1(*  Title:      HOL/Word/Word_Examples.thy
2    Authors:    Gerwin Klein and Thomas Sewell, NICTA
3
4Examples demonstrating and testing various word operations.
5*)
6
7section "Examples of word operations"
8
9theory Word_Examples
10  imports Word_Bitwise
11begin
12
13type_synonym word32 = "32 word"
14type_synonym word8 = "8 word"
15type_synonym byte = word8
16
17text "modulus"
18
19lemma "(27 :: 4 word) = -5" by simp
20
21lemma "(27 :: 4 word) = 11" by simp
22
23lemma "27 \<noteq> (11 :: 6 word)" by simp
24
25text "signed"
26
27lemma "(127 :: 6 word) = -1" by simp
28
29text "number ring simps"
30
31lemma
32  "27 + 11 = (38::'a::len word)"
33  "27 + 11 = (6::5 word)"
34  "7 * 3 = (21::'a::len word)"
35  "11 - 27 = (-16::'a::len word)"
36  "- (- 11) = (11::'a::len word)"
37  "-40 + 1 = (-39::'a::len word)"
38  by simp_all
39
40lemma "word_pred 2 = 1" by simp
41
42lemma "word_succ (- 3) = -2" by simp
43
44lemma "23 < (27::8 word)" by simp
45lemma "23 \<le> (27::8 word)" by simp
46lemma "\<not> 23 < (27::2 word)" by simp
47lemma "0 < (4::3 word)" by simp
48lemma "1 < (4::3 word)" by simp
49lemma "0 < (1::3 word)" by simp
50
51text "ring operations"
52
53lemma "a + 2 * b + c - b = (b + c) + (a :: 32 word)" by simp
54
55text "casting"
56
57lemma "uint (234567 :: 10 word) = 71" by simp
58lemma "uint (-234567 :: 10 word) = 953" by simp
59lemma "sint (234567 :: 10 word) = 71" by simp
60lemma "sint (-234567 :: 10 word) = -71" by simp
61lemma "uint (1 :: 10 word) = 1" by simp
62
63lemma "unat (-234567 :: 10 word) = 953" by simp
64lemma "unat (1 :: 10 word) = 1" by simp
65
66lemma "ucast (0b1010 :: 4 word) = (0b10 :: 2 word)" by simp
67lemma "ucast (0b1010 :: 4 word) = (0b1010 :: 10 word)" by simp
68lemma "scast (0b1010 :: 4 word) = (0b111010 :: 6 word)" by simp
69lemma "ucast (1 :: 4 word) = (1 :: 2 word)" by simp
70
71text "reducing goals to nat or int and arith:"
72lemma "i < x \<Longrightarrow> i < i + 1" for i x :: "'a::len word"
73  by unat_arith
74lemma "i < x \<Longrightarrow> i < i + 1" for i x :: "'a::len word"
75  by unat_arith
76
77text "bool lists"
78
79lemma "of_bl [True, False, True, True] = (0b1011::'a::len word)" by simp
80
81lemma "to_bl (0b110::4 word) = [False, True, True, False]" by simp
82
83text "this is not exactly fast, but bearable"
84lemma "of_bl (replicate 32 True) = (0xFFFFFFFF::32 word)" by simp
85
86text "this works only for replicate n True"
87lemma "of_bl (replicate 32 True) = (0xFFFFFFFF::32 word)"
88  by (unfold mask_bl [symmetric]) (simp add: mask_def)
89
90
91text "bit operations"
92
93lemma "0b110 AND 0b101 = (0b100 :: 32 word)" by simp
94lemma "0b110 OR 0b011 = (0b111 :: 8 word)" by simp
95lemma "0xF0 XOR 0xFF = (0x0F :: byte)" by simp
96lemma "NOT (0xF0 :: 16 word) = 0xFF0F" by simp
97lemma "0 AND 5 = (0 :: byte)" by simp
98lemma "1 AND 1 = (1 :: byte)" by simp
99lemma "1 AND 0 = (0 :: byte)" by simp
100lemma "1 AND 5 = (1 :: byte)" by simp
101lemma "1 OR 6 = (7 :: byte)" by simp
102lemma "1 OR 1 = (1 :: byte)" by simp
103lemma "1 XOR 7 = (6 :: byte)" by simp
104lemma "1 XOR 1 = (0 :: byte)" by simp
105lemma "NOT 1 = (254 :: byte)" by simp
106lemma "NOT 0 = (255 :: byte)" apply simp oops
107(* FIXME: "NOT 0" rewrites to "max_word" instead of "-1" *)
108
109lemma "(-1 :: 32 word) = 0xFFFFFFFF" by simp
110
111lemma "(0b0010 :: 4 word) !! 1" by simp
112lemma "\<not> (0b0010 :: 4 word) !! 0" by simp
113lemma "\<not> (0b1000 :: 3 word) !! 4" by simp
114lemma "\<not> (1 :: 3 word) !! 2" by simp
115
116lemma "(0b11000 :: 10 word) !! n = (n = 4 \<or> n = 3)"
117  by (auto simp add: bin_nth_Bit0 bin_nth_Bit1)
118
119lemma "set_bit 55 7 True = (183::'a::len0 word)" by simp
120lemma "set_bit 0b0010 7 True = (0b10000010::'a::len0 word)" by simp
121lemma "set_bit 0b0010 1 False = (0::'a::len0 word)" by simp
122lemma "set_bit 1 3 True = (0b1001::'a::len0 word)" by simp
123lemma "set_bit 1 0 False = (0::'a::len0 word)" by simp
124lemma "set_bit 0 3 True = (0b1000::'a::len0 word)" by simp
125lemma "set_bit 0 3 False = (0::'a::len0 word)" by simp
126
127lemma "lsb (0b0101::'a::len word)" by simp
128lemma "\<not> lsb (0b1000::'a::len word)" by simp
129lemma "lsb (1::'a::len word)" by simp
130lemma "\<not> lsb (0::'a::len word)" by simp
131
132lemma "\<not> msb (0b0101::4 word)" by simp
133lemma   "msb (0b1000::4 word)" by simp
134lemma "\<not> msb (1::4 word)" by simp
135lemma "\<not> msb (0::4 word)" by simp
136
137lemma "word_cat (27::4 word) (27::8 word) = (2843::'a::len word)" by simp
138lemma "word_cat (0b0011::4 word) (0b1111::6word) = (0b0011001111 :: 10 word)"
139  by simp
140
141lemma "0b1011 << 2 = (0b101100::'a::len0 word)" by simp
142lemma "0b1011 >> 2 = (0b10::8 word)" by simp
143lemma "0b1011 >>> 2 = (0b10::8 word)" by simp
144lemma "1 << 2 = (0b100::'a::len0 word)" apply simp? oops
145
146lemma "slice 3 (0b101111::6 word) = (0b101::3 word)" by simp
147lemma "slice 3 (1::6 word) = (0::3 word)" apply simp? oops
148
149lemma "word_rotr 2 0b0110 = (0b1001::4 word)" by simp
150lemma "word_rotl 1 0b1110 = (0b1101::4 word)" by simp
151lemma "word_roti 2 0b1110 = (0b1011::4 word)" by simp
152lemma "word_roti (- 2) 0b0110 = (0b1001::4 word)" by simp
153lemma "word_rotr 2 0 = (0::4 word)" by simp
154lemma "word_rotr 2 1 = (0b0100::4 word)" apply simp? oops
155lemma "word_rotl 2 1 = (0b0100::4 word)" apply simp? oops
156lemma "word_roti (- 2) 1 = (0b0100::4 word)" apply simp? oops
157
158lemma "(x AND 0xff00) OR (x AND 0x00ff) = (x::16 word)"
159proof -
160  have "(x AND 0xff00) OR (x AND 0x00ff) = x AND (0xff00 OR 0x00ff)"
161    by (simp only: word_ao_dist2)
162  also have "0xff00 OR 0x00ff = (-1::16 word)"
163    by simp
164  also have "x AND -1 = x"
165    by simp
166  finally show ?thesis .
167qed
168
169text "alternative proof using bitwise expansion"
170
171lemma "(x AND 0xff00) OR (x AND 0x00ff) = (x::16 word)"
172  by word_bitwise
173
174text "more proofs using bitwise expansion"
175
176lemma "(x AND NOT 3) >> 4 << 2 = ((x >> 2) AND NOT 3)"
177  for x :: "10 word"
178  by word_bitwise
179
180lemma "((x AND -8) >> 3) AND 7 = (x AND 56) >> 3"
181  for x :: "12 word"
182  by word_bitwise
183
184text "some problems require further reasoning after bit expansion"
185
186lemma "x \<le> 42 \<Longrightarrow> x \<le> 89"
187  for x :: "8 word"
188  apply word_bitwise
189  apply blast
190  done
191
192lemma "(x AND 1023) = 0 \<Longrightarrow> x \<le> -1024"
193  for x :: word32
194  apply word_bitwise
195  apply clarsimp
196  done
197
198text "operations like shifts by non-numerals will expose some internal list
199 representations but may still be easy to solve"
200
201lemma shiftr_overflow: "32 \<le> a \<Longrightarrow> b >> a = 0"
202  for b :: word32
203  apply word_bitwise
204  apply simp
205  done
206
207end
208