let A be Euclidean preIfWhileAlgebra; :: thesis: for X being non empty countable set
for s being Element of Funcs X,INT
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs X,INT ,(Funcs X,INT ) \ b,0
for x being Variable of g
for i being integer number holds
( ( s . x <= i implies (g . s,(x leq i)) . b = 1 ) & ( s . x > i implies (g . s,(x leq i)) . b = 0 ) & ( s . x >= i implies (g . s,(x geq i)) . b = 1 ) & ( s . x < i implies (g . s,(x geq i)) . b = 0 ) & ( for z being Element of X st z <> b holds
( (g . s,(x leq i)) . z = s . z & (g . s,(x geq i)) . z = s . z ) ) )
let X be non empty countable set ; :: thesis: for s being Element of Funcs X,INT
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs X,INT ,(Funcs X,INT ) \ b,0
for x being Variable of g
for i being integer number holds
( ( s . x <= i implies (g . s,(x leq i)) . b = 1 ) & ( s . x > i implies (g . s,(x leq i)) . b = 0 ) & ( s . x >= i implies (g . s,(x geq i)) . b = 1 ) & ( s . x < i implies (g . s,(x geq i)) . b = 0 ) & ( for z being Element of X st z <> b holds
( (g . s,(x leq i)) . z = s . z & (g . s,(x geq i)) . z = s . z ) ) )
let s be Element of Funcs X,INT ; :: thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs X,INT ,(Funcs X,INT ) \ b,0
for x being Variable of g
for i being integer number holds
( ( s . x <= i implies (g . s,(x leq i)) . b = 1 ) & ( s . x > i implies (g . s,(x leq i)) . b = 0 ) & ( s . x >= i implies (g . s,(x geq i)) . b = 1 ) & ( s . x < i implies (g . s,(x geq i)) . b = 0 ) & ( for z being Element of X st z <> b holds
( (g . s,(x leq i)) . z = s . z & (g . s,(x geq i)) . z = s . z ) ) )
let b be Element of X; :: thesis: for g being Euclidean ExecutionFunction of A, Funcs X,INT ,(Funcs X,INT ) \ b,0
for x being Variable of g
for i being integer number holds
( ( s . x <= i implies (g . s,(x leq i)) . b = 1 ) & ( s . x > i implies (g . s,(x leq i)) . b = 0 ) & ( s . x >= i implies (g . s,(x geq i)) . b = 1 ) & ( s . x < i implies (g . s,(x geq i)) . b = 0 ) & ( for z being Element of X st z <> b holds
( (g . s,(x leq i)) . z = s . z & (g . s,(x geq i)) . z = s . z ) ) )
let f be Euclidean ExecutionFunction of A, Funcs X,INT ,(Funcs X,INT ) \ b,0 ; :: thesis: for x being Variable of f
for i being integer number holds
( ( s . x <= i implies (f . s,(x leq i)) . b = 1 ) & ( s . x > i implies (f . s,(x leq i)) . b = 0 ) & ( s . x >= i implies (f . s,(x geq i)) . b = 1 ) & ( s . x < i implies (f . s,(x geq i)) . b = 0 ) & ( for z being Element of X st z <> b holds
( (f . s,(x leq i)) . z = s . z & (f . s,(x geq i)) . z = s . z ) ) )
reconsider b9 = b as Variable of f by Def2;
let x be Variable of f; :: thesis: for i being integer number holds
( ( s . x <= i implies (f . s,(x leq i)) . b = 1 ) & ( s . x > i implies (f . s,(x leq i)) . b = 0 ) & ( s . x >= i implies (f . s,(x geq i)) . b = 1 ) & ( s . x < i implies (f . s,(x geq i)) . b = 0 ) & ( for z being Element of X st z <> b holds
( (f . s,(x leq i)) . z = s . z & (f . s,(x geq i)) . z = s . z ) ) )
let i be integer number ; :: thesis: ( ( s . x <= i implies (f . s,(x leq i)) . b = 1 ) & ( s . x > i implies (f . s,(x leq i)) . b = 0 ) & ( s . x >= i implies (f . s,(x geq i)) . b = 1 ) & ( s . x < i implies (f . s,(x geq i)) . b = 0 ) & ( for z being Element of X st z <> b holds
( (f . s,(x leq i)) . z = s . z & (f . s,(x geq i)) . z = s . z ) ) )
reconsider x9 = x as Element of X ;
set v = ^ b9;
set t = leq (. x),(. i,A,f);
A1: (. i,A,f) . s = i by FUNCOP_1:13;
A2: (^ b9) . s = b by FUNCOP_1:13;
A3: ( (. x) . s < i implies IFGT i,((. x) . s),0 ,1 = 0 ) by XXREAL_0:def 11;
A4: ( (. x) . s >= i implies IFGT i,((. x) . s),0 ,1 = 1 ) by XXREAL_0:def 11;
A5: (leq (. i,A,f),(. x)) . s = IFGT ((. i,A,f) . s),((. x) . s),0 ,1 by Def31;
A6: ( (. x) . s > i implies IFGT ((. x) . s),i,0 ,1 = 0 ) by XXREAL_0:def 11;
A7: (leq (. x),(. i,A,f)) . s = IFGT ((. x) . s),((. i,A,f) . s),0 ,1 by Def31;
A8: ( (. x) . s <= i implies IFGT ((. x) . s),i,0 ,1 = 1 ) by XXREAL_0:def 11;
(. x) . s = s . ((^ x) . s) by Def19;
hence ( ( s . x <= i implies (f . s,(x leq i)) . b = 1 ) & ( s . x > i implies (f . s,(x leq i)) . b = 0 ) & ( s . x >= i implies (f . s,(x geq i)) . b = 1 ) & ( s . x < i implies (f . s,(x geq i)) . b = 0 ) & ( for z being Element of X st z <> b holds
( (f . s,(x leq i)) . z = s . z & (f . s,(x geq i)) . z = s . z ) ) ) by A1, A2, A8, A6, A4, A3, A7, A5, Th24, FUNCOP_1:13; :: thesis: verum