let n be Nat; :: thesis: for R being non trivial Ring

for a being Data-Location of R

for loc being Nat

for s1, s2 being State of (SCM R)

for P1, P2 being Instruction-Sequence of (SCM R)

for q being NAT -defined the InstructionsF of (SCM b_{1}) -valued finite non halt-free Function

for p being non empty b_{8} -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc & loc <> (IC (Comput (P1,s1,n))) + 1 holds

( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )

let R be non trivial Ring; :: thesis: for a being Data-Location of R

for loc being Nat

for s1, s2 being State of (SCM R)

for P1, P2 being Instruction-Sequence of (SCM R)

for q being NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function

for p being non empty b_{7} -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc & loc <> (IC (Comput (P1,s1,n))) + 1 holds

( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )

let a be Data-Location of R; :: thesis: for loc being Nat

for s1, s2 being State of (SCM R)

for P1, P2 being Instruction-Sequence of (SCM R)

for q being NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function

for p being non empty b_{6} -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc & loc <> (IC (Comput (P1,s1,n))) + 1 holds

( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )

let loc be Nat; :: thesis: for s1, s2 being State of (SCM R)

for P1, P2 being Instruction-Sequence of (SCM R)

for q being NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function

for p being non empty b_{5} -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc & loc <> (IC (Comput (P1,s1,n))) + 1 holds

( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )

let s1, s2 be State of (SCM R); :: thesis: for P1, P2 being Instruction-Sequence of (SCM R)

for q being NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function

for p being non empty b_{3} -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc & loc <> (IC (Comput (P1,s1,n))) + 1 holds

( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )

let P1, P2 be Instruction-Sequence of (SCM R); :: thesis: for q being NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function

for p being non empty b_{1} -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc & loc <> (IC (Comput (P1,s1,n))) + 1 holds

( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )

set Cs2i1 = Comput (P2,s2,(n + 1));

set Cs1i1 = Comput (P1,s1,(n + 1));

set I = CurInstr (P1,(Comput (P1,s1,n)));

let q be NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function; :: thesis: for p being non empty q -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc & loc <> (IC (Comput (P1,s1,n))) + 1 holds

( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )

let p be non empty q -autonomic FinPartState of (SCM R); :: thesis: ( p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc & loc <> (IC (Comput (P1,s1,n))) + 1 implies ( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R ) )

assume that

A1: ( p c= s1 & p c= s2 ) and

A2: ( q c= P1 & q c= P2 ) ; :: thesis: ( not CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc or not loc <> (IC (Comput (P1,s1,n))) + 1 or ( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R ) )

A3: CurInstr (P1,(Comput (P1,s1,n))) = CurInstr (P2,(Comput (P2,s2,n))) by A1, A2, AMISTD_5:7;

set Cs2i = Comput (P2,s2,n);

set Cs1i = Comput (P1,s1,n);

A4: Comput (P1,s1,(n + 1)) = Following (P1,(Comput (P1,s1,n))) by EXTPRO_1:3

.= Exec ((CurInstr (P1,(Comput (P1,s1,n)))),(Comput (P1,s1,n))) ;

A5: Comput (P2,s2,(n + 1)) = Following (P2,(Comput (P2,s2,n))) by EXTPRO_1:3

.= Exec ((CurInstr (P2,(Comput (P2,s2,n)))),(Comput (P2,s2,n))) ;

IC in dom p by AMISTD_5:6;

then A6: ( ((Comput (P1,s1,(n + 1))) | (dom p)) . (IC ) = (Comput (P1,s1,(n + 1))) . (IC ) & ((Comput (P2,s2,(n + 1))) | (dom p)) . (IC ) = (Comput (P2,s2,(n + 1))) . (IC ) ) by FUNCT_1:49;

assume that

A7: CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc and

A8: loc <> (IC (Comput (P1,s1,n))) + 1 ; :: thesis: ( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )

A9: IC (Comput (P1,s1,n)) = IC (Comput (P2,s2,n)) by A1, A2, AMISTD_5:7;

A10: (Comput (P2,s2,n)) . a = 0. R and

A11: (Comput (P1,s1,n)) . a <> 0. R ; :: thesis: contradiction

A12: (Comput (P1,s1,(n + 1))) . (IC ) = (IC (Comput (P1,s1,n))) + 1 by A4, A7, A11, SCMRING2:16;

(Comput (P2,s2,(n + 1))) . (IC ) = loc by A3, A5, A7, A10, SCMRING2:16;

hence contradiction by A1, A6, A8, A12, A2, EXTPRO_1:def 10; :: thesis: verum

for a being Data-Location of R

for loc being Nat

for s1, s2 being State of (SCM R)

for P1, P2 being Instruction-Sequence of (SCM R)

for q being NAT -defined the InstructionsF of (SCM b

for p being non empty b

( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )

let R be non trivial Ring; :: thesis: for a being Data-Location of R

for loc being Nat

for s1, s2 being State of (SCM R)

for P1, P2 being Instruction-Sequence of (SCM R)

for q being NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function

for p being non empty b

( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )

let a be Data-Location of R; :: thesis: for loc being Nat

for s1, s2 being State of (SCM R)

for P1, P2 being Instruction-Sequence of (SCM R)

for q being NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function

for p being non empty b

( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )

let loc be Nat; :: thesis: for s1, s2 being State of (SCM R)

for P1, P2 being Instruction-Sequence of (SCM R)

for q being NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function

for p being non empty b

( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )

let s1, s2 be State of (SCM R); :: thesis: for P1, P2 being Instruction-Sequence of (SCM R)

for q being NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function

for p being non empty b

( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )

let P1, P2 be Instruction-Sequence of (SCM R); :: thesis: for q being NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function

for p being non empty b

( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )

set Cs2i1 = Comput (P2,s2,(n + 1));

set Cs1i1 = Comput (P1,s1,(n + 1));

set I = CurInstr (P1,(Comput (P1,s1,n)));

let q be NAT -defined the InstructionsF of (SCM R) -valued finite non halt-free Function; :: thesis: for p being non empty q -autonomic FinPartState of (SCM R) st p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc & loc <> (IC (Comput (P1,s1,n))) + 1 holds

( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )

let p be non empty q -autonomic FinPartState of (SCM R); :: thesis: ( p c= s1 & p c= s2 & q c= P1 & q c= P2 & CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc & loc <> (IC (Comput (P1,s1,n))) + 1 implies ( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R ) )

assume that

A1: ( p c= s1 & p c= s2 ) and

A2: ( q c= P1 & q c= P2 ) ; :: thesis: ( not CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc or not loc <> (IC (Comput (P1,s1,n))) + 1 or ( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R ) )

A3: CurInstr (P1,(Comput (P1,s1,n))) = CurInstr (P2,(Comput (P2,s2,n))) by A1, A2, AMISTD_5:7;

set Cs2i = Comput (P2,s2,n);

set Cs1i = Comput (P1,s1,n);

A4: Comput (P1,s1,(n + 1)) = Following (P1,(Comput (P1,s1,n))) by EXTPRO_1:3

.= Exec ((CurInstr (P1,(Comput (P1,s1,n)))),(Comput (P1,s1,n))) ;

A5: Comput (P2,s2,(n + 1)) = Following (P2,(Comput (P2,s2,n))) by EXTPRO_1:3

.= Exec ((CurInstr (P2,(Comput (P2,s2,n)))),(Comput (P2,s2,n))) ;

IC in dom p by AMISTD_5:6;

then A6: ( ((Comput (P1,s1,(n + 1))) | (dom p)) . (IC ) = (Comput (P1,s1,(n + 1))) . (IC ) & ((Comput (P2,s2,(n + 1))) | (dom p)) . (IC ) = (Comput (P2,s2,(n + 1))) . (IC ) ) by FUNCT_1:49;

assume that

A7: CurInstr (P1,(Comput (P1,s1,n))) = a =0_goto loc and

A8: loc <> (IC (Comput (P1,s1,n))) + 1 ; :: thesis: ( (Comput (P1,s1,n)) . a = 0. R iff (Comput (P2,s2,n)) . a = 0. R )

A9: IC (Comput (P1,s1,n)) = IC (Comput (P2,s2,n)) by A1, A2, AMISTD_5:7;

hereby :: thesis: ( (Comput (P2,s2,n)) . a = 0. R implies (Comput (P1,s1,n)) . a = 0. R )

assume that assume
( (Comput (P1,s1,n)) . a = 0. R & (Comput (P2,s2,n)) . a <> 0. R )
; :: thesis: contradiction

then ( (Comput (P1,s1,(n + 1))) . (IC ) = loc & (Comput (P2,s2,(n + 1))) . (IC ) = (IC (Comput (P2,s2,n))) + 1 ) by A3, A4, A5, A7, SCMRING2:16;

hence contradiction by A1, A9, A6, A8, A2, EXTPRO_1:def 10; :: thesis: verum

end;then ( (Comput (P1,s1,(n + 1))) . (IC ) = loc & (Comput (P2,s2,(n + 1))) . (IC ) = (IC (Comput (P2,s2,n))) + 1 ) by A3, A4, A5, A7, SCMRING2:16;

hence contradiction by A1, A9, A6, A8, A2, EXTPRO_1:def 10; :: thesis: verum

A10: (Comput (P2,s2,n)) . a = 0. R and

A11: (Comput (P1,s1,n)) . a <> 0. R ; :: thesis: contradiction

A12: (Comput (P1,s1,(n + 1))) . (IC ) = (IC (Comput (P1,s1,n))) + 1 by A4, A7, A11, SCMRING2:16;

(Comput (P2,s2,(n + 1))) . (IC ) = loc by A3, A5, A7, A10, SCMRING2:16;

hence contradiction by A1, A6, A8, A12, A2, EXTPRO_1:def 10; :: thesis: verum