let A be Euclidean preIfWhileAlgebra; :: thesis: for X being non empty countable set

for T being Subset of (Funcs (X,INT))

for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T

for t being INT-Expression of A,f

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 st I is_terminating_wrt g holds

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

for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g

let X be non empty countable set ; :: thesis: for T being Subset of (Funcs (X,INT))

for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T

for t being INT-Expression of A,f

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 st I is_terminating_wrt g holds

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

for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g

let T be Subset of (Funcs (X,INT)); :: thesis: for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T

for t being INT-Expression of A,f

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 st I is_terminating_wrt g holds

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

for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g

let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T; :: thesis: for t being INT-Expression of A,f

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 st I is_terminating_wrt g holds

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

for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g

let t be INT-Expression of A,f; :: 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 st I is_terminating_wrt g holds

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

for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g

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 st I is_terminating_wrt g holds

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

for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g

let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); :: thesis: for I being Element of A st I is_terminating_wrt g holds

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

for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g

set S = Funcs (X,INT);

set T = (Funcs (X,INT)) \ (b,0);

let I be Element of A; :: thesis: ( I is_terminating_wrt g implies 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

for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g )

assume A1: I is_terminating_wrt g ; :: 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

for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g

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 for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g )

i += 1 is_terminating_wrt g by AOFA_000:104;

then A2: I \; (i += 1) is_terminating_wrt g by A1, AOFA_000:110;

set Q = while ((i leq n),(I \; (i += 1)));

given d being Function such that A3: d . b = 0 and

A4: d . n = 1 and

A5: 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 for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g )

assume A6: for s being Element of Funcs (X,INT) holds

( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ; :: thesis: for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g

let s be Element of Funcs (X,INT); :: according to AOFA_000:def 37 :: thesis: [s,(for-do ((i := t),(i leq n),(i += 1),I))] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g)

A7: [s,(i := t)] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g) by AOFA_000:def 36;

A8: i leq n is_terminating_wrt g by AOFA_000:104;

g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . ((g . (s,(i := t))),(i leq n)) by A3, A4, A5, A6, Th52;

then [(g . (s,(i := t))),(while ((i leq n),(I \; (i += 1))))] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g) by A8, A2, AOFA_000:114;

hence [s,(for-do ((i := t),(i leq n),(i += 1),I))] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g) by A7, AOFA_000:def 35; :: thesis: verum

for T being Subset of (Funcs (X,INT))

for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T

for t being INT-Expression of A,f

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 st I is_terminating_wrt g holds

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

for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g

let X be non empty countable set ; :: thesis: for T being Subset of (Funcs (X,INT))

for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T

for t being INT-Expression of A,f

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 st I is_terminating_wrt g holds

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

for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g

let T be Subset of (Funcs (X,INT)); :: thesis: for f being Euclidean ExecutionFunction of A, Funcs (X,INT),T

for t being INT-Expression of A,f

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 st I is_terminating_wrt g holds

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

for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g

let f be Euclidean ExecutionFunction of A, Funcs (X,INT),T; :: thesis: for t being INT-Expression of A,f

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 st I is_terminating_wrt g holds

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

for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g

let t be INT-Expression of A,f; :: 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 st I is_terminating_wrt g holds

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

for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g

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 st I is_terminating_wrt g holds

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

for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g

let g be Euclidean ExecutionFunction of A, Funcs (X,INT),(Funcs (X,INT)) \ (b,0); :: thesis: for I being Element of A st I is_terminating_wrt g holds

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

for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g

set S = Funcs (X,INT);

set T = (Funcs (X,INT)) \ (b,0);

let I be Element of A; :: thesis: ( I is_terminating_wrt g implies 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

for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g )

assume A1: I is_terminating_wrt g ; :: 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

for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g

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 for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g )

i += 1 is_terminating_wrt g by AOFA_000:104;

then A2: I \; (i += 1) is_terminating_wrt g by A1, AOFA_000:110;

set Q = while ((i leq n),(I \; (i += 1)));

given d being Function such that A3: d . b = 0 and

A4: d . n = 1 and

A5: 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 for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g )

assume A6: for s being Element of Funcs (X,INT) holds

( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ; :: thesis: for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g

let s be Element of Funcs (X,INT); :: according to AOFA_000:def 37 :: thesis: [s,(for-do ((i := t),(i leq n),(i += 1),I))] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g)

A7: [s,(i := t)] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g) by AOFA_000:def 36;

A8: i leq n is_terminating_wrt g by AOFA_000:104;

g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . ((g . (s,(i := t))),(i leq n)) by A3, A4, A5, A6, Th52;

then [(g . (s,(i := t))),(while ((i leq n),(I \; (i += 1))))] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g) by A8, A2, AOFA_000:114;

hence [s,(for-do ((i := t),(i leq n),(i += 1),I))] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g) by A7, AOFA_000:def 35; :: thesis: verum