:: A compiler of arithmetic expressions for { \bf SCM }
:: by Grzegorz Bancerek and Piotr Rudnicki
::
:: Received December 30, 1993
:: Copyright (c) 1993-2021 Association of Mizar Users

Lm1: 1 = { n where n is Nat : n < 1 }
by AXIOMS:4;

Lm2: 5 = { n where n is Nat : n < 5 }
by AXIOMS:4;

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 1;

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 Th1: :: SCM_COMP:1

for nt being NonTerminal of SCM-AE holds
not not nt = [0,0] & ... & not nt = [0,4]

proof end;

theorem :: SCM_COMP:2

[0,0] 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:3

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 ) );

registration

let s be State of SCM;
let t be Terminal of SCM-AE;

cluster 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;

deffunc H₁( NonTerminal of SCM-AE, set , set , Integer, Integer) -> Element of INT = (id INT) . ($1 -Meaning_on ($4,$5));

definition

let s be State of SCM;
let term be bin-term;

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 Th4: :: SCM_COMP:4

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

proof end;

theorem Th5: :: SCM_COMP:5

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:6

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 Nat;

func Selfwork (nt,n) -> XFinSequence of the InstructionsF 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 XFinSequence of the InstructionsF of SCM ) & ( nt = [0,1] implies <%(SubFrom ((dl. n),(dl. (n + 1))))%> is XFinSequence of the InstructionsF of SCM ) & ( nt = [0,2] implies <%(MultBy ((dl. n),(dl. (n + 1))))%> is XFinSequence of the InstructionsF of SCM ) & ( nt = [0,3] implies <%(Divide ((dl. n),(dl. (n + 1))))%> is XFinSequence of the InstructionsF of SCM ) & ( nt = [0,4] implies <%(Divide ((dl. n),(dl. (n + 1)))),((dl. n) := (dl. (n + 1)))%> is XFinSequence of the InstructionsF of SCM ) ) ;
consistency
for b₁ being XFinSequence of the InstructionsF 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 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

deffunc H₂( NonTerminal of SCM-AE, sequence of ( the InstructionsF of SCM ^omega), sequence of ( the InstructionsF of SCM ^omega), Nat) -> Element of the InstructionsF of SCM ^omega = (($2 . (In ($4,NAT))) ^ ($3 . (In (($4 + 1),NAT)))) ^ (Down (Selfwork ($1,$4)));
deffunc H₃( Terminal of SCM-AE, Nat) -> Element of the InstructionsF of SCM ^omega = Down <%((dl. $2) := (@ $1))%>;

func SCM-Compile -> Function of (TS SCM-AE),(Funcs (NAT,( the InstructionsF of SCM ^omega))) means :Def11: :: SCM_COMP:def 11
( ( for t being Terminal of SCM-AE ex g being sequence of ( the InstructionsF of SCM ^omega) st
( g = it . (root-tree t) & ( for n being 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 sequence of ( the InstructionsF of SCM ^omega) st
( g = it . (nt -tree (t1,t2)) & f1 = it . t1 & f2 = it . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ) ) ) );

existence
ex b₁ being Function of (TS SCM-AE),(Funcs (NAT,( the InstructionsF of SCM ^omega))) st
( ( for t being Terminal of SCM-AE ex g being sequence of ( the InstructionsF of SCM ^omega) st
( g = b₁ . (root-tree t) & ( for n being 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 sequence of ( the InstructionsF of SCM ^omega) st
( g = b₁ . (nt -tree (t1,t2)) & f1 = b₁ . t1 & f2 = b₁ . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ) ) ) )

proof end;

uniqueness
for b₁, b₂ being Function of (TS SCM-AE),(Funcs (NAT,( the InstructionsF of SCM ^omega))) st ( for t being Terminal of SCM-AE ex g being sequence of ( the InstructionsF of SCM ^omega) st
( g = b₁ . (root-tree t) & ( for n being 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 sequence of ( the InstructionsF of SCM ^omega) st
( g = b₁ . (nt -tree (t1,t2)) & f1 = b₁ . t1 & f2 = b₁ . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ) ) ) & ( for t being Terminal of SCM-AE ex g being sequence of ( the InstructionsF of SCM ^omega) st
( g = b₂ . (root-tree t) & ( for n being 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 sequence of ( the InstructionsF of SCM ^omega) st
( g = b₂ . (nt -tree (t1,t2)) & f1 = b₂ . t1 & f2 = b₂ . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def11 defines SCM-Compile SCM_COMP:def 11 :
for b₁ being Function of (TS SCM-AE),(Funcs (NAT,( the InstructionsF of SCM ^omega))) holds
( b₁ = SCM-Compile iff ( ( for t being Terminal of SCM-AE ex g being sequence of ( the InstructionsF of SCM ^omega) st
( g = b₁ . (root-tree t) & ( for n being 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 sequence of ( the InstructionsF of SCM ^omega) st
( g = b₁ . (nt -tree (t1,t2)) & f1 = b₁ . t1 & f2 = b₁ . t2 & ( for n being Nat holds g . n = ((f1 . (In (n,NAT))) ^ (f2 . (In ((n + 1),NAT)))) ^ (Selfwork (nt,n)) ) ) ) ) );

definition

let term be bin-term;
let aux be Nat;

func SCM-Compile (term,aux) -> XFinSequence of the InstructionsF of SCM equals :: SCM_COMP:def 12
(SCM-Compile . term) . aux;

coherence
(SCM-Compile . term) . aux is XFinSequence of the InstructionsF of SCM

proof end;

end;

:: deftheorem defines SCM-Compile SCM_COMP:def 12 :
for term being bin-term
for aux being Nat holds SCM-Compile (term,aux) = (SCM-Compile . term) . aux;

theorem Th7: :: SCM_COMP:7

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 Th8: :: SCM_COMP:8

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 :Def13: :: SCM_COMP:def 13
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_3:10;

end;

:: deftheorem Def13 defines d". SCM_COMP:def 13 :
for t being Terminal of SCM-AE
for b₂ being Element of NAT holds
( b₂ = d". t iff dl. b₂ = t );

definition

deffunc H₂( Terminal of SCM-AE) -> Element of NAT = d". $1;
deffunc H₃( NonTerminal of SCM-AE, set , set , Element of NAT , Element of NAT ) -> Element of NAT = max ($4,$5);
let term be bin-term;

func max_Data-Loc_in term -> Element of NAT means :Def14: :: SCM_COMP:def 14
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 Def14 defines max_Data-Loc_in SCM_COMP:def 14 :
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) ) ) );

set Term = the bin-term;

consider f being Function of (TS SCM-AE),NAT such that
max_Data-Loc_in the bin-term = f . the bin-term and
Lm4: for t being Terminal of SCM-AE holds f . (root-tree t) = d". t and
Lm5: 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 Def14;

theorem Th9: :: SCM_COMP:9

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 Th10: :: SCM_COMP:10

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 :: thesis: for s being Terminal of SCM-AE holds S₁[ root-tree s]

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 Th9;
then A2: s1 . (dl. (d". s)) = s2 . (dl. (d". s)) by A1;
A3: (root-tree s) @ s1 = s1 . s by Th4;
( s1 . s = s1 . (dl. (d". s)) & s2 . s = s2 . (dl. (d". s)) ) by Def13;
hence (root-tree s) @ s1 = (root-tree s) @ s2 by A2, A3, Th4; :: thesis: verum

end;

end;

Lm8: now :: thesis: for nt being NonTerminal of SCM-AE
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 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 that
nt ==> <*(root-label tl),(root-label tr)*> and
A1: S₁[tl] and
A2: 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 A3: 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 :: thesis: for dn being Element of NAT st dn <= max_Data-Loc_in tl holds
s1 . (dl. dn) = s2 . (dl. dn)

set ml = max_Data-Loc_in tl;
set mr = max_Data-Loc_in tr;
let dn be Element of NAT ; :: thesis: ( dn <= max_Data-Loc_in tl implies s1 . (dl. dn) = s2 . (dl. dn) )
A4: max_Data-Loc_in tl <= max ((max_Data-Loc_in tl),(max_Data-Loc_in tr)) by XXREAL_0:25;
assume dn <= max_Data-Loc_in tl ; :: thesis: s1 . (dl. dn) = s2 . (dl. dn)
then dn <= max ((max_Data-Loc_in tl),(max_Data-Loc_in tr)) by A4, XXREAL_0:2;
then dn <= max_Data-Loc_in (nt -tree (tl,tr)) by Th10;
hence s1 . (dl. dn) = s2 . (dl. dn) by A3; :: thesis: verum

end;

then A5: tl @ s1 = tl @ s2 by A1;

now :: thesis: for dn being Element of NAT st dn <= max_Data-Loc_in tr holds
s1 . (dl. dn) = s2 . (dl. dn)

set ml = max_Data-Loc_in tl;
set mr = max_Data-Loc_in tr;
let dn be Element of NAT ; :: thesis: ( dn <= max_Data-Loc_in tr implies s1 . (dl. dn) = s2 . (dl. dn) )
A6: max_Data-Loc_in tr <= max ((max_Data-Loc_in tl),(max_Data-Loc_in tr)) by XXREAL_0:25;
assume dn <= max_Data-Loc_in tr ; :: thesis: s1 . (dl. dn) = s2 . (dl. dn)
then dn <= max ((max_Data-Loc_in tl),(max_Data-Loc_in tr)) by A6, XXREAL_0:2;
then dn <= max_Data-Loc_in (nt -tree (tl,tr)) by Th10;
hence s1 . (dl. dn) = s2 . (dl. dn) by A3; :: thesis: verum

end;

then A7: tr @ s1 = tr @ s2 by A2;
(nt -tree (tl,tr)) @ s1 = nt -Meaning_on ((tl @ s1),(tr @ s1)) by Th5;
hence (nt -tree (tl,tr)) @ s1 = (nt -tree (tl,tr)) @ s2 by A5, A7, Th5; :: thesis: verum

end;

end;

theorem Th11: :: SCM_COMP:11

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;

defpred S₂[ bin-term] means for F being Instruction-Sequence of SCM
for aux, n being Element of NAT st Shift ((SCM-Compile ($1,aux)),n) c= F holds
for s being b₃ -started State of SCM st aux > max_Data-Loc_in $1 holds
ex i being Element of NAT ex u being State of SCM st
( u = Comput (F,s,(i + 1)) & i + 1 = len (SCM-Compile ($1,aux)) & IC (Comput (F,s,i)) = n + i & IC u = 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 Th12: :: SCM_COMP:12

for F being Instruction-Sequence of SCM
for term being bin-term
for aux, n being Element of NAT st Shift ((SCM-Compile (term,aux)),n) c= F holds
for s being b₄ -started State of SCM st aux > max_Data-Loc_in term holds
ex i being Element of NAT ex u being State of SCM st
( u = Comput (F,s,(i + 1)) & i + 1 = len (SCM-Compile (term,aux)) & IC (Comput (F,s,i)) = n + i & IC u = 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:13

for F being Instruction-Sequence of SCM
for term being bin-term
for aux, n being Element of NAT st Shift (((SCM-Compile (term,aux)) ^ <%(halt SCM)%>),n) c= F holds
for s being b₄ -started State of SCM st aux > max_Data-Loc_in term holds
( F halts_on s & (Result (F,s)) . (dl. aux) = term @ s & LifeSpan (F,s) = len (SCM-Compile (term,aux)) )

proof end;