9 imports Product_ord |
9 imports Product_ord |
10 begin |
10 begin |
11 |
11 |
12 text {* Conversions between nibbles and integers in [0..15]. *} |
12 text {* Conversions between nibbles and integers in [0..15]. *} |
13 |
13 |
14 consts |
14 fun |
15 nibble_to_int:: "nibble \<Rightarrow> int" |
15 nibble_to_int:: "nibble \<Rightarrow> int" where |
16 primrec |
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17 "nibble_to_int Nibble0 = 0" |
16 "nibble_to_int Nibble0 = 0" |
18 "nibble_to_int Nibble1 = 1" |
17 | "nibble_to_int Nibble1 = 1" |
19 "nibble_to_int Nibble2 = 2" |
18 | "nibble_to_int Nibble2 = 2" |
20 "nibble_to_int Nibble3 = 3" |
19 | "nibble_to_int Nibble3 = 3" |
21 "nibble_to_int Nibble4 = 4" |
20 | "nibble_to_int Nibble4 = 4" |
22 "nibble_to_int Nibble5 = 5" |
21 | "nibble_to_int Nibble5 = 5" |
23 "nibble_to_int Nibble6 = 6" |
22 | "nibble_to_int Nibble6 = 6" |
24 "nibble_to_int Nibble7 = 7" |
23 | "nibble_to_int Nibble7 = 7" |
25 "nibble_to_int Nibble8 = 8" |
24 | "nibble_to_int Nibble8 = 8" |
26 "nibble_to_int Nibble9 = 9" |
25 | "nibble_to_int Nibble9 = 9" |
27 "nibble_to_int NibbleA = 10" |
26 | "nibble_to_int NibbleA = 10" |
28 "nibble_to_int NibbleB = 11" |
27 | "nibble_to_int NibbleB = 11" |
29 "nibble_to_int NibbleC = 12" |
28 | "nibble_to_int NibbleC = 12" |
30 "nibble_to_int NibbleD = 13" |
29 | "nibble_to_int NibbleD = 13" |
31 "nibble_to_int NibbleE = 14" |
30 | "nibble_to_int NibbleE = 14" |
32 "nibble_to_int NibbleF = 15" |
31 | "nibble_to_int NibbleF = 15" |
33 |
32 |
34 definition |
33 definition |
35 int_to_nibble :: "int \<Rightarrow> nibble" where |
34 int_to_nibble :: "int \<Rightarrow> nibble" where |
36 "int_to_nibble x = (let y = x mod 16 in |
35 "int_to_nibble x = (let y = x mod 16 in |
37 if y = 0 then Nibble0 else |
36 if y = 0 then Nibble0 else |
49 if y = 12 then NibbleC else |
48 if y = 12 then NibbleC else |
50 if y = 13 then NibbleD else |
49 if y = 13 then NibbleD else |
51 if y = 14 then NibbleE else |
50 if y = 14 then NibbleE else |
52 NibbleF)" |
51 NibbleF)" |
53 |
52 |
54 lemma int_to_nibble_nibble_to_int: "int_to_nibble(nibble_to_int x) = x" |
53 lemma int_to_nibble_nibble_to_int: "int_to_nibble (nibble_to_int x) = x" |
55 by (cases x) (auto simp: int_to_nibble_def Let_def) |
54 by (cases x) (auto simp: int_to_nibble_def Let_def) |
56 |
55 |
57 lemma inj_nibble_to_int: "inj nibble_to_int" |
56 lemma inj_nibble_to_int: "inj nibble_to_int" |
58 by (rule inj_on_inverseI) (rule int_to_nibble_nibble_to_int) |
57 by (rule inj_on_inverseI) (rule int_to_nibble_nibble_to_int) |
59 |
58 |
65 lemma nibble_to_int_less_16: "nibble_to_int x < 16" |
64 lemma nibble_to_int_less_16: "nibble_to_int x < 16" |
66 by (cases x) auto |
65 by (cases x) auto |
67 |
66 |
68 text {* Conversion between chars and int pairs. *} |
67 text {* Conversion between chars and int pairs. *} |
69 |
68 |
70 consts |
69 fun |
71 char_to_int_pair :: "char \<Rightarrow> int \<times> int" |
70 char_to_int_pair :: "char \<Rightarrow> int \<times> int" where |
72 primrec |
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73 "char_to_int_pair (Char a b) = (nibble_to_int a, nibble_to_int b)" |
71 "char_to_int_pair (Char a b) = (nibble_to_int a, nibble_to_int b)" |
74 |
72 |
75 lemma inj_char_to_int_pair: "inj char_to_int_pair" |
73 lemma inj_char_to_int_pair: "inj char_to_int_pair" |
76 apply (rule inj_onI) |
74 apply (rule inj_onI) |
77 apply (case_tac x, case_tac y) |
75 apply (case_tac x, case_tac y) |
78 apply (auto simp: nibble_to_int_eq) |
76 apply (auto simp: nibble_to_int_eq) |
79 done |
77 done |
80 |
78 |
81 lemmas char_to_int_pair_eq = inj_char_to_int_pair [THEN inj_eq] |
79 lemmas char_to_int_pair_eq = inj_char_to_int_pair [THEN inj_eq] |
82 |
80 |
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81 |
83 text {* Instantiation of order classes *} |
82 text {* Instantiation of order classes *} |
84 |
83 |
85 instance char :: ord .. |
84 instance char :: ord |
86 |
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87 defs (overloaded) |
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88 char_le_def: "c \<le> d \<equiv> (char_to_int_pair c \<le> char_to_int_pair d)" |
85 char_le_def: "c \<le> d \<equiv> (char_to_int_pair c \<le> char_to_int_pair d)" |
89 char_less_def: "c < d \<equiv> (char_to_int_pair c < char_to_int_pair d)" |
86 char_less_def: "c < d \<equiv> (char_to_int_pair c < char_to_int_pair d)" .. |
90 |
87 |
91 lemmas char_ord_defs = char_less_def char_le_def |
88 lemmas char_ord_defs = char_less_def char_le_def |
92 |
89 |
93 instance char :: order |
90 instance char :: order |
94 by default (auto simp: char_ord_defs char_to_int_pair_eq order_less_le) |
91 by default (auto simp: char_ord_defs char_to_int_pair_eq order_less_le) |
95 |
92 |
96 instance char :: linorder |
93 instance char :: linorder |
97 by default (auto simp: char_le_def) |
94 by default (auto simp: char_le_def) |
98 |
95 |
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96 instance char :: distrib_lattice |
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97 "inf \<equiv> min" |
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98 "sup \<equiv> max" |
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99 by intro_classes |
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100 (auto simp add: inf_char_def sup_char_def min_max.sup_inf_distrib1) |
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101 |
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102 |
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103 text {* code generator setup *} |
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104 |
99 code_const char_to_int_pair |
105 code_const char_to_int_pair |
100 (SML "raise/ Fail/ \"char'_to'_int'_pair\"") |
106 (SML "raise/ Fail/ \"char'_to'_int'_pair\"") |
101 (OCaml "failwith \"char'_to'_int'_pair\"") |
107 (OCaml "failwith \"char'_to'_int'_pair\"") |
102 (Haskell "error/ \"char'_to'_int'_pair\"") |
108 (Haskell "error/ \"char'_to'_int'_pair\"") |
103 |
109 |