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Add AND and LE definitions

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Tobias Eidelpes 2021-06-03 10:07:37 +02:00
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Project2.fst
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@ -2,8 +2,8 @@ module Project2
open FStar.List.Tot open FStar.List.Tot
(* opcodes of the simplified bytecode fragment *) (* opcodes of the simplified bytecode fragment *)
type opcode = type opcode =
| ADD : opcode | ADD : opcode
| CALL : opcode | CALL : opcode
| AND : opcode | AND : opcode
@ -22,204 +22,206 @@ type opcode =
| JUMP : nat -> opcode | JUMP : nat -> opcode
| JUMPI : nat -> opcode | JUMPI : nat -> opcode
| RETURN : opcode | RETURN : opcode
| STOP : opcode | STOP : opcode
| FAIL : opcode | FAIL : opcode
(* Small step configurations. For simplicity we assume all values (in particular stack and memory values as well as memory and storage addresses) to be represented as integers *) (* Small step configurations. For simplicity we assume all values (in particular stack and memory values as well as memory and storage addresses) to be represented as integers *)
type address = int type address = int
(* a contract is a tuple of a an address and its code *) (* a contract is a tuple of a an address and its code *)
type contract = address * list opcode type contract = address * list opcode
(* accounts are tuples of the form (b, stor, code) where b is the account's balance, stor is the account's persistent storage, and code is it's opcode *) (* accounts are tuples of the form (b, stor, code) where b is the account's balance, stor is the account's persistent storage, and code is it's opcode *)
type account = nat * (int -> int) * list opcode type account = nat * (int -> int) * list opcode
(* the global state is a mapping from contract addresses to accounts *) (* the global state is a mapping from contract addresses to accounts *)
type gstate = address -> Tot account type gstate = address -> Tot account
(* execution environments are tuples of the form: (actor, input, code) where actor is the address of the active account, input is the input to the call and code is the code currently executed *) (* execution environments are tuples of the form: (actor, input, code) where actor is the address of the active account, input is the input to the call and code is the code currently executed *)
type exenv = address * int * list opcode type exenv = address * int * list opcode
(* machine states are tuples of the form: (gas, pc, m, s) where gas is the remaining gas, pc is the program counter, memory is the local memory and s is the machine stack *) (* machine states are tuples of the form: (gas, pc, m, s) where gas is the remaining gas, pc is the program counter, memory is the local memory and s is the machine stack *)
type mstate = nat * nat * (int -> int) * list int // does not carry gas type mstate = nat * nat * (int -> int) * list int // does not carry gas
(* a regular execution state is a tuple of the form (mu, iota, sigma) where mu is the machine state, iota is the execution environment and gstate is the global state *) (* a regular execution state is a tuple of the form (mu, iota, sigma) where mu is the machine state, iota is the execution environment and gstate is the global state *)
type regstate = mstate * exenv * gstate type regstate = mstate * exenv * gstate
(* the transaction environment only carries the blocktimestamp represented as an integer *) (* the transaction environment only carries the blocktimestamp represented as an integer *)
type tenv = int type tenv = int
(* terminating states are either exception states or halting states of the form (sigma, d, gas) where sigma is the global state at the point of halting, d the return value of the call and gas the remaining gas at the point of halting*) (* terminating states are either exception states or halting states of the form (sigma, d, gas) where sigma is the global state at the point of halting, d the return value of the call and gas the remaining gas at the point of halting*)
noeq type terstate = noeq type terstate =
| HaltState: gstate -> int -> nat -> terstate | HaltState: gstate -> int -> nat -> terstate
| ExcState: terstate | ExcState: terstate
(* callstacks *) (* callstacks *)
type plaincallstack = list regstate type plaincallstack = list regstate
noeq type callstack = noeq type callstack =
| Ter : terstate -> plaincallstack -> callstack | Ter : terstate -> plaincallstack -> callstack
| Exec : plaincallstack -> callstack | Exec : plaincallstack -> callstack
(* Small step function *) (* Small step function *)
(* Polymorphic update function *) (* Polymorphic update function *)
val update: (#a:eqtype) -> (f: a -> 'b) -> (p:a) -> (e: 'b) -> (x: a) -> Tot 'b val update: (#a:eqtype) -> (f: a -> 'b) -> (p:a) -> (e: 'b) -> (x: a) -> Tot 'b
let update (#a:eqtype) (f: a -> 'b) (p: a) (e: 'b) = let update (#a:eqtype) (f: a -> 'b) (p: a) (e: 'b) =
fun x -> if x = p then e else f x fun x -> if x = p then e else f x
(* size of callstacks *) (* size of callstacks *)
val size: callstack -> Tot nat val size: callstack -> Tot nat
let size (cs: callstack) = let size (cs: callstack) =
match cs with match cs with
| Exec ps -> length ps | Exec ps -> length ps
| Ter ts ps -> 1 + length ps | Ter ts ps -> 1 + length ps
(* a function that extracts the current opcode given the code and a pc *) (* a function that extracts the current opcode given the code and a pc *)
val getOpcode: list opcode -> nat -> Tot opcode val getOpcode: list opcode -> nat -> Tot opcode
let getOpcode code i = let getOpcode code i =
match (nth code i) with match (nth code i) with
| None -> STOP | None -> STOP
| Some oc -> oc | Some oc -> oc
(* a function checking whether a state is a call state. We characterize call states as state where CALL was executed and sufficiently many arguments where on the stack *) (* a function checking whether a state is a call state. We characterize call states as state where CALL was executed and sufficiently many arguments where on the stack *)
val isCallState: regstate -> Tot bool val isCallState: regstate -> Tot bool
let isCallState rs = let isCallState rs =
match rs with match rs with
| ((gas, pc, m, to:: v:: inp:: resaddr:: stack), (actor, input, code), sigma) -> getOpcode code pc = CALL | ((gas, pc, m, to:: v:: inp:: resaddr:: stack), (actor, input, code), sigma) -> getOpcode code pc = CALL
| _ -> false | _ -> false
(* Wellformedness definition: a callstack is well-formed if all of it's non top elements are call states *) (* Wellformedness definition: a callstack is well-formed if all of it's non top elements are call states *)
val wellformed: callstack -> Tot bool val wellformed: callstack -> Tot bool
let rec wellformed (cs: callstack) = let rec wellformed (cs: callstack) =
match cs with match cs with
| Exec [] -> false | Exec [] -> false
| Ter ts ps -> for_all (fun rs -> (isCallState rs)) ps | Ter ts ps -> for_all (fun rs -> (isCallState rs)) ps
| Exec (s::ps) -> for_all (fun rs -> (isCallState rs)) ps | Exec (s::ps) -> for_all (fun rs -> (isCallState rs)) ps
(* Type for the outcome of a single execution step: either the execution terminated (Stop) as a final state is reached or further execution steps are possible *) (* Type for the outcome of a single execution step: either the execution terminated (Stop) as a final state is reached or further execution steps are possible *)
noeq type step_outcome = noeq type step_outcome =
| Stop : (cs: callstack) -> step_outcome | Stop : (cs: callstack) -> step_outcome
| Next : (cs: callstack) -> step_outcome | Next : (cs: callstack) -> step_outcome
(* Auxiliary function for applying the effects of terminated states to the underneath execution states *) (* Auxiliary function for applying the effects of terminated states to the underneath execution states *)
val apply_returneffects: (ts: terstate) -> (rs: regstate{isCallState rs}) -> Tot regstate val apply_returneffects: (ts: terstate) -> (rs: regstate{isCallState rs}) -> Tot regstate
let apply_returneffects ts rs = let apply_returneffects ts rs =
let ((gas, pc, mem, to:: v:: imp:: resaddr:: stack), (actor, code, input), gs) = rs let ((gas, pc, mem, to:: v:: imp:: resaddr:: stack), (actor, code, input), gs) = rs
in assert (pc >= 0); in assert (pc >= 0);
match ts with match ts with
| ExcState -> ((0, pc+1, mem, 0::stack), (actor, code, input), gs) | ExcState -> ((0, pc+1, mem, 0::stack), (actor, code, input), gs)
| HaltState gs' res gas' -> ((gas', pc+1, update mem resaddr res, 1::stack), (actor, code, input), gs') | HaltState gs' res gas' -> ((gas', pc+1, update mem resaddr res, 1::stack), (actor, code, input), gs')
(* 3.1: Small-step semantics *) (* 3.1: Small-step semantics *)
(* Small step function that describes one step of execution. Replace all occurences of 'magic ()', by the definitions as specified in the paper semantics *) (* Small step function that describes one step of execution. Replace all occurences of 'magic ()', by the definitions as specified in the paper semantics *)
val step: tenv -> cs: callstack {wellformed cs} -> Tot step_outcome val step: tenv -> cs: callstack {wellformed cs} -> Tot step_outcome
let step te cs = let step te cs =
match cs with match cs with
| Ter ts [] -> Stop (Ter ts []) | Ter ts [] -> Stop (Ter ts [])
| Ter ts (s :: ps) -> Next (Exec ((apply_returneffects ts s)::ps)) | Ter ts (s :: ps) -> Next (Exec ((apply_returneffects ts s)::ps))
| Exec (s :: ps) -> | Exec (s :: ps) ->
let (((gas, pc, mem, stack), (actor, input, code), gs)) = s in let (((gas, pc, mem, stack), (actor, input, code), gs)) = s in
if gas < 1 then Next (Ter ExcState ps) if gas < 1 then Next (Ter ExcState ps)
else else
match (getOpcode code pc, stack) with match (getOpcode code pc, stack) with
| (ADD, a::b::stack') -> Next (Exec(((gas-1, pc+1, mem, (a+b):: stack'), (actor, input, code), gs) :: ps)) | (ADD, a::b::stack') -> Next (Exec(((gas-1, pc+1, mem, (a+b):: stack'), (actor, input, code), gs) :: ps))
| (AND, a::b::stack') -> magic () | (AND, a::b::stack') -> let c = (if a > 0 && b > 0 then 1 else 0) in
| (LE, a::b::stack') -> magic () Next (Exec(((gas-1, pc+1, mem, c::stack'), (actor, input, code), gs) :: ps))
| (LE, a::b::stack') -> let c = (if a <= b then 1 else 0) in
Next (Exec(((gas-1, pc+1, mem, c::stack'), (actor, input, code), gs) :: ps))
| (PUSH x, stack') -> Next (Exec(((gas-1, pc+1, mem, x::stack'), (actor, input, code), gs)::ps)) | (PUSH x, stack') -> Next (Exec(((gas-1, pc+1, mem, x::stack'), (actor, input, code), gs)::ps))
| (POP, x::stack') -> magic () | (POP, x::stack') -> magic ()
| (MSTORE, p::v::stack') -> Next (Exec(((gas-1, pc+1, update mem p v, stack'), (actor, input, code), gs)::ps)) | (MSTORE, p::v::stack') -> Next (Exec(((gas-1, pc+1, update mem p v, stack'), (actor, input, code), gs)::ps))
| (MLOAD, p::stack') -> magic () | (MLOAD, p::stack') -> magic ()
| (SSTORE, p::v::stack') -> magic () | (SSTORE, p::v::stack') -> magic ()
| (SLOAD, v::stack') -> Next (Exec(((gas-1, pc+1, mem, (let (bal, stor, code) = gs actor in stor v)::stack'), (actor, input, code), gs)::ps)) | (SLOAD, v::stack') -> Next (Exec(((gas-1, pc+1, mem, (let (bal, stor, code) = gs actor in stor v)::stack'), (actor, input, code), gs)::ps))
| (BALANCE, a::stack') -> magic () | (BALANCE, a::stack') -> magic ()
| (ADDRESS, stack') -> Next (Exec(((gas-1, pc+1, mem, actor::stack'), (actor, input, code), gs)::ps)) | (ADDRESS, stack') -> Next (Exec(((gas-1, pc+1, mem, actor::stack'), (actor, input, code), gs)::ps))
| (INPUT, stack') -> magic () | (INPUT, stack') -> magic ()
| (GAS, stack') -> Next (Exec(((gas-1, pc+1, mem, gas::stack'), (actor, input, code), gs)::ps)) | (GAS, stack') -> Next (Exec(((gas-1, pc+1, mem, gas::stack'), (actor, input, code), gs)::ps))
| (JUMP i, stack') -> Next (Exec((((gas-1, i, mem, stack'), (actor, input, code), gs))::ps)) | (JUMP i, stack') -> Next (Exec((((gas-1, i, mem, stack'), (actor, input, code), gs))::ps))
| (JUMPI i, b::stack') -> magic () | (JUMPI i, b::stack') -> magic ()
| (RETURN, v::stack') -> magic () | (RETURN, v::stack') -> magic ()
| (STOP, stack') -> Next (Ter (HaltState gs 0 (gas-1)) ps) | (STOP, stack') -> Next (Ter (HaltState gs 0 (gas-1)) ps)
| (TIMESTAMP, stack') -> magic () | (TIMESTAMP, stack') -> magic ()
| (CALL, to::v::inp::resaddr::stack') -> magic () | (CALL, to::v::inp::resaddr::stack') -> magic ()
| _ -> Next (Ter ExcState ps) | _ -> Next (Ter ExcState ps)
(* A simple wrapper for the step function that removes the execution outcome *) (* A simple wrapper for the step function that removes the execution outcome *)
val step_simp: (te: tenv) -> (cs: callstack {wellformed cs}) -> Tot (cs': callstack{wellformed cs'}) val step_simp: (te: tenv) -> (cs: callstack {wellformed cs}) -> Tot (cs': callstack{wellformed cs'})
let step_simp te cs = let step_simp te cs =
match (step te cs) with match (step te cs) with
| Next cs' -> cs' | Next cs' -> cs'
| Stop cs' -> cs' | Stop cs' -> cs'
(* Bounded step function that executes an execution state for (at most) n execution steps *) (* Bounded step function that executes an execution state for (at most) n execution steps *)
val nsteps: (n: nat) -> (te: tenv) -> (cs:callstack{wellformed cs}) -> Tot (cs:callstack{wellformed cs}) val nsteps: (n: nat) -> (te: tenv) -> (cs:callstack{wellformed cs}) -> Tot (cs:callstack{wellformed cs})
let rec nsteps n te cs = let rec nsteps n te cs =
if n=0 then cs if n=0 then cs
else else
nsteps (n-1) te (step_simp te cs) nsteps (n-1) te (step_simp te cs)
(* 3.2: Termination *) (* 3.2: Termination *)
(* We will define an interpreter function steps that executes the small step function till reaching a final state (indicated by Stop) *) (* We will define an interpreter function steps that executes the small step function till reaching a final state (indicated by Stop) *)
(* Our goal is to prove the termination of this function *) (* Our goal is to prove the termination of this function *)
(* To this end, define a function the following function on callstacks that assigns a measure (in terms of a list of naturals that gets lexicographically interpreted) to each call stack *) (* To this end, define a function the following function on callstacks that assigns a measure (in terms of a list of naturals that gets lexicographically interpreted) to each call stack *)
val getDecArgList: (cs: callstack {wellformed cs}) -> Tot (list nat) val getDecArgList: (cs: callstack {wellformed cs}) -> Tot (list nat)
let getDecArgList (cs: callstack {wellformed cs}) = let getDecArgList (cs: callstack {wellformed cs}) =
magic () magic ()
(* A simple helper function that converts a list to a lexicgraphical ordering *) (* A simple helper function that converts a list to a lexicgraphical ordering *)
val getLexFromList: (list nat) -> Tot (lex_t) val getLexFromList: (list nat) -> Tot (lex_t)
let rec getLexFromList ls = let rec getLexFromList ls =
match ls with match ls with
| [] -> LexTop | [] -> LexTop
| (l::ls') -> LexCons #nat l (getLexFromList ls') | (l::ls') -> LexCons #nat l (getLexFromList ls')
(* Interpreter function that executes the small step function till termination *) (* Interpreter function that executes the small step function till termination *)
(* Define the function getDecArg list, shuch that the given decreases clause is sufficient for proving the terminination of the function on all arguments *) (* Define the function getDecArg list, shuch that the given decreases clause is sufficient for proving the terminination of the function on all arguments *)
val steps: (te:tenv) -> (cs:callstack {wellformed cs}) -> Tot callstack (decreases (getLexFromList(getDecArgList cs))) val steps: (te:tenv) -> (cs:callstack {wellformed cs}) -> Tot callstack (decreases (getLexFromList(getDecArgList cs)))
let rec steps te cs = let rec steps te cs =
match (step te cs) with match (step te cs) with
| Next cs' -> steps te cs' | Next cs' -> steps te cs'
| Stop cs' -> cs' | Stop cs' -> cs'
(* 3.3: Final states *) (* 3.3: Final states *)
(* A syntactic characterization of final call stacks (similiar to stopping criterion in step) *) (* A syntactic characterization of final call stacks (similiar to stopping criterion in step) *)
val isFinal: (cs: callstack) -> Tot bool val isFinal: (cs: callstack) -> Tot bool
let isFinal cs = let isFinal cs =
match cs with match cs with
| Ter ts [] -> true | Ter ts [] -> true
| _ -> false | _ -> false
(* Prove that the syntactic characterization of final states implies a semantic characterization (namely that the execution of arbitrary steps does not affect the callstack anymore) *) (* Prove that the syntactic characterization of final states implies a semantic characterization (namely that the execution of arbitrary steps does not affect the callstack anymore) *)
val nsteps_stop: (n: nat) -> (te:tenv) -> (cs: callstack{wellformed cs}) -> val nsteps_stop: (n: nat) -> (te:tenv) -> (cs: callstack{wellformed cs}) ->
Lemma (requires (isFinal cs)) Lemma (requires (isFinal cs))
(ensures (nsteps n te cs == cs)) (ensures (nsteps n te cs == cs))
let rec nsteps_stop n te cs = let rec nsteps_stop n te cs =
admit () admit ()
(* Prove that if a call stack does not change within one step then it must be final. Formulate first the Lemma and then prove it *) (* Prove that if a call stack does not change within one step then it must be final. Formulate first the Lemma and then prove it *)
(* val progress: *) (* val progress: *)
(* 3.4: Uniqueness of callstack *) (* 3.4: Uniqueness of callstack *)
(* Prove that during an execution, every callstack is unique. To this end, first prove that callstacks are always decreasing within n > 0 execution steps (unless they are final) *) (* Prove that during an execution, every callstack is unique. To this end, first prove that callstacks are always decreasing within n > 0 execution steps (unless they are final) *)
(* Hint: Use the notion of 'smaller' that you used for proving the termination of steps *) (* Hint: Use the notion of 'smaller' that you used for proving the termination of steps *)
val order_decreases: (n: nat) -> (te: tenv) -> (cs: callstack{wellformed cs}) -> (cs': callstack) -> val order_decreases: (n: nat) -> (te: tenv) -> (cs: callstack{wellformed cs}) -> (cs': callstack) ->
Lemma (requires (nsteps n te cs == cs' /\ n > 0 /\ ~ (isFinal cs) )) Lemma (requires (nsteps n te cs == cs' /\ n > 0 /\ ~ (isFinal cs) ))
(ensures (getLexFromList(getDecArgList cs')<< getLexFromList(getDecArgList cs))) (ensures (getLexFromList(getDecArgList cs')<< getLexFromList(getDecArgList cs)))
let rec order_decreases n te cs cs' = let rec order_decreases n te cs cs' =
admit () admit ()
(* Use the previous Lemma to show that the callstacks during execution are unique *) (* Use the previous Lemma to show that the callstacks during execution are unique *)
val callstacks_unique: (n: nat) -> (te: tenv) -> (cs: callstack{wellformed cs}) -> (cs': callstack) -> val callstacks_unique: (n: nat) -> (te: tenv) -> (cs: callstack{wellformed cs}) -> (cs': callstack) ->
Lemma (requires (nsteps n te cs == cs' /\ n > 0 /\ ~ (isFinal cs) )) Lemma (requires (nsteps n te cs == cs' /\ n > 0 /\ ~ (isFinal cs) ))
(ensures (~ (cs == cs'))) (ensures (~ (cs == cs')))
let rec callstacks_unique n te cs cs' = let rec callstacks_unique n te cs cs' =
admit () admit ()
(* 3.5: Exception propagation *) (* 3.5: Exception propagation *)
(* Prove that when an exception occurs the execution will terminate within 2 * size cs steps *) (* Prove that when an exception occurs the execution will terminate within 2 * size cs steps *)
val exception_prop: (te:tenv) -> (ps:plaincallstack) -> val exception_prop: (te:tenv) -> (ps:plaincallstack) ->
Lemma (requires (wellformed (Ter ExcState ps))) Lemma (requires (wellformed (Ter ExcState ps)))
(ensures (nsteps (op_Multiply 2 (length ps)) te (Ter ExcState ps) == (Ter ExcState []))) (ensures (nsteps (op_Multiply 2 (length ps)) te (Ter ExcState ps) == (Ter ExcState [])))
let rec exception_prop te ps = let rec exception_prop te ps =
admit () admit ()