1 | include "Maybe.ma". |
---|
2 | include "Nat.ma". |
---|
3 | include "Util.ma". |
---|
4 | |
---|
5 | include "Plogic/equality.ma". |
---|
6 | |
---|
7 | ninductive List (A: Type[0]): Type[0] ≝ |
---|
8 | Empty: List A |
---|
9 | | Cons: A → List A → List A. |
---|
10 | |
---|
11 | notation "hvbox(hd break :: tl)" |
---|
12 | right associative with precedence 47 |
---|
13 | for @{ 'Cons $hd $tl }. |
---|
14 | |
---|
15 | interpretation "List empty" 'Empty = (Empty ?). |
---|
16 | interpretation "List cons" 'Cons = (Cons ?). |
---|
17 | |
---|
18 | notation "[ list0 x sep ; ]" |
---|
19 | non associative with precedence 90 |
---|
20 | for @{ fold right @'Empty rec acc @{ 'Cons $x $acc } }. |
---|
21 | |
---|
22 | nlet rec length (A: Type[0]) (l: List A) on l ≝ |
---|
23 | match l with |
---|
24 | [ Empty ⇒ Z |
---|
25 | | Cons hd tl ⇒ S $ length A tl |
---|
26 | ]. |
---|
27 | |
---|
28 | nlet rec append (A: Type[0]) (l: List A) (m: List A) on l ≝ |
---|
29 | match l with |
---|
30 | [ Empty ⇒ m |
---|
31 | | Cons hd tl ⇒ hd :: (append A tl m) |
---|
32 | ]. |
---|
33 | |
---|
34 | notation "hvbox(l break @ r)" |
---|
35 | right associative with precedence 47 |
---|
36 | for @{ 'append $l $r }. |
---|
37 | |
---|
38 | interpretation "List append" 'append = (append ?). |
---|
39 | |
---|
40 | nlet rec fold_right (A: Type[0]) (B: Type[0]) |
---|
41 | (f: A → B → B) (x: B) (l: List A) on l ≝ |
---|
42 | match l with |
---|
43 | [ Empty ⇒ x |
---|
44 | | Cons hd tl ⇒ f hd (fold_right A B f x tl) |
---|
45 | ]. |
---|
46 | |
---|
47 | nlet rec fold_left (A: Type[0]) (B: Type[0]) |
---|
48 | (f: A → B → A) (x: A) (l: List B) on l ≝ |
---|
49 | match l with |
---|
50 | [ Empty ⇒ x |
---|
51 | | Cons hd tl ⇒ f (fold_left A B f x tl) hd |
---|
52 | ]. |
---|
53 | |
---|
54 | nlet rec map (A: Type[0]) (B: Type[0]) |
---|
55 | (f: A → B) (l: List A) on l ≝ |
---|
56 | match l with |
---|
57 | [ Empty ⇒ Empty B |
---|
58 | | Cons hd tl ⇒ f hd :: map A B f tl |
---|
59 | ]. |
---|
60 | |
---|
61 | nlet rec null (A: Type[0]) (l: List A) on l ≝ |
---|
62 | match l with |
---|
63 | [ Empty ⇒ True |
---|
64 | | Cons hd tl ⇒ False |
---|
65 | ]. |
---|
66 | |
---|
67 | nlet rec reverse (A: Type[0]) (l: List A) on l ≝ |
---|
68 | match l with |
---|
69 | [ Empty ⇒ Empty A |
---|
70 | | Cons hd tl ⇒ reverse A tl @ (hd :: Empty A) |
---|
71 | ]. |
---|
72 | |
---|
73 | ndefinition head ≝ |
---|
74 | λA: Type[0]. |
---|
75 | λl: List A. |
---|
76 | match l with |
---|
77 | [ Empty ⇒ Nothing A |
---|
78 | | Cons hd tl ⇒ Just A hd |
---|
79 | ]. |
---|
80 | |
---|
81 | ndefinition tail ≝ |
---|
82 | λA: Type[0]. |
---|
83 | λl: List A. |
---|
84 | match l with |
---|
85 | [ Empty ⇒ Nothing (List A) |
---|
86 | | Cons hd tl ⇒ Just (List A) tl |
---|
87 | ]. |
---|
88 | |
---|
89 | nlet rec replicate (A: Type[0]) (n: Nat) (a: A) on n ≝ |
---|
90 | match n with |
---|
91 | [ Z ⇒ Empty A |
---|
92 | | S o ⇒ a :: replicate A o a |
---|
93 | ]. |
---|
94 | |
---|
95 | nlemma append_empty: |
---|
96 | ∀A: Type[0]. |
---|
97 | ∀l: List A. |
---|
98 | l @ (Empty A) = l. |
---|
99 | #A l. |
---|
100 | nelim l. |
---|
101 | nnormalize. |
---|
102 | @. |
---|
103 | #H L H2. |
---|
104 | nnormalize. |
---|
105 | nrewrite > H2. |
---|
106 | @. |
---|
107 | nqed. |
---|
108 | |
---|
109 | nlemma append_associative: |
---|
110 | ∀A: Type[0]. |
---|
111 | ∀l,m,n: List A. |
---|
112 | l @ (m @ n) = (l @ m) @ n. |
---|
113 | #A l m n. |
---|
114 | nelim l. |
---|
115 | nnormalize. |
---|
116 | @. |
---|
117 | #H L H2. |
---|
118 | nnormalize. |
---|
119 | nrewrite > H2. |
---|
120 | @. |
---|
121 | nqed. |
---|
122 | |
---|
123 | nlemma reverse_append: |
---|
124 | ∀A: Type[0]. |
---|
125 | ∀l, m: List A. |
---|
126 | reverse A (l @ m) = reverse A m @ reverse A l. |
---|
127 | #A l m. |
---|
128 | nelim l. |
---|
129 | nnormalize. |
---|
130 | napplyS append_empty. |
---|
131 | #H L A. |
---|
132 | nnormalize. |
---|
133 | nrewrite > A. |
---|
134 | napplyS append_associative. |
---|
135 | nqed. |
---|
136 | |
---|
137 | nlemma length_append: |
---|
138 | ∀A: Type[0]. |
---|
139 | ∀l, m: List A. |
---|
140 | length A (l @ m) = length A l + length A m. |
---|
141 | #A l m. |
---|
142 | nelim l. |
---|
143 | nnormalize. |
---|
144 | @. |
---|
145 | #H L H2. |
---|
146 | nnormalize. |
---|
147 | nrewrite > H2. |
---|
148 | @. |
---|
149 | nqed. |
---|
150 | |
---|
151 | (* |
---|
152 | nlemma length_reverse: |
---|
153 | ∀A: Type[0]. |
---|
154 | ∀l: List A. |
---|
155 | length A (reverse A l) = length A l. |
---|
156 | #A l. |
---|
157 | nelim l. |
---|
158 | nnormalize. |
---|
159 | @. |
---|
160 | #H L H2. |
---|
161 | nnormalize. |
---|
162 | napplyS length_append. |
---|
163 | |
---|
164 | nlemma reverse_reverse: |
---|
165 | ∀A: Type[0]. |
---|
166 | ∀l: List A. |
---|
167 | reverse A (reverse A l) = l. |
---|
168 | #A l. |
---|
169 | nelim l. |
---|
170 | nnormalize. |
---|
171 | @. |
---|
172 | #H L H2. |
---|
173 | nnormalize. |
---|
174 | *) |
---|