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 P being set
for I being Element of A st I is_terminating_wrt g,P 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) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) & P is_invariant_wrt i := t,g & P is_invariant_wrt I,g & P is_invariant_wrt i leq n,g & P is_invariant_wrt i += 1,g holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g,P
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 P being set
for I being Element of A st I is_terminating_wrt g,P 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) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) & P is_invariant_wrt i := t,g & P is_invariant_wrt I,g & P is_invariant_wrt i leq n,g & P is_invariant_wrt i += 1,g holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g,P
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 P being set
for I being Element of A st I is_terminating_wrt g,P 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) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) & P is_invariant_wrt i := t,g & P is_invariant_wrt I,g & P is_invariant_wrt i leq n,g & P is_invariant_wrt i += 1,g holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g,P
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 P being set
for I being Element of A st I is_terminating_wrt g,P 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) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) & P is_invariant_wrt i := t,g & P is_invariant_wrt I,g & P is_invariant_wrt i leq n,g & P is_invariant_wrt i += 1,g holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g,P
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 P being set
for I being Element of A st I is_terminating_wrt g,P 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) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) & P is_invariant_wrt i := t,g & P is_invariant_wrt I,g & P is_invariant_wrt i leq n,g & P is_invariant_wrt i += 1,g holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g,P
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 st I is_terminating_wrt g,P 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) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) & P is_invariant_wrt i := t,g & P is_invariant_wrt I,g & P is_invariant_wrt i leq n,g & P is_invariant_wrt i += 1,g holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g,P
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 st I is_terminating_wrt g,P 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) st s in P holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) & P is_invariant_wrt i := t,g & P is_invariant_wrt I,g & P is_invariant_wrt i leq n,g & P is_invariant_wrt i += 1,g holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g,P
set Z = Funcs (X,INT);
set T = (Funcs (X,INT)) \ (b,0);
let S be set ; :: thesis: for I being Element of A st I is_terminating_wrt g,S 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) st s in S holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) & S is_invariant_wrt i := t,g & S is_invariant_wrt I,g & S is_invariant_wrt i leq n,g & S is_invariant_wrt i += 1,g holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g,S
let I be Element of A; :: thesis: ( I is_terminating_wrt g,S 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) st s in S holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) & S is_invariant_wrt i := t,g & S is_invariant_wrt I,g & S is_invariant_wrt i leq n,g & S is_invariant_wrt i += 1,g holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g,S )
assume A1: I is_terminating_wrt g,S ; :: 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 S holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) & S is_invariant_wrt i := t,g & S is_invariant_wrt I,g & S is_invariant_wrt i leq n,g & S is_invariant_wrt i += 1,g holds
for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g,S
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 S holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) & S is_invariant_wrt i := t,g & S is_invariant_wrt I,g & S is_invariant_wrt i leq n,g & S is_invariant_wrt i += 1,g implies for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g,S )
given d being Function such that A2: d . b = 0 and
A3: d . n = 1 and
A4: d . i = 2 ; :: thesis: ( ex s being Element of Funcs (X,INT) st
( s in S & not ( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) ) or not S is_invariant_wrt i := t,g or not S is_invariant_wrt I,g or not S is_invariant_wrt i leq n,g or not S is_invariant_wrt i += 1,g or for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g,S )
set C = i leq n;
set J = I \; (i += 1);
assume that
A5: for s being Element of Funcs (X,INT) st s in S holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i ) and
A6: S is_invariant_wrt i := t,g and
A7: S is_invariant_wrt I,g and
A8: S is_invariant_wrt i leq n,g and
A9: S is_invariant_wrt i += 1,g ; :: thesis: for-do ((i := t),(i leq n),(i += 1),I) is_terminating_wrt g,S
let s be Element of Funcs (X,INT); :: according to AOFA_000:def 38 :: thesis: ( not s in S or [s,(for-do ((i := t),(i leq n),(i += 1),I))] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g) )
assume s in S ; :: thesis: [s,(for-do ((i := t),(i leq n),(i += 1),I))] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g)
then A10: g . (s,(i := t)) in S by A6;
for s being Element of Funcs (X,INT) st s in S holds
( (g . (s,I)) . n = s . n & (g . (s,I)) . i = s . i & g . (s,I) in S & g . (s,(i leq n)) in S & g . (s,(i += 1)) in S ) by A5, A7, A8, A9;
then A11: g iteration_terminates_for (I \; (i += 1)) \; (i leq n),g . ((g . (s,(i := t))),(i leq n)) by A2, A3, A4, A10, Th53;
set Q = while ((i leq n),(I \; (i += 1)));
A12: i leq n is_terminating_wrt g by AOFA_000:104;
A13: [s,(i := t)] in TerminatingPrograms (A,(Funcs (X,INT)),((Funcs (X,INT)) \ (b,0)),g) by AOFA_000:def 36;
S is_invariant_wrt I \; (i += 1),g by A7, A9, AOFA_000:109;
then A14: for s being Element of Funcs (X,INT) st s in S & g . ((g . (s,(I \; (i += 1)))),(i leq n)) in (Funcs (X,INT)) \ (b,0) holds
g . (s,(I \; (i += 1))) in S ;
i += 1 is_terminating_wrt g,S by AOFA_000:107;
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 A1, A7, A8, A10, A11, A12, A14, AOFA_000:111, AOFA_000:116;
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 A13, AOFA_000:def 35; :: thesis: verum