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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% FILE: zero_ineq.ml %
% EDITOR: Paul Curzon %
% DATE: July 1991 %
% DESCRIPTION: Inequalities about 0 %
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%********************************* HISTORY ********************************%
% %
% This file is based on the theories of %
% %
% Mike Benjamin %
% Rachel Cardell-Oliver %
% Paul Curzon %
% Wim Ploegaerts %
% %
%****************************************************************************%
% %
% PC 21/4/93 %
% Removed dependencies on several external files/theories %
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%****************************************************************************%
% %
% DEPENDANCIES : %
% %
% %
%****************************************************************************%
system `rm -f zero_ineq.th`;;
new_theory `zero_ineq`;;
% PC 22-4-92 These are no longer used%
%new_parent `ineq`;;%
%loadf `tools`;;%
%autoload_defs_and_thms `ineq`;;%
let autoload_defs_and_thms thy =
map (\name. autoload_theory(`definition`,thy,name))
(map fst (definitions thy));
map (\name. autoload_theory(`theorem`,thy,name))
(map fst (theorems thy)) in
autoload_defs_and_thms `prim_rec`;;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% AUTHOR: Wim Ploegaerts %
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%WP 4-9-90%
let M_LESS_0_LESS = prove_thm (
`M_LESS_0_LESS`,
"! m n . (m < n) ==> (0 < n)",
REPEAT GEN_TAC THEN
DISCH_THEN \thm . ASSUME_TAC
% PC 12/8/92 CONJ1 LESS_EQ_LESS_TRANS -> LESS_EQ_LESS_TRANS %
% (MATCH_MP (CONJ1 LESS_EQ_LESS_TRANS)%
(MATCH_MP (LESS_EQ_LESS_TRANS)
(CONJ (SPEC "m:num" ZERO_LESS_EQ) thm)) THEN
FIRST_ASSUM ACCEPT_TAC
);;
%<-------------------------------------------------------------------------->%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% AUTHOR: Paul Curzon %
% DATE: June 1991 %
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
let LESS1EQ0 = prove_thm (`LESS1EQ0`,
"!m . (m < 1) = (m = 0)",
GEN_TAC THEN
EQ_TAC THENL[
(CONV_TAC (DEPTH_CONV num_CONV)) THEN
(REPEAT STRIP_TAC) THEN
(REWRITE_TAC[REWRITE_RULE
[ASSUME "m < (SUC 0)"; NOT_LESS_0]
(SPECL ["m:num";"0"] LESS_LEMMA1)]);
DISCH_TAC THEN
(REWRITE_TAC[ASSUME "m = 0"]) THEN
(CONV_TAC (DEPTH_CONV num_CONV)) THEN
(REWRITE_TAC[LESS_SUC_REFL])]);;
% (POP_ASSUM (\th . ASSUME_TAC (REWRITE_RULE[th]%
% (SPECL ["m:num";"0"] LESS_LEMMA1)))) THEN%
% (POP_TAC (REWRITE_RULE[NOT_LESS_0])) THEN%
% POP_REWRITE_TAC;%
% DISCH_TAC THEN%
% POP_REWRITE_TAC THEN%
% (CONV_TAC (DEPTH_CONV num_CONV)) THEN%
% (REWRITE_TAC[LESS_SUC_REFL])]);;%
%<-------------------------------------------------------------------------->%
let NOT_EQ_0 = prove_thm(`NOT_EQ_0`,
"! m . ~ ( m = 0) ==> (m >= 1)",
INDUCT_TAC THENL
[
(REWRITE_TAC[]);
(REPEAT STRIP_TAC) THEN
(ASM_CASES_TAC "m = 0") THENL
[
REWRITE_TAC[ASSUME "m = 0"] THEN
% POP_REWRITE_TAC THEN%
(CONV_TAC (ONCE_DEPTH_CONV num_CONV)) THEN
(REWRITE_TAC[LESS_EQ_REFL;GREATER_EQ]);
(REWRITE_TAC[GREATER_EQ;ADD1]) THEN
(ONCE_REWRITE_TAC[ADD_SYM]) THEN
(REWRITE_TAC[LESS_EQ_ADD])
]
]
);;
%<-------------------------------------------------------------------------->%
let LESS_EQ_0_EQ = prove_thm(`LESS_EQ_0_EQ`,
"!m. m <= 0 ==> (m = 0)",
(REPEAT STRIP_TAC) THEN
(REWRITE_TAC [LESS_OR_EQ;
(REWRITE_RULE[ASSUME "m <= 0"]
(SPEC_ALL
(REWRITE_RULE[GSYM NOT_LESS_EQUAL]
LESS_0_CASES)) )]));;
% (REPEAT STRIP_TAC) THEN%
% (REWRITE_TAC [LESS_OR_EQ]) THEN%
% (ASSUME_TAC (SPEC_ALL (REWRITE_RULE[GSYM NOT_LESS_EQ_LESS] LESS_0_CASES))) THEN %
% PC 12/8/92 NOT_LESS_EQ_LESS -> NOT_LESS_EQUAL %
% (ASSUME_TAC (SPEC_ALL (REWRITE_RULE[GSYM NOT_LESS_EQUAL] LESS_0_CASES))) THEN%
% (REWRITE_A1_WITH_A2_THEN ASSUME_TAC) THEN%
% (ASM_REWRITE_TAC[]));;%
%<-------------------------------------------------------------------------->%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% AUTHOR: Mike Benjamin %
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% ***************************************************************************
* *
* If a number is greater than zero then it's not zero. *
* *
* GREATER_NOT_ZERO = |- !x. 0 < x ==> ~(x = 0) *
* *
**************************************************************************%
%let GREATER_NOT_ZERO = prove_thm (`GREATER_NOT_ZERO`,%
%"! (x:num). (0 < x) ==> ~(x = 0)",%
%GEN_TAC%
%THEN CUT_THM_TAC (SPEC "x:num" num_CASES)%
%THEN DISJ_LEFT_TAC%
%THENL [%
% ASM_REWRITE_TAC [LESS_REFL];%
% EXISTS_LEFT_TAC%
% THEN ASM_REWRITE_TAC []%
% THEN DISCH_TAC%
% THEN IMP_RES_TAC (SPEC "SUC (n:num)" (SPEC "0" LESS_NOT_EQ))%
% THEN N_REVERSE_TAC 1%
% THEN N_REWRITE_TAC 1]);;%
%New proof by PC 22-4-93%
let GREATER_NOT_ZERO = prove_thm (`GREATER_NOT_ZERO`,
"! (x:num). (0 < x) ==> ~(x = 0)",
(REPEAT STRIP_TAC) THEN
UNDISCH_TAC "0<x" THEN
ASM_REWRITE_TAC[LESS_REFL]);;
%****************************************************************************%
% %
% AUTHOR : Rachel Cardell-Oliver %
% DATE : 1990 %
% %
%****************************************************************************%
let NOT_0_AND_MORE =prove_thm(`NOT_0_AND_MORE`,
"!x:num. ~( (x=0) /\ (0<x) )",
REPEAT STRIP_TAC THEN UNDISCH_TAC "0<x" THEN ASM_REWRITE_TAC[LESS_REFL]);;
close_theory ();;
|
