let A be Euclidean preIfWhileAlgebra; :: thesis: for X being non empty countable set
for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs X,INT ,(Funcs X,INT ) \ b,0
for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
for s being Element of Funcs X,INT holds
( (g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) . x = s . y & (g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) . y = (s . x) mod (s . y) & ( for n, m being Element of NAT holds
not ( m = s . y & ( s in (Funcs X,INT ) \ b,0 implies m > 0 ) & ( m > 0 implies s in (Funcs X,INT ) \ b,0 ) & not g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 ),s ) ) )
let X be non empty countable set ; :: thesis: for b being Element of X
for g being Euclidean ExecutionFunction of A, Funcs X,INT ,(Funcs X,INT ) \ b,0
for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
for s being Element of Funcs X,INT holds
( (g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) . x = s . y & (g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) . y = (s . x) mod (s . y) & ( for n, m being Element of NAT holds
not ( m = s . y & ( s in (Funcs X,INT ) \ b,0 implies m > 0 ) & ( m > 0 implies s in (Funcs X,INT ) \ b,0 ) & not g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 ),s ) ) )
let b be Element of X; :: thesis: for g being Euclidean ExecutionFunction of A, Funcs X,INT ,(Funcs X,INT ) \ b,0
for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
for s being Element of Funcs X,INT holds
( (g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) . x = s . y & (g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) . y = (s . x) mod (s . y) & ( for n, m being Element of NAT holds
not ( m = s . y & ( s in (Funcs X,INT ) \ b,0 implies m > 0 ) & ( m > 0 implies s in (Funcs X,INT ) \ b,0 ) & not g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 ),s ) ) )
let g be Euclidean ExecutionFunction of A, Funcs X,INT ,(Funcs X,INT ) \ b,0 ; :: thesis: for x, y, z being Variable of g st ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) holds
for s being Element of Funcs X,INT holds
( (g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) . x = s . y & (g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) . y = (s . x) mod (s . y) & ( for n, m being Element of NAT holds
not ( m = s . y & ( s in (Funcs X,INT ) \ b,0 implies m > 0 ) & ( m > 0 implies s in (Funcs X,INT ) \ b,0 ) & not g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 ),s ) ) )
set h = g;
set S = Funcs X,INT ;
set T = (Funcs X,INT ) \ b,0 ;
let x, y, z be Variable of g; :: thesis: ( ex d being Function st
( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) implies for s being Element of Funcs X,INT holds
( (g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) . x = s . y & (g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) . y = (s . x) mod (s . y) & ( for n, m being Element of NAT holds
not ( m = s . y & ( s in (Funcs X,INT ) \ b,0 implies m > 0 ) & ( m > 0 implies s in (Funcs X,INT ) \ b,0 ) & not g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 ),s ) ) ) )
given d being Function such that A1: d . b = 0 and
A2: d . x = 1 and
A3: d . y = 2 and
A4: d . z = 3 ; :: thesis: for s being Element of Funcs X,INT holds
( (g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) . x = s . y & (g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) . y = (s . x) mod (s . y) & ( for n, m being Element of NAT holds
not ( m = s . y & ( s in (Funcs X,INT ) \ b,0 implies m > 0 ) & ( m > 0 implies s in (Funcs X,INT ) \ b,0 ) & not g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 ),s ) ) )
A5: z <> x by A2, A4;
let s be Element of Funcs X,INT ; :: thesis: ( (g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) . x = s . y & (g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) . y = (s . x) mod (s . y) & ( for n, m being Element of NAT holds
not ( m = s . y & ( s in (Funcs X,INT ) \ b,0 implies m > 0 ) & ( m > 0 implies s in (Funcs X,INT ) \ b,0 ) & not g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 ),s ) ) )
set I = (((z := x) \; (z %= y)) \; (x := y)) \; (y := z);
A6: y <> z by A3, A4;
A7: x <> y by A2, A3;

A8: now

let s be Element of Funcs X,INT ; :: thesis: ( (g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) . x = s . y & (g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) . y = (s . x) mod (s . y) )
reconsider s1 = g . s,(z := x) as Element of Funcs X,INT ;
reconsider s2 = g . s1,(z %= y) as Element of Funcs X,INT ;
reconsider s3 = g . s2,(x := y) as Element of Funcs X,INT ;
reconsider s4 = g . s3,(y := z) as Element of Funcs X,INT ;
A9: g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) = g . (g . s,(((z := x) \; (z %= y)) \; (x := y))),(y := z) by AOFA_000:def 29
.= g . (g . (g . s,((z := x) \; (z %= y))),(x := y)),(y := z) by AOFA_000:def 29
.= s4 by AOFA_000:def 29 ;
A10: s1 . z = s . x by Th27;
A11: s2 . y = s1 . y by A6, Th44;
A12: s2 . z = (s1 . z) mod (s1 . y) by Th44;
A13: s3 . z = s2 . z by A5, Th27;
A14: s3 . x = s2 . y by Th27;
s1 . y = s . y by A6, Th27;
hence ( (g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) . x = s . y & (g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) . y = (s . x) mod (s . y) ) by A7, A9, A10, A11, A12, A14, A13, Th27; :: thesis: verum

end;

hence ( (g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) . x = s . y & (g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) . y = (s . x) mod (s . y) ) ; :: thesis: for n, m being Element of NAT holds
not ( m = s . y & ( s in (Funcs X,INT ) \ b,0 implies m > 0 ) & ( m > 0 implies s in (Funcs X,INT ) \ b,0 ) & not g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 ),s )
deffunc H₁( Element of Funcs X,INT ) -> Element of NAT = In ($1 . y),NAT ;
defpred S₁[ Element of Funcs X,INT ] means $1 . y > 0 ;
set C = y gt 0 ;
A15: for s being Element of Funcs X,INT st S₁[s] holds
( ( S₁[g . s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 ))] implies g . s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 )) in (Funcs X,INT ) \ b,0 ) & ( g . s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 )) in (Funcs X,INT ) \ b,0 implies S₁[g . s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 ))] ) & H₁(g . s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 ))) < H₁(s) )

proof

let s be Element of Funcs X,INT ; :: thesis: ( S₁[s] implies ( ( S₁[g . s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 ))] implies g . s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 )) in (Funcs X,INT ) \ b,0 ) & ( g . s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 )) in (Funcs X,INT ) \ b,0 implies S₁[g . s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 ))] ) & H₁(g . s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 ))) < H₁(s) ) )
assume A16: s . y > 0 ; :: thesis: ( ( S₁[g . s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 ))] implies g . s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 )) in (Funcs X,INT ) \ b,0 ) & ( g . s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 )) in (Funcs X,INT ) \ b,0 implies S₁[g . s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 ))] ) & H₁(g . s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 ))) < H₁(s) )
reconsider s9 = g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) as Element of Funcs X,INT ;
A17: s9 . y = (s . x) mod (s . y) by A8;
then A18: 0 <= s9 . y by A16, NEWTON:78;
reconsider s99 = g . s9,(y gt 0 ) as Element of Funcs X,INT ;
A19: g . s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 )) = s99 by AOFA_000:def 29;
A20: ( s9 . y <= 0 implies s99 . b = 0 ) by Th38;
( s9 . y > 0 implies s99 . b = 1 ) by Th38;
hence ( S₁[g . s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 ))] iff g . s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 )) in (Funcs X,INT ) \ b,0 ) by A19, A20, Th2, Th38; :: thesis: H₁(g . s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 ))) < H₁(s)
s99 . y = s9 . y by A1, A3, Th38;
then A21: H₁(s99) = s9 . y by A18, FUNCT_7:def 1, INT_1:16;
A22: s9 . y < s . y by A16, A17, NEWTON:79;
then H₁(s) = s . y by A18, FUNCT_7:def 1, INT_1:16;
hence H₁(g . s,(((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 ))) < H₁(s) by A22, A21, AOFA_000:def 29; :: thesis: verum

end;

reconsider fin = g . s,(while (y gt 0 ),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) as Element of Funcs X,INT ;
reconsider s1 = g . s,(y gt 0 ) as Element of Funcs X,INT ;
let n, m be Element of NAT ; :: thesis: not ( m = s . y & ( s in (Funcs X,INT ) \ b,0 implies m > 0 ) & ( m > 0 implies s in (Funcs X,INT ) \ b,0 ) & not g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 ),s )
assume A23: m = s . y ; :: thesis: ( ( s in (Funcs X,INT ) \ b,0 & not m > 0 ) or ( m > 0 & not s in (Funcs X,INT ) \ b,0 ) or g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 ),s )
assume ( s in (Funcs X,INT ) \ b,0 iff m > 0 ) ; :: thesis: g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 ),s
then A24: ( s in (Funcs X,INT ) \ b,0 iff S₁[s] ) by A23;
g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 ),s from AOFA_000:sch 3(A24, A15);
hence g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 ),s ; :: thesis: verum