EVM uses bounded 256 bit integer words, and sometimes also bytes (8 bit words).
Here we provide the arithmetic of these words, as well as some data-structures over them.
Both are implemented using K's Int
.
module EVM-TYPES
imports INT
imports STRING
imports COLLECTIONS
imports BYTES
Some important numbers that are referred to often during execution.
These can be used for pattern-matching on the LHS of rules as well (macro
attribute expands all occurances of these in rules).
syntax Int ::= "pow256" /* 2 ^Int 256 */
| "pow255" /* 2 ^Int 255 */
| "pow160" /* 2 ^Int 160 */
| "pow16" /* 2 ^Int 16 */
// ----------------------------------------
rule pow256 => 115792089237316195423570985008687907853269984665640564039457584007913129639936 [macro]
rule pow255 => 57896044618658097711785492504343953926634992332820282019728792003956564819968 [macro]
rule pow160 => 1461501637330902918203684832716283019655932542976 [macro]
rule pow16 => 65536 [macro]
syntax Int ::= "minSInt128"
| "maxSInt128"
| "minUInt8"
| "maxUInt8"
| "minUInt16"
| "maxUInt16"
| "minUInt48"
| "maxUInt48"
| "minUInt128"
| "maxUInt128"
| "minUInt160"
| "maxUInt160"
| "minSInt256"
| "maxSInt256"
| "minUInt256"
| "maxUInt256"
| "minSFixed128x10"
| "maxSFixed128x10"
| "minUFixed128x10"
| "maxUFixed128x10"
// --------------------------------
rule minSInt128 => -170141183460469231731687303715884105728 [macro] /* -2^127 */
rule maxSInt128 => 170141183460469231731687303715884105727 [macro] /* 2^127 - 1 */
rule minSFixed128x10 => -1701411834604692317316873037158841057280000000000 [macro] /* (-2^127 ) * 10^10 */
rule maxSFixed128x10 => 1701411834604692317316873037158841057270000000000 [macro] /* ( 2^127 - 1) * 10^10 */
rule minSInt256 => -57896044618658097711785492504343953926634992332820282019728792003956564819968 [macro] /* -2^255 */
rule maxSInt256 => 57896044618658097711785492504343953926634992332820282019728792003956564819967 [macro] /* 2^255 - 1 */
rule minUInt8 => 0 [macro]
rule maxUInt8 => 255 [macro]
rule minUInt16 => 0 [macro]
rule maxUInt16 => 65535 [macro] /* 2^16 - 1 */
rule minUInt48 => 0 [macro]
rule maxUInt48 => 281474976710655 [macro] /* 2^48 - 1 */
rule minUInt128 => 0 [macro]
rule maxUInt128 => 340282366920938463463374607431768211455 [macro] /* 2^128 - 1 */
rule minUFixed128x10 => 0 [macro]
rule maxUFixed128x10 => 3402823669209384634633746074317682114550000000000 [macro] /* ( 2^128 - 1) * 10^10 */
rule minUInt160 => 0 [macro]
rule maxUInt160 => 1461501637330902918203684832716283019655932542975 [macro] /* 2^160 - 1 */
rule minUInt256 => 0 [macro]
rule maxUInt256 => 115792089237316195423570985008687907853269984665640564039457584007913129639935 [macro] /* 2^256 - 1 */
syntax Int ::= "eth"
// --------------------
rule eth => 1000000000000000000 [macro]
- Range of types
syntax Bool ::= #rangeSInt ( Int , Int )
| #rangeUInt ( Int , Int )
| #rangeSFixed ( Int , Int , Int )
| #rangeUFixed ( Int , Int , Int )
| #rangeAddress ( Int )
| #rangeBytes ( Int , Int )
// -------------------------------------------
rule #rangeSInt ( 128 , X ) => #range ( minSInt128 <= X <= maxSInt128 ) [macro]
rule #rangeSInt ( 256 , X ) => #range ( minSInt256 <= X <= maxSInt256 ) [macro]
rule #rangeUInt ( 8 , X ) => #range ( minUInt8 <= X <= maxUInt8 ) [macro]
rule #rangeUInt ( 16 , X ) => #range ( minUInt16 <= X <= maxUInt16 ) [macro]
rule #rangeUInt ( 48 , X ) => #range ( minUInt48 <= X <= maxUInt48 ) [macro]
rule #rangeUInt ( 128 , X ) => #range ( minUInt128 <= X <= maxUInt128 ) [macro]
rule #rangeUInt ( 256 , X ) => #range ( minUInt256 <= X <= maxUInt256 ) [macro]
rule #rangeSFixed ( 128 , 10 , X ) => #range ( minSFixed128x10 <= X <= maxSFixed128x10 ) [macro]
rule #rangeUFixed ( 128 , 10 , X ) => #range ( minUFixed128x10 <= X <= maxUFixed128x10 ) [macro]
rule #rangeAddress ( X ) => #range ( minUInt160 <= X <= maxUInt160 ) [macro]
rule #rangeBytes ( N , X ) => #range ( 0 <= X <= #nBytes(N) ) [macro]
syntax Bool ::= "#range" "(" Int "<" Int "<" Int ")"
| "#range" "(" Int "<" Int "<=" Int ")"
| "#range" "(" Int "<=" Int "<" Int ")"
| "#range" "(" Int "<=" Int "<=" Int ")"
// ------------------------------------------------------
rule #range ( LB < X < UB ) => LB <Int X andBool X <Int UB [macro]
rule #range ( LB < X <= UB ) => LB <Int X andBool X <=Int UB [macro]
rule #range ( LB <= X < UB ) => LB <=Int X andBool X <Int UB [macro]
rule #range ( LB <= X <= UB ) => LB <=Int X andBool X <=Int UB [macro]
-
chop
interprets an integer modulo$2^256$ .
syntax Int ::= chop ( Int ) [function, functional, smtlib(chop)]
// ----------------------------------------------------------------
rule chop ( I:Int ) => I modInt pow256 [concrete, smt-lemma]
Primitives provide the basic conversion from K's sorts Int
and Bool
to EVM's words.
bool2Word
interprets aBool
as aInt
.word2Bool
interprets aInt
as aBool
.
syntax Int ::= bool2Word ( Bool ) [function, functional]
// --------------------------------------------------------
rule bool2Word( B:Bool ) => 1 requires B
rule bool2Word( B:Bool ) => 0 requires notBool B
syntax Bool ::= word2Bool ( Int ) [function, functional]
// --------------------------------------------------------
rule word2Bool( W ) => false requires W ==Int 0
rule word2Bool( W ) => true requires W =/=Int 0
sgn
gives the twos-complement interperetation of the sign of a word.abs
gives the twos-complement interperetation of the magnitude of a word.
syntax Int ::= sgn ( Int ) [function, functional]
| abs ( Int ) [function, functional]
// -------------------------------------------------
rule sgn(I) => -1 requires I >=Int pow255
rule sgn(I) => 1 requires I <Int pow255
rule abs(I) => 0 -Word I requires sgn(I) ==Int -1
rule abs(I) => I requires sgn(I) ==Int 1
- #signed : uInt256 -> sInt256 (i.e., [minUInt256..maxUInt256] -> [minSInt256..maxSInt256])
- #unsigned : sInt256 -> uInt256 (i.e., [minSInt256..maxSInt256] -> [minUInt256..maxUInt256])
syntax Int ::= #signed ( Int ) [function, smtlib(signed), smt-prelude]
// -----------------------------------------
rule [#signed.positive]: #signed(DATA) => DATA
requires 0 <=Int DATA andBool DATA <=Int maxSInt256
rule [#signed.negative]: #signed(DATA) => DATA -Int pow256
requires maxSInt256 <Int DATA andBool DATA <=Int maxUInt256
syntax Int ::= #unsigned ( Int ) [function, smtlib(unsigned), smt-prelude]
// -------------------------------------------
rule [#unsigned.positive]: #unsigned(DATA) => DATA
requires 0 <=Int DATA andBool DATA <=Int maxSInt256
rule [#unsigned.negative]: #unsigned(DATA) => pow256 +Int DATA
requires minSInt256 <=Int DATA andBool DATA <Int 0
up/Int
performs integer division but rounds up instead of down.
NOTE: Here, we choose to add I2 -Int 1
to the numerator beforing doing the division to mimic the C++ implementation.
You could alternatively calculate I1 modInt I2
, then add one to the normal integer division afterward depending on the result.
syntax Int ::= Int "up/Int" Int [function]
// ------------------------------------------
rule I1 up/Int 0 => 0
rule I1 up/Int 1 => I1
rule I1 up/Int I2 => (I1 +Int (I2 -Int 1)) /Int I2 requires I2 >Int 1
log256Int
returns the log base 256 (floored) of an integer.
syntax Int ::= log256Int ( Int ) [function]
// -------------------------------------------
rule log256Int(N) => log2Int(N) /Int 8
The corresponding <op>Word
operations automatically perform the correct modulus for EVM words.
Warning: operands are assumed to be within the range of a 256 bit EVM word. Unbound integers may not return the correct result.
syntax Int ::= Int "+Word" Int [function, functional]
| Int "*Word" Int [function, functional]
| Int "-Word" Int [function, functional]
| Int "/Word" Int [function, functional]
| Int "%Word" Int [function, functional]
// -----------------------------------------------------
rule W0 +Word W1 => chop( W0 +Int W1 )
rule W0 -Word W1 => W0 -Int W1 requires W0 >=Int W1
rule W0 -Word W1 => chop( (W0 +Int pow256) -Int W1 ) requires W0 <Int W1
rule W0 *Word W1 => chop( W0 *Int W1 )
rule W0 /Word W1 => 0 requires W1 ==Int 0
rule W0 /Word W1 => W0 /Int W1 requires W1 =/=Int 0
rule W0 %Word W1 => 0 requires W1 ==Int 0
rule W0 %Word W1 => W0 modInt W1 requires W1 =/=Int 0
Care is needed for ^Word
to avoid big exponentiation.
The helper powmod
is a totalization of the operator _^%Int__
(which comes with K).
_^%Int__
is not defined when the modulus (third argument) is zero, but powmod
is.
syntax Int ::= Int "^Word" Int [function]
syntax Int ::= powmod(Int, Int, Int) [function, functional]
// -----------------------------------------------------------
rule W0 ^Word W1 => powmod(W0, W1, pow256)
rule [powmod.nonzero]: powmod(W0, W1, W2) => W0 ^%Int W1 W2 requires W2 =/=Int 0
rule [powmod.zero]: powmod(W0, W1, W2) => 0 requires W2 ==Int 0
/sWord
and %sWord
give the signed interperetations of /Word
and %Word
.
syntax Int ::= Int "/sWord" Int [function]
| Int "%sWord" Int [function]
// ------------------------------------------
rule W0 /sWord W1 => #sgnInterp(sgn(W0) *Int sgn(W1) , abs(W0) /Word abs(W1))
rule W0 %sWord W1 => #sgnInterp(sgn(W0) , abs(W0) %Word abs(W1))
syntax Int ::= #sgnInterp ( Int , Int ) [function, functional]
// --------------------------------------------------------------
rule #sgnInterp( W0 , W1 ) => 0 requires W0 ==Int 0
rule #sgnInterp( W0 , W1 ) => W1 requires W0 >Int 0
rule #sgnInterp( W0 , W1 ) => 0 -Word W1 requires W0 <Int 0
The <op>Word
comparisons similarly lift K operators to EVM ones:
syntax Int ::= Int "<Word" Int [function, functional]
| Int ">Word" Int [function, functional]
| Int "<=Word" Int [function, functional]
| Int ">=Word" Int [function, functional]
| Int "==Word" Int [function, functional]
// ------------------------------------------------------
rule W0 <Word W1 => bool2Word(W0 <Int W1)
rule W0 >Word W1 => bool2Word(W0 >Int W1)
rule W0 <=Word W1 => bool2Word(W0 <=Int W1)
rule W0 >=Word W1 => bool2Word(W0 >=Int W1)
rule W0 ==Word W1 => bool2Word(W0 ==Int W1)
s<Word
implements a less-than forWord
(with signed interperetation).
syntax Int ::= Int "s<Word" Int [function, functional]
// ------------------------------------------------------
rule [s<Word.pp]: W0 s<Word W1 => W0 <Word W1 requires sgn(W0) ==K 1 andBool sgn(W1) ==K 1
rule [s<Word.pn]: W0 s<Word W1 => bool2Word(false) requires sgn(W0) ==K 1 andBool sgn(W1) ==K -1
rule [s<Word.np]: W0 s<Word W1 => bool2Word(true) requires sgn(W0) ==K -1 andBool sgn(W1) ==K 1
rule [s<Word.nn]: W0 s<Word W1 => abs(W1) <Word abs(W0) requires sgn(W0) ==K -1 andBool sgn(W1) ==K -1
Bitwise logical operators are lifted from the integer versions.
syntax Int ::= "~Word" Int [function, functional]
| Int "|Word" Int [function, functional]
| Int "&Word" Int [function, functional]
| Int "xorWord" Int [function, functional]
| Int "<<Word" Int [function]
| Int ">>Word" Int [function]
| Int ">>sWord" Int [function]
// -------------------------------------------
rule ~Word W => W xorInt maxUInt256
rule W0 |Word W1 => W0 |Int W1
rule W0 &Word W1 => W0 &Int W1
rule W0 xorWord W1 => W0 xorInt W1
rule W0 <<Word W1 => chop( W0 <<Int W1 ) requires W1 <Int 256
rule W0 <<Word W1 => 0 requires W1 >=Int 256
rule W0 >>Word W1 => W0 >>Int W1
rule W0 >>sWord W1 => chop( (abs(W0) *Int sgn(W0)) >>Int W1 )
-
bit
gets bit$N$ (0 being MSB). -
byte
gets byte$N$ (0 being the MSB).
syntax Int ::= bit ( Int , Int ) [function]
| byte ( Int , Int ) [function]
// --------------------------------------------
rule bit (N, _) => 0 requires notBool (N >=Int 0 andBool N <Int 256)
rule byte(N, _) => 0 requires notBool (N >=Int 0 andBool N <Int 32)
rule bit (N, W) => bitRangeInt(W , (255 -Int N) , 1) requires N >=Int 0 andBool N <Int 256
rule byte(N, W) => bitRangeInt(W , ( 31 -Int N) *Int 8 , 8) requires N >=Int 0 andBool N <Int 32
-
#nBits
shifts in$N$ ones from the right. -
#nBytes
shifts in$N$ bytes of ones from the right. -
_<<Byte_
shifts an integer 8 bits to the left. -
_>>Byte_
shifts an integer 8 bits to the right.
syntax Int ::= #nBits ( Int ) [function]
| #nBytes ( Int ) [function]
| Int "<<Byte" Int [function]
| Int ">>Byte" Int [function]
// ------------------------------------------
rule #nBits(N) => (1 <<Int N) -Int 1 requires N >=Int 0
rule #nBytes(N) => #nBits(N *Int 8) requires N >=Int 0
rule N <<Byte M => N <<Int (8 *Int M)
rule N >>Byte M => N >>Int (8 *Int M)
-
signextend(N, W)
sign-extends from byte$N$ of$W$ (0 being MSB).
syntax Int ::= signextend( Int , Int ) [function, functional]
// -------------------------------------------------------------
rule [signextend.invalid]: signextend(N, W) => W requires N >=Int 32 orBool N <Int 0
rule [signextend.negative]: signextend(N, W) => chop( (#nBytes(31 -Int N) <<Byte (N +Int 1)) |Int W ) requires N <Int 32 andBool N >=Int 0 andBool word2Bool(bit(256 -Int (8 *Int (N +Int 1)), W))
rule [signextend.positive]: signextend(N, W) => chop( #nBytes(N +Int 1) &Int W ) requires N <Int 32 andBool N >=Int 0 andBool notBool word2Bool(bit(256 -Int (8 *Int (N +Int 1)), W))
A cons-list is used for the EVM wordstack.
.WordStack
serves as the empty worstack, and_:_
serves as the "cons" operator.
syntax WordStack [flatPredicate]
syntax WordStack ::= ".WordStack" [smtlib(_dotWS)]
| Int ":" WordStack [klabel(_:_WS), smtlib(_WS_)]
// --------------------------------------------------------------------
syntax Bytes ::= Int ":" Bytes [function]
// -----------------------------------------
rule I : BS => Int2Bytes(1, I, BE) +Bytes BS requires I <Int 256
-
#take(N , WS)
keeps the first$N$ elements of aWordStack
(passing with zeros as needed). -
#drop(N , WS)
removes the first$N$ elements of aWordStack
.
syntax WordStack ::= #take ( Int , WordStack ) [klabel(takeWordStack), function, functional]
// --------------------------------------------------------------------------------------------
rule [#take.base]: #take(N, WS) => .WordStack requires notBool N >Int 0
rule [#take.zero-pad]: #take(N, .WordStack) => 0 : #take(N -Int 1, .WordStack) requires N >Int 0
rule [#take.recursive]: #take(N, (W : WS):WordStack) => W : #take(N -Int 1, WS) requires N >Int 0
syntax WordStack ::= #drop ( Int , WordStack ) [klabel(dropWordStack), function, functional]
// --------------------------------------------------------------------------------------------
rule #drop(N, WS:WordStack) => WS requires notBool N >Int 0
rule #drop(N, .WordStack) => .WordStack requires N >Int 0
rule #drop(N, (W : WS):WordStack) => #drop(1, #drop(N -Int 1, (W : WS))) requires N >Int 1
rule #drop(1, (_ : WS):WordStack) => WS
syntax Bytes ::= #take ( Int , Bytes ) [klabel(takeBytes), function, functional]
// --------------------------------------------------------------------------------
rule #take(N, BS:Bytes) => .Bytes requires notBool N >Int 0
rule #take(N, BS:Bytes) => #padRightToWidth(N, .Bytes) requires notBool lengthBytes(BS) >Int 0 andBool N >Int 0
rule #take(N, BS:Bytes) => BS +Bytes #take(N -Int lengthBytes(BS), .Bytes) requires lengthBytes(BS) >Int 0 andBool notBool N >Int lengthBytes(BS)
rule #take(N, BS:Bytes) => BS [ 0 .. N ] requires lengthBytes(BS) >Int 0 andBool N >Int lengthBytes(BS)
syntax Bytes ::= #drop ( Int , Bytes ) [klabel(dropBytes), function, functional]
// --------------------------------------------------------------------------------
rule #drop(N, BS:Bytes) => BS requires notBool N >Int 0
rule #drop(N, BS:Bytes) => .Bytes requires notBool lengthBytes(BS) >Int 0 andBool N >Int 0
rule #drop(N, BS:Bytes) => .Bytes requires lengthBytes(BS) >Int 0 andBool N >Int lengthBytes(BS)
rule #drop(N, BS:Bytes) => substrBytes(BS, N, lengthBytes(BS)) requires lengthBytes(BS) >Int 0 andBool notBool N >Int lengthBytes(BS)
-
WS [ N ]
accesses element$N$ of$WS$ . -
WS [ N := W ]
sets element$N$ of$WS$ to$W$ (padding with zeros as needed).
syntax Int ::= WordStack "[" Int "]" [function]
// -----------------------------------------------
rule (W : _):WordStack [ N ] => W requires N ==Int 0
rule WS:WordStack [ N ] => #drop(N, WS) [ 0 ] requires N >Int 0
syntax WordStack ::= WordStack "[" Int ":=" Int "]" [function]
// --------------------------------------------------------------
rule (W0 : WS):WordStack [ N := W ] => W : WS requires N ==Int 0
rule (W0 : WS):WordStack [ N := W ] => W0 : (WS [ N -Int 1 := W ]) requires N >Int 0
- Definedness conditions for
WS [ N ]
andWS [ N := W ]
rule #Ceil(WS[N]) => {((0 <=Int N) andBool (N <Int #sizeWordStack(WS))) #Equals true} [anywhere]
rule #Ceil(WS[ N := W ]) => {((0 <=Int N) andBool (N <Int #sizeWordStack(WS))) #Equals true} [anywhere]
#sizeWordStack
calculates the size of aWordStack
._in_
determines if aInt
occurs in aWordStack
.
syntax Int ::= #sizeWordStack ( WordStack ) [function, functional, smtlib(sizeWordStack)]
// ----------------------------------------------------------------------------------------------------------------------------
rule #sizeWordStack ( .WordStack ) => 0
rule #sizeWordStack ( W : WS ) => #sizeWordStack(WS) +Int 1
syntax Bool ::= Int "in" WordStack [function]
// ---------------------------------------------
rule W in .WordStack => false
rule W in (W' : WS) => (W ==K W') orElseBool (W in WS)
#replicateAux
pushesN
copies ofA
onto aWordStack
.#replicate
is aWordStack
of lengthN
withA
the value of every element.
syntax WordStack ::= #replicate ( Int, Int ) [function, functional]
| #replicateAux ( Int, Int, WordStack ) [function, functional]
// ---------------------------------------------------------------------------------
rule #replicate ( N, A ) => #replicateAux(N, A, .WordStack)
rule #replicateAux( N, A, WS ) => #replicateAux(N -Int 1, A, A : WS) requires N >Int 0
rule #replicateAux( N, A, WS ) => WS requires notBool N >Int 0
WordStack2List
converts a term of sortWordStack
to a term of sortList
.
syntax List ::= WordStack2List ( WordStack ) [function, functional]
// -------------------------------------------------------------------
rule WordStack2List(.WordStack) => .List
rule WordStack2List(W : WS) => ListItem(W) WordStack2List(WS)
Most of EVM data is held in local memory.
-
WM [ N := WS ]
assigns a contiguous chunk of$WM$ to$WS$ starting at position$W$ . -
#range(M, START, WIDTH)
reads off$WIDTH$ elements from$WM$ beginning at position$START$ (padding with zeros as needed).
syntax Memory = Bytes
syntax Memory ::= Memory "[" Int ":=" ByteArray "]" [function, klabel(mapWriteBytes)]
// -------------------------------------------------------------------------------------
rule WS [ START := WS' ] => replaceAtBytes(padRightBytes(WS, START +Int #sizeByteArray(WS'), 0), START, WS') [concrete]
syntax ByteArray ::= #range ( Memory , Int , Int ) [function]
// -------------------------------------------------------------
rule #range(LM, START, WIDTH) => LM [ START .. WIDTH ] [concrete]
syntax Memory ::= ".Memory" [function]
// --------------------------------------
rule .Memory => .Bytes [macro]
syntax Memory ::= Memory "[" Int ":=" Int "]" [function]
// --------------------------------------------------------
rule WM [ IDX := VAL ] => padRightBytes(WM, IDX +Int 1, 0) [ IDX <- VAL ]
syntax Memory = Map
syntax Memory ::= Memory "[" Int ":=" ByteArray "]" [function, klabel(mapWriteBytes)]
// -------------------------------------------------------------------------------------
rule WM[ N := WS ] => WM [ N := WS, 0, #sizeByteArray(WS) ]
syntax Map ::= Map "[" Int ":=" ByteArray "," Int "," Int "]" [function]
// ------------------------------------------------------------------------
rule WM [ N := WS, I, I ] => WM
rule WM [ N := WS, I, J ] => (WM[N <- WS[I]])[ N +Int 1 := WS, I +Int 1, J ] [owise]
syntax ByteArray ::= #range ( Memory , Int , Int ) [function]
| #range ( Memory , Int , Int , Int , ByteArray ) [function, klabel(#rangeAux)]
// --------------------------------------------------------------------------------------------------
rule #range(WM, START, WIDTH) => #range(WM, START, 0, WIDTH, padLeftBytes(.Bytes, WIDTH, 0))
rule #range(WM, I, WIDTH, WIDTH, WS) => WS
rule #range(WM, I, J, WIDTH, WS) => #range(WM, I +Int 1, J +Int 1, WIDTH, WS [ J <- {WM[I] orDefault 0}:>Int ]) [owise]
syntax Memory ::= ".Memory" [function]
// --------------------------------------
rule .Memory => .Map [macro]
syntax Memory ::= Memory "[" Int ":=" Int "]" [function]
// --------------------------------------------------------
rule WM [ IDX := VAL:Int ] => WM [ IDX <- VAL ]
syntax Memory = Map
syntax Memory ::= Memory "[" Int ":=" ByteArray "]" [function, functional]
// --------------------------------------------------------------------------
rule [mapWriteBytes.base]: WM[ N := .WordStack ] => WM
rule [mapWriteBytes.recursive]: WM[ N := W : WS ] => (WM[N <- W])[N +Int 1 := WS]
syntax ByteArray ::= #range ( Memory , Int , Int ) [function, functional]
syntax ByteArray ::= #range ( Memory , Int , Int , ByteArray ) [function, functional, klabel(#rangeAux)]
// --------------------------------------------------------------------------------------------------------
rule [#range]: #range(WM, START, WIDTH) => #range(WM, START +Int WIDTH -Int 1, WIDTH, .WordStack)
rule [#rangeAux.base]: #range(WM, END, WIDTH, WS) => WS requires notBool WIDTH >Int 0
rule [#rangeAux.none]: #range(WM, END, WIDTH, WS) => #range(WM, END -Int 1, WIDTH -Int 1, 0 : WS) requires (WIDTH >Int 0) andBool notBool END in_keys(WM)
rule [#rangeAux.some]: #range(END |-> W WM, END, WIDTH, WS) => #range(WM, END -Int 1, WIDTH -Int 1, W : WS) requires (WIDTH >Int 0)
syntax Memory ::= ".Memory" [function]
// --------------------------------------
rule .Memory => .Map [macro]
syntax Memory ::= Memory "[" Int ":=" Int "]" [function]
// --------------------------------------------------------
rule WM [ IDX := VAL:Int ] => WM [ IDX <- VAL ]
The local memory of execution is a byte-array (instead of a word-array).
#asWord
will interperet a stack of bytes as a single word (with MSB first).#asInteger
will interperet a stack of bytes as a single arbitrary-precision integer (with MSB first).#asAccount
will interpret a stack of bytes as a single account id (with MSB first). Differs from#asWord
only in that an empty stack represents the empty account, not account zero.#asByteStack
will split a single word up into aByteArray
._++_
acts asByteArray
append.WS [ N .. W ]
access the range ofWS
beginning withN
of widthW
.#sizeByteArray
calculates the size of aByteArray
.#padToWidth(N, WS)
and#padRightToWidth
make sure that aWordStack
is the correct size.
syntax ByteArray = Bytes
syntax ByteArray ::= ".ByteArray" [function, functional]
// --------------------------------------------------------
rule .ByteArray => .Bytes [macro]
syntax Int ::= #asWord ( ByteArray ) [function, smtlib(asWord)]
// ---------------------------------------------------------------
rule #asWord(WS) => chop(Bytes2Int(WS, BE, Unsigned))
syntax Int ::= #asInteger ( ByteArray ) [function]
// --------------------------------------------------
rule #asInteger(WS) => Bytes2Int(WS, BE, Unsigned)
syntax Account ::= #asAccount ( ByteArray ) [function]
// ------------------------------------------------------
rule #asAccount(BS) => .Account requires lengthBytes(BS) ==Int 0
rule #asAccount(BS) => #asWord(BS) [owise]
syntax ByteArray ::= #asByteStack ( Int ) [function]
// ----------------------------------------------------
rule #asByteStack(W) => Int2Bytes(W, BE, Unsigned)
syntax ByteArray ::= ByteArray "++" ByteArray [function, right, klabel(_++_WS), smtlib(_plusWS_)]
// -------------------------------------------------------------------------------------------------
rule WS ++ WS' => WS +Bytes WS'
syntax ByteArray ::= ByteArray "[" Int ".." Int "]" [function]
// --------------------------------------------------------------
rule WS [ START .. WIDTH ] => substrBytes(padRightBytes(WS, START +Int WIDTH, 0), START, START +Int WIDTH) requires START <Int #sizeByteArray(WS)
rule WS [ START .. WIDTH ] => padRightBytes(.Bytes, WIDTH, 0) [owise]
syntax Int ::= #sizeByteArray ( ByteArray ) [function, functional]
// ------------------------------------------------------------------
rule #sizeByteArray ( WS ) => lengthBytes(WS) [concrete]
syntax ByteArray ::= #padToWidth ( Int , ByteArray ) [function]
| #padRightToWidth ( Int , ByteArray ) [function]
// --------------------------------------------------------------------
rule #padToWidth(N, BS) => padLeftBytes(BS, N, 0)
rule #padRightToWidth(N, BS) => padRightBytes(BS, N, 0)
syntax ByteArray = WordStack
syntax ByteArray ::= ".ByteArray" [function]
// --------------------------------------------
rule .ByteArray => .WordStack [macro]
syntax Int ::= #asWord ( ByteArray ) [function, functional, smtlib(asWord)]
// ---------------------------------------------------------------------------
rule [#asWord.base-empty]: #asWord( .WordStack ) => 0
rule [#asWord.base-single]: #asWord( W : .WordStack ) => W
rule [#asWord.recursive]: #asWord( W0 : W1 : WS ) => #asWord(((W0 *Word 256) +Word W1) : WS)
syntax Int ::= #asInteger ( ByteArray ) [function]
// --------------------------------------------------
rule #asInteger( .WordStack ) => 0
rule #asInteger( W : .WordStack ) => W
rule #asInteger( W0 : W1 : WS ) => #asInteger(((W0 *Int 256) +Int W1) : WS)
syntax Account ::= #asAccount ( ByteArray ) [function]
// ------------------------------------------------------
rule #asAccount( .WordStack ) => .Account
rule #asAccount( W : WS ) => #asWord(W : WS)
syntax ByteArray ::= #asByteStack ( Int ) [function, functional]
| #asByteStack ( Int , ByteArray ) [function, klabel(#asByteStackAux), smtlib(asByteStack)]
// --------------------------------------------------------------------------------------------------------------
rule [#asByteStack]: #asByteStack( W ) => #asByteStack( W , .WordStack )
rule [#asByteStackAux.base]: #asByteStack( 0 , WS ) => WS
rule [#asByteStackAux.recursive]: #asByteStack( W , WS ) => #asByteStack( W /Int 256 , W modInt 256 : WS ) requires W =/=K 0
syntax ByteArray ::= ByteArray "++" ByteArray [function, memo, right, klabel(_++_WS), smtlib(_plusWS_)]
// -------------------------------------------------------------------------------------------------------
rule .WordStack ++ WS' => WS'
rule (W : WS) ++ WS' => W : (WS ++ WS')
syntax ByteArray ::= ByteArray "[" Int ".." Int "]" [function, functional, memo]
// --------------------------------------------------------------------------------
rule WS [ START .. WIDTH ] => #take(WIDTH, #drop(START, WS))
syntax Int ::= #sizeByteArray ( ByteArray ) [function, functional, memo]
// ------------------------------------------------------------------------
rule #sizeByteArray ( WS ) => #sizeWordStack(WS) [concrete]
syntax ByteArray ::= #padToWidth ( Int , ByteArray ) [function, functional, memo]
| #padRightToWidth ( Int , ByteArray ) [function, memo]
// --------------------------------------------------------------------------------------
rule [#padToWidth]: #padToWidth(N, WS) => #replicateAux(N -Int #sizeByteArray(WS), 0, WS)
rule [#padRightToWidth]: #padRightToWidth(N, WS) => WS ++ #replicate(N -Int #sizeByteArray(WS), 0)
.Account
represents the case when an account ID is referenced in the yellowpaper, but the actual value of the account ID is the empty set. This is used, for example, when referring to the destination of a message which creates a new contract.
syntax Account ::= ".Account" | Int
// -----------------------------------
#addr
turns an Ethereum word into the corresponding Ethereum address (160 LSB).
syntax Int ::= #addr ( Int ) [function]
// ---------------------------------------
rule #addr(W) => W %Word pow160
#lookup
looks up a key in a map and returns 0 if the key doesn't exist, otherwise returning its value.
syntax Int ::= #lookup ( Map , Int ) [function]
// -----------------------------------------------
rule [#lookup.some]: #lookup( (KEY |-> VAL) M, KEY ) => VAL
rule [#lookup.none]: #lookup( M, KEY ) => 0 requires notBool KEY in_keys(M)
During execution of a transaction some things are recorded in the substate log (Section 6.1 in YellowPaper).
This is a right cons-list of SubstateLogEntry
(which contains the account ID along with the specified portions of the wordStack
and localMem
).
syntax SubstateLogEntry ::= "{" Int "|" List "|" ByteArray "}" [klabel(logEntry)]
// ---------------------------------------------------------------------------------
endmodule