:: TURING_1 semantic presentation

begin


definition
let A, B be non empty set ;
let f be Function of A,B;
let g be PartFunc of A,B;
:: original: +*
redefine funcf +* g ->  Function of A,B;
coherence
 f +* g is Function of A,B
proof
A1: now__::_thesis:_for_x_being_set_st_x_in_A_holds_
(f_+*_g)_._x_in_B
let x be set ; ::_thesis: ( x in A implies (f +* g) . b1 in B )
assume A2: x in A ; ::_thesis: (f +* g) . b1 in B
percases ( x in dom g or not x in dom g ) ;
supposeA3: x in dom g ; ::_thesis: (f +* g) . b1 in B
then  (f +* g) . x = g . x by FUNCT_4:13;
hence  (f +* g) . x in B by A3, PARTFUN1:4; ::_thesis: verum
end;
suppose not x in dom g ; ::_thesis: (f +* g) . b1 in B
then  (f +* g) . x = f . x by FUNCT_4:11;
hence  (f +* g) . x in B by A2, FUNCT_2:5; ::_thesis: verum
end;
end;
end;
A4: ( dom f = A & dom g c= A ) by FUNCT_2:def_1, RELAT_1:def_18;
dom (f +* g) = (dom f) \/ (dom g) by FUNCT_4:def_1
.= A by A4, XBOOLE_1:12 ;
hence  f +* g is Function of A,B by A1, FUNCT_2:3; ::_thesis: verum
end;
end;

definition
let X, Y be non empty set ;
let a be Element of X;
let b be Element of Y;
:: original: .-->
redefine funca .--> b ->  PartFunc of X,Y;
coherence
 a .--> b is PartFunc of X,Y
proof
set p = a .--> b;
 ( dom (a .--> b) = {a} & rng (a .--> b) = {b} ) by FUNCOP_1:8, FUNCOP_1:13;
hence  a .--> b is PartFunc of X,Y by RELSET_1:7; ::_thesis: verum
end;
end;

notation
let n be Nat;
synonym  SegM n for  succ n;
end;

definition
let n be Nat;
:: original: SegM
redefine func SegM n ->  Subset of NAT equals :: TURING_1:def 1
 {  k where k is Element of NAT : k <= n  }  ;
coherence
 SegM n is Subset of NAT
proof
 SegM n in NAT by ORDINAL1:def_12;
then  SegM n c= NAT by ORDINAL1:5;
hence  SegM n is Subset of NAT ; ::_thesis: verum
end;
compatibility
 for b1 being Subset of NAT holds
( b1 = SegM n iff b1 =  {  k where k is Element of NAT : k <= n  }  ) by NAT_1:54;
end;

:: deftheorem  defines SegM TURING_1:def_1_:_
 for n being Nat holds SegM n =  {  k where k is Element of NAT : k <= n  }  ;

registration
let n be Nat;
cluster SegM n ->  non empty finite ;
coherence
 ( SegM n is finite & not SegM n is empty )
proof
 n is Element of NAT by ORDINAL1:def_12;
hence  SegM n is finite by CQC_THE1:2; ::_thesis: not SegM n is empty
thus  not SegM n is empty ; ::_thesis: verum
end;
end;

theorem Th1: :: TURING_1:1
for k, n being Element of NAT holds
( k in SegM n iff k <= n )
proof
let k, n be Element of NAT ; ::_thesis: ( k in SegM n iff k <= n )
thus  ( k in SegM n implies k <= n ) ::_thesis: ( k <= n implies k in SegM n )
proof
assume  k in SegM n ; ::_thesis: k <= n
then  ex i being Element of NAT st
( k = i & i <= n ) ;
hence  k <= n ; ::_thesis: verum
end;
thus  ( k <= n implies k in SegM n ) ; ::_thesis: verum
end;

theorem Th2: :: TURING_1:2
for f being Function
for x, y, z, u, v being set st u <> x holds
(f +* ([x,y] .--> z)) . [u,v] = f . [u,v]
proof
let f be Function; ::_thesis: for x, y, z, u, v being set st u <> x holds
(f +* ([x,y] .--> z)) . [u,v] = f . [u,v]
let x, y, z, u, v be set ; ::_thesis: ( u <> x implies (f +* ([x,y] .--> z)) . [u,v] = f . [u,v] )
set p = [x,y] .--> z;
assume  u <> x ; ::_thesis: (f +* ([x,y] .--> z)) . [u,v] = f . [u,v]
then A1: [u,v] <> [x,y] by XTUPLE_0:1;
 dom ([x,y] .--> z) = {[x,y]} by FUNCOP_1:13;
then  not [u,v] in dom ([x,y] .--> z) by A1, TARSKI:def_1;
hence  (f +* ([x,y] .--> z)) . [u,v] = f . [u,v] by FUNCT_4:11; ::_thesis: verum
end;

theorem Th3: :: TURING_1:3
for f being Function
for x, y, z, u, v being set st v <> y holds
(f +* ([x,y] .--> z)) . [u,v] = f . [u,v]
proof
let f be Function; ::_thesis: for x, y, z, u, v being set st v <> y holds
(f +* ([x,y] .--> z)) . [u,v] = f . [u,v]
let x, y, z, u, v be set ; ::_thesis: ( v <> y implies (f +* ([x,y] .--> z)) . [u,v] = f . [u,v] )
set p = [x,y] .--> z;
assume  v <> y ; ::_thesis: (f +* ([x,y] .--> z)) . [u,v] = f . [u,v]
then A1: [u,v] <> [x,y] by XTUPLE_0:1;
 dom ([x,y] .--> z) = {[x,y]} by FUNCOP_1:13;
then  not [u,v] in dom ([x,y] .--> z) by A1, TARSKI:def_1;
hence  (f +* ([x,y] .--> z)) . [u,v] = f . [u,v] by FUNCT_4:11; ::_thesis: verum
end;

notation
let i be Nat;
let f be FinSequence;
synonym  Prefix (f,i) for f | i;
end;

definition
let i be Element of NAT ;
let f be FinSequence of NAT ;
:: original: Prefix
redefine func Prefix (f,i) ->  FinSequence of INT ;
coherence
 Prefix (f,i) is FinSequence of INT
proof
set p = f | (Seg i);
now__::_thesis:_for_j_being_Nat_st_j_in_dom_(f_|_(Seg_i))_holds_
(f_|_(Seg_i))_._j_in_INT
let j be Nat; ::_thesis: ( j in dom (f | (Seg i)) implies (f | (Seg i)) . j in INT )
assume A1: j in dom (f | (Seg i)) ; ::_thesis: (f | (Seg i)) . j in INT
then  j in dom f by RELAT_1:57;
then A2: f . j in NAT by FINSEQ_2:11;
 (f | (Seg i)) . j = f . j by A1, FUNCT_1:47;
hence  (f | (Seg i)) . j in INT by A2, NUMBERS:17; ::_thesis: verum
end;
hence  Prefix (f,i) is FinSequence of INT by FINSEQ_2:12; ::_thesis: verum
end;
end;

theorem Th4: :: TURING_1:4
for x1, x2 being Element of NAT holds
( Sum (Prefix (<*x1,x2*>,1)) = x1 & Sum (Prefix (<*x1,x2*>,2)) = x1 + x2 )
proof
let x1, x2 be Element of NAT ; ::_thesis: ( Sum (Prefix (<*x1,x2*>,1)) = x1 & Sum (Prefix (<*x1,x2*>,2)) = x1 + x2 )
reconsider y1 = x1 as Element of INT by INT_1:def_2;
thus  Sum (Prefix (<*x1,x2*>,1)) = Sum <*y1*> by FINSEQ_6:3
.= x1 by FINSOP_1:11 ; ::_thesis: Sum (Prefix (<*x1,x2*>,2)) = x1 + x2
reconsider y2 = x2 as Element of INT by INT_1:def_2;
 len <*x1,x2*> = 2 by FINSEQ_1:44;
hence  Sum (Prefix (<*x1,x2*>,2)) = Sum <*y1,y2*> by FINSEQ_3:49
.= x1 + x2 by RVSUM_1:77 ;
::_thesis: verum
end;

theorem Th5: :: TURING_1:5
for x1, x2, x3 being Element of NAT holds
( Sum (Prefix (<*x1,x2,x3*>,1)) = x1 & Sum (Prefix (<*x1,x2,x3*>,2)) = x1 + x2 & Sum (Prefix (<*x1,x2,x3*>,3)) = (x1 + x2) + x3 )
proof
let x1, x2, x3 be Element of NAT ; ::_thesis: ( Sum (Prefix (<*x1,x2,x3*>,1)) = x1 & Sum (Prefix (<*x1,x2,x3*>,2)) = x1 + x2 & Sum (Prefix (<*x1,x2,x3*>,3)) = (x1 + x2) + x3 )
reconsider y1 = x1 as Element of INT by INT_1:def_2;
thus  Sum (Prefix (<*x1,x2,x3*>,1)) = Sum <*y1*> by FINSEQ_6:4
.= x1 by FINSOP_1:11 ; ::_thesis: ( Sum (Prefix (<*x1,x2,x3*>,2)) = x1 + x2 & Sum (Prefix (<*x1,x2,x3*>,3)) = (x1 + x2) + x3 )
reconsider y2 = x2 as Element of INT by INT_1:def_2;
thus  Sum (Prefix (<*x1,x2,x3*>,2)) = Sum <*y1,y2*> by FINSEQ_6:5
.= x1 + x2 by RVSUM_1:77 ; ::_thesis: Sum (Prefix (<*x1,x2,x3*>,3)) = (x1 + x2) + x3
reconsider y3 = x3 as Element of INT by INT_1:def_2;
 len <*x1,x2,x3*> = 3 by FINSEQ_1:45;
hence  Sum (Prefix (<*x1,x2,x3*>,3)) = Sum <*y1,y2,y3*> by FINSEQ_3:49
.= (x1 + x2) + x3 by RVSUM_1:78 ;
::_thesis: verum
end;

begin

definition
attrc1 is strict ;
struct TuringStr  -> ;
aggrTuringStr(# Symbols, FStates, Tran, InitS, AcceptS #) ->  TuringStr ;
sel Symbols c1 ->  non empty finite set ;
sel FStates c1 ->  non empty finite set ;
sel Tran c1 ->  Function of [: the FStates of c1, the Symbols of c1:],[: the FStates of c1, the Symbols of c1,{(- 1),0,1}:];
sel InitS c1 ->  Element of the FStates of c1;
sel AcceptS c1 ->  Element of the FStates of c1;
end;

definition
let T be TuringStr ;
mode State of T is Element of the FStates of T;
mode Tape of T is Element of Funcs (INT, the Symbols of T);
mode Symbol of T is Element of the Symbols of T;
end;

definition
let T be TuringStr ;
let t be Tape of T;
let h be Integer;
let s be Symbol of T;
func Tape-Chg (t,h,s) ->  Tape of T equals :: TURING_1:def 2
t +* (h .--> s);
coherence
 t +* (h .--> s) is Tape of T
proof
set X = INT ;
set Y = the Symbols of T;
A1: ex f being Function st
( t = f & dom f = INT & rng f c= the Symbols of T ) by FUNCT_2:def_2;
 rng (h .--> s) = {s} by FUNCOP_1:8;
then  ( rng (t +* (h .--> s)) c= (rng t) \/ (rng (h .--> s)) & (rng t) \/ (rng (h .--> s)) c= the Symbols of T ) by A1, FUNCT_4:17, XBOOLE_1:8;
then A2: rng (t +* (h .--> s)) c= the Symbols of T by XBOOLE_1:1;
A3: h in INT by INT_1:def_2;
dom (t +* (h .--> s)) = (dom t) \/ (dom (h .--> s)) by FUNCT_4:def_1
.= (dom t) \/ {h} by FUNCOP_1:13
.= INT by A1, A3, ZFMISC_1:40 ;
hence  t +* (h .--> s) is Tape of T by A2, FUNCT_2:def_2; ::_thesis: verum
end;
end;

:: deftheorem  defines Tape-Chg TURING_1:def_2_:_
 for T being TuringStr
for t being Tape of T
for h being Integer
for s being Symbol of T holds Tape-Chg (t,h,s) = t +* (h .--> s);

definition
let T be TuringStr ;
mode All-State of T is Element of [: the FStates of T,INT,(Funcs (INT, the Symbols of T)):];
mode Tran-Source of T is Element of [: the FStates of T, the Symbols of T:];
mode Tran-Goal of T is Element of [: the FStates of T, the Symbols of T,{(- 1),0,1}:];
end;

definition
let T be TuringStr ;
let g be Tran-Goal of T;
func offset g ->  Integer equals :: TURING_1:def 3
g `3_3 ;
coherence
 g `3_3 is Integer by ENUMSET1:def_1;
end;

:: deftheorem  defines offset TURING_1:def_3_:_
 for T being TuringStr
for g being Tran-Goal of T holds offset g = g `3_3 ;

definition
let T be TuringStr ;
let s be All-State of T;
func Head s ->  Integer equals :: TURING_1:def 4
s `2_3 ;
coherence
 s `2_3 is Integer ;
end;

:: deftheorem  defines Head TURING_1:def_4_:_
 for T being TuringStr
for s being All-State of T holds Head s = s `2_3 ;

definition
let T be TuringStr ;
let s be All-State of T;
func TRAN s ->  Tran-Goal of T equals :: TURING_1:def 5
 the Tran of T . [(s `1_3),((s `3_3) . (Head s))];
correctness
coherence
the Tran of T . [(s `1_3),((s `3_3) . (Head s))] is Tran-Goal of T;
proof
reconsider x = Head s as Element of INT ;
 the Tran of T . [(s `1_3),((s `3_3) . x)] is Tran-Goal of T ;
hence  the Tran of T . [(s `1_3),((s `3_3) . (Head s))] is Tran-Goal of T ; ::_thesis: verum
end;
end;

:: deftheorem  defines TRAN TURING_1:def_5_:_
 for T being TuringStr
for s being All-State of T holds TRAN s = the Tran of T . [(s `1_3),((s `3_3) . (Head s))];

definition
let T be TuringStr ;
let s be All-State of T;
func Following s ->  All-State of T equals :Def6: :: TURING_1:def 6
[((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))] if s `1_3 <> the AcceptS of T
otherwise s;
correctness
coherence
( ( s `1_3 <> the AcceptS of T implies [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))] is All-State of T ) & ( not s `1_3 <> the AcceptS of T implies s is All-State of T ) );
consistency
for b1 being All-State of T holds verum;
proof
 (Head s) + (offset (TRAN s)) in INT by INT_1:def_2;
hence  ( ( s `1_3 <> the AcceptS of T implies [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))] is All-State of T ) & ( not s `1_3 <> the AcceptS of T implies s is All-State of T ) & ( for b1 being All-State of T holds verum ) ) by MCART_1:69; ::_thesis: verum
end;
end;

:: deftheorem Def6 defines Following TURING_1:def_6_:_
 for T being TuringStr
for s being All-State of T holds
( ( s `1_3 <> the AcceptS of T implies Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))] ) & ( not s `1_3 <> the AcceptS of T implies Following s = s ) );

definition
let T be TuringStr ;
let s be All-State of T;
func Computation s ->  Function of NAT,[: the FStates of T,INT,(Funcs (INT, the Symbols of T)):] means :Def7: :: TURING_1:def 7
( it . 0 = s & ( for i being Nat holds it . (i + 1) = Following (it . i) ) );
existence
 ex b1 being Function of NAT,[: the FStates of T,INT,(Funcs (INT, the Symbols of T)):] st
( b1 . 0 = s & ( for i being Nat holds b1 . (i + 1) = Following (b1 . i) ) )
proof
deffunc H1( set , All-State of T) ->  All-State of T = Following $2;
consider f being Function of NAT,[: the FStates of T,INT,(Funcs (INT, the Symbols of T)):] such that
A1: ( f . 0 = s & ( for n being Nat holds f . (n + 1) = H1(n,f . n) ) ) from NAT_1:sch_12();
take  f ; ::_thesis: ( f . 0 = s & ( for i being Nat holds f . (i + 1) = Following (f . i) ) )
thus  ( f . 0 = s & ( for i being Nat holds f . (i + 1) = Following (f . i) ) ) by A1; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Function of NAT,[: the FStates of T,INT,(Funcs (INT, the Symbols of T)):] st b1 . 0 = s & ( for i being Nat holds b1 . (i + 1) = Following (b1 . i) ) & b2 . 0 = s & ( for i being Nat holds b2 . (i + 1) = Following (b2 . i) ) holds
b1 = b2
proof
deffunc H1( set , All-State of T) ->  All-State of T = Following $2;
let F1, F2 be Function of NAT,[: the FStates of T,INT,(Funcs (INT, the Symbols of T)):]; ::_thesis: ( F1 . 0 = s & ( for i being Nat holds F1 . (i + 1) = Following (F1 . i) ) & F2 . 0 = s & ( for i being Nat holds F2 . (i + 1) = Following (F2 . i) ) implies F1 = F2 )
assume that
A2: F1 . 0 = s and
A3: for i being Nat holds F1 . (i + 1) = Following (F1 . i) and
A4: F2 . 0 = s and
A5: for i being Nat holds F2 . (i + 1) = Following (F2 . i) ; ::_thesis: F1 = F2
A6: for i being Nat holds F1 . (i + 1) = H1(i,F1 . i) by A3;
A7: for i being Nat holds F2 . (i + 1) = H1(i,F2 . i) by A5;
A8: F2 . 0 = s by A4;
A9: F1 . 0 = s by A2;
thus  F1 = F2 from NAT_1:sch_16(A9, A6, A8, A7); ::_thesis: verum
end;
end;

:: deftheorem Def7 defines Computation TURING_1:def_7_:_
 for T being TuringStr
for s being All-State of T
for b3 being Function of NAT,[: the FStates of T,INT,(Funcs (INT, the Symbols of T)):] holds
( b3 = Computation s iff ( b3 . 0 = s & ( for i being Nat holds b3 . (i + 1) = Following (b3 . i) ) ) );


theorem :: TURING_1:6
for T being TuringStr
for s being All-State of T st s `1_3 = the AcceptS of T holds
s = Following s by Def6;

theorem :: TURING_1:7
for T being TuringStr
for s being All-State of T holds (Computation s) . 0 = s by Def7;

theorem :: TURING_1:8
for k being Element of NAT
for T being TuringStr
for s being All-State of T holds (Computation s) . (k + 1) = Following ((Computation s) . k) by Def7;

theorem Th9: :: TURING_1:9
for T being TuringStr
for s being All-State of T holds (Computation s) . 1 = Following s
proof
let T be TuringStr ; ::_thesis: for s being All-State of T holds (Computation s) . 1 = Following s
let s be All-State of T; ::_thesis: (Computation s) . 1 = Following s
(Computation s) . (0 + 1) = Following ((Computation s) . 0) by Def7
.= Following s by Def7 ;
hence  (Computation s) . 1 = Following s ; ::_thesis: verum
end;

theorem Th10: :: TURING_1:10
for i, k being Element of NAT
for T being TuringStr
for s being All-State of T holds (Computation s) . (i + k) = (Computation ((Computation s) . i)) . k
proof
let i, k be Element of NAT ; ::_thesis: for T being TuringStr
for s being All-State of T holds (Computation s) . (i + k) = (Computation ((Computation s) . i)) . k
let T be TuringStr ; ::_thesis: for s being All-State of T holds (Computation s) . (i + k) = (Computation ((Computation s) . i)) . k
let s be All-State of T; ::_thesis: (Computation s) . (i + k) = (Computation ((Computation s) . i)) . k
defpred S1[ Element of NAT ] means (Computation s) . (i + $1) = (Computation ((Computation s) . i)) . $1;
A1: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A2: (Computation s) . (i + k) = (Computation ((Computation s) . i)) . k ; ::_thesis: S1[k + 1]
thus (Computation s) . (i + (k + 1)) = (Computation s) . ((i + k) + 1)
.= Following ((Computation s) . (i + k)) by Def7
.= (Computation ((Computation s) . i)) . (k + 1) by A2, Def7 ; ::_thesis: verum
end;
A3: S1[ 0 ] by Def7;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A3, A1);
hence  (Computation s) . (i + k) = (Computation ((Computation s) . i)) . k ; ::_thesis: verum
end;

theorem Th11: :: TURING_1:11
for i, j being Element of NAT
for T being TuringStr
for s being All-State of T st i <= j & Following ((Computation s) . i) = (Computation s) . i holds
(Computation s) . j = (Computation s) . i
proof
let i, j be Element of NAT ; ::_thesis: for T being TuringStr
for s being All-State of T st i <= j & Following ((Computation s) . i) = (Computation s) . i holds
(Computation s) . j = (Computation s) . i
let T be TuringStr ; ::_thesis: for s being All-State of T st i <= j & Following ((Computation s) . i) = (Computation s) . i holds
(Computation s) . j = (Computation s) . i
let s be All-State of T; ::_thesis: ( i <= j & Following ((Computation s) . i) = (Computation s) . i implies (Computation s) . j = (Computation s) . i )
assume that
A1: i <= j and
A2: Following ((Computation s) . i) = (Computation s) . i ; ::_thesis: (Computation s) . j = (Computation s) . i
consider k being Nat such that
A3: j = i + k by A1, NAT_1:10;
defpred S1[ Element of NAT ] means (Computation s) . (i + $1) = (Computation s) . i;
A4: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A5: (Computation s) . (i + k) = (Computation s) . i ; ::_thesis: S1[k + 1]
thus (Computation s) . (i + (k + 1)) = (Computation s) . ((i + k) + 1)
.= (Computation s) . i by A2, A5, Def7 ; ::_thesis: verum
end;
A6: S1[ 0 ] ;
A7: for k being Element of NAT holds S1[k] from NAT_1:sch_1(A6, A4);
 k in NAT by ORDINAL1:def_12;
hence  (Computation s) . j = (Computation s) . i by A3, A7; ::_thesis: verum
end;

theorem Th12: :: TURING_1:12
for i, j being Element of NAT
for T being TuringStr
for s being All-State of T st i <= j & ((Computation s) . i) `1_3 = the AcceptS of T holds
(Computation s) . j = (Computation s) . i
proof
let i, j be Element of NAT ; ::_thesis: for T being TuringStr
for s being All-State of T st i <= j & ((Computation s) . i) `1_3 = the AcceptS of T holds
(Computation s) . j = (Computation s) . i
let T be TuringStr ; ::_thesis: for s being All-State of T st i <= j & ((Computation s) . i) `1_3 = the AcceptS of T holds
(Computation s) . j = (Computation s) . i
let s be All-State of T; ::_thesis: ( i <= j & ((Computation s) . i) `1_3 = the AcceptS of T implies (Computation s) . j = (Computation s) . i )
assume that
A1: i <= j and
A2: ((Computation s) . i) `1_3 = the AcceptS of T ; ::_thesis: (Computation s) . j = (Computation s) . i
 Following ((Computation s) . i) = (Computation s) . i by A2, Def6;
hence  (Computation s) . j = (Computation s) . i by A1, Th11; ::_thesis: verum
end;

definition
let T be TuringStr ;
let s be All-State of T;
attrs is Accept-Halt  means :Def8: :: TURING_1:def 8
 ex k being Element of NAT st ((Computation s) . k) `1_3 = the AcceptS of T;
end;

:: deftheorem Def8 defines Accept-Halt TURING_1:def_8_:_
 for T being TuringStr
for s being All-State of T holds
( s is Accept-Halt iff ex k being Element of NAT st ((Computation s) . k) `1_3 = the AcceptS of T );

definition
let T be TuringStr ;
let s be All-State of T;
assume A1: s is Accept-Halt ;
func Result s ->  All-State of T means :Def9: :: TURING_1:def 9
 ex k being Element of NAT st
( it = (Computation s) . k & ((Computation s) . k) `1_3 = the AcceptS of T );
uniqueness
 for b1, b2 being All-State of T st ex k being Element of NAT st
( b1 = (Computation s) . k & ((Computation s) . k) `1_3 = the AcceptS of T ) & ex k being Element of NAT st
( b2 = (Computation s) . k & ((Computation s) . k) `1_3 = the AcceptS of T ) holds
b1 = b2
proof
let s1, s2 be All-State of T; ::_thesis: ( ex k being Element of NAT st
( s1 = (Computation s) . k & ((Computation s) . k) `1_3 = the AcceptS of T ) & ex k being Element of NAT st
( s2 = (Computation s) . k & ((Computation s) . k) `1_3 = the AcceptS of T ) implies s1 = s2 )
given k1 being Element of NAT  such that A1: ( s1 = (Computation s) . k1 & ((Computation s) . k1) `1_3 = the AcceptS of T ) ; ::_thesis: ( for k being Element of NAT holds
( not s2 = (Computation s) . k or not ((Computation s) . k) `1_3 = the AcceptS of T ) or s1 = s2 )
given k2 being Element of NAT  such that A2: ( s2 = (Computation s) . k2 & ((Computation s) . k2) `1_3 = the AcceptS of T ) ; ::_thesis: s1 = s2
 ( k1 <= k2 or k2 <= k1 ) ;
hence  s1 = s2 by A1, A2, Th12; ::_thesis: verum
end;
correctness
existence
ex b1 being All-State of T ex k being Element of NAT st
( b1 = (Computation s) . k & ((Computation s) . k) `1_3 = the AcceptS of T );
proof
 ex k being Element of NAT st ((Computation s) . k) `1_3 = the AcceptS of T by A1, Def8;
hence  ex b1 being All-State of T ex k being Element of NAT st
( b1 = (Computation s) . k & ((Computation s) . k) `1_3 = the AcceptS of T ) ; ::_thesis: verum
end;
end;

:: deftheorem Def9 defines Result TURING_1:def_9_:_
 for T being TuringStr
for s being All-State of T st s is Accept-Halt holds
for b3 being All-State of T holds
( b3 = Result s iff ex k being Element of NAT st
( b3 = (Computation s) . k & ((Computation s) . k) `1_3 = the AcceptS of T ) );

theorem Th13: :: TURING_1:13
for T being TuringStr
for s being All-State of T st s is Accept-Halt holds
ex k being Element of NAT st
( ((Computation s) . k) `1_3 = the AcceptS of T & Result s = (Computation s) . k & ( for i being Element of NAT st i < k holds
((Computation s) . i) `1_3 <> the AcceptS of T ) )
proof
let T be TuringStr ; ::_thesis: for s being All-State of T st s is Accept-Halt holds
ex k being Element of NAT st
( ((Computation s) . k) `1_3 = the AcceptS of T & Result s = (Computation s) . k & ( for i being Element of NAT st i < k holds
((Computation s) . i) `1_3 <> the AcceptS of T ) )
let s be All-State of T; ::_thesis: ( s is Accept-Halt implies ex k being Element of NAT st
( ((Computation s) . k) `1_3 = the AcceptS of T & Result s = (Computation s) . k & ( for i being Element of NAT st i < k holds
((Computation s) . i) `1_3 <> the AcceptS of T ) ) )
defpred S1[ Nat] means ((Computation s) . $1) `1_3 = the AcceptS of T;
assume A1: s is Accept-Halt ; ::_thesis: ex k being Element of NAT st
( ((Computation s) . k) `1_3 = the AcceptS of T & Result s = (Computation s) . k & ( for i being Element of NAT st i < k holds
((Computation s) . i) `1_3 <> the AcceptS of T ) )
then  ex k being Element of NAT st ((Computation s) . k) `1_3 = the AcceptS of T by Def8;
then A2: ex k being Nat st S1[k] ;
consider k being Nat such that
A3: ( S1[k] & ( for n being Nat st S1[n] holds
k <= n ) ) from NAT_1:sch_5(A2);
reconsider k = k as Element of NAT by ORDINAL1:def_12;
take  k ; ::_thesis: ( ((Computation s) . k) `1_3 = the AcceptS of T & Result s = (Computation s) . k & ( for i being Element of NAT st i < k holds
((Computation s) . i) `1_3 <> the AcceptS of T ) )
thus  S1[k] by A3; ::_thesis: ( Result s = (Computation s) . k & ( for i being Element of NAT st i < k holds
((Computation s) . i) `1_3 <> the AcceptS of T ) )
thus  Result s = (Computation s) . k by A1, A3, Def9; ::_thesis: for i being Element of NAT st i < k holds
((Computation s) . i) `1_3 <> the AcceptS of T
thus  for i being Element of NAT st i < k holds
((Computation s) . i) `1_3 <> the AcceptS of T by A3; ::_thesis: verum
end;

definition
let A, B be non empty set ;
let y be set ;
assume A1: y in B ;
func id (A,B,y) ->  Function of A,[:A,B:] means :: TURING_1:def 10
 for x being Element of A holds it . x = [x,y];
existence
 ex b1 being Function of A,[:A,B:] st
for x being Element of A holds b1 . x = [x,y]
proof
reconsider yy = y as Element of B by A1;
deffunc H1( Element of A) ->  Element of [:A,B:] = [$1,yy];
consider f being Function of A,[:A,B:] such that
A1: for x being Element of A holds f . x = H1(x) from FUNCT_2:sch_4();
take  f ; ::_thesis: for x being Element of A holds f . x = [x,y]
thus  for x being Element of A holds f . x = [x,y] by A1; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Function of A,[:A,B:] st ( for x being Element of A holds b1 . x = [x,y] ) & ( for x being Element of A holds b2 . x = [x,y] ) holds
b1 = b2
proof
deffunc H1( Element of A) ->  set = [$1,y];
thus  for f1, f2 being Function of A,[:A,B:] st ( for x being Element of A holds f1 . x = H1(x) ) & ( for x being Element of A holds f2 . x = H1(x) ) holds
f1 = f2 from BINOP_2:sch_1(); ::_thesis: verum
end;
end;

:: deftheorem  defines id TURING_1:def_10_:_
 for A, B being non empty set
for y being set st y in B holds
for b4 being Function of A,[:A,B:] holds
( b4 = id (A,B,y) iff for x being Element of A holds b4 . x = [x,y] );

definition
func Sum_Tran  ->  Function of [:(SegM 5),{0,1}:],[:(SegM 5),{0,1},{(- 1),0,1}:] equals :: TURING_1:def 11
(((((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([3,1] .--> [4,0,(- 1)])) +* ([4,1] .--> [4,1,(- 1)])) +* ([4,0] .--> [5,0,0]);
coherence
 (((((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([3,1] .--> [4,0,(- 1)])) +* ([4,1] .--> [4,1,(- 1)])) +* ([4,0] .--> [5,0,0]) is Function of [:(SegM 5),{0,1}:],[:(SegM 5),{0,1},{(- 1),0,1}:]
proof
reconsider p0 = 0 , p1 = 1, p2 = 2, p3 = 3, p4 = 4, p5 = 5 as Element of SegM 5 by Th1;
set A = [:(SegM 5),{0,1}:];
set B = {(- 1),0,1};
set C = [:(SegM 5),{0,1},{(- 1),0,1}:];
reconsider b0 = 0 , b1 = 1 as Element of {0,1} by TARSKI:def_2;
reconsider L = - 1 as Element of {(- 1),0,1} by ENUMSET1:def_1;
reconsider h = 0 , R = 1 as Element of {(- 1),0,1} by ENUMSET1:def_1;
 [:(SegM 5),{0,1},{(- 1),0,1}:] = [:[:(SegM 5),{0,1}:],{(- 1),0,1}:] by ZFMISC_1:def_3;
then reconsider OP = id ([:(SegM 5),{0,1}:],{(- 1),0,1},h) as Function of [:(SegM 5),{0,1}:],[:(SegM 5),{0,1},{(- 1),0,1}:] ;
 (((((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([3,1] .--> [4,0,(- 1)])) +* ([4,1] .--> [4,1,(- 1)])) +* ([4,0] .--> [5,0,0]) = ((((((((OP +* ([p0,b0] .--> [p0,b0,R])) +* ([p0,b1] .--> [p1,b0,R])) +* ([p1,b1] .--> [p1,b1,R])) +* ([p1,b0] .--> [p2,b1,R])) +* ([p2,b1] .--> [p2,b1,R])) +* ([p2,b0] .--> [p3,b0,L])) +* ([p3,b1] .--> [p4,b0,L])) +* ([p4,b1] .--> [p4,b1,L])) +* ([p4,b0] .--> [p5,b0,h]) ;
hence  (((((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([3,1] .--> [4,0,(- 1)])) +* ([4,1] .--> [4,1,(- 1)])) +* ([4,0] .--> [5,0,0]) is Function of [:(SegM 5),{0,1}:],[:(SegM 5),{0,1},{(- 1),0,1}:] ; ::_thesis: verum
end;
end;

:: deftheorem  defines Sum_Tran TURING_1:def_11_:_
 Sum_Tran = (((((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([3,1] .--> [4,0,(- 1)])) +* ([4,1] .--> [4,1,(- 1)])) +* ([4,0] .--> [5,0,0]);

theorem Th14: :: TURING_1:14
( Sum_Tran . [0,0] = [0,0,1] & Sum_Tran . [0,1] = [1,0,1] & Sum_Tran . [1,1] = [1,1,1] & Sum_Tran . [1,0] = [2,1,1] & Sum_Tran . [2,1] = [2,1,1] & Sum_Tran . [2,0] = [3,0,(- 1)] & Sum_Tran . [3,1] = [4,0,(- 1)] & Sum_Tran . [4,1] = [4,1,(- 1)] & Sum_Tran . [4,0] = [5,0,0] )
proof
set x = [0,1];
set x1 = [0,0];
set p1 = [0,0] .--> [0,0,1];
set p2 = [0,1] .--> [1,0,1];
set p3 = [1,1] .--> [1,1,1];
set p4 = [1,0] .--> [2,1,1];
set p5 = [2,1] .--> [2,1,1];
set p6 = [2,0] .--> [3,0,(- 1)];
set p7 = [3,1] .--> [4,0,(- 1)];
set p8 = [4,1] .--> [4,1,(- 1)];
set f = id ([:(SegM 5),{0,1}:],{(- 1),0,1},0);
thus Sum_Tran . [0,0] = (((((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([3,1] .--> [4,0,(- 1)])) +* ([4,1] .--> [4,1,(- 1)])) . [0,0] by Th2
.= ((((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([3,1] .--> [4,0,(- 1)])) . [0,0] by Th2
.= (((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) . [0,0] by Th2
.= ((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) . [0,0] by Th2
.= (((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) . [0,0] by Th2
.= ((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) . [0,0] by Th2
.= (((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) . [0,0] by Th2
.= ((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) . [0,0] by Th3
.= [0,0,1] by FUNCT_7:94 ; ::_thesis: ( Sum_Tran . [0,1] = [1,0,1] & Sum_Tran . [1,1] = [1,1,1] & Sum_Tran . [1,0] = [2,1,1] & Sum_Tran . [2,1] = [2,1,1] & Sum_Tran . [2,0] = [3,0,(- 1)] & Sum_Tran . [3,1] = [4,0,(- 1)] & Sum_Tran . [4,1] = [4,1,(- 1)] & Sum_Tran . [4,0] = [5,0,0] )
thus Sum_Tran . [0,1] = (((((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([3,1] .--> [4,0,(- 1)])) +* ([4,1] .--> [4,1,(- 1)])) . [0,1] by Th2
.= ((((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([3,1] .--> [4,0,(- 1)])) . [0,1] by Th2
.= (((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) . [0,1] by Th2
.= ((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) . [0,1] by Th2
.= (((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) . [0,1] by Th2
.= ((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) . [0,1] by Th2
.= (((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) . [0,1] by Th2
.= [1,0,1] by FUNCT_7:94 ; ::_thesis: ( Sum_Tran . [1,1] = [1,1,1] & Sum_Tran . [1,0] = [2,1,1] & Sum_Tran . [2,1] = [2,1,1] & Sum_Tran . [2,0] = [3,0,(- 1)] & Sum_Tran . [3,1] = [4,0,(- 1)] & Sum_Tran . [4,1] = [4,1,(- 1)] & Sum_Tran . [4,0] = [5,0,0] )
set x = [1,1];
thus Sum_Tran . [1,1] = (((((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([3,1] .--> [4,0,(- 1)])) +* ([4,1] .--> [4,1,(- 1)])) . [1,1] by Th2
.= ((((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([3,1] .--> [4,0,(- 1)])) . [1,1] by Th2
.= (((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) . [1,1] by Th2
.= ((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) . [1,1] by Th2
.= (((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) . [1,1] by Th2
.= ((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) . [1,1] by Th3
.= [1,1,1] by FUNCT_7:94 ; ::_thesis: ( Sum_Tran . [1,0] = [2,1,1] & Sum_Tran . [2,1] = [2,1,1] & Sum_Tran . [2,0] = [3,0,(- 1)] & Sum_Tran . [3,1] = [4,0,(- 1)] & Sum_Tran . [4,1] = [4,1,(- 1)] & Sum_Tran . [4,0] = [5,0,0] )
set x = [1,0];
thus Sum_Tran . [1,0] = (((((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([3,1] .--> [4,0,(- 1)])) +* ([4,1] .--> [4,1,(- 1)])) . [1,0] by Th2
.= ((((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([3,1] .--> [4,0,(- 1)])) . [1,0] by Th2
.= (((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) . [1,0] by Th2
.= ((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) . [1,0] by Th2
.= (((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) . [1,0] by Th2
.= [2,1,1] by FUNCT_7:94 ; ::_thesis: ( Sum_Tran . [2,1] = [2,1,1] & Sum_Tran . [2,0] = [3,0,(- 1)] & Sum_Tran . [3,1] = [4,0,(- 1)] & Sum_Tran . [4,1] = [4,1,(- 1)] & Sum_Tran . [4,0] = [5,0,0] )
set x = [2,1];
thus Sum_Tran . [2,1] = (((((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([3,1] .--> [4,0,(- 1)])) +* ([4,1] .--> [4,1,(- 1)])) . [2,1] by Th2
.= ((((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([3,1] .--> [4,0,(- 1)])) . [2,1] by Th2
.= (((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) . [2,1] by Th2
.= ((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) . [2,1] by Th3
.= [2,1,1] by FUNCT_7:94 ; ::_thesis: ( Sum_Tran . [2,0] = [3,0,(- 1)] & Sum_Tran . [3,1] = [4,0,(- 1)] & Sum_Tran . [4,1] = [4,1,(- 1)] & Sum_Tran . [4,0] = [5,0,0] )
set x = [2,0];
thus Sum_Tran . [2,0] = (((((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([3,1] .--> [4,0,(- 1)])) +* ([4,1] .--> [4,1,(- 1)])) . [2,0] by Th2
.= ((((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([3,1] .--> [4,0,(- 1)])) . [2,0] by Th2
.= (((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) . [2,0] by Th2
.= [3,0,(- 1)] by FUNCT_7:94 ; ::_thesis: ( Sum_Tran . [3,1] = [4,0,(- 1)] & Sum_Tran . [4,1] = [4,1,(- 1)] & Sum_Tran . [4,0] = [5,0,0] )
set x = [3,1];
thus Sum_Tran . [3,1] = (((((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([3,1] .--> [4,0,(- 1)])) +* ([4,1] .--> [4,1,(- 1)])) . [3,1] by Th2
.= ((((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([3,1] .--> [4,0,(- 1)])) . [3,1] by Th2
.= [4,0,(- 1)] by FUNCT_7:94 ; ::_thesis: ( Sum_Tran . [4,1] = [4,1,(- 1)] & Sum_Tran . [4,0] = [5,0,0] )
set x = [4,1];
thus Sum_Tran . [4,1] = (((((((((id ([:(SegM 5),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [0,0,1])) +* ([0,1] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([3,1] .--> [4,0,(- 1)])) +* ([4,1] .--> [4,1,(- 1)])) . [4,1] by Th3
.= [4,1,(- 1)] by FUNCT_7:94 ; ::_thesis: Sum_Tran . [4,0] = [5,0,0]
thus  Sum_Tran . [4,0] = [5,0,0] by FUNCT_7:94; ::_thesis: verum
end;

definition
let T be TuringStr ;
let t be Tape of T;
let i, j be Integer;
predt is_1_between i,j means :Def12: :: TURING_1:def 12
( t . i = 0 & t . j = 0 & ( for k being Integer st i < k & k < j holds
t . k = 1 ) );
end;

:: deftheorem Def12 defines is_1_between TURING_1:def_12_:_
 for T being TuringStr
for t being Tape of T
for i, j being Integer holds
( t is_1_between i,j iff ( t . i = 0 & t . j = 0 & ( for k being Integer st i < k & k < j holds
t . k = 1 ) ) );

definition
let f be FinSequence of NAT ;
let T be TuringStr ;
let t be Tape of T;
predt storeData f means :Def13: :: TURING_1:def 13
 for i being Element of NAT st 1 <= i & i < len f holds
t is_1_between (Sum (Prefix (f,i))) + (2 * (i - 1)),(Sum (Prefix (f,(i + 1)))) + (2 * i);
end;

:: deftheorem Def13 defines storeData TURING_1:def_13_:_
 for f being FinSequence of NAT
for T being TuringStr
for t being Tape of T holds
( t storeData f iff for i being Element of NAT st 1 <= i & i < len f holds
t is_1_between (Sum (Prefix (f,i))) + (2 * (i - 1)),(Sum (Prefix (f,(i + 1)))) + (2 * i) );

theorem Th15: :: TURING_1:15
for T being TuringStr
for t being Tape of T
for s, n being Element of NAT st t storeData <*s,n*> holds
t is_1_between s,(s + n) + 2
proof
let T be TuringStr ; ::_thesis: for t being Tape of T
for s, n being Element of NAT st t storeData <*s,n*> holds
t is_1_between s,(s + n) + 2
let t be Tape of T; ::_thesis: for s, n being Element of NAT st t storeData <*s,n*> holds
t is_1_between s,(s + n) + 2
let s, n be Element of NAT ; ::_thesis: ( t storeData <*s,n*> implies t is_1_between s,(s + n) + 2 )
set f = <*s,n*>;
assume A1: t storeData <*s,n*> ; ::_thesis: t is_1_between s,(s + n) + 2
A2: len <*s,n*> = 2 by FINSEQ_1:44;
 ( (Sum (Prefix (<*s,n*>,1))) + (2 * (1 - 1)) = s & (Sum (Prefix (<*s,n*>,(1 + 1)))) + (2 * 1) = (s + n) + 2 ) by Th4;
hence  t is_1_between s,(s + n) + 2 by A1, A2, Def13; ::_thesis: verum
end;

theorem Th16: :: TURING_1:16
for T being TuringStr
for t being Tape of T
for s, n being Element of NAT st t is_1_between s,(s + n) + 2 holds
t storeData <*s,n*>
proof
let T be TuringStr ; ::_thesis: for t being Tape of T
for s, n being Element of NAT st t is_1_between s,(s + n) + 2 holds
t storeData <*s,n*>
let t be Tape of T; ::_thesis: for s, n being Element of NAT st t is_1_between s,(s + n) + 2 holds
t storeData <*s,n*>
let s, n be Element of NAT ; ::_thesis: ( t is_1_between s,(s + n) + 2 implies t storeData <*s,n*> )
set f = <*s,n*>;
assume A1: t is_1_between s,(s + n) + 2 ; ::_thesis: t storeData <*s,n*>
A2: (Sum (Prefix (<*s,n*>,(1 + 1)))) + (2 * 1) = (s + n) + 2 by Th4;
now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<_len_<*s,n*>_holds_
t_is_1_between_(Sum_(Prefix_(<*s,n*>,i)))_+_(2_*_(i_-_1)),(Sum_(Prefix_(<*s,n*>,(i_+_1))))_+_(2_*_i)
let i be Element of NAT ; ::_thesis: ( 1 <= i & i < len <*s,n*> implies t is_1_between (Sum (Prefix (<*s,n*>,i))) + (2 * (i - 1)),(Sum (Prefix (<*s,n*>,(i + 1)))) + (2 * i) )
assume that
A3: 1 <= i and
A4: i < len <*s,n*> ; ::_thesis: t is_1_between (Sum (Prefix (<*s,n*>,i))) + (2 * (i - 1)),(Sum (Prefix (<*s,n*>,(i + 1)))) + (2 * i)
 len <*s,n*> = 2 by FINSEQ_1:44;
then  i + 1 <= 1 + 1 by A4, INT_1:7;
then  i <= 1 by XREAL_1:6;
then  i = 1 by A3, XXREAL_0:1;
hence  t is_1_between (Sum (Prefix (<*s,n*>,i))) + (2 * (i - 1)),(Sum (Prefix (<*s,n*>,(i + 1)))) + (2 * i) by A1, A2, Th4; ::_thesis: verum
end;
hence  t storeData <*s,n*> by Def13; ::_thesis: verum
end;

theorem Th17: :: TURING_1:17
for T being TuringStr
for t being Tape of T
for s, n being Element of NAT st t storeData <*s,n*> holds
( t . s = 0 & t . ((s + n) + 2) = 0 & ( for i being Integer st s < i & i < (s + n) + 2 holds
t . i = 1 ) )
proof
let T be TuringStr ; ::_thesis: for t being Tape of T
for s, n being Element of NAT st t storeData <*s,n*> holds
( t . s = 0 & t . ((s + n) + 2) = 0 & ( for i being Integer st s < i & i < (s + n) + 2 holds
t . i = 1 ) )
let t be Tape of T; ::_thesis: for s, n being Element of NAT st t storeData <*s,n*> holds
( t . s = 0 & t . ((s + n) + 2) = 0 & ( for i being Integer st s < i & i < (s + n) + 2 holds
t . i = 1 ) )
let s, n be Element of NAT ; ::_thesis: ( t storeData <*s,n*> implies ( t . s = 0 & t . ((s + n) + 2) = 0 & ( for i being Integer st s < i & i < (s + n) + 2 holds
t . i = 1 ) ) )
assume  t storeData <*s,n*> ; ::_thesis: ( t . s = 0 & t . ((s + n) + 2) = 0 & ( for i being Integer st s < i & i < (s + n) + 2 holds
t . i = 1 ) )
then A1: t is_1_between s,(s + n) + 2 by Th15;
hence  ( t . s = 0 & t . ((s + n) + 2) = 0 ) by Def12; ::_thesis: for i being Integer st s < i & i < (s + n) + 2 holds
t . i = 1
thus  for i being Integer st s < i & i < (s + n) + 2 holds
t . i = 1 by A1, Def12; ::_thesis: verum
end;

theorem Th18: :: TURING_1:18
for T being TuringStr
for t being Tape of T
for s, n1, n2 being Element of NAT st t storeData <*s,n1,n2*> holds
( t is_1_between s,(s + n1) + 2 & t is_1_between (s + n1) + 2,((s + n1) + n2) + 4 )
proof
let T be TuringStr ; ::_thesis: for t being Tape of T
for s, n1, n2 being Element of NAT st t storeData <*s,n1,n2*> holds
( t is_1_between s,(s + n1) + 2 & t is_1_between (s + n1) + 2,((s + n1) + n2) + 4 )
let t be Tape of T; ::_thesis: for s, n1, n2 being Element of NAT st t storeData <*s,n1,n2*> holds
( t is_1_between s,(s + n1) + 2 & t is_1_between (s + n1) + 2,((s + n1) + n2) + 4 )
let s, n1, n2 be Element of NAT ; ::_thesis: ( t storeData <*s,n1,n2*> implies ( t is_1_between s,(s + n1) + 2 & t is_1_between (s + n1) + 2,((s + n1) + n2) + 4 ) )
set f = <*s,n1,n2*>;
assume A1: t storeData <*s,n1,n2*> ; ::_thesis: ( t is_1_between s,(s + n1) + 2 & t is_1_between (s + n1) + 2,((s + n1) + n2) + 4 )
A2: len <*s,n1,n2*> = 3 by FINSEQ_1:45;
 ( (Sum (Prefix (<*s,n1,n2*>,1))) + (2 * (1 - 1)) = s & (Sum (Prefix (<*s,n1,n2*>,(1 + 1)))) + (2 * 1) = (s + n1) + 2 ) by Th5;
hence  t is_1_between s,(s + n1) + 2 by A1, A2, Def13; ::_thesis: t is_1_between (s + n1) + 2,((s + n1) + n2) + 4
 ( (Sum (Prefix (<*s,n1,n2*>,2))) + (2 * (2 - 1)) = (s + n1) + 2 & (Sum (Prefix (<*s,n1,n2*>,(2 + 1)))) + (2 * 2) = ((s + n1) + n2) + 4 ) by Th5;
hence  t is_1_between (s + n1) + 2,((s + n1) + n2) + 4 by A1, A2, Def13; ::_thesis: verum
end;

theorem Th19: :: TURING_1:19
for T being TuringStr
for t being Tape of T
for s, n1, n2 being Element of NAT st t storeData <*s,n1,n2*> holds
( t . s = 0 & t . ((s + n1) + 2) = 0 & t . (((s + n1) + n2) + 4) = 0 & ( for i being Integer st s < i & i < (s + n1) + 2 holds
t . i = 1 ) & ( for i being Integer st (s + n1) + 2 < i & i < ((s + n1) + n2) + 4 holds
t . i = 1 ) )
proof
let T be TuringStr ; ::_thesis: for t being Tape of T
for s, n1, n2 being Element of NAT st t storeData <*s,n1,n2*> holds
( t . s = 0 & t . ((s + n1) + 2) = 0 & t . (((s + n1) + n2) + 4) = 0 & ( for i being Integer st s < i & i < (s + n1) + 2 holds
t . i = 1 ) & ( for i being Integer st (s + n1) + 2 < i & i < ((s + n1) + n2) + 4 holds
t . i = 1 ) )
let t be Tape of T; ::_thesis: for s, n1, n2 being Element of NAT st t storeData <*s,n1,n2*> holds
( t . s = 0 & t . ((s + n1) + 2) = 0 & t . (((s + n1) + n2) + 4) = 0 & ( for i being Integer st s < i & i < (s + n1) + 2 holds
t . i = 1 ) & ( for i being Integer st (s + n1) + 2 < i & i < ((s + n1) + n2) + 4 holds
t . i = 1 ) )
let s, n1, n2 be Element of NAT ; ::_thesis: ( t storeData <*s,n1,n2*> implies ( t . s = 0 & t . ((s + n1) + 2) = 0 & t . (((s + n1) + n2) + 4) = 0 & ( for i being Integer st s < i & i < (s + n1) + 2 holds
t . i = 1 ) & ( for i being Integer st (s + n1) + 2 < i & i < ((s + n1) + n2) + 4 holds
t . i = 1 ) ) )
assume  t storeData <*s,n1,n2*> ; ::_thesis: ( t . s = 0 & t . ((s + n1) + 2) = 0 & t . (((s + n1) + n2) + 4) = 0 & ( for i being Integer st s < i & i < (s + n1) + 2 holds
t . i = 1 ) & ( for i being Integer st (s + n1) + 2 < i & i < ((s + n1) + n2) + 4 holds
t . i = 1 ) )
then A1: ( t is_1_between s,(s + n1) + 2 & t is_1_between (s + n1) + 2,((s + n1) + n2) + 4 ) by Th18;
hence  ( t . s = 0 & t . ((s + n1) + 2) = 0 & t . (((s + n1) + n2) + 4) = 0 ) by Def12; ::_thesis: ( ( for i being Integer st s < i & i < (s + n1) + 2 holds
t . i = 1 ) & ( for i being Integer st (s + n1) + 2 < i & i < ((s + n1) + n2) + 4 holds
t . i = 1 ) )
thus  ( ( for i being Integer st s < i & i < (s + n1) + 2 holds
t . i = 1 ) & ( for i being Integer st (s + n1) + 2 < i & i < ((s + n1) + n2) + 4 holds
t . i = 1 ) ) by A1, Def12; ::_thesis: verum
end;

theorem Th20: :: TURING_1:20
for f being FinSequence of NAT
for s being Element of NAT st len f >= 1 holds
( Sum (Prefix ((<*s*> ^ f),1)) = s & Sum (Prefix ((<*s*> ^ f),2)) = s + (f /. 1) )
proof
let f be FinSequence of NAT ; ::_thesis: for s being Element of NAT st len f >= 1 holds
( Sum (Prefix ((<*s*> ^ f),1)) = s & Sum (Prefix ((<*s*> ^ f),2)) = s + (f /. 1) )
let s be Element of NAT ; ::_thesis: ( len f >= 1 implies ( Sum (Prefix ((<*s*> ^ f),1)) = s & Sum (Prefix ((<*s*> ^ f),2)) = s + (f /. 1) ) )
set g = <*s*>;
set h = <*s*> ^ f;
reconsider x1 = s as Element of INT by INT_1:def_2;
reconsider x2 = f /. 1 as Element of INT by INT_1:def_2;
assume A1: len f >= 1 ; ::_thesis: ( Sum (Prefix ((<*s*> ^ f),1)) = s & Sum (Prefix ((<*s*> ^ f),2)) = s + (f /. 1) )
then consider n being Nat such that
A2: len f = 1 + n by NAT_1:10;
A3: len <*s*> = 1 by FINSEQ_1:39;
then  Seg 1 = dom <*s*> by FINSEQ_1:def_3;
hence  Sum (Prefix ((<*s*> ^ f),1)) = Sum <*x1*> by FINSEQ_1:21
.= s by FINSOP_1:11 ;
::_thesis: Sum (Prefix ((<*s*> ^ f),2)) = s + (f /. 1)
len (<*s*> ^ f) = 1 + (len f) by A3, FINSEQ_1:22
.= 2 + n by A2 ;
then consider p2, q2 being FinSequence of NAT  such that
A4: len p2 = 2 and
 len q2 = n and
A5: <*s*> ^ f = p2 ^ q2 by FINSEQ_2:23;
f /. 1 = f . 1 by A1, FINSEQ_4:15
.= (<*s*> ^ f) . (1 + 1) by A1, A3, FINSEQ_7:3 ;
then A6: p2 . 2 = f /. 1 by A4, A5, FINSEQ_1:64;
 Seg 2 = dom p2 by A4, FINSEQ_1:def_3;
then A7: p2 = Prefix ((<*s*> ^ f),2) by A5, FINSEQ_1:21;
 (<*s*> ^ f) . 1 = s by FINSEQ_1:41;
then  p2 . 1 = s by A4, A5, FINSEQ_1:64;
hence  Sum (Prefix ((<*s*> ^ f),2)) = Sum <*x1,x2*> by A4, A7, A6, FINSEQ_1:44
.= s + (f /. 1) by RVSUM_1:77 ;
::_thesis: verum
end;

theorem Th21: :: TURING_1:21
for f being FinSequence of NAT
for s being Element of NAT st len f >= 3 holds
( Sum (Prefix ((<*s*> ^ f),1)) = s & Sum (Prefix ((<*s*> ^ f),2)) = s + (f /. 1) & Sum (Prefix ((<*s*> ^ f),3)) = (s + (f /. 1)) + (f /. 2) & Sum (Prefix ((<*s*> ^ f),4)) = ((s + (f /. 1)) + (f /. 2)) + (f /. 3) )
proof
let f be FinSequence of NAT ; ::_thesis: for s being Element of NAT st len f >= 3 holds
( Sum (Prefix ((<*s*> ^ f),1)) = s & Sum (Prefix ((<*s*> ^ f),2)) = s + (f /. 1) & Sum (Prefix ((<*s*> ^ f),3)) = (s + (f /. 1)) + (f /. 2) & Sum (Prefix ((<*s*> ^ f),4)) = ((s + (f /. 1)) + (f /. 2)) + (f /. 3) )
let s be Element of NAT ; ::_thesis: ( len f >= 3 implies ( Sum (Prefix ((<*s*> ^ f),1)) = s & Sum (Prefix ((<*s*> ^ f),2)) = s + (f /. 1) & Sum (Prefix ((<*s*> ^ f),3)) = (s + (f /. 1)) + (f /. 2) & Sum (Prefix ((<*s*> ^ f),4)) = ((s + (f /. 1)) + (f /. 2)) + (f /. 3) ) )
set g = <*s*>;
set h = <*s*> ^ f;
reconsider x1 = s as Element of INT by INT_1:def_2;
reconsider x2 = f /. 1 as Element of INT by INT_1:def_2;
reconsider x3 = f /. 2 as Element of INT by INT_1:def_2;
reconsider x4 = f /. 3 as Element of INT by INT_1:def_2;
assume A1: len f >= 3 ; ::_thesis: ( Sum (Prefix ((<*s*> ^ f),1)) = s & Sum (Prefix ((<*s*> ^ f),2)) = s + (f /. 1) & Sum (Prefix ((<*s*> ^ f),3)) = (s + (f /. 1)) + (f /. 2) & Sum (Prefix ((<*s*> ^ f),4)) = ((s + (f /. 1)) + (f /. 2)) + (f /. 3) )
then consider n being Nat such that
A2: len f = 3 + n by NAT_1:10;
reconsider n = n as Element of NAT by ORDINAL1:def_12;
A3: len <*s*> = 1 by FINSEQ_1:39;
then A4: len (<*s*> ^ f) = 1 + (len f) by FINSEQ_1:22
.= 4 + n by A2 ;
then consider p4, q4 being FinSequence of NAT  such that
A5: len p4 = 4 and
 len q4 = n and
A6: <*s*> ^ f = p4 ^ q4 by FINSEQ_2:23;
f /. 3 = f . 3 by A1, FINSEQ_4:15
.= (<*s*> ^ f) . (1 + 3) by A1, A3, FINSEQ_7:3 ;
then A7: p4 . 4 = f /. 3 by A5, A6, FINSEQ_1:64;
 Seg 4 = dom p4 by A5, FINSEQ_1:def_3;
then A8: p4 = Prefix ((<*s*> ^ f),4) by A6, FINSEQ_1:21;
A9: 1 <= len f by A1, XXREAL_0:2;
hence  ( Sum (Prefix ((<*s*> ^ f),1)) = s & Sum (Prefix ((<*s*> ^ f),2)) = s + (f /. 1) ) by Th20; ::_thesis: ( Sum (Prefix ((<*s*> ^ f),3)) = (s + (f /. 1)) + (f /. 2) & Sum (Prefix ((<*s*> ^ f),4)) = ((s + (f /. 1)) + (f /. 2)) + (f /. 3) )
 len (<*s*> ^ f) = 3 + (1 + n) by A4;
then consider p3, q3 being FinSequence of NAT  such that
A10: len p3 = 3 and
 len q3 = 1 + n and
A11: <*s*> ^ f = p3 ^ q3 by FINSEQ_2:23;
A12: f /. 1 = f . 1 by A9, FINSEQ_4:15
.= (<*s*> ^ f) . (1 + 1) by A9, A3, FINSEQ_7:3 ;
then A13: p4 . 2 = f /. 1 by A5, A6, FINSEQ_1:64;
A14: 2 <= len f by A1, XXREAL_0:2;
then A15: f /. 2 = f . 2 by FINSEQ_4:15
.= (<*s*> ^ f) . (1 + 2) by A3, A14, FINSEQ_7:3 ;
then A16: p4 . 3 = f /. 2 by A5, A6, FINSEQ_1:64;
A17: p3 . 2 = f /. 1 by A12, A10, A11, FINSEQ_1:64;
 Seg 3 = dom p3 by A10, FINSEQ_1:def_3;
then A18: p3 = Prefix ((<*s*> ^ f),3) by A11, FINSEQ_1:21;
A19: p3 . 3 = f /. 2 by A15, A10, A11, FINSEQ_1:64;
A20: (<*s*> ^ f) . 1 = s by FINSEQ_1:41;
then  p3 . 1 = s by A10, A11, FINSEQ_1:64;
then  p3 = <*s,(f /. 1),(f /. 2)*> by A10, A17, A19, FINSEQ_1:45;
hence  Sum (Prefix ((<*s*> ^ f),3)) = (s + (f /. 1)) + (f /. 2) by A18, RVSUM_1:78; ::_thesis: Sum (Prefix ((<*s*> ^ f),4)) = ((s + (f /. 1)) + (f /. 2)) + (f /. 3)
 p4 . 1 = s by A20, A5, A6, FINSEQ_1:64;
then  p4 = <*s,(f /. 1),(f /. 2),(f /. 3)*> by A5, A13, A16, A7, FINSEQ_4:76;
hence  Sum (Prefix ((<*s*> ^ f),4)) = ((x1 + x2) + x3) + x4 by A8, RVSUM_1:142
.= ((s + (f /. 1)) + (f /. 2)) + (f /. 3) ;
::_thesis: verum
end;

theorem Th22: :: TURING_1:22
for T being TuringStr
for t being Tape of T
for s being Element of NAT
for f being FinSequence of NAT st len f >= 1 & t storeData <*s*> ^ f holds
t is_1_between s,(s + (f /. 1)) + 2
proof
let T be TuringStr ; ::_thesis: for t being Tape of T
for s being Element of NAT
for f being FinSequence of NAT st len f >= 1 & t storeData <*s*> ^ f holds
t is_1_between s,(s + (f /. 1)) + 2
let t be Tape of T; ::_thesis: for s being Element of NAT
for f being FinSequence of NAT st len f >= 1 & t storeData <*s*> ^ f holds
t is_1_between s,(s + (f /. 1)) + 2
let s be Element of NAT ; ::_thesis: for f being FinSequence of NAT st len f >= 1 & t storeData <*s*> ^ f holds
t is_1_between s,(s + (f /. 1)) + 2
let f be FinSequence of NAT ; ::_thesis: ( len f >= 1 & t storeData <*s*> ^ f implies t is_1_between s,(s + (f /. 1)) + 2 )
set g = <*s*> ^ f;
assume that
A1: len f >= 1 and
A2: t storeData <*s*> ^ f ; ::_thesis: t is_1_between s,(s + (f /. 1)) + 2
 len <*s*> = 1 by FINSEQ_1:39;
then  len (<*s*> ^ f) = 1 + (len f) by FINSEQ_1:22;
then  len (<*s*> ^ f) >= 1 + 1 by A1, XREAL_1:7;
then A3: 1 < len (<*s*> ^ f) by XXREAL_0:2;
 ( (Sum (Prefix ((<*s*> ^ f),1))) + (2 * (1 - 1)) = s & (Sum (Prefix ((<*s*> ^ f),(1 + 1)))) + (2 * 1) = (s + (f /. 1)) + 2 ) by A1, Th20;
hence  t is_1_between s,(s + (f /. 1)) + 2 by A2, A3, Def13; ::_thesis: verum
end;

theorem Th23: :: TURING_1:23
for T being TuringStr
for t being Tape of T
for s being Element of NAT
for f being FinSequence of NAT st len f >= 3 & t storeData <*s*> ^ f holds
( t is_1_between s,(s + (f /. 1)) + 2 & t is_1_between (s + (f /. 1)) + 2,((s + (f /. 1)) + (f /. 2)) + 4 & t is_1_between ((s + (f /. 1)) + (f /. 2)) + 4,(((s + (f /. 1)) + (f /. 2)) + (f /. 3)) + 6 )
proof
let T be TuringStr ; ::_thesis: for t being Tape of T
for s being Element of NAT
for f being FinSequence of NAT st len f >= 3 & t storeData <*s*> ^ f holds
( t is_1_between s,(s + (f /. 1)) + 2 & t is_1_between (s + (f /. 1)) + 2,((s + (f /. 1)) + (f /. 2)) + 4 & t is_1_between ((s + (f /. 1)) + (f /. 2)) + 4,(((s + (f /. 1)) + (f /. 2)) + (f /. 3)) + 6 )
let t be Tape of T; ::_thesis: for s being Element of NAT
for f being FinSequence of NAT st len f >= 3 & t storeData <*s*> ^ f holds
( t is_1_between s,(s + (f /. 1)) + 2 & t is_1_between (s + (f /. 1)) + 2,((s + (f /. 1)) + (f /. 2)) + 4 & t is_1_between ((s + (f /. 1)) + (f /. 2)) + 4,(((s + (f /. 1)) + (f /. 2)) + (f /. 3)) + 6 )
let s be Element of NAT ; ::_thesis: for f being FinSequence of NAT st len f >= 3 & t storeData <*s*> ^ f holds
( t is_1_between s,(s + (f /. 1)) + 2 & t is_1_between (s + (f /. 1)) + 2,((s + (f /. 1)) + (f /. 2)) + 4 & t is_1_between ((s + (f /. 1)) + (f /. 2)) + 4,(((s + (f /. 1)) + (f /. 2)) + (f /. 3)) + 6 )
let f be FinSequence of NAT ; ::_thesis: ( len f >= 3 & t storeData <*s*> ^ f implies ( t is_1_between s,(s + (f /. 1)) + 2 & t is_1_between (s + (f /. 1)) + 2,((s + (f /. 1)) + (f /. 2)) + 4 & t is_1_between ((s + (f /. 1)) + (f /. 2)) + 4,(((s + (f /. 1)) + (f /. 2)) + (f /. 3)) + 6 ) )
set g = <*s*> ^ f;
assume that
A1: len f >= 3 and
A2: t storeData <*s*> ^ f ; ::_thesis: ( t is_1_between s,(s + (f /. 1)) + 2 & t is_1_between (s + (f /. 1)) + 2,((s + (f /. 1)) + (f /. 2)) + 4 & t is_1_between ((s + (f /. 1)) + (f /. 2)) + 4,(((s + (f /. 1)) + (f /. 2)) + (f /. 3)) + 6 )
thus  t is_1_between s,(s + (f /. 1)) + 2 by A1, A2, Th22, XXREAL_0:2; ::_thesis: ( t is_1_between (s + (f /. 1)) + 2,((s + (f /. 1)) + (f /. 2)) + 4 & t is_1_between ((s + (f /. 1)) + (f /. 2)) + 4,(((s + (f /. 1)) + (f /. 2)) + (f /. 3)) + 6 )
 len <*s*> = 1 by FINSEQ_1:39;
then  len (<*s*> ^ f) = 1 + (len f) by FINSEQ_1:22;
then A3: len (<*s*> ^ f) >= 3 + 1 by A1, XREAL_1:7;
then A4: 2 < len (<*s*> ^ f) by XXREAL_0:2;
 ( (Sum (Prefix ((<*s*> ^ f),2))) + (2 * (2 - 1)) = (s + (f /. 1)) + 2 & (Sum (Prefix ((<*s*> ^ f),(2 + 1)))) + (2 * 2) = ((s + (f /. 1)) + (f /. 2)) + 4 ) by A1, Th21;
hence  t is_1_between (s + (f /. 1)) + 2,((s + (f /. 1)) + (f /. 2)) + 4 by A2, A4, Def13; ::_thesis: t is_1_between ((s + (f /. 1)) + (f /. 2)) + 4,(((s + (f /. 1)) + (f /. 2)) + (f /. 3)) + 6
A5: 3 < len (<*s*> ^ f) by A3, XXREAL_0:2;
 ( (Sum (Prefix ((<*s*> ^ f),3))) + (2 * (3 - 1)) = ((s + (f /. 1)) + (f /. 2)) + 4 & (Sum (Prefix ((<*s*> ^ f),(3 + 1)))) + (2 * 3) = (((s + (f /. 1)) + (f /. 2)) + (f /. 3)) + 6 ) by A1, Th21;
hence  t is_1_between ((s + (f /. 1)) + (f /. 2)) + 4,(((s + (f /. 1)) + (f /. 2)) + (f /. 3)) + 6 by A2, A5, Def13; ::_thesis: verum
end;

begin

definition
func SumTuring  ->  strict TuringStr  means :Def14: :: TURING_1:def 14
( the Symbols of it = {0,1} & the FStates of it = SegM 5 & the Tran of it = Sum_Tran & the InitS of it = 0 & the AcceptS of it = 5 );
existence
 ex b1 being strict TuringStr st
( the Symbols of b1 = {0,1} & the FStates of b1 = SegM 5 & the Tran of b1 = Sum_Tran & the InitS of b1 = 0 & the AcceptS of b1 = 5 )
proof
set St = SegM 5;
reconsider p0 = 0 , qF = 5 as Element of SegM 5 by Th1;
set Sym = {0,1};
take  TuringStr(# {0,1},(SegM 5),Sum_Tran,p0,qF #) ; ::_thesis: ( the Symbols of TuringStr(# {0,1},(SegM 5),Sum_Tran,p0,qF #) = {0,1} & the FStates of TuringStr(# {0,1},(SegM 5),Sum_Tran,p0,qF #) = SegM 5 & the Tran of TuringStr(# {0,1},(SegM 5),Sum_Tran,p0,qF #) = Sum_Tran & the InitS of TuringStr(# {0,1},(SegM 5),Sum_Tran,p0,qF #) = 0 & the AcceptS of TuringStr(# {0,1},(SegM 5),Sum_Tran,p0,qF #) = 5 )
thus  ( the Symbols of TuringStr(# {0,1},(SegM 5),Sum_Tran,p0,qF #) = {0,1} & the FStates of TuringStr(# {0,1},(SegM 5),Sum_Tran,p0,qF #) = SegM 5 & the Tran of TuringStr(# {0,1},(SegM 5),Sum_Tran,p0,qF #) = Sum_Tran & the InitS of TuringStr(# {0,1},(SegM 5),Sum_Tran,p0,qF #) = 0 & the AcceptS of TuringStr(# {0,1},(SegM 5),Sum_Tran,p0,qF #) = 5 ) ; ::_thesis: verum
end;
uniqueness
 for b1, b2 being strict TuringStr st the Symbols of b1 = {0,1} & the FStates of b1 = SegM 5 & the Tran of b1 = Sum_Tran & the InitS of b1 = 0 & the AcceptS of b1 = 5 & the Symbols of b2 = {0,1} & the FStates of b2 = SegM 5 & the Tran of b2 = Sum_Tran & the InitS of b2 = 0 & the AcceptS of b2 = 5 holds
b1 = b2 ;
end;

:: deftheorem Def14 defines SumTuring TURING_1:def_14_:_
 for b1 being strict TuringStr holds
( b1 = SumTuring iff ( the Symbols of b1 = {0,1} & the FStates of b1 = SegM 5 & the Tran of b1 = Sum_Tran & the InitS of b1 = 0 & the AcceptS of b1 = 5 ) );

Lm1:  for n being Element of NAT st n <= 5 holds
n is State of SumTuring
proof
let n be Element of NAT ; ::_thesis: ( n <= 5 implies n is State of SumTuring )
assume A1: n <= 5 ; ::_thesis: n is State of SumTuring
 the FStates of SumTuring = SegM 5 by Def14;
hence  n is State of SumTuring by A1, Th1; ::_thesis: verum
end;

theorem Th24: :: TURING_1:24
for T being TuringStr
for t being Tape of T
for h being Integer
for s being Symbol of T st t . h = s holds
Tape-Chg (t,h,s) = t
proof
let T be TuringStr ; ::_thesis: for t being Tape of T
for h being Integer
for s being Symbol of T st t . h = s holds
Tape-Chg (t,h,s) = t
let t be Tape of T; ::_thesis: for h being Integer
for s being Symbol of T st t . h = s holds
Tape-Chg (t,h,s) = t
let h be Integer; ::_thesis: for s being Symbol of T st t . h = s holds
Tape-Chg (t,h,s) = t
let s be Symbol of T; ::_thesis: ( t . h = s implies Tape-Chg (t,h,s) = t )
 ex f being Function st
( t = f & dom f = INT & rng f c= the Symbols of T ) by FUNCT_2:def_2;
then A1: h in dom t by INT_1:def_2;
assume  t . h = s ; ::_thesis: Tape-Chg (t,h,s) = t
hence  Tape-Chg (t,h,s) = t by A1, FUNCT_7:96; ::_thesis: verum
end;

Lm2:  ( 0 in the Symbols of SumTuring & 1 in the Symbols of SumTuring )
proof
 ( 0 in {0,1} & 1 in {0,1} ) by TARSKI:def_2;
hence  ( 0 in the Symbols of SumTuring & 1 in the Symbols of SumTuring ) by Def14; ::_thesis: verum
end;

theorem Th25: :: TURING_1:25
for T being TuringStr
for s being All-State of T
for p, h, t being set st s = [p,h,t] & p <> the AcceptS of T holds
Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))]
proof
let T be TuringStr ; ::_thesis: for s being All-State of T
for p, h, t being set st s = [p,h,t] & p <> the AcceptS of T holds
Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))]
let s be All-State of T; ::_thesis: for p, h, t being set st s = [p,h,t] & p <> the AcceptS of T holds
Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))]
let p, h, t be set ; ::_thesis: ( s = [p,h,t] & p <> the AcceptS of T implies Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))] )
assume that
A1: s = [p,h,t] and
A2: p <> the AcceptS of T ; ::_thesis: Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))]
 s `1_3 = p by A1, MCART_1:64;
hence  Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))] by A2, Def6; ::_thesis: verum
end;

Lm3:  for s being All-State of SumTuring
for p, h, t being set st s = [p,h,t] & p <> 5 holds
Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))]
proof
let s be All-State of SumTuring; ::_thesis: for p, h, t being set st s = [p,h,t] & p <> 5 holds
Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))]
let p, h, t be set ; ::_thesis: ( s = [p,h,t] & p <> 5 implies Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))] )
assume A1: ( s = [p,h,t] & p <> 5 ) ; ::_thesis: Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))]
 5 = the AcceptS of SumTuring by Def14;
hence  Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))] by A1, Th25; ::_thesis: verum
end;

theorem Th26: :: TURING_1:26
for T being TuringStr
for t being Tape of T
for h being Integer
for s being Symbol of T
for i being set holds
( (Tape-Chg (t,h,s)) . h = s & ( i <> h implies (Tape-Chg (t,h,s)) . i = t . i ) )
proof
let tm be TuringStr ; ::_thesis: for t being Tape of tm
for h being Integer
for s being Symbol of tm
for i being set holds
( (Tape-Chg (t,h,s)) . h = s & ( i <> h implies (Tape-Chg (t,h,s)) . i = t . i ) )
let t be Tape of tm; ::_thesis: for h being Integer
for s being Symbol of tm
for i being set holds
( (Tape-Chg (t,h,s)) . h = s & ( i <> h implies (Tape-Chg (t,h,s)) . i = t . i ) )
let h be Integer; ::_thesis: for s being Symbol of tm
for i being set holds
( (Tape-Chg (t,h,s)) . h = s & ( i <> h implies (Tape-Chg (t,h,s)) . i = t . i ) )
let s be Symbol of tm; ::_thesis: for i being set holds
( (Tape-Chg (t,h,s)) . h = s & ( i <> h implies (Tape-Chg (t,h,s)) . i = t . i ) )
let i be set ; ::_thesis: ( (Tape-Chg (t,h,s)) . h = s & ( i <> h implies (Tape-Chg (t,h,s)) . i = t . i ) )
set t1 = Tape-Chg (t,h,s);
set p = h .--> s;
thus  (Tape-Chg (t,h,s)) . h = s by FUNCT_7:94; ::_thesis: ( i <> h implies (Tape-Chg (t,h,s)) . i = t . i )
A1: dom (h .--> s) = {h} by FUNCOP_1:13;
assume  i <> h ; ::_thesis: (Tape-Chg (t,h,s)) . i = t . i
then  not i in dom (h .--> s) by A1, TARSKI:def_1;
hence  (Tape-Chg (t,h,s)) . i = t . i by FUNCT_4:11; ::_thesis: verum
end;

Lm4:  for tm being TuringStr
for s being All-State of tm
for p being State of tm
for h being Element of INT
for t being Tape of tm
for m, d being Element of NAT st d = h & 1 is Symbol of tm & s = [p,h,t] & the Tran of tm . [p,1] = [p,1,1] & p <> the AcceptS of tm & ( for i being Integer st d <= i & i < d + m holds
t . i = 1 ) holds
(Computation s) . m = [p,(d + m),t]
proof
let tm be TuringStr ; ::_thesis: for s being All-State of tm
for p being State of tm
for h being Element of INT
for t being Tape of tm
for m, d being Element of NAT st d = h & 1 is Symbol of tm & s = [p,h,t] & the Tran of tm . [p,1] = [p,1,1] & p <> the AcceptS of tm & ( for i being Integer st d <= i & i < d + m holds
t . i = 1 ) holds
(Computation s) . m = [p,(d + m),t]
let s be All-State of tm; ::_thesis: for p being State of tm
for h being Element of INT
for t being Tape of tm
for m, d being Element of NAT st d = h & 1 is Symbol of tm & s = [p,h,t] & the Tran of tm . [p,1] = [p,1,1] & p <> the AcceptS of tm & ( for i being Integer st d <= i & i < d + m holds
t . i = 1 ) holds
(Computation s) . m = [p,(d + m),t]
let p be State of tm; ::_thesis: for h being Element of INT
for t being Tape of tm
for m, d being Element of NAT st d = h & 1 is Symbol of tm & s = [p,h,t] & the Tran of tm . [p,1] = [p,1,1] & p <> the AcceptS of tm & ( for i being Integer st d <= i & i < d + m holds
t . i = 1 ) holds
(Computation s) . m = [p,(d + m),t]
let h be Element of INT ; ::_thesis: for t being Tape of tm
for m, d being Element of NAT st d = h & 1 is Symbol of tm & s = [p,h,t] & the Tran of tm . [p,1] = [p,1,1] & p <> the AcceptS of tm & ( for i being Integer st d <= i & i < d + m holds
t . i = 1 ) holds
(Computation s) . m = [p,(d + m),t]
let t be Tape of tm; ::_thesis: for m, d being Element of NAT st d = h & 1 is Symbol of tm & s = [p,h,t] & the Tran of tm . [p,1] = [p,1,1] & p <> the AcceptS of tm & ( for i being Integer st d <= i & i < d + m holds
t . i = 1 ) holds
(Computation s) . m = [p,(d + m),t]
let m, d be Element of NAT ; ::_thesis: ( d = h & 1 is Symbol of tm & s = [p,h,t] & the Tran of tm . [p,1] = [p,1,1] & p <> the AcceptS of tm & ( for i being Integer st d <= i & i < d + m holds
t . i = 1 ) implies (Computation s) . m = [p,(d + m),t] )
assume that
A1: d = h and
A2: 1 is Symbol of tm and
A3: s = [p,h,t] and
A4: the Tran of tm . [p,1] = [p,1,1] and
A5: p <> the AcceptS of tm and
A6: for i being Integer st d <= i & i < d + m holds
t . i = 1 ; ::_thesis: (Computation s) . m = [p,(d + m),t]
defpred S1[ Element of NAT ] means ( $1 <= m implies (Computation s) . $1 = [p,(d + $1),t] );
A7: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A8: S1[k] ; ::_thesis: S1[k + 1]
now__::_thesis:_(_k_+_1_<=_m_implies_(Computation_s)_._(k_+_1)_=_[p,(d_+_(k_+_1)),t]_)
reconsider T = 1 as Symbol of tm by A2;
set dk = d + k;
reconsider ik = d + k as Element of INT by INT_1:def_2;
set sk = [p,ik,t];
reconsider tt = [p,ik,t] `3_3 as Tape of tm ;
assume A9: k + 1 <= m ; ::_thesis: (Computation s) . (k + 1) = [p,(d + (k + 1)),t]
then  k < m by NAT_1:13;
then  d + k < d + m by XREAL_1:8;
then A10: t . ik = 1 by A6, NAT_1:11;
A11: TRAN [p,ik,t] = the Tran of tm . [p,(tt . (Head [p,ik,t]))]
.= the Tran of tm . [p,(t . (Head [p,ik,t]))]
.= [p,1,1] by A4, A10 ;
then A12: offset (TRAN [p,ik,t]) = 1 by MCART_1:64;
A13: Tape-Chg (([p,ik,t] `3_3),(Head [p,ik,t]),((TRAN [p,ik,t]) `2_3)) = Tape-Chg (t,(Head [p,ik,t]),((TRAN [p,ik,t]) `2_3))
.= Tape-Chg (t,(d + k),((TRAN [p,ik,t]) `2_3))
.= Tape-Chg (t,(d + k),T) by A11, MCART_1:64
.= t by A10, Th24 ;
thus (Computation s) . (k + 1) = Following [p,ik,t] by A8, A9, Def7, NAT_1:13
.= [((TRAN [p,ik,t]) `1_3),((Head [p,ik,t]) + (offset (TRAN [p,ik,t]))),t] by A5, A13, Th25
.= [p,((Head [p,ik,t]) + (offset (TRAN [p,ik,t]))),t] by A11, MCART_1:64
.= [p,((d + k) + 1),t] by A12
.= [p,(d + (k + 1)),t] ; ::_thesis: verum
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
A14: S1[ 0 ] by A1, A3, Def7;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A14, A7);
hence  (Computation s) . m = [p,(d + m),t] ; ::_thesis: verum
end;

theorem Th27: :: TURING_1:27
for s being All-State of SumTuring
for t being Tape of SumTuring
for head, n1, n2 being Element of NAT st s = [0,head,t] & t storeData <*head,n1,n2*> holds
( s is Accept-Halt & (Result s) `2_3 = 1 + head & (Result s) `3_3 storeData <*(1 + head),(n1 + n2)*> )
proof
reconsider F = 0 as Symbol of SumTuring by Lm2;
let s be All-State of SumTuring; ::_thesis: for t being Tape of SumTuring
for head, n1, n2 being Element of NAT st s = [0,head,t] & t storeData <*head,n1,n2*> holds
( s is Accept-Halt & (Result s) `2_3 = 1 + head & (Result s) `3_3 storeData <*(1 + head),(n1 + n2)*> )
let t be Tape of SumTuring; ::_thesis: for head, n1, n2 being Element of NAT st s = [0,head,t] & t storeData <*head,n1,n2*> holds
( s is Accept-Halt & (Result s) `2_3 = 1 + head & (Result s) `3_3 storeData <*(1 + head),(n1 + n2)*> )
let h, n1, n2 be Element of NAT ; ::_thesis: ( s = [0,h,t] & t storeData <*h,n1,n2*> implies ( s is Accept-Halt & (Result s) `2_3 = 1 + h & (Result s) `3_3 storeData <*(1 + h),(n1 + n2)*> ) )
assume that
A1: s = [0,h,t] and
A2: t storeData <*h,n1,n2*> ; ::_thesis: ( s is Accept-Halt & (Result s) `2_3 = 1 + h & (Result s) `3_3 storeData <*(1 + h),(n1 + n2)*> )
A3: t . ((h + n1) + 2) = 0 by A2, Th19;
set j3 = (((h + n1) + n2) + 4) - 1;
reconsider h1 = h + 1 as Element of INT by INT_1:def_2;
A4: h < h1 by XREAL_1:29;
set t1 = Tape-Chg (t,h1,F);
A5: ((h + 1) + 1) + n1 = (h + n1) + 2 ;
reconsider p4 = 4 as State of SumTuring by Lm1;
reconsider m3 = (((h + n1) + n2) + 4) - 1 as Element of INT by INT_1:def_2;
set j2 = ((((h + n1) + n2) + 4) - 1) - 1;
reconsider m2 = ((((h + n1) + n2) + 4) - 1) - 1 as Element of INT by INT_1:def_2;
set j1 = (n1 + n2) + 1;
set Rs = (Computation s) . ((((((n1 + 1) + (n2 + 1)) + 1) + 1) + (1 + 1)) + (((n1 + n2) + 1) + 1));
reconsider p2 = 2 as State of SumTuring by Lm1;
reconsider i2 = h1 + 1 as Element of INT by INT_1:def_2;
reconsider nk = (h1 + 1) + n1 as Element of INT by INT_1:def_2;
set i3 = (((h + 1) + 1) + n1) + 1;
reconsider n3 = (((h + 1) + 1) + n1) + 1 as Element of INT by INT_1:def_2;
A6: (((((h + n1) + n2) + 4) - 1) - 1) - 1 = h + ((n1 + n2) + 1) ;
reconsider T = 1 as Symbol of SumTuring by Lm2;
set t2 = Tape-Chg ((Tape-Chg (t,h1,F)),nk,T);
A7: ( h1 + 1 <= ((h + 1) + 1) + n1 & h1 < h1 + 1 ) by NAT_1:11, XREAL_1:29;
set i4 = ((h + n1) + n2) + 4;
reconsider p0 = 0 as State of SumTuring by Lm1;
set s1 = [p0,h1,t];
A8: t . h = 0 by A2, Th19;
 h <= h + n1 by NAT_1:11;
then A9: h + 2 <= (h + n1) + 2 by XREAL_1:7;
A10: t . (((h + n1) + n2) + 4) = 0 by A2, Th19;
 h <= h + (n1 + n2) by NAT_1:11;
then A11: h + 4 <= ((h + n1) + n2) + 4 by XREAL_1:7;
then A12: ( h1 < h + 3 & (h + 4) - 1 <= (((h + n1) + n2) + 4) - 1 ) by XREAL_1:8, XREAL_1:9;
A13: h1 < h + 2 by XREAL_1:8;
then A14: h1 < (h + n1) + 2 by A9, XXREAL_0:2;
A15: t . h = 0 by A2, Th19;
A16: ( (Tape-Chg (t,h1,F)) . h = 0 & (Tape-Chg (t,h1,F)) . ((h + n1) + 2) = 0 & (Tape-Chg (t,h1,F)) . (((h + n1) + n2) + 4) = 0 & ( for i being Integer st h1 < i & i < ((h + 1) + 1) + n1 holds
(Tape-Chg (t,h1,F)) . i = 1 ) & ( for i being Integer st (h + n1) + 2 < i & i < ((h + n1) + n2) + 4 holds
(Tape-Chg (t,h1,F)) . i = 1 ) )
proof
thus  (Tape-Chg (t,h1,F)) . h = 0 by A15, A4, Th26; ::_thesis: ( (Tape-Chg (t,h1,F)) . ((h + n1) + 2) = 0 & (Tape-Chg (t,h1,F)) . (((h + n1) + n2) + 4) = 0 & ( for i being Integer st h1 < i & i < ((h + 1) + 1) + n1 holds
(Tape-Chg (t,h1,F)) . i = 1 ) & ( for i being Integer st (h + n1) + 2 < i & i < ((h + n1) + n2) + 4 holds
(Tape-Chg (t,h1,F)) . i = 1 ) )
thus  (Tape-Chg (t,h1,F)) . ((h + n1) + 2) = 0 by A3, A9, A13, Th26; ::_thesis: ( (Tape-Chg (t,h1,F)) . (((h + n1) + n2) + 4) = 0 & ( for i being Integer st h1 < i & i < ((h + 1) + 1) + n1 holds
(Tape-Chg (t,h1,F)) . i = 1 ) & ( for i being Integer st (h + n1) + 2 < i & i < ((h + n1) + n2) + 4 holds
(Tape-Chg (t,h1,F)) . i = 1 ) )
 h1 < h + 4 by XREAL_1:8;
hence  (Tape-Chg (t,h1,F)) . (((h + n1) + n2) + 4) = 0 by A10, A11, Th26; ::_thesis: ( ( for i being Integer st h1 < i & i < ((h + 1) + 1) + n1 holds
(Tape-Chg (t,h1,F)) . i = 1 ) & ( for i being Integer st (h + n1) + 2 < i & i < ((h + n1) + n2) + 4 holds
(Tape-Chg (t,h1,F)) . i = 1 ) )
hereby  ::_thesis: for i being Integer st (h + n1) + 2 < i & i < ((h + n1) + n2) + 4 holds
(Tape-Chg (t,h1,F)) . i = 1
let i be Integer; ::_thesis: ( h1 < i & i < ((h + 1) + 1) + n1 implies (Tape-Chg (t,h1,F)) . i = 1 )
assume that
A17: h1 < i and
A18: i < ((h + 1) + 1) + n1 ; ::_thesis: (Tape-Chg (t,h1,F)) . i = 1
A19: h < i by A4, A17, XXREAL_0:2;
thus (Tape-Chg (t,h1,F)) . i = t . i by A17, Th26
.= 1 by A2, A5, A18, A19, Th19 ; ::_thesis: verum
end;
hereby  ::_thesis: verum
let i be Integer; ::_thesis: ( (h + n1) + 2 < i & i < ((h + n1) + n2) + 4 implies (Tape-Chg (t,h1,F)) . i = 1 )
assume that
A20: (h + n1) + 2 < i and
A21: i < ((h + n1) + n2) + 4 ; ::_thesis: (Tape-Chg (t,h1,F)) . i = 1
thus (Tape-Chg (t,h1,F)) . i = t . i by A14, A20, Th26
.= 1 by A2, A20, A21, Th19 ; ::_thesis: verum
end;
end;
A22: for i being Integer st (h + 1) + 1 <= i & i < ((h + 1) + 1) + n1 holds
(Tape-Chg (t,h1,F)) . i = 1
proof
let i be Integer; ::_thesis: ( (h + 1) + 1 <= i & i < ((h + 1) + 1) + n1 implies (Tape-Chg (t,h1,F)) . i = 1 )
assume that
A23: (h + 1) + 1 <= i and
A24: i < ((h + 1) + 1) + n1 ; ::_thesis: (Tape-Chg (t,h1,F)) . i = 1
 h1 < h1 + 1 by XREAL_1:29;
then  h1 < i by A23, XXREAL_0:2;
hence  (Tape-Chg (t,h1,F)) . i = 1 by A16, A24; ::_thesis: verum
end;
set t3 = Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F);
A25: (Tape-Chg (t,h1,F)) . h1 = 0 by Th26;
A26: ( (Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)) . h1 = 0 & (Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)) . (((h + n1) + n2) + 4) = 0 & ( for i being Integer st h1 < i & i < ((h + n1) + n2) + 4 holds
(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)) . i = 1 ) )
proof
thus  (Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)) . h1 = 0 by A25, A7, Th26; ::_thesis: ( (Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)) . (((h + n1) + n2) + 4) = 0 & ( for i being Integer st h1 < i & i < ((h + n1) + n2) + 4 holds
(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)) . i = 1 ) )
 h + n1 <= (h + n1) + n2 by NAT_1:11;
then A27: ((h + 1) + 1) + n1 <= ((h + n1) + n2) + 2 by A5, XREAL_1:7;
 ((h + n1) + n2) + 2 < ((h + n1) + n2) + 4 by XREAL_1:8;
hence  (Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)) . (((h + n1) + n2) + 4) = 0 by A16, A27, Th26; ::_thesis: for i being Integer st h1 < i & i < ((h + n1) + n2) + 4 holds
(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)) . i = 1
hereby  ::_thesis: verum
let i be Integer; ::_thesis: ( h1 < i & i < ((h + n1) + n2) + 4 implies (Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)) . b1 = 1 )
assume that
A28: h1 < i and
A29: i < ((h + n1) + n2) + 4 ; ::_thesis: (Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)) . b1 = 1
percases ( i < ((h + 1) + 1) + n1 or i = ((h + 1) + 1) + n1 or i > ((h + 1) + 1) + n1 ) by XXREAL_0:1;
supposeA30: i < ((h + 1) + 1) + n1 ; ::_thesis: (Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)) . b1 = 1
hence (Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)) . i = (Tape-Chg (t,h1,F)) . i by Th26
.= 1 by A16, A28, A30 ;
::_thesis: verum
end;
suppose i = ((h + 1) + 1) + n1 ; ::_thesis: (Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)) . b1 = 1
hence  (Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)) . i = 1 by Th26; ::_thesis: verum
end;
supposeA31: i > ((h + 1) + 1) + n1 ; ::_thesis: (Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)) . b1 = 1
hence (Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)) . i = (Tape-Chg (t,h1,F)) . i by Th26
.= 1 by A16, A29, A31 ;
::_thesis: verum
end;
end;
end;
end;
A32: ( (Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F)) . h1 = 0 & (Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F)) . ((((h + n1) + n2) + 4) - 1) = 0 & ( for i being Integer st h1 < i & i < (((h + n1) + n2) + 4) - 1 holds
(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F)) . i = 1 ) )
proof
thus  (Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F)) . h1 = 0 by A26, A12, Th26; ::_thesis: ( (Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F)) . ((((h + n1) + n2) + 4) - 1) = 0 & ( for i being Integer st h1 < i & i < (((h + n1) + n2) + 4) - 1 holds
(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F)) . i = 1 ) )
thus  (Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F)) . ((((h + n1) + n2) + 4) - 1) = 0 by Th26; ::_thesis: for i being Integer st h1 < i & i < (((h + n1) + n2) + 4) - 1 holds
(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F)) . i = 1
hereby  ::_thesis: verum
let i be Integer; ::_thesis: ( h1 < i & i < (((h + n1) + n2) + 4) - 1 implies (Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F)) . i = 1 )
assume that
A33: h1 < i and
A34: i < (((h + n1) + n2) + 4) - 1 ; ::_thesis: (Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F)) . i = 1
A35: i < ((h + n1) + n2) + 4 by A34, XREAL_1:146, XXREAL_0:2;
thus (Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F)) . i = (Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)) . i by A34, Th26
.= 1 by A26, A33, A35 ; ::_thesis: verum
end;
end;
then A36: Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F) is_1_between h1,(h1 + (n1 + n2)) + 2 by Def12;
reconsider p3 = 3 as State of SumTuring by Lm1;
set sm = [p2,n3,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))];
reconsider n4 = ((h + n1) + n2) + 4 as Element of INT by INT_1:def_2;
set sp2 = [p2,n4,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))];
set sp3 = [p3,m3,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))];
reconsider p1 = 1 as State of SumTuring by Lm1;
set s2 = [p1,i2,(Tape-Chg (t,h1,F))];
set sn = [p1,nk,(Tape-Chg (t,h1,F))];
reconsider sn3 = [p1,nk,(Tape-Chg (t,h1,F))] `3_3 as Tape of SumTuring ;
A37: TRAN [p1,nk,(Tape-Chg (t,h1,F))] = Sum_Tran . [([p1,nk,(Tape-Chg (t,h1,F))] `1_3),(sn3 . (Head [p1,nk,(Tape-Chg (t,h1,F))]))] by Def14
.= Sum_Tran . [p1,(sn3 . (Head [p1,nk,(Tape-Chg (t,h1,F))]))]
.= Sum_Tran . [p1,((Tape-Chg (t,h1,F)) . (Head [p1,nk,(Tape-Chg (t,h1,F))]))]
.= [2,1,1] by A16, Th14 ;
then A38: offset (TRAN [p1,nk,(Tape-Chg (t,h1,F))]) = 1 by MCART_1:64;
reconsider sn3 = [p2,n4,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))] `3_3 as Tape of SumTuring ;
A39: TRAN [p2,n4,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))] = Sum_Tran . [([p2,n4,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))] `1_3),(sn3 . (Head [p2,n4,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))]))] by Def14
.= Sum_Tran . [p2,(sn3 . (Head [p2,n4,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))]))]
.= Sum_Tran . [p2,((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)) . (Head [p2,n4,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))]))]
.= [3,0,(- 1)] by A26, Th14 ;
then A40: offset (TRAN [p2,n4,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))]) = - 1 by MCART_1:64;
Tape-Chg (([p2,n4,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))] `3_3),(Head [p2,n4,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))]),((TRAN [p2,n4,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))]) `2_3)) = Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),(Head [p2,n4,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))]),((TRAN [p2,n4,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))]) `2_3))
.= Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),(((h + n1) + n2) + 4),((TRAN [p2,n4,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))]) `2_3))
.= Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),(((h + n1) + n2) + 4),F) by A39, MCART_1:64
.= Tape-Chg ((Tape-Chg (t,h1,F)),nk,T) by A26, Th24 ;
then A41: Following [p2,n4,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))] = [((TRAN [p2,n4,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))]) `1_3),((Head [p2,n4,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))]) + (offset (TRAN [p2,n4,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))]))),(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))] by Lm3
.= [3,((Head [p2,n4,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))]) + (offset (TRAN [p2,n4,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))]))),(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))] by A39, MCART_1:64
.= [3,((((h + n1) + n2) + 4) - 1),(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))] by A40 ;
Tape-Chg (([p1,nk,(Tape-Chg (t,h1,F))] `3_3),(Head [p1,nk,(Tape-Chg (t,h1,F))]),((TRAN [p1,nk,(Tape-Chg (t,h1,F))]) `2_3)) = Tape-Chg ((Tape-Chg (t,h1,F)),(Head [p1,nk,(Tape-Chg (t,h1,F))]),((TRAN [p1,nk,(Tape-Chg (t,h1,F))]) `2_3))
.= Tape-Chg ((Tape-Chg (t,h1,F)),nk,((TRAN [p1,nk,(Tape-Chg (t,h1,F))]) `2_3))
.= Tape-Chg ((Tape-Chg (t,h1,F)),nk,T) by A37, MCART_1:64 ;
then A42: Following [p1,nk,(Tape-Chg (t,h1,F))] = [((TRAN [p1,nk,(Tape-Chg (t,h1,F))]) `1_3),((Head [p1,nk,(Tape-Chg (t,h1,F))]) + (offset (TRAN [p1,nk,(Tape-Chg (t,h1,F))]))),(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))] by Lm3
.= [2,((Head [p1,nk,(Tape-Chg (t,h1,F))]) + (offset (TRAN [p1,nk,(Tape-Chg (t,h1,F))]))),(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))] by A37, MCART_1:64
.= [p2,n3,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))] by A38 ;
reconsider s3 = s `3_3 as Tape of SumTuring ;
A43: TRAN s = Sum_Tran . [(s `1_3),(s3 . (Head s))] by Def14
.= Sum_Tran . [0,(s3 . (Head s))] by A1, MCART_1:64
.= Sum_Tran . [0,(t . (Head s))] by A1, MCART_1:64
.= Sum_Tran . [0,(t . h)] by A1, MCART_1:64
.= [0,0,1] by A2, Th14, Th19 ;
then A44: offset (TRAN s) = 1 by MCART_1:64;
A45: h1 < (h1 + 1) + n1 by A7, XXREAL_0:2;
A46: for i being Integer st (((h + 1) + 1) + n1) + 1 <= i & i < ((((h + 1) + 1) + n1) + 1) + (n2 + 1) holds
(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)) . i = 1
proof
let i be Integer; ::_thesis: ( (((h + 1) + 1) + n1) + 1 <= i & i < ((((h + 1) + 1) + n1) + 1) + (n2 + 1) implies (Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)) . i = 1 )
assume that
A47: (((h + 1) + 1) + n1) + 1 <= i and
A48: i < ((((h + 1) + 1) + n1) + 1) + (n2 + 1) ; ::_thesis: (Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)) . i = 1
 nk < (((h + 1) + 1) + n1) + 1 by XREAL_1:29;
then  h1 < (((h + 1) + 1) + n1) + 1 by A45, XXREAL_0:2;
then  h1 < i by A47, XXREAL_0:2;
hence  (Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)) . i = 1 by A26, A48; ::_thesis: verum
end;
set sp5 = [p4,h1,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))];
set sp4 = [p4,m2,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))];
defpred S1[ Element of NAT ] means ( h + $1 < ((((h + n1) + n2) + 4) - 1) - 1 implies (Computation [p4,m2,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]) . $1 = [4,((((((h + n1) + n2) + 4) - 1) - 1) - $1),(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))] );
 ( the Tran of SumTuring . [p2,1] = [p2,1,1] & p2 <> the AcceptS of SumTuring ) by Def14, Th14;
then A49: (Computation [p2,n3,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))]) . (n2 + 1) = [2,(((h + n1) + n2) + 4),(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))] by A46, Lm4;
 h1 < (((h + n1) + n2) + 4) - 1 by A12, XXREAL_0:2;
then A50: (Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)) . ((((h + n1) + n2) + 4) - 1) = 1 by A26, XREAL_1:146;
reconsider sn3 = [p3,m3,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))] `3_3 as Tape of SumTuring ;
A51: TRAN [p3,m3,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))] = Sum_Tran . [([p3,m3,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))] `1_3),(sn3 . (Head [p3,m3,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))]))] by Def14
.= Sum_Tran . [p3,(sn3 . (Head [p3,m3,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))]))]
.= Sum_Tran . [p3,((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)) . (Head [p3,m3,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))]))]
.= [4,0,(- 1)] by A50, Th14 ;
then A52: offset (TRAN [p3,m3,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))]) = - 1 by MCART_1:64;
A53: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A54: S1[k] ; ::_thesis: S1[k + 1]
now__::_thesis:_(_h_+_(k_+_1)_<_((((h_+_n1)_+_n2)_+_4)_-_1)_-_1_implies_(Computation_[p4,m2,(Tape-Chg_((Tape-Chg_((Tape-Chg_(t,h1,F)),nk,T)),((((h_+_n1)_+_n2)_+_4)_-_1),F))])_._(k_+_1)_=_[4,((((((h_+_n1)_+_n2)_+_4)_-_1)_-_1)_-_(k_+_1)),(Tape-Chg_((Tape-Chg_((Tape-Chg_(t,h1,F)),nk,T)),((((h_+_n1)_+_n2)_+_4)_-_1),F))]_)
reconsider ik = (((((h + n1) + n2) + 4) - 1) - 1) - k as Element of INT by INT_1:def_2;
set k3 = (((((h + n1) + n2) + 4) - 1) - 1) - k;
set sk = [p4,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))];
reconsider tt = [p4,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))] `3_3 as Tape of SumTuring ;
assume A55: h + (k + 1) < ((((h + n1) + n2) + 4) - 1) - 1 ; ::_thesis: (Computation [p4,m2,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]) . (k + 1) = [4,((((((h + n1) + n2) + 4) - 1) - 1) - (k + 1)),(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]
then  h1 + k < (((((h + n1) + n2) + 4) - 1) - 1) + 0 ;
then A56: h1 - 0 < (((((h + n1) + n2) + 4) - 1) - 1) - k by XREAL_1:21;
 h1 >= 0 by NAT_1:2;
then reconsider ii = (((((h + n1) + n2) + 4) - 1) - 1) - k as Element of NAT by A56, INT_1:3, XXREAL_0:2;
 ((((h + n1) + n2) + 4) - 1) - 1 <= (((((h + n1) + n2) + 4) - 1) - 1) + k by INT_1:6;
then  (((((h + n1) + n2) + 4) - 1) - 1) - k <= ((((h + n1) + n2) + 4) - 1) - 1 by XREAL_1:20;
then  (((((h + n1) + n2) + 4) - 1) - 1) - k < (((h + n1) + n2) + 4) - 1 by XREAL_1:146, XXREAL_0:2;
then A57: (Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F)) . ii = 1 by A32, A56;
A58: TRAN [p4,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))] = Sum_Tran . [([p4,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))] `1_3),(tt . (Head [p4,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]))] by Def14
.= Sum_Tran . [p4,(tt . (Head [p4,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]))]
.= Sum_Tran . [p4,((Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F)) . (Head [p4,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]))]
.= [4,1,(- 1)] by A57, Th14 ;
then A59: offset (TRAN [p4,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]) = - 1 by MCART_1:64;
A60: Tape-Chg (([p4,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))] `3_3),(Head [p4,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]),((TRAN [p4,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]) `2_3)) = Tape-Chg ((Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F)),(Head [p4,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]),((TRAN [p4,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]) `2_3))
.= Tape-Chg ((Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F)),((((((h + n1) + n2) + 4) - 1) - 1) - k),((TRAN [p4,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]) `2_3))
.= Tape-Chg ((Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F)),((((((h + n1) + n2) + 4) - 1) - 1) - k),T) by A58, MCART_1:64
.= Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F) by A57, Th24 ;
 h + k < (h + k) + 1 by XREAL_1:29;
hence (Computation [p4,m2,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]) . (k + 1) = Following [p4,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))] by A54, A55, Def7, XXREAL_0:2
.= [((TRAN [p4,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]) `1_3),((Head [p4,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]) + (offset (TRAN [p4,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]))),(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))] by A60, Lm3
.= [4,((Head [p4,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]) + (offset (TRAN [p4,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]))),(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))] by A58, MCART_1:64
.= [4,(((((((h + n1) + n2) + 4) - 1) - 1) - k) + (- 1)),(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))] by A59
.= [4,((((((h + n1) + n2) + 4) - 1) - 1) - (k + 1)),(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))] ;
::_thesis: verum
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
A61: S1[ 0 ] by Def7;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A61, A53);
then A62: (Computation [p4,m2,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]) . ((n1 + n2) + 1) = [4,((((((h + n1) + n2) + 4) - 1) - 1) - ((n1 + n2) + 1)),(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))] by A6, XREAL_1:146
.= [p4,h1,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))] ;
reconsider s3 = [p0,h1,t] `3_3 as Tape of SumTuring ;
A63: TRAN [p0,h1,t] = Sum_Tran . [([p0,h1,t] `1_3),(s3 . (Head [p0,h1,t]))] by Def14
.= Sum_Tran . [p0,(s3 . (Head [p0,h1,t]))]
.= Sum_Tran . [p0,(t . (Head [p0,h1,t]))]
.= Sum_Tran . [0,(t . h1)]
.= [1,0,1] by A2, A4, A14, Th14, Th19 ;
then A64: offset (TRAN [p0,h1,t]) = 1 by MCART_1:64;
Tape-Chg (([p3,m3,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))] `3_3),(Head [p3,m3,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))]),((TRAN [p3,m3,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))]) `2_3)) = Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),(Head [p3,m3,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))]),((TRAN [p3,m3,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))]) `2_3))
.= Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),((TRAN [p3,m3,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))]) `2_3))
.= Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F) by A51, MCART_1:64 ;
then A65: Following [p3,m3,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))] = [((TRAN [p3,m3,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))]) `1_3),((Head [p3,m3,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))]) + (offset (TRAN [p3,m3,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))]))),(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))] by Lm3
.= [4,((Head [p3,m3,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))]) + (offset (TRAN [p3,m3,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))]))),(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))] by A51, MCART_1:64
.= [p4,m2,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))] by A52 ;
A66: now__::_thesis:_Following_[p4,h1,(Tape-Chg_((Tape-Chg_((Tape-Chg_(t,h1,F)),nk,T)),((((h_+_n1)_+_n2)_+_4)_-_1),F))]_=_[5,(h1_+_0),(Tape-Chg_((Tape-Chg_((Tape-Chg_(t,h1,F)),nk,T)),((((h_+_n1)_+_n2)_+_4)_-_1),F))]
reconsider tt = [p4,h1,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))] `3_3 as Tape of SumTuring ;
A67: TRAN [p4,h1,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))] = Sum_Tran . [([p4,h1,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))] `1_3),(tt . (Head [p4,h1,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]))] by Def14
.= Sum_Tran . [4,(tt . (Head [p4,h1,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]))]
.= Sum_Tran . [4,((Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F)) . (Head [p4,h1,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]))]
.= [5,0,0] by A32, Th14 ;
then A68: offset (TRAN [p4,h1,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]) = 0 by MCART_1:64;
Tape-Chg (([p4,h1,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))] `3_3),(Head [p4,h1,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]),((TRAN [p4,h1,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]) `2_3)) = Tape-Chg ((Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F)),(Head [p4,h1,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]),((TRAN [p4,h1,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]) `2_3))
.= Tape-Chg ((Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F)),h1,((TRAN [p4,h1,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]) `2_3))
.= Tape-Chg ((Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F)),h1,F) by A67, MCART_1:64
.= Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F) by A32, Th24 ;
hence  Following [p4,h1,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))] = [((TRAN [p4,h1,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]) `1_3),((Head [p4,h1,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]) + (offset (TRAN [p4,h1,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]))),(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))] by Lm3
.= [5,((Head [p4,h1,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]) + (offset (TRAN [p4,h1,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))]))),(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))] by A67, MCART_1:64
.= [5,(h1 + 0),(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))] by A68 ;
::_thesis: verum
end;
Tape-Chg (([p0,h1,t] `3_3),(Head [p0,h1,t]),((TRAN [p0,h1,t]) `2_3)) = Tape-Chg (t,(Head [p0,h1,t]),((TRAN [p0,h1,t]) `2_3))
.= Tape-Chg (t,h1,((TRAN [p0,h1,t]) `2_3))
.= Tape-Chg (t,h1,F) by A63, MCART_1:64 ;
then A69: Following [p0,h1,t] = [((TRAN [p0,h1,t]) `1_3),((Head [p0,h1,t]) + (offset (TRAN [p0,h1,t]))),(Tape-Chg (t,h1,F))] by Lm3
.= [1,((Head [p0,h1,t]) + (offset (TRAN [p0,h1,t]))),(Tape-Chg (t,h1,F))] by A63, MCART_1:64
.= [p1,i2,(Tape-Chg (t,h1,F))] by A64 ;
Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)) = Tape-Chg (t,(Head s),((TRAN s) `2_3)) by A1, MCART_1:64
.= Tape-Chg (t,h,((TRAN s) `2_3)) by A1, MCART_1:64
.= Tape-Chg (t,h,F) by A43, MCART_1:64
.= t by A8, Th24 ;
then A70: Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),t] by A1, Lm3
.= [0,((Head s) + (offset (TRAN s))),t] by A43, MCART_1:64
.= [p0,h1,t] by A1, A44, MCART_1:64 ;
(Computation s) . (1 + 1) = Following ((Computation s) . 1) by Def7
.= [p1,i2,(Tape-Chg (t,h1,F))] by A70, A69, Th9 ;
then  (Computation s) . (((((n1 + 1) + (n2 + 1)) + 1) + 1) + (1 + 1)) = (Computation [p1,i2,(Tape-Chg (t,h1,F))]) . ((((n1 + 1) + (n2 + 1)) + 1) + 1) by Th10;
then (Computation s) . (((((n1 + 1) + (n2 + 1)) + 1) + 1) + (1 + 1)) = Following ((Computation [p1,i2,(Tape-Chg (t,h1,F))]) . (((n1 + 1) + (n2 + 1)) + 1)) by Def7
.= Following (Following ((Computation [p1,i2,(Tape-Chg (t,h1,F))]) . ((n1 + 1) + (n2 + 1)))) by Def7
.= Following (Following ((Computation ((Computation [p1,i2,(Tape-Chg (t,h1,F))]) . (n1 + 1))) . (n2 + 1))) by Th10 ;
then A71: (Computation s) . ((((((n1 + 1) + (n2 + 1)) + 1) + 1) + (1 + 1)) + (((n1 + n2) + 1) + 1)) = (Computation (Following (Following ((Computation ((Computation [p1,i2,(Tape-Chg (t,h1,F))]) . (n1 + 1))) . (n2 + 1))))) . (((n1 + n2) + 1) + 1) by Th10
.= (Computation (Following (Following ((Computation (Following ((Computation [p1,i2,(Tape-Chg (t,h1,F))]) . n1))) . (n2 + 1))))) . (((n1 + n2) + 1) + 1) by Def7 ;
 ( the Tran of SumTuring . [p1,1] = [p1,1,1] & p1 <> the AcceptS of SumTuring ) by Def14, Th14;
then  (Computation s) . ((((((n1 + 1) + (n2 + 1)) + 1) + 1) + (1 + 1)) + (((n1 + n2) + 1) + 1)) = (Computation (Following [p3,m3,(Tape-Chg ((Tape-Chg (t,h1,F)),nk,T))])) . (((n1 + n2) + 1) + 1) by A22, A42, A49, A41, A71, Lm4;
then A72: (Computation s) . ((((((n1 + 1) + (n2 + 1)) + 1) + 1) + (1 + 1)) + (((n1 + n2) + 1) + 1)) = [5,h1,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F))] by A65, A62, A66, Def7;
then A73: ((Computation s) . ((((((n1 + 1) + (n2 + 1)) + 1) + 1) + (1 + 1)) + (((n1 + n2) + 1) + 1))) `1_3 = 5 by MCART_1:64
.= the AcceptS of SumTuring by Def14 ;
hence  s is Accept-Halt by Def8; ::_thesis: ( (Result s) `2_3 = 1 + h & (Result s) `3_3 storeData <*(1 + h),(n1 + n2)*> )
then A74: Result s = (Computation s) . ((((((n1 + 1) + (n2 + 1)) + 1) + 1) + (1 + 1)) + (((n1 + n2) + 1) + 1)) by A73, Def9;
hence  (Result s) `2_3 = 1 + h by A72, MCART_1:64; ::_thesis: (Result s) `3_3 storeData <*(1 + h),(n1 + n2)*>
 (Result s) `3_3 = Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,F)),nk,T)),((((h + n1) + n2) + 4) - 1),F) by A72, A74, MCART_1:64;
hence  (Result s) `3_3 storeData <*(1 + h),(n1 + n2)*> by A36, Th16; ::_thesis: verum
end;

definition
let T be TuringStr ;
let F be Function;
predT computes F means :Def15: :: TURING_1:def 15
 for s being All-State of T
for t being Tape of T
for a being Element of NAT
for x being FinSequence of NAT st x in dom F & s = [ the InitS of T,a,t] & t storeData <*a*> ^ x holds
( s is Accept-Halt & ex b, y being Element of NAT st
( (Result s) `2_3 = b & y = F . x & (Result s) `3_3 storeData <*b*> ^ <*y*> ) );
end;

:: deftheorem Def15 defines computes TURING_1:def_15_:_
 for T being TuringStr
for F being Function holds
( T computes F iff for s being All-State of T
for t being Tape of T
for a being Element of NAT
for x being FinSequence of NAT st x in dom F & s = [ the InitS of T,a,t] & t storeData <*a*> ^ x holds
( s is Accept-Halt & ex b, y being Element of NAT st
( (Result s) `2_3 = b & y = F . x & (Result s) `3_3 storeData <*b*> ^ <*y*> ) ) );

theorem Th28: :: TURING_1:28
dom [+] c= 2 -tuples_on NAT
proof
 arity (1 proj 1) = 1 by COMPUT_1:37;
then  dom [+] c= (1 + 1) -tuples_on NAT by COMPUT_1:56;
hence  dom [+] c= 2 -tuples_on NAT ; ::_thesis: verum
end;

theorem :: TURING_1:29
SumTuring computes [+]
proof
now__::_thesis:_for_s_being_All-State_of_SumTuring
for_t_being_Tape_of_SumTuring
for_h1_being_Element_of_NAT_
for_x_being_FinSequence_of_NAT_st_x_in_dom_[+]_&_s_=_[_the_InitS_of_SumTuring,h1,t]_&_t_storeData_<*h1*>_^_x_holds_
(_s_is_Accept-Halt_&_ex_h2,_y_being_Element_of_NAT_st_
(_(Result_s)_`2_3_=_h2_&_y_=_[+]_._x_&_(Result_s)_`3_3_storeData_<*h2*>_^_<*y*>_)_)
let s be All-State of SumTuring; ::_thesis: for t being Tape of SumTuring
for h1 being Element of NAT
for x being FinSequence of NAT st x in dom [+] & s = [ the InitS of SumTuring,h1,t] & t storeData <*h1*> ^ x holds
( s is Accept-Halt & ex h2, y being Element of NAT st
( (Result s) `2_3 = h2 & y = [+] . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> ) )
let t be Tape of SumTuring; ::_thesis: for h1 being Element of NAT
for x being FinSequence of NAT st x in dom [+] & s = [ the InitS of SumTuring,h1,t] & t storeData <*h1*> ^ x holds
( s is Accept-Halt & ex h2, y being Element of NAT st
( (Result s) `2_3 = h2 & y = [+] . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> ) )
let h1 be Element of NAT ; ::_thesis: for x being FinSequence of NAT st x in dom [+] & s = [ the InitS of SumTuring,h1,t] & t storeData <*h1*> ^ x holds
( s is Accept-Halt & ex h2, y being Element of NAT st
( (Result s) `2_3 = h2 & y = [+] . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> ) )
let x be FinSequence of NAT ; ::_thesis: ( x in dom [+] & s = [ the InitS of SumTuring,h1,t] & t storeData <*h1*> ^ x implies ( s is Accept-Halt & ex h2, y being Element of NAT st
( (Result s) `2_3 = h2 & y = [+] . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> ) ) )
assume that
A1: x in dom [+] and
A2: s = [ the InitS of SumTuring,h1,t] and
A3: t storeData <*h1*> ^ x ; ::_thesis: ( s is Accept-Halt & ex h2, y being Element of NAT st
( (Result s) `2_3 = h2 & y = [+] . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> ) )
 x is Tuple of 2, NAT by A1, Th28, FINSEQ_2:131;
then consider i, j being Element of NAT  such that
A4: x = <*i,j*> by FINSEQ_2:100;
A5: s = [0,h1,t] by A2, Def14;
A6: <*h1*> ^ x = <*h1,i,j*> by A4, FINSEQ_1:43;
hence  s is Accept-Halt by A3, A5, Th27; ::_thesis: ex h2, y being Element of NAT st
( (Result s) `2_3 = h2 & y = [+] . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> )
take h2 = 1 + h1; ::_thesis: ex y being Element of NAT st
( (Result s) `2_3 = h2 & y = [+] . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> )
take y = i + j; ::_thesis: ( (Result s) `2_3 = h2 & y = [+] . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> )
 t storeData <*h1,i,j*> by A3, A4, FINSEQ_1:43;
hence  (Result s) `2_3 = h2 by A5, Th27; ::_thesis: ( y = [+] . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> )
thus  y = [+] . x by A4, COMPUT_1:85; ::_thesis: (Result s) `3_3 storeData <*h2*> ^ <*y*>
 (Result s) `3_3 storeData <*(1 + h1),(i + j)*> by A3, A5, A6, Th27;
hence  (Result s) `3_3 storeData <*h2*> ^ <*y*> ; ::_thesis: verum
end;
hence  SumTuring computes [+] by Def15; ::_thesis: verum
end;

begin

definition
func Succ_Tran  ->  Function of [:(SegM 4),{0,1}:],[:(SegM 4),{0,1},{(- 1),0,1}:] equals :: TURING_1:def 16
(((((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) +* ([3,1] .--> [3,1,(- 1)])) +* ([3,0] .--> [4,0,0]);
coherence
 (((((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) +* ([3,1] .--> [3,1,(- 1)])) +* ([3,0] .--> [4,0,0]) is Function of [:(SegM 4),{0,1}:],[:(SegM 4),{0,1},{(- 1),0,1}:]
proof
reconsider p0 = 0 , p1 = 1, p2 = 2, p3 = 3, p4 = 4 as Element of SegM 4 by Th1;
set A = [:(SegM 4),{0,1}:];
set B = {(- 1),0,1};
set C = [:(SegM 4),{0,1},{(- 1),0,1}:];
reconsider b0 = 0 , b1 = 1 as Element of {0,1} by TARSKI:def_2;
reconsider L = - 1 as Element of {(- 1),0,1} by ENUMSET1:def_1;
reconsider h = 0 , R = 1 as Element of {(- 1),0,1} by ENUMSET1:def_1;
 [:(SegM 4),{0,1},{(- 1),0,1}:] = [:[:(SegM 4),{0,1}:],{(- 1),0,1}:] by ZFMISC_1:def_3;
then reconsider OP = id ([:(SegM 4),{0,1}:],{(- 1),0,1},h) as Function of [:(SegM 4),{0,1}:],[:(SegM 4),{0,1},{(- 1),0,1}:] ;
 (((((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) +* ([3,1] .--> [3,1,(- 1)])) +* ([3,0] .--> [4,0,0]) = ((((((OP +* ([p0,b0] .--> [p1,b0,R])) +* ([p1,b1] .--> [p1,b1,R])) +* ([p1,b0] .--> [p2,b1,R])) +* ([p2,b0] .--> [p3,b0,L])) +* ([p2,b1] .--> [p3,b0,L])) +* ([p3,b1] .--> [p3,b1,L])) +* ([p3,b0] .--> [p4,b0,h]) ;
hence  (((((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) +* ([3,1] .--> [3,1,(- 1)])) +* ([3,0] .--> [4,0,0]) is Function of [:(SegM 4),{0,1}:],[:(SegM 4),{0,1},{(- 1),0,1}:] ; ::_thesis: verum
end;
end;

:: deftheorem  defines Succ_Tran TURING_1:def_16_:_
 Succ_Tran = (((((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) +* ([3,1] .--> [3,1,(- 1)])) +* ([3,0] .--> [4,0,0]);

theorem Th30: :: TURING_1:30
( Succ_Tran . [0,0] = [1,0,1] & Succ_Tran . [1,1] = [1,1,1] & Succ_Tran . [1,0] = [2,1,1] & Succ_Tran . [2,0] = [3,0,(- 1)] & Succ_Tran . [2,1] = [3,0,(- 1)] & Succ_Tran . [3,1] = [3,1,(- 1)] & Succ_Tran . [3,0] = [4,0,0] )
proof
set x = [1,1];
set x1 = [0,0];
set p1 = [0,0] .--> [1,0,1];
set p2 = [1,1] .--> [1,1,1];
set p3 = [1,0] .--> [2,1,1];
set p4 = [2,0] .--> [3,0,(- 1)];
set p5 = [2,1] .--> [3,0,(- 1)];
set p6 = [3,1] .--> [3,1,(- 1)];
set f = id ([:(SegM 4),{0,1}:],{(- 1),0,1},0);
thus Succ_Tran . [0,0] = (((((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) +* ([3,1] .--> [3,1,(- 1)])) . [0,0] by Th2
.= ((((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) . [0,0] by Th2
.= (((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) . [0,0] by Th2
.= ((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) . [0,0] by Th2
.= (((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) . [0,0] by Th2
.= ((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) . [0,0] by Th3
.= [1,0,1] by FUNCT_7:94 ; ::_thesis: ( Succ_Tran . [1,1] = [1,1,1] & Succ_Tran . [1,0] = [2,1,1] & Succ_Tran . [2,0] = [3,0,(- 1)] & Succ_Tran . [2,1] = [3,0,(- 1)] & Succ_Tran . [3,1] = [3,1,(- 1)] & Succ_Tran . [3,0] = [4,0,0] )
thus Succ_Tran . [1,1] = (((((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) +* ([3,1] .--> [3,1,(- 1)])) . [1,1] by Th2
.= ((((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) . [1,1] by Th2
.= (((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) . [1,1] by Th2
.= ((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) . [1,1] by Th2
.= (((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) . [1,1] by Th3
.= [1,1,1] by FUNCT_7:94 ; ::_thesis: ( Succ_Tran . [1,0] = [2,1,1] & Succ_Tran . [2,0] = [3,0,(- 1)] & Succ_Tran . [2,1] = [3,0,(- 1)] & Succ_Tran . [3,1] = [3,1,(- 1)] & Succ_Tran . [3,0] = [4,0,0] )
set x = [1,0];
thus Succ_Tran . [1,0] = (((((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) +* ([3,1] .--> [3,1,(- 1)])) . [1,0] by Th2
.= ((((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) . [1,0] by Th2
.= (((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) . [1,0] by Th2
.= ((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) . [1,0] by Th2
.= [2,1,1] by FUNCT_7:94 ; ::_thesis: ( Succ_Tran . [2,0] = [3,0,(- 1)] & Succ_Tran . [2,1] = [3,0,(- 1)] & Succ_Tran . [3,1] = [3,1,(- 1)] & Succ_Tran . [3,0] = [4,0,0] )
set x = [2,0];
thus Succ_Tran . [2,0] = (((((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) +* ([3,1] .--> [3,1,(- 1)])) . [2,0] by Th2
.= ((((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) . [2,0] by Th2
.= (((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) . [2,0] by Th3
.= [3,0,(- 1)] by FUNCT_7:94 ; ::_thesis: ( Succ_Tran . [2,1] = [3,0,(- 1)] & Succ_Tran . [3,1] = [3,1,(- 1)] & Succ_Tran . [3,0] = [4,0,0] )
set x = [2,1];
thus Succ_Tran . [2,1] = (((((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) +* ([3,1] .--> [3,1,(- 1)])) . [2,1] by Th2
.= ((((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) . [2,1] by Th2
.= [3,0,(- 1)] by FUNCT_7:94 ; ::_thesis: ( Succ_Tran . [3,1] = [3,1,(- 1)] & Succ_Tran . [3,0] = [4,0,0] )
set x = [3,1];
thus Succ_Tran . [3,1] = (((((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,1,1])) +* ([1,0] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) +* ([3,1] .--> [3,1,(- 1)])) . [3,1] by Th3
.= [3,1,(- 1)] by FUNCT_7:94 ; ::_thesis: Succ_Tran . [3,0] = [4,0,0]
thus  Succ_Tran . [3,0] = [4,0,0] by FUNCT_7:94; ::_thesis: verum
end;

definition
func SuccTuring  ->  strict TuringStr  means :Def17: :: TURING_1:def 17
( the Symbols of it = {0,1} & the FStates of it = SegM 4 & the Tran of it = Succ_Tran & the InitS of it = 0 & the AcceptS of it = 4 );
existence
 ex b1 being strict TuringStr st
( the Symbols of b1 = {0,1} & the FStates of b1 = SegM 4 & the Tran of b1 = Succ_Tran & the InitS of b1 = 0 & the AcceptS of b1 = 4 )
proof
set St = SegM 4;
reconsider p0 = 0 , qF = 4 as Element of SegM 4 by Th1;
set Sym = {0,1};
take  TuringStr(# {0,1},(SegM 4),Succ_Tran,p0,qF #) ; ::_thesis: ( the Symbols of TuringStr(# {0,1},(SegM 4),Succ_Tran,p0,qF #) = {0,1} & the FStates of TuringStr(# {0,1},(SegM 4),Succ_Tran,p0,qF #) = SegM 4 & the Tran of TuringStr(# {0,1},(SegM 4),Succ_Tran,p0,qF #) = Succ_Tran & the InitS of TuringStr(# {0,1},(SegM 4),Succ_Tran,p0,qF #) = 0 & the AcceptS of TuringStr(# {0,1},(SegM 4),Succ_Tran,p0,qF #) = 4 )
thus  ( the Symbols of TuringStr(# {0,1},(SegM 4),Succ_Tran,p0,qF #) = {0,1} & the FStates of TuringStr(# {0,1},(SegM 4),Succ_Tran,p0,qF #) = SegM 4 & the Tran of TuringStr(# {0,1},(SegM 4),Succ_Tran,p0,qF #) = Succ_Tran & the InitS of TuringStr(# {0,1},(SegM 4),Succ_Tran,p0,qF #) = 0 & the AcceptS of TuringStr(# {0,1},(SegM 4),Succ_Tran,p0,qF #) = 4 ) ; ::_thesis: verum
end;
uniqueness
 for b1, b2 being strict TuringStr st the Symbols of b1 = {0,1} & the FStates of b1 = SegM 4 & the Tran of b1 = Succ_Tran & the InitS of b1 = 0 & the AcceptS of b1 = 4 & the Symbols of b2 = {0,1} & the FStates of b2 = SegM 4 & the Tran of b2 = Succ_Tran & the InitS of b2 = 0 & the AcceptS of b2 = 4 holds
b1 = b2 ;
end;

:: deftheorem Def17 defines SuccTuring TURING_1:def_17_:_
 for b1 being strict TuringStr holds
( b1 = SuccTuring iff ( the Symbols of b1 = {0,1} & the FStates of b1 = SegM 4 & the Tran of b1 = Succ_Tran & the InitS of b1 = 0 & the AcceptS of b1 = 4 ) );

Lm5:  for n being Element of NAT st n <= 4 holds
n is State of SuccTuring
proof
let n be Element of NAT ; ::_thesis: ( n <= 4 implies n is State of SuccTuring )
assume A1: n <= 4 ; ::_thesis: n is State of SuccTuring
 the FStates of SuccTuring = SegM 4 by Def17;
hence  n is State of SuccTuring by A1, Th1; ::_thesis: verum
end;

Lm6:  ( 0 in the Symbols of SuccTuring & 1 in the Symbols of SuccTuring )
proof
 ( 0 in {0,1} & 1 in {0,1} ) by TARSKI:def_2;
hence  ( 0 in the Symbols of SuccTuring & 1 in the Symbols of SuccTuring ) by Def17; ::_thesis: verum
end;

Lm7:  for s being All-State of SuccTuring
for p, h, t being set st s = [p,h,t] & p <> 4 holds
Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))]
proof
let s be All-State of SuccTuring; ::_thesis: for p, h, t being set st s = [p,h,t] & p <> 4 holds
Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))]
let p, h, t be set ; ::_thesis: ( s = [p,h,t] & p <> 4 implies Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))] )
assume A1: ( s = [p,h,t] & p <> 4 ) ; ::_thesis: Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))]
 4 = the AcceptS of SuccTuring by Def17;
hence  Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))] by A1, Th25; ::_thesis: verum
end;

theorem Th31: :: TURING_1:31
for s being All-State of SuccTuring
for t being Tape of SuccTuring
for head, n being Element of NAT st s = [0,head,t] & t storeData <*head,n*> holds
( s is Accept-Halt & (Result s) `2_3 = head & (Result s) `3_3 storeData <*head,(n + 1)*> )
proof
reconsider F = 0 as Symbol of SuccTuring by Lm6;
let s be All-State of SuccTuring; ::_thesis: for t being Tape of SuccTuring
for head, n being Element of NAT st s = [0,head,t] & t storeData <*head,n*> holds
( s is Accept-Halt & (Result s) `2_3 = head & (Result s) `3_3 storeData <*head,(n + 1)*> )
let t be Tape of SuccTuring; ::_thesis: for head, n being Element of NAT st s = [0,head,t] & t storeData <*head,n*> holds
( s is Accept-Halt & (Result s) `2_3 = head & (Result s) `3_3 storeData <*head,(n + 1)*> )
let h, n be Element of NAT ; ::_thesis: ( s = [0,h,t] & t storeData <*h,n*> implies ( s is Accept-Halt & (Result s) `2_3 = h & (Result s) `3_3 storeData <*h,(n + 1)*> ) )
assume that
A1: s = [0,h,t] and
A2: t storeData <*h,n*> ; ::_thesis: ( s is Accept-Halt & (Result s) `2_3 = h & (Result s) `3_3 storeData <*h,(n + 1)*> )
A3: t . h = 0 by A2, Th17;
set i3 = (((h + 1) + 1) + n) + 1;
reconsider h1 = h + 1 as Element of INT by INT_1:def_2;
reconsider p1 = 1 as State of SuccTuring by Lm5;
A4: (h1 + 1) + n < (((h + 1) + 1) + n) + 1 by XREAL_1:29;
 h <= h + n by NAT_1:11;
then A5: h + 2 <= (h + n) + 2 by XREAL_1:7;
A6: h1 < h + 2 by XREAL_1:8;
then A7: h1 < (h + n) + 2 by A5, XXREAL_0:2;
reconsider p2 = 2 as State of SuccTuring by Lm5;
reconsider i2 = h1 + 1 as Element of INT by INT_1:def_2;
reconsider nk = (h1 + 1) + n as Element of INT by INT_1:def_2;
reconsider hh = h as Element of INT by INT_1:def_2;
reconsider n3 = (((h + 1) + 1) + n) + 1 as Element of INT by INT_1:def_2;
reconsider T = 1 as Symbol of SuccTuring by Lm6;
set t1 = Tape-Chg (t,h1,T);
A8: h < h1 by XREAL_1:29;
A9: t . ((h + n) + 2) = 0 by A2, Th17;
A10: ( (Tape-Chg (t,h1,T)) . h = 0 & (Tape-Chg (t,h1,T)) . ((h + n) + 2) = 0 & ( for i being Integer st h < i & i < (h + n) + 2 holds
(Tape-Chg (t,h1,T)) . i = 1 ) )
proof
thus  (Tape-Chg (t,h1,T)) . h = 0 by A3, A8, Th26; ::_thesis: ( (Tape-Chg (t,h1,T)) . ((h + n) + 2) = 0 & ( for i being Integer st h < i & i < (h + n) + 2 holds
(Tape-Chg (t,h1,T)) . i = 1 ) )
thus  (Tape-Chg (t,h1,T)) . ((h + n) + 2) = 0 by A9, A5, A6, Th26; ::_thesis: for i being Integer st h < i & i < (h + n) + 2 holds
(Tape-Chg (t,h1,T)) . i = 1
hereby  ::_thesis: verum
let i be Integer; ::_thesis: ( h < i & i < (h + n) + 2 implies (Tape-Chg (t,h1,T)) . b1 = 1 )
assume A11: ( h < i & i < (h + n) + 2 ) ; ::_thesis: (Tape-Chg (t,h1,T)) . b1 = 1
percases ( h1 = i or h1 <> i ) ;
suppose h1 = i ; ::_thesis: (Tape-Chg (t,h1,T)) . b1 = 1
hence  (Tape-Chg (t,h1,T)) . i = 1 by Th26; ::_thesis: verum
end;
suppose h1 <> i ; ::_thesis: (Tape-Chg (t,h1,T)) . b1 = 1
hence (Tape-Chg (t,h1,T)) . i = t . i by Th26
.= 1 by A2, A11, Th17 ;
::_thesis: verum
end;
end;
end;
end;
A12: for i being Integer st (h + 1) + 1 <= i & i < ((h + 1) + 1) + n holds
(Tape-Chg (t,h1,T)) . i = 1
proof
let i be Integer; ::_thesis: ( (h + 1) + 1 <= i & i < ((h + 1) + 1) + n implies (Tape-Chg (t,h1,T)) . i = 1 )
assume that
A13: (h + 1) + 1 <= i and
A14: i < ((h + 1) + 1) + n ; ::_thesis: (Tape-Chg (t,h1,T)) . i = 1
 h1 < h1 + 1 by XREAL_1:29;
then  h1 < i by A13, XXREAL_0:2;
then  h < i by A8, XXREAL_0:2;
hence  (Tape-Chg (t,h1,T)) . i = 1 by A10, A14; ::_thesis: verum
end;
reconsider s3 = s `3_3 as Tape of SuccTuring ;
A15: TRAN s = Succ_Tran . [(s `1_3),(s3 . (Head s))] by Def17
.= Succ_Tran . [0,(s3 . (Head s))] by A1, MCART_1:64
.= Succ_Tran . [0,(t . (Head s))] by A1, MCART_1:64
.= [1,0,1] by A1, A3, Th30, MCART_1:64 ;
then A16: offset (TRAN s) = 1 by MCART_1:64;
set s1 = [p1,h1,t];
reconsider s3 = [p1,h1,t] `3_3 as Tape of SuccTuring ;
Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)) = Tape-Chg (t,(Head s),((TRAN s) `2_3)) by A1, MCART_1:64
.= Tape-Chg (t,h,((TRAN s) `2_3)) by A1, MCART_1:64
.= Tape-Chg (t,h,F) by A15, MCART_1:64
.= t by A3, Th24 ;
then A17: Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),t] by A1, Lm7
.= [1,((Head s) + (offset (TRAN s))),t] by A15, MCART_1:64
.= [p1,h1,t] by A1, A16, MCART_1:64 ;
A18: TRAN [p1,h1,t] = Succ_Tran . [([p1,h1,t] `1_3),(s3 . (Head [p1,h1,t]))] by Def17
.= Succ_Tran . [p1,(s3 . (Head [p1,h1,t]))]
.= Succ_Tran . [p1,(t . (Head [p1,h1,t]))]
.= Succ_Tran . [1,(t . h1)]
.= [1,1,1] by A2, A8, A7, Th17, Th30 ;
then A19: offset (TRAN [p1,h1,t]) = 1 by MCART_1:64;
reconsider p1 = 1 as State of SuccTuring by Lm5;
set s2 = [p1,i2,(Tape-Chg (t,h1,T))];
Tape-Chg (([p1,h1,t] `3_3),(Head [p1,h1,t]),((TRAN [p1,h1,t]) `2_3)) = Tape-Chg (t,(Head [p1,h1,t]),((TRAN [p1,h1,t]) `2_3))
.= Tape-Chg (t,h1,((TRAN [p1,h1,t]) `2_3))
.= Tape-Chg (t,h1,T) by A18, MCART_1:64 ;
then A20: Following [p1,h1,t] = [((TRAN [p1,h1,t]) `1_3),((Head [p1,h1,t]) + (offset (TRAN [p1,h1,t]))),(Tape-Chg (t,h1,T))] by Lm7
.= [1,((Head [p1,h1,t]) + (offset (TRAN [p1,h1,t]))),(Tape-Chg (t,h1,T))] by A18, MCART_1:64
.= [p1,i2,(Tape-Chg (t,h1,T))] by A19 ;
reconsider p3 = 3 as State of SuccTuring by Lm5;
set sn = [p1,nk,(Tape-Chg (t,h1,T))];
set t2 = Tape-Chg ((Tape-Chg (t,h1,T)),nk,T);
set t3 = Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F);
 ( the Tran of SuccTuring . [p1,1] = [p1,1,1] & p1 <> the AcceptS of SuccTuring ) by Def17, Th30;
then A21: (Computation [p1,i2,(Tape-Chg (t,h1,T))]) . n = [p1,(((h + 1) + 1) + n),(Tape-Chg (t,h1,T))] by A12, Lm4;
 ( h1 + 1 <= ((h + 1) + 1) + n & h1 < h1 + 1 ) by NAT_1:11, XREAL_1:29;
then A22: h1 < (h1 + 1) + n by XXREAL_0:2;
A23: ( (Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)) . h = 0 & ( for i being Integer st h < i & i <= (h + n) + 2 holds
(Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)) . i = 1 ) )
proof
thus  (Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)) . h = 0 by A8, A10, A22, Th26; ::_thesis: for i being Integer st h < i & i <= (h + n) + 2 holds
(Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)) . i = 1
hereby  ::_thesis: verum
let i be Integer; ::_thesis: ( h < i & i <= (h + n) + 2 implies (Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)) . b1 = 1 )
assume that
A24: h < i and
A25: i <= (h + n) + 2 ; ::_thesis: (Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)) . b1 = 1
percases ( i <> (h + n) + 2 or i = (h + n) + 2 ) ;
supposeA26: i <> (h + n) + 2 ; ::_thesis: (Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)) . b1 = 1
then A27: i < (h + n) + 2 by A25, XXREAL_0:1;
thus (Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)) . i = (Tape-Chg (t,h1,T)) . i by A26, Th26
.= 1 by A10, A24, A27 ; ::_thesis: verum
end;
suppose i = (h + n) + 2 ; ::_thesis: (Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)) . b1 = 1
hence  (Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)) . i = 1 by Th26; ::_thesis: verum
end;
end;
end;
end;
set sp3 = [p3,nk,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))];
set sm = [p2,n3,(Tape-Chg ((Tape-Chg (t,h1,T)),nk,T))];
reconsider sm3 = [p2,n3,(Tape-Chg ((Tape-Chg (t,h1,T)),nk,T))] `3_3 as Tape of SuccTuring ;
A28: the Symbols of SuccTuring = {0,1} by Def17;
A29: now__::_thesis:_Succ_Tran_._[2,((Tape-Chg_((Tape-Chg_(t,h1,T)),nk,T))_._n3)]_=_[p3,0,(-_1)]
percases ( (Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)) . n3 = 1 or (Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)) . n3 = 0 ) by A28, TARSKI:def_2;
suppose (Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)) . n3 = 1 ; ::_thesis: Succ_Tran . [2,((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)) . n3)] = [p3,0,(- 1)]
hence  Succ_Tran . [2,((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)) . n3)] = [p3,0,(- 1)] by Th30; ::_thesis: verum
end;
suppose (Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)) . n3 = 0 ; ::_thesis: Succ_Tran . [2,((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)) . n3)] = [p3,0,(- 1)]
hence  Succ_Tran . [2,((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)) . n3)] = [p3,0,(- 1)] by Th30; ::_thesis: verum
end;
end;
end;
A30: TRAN [p2,n3,(Tape-Chg ((Tape-Chg (t,h1,T)),nk,T))] = Succ_Tran . [([p2,n3,(Tape-Chg ((Tape-Chg (t,h1,T)),nk,T))] `1_3),(sm3 . (Head [p2,n3,(Tape-Chg ((Tape-Chg (t,h1,T)),nk,T))]))] by Def17
.= Succ_Tran . [2,(sm3 . (Head [p2,n3,(Tape-Chg ((Tape-Chg (t,h1,T)),nk,T))]))]
.= Succ_Tran . [2,((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)) . (Head [p2,n3,(Tape-Chg ((Tape-Chg (t,h1,T)),nk,T))]))]
.= [p3,0,(- 1)] by A29 ;
then A31: offset (TRAN [p2,n3,(Tape-Chg ((Tape-Chg (t,h1,T)),nk,T))]) = - 1 by MCART_1:64;
set j1 = (1 + 1) + n;
set sp5 = [p3,hh,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))];
defpred S1[ Element of NAT ] means ( h + $1 <= nk implies (Computation [p3,nk,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]) . $1 = [3,(nk - $1),(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))] );
reconsider sn3 = [p1,nk,(Tape-Chg (t,h1,T))] `3_3 as Tape of SuccTuring ;
A32: h + ((1 + 1) + n) = nk ;
set Rs = (Computation s) . ((1 + 1) + (((n + 1) + 1) + (((1 + 1) + n) + 1)));
A33: TRAN [p1,nk,(Tape-Chg (t,h1,T))] = Succ_Tran . [([p1,nk,(Tape-Chg (t,h1,T))] `1_3),(sn3 . (Head [p1,nk,(Tape-Chg (t,h1,T))]))] by Def17
.= Succ_Tran . [p1,(sn3 . (Head [p1,nk,(Tape-Chg (t,h1,T))]))]
.= Succ_Tran . [p1,((Tape-Chg (t,h1,T)) . (Head [p1,nk,(Tape-Chg (t,h1,T))]))]
.= [2,1,1] by A10, Th30 ;
then A34: offset (TRAN [p1,nk,(Tape-Chg (t,h1,T))]) = 1 by MCART_1:64;
A35: h < (h1 + 1) + n by A8, A22, XXREAL_0:2;
A36: ( (Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F)) . h = 0 & (Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F)) . ((h + (n + 1)) + 2) = 0 & ( for k being Integer st h < k & k < (h + (n + 1)) + 2 holds
(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F)) . k = 1 ) )
proof
thus  (Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F)) . h = 0 by A35, A23, A4, Th26; ::_thesis: ( (Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F)) . ((h + (n + 1)) + 2) = 0 & ( for k being Integer st h < k & k < (h + (n + 1)) + 2 holds
(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F)) . k = 1 ) )
thus  (Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F)) . ((h + (n + 1)) + 2) = 0 by Th26; ::_thesis: for k being Integer st h < k & k < (h + (n + 1)) + 2 holds
(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F)) . k = 1
hereby  ::_thesis: verum
let i be Integer; ::_thesis: ( h < i & i < (h + (n + 1)) + 2 implies (Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F)) . i = 1 )
assume that
A37: h < i and
A38: i < (h + (n + 1)) + 2 ; ::_thesis: (Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F)) . i = 1
 i + 1 <= (h + (n + 1)) + 2 by A38, INT_1:7;
then A39: i <= ((h + (n + 1)) + 2) - 1 by XREAL_1:19;
thus (Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F)) . i = (Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)) . i by A38, Th26
.= 1 by A23, A37, A39 ; ::_thesis: verum
end;
end;
then A40: Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F) is_1_between h,(h + (n + 1)) + 2 by Def12;
Tape-Chg (([p2,n3,(Tape-Chg ((Tape-Chg (t,h1,T)),nk,T))] `3_3),(Head [p2,n3,(Tape-Chg ((Tape-Chg (t,h1,T)),nk,T))]),((TRAN [p2,n3,(Tape-Chg ((Tape-Chg (t,h1,T)),nk,T))]) `2_3)) = Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),(Head [p2,n3,(Tape-Chg ((Tape-Chg (t,h1,T)),nk,T))]),((TRAN [p2,n3,(Tape-Chg ((Tape-Chg (t,h1,T)),nk,T))]) `2_3))
.= Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,((TRAN [p2,n3,(Tape-Chg ((Tape-Chg (t,h1,T)),nk,T))]) `2_3))
.= Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F) by A30, MCART_1:64 ;
then A41: Following [p2,n3,(Tape-Chg ((Tape-Chg (t,h1,T)),nk,T))] = [((TRAN [p2,n3,(Tape-Chg ((Tape-Chg (t,h1,T)),nk,T))]) `1_3),((Head [p2,n3,(Tape-Chg ((Tape-Chg (t,h1,T)),nk,T))]) + (offset (TRAN [p2,n3,(Tape-Chg ((Tape-Chg (t,h1,T)),nk,T))]))),(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))] by Lm7
.= [p3,((Head [p2,n3,(Tape-Chg ((Tape-Chg (t,h1,T)),nk,T))]) + (offset (TRAN [p2,n3,(Tape-Chg ((Tape-Chg (t,h1,T)),nk,T))]))),(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))] by A30, MCART_1:64
.= [p3,nk,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))] by A31 ;
Tape-Chg (([p1,nk,(Tape-Chg (t,h1,T))] `3_3),(Head [p1,nk,(Tape-Chg (t,h1,T))]),((TRAN [p1,nk,(Tape-Chg (t,h1,T))]) `2_3)) = Tape-Chg ((Tape-Chg (t,h1,T)),(Head [p1,nk,(Tape-Chg (t,h1,T))]),((TRAN [p1,nk,(Tape-Chg (t,h1,T))]) `2_3))
.= Tape-Chg ((Tape-Chg (t,h1,T)),nk,((TRAN [p1,nk,(Tape-Chg (t,h1,T))]) `2_3))
.= Tape-Chg ((Tape-Chg (t,h1,T)),nk,T) by A33, MCART_1:64 ;
then A42: Following [p1,nk,(Tape-Chg (t,h1,T))] = [((TRAN [p1,nk,(Tape-Chg (t,h1,T))]) `1_3),((Head [p1,nk,(Tape-Chg (t,h1,T))]) + (offset (TRAN [p1,nk,(Tape-Chg (t,h1,T))]))),(Tape-Chg ((Tape-Chg (t,h1,T)),nk,T))] by Lm7
.= [2,((Head [p1,nk,(Tape-Chg (t,h1,T))]) + (offset (TRAN [p1,nk,(Tape-Chg (t,h1,T))]))),(Tape-Chg ((Tape-Chg (t,h1,T)),nk,T))] by A33, MCART_1:64
.= [p2,n3,(Tape-Chg ((Tape-Chg (t,h1,T)),nk,T))] by A34 ;
A43: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A44: S1[k] ; ::_thesis: S1[k + 1]
now__::_thesis:_(_h_+_(k_+_1)_<=_nk_implies_(Computation_[p3,nk,(Tape-Chg_((Tape-Chg_((Tape-Chg_(t,h1,T)),nk,T)),n3,F))])_._(k_+_1)_=_[3,(nk_-_(k_+_1)),(Tape-Chg_((Tape-Chg_((Tape-Chg_(t,h1,T)),nk,T)),n3,F))]_)
reconsider ik = nk - k as Element of INT by INT_1:def_2;
set k3 = nk - k;
A45: h + k < (h + k) + 1 by XREAL_1:29;
set sk = [p3,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))];
reconsider tt = [p3,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))] `3_3 as Tape of SuccTuring ;
 nk <= nk + k by INT_1:6;
then A46: nk - k <= nk by XREAL_1:20;
assume A47: h + (k + 1) <= nk ; ::_thesis: (Computation [p3,nk,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]) . (k + 1) = [3,(nk - (k + 1)),(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]
then  h + k < nk + 0 by A45, XXREAL_0:2;
then A48: h - 0 < nk - k by XREAL_1:21;
 h >= 0 by NAT_1:2;
then reconsider ii = nk - k as Element of NAT by A48, INT_1:3, XXREAL_0:2;
 (h + n) + 2 < ((h + n) + 2) + 1 by XREAL_1:29;
then  ii < (h + (n + 1)) + 2 by A46, XXREAL_0:2;
then A49: (Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F)) . ii = 1 by A36, A48;
A50: TRAN [p3,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))] = Succ_Tran . [([p3,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))] `1_3),(tt . (Head [p3,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]))] by Def17
.= Succ_Tran . [p3,(tt . (Head [p3,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]))]
.= Succ_Tran . [p3,((Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F)) . (Head [p3,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]))]
.= [3,1,(- 1)] by A49, Th30 ;
then A51: offset (TRAN [p3,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]) = - 1 by MCART_1:64;
A52: Tape-Chg (([p3,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))] `3_3),(Head [p3,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]),((TRAN [p3,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]) `2_3)) = Tape-Chg ((Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F)),(Head [p3,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]),((TRAN [p3,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]) `2_3))
.= Tape-Chg ((Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F)),(nk - k),((TRAN [p3,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]) `2_3))
.= Tape-Chg ((Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F)),(nk - k),T) by A50, MCART_1:64
.= Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F) by A49, Th24 ;
hereby  ::_thesis: verum
thus (Computation [p3,nk,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]) . (k + 1) = Following [p3,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))] by A44, A47, A45, Def7, XXREAL_0:2
.= [((TRAN [p3,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]) `1_3),((Head [p3,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]) + (offset (TRAN [p3,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]))),(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))] by A52, Lm7
.= [3,((Head [p3,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]) + (offset (TRAN [p3,ik,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]))),(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))] by A50, MCART_1:64
.= [3,((nk - k) + (- 1)),(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))] by A51
.= [3,(nk - (k + 1)),(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))] ; ::_thesis: verum
end;
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
A53: S1[ 0 ] by Def7;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A53, A43);
then A54: (Computation [p3,nk,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]) . ((1 + 1) + n) = [3,(nk - ((1 + 1) + n)),(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))] by A32
.= [p3,hh,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))] ;
A55: now__::_thesis:_Following_[p3,hh,(Tape-Chg_((Tape-Chg_((Tape-Chg_(t,h1,T)),nk,T)),n3,F))]_=_[4,(h_+_0),(Tape-Chg_((Tape-Chg_((Tape-Chg_(t,h1,T)),nk,T)),n3,F))]
reconsider tt = [p3,hh,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))] `3_3 as Tape of SuccTuring ;
A56: TRAN [p3,hh,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))] = Succ_Tran . [([p3,hh,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))] `1_3),(tt . (Head [p3,hh,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]))] by Def17
.= Succ_Tran . [3,(tt . (Head [p3,hh,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]))]
.= Succ_Tran . [3,((Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F)) . (Head [p3,hh,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]))]
.= [4,0,0] by A36, Th30 ;
then A57: offset (TRAN [p3,hh,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]) = 0 by MCART_1:64;
Tape-Chg (([p3,hh,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))] `3_3),(Head [p3,hh,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]),((TRAN [p3,hh,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]) `2_3)) = Tape-Chg ((Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F)),(Head [p3,hh,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]),((TRAN [p3,hh,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]) `2_3))
.= Tape-Chg ((Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F)),h,((TRAN [p3,hh,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]) `2_3))
.= Tape-Chg ((Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F)),h,F) by A56, MCART_1:64
.= Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F) by A36, Th24 ;
hence  Following [p3,hh,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))] = [((TRAN [p3,hh,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]) `1_3),((Head [p3,hh,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]) + (offset (TRAN [p3,hh,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]))),(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))] by Lm7
.= [4,((Head [p3,hh,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]) + (offset (TRAN [p3,hh,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]))),(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))] by A56, MCART_1:64
.= [4,(h + 0),(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))] by A57 ;
::_thesis: verum
end;
(Computation s) . ((1 + 1) + (((n + 1) + 1) + (((1 + 1) + n) + 1))) = (Computation ((Computation s) . (1 + 1))) . (((n + 1) + 1) + (((1 + 1) + n) + 1)) by Th10
.= (Computation (Following ((Computation s) . 1))) . (((n + 1) + 1) + (((1 + 1) + n) + 1)) by Def7
.= (Computation (Following [p1,h1,t])) . (((n + 1) + 1) + (((1 + 1) + n) + 1)) by A17, Th9
.= (Computation ((Computation [p1,i2,(Tape-Chg (t,h1,T))]) . ((n + 1) + 1))) . (((1 + 1) + n) + 1) by A20, Th10
.= (Computation (Following ((Computation [p1,i2,(Tape-Chg (t,h1,T))]) . (n + 1)))) . (((1 + 1) + n) + 1) by Def7 ;
then  (Computation s) . ((1 + 1) + (((n + 1) + 1) + (((1 + 1) + n) + 1))) = (Computation [p3,nk,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))]) . (((1 + 1) + n) + 1) by A21, A42, A41, Def7;
then A58: (Computation s) . ((1 + 1) + (((n + 1) + 1) + (((1 + 1) + n) + 1))) = [4,h,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F))] by A54, A55, Def7;
then A59: ((Computation s) . ((1 + 1) + (((n + 1) + 1) + (((1 + 1) + n) + 1)))) `1_3 = 4 by MCART_1:64
.= the AcceptS of SuccTuring by Def17 ;
hence  s is Accept-Halt by Def8; ::_thesis: ( (Result s) `2_3 = h & (Result s) `3_3 storeData <*h,(n + 1)*> )
then A60: Result s = (Computation s) . ((1 + 1) + (((n + 1) + 1) + (((1 + 1) + n) + 1))) by A59, Def9;
hence  (Result s) `2_3 = h by A58, MCART_1:64; ::_thesis: (Result s) `3_3 storeData <*h,(n + 1)*>
 (Result s) `3_3 = Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),nk,T)),n3,F) by A58, A60, MCART_1:64;
hence  (Result s) `3_3 storeData <*h,(n + 1)*> by A40, Th16; ::_thesis: verum
end;

theorem :: TURING_1:32
SuccTuring computes 1 succ 1
proof
now__::_thesis:_for_s_being_All-State_of_SuccTuring
for_t_being_Tape_of_SuccTuring
for_h_being_Element_of_NAT_
for_x_being_FinSequence_of_NAT_st_x_in_dom_(1_succ_1)_&_s_=_[_the_InitS_of_SuccTuring,h,t]_&_t_storeData_<*h*>_^_x_holds_
(_s_is_Accept-Halt_&_ex_h2_being_Element_of_NAT_ex_y_being_Element_of_NAT_st_
(_(Result_s)_`2_3_=_h2_&_y_=_(1_succ_1)_._x_&_(Result_s)_`3_3_storeData_<*h2*>_^_<*y*>_)_)
set sc = 1 succ 1;
let s be All-State of SuccTuring; ::_thesis: for t being Tape of SuccTuring
for h being Element of NAT
for x being FinSequence of NAT st x in dom (1 succ 1) & s = [ the InitS of SuccTuring,h,t] & t storeData <*h*> ^ x holds
( s is Accept-Halt & ex h2 being Element of NAT ex y being Element of NAT st
( (Result s) `2_3 = h2 & y = (1 succ 1) . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> ) )
let t be Tape of SuccTuring; ::_thesis: for h being Element of NAT
for x being FinSequence of NAT st x in dom (1 succ 1) & s = [ the InitS of SuccTuring,h,t] & t storeData <*h*> ^ x holds
( s is Accept-Halt & ex h2 being Element of NAT ex y being Element of NAT st
( (Result s) `2_3 = h2 & y = (1 succ 1) . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> ) )
let h be Element of NAT ; ::_thesis: for x being FinSequence of NAT st x in dom (1 succ 1) & s = [ the InitS of SuccTuring,h,t] & t storeData <*h*> ^ x holds
( s is Accept-Halt & ex h2 being Element of NAT ex y being Element of NAT st
( (Result s) `2_3 = h2 & y = (1 succ 1) . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> ) )
let x be FinSequence of NAT ; ::_thesis: ( x in dom (1 succ 1) & s = [ the InitS of SuccTuring,h,t] & t storeData <*h*> ^ x implies ( s is Accept-Halt & ex h2 being Element of NAT ex y being Element of NAT st
( (Result s) `2_3 = h2 & y = (1 succ 1) . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> ) ) )
assume that
A1: x in dom (1 succ 1) and
A2: s = [ the InitS of SuccTuring,h,t] and
A3: t storeData <*h*> ^ x ; ::_thesis: ( s is Accept-Halt & ex h2 being Element of NAT ex y being Element of NAT st
( (Result s) `2_3 = h2 & y = (1 succ 1) . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> ) )
A4: s = [0,h,t] by A2, Def17;
A5: dom (1 succ 1) = 1 -tuples_on NAT by COMPUT_1:def_7;
then  x is Tuple of 1, NAT by A1, FINSEQ_2:131;
then consider i being Element of NAT  such that
A6: x = <*i*> by FINSEQ_2:97;
A7: <*h*> ^ x = <*h,i*> by A6;
hence  s is Accept-Halt by A3, A4, Th31; ::_thesis: ex h2 being Element of NAT ex y being Element of NAT st
( (Result s) `2_3 = h2 & y = (1 succ 1) . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> )
take h2 = h; ::_thesis: ex y being Element of NAT st
( (Result s) `2_3 = h2 & y = (1 succ 1) . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> )
take y = i + 1; ::_thesis: ( (Result s) `2_3 = h2 & y = (1 succ 1) . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> )
thus  (Result s) `2_3 = h2 by A3, A4, A7, Th31; ::_thesis: ( y = (1 succ 1) . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> )
thus y = (x /. 1) + 1 by A6, FINSEQ_4:16
.= (1 succ 1) . x by A1, A5, COMPUT_1:def_7 ; ::_thesis: (Result s) `3_3 storeData <*h2*> ^ <*y*>
 (Result s) `3_3 storeData <*h2,(i + 1)*> by A3, A4, A7, Th31;
hence  (Result s) `3_3 storeData <*h2*> ^ <*y*> ; ::_thesis: verum
end;
hence  SuccTuring computes 1 succ 1 by Def15; ::_thesis: verum
end;

begin

definition
func Zero_Tran  ->  Function of [:(SegM 4),{0,1}:],[:(SegM 4),{0,1},{(- 1),0,1}:] equals :: TURING_1:def 18
(((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},1)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) +* ([3,1] .--> [4,1,(- 1)]);
coherence
 (((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},1)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) +* ([3,1] .--> [4,1,(- 1)]) is Function of [:(SegM 4),{0,1}:],[:(SegM 4),{0,1},{(- 1),0,1}:]
proof
reconsider p0 = 0 , p1 = 1, p2 = 2, p3 = 3, p4 = 4 as Element of SegM 4 by Th1;
set A = [:(SegM 4),{0,1}:];
set B = {(- 1),0,1};
set C = [:(SegM 4),{0,1},{(- 1),0,1}:];
reconsider b0 = 0 , b1 = 1 as Element of {0,1} by TARSKI:def_2;
reconsider L = - 1 as Element of {(- 1),0,1} by ENUMSET1:def_1;
reconsider R = 1 as Element of {(- 1),0,1} by ENUMSET1:def_1;
 [:(SegM 4),{0,1},{(- 1),0,1}:] = [:[:(SegM 4),{0,1}:],{(- 1),0,1}:] by ZFMISC_1:def_3;
then reconsider OP = id ([:(SegM 4),{0,1}:],{(- 1),0,1},R) as Function of [:(SegM 4),{0,1}:],[:(SegM 4),{0,1},{(- 1),0,1}:] ;
 (((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},1)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) +* ([3,1] .--> [4,1,(- 1)]) = ((((OP +* ([p0,b0] .--> [p1,b0,R])) +* ([p1,b1] .--> [p2,b1,R])) +* ([p2,b0] .--> [p3,b0,L])) +* ([p2,b1] .--> [p3,b0,L])) +* ([p3,b1] .--> [p4,b1,L]) ;
hence  (((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},1)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) +* ([3,1] .--> [4,1,(- 1)]) is Function of [:(SegM 4),{0,1}:],[:(SegM 4),{0,1},{(- 1),0,1}:] ; ::_thesis: verum
end;
end;

:: deftheorem  defines Zero_Tran TURING_1:def_18_:_
 Zero_Tran = (((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},1)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) +* ([3,1] .--> [4,1,(- 1)]);

theorem Th33: :: TURING_1:33
( Zero_Tran . [0,0] = [1,0,1] & Zero_Tran . [1,1] = [2,1,1] & Zero_Tran . [2,0] = [3,0,(- 1)] & Zero_Tran . [2,1] = [3,0,(- 1)] & Zero_Tran . [3,1] = [4,1,(- 1)] )
proof
set x = [1,1];
set x1 = [0,0];
set p1 = [0,0] .--> [1,0,1];
set p2 = [1,1] .--> [2,1,1];
set p3 = [2,0] .--> [3,0,(- 1)];
set p4 = [2,1] .--> [3,0,(- 1)];
set f = id ([:(SegM 4),{0,1}:],{(- 1),0,1},1);
thus Zero_Tran . [0,0] = (((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},1)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) . [0,0] by Th2
.= ((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},1)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) . [0,0] by Th2
.= (((id ([:(SegM 4),{0,1}:],{(- 1),0,1},1)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [2,1,1])) . [0,0] by Th2
.= ((id ([:(SegM 4),{0,1}:],{(- 1),0,1},1)) +* ([0,0] .--> [1,0,1])) . [0,0] by Th3
.= [1,0,1] by FUNCT_7:94 ; ::_thesis: ( Zero_Tran . [1,1] = [2,1,1] & Zero_Tran . [2,0] = [3,0,(- 1)] & Zero_Tran . [2,1] = [3,0,(- 1)] & Zero_Tran . [3,1] = [4,1,(- 1)] )
thus Zero_Tran . [1,1] = (((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},1)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) . [1,1] by Th2
.= ((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},1)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) . [1,1] by Th2
.= (((id ([:(SegM 4),{0,1}:],{(- 1),0,1},1)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [2,1,1])) . [1,1] by Th3
.= [2,1,1] by FUNCT_7:94 ; ::_thesis: ( Zero_Tran . [2,0] = [3,0,(- 1)] & Zero_Tran . [2,1] = [3,0,(- 1)] & Zero_Tran . [3,1] = [4,1,(- 1)] )
set x = [2,0];
thus Zero_Tran . [2,0] = (((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},1)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) . [2,0] by Th3
.= ((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},1)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) . [2,0] by Th3
.= [3,0,(- 1)] by FUNCT_7:94 ; ::_thesis: ( Zero_Tran . [2,1] = [3,0,(- 1)] & Zero_Tran . [3,1] = [4,1,(- 1)] )
set x = [2,1];
thus Zero_Tran . [2,1] = (((((id ([:(SegM 4),{0,1}:],{(- 1),0,1},1)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [2,1,1])) +* ([2,0] .--> [3,0,(- 1)])) +* ([2,1] .--> [3,0,(- 1)])) . [2,1] by Th2
.= [3,0,(- 1)] by FUNCT_7:94 ; ::_thesis: Zero_Tran . [3,1] = [4,1,(- 1)]
thus  Zero_Tran . [3,1] = [4,1,(- 1)] by FUNCT_7:94; ::_thesis: verum
end;

definition
func ZeroTuring  ->  strict TuringStr  means :Def19: :: TURING_1:def 19
( the Symbols of it = {0,1} & the FStates of it = SegM 4 & the Tran of it = Zero_Tran & the InitS of it = 0 & the AcceptS of it = 4 );
existence
 ex b1 being strict TuringStr st
( the Symbols of b1 = {0,1} & the FStates of b1 = SegM 4 & the Tran of b1 = Zero_Tran & the InitS of b1 = 0 & the AcceptS of b1 = 4 )
proof
set St = SegM 4;
reconsider qF = 4 as Element of SegM 4 by Th1;
reconsider p0 = 0 as Element of SegM 4 by Th1;
set Sym = {0,1};
take  TuringStr(# {0,1},(SegM 4),Zero_Tran,p0,qF #) ; ::_thesis: ( the Symbols of TuringStr(# {0,1},(SegM 4),Zero_Tran,p0,qF #) = {0,1} & the FStates of TuringStr(# {0,1},(SegM 4),Zero_Tran,p0,qF #) = SegM 4 & the Tran of TuringStr(# {0,1},(SegM 4),Zero_Tran,p0,qF #) = Zero_Tran & the InitS of TuringStr(# {0,1},(SegM 4),Zero_Tran,p0,qF #) = 0 & the AcceptS of TuringStr(# {0,1},(SegM 4),Zero_Tran,p0,qF #) = 4 )
thus  ( the Symbols of TuringStr(# {0,1},(SegM 4),Zero_Tran,p0,qF #) = {0,1} & the FStates of TuringStr(# {0,1},(SegM 4),Zero_Tran,p0,qF #) = SegM 4 & the Tran of TuringStr(# {0,1},(SegM 4),Zero_Tran,p0,qF #) = Zero_Tran & the InitS of TuringStr(# {0,1},(SegM 4),Zero_Tran,p0,qF #) = 0 & the AcceptS of TuringStr(# {0,1},(SegM 4),Zero_Tran,p0,qF #) = 4 ) ; ::_thesis: verum
end;
uniqueness
 for b1, b2 being strict TuringStr st the Symbols of b1 = {0,1} & the FStates of b1 = SegM 4 & the Tran of b1 = Zero_Tran & the InitS of b1 = 0 & the AcceptS of b1 = 4 & the Symbols of b2 = {0,1} & the FStates of b2 = SegM 4 & the Tran of b2 = Zero_Tran & the InitS of b2 = 0 & the AcceptS of b2 = 4 holds
b1 = b2 ;
end;

:: deftheorem Def19 defines ZeroTuring TURING_1:def_19_:_
 for b1 being strict TuringStr holds
( b1 = ZeroTuring iff ( the Symbols of b1 = {0,1} & the FStates of b1 = SegM 4 & the Tran of b1 = Zero_Tran & the InitS of b1 = 0 & the AcceptS of b1 = 4 ) );

Lm8:  for n being Element of NAT st n <= 4 holds
n is State of ZeroTuring
proof
let n be Element of NAT ; ::_thesis: ( n <= 4 implies n is State of ZeroTuring )
assume A1: n <= 4 ; ::_thesis: n is State of ZeroTuring
 the FStates of ZeroTuring = SegM 4 by Def19;
hence  n is State of ZeroTuring by A1, Th1; ::_thesis: verum
end;

Lm9:  ( 0 in the Symbols of ZeroTuring & 1 in the Symbols of ZeroTuring )
proof
 ( 0 in {0,1} & 1 in {0,1} ) by TARSKI:def_2;
hence  ( 0 in the Symbols of ZeroTuring & 1 in the Symbols of ZeroTuring ) by Def19; ::_thesis: verum
end;

Lm10:  for s being All-State of ZeroTuring
for p, h, t being set st s = [p,h,t] & p <> 4 holds
Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))]
proof
let s be All-State of ZeroTuring; ::_thesis: for p, h, t being set st s = [p,h,t] & p <> 4 holds
Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))]
let p, h, t be set ; ::_thesis: ( s = [p,h,t] & p <> 4 implies Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))] )
assume A1: ( s = [p,h,t] & p <> 4 ) ; ::_thesis: Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))]
 4 = the AcceptS of ZeroTuring by Def19;
hence  Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))] by A1, Th25; ::_thesis: verum
end;

theorem Th34: :: TURING_1:34
for s being All-State of ZeroTuring
for t being Tape of ZeroTuring
for head being Element of NAT
for f being FinSequence of NAT st len f >= 1 & s = [0,head,t] & t storeData <*head*> ^ f holds
( s is Accept-Halt & (Result s) `2_3 = head & (Result s) `3_3 storeData <*head,0*> )
proof
reconsider F = 0 as Symbol of ZeroTuring by Lm9;
let s be All-State of ZeroTuring; ::_thesis: for t being Tape of ZeroTuring
for head being Element of NAT
for f being FinSequence of NAT st len f >= 1 & s = [0,head,t] & t storeData <*head*> ^ f holds
( s is Accept-Halt & (Result s) `2_3 = head & (Result s) `3_3 storeData <*head,0*> )
let t be Tape of ZeroTuring; ::_thesis: for head being Element of NAT
for f being FinSequence of NAT st len f >= 1 & s = [0,head,t] & t storeData <*head*> ^ f holds
( s is Accept-Halt & (Result s) `2_3 = head & (Result s) `3_3 storeData <*head,0*> )
let h be Element of NAT ; ::_thesis: for f being FinSequence of NAT st len f >= 1 & s = [0,h,t] & t storeData <*h*> ^ f holds
( s is Accept-Halt & (Result s) `2_3 = h & (Result s) `3_3 storeData <*h,0*> )
let f be FinSequence of NAT ; ::_thesis: ( len f >= 1 & s = [0,h,t] & t storeData <*h*> ^ f implies ( s is Accept-Halt & (Result s) `2_3 = h & (Result s) `3_3 storeData <*h,0*> ) )
assume that
A1: len f >= 1 and
A2: s = [0,h,t] and
A3: t storeData <*h*> ^ f ; ::_thesis: ( s is Accept-Halt & (Result s) `2_3 = h & (Result s) `3_3 storeData <*h,0*> )
reconsider s3 = s `3_3 as Tape of ZeroTuring ;
reconsider h1 = h + 1 as Element of INT by INT_1:def_2;
set m = f /. 1;
set n = (h + (f /. 1)) + 2;
A4: h < h1 by XREAL_1:29;
reconsider p1 = 1 as State of ZeroTuring by Lm8;
set s1 = [p1,h1,t];
reconsider i2 = h1 + 1 as Element of INT by INT_1:def_2;
A5: h1 < i2 by XREAL_1:29;
reconsider T = 1 as Symbol of ZeroTuring by Lm9;
set t1 = Tape-Chg (t,h1,T);
set t2 = Tape-Chg ((Tape-Chg (t,h1,T)),i2,F);
set t3 = Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),i2,F)),h1,T);
A6: t is_1_between h,(h + (f /. 1)) + 2 by A1, A3, Th22;
then A7: t . h = 0 by Def12;
then  (Tape-Chg (t,h1,T)) . h = 0 by A4, Th26;
then  (Tape-Chg ((Tape-Chg (t,h1,T)),i2,F)) . h = 0 by A4, A5, Th26;
then A8: (Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),i2,F)),h1,T)) . h = 0 by A4, Th26;
A9: TRAN s = Zero_Tran . [(s `1_3),(s3 . (Head s))] by Def19
.= Zero_Tran . [0,(s3 . (Head s))] by A2, MCART_1:64
.= Zero_Tran . [0,(t . (Head s))] by A2, MCART_1:64
.= [1,0,1] by A2, A7, Th33, MCART_1:64 ;
then A10: offset (TRAN s) = 1 by MCART_1:64;
set Rs = (Computation s) . (((1 + 1) + 1) + 1);
reconsider p3 = 3 as State of ZeroTuring by Lm8;
A11: (h + 1) + 1 = (h + 0) + 2 ;
A12: (Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),i2,F)),h1,T)) . (h + 1) = 1 by Th26;
A13: now__::_thesis:_for_k_being_Integer_st_h_<_k_&_k_<_(h_+_0)_+_2_holds_
(Tape-Chg_((Tape-Chg_((Tape-Chg_(t,h1,T)),i2,F)),h1,T))_._k_=_1
let k be Integer; ::_thesis: ( h < k & k < (h + 0) + 2 implies (Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),i2,F)),h1,T)) . k = 1 )
assume  ( h < k & k < (h + 0) + 2 ) ; ::_thesis: (Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),i2,F)),h1,T)) . k = 1
then  ( h + 1 <= k & k <= h + 1 ) by A11, INT_1:7;
hence  (Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),i2,F)),h1,T)) . k = 1 by A12, XXREAL_0:1; ::_thesis: verum
end;
 (Tape-Chg (t,h1,T)) . (h + 1) = 1 by Th26;
then A14: (Tape-Chg ((Tape-Chg (t,h1,T)),i2,F)) . (h + 1) = 1 by A5, Th26;
set s3 = [p3,h1,(Tape-Chg ((Tape-Chg (t,h1,T)),i2,F))];
reconsider s33 = [p3,h1,(Tape-Chg ((Tape-Chg (t,h1,T)),i2,F))] `3_3 as Tape of ZeroTuring ;
A15: TRAN [p3,h1,(Tape-Chg ((Tape-Chg (t,h1,T)),i2,F))] = Zero_Tran . [([p3,h1,(Tape-Chg ((Tape-Chg (t,h1,T)),i2,F))] `1_3),(s33 . (Head [p3,h1,(Tape-Chg ((Tape-Chg (t,h1,T)),i2,F))]))] by Def19
.= Zero_Tran . [p3,(s33 . (Head [p3,h1,(Tape-Chg ((Tape-Chg (t,h1,T)),i2,F))]))]
.= Zero_Tran . [p3,((Tape-Chg ((Tape-Chg (t,h1,T)),i2,F)) . (Head [p3,h1,(Tape-Chg ((Tape-Chg (t,h1,T)),i2,F))]))]
.= [4,1,(- 1)] by A14, Th33 ;
then A16: offset (TRAN [p3,h1,(Tape-Chg ((Tape-Chg (t,h1,T)),i2,F))]) = - 1 by MCART_1:64;
reconsider p2 = 2 as State of ZeroTuring by Lm8;
reconsider s13 = [p1,h1,t] `3_3 as Tape of ZeroTuring ;
 h <= h + (f /. 1) by NAT_1:11;
then A17: h + 2 <= (h + (f /. 1)) + 2 by XREAL_1:7;
 h1 < h + 2 by XREAL_1:8;
then A18: h1 < (h + (f /. 1)) + 2 by A17, XXREAL_0:2;
A19: TRAN [p1,h1,t] = Zero_Tran . [([p1,h1,t] `1_3),(s13 . (Head [p1,h1,t]))] by Def19
.= Zero_Tran . [p1,(s13 . (Head [p1,h1,t]))]
.= Zero_Tran . [p1,(t . (Head [p1,h1,t]))]
.= Zero_Tran . [1,(t . h1)]
.= [2,1,1] by A6, A4, A18, Def12, Th33 ;
then A20: offset (TRAN [p1,h1,t]) = 1 by MCART_1:64;
A21: now__::_thesis:_Zero_Tran_._[2,((Tape-Chg_(t,h1,T))_._i2)]_=_[3,0,(-_1)]
A22: the Symbols of ZeroTuring = {0,1} by Def19;
percases ( (Tape-Chg (t,h1,T)) . i2 = 1 or (Tape-Chg (t,h1,T)) . i2 = 0 ) by A22, TARSKI:def_2;
suppose (Tape-Chg (t,h1,T)) . i2 = 1 ; ::_thesis: Zero_Tran . [2,((Tape-Chg (t,h1,T)) . i2)] = [3,0,(- 1)]
hence  Zero_Tran . [2,((Tape-Chg (t,h1,T)) . i2)] = [3,0,(- 1)] by Th33; ::_thesis: verum
end;
suppose (Tape-Chg (t,h1,T)) . i2 = 0 ; ::_thesis: Zero_Tran . [2,((Tape-Chg (t,h1,T)) . i2)] = [3,0,(- 1)]
hence  Zero_Tran . [2,((Tape-Chg (t,h1,T)) . i2)] = [3,0,(- 1)] by Th33; ::_thesis: verum
end;
end;
end;
set s2 = [p2,i2,(Tape-Chg (t,h1,T))];
reconsider s23 = [p2,i2,(Tape-Chg (t,h1,T))] `3_3 as Tape of ZeroTuring ;
A23: TRAN [p2,i2,(Tape-Chg (t,h1,T))] = Zero_Tran . [([p2,i2,(Tape-Chg (t,h1,T))] `1_3),(s23 . (Head [p2,i2,(Tape-Chg (t,h1,T))]))] by Def19
.= Zero_Tran . [p2,(s23 . (Head [p2,i2,(Tape-Chg (t,h1,T))]))]
.= Zero_Tran . [p2,((Tape-Chg (t,h1,T)) . (Head [p2,i2,(Tape-Chg (t,h1,T))]))]
.= [3,0,(- 1)] by A21 ;
then A24: offset (TRAN [p2,i2,(Tape-Chg (t,h1,T))]) = - 1 by MCART_1:64;
Tape-Chg (([p2,i2,(Tape-Chg (t,h1,T))] `3_3),(Head [p2,i2,(Tape-Chg (t,h1,T))]),((TRAN [p2,i2,(Tape-Chg (t,h1,T))]) `2_3)) = Tape-Chg ((Tape-Chg (t,h1,T)),(Head [p2,i2,(Tape-Chg (t,h1,T))]),((TRAN [p2,i2,(Tape-Chg (t,h1,T))]) `2_3))
.= Tape-Chg ((Tape-Chg (t,h1,T)),i2,((TRAN [p2,i2,(Tape-Chg (t,h1,T))]) `2_3))
.= Tape-Chg ((Tape-Chg (t,h1,T)),i2,F) by A23, MCART_1:64 ;
then A25: Following [p2,i2,(Tape-Chg (t,h1,T))] = [((TRAN [p2,i2,(Tape-Chg (t,h1,T))]) `1_3),((Head [p2,i2,(Tape-Chg (t,h1,T))]) + (offset (TRAN [p2,i2,(Tape-Chg (t,h1,T))]))),(Tape-Chg ((Tape-Chg (t,h1,T)),i2,F))] by Lm10
.= [3,((Head [p2,i2,(Tape-Chg (t,h1,T))]) + (offset (TRAN [p2,i2,(Tape-Chg (t,h1,T))]))),(Tape-Chg ((Tape-Chg (t,h1,T)),i2,F))] by A23, MCART_1:64
.= [p3,h1,(Tape-Chg ((Tape-Chg (t,h1,T)),i2,F))] by A24 ;
reconsider p3 = 3 as State of ZeroTuring by Lm8;
A26: Tape-Chg (([p3,h1,(Tape-Chg ((Tape-Chg (t,h1,T)),i2,F))] `3_3),(Head [p3,h1,(Tape-Chg ((Tape-Chg (t,h1,T)),i2,F))]),((TRAN [p3,h1,(Tape-Chg ((Tape-Chg (t,h1,T)),i2,F))]) `2_3)) = Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),i2,F)),(Head [p3,h1,(Tape-Chg ((Tape-Chg (t,h1,T)),i2,F))]),((TRAN [p3,h1,(Tape-Chg ((Tape-Chg (t,h1,T)),i2,F))]) `2_3))
.= Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),i2,F)),h1,((TRAN [p3,h1,(Tape-Chg ((Tape-Chg (t,h1,T)),i2,F))]) `2_3))
.= Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),i2,F)),h1,T) by A15, MCART_1:64 ;
set s3 = [p3,h1,(Tape-Chg ((Tape-Chg (t,h1,T)),i2,F))];
A27: Following [p3,h1,(Tape-Chg ((Tape-Chg (t,h1,T)),i2,F))] = [((TRAN [p3,h1,(Tape-Chg ((Tape-Chg (t,h1,T)),i2,F))]) `1_3),((Head [p3,h1,(Tape-Chg ((Tape-Chg (t,h1,T)),i2,F))]) + (offset (TRAN [p3,h1,(Tape-Chg ((Tape-Chg (t,h1,T)),i2,F))]))),(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),i2,F)),h1,T))] by A26, Lm10
.= [4,((Head [p3,h1,(Tape-Chg ((Tape-Chg (t,h1,T)),i2,F))]) + (offset (TRAN [p3,h1,(Tape-Chg ((Tape-Chg (t,h1,T)),i2,F))]))),(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),i2,F)),h1,T))] by A15, MCART_1:64
.= [4,h,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),i2,F)),h1,T))] by A16 ;
Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)) = Tape-Chg (t,(Head s),((TRAN s) `2_3)) by A2, MCART_1:64
.= Tape-Chg (t,h,((TRAN s) `2_3)) by A2, MCART_1:64
.= Tape-Chg (t,h,F) by A9, MCART_1:64
.= t by A7, Th24 ;
then A28: Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),t] by A2, Lm10
.= [1,((Head s) + (offset (TRAN s))),t] by A9, MCART_1:64
.= [p1,h1,t] by A2, A10, MCART_1:64 ;
Tape-Chg (([p1,h1,t] `3_3),(Head [p1,h1,t]),((TRAN [p1,h1,t]) `2_3)) = Tape-Chg (t,(Head [p1,h1,t]),((TRAN [p1,h1,t]) `2_3))
.= Tape-Chg (t,h1,((TRAN [p1,h1,t]) `2_3))
.= Tape-Chg (t,h1,T) by A19, MCART_1:64 ;
then A29: Following [p1,h1,t] = [((TRAN [p1,h1,t]) `1_3),((Head [p1,h1,t]) + (offset (TRAN [p1,h1,t]))),(Tape-Chg (t,h1,T))] by Lm10
.= [2,((Head [p1,h1,t]) + (offset (TRAN [p1,h1,t]))),(Tape-Chg (t,h1,T))] by A19, MCART_1:64
.= [p2,i2,(Tape-Chg (t,h1,T))] by A20 ;
A30: (Computation s) . (((1 + 1) + 1) + 1) = Following ((Computation s) . ((1 + 1) + 1)) by Def7
.= Following (Following ((Computation s) . (1 + 1))) by Def7
.= Following (Following (Following ((Computation s) . 1))) by Def7
.= [4,h,(Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),i2,F)),h1,T))] by A28, A29, A25, A27, Th9 ;
then A31: ((Computation s) . (((1 + 1) + 1) + 1)) `1_3 = 4 by MCART_1:64
.= the AcceptS of ZeroTuring by Def19 ;
hence  s is Accept-Halt by Def8; ::_thesis: ( (Result s) `2_3 = h & (Result s) `3_3 storeData <*h,0*> )
then A32: Result s = (Computation s) . (((1 + 1) + 1) + 1) by A31, Def9;
hence  (Result s) `2_3 = h by A30, MCART_1:64; ::_thesis: (Result s) `3_3 storeData <*h,0*>
A33: (Result s) `3_3 = Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),i2,F)),h1,T) by A30, A32, MCART_1:64;
 (Tape-Chg ((Tape-Chg (t,h1,T)),i2,F)) . i2 = 0 by Th26;
then  (Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),i2,F)),h1,T)) . i2 = 0 by A5, Th26;
then  Tape-Chg ((Tape-Chg ((Tape-Chg (t,h1,T)),i2,F)),h1,T) is_1_between h,(h + 0) + 2 by A8, A13, Def12;
hence  (Result s) `3_3 storeData <*h,0*> by A33, Th16; ::_thesis: verum
end;

theorem :: TURING_1:35
for n being Element of NAT st n >= 1 holds
ZeroTuring computes n const 0
proof
let n be Element of NAT ; ::_thesis: ( n >= 1 implies ZeroTuring computes n const 0 )
assume A1: n >= 1 ; ::_thesis: ZeroTuring computes n const 0
now__::_thesis:_for_s_being_All-State_of_ZeroTuring
for_t_being_Tape_of_ZeroTuring
for_h_being_Element_of_NAT_
for_x_being_FinSequence_of_NAT_st_x_in_dom_(n_const_0)_&_s_=_[_the_InitS_of_ZeroTuring,h,t]_&_t_storeData_<*h*>_^_x_holds_
(_s_is_Accept-Halt_&_ex_h2_being_Element_of_NAT_ex_y_being_Element_of_NAT_st_
(_(Result_s)_`2_3_=_h2_&_y_=_(n_const_0)_._x_&_(Result_s)_`3_3_storeData_<*h2*>_^_<*y*>_)_)
set cs = n const 0;
let s be All-State of ZeroTuring; ::_thesis: for t being Tape of ZeroTuring
for h being Element of NAT
for x being FinSequence of NAT st x in dom (n const 0) & s = [ the InitS of ZeroTuring,h,t] & t storeData <*h*> ^ x holds
( s is Accept-Halt & ex h2 being Element of NAT ex y being Element of NAT st
( (Result s) `2_3 = h2 & y = (n const 0) . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> ) )
let t be Tape of ZeroTuring; ::_thesis: for h being Element of NAT
for x being FinSequence of NAT st x in dom (n const 0) & s = [ the InitS of ZeroTuring,h,t] & t storeData <*h*> ^ x holds
( s is Accept-Halt & ex h2 being Element of NAT ex y being Element of NAT st
( (Result s) `2_3 = h2 & y = (n const 0) . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> ) )
let h be Element of NAT ; ::_thesis: for x being FinSequence of NAT st x in dom (n const 0) & s = [ the InitS of ZeroTuring,h,t] & t storeData <*h*> ^ x holds
( s is Accept-Halt & ex h2 being Element of NAT ex y being Element of NAT st
( (Result s) `2_3 = h2 & y = (n const 0) . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> ) )
let x be FinSequence of NAT ; ::_thesis: ( x in dom (n const 0) & s = [ the InitS of ZeroTuring,h,t] & t storeData <*h*> ^ x implies ( s is Accept-Halt & ex h2 being Element of NAT ex y being Element of NAT st
( (Result s) `2_3 = h2 & y = (n const 0) . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> ) ) )
assume that
A2: x in dom (n const 0) and
A3: s = [ the InitS of ZeroTuring,h,t] and
A4: t storeData <*h*> ^ x ; ::_thesis: ( s is Accept-Halt & ex h2 being Element of NAT ex y being Element of NAT st
( (Result s) `2_3 = h2 & y = (n const 0) . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> ) )
 arity (n const 0) = n by COMPUT_1:32;
then A5: dom (n const 0) c= n -tuples_on NAT by COMPUT_1:21;
then  x in n -tuples_on NAT by A2;
then  x in  {  f where f is Element of NAT * : len f = n  }  by FINSEQ_2:def_4;
then A6: ex f being Element of NAT * st
( x = f & len f = n ) ;
A7: s = [0,h,t] by A3, Def19;
hence  s is Accept-Halt by A1, A4, A6, Th34; ::_thesis: ex h2 being Element of NAT ex y being Element of NAT st
( (Result s) `2_3 = h2 & y = (n const 0) . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> )
take h2 = h; ::_thesis: ex y being Element of NAT st
( (Result s) `2_3 = h2 & y = (n const 0) . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> )
take y = 0 ; ::_thesis: ( (Result s) `2_3 = h2 & y = (n const 0) . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> )
thus  (Result s) `2_3 = h2 by A1, A4, A6, A7, Th34; ::_thesis: ( y = (n const 0) . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> )
thus  y = (n const 0) . x by A2, A5, FUNCOP_1:7; ::_thesis: (Result s) `3_3 storeData <*h2*> ^ <*y*>
 (Result s) `3_3 storeData <*h2,0*> by A1, A4, A6, A7, Th34;
hence  (Result s) `3_3 storeData <*h2*> ^ <*y*> ; ::_thesis: verum
end;
hence  ZeroTuring computes n const 0 by Def15; ::_thesis: verum
end;

begin

definition
func U3(n)Tran  ->  Function of [:(SegM 3),{0,1}:],[:(SegM 3),{0,1},{(- 1),0,1}:] equals :: TURING_1:def 20
(((((id ([:(SegM 3),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,0,1])) +* ([1,0] .--> [2,0,1])) +* ([2,1] .--> [2,0,1])) +* ([2,0] .--> [3,0,0]);
coherence
 (((((id ([:(SegM 3),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,0,1])) +* ([1,0] .--> [2,0,1])) +* ([2,1] .--> [2,0,1])) +* ([2,0] .--> [3,0,0]) is Function of [:(SegM 3),{0,1}:],[:(SegM 3),{0,1},{(- 1),0,1}:]
proof
reconsider p0 = 0 , p1 = 1, p2 = 2, p3 = 3 as Element of SegM 3 by Th1;
set A = [:(SegM 3),{0,1}:];
set B = {(- 1),0,1};
set C = [:(SegM 3),{0,1},{(- 1),0,1}:];
reconsider b0 = 0 , b1 = 1 as Element of {0,1} by TARSKI:def_2;
reconsider h = 0 , R = 1 as Element of {(- 1),0,1} by ENUMSET1:def_1;
 [:(SegM 3),{0,1},{(- 1),0,1}:] = [:[:(SegM 3),{0,1}:],{(- 1),0,1}:] by ZFMISC_1:def_3;
then reconsider OP = id ([:(SegM 3),{0,1}:],{(- 1),0,1},h) as Function of [:(SegM 3),{0,1}:],[:(SegM 3),{0,1},{(- 1),0,1}:] ;
 (((((id ([:(SegM 3),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,0,1])) +* ([1,0] .--> [2,0,1])) +* ([2,1] .--> [2,0,1])) +* ([2,0] .--> [3,0,0]) = ((((OP +* ([p0,b0] .--> [p1,b0,R])) +* ([p1,b1] .--> [p1,b0,R])) +* ([p1,b0] .--> [p2,b0,R])) +* ([p2,b1] .--> [p2,b0,R])) +* ([p2,b0] .--> [p3,b0,h]) ;
hence  (((((id ([:(SegM 3),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,0,1])) +* ([1,0] .--> [2,0,1])) +* ([2,1] .--> [2,0,1])) +* ([2,0] .--> [3,0,0]) is Function of [:(SegM 3),{0,1}:],[:(SegM 3),{0,1},{(- 1),0,1}:] ; ::_thesis: verum
end;
end;

:: deftheorem  defines U3(n)Tran TURING_1:def_20_:_
 U3(n)Tran = (((((id ([:(SegM 3),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,0,1])) +* ([1,0] .--> [2,0,1])) +* ([2,1] .--> [2,0,1])) +* ([2,0] .--> [3,0,0]);

theorem Th36: :: TURING_1:36
( U3(n)Tran . [0,0] = [1,0,1] & U3(n)Tran . [1,1] = [1,0,1] & U3(n)Tran . [1,0] = [2,0,1] & U3(n)Tran . [2,1] = [2,0,1] & U3(n)Tran . [2,0] = [3,0,0] )
proof
set x = [1,1];
set x1 = [0,0];
set p1 = [0,0] .--> [1,0,1];
set p2 = [1,1] .--> [1,0,1];
set p3 = [1,0] .--> [2,0,1];
set p4 = [2,1] .--> [2,0,1];
set f = id ([:(SegM 3),{0,1}:],{(- 1),0,1},0);
thus U3(n)Tran . [0,0] = (((((id ([:(SegM 3),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,0,1])) +* ([1,0] .--> [2,0,1])) +* ([2,1] .--> [2,0,1])) . [0,0] by Th2
.= ((((id ([:(SegM 3),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,0,1])) +* ([1,0] .--> [2,0,1])) . [0,0] by Th2
.= (((id ([:(SegM 3),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,0,1])) . [0,0] by Th2
.= ((id ([:(SegM 3),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) . [0,0] by Th3
.= [1,0,1] by FUNCT_7:94 ; ::_thesis: ( U3(n)Tran . [1,1] = [1,0,1] & U3(n)Tran . [1,0] = [2,0,1] & U3(n)Tran . [2,1] = [2,0,1] & U3(n)Tran . [2,0] = [3,0,0] )
thus U3(n)Tran . [1,1] = (((((id ([:(SegM 3),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,0,1])) +* ([1,0] .--> [2,0,1])) +* ([2,1] .--> [2,0,1])) . [1,1] by Th2
.= ((((id ([:(SegM 3),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,0,1])) +* ([1,0] .--> [2,0,1])) . [1,1] by Th2
.= (((id ([:(SegM 3),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,0,1])) . [1,1] by Th3
.= [1,0,1] by FUNCT_7:94 ; ::_thesis: ( U3(n)Tran . [1,0] = [2,0,1] & U3(n)Tran . [2,1] = [2,0,1] & U3(n)Tran . [2,0] = [3,0,0] )
set x = [1,0];
thus U3(n)Tran . [1,0] = (((((id ([:(SegM 3),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,0,1])) +* ([1,0] .--> [2,0,1])) +* ([2,1] .--> [2,0,1])) . [1,0] by Th2
.= ((((id ([:(SegM 3),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,0,1])) +* ([1,0] .--> [2,0,1])) . [1,0] by Th3
.= [2,0,1] by FUNCT_7:94 ; ::_thesis: ( U3(n)Tran . [2,1] = [2,0,1] & U3(n)Tran . [2,0] = [3,0,0] )
set x = [2,1];
thus U3(n)Tran . [2,1] = (((((id ([:(SegM 3),{0,1}:],{(- 1),0,1},0)) +* ([0,0] .--> [1,0,1])) +* ([1,1] .--> [1,0,1])) +* ([1,0] .--> [2,0,1])) +* ([2,1] .--> [2,0,1])) . [2,1] by Th3
.= [2,0,1] by FUNCT_7:94 ; ::_thesis: U3(n)Tran . [2,0] = [3,0,0]
thus  U3(n)Tran . [2,0] = [3,0,0] by FUNCT_7:94; ::_thesis: verum
end;

definition
func U3(n)Turing  ->  strict TuringStr  means :Def21: :: TURING_1:def 21
( the Symbols of it = {0,1} & the FStates of it = SegM 3 & the Tran of it = U3(n)Tran & the InitS of it = 0 & the AcceptS of it = 3 );
existence
 ex b1 being strict TuringStr st
( the Symbols of b1 = {0,1} & the FStates of b1 = SegM 3 & the Tran of b1 = U3(n)Tran & the InitS of b1 = 0 & the AcceptS of b1 = 3 )
proof
set St = SegM 3;
reconsider qF = 3 as Element of SegM 3 by Th1;
reconsider p0 = 0 as Element of SegM 3 by Th1;
set Sym = {0,1};
take  TuringStr(# {0,1},(SegM 3),U3(n)Tran,p0,qF #) ; ::_thesis: ( the Symbols of TuringStr(# {0,1},(SegM 3),U3(n)Tran,p0,qF #) = {0,1} & the FStates of TuringStr(# {0,1},(SegM 3),U3(n)Tran,p0,qF #) = SegM 3 & the Tran of TuringStr(# {0,1},(SegM 3),U3(n)Tran,p0,qF #) = U3(n)Tran & the InitS of TuringStr(# {0,1},(SegM 3),U3(n)Tran,p0,qF #) = 0 & the AcceptS of TuringStr(# {0,1},(SegM 3),U3(n)Tran,p0,qF #) = 3 )
thus  ( the Symbols of TuringStr(# {0,1},(SegM 3),U3(n)Tran,p0,qF #) = {0,1} & the FStates of TuringStr(# {0,1},(SegM 3),U3(n)Tran,p0,qF #) = SegM 3 & the Tran of TuringStr(# {0,1},(SegM 3),U3(n)Tran,p0,qF #) = U3(n)Tran & the InitS of TuringStr(# {0,1},(SegM 3),U3(n)Tran,p0,qF #) = 0 & the AcceptS of TuringStr(# {0,1},(SegM 3),U3(n)Tran,p0,qF #) = 3 ) ; ::_thesis: verum
end;
uniqueness
 for b1, b2 being strict TuringStr st the Symbols of b1 = {0,1} & the FStates of b1 = SegM 3 & the Tran of b1 = U3(n)Tran & the InitS of b1 = 0 & the AcceptS of b1 = 3 & the Symbols of b2 = {0,1} & the FStates of b2 = SegM 3 & the Tran of b2 = U3(n)Tran & the InitS of b2 = 0 & the AcceptS of b2 = 3 holds
b1 = b2 ;
end;

:: deftheorem Def21 defines U3(n)Turing TURING_1:def_21_:_
 for b1 being strict TuringStr holds
( b1 = U3(n)Turing iff ( the Symbols of b1 = {0,1} & the FStates of b1 = SegM 3 & the Tran of b1 = U3(n)Tran & the InitS of b1 = 0 & the AcceptS of b1 = 3 ) );

Lm11:  for n being Element of NAT st n <= 3 holds
n is State of U3(n)Turing
proof
let n be Element of NAT ; ::_thesis: ( n <= 3 implies n is State of U3(n)Turing )
assume A1: n <= 3 ; ::_thesis: n is State of U3(n)Turing
 the FStates of U3(n)Turing = SegM 3 by Def21;
hence  n is State of U3(n)Turing by A1, Th1; ::_thesis: verum
end;

Lm12:  ( 0 in the Symbols of U3(n)Turing & 1 in the Symbols of U3(n)Turing )
proof
 ( 0 in {0,1} & 1 in {0,1} ) by TARSKI:def_2;
hence  ( 0 in the Symbols of U3(n)Turing & 1 in the Symbols of U3(n)Turing ) by Def21; ::_thesis: verum
end;

Lm13:  for s being All-State of U3(n)Turing
for p, h, t being set st s = [p,h,t] & p <> 3 holds
Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))]
proof
let s be All-State of U3(n)Turing; ::_thesis: for p, h, t being set st s = [p,h,t] & p <> 3 holds
Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))]
let p, h, t be set ; ::_thesis: ( s = [p,h,t] & p <> 3 implies Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))] )
assume A1: ( s = [p,h,t] & p <> 3 ) ; ::_thesis: Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))]
 3 = the AcceptS of U3(n)Turing by Def21;
hence  Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),(Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)))] by A1, Th25; ::_thesis: verum
end;

Lm14:  for tm being TuringStr
for s being All-State of tm
for p being State of tm
for h being Element of INT
for t being Tape of tm
for m, d being Element of NAT st d = h & 0 is Symbol of tm & s = [p,h,t] & the Tran of tm . [p,1] = [p,0,1] & p <> the AcceptS of tm & ( for i being Integer st d <= i & i < d + m holds
t . i = 1 ) holds
ex t1 being Tape of tm st
( (Computation s) . m = [p,(d + m),t1] & ( for i being Integer st d <= i & i < d + m holds
t1 . i = 0 ) & ( for i being Integer st ( d > i or i >= d + m ) holds
t1 . i = t . i ) )
proof
let tm be TuringStr ; ::_thesis: for s being All-State of tm
for p being State of tm
for h being Element of INT
for t being Tape of tm
for m, d being Element of NAT st d = h & 0 is Symbol of tm & s = [p,h,t] & the Tran of tm . [p,1] = [p,0,1] & p <> the AcceptS of tm & ( for i being Integer st d <= i & i < d + m holds
t . i = 1 ) holds
ex t1 being Tape of tm st
( (Computation s) . m = [p,(d + m),t1] & ( for i being Integer st d <= i & i < d + m holds
t1 . i = 0 ) & ( for i being Integer st ( d > i or i >= d + m ) holds
t1 . i = t . i ) )
let s be All-State of tm; ::_thesis: for p being State of tm
for h being Element of INT
for t being Tape of tm
for m, d being Element of NAT st d = h & 0 is Symbol of tm & s = [p,h,t] & the Tran of tm . [p,1] = [p,0,1] & p <> the AcceptS of tm & ( for i being Integer st d <= i & i < d + m holds
t . i = 1 ) holds
ex t1 being Tape of tm st
( (Computation s) . m = [p,(d + m),t1] & ( for i being Integer st d <= i & i < d + m holds
t1 . i = 0 ) & ( for i being Integer st ( d > i or i >= d + m ) holds
t1 . i = t . i ) )
let p be State of tm; ::_thesis: for h being Element of INT
for t being Tape of tm
for m, d being Element of NAT st d = h & 0 is Symbol of tm & s = [p,h,t] & the Tran of tm . [p,1] = [p,0,1] & p <> the AcceptS of tm & ( for i being Integer st d <= i & i < d + m holds
t . i = 1 ) holds
ex t1 being Tape of tm st
( (Computation s) . m = [p,(d + m),t1] & ( for i being Integer st d <= i & i < d + m holds
t1 . i = 0 ) & ( for i being Integer st ( d > i or i >= d + m ) holds
t1 . i = t . i ) )
let h be Element of INT ; ::_thesis: for t being Tape of tm
for m, d being Element of NAT st d = h & 0 is Symbol of tm & s = [p,h,t] & the Tran of tm . [p,1] = [p,0,1] & p <> the AcceptS of tm & ( for i being Integer st d <= i & i < d + m holds
t . i = 1 ) holds
ex t1 being Tape of tm st
( (Computation s) . m = [p,(d + m),t1] & ( for i being Integer st d <= i & i < d + m holds
t1 . i = 0 ) & ( for i being Integer st ( d > i or i >= d + m ) holds
t1 . i = t . i ) )
let t be Tape of tm; ::_thesis: for m, d being Element of NAT st d = h & 0 is Symbol of tm & s = [p,h,t] & the Tran of tm . [p,1] = [p,0,1] & p <> the AcceptS of tm & ( for i being Integer st d <= i & i < d + m holds
t . i = 1 ) holds
ex t1 being Tape of tm st
( (Computation s) . m = [p,(d + m),t1] & ( for i being Integer st d <= i & i < d + m holds
t1 . i = 0 ) & ( for i being Integer st ( d > i or i >= d + m ) holds
t1 . i = t . i ) )
let m, d be Element of NAT ; ::_thesis: ( d = h & 0 is Symbol of tm & s = [p,h,t] & the Tran of tm . [p,1] = [p,0,1] & p <> the AcceptS of tm & ( for i being Integer st d <= i & i < d + m holds
t . i = 1 ) implies ex t1 being Tape of tm st
( (Computation s) . m = [p,(d + m),t1] & ( for i being Integer st d <= i & i < d + m holds
t1 . i = 0 ) & ( for i being Integer st ( d > i or i >= d + m ) holds
t1 . i = t . i ) ) )
assume that
A1: d = h and
A2: 0 is Symbol of tm and
A3: s = [p,h,t] and
A4: the Tran of tm . [p,1] = [p,0,1] and
A5: p <> the AcceptS of tm and
A6: for i being Integer st d <= i & i < d + m holds
t . i = 1 ; ::_thesis: ex t1 being Tape of tm st
( (Computation s) . m = [p,(d + m),t1] & ( for i being Integer st d <= i & i < d + m holds
t1 . i = 0 ) & ( for i being Integer st ( d > i or i >= d + m ) holds
t1 . i = t . i ) )
defpred S1[ Element of NAT ] means ( $1 <= m implies ex t1 being Tape of tm st
( (Computation s) . $1 = [p,(d + $1),t1] & ( for i being Integer st d <= i & i < d + $1 holds
t1 . i = 0 ) & ( for i being Integer st ( d > i or i >= d + $1 ) holds
t1 . i = t . i ) ) );
A7: for k being Element of NAT st S1[k] holds
S1[k + 1]
proof
let k be Element of NAT ; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A8: S1[k] ; ::_thesis: S1[k + 1]
now__::_thesis:_(_k_+_1_<=_m_implies_ex_t2_being_Tape_of_tm_st_
(_(Computation_s)_._(k_+_1)_=_[p,(d_+_(k_+_1)),t2]_&_(_for_i_being_Integer_st_d_<=_i_&_i_<_d_+_(k_+_1)_holds_
t2_._i_=_0_)_&_(_for_i_being_Integer_st_(_d_>_i_or_i_>=_d_+_(k_+_1)_)_holds_
t2_._i_=_t_._i_)_)_)
reconsider F = 0 as Symbol of tm by A2;
set dk = d + k;
reconsider ik = d + k as Element of INT by INT_1:def_2;
assume A9: k + 1 <= m ; ::_thesis: ex t2 being Tape of tm st
( (Computation s) . (k + 1) = [p,(d + (k + 1)),t2] & ( for i being Integer st d <= i & i < d + (k + 1) holds
t2 . i = 0 ) & ( for i being Integer st ( d > i or i >= d + (k + 1) ) holds
t2 . i = t . i ) )
then consider t1 being Tape of tm such that
A10: (Computation s) . k = [p,(d + k),t1] and
A11: for i being Integer st d <= i & i < d + k holds
t1 . i = 0 and
A12: for i being Integer st ( d > i or i >= d + k ) holds
t1 . i = t . i by A8, NAT_1:13;
 k < m by A9, NAT_1:13;
then A13: d + k < d + m by XREAL_1:8;
A14: t1 . ik = t . ik by A12
.= 1 by A6, A13, NAT_1:11 ;
take t2 = Tape-Chg (t1,(d + k),F); ::_thesis: ( (Computation s) . (k + 1) = [p,(d + (k + 1)),t2] & ( for i being Integer st d <= i & i < d + (k + 1) holds
t2 . i = 0 ) & ( for i being Integer st ( d > i or i >= d + (k + 1) ) holds
t2 . i = t . i ) )
set sk = [p,ik,t1];
reconsider tt = [p,ik,t1] `3_3 as Tape of tm ;
A15: TRAN [p,ik,t1] = the Tran of tm . [p,(tt . (Head [p,ik,t1]))]
.= the Tran of tm . [p,(t1 . (Head [p,ik,t1]))]
.= [p,0,1] by A4, A14 ;
then A16: offset (TRAN [p,ik,t1]) = 1 by MCART_1:64;
A17: Tape-Chg (([p,ik,t1] `3_3),(Head [p,ik,t1]),((TRAN [p,ik,t1]) `2_3)) = Tape-Chg (t1,(Head [p,ik,t1]),((TRAN [p,ik,t1]) `2_3))
.= Tape-Chg (t1,(d + k),((TRAN [p,ik,t1]) `2_3))
.= t2 by A15, MCART_1:64 ;
thus (Computation s) . (k + 1) = Following [p,ik,t1] by A10, Def7
.= [((TRAN [p,ik,t1]) `1_3),((Head [p,ik,t1]) + (offset (TRAN [p,ik,t1]))),t2] by A5, A17, Th25
.= [p,((Head [p,ik,t1]) + (offset (TRAN [p,ik,t1]))),t2] by A15, MCART_1:64
.= [p,((d + k) + 1),t2] by A16
.= [p,(d + (k + 1)),t2] ; ::_thesis: ( ( for i being Integer st d <= i & i < d + (k + 1) holds
t2 . i = 0 ) & ( for i being Integer st ( d > i or i >= d + (k + 1) ) holds
t2 . i = t . i ) )
hereby  ::_thesis: for i being Integer st ( d > i or i >= d + (k + 1) ) holds
t2 . i = t . i
let i be Integer; ::_thesis: ( d <= i & i < d + (k + 1) implies t2 . b1 = 0 )
assume that
A18: d <= i and
A19: i < d + (k + 1) ; ::_thesis: t2 . b1 = 0
percases ( i = d + k or i <> d + k ) ;
suppose i = d + k ; ::_thesis: t2 . b1 = 0
hence  t2 . i = 0 by Th26; ::_thesis: verum
end;
supposeA20: i <> d + k ; ::_thesis: t2 . b1 = 0
 i < (d + k) + 1 by A19;
then  i <= d + k by INT_1:7;
then A21: i < d + k by A20, XXREAL_0:1;
thus t2 . i = t1 . i by A20, Th26
.= 0 by A11, A18, A21 ; ::_thesis: verum
end;
end;
end;
hereby  ::_thesis: verum
let i be Integer; ::_thesis: ( ( d > i or i >= d + (k + 1) ) implies t2 . b1 = t . b1 )
assume A22: ( d > i or i >= d + (k + 1) ) ; ::_thesis: t2 . b1 = t . b1
percases ( d > i or i >= d + (k + 1) ) by A22;
supposeA23: d > i ; ::_thesis: t2 . b1 = t . b1
 d <= d + k by NAT_1:12;
hence t2 . i = t1 . i by A23, Th26
.= t . i by A12, A23 ;
::_thesis: verum
end;
supposeA24: i >= d + (k + 1) ; ::_thesis: t2 . b1 = t . b1
 k < k + 1 by NAT_1:13;
then A25: d + k < d + (k + 1) by XREAL_1:8;
then A26: i > d + k by A24, XXREAL_0:2;
thus t2 . i = t1 . i by A24, A25, Th26
.= t . i by A12, A26 ; ::_thesis: verum
end;
end;
end;
end;
hence  S1[k + 1] ; ::_thesis: verum
end;
A27: S1[ 0 ]
proof
assume  0 <= m ; ::_thesis: ex t1 being Tape of tm st
( (Computation s) . 0 = [p,(d + 0),t1] & ( for i being Integer st d <= i & i < d + 0 holds
t1 . i = 0 ) & ( for i being Integer st ( d > i or i >= d + 0 ) holds
t1 . i = t . i ) )
take t1 = t; ::_thesis: ( (Computation s) . 0 = [p,(d + 0),t1] & ( for i being Integer st d <= i & i < d + 0 holds
t1 . i = 0 ) & ( for i being Integer st ( d > i or i >= d + 0 ) holds
t1 . i = t . i ) )
thus  (Computation s) . 0 = [p,(d + 0),t1] by A1, A3, Def7; ::_thesis: ( ( for i being Integer st d <= i & i < d + 0 holds
t1 . i = 0 ) & ( for i being Integer st ( d > i or i >= d + 0 ) holds
t1 . i = t . i ) )
thus  for i being Integer st d <= i & i < d + 0 holds
t1 . i = 0 ; ::_thesis: for i being Integer st ( d > i or i >= d + 0 ) holds
t1 . i = t . i
thus  for i being Integer st ( d > i or i >= d + 0 ) holds
t1 . i = t . i ; ::_thesis: verum
end;
 for k being Element of NAT holds S1[k] from NAT_1:sch_1(A27, A7);
hence  ex t1 being Tape of tm st
( (Computation s) . m = [p,(d + m),t1] & ( for i being Integer st d <= i & i < d + m holds
t1 . i = 0 ) & ( for i being Integer st ( d > i or i >= d + m ) holds
t1 . i = t . i ) ) ; ::_thesis: verum
end;

theorem Th37: :: TURING_1:37
for s being All-State of U3(n)Turing
for t being Tape of U3(n)Turing
for head being Element of NAT
for f being FinSequence of NAT st len f >= 3 & s = [0,head,t] & t storeData <*head*> ^ f holds
( s is Accept-Halt & (Result s) `2_3 = ((head + (f /. 1)) + (f /. 2)) + 4 & (Result s) `3_3 storeData <*(((head + (f /. 1)) + (f /. 2)) + 4),(f /. 3)*> )
proof
reconsider F = 0 as Symbol of U3(n)Turing by Lm12;
let s be All-State of U3(n)Turing; ::_thesis: for t being Tape of U3(n)Turing
for head being Element of NAT
for f being FinSequence of NAT st len f >= 3 & s = [0,head,t] & t storeData <*head*> ^ f holds
( s is Accept-Halt & (Result s) `2_3 = ((head + (f /. 1)) + (f /. 2)) + 4 & (Result s) `3_3 storeData <*(((head + (f /. 1)) + (f /. 2)) + 4),(f /. 3)*> )
let t be Tape of U3(n)Turing; ::_thesis: for head being Element of NAT
for f being FinSequence of NAT st len f >= 3 & s = [0,head,t] & t storeData <*head*> ^ f holds
( s is Accept-Halt & (Result s) `2_3 = ((head + (f /. 1)) + (f /. 2)) + 4 & (Result s) `3_3 storeData <*(((head + (f /. 1)) + (f /. 2)) + 4),(f /. 3)*> )
let h be Element of NAT ; ::_thesis: for f being FinSequence of NAT st len f >= 3 & s = [0,h,t] & t storeData <*h*> ^ f holds
( s is Accept-Halt & (Result s) `2_3 = ((h + (f /. 1)) + (f /. 2)) + 4 & (Result s) `3_3 storeData <*(((h + (f /. 1)) + (f /. 2)) + 4),(f /. 3)*> )
let f be FinSequence of NAT ; ::_thesis: ( len f >= 3 & s = [0,h,t] & t storeData <*h*> ^ f implies ( s is Accept-Halt & (Result s) `2_3 = ((h + (f /. 1)) + (f /. 2)) + 4 & (Result s) `3_3 storeData <*(((h + (f /. 1)) + (f /. 2)) + 4),(f /. 3)*> ) )
assume that
A1: len f >= 3 and
A2: s = [0,h,t] and
A3: t storeData <*h*> ^ f ; ::_thesis: ( s is Accept-Halt & (Result s) `2_3 = ((h + (f /. 1)) + (f /. 2)) + 4 & (Result s) `3_3 storeData <*(((h + (f /. 1)) + (f /. 2)) + 4),(f /. 3)*> )
set n1 = (h + (f /. 1)) + 2;
set n2 = ((h + (f /. 1)) + (f /. 2)) + 4;
set n3 = (((h + (f /. 1)) + (f /. 2)) + (f /. 3)) + 6;
reconsider s03 = s `3_3 as Tape of U3(n)Turing ;
A4: t is_1_between h,(h + (f /. 1)) + 2 by A1, A3, Th23;
then A5: t . h = 0 by Def12;
A6: TRAN s = U3(n)Tran . [(s `1_3),(s03 . (Head s))] by Def21
.= U3(n)Tran . [0,(s03 . (Head s))] by A2, MCART_1:64
.= U3(n)Tran . [0,(t . (Head s))] by A2, MCART_1:64
.= [1,0,1] by A2, A5, Th36, MCART_1:64 ;
then A7: offset (TRAN s) = 1 by MCART_1:64;
reconsider p1 = 1 as State of U3(n)Turing by Lm11;
set m1 = (f /. 1) + 1;
reconsider h1 = h + 1 as Element of INT by INT_1:def_2;
set s1 = [p1,h1,t];
A8: t is_1_between (h + (f /. 1)) + 2,((h + (f /. 1)) + (f /. 2)) + 4 by A1, A3, Th23;
then A9: t . (((h + (f /. 1)) + (f /. 2)) + 4) = 0 by Def12;
Tape-Chg ((s `3_3),(Head s),((TRAN s) `2_3)) = Tape-Chg (t,(Head s),((TRAN s) `2_3)) by A2, MCART_1:64
.= Tape-Chg (t,h,((TRAN s) `2_3)) by A2, MCART_1:64
.= Tape-Chg (t,h,F) by A6, MCART_1:64
.= t by A5, Th24 ;
then A10: Following s = [((TRAN s) `1_3),((Head s) + (offset (TRAN s))),t] by A2, Lm13
.= [1,((Head s) + (offset (TRAN s))),t] by A6, MCART_1:64
.= [p1,h1,t] by A2, A7, MCART_1:64 ;
A11: t is_1_between ((h + (f /. 1)) + (f /. 2)) + 4,(((h + (f /. 1)) + (f /. 2)) + (f /. 3)) + 6 by A1, A3, Th23;
then A12: t . ((((h + (f /. 1)) + (f /. 2)) + (f /. 3)) + 6) = 0 by Def12;
reconsider p2 = 2 as State of U3(n)Turing by Lm11;
set s2 = (Computation [p1,h1,t]) . ((f /. 1) + 1);
reconsider s23 = ((Computation [p1,h1,t]) . ((f /. 1) + 1)) `3_3 as Tape of U3(n)Turing ;
set j1 = ((h + 1) + ((f /. 1) + 1)) + 1;
reconsider k1 = ((h + 1) + ((f /. 1) + 1)) + 1 as Element of INT by INT_1:def_2;
set m2 = (f /. 2) + 1;
set Rs = (Computation s) . ((((((f /. 2) + 1) + 1) + 1) + ((f /. 1) + 1)) + 1);
set m3 = ((((h + (f /. 1)) + (f /. 2)) + 4) + (f /. 3)) + 2;
A13: now__::_thesis:_for_i_being_Integer_st_h_+_1_<=_i_&_i_<_(h_+_1)_+_((f_/._1)_+_1)_holds_
t_._i_=_1
let i be Integer; ::_thesis: ( h + 1 <= i & i < (h + 1) + ((f /. 1) + 1) implies t . i = 1 )
assume that
A14: h + 1 <= i and
A15: i < (h + 1) + ((f /. 1) + 1) ; ::_thesis: t . i = 1
 h < h + 1 by XREAL_1:29;
then  h < i by A14, XXREAL_0:2;
hence  t . i = 1 by A4, A15, Def12; ::_thesis: verum
end;
 ( the Tran of U3(n)Turing . [p1,1] = [p1,0,1] & p1 <> the AcceptS of U3(n)Turing ) by Def21, Th36;
then consider t1 being Tape of U3(n)Turing such that
A16: (Computation [p1,h1,t]) . ((f /. 1) + 1) = [p1,((h + 1) + ((f /. 1) + 1)),t1] and
 for i being Integer st h + 1 <= i & i < (h + 1) + ((f /. 1) + 1) holds
t1 . i = 0 and
A17: for i being Integer st ( h + 1 > i or i >= (h + 1) + ((f /. 1) + 1) ) holds
t1 . i = t . i by A13, Lm12, Lm14;
 t . ((h + (f /. 1)) + 2) = 0 by A4, Def12;
then A18: t1 . ((h + 1) + ((f /. 1) + 1)) = 0 by A17;
A19: TRAN ((Computation [p1,h1,t]) . ((f /. 1) + 1)) = U3(n)Tran . [(((Computation [p1,h1,t]) . ((f /. 1) + 1)) `1_3),(s23 . (Head ((Computation [p1,h1,t]) . ((f /. 1) + 1))))] by Def21
.= U3(n)Tran . [p1,(s23 . (Head ((Computation [p1,h1,t]) . ((f /. 1) + 1))))] by A16, MCART_1:64
.= U3(n)Tran . [1,(t1 . (Head ((Computation [p1,h1,t]) . ((f /. 1) + 1))))] by A16, MCART_1:64
.= [2,0,1] by A16, A18, Th36, MCART_1:64 ;
then A20: offset (TRAN ((Computation [p1,h1,t]) . ((f /. 1) + 1))) = 1 by MCART_1:64;
set s3 = [p2,k1,t1];
Tape-Chg ((((Computation [p1,h1,t]) . ((f /. 1) + 1)) `3_3),(Head ((Computation [p1,h1,t]) . ((f /. 1) + 1))),((TRAN ((Computation [p1,h1,t]) . ((f /. 1) + 1))) `2_3)) = Tape-Chg (t1,(Head ((Computation [p1,h1,t]) . ((f /. 1) + 1))),((TRAN ((Computation [p1,h1,t]) . ((f /. 1) + 1))) `2_3)) by A16, MCART_1:64
.= Tape-Chg (t1,((h + 1) + ((f /. 1) + 1)),((TRAN ((Computation [p1,h1,t]) . ((f /. 1) + 1))) `2_3)) by A16, MCART_1:64
.= Tape-Chg (t1,((h + 1) + ((f /. 1) + 1)),F) by A19, MCART_1:64
.= t1 by A18, Th24 ;
then A21: Following ((Computation [p1,h1,t]) . ((f /. 1) + 1)) = [((TRAN ((Computation [p1,h1,t]) . ((f /. 1) + 1))) `1_3),((Head ((Computation [p1,h1,t]) . ((f /. 1) + 1))) + (offset (TRAN ((Computation [p1,h1,t]) . ((f /. 1) + 1))))),t1] by A16, Lm13
.= [2,((Head ((Computation [p1,h1,t]) . ((f /. 1) + 1))) + (offset (TRAN ((Computation [p1,h1,t]) . ((f /. 1) + 1))))),t1] by A19, MCART_1:64
.= [p2,k1,t1] by A16, A20, MCART_1:64 ;
A22: now__::_thesis:_for_i_being_Integer_st_((h_+_1)_+_((f_/._1)_+_1))_+_1_<=_i_&_i_<_(((h_+_1)_+_((f_/._1)_+_1))_+_1)_+_((f_/._2)_+_1)_holds_
t1_._i_=_1
let i be Integer; ::_thesis: ( ((h + 1) + ((f /. 1) + 1)) + 1 <= i & i < (((h + 1) + ((f /. 1) + 1)) + 1) + ((f /. 2) + 1) implies t1 . i = 1 )
assume that
A23: ((h + 1) + ((f /. 1) + 1)) + 1 <= i and
A24: i < (((h + 1) + ((f /. 1) + 1)) + 1) + ((f /. 2) + 1) ; ::_thesis: t1 . i = 1
 (h + 1) + ((f /. 1) + 1) < ((h + 1) + ((f /. 1) + 1)) + 1 by XREAL_1:29;
then A25: (h + 1) + ((f /. 1) + 1) < i by A23, XXREAL_0:2;
hence t1 . i = t . i by A17
.= 1 by A8, A24, A25, Def12 ;
::_thesis: verum
end;
set s4 = (Computation [p2,k1,t1]) . ((f /. 2) + 1);
reconsider s43 = ((Computation [p2,k1,t1]) . ((f /. 2) + 1)) `3_3 as Tape of U3(n)Turing ;
 ( the Tran of U3(n)Turing . [p2,1] = [p2,0,1] & p2 <> the AcceptS of U3(n)Turing ) by Def21, Th36;
then consider t2 being Tape of U3(n)Turing such that
A26: (Computation [p2,k1,t1]) . ((f /. 2) + 1) = [p2,((((h + 1) + ((f /. 1) + 1)) + 1) + ((f /. 2) + 1)),t2] and
 for i being Integer st ((h + 1) + ((f /. 1) + 1)) + 1 <= i & i < (((h + 1) + ((f /. 1) + 1)) + 1) + ((f /. 2) + 1) holds
t2 . i = 0 and
A27: for i being Integer st ( ((h + 1) + ((f /. 1) + 1)) + 1 > i or i >= (((h + 1) + ((f /. 1) + 1)) + 1) + ((f /. 2) + 1) ) holds
t2 . i = t1 . i by A22, Lm12, Lm14;
 2 <= (f /. 2) + 4 by NAT_1:12;
then A28: (h + (f /. 1)) + 2 <= (h + (f /. 1)) + ((f /. 2) + 4) by XREAL_1:7;
A29: now__::_thesis:_for_k_being_Integer_st_((h_+_(f_/._1))_+_(f_/._2))_+_4_<_k_&_k_<_((((h_+_(f_/._1))_+_(f_/._2))_+_4)_+_(f_/._3))_+_2_holds_
t2_._k_=_1
let k be Integer; ::_thesis: ( ((h + (f /. 1)) + (f /. 2)) + 4 < k & k < ((((h + (f /. 1)) + (f /. 2)) + 4) + (f /. 3)) + 2 implies t2 . k = 1 )
assume that
A30: ((h + (f /. 1)) + (f /. 2)) + 4 < k and
A31: k < ((((h + (f /. 1)) + (f /. 2)) + 4) + (f /. 3)) + 2 ; ::_thesis: t2 . k = 1
A32: (h + (f /. 1)) + 2 <= k by A28, A30, XXREAL_0:2;
thus t2 . k = t1 . k by A27, A30
.= t . k by A17, A32
.= 1 by A11, A30, A31, Def12 ; ::_thesis: verum
end;
A33: t2 . ((((h + 1) + ((f /. 1) + 1)) + 1) + ((f /. 2) + 1)) = t1 . ((((h + 1) + ((f /. 1) + 1)) + 1) + ((f /. 2) + 1)) by A27
.= 0 by A9, A17, A28 ;
A34: TRAN ((Computation [p2,k1,t1]) . ((f /. 2) + 1)) = U3(n)Tran . [(((Computation [p2,k1,t1]) . ((f /. 2) + 1)) `1_3),(s43 . (Head ((Computation [p2,k1,t1]) . ((f /. 2) + 1))))] by Def21
.= U3(n)Tran . [p2,(s43 . (Head ((Computation [p2,k1,t1]) . ((f /. 2) + 1))))] by A26, MCART_1:64
.= U3(n)Tran . [2,(t2 . (Head ((Computation [p2,k1,t1]) . ((f /. 2) + 1))))] by A26, MCART_1:64
.= [3,0,0] by A26, A33, Th36, MCART_1:64 ;
then A35: offset (TRAN ((Computation [p2,k1,t1]) . ((f /. 2) + 1))) = 0 by MCART_1:64;
Tape-Chg ((((Computation [p2,k1,t1]) . ((f /. 2) + 1)) `3_3),(Head ((Computation [p2,k1,t1]) . ((f /. 2) + 1))),((TRAN ((Computation [p2,k1,t1]) . ((f /. 2) + 1))) `2_3)) = Tape-Chg (t2,(Head ((Computation [p2,k1,t1]) . ((f /. 2) + 1))),((TRAN ((Computation [p2,k1,t1]) . ((f /. 2) + 1))) `2_3)) by A26, MCART_1:64
.= Tape-Chg (t2,((((h + 1) + ((f /. 1) + 1)) + 1) + ((f /. 2) + 1)),((TRAN ((Computation [p2,k1,t1]) . ((f /. 2) + 1))) `2_3)) by A26, MCART_1:64
.= Tape-Chg (t2,((((h + 1) + ((f /. 1) + 1)) + 1) + ((f /. 2) + 1)),F) by A34, MCART_1:64
.= t2 by A33, Th24 ;
then A36: Following ((Computation [p2,k1,t1]) . ((f /. 2) + 1)) = [((TRAN ((Computation [p2,k1,t1]) . ((f /. 2) + 1))) `1_3),((Head ((Computation [p2,k1,t1]) . ((f /. 2) + 1))) + (offset (TRAN ((Computation [p2,k1,t1]) . ((f /. 2) + 1))))),t2] by A26, Lm13
.= [3,((Head ((Computation [p2,k1,t1]) . ((f /. 2) + 1))) + (offset (TRAN ((Computation [p2,k1,t1]) . ((f /. 2) + 1))))),t2] by A34, MCART_1:64
.= [3,(((((h + 1) + ((f /. 1) + 1)) + 1) + ((f /. 2) + 1)) + 0),t2] by A26, A35, MCART_1:64 ;
(Computation s) . ((((((f /. 2) + 1) + 1) + 1) + ((f /. 1) + 1)) + 1) = (Computation ((Computation s) . 1)) . (((((f /. 2) + 1) + 1) + 1) + ((f /. 1) + 1)) by Th10
.= (Computation [p1,h1,t]) . (((((f /. 2) + 1) + 1) + 1) + ((f /. 1) + 1)) by A10, Th9
.= (Computation ((Computation [p1,h1,t]) . ((f /. 1) + 1))) . ((((f /. 2) + 1) + 1) + 1) by Th10 ;
then A37: (Computation s) . ((((((f /. 2) + 1) + 1) + 1) + ((f /. 1) + 1)) + 1) = (Computation ((Computation ((Computation [p1,h1,t]) . ((f /. 1) + 1))) . 1)) . (((f /. 2) + 1) + 1) by Th10
.= (Computation [p2,k1,t1]) . (((f /. 2) + 1) + 1) by A21, Th9
.= [3,((((h + 1) + ((f /. 1) + 1)) + 1) + ((f /. 2) + 1)),t2] by A36, Def7 ;
then A38: ((Computation s) . ((((((f /. 2) + 1) + 1) + 1) + ((f /. 1) + 1)) + 1)) `1_3 = 3 by MCART_1:64
.= the AcceptS of U3(n)Turing by Def21 ;
hence  s is Accept-Halt by Def8; ::_thesis: ( (Result s) `2_3 = ((h + (f /. 1)) + (f /. 2)) + 4 & (Result s) `3_3 storeData <*(((h + (f /. 1)) + (f /. 2)) + 4),(f /. 3)*> )
then A39: Result s = (Computation s) . ((((((f /. 2) + 1) + 1) + 1) + ((f /. 1) + 1)) + 1) by A38, Def9;
hence  (Result s) `2_3 = ((h + (f /. 1)) + (f /. 2)) + 4 by A37, MCART_1:64; ::_thesis: (Result s) `3_3 storeData <*(((h + (f /. 1)) + (f /. 2)) + 4),(f /. 3)*>
A40: (Result s) `3_3 = t2 by A37, A39, MCART_1:64;
A41: ((h + (f /. 1)) + (f /. 2)) + 4 <= (((h + (f /. 1)) + (f /. 2)) + 4) + ((f /. 3) + 2) by NAT_1:11;
then A42: (h + (f /. 1)) + 2 <= ((((h + (f /. 1)) + (f /. 2)) + 4) + (f /. 3)) + 2 by A28, XXREAL_0:2;
t2 . (((((h + (f /. 1)) + (f /. 2)) + 4) + (f /. 3)) + 2) = t1 . (((((h + (f /. 1)) + (f /. 2)) + 4) + (f /. 3)) + 2) by A27, A41
.= 0 by A12, A17, A42 ;
then  t2 is_1_between ((h + (f /. 1)) + (f /. 2)) + 4,((((h + (f /. 1)) + (f /. 2)) + 4) + (f /. 3)) + 2 by A33, A29, Def12;
hence  (Result s) `3_3 storeData <*(((h + (f /. 1)) + (f /. 2)) + 4),(f /. 3)*> by A40, Th16; ::_thesis: verum
end;

theorem :: TURING_1:38
for n being Element of NAT st n >= 3 holds
U3(n)Turing computes n proj 3
proof
let n be Element of NAT ; ::_thesis: ( n >= 3 implies U3(n)Turing computes n proj 3 )
assume A1: n >= 3 ; ::_thesis: U3(n)Turing computes n proj 3
now__::_thesis:_for_s_being_All-State_of_U3(n)Turing
for_t_being_Tape_of_U3(n)Turing
for_h_being_Element_of_NAT_
for_x_being_FinSequence_of_NAT_st_x_in_dom_(n_proj_3)_&_s_=_[_the_InitS_of_U3(n)Turing,h,t]_&_t_storeData_<*h*>_^_x_holds_
(_s_is_Accept-Halt_&_ex_h2_being_Element_of_NAT_ex_y_being_Element_of_NAT_st_
(_(Result_s)_`2_3_=_h2_&_y_=_(n_proj_3)_._x_&_(Result_s)_`3_3_storeData_<*h2*>_^_<*y*>_)_)
set pj = n proj 3;
let s be All-State of U3(n)Turing; ::_thesis: for t being Tape of U3(n)Turing
for h being Element of NAT
for x being FinSequence of NAT st x in dom (n proj 3) & s = [ the InitS of U3(n)Turing,h,t] & t storeData <*h*> ^ x holds
( s is Accept-Halt & ex h2 being Element of NAT ex y being Element of NAT st
( (Result s) `2_3 = h2 & y = (n proj 3) . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> ) )
let t be Tape of U3(n)Turing; ::_thesis: for h being Element of NAT
for x being FinSequence of NAT st x in dom (n proj 3) & s = [ the InitS of U3(n)Turing,h,t] & t storeData <*h*> ^ x holds
( s is Accept-Halt & ex h2 being Element of NAT ex y being Element of NAT st
( (Result s) `2_3 = h2 & y = (n proj 3) . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> ) )
let h be Element of NAT ; ::_thesis: for x being FinSequence of NAT st x in dom (n proj 3) & s = [ the InitS of U3(n)Turing,h,t] & t storeData <*h*> ^ x holds
( s is Accept-Halt & ex h2 being Element of NAT ex y being Element of NAT st
( (Result s) `2_3 = h2 & y = (n proj 3) . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> ) )
let x be FinSequence of NAT ; ::_thesis: ( x in dom (n proj 3) & s = [ the InitS of U3(n)Turing,h,t] & t storeData <*h*> ^ x implies ( s is Accept-Halt & ex h2 being Element of NAT ex y being Element of NAT st
( (Result s) `2_3 = h2 & y = (n proj 3) . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> ) ) )
assume that
A2: x in dom (n proj 3) and
A3: s = [ the InitS of U3(n)Turing,h,t] and
A4: t storeData <*h*> ^ x ; ::_thesis: ( s is Accept-Halt & ex h2 being Element of NAT ex y being Element of NAT st
( (Result s) `2_3 = h2 & y = (n proj 3) . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> ) )
 arity (n proj 3) = n by COMPUT_1:37;
then A5: dom (n proj 3) c= n -tuples_on NAT by COMPUT_1:21;
then  x in n -tuples_on NAT by A2;
then  x in  {  f where f is Element of NAT * : len f = n  }  by FINSEQ_2:def_4;
then A6: ex f being Element of NAT * st
( x = f & len f = n ) ;
A7: s = [0,h,t] by A3, Def21;
hence  s is Accept-Halt by A1, A4, A6, Th37; ::_thesis: ex h2 being Element of NAT ex y being Element of NAT st
( (Result s) `2_3 = h2 & y = (n proj 3) . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> )
take h2 = ((h + (x /. 1)) + (x /. 2)) + 4; ::_thesis: ex y being Element of NAT st
( (Result s) `2_3 = h2 & y = (n proj 3) . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> )
take y = x /. 3; ::_thesis: ( (Result s) `2_3 = h2 & y = (n proj 3) . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> )
thus  (Result s) `2_3 = h2 by A1, A4, A6, A7, Th37; ::_thesis: ( y = (n proj 3) . x & (Result s) `3_3 storeData <*h2*> ^ <*y*> )
thus y = x . 3 by A1, A6, FINSEQ_4:15
.= (n proj 3) . x by A2, A5, COMPUT_1:38 ; ::_thesis: (Result s) `3_3 storeData <*h2*> ^ <*y*>
 (Result s) `3_3 storeData <*h2,y*> by A1, A4, A6, A7, Th37;
hence  (Result s) `3_3 storeData <*h2*> ^ <*y*> ; ::_thesis: verum
end;
hence  U3(n)Turing computes n proj 3 by Def15; ::_thesis: verum
end;

begin

definition
let t1, t2 be TuringStr ;
func UnionSt (t1,t2) ->  non empty finite set  equals :: TURING_1:def 22
[: the FStates of t1,{ the InitS of t2}:] \/ [:{ the AcceptS of t1}, the FStates of t2:];
correctness
coherence
[: the FStates of t1,{ the InitS of t2}:] \/ [:{ the AcceptS of t1}, the FStates of t2:] is non empty finite set ;
;
end;

:: deftheorem  defines UnionSt TURING_1:def_22_:_
 for t1, t2 being TuringStr holds UnionSt (t1,t2) = [: the FStates of t1,{ the InitS of t2}:] \/ [:{ the AcceptS of t1}, the FStates of t2:];

theorem Th39: :: TURING_1:39
for t1, t2 being TuringStr holds
( [ the InitS of t1, the InitS of t2] in UnionSt (t1,t2) & [ the AcceptS of t1, the AcceptS of t2] in UnionSt (t1,t2) )
proof
let t1, t2 be TuringStr ; ::_thesis: ( [ the InitS of t1, the InitS of t2] in UnionSt (t1,t2) & [ the AcceptS of t1, the AcceptS of t2] in UnionSt (t1,t2) )
set p0 = the InitS of t1;
set q0 = the InitS of t2;
set p1 = the AcceptS of t1;
set q1 = the AcceptS of t2;
set A = [: the FStates of t1,{ the InitS of t2}:];
set B = [:{ the AcceptS of t1}, the FStates of t2:];
reconsider q = the InitS of t2 as Element of { the InitS of t2} by TARSKI:def_1;
reconsider p = the AcceptS of t1 as Element of { the AcceptS of t1} by TARSKI:def_1;
 [ the InitS of t1,q] in [: the FStates of t1,{ the InitS of t2}:] ;
hence  [ the InitS of t1, the InitS of t2] in UnionSt (t1,t2) by XBOOLE_0:def_3; ::_thesis: [ the AcceptS of t1, the AcceptS of t2] in UnionSt (t1,t2)
 [p, the AcceptS of t2] in [:{ the AcceptS of t1}, the FStates of t2:] ;
hence  [ the AcceptS of t1, the AcceptS of t2] in UnionSt (t1,t2) by XBOOLE_0:def_3; ::_thesis: verum
end;

theorem Th40: :: TURING_1:40
for s, t being TuringStr
for x being State of s holds [x, the InitS of t] in UnionSt (s,t)
proof
let s, t be TuringStr ; ::_thesis: for x being State of s holds [x, the InitS of t] in UnionSt (s,t)
let x be State of s; ::_thesis: [x, the InitS of t] in UnionSt (s,t)
set q0 = the InitS of t;
set A = [: the FStates of s,{ the InitS of t}:];
reconsider q = the InitS of t as Element of { the InitS of t} by TARSKI:def_1;
 [x,q] in [: the FStates of s,{ the InitS of t}:] ;
hence  [x, the InitS of t] in UnionSt (s,t) by XBOOLE_0:def_3; ::_thesis: verum
end;

theorem Th41: :: TURING_1:41
for s, t being TuringStr
for x being State of t holds [ the AcceptS of s,x] in UnionSt (s,t)
proof
let s, t be TuringStr ; ::_thesis: for x being State of t holds [ the AcceptS of s,x] in UnionSt (s,t)
let x be State of t; ::_thesis: [ the AcceptS of s,x] in UnionSt (s,t)
set p1 = the AcceptS of s;
set B = [:{ the AcceptS of s}, the FStates of t:];
reconsider p = the AcceptS of s as Element of { the AcceptS of s} by TARSKI:def_1;
 [p,x] in [:{ the AcceptS of s}, the FStates of t:] ;
hence  [ the AcceptS of s,x] in UnionSt (s,t) by XBOOLE_0:def_3; ::_thesis: verum
end;

theorem Th42: :: TURING_1:42
for s, t being TuringStr
for x being Element of UnionSt (s,t) ex x1 being State of s ex x2 being State of t st x = [x1,x2]
proof
let s, t be TuringStr ; ::_thesis: for x being Element of UnionSt (s,t) ex x1 being State of s ex x2 being State of t st x = [x1,x2]
let x be Element of UnionSt (s,t); ::_thesis: ex x1 being State of s ex x2 being State of t st x = [x1,x2]
set q0 = the InitS of t;
set p1 = the AcceptS of s;
set A = [: the FStates of s,{ the InitS of t}:];
set B = [:{ the AcceptS of s}, the FStates of t:];
percases ( x in [: the FStates of s,{ the InitS of t}:] or x in [:{ the AcceptS of s}, the FStates of t:] ) by XBOOLE_0:def_3;
suppose x in [: the FStates of s,{ the InitS of t}:] ; ::_thesis: ex x1 being State of s ex x2 being State of t st x = [x1,x2]
then consider x1 being State of s, x2 being Element of { the InitS of t} such that
A1: x = [x1,x2] by DOMAIN_1:1;
take  x1 ; ::_thesis: ex x2 being State of t st x = [x1,x2]
take  the InitS of t ; ::_thesis: x = [x1, the InitS of t]
thus  x = [x1, the InitS of t] by A1, TARSKI:def_1; ::_thesis: verum
end;
suppose x in [:{ the AcceptS of s}, the FStates of t:] ; ::_thesis: ex x1 being State of s ex x2 being State of t st x = [x1,x2]
then consider x1 being Element of { the AcceptS of s}, x2 being State of t such that
A2: x = [x1,x2] by DOMAIN_1:1;
take  the AcceptS of s ; ::_thesis: ex x2 being State of t st x = [ the AcceptS of s,x2]
take  x2 ; ::_thesis: x = [ the AcceptS of s,x2]
thus  x = [ the AcceptS of s,x2] by A2, TARSKI:def_1; ::_thesis: verum
end;
end;
end;

definition
let s, t be TuringStr ;
let x be Tran-Goal of s;
func FirstTuringTran (s,t,x) ->  Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t),{(- 1),0,1}:] equals :: TURING_1:def 23
[[(x `1_3), the InitS of t],(x `2_3),(x `3_3)];
coherence
 [[(x `1_3), the InitS of t],(x `2_3),(x `3_3)] is Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t),{(- 1),0,1}:]
proof
reconsider y1 = [(x `1_3), the InitS of t] as Element of UnionSt (s,t) by Th40;
set Sym = the Symbols of s \/ the Symbols of t;
reconsider y2 = x `2_3 as Element of the Symbols of s \/ the Symbols of t by XBOOLE_0:def_3;
 [y1,y2,(x `3_3)] in [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t),{(- 1),0,1}:] ;
hence  [[(x `1_3), the InitS of t],(x `2_3),(x `3_3)] is Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t),{(- 1),0,1}:] ; ::_thesis: verum
end;
end;

:: deftheorem  defines FirstTuringTran TURING_1:def_23_:_
 for s, t being TuringStr
for x being Tran-Goal of s holds FirstTuringTran (s,t,x) = [[(x `1_3), the InitS of t],(x `2_3),(x `3_3)];

definition
let s, t be TuringStr ;
let x be Tran-Goal of t;
func SecondTuringTran (s,t,x) ->  Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t),{(- 1),0,1}:] equals :: TURING_1:def 24
[[ the AcceptS of s,(x `1_3)],(x `2_3),(x `3_3)];
coherence
 [[ the AcceptS of s,(x `1_3)],(x `2_3),(x `3_3)] is Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t),{(- 1),0,1}:]
proof
reconsider y1 = [ the AcceptS of s,(x `1_3)] as Element of UnionSt (s,t) by Th41;
set Sym = the Symbols of s \/ the Symbols of t;
reconsider y2 = x `2_3 as Element of the Symbols of s \/ the Symbols of t by XBOOLE_0:def_3;
 [y1,y2,(x `3_3)] in [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t),{(- 1),0,1}:] ;
hence  [[ the AcceptS of s,(x `1_3)],(x `2_3),(x `3_3)] is Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t),{(- 1),0,1}:] ; ::_thesis: verum
end;
end;

:: deftheorem  defines SecondTuringTran TURING_1:def_24_:_
 for s, t being TuringStr
for x being Tran-Goal of t holds SecondTuringTran (s,t,x) = [[ the AcceptS of s,(x `1_3)],(x `2_3),(x `3_3)];

definition
let s, t be TuringStr ;
let x be Element of UnionSt (s,t);
:: original: `1
redefine funcx `1  ->  State of s;
coherence
 x `1 is State of s
proof
consider x1 being State of s, x2 being State of t such that
A1: x = [x1,x2] by Th42;
 [x1,x2] `1 = x1 ;
hence  x `1 is State of s by A1; ::_thesis: verum
end;
:: original: `2
redefine funcx `2  ->  State of t;
coherence
 x `2 is State of t
proof
consider x1 being State of s, x2 being State of t such that
A2: x = [x1,x2] by Th42;
 [x1,x2] `2 = x2 ;
hence  x `2 is State of t by A2; ::_thesis: verum
end;
end;

definition
let s, t be TuringStr ;
let x be Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):];
func FirstTuringState x ->  State of s equals :: TURING_1:def 25
(x `1) `1 ;
correctness
coherence
(x `1) `1 is State of s;
;
func SecondTuringState x ->  State of t equals :: TURING_1:def 26
(x `1) `2 ;
correctness
coherence
(x `1) `2 is State of t;
;
end;

:: deftheorem  defines FirstTuringState TURING_1:def_25_:_
 for s, t being TuringStr
for x being Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):] holds FirstTuringState x = (x `1) `1 ;

:: deftheorem  defines SecondTuringState TURING_1:def_26_:_
 for s, t being TuringStr
for x being Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):] holds SecondTuringState x = (x `1) `2 ;

definition
let X, Y, Z be non empty set ;
let x be Element of [:X,(Y \/ Z):];
given u being set , y being Element of Y such that A1: x = [u,y] ;
A2: [u,y] `2 = y ;
func FirstTuringSymbol x ->  Element of Y equals :Def27: :: TURING_1:def 27
x `2 ;
coherence
 x `2 is Element of Y by A1, A2;
end;

:: deftheorem Def27 defines FirstTuringSymbol TURING_1:def_27_:_
 for X, Y, Z being non empty set
for x being Element of [:X,(Y \/ Z):] st ex u being set ex y being Element of Y st x = [u,y] holds
FirstTuringSymbol x = x `2 ;

definition
let X, Y, Z be non empty set ;
let x be Element of [:X,(Y \/ Z):];
given u being set , z being Element of Z such that A1: x = [u,z] ;
A2: [u,z] `2 = z ;
func SecondTuringSymbol x ->  Element of Z equals :Def28: :: TURING_1:def 28
x `2 ;
coherence
 x `2 is Element of Z by A1, A2;
end;

:: deftheorem Def28 defines SecondTuringSymbol TURING_1:def_28_:_
 for X, Y, Z being non empty set
for x being Element of [:X,(Y \/ Z):] st ex u being set ex z being Element of Z st x = [u,z] holds
SecondTuringSymbol x = x `2 ;

definition
let s, t be TuringStr ;
let x be Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):];
func Uniontran (s,t,x) ->  Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t),{(- 1),0,1}:] equals :Def29: :: TURING_1:def 29
 FirstTuringTran (s,t,( the Tran of s . [(FirstTuringState x),(FirstTuringSymbol x)])) if  ex p being State of s ex y being Symbol of s st
( x = [[p, the InitS of t],y] & p <> the AcceptS of s )
SecondTuringTran (s,t,( the Tran of t . [(SecondTuringState x),(SecondTuringSymbol x)])) if  ex q being State of t ex y being Symbol of t st x = [[ the AcceptS of s,q],y]
otherwise [(x `1),(x `2),(- 1)];
consistency
 for b1 being Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t),{(- 1),0,1}:] st ex p being State of s ex y being Symbol of s st
( x = [[p, the InitS of t],y] & p <> the AcceptS of s ) & ex q being State of t ex y being Symbol of t st x = [[ the AcceptS of s,q],y] holds
( b1 = FirstTuringTran (s,t,( the Tran of s . [(FirstTuringState x),(FirstTuringSymbol x)])) iff b1 = SecondTuringTran (s,t,( the Tran of t . [(SecondTuringState x),(SecondTuringSymbol x)])) )
proof
let w be Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t),{(- 1),0,1}:]; ::_thesis: ( ex p being State of s ex y being Symbol of s st
( x = [[p, the InitS of t],y] & p <> the AcceptS of s ) & ex q being State of t ex y being Symbol of t st x = [[ the AcceptS of s,q],y] implies ( w = FirstTuringTran (s,t,( the Tran of s . [(FirstTuringState x),(FirstTuringSymbol x)])) iff w = SecondTuringTran (s,t,( the Tran of t . [(SecondTuringState x),(SecondTuringSymbol x)])) ) )
thus  ( ex p being State of s ex y being Symbol of s st
( x = [[p, the InitS of t],y] & p <> the AcceptS of s ) & ex q being State of t ex y being Symbol of t st x = [[ the AcceptS of s,q],y] implies ( w = FirstTuringTran (s,t,( the Tran of s . [(FirstTuringState x),(FirstTuringSymbol x)])) iff w = SecondTuringTran (s,t,( the Tran of t . [(SecondTuringState x),(SecondTuringSymbol x)])) ) ) ::_thesis: verum
proof
given p being State of s, y being Symbol of s such that A1: x = [[p, the InitS of t],y] and
A2: p <> the AcceptS of s ; ::_thesis: ( for q being State of t
for y being Symbol of t holds not x = [[ the AcceptS of s,q],y] or ( w = FirstTuringTran (s,t,( the Tran of s . [(FirstTuringState x),(FirstTuringSymbol x)])) iff w = SecondTuringTran (s,t,( the Tran of t . [(SecondTuringState x),(SecondTuringSymbol x)])) ) )
given q being State of t, z being Symbol of t such that A3: x = [[ the AcceptS of s,q],z] ; ::_thesis: ( w = FirstTuringTran (s,t,( the Tran of s . [(FirstTuringState x),(FirstTuringSymbol x)])) iff w = SecondTuringTran (s,t,( the Tran of t . [(SecondTuringState x),(SecondTuringSymbol x)])) )
 [p, the InitS of t] = [ the AcceptS of s,q] by A1, A3, XTUPLE_0:1;
hence  ( w = FirstTuringTran (s,t,( the Tran of s . [(FirstTuringState x),(FirstTuringSymbol x)])) iff w = SecondTuringTran (s,t,( the Tran of t . [(SecondTuringState x),(SecondTuringSymbol x)])) ) by A2, XTUPLE_0:1; ::_thesis: verum
end;
end;
coherence
 ( ( ex p being State of s ex y being Symbol of s st
( x = [[p, the InitS of t],y] & p <> the AcceptS of s ) implies FirstTuringTran (s,t,( the Tran of s . [(FirstTuringState x),(FirstTuringSymbol x)])) is Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t),{(- 1),0,1}:] ) & ( ex q being State of t ex y being Symbol of t st x = [[ the AcceptS of s,q],y] implies SecondTuringTran (s,t,( the Tran of t . [(SecondTuringState x),(SecondTuringSymbol x)])) is Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t),{(- 1),0,1}:] ) & ( ( for p being State of s
for y being Symbol of s holds
( not x = [[p, the InitS of t],y] or not p <> the AcceptS of s ) ) & ( for q being State of t
for y being Symbol of t holds not x = [[ the AcceptS of s,q],y] ) implies [(x `1),(x `2),(- 1)] is Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t),{(- 1),0,1}:] ) )
proof
reconsider l = - 1 as Element of {(- 1),0,1} by ENUMSET1:def_1;
 [(x `1),(x `2),l] in [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t),{(- 1),0,1}:] ;
hence  ( ( ex p being State of s ex y being Symbol of s st
( x = [[p, the InitS of t],y] & p <> the AcceptS of s ) implies FirstTuringTran (s,t,( the Tran of s . [(FirstTuringState x),(FirstTuringSymbol x)])) is Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t),{(- 1),0,1}:] ) & ( ex q being State of t ex y being Symbol of t st x = [[ the AcceptS of s,q],y] implies SecondTuringTran (s,t,( the Tran of t . [(SecondTuringState x),(SecondTuringSymbol x)])) is Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t),{(- 1),0,1}:] ) & ( ( for p being State of s
for y being Symbol of s holds
( not x = [[p, the InitS of t],y] or not p <> the AcceptS of s ) ) & ( for q being State of t
for y being Symbol of t holds not x = [[ the AcceptS of s,q],y] ) implies [(x `1),(x `2),(- 1)] is Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t),{(- 1),0,1}:] ) ) ; ::_thesis: verum
end;
end;

:: deftheorem Def29 defines Uniontran TURING_1:def_29_:_
 for s, t being TuringStr
for x being Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):] holds
( ( ex p being State of s ex y being Symbol of s st
( x = [[p, the InitS of t],y] & p <> the AcceptS of s ) implies Uniontran (s,t,x) = FirstTuringTran (s,t,( the Tran of s . [(FirstTuringState x),(FirstTuringSymbol x)])) ) & ( ex q being State of t ex y being Symbol of t st x = [[ the AcceptS of s,q],y] implies Uniontran (s,t,x) = SecondTuringTran (s,t,( the Tran of t . [(SecondTuringState x),(SecondTuringSymbol x)])) ) & ( ( for p being State of s
for y being Symbol of s holds
( not x = [[p, the InitS of t],y] or not p <> the AcceptS of s ) ) & ( for q being State of t
for y being Symbol of t holds not x = [[ the AcceptS of s,q],y] ) implies Uniontran (s,t,x) = [(x `1),(x `2),(- 1)] ) );

definition
let s, t be TuringStr ;
func UnionTran (s,t) ->  Function of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):],[:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t),{(- 1),0,1}:] means :Def30: :: TURING_1:def 30
 for x being Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):] holds it . x = Uniontran (s,t,x);
existence
 ex b1 being Function of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):],[:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t),{(- 1),0,1}:] st
for x being Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):] holds b1 . x = Uniontran (s,t,x)
proof
set Sm = the Symbols of s \/ the Symbols of t;
set X = [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):];
deffunc H1( Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):]) ->  Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t),{(- 1),0,1}:] = Uniontran (s,t,$1);
consider f being Function of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):],[:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t),{(- 1),0,1}:] such that
A1: for x being Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):] holds f . x = H1(x) from FUNCT_2:sch_4();
take  f ; ::_thesis: for x being Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):] holds f . x = Uniontran (s,t,x)
thus  for x being Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):] holds f . x = Uniontran (s,t,x) by A1; ::_thesis: verum
end;
uniqueness
 for b1, b2 being Function of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):],[:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t),{(- 1),0,1}:] st ( for x being Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):] holds b1 . x = Uniontran (s,t,x) ) & ( for x being Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):] holds b2 . x = Uniontran (s,t,x) ) holds
b1 = b2
proof
set Sm = the Symbols of s \/ the Symbols of t;
set X = [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):];
let f, g be Function of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):],[:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t),{(- 1),0,1}:]; ::_thesis: ( ( for x being Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):] holds f . x = Uniontran (s,t,x) ) & ( for x being Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):] holds g . x = Uniontran (s,t,x) ) implies f = g )
assume that
A2: for x being Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):] holds f . x = Uniontran (s,t,x) and
A3: for x being Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):] holds g . x = Uniontran (s,t,x) ; ::_thesis: f = g
now__::_thesis:_for_x_being_Element_of_[:(UnionSt_(s,t)),(_the_Symbols_of_s_\/_the_Symbols_of_t):]_holds_f_._x_=_g_._x
let x be Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):]; ::_thesis: f . x = g . x
thus f . x = Uniontran (s,t,x) by A2
.= g . x by A3 ; ::_thesis: verum
end;
hence  f = g by FUNCT_2:63; ::_thesis: verum
end;
end;

:: deftheorem Def30 defines UnionTran TURING_1:def_30_:_
 for s, t being TuringStr
for b3 being Function of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):],[:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t),{(- 1),0,1}:] holds
( b3 = UnionTran (s,t) iff for x being Element of [:(UnionSt (s,t)),( the Symbols of s \/ the Symbols of t):] holds b3 . x = Uniontran (s,t,x) );

definition
let T1, T2 be TuringStr ;
funcT1 ';' T2 ->  strict TuringStr  means :Def31: :: TURING_1:def 31
( the Symbols of it = the Symbols of T1 \/ the Symbols of T2 & the FStates of it = UnionSt (T1,T2) & the Tran of it = UnionTran (T1,T2) & the InitS of it = [ the InitS of T1, the InitS of T2] & the AcceptS of it = [ the AcceptS of T1, the AcceptS of T2] );
existence
 ex b1 being strict TuringStr st
( the Symbols of b1 = the Symbols of T1 \/ the Symbols of T2 & the FStates of b1 = UnionSt (T1,T2) & the Tran of b1 = UnionTran (T1,T2) & the InitS of b1 = [ the InitS of T1, the InitS of T2] & the AcceptS of b1 = [ the AcceptS of T1, the AcceptS of T2] )
proof
set St = UnionSt (T1,T2);
reconsider q1 = [ the AcceptS of T1, the AcceptS of T2] as Element of UnionSt (T1,T2) by Th39;
reconsider p0 = [ the InitS of T1, the InitS of T2] as Element of UnionSt (T1,T2) by Th39;
set Sym = the Symbols of T1 \/ the Symbols of T2;
take  TuringStr(# ( the Symbols of T1 \/ the Symbols of T2),(UnionSt (T1,T2)),(UnionTran (T1,T2)),p0,q1 #) ; ::_thesis: ( the Symbols of TuringStr(# ( the Symbols of T1 \/ the Symbols of T2),(UnionSt (T1,T2)),(UnionTran (T1,T2)),p0,q1 #) = the Symbols of T1 \/ the Symbols of T2 & the FStates of TuringStr(# ( the Symbols of T1 \/ the Symbols of T2),(UnionSt (T1,T2)),(UnionTran (T1,T2)),p0,q1 #) = UnionSt (T1,T2) & the Tran of TuringStr(# ( the Symbols of T1 \/ the Symbols of T2),(UnionSt (T1,T2)),(UnionTran (T1,T2)),p0,q1 #) = UnionTran (T1,T2) & the InitS of TuringStr(# ( the Symbols of T1 \/ the Symbols of T2),(UnionSt (T1,T2)),(UnionTran (T1,T2)),p0,q1 #) = [ the InitS of T1, the InitS of T2] & the AcceptS of TuringStr(# ( the Symbols of T1 \/ the Symbols of T2),(UnionSt (T1,T2)),(UnionTran (T1,T2)),p0,q1 #) = [ the AcceptS of T1, the AcceptS of T2] )
thus  ( the Symbols of TuringStr(# ( the Symbols of T1 \/ the Symbols of T2),(UnionSt (T1,T2)),(UnionTran (T1,T2)),p0,q1 #) = the Symbols of T1 \/ the Symbols of T2 & the FStates of TuringStr(# ( the Symbols of T1 \/ the Symbols of T2),(UnionSt (T1,T2)),(UnionTran (T1,T2)),p0,q1 #) = UnionSt (T1,T2) & the Tran of TuringStr(# ( the Symbols of T1 \/ the Symbols of T2),(UnionSt (T1,T2)),(UnionTran (T1,T2)),p0,q1 #) = UnionTran (T1,T2) & the InitS of TuringStr(# ( the Symbols of T1 \/ the Symbols of T2),(UnionSt (T1,T2)),(UnionTran (T1,T2)),p0,q1 #) = [ the InitS of T1, the InitS of T2] & the AcceptS of TuringStr(# ( the Symbols of T1 \/ the Symbols of T2),(UnionSt (T1,T2)),(UnionTran (T1,T2)),p0,q1 #) = [ the AcceptS of T1, the AcceptS of T2] ) ; ::_thesis: verum
end;
uniqueness
 for b1, b2 being strict TuringStr st the Symbols of b1 = the Symbols of T1 \/ the Symbols of T2 & the FStates of b1 = UnionSt (T1,T2) & the Tran of b1 = UnionTran (T1,T2) & the InitS of b1 = [ the InitS of T1, the InitS of T2] & the AcceptS of b1 = [ the AcceptS of T1, the AcceptS of T2] & the Symbols of b2 = the Symbols of T1 \/ the Symbols of T2 & the FStates of b2 = UnionSt (T1,T2) & the Tran of b2 = UnionTran (T1,T2) & the InitS of b2 = [ the InitS of T1, the InitS of T2] & the AcceptS of b2 = [ the AcceptS of T1, the AcceptS of T2] holds
b1 = b2 ;
end;

:: deftheorem Def31 defines ';' TURING_1:def_31_:_
 for T1, T2 being TuringStr
for b3 being strict TuringStr holds
( b3 = T1 ';' T2 iff ( the Symbols of b3 = the Symbols of T1 \/ the Symbols of T2 & the FStates of b3 = UnionSt (T1,T2) & the Tran of b3 = UnionTran (T1,T2) & the InitS of b3 = [ the InitS of T1, the InitS of T2] & the AcceptS of b3 = [ the AcceptS of T1, the AcceptS of T2] ) );

theorem Th43: :: TURING_1:43
for T1, T2 being TuringStr
for g being Tran-Goal of T1
for p being State of T1
for y being Symbol of T1 st p <> the AcceptS of T1 & g = the Tran of T1 . [p,y] holds
the Tran of (T1 ';' T2) . [[p, the InitS of T2],y] = [[(g `1_3), the InitS of T2],(g `2_3),(g `3_3)]
proof
let t1, t2 be TuringStr ; ::_thesis: for g being Tran-Goal of t1
for p being State of t1
for y being Symbol of t1 st p <> the AcceptS of t1 & g = the Tran of t1 . [p,y] holds
the Tran of (t1 ';' t2) . [[p, the InitS of t2],y] = [[(g `1_3), the InitS of t2],(g `2_3),(g `3_3)]
let g be Tran-Goal of t1; ::_thesis: for p being State of t1
for y being Symbol of t1 st p <> the AcceptS of t1 & g = the Tran of t1 . [p,y] holds
the Tran of (t1 ';' t2) . [[p, the InitS of t2],y] = [[(g `1_3), the InitS of t2],(g `2_3),(g `3_3)]
let p be State of t1; ::_thesis: for y being Symbol of t1 st p <> the AcceptS of t1 & g = the Tran of t1 . [p,y] holds
the Tran of (t1 ';' t2) . [[p, the InitS of t2],y] = [[(g `1_3), the InitS of t2],(g `2_3),(g `3_3)]
let y be Symbol of t1; ::_thesis: ( p <> the AcceptS of t1 & g = the Tran of t1 . [p,y] implies the Tran of (t1 ';' t2) . [[p, the InitS of t2],y] = [[(g `1_3), the InitS of t2],(g `2_3),(g `3_3)] )
assume that
A1: p <> the AcceptS of t1 and
A2: g = the Tran of t1 . [p,y] ; ::_thesis: the Tran of (t1 ';' t2) . [[p, the InitS of t2],y] = [[(g `1_3), the InitS of t2],(g `2_3),(g `3_3)]
set q0 = the InitS of t2;
set x = [[p, the InitS of t2],y];
 the InitS of t2 in { the InitS of t2} by TARSKI:def_1;
then  [p, the InitS of t2] in [: the FStates of t1,{ the InitS of t2}:] by ZFMISC_1:def_2;
then A3: [p, the InitS of t2] in [: the FStates of t1,{ the InitS of t2}:] \/ [:{ the AcceptS of t1}, the FStates of t2:] by XBOOLE_0:def_3;
 y in the Symbols of t1 \/ the Symbols of t2 by XBOOLE_0:def_3;
then reconsider xx = [[p, the InitS of t2],y] as Element of [:(UnionSt (t1,t2)),( the Symbols of t1 \/ the Symbols of t2):] by A3, ZFMISC_1:def_2;
A4: FirstTuringState xx = ([[p, the InitS of t2],y] `1) `1
.= [p, the InitS of t2] `1
.= p ;
A5: FirstTuringSymbol xx = [[p, the InitS of t2],y] `2 by Def27
.= y ;
thus  the Tran of (t1 ';' t2) . [[p, the InitS of t2],y] = (UnionTran (t1,t2)) . xx by Def31
.= Uniontran (t1,t2,xx) by Def30
.= FirstTuringTran (t1,t2,( the Tran of t1 . [p,y])) by A1, A4, A5, Def29
.= [[(g `1_3), the InitS of t2],(g `2_3),(g `3_3)] by A2 ; ::_thesis: verum
end;

theorem Th44: :: TURING_1:44
for T1, T2 being TuringStr
for g being Tran-Goal of T2
for q being State of T2
for y being Symbol of T2 st g = the Tran of T2 . [q,y] holds
the Tran of (T1 ';' T2) . [[ the AcceptS of T1,q],y] = [[ the AcceptS of T1,(g `1_3)],(g `2_3),(g `3_3)]
proof
let t1, t2 be TuringStr ; ::_thesis: for g being Tran-Goal of t2
for q being State of t2
for y being Symbol of t2 st g = the Tran of t2 . [q,y] holds
the Tran of (t1 ';' t2) . [[ the AcceptS of t1,q],y] = [[ the AcceptS of t1,(g `1_3)],(g `2_3),(g `3_3)]
let g be Tran-Goal of t2; ::_thesis: for q being State of t2
for y being Symbol of t2 st g = the Tran of t2 . [q,y] holds
the Tran of (t1 ';' t2) . [[ the AcceptS of t1,q],y] = [[ the AcceptS of t1,(g `1_3)],(g `2_3),(g `3_3)]
let q be State of t2; ::_thesis: for y being Symbol of t2 st g = the Tran of t2 . [q,y] holds
the Tran of (t1 ';' t2) . [[ the AcceptS of t1,q],y] = [[ the AcceptS of t1,(g `1_3)],(g `2_3),(g `3_3)]
let y be Symbol of t2; ::_thesis: ( g = the Tran of t2 . [q,y] implies the Tran of (t1 ';' t2) . [[ the AcceptS of t1,q],y] = [[ the AcceptS of t1,(g `1_3)],(g `2_3),(g `3_3)] )
assume A1: g = the Tran of t2 . [q,y] ; ::_thesis: the Tran of (t1 ';' t2) . [[ the AcceptS of t1,q],y] = [[ the AcceptS of t1,(g `1_3)],(g `2_3),(g `3_3)]
set pF = the AcceptS of t1;
set x = [[ the AcceptS of t1,q],y];
 the AcceptS of t1 in { the AcceptS of t1} by TARSKI:def_1;
then  [ the AcceptS of t1,q] in [:{ the AcceptS of t1}, the FStates of t2:] by ZFMISC_1:def_2;
then A2: [ the AcceptS of t1,q] in [: the FStates of t1,{ the InitS of t2}:] \/ [:{ the AcceptS of t1}, the FStates of t2:] by XBOOLE_0:def_3;
 y in the Symbols of t1 \/ the Symbols of t2 by XBOOLE_0:def_3;
then reconsider xx = [[ the AcceptS of t1,q],y] as Element of [:(UnionSt (t1,t2)),( the Symbols of t1 \/ the Symbols of t2):] by A2, ZFMISC_1:def_2;
A3: SecondTuringState xx = ([[ the AcceptS of t1,q],y] `1) `2
.= q ;
A4: SecondTuringSymbol xx = [[ the AcceptS of t1,q],y] `2 by Def28
.= y ;
thus  the Tran of (t1 ';' t2) . [[ the AcceptS of t1,q],y] = (UnionTran (t1,t2)) . xx by Def31
.= Uniontran (t1,t2,xx) by Def30
.= SecondTuringTran (t1,t2,( the Tran of t2 . [q,y])) by A3, A4, Def29
.= [[ the AcceptS of t1,(g `1_3)],(g `2_3),(g `3_3)] by A1 ; ::_thesis: verum
end;

theorem Th45: :: TURING_1:45
for T1, T2 being TuringStr
for s1 being All-State of T1
for h being Element of NAT
for t being Tape of T1
for s2 being All-State of T2
for s3 being All-State of (T1 ';' T2) st s1 is Accept-Halt & s1 = [ the InitS of T1,h,t] & s2 is Accept-Halt & s2 = [ the InitS of T2,((Result s1) `2_3),((Result s1) `3_3)] & s3 = [ the InitS of (T1 ';' T2),h,t] holds
( s3 is Accept-Halt & (Result s3) `2_3 = (Result s2) `2_3 & (Result s3) `3_3 = (Result s2) `3_3 )
proof
let tm1, tm2 be TuringStr ; ::_thesis: for s1 being All-State of tm1
for h being Element of NAT
for t being Tape of tm1
for s2 being All-State of tm2
for s3 being All-State of (tm1 ';' tm2) st s1 is Accept-Halt & s1 = [ the InitS of tm1,h,t] & s2 is Accept-Halt & s2 = [ the InitS of tm2,((Result s1) `2_3),((Result s1) `3_3)] & s3 = [ the InitS of (tm1 ';' tm2),h,t] holds
( s3 is Accept-Halt & (Result s3) `2_3 = (Result s2) `2_3 & (Result s3) `3_3 = (Result s2) `3_3 )
let s1 be All-State of tm1; ::_thesis: for h being Element of NAT
for t being Tape of tm1
for s2 being All-State of tm2
for s3 being All-State of (tm1 ';' tm2) st s1 is Accept-Halt & s1 = [ the InitS of tm1,h,t] & s2 is Accept-Halt & s2 = [ the InitS of tm2,((Result s1) `2_3),((Result s1) `3_3)] & s3 = [ the InitS of (tm1 ';' tm2),h,t] holds
( s3 is Accept-Halt & (Result s3) `2_3 = (Result s2) `2_3 & (Result s3) `3_3 = (Result s2) `3_3 )
let h be Element of NAT ; ::_thesis: for t being Tape of tm1
for s2 being All-State of tm2
for s3 being All-State of (tm1 ';' tm2) st s1 is Accept-Halt & s1 = [ the InitS of tm1,h,t] & s2 is Accept-Halt & s2 = [ the InitS of tm2,((Result s1) `2_3),((Result s1) `3_3)] & s3 = [ the InitS of (tm1 ';' tm2),h,t] holds
( s3 is Accept-Halt & (Result s3) `2_3 = (Result s2) `2_3 & (Result s3) `3_3 = (Result s2) `3_3 )
let t be Tape of tm1; ::_thesis: for s2 being All-State of tm2
for s3 being All-State of (tm1 ';' tm2) st s1 is Accept-Halt & s1 = [ the InitS of tm1,h,t] & s2 is Accept-Halt & s2 = [ the InitS of tm2,((Result s1) `2_3),((Result s1) `3_3)] & s3 = [ the InitS of (tm1 ';' tm2),h,t] holds
( s3 is Accept-Halt & (Result s3) `2_3 = (Result s2) `2_3 & (Result s3) `3_3 = (Result s2) `3_3 )
let s2 be All-State of tm2; ::_thesis: for s3 being All-State of (tm1 ';' tm2) st s1 is Accept-Halt & s1 = [ the InitS of tm1,h,t] & s2 is Accept-Halt & s2 = [ the InitS of tm2,((Result s1) `2_3),((Result s1) `3_3)] & s3 = [ the InitS of (tm1 ';' tm2),h,t] holds
( s3 is Accept-Halt & (Result s3) `2_3 = (Result s2) `2_3 & (Result s3) `3_3 = (Result s2) `3_3 )
let s3 be All-State of (tm1 ';' tm2); ::_thesis: ( s1 is Accept-Halt & s1 = [ the InitS of tm1,h,t] & s2 is Accept-Halt & s2 = [ the InitS of tm2,((Result s1) `2_3),((Result s1) `3_3)] & s3 = [ the InitS of (tm1 ';' tm2),h,t] implies ( s3 is Accept-Halt & (Result s3) `2_3 = (Result s2) `2_3 & (Result s3) `3_3 = (Result s2) `3_3 ) )
set p0 = the InitS of tm1;
set q0 = the InitS of tm2;
assume that
A1: s1 is Accept-Halt and
A2: s1 = [ the InitS of tm1,h,t] and
A3: s2 is Accept-Halt and
A4: s2 = [ the InitS of tm2,((Result s1) `2_3),((Result s1) `3_3)] and
A5: s3 = [ the InitS of (tm1 ';' tm2),h,t] ; ::_thesis: ( s3 is Accept-Halt & (Result s3) `2_3 = (Result s2) `2_3 & (Result s3) `3_3 = (Result s2) `3_3 )
set pF = the AcceptS of tm1;
set qF = the AcceptS of tm2;
consider k being Element of NAT  such that
A6: ((Computation s1) . k) `1_3 = the AcceptS of tm1 and
A7: Result s1 = (Computation s1) . k and
A8: for i being Element of NAT st i < k holds
((Computation s1) . i) `1_3 <> the AcceptS of tm1 by A1, Th13;
defpred S1[ Element of NAT ] means ( $1 <= k implies ( [(((Computation s1) . $1) `1_3), the InitS of tm2] = ((Computation s3) . $1) `1_3 & ((Computation s1) . $1) `2_3 = ((Computation s3) . $1) `2_3 & ((Computation s1) . $1) `3_3 = ((Computation s3) . $1) `3_3 ) );
A9: for i being Element of NAT st S1[i] holds
S1[i + 1]
proof
let i be Element of NAT ; ::_thesis: ( S1[i] implies S1[i + 1] )
assume A10: S1[i] ; ::_thesis: S1[i + 1]
now__::_thesis:_(_i_+_1_<=_k_implies_(_[(((Computation_s1)_._(i_+_1))_`1_3),_the_InitS_of_tm2]_=_((Computation_s3)_._(i_+_1))_`1_3_&_((Computation_s1)_._(i_+_1))_`2_3_=_((Computation_s3)_._(i_+_1))_`2_3_&_((Computation_s1)_._(i_+_1))_`3_3_=_((Computation_s3)_._(i_+_1))_`3_3_)_)
set s1i1 = (Computation s1) . (i + 1);
set s1i = (Computation s1) . i;
set s3i1 = (Computation s3) . (i + 1);
set s3i = (Computation s3) . i;
A11: i < i + 1 by XREAL_1:29;
set f = TRAN ((Computation s3) . i);
reconsider h = Head ((Computation s1) . i) as Element of INT ;
reconsider ss1 = ((Computation s1) . i) `3_3 as Tape of tm1 ;
reconsider y = ss1 . h as Symbol of tm1 ;
reconsider ss3 = ((Computation s3) . i) `3_3 as Tape of (tm1 ';' tm2) ;
set p = ((Computation s1) . i) `1_3 ;
set g = TRAN ((Computation s1) . i);
assume A12: i + 1 <= k ; ::_thesis: ( [(((Computation s1) . (i + 1)) `1_3), the InitS of tm2] = ((Computation s3) . (i + 1)) `1_3 & ((Computation s1) . (i + 1)) `2_3 = ((Computation s3) . (i + 1)) `2_3 & ((Computation s1) . (i + 1)) `3_3 = ((Computation s3) . (i + 1)) `3_3 )
then A13: i < k by A11, XXREAL_0:2;
then A14: ((Computation s1) . i) `1_3 <> the AcceptS of tm1 by A8;
A15: ((Computation s3) . i) `1_3 <> the AcceptS of (tm1 ';' tm2)
proof
assume  ((Computation s3) . i) `1_3 = the AcceptS of (tm1 ';' tm2) ; ::_thesis: contradiction
then  [(((Computation s1) . i) `1_3), the InitS of tm2] = [ the AcceptS of tm1, the AcceptS of tm2] by A10, A12, A11, Def31, XXREAL_0:2;
hence  contradiction by A14, XTUPLE_0:1; ::_thesis: verum
end;
A16: TRAN ((Computation s3) . i) = the Tran of (tm1 ';' tm2) . [[(((Computation s1) . i) `1_3), the InitS of tm2],y] by A10, A12, A11, XXREAL_0:2
.= [[((TRAN ((Computation s1) . i)) `1_3), the InitS of tm2],((TRAN ((Computation s1) . i)) `2_3),((TRAN ((Computation s1) . i)) `3_3)] by A8, A13, Th43 ;
then A17: (TRAN ((Computation s1) . i)) `2_3 = (TRAN ((Computation s3) . i)) `2_3 by MCART_1:def_6;
A18: (Computation s3) . (i + 1) = Following ((Computation s3) . i) by Def7
.= [((TRAN ((Computation s3) . i)) `1_3),((Head ((Computation s3) . i)) + (offset (TRAN ((Computation s3) . i)))),(Tape-Chg (ss3,(Head ((Computation s3) . i)),((TRAN ((Computation s3) . i)) `2_3)))] by A15, Def6 ;
A19: (Computation s1) . (i + 1) = Following ((Computation s1) . i) by Def7
.= [((TRAN ((Computation s1) . i)) `1_3),(h + (offset (TRAN ((Computation s1) . i)))),(Tape-Chg (ss1,h,((TRAN ((Computation s1) . i)) `2_3)))] by A14, Def6 ;
hence [(((Computation s1) . (i + 1)) `1_3), the InitS of tm2] = [((TRAN ((Computation s1) . i)) `1_3), the InitS of tm2] by MCART_1:def_5
.= (TRAN ((Computation s3) . i)) `1_3 by A16, MCART_1:def_5
.= ((Computation s3) . (i + 1)) `1_3 by A18, MCART_1:def_5 ;
::_thesis: ( ((Computation s1) . (i + 1)) `2_3 = ((Computation s3) . (i + 1)) `2_3 & ((Computation s1) . (i + 1)) `3_3 = ((Computation s3) . (i + 1)) `3_3 )
 offset (TRAN ((Computation s1) . i)) = offset (TRAN ((Computation s3) . i)) by A16, MCART_1:def_7;
hence ((Computation s1) . (i + 1)) `2_3 = (Head ((Computation s3) . i)) + (offset (TRAN ((Computation s3) . i))) by A10, A12, A11, A19, MCART_1:def_6, XXREAL_0:2
.= ((Computation s3) . (i + 1)) `2_3 by A18, MCART_1:def_6 ;
::_thesis: ((Computation s1) . (i + 1)) `3_3 = ((Computation s3) . (i + 1)) `3_3
thus ((Computation s1) . (i + 1)) `3_3 = ss3 +* (h .--> ((TRAN ((Computation s1) . i)) `2_3)) by A10, A12, A11, A19, MCART_1:def_7, XXREAL_0:2
.= ((Computation s3) . (i + 1)) `3_3 by A10, A12, A11, A17, A18, MCART_1:def_7, XXREAL_0:2 ; ::_thesis: verum
end;
hence  S1[i + 1] ; ::_thesis: verum
end;
set s1k = (Computation s1) . k;
set s3k = (Computation s3) . k;
A20: s3 = [[ the InitS of tm1, the InitS of tm2],h,t] by A5, Def31;
A21: S1[ 0 ]
proof
assume  0 <= k ; ::_thesis: ( [(((Computation s1) . 0) `1_3), the InitS of tm2] = ((Computation s3) . 0) `1_3 & ((Computation s1) . 0) `2_3 = ((Computation s3) . 0) `2_3 & ((Computation s1) . 0) `3_3 = ((Computation s3) . 0) `3_3 )
A22: ((Computation s3) . 0) `1_3 = s3 `1_3 by Def7
.= [ the InitS of tm1, the InitS of tm2] by A20, MCART_1:64 ;
((Computation s1) . 0) `1_3 = s1 `1_3 by Def7
.= the InitS of tm1 by A2, MCART_1:64 ;
hence  [(((Computation s1) . 0) `1_3), the InitS of tm2] = ((Computation s3) . 0) `1_3 by A22; ::_thesis: ( ((Computation s1) . 0) `2_3 = ((Computation s3) . 0) `2_3 & ((Computation s1) . 0) `3_3 = ((Computation s3) . 0) `3_3 )
thus ((Computation s1) . 0) `2_3 = s1 `2_3 by Def7
.= h by A2, MCART_1:64
.= s3 `2_3 by A5, MCART_1:64
.= ((Computation s3) . 0) `2_3 by Def7 ; ::_thesis: ((Computation s1) . 0) `3_3 = ((Computation s3) . 0) `3_3
thus ((Computation s1) . 0) `3_3 = s1 `3_3 by Def7
.= t by A2, MCART_1:64
.= s3 `3_3 by A5, MCART_1:64
.= ((Computation s3) . 0) `3_3 by Def7 ; ::_thesis: verum
end;
A23: for i being Element of NAT holds S1[i] from NAT_1:sch_1(A21, A9);
then A24: ((Computation s1) . k) `2_3 = ((Computation s3) . k) `2_3 ;
consider m being Element of NAT  such that
A25: ((Computation s2) . m) `1_3 = the AcceptS of tm2 and
A26: Result s2 = (Computation s2) . m and
A27: for i being Element of NAT st i < m holds
((Computation s2) . i) `1_3 <> the AcceptS of tm2 by A3, Th13;
defpred S2[ Element of NAT ] means ( $1 <= m implies ( [ the AcceptS of tm1,(((Computation s2) . $1) `1_3)] = ((Computation ((Computation s3) . k)) . $1) `1_3 & ((Computation s2) . $1) `2_3 = ((Computation ((Computation s3) . k)) . $1) `2_3 & ((Computation s2) . $1) `3_3 = ((Computation ((Computation s3) . k)) . $1) `3_3 ) );
A28: for i being Element of NAT st S2[i] holds
S2[i + 1]
proof
let i be Element of NAT ; ::_thesis: ( S2[i] implies S2[i + 1] )
assume A29: S2[i] ; ::_thesis: S2[i + 1]
now__::_thesis:_(_i_+_1_<=_m_implies_(_[_the_AcceptS_of_tm1,(((Computation_s2)_._(i_+_1))_`1_3)]_=_((Computation_((Computation_s3)_._k))_._(i_+_1))_`1_3_&_((Computation_s2)_._(i_+_1))_`2_3_=_((Computation_((Computation_s3)_._k))_._(i_+_1))_`2_3_&_((Computation_s2)_._(i_+_1))_`3_3_=_((Computation_((Computation_s3)_._k))_._(i_+_1))_`3_3_)_)
set s2i1 = (Computation s2) . (i + 1);
set s2i = (Computation s2) . i;
set ski1 = (Computation ((Computation s3) . k)) . (i + 1);
set ski = (Computation ((Computation s3) . k)) . i;
A30: i < i + 1 by XREAL_1:29;
reconsider ssk = ((Computation ((Computation s3) . k)) . i) `3_3 as Tape of (tm1 ';' tm2) ;
set f = TRAN ((Computation ((Computation s3) . k)) . i);
set q = ((Computation s2) . i) `1_3 ;
set g = TRAN ((Computation s2) . i);
reconsider h = Head ((Computation s2) . i) as Element of INT ;
reconsider ss2 = ((Computation s2) . i) `3_3 as Tape of tm2 ;
reconsider y = ss2 . h as Symbol of tm2 ;
assume A31: i + 1 <= m ; ::_thesis: ( [ the AcceptS of tm1,(((Computation s2) . (i + 1)) `1_3)] = ((Computation ((Computation s3) . k)) . (i + 1)) `1_3 & ((Computation s2) . (i + 1)) `2_3 = ((Computation ((Computation s3) . k)) . (i + 1)) `2_3 & ((Computation s2) . (i + 1)) `3_3 = ((Computation ((Computation s3) . k)) . (i + 1)) `3_3 )
then A32: TRAN ((Computation ((Computation s3) . k)) . i) = the Tran of (tm1 ';' tm2) . [[ the AcceptS of tm1,(((Computation s2) . i) `1_3)],y] by A29, A30, XXREAL_0:2
.= [[ the AcceptS of tm1,((TRAN ((Computation s2) . i)) `1_3)],((TRAN ((Computation s2) . i)) `2_3),((TRAN ((Computation s2) . i)) `3_3)] by Th44 ;
then A33: (TRAN ((Computation s2) . i)) `2_3 = (TRAN ((Computation ((Computation s3) . k)) . i)) `2_3 by MCART_1:def_6;
 i < m by A31, A30, XXREAL_0:2;
then A34: ((Computation s2) . i) `1_3 <> the AcceptS of tm2 by A27;
A35: ((Computation ((Computation s3) . k)) . i) `1_3 <> the AcceptS of (tm1 ';' tm2)
proof
assume  ((Computation ((Computation s3) . k)) . i) `1_3 = the AcceptS of (tm1 ';' tm2) ; ::_thesis: contradiction
then  [ the AcceptS of tm1,(((Computation s2) . i) `1_3)] = [ the AcceptS of tm1, the AcceptS of tm2] by A29, A31, A30, Def31, XXREAL_0:2;
hence  contradiction by A34, XTUPLE_0:1; ::_thesis: verum
end;
A36: (Computation ((Computation s3) . k)) . (i + 1) = Following ((Computation ((Computation s3) . k)) . i) by Def7
.= [((TRAN ((Computation ((Computation s3) . k)) . i)) `1_3),((Head ((Computation ((Computation s3) . k)) . i)) + (offset (TRAN ((Computation ((Computation s3) . k)) . i)))),(Tape-Chg (ssk,(Head ((Computation ((Computation s3) . k)) . i)),((TRAN ((Computation ((Computation s3) . k)) . i)) `2_3)))] by A35, Def6 ;
A37: (Computation s2) . (i + 1) = Following ((Computation s2) . i) by Def7
.= [((TRAN ((Computation s2) . i)) `1_3),(h + (offset (TRAN ((Computation s2) . i)))),(Tape-Chg (ss2,h,((TRAN ((Computation s2) . i)) `2_3)))] by A34, Def6 ;
hence [ the AcceptS of tm1,(((Computation s2) . (i + 1)) `1_3)] = [ the AcceptS of tm1,((TRAN ((Computation s2) . i)) `1_3)] by MCART_1:def_5
.= (TRAN ((Computation ((Computation s3) . k)) . i)) `1_3 by A32, MCART_1:def_5
.= ((Computation ((Computation s3) . k)) . (i + 1)) `1_3 by A36, MCART_1:def_5 ;
::_thesis: ( ((Computation s2) . (i + 1)) `2_3 = ((Computation ((Computation s3) . k)) . (i + 1)) `2_3 & ((Computation s2) . (i + 1)) `3_3 = ((Computation ((Computation s3) . k)) . (i + 1)) `3_3 )
 offset (TRAN ((Computation s2) . i)) = offset (TRAN ((Computation ((Computation s3) . k)) . i)) by A32, MCART_1:def_7;
hence ((Computation s2) . (i + 1)) `2_3 = (Head ((Computation ((Computation s3) . k)) . i)) + (offset (TRAN ((Computation ((Computation s3) . k)) . i))) by A29, A31, A30, A37, MCART_1:def_6, XXREAL_0:2
.= ((Computation ((Computation s3) . k)) . (i + 1)) `2_3 by A36, MCART_1:def_6 ;
::_thesis: ((Computation s2) . (i + 1)) `3_3 = ((Computation ((Computation s3) . k)) . (i + 1)) `3_3
thus ((Computation s2) . (i + 1)) `3_3 = ssk +* (h .--> ((TRAN ((Computation s2) . i)) `2_3)) by A29, A31, A30, A37, MCART_1:def_7, XXREAL_0:2
.= ((Computation ((Computation s3) . k)) . (i + 1)) `3_3 by A29, A31, A30, A33, A36, MCART_1:def_7, XXREAL_0:2 ; ::_thesis: verum
end;
hence  S2[i + 1] ; ::_thesis: verum
end;
A38: ((Computation s1) . k) `3_3 = ((Computation s3) . k) `3_3 by A23;
set s2m = (Computation s2) . m;
set skm = (Computation ((Computation s3) . k)) . m;
A39: (Computation s3) . (k + m) = (Computation ((Computation s3) . k)) . m by Th10;
A40: [(((Computation s1) . k) `1_3), the InitS of tm2] = ((Computation s3) . k) `1_3 by A23;
A41: S2[ 0 ]
proof
assume  0 <= m ; ::_thesis: ( [ the AcceptS of tm1,(((Computation s2) . 0) `1_3)] = ((Computation ((Computation s3) . k)) . 0) `1_3 & ((Computation s2) . 0) `2_3 = ((Computation ((Computation s3) . k)) . 0) `2_3 & ((Computation s2) . 0) `3_3 = ((Computation ((Computation s3) . k)) . 0) `3_3 )
thus [ the AcceptS of tm1,(((Computation s2) . 0) `1_3)] = [ the AcceptS of tm1,(s2 `1_3)] by Def7
.= [ the AcceptS of tm1, the InitS of tm2] by A4, MCART_1:64
.= ((Computation ((Computation s3) . k)) . 0) `1_3 by A6, A40, Def7 ; ::_thesis: ( ((Computation s2) . 0) `2_3 = ((Computation ((Computation s3) . k)) . 0) `2_3 & ((Computation s2) . 0) `3_3 = ((Computation ((Computation s3) . k)) . 0) `3_3 )
thus ((Computation s2) . 0) `2_3 = s2 `2_3 by Def7
.= ((Computation s3) . k) `2_3 by A4, A7, A24, MCART_1:64
.= ((Computation ((Computation s3) . k)) . 0) `2_3 by Def7 ; ::_thesis: ((Computation s2) . 0) `3_3 = ((Computation ((Computation s3) . k)) . 0) `3_3
thus ((Computation s2) . 0) `3_3 = s2 `3_3 by Def7
.= ((Computation s3) . k) `3_3 by A4, A7, A38, MCART_1:64
.= ((Computation ((Computation s3) . k)) . 0) `3_3 by Def7 ; ::_thesis: verum
end;
A42: for i being Element of NAT holds S2[i] from NAT_1:sch_1(A41, A28);
then  [ the AcceptS of tm1,(((Computation s2) . m) `1_3)] = ((Computation ((Computation s3) . k)) . m) `1_3 ;
then A43: ((Computation s3) . (k + m)) `1_3 = the AcceptS of (tm1 ';' tm2) by A25, A39, Def31;
hence A44: s3 is Accept-Halt by Def8; ::_thesis: ( (Result s3) `2_3 = (Result s2) `2_3 & (Result s3) `3_3 = (Result s2) `3_3 )
 ( ((Computation s2) . m) `2_3 = ((Computation ((Computation s3) . k)) . m) `2_3 & ((Computation s2) . m) `3_3 = ((Computation ((Computation s3) . k)) . m) `3_3 ) by A42;
hence  ( (Result s3) `2_3 = (Result s2) `2_3 & (Result s3) `3_3 = (Result s2) `3_3 ) by A26, A39, A43, A44, Def9; ::_thesis: verum
end;

theorem :: TURING_1:46
for tm1, tm2 being TuringStr
for t being Tape of tm1 st the Symbols of tm1 = the Symbols of tm2 holds
t is Tape of (tm1 ';' tm2)
proof
let tm1, tm2 be TuringStr ; ::_thesis: for t being Tape of tm1 st the Symbols of tm1 = the Symbols of tm2 holds
t is Tape of (tm1 ';' tm2)
let t be Tape of tm1; ::_thesis: ( the Symbols of tm1 = the Symbols of tm2 implies t is Tape of (tm1 ';' tm2) )
set S1 = the Symbols of tm1;
set S2 = the Symbols of tm2;
assume A1: the Symbols of tm1 = the Symbols of tm2 ; ::_thesis: t is Tape of (tm1 ';' tm2)
the Symbols of (tm1 ';' tm2) = the Symbols of tm1 \/ the Symbols of tm2 by Def31
.= the Symbols of tm1 by A1 ;
hence  t is Tape of (tm1 ';' tm2) ; ::_thesis: verum
end;

theorem :: TURING_1:47
for tm1, tm2 being TuringStr
for t being Tape of (tm1 ';' tm2) st the Symbols of tm1 = the Symbols of tm2 holds
( t is Tape of tm1 & t is Tape of tm2 )
proof
let tm1, tm2 be TuringStr ; ::_thesis: for t being Tape of (tm1 ';' tm2) st the Symbols of tm1 = the Symbols of tm2 holds
( t is Tape of tm1 & t is Tape of tm2 )
let t be Tape of (tm1 ';' tm2); ::_thesis: ( the Symbols of tm1 = the Symbols of tm2 implies ( t is Tape of tm1 & t is Tape of tm2 ) )
set S1 = the Symbols of tm1;
set S2 = the Symbols of tm2;
assume A1: the Symbols of tm1 = the Symbols of tm2 ; ::_thesis: ( t is Tape of tm1 & t is Tape of tm2 )
the Symbols of (tm1 ';' tm2) = the Symbols of tm1 \/ the Symbols of tm2 by Def31
.= the Symbols of tm1 by A1 ;
hence  ( t is Tape of tm1 & t is Tape of tm2 ) by A1; ::_thesis: verum
end;

theorem Th48: :: TURING_1:48
for f being FinSequence of NAT
for tm1, tm2 being TuringStr
for t1 being Tape of tm1
for t2 being Tape of tm2 st t1 = t2 & t1 storeData f holds
t2 storeData f
proof
let f be FinSequence of NAT ; ::_thesis: for tm1, tm2 being TuringStr
for t1 being Tape of tm1
for t2 being Tape of tm2 st t1 = t2 & t1 storeData f holds
t2 storeData f
let tm1, tm2 be TuringStr ; ::_thesis: for t1 being Tape of tm1
for t2 being Tape of tm2 st t1 = t2 & t1 storeData f holds
t2 storeData f
let t1 be Tape of tm1; ::_thesis: for t2 being Tape of tm2 st t1 = t2 & t1 storeData f holds
t2 storeData f
let t2 be Tape of tm2; ::_thesis: ( t1 = t2 & t1 storeData f implies t2 storeData f )
assume that
A1: t1 = t2 and
A2: t1 storeData f ; ::_thesis: t2 storeData f
now__::_thesis:_for_i_being_Element_of_NAT_st_1_<=_i_&_i_<_len_f_holds_
t2_is_1_between_(Sum_(Prefix_(f,i)))_+_(2_*_(i_-_1)),(Sum_(Prefix_(f,(i_+_1))))_+_(2_*_i)
let i be Element of NAT ; ::_thesis: ( 1 <= i & i < len f implies t2 is_1_between (Sum (Prefix (f,i))) + (2 * (i - 1)),(Sum (Prefix (f,(i + 1)))) + (2 * i) )
set m = (Sum (Prefix (f,i))) + (2 * (i - 1));
set n = (Sum (Prefix (f,(i + 1)))) + (2 * i);
assume  ( 1 <= i & i < len f ) ; ::_thesis: t2 is_1_between (Sum (Prefix (f,i))) + (2 * (i - 1)),(Sum (Prefix (f,(i + 1)))) + (2 * i)
then A3: t1 is_1_between (Sum (Prefix (f,i))) + (2 * (i - 1)),(Sum (Prefix (f,(i + 1)))) + (2 * i) by A2, Def13;
then A4: for k being Integer st (Sum (Prefix (f,i))) + (2 * (i - 1)) < k & k < (Sum (Prefix (f,(i + 1)))) + (2 * i) holds
t1 . k = 1 by Def12;
 ( t1 . ((Sum (Prefix (f,i))) + (2 * (i - 1))) = 0 & t1 . ((Sum (Prefix (f,(i + 1)))) + (2 * i)) = 0 ) by A3, Def12;
hence  t2 is_1_between (Sum (Prefix (f,i))) + (2 * (i - 1)),(Sum (Prefix (f,(i + 1)))) + (2 * i) by A1, A4, Def12; ::_thesis: verum
end;
hence  t2 storeData f by Def13; ::_thesis: verum
end;

Lm15:  for s being All-State of ZeroTuring
for t being Tape of ZeroTuring
for head, n being Element of NAT st s = [0,head,t] & t storeData <*head,n*> holds
( s is Accept-Halt & (Result s) `2_3 = head & (Result s) `3_3 storeData <*head,0*> )
proof
let s be All-State of ZeroTuring; ::_thesis: for t being Tape of ZeroTuring
for head, n being Element of NAT st s = [0,head,t] & t storeData <*head,n*> holds
( s is Accept-Halt & (Result s) `2_3 = head & (Result s) `3_3 storeData <*head,0*> )
let t be Tape of ZeroTuring; ::_thesis: for head, n being Element of NAT st s = [0,head,t] & t storeData <*head,n*> holds
( s is Accept-Halt & (Result s) `2_3 = head & (Result s) `3_3 storeData <*head,0*> )
let h, n be Element of NAT ; ::_thesis: ( s = [0,h,t] & t storeData <*h,n*> implies ( s is Accept-Halt & (Result s) `2_3 = h & (Result s) `3_3 storeData <*h,0*> ) )
 len <*n*> = 1 by FINSEQ_1:39;
hence  ( s = [0,h,t] & t storeData <*h,n*> implies ( s is Accept-Halt & (Result s) `2_3 = h & (Result s) `3_3 storeData <*h,0*> ) ) by Th34; ::_thesis: verum
end;

theorem :: TURING_1:49
for s being All-State of (ZeroTuring ';' SuccTuring)
for t being Tape of ZeroTuring
for head, n being Element of NAT st s = [[0,0],head,t] & t storeData <*head,n*> holds
( s is Accept-Halt & (Result s) `2_3 = head & (Result s) `3_3 storeData <*head,1*> )
proof
let s be All-State of (ZeroTuring ';' SuccTuring); ::_thesis: for t being Tape of ZeroTuring
for head, n being Element of NAT st s = [[0,0],head,t] & t storeData <*head,n*> holds
( s is Accept-Halt & (Result s) `2_3 = head & (Result s) `3_3 storeData <*head,1*> )
let t be Tape of ZeroTuring; ::_thesis: for head, n being Element of NAT st s = [[0,0],head,t] & t storeData <*head,n*> holds
( s is Accept-Halt & (Result s) `2_3 = head & (Result s) `3_3 storeData <*head,1*> )
let h, n be Element of NAT ; ::_thesis: ( s = [[0,0],h,t] & t storeData <*h,n*> implies ( s is Accept-Halt & (Result s) `2_3 = h & (Result s) `3_3 storeData <*h,1*> ) )
assume that
A1: s = [[0,0],h,t] and
A2: t storeData <*h,n*> ; ::_thesis: ( s is Accept-Halt & (Result s) `2_3 = h & (Result s) `3_3 storeData <*h,1*> )
reconsider h1 = h as Element of INT by INT_1:def_2;
set s1 = [ the InitS of ZeroTuring,h1,t];
A3: 0 = the InitS of ZeroTuring by Def19;
then A4: ( [ the InitS of ZeroTuring,h1,t] is Accept-Halt & (Result [ the InitS of ZeroTuring,h1,t]) `2_3 = h ) by A2, Lm15;
the Symbols of ZeroTuring = {0,1} by Def19
.= the Symbols of SuccTuring by Def17 ;
then reconsider t2 = (Result [ the InitS of ZeroTuring,h1,t]) `3_3 as Tape of SuccTuring ;
set s2 = [ the InitS of SuccTuring,h1,t2];
A5: 0 = the InitS of SuccTuring by Def17;
then A6: s = [ the InitS of (ZeroTuring ';' SuccTuring),h,t] by A1, A3, Def31;
 (Result [ the InitS of ZeroTuring,h1,t]) `3_3 storeData <*h,0*> by A2, A3, Lm15;
then A7: t2 storeData <*h,0*> by Th48;
then A8: (Result [ the InitS of SuccTuring,h1,t2]) `3_3 storeData <*h,(0 + 1)*> by A5, Th31;
A9: [ the InitS of SuccTuring,h1,t2] is Accept-Halt by A7, A5, Th31;
hence  s is Accept-Halt by A4, A6, Th45; ::_thesis: ( (Result s) `2_3 = h & (Result s) `3_3 storeData <*h,1*> )
 (Result [ the InitS of SuccTuring,h1,t2]) `2_3 = h by A7, A5, Th31;
hence  (Result s) `2_3 = h by A4, A9, A6, Th45; ::_thesis: (Result s) `3_3 storeData <*h,1*>
 (Result s) `3_3 = (Result [ the InitS of SuccTuring,h1,t2]) `3_3 by A4, A9, A6, Th45;
hence  (Result s) `3_3 storeData <*h,1*> by A8, Th48; ::_thesis: verum
end;