real.why 5.15 KB
Newer Older
1
2
3


theory Real 
MARCHE Claude's avatar
MARCHE Claude committed
4

MARCHE Claude's avatar
MARCHE Claude committed
5
  logic (< )(real, real)
6
  logic (<=)(real, real)
MARCHE Claude's avatar
MARCHE Claude committed
7
  logic (> )(real, real)
8
9
  logic (>=)(real, real)
  
MARCHE Claude's avatar
MARCHE Claude committed
10
  (* TODO : they are total order relations *)
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27

  logic zero : real = 0.0
  logic one  : real = 1.0
  
  clone export algebra.Field with 
     type t = real, logic zero = zero, logic one = one
end 

theory Abs

  use import Real

  logic abs(real) : real

  axiom Pos: forall x:real. x >= 0.0 -> abs(x) = x
  axiom Neg: forall x:real. x <= 0.0 -> abs(x) = -x

MARCHE Claude's avatar
MARCHE Claude committed
28
29
  lemma Abs_le: forall x,y:real. abs(x) <= y <-> -y <= x and x <= y

MARCHE Claude's avatar
MARCHE Claude committed
30
31
  lemma Abs_pos: forall x:real. abs(x) >= 0.0

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
end

theory MinMax

  use import Real

  logic min(real,real) : real
  logic max(real,real) : real

  axiom Max_is_ge   : forall x,y:real. max(x,y) >= x and max(x,y) >= y
  axiom Max_is_some : forall x,y:real. max(x,y) = x or max(x,y) = y
  axiom Min_is_le   : forall x,y:real. min(x,y) <= x and min(x,y) <= y
  axiom Min_is_some : forall x,y:real. min(x,y) = x or min(x,y) = y

end

theory FromInt

  use int.Int
  use import Real

  logic from_int(int) : real

  axiom Zero: from_int(0) = 0.0
  axiom One: from_int(1) = 1.0

  axiom Add: 
    forall x,y:int. from_int(Int.(+)(x,y)) = from_int(x) + from_int(y)
  axiom Sub: 
    forall x,y:int. from_int(Int.(-)(x,y)) = from_int(x) - from_int(y)
  axiom Mul: 
    forall x,y:int. from_int(Int.(*)(x,y)) = from_int(x) * from_int(y)
  axiom Neg: 
    forall x,y:int. from_int(Int.(-_)(x)) = - from_int(x)

MARCHE Claude's avatar
MARCHE Claude committed
67
68
  lemma Test: from_int(2) = 2.0

69
70
71
72
73
74
75
76
end

theory Truncate

  (* TODO: truncate, floor, ceil *)

end

MARCHE Claude's avatar
MARCHE Claude committed
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
theory Square

  use import Real

  logic sqr(x:real):real = x * x

  logic sqrt(real):real

  axiom Sqrt_positive: 
    forall x:real. x >= 0.0 -> sqrt(x) >= 0.0

  axiom Sqrt_square:  
    forall x:real. x >= 0.0 -> sqr(sqrt(x)) = x

  axiom Square_sqrt: 
    forall x:real. x >= 0.0 -> sqrt(x*x) = x
93
94
95
96
97

end

theory ExpLog

MARCHE Claude's avatar
MARCHE Claude committed
98
99
100
101
102
103
  use import Real

  logic exp(real) : real
  axiom Exp_zero : exp(0.0) = 1.0
  axiom Exp_sum : forall x,y:real. exp(x+y) = exp(x)*exp(y)

MARCHE Claude's avatar
MARCHE Claude committed
104
105
  logic e : real = exp(1.0)

MARCHE Claude's avatar
MARCHE Claude committed
106
107
108
109
110
111
112
  logic log(real) : real
  axiom Log_one : log(1.0) = 0.0
  axiom Log_mul : 
    forall x,y:real. x > 0.0 and y > 0.0 -> log(x*y) = log(x)+log(y)

  axiom Log_exp: forall x:real. log(exp(x)) = x

MARCHE Claude's avatar
MARCHE Claude committed
113
114
  lemma Log_e : log(e) = 1.0

MARCHE Claude's avatar
MARCHE Claude committed
115
116
  axiom Exp_log: forall x:real. x > 0.0 -> exp(log(x)) = x

MARCHE Claude's avatar
MARCHE Claude committed
117
  logic log2(x:real) : real = log(x)/log(2.0)
MARCHE Claude's avatar
MARCHE Claude committed
118
  logic log10(x:real) : real = log(x)/log(10.0)
119
120
121
122
123

end

theory Power 

MARCHE Claude's avatar
MARCHE Claude committed
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
  use import Real
  use import Square
  use import ExpLog

  logic pow(real,real) : real

  axiom Pow_zero_y:
    forall y:real. y <> 0.0 -> pow(0.0,y) = 0.0

  axiom Pow_x_zero:
    forall x:real. x <> 0.0 -> pow(x,0.0) = 1.0

  axiom Pow_x_one:
    forall x:real. pow(x,1.0) = x

  axiom Pow_one_y:
    forall y:real. pow(1.0,y) = 1.0

  axiom Pow_x_two:
    forall x:real. pow(x,2.0) = sqr(x)

  axiom Pow_half:
MARCHE Claude's avatar
MARCHE Claude committed
146
    forall x:real. x >= 0.0 -> pow(x,0.5) = sqrt(x)  
MARCHE Claude's avatar
MARCHE Claude committed
147
148

  axiom Pow_exp_log:
Andrei Paskevich's avatar
Andrei Paskevich committed
149
    forall x,y:real. x > 0.0 -> pow(x,y) = exp(y*log(x))
MARCHE Claude's avatar
MARCHE Claude committed
150

151
152
end

MARCHE Claude's avatar
MARCHE Claude committed
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
(** Trigonometric functions 
    [http://en.wikipedia.org/wiki/Trigonometric_function]
*)
theory Trigonometric

  use import Real
  use import Square
  use import Abs

  logic cos(real) : real
  logic sin(real) : real

  axiom Pythagorean_identity:
    forall x:real. sqr(cos(x))+sqr(sin(x)) = 1.0

  lemma Cos_le_one: forall x:real. abs(cos(x)) <= 1.0
  lemma Sin_le_one: forall x:real. abs(sin(x)) <= 1.0

  axiom Cos_0: cos(0.0) = 1.0
  axiom Sin_0: sin(0.0) = 0.0

  logic pi : real

  axiom Cos_pi: cos(pi) = -1.0
  axiom Sin_pi: sin(pi) = 0.0

  axiom Cos_pi2: cos(0.5*pi) = 0.0
  axiom Sin_pi2: sin(0.5*pi) = 1.0
181

MARCHE Claude's avatar
MARCHE Claude committed
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
  axiom Cos_plus_pi: forall x:real. cos(x+pi) = -cos(x)
  axiom Sin_plus_pi: forall x:real. sin(x+pi) = -sin(x)

  axiom Cos_plus_pi2: forall x:real. cos(x+0.5*pi) = -sin(x)
  axiom Sin_plus_pi2: forall x:real. sin(x+0.5*pi) = cos(x)

  axiom Cos_neg:
    forall x:real. cos(-x) = cos(x)
  axiom Sin_neg:
    forall x:real. sin(-x) = -sin(x)
 
  axiom Cos_sum:
    forall x,y:real. cos(x+y) = cos(x)*cos(y)-sin(x)*sin(y)
  axiom Sin_sum:
    forall x,y:real. sin(x+y) = sin(x)*cos(y)+cos(x)*sin(y)

  logic tan(x:real) : real = sin(x) / cos(x)
  logic atan(real) : real
  axiom Tan_atan:
    forall x:real. tan(atan(x)) = x
202
203
204

end

MARCHE Claude's avatar
MARCHE Claude committed
205
206
207
(** hyperbolic functions 
    [http://en.wikipedia.org/wiki/Hyperbolic_function]
*)
208
209
theory Hyperbolic

MARCHE Claude's avatar
MARCHE Claude committed
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
  use import Real
  use import Square
  use import ExpLog

  logic sinh(x:real) : real = 0.5*(exp(x) - exp(-x))
  logic cosh(x:real) : real = 0.5*(exp(x) + exp(-x))
  logic tanh(x:real) : real = sinh(x) / cosh(x)

  logic arsinh(x:real) : real = log(x+sqrt(sqr(x)+1.0))
  logic arcosh(x:real) : real
  axiom Arcosh_def:
    forall x:real. x >= 1.0 -> arcosh(x) = log(x+sqrt(sqr(x)-1.0))
  logic artanh(x:real) : real
  axiom Artanh_def:
    forall x:real. -1.0 < x and x < 1.0 -> artanh(x) = 0.5*log((1.0+x)/(1.0-x))
225
226
227

end

MARCHE Claude's avatar
MARCHE Claude committed
228
229
230
(** polar coordinates: atan2, hypot 
    [http://en.wikipedia.org/wiki/Atan2]
*)
231
232
theory Polar

MARCHE Claude's avatar
MARCHE Claude committed
233
234
  use import Real
  use import Square
MARCHE Claude's avatar
MARCHE Claude committed
235
  use import Trigonometric
MARCHE Claude's avatar
MARCHE Claude committed
236
237
238

  logic hypot(x:real,y:real) : real = sqrt(sqr(x)+sqr(y))
  logic atan2(real,real) : real
239
 
MARCHE Claude's avatar
MARCHE Claude committed
240
241
242
243
244
  axiom X_from_polar:
    forall x,y:real. x = hypot(x,y)*cos(atan2(y,x))

  axiom Y_from_polar:
    forall x,y:real. y = hypot(x,y)*sin(atan2(y,x))
245
246
end