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 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 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 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 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 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 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; :: 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 ) ) )
set v = ^ b9;
A2: (^ b9) . s = b ;
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 A2, A8, A6, A4, A3, A7, A5, Th24; :: thesis: verum