21 "llistD_Fun(r) == |
21 "llistD_Fun(r) == |
22 {(LNil,LNil)} Un |
22 {(LNil,LNil)} Un |
23 (UN x. (split(%l1 l2.(LCons(x,l1),LCons(x,l2))))``r)" |
23 (UN x. (split(%l1 l2.(LCons(x,l1),LCons(x,l2))))``r)" |
24 *) |
24 *) |
25 |
25 |
26 LList = Gfp + SList + |
26 LList = Main + SList + |
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 |
27 |
34 consts |
28 consts |
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 |
29 |
40 llist :: 'a item set => 'a item set |
30 llist :: 'a item set => 'a item set |
41 LListD :: "('a item * 'a item)set => ('a item * 'a item)set" |
31 LListD :: "('a item * 'a item)set => ('a item * 'a item)set" |
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 |
32 |
64 |
33 |
65 coinductive "llist(A)" |
34 coinductive "llist(A)" |
66 intrs |
35 intrs |
67 NIL_I "NIL: llist(A)" |
36 NIL_I "NIL: llist(A)" |
71 intrs |
40 intrs |
72 NIL_I "(NIL, NIL) : LListD(r)" |
41 NIL_I "(NIL, NIL) : LListD(r)" |
73 CONS_I "[| (a,b): r; (M,N) : LListD(r) |
42 CONS_I "[| (a,b): r; (M,N) : LListD(r) |
74 |] ==> (CONS a M, CONS b N) : LListD(r)" |
43 |] ==> (CONS a M, CONS b N) : LListD(r)" |
75 |
44 |
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45 |
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46 typedef (LList) |
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47 'a llist = "llist(range Leaf)" (llist.NIL_I) |
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48 |
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49 constdefs |
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50 (*Now used exclusively for abbreviating the coinduction rule*) |
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51 list_Fun :: ['a item set, 'a item set] => 'a item set |
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52 "list_Fun A X == {z. z = NIL | (? M a. z = CONS a M & a : A & M : X)}" |
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53 |
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54 LListD_Fun :: |
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55 "[('a item * 'a item)set, ('a item * 'a item)set] => |
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56 ('a item * 'a item)set" |
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57 "LListD_Fun r X == |
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58 {z. z = (NIL, NIL) | |
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59 (? M N a b. z = (CONS a M, CONS b N) & (a, b) : r & (M, N) : X)}" |
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60 |
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61 (*the abstract constructors*) |
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62 LNil :: 'a llist |
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63 "LNil == Abs_LList NIL" |
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64 |
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65 LCons :: ['a, 'a llist] => 'a llist |
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66 "LCons x xs == Abs_LList(CONS (Leaf x) (Rep_LList xs))" |
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67 |
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68 llist_case :: ['b, ['a, 'a llist]=>'b, 'a llist] => 'b |
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69 "llist_case c d l == |
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70 List_case c (%x y. d (inv Leaf x) (Abs_LList y)) (Rep_LList l)" |
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71 |
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72 LList_corec_fun :: "[nat, 'a=> ('b item * 'a) option, 'a] => 'b item" |
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73 "LList_corec_fun k f == |
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74 nat_rec (%x. {}) |
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75 (%j r x. case f x of None => NIL |
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76 | Some(z,w) => CONS z (r w)) |
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77 k" |
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78 |
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79 LList_corec :: "['a, 'a => ('b item * 'a) option] => 'b item" |
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80 "LList_corec a f == UN k. LList_corec_fun k f a" |
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81 |
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82 llist_corec :: "['a, 'a => ('b * 'a) option] => 'b llist" |
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83 "llist_corec a f == |
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84 Abs_LList(LList_corec a |
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85 (%z. case f z of None => None |
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86 | Some(v,w) => Some(Leaf(v), w)))" |
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87 |
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88 llistD_Fun :: "('a llist * 'a llist)set => ('a llist * 'a llist)set" |
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89 "llistD_Fun(r) == |
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90 prod_fun Abs_LList Abs_LList `` |
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91 LListD_Fun (diag(range Leaf)) |
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92 (prod_fun Rep_LList Rep_LList `` r)" |
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93 |
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94 |
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95 |
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96 (*The case syntax for type 'a llist*) |
76 translations |
97 translations |
77 "case p of LNil => a | LCons x l => b" == "llist_case a (%x l. b) p" |
98 "case p of LNil => a | LCons x l => b" == "llist_case a (%x l. b) p" |
78 |
99 |
79 |
100 |
80 defs |
101 (** Sample function definitions. Item-based ones start with L ***) |
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 |
102 |
85 LListD_Fun_def "LListD_Fun r X == |
103 constdefs |
86 {z. z = (NIL, NIL) | |
104 Lmap :: ('a item => 'b item) => ('a item => 'b item) |
87 (? M N a b. z = (CONS a M, CONS b N) & |
105 "Lmap f M == LList_corec M (List_case None (%x M'. Some((f(x), M'))))" |
88 (a, b) : r & (M, N) : X)}" |
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89 |
106 |
90 (*defining the abstract constructors*) |
107 lmap :: ('a=>'b) => ('a llist => 'b llist) |
91 LNil_def "LNil == Abs_llist(NIL)" |
108 "lmap f l == llist_corec l (%z. case z of LNil => None |
92 LCons_def "LCons x xs == Abs_llist(CONS (Leaf x) (Rep_llist xs))" |
109 | LCons y z => Some(f(y), z))" |
93 |
110 |
94 llist_case_def |
111 iterates :: ['a => 'a, 'a] => 'a llist |
95 "llist_case c d l == |
112 "iterates f a == llist_corec a (%x. Some((x, f(x))))" |
96 List_case c (%x y. d (inv Leaf x) (Abs_llist y)) (Rep_llist l)" |
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97 |
113 |
98 LList_corec_fun_def |
114 Lconst :: 'a item => 'a item |
99 "LList_corec_fun k f == |
115 "Lconst(M) == lfp(%N. CONS M N)" |
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 |
116 |
131 (*Append generates its result by applying f, where |
117 (*Append generates its result by applying f, where |
132 f((NIL,NIL)) = Inl(()) |
118 f((NIL,NIL)) = None |
133 f((NIL, CONS N1 N2)) = Inr((N1, (NIL,N2)) |
119 f((NIL, CONS N1 N2)) = Some((N1, (NIL,N2)) |
134 f((CONS M1 M2, N)) = Inr((M1, (M2,N)) |
120 f((CONS M1 M2, N)) = Some((M1, (M2,N)) |
135 *) |
121 *) |
136 |
122 |
137 Lappend_def |
123 Lappend :: ['a item, 'a item] => 'a item |
138 "Lappend M N == LList_corec (M,N) |
124 "Lappend M N == LList_corec (M,N) |
139 (split(List_case (List_case (Inl ()) (%N1 N2. Inr((N1, (NIL,N2))))) |
125 (split(List_case (List_case None (%N1 N2. Some((N1, (NIL,N2))))) |
140 (%M1 M2 N. Inr((M1, (M2,N))))))" |
126 (%M1 M2 N. Some((M1, (M2,N))))))" |
141 |
127 |
142 lappend_def |
128 lappend :: ['a llist, 'a llist] => 'a llist |
143 "lappend l n == llist_corec (l,n) |
129 "lappend l n == llist_corec (l,n) |
144 (split(llist_case (llist_case (Inl ()) (%n1 n2. Inr((n1, (LNil,n2))))) |
130 (split(llist_case (llist_case None (%n1 n2. Some((n1, (LNil,n2))))) |
145 (%l1 l2 n. Inr((l1, (l2,n))))))" |
131 (%l1 l2 n. Some((l1, (l2,n))))))" |
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 |
132 |
153 end |
133 end |