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) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y = |.((s . x) - (s . y)).| & ( for n, m being Element of NAT st n = s . x & m = s . y & ( s in (Funcs (X,INT)) \ (b,0) implies m > 0 ) & ( m > 0 implies s in (Funcs (X,INT)) \ (b,0) ) holds
g iteration_terminates_for ((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (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) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y = |.((s . x) - (s . y)).| & ( for n, m being Element of NAT st n = s . x & m = s . y & ( s in (Funcs (X,INT)) \ (b,0) implies m > 0 ) & ( m > 0 implies s in (Funcs (X,INT)) \ (b,0) ) holds
g iteration_terminates_for ((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (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) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y = |.((s . x) - (s . y)).| & ( for n, m being Element of NAT st n = s . x & m = s . y & ( s in (Funcs (X,INT)) \ (b,0) implies m > 0 ) & ( m > 0 implies s in (Funcs (X,INT)) \ (b,0) ) holds
g iteration_terminates_for ((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (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) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y = |.((s . x) - (s . y)).| & ( for n, m being Element of NAT st n = s . x & m = s . y & ( s in (Funcs (X,INT)) \ (b,0) implies m > 0 ) & ( m > 0 implies s in (Funcs (X,INT)) \ (b,0) ) holds
g iteration_terminates_for ((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0),s ) )
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) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y = |.((s . x) - (s . y)).| & ( for n, m being Element of NAT st n = s . x & m = s . y & ( s in (Funcs (X,INT)) \ (b,0) implies m > 0 ) & ( m > 0 implies s in (Funcs (X,INT)) \ (b,0) ) holds
g iteration_terminates_for ((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (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) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y = |.((s . x) - (s . y)).| & ( for n, m being Element of NAT st n = s . x & m = s . y & ( s in (Funcs (X,INT)) \ (b,0) implies m > 0 ) & ( m > 0 implies s in (Funcs (X,INT)) \ (b,0) ) holds
g iteration_terminates_for ((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0),s ) )
A5: y <> z by A3, A4;
let s be Element of Funcs (X,INT); :: thesis: ( (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y = |.((s . x) - (s . y)).| & ( for n, m being Element of NAT st n = s . x & m = s . y & ( s in (Funcs (X,INT)) \ (b,0) implies m > 0 ) & ( m > 0 implies s in (Funcs (X,INT)) \ (b,0) ) holds
g iteration_terminates_for ((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0),s ) )
set J = if-then ((z lt 0),(z *= (- 1)));
A6: g complies_with_if_wrt (Funcs (X,INT)) \ (b,0) by AOFA_000:def 32;
A7: z <> x by A2, A4;
set I = (((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z);
A8: x <> y by A2, A3;

A9: now :: thesis: for s being Element of Funcs (X,INT) holds
( (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y = |.((s . x) - (s . y)).| )

let s be Element of Funcs (X,INT); :: thesis: ( (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y = |.((s . x) - (s . y)).| )
set s1 = g . (s,(z := ((. x) - (. y))));
set s2 = g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0));
set q = g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))));
set qz = g . ((g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))),(z *= (- 1)));
A10: ((g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . z) * (- 1) = - ((g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . z) ;
set s3 = g . ((g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))))),(x := y));
set s4 = g . ((g . ((g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))))),(x := y))),(y := z));
A11: g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z))) = g . ((g . (s,(((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)))),(y := z)) by AOFA_000:def 29
.= g . ((g . ((g . (s,((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))))),(x := y))),(y := z)) by AOFA_000:def 29
.= g . ((g . ((g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))))),(x := y))),(y := z)) by AOFA_000:def 29 ;
( (g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . b = 1 implies g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0)) in (Funcs (X,INT)) \ (b,0) ) ;
then A12: ( (g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . b = 1 implies g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1))))) = g . ((g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))),(z *= (- 1))) ) by A6;
A13: (. x) . s = s . x by Th22;
A14: (g . ((g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))),(z *= (- 1)))) . y = (g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . y by A5, Th31;
A15: (. y) . s = s . y by Th22;
((. x) - (. y)) . s = ((. x) . s) - ((. y) . s) by Def11;
then A16: (g . (s,(z := ((. x) - (. y))))) . z = (s . x) - (s . y) by A13, A15, Th26;
A17: ( (g . (s,(z := ((. x) - (. y))))) . z < 0 implies (g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . b = 1 ) by Th38;
A18: (g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . z = (g . (s,(z := ((. x) - (. y))))) . z by A1, A4, Th38;
A19: (g . ((g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))),(z *= (- 1)))) . z = ((g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . z) * (- 1) by Th31;
A20: (g . ((g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))))),(x := y))) . z = (g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))))) . z by A7, Th27;
A21: (g . ((g . ((g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))))),(x := y))),(y := z))) . y = (g . ((g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))))),(x := y))) . z by Th27;
( (g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . b = 0 implies g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0)) nin (Funcs (X,INT)) \ (b,0) ) by Th2;
then A22: ( (g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . b = 0 implies g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1))))) = g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0)) ) by A6, AOFA_000:80;
A23: ( (g . (s,(z := ((. x) - (. y))))) . z >= 0 implies (g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . b = 0 ) by Th38;
A24: (g . (s,(z := ((. x) - (. y))))) . y = s . y by A5, Th26;
A25: (g . ((g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))))),(x := y))) . x = (g . ((g . (s,(z := ((. x) - (. y))))),(if-then ((z lt 0),(z *= (- 1)))))) . y by Th27;
(g . ((g . (s,(z := ((. x) - (. y))))),(z lt 0))) . y = (g . (s,(z := ((. x) - (. y))))) . y by A1, A3, Th38;
hence ( (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y = |.((s . x) - (s . y)).| ) by A8, A22, A12, A18, A17, A23, A14, A19, A10, A11, A24, A16, A25, A20, A21, Th27, ABSVALUE:def 1; :: thesis: verum

end;

hence ( (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . x = s . y & (g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)))) . y = |.((s . x) - (s . y)).| ) ; :: thesis: for n, m being Element of NAT st n = s . x & m = s . y & ( s in (Funcs (X,INT)) \ (b,0) implies m > 0 ) & ( m > 0 implies s in (Funcs (X,INT)) \ (b,0) ) holds
g iteration_terminates_for ((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0),s
deffunc H₁( Element of Funcs (X,INT)) -> Element of NAT = IFEQ (($1 . y),0,0,(IFEQ (($1 . x),0,2,(IFEQ (($1 . x),($1 . y),1,(In ((max ((2 * ($1 . x)),((2 * ($1 . y)) + 1))),NAT)))))));
defpred S₁[ Element of Funcs (X,INT)] means ( $1 . x >= 0 & $1 . y > 0 );
set C = y gt 0;
A26: for s being Element of Funcs (X,INT) st S₁[s] holds
( ( S₁[g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))] implies g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0))) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0))) in (Funcs (X,INT)) \ (b,0) implies S₁[g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))] ) & H₁(g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (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) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))] implies g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0))) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0))) in (Funcs (X,INT)) \ (b,0) implies S₁[g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))] ) & H₁(g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))) < H₁(s) ) )
assume that
A27: s . x >= 0 and
A28: s . y > 0 ; :: thesis: ( ( S₁[g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))] implies g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0))) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0))) in (Funcs (X,INT)) \ (b,0) implies S₁[g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))] ) & H₁(g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))) < H₁(s) )
reconsider s9 = g . (s,((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z))) as Element of Funcs (X,INT) ;
reconsider s99 = g . (s9,(y gt 0)) as Element of Funcs (X,INT) ;
A29: s9 . y = |.((s . x) - (s . y)).| by A9;
then reconsider nx = s . x, ny = s . y, nn = s99 . y as Element of NAT by A1, A3, A27, A28, Th38, INT_1:3;
A30: g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0))) = s99 by AOFA_000:def 29;
A31: s99 . x = s9 . x by A1, A2, Th38;
A32: ( s9 . y <= 0 implies s99 . b = 0 ) by Th38;
A33: ( s9 . y > 0 implies s99 . b = 1 ) by Th38;
hence ( S₁[g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))] iff g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0))) in (Funcs (X,INT)) \ (b,0) ) by A9, A28, A30, A31, A32, Th2, Th38; :: thesis: H₁(g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))) < H₁(s)
A34: s9 . x = s . y by A9;
A35: H₁(s99) = IFEQ (nn,0,0,(IFEQ (ny,0,2,(IFEQ (ny,nn,1,(max ((2 * ny),((2 * nn) + 1)))))))) by A31, A34;
(2 * ny) + 1 > 2 * ny by NAT_1:13;
then A36: max ((2 * nx),((2 * ny) + 1)) > 2 * ny by XXREAL_0:30;
A37: s99 . y = s9 . y by A1, A3, Th38;
A38: H₁(s) = IFEQ (ny,0,0,(IFEQ (nx,0,2,(IFEQ (nx,ny,1,(max ((2 * nx),((2 * ny) + 1))))))))
.= IFEQ (nx,0,2,(IFEQ (nx,ny,1,(max ((2 * nx),((2 * ny) + 1)))))) by A28, FUNCOP_1:def 8 ;

per cases ( nx = ny or nx > ny or ( nx < ny & nx > 0 ) or nx = 0 ) by XXREAL_0:1;

suppose A39: nx = ny ; :: thesis: H₁(g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))) < H₁(s)

then A40: IFEQ (nx,ny,1,(max ((2 * nx),((2 * ny) + 1)))) = 1 by FUNCOP_1:def 8;
A41: nn = 0 by A37, A29, A39, ABSVALUE:2;
H₁(s) = IFEQ (nx,ny,1,(max ((2 * nx),((2 * ny) + 1)))) by A28, A38, A39, FUNCOP_1:def 8;
hence H₁(g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))) < H₁(s) by A30, A40, A41, FUNCOP_1:def 8; :: thesis: verum

end;

suppose A42: nx > ny ; :: thesis: H₁(g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))) < H₁(s)

then IFEQ (nx,ny,1,(max ((2 * nx),((2 * ny) + 1)))) = max ((2 * nx),((2 * ny) + 1)) by FUNCOP_1:def 8;
then A43: H₁(s) = max ((2 * nx),((2 * ny) + 1)) by A38, A42, FUNCOP_1:def 8;
A44: nx - ny > 0 by A42, XREAL_1:50;
then A45: H₁(s99) = IFEQ (ny,0,2,(IFEQ (ny,nn,1,(max ((2 * ny),((2 * nn) + 1)))))) by A37, A33, A32, A29, A35, ABSVALUE:def 1, FUNCOP_1:def 8
.= IFEQ (ny,nn,1,(max ((2 * ny),((2 * nn) + 1)))) by A28, FUNCOP_1:def 8 ;
then A46: ( ny = nn implies H₁(s99) = 1 ) by FUNCOP_1:def 8;
nn = nx - ny by A37, A29, A44, ABSVALUE:def 1;
then nn < nx by A28, XREAL_1:44;
then nn + 1 <= nx by NAT_1:13;
then A47: 2 * (nn + 1) <= 2 * nx by XREAL_1:64;
A48: 1 <= (2 * nn) + 1 by NAT_1:11;
A49: ( ny <> nn implies H₁(s99) = max ((2 * ny),((2 * nn) + 1)) ) by A45, FUNCOP_1:def 8;
(2 * nn) + 2 = ((2 * nn) + 1) + 1 ;
then (2 * nn) + 1 < 2 * nx by A47, NAT_1:13;
then (2 * nn) + 1 < H₁(s) by A43, XXREAL_0:30;
hence H₁(g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))) < H₁(s) by A30, A36, A43, A46, A49, A48, XXREAL_0:2, XXREAL_0:29; :: thesis: verum

end;

suppose A50: ( nx < ny & nx > 0 ) ; :: thesis: H₁(g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))) < H₁(s)

A51: - (nx - ny) = ny - nx ;
A52: nx - ny < 0 by A50, XREAL_1:49;
then A53: nn = - (nx - ny) by A37, A29, ABSVALUE:def 1;
then A54: nn < ny by A50, A51, XREAL_1:44;
2 * nn < 2 * ny by A50, A53, A51, XREAL_1:44, XREAL_1:68;
then (2 * nn) + 1 < (2 * ny) + 1 by XREAL_1:6;
then A55: (2 * nn) + 1 < max ((2 * nx),((2 * ny) + 1)) by XXREAL_0:30;
nn > 0 by A37, A29, A52, A51, ABSVALUE:def 1;
then A56: H₁(s99) = IFEQ (ny,0,2,(IFEQ (ny,nn,1,(max ((2 * ny),((2 * nn) + 1)))))) by A35, FUNCOP_1:def 8
.= IFEQ (ny,nn,1,(max ((2 * ny),((2 * nn) + 1)))) by A28, FUNCOP_1:def 8
.= max ((2 * ny),((2 * nn) + 1)) by A54, FUNCOP_1:def 8 ;
H₁(s) = IFEQ (nx,ny,1,(max ((2 * nx),((2 * ny) + 1)))) by A38, A50, FUNCOP_1:def 8
.= max ((2 * nx),((2 * ny) + 1)) by A50, FUNCOP_1:def 8 ;
hence H₁(g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))) < H₁(s) by A30, A36, A55, A56, XXREAL_0:29; :: thesis: verum

end;

suppose A57: nx = 0 ; :: thesis: H₁(g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))) < H₁(s)

then A58: H₁(s) = 2 by A38, FUNCOP_1:def 8;
A59: nn = - (nx - ny) by A28, A37, A29, A57, ABSVALUE:def 1
.= ny by A57 ;
then H₁(s99) = IFEQ (ny,0,2,(IFEQ (ny,nn,1,(max ((2 * ny),((2 * nn) + 1)))))) by A28, A35, FUNCOP_1:def 8
.= IFEQ (ny,nn,1,(max ((2 * ny),((2 * nn) + 1)))) by A28, FUNCOP_1:def 8
.= 1 by A59, FUNCOP_1:def 8 ;
hence H₁(g . (s,(((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0)))) < H₁(s) by A30, A58; :: thesis: verum

end;

end;

end;

let n, m be Element of NAT ; :: thesis: ( n = s . x & m = s . y & ( s in (Funcs (X,INT)) \ (b,0) implies m > 0 ) & ( m > 0 implies s in (Funcs (X,INT)) \ (b,0) ) implies g iteration_terminates_for ((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0),s )
assume that
A60: n = s . x and
A61: 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) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (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) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0),s
then A62: ( s in (Funcs (X,INT)) \ (b,0) iff S₁[s] ) by A60, A61;
g iteration_terminates_for ((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0),s from AOFA_000:sch 3(A62, A26);
hence g iteration_terminates_for ((((z := ((. x) - (. y))) \; (if-then ((z lt 0),(z *= (- 1))))) \; (x := y)) \; (y := z)) \; (y gt 0),s ; :: thesis: verum