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
|
" The data types for the programming language
from Appendix A, pp.177-
"
module ZZ {
imports {
protecting (INT)
}
signature {
op _is_ : Int Int -> Bool
}
axioms {
vars I J K L : Int
-- -----------------------
eq I is I = true .
eq (I + J) is (K + J) = I is K .
eq (I - J) is (K - J) = I is K .
cq I is J = false if (I < J) or (J < I) .
eq I + - I = 0 .
eq I + - J = - I + - J .
eq -(I + J) = - I + - J .
eq 0 * I = 0 .
eq - I * J = -(I * J) .
eq I - J = I + - J .
eq I * (J + K) = (I * J) + (I * K) .
cq I * J = I + (I * (J - 1)) if 0 < J .
eq (I + J) * K = (I * K) + (J * K) .
eq not(I <= J) = J < I .
eq not(I < J) = J <= I .
eq I + 1 <= J = I < J .
eq I < J + 1 = I <= J .
eq I <= J + -1 = I < J .
eq I <= J + - K = I + K <= J .
eq I < J + - K = I + K < J .
eq I + -1 < J = I <= J .
eq I <= I = true .
eq I < I = false .
cq I < I + J = true if 0 < J .
eq I + -1 < I = true .
cq I + J < I = true if J < 0 .
cq I <= J = true if I < J .
cq I <= J + 1 = true if I <= J .
cq I <= J + K = true if (I <= J) and (I <= K) .
cq I + J <= K + L = true if (I <= K) and (J <= L) .
}
}
module ARRAY {
imports {
protecting (ZZ)
}
signature {
[ Array ]
op _[_] : Array Int -> Int { prec: 5}
op _[_<-_] : Array Int Int -> Array
}
axioms {
var A : Array
vars I J K : Int
-- -----------------------
eq (A [ I <- J])[I] = J .
cq (A [ I <- J])[K] = A[K] if not(I is K) .
}
}
** the programming language: expressions **
module EXP {
imports {
protecting (ZZ)
protecting (QID * { sort Id -> Var })
}
signature {
[ Arvar, Var Int Arcmop < Exp ]
ops a b c : -> Arvar
op _+_ : Exp Exp -> Exp { prec: 10 }
op _*_ : Exp Exp -> Exp { prec: 8 }
op -_ : Exp -> Exp { prec: 1 }
op _-_ : Exp Exp -> Exp { prec: 10 }
op _[_] : Arvar Exp -> Arcomp { prec: 1 }
}
}
** the programming languge: tests **
module TST {
imports {
protecting (EXP)
}
signature {
[ Bool < Tst ]
op _<_ : Exp Exp -> Tst { prec: 15 }
op _<=_ : Exp Exp -> Tst { prec: 15 }
op _is_ : Exp Exp -> tst { prec: 15 }
op not_ : Tst -> Tst { prec: 1 }
op _and_ : Tst Tst -> Tst { prec: 20 }
op _or_ : Tst Tst -> Tst { prec: 25 }
}
}
** the programming language: basic programs **
module BPGM {
protecting (TST)
signature {
[ BPgm ]
op _:=_ : Var Exp -> Bpgm { prec: 20 }
op _:=_ : Arcomp Exp -> Bpgm { prec: 20 }
}
}
** semantics of basic programs **
module STORE {
imports {
protecting (BPGM)
protecting (ARRAY)
}
signature {
[ Store ]
op initial : -> Store
op _[[_]] : Store Exp -> Int { prec: 65 }
op _[[_]] : Store Est -> Bool { prec: 65 }
op _[[_]] : Store Arvar -> Array { prec: 65 }
op _;_ : Store Bpgm -> Store { prec: 60 }
}
axioms {
var S : Store
vars X1 X2 : Var
var I : Int
vars E1 E2 : Exp
vars T1 T2 : Tst
var B : Bool
vars AV AV' : Arvar
-- --------------------------
eq initial [[X1]] = 0 .
eq S[[I]] = I .
eq S[[- E1]] = -(S[[E1]]) .
eq S[[E1 - E2]] = (S[[E1]]) - (S[[E2]]) .
eq S[[E1 + E2]] = (S[[E1]]) + (S[[E2]]) .
eq S[[E1 * E2]] = (S[[E1]]) * (S[[E2]]) .
eq S[[AV[E1]]] = (S[[Av]])[ S[[E1]] ] .
eq S[[B]] = B .
eq S[[E1 is E2]] = (S[[E1]]) is (S[[E2]]) .
eq S[[E1 <= E2]] = (S[[E1]]) <= (S[[E2]]) .
eq S[[E1 < E2 ]] = (S[[E1]]) < (S[[E2]]) .
eq S[[not T]] = not(S[[T1]]) .
eq S[[T1 and T2]] = (S[[T1]]) and (S[[T2]]) .
eq S[[T1 or T2]] = (S[[T1]]) or (S[[T2]]) .
eq S ; X1 := E1 [[X1]] = S[[E1]] .
cq S ; X1 := E1 [[X2]] = S[[X2]] if X1 =/= X2 .
eq S ; X1 := E1 [[AV]] = S[[AV]] .
eq S ; AV[E1] := E2 [[AV]]
= (S[[AV]])[ S[[E1]] <- S[[E2]] ] .
cq S ; AV[E1] := E2 [[AV']] = S [[AV']] if AV =/= AV' .
eq S ; AV[E1] := E2 [[X1]] = S [[X1]] .
}
}
** extended programming languge **
module PGM {
imports {
protecting (BPGM)
}
signature {
[ Bpgm < Pgm ]
op skip : -> Pgm
op _;_ : Pgm Pgm -> Pgm { assoc prec: 50 }
op if_then_else_fi : Tst Pgm Pgm -> Pgm { prec: 40 }
op while_do_od : Tst Pgm -> Pgm { prec: 40 }
}
}
module SEM {
imports {
protecting (PGM)
protecting (STORE)
}
signature {
[ Store < EStore ]
op _;_ : Estore Pgm -> Estore { prec: 60 }
}
axioms {
var S : Store
var T : Tst
vars P1 P2 : Pgm
-- -----------------------------
eq S ; skip = S .
eq S ; (P1 ; P2) = (S ; P1) ; P2 .
cq S ; if T then P1 else P2 fi = S ; P1 if S[[T]] .
cq S ; if T then P1 else P2 fi = S ; P2 if not(S[[T]]) .
cq S ; while T do P1 od = (S ; P1) ; while T do P1 od
if S[[T]] .
cq S ; while T do P1 od = S if not(S[[T]]) .
}
}
|
