#lang ivy1.7
# ---
# layout: page
# title: Proof of Classic Safety
# ---

include tendermint
include abstract_tendermint

# Here we prove the first accountability property: if two well-behaved nodes
# disagree, then there are two quorums Q1 and Q2 such that all members of the
# intersection of Q1 and Q2 have violated the accountability properties.

# The proof is done in two steps: first we prove the abstract specification
# satisfies the property, and then we show by refinement that this property
# also holds in the concrete specification.

# To see what is checked in the refinement proof, use `ivy_show isolate=accountable_safety_1 accountable_safety_1.ivy`
# To see what is checked in the abstract correctness proof, use `ivy_show isolate=abstract_accountable_safety_1 accountable_safety_1.ivy`
# To check the whole proof, use `ivy_check accountable_safety_1.ivy`.


# Proof of the accountability property in the abstract specification
# ==================================================================

# We prove with tactics (see `lemma_1` and `lemma_2`) that, if some basic
# invariants hold (see `invs` below), then the accountability property holds.

isolate abstract_accountable_safety = {

    instantiate abstract_tendermint

# The main property
# -----------------

# If there is disagreement, then there is evidence that a third of the nodes
# have violated the protocol:
    invariant [accountability] agreement | exists Q1,Q2 . nset.is_quorum(Q1) & nset.is_quorum(Q2) & (forall N . nset.member(N,Q1) & nset.member(N,Q2) -> observed_equivocation(N) | observed_unlawful_prevote(N))
    proof {
        apply lemma_2.thm # this reduces to goal to three subgoals: p1, p2, and p3 (see their definition below)
        proof [p1] {
            assume invs.inv1
        }
        proof [p2] {
            assume invs.inv2
        }
        proof [p3] {
            assume invs.inv3
        }
    }

# The invariants
# --------------

    isolate invs = {

        # well-behaved nodes observe their own actions faithfully:
        invariant [inv1] well_behaved(N) -> (observed_precommitted(N,R,V) = precommitted(N,R,V))
        # if a value is precommitted by a well-behaved node, then a quorum is observed to prevote it:
        invariant [inv2] (exists N . well_behaved(N) & precommitted(N,R,V)) & V ~= value.nil  -> exists Q . nset.is_quorum(Q) & forall N2 . nset.member(N2,Q) -> observed_prevoted(N2,R,V)
        # if a value is decided by a well-behaved node, then a quorum is observed to precommit it:
        invariant [inv3] (exists N . well_behaved(N) & decided(N,R,V)) -> 0 <= R & V ~= value.nil & exists Q . nset.is_quorum(Q) & forall N2 . nset.member(N2,Q) -> observed_precommitted(N2,R,V)
        private {
            invariant (precommitted(N,R,V) | prevoted(N,R,V)) -> 0 <= R
            invariant R < 0 -> left_round(N,R)
        }

    } with this, nset, round, accountable_bft.max_2f_byzantine

# The theorems proved with tactics
# --------------------------------

# Using complete induction on rounds, we prove that, assuming that the
# invariants inv1, inv2, and inv3 hold, the accountability property holds.

# For technical reasons, we separate the proof in two steps
    isolate lemma_1 = {

# complete induction is not built-in, so we introduce it with an axiom. Note that this only holds for a type where 0 is the smallest element
        axiom [complete_induction] {
            relation p(X:round)
            { # base case
                property p(0)
            }
            { # inductive step: show that if the property is true for all X lower or equal to x and y=x+1, then the property is true of y
                individual a:round
                individual b:round
                property (forall X. 0 <= X & X <= a -> p(X)) & round.succ(a,b) -> p(b)
            }
            #--------------------------
            property forall X . 0 <= X -> p(X)
        }

# the main lemma: if inv1 and inv2 below hold and a quorum is observed to
# precommit V1 at R1 and another quorum is observed to precommit V2~=V1 at
# R2>=R1, then the intersection of two quorums (i.e. f+1 nodes) is observed to
# violate the protocol
        theorem [thm] {
            property [p1] forall N,R,V . well_behaved(N) -> (observed_precommitted(N,R,V) = precommitted(N,R,V))
            property [p2] forall R,V . (exists N . well_behaved(N) & precommitted(N,R,V)) -> V = value.nil | exists Q . nset.is_quorum(Q) & forall N2 . nset.member(N2,Q) -> observed_prevoted(N2,R,V)
            #-----------------------------------------------------------------------------------------------------------------------
            property forall R2. 0 <= R2  -> ((exists V2,Q1,R1,V1,Q1 . V1 ~= value.nil & V2 ~= value.nil & V1 ~= V2 & 0 <= R1 & R1 <= R2 & nset.is_quorum(Q1) & (forall N . nset.member(N,Q1) -> observed_precommitted(N,R1,V1)) & (exists Q2 . nset.is_quorum(Q2) & forall N . nset.member(N,Q2) -> observed_prevoted(N,R2,V2))) -> exists Q1,Q2 . nset.is_quorum(Q1) & nset.is_quorum(Q2) & forall N . nset.member(N,Q1) & nset.member(N,Q2) -> observed_equivocation(N) | observed_unlawful_prevote(N))
        }
        proof {
            apply complete_induction # the two subgoals (base case and inductive case) are then discharged automatically
        }
    } with this, round, nset, accountable_bft.max_2f_byzantine, defs.observed_equivocation_def, defs.observed_unlawful_prevote_def

# Now we put lemma_1 in a form that matches exactly the accountability property
# we want to prove. This is a bit cumbersome and could probably be improved.
    isolate lemma_2 = {

        theorem [thm] {
            property [p1] forall N,R,V . well_behaved(N) -> (observed_precommitted(N,R,V) = precommitted(N,R,V))
            property [p2] forall R,V . (exists N . well_behaved(N) & precommitted(N,R,V)) & V ~= value.nil  -> exists Q . nset.is_quorum(Q) & forall N2 . nset.member(N2,Q) -> observed_prevoted(N2,R,V)
            property [p3] forall R,V. (exists N . well_behaved(N) & decided(N,R,V)) -> 0 <= R & V ~= value.nil & exists Q . nset.is_quorum(Q) & forall N2 . nset.member(N2,Q) -> observed_precommitted(N2,R,V)
            #-------------------------------------------------------------------------------------------------------------------------------------------
            property agreement | exists Q1,Q2 . nset.is_quorum(Q1) & nset.is_quorum(Q2) & forall N . nset.member(N,Q1) & nset.member(N,Q2) -> observed_equivocation(N) | observed_unlawful_prevote(N)
        }
        proof {
            assume lemma_1.thm
        }

    } with this, round,  defs.agreement_def, lemma_1, nset, accountable_bft.max_2f_byzantine

} with round

# The final proof
# ===============

isolate accountable_safety_1 = {

# First we instantiate the concrete protocol:
    instantiate tendermint(abstract_accountable_safety)

# We then define what we mean by agreement
    relation agreement
    definition [agreement_def] agreement = forall N1,N2. well_behaved(N1) & well_behaved(N2) & server.decision(N1) ~= value.nil & server.decision(N2) ~= value.nil -> server.decision(N1) = server.decision(N2)

    invariant abstract_accountable_safety.agreement -> agreement

    invariant [accountability] agreement | exists Q1,Q2 . nset.is_quorum(Q1) & nset.is_quorum(Q2) & forall N . nset.member(N,Q1) & nset.member(N,Q2) -> abstract_accountable_safety.observed_equivocation(N) | abstract_accountable_safety.observed_unlawful_prevote(N)

} with value, round, proposers, shim, abstract_accountable_safety, abstract_accountable_safety.defs.agreement_def, accountable_safety_1.agreement_def