:: Memory handling for SCM+FSA
:: by Piotr Rudnicki and Andrzej Trybulec
::
:: Received July 18, 1996
:: Copyright (c) 1996-2016 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 SCMFSA_2, XBOOLE_0, AMI_1, GRAPHSP, AMI_3, FINSEQ_1, FUNCT_1,
SUBSET_1, FINSUB_1, SETWISEO, CARD_1, TARSKI, RELAT_1, FINSET_1, NUMBERS,
FUNCT_4, VALUED_1, ARYTM_3, AMISTD_2, XXREAL_0, SCMFSA6A, FSM_1,
CIRCUIT2, STRUCT_0, ARYTM_1, INT_1, COMPLEX1, PARTFUN1, FINSEQ_2, NAT_1,
SF_MASTR, RELOC, COMPOS_1, EXTPRO_1, SCMPDS_4, TURING_1, AMISTD_1,
FUNCOP_1;
notations TARSKI, XBOOLE_0, ENUMSET1, SUBSET_1, CARD_1, ORDINAL1, NUMBERS,
XCMPLX_0, FINSET_1, FINSUB_1, NAT_1, INT_1, NAT_D, VALUED_1, STRUCT_0,
SETWISEO, SEQ_4, CARD_3, FINSEQ_1, FINSEQ_2, RELAT_1, FUNCT_1, RELSET_1,
PARTFUN1, FUNCT_2, FUNCOP_1, FUNCT_4, PBOOLE, FUNCT_7, MEMSTR_0,
COMPOS_0, COMPOS_1, COMPOS_2, AMI_2, EXTPRO_1, AMI_3, SCMFSA_2, AMISTD_2,
SCMFSA6A, INT_2, XXREAL_0, SCMFSA_M;
constructors SETWISEO, INT_2, SEQ_4, AMI_3, SCMFSA6A, RELSET_1, VALUED_1,
AMISTD_2, AMISTD_1, AMI_2, SCMFSA_1, CARD_5, PRE_POLY, DOMAIN_1, PBOOLE,
FUNCT_7, MEMSTR_0, SCMFSA_M, COMPOS_1, NAT_D, COMPOS_2;
registrations FUNCT_1, ORDINAL1, RELSET_1, FUNCT_2, FINSET_1, FINSUB_1,
XREAL_0, INT_1, SCMFSA_2, XBOOLE_0, VALUED_1, FINSEQ_1, AMISTD_2, AMI_3,
SCMFSA10, COMPOS_1, EXTPRO_1, CARD_1, CARD_2, COMPOS_0, SCM_INST, NAT_1,
SCMFSA6A, FUNCOP_1;
requirements NUMERALS, BOOLE, SUBSET, ARITHM, REAL;
definitions TARSKI, XBOOLE_0;
equalities MEMSTR_0, EXTPRO_1, COMPOS_1, AMI_3, SCMFSA_2, SCMFSA6A, COMPOS_2,
FUNCOP_1;
theorems TARSKI, ENUMSET1, ZFMISC_1, FINSEQ_1, FINSUB_1, NAT_1, RELAT_1,
GRFUNC_1, FUNCT_1, FUNCT_4, SCMFSA_2, SCMFSA_4, XBOOLE_0, XBOOLE_1,
AMI_2, MEMSTR_0, VALUED_1, PARTFUN1, COMPOS_1, EXTPRO_1, COMPOS_0,
SCMFSA_M, XTUPLE_0, NECKLACE, FINSET_1, SCMFSA6A, COMPOS_2, FUNCOP_1;
schemes NAT_1, FRAENKEL, SUBSET_1;
begin :: Uniqueness of instruction components
reserve a, b, c, a1, a2, b1, b2 for Int-Location,
l, l1, l2 for Nat,
f, g, f1, f2 for FinSeq-Location,
i, j for Instruction of SCM+FSA,
X, Y for set;
theorem Th1:
a1:=b1 = a2:=b2 implies a1 = a2 & b1 = b2
proof
assume
A1: a1:=b1 = a2:=b2;
consider A1, B1 being Data-Location such that
A2: a1 = A1 & b1 = B1 & a1:=b1 = A1:=B1 by SCMFSA_2:def 8;
consider A2, B2 being Data-Location such that
A3: a2 = A2 & b2 = B2 & a2:=b2 = A2:=B2 by SCMFSA_2:def 8;
A4: <*A2,B2*>.1 = A2 & <*A2,B2*>.2 = B2 by FINSEQ_1:44;
<*A1,B1*>.1 = A1 & <*A1,B1*>.2 = B1 by FINSEQ_1:44;
hence thesis by A1,A2,A3,A4,XTUPLE_0:3;
end;
theorem Th2:
AddTo(a1,b1) = AddTo(a2,b2) implies a1 = a2 & b1 = b2
proof
assume
A1: AddTo(a1,b1) = AddTo(a2,b2);
consider A1, B1 being Data-Location such that
A2: a1 = A1 & b1 = B1 & AddTo(a1,b1) = AddTo(A1,B1) by SCMFSA_2:def 9;
consider A2, B2 being Data-Location such that
A3: a2 = A2 & b2 = B2 & AddTo(a2,b2) = AddTo(A2,B2) by SCMFSA_2:def 9;
A4: <*A2,B2*>.1 = A2 & <*A2,B2*>.2 = B2 by FINSEQ_1:44;
<*A1,B1*>.1 = A1 & <*A1,B1*>.2 = B1 by FINSEQ_1:44;
hence thesis by A1,A2,A3,A4,XTUPLE_0:3;
end;
theorem Th3:
SubFrom(a1,b1) = SubFrom(a2,b2) implies a1 = a2 & b1 = b2
proof
assume
A1: SubFrom(a1,b1) = SubFrom(a2,b2);
consider A1, B1 being Data-Location such that
A2: a1 = A1 & b1 = B1 & SubFrom(a1,b1) = SubFrom(A1,B1) by SCMFSA_2:def 10;
consider A2, B2 being Data-Location such that
A3: a2 = A2 & b2 = B2 & SubFrom(a2,b2) = SubFrom(A2,B2) by SCMFSA_2:def 10;
A4: <*A2,B2*>.1 = A2 & <*A2,B2*>.2 = B2 by FINSEQ_1:44;
<*A1,B1*>.1 = A1 & <*A1,B1*>.2 = B1 by FINSEQ_1:44;
hence thesis by A1,A2,A3,A4,XTUPLE_0:3;
end;
theorem Th4:
MultBy(a1,b1) = MultBy(a2,b2) implies a1 = a2 & b1 = b2
proof
assume
A1: MultBy(a1,b1) = MultBy(a2,b2);
consider A1, B1 being Data-Location such that
A2: a1 = A1 & b1 = B1 & MultBy(a1,b1) = MultBy(A1,B1) by SCMFSA_2:def 11;
consider A2, B2 being Data-Location such that
A3: a2 = A2 & b2 = B2 & MultBy(a2,b2) = MultBy(A2,B2) by SCMFSA_2:def 11;
A4: <*A2,B2*>.1 = A2 & <*A2,B2*>.2 = B2 by FINSEQ_1:44;
<*A1,B1*>.1 = A1 & <*A1,B1*>.2 = B1 by FINSEQ_1:44;
hence thesis by A1,A2,A3,A4,XTUPLE_0:3;
end;
theorem Th5:
Divide(a1,b1) = Divide(a2,b2) implies a1 = a2 & b1 = b2
proof
assume
A1: Divide(a1,b1) = Divide(a2,b2);
consider A1, B1 being Data-Location such that
A2: a1 = A1 & b1 = B1 & Divide(a1,b1) = Divide(A1,B1) by SCMFSA_2:def 12;
consider A2, B2 being Data-Location such that
A3: a2 = A2 & b2 = B2 & Divide(a2,b2) = Divide(A2,B2) by SCMFSA_2:def 12;
A4: <*A2,B2*>.1 = A2 & <*A2,B2*>.2 = B2 by FINSEQ_1:44;
<*A1,B1*>.1 = A1 & <*A1,B1*>.2 = B1 by FINSEQ_1:44;
hence thesis by A1,A2,A3,A4,XTUPLE_0:3;
end;
theorem
goto l1 = goto l2 implies l1 = l2
proof
assume
A1: goto l1 = goto l2;
<*l1*>.1 = l1 & <*l2*>.1 = l2 by FINSEQ_1:40;
hence thesis by A1,XTUPLE_0:3;
end;
theorem Th7:
a1=0_goto l1 = a2=0_goto l2 implies a1 = a2 & l1 = l2
proof
assume
A1: a1=0_goto l1 = a2=0_goto l2;
consider A1 being Data-Location such
that
A2: a1 = A1 & a1=0_goto l1 = A1=0_goto l1 by SCMFSA_2:def 14;
consider A2 being Data-Location such
that
A3: a2 = A2 & a2=0_goto l2 = A2=0_goto l2 by SCMFSA_2:def 14;
A4: <*l2*>.1 = l2 & <*A2*>.1 = A2 by FINSEQ_1:40;
<*l1*>.1 = l1 & <*A1*>.1 = A1 by FINSEQ_1:40;
hence thesis by A1,A2,A3,A4,XTUPLE_0:3;
end;
theorem Th8:
a1>0_goto l1 = a2>0_goto l2 implies a1 = a2 & l1 = l2
proof
assume
A1: a1>0_goto l1 = a2>0_goto l2;
consider A1 being Data-Location such
that
A2: a1 = A1 & a1>0_goto l1 = A1>0_goto l1 by SCMFSA_2:def 15;
consider A2 being Data-Location such
that
A3: a2 = A2 & a2>0_goto l2 = A2>0_goto l2 by SCMFSA_2:def 15;
A4: <*l2*>.1 = l2 & <*A2*>.1 = A2 by FINSEQ_1:40;
<*l1*>.1 = l1 & <*A1*>.1 = A1 by FINSEQ_1:40;
hence thesis by A1,A2,A3,A4,XTUPLE_0:3;
end;
theorem Th9:
b1:=(f1, a1) = b2:=(f2, a2) implies a1 = a2 & b1 = b2 & f1 = f2
proof
A1: <*b1,f1,a1*>.1 = b1 & <*b1,f1,a1*>.2 = f1 by FINSEQ_1:45;
A2: <*b1,f1,a1*>.3 = a1 & <*b2,f2,a2*>.1 = b2 by FINSEQ_1:45;
A3: <*b2,f2,a2*>.2 = f2 & <*b2,f2,a2*>.3 = a2 by FINSEQ_1:45;
assume b1:=(f1, a1) = b2:=(f2, a2);
hence thesis by A1,A2,A3,XTUPLE_0:3;
end;
theorem Th10:
(f1, a1):=b1 = (f2, a2):=b2 implies a1 = a2 & b1 = b2 & f1 = f2
proof
A1: <*b1,f1,a1*>.1 = b1 & <*b1,f1,a1*>.2 = f1 by FINSEQ_1:45;
A2: <*b1,f1,a1*>.3 = a1 & <*b2,f2,a2*>.1 = b2 by FINSEQ_1:45;
A3: <*b2,f2,a2*>.2 = f2 & <*b2,f2,a2*>.3 = a2 by FINSEQ_1:45;
assume (f1, a1):=b1 = (f2, a2):=b2;
hence thesis by A1,A2,A3,XTUPLE_0:3;
end;
theorem Th11:
a1:=len f1 = a2:=len f2 implies a1 = a2 & f1 = f2
proof
A1: <*a1,f1*>.1 = a1 & <*a1,f1*>.2 = f1 by FINSEQ_1:44;
A2: <*a2,f2*>.1 = a2 & <*a2,f2*>.2 = f2 by FINSEQ_1:44;
assume a1:=len f1 = a2:=len f2;
hence thesis by A1,A2,XTUPLE_0:3;
end;
theorem Th12:
f1:=<0,...,0>a1 = f2:=<0,...,0>a2 implies a1 = a2 & f1 = f2
proof
A1: <*a1,f1*>.1 = a1 & <*a1,f1*>.2 = f1 by FINSEQ_1:44;
A2: <*a2,f2*>.1 = a2 & <*a2,f2*>.2 = f2 by FINSEQ_1:44;
assume f1:=<0,...,0>a1 = f2:=<0,...,0>a2;
hence thesis by A1,A2,XTUPLE_0:3;
end;
begin :: Integer locations used in a macro instruction
definition
let i be Instruction of SCM+FSA;
func UsedIntLoc i -> Element of Fin Int-Locations means
:Def1:
ex a, b being
Int-Location st (i = (a := b) or i = AddTo(a, b) or i = SubFrom(a, b) or i =
MultBy(a, b) or i = Divide(a, b)) & it = {a, b} if InsCode i in {1, 2, 3, 4, 5}
, ex a being Int-Location, l being Nat st (i = a
=0_goto l or i = a>0_goto l) & it = {a} if InsCode i = 7 or InsCode i = 8, ex a
, b being Int-Location, f being FinSeq-Location st (i = b := (f, a) or i = (f,
a) := b) & it = {a, b} if InsCode i = 9 or InsCode i = 10,
ex a being Int-Location, f being FinSeq-Location st
(i = a :=len f or i = f :=<0,...,0>a) & it = {a}
if InsCode i = 11 or InsCode i = 12
otherwise it = {};
existence
proof
hereby
assume InsCode i in {1, 2, 3, 4, 5};
then InsCode i = 1 or ... or InsCode i = 5 by ENUMSET1:def 3;
then consider a, b being Int-Location such that
A1: i = (a := b) or i = AddTo(a, b) or i = SubFrom(a, b) or i =
MultBy(a, b) or i = Divide(a, b) by SCMFSA_2:30,31,32,33,34;
reconsider a9 = a, b9 = b as Element of Int-Locations by AMI_2:def 16;
reconsider IT = {.a9, b9.} as Element of Fin Int-Locations;
take IT;
take a, b;
thus (i = (a := b) or i = AddTo(a, b) or i = SubFrom(a, b) or i = MultBy
(a, b) or i = Divide(a, b)) & IT = {a, b} by A1;
end;
hereby
assume InsCode i = 7 or InsCode i = 8;
then consider
l being Nat, a being Int-Location
such that
A2: i = a=0_goto l or i = a>0_goto l by SCMFSA_2:36,37;
reconsider a9 = a as Element of Int-Locations by AMI_2:def 16;
reconsider IT = {.a9.} as Element of Fin Int-Locations;
take IT;
take a, l;
thus (i = a=0_goto l or i = a>0_goto l) & IT = {a} by A2;
end;
hereby
assume InsCode i = 9 or InsCode i = 10;
then consider a, b being Int-Location, f being FinSeq-Location such that
A3: i = b := (f, a) or i = (f, a) := b by SCMFSA_2:38,39;
reconsider a9 = a, b9 = b as Element of Int-Locations by AMI_2:def 16;
reconsider IT = {.a9, b9.} as Element of Fin Int-Locations;
take IT;
take a, b, f;
thus (i = b := (f, a) or i = (f, a) := b) & IT = {a, b} by A3;
end;
hereby
assume InsCode i = 11 or InsCode i = 12;
then consider a being Int-Location, f being FinSeq-Location such that
A4: i = a :=len f or i = f :=<0,...,0>a by SCMFSA_2:40,41;
reconsider a9 = a as Element of Int-Locations by AMI_2:def 16;
reconsider IT = {.a9.} as Element of Fin Int-Locations;
take IT;
take a, f;
thus (i = a :=len f or i = f :=<0,...,0>a) & IT = {a} by A4;
end;
{} in Fin Int-Locations by FINSUB_1:7;
hence thesis;
end;
uniqueness
proof
let it1, it2 be Element of Fin Int-Locations;
hereby
assume InsCode i in {1, 2, 3, 4, 5};
given a1, b1 being Int-Location such that
A5: i = (a1 := b1) or i = AddTo(a1, b1) or i = SubFrom(a1, b1) or i
= MultBy(a1, b1) or i = Divide(a1, b1) and
A6: it1 = {a1, b1};
given a2, b2 being Int-Location such that
A7: i = (a2 := b2) or i = AddTo(a2, b2) or i = SubFrom(a2, b2) or i
= MultBy(a2, b2) or i = Divide(a2, b2) and
A8: it2 = {a2, b2};
A9: i = AddTo(a1, b1) or i = AddTo(a2, b2) implies InsCode i = 2 by
SCMFSA_2:19;
A10: i = Divide(a1, b1) or i = Divide(a2, b2) implies InsCode i = 5 by
SCMFSA_2:22;
A11: i = MultBy(a1, b1) or i = MultBy(a2, b2) implies InsCode i = 4 by
SCMFSA_2:21;
A12: i = SubFrom(a1, b1) or i = SubFrom(a2, b2) implies InsCode i = 3 by
SCMFSA_2:20;
per cases by A5,A7,A9,A12,A11,A10,SCMFSA_2:18;
suppose
A13: i = (a1 := b1) & i = (a2 := b2);
then a1 = a2 by Th1;
hence it1 = it2 by A6,A8,A13,Th1;
end;
suppose
A14: i = AddTo(a1, b1) & i = AddTo(a2, b2);
then a1 = a2 by Th2;
hence it1 = it2 by A6,A8,A14,Th2;
end;
suppose
A15: i = SubFrom(a1, b1) & i = SubFrom(a2, b2);
then a1 = a2 by Th3;
hence it1 = it2 by A6,A8,A15,Th3;
end;
suppose
A16: i = MultBy(a1, b1) & i = MultBy(a2, b2);
then a1 = a2 by Th4;
hence it1 = it2 by A6,A8,A16,Th4;
end;
suppose
A17: i = Divide(a1, b1) & i = Divide(a2, b2);
then a1 = a2 by Th5;
hence it1 = it2 by A6,A8,A17,Th5;
end;
end;
hereby
assume InsCode i = 7 or InsCode i = 8;
given a1 being Int-Location, l1 being Nat
such that
A18: i = a1=0_goto l1 or i = a1>0_goto l1 and
A19: it1 = {a1};
given a2 being Int-Location, l2 being Nat
such that
A20: i = a2=0_goto l2 or i = a2>0_goto l2 and
A21: it2 = {a2};
A22: i = a1>0_goto l1 or i = a2>0_goto l2 implies InsCode i = 8 by SCMFSA_2:25
;
per cases by A18,A20,A22,SCMFSA_2:24;
suppose
i = a1=0_goto l1 & i = a2=0_goto l2;
hence it1 = it2 by A19,A21,Th7;
end;
suppose
i = a1>0_goto l1 & i = a2>0_goto l2;
hence it1 = it2 by A19,A21,Th8;
end;
end;
hereby
assume InsCode i = 9 or InsCode i = 10;
given a1, b1 being Int-Location, f1 being FinSeq-Location such that
A23: i = b1 := (f1, a1) or i = (f1, a1) := b1 and
A24: it1 = {a1, b1};
given a2, b2 being Int-Location, f2 being FinSeq-Location such that
A25: i = b2 := (f2, a2) or i = (f2, a2) := b2 and
A26: it2 = {a2, b2};
A27: i = (f1, a1) := b1 or i = (f2, a2) := b2 implies InsCode i = 10 by
SCMFSA_2:27;
per cases by A23,A25,A27,SCMFSA_2:26;
suppose
A28: i = b1 := (f1, a1) & i = b2 := (f2, a2);
then a1 = a2 by Th9;
hence it1 = it2 by A24,A26,A28,Th9;
end;
suppose
A29: i = (f1, a1) := b1 & i = (f2, a2) := b2;
then a1 = a2 by Th10;
hence it1 = it2 by A24,A26,A29,Th10;
end;
end;
hereby
assume InsCode i = 11 or InsCode i = 12;
given a1 being Int-Location, f1 being FinSeq-Location such that
A30: i = a1 :=len f1 or i = f1 :=<0,...,0>a1 and
A31: it1 = {a1};
given a2 being Int-Location, f2 being FinSeq-Location such that
A32: i = a2 :=len f2 or i = f2 :=<0,...,0>a2 and
A33: it2 = {a2};
A34: i = f1 :=<0,...,0>a1 or i = f2 :=<0,...,0>a2 implies InsCode i = 12
by SCMFSA_2:29;
per cases by A30,A32,A34,SCMFSA_2:28;
suppose
i = a1 :=len f1 & i = a2 :=len f2;
hence it1 = it2 by A31,A33,Th11;
end;
suppose
i = f1 :=<0,...,0>a1 & i = f2 :=<0,...,0>a2;
hence it1 = it2 by A31,A33,Th12;
end;
end;
thus thesis;
end;
consistency by ENUMSET1:def 3;
end;
theorem Th13:
UsedIntLoc halt SCM+FSA = {}
proof
A1: InsCode halt SCM+FSA = 0 by COMPOS_1:70;
not 0 in {1, 2, 3, 4, 5};
hence thesis by Def1,A1;
end;
theorem Th14:
i = a:=b or i = AddTo(a, b) or i = SubFrom(a, b) or i = MultBy(a
, b) or i = Divide(a, b) implies UsedIntLoc i = {a, b}
proof
reconsider ab = {a, b} as Element of Fin Int-Locations
by FINSUB_1:def 5;
assume
A1: i = a:=b or i = AddTo(a, b) or i = SubFrom(a, b) or i = MultBy(a, b)
or i = Divide(a, b);
then InsCode i = 1 or ... or InsCode i = 5 by SCMFSA_2:18,19,20,21,22;
then InsCode i in {1, 2, 3, 4, 5} by ENUMSET1:def 3;
then UsedIntLoc i = ab by A1,Def1;
hence thesis;
end;
theorem Th15:
UsedIntLoc goto l = {}
proof
InsCode goto l = 6 & not 6 in {1, 2, 3, 4, 5} by ENUMSET1:def 3,SCMFSA_2:23;
hence thesis by Def1;
end;
theorem Th16:
i = a=0_goto l or i = a>0_goto l implies UsedIntLoc i = {a}
proof
reconsider ab = {a} as Element of Fin Int-Locations by FINSUB_1:def 5;
assume
A1: i = a=0_goto l or i = a>0_goto l;
then InsCode i = 7 or InsCode i = 8 by SCMFSA_2:24,25;
then UsedIntLoc i = ab by A1,Def1;
hence thesis;
end;
theorem Th17:
i = b := (f, a) or i = (f, a) := b implies UsedIntLoc i = {a, b}
proof
reconsider ab = {a, b} as Element of Fin Int-Locations
by FINSUB_1:def 5;
assume
A1: i = b := (f, a) or i = (f, a) := b;
then InsCode i = 9 or InsCode i = 10 by SCMFSA_2:26,27;
then UsedIntLoc i = ab by A1,Def1;
hence thesis;
end;
theorem Th18:
i = a :=len f or i = f :=<0,...,0>a implies UsedIntLoc i = {a}
proof
reconsider ab = {a} as Element of Fin Int-Locations by FINSUB_1:def 5;
assume
A1: i = a :=len f or i = f :=<0,...,0>a;
then InsCode i = 11 or InsCode i = 12 by SCMFSA_2:28,29;
then UsedIntLoc i = ab by A1,Def1;
hence thesis;
end;
definition let X be set;
func UsedILoc X -> Subset of Int-Locations equals
union { UsedIntLoc i : i in X };
coherence
proof set A = { UsedIntLoc i : i in X };
union A c= Int-Locations
proof let e be object;
assume e in union A;
then consider B being set such that
A1: e in B and
A2: B in A by TARSKI:def 4;
ex i st B = UsedIntLoc i & i in X by A2;
then B c= Int-Locations by FINSUB_1:def 5;
hence thesis by A1;
end;
hence thesis;
end;
end;
registration
let X be finite set;
cluster UsedILoc X -> finite;
coherence
proof set A = { UsedIntLoc i : i in X };
A1: X is finite;
A2: A is finite from FRAENKEL:sch 21(A1);
for Y being set st Y in A holds Y is finite
proof let Y be set;
assume Y in A;
then ex i st Y = UsedIntLoc i & i in X;
hence Y is finite;
end;
hence thesis by A2,FINSET_1:7;
end;
end;
Lm1:
i in X implies UsedIntLoc i c= UsedILoc X
proof
assume
A1: i in X;
let e be object;
assume
A2: e in UsedIntLoc i;
UsedIntLoc i in { UsedIntLoc j : j in X } by A1;
hence e in UsedILoc X by A2,TARSKI:def 4;
end;
Lm2:
X c= Y implies UsedILoc X c= UsedILoc Y
proof assume
A1: X c= Y;
let e be object;
assume
e in UsedILoc X;
then consider B being set such that
A2: e in B and
A3: B in { UsedIntLoc i : i in X } by TARSKI:def 4;
consider i such that
A4: B = UsedIntLoc i and
A5: i in X by A3;
i in Y by A1,A5;
then B in { UsedIntLoc j : j in Y } by A4;
hence e in UsedILoc Y by A2, TARSKI:def 4;
end;
Lm3:
UsedILoc(X \/ Y) = UsedILoc X \/ UsedILoc Y
proof
thus UsedILoc(X \/ Y) c= UsedILoc X \/ UsedILoc Y
proof let e be object;
assume e in UsedILoc(X \/ Y);
then consider B being set such that
A1: e in B and
A2: B in { UsedIntLoc i : i in X \/ Y } by TARSKI:def 4;
consider i such that
A3: B = UsedIntLoc i and
A4: i in X \/ Y by A2;
now per cases by A4,XBOOLE_0:def 3;
case i in X;
then B in { UsedIntLoc j : j in X } by A3;
hence e in UsedILoc X by A1,TARSKI:def 4;
end;
case i in Y;
then B in { UsedIntLoc j : j in Y } by A3;
hence e in UsedILoc Y by A1,TARSKI:def 4;
end;
end;
hence e in UsedILoc X \/ UsedILoc Y by XBOOLE_0:def 3;
end;
X c= X \/ Y & Y c= X \/ Y by XBOOLE_1:7;
then UsedILoc X c= UsedILoc(X \/ Y) & UsedILoc Y c= UsedILoc(X \/ Y) by Lm2;
hence UsedILoc X \/ UsedILoc Y c= UsedILoc(X \/ Y) by XBOOLE_1:8;
end;
definition
let p be Function;
func UsedILoc p -> Subset of Int-Locations equals
UsedILoc rng p;
coherence;
:: ex UIL being
:: Function of the InstructionsF of SCM+FSA, Fin Int-Locations st (for i being
:: Instruction of SCM+FSA holds UIL.i = UsedIntLoc i) & it = Union (UIL * p);
end;
registration
let p be preProgram of SCM+FSA;
cluster UsedILoc p -> finite;
coherence;
end;
reserve p, r for preProgram of SCM+FSA,
I, J for Program of SCM+FSA,
k, m, n for Nat;
theorem Th19:
i in rng p implies UsedIntLoc i c= UsedILoc p by Lm1;
theorem
UsedILoc(p +* r) c= UsedILoc p \/ UsedILoc r
proof
rng(p +* r) c= rng p \/ rng r by FUNCT_4:17;
then UsedILoc(p +* r) c= UsedILoc(rng p \/ rng r) by Lm2;
hence UsedILoc(p +* r) c= UsedILoc p \/ UsedILoc r by Lm3;
end;
theorem Th21:
dom p misses dom r implies
UsedILoc(p +* r) = UsedILoc p \/ UsedILoc r
proof
assume dom p misses dom r;
then rng(p +* r) = rng p \/ rng r by NECKLACE:6;
then UsedILoc(p +* r) = UsedILoc(rng p \/ rng r);
hence UsedILoc(p +* r) = UsedILoc p \/ UsedILoc r by Lm3;
end;
theorem Th22:
UsedILoc p = UsedILoc Shift(p, k)
proof
dom p c= NAT;
then rng p = rng Shift(p, k) by VALUED_1:26;
hence thesis;
end;
theorem Th23:
UsedIntLoc i = UsedIntLoc IncAddr(i, k)
proof
InsCode i <= 12 by SCMFSA_2:16;
then InsCode i = 0 or ... or InsCode i = 12;
then per cases;
suppose
InsCode i = 0;
then i = halt SCM+FSA by SCMFSA_2:95;
hence thesis by COMPOS_0:4;
end;
suppose
InsCode i = 1;
then ex a, b st i = a:=b by SCMFSA_2:30;
hence thesis by COMPOS_0:4;
end;
suppose
InsCode i = 2;
then ex a, b st i = AddTo(a,b) by SCMFSA_2:31;
hence thesis by COMPOS_0:4;
end;
suppose
InsCode i = 3;
then ex a, b st i = SubFrom(a, b) by SCMFSA_2:32;
hence thesis by COMPOS_0:4;
end;
suppose
InsCode i = 4;
then ex a, b st i = MultBy(a, b) by SCMFSA_2:33;
hence thesis by COMPOS_0:4;
end;
suppose
InsCode i = 5;
then ex a, b st i = Divide(a, b) by SCMFSA_2:34;
hence thesis by COMPOS_0:4;
end;
suppose
InsCode i = 6;
then consider l such that
A1: i = goto l by SCMFSA_2:35;
IncAddr(i, k) = goto (l+k) by A1,SCMFSA_4:1;
hence UsedIntLoc IncAddr(i, k) = {} by Th15
.= UsedIntLoc i by A1,Th15;
end;
suppose
InsCode i = 7;
then consider l, a such that
A2: i = a=0_goto l by SCMFSA_2:36;
IncAddr(i, k) = a=0_goto (l+k) by A2,SCMFSA_4:2;
hence UsedIntLoc IncAddr(i, k) = {a} by Th16
.= UsedIntLoc i by A2,Th16;
end;
suppose
InsCode i = 8;
then consider l, a such that
A3: i = a>0_goto l by SCMFSA_2:37;
IncAddr(i, k) = a>0_goto (l+k) by A3,SCMFSA_4:3;
hence UsedIntLoc IncAddr(i, k) = {a} by Th16
.= UsedIntLoc i by A3,Th16;
end;
suppose
InsCode i = 9;
then ex a, b, f st i = b:=(f,a) by SCMFSA_2:38;
hence thesis by COMPOS_0:4;
end;
suppose
InsCode i = 10;
then ex a, b, f st i = (f,a):=b by SCMFSA_2:39;
hence thesis by COMPOS_0:4;
end;
suppose
InsCode i = 11;
then ex a, f st i = a:=len f by SCMFSA_2:40;
hence thesis by COMPOS_0:4;
end;
suppose
InsCode i = 12;
then ex a,f st i = f:=<0,...,0>a by SCMFSA_2:41;
hence thesis by COMPOS_0:4;
end;
end;
theorem Th24:
UsedILoc p = UsedILoc IncAddr(p, k)
proof
set A = { UsedIntLoc i : i in rng p },
B = { UsedIntLoc i : i in rng IncAddr(p,k) };
A1: A c= B
proof let e be object;
assume e in A;
then consider i such that
A2: e = UsedIntLoc i and
A3: i in rng p;
consider x being object such that
A4: x in dom p and
A5: p.x = i by A3,FUNCT_1:def 3;
A6: x in dom IncAddr(p,k) by A4,COMPOS_1:def 21;
IncAddr(p,k).x = IncAddr(p/.x,k) by A4,COMPOS_1:def 21
.= IncAddr(i,k) by A4,A5,PARTFUN1:def 6;
then
A7: IncAddr(i,k) in rng IncAddr(p,k) by A6,FUNCT_1:3;
e = UsedIntLoc IncAddr(i,k) by A2,Th23;
hence e in B by A7;
end;
B c= A
proof let e be object;
assume e in B;
then consider i such that
A8: e = UsedIntLoc i and
A9: i in rng IncAddr(p,k);
consider x being object such that
A10: x in dom IncAddr(p,k) and
A11: IncAddr(p,k).x = i by A9,FUNCT_1:def 3;
A12: x in dom p by A10,COMPOS_1:def 21;
p/.x = p.x by A12,PARTFUN1:def 6;
then
A13: p/.x in rng p by A12,FUNCT_1:3;
i = IncAddr(p/.x,k) by A12,COMPOS_1:def 21,A11;
then e = UsedIntLoc(p/.x) by A8,Th23;
hence e in A by A13;
end;
hence thesis by A1,XBOOLE_0:def 10;
end;
theorem Th25:
UsedILoc I = UsedILoc Reloc(I, k)
proof
A1: Reloc(I,k) = IncAddr(Shift(I,k),k) by COMPOS_1:34;
UsedILoc Reloc(I, k) = UsedILoc Shift (I, k) by Th24,A1
.= UsedILoc I by Th22;
hence thesis;
end;
theorem Th26:
UsedILoc I = UsedILoc Directed I
proof
set A = { UsedIntLoc i : i in rng I },
B = { UsedIntLoc i : i in rng Directed I };
A1: A c= B
proof let e be object;
assume e in A;
then consider i such that
A2: e = UsedIntLoc i and
A3: i in rng I;
consider x being object such that
A4: x in dom I and
A5: I.x = i by A3,FUNCT_1:def 3;
set j = (Directed I).x;
x in dom Directed I by A4,FUNCT_4:99;
then
A6: j in rng Directed I by FUNCT_1:3;
reconsider j as Instruction of SCM+FSA by A6;
now per cases;
suppose
A7: i = halt SCM+FSA;
then j = goto card I by A4,A5,FUNCT_4:106;
hence UsedIntLoc i = UsedIntLoc j by Th15,A7,Th13;
end;
suppose i <> halt SCM+FSA;
hence UsedIntLoc i = UsedIntLoc j by A5,FUNCT_4:105;
end;
end;
hence e in B by A2,A6;
end;
B c= A
proof let e be object;
assume e in B;
then consider i such that
A8: e = UsedIntLoc i and
A9: i in rng Directed I;
consider x being object such that
A10: x in dom Directed I and
A11: (Directed I).x = i by A9,FUNCT_1:def 3;
set j = I.x;
A12: x in dom I by A10,FUNCT_4:99;
then
A13: j in rng I by FUNCT_1:3;
reconsider j as Instruction of SCM+FSA by A13;
now per cases;
suppose
A14: j = halt SCM+FSA;
then
i = goto card I by A11,FUNCT_4:106,A12;
hence UsedIntLoc i = UsedIntLoc j by A14,Th13,Th15;
end;
suppose j <> halt SCM+FSA;
hence UsedIntLoc i = UsedIntLoc j by A11,FUNCT_4:105;
end;
end;
hence e in A by A8,A13;
end;
hence thesis by A1,XBOOLE_0:def 10;
end;
theorem Th27:
UsedILoc (I ";" J) = (UsedILoc I) \/ (UsedILoc J)
proof
A1: card(stop Directed I) -' 1 = card Directed I by COMPOS_1:71
.= card I by SCMFSA6A:36;
dom I = dom Directed I by FUNCT_4:99;
then
A2: dom Directed I misses dom (Reloc(J, card I)) by COMPOS_1:48;
thus UsedILoc (I ";" J)
= UsedILoc(stop Directed I ';' J)
.= UsedILoc Directed I \/ UsedILoc Reloc(J, card I) by A2,Th21,A1
.= (UsedILoc I) \/ UsedILoc Reloc(J, card I) by Th26
.= (UsedILoc I) \/ (UsedILoc J) by Th25;
end;
theorem Th28:
UsedILoc Macro i = UsedIntLoc i
proof
rng Macro i = {i, halt SCM+FSA} by COMPOS_1:67;
hence UsedILoc Macro i
= union { UsedIntLoc j : j in {i, halt SCM+FSA}}
.= UsedIntLoc i \/ UsedIntLoc halt SCM+FSA from SUBSET_1:sch 6
.= UsedIntLoc i by Th13;
end;
theorem
UsedILoc (i ";" J) = (UsedIntLoc i) \/ UsedILoc J
proof
thus UsedILoc (i ";" J) = (UsedILoc Macro i) \/ UsedILoc J by Th27
.= (UsedIntLoc i) \/ UsedILoc J by Th28;
end;
theorem
UsedILoc (I ";" j) = (UsedILoc I) \/ UsedIntLoc j
proof
thus UsedILoc (I ";" j) = (UsedILoc I) \/ UsedILoc Macro j by Th27
.= (UsedILoc I) \/ UsedIntLoc j by Th28;
end;
theorem
UsedILoc (i ";" j) = (UsedIntLoc i) \/ UsedIntLoc j
proof
thus UsedILoc (i ";" j) = (UsedILoc Macro i) \/ UsedILoc Macro j by
Th27
.= (UsedILoc Macro i) \/ UsedIntLoc j by Th28
.= (UsedIntLoc i) \/ UsedIntLoc j by Th28;
end;
begin :: Finite sequence locations used in macro instructions
definition
let i be Instruction of SCM+FSA;
func UsedInt*Loc i -> Element of Fin FinSeq-Locations means
:Def4:
ex a, b
being Int-Location, f being FinSeq-Location st (i = b := (f, a) or i = (f, a)
:= b) & it = {f} if InsCode i = 9 or InsCode i = 10, ex a being Int-Location, f
being FinSeq-Location st (i = a :=len f or i = f :=<0,...,0>a) & it = {f} if
InsCode i = 11 or InsCode i = 12 otherwise it = {};
existence
proof
hereby
assume InsCode i = 9 or InsCode i = 10;
then consider a, b being Int-Location, f being FinSeq-Location such that
A1: i = b := (f, a) or i = (f, a) := b by SCMFSA_2:38,39;
reconsider f9 = f as Element of FinSeq-Locations by SCMFSA_2:def 5;
reconsider IT = {.f9.} as Element of Fin FinSeq-Locations;
take IT;
take a, b, f;
thus (i = b := (f, a) or i = (f, a) := b) & IT = {f} by A1;
end;
hereby
assume InsCode i = 11 or InsCode i = 12;
then consider a being Int-Location, f being FinSeq-Location such that
A2: i = a :=len f or i = f :=<0,...,0>a by SCMFSA_2:40,41;
reconsider f9 = f as Element of FinSeq-Locations by SCMFSA_2:def 5;
reconsider IT = {.f9.} as Element of Fin FinSeq-Locations;
take IT;
take a, f;
thus (i = a :=len f or i = f :=<0,...,0>a) & IT = {f} by A2;
end;
{} in Fin FinSeq-Locations by FINSUB_1:7;
hence thesis;
end;
uniqueness
proof
let it1, it2 be Element of Fin FinSeq-Locations;
hereby
assume InsCode i = 9 or InsCode i = 10;
given a1, b1 being Int-Location, f1 being FinSeq-Location such that
A3: i = b1 := (f1, a1) or i = (f1, a1) := b1 and
A4: it1 = {f1};
given a2, b2 being Int-Location, f2 being FinSeq-Location such that
A5: i = b2 := (f2, a2) or i = (f2, a2) := b2 and
A6: it2 = {f2};
A7: i = (f1, a1) := b1 or i = (f2, a2) := b2 implies InsCode i = 10 by
SCMFSA_2:27;
per cases by A3,A5,A7,SCMFSA_2:26;
suppose
i = b1 := (f1, a1) & i = b2 := (f2, a2);
hence it1 = it2 by A4,A6,Th9;
end;
suppose
i = (f1, a1) := b1 & i = (f2, a2) := b2;
hence it1 = it2 by A4,A6,Th10;
end;
end;
hereby
assume InsCode i = 11 or InsCode i = 12;
given a1 being Int-Location, f1 being FinSeq-Location such that
A8: i = a1 :=len f1 or i = f1 :=<0,...,0>a1 and
A9: it1 = {f1};
given a2 being Int-Location, f2 being FinSeq-Location such that
A10: i = a2 :=len f2 or i = f2 :=<0,...,0>a2 and
A11: it2 = {f2};
A12: i = f1 :=<0,...,0>a1 or i = f2 :=<0,...,0>a2 implies InsCode i = 12
by SCMFSA_2:29;
per cases by A8,A10,A12,SCMFSA_2:28;
suppose
i = a1 :=len f1 & i = a2 :=len f2;
hence it1 = it2 by A9,A11,Th11;
end;
suppose
i = f1 :=<0,...,0>a1 & i = f2 :=<0,...,0>a2;
hence it1 = it2 by A9,A11,Th12;
end;
end;
thus thesis;
end;
consistency;
end;
theorem Th32:
i = halt SCM+FSA or i = a:=b or i = AddTo(a, b) or i = SubFrom(a
, b) or i = MultBy(a, b) or i = Divide(a, b) or i = goto l or i = a=0_goto l or
i = a>0_goto l implies UsedInt*Loc i = {}
proof
assume i = halt SCM+FSA or i = a:=b or i = AddTo(a, b) or i = SubFrom(a, b)
or i = MultBy(a, b) or i = Divide(a, b) or i = goto l or i = a=0_goto l or i =
a>0_goto l;
then InsCode i = 0 or ... or InsCode i =
8 by COMPOS_1:70,SCMFSA_2:18,19,20,21,22,23,24,25;
hence thesis by Def4;
end;
theorem Th33:
i = b := (f, a) or i = (f, a) := b implies UsedInt*Loc i = {f}
proof
f in FinSeq-Locations by SCMFSA_2:def 5;
then {f} c= FinSeq-Locations by ZFMISC_1:31;
then reconsider ab = {f} as Element of Fin FinSeq-Locations by FINSUB_1:def 5
;
assume
A1: i = b := (f, a) or i = (f, a) := b;
then InsCode i = 9 or InsCode i = 10 by SCMFSA_2:26,27;
then UsedInt*Loc i = ab by A1,Def4;
hence thesis;
end;
theorem Th34:
i = a :=len f or i = f :=<0,...,0>a implies UsedInt*Loc i = {f}
proof
f in FinSeq-Locations by SCMFSA_2:def 5;
then {f} c= FinSeq-Locations by ZFMISC_1:31;
then reconsider ab = {f} as Element of Fin FinSeq-Locations by FINSUB_1:def 5
;
assume
A1: i = a :=len f or i = f :=<0,...,0>a;
then InsCode i = 11 or InsCode i = 12 by SCMFSA_2:28,29;
then UsedInt*Loc i = ab by A1,Def4;
hence thesis;
end;
definition let X be set;
func UsedI*Loc X -> Subset of FinSeq-Locations equals
union { UsedInt*Loc i : i in X };
coherence
proof set A = { UsedInt*Loc i : i in X };
union A c= FinSeq-Locations
proof let e be object;
assume e in union A;
then consider B being set such that
A1: e in B and
A2: B in A by TARSKI:def 4;
ex i st B = UsedInt*Loc i & i in X by A2;
then B c= FinSeq-Locations by FINSUB_1:def 5;
hence thesis by A1;
end;
hence thesis;
end;
end;
Lm4:
i in X implies UsedInt*Loc i c= UsedI*Loc X
proof
assume
A1: i in X;
let e be object;
assume
A2: e in UsedInt*Loc i;
UsedInt*Loc i in { UsedInt*Loc j : j in X } by A1;
hence e in UsedI*Loc X by A2,TARSKI:def 4;
end;
Lm5:
X c= Y implies UsedI*Loc X c= UsedI*Loc Y
proof assume
A1: X c= Y;
let e be object;
assume
e in UsedI*Loc X;
then consider B being set such that
A2: e in B and
A3: B in { UsedInt*Loc i : i in X } by TARSKI:def 4;
consider i such that
A4: B = UsedInt*Loc i and
A5: i in X by A3;
i in Y by A1,A5;
then B in { UsedInt*Loc j : j in Y } by A4;
hence e in UsedI*Loc Y by A2, TARSKI:def 4;
end;
Lm6:
UsedI*Loc(X \/ Y) = UsedI*Loc X \/ UsedI*Loc Y
proof
thus UsedI*Loc(X \/ Y) c= UsedI*Loc X \/ UsedI*Loc Y
proof let e be object;
assume e in UsedI*Loc(X \/ Y);
then consider B being set such that
A1: e in B and
A2: B in { UsedInt*Loc i : i in X \/ Y } by TARSKI:def 4;
consider i such that
A3: B = UsedInt*Loc i and
A4: i in X \/ Y by A2;
now per cases by A4,XBOOLE_0:def 3;
case i in X;
then B in { UsedInt*Loc j : j in X } by A3;
hence e in UsedI*Loc X by A1,TARSKI:def 4;
end;
case i in Y;
then B in { UsedInt*Loc j : j in Y } by A3;
hence e in UsedI*Loc Y by A1,TARSKI:def 4;
end;
end;
hence e in UsedI*Loc X \/ UsedI*Loc Y by XBOOLE_0:def 3;
end;
X c= X \/ Y & Y c= X \/ Y by XBOOLE_1:7;
then UsedI*Loc X c= UsedI*Loc(X \/ Y) & UsedI*Loc Y c= UsedI*Loc(X \/ Y)
by Lm5;
hence UsedI*Loc X \/ UsedI*Loc Y c= UsedI*Loc(X \/ Y) by XBOOLE_1:8;
end;
definition
let p be Function;
func UsedI*Loc p -> Subset of FinSeq-Locations equals
UsedI*Loc rng p;
coherence;
end;
registration
let X be finite set;
cluster UsedI*Loc X -> finite;
coherence
proof set A = { UsedInt*Loc i : i in X };
A1: X is finite;
A2: A is finite from FRAENKEL:sch 21(A1);
for Y being set st Y in A holds Y is finite
proof let Y be set;
assume Y in A;
then ex i st Y = UsedInt*Loc i & i in X;
hence Y is finite;
end;
hence thesis by A2,FINSET_1:7;
end;
end;
registration
let p be preProgram of SCM+FSA;
cluster UsedI*Loc p -> finite;
coherence;
end;
theorem Th35:
i in rng p implies UsedInt*Loc i c= UsedI*Loc p by Lm4;
theorem
UsedI*Loc (p +* r) c= UsedI*Loc p \/ UsedI*Loc r
proof
rng(p +* r) c= rng p \/ rng r by FUNCT_4:17;
then UsedI*Loc(p +* r) c= UsedI*Loc(rng p \/ rng r) by Lm5;
hence UsedI*Loc(p +* r) c= UsedI*Loc p \/ UsedI*Loc r by Lm6;
end;
theorem Th37:
dom p misses dom r implies UsedI*Loc (p +* r) = (UsedI*Loc p
) \/ (UsedI*Loc r)
proof
assume dom p misses dom r;
then rng(p +* r) = rng p \/ rng r by NECKLACE:6;
then UsedI*Loc(p +* r) = UsedI*Loc(rng p \/ rng r);
hence UsedI*Loc(p +* r) = UsedI*Loc p \/ UsedI*Loc r by Lm6;
end;
theorem Th38:
UsedI*Loc p = UsedI*Loc Shift(p, k)
proof
dom p c= NAT;
then rng p = rng Shift(p, k) by VALUED_1:26;
hence thesis;
end;
theorem Th39:
UsedInt*Loc i = UsedInt*Loc IncAddr(i, k)
proof
InsCode i <= 12 by SCMFSA_2:16;
then InsCode i = 0 or ... or InsCode i = 12;
then per cases;
suppose
InsCode i = 0;
then i = halt SCM+FSA by SCMFSA_2:95;
hence thesis by COMPOS_0:4;
end;
suppose
InsCode i = 1;
then ex a, b st i = a:=b by SCMFSA_2:30;
hence thesis by COMPOS_0:4;
end;
suppose
InsCode i = 2;
then ex a, b st i = AddTo(a,b) by SCMFSA_2:31;
hence thesis by COMPOS_0:4;
end;
suppose
InsCode i = 3;
then ex a, b st i = SubFrom(a, b) by SCMFSA_2:32;
hence thesis by COMPOS_0:4;
end;
suppose
InsCode i = 4;
then ex a, b st i = MultBy(a, b) by SCMFSA_2:33;
hence thesis by COMPOS_0:4;
end;
suppose
InsCode i = 5;
then ex a, b st i = Divide(a, b) by SCMFSA_2:34;
hence thesis by COMPOS_0:4;
end;
suppose
InsCode i = 6;
then consider l such that
A1: i = goto l by SCMFSA_2:35;
IncAddr(i, k) = goto (l+k) by A1,SCMFSA_4:1;
hence UsedInt*Loc IncAddr(i, k) = {} by Th32
.= UsedInt*Loc i by A1,Th32;
end;
suppose
InsCode i = 7;
then consider l, a such that
A2: i = a=0_goto l by SCMFSA_2:36;
IncAddr(i, k) = a=0_goto (l+k) by A2,SCMFSA_4:2;
hence UsedInt*Loc IncAddr(i, k) = {} by Th32
.= UsedInt*Loc i by A2,Th32;
end;
suppose
InsCode i = 8;
then consider l, a such that
A3: i = a>0_goto l by SCMFSA_2:37;
IncAddr(i, k) = a>0_goto (l+k) by A3,SCMFSA_4:3;
hence UsedInt*Loc IncAddr(i, k) = {} by Th32
.= UsedInt*Loc i by A3,Th32;
end;
suppose
InsCode i = 9;
then ex a, b, f st i = b:=(f,a) by SCMFSA_2:38;
hence thesis by COMPOS_0:4;
end;
suppose
InsCode i = 10;
then ex a, b, f st i = (f,a):=b by SCMFSA_2:39;
hence thesis by COMPOS_0:4;
end;
suppose
InsCode i = 11;
then ex a, f st i = a:=len f by SCMFSA_2:40;
hence thesis by COMPOS_0:4;
end;
suppose
InsCode i = 12;
then ex a,f st i = f:=<0,...,0>a by SCMFSA_2:41;
hence thesis by COMPOS_0:4;
end;
end;
theorem Th40:
UsedI*Loc p = UsedI*Loc IncAddr(p, k)
proof
set A = { UsedInt*Loc i : i in rng p },
B = { UsedInt*Loc i : i in rng IncAddr(p,k) };
A1: A c= B
proof let e be object;
assume e in A;
then consider i such that
A2: e = UsedInt*Loc i and
A3: i in rng p;
consider x being object such that
A4: x in dom p and
A5: p.x = i by A3,FUNCT_1:def 3;
A6: x in dom IncAddr(p,k) by A4,COMPOS_1:def 21;
IncAddr(p,k).x = IncAddr(p/.x,k) by A4,COMPOS_1:def 21
.= IncAddr(i,k) by A4,A5,PARTFUN1:def 6;
then
A7: IncAddr(i,k) in rng IncAddr(p,k) by A6,FUNCT_1:3;
e = UsedInt*Loc IncAddr(i,k) by A2,Th39;
hence e in B by A7;
end;
B c= A
proof let e be object;
assume e in B;
then consider i such that
A8: e = UsedInt*Loc i and
A9: i in rng IncAddr(p,k);
consider x being object such that
A10: x in dom IncAddr(p,k) and
A11: IncAddr(p,k).x = i by A9,FUNCT_1:def 3;
A12: x in dom p by A10,COMPOS_1:def 21;
p/.x = p.x by A12,PARTFUN1:def 6;
then
A13: p/.x in rng p by A12,FUNCT_1:3;
i = IncAddr(p/.x,k) by A12,COMPOS_1:def 21,A11;
then e = UsedInt*Loc(p/.x) by A8,Th39;
hence e in A by A13;
end;
hence thesis by A1,XBOOLE_0:def 10;
end;
theorem Th41:
UsedI*Loc I = UsedI*Loc Reloc(I, k)
proof
A1: Reloc(I,k) = IncAddr(Shift(I,k),k) by COMPOS_1:34;
UsedI*Loc Reloc(I, k) = UsedI*Loc Reloc(I,k)
.= UsedI*Loc Shift (I, k) by Th40,A1
.= UsedI*Loc I by Th38;
hence thesis;
end;
theorem Th42:
UsedI*Loc I = UsedI*Loc Directed I
proof
set A = { UsedInt*Loc i : i in rng I },
B = { UsedInt*Loc i : i in rng Directed I };
A1: A c= B
proof let e be object;
assume e in A;
then consider i such that
A2: e = UsedInt*Loc i and
A3: i in rng I;
consider x being object such that
A4: x in dom I and
A5: I.x = i by A3,FUNCT_1:def 3;
set j = (Directed I).x;
x in dom Directed I by A4,FUNCT_4:99;
then
A6: j in rng Directed I by FUNCT_1:3;
reconsider j as Instruction of SCM+FSA by A6;
now per cases;
suppose
A7: i = halt SCM+FSA;
then
A8: j = goto card I by A4,A5,FUNCT_4:106;
thus UsedInt*Loc i = {} by A7,Th32
.= UsedInt*Loc j by A8,Th32;
end;
suppose i <> halt SCM+FSA;
hence UsedInt*Loc i = UsedInt*Loc j by A5,FUNCT_4:105;
end;
end;
hence e in B by A2,A6;
end;
B c= A
proof let e be object;
assume e in B;
then consider i such that
A9: e = UsedInt*Loc i and
A10: i in rng Directed I;
consider x being object such that
A11: x in dom Directed I and
A12: (Directed I).x = i by A10,FUNCT_1:def 3;
set j = I.x;
A13: x in dom I by A11,FUNCT_4:99;
then
A14: j in rng I by FUNCT_1:3;
reconsider j as Instruction of SCM+FSA by A14;
now per cases;
suppose
A15: j = halt SCM+FSA;
then
i = goto card I by A12,FUNCT_4:106,A13;
hence UsedInt*Loc i = {} by Th32
.= UsedInt*Loc j by A15,Th32;
end;
suppose j <> halt SCM+FSA;
hence UsedInt*Loc i = UsedInt*Loc j by A12,FUNCT_4:105;
end;
end;
hence e in A by A9,A14;
end;
hence thesis by A1,XBOOLE_0:def 10;
end;
theorem Th43:
UsedI*Loc (I ";" J) = UsedI*Loc I \/ UsedI*Loc J
proof
A1: card(stop Directed I) -' 1 = card Directed I by COMPOS_1:71
.= card I by SCMFSA6A:36;
dom I = dom Directed I by FUNCT_4:99;
then
A2: dom Directed I misses dom (Reloc(J, card I)) by COMPOS_1:48;
thus UsedI*Loc (I ";" J)
= UsedI*Loc(stop Directed I ';' J)
.= UsedI*Loc Directed I \/ UsedI*Loc Reloc(J, card I) by A2,Th37,A1
.= (UsedI*Loc I) \/ UsedI*Loc Reloc(J, card I) by Th42
.= (UsedI*Loc I) \/ (UsedI*Loc J) by Th41;
end;
theorem Th44:
UsedI*Loc Macro i = UsedInt*Loc i
proof
rng Macro i = {i, halt SCM+FSA} by COMPOS_1:67;
hence UsedI*Loc Macro i
= union { UsedInt*Loc j : j in {i, halt SCM+FSA}}
.= UsedInt*Loc i \/ UsedInt*Loc halt SCM+FSA from SUBSET_1:sch 6
.= UsedInt*Loc i \/ {} by Th32
.= UsedInt*Loc i;
end;
theorem
UsedI*Loc (i ";" J) = (UsedInt*Loc i) \/ UsedI*Loc J
proof
thus UsedI*Loc (i ";" J) = (UsedI*Loc Macro i) \/ UsedI*Loc J by Th43
.= (UsedInt*Loc i) \/ UsedI*Loc J by Th44;
end;
theorem
UsedI*Loc (I ";" j) = (UsedI*Loc I) \/ UsedInt*Loc j
proof
thus UsedI*Loc (I ";" j) = (UsedI*Loc I) \/ UsedI*Loc Macro j by Th43
.= (UsedI*Loc I) \/ UsedInt*Loc j by Th44;
end;
theorem
UsedI*Loc (i ";" j) = (UsedInt*Loc i) \/ UsedInt*Loc j
proof
thus UsedI*Loc (i ";" j) = (UsedI*Loc Macro i) \/ UsedI*Loc Macro j by
Th43
.= (UsedI*Loc Macro i) \/ UsedInt*Loc j by Th44
.= (UsedInt*Loc i) \/ UsedInt*Loc j by Th44;
end;
begin :: Choosing integer location not used in a macro instruction
reserve L for finite Subset of Int-Locations;
::$CD
theorem
not FirstNotIn L in L by SCMFSA_M:14;
theorem
FirstNotIn L = intloc m & not intloc n in L implies m <= n by SCMFSA_M:15;
definition
let p be preProgram of SCM+FSA;
func FirstNotUsed p -> Int-Location means
:Def7:
ex sil being finite Subset
of Int-Locations st sil = UsedILoc p \/ {intloc 0} & it = FirstNotIn sil;
existence
proof
reconsider i0 = {intloc 0} as finite Subset of Int-Locations;
reconsider sil = UsedILoc p \/ i0 as finite Subset of Int-Locations;
take FirstNotIn sil, sil;
thus thesis;
end;
uniqueness;
end;
registration
let p be preProgram of SCM+FSA;
cluster FirstNotUsed p -> read-write;
coherence
proof
consider sil being finite Subset of Int-Locations such that
A1: sil = UsedILoc p \/ {intloc 0} and
A2: FirstNotUsed p = FirstNotIn sil by Def7;
now
assume FirstNotIn sil = intloc 0;
then not intloc 0 in sil by SCMFSA_M:14;
hence contradiction by A1,ZFMISC_1:136;
end;
hence thesis by A2,SCMFSA_M:def 2;
end;
end;
theorem Th50:
not FirstNotUsed p in UsedILoc p
proof
consider sil being finite Subset of Int-Locations such that
A1: sil = UsedILoc p \/ {intloc 0} and
A2: FirstNotUsed p = FirstNotIn sil by Def7;
not FirstNotUsed p in sil by A2,SCMFSA_M:14;
hence thesis by A1,XBOOLE_0:def 3;
end;
theorem
a:=b in rng p or AddTo(a, b) in rng p or SubFrom(a, b) in rng p or
MultBy(a, b) in rng p or Divide(a, b) in rng p implies FirstNotUsed p <> a &
FirstNotUsed p <> b
proof
assume a:=b in rng p or AddTo(a, b) in rng p or SubFrom(a, b) in rng p or
MultBy(a, b) in rng p or Divide(a, b) in rng p;
then consider i being Instruction of SCM+FSA such that
A1: i in rng p and
A2: i = a:=b or i = AddTo(a, b) or i = SubFrom(a, b) or i = MultBy(a, b
) or i = Divide(a, b);
UsedIntLoc i = {a, b} by A2,Th14;
then
A3: {a, b} c= UsedILoc p by A1,Th19;
not FirstNotUsed p in UsedILoc p by Th50;
hence thesis by A3,ZFMISC_1:32;
end;
theorem
a=0_goto l in rng p or a>0_goto l in rng p implies FirstNotUsed p <> a
proof
assume a=0_goto l in rng p or a>0_goto l in rng p;
then consider i being Instruction of SCM+FSA such that
A1: i in rng p and
A2: i = a=0_goto l or i = a>0_goto l;
UsedIntLoc i = {a} by A2,Th16;
then
A3: {a} c= UsedILoc p by A1,Th19;
not FirstNotUsed p in UsedILoc p by Th50;
hence thesis by A3,ZFMISC_1:31;
end;
theorem
b := (f, a) in rng p or (f, a) := b in rng p implies FirstNotUsed p <>
a & FirstNotUsed p <> b
proof
assume b := (f, a) in rng p or (f, a) := b in rng p;
then consider i being Instruction of SCM+FSA such that
A1: i in rng p and
A2: i = b := (f, a) or i = (f, a) := b;
UsedIntLoc i = {a, b} by A2,Th17;
then
A3: {a, b} c= UsedILoc p by A1,Th19;
not FirstNotUsed p in UsedILoc p by Th50;
hence thesis by A3,ZFMISC_1:32;
end;
theorem
a :=len f in rng p or f :=<0,...,0>a in rng p implies FirstNotUsed p
<> a
proof
assume a :=len f in rng p or f :=<0,...,0>a in rng p;
then consider i being Instruction of SCM+FSA such that
A1: i in rng p and
A2: i = a :=len f or i = f :=<0,...,0>a;
UsedIntLoc i = {a} by A2,Th18;
then
A3: {a} c= UsedILoc p by A1,Th19;
not FirstNotUsed p in UsedILoc p by Th50;
hence thesis by A3,ZFMISC_1:31;
end;
begin :: Choosing finite sequence location not used in a macro instruction
reserve L for finite Subset of FinSeq-Locations;
definition
::$CD
end;
theorem
not First*NotIn L in L by SCMFSA_M:16;
theorem
First*NotIn L = fsloc m & not fsloc n in L implies m <= n by SCMFSA_M:17;
definition
let p be preProgram of SCM+FSA;
func First*NotUsed p -> FinSeq-Location means
:Def8:
ex sil being finite
Subset of FinSeq-Locations st sil = UsedI*Loc p & it = First*NotIn sil;
existence
proof
take First*NotIn UsedI*Loc p, UsedI*Loc p;
thus thesis;
end;
uniqueness;
end;
theorem Th57:
not First*NotUsed p in UsedI*Loc p
proof
ex sil being finite Subset of FinSeq-Locations st sil = UsedI*Loc p &
First*NotUsed p = First*NotIn sil by Def8;
hence thesis by SCMFSA_M:16;
end;
theorem
b := (f, a) in rng p or (f, a) := b in rng p implies First*NotUsed p
<> f
proof
assume b := (f, a) in rng p or (f, a) := b in rng p;
then consider i being Instruction of SCM+FSA such that
A1: i in rng p and
A2: i = b := (f, a) or i = (f, a) := b;
UsedInt*Loc i = {f} by A2,Th33;
then
A3: {f} c= UsedI*Loc p by A1,Th35;
not First*NotUsed p in UsedI*Loc p by Th57;
hence thesis by A3,ZFMISC_1:31;
end;
theorem
a :=len f in rng p or f :=<0,...,0>a in rng p implies First*NotUsed p
<> f
proof
assume a :=len f in rng p or f :=<0,...,0>a in rng p;
then consider i being Instruction of SCM+FSA such that
A1: i in rng p and
A2: i = a :=len f or i = f :=<0,...,0>a;
UsedInt*Loc i = {f} by A2,Th34;
then
A3: {f} c= UsedI*Loc p by A1,Th35;
not First*NotUsed p in UsedI*Loc p by Th57;
hence thesis by A3,ZFMISC_1:31;
end;
begin :: Semantics
reserve s, t for State of SCM+FSA;
reserve P for Instruction-Sequence of SCM+FSA;
theorem Th60:
not c in UsedIntLoc i implies Exec(i, s).c = s.c
proof
assume
A1: not c in UsedIntLoc i;
InsCode i <= 12 by SCMFSA_2:16;
then InsCode i = 0 or ... or InsCode i = 12;
then per cases;
suppose
InsCode i = 0;
then i = halt SCM+FSA by SCMFSA_2:95;
hence thesis by EXTPRO_1:def 3;
end;
suppose
InsCode i = 1;
then consider a, b such that
A2: i = a:=b by SCMFSA_2:30;
UsedIntLoc i = {a, b} by A2,Th14;
then c <> a by A1,TARSKI:def 2;
hence thesis by A2,SCMFSA_2:63;
end;
suppose
InsCode i = 2;
then consider a, b such that
A3: i = AddTo(a,b) by SCMFSA_2:31;
UsedIntLoc i = {a, b} by A3,Th14;
then c <> a by A1,TARSKI:def 2;
hence thesis by A3,SCMFSA_2:64;
end;
suppose
InsCode i = 3;
then consider a, b such that
A4: i = SubFrom(a, b) by SCMFSA_2:32;
UsedIntLoc i = {a, b} by A4,Th14;
then c <> a by A1,TARSKI:def 2;
hence thesis by A4,SCMFSA_2:65;
end;
suppose
InsCode i = 4;
then consider a, b such that
A5: i = MultBy(a, b) by SCMFSA_2:33;
UsedIntLoc i = {a, b} by A5,Th14;
then c <> a by A1,TARSKI:def 2;
hence thesis by A5,SCMFSA_2:66;
end;
suppose
InsCode i = 5;
then consider a, b such that
A6: i = Divide(a, b) by SCMFSA_2:34;
UsedIntLoc i = {a, b} by A6,Th14;
then c <> a & c <> b by A1,TARSKI:def 2;
hence thesis by A6,SCMFSA_2:67;
end;
suppose
InsCode i = 6;
then ex l st i = goto l by SCMFSA_2:35;
hence thesis by SCMFSA_2:69;
end;
suppose
InsCode i = 7;
then ex l, a st i = a=0_goto l by SCMFSA_2:36;
hence thesis by SCMFSA_2:70;
end;
suppose
InsCode i = 8;
then ex l, a st i = a>0_goto l by SCMFSA_2:37;
hence thesis by SCMFSA_2:71;
end;
suppose
InsCode i = 9;
then consider a, b, f such that
A7: i = b:=(f,a) by SCMFSA_2:38;
UsedIntLoc i = {a, b} by A7,Th17;
then c <> b by A1,TARSKI:def 2;
hence thesis by A7,SCMFSA_2:72;
end;
suppose
InsCode i = 10;
then ex a, b, f st i = (f,a):=b by SCMFSA_2:39;
hence thesis by SCMFSA_2:73;
end;
suppose
InsCode i = 11;
then consider a, f such that
A8: i = a:=len f by SCMFSA_2:40;
UsedIntLoc i = {a} by A8,Th18;
then c <> a by A1,TARSKI:def 1;
hence thesis by A8,SCMFSA_2:74;
end;
suppose
InsCode i = 12;
then ex a,f st i = f:=<0,...,0>a by SCMFSA_2:41;
hence thesis by SCMFSA_2:75;
end;
end;
theorem
I c= P &
(for m st m < n holds IC Comput(P,s,m) in dom I) &
not a in UsedILoc I implies Comput(P,s,n).a = s.a
proof
assume that
A1: I c= P and
A2: for m st m < n holds IC Comput(P,s,m) in dom I and
A3: not a in UsedILoc I;
defpred X[Nat] means $1 <= n implies Comput(P,s,$1).a = s.a;
A4: for m st X[m] holds X[m+1]
proof
let m;
set sm = Comput(P,s,m);
assume
A5: m <= n implies sm.a = s.a;
assume
A6: m+1 <= n;
then m < n by NAT_1:13;
then
A7: IC sm in dom I by A2;
then
A8: I.IC sm in rng I by FUNCT_1:def 3;
dom P = NAT by PARTFUN1:def 2;
then
A9: (P)/.IC sm = P.IC sm by PARTFUN1:def 6;
I.IC sm = P.IC sm by A7,A1,GRFUNC_1:2;
then UsedIntLoc P.IC sm c= UsedILoc I by A8,Th19;
then
A10: not a in UsedIntLoc P.IC sm by A3;
thus Comput(P,s,m+1).a = (Following(P,sm)).a by EXTPRO_1:3
.= s.a by A5,A6,A10,Th60,A9,NAT_1:13;
end;
A11: X[0];
for m holds X[m] from NAT_1:sch 2(A11, A4);
hence thesis;
end;
theorem Th62:
not f in UsedInt*Loc i implies Exec(i, s).f = s.f
proof
assume
A1: not f in UsedInt*Loc i;
InsCode i <= 12 by SCMFSA_2:16;
then InsCode i = 0 or ... or InsCode i = 12;
then per cases;
suppose
InsCode i = 0;
then i = halt SCM+FSA by SCMFSA_2:95;
hence thesis by EXTPRO_1:def 3;
end;
suppose
InsCode i = 1;
then ex a, b st i = a:=b by SCMFSA_2:30;
hence thesis by SCMFSA_2:63;
end;
suppose
InsCode i = 2;
then ex a, b st i = AddTo(a,b) by SCMFSA_2:31;
hence thesis by SCMFSA_2:64;
end;
suppose
InsCode i = 3;
then ex a, b st i = SubFrom(a, b) by SCMFSA_2:32;
hence thesis by SCMFSA_2:65;
end;
suppose
InsCode i = 4;
then ex a, b st i = MultBy(a, b) by SCMFSA_2:33;
hence thesis by SCMFSA_2:66;
end;
suppose
InsCode i = 5;
then ex a, b st i = Divide(a, b) by SCMFSA_2:34;
hence thesis by SCMFSA_2:67;
end;
suppose
InsCode i = 6;
then ex l st i = goto l by SCMFSA_2:35;
hence thesis by SCMFSA_2:69;
end;
suppose
InsCode i = 7;
then ex l, a st i = a=0_goto l by SCMFSA_2:36;
hence thesis by SCMFSA_2:70;
end;
suppose
InsCode i = 8;
then ex l, a st i = a>0_goto l by SCMFSA_2:37;
hence thesis by SCMFSA_2:71;
end;
suppose
InsCode i = 9;
then ex a, b, g st i = b:=(g,a) by SCMFSA_2:38;
hence thesis by SCMFSA_2:72;
end;
suppose
InsCode i = 10;
then consider a, b, g such that
A2: i = (g,a):=b by SCMFSA_2:39;
UsedInt*Loc i = {g} by A2,Th33;
then f <> g by A1,TARSKI:def 1;
hence thesis by A2,SCMFSA_2:73;
end;
suppose
InsCode i = 11;
then ex a, g st i = a:=len g by SCMFSA_2:40;
hence thesis by SCMFSA_2:74;
end;
suppose
InsCode i = 12;
then consider a,g such that
A3: i = g:=<0,...,0>a by SCMFSA_2:41;
UsedInt*Loc i = {g} by A3,Th34;
then f <> g by A1,TARSKI:def 1;
hence thesis by A3,SCMFSA_2:75;
end;
end;
theorem
I c= P &
(for m st m < n holds IC Comput(P,s,m) in dom I) &
not f in UsedI*Loc I implies Comput(P,s,n).f = s.f
proof
assume that
A1: I c= P and
A2: for m st m < n holds IC Comput(P,s,m) in dom I and
A3: not f in UsedI*Loc I;
defpred X[Nat] means $1 <= n implies Comput(P,s,$1).f = s.f;
A4: for m st X[m] holds X[m+1]
proof
let m;
set sm = Comput(P,s,m);
assume
A5: m <= n implies sm.f = s.f;
assume
A6: m+1 <= n;
then m < n by NAT_1:13;
then
A7: IC sm in dom I by A2;
then
A8: I.IC sm in rng I by FUNCT_1:def 3;
dom P = NAT by PARTFUN1:def 2;
then
A9: (P)/.IC sm
= P.IC sm by PARTFUN1:def 6;
I.IC sm = P.IC sm by A7,A1,GRFUNC_1:2;
then UsedInt*Loc P.IC sm c= UsedI*Loc I by A8,Th35;
then
A10: not f in UsedInt*Loc P.IC sm by A3;
thus Comput(P,s,m+1).f = (Following(P,sm)).f by EXTPRO_1:3
.= s.f by A5,A6,A10,Th62,A9,NAT_1:13;
end;
A11: X[0];
for m holds X[m] from NAT_1:sch 2(A11, A4);
hence thesis;
end;
theorem Th64:
s | UsedIntLoc i = t | UsedIntLoc i & s | UsedInt*Loc i = t |
UsedInt*Loc i & IC s = IC t implies IC Exec(i, s) = IC Exec(i, t) & Exec(i, s)
| UsedIntLoc i = Exec(i, t) | UsedIntLoc i & Exec(i, s) | UsedInt*Loc i = Exec(
i, t) | UsedInt*Loc i
proof
assume that
A1: s | UsedIntLoc i = t | UsedIntLoc i and
A2: s | UsedInt*Loc i = t | UsedInt*Loc i and
A3: IC s = IC t;
set UFi = UsedInt*Loc i;
set Ui = UsedIntLoc i;
set Et = Exec(i, t);
set Es = Exec(i, s);
A4: dom Es = the carrier of SCM+FSA by PARTFUN1:def 2
.= dom Et by PARTFUN1:def 2;
InsCode i <= 12 by SCMFSA_2:16;
then InsCode i = 0 or ... or InsCode i = 12;
then per cases;
suppose
InsCode i = 0;
then
A5: i = halt SCM+FSA by SCMFSA_2:95;
then Exec(i, s) = s by EXTPRO_1:def 3;
hence thesis by A1,A2,A3,A5,EXTPRO_1:def 3;
end;
suppose
InsCode i = 1;
then consider a, b such that
A6: i = a:=b by SCMFSA_2:30;
A7: Ui = {a, b} by A6,Th14;
then
A8: b in Ui by TARSKI:def 2;
then s.b = (s | Ui).b by FUNCT_1:49;
then
A9: s.b = t.b by A1,A8,FUNCT_1:49;
thus IC Es = IC t + 1 by A3,A6,SCMFSA_2:63
.= IC Et by A6,SCMFSA_2:63;
a = b or a <> b;
then
A10: Es.b = s.b & Et.b = t.b by A6,SCMFSA_2:63;
Es.a = s.b & Et.a = t.b by A6,SCMFSA_2:63;
hence Es | Ui = Et | Ui by A4,A7,A9,A10,GRFUNC_1:30;
A11: UFi = {} by A6,Th32;
hence Es | UFi = {} by RELAT_1:81
.= Et | UFi by A11,RELAT_1:81;
end;
suppose
InsCode i = 2;
then consider a, b such that
A12: i = AddTo(a,b) by SCMFSA_2:31;
thus IC Es = IC t + 1 by A3,A12,SCMFSA_2:64
.= IC Et by A12,SCMFSA_2:64;
A13: Ui = {a, b} by A12,Th14;
then
A14: a in Ui by TARSKI:def 2;
then s.a = (s | Ui).a by FUNCT_1:49;
then
A15: s.a = t.a by A1,A14,FUNCT_1:49;
A16: now
per cases;
case
a = b;
hence Es.b = s.a + s.b & Et.b = t.a + t.b by A12,SCMFSA_2:64;
end;
case
a<> b;
hence Es.b = s.b & Et.b = t.b by A12,SCMFSA_2:64;
end;
end;
A17: b in Ui by A13,TARSKI:def 2;
then s.b = (s | Ui).b by FUNCT_1:49;
then
A18: s.b = t.b by A1,A17,FUNCT_1:49;
Es.a = s.a + s.b & Et.a = t.a + t.b by A12,SCMFSA_2:64;
hence Es | Ui = Et | Ui by A4,A13,A15,A18,A16,GRFUNC_1:30;
A19: UFi = {} by A12,Th32;
hence Es | UFi = {} by RELAT_1:81
.= Et | UFi by A19,RELAT_1:81;
end;
suppose
InsCode i = 3;
then consider a, b such that
A20: i = SubFrom(a, b) by SCMFSA_2:32;
thus IC Es = IC t + 1 by A3,A20,SCMFSA_2:65
.= IC Et by A20,SCMFSA_2:65;
A21: Ui = {a, b} by A20,Th14;
then
A22: a in Ui by TARSKI:def 2;
then s.a = (s | Ui).a by FUNCT_1:49;
then
A23: s.a = t.a by A1,A22,FUNCT_1:49;
A24: now
per cases;
case
a = b;
hence Es.b = s.a - s.b & Et.b = t.a - t.b by A20,SCMFSA_2:65;
end;
case
a<> b;
hence Es.b = s.b & Et.b = t.b by A20,SCMFSA_2:65;
end;
end;
A25: b in Ui by A21,TARSKI:def 2;
then s.b = (s | Ui).b by FUNCT_1:49;
then
A26: s.b = t.b by A1,A25,FUNCT_1:49;
Es.a = s.a - s.b & Et.a = t.a - t.b by A20,SCMFSA_2:65;
hence Es | Ui = Et | Ui by A4,A21,A23,A26,A24,GRFUNC_1:30;
A27: UFi = {} by A20,Th32;
hence Es | UFi = {} by RELAT_1:81
.= Et | UFi by A27,RELAT_1:81;
end;
suppose
InsCode i = 4;
then consider a, b such that
A28: i = MultBy(a, b) by SCMFSA_2:33;
thus IC Es = IC t + 1 by A3,A28,SCMFSA_2:66
.= IC Et by A28,SCMFSA_2:66;
A29: Ui = {a, b} by A28,Th14;
then
A30: a in Ui by TARSKI:def 2;
then s.a = (s | Ui).a by FUNCT_1:49;
then
A31: s.a = t.a by A1,A30,FUNCT_1:49;
A32: now
per cases;
case
a = b;
hence Es.b = s.a * s.b & Et.b = t.a * t.b by A28,SCMFSA_2:66;
end;
case
a<> b;
hence Es.b = s.b & Et.b = t.b by A28,SCMFSA_2:66;
end;
end;
A33: b in Ui by A29,TARSKI:def 2;
then s.b = (s | Ui).b by FUNCT_1:49;
then
A34: s.b = t.b by A1,A33,FUNCT_1:49;
Es.a = s.a * s.b & Et.a = t.a * t.b by A28,SCMFSA_2:66;
hence Es | Ui = Et | Ui by A4,A29,A31,A34,A32,GRFUNC_1:30;
A35: UFi = {} by A28,Th32;
hence Es | UFi = {} by RELAT_1:81
.= Et | UFi by A35,RELAT_1:81;
end;
suppose
InsCode i = 5;
then consider a, b such that
A36: i = Divide(a, b) by SCMFSA_2:34;
A37: Ui = {a, b} by A36,Th14;
then
A38: a in Ui by TARSKI:def 2;
then s.a = (s | Ui).a by FUNCT_1:49;
then
A39: s.a = t.a by A1,A38,FUNCT_1:49;
A40: UFi = {} by A36,Th32;
A41: b in Ui by A37,TARSKI:def 2;
then s.b = (s | Ui).b by FUNCT_1:49;
then
A42: s.b = t.b by A1,A41,FUNCT_1:49;
hereby
per cases;
suppose
A43: a = b;
hence IC Es = IC t + 1 by A3,A36,SCMFSA_2:68
.= IC Et by A36,A43,SCMFSA_2:68;
Es.a = s.a mod s.a & Et.a = t.a mod t.b by A36,A43,SCMFSA_2:68;
hence Es | Ui = Et | Ui by A4,A37,A39,A43,GRFUNC_1:30;
thus Es | UFi = {} by A40,RELAT_1:81
.= Et | UFi by A40,RELAT_1:81;
end;
suppose
A44: a <> b;
thus IC Es = IC t + 1 by A3,A36,SCMFSA_2:67
.= IC Et by A36,SCMFSA_2:67;
A45: Es.b = s.a mod s.b & Et.b = t.a mod t.b by A36,SCMFSA_2:67;
Es.a = s.a div s.b & Et.a = t.a div t.b by A36,A44,SCMFSA_2:67;
hence Es | Ui = Et | Ui by A4,A37,A39,A42,A45,GRFUNC_1:30;
thus Es | UFi = {} by A40,RELAT_1:81
.= Et | UFi by A40,RELAT_1:81;
end;
end;
end;
suppose
InsCode i = 6;
then consider l such that
A46: i = goto l by SCMFSA_2:35;
thus IC Es = l by A46,SCMFSA_2:69
.= IC Et by A46,SCMFSA_2:69;
A47: Ui = {} by A46,Th15;
hence Es | Ui = {} by RELAT_1:81
.= Et | Ui by A47,RELAT_1:81;
A48: UFi = {} by A46,Th32;
hence Es | UFi = {} by RELAT_1:81
.= Et | UFi by A48,RELAT_1:81;
end;
suppose
InsCode i = 7;
then consider l, a such that
A49: i = a=0_goto l by SCMFSA_2:36;
A50: Ui = {a} by A49,Th16;
then
A51: a in Ui by TARSKI:def 1;
then
A52: s.a = (s | Ui).a by FUNCT_1:49;
then
A53: s.a = t.a by A1,A51,FUNCT_1:49;
hereby
per cases;
suppose
A54: s.a = 0;
hence IC Es = l by A49,SCMFSA_2:70
.= IC Et by A49,A53,A54,SCMFSA_2:70;
end;
suppose
A55: s.a <> 0;
hence IC Es = IC s + 1 by A49,SCMFSA_2:70
.= IC Et by A3,A49,A53,A55,SCMFSA_2:70;
end;
end;
Es.a = s.a & Et.a = t.a by A49,SCMFSA_2:70;
hence Es | Ui = Et | Ui by A1,A4,A50,A51,A52,FUNCT_1:49,GRFUNC_1:29;
A56: UFi = {} by A49,Th32;
hence Es | UFi = {} by RELAT_1:81
.= Et | UFi by A56,RELAT_1:81;
end;
suppose
InsCode i = 8;
then consider l, a such that
A57: i = a>0_goto l by SCMFSA_2:37;
A58: Ui = {a} by A57,Th16;
then
A59: a in Ui by TARSKI:def 1;
then
A60: s.a = (s | Ui).a by FUNCT_1:49;
then
A61: s.a = t.a by A1,A59,FUNCT_1:49;
hereby
per cases;
suppose
A62: s.a > 0;
hence IC Es = l by A57,SCMFSA_2:71
.= IC Et by A57,A61,A62,SCMFSA_2:71;
end;
suppose
A63: s.a <= 0;
hence IC Es = IC s + 1 by A57,SCMFSA_2:71
.= IC Et by A3,A57,A61,A63,SCMFSA_2:71;
end;
end;
Es.a = s.a & Et.a = t.a by A57,SCMFSA_2:71;
hence Es | Ui = Et | Ui by A1,A4,A58,A59,A60,FUNCT_1:49,GRFUNC_1:29;
A64: UFi = {} by A57,Th32;
hence Es | UFi = {} by RELAT_1:81
.= Et | UFi by A64,RELAT_1:81;
end;
suppose
InsCode i = 9;
then consider a, b, f such that
A65: i = b:=(f,a) by SCMFSA_2:38;
A66: Ui = {a, b} by A65,Th17;
then
A67: a in Ui by TARSKI:def 2;
then s.a = (s | Ui).a by FUNCT_1:49;
then
A68: s.a = t.a by A1,A67,FUNCT_1:49;
thus IC Es = IC t + 1 by A3,A65,SCMFSA_2:72
.= IC Et by A65,SCMFSA_2:72;
A69: UFi = {f} by A65,Th33;
then
A70: f in UFi by TARSKI:def 1;
then
A71: s.f = (s | UFi).f by FUNCT_1:49;
A72: (ex m st m = |.s.a.| & Es.b = (s.f)/.m )& ex n st n = |.t.a.| & Et
.b = (t.f)/.n by A65,SCMFSA_2:72;
A73: now
per cases;
case
a = b;
thus Es.b = Et.b by A2,A70,A71,A68,A72,FUNCT_1:49;
end;
case
a <> b;
hence Es.a = s.a & Et.a = t.a by A65,SCMFSA_2:72;
end;
end;
s.f = t.f by A2,A70,A71,FUNCT_1:49;
hence Es | Ui = Et | Ui by A4,A66,A68,A72,A73,GRFUNC_1:30;
Es.f = s.f & Et.f = t.f by A65,SCMFSA_2:72;
hence thesis by A2,A4,A69,A70,A71,FUNCT_1:49,GRFUNC_1:29;
end;
suppose
InsCode i = 10;
then consider a, b, f such that
A74: i = (f,a):=b by SCMFSA_2:39;
thus IC Es = IC t + 1 by A3,A74,SCMFSA_2:73
.= IC Et by A74,SCMFSA_2:73;
A75: Et.a = t.a & Et.b = t.b by A74,SCMFSA_2:73;
A76: Ui = {a, b} by A74,Th17;
then
A77: a in Ui by TARSKI:def 2;
then s.a = (s | Ui).a by FUNCT_1:49;
then
A78: s.a = t.a by A1,A77,FUNCT_1:49;
A79: b in Ui by A76,TARSKI:def 2;
then s.b = (s | Ui).b by FUNCT_1:49;
then
A80: s.b = t.b by A1,A79,FUNCT_1:49;
A81: UFi = {f} by A74,Th33;
then
A82: f in UFi by TARSKI:def 1;
then s.f = (s | UFi).f by FUNCT_1:49;
then
A83: s.f = t.f by A2,A82,FUNCT_1:49;
Es.a = s.a & Es.b = s.b by A74,SCMFSA_2:73;
hence Es | Ui = Et | Ui by A4,A76,A78,A80,A75,GRFUNC_1:30;
(ex m st m = |.s.a.| & Es.f = s.f+*(m,s.b) )& ex n st n = |.t.a.|
& Et. f = t.f+*(n,t.b) by A74,SCMFSA_2:73;
hence thesis by A4,A81,A78,A80,A83,GRFUNC_1:29;
end;
suppose
InsCode i = 11;
then consider a, f such that
A84: i = a:=len f by SCMFSA_2:40;
thus IC Es = IC t + 1 by A3,A84,SCMFSA_2:74
.= IC Et by A84,SCMFSA_2:74;
A85: Et.a = len(t.f) by A84,SCMFSA_2:74;
A86: Ui = {a} & Es.a = len(s.f) by A84,Th18,SCMFSA_2:74;
A87: UFi = {f} by A84,Th34;
then
A88: f in UFi by TARSKI:def 1;
then
A89: s.f = (s | UFi).f by FUNCT_1:49;
then s.f = t.f by A2,A88,FUNCT_1:49;
hence Es | Ui = Et | Ui by A4,A86,A85,GRFUNC_1:29;
Es.f = s.f & Et.f = t.f by A84,SCMFSA_2:74;
hence thesis by A2,A4,A87,A88,A89,FUNCT_1:49,GRFUNC_1:29;
end;
suppose
InsCode i = 12;
then consider a,f such that
A90: i = f:=<0,...,0>a by SCMFSA_2:41;
thus IC Es = IC t + 1 by A3,A90,SCMFSA_2:75
.= IC Et by A90,SCMFSA_2:75;
A91: Ui = {a} by A90,Th18;
then
A92: a in Ui by TARSKI:def 1;
then
A93: s.a = (s | Ui).a by FUNCT_1:49;
A94: UFi = {f} & ex m st m = |.s.a.| & Es.f = m |-> 0
by A90,Th34,SCMFSA_2:75;
Es.a = s.a & Et.a = t.a by A90,SCMFSA_2:75;
hence Es | Ui = Et | Ui by A1,A4,A91,A92,A93,FUNCT_1:49,GRFUNC_1:29;
A95: ex n st n = |.t.a.| & Et.f = n |-> 0 by A90,SCMFSA_2:75;
s.a = t.a by A1,A92,A93,FUNCT_1:49;
hence thesis by A4,A94,A95,GRFUNC_1:29;
end;
end;
theorem
for P,Q being Instruction-Sequence of SCM+FSA
st I c= P & I c= Q &
Start-At(0,SCM+FSA) c= s & Start-At(0,SCM+FSA) c= t &
s | UsedILoc I = t | UsedILoc I &
s | UsedI*Loc I = t | UsedI*Loc I &
(for m st m < n holds IC Comput(P,s,m) in dom I)
holds
(for m st m < n holds IC Comput(Q,t,m) in dom I) &
for m st m <= n holds IC Comput(P,s,m) = IC Comput(Q,t,m) &
(for a st a in UsedILoc I holds Comput(P,s,m).a = Comput(Q,t,m).a) &
for f st f in UsedI*Loc I holds Comput(P,s,m).f = Comput(Q,t,m).f
proof
let P,Q be Instruction-Sequence of SCM+FSA such that
A1: I c= P & I c= Q;
assume that
A2: Start-At(0,SCM+FSA) c= s and
A3: Start-At(0,SCM+FSA) c= t and
A4: s | UsedILoc I = t | UsedILoc I and
A5: s | UsedI*Loc I = t | UsedI*Loc I and
A6: for m st m < n holds IC Comput(P,s,m) in dom I;
defpred P[Nat] means $1 < n implies IC Comput(Q,t,$1) in dom I
& IC Comput(P,s,$1) = IC Comput(Q,t,$1) & (for a st
a in UsedILoc I
holds Comput(P,s,$1).a = Comput(Q,t,$1).a) & for f st
f in UsedI*Loc I
holds Comput(P,s,$1).f = Comput(Q,t,$1).f;
A7: now
let m;
assume
A8: P[m];
thus P[m+1]
proof
set i = P.IC Comput(P,s,m);
dom P = NAT by PARTFUN1:def 2;
then
A9: P/.IC Comput(P,s,m) = P.IC Comput(P,s,m) by PARTFUN1:def 6;
set m1 = m+1;
A10: Comput(P,s,m1) = Following(P,Comput(P,s,m)) by EXTPRO_1:3
.= Exec(P.IC Comput(P,s,m),Comput(P,s,m))
by A9;
assume
A11: m1 < n;
now
thus dom ( Comput(P,s,m) | UsedI*Loc I) = dom ( Comput(
P,s,m))
/\ UsedI*Loc I by RELAT_1:61
.= (the carrier of SCM+FSA) /\ UsedI*Loc I
by PARTFUN1:def 2
.= dom ( Comput(Q,t,m)) /\ UsedI*Loc I by PARTFUN1:def 2;
let x be object;
assume x in dom ( Comput(P,s,m) | UsedI*Loc I);
then
A12: x in UsedI*Loc I by RELAT_1:57;
then x in FinSeq-Locations;
then reconsider x9 = x as FinSeq-Location by SCMFSA_2:def 5;
thus ( Comput(P,s,m) | UsedI*Loc I).x = Comput(
P,s,m).x9 by A12,FUNCT_1:49
.= Comput(Q,t,m).x by A8,A11,A12,NAT_1:13;
end;
then
A13: Comput(P,s,m) | UsedI*Loc I = Comput(Q,t,m) | UsedI*Loc I
by FUNCT_1:46;
A14: P.IC Comput(P,s,m) = I.IC Comput(P,s,m)
by A8,A11,A1,GRFUNC_1:2,NAT_1:13;
then
A15: P.IC Comput(P,s,m) = Q.IC Comput(Q,t,m)
by A8,A11,A1,GRFUNC_1:2,NAT_1:13;
now
thus dom ( Comput(P,s,m) | UsedILoc I) = dom ( Comput(
P,s,m))
/\ UsedILoc I by RELAT_1:61
.= (the carrier of SCM+FSA) /\ UsedILoc I
by PARTFUN1:def 2
.= dom ( Comput(Q,t,m)) /\ UsedILoc I by PARTFUN1:def 2;
let x be object;
assume x in dom ( Comput(P,s,m) | UsedILoc I);
then
A16: x in UsedILoc I by RELAT_1:57;
then x in Int-Locations;
then reconsider x9 = x as Int-Location by AMI_2:def 16;
thus ( Comput(P,s,m) | UsedILoc I).x = Comput(
P,s,m).x9 by A16,FUNCT_1:49
.= Comput(Q,t,m).x by A8,A11,A16,NAT_1:13;
end;
then
A17: Comput(P,s,m) | UsedILoc I = Comput(Q,t,m) | UsedILoc I
by FUNCT_1:46;
dom Q = NAT by PARTFUN1:def 2;
then
A18: Q/.IC Comput(Q,t,m) = Q.IC Comput(Q,t,m) by PARTFUN1:def 6;
A19: Comput(Q,t,m1) =
Following(Q,Comput(Q,t,m)) by EXTPRO_1:3
.= Exec(Q.IC Comput(Q,t,m),Comput(Q,t,m))
by A18;
m < n by A11,NAT_1:13;
then IC Comput(P,s,m) in dom I by A6;
then
A20: i in rng I by A14,FUNCT_1:def 3;
then
A21: Comput(P,s,m) | UsedInt*Loc i = ( Comput(P,s,m)
| UsedI*Loc
I) | UsedInt*Loc i by Th35,RELAT_1:74
.= Comput(Q,t,m) | UsedInt*Loc i by A20,A13,Th35,RELAT_1:74;
A22: Comput(P,s,m) | UsedIntLoc i = ( Comput(P,s,m)
| UsedILoc I
) | UsedIntLoc i by A20,Th19,RELAT_1:74
.= Comput(Q,t,m) | UsedIntLoc i by A20,A17,Th19,RELAT_1:74;
then
A23: Exec(i, Comput(P,s,m)) | UsedInt*Loc i
= Exec(i, Comput(Q,t,m)) | UsedInt*Loc i
by A8,A11,A21,Th64,NAT_1:13;
A24: IC Exec(i, Comput(P,s,m)) = IC Exec(i, Comput(Q,t,m))
by A8,A11,A22,A21,Th64,NAT_1:13;
hence IC Comput(Q,t,m1) in dom I by A6,A11,A10,A19,A15;
thus IC Comput(P,s,m1) = IC Comput(Q,t,m1) by A8,A11,A10,A19,A14,A24,A1,
GRFUNC_1:2,NAT_1:13;
A25: Exec(i, Comput(P,s,m)) | UsedIntLoc i = Exec(i, Comput(
Q,t,m)
) | UsedIntLoc i by A8,A11,A22,A21,Th64,NAT_1:13;
hereby
let a;
assume
A26: a in UsedILoc I;
per cases;
suppose
A27: a in UsedIntLoc i;
hence
Comput(P,s,m1).a = (Exec(i, Comput(P,s,m))
| UsedIntLoc i
).a by A10,FUNCT_1:49
.= Comput(Q,t,m1).a by A19,A15,A25,A27,FUNCT_1:49;
end;
suppose
A28: not a in UsedIntLoc i;
hence Comput(P,s,m1).a = Comput(P,s,m).a
by A10,Th60
.= Comput(Q,t,m).a by A8,A11,A26,NAT_1:13
.= Comput(Q,t,m1).a by A19,A15,A28,Th60;
end;
end;
let f;
assume
A29: f in UsedI*Loc I;
per cases;
suppose
A30: f in UsedInt*Loc i;
hence
Comput(P,s,m1).f = (Exec(i, Comput(P,s,m)) |
UsedInt*Loc i)
.f by A10,FUNCT_1:49
.= Comput(Q,t,m1).f by A19,A15,A23,A30,FUNCT_1:49;
end;
suppose
A31: not f in UsedInt*Loc i;
hence Comput(P,s,m1).f = Comput(P,s,m).f by A10,Th62
.= Comput(Q,t,m).f by A8,A11,A29,NAT_1:13
.= Comput(Q,t,m1).f by A19,A15,A31,Th62;
end;
end;
end;
A32: P[0]
proof
A33: IC Comput(Q,t,0) = IC t
.= 0 by A3,MEMSTR_0:39;
A34: IC Comput(P,s,0) = IC s
.= 0 by A2,MEMSTR_0:39;
assume 0 < n;
hence IC Comput(Q,t,0) in dom I by A6,A34,A33;
thus IC Comput(P,s,0) = IC Comput(Q,t,0) by A34,A33;
hereby
let a;
assume
A35: a in UsedILoc I;
thus Comput(P,s,0).a = s.a
.= (s | UsedILoc I).a by A35,FUNCT_1:49
.= t.a by A4,A35,FUNCT_1:49
.= Comput(Q,t,0).a;
end;
let f;
assume
A36: f in UsedI*Loc I;
thus Comput(P,s,0).f = s.f
.= (s | UsedI*Loc I).f by A36,FUNCT_1:49
.= t.f by A5,A36,FUNCT_1:49
.= Comput(Q,t,0).f;
end;
A37: for m holds P[m] from NAT_1:sch 2(A32, A7);
hence for m st m < n holds IC Comput(Q,t,m) in dom I;
let m;
assume
A38: m <= n;
per cases by NAT_1:6;
suppose
A39: m = 0;
A40: IC Comput(Q,t,0) = IC t
.= 0 by A3,MEMSTR_0:39;
IC Comput(P,s,0) = IC s
.= 0 by A2,MEMSTR_0:39;
hence IC Comput(P,s,m) = IC Comput(Q,t,m) by A39,A40;
hereby
let a;
assume
A41: a in UsedILoc I;
thus Comput(P,s,m).a = s.a by A39
.= (s | UsedILoc I).a by A41,FUNCT_1:49
.= t.a by A4,A41,FUNCT_1:49
.= Comput(Q,t,m).a by A39;
end;
let f;
assume
A42: f in UsedI*Loc I;
thus Comput(P,s,m).f = s.f by A39
.= (s | UsedI*Loc I).f by A42,FUNCT_1:49
.= t.f by A5,A42,FUNCT_1:49
.= Comput(Q,t,m).f by A39;
end;
suppose
ex p being Nat st m = p+1;
then consider p being Nat such that
A43: m = p+1;
reconsider p as Nat;
A44: p < n by A38,A43,NAT_1:13;
then
A45: IC Comput(P,s,p) in dom I by A6;
now
thus dom ( Comput(P,s,p) | UsedI*Loc I) = dom ( Comput(
P,s,p))
/\ UsedI*Loc I by RELAT_1:61
.= (the carrier of SCM+FSA) /\ UsedI*Loc I by PARTFUN1:def 2
.= dom ( Comput(Q,t,p)) /\ UsedI*Loc I by PARTFUN1:def 2;
let x be object;
assume x in dom ( Comput(P,s,p) | UsedI*Loc I);
then
A46: x in UsedI*Loc I by RELAT_1:57;
then x in FinSeq-Locations;
then reconsider x9 = x as FinSeq-Location by SCMFSA_2:def 5;
thus ( Comput(P,s,p) | UsedI*Loc I).x
= Comput(P,s,p).x9 by A46,FUNCT_1:49
.= Comput(Q,t,p).x by A37,A44,A46;
end;
then
A47: Comput(P,s,p) | UsedI*Loc I = Comput(Q,t,p) |
UsedI*Loc I
by FUNCT_1:46;
set i = P.IC Comput(P,s,p);
set p1 = p+1;
dom P = NAT by PARTFUN1:def 2;
then
A48: P/.IC Comput(P,s,p) = P.IC Comput(P,s,p) by PARTFUN1:def 6;
A49: Comput(P,s,p1) = Following(P,Comput(P,s,p)) by EXTPRO_1:3
.= Exec(P.IC Comput(P,s,p), Comput(P,s,p))
by A48;
now
thus dom ( Comput(P,s,p) | UsedILoc I) = dom ( Comput(
P,s,p)) /\
UsedILoc I by RELAT_1:61
.= (the carrier of SCM+FSA) /\ UsedILoc I by PARTFUN1:def 2
.= dom ( Comput(Q,t,p)) /\ UsedILoc I by PARTFUN1:def 2;
let x be object;
assume x in dom ( Comput(P,s,p) | UsedILoc I);
then
A50: x in UsedILoc I by RELAT_1:57;
then x in Int-Locations;
then reconsider x9 = x as Int-Location by AMI_2:def 16;
thus ( Comput(P,s,p) | UsedILoc I).x
= Comput(P,s,p).x9 by A50,FUNCT_1:49
.= Comput(Q,t,p).x by A37,A44,A50;
end;
then
A51: Comput(P,s,p) | UsedILoc I = Comput(Q,t,p) |
UsedILoc I by FUNCT_1:46;
A52: IC Comput(P,s,p) = IC Comput(Q,t,p) by A37,A44;
A53: P.IC Comput(P,s,p) = I.IC Comput(P,s,p) by A45,A1,GRFUNC_1:2;
dom Q = NAT by PARTFUN1:def 2;
then
A54: Q/.IC Comput(Q,t,p) = Q.IC Comput(Q,t,p) by PARTFUN1:def 6;
A55: Comput(Q,t,p1) = Following(Q,Comput(Q,t,p)) by EXTPRO_1:3
.= Exec(Q.IC Comput(Q,t,p), Comput(Q,t,p))
by A54;
A56: P.IC Comput(P,s,p) = Q.IC Comput(Q,t,p) by A52,A45,A53,A1,GRFUNC_1:2;
IC Comput(P,s,p) in dom I by A6,A44;
then
A57: i in rng I by A53,FUNCT_1:def 3;
then
A58: Comput(P,s,p) | UsedInt*Loc i = ( Comput(P,s,p)
| UsedI*Loc I
) | UsedInt*Loc i by Th35,RELAT_1:74
.= Comput(Q,t,p) | UsedInt*Loc i by A57,A47,Th35,RELAT_1:74;
A59: Comput(P,s,p) | UsedIntLoc i = ( Comput(P,s,p) |
UsedILoc I)
| UsedIntLoc i by A57,Th19,RELAT_1:74
.= Comput(Q,t,p) | UsedIntLoc i by A57,A51,Th19,RELAT_1:74;
hence IC Comput(P,s,m) = IC Comput(Q,t,m) by A43,A49,A55,A52,A56,A58,Th64;
A60: Exec(i, Comput(P,s,p)) | UsedIntLoc i = Exec(i, Comput(
Q,t,p))
| UsedIntLoc i by A52,A59,A58,Th64;
hereby
let a;
assume
A61: a in UsedILoc I;
per cases;
suppose
A62: a in UsedIntLoc i;
hence
Comput(P,s,m).a = (Exec(i, Comput(P,s,p)) |
UsedIntLoc i).a
by A43,A49,FUNCT_1:49
.= Comput(Q,t,m).a by A43,A55,A56,A60,A62,FUNCT_1:49;
end;
suppose
A63: not a in UsedIntLoc i;
hence Comput(P,s,m).a = Comput(P,s,p).a by A43,A49,Th60
.= Comput(Q,t,p).a by A37,A44,A61
.= Comput(Q,t,m).a by A43,A55,A56,A63,Th60;
end;
end;
A64: Exec(i, Comput(P,s,p)) | UsedInt*Loc i = Exec(i, Comput(Q,t,p))
| UsedInt*Loc i by A52,A59,A58,Th64;
hereby
let f;
assume
A65: f in UsedI*Loc I;
per cases;
suppose
A66: f in UsedInt*Loc i;
hence
Comput(P,s,m).f = (Exec(i, Comput(P,s,p)) |UsedInt*Loc i).f
by A43,A49,FUNCT_1:49
.= Comput(Q,t,m).f by A43,A55,A56,A64,A66,FUNCT_1:49;
end;
suppose
A67: not f in UsedInt*Loc i;
hence Comput(P,s,m).f = Comput(P,s,p).f by A43,A49,Th62
.= Comput(Q,t,p).f by A37,A44,A65
.= Comput(Q,t,m).f by A43,A55,A56,A67,Th62;
end;
end;
end;
end;
reserve i1 for Instruction of SCM+FSA;
:: from SCMFSA9A
Lm7: :: singleUsed
UsedILoc(l .--> i) = UsedIntLoc i
proof rng(l .--> i) = {i} by FUNCOP_1:8;
hence UsedILoc(l .--> i) = union { UsedIntLoc i1 : i1 in {i} }
.= UsedIntLoc i from SUBSET_1:sch 5;
end;
Lm8: :: singleUsedF:
UsedI*Loc (l .--> i) = UsedInt*Loc i
proof rng(l .--> i) = {i} by FUNCOP_1:8;
hence UsedI*Loc(l .--> i) = union { UsedInt*Loc i1 : i1 in {i} }
.= UsedInt*Loc i from SUBSET_1:sch 5;
end;
theorem
for I being MacroInstruction of SCM+FSA, k being Nat
holds UsedILoc(I ';' goto k) = UsedILoc I
proof let I be MacroInstruction of SCM+FSA, k be Nat;
A1: dom (LastLoc I .--> halt SCM+FSA) = {LastLoc I} by FUNCOP_1:13;
not LastLoc I in dom CutLastLoc I by COMPOS_2:17;
then {LastLoc I} misses dom CutLastLoc I by ZFMISC_1:50;
then
A2: dom CutLastLoc I misses dom (LastLoc I .--> halt SCM+FSA)
by A1;
A3: dom CutLastLoc I misses dom Reloc(Macro goto k, card I -' 1)
by COMPOS_1:18;
thus UsedILoc(I ';' goto k) = UsedILoc(I ';' Macro goto k)
.= UsedILoc(CutLastLoc I +* Reloc(Macro goto k,card I -' 1))
.= UsedILoc CutLastLoc I \/ UsedILoc Reloc(Macro goto k,card I -' 1)
by A3,Th21
.= UsedILoc CutLastLoc I \/ UsedILoc Macro goto k by Th25
.= UsedILoc CutLastLoc I \/ UsedIntLoc goto k by Th28
.= UsedILoc CutLastLoc I \/ {} by Th15
.= UsedILoc CutLastLoc I \/ UsedIntLoc halt SCM+FSA by Th13
.= UsedILoc CutLastLoc I \/ UsedILoc (LastLoc I .--> halt SCM+FSA) by Lm7
.= UsedILoc(CutLastLoc I +* (LastLoc I .--> halt SCM+FSA)) by A2,Th21
.= UsedILoc I by COMPOS_2:20;
end;
theorem
for I being MacroInstruction of SCM+FSA, k being Nat
holds UsedI*Loc(I ';' goto k) = UsedI*Loc I
proof let I be MacroInstruction of SCM+FSA, k be Nat;
A1: dom (LastLoc I .--> halt SCM+FSA) = {LastLoc I} by FUNCOP_1:13;
not LastLoc I in dom CutLastLoc I by COMPOS_2:17;
then {LastLoc I} misses dom CutLastLoc I by ZFMISC_1:50;
then
A2: dom CutLastLoc I misses dom (LastLoc I .--> halt SCM+FSA)
by A1;
A3: dom CutLastLoc I misses dom Reloc(Macro goto k, card I -' 1)
by COMPOS_1:18;
thus UsedI*Loc(I ';' goto k) = UsedI*Loc(I ';' Macro goto k)
.= UsedI*Loc(CutLastLoc I +* Reloc(Macro goto k,card I -' 1))
.= UsedI*Loc CutLastLoc I \/ UsedI*Loc Reloc(Macro goto k,card I -' 1)
by A3,Th37
.= UsedI*Loc CutLastLoc I \/ UsedI*Loc Macro goto k by Th41
.= UsedI*Loc CutLastLoc I \/ UsedInt*Loc goto k by Th44
.= UsedI*Loc CutLastLoc I \/ {} by Th32
.= UsedI*Loc CutLastLoc I \/ UsedInt*Loc halt SCM+FSA by Th32
.= UsedI*Loc CutLastLoc I \/ UsedI*Loc (LastLoc I .--> halt SCM+FSA) by Lm8
.= UsedI*Loc(CutLastLoc I +* (LastLoc I .--> halt SCM+FSA)) by A2,Th37
.= UsedI*Loc I by COMPOS_2:20;
end;