:: Some Remarks on Simple Concrete Model of Computer :: by Andrzej Trybulec and Yatsuka Nakamura environ vocabularies NUMBERS, SUBSET_1, STRUCT_0, AMI_1, AMI_2, FUNCT_7, XBOOLE_0, RELAT_1, TARSKI, CAT_1, FSM_1, FUNCT_1, INT_1, NAT_1, GRAPHSP, FINSEQ_1, CARD_1, ARYTM_3, ARYTM_1, FUNCOP_1, XXREAL_0, GLIB_000, FUNCT_4, AMI_3, RECDEF_2, QUANTAL1, XCMPLX_0, MEMSTR_0, GOBRD13; notations TARSKI, XBOOLE_0, ENUMSET1, XTUPLE_0, SUBSET_1, ORDINAL1, XCMPLX_0, RELAT_1, FUNCT_1, XXREAL_0, INT_1, FUNCOP_1, CARD_1, FUNCT_4, FUNCT_7, FINSEQ_1, RECDEF_2, NUMBERS, MEASURE6, STRUCT_0, MEMSTR_0, COMPOS_0, COMPOS_1, EXTPRO_1, SCM_INST, AMI_2; constructors DOMAIN_1, FINSEQ_4, CAT_2, AMI_2, RELSET_1, EXTPRO_1, FUNCT_7, MEASURE6, XTUPLE_0, FUNCT_4; registrations XBOOLE_0, SETFAM_1, ORDINAL1, FUNCOP_1, XREAL_0, INT_1, CARD_3, AMI_2, FUNCT_1, FINSEQ_1, EXTPRO_1, FUNCT_4, MEMSTR_0, RELAT_1, COMPOS_0, SCM_INST, XTUPLE_0, ORDINAL2, CARD_1, FACIRC_1; requirements REAL, NUMERALS, SUBSET, BOOLE, ARITHM; definitions EXTPRO_1; equalities TARSKI, COMPOS_1, EXTPRO_1, FUNCOP_1, AMI_2, STRUCT_0, MEMSTR_0, COMPOS_0, SCM_INST; expansions EXTPRO_1, AMI_2, STRUCT_0, MEMSTR_0; theorems TARSKI, ZFMISC_1, ENUMSET1, AMI_2, FUNCOP_1, FUNCT_4, CARD_3, INT_1, ORDINAL1, XBOOLE_0, XBOOLE_1, XXREAL_0, NAT_1, RELAT_1, FUNCT_1, PARTFUN1, SCM_INST, XTUPLE_0, SUBSET_1; begin :: A small concrete machine reserve i,j,k for Nat; :: registration :: cluster -> with_zero for non zero Nat; :: coherence :: proof let n be non zero Nat; :: {} in n by ORDINAL3:8; :: hence thesis; :: end; :: end; reserve I,J,K for Element of Segm 9, a,a1,a2 for Nat, b,b1,b2,c,c1 for Element of SCM-Data-Loc; reserve T for InsType of SCM-Instr, I for Element of SCM-Instr; registration let n be non zero Nat; cluster Segm n -> with_zero; coherence; end; definition func SCM -> strict AMI-Struct over Segm 2 equals AMI-Struct(# SCM-Memory,In(NAT,SCM-Memory),SCM-Instr,SCM-OK,SCM-VAL, SCM-Exec#); coherence; end; registration cluster SCM -> non empty; coherence; end; Lm1: the_Values_of SCM = (the ValuesF of SCM)*(the Object-Kind of SCM) .= SCM-VAL*SCM-OK; registration cluster SCM -> with_non-empty_values; coherence; end; registration cluster SCM -> IC-Ins-separated; coherence by AMI_2:22,SUBSET_1:def 8,AMI_2:6; end; registration cluster Int-like for Object of SCM; existence proof reconsider x = the Element of SCM-Data-Loc as Object of SCM; take x; thus thesis; end; end; definition mode Data-Location is Int-like Object of SCM; end; registration let s be State of SCM, d be Data-Location; cluster s.d -> integer; coherence proof reconsider D = d as Element of SCM-Data-Loc by AMI_2:def 16; reconsider S = s as SCM-State by CARD_3:107; S.D = s.d; hence thesis; end; end; reserve a,b,c for Data-Location, loc for Nat, I for Instruction of SCM; definition ::$CD let a,b; func a := b -> Instruction of SCM equals [ 1, {}, <*a, b*>]; correctness proof reconsider mk = a, ml = b as Element of SCM-Data-Loc by AMI_2:def 16; 1 in { 1,2,3,4,5} by ENUMSET1:def 3; then [ 1, {}, <*mk, ml*>] in SCM-Instr by SCM_INST:4; hence thesis; end; func AddTo(a,b) -> Instruction of SCM equals [ 2, {}, <*a, b*>]; correctness proof reconsider mk = a, ml = b as Element of SCM-Data-Loc by AMI_2:def 16; 2 in { 1,2,3,4,5} by ENUMSET1:def 3; then [ 2, {}, <*mk, ml*>] in SCM-Instr by SCM_INST:4; hence thesis; end; func SubFrom(a,b) -> Instruction of SCM equals [ 3, {}, <*a, b*>]; correctness proof reconsider mk = a, ml = b as Element of SCM-Data-Loc by AMI_2:def 16; 3 in { 1,2,3,4,5} by ENUMSET1:def 3; then [ 3, {}, <*mk, ml*>] in SCM-Instr by SCM_INST:4; hence thesis; end; func MultBy(a,b) -> Instruction of SCM equals [ 4, {}, <*a, b*>]; correctness proof reconsider mk = a, ml = b as Element of SCM-Data-Loc by AMI_2:def 16; 4 in { 1,2,3,4,5} by ENUMSET1:def 3; then [ 4, {}, <*mk, ml*>] in SCM-Instr by SCM_INST:4; hence thesis; end; func Divide(a,b) -> Instruction of SCM equals [ 5, {}, <*a, b*>]; correctness proof reconsider mk = a, ml = b as Element of SCM-Data-Loc by AMI_2:def 16; 5 in { 1,2,3,4,5} by ENUMSET1:def 3; then [ 5, {}, <*mk, ml*>] in SCM-Instr by SCM_INST:4; hence thesis; end; end; definition let loc; func SCM-goto loc -> Instruction of SCM equals [ 6, <*loc*>, {} ]; correctness by SCM_INST:2; let a; func a=0_goto loc -> Instruction of SCM equals [ 7, <*loc*>, <*a*>]; correctness proof reconsider a as Element of SCM-Data-Loc by AMI_2:def 16; reconsider loc as Nat; 7 in { 7,8 } by TARSKI:def 2; then [ 7, <*loc*>, <*a*>] in SCM-Instr by SCM_INST:3; hence thesis; end; func a>0_goto loc -> Instruction of SCM equals [ 8, <*loc*>, <*a*>]; correctness proof reconsider a as Element of SCM-Data-Loc by AMI_2:def 16; reconsider loc as Nat; 8 in { 7,8 } by TARSKI:def 2; then [ 8, <*loc*>, <*a*>] in SCM-Instr by SCM_INST:3; hence thesis; end; end; reserve s for State of SCM; theorem Th1: IC SCM = NAT by AMI_2:22,SUBSET_1:def 8; begin :: Users guide theorem Th2: Exec(a:=b, s).IC SCM = IC s + 1 & Exec(a:=b, s).a = s.b & for c st c <> a holds Exec(a:=b, s).c = s.c proof reconsider S = s as SCM-State by CARD_3:107; reconsider mk = a, ml = b as Element of SCM-Data-Loc by AMI_2:def 16; reconsider I = a:=b as Element of SCM-Instr; set S1 = SCM-Chg(S, I address_1,S.(I address_2)); reconsider i = 1 as Element of Segm 9 by NAT_1:44; A1: I = [ i, {}, <*mk, ml*>]; then A2: I address_1 = mk by SCM_INST:5; A3: I address_2 = ml by A1,SCM_INST:5; A4: Exec(a:=b, s) = SCM-Exec-Res(I,S) by AMI_2:def 15 .= (SCM-Chg(S1, IC S + 1)) by A1,AMI_2:def 14; hence Exec(a:=b, s).IC SCM = IC s + 1 by Th1,AMI_2:11; thus Exec(a:=b, s).a = S1.mk by A4,AMI_2:12 .= s.b by A2,A3,AMI_2:15; let c; reconsider mn = c as Element of SCM-Data-Loc by AMI_2:def 16; assume A5: c <> a; thus Exec(a:=b, s).c = S1.mn by A4,AMI_2:12 .= s.c by A2,A5,AMI_2:16; end; theorem Th3: Exec(AddTo(a,b), s).IC SCM = IC s + 1 & Exec(AddTo(a,b), s).a = s.a + s.b & for c st c <> a holds Exec(AddTo(a,b), s).c = s.c proof reconsider S = s as SCM-State by CARD_3:107; reconsider mk = a, ml = b as Element of SCM-Data-Loc by AMI_2:def 16; reconsider I = AddTo(a,b) as Element of SCM-Instr; set S1 = SCM-Chg(S, I address_1,S.(I address_1)+S.(I address_2)); reconsider i = 2 as Element of Segm 9 by NAT_1:44; A1: I = [ i, {}, <*mk, ml*>]; then A2: I address_1 = mk by SCM_INST:5; A3: I address_2 = ml by A1,SCM_INST:5; A4: Exec(AddTo(a,b), s) = SCM-Exec-Res(I,S) by AMI_2:def 15 .= (SCM-Chg(S1, IC S + 1)) by A1,AMI_2:def 14; hence Exec(AddTo(a,b), s).IC SCM = IC s + 1 by Th1,AMI_2:11; thus Exec(AddTo(a,b), s).a = S1.mk by A4,AMI_2:12 .= s.a + s.b by A2,A3,AMI_2:15; let c; reconsider mn = c as Element of SCM-Data-Loc by AMI_2:def 16; assume A5: c <> a; thus Exec(AddTo(a,b), s).c = S1.mn by A4,AMI_2:12 .= s.c by A2,A5,AMI_2:16; end; theorem Th4: Exec(SubFrom(a,b), s).IC SCM = IC s + 1 & Exec(SubFrom(a,b), s) .a = s.a - s.b & for c st c <> a holds Exec(SubFrom(a,b), s).c = s.c proof reconsider S = s as SCM-State by CARD_3:107; reconsider mk = a, ml = b as Element of SCM-Data-Loc by AMI_2:def 16; reconsider I = SubFrom(a,b) as Element of SCM-Instr; set S1 = SCM-Chg(S, I address_1,S.(I address_1)-S.(I address_2)); reconsider i = 3 as Element of Segm 9 by NAT_1:44; A1: I = [ i, {}, <*mk, ml*>]; then A2: I address_1 = mk by SCM_INST:5; A3: I address_2 = ml by A1,SCM_INST:5; A4: Exec(SubFrom(a,b), s) = SCM-Exec-Res(I,S) by AMI_2:def 15 .= (SCM-Chg(S1, IC S + 1)) by A1,AMI_2:def 14; hence Exec(SubFrom(a,b), s).IC SCM = IC s + 1 by Th1,AMI_2:11; thus Exec(SubFrom(a,b), s).a = S1.mk by A4,AMI_2:12 .= s.a - s.b by A2,A3,AMI_2:15; let c; reconsider mn = c as Element of SCM-Data-Loc by AMI_2:def 16; assume A5: c <> a; thus Exec(SubFrom(a,b), s).c = S1.mn by A4,AMI_2:12 .= s.c by A2,A5,AMI_2:16; end; theorem Th5: Exec(MultBy(a,b), s).IC SCM = IC s + 1 & Exec(MultBy(a,b), s).a = s.a * s.b & for c st c <> a holds Exec(MultBy(a,b), s).c = s.c proof reconsider S = s as SCM-State by CARD_3:107; reconsider mk = a, ml = b as Element of SCM-Data-Loc by AMI_2:def 16; reconsider I = MultBy(a,b) as Element of SCM-Instr; set S1 = SCM-Chg(S, I address_1,S.(I address_1)*S.(I address_2)); reconsider i = 4 as Element of Segm 9 by NAT_1:44; A1: I = [ i, {}, <*mk, ml*>]; then A2: I address_1 = mk by SCM_INST:5; A3: I address_2 = ml by A1,SCM_INST:5; A4: Exec(MultBy(a,b), s) = SCM-Exec-Res(I,S) by AMI_2:def 15 .= (SCM-Chg(S1, IC S + 1)) by A1,AMI_2:def 14; hence Exec(MultBy(a,b), s).IC SCM = IC s + 1 by Th1,AMI_2:11; thus Exec(MultBy(a,b), s).a = S1.mk by A4,AMI_2:12 .= s.a * s.b by A2,A3,AMI_2:15; let c; reconsider mn = c as Element of SCM-Data-Loc by AMI_2:def 16; assume A5: c <> a; thus Exec(MultBy(a,b), s).c = S1.mn by A4,AMI_2:12 .= s.c by A2,A5,AMI_2:16; end; theorem Th6: Exec(Divide(a,b), s).IC SCM = IC s + 1 & (a <> b implies Exec( Divide(a,b), s).a = s.a div s.b) & Exec(Divide(a,b), s).b = s.a mod s.b & for c st c <> a & c <> b holds Exec(Divide(a,b), s).c = s.c proof reconsider S = s as SCM-State by CARD_3:107; reconsider mk = a, ml = b as Element of SCM-Data-Loc by AMI_2:def 16; reconsider I = Divide(a,b) as Element of SCM-Instr; set S1 = SCM-Chg(S, I address_1,S.(I address_1) div S.(I address_2)); set S19 = SCM-Chg(S1, I address_2,S.(I address_1) mod S.(I address_2)); reconsider i = 5 as Element of Segm 9 by NAT_1:44; A1: I = [ i, {}, <*mk, ml*>]; then A2: I address_1 = mk by SCM_INST:5; A3: Exec(Divide(a,b), s) = SCM-Exec-Res(I,S) by AMI_2:def 15 .= (SCM-Chg(S19, IC S + 1)) by A1,AMI_2:def 14; hence Exec(Divide(a,b), s).IC SCM = IC s + 1 by Th1,AMI_2:11; A4: I address_2 = ml by A1,SCM_INST:5; hereby assume A5: a <> b; thus Exec(Divide(a,b), s).a = S19.mk by A3,AMI_2:12 .= S1.mk by A4,A5,AMI_2:16 .= s.a div s.b by A2,A4,AMI_2:15; end; thus Exec(Divide(a,b), s).b = S19.ml by A3,AMI_2:12 .= s.a mod s.b by A2,A4,AMI_2:15; let c; reconsider mn = c as Element of SCM-Data-Loc by AMI_2:def 16; assume that A6: c <> a and A7: c <> b; thus Exec(Divide(a,b), s).c = S19.mn by A3,AMI_2:12 .= S1.mn by A4,A7,AMI_2:16 .= s.c by A2,A6,AMI_2:16; end; theorem Exec(SCM-goto loc, s).IC SCM = loc & Exec(SCM-goto loc, s).c = s.c proof reconsider mn = c as Element of SCM-Data-Loc by AMI_2:def 16; reconsider mj = loc as Element of NAT by ORDINAL1:def 12; reconsider I = SCM-goto loc as Element of SCM-Instr; reconsider S = s as SCM-State by CARD_3:107; reconsider i = 6 as Element of Segm 9 by NAT_1:44; A1: I = [ i, <*mj*>, {} ]; A2: Exec(SCM-goto loc, s) = SCM-Exec-Res(I,S) by AMI_2:def 15 .= (SCM-Chg(S,I jump_address)) by AMI_2:def 14; I jump_address = mj by A1,SCM_INST:6; hence Exec(SCM-goto loc, s).IC SCM = loc by A2,Th1,AMI_2:11; thus Exec(SCM-goto loc, s).c = S.mn by A2,AMI_2:12 .= s.c; end; theorem Th8: (s.a = 0 implies Exec(a =0_goto loc, s).IC SCM = loc) & (s.a <> 0 implies Exec(a=0_goto loc, s).IC SCM = IC s + 1) & Exec(a=0_goto loc, s).c = s.c proof reconsider mn = c as Element of SCM-Data-Loc by AMI_2:def 16; reconsider I = a=0_goto loc as Element of SCM-Instr; reconsider S = s as SCM-State by CARD_3:107; reconsider i = 7 as Element of Segm 9 by NAT_1:44; reconsider a9 = a as Element of SCM-Data-Loc by AMI_2:def 16; reconsider mj = loc as Element of NAT by ORDINAL1:def 12; A1: I = [ i, <*mj*>, <*a9*>]; A2: Exec(a=0_goto loc, s) = SCM-Exec-Res(I,S) by AMI_2:def 15 .= SCM-Chg(S,IFEQ(S.(I cond_address),0,I cjump_address,IC S + 1)) by A1, AMI_2:def 14; thus s.a = 0 implies Exec(a=0_goto loc, s).IC SCM = loc proof assume s.a = 0; then A3: S.(I cond_address)=0 by A1,SCM_INST:7; thus Exec(a=0_goto loc, s).IC SCM = IFEQ(S.(I cond_address),0,I cjump_address,IC S + 1) by A2,Th1,AMI_2:11 .= I cjump_address by A3,FUNCOP_1:def 8 .= loc by A1,SCM_INST:7; end; thus s.a <> 0 implies Exec(a=0_goto loc, s).IC SCM = IC s + 1 proof assume s.a <> 0; then A4: S.(I cond_address) <> 0 by A1,SCM_INST:7; thus Exec(a=0_goto loc, s).IC SCM = IFEQ(S.(I cond_address),0,I cjump_address,IC S + 1) by A2,Th1,AMI_2:11 .= IC s + 1 by A4,Th1,FUNCOP_1:def 8; end; thus Exec(a=0_goto loc, s).c = S.mn by A2,AMI_2:12 .= s.c; end; theorem Th9: (s.a > 0 implies Exec(a >0_goto loc, s).IC SCM = loc) & (s.a <= 0 implies Exec(a>0_goto loc, s).IC SCM = IC s + 1) & Exec(a>0_goto loc, s).c = s.c proof reconsider mn = c as Element of SCM-Data-Loc by AMI_2:def 16; reconsider I = a>0_goto loc as Element of SCM-Instr; reconsider S = s as SCM-State by CARD_3:107; reconsider i = 8 as Element of Segm 9 by NAT_1:44; reconsider a9 = a as Element of SCM-Data-Loc by AMI_2:def 16; reconsider mj = loc as Nat; A1: I = [ i, <*mj*>, <*a9*>]; A2: Exec(a>0_goto loc, s) = SCM-Exec-Res(I,S) by AMI_2:def 15 .= SCM-Chg(S,IFGT(S.(I cond_address),0,I cjump_address,IC S + 1)) by A1, AMI_2:def 14; thus s.a > 0 implies Exec(a>0_goto loc, s).IC SCM = loc proof assume s.a > 0; then A3: S.(I cond_address) > 0 by A1,SCM_INST:7; thus Exec(a>0_goto loc, s).IC SCM = IFGT(S.(I cond_address),0,I cjump_address,IC S + 1) by A2,Th1,AMI_2:11 .= I cjump_address by A3,XXREAL_0:def 11 .= loc by A1,SCM_INST:7; end; thus s.a <= 0 implies Exec(a>0_goto loc, s).IC SCM = IC s + 1 proof assume s.a <= 0; then A4: S.(I cond_address) <= 0 by A1,SCM_INST:7; thus Exec(a>0_goto loc, s).IC SCM = IFGT(S.(I cond_address),0,I cjump_address,IC S + 1) by A2,Th1,AMI_2:11 .= IC s + 1 by A4,Th1,XXREAL_0:def 11; end; thus Exec(a>0_goto loc, s).c = S.mn by A2,AMI_2:12 .= s.c; end; reserve Y,K,T for Element of Segm 9, a1,a2,a3 for Nat, b1,b2,c1,c2, c3 for Element of SCM-Data-Loc; Lm2: for I being Instruction of SCM st ex s st Exec(I,s).IC SCM = IC s + 1 holds I is non halting proof let I be Instruction of SCM; given s such that A1: Exec(I, s).IC SCM = IC s + 1; assume I is halting; then Exec(I,s).IC SCM = s.NAT by Th1; hence contradiction by A1,Th1; IC s = s.NAT by AMI_2:22,SUBSET_1:def 8; then reconsider w = s.NAT as Nat; end; Lm3: for I being Instruction of SCM st I = [0,{},{}] holds I is halting proof let I be Instruction of SCM; assume A1: I = [0,{},{}]; then A2: I`3_3 = {}; then A3: ( not(ex mk, ml being Element of SCM-Data-Loc st I = [ 1, {}, <*mk, ml*>])) & not( ex mk, ml being Element of SCM-Data-Loc st I = [ 2, {}, <*mk, ml*>]); A4: ( not(ex mk being Nat, ml being Element of SCM-Data-Loc st I = [ 7, <*mk*>, <*ml*>]))& not(ex mk being Nat, ml being Element of SCM-Data-Loc st I = [ 8, <*mk*>, <*ml*>]) by A2; I`2_3 = {} by A1; then A5: ( not(ex mk, ml being Element of SCM-Data-Loc st I = [ 5, {}, <*mk, ml*>])) & not( ex mk being Nat st I = [ 6, <*mk*>, {}]) by A2; reconsider L = I as Element of SCM-Instr; let s be State of SCM; reconsider t = s as SCM-State by CARD_3:107; A6: ( not(ex mk, ml being Element of SCM-Data-Loc st I = [ 3, {}, <*mk, ml*>])) & not( ex mk, ml being Element of SCM-Data-Loc st I = [ 4, {}, <*mk, ml*>]) by A2; thus Exec(I,s) = SCM-Exec-Res(L,t) by AMI_2:def 15 .= s by A3,A6,A5,A4,AMI_2:def 14; end; Lm4: a := b is non halting proof set s =the State of SCM; Exec(a:=b,s).IC SCM = IC s + 1 by Th2; hence thesis by Lm2; end; Lm5: AddTo(a,b) is non halting proof set s =the State of SCM; Exec(AddTo(a,b),s).IC SCM = IC s + 1 by Th3; hence thesis by Lm2; end; Lm6: SubFrom(a,b) is non halting proof set s =the State of SCM; Exec(SubFrom(a,b),s).IC SCM = IC s + 1 by Th4; hence thesis by Lm2; end; Lm7: MultBy(a,b) is non halting proof set s =the State of SCM; Exec(MultBy(a,b),s).IC SCM = IC s + 1 by Th5; hence thesis by Lm2; end; Lm8: Divide(a,b) is non halting proof set s =the State of SCM; Exec(Divide(a,b),s).IC SCM = IC s + 1 by Th6; hence thesis by Lm2; end; Lm9: SCM-goto loc is non halting proof set f = the_Values_of SCM; set s =the SCM-State; assume A1: SCM-goto loc is halting; reconsider a3 = loc as Nat; reconsider V = SCM-goto loc as Element of SCM-Instr; set t = s +* (NAT.--> (a3+1)); A2: dom s = the carrier of SCM by AMI_2:28; NAT in dom (NAT.--> (a3+1)) by TARSKI:def 1; then A4: t.NAT = (NAT.--> (a3+1)).NAT by FUNCT_4:13 .= a3+1 by FUNCOP_1:72; A5: for x being object st x in dom f holds t.x in f.x proof let x be object such that A6: x in dom f; per cases; suppose A7: x = NAT; then f.x = NAT by AMI_2:6; hence thesis by A4,A7,ORDINAL1:def 12; end; suppose x <> NAT; then not x in dom (NAT.--> (a3+1)) by TARSKI:def 1; then t.x = s.x by FUNCT_4:11; hence thesis by A6,CARD_3:9; end; end; A8: {NAT} c= SCM-Memory by AMI_2:22,ZFMISC_1:31; A9: dom t = dom s \/ dom (NAT.--> (a3+1)) by FUNCT_4:def 1 .= SCM-Memory \/ dom (NAT.--> (a3+1)) by A2 .= SCM-Memory \/ {NAT} .= SCM-Memory by A8,XBOOLE_1:12; dom f = SCM-Memory by AMI_2:27; then reconsider t as State of SCM by A9,A5,FUNCT_1:def 14,PARTFUN1:def 2 ,RELAT_1:def 18; reconsider w = t as SCM-State by CARD_3:107; NAT in dom (NAT .--> loc) by TARSKI:def 1; then A10: (w +* (NAT .--> loc)).NAT = (NAT .--> loc).NAT by FUNCT_4:13 .= loc by FUNCOP_1:72; 6 is Element of Segm 9 by NAT_1:44; then w +* (NAT .--> loc) = SCM-Chg(w,V jump_address) by SCM_INST:6 .= SCM-Exec-Res(V,w) by AMI_2:def 14 .= Exec(SCM-goto loc,t) by AMI_2:def 15 .= t by A1; hence contradiction by A4,A10; end; Lm10: a=0_goto loc is non halting proof set f = the_Values_of SCM; set s =the SCM-State; reconsider V = a=0_goto loc as Element of SCM-Instr; reconsider a3 = loc as Nat; set t = s +* (NAT.--> (a3+1)); A1: {NAT} c= SCM-Memory by AMI_2:22,ZFMISC_1:31; A2: dom s = the carrier of SCM by AMI_2:28; A3: dom t = dom s \/ dom (NAT.--> (a3+1)) by FUNCT_4:def 1 .= SCM-Memory \/ dom (NAT.--> (a3+1)) by A2 .= SCM-Memory \/ {NAT} .= SCM-Memory by A1,XBOOLE_1:12; A4: 7 is Element of Segm 9 by NAT_1:44; NAT in dom (NAT.--> (a3+1)) by TARSKI:def 1; then A6: t.NAT = (NAT.--> (a3+1)).NAT by FUNCT_4:13 .= a3+1 by FUNCOP_1:72; A7: for x being object st x in dom f holds t.x in f.x proof let x be object such that A8: x in dom f; per cases; suppose A9: x = NAT; then f.x = NAT by AMI_2:6; hence thesis by A6,A9,ORDINAL1:def 12; end; suppose x <> NAT; then not x in dom (NAT.--> (a3+1)) by TARSKI:def 1; then t.x = s.x by FUNCT_4:11; hence thesis by A8,CARD_3:9; end; end; dom f = SCM-Memory by AMI_2:27; then reconsider t as State of SCM by A3,A7,FUNCT_1:def 14,PARTFUN1:def 2 ,RELAT_1:def 18; reconsider w = t as SCM-State by CARD_3:107; NAT in dom (NAT .--> loc) by TARSKI:def 1; then A10: (w +* (NAT .--> loc)).NAT = (NAT .--> loc).NAT by FUNCT_4:13 .= loc by FUNCOP_1:72; assume A11: a=0_goto loc is halting; A12: a is Element of SCM-Data-Loc by AMI_2:def 16; per cases; suppose A13: w.(V cond_address) <> 0; IC w = w.NAT; then reconsider e = w.NAT as Nat; IC t = IC w & t.a <> 0 by A4,A12,A13,AMI_2:22,SCM_INST:7,SUBSET_1:def 8; then A14: Exec(a=0_goto loc,t).IC SCM = e+1 by Th8; Exec(a=0_goto loc,t).IC SCM = w.NAT by A11,Th1; hence contradiction by A14; end; suppose w.(V cond_address) = 0; then IFEQ(w.(V cond_address),0,V cjump_address,IC w + 1) = V cjump_address by FUNCOP_1:def 8; then w +* (NAT .--> loc) = SCM-Chg(w,IFEQ(w.(V cond_address),0,V cjump_address,IC w + 1)) by A4,A12,SCM_INST:7 .= SCM-Exec-Res(V,w) by A12,AMI_2:def 14 .= Exec(a=0_goto loc,t) by AMI_2:def 15 .= t by A11; hence contradiction by A6,A10; end; end; Lm11: a>0_goto loc is non halting proof set f = the_Values_of SCM; set s =the SCM-State; reconsider V = a>0_goto loc as Element of SCM-Instr; reconsider a3 = loc as Nat; set t = s +* (NAT.--> (a3+1)); A1: {NAT} c= SCM-Memory by AMI_2:22,ZFMISC_1:31; A2: dom s = the carrier of SCM by AMI_2:28; A3: dom t = dom s \/ dom (NAT.--> (a3+1)) by FUNCT_4:def 1 .= SCM-Memory \/ dom (NAT.--> (a3+1)) by A2 .= SCM-Memory \/ {NAT} .= SCM-Memory by A1,XBOOLE_1:12; A4: 8 is Element of Segm 9 by NAT_1:44; NAT in dom (NAT.--> (a3+1)) by TARSKI:def 1; then A6: t.NAT = (NAT.--> (a3+1)).NAT by FUNCT_4:13 .= a3 + 1 by FUNCOP_1:72; A7: for x being object st x in dom f holds t.x in f.x proof let x be object such that A8: x in dom f; per cases; suppose A9: x = NAT; then f.x = NAT by AMI_2:6; hence thesis by A6,A9,ORDINAL1:def 12; end; suppose x <> NAT; then not x in dom (NAT.--> (a3+1)) by TARSKI:def 1; then t.x = s.x by FUNCT_4:11; hence thesis by A8,CARD_3:9; end; end; dom f = SCM-Memory by AMI_2:27; then reconsider t as State of SCM by A3,A7,FUNCT_1:def 14,PARTFUN1:def 2 ,RELAT_1:def 18; reconsider w = t as SCM-State by CARD_3:107; NAT in dom (NAT .--> loc) by TARSKI:def 1; then A10: (w +* (NAT .--> loc)).NAT = (NAT .--> loc).NAT by FUNCT_4:13 .= loc by FUNCOP_1:72; assume A11: a>0_goto loc is halting; A12: a is Element of SCM-Data-Loc by AMI_2:def 16; per cases; suppose A13: w.(V cond_address) <= 0; IC w = w.NAT; then reconsider e = w.NAT as Nat; IC t = IC w & t.a <= 0 by A4,A12,A13,AMI_2:22,SCM_INST:7,SUBSET_1:def 8; then A14: Exec(a>0_goto loc,t).IC SCM = e+1 by Th9; Exec(a>0_goto loc,t).IC SCM = w.NAT by A11,Th1; hence contradiction by A14; end; suppose w.(V cond_address) > 0; then IFGT(w.(V cond_address),0,V cjump_address,IC w + 1) = V cjump_address by XXREAL_0:def 11; then w +* (NAT .--> loc) = SCM-Chg(w,IFGT(w.(V cond_address),0,V cjump_address,IC w + 1)) by A4,A12,SCM_INST:7 .= SCM-Exec-Res(V,w) by A12,AMI_2:def 14 .= Exec(a>0_goto loc,t) by AMI_2:def 15 .= t by A11; hence contradiction by A6,A10; end; end; Lm12: for I being set holds I is Instruction of SCM iff I = [0,{},{}] or (ex a,b st I = a:=b) or (ex a,b st I = AddTo(a,b)) or (ex a,b st I = SubFrom(a,b)) or ( ex a,b st I = MultBy(a,b)) or (ex a,b st I = Divide(a,b)) or (ex loc st I = SCM-goto loc) or (ex a,loc st I = a=0_goto loc) or ex a,loc st I = a>0_goto loc proof let I be set; thus I is Instruction of SCM implies I = [0,{},{}] or (ex a,b st I = a:=b) or ( ex a,b st I = AddTo(a,b)) or (ex a,b st I = SubFrom(a,b)) or (ex a,b st I = MultBy(a,b)) or (ex a,b st I = Divide(a,b)) or (ex loc st I = SCM-goto loc) or (ex a,loc st I = a=0_goto loc) or ex a,loc st I = a>0_goto loc proof assume I is Instruction of SCM; then I in { [SCM-Halt,{},{}] } \/ { [Y,<*a3*>,{}] : Y = 6 } \/ { [K,<*a1*>, <*b1*>] : K in { 7,8 } } or I in { [T,{},<*c2,c3*>] : T in { 1,2,3,4,5 } } by XBOOLE_0:def 3; then A1: I in { [SCM-Halt,{},{}] } \/ { [Y,<*a3*>,{}] : Y = 6 } or I in { [K,<*a1*>,<*b1*>]: K in { 7,8 } } or I in { [T,{},<*c2,c3*>] : T in { 1,2,3,4,5 } } by XBOOLE_0:def 3; per cases by A1,XBOOLE_0:def 3; suppose I in { [SCM-Halt,{},{}] }; hence thesis by TARSKI:def 1; end; suppose I in { [Y,<*a3*>,{}] : Y = 6 }; then consider Y, a3 such that A2: I = [Y,<*a3*>,{}] & Y = 6; I = SCM-goto a3 by A2; hence thesis; end; suppose I in { [K,<*a1*>,<*b1*>]: K in { 7,8 } }; then consider K, a1, b1 such that A3: I = [K,<*a1*>,<*b1*>]& K in { 7,8 }; reconsider a = b1 as Data-Location by AMI_2:def 16; reconsider loc = a1 as Nat; I = a=0_goto a1 or I = a>0_goto a1 by A3,TARSKI:def 2; hence thesis; end; suppose I in { [T,{},<*c2,c3*>] : T in { 1,2,3,4,5 } }; then consider T, c2, c3 such that A4: I = [T,{},<*c2,c3*>] & T in { 1,2,3,4,5 }; reconsider a = c2, b = c3 as Data-Location by AMI_2:def 16; I = a:=b or I = AddTo(a,b) or I = SubFrom(a,b) or I = MultBy(a,b) or I = Divide(a,b) by A4,ENUMSET1:def 3; hence thesis; end; end; thus thesis by SCM_INST:1; end; Lm13: for W being Instruction of SCM st W is halting holds W = [0,{},{}] proof set I = [0,{},{}]; let W be Instruction of SCM such that A1: W is halting; assume A2: I <> W; per cases by Lm12; suppose W = [0,{},{}]; hence thesis by A2; end; suppose ex a,b st W = a:=b; hence thesis by A1,Lm4; end; suppose ex a,b st W = AddTo(a,b); hence thesis by A1,Lm5; end; suppose ex a,b st W = SubFrom(a,b); hence thesis by A1,Lm6; end; suppose ex a,b st W = MultBy(a,b); hence thesis by A1,Lm7; end; suppose ex a,b st W = Divide(a,b); hence thesis by A1,Lm8; end; suppose ex loc st W = SCM-goto loc; hence thesis by A1,Lm9; end; suppose ex a,loc st W = a=0_goto loc; hence thesis by A1,Lm10; end; suppose ex a,loc st W = a>0_goto loc; hence thesis by A1,Lm11; end; end; registration cluster SCM -> halting; coherence by Lm3; end; begin :: Small concrete model definition let k be Nat; func dl.k -> Data-Location equals [1,k]; coherence proof reconsider k as Element of NAT by ORDINAL1:def 12; 1 in {1} by TARSKI:def 1; then [1,k] in SCM-Data-Loc by ZFMISC_1:87; hence thesis by AMI_2:def 16; end; end; reserve i,j,k for Nat; theorem i <> j implies dl.i <> dl.j by XTUPLE_0:1; theorem Th11: for l being Data-Location holds Values l = INT by AMI_2:def 16,AMI_2:7; definition let la be Data-Location; let a be Integer; redefine func la .--> a -> PartState of SCM; coherence proof A1: a is Element of INT & Values la = INT by Th11,INT_1:def 2; then reconsider a as Element of (the_Values_of SCM).la; la .--> a is PartState of SCM by A1; hence thesis; end; end; definition let la,lb be Data-Location; let a, b be Integer; redefine func (la,lb) --> (a,b) -> PartState of SCM; coherence proof A1: a is Element of INT & b is Element of INT by INT_1:def 2; A2: Values la = INT & Values lb = INT by Th11; then reconsider a as Element of Values la by A1; reconsider b as Element of Values lb by A1,A2; (la,lb) --> (a,b) is PartState of SCM; hence thesis; end; end; theorem dl.i <> j; theorem IC SCM <> dl.i & IC SCM <> i by Th1; begin :: Halt Instruction theorem for I being Instruction of SCM st ex s st Exec(I,s).IC SCM = IC s + 1 holds I is non halting by Lm2; theorem for I being Instruction of SCM st I = [0,{},{}] holds I is halting by Lm3; theorem a := b is non halting by Lm4; theorem AddTo(a,b) is non halting by Lm5; theorem SubFrom(a,b) is non halting by Lm6; theorem MultBy(a,b) is non halting by Lm7; theorem Divide(a,b) is non halting by Lm8; theorem SCM-goto loc is non halting by Lm9; theorem a=0_goto loc is non halting by Lm10; theorem a>0_goto loc is non halting by Lm11; theorem for I being set holds I is Instruction of SCM iff I = [0,{},{}] or (ex a, b st I = a:=b) or (ex a,b st I = AddTo(a,b)) or (ex a,b st I = SubFrom(a,b)) or (ex a,b st I = MultBy(a,b)) or (ex a,b st I = Divide(a,b)) or (ex loc st I = SCM-goto loc) or (ex a,loc st I = a=0_goto loc) or ex a,loc st I = a>0_goto loc by Lm12; theorem for I being Instruction of SCM st I is halting holds I = halt SCM by Lm13; theorem halt SCM = [0,{},{}]; theorem Th27: Data-Locations SCM = SCM-Data-Loc proof SCM-Data-Loc misses {NAT} by AMI_2:20,ZFMISC_1:50; then A1: SCM-Data-Loc misses {NAT}; thus Data-Locations SCM = {NAT} \/ SCM-Data-Loc \ ({NAT}) by AMI_2:22 ,SUBSET_1:def 8 .= SCM-Data-Loc \/ ({NAT}) \ ({NAT}) .= SCM-Data-Loc \ ({NAT}) by XBOOLE_1:40 .= SCM-Data-Loc by A1,XBOOLE_1:83; end; theorem for d being Data-Location holds d in Data-Locations SCM by Th27,AMI_2:def 16; theorem for s being SCM-State holds s is State of SCM by Lm1; theorem for l being Element of SCM-Instr holds InsCode l <= 8 by SCM_INST:10;