let i, j be Nat; :: thesis: ( i <= j implies for N being non empty with_non-empty_elements set
for S being non empty halting IC-Ins-separated AMI-Struct of NAT ,N
for f being finite PartFunc of NAT ,the Instructions of S
for s being State of S st CurInstr f,(Comput f,s,i) = halt S holds
Comput f,s,j = Comput f,s,i )
assume
i <= j
; :: thesis: for N being non empty with_non-empty_elements set
for S being non empty halting IC-Ins-separated AMI-Struct of NAT ,N
for f being finite PartFunc of NAT ,the Instructions of S
for s being State of S st CurInstr f,(Comput f,s,i) = halt S holds
Comput f,s,j = Comput f,s,i
then consider k being Nat such that
A1:
j = i + k
by NAT_1:10;
reconsider k = k as Element of NAT by ORDINAL1:def 13;
A2:
j = i + k
by A1;
let N be non empty with_non-empty_elements set ; :: thesis: for S being non empty halting IC-Ins-separated AMI-Struct of NAT ,N
for f being finite PartFunc of NAT ,the Instructions of S
for s being State of S st CurInstr f,(Comput f,s,i) = halt S holds
Comput f,s,j = Comput f,s,i
let S be non empty halting IC-Ins-separated AMI-Struct of NAT ,N; :: thesis: for f being finite PartFunc of NAT ,the Instructions of S
for s being State of S st CurInstr f,(Comput f,s,i) = halt S holds
Comput f,s,j = Comput f,s,i
let f be finite PartFunc of NAT ,the Instructions of S; :: thesis: for s being State of S st CurInstr f,(Comput f,s,i) = halt S holds
Comput f,s,j = Comput f,s,i
let s be State of S; :: thesis: ( CurInstr f,(Comput f,s,i) = halt S implies Comput f,s,j = Comput f,s,i )
assume A3:
CurInstr f,(Comput f,s,i) = halt S
; :: thesis: Comput f,s,j = Comput f,s,i
defpred S1[ Element of NAT ] means Comput f,s,(i + $1) = Comput f,s,i;
A4:
S1[ 0 ]
;
A5:
now let k be
Element of
NAT ;
:: thesis: ( S1[k] implies S1[k + 1] )assume A6:
S1[
k]
;
:: thesis: S1[k + 1] Comput f,
s,
(i + (k + 1)) =
Comput f,
s,
((i + k) + 1)
.=
Following f,
(Comput f,s,(i + k))
by Tx1
.=
Comput f,
s,
i
by A3, A6, AMI_1:def 8
;
hence
S1[
k + 1]
;
:: thesis: verum end;
for k being Element of NAT holds S1[k]
from NAT_1:sch 1(A4, A5);
hence
Comput f,s,j = Comput f,s,i
by A2; :: thesis: verum