BoardroomVoting.v

From Coq Require Import List. Import ListNotations.
From Coq Require Import ZArith.
From Coq Require Import Znumtheory.
From Coq Require Import Permutation.
From Coq Require Import Lia.
From ConCert.Utils Require Import Automation.
From ConCert.Utils Require Import Extras.
From ConCert.Utils Require Import RecordUpdate.
From ConCert.Execution Require Import Blockchain.
From ConCert.Execution Require Import Containers.
From ConCert.Execution Require Import Monad.
From ConCert.Execution Require Import ResultMonad.
From ConCert.Execution Require Import Serializable.
From ConCert.Execution Require Import ContractMonads.
From ConCert.Execution Require Import ContractCommon. Import AddressMap.
From ConCert.Examples.BoardroomVoting Require Import BoardroomMath.



Module Type BoardroomParams.
  Parameter A : Type.
  Parameter H : list positive -> positive.
  Parameter ser : Serializable A.
  Parameter axioms : BoardroomAxioms A.
  Parameter gen : Generator axioms.
  Parameter discr_log : DiscreteLog axioms gen.
  Parameter Base : ChainBase.
End BoardroomParams.

Module BoardroomVoting (Params : BoardroomParams).
  Import Params.
  #[local]
  Existing Instance ser.
  #[local]
  Existing Instance axioms.
  #[local]
  Existing Instance gen.
  #[local]
  Existing Instance discr_log.
  #[local]
  Existing Instance Base.

  (* Allow us to automatically derive Serializable instances *)
  Set Nonrecursive Elimination Schemes.
  Record Setup :=
    build_setup {
      eligible_voters : AddrMap unit;
      finish_registration_by : nat;
      finish_commit_by : option nat;
      finish_vote_by : nat;
      registration_deposit : Amount;
    }.

  Record VoterInfo :=
    build_voter_info {
      voter_index : nat;
      vote_hash : positive;
      public_vote : A;
    }.

  Record State :=
    build_state {
      owner : Address;
      registered_voters : AddrMap VoterInfo;
      public_keys : list A;
      setup : Setup;
      tally : option nat;
    }.

  Definition Error : Type := nat.
  Definition default_error : Error := 1%nat.

  (* w, a1, b1, a2, b2, d1, d2 *)
  Definition VoteProof := (Z * A * A * A * A * Z * Z * Z * Z)%type.

  Inductive Msg :=
  | signup (pk : A) (proof : A * Z)
  | commit_to_vote (hash : positive)
  | submit_vote (v : A) (proof : VoteProof)
  | tally_votes.

  (* begin hide *)
  MetaCoq Run (make_setters VoterInfo).
  MetaCoq Run (make_setters State).
  (* end hide *)

  Section Serialization.
    Global Instance Setup_serializable : Serializable Setup :=
      Derive Serializable Setup_rect<build_setup>.

    Global Instance VoterInfo_serializable : Serializable VoterInfo :=
      Derive Serializable VoterInfo_rect<build_voter_info>.

    Global Instance State_serializable : Serializable State :=
      Derive Serializable State_rect<build_state>.

    Global Instance Msg_serializable : Serializable Msg :=
      Derive Serializable Msg_rect<signup, commit_to_vote, submit_vote, tally_votes>.
  End Serialization.

  Local Open Scope broom.

  Definition encodeA : A -> positive := countable.encode.
  Definition encodeNat : nat -> positive := countable.encode.

  Definition hash_sk_data (gv pk : A) (i : nat) : positive :=
    H [encodeA (generator : A); encodeA gv; encodeA pk; encodeNat i].

  (* This follows the original open vote protocol paper. It is a schnorr signature
      with the fiat-shamir heuristic applied. *)
  Definition secret_key_proof (sk : Z) (v : Z) (i : nat) : A * Z :=
    let gv : A := generator^v in
    let pk := compute_public_key sk in
    let z := Zpos (hash_sk_data gv pk i) in
    let r := (v - sk * z)%Z in
    (gv, r).

  Definition verify_secret_key_proof (pk : A) (i : nat) (proof : A * Z) : bool :=
    let (gv, r) := proof in
    let z := Zpos (hash_sk_data gv pk i) in
    elmeqb gv (generator^r * (pk^z)).

  Definition hash_sv_data (i : nat) (pk rk a1 b1 a2 b2 : A) : positive :=
    H (encodeNat i :: map encodeA [pk; rk; a1; b1; a2; b2]).

  Definition secret_vote_proof (sk : Z) (rk : A) (sv : bool) (i : nat) (w r d : Z) : VoteProof :=
    let pk : A := compute_public_key sk in
    let pv : A := compute_public_vote rk sk sv in
    if sv then
      let a1 : A := generator^r * pk^d in
      let b1 : A := rk^r * pv^d in
      let a2 : A := generator^w in
      let b2 : A := rk^w in
      let c := Zpos (hash_sv_data i pk rk a1 b1 a2 b2) in
      let d2 := c - d in
      let r2 := w - sk*d2 in
      (w, a1, b1, a2, b2, d, d2, r, r2)
    else
      let a1 := generator^w in
      let b1 := rk^w in
      let a2 := generator^r * pk^d in
      let b2 := rk^r * (pv * inv generator)^d in
      let c := Zpos (hash_sv_data i pk rk a1 b1 a2 b2) in
      let d1 := c - d in
      let r1 := w - sk*d1 in
      (w, a1, b1, a2, b2, d1, d, r1, r).

  Local Open Scope bool.
  Definition verify_secret_vote_proof (pk rk pv : A) (i : nat) (proof : VoteProof) : bool :=
    let '(w, a1, b1, a2, b2, d1, d2, r1, r2) := proof in
    let c := hash_sv_data i pk rk a1 b1 a2 b2 in
    (Zpos c =? d1 + d2)%Z &&
    (a1 =? generator^r1 * pk^d1)%broom &&
    (b1 =? rk^r1 * pv^d1)%broom &&
    (a2 =? generator^r2 * pk^d2)%broom &&
    (b2 =? rk^r2 * (pv * inv generator)^d2)%broom.

  Definition make_signup_msg (sk : Z) (v : Z) (i : nat) : Msg :=
    signup (compute_public_key sk) (secret_key_proof sk v i).

  Definition make_commit_msg (pks : list A) (my_index : nat) (sk : Z) (sv : bool) : Msg :=
    let pv := compute_public_vote (reconstructed_key pks my_index) sk sv in
    commit_to_vote (H [encodeA pv]).

  Definition make_vote_msg (pks : list A) (my_index : nat) (sk : Z) (sv : bool) (w r d : Z) : Msg :=
    let rk := reconstructed_key pks my_index in
    submit_vote (compute_public_vote rk sk sv)
                (secret_vote_proof sk rk sv my_index w r d).

  Definition assert_true_init (check : bool) : ContractIniter Setup Error unit :=
    @lift _ (fun T => result T Error) _ _ (if check then Ok tt else Err default_error).
  Definition assert_true (check : bool) : ContractReceiver State Msg Error unit :=
    @lift _ (fun T => result T Error) _ _ (if check then Ok tt else Err default_error).
  Definition assert_false (check : bool) : ContractReceiver State Msg Error unit :=
    @lift _ (fun T => result T Error) _ _ (if check then Err default_error else Ok tt).
  Definition assert_some {A : Type} (check : option A) : ContractReceiver State Msg Error unit :=
    @lift _ (fun T => result T Error) _ _ (if check then Ok tt else Err default_error).
  Definition assert_none {A : Type} (check : option A) : ContractReceiver State Msg Error unit :=
    @lift _ (fun T => result T Error) _ _ (if check then Err default_error else Ok tt).

  (* A necessary aliasing to make extraction work *)
  Definition ContractIniterSetupState := ContractIniter Setup Error State.

  Definition init : ContractIniterSetupState :=
    do owner <- lift caller_addr;
    do setup <- deployment_setup;
    do assert_true_init
      (finish_registration_by setup <? finish_vote_by setup)%nat;
    accept_deployment
      {| owner := owner;
        registered_voters := AddressMap.empty;
        public_keys := [];
        setup := setup;
        tally := None; |}.

  Definition ContractReceiverStateMsgState := ContractReceiver State Msg Error State.

  Definition handle_signup pk prf state caller cur_slot : ContractReceiverStateMsgState :=
    do assert_false (finish_registration_by (setup state) <? cur_slot)%nat;
    do assert_some (AddressMap.find caller (eligible_voters (setup state)));
    do assert_none (AddressMap.find caller (registered_voters state));
    do amt <- lift call_amount;
    do assert_true (amt =? (registration_deposit (setup state)))%Z;
    do assert_true (Z.of_nat (length (public_keys state)) <? order - 2);
    let index := length (public_keys state) in
    do assert_true (verify_secret_key_proof pk index prf);
    let inf := {| voter_index := index;
                  vote_hash := 1%positive;
                  public_vote := zero; |} in
    let new_state := state<|registered_voters ::= AddressMap.add caller inf|>
                          <|public_keys ::= fun l => l ++ [pk]|> in
    accept_call new_state.

  Definition handle_commit_to_vote hash state caller cur_slot : ContractReceiverStateMsgState :=
    do commit_by <- lift (result_of_option (finish_commit_by (setup state)) default_error);
    do assert_false (commit_by <? cur_slot)%nat;
    do inf <- lift (result_of_option (AddressMap.find caller (registered_voters state)) default_error);
    let inf := inf<|vote_hash := hash|> in
    accept_call (state<|registered_voters ::= AddressMap.add caller inf|>).

  Definition handle_submit_vote v proof state caller cur_slot : ContractReceiverStateMsgState :=
    do assert_false (finish_vote_by (setup state) <? cur_slot)%nat;
    do inf <- lift (result_of_option (AddressMap.find caller (registered_voters state)) default_error);
    do @lift _ (fun T => result T Error) _ _ (if finish_commit_by (setup state) then
              if (H [encodeA v] =? vote_hash inf)%positive then Ok tt else Err default_error
            else
              Ok tt);
    do @lift _ (fun T => result T Error) _ _ (if verify_secret_vote_proof
                  (nth (voter_index inf) (public_keys state) 0)
                  (reconstructed_key (public_keys state) (voter_index inf))
                  v
                  (voter_index inf)
                  proof then Ok tt else Err default_error);
    let inf := inf<|public_vote := v|> in
    accept_call (state<|registered_voters ::= AddressMap.add caller inf|>).

  Definition handle_tally_votes state cur_slot : ContractReceiverStateMsgState :=
    do assert_false (cur_slot <? finish_vote_by (setup state))%nat;
    do assert_none (tally state);
    let voters := AddressMap.values (registered_voters state) in
    do assert_false (existsb
                  (fun vi => if elmeqb (public_vote vi) zero then true else false)
                  voters);
    let votes := map public_vote voters in
    do res <- @lift _ (fun T => result T Error) _ _ (result_of_option (bruteforce_tally votes) default_error);
    accept_call (state<|tally := Some res|>).

  Definition receive : ContractReceiverStateMsgState :=
    do state <- my_state;
    do caller <- lift caller_addr;
    do cur_slot <- lift current_slot;
    do msg <- call_msg default_error;
    match msg with
    | signup pk prf => handle_signup pk prf state caller cur_slot
    | commit_to_vote hash => handle_commit_to_vote hash state caller cur_slot
    | submit_vote v proof => handle_submit_vote v proof state caller cur_slot
    | tally_votes => handle_tally_votes state cur_slot
    end.

  Definition boardroom_voting : Contract Setup Msg State Error :=
    build_contract init receive.

  Section Theories.

    Record SecretVoterInfo :=
      build_secret_voter_info {
        svi_index : nat;
        (* Secret key *)
        svi_sk : Z;
        (* Chosen randomness for knowledge of secret key proof *)
        svi_sk_r : Z;
        (* Secret vote *)
        svi_sv : bool;
        (* Chosen random w for vote proof *)
        svi_sv_w : Z;
        (* Chosen random r for vote proof *)
        svi_sv_r : Z;
        (* Chosen random d for vote proof *)
        svi_sv_d : Z;
      }.

    (* begin hide *)
    MetaCoq Run (make_setters SecretVoterInfo).
    (* end hide *)

    (* For correctness we assume that all signups and vote messages were
      created using the make_signup_msg and make_vote_msg functions from
      the contract *)
    Fixpoint MsgAssumption
            (pks : list A)
            (parties : Address -> SecretVoterInfo)
            (calls : list (ContractCallInfo Msg)) : Prop :=
      match calls with
      | call :: calls =>
        let party := parties (Blockchain.call_from call) in
        match Blockchain.call_msg call with
        | Some (signup pk prf as m) => m = make_signup_msg (svi_sk party) (svi_sk_r party)
                                                          (svi_index party)
        | Some (submit_vote _ _ as m) =>
          m = make_vote_msg
                pks
                (svi_index party)
                (svi_sk party)
                (svi_sv party)
                (svi_sv_w party)
                (svi_sv_r party)
                (svi_sv_d party)
        | _ => True
        end /\ MsgAssumption pks parties calls
      | [] => True
      end.

    Definition signups (calls : list (ContractCallInfo Msg)) : list (Address * A) :=
      (* reverse the signups since the calls will have the last one at the head *)
      rev (map_option (fun call => match Blockchain.call_msg call with
                                  | Some (signup pk prf) => Some (Blockchain.call_from call, pk)
                                  | _ => None
                                  end) calls).

    (* The index map and public keys list provided also needs to match the
      order in which parties signed up in the contract. *)
    Definition SignupOrderAssumption
              (pks : list A)
              (parties : Address -> SecretVoterInfo)
              (calls : list (ContractCallInfo Msg)) : Prop :=
      All (fun '((addr, pk), i) => svi_index (parties addr) = i /\ nth_error pks i = Some pk)
          (zip (signups calls) (seq 0 (length (signups calls)))).

    Local Open Scope nat.

    Lemma no_outgoing bstate caddr :
      reachable bstate ->
      env_contracts bstate caddr = Some (boardroom_voting : WeakContract) ->
      outgoing_acts bstate caddr = [].
    Proof.
      intros.
      apply (lift_outgoing_acts_nil boardroom_voting); try easy.
      intros.
      destruct msg as [msg|]; cbn -[Nat.ltb] in *; try congruence.
      destruct msg.
      - destruct (_ <? _); cbn in *; try congruence.
        destruct (AddressMap.find _ _); cbn in *; try congruence.
        destruct (AddressMap.find _ _); cbn in *; try congruence.
        destruct (_ =? _)%Z; cbn in *; try congruence.
        destruct (_ <? _)%Z; cbn in *; try congruence.
        destruct (verify_secret_key_proof _ _ _); cbn in *; congruence.
      - destruct (finish_commit_by _); cbn -[Nat.ltb] in *; try congruence.
        destruct (_ <? _); cbn in *; try congruence.
        destruct (AddressMap.find _ _); cbn in *; congruence.
      - destruct (_ <? _); cbn in *; try congruence.
        destruct (AddressMap.find _ _); cbn in *; try congruence.
        destruct (if finish_commit_by _ then _ else _); cbn in *; try congruence.
        destruct (verify_secret_vote_proof _ _ _ _); cbn in *; congruence.
      - destruct (_ <? _); cbn in *; try congruence.
        destruct (tally _); cbn in *; try congruence.
        destruct (existsb _ _); cbn in *; try congruence.
        destruct (bruteforce_tally _); cbn in *; congruence.
    Qed.

    Lemma Permutation_modify k vold vnew (m : AddrMap VoterInfo) :
      FMap.find k m = Some vold ->
      voter_index vold = voter_index vnew ->
      Permutation (map (fun '(_, v) => voter_index v)
                      (FMap.elements m))
                  (seq 0 (FMap.size m)) ->
      Permutation
        (map (fun '(_, v0) => voter_index v0)
            (FMap.elements (FMap.add k vnew m)))
        (seq 0 (FMap.size m)).
    Proof.
      intros find_some index old_perm.
      rewrite <- old_perm.
      rewrite <- (FMap.add_id _ _ _ find_some) at 2.
      rewrite <- (FMap.add_remove k vold).
      rewrite (FMap.elements_add_existing k vold vnew) by auto.
      rewrite FMap.elements_add by auto.
      cbn.
      now rewrite index.
    Qed.

    Lemma all_signups pks parties calls :
      SignupOrderAssumption pks parties calls ->
      length (signups calls) = length pks ->
      map snd (signups calls) = pks.
    Proof.
      intros order len_signups.
      unfold SignupOrderAssumption in order.
      revert parties pks len_signups order.
      induction (signups calls) as [|[addr pk] xs IH]; intros parties pks len_signups order.
      - destruct pks; cbn in *; congruence.
      - cbn in *.
        destruct pks as [|pk' pks]; cbn in *; try lia.
        destruct order as [[index_eq nth_eq] all].
        f_equal; try congruence.
        apply (IH (fun addr => (parties addr)<|svi_index ::= fun i => i - 1|>));
          [lia|].
        clear -all.
        rewrite <- (map_id xs) in all at 1.
        rewrite <- seq_shift in all.
        rewrite zip_map in all.
        apply All_map in all.
        apply (All_ext_in _ _ _ all).
        intros.
        destruct a, p.
        cbn in *.
        split; [|tauto].
        destruct H1; lia.
    Qed.

    Local Open Scope broom.
    Lemma elmeqb_eq (a a' : A) :
      (a =? a') = true <-> a == a'.
    Proof.
      destruct (elmeqb_spec a a'); [tauto|].
      split; congruence.
    Qed.

    Hint Resolve
        pow_nonzero generator_nonzero int_domain generator_nonzero compute_public_key_unit
        reconstructed_key_unit
      : broom.
    Lemma verify_secret_key_proof_spec sk v i :
      verify_secret_key_proof (compute_public_key sk) i (secret_key_proof sk v i) = true.
    Proof with auto with broom.
      cbn.
      apply elmeqb_eq.
      apply log_both...
      rewrite log_pow...
      rewrite log_mul...
      unfold "exp=".
      assert (order - 1 <> 0)%Z by (pose proof order_ge_2; lia).
      rewrite Z.add_mod...
      rewrite !log_pow...
      rewrite log_generator.
      rewrite !Z.mul_1_r.
      unfold compute_public_key.
      rewrite <- Z.mul_mod_idemp_r...
      rewrite log_pow...
      rewrite log_generator.
      rewrite Z.mul_1_r.
      rewrite Z.mul_mod_idemp_r...
      rewrite <- Z.add_mod...
      f_equal.
      lia.
    Qed.

    Lemma verify_secret_vote_proof_spec sk pks sv i w r d :
      All (fun pk => pk !== 0) pks ->
      verify_secret_vote_proof
        (compute_public_key sk)
        (reconstructed_key pks i)
        (compute_public_vote (reconstructed_key pks i) sk sv)
        i
        (secret_vote_proof sk (reconstructed_key pks i) sv i w r d) = true.
    Proof.
      intros all_units.
      set (rk := reconstructed_key pks i).
      unfold verify_secret_vote_proof, secret_vote_proof.
      cbn.
      destruct sv.
      - set (h := hash_sv_data _ _ _ _ _ _ _).
        rewrite Zplus_minus.
        rewrite Pos.eqb_refl, !elmeqb_refl.
        cbn.
        unfold compute_public_key.
        rewrite pow_pow by (auto with broom).
        rewrite <- pow_plus by (auto with broom).
        rewrite Z.sub_add.
        rewrite elmeqb_refl.
        cbn.
        unfold compute_public_vote.
        rewrite <- (mul_assoc (rk^sk)).
        rewrite (mul_comm generator).
        rewrite inv_inv_l by (auto with broom).
        rewrite (mul_comm (rk^sk)), mul_1_l.
        rewrite pow_pow by (subst rk; auto with broom).
        rewrite <- pow_plus by (subst rk; auto with broom).
        rewrite Z.sub_add.
        now rewrite elmeqb_refl.
      - set (h := hash_sv_data _ _ _ _ _ _ _).
        rewrite Z.sub_add.
        rewrite Pos.eqb_refl, !elmeqb_refl.
        cbn.
        unfold compute_public_key.
        rewrite pow_pow by (auto with broom).
        rewrite <- pow_plus by (auto with broom).
        rewrite Z.sub_add.
        rewrite elmeqb_refl.
        cbn.
        unfold compute_public_vote.
        rewrite (mul_comm (rk^sk)), mul_1_l.
        rewrite pow_pow by (subst rk; auto with broom).
        rewrite <- pow_plus by (subst rk; auto with broom).
        rewrite Z.sub_add.
        now rewrite elmeqb_refl.
    Qed.

    Local Set Keyed Unification.

    Definition has_tallied (calls : list (ContractCallInfo Msg)) : bool :=
      existsb (fun c => match Blockchain.call_msg c with
                        | Some tally_votes => true
                        | _ => false
                        end) calls.

    Theorem boardroom_voting_correct_strong
            (bstate : ChainState)
            (caddr : Address)
            (trace : ChainTrace empty_state bstate)
            (parties : Address -> SecretVoterInfo)
            (pks : list A) :
        env_contracts bstate caddr = Some (boardroom_voting : WeakContract) ->
        exists (cstate : State)
              (depinfo : DeploymentInfo Setup)
              (inc_calls : list (ContractCallInfo Msg)),
          deployment_info Setup trace caddr = Some depinfo /\
          contract_state bstate caddr = Some cstate /\
          incoming_calls Msg trace caddr = Some inc_calls /\

          finish_registration_by (setup cstate) < finish_vote_by (setup cstate) /\

          (Blockchain.current_slot bstate < finish_vote_by (setup cstate) ->
          has_tallied inc_calls = false) /\

          length (public_keys cstate) = FMap.size (registered_voters cstate) /\
          public_keys cstate = map snd (signups inc_calls) /\

          (Z.of_nat (length (public_keys cstate)) < order - 1)%Z /\

          (MsgAssumption pks parties inc_calls ->
          SignupOrderAssumption pks parties inc_calls ->
          (finish_registration_by (setup cstate) < Blockchain.current_slot bstate ->
            length pks = length (signups inc_calls)) ->

          Permutation (map (fun '(_, v) => voter_index v)
                            (FMap.elements (registered_voters cstate)))
                      (seq 0 (length (public_keys cstate))) /\

          Permutation (FMap.keys (registered_voters cstate))
                      (map fst (signups inc_calls)) /\

          (forall addr inf,
              FMap.find addr (registered_voters cstate) = Some inf ->
              voter_index inf < length (public_keys cstate) /\
              voter_index inf = svi_index (parties addr) /\
              nth_error (public_keys cstate) (voter_index inf) =
              Some (compute_public_key (svi_sk (parties addr))) /\
              (public_vote inf == zero \/
                public_vote inf = compute_public_vote
                                    (reconstructed_key pks (voter_index inf))
                                    (svi_sk (parties addr))
                                    (svi_sv (parties addr)))) /\
          ((has_tallied inc_calls = false ->
            tally cstate = None) /\
            (has_tallied inc_calls = true ->
            tally cstate = Some (sumnat (fun party => if svi_sv (parties party) then 1 else 0)%nat
                                        (map fst (signups inc_calls)))))).
    Proof.
      contract_induction; intros.
      - [AddBlockFacts]: exact (fun _ old_slot _ _ new_slot _ => old_slot < new_slot).
        subst AddBlockFacts.
        cbn in facts.
        destruct_and_split; try tauto.
        + eauto with lia.
        + intros; eauto with lia.
      - cbn -[Nat.ltb] in *.
        destruct (_ <? _) eqn:ltb; [|congruence].
        apply Nat.ltb_lt in ltb.
        inversion_clear init_some.
        cbn.
        split; auto.
        split; auto.
        split; [symmetry; apply FMap.size_empty|].
        split; [auto|].
        pose proof order_ge_2.
        split; [lia|].
        intros _ _ _.
        unfold FMap.keys.
        unfold AddressMap.empty in *.
        rewrite @FMap.elements_empty.
        split; [auto|].
        split; [auto|].
        split; [|easy].
        intros ? ? find.
        now rewrite @FMap.find_empty in find.
      - auto.
      - cbn -[Nat.ltb] in receive_some.
        destruct msg as [msg|]; cbn -[Nat.ltb] in *; [|congruence].
        destruct msg.
        unfold AddressMap.add in *. unfold AddressMap.find in *.
        + (* signup *)
          destruct (_ <? _)%nat eqn:intime in receive_some; cbn -[Nat.ltb] in *; [congruence|].
          apply Nat.ltb_ge in intime.
          destruct (FMap.find _ _) in receive_some; cbn in *; [|congruence].
          destruct (FMap.find _ _) eqn:new in receive_some; cbn in *; [congruence|].
          destruct (_ =? _)%Z in receive_some; cbn in *; [|congruence].
          destruct (_ <? _)%Z eqn:lt in receive_some; cbn in *; [|congruence].
          destruct (verify_secret_key_proof _ _ _) eqn:verify_zkp in receive_some;
            cbn in *; [|congruence].
          inversion_clear receive_some.
          cbn.
          split; [lia|].
          split; [tauto|].
          split.
          { rewrite app_length, FMap.size_add_new by auto; cbn; lia. }
          apply Z.ltb_lt in lt.
          rewrite app_length in *.
          cbn.
          fold (has_tallied prev_inc_calls).
          fold (signups prev_inc_calls).
          rewrite app_length, map_app; cbn.
          split; [destruct_and_split; congruence|].
          split; [lia|].
          intros [signup_assum msg_assum] order_assum num_signups_assum.
          destruct IH as (reg_lt & cur_lt & _ & pks_signups & _ & IH).
          unshelve epose proof (IH _ _ _) as IH.
          * auto.
          * rewrite seq_app in order_assum.
            rewrite zip_app in order_assum by (now rewrite seq_length).
            apply All_app in order_assum.
            tauto.
          * intros.
            lia.
          * split.
            { destruct IH as (perm & _).
              cbn.
              rewrite FMap.elements_add by auto.
              cbn.
              rewrite seq_app.
              cbn.
              perm_simplify. }
            split.
            { destruct IH as (_ & perm & _).
              rewrite map_app.
              unfold FMap.keys.
              rewrite FMap.elements_add by auto.
              cbn.
              now perm_simplify. }

            split; cycle 1.
            {
              split; [easy|].
              intros tallied.
              specialize (cur_lt ltac:(lia)).
              congruence.
            }
            intros addr inf find_add.
            destruct (address_eqb_spec addr (ctx_from ctx)) as [->|].
            -- rewrite (FMap.find_add (ctx_from ctx)) in find_add.
              inversion_clear find_add.
              cbn.
              unfold make_signup_msg in signup_assum.
              rewrite nth_error_snoc.
              rewrite seq_app, zip_app in order_assum by (now rewrite seq_length).
              apply All_app in order_assum.
              cbn in order_assum.
              destruct order_assum as [_ []].
              split; [lia|].
              rewrite pks_signups, map_length.
              split; [symmetry; tauto|].
              split; [congruence|].
              left; easy.
            -- rewrite FMap.find_add_ne in find_add by auto.
              destruct IH as (_ & _ & IH & _).
              specialize (IH _ _ find_add).
              split; [lia|].
              now rewrite nth_error_app1 by lia.
        + (* commit_to_vote *)
          destruct (finish_commit_by _); cbn -[Nat.ltb] in *; [|congruence].
          destruct (_ <? _); cbn in *; [congruence|].
          unfold AddressMap.find in *.
          destruct (FMap.find _ _) eqn:found; cbn in *; [|congruence].
          inversion_clear receive_some; cbn.
          split; [lia|].
          split; [tauto|].
          split.
          unfold AddressMap.add.
          { rewrite FMap.size_add_existing by congruence; tauto. }
          split; [tauto|].
          split; [tauto|].
          intros [_ msg_assum] order_assum num_signups_assum.
          destruct IH as (_ & _ & len_pks & _ & _ & IH).
          specialize (IH msg_assum order_assum num_signups_assum).
          setoid_rewrite (FMap.keys_already _ _ _ _ found).
          split.
          {
            destruct IH as (perm & _).
            rewrite len_pks in *.
            apply Permutation_modify with (vold := v); auto.
          }
          split; [tauto|].
          split; [|tauto].
          intros addr inf find_add.
          unfold AddressMap.add in *.
          destruct IH as (_ & _ & IH & _).
          destruct (address_eqb_spec addr (ctx_from ctx)) as [->|].
          * rewrite FMap.find_add in find_add.
            inversion_clear find_add; cbn.
            auto.
          * rewrite FMap.find_add_ne in find_add by auto.
            auto.
        + (* submit_vote *)
          destruct (_ <? _); cbn -[Nat.ltb] in *; [congruence|].
          unfold AddressMap.find in *.
          destruct (FMap.find _ _) eqn:found; cbn in *; [|congruence].
          destruct (if finish_commit_by _ then _ else _); cbn in *; [|congruence].
          destruct (verify_secret_vote_proof _ _ _ _); cbn in *; [|congruence].
          inversion_clear receive_some; cbn.
          split; [lia|].
          split; [tauto|].
          rewrite FMap.size_add_existing by congruence.
          split; [tauto|].
          split; [tauto|].
          split; [tauto|].
          intros [vote_assum msg_assum] order_assum num_signups_assum.
          destruct IH as (_ & _ & len_pks & _ & _ & IH).
          specialize (IH msg_assum order_assum num_signups_assum).
          setoid_rewrite (FMap.keys_already _ _ _ _ found).
          split.
          { destruct IH as (perm & _).
            rewrite len_pks in *.
            apply Permutation_modify with (vold := v0); auto. }
          split; [tauto|].
          split; [|tauto].
          intros addr inf find_add.
          destruct IH as (_ & _ & IH & _).
          destruct (address_eqb_spec addr (ctx_from ctx)) as [->|].
          * rewrite FMap.find_add in find_add.
            inversion_clear find_add; cbn.
            specialize (IH _ _ found).
            repeat split; try tauto.
            right.
            unfold make_vote_msg in *.
            inversion vote_assum.
            destruct_hyps.
            replace (svi_index (parties (ctx_from ctx))) with (voter_index v0) by congruence.
            easy.
          * rewrite FMap.find_add_ne in find_add by auto.
            auto.
        + (* tally_votes *)
          destruct (_ <? _) eqn:intime; cbn in *; [congruence|].
          destruct (tally prev_state); cbn in *; [congruence|].
          destruct (existsb _ _) eqn:all_voted; cbn in *; [congruence|].
          destruct (bruteforce_tally _) eqn:bruteforce; cbn -[Nat.ltb] in *; [|congruence].
          inversion_clear receive_some; cbn.
          apply Nat.ltb_ge in intime.
          split; [lia|].
          split; [intros; lia|].
          split; [tauto|].
          split; [tauto|].
          split; [tauto|].
          intros (_ & msg_assum) order_assum num_signups_assum.
          split; [tauto|].
          split; [tauto|].
          split; [tauto|].
          split; [easy|].
          intros _.
          apply f_equal.
          destruct IH as (finish_before_vote & _ & len_pks & pks_signups & party_count & IH).
          specialize (IH msg_assum order_assum num_signups_assum).
          destruct IH as (perm & perm' & addrs & _).
          unfold AddressMap.values in *.
          unfold FMap.values in bruteforce.
          rewrite map_map in bruteforce.
          rewrite (map_ext_in _ (fun '(_, v) => public_vote v)) in bruteforce
            by (now intros []).
          rewrite (bruteforce_tally_correct
                        (FMap.elements (registered_voters prev_state))
                        (fun '(_, v) => voter_index v)
                        (fun '(addr, _) => svi_sk (parties addr))
                        (public_keys prev_state)
                        (fun kvp => svi_sv (parties (fst kvp)))
                        (fun '(_, v) => public_vote v)) in bruteforce.
          * inversion bruteforce.
            rewrite <- (sumnat_map fst (fun a => if svi_sv (parties a) then 1 else 0))%nat.
            now setoid_rewrite perm'.
          * now rewrite FMap.length_elements, <- len_pks.
          * now rewrite FMap.length_elements, <- len_pks.
          * now rewrite FMap.length_elements, <- len_pks.
          * intros [k v] kvpin.
            apply FMap.In_elements in kvpin.
            specialize (addrs _ _ kvpin).
            tauto.
          * intros [k v] kvpin.
            rewrite existsb_forallb in all_voted.
            apply Bool.negb_false_iff in all_voted.
            rewrite forallb_forall in all_voted.
            unshelve epose proof (all_voted v _) as all_voted.
            {
              apply in_map_iff.
              exists (k, v).
              tauto.
            }
            apply Bool.negb_true_iff in all_voted.
            destruct (elmeqb_spec (public_vote v) zero); [congruence|].
            apply FMap.In_elements in kvpin.
            specialize (addrs _ _ kvpin).
            cbn.
            destruct addrs as (_ & _ & _ & []); [easy|].
            fold (signups prev_inc_calls) (SignupOrderAssumption pks parties prev_inc_calls) in *.
            rewrite pks_signups.
            specialize (num_signups_assum ltac:(lia)).
            now rewrite (all_signups pks parties) by auto.
      - [CallFacts]: exact (fun _ ctx _ _ _ => ctx_from ctx <> ctx_contract_address ctx).
        subst CallFacts; cbn in *; congruence.
      - auto.
      - [DeployFacts]: exact (fun _ _ => True).
        unset_all; subst; cbn in *.
        destruct_chain_step; auto.
        + destruct valid_header; auto.
        + destruct_action_eval; auto.
          intros.
          apply trace_reachable in from_reachable.
          pose proof (no_outgoing _ _ from_reachable H0).
          unfold outgoing_acts in H2.
          rewrite queue_prev in H2.
          cbn in H2.
          destruct (address_eqb_spec (act_from act) to_addr); cbn in *; try congruence.
          subst.
          cbn in *. congruence.
    Qed.

    Theorem boardroom_voting_correct
            (bstate : ChainState)
            (caddr : Address)
            (trace : ChainTrace empty_state bstate)
            (* list of all public keys, in the order of signups *)
            (pks : list A)
            (* function mapping a party to information about him *)
            (parties : Address -> SecretVoterInfo) :
        env_contracts bstate caddr = Some (boardroom_voting : WeakContract) ->
        exists (cstate : State)
              (depinfo : DeploymentInfo Setup)
              (inc_calls : list (ContractCallInfo Msg)),
          deployment_info Setup trace caddr = Some depinfo /\
          contract_state bstate caddr = Some cstate /\
          incoming_calls Msg trace caddr = Some inc_calls /\

          (* assuming that the message sent were created with the
              functions provided by this smart contract *)
          MsgAssumption pks parties inc_calls ->

          (* ..and that people signed up in the order given by 'index'
              and 'pks' *)
          SignupOrderAssumption pks parties inc_calls ->

          (* ..and that the correct number of people register *)
          (finish_registration_by (setup cstate) < Blockchain.current_slot bstate ->
          length pks = length (signups inc_calls)) ->

          (* then if we have not tallied yet, the tally is none *)
          ((has_tallied inc_calls = false -> tally cstate = None) /\
          (* or if we have tallied yet, the tally is correct *)
          (has_tallied inc_calls = true ->
            tally cstate = Some (sumnat (fun party => if svi_sv (parties party) then 1 else 0)%nat
                                        (map fst (signups inc_calls))))).
    Proof.
      intros deployed.
      destruct (boardroom_voting_correct_strong bstate caddr trace parties pks deployed)
        as (cstate & depinfo & inc_calls & P).
      exists cstate, depinfo, inc_calls.
      tauto.
    Qed.

  End Theories.

End BoardroomVoting.