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
while (y gt 0 ),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) is_terminating_wrt g,{ s where s is Element of Funcs X,INT : ( s . x > s . y & s . y >= 0 ) }
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
while (y gt 0 ),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) is_terminating_wrt g,{ s where s is Element of Funcs X,INT : ( s . x > s . y & s . y >= 0 ) }
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
while (y gt 0 ),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) is_terminating_wrt g,{ s where s is Element of Funcs X,INT : ( s . x > s . y & s . y >= 0 ) }
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
while (y gt 0 ),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) is_terminating_wrt g,{ s where s is Element of Funcs X,INT : ( s . x > s . y & s . y >= 0 ) }
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 while (y gt 0 ),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) is_terminating_wrt g,{ s where s is Element of Funcs X,INT : ( s . x > s . y & s . y >= 0 ) } )
set P = { s where s is Element of Funcs X,INT : ( s . x > s . y & s . y >= 0 ) } ;
given d being Function such that 00: ( d . b = 0 & d . x = 1 & d . y = 2 & d . z = 3 ) ; :: thesis: while (y gt 0 ),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) is_terminating_wrt g,{ s where s is Element of Funcs X,INT : ( s . x > s . y & s . y >= 0 ) }
set C = y gt 0 ;
set I = (((z := x) \; (z %= y)) \; (x := y)) \; (y := z);
01: { s where s is Element of Funcs X,INT : ( s . x > s . y & s . y >= 0 ) } is_invariant_wrt y gt 0 ,g

proof

let s be Element of Funcs X,INT ; :: according to AOFA_000:def 39 :: thesis: ( not s in { s where s is Element of Funcs X,INT : ( s . x > s . y & s . y >= 0 ) } or g . s,(y gt 0 ) in { s where s is Element of Funcs X,INT : ( s . x > s . y & s . y >= 0 ) } )
assume s in { s where s is Element of Funcs X,INT : ( s . x > s . y & s . y >= 0 ) } ; :: thesis: g . s,(y gt 0 ) in { s where s is Element of Funcs X,INT : ( s . x > s . y & s . y >= 0 ) }
then A0: ex s' being Element of Funcs X,INT st
( s' = s & s' . x > s' . y & s' . y >= 0 ) ;
set s1 = g . s,(y gt 0 );
( (g . s,(y gt 0 )) . x = s . x & (g . s,(y gt 0 )) . y = s . y ) by 00, Th015;
hence g . s,(y gt 0 ) in { s where s is Element of Funcs X,INT : ( s . x > s . y & s . y >= 0 ) } by A0; :: thesis: verum

end;

02: ( y gt 0 is_terminating_wrt g & (((z := x) \; (z %= y)) \; (x := y)) \; (y := z) is_terminating_wrt g,{ s where s is Element of Funcs X,INT : ( s . x > s . y & s . y >= 0 ) } ) by AOFA_000:104, AOFA_000:107;

03: now

let s be Element of Funcs X,INT ; :: thesis: ( s in { s where s is Element of Funcs X,INT : ( s . x > s . y & s . y >= 0 ) } & g . (g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))),(y gt 0 ) in (Funcs X,INT ) \ b,0 implies g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) in { s where s is Element of Funcs X,INT : ( s . x > s . y & s . y >= 0 ) } )
assume s in { s where s is Element of Funcs X,INT : ( s . x > s . y & s . y >= 0 ) } ; :: thesis: ( g . (g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))),(y gt 0 ) in (Funcs X,INT ) \ b,0 implies g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) in { s where s is Element of Funcs X,INT : ( s . x > s . y & s . y >= 0 ) } )
then ex s' being Element of Funcs X,INT st
( s' = s & s' . x > s' . y & s' . y >= 0 ) ;
then reconsider n = s . x, m = s . y as Element of NAT by INT_1:16;
assume g . (g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))),(y gt 0 ) in (Funcs X,INT ) \ b,0 ; :: thesis: g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) in { s where s is Element of Funcs X,INT : ( s . x > s . y & s . y >= 0 ) }
then A2: ( (g . (g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))),(y gt 0 )) . b <> 0 & (g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) . y = (s . x) mod (s . y) & (g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) . x = s . y ) by 00, LemTS, LemEAterm;
then A1: (g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) . y > 0 by Th015;
then m <> 0 by A2, INT_1:def 8;
then m > 0 ;
then (g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) . x > (g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z))) . y by A2, NEWTON:79;
hence g . s,((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) in { s where s is Element of Funcs X,INT : ( s . x > s . y & s . y >= 0 ) } by A1; :: thesis: verum

end;

now

let s be Element of Funcs X,INT ; :: thesis: ( g . s,(y gt 0 ) in { s where s is Element of Funcs X,INT : ( s . x > s . y & s . y >= 0 ) } implies g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 ),g . s,(y gt 0 ) )
set s1 = g . s,(y gt 0 );
assume g . s,(y gt 0 ) in { s where s is Element of Funcs X,INT : ( s . x > s . y & s . y >= 0 ) } ; :: thesis: g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 ),g . s,(y gt 0 )
then A0: ex s' being Element of Funcs X,INT st
( s' = g . s,(y gt 0 ) & s' . x > s' . y & s' . y >= 0 ) ;
then reconsider n = (g . s,(y gt 0 )) . x, m = (g . s,(y gt 0 )) . y as Element of NAT by INT_1:16;
( (g . s,(y gt 0 )) . x = s . x & (g . s,(y gt 0 )) . y = s . y & ( s . y > 0 implies (g . s,(y gt 0 )) . b = 1 ) & ( s . y <= 0 implies (g . s,(y gt 0 )) . b = 0 ) ) by 00, Th015;
then ( n > m & ( g . s,(y gt 0 ) in (Funcs X,INT ) \ b,0 implies m > 0 ) & ( m > 0 implies g . s,(y gt 0 ) in (Funcs X,INT ) \ b,0 ) ) by A0, LemTS;
hence g iteration_terminates_for ((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) \; (y gt 0 ),g . s,(y gt 0 ) by 00, LemEAterm; :: thesis: verum

end;

hence while (y gt 0 ),((((z := x) \; (z %= y)) \; (x := y)) \; (y := z)) is_terminating_wrt g,{ s where s is Element of Funcs X,INT : ( s . x > s . y & s . y >= 0 ) } by 01, 02, 03, AOFA_000:118; :: thesis: verum