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
|
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% FILE: suc.ml %
% EDITOR: Paul Curzon %
% DATE: July 1991 %
% DESCRIPTION : Theorems about SUC %
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%********************************* HISTORY ********************************%
% %
% This file is based on the theories of %
% %
% Richard J.Boulton %
% Rachel Cardell-Oliver %
% Paul Curzon %
% Jeff Joyce %
% Philippe Leveilley %
% Wim Ploegaerts %
% %
%****************************************************************************%
% %
% PC 5/7/92 NOT_SUC_LESS_EQ --> NOT_FORALL_SUC_LESS_EQ %
% %
%****************************************************************************%
% %
% PC 21/4/93 %
% Removed dependencies on several external files/theories %
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%****************************************************************************%
% %
% DEPENDANCIES : %
% %
% %
%****************************************************************************%
system `rm -f suc.th`;;
new_theory `suc`;;
% PC 22-4-92 These are no longer used%
%new_parent `ineq`;;%
%loadf `tools`;;%
%autoload_defs_and_thms `ineq`;;%
%<-------------------------------------------------------------------------->%
%<PL>%
% Moved to main system by RJB on 92.09.28 %
%let NOT_SUC_LESS_EQ_0 =%
% prove_thm (`NOT_SUC_LESS_EQ_0`,%
% "! n . ~ (SUC n) <= 0",%
% PC 12/8/92 NOT_LESS_EQ_LESS -> NOT_LESS_EQUAL %
% REWRITE_TAC[NOT_LESS_EQ_LESS;LESS_0]);;%
% REWRITE_TAC[NOT_LESS_EQUAL;LESS_0]);;%
%<-------------------------------------------------------------------------->%
%<PL>%
% Name changed from NOT_SUC_LESS_EQ to avoid clash. PC 5/7/92 %
let NOT_FORALL_SUC_LESS_EQ =
save_thm (`NOT_FORALL_SUC_LESS_EQ`,
NOT_INTRO
(DISCH_ALL
(REWRITE_RULE [NOT_SUC_LESS_EQ_0]
(SPEC "0" (ASSUME "(!n m. (SUC m) <= n)")))));;
%<-------------------------------------------------------------------------->%
% NOT_0_GREATER_EQ_SUC |- ~0 >= (SUC n)%
let NOT_0_GREATER_EQ_SUC = save_thm(`NOT_0_GREATER_EQ_SUC`,
GEN_ALL
(REWRITE_RULE[LESS_0;GSYM GREATER_EQ]
(SPEC "SUC n" (SPEC "0" LESS_EQ_ANTISYM))));;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% AUTHOR: Paul Curzon %
% DATE: June 1991 %
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%SUC_GREATER_EQ_SUC = |- !m n. (SUC m) >= (SUC n) = m >= n %
let SUC_GREATER_EQ_SUC = save_thm (`SUC_GREATER_EQ_SUC`,
REWRITE_RULE[GSYM GREATER_EQ] LESS_EQ_MONO);;
%****************************************************************************%
% %
% AUTHOR : R.J.Boulton %
% DATE : 1990 %
% %
%****************************************************************************%
%----------------------------------------------------------------------------%
% LESS_EQ_MONO_EQ = |- !m n. (SUC m) <= (SUC n) = m <= n %
%----------------------------------------------------------------------------%
let LESS_EQ_MONO_EQ = LESS_EQ_MONO;;
%----------------------------------------------------------------------------%
% LESS_EQ_LESS_SUC = |- !m n. m <= n = m < (SUC n) %
%----------------------------------------------------------------------------%
let LESS_EQ_LESS_SUC =
prove_thm
(`LESS_EQ_LESS_SUC`,
"!m n. (m <= n) = (m < (SUC n))",
REWRITE_TAC [LESS_OR_EQ] THEN
REPEAT GEN_TAC THEN
EQ_TAC THENL
[STRIP_TAC THENL
[IMP_RES_TAC LESS_SUC;
ASM_REWRITE_TAC [LESS_SUC_REFL]];
REWRITE_TAC [LESS_THM] THEN
STRIP_TAC THEN
ASM_REWRITE_TAC []]);;
%----------------------------------------------------------------------------%
let SUC_LESS_EQ = prove_thm(`SUC_LESS_EQ`,
"!m n. ((m <= n) /\ ~(m = n)) ==> (SUC m) <= n",
REPEAT STRIP_TAC THEN
UNDISCH_TAC "m <= n" THEN
SUBST_TAC [SPEC_ALL LESS_OR_EQ] THEN
ASM_REWRITE_TAC [LESS_EQ]);;
%----------------------------------------------------------------------------%
let NOT_SUC_LESS_EQ_SELF = prove_thm(`NOT_SUC_LESS_EQ_SELF`,
"!n. ~((SUC n) <= n)",
GEN_TAC THEN
REWRITE_TAC [LESS_OR_EQ;DE_MORGAN_THM] THEN
REWRITE_TAC [SUC_ID;NOT_LESS;LESS_EQ_SUC_REFL]);;
%----------------------------------------------------------------------------%
%****************************************************************************%
% %
% AUTHOR : Rachel Cardell-Oliver %
% DATE : 1990 %
% %
%****************************************************************************%
let SUC_0=prove_thm(`SUC_0`,
"1 = (SUC 0)",
ASM_REWRITE_TAC[ADD1;ADD_CLAUSES]);;
let SUC_NOT_0 = save_thm(`SUC_NOT_0`, GSYM SUC_NOT);;
close_theory();;
|
