:: A compiler of arithmetic expressions for { \bf SCM }
:: by Grzegorz Bancerek and Piotr Rudnicki
::
:: Received December 30, 1993
:: Copyright (c) 1993 Association of Mizar Users

theorem Th1: :: SCM_COMP:1

for I1, I2 being FinSequence of the Instructions of SCM
for D being FinSequence of INT
for il, ps, ds being Element of NAT
for s being State-consisting of il,ps,ds,I1 ^ I2,D holds
( s is State-consisting of il,ps,ds,I1,D & s is State-consisting of il,ps + (len I1),ds,I2,D )

proof end;

theorem Th2: :: SCM_COMP:2

for I1, I2 being FinSequence of the Instructions of SCM
for il, ps, ds, k, il1 being Element of NAT
for s being State-consisting of il,ps,ds,I1 ^ I2, <*> INT
for u being State of SCM st u = Computation s,k & il. il1 = IC u holds
u is State-consisting of il1,ps + (len I1),ds,I2, <*> INT

proof end;

Lm1: 1 = { n where n is Element of NAT : n < 1 }
by AXIOMS:30;

Lm2: 5 = { n where n is Element of NAT : n < 5 }
by AXIOMS:30;

definition

func SCM-AE -> non empty strict with_terminals with_nonterminals with_useful_nonterminals binary DTConstrStr means :Def1: :: SCM_COMP:def 1
( Terminals it = SCM-Data-Loc & NonTerminals it = [:1,5:] & ( for x, y, z being Symbol of it holds
( x ==> <*y,z*> iff x in [:1,5:] ) ) );
existence
ex b₁ being non empty strict with_terminals with_nonterminals with_useful_nonterminals binary DTConstrStr st
( Terminals b₁ = SCM-Data-Loc & NonTerminals b₁ = [:1,5:] & ( for x, y, z being Symbol of b₁ holds
( x ==> <*y,z*> iff x in [:1,5:] ) ) )

proof end;

uniqueness
for b₁, b₂ being non empty strict with_terminals with_nonterminals with_useful_nonterminals binary DTConstrStr st Terminals b₁ = SCM-Data-Loc & NonTerminals b₁ = [:1,5:] & ( for x, y, z being Symbol of b₁ holds
( x ==> <*y,z*> iff x in [:1,5:] ) ) & Terminals b₂ = SCM-Data-Loc & NonTerminals b₂ = [:1,5:] & ( for x, y, z being Symbol of b₂ holds
( x ==> <*y,z*> iff x in [:1,5:] ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def1 defines SCM-AE SCM_COMP:def 1 :

for b₁ being non empty strict with_terminals with_nonterminals with_useful_nonterminals binary DTConstrStr holds
( b₁ = SCM-AE iff ( Terminals b₁ = SCM-Data-Loc & NonTerminals b₁ = [:1,5:] & ( for x, y, z being Symbol of b₁ holds
( x ==> <*y,z*> iff x in [:1,5:] ) ) ) );

definition

mode bin-term is Element of TS SCM-AE ;

end;

Lm3: NonTerminals SCM-AE = [:1,5:]
by Def1;

definition

let nt be NonTerminal of SCM-AE ;
let tl, tr be bin-term;
:: original: -tree
redefine func nt -tree tl,tr -> bin-term;
coherence
nt -tree tl,tr is bin-term

proof end;

end;

definition

let t be Terminal of SCM-AE ;
:: original: root-tree
redefine func root-tree t -> bin-term;
coherence
root-tree t is bin-term by DTCONSTR:def 4;

end;

definition

let t be Terminal of SCM-AE ;
func @ t -> Data-Location equals :: SCM_COMP:def 2
t;
coherence
t is Data-Location

proof end;

end;

:: deftheorem defines @ SCM_COMP:def 2 :

for t being Terminal of SCM-AE holds @ t = t;

theorem Th3: :: SCM_COMP:3

for nt being NonTerminal of SCM-AE holds
( nt = [0 ,0 ] or nt = [0 ,1] or nt = [0 ,2] or nt = [0 ,3] or nt = [0 ,4] )

proof end;

theorem :: SCM_COMP:4

( [0 ,0 ] is NonTerminal of SCM-AE & [0 ,1] is NonTerminal of SCM-AE & [0 ,2] is NonTerminal of SCM-AE & [0 ,3] is NonTerminal of SCM-AE & [0 ,4] is NonTerminal of SCM-AE )

proof end;

then reconsider nt0 = [0 ,0 ], nt1 = [0 ,1], nt2 = [0 ,2], nt3 = [0 ,3], nt4 = [0 ,4] as NonTerminal of SCM-AE ;

definition

let t1, t2 be bin-term;
func t1 + t2 -> bin-term equals :: SCM_COMP:def 3
[0 ,0 ] -tree t1,t2;
coherence
[0 ,0 ] -tree t1,t2 is bin-term

proof end;

func t1 - t2 -> bin-term equals :: SCM_COMP:def 4
[0 ,1] -tree t1,t2;
coherence
[0 ,1] -tree t1,t2 is bin-term

proof end;

func t1 * t2 -> bin-term equals :: SCM_COMP:def 5
[0 ,2] -tree t1,t2;
coherence
[0 ,2] -tree t1,t2 is bin-term

proof end;

func t1 div t2 -> bin-term equals :: SCM_COMP:def 6
[0 ,3] -tree t1,t2;
coherence
[0 ,3] -tree t1,t2 is bin-term

proof end;

func t1 mod t2 -> bin-term equals :: SCM_COMP:def 7
[0 ,4] -tree t1,t2;
coherence
[0 ,4] -tree t1,t2 is bin-term

proof end;

end;

:: deftheorem defines + SCM_COMP:def 3 :

for t1, t2 being bin-term holds t1 + t2 = [0 ,0 ] -tree t1,t2;

:: deftheorem defines - SCM_COMP:def 4 :

for t1, t2 being bin-term holds t1 - t2 = [0 ,1] -tree t1,t2;

:: deftheorem defines * SCM_COMP:def 5 :

for t1, t2 being bin-term holds t1 * t2 = [0 ,2] -tree t1,t2;

:: deftheorem defines div SCM_COMP:def 6 :

for t1, t2 being bin-term holds t1 div t2 = [0 ,3] -tree t1,t2;

:: deftheorem defines mod SCM_COMP:def 7 :

for t1, t2 being bin-term holds t1 mod t2 = [0 ,4] -tree t1,t2;

theorem :: SCM_COMP:5

for term being bin-term holds
( ex t being Terminal of SCM-AE st term = root-tree t or ex tl, tr being bin-term st
( term = tl + tr or term = tl - tr or term = tl * tr or term = tl div tr or term = tl mod tr ) )

proof end;

definition

let o be NonTerminal of SCM-AE ;
let i, j be Integer;
func o -Meaning_on i,j -> Integer equals :Def8: :: SCM_COMP:def 8
i + j if o = [0 ,0 ]
i - j if o = [0 ,1]
i * j if o = [0 ,2]
i div j if o = [0 ,3]
i mod j if o = [0 ,4]
;
coherence
( ( o = [0 ,0 ] implies i + j is Integer ) & ( o = [0 ,1] implies i - j is Integer ) & ( o = [0 ,2] implies i * j is Integer ) & ( o = [0 ,3] implies i div j is Integer ) & ( o = [0 ,4] implies i mod j is Integer ) ) ;
consistency
for b₁ being Integer holds
( ( o = [0 ,0 ] & o = [0 ,1] implies ( b₁ = i + j iff b₁ = i - j ) ) & ( o = [0 ,0 ] & o = [0 ,2] implies ( b₁ = i + j iff b₁ = i * j ) ) & ( o = [0 ,0 ] & o = [0 ,3] implies ( b₁ = i + j iff b₁ = i div j ) ) & ( o = [0 ,0 ] & o = [0 ,4] implies ( b₁ = i + j iff b₁ = i mod j ) ) & ( o = [0 ,1] & o = [0 ,2] implies ( b₁ = i - j iff b₁ = i * j ) ) & ( o = [0 ,1] & o = [0 ,3] implies ( b₁ = i - j iff b₁ = i div j ) ) & ( o = [0 ,1] & o = [0 ,4] implies ( b₁ = i - j iff b₁ = i mod j ) ) & ( o = [0 ,2] & o = [0 ,3] implies ( b₁ = i * j iff b₁ = i div j ) ) & ( o = [0 ,2] & o = [0 ,4] implies ( b₁ = i * j iff b₁ = i mod j ) ) & ( o = [0 ,3] & o = [0 ,4] implies ( b₁ = i div j iff b₁ = i mod j ) ) )

proof end;

end;

:: deftheorem Def8 defines -Meaning_on SCM_COMP:def 8 :

for o being NonTerminal of SCM-AE
for i, j being Integer holds
( ( o = [0 ,0 ] implies o -Meaning_on i,j = i + j ) & ( o = [0 ,1] implies o -Meaning_on i,j = i - j ) & ( o = [0 ,2] implies o -Meaning_on i,j = i * j ) & ( o = [0 ,3] implies o -Meaning_on i,j = i div j ) & ( o = [0 ,4] implies o -Meaning_on i,j = i mod j ) );

definition

let s be State of SCM ;
let t be Terminal of SCM-AE ;
:: original: .
redefine func s . t -> Integer;
coherence
s . t is Integer

proof end;

end;

definition

let D be non empty set ;
let f be Function of INT ,D;
let x be Integer;
:: original: .
redefine func f . x -> Element of D;
coherence
f . x is Element of D

proof end;

end;

set i2i = id INT ;

definition

let s be State of SCM ;
let term be bin-term;
deffunc H₁( NonTerminal of SCM-AE , set , set , Integer, Integer) -> Element of INT = (id INT ) . ($1 -Meaning_on $4,$5);
deffunc H₂( Terminal of SCM-AE ) -> Element of INT = (id INT ) . (s . $1);
func term @ s -> Integer means :Def9: :: SCM_COMP:def 9
ex f being Function of TS SCM-AE , INT st
( it = f . term & ( for t being Terminal of SCM-AE holds f . (root-tree t) = s . t ) & ( for nt being NonTerminal of SCM-AE
for tl, tr being bin-term
for rtl, rtr being Symbol of SCM-AE st rtl = root-label tl & rtr = root-label tr & nt ==> <*rtl,rtr*> holds
for xl, xr being Element of INT st xl = f . tl & xr = f . tr holds
f . (nt -tree tl,tr) = nt -Meaning_on xl,xr ) );
existence
ex b₁ being Integer ex f being Function of TS SCM-AE , INT st
( b₁ = f . term & ( for t being Terminal of SCM-AE holds f . (root-tree t) = s . t ) & ( for nt being NonTerminal of SCM-AE
for tl, tr being bin-term
for rtl, rtr being Symbol of SCM-AE st rtl = root-label tl & rtr = root-label tr & nt ==> <*rtl,rtr*> holds
for xl, xr being Element of INT st xl = f . tl & xr = f . tr holds
f . (nt -tree tl,tr) = nt -Meaning_on xl,xr ) )

proof end;

uniqueness
for b₁, b₂ being Integer st ex f being Function of TS SCM-AE , INT st
( b₁ = f . term & ( for t being Terminal of SCM-AE holds f . (root-tree t) = s . t ) & ( for nt being NonTerminal of SCM-AE
for tl, tr being bin-term
for rtl, rtr being Symbol of SCM-AE st rtl = root-label tl & rtr = root-label tr & nt ==> <*rtl,rtr*> holds
for xl, xr being Element of INT st xl = f . tl & xr = f . tr holds
f . (nt -tree tl,tr) = nt -Meaning_on xl,xr ) ) & ex f being Function of TS SCM-AE , INT st
( b₂ = f . term & ( for t being Terminal of SCM-AE holds f . (root-tree t) = s . t ) & ( for nt being NonTerminal of SCM-AE
for tl, tr being bin-term
for rtl, rtr being Symbol of SCM-AE st rtl = root-label tl & rtr = root-label tr & nt ==> <*rtl,rtr*> holds
for xl, xr being Element of INT st xl = f . tl & xr = f . tr holds
f . (nt -tree tl,tr) = nt -Meaning_on xl,xr ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def9 defines @ SCM_COMP:def 9 :

for s being State of SCM
for term being bin-term
for b₃ being Integer holds
( b₃ = term @ s iff ex f being Function of TS SCM-AE , INT st
( b₃ = f . term & ( for t being Terminal of SCM-AE holds f . (root-tree t) = s . t ) & ( for nt being NonTerminal of SCM-AE
for tl, tr being bin-term
for rtl, rtr being Symbol of SCM-AE st rtl = root-label tl & rtr = root-label tr & nt ==> <*rtl,rtr*> holds
for xl, xr being Element of INT st xl = f . tl & xr = f . tr holds
f . (nt -tree tl,tr) = nt -Meaning_on xl,xr ) ) );

theorem Th6: :: SCM_COMP:6

for s being State of SCM
for t being Terminal of SCM-AE holds (root-tree t) @ s = s . t

proof end;

theorem Th7: :: SCM_COMP:7

for s being State of SCM
for nt being NonTerminal of SCM-AE
for tl, tr being bin-term holds (nt -tree tl,tr) @ s = nt -Meaning_on (tl @ s),(tr @ s)

proof end;

theorem :: SCM_COMP:8

for s being State of SCM
for tl, tr being bin-term holds
( (tl + tr) @ s = (tl @ s) + (tr @ s) & (tl - tr) @ s = (tl @ s) - (tr @ s) & (tl * tr) @ s = (tl @ s) * (tr @ s) & (tl div tr) @ s = (tl @ s) div (tr @ s) & (tl mod tr) @ s = (tl @ s) mod (tr @ s) )

proof end;

definition

let nt be NonTerminal of SCM-AE ;
let n be Element of NAT ;
func Selfwork nt,n -> Element of the Instructions of SCM * equals :Def10: :: SCM_COMP:def 10
<*(AddTo (dl. n),(dl. (n + 1)))*> if nt = [0 ,0 ]
<*(SubFrom (dl. n),(dl. (n + 1)))*> if nt = [0 ,1]
<*(MultBy (dl. n),(dl. (n + 1)))*> if nt = [0 ,2]
<*(Divide (dl. n),(dl. (n + 1)))*> if nt = [0 ,3]
<*(Divide (dl. n),(dl. (n + 1))),((dl. n) := (dl. (n + 1)))*> if nt = [0 ,4]
;
coherence
( ( nt = [0 ,0 ] implies <*(AddTo (dl. n),(dl. (n + 1)))*> is Element of the Instructions of SCM * ) & ( nt = [0 ,1] implies <*(SubFrom (dl. n),(dl. (n + 1)))*> is Element of the Instructions of SCM * ) & ( nt = [0 ,2] implies <*(MultBy (dl. n),(dl. (n + 1)))*> is Element of the Instructions of SCM * ) & ( nt = [0 ,3] implies <*(Divide (dl. n),(dl. (n + 1)))*> is Element of the Instructions of SCM * ) & ( nt = [0 ,4] implies <*(Divide (dl. n),(dl. (n + 1))),((dl. n) := (dl. (n + 1)))*> is Element of the Instructions of SCM * ) ) ;
consistency
for b₁ being Element of the Instructions of SCM * holds
( ( nt = [0 ,0 ] & nt = [0 ,1] implies ( b₁ = <*(AddTo (dl. n),(dl. (n + 1)))*> iff b₁ = <*(SubFrom (dl. n),(dl. (n + 1)))*> ) ) & ( nt = [0 ,0 ] & nt = [0 ,2] implies ( b₁ = <*(AddTo (dl. n),(dl. (n + 1)))*> iff b₁ = <*(MultBy (dl. n),(dl. (n + 1)))*> ) ) & ( nt = [0 ,0 ] & nt = [0 ,3] implies ( b₁ = <*(AddTo (dl. n),(dl. (n + 1)))*> iff b₁ = <*(Divide (dl. n),(dl. (n + 1)))*> ) ) & ( nt = [0 ,0 ] & nt = [0 ,4] implies ( b₁ = <*(AddTo (dl. n),(dl. (n + 1)))*> iff b₁ = <*(Divide (dl. n),(dl. (n + 1))),((dl. n) := (dl. (n + 1)))*> ) ) & ( nt = [0 ,1] & nt = [0 ,2] implies ( b₁ = <*(SubFrom (dl. n),(dl. (n + 1)))*> iff b₁ = <*(MultBy (dl. n),(dl. (n + 1)))*> ) ) & ( nt = [0 ,1] & nt = [0 ,3] implies ( b₁ = <*(SubFrom (dl. n),(dl. (n + 1)))*> iff b₁ = <*(Divide (dl. n),(dl. (n + 1)))*> ) ) & ( nt = [0 ,1] & nt = [0 ,4] implies ( b₁ = <*(SubFrom (dl. n),(dl. (n + 1)))*> iff b₁ = <*(Divide (dl. n),(dl. (n + 1))),((dl. n) := (dl. (n + 1)))*> ) ) & ( nt = [0 ,2] & nt = [0 ,3] implies ( b₁ = <*(MultBy (dl. n),(dl. (n + 1)))*> iff b₁ = <*(Divide (dl. n),(dl. (n + 1)))*> ) ) & ( nt = [0 ,2] & nt = [0 ,4] implies ( b₁ = <*(MultBy (dl. n),(dl. (n + 1)))*> iff b₁ = <*(Divide (dl. n),(dl. (n + 1))),((dl. n) := (dl. (n + 1)))*> ) ) & ( nt = [0 ,3] & nt = [0 ,4] implies ( b₁ = <*(Divide (dl. n),(dl. (n + 1)))*> iff b₁ = <*(Divide (dl. n),(dl. (n + 1))),((dl. n) := (dl. (n + 1)))*> ) ) )

proof end;

end;

:: deftheorem Def10 defines Selfwork SCM_COMP:def 10 :

for nt being NonTerminal of SCM-AE
for n being Element of NAT holds
( ( nt = [0 ,0 ] implies Selfwork nt,n = <*(AddTo (dl. n),(dl. (n + 1)))*> ) & ( nt = [0 ,1] implies Selfwork nt,n = <*(SubFrom (dl. n),(dl. (n + 1)))*> ) & ( nt = [0 ,2] implies Selfwork nt,n = <*(MultBy (dl. n),(dl. (n + 1)))*> ) & ( nt = [0 ,3] implies Selfwork nt,n = <*(Divide (dl. n),(dl. (n + 1)))*> ) & ( nt = [0 ,4] implies Selfwork nt,n = <*(Divide (dl. n),(dl. (n + 1))),((dl. n) := (dl. (n + 1)))*> ) );

definition

let term be bin-term;
let aux be Element of NAT ;
deffunc H₁( Terminal of SCM-AE , Element of NAT ) -> Element of the Instructions of SCM * = <*((dl. $2) := (@ $1))*>;
deffunc H₂( NonTerminal of SCM-AE , Function of NAT ,the Instructions of SCM * , Function of NAT ,the Instructions of SCM * , Element of NAT ) -> Element of the Instructions of SCM * = (($2 . $4) ^ ($3 . ($4 + 1))) ^ (Selfwork $1,$4);
func SCM-Compile term,aux -> FinSequence of the Instructions of SCM means :Def11: :: SCM_COMP:def 11
ex f being Function of TS SCM-AE , Funcs NAT ,(the Instructions of SCM * ) st
( it = (f . term) . aux & ( for t being Terminal of SCM-AE ex g being Function of NAT ,the Instructions of SCM * st
( g = f . (root-tree t) & ( for n being Element of NAT holds g . n = <*((dl. n) := (@ t))*> ) ) ) & ( for nt being NonTerminal of SCM-AE
for t1, t2 being bin-term
for rtl, rtr being Symbol of SCM-AE st rtl = root-label t1 & rtr = root-label t2 & nt ==> <*rtl,rtr*> holds
ex g, f1, f2 being Function of NAT ,the Instructions of SCM * st
( g = f . (nt -tree t1,t2) & f1 = f . t1 & f2 = f . t2 & ( for n being Element of NAT holds g . n = ((f1 . n) ^ (f2 . (n + 1))) ^ (Selfwork nt,n) ) ) ) );
existence
ex b₁ being FinSequence of the Instructions of SCM ex f being Function of TS SCM-AE , Funcs NAT ,(the Instructions of SCM * ) st
( b₁ = (f . term) . aux & ( for t being Terminal of SCM-AE ex g being Function of NAT ,the Instructions of SCM * st
( g = f . (root-tree t) & ( for n being Element of NAT holds g . n = <*((dl. n) := (@ t))*> ) ) ) & ( for nt being NonTerminal of SCM-AE
for t1, t2 being bin-term
for rtl, rtr being Symbol of SCM-AE st rtl = root-label t1 & rtr = root-label t2 & nt ==> <*rtl,rtr*> holds
ex g, f1, f2 being Function of NAT ,the Instructions of SCM * st
( g = f . (nt -tree t1,t2) & f1 = f . t1 & f2 = f . t2 & ( for n being Element of NAT holds g . n = ((f1 . n) ^ (f2 . (n + 1))) ^ (Selfwork nt,n) ) ) ) )

proof end;

uniqueness
for b₁, b₂ being FinSequence of the Instructions of SCM st ex f being Function of TS SCM-AE , Funcs NAT ,(the Instructions of SCM * ) st
( b₁ = (f . term) . aux & ( for t being Terminal of SCM-AE ex g being Function of NAT ,the Instructions of SCM * st
( g = f . (root-tree t) & ( for n being Element of NAT holds g . n = <*((dl. n) := (@ t))*> ) ) ) & ( for nt being NonTerminal of SCM-AE
for t1, t2 being bin-term
for rtl, rtr being Symbol of SCM-AE st rtl = root-label t1 & rtr = root-label t2 & nt ==> <*rtl,rtr*> holds
ex g, f1, f2 being Function of NAT ,the Instructions of SCM * st
( g = f . (nt -tree t1,t2) & f1 = f . t1 & f2 = f . t2 & ( for n being Element of NAT holds g . n = ((f1 . n) ^ (f2 . (n + 1))) ^ (Selfwork nt,n) ) ) ) ) & ex f being Function of TS SCM-AE , Funcs NAT ,(the Instructions of SCM * ) st
( b₂ = (f . term) . aux & ( for t being Terminal of SCM-AE ex g being Function of NAT ,the Instructions of SCM * st
( g = f . (root-tree t) & ( for n being Element of NAT holds g . n = <*((dl. n) := (@ t))*> ) ) ) & ( for nt being NonTerminal of SCM-AE
for t1, t2 being bin-term
for rtl, rtr being Symbol of SCM-AE st rtl = root-label t1 & rtr = root-label t2 & nt ==> <*rtl,rtr*> holds
ex g, f1, f2 being Function of NAT ,the Instructions of SCM * st
( g = f . (nt -tree t1,t2) & f1 = f . t1 & f2 = f . t2 & ( for n being Element of NAT holds g . n = ((f1 . n) ^ (f2 . (n + 1))) ^ (Selfwork nt,n) ) ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def11 defines SCM-Compile SCM_COMP:def 11 :

for term being bin-term
for aux being Element of NAT
for b₃ being FinSequence of the Instructions of SCM holds
( b₃ = SCM-Compile term,aux iff ex f being Function of TS SCM-AE , Funcs NAT ,(the Instructions of SCM * ) st
( b₃ = (f . term) . aux & ( for t being Terminal of SCM-AE ex g being Function of NAT ,the Instructions of SCM * st
( g = f . (root-tree t) & ( for n being Element of NAT holds g . n = <*((dl. n) := (@ t))*> ) ) ) & ( for nt being NonTerminal of SCM-AE
for t1, t2 being bin-term
for rtl, rtr being Symbol of SCM-AE st rtl = root-label t1 & rtr = root-label t2 & nt ==> <*rtl,rtr*> holds
ex g, f1, f2 being Function of NAT ,the Instructions of SCM * st
( g = f . (nt -tree t1,t2) & f1 = f . t1 & f2 = f . t2 & ( for n being Element of NAT holds g . n = ((f1 . n) ^ (f2 . (n + 1))) ^ (Selfwork nt,n) ) ) ) ) );

consider term' being bin-term;

consider f being Function of TS SCM-AE , Funcs NAT ,(the Instructions of SCM * ) such that
SCM-Compile term',0 = (f . term') . 0 and
Lm4: ( ( for t being Terminal of SCM-AE ex g being Function of NAT ,the Instructions of SCM * st
( g = f . (root-tree t) & ( for n being Element of NAT holds g . n = <*((dl. n) := (@ t))*> ) ) ) & ( for nt being NonTerminal of SCM-AE
for t1, t2 being bin-term
for rtl, rtr being Symbol of SCM-AE st rtl = root-label t1 & rtr = root-label t2 & nt ==> <*rtl,rtr*> holds
ex g, f1, f2 being Function of NAT ,the Instructions of SCM * st
( g = f . (nt -tree t1,t2) & f1 = f . t1 & f2 = f . t2 & ( for n being Element of NAT holds g . n = ((f1 . n) ^ (f2 . (n + 1))) ^ (Selfwork nt,n) ) ) ) ) by Def11;

theorem Th9: :: SCM_COMP:9

for t being Terminal of SCM-AE
for n being Element of NAT holds SCM-Compile (root-tree t),n = <*((dl. n) := (@ t))*>

proof end;

theorem Th10: :: SCM_COMP:10

for nt being NonTerminal of SCM-AE
for t1, t2 being bin-term
for n being Element of NAT
for rtl, rtr being Symbol of SCM-AE st rtl = root-label t1 & rtr = root-label t2 & nt ==> <*rtl,rtr*> holds
SCM-Compile (nt -tree t1,t2),n = ((SCM-Compile t1,n) ^ (SCM-Compile t2,(n + 1))) ^ (Selfwork nt,n)

proof end;

definition

let t be Terminal of SCM-AE ;
func d". t -> Element of NAT means :Def12: :: SCM_COMP:def 12
dl. it = t;
existence
ex b₁ being Element of NAT st dl. b₁ = t

proof end;

uniqueness
for b₁, b₂ being Element of NAT st dl. b₁ = t & dl. b₂ = t holds
b₁ = b₂ by AMI_3:52;

end;

:: deftheorem Def12 defines d". SCM_COMP:def 12 :

for t being Terminal of SCM-AE
for b₂ being Element of NAT holds
( b₂ = d". t iff dl. b₂ = t );

definition

let term be bin-term;
deffunc H₁( NonTerminal of SCM-AE , set , set , Element of NAT , Element of NAT ) -> Element of NAT = max $4,$5;
deffunc H₂( Terminal of SCM-AE ) -> Element of NAT = d". $1;
func max_Data-Loc_in term -> Element of NAT means :Def13: :: SCM_COMP:def 13
ex f being Function of TS SCM-AE , NAT st
( it = f . term & ( for t being Terminal of SCM-AE holds f . (root-tree t) = d". t ) & ( for nt being NonTerminal of SCM-AE
for tl, tr being bin-term
for rtl, rtr being Symbol of SCM-AE st rtl = root-label tl & rtr = root-label tr & nt ==> <*rtl,rtr*> holds
for xl, xr being Element of NAT st xl = f . tl & xr = f . tr holds
f . (nt -tree tl,tr) = max xl,xr ) );
existence
ex b₁ being Element of NAT ex f being Function of TS SCM-AE , NAT st
( b₁ = f . term & ( for t being Terminal of SCM-AE holds f . (root-tree t) = d". t ) & ( for nt being NonTerminal of SCM-AE
for tl, tr being bin-term
for rtl, rtr being Symbol of SCM-AE st rtl = root-label tl & rtr = root-label tr & nt ==> <*rtl,rtr*> holds
for xl, xr being Element of NAT st xl = f . tl & xr = f . tr holds
f . (nt -tree tl,tr) = max xl,xr ) )

proof end;

uniqueness
for b₁, b₂ being Element of NAT st ex f being Function of TS SCM-AE , NAT st
( b₁ = f . term & ( for t being Terminal of SCM-AE holds f . (root-tree t) = d". t ) & ( for nt being NonTerminal of SCM-AE
for tl, tr being bin-term
for rtl, rtr being Symbol of SCM-AE st rtl = root-label tl & rtr = root-label tr & nt ==> <*rtl,rtr*> holds
for xl, xr being Element of NAT st xl = f . tl & xr = f . tr holds
f . (nt -tree tl,tr) = max xl,xr ) ) & ex f being Function of TS SCM-AE , NAT st
( b₂ = f . term & ( for t being Terminal of SCM-AE holds f . (root-tree t) = d". t ) & ( for nt being NonTerminal of SCM-AE
for tl, tr being bin-term
for rtl, rtr being Symbol of SCM-AE st rtl = root-label tl & rtr = root-label tr & nt ==> <*rtl,rtr*> holds
for xl, xr being Element of NAT st xl = f . tl & xr = f . tr holds
f . (nt -tree tl,tr) = max xl,xr ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def13 defines max_Data-Loc_in SCM_COMP:def 13 :

for term being bin-term
for b₂ being Element of NAT holds
( b₂ = max_Data-Loc_in term iff ex f being Function of TS SCM-AE , NAT st
( b₂ = f . term & ( for t being Terminal of SCM-AE holds f . (root-tree t) = d". t ) & ( for nt being NonTerminal of SCM-AE
for tl, tr being bin-term
for rtl, rtr being Symbol of SCM-AE st rtl = root-label tl & rtr = root-label tr & nt ==> <*rtl,rtr*> holds
for xl, xr being Element of NAT st xl = f . tl & xr = f . tr holds
f . (nt -tree tl,tr) = max xl,xr ) ) );

consider Term being bin-term;

consider f being Function of TS SCM-AE , NAT such that
max_Data-Loc_in Term = f . Term and
Lm5: ( ( for t being Terminal of SCM-AE holds f . (root-tree t) = d". t ) & ( for nt being NonTerminal of SCM-AE
for tl, tr being bin-term
for rtl, rtr being Symbol of SCM-AE st rtl = root-label tl & rtr = root-label tr & nt ==> <*rtl,rtr*> holds
for xl, xr being Element of NAT st xl = f . tl & xr = f . tr holds
f . (nt -tree tl,tr) = max xl,xr ) ) by Def13;

theorem Th11: :: SCM_COMP:11

for t being Terminal of SCM-AE holds max_Data-Loc_in (root-tree t) = d". t

proof end;

Lm6: NonTerminals SCM-AE = [:1,5:]
by Def1;

theorem Th12: :: SCM_COMP:12

for nt being NonTerminal of SCM-AE
for tl, tr being bin-term holds max_Data-Loc_in (nt -tree tl,tr) = max (max_Data-Loc_in tl),(max_Data-Loc_in tr)

proof end;

defpred S₁[ bin-term] means for s1, s2 being State of SCM st ( for dn being Element of NAT st dn <= max_Data-Loc_in $1 holds
s1 . (dl. dn) = s2 . (dl. dn) ) holds
$1 @ s1 = $1 @ s2;

Lm7: now

let s be Terminal of SCM-AE ; :: thesis: S₁[ root-tree s]
thus S₁[ root-tree s] :: thesis: verum

proof

let s1, s2 be State of SCM ; :: thesis: ( ( for dn being Element of NAT st dn <= max_Data-Loc_in (root-tree s) holds
s1 . (dl. dn) = s2 . (dl. dn) ) implies (root-tree s) @ s1 = (root-tree s) @ s2 )
assume A1: for dn being Element of NAT st dn <= max_Data-Loc_in (root-tree s) holds
s1 . (dl. dn) = s2 . (dl. dn) ; :: thesis: (root-tree s) @ s1 = (root-tree s) @ s2
d". s <= max_Data-Loc_in (root-tree s) by Th11;
then A2: ( s1 . (dl. (d". s)) = s2 . (dl. (d". s)) & s1 . s = s1 . (dl. (d". s)) & s2 . s = s2 . (dl. (d". s)) ) by A1, Def12;
(root-tree s) @ s1 = s1 . s by Th6;
hence (root-tree s) @ s1 = (root-tree s) @ s2 by A2, Th6; :: thesis: verum

end;

end;

Lm8: now

let nt be NonTerminal of SCM-AE ; :: thesis: for tl, tr being Element of TS SCM-AE st nt ==> <*(root-label tl),(root-label tr)*> & S₁[tl] & S₁[tr] holds
S₁[nt -tree tl,tr]
let tl, tr be Element of TS SCM-AE ; :: thesis: ( nt ==> <*(root-label tl),(root-label tr)*> & S₁[tl] & S₁[tr] implies S₁[nt -tree tl,tr] )
assume A1: ( nt ==> <*(root-label tl),(root-label tr)*> & S₁[tl] & S₁[tr] ) ; :: thesis: S₁[nt -tree tl,tr]
thus S₁[nt -tree tl,tr] :: thesis: verum

proof

let s1, s2 be State of SCM ; :: thesis: ( ( for dn being Element of NAT st dn <= max_Data-Loc_in (nt -tree tl,tr) holds
s1 . (dl. dn) = s2 . (dl. dn) ) implies (nt -tree tl,tr) @ s1 = (nt -tree tl,tr) @ s2 )
assume A2: for dn being Element of NAT st dn <= max_Data-Loc_in (nt -tree tl,tr) holds
s1 . (dl. dn) = s2 . (dl. dn) ; :: thesis: (nt -tree tl,tr) @ s1 = (nt -tree tl,tr) @ s2

now

let dn be Element of NAT ; :: thesis: ( dn <= max_Data-Loc_in tl implies s1 . (dl. dn) = s2 . (dl. dn) )
assume A3: dn <= max_Data-Loc_in tl ; :: thesis: s1 . (dl. dn) = s2 . (dl. dn)
set ml = max_Data-Loc_in tl;
set mr = max_Data-Loc_in tr;
max_Data-Loc_in tl <= max (max_Data-Loc_in tl),(max_Data-Loc_in tr) by XXREAL_0:25;
then dn <= max (max_Data-Loc_in tl),(max_Data-Loc_in tr) by A3, XXREAL_0:2;
then dn <= max_Data-Loc_in (nt -tree tl,tr) by Th12;
hence s1 . (dl. dn) = s2 . (dl. dn) by A2; :: thesis: verum

end;

then A4: tl @ s1 = tl @ s2 by A1;

now

let dn be Element of NAT ; :: thesis: ( dn <= max_Data-Loc_in tr implies s1 . (dl. dn) = s2 . (dl. dn) )
assume A5: dn <= max_Data-Loc_in tr ; :: thesis: s1 . (dl. dn) = s2 . (dl. dn)
set ml = max_Data-Loc_in tl;
set mr = max_Data-Loc_in tr;
max_Data-Loc_in tr <= max (max_Data-Loc_in tl),(max_Data-Loc_in tr) by XXREAL_0:25;
then dn <= max (max_Data-Loc_in tl),(max_Data-Loc_in tr) by A5, XXREAL_0:2;
then dn <= max_Data-Loc_in (nt -tree tl,tr) by Th12;
hence s1 . (dl. dn) = s2 . (dl. dn) by A2; :: thesis: verum

end;

then A6: tr @ s1 = tr @ s2 by A1;
(nt -tree tl,tr) @ s1 = nt -Meaning_on (tl @ s1),(tr @ s1) by Th7;
hence (nt -tree tl,tr) @ s1 = (nt -tree tl,tr) @ s2 by A4, A6, Th7; :: thesis: verum

end;

end;

theorem Th13: :: SCM_COMP:13

for term being bin-term
for s1, s2 being State of SCM st ( for dn being Element of NAT st dn <= max_Data-Loc_in term holds
s1 . (dl. dn) = s2 . (dl. dn) ) holds
term @ s1 = term @ s2

proof end;

set D = <*> INT ;

defpred S₂[ bin-term] means for aux, n, k being Element of NAT
for s being State-consisting of n,n,k, SCM-Compile $1,aux, <*> INT st aux > max_Data-Loc_in $1 holds
ex i being Element of NAT ex u being State of SCM st
( u = Computation s,(i + 1) & i + 1 = len (SCM-Compile $1,aux) & IC (Computation s,i) = il. (n + i) & IC u = il. (n + (i + 1)) & u . (dl. aux) = $1 @ s & ( for dn being Element of NAT st dn < aux holds
s . (dl. dn) = u . (dl. dn) ) );

theorem Th14: :: SCM_COMP:14

for term being bin-term
for aux, n, k being Element of NAT
for s being State-consisting of n,n,k, SCM-Compile term,aux, <*> INT st aux > max_Data-Loc_in term holds
ex i being Element of NAT ex u being State of SCM st
( u = Computation s,(i + 1) & i + 1 = len (SCM-Compile term,aux) & IC (Computation s,i) = il. (n + i) & IC u = il. (n + (i + 1)) & u . (dl. aux) = term @ s & ( for dn being Element of NAT st dn < aux holds
s . (dl. dn) = u . (dl. dn) ) )

proof end;

theorem :: SCM_COMP:15

for term being bin-term
for aux, n, k being Element of NAT
for s being State-consisting of n,n,k,(SCM-Compile term,aux) ^ <*(halt SCM )*>, <*> INT st aux > max_Data-Loc_in term holds
( s is halting & (Result s) . (dl. aux) = term @ s & LifeSpan s = len (SCM-Compile term,aux) )

proof end;