:: A compiler of arithmetic expressions for { \bf SCM }
:: by Grzegorz Bancerek and Piotr Rudnicki
::
:: 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 end;

Lm3:
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 by DTCONSTR:def 1;
end;

definition
let t be Terminal of SCM-AE;
func @ t -> Data-Location equals :: SCM_COMP:def 2
t;
coherence
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 = () . ($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 (),INT st ( it = f . term & ( for t being Terminal of SCM-AE holds f . () = 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 (),INT st ( b1 = f . term & ( for t being Terminal of SCM-AE holds f . () = 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 (),INT st ( b1 = f . term & ( for t being Terminal of SCM-AE holds f . () = 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 (),INT st ( b2 = f . term & ( for t being Terminal of SCM-AE holds f . () = 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 (),INT st ( b3 = f . term & ( for t being Terminal of SCM-AE holds f . () = 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 () @ 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 (), sequence of (), 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 (),(Funcs (NAT,())) means :Def11: :: SCM_COMP:def 11 ( ( for t being Terminal of SCM-AE ex g being sequence of () st ( g = it . () & ( 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 () 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 (),(Funcs (NAT,())) st ( ( for t being Terminal of SCM-AE ex g being sequence of () st ( g = b1 . () & ( 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 () 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 (),(Funcs (NAT,())) st ( for t being Terminal of SCM-AE ex g being sequence of () st ( g = b1 . () & ( 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 () 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 () st ( g = b2 . () & ( 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 () 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 (),(Funcs (NAT,())) holds ( b1 = SCM-Compile iff ( ( for t being Terminal of SCM-AE ex g being sequence of () st ( g = b1 . () & ( 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 () 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 ((),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 (),NAT st
( it = f . term & ( for t being Terminal of SCM-AE holds f . () = 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 (),NAT st
( b1 = f . term & ( for t being Terminal of SCM-AE holds f . () = 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 (),NAT st
( b1 = f . term & ( for t being Terminal of SCM-AE holds f . () = 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 (),NAT st
( b2 = f . term & ( for t being Terminal of SCM-AE holds f . () = 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 (),NAT st
( b2 = f . term & ( for t being Terminal of SCM-AE holds f . () = 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 (),NAT such that
and
Lm4: for t being Terminal of SCM-AE holds f . () = 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 () = d". t
proof end;

Lm6:
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 ((),())
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 () holds s1 . (dl. dn) = s2 . (dl. dn) ) implies () @ s1 = () @ s2 ) assume A1: for dn being Element of NAT st dn <= max_Data-Loc_in () holds s1 . (dl. dn) = s2 . (dl. dn) ; :: thesis: () @ s1 = () @ s2 d". s <= max_Data-Loc_in () by Th9; then A2: s1 . (dl. (d". s)) = s2 . (dl. (d". s)) by A1; A3: () @ s1 = s1 . s by Th4; ( s1 . s = s1 . (dl. (d". s)) & s2 . s = s2 . (dl. (d". s)) ) by Def13; hence () @ s1 = () @ 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 ==> <*(),()*> & 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 ==> <*(),()*> & S1[tl] & S1[tr] holds S1[nt -tree (tl,tr)] let tl, tr be Element of TS SCM-AE; :: thesis: ( nt ==> <*(),()*> & S1[tl] & S1[tr] implies S1[nt -tree (tl,tr)] ) assume that nt ==> <*(),()*> 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 ((),()) by XXREAL_0:25; assume dn <= max_Data-Loc_in tl ; :: thesis: s1 . (dl. dn) = s2 . (dl. dn) then dn <= max ((),()) by ; 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 ((),()) by XXREAL_0:25; assume dn <= max_Data-Loc_in tr ; :: thesis: s1 . (dl. dn) = s2 . (dl. dn) then dn <= max ((),()) by ; 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)) ^ <%()%>),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;