let A be Element of LTLB_WFF ; :: thesis: for F being Subset of LTLB_WFF st F |- A holds

F |= A

let F be Subset of LTLB_WFF; :: thesis: ( F |- A implies F |= A )

assume F |- A ; :: thesis: F |= A

then consider f being FinSequence of LTLB_WFF such that

A1: f . (len f) = A and

A2: 1 <= len f and

A3: for i being Nat st 1 <= i & i <= len f holds

prc f,F,i ;

defpred S_{1}[ Nat] means ( 1 <= $1 & $1 <= len f implies F |= f /. $1 );

A4: for i being Nat st ( for j being Nat st j < i holds

S_{1}[j] ) holds

S_{1}[i]
_{1}[i]
from NAT_1:sch 4(A4);

f /. (len f) = A by A1, A2, Lm1;

hence F |= A by A2, A22; :: thesis: verum

F |= A

let F be Subset of LTLB_WFF; :: thesis: ( F |- A implies F |= A )

assume F |- A ; :: thesis: F |= A

then consider f being FinSequence of LTLB_WFF such that

A1: f . (len f) = A and

A2: 1 <= len f and

A3: for i being Nat st 1 <= i & i <= len f holds

prc f,F,i ;

defpred S

A4: for i being Nat st ( for j being Nat st j < i holds

S

S

proof

A22:
for i being Nat holds S
let i be Nat; :: thesis: ( ( for j being Nat st j < i holds

S_{1}[j] ) implies S_{1}[i] )

assume A5: for j being Nat st j < i holds

S_{1}[j]
; :: thesis: S_{1}[i]

end;S

assume A5: for j being Nat st j < i holds

S

per cases
( i = 0 or not i < 1 )
by NAT_1:14;

end;

suppose
not i < 1
; :: thesis: S_{1}[i]

assume that

A6: 1 <= i and

A7: i <= len f ; :: thesis: F |= f /. i

end;A6: 1 <= i and

A7: i <= len f ; :: thesis: F |= f /. i

per cases
( f . i in LTL_axioms or f . i in F or ex j, k being Nat st

( 1 <= j & j < i & 1 <= k & k < i & ( f /. j,f /. k MP_rule f /. i or f /. j,f /. k IND_rule f /. i ) ) or ex j being Nat st

( 1 <= j & j < i & f /. j NEX_rule f /. i ) ) by A3, A6, A7, Def29;

end;

( 1 <= j & j < i & 1 <= k & k < i & ( f /. j,f /. k MP_rule f /. i or f /. j,f /. k IND_rule f /. i ) ) or ex j being Nat st

( 1 <= j & j < i & f /. j NEX_rule f /. i ) ) by A3, A6, A7, Def29;

suppose
f . i in LTL_axioms
; :: thesis: F |= f /. i

then A8:
f /. i in LTL_axioms
by A6, A7, Lm1;

end;suppose
f . i in F
; :: thesis: F |= f /. i

then A9:
f /. i in F
by A6, A7, Lm1;

thus F |= f /. i :: thesis: verum

end;thus F |= f /. i :: thesis: verum

proof

let M be LTLModel; :: according to LTLAXIO1:def 14 :: thesis: ( M |= F implies M |= f /. i )

assume M |= F ; :: thesis: M |= f /. i

hence M |= f /. i by A9; :: thesis: verum

end;assume M |= F ; :: thesis: M |= f /. i

hence M |= f /. i by A9; :: thesis: verum

suppose
ex j, k being Nat st

( 1 <= j & j < i & 1 <= k & k < i & ( f /. j,f /. k MP_rule f /. i or f /. j,f /. k IND_rule f /. i ) ) ; :: thesis: F |= f /. i

( 1 <= j & j < i & 1 <= k & k < i & ( f /. j,f /. k MP_rule f /. i or f /. j,f /. k IND_rule f /. i ) ) ; :: thesis: F |= f /. i

then consider j, k being Nat such that

A10: 1 <= j and

A11: j < i and

A12: 1 <= k and

A13: k < i and

A14: ( f /. j,f /. k MP_rule f /. i or f /. j,f /. k IND_rule f /. i ) ;

k <= len f by A7, A13, XXREAL_0:2;

then A15: F |= f /. k by A5, A12, A13;

A16: j <= len f by A7, A11, XXREAL_0:2;

end;A10: 1 <= j and

A11: j < i and

A12: 1 <= k and

A13: k < i and

A14: ( f /. j,f /. k MP_rule f /. i or f /. j,f /. k IND_rule f /. i ) ;

k <= len f by A7, A13, XXREAL_0:2;

then A15: F |= f /. k by A5, A12, A13;

A16: j <= len f by A7, A11, XXREAL_0:2;

suppose
ex j being Nat st

( 1 <= j & j < i & f /. j NEX_rule f /. i ) ; :: thesis: F |= f /. i

( 1 <= j & j < i & f /. j NEX_rule f /. i ) ; :: thesis: F |= f /. i

then consider j being Nat such that

A19: 1 <= j and

A20: j < i and

A21: f /. j NEX_rule f /. i ;

S_{1}[j]
by A5, A20;

then F |= 'X' (f /. j) by A7, A19, A20, Th25, XXREAL_0:2;

hence F |= f /. i by A21; :: thesis: verum

end;A19: 1 <= j and

A20: j < i and

A21: f /. j NEX_rule f /. i ;

S

then F |= 'X' (f /. j) by A7, A19, A20, Th25, XXREAL_0:2;

hence F |= f /. i by A21; :: thesis: verum

f /. (len f) = A by A1, A2, Lm1;

hence F |= A by A2, A22; :: thesis: verum