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
|
open HolKernel boolLib bossLib Parse
open boolTheory arithmeticTheory integerTheory intLib listTheory
open primitivesBaseTacLib
val _ = new_theory "primitivesArith"
(* TODO: we need a better library of lemmas about arithmetic *)
val not_le_eq_gt = store_thm("not_le_eq_gt", “!(x y: int). ~(x <= y) <=> x > y”, cooper_tac)
val not_lt_eq_ge = store_thm("not_lt_eq_ge", “!(x y: int). ~(x < y) <=> x >= y”, cooper_tac)
(* TODO: add gsymed versions of those as rewriting theorems by default (we only want to
manipulate ‘<’ and ‘≤’) *)
val not_ge_eq_lt = store_thm("not_ge_eq_lt", “!(x y: int). ~(x >= y) <=> x < y”, cooper_tac)
val not_gt_eq_le = store_thm("not_gt_eq_le", “!(x y: int). ~(x > y) <=> x <= y”, cooper_tac)
val ge_eq_le = store_thm("ge_eq_le", “!(x y : int). x >= y <=> y <= x”, cooper_tac)
val le_eq_ge = store_thm("le_eq_ge", “!(x y : int). x <= y <=> y >= x”, cooper_tac)
val gt_eq_lt = store_thm("gt_eq_lt", “!(x y : int). x > y <=> y < x”, cooper_tac)
val lt_eq_gt = store_thm("lt_eq_gt", “!(x y : int). x < y <=> y > x”, cooper_tac)
Theorem int_add_minus_same_eq:
∀ (i j : int). i + j - j = i
Proof
int_tac
QED
(*
* We generate and save an induction theorem for positive integers
*)
(* This is the induction theorem we want.
Unfortunately, it often can't be applied by [Induct_on].
*)
Theorem int_induction_ideal:
!(P : int -> bool). P 0 /\ (!i. 0 <= i /\ P i ==> P (i+1)) ==> !i. 0 <= i ==> P i
Proof
ntac 4 strip_tac >>
Induct_on ‘Num i’ >> rpt strip_tac
>-(sg ‘i = 0’ >- cooper_tac >> fs []) >>
last_assum (qspec_assume ‘i-1’) >>
fs [] >> pop_assum irule >>
rw [] >> try_tac cooper_tac >>
first_assum (qspec_assume ‘i - 1’) >>
pop_assum irule >>
rw [] >> try_tac cooper_tac
QED
(* This induction theorem works well with [Induct_on] *)
Theorem int_induction:
!(P : int -> bool). (∀i. i < 0 ⇒ P i) ∧ P 0 /\ (!i. P i ==> P (i+1)) ==> !i. P i
Proof
ntac 3 strip_tac >>
Cases_on ‘i < 0’ >- (last_assum irule >> fs []) >>
fs [not_lt_eq_ge] >>
Induct_on ‘Num i’ >> rpt strip_tac >> pop_last_assum ignore_tac
>-(sg ‘i = 0’ >- cooper_tac >> fs []) >>
last_assum (qspec_assume ‘i-1’) >>
fs [] >> pop_assum irule >>
rw [] >> try_tac cooper_tac >>
first_assum (qspec_assume ‘i - 1’) >>
pop_assum irule >>
rw [] >> try_tac cooper_tac
QED
val _ = TypeBase.update_induction int_induction
Theorem int_of_num_id:
∀i. 0 ≤ i ⇒ &Num i = i
Proof
fs [INT_OF_NUM]
QED
Theorem int_add:
∀m n. &(m + n) = &m + &n
Proof
fs [INT_ADD]
QED
Theorem int_of_num_inj:
!n m. &n = &m ==> n = m
Proof
rpt strip_tac >>
fs [Num]
QED
(* TODO: use as rewriting theorem by default? *)
Theorem add_sub_same_eq:
∀(i j : int). i + j - j = i
Proof
cooper_tac
QED
Theorem num_sub_eq:
!(x y z : int). x = y - z ==> 0 <= x ==> 0 <= z ==> Num y = Num z + Num x
Proof
rpt strip_tac >>
sg ‘0 <= y’ >- cooper_tac >>
rfs [] >>
(* Convert to integers *)
irule int_of_num_inj >>
imp_res_tac int_of_num >>
(* Associativity of & *)
pure_rewrite_tac [int_add] >>
cooper_tac
QED
Theorem num_sub_1_eq:
!(x y : int). x = y - 1 ==> 0 <= x ==> Num y = SUC (Num x)
Proof
rpt strip_tac >>
(* Get rid of the SUC *)
sg ‘SUC (Num x) = 1 + Num x’ >-(rw [ADD]) >> rw [] >>
(* Massage a bit the goal *)
qsuff_tac ‘Num y = Num (y − 1) + Num 1’ >- cooper_tac >>
(* Apply the general theorem *)
irule num_sub_eq >>
cooper_tac
QED
Theorem pos_mul_pos_is_pos:
!(x y : int). 0 <= x ==> 0 <= y ==> 0 <= x * y
Proof
rpt strip_tac >>
sg ‘0 <= &(Num x) * &(Num y)’ >- (rw [INT_MUL_CALCULATE] >> cooper_tac) >>
sg_dep_rewrite_all_tac int_of_num >> try_tac cooper_tac >> fs []
QED
Theorem pos_div_pos_is_pos:
!(x y : int). 0 <= x ==> 0 < y ==> 0 <= x / y
Proof
rpt strip_tac >>
rw [le_eq_ge] >>
sg ‘y <> 0’ >- cooper_tac >>
qspecl_then [‘\x. x >= 0’, ‘x’, ‘y’] ASSUME_TAC INT_DIV_FORALL_P >>
fs [] >> pop_ignore_tac >> rw [] >- cooper_tac >>
fs [not_lt_eq_ge] >>
(* Proof by contradiction: assume k < 0 *)
spose_not_then assume_tac >>
fs [not_ge_eq_lt] >>
sg ‘k * y = (k + 1) * y + - y’ >- (fs [INT_RDISTRIB] >> cooper_tac) >>
sg ‘0 <= (-(k + 1)) * y’ >- (irule pos_mul_pos_is_pos >> cooper_tac) >>
cooper_tac
QED
Theorem pos_div_pos_le:
!(x y d : int). 0 <= x ==> 0 <= y ==> 0 < d ==> x <= y ==> x / d <= y / d
Proof
rpt strip_tac >>
sg ‘d <> 0’ >- cooper_tac >>
qspecl_assume [‘\k. k = x / d’, ‘x’, ‘d’] INT_DIV_P >>
qspecl_assume [‘\k. k = y / d’, ‘y’, ‘d’] INT_DIV_P >>
rfs [not_lt_eq_ge] >> try_tac cooper_tac >>
sg ‘y = (x / d) * d + (r' + y - x)’ >- cooper_tac >>
qspecl_assume [‘(x / d) * d’, ‘r' + y - x’, ‘d’] INT_ADD_DIV >>
rfs [] >>
Cases_on ‘x = y’ >- fs [] >>
sg ‘r' + y ≠ x’ >- cooper_tac >> fs [] >>
sg ‘((x / d) * d) / d = x / d’ >- (irule INT_DIV_RMUL >> cooper_tac) >>
fs [] >>
sg ‘0 <= (r' + y − x) / d’ >- (irule pos_div_pos_is_pos >> cooper_tac) >>
metis_tac [INT_LE_ADDR]
QED
Theorem pos_div_pos_le_init:
!(x y : int). 0 <= x ==> 0 < y ==> x / y <= x
Proof
rpt strip_tac >>
sg ‘y <> 0’ >- cooper_tac >>
qspecl_assume [‘\k. k = x / y’, ‘x’, ‘y’] INT_DIV_P >>
rfs [not_lt_eq_ge] >- cooper_tac >>
sg ‘y = (y - 1) + 1’ >- rw [] >>
sg ‘x = x / y + ((x / y) * (y - 1) + r)’ >-(
qspecl_assume [‘1’, ‘y-1’, ‘x / y’] INT_LDISTRIB >>
rfs [] >>
cooper_tac
) >>
sg ‘!a b c. 0 <= c ==> a = b + c ==> b <= a’ >- cooper_tac >>
pop_assum irule >>
exists_tac “x / y * (y − 1) + r” >>
sg ‘0 <= x / y’ >- (irule pos_div_pos_is_pos >> cooper_tac) >>
sg ‘0 <= (x / y) * (y - 1)’ >- (irule pos_mul_pos_is_pos >> cooper_tac) >>
cooper_tac
QED
Theorem pos_mod_pos_is_pos:
!(x y : int). 0 <= x ==> 0 < y ==> 0 <= x % y
Proof
rpt strip_tac >>
sg ‘y <> 0’ >- cooper_tac >>
imp_res_tac INT_DIVISION >>
first_x_assum (qspec_then ‘x’ assume_tac) >>
first_x_assum (qspec_then ‘x’ assume_tac) >>
sg ‘~(y < 0)’ >- cooper_tac >> fs []
QED
Theorem pos_mod_pos_le_init:
!(x y : int). 0 <= x ==> 0 < y ==> x % y <= x
Proof
rpt strip_tac >>
sg ‘y <> 0’ >- cooper_tac >>
imp_res_tac INT_DIVISION >>
first_x_assum (qspec_then ‘x’ assume_tac) >>
first_x_assum (qspec_then ‘x’ assume_tac) >>
sg ‘~(y < 0)’ >- cooper_tac >> fs [] >>
sg ‘0 <= x % y’ >- (irule pos_mod_pos_is_pos >> cooper_tac) >>
sg ‘0 <= x / y’ >- (irule pos_div_pos_is_pos >> cooper_tac) >>
sg ‘0 <= (x / y) * y’ >- (irule pos_mul_pos_is_pos >> cooper_tac) >>
cooper_tac
QED
val _ = export_theory ()
|
