1(* Author: Tobias Nipkow *)
2
3theory Abs_Int1_const
4imports Abs_Int1
5begin
6
7subsection "Constant Propagation"
8
9datatype const = Const val | Any
10
11fun \<gamma>_const where
12"\<gamma>_const (Const i) = {i}" |
13"\<gamma>_const (Any) = UNIV"
14
15fun plus_const where
16"plus_const (Const i) (Const j) = Const(i+j)" |
17"plus_const _ _ = Any"
18
19lemma plus_const_cases: "plus_const a1 a2 =
20  (case (a1,a2) of (Const i, Const j) \<Rightarrow> Const(i+j) | _ \<Rightarrow> Any)"
21by(auto split: prod.split const.split)
22
23instantiation const :: semilattice_sup_top
24begin
25
26fun less_eq_const where "x \<le> y = (y = Any | x=y)"
27
28definition "x < (y::const) = (x \<le> y & \<not> y \<le> x)"
29
30fun sup_const where "x \<squnion> y = (if x=y then x else Any)"
31
32definition "\<top> = Any"
33
34instance
35proof (standard, goal_cases)
36  case 1 thus ?case by (rule less_const_def)
37next
38  case (2 x) show ?case by (cases x) simp_all
39next
40  case (3 x y z) thus ?case by(cases z, cases y, cases x, simp_all)
41next
42  case (4 x y) thus ?case by(cases x, cases y, simp_all, cases y, simp_all)
43next
44  case (6 x y) thus ?case by(cases x, cases y, simp_all)
45next
46  case (5 x y) thus ?case by(cases y, cases x, simp_all)
47next
48  case (7 x y z) thus ?case by(cases z, cases y, cases x, simp_all)
49next
50  case 8 thus ?case by(simp add: top_const_def)
51qed
52
53end
54
55
56global_interpretation Val_semilattice
57where \<gamma> = \<gamma>_const and num' = Const and plus' = plus_const
58proof (standard, goal_cases)
59  case (1 a b) thus ?case
60    by(cases a, cases b, simp, simp, cases b, simp, simp)
61next
62  case 2 show ?case by(simp add: top_const_def)
63next
64  case 3 show ?case by simp
65next
66  case 4 thus ?case by(auto simp: plus_const_cases split: const.split)
67qed
68
69global_interpretation Abs_Int
70where \<gamma> = \<gamma>_const and num' = Const and plus' = plus_const
71defines AI_const = AI and step_const = step' and aval'_const = aval'
72..
73
74
75subsubsection "Tests"
76
77definition "steps c i = (step_const \<top> ^^ i) (bot c)"
78
79value "show_acom (steps test1_const 0)"
80value "show_acom (steps test1_const 1)"
81value "show_acom (steps test1_const 2)"
82value "show_acom (steps test1_const 3)"
83value "show_acom (the(AI_const test1_const))"
84
85value "show_acom (the(AI_const test2_const))"
86value "show_acom (the(AI_const test3_const))"
87
88value "show_acom (steps test4_const 0)"
89value "show_acom (steps test4_const 1)"
90value "show_acom (steps test4_const 2)"
91value "show_acom (steps test4_const 3)"
92value "show_acom (steps test4_const 4)"
93value "show_acom (the(AI_const test4_const))"
94
95value "show_acom (steps test5_const 0)"
96value "show_acom (steps test5_const 1)"
97value "show_acom (steps test5_const 2)"
98value "show_acom (steps test5_const 3)"
99value "show_acom (steps test5_const 4)"
100value "show_acom (steps test5_const 5)"
101value "show_acom (steps test5_const 6)"
102value "show_acom (the(AI_const test5_const))"
103
104value "show_acom (steps test6_const 0)"
105value "show_acom (steps test6_const 1)"
106value "show_acom (steps test6_const 2)"
107value "show_acom (steps test6_const 3)"
108value "show_acom (steps test6_const 4)"
109value "show_acom (steps test6_const 5)"
110value "show_acom (steps test6_const 6)"
111value "show_acom (steps test6_const 7)"
112value "show_acom (steps test6_const 8)"
113value "show_acom (steps test6_const 9)"
114value "show_acom (steps test6_const 10)"
115value "show_acom (steps test6_const 11)"
116value "show_acom (steps test6_const 12)"
117value "show_acom (steps test6_const 13)"
118value "show_acom (the(AI_const test6_const))"
119
120
121text\<open>Monotonicity:\<close>
122
123global_interpretation Abs_Int_mono
124where \<gamma> = \<gamma>_const and num' = Const and plus' = plus_const
125proof (standard, goal_cases)
126  case 1 thus ?case by(auto simp: plus_const_cases split: const.split)
127qed
128
129text\<open>Termination:\<close>
130
131definition m_const :: "const \<Rightarrow> nat" where
132"m_const x = (if x = Any then 0 else 1)"
133
134global_interpretation Abs_Int_measure
135where \<gamma> = \<gamma>_const and num' = Const and plus' = plus_const
136and m = m_const and h = "1"
137proof (standard, goal_cases)
138  case 1 thus ?case by(auto simp: m_const_def split: const.splits)
139next
140  case 2 thus ?case by(auto simp: m_const_def less_const_def split: const.splits)
141qed
142
143thm AI_Some_measure
144
145end
146