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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 defs |
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77 (*Now used exclusively for abbreviating the coinduction rule*) |
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78 list_Fun_def "list_Fun A X == \ |
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79 \ {z. z = NIL | (? M a. z = CONS a M & a : A & M : X)}" |
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80 |
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81 LListD_Fun_def "LListD_Fun r X == \ |
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82 \ {z. z = <NIL, NIL> | \ |
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83 \ (? M N a b. z = <CONS a M, CONS b N> & \ |
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84 \ <a, b> : r & <M, N> : X)}" |
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85 |
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86 (*defining the abstract constructors*) |
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87 LNil_def "LNil == Abs_llist(NIL)" |
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88 LCons_def "LCons x xs == Abs_llist(CONS (Leaf x) (Rep_llist xs))" |
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89 |
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90 llist_case_def |
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91 "llist_case c d l == \ |
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92 \ List_case c (%x y. d (Inv Leaf x) (Abs_llist y)) (Rep_llist l)" |
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93 |
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94 LList_corec_fun_def |
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95 "LList_corec_fun k f == \ |
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96 \ nat_rec k (%x. {}) \ |
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97 \ (%j r x. sum_case (%u.NIL) (split(%z w. CONS z (r w))) (f x))" |
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98 |
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99 LList_corec_def |
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100 "LList_corec a f == UN k. LList_corec_fun k f a" |
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101 |
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102 llist_corec_def |
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103 "llist_corec a f == \ |
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104 \ Abs_llist(LList_corec a (%z.sum_case (%x.Inl(x)) \ |
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105 \ (split(%v w. Inr(<Leaf(v), w>))) (f z)))" |
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106 |
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107 llistD_Fun_def |
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108 "llistD_Fun(r) == \ |
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109 \ prod_fun Abs_llist Abs_llist `` \ |
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110 \ LListD_Fun (diag(range Leaf)) \ |
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111 \ (prod_fun Rep_llist Rep_llist `` r)" |
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112 |
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113 Lconst_def "Lconst(M) == lfp(%N. CONS M N)" |
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114 |
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115 Lmap_def |
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116 "Lmap f M == LList_corec M (List_case (Inl Unity) (%x M'. Inr(<f(x), M'>)))" |
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117 |
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118 lmap_def |
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119 "lmap f l == llist_corec l (llist_case (Inl Unity) (%y z. Inr(<f(y), z>)))" |
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120 |
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121 iterates_def "iterates f a == llist_corec a (%x. Inr(<x, f(x)>))" |
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122 |
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123 (*Append generates its result by applying f, where |
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124 f(<NIL,NIL>) = Inl(Unity) |
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125 f(<NIL, CONS N1 N2>) = Inr(<N1, <NIL,N2>) |
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126 f(<CONS M1 M2, N>) = Inr(<M1, <M2,N>) |
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127 *) |
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128 |
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129 Lappend_def |
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130 "Lappend M N == LList_corec <M,N> \ |
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131 \ (split(List_case (List_case (Inl Unity) (%N1 N2. Inr(<N1, <NIL,N2>>))) \ |
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132 \ (%M1 M2 N. Inr(<M1, <M2,N>>))))" |
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133 |
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134 lappend_def |
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135 "lappend l n == llist_corec <l,n> \ |
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136 \ (split(llist_case (llist_case (Inl Unity) (%n1 n2. Inr(<n1, <LNil,n2>>))) \ |
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137 \ (%l1 l2 n. Inr(<l1, <l2,n>>))))" |
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138 |
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139 rules |
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140 (*faking a type definition...*) |
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141 Rep_llist "Rep_llist(xs): llist(range(Leaf))" |
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142 Rep_llist_inverse "Abs_llist(Rep_llist(xs)) = xs" |
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143 Abs_llist_inverse "M: llist(range(Leaf)) ==> Rep_llist(Abs_llist(M)) = M" |
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144 |
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145 end |