:: AMI_6 semantic presentation
begin
theorem :: AMI_6:1
for T being InsType of the InstructionsF of SCM holds
( T = 0 or T = 1 or T = 2 or T = 3 or T = 4 or T = 5 or T = 6 or T = 7 or T = 8 )
proof
let T be InsType of the InstructionsF of SCM; ::_thesis: ( T = 0 or T = 1 or T = 2 or T = 3 or T = 4 or T = 5 or T = 6 or T = 7 or T = 8 )
consider y being set such that
A1: [T,y] in proj1 the InstructionsF of SCM by XTUPLE_0:def_12;
consider x being set such that
A2: [[T,y],x] in the InstructionsF of SCM by A1, XTUPLE_0:def_12;
reconsider I = [T,y,x] as Instruction of SCM by A2;
T = InsCode I by RECDEF_2:def_1;
hence ( T = 0 or T = 1 or T = 2 or T = 3 or T = 4 or T = 5 or T = 6 or T = 7 or T = 8 ) by AMI_5:5, NAT_1:32; ::_thesis: verum
end;
theorem Th2: :: AMI_6:2
JumpPart (halt SCM) = {} ;
theorem :: AMI_6:3
for T being InsType of the InstructionsF of SCM st T = 0 holds
JumpParts T = {0}
proof
let T be InsType of the InstructionsF of SCM; ::_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 such that
A2: a = JumpPart I and
A3: InsCode I = T ;
I = halt SCM by A1, A3, AMI_5:7;
hence a in {0} by A2, Th2, 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 A: a = 0 by TARSKI:def_1;
InsCode (halt SCM) = 0 by COMPOS_1:70;
hence a in JumpParts T by A1, Th2, A; ::_thesis: verum
end;
theorem :: AMI_6:4
for T being InsType of the InstructionsF of SCM st T = 1 holds
JumpParts T = {{}}
proof
let T be InsType of the InstructionsF of SCM; ::_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 such that
A2: x = JumpPart I and
A3: InsCode I = T ;
consider a, b being Data-Location such that
A4: I = a := b by A1, A3, AMI_5:8;
x = {} by A2, A4, RECDEF_2:def_2;
hence x in {{}} by TARSKI:def_1; ::_thesis: verum
end;
set a = the Data-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 Data-Location := the Data-Location) by RECDEF_2:def_2;
InsCode ( the Data-Location := the Data-Location) = 1 by RECDEF_2:def_1;
hence x in JumpParts T by A5, A1; ::_thesis: verum
end;
theorem :: AMI_6:5
for T being InsType of the InstructionsF of SCM st T = 2 holds
JumpParts T = {{}}
proof
let T be InsType of the InstructionsF of SCM; ::_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 such that
A2: x = JumpPart I and
A3: InsCode I = T ;
consider a, b being Data-Location such that
A4: I = AddTo (a,b) by A1, A3, AMI_5:9;
x = {} by A2, A4, RECDEF_2:def_2;
hence x in {{}} by TARSKI:def_1; ::_thesis: verum
end;
set a = the Data-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 Data-Location, the Data-Location)) by RECDEF_2:def_2;
InsCode (AddTo ( the Data-Location, the Data-Location)) = 2 by RECDEF_2:def_1;
hence x in JumpParts T by A5, A1; ::_thesis: verum
end;
theorem :: AMI_6:6
for T being InsType of the InstructionsF of SCM st T = 3 holds
JumpParts T = {{}}
proof
let T be InsType of the InstructionsF of SCM; ::_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 such that
A2: x = JumpPart I and
A3: InsCode I = T ;
consider a, b being Data-Location such that
A4: I = SubFrom (a,b) by A1, A3, AMI_5:10;
x = {} by A2, A4, RECDEF_2:def_2;
hence x in {{}} by TARSKI:def_1; ::_thesis: verum
end;
set a = the Data-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 Data-Location, the Data-Location)) by RECDEF_2:def_2;
InsCode (SubFrom ( the Data-Location, the Data-Location)) = 3 by RECDEF_2:def_1;
hence x in JumpParts T by A5, A1; ::_thesis: verum
end;
theorem :: AMI_6:7
for T being InsType of the InstructionsF of SCM st T = 4 holds
JumpParts T = {{}}
proof
let T be InsType of the InstructionsF of SCM; ::_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 such that
A2: x = JumpPart I and
A3: InsCode I = T ;
consider a, b being Data-Location such that
A4: I = MultBy (a,b) by A1, A3, AMI_5:11;
x = {} by A2, A4, RECDEF_2:def_2;
hence x in {{}} by TARSKI:def_1; ::_thesis: verum
end;
set a = the Data-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 Data-Location, the Data-Location)) by RECDEF_2:def_2;
InsCode (MultBy ( the Data-Location, the Data-Location)) = 4 by RECDEF_2:def_1;
hence x in JumpParts T by A5, A1; ::_thesis: verum
end;
theorem :: AMI_6:8
for T being InsType of the InstructionsF of SCM st T = 5 holds
JumpParts T = {{}}
proof
let T be InsType of the InstructionsF of SCM; ::_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 such that
A2: x = JumpPart I and
A3: InsCode I = T ;
consider a, b being Data-Location such that
A4: I = Divide (a,b) by A1, A3, AMI_5:12;
x = {} by A2, A4, RECDEF_2:def_2;
hence x in {{}} by TARSKI:def_1; ::_thesis: verum
end;
set a = the Data-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 Data-Location, the Data-Location)) by RECDEF_2:def_2;
InsCode (Divide ( the Data-Location, the Data-Location)) = 5 by RECDEF_2:def_1;
hence x in JumpParts T by A5, A1; ::_thesis: verum
end;
theorem Th9: :: AMI_6:9
for T being InsType of the InstructionsF of SCM st T = 6 holds
dom (product" (JumpParts T)) = {1}
proof
let T be InsType of the InstructionsF of SCM; ::_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 (SCM-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 (SCM-goto the Element of NAT ) = 6 by RECDEF_2:def_1;
then A3: JumpPart (SCM-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 (SCM-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 such that
A5: f = JumpPart I and
A6: InsCode I = T ;
consider i1 being Element of NAT such that
A7: I = SCM-goto i1 by A1, A6, AMI_5:13;
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 Th10: :: AMI_6:10
for T being InsType of the InstructionsF of SCM st T = 7 holds
dom (product" (JumpParts T)) = {1}
proof
let T be InsType of the InstructionsF of SCM; ::_thesis: ( T = 7 implies dom (product" (JumpParts T)) = {1} )
set i1 = the Element of NAT ;
set a = the Data-Location;
assume A1: T = 7 ; ::_thesis: dom (product" (JumpParts T)) = {1}
A2: JumpPart ( the Data-Location =0_goto the Element of NAT ) = <* the Element of NAT *> by RECDEF_2:def_2;
hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: {1} c= dom (product" (JumpParts T))
let x be set ; ::_thesis: ( x in dom (product" (JumpParts T)) implies x in {1} )
InsCode ( the Data-Location =0_goto the Element of NAT ) = 7 by RECDEF_2:def_1;
then A3: JumpPart ( the Data-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 Data-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 such that
A5: f = JumpPart I and
A6: InsCode I = T ;
consider i1 being Element of NAT , a being Data-Location such that
A7: I = a =0_goto i1 by A1, A6, AMI_5:14;
f = <*i1*> by A5, A7, RECDEF_2:def_2;
hence x in dom f by A4, FINSEQ_1:2, FINSEQ_1:38; ::_thesis: verum
end;
then x in DOM (JumpParts T) by CARD_3:109;
hence x in dom (product" (JumpParts T)) by CARD_3:def_12; ::_thesis: verum
end;
theorem Th11: :: AMI_6:11
for T being InsType of the InstructionsF of SCM st T = 8 holds
dom (product" (JumpParts T)) = {1}
proof
let T be InsType of the InstructionsF of SCM; ::_thesis: ( T = 8 implies dom (product" (JumpParts T)) = {1} )
set i1 = the Element of NAT ;
set a = the Data-Location;
assume A1: T = 8 ; ::_thesis: dom (product" (JumpParts T)) = {1}
A2: JumpPart ( the Data-Location >0_goto the Element of NAT ) = <* the Element of NAT *> by RECDEF_2:def_2;
hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: {1} c= dom (product" (JumpParts T))
let x be set ; ::_thesis: ( x in dom (product" (JumpParts T)) implies x in {1} )
InsCode ( the Data-Location >0_goto the Element of NAT ) = 8 by RECDEF_2:def_1;
then A3: JumpPart ( the Data-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 Data-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 such that
A5: f = JumpPart I and
A6: InsCode I = T ;
consider i1 being Element of NAT , a being Data-Location such that
A7: I = a >0_goto i1 by A1, A6, AMI_5:15;
f = <*i1*> by A5, A7, RECDEF_2:def_2;
hence x in dom f by A4, FINSEQ_1:2, FINSEQ_1:38; ::_thesis: verum
end;
then x in DOM (JumpParts T) by CARD_3:109;
hence x in dom (product" (JumpParts T)) by CARD_3:def_12; ::_thesis: verum
end;
theorem :: AMI_6:12
for k1 being Nat holds (product" (JumpParts (InsCode (SCM-goto k1)))) . 1 = NAT
proof
let k1 be Nat; ::_thesis: (product" (JumpParts (InsCode (SCM-goto k1)))) . 1 = NAT
InsCode (SCM-goto k1) = 6 by RECDEF_2:def_1;
then dom (product" (JumpParts (InsCode (SCM-goto k1)))) = {1} by Th9;
then A1: 1 in dom (product" (JumpParts (InsCode (SCM-goto k1)))) by TARSKI:def_1;
hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: NAT c= (product" (JumpParts (InsCode (SCM-goto k1)))) . 1
let x be set ; ::_thesis: ( x in (product" (JumpParts (InsCode (SCM-goto k1)))) . 1 implies x in NAT )
assume x in (product" (JumpParts (InsCode (SCM-goto k1)))) . 1 ; ::_thesis: x in NAT
then x in pi ((JumpParts (InsCode (SCM-goto k1))),1) by A1, CARD_3:def_12;
then consider g being Function such that
A2: g in JumpParts (InsCode (SCM-goto k1)) and
A3: x = g . 1 by CARD_3:def_6;
consider I being Instruction of SCM such that
A4: g = JumpPart I and
A5: InsCode I = InsCode (SCM-goto k1) by A2;
InsCode I = 6 by A5, RECDEF_2:def_1;
then consider i2 being Element of NAT such that
A6: I = SCM-goto i2 by AMI_5:13;
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 (SCM-goto k1)))) . 1 )
assume x in NAT ; ::_thesis: x in (product" (JumpParts (InsCode (SCM-goto k1)))) . 1
then reconsider x = x as Element of NAT ;
InsCode (SCM-goto k1) = 6 by RECDEF_2:def_1;
then ( JumpPart (SCM-goto x) = <*x*> & InsCode (SCM-goto k1) = InsCode (SCM-goto x) ) by RECDEF_2:def_1, RECDEF_2:def_2;
then A7: <*x*> in JumpParts (InsCode (SCM-goto k1)) ;
<*x*> . 1 = x by FINSEQ_1:def_8;
then x in pi ((JumpParts (InsCode (SCM-goto k1))),1) by A7, CARD_3:def_6;
hence x in (product" (JumpParts (InsCode (SCM-goto k1)))) . 1 by A1, CARD_3:def_12; ::_thesis: verum
end;
theorem :: AMI_6:13
for a being Data-Location
for k1 being Nat holds (product" (JumpParts (InsCode (a =0_goto k1)))) . 1 = NAT
proof
let a be Data-Location; ::_thesis: for k1 being Nat holds (product" (JumpParts (InsCode (a =0_goto k1)))) . 1 = NAT
let k1 be Nat; ::_thesis: (product" (JumpParts (InsCode (a =0_goto k1)))) . 1 = NAT
InsCode (a =0_goto k1) = 7 by RECDEF_2:def_1;
then dom (product" (JumpParts (InsCode (a =0_goto k1)))) = {1} by Th10;
then A1: 1 in dom (product" (JumpParts (InsCode (a =0_goto k1)))) by TARSKI:def_1;
hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: NAT c= (product" (JumpParts (InsCode (a =0_goto k1)))) . 1
let x be set ; ::_thesis: ( x in (product" (JumpParts (InsCode (a =0_goto k1)))) . 1 implies x in NAT )
assume x in (product" (JumpParts (InsCode (a =0_goto k1)))) . 1 ; ::_thesis: x in NAT
then x in pi ((JumpParts (InsCode (a =0_goto k1))),1) by A1, CARD_3:def_12;
then consider g being Function such that
A2: g in JumpParts (InsCode (a =0_goto k1)) and
A3: x = g . 1 by CARD_3:def_6;
consider I being Instruction of SCM such that
A4: g = JumpPart I and
A5: InsCode I = InsCode (a =0_goto k1) by A2;
InsCode I = 7 by A5, RECDEF_2:def_1;
then consider i2 being Element of NAT , b being Data-Location such that
A6: I = b =0_goto i2 by AMI_5:14;
g = <*i2*> by A4, A6, RECDEF_2:def_2;
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 k1)))) . 1 )
assume x in NAT ; ::_thesis: x in (product" (JumpParts (InsCode (a =0_goto k1)))) . 1
then reconsider x = x as Element of NAT ;
InsCode (a =0_goto k1) = 7 by RECDEF_2:def_1;
then ( JumpPart (a =0_goto x) = <*x*> & InsCode (a =0_goto k1) = InsCode (a =0_goto x) ) by RECDEF_2:def_1, RECDEF_2:def_2;
then A7: <*x*> in JumpParts (InsCode (a =0_goto k1)) ;
<*x*> . 1 = x by FINSEQ_1:40;
then x in pi ((JumpParts (InsCode (a =0_goto k1))),1) by A7, CARD_3:def_6;
hence x in (product" (JumpParts (InsCode (a =0_goto k1)))) . 1 by A1, CARD_3:def_12; ::_thesis: verum
end;
theorem :: AMI_6:14
for a being Data-Location
for k1 being Nat holds (product" (JumpParts (InsCode (a >0_goto k1)))) . 1 = NAT
proof
let a be Data-Location; ::_thesis: for k1 being Nat holds (product" (JumpParts (InsCode (a >0_goto k1)))) . 1 = NAT
let k1 be Nat; ::_thesis: (product" (JumpParts (InsCode (a >0_goto k1)))) . 1 = NAT
InsCode (a >0_goto k1) = 8 by RECDEF_2:def_1;
then dom (product" (JumpParts (InsCode (a >0_goto k1)))) = {1} by Th11;
then A1: 1 in dom (product" (JumpParts (InsCode (a >0_goto k1)))) by TARSKI:def_1;
hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: NAT c= (product" (JumpParts (InsCode (a >0_goto k1)))) . 1
let x be set ; ::_thesis: ( x in (product" (JumpParts (InsCode (a >0_goto k1)))) . 1 implies x in NAT )
assume x in (product" (JumpParts (InsCode (a >0_goto k1)))) . 1 ; ::_thesis: x in NAT
then x in pi ((JumpParts (InsCode (a >0_goto k1))),1) by A1, CARD_3:def_12;
then consider g being Function such that
A2: g in JumpParts (InsCode (a >0_goto k1)) and
A3: x = g . 1 by CARD_3:def_6;
consider I being Instruction of SCM such that
A4: g = JumpPart I and
A5: InsCode I = InsCode (a >0_goto k1) by A2;
InsCode I = 8 by A5, RECDEF_2:def_1;
then consider i2 being Element of NAT , b being Data-Location such that
A6: I = b >0_goto i2 by AMI_5:15;
g = <*i2*> by A4, A6, RECDEF_2:def_2;
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 k1)))) . 1 )
assume x in NAT ; ::_thesis: x in (product" (JumpParts (InsCode (a >0_goto k1)))) . 1
then reconsider x = x as Element of NAT ;
InsCode (a >0_goto k1) = 8 by RECDEF_2:def_1;
then ( JumpPart (a >0_goto x) = <*x*> & InsCode (a >0_goto k1) = InsCode (a >0_goto x) ) by RECDEF_2:def_1, RECDEF_2:def_2;
then A7: <*x*> in JumpParts (InsCode (a >0_goto k1)) ;
<*x*> . 1 = x by FINSEQ_1:40;
then x in pi ((JumpParts (InsCode (a >0_goto k1))),1) by A7, CARD_3:def_6;
hence x in (product" (JumpParts (InsCode (a >0_goto k1)))) . 1 by A1, CARD_3:def_12; ::_thesis: verum
end;
Lm1: for i being Instruction of SCM st ( for l being Element of NAT holds NIC (i,l) = {(succ l)} ) holds
JUMP i is empty
proof
set p = 1;
set q = 2;
let i be Instruction of SCM; ::_thesis: ( ( for l being Element of NAT holds NIC (i,l) = {(succ l)} ) implies JUMP i is empty )
assume A1: for l being Element of NAT holds NIC (i,l) = {(succ l)} ; ::_thesis: JUMP i is empty
set X = { (NIC (i,f)) where f is Element of NAT : verum } ;
reconsider p = 1, q = 2 as Element of NAT ;
assume not JUMP i is empty ; ::_thesis: contradiction
then consider x being set such that
A2: x in meet { (NIC (i,f)) where f is Element of NAT : verum } by XBOOLE_0:def_1;
NIC (i,p) = {(succ p)} by A1;
then {(succ p)} in { (NIC (i,f)) where f is Element of NAT : verum } ;
then x in {(succ p)} by A2, SETFAM_1:def_1;
then A3: x = succ p by TARSKI:def_1;
NIC (i,q) = {(succ q)} by A1;
then {(succ q)} in { (NIC (i,f)) where f is Element of NAT : verum } ;
then x in {(succ q)} by A2, SETFAM_1:def_1;
hence contradiction by A3, TARSKI:def_1; ::_thesis: verum
end;
registration
cluster JUMP (halt SCM) -> empty ;
coherence
JUMP (halt SCM) is empty ;
end;
registration
let a, b be Data-Location;
clustera := b -> sequential ;
coherence
a := b is sequential
proof
let s be State of SCM; :: 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 AMI_3:2; ::_thesis: verum
end;
cluster AddTo (a,b) -> sequential ;
coherence
AddTo (a,b) is sequential
proof
let s be State of SCM; :: 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 AMI_3:3; ::_thesis: verum
end;
cluster SubFrom (a,b) -> sequential ;
coherence
SubFrom (a,b) is sequential
proof
let s be State of SCM; :: 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 AMI_3:4; ::_thesis: verum
end;
cluster MultBy (a,b) -> sequential ;
coherence
MultBy (a,b) is sequential
proof
let s be State of SCM; :: 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 AMI_3:5; ::_thesis: verum
end;
cluster Divide (a,b) -> sequential ;
coherence
Divide (a,b) is sequential
proof
let s be State of SCM; :: 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 AMI_3:6; ::_thesis: verum
end;
end;
registration
let a, b be Data-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;
end;
registration
let a, b be Data-Location;
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;
end;
registration
let a, b be Data-Location;
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;
end;
registration
let a, b be Data-Location;
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;
end;
registration
let a, b be Data-Location;
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 Th15: :: AMI_6:15
for il being Element of NAT
for k being Nat holds NIC ((SCM-goto k),il) = {k}
proof
let il be Element of NAT ; ::_thesis: for k being Nat holds NIC ((SCM-goto k),il) = {k}
let k be Nat; ::_thesis: NIC ((SCM-goto k),il) = {k}
now__::_thesis:_for_x_being_set_holds_
(_x_in_{k}_iff_x_in__{__(IC_(Exec_((SCM-goto_k),s)))_where_s_is_Element_of_product_(the_Values_of_SCM)_:_IC_s_=_il__}__)
let x be set ; ::_thesis: ( x in {k} iff x in { (IC (Exec ((SCM-goto k),s))) where s is Element of product (the_Values_of SCM) : IC s = il } )
A1: now__::_thesis:_(_x_=_k_implies_x_in__{__(IC_(Exec_((SCM-goto_k),s)))_where_s_is_Element_of_product_(the_Values_of_SCM)_:_IC_s_=_il__}__)
reconsider il1 = il as Element of Values (IC ) by MEMSTR_0:def_6;
set I = SCM-goto k;
set t = the State of SCM;
set Q = the Instruction-Sequence of SCM;
assume A2: x = k ; ::_thesis: x in { (IC (Exec ((SCM-goto k),s))) where s is Element of product (the_Values_of SCM) : IC s = il }
reconsider n = il as Element of NAT ;
reconsider u = the State of SCM +* ((IC ),il1) as Element of product (the_Values_of SCM) by CARD_3:107;
reconsider P = the Instruction-Sequence of SCM +* (il,(SCM-goto k)) as Instruction-Sequence of SCM ;
A3: P /. il = P . il by PBOOLE:143;
IC in dom the State of SCM by MEMSTR_0:2;
then A4: IC u = n by FUNCT_7:31;
il in NAT ;
then il in dom the Instruction-Sequence of SCM by PARTFUN1:def_2;
then A5: P . n = SCM-goto k by FUNCT_7:31;
then IC (Following (P,u)) = k by A3, A4, AMI_3:7;
hence x in { (IC (Exec ((SCM-goto k),s))) where s is Element of product (the_Values_of SCM) : IC s = il } by A2, A4, A3, A5; ::_thesis: verum
end;
now__::_thesis:_(_x_in__{__(IC_(Exec_((SCM-goto_k),s)))_where_s_is_Element_of_product_(the_Values_of_SCM)_:_IC_s_=_il__}__implies_x_=_k_)
assume x in { (IC (Exec ((SCM-goto k),s))) where s is Element of product (the_Values_of SCM) : IC s = il } ; ::_thesis: x = k
then ex s being Element of product (the_Values_of SCM) st
( x = IC (Exec ((SCM-goto k),s)) & IC s = il ) ;
hence x = k by AMI_3:7; ::_thesis: verum
end;
hence ( x in {k} iff x in { (IC (Exec ((SCM-goto k),s))) where s is Element of product (the_Values_of SCM) : IC s = il } ) by A1, TARSKI:def_1; ::_thesis: verum
end;
hence NIC ((SCM-goto k),il) = {k} by TARSKI:1; ::_thesis: verum
end;
theorem Th16: :: AMI_6:16
for k being Nat holds JUMP (SCM-goto k) = {k}
proof
let k be Nat; ::_thesis: JUMP (SCM-goto k) = {k}
set X = { (NIC ((SCM-goto k),il)) where il is Element of NAT : verum } ;
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_meet__{__(NIC_((SCM-goto_k),il))_where_il_is_Element_of_NAT_:_verum__}__implies_x_in_{k}_)_&_(_x_in_{k}_implies_x_in_meet__{__(NIC_((SCM-goto_k),il))_where_il_is_Element_of_NAT_:_verum__}__)_)
let x be set ; ::_thesis: ( ( x in meet { (NIC ((SCM-goto k),il)) where il is Element of NAT : verum } implies x in {k} ) & ( x in {k} implies x in meet { (NIC ((SCM-goto k),il)) where il is Element of NAT : verum } ) )
hereby ::_thesis: ( x in {k} implies x in meet { (NIC ((SCM-goto k),il)) where il is Element of NAT : verum } )
set il1 = 1;
A1: NIC ((SCM-goto k),1) in { (NIC ((SCM-goto k),il)) where il is Element of NAT : verum } ;
assume x in meet { (NIC ((SCM-goto k),il)) where il is Element of NAT : verum } ; ::_thesis: x in {k}
then x in NIC ((SCM-goto k),1) by A1, SETFAM_1:def_1;
hence x in {k} by Th15; ::_thesis: verum
end;
assume x in {k} ; ::_thesis: x in meet { (NIC ((SCM-goto k),il)) where il is Element of NAT : verum }
then A2: x = k by TARSKI:def_1;
A3: now__::_thesis:_for_Y_being_set_st_Y_in__{__(NIC_((SCM-goto_k),il))_where_il_is_Element_of_NAT_:_verum__}__holds_
k_in_Y
let Y be set ; ::_thesis: ( Y in { (NIC ((SCM-goto k),il)) where il is Element of NAT : verum } implies k in Y )
assume Y in { (NIC ((SCM-goto k),il)) where il is Element of NAT : verum } ; ::_thesis: k in Y
then consider il being Element of NAT such that
A4: Y = NIC ((SCM-goto k),il) ;
NIC ((SCM-goto k),il) = {k} by Th15;
hence k in Y by A4, TARSKI:def_1; ::_thesis: verum
end;
reconsider k = k as Element of NAT by ORDINAL1:def_12;
NIC ((SCM-goto k),k) in { (NIC ((SCM-goto k),il)) where il is Element of NAT : verum } ;
hence x in meet { (NIC ((SCM-goto k),il)) where il is Element of NAT : verum } by A2, A3, SETFAM_1:def_1; ::_thesis: verum
end;
hence JUMP (SCM-goto k) = {k} by TARSKI:1; ::_thesis: verum
end;
registration
let i1 be Element of NAT ;
cluster JUMP (SCM-goto i1) -> 1 -element ;
coherence
JUMP (SCM-goto i1) is 1 -element
proof
JUMP (SCM-goto i1) = {i1} by Th16;
hence JUMP (SCM-goto i1) is 1 -element ; ::_thesis: verum
end;
end;
theorem Th17: :: AMI_6:17
for a being Data-Location
for il being Element of NAT
for k being Nat holds NIC ((a =0_goto k),il) = {k,(succ il)}
proof
let a be Data-Location; ::_thesis: for il being Element of NAT
for k being Nat holds NIC ((a =0_goto k),il) = {k,(succ il)}
let il be Element of NAT ; ::_thesis: for k being Nat holds NIC ((a =0_goto k),il) = {k,(succ il)}
let k be Nat; ::_thesis: NIC ((a =0_goto k),il) = {k,(succ il)}
set t = the State of SCM;
set Q = the Instruction-Sequence of SCM;
hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: {k,(succ il)} c= NIC ((a =0_goto k),il)
let x be set ; ::_thesis: ( x in NIC ((a =0_goto k),il) implies b1 in {k,(succ il)} )
assume x in NIC ((a =0_goto k),il) ; ::_thesis: b1 in {k,(succ il)}
then consider s being Element of product (the_Values_of SCM) such that
A1: ( x = IC (Exec ((a =0_goto k),s)) & IC s = il ) ;
percases ( s . a = 0 or s . a <> 0 ) ;
suppose s . a = 0 ; ::_thesis: b1 in {k,(succ il)}
then x = k by A1, AMI_3:8;
hence x in {k,(succ il)} by TARSKI:def_2; ::_thesis: verum
end;
suppose s . a <> 0 ; ::_thesis: b1 in {k,(succ il)}
then x = succ il by A1, AMI_3:8;
hence x in {k,(succ il)} by TARSKI:def_2; ::_thesis: verum
end;
end;
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {k,(succ il)} or x in NIC ((a =0_goto k),il) )
set I = a =0_goto k;
A2: IC <> a by AMI_5:2;
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 +* ((IC ),il1) as Element of product (the_Values_of SCM) by CARD_3:107;
reconsider P = the Instruction-Sequence of SCM +* (il,(a =0_goto k)) as Instruction-Sequence of SCM ;
assume A3: x in {k,(succ il)} ; ::_thesis: x in NIC ((a =0_goto k),il)
percases ( x = k or x = succ il ) by A3, TARSKI:def_2;
supposeA4: x = k ; ::_thesis: x in NIC ((a =0_goto k),il)
reconsider v = u +* (a .--> 0) as Element of product (the_Values_of SCM) by CARD_3:107;
A5: IC in dom the State of SCM by MEMSTR_0:2;
A6: dom (a .--> 0) = {a} by FUNCOP_1:13;
then not IC in dom (a .--> 0) by A2, TARSKI:def_1;
then A7: IC v = IC u by FUNCT_4:11
.= n by A5, FUNCT_7:31 ;
A8: P /. il = P . il by PBOOLE:143;
il in NAT ;
then il in dom the Instruction-Sequence of SCM by PARTFUN1:def_2;
then A9: P . il = a =0_goto k by FUNCT_7:31;
a in dom (a .--> 0) by A6, TARSKI:def_1;
then v . a = (a .--> 0) . a by FUNCT_4:13
.= 0 by FUNCOP_1:72 ;
then IC (Following (P,v)) = k by A7, A9, A8, AMI_3:8;
hence x in NIC ((a =0_goto k),il) by A4, A7, A9, A8; ::_thesis: verum
end;
supposeA10: x = succ il ; ::_thesis: x in NIC ((a =0_goto k),il)
reconsider v = u +* (a .--> 1) as Element of product (the_Values_of SCM) by CARD_3:107;
A11: IC in dom the State of SCM by MEMSTR_0:2;
A12: dom (a .--> 1) = {a} by FUNCOP_1:13;
then not IC in dom (a .--> 1) by A2, TARSKI:def_1;
then A13: IC v = IC u by FUNCT_4:11
.= n by A11, FUNCT_7:31 ;
A14: P /. il = P . il by PBOOLE:143;
il in NAT ;
then il in dom the Instruction-Sequence of SCM by PARTFUN1:def_2;
then A15: P . il = a =0_goto k by FUNCT_7:31;
a in dom (a .--> 1) by A12, 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 A13, A15, A14, AMI_3:8;
hence x in NIC ((a =0_goto k),il) by A10, A13, A15, A14; ::_thesis: verum
end;
end;
end;
theorem Th18: :: AMI_6:18
for a being Data-Location
for k being Nat holds JUMP (a =0_goto k) = {k}
proof
let a be Data-Location; ::_thesis: for k being Nat holds JUMP (a =0_goto k) = {k}
let k be Nat; ::_thesis: JUMP (a =0_goto k) = {k}
set X = { (NIC ((a =0_goto k),il)) where il is Element of NAT : verum } ;
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_meet__{__(NIC_((a_=0_goto_k),il))_where_il_is_Element_of_NAT_:_verum__}__implies_x_in_{k}_)_&_(_x_in_{k}_implies_x_in_meet__{__(NIC_((a_=0_goto_k),il))_where_il_is_Element_of_NAT_:_verum__}__)_)
let x be set ; ::_thesis: ( ( x in meet { (NIC ((a =0_goto k),il)) where il is Element of NAT : verum } implies x in {k} ) & ( x in {k} implies x in meet { (NIC ((a =0_goto k),il)) where il is Element of NAT : verum } ) )
A1: now__::_thesis:_for_Y_being_set_st_Y_in__{__(NIC_((a_=0_goto_k),il))_where_il_is_Element_of_NAT_:_verum__}__holds_
k_in_Y
let Y be set ; ::_thesis: ( Y in { (NIC ((a =0_goto k),il)) where il is Element of NAT : verum } implies k in Y )
assume Y in { (NIC ((a =0_goto k),il)) where il is Element of NAT : verum } ; ::_thesis: k in Y
then consider il being Element of NAT such that
A2: Y = NIC ((a =0_goto k),il) ;
NIC ((a =0_goto k),il) = {k,(succ il)} by Th17;
hence k in Y by A2, TARSKI:def_2; ::_thesis: verum
end;
hereby ::_thesis: ( x in {k} implies x in meet { (NIC ((a =0_goto k),il)) where il is Element of NAT : verum } )
set il1 = 1;
set il2 = 2;
assume A3: x in meet { (NIC ((a =0_goto k),il)) where il is Element of NAT : verum } ; ::_thesis: x in {k}
A4: NIC ((a =0_goto k),2) = {k,(succ 2)} by Th17;
NIC ((a =0_goto k),2) in { (NIC ((a =0_goto k),il)) where il is Element of NAT : verum } ;
then x in NIC ((a =0_goto k),2) by A3, SETFAM_1:def_1;
then A5: ( x = k or x = succ 2 ) by A4, TARSKI:def_2;
A6: NIC ((a =0_goto k),1) = {k,(succ 1)} by Th17;
NIC ((a =0_goto k),1) in { (NIC ((a =0_goto k),il)) where il is Element of NAT : verum } ;
then x in NIC ((a =0_goto k),1) by A3, SETFAM_1:def_1;
then ( x = k or x = succ 1 ) by A6, TARSKI:def_2;
hence x in {k} by A5, TARSKI:def_1; ::_thesis: verum
end;
assume x in {k} ; ::_thesis: x in meet { (NIC ((a =0_goto k),il)) where il is Element of NAT : verum }
then A7: x = k by TARSKI:def_1;
reconsider k = k as Element of NAT by ORDINAL1:def_12;
NIC ((a =0_goto k),k) in { (NIC ((a =0_goto k),il)) where il is Element of NAT : verum } ;
hence x in meet { (NIC ((a =0_goto k),il)) where il is Element of NAT : verum } by A7, A1, SETFAM_1:def_1; ::_thesis: verum
end;
hence JUMP (a =0_goto k) = {k} by TARSKI:1; ::_thesis: verum
end;
registration
let a be Data-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 Th18;
hence JUMP (a =0_goto i1) is 1 -element ; ::_thesis: verum
end;
end;
theorem Th19: :: AMI_6:19
for a being Data-Location
for il being Element of NAT
for k being Nat holds NIC ((a >0_goto k),il) = {k,(succ il)}
proof
let a be Data-Location; ::_thesis: for il being Element of NAT
for k being Nat holds NIC ((a >0_goto k),il) = {k,(succ il)}
let il be Element of NAT ; ::_thesis: for k being Nat holds NIC ((a >0_goto k),il) = {k,(succ il)}
let k be Nat; ::_thesis: NIC ((a >0_goto k),il) = {k,(succ il)}
set t = the State of SCM;
set Q = the Instruction-Sequence of SCM;
hereby :: according to TARSKI:def_3,XBOOLE_0:def_10 ::_thesis: {k,(succ il)} c= NIC ((a >0_goto k),il)
let x be set ; ::_thesis: ( x in NIC ((a >0_goto k),il) implies b1 in {k,(succ il)} )
assume x in NIC ((a >0_goto k),il) ; ::_thesis: b1 in {k,(succ il)}
then consider s being Element of product (the_Values_of SCM) such that
A1: ( x = IC (Exec ((a >0_goto k),s)) & IC s = il ) ;
percases ( s . a > 0 or s . a <= 0 ) ;
suppose s . a > 0 ; ::_thesis: b1 in {k,(succ il)}
then x = k by A1, AMI_3:9;
hence x in {k,(succ il)} by TARSKI:def_2; ::_thesis: verum
end;
suppose s . a <= 0 ; ::_thesis: b1 in {k,(succ il)}
then x = succ il by A1, AMI_3:9;
hence x in {k,(succ il)} by TARSKI:def_2; ::_thesis: verum
end;
end;
end;
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in {k,(succ il)} or x in NIC ((a >0_goto k),il) )
set I = a >0_goto k;
A2: IC <> a by AMI_5:2;
reconsider il1 = il as Element of Values (IC ) by MEMSTR_0:def_6;
reconsider u = the State of SCM +* ((IC ),il1) as Element of product (the_Values_of SCM) by CARD_3:107;
reconsider P = the Instruction-Sequence of SCM +* (il,(a >0_goto k)) as Instruction-Sequence of SCM ;
reconsider n = il1 as Element of NAT ;
assume A3: x in {k,(succ il)} ; ::_thesis: x in NIC ((a >0_goto k),il)
percases ( x = k or x = succ il ) by A3, TARSKI:def_2;
supposeA4: x = k ; ::_thesis: x in NIC ((a >0_goto k),il)
reconsider v = u +* (a .--> 1) as Element of product (the_Values_of SCM) by CARD_3:107;
A5: IC in dom the State of SCM by MEMSTR_0:2;
A6: dom (a .--> 1) = {a} by FUNCOP_1:13;
then not IC in dom (a .--> 1) by A2, TARSKI:def_1;
then A7: IC v = IC u by FUNCT_4:11
.= n by A5, FUNCT_7:31 ;
A8: P /. il = P . il by PBOOLE:143;
il in NAT ;
then il in dom the Instruction-Sequence of SCM by PARTFUN1:def_2;
then A9: P . il = a >0_goto k by FUNCT_7:31;
a in dom (a .--> 1) by A6, TARSKI:def_1;
then v . a = (a .--> 1) . a by FUNCT_4:13
.= 1 by FUNCOP_1:72 ;
then IC (Following (P,v)) = k by A7, A9, A8, AMI_3:9;
hence x in NIC ((a >0_goto k),il) by A4, A7, A9, A8; ::_thesis: verum
end;
supposeA10: x = succ il ; ::_thesis: x in NIC ((a >0_goto k),il)
reconsider v = u +* (a .--> 0) as Element of product (the_Values_of SCM) by CARD_3:107;
A11: IC in dom the State of SCM by MEMSTR_0:2;
A12: dom (a .--> 0) = {a} by FUNCOP_1:13;
then not IC in dom (a .--> 0) by A2, TARSKI:def_1;
then A13: IC v = IC u by FUNCT_4:11
.= n by A11, FUNCT_7:31 ;
A14: P /. il = P . il by PBOOLE:143;
il in NAT ;
then il in dom the Instruction-Sequence of SCM by PARTFUN1:def_2;
then A15: P . il = a >0_goto k by FUNCT_7:31;
a in dom (a .--> 0) by A12, 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 A13, A15, A14, AMI_3:9;
hence x in NIC ((a >0_goto k),il) by A10, A13, A15, A14; ::_thesis: verum
end;
end;
end;
theorem Th20: :: AMI_6:20
for a being Data-Location
for k being Nat holds JUMP (a >0_goto k) = {k}
proof
let a be Data-Location; ::_thesis: for k being Nat holds JUMP (a >0_goto k) = {k}
let k be Nat; ::_thesis: JUMP (a >0_goto k) = {k}
set X = { (NIC ((a >0_goto k),il)) where il is Element of NAT : verum } ;
now__::_thesis:_for_x_being_set_holds_
(_(_x_in_meet__{__(NIC_((a_>0_goto_k),il))_where_il_is_Element_of_NAT_:_verum__}__implies_x_in_{k}_)_&_(_x_in_{k}_implies_x_in_meet__{__(NIC_((a_>0_goto_k),il))_where_il_is_Element_of_NAT_:_verum__}__)_)
let x be set ; ::_thesis: ( ( x in meet { (NIC ((a >0_goto k),il)) where il is Element of NAT : verum } implies x in {k} ) & ( x in {k} implies x in meet { (NIC ((a >0_goto k),il)) where il is Element of NAT : verum } ) )
A1: now__::_thesis:_for_Y_being_set_st_Y_in__{__(NIC_((a_>0_goto_k),il))_where_il_is_Element_of_NAT_:_verum__}__holds_
k_in_Y
let Y be set ; ::_thesis: ( Y in { (NIC ((a >0_goto k),il)) where il is Element of NAT : verum } implies k in Y )
assume Y in { (NIC ((a >0_goto k),il)) where il is Element of NAT : verum } ; ::_thesis: k in Y
then consider il being Element of NAT such that
A2: Y = NIC ((a >0_goto k),il) ;
NIC ((a >0_goto k),il) = {k,(succ il)} by Th19;
hence k in Y by A2, TARSKI:def_2; ::_thesis: verum
end;
hereby ::_thesis: ( x in {k} implies x in meet { (NIC ((a >0_goto k),il)) where il is Element of NAT : verum } )
set il1 = 1;
set il2 = 2;
assume A3: x in meet { (NIC ((a >0_goto k),il)) where il is Element of NAT : verum } ; ::_thesis: x in {k}
A4: NIC ((a >0_goto k),2) = {k,(succ 2)} by Th19;
NIC ((a >0_goto k),2) in { (NIC ((a >0_goto k),il)) where il is Element of NAT : verum } ;
then x in NIC ((a >0_goto k),2) by A3, SETFAM_1:def_1;
then A5: ( x = k or x = succ 2 ) by A4, TARSKI:def_2;
A6: NIC ((a >0_goto k),1) = {k,(succ 1)} by Th19;
NIC ((a >0_goto k),1) in { (NIC ((a >0_goto k),il)) where il is Element of NAT : verum } ;
then x in NIC ((a >0_goto k),1) by A3, SETFAM_1:def_1;
then ( x = k or x = succ 1 ) by A6, TARSKI:def_2;
hence x in {k} by A5, TARSKI:def_1; ::_thesis: verum
end;
assume x in {k} ; ::_thesis: x in meet { (NIC ((a >0_goto k),il)) where il is Element of NAT : verum }
then A7: x = k by TARSKI:def_1;
reconsider k = k as Element of NAT by ORDINAL1:def_12;
NIC ((a >0_goto k),k) in { (NIC ((a >0_goto k),il)) where il is Element of NAT : verum } ;
hence x in meet { (NIC ((a >0_goto k),il)) where il is Element of NAT : verum } by A7, A1, SETFAM_1:def_1; ::_thesis: verum
end;
hence JUMP (a >0_goto k) = {k} by TARSKI:1; ::_thesis: verum
end;
registration
let a be Data-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 Th20;
hence JUMP (a >0_goto i1) is 1 -element ; ::_thesis: verum
end;
end;
theorem Th21: :: AMI_6:21
for il being Element of NAT holds SUCC (il,SCM) = {il,(succ il)}
proof
let il be Element of NAT ; ::_thesis: SUCC (il,SCM) = {il,(succ il)}
set X = { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of SCM : 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_:_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_:_verum__}__)_)
let x be set ; ::_thesis: ( ( x in union { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of SCM : 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 : 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 : verum } )
assume x in union { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of SCM : 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 : verum } by TARSKI:def_4;
consider i being Element of the InstructionsF of SCM such that
A3: Y = (NIC (i,il)) \ (JUMP i) by A2;
percases ( i = [0,{},{}] or ex a, b being Data-Location st i = a := b or ex a, b being Data-Location st i = AddTo (a,b) or ex a, b being Data-Location st i = SubFrom (a,b) or ex a, b being Data-Location st i = MultBy (a,b) or ex a, b being Data-Location st i = Divide (a,b) or ex k being Nat st i = SCM-goto k or ex a being Data-Location ex k being Nat st i = a =0_goto k or ex a being Data-Location ex k being Nat st i = a >0_goto k ) by AMI_3:24;
suppose i = [0,{},{}] ; ::_thesis: x in {il,(succ il)}
then x in {il} \ (JUMP (halt SCM)) by A1, A3, AMISTD_1:2;
then x = il by TARSKI:def_1;
hence x in {il,(succ il)} by TARSKI:def_2; ::_thesis: verum
end;
suppose ex a, b being Data-Location st i = a := b ; ::_thesis: x in {il,(succ il)}
then consider a, b being Data-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 Data-Location st i = AddTo (a,b) ; ::_thesis: x in {il,(succ il)}
then consider a, b being Data-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 Data-Location st i = SubFrom (a,b) ; ::_thesis: x in {il,(succ il)}
then consider a, b being Data-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 Data-Location st i = MultBy (a,b) ; ::_thesis: x in {il,(succ il)}
then consider a, b being Data-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 Data-Location st i = Divide (a,b) ; ::_thesis: x in {il,(succ il)}
then consider a, b being Data-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 k being Nat st i = SCM-goto k ; ::_thesis: x in {il,(succ il)}
then consider k being Nat such that
A9: i = SCM-goto k ;
x in {k} \ (JUMP i) by A1, A3, A9, Th15;
then x in {k} \ {k} by A9, Th16;
hence x in {il,(succ il)} by XBOOLE_1:37; ::_thesis: verum
end;
suppose ex a being Data-Location ex k being Nat st i = a =0_goto k ; ::_thesis: x in {il,(succ il)}
then consider a being Data-Location, k being Nat such that
A10: i = a =0_goto k ;
A11: NIC (i,il) = {k,(succ il)} by A10, Th17;
x in NIC (i,il) by A1, A3, XBOOLE_0:def_5;
then A12: ( x = k or x = succ il ) by A11, TARSKI:def_2;
x in (NIC (i,il)) \ {k} by A1, A3, A10, Th18;
then not x in {k} by XBOOLE_0:def_5;
hence x in {il,(succ il)} by A12, TARSKI:def_1, TARSKI:def_2; ::_thesis: verum
end;
suppose ex a being Data-Location ex k being Nat st i = a >0_goto k ; ::_thesis: x in {il,(succ il)}
then consider a being Data-Location, k being Nat such that
A13: i = a >0_goto k ;
A14: NIC (i,il) = {k,(succ il)} by A13, Th19;
x in NIC (i,il) by A1, A3, XBOOLE_0:def_5;
then A15: ( x = k or x = succ il ) by A14, TARSKI:def_2;
x in (NIC (i,il)) \ {k} by A1, A3, A13, Th20;
then not x in {k} by XBOOLE_0:def_5;
hence x in {il,(succ il)} by A15, TARSKI:def_1, TARSKI:def_2; ::_thesis: verum
end;
end;
end;
assume A16: x in {il,(succ il)} ; ::_thesis: b1 in union { ((NIC (b2,il)) \ (JUMP b2)) where I is Element of the InstructionsF of SCM : verum }
percases ( x = il or x = succ il ) by A16, TARSKI:def_2;
supposeA17: x = il ; ::_thesis: b1 in union { ((NIC (b2,il)) \ (JUMP b2)) where I is Element of the InstructionsF of SCM : verum }
set i = halt SCM;
(NIC ((halt SCM),il)) \ (JUMP (halt SCM)) = {il} by AMISTD_1:2;
then A18: {il} in { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of SCM : verum } ;
x in {il} by A17, TARSKI:def_1;
hence x in union { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of SCM : verum } by A18, TARSKI:def_4; ::_thesis: verum
end;
supposeA19: x = succ il ; ::_thesis: b1 in union { ((NIC (b2,il)) \ (JUMP b2)) where I is Element of the InstructionsF of SCM : verum }
set a = the Data-Location;
set i = AddTo ( the Data-Location, the Data-Location);
(NIC ((AddTo ( the Data-Location, the Data-Location)),il)) \ (JUMP (AddTo ( the Data-Location, the Data-Location))) = {(succ il)} by AMISTD_1:12;
then A20: {(succ il)} in { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of SCM : verum } ;
x in {(succ il)} by A19, TARSKI:def_1;
hence x in union { ((NIC (I,il)) \ (JUMP I)) where I is Element of the InstructionsF of SCM : verum } by A20, TARSKI:def_4; ::_thesis: verum
end;
end;
end;
hence SUCC (il,SCM) = {il,(succ il)} by TARSKI:1; ::_thesis: verum
end;
theorem Th22: :: AMI_6:22
for k being Element of NAT holds
( k + 1 in SUCC (k,SCM) & ( for j being Element of NAT st j in SUCC (k,SCM) holds
k <= j ) )
proof
let k be Element of NAT ; ::_thesis: ( k + 1 in SUCC (k,SCM) & ( for j being Element of NAT st j in SUCC (k,SCM) holds
k <= j ) )
reconsider fk = k as Element of NAT ;
A1: SUCC (k,SCM) = {k,(succ fk)} by Th21;
hence k + 1 in SUCC (k,SCM) by TARSKI:def_2; ::_thesis: for j being Element of NAT st j in SUCC (k,SCM) holds
k <= j
let j be Element of NAT ; ::_thesis: ( j in SUCC (k,SCM) implies k <= j )
assume A2: j in SUCC (k,SCM) ; ::_thesis: k <= j
reconsider fk = k as Element of NAT ;
percases ( j = k or j = succ fk ) by A1, A2, TARSKI:def_2;
suppose j = k ; ::_thesis: k <= j
hence k <= j ; ::_thesis: verum
end;
suppose j = succ fk ; ::_thesis: k <= j
hence k <= j by NAT_1:11; ::_thesis: verum
end;
end;
end;
registration
cluster SCM -> standard ;
coherence
SCM is standard by Th22, AMISTD_1:3;
end;
registration
cluster InsCode (halt SCM) -> jump-only for InsType of the InstructionsF of SCM;
coherence
for b1 being InsType of the InstructionsF of SCM st b1 = InsCode (halt SCM) holds
b1 is jump-only
proof
now__::_thesis:_for_s_being_State_of_SCM
for_o_being_Object_of_SCM
for_I_being_Instruction_of_SCM_st_InsCode_I_=_InsCode_(halt_SCM)_&_o_in_Data-Locations_holds_
(Exec_(I,s))_._o_=_s_._o
let s be State of SCM; ::_thesis: for o being Object of SCM
for I being Instruction of SCM st InsCode I = InsCode (halt SCM) & o in Data-Locations holds
(Exec (I,s)) . o = s . o
let o be Object of SCM; ::_thesis: for I being Instruction of SCM st InsCode I = InsCode (halt SCM) & o in Data-Locations holds
(Exec (I,s)) . o = s . o
let I be Instruction of SCM; ::_thesis: ( InsCode I = InsCode (halt SCM) & o in Data-Locations implies (Exec (I,s)) . o = s . o )
assume that
A1: InsCode I = InsCode (halt SCM) and
o in Data-Locations ; ::_thesis: (Exec (I,s)) . o = s . o
I = halt SCM by A1, AMI_5:7, 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 st b1 = InsCode (halt SCM) holds
b1 is jump-only by AMISTD_1:def_1; ::_thesis: verum
end;
end;
registration
cluster halt SCM -> jump-only ;
coherence
halt SCM is jump-only
proof
thus InsCode (halt SCM) is jump-only ; :: according to AMISTD_1:def_2 ::_thesis: verum
end;
end;
registration
let i1 be Element of NAT ;
cluster InsCode (SCM-goto i1) -> jump-only for InsType of the InstructionsF of SCM;
coherence
for b1 being InsType of the InstructionsF of SCM st b1 = InsCode (SCM-goto i1) holds
b1 is jump-only
proof
let T be InsType of the InstructionsF of SCM; ::_thesis: ( T = InsCode (SCM-goto i1) implies T is jump-only )
assume A1: T = InsCode (SCM-goto i1) ; ::_thesis: T is jump-only
let s be State of SCM; :: according to AMISTD_1:def_1 ::_thesis: for b1 being Element of the U1 of SCM
for b2 being Element of the InstructionsF of SCM holds
( not InsCode b2 = T or not b1 in Data-Locations or (Exec (b2,s)) . b1 = s . b1 )
let o be Object of SCM; ::_thesis: for b1 being Element of the InstructionsF of SCM holds
( not InsCode b1 = T or not o in Data-Locations or (Exec (b1,s)) . o = s . o )
let I be Instruction of SCM; ::_thesis: ( not InsCode I = T or not o in Data-Locations or (Exec (I,s)) . o = s . o )
assume that
A2: InsCode I = T and
A3: o in Data-Locations ; ::_thesis: (Exec (I,s)) . o = s . o
InsCode I = 6 by A2, A1, RECDEF_2:def_1;
then A4: ex i2 being Element of NAT st I = SCM-goto i2 by AMI_5:13;
o is Data-Location by A3, AMI_2:def_16, AMI_3:27;
hence (Exec (I,s)) . o = s . o by A4, AMI_3:7; ::_thesis: verum
end;
end;
registration
let i1 be Element of NAT ;
cluster SCM-goto i1 -> non ins-loc-free jump-only non sequential ;
coherence
( SCM-goto i1 is jump-only & not SCM-goto i1 is sequential & not SCM-goto i1 is ins-loc-free )
proof
thus InsCode (SCM-goto i1) is jump-only ; :: according to AMISTD_1:def_2 ::_thesis: ( not SCM-goto i1 is sequential & not SCM-goto i1 is ins-loc-free )
JUMP (SCM-goto i1) <> {} ;
hence not SCM-goto i1 is sequential by AMISTD_1:13; ::_thesis: not SCM-goto i1 is ins-loc-free
dom (JumpPart (SCM-goto i1)) = dom <*i1*> by RECDEF_2:def_2
.= {1} by FINSEQ_1:2, FINSEQ_1:def_8 ;
hence not JumpPart (SCM-goto i1) is empty ; :: according to COMPOS_0:def_8 ::_thesis: verum
end;
end;
registration
let a be Data-Location;
let i1 be Element of NAT ;
cluster InsCode (a =0_goto i1) -> jump-only for InsType of the InstructionsF of SCM;
coherence
for b1 being InsType of the InstructionsF of SCM st b1 = InsCode (a =0_goto i1) holds
b1 is jump-only
proof
set S = SCM ;
now__::_thesis:_for_s_being_State_of_SCM
for_o_being_Object_of_SCM
for_I_being_Instruction_of_SCM_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; ::_thesis: for o being Object of SCM
for I being Instruction of SCM st InsCode I = InsCode (a =0_goto i1) & o in Data-Locations holds
(Exec (I,s)) . o = s . o
let o be Object of SCM; ::_thesis: for I being Instruction of SCM st InsCode I = InsCode (a =0_goto i1) & o in Data-Locations holds
(Exec (I,s)) . o = s . o
let I be Instruction of SCM; ::_thesis: ( InsCode I = InsCode (a =0_goto i1) & o in Data-Locations implies (Exec (I,s)) . o = s . o )
assume that
A1: InsCode I = InsCode (a =0_goto i1) and
A2: o in Data-Locations ; ::_thesis: (Exec (I,s)) . o = s . o
InsCode I = 7 by A1, RECDEF_2:def_1;
then A3: ex i2 being Element of NAT ex b being Data-Location st I = b =0_goto i2 by AMI_5:14;
o is Data-Location by A2, AMI_2:def_16, AMI_3:27;
hence (Exec (I,s)) . o = s . o by A3, AMI_3:8; ::_thesis: verum
end;
hence for b1 being InsType of the InstructionsF of SCM st b1 = InsCode (a =0_goto i1) holds
b1 is jump-only by AMISTD_1:def_1; ::_thesis: verum
end;
cluster InsCode (a >0_goto i1) -> jump-only for InsType of the InstructionsF of SCM;
coherence
for b1 being InsType of the InstructionsF of SCM st b1 = InsCode (a >0_goto i1) holds
b1 is jump-only
proof
set S = SCM ;
now__::_thesis:_for_s_being_State_of_SCM
for_o_being_Object_of_SCM
for_I_being_Instruction_of_SCM_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; ::_thesis: for o being Object of SCM
for I being Instruction of SCM st InsCode I = InsCode (a >0_goto i1) & o in Data-Locations holds
(Exec (I,s)) . o = s . o
let o be Object of SCM; ::_thesis: for I being Instruction of SCM st InsCode I = InsCode (a >0_goto i1) & o in Data-Locations holds
(Exec (I,s)) . o = s . o
let I be Instruction of SCM; ::_thesis: ( InsCode I = InsCode (a >0_goto i1) & o in Data-Locations implies (Exec (I,s)) . o = s . o )
assume that
A4: InsCode I = InsCode (a >0_goto i1) and
A5: o in Data-Locations ; ::_thesis: (Exec (I,s)) . o = s . o
InsCode I = 8 by A4, RECDEF_2:def_1;
then A6: ex i2 being Element of NAT ex b being Data-Location st I = b >0_goto i2 by AMI_5:15;
o is Data-Location by A5, AMI_2:def_16, AMI_3:27;
hence (Exec (I,s)) . o = s . o by A6, AMI_3:9; ::_thesis: verum
end;
hence for b1 being InsType of the InstructionsF of SCM 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 Data-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 RECDEF_2:def_2
.= {1} by FINSEQ_1:2, FINSEQ_1:38 ;
hence not JumpPart (a =0_goto i1) is empty ; :: according to COMPOS_0:def_8 ::_thesis: verum
end;
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 RECDEF_2:def_2
.= {1} by FINSEQ_1:2, FINSEQ_1:38 ;
hence not JumpPart (a >0_goto i1) is empty ; :: according to COMPOS_0:def_8 ::_thesis: verum
end;
end;
Lm2: dl. 0 <> dl. 1
by AMI_3:10;
registration
let a, b be Data-Location;
cluster InsCode (a := b) -> non jump-only for InsType of the InstructionsF of SCM;
coherence
for b1 being InsType of the InstructionsF of SCM st b1 = InsCode (a := b) holds
not b1 is jump-only
proof
set w = the State of SCM;
set t = the State of SCM +* (((dl. 0),(dl. 1)) --> (0,1));
A1: InsCode (a := b) = 1 by RECDEF_2:def_1
.= InsCode ((dl. 0) := (dl. 1)) by RECDEF_2:def_1 ;
A2: dl. 0 in Data-Locations by AMI_3:28;
A3: dom (((dl. 0),(dl. 1)) --> (0,1)) = {(dl. 0),(dl. 1)} by FUNCT_4:62;
then A4: dl. 1 in dom (((dl. 0),(dl. 1)) --> (0,1)) by TARSKI:def_2;
dl. 0 in dom (((dl. 0),(dl. 1)) --> (0,1)) by A3, TARSKI:def_2;
then A5: ( the State of SCM +* (((dl. 0),(dl. 1)) --> (0,1))) . (dl. 0) = (((dl. 0),(dl. 1)) --> (0,1)) . (dl. 0) by FUNCT_4:13
.= 0 by AMI_3:10, FUNCT_4:63 ;
(Exec (((dl. 0) := (dl. 1)),( the State of SCM +* (((dl. 0),(dl. 1)) --> (0,1))))) . (dl. 0) = ( the State of SCM +* (((dl. 0),(dl. 1)) --> (0,1))) . (dl. 1) by AMI_3:2
.= (((dl. 0),(dl. 1)) --> (0,1)) . (dl. 1) by A4, FUNCT_4:13
.= 1 by FUNCT_4:63 ;
hence for b1 being InsType of the InstructionsF of SCM st b1 = InsCode (a := b) holds
not b1 is jump-only by A1, A2, A5, AMISTD_1:def_1; ::_thesis: verum
end;
cluster InsCode (AddTo (a,b)) -> non jump-only for InsType of the InstructionsF of SCM;
coherence
for b1 being InsType of the InstructionsF of SCM st b1 = InsCode (AddTo (a,b)) holds
not b1 is jump-only
proof
set w = the State of SCM;
set t = the State of SCM +* (((dl. 0),(dl. 1)) --> (0,1));
A6: InsCode (AddTo (a,b)) = 2 by RECDEF_2:def_1
.= InsCode (AddTo ((dl. 0),(dl. 1))) by RECDEF_2:def_1 ;
A7: dom (((dl. 0),(dl. 1)) --> (0,1)) = {(dl. 0),(dl. 1)} by FUNCT_4:62;
then dl. 0 in dom (((dl. 0),(dl. 1)) --> (0,1)) by TARSKI:def_2;
then A8: ( the State of SCM +* (((dl. 0),(dl. 1)) --> (0,1))) . (dl. 0) = (((dl. 0),(dl. 1)) --> (0,1)) . (dl. 0) by FUNCT_4:13
.= 0 by AMI_3:10, FUNCT_4:63 ;
A9: dl. 0 in Data-Locations by AMI_3:28;
dl. 1 in dom (((dl. 0),(dl. 1)) --> (0,1)) by A7, TARSKI:def_2;
then ( the State of SCM +* (((dl. 0),(dl. 1)) --> (0,1))) . (dl. 1) = (((dl. 0),(dl. 1)) --> (0,1)) . (dl. 1) by FUNCT_4:13
.= 1 by FUNCT_4:63 ;
then ( dl. 0 <> IC & (Exec ((AddTo ((dl. 0),(dl. 1))),( the State of SCM +* (((dl. 0),(dl. 1)) --> (0,1))))) . (dl. 0) = 0 + 1 ) by A8, AMI_3:3, AMI_3:13;
hence for b1 being InsType of the InstructionsF of SCM st b1 = InsCode (AddTo (a,b)) holds
not b1 is jump-only by A6, A8, A9, AMISTD_1:def_1; ::_thesis: verum
end;
cluster InsCode (SubFrom (a,b)) -> non jump-only for InsType of the InstructionsF of SCM;
coherence
for b1 being InsType of the InstructionsF of SCM st b1 = InsCode (SubFrom (a,b)) holds
not b1 is jump-only
proof
set w = the State of SCM;
set t = the State of SCM +* (((dl. 0),(dl. 1)) --> (0,1));
A10: InsCode (SubFrom (a,b)) = 3 by RECDEF_2:def_1
.= InsCode (SubFrom ((dl. 0),(dl. 1))) by RECDEF_2:def_1 ;
A11: dom (((dl. 0),(dl. 1)) --> (0,1)) = {(dl. 0),(dl. 1)} by FUNCT_4:62;
then dl. 0 in dom (((dl. 0),(dl. 1)) --> (0,1)) by TARSKI:def_2;
then A12: ( the State of SCM +* (((dl. 0),(dl. 1)) --> (0,1))) . (dl. 0) = (((dl. 0),(dl. 1)) --> (0,1)) . (dl. 0) by FUNCT_4:13
.= 0 by AMI_3:10, FUNCT_4:63 ;
A13: dl. 0 in Data-Locations by AMI_3:28;
dl. 1 in dom (((dl. 0),(dl. 1)) --> (0,1)) by A11, TARSKI:def_2;
then A14: ( the State of SCM +* (((dl. 0),(dl. 1)) --> (0,1))) . (dl. 1) = (((dl. 0),(dl. 1)) --> (0,1)) . (dl. 1) by FUNCT_4:13
.= 1 by FUNCT_4:63 ;
(Exec ((SubFrom ((dl. 0),(dl. 1))),( the State of SCM +* (((dl. 0),(dl. 1)) --> (0,1))))) . (dl. 0) = (( the State of SCM +* (((dl. 0),(dl. 1)) --> (0,1))) . (dl. 0)) - (( the State of SCM +* (((dl. 0),(dl. 1)) --> (0,1))) . (dl. 1)) by AMI_3:4
.= - 1 by A12, A14 ;
hence for b1 being InsType of the InstructionsF of SCM st b1 = InsCode (SubFrom (a,b)) holds
not b1 is jump-only by A10, A12, A13, AMISTD_1:def_1; ::_thesis: verum
end;
cluster InsCode (MultBy (a,b)) -> non jump-only for InsType of the InstructionsF of SCM;
coherence
for b1 being InsType of the InstructionsF of SCM st b1 = InsCode (MultBy (a,b)) holds
not b1 is jump-only
proof
set w = the State of SCM;
set t = the State of SCM +* (((dl. 0),(dl. 1)) --> (1,0));
A15: InsCode (MultBy (a,b)) = 4 by RECDEF_2:def_1
.= InsCode (MultBy ((dl. 0),(dl. 1))) by RECDEF_2:def_1 ;
A16: dom (((dl. 0),(dl. 1)) --> (1,0)) = {(dl. 0),(dl. 1)} by FUNCT_4:62;
then dl. 0 in dom (((dl. 0),(dl. 1)) --> (1,0)) by TARSKI:def_2;
then A17: ( the State of SCM +* (((dl. 0),(dl. 1)) --> (1,0))) . (dl. 0) = (((dl. 0),(dl. 1)) --> (1,0)) . (dl. 0) by FUNCT_4:13
.= 1 by AMI_3:10, FUNCT_4:63 ;
A18: dl. 0 in Data-Locations by AMI_3:28;
dl. 1 in dom (((dl. 0),(dl. 1)) --> (1,0)) by A16, TARSKI:def_2;
then A19: ( the State of SCM +* (((dl. 0),(dl. 1)) --> (1,0))) . (dl. 1) = (((dl. 0),(dl. 1)) --> (1,0)) . (dl. 1) by FUNCT_4:13
.= 0 by FUNCT_4:63 ;
(Exec ((MultBy ((dl. 0),(dl. 1))),( the State of SCM +* (((dl. 0),(dl. 1)) --> (1,0))))) . (dl. 0) = (( the State of SCM +* (((dl. 0),(dl. 1)) --> (1,0))) . (dl. 0)) * (( the State of SCM +* (((dl. 0),(dl. 1)) --> (1,0))) . (dl. 1)) by AMI_3:5
.= 0 by A19 ;
hence for b1 being InsType of the InstructionsF of SCM st b1 = InsCode (MultBy (a,b)) holds
not b1 is jump-only by A15, A17, A18, AMISTD_1:def_1; ::_thesis: verum
end;
cluster InsCode (Divide (a,b)) -> non jump-only for InsType of the InstructionsF of SCM;
coherence
for b1 being InsType of the InstructionsF of SCM st b1 = InsCode (Divide (a,b)) holds
not b1 is jump-only
proof
set w = the State of SCM;
set t = the State of SCM +* (((dl. 0),(dl. 1)) --> (7,3));
A20: InsCode (Divide (a,b)) = 5 by RECDEF_2:def_1
.= InsCode (Divide ((dl. 0),(dl. 1))) by RECDEF_2:def_1 ;
A21: dom (((dl. 0),(dl. 1)) --> (7,3)) = {(dl. 0),(dl. 1)} by FUNCT_4:62;
then dl. 0 in dom (((dl. 0),(dl. 1)) --> (7,3)) by TARSKI:def_2;
then A22: ( the State of SCM +* (((dl. 0),(dl. 1)) --> (7,3))) . (dl. 0) = (((dl. 0),(dl. 1)) --> (7,3)) . (dl. 0) by FUNCT_4:13
.= 7 by AMI_3:10, FUNCT_4:63 ;
A23: 7 = (2 * 3) + 1 ;
A24: dl. 0 in Data-Locations by AMI_3:28;
dl. 1 in dom (((dl. 0),(dl. 1)) --> (7,3)) by A21, TARSKI:def_2;
then ( the State of SCM +* (((dl. 0),(dl. 1)) --> (7,3))) . (dl. 1) = (((dl. 0),(dl. 1)) --> (7,3)) . (dl. 1) by FUNCT_4:13
.= 3 by FUNCT_4:63 ;
then (Exec ((Divide ((dl. 0),(dl. 1))),( the State of SCM +* (((dl. 0),(dl. 1)) --> (7,3))))) . (dl. 0) = 7 div 3 by A22, Lm2, AMI_3:6
.= 2 by A23, NAT_D:def_1 ;
hence for b1 being InsType of the InstructionsF of SCM st b1 = InsCode (Divide (a,b)) holds
not b1 is jump-only by A20, A22, A24, AMISTD_1:def_1; ::_thesis: verum
end;
end;
registration
let a, b be Data-Location;
clustera := b -> non jump-only ;
coherence
not a := b is jump-only
proof
thus not InsCode (a := b) is jump-only ; :: according to AMISTD_1:def_2 ::_thesis: verum
end;
cluster AddTo (a,b) -> non jump-only ;
coherence
not AddTo (a,b) is jump-only
proof
thus not InsCode (AddTo (a,b)) is jump-only ; :: according to AMISTD_1:def_2 ::_thesis: verum
end;
cluster SubFrom (a,b) -> non jump-only ;
coherence
not SubFrom (a,b) is jump-only
proof
thus not InsCode (SubFrom (a,b)) is jump-only ; :: according to AMISTD_1:def_2 ::_thesis: verum
end;
cluster MultBy (a,b) -> non jump-only ;
coherence
not MultBy (a,b) is jump-only
proof
thus not InsCode (MultBy (a,b)) is jump-only ; :: according to AMISTD_1:def_2 ::_thesis: verum
end;
cluster Divide (a,b) -> non jump-only ;
coherence
not Divide (a,b) is jump-only
proof
thus not InsCode (Divide (a,b)) is jump-only ; :: according to AMISTD_1:def_2 ::_thesis: verum
end;
end;
registration
cluster SCM -> with_explicit_jumps ;
coherence
SCM is with_explicit_jumps
proof
let I be Instruction of SCM; :: according to AMISTD_2:def_2 ::_thesis: I is with_explicit_jumps
thus JUMP I c= rng (JumpPart I) :: according to AMISTD_2:def_1,XBOOLE_0:def_10 ::_thesis: proj2 (JumpPart I) c= JUMP I
proof
let f be set ; :: according to TARSKI:def_3 ::_thesis: ( not f in JUMP I or f in rng (JumpPart I) )
assume A1: f in JUMP I ; ::_thesis: f in rng (JumpPart I)
percases ( I = [0,{},{}] or ex a, b being Data-Location st I = a := b or ex a, b being Data-Location st I = AddTo (a,b) or ex a, b being Data-Location st I = SubFrom (a,b) or ex a, b being Data-Location st I = MultBy (a,b) or ex a, b being Data-Location st I = Divide (a,b) or ex k being Nat st I = SCM-goto k or ex a being Data-Location ex k1 being Nat st I = a =0_goto k1 or ex a being Data-Location ex k1 being Nat st I = a >0_goto k1 ) by AMI_3:24;
suppose I = [0,{},{}] ; ::_thesis: f in rng (JumpPart I)
hence f in rng (JumpPart I) by A1, AMI_3:26; ::_thesis: verum
end;
suppose ex a, b being Data-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 Data-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 Data-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 Data-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 Data-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 k being Nat st I = SCM-goto k ; ::_thesis: f in rng (JumpPart I)
consider k1 being Nat such that
A3: I = SCM-goto k1 by A2;
A4: JumpPart (SCM-goto k1) = <*k1*> by RECDEF_2:def_2;
A5: rng <*k1*> = {k1} by FINSEQ_1:39;
JUMP (SCM-goto k1) = {k1} by Th16;
hence f in rng (JumpPart I) by A1, A3, A4, A5; ::_thesis: verum
end;
supposeA6: ex a being Data-Location ex k1 being Nat st I = a =0_goto k1 ; ::_thesis: f in rng (JumpPart I)
consider a being Data-Location, k1 being Nat such that
A7: I = a =0_goto k1 by A6;
A8: JumpPart (a =0_goto k1) = <*k1*> by RECDEF_2:def_2;
A9: rng <*k1*> = {k1} by FINSEQ_1:39;
JUMP (a =0_goto k1) = {k1} by Th18;
hence f in rng (JumpPart I) by A1, A7, A8, A9; ::_thesis: verum
end;
supposeA10: ex a being Data-Location ex k1 being Nat st I = a >0_goto k1 ; ::_thesis: f in rng (JumpPart I)
consider a being Data-Location, k1 being Nat such that
A11: I = a >0_goto k1 by A10;
A12: JumpPart (a >0_goto k1) = <*k1*> by RECDEF_2:def_2;
A13: rng <*k1*> = {k1} by FINSEQ_1:39;
JUMP (a >0_goto k1) = {k1} by Th20;
hence f in rng (JumpPart I) by A1, A11, A12, A13; ::_thesis: verum
end;
end;
end;
let f be set ; :: according to TARSKI:def_3 ::_thesis: ( not f in proj2 (JumpPart I) or f in JUMP I )
assume f in rng (JumpPart I) ; ::_thesis: f in JUMP I
then consider k being set such that
A14: k in dom (JumpPart I) and
A15: f = (JumpPart I) . k by FUNCT_1:def_3;
percases ( I = [0,{},{}] or ex a, b being Data-Location st I = a := b or ex a, b being Data-Location st I = AddTo (a,b) or ex a, b being Data-Location st I = SubFrom (a,b) or ex a, b being Data-Location st I = MultBy (a,b) or ex a, b being Data-Location st I = Divide (a,b) or ex k being Nat st I = SCM-goto k or ex a being Data-Location ex k being Nat st I = a =0_goto k or ex a being Data-Location ex k1 being Nat st I = a >0_goto k1 ) by AMI_3:24;
suppose I = [0,{},{}] ; ::_thesis: f in JUMP I
then dom (JumpPart I) = dom {} by RECDEF_2:def_2;
hence f in JUMP I by A14; ::_thesis: verum
end;
suppose ex a, b being Data-Location st I = a := b ; ::_thesis: f in JUMP I
then consider a, b being Data-Location such that
A16: I = a := b ;
k in dom {} by A14, A16, RECDEF_2:def_2;
hence f in JUMP I ; ::_thesis: verum
end;
suppose ex a, b being Data-Location st I = AddTo (a,b) ; ::_thesis: f in JUMP I
then consider a, b being Data-Location such that
A17: I = AddTo (a,b) ;
k in dom {} by A14, A17, RECDEF_2:def_2;
hence f in JUMP I ; ::_thesis: verum
end;
suppose ex a, b being Data-Location st I = SubFrom (a,b) ; ::_thesis: f in JUMP I
then consider a, b being Data-Location such that
A18: I = SubFrom (a,b) ;
k in dom {} by A14, A18, RECDEF_2:def_2;
hence f in JUMP I ; ::_thesis: verum
end;
suppose ex a, b being Data-Location st I = MultBy (a,b) ; ::_thesis: f in JUMP I
then consider a, b being Data-Location such that
A19: I = MultBy (a,b) ;
k in dom {} by A14, A19, RECDEF_2:def_2;
hence f in JUMP I ; ::_thesis: verum
end;
suppose ex a, b being Data-Location st I = Divide (a,b) ; ::_thesis: f in JUMP I
then consider a, b being Data-Location such that
A20: I = Divide (a,b) ;
k in dom {} by A14, A20, RECDEF_2:def_2;
hence f in JUMP I ; ::_thesis: verum
end;
suppose ex k being Nat st I = SCM-goto k ; ::_thesis: f in JUMP I
then consider k1 being Nat such that
A21: I = SCM-goto k1 ;
A22: JumpPart I = <*k1*> by A21, RECDEF_2:def_2;
then k = 1 by A14, FINSEQ_1:90;
then A23: f = k1 by A22, A15, FINSEQ_1:def_8;
JUMP I = {k1} by A21, Th16;
hence f in JUMP I by A23, TARSKI:def_1; ::_thesis: verum
end;
suppose ex a being Data-Location ex k being Nat st I = a =0_goto k ; ::_thesis: f in JUMP I
then consider a being Data-Location, k1 being Nat such that
A24: I = a =0_goto k1 ;
A25: JumpPart I = <*k1*> by A24, RECDEF_2:def_2;
then k = 1 by A14, FINSEQ_1:90;
then A26: f = k1 by A25, A15, FINSEQ_1:40;
JUMP I = {k1} by A24, Th18;
hence f in JUMP I by A26, TARSKI:def_1; ::_thesis: verum
end;
suppose ex a being Data-Location ex k1 being Nat st I = a >0_goto k1 ; ::_thesis: f in JUMP I
then consider a being Data-Location, k1 being Nat such that
A27: I = a >0_goto k1 ;
A28: JumpPart I = <*k1*> by A27, RECDEF_2:def_2;
then k = 1 by A14, FINSEQ_1:90;
then A29: f = k1 by A28, A15, FINSEQ_1:40;
JUMP I = {k1} by A27, Th20;
hence f in JUMP I by A29, TARSKI:def_1; ::_thesis: verum
end;
end;
end;
end;
theorem Th23: :: AMI_6:23
for i1 being Element of NAT
for k being Nat holds IncAddr ((SCM-goto i1),k) = SCM-goto (i1 + k)
proof
let i1 be Element of NAT ; ::_thesis: for k being Nat holds IncAddr ((SCM-goto i1),k) = SCM-goto (i1 + k)
let k be Nat; ::_thesis: IncAddr ((SCM-goto i1),k) = SCM-goto (i1 + k)
A1: JumpPart (IncAddr ((SCM-goto i1),k)) = k + (JumpPart (SCM-goto i1)) by COMPOS_0:def_9;
then A2: dom (JumpPart (IncAddr ((SCM-goto i1),k))) = dom (JumpPart (SCM-goto i1)) by VALUED_1:def_2;
A3: dom (JumpPart (SCM-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 (SCM-goto i1)) by RECDEF_2:def_2 ;
A4: for x being set st x in dom (JumpPart (SCM-goto i1)) holds
(JumpPart (IncAddr ((SCM-goto i1),k))) . x = (JumpPart (SCM-goto (i1 + k))) . x
proof
let x be set ; ::_thesis: ( x in dom (JumpPart (SCM-goto i1)) implies (JumpPart (IncAddr ((SCM-goto i1),k))) . x = (JumpPart (SCM-goto (i1 + k))) . x )
assume A5: x in dom (JumpPart (SCM-goto i1)) ; ::_thesis: (JumpPart (IncAddr ((SCM-goto i1),k))) . x = (JumpPart (SCM-goto (i1 + k))) . x
then x in dom <*i1*> by RECDEF_2:def_2;
then A6: x = 1 by FINSEQ_1:90;
set f = (JumpPart (SCM-goto i1)) . x;
A7: (JumpPart (IncAddr ((SCM-goto i1),k))) . x = k + ((JumpPart (SCM-goto i1)) . x) by A5, A2, A1, VALUED_1:def_2;
(JumpPart (SCM-goto i1)) . x = <*i1*> . x by RECDEF_2:def_2
.= i1 by A6, FINSEQ_1:def_8 ;
hence (JumpPart (IncAddr ((SCM-goto i1),k))) . x = <*(i1 + k)*> . x by A6, A7, FINSEQ_1:def_8
.= (JumpPart (SCM-goto (i1 + k))) . x by RECDEF_2:def_2 ;
::_thesis: verum
end;
A8: AddressPart (IncAddr ((SCM-goto i1),k)) = AddressPart (SCM-goto i1) by COMPOS_0:def_9
.= {} by RECDEF_2:def_3
.= AddressPart (SCM-goto (i1 + k)) by RECDEF_2:def_3 ;
A9: InsCode (IncAddr ((SCM-goto i1),k)) = InsCode (SCM-goto i1) by COMPOS_0:def_9
.= 6 by RECDEF_2:def_1
.= InsCode (SCM-goto (i1 + k)) by RECDEF_2:def_1 ;
JumpPart (IncAddr ((SCM-goto i1),k)) = JumpPart (SCM-goto (i1 + k)) by A2, A3, A4, FUNCT_1:2;
hence IncAddr ((SCM-goto i1),k) = SCM-goto (i1 + k) by A8, A9, COMPOS_0:1; ::_thesis: verum
end;
theorem Th24: :: AMI_6:24
for a being Data-Location
for i1 being Element of NAT
for k being Nat holds IncAddr ((a =0_goto i1),k) = a =0_goto (i1 + k)
proof
let a be Data-Location; ::_thesis: for i1 being Element of NAT
for k being Nat holds IncAddr ((a =0_goto i1),k) = a =0_goto (i1 + k)
let i1 be Element of NAT ; ::_thesis: for k being Nat holds IncAddr ((a =0_goto i1),k) = a =0_goto (i1 + k)
let k be Nat; ::_thesis: IncAddr ((a =0_goto i1),k) = a =0_goto (i1 + k)
A1: JumpPart (IncAddr ((a =0_goto i1),k)) = k + (JumpPart (a =0_goto i1)) by COMPOS_0:def_9;
then A2: dom (JumpPart (IncAddr ((a =0_goto i1),k))) = dom (JumpPart (a =0_goto i1)) by VALUED_1:def_2;
A3: dom (JumpPart (a =0_goto (i1 + k))) = dom <*(i1 + k)*> by RECDEF_2:def_2
.= Seg 1 by FINSEQ_1:38
.= dom <*i1*> by FINSEQ_1:38
.= dom (JumpPart (a =0_goto i1)) by RECDEF_2:def_2 ;
A4: for x being set st x in dom (JumpPart (a =0_goto i1)) holds
(JumpPart (IncAddr ((a =0_goto i1),k))) . x = (JumpPart (a =0_goto (i1 + k))) . x
proof
let x be set ; ::_thesis: ( x in dom (JumpPart (a =0_goto i1)) implies (JumpPart (IncAddr ((a =0_goto i1),k))) . x = (JumpPart (a =0_goto (i1 + k))) . x )
assume A5: x in dom (JumpPart (a =0_goto i1)) ; ::_thesis: (JumpPart (IncAddr ((a =0_goto i1),k))) . x = (JumpPart (a =0_goto (i1 + k))) . x
then x in dom <*i1*> by RECDEF_2:def_2;
then A6: x = 1 by FINSEQ_1:90;
set f = (JumpPart (a =0_goto i1)) . x;
A7: (JumpPart (IncAddr ((a =0_goto i1),k))) . x = k + ((JumpPart (a =0_goto i1)) . x) by A1, A2, A5, VALUED_1:def_2;
(JumpPart (a =0_goto i1)) . x = <*i1*> . x by RECDEF_2:def_2
.= i1 by A6, FINSEQ_1:40 ;
hence (JumpPart (IncAddr ((a =0_goto i1),k))) . x = <*(i1 + k)*> . x by A6, A7, FINSEQ_1:40
.= (JumpPart (a =0_goto (i1 + k))) . x by RECDEF_2:def_2 ;
::_thesis: verum
end;
A8: AddressPart (IncAddr ((a =0_goto i1),k)) = AddressPart (a =0_goto i1) by COMPOS_0:def_9
.= <*a*> by RECDEF_2:def_3
.= AddressPart (a =0_goto (i1 + k)) by RECDEF_2:def_3 ;
A9: InsCode (IncAddr ((a =0_goto i1),k)) = InsCode (a =0_goto i1) by COMPOS_0:def_9
.= 7 by RECDEF_2:def_1
.= InsCode (a =0_goto (i1 + k)) by RECDEF_2:def_1 ;
JumpPart (IncAddr ((a =0_goto i1),k)) = JumpPart (a =0_goto (i1 + k)) by A2, A3, A4, FUNCT_1:2;
hence IncAddr ((a =0_goto i1),k) = a =0_goto (i1 + k) by A8, A9, COMPOS_0:1; ::_thesis: verum
end;
theorem Th25: :: AMI_6:25
for a being Data-Location
for i1 being Element of NAT
for k being Nat holds IncAddr ((a >0_goto i1),k) = a >0_goto (i1 + k)
proof
let a be Data-Location; ::_thesis: for i1 being Element of NAT
for k being Nat holds IncAddr ((a >0_goto i1),k) = a >0_goto (i1 + k)
let i1 be Element of NAT ; ::_thesis: for k being Nat holds IncAddr ((a >0_goto i1),k) = a >0_goto (i1 + k)
let k be Nat; ::_thesis: IncAddr ((a >0_goto i1),k) = a >0_goto (i1 + k)
A1: JumpPart (IncAddr ((a >0_goto i1),k)) = k + (JumpPart (a >0_goto i1)) by COMPOS_0:def_9;
then A2: dom (JumpPart (IncAddr ((a >0_goto i1),k))) = dom (JumpPart (a >0_goto i1)) by VALUED_1:def_2;
A3: dom (JumpPart (a >0_goto (i1 + k))) = dom <*(i1 + k)*> by RECDEF_2:def_2
.= Seg 1 by FINSEQ_1:38
.= dom <*i1*> by FINSEQ_1:38
.= dom (JumpPart (a >0_goto i1)) by RECDEF_2:def_2 ;
A4: for x being set st x in dom (JumpPart (a >0_goto i1)) holds
(JumpPart (IncAddr ((a >0_goto i1),k))) . x = (JumpPart (a >0_goto (i1 + k))) . x
proof
let x be set ; ::_thesis: ( x in dom (JumpPart (a >0_goto i1)) implies (JumpPart (IncAddr ((a >0_goto i1),k))) . x = (JumpPart (a >0_goto (i1 + k))) . x )
assume A5: x in dom (JumpPart (a >0_goto i1)) ; ::_thesis: (JumpPart (IncAddr ((a >0_goto i1),k))) . x = (JumpPart (a >0_goto (i1 + k))) . x
then x in dom <*i1*> by RECDEF_2:def_2;
then A6: x = 1 by FINSEQ_1:90;
set f = (JumpPart (a >0_goto i1)) . x;
A7: (JumpPart (IncAddr ((a >0_goto i1),k))) . x = k + ((JumpPart (a >0_goto i1)) . x) by A1, A2, A5, VALUED_1:def_2;
(JumpPart (a >0_goto i1)) . x = <*i1*> . x by RECDEF_2:def_2
.= i1 by A6, FINSEQ_1:40 ;
hence (JumpPart (IncAddr ((a >0_goto i1),k))) . x = <*(i1 + k)*> . x by A6, A7, FINSEQ_1:40
.= (JumpPart (a >0_goto (i1 + k))) . x by RECDEF_2:def_2 ;
::_thesis: verum
end;
A8: AddressPart (IncAddr ((a >0_goto i1),k)) = AddressPart (a >0_goto i1) by COMPOS_0:def_9
.= <*a*> by RECDEF_2:def_3
.= AddressPart (a >0_goto (i1 + k)) by RECDEF_2:def_3 ;
A9: InsCode (IncAddr ((a >0_goto i1),k)) = InsCode (a >0_goto i1) by COMPOS_0:def_9
.= 8 by RECDEF_2:def_1
.= InsCode (a >0_goto (i1 + k)) by RECDEF_2:def_1 ;
JumpPart (IncAddr ((a >0_goto i1),k)) = JumpPart (a >0_goto (i1 + k)) by A2, A3, A4, FUNCT_1:2;
hence IncAddr ((a >0_goto i1),k) = a >0_goto (i1 + k) by A8, A9, COMPOS_0:1; ::_thesis: verum
end;
registration
cluster SCM -> IC-relocable ;
coherence
SCM is IC-relocable
proof
thus SCM is IC-relocable ::_thesis: verum
proof
let I be Instruction of SCM; :: according to AMISTD_2:def_4 ::_thesis: I is IC-relocable
percases ( I = [0,{},{}] or ex a, b being Data-Location st I = a := b or ex a, b being Data-Location st I = AddTo (a,b) or ex a, b being Data-Location st I = SubFrom (a,b) or ex a, b being Data-Location st I = MultBy (a,b) or ex a, b being Data-Location st I = Divide (a,b) or ex k being Nat st I = SCM-goto k or ex a being Data-Location ex k being Nat st I = a =0_goto k or ex a being Data-Location ex k being Nat st I = a >0_goto k ) by AMI_3:24;
suppose I = [0,{},{}] ; ::_thesis: I is IC-relocable
hence I is IC-relocable by AMI_3:26; ::_thesis: verum
end;
suppose ex a, b being Data-Location st I = a := b ; ::_thesis: I is IC-relocable
hence I is IC-relocable ; ::_thesis: verum
end;
suppose ex a, b being Data-Location st I = AddTo (a,b) ; ::_thesis: I is IC-relocable
hence I is IC-relocable ; ::_thesis: verum
end;
suppose ex a, b being Data-Location st I = SubFrom (a,b) ; ::_thesis: I is IC-relocable
hence I is IC-relocable ; ::_thesis: verum
end;
suppose ex a, b being Data-Location st I = MultBy (a,b) ; ::_thesis: I is IC-relocable
hence I is IC-relocable ; ::_thesis: verum
end;
suppose ex a, b being Data-Location st I = Divide (a,b) ; ::_thesis: I is IC-relocable
hence I is IC-relocable ; ::_thesis: verum
end;
supposeA1: ex k being Nat st I = SCM-goto k ; ::_thesis: I is IC-relocable
let j, k be Nat; :: according to AMISTD_2:def_3 ::_thesis: for b1 being set holds K357((IC (Exec ((IncAddr (I,j)),b1))),k) = IC (Exec ((IncAddr (I,(j + k))),(IncIC (b1,k))))
let s1 be State of SCM; ::_thesis: K357((IC (Exec ((IncAddr (I,j)),s1))),k) = IC (Exec ((IncAddr (I,(j + k))),(IncIC (s1,k))))
set s2 = IncIC (s1,k);
consider k1 being Nat such that
A2: I = SCM-goto k1 by A1;
reconsider i1 = k1 as Element of NAT by ORDINAL1:def_12;
IC (Exec (I,s1)) = k1 by A2, AMI_3:7;
hence (IC (Exec ((IncAddr (I,j)),s1))) + k = (IC (Exec ((SCM-goto (j + k1)),s1))) + k by A2, Th23
.= (j + k1) + k by AMI_3:7
.= IC (Exec ((SCM-goto ((j + i1) + k)),(IncIC (s1,k)))) by AMI_3:7
.= IC (Exec ((SCM-goto ((j + k) + i1)),(IncIC (s1,k))))
.= IC (Exec ((IncAddr (I,(j + k))),(IncIC (s1,k)))) by A2, Th23 ;
::_thesis: verum
end;
suppose ex a being Data-Location ex k being Nat st I = a =0_goto k ; ::_thesis: I is IC-relocable
then consider a being Data-Location, k1 being Nat such that
A3: I = a =0_goto k1 ;
reconsider i1 = k1 as Element of NAT by ORDINAL1:def_12;
let j, k be Nat; :: according to AMISTD_2:def_3 ::_thesis: for b1 being set holds K357((IC (Exec ((IncAddr (I,j)),b1))),k) = IC (Exec ((IncAddr (I,(j + k))),(IncIC (b1,k))))
let s1 be State of SCM; ::_thesis: K357((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 AMI_5:2, FUNCOP_1:13;
then not a in dom ((IC ) .--> ((IC s1) + k)) by TARSKI:def_1;
then A4: s1 . a = (IncIC (s1,k)) . a by FUNCT_4:11;
now__::_thesis:_(IC_(Exec_((IncAddr_(I,j)),s1)))_+_k_=_IC_(Exec_((IncAddr_(I,(j_+_k))),(IncIC_(s1,k))))
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))))
then IC (Exec (I,s1)) = k1 by A3, AMI_3:8;
hence (IC (Exec ((IncAddr (I,j)),s1))) + k = (IC (Exec ((a =0_goto (j + k1)),s1))) + k by A3, Th24
.= (j + k1) + k by A5, AMI_3:8
.= IC (Exec ((a =0_goto ((j + i1) + k)),(IncIC (s1,k)))) by A4, A5, AMI_3:8
.= IC (Exec ((a =0_goto ((j + k) + i1)),(IncIC (s1,k))))
.= IC (Exec ((IncAddr (I,(j + k))),(IncIC (s1,k)))) by A3, Th24 ;
::_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, Th24;
A8: IncAddr (I,(j + k)) = a =0_goto (i1 + (j + k)) by A3, Th24;
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, AMI_3:8
.= ((IC s1) + 1) + k
.= succ (IC (IncIC (s1,k))) by A9
.= IC (Exec ((IncAddr (I,(j + k))),(IncIC (s1,k)))) by A8, A6, A4, AMI_3:8 ; ::_thesis: verum
end;
end;
end;
hence K357((IC (Exec ((IncAddr (I,j)),s1))),k) = IC (Exec ((IncAddr (I,(j + k))),(IncIC (s1,k)))) ; ::_thesis: verum
end;
suppose ex a being Data-Location ex k being Nat st I = a >0_goto k ; ::_thesis: I is IC-relocable
then consider a being Data-Location, k1 being Nat such that
A10: I = a >0_goto k1 ;
reconsider i1 = k1 as Element of NAT by ORDINAL1:def_12;
let j, k be Nat; :: according to AMISTD_2:def_3 ::_thesis: for b1 being set holds K357((IC (Exec ((IncAddr (I,j)),b1))),k) = IC (Exec ((IncAddr (I,(j + k))),(IncIC (b1,k))))
let s1 be State of SCM; ::_thesis: K357((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 AMI_5:2, FUNCOP_1:13;
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: K357((IC (Exec ((IncAddr (I,j)),s1))),k) = IC (Exec ((IncAddr (I,(j + k))),(IncIC (s1,k))))
then IC (Exec (I,s1)) = k1 by A10, AMI_3:9;
hence (IC (Exec ((IncAddr (I,j)),s1))) + k = (IC (Exec ((a >0_goto (j + k1)),s1))) + k by A10, Th25
.= (j + k1) + k by A12, AMI_3:9
.= IC (Exec ((a >0_goto ((j + i1) + k)),(IncIC (s1,k)))) by A11, A12, AMI_3:9
.= IC (Exec ((a >0_goto ((j + k) + i1)),(IncIC (s1,k))))
.= IC (Exec ((IncAddr (I,(j + k))),(IncIC (s1,k)))) by A10, Th25 ;
::_thesis: verum
end;
supposeA13: s1 . a <= 0 ; ::_thesis: K357((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, Th25;
A15: IncAddr (I,(j + k)) = a >0_goto (i1 + (j + k)) by A10, Th25;
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, AMI_3:9
.= ((IC s1) + 1) + k
.= succ (IC (IncIC (s1,k))) by A16
.= IC (Exec ((IncAddr (I,(j + k))),(IncIC (s1,k)))) by A15, A13, A11, AMI_3:9 ; ::_thesis: verum
end;
end;
end;
end;
end;
end;
end;