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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% FILE: odd_even.ml %
% EDITOR: Paul Curzon %
% DATE: July 1991 %
% DESCRIPTION : Odd and Even %
% %
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%********************************* HISTORY ********************************%
% %
% %
% Author: Wim Ploegaerts (ploegaer@imec.be) %
% %
% Date: Fri Feb 8 1991 %
% %
% Organization: Imec vzw. %
% Kapeldreef 75 %
% 3001 Leuven - Belgium %
% %
%****************************************************************************%
% %
% Rewritten by PC 14/8/92 to take account of ODD and EVEN definitions %
% placed in main system by JRH. The definitions ODD_NUM and EVEN_NUM are %
% now called ODD and EVEN, respectively. Replicated theorems have been %
% removed from the library (causing name changes). The names of others %
% have been changed for consistency with the system theorems %
% %
% The theorem name changes are: %
% %
% EVEN_NUM --> EVEN_EXISTS %
% NUM_EVEN_MULT --> EVEN_IMPL_MULT %
% NUM_EVEN_ODD_PLUS_CASES --> EVEN_ODD_PLUS_CASES %
% NUM_EVEN_ODD_SUC --> EVEN_ODD_SUC %
% NUM_EVEN_OR_ODD --> EVEN_OR_ODD %
% NUM_MULT_EVEN --> MULT_EVEN %
% NUM_MULT_ODD --> MULT_ODD %
% NUM_NOT_EVEN_AND_ODD --> EVEN_AND_ODD %
% NUM_NOT_EVEN_NOT_ODD --> EVEN_ODD / ODD_EVEN %
% (conjunct split and GSYM'd) %
% NUM_NOT_EVEN_ODD_SUC_EVEN_ODD --> NOT_EVEN_ODD_SUC_EVEN_ODD %
% NUM_ODD_MULT --> ODD_IMPL_MULT %
% ODD_NUM --> ODD_EXISTS %
% %
%****************************************************************************%
% %
% PC 21/4/93 %
% Removed dependencies on several external files/theories %
% %
%****************************************************************************%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% %
% DEPENDANCIES : %
% %
% %
%****************************************************************************%
system `rm -f odd_even.th`;;
new_theory `odd_even`;;
%<--------------------------------------------------------------------------->%
let EVEN_ODD_0 = prove_thm (
`EVEN_ODD_0`,
"(EVEN 0) /\ ~(ODD 0)",
REWRITE_TAC [ODD;EVEN]
);;
%<--------------------------------------------------------------------------->%
%
|- !n. (~EVEN(SUC n) = EVEN n) /\ (~ODD(SUC n) = ODD n)
%
let NOT_EVEN_ODD_SUC_EVEN_ODD = prove_thm (
`NOT_EVEN_ODD_SUC_EVEN_ODD`,
"!n. (~EVEN(SUC n) = EVEN n) /\ (~ODD(SUC n) = ODD n)",
REWRITE_TAC [ODD;EVEN]
);;
%<--------------------------------------------------------------------------->%
let EVEN_ODD_SUC = prove_thm (
`EVEN_ODD_SUC`,
"!n. (EVEN(SUC n) = ODD n) /\ (ODD(SUC n) = EVEN n)",
REWRITE_TAC [ODD;EVEN;GSYM ODD_EVEN;GSYM EVEN_ODD]
);;
%<--------------------------------------------------------------------------->%
let EVEN_ODD_PLUS_CASES = prove_thm (
`EVEN_ODD_PLUS_CASES`,
"!n m . (((ODD n) /\ (ODD m)) ==> (EVEN (n + m))) /\
(((ODD n) /\ (EVEN m)) ==> (ODD (n + m))) /\
(((EVEN n) /\ (EVEN m)) ==> (EVEN (n + m)))",
REPEAT STRIP_TAC THEN
REWRITE_TAC [EVEN_ADD;ODD_ADD] THEN
ASM_REWRITE_TAC [GSYM EVEN_ODD] THEN
ASM_REWRITE_TAC [EVEN_ODD]
);;
%<--------------------------------------------------------------------------->%
let EVEN_IMPL_MULT = prove_thm (
`EVEN_IMPL_MULT`,
"! n m . (EVEN n) \/ (EVEN m) ==> (EVEN (n * m))",
REWRITE_TAC [EVEN_MULT]
);;
%<--------------------------------------------------------------------------->%
let ODD_IMPL_MULT = prove_thm (
`ODD_IMPL_MULT`,
"! n m . (ODD n) /\ (ODD m) ==> (ODD (n * m))",
REWRITE_TAC [ODD_MULT]
);;
%<--------------------------------------------------------------------------->%
%
MULT_ODD = |- !n m. ODD(n * m) ==> ODD n /\ ODD m
%
let MULT_ODD = prove_thm (
`MULT_ODD`,
"!n m. ODD(n * m) ==> ODD n /\ ODD m",
REWRITE_TAC [ODD_MULT]);;
%<--------------------------------------------------------------------------->%
%
MULT_EVEN =
|- !n m. EVEN(n * m) ==> EVEN n \/ EVEN m
%
let MULT_EVEN = prove_thm (
`MULT_EVEN`,
"!n m. EVEN(n * m) ==> EVEN n \/ EVEN m",
REWRITE_TAC [EVEN_MULT]);;
close_theory();;
|
