:: Development of Terminology for {\bf SCM}
:: by Grzegorz Bancerek and Piotr Rudnicki
::
:: Received October 8, 1993
:: Copyright (c) 1993-2017 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, INT_1, FINSEQ_1, SUBSET_1, FSM_1, AMI_3, PARTFUN1,
AMI_1, EXTPRO_1, RELAT_1, FUNCT_1, CIRCUIT2, MSUALG_1, ARYTM_3, XXREAL_0,
CARD_1, TARSKI, AFINSQ_1, ORDINAL4, GRAPHSP, ARYTM_1, SCM_1, STRUCT_0,
RECDEF_2, FUNCT_4, CAT_1, GOBRD13, MEMSTR_0, NAT_1;
notations TARSKI, XTUPLE_0, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, NAT_1,
INT_1, RELAT_1, FUNCT_1, PARTFUN1, FINSEQ_1, XXREAL_0, AFINSQ_1, FUNCT_4,
RECDEF_2, STRUCT_0, MEMSTR_0, COMPOS_1, EXTPRO_1, AMI_3;
constructors AMI_3, RELSET_1, PRE_POLY, XTUPLE_0;
registrations ORDINAL1, NUMBERS, XXREAL_0, XREAL_0, NAT_1, INT_1, AMI_3,
AFINSQ_1, RELAT_1, PBOOLE, FINSEQ_1, STRUCT_0, VALUED_0, FUNCT_4,
COMPOS_0, XTUPLE_0, MEMSTR_0;
requirements NUMERALS, REAL, SUBSET, BOOLE, ARITHM;
definitions TARSKI, RELAT_1, FUNCT_1, MEMSTR_0;
equalities AFINSQ_1, EXTPRO_1, NAT_1, AMI_3, FUNCOP_1, MEMSTR_0;
theorems NAT_1, INT_1, AMI_3, AFINSQ_1, PARTFUN1, FUNCT_4, FUNCOP_1, TARSKI,
EXTPRO_1, GRFUNC_1, CARD_1, MEMSTR_0;
schemes PARTFUN1;
begin
registration let i1 be Integer;
cluster <%i1%> -> INT-valued;
coherence;
let i2 be Integer;
cluster <%i1,i2%> -> INT-valued;
coherence;
let i3 be Integer;
cluster <%i1,i2,i3%> -> INT-valued;
coherence;
let i4 be Integer;
cluster <%i1,i2,i3,i4%> -> INT-valued;
coherence;
end;
definition
let D be XFinSequence of INT;
mode State-consisting of D -> State of SCM means
:Def1:
for k being Element of NAT st k < len D holds it.dl.k = D.k;
existence
proof
defpred P[object,object] means
ex k being Element of NAT st $1 = dl.k & $2 = D.k;
A1: for x,y being object
st x in the carrier of SCM & P[x,y] holds y in INT by INT_1:def 2;
A2: for x,y1,y2 being object
st x in the carrier of SCM & P[x,y1] & P[x,y2] holds y1 = y2 by AMI_3:10;
consider p being PartFunc of SCM,INT such that
A3: for x being object
holds x in dom p iff x in the carrier of SCM &
ex y being object st P[x,y] and
A4: for x being object st x in dom p holds P[x,p.x]
from PARTFUN1:sch 2(A1,A2);
A5: p is (the carrier of SCM)-defined
proof
let e be object;
thus thesis by A3;
end;
p is (the_Values_of SCM)-compatible
proof let e be object;
assume
A6: e in dom p;
then
A7: ex y being object st P[e,y] by A3;
reconsider o = e as Object of SCM by A6,A3;
A8: Values o = INT by A7,AMI_3:11;
consider k being Element of NAT such that
A9: o = dl.k & p.o = D.k by A4,A6;
thus p.e in (the_Values_of SCM).e by A8,A9,INT_1:def 2;
end;
then reconsider p as PartState of SCM by A5;
take s = (the State of SCM)+*p;
let k be Element of NAT;
assume k < len D;
ex i being Element of NAT st dl.k = dl.i & D.k = D.i;
then
A10: dl.k in dom p by A3;
then consider i being Element of NAT such that
A11: dl.k = dl.i & p.dl.k = D.i by A4;
A12: k = i by A11,AMI_3:10;
p c= s by FUNCT_4:25;
hence s.dl.k = D.k by A12,A11,A10,GRFUNC_1:2;
end;
end;
registration
let D be XFinSequence of INT, il be Element of NAT;
cluster il-started for State-consisting of D;
existence
proof
set s = the State-consisting of D;
set s1 = s +* Start-At(il,SCM);
for k being Element of NAT st k < len D holds s1.dl.k = D.k
proof let k be Element of NAT;
assume
A1: k < len D;
A2: dom Start-At(il,SCM) = {IC SCM} by FUNCOP_1:13;
dl.k <> IC SCM by AMI_3:13;
then not dl.k in dom Start-At(il,SCM) by A2,TARSKI:def 1;
hence s1.dl.k = s.dl.k by FUNCT_4:11
.= D.k by A1,Def1;
end;
then reconsider s1 as State-consisting of D by Def1;
take s1;
thus IC s1 = il by MEMSTR_0:16;
end;
end;
theorem
for i1, i2, i3, i4 being Integer, il being Element of NAT,
s being il-started State-consisting of <%i1,i2,i3,i4%>
holds
s.dl.0 = i1 & s.dl.1 = i2 & s.dl.2 = i3 & s.dl.3 = i4
proof
let i1, i2, i3, i4 be Integer, il be Element of NAT,
s be il-started State-consisting of <%i1,i2,i3,i4%>;
set D = <%i1,i2,i3,i4%>;
A1: D.2 = i3 & D.3 = i4;
A2: len D = 4 & 0+0=0 by AFINSQ_1:84;
D.0 = i1 & D.1 = i2;
hence thesis by A1,A2,Def1;
end;
theorem Th2:
for i1, i2 being Integer, il being Element of NAT,
s being il-started State-consisting of <%i1,i2%> holds
s.dl.0 = i1 & s.dl.1 = i2
proof
let i1, i2 be Integer, il be Element of NAT,
s be il-started State-consisting of <%i1,i2%>;
set data = <%i1,i2%>;
A1: len data = 2 & data.0 = i1 by AFINSQ_1:38;
data.1 = i2;
hence thesis by A1,Def1;
end;
theorem Th3:
for I1, I2 being Instruction of SCM
for F being total
NAT-defined (the InstructionsF of SCM)-valued Function
st <%I1%>^<%I2%> c= F
holds F.0 = I1 & F.1 = I2
proof
let I1, I2 be Instruction of SCM;
let F be total
NAT-defined the InstructionsF of SCM-valued Function such that
A1: <%I1%>^<%I2%> c= F;
set ins = <%I1%>^<%I2%>;
A2: ins = <%I1, I2%>;
then
A3: ins.1 = I2;
A4: ins.0 = I1 by A2;
len ins = 2 by A2,AFINSQ_1:38;
then 0 in dom ins & 1 in dom ins by CARD_1:50,TARSKI:def 2;
hence thesis by A1,A3,A4,GRFUNC_1:2;
end;
reserve F for total
NAT-defined (the InstructionsF of SCM)-valued Function;
Lm1:
for k being Nat, s being State of SCM
holds Comput(F,s,k+1) = Exec(CurInstr(F,Comput(F,s,k)),Comput(F,s,k))
proof
let k be Nat, s be State of SCM;
thus Comput(F,s,k+1) = Following(F,Comput(F,s,k)) by EXTPRO_1:3
.= Exec(CurInstr(F,Comput(F,s,k)),Comput(F,s,k));
end;
Lm2: now
let F be total
NAT-defined (the InstructionsF of SCM)-valued Function;
let k, n be Element of NAT, s be State of SCM, a, b be Data-Location;
assume
A1: IC Comput(F,s,k) = n;
set csk1 = Comput(F,s,k+1);
set csk = Comput(F,s,k);
assume
A2: F.n = a := b or F.n = AddTo(a,b) or F.n = SubFrom(a, b) or
F.n = MultBy(a, b) or a<>b & F.n = Divide(a,b);
A3: dom F = NAT by PARTFUN1:def 2;
thus csk1 = (Exec(CurInstr(F,csk), csk)) by Lm1
.= Exec(F.n, csk) by A1,A3,PARTFUN1:def 6;
hence IC csk1 = IC csk + 1 by A2,AMI_3:2,3,4,5,6
.= n+1 by A1;
end;
theorem Th4:
for k, n being Element of NAT, s being State of SCM,
a, b being Data-Location st
IC Comput(F,s,k) = n & F.n = a := b
holds IC Comput(F,s,k+1) = n+1 &
Comput(F,s,k+1).a = Comput(F,s,k).b &
for d being Data-Location st d <> a
holds Comput(F,s,k+1).d = Comput(F,s,k).d
proof
let k, n be Element of NAT, s be State of SCM, a, b be Data-Location;
assume
A1: IC Comput(F,s,k) = n;
assume
F.n = a := b;
then Comput(F,s,k+1)
= Exec(a:=b, Comput(F,s,k)) by A1,Lm2;
hence thesis by A1,AMI_3:2;
end;
theorem Th5:
for k, n being Element of NAT, s being State of SCM, a, b being
Data-Location st IC Comput(F,s,k) = n & F.n = AddTo(a,b)
holds IC Comput(F,s,k+1) = n+1 &
Comput(F,s,k+1).a = Comput(F,s,k).a+
Comput(F,s,k).b & for d being Data-Location st d <> a holds Comput
(F,s,k+
1).d = Comput(F,s,k).d
proof
let k, n be Element of NAT, s be State of SCM, a, b be Data-Location;
assume
A1: IC Comput(F,s,k) = n;
assume
F.n = AddTo(a,b);
then Comput(F,s,k+1)
= Exec(AddTo(a,b), Comput(F,s,k)) by A1,Lm2;
hence thesis by A1,AMI_3:3;
end;
theorem Th6:
for k, n being Element of NAT, s being State of SCM, a, b being
Data-Location st IC Comput(F,s,k) = n & F.n = SubFrom(a,b)
holds IC Comput(F,s,k+1) = n+1 &
Comput(F,s,k+1).a =
Comput(F,s,k).a-
Comput(F,s,k).b & for d being Data-Location st d <> a holds Comput
(F,s,k+
1).d = Comput(F,s,k).d
proof
let k, n be Element of NAT, s be State of SCM, a, b be Data-Location;
assume
A1: IC Comput(F,s,k)= n;
assume
F.n = SubFrom(a,b);
then Comput(F,s,k+1) =
Exec(SubFrom(a,b), Comput(F,s,k)) by A1,Lm2;
hence thesis by A1,AMI_3:4;
end;
theorem Th7:
for k, n being Element of NAT, s being State of SCM, a, b being
Data-Location st IC Comput(F,s,k) = n & F.n = MultBy(a,b) holds
IC
Comput(F,s,k+1) = (n+1) & Comput(F,s,k+1).a =
Comput(F,s,k).a*
Comput(F,s,k).b & for d being Data-Location st d <> a holds Comput
(F,s,k+
1).d = Comput(F,s,k).d
proof
let k, n be Element of NAT, s be State of SCM, a, b be Data-Location;
assume
A1: IC Comput(F,s,k) = n;
assume
F.n = MultBy(a,b);
then Comput(F,s,k+1)
= Exec(MultBy(a,b), Comput(F,s,k)) by A1,Lm2;
hence thesis by A1,AMI_3:5;
end;
theorem Th8:
for k, n being Element of NAT, s being State of SCM, a, b being
Data-Location st IC Comput(F,s,k) = n & F.n = Divide(a,b) & a<>b
holds
IC Comput(F,s,k+1) = (n+1) & Comput(F,s,k+1).a =
Comput(F,s,k).a
div Comput(F,s,k).b & Comput(F,s,k+1).b = Comput(
F,s,k).a mod
Comput(F,s,k).b & for d being Data-Location st d <> a & d <> b
holds
Comput(F,s,k+1).d = Comput(F,s,k).d
proof
let k, n be Element of NAT, s be State of SCM, a, b be Data-Location;
assume
A1: IC Comput(F,s,k) = n;
assume
A2: F.n = Divide(a,b) & a <> b;
then Comput(F,s,k+1)
= Exec(Divide(a,b), Comput(F,s,k)) by A1,Lm2;
hence thesis by A1,A2,AMI_3:6;
end;
theorem Th9:
for k, n being Element of NAT, s being State of SCM, il being
Element of NAT st IC Comput(F,s,k) = n &
F.n = SCM-goto il
holds IC Comput(F,s,k+1) = il & for d being Data-Location holds
Comput(F,s,k+1).d = Comput(F,s,k).d
proof
let k, n be Element of NAT, s be State of SCM, il be Element of NAT;
assume
A1: IC Comput(F,s,k) = n & F.n = SCM-goto il;
set csk1 = Comput(F,s,k+1);
set csk = Comput(F,s,k);
A2: dom F = NAT by PARTFUN1:def 2;
A3: csk1 = Exec(CurInstr(F,csk), csk) by Lm1
.= Exec(SCM-goto il, csk) by A1,A2,PARTFUN1:def 6;
hence IC csk1 = il by AMI_3:7;
thus thesis by A3,AMI_3:7;
end;
theorem Th10:
for k, n being Nat, s being State of SCM, a being
Data-Location, il being Element of NAT st IC Comput(F,s,k) =
n & F.n = a =0_goto il holds ( Comput(F,s,k).a = 0 implies IC
Comput(F,s,k+1) = il) & ( Comput(F,s,k).a <>0 implies
IC Comput(F,s,k+
1) = (n+1)) & for d being Data-Location holds Comput(F,s,k+1).d
=
Comput(F,s,k).d
proof
let k, n be Nat, s be State of SCM, a be Data-Location, il be
Element of NAT;
assume that
A1: IC Comput(F,s,k) = n and
A2: F.n = a =0_goto il;
set csk1 = Comput(F,s,k+1);
set csk = Comput(F,s,k);
A3: dom F = NAT by PARTFUN1:def 2;
A4: csk1 = Exec(CurInstr(F,csk), csk) by Lm1
.= Exec(a =0_goto il, csk) by A1,A2,A3,PARTFUN1:def 6;
hence csk.a = 0 implies IC csk1 = il by AMI_3:8;
thus thesis by A1,A4,AMI_3:8;
end;
theorem Th11:
for k, n being Element of NAT, s being State of SCM, a being
Data-Location, il being Element of NAT st IC Comput(F,s,k) =
n & F.n = a >0_goto il holds ( Comput(F,s,k).a > 0 implies IC
Comput(F,s,k+1) = il) & ( Comput(F,s,k).a <= 0
implies IC Comput(F,s,k
+1) = (n+1)) & for d being Data-Location holds Comput(F,s,k+1).d
=
Comput(F,s,k).d
proof
let k, n be Element of NAT, s be State of SCM, a be Data-Location, il be
Element of NAT;
assume that
A1: IC Comput(F,s,k) = n and
A2: F.n = a >0_goto il;
set csk1 = Comput(F,s,k+1);
set csk = Comput(F,s,k);
A3: dom F = NAT by PARTFUN1:def 2;
csk1 = Exec(CurInstr(F,csk), csk) by Lm1
.= Exec(a >0_goto il, csk) by A1,A2,A3,PARTFUN1:def 6;
hence thesis by A1,AMI_3:9;
end;
theorem Th12:
(halt SCM)`1_3 = 0 &
(for a, b being Data-Location holds (a := b)`1_3 = 1) &
(for a, b being Data-Location holds (AddTo(a,b))`1_3 = 2) &
(for a, b being Data-Location holds (SubFrom(a,b))`1_3 = 3) &
(for a, b being Data-Location holds (MultBy(a,b))`1_3 = 4) &
(for a, b being Data-Location holds (Divide(a,b))`1_3 = 5) &
(for i being Element of NAT holds (SCM-goto i)`1_3 = 6) &
(for a being Data-Location, i being Element of NAT
holds (a =0_goto i)`1_3 = 7) &
for a being Data-Location, i being Element of NAT holds
(a >0_goto i)`1_3 = 8 by AMI_3:26;
theorem
for i1, i2, i3, i4 being Integer,
s being State of SCM st s.dl.0 = i1 & s.dl.1 = i2 & s.dl.2 = i3 & s.dl.3 = i4
holds s is State-consisting of <%i1,i2,i3,i4%>
proof
let i1, i2, i3, i4 be Integer, s be State of SCM such that
A1: s.dl.0 = i1 & s.dl.1 = i2 & s.dl.2 = i3 & s.dl.3 = i4;
set D = <%i1,i2,i3,i4%>;
now
let k be Element of NAT;
A2: len D=4 & 4=3+1 by AFINSQ_1:84;
assume k < len D;
then k <= 3 by A2,NAT_1:13;
then k=0 or ... or k=3;
hence s.dl.k=D.k by A1;
end;
hence thesis by Def1;
end;
:: Empty program
theorem
for F being total
NAT-defined (the InstructionsF of SCM)-valued Function
st <%halt SCM%> c= F
for s being 0-started State-consisting of <*>INT
holds F halts_on s & LifeSpan(F,s) = 0 & Result(F,s) = s
proof
let F be total
NAT-defined (the InstructionsF of SCM)-valued Function such that
A1: <%halt SCM%> c= F;
let s be 0-started State-consisting of <*>INT;
1 = len <%halt SCM%> by AFINSQ_1:34;
then
0 in dom<%halt SCM%> by CARD_1:49,TARSKI:def 1;
then
A2: F.(0+0) = <%halt SCM%>.0 by A1,GRFUNC_1:2
.= halt SCM;
A3: s = Comput(F,s,0) by EXTPRO_1:2;
F.IC s = halt SCM by A2,MEMSTR_0:def 11;
hence
A4: F halts_on s by A3,EXTPRO_1:30;
dom F = NAT by PARTFUN1:def 2;
then CurInstr(F,s) = F.IC s by PARTFUN1:def 6
.= halt SCM by A2,MEMSTR_0:def 11;
hence LifeSpan(F,s) = 0 by A4,A3,EXTPRO_1:def 15;
IC s = 0 by MEMSTR_0:def 11;
hence thesis by A2,A3,EXTPRO_1:7;
end;
:: Assignment
theorem
for F being total
NAT-defined (the InstructionsF of SCM)-valued Function
st <%dl.0 := dl.1%>^<%halt SCM%> c= F
for i1, i2 being Integer,
s being 0-started State-consisting of <%i1,i2%>
holds F halts_on s & LifeSpan(F,s) = 1 & (Result(F,s)).dl.0 = i2 &
for d being Data-Location st d<>dl.0
holds (Result(F,s)).d = s.d
proof
let F be total
NAT-defined (the InstructionsF of SCM)-valued Function such that
A1: <%dl.0 := dl.1%>^<%halt SCM%> c= F;
let i1, i2 be Integer,
s be 0-started State-consisting of <%i1,i2%>;
set s1 = Comput(F,s,0+1);
A2: s.dl.1 = i2 by Th2;
A3: IC s = 0 & s = Comput(F,s,0) by EXTPRO_1:2,MEMSTR_0:def 11;
A4: F.0 = dl.0 := dl.1 by A1,Th3;
then
A5: IC s1 = 0+1 by A3,Th4;
A6: F.1 = halt SCM by A1,Th3;
hence F halts_on s by A5,EXTPRO_1:30;
thus LifeSpan(F,s) = 1 by A6,A3,A5,EXTPRO_1:33;
s1.dl.0 = s.dl.1 by A4,A3,Th4;
hence (Result(F,s)).dl.0 = i2 by A6,A2,A5,EXTPRO_1:7;
let d be Data-Location;
assume
A7: d<>dl.0;
thus (Result(F,s)).d = s1.d by A6,A5,EXTPRO_1:7
.= s.d by A4,A3,A7,Th4;
end;
:: Adding two integers
theorem
for F being total
NAT-defined (the InstructionsF of SCM)-valued Function
st <%AddTo(dl.0,dl.1)%>^<%halt SCM%> c= F
for i1, i2 being Integer,
s being 0-started State-consisting of <%i1,i2%>
holds F halts_on s & LifeSpan(F,s) = 1 & (Result(F,s)).dl.0 = i1 + i2 &
for d being Data-Location st d<>dl.0 holds Result(F,s).d = s.d
proof
let F be total
NAT-defined (the InstructionsF of SCM)-valued Function such that
A1: <%AddTo(dl.0,dl.1)%>^<%halt SCM%> c= F;
let i1, i2 be Integer,
s be 0-started State-consisting of <%i1,i2%>;
set s0 = Comput(F,s,0);
set s1 = Comput(F,s,0+1);
A2: s = s0 by EXTPRO_1:2;
A3: s.dl.0 = i1 & s.dl.1 = i2 by Th2;
A4: IC s = 0 by MEMSTR_0:def 11;
A5: F.0 = AddTo(dl.0,dl.1) by A1,Th3;
then
A6: IC s1 = (0+1) by A4,A2,Th5;
A7: F.1 = halt SCM by A1,Th3;
hence F halts_on s by A6,EXTPRO_1:30;
thus LifeSpan(F,s) = 1 by A4,A7,A2,A6,EXTPRO_1:33;
s1.dl.0 = s0.dl.0 + s0.dl.1 by A4,A5,A2,Th5;
hence (Result(F,s)).dl.0 = i1 + i2 by A7,A3,A2,A6,EXTPRO_1:7;
let d be Data-Location;
assume
A8: d<>dl.0;
thus (Result(F,s)).d = s1.d by A7,A6,EXTPRO_1:7
.= s.d by A4,A5,A2,A8,Th5;
end;
:: Subtracting two integers
theorem
for F being total
NAT-defined (the InstructionsF of SCM)-valued Function
st <%SubFrom(dl.0,dl.1)%>^<%halt SCM%> c= F
for i1, i2 being Integer,
s being 0-started State-consisting of <%i1,i2%>
holds F halts_on s
& LifeSpan(F,s) = 1 &
(Result(F,s)).dl.0 = i1 - i2 & for d being Data-Location
st d<>dl.0 holds
(Result(F,s)).d = s.d
proof
let F being total
NAT-defined (the InstructionsF of SCM)-valued Function such that
A1: <%SubFrom(dl.0,dl.1)%>^<%halt SCM%> c= F;
let i1, i2 be Integer,
s be 0-started State-consisting of <%i1,i2%>;
set s0 = Comput(F,s,0);
set s1 = Comput(F,s,0+1);
A2: s = s0 by EXTPRO_1:2;
A3: s.dl.0 = i1 & s.dl.1 = i2 by Th2;
A4: IC s = 0 by MEMSTR_0:def 11;
A5: F.0 = SubFrom(dl.0,dl.1) by A1,Th3;
then
A6: IC s1 = (0+1) by A4,A2,Th6;
A7: F.1 = halt SCM by A1,Th3;
hence F halts_on s by A6,EXTPRO_1:30;
thus LifeSpan(F,s) = 1 by A4,A7,A2,A6,EXTPRO_1:33;
s1.dl.0 = s0.dl.0 - s0.dl.1 by A4,A5,A2,Th6;
hence (Result(F,s)).dl.0 = i1 - i2 by A7,A3,A2,A6,EXTPRO_1:7;
let d be Data-Location;
assume
A8: d<>dl.0;
thus (Result(F,s)).d = s1.d by A7,A6,EXTPRO_1:7
.= s.d by A4,A5,A2,A8,Th6;
end;
:: Multiplying two integers
theorem
for F being total
NAT-defined (the InstructionsF of SCM)-valued Function
st <%MultBy(dl.0,dl.1)%>^<%halt SCM%> c= F
for i1, i2 being Integer,
s being 0-started State-consisting of <%i1,i2%>
holds F halts_on s & LifeSpan(F,s) = 1 & (Result(F,s)).dl.0 = i1 * i2 &
for d being Data-Location st d<>dl.0 holds (Result(F,s)).d = s.d
proof
let F being total
NAT-defined (the InstructionsF of SCM)-valued Function such that
A1: <%MultBy(dl.0,dl.1)%>^<%halt SCM%> c= F;
let i1, i2 be Integer,
s be 0-started State-consisting of <%i1,i2%>;
set s0 = Comput(F,s,0);
set s1 = Comput(F,s,0+1);
A2: s = s0 by EXTPRO_1:2;
A3: s.dl.0 = i1 & s.dl.1 = i2 by Th2;
A4: IC s = 0 by MEMSTR_0:def 11;
A5: F.0 = MultBy(dl.0,dl.1) by A1,Th3;
then
A6: IC s1 = (0+1) by A4,A2,Th7;
A7: F.1 = halt SCM by A1,Th3;
hence F halts_on s by A6,EXTPRO_1:30;
thus LifeSpan(F,s) = 1 by A4,A7,A2,A6,EXTPRO_1:33;
s1.dl.0 = s0.dl.0 * s0.dl.1 by A4,A5,A2,Th7;
hence (Result(F,s)).dl.0 = i1 * i2 by A7,A3,A2,A6,EXTPRO_1:7;
let d be Data-Location;
assume
A8: d<>dl.0;
thus (Result(F,s)).d = s1.d by A7,A6,EXTPRO_1:7
.= s.d by A4,A5,A2,A8,Th7;
end;
:: Dividing two integers
theorem
for F being total
NAT-defined (the InstructionsF of SCM)-valued Function
st <%Divide(dl.0,dl.1)%>^<%halt SCM%> c= F
for i1, i2 being Integer,
s being 0-started State-consisting of <%i1,i2%>
holds F halts_on s
& LifeSpan(F,s)
= 1 & (Result(F,s)).dl.0 = i1 div i2 & (Result(F,s)
).dl.1 = i1 mod i2 & for d being
Data-Location st d<>dl.0 & d<>dl.1 holds (Result(F,s)).d = s.d
proof
let F being total
NAT-defined (the InstructionsF of SCM)-valued Function such that
A1: <%Divide(dl.0,dl.1)%>^<%halt SCM%> c= F;
let i1, i2 be Integer,
s be 0-started State-consisting of <%i1,i2%>;
set s1 = Comput(F,s,0+1);
A2: dl.0 <> dl.1 by AMI_3:10;
A3: IC s = 0 & F.0 = Divide(dl.0,dl.1) by A1,Th3,MEMSTR_0:def 11;
A4: s.dl.0 = i1 & s.dl.1 = i2 by Th2;
A5: s = Comput(F,s,0) by EXTPRO_1:2;
F.1 = halt SCM by A1,Th3;
then
A6: F.(IC s1) = halt SCM by A3,A2,A5,Th8;
hence F halts_on s by EXTPRO_1:30;
Divide(dl.0, dl.1) <> halt SCM by Th12;
hence LifeSpan(F,s) = 1 by A3,A5,A6,EXTPRO_1:32;
thus (Result(F,s)).dl.0 = s1.dl.0 by A6,EXTPRO_1:7
.= i1 div i2 by A3,A4,A2,A5,Th8;
thus (Result(F,s)).dl.1 = s1.dl.1 by A6,EXTPRO_1:7
.= i1 mod i2 by A3,A4,A2,A5,Th8;
let d be Data-Location;
assume
A7: d<>dl.0 & d<>dl.1;
thus (Result(F,s)).d = s1.d by A6,EXTPRO_1:7
.= s.d by A3,A2,A5,A7,Th8;
end;
:: Unconditional jump
theorem
for F being total
NAT-defined (the InstructionsF of SCM)-valued Function
st <%SCM-goto 1%>^<%halt SCM%> c= F
for i1, i2 being Integer,
s being 0-started State-consisting of <%i1,i2%>
holds F halts_on s &
LifeSpan(F,s) = 1 & for d
being Data-Location holds (Result(F,s)).d = s.d
proof
let F being total
NAT-defined (the InstructionsF of SCM)-valued Function such that
A1: <%SCM-goto 1%>^<%halt SCM%> c= F;
let i1, i2 be Integer,
s be 0-started State-consisting of <%i1,i2%>;
set s1 = Comput(F,s,0+1);
A2: IC s = 0 & s = Comput(F,s,0) by EXTPRO_1:2,MEMSTR_0:def 11;
A3: F.0 = SCM-goto 1 by A1,Th3;
then
A4: IC s1 = (0+1) by A2,Th9;
A5: F.1 = halt SCM by A1,Th3;
hence F halts_on s by A4,EXTPRO_1:30;
thus LifeSpan(F,s) = 1 by A5,A2,A4,EXTPRO_1:33;
let d be Data-Location;
thus (Result(F,s)).d = s1.d by A5,A4,EXTPRO_1:7
.= s.d by A3,A2,Th9;
end;
:: Jump at zero
theorem
for F being total
NAT-defined (the InstructionsF of SCM)-valued Function
st <%dl.0=0_goto 1%>^<%halt SCM%> c= F
for i1, i2 being Integer,
s being 0-started State-consisting of <%i1,i2%>
holds F halts_on s
& LifeSpan(F,s) = 1
& for d being Data-Location holds (Result(F,s)).d = s.d
proof
let F being total
NAT-defined (the InstructionsF of SCM)-valued Function such that
A1: <%dl.0=0_goto 1%>^<%halt SCM%> c= F;
let i1, i2 be Integer,
s be 0-started State-consisting of <%i1,i2%>;
set s1 = Comput(F,s,0+1);
A2: F.0 = dl.0 =0_goto 1 by A1,Th3;
A3: F.1 = halt SCM by A1,Th3;
A4: IC s = 0 & s = Comput(F,s,0) by EXTPRO_1:2,MEMSTR_0:def 11;
s.dl.0 = i1 by Th2;
then
A5: IC s1 = (0+1) by A2,A4,Th10;
hence F halts_on s by A3,EXTPRO_1:30;
thus LifeSpan(F,s) = 1 by A3,A4,A5,EXTPRO_1:33;
let d be Data-Location;
thus (Result(F,s)).d = s1.d by A3,A5,EXTPRO_1:7
.= s.d by A2,A4,Th10;
end;
:: Jump at greater than zero
theorem
for F being total
NAT-defined (the InstructionsF of SCM)-valued Function
st <%dl.0>0_goto 1%>^<%halt SCM%> c= F
for i1, i2 being Integer,
s being 0-started State-consisting of <%i1,i2%>
holds F halts_on s
& LifeSpan(F,s) = 1
& for d being Data-Location holds (Result(F,s)).d = s.d
proof
let F being total
NAT-defined (the InstructionsF of SCM)-valued Function such that
A1: <%dl.0>0_goto 1%>^<%halt SCM%> c= F;
let i1, i2 be Integer,
s be 0-started State-consisting of <%i1,i2%>;
set s1 = Comput(F,s,0+1);
A2: F.0 = dl.0 >0_goto 1 by A1,Th3;
A3: F.1 = halt SCM by A1,Th3;
A4: IC s = 0 & s = Comput(F,s,0) by EXTPRO_1:2,MEMSTR_0:def 11;
s.dl.0 = i1 by Th2;
then
A5: IC s1 = (0+1) by A2,A4,Th11;
hence F halts_on s by A3,EXTPRO_1:30;
thus LifeSpan(F,s) = 1 by A3,A4,A5,EXTPRO_1:33;
let d be Data-Location;
thus (Result(F,s)).d = s1.d by A3,A5,EXTPRO_1:7
.= s.d by A2,A4,Th11;
end;
theorem
for s being State of SCM holds s is State-consisting of <*>INT
proof let s be State of SCM;
let k be Element of NAT;
thus thesis;
end;