Theory Word_8
section "Words of Length 8"
theory Word_8
imports
More_Word
Enumeration_Word
Even_More_List
Signed_Words
Word_Lemmas
begin
context
includes bit_operations_syntax
begin
lemma len8: "len_of (x :: 8 itself) = 8" by simp
lemma word8_and_max_simp:
‹x AND 0xFF = x› for x :: ‹8 word›
using word_and_full_mask_simp [of x]
by (simp add: numeral_eq_Suc mask_Suc_exp)
lemma enum_word8_eq:
‹enum = [0 :: 8 word, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36,
37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70,
71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87,
88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103,
104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117,
118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131,
132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145,
146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159,
160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173,
174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187,
188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201,
202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215,
216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229,
230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243,
244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255]› (is ‹?lhs = ?rhs›)
proof -
have ‹map unat ?lhs = [0..<256]›
by (simp add: enum_word_def comp_def take_bit_nat_eq_self map_idem_upt_eq unsigned_of_nat)
also have ‹… = map unat ?rhs›
by (simp add: upt_zero_numeral_unfold)
finally show ?thesis
using unat_inj by (rule map_injective)
qed
lemma set_enum_word8_def:
"(set enum :: 8 word set) = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36,
37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70,
71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87,
88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103,
104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117,
118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131,
132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145,
146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159,
160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173,
174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187,
188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201,
202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215,
216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229,
230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243,
244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255}"
by (simp add: enum_word8_eq)
lemma set_strip_insert: "⟦ x ∈ insert a S; x ≠ a ⟧ ⟹ x ∈ S"
by simp
lemma word8_exhaust:
fixes x :: ‹8 word›
shows "⟦x ≠ 0; x ≠ 1; x ≠ 2; x ≠ 3; x ≠ 4; x ≠ 5; x ≠ 6; x ≠ 7; x ≠ 8; x ≠ 9; x ≠ 10; x ≠ 11; x ≠
12; x ≠ 13; x ≠ 14; x ≠ 15; x ≠ 16; x ≠ 17; x ≠ 18; x ≠ 19; x ≠ 20; x ≠ 21; x ≠ 22; x ≠
23; x ≠ 24; x ≠ 25; x ≠ 26; x ≠ 27; x ≠ 28; x ≠ 29; x ≠ 30; x ≠ 31; x ≠ 32; x ≠ 33; x ≠
34; x ≠ 35; x ≠ 36; x ≠ 37; x ≠ 38; x ≠ 39; x ≠ 40; x ≠ 41; x ≠ 42; x ≠ 43; x ≠ 44; x ≠
45; x ≠ 46; x ≠ 47; x ≠ 48; x ≠ 49; x ≠ 50; x ≠ 51; x ≠ 52; x ≠ 53; x ≠ 54; x ≠ 55; x ≠
56; x ≠ 57; x ≠ 58; x ≠ 59; x ≠ 60; x ≠ 61; x ≠ 62; x ≠ 63; x ≠ 64; x ≠ 65; x ≠ 66; x ≠
67; x ≠ 68; x ≠ 69; x ≠ 70; x ≠ 71; x ≠ 72; x ≠ 73; x ≠ 74; x ≠ 75; x ≠ 76; x ≠ 77; x ≠
78; x ≠ 79; x ≠ 80; x ≠ 81; x ≠ 82; x ≠ 83; x ≠ 84; x ≠ 85; x ≠ 86; x ≠ 87; x ≠ 88; x ≠
89; x ≠ 90; x ≠ 91; x ≠ 92; x ≠ 93; x ≠ 94; x ≠ 95; x ≠ 96; x ≠ 97; x ≠ 98; x ≠ 99; x ≠
100; x ≠ 101; x ≠ 102; x ≠ 103; x ≠ 104; x ≠ 105; x ≠ 106; x ≠ 107; x ≠ 108; x ≠ 109; x ≠
110; x ≠ 111; x ≠ 112; x ≠ 113; x ≠ 114; x ≠ 115; x ≠ 116; x ≠ 117; x ≠ 118; x ≠ 119; x ≠
120; x ≠ 121; x ≠ 122; x ≠ 123; x ≠ 124; x ≠ 125; x ≠ 126; x ≠ 127; x ≠ 128; x ≠ 129; x ≠
130; x ≠ 131; x ≠ 132; x ≠ 133; x ≠ 134; x ≠ 135; x ≠ 136; x ≠ 137; x ≠ 138; x ≠ 139; x ≠
140; x ≠ 141; x ≠ 142; x ≠ 143; x ≠ 144; x ≠ 145; x ≠ 146; x ≠ 147; x ≠ 148; x ≠ 149; x ≠
150; x ≠ 151; x ≠ 152; x ≠ 153; x ≠ 154; x ≠ 155; x ≠ 156; x ≠ 157; x ≠ 158; x ≠ 159; x ≠
160; x ≠ 161; x ≠ 162; x ≠ 163; x ≠ 164; x ≠ 165; x ≠ 166; x ≠ 167; x ≠ 168; x ≠ 169; x ≠
170; x ≠ 171; x ≠ 172; x ≠ 173; x ≠ 174; x ≠ 175; x ≠ 176; x ≠ 177; x ≠ 178; x ≠ 179; x ≠
180; x ≠ 181; x ≠ 182; x ≠ 183; x ≠ 184; x ≠ 185; x ≠ 186; x ≠ 187; x ≠ 188; x ≠ 189; x ≠
190; x ≠ 191; x ≠ 192; x ≠ 193; x ≠ 194; x ≠ 195; x ≠ 196; x ≠ 197; x ≠ 198; x ≠ 199; x ≠
200; x ≠ 201; x ≠ 202; x ≠ 203; x ≠ 204; x ≠ 205; x ≠ 206; x ≠ 207; x ≠ 208; x ≠ 209; x ≠
210; x ≠ 211; x ≠ 212; x ≠ 213; x ≠ 214; x ≠ 215; x ≠ 216; x ≠ 217; x ≠ 218; x ≠ 219; x ≠
220; x ≠ 221; x ≠ 222; x ≠ 223; x ≠ 224; x ≠ 225; x ≠ 226; x ≠ 227; x ≠ 228; x ≠ 229; x ≠
230; x ≠ 231; x ≠ 232; x ≠ 233; x ≠ 234; x ≠ 235; x ≠ 236; x ≠ 237; x ≠ 238; x ≠ 239; x ≠
240; x ≠ 241; x ≠ 242; x ≠ 243; x ≠ 244; x ≠ 245; x ≠ 246; x ≠ 247; x ≠ 248; x ≠ 249; x ≠
250; x ≠ 251; x ≠ 252; x ≠ 253; x ≠ 254; x ≠ 255⟧ ⟹ P"
apply (subgoal_tac "x ∈ set enum", subst (asm) set_enum_word8_def)
apply (drule set_strip_insert, assumption)+
apply (erule emptyE)
apply (subst enum_UNIV, rule UNIV_I)
done
end
end