### The Mizar article:

### On the Decomposition of the States of SCM

**by****Yasushi Tanaka**

- Received November 23, 1993
Copyright (c) 1993 Association of Mizar Users

- MML identifier: AMI_5
- [ MML identifier index ]

environ vocabulary BOOLE, NAT_1, ARYTM_1, FUNCT_1, RELAT_1, FUNCT_4, PARTFUN1, AMI_3, AMI_1, AMI_2, GR_CY_1, FINSEQ_1, CARD_3, FINSET_1, TARSKI, CAT_1, FUNCOP_1, MCART_1, ORDINAL2, QC_LANG1, AMI_4, AMI_5, ARYTM; notation TARSKI, XBOOLE_0, SUBSET_1, ORDINAL1, ORDINAL2, NUMBERS, XCMPLX_0, XREAL_0, CARD_3, MCART_1, DOMAIN_1, RELAT_1, FUNCT_1, FUNCT_2, FUNCT_4, INT_1, NAT_1, PARTFUN1, STRUCT_0, GR_CY_1, CQC_LANG, FINSET_1, FINSEQ_1, CAT_3, AMI_1, AMI_2, AMI_3, AMI_4, BINARITH; constructors WELLORD2, DOMAIN_1, PARTFUN1, AMI_2, AMI_4, BINARITH, FINSEQ_4, CAT_3, MEMBERED, XBOOLE_0; clusters AMI_1, AMI_2, AMI_3, INT_1, FUNCT_1, RELSET_1, XBOOLE_0, FINSEQ_1, FRAENKEL, MEMBERED, ZFMISC_1, NUMBERS, ORDINAL2, ARYTM_3; requirements NUMERALS, REAL, SUBSET, BOOLE, ARITHM; definitions AMI_1, TARSKI, AMI_3, WELLORD2, FUNCT_1, XBOOLE_0; theorems AMI_1, AMI_2, AMI_3, GRFUNC_1, NAT_1, SCM_1, CQC_LANG, TARSKI, FUNCOP_1, FUNCT_4, FUNCT_1, PARTFUN1, MCART_1, GR_CY_1, FUNCT_2, SUBSET_1, BINARITH, CARD_3, FINSET_1, ZFMISC_1, AMI_4, ALGSEQ_1, ENUMSET1, CQC_THE1, CARD_1, CARD_4, RELAT_1, ORDINAL2, XBOOLE_0, XBOOLE_1, XCMPLX_1; schemes FUNCT_2; begin canceled 2; theorem for m,k being Nat st k <> 0 holds m * k div k = m proof let m,k be Nat; assume k <> 0; then m*k = k * m + 0 & 0 < k or m = 0 & k = 0 by NAT_1:19; hence m*k div k = m by NAT_1:def 1; end; theorem for i,j being natural number st i >= j holds i -' j + j = i proof let i,j be natural number; assume i >= j; then ex m being Nat st i = j + m by NAT_1:28; hence i -' j + j = i by BINARITH:39; end; theorem for f,g being Function, A,B being set st A c= B & f|B = g|B holds f|A = g|A proof let f,g be Function, A,B be set; assume that A1: A c= B and A2: f|B = g|B; A3: A = B /\ A by A1,XBOOLE_1:28; hence f|A = (f|B)|A by RELAT_1:100 .= g|A by A2,A3,RELAT_1:100; end; theorem Th6: for p,q being Function , A being set holds (p +* q)|A = p|A +* q|A proof let p,q be Function , A be set; A1: dom ((p +* q)|A) = dom (p +* q) /\ A by RELAT_1:90 .= (dom p \/ dom q) /\ A by FUNCT_4:def 1 .= (dom p /\ A) \/ (dom q /\ A) by XBOOLE_1:23 .= dom (p|A) \/ (dom q /\ A) by RELAT_1:90 .= dom (p|A) \/ dom (q|A) by RELAT_1:90; for x being set st x in dom (p|A) \/ dom (q|A) holds (x in dom (q|A) implies ((p +* q)|A).x = (q|A).x) & (not x in dom (q|A) implies ((p +* q)|A).x = (p|A).x) proof let x be set; assume A2:x in dom (p|A) \/ dom (q|A); then x in dom (p|A) or x in dom (q|A) by XBOOLE_0:def 2; then x in (dom p /\ A) or (x in dom q /\ A) by RELAT_1:90; then A3: x in A by XBOOLE_0:def 3; hereby assume A4: x in dom (q|A); then x in (dom q /\ A) by RELAT_1:90; then A5: x in dom q by XBOOLE_0:def 3; thus ((p +* q)|A).x = (p +* q).x by A1,A2,FUNCT_1:70 .= q.x by A5,FUNCT_4:14 .= (q|A).x by A4,FUNCT_1:70; end; ::hereby assume A6: not x in dom (q|A); then not x in (dom q /\ A) by RELAT_1:90; then A7: not x in dom q by A3,XBOOLE_0:def 3; A8: x in dom (p|A) by A2,A6,XBOOLE_0:def 2; then x in dom p /\ A by RELAT_1:90; then x in dom p by XBOOLE_0:def 3; then x in dom (p +* q) by FUNCT_4:13; then x in dom (p +* q) /\ A by A3,XBOOLE_0:def 3; then x in dom ((p +* q)|A) by RELAT_1:90; hence ((p +* q)|A).x = (p +* q).x by FUNCT_1:70 .= p.x by A7,FUNCT_4:12 .= (p|A).x by A8,FUNCT_1:70; end; hence thesis by A1,FUNCT_4:def 1; end; theorem Th7: for f,g being Function, A being set st A misses dom g holds (f +* g)|A = f|A proof let f,g be Function, A be set; assume A misses dom g; then dom g /\ A = {} by XBOOLE_0:def 7; then dom (g|A) = {} by RELAT_1:90; then g|A is Function of {},{} by FUNCT_2:55; then g|A = {} by PARTFUN1:57; hence (f +* g)|A = f|A +* {} by Th6 .= f|A by FUNCT_4:22; end; theorem for f,g being Function , A being set holds dom f misses A implies (f +* g)|A = g|A proof let f,g be Function , A be set; assume dom f misses A; then dom f /\ A = {} by XBOOLE_0:def 7; then dom (f|A) = {} by RELAT_1:90; then f|A is Function of {},{} by FUNCT_2:55; then f|A = {} by PARTFUN1:57; hence (f +* g)|A = {} +* g|A by Th6 .= g|A by FUNCT_4:21; end; theorem for f,g,h being Function st dom g = dom h holds f +* g +* h = f +* h proof let f,g,h be Function; assume A1:dom g = dom h; thus f +* g +* h = f +* (g +* h) by FUNCT_4:15 .= f +* h by A1,FUNCT_4:20; end; theorem Th10: for f,g being Function holds f c= g implies f +* g = g & g +* f = g proof let f,g be Function; assume A1: f c= g; then dom f c= dom g by GRFUNC_1:8; hence A2: f +* g = g by FUNCT_4:20; f tolerates g by A1,PARTFUN1:135; hence g +* f = g by A2,FUNCT_4:35; end; theorem for f being Function, A being set holds f +* f|A = f proof let f be Function, A be set; (f|A) c= f by RELAT_1:88; hence f +* f|A = f by Th10; end; theorem for f,g being Function, B,C being set st dom f c= B & dom g c= C & B misses C holds (f +* g)|B = f & (f +* g)|C = g proof let f,g be Function, B,C be set; assume that A1: dom f c= B and A2: dom g c= C and A3: B misses C; dom f misses C by A1,A3,XBOOLE_1:63; then A4: dom f /\ C = {} by XBOOLE_0:def 7; dom g misses B by A2,A3,XBOOLE_1:63; then dom g /\ B = {} by XBOOLE_0:def 7; then dom (g|B) = {} by RELAT_1:90; then g|B is Function of {},{} by FUNCT_2:55; then A5: g|B = {} by PARTFUN1:57; dom (f|C) = {} by A4,RELAT_1:90; then f|C is Function of {},{} by FUNCT_2:55; then A6: f|C = {} by PARTFUN1:57; thus (f +* g)|B = f|B +* g|B by Th6 .= f|B by A5,FUNCT_4:22 .= f by A1,RELAT_1:97; thus (f +* g)|C = f|C +* g|C by Th6 .= g|C by A6,FUNCT_4:21 .= g by A2,RELAT_1:97; end; theorem for p,q being Function, A being set holds dom p c= A & dom q misses A implies (p +* q)|A = p proof let p,q be Function, A be set; assume that A1: dom p c= A and A2: dom q misses A; thus (p +* q )|A = p|A by A2,Th7 .= p by A1,RELAT_1:97; end; theorem Th14: for f being Function, A,B being set holds f|(A \/ B) = f|A +* f|B proof let f be Function, A,B be set; A1: f|(A \/ B)|A = f|((A \/ B) /\ A) by RELAT_1:100 .= f|A by XBOOLE_1:21; A2: f|(A \/ B)|B = f|((A \/ B) /\ B) by RELAT_1:100 .= f|B by XBOOLE_1:21; dom (f|(A \/ B)) c= A \/ B by RELAT_1:87; hence f|(A \/ B) = f|A +* f|B by A1,A2,AMI_1:16; end; begin :: Total states of SCM :: AMI_1:48' theorem Th15: for a being Data-Location, s being State of SCM holds Exec(Divide(a,a), s).IC SCM = Next IC s & Exec(Divide(a,a), s).a = s.a mod s.a & for c being Data-Location st c <> a holds Exec(Divide(a,a), s).c = s.c proof let a be Data-Location, s be State of SCM; reconsider mk = a as Element of SCM-Data-Loc by AMI_3:def 2; reconsider I = Divide(a,a) as Element of SCM-Instr by AMI_3:def 1; reconsider S = s as SCM-State by AMI_3:def 1; set S1 = SCM-Chg(S, I address_1,S.(I address_1) div S.(I address_2)); set S1' = SCM-Chg(S1, I address_2,S.(I address_1) mod S.(I address_2)); reconsider i = 5 as Element of Segm 9 by GR_CY_1:10; A1: I = [ i, <*mk, mk*>] by AMI_3:def 7; A2: Exec(Divide(a,a), s) = SCM-Exec-Res(I,S) by AMI_3:7 .= (SCM-Chg(S1', Next IC S)) by A1,AMI_2:def 16; A3: I address_1 = mk & I address_2 = mk by A1,AMI_2:23; thus Exec(Divide(a,a), s).IC SCM = Next IC s by AMI_3:12; thus Exec(Divide(a,a), s).a = S1'.mk by A2,AMI_2:17 .= s.a mod s.a by A3,AMI_2:20; thus thesis by AMI_3:12; end; theorem Th16: for x being set st x in SCM-Data-Loc holds x is Data-Location proof let y be set such that A1: y in SCM-Data-Loc; thus y is Data-Location by A1,AMI_3:def 1,def 2; end; canceled; theorem Th18: for dl being Data-Location ex i being Nat st dl = dl.i proof let dl be Data-Location; dl in SCM-Data-Loc by AMI_3:def 2; then consider k being Nat such that A1: dl = 2*k +1 by AMI_2:def 2; consider i being Nat such that A2: k = i; take i; thus dl = dl.i by A1,A2,AMI_3:def 19; end; theorem Th19: for il being Instruction-Location of SCM ex i being Nat st il = il.i proof let il be Instruction-Location of SCM; il in SCM-Instr-Loc by AMI_3:def 1; then consider k being Nat such that A1: il = 2*k & k > 0 by AMI_2:def 3; consider i being Nat such that A2: k = i + 1 by A1,NAT_1:22; take i; thus il = 2*i + 2*1 by A1,A2,XCMPLX_1:8 .= il.i by AMI_3:def 20; end; theorem Th20: for dl being Data-Location holds dl <> IC SCM proof let dl be Data-Location; consider i being Nat such that A1: dl = dl.i by Th18; thus thesis by A1,AMI_3:57; end; reserve N for with_non-empty_elements set, S for IC-Ins-separated definite (non empty non void AMI-Struct over N); 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 Nat such that A1: il = il.i by Th19; consider j being Nat such that A2: dl = dl.j by Th18; thus il <> dl by A1,A2,AMI_3:56; end; reserve i, j, k for Nat; theorem Th23: the carrier of SCM = {IC SCM} \/ SCM-Data-Loc \/ SCM-Instr-Loc proof A1: NAT c= {0} \/ { 2*k + 1: not contradiction } \/ { 2*j : j > 0} proof let x be set; assume x in NAT; then reconsider n = x as Nat; A2: n div 2 = 0 or n div 2 > 0 by NAT_1:19; n mod 2 < 2 by NAT_1:46; then n mod 2 = 0 or n mod 2 =1 by ALGSEQ_1:4; then A3: n = 2 * (n div 2) + 0 or n = 2 * (n div 2) + 1 by NAT_1:47; per cases by A2,A3; suppose x = 0; then x in {0} by TARSKI:def 1; then x in {0} \/ { 2*k + 1: not contradiction } by XBOOLE_0:def 2; hence x in {0} \/ { 2*k + 1: not contradiction } \/ { 2*j : j > 0} by XBOOLE_0:def 2; suppose ex k st x = 2*k +1; then x in { 2*k +1: not contradiction}; then x in {0} \/ { 2*k +1: not contradiction} by XBOOLE_0:def 2; hence x in {0} \/ { 2*k + 1: not contradiction } \/ { 2*j : j > 0} by XBOOLE_0:def 2; suppose ex k st x = 2*k & k > 0; then x in { 2*j : j > 0}; hence x in {0} \/ { 2*k + 1: not contradiction } \/ { 2*j : j > 0} by XBOOLE_0:def 2; end; {IC SCM} c= NAT by AMI_3:4,ZFMISC_1:37; then {IC SCM} \/ SCM-Data-Loc c= NAT by XBOOLE_1:8; then {IC SCM} \/ SCM-Data-Loc \/ SCM-Instr-Loc c= NAT by XBOOLE_1:8; hence thesis by A1,AMI_2:def 2,def 3,AMI_3:4,def 1,XBOOLE_0:def 10; end; 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_3:36; l in {IC SCM} \/ SCM-Data-Loc \/ SCM-Instr-Loc by AMI_3:def 1,XBOOLE_0:def 2; hence l in dom s by Th23,AMI_3:36; end; theorem Th25: for s being State of S holds IC S in dom s proof let s be State of S; dom s = the carrier of S by AMI_3:36; hence IC S in dom s; end; 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 AMI_3:def 1, 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 AMI_3:def 1,CARD_3:def 5; A6: NAT = dom g1 & NAT = dom g2 by A4,A5,FUNCT_2:def 1; now let x be set such that A7: x in NAT; A8: x in {IC SCM} \/ SCM-Data-Loc or x in SCM-Instr-Loc by A7,Th23,AMI_3:def 1,XBOOLE_0:def 2; per cases by A8,XBOOLE_0:def 2; suppose x in {IC SCM}; then A9: x = IC SCM by TARSKI:def 1; s1.IC SCM = IC(s2) by A1,AMI_1:def 15 .= s2.IC SCM by AMI_1:def 15; hence g1.x = g2.x by A4,A5,A9; suppose x in SCM-Data-Loc; then x is Data-Location by Th16; hence g1.x = g2.x by A2,A4,A5; suppose x in SCM-Instr-Loc; hence g1.x = g2.x by A3,A4,A5,AMI_3:def 1; 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_3:36; 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_3:36; 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) = 2*$1 + 1; consider f being Function of NAT, NAT such that A1: for x being Element of NAT holds f.x = F(x) from LambdaD; 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 Nat by A3,A4,FUNCT_2:def 1; dl.k1 = 2 * k1 + 1 by AMI_3:def 19 .= f.k1 by A1 .= 2 * k2 + 1 by A1,A5 .= dl.k2 by AMI_3:def 19; 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 Nat by A6,FUNCT_2:def 1; y = 2 * x + 1 by A1,A7 .= dl.x by AMI_3:def 19; 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 1,def 2; consider k being Nat such that A9: d = dl.k by Th18; y = 2 * k + 1 by A9,AMI_3:def 19 .= f.k by 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) = 2*$1 + 2; consider f being Function of NAT, NAT such that A1: for x being Element of NAT holds f.x = F(x) from LambdaD; A2: dom f = NAT by FUNCT_2:def 1; NAT,SCM-Instr-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 Nat by A3,A4,FUNCT_2:def 1; il.k1 = 2 * k1 + 2 by AMI_3:def 20 .= f.k1 by A1 .= 2 * k2 + 2 by A1,A5 .= il.k2 by AMI_3:def 20; hence x1 = x2 by AMI_3:53; end; thus dom f = NAT by FUNCT_2:def 1; thus rng f c= SCM-Instr-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 Nat by A6,FUNCT_2:def 1; y = 2 * x + 2 by A1,A7 .= il.x by AMI_3:def 20; hence y in SCM-Instr-Loc by AMI_3:def 1; end; thus SCM-Instr-Loc c= rng f proof let y be set; assume y in SCM-Instr-Loc; then reconsider d = y as Instruction-Location of SCM by AMI_3:def 1; consider k being Nat such that A8: d = il.k by Th19; y = 2 * k + 2 by A8,AMI_3:def 20 .= f.k by A1; hence y in rng f by A2,FUNCT_1:def 5; end; end; hence the Instruction-Locations of SCM is not finite by AMI_3:def 1,CARD_1: 68,CARD_4:15; end; definition cluster SCM-Data-Loc -> infinite; coherence by Th31; cluster the Instruction-Locations of SCM -> infinite; coherence by Th32; end; theorem Th33: SCM-Data-Loc misses SCM-Instr-Loc proof { 2*i + 1: not contradiction } /\ { 2*k : k > 0 } = {} proof consider x being Element of { 2*i + 1: not contradiction } /\ { 2*k : k > 0 }; assume { 2*i + 1: not contradiction } /\ { 2*k : k > 0 } <> {}; then A1: x in { 2*i + 1: not contradiction } & x in { 2*k : k > 0 } by XBOOLE_0:def 3; then consider i such that A2: x = 2*i + 1; consider k such that A3: x = 2*k & k > 0 by A1; consider l being Nat such that A4: k = l + 1 by A3,NAT_1:22; x = 2*l + 2*1 by A3,A4,XCMPLX_1:8 .= 2*l + 2; then x = dl.i & x = il.l by A2,AMI_3:def 19,def 20; hence contradiction by SCM_1:7; end; hence SCM-Data-Loc misses SCM-Instr-Loc by AMI_2:def 2,def 3,XBOOLE_0:def 7; end; theorem for s being State of S holds Start-At(IC s) = s | {IC S} proof let s be State of S; A1: IC S in dom s by Th25; thus Start-At(IC s) = IC S .--> IC s by AMI_3:def 12 .= IC S .--> s.IC S by AMI_1:def 15 .= {[IC S,s.IC S]} by AMI_1:19 .= s | {IC S} by A1,AMI_3:22; end; theorem Th35: for l be Instruction-Location of S holds Start-At l = {[IC S,l]} proof let l be Instruction-Location of S; thus Start-At l = IC S .--> l by AMI_3:def 12 .= {IC S} --> l by CQC_LANG:def 2 .= [:{IC S}, { l }:] by FUNCOP_1:def 2 .= {[IC S, l ]} by ZFMISC_1:35; end; definition let N be set, A be AMI-Struct over N; let I be Element of the Instructions of A; func InsCode I -> InsType of A equals :Def1: I `1; coherence proof reconsider I as Instruction of A; I`1 in the Instruction-Codes of A; hence thesis; end; end; definition let I be Instruction of SCM; cluster InsCode I -> natural; coherence proof InsCode I in Segm 9 by AMI_3:def 1; hence thesis by ORDINAL2:def 21; end; end; definition let I be Instruction of SCM; func @I -> Element of SCM-Instr equals :Def2: I; coherence by AMI_3:def 1; end; definition let loc be Element of SCM-Instr-Loc; func loc@ -> Instruction-Location of SCM equals :Def3: loc; coherence by AMI_3:def 1; end; definition let loc be Element of SCM-Data-Loc; func loc@ -> Data-Location equals :Def4: loc; coherence proof thus thesis by AMI_3:def 1,def 2; end; end; theorem Th36: for l being Instruction of SCM holds InsCode(l) <= 8 proof let l be Instruction of SCM; InsCode(l) < 8+1 by AMI_3:def 1,GR_CY_1:10; hence InsCode(l) <= 8 by NAT_1:38; end; reserve a, b for Data-Location, loc for Instruction-Location of SCM; theorem Th37: InsCode (halt SCM) = 0 proof thus InsCode (halt SCM) = (halt SCM)`1 by Def1 .= 0 by AMI_3:71,MCART_1:7; end; theorem InsCode (a:=b) = 1 proof A1: a:=b = [ 1, <*a, b*>] by AMI_3:def 3; thus InsCode (a:=b) = (a:=b)`1 by Def1 .= 1 by A1,MCART_1:7; end; theorem InsCode (AddTo(a,b)) = 2 proof A1: AddTo(a,b) = [ 2, <*a, b*>] by AMI_3:def 4; thus InsCode (AddTo(a,b)) = (AddTo(a,b))`1 by Def1 .= 2 by A1,MCART_1:7; end; theorem InsCode (SubFrom(a,b)) = 3 proof A1: SubFrom(a,b) = [ 3, <*a, b*>] by AMI_3:def 5; thus InsCode (SubFrom(a,b)) = (SubFrom(a,b))`1 by Def1 .= 3 by A1,MCART_1:7; end; theorem InsCode (MultBy(a,b)) = 4 proof A1: MultBy(a,b) = [ 4, <*a, b*>] by AMI_3:def 6; thus InsCode (MultBy(a,b)) = (MultBy(a,b))`1 by Def1 .= 4 by A1,MCART_1:7; end; theorem InsCode (Divide(a,b)) = 5 proof A1: Divide(a,b) = [ 5, <*a, b*>] by AMI_3:def 7; thus InsCode (Divide(a,b)) = (Divide(a,b))`1 by Def1 .= 5 by A1,MCART_1:7; end; theorem InsCode (goto loc) = 6 proof A1: goto loc = [ 6, <*loc*>] by AMI_3:def 8; thus InsCode (goto loc) = (goto loc)`1 by Def1 .= 6 by A1,MCART_1:7; end; theorem InsCode (a=0_goto loc) = 7 proof A1: a=0_goto loc = [ 7, <*loc,a*>] by AMI_3:def 9; thus InsCode (a=0_goto loc) = (a=0_goto loc)`1 by Def1 .= 7 by A1,MCART_1:7; end; theorem InsCode (a>0_goto loc) = 8 proof A1: a>0_goto loc = [ 8, <*loc,a*>] by AMI_3:def 10; thus InsCode (a>0_goto loc) = (a>0_goto loc)`1 by Def1 .= 8 by A1,MCART_1:7; 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, da,db for Data-Location, loc for Instruction-Location of SCM; 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 = ins `1 by Def1 .= I by A3,MCART_1:7; hence contradiction by A1,A4,ENUMSET1:23; 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 = ins `1 by Def1 .= 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; InsCode ins = ins `1 by Def1 .= J by A9,MCART_1:7; hence contradiction by A1,A10; end; ins in { [SCM-Halt,{}] } \/ { [J,<*a*>] : J = 6 } \/ { [K,<*a1,b1*>] : K in { 7,8 } } by A2,AMI_2:def 4,AMI_3:def 1, 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_2:def 1,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_2:def 1,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 = ins `1 by Def1 .= 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; InsCode ins = ins `1 by Def1 .= J by A6,MCART_1:7; hence contradiction by A1,A7; 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 AMI_2:def 4,AMI_3:def 1 ,XBOOLE_0:def 2; then consider I,b,c such that A8: ins = [I,<*b,c*>] and I in { 1,2,3,4,5}; A9: InsCode ins = ins `1 by Def1 .= I by A8,MCART_1:7; reconsider da = b@ ,db = c@ as Data-Location; take da,db; b = b@ & c = c@ by Def4; hence ins = da:=db by A1,A8,A9,AMI_3:def 3; 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_2:def 1,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 = ins `1 by Def1 .= 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; InsCode ins = ins `1 by Def1 .= J by A6,MCART_1:7; hence contradiction by A1,A7; 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 AMI_2:def 4,AMI_3:def 1 ,XBOOLE_0:def 2; then consider I,b,c such that A8: ins = [I,<*b,c*>] and I in { 1,2,3,4,5}; A9: InsCode ins = ins `1 by Def1 .= I by A8,MCART_1:7; reconsider da = b@ ,db = c@ as Data-Location; take da,db; b = b@ & c = c@ by Def4; hence ins = AddTo(da,db) by A1,A8,A9,AMI_3:def 4; 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_2:def 1,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 = ins `1 by Def1 .= 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; InsCode ins = ins `1 by Def1 .= J by A6,MCART_1:7; hence contradiction by A1,A7; 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 AMI_2:def 4,AMI_3:def 1 ,XBOOLE_0:def 2; then consider I,b,c such that A8: ins = [I,<*b,c*>] and I in { 1,2,3,4,5}; A9: InsCode ins = ins `1 by Def1 .= I by A8,MCART_1:7; reconsider da = b@ ,db = c@ as Data-Location; take da,db; b = b@ & c = c@ by Def4; hence ins = SubFrom(da,db) by A1,A8,A9,AMI_3:def 5; 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_2:def 1,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 = ins `1 by Def1 .= 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; InsCode ins = ins `1 by Def1 .= J by A6,MCART_1:7; hence contradiction by A1,A7; 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 AMI_2:def 4,AMI_3:def 1 ,XBOOLE_0:def 2; then consider I,b,c such that A8: ins = [I,<*b,c*>] and I in { 1,2,3,4,5}; A9: InsCode ins = ins `1 by Def1 .= I by A8,MCART_1:7; reconsider da = b@ ,db = c@ as Data-Location; take da,db; b = b@ & c = c@ by Def4; hence ins = MultBy(da,db) by A1,A8,A9,AMI_3:def 6; 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_2:def 1,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 = ins `1 by Def1 .= 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; InsCode ins = ins `1 by Def1 .= J by A6,MCART_1:7; hence contradiction by A1,A7; 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 AMI_2:def 4,AMI_3:def 1 ,XBOOLE_0:def 2; then consider I,b,c such that A8: ins = [I,<*b,c*>] and I in { 1,2,3,4,5}; A9: InsCode ins = ins `1 by Def1 .= I by A8,MCART_1:7; reconsider da = b@ ,db = c@ as Data-Location; take da,db; b = b@ & c = c@ by Def4; hence ins = Divide (da,db) by A1,A8,A9,AMI_3:def 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_2:def 1,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 = ins `1 by Def1 .= 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 = ins `1 by Def1 .= I by A6,MCART_1:7; hence contradiction by A1,A7,ENUMSET1:23; end; then ins in { [SCM-Halt,{}] } \/ { [J,<*a*>] : J = 6 } \/ { [K,<*a1,b1*>] : K in { 7,8 } } by AMI_2:def 4,AMI_3:def 1,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; ins = [6,<*a@*>] by A8,A9,Def3; hence ins = goto loc by AMI_3:def 8; 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_2:def 1,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; InsCode ins = ins `1 by Def1 .= J by A4,MCART_1:7; hence contradiction by A1,A5; 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 = ins `1 by Def1 .= I by A7,MCART_1:7; hence contradiction by A1,A8,ENUMSET1:23; 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,AMI_2:def 4,AMI_3:def 1, 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 }; A11: InsCode ins = ins `1 by Def1 .= K by A10,MCART_1:7; reconsider loc = a1@ as Instruction-Location of SCM; reconsider da = b1@ as Data-Location; take loc,da; a1 = a1@ & b1 = b1@ by Def3,Def4; hence ins = da=0_goto loc by A1,A10,A11,AMI_3:def 9; 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_2:def 1,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; InsCode ins = ins `1 by Def1 .= J by A4,MCART_1:7; hence contradiction by A1,A5; 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 = ins `1 by Def1 .= I by A7,MCART_1:7; hence contradiction by A1,A8,ENUMSET1:23; 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,AMI_2:def 4,AMI_3:def 1, 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 }; A11: InsCode ins = ins `1 by Def1 .= K by A10,MCART_1:7; reconsider loc = a1@ as Instruction-Location of SCM; reconsider da = b1@ as Data-Location; take loc,da; a1 = a1@ & b1 = b1@ by Def3,Def4; hence ins = da>0_goto loc by A1,A10,A11,AMI_3:def 10; 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 by AMI_3:def 1; @(goto loc) = goto loc by Def2 .= [ roku, <*mk*>] by AMI_3:def 8; 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 by AMI_3:def 1; reconsider aa=a as Element of SCM-Data-Loc by AMI_3:def 2; @(a=0_goto loc) = a=0_goto loc by Def2 .= [ nana, <*mk,aa*>] by AMI_3:def 9; 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 by AMI_3:def 1; reconsider aa=a as Element of SCM-Data-Loc by AMI_3:def 2; @(a>0_goto loc) = a>0_goto loc by Def2 .= [ hachi, <*mk,aa*>] by AMI_3:def 10; 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_3:36; 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_3:36; then A5: IC SCM in dom (s2 | (SCM-Data-Loc \/ {IC SCM})) by A2,RELAT_1:91; A6: IC s1 = s1.IC SCM by AMI_1:def 15 .= (s2 | (SCM-Data-Loc \/ {IC SCM})).IC SCM by A1,A4,FUNCT_1:70 .= s2.IC SCM by A5,FUNCT_1:70 .= IC s2 by AMI_1:def 15; let l be Instruction of SCM; A7: dom Exec(l,s1) = the carrier of SCM by AMI_3:36; A8: dom Exec(l,s2) = the carrier of SCM by AMI_3:36; A9: SCM-Data-Loc c= (SCM-Data-Loc \/ {IC SCM}) by XBOOLE_1:7; then A10: SCM-Data-Loc c= dom(Exec (l,s1)) by A3,A7,XBOOLE_1:1; A11: SCM-Data-Loc c= dom(Exec (l,s2)) by A3,A8,A9,XBOOLE_1:1; A12: InsCode(l) <= 8 by Th36; per cases by A12,CQC_THE1:9; suppose InsCode (l) = 0; then A13: 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 A13,AMI_1:def 8; suppose InsCode (l) = 1; then consider da,db such that A14: l = da:=db by Th47; da in SCM-Data-Loc by AMI_3:def 2; then A15: SCM-Data-Loc = SCM-Data-Loc \/ {da} by ZFMISC_1:46 .= (SCM-Data-Loc \ {da} ) \/ {da} by XBOOLE_1:39; SCM-Data-Loc \ {da} c= SCM-Data-Loc by XBOOLE_1:36; then SCM-Data-Loc \ {da} c= dom(Exec (l,s1)) by A10,XBOOLE_1:1; then A16: dom ((Exec (l,s1)) | (SCM-Data-Loc \ {da})) = (SCM-Data-Loc \ {da}) by RELAT_1:91; SCM-Data-Loc \ {da} c= SCM-Data-Loc by XBOOLE_1:36; then SCM-Data-Loc \ {da} c= dom(Exec (l,s2)) by A11,XBOOLE_1:1; then A17: dom ((Exec (l,s2)) | (SCM-Data-Loc \ {da})) = (SCM-Data-Loc \ {da}) by 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 A18: x in ((SCM-Data-Loc) \ {da}); then A19: x in SCM-Data-Loc by XBOOLE_0:def 4; A20: not x in {da} by A18,XBOOLE_0:def 4; reconsider a = x as Data-Location by A19,AMI_3:def 1,def 2; A21: a <> da by A20,TARSKI:def 1; A22: a in (SCM-Data-Loc \/ {IC SCM}) by A19,XBOOLE_0:def 2; thus (Exec (l,s1) | (SCM-Data-Loc \ {da})).x = (Exec (l,s1)).a by A18,FUNCT_1:72 .= s1.a by A14,A21,AMI_3:8 .= (s1 | (SCM-Data-Loc \/ {IC SCM})).a by A22,FUNCT_1:72 .= s2.a by A1,A22,FUNCT_1:72 .= (Exec (l,s2)).a by A14,A21,AMI_3:8 .= (Exec (l,s2) | (SCM-Data-Loc \ {da})).x by A18,FUNCT_1:72; end; then A23: Exec (l,s1) | (SCM-Data-Loc \ {da} ) = Exec (l,s2) | (SCM-Data-Loc \ {da} ) by A16,A17,FUNCT_1:9; A24: db in SCM-Data-Loc by AMI_3:def 2; Exec (l,s1).da = s1.db by A14,AMI_3:8 .= (s1 | (SCM-Data-Loc \/ {IC SCM})).db by A9,A24,FUNCT_1:72 .= s2.db by A1,A9,A24,FUNCT_1:72 .= Exec (l,s2).da by A14,AMI_3:8; then Exec (l,s1) | {da} = Exec(l,s2) | {da} by A7,A8,AMI_3:24; then A25: Exec (l,s1) | SCM-Data-Loc = Exec (l,s2) | SCM-Data-Loc by A15,A23,AMI_3:20; Exec (l,s1).IC SCM = Next IC s1 by A14,AMI_3:8 .= Exec (l,s2).IC SCM by A6,A14,AMI_3:8; then Exec (l,s1) | {IC SCM} = Exec (l,s2) | {IC SCM} by A7,A8,AMI_3:24; hence Exec (l,s1) | (SCM-Data-Loc \/ {IC SCM}) = Exec (l,s2) | (SCM-Data-Loc \/ {IC SCM}) by A25,AMI_3:20; suppose InsCode (l) = 2; then consider da,db such that A26: l = AddTo(da,db) by Th48; da in SCM-Data-Loc by AMI_3:def 2; then A27: SCM-Data-Loc = SCM-Data-Loc \/ {da} by ZFMISC_1:46 .= (SCM-Data-Loc \ {da} ) \/ {da} by XBOOLE_1:39; SCM-Data-Loc \ {da} c= SCM-Data-Loc by XBOOLE_1:36; then SCM-Data-Loc \ {da} c= dom(Exec (l,s1)) by A10,XBOOLE_1:1; then A28: dom ((Exec (l,s1)) | (SCM-Data-Loc \ {da})) = (SCM-Data-Loc \ {da}) by RELAT_1:91; SCM-Data-Loc \ {da} c= SCM-Data-Loc by XBOOLE_1:36; then SCM-Data-Loc \ {da} c= dom(Exec (l,s2)) by A11,XBOOLE_1:1; then A29: dom ((Exec (l,s2)) | (SCM-Data-Loc \ {da})) = (SCM-Data-Loc \ {da}) by 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 A30: x in ((SCM-Data-Loc) \ {da}); then A31: x in SCM-Data-Loc by XBOOLE_0:def 4; A32: not x in {da} by A30,XBOOLE_0:def 4; reconsider a = x as Data-Location by A31,AMI_3:def 1,def 2; A33: a <> da by A32,TARSKI:def 1; A34: a in (SCM-Data-Loc \/ {IC SCM}) by A31,XBOOLE_0:def 2; thus (Exec (l,s1) | (SCM-Data-Loc \ {da})).x = (Exec (l,s1)).a by A30,FUNCT_1:72 .= s1.a by A26,A33,AMI_3:9 .= (s1 | (SCM-Data-Loc \/ {IC SCM})).a by A34,FUNCT_1:72 .= s2.a by A1,A34,FUNCT_1:72 .= (Exec (l,s2)).a by A26,A33,AMI_3:9 .= (Exec (l,s2) | (SCM-Data-Loc \ {da})).x by A30,FUNCT_1:72; end; then A35: Exec (l,s1) | (SCM-Data-Loc \ {da} ) = Exec (l,s2) | (SCM-Data-Loc \ {da} ) by A28,A29,FUNCT_1:9; A36: db in SCM-Data-Loc by AMI_3:def 2; A37: da in SCM-Data-Loc by AMI_3:def 2; then A38: s1.da = (s1 | (SCM-Data-Loc \/ {IC SCM})).da by A9,FUNCT_1:72 .= s2.da by A1,A9,A37,FUNCT_1:72; A39: s1.db = (s1 | (SCM-Data-Loc \/ {IC SCM})).db by A9,A36,FUNCT_1:72 .= s2.db by A1,A9,A36,FUNCT_1:72; Exec (l,s1).da = s1.da + s1.db by A26,AMI_3:9 .= Exec (l,s2).da by A26,A38,A39,AMI_3:9; then Exec (l,s1) | {da} = Exec(l,s2) | {da} by A7,A8,AMI_3:24; then A40: Exec (l,s1) | SCM-Data-Loc = Exec (l,s2) | SCM-Data-Loc by A27,A35,AMI_3:20; Exec (l,s1).IC SCM = Next IC s1 by A26,AMI_3:9 .= Exec (l,s2).IC SCM by A6,A26,AMI_3:9; then Exec (l,s1) | {IC SCM} = Exec (l,s2) | {IC SCM} by A7,A8,AMI_3:24; hence Exec (l,s1) | (SCM-Data-Loc \/ {IC SCM}) = Exec (l,s2) | (SCM-Data-Loc \/ {IC SCM}) by A40,AMI_3:20; suppose InsCode (l) = 3; then consider da,db such that A41: l = SubFrom(da,db) by Th49; da in SCM-Data-Loc by AMI_3:def 2; then A42: SCM-Data-Loc = SCM-Data-Loc \/ {da} by ZFMISC_1:46 .= (SCM-Data-Loc \ {da} ) \/ {da} by XBOOLE_1:39; SCM-Data-Loc \ {da} c= SCM-Data-Loc by XBOOLE_1:36; then SCM-Data-Loc \ {da} c= dom(Exec (l,s1)) by A10,XBOOLE_1:1; then A43: dom ((Exec (l,s1)) | (SCM-Data-Loc \ {da})) = (SCM-Data-Loc \ {da}) by RELAT_1:91; SCM-Data-Loc \ {da} c= SCM-Data-Loc by XBOOLE_1:36; then SCM-Data-Loc \ {da} c= dom(Exec (l,s2)) by A11,XBOOLE_1:1; then A44: dom ((Exec (l,s2)) | (SCM-Data-Loc \ {da})) = (SCM-Data-Loc \ {da}) by 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 A45: x in ((SCM-Data-Loc) \ {da}); then A46: x in SCM-Data-Loc by XBOOLE_0:def 4; A47: not x in {da} by A45,XBOOLE_0:def 4; reconsider a = x as Data-Location by A46,AMI_3:def 1,def 2; A48: a <> da by A47,TARSKI:def 1; A49: a in (SCM-Data-Loc \/ {IC SCM}) by A46,XBOOLE_0:def 2; thus (Exec (l,s1) | (SCM-Data-Loc \ {da})).x = (Exec (l,s1)).a by A45,FUNCT_1:72 .= s1.a by A41,A48,AMI_3:10 .= (s1 | (SCM-Data-Loc \/ {IC SCM})).a by A49,FUNCT_1:72 .= s2.a by A1,A49,FUNCT_1:72 .= (Exec (l,s2)).a by A41,A48,AMI_3:10 .= (Exec (l,s2) | (SCM-Data-Loc \ {da})).x by A45,FUNCT_1:72; end; then A50: Exec (l,s1) | (SCM-Data-Loc \ {da} ) = Exec (l,s2) | (SCM-Data-Loc \ {da} ) by A43,A44,FUNCT_1:9; A51: db in SCM-Data-Loc by AMI_3:def 2; A52: da in SCM-Data-Loc by AMI_3:def 2; then A53: s1.da = (s1 | (SCM-Data-Loc \/ {IC SCM})).da by A9,FUNCT_1:72 .= s2.da by A1,A9,A52,FUNCT_1:72; A54: s1.db = (s1 | (SCM-Data-Loc \/ {IC SCM})).db by A9,A51,FUNCT_1:72 .= s2.db by A1,A9,A51,FUNCT_1:72; Exec (l,s1).da = s1.da - s1.db by A41,AMI_3:10 .= Exec (l,s2).da by A41,A53,A54,AMI_3:10; then Exec (l,s1) | {da} = Exec(l,s2) | {da} by A7,A8,AMI_3:24; then A55: Exec (l,s1) | SCM-Data-Loc = Exec (l,s2) | SCM-Data-Loc by A42,A50,AMI_3:20; Exec (l,s1).IC SCM = Next IC s1 by A41,AMI_3:10 .= Exec (l,s2).IC SCM by A6,A41,AMI_3:10; then Exec (l,s1) | {IC SCM} = Exec (l,s2) | {IC SCM} by A7,A8,AMI_3:24; hence Exec (l,s1) | (SCM-Data-Loc \/ {IC SCM}) = Exec (l,s2) | (SCM-Data-Loc \/ {IC SCM}) by A55,AMI_3:20; suppose InsCode (l) = 4; then consider da,db such that A56: l = MultBy(da,db) by Th50; da in SCM-Data-Loc by AMI_3:def 2; then A57: SCM-Data-Loc = SCM-Data-Loc \/ {da} by ZFMISC_1:46 .= (SCM-Data-Loc \ {da} ) \/ {da} by XBOOLE_1:39; SCM-Data-Loc \ {da} c= SCM-Data-Loc by XBOOLE_1:36; then SCM-Data-Loc \ {da} c= dom(Exec (l,s1)) by A10,XBOOLE_1:1; then A58: dom ((Exec (l,s1)) | (SCM-Data-Loc \ {da})) = (SCM-Data-Loc \ {da}) by RELAT_1:91; SCM-Data-Loc \ {da} c= SCM-Data-Loc by XBOOLE_1:36; then SCM-Data-Loc \ {da} c= dom(Exec (l,s2)) by A11,XBOOLE_1:1; then A59: dom ((Exec (l,s2)) | (SCM-Data-Loc \ {da})) = (SCM-Data-Loc \ {da}) by 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 A60: x in ((SCM-Data-Loc) \ {da}); then A61: x in SCM-Data-Loc by XBOOLE_0:def 4; A62: not x in {da} by A60,XBOOLE_0:def 4; reconsider a = x as Data-Location by A61,AMI_3:def 1,def 2; A63: a <> da by A62,TARSKI:def 1; A64: a in (SCM-Data-Loc \/ {IC SCM}) by A61,XBOOLE_0:def 2; thus (Exec (l,s1) | (SCM-Data-Loc \ {da})).x = (Exec (l,s1)).a by A60,FUNCT_1:72 .= s1.a by A56,A63,AMI_3:11 .= (s1 | (SCM-Data-Loc \/ {IC SCM})).a by A64,FUNCT_1:72 .= s2.a by A1,A64,FUNCT_1:72 .= (Exec (l,s2)).a by A56,A63,AMI_3:11 .= (Exec (l,s2) | (SCM-Data-Loc \ {da})).x by A60,FUNCT_1:72; end; then A65: Exec (l,s1) | (SCM-Data-Loc \ {da} ) = Exec (l,s2) | (SCM-Data-Loc \ {da} ) by A58,A59,FUNCT_1:9; A66: db in SCM-Data-Loc by AMI_3:def 2; A67: da in SCM-Data-Loc by AMI_3:def 2; then A68: s1.da = (s1 | (SCM-Data-Loc \/ {IC SCM})).da by A9,FUNCT_1:72 .= s2.da by A1,A9,A67,FUNCT_1:72; A69: s1.db = (s1 | (SCM-Data-Loc \/ {IC SCM})).db by A9,A66,FUNCT_1:72 .= s2.db by A1,A9,A66,FUNCT_1:72; Exec (l,s1).da = s1.da * s1.db by A56,AMI_3:11 .= Exec (l,s2).da by A56,A68,A69,AMI_3:11; then Exec (l,s1) | {da} = Exec(l,s2) | {da} by A7,A8,AMI_3:24; then A70: Exec (l,s1) | SCM-Data-Loc = Exec (l,s2) | SCM-Data-Loc by A57,A65,AMI_3:20; Exec (l,s1).IC SCM = Next IC s1 by A56,AMI_3:11 .= Exec (l,s2).IC SCM by A6,A56,AMI_3:11; then Exec (l,s1) | {IC SCM} = Exec (l,s2) | {IC SCM} by A7,A8,AMI_3:24; hence Exec (l,s1) | (SCM-Data-Loc \/ {IC SCM}) = Exec (l,s2) | (SCM-Data-Loc \/ {IC SCM}) by A70,AMI_3:20; suppose InsCode (l) = 5; then consider da,db such that A71: l = Divide(da,db) by Th51; thus thesis proof per cases; suppose A72: da=db; da in SCM-Data-Loc by AMI_3:def 2; then A73: SCM-Data-Loc = SCM-Data-Loc \/ {da} by ZFMISC_1:46 .= (SCM-Data-Loc \ {da} ) \/ {da} by XBOOLE_1:39; SCM-Data-Loc \ {da} c= SCM-Data-Loc by XBOOLE_1:36; then SCM-Data-Loc \ {da} c= dom(Exec (l,s1)) by A10,XBOOLE_1:1; then A74: dom ((Exec (l,s1)) | (SCM-Data-Loc \ {da})) = (SCM-Data-Loc \ {da}) by RELAT_1:91; SCM-Data-Loc \ {da} c= SCM-Data-Loc by XBOOLE_1:36; then SCM-Data-Loc \ {da} c= dom(Exec (l,s2)) by A11,XBOOLE_1:1; then A75: dom ((Exec (l,s2)) | (SCM-Data-Loc \ {da})) = (SCM-Data-Loc \ {da}) by 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 A76: x in ((SCM-Data-Loc) \ {da}); then A77: x in SCM-Data-Loc by XBOOLE_0:def 4; A78: not x in {da} by A76,XBOOLE_0:def 4; reconsider a = x as Data-Location by A77,AMI_3:def 1,def 2; A79: a <> da by A78,TARSKI:def 1; A80: a in (SCM-Data-Loc \/ {IC SCM}) by A77,XBOOLE_0:def 2; thus (Exec (l,s1) | (SCM-Data-Loc \ {da})).x = (Exec (l,s1)).a by A76,FUNCT_1:72 .= s1.a by A71,A72,A79,Th15 .= (s1 | (SCM-Data-Loc \/ {IC SCM})).a by A80,FUNCT_1:72 .= s2.a by A1,A80,FUNCT_1:72 .= (Exec (l,s2)).a by A71,A72,A79,Th15 .= (Exec (l,s2) | (SCM-Data-Loc \ {da})).x by A76,FUNCT_1:72; end; then A81: Exec (l,s1) | (SCM-Data-Loc \ {da} ) = Exec (l,s2) | (SCM-Data-Loc \ {da} ) by A74,A75,FUNCT_1:9; A82: da in SCM-Data-Loc by AMI_3:def 2; then A83: s1.da = (s1 | (SCM-Data-Loc \/ {IC SCM})).da by A9,FUNCT_1:72 .= s2.da by A1,A9,A82,FUNCT_1:72; Exec (l,s1).da = s1.da mod s1.da by A71,A72,Th15 .= Exec (l,s2).da by A71,A72,A83,Th15; then Exec (l,s1) | {da} = Exec(l,s2) | {da} by A7,A8,AMI_3:24; then A84: Exec (l,s1) | SCM-Data-Loc = Exec (l,s2) | SCM-Data-Loc by A73,A81,AMI_3:20; Exec (l,s1).IC SCM = Next IC s1 by A71,A72,Th15 .= Exec (l,s2).IC SCM by A6,A71,A72,Th15; then Exec (l,s1) | {IC SCM} = Exec (l,s2) | {IC SCM} by A7,A8,AMI_3:24; hence Exec (l,s1) | (SCM-Data-Loc \/ {IC SCM}) = Exec (l,s2) | (SCM-Data-Loc \/ {IC SCM}) by A84,AMI_3:20; suppose A85: da <> db; A86: da in SCM-Data-Loc by AMI_3:def 2; db in SCM-Data-Loc by AMI_3:def 2; then A87: SCM-Data-Loc = SCM-Data-Loc \/ {da,db} by A86,ZFMISC_1:48 .= (SCM-Data-Loc \ {da,db} ) \/ {da,db} by XBOOLE_1:39; SCM-Data-Loc \ {da,db} c= SCM-Data-Loc by XBOOLE_1:36; then SCM-Data-Loc \ {da,db} c= dom(Exec (l,s1)) by A10,XBOOLE_1:1; then A88: dom ((Exec (l,s1)) | (SCM-Data-Loc \ {da,db})) = (SCM-Data-Loc \ {da ,db}) by RELAT_1:91; SCM-Data-Loc \ {da,db} c= SCM-Data-Loc by XBOOLE_1:36; then SCM-Data-Loc \ {da,db} c= dom(Exec (l,s2)) by A11,XBOOLE_1:1; then A89: dom ((Exec (l,s2)) | (SCM-Data-Loc \ {da,db})) = (SCM-Data-Loc \ {da ,db}) by 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 A90: x in ((SCM-Data-Loc) \ {da,db}); then A91: x in SCM-Data-Loc by XBOOLE_0:def 4; A92: not x in {da,db} by A90,XBOOLE_0:def 4; reconsider a = x as Data-Location by A91,AMI_3:def 1,def 2; A93: a <> da & a <> db by A92,TARSKI:def 2; A94: a in (SCM-Data-Loc \/ {IC SCM}) by A91,XBOOLE_0:def 2; thus (Exec (l,s1) | (SCM-Data-Loc \ {da,db})).x = (Exec (l,s1)).a by A90,FUNCT_1:72 .= s1.a by A71,A93,AMI_3:12 .= (s1 | (SCM-Data-Loc \/ {IC SCM})).a by A94,FUNCT_1:72 .= s2.a by A1,A94,FUNCT_1:72 .= (Exec (l,s2)).a by A71,A93,AMI_3:12 .= (Exec (l,s2) | (SCM-Data-Loc \ {da,db})).x by A90,FUNCT_1:72; end; then A95: Exec (l,s1) | (SCM-Data-Loc \ {da,db} ) = Exec (l,s2) | (SCM-Data-Loc \ {da,db} ) by A88,A89,FUNCT_1:9; A96: db in SCM-Data-Loc by AMI_3:def 2; A97: da in SCM-Data-Loc by AMI_3:def 2; then A98: s1.da = (s1 | (SCM-Data-Loc \/ {IC SCM})).da by A9,FUNCT_1:72 .= s2.da by A1,A9,A97,FUNCT_1:72; A99: s1.db = (s1 | (SCM-Data-Loc \/ {IC SCM})).db by A9,A96,FUNCT_1:72 .= s2.db by A1,A9,A96,FUNCT_1:72; A100: Exec (l,s1).da = s1.da div s1.db by A71,A85,AMI_3:12 .= Exec (l,s2).da by A71,A85,A98,A99,AMI_3:12; Exec (l,s1).db = s1.da mod s1.db by A71,AMI_3:12 .= Exec (l,s2).db by A71,A98,A99,AMI_3:12; then Exec (l,s1) | {da,db} = Exec(l,s2) | {da,db} by A7,A8,A100,AMI_3:25; then A101: Exec (l,s1) | SCM-Data-Loc = Exec (l,s2) | SCM-Data-Loc by A87,A95,AMI_3:20; Exec (l,s1).IC SCM = Next IC s1 by A71,AMI_3:12 .= Exec (l,s2).IC SCM by A6,A71,AMI_3:12; then Exec (l,s1) | {IC SCM} = Exec (l,s2) | {IC SCM} by A7,A8,AMI_3:24; hence Exec (l,s1) | (SCM-Data-Loc \/ {IC SCM}) = Exec (l,s2) | (SCM-Data-Loc \/ {IC SCM}) by A101,AMI_3:20; end; suppose InsCode (l) = 6; then consider loc such that A102: l = goto loc by Th52; A103: dom ((Exec (l,s1)) | SCM-Data-Loc) = SCM-Data-Loc by A10,RELAT_1:91; A104: dom ((Exec (l,s2)) | SCM-Data-Loc) = SCM-Data-Loc by A11,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 A105: x in SCM-Data-Loc; then reconsider a = x as Data-Location by AMI_3:def 1,def 2; A106: a in (SCM-Data-Loc \/ {IC SCM}) by A105,XBOOLE_0:def 2; thus (Exec (l,s1) | SCM-Data-Loc ).x = (Exec (l,s1)).a by A105,FUNCT_1:72 .= s1.a by A102,AMI_3:13 .= (s1 | (SCM-Data-Loc \/ {IC SCM})).a by A106,FUNCT_1:72 .= s2.a by A1,A106,FUNCT_1:72 .= (Exec (l,s2)).a by A102,AMI_3:13 .= (Exec (l,s2) | SCM-Data-Loc ).x by A105,FUNCT_1:72; end; then A107: Exec (l,s1) | (SCM-Data-Loc ) = Exec (l,s2) | (SCM-Data-Loc ) by A103,A104,FUNCT_1:9; Exec (l,s1).IC SCM = loc by A102,AMI_3:13 .= Exec (l,s2).IC SCM by A102,AMI_3:13; then Exec (l,s1) | {IC SCM} = Exec (l,s2) | {IC SCM} by A7,A8,AMI_3:24; hence Exec (l,s1) | (SCM-Data-Loc \/ {IC SCM}) = Exec (l,s2) | (SCM-Data-Loc \/ {IC SCM}) by A107,AMI_3:20; suppose InsCode (l) = 7; then consider loc,da such that A108: l = da=0_goto loc by Th53; A109: dom ((Exec (l,s1)) | SCM-Data-Loc) = SCM-Data-Loc by A10,RELAT_1:91; A110: dom ((Exec (l,s2)) | SCM-Data-Loc) = SCM-Data-Loc by A11,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 A111: x in SCM-Data-Loc; then reconsider a = x as Data-Location by AMI_3:def 1,def 2; A112: a in (SCM-Data-Loc \/ {IC SCM}) by A111,XBOOLE_0:def 2; thus (Exec (l,s1) | SCM-Data-Loc ).x = (Exec (l,s1)).a by A111,FUNCT_1:72 .= s1.a by A108,AMI_3:14 .= (s1 | (SCM-Data-Loc \/ {IC SCM})).a by A112,FUNCT_1:72 .= s2.a by A1,A112,FUNCT_1:72 .= (Exec (l,s2)).a by A108,AMI_3:14 .= (Exec (l,s2) | SCM-Data-Loc ).x by A111,FUNCT_1:72; end; then A113: Exec (l,s1) | (SCM-Data-Loc ) = Exec (l,s2) | (SCM-Data-Loc ) by A109,A110,FUNCT_1:9; Exec (l,s1).IC SCM = Exec (l,s2).IC SCM proof A114: da in SCM-Data-Loc by AMI_3:def 2; then A115: s1.da = (s1 | (SCM-Data-Loc \/ {IC SCM})).da by A9,FUNCT_1:72 .= s2.da by A1,A9,A114,FUNCT_1:72; per cases; suppose A116: s1.da = 0; hence Exec (l,s1).IC SCM = loc by A108,AMI_3:14 .= Exec (l,s2).IC SCM by A108,A115,A116,AMI_3 :14; suppose A117: s1.da <> 0; hence Exec (l,s1).IC SCM = Next IC s1 by A108,AMI_3:14 .= Exec (l,s2).IC SCM by A6,A108,A115,A117, AMI_3:14; end; then Exec (l,s1) | {IC SCM} = Exec (l,s2) | {IC SCM} by A7,A8,AMI_3:24; hence Exec (l,s1) | (SCM-Data-Loc \/ {IC SCM}) = Exec (l,s2) | (SCM-Data-Loc \/ {IC SCM}) by A113,AMI_3:20; suppose InsCode (l) = 8; then consider loc,da such that A118: l = da>0_goto loc by Th54; A119: dom ((Exec (l,s1)) | SCM-Data-Loc) = SCM-Data-Loc by A10,RELAT_1:91; A120: dom ((Exec (l,s2)) | SCM-Data-Loc) = SCM-Data-Loc by A11,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 A121: x in SCM-Data-Loc; then reconsider a = x as Data-Location by AMI_3:def 1,def 2; A122: a in (SCM-Data-Loc \/ {IC SCM}) by A121,XBOOLE_0:def 2; thus (Exec (l,s1) | SCM-Data-Loc ).x = (Exec (l,s1)).a by A121,FUNCT_1:72 .= s1.a by A118,AMI_3:15 .= (s1 | (SCM-Data-Loc \/ {IC SCM})).a by A122,FUNCT_1:72 .= s2.a by A1,A122,FUNCT_1:72 .= (Exec (l,s2)).a by A118,AMI_3:15 .= (Exec (l,s2) | SCM-Data-Loc ).x by A121,FUNCT_1:72; end; then A123: Exec (l,s1) | (SCM-Data-Loc ) = Exec (l,s2) | (SCM-Data-Loc ) by A119,A120,FUNCT_1:9; Exec (l,s1).IC SCM = Exec (l,s2).IC SCM proof A124: da in SCM-Data-Loc by AMI_3:def 2; then A125: s1.da = (s1 | (SCM-Data-Loc \/ {IC SCM})).da by A9,FUNCT_1:72 .= s2.da by A1,A9,A124,FUNCT_1:72; per cases; suppose A126: s1.da > 0; hence Exec (l,s1).IC SCM = loc by A118,AMI_3:15 .= Exec (l,s2).IC SCM by A118,A125,A126,AMI_3 :15; suppose A127: s1.da <= 0; hence Exec (l,s1).IC SCM = Next IC s1 by A118,AMI_3:15 .= Exec (l,s2).IC SCM by A6,A118,A125,A127, AMI_3:15; end; then Exec (l,s1) | {IC SCM} = Exec (l,s2) | {IC SCM} by A7,A8,AMI_3:24; hence Exec (l,s1) | (SCM-Data-Loc \/ {IC SCM}) = Exec (l,s2) | (SCM-Data-Loc \/ {IC SCM}) by A123,AMI_3:20; 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_3:36; then A1: dom (Exec (i, s) | SCM-Instr-Loc) = SCM-Instr-Loc by AMI_3:def 1, RELAT_1:91; dom s = the carrier of SCM by AMI_3:36; then A2: dom (s | SCM-Instr-Loc) = SCM-Instr-Loc by AMI_3:def 1,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 A3: x in SCM-Instr-Loc; then reconsider l = x as Instruction-Location of SCM by AMI_3:def 1; thus (Exec (i, s) | SCM-Instr-Loc).x = (Exec (i, s)).l by A3,FUNCT_1:72 .= s.l by AMI_1:def 13 .= (s | SCM-Instr-Loc).x by A3,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 theorem Th60: for p being FinPartState of S, s being State of S st IC S in dom p & p c= s holds IC p = IC s proof let p be FinPartState of S, s be State of S; assume that A1: IC S in dom p and A2: p c= s; thus IC p = p.IC S by A1,AMI_3:def 16 .= s.IC S by A1,A2,GRFUNC_1:8 .= IC s by AMI_1:def 15; end; definition let N,S; let p be FinPartState of S, loc be Instruction-Location of S; assume A1: loc in dom p; func pi (p , loc) -> Instruction of S equals p.loc; coherence proof consider s be State of S such that A2: p c= s by AMI_3:39; s.loc = p.loc by A1,A2,GRFUNC_1:8; hence thesis; end; end; theorem Th61: for N being set, S being AMI-Struct over N for x being set, p being FinPartState of S st x c= p holds x is FinPartState of S proof let N be set, S be AMI-Struct over N; let x be set, p be FinPartState of S; A1: p in sproduct the Object-Kind of S & p is finite by AMI_1:def 24; assume A2: x c= p; then reconsider f = x as Function by GRFUNC_1:6; f in sproduct the Object-Kind of S & f is finite by A1,A2,AMI_1:40,FINSET_1:13; hence x is FinPartState of S by AMI_1:def 24; end; definition let N be set; let S be AMI-Struct over N; let p be FinPartState of S; func ProgramPart p -> programmed FinPartState of S equals :Def6: p | the Instruction-Locations of S; coherence proof set q = p | the Instruction-Locations of S; q c= p by RELAT_1:88; then reconsider q as FinPartState of S by Th61; dom q = dom p /\ the Instruction-Locations of S by RELAT_1:90; then dom q c= the Instruction-Locations of S by XBOOLE_1:17; hence thesis by AMI_3:def 13; end; end; definition let N be set; let S be non empty AMI-Struct over N; let p be FinPartState of S; func DataPart p -> FinPartState of S equals :Def7: p | ((the carrier of S) \ ({IC S} \/ the Instruction-Locations of S)); coherence proof p | ((the carrier of S) \ ({IC S} \/ the Instruction-Locations of S)) c= p by RELAT_1:88; hence thesis by Th61; end; end; definition let N be set, S be non empty AMI-Struct over N; let IT be FinPartState of S; attr IT is data-only means :Def8: dom IT misses {IC S} \/ the Instruction-Locations of S; end; 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 Th16; 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 Th33,XBOOLE_1:70; the carrier of SCM = {IC SCM} \/ (the Instruction-Locations of SCM) \/ SCM-Data-Loc by Th23,AMI_3:def 1,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,AMI_3:def 1,XBOOLE_1:83; hence thesis by Def7; 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_3:37; 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 Th16; 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 Th33,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 Def8,AMI_3:def 1; end; definition let N be set, S be non empty AMI-Struct over N; cluster data-only FinPartState of S; existence proof consider p being FinPartState of S; {} c= p by XBOOLE_1:2; then reconsider p = {} as FinPartState of S by Th61; take p; thus dom p misses {IC S} \/ the Instruction-Locations of S by RELAT_1:60, XBOOLE_1:65; end; end; theorem Th62: for N being set, S being non empty AMI-Struct over N for p being FinPartState of S holds DataPart p c= p proof let N be set, S be non empty AMI-Struct over N; let p be FinPartState of S; DataPart p = p| ((the carrier of S) \ ({IC S} \/ the Instruction-Locations of S)) by Def7; hence DataPart p c= p by RELAT_1:88; end; theorem Th63: for N being set, S being AMI-Struct over N for p being FinPartState of S holds ProgramPart p c= p proof let N be set, S be AMI-Struct over N; let p be FinPartState of S; ProgramPart p = p|the Instruction-Locations of S by Def6; hence ProgramPart p c= p by RELAT_1:88; end; theorem for S being steady-programmed IC-Ins-separated definite (non empty non void AMI-Struct over N) for p being FinPartState of S, s being State of S st p c= s for i being Nat holds ProgramPart p c= (Computation (s)).i proof let S be steady-programmed IC-Ins-separated definite (non empty non void AMI-Struct over N); let p be FinPartState of S, s be State of S such that A1: p c= s; let i be Nat; ProgramPart p c= p by Th63; then ProgramPart p c= s by A1,XBOOLE_1:1; hence ProgramPart p c= (Computation (s)).i by AMI_3:38; end; theorem Th65: for N being set, S being non empty AMI-Struct over N for p being FinPartState of S holds not IC S in dom (DataPart p) proof let N be set, S be non empty AMI-Struct over N; let p be FinPartState of S; DataPart p = p | ((the carrier of S) \ ({IC S} \/ the Instruction-Locations of S)) by Def7; then A1: dom(DataPart p) c= ((the carrier of S) \ ({IC S} \/ the Instruction-Locations of S)) by RELAT_1:87; assume IC S in dom (DataPart p); then not IC S in {IC S} \/ the Instruction-Locations of S by A1,XBOOLE_0:def 4; then not IC S in {IC S} by XBOOLE_0:def 2; hence contradiction by TARSKI:def 1; end; theorem Th66: for S being IC-Ins-separated definite realistic (non empty non void AMI-Struct over N) for p being FinPartState of S holds not IC S in dom (ProgramPart p) proof let S be IC-Ins-separated definite realistic (non empty non void AMI-Struct over N); let p be FinPartState of S; ProgramPart p = p | the Instruction-Locations of S by Def6; then A1: dom(ProgramPart p) c= the Instruction-Locations of S by RELAT_1:87; assume IC S in dom (ProgramPart p); hence contradiction by A1,AMI_1:48; end; theorem for N being set, S being non empty AMI-Struct over N for p being FinPartState of S holds {IC S} misses dom (DataPart p) proof let N be set, S be non empty AMI-Struct over N; let p be FinPartState of S; not IC S in dom (DataPart p) by Th65; hence {IC S} misses dom (DataPart p) by ZFMISC_1:56; end; theorem for S being IC-Ins-separated definite realistic (non empty non void AMI-Struct over N) for p being FinPartState of S holds {IC S} misses dom (ProgramPart p) proof let S be IC-Ins-separated definite realistic (non empty non void AMI-Struct over N); let p be FinPartState of S; not IC S in dom (ProgramPart p) by Th66; hence {IC S} misses dom (ProgramPart p) by ZFMISC_1:56; end; 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; theorem for p being FinPartState of SCM holds dom ProgramPart p c= SCM-Instr-Loc proof let p be FinPartState of SCM; ProgramPart p = p | the Instruction-Locations of SCM by Def6; hence dom ProgramPart p c= SCM-Instr-Loc by AMI_3:def 1,RELAT_1:87; end; theorem Th71: for p,q being FinPartState of S holds dom DataPart p misses dom ProgramPart q proof let p,q be FinPartState of S; the Instruction-Locations of S c= {IC S} \/ the Instruction-Locations of S by XBOOLE_1:7; then A1: ((the carrier of S) \ ({IC S} \/ the Instruction-Locations of S)) c= (the carrier of S) \ the Instruction-Locations of S by XBOOLE_1:34; DataPart p = p | ((the carrier of S) \ ({IC S} \/ the Instruction-Locations of S)) by Def7; then dom(DataPart p) c= ((the carrier of S) \ ({IC S} \/ the Instruction-Locations of S)) by RELAT_1:87; then A2: dom(DataPart p) c= (the carrier of S) \ the Instruction-Locations of S by A1,XBOOLE_1:1; ProgramPart q = q | the Instruction-Locations of S by Def6; then A3: dom ProgramPart q c= the Instruction-Locations of S by RELAT_1:87; (the Instruction-Locations of S) misses ((the carrier of S) \ the Instruction-Locations of S) by XBOOLE_1:79; hence dom DataPart p misses dom ProgramPart q by A2,A3,XBOOLE_1:64; end; theorem Th72: for p being programmed FinPartState of S holds ProgramPart p = p proof let p be programmed FinPartState of S; A1: dom p c= dom ProgramPart p proof let x be set; assume A2: x in dom p; dom ProgramPart p = dom (p | the Instruction-Locations of S) by Def6; then A3: dom ProgramPart p = dom p /\ the Instruction-Locations of S by RELAT_1:90; dom p c= the Instruction-Locations of S by AMI_3:def 13; hence x in dom ProgramPart p by A2,A3,XBOOLE_0:def 3; end; A4: ProgramPart p c= p by Th63; then dom ProgramPart p c= dom p by GRFUNC_1:8; then dom ProgramPart p = dom p by A1,XBOOLE_0:def 10; hence ProgramPart p = p by A4,GRFUNC_1:9; end; theorem for p being FinPartState of S, l being Instruction-Location of S st l in dom p holds l in dom ProgramPart p proof let p be FinPartState of S, l be Instruction-Location of S; assume A1: l in dom p; dom ProgramPart p = dom (p | the Instruction-Locations of S) by Def6; then dom ProgramPart p = dom p /\ the Instruction-Locations of S by RELAT_1:90; hence l in dom ProgramPart p by A1,XBOOLE_0:def 3; end; theorem for p being data-only FinPartState of S, q being FinPartState of S holds p c= q iff p c= DataPart(q) proof let p be data-only FinPartState of S, q be FinPartState of S; set X = (the carrier of S) \ ({IC S} \/ the Instruction-Locations of S); hereby assume p c= q; then p |X c= q | X by RELAT_1:105; then A1: p |X c= DataPart(q) by Def7; A2: X \/ ({IC S} \/ the Instruction-Locations of S) = (the carrier of S) \/ ({IC S} \/ the Instruction-Locations of S) by XBOOLE_1:39 .= the carrier of S by XBOOLE_1:12; A3: dom p misses {IC S} \/ the Instruction-Locations of S by Def8; dom p c= the carrier of S by AMI_3:37; then dom p c= X by A2,A3,XBOOLE_1:73; hence p c= DataPart(q) by A1,RELAT_1:97; end; assume p c= DataPart(q); then A4: p c= q | X by Def7; q | X c= q by RELAT_1:88; hence p c= q by A4,XBOOLE_1:1; end; theorem for S being IC-Ins-separated definite realistic (non empty non void AMI-Struct over N) for p being FinPartState of S st IC S in dom p holds p = Start-At(IC p) +* ProgramPart p +* DataPart p proof let S be IC-Ins-separated definite realistic (non empty non void AMI-Struct over N); let p be FinPartState of S; assume A1: IC S in dom p; then A2: {IC S} is Subset of dom p by SUBSET_1:63; A3: dom p c= the carrier of S by AMI_3:37; A4: ({IC S} \/ (the Instruction-Locations of S) \/ ((the carrier of S) \ ({IC S} \/ the Instruction-Locations of S))) = ((the carrier of S) \/ ({IC S} \/ the Instruction-Locations of S)) by XBOOLE_1:39 .= ((the carrier of S) \/ {IC S} \/ the Instruction-Locations of S) by XBOOLE_1:4 .= ((the carrier of S) \/ the Instruction-Locations of S) by XBOOLE_1:12 .= the carrier of S by XBOOLE_1:12; A5: dom (Start-At(IC p) +* ProgramPart p +* DataPart p) = dom (Start-At(IC p) +* ProgramPart p) \/ dom (DataPart p) by FUNCT_4:def 1 .= dom (Start-At(IC p)) \/ dom (ProgramPart p) \/ dom (DataPart p) by FUNCT_4:def 1 .= {IC S} \/ dom (ProgramPart p) \/ dom (DataPart p) by AMI_3:34 .= {IC S} \/ dom (p | the Instruction-Locations of S) \/ dom(DataPart p) by Def6 .= {IC S} \/ dom (p|the Instruction-Locations of S) \/ dom(p|((the carrier of S) \ ({IC S} \/ the Instruction-Locations of S))) by Def7 .= dom p /\ {IC S} \/ dom (p|the Instruction-Locations of S) \/ dom(p|((the carrier of S) \ ({IC S} \/ the Instruction-Locations of S))) by A2,XBOOLE_1: 28 .= dom p /\ {IC S} \/ dom p /\ (the Instruction-Locations of S) \/ dom(p|((the carrier of S) \ ({IC S} \/ the Instruction-Locations of S))) by RELAT_1:90 .= dom p /\ {IC S} \/ dom p /\ (the Instruction-Locations of S) \/ dom p /\ ((the carrier of S) \ ({IC S} \/ the Instruction-Locations of S)) by RELAT_1:90 .= dom p /\ ({IC S} \/ (the Instruction-Locations of S)) \/ dom p /\ ((the carrier of S) \ ({IC S} \/ the Instruction-Locations of S)) by XBOOLE_1:23 .= dom p /\ the carrier of S by A4,XBOOLE_1:23 .= dom p by A3,XBOOLE_1:28; now let x be set; assume A6: x in dom p; then A7: x in {IC S} \/ (the Instruction-Locations of S) or x in (the carrier of S) \ ({IC S} \/ the Instruction-Locations of S) by A3,A4,XBOOLE_0:def 2; per cases by A7,XBOOLE_0:def 2; suppose A8: x in {IC S}; then A9: x = IC S by TARSKI:def 1; {IC S} = dom Start-At (IC p) by AMI_3:34; then IC S in dom Start-At(IC p) by TARSKI:def 1; then A10: IC S in dom Start-At(IC p) \/ dom ProgramPart p by XBOOLE_0:def 2; then IC S in dom (Start-At(IC p) +* ProgramPart p) by FUNCT_4:def 1; then A11: IC S in dom (Start-At(IC p) +* ProgramPart p) \/ dom DataPart p by XBOOLE_0:def 2; A12: not IC S in dom (ProgramPart p) by Th66; not IC S in dom (DataPart p) by Th65; then (Start-At(IC p) +* ProgramPart p +* DataPart p).x = (Start-At(IC p) +* ProgramPart p).x by A9,A11,FUNCT_4:def 1 .= (Start-At(IC p)).x by A9,A10,A12,FUNCT_4:def 1 .= IC p by A9,AMI_3:50 .= p.IC S by A1,AMI_3:def 16; hence p.x = (Start-At(IC p) +* ProgramPart p +* DataPart p).x by A8,TARSKI :def 1; suppose x in (the carrier of S) \ ({IC S} \/ the Instruction-Locations of S); then x in dom p /\ ((the carrier of S) \ ({IC S} \/ the Instruction-Locations of S)) by A6,XBOOLE_0:def 3; then A13: x in dom (p | ((the carrier of S) \ ({IC S} \/ the Instruction-Locations of S))) by RELAT_1:90; then A14: x in dom (Start-At(IC p) +* ProgramPart p) \/ dom (p | ((the carrier of S) \ ({IC S} \/ the Instruction-Locations of S))) by XBOOLE_0:def 2; (Start-At(IC p) +* ProgramPart p +* DataPart p).x = (Start-At(IC p) +* ProgramPart p +* p | ((the carrier of S) \ ({IC S} \/ the Instruction-Locations of S))).x by Def7 .= (p | ((the carrier of S) \ ({IC S} \/ the Instruction-Locations of S))).x by A13,A14,FUNCT_4:def 1 .= p.x by A13,FUNCT_1:70; hence p.x = (Start-At(IC p) +* ProgramPart p +* DataPart p).x; suppose x in the Instruction-Locations of S; then x in dom p /\ the Instruction-Locations of S by A6,XBOOLE_0:def 3; then A15: x in dom (p | the Instruction-Locations of S) by RELAT_1:90; then A16: x in dom (ProgramPart p) by Def6; then A17: x in dom (Start-At(IC p)) \/ dom (ProgramPart p) by XBOOLE_0:def 2; then x in dom (Start-At(IC p) +* ProgramPart p) by FUNCT_4:def 1; then A18: x in dom (Start-At(IC p) +* ProgramPart p) \/ dom (DataPart p) by XBOOLE_0:def 2; dom (DataPart p) misses dom (ProgramPart p) by Th71; then not x in dom (DataPart p) by A16,XBOOLE_0:3; then (Start-At(IC p) +* ProgramPart p +* DataPart p).x = (Start-At(IC p) +* ProgramPart p).x by A18,FUNCT_4:def 1 .= (ProgramPart p).x by A16,A17,FUNCT_4:def 1 .= (p | the Instruction-Locations of S ).x by Def6 .= p.x by A15,FUNCT_1:70; hence p.x = (Start-At(IC p) +* ProgramPart p +* DataPart p).x; end; hence p = Start-At(IC p) +* ProgramPart p +* DataPart p by A5,FUNCT_1:9; end; definition let N,S;let IT be PartFunc of FinPartSt S,FinPartSt S; attr IT is data-only means for p being FinPartState of S st p in dom IT holds p is data-only & for q being FinPartState of S st q = IT.p holds q is data-only; end; theorem for S being IC-Ins-separated definite realistic (non empty non void AMI-Struct over N) for p being FinPartState of S st IC S in dom p holds p is not programmed proof let S be IC-Ins-separated definite realistic (non empty non void AMI-Struct over N); let p be FinPartState of S; assume A1: IC S in dom p; assume p is programmed; then dom p = dom ProgramPart p by Th72; hence contradiction by A1,Th66; end; definition let N; let S be non void AMI-Struct over N; let s be State of S; let p be FinPartState of S; redefine func s +* p -> State of S; coherence proof sproduct the Object-Kind of S <> {}; hence thesis by AMI_1:29; end; end; 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_3:def 13; 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; x = IC SCM by A2,TARSKI:def 1; hence contradiction by A3,AMI_1:48,AMI_3:def 1; end; then SCM-Data-Loc \/ {IC SCM} misses SCM-Instr-Loc by Th33,XBOOLE_1:70; then A4: SCM-Data-Loc \/ {IC SCM} misses dom p by A1,AMI_3:def 1, XBOOLE_1:63; then A5: s|(SCM-Data-Loc \/ {IC SCM}) = (s +* p) | (SCM-Data-Loc \/ {IC SCM}) by Th7; A6: (Exec(i,s) +* p)|(SCM-Data-Loc \/ {IC SCM}) = Exec(i,s)|(SCM-Data-Loc \/ {IC SCM}) by A4,Th7 .= 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 Th6 .= Exec (i,s) |SCM-Instr-Loc +* p|SCM-Instr-Loc by Th59 .= (Exec (i, s) +* p)|SCM-Instr-Loc by Th6; 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_3:36 .= (Exec (i, s) +* p)| ({IC SCM} \/ SCM-Data-Loc) +* (Exec (i, s) +* p)|SCM-Instr-Loc by A6,A7,Th14 .= (Exec (i,s) +* p)| the carrier of SCM by Th14,Th23 .= (Exec (i,s) +* p)| dom(Exec (i, s) +* p) by AMI_3:36 .= Exec (i,s) +* p by RELAT_1:97; end; theorem for p being FinPartState of S st IC S in dom p holds Start-At (IC p) c= p proof let p be FinPartState of S; assume A1: IC S in dom p; then A2: IC p = p.IC S by AMI_3:def 16; [IC S, p.IC S] in p by A1,FUNCT_1:8; then {[IC S, p.IC S]} c= p by ZFMISC_1:37; hence Start-At (IC p) c= p by A2,Th35; end; theorem for s being State of S, iloc being Instruction-Location of S holds IC (s +* Start-At iloc ) = iloc proof let s be State of S, iloc be Instruction-Location of S; A1: dom (Start-At iloc) = {IC S} & IC S in {IC S} by AMI_3:34,TARSKI:def 1; then A2: IC S in dom s \/ {IC S} by XBOOLE_0:def 2; thus IC (s +* Start-At iloc ) = (s +* Start-At iloc ).IC S by AMI_1:def 15 .= (Start-At iloc).IC S by A1,A2,FUNCT_4:def 1 .= iloc by AMI_3:50; end; 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 AMI_3:34; a in the carrier of SCM; then a in dom s by AMI_3:36; 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; theorem for S being IC-Ins-separated definite realistic (non empty non void AMI-Struct over N) for s being State of S, iloc being Instruction-Location of S, a being Instruction-Location of S holds s.a = (s +* Start-At iloc).a proof let S be IC-Ins-separated definite realistic (non empty non void AMI-Struct over N); let s be State of S, iloc be Instruction-Location of S, a be Instruction-Location of S; A1: dom (Start-At iloc) = {IC S} by AMI_3:34; a in the carrier of S; then a in dom s by AMI_3:36; then A2: a in dom s \/ dom (Start-At iloc) by XBOOLE_0:def 2; a <> IC S by AMI_1:48; then not a in {IC S} by TARSKI:def 1; hence s.a = (s +* Start-At iloc).a by A1,A2,FUNCT_4:def 1; end; theorem for s, t being State of S, A be set holds s +* t|A is State of S proof let s, t be State of S, A be set; A1: t in product the Object-Kind of S; product the Object-Kind of S c= sproduct the Object-Kind of S by AMI_1:27; then t|A in sproduct the Object-Kind of S by A1,AMI_1:41; hence s +* t|A is State of S by AMI_1:29; end; begin :: Autonomic finite partial states of SCM 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_3:37; 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; p is finite by AMI_1:def 24; then dom p is finite by AMI_1:21; then 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 1,def 2; consider il being Element of (the Instruction-Locations of SCM) \ dom p; p is finite by AMI_1:def 24; then dom p is finite by AMI_1:21; then 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 reconsider il as Instruction-Location of SCM by XBOOLE_0: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_3:39; consider s2 being State of SCM such that A7: p2 c= s2 by AMI_3:39; 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 AMI_3:34 .= 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 AMI_3:34 .= 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 Th62; then A16: dom DataPart p c= dom p by RELAT_1:25; dom ((Computation s1).1) = the carrier of SCM by AMI_3:36; then dom p c= dom ((Computation s1).1) by AMI_3:37; then A17: dom ((Computation s1).1|dom p) = dom p by RELAT_1:91; A18: dom(Start-At il) = {IC SCM} by AMI_3:34; 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 A23: IC SCM in dom p1 by A21,XBOOLE_0:def 2; A24: IC s1 = s1.IC SCM by AMI_1:def 15 .= p1.IC SCM by A6,A23,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 AMI_3:50; dom (il .--> (d1:=d2)) = {il} by CQC_LANG:5; then A25: il in dom (il .--> (d1:=d2)) by TARSKI:def 1; A26: dom (d2 .--> 0) = {d2} by CQC_LANG:5; il <> d2 by Th22; then A27: not il in dom (d2 .--> 0) by A26,TARSKI:def 1; A28: dom ((il .--> (d1:=d2)) +* ( d2.--> 0)) = dom (il .--> (d1:=d2)) \/ dom ( d2.--> 0) by FUNCT_4:def 1; then A29: il in dom ((il .--> (d1:=d2)) +* ( d2.--> 0)) by A25,XBOOLE_0:def 2; il <> IC SCM by AMI_1:48; then A30: not il in dom (Start-At il) by A18,TARSKI:def 1; A31: il in dom ((il .--> (d1:=d2)) +* ( d2.--> 0) +* Start-At il) by A20,A29,XBOOLE_0:def 2; then il in dom p1 by A22,XBOOLE_0:def 2; then A32: s1.il = p1.il by A6,GRFUNC_1:8 .= ((il .--> (d1:=d2)) +* ( d2.--> 0) +* Start-At il).il by A31,FUNCT_4:14 .= ((il .--> (d1:=d2)) +* ( d2.--> 0)).il by A30,FUNCT_4:12 .= (il .--> (d1:=d2)).il by A27,FUNCT_4:12 .=(d1:=d2) by CQC_LANG:6; A33: d2 in dom (d2 .--> 0) by A26,TARSKI:def 1; then A34: d2 in dom ((il .--> (d1:=d2)) +* ( d2.--> 0)) by A28,XBOOLE_0:def 2; d2 <> IC SCM by Th20; then A35: not d2 in dom (Start-At il) by A18,TARSKI:def 1; A36: d2 in dom ((il .--> (d1:=d2)) +* ( d2.--> 0) +* Start-At il) by A20,A34,XBOOLE_0:def 2; then d2 in dom p1 by A22,XBOOLE_0:def 2; then A37: s1.d2 = p1.d2 by A6,GRFUNC_1:8 .= ((il .--> (d1:=d2)) +* ( d2.--> 0) +* Start-At il).d2 by A36,FUNCT_4:14 .= ((il .--> (d1:=d2)) +* ( d2.--> 0)).d2 by A35,FUNCT_4:12 .= (d2.--> 0).d2 by A33,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 .= Exec(CurInstr s1,s1).d1 by AMI_1:def 18 .= Exec(s1.il,s1).d1 by A24,AMI_1:def 17 .= 0 by A32,A37,AMI_3:8; then A38: ((Computation s1).1|dom p).d1 = 0 by A3,A16,A17,FUNCT_1:70; dom ((Computation s2).1) = the carrier of SCM by AMI_3:36; then dom p c= dom ((Computation s2).1) by AMI_3:37; then A39: dom ((Computation s2).1|dom p) = dom p by RELAT_1:91; A40: dom(Start-At il) = {IC SCM} by AMI_3:34; then A41: IC SCM in dom (Start-At il) by TARSKI:def 1; A42: 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 A43: IC SCM in dom ((il .--> (d1:=d2)) +* ( d2.--> 1) +* Start-At il) by A41,XBOOLE_0:def 2; A44: dom p2 = dom p \/ dom ((il .--> (d1:=d2)) +* ( d2.--> 1) +* Start-At il) by FUNCT_4:def 1; then A45: IC SCM in dom p2 by A43,XBOOLE_0:def 2; A46: IC s2 = s2.IC SCM by AMI_1:def 15 .= p2.IC SCM by A7,A45,GRFUNC_1:8 .= ((il .--> (d1:=d2)) +* ( d2.--> 1) +* Start-At il).IC SCM by A43,FUNCT_4:14 .= (Start-At il).IC SCM by A41,FUNCT_4:14 .= il by AMI_3:50; dom (il .--> (d1:=d2)) = {il} by CQC_LANG:5; then A47: il in dom (il .--> (d1:=d2)) by TARSKI:def 1; A48: dom (d2 .--> 1) = {d2} by CQC_LANG:5; il <> d2 by Th22; then A49: not il in dom (d2 .--> 1) by A48,TARSKI:def 1; A50: dom ((il .--> (d1:=d2)) +* ( d2.--> 1)) = dom (il .--> (d1:=d2)) \/ dom ( d2.--> 1) by FUNCT_4:def 1; then A51: il in dom ((il .--> (d1:=d2)) +* ( d2.--> 1)) by A47,XBOOLE_0:def 2; il <> IC SCM by AMI_1:48; then A52: not il in dom (Start-At il) by A40,TARSKI:def 1; A53: il in dom ((il .--> (d1:=d2)) +* ( d2.--> 1) +* Start-At il) by A42,A51,XBOOLE_0:def 2; then il in dom p2 by A44,XBOOLE_0:def 2; then A54: s2.il = p2.il by A7,GRFUNC_1:8 .= ((il .--> (d1:=d2)) +* ( d2.--> 1) +* Start-At il).il by A53,FUNCT_4:14 .= ((il .--> (d1:=d2)) +* ( d2.--> 1)).il by A52,FUNCT_4:12 .= (il .--> (d1:=d2)).il by A49,FUNCT_4:12 .=(d1:=d2) by CQC_LANG:6; A55: d2 in dom (d2 .--> 1) by A48,TARSKI:def 1; then A56: d2 in dom ((il .--> (d1:=d2)) +* ( d2.--> 1)) by A50,XBOOLE_0:def 2; d2 <> IC SCM by Th20; then A57: not d2 in dom (Start-At il) by A40,TARSKI:def 1; A58: d2 in dom ((il .--> (d1:=d2)) +* ( d2.--> 1) +* Start-At il) by A42,A56,XBOOLE_0:def 2; then d2 in dom p2 by A44,XBOOLE_0:def 2; then A59: s2.d2 = p2.d2 by A7,GRFUNC_1:8 .= ((il .--> (d1:=d2)) +* ( d2.--> 1) +* Start-At il).d2 by A58,FUNCT_4:14 .= ((il .--> (d1:=d2)) +* ( d2.--> 1)).d2 by A57,FUNCT_4:12 .= (d2.--> 1).d2 by A55,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 .= Exec(CurInstr s2,s2).d1 by AMI_1:def 18 .= Exec(s2.il,s2).d1 by A46,AMI_1:def 17 .= 1 by A54,A59,AMI_3:8; hence (Computation s1).1|dom p <> (Computation s2).1|dom p by A3,A16,A38, A39,FUNCT_1:70; end; hence contradiction; end; definition 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 AMI_3:34; 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; 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_3:def 1,def 13; dom p c= the carrier of SCM by AMI_3:37; 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; 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; 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_3:39; consider s2 being State of SCM such that A4: p2 c= s2 by AMI_3:39; 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 (CurInstr s1,s1)).IC SCM by AMI_1:def 18 .= Exec (s1.IC s1,s1).IC SCM by AMI_1:def 17 .= Exec (goto il.0,s1).IC SCM by A1,A7,A10,Th60 .= il.0 by AMI_3:13; A13: (Following s2).IC SCM = (Exec (CurInstr s2,s2)).IC SCM by AMI_1:def 18 .= Exec (s2.IC s2,s2).IC SCM by AMI_1:def 17 .= Exec (goto il.1,s2).IC SCM by A1,A7,A11,Th60 .= 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 by AMI_3:53; 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 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 Nat; set Csi = (Computation s).i; set loc = IC Csi; consider ll being Nat such that A2: loc = il.ll by Th19; set loc1 = il.(ll+1); A3: loc <> loc1 proof assume loc = loc1; then ll + 0 = ll + 1 by A2,AMI_3:53; hence contradiction by XCMPLX_1:2; end; A4: loc = Csi.IC SCM by AMI_1:def 15; assume A5: not IC (Computation s).i in dom ProgramPart(p); ProgramPart p = p|SCM-Instr-Loc by Def6,AMI_3:def 1; then loc in dom ProgramPart p iff loc in dom p /\ SCM-Instr-Loc by FUNCT_1: 68 ; then A6:not loc in dom p by A5,AMI_3:def 1,XBOOLE_0:def 3; set p1 = p +* (loc .--> goto loc); set p2 = p +* (loc .--> goto loc1); A7: dom p1 = dom p \/ dom (loc .--> goto loc) & dom p2 = dom p \/ dom (loc .--> goto loc1) by FUNCT_4:def 1; A8: dom (loc .--> goto loc) = {loc} & dom (loc .--> goto loc1) = {loc} by CQC_LANG:5; then A9: loc in dom (loc .--> goto loc) & loc in dom (loc .--> goto loc1) by TARSKI:def 1; then A10: loc in dom p1 & loc in dom p2 by A7,XBOOLE_0:def 2; consider s1 being State of SCM such that A11: p1 c= s1 by AMI_3:39; consider s2 being State of SCM such that A12: p2 c= s2 by AMI_3:39; set Cs1i = (Computation s1).i; set Cs2i = (Computation s2).i; A13: 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_3:36; then A14: dom p c= dom s1 & dom p c= dom s2 by AMI_3:37; now let x be set; assume A15: x in dom p; then dom p misses dom (loc .--> goto loc) & x in dom p1 by A6,A7,A8,XBOOLE_0:def 2,ZFMISC_1:56; then p.x = p1.x & p1.x = s1.x by A11,A15,FUNCT_4:17,GRFUNC_1:8; hence p.x = s1.x; end; hence A16: p c= s1 by A14,GRFUNC_1:8; now let x be set; assume A17: x in dom p; then dom p misses dom (loc .--> goto loc1) & x in dom p2 by A6,A7,A8,XBOOLE_0:def 2,ZFMISC_1:56; then p.x = p2.x & p2.x = s2.x by A12,A17,FUNCT_4:17,GRFUNC_1:8; hence p.x = s2.x; end; hence A18: p c= s2 by A14,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 A9,FUNCT_4:14; then A19: s1.loc = goto loc & s2.loc = goto loc1 by A10,A11,A12,GRFUNC_1:8; take k = i+1; set Cs1k = (Computation s1).k; set Cs2k = (Computation s2).k; A20: Cs1k = Following Cs1i by AMI_1:def 19 .= Exec (CurInstr Cs1i, Cs1i) by AMI_1:def 18; A21: Cs2k = Following Cs2i by AMI_1:def 19 .= Exec (CurInstr Cs2i, Cs2i) by AMI_1:def 18; A22: Cs1i.loc = goto loc & Cs2i.loc = goto loc1 by A19,AMI_1:54; A23: (Cs1i|dom p) = (Csi|dom p) by A1,A16,AMI_1:def 25; A24: Cs1i.IC SCM = (Cs1i|dom p).IC SCM & Csi.IC SCM = (Csi|dom p).IC SCM by A13,FUNCT_1:72; (Cs1i|dom p) = (Cs2i|dom p) by A16,A18,AMI_1:def 25; then A25: Cs1i.IC SCM = loc & Cs2i.IC SCM = loc by A4,A13,A23,A24,FUNCT_1:72 ; IC Cs1i = Cs1i.IC SCM & IC Cs2i = Cs2i.IC SCM by AMI_1:def 15; then CurInstr Cs1i = goto loc & CurInstr Cs2i = goto loc1 by A22,A25,AMI_1:def 17; then A26: Cs1k.IC SCM = loc & Cs2k.IC SCM = loc1 by A20,A21,AMI_3:13; (Cs1k|dom p).IC SCM = Cs1k.IC SCM & (Cs2k|dom p).IC SCM = Cs2k.IC SCM by A13,FUNCT_1:72; hence Cs1k|dom p <> Cs2k|dom p by A3,A26; 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 Nat, da, db being Data-Location, loc being Instruction-Location of SCM, 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 Nat, da, db 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; thus A4: IC Cs1i = IC Cs2i proof assume A5: IC (Computation s1).i <> IC (Computation s2).i; A6: IC Cs1i = Cs1i.IC SCM & IC Cs2i = Cs2i.IC SCM by AMI_1:def 15; (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,A6,AMI_1:def 25; end; thus I = CurInstr ((Computation s2).i) proof assume A7: I <> CurInstr ((Computation s2).i); A8: Cs1i.IC Cs1i = I & Cs2i.IC Cs2i = CurInstr Cs2i by A2,AMI_1:def 17; A9: IC Cs1i in dom ProgramPart p & IC Cs2i in dom ProgramPart p by A1,Th86; ProgramPart p c= p by Th63; 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 A9,FUNCT_1:72; hence contradiction by A1,A4,A7,A8,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 Nat, da, db being Data-Location, loc being Instruction-Location of SCM, 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 Nat, da, db 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: 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) by AMI_1:def 18; A5: Cs2i1 = Following Cs2i by AMI_1:def 19 .= Exec (CurInstr Cs2i, Cs2i) by AMI_1:def 18; 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 Nat, da, db being Data-Location, loc being Instruction-Location of SCM, 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 Nat, da, db 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: 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) by AMI_1:def 18; A5: Cs2i1 = Following Cs2i by AMI_1:def 19 .= Exec (CurInstr Cs2i, Cs2i) by AMI_1:def 18; 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 Nat, da, db being Data-Location, loc being Instruction-Location of SCM, 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 Nat, da, db 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: 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) by AMI_1:def 18; A5: Cs2i1 = Following Cs2i by AMI_1:def 19 .= Exec (CurInstr Cs2i, Cs2i) by AMI_1:def 18; 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 Nat, da, db being Data-Location, loc being Instruction-Location of SCM, 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 Nat, da, db 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: 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) by AMI_1:def 18; A5: Cs2i1 = Following Cs2i by AMI_1:def 19 .= Exec (CurInstr Cs2i, Cs2i) by AMI_1:def 18; 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 Nat, da, db being Data-Location, loc being Instruction-Location of SCM, 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 Nat, da, db 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: 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) by AMI_1:def 18; A5: Cs2i1 = Following Cs2i by AMI_1:def 19 .= Exec (CurInstr Cs2i, Cs2i) by AMI_1:def 18; 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 Nat, da, db being Data-Location, loc being Instruction-Location of SCM, 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 Nat, da, db 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: 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) by AMI_1:def 18; A5: Cs2i1 = Following Cs2i by AMI_1:def 19 .= Exec (CurInstr Cs2i, Cs2i) by AMI_1:def 18; 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 Nat, da, db 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 Nat, da, db 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) by AMI_1:def 18; A6: Cs2i1 = Following Cs2i by AMI_1:def 19 .= Exec (CurInstr Cs2i, Cs2i) by AMI_1:def 18; 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 Nat, da, db 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 Nat, da, db 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) by AMI_1:def 18; A7: Cs2i1 = Following Cs2i by AMI_1:def 19 .= Exec (CurInstr Cs2i, Cs2i) by AMI_1:def 18; 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;

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