b): r; (M,N) : LListD(r) |
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74 |] ==> (CONS a M, CONS b N) : LListD(r)" |
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75 |
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76 translations |
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77 "case p of LNil => a | LCons x l => b" == "llist_case a (%x l.b) p" |
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78 |
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79 |
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80 defs |
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81 (*Now used exclusively for abbreviating the coinduction rule*) |
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82 list_Fun_def "list_Fun A X == |
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83 {z. z = NIL | (? M a. z = CONS a M & a : A & M : X)}" |
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84 |
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85 LListD_Fun_def "LListD_Fun r X == |
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86 {z. z = (NIL, NIL) | |
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87 (? M N a b. z = (CONS a M, CONS b N) & |
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88 (a, b) : r & (M, N) : X)}" |
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89 |
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90 (*defining the abstract constructors*) |
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91 LNil_def "LNil == Abs_llist(NIL)" |
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92 LCons_def "LCons x xs == Abs_llist(CONS (Leaf x) (Rep_llist xs))" |
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93 |
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94 llist_case_def |
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95 "llist_case c d l == |
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96 List_case c (%x y. d (inv Leaf x) (Abs_llist y)) (Rep_llist l)" |
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97 |
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98 LList_corec_fun_def |
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99 "LList_corec_fun k f == |
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100 nat_rec (%x. {}) |
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101 (%j r x. case f x of Inl u => NIL |
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102 | Inr(z,w) => CONS z (r w)) |
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103 k" |
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104 |
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105 LList_corec_def |
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106 "LList_corec a f == UN k. LList_corec_fun k f a" |
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107 |
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108 llist_corec_def |
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109 "llist_corec a f == |
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110 Abs_llist(LList_corec a |
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111 (%z.case f z of Inl x => Inl(x) |
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112 | Inr(v,w) => Inr(Leaf(v), w)))" |
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113 |
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114 llistD_Fun_def |
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115 "llistD_Fun(r) == |
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116 prod_fun Abs_llist Abs_llist `` |
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117 LListD_Fun (diag(range Leaf)) |
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118 (prod_fun Rep_llist Rep_llist `` r)" |
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119 |
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120 Lconst_def "Lconst(M) == lfp(%N. CONS M N)" |
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121 |
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122 Lmap_def |
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123 "Lmap f M == LList_corec M (List_case (Inl ()) (%x M'. Inr((f(x), M'))))" |
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124 |
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125 lmap_def |
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126 "lmap f l == llist_corec l (%z. case z of LNil => (Inl ()) |
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127 | LCons y z => Inr(f(y), z))" |
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128 |
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129 iterates_def "iterates f a == llist_corec a (%x. Inr((x, f(x))))" |
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130 |
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131 (*Append generates its result by applying f, where |
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132 f((NIL,NIL)) = Inl(()) |
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133 f((NIL, CONS N1 N2)) = Inr((N1, (NIL,N2)) |
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134 f((CONS M1 M2, N)) = Inr((M1, (M2,N)) |
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135 *) |
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136 |
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137 Lappend_def |
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138 "Lappend M N == LList_corec (M,N) |
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139 (split(List_case (List_case (Inl ()) (%N1 N2. Inr((N1, (NIL,N2))))) |
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140 (%M1 M2 N. Inr((M1, (M2,N))))))" |
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141 |
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142 lappend_def |
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143 "lappend l n == llist_corec (l,n) |
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144 (split(llist_case (llist_case (Inl ()) (%n1 n2. Inr((n1, (LNil,n2))))) |
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145 (%l1 l2 n. Inr((l1, (l2,n))))))" |
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146 |
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147 rules |
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148 (*faking a type definition...*) |
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149 Rep_llist "Rep_llist(xs): llist(range(Leaf))" |
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150 Rep_llist_inverse "Abs_llist(Rep_llist(xs)) = xs" |
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151 Abs_llist_inverse "M: llist(range(Leaf)) ==> Rep_llist(Abs_llist(M)) = M" |
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152 |
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153 end |