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 I being Element of A
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) holds
g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n))
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 I being Element of A
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) holds
g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n))
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 I being Element of A
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) holds
g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n))
let b be Element of X; :: thesis: for g being Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0)
for I being Element of A
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) holds
g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n))
let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); :: thesis: for I being Element of A
for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) holds
g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n))
let I be Element of A; :: thesis: for i, n being Variable of g st ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) holds
g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n))
let i, n be Variable of g; :: thesis: ( ex d being Function st
( d . b = 0 & d . n = 1 & d . i = 2 ) & ( for s being Element of Funcs (X,INT) holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) implies g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n)) )
given d being Function such that A1: d . b = 0 and
A2: d . n = 1 and
A3: d . i = 2 ; :: thesis: ( ex s being Element of Funcs (X,INT) st
( (g . (s,I)) . n = s . n implies not (g . (s,I)) . i = s . i ) or g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n)) )
set J = i += 1;
set C = i leq n;
set S = Funcs (X,INT);
set h = g;
assume A4: for s being Element of Funcs (X,INT) holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ; :: thesis: g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n))
deffunc H₁( Element of Funcs (X,INT)) -> Element of NAT = In (((($1 . n) + 1) - ($1 . i)),NAT);
defpred S₁[ Element of Funcs (X,INT)] means $1 . i <= $1 . n;
set T = (Funcs (X,INT)) \ (b,0);
A5: i <> n by A2, A3;
A6: for s being Element of Funcs (X,INT) st S₁[s] holds
( ( S₁[g . (s,((I \; (i += 1)) \; (i leq n)))] implies g . (s,((I \; (i += 1)) \; (i leq n))) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,((I \; (i += 1)) \; (i leq n))) in (Funcs (X,INT)) \ (b,0) implies S₁[g . (s,((I \; (i += 1)) \; (i leq n)))] ) & H₁(g . (s,((I \; (i += 1)) \; (i leq n)))) < H₁(s) )

proof

let s be Element of Funcs (X,INT); :: thesis: ( S₁[s] implies ( ( S₁[g . (s,((I \; (i += 1)) \; (i leq n)))] implies g . (s,((I \; (i += 1)) \; (i leq n))) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,((I \; (i += 1)) \; (i leq n))) in (Funcs (X,INT)) \ (b,0) implies S₁[g . (s,((I \; (i += 1)) \; (i leq n)))] ) & H₁(g . (s,((I \; (i += 1)) \; (i leq n)))) < H₁(s) ) )
set s1 = g . (s,I);
set q = g . (s,(I \; (i += 1)));
set q1 = g . ((g . (s,(I \; (i += 1)))),(i leq n));
A7: g . (s,(I \; (i += 1))) = g . ((g . (s,I)),(i += 1)) by AOFA_000:def 29;
(g . (s,I)) . i = s . i by A4;
then (g . (s,(I \; (i += 1)))) . i = (s . i) + 1 by A7, Th28;
then A8: (g . ((g . (s,(I \; (i += 1)))),(i leq n))) . i = (s . i) + 1 by A1, A3, Th35;
A9: ( (g . (s,(I \; (i += 1)))) . i > (g . (s,(I \; (i += 1)))) . n implies (g . ((g . (s,(I \; (i += 1)))),(i leq n))) . b = 0 ) by Th35;
assume S₁[s] ; :: thesis: ( ( S₁[g . (s,((I \; (i += 1)) \; (i leq n)))] implies g . (s,((I \; (i += 1)) \; (i leq n))) in (Funcs (X,INT)) \ (b,0) ) & ( g . (s,((I \; (i += 1)) \; (i leq n))) in (Funcs (X,INT)) \ (b,0) implies S₁[g . (s,((I \; (i += 1)) \; (i leq n)))] ) & H₁(g . (s,((I \; (i += 1)) \; (i leq n)))) < H₁(s) )
then reconsider ni = (s . n) - (s . i) as Element of NAT by INT_1:3, XREAL_1:48;
A10: g . ((g . (s,(I \; (i += 1)))),(i leq n)) = g . (s,((I \; (i += 1)) \; (i leq n))) by AOFA_000:def 29;
A11: ( (g . (s,(I \; (i += 1)))) . i <= (g . (s,(I \; (i += 1)))) . n implies (g . ((g . (s,(I \; (i += 1)))),(i leq n))) . b = 1 ) by Th35;
(g . ((g . (s,(I \; (i += 1)))),(i leq n))) . i = (g . (s,(I \; (i += 1)))) . i by A1, A3, Th35;
hence ( S₁[g . (s,((I \; (i += 1)) \; (i leq n)))] iff g . (s,((I \; (i += 1)) \; (i leq n))) in (Funcs (X,INT)) \ (b,0) ) by A1, A2, A9, A11, A10, Th2, Th35; :: thesis: H₁(g . (s,((I \; (i += 1)) \; (i leq n)))) < H₁(s)
A12: H₁(s) = ni + 1 ;
(g . (s,I)) . n = s . n by A4;
then (g . (s,(I \; (i += 1)))) . n = s . n by A5, A7, Th28;
then (g . ((g . (s,(I \; (i += 1)))),(i leq n))) . n = s . n by A1, A2, Th35;
then H₁(g . ((g . (s,(I \; (i += 1)))),(i leq n))) = ni by A8;
hence H₁(g . (s,((I \; (i += 1)) \; (i leq n)))) < H₁(s) by A10, A12, NAT_1:13; :: thesis: verum

end;

reconsider s1 = g . (s,(i leq n)) as Element of Funcs (X,INT) ;
A13: ( s . i > s . n implies s1 . b = 0 ) by Th35;
A14: ( s . i <= s . n implies s1 . b = 1 ) by Th35;
s1 . i = s . i by A1, A3, Th35;
then A15: ( s1 in (Funcs (X,INT)) \ (b,0) iff S₁[s1] ) by A1, A2, A13, A14, Th2, Th35;
g iteration_terminates_for (I \; (i += 1)) \; (i leq n),s1 from AOFA_000:sch 3(A15, A6);
hence g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n)) ; :: thesis: verum