231 lines
11 KiB
Plaintext
231 lines
11 KiB
Plaintext
module Project2
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open FStar.List.Tot
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(* opcodes of the simplified bytecode fragment *)
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type opcode =
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| ADD : opcode
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| CALL : opcode
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| AND : opcode
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| LE : opcode
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| PUSH : int -> opcode
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| POP : opcode
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| MLOAD : opcode
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| MSTORE : opcode
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| SLOAD : opcode
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| SSTORE : opcode
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| TIMESTAMP : opcode
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| BALANCE : opcode
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| INPUT : opcode
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| ADDRESS : opcode
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| GAS : opcode
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| JUMP : nat -> opcode
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| JUMPI : nat -> opcode
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| RETURN : opcode
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| STOP : opcode
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| FAIL : opcode
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(* 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 *)
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type address = int
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(* a contract is a tuple of a an address and its code *)
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type contract = address * list opcode
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(* 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 *)
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type account = nat * (int -> int) * list opcode
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(* the global state is a mapping from contract addresses to accounts *)
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type gstate = address -> Tot account
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(* 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 *)
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type exenv = address * int * list opcode
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(* 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 *)
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type mstate = nat * nat * (int -> int) * list int // does not carry gas
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(* 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 *)
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type regstate = mstate * exenv * gstate
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(* the transaction environment only carries the blocktimestamp represented as an integer *)
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type tenv = int
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(* 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*)
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noeq type terstate =
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| HaltState: gstate -> int -> nat -> terstate
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| ExcState: terstate
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(* callstacks *)
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type plaincallstack = list regstate
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noeq type callstack =
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| Ter : terstate -> plaincallstack -> callstack
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| Exec : plaincallstack -> callstack
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(* Small step function *)
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(* Polymorphic update function *)
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val update: (#a:eqtype) -> (f: a -> 'b) -> (p:a) -> (e: 'b) -> (x: a) -> Tot 'b
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let update (#a:eqtype) (f: a -> 'b) (p: a) (e: 'b) =
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fun x -> if x = p then e else f x
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(* size of callstacks *)
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val size: callstack -> Tot nat
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let size (cs: callstack) =
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match cs with
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| Exec ps -> length ps
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| Ter ts ps -> 1 + length ps
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(* a function that extracts the current opcode given the code and a pc *)
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val getOpcode: list opcode -> nat -> Tot opcode
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let getOpcode code i =
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match (nth code i) with
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| None -> STOP
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| Some oc -> oc
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(* 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 *)
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val isCallState: regstate -> Tot bool
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let isCallState rs =
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match rs with
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| ((gas, pc, m, to:: v:: inp:: resaddr:: stack), (actor, input, code), sigma) -> getOpcode code pc = CALL
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| _ -> false
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(* Wellformedness definition: a callstack is well-formed if all of it's non top elements are call states *)
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val wellformed: callstack -> Tot bool
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let rec wellformed (cs: callstack) =
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match cs with
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| Exec [] -> false
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| Ter ts ps -> for_all (fun rs -> (isCallState rs)) ps
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| Exec (s::ps) -> for_all (fun rs -> (isCallState rs)) ps
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(* 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 *)
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noeq type step_outcome =
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| Stop : (cs: callstack) -> step_outcome
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| Next : (cs: callstack) -> step_outcome
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(* Auxiliary function for applying the effects of terminated states to the underneath execution states *)
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val apply_returneffects: (ts: terstate) -> (rs: regstate{isCallState rs}) -> Tot regstate
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let apply_returneffects ts rs =
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let ((gas, pc, mem, to:: v:: imp:: resaddr:: stack), (actor, code, input), gs) = rs
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in assert (pc >= 0);
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match ts with
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| ExcState -> ((0, pc+1, mem, 0::stack), (actor, code, input), gs)
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| HaltState gs' res gas' -> ((gas', pc+1, update mem resaddr res, 1::stack), (actor, code, input), gs')
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(* 3.1: Small-step semantics *)
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(* Small step function that describes one step of execution. Replace all occurences of 'magic ()', by the definitions as specified in the paper semantics *)
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val step: tenv -> cs: callstack {wellformed cs} -> Tot step_outcome
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let step te cs =
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match cs with
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| Ter ts [] -> Stop (Ter ts [])
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| Ter ts (s :: ps) -> Next (Exec ((apply_returneffects ts s)::ps))
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| Exec (s :: ps) ->
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let (((gas, pc, mem, stack), (actor, input, code), gs)) = s in
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if gas < 1 then Next (Ter ExcState ps)
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else
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match (getOpcode code pc, stack) with
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| (ADD, a::b::stack') -> Next (Exec(((gas-1, pc+1, mem, (a+b):: stack'), (actor, input, code), gs) :: ps))
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| (AND, a::b::stack') -> let c = (if a > 0 && b > 0 then 1 else 0) in
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Next (Exec(((gas-1, pc+1, mem, c::stack'), (actor, input, code), gs) :: ps))
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| (LE, a::b::stack') -> let c = (if a <= b then 1 else 0) in
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Next (Exec(((gas-1, pc+1, mem, c::stack'), (actor, input, code), gs) :: ps))
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| (PUSH x, stack') -> Next (Exec(((gas-1, pc+1, mem, x::stack'), (actor, input, code), gs)::ps))
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| (POP, x::stack') -> Next (Exec(((gas-1, pc+1, mem, stack') (actor, input, code), gs) :: ps))
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| (MSTORE, p::v::stack') -> Next (Exec(((gas-1, pc+1, update mem p v, stack'), (actor, input, code), gs)::ps))
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| (MLOAD, p::stack') -> let v = (mem p) in
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Next (Exec(((gas-1, pc+1, mem, v::stack'), (actor, input, code), gs') :: ps))
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| (SSTORE, p::v::stack') -> let acc = (let (bal, stor, code) = gs actor in (bal, update stor p v, code)) in
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let gs' = update gs actor acc in
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Next (Exec(((gas-1, pc+1, mem, stack'), (actor, input, code), gs') :: ps))
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| (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))
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| (BALANCE, a::stack') -> magic ()
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| (ADDRESS, stack') -> Next (Exec(((gas-1, pc+1, mem, actor::stack'), (actor, input, code), gs)::ps))
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| (INPUT, stack') -> magic ()
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| (GAS, stack') -> Next (Exec(((gas-1, pc+1, mem, gas::stack'), (actor, input, code), gs)::ps))
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| (JUMP i, stack') -> Next (Exec((((gas-1, i, mem, stack'), (actor, input, code), gs))::ps))
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| (JUMPI i, b::stack') -> magic ()
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| (RETURN, v::stack') -> magic ()
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| (STOP, stack') -> Next (Ter (HaltState gs 0 (gas-1)) ps)
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| (TIMESTAMP, stack') -> magic ()
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| (CALL, to::v::inp::resaddr::stack') -> magic ()
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| _ -> Next (Ter ExcState ps)
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(* A simple wrapper for the step function that removes the execution outcome *)
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val step_simp: (te: tenv) -> (cs: callstack {wellformed cs}) -> Tot (cs': callstack{wellformed cs'})
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let step_simp te cs =
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match (step te cs) with
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| Next cs' -> cs'
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| Stop cs' -> cs'
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(* Bounded step function that executes an execution state for (at most) n execution steps *)
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val nsteps: (n: nat) -> (te: tenv) -> (cs:callstack{wellformed cs}) -> Tot (cs:callstack{wellformed cs})
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let rec nsteps n te cs =
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if n=0 then cs
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else
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nsteps (n-1) te (step_simp te cs)
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(* 3.2: Termination *)
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(* We will define an interpreter function steps that executes the small step function till reaching a final state (indicated by Stop) *)
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(* Our goal is to prove the termination of this function *)
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(* 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 *)
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val getDecArgList: (cs: callstack {wellformed cs}) -> Tot (list nat)
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let getDecArgList (cs: callstack {wellformed cs}) =
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magic ()
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(* A simple helper function that converts a list to a lexicgraphical ordering *)
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val getLexFromList: (list nat) -> Tot (lex_t)
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let rec getLexFromList ls =
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match ls with
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| [] -> LexTop
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| (l::ls') -> LexCons #nat l (getLexFromList ls')
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(* Interpreter function that executes the small step function till termination *)
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(* Define the function getDecArg list, shuch that the given decreases clause is sufficient for proving the terminination of the function on all arguments *)
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val steps: (te:tenv) -> (cs:callstack {wellformed cs}) -> Tot callstack (decreases (getLexFromList(getDecArgList cs)))
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let rec steps te cs =
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match (step te cs) with
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| Next cs' -> steps te cs'
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| Stop cs' -> cs'
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(* 3.3: Final states *)
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(* A syntactic characterization of final call stacks (similiar to stopping criterion in step) *)
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val isFinal: (cs: callstack) -> Tot bool
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let isFinal cs =
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match cs with
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| Ter ts [] -> true
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| _ -> false
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(* 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) *)
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val nsteps_stop: (n: nat) -> (te:tenv) -> (cs: callstack{wellformed cs}) ->
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Lemma (requires (isFinal cs))
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(ensures (nsteps n te cs == cs))
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let rec nsteps_stop n te cs =
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admit ()
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(* 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 *)
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(* val progress: *)
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(* 3.4: Uniqueness of callstack *)
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(* 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) *)
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(* Hint: Use the notion of 'smaller' that you used for proving the termination of steps *)
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val order_decreases: (n: nat) -> (te: tenv) -> (cs: callstack{wellformed cs}) -> (cs': callstack) ->
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Lemma (requires (nsteps n te cs == cs' /\ n > 0 /\ ~ (isFinal cs) ))
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(ensures (getLexFromList(getDecArgList cs')<< getLexFromList(getDecArgList cs)))
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let rec order_decreases n te cs cs' =
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admit ()
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(* Use the previous Lemma to show that the callstacks during execution are unique *)
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val callstacks_unique: (n: nat) -> (te: tenv) -> (cs: callstack{wellformed cs}) -> (cs': callstack) ->
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Lemma (requires (nsteps n te cs == cs' /\ n > 0 /\ ~ (isFinal cs) ))
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(ensures (~ (cs == cs')))
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let rec callstacks_unique n te cs cs' =
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admit ()
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(* 3.5: Exception propagation *)
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(* Prove that when an exception occurs the execution will terminate within 2 * size cs steps *)
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val exception_prop: (te:tenv) -> (ps:plaincallstack) ->
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Lemma (requires (wellformed (Ter ExcState ps)))
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(ensures (nsteps (op_Multiply 2 (length ps)) te (Ter ExcState ps) == (Ter ExcState [])))
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let rec exception_prop te ps =
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admit ()
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