EVM Words

requires "krypto.k"

JSON Formatting

The JSON format is used extensively for communication in the Ethereum circles. Writing a JSON-ish parser in K takes 6 lines.

module JSON
    imports INT
    imports STRING
    imports BOOL

    syntax JSONs   ::= List{JSON,","}      [klabel(JSONs)      , symbol]
    syntax JSONKey ::= String
    syntax JSON    ::= "null"              [klabel(JSONnull)   , symbol]
                     | String | Int | Bool
                     | JSONKey ":" JSON    [klabel(JSONEntry)  , symbol]
                     | "{" JSONs "}"       [klabel(JSONObject) , symbol]
                     | "[" JSONs "]"       [klabel(JSONList)   , symbol]
 // --------------------------------------------------------------------
endmodule

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-DATA
    imports KRYPTO
    imports STRING-BUFFER
    imports MAP-SYMBOLIC
    imports COLLECTIONS
    imports JSON

```{.k .concrete .bytes} imports BYTES

**TODO**: Adding `Int` to `JSONKey` is a hack to make certain parts of semantics easier.

```k
    syntax JSONKey ::= Int
 // ----------------------

Utilities

Important Powers

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]

Boolean Conversions

Primitives provide the basic conversion from K’s sorts Int and Bool to EVM’s words.

bool2Word interprets a Bool as a Int.
word2Bool interprets a Int as a Bool.

    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]
 // -----------------------------------------
    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]
 // -------------------------------------------
    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

Empty Account

.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
 // -----------------------------------

Word Operations

Low-Level

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

Word Comparison

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 for Word (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 Operators

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))

keccak serves as a wrapper around the Keccak256 in KRYPTO.

    syntax Int ::= keccak ( ByteArray ) [function, smtlib(smt_keccak)]
 // ------------------------------------------------------------------
    rule [keccak]: keccak(WS) => #parseHexWord(Keccak256(#unparseByteStack(WS)))

Data-Structures over `Word`

A WordStack for EVM

As a cons-list

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_)]
 // --------------------------------------------------------------------

#take(N , WS) keeps the first $N$ elements of a WordStack (passing with zeros as needed).
#drop(N , WS) removes the first $N$ elements of a WordStack.

    syntax WordStack ::= #take ( Int , WordStack ) [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))   => W : #take(N -Int 1, WS)         requires N >Int 0

    syntax WordStack ::= #drop ( Int , WordStack ) [function, functional]
 // ---------------------------------------------------------------------
    rule #drop(N, WS)         => WS                  requires notBool N >Int 0
    rule #drop(N, .WordStack) => .WordStack
    rule #drop(N, (W : WS))   => #drop(1, #drop(N -Int 1, (W : WS))) requires N >Int 1
    rule #drop(1, (_ : WS))   => WS

Element Access

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 : _) [ N ] => W                  requires N ==Int 0
    rule WS      [ N ] => #drop(N, WS) [ 0 ] requires N >Int 0

    syntax WordStack ::= WordStack "[" Int ":=" Int "]" [function]
 // --------------------------------------------------------------
    rule (W0 : WS)  [ N := W ] => W  : WS                             requires N ==Int 0
    rule (W0 : WS)  [ N := W ] => W0 : (WS [ N -Int 1 := W ])         requires N >Int 0

Definedness conditions for WS [ N ] and WS [ N := W ]

```{.k .symbolic} 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 a `WordStack`.
-   `_in_` determines if a `Int` occurs in a `WordStack`.

```k
    syntax Int ::= #sizeWordStack ( WordStack )       [function, functional, smtlib(sizeWordStack)]
                 | #sizeWordStack ( WordStack , Int ) [function, functional, klabel(sizeWordStackAux), smtlib(sizeWordStackAux)]
 // ----------------------------------------------------------------------------------------------------------------------------
    rule #sizeWordStack ( WS ) => #sizeWordStack(WS, 0)
    rule #sizeWordStack ( .WordStack, SIZE ) => SIZE
    rule #sizeWordStack ( W : WS, SIZE )     => #sizeWordStack(WS, SIZE +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 pushes N copies of A onto a WordStack.
#replicate is a WordStack of length N with A 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 sort WordStack to a term of sort List.

    syntax List ::= WordStack2List ( WordStack ) [function, functional]
 // -------------------------------------------------------------------
    rule WordStack2List(.WordStack) => .List
    rule WordStack2List(W : WS) => ListItem(W) WordStack2List(WS)

Byte Arrays

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 a ByteArray.
_++_ acts as ByteArray append.
WS [ N .. W ] access the range of WS beginning with N of width W.
#sizeByteArray calculates the size of a ByteArray.
#padToWidth(N, WS) and #padRightToWidth make sure that a WordStack is the correct size.

```{.k .bytes} syntax ByteArray = Bytes syntax ByteArray ::= “.ByteArray” [function, functional] // ——————————————————– rule .ByteArray => .Bytes

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 String ::= #asString ( ByteArray ) [function]  // ----------------------------------------------------
rule #asString(WS) => Bytes2String(WS)

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)

syntax ByteArray ::= #padToWidth ( Int , ByteArray ) [function]  // ---------------------------------------------------------------
rule #padToWidth(N, WS) => padLeftBytes(WS, N, 0) ```

```{.k .nobytes} syntax ByteArray = WordStack syntax ByteArray ::= “.ByteArray” [function] // ——————————————– rule .ByteArray => .WordStack

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 String ::= #asString ( ByteArray ) [function]  // ----------------------------------------------------
rule #asString( .WordStack     ) => ""
rule #asString( W : .WordStack ) => chrChar( W )
rule #asString( W0 : WS        ) => chrChar( W0 ) +String #asString( 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)

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) ```

Addresses

#addr turns an Ethereum word into the corresponding Ethereum address (160 LSB).

    syntax Int ::= #addr ( Int ) [function]
 // ---------------------------------------
    rule #addr(W) => W %Word pow160

#newAddr computes the address of a new account given the address and nonce of the creating account.
#sender computes the sender of the transaction from its data and signature.
#addrFromPrivateKey computes the address of an account given its private key

    syntax Int ::= #newAddr ( Int , Int ) [function]
                 | #newAddr ( Int , Int , ByteArray ) [function, klabel(#newAddrCreate2)]
 // -------------------------------------------------------------------------------------
    rule [#newAddr]:        #newAddr(ACCT, NONCE) => #addr(#parseHexWord(Keccak256(#rlpEncodeLength(#rlpEncodeBytes(ACCT, 20) +String #rlpEncodeWord(NONCE), 192))))
    rule [#newAddrCreate2]: #newAddr(ACCT, SALT, INITCODE) => #addr(#parseHexWord(Keccak256("\xff" +String #unparseByteStack(#padToWidth(20, #asByteStack(ACCT))) +String #unparseByteStack(#padToWidth(32, #asByteStack(SALT))) +String #unparseByteStack(#parseHexBytes(Keccak256(#unparseByteStack(INITCODE)))))))

    syntax Account ::= #sender ( Int , Int , Int , Account , Int , String , Int , ByteArray , ByteArray ) [function]
                     | #sender ( String , Int , String , String )                                         [function, klabel(#senderAux)]
                     | #sender ( String )                                                                 [function, klabel(#senderAux2)]
 // -------------------------------------------------------------------------------------------------------------------------------------
    rule #sender(TN, TP, TG, TT, TV, DATA, TW, TR, TS)
      => #sender(#unparseByteStack(#parseHexBytes(Keccak256(#rlpEncodeLength(#rlpEncodeWordStack(TN : TP : TG : .WordStack) +String #rlpEncodeAccount(TT) +String #rlpEncodeWord(TV) +String #rlpEncodeString(DATA), 192)))), TW, #unparseByteStack(TR), #unparseByteStack(TS))

    rule #sender(HT, TW, TR, TS) => #sender(ECDSARecover(HT, TW, TR, TS))

    rule #sender("")  => .Account
    rule #sender(STR) => #addr(#parseHexWord(Keccak256(STR))) requires STR =/=String ""

    syntax Int ::= #addrFromPrivateKey ( String ) [function]
 // --------------------------------------------------------
    rule #addrFromPrivateKey ( KEY ) => #addr( #parseHexWord( Keccak256 ( Hex2Raw( ECDSAPubKey( Hex2Raw( KEY ) ) ) ) ) )

#blockHeaderHash computes the hash of a block header given all the block data.

    syntax Int ::= #blockHeaderHash( Int , Int , Int , Int , Int , Int , ByteArray , Int , Int , Int , Int , Int , ByteArray , Int , Int ) [function, klabel(blockHeaderHash), symbol]
                 | #blockHeaderHash(String, String, String, String, String, String, String, String, String, String, String, String, String, String, String) [function, klabel(#blockHashHeaderStr), symbol]
 // -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
   rule #blockHeaderHash(HP, HO, HC, HR, HT, HE, HB, HD, HI, HL, HG, HS, HX, HM, HN)
         => #blockHeaderHash(#asWord(#parseByteStackRaw(HP)),
                             #asWord(#parseByteStackRaw(HO)),
                             #asWord(#parseByteStackRaw(HC)),
                             #asWord(#parseByteStackRaw(HR)),
                             #asWord(#parseByteStackRaw(HT)),
                             #asWord(#parseByteStackRaw(HE)),
                                     #parseByteStackRaw(HB) ,
                             #asWord(#parseByteStackRaw(HD)),
                             #asWord(#parseByteStackRaw(HI)),
                             #asWord(#parseByteStackRaw(HL)),
                             #asWord(#parseByteStackRaw(HG)),
                             #asWord(#parseByteStackRaw(HS)),
                                     #parseByteStackRaw(HX) ,
                             #asWord(#parseByteStackRaw(HM)),
                             #asWord(#parseByteStackRaw(HN)))

    rule #blockHeaderHash(HP, HO, HC, HR, HT, HE, HB, HD, HI, HL, HG, HS, HX, HM, HN)
         => #parseHexWord(Keccak256(#rlpEncodeLength(         #rlpEncodeBytes(HP, 32)
                                                      +String #rlpEncodeBytes(HO, 32)
                                                      +String #rlpEncodeBytes(HC, 20)
                                                      +String #rlpEncodeBytes(HR, 32)
                                                      +String #rlpEncodeBytes(HT, 32)
                                                      +String #rlpEncodeBytes(HE, 32)
                                                      +String #rlpEncodeString(#unparseByteStack(HB))
                                                      +String #rlpEncodeWordStack(HD : HI : HL : HG : HS : .WordStack)
                                                      +String #rlpEncodeString(#unparseByteStack(HX))
                                                      +String #rlpEncodeBytes(HM, 32)
                                                      +String #rlpEncodeBytes(HN, 8),
                                                    192)))

M3:2048 computes the 2048-bit hash of a log entry in which exactly 3 bits are set. This is used to compute the Bloom filter of a log entry.

    syntax Int ::= "M3:2048" "(" ByteArray ")" [function]
 // -----------------------------------------------------
    rule M3:2048(WS) => setBloomFilterBits(#parseByteStack(Keccak256(#unparseByteStack(WS))))

    syntax Int ::= setBloomFilterBits(ByteArray) [function]
 // -------------------------------------------------------
    rule setBloomFilterBits(HASH) => (1 <<Int getBloomFilterBit(HASH, 0)) |Int (1 <<Int getBloomFilterBit(HASH, 2)) |Int (1 <<Int getBloomFilterBit(HASH, 4))

    syntax Int ::= getBloomFilterBit(ByteArray, Int) [function]
 // -----------------------------------------------------------
    rule getBloomFilterBit(X, I) => #asInteger(X [ I .. 2 ]) %Int 2048

Word Map

Most of EVM data is held in finite maps. We are using the polymorphic Map sort for these word maps.

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).

```{.k .bytes} syntax Map ::= Map “[” 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 ( Map , Int , Int )                   [function]
                   | #range ( Map , 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] ```

```{.k .nobytes} syntax Map ::= Map “[” 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 ( Map , Int , Int )             [function, functional]
syntax ByteArray ::= #range ( Map , 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) ```

#removeZeros removes any entries in a map with zero values.

    syntax Map ::= #removeZeros ( Map ) [function]
                 | #removeZeros ( List , Map ) [function, klabel(#removeZerosAux)]
 // ------------------------------------------------------------------------------
    rule #removeZeros( M )                                   => #removeZeros(Set2List(keys(M)), M)
    rule #removeZeros( .List, .Map )                         => .Map
    rule #removeZeros( ListItem(KEY) L, KEY |-> 0 REST )     => #removeZeros(L, REST)
    rule #removeZeros( ListItem(KEY) L, KEY |-> VALUE REST ) => KEY |-> VALUE #removeZeros(L, REST) requires VALUE =/=K 0

#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)

Parsing/Unparsing

The EVM test-sets are represented in JSON format with hex-encoding of the data and programs. Here we provide some standard parser/unparser functions for that format.

Parsing

These parsers can interperet hex-encoded strings as Ints, ByteArrays, and Maps.

#parseHexWord interprets a string as a single hex-encoded Word.
#parseHexBytes interprets a string as a hex-encoded stack of bytes.
#alignHexString makes sure that the length of a (hex)string is even.
#parseByteStack interprets a string as a hex-encoded stack of bytes, but makes sure to remove the leading “0x”.
#parseByteStackRaw casts a string as a stack of bytes, ignoring any encoding.
#parseWordStack interprets a JSON list as a stack of Word.
#parseMap interprets a JSON key/value object as a map from Word to Word.
#parseAddr interprets a string as a 160 bit hex-endcoded address.

    syntax Int ::= #parseHexWord ( String ) [function]
                 | #parseWord    ( String ) [function]
 // --------------------------------------------------
    rule #parseHexWord("")   => 0
    rule #parseHexWord("0x") => 0
    rule #parseHexWord(S)    => String2Base(replaceAll(S, "0x", ""), 16) requires (S =/=String "") andBool (S =/=String "0x")

    rule #parseWord("") => 0
    rule #parseWord(S)  => #parseHexWord(S) requires lengthString(S) >=Int 2 andBool substrString(S, 0, 2) ==String "0x"
    rule #parseWord(S)  => String2Int(S) [owise]

    syntax String ::= #alignHexString ( String ) [function, functional]
 // -------------------------------------------------------------------
    rule #alignHexString(S) => S             requires         lengthString(S) modInt 2 ==Int 0
    rule #alignHexString(S) => "0" +String S requires notBool lengthString(S) modInt 2 ==Int 0

```{.k .bytes} syntax ByteArray ::= #parseHexBytes ( String ) [function] | #parseHexBytesAux ( String ) [function] | #parseByteStack ( String ) [function, memo] | #parseByteStackRaw ( String ) [function] // ——————————————————————- rule #parseByteStack(S) => #parseHexBytes(replaceAll(S, “0x”, “”))

rule #parseHexBytes(S)  => #parseHexBytesAux(#alignHexString(S))
rule #parseHexBytesAux("") => .ByteArray
rule #parseHexBytesAux(S)  => Int2Bytes(1, String2Base(substrString(S, 0, 2), 16), BE) +Bytes #parseHexBytesAux(substrString(S, 2, lengthString(S)))
  requires lengthString(S) >=Int 2

rule #parseByteStackRaw(S) => String2Bytes(S) ```

```{.k .nobytes} syntax ByteArray ::= #parseHexBytes ( String ) [function] | #parseHexBytesAux ( String ) [function] | #parseByteStack ( String ) [function] | #parseByteStackRaw ( String ) [function] // ————————————————————- rule #parseByteStack(S) => #parseHexBytes(replaceAll(S, “0x”, “”))

rule #parseHexBytes(S)  => #parseHexBytesAux(#alignHexString(S))
rule #parseHexBytesAux("") => .WordStack
rule #parseHexBytesAux(S)  => #parseHexWord(substrString(S, 0, 2)) : #parseHexBytesAux(substrString(S, 2, lengthString(S)))
   requires lengthString(S) >=Int 2

rule #parseByteStackRaw(S) => ordChar(substrString(S, 0, 1)) : #parseByteStackRaw(substrString(S, 1, lengthString(S))) requires lengthString(S) >=Int 1
rule #parseByteStackRaw("") => .WordStack ```

    syntax Map ::= #parseMap ( JSON ) [function]
 // --------------------------------------------
    rule #parseMap( { .JSONs                      } ) => .Map
    rule #parseMap( { _   : (VALUE:String) , REST } ) => #parseMap({ REST })                                                requires #parseHexWord(VALUE) ==K 0
    rule #parseMap( { KEY : (VALUE:String) , REST } ) => #parseMap({ REST }) [ #parseHexWord(KEY) <- #parseHexWord(VALUE) ] requires #parseHexWord(VALUE) =/=K 0

    syntax Int ::= #parseAddr ( String ) [function]
 // -----------------------------------------------
    rule #parseAddr(S) => #addr(#parseHexWord(S))

Unparsing

We need to interperet a ByteArray as a String again so that we can call Keccak256 on it from KRYPTO.

#unparseByteStack turns a stack of bytes (as a ByteArray) into a String.
#padByte ensures that the String interperetation of a Int is wide enough.

```{.k .bytes} syntax String ::= #unparseByteStack ( ByteArray ) [function, klabel(unparseByteStack), symbol] // ———————————————————————————————- rule #unparseByteStack(WS) => Bytes2String(WS)

```{.k .nobytes}
    syntax String ::= #unparseByteStack ( ByteArray )                [function, klabel(unparseByteStack), symbol]
                    | #unparseByteStack ( ByteArray , StringBuffer ) [function, klabel(#unparseByteStackAux)]
 // ---------------------------------------------------------------------------------------------------------
    rule #unparseByteStack ( WS ) => #unparseByteStack(WS, .StringBuffer)

    rule #unparseByteStack( .WordStack, BUFFER ) => StringBuffer2String(BUFFER)
    rule #unparseByteStack( W : WS, BUFFER )     => #unparseByteStack(WS, BUFFER +String chrChar(W modInt (2 ^Int 8)))

    syntax String ::= #padByte( String ) [function]
 // -----------------------------------------------
    rule #padByte( S ) => S             requires lengthString(S) ==K 2
    rule #padByte( S ) => "0" +String S requires lengthString(S) ==K 1

    syntax String ::= #unparseQuantity( Int ) [function]
 // ----------------------------------------------------
    rule #unparseQuantity( I ) => "0x" +String Base2String(I, 16)

    syntax String ::= #unparseData          ( Int, Int  ) [function]
                    | #unparseDataByteArray ( ByteArray ) [function]
 // ----------------------------------------------------------------
    rule #unparseData( DATA, LENGTH ) => #unparseDataByteArray(#padToWidth(LENGTH,#asByteStack(DATA)))

    rule #unparseDataByteArray( DATA ) => replaceFirst(Base2String(#asInteger(#asByteStack(1) ++ DATA), 16), "1", "0x")

String Helper Functions

Hex2Raw Takes a string of hex encoded bytes and converts it to a raw bytestring
Raw2Hex Takes a string of raw bytes and converts it to a hex representation

    syntax String ::= Hex2Raw ( String ) [function]
                    | Raw2Hex ( String ) [function]
 // -----------------------------------------------
    rule Hex2Raw ( S ) => #unparseByteStack( #parseByteStack ( S ) )
    rule Raw2Hex ( S ) => #unparseDataByteArray( #parseByteStackRaw ( S ) )

Recursive Length Prefix (RLP)

RLP encoding is used extensively for executing the blocks of a transaction. For details about RLP encoding, see the YellowPaper Appendix B.

Encoding

#rlpEncodeWord RLP encodes a single EVM word.
#rlpEncodeString RLP encodes a single String.

    syntax String ::= #rlpEncodeWord ( Int )            [function]
                    | #rlpEncodeBytes ( Int , Int )     [function]
                    | #rlpEncodeWordStack ( WordStack ) [function]
                    | #rlpEncodeString ( String )       [function]
                    | #rlpEncodeAccount ( Account )     [function]
 // --------------------------------------------------------------
    rule #rlpEncodeWord(0) => "\x80"
    rule #rlpEncodeWord(WORD) => chrChar(WORD) requires WORD >Int 0 andBool WORD <Int 128
    rule #rlpEncodeWord(WORD) => #rlpEncodeLength(#unparseByteStack(#asByteStack(WORD)), 128) requires WORD >=Int 128

    rule #rlpEncodeBytes(WORD, LEN) => #rlpEncodeString(#unparseByteStack(#padToWidth(LEN, #asByteStack(WORD))))

    rule #rlpEncodeWordStack(.WordStack) => ""
    rule #rlpEncodeWordStack(W : WS)     => #rlpEncodeWord(W) +String #rlpEncodeWordStack(WS)

    rule #rlpEncodeString(STR) => STR                        requires           lengthString(STR) ==Int 1 andBool ordChar(STR) <Int 128
    rule #rlpEncodeString(STR) => #rlpEncodeLength(STR, 128) requires notBool ( lengthString(STR) ==Int 1 andBool ordChar(STR) <Int 128 )

    rule #rlpEncodeAccount(.Account) => "\x80"
    rule #rlpEncodeAccount(ACCT)     => #rlpEncodeBytes(ACCT, 20) requires ACCT =/=K .Account

    syntax String ::= #rlpEncodeLength ( String , Int )          [function]
                    | #rlpEncodeLength ( String , Int , String ) [function, klabel(#rlpEncodeLengthAux)]
 // ----------------------------------------------------------------------------------------------------
    rule #rlpEncodeLength(STR, OFFSET) => chrChar(lengthString(STR) +Int OFFSET) +String STR                                requires           lengthString(STR) <Int 56
    rule #rlpEncodeLength(STR, OFFSET) => #rlpEncodeLength(STR, OFFSET, #unparseByteStack(#asByteStack(lengthString(STR)))) requires notBool ( lengthString(STR) <Int 56 )
    rule #rlpEncodeLength(STR, OFFSET, BL) => chrChar(lengthString(BL) +Int OFFSET +Int 55) +String BL +String STR

    syntax String ::= #rlpEncodeMerkleTree ( MerkleTree ) [function]
 // ----------------------------------------------------------------
    rule #rlpEncodeMerkleTree ( .MerkleTree ) => "\x80"

    rule #rlpEncodeMerkleTree ( MerkleLeaf ( PATH, VALUE ) )
      => #rlpEncodeLength(         #rlpEncodeString( #asString( #HPEncode( PATH, 1 ) ) )
                           +String #rlpEncodeString( VALUE )
                         , 192
                         )

    rule #rlpEncodeMerkleTree ( MerkleExtension ( PATH, TREE ) )
      => #rlpEncodeLength(         #rlpEncodeString( #asString( #HPEncode( PATH, 0 ) ) )
                           +String #rlpMerkleH( #rlpEncodeMerkleTree( TREE ) )
                         , 192
                         )

    rule #rlpEncodeMerkleTree ( MerkleBranch (  0 |->  P0:MerkleTree  1 |->  P1:MerkleTree  2 |->  P2:MerkleTree  3 |->  P3:MerkleTree
                                                4 |->  P4:MerkleTree  5 |->  P5:MerkleTree  6 |->  P6:MerkleTree  7 |->  P7:MerkleTree
                                                8 |->  P8:MerkleTree  9 |->  P9:MerkleTree 10 |-> P10:MerkleTree 11 |-> P11:MerkleTree
                                               12 |-> P12:MerkleTree 13 |-> P13:MerkleTree 14 |-> P14:MerkleTree 15 |-> P15:MerkleTree
                                             , VALUE
                                             )
                              )
      => #rlpEncodeLength(         #rlpMerkleH( #rlpEncodeMerkleTree(  P0 ) ) +String #rlpMerkleH( #rlpEncodeMerkleTree(  P1 ) )
                           +String #rlpMerkleH( #rlpEncodeMerkleTree(  P2 ) ) +String #rlpMerkleH( #rlpEncodeMerkleTree(  P3 ) )
                           +String #rlpMerkleH( #rlpEncodeMerkleTree(  P4 ) ) +String #rlpMerkleH( #rlpEncodeMerkleTree(  P5 ) )
                           +String #rlpMerkleH( #rlpEncodeMerkleTree(  P6 ) ) +String #rlpMerkleH( #rlpEncodeMerkleTree(  P7 ) )
                           +String #rlpMerkleH( #rlpEncodeMerkleTree(  P8 ) ) +String #rlpMerkleH( #rlpEncodeMerkleTree(  P9 ) )
                           +String #rlpMerkleH( #rlpEncodeMerkleTree( P10 ) ) +String #rlpMerkleH( #rlpEncodeMerkleTree( P11 ) )
                           +String #rlpMerkleH( #rlpEncodeMerkleTree( P12 ) ) +String #rlpMerkleH( #rlpEncodeMerkleTree( P13 ) )
                           +String #rlpMerkleH( #rlpEncodeMerkleTree( P14 ) ) +String #rlpMerkleH( #rlpEncodeMerkleTree( P15 ) )
                           +String #rlpEncodeString( VALUE )
                         , 192
                         )

    syntax String ::= #rlpMerkleH ( String ) [function,klabel(MerkleRLPAux)]
 // ------------------------------------------------------------------------
    rule #rlpMerkleH ( X ) => #rlpEncodeString( Hex2Raw( Keccak256( X ) ) )
      requires lengthString(X) >=Int 32

    rule #rlpMerkleH ( X ) => X
      requires notBool lengthString(X) >=Int 32

Decoding

#rlpDecode RLP decodes a single String into a JSON.
#rlpDecodeList RLP decodes a single String into a JSONs, interpereting the string as the RLP encoding of a list.

    syntax JSON ::= #rlpDecode(String)               [function]
                  | #rlpDecode(String, LengthPrefix) [function, klabel(#rlpDecodeAux)]
 // ----------------------------------------------------------------------------------
    rule #rlpDecode(STR) => #rlpDecode(STR, #decodeLengthPrefix(STR, 0))
    rule #rlpDecode(STR, #str(LEN, POS))  => substrString(STR, POS, POS +Int LEN)
    rule #rlpDecode(STR, #list(LEN, POS)) => [#rlpDecodeList(STR, POS)]

    syntax JSONs ::= #rlpDecodeList(String, Int)               [function]
                   | #rlpDecodeList(String, Int, LengthPrefix) [function, klabel(#rlpDecodeListAux)]
 // ------------------------------------------------------------------------------------------------
    rule #rlpDecodeList(STR, POS) => #rlpDecodeList(STR, POS, #decodeLengthPrefix(STR, POS)) requires POS <Int lengthString(STR)
    rule #rlpDecodeList(STR, POS) => .JSONs [owise]
    rule #rlpDecodeList(STR, POS, _:LengthPrefixType(L, P)) => #rlpDecode(substrString(STR, POS, L +Int P)) , #rlpDecodeList(STR, L +Int P)

    syntax LengthPrefixType ::= "#str" | "#list"
    syntax LengthPrefix ::= LengthPrefixType "(" Int "," Int ")"
                          | #decodeLengthPrefix ( String , Int )                                [function]
                          | #decodeLengthPrefix ( String , Int , Int )                          [function, klabel(#decodeLengthPrefixAux)]
                          | #decodeLengthPrefixLength ( LengthPrefixType , String , Int , Int ) [function]
                          | #decodeLengthPrefixLength ( LengthPrefixType , Int    , Int , Int ) [function, klabel(#decodeLengthPrefixLengthAux)]
 // --------------------------------------------------------------------------------------------------------------------------------------------
    rule #decodeLengthPrefix(STR, START) => #decodeLengthPrefix(STR, START, ordChar(substrString(STR, START, START +Int 1)))

    rule #decodeLengthPrefix(STR, START, B0) => #str(1, START)                                   requires B0 <Int 128
    rule #decodeLengthPrefix(STR, START, B0) => #str(B0 -Int 128, START +Int 1)                  requires B0 >=Int 128 andBool B0 <Int (128 +Int 56)
    rule #decodeLengthPrefix(STR, START, B0) => #decodeLengthPrefixLength(#str, STR, START, B0)  requires B0 >=Int (128 +Int 56) andBool B0 <Int 192
    rule #decodeLengthPrefix(STR, START, B0) => #list(B0 -Int 192, START +Int 1)                 requires B0 >=Int 192 andBool B0 <Int 192 +Int 56
    rule #decodeLengthPrefix(STR, START, B0) => #decodeLengthPrefixLength(#list, STR, START, B0) [owise]

    rule #decodeLengthPrefixLength(#str,  STR, START, B0) => #decodeLengthPrefixLength(#str,  START, B0 -Int 128 -Int 56 +Int 1, #asWord(#parseByteStackRaw(substrString(STR, START +Int 1, START +Int 1 +Int (B0 -Int 128 -Int 56 +Int 1)))))
    rule #decodeLengthPrefixLength(#list, STR, START, B0) => #decodeLengthPrefixLength(#list, START, B0 -Int 192 -Int 56 +Int 1, #asWord(#parseByteStackRaw(substrString(STR, START +Int 1, START +Int 1 +Int (B0 -Int 192 -Int 56 +Int 1)))))
    rule #decodeLengthPrefixLength(TYPE, START, LL, L) => TYPE(L, START +Int 1 +Int LL)

Merkle Patricia Tree

Appendix C and D from the Ethereum Yellow Paper
https://github.com/ethereum/wiki/wiki/Patricia-Tree

    syntax KItem ::= Int | MerkleTree // For testing purposes

    syntax MerkleTree ::= MerkleBranch    ( Map, String )
                        | MerkleExtension ( ByteArray, MerkleTree )
                        | MerkleLeaf      ( ByteArray, String )
                        | ".MerkleTree"
                        | ".MerkleBranch"            [function]
 // -----------------------------------------------------------
    rule .MerkleBranch
      => MerkleBranch (  0 |-> .MerkleTree  1 |-> .MerkleTree  2 |-> .MerkleTree  3 |-> .MerkleTree
                         4 |-> .MerkleTree  5 |-> .MerkleTree  6 |-> .MerkleTree  7 |-> .MerkleTree
                         8 |-> .MerkleTree  9 |-> .MerkleTree 10 |-> .MerkleTree 11 |-> .MerkleTree
                        12 |-> .MerkleTree 13 |-> .MerkleTree 14 |-> .MerkleTree 15 |-> .MerkleTree
                      , ""
                      )

    syntax MerkleTree ::= MerkleUpdate ( MerkleTree,     String, String ) [function]
                        | MerkleUpdate ( MerkleTree,  ByteArray, String ) [function,klabel(MerkleUpdateAux)]
 // --------------------------------------------------------------------------------------------------------
    rule MerkleUpdate ( TREE, S:String, VALUE ) => MerkleUpdate ( TREE, #nibbleize ( #parseByteStackRaw( S ) ), VALUE )

    rule MerkleUpdate ( .MerkleTree, PATH:ByteArray, VALUE ) => MerkleLeaf ( PATH, VALUE )

    rule MerkleUpdate ( MerkleLeaf ( LEAFPATH, _ ), PATH, VALUE )
      => MerkleLeaf( LEAFPATH, VALUE )
      requires #asString( LEAFPATH ) ==String #asString( PATH )

    rule MerkleUpdate ( MerkleLeaf ( LEAFPATH, LEAFVALUE ), PATH, VALUE )
      => MerkleUpdate ( MerkleUpdate ( .MerkleBranch, LEAFPATH, LEAFVALUE ), PATH, VALUE )
      requires #sizeByteArray( LEAFPATH ) >Int 0
       andBool #sizeByteArray( PATH ) >Int 0
       andBool LEAFPATH[0] =/=Int PATH[0]

    rule MerkleUpdate ( MerkleLeaf ( LEAFPATH, LEAFVALUE ), PATH, VALUE )
      => #merkleExtensionBuilder( .ByteArray, LEAFPATH, LEAFVALUE, PATH, VALUE ) [owise]

    rule MerkleUpdate ( MerkleExtension ( EXTPATH, EXTTREE ), PATH, VALUE )
      => MerkleExtension ( EXTPATH, MerkleUpdate ( EXTTREE, .ByteArray, VALUE ) )
      requires #asString( EXTPATH ) ==String #asString( PATH )

    rule MerkleUpdate ( MerkleExtension ( EXTPATH, EXTTREE ), PATH, VALUE )
      => #merkleExtensionBrancher( MerkleUpdate( .MerkleBranch, PATH, VALUE ), EXTPATH, EXTTREE )
      requires #sizeByteArray( EXTPATH ) >Int 0
       andBool #sizeByteArray( PATH ) >Int 0
       andBool EXTPATH[0] =/=Int PATH[0]

    rule MerkleUpdate ( MerkleExtension ( EXTPATH, EXTTREE ), PATH, VALUE )
      => #merkleExtensionSplitter( .ByteArray, EXTPATH, EXTTREE, PATH, VALUE ) [owise]

    rule MerkleUpdate ( MerkleBranch( M, _ ), PATH, VALUE )
      => MerkleBranch( M, VALUE )
      requires #sizeByteArray( PATH ) ==Int 0

    rule MerkleUpdate ( MerkleBranch( M, BRANCHVALUE ), PATH, VALUE )
      => #merkleBrancher ( M, BRANCHVALUE, PATH[0], PATH[1 .. #sizeByteArray(PATH) -Int 1], VALUE ) [owise]

MerkleUpdateMap Takes a mapping of ByteArray |-> String and generates a trie

    syntax MerkleTree ::= MerkleUpdateMap    ( MerkleTree , Map        ) [function]
                        | MerkleUpdateMapAux ( MerkleTree , Map , List ) [function]
 // -------------------------------------------------------------------------------
    rule MerkleUpdateMap(TREE, MMAP) => MerkleUpdateMapAux(TREE, MMAP, keys_list(MMAP))

    rule MerkleUpdateMapAux(TREE, _, .List ) => TREE
    rule MerkleUpdateMapAux(TREE                                                    , MMAP, ListItem(KEY) REST)
      => MerkleUpdateMapAux(MerkleUpdate(TREE, #nibbleize(KEY), {MMAP[KEY]}:>String), MMAP,               REST)

Merkle Tree Aux Functions

    syntax ByteArray ::= #nibbleize ( ByteArray ) [function]
                       | #byteify   ( ByteArray ) [function]
 // --------------------------------------------------------
    rule #nibbleize ( B ) => (      #asByteStack ( B [ 0 ] /Int 16 )[0 .. 1]
                               ++ ( #asByteStack ( B [ 0 ] %Int 16 )[0 .. 1] )
                             ) ++ #nibbleize ( B[1 .. #sizeByteArray(B) -Int 1] )
      requires #sizeByteArray(B) >Int 0

    rule #nibbleize ( B ) => .ByteArray
      requires notBool #sizeByteArray(B) >Int 0

    rule #byteify ( B ) =>    #asByteStack ( B[0] *Int 16 +Int B[1] )[0 .. 1]
                           ++ #byteify ( B[2 .. #sizeByteArray(B) -Int 2] )
      requires #sizeByteArray(B) >Int 0

    rule #byteify ( B ) => .ByteArray
      requires notBool #sizeByteArray(B) >Int 0

    syntax ByteArray ::= #HPEncode ( ByteArray, Int ) [function]
 // ------------------------------------------------------------
    rule #HPEncode ( X, T ) => #asByteStack ( ( HPEncodeAux(T) +Int 1 ) *Int 16 +Int X[0] ) ++ #byteify( X[1 .. #sizeByteArray(X) -Int 1] )
      requires #sizeByteArray(X) %Int 2 =/=Int 0

    rule #HPEncode ( X, T ) => #asByteStack ( HPEncodeAux(T) *Int 16 )[0 .. 1] ++ #byteify( X )
      requires notBool #sizeByteArray(X) %Int 2 =/=Int 0

    syntax Int ::= HPEncodeAux ( Int ) [function]
 // ---------------------------------------------
    rule HPEncodeAux ( X ) => 0 requires         X ==Int 0
    rule HPEncodeAux ( X ) => 2 requires notBool X ==Int 0

    syntax MerkleTree ::= #merkleBrancher ( Map, String, Int, ByteArray, String ) [function]
 // ----------------------------------------------------------------------------------------
    rule #merkleBrancher ( X |-> TREE M, BRANCHVALUE, X, PATH, VALUE )
      => MerkleBranch( M[X <- MerkleUpdate( TREE, PATH, VALUE )], BRANCHVALUE )

    syntax MerkleTree ::= #merkleExtensionBuilder( ByteArray, ByteArray, String, ByteArray, String ) [function]
 // -----------------------------------------------------------------------------------------------------------
    rule #merkleExtensionBuilder( PATH, P1, V1, P2, V2 )
      => #merkleExtensionBuilder( PATH ++ ( #asByteStack( P1[0] )[0 .. 1] )
                                , P1[1 .. #sizeByteArray(P1) -Int 1], V1
                                , P2[1 .. #sizeByteArray(P2) -Int 1], V2
                                )
      [owise]

    rule #merkleExtensionBuilder( PATH, P1, V1, P2, V2 )
      => MerkleExtension( PATH, MerkleUpdate( MerkleUpdate( .MerkleBranch, P1, V1 ), P2, V2 ) )
      requires #sizeByteArray(P1) >Int 0
       andBool #sizeByteArray(P2) >Int 0
       andBool P1[0] =/=Int P2[0]

    rule #merkleExtensionBuilder( PATH, P1, V1, P2, V2 )
      => MerkleExtension( PATH, MerkleUpdate( MerkleUpdate( .MerkleBranch, P1, V1 ), P2, V2 ) )
      requires #sizeByteArray(P1) ==Int 0
        orBool #sizeByteArray(P2) ==Int 0

    syntax MerkleTree ::= #merkleExtensionBrancher ( MerkleTree, ByteArray, MerkleTree )                   [function]
                        | #merkleExtensionSplitter ( ByteArray, ByteArray, MerkleTree, ByteArray, String ) [function]
 // -----------------------------------------------------------------------------------------------------------------
    rule #merkleExtensionBrancher( MerkleBranch(M, VALUE), PATH, EXTTREE )
      => MerkleBranch( M[PATH[0] <- MerkleExtension( PATH[1 .. #sizeByteArray(PATH) -Int 1], EXTTREE )], VALUE )

    rule #merkleExtensionSplitter( PATH, P1, TREE, P2, VALUE )
      => #merkleExtensionSplitter( PATH ++ ( #asByteStack( P1[0] )[0 .. 1] )
                                 , P1[1 .. #sizeByteArray(P1) -Int 1], TREE
                                 , P2[1 .. #sizeByteArray(P2) -Int 1], VALUE
                                 )
      [owise]

    rule #merkleExtensionSplitter( PATH, P1, TREE, P2, VALUE )
      => MerkleExtension( PATH, #merkleExtensionBrancher( MerkleUpdate( .MerkleBranch, P2, VALUE ), P1, TREE ) )
      requires #sizeByteArray(P1) >Int 0
       andBool #sizeByteArray(P2) >Int 0
       andBool P1[0] =/=Int P2[0]

    rule #merkleExtensionSplitter( PATH, P1, TREE, P2, VALUE )
      => MerkleExtension( PATH, MerkleUpdate( TREE, P2, VALUE ) )
      requires #sizeByteArray(P1) ==Int 0

    rule #merkleExtensionSplitter( PATH, P1, TREE, P2, VALUE )
      => MerkleExtension( PATH, #merkleExtensionBrancher( MerkleUpdate( .MerkleBranch, P2, VALUE ), P1, TREE ) )
      requires #sizeByteArray(P2) ==Int 0

Tree Root Helper Functions

Storage Root

    syntax Map ::= #intMap2StorageMap( Map ) [function]
 // ---------------------------------------------------
    rule #intMap2StorageMap( .Map          ) => .Map
    rule #intMap2StorageMap( KEY |-> VAL M ) => #padToWidth( 32, #asByteStack( KEY ) ) |-> #rlpEncodeWord( VAL ) #intMap2StorageMap(M)

    syntax MerkleTree ::= #storageRoot( Map ) [function]
 // ----------------------------------------------------
    rule #storageRoot( STORAGE ) => MerkleUpdateMap( .MerkleTree, #intMap2StorageMap( STORAGE ) )

State Root

    syntax Map ::= "#precompiledContracts" [function]
 // -------------------------------------------------
    rule #precompiledContracts
      => #parseByteStackRaw( Hex2Raw( #unparseData( 1, 20 ) ) ) |-> #emptyContractRLP
         #parseByteStackRaw( Hex2Raw( #unparseData( 2, 20 ) ) ) |-> #emptyContractRLP
         #parseByteStackRaw( Hex2Raw( #unparseData( 3, 20 ) ) ) |-> #emptyContractRLP
         #parseByteStackRaw( Hex2Raw( #unparseData( 4, 20 ) ) ) |-> #emptyContractRLP
         #parseByteStackRaw( Hex2Raw( #unparseData( 5, 20 ) ) ) |-> #emptyContractRLP
         #parseByteStackRaw( Hex2Raw( #unparseData( 6, 20 ) ) ) |-> #emptyContractRLP
         #parseByteStackRaw( Hex2Raw( #unparseData( 7, 20 ) ) ) |-> #emptyContractRLP
         #parseByteStackRaw( Hex2Raw( #unparseData( 8, 20 ) ) ) |-> #emptyContractRLP

    syntax String ::= "#emptyContractRLP" [function]
 // ------------------------------------------------
    rule #emptyContractRLP => #rlpEncodeLength(         #rlpEncodeWord(0)
                                                +String #rlpEncodeWord(0)
                                                +String #rlpEncodeString( Hex2Raw( Keccak256("\x80") ) )
                                                +String #rlpEncodeString( Hex2Raw( Keccak256("") ) )
                                              , 192
                                              )

    syntax AccountData ::= AcctData ( nonce: Int, balance: Int, store: Map, code: ByteArray )
 // -----------------------------------------------------------------------------------------

    syntax String      ::= #rlpEncodeFullAccount( AccountData ) [function]
 // ----------------------------------------------------------------------
    rule  #rlpEncodeFullAccount( AcctData( NONCE, BAL, STORAGE, CODE ) )
         => #rlpEncodeLength(         #rlpEncodeWord(NONCE)
                              +String #rlpEncodeWord(BAL)
                              +String #rlpEncodeString( Hex2Raw( Keccak256( #rlpEncodeMerkleTree( #storageRoot( STORAGE ) ) ) ) )
                              +String #rlpEncodeString( Hex2Raw( Keccak256( #asString( CODE ) ) ) )
                            , 192
                            )

endmodule

evm-semantics

K Semantics of the Ethereum Virtual Machine (EVM)

EVM Words

JSON Formatting

Utilities

Important Powers

Boolean Conversions

Empty Account

Word Operations

Low-Level

Word Comparison

Bitwise Operators

Data-Structures over `Word`

A WordStack for EVM

As a cons-list

Element Access

Byte Arrays

Addresses

Word Map

Parsing/Unparsing

Parsing

Unparsing

String Helper Functions

Recursive Length Prefix (RLP)

Encoding

Decoding

Merkle Patricia Tree

Merkle Tree Aux Functions

Tree Root Helper Functions

Storage Root

State Root

EVM Words

JSON Formatting

Utilities

Important Powers

Boolean Conversions

Empty Account

Word Operations

Low-Level

Word Comparison

Bitwise Operators

Data-Structures over Word

A WordStack for EVM

As a cons-list

Element Access

Byte Arrays

Addresses

Word Map

Parsing/Unparsing

Parsing

Unparsing

String Helper Functions

Recursive Length Prefix (RLP)

Encoding

Decoding

Merkle Patricia Tree

Merkle Tree Aux Functions

Tree Root Helper Functions

Storage Root

State Root

Data-Structures over `Word`