:: On the Decomposition of the States of SCM :: by Yasushi Tanaka :: :: Received November 23, 1993 :: Copyright (c) 1993 Association of Mizar Users environ vocabularies BOOLE, NAT_1, ARYTM_1, FUNCT_1, RELAT_1, FUNCT_4, AMI_3, AMI_1, AMI_2, GR_CY_1, FINSEQ_1, FINSET_1, TARSKI, CAT_1, FUNCOP_1, MCART_1, ORDINAL2, QC_LANG1, AMI_4, AMI_5, INT_1; notations TARSKI, XBOOLE_0, SETFAM_1, SUBSET_1, ORDINAL1, NUMBERS, XCMPLX_0, CARD_1, ZFMISC_1, CARD_3, MCART_1, DOMAIN_1, RELAT_1, FUNCT_1, FUNCT_2, FUNCT_4, INT_1, NAT_1, NAT_D, STRUCT_0, FINSET_1, FINSEQ_1, AMI_1, AMI_2, AMI_3, AMI_4, XXREAL_0; constructors WELLORD2, DOMAIN_1, XXREAL_0, NAT_1, NAT_D, FINSEQ_4, AMI_2, AMI_4; registrations XBOOLE_0, SUBSET_1, SETFAM_1, RELAT_1, FUNCT_1, ORDINAL1, ARYTM_3, FRAENKEL, NUMBERS, XREAL_0, INT_1, FINSEQ_1, CARD_3, STRUCT_0, AMI_1, AMI_2, AMI_3, FINSET_1; requirements NUMERALS, REAL, SUBSET, BOOLE, ARITHM; definitions AMI_1, TARSKI, AMI_3, WELLORD2, FUNCT_1, XBOOLE_0, FUNCOP_1, CARD_1, AMI_2; theorems AMI_1, AMI_2, AMI_3, GRFUNC_1, NAT_1, SCM_1, CQC_LANG, TARSKI, FUNCOP_1, FUNCT_4, FUNCT_1, MCART_1, GR_CY_1, FUNCT_2, CARD_3, FINSET_1, ZFMISC_1, AMI_4, ENUMSET1, CARD_1, CARD_4, RELAT_1, ORDINAL1, XBOOLE_0, XBOOLE_1, NAT_D, SYSREL; schemes FUNCT_2; begin canceled 17; reserve x,y for set; theorem Th18: for dl being Data-Location ex i being Element of NAT st dl = dl.i proof let dl be Data-Location; dl in SCM-Data-Loc by AMI_3:def 2; then consider x,y such that W1: x in {1} and W2: y in NAT and W3: dl = [x,y] by ZFMISC_1:103; reconsider k = y as Element of NAT by W2; A1: dl = [1,k] by W1,W3,TARSKI:def 1; take k; thus dl = dl.k by A1; end; theorem Th19: for il being Instruction-Location of SCM ex i being Element of NAT st il = il.i proof let il be Instruction-Location of SCM; consider k being Element of NAT such that A1: il = k; take k; thus il = il.k by A1; end; theorem Th20: for dl being Data-Location holds dl <> IC SCM proof let dl be Data-Location; consider i being Element of NAT such that A1: dl = dl.i by Th18; thus thesis by A1,AMI_3:57; end; canceled; theorem Th22: for il being Instruction-Location of SCM, dl being Data-Location holds il <> dl proof let il be Instruction-Location of SCM, dl be Data-Location; consider i being Element of NAT such that A1: il = il.i by Th19; consider j being Element of NAT such that A2: dl = dl.j by Th18; thus il <> dl by A1,A2,AMI_3:56; end; reserve i, j, k for Element of NAT; theorem Th23: the carrier of SCM = {IC SCM} \/ SCM-Data-Loc \/ SCM-Instr-Loc by AMI_3:4; theorem for s being State of SCM, d being Data-Location, l being Instruction-Location of SCM holds d in dom s & l in dom s proof let s be State of SCM, d be Data-Location, l be Instruction-Location of SCM; d in SCM-Data-Loc by AMI_3:def 2; then d in {IC SCM} \/ SCM-Data-Loc by XBOOLE_0:def 2; then d in {IC SCM} \/ SCM-Data-Loc \/ SCM-Instr-Loc by XBOOLE_0:def 2; hence d in dom s by Th23,AMI_1:79; l in {IC SCM} \/ SCM-Data-Loc \/ SCM-Instr-Loc by XBOOLE_0:def 2; hence l in dom s by Th23,AMI_1:79; end; canceled; theorem for s1,s2 being State of SCM st IC(s1) = IC(s2) & (for a being Data-Location holds s1.a = s2.a) & (for i being Instruction-Location of SCM holds s1.i = s2.i) holds s1 = s2 proof let s1,s2 be State of SCM such that A1: IC(s1) = IC(s2) and A2: (for a being Data-Location holds s1.a = s2.a) and A3: (for i being Instruction-Location of SCM holds s1.i = s2.i); consider g1 being Function such that A4: s1 = g1 & dom g1 = dom SCM-OK & for x being set st x in dom SCM-OK holds g1.x in SCM-OK.x by CARD_3:def 5; consider g2 being Function such that A5: s2 = g2 & dom g2 = dom SCM-OK & for x being set st x in dom SCM-OK holds g2.x in SCM-OK.x by CARD_3:def 5; A6: SCM-Memory = dom g1 & SCM-Memory = dom g2 by A4,A5,FUNCT_2:def 1; now let x be set such that A7: x in SCM-Memory; A8: x in {IC SCM} \/ SCM-Data-Loc or x in SCM-Instr-Loc by A7,Th23,XBOOLE_0:def 2; per cases by A8,XBOOLE_0:def 2; suppose x in {IC SCM}; then x = IC SCM by TARSKI:def 1; hence g1.x = g2.x by A4,A5,A1; end; suppose x in SCM-Data-Loc; then x is Data-Location by AMI_3:def 2; hence g1.x = g2.x by A2,A4,A5; end; suppose x in SCM-Instr-Loc; then reconsider l = x as Instruction-Location of SCM by AMI_1:def 4; g1.l = g2.l by A3,A4,A5; hence g1.x = g2.x; end; end; hence s1 = s2 by A4,A5,A6,FUNCT_1:9; end; theorem Th27: for s being State of SCM holds SCM-Data-Loc c= dom s proof let s be State of SCM; SCM-Data-Loc c= SCM-Data-Loc \/ SCM-Instr-Loc by XBOOLE_1:10; then SCM-Data-Loc c= {IC SCM} \/ (SCM-Data-Loc \/ SCM-Instr-Loc) by XBOOLE_1:10; then SCM-Data-Loc c= {IC SCM} \/ SCM-Data-Loc \/ SCM-Instr-Loc by XBOOLE_1: 4 ; hence SCM-Data-Loc c= dom s by Th23,AMI_1:79; end; theorem Th28: for s being State of SCM holds SCM-Instr-Loc c= dom s proof let s be State of SCM; SCM-Instr-Loc c= {IC SCM} \/ SCM-Data-Loc \/ SCM-Instr-Loc by XBOOLE_1: 10 ; hence SCM-Instr-Loc c= dom s by Th23,AMI_1:79; end; theorem for s being State of SCM holds dom (s|SCM-Data-Loc) = SCM-Data-Loc proof let s be State of SCM; SCM-Data-Loc c= dom s by Th27; hence dom (s|SCM-Data-Loc) = SCM-Data-Loc by RELAT_1:91; end; theorem for s being State of SCM holds dom (s|SCM-Instr-Loc) = SCM-Instr-Loc proof let s be State of SCM; SCM-Instr-Loc c= dom s by Th28; hence dom (s|SCM-Instr-Loc) = SCM-Instr-Loc by RELAT_1:91; end; theorem Th31: SCM-Data-Loc is not finite proof deffunc F(Element of NAT) = [1,$1]; 1 in {1} by TARSKI:def 1; then A: for x being Element of NAT holds F(x) is Element of SCM-Data-Loc by ZFMISC_1:106; consider f being Function of NAT, SCM-Data-Loc such that A1: for x being Element of NAT holds f.x = F(x) from FUNCT_2:sch 9(A); A2: dom f = NAT by FUNCT_2:def 1; NAT,SCM-Data-Loc are_equipotent proof take f; thus f is one-to-one proof let x1,x2 be set such that A3: x1 in dom f and A4: x2 in dom f and A5: f.x1 = f.x2; reconsider k1 = x1 ,k2 = x2 as Element of NAT by A3,A4,FUNCT_2:def 1; dl.k1 = f.k1 by A1 .= dl.k2 by A1,A5; hence x1 = x2 by AMI_3:52; end; thus dom f = NAT by FUNCT_2:def 1; thus rng f c= SCM-Data-Loc proof let y be set; assume y in rng f; then consider x be set such that A6: x in dom f and A7: y = f.x by FUNCT_1:def 5; reconsider x as Element of NAT by A6,FUNCT_2:def 1; y = dl.x by A1,A7; hence y in SCM-Data-Loc by AMI_3:def 2; end; thus SCM-Data-Loc c= rng f proof let y be set such that A8: y in SCM-Data-Loc; reconsider d = y as Data-Location by A8,AMI_3:def 2; consider k being Element of NAT such that A9: d = dl.k by Th18; y = f.k by A9,A1; hence y in rng f by A2,FUNCT_1:def 5; end; end; hence SCM-Data-Loc is not finite by CARD_1:68,CARD_4:15; end; theorem Th32: the Instruction-Locations of SCM is not finite proof deffunc F(Element of NAT) = $1; consider f being Function of NAT, NAT; thus the Instruction-Locations of SCM is not finite by CARD_4:15; end; registration cluster SCM-Data-Loc -> infinite; coherence by Th31; cluster the Instruction-Locations of SCM -> infinite; coherence by Th32; end; registration let I be Instruction of SCM; cluster InsCode I -> natural; coherence proof dom [: NAT, (union {INT} \/ SCM-Memory)* :] = NAT by SYSREL:12; then A1: dom the Instructions of SCM c= NAT by RELAT_1:25; InsCode I in dom the Instructions of SCM; hence thesis by A1,ORDINAL1:def 13; end; end; definition canceled; let I be Instruction of SCM; func @I -> Element of SCM-Instr equals I; coherence; end; definition let loc be Element of SCM-Instr-Loc; func loc@ -> Instruction-Location of SCM equals loc; coherence by AMI_1:def 4; end; definition let loc be Element of SCM-Data-Loc; func loc@ -> Data-Location equals loc; coherence by AMI_3:def 2; end; reserve I,J,K for Element of Segm 9, a,a1 for Element of SCM-Instr-Loc, b,b1,c for Element of SCM-Data-Loc; canceled 3; theorem Th36: for l being Instruction of SCM holds InsCode(l) <= 8 proof let l be Instruction of SCM; l in { [SCM-Halt,{}] } \/ { [J,<*a*>] : J = 6 } \/ { [K,<*a1,b1*>] : K in { 7,8 } } or l in { [I,<*b,c*>] : I in { 1,2,3,4,5} } by XBOOLE_0:def 2; then A1: l in { [SCM-Halt,{}] } \/ { [J,<*a*>] : J = 6 } or l in { [K,<*a1,b1*>] : K in { 7,8 } } or l in { [I,<*b,c*>] : I in { 1,2,3,4,5} } by XBOOLE_0:def 2; per cases by A1, XBOOLE_0:def 2; suppose l in { [SCM-Halt,{}] }; then l = [SCM-Halt,{}] by TARSKI:def 1; then l`1 = 0 by MCART_1:7; hence InsCode(l) <= 8; end; suppose l in { [J,<*a*>] : J = 6 }; then ex J,a st l = [J,<*a*>] & J = 6; then l`1 = 6 by MCART_1:7; hence InsCode(l) <= 8; end; suppose l in { [K,<*a1,b1*>] : K in { 7,8 } }; then ex K,a1,b1 st l = [K,<*a1,b1*>] & K in { 7,8 }; then l`1 in { 7,8 } by MCART_1:7; then l`1 = 7 or l`1 = 8 by TARSKI:def 2; hence InsCode(l) <= 8; end; suppose l in { [I,<*b,c*>] : I in { 1,2,3,4,5} }; then ex I,b,c st l = [I,<*b,c*>] & I in { 1,2,3,4,5}; then l`1 in { 1,2,3,4,5} by MCART_1:7; then l`1 = 1 or l`1 = 2 or l`1 = 3 or l`1 = 4 or l`1 = 5 by ENUMSET1:def 3; hence InsCode(l) <= 8; end; end; reserve a, b for Data-Location, loc for Instruction-Location of SCM; theorem Th37: InsCode (halt SCM) = 0 by AMI_3:71,MCART_1:7; reserve I,J,K for Element of Segm 9, a,a1 for Element of SCM-Instr-Loc, b,b1,c for Element of SCM-Data-Loc, da,db for Data-Location, loc for Instruction-Location of SCM; canceled 8; theorem Th46: for ins being Instruction of SCM st InsCode ins = 0 holds ins = halt SCM proof let ins be Instruction of SCM such that A1: InsCode ins = 0; A2: now assume ins in { [I,<*b,c*>] : I in { 1,2,3,4,5} }; then consider I,b,c such that A3: ins = [I,<*b,c*>] and A4: I in { 1,2,3,4,5}; InsCode ins = I by A3,MCART_1:7; hence contradiction by A1,A4,ENUMSET1:def 3; end; A5: now assume ins in { [K,<*a1,b1*>] : K in { 7,8 } }; then consider K,a1,b1 such that A6: ins = [K,<*a1,b1*>] and A7: K in { 7,8 }; InsCode ins = K by A6,MCART_1:7; hence contradiction by A1,A7,TARSKI:def 2; end; A8: now assume ins in { [J,<*a*>] : J = 6 }; then consider J,a such that A9: ins = [J,<*a*>] and A10: J = 6; thus contradiction by A1,A9,A10, MCART_1:7; end; ins in { [SCM-Halt,{}] } \/ { [J,<*a*>] : J = 6 } \/ { [K,<*a1,b1*>] : K in { 7,8 } } by A2,XBOOLE_0:def 2; then ins in { [SCM-Halt,{}] } \/ { [J,<*a*>] : J = 6 } by A5,XBOOLE_0:def 2; then ins in {[SCM-Halt,{}]} by A8,XBOOLE_0:def 2; hence ins = halt SCM by AMI_3:71,TARSKI:def 1; end; theorem Th47: for ins being Instruction of SCM st InsCode ins = 1 holds ex da,db st ins = da:=db proof let ins be Instruction of SCM such that A1: InsCode ins = 1; A2: not ins in { [SCM-Halt,{}] } by A1,Th37,AMI_3:71,TARSKI:def 1; A3: now assume ins in { [K,<*a1,b1*>] : K in { 7,8 } }; then consider K,a1,b1 such that A4: ins = [K,<*a1,b1*>] and A5: K in { 7,8 }; InsCode ins = K by A4,MCART_1:7; hence contradiction by A1,A5,TARSKI:def 2; end; now assume ins in { [J,<*a*>] : J = 6 }; then consider J,a such that A6: ins = [J,<*a*>] and A7: J = 6; thus contradiction by A1,A6,A7, MCART_1:7; end; then not ins in { [SCM-Halt,{}] } \/ { [J,<*a*>] : J = 6 } by A2,XBOOLE_0:def 2; then not ins in { [SCM-Halt,{}] } \/ { [J,<*a*>] : J = 6 } \/ { [K,<*a1,b1*>] : K in { 7,8 } } by A3,XBOOLE_0:def 2; then ins in { [I,<*b,c*>] : I in { 1,2,3,4,5} } by XBOOLE_0: def 2; then consider I,b,c such that A8: ins = [I,<*b,c*>] and I in { 1,2,3,4,5}; reconsider da = b@ ,db = c@ as Data-Location; take da,db; thus ins = da:=db by A1,A8,MCART_1:7; end; theorem Th48: for ins being Instruction of SCM st InsCode ins = 2 holds ex da,db st ins = AddTo(da,db) proof let ins be Instruction of SCM such that A1: InsCode ins = 2; A2: not ins in { [SCM-Halt,{}] } by A1,Th37,AMI_3:71,TARSKI:def 1; A3: now assume ins in { [K,<*a1,b1*>] : K in { 7,8 } }; then consider K,a1,b1 such that A4: ins = [K,<*a1,b1*>] and A5: K in { 7,8 }; InsCode ins = K by A4,MCART_1:7; hence contradiction by A1,A5,TARSKI:def 2; end; now assume ins in { [J,<*a*>] : J = 6 }; then consider J,a such that A6: ins = [J,<*a*>] and A7: J = 6; thus contradiction by A1,A6,A7, MCART_1:7; end; then not ins in { [SCM-Halt,{}] } \/ { [J,<*a*>] : J = 6 } by A2,XBOOLE_0:def 2; then not ins in { [SCM-Halt,{}] } \/ { [J,<*a*>] : J = 6 } \/ { [K,<*a1,b1*>] : K in { 7,8 } } by A3,XBOOLE_0:def 2; then ins in { [I,<*b,c*>] : I in { 1,2,3,4,5} } by XBOOLE_0: def 2; then consider I,b,c such that A8: ins = [I,<*b,c*>] and I in { 1,2,3,4,5}; reconsider da = b@ ,db = c@ as Data-Location; take da,db; thus ins = AddTo(da,db) by A1,A8,MCART_1:7; end; theorem Th49: for ins being Instruction of SCM st InsCode ins = 3 holds ex da,db st ins = SubFrom(da,db) proof let ins be Instruction of SCM such that A1: InsCode ins = 3; A2: not ins in { [SCM-Halt,{}] } by A1,Th37,AMI_3:71,TARSKI:def 1; A3: now assume ins in { [K,<*a1,b1*>] : K in { 7,8 } }; then consider K,a1,b1 such that A4: ins = [K,<*a1,b1*>] and A5: K in { 7,8 }; InsCode ins = K by A4,MCART_1:7; hence contradiction by A1,A5,TARSKI:def 2; end; now assume ins in { [J,<*a*>] : J = 6 }; then consider J,a such that A6: ins = [J,<*a*>] and A7: J = 6; thus contradiction by A1,A6,A7, MCART_1:7; end; then not ins in { [SCM-Halt,{}] } \/ { [J,<*a*>] : J = 6 } by A2,XBOOLE_0:def 2; then not ins in { [SCM-Halt,{}] } \/ { [J,<*a*>] : J = 6 } \/ { [K,<*a1,b1*>] : K in { 7,8 } } by A3,XBOOLE_0:def 2; then ins in { [I,<*b,c*>] : I in { 1,2,3,4,5} } by XBOOLE_0: def 2; then consider I,b,c such that A8: ins = [I,<*b,c*>] and I in { 1,2,3,4,5}; reconsider da = b@ ,db = c@ as Data-Location; take da,db; thus ins = SubFrom(da,db) by A1,A8,MCART_1:7; end; theorem Th50: for ins being Instruction of SCM st InsCode ins = 4 holds ex da,db st ins = MultBy(da,db) proof let ins be Instruction of SCM such that A1: InsCode ins = 4; A2: not ins in { [SCM-Halt,{}] } by A1,Th37,AMI_3:71,TARSKI:def 1; A3: now assume ins in { [K,<*a1,b1*>] : K in { 7,8 } }; then consider K,a1,b1 such that A4: ins = [K,<*a1,b1*>] and A5: K in { 7,8 }; InsCode ins = K by A4,MCART_1:7; hence contradiction by A1,A5,TARSKI:def 2; end; now assume ins in { [J,<*a*>] : J = 6 }; then consider J,a such that A6: ins = [J,<*a*>] and A7: J = 6; thus contradiction by A1,A6,A7, MCART_1:7; end; then not ins in { [SCM-Halt,{}] } \/ { [J,<*a*>] : J = 6 } by A2,XBOOLE_0:def 2; then not ins in { [SCM-Halt,{}] } \/ { [J,<*a*>] : J = 6 } \/ { [K,<*a1,b1*>] : K in { 7,8 } } by A3,XBOOLE_0:def 2; then ins in { [I,<*b,c*>] : I in { 1,2,3,4,5} } by XBOOLE_0: def 2; then consider I,b,c such that A8: ins = [I,<*b,c*>] and I in { 1,2,3,4,5}; reconsider da = b@ ,db = c@ as Data-Location; take da,db; thus ins = MultBy(da,db) by A1,A8,MCART_1:7; end; theorem Th51: for ins being Instruction of SCM st InsCode ins = 5 holds ex da,db st ins = Divide(da,db) proof let ins be Instruction of SCM such that A1: InsCode ins = 5; A2: not ins in { [SCM-Halt,{}] } by A1,Th37,AMI_3:71,TARSKI:def 1; A3: now assume ins in { [K,<*a1,b1*>] : K in { 7,8 } }; then consider K,a1,b1 such that A4: ins = [K,<*a1,b1*>] and A5: K in { 7,8 }; InsCode ins = K by A4,MCART_1:7; hence contradiction by A1,A5,TARSKI:def 2; end; now assume ins in { [J,<*a*>] : J = 6 }; then consider J,a such that A6: ins = [J,<*a*>] and A7: J = 6; thus contradiction by A1,A6,A7, MCART_1:7; end; then not ins in { [SCM-Halt,{}] } \/ { [J,<*a*>] : J = 6 } by A2,XBOOLE_0:def 2; then not ins in { [SCM-Halt,{}] } \/ { [J,<*a*>] : J = 6 } \/ { [K,<*a1,b1*>] : K in { 7,8 } } by A3,XBOOLE_0:def 2; then ins in { [I,<*b,c*>] : I in { 1,2,3,4,5} } by XBOOLE_0: def 2; then consider I,b,c such that A8: ins = [I,<*b,c*>] and I in { 1,2,3,4,5}; reconsider da = b@ ,db = c@ as Data-Location; take da,db; thus ins = Divide (da,db) by A1,A8,MCART_1:7; end; theorem Th52: for ins being Instruction of SCM st InsCode ins = 6 holds ex loc st ins = goto loc proof let ins be Instruction of SCM such that A1: InsCode ins = 6; A2: not ins in { [SCM-Halt,{}] } by A1,Th37,AMI_3:71,TARSKI:def 1; A3: now assume ins in { [K,<*a1,b1*>] : K in { 7,8 } }; then consider K,a1,b1 such that A4: ins = [K,<*a1,b1*>] and A5: K in { 7,8 }; InsCode ins = K by A4,MCART_1:7; hence contradiction by A1,A5,TARSKI:def 2; end; now assume ins in { [I,<*b,c*>] : I in { 1,2,3,4,5} }; then consider I,b,c such that A6: ins = [I,<*b,c*>] and A7: I in { 1,2,3,4,5}; InsCode ins = I by A6,MCART_1:7; hence contradiction by A1,A7,ENUMSET1:def 3; end; then ins in { [SCM-Halt,{}] } \/ { [J,<*a*>] : J = 6 } \/ { [K,<*a1,b1*>] : K in { 7,8 } } by XBOOLE_0:def 2; then ins in { [SCM-Halt,{}] } \/ { [J,<*a*>] : J = 6 } by A3,XBOOLE_0:def 2; then ins in { [J,<*a*>] : J = 6 } by A2,XBOOLE_0:def 2; then consider J,a such that A8: ins = [J,<*a*>] and A9: J = 6; reconsider loc = a@ as Instruction-Location of SCM; take loc; thus ins = goto loc by A8,A9; end; theorem Th53: for ins being Instruction of SCM st InsCode ins = 7 holds ex loc,da st ins = da=0_goto loc proof let ins be Instruction of SCM such that A1: InsCode ins = 7; A2: not ins in { [SCM-Halt,{}] } by A1,Th37,AMI_3:71,TARSKI:def 1; A3: now assume ins in { [J,<*a*>] : J = 6 }; then consider J,a such that A4: ins = [J,<*a*>] and A5: J = 6; thus contradiction by A1,A4,A5, MCART_1:7; end; A6: now assume ins in { [I,<*b,c*>] : I in { 1,2,3,4,5} }; then consider I,b,c such that A7: ins = [I,<*b,c*>] and A8: I in { 1,2,3,4,5}; InsCode ins = I by A7,MCART_1:7; hence contradiction by A1,A8,ENUMSET1:def 3; end; A9: not ins in { [SCM-Halt,{}] } \/ { [J,<*a*>] : J = 6 } by A2,A3,XBOOLE_0:def 2; ins in { [SCM-Halt,{}] } \/ { [J,<*a*>] : J = 6 } \/ { [K,<*a1,b1*>] : K in { 7,8 } } by A6,XBOOLE_0:def 2; then ins in { [K,<*a1,b1*>] : K in { 7,8 } } by A9,XBOOLE_0:def 2; then consider K,a1,b1 such that A10: ins = [K,<*a1,b1*>] and K in { 7,8 }; reconsider loc = a1@ as Instruction-Location of SCM; reconsider da = b1@ as Data-Location; take loc,da; thus ins = da=0_goto loc by A1,A10,MCART_1:7; end; theorem Th54: for ins being Instruction of SCM st InsCode ins = 8 holds ex loc,da st ins = da>0_goto loc proof let ins be Instruction of SCM such that A1: InsCode ins = 8; A2: not ins in { [SCM-Halt,{}] } by A1,Th37,AMI_3:71,TARSKI:def 1; A3: now assume ins in { [J,<*a*>] : J = 6 }; then consider J,a such that A4: ins = [J,<*a*>] and A5: J = 6; thus contradiction by A1,A4,A5, MCART_1:7; end; A6: now assume ins in { [I,<*b,c*>] : I in { 1,2,3,4,5} }; then consider I,b,c such that A7: ins = [I,<*b,c*>] and A8: I in { 1,2,3,4,5}; InsCode ins = I by A7,MCART_1:7; hence contradiction by A1,A8,ENUMSET1:def 3; end; A9: not ins in { [SCM-Halt,{}] } \/ { [J,<*a*>] : J = 6 } by A2,A3,XBOOLE_0:def 2; ins in { [SCM-Halt,{}] } \/ { [J,<*a*>] : J = 6 } \/ { [K,<*a1,b1*>] : K in { 7,8 } } by A6,XBOOLE_0:def 2; then ins in { [K,<*a1,b1*>] : K in { 7,8 } } by A9,XBOOLE_0:def 2; then consider K,a1,b1 such that A10: ins = [K,<*a1,b1*>] and K in { 7,8 }; reconsider loc = a1@ as Instruction-Location of SCM; reconsider da = b1@ as Data-Location; take loc,da; thus ins = da>0_goto loc by A1,A10,MCART_1:7; end; theorem for loc being Instruction-Location of SCM holds (@(goto loc)) jump_address = loc proof let loc be Instruction-Location of SCM; reconsider roku=6 as Element of Segm 9 by GR_CY_1:10; reconsider mk=loc as Element of SCM-Instr-Loc; @(goto loc) = [ roku, <*mk*>]; hence (@(goto loc)) jump_address = loc by AMI_2:24; end; theorem for loc being Instruction-Location of SCM, a being Data-Location holds (@(a=0_goto loc)) cjump_address = loc & (@(a=0_goto loc)) cond_address = a proof let loc be Instruction-Location of SCM, a be Data-Location; reconsider nana=7 as Element of Segm 9 by GR_CY_1:10; reconsider mk=loc as Element of SCM-Instr-Loc; reconsider aa=a as Element of SCM-Data-Loc by AMI_3:def 2; @(a=0_goto loc) = [ nana, <*mk,aa*>]; hence (@(a=0_goto loc)) cjump_address = loc & (@(a=0_goto loc)) cond_address = a by AMI_2:25; end; theorem for loc being Instruction-Location of SCM, a being Data-Location holds (@(a>0_goto loc)) cjump_address = loc & (@(a>0_goto loc)) cond_address = a proof let loc be Instruction-Location of SCM, a be Data-Location; reconsider hachi=8 as Element of Segm 9 by GR_CY_1:10; reconsider mk=loc as Element of SCM-Instr-Loc; reconsider aa=a as Element of SCM-Data-Loc by AMI_3:def 2; @(a>0_goto loc) = [ hachi, <*mk,aa*>]; hence (@(a>0_goto loc)) cjump_address = loc & (@(a>0_goto loc)) cond_address = a by AMI_2:25; end; theorem Th58: for s1,s2 being State of SCM st (s1 | (SCM-Data-Loc \/ {IC SCM})) = (s2 | (SCM-Data-Loc \/ {IC SCM})) for l being Instruction of SCM holds Exec (l,s1) | (SCM-Data-Loc \/ {IC SCM}) = Exec (l,s2) | (SCM-Data-Loc \/ {IC SCM}) proof let s1,s2 be State of SCM such that A1: (s1 | (SCM-Data-Loc \/ {IC SCM})) = (s2 | (SCM-Data-Loc \/ {IC SCM})); IC SCM in {IC SCM} by TARSKI:def 1; then A2: IC SCM in (SCM-Data-Loc \/ {IC SCM}) by XBOOLE_0:def 2; A3: (SCM-Data-Loc \/ {IC SCM}) c= the carrier of SCM by Th23,XBOOLE_1:7 ; then (SCM-Data-Loc \/ {IC SCM}) c= dom s1 by AMI_1:79; then A4: IC SCM in dom (s1 | (SCM-Data-Loc \/ {IC SCM})) by A2,RELAT_1:91; (SCM-Data-Loc \/ {IC SCM}) c= dom s2 by A3,AMI_1:79; then A5: IC SCM in dom (s2 | (SCM-Data-Loc \/ {IC SCM})) by A2,RELAT_1:91; A6: IC s1 = (s2 | (SCM-Data-Loc \/ {IC SCM})).IC SCM by A1,A4,FUNCT_1:70 .= IC s2 by A5,FUNCT_1:70; let l be Instruction of SCM; A7: dom Exec(l,s1) = the carrier of SCM by AMI_1:79; A8: dom Exec(l,s2) = the carrier of SCM by AMI_1:79; A9: SCM-Data-Loc c= (SCM-Data-Loc \/ {IC SCM}) by XBOOLE_1:7; A10: InsCode(l) <= 8 by Th36; per cases by A10,NAT_1:33; suppose InsCode (l) = 0; then A11: l = halt SCM by Th46; hence Exec (l,s1) | (SCM-Data-Loc \/ {IC SCM}) = s2 | (SCM-Data-Loc \/ {IC SCM}) by A1,AMI_1:def 8 .= Exec (l,s2) | (SCM-Data-Loc \/ {IC SCM}) by A11,AMI_1:def 8; end; suppose InsCode (l) = 1; then consider da,db such that A12: l = da:=db by Th47; da in SCM-Data-Loc by AMI_3:def 2; then A13: SCM-Data-Loc = SCM-Data-Loc \/ {da} by ZFMISC_1:46 .= (SCM-Data-Loc \ {da} ) \/ {da} by XBOOLE_1:39; A14: dom ((Exec (l,s1)) | (SCM-Data-Loc \ {da})) = (SCM-Data-Loc \ {da}) by A7,RELAT_1:91; A15: dom ((Exec (l,s2)) | (SCM-Data-Loc \ {da})) = (SCM-Data-Loc \ {da}) by A8,RELAT_1:91; for x being set st x in ((SCM-Data-Loc) \ {da}) holds (Exec (l,s1) | (SCM-Data-Loc \ {da})).x = (Exec (l,s2) | (SCM-Data-Loc \ {da})).x proof let x be set; assume A16: x in ((SCM-Data-Loc) \ {da}); then A17: x in SCM-Data-Loc by XBOOLE_0:def 4; A18: not x in {da} by A16,XBOOLE_0:def 4; reconsider a = x as Data-Location by A17,AMI_3:def 2; A19: a <> da by A18,TARSKI:def 1; A20: a in (SCM-Data-Loc \/ {IC SCM}) by A17,XBOOLE_0:def 2; thus (Exec (l,s1) | (SCM-Data-Loc \ {da})).x = (Exec (l,s1)).a by A16,FUNCT_1:72 .= s1.a by A12,A19,AMI_3:8 .= (s1 | (SCM-Data-Loc \/ {IC SCM})).a by A20,FUNCT_1:72 .= s2.a by A1,A20,FUNCT_1:72 .= (Exec (l,s2)).a by A12,A19,AMI_3:8 .= (Exec (l,s2) | (SCM-Data-Loc \ {da})).x by A16,FUNCT_1:72; end; then A21: Exec (l,s1) | (SCM-Data-Loc \ {da} ) = Exec (l,s2) | (SCM-Data-Loc \ {da} ) by A14,A15,FUNCT_1:9; A22: db in SCM-Data-Loc by AMI_3:def 2; Exec (l,s1).da = s1.db by A12,AMI_3:8 .= (s1 | (SCM-Data-Loc \/ {IC SCM})).db by A9,A22,FUNCT_1:72 .= s2.db by A1,A9,A22,FUNCT_1:72 .= Exec (l,s2).da by A12,AMI_3:8; then Exec (l,s1) | {da} = Exec(l,s2) | {da} by A7,A8,GRFUNC_1:90; then A23: Exec (l,s1) | SCM-Data-Loc = Exec (l,s2) | SCM-Data-Loc by A13,A21,RELAT_1:185; Exec (l,s1).IC SCM = Next IC s1 by A12,AMI_3:8 .= Exec (l,s2).IC SCM by A6,A12,AMI_3:8; then Exec (l,s1) | {IC SCM} = Exec (l,s2) | {IC SCM} by A7,A8,GRFUNC_1:90; hence Exec (l,s1) | (SCM-Data-Loc \/ {IC SCM}) = Exec (l,s2) | (SCM-Data-Loc \/ {IC SCM}) by A23,RELAT_1:185; end; suppose InsCode (l) = 2; then consider da,db such that A24: l = AddTo(da,db) by Th48; da in SCM-Data-Loc by AMI_3:def 2; then A25: SCM-Data-Loc = SCM-Data-Loc \/ {da} by ZFMISC_1:46 .= (SCM-Data-Loc \ {da} ) \/ {da} by XBOOLE_1:39; A26: dom ((Exec (l,s1)) | (SCM-Data-Loc \ {da})) = (SCM-Data-Loc \ {da}) by A7,RELAT_1:91; A27: dom ((Exec (l,s2)) | (SCM-Data-Loc \ {da})) = (SCM-Data-Loc \ {da}) by A8,RELAT_1:91; for x being set st x in ((SCM-Data-Loc) \ {da}) holds (Exec (l,s1) | (SCM-Data-Loc \ {da})).x = (Exec (l,s2) | (SCM-Data-Loc \ {da})).x proof let x be set; assume A28: x in ((SCM-Data-Loc) \ {da}); then A29: x in SCM-Data-Loc by XBOOLE_0:def 4; A30: not x in {da} by A28,XBOOLE_0:def 4; reconsider a = x as Data-Location by A29,AMI_3:def 2; A31: a <> da by A30,TARSKI:def 1; A32: a in (SCM-Data-Loc \/ {IC SCM}) by A29,XBOOLE_0:def 2; thus (Exec (l,s1) | (SCM-Data-Loc \ {da})).x = (Exec (l,s1)).a by A28,FUNCT_1:72 .= s1.a by A24,A31,AMI_3:9 .= (s1 | (SCM-Data-Loc \/ {IC SCM})).a by A32,FUNCT_1:72 .= s2.a by A1,A32,FUNCT_1:72 .= (Exec (l,s2)).a by A24,A31,AMI_3:9 .= (Exec (l,s2) | (SCM-Data-Loc \ {da})).x by A28,FUNCT_1:72; end; then A33: Exec (l,s1) | (SCM-Data-Loc \ {da} ) = Exec (l,s2) | (SCM-Data-Loc \ {da} ) by A26,A27,FUNCT_1:9; A34: db in SCM-Data-Loc by AMI_3:def 2; A35: da in SCM-Data-Loc by AMI_3:def 2; then A36: s1.da = (s1 | (SCM-Data-Loc \/ {IC SCM})).da by A9,FUNCT_1:72 .= s2.da by A1,A9,A35,FUNCT_1:72; A37: s1.db = (s1 | (SCM-Data-Loc \/ {IC SCM})).db by A9,A34,FUNCT_1:72 .= s2.db by A1,A9,A34,FUNCT_1:72; Exec (l,s1).da = s1.da + s1.db by A24,AMI_3:9 .= Exec (l,s2).da by A24,A36,A37,AMI_3:9; then Exec (l,s1) | {da} = Exec(l,s2) | {da} by A7,A8,GRFUNC_1:90; then A38: Exec (l,s1) | SCM-Data-Loc = Exec (l,s2) | SCM-Data-Loc by A25,A33,RELAT_1:185; Exec (l,s1).IC SCM = Next IC s1 by A24,AMI_3:9 .= Exec (l,s2).IC SCM by A6,A24,AMI_3:9; then Exec (l,s1) | {IC SCM} = Exec (l,s2) | {IC SCM} by A7,A8,GRFUNC_1:90; hence Exec (l,s1) | (SCM-Data-Loc \/ {IC SCM}) = Exec (l,s2) | (SCM-Data-Loc \/ {IC SCM}) by A38,RELAT_1:185; end; suppose InsCode (l) = 3; then consider da,db such that A39: l = SubFrom(da,db) by Th49; da in SCM-Data-Loc by AMI_3:def 2; then A40: SCM-Data-Loc = SCM-Data-Loc \/ {da} by ZFMISC_1:46 .= (SCM-Data-Loc \ {da} ) \/ {da} by XBOOLE_1:39; A41: dom ((Exec (l,s1)) | (SCM-Data-Loc \ {da})) = (SCM-Data-Loc \ {da}) by A7,RELAT_1:91; A42: dom ((Exec (l,s2)) | (SCM-Data-Loc \ {da})) = (SCM-Data-Loc \ {da}) by A8,RELAT_1:91; for x being set st x in ((SCM-Data-Loc) \ {da}) holds (Exec (l,s1) | (SCM-Data-Loc \ {da})).x = (Exec (l,s2) | (SCM-Data-Loc \ {da})).x proof let x be set; assume A43: x in ((SCM-Data-Loc) \ {da}); then A44: x in SCM-Data-Loc by XBOOLE_0:def 4; A45: not x in {da} by A43,XBOOLE_0:def 4; reconsider a = x as Data-Location by A44,AMI_3:def 2; A46: a <> da by A45,TARSKI:def 1; A47: a in (SCM-Data-Loc \/ {IC SCM}) by A44,XBOOLE_0:def 2; thus (Exec (l,s1) | (SCM-Data-Loc \ {da})).x = (Exec (l,s1)).a by A43,FUNCT_1:72 .= s1.a by A39,A46,AMI_3:10 .= (s1 | (SCM-Data-Loc \/ {IC SCM})).a by A47,FUNCT_1:72 .= s2.a by A1,A47,FUNCT_1:72 .= (Exec (l,s2)).a by A39,A46,AMI_3:10 .= (Exec (l,s2) | (SCM-Data-Loc \ {da})).x by A43,FUNCT_1:72; end; then A48: Exec (l,s1) | (SCM-Data-Loc \ {da} ) = Exec (l,s2) | (SCM-Data-Loc \ {da} ) by A41,A42,FUNCT_1:9; A49: db in SCM-Data-Loc by AMI_3:def 2; A50: da in SCM-Data-Loc by AMI_3:def 2; then A51: s1.da = (s1 | (SCM-Data-Loc \/ {IC SCM})).da by A9,FUNCT_1:72 .= s2.da by A1,A9,A50,FUNCT_1:72; A52: s1.db = (s1 | (SCM-Data-Loc \/ {IC SCM})).db by A9,A49,FUNCT_1:72 .= s2.db by A1,A9,A49,FUNCT_1:72; Exec (l,s1).da = s1.da - s1.db by A39,AMI_3:10 .= Exec (l,s2).da by A39,A51,A52,AMI_3:10; then Exec (l,s1) | {da} = Exec(l,s2) | {da} by A7,A8,GRFUNC_1:90; then A53: Exec (l,s1) | SCM-Data-Loc = Exec (l,s2) | SCM-Data-Loc by A40,A48,RELAT_1:185; Exec (l,s1).IC SCM = Next IC s1 by A39,AMI_3:10 .= Exec (l,s2).IC SCM by A6,A39,AMI_3:10; then Exec (l,s1) | {IC SCM} = Exec (l,s2) | {IC SCM} by A7,A8,GRFUNC_1:90; hence Exec (l,s1) | (SCM-Data-Loc \/ {IC SCM}) = Exec (l,s2) | (SCM-Data-Loc \/ {IC SCM}) by A53,RELAT_1:185; end; suppose InsCode (l) = 4; then consider da,db such that A54: l = MultBy(da,db) by Th50; da in SCM-Data-Loc by AMI_3:def 2; then A55: SCM-Data-Loc = SCM-Data-Loc \/ {da} by ZFMISC_1:46 .= (SCM-Data-Loc \ {da} ) \/ {da} by XBOOLE_1:39; A56: dom ((Exec (l,s1)) | (SCM-Data-Loc \ {da})) = (SCM-Data-Loc \ {da}) by A7,RELAT_1:91; A57: dom ((Exec (l,s2)) | (SCM-Data-Loc \ {da})) = (SCM-Data-Loc \ {da}) by A8,RELAT_1:91; for x being set st x in ((SCM-Data-Loc) \ {da}) holds (Exec (l,s1) | (SCM-Data-Loc \ {da})).x = (Exec (l,s2) | (SCM-Data-Loc \ {da})).x proof let x be set; assume A58: x in ((SCM-Data-Loc) \ {da}); then A59: x in SCM-Data-Loc by XBOOLE_0:def 4; A60: not x in {da} by A58,XBOOLE_0:def 4; reconsider a = x as Data-Location by A59,AMI_3:def 2; A61: a <> da by A60,TARSKI:def 1; A62: a in (SCM-Data-Loc \/ {IC SCM}) by A59,XBOOLE_0:def 2; thus (Exec (l,s1) | (SCM-Data-Loc \ {da})).x = (Exec (l,s1)).a by A58,FUNCT_1:72 .= s1.a by A54,A61,AMI_3:11 .= (s1 | (SCM-Data-Loc \/ {IC SCM})).a by A62,FUNCT_1:72 .= s2.a by A1,A62,FUNCT_1:72 .= (Exec (l,s2)).a by A54,A61,AMI_3:11 .= (Exec (l,s2) | (SCM-Data-Loc \ {da})).x by A58,FUNCT_1:72; end; then A63: Exec (l,s1) | (SCM-Data-Loc \ {da} ) = Exec (l,s2) | (SCM-Data-Loc \ {da} ) by A56,A57,FUNCT_1:9; A64: db in SCM-Data-Loc by AMI_3:def 2; A65: da in SCM-Data-Loc by AMI_3:def 2; then A66: s1.da = (s1 | (SCM-Data-Loc \/ {IC SCM})).da by A9,FUNCT_1:72 .= s2.da by A1,A9,A65,FUNCT_1:72; A67: s1.db = (s1 | (SCM-Data-Loc \/ {IC SCM})).db by A9,A64,FUNCT_1:72 .= s2.db by A1,A9,A64,FUNCT_1:72; Exec (l,s1).da = s1.da * s1.db by A54,AMI_3:11 .= Exec (l,s2).da by A54,A66,A67,AMI_3:11; then Exec (l,s1) | {da} = Exec(l,s2) | {da} by A7,A8,GRFUNC_1:90; then A68: Exec (l,s1) | SCM-Data-Loc = Exec (l,s2) | SCM-Data-Loc by A55,A63,RELAT_1:185; Exec (l,s1).IC SCM = Next IC s1 by A54,AMI_3:11 .= Exec (l,s2).IC SCM by A6,A54,AMI_3:11; then Exec (l,s1) | {IC SCM} = Exec (l,s2) | {IC SCM} by A7,A8,GRFUNC_1:90; hence Exec (l,s1) | (SCM-Data-Loc \/ {IC SCM}) = Exec (l,s2) | (SCM-Data-Loc \/ {IC SCM}) by A68,RELAT_1:185; end; suppose InsCode (l) = 5; then consider da,db such that A69: l = Divide(da,db) by Th51; thus thesis proof per cases; suppose A70: da=db; da in SCM-Data-Loc by AMI_3:def 2; then A71: SCM-Data-Loc = SCM-Data-Loc \/ {da} by ZFMISC_1:46 .= (SCM-Data-Loc \ {da} ) \/ {da} by XBOOLE_1:39; A72: dom ((Exec (l,s1)) | (SCM-Data-Loc \ {da})) = (SCM-Data-Loc \ {da}) by A7,RELAT_1:91; A73: dom ((Exec (l,s2)) | (SCM-Data-Loc \ {da})) = (SCM-Data-Loc \ {da}) by A8,RELAT_1:91; for x being set st x in ((SCM-Data-Loc) \ {da}) holds (Exec (l,s1) | (SCM-Data-Loc \ {da})).x = (Exec (l,s2) | (SCM-Data-Loc \ {da})).x proof let x be set; assume A74: x in ((SCM-Data-Loc) \ {da}); then A75: x in SCM-Data-Loc by XBOOLE_0:def 4; A76: not x in {da} by A74,XBOOLE_0:def 4; reconsider a = x as Data-Location by A75,AMI_3:def 2; A77: a <> da by A76,TARSKI:def 1; A78: a in (SCM-Data-Loc \/ {IC SCM}) by A75,XBOOLE_0:def 2; thus (Exec (l,s1) | (SCM-Data-Loc \ {da})).x = (Exec (l,s1)).a by A74,FUNCT_1:72 .= s1.a by A69,A70,A77,AMI_3:12 .= (s1 | (SCM-Data-Loc \/ {IC SCM})).a by A78,FUNCT_1:72 .= s2.a by A1,A78,FUNCT_1:72 .= (Exec (l,s2)).a by A69,A70,A77,AMI_3:12 .= (Exec (l,s2) | (SCM-Data-Loc \ {da})).x by A74,FUNCT_1:72; end; then A79: Exec (l,s1) | (SCM-Data-Loc \ {da} ) = Exec (l,s2) | (SCM-Data-Loc \ {da} ) by A72,A73,FUNCT_1:9; A80: da in SCM-Data-Loc by AMI_3:def 2; then A81: s1.da = (s1 | (SCM-Data-Loc \/ {IC SCM})).da by A9,FUNCT_1:72 .= s2.da by A1,A9,A80,FUNCT_1:72; Exec (l,s1).da = s1.da mod s1.da by A69,A70,AMI_3:12 .= Exec (l,s2).da by A69,A70,A81,AMI_3:12; then Exec (l,s1) | {da} = Exec(l,s2) | {da} by A7,A8,GRFUNC_1:90; then A82: Exec (l,s1) | SCM-Data-Loc = Exec (l,s2) | SCM-Data-Loc by A71,A79,RELAT_1:185; Exec (l,s1).IC SCM = Next IC s1 by A69,AMI_3:12 .= Exec (l,s2).IC SCM by A6,A69,AMI_3:12; then Exec (l,s1) | {IC SCM} = Exec (l,s2) | {IC SCM} by A7,A8,GRFUNC_1:90; hence Exec (l,s1) | (SCM-Data-Loc \/ {IC SCM}) = Exec (l,s2) | (SCM-Data-Loc \/ {IC SCM}) by A82,RELAT_1:185; end; suppose A83: da <> db; A84: da in SCM-Data-Loc by AMI_3:def 2; db in SCM-Data-Loc by AMI_3:def 2; then A85: SCM-Data-Loc = SCM-Data-Loc \/ {da,db} by A84,ZFMISC_1:48 .= (SCM-Data-Loc \ {da,db} ) \/ {da,db} by XBOOLE_1:39; A86: dom ((Exec (l,s1)) | (SCM-Data-Loc \ {da,db})) = (SCM-Data-Loc \ {da ,db}) by A7,RELAT_1:91; A87: dom ((Exec (l,s2)) | (SCM-Data-Loc \ {da,db})) = (SCM-Data-Loc \ {da ,db}) by A8,RELAT_1:91; for x being set st x in ((SCM-Data-Loc) \ {da,db}) holds (Exec (l,s1) | (SCM-Data-Loc \ {da,db})).x = (Exec (l,s2) | (SCM-Data-Loc \ {da,db})).x proof let x be set; assume A88: x in ((SCM-Data-Loc) \ {da,db}); then A89: x in SCM-Data-Loc by XBOOLE_0:def 4; A90: not x in {da,db} by A88,XBOOLE_0:def 4; reconsider a = x as Data-Location by A89,AMI_3:def 2; A91: a <> da & a <> db by A90,TARSKI:def 2; A92: a in (SCM-Data-Loc \/ {IC SCM}) by A89,XBOOLE_0:def 2; thus (Exec (l,s1) | (SCM-Data-Loc \ {da,db})).x = (Exec (l,s1)).a by A88,FUNCT_1:72 .= s1.a by A69,A91,AMI_3:12 .= (s1 | (SCM-Data-Loc \/ {IC SCM})).a by A92,FUNCT_1:72 .= s2.a by A1,A92,FUNCT_1:72 .= (Exec (l,s2)).a by A69,A91,AMI_3:12 .= (Exec (l,s2) | (SCM-Data-Loc \ {da,db})).x by A88,FUNCT_1:72; end; then A93: Exec (l,s1) | (SCM-Data-Loc \ {da,db} ) = Exec (l,s2) | (SCM-Data-Loc \ {da,db} ) by A86,A87,FUNCT_1:9; A94: db in SCM-Data-Loc by AMI_3:def 2; A95: da in SCM-Data-Loc by AMI_3:def 2; then A96: s1.da = (s1 | (SCM-Data-Loc \/ {IC SCM})).da by A9,FUNCT_1:72 .= s2.da by A1,A9,A95,FUNCT_1:72; A97: s1.db = (s1 | (SCM-Data-Loc \/ {IC SCM})).db by A9,A94,FUNCT_1:72 .= s2.db by A1,A9,A94,FUNCT_1:72; A98: Exec (l,s1).da = s1.da div s1.db by A69,A83,AMI_3:12 .= Exec (l,s2).da by A69,A83,A96,A97,AMI_3:12; Exec (l,s1).db = s1.da mod s1.db by A69,AMI_3:12 .= Exec (l,s2).db by A69,A96,A97,AMI_3:12; then Exec (l,s1) | {da,db} = Exec(l,s2) | {da,db} by A7,A8,A98,GRFUNC_1:91; then A99: Exec (l,s1) | SCM-Data-Loc = Exec (l,s2) | SCM-Data-Loc by A85,A93,RELAT_1:185; Exec (l,s1).IC SCM = Next IC s1 by A69,AMI_3:12 .= Exec (l,s2).IC SCM by A6,A69,AMI_3:12; then Exec (l,s1) | {IC SCM} = Exec (l,s2) | {IC SCM} by A7,A8,GRFUNC_1:90; hence Exec (l,s1) | (SCM-Data-Loc \/ {IC SCM}) = Exec (l,s2) | (SCM-Data-Loc \/ {IC SCM}) by A99,RELAT_1:185; end; end; end; suppose InsCode (l) = 6; then consider loc such that A100: l = goto loc by Th52; A101: dom ((Exec (l,s1)) | SCM-Data-Loc) = SCM-Data-Loc by A7,RELAT_1:91; A102: dom ((Exec (l,s2)) | SCM-Data-Loc) = SCM-Data-Loc by A8,RELAT_1:91; for x being set st x in SCM-Data-Loc holds (Exec (l,s1) | SCM-Data-Loc ).x = (Exec (l,s2) | SCM-Data-Loc ).x proof let x be set; assume A103: x in SCM-Data-Loc; then reconsider a = x as Data-Location by AMI_3:def 2; A104: a in (SCM-Data-Loc \/ {IC SCM}) by A103,XBOOLE_0:def 2; thus (Exec (l,s1) | SCM-Data-Loc ).x = (Exec (l,s1)).a by A103,FUNCT_1:72 .= s1.a by A100,AMI_3:13 .= (s1 | (SCM-Data-Loc \/ {IC SCM})).a by A104,FUNCT_1:72 .= s2.a by A1,A104,FUNCT_1:72 .= (Exec (l,s2)).a by A100,AMI_3:13 .= (Exec (l,s2) | SCM-Data-Loc ).x by A103,FUNCT_1:72; end; then A105: Exec (l,s1) | (SCM-Data-Loc ) = Exec (l,s2) | (SCM-Data-Loc ) by A101,A102,FUNCT_1:9; Exec (l,s1).IC SCM = loc by A100,AMI_3:13 .= Exec (l,s2).IC SCM by A100,AMI_3:13; then Exec (l,s1) | {IC SCM} = Exec (l,s2) | {IC SCM} by A7,A8,GRFUNC_1:90; hence Exec (l,s1) | (SCM-Data-Loc \/ {IC SCM}) = Exec (l,s2) | (SCM-Data-Loc \/ {IC SCM}) by A105,RELAT_1:185; end; suppose InsCode (l) = 7; then consider loc,da such that A106: l = da=0_goto loc by Th53; A107: dom ((Exec (l,s1)) | SCM-Data-Loc) = SCM-Data-Loc by A7,RELAT_1:91; A108: dom ((Exec (l,s2)) | SCM-Data-Loc) = SCM-Data-Loc by A8,RELAT_1:91; for x being set st x in SCM-Data-Loc holds (Exec (l,s1) | SCM-Data-Loc ).x = (Exec (l,s2) | SCM-Data-Loc ).x proof let x be set; assume A109: x in SCM-Data-Loc; then reconsider a = x as Data-Location by AMI_3:def 2; A110: a in (SCM-Data-Loc \/ {IC SCM}) by A109,XBOOLE_0:def 2; thus (Exec (l,s1) | SCM-Data-Loc ).x = (Exec (l,s1)).a by A109,FUNCT_1:72 .= s1.a by A106,AMI_3:14 .= (s1 | (SCM-Data-Loc \/ {IC SCM})).a by A110,FUNCT_1:72 .= s2.a by A1,A110,FUNCT_1:72 .= (Exec (l,s2)).a by A106,AMI_3:14 .= (Exec (l,s2) | SCM-Data-Loc ).x by A109,FUNCT_1:72; end; then A111: Exec (l,s1) | (SCM-Data-Loc ) = Exec (l,s2) | (SCM-Data-Loc ) by A107,A108,FUNCT_1:9; Exec (l,s1).IC SCM = Exec (l,s2).IC SCM proof A112: da in SCM-Data-Loc by AMI_3:def 2; then A113: s1.da = (s1 | (SCM-Data-Loc \/ {IC SCM})).da by A9,FUNCT_1:72 .= s2.da by A1,A9,A112,FUNCT_1:72; per cases; suppose A114: s1.da = 0; hence Exec (l,s1).IC SCM = loc by A106,AMI_3:14 .= Exec (l,s2).IC SCM by A106,A113,A114, AMI_3:14; end; suppose A115: s1.da <> 0; hence Exec (l,s1).IC SCM = Next IC s1 by A106,AMI_3:14 .= Exec (l,s2).IC SCM by A6,A106,A113,A115, AMI_3:14; end; end; then Exec (l,s1) | {IC SCM} = Exec (l,s2) | {IC SCM} by A7,A8,GRFUNC_1:90; hence Exec (l,s1) | (SCM-Data-Loc \/ {IC SCM}) = Exec (l,s2) | (SCM-Data-Loc \/ {IC SCM}) by A111,RELAT_1:185; end; suppose InsCode (l) = 8; then consider loc,da such that A116: l = da>0_goto loc by Th54; A117: dom ((Exec (l,s1)) | SCM-Data-Loc) = SCM-Data-Loc by A7,RELAT_1:91; A118: dom ((Exec (l,s2)) | SCM-Data-Loc) = SCM-Data-Loc by A8,RELAT_1:91; for x being set st x in SCM-Data-Loc holds (Exec (l,s1) | SCM-Data-Loc ).x = (Exec (l,s2) | SCM-Data-Loc ).x proof let x be set; assume A119: x in SCM-Data-Loc; then reconsider a = x as Data-Location by AMI_3:def 2; A120: a in (SCM-Data-Loc \/ {IC SCM}) by A119,XBOOLE_0:def 2; thus (Exec (l,s1) | SCM-Data-Loc ).x = (Exec (l,s1)).a by A119,FUNCT_1:72 .= s1.a by A116,AMI_3:15 .= (s1 | (SCM-Data-Loc \/ {IC SCM})).a by A120,FUNCT_1:72 .= s2.a by A1,A120,FUNCT_1:72 .= (Exec (l,s2)).a by A116,AMI_3:15 .= (Exec (l,s2) | SCM-Data-Loc ).x by A119,FUNCT_1:72; end; then A121: Exec (l,s1) | (SCM-Data-Loc ) = Exec (l,s2) | (SCM-Data-Loc ) by A117,A118,FUNCT_1:9; Exec (l,s1).IC SCM = Exec (l,s2).IC SCM proof A122: da in SCM-Data-Loc by AMI_3:def 2; then A123: s1.da = (s1 | (SCM-Data-Loc \/ {IC SCM})).da by A9,FUNCT_1:72 .= s2.da by A1,A9,A122,FUNCT_1:72; per cases; suppose A124: s1.da > 0; hence Exec (l,s1).IC SCM = loc by A116,AMI_3:15 .= Exec (l,s2).IC SCM by A116,A123,A124, AMI_3:15; end; suppose A125: s1.da <= 0; hence Exec (l,s1).IC SCM = Next IC s1 by A116,AMI_3:15 .= Exec (l,s2).IC SCM by A6,A116,A123,A125, AMI_3:15; end; end; then Exec (l,s1) | {IC SCM} = Exec (l,s2) | {IC SCM} by A7,A8,GRFUNC_1:90; hence Exec (l,s1) | (SCM-Data-Loc \/ {IC SCM}) = Exec (l,s2) | (SCM-Data-Loc \/ {IC SCM}) by A121,RELAT_1:185; end; end; theorem Th59: for i being Instruction of SCM, s being State of SCM holds Exec (i, s) | SCM-Instr-Loc = s | SCM-Instr-Loc proof let i be Instruction of SCM, s be State of SCM; dom (Exec (i,s)) = the carrier of SCM by AMI_1:79; then A1: dom (Exec (i, s) | SCM-Instr-Loc) = SCM-Instr-Loc by RELAT_1:91; dom s = the carrier of SCM by AMI_1:79; then A2: dom (s | SCM-Instr-Loc) = SCM-Instr-Loc by RELAT_1:91; for x being set st x in SCM-Instr-Loc holds (Exec (i, s) | SCM-Instr-Loc).x = (s | SCM-Instr-Loc).x proof let x be set; assume x in SCM-Instr-Loc; then reconsider l = x as Instruction-Location of SCM by AMI_1:def 4; thus (Exec (i, s) | SCM-Instr-Loc).x = (Exec (i, s)).l by FUNCT_1:72 .= s.l by AMI_1:def 13 .= (s | SCM-Instr-Loc).x by FUNCT_1:72; end; hence Exec (i, s) | SCM-Instr-Loc = s | SCM-Instr-Loc by A1,A2,FUNCT_1:9; end; begin :: Finite partial states of SCM Lm1: for p being FinPartState of SCM holds DataPart p = p | SCM-Data-Loc proof now assume IC SCM in SCM-Data-Loc; then IC SCM is Data-Location by AMI_3:def 2; hence contradiction by Th20; end; then SCM-Data-Loc misses {IC SCM} by ZFMISC_1:56; then A1: SCM-Data-Loc misses {IC SCM} \/ SCM-Instr-Loc by AMI_2:29,XBOOLE_1:70; the carrier of SCM = {IC SCM} \/ (the Instruction-Locations of SCM) \/ SCM-Data-Loc by Th23,XBOOLE_1:4; then (the carrier of SCM) \ ({IC SCM} \/ the Instruction-Locations of SCM) = SCM-Data-Loc \ ({IC SCM} \/ the Instruction-Locations of SCM) by XBOOLE_1:40 .= SCM-Data-Loc by A1,XBOOLE_1:83; hence thesis; end; Lm2: for f being FinPartState of SCM holds f is data-only iff dom f c= SCM-Data-Loc proof let f be FinPartState of SCM; dom f c= the carrier of SCM by AMI_1:80; then A1: dom f c= {IC SCM} \/ SCM-Instr-Loc \/ SCM-Data-Loc by Th23,XBOOLE_1:4 ; now assume IC SCM in SCM-Data-Loc; then IC SCM is Data-Location by AMI_3:def 2; hence contradiction by Th20; end; then SCM-Data-Loc misses {IC SCM} by ZFMISC_1:56; then SCM-Data-Loc misses {IC SCM} \/ SCM-Instr-Loc by AMI_2:29,XBOOLE_1:70; then dom f misses {IC SCM} \/ SCM-Instr-Loc iff dom f c= SCM-Data-Loc by A1,XBOOLE_1:63,73; hence thesis by AMI_1:def 50; end; canceled 9; theorem for p being FinPartState of SCM holds dom DataPart p c= SCM-Data-Loc proof let p be FinPartState of SCM; DataPart p = p|SCM-Data-Loc by Lm1; hence dom DataPart p c= SCM-Data-Loc by RELAT_1:87; end; canceled 7; theorem for i being Instruction of SCM, s being State of SCM, p being programmed FinPartState of SCM holds Exec (i, s +* p) = Exec (i,s) +* p proof let i be Instruction of SCM, s be State of SCM, p be programmed FinPartState of SCM; A1: dom p c= the Instruction-Locations of SCM by AMI_1:def 40; now assume {IC SCM} meets SCM-Instr-Loc; then consider x being set such that A2: x in {IC SCM} and A3: x in SCM-Instr-Loc by XBOOLE_0:3; reconsider l = x as Instruction-Location of SCM by AMI_1:def 4,A3; l = IC SCM by A2,TARSKI:def 1; hence contradiction by AMI_1:48; end; then SCM-Data-Loc \/ {IC SCM} misses SCM-Instr-Loc by AMI_2:29,XBOOLE_1:70; then A4: SCM-Data-Loc \/ {IC SCM} misses dom p by A1,XBOOLE_1:63; then A5: s|(SCM-Data-Loc \/ {IC SCM}) = (s +* p) | (SCM-Data-Loc \/ {IC SCM}) by FUNCT_4:76; A6: (Exec(i,s) +* p)|(SCM-Data-Loc \/ {IC SCM}) = Exec(i,s)|(SCM-Data-Loc \/ {IC SCM}) by A4,FUNCT_4:76 .= Exec(i,s +* p) | (SCM-Data-Loc \/ {IC SCM}) by A5,Th58; A7: Exec (i, s +* p)|SCM-Instr-Loc = (s +* p)|SCM-Instr-Loc by Th59 .= s |SCM-Instr-Loc +* p|SCM-Instr-Loc by FUNCT_4:75 .= Exec (i,s) |SCM-Instr-Loc +* p|SCM-Instr-Loc by Th59 .= (Exec (i, s) +* p)|SCM-Instr-Loc by FUNCT_4:75; thus Exec (i, s +* p) = Exec (i, s +* p)| dom(Exec (i, s +* p)) by RELAT_1:97 .= Exec (i, s +* p)| ({IC SCM} \/ SCM-Data-Loc \/ SCM-Instr-Loc) by Th23,AMI_1:79 .= (Exec (i, s) +* p)| ({IC SCM} \/ SCM-Data-Loc) +* (Exec (i, s) +* p)|SCM-Instr-Loc by A6,A7,FUNCT_4:83 .= (Exec (i,s) +* p)| the carrier of SCM by FUNCT_4:83,Th23 .= (Exec (i,s) +* p)| dom(Exec (i, s) +* p) by AMI_1:79 .= Exec (i,s) +* p by RELAT_1:97; end; canceled 2; theorem for s being State of SCM, iloc being Instruction-Location of SCM, a being Data-Location holds s.a = (s +* Start-At iloc).a proof let s be State of SCM, iloc be Instruction-Location of SCM, a be Data-Location; A1: dom (Start-At iloc) = {IC SCM} by FUNCOP_1:19; a in the carrier of SCM; then a in dom s by AMI_1:79; then A2: a in dom s \/ dom (Start-At iloc) by XBOOLE_0:def 2; a <> IC SCM by Th20; then not a in {IC SCM} by TARSKI:def 1; hence s.a = (s +* Start-At iloc).a by A1,A2,FUNCT_4:def 1; end; begin :: Autonomic finite partial states of SCM canceled 2; theorem Th83: for p being autonomic FinPartState of SCM st DataPart p <> {} holds IC SCM in dom p proof let p be autonomic FinPartState of SCM; assume DataPart p <> {}; then A1: dom DataPart p <> {} by RELAT_1:64; assume A2: not IC SCM in dom p; p is not autonomic proof consider d1 being Element of dom DataPart p; A3: d1 in dom DataPart p by A1; dom DataPart p c= the carrier of SCM by AMI_1:80; then reconsider d1 as Element of SCM by A3; DataPart p = p | SCM-Data-Loc by Lm1; then dom DataPart p c= SCM-Data-Loc by RELAT_1:87; then reconsider d1 as Data-Location by A3,AMI_3:def 2; consider d2 being Element of SCM-Data-Loc \ dom p; not SCM-Data-Loc c= dom p by FINSET_1:13; then A4: SCM-Data-Loc \ dom p <> {} by XBOOLE_1:37; then d2 in SCM-Data-Loc by XBOOLE_0:def 4; then reconsider d2 as Data-Location by AMI_3:def 2; consider il being Element of (the Instruction-Locations of SCM) \ dom p; not the Instruction-Locations of SCM c= dom p by FINSET_1:13; then A5: (the Instruction-Locations of SCM) \ dom p <> {} by XBOOLE_1:37; then il is Element of the Instruction-Locations of SCM by XBOOLE_0:def 4; then reconsider il as Instruction-Location of SCM by AMI_1:def 4; set p1 = p +* ((il .--> (d1:=d2)) +* ( d2.--> 0) +* Start-At il); set p2 = p +* ((il .--> (d1:=d2)) +* ( d2.--> 1) +* Start-At il); consider s1 being State of SCM such that A6: p1 c= s1 by AMI_1:82; consider s2 being State of SCM such that A7: p2 c= s2 by AMI_1:82; take s1,s2; A8: not d2 in dom p by A4,XBOOLE_0:def 4; A9: not il in dom p by A5,XBOOLE_0:def 4; dom p misses {IC SCM} by A2,ZFMISC_1:56; then A10: dom p /\ {IC SCM} = {} by XBOOLE_0:def 7; dom p misses {d2} by A8,ZFMISC_1:56; then A11: dom p /\ {d2} = {} by XBOOLE_0:def 7; A12: dom p misses {il} by A9,ZFMISC_1:56; dom ((il .--> (d1:=d2)) +* ( d2.--> 0) +* Start-At il) = dom((il .--> (d1:=d2)) +* ( d2.--> 0)) \/ dom(Start-At il) by FUNCT_4:def 1 .= dom((il .--> (d1:=d2)) +* ( d2.--> 0)) \/ {IC SCM} by FUNCOP_1:19 .= dom(il .--> (d1:=d2)) \/ dom ( d2.--> 0) \/ {IC SCM} by FUNCT_4:def 1 .= {il} \/ dom ( d2.--> 0) \/ {IC SCM} by CQC_LANG:5 .= {il} \/ {d2} \/ {IC SCM} by CQC_LANG:5; then dom p /\ dom ((il .--> (d1:=d2)) +* ( d2.--> 0) +* Start-At il) = dom p /\ ({il} \/ {d2}) \/ {} by A10,XBOOLE_1:23 .= dom p /\ {il} \/ {} by A11,XBOOLE_1:23 .= {} by A12,XBOOLE_0:def 7; then dom p misses dom ((il .--> (d1:=d2)) +* ( d2.--> 0) +* Start-At il) by XBOOLE_0:def 7; then p c= p1 by FUNCT_4:33; hence p c= s1 by A6,XBOOLE_1:1; dom p misses {IC SCM} by A2,ZFMISC_1:56; then A13: dom p /\ {IC SCM} = {} by XBOOLE_0:def 7; dom p misses {d2} by A8,ZFMISC_1:56; then A14: dom p /\ {d2} = {} by XBOOLE_0:def 7; A15: dom p misses {il} by A9,ZFMISC_1:56; dom ((il .--> (d1:=d2)) +* (d2.--> 1) +* Start-At il) = dom((il .--> (d1:=d2)) +* ( d2.--> 1)) \/ dom(Start-At il) by FUNCT_4:def 1 .= dom((il .--> (d1:=d2)) +* ( d2.--> 1)) \/ {IC SCM} by FUNCOP_1:19 .= dom(il .--> (d1:=d2)) \/ dom ( d2.--> 1) \/ {IC SCM} by FUNCT_4:def 1 .= {il} \/ dom ( d2.--> 1) \/ {IC SCM} by CQC_LANG:5 .= {il} \/ {d2} \/ {IC SCM} by CQC_LANG:5; then dom p /\ dom ((il .--> (d1:=d2)) +* ( d2.--> 1) +* Start-At il) = dom p /\ ({il} \/ {d2}) \/ {} by A13,XBOOLE_1:23 .= dom p /\ {il} \/ {} by A14,XBOOLE_1:23 .= {} by A15,XBOOLE_0:def 7; then dom p misses dom ((il .--> (d1:=d2)) +* ( d2.--> 1) +* Start-At il) by XBOOLE_0:def 7; then p c= p2 by FUNCT_4:33; hence p c= s2 by A7,XBOOLE_1:1; take 1; DataPart p c= p by RELAT_1:88; then A16: dom DataPart p c= dom p by RELAT_1:25; dom ((Computation s1).1) = the carrier of SCM by AMI_1:79; then dom p c= dom ((Computation s1).1) by AMI_1:80; then A17: dom ((Computation s1).1|dom p) = dom p by RELAT_1:91; A18: dom(Start-At il) = {IC SCM} by FUNCOP_1:19; then A19: IC SCM in dom (Start-At il) by TARSKI:def 1; A20: dom ((il .--> (d1:=d2)) +* ( d2.--> 0) +* Start-At il) = dom ((il .--> (d1:=d2)) +* ( d2.--> 0)) \/ dom(Start-At il) by FUNCT_4:def 1; then A21: IC SCM in dom ((il .--> (d1:=d2)) +* ( d2.--> 0) +* Start-At il) by A19,XBOOLE_0:def 2; A22: dom p1 = dom p \/ dom ((il .--> (d1:=d2)) +* ( d2.--> 0) +* Start-At il) by FUNCT_4:def 1; then IC SCM in dom p1 by A21,XBOOLE_0:def 2; then A23: IC s1 = p1.IC SCM by A6,GRFUNC_1:8 .= ((il .--> (d1:=d2)) +* ( d2.--> 0) +* Start-At il).IC SCM by A21,FUNCT_4:14 .= (Start-At il).IC SCM by A19,FUNCT_4:14 .= il by CQC_LANG:6; dom (il .--> (d1:=d2)) = {il} by CQC_LANG:5; then A24: il in dom (il .--> (d1:=d2)) by TARSKI:def 1; A25: dom (d2 .--> 0) = {d2} by CQC_LANG:5; il <> d2 by Th22; then A26: not il in dom (d2 .--> 0) by A25,TARSKI:def 1; A27: dom ((il .--> (d1:=d2)) +* ( d2.--> 0)) = dom (il .--> (d1:=d2)) \/ dom ( d2.--> 0) by FUNCT_4:def 1; then A28: il in dom ((il .--> (d1:=d2)) +* ( d2.--> 0)) by A24,XBOOLE_0:def 2; il <> IC SCM by AMI_1:48; then A29: not il in dom (Start-At il) by A18,TARSKI:def 1; A30: il in dom ((il .--> (d1:=d2)) +* ( d2.--> 0) +* Start-At il) by A20,A28,XBOOLE_0:def 2; then il in dom p1 by A22,XBOOLE_0:def 2; then A31: s1.il = p1.il by A6,GRFUNC_1:8 .= ((il .--> (d1:=d2)) +* ( d2.--> 0) +* Start-At il).il by A30,FUNCT_4:14 .= ((il .--> (d1:=d2)) +* ( d2.--> 0)).il by A29,FUNCT_4:12 .= (il .--> (d1:=d2)).il by A26,FUNCT_4:12 .=(d1:=d2) by CQC_LANG:6; A32: d2 in dom (d2 .--> 0) by A25,TARSKI:def 1; then A33: d2 in dom ((il .--> (d1:=d2)) +* ( d2.--> 0)) by A27,XBOOLE_0:def 2; d2 <> IC SCM by Th20; then A34: not d2 in dom (Start-At il) by A18,TARSKI:def 1; A35: d2 in dom ((il .--> (d1:=d2)) +* ( d2.--> 0) +* Start-At il) by A20,A33,XBOOLE_0:def 2; then d2 in dom p1 by A22,XBOOLE_0:def 2; then A36: s1.d2 = p1.d2 by A6,GRFUNC_1:8 .= ((il .--> (d1:=d2)) +* ( d2.--> 0) +* Start-At il).d2 by A35,FUNCT_4:14 .= ((il .--> (d1:=d2)) +* ( d2.--> 0)).d2 by A34,FUNCT_4:12 .= (d2.--> 0).d2 by A32,FUNCT_4:14 .= 0 by CQC_LANG:6; (Computation s1).(0+1).d1 = (Following (Computation s1).0).d1 by AMI_1:def 19 .= (Following s1).d1 by AMI_1:def 19 .= 0 by A23,A31,A36,AMI_3:8; then A37: ((Computation s1).1|dom p).d1 = 0 by A3,A16,A17,FUNCT_1:70; dom ((Computation s2).1) = the carrier of SCM by AMI_1:79; then dom p c= dom ((Computation s2).1) by AMI_1:80; then A38: dom ((Computation s2).1|dom p) = dom p by RELAT_1:91; A39: dom(Start-At il) = {IC SCM} by FUNCOP_1:19; then A40: IC SCM in dom (Start-At il) by TARSKI:def 1; A41: dom ((il .--> (d1:=d2)) +* ( d2.--> 1) +* Start-At il) = dom ((il .--> (d1:=d2)) +* ( d2.--> 1)) \/ dom(Start-At il) by FUNCT_4:def 1; then A42: IC SCM in dom ((il .--> (d1:=d2)) +* ( d2.--> 1) +* Start-At il) by A40,XBOOLE_0:def 2; A43: dom p2 = dom p \/ dom ((il .--> (d1:=d2)) +* ( d2.--> 1) +* Start-At il) by FUNCT_4:def 1; then IC SCM in dom p2 by A42,XBOOLE_0:def 2; then A44: IC s2 = p2.IC SCM by A7,GRFUNC_1:8 .= ((il .--> (d1:=d2)) +* ( d2.--> 1) +* Start-At il).IC SCM by A42,FUNCT_4:14 .= (Start-At il).IC SCM by A40,FUNCT_4:14 .= il by CQC_LANG:6; dom (il .--> (d1:=d2)) = {il} by CQC_LANG:5; then A45: il in dom (il .--> (d1:=d2)) by TARSKI:def 1; A46: dom (d2 .--> 1) = {d2} by CQC_LANG:5; il <> d2 by Th22; then A47: not il in dom (d2 .--> 1) by A46,TARSKI:def 1; A48: dom ((il .--> (d1:=d2)) +* ( d2.--> 1)) = dom (il .--> (d1:=d2)) \/ dom ( d2.--> 1) by FUNCT_4:def 1; then A49: il in dom ((il .--> (d1:=d2)) +* ( d2.--> 1)) by A45,XBOOLE_0:def 2; il <> IC SCM by AMI_1:48; then A50: not il in dom (Start-At il) by A39,TARSKI:def 1; A51: il in dom ((il .--> (d1:=d2)) +* ( d2.--> 1) +* Start-At il) by A41,A49,XBOOLE_0:def 2; then il in dom p2 by A43,XBOOLE_0:def 2; then A52: s2.il = p2.il by A7,GRFUNC_1:8 .= ((il .--> (d1:=d2)) +* ( d2.--> 1) +* Start-At il).il by A51,FUNCT_4:14 .= ((il .--> (d1:=d2)) +* ( d2.--> 1)).il by A50,FUNCT_4:12 .= (il .--> (d1:=d2)).il by A47,FUNCT_4:12 .=(d1:=d2) by CQC_LANG:6; A53: d2 in dom (d2 .--> 1) by A46,TARSKI:def 1; then A54: d2 in dom ((il .--> (d1:=d2)) +* ( d2.--> 1)) by A48,XBOOLE_0:def 2; d2 <> IC SCM by Th20; then A55: not d2 in dom (Start-At il) by A39,TARSKI:def 1; A56: d2 in dom ((il .--> (d1:=d2)) +* ( d2.--> 1) +* Start-At il) by A41,A54,XBOOLE_0:def 2; then d2 in dom p2 by A43,XBOOLE_0:def 2; then A57: s2.d2 = p2.d2 by A7,GRFUNC_1:8 .= ((il .--> (d1:=d2)) +* ( d2.--> 1) +* Start-At il).d2 by A56,FUNCT_4:14 .= ((il .--> (d1:=d2)) +* ( d2.--> 1)).d2 by A55,FUNCT_4:12 .= (d2.--> 1).d2 by A53,FUNCT_4:14 .= 1 by CQC_LANG:6; (Computation s2).(0+1).d1 = (Following (Computation s2).0).d1 by AMI_1:def 19 .= (Following s2).d1 by AMI_1:def 19 .= 1 by A44,A52,A57,AMI_3:8; hence (Computation s1).1|dom p <> (Computation s2).1|dom p by A3,A16,A37 ,A38,FUNCT_1:70; end; hence contradiction; end; registration cluster autonomic non programmed FinPartState of SCM; existence proof take p = (Start-At il.0) +* Euclide-Algorithm +* (dl.0,dl.1) --> (1,1); (dl.0,dl.1) --> (1,1) in dom Euclide-Function by AMI_4:11; then consider s being FinPartState of SCM such that A1: (dl.0,dl.1) --> (1,1) = s and A2: (Start-At il.0) +* Euclide-Algorithm +* s is pre-program of SCM and Euclide-Function.s c= Result((Start-At il.0) +* Euclide-Algorithm +* s ) by AMI_1:def 29,AMI_4:13; thus p is autonomic by A1,A2; take IC SCM; A3: dom p = dom ((Start-At il.0) +* Euclide-Algorithm) \/ dom((dl.0,dl.1) --> (1,1)) by FUNCT_4:def 1; A4: dom ((Start-At il.0) +* Euclide-Algorithm) = dom (Start-At il.0) \/ dom (Euclide-Algorithm) by FUNCT_4:def 1; dom (Start-At il.0) = {IC SCM} by FUNCOP_1:19; then IC SCM in dom (Start-At il.0) by TARSKI:def 1; then IC SCM in dom ((Start-At il.0) +* Euclide-Algorithm) by A4, XBOOLE_0:def 2; hence IC SCM in dom p by A3,XBOOLE_0:def 2; assume IC SCM in the Instruction-Locations of SCM; then reconsider il = IC SCM as Instruction-Location of SCM by AMI_1:def 4; il in the Instruction-Locations of SCM; hence contradiction by Th19,SCM_1:7; end; end; theorem Th84: for p being autonomic non programmed FinPartState of SCM holds IC SCM in dom p proof let p be autonomic non programmed FinPartState of SCM; A1: not dom p c= SCM-Instr-Loc by AMI_1:def 40; dom p c= the carrier of SCM by AMI_1:80; then dom p = dom p /\ the carrier of SCM by XBOOLE_1:28 .= dom p /\ ({IC SCM} \/ SCM-Data-Loc) \/ dom p /\ SCM-Instr-Loc by Th23,XBOOLE_1:23; then dom p /\ ({IC SCM} \/ SCM-Data-Loc) <> {} by A1,XBOOLE_1:17; then A2: dom p /\ {IC SCM} \/ dom p /\ SCM-Data-Loc <> {} by XBOOLE_1:23; per cases by A2; suppose dom p /\ {IC SCM} <> {}; then dom p meets {IC SCM} by XBOOLE_0:def 7; hence IC SCM in dom p by ZFMISC_1:56; end; suppose A3: dom p /\ SCM-Data-Loc <> {}; DataPart p = p | SCM-Data-Loc by Lm1; then DataPart p <> {} by A3,RELAT_1:60,90; hence IC SCM in dom p by Th83; end; end; theorem for p being autonomic FinPartState of SCM st IC SCM in dom p holds IC p in dom p proof let p be autonomic FinPartState of SCM; assume A1: IC SCM in dom p; assume A2: not IC p in dom p; set il = IC p; set p1 = p +* ((il .--> goto il.0)); set p2 = p +* ((il .--> goto il.1)); consider s1 being State of SCM such that A3: p1 c= s1 by AMI_1:82; consider s2 being State of SCM such that A4: p2 c= s2 by AMI_1:82; p is not autonomic proof A5: dom (il .--> (goto il.1)) = {il} by CQC_LANG:5; A6: dom (il .--> (goto il.0)) = {il} by CQC_LANG:5; take s1,s2; dom p misses {il} by A2,ZFMISC_1:56; then p c= p1 & p c= p2 by A5,A6,FUNCT_4:33; hence A7: p c= s1 & p c= s2 by A3,A4,XBOOLE_1:1; take 1; A8: il in dom (il .--> (goto il.1)) by A5,TARSKI:def 1; A9: il in dom (il .--> (goto il.0)) by A6,TARSKI:def 1; dom p1 = dom p \/ dom ((il .--> goto il.0)) by FUNCT_4:def 1; then il in dom p1 by A9,XBOOLE_0:def 2; then A10: s1.il = p1.il by A3,GRFUNC_1:8 .= ((il .--> goto il.0)).il by A9,FUNCT_4:14 .= goto il.0 by CQC_LANG:6; dom p2 = dom p \/ dom ((il .--> goto il.1)) by FUNCT_4:def 1; then il in dom p2 by A8,XBOOLE_0:def 2; then A11: s2.il = p2.il by A4,GRFUNC_1:8 .= ((il .--> goto il.1)).il by A8,FUNCT_4:14 .= goto il.1 by CQC_LANG:6; A12: (Following s1).IC SCM = Exec (goto il.0,s1).IC SCM by A1,A7,A10,AMI_1:97 .= il.0 by AMI_3:13; A13: (Following s2).IC SCM = Exec (goto il.1,s2).IC SCM by A1,A7,A11,AMI_1:97 .= il.1 by AMI_3:13; assume A14: (Computation s1).1|dom p = (Computation s2).1|dom p; A15: (Following(s1))|dom p = (Following ((Computation s1).0))|dom p by AMI_1:def 19 .= (Computation s1).(0+1)|dom p by AMI_1:def 19 .= (Following ((Computation s2).0))|dom p by A14, AMI_1:def 19 .= (Following(s2))|dom p by AMI_1:def 19; il.0 = ((Following(s1))|dom p).IC SCM by A1,A12,FUNCT_1:72 .= il.1 by A1,A13,A15,FUNCT_1:72; hence contradiction; end; hence contradiction; end; theorem Th86: for p being autonomic non programmed FinPartState of SCM, s being State of SCM st p c= s for i being Element of NAT holds IC (Computation s).i in dom ProgramPart(p) proof let p be autonomic non programmed FinPartState of SCM, s be State of SCM such that A1: p c= s; let i be Element of NAT; set Csi = (Computation s).i; set loc = IC Csi; consider ll being Element of NAT such that A2: loc = il.ll by Th19; set loc1 = il.(ll+1); assume A3: not IC (Computation s).i in dom ProgramPart(p); loc in dom ProgramPart p iff loc in dom p /\ SCM-Instr-Loc by FUNCT_1:68; then A4:not loc in dom p by A3,XBOOLE_0:def 3; set p1 = p +* (loc .--> goto loc); set p2 = p +* (loc .--> goto loc1); A5: dom p1 = dom p \/ dom (loc .--> goto loc) & dom p2 = dom p \/ dom (loc .--> goto loc1) by FUNCT_4:def 1; A6: dom (loc .--> goto loc) = {loc} & dom (loc .--> goto loc1) = {loc} by CQC_LANG:5; then A7: loc in dom (loc .--> goto loc) & loc in dom (loc .--> goto loc1) by TARSKI:def 1; then A8: loc in dom p1 & loc in dom p2 by A5,XBOOLE_0:def 2; consider s1 being State of SCM such that A9: p1 c= s1 by AMI_1:82; consider s2 being State of SCM such that A10: p2 c= s2 by AMI_1:82; set Cs1i = (Computation s1).i; set Cs2i = (Computation s2).i; A11: IC SCM in dom p by Th84; p is not autonomic proof take s1, s2; dom s1 = the carrier of SCM & dom s2 = the carrier of SCM by AMI_1:79; then A12: dom p c= dom s1 & dom p c= dom s2 by AMI_1:80; now let x be set; assume A13: x in dom p; then dom p misses dom (loc .--> goto loc) & x in dom p1 by A4,A5,A6,XBOOLE_0:def 2,ZFMISC_1:56; then p.x = p1.x & p1.x = s1.x by A9,A13,FUNCT_4:17,GRFUNC_1:8; hence p.x = s1.x; end; hence A14: p c= s1 by A12,GRFUNC_1:8; now let x be set; assume A15: x in dom p; then dom p misses dom (loc .--> goto loc1) & x in dom p2 by A4,A5,A6,XBOOLE_0:def 2,ZFMISC_1:56; then p.x = p2.x & p2.x = s2.x by A10,A15,FUNCT_4:17,GRFUNC_1:8; hence p.x = s2.x; end; hence A16: p c= s2 by A12,GRFUNC_1:8; (loc .--> goto loc).loc = goto loc & (loc .--> goto loc1).loc = goto loc1 by CQC_LANG:6; then p1.loc = goto loc & p2.loc = goto loc1 by A7,FUNCT_4:14; then A17: s1.loc = goto loc & s2.loc = goto loc1 by A8,A9,A10,GRFUNC_1:8; take k = i+1; set Cs1k = (Computation s1).k; set Cs2k = (Computation s2).k; A18: Cs1k = Following Cs1i by AMI_1:def 19 .= Exec (CurInstr Cs1i, Cs1i); A19: Cs2k = Following Cs2i by AMI_1:def 19 .= Exec (CurInstr Cs2i, Cs2i); A20: Cs1i.loc = goto loc & Cs2i.loc = goto loc1 by A17,AMI_1:54; A21: (Cs1i|dom p) = (Csi|dom p) by A1,A14,AMI_1:def 25; A22: Cs1i.IC SCM = (Cs1i|dom p).IC SCM & Csi.IC SCM = (Csi|dom p).IC SCM by A11,FUNCT_1:72; (Cs1i|dom p) = (Cs2i|dom p) by A14,A16,AMI_1:def 25; then Cs1i.IC SCM = loc & Cs2i.IC SCM = loc by A11,A21,A22,FUNCT_1:72; then A23: Cs1k.IC SCM = loc & Cs2k.IC SCM = loc1 by A18,A19,A20,AMI_3:13; (Cs1k|dom p).IC SCM = Cs1k.IC SCM & (Cs2k|dom p).IC SCM = Cs2k.IC SCM by A11,FUNCT_1:72; hence Cs1k|dom p <> Cs2k|dom p by A2,A23; end; hence contradiction; end; theorem Th87: for p being autonomic non programmed FinPartState of SCM, s1, s2 being State of SCM st p c= s1 & p c= s2 for i being Element of NAT, I being Instruction of SCM st I = CurInstr ((Computation s1).i) holds IC (Computation s1).i = IC (Computation s2).i & I = CurInstr ((Computation s2).i) proof let p be autonomic non programmed FinPartState of SCM, s1, s2 be State of SCM such that A1: p c= s1 & p c= s2; let i be Element of NAT, I be Instruction of SCM such that A2: I = CurInstr ((Computation s1).i); set Cs1i = (Computation s1).i; set Cs2i = (Computation s2).i; A3: IC SCM in dom p by Th84; thus A4: IC Cs1i = IC Cs2i proof assume A5: IC (Computation s1).i <> IC (Computation s2).i; (Cs1i|dom p).IC SCM = Cs1i.IC SCM & (Cs2i|dom p).IC SCM = Cs2i.IC SCM by A3,FUNCT_1:72; hence contradiction by A1,A5,AMI_1:def 25; end; thus I = CurInstr ((Computation s2).i) proof assume A6: I <> CurInstr ((Computation s2).i); A7: IC Cs1i in dom ProgramPart p & IC Cs2i in dom ProgramPart p by A1,Th86; ProgramPart p c= p by RELAT_1:88; then dom ProgramPart p c= dom p by GRFUNC_1:8; then (Cs1i|dom p).IC Cs1i = Cs1i.IC Cs1i & (Cs2i|dom p).IC Cs2i = Cs2i.IC Cs2i by A7,FUNCT_1:72; hence contradiction by A1,A2,A4,A6,AMI_1:def 25; end; end; theorem for p being autonomic non programmed FinPartState of SCM, s1, s2 being State of SCM st p c= s1 & p c= s2 for i being Element of NAT, da, db being Data-Location, I being Instruction of SCM st I = CurInstr ((Computation s1).i) holds I = da := db & da in dom p implies (Computation s1).i.db = (Computation s2).i.db proof let p be autonomic non programmed FinPartState of SCM, s1, s2 be State of SCM such that A1: p c= s1 & p c= s2; let i be Element of NAT, da, db be Data-Location, I be Instruction of SCM such that A2: I = CurInstr ((Computation s1).i); set Cs1i = (Computation s1).i; set Cs2i = (Computation s2).i; A3: I = CurInstr ((Computation s2).i) by A1,A2,Th87; set Cs1i1 = (Computation s1).(i+1); set Cs2i1 = (Computation s2).(i+1); A4: Cs1i1 = Following Cs1i by AMI_1:def 19 .= Exec (CurInstr Cs1i, Cs1i); A5: Cs2i1 = Following Cs2i by AMI_1:def 19 .= Exec (CurInstr Cs2i, Cs2i); A6: da in dom p implies (Cs1i1|dom p).da = Cs1i1.da & (Cs2i1|dom p).da = Cs2i1.da by FUNCT_1:72; assume A7: I = da := db & da in dom p & (Computation s1).i.db <> (Computation s2).i.db; then Cs1i1.da = Cs1i.db & Cs2i1.da = Cs2i.db by A2,A3,A4,A5,AMI_3:8; hence contradiction by A1,A6,A7,AMI_1:def 25; end; theorem for p being autonomic non programmed FinPartState of SCM, s1, s2 being State of SCM st p c= s1 & p c= s2 for i being Element of NAT, da, db being Data-Location, I being Instruction of SCM st I = CurInstr ((Computation s1).i) holds I = AddTo(da, db) & da in dom p implies (Computation s1).i.da + (Computation s1).i.db = (Computation s2).i.da + (Computation s2).i.db proof let p be autonomic non programmed FinPartState of SCM, s1, s2 be State of SCM such that A1: p c= s1 & p c= s2; let i be Element of NAT, da, db be Data-Location, I be Instruction of SCM such that A2: I = CurInstr ((Computation s1).i); set Cs1i = (Computation s1).i; set Cs2i = (Computation s2).i; A3: I = CurInstr ((Computation s2).i) by A1,A2,Th87; set Cs1i1 = (Computation s1).(i+1); set Cs2i1 = (Computation s2).(i+1); A4: Cs1i1 = Following Cs1i by AMI_1:def 19 .= Exec (CurInstr Cs1i, Cs1i); A5: Cs2i1 = Following Cs2i by AMI_1:def 19 .= Exec (CurInstr Cs2i, Cs2i); A6: da in dom p implies (Cs1i1|dom p).da = Cs1i1.da & (Cs2i1|dom p).da = Cs2i1.da by FUNCT_1:72; assume A7: I = AddTo(da, db) & da in dom p & (Computation s1).i.da + (Computation s1).i.db <> (Computation s2).i.da + (Computation s2).i.db; then Cs1i1.da = Cs1i.da + Cs1i.db & Cs2i1.da = Cs2i.da + Cs2i.db by A2,A3,A4,A5,AMI_3:9; hence contradiction by A1,A6,A7,AMI_1:def 25; end; theorem for p being autonomic non programmed FinPartState of SCM, s1, s2 being State of SCM st p c= s1 & p c= s2 for i being Element of NAT, da, db being Data-Location, I being Instruction of SCM st I = CurInstr ((Computation s1).i) holds I = SubFrom(da, db) & da in dom p implies (Computation s1).i.da - (Computation s1).i.db = (Computation s2).i.da - (Computation s2).i.db proof let p be autonomic non programmed FinPartState of SCM, s1, s2 be State of SCM such that A1: p c= s1 & p c= s2; let i be Element of NAT, da, db be Data-Location, I be Instruction of SCM such that A2: I = CurInstr ((Computation s1).i); set Cs1i = (Computation s1).i; set Cs2i = (Computation s2).i; A3: I = CurInstr ((Computation s2).i) by A1,A2,Th87; set Cs1i1 = (Computation s1).(i+1); set Cs2i1 = (Computation s2).(i+1); A4: Cs1i1 = Following Cs1i by AMI_1:def 19 .= Exec (CurInstr Cs1i, Cs1i); A5: Cs2i1 = Following Cs2i by AMI_1:def 19 .= Exec (CurInstr Cs2i, Cs2i); A6: da in dom p implies (Cs1i1|dom p).da = Cs1i1.da & (Cs2i1|dom p).da = Cs2i1.da by FUNCT_1:72; assume A7: I = SubFrom(da, db) & da in dom p & (Computation s1).i.da - (Computation s1).i.db <> (Computation s2).i.da - (Computation s2).i.db; then Cs1i1.da = Cs1i.da - Cs1i.db & Cs2i1.da = Cs2i.da - Cs2i.db by A2,A3,A4,A5,AMI_3:10; hence contradiction by A1,A6,A7,AMI_1:def 25; end; theorem for p being autonomic non programmed FinPartState of SCM, s1, s2 being State of SCM st p c= s1 & p c= s2 for i being Element of NAT, da, db being Data-Location, I being Instruction of SCM st I = CurInstr ((Computation s1).i) holds I = MultBy(da, db) & da in dom p implies (Computation s1).i.da * (Computation s1).i.db = (Computation s2).i.da * (Computation s2).i.db proof let p be autonomic non programmed FinPartState of SCM, s1, s2 be State of SCM such that A1: p c= s1 & p c= s2; let i be Element of NAT, da, db be Data-Location, I be Instruction of SCM such that A2: I = CurInstr ((Computation s1).i); set Cs1i = (Computation s1).i; set Cs2i = (Computation s2).i; A3: I = CurInstr ((Computation s2).i) by A1,A2,Th87; set Cs1i1 = (Computation s1).(i+1); set Cs2i1 = (Computation s2).(i+1); A4: Cs1i1 = Following Cs1i by AMI_1:def 19 .= Exec (CurInstr Cs1i, Cs1i); A5: Cs2i1 = Following Cs2i by AMI_1:def 19 .= Exec (CurInstr Cs2i, Cs2i); A6: da in dom p implies (Cs1i1|dom p).da = Cs1i1.da & (Cs2i1|dom p).da = Cs2i1.da by FUNCT_1:72; assume A7: I = MultBy(da, db) & da in dom p & (Computation s1).i.da * (Computation s1).i.db <> (Computation s2).i.da * (Computation s2).i.db; then Cs1i1.da = Cs1i.da * Cs1i.db & Cs2i1.da = Cs2i.da * Cs2i.db by A2,A3,A4,A5,AMI_3:11; hence contradiction by A1,A6,A7,AMI_1:def 25; end; theorem for p being autonomic non programmed FinPartState of SCM, s1, s2 being State of SCM st p c= s1 & p c= s2 for i being Element of NAT, da, db being Data-Location, I being Instruction of SCM st I = CurInstr ((Computation s1).i) holds I = Divide(da, db) & da in dom p & da <> db implies (Computation s1).i.da div (Computation s1).i.db = (Computation s2).i.da div (Computation s2).i.db proof let p be autonomic non programmed FinPartState of SCM, s1, s2 be State of SCM such that A1: p c= s1 & p c= s2; let i be Element of NAT, da, db be Data-Location, I be Instruction of SCM such that A2: I = CurInstr ((Computation s1).i); set Cs1i = (Computation s1).i; set Cs2i = (Computation s2).i; A3: I = CurInstr ((Computation s2).i) by A1,A2,Th87; set Cs1i1 = (Computation s1).(i+1); set Cs2i1 = (Computation s2).(i+1); A4: Cs1i1 = Following Cs1i by AMI_1:def 19 .= Exec (CurInstr Cs1i, Cs1i); A5: Cs2i1 = Following Cs2i by AMI_1:def 19 .= Exec (CurInstr Cs2i, Cs2i); A6: da in dom p implies (Cs1i1|dom p).da = Cs1i1.da & (Cs2i1|dom p).da = Cs2i1.da by FUNCT_1:72; assume A7: I = Divide(da, db) & da in dom p & da <> db & (Computation s1).i.da div (Computation s1).i.db <> (Computation s2).i.da div (Computation s2).i.db; then Cs1i1.da = Cs1i.da div Cs1i.db & Cs2i1.da = Cs2i.da div Cs2i.db by A2,A3,A4,A5,AMI_3:12; hence contradiction by A1,A6,A7,AMI_1:def 25; end; theorem for p being autonomic non programmed FinPartState of SCM, s1, s2 being State of SCM st p c= s1 & p c= s2 for i being Element of NAT, da, db being Data-Location, I being Instruction of SCM st I = CurInstr ((Computation s1).i) holds I = Divide(da, db) & db in dom p & da <> db implies (Computation s1).i.da mod (Computation s1).i.db = (Computation s2).i.da mod (Computation s2).i.db proof let p be autonomic non programmed FinPartState of SCM, s1, s2 be State of SCM such that A1: p c= s1 & p c= s2; let i be Element of NAT, da, db be Data-Location, I be Instruction of SCM such that A2: I = CurInstr ((Computation s1).i); set Cs1i = (Computation s1).i; set Cs2i = (Computation s2).i; A3: I = CurInstr ((Computation s2).i) by A1,A2,Th87; set Cs1i1 = (Computation s1).(i+1); set Cs2i1 = (Computation s2).(i+1); A4: Cs1i1 = Following Cs1i by AMI_1:def 19 .= Exec (CurInstr Cs1i, Cs1i); A5: Cs2i1 = Following Cs2i by AMI_1:def 19 .= Exec (CurInstr Cs2i, Cs2i); assume A6: I = Divide(da, db) & db in dom p & da <> db & (Computation s1).i.da mod (Computation s1).i.db <> (Computation s2).i.da mod (Computation s2).i.db; then A7: (Cs1i1|dom p).db = Cs1i1.db & (Cs2i1|dom p).db = Cs2i1.db by FUNCT_1:72; Cs1i1.db = Cs1i.da mod Cs1i.db & Cs2i1.db = Cs2i.da mod Cs2i.db by A2,A3,A4,A5,A6,AMI_3:12; hence contradiction by A1,A6,A7,AMI_1:def 25; end; theorem for p being autonomic non programmed FinPartState of SCM, s1, s2 being State of SCM st p c= s1 & p c= s2 for i being Element of NAT, da being Data-Location, loc being Instruction-Location of SCM, I being Instruction of SCM st I = CurInstr ((Computation s1).i) holds I = da=0_goto loc & loc <> Next (IC (Computation s1).i) implies ((Computation s1).i.da = 0 iff (Computation s2).i.da = 0) proof let p be autonomic non programmed FinPartState of SCM, s1, s2 be State of SCM such that A1: p c= s1 & p c= s2; let i be Element of NAT, da be Data-Location, loc be Instruction-Location of SCM, I be Instruction of SCM such that A2: I = CurInstr ((Computation s1).i); set Cs1i = (Computation s1).i; set Cs2i = (Computation s2).i; A3: IC SCM in dom p by Th84; A4: I = CurInstr ((Computation s2).i) by A1,A2,Th87; set Cs1i1 = (Computation s1).(i+1); set Cs2i1 = (Computation s2).(i+1); A5: Cs1i1 = Following Cs1i by AMI_1:def 19 .= Exec (CurInstr Cs1i, Cs1i); A6: Cs2i1 = Following Cs2i by AMI_1:def 19 .= Exec (CurInstr Cs2i, Cs2i); A7:(Cs1i1|dom p).IC SCM = Cs1i1.IC SCM & (Cs2i1|dom p).IC SCM = Cs2i1.IC SCM by A3,FUNCT_1:72; A8: (Cs1i1|dom p) = (Cs2i1|dom p) by A1,AMI_1:def 25; assume A9: I = da=0_goto loc & loc <> Next (IC (Computation s1).i); A10: now assume (Computation s1).i.da = 0 & (Computation s2).i.da <> 0; then Cs1i1.IC SCM = loc & Cs2i1.IC SCM = Next IC Cs2i by A2,A4,A5,A6,A9,AMI_3:14; hence contradiction by A1,A2,A7,A8,A9,Th87; end; now assume (Computation s2).i.da = 0 & (Computation s1).i.da <> 0; then Cs2i1.IC SCM = loc & Cs1i1.IC SCM = Next IC Cs1i by A2,A4,A5,A6,A9,AMI_3:14; hence contradiction by A1,A7,A9,AMI_1:def 25; end; hence (Computation s1).i.da = 0 iff (Computation s2).i.da = 0 by A10; end; theorem for p being autonomic non programmed FinPartState of SCM, s1, s2 being State of SCM st p c= s1 & p c= s2 for i being Element of NAT, da being Data-Location, loc being Instruction-Location of SCM, I being Instruction of SCM st I = CurInstr ((Computation s1).i) holds I = da>0_goto loc & loc <> Next (IC (Computation s1).i) implies ((Computation s1).i.da > 0 iff (Computation s2).i.da > 0) proof let p be autonomic non programmed FinPartState of SCM, s1, s2 be State of SCM such that A1: p c= s1 & p c= s2; let i be Element of NAT, da be Data-Location, loc be Instruction-Location of SCM, I be Instruction of SCM such that A2: I = CurInstr ((Computation s1).i); set Cs1i = (Computation s1).i; set Cs2i = (Computation s2).i; A3: IC SCM in dom p by Th84; A4: IC Cs1i = IC Cs2i by A1,A2,Th87; A5: I = CurInstr ((Computation s2).i) by A1,A2,Th87; set Cs1i1 = (Computation s1).(i+1); set Cs2i1 = (Computation s2).(i+1); A6: Cs1i1 = Following Cs1i by AMI_1:def 19 .= Exec (CurInstr Cs1i, Cs1i); A7: Cs2i1 = Following Cs2i by AMI_1:def 19 .= Exec (CurInstr Cs2i, Cs2i); A8: (Cs1i1|dom p).IC SCM = Cs1i1.IC SCM & (Cs2i1|dom p).IC SCM = Cs2i1.IC SCM by A3,FUNCT_1:72; A9: (Cs1i1|dom p) = (Cs2i1|dom p) by A1,AMI_1:def 25; assume A10: I = da>0_goto loc & loc <> Next (IC (Computation s1).i); A11: now assume A12: (Computation s1).i.da > 0 & (Computation s2).i.da <= 0; then Cs1i1.IC SCM = loc by A2,A6,A10,AMI_3:15; hence contradiction by A4,A5,A7,A8,A9,A10,A12,AMI_3:15; end; now assume A13: (Computation s2).i.da > 0 & (Computation s1).i.da <= 0; then Cs2i1.IC SCM = loc by A5,A7,A10,AMI_3:15; hence contradiction by A2,A6,A8,A9,A10,A13,AMI_3:15; end; hence (Computation s1).i.da > 0 iff (Computation s2).i.da > 0 by A11; end; theorem for p being FinPartState of SCM holds DataPart p = p | SCM-Data-Loc by Lm1; theorem for f being FinPartState of SCM holds f is data-only iff dom f c= SCM-Data-Loc by Lm2;