:: SCMRING3 semantic presentation begin theorem Th1: :: SCMRING3:1 for R being Ring for a being Data-Location of R holds Values a = the carrier of R proof let R be Ring; ::_thesis: for a being Data-Location of R holds Values a = the carrier of R let a be Data-Location of R; ::_thesis: Values a = the carrier of R ( a in Data-Locations & the_Values_of (SCM R) = (SCM-VAL R) * SCM-OK ) by SCMRING2:1, SCMRING2:24; hence Values a = the carrier of R by AMI_3:27, SCMRING1:4; ::_thesis: verum end; definition let R be Ring; let la, lb be Data-Location of R; let a, b be Element of R; :: original: --> redefine func(la,lb) --> (a,b) -> FinPartState of (SCM R); coherence (la,lb) --> (a,b) is FinPartState of (SCM R) proof reconsider b9 = b as Element of Values lb by Th1; reconsider a9 = a as Element of Values la by Th1; (la,lb) --> (a9,b9) is FinPartState of (SCM R) ; hence (la,lb) --> (a,b) is FinPartState of (SCM R) ; ::_thesis: verum end; end; theorem Th2: :: SCMRING3:2 for R being Ring for a being Data-Location of R holds a <> IC proof let R be Ring; ::_thesis: for a being Data-Location of R holds a <> IC let a be Data-Location of R; ::_thesis: a <> IC a in SCM-Data-Loc by AMI_2:def_16; then a <> NAT by AMI_2:20; hence a <> IC by SCMRING2:8; ::_thesis: verum end; theorem :: SCMRING3:3 for R being Ring for o being Object of (SCM R) holds ( o = IC or o is Data-Location of R ) proof let R be Ring; ::_thesis: for o being Object of (SCM R) holds ( o = IC or o is Data-Location of R ) let o be Object of (SCM R); ::_thesis: ( o = IC or o is Data-Location of R ) assume o <> IC ; ::_thesis: o is Data-Location of R then not o in {(IC )} by TARSKI:def_1; then A1: not o in {NAT} by SCMRING2:8; not o in {NAT} by A1; then o in the carrier of (SCM R) \ {NAT} by XBOOLE_0:def_5; then o in SCM-Data-Loc by SCMRING2:25; hence o is Data-Location of R by AMI_2:def_16; ::_thesis: verum end; theorem :: SCMRING3:4 canceled; theorem :: SCMRING3:5 for R being Ring for a, b being Data-Location of R holds InsCode (a := b) = 1 by RECDEF_2:def_1; theorem :: SCMRING3:6 for R being Ring for a, b being Data-Location of R holds InsCode (AddTo (a,b)) = 2 by RECDEF_2:def_1; theorem :: SCMRING3:7 for R being Ring for a, b being Data-Location of R holds InsCode (SubFrom (a,b)) = 3 by RECDEF_2:def_1; theorem :: SCMRING3:8 for R being Ring for a, b being Data-Location of R holds InsCode (MultBy (a,b)) = 4 by RECDEF_2:def_1; theorem :: SCMRING3:9 for R being Ring for r being Element of R for a being Data-Location of R holds InsCode (a := r) = 5 by RECDEF_2:def_1; theorem Th10: :: SCMRING3:10 for R being Ring for i1 being Element of NAT holds InsCode (goto (i1,R)) = 6 by RECDEF_2:def_1; theorem Th11: :: SCMRING3:11 for R being Ring for a being Data-Location of R for i1 being Element of NAT holds InsCode (a =0_goto i1) = 7 by RECDEF_2:def_1; theorem Th12: :: SCMRING3:12 for R being Ring for I being Instruction of (SCM R) st InsCode I = 0 holds I = halt (SCM R) proof let R be Ring; ::_thesis: for I being Instruction of (SCM R) st InsCode I = 0 holds I = halt (SCM R) let I be Instruction of (SCM R); ::_thesis: ( InsCode I = 0 implies I = halt (SCM R) ) A1: ( I = [0,{},{}] or ex a, b being Data-Location of R st I = a := b or ex a, b being Data-Location of R st I = AddTo (a,b) or ex a, b being Data-Location of R st I = SubFrom (a,b) or ex a, b being Data-Location of R st I = MultBy (a,b) or ex i1 being Element of NAT st I = goto (i1,R) or ex a being Data-Location of R ex i1 being Element of NAT st I = a =0_goto i1 or ex a being Data-Location of R ex r being Element of R st I = a := r ) by SCMRING2:7; assume InsCode I = 0 ; ::_thesis: I = halt (SCM R) hence I = halt (SCM R) by A1, RECDEF_2:def_1; ::_thesis: verum end; theorem Th13: :: SCMRING3:13 for R being Ring for I being Instruction of (SCM R) st InsCode I = 1 holds ex a, b being Data-Location of R st I = a := b proof let R be Ring; ::_thesis: for I being Instruction of (SCM R) st InsCode I = 1 holds ex a, b being Data-Location of R st I = a := b let I be Instruction of (SCM R); ::_thesis: ( InsCode I = 1 implies ex a, b being Data-Location of R st I = a := b ) A1: ( I = [0,{},{}] or ex a, b being Data-Location of R st I = a := b or ex a, b being Data-Location of R st I = AddTo (a,b) or ex a, b being Data-Location of R st I = SubFrom (a,b) or ex a, b being Data-Location of R st I = MultBy (a,b) or ex i1 being Element of NAT st I = goto (i1,R) or ex a being Data-Location of R ex i1 being Element of NAT st I = a =0_goto i1 or ex a being Data-Location of R ex r being Element of R st I = a := r ) by SCMRING2:7; assume InsCode I = 1 ; ::_thesis: ex a, b being Data-Location of R st I = a := b hence ex a, b being Data-Location of R st I = a := b by A1, RECDEF_2:def_1; ::_thesis: verum end; theorem Th14: :: SCMRING3:14 for R being Ring for I being Instruction of (SCM R) st InsCode I = 2 holds ex a, b being Data-Location of R st I = AddTo (a,b) proof let R be Ring; ::_thesis: for I being Instruction of (SCM R) st InsCode I = 2 holds ex a, b being Data-Location of R st I = AddTo (a,b) let I be Instruction of (SCM R); ::_thesis: ( InsCode I = 2 implies ex a, b being Data-Location of R st I = AddTo (a,b) ) A1: ( I = [0,{},{}] or ex a, b being Data-Location of R st I = a := b or ex a, b being Data-Location of R st I = AddTo (a,b) or ex a, b being Data-Location of R st I = SubFrom (a,b) or ex a, b being Data-Location of R st I = MultBy (a,b) or ex i1 being Element of NAT st I = goto (i1,R) or ex a being Data-Location of R ex i1 being Element of NAT st I = a =0_goto i1 or ex a being Data-Location of R ex r being Element of R st I = a := r ) by SCMRING2:7; assume InsCode I = 2 ; ::_thesis: ex a, b being Data-Location of R st I = AddTo (a,b) hence ex a, b being Data-Location of R st I = AddTo (a,b) by A1, RECDEF_2:def_1; ::_thesis: verum end; theorem Th15: :: SCMRING3:15 for R being Ring for I being Instruction of (SCM R) st InsCode I = 3 holds ex a, b being Data-Location of R st I = SubFrom (a,b) proof let R be Ring; ::_thesis: for I being Instruction of (SCM R) st InsCode I = 3 holds ex a, b being Data-Location of R st I = SubFrom (a,b) let I be Instruction of (SCM R); ::_thesis: ( InsCode I = 3 implies ex a, b being Data-Location of R st I = SubFrom (a,b) ) A1: ( I = [0,{},{}] or ex a, b being Data-Location of R st I = a := b or ex a, b being Data-Location of R st I = AddTo (a,b) or ex a, b being Data-Location of R st I = SubFrom (a,b) or ex a, b being Data-Location of R st I = MultBy (a,b) or ex i1 being Element of NAT st I = goto (i1,R) or ex a being Data-Location of R ex i1 being Element of NAT st I = a =0_goto i1 or ex a being Data-Location of R ex r being Element of R st I = a := r ) by SCMRING2:7; assume InsCode I = 3 ; ::_thesis: ex a, b being Data-Location of R st I = SubFrom (a,b) hence ex a, b being Data-Location of R st I = SubFrom (a,b) by A1, RECDEF_2:def_1; ::_thesis: verum end; theorem Th16: :: SCMRING3:16 for R being Ring for I being Instruction of (SCM R) st InsCode I = 4 holds ex a, b being Data-Location of R st I = MultBy (a,b) proof let R be Ring; ::_thesis: for I being Instruction of (SCM R) st InsCode I = 4 holds ex a, b being Data-Location of R st I = MultBy (a,b) let I be Instruction of (SCM R); ::_thesis: ( InsCode I = 4 implies ex a, b being Data-Location of R st I = MultBy (a,b) ) A1: ( I = [0,{},{}] or ex a, b being Data-Location of R st I = a := b or ex a, b being Data-Location of R st I = AddTo (a,b) or ex a, b being Data-Location of R st I = SubFrom (a,b) or ex a, b being Data-Location of R st I = MultBy (a,b) or ex i1 being Element of NAT st I = goto (i1,R) or ex a being Data-Location of R ex i1 being Element of NAT st I = a =0_goto i1 or ex a being Data-Location of R ex r being Element of R st I = a := r ) by SCMRING2:7; assume InsCode I = 4 ; ::_thesis: ex a, b being Data-Location of R st I = MultBy (a,b) hence ex a, b being Data-Location of R st I = MultBy (a,b) by A1, RECDEF_2:def_1; ::_thesis: verum end; theorem Th17: :: SCMRING3:17 for R being Ring for I being Instruction of (SCM R) st InsCode I = 5 holds ex a being Data-Location of R ex r being Element of R st I = a := r proof let R be Ring; ::_thesis: for I being Instruction of (SCM R) st InsCode I = 5 holds ex a being Data-Location of R ex r being Element of R st I = a := r let I be Instruction of (SCM R); ::_thesis: ( InsCode I = 5 implies ex a being Data-Location of R ex r being Element of R st I = a := r ) A1: ( I = [0,{},{}] or ex a, b being Data-Location of R st I = a := b or ex a, b being Data-Location of R st I = AddTo (a,b) or ex a, b being Data-Location of R st I = SubFrom (a,b) or ex a, b being Data-Location of R st I = MultBy (a,b) or ex i1 being Element of NAT st I = goto (i1,R) or ex a being Data-Location of R ex i1 being Element of NAT st I = a =0_goto i1 or ex a being Data-Location of R ex r being Element of R st I = a := r ) by SCMRING2:7; assume InsCode I = 5 ; ::_thesis: ex a being Data-Location of R ex r being Element of R st I = a := r hence ex a being Data-Location of R ex r being Element of R st I = a := r by A1, RECDEF_2:def_1; ::_thesis: verum end; theorem Th18: :: SCMRING3:18 for R being Ring for I being Instruction of (SCM R) st InsCode I = 6 holds ex i2 being Element of NAT st I = goto (i2,R) proof let R be Ring; ::_thesis: for I being Instruction of (SCM R) st InsCode I = 6 holds ex i2 being Element of NAT st I = goto (i2,R) let I be Instruction of (SCM R); ::_thesis: ( InsCode I = 6 implies ex i2 being Element of NAT st I = goto (i2,R) ) A1: ( I = [0,{},{}] or ex a, b being Data-Location of R st I = a := b or ex a, b being Data-Location of R st I = AddTo (a,b) or ex a, b being Data-Location of R st I = SubFrom (a,b) or ex a, b being Data-Location of R st I = MultBy (a,b) or ex i1 being Element of NAT st I = goto (i1,R) or ex a being Data-Location of R ex i1 being Element of NAT st I = a =0_goto i1 or ex a being Data-Location of R ex r being Element of R st I = a := r ) by SCMRING2:7; assume InsCode I = 6 ; ::_thesis: ex i2 being Element of NAT st I = goto (i2,R) hence ex i2 being Element of NAT st I = goto (i2,R) by A1, RECDEF_2:def_1; ::_thesis: verum end; theorem Th19: :: SCMRING3:19 for R being Ring for I being Instruction of (SCM R) st InsCode I = 7 holds ex a being Data-Location of R ex i1 being Element of NAT st I = a =0_goto i1 proof let R be Ring; ::_thesis: for I being Instruction of (SCM R) st InsCode I = 7 holds ex a being Data-Location of R ex i1 being Element of NAT st I = a =0_goto i1 let I be Instruction of (SCM R); ::_thesis: ( InsCode I = 7 implies ex a being Data-Location of R ex i1 being Element of NAT st I = a =0_goto i1 ) A1: ( I = [0,{},{}] or ex a, b being Data-Location of R st I = a := b or ex a, b being Data-Location of R st I = AddTo (a,b) or ex a, b being Data-Location of R st I = SubFrom (a,b) or ex a, b being Data-Location of R st I = MultBy (a,b) or ex i1 being Element of NAT st I = goto (i1,R) or ex a being Data-Location of R ex i1 being Element of NAT st I = a =0_goto i1 or ex a being Data-Location of R ex r being Element of R st I = a := r ) by SCMRING2:7; assume InsCode I = 7 ; ::_thesis: ex a being Data-Location of R ex i1 being Element of NAT st I = a =0_goto i1 hence ex a being Data-Location of R ex i1 being Element of NAT st I = a =0_goto i1 by A1, RECDEF_2:def_1; ::_thesis: verum end; Lm1: for x, y being set st x in dom <*y*> holds x = 1 proof let x, y be set ; ::_thesis: ( x in dom <*y*> implies x = 1 ) assume x in dom <*y*> ; ::_thesis: x = 1 then x in Seg 1 by FINSEQ_1:def_8; hence x = 1 by FINSEQ_1:2, TARSKI:def_1; ::_thesis: verum end; Lm2: for R being Ring for T being InsType of the InstructionsF of (SCM R) holds ( T = 0 or T = 1 or T = 2 or T = 3 or T = 4 or T = 5 or T = 6 or T = 7 ) proof let R be Ring; ::_thesis: for T being InsType of the InstructionsF of (SCM R) holds ( T = 0 or T = 1 or T = 2 or T = 3 or T = 4 or T = 5 or T = 6 or T = 7 ) let T be InsType of the InstructionsF of (SCM R); ::_thesis: ( T = 0 or T = 1 or T = 2 or T = 3 or T = 4 or T = 5 or T = 6 or T = 7 ) consider y being set such that A1: [T,y] in proj1 the InstructionsF of (SCM R) by XTUPLE_0:def_12; consider x being set such that A2: [[T,y],x] in the InstructionsF of (SCM R) by A1, XTUPLE_0:def_12; [T,y,x] in SCM-Instr R by A2, SCMRING2:def_1; then ( [T,y,x] in (({[0,{},{}]} \/ { [I,{},<*a,b*>] where I is Element of Segm 8, a, b is Element of Data-Locations : I in {1,2,3,4} } ) \/ { [6,<*i*>,{}] where i is Element of NAT : verum } ) \/ { [7,<*i*>,<*a*>] where i is Element of NAT , a is Element of Data-Locations : verum } or [T,y,x] in { [5,{},<*a,r*>] where a is Element of Data-Locations , r is Element of R : verum } ) by AMI_3:27, XBOOLE_0:def_3; then ( [T,y,x] in ({[0,{},{}]} \/ { [I,{},<*a,b*>] where I is Element of Segm 8, a, b is Element of Data-Locations : I in {1,2,3,4} } ) \/ { [6,<*i*>,{}] where i is Element of NAT : verum } or [T,y,x] in { [7,<*i*>,<*a*>] where i is Element of NAT , a is Element of Data-Locations : verum } or [T,y,x] in { [5,{},<*a,r*>] where a is Element of Data-Locations , r is Element of R : verum } ) by XBOOLE_0:def_3; then A3: ( [T,y,x] in {[0,{},{}]} \/ { [I,{},<*a,b*>] where I is Element of Segm 8, a, b is Element of Data-Locations : I in {1,2,3,4} } or [T,y,x] in { [6,<*i*>,{}] where i is Element of NAT : verum } or [T,y,x] in { [7,<*i*>,<*a*>] where i is Element of NAT , a is Element of Data-Locations : verum } or [T,y,x] in { [5,{},<*a,r*>] where a is Element of Data-Locations , r is Element of R : verum } ) by XBOOLE_0:def_3; percases ( [T,y,x] in {[0,{},{}]} or [T,y,x] in { [I,{},<*a,b*>] where I is Element of Segm 8, a, b is Element of Data-Locations : I in {1,2,3,4} } or [T,y,x] in { [6,<*i*>,{}] where i is Element of NAT : verum } or [T,y,x] in { [7,<*i*>,<*a*>] where i is Element of NAT , a is Element of Data-Locations : verum } or [T,y,x] in { [5,{},<*a,r*>] where a is Element of Data-Locations , r is Element of R : verum } ) by A3, XBOOLE_0:def_3; suppose [T,y,x] in {[0,{},{}]} ; ::_thesis: ( T = 0 or T = 1 or T = 2 or T = 3 or T = 4 or T = 5 or T = 6 or T = 7 ) then [T,y,x] = [0,{},{}] by TARSKI:def_1; hence ( T = 0 or T = 1 or T = 2 or T = 3 or T = 4 or T = 5 or T = 6 or T = 7 ) by XTUPLE_0:3; ::_thesis: verum end; suppose [T,y,x] in { [I,{},<*a,b*>] where I is Element of Segm 8, a, b is Element of Data-Locations : I in {1,2,3,4} } ; ::_thesis: ( T = 0 or T = 1 or T = 2 or T = 3 or T = 4 or T = 5 or T = 6 or T = 7 ) then ex I being Element of Segm 8 ex a, b being Element of Data-Locations st ( [T,y,x] = [I,{},<*a,b*>] & I in {1,2,3,4} ) ; then T in {1,2,3,4} by XTUPLE_0:3; hence ( T = 0 or T = 1 or T = 2 or T = 3 or T = 4 or T = 5 or T = 6 or T = 7 ) by ENUMSET1:def_2; ::_thesis: verum end; suppose [T,y,x] in { [6,<*i*>,{}] where i is Element of NAT : verum } ; ::_thesis: ( T = 0 or T = 1 or T = 2 or T = 3 or T = 4 or T = 5 or T = 6 or T = 7 ) then ex i being Element of NAT st [T,y,x] = [6,<*i*>,{}] ; hence ( T = 0 or T = 1 or T = 2 or T = 3 or T = 4 or T = 5 or T = 6 or T = 7 ) by XTUPLE_0:3; ::_thesis: verum end; suppose [T,y,x] in { [7,<*i*>,<*a*>] where i is Element of NAT , a is Element of Data-Locations : verum } ; ::_thesis: ( T = 0 or T = 1 or T = 2 or T = 3 or T = 4 or T = 5 or T = 6 or T = 7 ) then ex i being Element of NAT ex a being Element of Data-Locations st [T,y,x] = [7,<*i*>,<*a*>] ; hence ( T = 0 or T = 1 or T = 2 or T = 3 or T = 4 or T = 5 or T = 6 or T = 7 ) by XTUPLE_0:3; ::_thesis: verum end; suppose [T,y,x] in { [5,{},<*a,r*>] where a is Element of Data-Locations , r is Element of R : verum } ; ::_thesis: ( T = 0 or T = 1 or T = 2 or T = 3 or T = 4 or T = 5 or T = 6 or T = 7 ) then ex a being Element of Data-Locations ex r being Element of R st [T,y,x] = [5,{},<*a,r*>] ; hence ( T = 0 or T = 1 or T = 2 or T = 3 or T = 4 or T = 5 or T = 6 or T = 7 ) by XTUPLE_0:3; ::_thesis: verum end; end; end; theorem :: SCMRING3:20 for R being Ring for T being InsType of the InstructionsF of (SCM R) st T = 0 holds JumpParts T = {0} proof let R be Ring; ::_thesis: for T being InsType of the InstructionsF of (SCM R) st T = 0 holds JumpParts T = {0} let T be InsType of the InstructionsF of (SCM R); ::_thesis: ( T = 0 implies JumpParts T = {0} ) assume A1: T = 0 ; ::_thesis: JumpParts T = {0} hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: {0} c= JumpParts T let a be set ; ::_thesis: ( a in JumpParts T implies a in {0} ) assume a in JumpParts T ; ::_thesis: a in {0} then consider I being Instruction of (SCM R) such that A2: a = JumpPart I and A3: InsCode I = T ; I = halt (SCM R) by A1, A3, Th12; then a = {} by A2; hence a in {0} by TARSKI:def_1; ::_thesis: verum end; let a be set ; :: according to TARSKI:def_3 ::_thesis: ( not a in {0} or a in JumpParts T ) assume a in {0} ; ::_thesis: a in JumpParts T then A4: a = 0 by TARSKI:def_1; ( InsCode (halt (SCM R)) = 0 & JumpPart (halt (SCM R)) = 0 ) by COMPOS_1:70; hence a in JumpParts T by A1, A4; ::_thesis: verum end; theorem :: SCMRING3:21 for R being Ring for T being InsType of the InstructionsF of (SCM R) st T = 1 holds JumpParts T = {{}} proof let R be Ring; ::_thesis: for T being InsType of the InstructionsF of (SCM R) st T = 1 holds JumpParts T = {{}} let T be InsType of the InstructionsF of (SCM R); ::_thesis: ( T = 1 implies JumpParts T = {{}} ) assume A1: T = 1 ; ::_thesis: JumpParts T = {{}} hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: {{}} c= JumpParts T let x be set ; ::_thesis: ( x in JumpParts T implies x in {{}} ) assume x in JumpParts T ; ::_thesis: x in {{}} then consider I being Instruction of (SCM R) such that A2: x = JumpPart I and A3: InsCode I = T ; consider a, b being Data-Location of R such that A4: I = a := b by A1, A3, Th13; x = {} by A2, A4, RECDEF_2:def_2; hence x in {{}} by TARSKI:def_1; ::_thesis: verum end; set a = the Data-Location of R; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {{}} or x in JumpParts T ) assume x in {{}} ; ::_thesis: x in JumpParts T then x = {} by TARSKI:def_1; then A5: x = JumpPart ( the Data-Location of R := the Data-Location of R) by RECDEF_2:def_2; InsCode ( the Data-Location of R := the Data-Location of R) = 1 by RECDEF_2:def_1; hence x in JumpParts T by A5, A1; ::_thesis: verum end; theorem :: SCMRING3:22 for R being Ring for T being InsType of the InstructionsF of (SCM R) st T = 2 holds JumpParts T = {{}} proof let R be Ring; ::_thesis: for T being InsType of the InstructionsF of (SCM R) st T = 2 holds JumpParts T = {{}} let T be InsType of the InstructionsF of (SCM R); ::_thesis: ( T = 2 implies JumpParts T = {{}} ) assume A1: T = 2 ; ::_thesis: JumpParts T = {{}} hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: {{}} c= JumpParts T let x be set ; ::_thesis: ( x in JumpParts T implies x in {{}} ) assume x in JumpParts T ; ::_thesis: x in {{}} then consider I being Instruction of (SCM R) such that A2: x = JumpPart I and A3: InsCode I = T ; consider a, b being Data-Location of R such that A4: I = AddTo (a,b) by A1, A3, Th14; x = {} by A2, A4, RECDEF_2:def_2; hence x in {{}} by TARSKI:def_1; ::_thesis: verum end; set a = the Data-Location of R; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {{}} or x in JumpParts T ) assume x in {{}} ; ::_thesis: x in JumpParts T then x = {} by TARSKI:def_1; then A5: x = JumpPart (AddTo ( the Data-Location of R, the Data-Location of R)) by RECDEF_2:def_2; InsCode (AddTo ( the Data-Location of R, the Data-Location of R)) = 2 by RECDEF_2:def_1; hence x in JumpParts T by A5, A1; ::_thesis: verum end; theorem :: SCMRING3:23 for R being Ring for T being InsType of the InstructionsF of (SCM R) st T = 3 holds JumpParts T = {{}} proof let R be Ring; ::_thesis: for T being InsType of the InstructionsF of (SCM R) st T = 3 holds JumpParts T = {{}} let T be InsType of the InstructionsF of (SCM R); ::_thesis: ( T = 3 implies JumpParts T = {{}} ) assume A1: T = 3 ; ::_thesis: JumpParts T = {{}} hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: {{}} c= JumpParts T let x be set ; ::_thesis: ( x in JumpParts T implies x in {{}} ) assume x in JumpParts T ; ::_thesis: x in {{}} then consider I being Instruction of (SCM R) such that A2: x = JumpPart I and A3: InsCode I = T ; consider a, b being Data-Location of R such that A4: I = SubFrom (a,b) by A1, A3, Th15; x = {} by A2, A4, RECDEF_2:def_2; hence x in {{}} by TARSKI:def_1; ::_thesis: verum end; set a = the Data-Location of R; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {{}} or x in JumpParts T ) assume x in {{}} ; ::_thesis: x in JumpParts T then x = {} by TARSKI:def_1; then A5: x = JumpPart (SubFrom ( the Data-Location of R, the Data-Location of R)) by RECDEF_2:def_2; InsCode (SubFrom ( the Data-Location of R, the Data-Location of R)) = 3 by RECDEF_2:def_1; hence x in JumpParts T by A5, A1; ::_thesis: verum end; theorem :: SCMRING3:24 for R being Ring for T being InsType of the InstructionsF of (SCM R) st T = 4 holds JumpParts T = {{}} proof let R be Ring; ::_thesis: for T being InsType of the InstructionsF of (SCM R) st T = 4 holds JumpParts T = {{}} let T be InsType of the InstructionsF of (SCM R); ::_thesis: ( T = 4 implies JumpParts T = {{}} ) assume A1: T = 4 ; ::_thesis: JumpParts T = {{}} hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: {{}} c= JumpParts T let x be set ; ::_thesis: ( x in JumpParts T implies x in {{}} ) assume x in JumpParts T ; ::_thesis: x in {{}} then consider I being Instruction of (SCM R) such that A2: x = JumpPart I and A3: InsCode I = T ; consider a, b being Data-Location of R such that A4: I = MultBy (a,b) by A1, A3, Th16; x = {} by A2, A4, RECDEF_2:def_2; hence x in {{}} by TARSKI:def_1; ::_thesis: verum end; set a = the Data-Location of R; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {{}} or x in JumpParts T ) assume x in {{}} ; ::_thesis: x in JumpParts T then x = {} by TARSKI:def_1; then A5: x = JumpPart (MultBy ( the Data-Location of R, the Data-Location of R)) by RECDEF_2:def_2; InsCode (MultBy ( the Data-Location of R, the Data-Location of R)) = 4 by RECDEF_2:def_1; hence x in JumpParts T by A5, A1; ::_thesis: verum end; theorem :: SCMRING3:25 for R being Ring for T being InsType of the InstructionsF of (SCM R) st T = 5 holds JumpParts T = {{}} proof let R be Ring; ::_thesis: for T being InsType of the InstructionsF of (SCM R) st T = 5 holds JumpParts T = {{}} let T be InsType of the InstructionsF of (SCM R); ::_thesis: ( T = 5 implies JumpParts T = {{}} ) assume A1: T = 5 ; ::_thesis: JumpParts T = {{}} hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: {{}} c= JumpParts T let x be set ; ::_thesis: ( x in JumpParts T implies x in {{}} ) assume x in JumpParts T ; ::_thesis: x in {{}} then consider I being Instruction of (SCM R) such that A2: x = JumpPart I and A3: InsCode I = T ; consider a being Data-Location of R, r being Element of R such that A4: I = a := r by A1, A3, Th17; x = {} by A2, A4, RECDEF_2:def_2; hence x in {{}} by TARSKI:def_1; ::_thesis: verum end; set a = the Data-Location of R; set r = the Element of R; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {{}} or x in JumpParts T ) assume x in {{}} ; ::_thesis: x in JumpParts T then x = {} by TARSKI:def_1; then A5: x = JumpPart ( the Data-Location of R := the Element of R) by RECDEF_2:def_2; InsCode ( the Data-Location of R := the Element of R) = 5 by RECDEF_2:def_1; hence x in JumpParts T by A5, A1; ::_thesis: verum end; theorem Th26: :: SCMRING3:26 for R being Ring for T being InsType of the InstructionsF of (SCM R) st T = 6 holds dom (product" (JumpParts T)) = {1} proof let R be Ring; ::_thesis: for T being InsType of the InstructionsF of (SCM R) st T = 6 holds dom (product" (JumpParts T)) = {1} let T be InsType of the InstructionsF of (SCM R); ::_thesis: ( T = 6 implies dom (product" (JumpParts T)) = {1} ) set i1 = the Element of NAT ; assume A1: T = 6 ; ::_thesis: dom (product" (JumpParts T)) = {1} A2: JumpPart (goto ( the Element of NAT ,R)) = <* the Element of NAT *> by RECDEF_2:def_2; hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: {1} c= dom (product" (JumpParts T)) let x be set ; ::_thesis: ( x in dom (product" (JumpParts T)) implies x in {1} ) InsCode (goto ( the Element of NAT ,R)) = 6 by RECDEF_2:def_1; then A3: JumpPart (goto ( the Element of NAT ,R)) in JumpParts T by A1; assume x in dom (product" (JumpParts T)) ; ::_thesis: x in {1} then x in DOM (JumpParts T) by CARD_3:def_12; then x in dom (JumpPart (goto ( the Element of NAT ,R))) by A3, CARD_3:108; hence x in {1} by A2, FINSEQ_1:2, FINSEQ_1:def_8; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {1} or x in dom (product" (JumpParts T)) ) assume A4: x in {1} ; ::_thesis: x in dom (product" (JumpParts T)) for f being Function st f in JumpParts T holds x in dom f proof let f be Function; ::_thesis: ( f in JumpParts T implies x in dom f ) assume f in JumpParts T ; ::_thesis: x in dom f then consider I being Instruction of (SCM R) such that A5: f = JumpPart I and A6: InsCode I = T ; consider i1 being Element of NAT such that A7: I = goto (i1,R) by A1, A6, Th18; f = <*i1*> by A5, A7, RECDEF_2:def_2; hence x in dom f by A4, FINSEQ_1:2, FINSEQ_1:def_8; ::_thesis: verum end; then x in DOM (JumpParts T) by CARD_3:109; hence x in dom (product" (JumpParts T)) by CARD_3:def_12; ::_thesis: verum end; theorem Th27: :: SCMRING3:27 for R being Ring for T being InsType of the InstructionsF of (SCM R) st T = 7 holds dom (product" (JumpParts T)) = {1} proof let R be Ring; ::_thesis: for T being InsType of the InstructionsF of (SCM R) st T = 7 holds dom (product" (JumpParts T)) = {1} let T be InsType of the InstructionsF of (SCM R); ::_thesis: ( T = 7 implies dom (product" (JumpParts T)) = {1} ) set i1 = the Element of NAT ; set a = the Data-Location of R; assume A1: T = 7 ; ::_thesis: dom (product" (JumpParts T)) = {1} A2: JumpPart ( the Data-Location of R =0_goto the Element of NAT ) = <* the Element of NAT *> by RECDEF_2:def_2; hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: {1} c= dom (product" (JumpParts T)) let x be set ; ::_thesis: ( x in dom (product" (JumpParts T)) implies x in {1} ) InsCode ( the Data-Location of R =0_goto the Element of NAT ) = 7 by RECDEF_2:def_1; then A3: JumpPart ( the Data-Location of R =0_goto the Element of NAT ) in JumpParts T by A1; assume x in dom (product" (JumpParts T)) ; ::_thesis: x in {1} then x in DOM (JumpParts T) by CARD_3:def_12; then x in dom (JumpPart ( the Data-Location of R =0_goto the Element of NAT )) by A3, CARD_3:108; hence x in {1} by A2, FINSEQ_1:2, FINSEQ_1:38; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {1} or x in dom (product" (JumpParts T)) ) assume A4: x in {1} ; ::_thesis: x in dom (product" (JumpParts T)) for f being Function st f in JumpParts T holds x in dom f proof let f be Function; ::_thesis: ( f in JumpParts T implies x in dom f ) assume f in JumpParts T ; ::_thesis: x in dom f then consider I being Instruction of (SCM R) such that A5: f = JumpPart I and A6: InsCode I = T ; consider a being Data-Location of R, i1 being Element of NAT such that A7: I = a =0_goto i1 by A1, A6, Th19; f = <*i1*> by A5, A7, RECDEF_2:def_2; hence x in dom f by A4, FINSEQ_1:2, FINSEQ_1:38; ::_thesis: verum end; then x in DOM (JumpParts T) by CARD_3:109; hence x in dom (product" (JumpParts T)) by CARD_3:def_12; ::_thesis: verum end; theorem :: SCMRING3:28 for R being Ring for i1 being Element of NAT holds (product" (JumpParts (InsCode (goto (i1,R))))) . 1 = NAT proof let R be Ring; ::_thesis: for i1 being Element of NAT holds (product" (JumpParts (InsCode (goto (i1,R))))) . 1 = NAT let i1 be Element of NAT ; ::_thesis: (product" (JumpParts (InsCode (goto (i1,R))))) . 1 = NAT dom (product" (JumpParts (InsCode (goto (i1,R))))) = {1} by Th10, Th26; then A1: 1 in dom (product" (JumpParts (InsCode (goto (i1,R))))) by TARSKI:def_1; A2: InsCode (goto (i1,R)) = 6 by RECDEF_2:def_1; hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: NAT c= (product" (JumpParts (InsCode (goto (i1,R))))) . 1 let x be set ; ::_thesis: ( x in (product" (JumpParts (InsCode (goto (i1,R))))) . 1 implies x in NAT ) assume x in (product" (JumpParts (InsCode (goto (i1,R))))) . 1 ; ::_thesis: x in NAT then x in pi ((JumpParts (InsCode (goto (i1,R)))),1) by A1, CARD_3:def_12; then consider g being Function such that A3: g in JumpParts (InsCode (goto (i1,R))) and A4: x = g . 1 by CARD_3:def_6; consider I being Instruction of (SCM R) such that A5: g = JumpPart I and A6: InsCode I = InsCode (goto (i1,R)) by A3; consider i2 being Element of NAT such that A7: I = goto (i2,R) by A2, A6, Th18; g = <*i2*> by A5, A7, RECDEF_2:def_2; then x = i2 by A4, FINSEQ_1:def_8; hence x in NAT ; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in NAT or x in (product" (JumpParts (InsCode (goto (i1,R))))) . 1 ) assume x in NAT ; ::_thesis: x in (product" (JumpParts (InsCode (goto (i1,R))))) . 1 then reconsider x = x as Element of NAT ; ( JumpPart (goto (x,R)) = <*x*> & InsCode (goto (i1,R)) = InsCode (goto (x,R)) ) by A2, RECDEF_2:def_1, RECDEF_2:def_2; then A8: <*x*> in JumpParts (InsCode (goto (i1,R))) ; <*x*> . 1 = x by FINSEQ_1:def_8; then x in pi ((JumpParts (InsCode (goto (i1,R)))),1) by A8, CARD_3:def_6; hence x in (product" (JumpParts (InsCode (goto (i1,R))))) . 1 by A1, CARD_3:def_12; ::_thesis: verum end; theorem :: SCMRING3:29 for R being Ring for a being Data-Location of R for i1 being Element of NAT holds (product" (JumpParts (InsCode (a =0_goto i1)))) . 1 = NAT proof let R be Ring; ::_thesis: for a being Data-Location of R for i1 being Element of NAT holds (product" (JumpParts (InsCode (a =0_goto i1)))) . 1 = NAT let a be Data-Location of R; ::_thesis: for i1 being Element of NAT holds (product" (JumpParts (InsCode (a =0_goto i1)))) . 1 = NAT let i1 be Element of NAT ; ::_thesis: (product" (JumpParts (InsCode (a =0_goto i1)))) . 1 = NAT dom (product" (JumpParts (InsCode (a =0_goto i1)))) = {1} by Th11, Th27; then A1: 1 in dom (product" (JumpParts (InsCode (a =0_goto i1)))) by TARSKI:def_1; A2: InsCode (a =0_goto i1) = 7 by RECDEF_2:def_1; hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: NAT c= (product" (JumpParts (InsCode (a =0_goto i1)))) . 1 let x be set ; ::_thesis: ( x in (product" (JumpParts (InsCode (a =0_goto i1)))) . 1 implies x in NAT ) assume x in (product" (JumpParts (InsCode (a =0_goto i1)))) . 1 ; ::_thesis: x in NAT then x in pi ((JumpParts (InsCode (a =0_goto i1))),1) by A1, CARD_3:def_12; then consider g being Function such that A3: g in JumpParts (InsCode (a =0_goto i1)) and A4: x = g . 1 by CARD_3:def_6; consider I being Instruction of (SCM R) such that A5: g = JumpPart I and A6: InsCode I = InsCode (a =0_goto i1) by A3; consider b being Data-Location of R, i2 being Element of NAT such that A7: I = b =0_goto i2 by A2, A6, Th19; g = <*i2*> by A5, A7, RECDEF_2:def_2; then x = i2 by A4, FINSEQ_1:40; hence x in NAT ; ::_thesis: verum end; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in NAT or x in (product" (JumpParts (InsCode (a =0_goto i1)))) . 1 ) assume x in NAT ; ::_thesis: x in (product" (JumpParts (InsCode (a =0_goto i1)))) . 1 then reconsider x = x as Element of NAT ; ( JumpPart (a =0_goto x) = <*x*> & InsCode (a =0_goto i1) = InsCode (a =0_goto x) ) by A2, RECDEF_2:def_1, RECDEF_2:def_2; then A8: <*x*> in JumpParts (InsCode (a =0_goto i1)) ; <*x*> . 1 = x by FINSEQ_1:40; then x in pi ((JumpParts (InsCode (a =0_goto i1))),1) by A8, CARD_3:def_6; hence x in (product" (JumpParts (InsCode (a =0_goto i1)))) . 1 by A1, CARD_3:def_12; ::_thesis: verum end; Lm3: for R being Ring for i being Instruction of (SCM R) st ( for l being Element of NAT holds NIC (i,l) = {(succ l)} ) holds JUMP i is empty proof let R be Ring; ::_thesis: for i being Instruction of (SCM R) st ( for l being Element of NAT holds NIC (i,l) = {(succ l)} ) holds JUMP i is empty set p = 1; set q = 2; let i be Instruction of (SCM R); ::_thesis: ( ( for l being Element of NAT holds NIC (i,l) = {(succ l)} ) implies JUMP i is empty ) assume A1: for l being Element of NAT holds NIC (i,l) = {(succ l)} ; ::_thesis: JUMP i is empty set X = { (NIC (i,f)) where f is Element of NAT : verum } ; reconsider p = 1, q = 2 as Element of NAT ; assume not JUMP i is empty ; ::_thesis: contradiction then consider x being set such that A2: x in meet { (NIC (i,f)) where f is Element of NAT : verum } by XBOOLE_0:def_1; NIC (i,p) = {(succ p)} by A1; then {(succ p)} in { (NIC (i,f)) where f is Element of NAT : verum } ; then x in {(succ p)} by A2, SETFAM_1:def_1; then A3: x = succ p by TARSKI:def_1; NIC (i,q) = {(succ q)} by A1; then {(succ q)} in { (NIC (i,f)) where f is Element of NAT : verum } ; then x in {(succ q)} by A2, SETFAM_1:def_1; hence contradiction by A3, TARSKI:def_1; ::_thesis: verum end; registration let R be Ring; cluster JUMP (halt (SCM R)) -> empty ; coherence JUMP (halt (SCM R)) is empty ; end; registration let R be Ring; let a, b be Data-Location of R; clustera := b -> sequential ; coherence a := b is sequential proof let s be State of (SCM R); :: according to AMISTD_1:def_8 ::_thesis: (Exec ((a := b),s)) . (0. (SCM R)) = succ (IC s) thus (Exec ((a := b),s)) . (0. (SCM R)) = succ (IC s) by SCMRING2:11; ::_thesis: verum end; cluster AddTo (a,b) -> sequential ; coherence AddTo (a,b) is sequential proof let s be State of (SCM R); :: according to AMISTD_1:def_8 ::_thesis: (Exec ((AddTo (a,b)),s)) . (0. (SCM R)) = succ (IC s) thus (Exec ((AddTo (a,b)),s)) . (0. (SCM R)) = succ (IC s) by SCMRING2:12; ::_thesis: verum end; cluster SubFrom (a,b) -> sequential ; coherence SubFrom (a,b) is sequential proof let s be State of (SCM R); :: according to AMISTD_1:def_8 ::_thesis: (Exec ((SubFrom (a,b)),s)) . (0. (SCM R)) = succ (IC s) thus (Exec ((SubFrom (a,b)),s)) . (0. (SCM R)) = succ (IC s) by SCMRING2:13; ::_thesis: verum end; cluster MultBy (a,b) -> sequential ; coherence MultBy (a,b) is sequential proof let s be State of (SCM R); :: according to AMISTD_1:def_8 ::_thesis: (Exec ((MultBy (a,b)),s)) . (0. (SCM R)) = succ (IC s) thus (Exec ((MultBy (a,b)),s)) . (0. (SCM R)) = succ (IC s) by SCMRING2:14; ::_thesis: verum end; end; registration let R be Ring; let a be Data-Location of R; let r be Element of R; clustera := r -> sequential ; coherence a := r is sequential proof let s be State of (SCM R); :: according to AMISTD_1:def_8 ::_thesis: (Exec ((a := r),s)) . (0. (SCM R)) = succ (IC s) thus (Exec ((a := r),s)) . (0. (SCM R)) = succ (IC s) by SCMRING2:17; ::_thesis: verum end; end; registration let R be Ring; let a, b be Data-Location of R; cluster JUMP (a := b) -> empty ; coherence JUMP (a := b) is empty proof for l being Element of NAT holds NIC ((a := b),l) = {(succ l)} by AMISTD_1:12; hence JUMP (a := b) is empty by Lm3; ::_thesis: verum end; end; registration let R be Ring; let a, b be Data-Location of R; cluster JUMP (AddTo (a,b)) -> empty ; coherence JUMP (AddTo (a,b)) is empty proof for l being Element of NAT holds NIC ((AddTo (a,b)),l) = {(succ l)} by AMISTD_1:12; hence JUMP (AddTo (a,b)) is empty by Lm3; ::_thesis: verum end; end; registration let R be Ring; let a, b be Data-Location of R; cluster JUMP (SubFrom (a,b)) -> empty ; coherence JUMP (SubFrom (a,b)) is empty proof for l being Element of NAT holds NIC ((SubFrom (a,b)),l) = {(succ l)} by AMISTD_1:12; hence JUMP (SubFrom (a,b)) is empty by Lm3; ::_thesis: verum end; end; registration let R be Ring; let a, b be Data-Location of R; cluster JUMP (MultBy (a,b)) -> empty ; coherence JUMP (MultBy (a,b)) is empty proof for l being Element of NAT holds NIC ((MultBy (a,b)),l) = {(succ l)} by AMISTD_1:12; hence JUMP (MultBy (a,b)) is empty by Lm3; ::_thesis: verum end; end; registration let R be Ring; let a be Data-Location of R; let r be Element of R; cluster JUMP (a := r) -> empty ; coherence JUMP (a := r) is empty proof for l being Element of NAT holds NIC ((a := r),l) = {(succ l)} by AMISTD_1:12; hence JUMP (a := r) is empty by Lm3; ::_thesis: verum end; end; theorem Th30: :: SCMRING3:30 for R being Ring for i1, il being Element of NAT holds NIC ((goto (i1,R)),il) = {i1} proof let R be Ring; ::_thesis: for i1, il being Element of NAT holds NIC ((goto (i1,R)),il) = {i1} let i1, il be Element of NAT ; ::_thesis: NIC ((goto (i1,R)),il) = {i1} now__::_thesis:_for_x_being_set_holds_ (_x_in_{i1}_iff_x_in__{__(IC_(Exec_((goto_(i1,R)),s)))_where_s_is_Element_of_product_(the_Values_of_(SCM_R))_:_IC_s_=_il__}__) let x be set ; ::_thesis: ( x in {i1} iff x in { (IC (Exec ((goto (i1,R)),s))) where s is Element of product (the_Values_of (SCM R)) : IC s = il } ) A1: now__::_thesis:_(_x_=_i1_implies_x_in__{__(IC_(Exec_((goto_(i1,R)),s)))_where_s_is_Element_of_product_(the_Values_of_(SCM_R))_:_IC_s_=_il__}__) reconsider il1 = il as Element of Values (IC ) by MEMSTR_0:def_6; set I = goto (i1,R); set t = the State of (SCM R); set Q = the Instruction-Sequence of (SCM R); assume A2: x = i1 ; ::_thesis: x in { (IC (Exec ((goto (i1,R)),s))) where s is Element of product (the_Values_of (SCM R)) : IC s = il } reconsider u = the State of (SCM R) +* ((IC ),il1) as Element of product (the_Values_of (SCM R)) by CARD_3:107; reconsider P = the Instruction-Sequence of (SCM R) +* (il,(goto (i1,R))) as Instruction-Sequence of (SCM R) ; A3: P /. il = P . il by PBOOLE:143; IC in dom the State of (SCM R) by MEMSTR_0:2; then A4: IC u = il by FUNCT_7:31; il in NAT ; then il in dom the Instruction-Sequence of (SCM R) by PARTFUN1:def_2; then A5: P . il = goto (i1,R) by FUNCT_7:31; then IC (Following (P,u)) = i1 by A4, A3, SCMRING2:15; hence x in { (IC (Exec ((goto (i1,R)),s))) where s is Element of product (the_Values_of (SCM R)) : IC s = il } by A2, A3, A4, A5; ::_thesis: verum end; now__::_thesis:_(_x_in__{__(IC_(Exec_((goto_(i1,R)),s)))_where_s_is_Element_of_product_(the_Values_of_(SCM_R))_:_IC_s_=_il__}__implies_x_=_i1_) assume x in { (IC (Exec ((goto (i1,R)),s))) where s is Element of product (the_Values_of (SCM R)) : IC s = il } ; ::_thesis: x = i1 then ex s being Element of product (the_Values_of (SCM R)) st ( x = IC (Exec ((goto (i1,R)),s)) & IC s = il ) ; hence x = i1 by SCMRING2:15; ::_thesis: verum end; hence ( x in {i1} iff x in { (IC (Exec ((goto (i1,R)),s))) where s is Element of product (the_Values_of (SCM R)) : IC s = il } ) by A1, TARSKI:def_1; ::_thesis: verum end; hence NIC ((goto (i1,R)),il) = {i1} by TARSKI:1; ::_thesis: verum end; theorem Th31: :: SCMRING3:31 for R being Ring for i1 being Element of NAT holds JUMP (goto (i1,R)) = {i1} proof let R be Ring; ::_thesis: for i1 being Element of NAT holds JUMP (goto (i1,R)) = {i1} let i1 be Element of NAT ; ::_thesis: JUMP (goto (i1,R)) = {i1} set X = { (NIC ((goto (i1,R)),il)) where il is Element of NAT : verum } ; now__::_thesis:_for_x_being_set_holds_ (_(_x_in_meet__{__(NIC_((goto_(i1,R)),il))_where_il_is_Element_of_NAT_:_verum__}__implies_x_in_{i1}_)_&_(_x_in_{i1}_implies_x_in_meet__{__(NIC_((goto_(i1,R)),il))_where_il_is_Element_of_NAT_:_verum__}__)_) let x be set ; ::_thesis: ( ( x in meet { (NIC ((goto (i1,R)),il)) where il is Element of NAT : verum } implies x in {i1} ) & ( x in {i1} implies x in meet { (NIC ((goto (i1,R)),il)) where il is Element of NAT : verum } ) ) hereby ::_thesis: ( x in {i1} implies x in meet { (NIC ((goto (i1,R)),il)) where il is Element of NAT : verum } ) reconsider il1 = 1 as Element of NAT ; A1: NIC ((goto (i1,R)),il1) in { (NIC ((goto (i1,R)),il)) where il is Element of NAT : verum } ; assume x in meet { (NIC ((goto (i1,R)),il)) where il is Element of NAT : verum } ; ::_thesis: x in {i1} then x in NIC ((goto (i1,R)),il1) by A1, SETFAM_1:def_1; hence x in {i1} by Th30; ::_thesis: verum end; assume x in {i1} ; ::_thesis: x in meet { (NIC ((goto (i1,R)),il)) where il is Element of NAT : verum } then A2: x = i1 by TARSKI:def_1; A3: now__::_thesis:_for_Y_being_set_st_Y_in__{__(NIC_((goto_(i1,R)),il))_where_il_is_Element_of_NAT_:_verum__}__holds_ i1_in_Y let Y be set ; ::_thesis: ( Y in { (NIC ((goto (i1,R)),il)) where il is Element of NAT : verum } implies i1 in Y ) assume Y in { (NIC ((goto (i1,R)),il)) where il is Element of NAT : verum } ; ::_thesis: i1 in Y then consider il being Element of NAT such that A4: Y = NIC ((goto (i1,R)),il) ; NIC ((goto (i1,R)),il) = {i1} by Th30; hence i1 in Y by A4, TARSKI:def_1; ::_thesis: verum end; NIC ((goto (i1,R)),i1) in { (NIC ((goto (i1,R)),il)) where il is Element of NAT : verum } ; hence x in meet { (NIC ((goto (i1,R)),il)) where il is Element of NAT : verum } by A2, A3, SETFAM_1:def_1; ::_thesis: verum end; hence JUMP (goto (i1,R)) = {i1} by TARSKI:1; ::_thesis: verum end; registration let R be Ring; let i1 be Element of NAT ; cluster JUMP (goto (i1,R)) -> 1 -element ; coherence JUMP (goto (i1,R)) is 1 -element proof JUMP (goto (i1,R)) = {i1} by Th31; hence JUMP (goto (i1,R)) is 1 -element ; ::_thesis: verum end; end; theorem Th32: :: SCMRING3:32 for R being Ring for a being Data-Location of R for i1, il being Element of NAT holds ( i1 in NIC ((a =0_goto i1),il) & NIC ((a =0_goto i1),il) c= {i1,(succ il)} ) proof let R be Ring; ::_thesis: for a being Data-Location of R for i1, il being Element of NAT holds ( i1 in NIC ((a =0_goto i1),il) & NIC ((a =0_goto i1),il) c= {i1,(succ il)} ) let a be Data-Location of R; ::_thesis: for i1, il being Element of NAT holds ( i1 in NIC ((a =0_goto i1),il) & NIC ((a =0_goto i1),il) c= {i1,(succ il)} ) let i1, il be Element of NAT ; ::_thesis: ( i1 in NIC ((a =0_goto i1),il) & NIC ((a =0_goto i1),il) c= {i1,(succ il)} ) set t = the State of (SCM R); set Q = the Instruction-Sequence of (SCM R); set I = a =0_goto i1; reconsider a9 = a as Element of Data-Locations by SCMRING2:1; reconsider il1 = il as Element of Values (IC ) by MEMSTR_0:def_6; Values a = ((SCM-VAL R) * SCM-OK) . a9 by SCMRING2:24 .= the carrier of R by AMI_3:27, SCMRING1:4 ; then reconsider 0R = 0. R as Element of Values a ; reconsider u = the State of (SCM R) +* ((IC ),il1) as Element of product (the_Values_of (SCM R)) by CARD_3:107; reconsider P = the Instruction-Sequence of (SCM R) +* (il,(a =0_goto i1)) as Instruction-Sequence of (SCM R) ; reconsider v = u +* (a .--> 0R) as Element of product (the_Values_of (SCM R)) by CARD_3:107; A1: IC in dom the State of (SCM R) by MEMSTR_0:2; A2: dom (a .--> 0R) = {a} by FUNCOP_1:13; IC <> a by Th2; then not IC in dom (a .--> 0R) by A2, TARSKI:def_1; then A3: IC v = IC u by FUNCT_4:11 .= il by A1, FUNCT_7:31 ; A4: P /. il = P . il by PBOOLE:143; il in NAT ; then il in dom the Instruction-Sequence of (SCM R) by PARTFUN1:def_2; then A5: P . il = a =0_goto i1 by FUNCT_7:31; a in dom (a .--> 0R) by A2, TARSKI:def_1; then v . a = (a .--> 0R) . a by FUNCT_4:13 .= 0. R by FUNCOP_1:72 ; then IC (Following (P,v)) = i1 by A3, A5, A4, SCMRING2:16; hence i1 in NIC ((a =0_goto i1),il) by A3, A5, A4; ::_thesis: NIC ((a =0_goto i1),il) c= {i1,(succ il)} let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in NIC ((a =0_goto i1),il) or x in {i1,(succ il)} ) assume x in NIC ((a =0_goto i1),il) ; ::_thesis: x in {i1,(succ il)} then consider s being Element of product (the_Values_of (SCM R)) such that A6: ( x = IC (Exec ((a =0_goto i1),s)) & IC s = il ) ; percases ( s . a = 0. R or s . a <> 0. R ) ; suppose s . a = 0. R ; ::_thesis: x in {i1,(succ il)} then x = i1 by A6, SCMRING2:16; hence x in {i1,(succ il)} by TARSKI:def_2; ::_thesis: verum end; suppose s . a <> 0. R ; ::_thesis: x in {i1,(succ il)} then x = succ il by A6, SCMRING2:16; hence x in {i1,(succ il)} by TARSKI:def_2; ::_thesis: verum end; end; end; theorem :: SCMRING3:33 for R being non trivial Ring for a being Data-Location of R for il, i1 being Element of NAT holds NIC ((a =0_goto i1),il) = {i1,(succ il)} proof let R be non trivial Ring; ::_thesis: for a being Data-Location of R for il, i1 being Element of NAT holds NIC ((a =0_goto i1),il) = {i1,(succ il)} let a be Data-Location of R; ::_thesis: for il, i1 being Element of NAT holds NIC ((a =0_goto i1),il) = {i1,(succ il)} let il, i1 be Element of NAT ; ::_thesis: NIC ((a =0_goto i1),il) = {i1,(succ il)} set t = the State of (SCM R); set Q = the Instruction-Sequence of (SCM R); set I = a =0_goto i1; reconsider a9 = a as Element of Data-Locations by SCMRING2:1; A1: Values a = ((SCM-VAL R) * SCM-OK) . a9 by SCMRING2:24 .= the carrier of R by AMI_3:27, SCMRING1:4 ; reconsider il1 = il as Element of Values (IC ) by MEMSTR_0:def_6; thus NIC ((a =0_goto i1),il) c= {i1,(succ il)} by Th32; :: according to XBOOLE_0:def_10 ::_thesis: {i1,(succ il)} c= NIC ((a =0_goto i1),il) reconsider u = the State of (SCM R) +* ((IC ),il1) as Element of product (the_Values_of (SCM R)) by CARD_3:107; reconsider P = the Instruction-Sequence of (SCM R) +* (il,(a =0_goto i1)) as Instruction-Sequence of (SCM R) ; let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {i1,(succ il)} or x in NIC ((a =0_goto i1),il) ) A2: IC <> a by Th2; A3: IC in dom the State of (SCM R) by MEMSTR_0:2; assume A4: x in {i1,(succ il)} ; ::_thesis: x in NIC ((a =0_goto i1),il) percases ( x = i1 or x = succ il ) by A4, TARSKI:def_2; supposeA5: x = i1 ; ::_thesis: x in NIC ((a =0_goto i1),il) reconsider 0R = 0. R as Element of Values a by A1; reconsider v = u +* (a .--> 0R) as Element of product (the_Values_of (SCM R)) by CARD_3:107; A6: dom (a .--> 0R) = {a} by FUNCOP_1:13; then not IC in dom (a .--> 0R) by A2, TARSKI:def_1; then A7: IC v = IC u by FUNCT_4:11 .= il by A3, FUNCT_7:31 ; A8: P /. il = P . il by PBOOLE:143; il in NAT ; then il in dom the Instruction-Sequence of (SCM R) by PARTFUN1:def_2; then A9: P . il = a =0_goto i1 by FUNCT_7:31; a in dom (a .--> 0R) by A6, TARSKI:def_1; then v . a = (a .--> 0R) . a by FUNCT_4:13 .= 0. R by FUNCOP_1:72 ; then IC (Following (P,v)) = i1 by A7, A8, A9, SCMRING2:16; hence x in NIC ((a =0_goto i1),il) by A5, A7, A8, A9; ::_thesis: verum end; supposeA10: x = succ il ; ::_thesis: x in NIC ((a =0_goto i1),il) consider e being Element of R such that A11: e <> 0. R by STRUCT_0:def_18; reconsider E = e as Element of Values a by A1; reconsider v = u +* (a .--> E) as Element of product (the_Values_of (SCM R)) by CARD_3:107; A12: dom (a .--> E) = {a} by FUNCOP_1:13; then not IC in dom (a .--> E) by A2, TARSKI:def_1; then A13: IC v = IC u by FUNCT_4:11 .= il by A3, FUNCT_7:31 ; A14: P /. il = P . il by PBOOLE:143; il in NAT ; then il in dom the Instruction-Sequence of (SCM R) by PARTFUN1:def_2; then A15: P . il = a =0_goto i1 by FUNCT_7:31; a in dom (a .--> E) by A12, TARSKI:def_1; then v . a = (a .--> E) . a by FUNCT_4:13 .= E by FUNCOP_1:72 ; then IC (Following (P,v)) = succ il by A11, A13, A14, A15, SCMRING2:16; hence x in NIC ((a =0_goto i1),il) by A10, A13, A14, A15; ::_thesis: verum end; end; end; theorem Th34: :: SCMRING3:34 for R being Ring for a being Data-Location of R for i1 being Element of NAT holds JUMP (a =0_goto i1) = {i1} proof let R be Ring; ::_thesis: for a being Data-Location of R for i1 being Element of NAT holds JUMP (a =0_goto i1) = {i1} let a be Data-Location of R; ::_thesis: for i1 being Element of NAT holds JUMP (a =0_goto i1) = {i1} let i1 be Element of NAT ; ::_thesis: JUMP (a =0_goto i1) = {i1} set X = { (NIC ((a =0_goto i1),il)) where il is Element of NAT : verum } ; now__::_thesis:_for_x_being_set_holds_ (_(_x_in_meet__{__(NIC_((a_=0_goto_i1),il))_where_il_is_Element_of_NAT_:_verum__}__implies_x_in_{i1}_)_&_(_x_in_{i1}_implies_x_in_meet__{__(NIC_((a_=0_goto_i1),il))_where_il_is_Element_of_NAT_:_verum__}__)_) let x be set ; ::_thesis: ( ( x in meet { (NIC ((a =0_goto i1),il)) where il is Element of NAT : verum } implies x in {i1} ) & ( x in {i1} implies x in meet { (NIC ((a =0_goto i1),il)) where il is Element of NAT : verum } ) ) A1: now__::_thesis:_for_Y_being_set_st_Y_in__{__(NIC_((a_=0_goto_i1),il))_where_il_is_Element_of_NAT_:_verum__}__holds_ i1_in_Y let Y be set ; ::_thesis: ( Y in { (NIC ((a =0_goto i1),il)) where il is Element of NAT : verum } implies i1 in Y ) assume Y in { (NIC ((a =0_goto i1),il)) where il is Element of NAT : verum } ; ::_thesis: i1 in Y then ex il being Element of NAT st Y = NIC ((a =0_goto i1),il) ; hence i1 in Y by Th32; ::_thesis: verum end; hereby ::_thesis: ( x in {i1} implies x in meet { (NIC ((a =0_goto i1),il)) where il is Element of NAT : verum } ) reconsider il1 = 1, il2 = 2 as Element of NAT ; assume A2: x in meet { (NIC ((a =0_goto i1),il)) where il is Element of NAT : verum } ; ::_thesis: x in {i1} A3: NIC ((a =0_goto i1),il2) c= {i1,(succ il2)} by Th32; NIC ((a =0_goto i1),il2) in { (NIC ((a =0_goto i1),il)) where il is Element of NAT : verum } ; then x in NIC ((a =0_goto i1),il2) by A2, SETFAM_1:def_1; then A4: ( x = i1 or x = succ il2 ) by A3, TARSKI:def_2; A5: NIC ((a =0_goto i1),il1) c= {i1,(succ il1)} by Th32; NIC ((a =0_goto i1),il1) in { (NIC ((a =0_goto i1),il)) where il is Element of NAT : verum } ; then x in NIC ((a =0_goto i1),il1) by A2, SETFAM_1:def_1; then ( x = i1 or x = succ il1 ) by A5, TARSKI:def_2; hence x in {i1} by A4, TARSKI:def_1; ::_thesis: verum end; assume x in {i1} ; ::_thesis: x in meet { (NIC ((a =0_goto i1),il)) where il is Element of NAT : verum } then A6: x = i1 by TARSKI:def_1; NIC ((a =0_goto i1),i1) in { (NIC ((a =0_goto i1),il)) where il is Element of NAT : verum } ; hence x in meet { (NIC ((a =0_goto i1),il)) where il is Element of NAT : verum } by A6, A1, SETFAM_1:def_1; ::_thesis: verum end; hence JUMP (a =0_goto i1) = {i1} by TARSKI:1; ::_thesis: verum end; registration let R be Ring; let a be Data-Location of R; let i1 be Element of NAT ; cluster JUMP (a =0_goto i1) -> 1 -element ; coherence JUMP (a =0_goto i1) is 1 -element proof JUMP (a =0_goto i1) = {i1} by Th34; hence JUMP (a =0_goto i1) is 1 -element ; ::_thesis: verum end; end; theorem Th35: :: SCMRING3:35 for R being Ring for il being Element of NAT holds SUCC (il,(SCM R)) = {il,(succ il)} proof let R be Ring; ::_thesis: for il being Element of NAT holds SUCC (il,(SCM R)) = {il,(succ il)} let il be Element of NAT ; ::_thesis: SUCC (il,(SCM R)) = {il,(succ il)} set X = { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of (SCM R) : verum } ; set N = {il,(succ il)}; now__::_thesis:_for_x_being_set_holds_ (_(_x_in_union__{__((NIC_(I,il))_\_(JUMP_I))_where_I_is_Element_of_the_InstructionsF_of_(SCM_R)_:_verum__}__implies_x_in_{il,(succ_il)}_)_&_(_x_in_{il,(succ_il)}_implies_x_in_union__{__((NIC_(I,il))_\_(JUMP_I))_where_I_is_Element_of_the_InstructionsF_of_(SCM_R)_:_verum__}__)_) let x be set ; ::_thesis: ( ( x in union { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of (SCM R) : verum } implies x in {il,(succ il)} ) & ( x in {il,(succ il)} implies b1 in union { ((NIC (b2,il)) \ (JUMP b2)) where I is Element of the InstructionsF of (SCM R) : verum } ) ) hereby ::_thesis: ( x in {il,(succ il)} implies b1 in union { ((NIC (b2,il)) \ (JUMP b2)) where I is Element of the InstructionsF of (SCM R) : verum } ) assume x in union { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of (SCM R) : verum } ; ::_thesis: x in {il,(succ il)} then consider Y being set such that A1: x in Y and A2: Y in { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of (SCM R) : verum } by TARSKI:def_4; consider i being Element of the InstructionsF of (SCM R) such that A3: Y = (NIC (i,il)) \ (JUMP i) by A2; percases ( i = [0,{},{}] or ex a, b being Data-Location of R st i = a := b or ex a, b being Data-Location of R st i = AddTo (a,b) or ex a, b being Data-Location of R st i = SubFrom (a,b) or ex a, b being Data-Location of R st i = MultBy (a,b) or ex i1 being Element of NAT st i = goto (i1,R) or ex a being Data-Location of R ex i1 being Element of NAT st i = a =0_goto i1 or ex a being Data-Location of R ex r being Element of R st i = a := r ) by SCMRING2:7; suppose i = [0,{},{}] ; ::_thesis: x in {il,(succ il)} then i = halt (SCM R) ; then x in {il} \ (JUMP (halt (SCM R))) by A1, A3, AMISTD_1:2; then x = il by TARSKI:def_1; hence x in {il,(succ il)} by TARSKI:def_2; ::_thesis: verum end; suppose ex a, b being Data-Location of R st i = a := b ; ::_thesis: x in {il,(succ il)} then consider a, b being Data-Location of R such that A4: i = a := b ; x in {(succ il)} \ (JUMP (a := b)) by A1, A3, A4, AMISTD_1:12; then x = succ il by TARSKI:def_1; hence x in {il,(succ il)} by TARSKI:def_2; ::_thesis: verum end; suppose ex a, b being Data-Location of R st i = AddTo (a,b) ; ::_thesis: x in {il,(succ il)} then consider a, b being Data-Location of R such that A5: i = AddTo (a,b) ; x in {(succ il)} \ (JUMP (AddTo (a,b))) by A1, A3, A5, AMISTD_1:12; then x = succ il by TARSKI:def_1; hence x in {il,(succ il)} by TARSKI:def_2; ::_thesis: verum end; suppose ex a, b being Data-Location of R st i = SubFrom (a,b) ; ::_thesis: x in {il,(succ il)} then consider a, b being Data-Location of R such that A6: i = SubFrom (a,b) ; x in {(succ il)} \ (JUMP (SubFrom (a,b))) by A1, A3, A6, AMISTD_1:12; then x = succ il by TARSKI:def_1; hence x in {il,(succ il)} by TARSKI:def_2; ::_thesis: verum end; suppose ex a, b being Data-Location of R st i = MultBy (a,b) ; ::_thesis: x in {il,(succ il)} then consider a, b being Data-Location of R such that A7: i = MultBy (a,b) ; x in {(succ il)} \ (JUMP (MultBy (a,b))) by A1, A3, A7, AMISTD_1:12; then x = succ il by TARSKI:def_1; hence x in {il,(succ il)} by TARSKI:def_2; ::_thesis: verum end; suppose ex i1 being Element of NAT st i = goto (i1,R) ; ::_thesis: x in {il,(succ il)} then consider i1 being Element of NAT such that A8: i = goto (i1,R) ; x in {i1} \ (JUMP i) by A1, A3, A8, Th30; then x in {i1} \ {i1} by A8, Th31; hence x in {il,(succ il)} by XBOOLE_1:37; ::_thesis: verum end; suppose ex a being Data-Location of R ex i1 being Element of NAT st i = a =0_goto i1 ; ::_thesis: x in {il,(succ il)} then consider a being Data-Location of R, i1 being Element of NAT such that A9: i = a =0_goto i1 ; A10: NIC (i,il) c= {i1,(succ il)} by A9, Th32; x in NIC (i,il) by A1, A3, XBOOLE_0:def_5; then A11: ( x = i1 or x = succ il ) by A10, TARSKI:def_2; x in (NIC (i,il)) \ {i1} by A1, A3, A9, Th34; then not x in {i1} by XBOOLE_0:def_5; hence x in {il,(succ il)} by A11, TARSKI:def_1, TARSKI:def_2; ::_thesis: verum end; suppose ex a being Data-Location of R ex r being Element of R st i = a := r ; ::_thesis: x in {il,(succ il)} then consider a being Data-Location of R, r being Element of R such that A12: i = a := r ; x in {(succ il)} \ (JUMP (a := r)) by A1, A3, A12, AMISTD_1:12; then x = succ il by TARSKI:def_1; hence x in {il,(succ il)} by TARSKI:def_2; ::_thesis: verum end; end; end; assume A13: x in {il,(succ il)} ; ::_thesis: b1 in union { ((NIC (b2,il)) \ (JUMP b2)) where I is Element of the InstructionsF of (SCM R) : verum } percases ( x = il or x = succ il ) by A13, TARSKI:def_2; supposeA14: x = il ; ::_thesis: b1 in union { ((NIC (b2,il)) \ (JUMP b2)) where I is Element of the InstructionsF of (SCM R) : verum } set i = halt (SCM R); (NIC ((halt (SCM R)),il)) \ (JUMP (halt (SCM R))) = {il} by AMISTD_1:2; then A15: {il} in { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of (SCM R) : verum } ; x in {il} by A14, TARSKI:def_1; hence x in union { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of (SCM R) : verum } by A15, TARSKI:def_4; ::_thesis: verum end; supposeA16: x = succ il ; ::_thesis: b1 in union { ((NIC (b2,il)) \ (JUMP b2)) where I is Element of the InstructionsF of (SCM R) : verum } set a = the Data-Location of R; set i = AddTo ( the Data-Location of R, the Data-Location of R); (NIC ((AddTo ( the Data-Location of R, the Data-Location of R)),il)) \ (JUMP (AddTo ( the Data-Location of R, the Data-Location of R))) = {(succ il)} by AMISTD_1:12; then A17: {(succ il)} in { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of (SCM R) : verum } ; x in {(succ il)} by A16, TARSKI:def_1; hence x in union { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of (SCM R) : verum } by A17, TARSKI:def_4; ::_thesis: verum end; end; end; hence SUCC (il,(SCM R)) = {il,(succ il)} by TARSKI:1; ::_thesis: verum end; theorem Th36: :: SCMRING3:36 for R being Ring for k being Element of NAT holds ( k + 1 in SUCC (k,(SCM R)) & ( for j being Element of NAT st j in SUCC (k,(SCM R)) holds k <= j ) ) proof let R be Ring; ::_thesis: for k being Element of NAT holds ( k + 1 in SUCC (k,(SCM R)) & ( for j being Element of NAT st j in SUCC (k,(SCM R)) holds k <= j ) ) let k be Element of NAT ; ::_thesis: ( k + 1 in SUCC (k,(SCM R)) & ( for j being Element of NAT st j in SUCC (k,(SCM R)) holds k <= j ) ) reconsider fk = k as Element of NAT ; A1: SUCC (k,(SCM R)) = {k,(succ fk)} by Th35; hence k + 1 in SUCC (k,(SCM R)) by TARSKI:def_2; ::_thesis: for j being Element of NAT st j in SUCC (k,(SCM R)) holds k <= j let j be Element of NAT ; ::_thesis: ( j in SUCC (k,(SCM R)) implies k <= j ) assume A2: j in SUCC (k,(SCM R)) ; ::_thesis: k <= j reconsider fk = k as Element of NAT ; percases ( j = k or j = succ fk ) by A1, A2, TARSKI:def_2; suppose j = k ; ::_thesis: k <= j hence k <= j ; ::_thesis: verum end; suppose j = succ fk ; ::_thesis: k <= j hence k <= j by NAT_1:11; ::_thesis: verum end; end; end; registration let R be Ring; cluster SCM R -> standard ; coherence SCM R is standard proof deffunc H1( Element of NAT ) -> Element of NAT = R; for k being Element of NAT holds ( k + 1 in SUCC (k,(SCM R)) & ( for j being Element of NAT st j in SUCC (k,(SCM R)) holds k <= j ) ) by Th36; hence SCM R is standard by AMISTD_1:3; ::_thesis: verum end; end; definition let R be Ring; let k be Element of NAT ; func dl. (R,k) -> Data-Location of R equals :: SCMRING3:def 1 dl. k; coherence dl. k is Data-Location of R proof dl. k in Data-Locations by AMI_2:def_16, AMI_3:27; hence dl. k is Data-Location of R by SCMRING2:1; ::_thesis: verum end; end; :: deftheorem defines dl. SCMRING3:def_1_:_ for R being Ring for k being Element of NAT holds dl. (R,k) = dl. k; registration let R be Ring; cluster InsCode (halt (SCM R)) -> jump-only for InsType of the InstructionsF of (SCM R); coherence for b1 being InsType of the InstructionsF of (SCM R) st b1 = InsCode (halt (SCM R)) holds b1 is jump-only proof now__::_thesis:_for_s_being_State_of_(SCM_R) for_o_being_Object_of_(SCM_R) for_I_being_Instruction_of_(SCM_R)_st_InsCode_I_=_InsCode_(halt_(SCM_R))_&_o_in_Data-Locations_holds_ (Exec_(I,s))_._o_=_s_._o let s be State of (SCM R); ::_thesis: for o being Object of (SCM R) for I being Instruction of (SCM R) st InsCode I = InsCode (halt (SCM R)) & o in Data-Locations holds (Exec (I,s)) . o = s . o let o be Object of (SCM R); ::_thesis: for I being Instruction of (SCM R) st InsCode I = InsCode (halt (SCM R)) & o in Data-Locations holds (Exec (I,s)) . o = s . o let I be Instruction of (SCM R); ::_thesis: ( InsCode I = InsCode (halt (SCM R)) & o in Data-Locations implies (Exec (I,s)) . o = s . o ) assume that A1: InsCode I = InsCode (halt (SCM R)) and o in Data-Locations ; ::_thesis: (Exec (I,s)) . o = s . o I = halt (SCM R) by A1, Th12, COMPOS_1:70; hence (Exec (I,s)) . o = s . o by EXTPRO_1:def_3; ::_thesis: verum end; hence for b1 being InsType of the InstructionsF of (SCM R) st b1 = InsCode (halt (SCM R)) holds b1 is jump-only by AMISTD_1:def_1; ::_thesis: verum end; end; registration let R be Ring; cluster halt (SCM R) -> jump-only ; coherence halt (SCM R) is jump-only proof thus InsCode (halt (SCM R)) is jump-only ; :: according to AMISTD_1:def_2 ::_thesis: verum end; end; registration let R be Ring; let i1 be Element of NAT ; cluster InsCode (goto (i1,R)) -> jump-only for InsType of the InstructionsF of (SCM R); coherence for b1 being InsType of the InstructionsF of (SCM R) st b1 = InsCode (goto (i1,R)) holds b1 is jump-only proof set S = SCM R; now__::_thesis:_for_s_being_State_of_(SCM_R) for_o_being_Object_of_(SCM_R) for_I_being_Instruction_of_(SCM_R)_st_InsCode_I_=_InsCode_(goto_(i1,R))_&_o_in_Data-Locations_holds_ (Exec_(I,s))_._o_=_s_._o let s be State of (SCM R); ::_thesis: for o being Object of (SCM R) for I being Instruction of (SCM R) st InsCode I = InsCode (goto (i1,R)) & o in Data-Locations holds (Exec (I,s)) . o = s . o let o be Object of (SCM R); ::_thesis: for I being Instruction of (SCM R) st InsCode I = InsCode (goto (i1,R)) & o in Data-Locations holds (Exec (I,s)) . o = s . o let I be Instruction of (SCM R); ::_thesis: ( InsCode I = InsCode (goto (i1,R)) & o in Data-Locations implies (Exec (I,s)) . o = s . o ) assume that A1: InsCode I = InsCode (goto (i1,R)) and A2: o in Data-Locations ; ::_thesis: (Exec (I,s)) . o = s . o InsCode (goto (i1,R)) = 6 by RECDEF_2:def_1; then A3: ex i2 being Element of NAT st I = goto (i2,R) by A1, Th18; o in Data-Locations by A2, SCMRING2:22; then o is Data-Location of R by SCMRING2:1; hence (Exec (I,s)) . o = s . o by A3, SCMRING2:15; ::_thesis: verum end; hence for b1 being InsType of the InstructionsF of (SCM R) st b1 = InsCode (goto (i1,R)) holds b1 is jump-only by AMISTD_1:def_1; ::_thesis: verum end; end; registration let R be Ring; let i1 be Element of NAT ; cluster goto (i1,R) -> jump-only ; coherence goto (i1,R) is jump-only proof thus InsCode (goto (i1,R)) is jump-only ; :: according to AMISTD_1:def_2 ::_thesis: verum end; end; registration let R be Ring; let a be Data-Location of R; let i1 be Element of NAT ; cluster InsCode (a =0_goto i1) -> jump-only for InsType of the InstructionsF of (SCM R); coherence for b1 being InsType of the InstructionsF of (SCM R) st b1 = InsCode (a =0_goto i1) holds b1 is jump-only proof set S = SCM R; now__::_thesis:_for_s_being_State_of_(SCM_R) for_o_being_Object_of_(SCM_R) for_I_being_Instruction_of_(SCM_R)_st_InsCode_I_=_InsCode_(a_=0_goto_i1)_&_o_in_Data-Locations_holds_ (Exec_(I,s))_._o_=_s_._o let s be State of (SCM R); ::_thesis: for o being Object of (SCM R) for I being Instruction of (SCM R) st InsCode I = InsCode (a =0_goto i1) & o in Data-Locations holds (Exec (I,s)) . o = s . o let o be Object of (SCM R); ::_thesis: for I being Instruction of (SCM R) st InsCode I = InsCode (a =0_goto i1) & o in Data-Locations holds (Exec (I,s)) . o = s . o let I be Instruction of (SCM R); ::_thesis: ( InsCode I = InsCode (a =0_goto i1) & o in Data-Locations implies (Exec (I,s)) . o = s . o ) assume that A1: InsCode I = InsCode (a =0_goto i1) and A2: o in Data-Locations ; ::_thesis: (Exec (I,s)) . o = s . o InsCode (a =0_goto i1) = 7 by RECDEF_2:def_1; then A3: ex b being Data-Location of R ex i2 being Element of NAT st I = b =0_goto i2 by A1, Th19; o in Data-Locations by A2, SCMRING2:22; then o is Data-Location of R by SCMRING2:1; hence (Exec (I,s)) . o = s . o by A3, SCMRING2:16; ::_thesis: verum end; hence for b1 being InsType of the InstructionsF of (SCM R) st b1 = InsCode (a =0_goto i1) holds b1 is jump-only by AMISTD_1:def_1; ::_thesis: verum end; end; registration let R be Ring; let a be Data-Location of R; let i1 be Element of NAT ; clustera =0_goto i1 -> jump-only ; coherence a =0_goto i1 is jump-only proof thus InsCode (a =0_goto i1) is jump-only ; :: according to AMISTD_1:def_2 ::_thesis: verum end; end; registration let S be non trivial Ring; let p, q be Data-Location of S; cluster InsCode (p := q) -> non jump-only for InsType of the InstructionsF of (SCM S); coherence for b1 being InsType of the InstructionsF of (SCM S) st b1 = InsCode (p := q) holds not b1 is jump-only proof set w = the State of (SCM S); consider e being Element of S such that A1: e <> 0. S by STRUCT_0:def_18; reconsider e = e as Element of S ; set t = the State of (SCM S) +* (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e)); A2: dom (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e)) = {(dl. (S,0)),(dl. (S,1))} by FUNCT_4:62; then A3: dl. (S,1) in dom (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e)) by TARSKI:def_2; A4: InsCode (p := q) = 1 by RECDEF_2:def_1 .= InsCode ((dl. (S,0)) := (dl. (S,1))) by RECDEF_2:def_1 ; dl. (S,0) in Data-Locations by SCMRING2:1; then A5: dl. (S,0) in Data-Locations by SCMRING2:22; dl. (S,0) in dom (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e)) by A2, TARSKI:def_2; then A6: ( the State of (SCM S) +* (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e))) . (dl. (S,0)) = (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e)) . (dl. (S,0)) by FUNCT_4:13 .= 0. S by AMI_3:10, FUNCT_4:63 ; (Exec (((dl. (S,0)) := (dl. (S,1))),( the State of (SCM S) +* (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e))))) . (dl. (S,0)) = ( the State of (SCM S) +* (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e))) . (dl. (S,1)) by SCMRING2:11 .= (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e)) . (dl. (S,1)) by A3, FUNCT_4:13 .= e by FUNCT_4:63 ; hence for b1 being InsType of the InstructionsF of (SCM S) st b1 = InsCode (p := q) holds not b1 is jump-only by A1, A4, A6, A5, AMISTD_1:def_1; ::_thesis: verum end; end; registration let S be non trivial Ring; let p, q be Data-Location of S; clusterp := q -> non jump-only ; coherence not p := q is jump-only proof thus not InsCode (p := q) is jump-only ; :: according to AMISTD_1:def_2 ::_thesis: verum end; end; registration let S be non trivial Ring; let p, q be Data-Location of S; cluster InsCode (AddTo (p,q)) -> non jump-only for InsType of the InstructionsF of (SCM S); coherence for b1 being InsType of the InstructionsF of (SCM S) st b1 = InsCode (AddTo (p,q)) holds not b1 is jump-only proof set w = the State of (SCM S); consider e being Element of S such that A1: e <> 0. S by STRUCT_0:def_18; reconsider e = e as Element of S ; set t = the State of (SCM S) +* (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e)); A2: dom (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e)) = {(dl. (S,0)),(dl. (S,1))} by FUNCT_4:62; then dl. (S,0) in dom (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e)) by TARSKI:def_2; then A3: ( the State of (SCM S) +* (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e))) . (dl. (S,0)) = (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e)) . (dl. (S,0)) by FUNCT_4:13 .= 0. S by AMI_3:10, FUNCT_4:63 ; A4: InsCode (AddTo (p,q)) = 2 by RECDEF_2:def_1 .= InsCode (AddTo ((dl. (S,0)),(dl. (S,1)))) by RECDEF_2:def_1 ; dl. (S,1) in dom (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e)) by A2, TARSKI:def_2; then A5: ( the State of (SCM S) +* (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e))) . (dl. (S,1)) = (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e)) . (dl. (S,1)) by FUNCT_4:13 .= e by FUNCT_4:63 ; dl. (S,0) in Data-Locations by SCMRING2:1; then A6: dl. (S,0) in Data-Locations by SCMRING2:22; (Exec ((AddTo ((dl. (S,0)),(dl. (S,1)))),( the State of (SCM S) +* (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e))))) . (dl. (S,0)) = (( the State of (SCM S) +* (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e))) . (dl. (S,0))) + (( the State of (SCM S) +* (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e))) . (dl. (S,1))) by SCMRING2:12 .= e by A3, A5, RLVECT_1:4 ; hence for b1 being InsType of the InstructionsF of (SCM S) st b1 = InsCode (AddTo (p,q)) holds not b1 is jump-only by A1, A4, A3, A6, AMISTD_1:def_1; ::_thesis: verum end; end; registration let S be non trivial Ring; let p, q be Data-Location of S; cluster AddTo (p,q) -> non jump-only ; coherence not AddTo (p,q) is jump-only proof thus not InsCode (AddTo (p,q)) is jump-only ; :: according to AMISTD_1:def_2 ::_thesis: verum end; end; registration let S be non trivial Ring; let p, q be Data-Location of S; cluster InsCode (SubFrom (p,q)) -> non jump-only for InsType of the InstructionsF of (SCM S); coherence for b1 being InsType of the InstructionsF of (SCM S) st b1 = InsCode (SubFrom (p,q)) holds not b1 is jump-only proof set w = the State of (SCM S); consider e being Element of S such that A1: e <> 0. S by STRUCT_0:def_18; reconsider e = e as Element of S ; A2: now__::_thesis:_not_-_e_=_0._S assume - e = 0. S ; ::_thesis: contradiction then e = - (0. S) by RLVECT_1:17; hence contradiction by A1, RLVECT_1:12; ::_thesis: verum end; set t = the State of (SCM S) +* (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e)); A3: InsCode (SubFrom (p,q)) = 3 by RECDEF_2:def_1 .= InsCode (SubFrom ((dl. (S,0)),(dl. (S,1)))) by RECDEF_2:def_1 ; A4: dom (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e)) = {(dl. (S,0)),(dl. (S,1))} by FUNCT_4:62; then dl. (S,0) in dom (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e)) by TARSKI:def_2; then A5: ( the State of (SCM S) +* (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e))) . (dl. (S,0)) = (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e)) . (dl. (S,0)) by FUNCT_4:13 .= 0. S by AMI_3:10, FUNCT_4:63 ; dl. (S,1) in dom (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e)) by A4, TARSKI:def_2; then A6: ( the State of (SCM S) +* (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e))) . (dl. (S,1)) = (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e)) . (dl. (S,1)) by FUNCT_4:13 .= e by FUNCT_4:63 ; dl. (S,0) in Data-Locations by SCMRING2:1; then A7: dl. (S,0) in Data-Locations by SCMRING2:22; (Exec ((SubFrom ((dl. (S,0)),(dl. (S,1)))),( the State of (SCM S) +* (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e))))) . (dl. (S,0)) = (( the State of (SCM S) +* (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e))) . (dl. (S,0))) - (( the State of (SCM S) +* (((dl. (S,0)),(dl. (S,1))) --> ((0. S),e))) . (dl. (S,1))) by SCMRING2:13 .= - e by A5, A6, RLVECT_1:14 ; hence for b1 being InsType of the InstructionsF of (SCM S) st b1 = InsCode (SubFrom (p,q)) holds not b1 is jump-only by A3, A5, A2, A7, AMISTD_1:def_1; ::_thesis: verum end; end; registration let S be non trivial Ring; let p, q be Data-Location of S; cluster SubFrom (p,q) -> non jump-only ; coherence not SubFrom (p,q) is jump-only proof thus not InsCode (SubFrom (p,q)) is jump-only ; :: according to AMISTD_1:def_2 ::_thesis: verum end; end; registration let S be non trivial Ring; let p, q be Data-Location of S; cluster InsCode (MultBy (p,q)) -> non jump-only for InsType of the InstructionsF of (SCM S); coherence for b1 being InsType of the InstructionsF of (SCM S) st b1 = InsCode (MultBy (p,q)) holds not b1 is jump-only proof IC = IC by AMI_3:1, SCMRING2:8; then A1: ( 0. S <> 1_ S & dl. (S,0) <> IC ) by AMI_3:13, LMOD_6:def_1; set w = the State of (SCM S); set t = the State of (SCM S) +* (((dl. (S,0)),(dl. (S,1))) --> ((1_ S),(0. S))); A2: InsCode (MultBy (p,q)) = 4 by RECDEF_2:def_1 .= InsCode (MultBy ((dl. (S,0)),(dl. (S,1)))) by RECDEF_2:def_1 ; A3: dom (((dl. (S,0)),(dl. (S,1))) --> ((1_ S),(0. S))) = {(dl. (S,0)),(dl. (S,1))} by FUNCT_4:62; then dl. (S,0) in dom (((dl. (S,0)),(dl. (S,1))) --> ((1_ S),(0. S))) by TARSKI:def_2; then A4: ( the State of (SCM S) +* (((dl. (S,0)),(dl. (S,1))) --> ((1_ S),(0. S)))) . (dl. (S,0)) = (((dl. (S,0)),(dl. (S,1))) --> ((1_ S),(0. S))) . (dl. (S,0)) by FUNCT_4:13 .= 1_ S by AMI_3:10, FUNCT_4:63 ; dl. (S,1) in dom (((dl. (S,0)),(dl. (S,1))) --> ((1_ S),(0. S))) by A3, TARSKI:def_2; then A5: ( the State of (SCM S) +* (((dl. (S,0)),(dl. (S,1))) --> ((1_ S),(0. S)))) . (dl. (S,1)) = (((dl. (S,0)),(dl. (S,1))) --> ((1_ S),(0. S))) . (dl. (S,1)) by FUNCT_4:13 .= 0. S by FUNCT_4:63 ; dl. (S,0) in Data-Locations by SCMRING2:1; then A6: dl. (S,0) in Data-Locations by SCMRING2:22; (Exec ((MultBy ((dl. (S,0)),(dl. (S,1)))),( the State of (SCM S) +* (((dl. (S,0)),(dl. (S,1))) --> ((1_ S),(0. S)))))) . (dl. (S,0)) = (( the State of (SCM S) +* (((dl. (S,0)),(dl. (S,1))) --> ((1_ S),(0. S)))) . (dl. (S,0))) * (( the State of (SCM S) +* (((dl. (S,0)),(dl. (S,1))) --> ((1_ S),(0. S)))) . (dl. (S,1))) by SCMRING2:14 .= 0. S by A5, VECTSP_1:6 ; hence for b1 being InsType of the InstructionsF of (SCM S) st b1 = InsCode (MultBy (p,q)) holds not b1 is jump-only by A2, A1, A4, A6, AMISTD_1:def_1; ::_thesis: verum end; end; registration let S be non trivial Ring; let p, q be Data-Location of S; cluster MultBy (p,q) -> non jump-only ; coherence not MultBy (p,q) is jump-only proof thus not InsCode (MultBy (p,q)) is jump-only ; :: according to AMISTD_1:def_2 ::_thesis: verum end; end; registration let S be non trivial Ring; let p be Data-Location of S; let w be Element of S; cluster InsCode (p := w) -> non jump-only for InsType of the InstructionsF of (SCM S); coherence for b1 being InsType of the InstructionsF of (SCM S) st b1 = InsCode (p := w) holds not b1 is jump-only proof set j = the State of (SCM S); A1: InsCode (p := w) = 5 by RECDEF_2:def_1 .= InsCode ((dl. (S,0)) := w) by RECDEF_2:def_1 ; the carrier of S <> {w} ; then consider e being set such that A2: e in the carrier of S and A3: e <> w by ZFMISC_1:35; Values (dl. (S,0)) = the carrier of S by Th1; then reconsider e = e as Element of Values (dl. (S,0)) by A2; reconsider v = (dl. (S,0)) .--> e as PartState of (SCM S) ; set t = the State of (SCM S) +* v; dom ((dl. (S,0)) .--> e) = {(dl. (S,0))} by FUNCOP_1:13; then dl. (S,0) in dom ((dl. (S,0)) .--> e) by TARSKI:def_1; then A4: ( the State of (SCM S) +* v) . (dl. (S,0)) = ((dl. (S,0)) .--> e) . (dl. (S,0)) by FUNCT_4:13 .= e by FUNCOP_1:72 ; dl. (S,0) in Data-Locations by SCMRING2:1; then A5: dl. (S,0) in Data-Locations by SCMRING2:22; (Exec (((dl. (S,0)) := w),( the State of (SCM S) +* v))) . (dl. (S,0)) = w by SCMRING2:17; hence for b1 being InsType of the InstructionsF of (SCM S) st b1 = InsCode (p := w) holds not b1 is jump-only by A3, A1, A4, A5, AMISTD_1:def_1; ::_thesis: verum end; end; registration let S be non trivial Ring; let p be Data-Location of S; let w be Element of S; clusterp := w -> non jump-only ; coherence not p := w is jump-only proof thus not InsCode (p := w) is jump-only ; :: according to AMISTD_1:def_2 ::_thesis: verum end; end; registration let R be Ring; let i1 be Element of NAT ; cluster goto (i1,R) -> non sequential ; coherence not goto (i1,R) is sequential proof JUMP (goto (i1,R)) <> {} ; hence not goto (i1,R) is sequential by AMISTD_1:13; ::_thesis: verum end; end; registration let R be Ring; let a be Data-Location of R; let i1 be Element of NAT ; clustera =0_goto i1 -> non sequential ; coherence not a =0_goto i1 is sequential proof JUMP (a =0_goto i1) <> {} ; hence not a =0_goto i1 is sequential by AMISTD_1:13; ::_thesis: verum end; end; registration let R be Ring; let i1 be Element of NAT ; cluster goto (i1,R) -> non ins-loc-free ; coherence not goto (i1,R) is ins-loc-free proof dom (JumpPart (goto (i1,R))) = dom <*i1*> by RECDEF_2:def_2 .= {1} by FINSEQ_1:2, FINSEQ_1:def_8 ; hence not JumpPart (goto (i1,R)) is empty ; :: according to COMPOS_0:def_8 ::_thesis: verum end; end; registration let R be Ring; let a be Data-Location of R; let i1 be Element of NAT ; clustera =0_goto i1 -> non ins-loc-free ; coherence not a =0_goto i1 is ins-loc-free proof dom (JumpPart (a =0_goto i1)) = dom <*i1*> by RECDEF_2:def_2 .= {1} by FINSEQ_1:2, FINSEQ_1:38 ; hence not JumpPart (a =0_goto i1) is empty ; :: according to COMPOS_0:def_8 ::_thesis: verum end; end; registration let R be Ring; cluster SCM R -> with_explicit_jumps ; coherence SCM R is with_explicit_jumps proof let I be Instruction of (SCM R); :: according to AMISTD_2:def_2 ::_thesis: I is with_explicit_jumps thus JUMP I c= rng (JumpPart I) :: according to AMISTD_2:def_1,XBOOLE_0:def_10 ::_thesis: proj2 (JumpPart I) c= JUMP I proof let f be set ; :: according to TARSKI:def_3 ::_thesis: ( not f in JUMP I or f in rng (JumpPart I) ) assume A1: f in JUMP I ; ::_thesis: f in rng (JumpPart I) percases ( I = [0,{},{}] or ex a, b being Data-Location of R st I = a := b or ex a, b being Data-Location of R st I = AddTo (a,b) or ex a, b being Data-Location of R st I = SubFrom (a,b) or ex a, b being Data-Location of R st I = MultBy (a,b) or ex i1 being Element of NAT st I = goto (i1,R) or ex a being Data-Location of R ex i1 being Element of NAT st I = a =0_goto i1 or ex a being Data-Location of R ex r being Element of R st I = a := r ) by SCMRING2:7; supposeA2: I = [0,{},{}] ; ::_thesis: f in rng (JumpPart I) JUMP (halt (SCM R)) is empty ; hence f in rng (JumpPart I) by A1, A2; ::_thesis: verum end; suppose ex a, b being Data-Location of R st I = a := b ; ::_thesis: f in rng (JumpPart I) hence f in rng (JumpPart I) by A1; ::_thesis: verum end; suppose ex a, b being Data-Location of R st I = AddTo (a,b) ; ::_thesis: f in rng (JumpPart I) hence f in rng (JumpPart I) by A1; ::_thesis: verum end; suppose ex a, b being Data-Location of R st I = SubFrom (a,b) ; ::_thesis: f in rng (JumpPart I) hence f in rng (JumpPart I) by A1; ::_thesis: verum end; suppose ex a, b being Data-Location of R st I = MultBy (a,b) ; ::_thesis: f in rng (JumpPart I) hence f in rng (JumpPart I) by A1; ::_thesis: verum end; supposeA3: ex i1 being Element of NAT st I = goto (i1,R) ; ::_thesis: f in rng (JumpPart I) consider i1 being Element of NAT such that A4: I = goto (i1,R) by A3; A5: JumpPart (goto (i1,R)) = <*i1*> by RECDEF_2:def_2; rng <*i1*> = {i1} by FINSEQ_1:39; hence f in rng (JumpPart I) by A1, A4, A5, Th31; ::_thesis: verum end; supposeA6: ex a being Data-Location of R ex i1 being Element of NAT st I = a =0_goto i1 ; ::_thesis: f in rng (JumpPart I) consider a being Data-Location of R, i1 being Element of NAT such that A7: I = a =0_goto i1 by A6; A8: JumpPart (a =0_goto i1) = <*i1*> by RECDEF_2:def_2; rng <*i1*> = {i1} by FINSEQ_1:39; hence f in rng (JumpPart I) by A1, A7, A8, Th34; ::_thesis: verum end; suppose ex a being Data-Location of R ex r being Element of R st I = a := r ; ::_thesis: f in rng (JumpPart I) hence f in rng (JumpPart I) by A1; ::_thesis: verum end; end; end; let f be set ; :: according to TARSKI:def_3 ::_thesis: ( not f in proj2 (JumpPart I) or f in JUMP I ) assume f in rng (JumpPart I) ; ::_thesis: f in JUMP I then consider k being set such that A9: k in dom (JumpPart I) and A10: f = (JumpPart I) . k by FUNCT_1:def_3; percases ( I = [0,{},{}] or ex a, b being Data-Location of R st I = a := b or ex a, b being Data-Location of R st I = AddTo (a,b) or ex a, b being Data-Location of R st I = SubFrom (a,b) or ex a, b being Data-Location of R st I = MultBy (a,b) or ex i1 being Element of NAT st I = goto (i1,R) or ex a being Data-Location of R ex i1 being Element of NAT st I = a =0_goto i1 or ex a being Data-Location of R ex r being Element of R st I = a := r ) by SCMRING2:7; suppose I = [0,{},{}] ; ::_thesis: f in JUMP I then I = halt (SCM R) ; hence f in JUMP I by A9; ::_thesis: verum end; suppose ex a, b being Data-Location of R st I = a := b ; ::_thesis: f in JUMP I then consider a, b being Data-Location of R such that A11: I = a := b ; k in dom {} by A9, A11, RECDEF_2:def_2; hence f in JUMP I ; ::_thesis: verum end; suppose ex a, b being Data-Location of R st I = AddTo (a,b) ; ::_thesis: f in JUMP I then consider a, b being Data-Location of R such that A12: I = AddTo (a,b) ; k in dom {} by A9, A12, RECDEF_2:def_2; hence f in JUMP I ; ::_thesis: verum end; suppose ex a, b being Data-Location of R st I = SubFrom (a,b) ; ::_thesis: f in JUMP I then consider a, b being Data-Location of R such that A13: I = SubFrom (a,b) ; k in dom {} by A9, A13, RECDEF_2:def_2; hence f in JUMP I ; ::_thesis: verum end; suppose ex a, b being Data-Location of R st I = MultBy (a,b) ; ::_thesis: f in JUMP I then consider a, b being Data-Location of R such that A14: I = MultBy (a,b) ; k in dom {} by A9, A14, RECDEF_2:def_2; hence f in JUMP I ; ::_thesis: verum end; suppose ex i1 being Element of NAT st I = goto (i1,R) ; ::_thesis: f in JUMP I then consider i1 being Element of NAT such that A15: I = goto (i1,R) ; A16: JumpPart I = <*i1*> by A15, RECDEF_2:def_2; then k = 1 by A9, Lm1; then A17: f = i1 by A16, A10, FINSEQ_1:def_8; JUMP I = {i1} by A15, Th31; hence f in JUMP I by A17, TARSKI:def_1; ::_thesis: verum end; suppose ex a being Data-Location of R ex i1 being Element of NAT st I = a =0_goto i1 ; ::_thesis: f in JUMP I then consider a being Data-Location of R, i1 being Element of NAT such that A18: I = a =0_goto i1 ; A19: JumpPart I = <*i1*> by A18, RECDEF_2:def_2; then k = 1 by A9, Lm1; then A20: f = i1 by A19, A10, FINSEQ_1:40; JUMP I = {i1} by A18, Th34; hence f in JUMP I by A20, TARSKI:def_1; ::_thesis: verum end; suppose ex a being Data-Location of R ex r being Element of R st I = a := r ; ::_thesis: f in JUMP I then consider a being Data-Location of R, r being Element of R such that A21: I = a := r ; k in dom {} by A9, A21, RECDEF_2:def_2; hence f in JUMP I ; ::_thesis: verum end; end; end; end; theorem Th37: :: SCMRING3:37 for R being Ring for i1 being Element of NAT for k being Nat holds IncAddr ((goto (i1,R)),k) = goto ((i1 + k),R) proof let R be Ring; ::_thesis: for i1 being Element of NAT for k being Nat holds IncAddr ((goto (i1,R)),k) = goto ((i1 + k),R) let i1 be Element of NAT ; ::_thesis: for k being Nat holds IncAddr ((goto (i1,R)),k) = goto ((i1 + k),R) let k be Nat; ::_thesis: IncAddr ((goto (i1,R)),k) = goto ((i1 + k),R) A1: JumpPart (IncAddr ((goto (i1,R)),k)) = k + (JumpPart (goto (i1,R))) by COMPOS_0:def_9; then A2: dom (JumpPart (IncAddr ((goto (i1,R)),k))) = dom (JumpPart (goto (i1,R))) by VALUED_1:def_2; A3: dom (JumpPart (goto ((i1 + k),R))) = dom <*(i1 + k)*> by RECDEF_2:def_2 .= Seg 1 by FINSEQ_1:def_8 .= dom <*i1*> by FINSEQ_1:def_8 .= dom (JumpPart (goto (i1,R))) by RECDEF_2:def_2 ; A4: for x being set st x in dom (JumpPart (goto (i1,R))) holds (JumpPart (IncAddr ((goto (i1,R)),k))) . x = (JumpPart (goto ((i1 + k),R))) . x proof let x be set ; ::_thesis: ( x in dom (JumpPart (goto (i1,R))) implies (JumpPart (IncAddr ((goto (i1,R)),k))) . x = (JumpPart (goto ((i1 + k),R))) . x ) assume A5: x in dom (JumpPart (goto (i1,R))) ; ::_thesis: (JumpPart (IncAddr ((goto (i1,R)),k))) . x = (JumpPart (goto ((i1 + k),R))) . x then x in dom <*i1*> by RECDEF_2:def_2; then A6: x = 1 by Lm1; reconsider f = (JumpPart (goto (i1,R))) . x as Element of NAT by ORDINAL1:def_12; A7: (JumpPart (IncAddr ((goto (i1,R)),k))) . x = k + f by A5, A1, A2, VALUED_1:def_2; f = <*i1*> . x by RECDEF_2:def_2 .= i1 by A6, FINSEQ_1:def_8 ; hence (JumpPart (IncAddr ((goto (i1,R)),k))) . x = <*(i1 + k)*> . x by A6, A7, FINSEQ_1:def_8 .= (JumpPart (goto ((i1 + k),R))) . x by RECDEF_2:def_2 ; ::_thesis: verum end; A8: InsCode (IncAddr ((goto (i1,R)),k)) = InsCode (goto (i1,R)) by COMPOS_0:def_9 .= 6 by RECDEF_2:def_1 .= InsCode (goto ((i1 + k),R)) by RECDEF_2:def_1 ; A9: AddressPart (IncAddr ((goto (i1,R)),k)) = AddressPart (goto (i1,R)) by COMPOS_0:def_9 .= {} by RECDEF_2:def_3 .= AddressPart (goto ((i1 + k),R)) by RECDEF_2:def_3 ; JumpPart (IncAddr ((goto (i1,R)),k)) = JumpPart (goto ((i1 + k),R)) by A2, A3, A4, FUNCT_1:2; hence IncAddr ((goto (i1,R)),k) = goto ((i1 + k),R) by A8, A9, COMPOS_0:1; ::_thesis: verum end; theorem Th38: :: SCMRING3:38 for R being Ring for a being Data-Location of R for i1 being Element of NAT for k being Nat holds IncAddr ((a =0_goto i1),k) = a =0_goto (i1 + k) proof let R be Ring; ::_thesis: for a being Data-Location of R for i1 being Element of NAT for k being Nat holds IncAddr ((a =0_goto i1),k) = a =0_goto (i1 + k) let a be Data-Location of R; ::_thesis: for i1 being Element of NAT for k being Nat holds IncAddr ((a =0_goto i1),k) = a =0_goto (i1 + k) let i1 be Element of NAT ; ::_thesis: for k being Nat holds IncAddr ((a =0_goto i1),k) = a =0_goto (i1 + k) let k be Nat; ::_thesis: IncAddr ((a =0_goto i1),k) = a =0_goto (i1 + k) A1: JumpPart (IncAddr ((a =0_goto i1),k)) = k + (JumpPart (a =0_goto i1)) by COMPOS_0:def_9; then A2: dom (JumpPart (IncAddr ((a =0_goto i1),k))) = dom (JumpPart (a =0_goto i1)) by VALUED_1:def_2; A3: dom (JumpPart (a =0_goto (i1 + k))) = dom <*(i1 + k)*> by RECDEF_2:def_2 .= Seg 1 by FINSEQ_1:38 .= dom <*i1*> by FINSEQ_1:38 .= dom (JumpPart (a =0_goto i1)) by RECDEF_2:def_2 ; A4: for x being set st x in dom (JumpPart (a =0_goto i1)) holds (JumpPart (IncAddr ((a =0_goto i1),k))) . x = (JumpPart (a =0_goto (i1 + k))) . x proof let x be set ; ::_thesis: ( x in dom (JumpPart (a =0_goto i1)) implies (JumpPart (IncAddr ((a =0_goto i1),k))) . x = (JumpPart (a =0_goto (i1 + k))) . x ) assume A5: x in dom (JumpPart (a =0_goto i1)) ; ::_thesis: (JumpPart (IncAddr ((a =0_goto i1),k))) . x = (JumpPart (a =0_goto (i1 + k))) . x then x in dom <*i1*> by RECDEF_2:def_2; then A6: x = 1 by FINSEQ_1:90; reconsider f = (JumpPart (a =0_goto i1)) . x as Element of NAT by ORDINAL1:def_12; A7: (JumpPart (IncAddr ((a =0_goto i1),k))) . x = k + f by A5, A1, A2, VALUED_1:def_2; f = <*i1*> . x by RECDEF_2:def_2 .= i1 by A6, FINSEQ_1:40 ; hence (JumpPart (IncAddr ((a =0_goto i1),k))) . x = <*(i1 + k)*> . x by A6, A7, FINSEQ_1:40 .= (JumpPart (a =0_goto (i1 + k))) . x by RECDEF_2:def_2 ; ::_thesis: verum end; A8: InsCode (IncAddr ((a =0_goto i1),k)) = InsCode (a =0_goto i1) by COMPOS_0:def_9 .= 7 by RECDEF_2:def_1 .= InsCode (a =0_goto (i1 + k)) by RECDEF_2:def_1 ; A9: AddressPart (IncAddr ((a =0_goto i1),k)) = AddressPart (a =0_goto i1) by COMPOS_0:def_9 .= <*a*> by RECDEF_2:def_3 .= AddressPart (a =0_goto (i1 + k)) by RECDEF_2:def_3 ; JumpPart (IncAddr ((a =0_goto i1),k)) = JumpPart (a =0_goto (i1 + k)) by A2, A3, A4, FUNCT_1:2; hence IncAddr ((a =0_goto i1),k) = a =0_goto (i1 + k) by A8, A9, COMPOS_0:1; ::_thesis: verum end; registration let R be Ring; cluster SCM R -> IC-relocable ; coherence SCM R is IC-relocable proof thus SCM R is IC-relocable ::_thesis: verum proof let I be Instruction of (SCM R); :: according to AMISTD_2:def_4 ::_thesis: I is IC-relocable percases ( I = [0,{},{}] or ex a, b being Data-Location of R st I = a := b or ex a, b being Data-Location of R st I = AddTo (a,b) or ex a, b being Data-Location of R st I = SubFrom (a,b) or ex a, b being Data-Location of R st I = MultBy (a,b) or ex i1 being Element of NAT st I = goto (i1,R) or ex a being Data-Location of R ex i1 being Element of NAT st I = a =0_goto i1 or ex a being Data-Location of R ex r being Element of R st I = a := r ) by SCMRING2:7; suppose I = [0,{},{}] ; ::_thesis: I is IC-relocable then I = halt (SCM R) ; hence I is IC-relocable ; ::_thesis: verum end; suppose ex a, b being Data-Location of R st I = a := b ; ::_thesis: I is IC-relocable hence I is IC-relocable ; ::_thesis: verum end; suppose ex a, b being Data-Location of R st I = AddTo (a,b) ; ::_thesis: I is IC-relocable hence I is IC-relocable ; ::_thesis: verum end; suppose ex a, b being Data-Location of R st I = SubFrom (a,b) ; ::_thesis: I is IC-relocable hence I is IC-relocable ; ::_thesis: verum end; suppose ex a, b being Data-Location of R st I = MultBy (a,b) ; ::_thesis: I is IC-relocable hence I is IC-relocable ; ::_thesis: verum end; supposeA1: ex i1 being Element of NAT st I = goto (i1,R) ; ::_thesis: I is IC-relocable let j, k be Nat; :: according to AMISTD_2:def_3 ::_thesis: for b1 being set holds (IC (Exec ((IncAddr (I,j)),b1))) + k = IC (Exec ((IncAddr (I,(j + k))),(IncIC (b1,k)))) let s1 be State of (SCM R); ::_thesis: (IC (Exec ((IncAddr (I,j)),s1))) + k = IC (Exec ((IncAddr (I,(j + k))),(IncIC (s1,k)))) set s2 = IncIC (s1,k); consider i1 being Element of NAT such that A2: I = goto (i1,R) by A1; thus (IC (Exec ((IncAddr (I,j)),s1))) + k = (IC (Exec ((goto ((j + i1),R)),s1))) + k by A2, Th37 .= (j + i1) + k by SCMRING2:15 .= IC (Exec ((goto (((j + i1) + k),R)),(IncIC (s1,k)))) by SCMRING2:15 .= IC (Exec ((goto (((j + k) + i1),R)),(IncIC (s1,k)))) .= IC (Exec ((IncAddr (I,(j + k))),(IncIC (s1,k)))) by A2, Th37 ; ::_thesis: verum end; suppose ex a being Data-Location of R ex i1 being Element of NAT st I = a =0_goto i1 ; ::_thesis: I is IC-relocable then consider a being Data-Location of R, i1 being Element of NAT such that A3: I = a =0_goto i1 ; let j, k be Nat; :: according to AMISTD_2:def_3 ::_thesis: for b1 being set holds (IC (Exec ((IncAddr (I,j)),b1))) + k = IC (Exec ((IncAddr (I,(j + k))),(IncIC (b1,k)))) let s1 be State of (SCM R); ::_thesis: (IC (Exec ((IncAddr (I,j)),s1))) + k = IC (Exec ((IncAddr (I,(j + k))),(IncIC (s1,k)))) set s2 = IncIC (s1,k); ( a <> IC & dom ((IC ) .--> ((IC s1) + k)) = {(IC )} ) by Th2, FUNCOP_1:13; then not a in dom ((IC ) .--> ((IC s1) + k)) by TARSKI:def_1; then A4: s1 . a = (IncIC (s1,k)) . a by FUNCT_4:11; percases ( s1 . a = 0. R or s1 . a <> 0. R ) ; supposeA5: s1 . a = 0. R ; ::_thesis: (IC (Exec ((IncAddr (I,j)),s1))) + k = IC (Exec ((IncAddr (I,(j + k))),(IncIC (s1,k)))) thus (IC (Exec ((IncAddr (I,j)),s1))) + k = (IC (Exec ((a =0_goto (j + i1)),s1))) + k by A3, Th38 .= (j + i1) + k by A5, SCMRING2:16 .= IC (Exec ((a =0_goto ((j + i1) + k)),(IncIC (s1,k)))) by A4, A5, SCMRING2:16 .= IC (Exec ((a =0_goto ((j + k) + i1)),(IncIC (s1,k)))) .= IC (Exec ((IncAddr (I,(j + k))),(IncIC (s1,k)))) by A3, Th38 ; ::_thesis: verum end; supposeA6: s1 . a <> 0. R ; ::_thesis: (IC (Exec ((IncAddr (I,j)),s1))) + k = IC (Exec ((IncAddr (I,(j + k))),(IncIC (s1,k)))) A7: IncAddr (I,j) = a =0_goto (i1 + j) by A3, Th38; A8: IncAddr (I,(j + k)) = a =0_goto (i1 + (j + k)) by A3, Th38; dom ((IC ) .--> ((IC s1) + k)) = {(IC )} by FUNCOP_1:13; then IC in dom ((IC ) .--> ((IC s1) + k)) by TARSKI:def_1; then A9: IC (IncIC (s1,k)) = ((IC ) .--> ((IC s1) + k)) . (IC ) by FUNCT_4:13 .= (IC s1) + k by FUNCOP_1:72 ; thus (IC (Exec ((IncAddr (I,j)),s1))) + k = (succ (IC s1)) + k by A7, A6, SCMRING2:16 .= ((IC s1) + 1) + k .= succ (IC (IncIC (s1,k))) by A9 .= IC (Exec ((IncAddr (I,(j + k))),(IncIC (s1,k)))) by A8, A6, A4, SCMRING2:16 ; ::_thesis: verum end; end; end; suppose ex a being Data-Location of R ex r being Element of R st I = a := r ; ::_thesis: I is IC-relocable hence I is IC-relocable ; ::_thesis: verum end; end; end; end; end; theorem :: SCMRING3:39 for R being Ring for I being Instruction of (SCM R) holds InsCode I <= 7 proof let R be Ring; ::_thesis: for I being Instruction of (SCM R) holds InsCode I <= 7 let I be Instruction of (SCM R); ::_thesis: InsCode I <= 7 set T = InsCode I; ( InsCode I = 0 or InsCode I = 1 or InsCode I = 2 or InsCode I = 3 or InsCode I = 4 or InsCode I = 5 or InsCode I = 6 or InsCode I = 7 ) by Lm2; hence InsCode I <= 7 ; ::_thesis: verum end;