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, AOFA_000:def 39;
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, AOFA_000:def 39;
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 by AOFA_000:def 39;
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
set P = for-do (i := t),(i leq n),(i += 1),I;