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 P being set
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) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i & g . (s,I) in P & g . (s,(i leq n)) in P & g . (s,(i += 1)) in P ) ) & s in P 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 P being set
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) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i & g . (s,I) in P & g . (s,(i leq n)) in P & g . (s,(i += 1)) in P ) ) & s in P 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 P being set
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) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i & g . (s,I) in P & g . (s,(i leq n)) in P & g . (s,(i += 1)) in P ) ) & s in P 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 P being set
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) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i & g . (s,I) in P & g . (s,(i leq n)) in P & g . (s,(i += 1)) in P ) ) & s in P 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 P being set
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) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i & g . (s,I) in P & g . (s,(i leq n)) in P & g . (s,(i += 1)) in P ) ) & s in P holds
g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n))
let P be set ; :: 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) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i & g . (s,I) in P & g . (s,(i leq n)) in P & g . (s,(i += 1)) in P ) ) & s in P 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) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i & g . (s,I) in P & g . (s,(i leq n)) in P & g . (s,(i += 1)) in P ) ) & s in P 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) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i & g . (s,I) in P & g . (s,(i leq n)) in P & g . (s,(i += 1)) in P ) ) & s in P 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
( s in P & not ( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i & g . (s,I) in P & g . (s,(i leq n)) in P & g . (s,(i += 1)) in P ) ) or not s in P 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 that
A4: for s being Element of Funcs (X,INT) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i & g . (s,I) in P & g . (s,(i leq n)) in P & g . (s,(i += 1)) in P ) and
A5: s in P ; :: thesis: g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n))
defpred S1[ Element of Funcs (X,INT)] means $1 in P;
reconsider s1 = g . (s,(i leq n)) as Element of Funcs (X,INT) ;
A6: S1[s1] by A4, A5;
deffunc H1( Element of Funcs (X,INT)) -> Element of NAT = In (((($1 . n) + 1) - ($1 . i)),NAT);
defpred S2[ Element of Funcs (X,INT)] means $1 . i <= $1 . n;
set T = (Funcs (X,INT)) \ (b,0);
A7: i <> n by A2, A3;
A8: for s being Element of Funcs (X,INT) st S1[s] & s in (Funcs (X,INT)) \ (b,0) & S2[s] holds
( S1[g . (s,((I \; (i += 1)) \; (i leq n)))] & ( S2[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 S2[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] & s in (Funcs (X,INT)) \ (b,0) & S2[s] implies ( S1[g . (s,((I \; (i += 1)) \; (i leq n)))] & ( S2[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 S2[g . (s,((I \; (i += 1)) \; (i leq n)))] ) & H1(g . (s,((I \; (i += 1)) \; (i leq n)))) < H1(s) ) )
assume that
A9: S1[s] and
s in (Funcs (X,INT)) \ (b,0) and
A10: S2[s] ; :: thesis: ( S1[g . (s,((I \; (i += 1)) \; (i leq n)))] & ( S2[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 S2[g . (s,((I \; (i += 1)) \; (i leq n)))] ) & H1(g . (s,((I \; (i += 1)) \; (i leq n)))) < H1(s) )
set q = g . (s,(I \; (i += 1)));
set s1 = g . (s,I);
set q1 = g . ((g . (s,(I \; (i += 1)))),(i leq n));
A11: g . (s,(I \; (i += 1))) = g . ((g . (s,I)),(i += 1)) by AOFA_000:def 29;
(g . (s,I)) . i = s . i by A4, A9;
then (g . (s,(I \; (i += 1)))) . i = (s . i) + 1 by A11, Th28;
then A12: (g . ((g . (s,(I \; (i += 1)))),(i leq n))) . i = (s . i) + 1 by A1, A3, Th35;
reconsider ni = (s . n) - (s . i) as Element of NAT by A10, INT_1:3, XREAL_1:48;
A13: H1(s) = ni + 1 ;
A14: ( (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;
A15: g . ((g . (s,(I \; (i += 1)))),(i leq n)) = g . (s,((I \; (i += 1)) \; (i leq n))) by AOFA_000:def 29;
S1[g . (s,I)] by A4, A9;
then S1[g . (s,(I \; (i += 1)))] by A4, A11;
hence S1[g . (s,((I \; (i += 1)) \; (i leq n)))] by A4, A15; :: thesis: ( ( S2[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 S2[g . (s,((I \; (i += 1)) \; (i leq n)))] ) & H1(g . (s,((I \; (i += 1)) \; (i leq n)))) < H1(s) )
A16: ( (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;
(g . ((g . (s,(I \; (i += 1)))),(i leq n))) . i = (g . (s,(I \; (i += 1)))) . i by A1, A3, Th35;
hence ( S2[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, A16, A14, A15, Th2, Th35; :: thesis: H1(g . (s,((I \; (i += 1)) \; (i leq n)))) < H1(s)
(g . (s,I)) . n = s . n by A4, A9;
then (g . (s,(I \; (i += 1)))) . n = s . n by A7, A11, 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 A12;
hence H1(g . (s,((I \; (i += 1)) \; (i leq n)))) < H1(s) by A15, A13, NAT_1:13; :: thesis: verum
end;
A17: ( s . i > s . n implies s1 . b = 0 ) by Th35;
A18: ( s . i <= s . n implies s1 . b = 1 ) by Th35;
s1 . i = s . i by A1, A3, Th35;
then A19: ( s1 in (Funcs (X,INT)) \ (b,0) iff S2[s1] ) by A1, A2, A17, A18, Th2, Th35;
g iteration_terminates_for (I \; (i += 1)) \; (i leq n),s1 from AOFA_000:sch 4(A6, A19, A8);
 hence g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . (s,(i leq n)) ; :: thesis: verum