:: SCMRING2 semantic presentation

definition

let R be good Ring;
func SCM R -> strict AMI-Struct of {the carrier of R} means :Def1: :: SCMRING2:def 1
( the carrier of it = NAT & the Instruction-Counter of it = 0 & the Instruction-Locations of it = SCM-Instr-Loc & the Instruction-Codes of it = Segm 8 & the Instructions of it = SCM-Instr R & the Object-Kind of it = SCM-OK R & the Execution of it = SCM-Exec R );
existence

proof end;

uniqueness ;

end;

:: deftheorem Def1 defines SCM SCMRING2:def 1 :

registration

let R be good Ring;
cluster SCM R -> non empty strict non void ;
coherence

proof end;

end;

definition

let R be good Ring;
let s be State of (SCM R);
let a be Element of SCM-Data-Loc ;
:: original: .
redefine func s . a -> Element of R;
coherence

proof end;

end;

definition

let R be good Ring;
mode Data-Location of R -> Object of (SCM R) means :Def2: :: SCMRING2:def 2
it in the carrier of (SCM R) \ (SCM-Instr-Loc \/ {0});
existence

proof end;

end;

:: deftheorem Def2 defines Data-Location SCMRING2:def 2 :

theorem Th1: :: SCMRING2:1

for x being set
for R being good Ring holds
( x is Data-Location of R iff x in SCM-Data-Loc )

proof end;

definition

let R be good Ring;
let s be State of (SCM R);
let a be Data-Location of R;
:: original: .
redefine func s . a -> Element of R;
coherence

proof end;

end;

theorem Th2: :: SCMRING2:2

for S being non empty 1-sorted holds [0,{} ] in SCM-Instr S

proof end;

theorem Th3: :: SCMRING2:3

for R being good Ring holds [0,{} ] is Instruction of (SCM R)

proof end;

theorem Th4: :: SCMRING2:4

for S being non empty 1-sorted
for x being set
for R being good Ring
for d1, d2 being Data-Location of R st x in {1,2,3,4} holds
[x,<*d1,d2*>] in SCM-Instr S

proof end;

theorem Th5: :: SCMRING2:5

for S being non empty 1-sorted
for t being Element of S
for R being good Ring
for d1 being Data-Location of R holds [5,<*d1,t*>] in SCM-Instr S

proof end;

theorem Th6: :: SCMRING2:6

for S being non empty 1-sorted
for R being good Ring
for i1 being Instruction-Location of (SCM R) holds [6,<*i1*>] in SCM-Instr S

proof end;

theorem Th7: :: SCMRING2:7

for S being non empty 1-sorted
for R being good Ring
for d1 being Data-Location of R
for i1 being Instruction-Location of (SCM R) holds [7,<*i1,d1*>] in SCM-Instr S

proof end;

definition

let R be good Ring;
let a, b be Data-Location of R;
func a := b -> Instruction of (SCM R) equals :: SCMRING2:def 3
[1,<*a,b*>];
coherence

proof end;

func AddTo a,b -> Instruction of (SCM R) equals :: SCMRING2:def 4
[2,<*a,b*>];
coherence

proof end;

func SubFrom a,b -> Instruction of (SCM R) equals :: SCMRING2:def 5
[3,<*a,b*>];
coherence

proof end;

func MultBy a,b -> Instruction of (SCM R) equals :: SCMRING2:def 6
[4,<*a,b*>];
coherence

proof end;

end;

:: deftheorem defines := SCMRING2:def 3 :

:: deftheorem defines AddTo SCMRING2:def 4 :

:: deftheorem defines SubFrom SCMRING2:def 5 :

:: deftheorem defines MultBy SCMRING2:def 6 :

definition

let R be good Ring;
let a be Data-Location of R;
let r be Element of R;
func a := r -> Instruction of (SCM R) equals :: SCMRING2:def 7
[5,<*a,r*>];
coherence

proof end;

end;

:: deftheorem defines := SCMRING2:def 7 :

definition

let R be good Ring;
let l be Instruction-Location of (SCM R);
func goto l -> Instruction of (SCM R) equals :: SCMRING2:def 8
[6,<*l*>];
coherence

proof end;

end;

:: deftheorem defines goto SCMRING2:def 8 :

definition

let R be good Ring;
let l be Instruction-Location of (SCM R);
let a be Data-Location of R;
func a =0_goto l -> Instruction of (SCM R) equals :: SCMRING2:def 9
[7,<*l,a*>];
coherence

proof end;

end;

:: deftheorem defines =0_goto SCMRING2:def 9 :

theorem Th8: :: SCMRING2:8

for R being good Ring
for I being set holds
( I is Instruction of (SCM R) iff ( I = [0,{} ] or ex a, b being Data-Location of R st I = a := b or ex a, b being Data-Location of R st I = AddTo a,b or ex a, b being Data-Location of R st I = SubFrom a,b or ex a, b being Data-Location of R st I = MultBy a,b or ex i1 being Instruction-Location of (SCM R) st I = goto i1 or ex a being Data-Location of R ex i1 being Instruction-Location of (SCM R) st I = a =0_goto i1 or ex a being Data-Location of R ex r being Element of R st I = a := r ) )

proof end;

registration

let R be good Ring;
cluster SCM R -> non empty strict non void IC-Ins-separated ;
coherence

proof end;

end;

theorem Th9: :: SCMRING2:9

for R being good Ring holds IC (SCM R) = 0 by Def1;

theorem Th10: :: SCMRING2:10

for R being good Ring
for s being State of (SCM R)
for S being SCM-State of R st S = s holds
IC s = IC S

proof end;

definition

let R be good Ring;
let i1 be Instruction-Location of (SCM R);
func Next i1 -> Instruction-Location of (SCM R) means :Def10: :: SCMRING2:def 10
ex mj being Element of SCM-Instr-Loc st
( mj = i1 & it = Next mj );
existence

proof end;

uniqueness ;

end;

:: deftheorem Def10 defines Next SCMRING2:def 10 :

theorem Th11: :: SCMRING2:11

for R being good Ring
for i1 being Instruction-Location of (SCM R)
for mj being Element of SCM-Instr-Loc st mj = i1 holds
Next mj = Next i1

proof end;

theorem Th12: :: SCMRING2:12

for R being good Ring
for s being State of (SCM R)
for I being Instruction of (SCM R)
for i being Element of SCM-Instr R st i = I holds
for S being SCM-State of R st S = s holds
Exec I,s = SCM-Exec-Res i,S

proof end;

theorem Th13: :: SCMRING2:13

for R being good Ring
for a, b being Data-Location of R
for s being State of (SCM R) holds
( (Exec (a := b),s) . (IC (SCM R)) = Next (IC s) & (Exec (a := b),s) . a = s . b & ( for c being Data-Location of R st c <> a holds
(Exec (a := b),s) . c = s . c ) )

proof end;

theorem Th14: :: SCMRING2:14

for R being good Ring
for a, b being Data-Location of R
for s being State of (SCM R) holds
( (Exec (AddTo a,b),s) . (IC (SCM R)) = Next (IC s) & (Exec (AddTo a,b),s) . a = (s . a) + (s . b) & ( for c being Data-Location of R st c <> a holds
(Exec (AddTo a,b),s) . c = s . c ) )

proof end;

theorem Th15: :: SCMRING2:15

for R being good Ring
for a, b being Data-Location of R
for s being State of (SCM R) holds
( (Exec (SubFrom a,b),s) . (IC (SCM R)) = Next (IC s) & (Exec (SubFrom a,b),s) . a = (s . a) - (s . b) & ( for c being Data-Location of R st c <> a holds
(Exec (SubFrom a,b),s) . c = s . c ) )

proof end;

theorem Th16: :: SCMRING2:16

for R being good Ring
for a, b being Data-Location of R
for s being State of (SCM R) holds
( (Exec (MultBy a,b),s) . (IC (SCM R)) = Next (IC s) & (Exec (MultBy a,b),s) . a = (s . a) * (s . b) & ( for c being Data-Location of R st c <> a holds
(Exec (MultBy a,b),s) . c = s . c ) )

proof end;

theorem :: SCMRING2:17

for R being good Ring
for c being Data-Location of R
for i1 being Instruction-Location of (SCM R)
for s being State of (SCM R) holds
( (Exec (goto i1),s) . (IC (SCM R)) = i1 & (Exec (goto i1),s) . c = s . c )

proof end;

theorem Th18: :: SCMRING2:18

for R being good Ring
for a, c being Data-Location of R
for i1 being Instruction-Location of (SCM R)
for s being State of (SCM R) holds
( ( s . a = 0. R implies (Exec (a =0_goto i1),s) . (IC (SCM R)) = i1 ) & ( s . a <> 0. R implies (Exec (a =0_goto i1),s) . (IC (SCM R)) = Next (IC s) ) & (Exec (a =0_goto i1),s) . c = s . c )

proof end;

theorem Th19: :: SCMRING2:19

for R being good Ring
for r being Element of R
for a being Data-Location of R
for s being State of (SCM R) holds
( (Exec (a := r),s) . (IC (SCM R)) = Next (IC s) & (Exec (a := r),s) . a = r & ( for c being Data-Location of R st c <> a holds
(Exec (a := r),s) . c = s . c ) )

proof end;

theorem Th20: :: SCMRING2:20

for R being good Ring
for I being Instruction of (SCM R) st ex s being State of (SCM R) st (Exec I,s) . (IC (SCM R)) = Next (IC s) holds
not I is halting

proof end;

theorem Th21: :: SCMRING2:21

for R being good Ring
for I being Instruction of (SCM R) st I = [0,{} ] holds
I is halting

proof end;

Lm1: for R being good Ring
for a, b being Data-Location of R holds not a := b is halting

proof end;

Lm2: for R being good Ring
for a, b being Data-Location of R holds not AddTo a,b is halting

proof end;

Lm3: for R being good Ring
for a, b being Data-Location of R holds not SubFrom a,b is halting

proof end;

Lm4: for R being good Ring
for a, b being Data-Location of R holds not MultBy a,b is halting

proof end;

Lm5: for R being good Ring
for i1 being Instruction-Location of (SCM R) holds not goto i1 is halting

proof end;

Lm6: for R being good Ring
for a being Data-Location of R
for i1 being Instruction-Location of (SCM R) holds not a =0_goto i1 is halting

proof end;

Lm7: for R being good Ring
for r being Element of R
for a being Data-Location of R holds not a := r is halting

proof end;

registration

let R be good Ring;
let a, b be Data-Location of R;
cluster a := b -> non halting ;
coherence by Lm1;
cluster AddTo a,b -> non halting ;
coherence by Lm2;
cluster SubFrom a,b -> non halting ;
coherence by Lm3;
cluster MultBy a,b -> non halting ;
coherence by Lm4;

end;

registration

let R be good Ring;
let i1 be Instruction-Location of (SCM R);
cluster goto i1 -> non halting ;
coherence by Lm5;

end;

registration

let R be good Ring;
let a be Data-Location of R;
let i1 be Instruction-Location of (SCM R);
cluster a =0_goto i1 -> non halting ;
coherence by Lm6;

end;

registration

let R be good Ring;
let a be Data-Location of R;
let r be Element of R;
cluster a := r -> non halting ;
coherence by Lm7;

end;

registration

let R be good Ring;
cluster SCM R -> non empty strict non void halting IC-Ins-separated data-oriented steady-programmed definite realistic ;
coherence

proof end;

end;

theorem :: SCMRING2:22

canceled;

theorem :: SCMRING2:23

canceled;

theorem :: SCMRING2:24

canceled;

theorem :: SCMRING2:25

canceled;

theorem :: SCMRING2:26

canceled;

theorem :: SCMRING2:27

canceled;

theorem :: SCMRING2:28

canceled;

theorem Th29: :: SCMRING2:29

for R being good Ring
for I being Instruction of (SCM R) st I is halting holds
I = halt (SCM R)

proof end;

theorem :: SCMRING2:30

for R being good Ring holds halt (SCM R) = [0,{} ]

proof end;