:: SCMFSA10 semantic presentation begin definition let la, lb be Int-Location; let a, b be Integer; :: original: --> redefine func(la,lb) --> (a,b) -> PartState of SCM+FSA; coherence (la,lb) --> (a,b) is PartState of SCM+FSA proof A1: Values lb = INT by SCMFSA_2:11; b is Element of INT by INT_1:def_2; then reconsider b = b as Element of Values lb by A1; A2: Values la = INT by SCMFSA_2:11; a is Element of INT by INT_1:def_2; then reconsider a = a as Element of Values la by A2; (la,lb) --> (a,b) is PartState of SCM+FSA ; hence (la,lb) --> (a,b) is PartState of SCM+FSA ; ::_thesis: verum end; end; theorem Th1: :: SCMFSA10:1 for o being Object of SCM+FSA holds ( not o in Data-Locations or o is Int-Location or o is FinSeq-Location ) proof let o be Object of SCM+FSA; ::_thesis: ( not o in Data-Locations or o is Int-Location or o is FinSeq-Location ) assume o in Data-Locations ; ::_thesis: ( o is Int-Location or o is FinSeq-Location ) then ( o in SCM-Data-Loc or o in SCM+FSA-Data*-Loc ) by SCMFSA_2:100, XBOOLE_0:def_3; hence ( o is Int-Location or o is FinSeq-Location ) by AMI_2:def_16, SCMFSA_2:def_5; ::_thesis: verum end; theorem Th2: :: SCMFSA10:2 for a, b being Int-Location holds a := b = [1,{},<*a,b*>] proof let a, b be Int-Location; ::_thesis: a := b = [1,{},<*a,b*>] ex A, B being Data-Location st ( a = A & b = B & a := b = A := B ) by SCMFSA_2:def_8; hence a := b = [1,{},<*a,b*>] ; ::_thesis: verum end; theorem Th3: :: SCMFSA10:3 for a, b being Int-Location holds AddTo (a,b) = [2,{},<*a,b*>] proof let a, b be Int-Location; ::_thesis: AddTo (a,b) = [2,{},<*a,b*>] ex A, B being Data-Location st ( a = A & b = B & AddTo (a,b) = AddTo (A,B) ) by SCMFSA_2:def_9; hence AddTo (a,b) = [2,{},<*a,b*>] ; ::_thesis: verum end; theorem Th4: :: SCMFSA10:4 for a, b being Int-Location holds SubFrom (a,b) = [3,{},<*a,b*>] proof let a, b be Int-Location; ::_thesis: SubFrom (a,b) = [3,{},<*a,b*>] ex A, B being Data-Location st ( a = A & b = B & SubFrom (a,b) = SubFrom (A,B) ) by SCMFSA_2:def_10; hence SubFrom (a,b) = [3,{},<*a,b*>] ; ::_thesis: verum end; theorem Th5: :: SCMFSA10:5 for a, b being Int-Location holds MultBy (a,b) = [4,{},<*a,b*>] proof let a, b be Int-Location; ::_thesis: MultBy (a,b) = [4,{},<*a,b*>] ex A, B being Data-Location st ( a = A & b = B & MultBy (a,b) = MultBy (A,B) ) by SCMFSA_2:def_11; hence MultBy (a,b) = [4,{},<*a,b*>] ; ::_thesis: verum end; theorem Th6: :: SCMFSA10:6 for a, b being Int-Location holds Divide (a,b) = [5,{},<*a,b*>] proof let a, b be Int-Location; ::_thesis: Divide (a,b) = [5,{},<*a,b*>] ex A, B being Data-Location st ( a = A & b = B & Divide (a,b) = Divide (A,B) ) by SCMFSA_2:def_12; hence Divide (a,b) = [5,{},<*a,b*>] ; ::_thesis: verum end; theorem Th7: :: SCMFSA10:7 for a being Int-Location for il being Element of NAT holds a =0_goto il = [7,<*il*>,<*a*>] proof let a be Int-Location; ::_thesis: for il being Element of NAT holds a =0_goto il = [7,<*il*>,<*a*>] let il be Element of NAT ; ::_thesis: a =0_goto il = [7,<*il*>,<*a*>] ex A being Data-Location st ( A = a & A =0_goto il = a =0_goto il ) by SCMFSA_2:def_14; hence a =0_goto il = [7,<*il*>,<*a*>] ; ::_thesis: verum end; theorem Th8: :: SCMFSA10:8 for a being Int-Location for il being Element of NAT holds a >0_goto il = [8,<*il*>,<*a*>] proof let a be Int-Location; ::_thesis: for il being Element of NAT holds a >0_goto il = [8,<*il*>,<*a*>] let il be Element of NAT ; ::_thesis: a >0_goto il = [8,<*il*>,<*a*>] ex A being Data-Location st ( A = a & A >0_goto il = a >0_goto il ) by SCMFSA_2:def_15; hence a >0_goto il = [8,<*il*>,<*a*>] ; ::_thesis: verum end; theorem Th9: :: SCMFSA10:9 JumpPart (halt SCM+FSA) = {} ; theorem Th10: :: SCMFSA10:10 for a, b being Int-Location holds JumpPart (a := b) = {} proof let a, b be Int-Location; ::_thesis: JumpPart (a := b) = {} thus JumpPart (a := b) = [1,{},<*a,b*>] `2_3 by Th2 .= {} ; ::_thesis: verum end; theorem Th11: :: SCMFSA10:11 for a, b being Int-Location holds JumpPart (AddTo (a,b)) = {} proof let a, b be Int-Location; ::_thesis: JumpPart (AddTo (a,b)) = {} thus JumpPart (AddTo (a,b)) = [2,{},<*a,b*>] `2_3 by Th3 .= {} ; ::_thesis: verum end; theorem Th12: :: SCMFSA10:12 for a, b being Int-Location holds JumpPart (SubFrom (a,b)) = {} proof let a, b be Int-Location; ::_thesis: JumpPart (SubFrom (a,b)) = {} thus JumpPart (SubFrom (a,b)) = [3,{},<*a,b*>] `2_3 by Th4 .= {} ; ::_thesis: verum end; theorem Th13: :: SCMFSA10:13 for a, b being Int-Location holds JumpPart (MultBy (a,b)) = {} proof let a, b be Int-Location; ::_thesis: JumpPart (MultBy (a,b)) = {} thus JumpPart (MultBy (a,b)) = [4,{},<*a,b*>] `2_3 by Th5 .= {} ; ::_thesis: verum end; theorem Th14: :: SCMFSA10:14 for a, b being Int-Location holds JumpPart (Divide (a,b)) = {} proof let a, b be Int-Location; ::_thesis: JumpPart (Divide (a,b)) = {} thus JumpPart (Divide (a,b)) = [5,{},<*a,b*>] `2_3 by Th6 .= {} ; ::_thesis: verum end; theorem Th15: :: SCMFSA10:15 for i1 being Element of NAT for a being Int-Location holds JumpPart (a =0_goto i1) = <*i1*> proof let i1 be Element of NAT ; ::_thesis: for a being Int-Location holds JumpPart (a =0_goto i1) = <*i1*> let a be Int-Location; ::_thesis: JumpPart (a =0_goto i1) = <*i1*> thus JumpPart (a =0_goto i1) = [7,<*i1*>,<*a*>] `2_3 by Th7 .= <*i1*> ; ::_thesis: verum end; theorem Th16: :: SCMFSA10:16 for i1 being Element of NAT for a being Int-Location holds JumpPart (a >0_goto i1) = <*i1*> proof let i1 be Element of NAT ; ::_thesis: for a being Int-Location holds JumpPart (a >0_goto i1) = <*i1*> let a be Int-Location; ::_thesis: JumpPart (a >0_goto i1) = <*i1*> thus JumpPart (a >0_goto i1) = [8,<*i1*>,<*a*>] `2_3 by Th8 .= <*i1*> ; ::_thesis: verum end; theorem :: SCMFSA10:17 for T being InsType of the InstructionsF of SCM+FSA st T = 0 holds JumpParts T = {0} proof let T be InsType of the InstructionsF of SCM+FSA; ::_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+FSA such that A2: a = JumpPart I and A3: InsCode I = T ; I = halt SCM+FSA by A1, A3, SCMFSA_2:95; hence a in {0} by A2, Th9, 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+FSA) = 0 by COMPOS_1:70; hence a in JumpParts T by A1, Th9, A4; ::_thesis: verum end; theorem :: SCMFSA10:18 for T being InsType of the InstructionsF of SCM+FSA st T = 1 holds JumpParts T = {{}} proof let T be InsType of the InstructionsF of SCM+FSA; ::_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+FSA such that A2: x = JumpPart I and A3: InsCode I = T ; consider a, b being Int-Location such that A4: I = a := b by A1, A3, SCMFSA_2:30; x = {} by A2, Th10, A4; hence x in {{}} by TARSKI:def_1; ::_thesis: verum end; set a = the Int-Location; 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 Int-Location := the Int-Location) by Th10; InsCode ( the Int-Location := the Int-Location) = 1 by SCMFSA_2:18; hence x in JumpParts T by A5, A1; ::_thesis: verum end; theorem :: SCMFSA10:19 for T being InsType of the InstructionsF of SCM+FSA st T = 2 holds JumpParts T = {{}} proof let T be InsType of the InstructionsF of SCM+FSA; ::_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+FSA such that A2: x = JumpPart I and A3: InsCode I = T ; consider a, b being Int-Location such that A4: I = AddTo (a,b) by A1, A3, SCMFSA_2:31; x = {} by A2, Th11, A4; hence x in {{}} by TARSKI:def_1; ::_thesis: verum end; set a = the Int-Location; 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 Int-Location, the Int-Location)) by Th11; InsCode (AddTo ( the Int-Location, the Int-Location)) = 2 by SCMFSA_2:19; hence x in JumpParts T by A5, A1; ::_thesis: verum end; theorem :: SCMFSA10:20 for T being InsType of the InstructionsF of SCM+FSA st T = 3 holds JumpParts T = {{}} proof let T be InsType of the InstructionsF of SCM+FSA; ::_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+FSA such that A2: x = JumpPart I and A3: InsCode I = T ; consider a, b being Int-Location such that A4: I = SubFrom (a,b) by A1, A3, SCMFSA_2:32; x = {} by A2, Th12, A4; hence x in {{}} by TARSKI:def_1; ::_thesis: verum end; set a = the Int-Location; 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 Int-Location, the Int-Location)) by Th12; InsCode (SubFrom ( the Int-Location, the Int-Location)) = 3 by SCMFSA_2:20; hence x in JumpParts T by A5, A1; ::_thesis: verum end; theorem :: SCMFSA10:21 for T being InsType of the InstructionsF of SCM+FSA st T = 4 holds JumpParts T = {{}} proof let T be InsType of the InstructionsF of SCM+FSA; ::_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+FSA such that A2: x = JumpPart I and A3: InsCode I = T ; consider a, b being Int-Location such that A4: I = MultBy (a,b) by A1, A3, SCMFSA_2:33; x = {} by A2, Th13, A4; hence x in {{}} by TARSKI:def_1; ::_thesis: verum end; set a = the Int-Location; 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 Int-Location, the Int-Location)) by Th13; InsCode (MultBy ( the Int-Location, the Int-Location)) = 4 by SCMFSA_2:21; hence x in JumpParts T by A5, A1; ::_thesis: verum end; theorem :: SCMFSA10:22 for T being InsType of the InstructionsF of SCM+FSA st T = 5 holds JumpParts T = {{}} proof let T be InsType of the InstructionsF of SCM+FSA; ::_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+FSA such that A2: x = JumpPart I and A3: InsCode I = T ; consider a, b being Int-Location such that A4: I = Divide (a,b) by A1, A3, SCMFSA_2:34; x = {} by A2, Th14, A4; hence x in {{}} by TARSKI:def_1; ::_thesis: verum end; set a = the Int-Location; 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 (Divide ( the Int-Location, the Int-Location)) by Th14; InsCode (Divide ( the Int-Location, the Int-Location)) = 5 by SCMFSA_2:22; hence x in JumpParts T by A5, A1; ::_thesis: verum end; theorem Th23: :: SCMFSA10:23 for T being InsType of the InstructionsF of SCM+FSA st T = 6 holds dom (product" (JumpParts T)) = {1} proof let T be InsType of the InstructionsF of SCM+FSA; ::_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 ) = <* 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 ) = 6 by SCMFSA_2:23; then A3: JumpPart (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 (goto the Element of NAT )) 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+FSA such that A5: f = JumpPart I and A6: InsCode I = T ; consider i1 being Element of NAT such that A7: I = goto i1 by A1, A6, SCMFSA_2:35; 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 Th24: :: SCMFSA10:24 for T being InsType of the InstructionsF of SCM+FSA st T = 7 holds dom (product" (JumpParts T)) = {1} proof let T be InsType of the InstructionsF of SCM+FSA; ::_thesis: ( T = 7 implies dom (product" (JumpParts T)) = {1} ) set i1 = the Element of NAT ; set a = the Int-Location; assume A1: T = 7 ; ::_thesis: dom (product" (JumpParts T)) = {1} A2: JumpPart ( the Int-Location =0_goto the Element of NAT ) = <* the Element of NAT *> by Th15; 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 Int-Location =0_goto the Element of NAT ) = 7 by SCMFSA_2:24; then A3: JumpPart ( the Int-Location =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 Int-Location =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+FSA such that A5: f = JumpPart I and A6: InsCode I = T ; consider i1 being Element of NAT , a being Int-Location such that A7: I = a =0_goto i1 by A1, A6, SCMFSA_2:36; f = <*i1*> by A5, A7, Th15; 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 Th25: :: SCMFSA10:25 for T being InsType of the InstructionsF of SCM+FSA st T = 8 holds dom (product" (JumpParts T)) = {1} proof let T be InsType of the InstructionsF of SCM+FSA; ::_thesis: ( T = 8 implies dom (product" (JumpParts T)) = {1} ) set i1 = the Element of NAT ; set a = the Int-Location; assume A1: T = 8 ; ::_thesis: dom (product" (JumpParts T)) = {1} A2: JumpPart ( the Int-Location >0_goto the Element of NAT ) = <* the Element of NAT *> by Th16; 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 Int-Location >0_goto the Element of NAT ) = 8 by SCMFSA_2:25; then A3: JumpPart ( the Int-Location >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 Int-Location >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+FSA such that A5: f = JumpPart I and A6: InsCode I = T ; consider i1 being Element of NAT , a being Int-Location such that A7: I = a >0_goto i1 by A1, A6, SCMFSA_2:37; f = <*i1*> by A5, A7, Th16; 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 :: SCMFSA10:26 for T being InsType of the InstructionsF of SCM+FSA st T = 9 holds JumpParts T = {{}} proof let T be InsType of the InstructionsF of SCM+FSA; ::_thesis: ( T = 9 implies JumpParts T = {{}} ) assume A1: T = 9 ; ::_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+FSA such that A2: x = JumpPart I and A3: InsCode I = T ; consider a, b being Int-Location, f being FinSeq-Location such that A4: I = b := (f,a) by A1, A3, SCMFSA_2:38; x = {} by A2, A4, RECDEF_2:def_2; hence x in {{}} by TARSKI:def_1; ::_thesis: verum end; set a = the Int-Location; set f = the FinSeq-Location ; 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 Int-Location := ( the FinSeq-Location , the Int-Location)) by RECDEF_2:def_2; InsCode ( the Int-Location := ( the FinSeq-Location , the Int-Location)) = 9 by SCMFSA_2:26; hence x in JumpParts T by A5, A1; ::_thesis: verum end; theorem :: SCMFSA10:27 for T being InsType of the InstructionsF of SCM+FSA st T = 10 holds JumpParts T = {{}} proof let T be InsType of the InstructionsF of SCM+FSA; ::_thesis: ( T = 10 implies JumpParts T = {{}} ) assume A1: T = 10 ; ::_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+FSA such that A2: x = JumpPart I and A3: InsCode I = T ; consider a, b being Int-Location, f being FinSeq-Location such that A4: I = (f,a) := b by A1, A3, SCMFSA_2:39; x = {} by A2, A4, RECDEF_2:def_2; hence x in {{}} by TARSKI:def_1; ::_thesis: verum end; set a = the Int-Location; set f = the FinSeq-Location ; 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 FinSeq-Location , the Int-Location) := the Int-Location) by RECDEF_2:def_2; InsCode (( the FinSeq-Location , the Int-Location) := the Int-Location) = 10 by SCMFSA_2:27; hence x in JumpParts T by A5, A1; ::_thesis: verum end; theorem :: SCMFSA10:28 for T being InsType of the InstructionsF of SCM+FSA st T = 11 holds JumpParts T = {{}} proof let T be InsType of the InstructionsF of SCM+FSA; ::_thesis: ( T = 11 implies JumpParts T = {{}} ) assume A1: T = 11 ; ::_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+FSA such that A2: x = JumpPart I and A3: InsCode I = T ; consider a being Int-Location, f being FinSeq-Location such that A4: I = a :=len f by A1, A3, SCMFSA_2:40; x = {} by A2, A4, RECDEF_2:def_2; hence x in {{}} by TARSKI:def_1; ::_thesis: verum end; set a = the Int-Location; set f = the FinSeq-Location ; 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 Int-Location :=len the FinSeq-Location ) by RECDEF_2:def_2; InsCode ( the Int-Location :=len the FinSeq-Location ) = 11 by SCMFSA_2:28; hence x in JumpParts T by A5, A1; ::_thesis: verum end; theorem :: SCMFSA10:29 for T being InsType of the InstructionsF of SCM+FSA st T = 12 holds JumpParts T = {{}} proof let T be InsType of the InstructionsF of SCM+FSA; ::_thesis: ( T = 12 implies JumpParts T = {{}} ) assume A1: T = 12 ; ::_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+FSA such that A2: x = JumpPart I and A3: InsCode I = T ; consider a being Int-Location, f being FinSeq-Location such that A4: I = f :=<0,...,0> a by A1, A3, SCMFSA_2:41; x = {} by A2, A4, RECDEF_2:def_2; hence x in {{}} by TARSKI:def_1; ::_thesis: verum end; set a = the Int-Location; set f = the FinSeq-Location ; 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 FinSeq-Location :=<0,...,0> the Int-Location) by RECDEF_2:def_2; InsCode ( the FinSeq-Location :=<0,...,0> the Int-Location) = 12 by SCMFSA_2:29; hence x in JumpParts T by A5, A1; ::_thesis: verum end; theorem :: SCMFSA10:30 for i1 being Element of NAT holds (product" (JumpParts (InsCode (goto i1)))) . 1 = NAT proof let i1 be Element of NAT ; ::_thesis: (product" (JumpParts (InsCode (goto i1)))) . 1 = NAT dom (product" (JumpParts (InsCode (goto i1)))) = {1} by Th23, SCMFSA_2:23; then A1: 1 in dom (product" (JumpParts (InsCode (goto i1)))) by TARSKI:def_1; hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: NAT c= (product" (JumpParts (InsCode (goto i1)))) . 1 let x be set ; ::_thesis: ( x in (product" (JumpParts (InsCode (goto i1)))) . 1 implies x in NAT ) assume x in (product" (JumpParts (InsCode (goto i1)))) . 1 ; ::_thesis: x in NAT then x in pi ((JumpParts (InsCode (goto i1))),1) by A1, CARD_3:def_12; then consider g being Function such that A2: g in JumpParts (InsCode (goto i1)) and A3: x = g . 1 by CARD_3:def_6; consider I being Instruction of SCM+FSA such that A4: g = JumpPart I and A5: InsCode I = InsCode (goto i1) by A2; consider i2 being Element of NAT such that A6: I = goto i2 by A5, SCMFSA_2:23, SCMFSA_2:35; g = <*i2*> by A4, A6, RECDEF_2:def_2; then x = i2 by A3, 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)))) . 1 ) assume x in NAT ; ::_thesis: x in (product" (JumpParts (InsCode (goto i1)))) . 1 then reconsider x = x as Element of NAT ; A7: <*x*> . 1 = x by FINSEQ_1:def_8; InsCode (goto i1) = 6 by SCMFSA_2:23; then A8: InsCode (goto i1) = InsCode (goto x) by SCMFSA_2:23; JumpPart (goto x) = <*x*> by RECDEF_2:def_2; then <*x*> in JumpParts (InsCode (goto i1)) by A8; then x in pi ((JumpParts (InsCode (goto i1))),1) by A7, CARD_3:def_6; hence x in (product" (JumpParts (InsCode (goto i1)))) . 1 by A1, CARD_3:def_12; ::_thesis: verum end; theorem :: SCMFSA10:31 for i1 being Element of NAT for a being Int-Location holds (product" (JumpParts (InsCode (a =0_goto i1)))) . 1 = NAT proof let i1 be Element of NAT ; ::_thesis: for a being Int-Location holds (product" (JumpParts (InsCode (a =0_goto i1)))) . 1 = NAT let a be Int-Location; ::_thesis: (product" (JumpParts (InsCode (a =0_goto i1)))) . 1 = NAT dom (product" (JumpParts (InsCode (a =0_goto i1)))) = {1} by Th24, SCMFSA_2:24; then A1: 1 in dom (product" (JumpParts (InsCode (a =0_goto i1)))) by TARSKI: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 A2: g in JumpParts (InsCode (a =0_goto i1)) and A3: x = g . 1 by CARD_3:def_6; consider I being Instruction of SCM+FSA such that A4: g = JumpPart I and A5: InsCode I = InsCode (a =0_goto i1) by A2; consider i2 being Element of NAT , b being Int-Location such that A6: I = b =0_goto i2 by A5, SCMFSA_2:24, SCMFSA_2:36; g = <*i2*> by A4, A6, Th15; then x = i2 by A3, 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 ; A7: <*x*> . 1 = x by FINSEQ_1:40; InsCode (a =0_goto i1) = 7 by SCMFSA_2:24; then A8: InsCode (a =0_goto i1) = InsCode (a =0_goto x) by SCMFSA_2:24; JumpPart (a =0_goto x) = <*x*> by Th15; then <*x*> in JumpParts (InsCode (a =0_goto i1)) by A8; then x in pi ((JumpParts (InsCode (a =0_goto i1))),1) by A7, CARD_3:def_6; hence x in (product" (JumpParts (InsCode (a =0_goto i1)))) . 1 by A1, CARD_3:def_12; ::_thesis: verum end; theorem :: SCMFSA10:32 for i1 being Element of NAT for a being Int-Location holds (product" (JumpParts (InsCode (a >0_goto i1)))) . 1 = NAT proof let i1 be Element of NAT ; ::_thesis: for a being Int-Location holds (product" (JumpParts (InsCode (a >0_goto i1)))) . 1 = NAT let a be Int-Location; ::_thesis: (product" (JumpParts (InsCode (a >0_goto i1)))) . 1 = NAT dom (product" (JumpParts (InsCode (a >0_goto i1)))) = {1} by Th25, SCMFSA_2:25; then A1: 1 in dom (product" (JumpParts (InsCode (a >0_goto i1)))) by TARSKI: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 A2: g in JumpParts (InsCode (a >0_goto i1)) and A3: x = g . 1 by CARD_3:def_6; consider I being Instruction of SCM+FSA such that A4: g = JumpPart I and A5: InsCode I = InsCode (a >0_goto i1) by A2; consider i2 being Element of NAT , b being Int-Location such that A6: I = b >0_goto i2 by A5, SCMFSA_2:25, SCMFSA_2:37; g = <*i2*> by A4, A6, Th16; then x = i2 by A3, 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 ; A7: <*x*> . 1 = x by FINSEQ_1:40; InsCode (a >0_goto i1) = 8 by SCMFSA_2:25; then A8: InsCode (a >0_goto i1) = InsCode (a >0_goto x) by SCMFSA_2:25; JumpPart (a >0_goto x) = <*x*> by Th16; then <*x*> in JumpParts (InsCode (a >0_goto i1)) by A8; then x in pi ((JumpParts (InsCode (a >0_goto i1))),1) by A7, CARD_3:def_6; hence x in (product" (JumpParts (InsCode (a >0_goto i1)))) . 1 by A1, CARD_3:def_12; ::_thesis: verum end; Lm1: for i being Instruction of SCM+FSA st ( for l being Element of NAT holds NIC (i,l) = {(succ l)} ) holds JUMP i is empty proof reconsider p = 0 , q = 1 as Element of NAT ; let i be Instruction of SCM+FSA; ::_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 = p, q = q 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 cluster JUMP (halt SCM+FSA) -> empty ; coherence JUMP (halt SCM+FSA) is empty ; end; registration let a, b be Int-Location; clustera := b -> sequential ; coherence a := b is sequential proof let s be State of SCM+FSA; :: according to AMISTD_1:def_8 ::_thesis: (Exec ((a := b),s)) . (IC ) = succ (IC s) thus (Exec ((a := b),s)) . (IC ) = succ (IC s) by SCMFSA_2:63; ::_thesis: verum end; cluster AddTo (a,b) -> sequential ; coherence AddTo (a,b) is sequential proof let s be State of SCM+FSA; :: according to AMISTD_1:def_8 ::_thesis: (Exec ((AddTo (a,b)),s)) . (IC ) = succ (IC s) thus (Exec ((AddTo (a,b)),s)) . (IC ) = succ (IC s) by SCMFSA_2:64; ::_thesis: verum end; cluster SubFrom (a,b) -> sequential ; coherence SubFrom (a,b) is sequential proof let s be State of SCM+FSA; :: according to AMISTD_1:def_8 ::_thesis: (Exec ((SubFrom (a,b)),s)) . (IC ) = succ (IC s) thus (Exec ((SubFrom (a,b)),s)) . (IC ) = succ (IC s) by SCMFSA_2:65; ::_thesis: verum end; cluster MultBy (a,b) -> sequential ; coherence MultBy (a,b) is sequential proof let s be State of SCM+FSA; :: according to AMISTD_1:def_8 ::_thesis: (Exec ((MultBy (a,b)),s)) . (IC ) = succ (IC s) thus (Exec ((MultBy (a,b)),s)) . (IC ) = succ (IC s) by SCMFSA_2:66; ::_thesis: verum end; cluster Divide (a,b) -> sequential ; coherence Divide (a,b) is sequential proof let s be State of SCM+FSA; :: according to AMISTD_1:def_8 ::_thesis: (Exec ((Divide (a,b)),s)) . (IC ) = succ (IC s) thus (Exec ((Divide (a,b)),s)) . (IC ) = succ (IC s) by SCMFSA_2:67; ::_thesis: verum end; end; registration let a, b be Int-Location; 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 Lm1; ::_thesis: verum end; 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 Lm1; ::_thesis: verum end; 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 Lm1; ::_thesis: verum end; 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 Lm1; ::_thesis: verum end; cluster JUMP (Divide (a,b)) -> empty ; coherence JUMP (Divide (a,b)) is empty proof for l being Element of NAT holds NIC ((Divide (a,b)),l) = {(succ l)} by AMISTD_1:12; hence JUMP (Divide (a,b)) is empty by Lm1; ::_thesis: verum end; end; theorem Th33: :: SCMFSA10:33 for i1, il being Element of NAT holds NIC ((goto i1),il) = {i1} proof let i1, il be Element of NAT ; ::_thesis: NIC ((goto i1),il) = {i1} now__::_thesis:_for_x_being_set_holds_ (_x_in_{i1}_iff_x_in__{__(IC_(Exec_((goto_i1),s)))_where_s_is_Element_of_product_(the_Values_of_SCM+FSA)_:_IC_s_=_il__}__) let x be set ; ::_thesis: ( x in {i1} iff x in { (IC (Exec ((goto i1),s))) where s is Element of product (the_Values_of SCM+FSA) : IC s = il } ) A1: now__::_thesis:_(_x_=_i1_implies_x_in__{__(IC_(Exec_((goto_i1),s)))_where_s_is_Element_of_product_(the_Values_of_SCM+FSA)_:_IC_s_=_il__}__) reconsider il1 = il as Element of Values (IC ) by MEMSTR_0:def_6; reconsider n = il1 as Element of NAT ; set I = goto i1; set t = the State of SCM+FSA; set Q = the Instruction-Sequence of SCM+FSA; assume A2: x = i1 ; ::_thesis: x in { (IC (Exec ((goto i1),s))) where s is Element of product (the_Values_of SCM+FSA) : IC s = il } reconsider u = the State of SCM+FSA +* ((IC ),il1) as Element of product (the_Values_of SCM+FSA) by CARD_3:107; reconsider P = the Instruction-Sequence of SCM+FSA +* (il,(goto i1)) as Instruction-Sequence of SCM+FSA ; IC in dom the State of SCM+FSA by MEMSTR_0:2; then A3: IC u = n by FUNCT_7:31; A4: P /. il = P . il by PBOOLE:143; il in NAT ; then il in dom the Instruction-Sequence of SCM+FSA by PARTFUN1:def_2; then A5: P . n = goto i1 by FUNCT_7:31; then IC (Following (P,u)) = i1 by A3, A4, SCMFSA_2:69; hence x in { (IC (Exec ((goto i1),s))) where s is Element of product (the_Values_of SCM+FSA) : IC s = il } by A2, A3, A5, A4; ::_thesis: verum end; now__::_thesis:_(_x_in__{__(IC_(Exec_((goto_i1),s)))_where_s_is_Element_of_product_(the_Values_of_SCM+FSA)_:_IC_s_=_il__}__implies_x_=_i1_) assume x in { (IC (Exec ((goto i1),s))) where s is Element of product (the_Values_of SCM+FSA) : IC s = il } ; ::_thesis: x = i1 then ex s being Element of product (the_Values_of SCM+FSA) st ( x = IC (Exec ((goto i1),s)) & IC s = il ) ; hence x = i1 by SCMFSA_2:69; ::_thesis: verum end; hence ( x in {i1} iff x in { (IC (Exec ((goto i1),s))) where s is Element of product (the_Values_of SCM+FSA) : IC s = il } ) by A1, TARSKI:def_1; ::_thesis: verum end; hence NIC ((goto i1),il) = {i1} by TARSKI:1; ::_thesis: verum end; theorem Th34: :: SCMFSA10:34 for i1 being Element of NAT holds JUMP (goto i1) = {i1} proof let i1 be Element of NAT ; ::_thesis: JUMP (goto i1) = {i1} set X = { (NIC ((goto i1),il)) where il is Element of NAT : verum } ; now__::_thesis:_for_x_being_set_holds_ (_(_x_in_meet__{__(NIC_((goto_i1),il))_where_il_is_Element_of_NAT_:_verum__}__implies_x_in_{i1}_)_&_(_x_in_{i1}_implies_x_in_meet__{__(NIC_((goto_i1),il))_where_il_is_Element_of_NAT_:_verum__}__)_) let x be set ; ::_thesis: ( ( x in meet { (NIC ((goto i1),il)) where il is Element of NAT : verum } implies x in {i1} ) & ( x in {i1} implies x in meet { (NIC ((goto i1),il)) where il is Element of NAT : verum } ) ) hereby ::_thesis: ( x in {i1} implies x in meet { (NIC ((goto i1),il)) where il is Element of NAT : verum } ) set il1 = 1; A1: NIC ((goto i1),1) in { (NIC ((goto i1),il)) where il is Element of NAT : verum } ; assume x in meet { (NIC ((goto i1),il)) where il is Element of NAT : verum } ; ::_thesis: x in {i1} then x in NIC ((goto i1),1) by A1, SETFAM_1:def_1; hence x in {i1} by Th33; ::_thesis: verum end; assume x in {i1} ; ::_thesis: x in meet { (NIC ((goto i1),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),il))_where_il_is_Element_of_NAT_:_verum__}__holds_ i1_in_Y let Y be set ; ::_thesis: ( Y in { (NIC ((goto i1),il)) where il is Element of NAT : verum } implies i1 in Y ) assume Y in { (NIC ((goto i1),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),il) ; NIC ((goto i1),il) = {i1} by Th33; hence i1 in Y by A4, TARSKI:def_1; ::_thesis: verum end; NIC ((goto i1),i1) in { (NIC ((goto i1),il)) where il is Element of NAT : verum } ; hence x in meet { (NIC ((goto i1),il)) where il is Element of NAT : verum } by A2, A3, SETFAM_1:def_1; ::_thesis: verum end; hence JUMP (goto i1) = {i1} by TARSKI:1; ::_thesis: verum end; registration let i1 be Element of NAT ; cluster JUMP (goto i1) -> 1 -element ; coherence JUMP (goto i1) is 1 -element proof JUMP (goto i1) = {i1} by Th34; hence JUMP (goto i1) is 1 -element ; ::_thesis: verum end; end; theorem Th35: :: SCMFSA10:35 for i1, il being Element of NAT for a being Int-Location holds NIC ((a =0_goto i1),il) = {i1,(succ il)} proof let i1, il be Element of NAT ; ::_thesis: for a being Int-Location holds NIC ((a =0_goto i1),il) = {i1,(succ il)} let a be Int-Location; ::_thesis: NIC ((a =0_goto i1),il) = {i1,(succ il)} set t = the State of SCM+FSA; set Q = the Instruction-Sequence of SCM+FSA; hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: {i1,(succ il)} c= NIC ((a =0_goto i1),il) let x be set ; ::_thesis: ( x in NIC ((a =0_goto i1),il) implies b1 in {i1,(succ il)} ) assume x in NIC ((a =0_goto i1),il) ; ::_thesis: b1 in {i1,(succ il)} then consider s being Element of product (the_Values_of SCM+FSA) such that A1: x = IC (Exec ((a =0_goto i1),s)) and A2: IC s = il ; percases ( s . a = 0 or s . a <> 0 ) ; suppose s . a = 0 ; ::_thesis: b1 in {i1,(succ il)} then x = i1 by A1, SCMFSA_2:70; hence x in {i1,(succ il)} by TARSKI:def_2; ::_thesis: verum end; suppose s . a <> 0 ; ::_thesis: b1 in {i1,(succ il)} then x = succ il by A1, A2, SCMFSA_2:70; hence x in {i1,(succ il)} by TARSKI:def_2; ::_thesis: verum end; end; end; 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) ) set I = a =0_goto i1; A3: IC <> a by SCMFSA_2:56; reconsider il1 = il as Element of Values (IC ) by MEMSTR_0:def_6; reconsider u = the State of SCM+FSA +* ((IC ),il1) as Element of product (the_Values_of SCM+FSA) by CARD_3:107; reconsider P = the Instruction-Sequence of SCM+FSA +* (il,(a =0_goto i1)) as Instruction-Sequence of SCM+FSA ; reconsider n = il as Element of NAT ; 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 v = u +* (a .--> 0) as Element of product (the_Values_of SCM+FSA) by CARD_3:107; A6: IC in dom the State of SCM+FSA by MEMSTR_0:2; A7: dom (a .--> 0) = {a} by FUNCOP_1:13; then not IC in dom (a .--> 0) by A3, TARSKI:def_1; then A8: IC v = IC u by FUNCT_4:11 .= n by A6, FUNCT_7:31 ; A9: P /. il = P . il by PBOOLE:143; il in NAT ; then il in dom the Instruction-Sequence of SCM+FSA by PARTFUN1:def_2; then A10: P . il = a =0_goto i1 by FUNCT_7:31; a in dom (a .--> 0) by A7, TARSKI:def_1; then v . a = (a .--> 0) . a by FUNCT_4:13 .= 0 by FUNCOP_1:72 ; then IC (Following (P,v)) = i1 by A8, A10, A9, SCMFSA_2:70; hence x in NIC ((a =0_goto i1),il) by A5, A8, A10, A9; ::_thesis: verum end; supposeA11: x = succ il ; ::_thesis: x in NIC ((a =0_goto i1),il) reconsider v = u +* (a .--> 1) as Element of product (the_Values_of SCM+FSA) by CARD_3:107; A12: IC in dom the State of SCM+FSA by MEMSTR_0:2; A13: dom (a .--> 1) = {a} by FUNCOP_1:13; then not IC in dom (a .--> 1) by A3, TARSKI:def_1; then A14: IC v = IC u by FUNCT_4:11 .= n by A12, FUNCT_7:31 ; A15: P /. il = P . il by PBOOLE:143; il in NAT ; then il in dom the Instruction-Sequence of SCM+FSA by PARTFUN1:def_2; then A16: P . il = a =0_goto i1 by FUNCT_7:31; a in dom (a .--> 1) by A13, TARSKI:def_1; then v . a = (a .--> 1) . a by FUNCT_4:13 .= 1 by FUNCOP_1:72 ; then IC (Following (P,v)) = succ il by A14, A16, A15, SCMFSA_2:70; hence x in NIC ((a =0_goto i1),il) by A11, A14, A16, A15; ::_thesis: verum end; end; end; theorem Th36: :: SCMFSA10:36 for i1 being Element of NAT for a being Int-Location holds JUMP (a =0_goto i1) = {i1} proof let i1 be Element of NAT ; ::_thesis: for a being Int-Location holds JUMP (a =0_goto i1) = {i1} let a be Int-Location; ::_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 consider il being Element of NAT such that A2: Y = NIC ((a =0_goto i1),il) ; NIC ((a =0_goto i1),il) = {i1,(succ il)} by Th35; hence i1 in Y by A2, TARSKI:def_2; ::_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 } ) set il1 = 1; set il2 = 2; assume A3: x in meet { (NIC ((a =0_goto i1),il)) where il is Element of NAT : verum } ; ::_thesis: x in {i1} A4: NIC ((a =0_goto i1),2) = {i1,(succ 2)} by Th35; NIC ((a =0_goto i1),2) in { (NIC ((a =0_goto i1),il)) where il is Element of NAT : verum } ; then x in NIC ((a =0_goto i1),2) by A3, SETFAM_1:def_1; then A5: ( x = i1 or x = succ 2 ) by A4, TARSKI:def_2; A6: NIC ((a =0_goto i1),1) = {i1,(succ 1)} by Th35; NIC ((a =0_goto i1),1) in { (NIC ((a =0_goto i1),il)) where il is Element of NAT : verum } ; then x in NIC ((a =0_goto i1),1) by A3, SETFAM_1:def_1; then ( x = i1 or x = succ 1 ) by A6, TARSKI:def_2; hence x in {i1} by A5, 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 A7: 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 A7, A1, SETFAM_1:def_1; ::_thesis: verum end; hence JUMP (a =0_goto i1) = {i1} by TARSKI:1; ::_thesis: verum end; registration let a be Int-Location; 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 Th36; hence JUMP (a =0_goto i1) is 1 -element ; ::_thesis: verum end; end; theorem Th37: :: SCMFSA10:37 for i1, il being Element of NAT for a being Int-Location holds NIC ((a >0_goto i1),il) = {i1,(succ il)} proof let i1, il be Element of NAT ; ::_thesis: for a being Int-Location holds NIC ((a >0_goto i1),il) = {i1,(succ il)} let a be Int-Location; ::_thesis: NIC ((a >0_goto i1),il) = {i1,(succ il)} set t = the State of SCM+FSA; set Q = the Instruction-Sequence of SCM+FSA; hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: {i1,(succ il)} c= NIC ((a >0_goto i1),il) let x be set ; ::_thesis: ( x in NIC ((a >0_goto i1),il) implies b1 in {i1,(succ il)} ) assume x in NIC ((a >0_goto i1),il) ; ::_thesis: b1 in {i1,(succ il)} then consider s being Element of product (the_Values_of SCM+FSA) such that A1: x = IC (Exec ((a >0_goto i1),s)) and A2: IC s = il ; percases ( s . a > 0 or s . a <= 0 ) ; suppose s . a > 0 ; ::_thesis: b1 in {i1,(succ il)} then x = i1 by A1, SCMFSA_2:71; hence x in {i1,(succ il)} by TARSKI:def_2; ::_thesis: verum end; suppose s . a <= 0 ; ::_thesis: b1 in {i1,(succ il)} then x = succ il by A1, A2, SCMFSA_2:71; hence x in {i1,(succ il)} by TARSKI:def_2; ::_thesis: verum end; end; end; 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) ) set I = a >0_goto i1; A3: IC <> a by SCMFSA_2:56; reconsider il1 = il as Element of Values (IC ) by MEMSTR_0:def_6; reconsider n = il as Element of NAT ; reconsider u = the State of SCM+FSA +* ((IC ),il1) as Element of product (the_Values_of SCM+FSA) by CARD_3:107; reconsider P = the Instruction-Sequence of SCM+FSA +* (il,(a >0_goto i1)) as Instruction-Sequence of SCM+FSA ; 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 v = u +* (a .--> 1) as Element of product (the_Values_of SCM+FSA) by CARD_3:107; A6: IC in dom the State of SCM+FSA by MEMSTR_0:2; A7: dom (a .--> 1) = {a} by FUNCOP_1:13; then not IC in dom (a .--> 1) by A3, TARSKI:def_1; then A8: IC v = IC u by FUNCT_4:11 .= n by A6, FUNCT_7:31 ; A9: P /. il = P . il by PBOOLE:143; il in NAT ; then il in dom the Instruction-Sequence of SCM+FSA by PARTFUN1:def_2; then A10: P . il = a >0_goto i1 by FUNCT_7:31; a in dom (a .--> 1) by A7, TARSKI:def_1; then v . a = (a .--> 1) . a by FUNCT_4:13 .= 1 by FUNCOP_1:72 ; then IC (Following (P,v)) = i1 by A8, A10, A9, SCMFSA_2:71; hence x in NIC ((a >0_goto i1),il) by A5, A8, A10, A9; ::_thesis: verum end; supposeA11: x = succ il ; ::_thesis: x in NIC ((a >0_goto i1),il) reconsider v = u +* (a .--> 0) as Element of product (the_Values_of SCM+FSA) by CARD_3:107; A12: IC in dom the State of SCM+FSA by MEMSTR_0:2; A13: dom (a .--> 0) = {a} by FUNCOP_1:13; then not IC in dom (a .--> 0) by A3, TARSKI:def_1; then A14: IC v = IC u by FUNCT_4:11 .= n by A12, FUNCT_7:31 ; A15: P /. il = P . il by PBOOLE:143; il in NAT ; then il in dom the Instruction-Sequence of SCM+FSA by PARTFUN1:def_2; then A16: P . il = a >0_goto i1 by FUNCT_7:31; a in dom (a .--> 0) by A13, TARSKI:def_1; then v . a = (a .--> 0) . a by FUNCT_4:13 .= 0 by FUNCOP_1:72 ; then IC (Following (P,v)) = succ il by A14, A16, A15, SCMFSA_2:71; hence x in NIC ((a >0_goto i1),il) by A11, A14, A16, A15; ::_thesis: verum end; end; end; theorem Th38: :: SCMFSA10:38 for i1 being Element of NAT for a being Int-Location holds JUMP (a >0_goto i1) = {i1} proof let i1 be Element of NAT ; ::_thesis: for a being Int-Location holds JUMP (a >0_goto i1) = {i1} let a be Int-Location; ::_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 consider il being Element of NAT such that A2: Y = NIC ((a >0_goto i1),il) ; NIC ((a >0_goto i1),il) = {i1,(succ il)} by Th37; hence i1 in Y by A2, TARSKI:def_2; ::_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 } ) set il1 = 1; set il2 = 2; assume A3: x in meet { (NIC ((a >0_goto i1),il)) where il is Element of NAT : verum } ; ::_thesis: x in {i1} A4: NIC ((a >0_goto i1),2) = {i1,(succ 2)} by Th37; NIC ((a >0_goto i1),2) in { (NIC ((a >0_goto i1),il)) where il is Element of NAT : verum } ; then x in NIC ((a >0_goto i1),2) by A3, SETFAM_1:def_1; then A5: ( x = i1 or x = succ 2 ) by A4, TARSKI:def_2; A6: NIC ((a >0_goto i1),1) = {i1,(succ 1)} by Th37; NIC ((a >0_goto i1),1) in { (NIC ((a >0_goto i1),il)) where il is Element of NAT : verum } ; then x in NIC ((a >0_goto i1),1) by A3, SETFAM_1:def_1; then ( x = i1 or x = succ 1 ) by A6, TARSKI:def_2; hence x in {i1} by A5, 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 A7: 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 A7, A1, SETFAM_1:def_1; ::_thesis: verum end; hence JUMP (a >0_goto i1) = {i1} by TARSKI:1; ::_thesis: verum end; registration let a be Int-Location; 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 Th38; hence JUMP (a >0_goto i1) is 1 -element ; ::_thesis: verum end; end; registration let a, b be Int-Location; let f be FinSeq-Location ; clustera := (f,b) -> sequential ; coherence a := (f,b) is sequential proof let s be State of SCM+FSA; :: according to AMISTD_1:def_8 ::_thesis: (Exec ((a := (f,b)),s)) . (IC ) = succ (IC s) thus (Exec ((a := (f,b)),s)) . (IC ) = succ (IC s) by SCMFSA_2:72; ::_thesis: verum end; end; registration let a, b be Int-Location; let f be FinSeq-Location ; cluster JUMP (a := (f,b)) -> empty ; coherence JUMP (a := (f,b)) is empty proof for l being Element of NAT holds NIC ((a := (f,b)),l) = {(succ l)} by AMISTD_1:12; hence JUMP (a := (f,b)) is empty by Lm1; ::_thesis: verum end; end; registration let a, b be Int-Location; let f be FinSeq-Location ; cluster(f,b) := a -> sequential ; coherence (f,b) := a is sequential proof let s be State of SCM+FSA; :: according to AMISTD_1:def_8 ::_thesis: (Exec (((f,b) := a),s)) . (IC ) = succ (IC s) thus (Exec (((f,b) := a),s)) . (IC ) = succ (IC s) by SCMFSA_2:73; ::_thesis: verum end; end; registration let a, b be Int-Location; let f be FinSeq-Location ; cluster JUMP ((f,b) := a) -> empty ; coherence JUMP ((f,b) := a) is empty proof for l being Element of NAT holds NIC (((f,b) := a),l) = {(succ l)} by AMISTD_1:12; hence JUMP ((f,b) := a) is empty by Lm1; ::_thesis: verum end; end; registration let a be Int-Location; let f be FinSeq-Location ; clustera :=len f -> sequential ; coherence a :=len f is sequential proof let s be State of SCM+FSA; :: according to AMISTD_1:def_8 ::_thesis: (Exec ((a :=len f),s)) . (IC ) = succ (IC s) thus (Exec ((a :=len f),s)) . (IC ) = succ (IC s) by SCMFSA_2:74; ::_thesis: verum end; end; registration let a be Int-Location; let f be FinSeq-Location ; cluster JUMP (a :=len f) -> empty ; coherence JUMP (a :=len f) is empty proof for l being Element of NAT holds NIC ((a :=len f),l) = {(succ l)} by AMISTD_1:12; hence JUMP (a :=len f) is empty by Lm1; ::_thesis: verum end; end; registration let a be Int-Location; let f be FinSeq-Location ; clusterf :=<0,...,0> a -> sequential ; coherence f :=<0,...,0> a is sequential proof let s be State of SCM+FSA; :: according to AMISTD_1:def_8 ::_thesis: (Exec ((f :=<0,...,0> a),s)) . (IC ) = succ (IC s) thus (Exec ((f :=<0,...,0> a),s)) . (IC ) = succ (IC s) by SCMFSA_2:75; ::_thesis: verum end; end; registration let a be Int-Location; let f be FinSeq-Location ; cluster JUMP (f :=<0,...,0> a) -> empty ; coherence JUMP (f :=<0,...,0> a) is empty proof for l being Element of NAT holds NIC ((f :=<0,...,0> a),l) = {(succ l)} by AMISTD_1:12; hence JUMP (f :=<0,...,0> a) is empty by Lm1; ::_thesis: verum end; end; theorem Th39: :: SCMFSA10:39 for il being Element of NAT holds SUCC (il,SCM+FSA) = {il,(succ il)} proof let il be Element of NAT ; ::_thesis: SUCC (il,SCM+FSA) = {il,(succ il)} set X = { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of SCM+FSA : 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+FSA_:_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+FSA_:_verum__}__)_) let x be set ; ::_thesis: ( ( x in union { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of SCM+FSA : 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+FSA : 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+FSA : verum } ) assume x in union { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of SCM+FSA : 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+FSA : verum } by TARSKI:def_4; consider i being Element of the InstructionsF of SCM+FSA such that A3: Y = (NIC (i,il)) \ (JUMP i) by A2; percases ( i = [0,{},{}] or ex a, b being Int-Location st i = a := b or ex a, b being Int-Location st i = AddTo (a,b) or ex a, b being Int-Location st i = SubFrom (a,b) or ex a, b being Int-Location st i = MultBy (a,b) or ex a, b being Int-Location st i = Divide (a,b) or ex i1 being Element of NAT st i = goto i1 or ex i1 being Element of NAT ex a being Int-Location st i = a =0_goto i1 or ex i1 being Element of NAT ex a being Int-Location st i = a >0_goto i1 or ex a, b being Int-Location ex f being FinSeq-Location st i = b := (f,a) or ex a, b being Int-Location ex f being FinSeq-Location st i = (f,a) := b or ex a being Int-Location ex f being FinSeq-Location st i = a :=len f or ex a being Int-Location ex f being FinSeq-Location st i = f :=<0,...,0> a ) by SCMFSA_2:93; suppose i = [0,{},{}] ; ::_thesis: x in {il,(succ il)} then x in {il} \ (JUMP (halt SCM+FSA)) 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 Int-Location st i = a := b ; ::_thesis: x in {il,(succ il)} then consider a, b being Int-Location 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 Int-Location st i = AddTo (a,b) ; ::_thesis: x in {il,(succ il)} then consider a, b being Int-Location 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 Int-Location st i = SubFrom (a,b) ; ::_thesis: x in {il,(succ il)} then consider a, b being Int-Location 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 Int-Location st i = MultBy (a,b) ; ::_thesis: x in {il,(succ il)} then consider a, b being Int-Location 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 a, b being Int-Location st i = Divide (a,b) ; ::_thesis: x in {il,(succ il)} then consider a, b being Int-Location such that A8: i = Divide (a,b) ; x in {(succ il)} \ (JUMP (Divide (a,b))) by A1, A3, A8, 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 ; ::_thesis: x in {il,(succ il)} then consider i1 being Element of NAT such that A9: i = goto i1 ; x in {i1} \ (JUMP i) by A1, A3, A9, Th33; then x in {i1} \ {i1} by A9, Th34; hence x in {il,(succ il)} by XBOOLE_1:37; ::_thesis: verum end; suppose ex i1 being Element of NAT ex a being Int-Location st i = a =0_goto i1 ; ::_thesis: x in {il,(succ il)} then consider i1 being Element of NAT , a being Int-Location such that A10: i = a =0_goto i1 ; A11: NIC (i,il) = {i1,(succ il)} by A10, Th35; x in NIC (i,il) by A1, A3, XBOOLE_0:def_5; then A12: ( x = i1 or x = succ il ) by A11, TARSKI:def_2; x in (NIC (i,il)) \ {i1} by A1, A3, A10, Th36; then not x in {i1} by XBOOLE_0:def_5; hence x in {il,(succ il)} by A12, TARSKI:def_1, TARSKI:def_2; ::_thesis: verum end; suppose ex i1 being Element of NAT ex a being Int-Location st i = a >0_goto i1 ; ::_thesis: x in {il,(succ il)} then consider i1 being Element of NAT , a being Int-Location such that A13: i = a >0_goto i1 ; A14: NIC (i,il) = {i1,(succ il)} by A13, Th37; x in NIC (i,il) by A1, A3, XBOOLE_0:def_5; then A15: ( x = i1 or x = succ il ) by A14, TARSKI:def_2; x in (NIC (i,il)) \ {i1} by A1, A3, A13, Th38; then not x in {i1} by XBOOLE_0:def_5; hence x in {il,(succ il)} by A15, TARSKI:def_1, TARSKI:def_2; ::_thesis: verum end; suppose ex a, b being Int-Location ex f being FinSeq-Location st i = b := (f,a) ; ::_thesis: x in {il,(succ il)} then consider a, b being Int-Location, f being FinSeq-Location such that A16: i = b := (f,a) ; x in {(succ il)} \ (JUMP (b := (f,a))) by A1, A3, A16, 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 Int-Location ex f being FinSeq-Location st i = (f,a) := b ; ::_thesis: x in {il,(succ il)} then consider a, b being Int-Location, f being FinSeq-Location such that A17: i = (f,a) := b ; x in {(succ il)} \ (JUMP ((f,a) := b)) by A1, A3, A17, 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 being Int-Location ex f being FinSeq-Location st i = a :=len f ; ::_thesis: x in {il,(succ il)} then consider a being Int-Location, f being FinSeq-Location such that A18: i = a :=len f ; x in {(succ il)} \ (JUMP (a :=len f)) by A1, A3, A18, 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 being Int-Location ex f being FinSeq-Location st i = f :=<0,...,0> a ; ::_thesis: x in {il,(succ il)} then consider a being Int-Location, f being FinSeq-Location such that A19: i = f :=<0,...,0> a ; x in {(succ il)} \ (JUMP (f :=<0,...,0> a)) by A1, A3, A19, 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 A20: x in {il,(succ il)} ; ::_thesis: b1 in union { ((NIC (b2,il)) \ (JUMP b2)) where I is Element of the InstructionsF of SCM+FSA : verum } percases ( x = il or x = succ il ) by A20, TARSKI:def_2; supposeA21: x = il ; ::_thesis: b1 in union { ((NIC (b2,il)) \ (JUMP b2)) where I is Element of the InstructionsF of SCM+FSA : verum } set i = halt SCM+FSA; (NIC ((halt SCM+FSA),il)) \ (JUMP (halt SCM+FSA)) = {il} by AMISTD_1:2; then A22: {il} in { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of SCM+FSA : verum } ; x in {il} by A21, TARSKI:def_1; hence x in union { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of SCM+FSA : verum } by A22, TARSKI:def_4; ::_thesis: verum end; supposeA23: x = succ il ; ::_thesis: b1 in union { ((NIC (b2,il)) \ (JUMP b2)) where I is Element of the InstructionsF of SCM+FSA : verum } set a = the Int-Location; set i = AddTo ( the Int-Location, the Int-Location); (NIC ((AddTo ( the Int-Location, the Int-Location)),il)) \ (JUMP (AddTo ( the Int-Location, the Int-Location))) = {(succ il)} by AMISTD_1:12; then A24: {(succ il)} in { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of SCM+FSA : verum } ; x in {(succ il)} by A23, TARSKI:def_1; hence x in union { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of SCM+FSA : verum } by A24, TARSKI:def_4; ::_thesis: verum end; end; end; hence SUCC (il,SCM+FSA) = {il,(succ il)} by TARSKI:1; ::_thesis: verum end; theorem Th40: :: SCMFSA10:40 for k being Element of NAT holds ( k + 1 in SUCC (k,SCM+FSA) & ( for j being Element of NAT st j in SUCC (k,SCM+FSA) holds k <= j ) ) proof let k be Element of NAT ; ::_thesis: ( k + 1 in SUCC (k,SCM+FSA) & ( for j being Element of NAT st j in SUCC (k,SCM+FSA) holds k <= j ) ) A1: SUCC (k,SCM+FSA) = {k,(succ k)} by Th39; hence k + 1 in SUCC (k,SCM+FSA) by TARSKI:def_2; ::_thesis: for j being Element of NAT st j in SUCC (k,SCM+FSA) holds k <= j let j be Element of NAT ; ::_thesis: ( j in SUCC (k,SCM+FSA) implies k <= j ) assume A2: j in SUCC (k,SCM+FSA) ; ::_thesis: k <= j percases ( j = k or j = succ k ) by A1, A2, TARSKI:def_2; suppose j = k ; ::_thesis: k <= j hence k <= j ; ::_thesis: verum end; suppose j = succ k ; ::_thesis: k <= j hence k <= j by NAT_1:11; ::_thesis: verum end; end; end; registration cluster SCM+FSA -> standard ; coherence SCM+FSA is standard by Th40, AMISTD_1:3; end; registration cluster(halt SCM+FSA) `1_3 -> jump-only for InsType of the InstructionsF of SCM+FSA; coherence for b1 being InsType of the InstructionsF of SCM+FSA st b1 = InsCode (halt SCM+FSA) holds b1 is jump-only proof now__::_thesis:_for_s_being_State_of_SCM+FSA for_o_being_Object_of_SCM+FSA for_I_being_Instruction_of_SCM+FSA_st_InsCode_I_=_InsCode_(halt_SCM+FSA)_&_o_in_Data-Locations_holds_ (Exec_(I,s))_._o_=_s_._o let s be State of SCM+FSA; ::_thesis: for o being Object of SCM+FSA for I being Instruction of SCM+FSA st InsCode I = InsCode (halt SCM+FSA) & o in Data-Locations holds (Exec (I,s)) . o = s . o let o be Object of SCM+FSA; ::_thesis: for I being Instruction of SCM+FSA st InsCode I = InsCode (halt SCM+FSA) & o in Data-Locations holds (Exec (I,s)) . o = s . o let I be Instruction of SCM+FSA; ::_thesis: ( InsCode I = InsCode (halt SCM+FSA) & o in Data-Locations implies (Exec (I,s)) . o = s . o ) assume that A1: InsCode I = InsCode (halt SCM+FSA) and o in Data-Locations ; ::_thesis: (Exec (I,s)) . o = s . o I = halt SCM+FSA by A1, SCMFSA_2:95, 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+FSA st b1 = InsCode (halt SCM+FSA) holds b1 is jump-only by AMISTD_1:def_1; ::_thesis: verum end; end; registration cluster halt SCM+FSA -> jump-only ; coherence halt SCM+FSA is jump-only proof thus InsCode (halt SCM+FSA) is jump-only ; :: according to AMISTD_1:def_2 ::_thesis: verum end; end; registration let i1 be Element of NAT ; cluster(goto i1) `1_3 -> jump-only for InsType of the InstructionsF of SCM+FSA; coherence for b1 being InsType of the InstructionsF of SCM+FSA st b1 = InsCode (goto i1) holds b1 is jump-only proof set S = SCM+FSA ; now__::_thesis:_for_s_being_State_of_SCM+FSA for_o_being_Object_of_SCM+FSA for_I_being_Instruction_of_SCM+FSA_st_InsCode_I_=_InsCode_(goto_i1)_&_o_in_Data-Locations_holds_ (Exec_(I,s))_._o_=_s_._o let s be State of SCM+FSA; ::_thesis: for o being Object of SCM+FSA for I being Instruction of SCM+FSA st InsCode I = InsCode (goto i1) & o in Data-Locations holds (Exec (b5,b3)) . b4 = b3 . b4 let o be Object of SCM+FSA; ::_thesis: for I being Instruction of SCM+FSA st InsCode I = InsCode (goto i1) & o in Data-Locations holds (Exec (b4,b2)) . b3 = b2 . b3 let I be Instruction of SCM+FSA; ::_thesis: ( InsCode I = InsCode (goto i1) & o in Data-Locations implies (Exec (b3,b1)) . b2 = b1 . b2 ) assume that A1: InsCode I = InsCode (goto i1) and A2: o in Data-Locations ; ::_thesis: (Exec (b3,b1)) . b2 = b1 . b2 A3: ex i2 being Element of NAT st I = goto i2 by A1, SCMFSA_2:23, SCMFSA_2:35; percases ( o is Int-Location or o is FinSeq-Location ) by A2, Th1; suppose o is Int-Location ; ::_thesis: (Exec (b3,b1)) . b2 = b1 . b2 hence (Exec (I,s)) . o = s . o by A3, SCMFSA_2:69; ::_thesis: verum end; suppose o is FinSeq-Location ; ::_thesis: (Exec (b3,b1)) . b2 = b1 . b2 hence (Exec (I,s)) . o = s . o by A3, SCMFSA_2:69; ::_thesis: verum end; end; end; hence for b1 being InsType of the InstructionsF of SCM+FSA st b1 = InsCode (goto i1) holds b1 is jump-only by AMISTD_1:def_1; ::_thesis: verum end; end; registration let i1 be Element of NAT ; cluster goto i1 -> non ins-loc-free jump-only non sequential ; coherence ( goto i1 is jump-only & not goto i1 is sequential & not goto i1 is ins-loc-free ) proof thus InsCode (goto i1) is jump-only ; :: according to AMISTD_1:def_2 ::_thesis: ( not goto i1 is sequential & not goto i1 is ins-loc-free ) JUMP (goto i1) <> {} ; hence not goto i1 is sequential by AMISTD_1:13; ::_thesis: not goto i1 is ins-loc-free dom (JumpPart (goto i1)) = dom <*i1*> by RECDEF_2:def_2 .= {1} by FINSEQ_1:2, FINSEQ_1:def_8 ; hence not JumpPart (goto i1) is empty ; :: according to COMPOS_0:def_8 ::_thesis: verum end; end; registration let a be Int-Location; let i1 be Element of NAT ; cluster(a =0_goto i1) `1_3 -> jump-only for InsType of the InstructionsF of SCM+FSA; coherence for b1 being InsType of the InstructionsF of SCM+FSA st b1 = InsCode (a =0_goto i1) holds b1 is jump-only proof set S = SCM+FSA ; now__::_thesis:_for_s_being_State_of_SCM+FSA for_o_being_Object_of_SCM+FSA for_I_being_Instruction_of_SCM+FSA_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+FSA; ::_thesis: for o being Object of SCM+FSA for I being Instruction of SCM+FSA st InsCode I = InsCode (a =0_goto i1) & o in Data-Locations holds (Exec (b5,b3)) . b4 = b3 . b4 let o be Object of SCM+FSA; ::_thesis: for I being Instruction of SCM+FSA st InsCode I = InsCode (a =0_goto i1) & o in Data-Locations holds (Exec (b4,b2)) . b3 = b2 . b3 let I be Instruction of SCM+FSA; ::_thesis: ( InsCode I = InsCode (a =0_goto i1) & o in Data-Locations implies (Exec (b3,b1)) . b2 = b1 . b2 ) assume that A1: InsCode I = InsCode (a =0_goto i1) and A2: o in Data-Locations ; ::_thesis: (Exec (b3,b1)) . b2 = b1 . b2 A3: ex i2 being Element of NAT ex b being Int-Location st I = b =0_goto i2 by A1, SCMFSA_2:24, SCMFSA_2:36; percases ( o is Int-Location or o is FinSeq-Location ) by A2, Th1; suppose o is Int-Location ; ::_thesis: (Exec (b3,b1)) . b2 = b1 . b2 hence (Exec (I,s)) . o = s . o by A3, SCMFSA_2:70; ::_thesis: verum end; suppose o is FinSeq-Location ; ::_thesis: (Exec (b3,b1)) . b2 = b1 . b2 hence (Exec (I,s)) . o = s . o by A3, SCMFSA_2:70; ::_thesis: verum end; end; end; hence for b1 being InsType of the InstructionsF of SCM+FSA st b1 = InsCode (a =0_goto i1) holds b1 is jump-only by AMISTD_1:def_1; ::_thesis: verum end; cluster(a >0_goto i1) `1_3 -> jump-only for InsType of the InstructionsF of SCM+FSA; coherence for b1 being InsType of the InstructionsF of SCM+FSA st b1 = InsCode (a >0_goto i1) holds b1 is jump-only proof set S = SCM+FSA ; now__::_thesis:_for_s_being_State_of_SCM+FSA for_o_being_Object_of_SCM+FSA for_I_being_Instruction_of_SCM+FSA_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+FSA; ::_thesis: for o being Object of SCM+FSA for I being Instruction of SCM+FSA st InsCode I = InsCode (a >0_goto i1) & o in Data-Locations holds (Exec (b5,b3)) . b4 = b3 . b4 let o be Object of SCM+FSA; ::_thesis: for I being Instruction of SCM+FSA st InsCode I = InsCode (a >0_goto i1) & o in Data-Locations holds (Exec (b4,b2)) . b3 = b2 . b3 let I be Instruction of SCM+FSA; ::_thesis: ( InsCode I = InsCode (a >0_goto i1) & o in Data-Locations implies (Exec (b3,b1)) . b2 = b1 . b2 ) assume that A4: InsCode I = InsCode (a >0_goto i1) and A5: o in Data-Locations ; ::_thesis: (Exec (b3,b1)) . b2 = b1 . b2 A6: ex i2 being Element of NAT ex b being Int-Location st I = b >0_goto i2 by A4, SCMFSA_2:25, SCMFSA_2:37; percases ( o is Int-Location or o is FinSeq-Location ) by A5, Th1; suppose o is Int-Location ; ::_thesis: (Exec (b3,b1)) . b2 = b1 . b2 hence (Exec (I,s)) . o = s . o by A6, SCMFSA_2:71; ::_thesis: verum end; suppose o is FinSeq-Location ; ::_thesis: (Exec (b3,b1)) . b2 = b1 . b2 hence (Exec (I,s)) . o = s . o by A6, SCMFSA_2:71; ::_thesis: verum end; end; end; hence for b1 being InsType of the InstructionsF of SCM+FSA st b1 = InsCode (a >0_goto i1) holds b1 is jump-only by AMISTD_1:def_1; ::_thesis: verum end; end; registration let a be Int-Location; let i1 be Element of NAT ; clustera =0_goto i1 -> non ins-loc-free jump-only non sequential ; coherence ( a =0_goto i1 is jump-only & not a =0_goto i1 is sequential & not a =0_goto i1 is ins-loc-free ) proof thus InsCode (a =0_goto i1) is jump-only ; :: according to AMISTD_1:def_2 ::_thesis: ( not a =0_goto i1 is sequential & not a =0_goto i1 is ins-loc-free ) JUMP (a =0_goto i1) <> {} ; hence not a =0_goto i1 is sequential by AMISTD_1:13; ::_thesis: not a =0_goto i1 is ins-loc-free dom (JumpPart (a =0_goto i1)) = dom <*i1*> by Th15 .= {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; clustera >0_goto i1 -> non ins-loc-free jump-only non sequential ; coherence ( a >0_goto i1 is jump-only & not a >0_goto i1 is sequential & not a >0_goto i1 is ins-loc-free ) proof thus InsCode (a >0_goto i1) is jump-only ; :: according to AMISTD_1:def_2 ::_thesis: ( not a >0_goto i1 is sequential & not a >0_goto i1 is ins-loc-free ) JUMP (a >0_goto i1) <> {} ; hence not a >0_goto i1 is sequential by AMISTD_1:13; ::_thesis: not a >0_goto i1 is ins-loc-free dom (JumpPart (a >0_goto i1)) = dom <*i1*> by Th16 .= {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; Lm2: intloc 0 <> intloc 1 by AMI_3:10; registration let a, b be Int-Location; cluster(a := b) `1_3 -> non jump-only for InsType of the InstructionsF of SCM+FSA; coherence for b1 being InsType of the InstructionsF of SCM+FSA st b1 = InsCode (a := b) holds not b1 is jump-only proof set w = the State of SCM+FSA; set t = the State of SCM+FSA +* (((intloc 0),(intloc 1)) --> (0,1)); A1: InsCode (a := b) = 1 by SCMFSA_2:18 .= InsCode ((intloc 0) := (intloc 1)) by SCMFSA_2:18 ; A2: dom (((intloc 0),(intloc 1)) --> (0,1)) = {(intloc 0),(intloc 1)} by FUNCT_4:62; then A3: intloc 1 in dom (((intloc 0),(intloc 1)) --> (0,1)) by TARSKI:def_2; intloc 0 in dom (((intloc 0),(intloc 1)) --> (0,1)) by A2, TARSKI:def_2; then A4: ( the State of SCM+FSA +* (((intloc 0),(intloc 1)) --> (0,1))) . (intloc 0) = (((intloc 0),(intloc 1)) --> (0,1)) . (intloc 0) by FUNCT_4:13 .= 0 by AMI_3:10, FUNCT_4:63 ; intloc 0 in Int-Locations by AMI_2:def_16; then A5: intloc 0 in Data-Locations by SCMFSA_2:100, XBOOLE_0:def_3; (Exec (((intloc 0) := (intloc 1)),( the State of SCM+FSA +* (((intloc 0),(intloc 1)) --> (0,1))))) . (intloc 0) = ( the State of SCM+FSA +* (((intloc 0),(intloc 1)) --> (0,1))) . (intloc 1) by SCMFSA_2:63 .= (((intloc 0),(intloc 1)) --> (0,1)) . (intloc 1) by A3, FUNCT_4:13 .= 1 by FUNCT_4:63 ; hence for b1 being InsType of the InstructionsF of SCM+FSA st b1 = InsCode (a := b) holds not b1 is jump-only by A1, A4, A5, AMISTD_1:def_1; ::_thesis: verum end; cluster(AddTo (a,b)) `1_3 -> non jump-only for InsType of the InstructionsF of SCM+FSA; coherence for b1 being InsType of the InstructionsF of SCM+FSA st b1 = InsCode (AddTo (a,b)) holds not b1 is jump-only proof set w = the State of SCM+FSA; set t = the State of SCM+FSA +* (((intloc 0),(intloc 1)) --> (0,1)); A6: InsCode (AddTo (a,b)) = 2 by SCMFSA_2:19 .= InsCode (AddTo ((intloc 0),(intloc 1))) by SCMFSA_2:19 ; A7: dom (((intloc 0),(intloc 1)) --> (0,1)) = {(intloc 0),(intloc 1)} by FUNCT_4:62; then intloc 0 in dom (((intloc 0),(intloc 1)) --> (0,1)) by TARSKI:def_2; then A8: ( the State of SCM+FSA +* (((intloc 0),(intloc 1)) --> (0,1))) . (intloc 0) = (((intloc 0),(intloc 1)) --> (0,1)) . (intloc 0) by FUNCT_4:13 .= 0 by AMI_3:10, FUNCT_4:63 ; intloc 0 in Int-Locations by AMI_2:def_16; then A9: intloc 0 in Data-Locations by SCMFSA_2:100, XBOOLE_0:def_3; intloc 1 in dom (((intloc 0),(intloc 1)) --> (0,1)) by A7, TARSKI:def_2; then ( the State of SCM+FSA +* (((intloc 0),(intloc 1)) --> (0,1))) . (intloc 1) = (((intloc 0),(intloc 1)) --> (0,1)) . (intloc 1) by FUNCT_4:13 .= 1 by FUNCT_4:63 ; then (Exec ((AddTo ((intloc 0),(intloc 1))),( the State of SCM+FSA +* (((intloc 0),(intloc 1)) --> (0,1))))) . (intloc 0) = 0 + 1 by A8, SCMFSA_2:64; hence for b1 being InsType of the InstructionsF of SCM+FSA st b1 = InsCode (AddTo (a,b)) holds not b1 is jump-only by A6, A8, A9, AMISTD_1:def_1; ::_thesis: verum end; cluster(SubFrom (a,b)) `1_3 -> non jump-only for InsType of the InstructionsF of SCM+FSA; coherence for b1 being InsType of the InstructionsF of SCM+FSA st b1 = InsCode (SubFrom (a,b)) holds not b1 is jump-only proof set w = the State of SCM+FSA; set t = the State of SCM+FSA +* (((intloc 0),(intloc 1)) --> (0,1)); A10: InsCode (SubFrom (a,b)) = 3 by SCMFSA_2:20 .= InsCode (SubFrom ((intloc 0),(intloc 1))) by SCMFSA_2:20 ; A11: dom (((intloc 0),(intloc 1)) --> (0,1)) = {(intloc 0),(intloc 1)} by FUNCT_4:62; then intloc 0 in dom (((intloc 0),(intloc 1)) --> (0,1)) by TARSKI:def_2; then A12: ( the State of SCM+FSA +* (((intloc 0),(intloc 1)) --> (0,1))) . (intloc 0) = (((intloc 0),(intloc 1)) --> (0,1)) . (intloc 0) by FUNCT_4:13 .= 0 by AMI_3:10, FUNCT_4:63 ; intloc 0 in Int-Locations by AMI_2:def_16; then A13: intloc 0 in Data-Locations by SCMFSA_2:100, XBOOLE_0:def_3; intloc 1 in dom (((intloc 0),(intloc 1)) --> (0,1)) by A11, TARSKI:def_2; then A14: ( the State of SCM+FSA +* (((intloc 0),(intloc 1)) --> (0,1))) . (intloc 1) = (((intloc 0),(intloc 1)) --> (0,1)) . (intloc 1) by FUNCT_4:13 .= 1 by FUNCT_4:63 ; (Exec ((SubFrom ((intloc 0),(intloc 1))),( the State of SCM+FSA +* (((intloc 0),(intloc 1)) --> (0,1))))) . (intloc 0) = (( the State of SCM+FSA +* (((intloc 0),(intloc 1)) --> (0,1))) . (intloc 0)) - (( the State of SCM+FSA +* (((intloc 0),(intloc 1)) --> (0,1))) . (intloc 1)) by SCMFSA_2:65 .= - 1 by A12, A14 ; hence for b1 being InsType of the InstructionsF of SCM+FSA st b1 = InsCode (SubFrom (a,b)) holds not b1 is jump-only by A10, A12, A13, AMISTD_1:def_1; ::_thesis: verum end; cluster(MultBy (a,b)) `1_3 -> non jump-only for InsType of the InstructionsF of SCM+FSA; coherence for b1 being InsType of the InstructionsF of SCM+FSA st b1 = InsCode (MultBy (a,b)) holds not b1 is jump-only proof set w = the State of SCM+FSA; set t = the State of SCM+FSA +* (((intloc 0),(intloc 1)) --> (1,0)); A15: InsCode (MultBy (a,b)) = 4 by SCMFSA_2:21 .= InsCode (MultBy ((intloc 0),(intloc 1))) by SCMFSA_2:21 ; A16: dom (((intloc 0),(intloc 1)) --> (1,0)) = {(intloc 0),(intloc 1)} by FUNCT_4:62; then intloc 0 in dom (((intloc 0),(intloc 1)) --> (1,0)) by TARSKI:def_2; then A17: ( the State of SCM+FSA +* (((intloc 0),(intloc 1)) --> (1,0))) . (intloc 0) = (((intloc 0),(intloc 1)) --> (1,0)) . (intloc 0) by FUNCT_4:13 .= 1 by AMI_3:10, FUNCT_4:63 ; intloc 0 in Int-Locations by AMI_2:def_16; then A18: intloc 0 in Data-Locations by SCMFSA_2:100, XBOOLE_0:def_3; intloc 1 in dom (((intloc 0),(intloc 1)) --> (1,0)) by A16, TARSKI:def_2; then A19: ( the State of SCM+FSA +* (((intloc 0),(intloc 1)) --> (1,0))) . (intloc 1) = (((intloc 0),(intloc 1)) --> (1,0)) . (intloc 1) by FUNCT_4:13 .= 0 by FUNCT_4:63 ; (Exec ((MultBy ((intloc 0),(intloc 1))),( the State of SCM+FSA +* (((intloc 0),(intloc 1)) --> (1,0))))) . (intloc 0) = (( the State of SCM+FSA +* (((intloc 0),(intloc 1)) --> (1,0))) . (intloc 0)) * (( the State of SCM+FSA +* (((intloc 0),(intloc 1)) --> (1,0))) . (intloc 1)) by SCMFSA_2:66 .= 0 by A19 ; hence for b1 being InsType of the InstructionsF of SCM+FSA st b1 = InsCode (MultBy (a,b)) holds not b1 is jump-only by A15, A17, A18, AMISTD_1:def_1; ::_thesis: verum end; cluster(Divide (a,b)) `1_3 -> non jump-only for InsType of the InstructionsF of SCM+FSA; coherence for b1 being InsType of the InstructionsF of SCM+FSA st b1 = InsCode (Divide (a,b)) holds not b1 is jump-only proof set w = the State of SCM+FSA; set t = the State of SCM+FSA +* (((intloc 0),(intloc 1)) --> (7,3)); A20: InsCode (Divide (a,b)) = 5 by SCMFSA_2:22 .= InsCode (Divide ((intloc 0),(intloc 1))) by SCMFSA_2:22 ; A21: dom (((intloc 0),(intloc 1)) --> (7,3)) = {(intloc 0),(intloc 1)} by FUNCT_4:62; then intloc 0 in dom (((intloc 0),(intloc 1)) --> (7,3)) by TARSKI:def_2; then A22: ( the State of SCM+FSA +* (((intloc 0),(intloc 1)) --> (7,3))) . (intloc 0) = (((intloc 0),(intloc 1)) --> (7,3)) . (intloc 0) by FUNCT_4:13 .= 7 by AMI_3:10, FUNCT_4:63 ; A23: 7 = (2 * 3) + 1 ; intloc 0 in Int-Locations by AMI_2:def_16; then A24: intloc 0 in Data-Locations by SCMFSA_2:100, XBOOLE_0:def_3; intloc 1 in dom (((intloc 0),(intloc 1)) --> (7,3)) by A21, TARSKI:def_2; then ( the State of SCM+FSA +* (((intloc 0),(intloc 1)) --> (7,3))) . (intloc 1) = (((intloc 0),(intloc 1)) --> (7,3)) . (intloc 1) by FUNCT_4:13 .= 3 by FUNCT_4:63 ; then (Exec ((Divide ((intloc 0),(intloc 1))),( the State of SCM+FSA +* (((intloc 0),(intloc 1)) --> (7,3))))) . (intloc 0) = 7 div 3 by A22, Lm2, SCMFSA_2:67 .= 2 by A23, NAT_D:def_1 ; hence for b1 being InsType of the InstructionsF of SCM+FSA st b1 = InsCode (Divide (a,b)) holds not b1 is jump-only by A20, A22, A24, AMISTD_1:def_1; ::_thesis: verum end; end; Lm3: fsloc 0 <> intloc 0 by SCMFSA_2:99; Lm4: fsloc 0 <> intloc 1 by SCMFSA_2:99; registration let a, b be Int-Location; let f be FinSeq-Location ; cluster(b := (f,a)) `1_3 -> non jump-only for InsType of the InstructionsF of SCM+FSA; coherence for b1 being InsType of the InstructionsF of SCM+FSA st b1 = InsCode (b := (f,a)) holds not b1 is jump-only proof Values (intloc 1) = INT by SCMFSA_2:11; then reconsider E = 1 as Element of Values (intloc 1) by INT_1:def_1; Values (intloc 0) = INT by SCMFSA_2:11; then reconsider D = 1 as Element of Values (intloc 0) by INT_1:def_1; reconsider DWA = 2 as Element of INT by INT_1:def_1; set w = the State of SCM+FSA; <*DWA*> in INT * by FINSEQ_1:def_11; then reconsider F = <*2*> as Element of Values (fsloc 0) by SCMFSA_2:12; reconsider t = (( the State of SCM+FSA +* ((fsloc 0) .--> F)) +* ((intloc 0) .--> D)) +* ((intloc 1) .--> E) as State of SCM+FSA ; A1: t . (intloc 0) = D by AMI_3:10, BVFUNC14:12; A2: t . (fsloc 0) = F by Lm3, Lm4, FUNCT_7:114; then dom (t . (fsloc 0)) = {1} by FINSEQ_1:2, FINSEQ_1:def_8; then A3: 1 in dom (t . (fsloc 0)) by TARSKI:def_1; consider k being Element of NAT such that A4: k = abs (t . (intloc 1)) and A5: (Exec (((intloc 0) := ((fsloc 0),(intloc 1))),t)) . (intloc 0) = (t . (fsloc 0)) /. k by SCMFSA_2:72; intloc 0 in Int-Locations by AMI_2:def_16; then A6: intloc 0 in Data-Locations by SCMFSA_2:100, XBOOLE_0:def_3; t . (intloc 1) = E by FUNCT_7:94; then k = 1 by A4, ABSVALUE:def_1; then A7: (Exec (((intloc 0) := ((fsloc 0),(intloc 1))),t)) . (intloc 0) = (t . (fsloc 0)) . 1 by A5, A3, PARTFUN1:def_6 .= 2 by A2, FINSEQ_1:def_8 ; InsCode (b := (f,a)) = 9 by SCMFSA_2:26 .= InsCode ((intloc 0) := ((fsloc 0),(intloc 1))) by SCMFSA_2:26 ; hence for b1 being InsType of the InstructionsF of SCM+FSA st b1 = InsCode (b := (f,a)) holds not b1 is jump-only by A1, A7, A6, AMISTD_1:def_1; ::_thesis: verum end; cluster((f,a) := b) `1_3 -> non jump-only for InsType of the InstructionsF of SCM+FSA; coherence for b1 being InsType of the InstructionsF of SCM+FSA st b1 = InsCode ((f,a) := b) holds not b1 is jump-only proof Values (intloc 0) = INT by SCMFSA_2:11; then reconsider D = 1 as Element of Values (intloc 0) by INT_1:def_1; reconsider DWA = 2 as Element of INT by INT_1:def_1; set w = the State of SCM+FSA; A8: InsCode ((f,a) := b) = 10 by SCMFSA_2:27 .= InsCode (((fsloc 0),(intloc 1)) := (intloc 0)) by SCMFSA_2:27 ; Values (intloc 1) = INT by SCMFSA_2:11; then reconsider E = 1 as Element of Values (intloc 1) by INT_1:def_1; <*DWA*> in INT * by FINSEQ_1:def_11; then reconsider F = <*2*> as Element of Values (fsloc 0) by SCMFSA_2:12; reconsider t = (( the State of SCM+FSA +* ((fsloc 0) .--> F)) +* ((intloc 0) .--> D)) +* ((intloc 1) .--> E) as State of SCM+FSA ; consider k being Element of NAT such that A9: k = abs (t . (intloc 1)) and A10: (Exec ((((fsloc 0),(intloc 1)) := (intloc 0)),t)) . (fsloc 0) = (t . (fsloc 0)) +* (k,(t . (intloc 0))) by SCMFSA_2:73; t . (intloc 1) = E by FUNCT_7:94; then A11: k = 1 by A9, ABSVALUE:def_1; fsloc 0 in FinSeq-Locations by SCMFSA_2:def_5; then A12: fsloc 0 in Data-Locations by SCMFSA_2:100, XBOOLE_0:def_3; A13: F <> <*D*> by FINSEQ_1:76; A14: t . (fsloc 0) = F by Lm3, Lm4, FUNCT_7:114; t . (intloc 0) = D by AMI_3:10, BVFUNC14:12; then (Exec ((((fsloc 0),(intloc 1)) := (intloc 0)),t)) . (fsloc 0) = <*D*> by A14, A10, A11, FUNCT_7:95; hence for b1 being InsType of the InstructionsF of SCM+FSA st b1 = InsCode ((f,a) := b) holds not b1 is jump-only by A8, A14, A13, A12, AMISTD_1:def_1; ::_thesis: verum end; end; registration let a, b be Int-Location; let f be FinSeq-Location ; clusterb := (f,a) -> non jump-only ; coherence not b := (f,a) is jump-only proof thus not InsCode (b := (f,a)) is jump-only ; :: according to AMISTD_1:def_2 ::_thesis: verum end; cluster(f,a) := b -> non jump-only ; coherence not (f,a) := b is jump-only proof thus not InsCode ((f,a) := b) is jump-only ; :: according to AMISTD_1:def_2 ::_thesis: verum end; end; registration let a be Int-Location; let f be FinSeq-Location ; cluster(a :=len f) `1_3 -> non jump-only for InsType of the InstructionsF of SCM+FSA; coherence for b1 being InsType of the InstructionsF of SCM+FSA st b1 = InsCode (a :=len f) holds not b1 is jump-only proof Values (intloc 0) = INT by SCMFSA_2:11; then reconsider D = 3 as Element of Values (intloc 0) by INT_1:def_1; reconsider DWA = 2 as Element of INT by INT_1:def_1; set w = the State of SCM+FSA; A1: InsCode (a :=len f) = 11 by SCMFSA_2:28 .= InsCode ((intloc 0) :=len (fsloc 0)) by SCMFSA_2:28 ; <*DWA*> in INT * by FINSEQ_1:def_11; then reconsider F = <*2*> as Element of Values (fsloc 0) by SCMFSA_2:12; reconsider t = ( the State of SCM+FSA +* ((fsloc 0) .--> F)) +* ((intloc 0) .--> D) as State of SCM+FSA ; A2: t . (fsloc 0) = F by BVFUNC14:12, SCMFSA_2:99; intloc 0 in Int-Locations by AMI_2:def_16; then A3: intloc 0 in Data-Locations by SCMFSA_2:100, XBOOLE_0:def_3; A4: t . (intloc 0) = D by FUNCT_7:94; (Exec (((intloc 0) :=len (fsloc 0)),t)) . (intloc 0) = len (t . (fsloc 0)) by SCMFSA_2:74 .= 1 by A2, FINSEQ_1:39 ; hence for b1 being InsType of the InstructionsF of SCM+FSA st b1 = InsCode (a :=len f) holds not b1 is jump-only by A1, A4, A3, AMISTD_1:def_1; ::_thesis: verum end; cluster(f :=<0,...,0> a) `1_3 -> non jump-only for InsType of the InstructionsF of SCM+FSA; coherence for b1 being InsType of the InstructionsF of SCM+FSA st b1 = InsCode (f :=<0,...,0> a) holds not b1 is jump-only proof Values (intloc 0) = INT by SCMFSA_2:11; then reconsider D = 1 as Element of Values (intloc 0) by INT_1:def_1; reconsider DWA = 2 as Element of INT by INT_1:def_1; set w = the State of SCM+FSA; <*DWA*> in INT * by FINSEQ_1:def_11; then reconsider F = <*2*> as Element of Values (fsloc 0) by SCMFSA_2:12; reconsider t = ( the State of SCM+FSA +* ((fsloc 0) .--> F)) +* ((intloc 0) .--> D) as State of SCM+FSA ; A5: t . (fsloc 0) = F by BVFUNC14:12, SCMFSA_2:99; A6: F <> <*0*> by FINSEQ_1:76; consider k being Element of NAT such that A7: k = abs (t . (intloc 0)) and A8: (Exec (((fsloc 0) :=<0,...,0> (intloc 0)),t)) . (fsloc 0) = k |-> 0 by SCMFSA_2:75; fsloc 0 in FinSeq-Locations by SCMFSA_2:def_5; then A9: fsloc 0 in Data-Locations by SCMFSA_2:100, XBOOLE_0:def_3; t . (intloc 0) = D by FUNCT_7:94; then k = 1 by A7, ABSVALUE:def_1; then A10: (Exec (((fsloc 0) :=<0,...,0> (intloc 0)),t)) . (fsloc 0) = <*0*> by A8, FINSEQ_2:59; InsCode (f :=<0,...,0> a) = 12 by SCMFSA_2:29 .= InsCode ((fsloc 0) :=<0,...,0> (intloc 0)) by SCMFSA_2:29 ; hence for b1 being InsType of the InstructionsF of SCM+FSA st b1 = InsCode (f :=<0,...,0> a) holds not b1 is jump-only by A5, A6, A10, A9, AMISTD_1:def_1; ::_thesis: verum end; end; registration let a be Int-Location; let f be FinSeq-Location ; clustera :=len f -> non jump-only ; coherence not a :=len f is jump-only proof thus not InsCode (a :=len f) is jump-only ; :: according to AMISTD_1:def_2 ::_thesis: verum end; clusterf :=<0,...,0> a -> non jump-only ; coherence not f :=<0,...,0> a is jump-only proof thus not InsCode (f :=<0,...,0> a) is jump-only ; :: according to AMISTD_1:def_2 ::_thesis: verum end; end; registration cluster SCM+FSA -> with_explicit_jumps ; coherence SCM+FSA is with_explicit_jumps proof let I be Instruction of SCM+FSA; :: 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 (I `2_3) 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 Int-Location st I = a := b or ex a, b being Int-Location st I = AddTo (a,b) or ex a, b being Int-Location st I = SubFrom (a,b) or ex a, b being Int-Location st I = MultBy (a,b) or ex a, b being Int-Location st I = Divide (a,b) or ex i1 being Element of NAT st I = goto i1 or ex i1 being Element of NAT ex a being Int-Location st I = a =0_goto i1 or ex i1 being Element of NAT ex a being Int-Location st I = a >0_goto i1 or ex a, b being Int-Location ex f being FinSeq-Location st I = b := (f,a) or ex a, b being Int-Location ex f being FinSeq-Location st I = (f,a) := b or ex a being Int-Location ex f being FinSeq-Location st I = a :=len f or ex a being Int-Location ex f being FinSeq-Location st I = f :=<0,...,0> a ) by SCMFSA_2:93; suppose I = [0,{},{}] ; ::_thesis: f in rng (JumpPart I) hence f in rng (JumpPart I) by A1, SCMFSA_2:96; ::_thesis: verum end; suppose ex a, b being Int-Location 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 Int-Location 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 Int-Location 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 Int-Location st I = MultBy (a,b) ; ::_thesis: f in rng (JumpPart I) hence f in rng (JumpPart I) by A1; ::_thesis: verum end; suppose ex a, b being Int-Location st I = Divide (a,b) ; ::_thesis: f in rng (JumpPart I) hence f in rng (JumpPart I) by A1; ::_thesis: verum end; supposeA2: ex i1 being Element of NAT st I = goto i1 ; ::_thesis: f in rng (JumpPart I) consider i1 being Element of NAT such that A3: I = goto i1 by A2; A4: JumpPart (goto i1) = <*i1*> by RECDEF_2:def_2; rng <*i1*> = {i1} by FINSEQ_1:39; hence f in rng (JumpPart I) by A1, A3, A4, Th34; ::_thesis: verum end; supposeA5: ex i1 being Element of NAT ex a being Int-Location st I = a =0_goto i1 ; ::_thesis: f in rng (JumpPart I) consider a being Int-Location, i1 being Element of NAT such that A6: I = a =0_goto i1 by A5; A7: JumpPart (a =0_goto i1) = <*i1*> by Th15; rng <*i1*> = {i1} by FINSEQ_1:39; hence f in rng (JumpPart I) by A1, A6, A7, Th36; ::_thesis: verum end; supposeA8: ex i1 being Element of NAT ex a being Int-Location st I = a >0_goto i1 ; ::_thesis: f in rng (JumpPart I) consider a being Int-Location, i1 being Element of NAT such that A9: I = a >0_goto i1 by A8; A10: JumpPart (a >0_goto i1) = <*i1*> by Th16; rng <*i1*> = {i1} by FINSEQ_1:39; hence f in rng (JumpPart I) by A1, A9, A10, Th38; ::_thesis: verum end; suppose ex a, b being Int-Location ex f being FinSeq-Location st I = b := (f,a) ; ::_thesis: f in rng (JumpPart I) hence f in rng (JumpPart I) by A1; ::_thesis: verum end; suppose ex a, b being Int-Location ex f being FinSeq-Location st I = (f,a) := b ; ::_thesis: f in rng (JumpPart I) hence f in rng (JumpPart I) by A1; ::_thesis: verum end; suppose ex a being Int-Location ex f being FinSeq-Location st I = a :=len f ; ::_thesis: f in rng (JumpPart I) hence f in rng (JumpPart I) by A1; ::_thesis: verum end; suppose ex a being Int-Location ex f being FinSeq-Location st I = f :=<0,...,0> a ; ::_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 (I `2_3) or f in JUMP I ) assume f in rng (JumpPart I) ; ::_thesis: f in JUMP I then consider k being set such that A11: k in dom (JumpPart I) and A12: f = (JumpPart I) . k by FUNCT_1:def_3; percases ( I = [0,{},{}] or ex a, b being Int-Location st I = a := b or ex a, b being Int-Location st I = AddTo (a,b) or ex a, b being Int-Location st I = SubFrom (a,b) or ex a, b being Int-Location st I = MultBy (a,b) or ex a, b being Int-Location st I = Divide (a,b) or ex i1 being Element of NAT st I = goto i1 or ex i1 being Element of NAT ex a being Int-Location st I = a =0_goto i1 or ex i1 being Element of NAT ex a being Int-Location st I = a >0_goto i1 or ex a, b being Int-Location ex f being FinSeq-Location st I = b := (f,a) or ex a, b being Int-Location ex f being FinSeq-Location st I = (f,a) := b or ex a being Int-Location ex f being FinSeq-Location st I = a :=len f or ex a being Int-Location ex f being FinSeq-Location st I = f :=<0,...,0> a ) by SCMFSA_2:93; suppose I = [0,{},{}] ; ::_thesis: f in JUMP I then dom (JumpPart I) = dom {} by RECDEF_2:def_2; hence f in JUMP I by A11; ::_thesis: verum end; suppose ex a, b being Int-Location st I = a := b ; ::_thesis: f in JUMP I then consider a, b being Int-Location such that A13: I = a := b ; k in dom {} by A11, A13, Th10; hence f in JUMP I ; ::_thesis: verum end; suppose ex a, b being Int-Location st I = AddTo (a,b) ; ::_thesis: f in JUMP I then consider a, b being Int-Location such that A14: I = AddTo (a,b) ; k in dom {} by A11, A14, Th11; hence f in JUMP I ; ::_thesis: verum end; suppose ex a, b being Int-Location st I = SubFrom (a,b) ; ::_thesis: f in JUMP I then consider a, b being Int-Location such that A15: I = SubFrom (a,b) ; k in dom {} by A11, A15, Th12; hence f in JUMP I ; ::_thesis: verum end; suppose ex a, b being Int-Location st I = MultBy (a,b) ; ::_thesis: f in JUMP I then consider a, b being Int-Location such that A16: I = MultBy (a,b) ; k in dom {} by A11, A16, Th13; hence f in JUMP I ; ::_thesis: verum end; suppose ex a, b being Int-Location st I = Divide (a,b) ; ::_thesis: f in JUMP I then consider a, b being Int-Location such that A17: I = Divide (a,b) ; k in dom {} by A11, A17, Th14; hence f in JUMP I ; ::_thesis: verum end; suppose ex i1 being Element of NAT st I = goto i1 ; ::_thesis: f in JUMP I then consider i1 being Element of NAT such that A18: I = goto i1 ; A19: JumpPart I = <*i1*> by A18, RECDEF_2:def_2; then k = 1 by A11, FINSEQ_1:90; then A20: f = i1 by A19, A12, FINSEQ_1:def_8; JUMP I = {i1} by A18, Th34; hence f in JUMP I by A20, TARSKI:def_1; ::_thesis: verum end; suppose ex i1 being Element of NAT ex a being Int-Location st I = a =0_goto i1 ; ::_thesis: f in JUMP I then consider a being Int-Location, i1 being Element of NAT such that A21: I = a =0_goto i1 ; A22: JumpPart I = <*i1*> by A21, Th15; then k = 1 by A11, FINSEQ_1:90; then A23: f = i1 by A22, A12, FINSEQ_1:def_8; JUMP I = {i1} by A21, Th36; hence f in JUMP I by A23, TARSKI:def_1; ::_thesis: verum end; suppose ex i1 being Element of NAT ex a being Int-Location st I = a >0_goto i1 ; ::_thesis: f in JUMP I then consider a being Int-Location, i1 being Element of NAT such that A24: I = a >0_goto i1 ; A25: JumpPart I = <*i1*> by A24, Th16; then k = 1 by A11, FINSEQ_1:90; then A26: f = i1 by A25, A12, FINSEQ_1:def_8; JUMP I = {i1} by A24, Th38; hence f in JUMP I by A26, TARSKI:def_1; ::_thesis: verum end; suppose ex a, b being Int-Location ex f being FinSeq-Location st I = b := (f,a) ; ::_thesis: f in JUMP I then consider a, b being Int-Location, f being FinSeq-Location such that A27: I = b := (f,a) ; k in dom {} by A11, A27, RECDEF_2:def_2; hence f in JUMP I ; ::_thesis: verum end; suppose ex a, b being Int-Location ex f being FinSeq-Location st I = (f,a) := b ; ::_thesis: f in JUMP I then consider a, b being Int-Location, f being FinSeq-Location such that A28: I = (f,a) := b ; k in dom {} by A11, A28, RECDEF_2:def_2; hence f in JUMP I ; ::_thesis: verum end; suppose ex a being Int-Location ex f being FinSeq-Location st I = a :=len f ; ::_thesis: f in JUMP I then consider a being Int-Location, f being FinSeq-Location such that A29: I = a :=len f ; k in dom {} by A11, A29, RECDEF_2:def_2; hence f in JUMP I ; ::_thesis: verum end; suppose ex a being Int-Location ex f being FinSeq-Location st I = f :=<0,...,0> a ; ::_thesis: f in JUMP I then consider a being Int-Location, f being FinSeq-Location such that A30: I = f :=<0,...,0> a ; k in dom {} by A11, A30, RECDEF_2:def_2; hence f in JUMP I ; ::_thesis: verum end; end; end; end; theorem Th41: :: SCMFSA10:41 for i1 being Element of NAT for k being Nat holds IncAddr ((goto i1),k) = goto (i1 + k) proof let i1 be Element of NAT ; ::_thesis: for k being Nat holds IncAddr ((goto i1),k) = goto (i1 + k) let k be Nat; ::_thesis: IncAddr ((goto i1),k) = goto (i1 + k) A1: InsCode (IncAddr ((goto i1),k)) = InsCode (goto i1) by COMPOS_0:def_9 .= 6 by SCMFSA_2:23 .= InsCode (goto (i1 + k)) by SCMFSA_2:23 ; A2: AddressPart (IncAddr ((goto i1),k)) = AddressPart (goto i1) by COMPOS_0:def_9 .= {} by RECDEF_2:def_3 .= AddressPart (goto (i1 + k)) by RECDEF_2:def_3 ; A3: JumpPart (IncAddr ((goto i1),k)) = k + (JumpPart (goto i1)) by COMPOS_0:def_9; then A4: dom (JumpPart (IncAddr ((goto i1),k))) = dom (JumpPart (goto i1)) by VALUED_1:def_2; A5: for x being set st x in dom (JumpPart (goto i1)) holds (JumpPart (IncAddr ((goto i1),k))) . x = (JumpPart (goto (i1 + k))) . x proof let x be set ; ::_thesis: ( x in dom (JumpPart (goto i1)) implies (JumpPart (IncAddr ((goto i1),k))) . x = (JumpPart (goto (i1 + k))) . x ) assume A6: x in dom (JumpPart (goto i1)) ; ::_thesis: (JumpPart (IncAddr ((goto i1),k))) . x = (JumpPart (goto (i1 + k))) . x then x in dom <*i1*> by RECDEF_2:def_2; then A7: x = 1 by FINSEQ_1:90; set f = (JumpPart (goto i1)) . x; A8: (JumpPart (IncAddr ((goto i1),k))) . x = k + ((JumpPart (goto i1)) . x) by A4, A3, A6, VALUED_1:def_2; (JumpPart (goto i1)) . x = <*i1*> . x by RECDEF_2:def_2 .= i1 by A7, FINSEQ_1:def_8 ; hence (JumpPart (IncAddr ((goto i1),k))) . x = <*(i1 + k)*> . x by A7, A8, FINSEQ_1:def_8 .= (JumpPart (goto (i1 + k))) . x by RECDEF_2:def_2 ; ::_thesis: verum end; dom (JumpPart (goto (i1 + k))) = 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)) by RECDEF_2:def_2 ; then JumpPart (IncAddr ((goto i1),k)) = JumpPart (goto (i1 + k)) by A4, A5, FUNCT_1:2; hence IncAddr ((goto i1),k) = goto (i1 + k) by A1, A2, COMPOS_0:1; ::_thesis: verum end; theorem Th42: :: SCMFSA10:42 for i1 being Element of NAT for k being Nat for a being Int-Location holds IncAddr ((a =0_goto i1),k) = a =0_goto (i1 + k) proof let i1 be Element of NAT ; ::_thesis: for k being Nat for a being Int-Location holds IncAddr ((a =0_goto i1),k) = a =0_goto (i1 + k) let k be Nat; ::_thesis: for a being Int-Location holds IncAddr ((a =0_goto i1),k) = a =0_goto (i1 + k) let a be Int-Location; ::_thesis: IncAddr ((a =0_goto i1),k) = a =0_goto (i1 + k) A1: InsCode (IncAddr ((a =0_goto i1),k)) = InsCode (a =0_goto i1) by COMPOS_0:def_9 .= 7 by SCMFSA_2:24 .= InsCode (a =0_goto (i1 + k)) by SCMFSA_2:24 ; A2: a =0_goto i1 = [7,<*i1*>,<*a*>] by Th7; A3: a =0_goto (i1 + k) = [7,<*(i1 + k)*>,<*a*>] by Th7; A4: AddressPart (IncAddr ((a =0_goto i1),k)) = AddressPart (a =0_goto i1) by COMPOS_0:def_9 .= <*a*> by A2, RECDEF_2:def_3 .= AddressPart (a =0_goto (i1 + k)) by A3, RECDEF_2:def_3 ; A5: JumpPart (IncAddr ((a =0_goto i1),k)) = k + (JumpPart (a =0_goto i1)) by COMPOS_0:def_9; then A6: dom (JumpPart (IncAddr ((a =0_goto i1),k))) = dom (JumpPart (a =0_goto i1)) by VALUED_1:def_2; A7: 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 A8: 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 Th15; then A9: x = 1 by FINSEQ_1:90; set f = (JumpPart (a =0_goto i1)) . x; A10: (JumpPart (IncAddr ((a =0_goto i1),k))) . x = k + ((JumpPart (a =0_goto i1)) . x) by A6, A5, A8, VALUED_1:def_2; (JumpPart (a =0_goto i1)) . x = <*i1*> . x by Th15 .= i1 by A9, FINSEQ_1:40 ; hence (JumpPart (IncAddr ((a =0_goto i1),k))) . x = <*(i1 + k)*> . x by A9, A10, FINSEQ_1:40 .= (JumpPart (a =0_goto (i1 + k))) . x by Th15 ; ::_thesis: verum end; dom (JumpPart (a =0_goto (i1 + k))) = dom <*(i1 + k)*> by Th15 .= Seg 1 by FINSEQ_1:38 .= dom <*i1*> by FINSEQ_1:38 .= dom (JumpPart (a =0_goto i1)) by Th15 ; then JumpPart (IncAddr ((a =0_goto i1),k)) = JumpPart (a =0_goto (i1 + k)) by A6, A7, FUNCT_1:2; hence IncAddr ((a =0_goto i1),k) = a =0_goto (i1 + k) by A1, A4, COMPOS_0:1; ::_thesis: verum end; theorem Th43: :: SCMFSA10:43 for i1 being Element of NAT for k being Nat for a being Int-Location holds IncAddr ((a >0_goto i1),k) = a >0_goto (i1 + k) proof let i1 be Element of NAT ; ::_thesis: for k being Nat for a being Int-Location holds IncAddr ((a >0_goto i1),k) = a >0_goto (i1 + k) let k be Nat; ::_thesis: for a being Int-Location holds IncAddr ((a >0_goto i1),k) = a >0_goto (i1 + k) let a be Int-Location; ::_thesis: IncAddr ((a >0_goto i1),k) = a >0_goto (i1 + k) A1: InsCode (IncAddr ((a >0_goto i1),k)) = InsCode (a >0_goto i1) by COMPOS_0:def_9 .= 8 by SCMFSA_2:25 .= InsCode (a >0_goto (i1 + k)) by SCMFSA_2:25 ; A2: a >0_goto i1 = [8,<*i1*>,<*a*>] by Th8; A3: a >0_goto (i1 + k) = [8,<*(i1 + k)*>,<*a*>] by Th8; A4: AddressPart (IncAddr ((a >0_goto i1),k)) = AddressPart (a >0_goto i1) by COMPOS_0:def_9 .= <*a*> by A2, RECDEF_2:def_3 .= AddressPart (a >0_goto (i1 + k)) by A3, RECDEF_2:def_3 ; A5: JumpPart (IncAddr ((a >0_goto i1),k)) = k + (JumpPart (a >0_goto i1)) by COMPOS_0:def_9; then A6: dom (JumpPart (IncAddr ((a >0_goto i1),k))) = dom (JumpPart (a >0_goto i1)) by VALUED_1:def_2; A7: 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 A8: 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 Th16; then A9: x = 1 by FINSEQ_1:90; set f = (JumpPart (a >0_goto i1)) . 1; A10: (JumpPart (IncAddr ((a >0_goto i1),k))) . 1 = k + ((JumpPart (a >0_goto i1)) . 1) by A9, A6, A5, A8, VALUED_1:def_2; (JumpPart (a >0_goto i1)) . 1 = <*i1*> . x by Th16, A9 .= i1 by A9, FINSEQ_1:40 ; hence (JumpPart (IncAddr ((a >0_goto i1),k))) . x = <*(i1 + k)*> . x by A9, A10, FINSEQ_1:40 .= (JumpPart (a >0_goto (i1 + k))) . x by Th16 ; ::_thesis: verum end; dom (JumpPart (a >0_goto (i1 + k))) = dom <*(i1 + k)*> by Th16 .= Seg 1 by FINSEQ_1:38 .= dom <*i1*> by FINSEQ_1:38 .= dom (JumpPart (a >0_goto i1)) by Th16 ; then JumpPart (IncAddr ((a >0_goto i1),k)) = JumpPart (a >0_goto (i1 + k)) by A6, A7, FUNCT_1:2; hence IncAddr ((a >0_goto i1),k) = a >0_goto (i1 + k) by A1, A4, COMPOS_0:1; ::_thesis: verum end; registration cluster SCM+FSA -> IC-relocable ; coherence SCM+FSA is IC-relocable proof let I be Instruction of SCM+FSA; :: according to AMISTD_2:def_4 ::_thesis: I is IC-relocable percases ( I = [0,{},{}] or ex a, b being Int-Location st I = a := b or ex a, b being Int-Location st I = AddTo (a,b) or ex a, b being Int-Location st I = SubFrom (a,b) or ex a, b being Int-Location st I = MultBy (a,b) or ex a, b being Int-Location st I = Divide (a,b) or ex i1 being Element of NAT st I = goto i1 or ex i1 being Element of NAT ex a being Int-Location st I = a =0_goto i1 or ex i1 being Element of NAT ex a being Int-Location st I = a >0_goto i1 or ex a, b being Int-Location ex f being FinSeq-Location st I = b := (f,a) or ex a, b being Int-Location ex f being FinSeq-Location st I = (f,a) := b or ex a being Int-Location ex f being FinSeq-Location st I = a :=len f or ex a being Int-Location ex f being FinSeq-Location st I = f :=<0,...,0> a ) by SCMFSA_2:93; suppose I = [0,{},{}] ; ::_thesis: I is IC-relocable hence I is IC-relocable by SCMFSA_2:96; ::_thesis: verum end; suppose ex a, b being Int-Location st I = a := b ; ::_thesis: I is IC-relocable hence I is IC-relocable ; ::_thesis: verum end; suppose ex a, b being Int-Location st I = AddTo (a,b) ; ::_thesis: I is IC-relocable hence I is IC-relocable ; ::_thesis: verum end; suppose ex a, b being Int-Location st I = SubFrom (a,b) ; ::_thesis: I is IC-relocable hence I is IC-relocable ; ::_thesis: verum end; suppose ex a, b being Int-Location st I = MultBy (a,b) ; ::_thesis: I is IC-relocable hence I is IC-relocable ; ::_thesis: verum end; suppose ex a, b being Int-Location st I = Divide (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 ; ::_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+FSA; ::_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 by A1; thus (IC (Exec ((IncAddr (I,j)),s1))) + k = (IC (Exec ((goto (j + i1)),s1))) + k by A2, Th41 .= (j + i1) + k by SCMFSA_2:69 .= IC (Exec ((goto ((j + k) + i1)),(IncIC (s1,k)))) by SCMFSA_2:69 .= IC (Exec ((IncAddr (I,(j + k))),(IncIC (s1,k)))) by A2, Th41 ; ::_thesis: verum end; suppose ex i1 being Element of NAT ex a being Int-Location st I = a =0_goto i1 ; ::_thesis: I is IC-relocable then consider a being Int-Location, 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+FSA; ::_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 FUNCOP_1:13, SCMFSA_2:56; 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 or s1 . a <> 0 ) ; supposeA5: s1 . a = 0 ; ::_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, Th42 .= (j + i1) + k by A5, SCMFSA_2:70 .= IC (Exec ((a =0_goto ((j + k) + i1)),(IncIC (s1,k)))) by A4, A5, SCMFSA_2:70 .= IC (Exec ((IncAddr (I,(j + k))),(IncIC (s1,k)))) by A3, Th42 ; ::_thesis: verum end; supposeA6: s1 . a <> 0 ; ::_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, Th42; A8: IncAddr (I,(j + k)) = a =0_goto (i1 + (j + k)) by A3, Th42; 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, SCMFSA_2:70 .= succ (IC (IncIC (s1,k))) by A9 .= IC (Exec ((IncAddr (I,(j + k))),(IncIC (s1,k)))) by A8, A6, A4, SCMFSA_2:70 ; ::_thesis: verum end; end; end; suppose ex i1 being Element of NAT ex a being Int-Location st I = a >0_goto i1 ; ::_thesis: I is IC-relocable then consider i1 being Element of NAT , a being Int-Location such that A10: 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+FSA; ::_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 FUNCOP_1:13, SCMFSA_2:56; then not a in dom ((IC ) .--> ((IC s1) + k)) by TARSKI:def_1; then A11: s1 . a = (IncIC (s1,k)) . a by FUNCT_4:11; percases ( s1 . a > 0 or s1 . a <= 0 ) ; supposeA12: s1 . a > 0 ; ::_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 A10, Th43 .= (j + i1) + k by A12, SCMFSA_2:71 .= IC (Exec ((a >0_goto ((j + k) + i1)),(IncIC (s1,k)))) by A11, A12, SCMFSA_2:71 .= IC (Exec ((IncAddr (I,(j + k))),(IncIC (s1,k)))) by A10, Th43 ; ::_thesis: verum end; supposeA13: s1 . a <= 0 ; ::_thesis: (IC (Exec ((IncAddr (I,j)),s1))) + k = IC (Exec ((IncAddr (I,(j + k))),(IncIC (s1,k)))) A14: IncAddr (I,j) = a >0_goto (i1 + j) by A10, Th43; A15: IncAddr (I,(j + k)) = a >0_goto (i1 + (j + k)) by A10, Th43; dom ((IC ) .--> ((IC s1) + k)) = {(IC )} by FUNCOP_1:13; then IC in dom ((IC ) .--> ((IC s1) + k)) by TARSKI:def_1; then A16: 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 A14, A13, SCMFSA_2:71 .= succ (IC (IncIC (s1,k))) by A16 .= IC (Exec ((IncAddr (I,(j + k))),(IncIC (s1,k)))) by A15, A13, A11, SCMFSA_2:71 ; ::_thesis: verum end; end; end; suppose ex a, b being Int-Location ex f being FinSeq-Location st I = b := (f,a) ; ::_thesis: I is IC-relocable hence I is IC-relocable ; ::_thesis: verum end; suppose ex a, b being Int-Location ex f being FinSeq-Location st I = (f,a) := b ; ::_thesis: I is IC-relocable hence I is IC-relocable ; ::_thesis: verum end; suppose ex a being Int-Location ex f being FinSeq-Location st I = a :=len f ; ::_thesis: I is IC-relocable hence I is IC-relocable ; ::_thesis: verum end; suppose ex a being Int-Location ex f being FinSeq-Location st I = f :=<0,...,0> a ; ::_thesis: I is IC-relocable hence I is IC-relocable ; ::_thesis: verum end; end; end; end;