    --
    -- Copyright (c) 2014 Joachim Breitner
    --

    module CallArity
        ( callArityAnalProgram
        , callArityRHS -- for testing
        ) where

    import VarSet
    import VarEnv
    import DynFlags ( DynFlags )

    import BasicTypes
    import CoreSyn
    import Id
    import CoreArity ( typeArity )
    import CoreUtils ( exprIsHNF )
    --import Outputable
    import UnVarGraph
    import Demand

    import Control.Arrow ( first, second )

    {-
    %************************************************************************
    %*                                                                      *
                  Call Arity Analyis
    %*                                                                      *
    %************************************************************************

    Note [Call Arity: The goal]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~

    The goal of this analysis is to find out if we can eta-expand a local function,
    based on how it is being called. The motivating example is this code,
    which comes up when we implement foldl using foldr, and do list fusion:

        let go = \x -> let d = case ... of
                                  False -> go (x+1)
                                  True  -> id
                       in \z -> d (x + z)
        in go 1 0

    If we do not eta-expand `go` to have arity 2, we are going to allocate a lot of
    partial function applications, which would be bad.

    The function `go` has a type of arity two, but only one lambda is manifest.
    Furthermore, an analysis that only looks at the RHS of go cannot be sufficient
    to eta-expand go: If `go` is ever called with one argument (and the result used
    multiple times), we would be doing the work in `...` multiple times.

    So `callArityAnalProgram` looks at the whole let expression to figure out if
    all calls are nice, i.e. have a high enough arity. It then stores the result in
    the `calledArity` field of the `IdInfo` of `go`, which the next simplifier
    phase will eta-expand.

    The specification of the `calledArity` field is:

        No work will be lost if you eta-expand me to the arity in `calledArity`.

    What we want to know for a variable
    -----------------------------------

    For every let-bound variable we'd like to know:
      1. A lower bound on the arity of all calls to the variable, and
      2. whether the variable is being called at most once or possible multiple
         times.

    It is always ok to lower the arity, or pretend that there are multiple calls.
    In particular, "Minimum arity 0 and possible called multiple times" is always
    correct.

    What we want to know from an expression
    ---------------------------------------

    In order to obtain that information for variables, we analyize expression and
    obtain bits of information:

     I.  The arity analysis:
         For every variable, whether it is absent, or called,
         and if called, which what arity.

     II. The Co-Called analysis:
         For every two variables, whether there is a possibility that both are being
         called.
         We obtain as a special case: For every variables, whether there is a
         possibility that it is being called twice.

    For efficiency reasons, we gather this information only for a set of
    *interesting variables*, to avoid spending time on, e.g., variables from pattern matches.

    The two analysis are not completely independent, as a higher arity can improve
    the information about what variables are being called once or multiple times.

    Note [Analysis I: The arity analyis]
    ------------------------------------

    The arity analysis is quite straight forward: The information about an
    expression is an
        VarEnv Arity
    where absent variables are bound to Nothing and otherwise to a lower bound to
    their arity.

    When we analyize an expression, we analyize it with a given context arity.
    Lambdas decrease and applications increase the incoming arity. Analysizing a
    variable will put that arity in the environment. In lets or cases all the
    results from the various subexpressions are lubed, which takes the point-wise
    minimum (considering Nothing an infinity).

    Note [Analysis II: The Co-Called analysis]
    ------------------------------------------

    The second part is more sophisticated. For reasons explained below, it is not
    sufficient to simply know how often an expression evalutes a variable. Instead
    we need to know which variables are possibly called together.

    The data structure here is an undirected graph of variables, which is provided
    by the abstract
        UnVarGraph

    It is safe to return a larger graph, i.e. one with more edges. The worst case
    (i.e. the least useful and always correct result) is the complete graph on all
    free variables, which means that anything can be called together with anything
    (including itself).

    Notation for the following:
    C(e)  is the co-called result for e.
    G₁∪G₂ is the union of two graphs
    fv    is the set of free variables (conveniently the domain of the arity analysis result)
    S₁×S₂ is the complete bipartite graph { {a,b} | a ∈ S₁, b ∈ S₂ }
    S²    is the complete graph on the set of variables S, S² = S×S
    C'(e) is a variant for bound expression:
          If e is called at most once, or it is and stays a thunk (after the analysis),
          it is simply C(e). Otherwise, the expression can be called multiple times
          and we return (fv e)²

    The interesting cases of the analysis:
     * Var v:
       No other variables are being called.
       Return {} (the empty graph)
     * Lambda v e, under arity 0:
       This means that e can be evaluated many times and we cannot get
       any useful co-call information.
       Return (fv e)²
     * Case alternatives alt₁,alt₂,...:
       Only one can be execuded, so
       Return (alt₁ ∪ alt₂ ∪...)
     * App e₁ e₂ (and analogously Case scrut alts):
       We get the results from both sides. Additionally, anything called by e₁ can
       possibly called with anything from e₂.
       Return: C(e₁) ∪ C(e₂) ∪ (fv e₁) × (fv e₂)
     * Let v = rhs in body:
       In addition to the results from the subexpressions, add all co-calls from
       everything that the body calls together with v to everthing that is called
       by v.
       Return: C'(rhs) ∪ C(body) ∪ (fv rhs) × {v'| {v,v'} ∈ C(body)}
     * Letrec v₁ = rhs₁ ... vₙ = rhsₙ in body
       Tricky.
       We assume that it is really mutually recursive, i.e. that every variable
       calls one of the others, and that this is strongly connected (otherwise we
       return an over-approximation, so that's ok), see note [Recursion and fixpointing].

       Let V = {v₁,...vₙ}.
       Assume that the vs have been analysed with an incoming demand and
       cardinality consistent with the final result (this is the fixed-pointing).
       Again we can use the results from all subexpressions.
       In addition, for every variable vᵢ, we need to find out what it is called
       with (call this set Sᵢ). There are two cases:
        * If vᵢ is a function, we need to go through all right-hand-sides and bodies,
          and collect every variable that is called together with any variable from V:
          Sᵢ = {v' | j ∈ {1,...,n},      {v',vⱼ} ∈ C'(rhs₁) ∪ ... ∪ C'(rhsₙ) ∪ C(body) }
        * If vᵢ is a thunk, then its rhs is evaluated only once, so we need to
          exclude it from this set:
          Sᵢ = {v' | j ∈ {1,...,n}, j≠i, {v',vⱼ} ∈ C'(rhs₁) ∪ ... ∪ C'(rhsₙ) ∪ C(body) }
       Finally, combine all this:
       Return: C(body) ∪
               C'(rhs₁) ∪ ... ∪ C'(rhsₙ) ∪
               (fv rhs₁) × S₁) ∪ ... ∪ (fv rhsₙ) × Sₙ)

    Using the result: Eta-Expansion
    -------------------------------

    We use the result of these two analyses to decide whether we can eta-expand the
    rhs of a let-bound variable.

    If the variable is already a function (exprIsHNF), and all calls to the
    variables have a higher arity than the current manifest arity (i.e. the number
    of lambdas), expand.

    If the variable is a thunk we must be careful: Eta-Expansion will prevent
    sharing of work, so this is only safe if there is at most one call to the
    function. Therefore, we check whether {v,v} ∈ G.

        Example:

            let n = case .. of .. -- A thunk!
            in n 0 + n 1

        vs.

            let n = case .. of ..
            in case .. of T -> n 0
                          F -> n 1

        We are only allowed to eta-expand `n` if it is going to be called at most
        once in the body of the outer let. So we need to know, for each variable
        individually, that it is going to be called at most once.

    Why the co-call graph?
    ----------------------

    Why is it not sufficient to simply remember which variables are called once and
    which are called multiple times? It would be in the previous example, but consider

            let n = case .. of ..
            in case .. of
                True -> let go = \y -> case .. of
                                         True -> go (y + n 1)
                                         False > n
                        in go 1
                False -> n

    vs.

            let n = case .. of ..
            in case .. of
                True -> let go = \y -> case .. of
                                         True -> go (y+1)
                                         False > n
                        in go 1
                False -> n

    In both cases, the body and the rhs of the inner let call n at most once.
    But only in the second case that holds for the whole expression! The
    crucial difference is that in the first case, the rhs of `go` can call
    *both* `go` and `n`, and hence can call `n` multiple times as it recurses,
    while in the second case find out that `go` and `n` are not called together.

    Why co-call information for functions?
    --------------------------------------

    Although for eta-expansion we need the information only for thunks, we still
    need to know whether functions are being called once or multiple times, and
    together with what other functions.

        Example:

            let n = case .. of ..
                f x = n (x+1)
            in f 1 + f 2

        vs.

            let n = case .. of ..
                f x = n (x+1)
            in case .. of T -> f 0
                          F -> f 1

        Here, the body of f calls n exactly once, but f itself is being called
        multiple times, so eta-expansion is not allowed.

    Note [Analysis type signature]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    The work-hourse of the analysis is the function `callArityAnal`, with the
    following type:

        type CallArityRes = (UnVarGraph, VarEnv Arity)
        callArityAnal ::
            Arity ->  -- The arity this expression is called with
            VarSet -> -- The set of interesting variables
            CoreExpr ->  -- The expression to analyse
            (CallArityRes, CoreExpr)

    and the following specification:

      ((coCalls, callArityEnv), expr') = callArityEnv arity interestingIds expr

                                <=>

      Assume the expression `expr` is being passed `arity` arguments. Then it holds that
        * The domain of `callArityEnv` is a subset of `interestingIds`.
        * Any variable from `interestingIds` that is not mentioned in the `callArityEnv`
          is absent, i.e. not called at all.
        * Every call from `expr` to a variable bound to n in `callArityEnv` has at
          least n value arguments.
        * For two interesting variables `v1` and `v2`, they are not adjacent in `coCalls`,
          then in no execution of `expr` both are being called.
      Furthermore, expr' is expr with the callArity field of the `IdInfo` updated.

    Note [Which variables are interesting]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    The analysis would quickly become prohibitive expensive if we would analyse all
    variables; for most variables we simply do not care about how often they are
    called, i.e. variables bound in a pattern match. So interesting are variables that are
     * top-level or let bound
     * and possibly functions (typeArity > 0)

    Note [Taking boring variables into account]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    If we decide that the variable bound in `let x = e1 in e2` is not interesting,
    the analysis of `e2` will not report anything about `x`. To ensure that
    `callArityBind` does still do the right thing we have to take that into account
    everytime we would be lookup up `x` in the analysis result of `e2`.
      * Instead of calling lookupCallArityRes, we return (0, True), indicating
        that this variable might be called many times with no variables.
      * Instead of checking `calledWith x`, we assume that everything can be called
        with it.
      * In the recursive case, when calclulating the `cross_calls`, if there is
        any boring variable in the recursive group, we ignore all co-call-results
        and directly go to a very conservative assumption.

    The last point has the nice side effect that the relatively expensive
    integration of co-call results in a recursive groups is often skipped. This
    helped to avoid the compile time blowup in some real-world code with large
    recursive groups (#10293).

    Note [Recursion and fixpointing]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    For a mutually recursive let, we begin by
     1. analysing the body, using the same incoming arity as for the whole expression.
     2. Then we iterate, memoizing for each of the bound variables the last
        analysis call, i.e. incoming arity, whether it is called once, and the CallArityRes.
     3. We combine the analysis result from the body and the memoized results for
        the arguments (if already present).
     4. For each variable, we find out the incoming arity and whether it is called
        once, based on the the current analysis result. If this differs from the
        memoized results, we re-analyse the rhs and update the memoized table.
     5. If nothing had to be reanalized, we are done.
        Otherwise, repeat from step 3.

    Note [Thunks in recursive groups]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    We never eta-expand a thunk in a recursive group, on the grounds that if it is
    part of a recursive group, then it will be called multipe times.

    This is not necessarily true, e.g.  it would be safe to eta-expand t2 (but not
    t1) in the follwing code:

      let go x = t1
          t1 = if ... then t2 else ...
          t2 = if ... then go 1 else ...
      in go 0

    Detecting this would reqiure finding out what variables are only ever called
    from thunks. While this is certainly possible, we yet have to see this to be
    relevant in the wild.

    Note [Analysing top-level binds]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    We can eta-expand top-level-binds if they are not exported, as we see all calls
    to them. The plan is as follows: Treat the top-level binds as nested lets around
    a body representing “all external calls”, which returns a pessimistic
    CallArityRes (the co-call graph is the complete graph, all arityies 0).

    Note [Trimming arity]
    ~~~~~~~~~~~~~~~~~~~~~~~~~~~~

    In the Call Arity papers, we are working on an untyped lambda calculus with no
    other id annotations, where eta-expansion is always possible. But this is not
    the case for Core!
     1. We need to ensure the invariant
          callArity e <= typeArity (exprType e)
        for the same reasons that exprArity needs this invariant (see Note
        [exprArity invariant] in CoreArity).

        If we are not doing that, a too-high arity annotation will be stored with
        the id, confusing the simplifier later on.

     2. Eta-expanding a right hand side might invalidate existing annotations. In
        particular, if an id has a strictness annotation of <...><...>b, then
        passing one argument to it will definitely bottom out, so the simplifier
        will throw away additional parameters. This conflicts with Call Arity! So
        we ensure that we never eta-expand such a value beyond the number of
        arguments mentioned in the strictness signature.
        See #10176 for a real-world-example.

    -}

    -- Main entry point

    callArityAnalProgram :: DynFlags -> CoreProgram -> CoreProgram
    callArityAnalProgram _dflags binds = binds'
      where
        (_, binds') = callArityTopLvl [] emptyVarSet binds

    -- See Note [Analysing top-level-binds]
    callArityTopLvl :: [Var] -> VarSet -> [CoreBind] -> (CallArityRes, [CoreBind])
    callArityTopLvl exported _ []
        = ( calledMultipleTimes $ (emptyUnVarGraph, mkVarEnv $ [(v, 0) | v <- exported])
          , [] )
    callArityTopLvl exported int1 (b:bs)
        = (ae2, b':bs')
      where
        int2 = bindersOf b
        exported' = filter isExportedId int2 ++ exported
        int' = int1 `addInterestingBinds` b
        (ae1, bs') = callArityTopLvl exported' int' bs
        (ae2, b')  = callArityBind (boringBinds b) ae1 int1 b

    callArityRHS :: CoreExpr -> CoreExpr
    callArityRHS = snd . callArityAnal 0 emptyVarSet

    -- The main analysis function. See Note [Analysis type signature]
    callArityAnal ::
        Arity ->  -- The arity this expression is called with
        VarSet -> -- The set of interesting variables
        CoreExpr ->  -- The expression to analyse
        (CallArityRes, CoreExpr)
            -- How this expression uses its interesting variables
            -- and the expression with IdInfo updated

    -- The trivial base cases
    callArityAnal _     _   e@(Lit _)
        = (emptyArityRes, e)
    callArityAnal _     _   e@(Type _)
        = (emptyArityRes, e)
    callArityAnal _     _   e@(Coercion _)
        = (emptyArityRes, e)
    -- The transparent cases
    callArityAnal arity int (Tick t e)
        = second (Tick t) $ callArityAnal arity int e
    callArityAnal arity int (Cast e co)
        = second (\e -> Cast e co) $ callArityAnal arity int e

    -- The interesting case: Variables, Lambdas, Lets, Applications, Cases
    callArityAnal arity int e@(Var v)
        | v `elemVarSet` int
        = (unitArityRes v arity, e)
        | otherwise
        = (emptyArityRes, e)

    -- Non-value lambdas are ignored
    callArityAnal arity int (Lam v e) | not (isId v)
        = second (Lam v) $ callArityAnal arity (int `delVarSet` v) e

    -- We have a lambda that may be called multiple times, so its free variables
    -- can all be co-called.
    callArityAnal 0     int (Lam v e)
        = (ae', Lam v e')
      where
        (ae, e') = callArityAnal 0 (int `delVarSet` v) e
        ae' = calledMultipleTimes ae
    -- We have a lambda that we are calling. decrease arity.
    callArityAnal arity int (Lam v e)
        = (ae, Lam v e')
      where
        (ae, e') = callArityAnal (arity - 1) (int `delVarSet` v) e

    -- Application. Increase arity for the called expresion, nothing to know about
    -- the second
    callArityAnal arity int (App e (Type t))
        = second (\e -> App e (Type t)) $ callArityAnal arity int e
    callArityAnal arity int (App e1 e2)
        = (final_ae, App e1' e2')
      where
        (ae1, e1') = callArityAnal (arity + 1) int e1
        (ae2, e2') = callArityAnal 0           int e2
        -- See Note [Case and App: Which side to take?]
        final_ae = ae1 `both` ae2

    -- Case expression.
    callArityAnal arity int (Case scrut bndr ty alts)
        = -- pprTrace "callArityAnal:Case"
          --          (vcat [ppr scrut, ppr final_ae])
          (final_ae, Case scrut' bndr ty alts')
      where
        (alt_aes, alts') = unzip $ map go alts
        go (dc, bndrs, e) = let (ae, e') = callArityAnal arity int e
                            in  (ae, (dc, bndrs, e'))
        alt_ae = lubRess alt_aes
        (scrut_ae, scrut') = callArityAnal 0 int scrut
        -- See Note [Case and App: Which side to take?]
        final_ae = scrut_ae `both` alt_ae

    -- For lets, use callArityBind
    callArityAnal arity int (Let bind e)
      = -- pprTrace "callArityAnal:Let"
        --          (vcat [ppr v, ppr arity, ppr n, ppr final_ae ])
        (final_ae, Let bind' e')
      where
        int_body = int `addInterestingBinds` bind
        (ae_body, e') = callArityAnal arity int_body e
        (final_ae, bind') = callArityBind (boringBinds bind) ae_body int bind

    -- Which bindings should we look at?
    -- See Note [Which variables are interesting]
    isInteresting :: Var -> Bool
    isInteresting v = 0 < length (typeArity (idType v))

    interestingBinds :: CoreBind -> [Var]
    interestingBinds = filter isInteresting . bindersOf

    boringBinds :: CoreBind -> VarSet
    boringBinds = mkVarSet . filter (not . isInteresting) . bindersOf

    addInterestingBinds :: VarSet -> CoreBind -> VarSet
    addInterestingBinds int bind
        = int `delVarSetList`    bindersOf bind -- Possible shadowing
              `extendVarSetList` interestingBinds bind

    -- Used for both local and top-level binds
    -- Second argument is the demand from the body
    callArityBind :: VarSet -> CallArityRes -> VarSet -> CoreBind -> (CallArityRes, CoreBind)
    -- Non-recursive let
    callArityBind boring_vars ae_body int (NonRec v rhs)
      | otherwise
      = -- pprTrace "callArityBind:NonRec"
        --          (vcat [ppr v, ppr ae_body, ppr int, ppr ae_rhs, ppr safe_arity])
        (final_ae, NonRec v' rhs')
      where
        is_thunk = not (exprIsHNF rhs)
        -- If v is boring, we will not find it in ae_body, but always assume (0, False)
        boring = v `elemVarSet` boring_vars

        (arity, called_once)
            | boring    = (0, False) -- See Note [Taking boring variables into account]
            | otherwise = lookupCallArityRes ae_body v
        safe_arity | called_once = arity
                   | is_thunk    = 0      -- A thunk! Do not eta-expand
                   | otherwise   = arity

        -- See Note [Trimming arity]
        trimmed_arity = trimArity v safe_arity

        (ae_rhs, rhs') = callArityAnal trimmed_arity int rhs

        ae_rhs'| called_once     = ae_rhs
               | safe_arity == 0 = ae_rhs -- If it is not a function, its body is evaluated only once
               | otherwise       = calledMultipleTimes ae_rhs

        called_by_v = domRes ae_rhs'
        called_with_v
            | boring    = domRes ae_body
            | otherwise = calledWith ae_body v `delUnVarSet` v
        final_ae = addCrossCoCalls called_by_v called_with_v $ ae_rhs' `lubRes` resDel v ae_body

        v' = v `setIdCallArity` trimmed_arity

    -- Recursive let. See Note [Recursion and fixpointing]
    callArityBind boring_vars ae_body int b@(Rec binds)
      = -- (if length binds > 300 then
        -- pprTrace "callArityBind:Rec"
        --           (vcat [ppr (Rec binds'), ppr ae_body, ppr int, ppr ae_rhs]) else id) $
        (final_ae, Rec binds')
      where
        -- See Note [Taking boring variables into account]
        any_boring = any (`elemVarSet` boring_vars) [ i | (i, _) <- binds]

        int_body = int `addInterestingBinds` b
        (ae_rhs, binds') = fix initial_binds
        final_ae = bindersOf b `resDelList` ae_rhs

        initial_binds = [(i,Nothing,e) | (i,e) <- binds]

        fix :: [(Id, Maybe (Bool, Arity, CallArityRes), CoreExpr)] -> (CallArityRes, [(Id, CoreExpr)])
        fix ann_binds
            | -- pprTrace "callArityBind:fix" (vcat [ppr ann_binds, ppr any_change, ppr ae]) $
              any_change
            = fix ann_binds'
            | otherwise
            = (ae, map (\(i, _, e) -> (i, e)) ann_binds')
          where
            aes_old = [ (i,ae) | (i, Just (_,_,ae), _) <- ann_binds ]
            ae = callArityRecEnv any_boring aes_old ae_body

            rerun (i, mbLastRun, rhs)
                | i `elemVarSet` int_body && not (i `elemUnVarSet` domRes ae)
                -- No call to this yet, so do nothing
                = (False, (i, Nothing, rhs))

                | Just (old_called_once, old_arity, _) <- mbLastRun
                , called_once == old_called_once
                , new_arity == old_arity
                -- No change, no need to re-analize
                = (False, (i, mbLastRun, rhs))

                | otherwise
                -- We previously analized this with a different arity (or not at all)
                = let is_thunk = not (exprIsHNF rhs)

                      safe_arity | is_thunk    = 0  -- See Note [Thunks in recursive groups]
                                 | otherwise   = new_arity

                      -- See Note [Trimming arity]
                      trimmed_arity = trimArity i safe_arity

                      (ae_rhs, rhs') = callArityAnal trimmed_arity int_body rhs

                      ae_rhs' | called_once     = ae_rhs
                              | safe_arity == 0 = ae_rhs -- If it is not a function, its body is evaluated only once
                              | otherwise       = calledMultipleTimes ae_rhs

                  in (True, (i `setIdCallArity` trimmed_arity, Just (called_once, new_arity, ae_rhs'), rhs'))
              where
                -- See Note [Taking boring variables into account]
                (new_arity, called_once) | i `elemVarSet` boring_vars = (0, False)
                                         | otherwise                  = lookupCallArityRes ae i

            (changes, ann_binds') = unzip $ map rerun ann_binds
            any_change = or changes

    -- Combining the results from body and rhs, (mutually) recursive case
    -- See Note [Analysis II: The Co-Called analysis]
    callArityRecEnv :: Bool -> [(Var, CallArityRes)] -> CallArityRes -> CallArityRes
    callArityRecEnv any_boring ae_rhss ae_body
        = -- (if length ae_rhss > 300 then pprTrace "callArityRecEnv" (vcat [ppr ae_rhss, ppr ae_body, ppr ae_new]) else id) $
          ae_new
      where
        vars = map fst ae_rhss

        ae_combined = lubRess (map snd ae_rhss) `lubRes` ae_body

        cross_calls
            -- See Note [Taking boring variables into account]
            | any_boring          = completeGraph (domRes ae_combined)
            -- Also, calculating cross_calls is expensive. Simply be conservative
            -- if the mutually recursive group becomes too large.
            | length ae_rhss > 25 = completeGraph (domRes ae_combined)
            | otherwise           = unionUnVarGraphs $ map cross_call ae_rhss
        cross_call (v, ae_rhs) = completeBipartiteGraph called_by_v called_with_v
          where
            is_thunk = idCallArity v == 0
            -- What rhs are relevant as happening before (or after) calling v?
            --    If v is a thunk, everything from all the _other_ variables
            --    If v is not a thunk, everything can happen.
            ae_before_v | is_thunk  = lubRess (map snd $ filter ((/= v) . fst) ae_rhss) `lubRes` ae_body
                        | otherwise = ae_combined
            -- What do we want to know from these?
            -- Which calls can happen next to any recursive call.
            called_with_v
                = unionUnVarSets $ map (calledWith ae_before_v) vars
            called_by_v = domRes ae_rhs

        ae_new = first (cross_calls `unionUnVarGraph`) ae_combined

    -- See Note [Trimming arity]
    trimArity :: Id -> Arity -> Arity
    trimArity v a = minimum [a, max_arity_by_type, max_arity_by_strsig]
      where
        max_arity_by_type = length (typeArity (idType v))
        max_arity_by_strsig
            | isBotRes result_info = length demands
            | otherwise = a

        (demands, result_info) = splitStrictSig (idStrictness v)

    ---------------------------------------
    -- Functions related to CallArityRes --
    ---------------------------------------

    -- Result type for the two analyses.
    -- See Note [Analysis I: The arity analyis]
    -- and Note [Analysis II: The Co-Called analysis]
    type CallArityRes = (UnVarGraph, VarEnv Arity)

    emptyArityRes :: CallArityRes
    emptyArityRes = (emptyUnVarGraph, emptyVarEnv)

    unitArityRes :: Var -> Arity -> CallArityRes
    unitArityRes v arity = (emptyUnVarGraph, unitVarEnv v arity)

    resDelList :: [Var] -> CallArityRes -> CallArityRes
    resDelList vs ae = foldr resDel ae vs

    resDel :: Var -> CallArityRes -> CallArityRes
    resDel v (g, ae) = (g `delNode` v, ae `delVarEnv` v)

    domRes :: CallArityRes -> UnVarSet
    domRes (_, ae) = varEnvDom ae

    -- In the result, find out the minimum arity and whether the variable is called
    -- at most once.
    lookupCallArityRes :: CallArityRes -> Var -> (Arity, Bool)
    lookupCallArityRes (g, ae) v
        = case lookupVarEnv ae v of
            Just a -> (a, not (v `elemUnVarSet` (neighbors g v)))
            Nothing -> (0, False)

    calledWith :: CallArityRes -> Var -> UnVarSet
    calledWith (g, _) v = neighbors g v

    addCrossCoCalls :: UnVarSet -> UnVarSet -> CallArityRes -> CallArityRes
    addCrossCoCalls set1 set2 = first (completeBipartiteGraph set1 set2 `unionUnVarGraph`)

    -- Replaces the co-call graph by a complete graph (i.e. no information)
    calledMultipleTimes :: CallArityRes -> CallArityRes
    calledMultipleTimes res = first (const (completeGraph (domRes res))) res

    -- Used for application and cases
    both :: CallArityRes -> CallArityRes -> CallArityRes
    both r1 r2 = addCrossCoCalls (domRes r1) (domRes r2) $ r1 `lubRes` r2

    -- Used when combining results from alternative cases; take the minimum
    lubRes :: CallArityRes -> CallArityRes -> CallArityRes
    lubRes (g1, ae1) (g2, ae2) = (g1 `unionUnVarGraph` g2, ae1 `lubArityEnv` ae2)

    lubArityEnv :: VarEnv Arity -> VarEnv Arity -> VarEnv Arity
    lubArityEnv = plusVarEnv_C min

    lubRess :: [CallArityRes] -> CallArityRes
    lubRess = foldl lubRes emptyArityRes