:: The Instructions for SCM+FSA Computer
:: by Andrzej Trybulec , Yatsuka Nakamura and Piotr Rudnicki
::
:: Received February 3, 1996
:: Copyright (c) 1996-2017 Association of Mizar Users
:: (Stowarzyszenie Uzytkownikow Mizara, Bialystok, Poland).
:: This code can be distributed under the GNU General Public Licence
:: version 3.0 or later, or the Creative Commons Attribution-ShareAlike
:: License version 3.0 or later, subject to the binding interpretation
:: detailed in file COPYING.interpretation.
:: See COPYING.GPL and COPYING.CC-BY-SA for the full text of these
:: licenses, or see http://www.gnu.org/licenses/gpl.html and
:: http://creativecommons.org/licenses/by-sa/3.0/.
environ
vocabularies NUMBERS, SUBSET_1, AMI_2, XBOOLE_0, TARSKI, CARD_1, FINSEQ_1,
ZFMISC_1, RELAT_1, AMI_1, XXREAL_0, FUNCT_1, PARTFUN1, SCMFSA_1,
RECDEF_2, UNIALG_1, AMISTD_2, VALUED_0, COMPOS_0;
notations TARSKI, XBOOLE_0, ZFMISC_1, XTUPLE_0, XFAMILY, SUBSET_1, ORDINAL1,
CARD_1, NUMBERS, MCART_1, RELAT_1, FUNCT_1, PARTFUN1, VALUED_0, FINSEQ_1,
XXREAL_0, RECDEF_2, COMPOS_0, SCM_INST;
constructors AMI_3, VALUED_0, XTUPLE_0, NUMBERS, XFAMILY;
registrations XBOOLE_0, FUNCT_1, ORDINAL1, XREAL_0, FINSEQ_1, GR_CY_1,
RELAT_1, COMPOS_0, SCM_INST, VALUED_0, XTUPLE_0, CARD_1;
requirements NUMERALS, REAL, BOOLE, SUBSET;
definitions TARSKI, XBOOLE_0, FINSEQ_1, COMPOS_0;
equalities COMPOS_0, XTUPLE_0, ORDINAL1;
expansions TARSKI, COMPOS_0;
theorems TARSKI, FINSEQ_1, FINSEQ_4, MCART_1, XBOOLE_0, XBOOLE_1, NUMBERS,
NAT_1, COMPOS_0, SCM_INST, XTUPLE_0;
begin
reserve x,y,z for set,
k for Element of NAT;
definition
func SCM+FSA-Data*-Loc -> set equals
INT \ NAT;
coherence;
end;
registration
cluster SCM+FSA-Data*-Loc -> non empty;
coherence
proof
not INT c= NAT by NUMBERS:7,17,XBOOLE_0:def 10;
hence thesis by XBOOLE_1:37;
end;
end;
reserve J,J1,K for Element of Segm 13,
a for Element of NAT,
b,b1,b2,c,c1,c2 for Element of SCM-Data-Loc,
f,f1,f2 for Element of SCM+FSA-Data*-Loc;
definition
func SCM+FSA-Instr -> non empty set equals
SCM-Instr \/ { [J,{},<*c,f,b*>] : J in {9,10} }
\/ { [K,{},<*c1,f1*>] : K in {11,12} };
coherence;
end;
theorem Th1:
SCM-Instr c= SCM+FSA-Instr
proof
A1: SCM-Instr c= SCM-Instr \/ { [J,{},<*c,f,b*>] : J in {9,10} }
by XBOOLE_1:7;
SCM-Instr \/ { [J,{},<*c,f,b*>] : J in {9,10} }
c= SCM-Instr \/ { [J1,{},<*c2,f2,b2*>] : J1 in {9,10} }
\/ { [K,{},<*c1,f1*>] : K in {11,12} } by XBOOLE_1:7;
then
A2: SCM-Instr
c= SCM-Instr \/ { [J,{},<*c,f,b*>] : J in {9,10} }
\/ { [K,{},<*c1,f1*>] : K in {11,12} } by A1;
thus thesis by A2;
end;
Lm1:
SCM+FSA-Instr c= [:NAT,NAT*,proj2 SCM+FSA-Instr:]
proof let x be object;
assume
A1: x in SCM+FSA-Instr;
per cases by A1;
suppose
A2: x in SCM-Instr \/ { [J,{},<*c,f,b*>] : J in {9,10} }
\/ { [K,{},<*c1,f1*>] : K in {11,12} };
per cases by A2,XBOOLE_0:def 3;
suppose
A3: x in SCM-Instr \/ { [J,{},<*c,f,b*>] : J in {9,10} };
per cases by A3,XBOOLE_0:def 3;
suppose x in SCM-Instr;
then
A4: x in [:NAT,NAT*,proj2 SCM-Instr:] by SCM_INST:8;
proj2 SCM-Instr c= proj2 SCM+FSA-Instr by Th1,XTUPLE_0:9;
then [:NAT,NAT*,proj2 SCM-Instr:] c= [:NAT,NAT*,proj2 SCM+FSA-Instr:]
by MCART_1:73;
hence x in [:NAT,NAT*,proj2 SCM+FSA-Instr:] by A4;
end;
suppose x in { [J,{},<*c,f,b*>] : J in {9,10}};
then consider J,b,c,f such that
A5: x = [J,{},<*c,f,b*>] & J in {9,10};
A6: {} in NAT* by FINSEQ_1:49;
J in NAT & <*c,f,b*> in proj2 SCM+FSA-Instr by A1,A5,XTUPLE_0:def 13;
hence x in [:NAT,NAT*,proj2 SCM+FSA-Instr:] by A5,A6,MCART_1:69;
end;
end;
suppose x in { [K,{},<*c1,f1*>] : K in {11,12} };
then consider K,c1,f1 such that
A7: x = [K,{},<*c1,f1*>] & K in {11,12};
A8: {} in NAT* by FINSEQ_1:49;
K in NAT & <*c1,f1*> in proj2 SCM+FSA-Instr by A1,A7,XTUPLE_0:def 13;
hence x in [:NAT,NAT*,proj2 SCM+FSA-Instr:] by A7,A8,MCART_1:69;
end;
end;
suppose x in the set of all [13,{},<*b1*>] ;
then consider b1 such that
A9: x = [13,{},<*b1*>];
A10: {} in NAT* by FINSEQ_1:49;
K in NAT & <*b1*> in proj2 SCM+FSA-Instr by A1,A9,XTUPLE_0:def 13;
hence x in [:NAT,NAT*,proj2 SCM+FSA-Instr:] by A9,A10,MCART_1:69;
end;
end;
registration
cluster proj2 SCM+FSA-Instr -> FinSequence-membered;
coherence
proof let f be object;
assume f in proj2 SCM+FSA-Instr;
then consider y being object such that
A1: [y,f] in SCM+FSA-Instr by XTUPLE_0:def 13;
set x = [y,f];
per cases by A1;
suppose
A2: x in SCM-Instr \/ { [J,{},<*c,f2,b*>] : J in {9,10} }
\/ { [K,{},<*c1,f1*>] : K in {11,12} };
per cases by A2,XBOOLE_0:def 3;
suppose
A3: x in SCM-Instr \/ { [J,{},<*c,f1,b*>] : J in {9,10} };
per cases by A3,XBOOLE_0:def 3;
suppose x in SCM-Instr;
then f in proj2 SCM-Instr by XTUPLE_0:def 13;
hence f is FinSequence;
end;
suppose x in { [J,{},<*c,f1,b*>] : J in {9,10}};
then consider J,b,c,f1 such that
A4: x = [J,{},<*c,f1,b*>] & J in {9,10};
f = <*c,f1,b*> by A4,XTUPLE_0:1;
hence f is FinSequence;
end;
end;
suppose x in { [K,{},<*c1,f1*>] : K in {11,12} };
then consider K,c1,f1 such that
A5: x = [K,{},<*c1,f1*>] & K in {11,12};
f = <*c1,f1*> by A5,XTUPLE_0:1;
hence f is FinSequence;
end;
end;
suppose x in the set of all [13,{},<*b1*>] ;
then consider b1 such that
A6: x = [13,{},<*b1*>];
f = <*b1*> by A6,XTUPLE_0:1;
hence f is FinSequence;
end;
end;
end;
registration
cluster SCM+FSA-Instr -> standard-ins non empty;
coherence
proof
thus SCM+FSA-Instr is standard-ins
proof
consider X being non empty set such that
A1: proj2 SCM+FSA-Instr c= X* by FINSEQ_1:85;
take X;
[:NAT,NAT*,proj2 SCM+FSA-Instr:] c= [: NAT,NAT*,X*:] by A1,MCART_1:73;
hence SCM+FSA-Instr c= [:NAT,NAT*,X*:] by Lm1;
end;
thus thesis;
end;
end;
theorem Th2:
for I being Element of SCM+FSA-Instr st I`1_3 <= 8 holds I in SCM-Instr
proof
let I be Element of SCM+FSA-Instr such that
A1: I`1_3 <= 8;
A2: now
assume I in { [K,{},<*c1,f1*>] : K in {11,12} };
then consider K,c,f such that
A3: I = [K,{},<*c,f*>] and
A4: K in {11,12};
I`1_3 = K by A3;
then I`1_3 = 11 or I`1_3 = 12 by A4,TARSKI:def 2;
hence contradiction by A1;
end;
A5: now
assume I in { [J,{},<*c,f,b*>] : J in {9,10} };
then consider J,b,c,f such that
A6: I = [J,{},<*c,f,b*>] and
A7: J in {9,10};
I`1_3 = J by A6;
then I`1_3 = 9 or I`1_3 = 10 by A7,TARSKI:def 2;
hence contradiction by A1;
end;
A8: now
assume I in the set of all [13,{},<*b1*>] ;
then consider b1 such that
A9: I = [13,{},<*b1*>];
I`1_3 = 13 by A9;
hence contradiction by A1;
end;
I in SCM-Instr \/ { [J,{},<*c,f,b*>] : J in {9,10} }
or I in { [K,{},<*c1,f1*>] : K in {11,12} }
or I in the set of all [13,{},<*b1*>]
by XBOOLE_0:def 3;
hence thesis by A2,A5,A8,XBOOLE_0:def 3;
end;
theorem Th3:
[0,{},{}] in SCM+FSA-Instr by Th1,SCM_INST:1;
theorem Th4:
x in {9,10} implies [x,{},<*c,f,b*>] in SCM+FSA-Instr
proof
assume
A1: x in {9,10};
then x = 9 or x = 10 by TARSKI:def 2;
then reconsider x as Element of Segm 13 by NAT_1:44;
[x,{},<*c,f,b*>] in { [K,{},<*c1,f1,b1*>] : K in {9,10}} by A1;
then [x,{},<*c,f,b*>] in SCM-Instr \/ { [J,{},<*c1,f1,b1*>] : J in {9,10}}
by XBOOLE_0:def 3;
then [x,{},<*c,f,b*>] in SCM-Instr \/ { [J,{},<*c1,f1,b1*>] : J in {9,10}}
\/ { [K,{},<*c2,f2*>] : K in {11,12} }
by XBOOLE_0:def 3;
hence thesis;
end;
theorem Th5:
x in {11,12} implies [x,{},<*c,f*>] in SCM+FSA-Instr
proof
assume
A1: x in {11,12};
then x = 11 or x = 12 by TARSKI:def 2;
then reconsider x as Element of Segm 13 by NAT_1:44;
[x,{},<*c,f*>] in { [K,{},<*c1,f1*>] : K in {11,12}} by A1;
then [x,{},<*c,f*>] in SCM-Instr \/ { [J,{},<*c2,f2,b*>] : J in {9,10} }
\/ { [K,{},<*c1,f1*>] : K in {11,12} }
by XBOOLE_0:def 3;
then [x,{},<*c,f*>] in SCM-Instr \/ { [J,{},<*c2,f2,b*>] : J in {9,10} }
\/ { [K,{},<*c1,f1*>] : K in {11,12} };
hence thesis;
end;
definition
let x be Element of SCM+FSA-Instr;
given c,f,b,J such that
A1: x = [J,{},<*c,f,b*>];
func x int_addr1 -> Element of SCM-Data-Loc means
ex c,f,b st <*c,f,b*> = x`3_3 & it = c;
existence
by A1;
uniqueness
proof
let a1,a2 be Element of SCM-Data-Loc;
given c1,f1,b1 such that
A2: <*c1,f1,b1*> = x`3_3 and
A3: a1 = c1;
given c2,f2,b2 such that
A4: <*c2,f2,b2*> = x`3_3 & a2 = c2;
thus a1 = <*c1,f1,b1*>.1 by A3,FINSEQ_1:45
.= a2 by A2,A4,FINSEQ_1:45;
end;
func x int_addr2 -> Element of SCM-Data-Loc means
ex c,f,b st <*c,f,b*> = x`3_3 & it = b;
existence
by A1;
correctness
proof
let a1,a2 be Element of SCM-Data-Loc;
given c1,f1,b1 such that
A5: <*c1,f1,b1*> = x`3_3 and
A6: a1 = b1;
given c2,f2,b2 such that
A7: <*c2,f2,b2*> = x`3_3 & a2 = b2;
thus a1 = <*c1,f1,b1*>.3 by A6,FINSEQ_1:45
.= a2 by A5,A7,FINSEQ_1:45;
end;
func x coll_addr1 -> Element of SCM+FSA-Data*-Loc means
ex c,f,b st <*c,f,b*> = x`3_3 & it = f;
existence
by A1;
correctness
proof
let a1,a2 be Element of SCM+FSA-Data*-Loc;
given c1,f1,b1 such that
A8: <*c1,f1,b1*> = x`3_3 and
A9: a1 = f1;
given c2,f2,b2 such that
A10: <*c2,f2,b2*> = x`3_3 & a2 = f2;
thus a1 = <*c1,f1,b1*>.2 by A9,FINSEQ_1:45
.= a2 by A8,A10,FINSEQ_1:45;
end;
end;
definition
let x be Element of SCM+FSA-Instr;
given c such that
A1: x = [ 13,{}, <*c*>];
func x int_addr -> Element of SCM-Data-Loc means
ex c st <*c*> = x`3_3 & it = c;
existence
by A1;
uniqueness
proof
let a1,a2 be Element of SCM-Data-Loc;
given c1 such that
A2: <*c1*> = x`3_3 and
A3: a1 = c1;
given c2 such that
A4: <*c2*> = x`3_3 & a2 = c2;
thus a1 = <*c1*>/.1 by A3,FINSEQ_4:16
.= a2 by A2,A4,FINSEQ_4:16;
end;
end;
definition
let x be Element of SCM+FSA-Instr;
given c,f,J such that
A1: x = [ J,{}, <*c,f*>];
func x int_addr3 -> Element of SCM-Data-Loc means
ex c,f st <*c,f*> = x`3_3 & it = c;
existence
by A1;
uniqueness
proof
let a1,a2 be Element of SCM-Data-Loc;
given c1,f1 such that
A2: <*c1,f1*> = x`3_3 and
A3: a1 = c1;
given c2,f2 such that
A4: <*c2,f2*> = x`3_3 & a2 = c2;
thus a1 = <*c1,f1*>.1 by A3,FINSEQ_1:44
.= a2 by A2,A4,FINSEQ_1:44;
end;
func x coll_addr2 -> Element of SCM+FSA-Data*-Loc means
ex c,f st <*c,f*> = x`3_3 & it = f;
existence
by A1;
correctness
proof
let a1,a2 be Element of SCM+FSA-Data*-Loc;
given c1,f1 such that
A5: <*c1,f1*> = x`3_3 and
A6: a1 = f1;
given c2,f2 such that
A7: <*c2,f2*> = x`3_3 & a2 = f2;
thus a1 = <*c1,f1*>.2 by A6,FINSEQ_1:44
.= a2 by A5,A7,FINSEQ_1:44;
end;
end;
theorem
SCM+FSA-Instr c= [:NAT,NAT*,proj2 SCM+FSA-Instr:] by Lm1;
theorem Th7:
for x being Element of SCM+FSA-Instr holds
x in SCM-Instr &
:: (InsCode x = 0 or
:: InsCode x = 1 or InsCode x = 2 or InsCode x = 3 or
:: InsCode x = 4 or InsCode x = 5 or InsCode x = 6 or InsCode x = 7 or
:: InsCode x = 8)
(InsCode x = 0 or ... or InsCode x = 8)
or
x in { [J,{},<*c,f,b*>] : J in {9,10} }
& (InsCode x = 9 or InsCode x = 10) or
x in { [K,{},<*c1,f1*>] : K in {11,12} }
& (InsCode x = 11 or InsCode x = 12)
proof
let x be Element of SCM+FSA-Instr;
x in SCM-Instr \/ { [J,{},<*c,f,b*>] : J in {9,10} } or
x in { [K,{},<*c1,f1*>] : K in {11,12} } by XBOOLE_0:def 3;
then per cases by XBOOLE_0:def 3;
case x in SCM-Instr;
then InsCode x <= 8 by SCM_INST:10;
then InsCode x = 0 or ... or InsCode x = 8 by NAT_1:60;
hence thesis;
end;
case x in { [J,{},<*c,f,b*>] : J in {9,10} };
then consider J,b,c,f such that
A1: x = [J,{},<*c,f,b*>] and
A2: J in { 9,10 };
InsCode x = J by A1;
hence thesis by A2,TARSKI:def 2;
end;
case x in { [K,{},<*c1,f1*>] : K in {11,12} };
then consider K,c1,f1 such that
A3: x = [K,{},<*c1,f1*>] and
A4: K in { 11,12};
InsCode x = K by A3;
hence thesis by A4,TARSKI:def 2;
end;
end;
Lm2:
for i being Element of SCM+FSA-Instr holds InsCode i <= 12
proof let i be Element of SCM+FSA-Instr;
(InsCode i = 0 or ... or InsCode i = 8) or
(InsCode i = 9 or ... or InsCode i = 12) by Th7;
hence thesis;
end;
Lm3:
for i being Element of SCM+FSA-Instr
st InsCode i = 9 or InsCode i = 10
holds JumpPart i = {}
proof let i be Element of SCM+FSA-Instr;
assume
A1: InsCode i = 9 or InsCode i = 10;
then not(InsCode i = 0 or ... or InsCode i = 8);
then i in { [J,{},<*c,f,b*>] : J in {9,10} } by A1,Th7;
then ex J,b,c,f st i = [J,{},<*c,f,b*>] & J in { 9,10 };
hence thesis;
end;
Lm4:
for i being Element of SCM+FSA-Instr
st InsCode i = 11 or InsCode i = 12
holds JumpPart i = {}
proof let i be Element of SCM+FSA-Instr;
assume
A1: InsCode i = 11 or InsCode i = 12;
then not(InsCode i = 0 or ... or InsCode i = 8);
then i in { [K,{},<*c1,f1*>] : K in {11,12} } by A1,Th7;
then ex K,c1,f1 st i = [K,{},<*c1,f1*>] & K in { 11,12 };
hence thesis;
end;
registration
cluster SCM+FSA-Instr -> homogeneous;
coherence
proof
let i, j be Element of SCM+FSA-Instr such that
A1: InsCode i = InsCode j;
InsCode i <= 12 by Lm2;
then InsCode i = 0 or ... or InsCode i = 12 by NAT_1:60;
then per cases;
suppose InsCode i = 0 or InsCode i = 1 or InsCode i = 2 or InsCode i = 3
or InsCode i = 4 or InsCode i = 5
or InsCode i = 6 or InsCode i = 7 or InsCode i = 8;
then i in SCM-Instr & j in SCM-Instr by A1,Th7;
hence thesis by A1,COMPOS_0:def 5;
end;
suppose
InsCode i = 9 or InsCode i = 10;
then JumpPart i = {} & JumpPart j = {} by A1,Lm3;
hence thesis;
end;
suppose InsCode i = 11 or InsCode i = 12;
then JumpPart i = {} & JumpPart j = {} by A1,Lm4;
hence thesis;
end;
end;
end;
Lm5:
for i being Element of SCM+FSA-Instr, ii being Element of SCM-Instr
st i = ii
holds JumpParts InsCode i = JumpParts InsCode ii
proof let i be Element of SCM+FSA-Instr, ii be Element of SCM-Instr such that
A1: i = ii;
thus JumpParts InsCode i c= JumpParts InsCode ii
proof let e be object;
assume e in JumpParts InsCode i;
then consider I being Element of SCM+FSA-Instr such that
A2: e = JumpPart I and
A3: InsCode I = InsCode i;
InsCode I <= 8 by A1,A3,SCM_INST:10;
then reconsider II = I as Element of SCM-Instr by Th2;
InsCode II = InsCode ii by A1,A3;
hence e in JumpParts InsCode ii by A2;
end;
let e be object;
assume e in JumpParts InsCode ii;
then consider II being Element of SCM-Instr such that
A4: e = JumpPart II and
A5: InsCode II = InsCode ii;
A6: SCM-Instr c= SCM+FSA-Instr by Th1;
II in SCM+FSA-Instr by A6;
then reconsider I = II as Element of SCM+FSA-Instr;
InsCode I = InsCode i by A1,A5;
hence e in JumpParts InsCode i by A4;
end;
reserve T for InsType of SCM+FSA-Instr;
theorem Th8:
T = 9 or T =10 implies JumpParts T = {{}}
proof
assume
A1: T = 9 or T =10;
then
A2: not(T = 0 or ... or T = 8);
hereby
let x be object;
assume x in JumpParts T;
then consider I being Element of SCM+FSA-Instr such that
A3: x = JumpPart I and
A4: InsCode I = T;
I in { [J,{},<*c,f,b*>]
where J is Element of Segm 13, b,c is Element of SCM-Data-Loc,
f is Element of SCM+FSA-Data*-Loc : J in {9,10} } by A1,A4,Th7,A2;
then consider
J being Element of Segm 13, b,c being Element of SCM-Data-Loc,
f being Element of SCM+FSA-Data*-Loc such that
A5: I = [J,{},<*c,f,b*>] & J in {9,10};
x = {} by A3,A5;
hence x in {{}} by TARSKI:def 1;
end;
set a = the Element of SCM-Data-Loc,
f = the Element of SCM+FSA-Data*-Loc;
let x be object;
T in {9,10} by A1,TARSKI:def 2;
then
A6: [T,{},<*a,f,a*>] in SCM+FSA-Instr by Th4;
assume x in {{}};
then x = {} by TARSKI:def 1;
then
A7: x = JumpPart[T,{},<*a,f,a*>];
InsCode[T,{},<*a,f,a*>] = T;
hence thesis by A7,A6;
end;
theorem Th9:
T = 11 or T = 12 implies JumpParts T = {{}}
proof
assume
A1: T = 11 or T = 12;
then
A2: not(T = 0 or ... or T = 8);
hereby
let x be object;
assume x in JumpParts T;
then consider I being Element of SCM+FSA-Instr such that
A3: x = JumpPart I and
A4: InsCode I = T;
I in { [K,{},<*c1,f1*>]
where K is Element of Segm 13, c1 is Element of SCM-Data-Loc,
f1 is Element of SCM+FSA-Data*-Loc
: K in {11,12} } by A1,A4,Th7,A2;
then consider K being Element of Segm 13,
c1 being Element of SCM-Data-Loc,
f1 being Element of SCM+FSA-Data*-Loc
such that
A5: I = [K,{},<*c1,f1*>] & K in {11,12};
x = {} by A3,A5;
hence x in {{}} by TARSKI:def 1;
end;
set a = the Element of SCM-Data-Loc, f = the Element of SCM+FSA-Data*-Loc;
let x be object;
T in {11,12} by A1,TARSKI:def 2;
then
A6: [T,{},<*a,f*>] in SCM+FSA-Instr by Th5;
assume x in {{}};
then x = {} by TARSKI:def 1;
then
A7: x = JumpPart[T,{},<*a,f*>];
InsCode[T,{},<*a,f*>] = T;
hence thesis by A7,A6;
end;
registration
cluster SCM+FSA-Instr -> J/A-independent;
coherence
proof
let T be InsType of SCM+FSA-Instr,
f1,f2 be natural-valued Function such that
A1: f1 in JumpParts T and
A2: dom f1 = dom f2;
let p be object such that
A3: [T,f1,p] in SCM+FSA-Instr;
reconsider II = [T,f1,p] as Element of SCM+FSA-Instr by A3;
InsCode II <= 12 by Lm2;
then InsCode II = 0 or ... or InsCode II = 12 by NAT_1:60;
then per cases;
suppose InsCode II = 0 or InsCode II = 1 or InsCode II = 2 or
InsCode II = 3 or InsCode II = 4 or InsCode II = 5
or InsCode II = 6 or InsCode II = 7 or InsCode II = 8;
then
A4: InsCode II <= 8;
then reconsider ii = II as Element of SCM-Instr by Th2;
A5: T = InsCode ii;
then T in InsCodes SCM-Instr;
then reconsider t = T as InsType of SCM-Instr;
A6: [t,f1,p] in SCM-Instr by A4,Th2;
JumpParts t = JumpParts T by A5,Lm5;
then [t,f2,p] in SCM-Instr by A1,A2,A6,COMPOS_0:def 7;
then [T,f2,p] in SCM-Instr;
hence [T,f2,p] in SCM+FSA-Instr by Th1;
end;
suppose T = 9 or T = 10 or T = 11 or T = 12;
then JumpParts T = {0} by Th8,Th9;
then f1 = 0 by A1,TARSKI:def 1;
then f1 = f2 by A2;
hence [T,f2,p] in SCM+FSA-Instr by A3;
end;
end;
end;
registration
cluster SCM+FSA-Instr -> with_halt;
coherence
by Th3;
end;