content: 
  {-# LANGUAGE ImplicitParams             #-}
  {-# LANGUAGE NondecreasingIndentation   #-}
  
  {- Checking for Structural recursion
     Authors: Andreas Abel, Nils Anders Danielsson, Ulf Norell,
                Karl Mehltretter and others
     Created: 2007-05-28
     Source : TypeCheck.Rules.Decl
   -}
  
  module Agda.Termination.TermCheck
      ( termDecl
      , termMutual
      , Result
      ) where
  
  import Prelude hiding ( null, zip, zipWith )
  
  import Control.Applicative  ( liftA2 )
  import Control.Monad        ( (<=<), filterM, forM, forM_, zipWithM )
  
  import Data.Foldable (toList)
  import qualified Data.List as List
  import Data.Monoid hiding ((<>))
  import Data.Set (Set)
  import qualified Data.Set as Set
  
  import qualified Agda.Syntax.Abstract as A
  import Agda.Syntax.Internal as I
  import Agda.Syntax.Internal.Pattern as I
  import Agda.Syntax.Internal.Generic
  import qualified Agda.Syntax.Info as Info
  import Agda.Syntax.Position
  import Agda.Syntax.Common
  import Agda.Syntax.Translation.InternalToAbstract (NamedClause(..))
  
  import Agda.Termination.CutOff
  import Agda.Termination.Monad
  import Agda.Termination.CallGraph hiding (toList)
  import qualified Agda.Termination.CallGraph as CallGraph
  import Agda.Termination.CallMatrix hiding (toList)
  import Agda.Termination.Order     as Order
  import qualified Agda.Termination.SparseMatrix as Matrix
  import Agda.Termination.Termination (Terminates(..), GuardednessHelps(..), endos, idempotent)
  import qualified Agda.Termination.Termination  as Term
  import Agda.Termination.RecCheck
  
  import Agda.TypeChecking.Datatypes
  import Agda.TypeChecking.Functions
  import Agda.TypeChecking.Monad
  import Agda.TypeChecking.Pretty
  import Agda.TypeChecking.Forcing
  import Agda.TypeChecking.Records -- (isRecordConstructor, isInductiveRecord)
  import Agda.TypeChecking.Reduce (reduce, normalise, instantiate, instantiateFull, appDefE_)
  import Agda.TypeChecking.SizedTypes
  import Agda.TypeChecking.Substitute
  import Agda.TypeChecking.Telescope
  
  import qualified Agda.Benchmarking as Benchmark
  import Agda.TypeChecking.Monad.Benchmark (billTo, billPureTo)
  
  import Agda.Interaction.Options
  
  import Agda.Utils.Either
  import Agda.Utils.Function
  import Agda.Utils.Functor
  import Agda.Utils.List
  import Agda.Utils.ListInf ( pattern (:<) )
  import Agda.Utils.ListInf qualified as ListInf
  import Agda.Utils.Maybe
  import Agda.Utils.Monad -- (mapM', forM', ifM, or2M, and2M)
  import Agda.Utils.Null
  import Agda.Syntax.Common.Pretty (prettyShow)
  import Agda.Utils.Singleton
  import Agda.Utils.Size
  -- import Agda.Utils.SmallSet (SmallSet)
  import qualified Agda.Utils.SmallSet as SmallSet
  import qualified Agda.Utils.VarSet as VarSet
  import Agda.Utils.Zip
  
  import Agda.Utils.Impossible
  
  -- | Call graph with call info for composed calls.
  
  type Calls = CallGraph CallPath
  
  -- | The result of termination checking a module.
  --   Must be a 'Monoid' and have 'Singleton'.
  
  type Result = [TerminationError]
  
  -- | Entry point: Termination check a single declaration.
  --
  --   Precondition: 'envMutualBlock' must be set correctly.
  
  termDecl :: A.Declaration -> TCM Result
  termDecl d = inTopContext $ termDecl' d
  
  
  -- | Termination check a single declaration
  --   (without necessarily ignoring @abstract@).
  
  termDecl' :: A.Declaration -> TCM Result
  termDecl' = \case
      A.Axiom {}            -> return mempty
      A.Field {}            -> return mempty
      A.Primitive {}        -> return mempty
      A.Mutual i ds         -> termMutual $ getNames ds
      A.Section _ _ _ _ ds  -> termDecls ds
          -- section structure can be ignored as we are termination checking
          -- definitions lifted to the top-level
      A.Apply {}            -> return mempty
      A.Import {}           -> return mempty
      A.Pragma {}           -> return mempty
      A.Open {}             -> return mempty
      A.PatternSynDef {}    -> return mempty
      A.UnfoldingDecl{}     -> return mempty
      A.Generalize {}       -> return mempty
          -- open, pattern synonym and generalize defs are just artifacts from the concrete syntax
      A.ScopedDecl scope ds -> {- withScope_ scope $ -} termDecls ds
          -- scope is irrelevant as we are termination checking Syntax.Internal
      A.RecSig{}            -> return mempty
      A.RecDef _ x _ _ _ _ ds -> termMutual [x] <> termDecls ds
          -- Andreas, 2022-10-23, issue #5823
          -- Also check record types for termination.
          -- They are unfolded during construction of unique inhabitants of eta-records.
      -- These should all be wrapped in mutual blocks:
      A.FunDef{}      -> __IMPOSSIBLE__
      A.DataSig{}     -> __IMPOSSIBLE__
      A.DataDef{}     -> __IMPOSSIBLE__
      A.UnquoteDecl{} -> __IMPOSSIBLE__
      A.UnquoteDef{}  -> __IMPOSSIBLE__
      A.UnquoteData{} -> __IMPOSSIBLE__
    where
      termDecls ds = concat <$> mapM termDecl' ds
  
      -- The mutual names mentioned in the abstract syntax
      -- for symbols that need to be termination-checked.
      getNames = concatMap getName
      getName (A.FunDef i x cs)   = [x]
      getName (A.RecDef _ x _ _ _ _ ds)   = x : getNames ds
      getName (A.Mutual _ ds)             = getNames ds
      getName (A.Section _ _ _ _ ds)      = getNames ds
      getName (A.ScopedDecl _ ds)         = getNames ds
      getName (A.UnquoteDecl _ _ xs _)    = xs
      getName (A.UnquoteDef _ xs _)       = xs
      getName _                           = []
  
  
  -- | Entry point: Termination check the current mutual block.
  
  termMutual
    :: [QName]
       -- ^ The function names defined in this block on top-level.
       --   (For error-reporting only.)
    -> TCM Result
  termMutual names0 = ifNotM (optTerminationCheck <$> pragmaOptions) (return mempty) $ {-else-}
   inTopContext $ do
  
    -- Get set of mutually defined names from the TCM.
    -- This includes local and auxiliary functions introduced
    -- during type-checking.
    mid <- fromMaybe __IMPOSSIBLE__ <$> asksTC envMutualBlock
    mutualBlock <- lookupMutualBlock mid
    let allNames = Set.filter (not . isAbsurdLambdaName) $
                   mutualNames mutualBlock
        names    = if null names0 then allNames else Set.fromList names0
        i        = mutualInfo mutualBlock
  
    -- We set the range to avoid panics when printing error messages.
    setCurrentRange i $ do
  
    -- The following debug statement is part of a test case for Issue
    -- #3590.
    reportSLn "term.mutual.id" 40 $
      "Termination checking mutual block " ++ prettyShow mid
    reportSLn "term.mutual" 10 $ "Termination checking " ++ prettyShow allNames
  
    -- NO_TERMINATION_CHECK
    if (Info.mutualTerminationCheck i `elem` [ NoTerminationCheck, Terminating ]) then do
        reportSLn "term.warn.yes" 10 $ "Skipping termination check for " ++ prettyShow names
        forM_ allNames $ \ q -> setTerminates q $ Just True -- considered terminating!
        return mempty
    -- NON_TERMINATING
    else if (Info.mutualTerminationCheck i == NonTerminating) then do
        reportSLn "term.warn.yes" 10 $ "Considering as non-terminating: " ++ prettyShow names
        forM_ allNames $ \ q -> setTerminates q $ Just False
        return mempty
    else do
      sccs <- do
        -- Andreas, 2016-10-01 issue #2231
        -- Recursivity checker has to see through abstract definitions!
        ignoreAbstractMode $ do
          billTo [Benchmark.Termination, Benchmark.RecCheck] $ recursive allNames
        -- -- Andreas, 2017-03-24, use positivity info to skip non-recursive functions
        -- skip = ignoreAbstractMode $ forallM allNames $ \ x -> do
        --   null <$> getMutual x
        -- PROBLEMS with test/Succeed/AbstractCoinduction.agda
  
      -- Trivially terminating (non-recursive)?
      when (null sccs) $
        reportSLn "term.warn.yes" 10 $ "Trivially terminating: " ++ prettyShow names
  
      -- Actual termination checking needed: go through SCCs.
      concat <$> do
       forM sccs $ \ allNames -> do
  
       -- Andreas, 2025-05-31, AIM XL, re issue #7906:
       -- Clear previous information about termination to avoid loops in the termination checker.
       forM_ allNames $ \ q -> setTerminates q Nothing
  
       -- Set the mutual names in the termination environment.
       let namesSCC = Set.filter (`Set.member` allNames) names
       let setNames e = e
             { terMutual    = allNames
             , terUserNames = namesSCC
             }
           runTerm cont = runTerDefault $ do
             cutoff <- terGetCutOff
             reportSLn "term.top" 10 $ "Termination checking " ++ prettyShow namesSCC ++
               " with cutoff=" ++ show cutoff ++ "..."
             terLocal setNames cont
  
       -- New check currently only makes a difference for copatterns and record types.
       -- Since it is slow, only invoke it if
       -- any of the definitions uses copatterns or is a record type.
       ifM (existsM allNames $ \ q -> usesCopatterns q `or2M` (isJust <$> isRecord q))
           -- Then: New check, one after another.
           (runTerm $ forM' allNames $ termFunction)
           -- Else: Old check, all at once.
           (runTerm $ termMutual')
  
  -- | Run the termination checker possibly twice and take the best result.
  --   Run it first without extracting descent information from dot patterns.
  --   If this proves termination, we are done.
  --   If this did not manage to prove termination, try with dot patterns.
  --   If this did not manage to prove termination either, return the offending paths.
  --   Otherwise, if the first run proved termination subject to (deactivated) guardedness,
  --   return this result.
  --   Otherwise, return the second result.
  withOrWithoutDotPatterns ::
       (Node -> Bool)
    -> TerM Calls
    -> TerM (Terminates CallPath)
  withOrWithoutDotPatterns filt collect = do
      useGuardedness <- liftTCM guardednessOption
      cutoff <- terGetCutOff
      let ?cutoff = cutoff
      -- Run the continuation @k@ with the result of @m@ unless @m@ already certifies termination.
      let unlessTerminates m k = m >>= \case
            Terminates
              -> return Terminates
            TerminatesNot GuardednessHelpsYes _ | useGuardedness
              -> return Terminates
            r -> k r
  
      -- First try to termination check ignoring the dot patterns
      calls1 <- terSetUseDotPatterns False collect
      reportCalls "no " calls1
      unlessTerminates (billToTerGraph $ Term.terminatesFilter filt calls1) \ r1 -> do
        -- Try again, but include the dot patterns this time.
        calls2 <- terSetUseDotPatterns True collect
        reportCalls "" calls2
        unlessTerminates (billToTerGraph $ Term.terminatesFilter filt calls2) \ r2 -> do
          case r1 of
            TerminatesNot GuardednessHelpsNot _ -> return r2
              -- We might terminate with guardedness and dot patterns (r2).
            _              -> return r1
              -- Since r2 did not certify termination, the simpler r1 is preferable.
  
  -- | @termMutual'@ checks all names of the current mutual block,
  --   henceforth called @allNames@, for termination.
  --
  --   @allNames@ is taken from 'Internal' syntax, it contains also
  --   the definitions created by the type checker (e.g., with-functions).
  
  termMutual' :: TerM Result
  termMutual' = do
  
    -- collect all recursive calls in the block
    allNames <- terGetMutual
    let collect :: TerM Calls
        collect = forM' allNames termDef
  
    r <- withOrWithoutDotPatterns (const True) collect
  
    -- @names@ is taken from the 'Abstract' syntax, so it contains only
    -- the names the user has declared.  This is for error reporting.
    names <- terGetUserNames
    case r of
  
      TerminatesNot guardednessHelps calls -> do
        mapM_ (`setTerminates` Just False) allNames
        return $ singleton $ terminationError names calls guardednessHelps
  
      Terminates -> do
        liftTCM $ reportSLn "term.warn.yes" 2 $
          prettyShow (names) ++ " does termination check"
        mapM_ (`setTerminates` Just True) allNames
        return mempty
  
  -- | Smart constructor for 'TerminationError'.
  --   Removes 'termErrFunctions' that are not mentioned in 'termErrCalls'.
  terminationError :: Set QName -> CallPath -> GuardednessHelps -> TerminationError
  terminationError names calls guardednessHelps = TerminationError names' calls' guardednessHelps
    where
    calls'    = callInfos calls
    mentioned = map callInfoTarget calls'
    names'    = filter (hasElem mentioned) $ toList names
  
  billToTerGraph :: a -> TerM a
  billToTerGraph a = liftTCM $ billPureTo [Benchmark.Termination, Benchmark.Graph] a
  
  -- | @reportCalls@ for debug printing.
  --
  --   Replays the call graph completion for debugging.
  
  reportCalls :: String -> Calls -> TerM ()
  reportCalls no calls = do
    cutoff <- terGetCutOff
    let ?cutoff = cutoff
  
    -- We work in TCM exclusively.
    liftTCM $ do
  
      reportS "term.lex" 20
        [ "Calls (" ++ no ++ "dot patterns): " ++ prettyShow calls
        ]
  
      -- Print the whole completion phase.
      verboseS "term.matrices" 40 $ do
        let header s = unlines
              [ replicate n '='
              , replicate k '=' ++ s ++ replicate k' '='
              , replicate n '='
              ]
              where n  = 70
                    r  = n - length s
                    k  = r `div` 2
                    k' = r - k
        let report s cs = reportSDoc "term.matrices" 40 $ vcat
              [ text   $ header s
              , nest 2 $ pretty cs
              ]
            cs0     = calls
            step cs = do
              let (new, cs') = completionStep cs0 cs
              report " New call matrices " new
              return $ if null new then Left () else Right cs'
        report " Initial call matrices " cs0
        trampolineM step cs0
  
      -- Print the result of completion.
      let calls' = CallGraph.complete calls
          idems = filter idempotent $ endos $ CallGraph.toList calls'
      -- TODO
      -- reportSDoc "term.behaviours" 20 $ vcat
      --   [ text $ "Recursion behaviours (" ++ no ++ "dot patterns):"
      --   , nest 2 $ return $ Term.prettyBehaviour calls'
      --   ]
      reportSDoc "term.matrices" 30 $ vcat
        [ text $ "Idempotent call matrices (" ++ no ++ "dot patterns):\n"
        , nest 2 $ vcat $ punctuate "\n" $ map pretty idems
        ]
      -- reportSDoc "term.matrices" 30 $ vcat
      --   [ text $ "Other call matrices (" ++ no ++ "dot patterns):"
      --   , nest 2 $ pretty $ CallGraph.fromList others
      --   ]
      return ()
  
  -- | @termFunction name@ checks @name@ for termination.
  -- If it passes the termination check it is marked as "terminates" in the signature.
  
  termFunction :: QName -> TerM Result
  termFunction name = inConcreteOrAbstractMode name $ \ def -> do
  
    -- Function @name@ is henceforth referred to by its @index@
    -- in the list of @allNames@ of the mutual block.
  
    allNames <- terGetMutual
    let index = fromMaybe __IMPOSSIBLE__ $ Set.lookupIndex name allNames
  
    -- Retrieve the target type of the function to check.
    -- #4256: Don't use typeOfConst (which instantiates type with module params), since termination
    -- checking is running in the empty context, but with the current module unchanged.
    target <- case theDef def of
      -- We are termination-checking a record (calls to record will not be guarding):
      Record{} -> return TargetRecord
      -- We are termination-checking a definition:
      _ -> typeEndsInDef (defType def) <&> \case
             Just d  -> TargetDef d
             Nothing -> TargetOther
    reportTarget target
    terSetTarget target $ do
  
      -- Collect the recursive calls in the block which (transitively)
      -- involve @name@,
      -- taking the target of @name@ into account for computing guardedness.
  
      let collect = (`trampolineM` (Set.singleton index, mempty, mempty)) $ \ (todo, done, calls) -> do
            if null todo then return $ Left calls else do
              -- Extract calls originating from indices in @todo@.
              new <- forM' todo $ \ i ->
                termDef $
                if i < 0 || i >= Set.size allNames
                then __IMPOSSIBLE__
                else Set.elemAt i allNames
              -- Mark those functions as processed and add the calls to the result.
              let done'  = done `mappend` todo
                  calls' = new  `mappend` calls
              -- Compute the new todo list:
                  todo' = CallGraph.targetNodes new Set.\\ done'
              -- Jump the trampoline.
              return $ Right (todo', done', calls')
  
      r <- withOrWithoutDotPatterns (== index) collect
  
      names <- terGetUserNames
      case r of
  
        TerminatesNot guardednessHelps callpaths -> do
          let calls = callInfos callpaths
          -- Mark as non-terminating.
          setTerminates name $ Just False
  
          -- Functions must be terminating, records types need not...
          case theDef def of
  
            -- Records need not terminate, so we just put the error on the debug log.
            Record{} -> do
              reportSDoc "term.warn.no" 10 $ vcat $
                hsep [ "Record type", prettyTCM name, "does not termination check.", "Problematic calls:" ] :
                map (nest 2 . prettyTCM) (List.sortOn getRange calls)
              mempty
  
            -- Functions must terminate, so we report the error.
            _ -> do
              let err = TerminationError [name | name `elem` names] calls guardednessHelps
              return $ singleton err
  
        Terminates -> do
          reportSLn "term.warn.yes" 2 $
            prettyShow name ++ " does termination check"
          setTerminates name $ Just True
          return mempty
     where
       reportTarget :: MonadDebug m => Target -> m ()
       reportTarget tgt = reportSLn "term.target" 20 $ ("  " ++) $
         case tgt of
           TargetRecord -> "termination checking a record type"
           TargetDef q  -> unwords [ "target type ends in", prettyShow q ]
           TargetOther  -> "target type not recognized"
  
  -- | To process the target type.
  typeEndsInDef :: MonadTCM tcm => Type -> tcm (Maybe QName)
  typeEndsInDef t = liftTCM $ do
    TelV _ core <- telViewPath t
    case unEl core of
      Def d vs -> return $ Just d
      _        -> return Nothing
  
  -- | Termination check a definition by pattern matching.
  --
  --   TODO: Refactor!
  --   As this function may be called twice,
  --   once disregarding dot patterns,
  --   the second time regarding dot patterns,
  --   it is better if we separated bare call extraction
  --   from computing the change in structural order.
  --   Only the latter depends on the choice whether we
  --   consider dot patterns or not.
  termDef :: QName -> TerM Calls
  termDef name = terSetCurrent name $ inConcreteOrAbstractMode name $ \ def -> do
  
   -- Skip calls to record types unless we are checking a record type in the first place.
   let isRecord_ = case theDef def of { Record{} -> True; _ -> False }
   let notTargetRecord = terGetTarget <&> \case
         TargetRecord -> False
         _ -> True
   ifM (pure isRecord_ `and2M` notTargetRecord) mempty {-else-} $ do
  
    -- Retrieve definition
    let t = defType def
  
    liftTCM $ reportSDoc "term.def.fun" 5 $
      sep [ "termination checking type of" <+> prettyTCM name
          , nest 2 $ ":" <+> prettyTCM t
          ]
  
    termType t `mappend` do
  
    liftTCM $ reportSDoc "term.def.fun" 5 $
      sep [ "termination checking body of" <+> prettyTCM name
          , nest 2 $ ":" <+> prettyTCM t
          ]
  
    -- If --without-K, we disregard all arguments (and result)
    -- which are not of data or record type.
  
    withoutKEnabled <- liftTCM withoutKOption
    applyWhen withoutKEnabled (setMasks t) $ do
  
      -- If the result should be disregarded, set all calls to unguarded.
      applyWhenM terGetMaskResult terUnguarded $ do
  
        case theDef def of
          Function{ funClauses = cls  } -> forM' cls $ \ cl -> do
            if hasDefP (namedClausePats cl) -- generated hcomp clause, should be safe.
                                            -- TODO find proper strategy.
              then return empty
              else termClause cl
  
          -- @record R pars : Set where field tel@
          -- is treated like function @R pars = tel@.
          Record{ recPars, recTel } -> termRecTel recPars recTel
  
          _ -> return empty
  
  -- | Extract "calls" to the field types from a record constructor telescope.
  -- Does not extract from the parameters, but treats these as the "pattern variables"
  -- (the lhs of the "function").
  termRecTel :: Nat -> Telescope -> TerM Calls
  termRecTel npars tel = do
    -- Set up the record parameters like function parameters.
    let (pars, fields) = splitAt npars $ telToList tel
    addContext pars $ do
      ps <- mkPats npars
      terSetPatterns ps $ terSetSizeDepth pars $ do
        -- Treat the record fields like the body of a function.
        extract $ telFromList fields
    where
    -- create n variable patterns
    mkPats n  = map mkPat <$> getContextVars
    mkPat (i, x) = notMasked $ VarP defaultPatternInfo $ DBPatVar (prettyShow x) i
  
  -- | Collect calls in type signature @f : (x1:A1)...(xn:An) -> B@.
  --   It is treated as if there were the additional function clauses.
  --   @@
  --      f = A1
  --      f x1 = A2
  --      f x1 x2 = A3
  --      ...
  --      f x1 ... xn = B
  --   @@
  
  termType :: Type -> TerM Calls
  termType = return mempty
  -- termType = loop 0  -- Andreas, 2019-04-10 deactivate for backwards-compatibility in 2.6.0 #1556
    where
    loop n t = do
      ps <- mkPats n
      reportSDoc "term.type" 60 $ vcat
        [ text $ "termType " ++ show n ++ " with " ++ show (length ps) ++ " patterns"
        , nest 2 $ "looking at type " <+> prettyTCM t
        ]
      tel <- getContextTelescope  -- Andreas, 2018-11-15, issue #3394, forgotten initialization of terSizeDepth
      terSetPatterns ps $ terSetSizeDepth tel $ do
        ifNotPiType t {-then-} extract {-else-} $ \ dom absB -> do
          extract dom `mappend` underAbstractionAbs dom absB (loop $! n + 1)
  
    -- create n variable patterns
    mkPats n  = map mkPat <$> getContextVars
    mkPat (i, x) = notMasked $ VarP defaultPatternInfo $ DBPatVar (prettyShow x) i
  
  -- | Mask arguments and result for termination checking
  --   according to type of function.
  --   Only arguments of types ending in data/record or Size are counted in.
  setMasks :: Type -> TerM a -> TerM a
  setMasks t cont = do
    (ds, d) <- liftTCM $ do
      TelV tel core <- telViewPath t
      -- Check argument types
      ds <- checkArgumentTypes tel
      -- Check result types
      d  <- addContext tel $ isNothing <.> isDataOrRecord . unEl $ core
      when d $
        reportSLn "term.mask" 20 $ "result type is not data or record type, ignoring guardedness for --without-K"
      return (ds, d)
    terSetMaskArgs (ListInf.pad ds True) $ terSetMaskResult d $ cont
  
    where
      checkArgumentTypes :: Telescope -> TCM [Bool]
      checkArgumentTypes EmptyTel = return []
      checkArgumentTypes (ExtendTel dom atel) = do
        TelV tel2 t <- telViewPath $ unDom dom
        d <- addContext tel2 $
          (isNothing <$> isDataOrRecord (unEl t)) `or2M` (isJust <$> isSizeType t)
        when d $
          reportSDoc "term.mask" 20 $ do
            "argument type "
              <+> prettyTCM t
              <+> " is not data or record type, ignoring structural descent for --without-K"
        underAbstraction dom atel $ \tel -> (d:) <$> checkArgumentTypes tel
  
  -- | Is the current target type among the given ones?
  
  targetElem :: [QName] -> TerM Bool
  targetElem ds = terGetTarget <&> \case
    TargetDef d  -> d `elem` ds
    TargetRecord -> False
    TargetOther  -> False
  
  
  -- | Convert a term (from a dot pattern) to a DeBruijn pattern.
  --
  --   The term is first normalized and stripped of all non-coinductive projections.
  
  termToDBP :: Term -> TerM DeBruijnPattern
  termToDBP t = ifNotM terGetUseDotPatterns (return unusedVar) $ {- else -} do
    termToPattern =<< do liftTCM $ stripAllProjections =<< normalise t
  
  -- | Convert a term (from a dot pattern) to a pattern for the purposes of the termination checker.
  --
  --   @SIZESUC@ is treated as a constructor.
  
  class TermToPattern a b where
    termToPattern :: a -> TerM b
  
    default termToPattern :: (TermToPattern a' b', Traversable f, a ~ f a', b ~ f b') => a -> TerM b
    termToPattern = traverse termToPattern
  
  instance TermToPattern a b => TermToPattern [a] [b] where
  instance TermToPattern a b => TermToPattern (Arg a) (Arg b) where
  instance TermToPattern a b => TermToPattern (Named c a) (Named c b) where
  
  -- OVERLAPPING
  -- instance TermToPattern a b => TermToPattern a (Named c b) where
  --   termToPattern t = unnamed <$> termToPattern t
  
  instance TermToPattern Term DeBruijnPattern where
    termToPattern t = liftTCM (constructorForm t) >>= \case
      -- Constructors.
      Con c _ args -> ifNotConsOfHIT c $
        ConP c noConPatternInfo . map (fmap unnamed) <$> termToPattern (fromMaybe __IMPOSSIBLE__ $ allApplyElims args)
      Def s [Apply arg] -> do
        suc <- terGetSizeSuc
        if Just s == suc then ConP (ConHead s IsData Inductive []) noConPatternInfo . map (fmap unnamed) <$> termToPattern [arg]
         else fallback
      DontCare t  -> termToPattern t -- OR: __IMPOSSIBLE__  -- removed by stripAllProjections
      -- Leaves.
      Var i []    -> varP . (`DBPatVar` i) . prettyShow <$> nameOfBV i
      Lit l       -> return $ litP l
      Dummy s _   -> __IMPOSSIBLE_VERBOSE__ s
      t           -> fallback
      where
      -- Andrea: 22/04/2020.
      -- With cubical we will always have a clause where the dot
      -- patterns are instead replaced with a variable, so they
      -- cannot be relied on for termination.
      -- See issue #4606 for a counterexample involving HITs.
      --
      -- Without the presence of HITs I conjecture that dot patterns
      -- could be turned into actual splits, because no-confusion
      -- would make the other cases impossible, so I do not disable
      -- this for --without-K entirely.
      --
      -- Szumi, 2025-03-11:
      -- Instead of completely turning off dot-pattern termination for cubical,
      -- it should be enough to only ignore constructors of HITs in dot patterns.
      -- This way, the issues #5953 and #4725 are also avoided.
      ifNotConsOfHIT c    = ifM (consOfHIT (conName c)) fallback
      fallback            = return $ dotP t
  
  -- | Masks all non-data/record type patterns if --without-K.
  --   See issue #1023.
  maskNonDataArgs :: [DeBruijnPattern] -> TerM [Masked DeBruijnPattern]
  maskNonDataArgs ps = zipWith mask ps <$> terGetMaskArgs
    where
      mask p@ProjP{} _ = Masked False p
      mask p         d = Masked d     p
  
  -- | Drop elements of the list which correspond to arguments forced by
  -- the constructor with the given QName.
  mapForcedArguments :: QName -> [a] -> (IsForced -> a -> Maybe b) -> TerM [b]
  mapForcedArguments c xs k = do
    forcedArgs <- getForcedArgs c
    let go xs (p:ps) = do
          let (f, xs') = nextIsForced xs
          case k f p of
            Just b  -> b:go xs' ps
            Nothing -> go xs' ps
        go _ [] = []
    pure $ go forcedArgs xs
  
  -- | Extract recursive calls from one clause.
  termClause :: Clause -> TerM Calls
  termClause clause = do
    Clause{ clauseTel = tel, namedClausePats = ps, clauseBody = body } <- etaExpandClause clause
    liftTCM $ reportSDoc "term.check.clause" 25 $ vcat
      [ "termClause"
      , nest 2 $ "tel =" <+> prettyTCM tel
      , nest 2 $ "ps  =" <+> do addContext tel $ prettyTCMPatternList ps
      ]
    forM' body $ \ v -> addContext tel $ do
      -- TODO: combine the following two traversals, avoid full normalisation.
      -- Parse dot patterns as patterns as far as possible.
      ps <- postTraversePatternM parseDotP ps
      -- Blank out coconstructors.
      ps <- preTraversePatternM stripCoCon ps
      -- Mask non-data arguments.
      mdbpats <- maskNonDataArgs $ map namedArg ps
      terSetPatterns mdbpats $ do
        terSetSizeDepth tel $ do
          reportBody v
          extract v
  
    where
      parseDotP = \case
        DotP o t -> termToDBP t
        p        -> return p
      stripCoCon = \case
        ConP (ConHead c _ CoInductive _) _ _ -> return unusedVar
        p -> return p
      reportBody :: Term -> TerM ()
      reportBody v = verboseS "term.check.clause" 6 $ do
        f       <- terGetCurrent
        pats    <- terGetPatterns
        liftTCM $ reportSDoc "term.check.clause" 6 $ do
          sep [ text ("termination checking clause of")
                  <+> prettyTCM f
              , nest 2 $ "lhs:" <+> sep (map prettyTCM pats)
              , nest 2 $ "rhs:" <+> prettyTCM v
              ]
  
  
  -- | Extract recursive calls from expressions.
  class ExtractCalls a where
    extract :: a -> TerM Calls
  
  instance ExtractCalls a => ExtractCalls (Abs a) where
    extract (NoAbs _ a) = extract a
    extract (Abs x a)   = addContext x $ terRaise $ extract a
  
  instance ExtractCalls a => ExtractCalls (Arg a) where
    extract = extract . unArg
  
  instance ExtractCalls a => ExtractCalls (Dom a) where
    extract = extract . unDom
  
  instance ExtractCalls a => ExtractCalls (Elim' a) where
    extract Proj{}    = return empty
    extract (Apply a) = extract $ unArg a
    extract (IApply x y a) = extract (x,(y,a)) -- TODO Andrea: conservative
  
  instance ExtractCalls a => ExtractCalls [a] where
    extract = mapM' extract
  
  instance (ExtractCalls a, ExtractCalls b) => ExtractCalls (a,b) where
    extract (a, b) = CallGraph.union <$> extract a <*> extract b
  
  instance (ExtractCalls a, ExtractCalls b, ExtractCalls c) => ExtractCalls (a,b,c) where
    extract (a, b, c) = extract (a, (b, c))
  
  -- | Sorts can contain arbitrary terms of type @Level@,
  --   so look for recursive calls also in sorts.
  --   Ideally, 'Sort' would not be its own datatype but just
  --   a subgrammar of 'Term', then we would not need this boilerplate.
  
  instance ExtractCalls Sort where
    extract s = do
      liftTCM $ do
        reportSDoc "term.sort" 20 $
          "extracting calls from sort" <+> prettyTCM s
        reportSDoc "term.sort" 50 $
          text ("s = " ++ show s)
      case s of
        Inf _ _    -> return empty
        SizeUniv   -> return empty
        LockUniv   -> return empty
        LevelUniv  -> return empty
        IntervalUniv -> return empty
        Univ _ t       -> terUnguarded $ extract t  -- no guarded levels
        PiSort a s1 s2 -> extract (a, s1, s2)
        FunSort s1 s2 -> extract (s1, s2)
        UnivSort s -> extract s
        MetaS x es -> return empty
        DefS d es  -> return empty
        DummyS{}   -> return empty
  
  -- | Extract recursive calls from a type.
  
  instance ExtractCalls Type where
    extract (El s t) = extract (s, t)
  
  instance ExtractCalls a => ExtractCalls (Tele a) where
    extract = \case
      EmptyTel        -> mempty
      ExtendTel a tel -> extract a <> extract tel
  
  -- | Extract recursive calls from a constructor application.
  
  constructor
    :: QName
      -- ^ Constructor name.
    -> Induction
      -- ^ Should the constructor be treated as inductive or coinductive?
    -> [(Arg Term, Bool)]
      -- ^ All the arguments,
      --   and for every argument a boolean which is 'True' iff the
      --   argument should be viewed as preserving guardedness.
    -> TerM Calls
  constructor c ind args = do
    cutoff <- terGetCutOff
    let ?cutoff = cutoff
    forM' args $ \ (arg, preserves) -> do
      let g' = case (preserves, ind) of
               (True,  Inductive)   -> id
               (True,  CoInductive) -> (Order.lt .*.)
               (False, _)           -> const Order.unknown
      terModifyGuarded g' $ extract arg
  
  -- | Handles function applications @g es@.
  
  function :: QName -> Elims -> TerM Calls
  function g es0 = do
  
      f       <- terGetCurrent
      names   <- terGetMutual
      guarded <- terGetGuarded
  
      -- let gArgs = Def g es0
      liftTCM $ reportSDoc "term.function" 30 $
        "termination checking function call " <+> prettyTCM (Def g es0)
  
      -- First, look for calls in the arguments of the call gArgs.
  
      -- If the function is a projection but not for a coinductive record,
      -- then preserve guardedness for its principal argument.
      isProj <- isProjectionButNotCoinductive g
      let unguards = ListInf.repeat Order.unknown
      let guards = applyWhen isProj (guarded :<) unguards
      -- Collect calls in the arguments of this call.
      let args = map unArg $ argsFromElims es0
      calls <- forM' (zip guards args) $ \ (guard, a) -> do
        terSetGuarded guard $ extract a
  
      -- Then, consider call gArgs itself.
  
      liftTCM $ reportSDoc "term.found.call" 20 $
        sep [ "found call from" <+> prettyTCM f
            , nest 2 $ "to" <+> prettyTCM g
            ]
  
      -- insert this call into the call list
      case Set.lookupIndex g names of
  
         -- call leads outside the mutual block and can be ignored
         Nothing   -> return calls
  
         -- call is to one of the mutally recursive functions/record
         Just gInd -> do
           cutoff <- terGetCutOff
           let ?cutoff = cutoff
  
           -- Andreas, 2017-02-14, issue #2458:
           -- If we have inlined with-functions, we could be illtyped,
           -- hence, do not reduce anything.
           -- Andreas, 2017-06-20 issue #2613:
           -- We still need to reduce constructors, even when with-inlining happened.
           es <- -- ifM terGetHaveInlinedWith (return es0) {-else-} $
             liftTCM $ forM es0 $
               -- 2017-09-09, re issue #2732
               -- The eta-contraction that was here does not seem necessary to make structural order
               -- comparison not having to worry about eta.
               -- Maybe we thought an eta redex could come from a meta instantiation.
               -- However, eta-contraction is already performed by instantiateFull.
               -- See test/Succeed/Issue2732-termination.agda.
               traverse reduceCon <=< instantiateFull
  
             -- 2017-05-16, issue #2403: Argument normalization is too expensive,
             -- even if we only expand non-recursive functions.
             -- Argument normalization TURNED OFF.
             -- liftTCM $ billTo [Benchmark.Termination, Benchmark.Reduce] $ do
             --  -- Andreas, 2017-01-13, issue #2403, normalize arguments for the structural ordering.
             --  -- Andreas, 2017-03-25, issue #2495, restrict this to non-recursive functions
             --  -- otherwise, the termination checking may run forever.
             --  reportSLn "term.reduce" 90 $ "normalizing call arguments"
             --  modifyAllowedReductions (List.\\ [UnconfirmedReductions,RecursiveReductions]) $
             --    forM es0 $ \ e -> do
             --      reportSDoc "term.reduce" 95 $ "normalizing " <+> prettyTCM e
             --      etaContract =<< normalise e
  
           -- Compute the call matrix.
  
           -- Andreas, 2014-03-26 only 6% of termination time for library test
           -- spent on call matrix generation
           (nrows, ncols, matrix) <- billTo [Benchmark.Termination, Benchmark.Compare] $
             compareArgs es
  
           -- Andreas, 2022-03-21, #5823:
           -- If we are "calling" a record type we are guarded unless the origin
           -- of the termination analysis is itself a record.
           -- This is because we usually do not "unfold" record types into their
           -- field telescope.  We only do so when trying to construct the
           -- unique inhabitant of record type (singleton analysis).
           -- In the latter case, a call to a record type is not guarding.
           guarded' <- isRecord g >>= \case
             Just{} -> terGetTarget >>= \case
               TargetRecord
                 -> return guarded
               _ -> return (guarded .*. Order.lt)
                      -- guarding when we call a record and not termination checking a record
             Nothing
               -- only a delayed definition can be guarded
               | Order.decreasing guarded
                 -> return Order.le
               | otherwise
                 -> return guarded
           liftTCM $ reportSLn "term.guardedness" 20 $
             "composing with guardedness " ++ prettyShow guarded ++
             " counting as " ++ prettyShow guarded'
           let matrix' = composeGuardedness guarded' matrix
  
           -- Andreas, 2013-04-26 FORBIDDINGLY expensive!
           -- This PrettyTCM QName cost 50% of the termination time for std-lib!!
           -- gPretty <-liftTCM $ billTo [Benchmark.Termination, Benchmark.Level] $
           --   render <$> prettyTCM g
  
           -- Andreas, 2013-05-19 as pointed out by Andrea Vezzosi,
           -- printing the call eagerly is forbiddingly expensive.
           -- So we build a closure such that we can print the call
           -- whenever we really need to.
           -- This saves 30s (12%) on the std-lib!
           -- Andreas, 2015-01-21 Issue 1410: Go to the module where g is defined
           -- otherwise its free variables with be prepended to the call
           -- in the error message.
           doc <- liftTCM $ withCurrentModule (qnameModule g) $ buildClosure $
             Def g $ List.dropWhileEnd ((Inserted ==) . getOrigin) es0
             -- Andreas, 2018-07-22, issue #3136
             -- Dropping only inserted arguments at the end, since
             -- dropping arguments in the middle might make the printer crash.
             -- Def g $ filter ((/= Inserted) . getOrigin) es0
             -- Andreas, 2017-01-05, issue #2376
             -- Remove arguments inserted by etaExpandClause.
  
           let src  = fromMaybe __IMPOSSIBLE__ $ Set.lookupIndex f names
               tgt  = gInd
               cm   = makeCM ncols nrows matrix'
               info = CallPath $ singleton $
                      CallInfo
                        { callInfoTarget = g
                        , callInfoCall   = doc
                        }
           verboseS "term.kept.call" 5 $ do
             pats <- terGetPatterns
             reportSDoc "term.kept.call" 5 $ vcat
               [ "kept call from" <+> text (prettyShow f) <+> hsep (map prettyTCM pats)
               , nest 2 $ "to" <+> text (prettyShow g) <+>
                           hsep (map (parens . prettyTCM) args)
               , nest 2 $ "call matrix (with guardedness): "
               , nest 2 $ pretty cm
               ]
           return $ CallGraph.insert src tgt cm info calls
  
    where
      -- We have to reduce constructors in case they're reexported.
      -- Andreas, Issue 1530: constructors have to be reduced deep inside terms,
      -- thus, we need to use traverseTermM.
      reduceCon :: Term -> TCM Term
      reduceCon = traverseTermM $ \case
        Con c ci vs -> (`applyE` vs) <$> reduce (Con c ci [])  -- make sure we don't reduce the arguments
        t -> return t
  
  
  -- | Try to get rid of a function call targeting the current SCC
  --   using a non-recursive clause.
  --
  --   This can help copattern definitions of dependent records.
  tryReduceNonRecursiveClause
    :: QName                 -- ^ Function
    -> Elims                 -- ^ Arguments
    -> (Term -> TerM Calls)  -- ^ Continue here if we managed to reduce.
    -> TerM Calls            -- ^ Otherwise, continue here.
    -> TerM Calls
  tryReduceNonRecursiveClause g es continue fallback = do
    -- Andreas, 2020-02-06, re: issue #906
    let v0 = Def g es
    reportSDoc "term.reduce" 40 $ "Trying to reduce away call: " <+> prettyTCM v0
  
    -- First, make sure the function is in the current SCC.
    ifM (notElem g <$> terGetMutual) fallback {-else-} $ do
    reportSLn "term.reduce" 40 $ "This call is in the current SCC!"
  
    def <- getConstInfo g
    -- -- Then, collect its clauses.
    -- cls <- defClauses <$> getConstInfo g
    -- reportSLn "term.reduce" 40 $ unwords [ "Function has", show (length cls), "clauses"]
    -- reportSDoc "term.reduce" 80 $ vcat $ map (prettyTCM . NamedClause g True) cls
    -- reportSLn  "term.reduce" 80 . ("allowed reductions = " ++) . show . SmallSet.elems
    --   =<< asksTC envAllowedReductions
  
    -- Finally, try to reduce with the non-recursive clauses (and no rewrite rules).
    r <- liftTCM $
      modifyAllowedReductions (SmallSet.delete UnconfirmedReductions) $
      runReduceM $ appDefE_ g v0 (defClauses def) (defCompiled def) [] (map notReduced es)
    case r of
      NoReduction{}    -> fallback
      YesReduction _ v -> do
        reportSDoc "term.reduce" 30 $ vcat
          [ "Termination checker: Successfully reduced away call:"
          , nest 2 $ prettyTCM v0
          ]
        verboseS "term.reduce" 5 $ tick "termination-checker-reduced-nonrecursive-call"
        continue v
  
  -- | Extract recursive calls from a term.
  
  instance ExtractCalls Term where
    extract t = do
      reportSDoc "term.check.term" 50 $ do
        "looking for calls in" <+> prettyTCM t
  
      -- Instantiate top-level MetaVar.
      instantiate t >>= \case
  
        -- Constructed value.
        Con ConHead{conName = c, conDataRecord = dataOrRec} _ es -> do
          let args = fromMaybe __IMPOSSIBLE__ $ allApplyElims es
          -- A constructor preserves the guardedness of all its arguments.
          -- Andreas, 2022-09-19, issue #6108:
          -- A higher constructor does not.  So check if there is an @IApply@ amoung @es@.
          let noIApply = all isProperApplyElim es
          let argsg = map (,noIApply) args
  
          -- If we encounter a coinductive record constructor
          -- in a type mutual with the current target
          -- then we count it as guarding.
          let inductive   = return Inductive    -- not guarding, but preserving
              coinductive = return CoInductive  -- guarding
          -- ♯ is guarding
          ind <- ifM ((Just c ==) <$> terGetSharp) coinductive $ {-else-} do
            -- data constructors are not guarding
            if dataOrRec == IsData then inductive else do
            -- abstract constructors are not guarding
            caseMaybeM (isRecordConstructor c) inductive $ \ (q, def) -> do
              reportSLn "term.check.term" 50 $ "constructor " ++ prettyShow c ++ " has record type " ++ prettyShow q
              -- inductive record constructors are not guarding
              if _recInduction def /= Just CoInductive then inductive else do
              -- coinductive constructors unrelated to the mutually
              -- constructed inhabitants of coinductive types are not guarding
              ifM (targetElem . fromMaybe __IMPOSSIBLE__ $ _recMutual def)
                 {-then-} coinductive
                 {-else-} inductive
          constructor c ind argsg
  
        -- Function, data, or record type.
        Def g es -> tryReduceNonRecursiveClause g es extract $ function g es
  
        -- Abstraction. Preserves guardedness.
        Lam h b -> extract b
  
        -- Neutral term. Destroys guardedness.
        Var i es -> terUnguarded $ extract es
  
        -- Dependent function space.
        Pi a (Abs x b) ->
          CallGraph.union <$>
          extract a <*> do
            a <- maskSizeLt a  -- OR: just do not add a to the context!
            addContext (x, a) $ terRaise $ extract b
  
        -- Non-dependent function space.
        Pi a (NoAbs _ b) ->
          CallGraph.union <$> extract a <*> extract b
  
        -- Literal.
        Lit l -> return empty
  
        -- Sort.
        Sort s -> extract s
  
        -- Unsolved metas are not considered termination problems, there
        -- will be a warning for them anyway.
        MetaV x args -> return empty
  
        -- Erased and not-yet-erased proof.
        DontCare t -> extract t
  
        -- Level.
        Level l -> -- billTo [Benchmark.Termination, Benchmark.Level] $ do
          -- Andreas, 2014-03-26 Benchmark discontinued, < 0.3% spent on levels.
          extract l
  
        -- Dummy.
        Dummy{} -> return empty
  
  -- | Extract recursive calls from level expressions.
  
  instance ExtractCalls Level where
    extract (Max n as) = extract as
  
  instance ExtractCalls PlusLevel where
    extract (Plus n l) = extract l
  
  -- | Rewrite type @tel -> Size< u@ to @tel -> Size@.
  maskSizeLt :: MonadTCM tcm => Dom Type -> tcm (Dom Type)
  maskSizeLt !dom = liftTCM $ do
    let a = unDom dom
    (msize, msizelt) <- getBuiltinSize
    case (msize, msizelt) of
      (_ , Nothing) -> return dom
      (Nothing, _)  -> __IMPOSSIBLE__
      (Just size, Just sizelt) -> do
        TelV tel c <- telView a
        case a of
          El s (Def d [v]) | d == sizelt -> return $
            abstract tel (El s $ Def size []) <$ dom
          _ -> return dom
  
  {- | @compareArgs es@
  
       Compare the list of de Bruijn patterns (=parameters) @pats@
       with a list of arguments @es@ and create a call maxtrix
       with |es| rows and |pats| columns.
  
       The guardedness is the number of projection patterns in @pats@
       minus the number of projections in @es@.
   -}
  compareArgs :: [Elim] -> TerM (Int, Int, [[Order]])
  compareArgs es = do
    pats <- terGetPatterns
    liftTCM $ reportSDoc "term.compareArgs" 90 $ vcat
      [ text $ "comparing " ++ show (length es) ++ " args to " ++ show (length pats) ++ " patterns"
      ]
    -- apats <- annotatePatsWithUseSizeLt pats
    -- reportSDoc "term.compare" 20 $
    --   "annotated patterns = " <+> sep (map prettyTCM apats)
    -- matrix <- forM es $ \ e -> forM apats $ \ (b, p) -> terSetUseSizeLt b $ compareElim e p
    matrix <- withUsableVars pats $ forM es $ \ e -> forM pats $ \ p -> compareElim e p
  
    -- Count the number of coinductive projection(pattern)s in caller and callee.
    -- Only recursive coinductive projections are eligible (Issue 1209).
    projsCaller <- length <$> do
      filterM (isCoinductiveProjection True) $ mapMaybe (fmap (headAmbQ . snd) . isProjP . getMasked) pats
    projsCallee <- length <$> do
      filterM (isCoinductiveProjection True) $ mapMaybe (fmap snd . isProjElim) es
    cutoff <- terGetCutOff
    let ?cutoff = cutoff
    let guardedness = decr True $ projsCaller - projsCallee
    liftTCM $ reportSDoc "term.guardedness" 30 $ sep
      [ "compareArgs:"
      , nest 2 $ text $ "projsCaller = " ++ prettyShow projsCaller
      , nest 2 $ text $ "projsCallee = " ++ prettyShow projsCallee
      , nest 2 $ text $ "guardedness of call: " ++ prettyShow guardedness
      ]
    return $ addGuardedness guardedness (size es, size pats, matrix)
  
  -- | Traverse patterns from left to right.
  --   When we come to a projection pattern,
  --   switch usage of SIZELT constraints:
  --   on, if coinductive,
  --   off, if inductive.
  --
  --   UNUSED
  --annotatePatsWithUseSizeLt :: [DeBruijnPattern] -> TerM [(Bool,DeBruijnPattern)]
  --annotatePatsWithUseSizeLt = loop where
  --  loop [] = return []
  --  loop (p@(ProjP _ q) : pats) = ((False,p) :) <$> do projUseSizeLt q $ loop pats
  --  loop (p : pats) = (\ b ps -> (b,p) : ps) <$> terGetUseSizeLt <*> loop pats
  
  
  -- | @compareElim e dbpat@
  
  compareElim :: Elim -> Masked DeBruijnPattern -> TerM Order
  compareElim e p = do
    liftTCM $ do
      reportSDoc "term.compare" 30 $ sep
        [ "compareElim"
        , nest 2 $ "e = " <> prettyTCM e
        , nest 2 $ "p = " <> prettyTCM p
        ]
      reportSDoc "term.compare" 50 $ sep
        [ nest 2 $ text $ "e = " ++ show e
        , nest 2 $ text $ "p = " ++ show p
        ]
    case (e, getMasked p) of
      (Proj _ d, ProjP _ d') -> do
        d  <- getOriginalProjection d
        d' <- getOriginalProjection d'
        o  <- compareProj d d'
        reportSDoc "term.compare" 30 $ sep
          [ text $ "comparing callee projection " ++ prettyShow d
          , text $ "against caller projection " ++ prettyShow d'
          , text $ "yields order " ++ prettyShow o
          ]
        return o
      (Proj{}, _)            -> return Order.unknown
      (Apply{}, ProjP{})     -> return Order.unknown
      (Apply arg, _)         -> compareTerm (unArg arg) p
      -- TODO Andrea: making sense?
      (IApply{}, ProjP{})  -> return Order.unknown
      (IApply _ _ arg, _)    -> compareTerm arg p
  
  -- | In dependent records, the types of later fields may depend on the
  --   values of earlier fields.  Thus when defining an inhabitant of a
  --   dependent record type such as Σ by copattern matching,
  --   a recursive call eliminated by an earlier projection (proj₁) might
  --   occur in the definition at a later projection (proj₂).
  --   Thus, earlier projections are considered "smaller" when
  --   comparing copattern spines.  This is an ok approximation
  --   of the actual dependency order.
  --   See issues 906, 942.
  compareProj :: MonadTCM tcm => QName -> QName -> tcm Order
  compareProj d d'
    | d == d' = return Order.le
    | otherwise = liftTCM $ do
        -- different projections
        mr  <- getRecordOfField d
        mr' <- getRecordOfField d'
        case (mr, mr') of
          (Just r, Just r') | r == r' -> do
            -- of same record
            def <- theDef <$> getConstInfo r
            case def of
              Record{ recFields = fs } -> do
                fs <- return $ map unDom fs
                case (List.find (d ==) fs, List.find (d' ==) fs) of
                  (Just i, Just i')
                    -- earlier field is smaller
                    | i < i'    -> return Order.lt
                    | i == i'   -> do
                       __IMPOSSIBLE__
                    | otherwise -> return Order.unknown
                  _ -> __IMPOSSIBLE__
              _ -> __IMPOSSIBLE__
          _ -> return Order.unknown
  
  -- | 'makeCM' turns the result of 'compareArgs' into a proper call matrix
  makeCM :: Int -> Int -> [[Order]] -> CallMatrix
  makeCM ncols nrows matrix = CallMatrix $
    Matrix.fromLists (Matrix.Size nrows ncols) matrix
  
  -- | 'addGuardedness' adds guardedness flag in the upper left corner
  -- (0,0).
  addGuardedness :: Order -> (Int, Int, [[Order]]) -> (Int, Int, [[Order]])
  addGuardedness o (nrows, ncols, m) =
    (nrows + 1, ncols + 1,
     (o : replicate ncols Order.unknown) : map (Order.unknown :) m)
  
  -- | Compose something with the upper-left corner of a call matrix
  composeGuardedness :: (?cutoff :: CutOff) => Order -> [[Order]] -> [[Order]]
  composeGuardedness o ((corner : row) : rows) = ((o .*. corner) : row) : rows
  composeGuardedness _ _ = __IMPOSSIBLE__
  
  -- | Stripping off a record constructor is not counted as decrease, in
  --   contrast to a data constructor.
  --   A record constructor increases/decreases by 0, a data constructor by 1.
  offsetFromConstructor :: HasConstInfo tcm => QName -> tcm Int
  offsetFromConstructor c =
    ifM (isEtaOrCoinductiveRecordConstructor c) (return 0) (return 1)
  
  --UNUSED Liang-Ting 2019-07-16
  ---- | Compute the proper subpatterns of a 'DeBruijnPattern'.
  --subPatterns :: DeBruijnPattern -> [DeBruijnPattern]
  --subPatterns = foldPattern $ \case
  --  ConP _ _ ps -> map namedArg ps
  --  DefP _ _ ps -> map namedArg ps -- TODO check semantics
  --  VarP _ _    -> mempty
  --  LitP _      -> mempty
  --  DotP _ _    -> mempty
  --  ProjP _ _   -> mempty
  --  IApplyP{}   -> mempty
  
  
  compareTerm :: Term -> Masked DeBruijnPattern -> TerM Order
  compareTerm t p = do
  --   reportSDoc "term.compare" 25 $
  --     " comparing term " <+> prettyTCM t <+>
  --     " to pattern " <+> prettyTCM p
    t <- liftTCM $ stripAllProjections t
    o <- compareTerm' t p
    liftTCM $ reportSDoc "term.compare" 25 $
      " comparing term " <+> prettyTCM t <+>
      " to pattern " <+> prettyTCM p <+>
      text (" results in " ++ prettyShow o)
    return o
  
  
  -- | Remove all non-coinductive projections from an algebraic term
  --   (not going under binders).
  --   Also, remove 'DontCare's.
  --
  class StripAllProjections a where
    stripAllProjections :: a -> TCM a
  
  instance StripAllProjections a => StripAllProjections (Arg a) where
    stripAllProjections = traverse stripAllProjections
  
  instance StripAllProjections Elims where
    stripAllProjections es =
      case es of
        []             -> return []
        (Apply a : es) -> do
          (:) <$> (Apply <$> stripAllProjections a) <*> stripAllProjections es
        (IApply x y a : es) -> do
          -- TODO Andrea: are we doind extra work?
          (:) <$> (IApply <$> stripAllProjections x
                          <*> stripAllProjections y
                          <*> stripAllProjections a)
              <*> stripAllProjections es
        (Proj o p  : es) -> do
          isP <- isProjectionButNotCoinductive p
          applyUnless isP (Proj o p :) <$> stripAllProjections es
  
  instance StripAllProjections Args where
    stripAllProjections = mapM stripAllProjections
  
  instance StripAllProjections Term where
    stripAllProjections t = do
      case t of
        Var i es   -> Var i <$> stripAllProjections es
        Con c ci ts -> do
          -- Andreas, 2019-02-23, re #2613.  This is apparently not necessary:
          -- c <- fromRightM (\ err -> return c) $ getConForm (conName c)
          Con c ci <$> stripAllProjections ts
        Def d es   -> Def d <$> stripAllProjections es
        DontCare t -> stripAllProjections t
        _ -> return t
  
  -- | Normalize outermost constructor name in a pattern.
  
  reduceConPattern :: DeBruijnPattern -> TCM DeBruijnPattern
  reduceConPattern = \case
    ConP c i ps -> fromRightM (\ err -> return c) (getConForm (conName c)) <&> \ c' ->
      ConP c' i ps
    p -> return p
  
  -- | @compareTerm' t dbpat@
  
  compareTerm' :: Term -> Masked DeBruijnPattern -> TerM Order
  compareTerm' v mp@(Masked m p) = do
    suc  <- terGetSizeSuc
    cutoff <- terGetCutOff
    let ?cutoff = cutoff
    v <- liftTCM (instantiate v)
    p <- liftTCM $ reduceConPattern p
    case (v, p) of
  
      -- Andreas, 2013-11-20 do not drop projections,
      -- in any case not coinductive ones!:
      (Var i es, _) | Just{} <- allApplyElims es ->
        compareVar i mp
  
      (DontCare t, _) ->
        compareTerm' t mp
  
      -- Andreas, 2014-09-22, issue 1281:
      -- For metas, termination checking should be optimistic.
      -- If there is any instance of the meta making termination
      -- checking succeed, then we should not fail.
      -- Thus, we assume the meta will be instantiated with the
      -- deepest variable in @p@.
      -- For sized types, the depth is maximally
      -- the number of SIZELT hypotheses one can have in a context.
      (MetaV{}, p) -> Order.decr True . max (if m then 0 else patternDepth p) . pred <$>
         terAsks _terSizeDepth
  
      -- Successor on both sides cancel each other.
      -- We ignore the mask for sizes.
      (Def s [Apply t], ConP s' _ [p]) | s == conName s' && Just s == suc ->
        compareTerm' (unArg t) (notMasked $ namedArg p)
  
      -- Register also size increase.
      (Def s [Apply t], p) | Just s == suc ->
        -- Andreas, 2012-10-19 do not cut off here
        increase 1 <$> compareTerm' (unArg t) mp
  
      -- In all cases that do not concern sizes,
      -- we cannot continue if pattern is masked.
  
      _ | m -> return Order.unknown
  
      (Lit l, LitP _ l')
        | l == l'     -> return Order.le
        | otherwise   -> return Order.unknown
  
      (Lit l, _) -> do
        v <- liftTCM $ constructorForm v
        case v of
          Lit{}       -> return Order.unknown
          v           -> compareTerm' v mp
  
      -- Andreas, 2011-04-19 give subterm priority over matrix order
  
      (Con{}, ConP c _ ps) | any (isSubTerm v . namedArg) ps ->
        decr True <$> offsetFromConstructor (conName c)
  
      (Con c _ es, ConP c' _ ps) | conName c == conName c'->
        let ts = fromMaybe __IMPOSSIBLE__ $ allApplyElims es in
        compareConArgs ts ps
  
      (Con _ _ [], _) -> return Order.le
  
      -- new case for counting constructors / projections
      -- register also increase
      (Con c _ es, _) -> do
        let ts = fromMaybe __IMPOSSIBLE__ $ allApplyElims es
        increase <$> offsetFromConstructor (conName c)
                 <*> (infimum <$> mapM (\ t -> compareTerm' (unArg t) mp) ts)
  
      (t, p) -> return $ subTerm t p
  
  -- | @subTerm@ computes a size difference (Order)
  subTerm :: (?cutoff :: CutOff) => Term -> DeBruijnPattern -> Order
  subTerm t p = if equal t p then Order.le else properSubTerm t p
    where
      equal (Con c _ es) (ConP c' _ ps) =
        let ts = fromMaybe __IMPOSSIBLE__ $ allApplyElims es in
        and $ (conName c == conName c')
            : (length ts == length ps)
            : zipWith (\ t p -> equal (unArg t) (namedArg p)) ts ps
      equal (Var i []) (VarP _ x) = i == dbPatVarIndex x
      equal (Lit l)    (LitP _ l') = l == l'
      -- Terms.
      -- Checking for identity here is very fragile.
      -- However, we cannot do much more, as we are not allowed to normalize t.
      -- (It might diverge, and we are just in the process of termination checking.)
      equal t         (DotP _ t') = t == t'
      equal _ _ = False
  
      properSubTerm t (ConP _ _ ps) =
        setUsability True $ decrease 1 $ supremum $ map (subTerm t . namedArg) ps
      properSubTerm _ _ = Order.unknown
  
  isSubTerm :: (?cutoff :: CutOff) => Term -> DeBruijnPattern -> Bool
  isSubTerm t p = nonIncreasing $ subTerm t p
  
  compareConArgs :: Args -> [NamedArg DeBruijnPattern] -> TerM Order
  compareConArgs ts ps = do
    cutoff <- terGetCutOff
    let ?cutoff = cutoff
    case compare (length ts) (length ps) of
  
      -- We may assume |ps| >= |ts|, otherwise c ps would be of functional type
      -- which is impossible.
      GT -> __IMPOSSIBLE__
  
      -- Andreas, 2022-08-31, issue #6059: doing anything smarter than
      -- @unknown@ here can lead to non-termination.
      LT -> return Order.unknown
  
      EQ -> List.foldl' (Order..*.) Order.le <$>
                 zipWithM compareTerm' (map unArg ts) (map (notMasked . namedArg) ps)
         -- corresponds to taking the size, not the height
         -- allows examples like (x, y) < (Succ x, y)
  {- version which does an "order matrix"
     -- Andreas, 2013-02-18 disabled because it is unclear
     -- how to scale idempotency test to matrix-shaped orders (need thinking/researcH)
     -- Trigges issue 787.
          (_,_) -> do -- build "call matrix"
            m <- mapM (\t -> mapM (compareTerm' suc (unArg t)) ps) ts
            let m2 = makeCM (length ps) (length ts) m
            return $ Order.orderMat (Order.mat m2)
  -}
  {- version which takes height
  --    if null ts then Order.Le
  --               else Order.infimum (zipWith compareTerm' (map unArg ts) ps)
  -}
  
  compareVar :: Nat -> Masked DeBruijnPattern -> TerM Order
  compareVar i (Masked m p) = do
    suc    <- terGetSizeSuc
    cutoff <- terGetCutOff
    let ?cutoff = cutoff
    let no = return Order.unknown
    case p of
      ProjP{}   -> no
      IApplyP _ _ _ x  -> compareVarVar i (Masked m x)
      LitP{}    -> no
      DotP{}   -> no
      VarP _ x  -> compareVarVar i (Masked m x)
  
      ConP s _ [p] | Just (conName s) == suc ->
        setUsability True . decrease 1 <$> compareVar i (notMasked $ namedArg p)
  
      ConP c pi ps -> if m then no else setUsability True <$> do
        let
          dropit Forced _ = Nothing
          dropit NotForced x = Just x
        ps <- ifM (optForcedArgumentRecursion <$> pragmaOptions)
          {- then -} (pure ps)
          {- else -} (mapForcedArguments (conName c) ps dropit)
        decrease <$> offsetFromConstructor (conName c)
                 <*> (Order.supremum <$> mapM (compareVar i . notMasked . namedArg) ps)
      DefP _ c ps -> if m then no else setUsability True <$> do
        decrease <$> offsetFromConstructor c
                 <*> (Order.supremum <$> mapM (compareVar i . notMasked . namedArg) ps)
        -- This should be fine for c == hcomp
  
  -- | Compare two variables.
  --
  --   The first variable comes from a term, the second from a pattern.
  compareVarVar :: Nat -> Masked DBPatVar -> TerM Order
  compareVarVar i (Masked m x@(DBPatVar _ j))
    | i == j = if not m then return Order.le else liftTCM $
        -- If j is a size, we ignore the mask.
        ifM (isJust <$> do isSizeType =<< reduce =<< typeOfBV j)
          {- then -} (return Order.le)
          {- else -} (return Order.unknown)
    | otherwise = do
        -- record usability of variable
        u <- (i `VarSet.member`) <$> terGetUsableVars
        -- Andreas, 2017-07-26, issue #2331.
        -- The usability logic is refuted by bounded size quantification in terms.
        -- Thus, it is switched off (the infrastructure remains in place for now).
        if not u then return Order.unknown else do
        -- Only if usable:
        res <- isBounded i
        case res of
          BoundedNo  -> return Order.unknown
          BoundedLt v -> setUsability u . decrease 1 <$> compareTerm' v (Masked m $ varP x)
  