let q be NAT -defined the InstructionsF of (STC N) -valued finite non halt-free Function; :: according to AMISTD_5:def 4 :: thesis: for p being non empty q -autonomic FinPartState of (STC N)

for s being State of (STC N) st p c= s holds

for P being Instruction-Sequence of (STC N) st q c= P holds

for i being Nat holds IC (Comput (P,s,i)) in dom q

let p be non empty q -autonomic FinPartState of (STC N); :: thesis: for s being State of (STC N) st p c= s holds

for P being Instruction-Sequence of (STC N) st q c= P holds

for i being Nat holds IC (Comput (P,s,i)) in dom q

let s be State of (STC N); :: thesis: ( p c= s implies for P being Instruction-Sequence of (STC N) st q c= P holds

for i being Nat holds IC (Comput (P,s,i)) in dom q )

assume A1: p c= s ; :: thesis: for P being Instruction-Sequence of (STC N) st q c= P holds

for i being Nat holds IC (Comput (P,s,i)) in dom q

let P be Instruction-Sequence of (STC N); :: thesis: ( q c= P implies for i being Nat holds IC (Comput (P,s,i)) in dom q )

assume A2: q c= P ; :: thesis: for i being Nat holds IC (Comput (P,s,i)) in dom q

let i be Nat; :: thesis: IC (Comput (P,s,i)) in dom q

set Csi = Comput (P,s,i);

set loc = IC (Comput (P,s,i));

set loc1 = (IC (Comput (P,s,i))) + 1;

assume A3: not IC (Comput (P,s,i)) in dom q ; :: thesis: contradiction

the InstructionsF of (STC N) = {[0,0,0],[1,0,0]} by AMISTD_1:def 7;

then reconsider I = [1,0,0] as Instruction of (STC N) by TARSKI:def 2;

set p1 = q +* ((IC (Comput (P,s,i))) .--> I);

set p2 = q +* ((IC (Comput (P,s,i))) .--> (halt (STC N)));

reconsider P1 = P +* ((IC (Comput (P,s,i))) .--> I) as Instruction-Sequence of (STC N) ;

reconsider P2 = P +* ((IC (Comput (P,s,i))) .--> (halt (STC N))) as Instruction-Sequence of (STC N) ;

A5: IC (Comput (P,s,i)) in dom ((IC (Comput (P,s,i))) .--> (halt (STC N))) by TARSKI:def 1;

A7: IC (Comput (P,s,i)) in dom ((IC (Comput (P,s,i))) .--> I) by TARSKI:def 1;

A8: dom q misses dom ((IC (Comput (P,s,i))) .--> (halt (STC N))) by A3, ZFMISC_1:50;

A9: dom q misses dom ((IC (Comput (P,s,i))) .--> I) by A3, ZFMISC_1:50;

A10: q +* ((IC (Comput (P,s,i))) .--> I) c= P1 by A2, FUNCT_4:123;

A11: q +* ((IC (Comput (P,s,i))) .--> (halt (STC N))) c= P2 by A2, FUNCT_4:123;

set Cs2i = Comput (P2,s,i);

set Cs1i = Comput (P1,s,i);

not p is q -autonomic

for s being State of (STC N) st p c= s holds

for P being Instruction-Sequence of (STC N) st q c= P holds

for i being Nat holds IC (Comput (P,s,i)) in dom q

let p be non empty q -autonomic FinPartState of (STC N); :: thesis: for s being State of (STC N) st p c= s holds

for P being Instruction-Sequence of (STC N) st q c= P holds

for i being Nat holds IC (Comput (P,s,i)) in dom q

let s be State of (STC N); :: thesis: ( p c= s implies for P being Instruction-Sequence of (STC N) st q c= P holds

for i being Nat holds IC (Comput (P,s,i)) in dom q )

assume A1: p c= s ; :: thesis: for P being Instruction-Sequence of (STC N) st q c= P holds

for i being Nat holds IC (Comput (P,s,i)) in dom q

let P be Instruction-Sequence of (STC N); :: thesis: ( q c= P implies for i being Nat holds IC (Comput (P,s,i)) in dom q )

assume A2: q c= P ; :: thesis: for i being Nat holds IC (Comput (P,s,i)) in dom q

let i be Nat; :: thesis: IC (Comput (P,s,i)) in dom q

set Csi = Comput (P,s,i);

set loc = IC (Comput (P,s,i));

set loc1 = (IC (Comput (P,s,i))) + 1;

assume A3: not IC (Comput (P,s,i)) in dom q ; :: thesis: contradiction

the InstructionsF of (STC N) = {[0,0,0],[1,0,0]} by AMISTD_1:def 7;

then reconsider I = [1,0,0] as Instruction of (STC N) by TARSKI:def 2;

set p1 = q +* ((IC (Comput (P,s,i))) .--> I);

set p2 = q +* ((IC (Comput (P,s,i))) .--> (halt (STC N)));

reconsider P1 = P +* ((IC (Comput (P,s,i))) .--> I) as Instruction-Sequence of (STC N) ;

reconsider P2 = P +* ((IC (Comput (P,s,i))) .--> (halt (STC N))) as Instruction-Sequence of (STC N) ;

A5: IC (Comput (P,s,i)) in dom ((IC (Comput (P,s,i))) .--> (halt (STC N))) by TARSKI:def 1;

A7: IC (Comput (P,s,i)) in dom ((IC (Comput (P,s,i))) .--> I) by TARSKI:def 1;

A8: dom q misses dom ((IC (Comput (P,s,i))) .--> (halt (STC N))) by A3, ZFMISC_1:50;

A9: dom q misses dom ((IC (Comput (P,s,i))) .--> I) by A3, ZFMISC_1:50;

A10: q +* ((IC (Comput (P,s,i))) .--> I) c= P1 by A2, FUNCT_4:123;

A11: q +* ((IC (Comput (P,s,i))) .--> (halt (STC N))) c= P2 by A2, FUNCT_4:123;

set Cs2i = Comput (P2,s,i);

set Cs1i = Comput (P1,s,i);

not p is q -autonomic

proof

hence
contradiction
; :: thesis: verum
((IC (Comput (P,s,i))) .--> (halt (STC N))) . (IC (Comput (P,s,i))) = halt (STC N)
by FUNCOP_1:72;

then A12: P2 . (IC (Comput (P,s,i))) = halt (STC N) by A5, FUNCT_4:13;

((IC (Comput (P,s,i))) .--> I) . (IC (Comput (P,s,i))) = I by FUNCOP_1:72;

then A13: P1 . (IC (Comput (P,s,i))) = I by A7, FUNCT_4:13;

take P1 ; :: according to EXTPRO_1:def 10 :: thesis: ex b_{1} being set st

( q c= P1 & q c= b_{1} & ex b_{2}, b_{3} being set st

( p c= b_{2} & p c= b_{3} & not for b_{4} being set holds (Comput (P1,b_{2},b_{4})) | (proj1 p) = (Comput (b_{1},b_{3},b_{4})) | (proj1 p) ) )

take P2 ; :: thesis: ( q c= P1 & q c= P2 & ex b_{1}, b_{2} being set st

( p c= b_{1} & p c= b_{2} & not for b_{3} being set holds (Comput (P1,b_{1},b_{3})) | (proj1 p) = (Comput (P2,b_{2},b_{3})) | (proj1 p) ) )

q c= q +* ((IC (Comput (P,s,i))) .--> I) by A9, FUNCT_4:32;

hence A14: q c= P1 by A10, XBOOLE_1:1; :: thesis: ( q c= P2 & ex b_{1}, b_{2} being set st

( p c= b_{1} & p c= b_{2} & not for b_{3} being set holds (Comput (P1,b_{1},b_{3})) | (proj1 p) = (Comput (P2,b_{2},b_{3})) | (proj1 p) ) )

q c= q +* ((IC (Comput (P,s,i))) .--> (halt (STC N))) by A8, FUNCT_4:32;

hence A15: q c= P2 by A11, XBOOLE_1:1; :: thesis: ex b_{1}, b_{2} being set st

( p c= b_{1} & p c= b_{2} & not for b_{3} being set holds (Comput (P1,b_{1},b_{3})) | (proj1 p) = (Comput (P2,b_{2},b_{3})) | (proj1 p) )

take s ; :: thesis: ex b_{1} being set st

( p c= s & p c= b_{1} & not for b_{2} being set holds (Comput (P1,s,b_{2})) | (proj1 p) = (Comput (P2,b_{1},b_{2})) | (proj1 p) )

take s ; :: thesis: ( p c= s & p c= s & not for b_{1} being set holds (Comput (P1,s,b_{1})) | (proj1 p) = (Comput (P2,s,b_{1})) | (proj1 p) )

thus p c= s by A1; :: thesis: ( p c= s & not for b_{1} being set holds (Comput (P1,s,b_{1})) | (proj1 p) = (Comput (P2,s,b_{1})) | (proj1 p) )

A16: (Comput (P1,s,i)) | (dom p) = (Comput (P,s,i)) | (dom p) by A14, A2, A1, EXTPRO_1:def 10;

thus p c= s by A1; :: thesis: not for b_{1} being set holds (Comput (P1,s,b_{1})) | (proj1 p) = (Comput (P2,s,b_{1})) | (proj1 p)

A17: (Comput (P1,s,i)) | (dom p) = (Comput (P2,s,i)) | (dom p) by A14, A15, A1, EXTPRO_1:def 10;

take k = i + 1; :: thesis: not (Comput (P1,s,k)) | (proj1 p) = (Comput (P2,s,k)) | (proj1 p)

set Cs1k = Comput (P1,s,k);

A18: Comput (P1,s,k) = Following (P1,(Comput (P1,s,i))) by EXTPRO_1:3

.= Exec ((CurInstr (P1,(Comput (P1,s,i)))),(Comput (P1,s,i))) ;

InsCode I = 1 ;

then A19: IC (Exec (I,(Comput (P1,s,i)))) = (IC (Comput (P1,s,i))) + 1 by AMISTD_1:9;

A20: IC (Comput (P,s,i)) = IC ((Comput (P,s,i)) | (dom p)) by Th6, FUNCT_1:49;

then IC (Comput (P1,s,i)) = IC (Comput (P,s,i)) by A16, Th6, FUNCT_1:49;

then A21: IC (Comput (P1,s,k)) = (IC (Comput (P,s,i))) + 1 by A19, A18, A13, PBOOLE:143;

set Cs2k = Comput (P2,s,k);

A22: Comput (P2,s,k) = Following (P2,(Comput (P2,s,i))) by EXTPRO_1:3

.= Exec ((CurInstr (P2,(Comput (P2,s,i)))),(Comput (P2,s,i))) ;

A23: P2 /. (IC (Comput (P2,s,i))) = P2 . (IC (Comput (P2,s,i))) by PBOOLE:143;

IC (Comput (P2,s,i)) = IC (Comput (P,s,i)) by A16, A20, A17, Th6, FUNCT_1:49;

then A24: IC (Comput (P2,s,k)) = IC (Comput (P,s,i)) by A22, A12, A23, EXTPRO_1:def 3;

( IC ((Comput (P1,s,k)) | (dom p)) = IC (Comput (P1,s,k)) & IC ((Comput (P2,s,k)) | (dom p)) = IC (Comput (P2,s,k)) ) by Th6, FUNCT_1:49;

hence not (Comput (P1,s,k)) | (proj1 p) = (Comput (P2,s,k)) | (proj1 p) by A21, A24; :: thesis: verum

end;then A12: P2 . (IC (Comput (P,s,i))) = halt (STC N) by A5, FUNCT_4:13;

((IC (Comput (P,s,i))) .--> I) . (IC (Comput (P,s,i))) = I by FUNCOP_1:72;

then A13: P1 . (IC (Comput (P,s,i))) = I by A7, FUNCT_4:13;

take P1 ; :: according to EXTPRO_1:def 10 :: thesis: ex b

( q c= P1 & q c= b

( p c= b

take P2 ; :: thesis: ( q c= P1 & q c= P2 & ex b

( p c= b

q c= q +* ((IC (Comput (P,s,i))) .--> I) by A9, FUNCT_4:32;

hence A14: q c= P1 by A10, XBOOLE_1:1; :: thesis: ( q c= P2 & ex b

( p c= b

q c= q +* ((IC (Comput (P,s,i))) .--> (halt (STC N))) by A8, FUNCT_4:32;

hence A15: q c= P2 by A11, XBOOLE_1:1; :: thesis: ex b

( p c= b

take s ; :: thesis: ex b

( p c= s & p c= b

take s ; :: thesis: ( p c= s & p c= s & not for b

thus p c= s by A1; :: thesis: ( p c= s & not for b

A16: (Comput (P1,s,i)) | (dom p) = (Comput (P,s,i)) | (dom p) by A14, A2, A1, EXTPRO_1:def 10;

thus p c= s by A1; :: thesis: not for b

A17: (Comput (P1,s,i)) | (dom p) = (Comput (P2,s,i)) | (dom p) by A14, A15, A1, EXTPRO_1:def 10;

take k = i + 1; :: thesis: not (Comput (P1,s,k)) | (proj1 p) = (Comput (P2,s,k)) | (proj1 p)

set Cs1k = Comput (P1,s,k);

A18: Comput (P1,s,k) = Following (P1,(Comput (P1,s,i))) by EXTPRO_1:3

.= Exec ((CurInstr (P1,(Comput (P1,s,i)))),(Comput (P1,s,i))) ;

InsCode I = 1 ;

then A19: IC (Exec (I,(Comput (P1,s,i)))) = (IC (Comput (P1,s,i))) + 1 by AMISTD_1:9;

A20: IC (Comput (P,s,i)) = IC ((Comput (P,s,i)) | (dom p)) by Th6, FUNCT_1:49;

then IC (Comput (P1,s,i)) = IC (Comput (P,s,i)) by A16, Th6, FUNCT_1:49;

then A21: IC (Comput (P1,s,k)) = (IC (Comput (P,s,i))) + 1 by A19, A18, A13, PBOOLE:143;

set Cs2k = Comput (P2,s,k);

A22: Comput (P2,s,k) = Following (P2,(Comput (P2,s,i))) by EXTPRO_1:3

.= Exec ((CurInstr (P2,(Comput (P2,s,i)))),(Comput (P2,s,i))) ;

A23: P2 /. (IC (Comput (P2,s,i))) = P2 . (IC (Comput (P2,s,i))) by PBOOLE:143;

IC (Comput (P2,s,i)) = IC (Comput (P,s,i)) by A16, A20, A17, Th6, FUNCT_1:49;

then A24: IC (Comput (P2,s,k)) = IC (Comput (P,s,i)) by A22, A12, A23, EXTPRO_1:def 3;

( IC ((Comput (P1,s,k)) | (dom p)) = IC (Comput (P1,s,k)) & IC ((Comput (P2,s,k)) | (dom p)) = IC (Comput (P2,s,k)) ) by Th6, FUNCT_1:49;

hence not (Comput (P1,s,k)) | (proj1 p) = (Comput (P2,s,k)) | (proj1 p) by A21, A24; :: thesis: verum