SCMRING1: The Construction of { \bf SCM } over Ring

:: The Construction of { \bf SCM } over Ring
:: by Artur Korni{\l}owicz
::
:: Received November 29, 1998
:: Copyright (c) 1998-2021 Association of Mizar Users

definition

let S be non empty 1-sorted ;

func SCM-VAL S -> ManySortedSet of Segm 2 equals :: SCMRING1:def 3
<%NAT, the carrier of S%>;

end;

:: deftheorem SCMRING1:def 1 :

:: deftheorem SCMRING1:def 2 :

:: deftheorem defines SCM-VAL SCMRING1:def 3 :

Lm1: for S being non empty 1-sorted holds dom ((SCM-VAL S) * SCM-OK) = SCM-Memory

proof end;

theorem :: SCMRING1:1

canceled;

::$CT

theorem Th1: :: SCMRING1:2

for G being non empty 1-sorted holds ((SCM-VAL G) * SCM-OK) . NAT = NAT

proof end;

theorem Th2: :: SCMRING1:3

for G being non empty 1-sorted
for i being Element of SCM-Memory st i in SCM-Data-Loc holds
((SCM-VAL G) * SCM-OK) . i = the carrier of G

proof end;

registration

let G be non empty 1-sorted ;

cluster SCM-OK * (SCM-VAL G) -> non-empty ;

proof end;

end;

definition

let S be non empty 1-sorted ;
mode SCM-State of S is Element of product ((SCM-VAL S) * SCM-OK);

end;

theorem :: SCMRING1:4

for d1 being Element of SCM-Data-Loc
for G being non empty 1-sorted holds ((SCM-VAL G) * SCM-OK) . d1 = the carrier of G by Th2;

theorem Th4: :: SCMRING1:5

for G being non empty 1-sorted holds pi ((product ((SCM-VAL G) * SCM-OK)),NAT) = NAT

proof end;

theorem Th5: :: SCMRING1:6

for G being non empty 1-sorted
for a being Element of SCM-Data-Loc holds pi ((product ((SCM-VAL G) * SCM-OK)),a) = the carrier of G

proof end;

definition

let G be non empty 1-sorted ;
let s be SCM-State of G;

func IC s -> Element of NAT equals :: SCMRING1:def 4
s . NAT;

proof end;

end;

:: deftheorem defines IC SCMRING1:def 4 :

definition

let R be non empty 1-sorted ;
let s be SCM-State of R;
let u be natural Number ;

func SCM-Chg (s,u) -> SCM-State of R equals :: SCMRING1:def 5
s +* (NAT .--> u);

proof end;

end;

:: deftheorem defines SCM-Chg SCMRING1:def 5 :

theorem :: SCMRING1:7

for G being non empty 1-sorted
for s being SCM-State of G
for u being natural Number holds (SCM-Chg (s,u)) . NAT = u

proof end;

theorem :: SCMRING1:8

for G being non empty 1-sorted
for s being SCM-State of G
for u being natural Number
for mk being Element of SCM-Data-Loc holds (SCM-Chg (s,u)) . mk = s . mk

proof end;

theorem :: SCMRING1:9

for G being non empty 1-sorted
for s being SCM-State of G
for u, v being natural Number holds (SCM-Chg (s,u)) . v = s . v

proof end;

definition

let R be non empty 1-sorted ;
let s be SCM-State of R;
let t be Element of SCM-Data-Loc ;
let u be Element of R;

func SCM-Chg (s,t,u) -> SCM-State of R equals :: SCMRING1:def 6
s +* (t .--> u);

proof end;

end;

:: deftheorem defines SCM-Chg SCMRING1:def 6 :

theorem :: SCMRING1:10

for G being non empty 1-sorted
for s being SCM-State of G
for t being Element of SCM-Data-Loc
for u being Element of G holds (SCM-Chg (s,t,u)) . NAT = s . NAT

proof end;

theorem :: SCMRING1:11

for G being non empty 1-sorted
for s being SCM-State of G
for t being Element of SCM-Data-Loc
for u being Element of G holds (SCM-Chg (s,t,u)) . t = u

proof end;

theorem :: SCMRING1:12

for G being non empty 1-sorted
for s being SCM-State of G
for t being Element of SCM-Data-Loc
for u being Element of G
for mk being Element of SCM-Data-Loc st mk <> t holds
(SCM-Chg (s,t,u)) . mk = s . mk

proof end;

definition

let R be non empty 1-sorted ;
let s be SCM-State of R;
let a be Element of SCM-Data-Loc ;
:: original: .
redefine func s . a -> Element of R;
coherence

proof end;

end;

definition

let S be non empty 1-sorted ;
let d be Element of SCM-Data-Loc ;
let s be Element of S;
:: original: <*
redefine func <*d,s*> -> FinSequence of SCM-Data-Loc \/ the carrier of S;
coherence

proof end;

end;

definition

let R be Ring;
let x be Element of SCM-Instr R;
let s be SCM-State of R;

func SCM-Exec-Res (x,s) -> SCM-State of R equals :: SCMRING1:def 14
SCM-Chg ((SCM-Chg (s,(x address_1),(s . (x address_2)))),((IC s) + 1)) if ex mk, ml being Element of SCM-Data-Loc st x = [1,{},<*mk,ml*>]
SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) + (s . (x address_2))))),((IC s) + 1)) if ex mk, ml being Element of SCM-Data-Loc st x = [2,{},<*mk,ml*>]
SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) - (s . (x address_2))))),((IC s) + 1)) if ex mk, ml being Element of SCM-Data-Loc st x = [3,{},<*mk,ml*>]
SCM-Chg ((SCM-Chg (s,(x address_1),((s . (x address_1)) * (s . (x address_2))))),((IC s) + 1)) if ex mk, ml being Element of SCM-Data-Loc st x = [4,{},<*mk,ml*>]
SCM-Chg (s,(x jump_address)) if ex mk being Element of NAT st x = [6,<*mk*>,{}]
SCM-Chg (s,(IFEQ ((s . (x cond_address)),(0. R),(x cjump_address),((IC s) + 1)))) if ex mk being Element of NAT ex ml being Element of SCM-Data-Loc st x = [7,<*mk*>,<*ml*>]
SCM-Chg ((SCM-Chg (s,(x const_address),(x const_value))),((IC s) + 1)) if ex mk being Element of SCM-Data-Loc ex r being Element of R st x = [5,{},<*mk,r*>]
otherwise s;

coherence ;
consistency by XTUPLE_0:3;

end;

:: deftheorem SCMRING1:def 7 :

:: deftheorem SCMRING1:def 8 :

:: deftheorem SCMRING1:def 9 :

:: deftheorem SCMRING1:def 10 :

:: deftheorem SCMRING1:def 11 :

:: deftheorem SCMRING1:def 12 :

:: deftheorem SCMRING1:def 13 :

:: deftheorem defines SCM-Exec-Res SCMRING1:def 14 :

definition

let R be Ring;

func SCM-Exec R -> Action of (SCM-Instr R),(product ((SCM-VAL R) * SCM-OK)) means :: SCMRING1:def 15
for x being Element of SCM-Instr R
for y being SCM-State of R holds (it . x) . y = SCM-Exec-Res (x,y);

proof end;

proof end;

end;

:: deftheorem defines SCM-Exec SCMRING1:def 15 :

theorem :: SCMRING1:13

canceled;

theorem :: SCMRING1:14

canceled;

theorem :: SCMRING1:15

canceled;

theorem :: SCMRING1:16

canceled;

theorem :: SCMRING1:17

canceled;

theorem :: SCMRING1:18

canceled;

::$CT 6

theorem :: SCMRING1:19

for S being non empty 1-sorted
for s being SCM-State of S holds dom s = SCM-Memory

proof end;