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lemmas.k
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lemmas.k
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requires "evm.md"
requires "foundry.md"
module FV-LEMMAS
imports BOOL
imports FOUNDRY
imports INFINITE-GAS
imports INT-SYMBOLIC
imports MAP-SYMBOLIC
imports SET-SYMBOLIC
syntax StepSort ::= Int
| Bool
| Bytes
| Set
// -------------------------
syntax KItem ::= runLemma ( StepSort )
| doneLemma( StepSort )
rule <k> runLemma(T) => doneLemma(T) ... </k>
// ---------------------------------------------
syntax Bool ::= #notEq ( KItem, KItem ) [function, no-evaluators]
// ----------------------------------------------------------------------------------------------------
//
// Bool
//
rule X ==Bool true => X [simplification]
rule true ==Bool X => X [simplification]
rule false ==Bool X => notBool X [simplification]
rule X ==Bool false => notBool X [simplification]
rule notBool notBool X => X [simplification]
rule notBool X ==Bool notBool Y => X ==Bool Y [simplification]
rule { notBool X #Equals notBool Y } => { X #Equals Y } [simplification]
rule bool2Word ( X ) => 1 requires X [simplification]
rule bool2Word ( X ) => 0 requires notBool X [simplification]
rule bool2Word ( X ) ==Int bool2Word ( Y ) => X ==Bool Y [simplification]
rule { bool2Word ( X ) #Equals bool2Word ( Y ) } => { X #Equals Y } [simplification]
//
// ML
//
rule { true #Equals X ==K Y } => { X #Equals Y } [simplification]
rule { true #Equals X:Int ==Int Y:Int } => { X #Equals Y } [simplification]
rule { false #Equals X ==K Y } => #Not ( { X #Equals Y } ) [simplification]
rule { false #Equals X:Int ==Int Y:Int } => #Not ( { X #Equals Y } ) [simplification]
rule { true #Equals notBool X:Bool } => { false #Equals X } [simplification]
rule { false #Equals notBool X:Bool } => { true #Equals X } [simplification]
rule { X ==K Y #Equals true } => { X #Equals Y } [simplification]
rule { X:Int ==Int Y:Int #Equals true } => { X #Equals Y } [simplification]
rule { X ==K Y #Equals false } => #Not ( { X #Equals Y } ) [simplification]
rule { X:Int ==Int Y:Int #Equals false } => #Not ( { X #Equals Y } ) [simplification]
rule { notBool X:Bool #Equals true } => { false #Equals X } [simplification]
rule { notBool X:Bool #Equals false } => { true #Equals X } [simplification]
//
// Bitwise simplifications
//
// Concrete to the left
rule A &Int B => B &Int A [simplification(40), concrete(B), symbolic(A)]
rule A |Int B => B |Int A [simplification(40), concrete(B), symbolic(A)]
// Non-zeroedness of |Int
rule X |Int _ ==Int 0 => false
requires 0 <Int X
[simplification]
// Moving from &Int to modInt
rule 1 &Int X => X modInt 2 [simplification]
// &Int yields zero for notMax and operand in appropriate range
rule [bitwise-and-zero]:
X &Int Y => 0
requires 0 <=Int X
andBool pow256 -Int X ==Int 2 ^Int log2Int(pow256 -Int X)
andBool 0 <=Int Y andBool Y <Int 2 ^Int log2Int(pow256 -Int X)
[concrete(X), simplification, comm]
// Deconstruction of <<Int into #buf
rule X <<Int Y => #asWord ( #buf ( 32 -Int (Y /Int 8) , X ) +Bytes #buf ( Y /Int 8 , 0 ) )
requires 0 <=Int X andBool X <Int 2 ^Int (256 -Int Y)
andBool 0 <=Int Y andBool Y <=Int 256 andBool Y modInt 8 ==Int 0
[simplification, concrete(Y)]
rule Z <Int X &Int Y => false
requires #rangeUInt(256, X)
andBool #rangeUInt(256, Y)
andBool #rangeUInt(256, Z)
andBool ((Y <Int Z) orBool (X <Int Z))
[simplification]
rule X &Int Y <Int Z => true
requires #rangeUInt(256, X)
andBool #rangeUInt(256, Y)
andBool #rangeUInt(256, Z)
andBool ((Y <Int Z) orBool (X <Int Z))
[simplification]
rule X &Int #asWord ( _Y +Bytes Z ) => X &Int #asWord ( Z )
requires X <Int 2 ^Int (8 *Int lengthBytes(Z))
[concrete(X, Z), simplification]
rule X &Int #asWord ( _ +Bytes Z ) >>Int T => X &Int #asWord ( Z ) >>Int T
requires X <Int 2 ^Int (8 *Int lengthBytes(Z) -Int T)
[concrete(X, Z, T), simplification]
// |Int distributivity over #asWord and +Bytes, v1
rule A |Int #asWord ( BA1 +Bytes BA2 ) =>
#asWord ( BA1 +Bytes #buf ( lengthBytes(BA2), A |Int #asWord ( BA2 ) ) )
requires A <Int 2 ^Int (8 *Int lengthBytes(BA2))
[concrete(A), simplification]
// |Int distributivity over #asWord and +Bytes, v2
rule A |Int #asWord ( BA1 +Bytes BA2 ) =>
#asWord (
#buf ( lengthBytes(BA1), (A >>Int (8 *Int lengthBytes(BA2))) |Int #asWord ( BA1 ) )
+Bytes
#buf ( lengthBytes(BA2), (A modInt (2 ^Int (8 *Int lengthBytes(BA2)))) |Int #asWord ( BA2 ) )
)
requires 0 <=Int A
[simplification(40), concrete(A, BA1)]
//
// &Int
//
// Commutativity
rule A &Int B ==Int B &Int A => true [simplification, smt-lemma]
rule { A &Int B #Equals B &Int A } => #Top [simplification]
// Distributivity of &Int and |Int
rule A &Int (B |Int C) => (A &Int B) |Int (A &Int C)
[concrete(A, B), simplification]
rule A &Int (B |Int C) => (A &Int B) |Int (A &Int C)
[concrete(A, C), simplification]
// &Int on non-negative integers remains non-negative
rule 0 <=Int (X &Int Y) => true
requires 0 <=Int X
andBool 0 <=Int Y
[simplification, smt-lemma]
// Result of &Int cannot be greater than the operands
rule (X &Int Y) <=Int Z => true
requires 0 <=Int X
andBool 0 <=Int Y
andBool (X <=Int Z orBool Y <=Int Z)
[simplification]
// Anything negative is <Int than &Int
rule A <Int X &Int Y => true
requires 0 <=Int X andBool 0 <=Int Y
andBool A <Int 0
[simplification, concrete(A)]
// Deconstruction of (maxUInt &Int ...)
rule maxUInt8 &Int #asWord ( BA ) => #asWord ( #range(BA, 31, 1) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt16 &Int #asWord ( BA ) => #asWord ( #range(BA, 30, 2) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt24 &Int #asWord ( BA ) => #asWord ( #range(BA, 29, 3) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt32 &Int #asWord ( BA ) => #asWord ( #range(BA, 28, 4) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt40 &Int #asWord ( BA ) => #asWord ( #range(BA, 27, 5) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt48 &Int #asWord ( BA ) => #asWord ( #range(BA, 26, 6) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt56 &Int #asWord ( BA ) => #asWord ( #range(BA, 25, 7) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt64 &Int #asWord ( BA ) => #asWord ( #range(BA, 24, 8) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt72 &Int #asWord ( BA ) => #asWord ( #range(BA, 23, 9) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt80 &Int #asWord ( BA ) => #asWord ( #range(BA, 22, 10) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt88 &Int #asWord ( BA ) => #asWord ( #range(BA, 21, 11) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt96 &Int #asWord ( BA ) => #asWord ( #range(BA, 20, 12) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt104 &Int #asWord ( BA ) => #asWord ( #range(BA, 19, 13) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt112 &Int #asWord ( BA ) => #asWord ( #range(BA, 18, 14) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt120 &Int #asWord ( BA ) => #asWord ( #range(BA, 17, 15) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt128 &Int #asWord ( BA ) => #asWord ( #range(BA, 16, 16) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt136 &Int #asWord ( BA ) => #asWord ( #range(BA, 15, 17) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt144 &Int #asWord ( BA ) => #asWord ( #range(BA, 14, 18) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt152 &Int #asWord ( BA ) => #asWord ( #range(BA, 13, 19) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt160 &Int #asWord ( BA ) => #asWord ( #range(BA, 12, 20) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt168 &Int #asWord ( BA ) => #asWord ( #range(BA, 11, 21) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt176 &Int #asWord ( BA ) => #asWord ( #range(BA, 10, 22) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt184 &Int #asWord ( BA ) => #asWord ( #range(BA, 9, 23) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt192 &Int #asWord ( BA ) => #asWord ( #range(BA, 8, 24) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt200 &Int #asWord ( BA ) => #asWord ( #range(BA, 7, 25) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt208 &Int #asWord ( BA ) => #asWord ( #range(BA, 6, 26) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt216 &Int #asWord ( BA ) => #asWord ( #range(BA, 5, 27) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt224 &Int #asWord ( BA ) => #asWord ( #range(BA, 4, 28) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt232 &Int #asWord ( BA ) => #asWord ( #range(BA, 3, 29) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt240 &Int #asWord ( BA ) => #asWord ( #range(BA, 2, 30) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt248 &Int #asWord ( BA ) => #asWord ( #range(BA, 1, 31) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule maxUInt256 &Int #asWord ( BA ) => #asWord ( #range(BA, 0, 32) ) requires lengthBytes(BA) ==Int 32 [simplification]
// Deconstruction of (notMaxUInt &Int ...)
rule notMaxUInt8 &Int #asWord ( BA ) => #asWord ( #range(BA, 0, 31) +Bytes #buf ( 1, 0 ) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule notMaxUInt16 &Int #asWord ( BA ) => #asWord ( #range(BA, 0, 30) +Bytes #buf ( 2, 0 ) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule notMaxUInt32 &Int #asWord ( BA ) => #asWord ( #range(BA, 0, 28) +Bytes #buf ( 4, 0 ) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule notMaxUInt64 &Int #asWord ( BA ) => #asWord ( #range(BA, 0, 24) +Bytes #buf ( 8, 0 ) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule notMaxUInt96 &Int #asWord ( BA ) => #asWord ( #range(BA, 0, 20) +Bytes #buf ( 12, 0 ) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule notMaxUInt128 &Int #asWord ( BA ) => #asWord ( #range(BA, 0, 16) +Bytes #buf ( 16, 0 ) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule notMaxUInt160 &Int #asWord ( BA ) => #asWord ( #range(BA, 0, 12) +Bytes #buf ( 20, 0 ) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule notMaxUInt192 &Int #asWord ( BA ) => #asWord ( #range(BA, 0, 8) +Bytes #buf ( 24, 0 ) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule notMaxUInt208 &Int #asWord ( BA ) => #asWord ( #range(BA, 0, 6) +Bytes #buf ( 26, 0 ) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule notMaxUInt224 &Int #asWord ( BA ) => #asWord ( #range(BA, 0, 4) +Bytes #buf ( 28, 0 ) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule notMaxUInt240 &Int #asWord ( BA ) => #asWord ( #range(BA, 0, 2) +Bytes #buf ( 30, 0 ) ) requires lengthBytes(BA) ==Int 32 [simplification]
rule notMaxUInt248 &Int #asWord ( BA ) => #asWord ( #range(BA, 0, 1) +Bytes #buf ( 31, 0 ) ) requires lengthBytes(BA) ==Int 32 [simplification]
// Irrelevance of lower bits
rule notMaxUInt8 &Int (X |Int (maxUInt8 &Int _)) => notMaxUInt8 &Int X [simplification]
rule notMaxUInt16 &Int (X |Int (maxUInt16 &Int _)) => notMaxUInt16 &Int X [simplification]
rule notMaxUInt32 &Int (X |Int (maxUInt32 &Int _)) => notMaxUInt32 &Int X [simplification]
rule notMaxUInt64 &Int (X |Int (maxUInt64 &Int _)) => notMaxUInt64 &Int X [simplification]
rule notMaxUInt96 &Int (X |Int (maxUInt96 &Int _)) => notMaxUInt96 &Int X [simplification]
rule notMaxUInt128 &Int (X |Int (maxUInt128 &Int _)) => notMaxUInt128 &Int X [simplification]
rule notMaxUInt160 &Int (X |Int (maxUInt160 &Int _)) => notMaxUInt160 &Int X [simplification]
rule notMaxUInt192 &Int (X |Int (maxUInt192 &Int _)) => notMaxUInt192 &Int X [simplification]
rule notMaxUInt208 &Int (X |Int (maxUInt208 &Int _)) => notMaxUInt208 &Int X [simplification]
rule notMaxUInt224 &Int (X |Int (maxUInt224 &Int _)) => notMaxUInt224 &Int X [simplification]
rule notMaxUInt240 &Int (X |Int (maxUInt240 &Int _)) => notMaxUInt240 &Int X [simplification]
rule notMaxUInt248 &Int (X |Int (maxUInt248 &Int _)) => notMaxUInt248 &Int X [simplification]
// Prepend 4 bytes (used for function selectors)
rule A |Int #asWord ( BUF ) => #asWord ( #range ( #buf ( 32 , A ) , 0 , 4 ) +Bytes BUF )
requires notMaxUInt224 &Int A ==Int A
andBool lengthBytes ( BUF ) ==Int 28
[simplification, concrete(A)]
rule #buf ( 32, A |Int #asWord ( B:Bytes ) ) => #buf (32 -Int lengthBytes(B), A >>Int (8 *Int lengthBytes(B)) ) +Bytes B
requires 0 <=Int A andBool A <Int pow256
andBool lengthBytes(B) <=Int 32
andBool A modInt (2 ^Int lengthBytes(B)) ==Int 0
[simplification, concrete(A)]
//
// Arithmetic
//
rule chop ( X ) => X
requires 0 <=Int X andBool X <Int pow256
[simplification]
rule 0 <=Int A +Int B => true
requires 0 <=Int A andBool 0 <=Int B
[simplification]
//
// Sets
//
// Empty sets has no elements
rule _ in .Set => false [simplification]
rule { true #Equals _ in .Set} => #Bottom [simplification]
rule { false #Equals _ in .Set} => #Top [simplification]
rule S:Set |Set SetItem( X ) => S requires X in S [simplification]
rule X in _:Set SetItem( Y ) => true requires X ==Int Y [simplification]
rule X in S:Set SetItem( Y ) => X in S requires X =/=Int Y [simplification]
rule (S1:Set |Set SetItem( X )) |Set S2:Set => S1 |Set S2 requires X in S2 [simplification, concrete(X, S2)]
rule (S1:Set |Set SetItem( X )) |Set S2:Set => S1 |Set ( SetItem(X) S2 ) requires notBool X in S2 [simplification, concrete(X, S2)]
//
// #lookup
//
rule #lookup(.Map, _) => 0
[simplification]
rule #lookup((K:Int |-> _:Int) M:Map, X:Int) => #lookup(M, X)
requires X =/=Int K
[simplification]
rule #lookup(M:Map [K:Int <- _], X:Int) => #lookup(M, X)
requires X =/=Int K
[simplification]
rule M:Map [ K:Int <- V:Int ] => M
requires V ==Int #lookup(M, K)
andBool V =/=Int 0
[simplification]
//
// keccak assumptions: these assumptions are not sound in principle, but are
// required for verification - they should be collected at the end of execution
//
rule 0 <=Int keccak( _ ) => true [simplification, smt-lemma]
rule keccak( _ ) <Int pow256 => true [simplification, smt-lemma]
// keccak does not equal a concrete value
rule [keccak-eq-conc-false]: keccak(A) ==Int B => false [symbolic(A), concrete(B), simplification, comm]
rule [keccak-neq-conc-true]: keccak(A) =/=Int B => true [symbolic(A), concrete(B), simplification, comm]
rule [keccak-eq-conc-false-ml]: { keccak(_A) #Equals _B } => #Bottom [symbolic(_A), concrete(_B), simplification, comm]
// keccak is injective
rule [keccak-inj]: keccak(A) ==Int keccak(B) => A ==K B [simplification]
// keccak has no "fixpoint"
rule [keccak-no-fix-eq-false]: #buf(32, keccak(X)) ==K X => false [simplification]
rule [keccak-no-fix-neq-true]: #buf(32, keccak(X)) =/=K X => true [simplification]
// chop of negative keccak
rule chop (0 -Int keccak(BA)) => pow256 -Int keccak(BA)
[simplification]
// keccak cannot equal a number outside of its range
rule { X #Equals keccak (_) } => #Bottom
requires X <Int 0 orBool X >=Int pow256
[concrete(X), simplification]
// anything negative is smaller than a keccak
rule X <Int keccak ( _ ) => true
requires X <Int 0
[concrete(X), simplification]
// a keccak is smaller than anything greater than pow256 - 32
rule keccak ( _ ) <Int X => true
requires (pow256 -Int 32) <Int X
[concrete(X), simplification]
endmodule