let i, j be Nat; :: thesis: ( i <= j implies for N being non empty with_zero set

for S being non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N

for p being NAT -defined the InstructionsF of b_{2} -valued Function

for s being State of S st CurInstr (p,(Comput (p,s,i))) = halt S holds

Comput (p,s,j) = Comput (p,s,i) )

assume i <= j ; :: thesis: for N being non empty with_zero set

for S being non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N

for p being NAT -defined the InstructionsF of b_{2} -valued Function

for s being State of S st CurInstr (p,(Comput (p,s,i))) = halt S holds

Comput (p,s,j) = Comput (p,s,i)

then consider k being Nat such that

A1: j = i + k by NAT_1:10;

reconsider k = k as Nat ;

A2: j = i + k by A1;

let N be non empty with_zero set ; :: thesis: for S being non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N

for p being NAT -defined the InstructionsF of b_{1} -valued Function

for s being State of S st CurInstr (p,(Comput (p,s,i))) = halt S holds

Comput (p,s,j) = Comput (p,s,i)

let S be non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N; :: thesis: for p being NAT -defined the InstructionsF of S -valued Function

for s being State of S st CurInstr (p,(Comput (p,s,i))) = halt S holds

Comput (p,s,j) = Comput (p,s,i)

let p be NAT -defined the InstructionsF of S -valued Function; :: thesis: for s being State of S st CurInstr (p,(Comput (p,s,i))) = halt S holds

Comput (p,s,j) = Comput (p,s,i)

let s be State of S; :: thesis: ( CurInstr (p,(Comput (p,s,i))) = halt S implies Comput (p,s,j) = Comput (p,s,i) )

assume A3: CurInstr (p,(Comput (p,s,i))) = halt S ; :: thesis: Comput (p,s,j) = Comput (p,s,i)

defpred S_{1}[ Nat] means Comput (p,s,(i + $1)) = Comput (p,s,i);

_{1}[ 0 ]
;

for k being Nat holds S_{1}[k]
from NAT_1:sch 2(A6, A4);

hence Comput (p,s,j) = Comput (p,s,i) by A2; :: thesis: verum

for S being non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N

for p being NAT -defined the InstructionsF of b

for s being State of S st CurInstr (p,(Comput (p,s,i))) = halt S holds

Comput (p,s,j) = Comput (p,s,i) )

assume i <= j ; :: thesis: for N being non empty with_zero set

for S being non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N

for p being NAT -defined the InstructionsF of b

for s being State of S st CurInstr (p,(Comput (p,s,i))) = halt S holds

Comput (p,s,j) = Comput (p,s,i)

then consider k being Nat such that

A1: j = i + k by NAT_1:10;

reconsider k = k as Nat ;

A2: j = i + k by A1;

let N be non empty with_zero set ; :: thesis: for S being non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N

for p being NAT -defined the InstructionsF of b

for s being State of S st CurInstr (p,(Comput (p,s,i))) = halt S holds

Comput (p,s,j) = Comput (p,s,i)

let S be non empty with_non-empty_values IC-Ins-separated halting AMI-Struct over N; :: thesis: for p being NAT -defined the InstructionsF of S -valued Function

for s being State of S st CurInstr (p,(Comput (p,s,i))) = halt S holds

Comput (p,s,j) = Comput (p,s,i)

let p be NAT -defined the InstructionsF of S -valued Function; :: thesis: for s being State of S st CurInstr (p,(Comput (p,s,i))) = halt S holds

Comput (p,s,j) = Comput (p,s,i)

let s be State of S; :: thesis: ( CurInstr (p,(Comput (p,s,i))) = halt S implies Comput (p,s,j) = Comput (p,s,i) )

assume A3: CurInstr (p,(Comput (p,s,i))) = halt S ; :: thesis: Comput (p,s,j) = Comput (p,s,i)

defpred S

A4: now :: thesis: for k being Nat st S_{1}[k] holds

S_{1}[k + 1]

A6:
SS

let k be Nat; :: thesis: ( S_{1}[k] implies S_{1}[k + 1] )

assume A5: S_{1}[k]
; :: thesis: S_{1}[k + 1]

Comput (p,s,(i + (k + 1))) = Comput (p,s,((i + k) + 1))

.= Following (p,(Comput (p,s,(i + k)))) by Th3

.= Comput (p,s,i) by A3, A5, Def3 ;

hence S_{1}[k + 1]
; :: thesis: verum

end;assume A5: S

Comput (p,s,(i + (k + 1))) = Comput (p,s,((i + k) + 1))

.= Following (p,(Comput (p,s,(i + k)))) by Th3

.= Comput (p,s,i) by A3, A5, Def3 ;

hence S

for k being Nat holds S

hence Comput (p,s,j) = Comput (p,s,i) by A2; :: thesis: verum