Compiler Principles Fall Compiler Principles Lecture 7: Lowering Correctness Roman Manevich Ben-Gurion University of the Negev

Tentative syllabus Front End Scanning Top-down Parsing (LL) Bottom-up Parsing (LR) Intermediate Representation Lowering Lowering Correctness Optimizations Dataflow Analysis Loop Optimizations Code Generation Register Allocation Instruction Selection 2

Previously The role of intermediate representations Two example languages – A high-level language – An intermediate language Lowering Correctness – Formal meaning of programs 3

While syntax A  n | x | A ArithOp A | ( A ) ArithOp  - | + | * | / B  true | false | A = A | A  A |  B | B  B | ( B ) S  x := A | skip | S ; S | { S } | if B then S else S | while B S 4 n  Numnumerals x  Varprogram variables

IL syntax V  n | x R  V Op V | V Op  - | + | * | / | = |  | > | … C  l : skip | l : x := R | l : Goto l’ | l : IfZ x Goto l’ | l : IfNZ x Goto l’ IR  C + 5 n  NumNumerals l  Num Labels x  Temp  VarTemporaries and variables

Translation rules for expressions 6 cgen(n) = (l: t:=n, t)where l and t are fresh cgen(x) = (l: t:=x, t)where l and t are fresh cgen(e 1 ) = (P 1, t 1 ) cgen(e 2 ) = (P 2, t 2 ) cgen(e 1 op e 2 ) = (P 1 · P 2 · l: t:=t 1 op t 2, t) where l and t are fresh

Translation rules for statements 7 cgen(e) = (P, t) cgen( x := e) = P · l: x :=t where l is fresh cgen( S 1 ) = P 1, cgen( S 2 ) = P 2 cgen( S 1 ; S 2 ) = P 1 · P 2 cgen( skip ) = l: skip where l is fresh

Translation rules for conditions 8 cgen( b ) = (Pb, t), cgen( S 1 ) = P 1, cgen( S 2 ) = P 2 cgen( if b then S 1 else S 2 ) = Pb lb: IfZ t Goto label(P 2 ) P 1 l finish : Goto L after P 2 l after : skip where lb, l finish, l after are fresh

Translation rules for loops 9 cgen( b ) = (Pb, t), cgen( S ) = P cgen( while b S ) = l before : skip Pb IfZ t Goto l after P l loop : Goto L before l after : skip where l after, l before, l loop are fresh

Translation example 10 1: t1 := 137 2: t2 := 3 3: t3 := t1 + t2 4: y := t3 5: t4 := x 6: t5 := 0 7: t6 := t4=t5 8: IfZ t6 Goto 12 9: t7 := y 10: z := t7 11: Goto 14 12: t8 := y 13: x := t8 14: skip y := 137+3; if x=0 z := y; else x := y;

agenda Operational semantics of While Operational semantics of IL Formalizing the correctness of lowering Proving correctness 11

Correctness 12

Compiler correctness Intuitively, a compiler translates programs in one language (usually high) to another language (usually lower) such that they are bot equivalent Our goal is to formally define the meaning of this equivalence But first, we must define the meaning of a programming language 13

Formal semantics 14

Operational semantics of while 15

While syntax reminder A  n | x | A ArithOp A | ( A ) ArithOp  - | + | * | / B  true | false | A = A | A  A |  B | B  B | ( B ) S  x := A | skip | S ; S | { S } | if B then S else S | while B S 16 n  Numnumerals x  Varprogram variables

Semantic categories Z Integers {0, 1, -1, 2, -2, …} T Truth values { ff, tt } State Var  Z Example state:  =[ x  5, y  7, z  0] Lookup:  ( x) = 5 Update:  [ x  6] = [ x  6, y  7, z  0] 17

Semantics of expressions 18

Semantics of arithmetic expressions Semantic function  A  : State  Z Defined by induction on the syntax tree  n   = n  x   =  (x)  a 1 + a 2   =  a 1   +  a 2    a 1 - a 2   =  a 1   -  a 2    a 1 * a 2   =  a 1     a 2    (a 1 )   =  a 1   --- not needed  - a   = 0 -  a 1   Compositional Expressions in While are side-effect free 19

Semantics of boolean expressions Semantic function  B  : State  T Defined by induction on the syntax tree  true   = tt  false   = ff  a 1 = a 2   =  a 1  a 2   =  b 1  b 2   =  b   = Compositional Expressions in While are side-effect free 20

Natural operating semantics Developed by Gilles Kahn [STACS 1987]STACS 1987 Configurations  S,  Statement S is about to execute on state   Terminal (final) state Transitions  S,    ’ Execution of S from  will terminate with the result state  ’ – Ignores non-terminating computations 21

Natural operating semantics  defined by rules of the form The meaning of compound statements is defined using the meaning immediate constituent statements 22  S 1,  1    1 ’, …,  S n,  n    n ’  S,    ’ if… premise conclusion side condition

Natural semantics for While 23  x := a,    [x  a   ] [ass ns ]  skip,    [skip ns ]  S 1,    ’,  S 2,  ’    ’’  S 1 ; S 2,    ’’ [comp ns ]  S 1,    ’  if b then S 1 else S 2,    ’ if  b   = tt [if tt ns ]  S 2,    ’  if b then S 1 else S 2,    ’ if  b   = ff [if ff ns ] axioms

Natural semantics for While 24  S,     ’,  while b S,  ’    ’’  while b S,    ’’ if  b   = tt [while tt ns ]  while b S,    if  b   = ff [while ff ns ] Non-compositional

Executing the semantics 25

Example Let  0 be the state which assigns zero to all program variables 26  x:=x+1,  0    skip,  0    0  0 [x  1]  skip,  0    0,  x:=x+1,  0    0 [x  1]  skip ; x:=x+1,  0    0 [x  1]  x:=x+1,  0    0 [x  1]  if x=0 then x:=x+1 else skip,  0    0 [x  1]

Derivation trees Using axioms and rules to derive a transition  S,    ’ gives a derivation tree – Root:  S,    ’ – Leaves: axioms – Internal nodes: conclusions of rules Immediate children: matching rule premises 27

Derivation tree example 1 Assume  0 =[x  5, y  7, z  0]  1 =[x  5, y  7, z  5]  2 =[x  7, y  7, z  5]  3 =[x  7, y  5, z  5] 28  ( z:=x; x:=y); y:=z,  0    3  ( z:=x; x:=y),  0    2  y:=z,  2    3  z:=x,  0    1  x:=y,  1    2 [ass ns ] [comp ns ]

Derivation tree example 1 Assume  0 =[x  5, y  7, z  0]  1 =[x  5, y  7, z  5]  2 =[x  7, y  7, z  5]  3 =[x  7, y  5, z  5] 29  ( z:=x; x:=y); y:=z,  0    3  ( z:=x; x:=y),  0    2  y:=z,  2    3  z:=x,  0    1  x:=y,  1    2 [ass ns ] [comp ns ]

Top-down evaluation via derivation trees Given a statement S and an input state  find an output state  ’ such that  S,    ’ Start with the root and repeatedly apply rules until the axioms are reached – Inspect different alternatives in order Theorem: In While,  ’ and the derivation tree are unique 30

Top-down evaluation example Factorial program with  x = 2 Shorthand: W= while  (x=1) { y:=y*x; x:=x-1 } 31  y:=1; while  (x=1) { y:=y*x; x:=x-1 },    [y  2][x  1]  y:=1,    [y  1]  W,  [y  1]    [y  2, x  1]  y:=y*x; x:=x-1,  [y  1]    [y  2][x  1]  W,  [y  2][x  1]    [y  2, x  1]  y:=y*x,  [y  1]    [y  2]  x:=x-1,  [y  2]    [y  2][x  1] [ass ns ] [comp ns ] [ass ns ] [comp ns ] [while ff ns ] [while tt ns ] [ass ns ]

Properties of natural semantics 32

Program termination Given a statement S and input  – S terminates on  if there exists a state  ’ such that  S,    ’ – S loops on  if there is no state  ’ such that  S,    ’ Given a statement S – S always terminates if for every input state , S terminates on  – S always loops if for every input state , S loops on  33

Semantic equivalence S 1 and S 2 are semantically equivalent if for all  and  ’  S 1,    ’ if and only if  S 2,    ’ Simple example while b do S is semantically equivalent to: if b then (S; while b S) else skip – Read proof in pages

Properties of natural semantics Equivalence of program constructs – skip; skip is semantically equivalent to skip – ((S 1 ; S 2 ); S 3 ) is semantically equivalent to (S 1 ; (S 2 ; S 3 )) – (x:=5; y:=x*8) is semantically equivalent to (x:=5; y:=40) 35

Equivalence of {S 1 ; S 2 }; S 3 and S 1 ; {S 2 ; S 3 } 36

Equivalence of {S 1 ; S 2 }; S 3 and S 1 ; {S 2 ; S 3 } 37  (S 1 ; S 2 ),    12,  S 3,  12    ’  {S 1 ; S 2 }; S 3,    ’  S 1,    1,  S 2,  1    12  S 1,    1,  {S 2 ; S 3 },  1    ’  S 1 ; {S 2 ; S 3 },    ’  S 2,  1    12,  S 3,  12    ’ Assume  (S 1 ; S 2 ); S 3,     ’ then the following unique derivation tree exists: Using the rule applications above, we can construct the following derivation tree: And vice versa.

Deterministic semantics for While Theorem: for all statements S and states  1,  2 if  S,    1 and  S,    2 then  1 =  2 38

The semantics of statements The meaning of a statement S is defined as Examples:  skip   =   x:=1   =  [x  1]  while true do skip   = undefined 39  S   =  ’ if  S,    ’  else

Operational semantics of IL 40

IL syntax reminder V  n | x R  V Op V | V Op  - | + | * | / | = |  | > | … C  l : skip | l : x := R | l : Goto l’ | l : IfZ x Goto l’ IR  C + 41 n  NumNumerals l  Num Labels x  Var  TempVariables and temporaries

Intermediate program states Z Integers {0, 1, -1, 2, -2, …} IState (Var  Temp  {pc})  Z – Var, Temp, and {pc} are all disjoint – For every state m and program P=1:c 1,…,n:c n we have that 1  m(pc)  n+1 We can check that the labels used in P are within the range 1..n 42

Rules for executing commands We will use rules of the following form Here m is the pre-state, which is scheduled to be executed as the program counter indicates, and m’ is the post-state The rules specialize for the particular type of command C and possibly other conditions 43 m(pc) = l P( l ) = C m  m’

Rules for executing commands 44 m(pc) = l P( l ) = skip m  m[pc  l +1] m(pc) = l P( l ) = Goto l’ m  m[pc  l’ ] m(pc) = l P( l ) = x := v 1 op v 2 m  m[pc  l +1, x  M( v 1 ) op M( v 2 )] M(v)= m(v) v  Var  Temp v else m(pc) = l P( l ) = x := v m  m[pc  l +1, x  M( v )] M(v)= m(v) v  Var  Temp v else

Rules for executing commands 45 m(pc) = l P( l ) = IfZ x Goto l’ m( x )=0 m  m[pc  l’ ] m(pc) = l P( l ) = IfZ x Goto l’ m( x )  0 m  m[pc  l +1] m(pc) = l P( l ) = IfNZ x Goto l’ m( x )  0 m  m[pc  l’ ] m(pc) = l P( l ) = IfNZ x Goto l’ m( x )=0 m  m[pc  l +1]

Executing programs For a program P=1:c 1,…,n:c n we define executions as finite or infinite sequences m 1  m 2  …  m n  … We write m  * m’ if there is a finite execution starting at m and ending at m’: m = m 1  m 2  …  m n = m’ 46

Semantics of a program For a program P=1:c 1,…,n:c n and a state m s.t. m(pc)=1 we define the result of executing P on m as Lemma: the function is well-defined (i.e., at most one output state) 47 P m=P m= m’if m  * m’ and m’(pc)=n+1  else

Execution example Execute the following intermediate language program on a state where all variables evaluate to : t1 := 137 2: y := t : IfZ x Goto 7 4: t2 := y 5: z := t2 6: Goto 9 7: t3 := y 8: x := t3 9: skip m = [pc  1, t1  0, t2  0, t3  0, x  0, y  0]  ?

Execution example Execute the following intermediate language program on a state where all variables evaluate to : t1 := 137 2: y := t : IfZ x Goto 7 4: t2 := y 5: z := t2 6: Goto 9 7: t3 := y 8: x := t3 9: skip m = [pc  1, t1  0, t2  0, t3  0, x  0, y  0]  m[pc  2, t1  137]  m[pc  3, t1  137, y  140]  m[pc  7, t1  137, y  140]  m[pc  8, t1  137, t3  140, y  140]  m[pc  9, t1  137, t3  140,, x  140, y  140]  m[pc  10, t1  137, t3  140,, x  140, y  140]

Proof methodology 50

Structural induction To prove a property of a derivation tree – Prove property holds for leaves – Assume property holds on all sub-trees of a given node and establish that it holds for the node Conclude that the property holds for every derivation tree 51

Defining and Proving equivalence for expressions 52

Exercise 1 Are the following equivalent in your opinion? 53 x := 1371: t1 := 137 2: x := t1 ILWhile

Exercise 2 Are the following equivalent in your opinion? 54 x := 1371: y := 137 2: x := y ILWhile

Exercise 3 Are the following equivalent in your opinion? 55 x := 1372: t2 := 138 3: t1 := 137 4: x := t1 ILWhile

Exercise 4 Are the following equivalent in your opinion? 56 x := 1372: t2 := 138 3: t1 := 137 4: x := t1 5: t1 := 138 ILWhile

Equivalence of arithmetic expressions Define TVar = Var  Temp While state:   Var  Z IL state:  m  (Var  Temp  {pc})  Z An arithmetic While expression a is equivalent to an IL program P and x  TVar iff for every input state m such that m(pc)=label(P):  a  m| Var = (  P  m) x 57

Defining equivalence for expressions 58 ILWhile (P, t)a (  P  m) t  a  m| Var cgen = (P, t)a  if  m: m(pc)=label(P) Definition:

Equivalence of Boolean expressions A Boolean While expression b is equivalent to an IL program P and x  TVar iff for every input state m such that m(pc)=label(P):  b  m| Var = tt and (  P  m) x = 1 or  b  m| Var = ff and (  P  m) x = 0 59

Equivalence of atomic expressions 60 cgen(n) = (l: t:=n, t)where l and t are fresh cgen(x) = (l: t:=x, t)where l and t are fresh Claim: n  (l: t:=n, t) Proof: choose w.l.o.g m such that m(pc)=l  n  m| Var   = n (  l: t:=n  m) t = m[pc  l+1, t  n] t = n Q.E.D Claim: n  (l: t:=x, t) Proof: choose w.l.o.g m such that m(pc)=l  x  m| Var   = m(x) since x  Var (  l: t:=x  m) t = m[pc  l+1, t  m(x)] t = m(x) Q.E.D

Lemmas for expressions 61 cgen(e 1 ) = (P 1, t 1 ) cgen(e 2 ) = (P 2, t 2 ) cgen(e 1 op e 2 ) = (P 1 · P 2 · l: t:=t 1 op t 2, t) where l and t are fresh Lemma 1: Let cgen(e 1 op e 2 )=(P, t) then P always terminates Lemma 2 (sequential execution): Let P e and P be two IL such that (P e, t)=cgen(e) and labels(P e )  labels(P) =  and Temps(P e )   Temps(P) = {t} Then for every m, such that m(pc)=label(P 1 ) (  P e  m) = m’ such that m’(pc) = label(P). Moreover, let  =m| Var then  P e · P  m| Var = (  P  m[t  e   ])| Var. Lemma 3: Let Temps(P) be the set of temporaries appearing in P. Let P 1 and P 2 be two programs such that Temps(P 1 )   Temps(P 2 ) =  and. Then for every m, such that m(pc)=label(P) (  P 1 · P 2  m)| Var = (  P 1  m 1 ) t 1

Lemmas for expressions 62 cgen(e 1 ) = (P 1, t 1 ) cgen(e 2 ) = (P 2, t 2 ) cgen(e 1 op e 2 ) = (P 1 · P 2 · l: t:=t 1 op t 2, t) where l and t are fresh Lemma 4: Let P 1 and P 2 be two IL programs such that cgen(e 1 op e 2 ) = (P 1 · P 2 · l: t:=t 1 op t 2, t) Then for every m 1, m 2 such that m 1 (pc)=label(P 1 ) and m 2 (pc)=label(P 2 ) (  P 1 · P 2  m 1 ) t 1 = (  P 1  m 1 ) t 1 (  P 1 · P 2  m 1 ) t 2 = (  P 2  m 1 ) t 2 Lemma 3: Let Temps(P) stand for the set of temporaries appearing in P. If cgen(e 1 op e 2 ) = (P 1 · P 2 · l: t:=t 1 op t 2, t) Then Temps(P 1 )   Temps(P 2 ) = 

Equivalence of compound expressions 63 cgen(e 1 ) = (P 1, t 1 ) cgen(e 2 ) = (P 2, t 2 ) cgen(e 1 op e 2 ) = (P 1 · P 2 · l: t:=t 1 op t 2, t) where l and t are fresh Claim: Let P= P 1 · P 2 · l: t:=t 1 op t 2 then e 1 op e 2  (P, t) Proof: choose w.l.o.g m such that m(pc)=label(P 1 ) let  =m| Var  e 1 + e 2   =  e 1   +  e 2   by the induction hypothesis: e 1  (P 1, t 1 ) and e 2  (P 2, t 2 ) Denote m’ =  P 1 · P 2  m By Lemma 4, we have that (  P 1 · P 2  m) t 1 = (  P 1  m) t 1 =  e 1   = m’ t 1 (  P 1 · P 2  m) t 2 = (  P 2  m) t 2 =  e 2   = m’ t 2 Therefore, by Lemma 2 (  P  m) t = (  l: t:=t 1 op t 2  m’) t = m’[pc  l+1, t  m’(t 1 ) op m’(t 2 )  m) t =  e 1   +  e 2   =  e 1 + e 2   Q.E.D

Conclusion Missing: proof for Boolean expression (exercise for home) Theorem 1: for each While expression e we have that cgen(e)  e 64

Defining and Proving equivalence for statements 65

State equivalence Define  m iff  =m| Var – That is, for each x  Var  (x)=m(x) 66

Statement equivalence A While statement S is equivalent to an IL program P iff for every input state m such that m(pc)=label(P):  S  m| Var  (  P   )| Var 67

Defining equivalence for statements 68 IL While PS cgen P mP m S S  P S if  m,  : m(pc)=label(P) and  m Definition:

Equivalence of skip 69 cgen( skip ) = l: skip where l is fresh Claim: skip  l: skip Proof: choose w.l.o.g m such that m(pc)=l and  =m| Var  skip  m| Var  = m| Var (  l: skip  m)| Var = m[pc  l+1]| Var = m| Var Q.E.D

Equivalence of assignments 70 Lemma 5: let DefVars(P) denote the variables being assigned-to in P. If DefVars(P)=  Then for every m such that m(pc)=label(P) (  P  m)| Var = m| Var cgen(a) = (P, t) cgen( x := a) = P · l: x := t where l is fresh

Equivalence of assignments 71 Claim: x := a  P · l: x := t Proof: choose w.l.o.g m such that m(pc)=l and  =m| Var  x := a   =  [ x  a   ] Let m’ =  P  m. Then from Theorem 1: a  (P,t). That is,  a   = m’(t) From Lemma 2:  P · l: x := t  m =  l: x := t  m’ = m’[pc  l+1, x  m’(t)] = m’[pc  l+1, x  a   ] Now, since DefVars(P)=  by Lemma 5, we have that m’| Var = m| Var Therefore,  [ x  a   ]| Var =  P · l: x := t  | Var Q.E.D cgen(a) = (P, t) cgen( x := a) = P · l: x := t where l is fresh

Natural semantics for sequencing 72  S 1,    ’,  S 2,  ’    ’’  S 1 ; S 2,    ’’ [comp ns ] Lemma 6:  S 1 ; S 2   =  S 2  (  S 1   )

Helper lemmas for sequencing 73 Lemma 7: Let P 1 and P 1 be two IL programs such that such that cgen( S 1 ; S 2 ) = P 1 · P 2 Then labels(P 1 )  labels(P 2 ) =  and Temps(P 1 )   Temps(P 2 ) =  and for every m, such that m(pc)=label(P 1 ) (  P 1  m) = m’ such that m’(pc) = label(P 2 ). Moreover,  P 1 · P 2  m =  P 2  m’ cgen( S 1 ) = P 1, cgen( S 2 ) = P 2 cgen( S 1 ; S 2 ) = P 1 · P 2

Equivalence of sequencing 74 Claim: assume S 1  P 1 and S 2  P 2 then S 1 ; S 2  P 1 · P 2 Proof: choose w.l.o.g m such that m(pc)=l and  =m| Var By Lemma 7, we have that (  P 1  m) = m’ such that m’(pc) = label(P 2 ) By the induction hypothesis m’| Var =  S 1  . By Lemma 7, we have that  P 1 · P 2  m =  P 2  m’ Let  S 1   =  ’. By the induction hypothesis, since m’   ’, we have that  P 2  m’   S 2   ’ By the definition of the natural semantics (lemma 6), we have that  S 1 ; S 2     P 1 · P 2  m Q.E.D cgen( S 1 ) = P 1, cgen( S 2 ) = P 2 cgen( S 1 ; S 2 ) = P 1 · P 2

Equivalence of conditions 75 cgen( b ) = (Pb, t), cgen( S 1 ) = P 1, cgen( S 2 ) = P 2 cgen( if b then S 1 else S 2 ) = Pb lb: IfZ t Goto label(P 2 ) P 1 l finish : Goto L after P 2 l after : skip where lb, l finish, l after are fresh Lemma 8: for all states  we have that If  b   = tt then,  if b then S 1 else S 2   =  S 1   If  b   = ff then,  if b then S 1 else S 2   =  S 2  

Helper lemmas 76 cgen( b ) = (Pb, t), cgen( S 1 ) = P 1, cgen( S 2 ) = P 2 cgen( if b then S 1 else S 2 ) = Pb lb: IfZ t Goto label(P 2 ) P 1 l finish : Goto L after P 2 l after : skip where lb, l finish, l after are fresh

Equivalence of conditions: ff 77 Claim: Let P=cgen( if b then S 1 else S 2 ) if b then S 1 else S 2  P Proof: choose w.l.o.g m such that m(pc)=l and  =m| Var From Theorem 1: b  (P,t) therefore  Pb  m t = 0 Case ff: assume  b   = ff. By Lemma 8:  if b then S 1 else S 2   =  S 2   =  2 By Lemma 2  Pb · lb: IfZ t Goto label(P 2 ) · P 1 · l finish : Goto L after · P 2 · l after : skip  m   IfZ t Goto label(P 2 ) · P 1 · l finish : Goto L after · P 2 · l after : skip  (m[t  0])   IfZ t Goto label(P 2 ) · P 1 · l finish : Goto L after · P 2 · l after : skip  (m[pc  label(P 2 )])  By the induction hypothesis and lemma 2: Let  P 2  m[pc  label(P 2 )] = m 2 then m 2   2  IfZ t Goto label(P 2 ) · P 1 · l finish : Goto L after · P 2 · l after : skip  (m 2 [pc  l after ]) = m 2 [pc  l after +1]  m 2

Equivalence of conditions: tt 78 Claim: Let P=cgen( if b then S 1 else S 2 ) if b then S 1 else S 2  P Proof: choose w.l.o.g m such that m(pc)=l and  =m| Var From Theorem 1: b  (P,t) therefore  Pb  m t = 0 Case ff: assume  b   = tt. By Lemma 8:  if b then S 1 else S 2   =  S 1   =  1 By Lemma 2  Pb · lb: IfZ t Goto label(P 2 ) · P 1 · l finish : Goto L after · P 2 · l after : skip  m   IfZ t Goto label(P 2 ) · P 1 · l finish : Goto L after · P 2 · l after : skip  (m[t  1])   IfZ t Goto label(P 2 ) · P 1 · l finish : Goto L after · P 2 · l after : skip  (m[pc  label(P 1 )])  By the induction hypothesis and lemma 2: Let  P 1  m[pc  label(P 1 )] = m 1 then m 1   1  IfZ t Goto label(P 2 ) · P 1 · l finish : Goto L after · P 2 · l after : skip  (m 1 [pc  l finish ]) = m 1 [pc  l after ] = m 1 [pc  l after +1]  m 1

Equivalence for loops 79 cgen( b ) = (Pb, t), cgen( S ) = P cgen( while b S ) = l before : skip Pb IfZ t Goto l after P l loop : Goto L before l after : skip where l after, l before, l loop are fresh

Proof outline Let  be a state We will split the proof into two cases: 1.  while b S   terminates (there is a derivation tree) 2.  while b S   loops (no derivation tree) 80

Case 1:  while b S   terminates 81

Case 2:  while b S   loops 82

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