:: AMI_6 semantic presentation

theorem Th1: :: AMI_6:1

for a being Data-Location holds not a in the Instruction-Locations of SCM

proof end;

theorem Th2: :: AMI_6:2

SCM-Data-Loc <> the Instruction-Locations of SCM

proof end;

theorem Th3: :: AMI_6:3

for o being Object of SCM holds
( o = IC SCM or o in the Instruction-Locations of SCM or o is Data-Location )

proof end;

theorem Th4: :: AMI_6:4

for i1, i2 being Instruction-Location of SCM st i1 <> i2 holds
Next i1 <> Next i2

proof end;

theorem Th5: :: AMI_6:5

for a being Data-Location
for s1, s2 being State of SCM st s1,s2 equal_outside the Instruction-Locations of SCM holds
s1 . a = s2 . a

proof end;

theorem Th6: :: AMI_6:6

for N being with_non-empty_elements set
for S being non empty non void IC-Ins-separated definite realistic AMI-Struct of N
for t, u being State of S
for il being Instruction-Location of S
for e being Element of ObjectKind (IC S)
for I being Element of ObjectKind il st e = il & u = t +* ((IC S),il --> e,I) holds
( u . il = I & IC u = il & IC (Following u) = (Exec (u . (IC u)),u) . (IC S) )

proof end;

Lm1: for x, y being set st x in dom <*y*> holds
x = 1

proof end;

Lm2: for x, y, z being set holds
( not x in dom <*y,z*> or x = 1 or x = 2 )

proof end;

Lm3: for T being InsType 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 end;

theorem Th7: :: AMI_6:7

AddressPart (halt SCM ) = {} by AMI_3:71, MCART_1:def 2;

theorem Th8: :: AMI_6:8

for a, b being Data-Location holds AddressPart (a := b) = <*a,b*>

proof end;

theorem Th9: :: AMI_6:9

for a, b being Data-Location holds AddressPart (AddTo a,b) = <*a,b*>

proof end;

theorem Th10: :: AMI_6:10

for a, b being Data-Location holds AddressPart (SubFrom a,b) = <*a,b*>

proof end;

theorem Th11: :: AMI_6:11

for a, b being Data-Location holds AddressPart (MultBy a,b) = <*a,b*>

proof end;

theorem Th12: :: AMI_6:12

for a, b being Data-Location holds AddressPart (Divide a,b) = <*a,b*>

proof end;

theorem Th13: :: AMI_6:13

for i1 being Instruction-Location of SCM holds AddressPart (goto i1) = <*i1*>

proof end;

theorem Th14: :: AMI_6:14

for a being Data-Location
for i1 being Instruction-Location of SCM holds AddressPart (a =0_goto i1) = <*i1,a*>

proof end;

theorem Th15: :: AMI_6:15

for a being Data-Location
for i1 being Instruction-Location of SCM holds AddressPart (a >0_goto i1) = <*i1,a*>

proof end;

theorem Th16: :: AMI_6:16

for T being InsType of SCM st T = 0 holds
AddressParts T = {0}

proof end;

registration

let T be InsType of SCM ;
cluster AddressParts T -> non empty ;
coherence

proof end;

end;

theorem Th17: :: AMI_6:17

for T being InsType of SCM st T = 1 holds
dom (PA (AddressParts T)) = {1,2}

proof end;

theorem Th18: :: AMI_6:18

for T being InsType of SCM st T = 2 holds
dom (PA (AddressParts T)) = {1,2}

proof end;

theorem Th19: :: AMI_6:19

for T being InsType of SCM st T = 3 holds
dom (PA (AddressParts T)) = {1,2}

proof end;

theorem Th20: :: AMI_6:20

for T being InsType of SCM st T = 4 holds
dom (PA (AddressParts T)) = {1,2}

proof end;

theorem Th21: :: AMI_6:21

for T being InsType of SCM st T = 5 holds
dom (PA (AddressParts T)) = {1,2}

proof end;

theorem Th22: :: AMI_6:22

for T being InsType of SCM st T = 6 holds
dom (PA (AddressParts T)) = {1}

proof end;

theorem Th23: :: AMI_6:23

for T being InsType of SCM st T = 7 holds
dom (PA (AddressParts T)) = {1,2}

proof end;

theorem Th24: :: AMI_6:24

for T being InsType of SCM st T = 8 holds
dom (PA (AddressParts T)) = {1,2}

proof end;

theorem Th25: :: AMI_6:25

for a, b being Data-Location holds (PA (AddressParts (InsCode (a := b)))) . 1 = SCM-Data-Loc

proof end;

theorem Th26: :: AMI_6:26

for a, b being Data-Location holds (PA (AddressParts (InsCode (a := b)))) . 2 = SCM-Data-Loc

proof end;

theorem Th27: :: AMI_6:27

for a, b being Data-Location holds (PA (AddressParts (InsCode (AddTo a,b)))) . 1 = SCM-Data-Loc

proof end;

theorem Th28: :: AMI_6:28

for a, b being Data-Location holds (PA (AddressParts (InsCode (AddTo a,b)))) . 2 = SCM-Data-Loc

proof end;

theorem Th29: :: AMI_6:29

for a, b being Data-Location holds (PA (AddressParts (InsCode (SubFrom a,b)))) . 1 = SCM-Data-Loc

proof end;

theorem Th30: :: AMI_6:30

for a, b being Data-Location holds (PA (AddressParts (InsCode (SubFrom a,b)))) . 2 = SCM-Data-Loc

proof end;

theorem Th31: :: AMI_6:31

for a, b being Data-Location holds (PA (AddressParts (InsCode (MultBy a,b)))) . 1 = SCM-Data-Loc

proof end;

theorem Th32: :: AMI_6:32

for a, b being Data-Location holds (PA (AddressParts (InsCode (MultBy a,b)))) . 2 = SCM-Data-Loc

proof end;

theorem Th33: :: AMI_6:33

for a, b being Data-Location holds (PA (AddressParts (InsCode (Divide a,b)))) . 1 = SCM-Data-Loc

proof end;

theorem Th34: :: AMI_6:34

for a, b being Data-Location holds (PA (AddressParts (InsCode (Divide a,b)))) . 2 = SCM-Data-Loc

proof end;

theorem Th35: :: AMI_6:35

for i1 being Instruction-Location of SCM holds (PA (AddressParts (InsCode (goto i1)))) . 1 = the Instruction-Locations of SCM

proof end;

theorem Th36: :: AMI_6:36

for a being Data-Location
for i1 being Instruction-Location of SCM holds (PA (AddressParts (InsCode (a =0_goto i1)))) . 1 = the Instruction-Locations of SCM

proof end;

theorem Th37: :: AMI_6:37

for a being Data-Location
for i1 being Instruction-Location of SCM holds (PA (AddressParts (InsCode (a =0_goto i1)))) . 2 = SCM-Data-Loc

proof end;

theorem Th38: :: AMI_6:38

for a being Data-Location
for i1 being Instruction-Location of SCM holds (PA (AddressParts (InsCode (a >0_goto i1)))) . 1 = the Instruction-Locations of SCM

proof end;

theorem Th39: :: AMI_6:39

for a being Data-Location
for i1 being Instruction-Location of SCM holds (PA (AddressParts (InsCode (a >0_goto i1)))) . 2 = SCM-Data-Loc

proof end;

Lm4: for l being Instruction-Location of SCM
for i being Instruction of SCM st ( for s being State of SCM st IC s = l & s . l = i holds
(Exec i,s) . (IC SCM ) = Next (IC s) ) holds
NIC i,l = {(Next l)}

proof end;

Lm5: for i being Instruction of SCM st ( for l being Instruction-Location of SCM holds NIC i,l = {(Next l)} ) holds
JUMP i is empty

proof end;

theorem Th40: :: AMI_6:40

for il being Instruction-Location of SCM holds NIC (halt SCM ),il = {il}

proof end;

registration

cluster JUMP (halt SCM ) -> empty ;
coherence

proof end;

end;

theorem Th41: :: AMI_6:41

for a, b being Data-Location
for il being Instruction-Location of SCM holds NIC (a := b),il = {(Next il)}

proof end;

registration

let a, b be Data-Location ;
cluster JUMP (a := b) -> empty ;
coherence

proof end;

end;

theorem Th42: :: AMI_6:42

for a, b being Data-Location
for il being Instruction-Location of SCM holds NIC (AddTo a,b),il = {(Next il)}

proof end;

registration

let a, b be Data-Location ;
cluster JUMP (AddTo a,b) -> empty ;
coherence

proof end;

end;

theorem Th43: :: AMI_6:43

for a, b being Data-Location
for il being Instruction-Location of SCM holds NIC (SubFrom a,b),il = {(Next il)}

proof end;

registration

let a, b be Data-Location ;
cluster JUMP (SubFrom a,b) -> empty ;
coherence

proof end;

end;

theorem Th44: :: AMI_6:44

for a, b being Data-Location
for il being Instruction-Location of SCM holds NIC (MultBy a,b),il = {(Next il)}

proof end;

registration

let a, b be Data-Location ;
cluster JUMP (MultBy a,b) -> empty ;
coherence

proof end;

end;

theorem Th45: :: AMI_6:45

for a, b being Data-Location
for il being Instruction-Location of SCM holds NIC (Divide a,b),il = {(Next il)}

proof end;

registration

let a, b be Data-Location ;
cluster JUMP (Divide a,b) -> empty ;
coherence

proof end;

end;

theorem Th46: :: AMI_6:46

for i1, il being Instruction-Location of SCM holds NIC (goto i1),il = {i1}

proof end;

theorem Th47: :: AMI_6:47

for i1 being Instruction-Location of SCM holds JUMP (goto i1) = {i1}

proof end;

registration

let i1 be Instruction-Location of SCM ;
cluster JUMP (goto i1) -> non empty trivial ;
coherence

proof end;

end;

theorem Th48: :: AMI_6:48

for a being Data-Location
for i1, il being Instruction-Location of SCM holds NIC (a =0_goto i1),il = {i1,(Next il)}

proof end;

theorem Th49: :: AMI_6:49

for a being Data-Location
for i1 being Instruction-Location of SCM holds JUMP (a =0_goto i1) = {i1}

proof end;

registration

let a be Data-Location ;
let i1 be Instruction-Location of SCM ;
cluster JUMP (a =0_goto i1) -> non empty trivial ;
coherence

proof end;

end;

theorem Th50: :: AMI_6:50

for a being Data-Location
for i1, il being Instruction-Location of SCM holds NIC (a >0_goto i1),il = {i1,(Next il)}

proof end;

theorem Th51: :: AMI_6:51

for a being Data-Location
for i1 being Instruction-Location of SCM holds JUMP (a >0_goto i1) = {i1}

proof end;

registration

let a be Data-Location ;
let i1 be Instruction-Location of SCM ;
cluster JUMP (a >0_goto i1) -> non empty trivial ;
coherence

proof end;

end;

theorem Th52: :: AMI_6:52

for il being Instruction-Location of SCM holds SUCC il = {il,(Next il)}

proof end;

theorem Th53: :: AMI_6:53

for f being Function of NAT ,the Instruction-Locations of SCM st ( for k being Nat holds f . k = il. k ) holds
( f is bijective & ( for k being Nat holds
( f . (k + 1) in SUCC (f . k) & ( for j being Nat st f . j in SUCC (f . k) holds
k <= j ) ) ) )

proof end;

registration

cluster SCM -> standard ;
coherence

proof end;

end;

theorem Th54: :: AMI_6:54