diff --git a/COPYING b/COPYING
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+++ b/COPYING
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+ GNU GENERAL PUBLIC LICENSE
+ Version 3, 29 June 2007
+
+ Copyright (C) 2007 Free Software Foundation, Inc.
+ Everyone is permitted to copy and distribute verbatim copies
+ of this license document, but changing it is not allowed.
+
+ Preamble
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+Program, unless a warranty or assumption of liability accompanies a
+copy of the Program in return for a fee.
+
+ END OF TERMS AND CONDITIONS
+
+ How to Apply These Terms to Your New Programs
+
+ If you develop a new program, and you want it to be of the greatest
+possible use to the public, the best way to achieve this is to make it
+free software which everyone can redistribute and change under these terms.
+
+ To do so, attach the following notices to the program. It is safest
+to attach them to the start of each source file to most effectively
+state the exclusion of warranty; and each file should have at least
+the "copyright" line and a pointer to where the full notice is found.
+
+
+ Copyright (C)
+
+ This program is free software: you can redistribute it and/or modify
+ it under the terms of the GNU General Public License as published by
+ the Free Software Foundation, either version 3 of the License, or
+ (at your option) any later version.
+
+ This program is distributed in the hope that it will be useful,
+ but WITHOUT ANY WARRANTY; without even the implied warranty of
+ MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
+ GNU General Public License for more details.
+
+ You should have received a copy of the GNU General Public License
+ along with this program. If not, see .
+
+Also add information on how to contact you by electronic and paper mail.
+
+ If the program does terminal interaction, make it output a short
+notice like this when it starts in an interactive mode:
+
+ Copyright (C)
+ This program comes with ABSOLUTELY NO WARRANTY; for details type `show w'.
+ This is free software, and you are welcome to redistribute it
+ under certain conditions; type `show c' for details.
+
+The hypothetical commands `show w' and `show c' should show the appropriate
+parts of the General Public License. Of course, your program's commands
+might be different; for a GUI interface, you would use an "about box".
+
+ You should also get your employer (if you work as a programmer) or school,
+if any, to sign a "copyright disclaimer" for the program, if necessary.
+For more information on this, and how to apply and follow the GNU GPL, see
+.
+
+ The GNU General Public License does not permit incorporating your program
+into proprietary programs. If your program is a subroutine library, you
+may consider it more useful to permit linking proprietary applications with
+the library. If this is what you want to do, use the GNU Lesser General
+Public License instead of this License. But first, please read
+.
diff --git a/Dedukti.hs b/Dedukti.hs
new file mode 100644
--- /dev/null
+++ b/Dedukti.hs
@@ -0,0 +1,85 @@
+-- |
+-- Copyright : © 2009 CNRS - École Polytechnique - INRIA
+-- License : GPL
+--
+-- This module is the entry point for europa. Based on the command line
+-- arguments the appropriate driver is invoked. Everything is coordinated by
+-- the driver. This module is also the place where global configuration data
+-- is initialized.
+module Main where
+
+import System.Environment
+import qualified Dedukti.Config as Config
+import Dedukti.DkM
+import Dedukti.Module
+import Dedukti.Driver.Batch
+import Dedukti.Driver.Compile
+import Text.PrettyPrint.Leijen
+import Control.Monad (unless, when)
+import System.Console.GetOpt
+import System.Exit
+import System.IO
+import qualified Data.Text.Lazy.Encoding as T
+import qualified Data.Text.Lazy as T
+import qualified Data.ByteString.Lazy as B
+
+
+data Flag = FlagMake | FlagHelp | FlagVersion | FlagVerbose | FlagVeryVerbose
+ deriving (Eq, Ord, Show)
+
+options = [ Option [] ["make"] (NoArg FlagMake)
+ "Build MODULE and all its dependencies in one go."
+ , Option ['v'] [] (OptArg verb "v")
+ "Be verbose. -vv to be even more verbose."
+ , Option ['h'] ["help"] (NoArg FlagHelp) "This usage information."
+ , Option [] ["version"] (NoArg FlagVersion) "Output version information then exit." ]
+ where verb Nothing = FlagVerbose
+ verb (Just "v") = FlagVeryVerbose
+ verb _ = error "Unrecognized verbosity level."
+
+printUsage = do
+ self <- parameter Config.imageName
+ let header = show $ text "Usage:" <+>
+ (text self <+> text "[OPTION]..." <+> text "MODULE")
+ io $ hPutStrLn stderr header
+
+printHelp = do
+ self <- parameter Config.imageName
+ let header = show $ text "Usage:" <+>
+ (text self <+> text "[OPTION]..." <+> text "MODULE")
+ <$> text "Options:"
+ io $ putStrLn (usageInfo header options)
+
+bailout = printUsage >> io exitFailure
+
+printVersion = do
+ self <- parameter Config.imageName
+ version <- parameter Config.version
+ io $ B.putStrLn $ T.encodeUtf8 $ T.pack $ flip displayS "" $ renderPretty 0.70 100 $
+ text "Dedukti" <+> text version <> line <> line <>
+ text "Copyright (c) 2009 CNRS - École Polytechnique - INRIA." <> line <> line <>
+ fillText "You may redistribute copies of Dedukti under the terms of \
+ \the GNU General Public License. For more information about \
+ \these matters, see the file named COPYING."
+
+initializeConfiguration = foldr aux Config.defaultConfig
+ where aux FlagVerbose c = c { Config.verbosity = Verbose }
+ aux FlagVeryVerbose c = c { Config.verbosity = Debug }
+ aux _ c = c
+
+main = do
+ args <- getArgs
+ let (opts, files, errs) = getOpt RequireOrder options args
+ when (not (null errs)) $ do
+ hPutDoc stderr (vsep (map text errs))
+ exitFailure
+ runDkM (initializeConfiguration opts) $
+ case undefined of
+ _ | FlagHelp `elem` opts -> printHelp
+ | FlagVersion `elem` opts -> printVersion
+ _ | FlagMake `elem` opts -> do
+ unless (length files == 1) bailout
+ make [moduleFromPath (head files)]
+ | otherwise -> do
+ unless (length files == 1) bailout
+ compile (moduleFromPath (head files))
diff --git a/Dedukti/Analysis/Dependency.hs b/Dedukti/Analysis/Dependency.hs
new file mode 100644
--- /dev/null
+++ b/Dedukti/Analysis/Dependency.hs
@@ -0,0 +1,17 @@
+-- |
+-- Copyright : © 2009 CNRS - École Polytechnique - INRIA
+-- License : GPL
+--
+-- Find all direct dependencies of the current module.
+
+module Dedukti.Analysis.Dependency where
+
+import Dedukti.Core
+import Dedukti.Module
+import qualified Data.Set as Set
+
+
+collectDependencies :: Module Qid a -> [MName]
+collectDependencies drs =
+ Set.toList $ Set.fromList [ m | Var x _ <- everyone drs
+ , Just m <- return (provenance x) ]
diff --git a/Dedukti/Analysis/Rule.hs b/Dedukti/Analysis/Rule.hs
new file mode 100644
--- /dev/null
+++ b/Dedukti/Analysis/Rule.hs
@@ -0,0 +1,39 @@
+module Dedukti.Analysis.Rule where
+
+import Dedukti.Core
+import Dedukti.Module
+import Dedukti.DkM
+import qualified Dedukti.Rule as Rule
+import Dedukti.Pretty ()
+import Text.PrettyPrint.Leijen hiding (group)
+import Data.List (group, sort)
+
+
+newtype NonContiguousRules = NonContiguousRules Qid
+ deriving (Eq, Ord, Typeable)
+
+instance Show NonContiguousRules where
+ show (NonContiguousRules id) =
+ show (text "Rules for" <+> pretty id <+> text "should be given contiguously.")
+
+instance Exception NonContiguousRules
+
+checkOrdering :: [TyRule Qid a] -> DkM ()
+checkOrdering rules = do
+ mapM_ (\x -> when (length x > 1) (throw $ NonContiguousRules (head x))) $
+ group $ sort $ map head $ group $ map Rule.headConstant rules
+
+newtype BadPattern = BadPattern [Qid]
+ deriving (Eq, Ord, Typeable)
+
+instance Show BadPattern where
+ show (BadPattern ids) =
+ let ppvars = sep (punctuate comma (map pretty ids))
+ in show (text "Pattern variables" <+> ppvars <+> text "cannot appear in constructor position.")
+
+instance Exception BadPattern
+
+checkHead :: TyRule Qid a -> DkM ()
+checkHead (env :@ lhs :--> rhs) =
+ let bad = [ x | App (Var x _) _ _ <- everyone lhs, x `isin` env ]
+ in when (not (null bad)) $ throw (BadPattern bad)
diff --git a/Dedukti/Analysis/Scope.hs b/Dedukti/Analysis/Scope.hs
new file mode 100644
--- /dev/null
+++ b/Dedukti/Analysis/Scope.hs
@@ -0,0 +1,76 @@
+-- |
+-- Copyright : © 2009 CNRS - École Polytechnique - INRIA
+-- License : GPL
+--
+-- Check that all occurrences of variables are in scope of their definitions.
+-- Other well-formedness checks can also be found here, such as rejecting
+-- duplicate top-level definitions and enforcing contiguity of rule
+-- defnitions.
+
+module Dedukti.Analysis.Scope where
+
+import Dedukti.Core
+import Dedukti.Module
+import qualified Dedukti.Rule as Rule
+import Dedukti.Pretty ()
+import Dedukti.DkM
+import Data.List (sort, group)
+import qualified Data.Set as Set
+
+
+newtype DuplicateDefinition = DuplicateDefinition Qid
+ deriving (Eq, Ord, Typeable)
+
+instance Show DuplicateDefinition where
+ show (DuplicateDefinition id) =
+ show (text "duplicate definition" <+> pretty id)
+
+instance Exception DuplicateDefinition
+
+newtype ScopeError = ScopeError Qid
+ deriving (Eq, Ord, Typeable)
+
+instance Show ScopeError where
+ show (ScopeError id) = show (pretty id <+> text "not in scope.")
+
+instance Exception ScopeError
+
+newtype IllegalEnvironment = IllegalEnvironment Qid
+ deriving (Eq, Ord, Typeable)
+
+instance Show IllegalEnvironment where
+ show (IllegalEnvironment id) = show (pretty id <+> text "appears in environment but not in head of rule.")
+
+instance Exception IllegalEnvironment
+
+checkUniqueness :: Module Qid a -> DkM ()
+checkUniqueness (decls, rules) = do
+ chk decls
+ mapM_ (\(env :@ _) -> chk (env_bindings env)) rules
+ where chk bs = mapM_ (\x -> when (length x > 1)
+ (throw $ DuplicateDefinition (head x))) $
+ group $ sort $ map bind_name bs
+
+checkScopes :: forall a. Show a => Set.Set Qid -> Module Qid a -> DkM ()
+checkScopes env (decls, rules) = do
+ topenv <- foldM chkBinding env decls
+ mapM_ (chkRule topenv) rules
+ where chkBinding env (x ::: ty) = do
+ chkExpr env ty
+ return $ Set.insert x env
+ chkRule topenv r@(env :@ rule) = do
+ let lhsvars = Set.fromList [ x | Var x _ <- everyone (Rule.head r) ]
+ mapM_ (\x -> when (x `Set.notMember` lhsvars) $
+ throw (IllegalEnvironment x)) (map bind_name $ env_bindings env)
+ ruleenv <- foldM chkBinding topenv $ env_bindings env
+ descendM (chkExpr (topenv `Set.union` ruleenv)) rule
+ chkExpr env t@(Var x _) = do
+ when (x `Set.notMember` env) (throw $ ScopeError x)
+ return (t :: Expr Qid a)
+ chkExpr env (Lam (x ::: ty) t _) = do
+ chkExpr env ty
+ chkExpr (Set.insert x env) t
+ chkExpr env (Pi (x ::: ty) t _) = do
+ chkExpr env ty
+ chkExpr (Set.insert x env) t
+ chkExpr env t = descendM (chkExpr env) t
diff --git a/Dedukti/CodeGen.hs b/Dedukti/CodeGen.hs
new file mode 100644
--- /dev/null
+++ b/Dedukti/CodeGen.hs
@@ -0,0 +1,26 @@
+-- |
+-- Copyright : © 2009 CNRS - École Polytechnique - INRIA
+-- License : GPL
+--
+-- Interface for all code generators.
+module Dedukti.CodeGen (CodeGen(..)) where
+
+import Dedukti.Core
+import Dedukti.Module
+import qualified Data.Text.Lazy as T
+
+
+class CodeGen o where
+ data Bundle o
+
+ -- | Emit code corresponding to an individual rule set.
+ emit :: RuleSet (Id o) (A o) -> o
+
+ coalesce :: [o] -> Bundle o
+
+ -- | Produce the byte sequence to write to a file, given the code
+ -- for all the rule sets.
+ serialize :: MName -- ^ The module name
+ -> [MName] -- ^ Dependencies
+ -> Bundle o -- ^ Code
+ -> T.Text
diff --git a/Dedukti/CodeGen/Exts.hs b/Dedukti/CodeGen/Exts.hs
new file mode 100644
--- /dev/null
+++ b/Dedukti/CodeGen/Exts.hs
@@ -0,0 +1,231 @@
+-- |
+-- Copyright : © 2009 CNRS - École Polytechnique - INRIA
+-- License : GPL
+--
+-- A code generator based on the haskell-src-exts package by Niklas Broberg.
+
+module Dedukti.CodeGen.Exts
+ (module Dedukti.CodeGen, Code) where
+
+import Dedukti.CodeGen
+import Dedukti.Core
+import Dedukti.Module
+import Dedukti.Pretty
+import qualified Dedukti.Rule as Rule
+import qualified Language.Haskell.Exts.Syntax as Hs
+import Language.Haskell.Exts.Pretty
+import qualified Data.Text.Lazy as T
+import Data.Char (toUpper)
+import qualified Data.Stream as Stream
+import Prelude hiding ((*))
+
+
+type Em a = a (Id Record) (A Record)
+
+type instance Id Record = Qid
+type instance A Record = Unannot
+
+-- External view of record.
+type Code = Record
+
+-- Create a record for each declaration in the source.
+data Record = Rec { rec_name :: Qid
+ , rec_rules :: Int -- ^ Number of rules associated with qid.
+ , rec_code :: [Hs.Decl] }
+
+instance CodeGen Record where
+ data Bundle Record = Bundle [Hs.Decl]
+
+ emit rs@(RS x ty rules) =
+ Rec x (length rules) (function rs : def_ty : def_box : zipWith defs_rule [0..] rules)
+ where def_ty = value (x .$ "ty") (code ty)
+ def_box = value (x .$ "box")
+ (primbbox (term ty) (var (x .$ "ty")) (var x))
+ -- Checking rules involves much of the same work as checking all
+ -- declarations at top-level, so let's just call the code
+ -- generation functions recursively.
+ defs_rule n (env :@ lhs :--> rhs) =
+ let rec (x ::: ty) rs = (emit (RS x ty []) :: Record) : rs
+ Bundle decls = coalesce $ foldr rec [ruleCheck] (env_bindings env)
+ in Hs.FunBind [Hs.Match (*) (varName (x .$ "rule" .$ T.pack (show n)))
+ []
+ Nothing
+ (Hs.UnGuardedRhs (primitiveVar "main" []))
+ (Hs.BDecls decls)]
+ where ruleCheck = Rec (qid "rule") 0
+ [value (qid "rule" .$ "box")
+ (primitiveVar "checkRule" [term lhs, term rhs])]
+
+ coalesce records = Bundle $ concatMap rec_code records ++ [main]
+ where main = Hs.FunBind [Hs.Match (*) (Hs.Ident "main") []
+ Nothing (Hs.UnGuardedRhs checks) (Hs.BDecls [])]
+ checks = Hs.Do (concatMap rules records ++ map declaration records)
+ declaration rec = Hs.Qualifier (primitiveVar "checkDeclaration"
+ [ Hs.Lit $ Hs.String $ show $ pretty $ unqualify $ rec_name rec
+ , var (rec_name rec .$ "box") ])
+ rules (Rec _ 0 _) = []
+ rules (Rec x nr _) =
+ [Hs.Qualifier $ primitiveVar "putStrLn" [Hs.Lit $ Hs.String ("Starting rule " ++ show (pretty (unqualify x)))]] ++
+ map (\n -> Hs.Qualifier $ var (x .$ "rule" .$ T.pack (show n))) [0..nr-1] ++
+ [Hs.Qualifier $ primitiveVar "putStrLn" [Hs.Lit $ Hs.String ("Finished rule " ++ show (pretty (unqualify x)))]]
+
+ serialize mod deps (Bundle decls) =
+ T.pack $ prettyPrintWithMode defaultMode {layout = PPInLine} $
+ Hs.Module (*) (modname mod) [] Nothing Nothing imports decls
+ where imports = runtime : map (\m -> Hs.ImportDecl (*) (modname m) True False Nothing Nothing Nothing) deps
+ runtime = Hs.ImportDecl (*) (Hs.ModuleName "Dedukti.Runtime") False False Nothing Nothing Nothing
+ modname m = Hs.ModuleName $ T.unpack $ T.intercalate "." $ map capitalize $ toList m
+
+-- | A similar encoding of names as the z-encoding of GHC. Non-letter
+-- characters are escaped with an x.
+xencode :: Qid -> String
+xencode qid =
+ T.unpack $
+ joinQ (qid_qualifier qid) `T.append`
+ -- Prepend all idents with an x to avoid clash with runtime functions.
+ T.cons 'x' (enc (qid_stem qid)) `T.append`
+ joinS (qid_suffix qid)
+ where joinQ Root = ""
+ joinQ (h :. x) = joinQ h `T.append` capitalize x `T.append` "."
+ joinS Root = ""
+ joinS (h :. x) = joinS h `T.append` "_" `T.append` x
+ enc = T.concatMap f where
+ f 'x' = "xx"
+ f '\'' = "xq"
+ f '_' = "xu"
+ f x | x >= '0', x <= '9' = 'x' `T.cons` T.singleton x
+ | otherwise = T.singleton x
+
+function :: Em RuleSet -> Hs.Decl
+function (RS x _ []) =
+ Hs.FunBind [Hs.Match (*) (varName x) [] Nothing (Hs.UnGuardedRhs (primCon x)) (Hs.BDecls [])]
+function (RS x _ rs) =
+ Hs.FunBind [Hs.Match (*) (varName x) [] Nothing (Hs.UnGuardedRhs rhs) (Hs.BDecls [f])]
+ where n = Rule.arity (head rs)
+ rhs = foldr primLam
+ (application (Hs.Var (Hs.UnQual (Hs.Ident "__")) : Stream.take n variables))
+ (Stream.take n pvariables)
+ f | n > 0 = Hs.FunBind (map clause rs ++ [defaultClause x n])
+ | otherwise = Hs.FunBind (map clause rs)
+
+clause :: Em TyRule -> Hs.Match
+clause rule =
+ let (lrule@(env :@ _ :--> rhs), constraints) = Rule.linearize qids rule
+ in if null constraints
+ then Hs.Match (*) (Hs.Ident "__") (map (pattern env) (Rule.patterns lrule))
+ Nothing (Hs.UnGuardedRhs (code rhs)) (Hs.BDecls [])
+ else Hs.Match (*) (Hs.Ident "__") (map (pattern env) (Rule.patterns lrule))
+ Nothing (Hs.GuardedRhss [Hs.GuardedRhs (*) (guards constraints) (code rhs)]) (Hs.BDecls [])
+ where guards constraints =
+ map (\(x, x') -> Hs.Qualifier $
+ primitiveVar "convertible" [Hs.Lit (Hs.Int 0), var x, var x']) constraints
+ qids = Stream.unfold (\i -> ((qid $ T.pack $ show i) .$ "fresh", i + 1)) 0
+
+defaultClause :: Id Record -> Int -> Hs.Match
+defaultClause x n =
+ Hs.Match (*) (Hs.Ident "__") (Stream.take n pvariables) Nothing
+ (Hs.UnGuardedRhs (primApps x (Stream.take n variables))) (Hs.BDecls [])
+
+value :: Id Record -> Hs.Exp -> Hs.Decl
+value x rhs =
+ Hs.FunBind [Hs.Match (*) (varName x) [] Nothing (Hs.UnGuardedRhs rhs) (Hs.BDecls [])]
+
+
+pattern :: Em Env -> Em Expr -> Hs.Pat
+pattern env (Var x _) | x `isin` env = pvar x
+pattern env expr = case unapply expr of
+ Var x _ : xs -> primAppsP x (map (pattern env) xs)
+
+-- | Turn an expression into object code with types erased.
+code :: Em Expr -> Hs.Exp
+code (Var x _) = var x
+code (Lam (x ::: ty) t _) = primLam (pvar x) (code t)
+code (Lam (Hole ty) t _) = primLam Hs.PWildCard (code t)
+code (Pi (x ::: ty) t _) = primPi (code ty) (pvar x) (code t)
+code (Pi (Hole ty) t _) = primPi (code ty) Hs.PWildCard (code t)
+code (App t1 t2 _) = primap (code t1) (code t2)
+code Type = primType
+
+-- | Turn a term into its Haskell representation, including all types.
+term :: Em Expr -> Hs.Exp
+term (Var x _) = var (x .$ "box")
+term (Lam b t _) = primTLam b (term t)
+term (Pi b t _) = primTPi b (term t)
+term (App t1 t2 _) = primTApp (term t1) (primUBox (term t2) (code t2))
+term Type = primTType
+term Kind = primTKind
+
+(*) :: Hs.SrcLoc
+(*) = Hs.SrcLoc "" 0 0
+
+varName :: Id Record -> Hs.Name
+varName = Hs.Ident . xencode . unqualify
+
+-- | Smart variable constructor.
+var :: Id Record -> Hs.Exp
+var = Hs.Var . Hs.UnQual . Hs.Ident . xencode
+
+pvar :: Id Record -> Hs.Pat
+pvar = Hs.PVar . varName
+
+-- | Produce a set of variables y1, ..., yn
+variables =
+ Stream.unfold (\i -> (Hs.Var $ Hs.UnQual $ Hs.Ident $ ('y':) $ show i, i + 1)) 0
+
+pvariables =
+ Stream.unfold (\i -> (Hs.PVar $ Hs.Ident $ ('y':) $ show i, i + 1)) 0
+
+application :: [Hs.Exp] -> Hs.Exp
+application = foldl1 Hs.App
+
+-- Primitives
+
+primitiveVar s [] = Hs.Var $ Hs.UnQual $ Hs.Ident s
+primitiveVar s xs = Hs.Paren $ application $ (Hs.Var $ Hs.UnQual $ Hs.Ident s) : xs
+
+primitiveCon s [] = Hs.Con $ Hs.UnQual $ Hs.Ident s
+primitiveCon s xs = Hs.Paren $ application $ (Hs.Con $ Hs.UnQual $ Hs.Ident s) : xs
+
+primap t1 t2 = primitiveVar "ap" [t1, t2]
+primApp t1 t2 = primitiveCon "App" [t1, t2]
+primCon c = primitiveCon "Con" [Hs.Lit (Hs.String (show (pretty c)))]
+primType = primitiveCon "Type" []
+
+primLam pat t = primitiveCon "Lam" [Hs.Paren (Hs.Lambda (*) [pat] t)]
+primPi dom pat range = primitiveCon "Pi" [dom, Hs.Paren (Hs.Lambda (*) [pat] range)]
+
+primApps c = foldl primApp (primCon c)
+
+typedAbstraction c b t =
+ let (pat, ty, ran) =
+ case b of
+ x ::: ty -> ( pvar (x .$ "box")
+ , ty
+ , Hs.Let (Hs.BDecls [value x (primobj (var (x .$ "box")))]) t )
+ Hole ty -> (Hs.PWildCard, ty, t)
+ dom = if isVariable ty
+ then term ty else primsbox (term ty) primType (code ty)
+ in primitiveCon c [dom, Hs.Paren (Hs.Lambda (*) [pat] ran)]
+
+primTLam = typedAbstraction "TLam"
+primTPi = typedAbstraction "TPi"
+primTApp t1 t2 = primitiveCon "TApp" [t1, t2]
+primTType = primitiveCon "TType" []
+primTKind = primitiveCon "TKind" []
+
+primUBox ty obj_code = primitiveCon "UBox" [ty, obj_code]
+primbbox ty ty_code obj_code = primitiveVar "bbox" [ty, ty_code, obj_code]
+primsbox ty ty_code obj_code = primitiveVar "sbox" [ty, ty_code, obj_code]
+
+primobj t = primitiveVar "obj" [t]
+
+-- | Build a pattern matching a constant.
+primConP c = Hs.PParen $ Hs.PApp (Hs.UnQual $ Hs.Ident "Con") [Hs.PLit (Hs.String (show (pretty c)))]
+primAppP t1 t2 = Hs.PParen $ Hs.PApp (Hs.UnQual $ Hs.Ident "App") [t1, t2]
+primAppsP c = foldl primAppP (primConP c)
+
+-- | Capitalize a word.
+capitalize :: T.Text -> T.Text
+capitalize s = case T.uncons s of
+ Nothing -> T.empty
+ Just (x, xs) -> toUpper x `T.cons` xs
diff --git a/Dedukti/Config.hs b/Dedukti/Config.hs
new file mode 100644
--- /dev/null
+++ b/Dedukti/Config.hs
@@ -0,0 +1,27 @@
+-- |
+-- Copyright : © 2009 CNRS - École Polytechnique - INRIA
+-- License : GPL
+--
+-- Global site-specific configuration variables.
+
+module Dedukti.Config where
+
+
+data Verbosity = Quiet | Verbose | Debug
+ deriving (Eq, Ord, Show)
+
+data Config = Config
+ { homeDir :: FilePath
+ , imageName :: FilePath
+ , version :: String
+ , hsCompiler :: FilePath
+ , verbosity :: Verbosity
+ }
+
+defaultConfig =
+ Config { homeDir = "."
+ , imageName = "dedukti"
+ , hsCompiler = "ghc"
+ , version = "0.1"
+ , verbosity = Quiet }
+
diff --git a/Dedukti/Core.hs b/Dedukti/Core.hs
new file mode 100644
--- /dev/null
+++ b/Dedukti/Core.hs
@@ -0,0 +1,281 @@
+-- |
+-- Copyright : © 2009 CNRS - École Polytechnique - INRIA
+-- License : GPL
+
+module Dedukti.Core
+ ( -- * Terms
+ Expr(..), Binding(..)
+ -- * Rules
+ , Rule(..), Env, TyRule(..), RuleSet(..)
+ -- * Type Synonyms
+ , Module
+ -- * Type functions
+ , Id, A
+ -- * Convenience functions
+ , (.->)
+ , bind_name, bind_type
+ , isAbstraction, isApplication, isVariable, isAtomic, isApplicative
+ -- * Environments
+ , emptyEnv, env_bindings, env_domain, env_codomain, (&), (!), isin
+ -- * Annotations
+ , Unannot, nann, (%%), (%%%), (<%%>), (<%%%>)
+ -- * Smart constructors
+ , abstract, apply, unapply
+ -- * Transformations
+ , Transform(..), transform, descend
+ -- * Query
+ , everyone
+ ) where
+
+import Control.Applicative
+import Control.Monad.Identity
+import Control.Monad.State
+import qualified Data.Map as Map
+
+
+data Expr id a = Lam (Binding id a) (Expr id a) a
+ | Pi (Binding id a) (Expr id a) a
+ | App (Expr id a) (Expr id a) a
+ | Var id a
+ | Type
+ | Kind
+ deriving (Eq, Ord, Show)
+
+infix 2 :::
+
+-- | A type decorating a variable, or a type on its own.
+data Binding id a = id ::: Expr id a
+ | Hole (Expr id a)
+ deriving (Eq, Ord, Show)
+
+-- | A rewrite rule.
+data Rule id a = Expr id a :--> Expr id a
+ deriving (Eq, Ord, Show)
+infix 9 :-->
+
+-- | An environment is and *ordered* list of bindings, since types can depend
+-- on items defined earlier in environment. We opt for a hybrid
+-- representation, for both fast membership tests and conservation of order.
+data Env id a = Env [Binding id a] (Map.Map id (Expr id a))
+ deriving (Eq, Ord, Show)
+
+-- | A rewrite rule paired with a typing environment.
+data TyRule id a = Env id a :@ Rule id a
+ deriving (Eq, Ord, Show)
+infix 8 :@
+
+-- | A set of rewrite rules sharing a same head constant.
+-- Invariant:
+--
+-- > all ((== rs_name ruleset) . headConstant) (rs_rules ruleset)
+data RuleSet id a = RS { rs_name :: id
+ , rs_type :: Expr id a
+ , rs_rules :: [TyRule id a] }
+ deriving (Eq, Ord, Show)
+
+type Module id a = ([Binding id a], [TyRule id a])
+
+type family Id t
+type family A t
+
+type instance Id (Module id a) = id
+type instance Id (Binding id a) = id
+type instance Id (Rule id a) = id
+type instance Id (TyRule id a) = id
+type instance Id (RuleSet id a) = id
+type instance Id (Expr id a) = id
+
+type instance A (Module id a) = a
+type instance A (Binding id a) = a
+type instance A (Rule id a) = a
+type instance A (TyRule id a) = a
+type instance A (RuleSet id a) = a
+type instance A (Expr id a) = a
+
+x .-> y = Pi (Hole x) y
+infixr .->
+
+bind_type (x ::: ty) = ty
+bind_type (Hole ty) = ty
+
+bind_name (x ::: _) = x
+bind_name (Hole _) = error "Binding has no name."
+
+isAbstraction (Lam _ _ _) = True
+isAbstraction (Pi _ _ _) = True
+isAbstraction _ = False
+
+isApplication (App _ _ _) = True
+isApplication _ = False
+
+isVariable (Var _ _) = True
+isVariable _ = False
+
+isAtomic (Var _ _) = True
+isAtomic Type = True
+isAtomic Kind = True
+isAtomic _ = False
+
+isApplicative x = isAtomic x || isApplication x
+
+env_bindings (Env bs _) = bs
+env_domain (Env bs map) = Map.keys map
+env_codomain (Env bs map) = Map.elems map
+
+infix 1 &
+
+emptyEnv = Env [] Map.empty
+
+-- | Extend an environment with a new binding.
+(&) :: Ord id => Binding id a -> Env id a -> Env id a
+x ::: ty & Env bs map = Env ((x ::: ty) : bs) (Map.insert x ty map)
+
+(!) :: Ord id => Env id a -> id -> Expr id a
+Env _ map ! x = map Map.! x
+
+isin :: Ord id => id -> Env id a -> Bool
+isin x (Env _ map) = Map.member x map
+
+fromBindings :: Ord id => [Binding id a] -> Env id a
+fromBindings = foldr (&) (Env [] Map.empty)
+
+-- | Phantom type used to express no annotation.
+data Unannot = Unannot deriving (Eq, Ord, Show)
+
+-- | Unannot should stay abstract. |nann| constructs a value of type |Unannot|.
+nann = Unannot
+
+-- | Annotation operator.
+(%%) :: (a -> Expr id a) -> a -> Expr id a
+(%%) = ($)
+
+-- | Apply annotations to an annotation expecting context.
+(%%%) :: ([a] -> Expr id a) -> [a] -> Expr id a
+(%%%) = ($)
+
+-- | Applicative annotation operator.
+(<%%>) :: Applicative f => f (a -> Expr id a) -> a -> f (Expr id a)
+x <%%> a = x <*> pure a
+
+-- | Applicative multi-annotation operator.
+(<%%%>) :: Applicative f => f ([a] -> Expr id a) -> [a] -> f (Expr id a)
+x <%%%> a = x <*> pure a
+
+infixl 1 %%
+infixl 1 %%%
+infixl 1 <%%>
+infixl 1 <%%%>
+
+-- | Invariant: in abstract xs t annots, length annots == length xs.
+abstract :: [Binding id a] -> Expr id a -> [a] -> Expr id a
+abstract [] t _ = t
+abstract (x:xs) t (a:annots) = Lam x (abstract xs t annots) %% a
+abstract _ _ _ = error "Fewer annotations than number of variables."
+
+-- | Invariant: in apply ts annots, length annots == length ts - 1.
+apply :: Expr id a -> [Expr id a] -> [a] -> Expr id a
+apply t [] _ = t
+apply t (x:xs) (a:annots) = apply (App t x %% a) xs annots
+apply _ _ _= error "Fewer annotations than number of applications."
+
+-- | Turn nested applications into a list.
+unapply :: Expr id a -> [Expr id a]
+unapply = reverse . aux where
+ aux (App t1 t2 _) = t2 : aux t1
+ aux t = [t]
+
+class Ord (Id t) => Transform t where
+ -- | Effectful bottom-up transformation on terms.
+ transformM :: (Monad m, Ord (Id t)) => (Expr (Id t) (A t) -> m (Expr (Id t) (A t))) -> t -> m t
+
+ -- | Helper function for top-down transformations.
+ descendM :: (Monad m, Ord (Id t)) => (Expr (Id t) (A t) -> m (Expr (Id t) (A t))) -> t -> m t
+
+instance Ord id => Transform (Module id a) where
+ transformM f (decls, rules) =
+ return (,) `ap` mapM (transformM f) decls `ap` mapM (transformM f) rules
+
+ descendM f (decls, rules) =
+ return (,) `ap` mapM (descendM f) decls `ap` mapM (descendM f) rules
+
+instance Ord id => Transform (Binding id a) where
+ transformM f (x ::: ty) = return (x :::) `ap` transformM f ty
+ transformM f (Hole ty) = return Hole `ap` transformM f ty
+
+ descendM f (x ::: ty) = return (x :::) `ap` f ty
+ descendM f (Hole ty) = return Hole `ap` f ty
+
+instance Ord id => Transform (TyRule id a) where
+ transformM f (env :@ rule) =
+ return (:@) `ap` (return fromBindings `ap` mapM (transformM f) (env_bindings env)) `ap`
+ transformM f rule
+
+ descendM f (env :@ rule) =
+ return (:@) `ap` (return fromBindings `ap` mapM (descendM f) (env_bindings env)) `ap`
+ descendM f rule
+
+instance Ord id => Transform (Rule id a) where
+ transformM f (lhs :--> rhs) =
+ return (:-->) `ap` transformM f lhs `ap` transformM f rhs
+ descendM f (lhs :--> rhs) = return (:-->) `ap` f lhs `ap` f rhs
+
+instance Ord id => Transform (RuleSet id a) where
+ transformM f RS{..} =
+ return RS `ap` return rs_name `ap` transformM f rs_type `ap` mapM (transformM f) rs_rules
+
+ descendM f RS{..} =
+ return RS `ap` return rs_name `ap` descendM f rs_type `ap` mapM (descendM f) rs_rules
+
+instance Ord id => Transform (Expr id a) where
+ transformM f (Lam (x ::: ty) t a) = do
+ ty' <- transformM f ty
+ t' <- transformM f t
+ f $ Lam (x ::: ty') t' a
+ transformM f (Lam (Hole ty) t a) = do
+ ty' <- transformM f ty
+ t' <- transformM f t
+ f $ Lam (Hole ty') t' a
+ transformM f (Pi (x ::: ty) t a) = do
+ ty' <- transformM f ty
+ t' <- transformM f t
+ f $ Pi (x ::: ty') t' a
+ transformM f (Pi (Hole ty) t a) = do
+ ty' <- transformM f ty
+ t' <- transformM f t
+ f $ Pi (Hole ty') t' a
+ transformM f (App t1 t2 a) = do
+ f =<< return App `ap` transformM f t1 `ap` transformM f t2 `ap` return a
+ transformM f t = f t
+
+ descendM f (Lam (x ::: ty) t a) = do
+ ty' <- f ty
+ t' <- f t
+ return $ Lam (x ::: ty') t' a
+ descendM f (Lam (Hole ty) t a) = do
+ ty' <- f ty
+ t' <- f t
+ return $ Lam (Hole ty') t' a
+ descendM f (Pi (x ::: ty) t a) = do
+ ty' <- f ty
+ t' <- f t
+ return $ Pi (x ::: ty') t' a
+ descendM f (Pi (Hole ty) t a) = do
+ ty' <- f ty
+ t' <- f t
+ return $ Pi (Hole ty') t' a
+ descendM f (App t1 t2 a) = do
+ return App `ap` f t1 `ap` f t2 `ap` return a
+ descendM f t = return t
+
+-- | Pure bottom-up transformation on terms.
+transform :: Transform t => (Expr (Id t) (A t) -> Expr (Id t) (A t)) -> t -> t
+transform f = runIdentity . transformM (return . f)
+
+descend :: Transform t => (Expr (Id t) (A t) -> Expr (Id t) (A t)) -> t -> t
+descend f = runIdentity . descendM (return . f)
+
+-- | Produces all substructures of the given term. Often useful as a generator
+-- in a list comprehension.
+everyone :: Transform t => t -> [Expr (Id t) (A t)]
+everyone t = execState (transformM f t) [] where
+ f t = withState (t:) (return t)
diff --git a/Dedukti/DkM.hs b/Dedukti/DkM.hs
new file mode 100644
--- /dev/null
+++ b/Dedukti/DkM.hs
@@ -0,0 +1,81 @@
+-- |
+-- Copyright : © 2009 CNRS - École Polytechnique - INRIA
+-- License : GPL
+--
+-- The dedukti monad. This provides various facilities such as accumulating
+-- warning messages or displaying error messages to the screen. Debugging
+-- facilities and an interface to the system are also provided.
+
+module Dedukti.DkM ( module Control.Monad
+ , DkM, runDkM, warn, warnings, say
+ , Verbosity(..)
+ , configuration, parameter
+ , command
+ -- pretty-printing combinators.
+ , Pretty(..), text, (<+>), (<>), int
+ , fillText
+ , E.Exception(..), Typeable, E.throw, io
+ , onException) where
+
+import Dedukti.Config as Config
+import Control.Monad
+import Control.Monad.Reader
+import qualified Control.Exception as E
+import Control.Applicative
+import Data.Typeable (Typeable) -- for exceptions
+import System.IO
+import System.Cmd
+import System.Exit
+import Text.PrettyPrint.Leijen hiding ((<$>))
+
+
+instance Applicative (ReaderT Config IO) where
+ pure = return
+ (<*>) = ap
+
+newtype DkM a = DkM (ReaderT Config IO a)
+ deriving (Monad, MonadIO, Functor, Applicative, MonadReader Config)
+
+runDkM :: Config -> DkM a -> IO a
+runDkM conf (DkM m) = runReaderT m conf
+
+-- | Get all global parameters.
+configuration :: DkM Config
+configuration = ask
+
+-- | Select one parameter.
+parameter :: (Config -> a) -> DkM a
+parameter sel = sel <$> ask
+
+-- | Wrapper around 'rawSystem'.
+command :: String -> [String] -> DkM ExitCode
+command exe args = do
+ say Verbose $ text "**" <+> text exe <+> hsep (map (squotes . text) args)
+ io $ rawSystem exe args
+
+-- | A pretty-printing combinator that outputs filled text with wrapping on
+-- word boundaries.
+fillText :: String -> Doc
+fillText = fillSep . map text . words
+
+-- | Register a new warning.
+warn :: String -> DkM ()
+warn = undefined
+
+-- | Get the list of warnings so far.
+warnings :: DkM [Doc]
+warnings = undefined
+
+-- | Write message only if verbosity level is at least the given level.
+say :: Verbosity -> Doc -> DkM ()
+say v msg = do v' <- parameter Config.verbosity
+ when (v <= v') $ io $ hPutDoc stderr (msg <> line)
+
+-- | Shorter name for the oft used 'liftIO'.
+io :: IO a -> DkM a
+io = liftIO
+
+onException :: DkM a -> DkM b -> DkM a
+onException x y = do
+ conf <- configuration
+ io $ runDkM conf x `E.onException` runDkM conf y
diff --git a/Dedukti/Driver/Batch.hs b/Dedukti/Driver/Batch.hs
new file mode 100644
--- /dev/null
+++ b/Dedukti/Driver/Batch.hs
@@ -0,0 +1,115 @@
+-- |
+-- Copyright : © 2009 CNRS - École Polytechnique - INRIA
+-- License : GPL
+--
+-- The batch driver. It compiles all given targets and all their dependencies,
+-- also invoking the Haskell compiler on the generated source code.
+module Dedukti.Driver.Batch (make) where
+
+import Dedukti.Driver.Compile
+import Dedukti.Analysis.Dependency
+import Dedukti.Module
+import Dedukti.Parser
+import Dedukti.DkM
+import qualified Dedukti.Config as Config
+import qualified Control.Hmk.IO as IO
+import Control.Hmk
+import qualified Data.Text.Lazy.Encoding as T
+import qualified Data.ByteString.Lazy as B
+import qualified Data.Map as Map
+import Control.Monad.State
+import Control.Applicative
+import Data.Typeable (Typeable)
+import System.Directory (copyFile)
+import Data.Char (toUpper)
+
+
+cmp x y = do
+ s <- io $ IO.isStale x y
+ say Debug $ text "Compared" <+> text x <+> text y <> text ":" <+> text (show s)
+ return s
+
+-- | Trace a dependency graph in the form of a set of rules, starting from the
+-- given root modules. Finding the dependencies of a module requires parsing
+-- the corresponding source file. To avoid parsing each file twice, the AST is
+-- kept in-memory in case it is needed later during compilation.
+rules :: [MName] -> DkM [Rule DkM FilePath]
+rules targets = evalStateT (rules' targets) Map.empty
+
+-- Maintain a list of already seen modules to avoid parsing same modules twice
+-- when the dependency graph is not a tree.
+rules' targets = concat <$> mapM f targets where
+ -- Collect dependencies.
+ f mod = do
+ seen <- get
+ case Map.lookup mod seen of
+ Just deps -> return deps
+ Nothing -> do
+ lift $ say Verbose $ text "Parsing" <+> text (show mod) <+> text "..."
+ let path = srcPathFromModule mod
+ src <- lift (parse path <$> io (liftM T.decodeUtf8 (B.readFile path)))
+ let dependencies = collectDependencies src
+ rs = g mod dependencies (task_compile mod src)
+ -- Recursively construct rules for dependent modules.
+ rsdeps <- rules' dependencies
+ lift $ say Verbose $ text "Dependencies of" <+> text (show mod) <+> text ":"
+ <+> text (show dependencies)
+ put (Map.insert mod (rs ++ rsdeps) seen)
+ return $ rs ++ rsdeps
+ -- Now that we have the dependencies of the module, we can enounce a few
+ -- build rules concerning the module.
+ g mod ds compile = let capitalize x = toUpper (head x) : tail x
+ eu = srcPathFromModule mod
+ euo = objPathFromModule mod
+ eui = ifacePathFromModule mod
+ hi = pathFromModule ".hi" mod
+ chi = capitalize hi
+ o = pathFromModule ".o" mod
+ dephis = map (capitalize . pathFromModule ".hi") ds
+ in [ Rule eu [] Nothing cmp
+ , Rule euo [eu] (Just compile) cmp
+ , Rule eui [eu] (Just compile) cmp
+ , Rule hi (euo:eui:dephis) (Just $ task_hscomp euo) cmp
+ , Rule o (euo:eui:dephis) (Just $ task_hscomp euo) cmp
+ , Rule chi [hi] (Just $ task_himv hi chi) cmp ]
+ task_hscomp euo _ = do
+ hscomp <- parameter Config.hsCompiler
+ io . IO.testExitCode =<< command hscomp [ "-c", "-w", "-x", "hs", euo
+ , "-XOverloadedStrings"
+ , "-XPatternGuards" ]
+ -- GHC won't find the interface files if their names don't start with a
+ -- capital letter. So alias the interface file with a capitalized name.
+ task_himv hi chi _ = do
+ io $ copyFile hi chi
+ return TaskSuccess
+ task_compile mod src _ = do
+ compileAST mod src `onException`
+ (say Quiet $ text "In module" <+> pretty mod <> text ":")
+ return TaskSuccess
+
+data CommandError = CommandError
+ deriving Typeable
+
+instance Show CommandError where
+ show CommandError = "Command returned non-zero exit status."
+
+instance Exception CommandError
+
+-- | Perform each system action, aborting if an action returns
+-- non-zero exit code.
+abortOnError :: [DkM Result] -> DkM ()
+abortOnError = mapM_ f where
+ f cmd = do code <- cmd
+ case code of
+ TaskSuccess -> return ()
+ TaskFailure -> throw CommandError
+
+-- | Compile each of the modules given as input and all of their
+-- dependencies, if necessary.
+make :: [MName] -> DkM ()
+make modules = do
+ let targets = map (pathFromModule ".o") modules
+ rs <- process cmp <$> rules modules
+ schedule <- mk rs targets
+ say Debug $ text "Tasks to execute:" <+> int (length schedule)
+ abortOnError schedule
diff --git a/Dedukti/Driver/Compile.hs b/Dedukti/Driver/Compile.hs
new file mode 100644
--- /dev/null
+++ b/Dedukti/Driver/Compile.hs
@@ -0,0 +1,80 @@
+-- |
+-- Copyright : © 2009 CNRS - École Polytechnique - INRIA
+-- License : GPL
+--
+-- Compile one file to Haskell source code.
+
+module Dedukti.Driver.Compile (compile, compileAST) where
+
+import Dedukti.Module
+import Dedukti.Parser
+import Dedukti.DkM
+import Dedukti.Core
+import Dedukti.Analysis.Dependency
+import Dedukti.Analysis.Scope
+import qualified Dedukti.CodeGen.Exts as CG
+import qualified Dedukti.Rule as Rule
+import qualified Dedukti.Analysis.Rule as Rule
+import qualified Data.Text.Lazy as T
+import qualified Data.Text.Lazy.Encoding as T
+import qualified Data.ByteString.Lazy as B
+import qualified Data.Set as Set
+
+
+readT = io . liftM T.decodeUtf8 . B.readFile
+writeT path = io . B.writeFile path . T.encodeUtf8
+
+-- | Qualify all occurrences of identifiers defined in current module.
+selfQualify :: MName -> [Pa RuleSet] -> [Pa RuleSet]
+selfQualify mod rsets = let defs = Set.fromList (map rs_name rsets)
+ in map (descend (f defs))
+ (map (\RS{..} -> RS{rs_name = rs_name{qid_qualifier = mod}, ..}) rsets)
+ where f defs (Var x a) | Nothing <- provenance x
+ , x `Set.member` defs = Var x{qid_qualifier = mod} a
+ f defs (Lam (x ::: ty) t a) =
+ Lam (x ::: f defs ty) (f (Set.delete x defs) t) a
+ f defs (Pi (x ::: ty) t a) =
+ Pi (x ::: f defs ty) (f (Set.delete x defs) t) a
+ f defs t = descend (f defs) (t :: Pa Expr)
+
+-- | Read the interface file of each module name to collect the declarations
+-- exported by the module.
+populateInitialEnvironment :: [MName] -> DkM (Set.Set Qid)
+populateInitialEnvironment =
+ liftM Set.unions .
+ mapM (\dep -> let path = ifacePathFromModule dep
+ in liftM (Set.fromList . map (qual dep) . parseIface path) $
+ readT path)
+ where qual mod qid = qid{qid_qualifier = mod}
+
+-- | Generate the content of an interface file.
+interface :: Pa Module -> T.Text
+interface (decls, _) = T.unlines (map (qid_stem . bind_name) decls)
+
+-- | Emit Haskell code for one module.
+compile :: MName -> DkM ()
+compile mod = do
+ say Verbose $ text "Parsing" <+> text (show mod) <+> text "..."
+ let path = srcPathFromModule mod
+ compileAST mod =<< return (parse path) `ap` readT path
+
+-- | Emit Haskell code for one module, starting from the AST.
+compileAST :: MName -> Pa Module -> DkM ()
+compileAST mod src@(decls, rules) = do
+ let deps = collectDependencies src
+ -- For the purposes of scope checking it is necessary to load in the
+ -- environment all those declarations from immediate dependencies. For this
+ -- we read an interface file, much faster to parse than the actual
+ -- dependencies themselves.
+ say Verbose $ text "Populating environment for" <+> text (show mod) <+> text "..."
+ extdecls <- populateInitialEnvironment deps
+ say Verbose $ text "Checking" <+> text (show mod) <+> text "..."
+ checkUniqueness src
+ checkScopes extdecls src
+ Rule.checkOrdering rules
+ mapM_ Rule.checkHead rules
+ say Debug $ pretty (concatMap rs_rules (Rule.ruleSets decls rules))
+ say Verbose $ text "Compiling" <+> text (show mod) <+> text "..."
+ let code = map CG.emit (selfQualify mod (Rule.ruleSets decls rules)) :: [CG.Code]
+ writeT (objPathFromModule mod) $ CG.serialize mod deps $ CG.coalesce code
+ writeT (ifacePathFromModule mod) $ interface src
diff --git a/Dedukti/Driver/Interactive.hs b/Dedukti/Driver/Interactive.hs
new file mode 100644
--- /dev/null
+++ b/Dedukti/Driver/Interactive.hs
@@ -0,0 +1,20 @@
+-- |
+-- Copyright : © 2009 CNRS - École Polytechnique - INRIA
+-- License : GPL
+
+module Dedukti.Driver.Interactive (eval) where
+
+import Dedukti.Module
+import Dedukti.Parser
+import qualified Dedukti.CodeGen.Exts as CG
+import qualified Dedukti.Rule as Rule
+import Data.ByteString.Lazy (ByteString)
+
+
+-- | Emit Haskell code for one module.
+eval :: String -> ByteString
+eval input =
+ let (decls, rules) = parse "" input
+ code = map CG.emit (Rule.ruleSets decls rules) :: [CG.Code]
+ mod = Module (hierarchy ["interactive"])
+ in CG.serialize (undefined :: CG.Code) mod $ CG.coalesce code
diff --git a/Dedukti/Module.hs b/Dedukti/Module.hs
new file mode 100644
--- /dev/null
+++ b/Dedukti/Module.hs
@@ -0,0 +1,98 @@
+-- |
+-- Copyright : © 2009 CNRS - École Polytechnique - INRIA
+-- License : GPL
+--
+-- A representation of module names and associated functions to map module
+-- names to source files and vice-versa. Qualified names, as required in the
+-- presence of modules, are also defined here.
+module Dedukti.Module
+ ( -- * Data types
+ Hierarchy(..), MName
+ -- * Exceptions
+ , InvalidModuleName(..)
+ -- * Functions
+ , hierarchy, toList
+ , pathFromModule, moduleFromPath
+ , srcPathFromModule, objPathFromModule, ifacePathFromModule
+ -- * Qualified names.
+ , Qid(..), qid, (.$), provenance, unqualify
+ ) where
+
+import Dedukti.DkM
+import System.FilePath
+import Data.Char (isAlpha, isAlphaNum)
+import qualified Data.Text.Lazy as T
+import Text.PrettyPrint.Leijen
+
+
+data Hierarchy = !Hierarchy :. !T.Text | Root
+ deriving (Eq, Ord, Show)
+
+type MName = Hierarchy
+
+newtype InvalidModuleName = InvalidModuleName String
+ deriving (Eq, Ord, Typeable)
+
+instance Show InvalidModuleName where
+ show (InvalidModuleName name) = "invalid character in " ++ name
+
+instance Exception InvalidModuleName
+
+instance Pretty MName where
+ pretty (Root :. x) = text (T.unpack x)
+ pretty (xs :. x) = pretty xs <> char '.' <> text (T.unpack x)
+
+hierarchy :: [T.Text] -> Hierarchy
+hierarchy = f . reverse where
+ f [] = Root
+ f (x:xs) = f xs :. x
+
+toList :: Hierarchy -> [T.Text]
+toList = reverse . f where
+ f Root = []
+ f (xs :. x) = x : f xs
+
+-- | Raise an exception if module name component is a valid identifier.
+check :: String -> String
+check cmpt@(x:xs) | isAlpha x, and (map isAlphaNum xs) = cmpt
+ | otherwise = throw $ InvalidModuleName cmpt
+
+pathFromModule :: String -> MName -> FilePath
+pathFromModule ext mod =
+ addExtension (joinPath $ map T.unpack $ toList mod) ext
+
+moduleFromPath :: FilePath -> MName
+moduleFromPath =
+ hierarchy . map (T.pack . check) . splitDirectories . dropExtension
+
+srcPathFromModule :: MName -> FilePath
+srcPathFromModule = pathFromModule ".eu"
+
+objPathFromModule :: MName -> FilePath
+objPathFromModule = pathFromModule ".euo"
+
+ifacePathFromModule :: MName -> FilePath
+ifacePathFromModule = pathFromModule ".eui"
+
+-- | The datatype of qualified names.
+data Qid = Qid { qid_qualifier :: !Hierarchy
+ , qid_stem :: !T.Text
+ , qid_suffix :: !Hierarchy }
+ deriving (Eq, Ord, Show)
+
+-- | Shorthand qid introduction.
+qid :: T.Text -> Qid
+qid x = Qid Root x Root
+
+-- | Append suffix.
+(.$) :: Qid -> T.Text -> Qid
+(Qid qual x sufs) .$ suf = Qid qual x (sufs :. suf)
+
+-- | Get the module where the qid is defined, based on its qualifier.
+provenance :: Qid -> Maybe MName
+provenance (Qid Root _ _) = Nothing
+provenance (Qid qual _ _) = Just qual
+
+-- | Remove any qualifier.
+unqualify :: Qid -> Qid
+unqualify qid = qid{qid_qualifier = Root}
diff --git a/Dedukti/Parser.hs b/Dedukti/Parser.hs
new file mode 100644
--- /dev/null
+++ b/Dedukti/Parser.hs
@@ -0,0 +1,181 @@
+-- |
+-- Copyright : © 2009 CNRS - École Polytechnique - INRIA
+-- License : GPL
+
+{-# OPTIONS_GHC -fno-warn-unused-binds #-}
+module Dedukti.Parser (Pa, Dedukti.Parser.parse, parseIface) where
+
+import Dedukti.Core
+import Dedukti.Module
+import Text.Parsec hiding (ParseError, parse)
+import qualified Text.Parsec.Token as Token
+import Control.Applicative hiding ((<|>), many)
+import Control.Monad.Identity
+import qualified Control.Exception as Exception
+import qualified Data.Text.Lazy as T
+import Data.Typeable (Typeable)
+
+
+-- The AST type as returned by the Parser.
+type Pa t = t Qid Unannot
+
+-- The parsing monad.
+type P = Parsec String [Pa TyRule]
+
+newtype ParseError = ParseError String
+ deriving Typeable
+
+instance Show ParseError where
+ show (ParseError e) = e
+
+instance Exception.Exception ParseError
+
+newtype IfaceError = IfaceError String
+ deriving Typeable
+
+instance Show IfaceError where
+ show (IfaceError f) = "Broken interface file " ++ f ++ "."
+
+instance Exception.Exception IfaceError
+
+parse :: SourceName -> T.Text -> Pa Module
+parse name input =
+ -- At the toplevel, a source file is a list of declarations and rule
+ -- definitions. Here rules are accumulated by side-effect, added to the
+ -- parser state as we encounter them.
+ case runParser ((,) <$> toplevel <*> allRules) [] name (T.unpack input) of
+ Left e -> Exception.throw (ParseError (show e))
+ Right x -> x
+
+-- | Parser for interface files.
+parseIface :: SourceName -> T.Text -> [Qid]
+parseIface _ = map qid . T.lines
+
+addRule :: Pa TyRule -> P ()
+addRule rule = modifyState (rule:)
+
+-- | Retrieve all rules encountered so far from the parser state.
+allRules :: P [Pa TyRule]
+allRules = liftM reverse getState
+
+lexDef = Token.LanguageDef
+ { Token.commentStart = "(;"
+ , Token.commentEnd = ";)"
+ , Token.commentLine = ";"
+ , Token.nestedComments = False
+ , Token.identStart = alphaNum <|> char '_' <|> char '\''
+ , Token.identLetter = alphaNum <|> char '_' <|> char '\''
+ , Token.opStart = parserFail "No user defined operators yet."
+ , Token.opLetter = parserFail "No user defined operators yet."
+ , Token.reservedNames = ["Type", "Kind"]
+ , Token.reservedOpNames = [":", "=>", "->", "-->"]
+ , Token.caseSensitive = True
+ }
+
+Token.LanguageDef{..} = lexDef
+Token.TokenParser{..} = Token.makeTokenParser lexDef
+
+-- | Qualified or unqualified name.
+--
+-- > qid ::= id.id | id
+qident = ident "qid" where
+ ident = do
+ c <- identStart
+ cs <- many identLetter
+ x <- (do let qualifier = T.pack (c:cs)
+ c <- try $ do char '.'; identStart
+ cs <- many identLetter
+ let name = T.pack (c:cs)
+ return $ Qid (Root :. qualifier) name Root)
+ <|> return (qid (T.pack (c:cs)))
+ whiteSpace
+ return (Var x nann)
+
+-- | Unqualified name.
+ident = qid . T.pack <$> identifier
+
+-- | Root production rule of the grammar.
+--
+-- > toplevel ::= declaration toplevel
+-- > | rule toplevel
+-- > | eof
+toplevel =
+ whiteSpace *>
+ ( (rule *> toplevel) -- Rules are accumulated by side-effect.
+ <|> ((:) <$> declaration <*> toplevel)
+ <|> (eof *> return []))
+
+-- | Binding construct.
+--
+-- > binding ::= id : term
+binding = ((:::) <$> ident <* reservedOp ":" <*> term)
+ "binding"
+
+-- | Top-level declarations.
+--
+-- > declaration ::= id ":" term "."
+declaration = (binding <* dot)
+ "declaration"
+
+-- | Left hand side of an abstraction or a product.
+--
+-- > domain ::= id ":" applicative
+-- > | applicative
+domain = ( ((:::) <$> try (ident <* reservedOp ":") <*> applicative)
+ <|> (Hole <$> applicative))
+ "domain"
+
+-- |
+-- > sort ::= Type
+sort = Type <$ reserved "Type"
+
+-- | Terms and types.
+--
+-- We first try to parse as the domain of a lambda or pi. If we
+-- later find out there was no arrow after the domain, then we take
+-- the domain to be an expression, and return that.
+--
+-- > term ::= domain "->" term
+-- > | domain "=>" term
+-- > | applicative
+term = do
+ d <- domain
+ choice [ pi d "pi"
+ , lambda d "lambda"
+ , return (bind_type d)]
+ where pi d = Pi <$> pure d <* reservedOp "->" <*> term <%%> nann
+ lambda d = Lam <$> pure d <* reservedOp "=>" <*> term <%%> nann
+
+-- | Constituents of an applicative form.
+--
+-- > simple ::= sort
+-- > | qid
+-- > | "(" term ")"
+simple = sort <|> qident <|> parens term
+
+-- | Expressions that are either a name or an application of a
+-- expression to one or more arguments.
+--
+-- > applicative ::= simple
+-- > | applicative simple
+-- >
+applicative = (\xs -> case xs of
+ [t] -> t
+ (f:ts) -> apply f ts (repeat nann))
+ <$> many1 simple
+ "applicative"
+
+-- | A rule.
+--
+-- > rule ::= env term "-->" term
+-- > env ::= "[]"
+-- > | "[" env2 "]"
+-- > env2 ::= binding
+-- > | binding "," env2
+rule = ((\env lhs rhs -> foldr (&) emptyEnv env :@ lhs :--> rhs)
+ <$> brackets (sepBy binding comma)
+ <*> term
+ <* reservedOp "-->"
+ <*> term
+ <* dot) >>= addRule
+ "rule"
diff --git a/Dedukti/Pretty.hs b/Dedukti/Pretty.hs
new file mode 100644
--- /dev/null
+++ b/Dedukti/Pretty.hs
@@ -0,0 +1,58 @@
+-- |
+-- Copyright : © 2009 CNRS - École Polytechnique - INRIA
+-- License : GPL
+--
+-- Pretty-printing of various core data types. This is not meant as a
+-- replacement of the |Show| class but rather an alternative. You
+-- should derive Show for all data types, then declare them instances
+-- of |Pretty| where appropriate.
+
+module Dedukti.Pretty (pretty) where
+
+import Dedukti.Core
+import Dedukti.Module
+import Text.PrettyPrint.Leijen
+import qualified Data.Text.Lazy as T
+
+
+textT = text . T.unpack
+
+instance Pretty id => Pretty (Expr id a) where
+ pretty (Lam x t _) = pretty x <+> text "=>" <+> pretty t
+ pretty (Pi x t _) | Pi _ _ _ <- bind_type x = parens (pretty x) <+> text "->" <+> pretty t
+ | otherwise = pretty x <+> text "->" <+> pretty t
+ pretty (App t1 t2 _) =
+ let f = if isApplicative t1 then id else parens
+ g = if isAtomic t2 then id else parens
+ in f (pretty t1) <+> g (pretty t2)
+ pretty (Var x _) = pretty x
+ pretty Type = text "Type"
+ pretty Kind = text "Kind"
+
+instance Pretty id => Pretty (Binding id a) where
+ pretty (x ::: ty) = pretty x <+> char ':' <+> pretty ty
+ pretty (Hole ty) = pretty ty
+
+ prettyList = vcat . map (\x -> pretty x <> dot)
+
+instance Pretty id => Pretty (Rule id a) where
+ pretty (lhs :--> rhs) = pretty lhs <+> text "-->" <+> pretty rhs
+
+instance (Eq a, Ord id, Pretty id) => Pretty (TyRule id a) where
+ pretty (env :@ rule)
+ | env == emptyEnv = text "[]" <+> pretty rule
+ | otherwise =
+ encloseSep (text "[ ") (text " ]") (text ", ")
+ (map pretty (env_bindings env))
+ <+> pretty rule
+
+ prettyList = vcat . map (\x -> pretty x <> dot)
+
+instance Pretty Qid where
+ pretty qid = joinQ (qid_qualifier qid) <>
+ textT (qid_stem qid) <>
+ joinS (qid_suffix qid)
+ where joinQ Root = empty
+ joinQ (h :. x) = joinQ h <> textT x <> dot
+ joinS Root = empty
+ joinS (h :. x) = joinS h <> char '_' <> textT x
diff --git a/Dedukti/Rule.hs b/Dedukti/Rule.hs
new file mode 100644
--- /dev/null
+++ b/Dedukti/Rule.hs
@@ -0,0 +1,69 @@
+-- |
+-- Copyright : © 2009 CNRS - École Polytechnique - INRIA
+-- License : GPL
+--
+-- Various utility functions over rewrite rules and rule sets.
+
+module Dedukti.Rule where
+
+import Dedukti.Core
+import Data.List (groupBy, sortBy)
+import qualified Data.Stream as Stream
+import Control.Monad.State
+import qualified Data.Set as Set
+import qualified Data.Map as Map
+import Prelude hiding (head)
+import qualified Prelude
+
+
+head :: TyRule id a -> Expr id a
+head (_ :@ (lhs :--> _)) = lhs
+
+headConstant :: TyRule id a -> id
+headConstant = unvar . Prelude.head . unapply . head where
+ unvar (Var x _) = x
+
+patterns :: TyRule id a -> [Expr id a]
+patterns = Prelude.tail . unapply . head
+
+-- | Group set of rules by head constant.
+group :: Eq id => [TyRule id a] -> [[TyRule id a]]
+group = groupBy f where
+ f x y = headConstant x == headConstant y
+
+arity :: TyRule id a -> Int
+arity (_ :@ lhs :--> _) = length (unapply lhs) - 1
+
+-- | Combine declarations with their associated rules, if any.
+ruleSets :: (Show id, Show a, Ord id) => [Binding id a] -> [TyRule id a] -> [RuleSet id a]
+ruleSets ds rs = snd $ foldl aux (sortBy cmp (group rs), []) ds where
+ aux ([], rsets) (x ::: ty) = ([], RS x ty [] : rsets)
+ aux (rs : rss, rsets) (x ::: ty)
+ | x == headConstant (Prelude.head rs) = (rss, RS x ty rs : rsets)
+ | otherwise = (rs : rss, RS x ty [] : rsets)
+ -- We cannot change the order of the declarations, but we need rules to be
+ -- in the same order as the declarations.
+ ordering = Map.fromList (zip (map bind_name ds) [0..])
+ cmp x y = let xi = ordering Map.! headConstant (Prelude.head x)
+ yi = ordering Map.! headConstant (Prelude.head y)
+ in compare xi yi
+
+-- | Make the rule left-linear and return collected unification constraints.
+-- This function must be provided with an infinite supply of fresh variable
+-- names.
+linearize :: Ord id => Stream.Stream id -> TyRule id a -> (TyRule id a, [(id, id)])
+linearize xs (env :@ lhs :--> rhs) =
+ let (lhs', (_, _, constraints)) = runState (transformM f lhs) (xs, Set.empty, [])
+ -- Add new variables to the environment, with same type as
+ -- that of the variables they are unified to.
+ env' = foldr (\(x,x') env -> x' ::: (env ! x) & env) env constraints
+ in (env' :@ lhs' :--> rhs, constraints)
+ where f t@(Var x a) | x `isin` env = do
+ (xs, seen, constraints) <- get
+ if x `Set.member` seen then
+ do let Stream.Cons x' xs' = xs
+ put (xs', Set.insert x seen, (x, x'):constraints)
+ return $ Var x' a else
+ do put (xs, Set.insert x seen, constraints)
+ return t
+ f t = return t
diff --git a/Dedukti/Runtime.hs b/Dedukti/Runtime.hs
new file mode 100644
--- /dev/null
+++ b/Dedukti/Runtime.hs
@@ -0,0 +1,139 @@
+-- Copyright © 2009 CNRS - École Polytechnique - INRIA
+--
+-- Permission to use, copy, modify, and distribute this file for any
+-- purpose with or without fee is hereby granted, provided that the above
+-- copyright notice and this permission notice appear in all copies.
+--
+-- THE SOFTWARE IS PROVIDED "AS IS" AND THE AUTHOR DISCLAIMS ALL WARRANTIES
+-- WITH REGARD TO THIS SOFTWARE INCLUDING ALL IMPLIED WARRANTIES OF
+-- MERCHANTABILITY AND FITNESS. IN NO EVENT SHALL THE AUTHOR BE LIABLE FOR
+-- ANY SPECIAL, DIRECT, INDIRECT, OR CONSEQUENTIAL DAMAGES OR ANY DAMAGES
+-- WHATSOEVER RESULTING FROM LOSS OF USE, DATA OR PROFITS, WHETHER IN AN
+-- ACTION OF CONTRACT, NEGLIGENCE OR OTHER TORTIOUS ACTION, ARISING OUT OF
+-- OR IN CONNECTION WITH THE USE OR PERFORMANCE OF THIS SOFTWARE.
+
+-- | All generated Haskell files import this module. The data type
+-- declarations are given here, along with the conversion relation and type
+-- inference function.
+
+module Dedukti.Runtime
+ ( Code(..), Term(..), ap
+ , convertible
+ , bbox, sbox, obj
+ , start, stop
+ , checkDeclaration
+ , checkRule) where
+
+import qualified Data.ByteString.Char8 as B
+import Control.Exception
+import Text.Show.Functions ()
+import Data.Typeable hiding (typeOf)
+import Prelude hiding (pi, catch)
+import System.IO
+import Data.Time.Clock
+
+
+-- Exceptions
+
+data SortError = SortError
+ deriving (Show, Typeable)
+
+data TypeError = TypeError
+ deriving (Show, Typeable)
+
+data RuleError = RuleError
+ deriving (Show, Typeable)
+
+instance Exception SortError
+instance Exception TypeError
+instance Exception RuleError
+
+-- Convertible and static terms.
+
+data Code = Var !Int
+ | Con !B.ByteString
+ | Lam !(Code -> Code)
+ | Pi Code !(Code -> Code)
+ | App Code Code
+ | Type
+ | Kind
+ deriving (Eq, Show)
+
+data Term = TLam !Term !(Term -> Term)
+ | TPi !Term !(Term -> Term)
+ | TApp !Term !Term
+ | TType
+ | Box Code Code
+ | UBox Term Code
+ deriving Show
+
+instance Eq (Code -> Code)
+
+ap :: Code -> Code -> Code
+ap (Lam f) t = f t
+ap t1 t2 = App t1 t2
+
+obj :: Term -> Code
+obj (Box _ obj) = obj
+
+convertible :: Int -> Code -> Code -> Bool
+convertible n (Var x) (Var x') = x == x'
+convertible n (Con c) (Con c') = c == c'
+convertible n (Lam t) (Lam t') =
+ convertible (n + 1) (t (Var n)) (t' (Var n))
+convertible n (Pi ty1 ty2) (Pi ty3 ty4) =
+ convertible n ty1 ty3 && convertible (n + 1) (ty2 (Var n)) (ty4 (Var n))
+convertible n (App t1 t2) (App t3 t4) =
+ convertible n t1 t3 && convertible n t2 t4
+convertible n Type Type = True
+convertible n Kind Kind = True
+convertible n _ _ = False
+
+-- | A box in which we didn't put anything.
+emptyBox = Box undefined undefined
+
+bbox, sbox :: Term -> Code -> Code -> Term
+
+-- | A big box holds terms of sort Type or Kind
+bbox = box [Type, Kind]
+
+-- | A small box holds terms of sort Type.
+sbox = box [Type]
+
+box sorts ty ty_code obj_code
+ | typeOf 0 ty `elem` sorts = Box ty_code obj_code
+ | otherwise = throw SortError
+
+typeOf :: Int -> Term -> Code
+typeOf n (Box ty _) = ty
+typeOf n (TLam bx@(Box Type a) f) = Pi a (\x -> typeOf n (f (Box a x)))
+typeOf n (TPi bx@(Box Type a) f) = typeOf (n + 1) (f (Box a (Var n)))
+typeOf n (TApp t1 bx@(Box ty2 t2))
+ | Pi tya f <- typeOf n t1, convertible n tya ty2 = f t2
+typeOf n (TApp t1 bx@(UBox tty2 t2))
+ | Pi tya f <- typeOf n t1, ty2 <- typeOf n tty2,
+ convertible n tya ty2 = f t2
+typeOf n TType = Kind
+typeOf n t = throw TypeError
+
+checkDeclaration :: String -> Term -> IO ()
+checkDeclaration x t = catch (evaluate t >> putStrLn "Check") handler
+ where handler (SomeException e) = do
+ putStrLn $ "Error during checking of " ++ x
+ throw e
+
+checkRule :: Term -> Term -> Term
+checkRule lhs rhs | convertible 0 (typeOf 0 lhs) (typeOf 0 rhs) = emptyBox
+ | otherwise = throw RuleError
+
+start :: IO UTCTime
+start = do
+ putStrLn "Start."
+ getCurrentTime
+
+
+stop :: UTCTime -> IO ()
+stop t = do
+ t' <- getCurrentTime
+ let total = diffUTCTime t' t
+ putStrLn $ "Stop. Runtime: " ++ show total
diff --git a/Setup.lhs b/Setup.lhs
new file mode 100644
--- /dev/null
+++ b/Setup.lhs
@@ -0,0 +1,3 @@
+#!/usr/bin/env runhaskell
+> import Distribution.Simple
+> main = defaultMain
diff --git a/dedukti.cabal b/dedukti.cabal
new file mode 100644
--- /dev/null
+++ b/dedukti.cabal
@@ -0,0 +1,58 @@
+name: dedukti
+version: 1.0.0
+author: Mathieu Boespflug
+maintainer: Mathieu Boespflug
+copyright: © 2009 CNRS - École Polytechnique - INRIA
+homepage: http://www.lix.polytechnique.fr/~mboes/src/dedukti.git
+synopsis: A type-checker for the λΠ-modulo calculus.
+description:
+ Dedukti is a proof checker for the λΠ-modulo calculus, a
+ dependently typed λ-calculus with the addition of typed rewrite
+ rules, capable of expressing proofs in Deduction Modulo [1].
+ .
+ [1] G. Dowek, Th. Hardin, C. Kirchner, Theorem proving modulo,
+ /Journal of Automated Reasoning/, 31, 2003, pp. 33-72.
+category: Theorem Provers, Compilers/Interpreters
+license: GPL
+license-file: COPYING
+cabal-version: >= 1.6.0
+build-type: Simple
+tested-with: GHC ==6.10
+data-files: t/*.eu
+
+
+executable dedukti
+ main-is: Dedukti.hs
+ other-modules: Dedukti.Core,
+ Dedukti.Parser,
+ Dedukti.Pretty,
+ Dedukti.Driver.Interactive,
+ Dedukti.Driver.Batch,
+ Dedukti.Driver.Compile,
+ Dedukti.Rule,
+ Dedukti.DkM,
+ Dedukti.Config,
+ Dedukti.Module,
+ Dedukti.CodeGen
+ Dedukti.CodeGen.Exts,
+ Dedukti.Analysis.Rule,
+ Dedukti.Analysis.Scope,
+ Dedukti.Analysis.Dependency
+
+ build-depends: base >= 4 && < 5, mtl >= 1.1, containers >= 0.2,
+ directory, filepath, process,
+ parsec >= 3.0.0, wl-pprint >= 1.0, bytestring >= 0.9.1.0,
+ haskell-src-exts >= 1.1.0, Stream >= 0.3, text >= 0.3,
+ hmk >= 0.9
+ extensions: EmptyDataDecls, PatternGuards, GeneralizedNewtypeDeriving
+ DeriveDataTypeable, TypeFamilies, LiberalTypeSynonyms,
+ FlexibleInstances, FlexibleContexts, OverloadedStrings,
+ RecordWildCards, TypeSynonymInstances, ScopedTypeVariables
+ MultiParamTypeClasses
+ ghc-options: -fwarn-unused-binds -fwarn-unused-imports
+
+library
+ exposed-modules: Dedukti.Runtime
+ build-depends: time >= 1.1
+ extensions: DeriveDataTypeable, PatternGuards, FlexibleInstances
+ ghc-options: -fwarn-unused-binds -fwarn-unused-imports
diff --git a/t/Coq1univ.eu b/t/Coq1univ.eu
new file mode 100644
--- /dev/null
+++ b/t/Coq1univ.eu
@@ -0,0 +1,70 @@
+Uset : Type.
+Uprop : Type.
+Utype : Type.
+
+eprop : x : Uprop -> Type.
+eset : x : Uset -> Type.
+etype : x : Utype -> Type.
+
+dotset : Utype.
+dotprop : Utype.
+
+; /!\ type : type /!\, should use universes
+dottype : Utype.
+
+; /!\ subtyping in coq, should be unidirectional /!\
+[] Uprop --> Utype.
+[] Uset --> Utype.
+
+dotpipp : x : Uprop -> y : (eprop x -> Uprop) -> Uprop.
+dotpips : x : Uprop -> y : (eprop x -> Uset) -> Uset.
+dotpipt : x : Uprop -> y : (eprop x -> Utype) -> Utype.
+dotpisp : x : Uset -> y : (eset x -> Uprop) -> Uprop.
+dotpitp : x : Utype -> y : (etype x -> Uprop) -> Uprop.
+dotpist : x : Uset -> y : (eset x -> Utype) -> Utype.
+dotpits : x : Utype -> y : (etype x -> Uset) -> Uset.
+dotpiss : x : Uset -> y : (eset x -> Uset) -> Uset.
+dotpitt : x : Utype -> y : (etype x -> Utype) -> Utype.
+
+
+[x:Uprop, y : eprop x -> Uprop]
+ eprop (dotpipp x y) --> w : eprop x -> eprop (y w).
+
+[x:Uset, y : eset x -> Uprop]
+ eprop (dotpisp x y) --> w : eset x -> eprop (y w).
+
+[x:Utype, y : etype x -> Uprop]
+ eprop (dotpitp x y) --> w : etype x -> eprop (y w).
+
+; /!\
+[P : Uprop] eprop P --> etype P.
+
+[x:Uprop, y : eprop x -> Uset]
+ eset (dotpips x y) --> w : eprop x -> eset (y w).
+
+[x:Utype, y : etype x -> Uset]
+ eset (dotpits x y) --> w : etype x -> eset (y w).
+
+[x:Uset, y : eset x -> Uset]
+ eset (dotpiss x y) --> w : eset x -> eset (y w).
+
+; /!\
+[P : Uset] eset P --> etype P.
+
+[x:Uset, y : eset x -> Utype]
+ etype (dotpist x y) --> w : eset x -> etype (y w).
+
+[x:Utype, y : etype x -> Utype]
+ etype (dotpitt x y) --> w : etype x -> etype (y w).
+
+[x:Uprop, y : eprop x -> Utype]
+ etype (dotpipt x y) --> w : eprop x -> etype (y w).
+
+
+[] (etype dotset) --> Uset.
+[] (etype dotprop) --> Uprop.
+; /!\
+[] (etype dottype) --> Utype.
+
+; end of Coq1univ
+
diff --git a/t/Logic.eu b/t/Logic.eu
new file mode 100644
--- /dev/null
+++ b/t/Logic.eu
@@ -0,0 +1,290 @@
+True : Uprop.
+I : (eprop True) .
+case_0 : (P : Utype -> (f : (etype P) -> (t : (eprop True) -> (_0 : (eprop True) -> (etype P) ) ) ) ) .
+I_case_0 : (P : Utype -> (f : (etype P) -> (t : (eprop True) -> (eprop True) ) ) ) .
+[P : Utype, f : (etype P) , t : (eprop True) ] ( ( (I_case_0 P) f) t) --> I.
+[P : Utype, f : (etype P) , t : (eprop True) ] ( ( ( (case_0 P) f) t) ( ( (I_case_0 P) f) t) ) --> f.
+True_rect : (P : Utype -> (f : (etype P) -> (t : (eprop True) -> (etype P) ) ) ) .
+[] True_rect --> (P : (etype dottype) => (f : (etype P) => (t : (eprop True) => ( ( ( (case_0 P) f) t) t) ) ) ) .
+True_ind : (P : Uprop -> (f : (eprop P) -> (t : (eprop True) -> (eprop P) ) ) ) .
+[] True_ind --> (P : (etype dotprop) => (True_rect P) ) .
+True_rec : (P : Uset -> (f : (eset P) -> (t : (eprop True) -> (eset P) ) ) ) .
+[] True_rec --> (P : (etype dotset) => (True_rect P) ) .
+False : Uprop.
+case_1 : (P : Utype -> (f : (eprop False) -> (_1 : (eprop False) -> (etype P) ) ) ) .
+False_rect : (P : Utype -> (f : (eprop False) -> (etype P) ) ) .
+[] False_rect --> (P : (etype dottype) => (f : (eprop False) => ( ( (case_1 P) f) f) ) ) .
+False_ind : (P : Uprop -> (f : (eprop False) -> (eprop P) ) ) .
+[] False_ind --> (P : (etype dotprop) => (False_rect P) ) .
+False_rec : (P : Uset -> (f : (eprop False) -> (eset P) ) ) .
+[] False_rec --> (P : (etype dotset) => (False_rect P) ) .
+not : (A : Uprop -> Uprop) .
+[] not --> (A : (etype dotprop) => ( (dotpipp A) (_2 : (eprop A) => False) ) ) .
+and : (A : Uprop -> (B : Uprop -> Uprop) ) .
+conj : (A : Uprop -> (B : Uprop -> (_4 : (eprop A) -> (_3 : (eprop B) -> (eprop ( (and A) B) ) ) ) ) ) .
+case_2 : (A : Uprop -> (B : Uprop -> (P : Utype -> (f : (_6 : (eprop A) -> (_5 : (eprop B) -> (etype P) ) ) -> (a : (eprop ( (and A) B) ) -> (_7 : (eprop ( (and A) B) ) -> (etype P) ) ) ) ) ) ) .
+conj_case_2 : (A : Uprop -> (B : Uprop -> (P : Utype -> (f : (_11 : (eprop A) -> (_10 : (eprop B) -> (etype P) ) ) -> (a : (eprop ( (and A) B) ) -> (_13 : (eprop A) -> (_12 : (eprop B) -> (eprop ( (and A) B) ) ) ) ) ) ) ) ) .
+[A : Uprop, B : Uprop, P : Utype, f : (_15 : (eprop A) -> (_14 : (eprop B) -> (etype P) ) ) , a : (eprop ( (and A) B) ) ] ( ( ( ( (conj_case_2 A) B) P) f) a) --> ( (conj A) B) .
+[A : Uprop, B : Uprop, P : Utype, f : (_9 : (eprop A) -> (_8 : (eprop B) -> (etype P) ) ) , a : (eprop ( (and A) B) ) , var_0 : (eprop A) , var_1 : (eprop B) ] ( ( ( ( ( (case_2 A) B) P) f) a) ( ( ( ( ( ( (conj_case_2 A) B) P) f) a) var_0) var_1) ) --> ( (f var_0) var_1) .
+and_rect : (A : Uprop -> (B : Uprop -> (P : Utype -> (f : (_19 : (eprop A) -> (_18 : (eprop B) -> (etype P) ) ) -> (a : (eprop ( (and A) B) ) -> (etype P) ) ) ) ) ) .
+[] and_rect --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dottype) => (f : (etype ( (dotpipt A) (_17 : (eprop A) => ( (dotpipt B) (_16 : (eprop B) => P) ) ) ) ) => (a : (eprop ( (and A) B) ) => ( ( ( ( ( (case_2 A) B) P) f) a) a) ) ) ) ) ) .
+and_ind : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_21 : (eprop A) -> (_20 : (eprop B) -> (eprop P) ) ) -> (a : (eprop ( (and A) B) ) -> (eprop P) ) ) ) ) ) .
+[] and_ind --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dotprop) => ( ( (and_rect A) B) P) ) ) ) .
+and_rec : (A : Uprop -> (B : Uprop -> (P : Uset -> (f : (_23 : (eprop A) -> (_22 : (eprop B) -> (eset P) ) ) -> (a : (eprop ( (and A) B) ) -> (eset P) ) ) ) ) ) .
+[] and_rec --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dotset) => ( ( (and_rect A) B) P) ) ) ) .
+case_3 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (and A) B) ) -> (_24 : (eprop ( (and A) B) ) -> (eprop A) ) ) ) ) .
+conj_case_3 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (and A) B) ) -> (_26 : (eprop A) -> (_25 : (eprop B) -> (eprop ( (and A) B) ) ) ) ) ) ) .
+[A : Uprop, B : Uprop, H : (eprop ( (and A) B) ) ] ( ( (conj_case_3 A) B) H) --> ( (conj A) B) .
+[A : Uprop, B : Uprop, H : (eprop ( (and A) B) ) , var_2 : (eprop A) , var_3 : (eprop B) ] ( ( ( (case_3 A) B) H) ( ( ( ( (conj_case_3 A) B) H) var_2) var_3) ) --> ( ( (H欧0 : (eprop A) => (H0 : (eprop B) => H欧0) ) var_2) var_3) .
+proj1 : (A : Uprop -> (B : Uprop -> (_27 : (eprop ( (and A) B) ) -> (eprop A) ) ) ) .
+[] proj1 --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (and A) B) ) => ( ( ( (case_3 A) B) H) H) ) ) ) .
+case_4 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (and A) B) ) -> (_28 : (eprop ( (and A) B) ) -> (eprop B) ) ) ) ) .
+conj_case_4 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (and A) B) ) -> (_30 : (eprop A) -> (_29 : (eprop B) -> (eprop ( (and A) B) ) ) ) ) ) ) .
+[A : Uprop, B : Uprop, H : (eprop ( (and A) B) ) ] ( ( (conj_case_4 A) B) H) --> ( (conj A) B) .
+[A : Uprop, B : Uprop, H : (eprop ( (and A) B) ) , var_4 : (eprop A) , var_5 : (eprop B) ] ( ( ( (case_4 A) B) H) ( ( ( ( (conj_case_4 A) B) H) var_4) var_5) ) --> ( ( (H欧0 : (eprop A) => (H0 : (eprop B) => H0) ) var_4) var_5) .
+proj2 : (A : Uprop -> (B : Uprop -> (_31 : (eprop ( (and A) B) ) -> (eprop B) ) ) ) .
+[] proj2 --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (and A) B) ) => ( ( ( (case_4 A) B) H) H) ) ) ) .
+or : (A : Uprop -> (B : Uprop -> Uprop) ) .
+or_introl : (A : Uprop -> (B : Uprop -> (_32 : (eprop A) -> (eprop ( (or A) B) ) ) ) ) .
+or_intror : (A : Uprop -> (B : Uprop -> (_33 : (eprop B) -> (eprop ( (or A) B) ) ) ) ) .
+case_5 : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_34 : (eprop A) -> (eprop P) ) -> (f欧0 : (_35 : (eprop B) -> (eprop P) ) -> (o : (eprop ( (or A) B) ) -> (_36 : (eprop ( (or A) B) ) -> (eprop P) ) ) ) ) ) ) ) .
+or_introl_case_5 : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_39 : (eprop A) -> (eprop P) ) -> (f欧0 : (_40 : (eprop B) -> (eprop P) ) -> (o : (eprop ( (or A) B) ) -> (_41 : (eprop A) -> (eprop ( (or A) B) ) ) ) ) ) ) ) ) .
+[A : Uprop, B : Uprop, P : Uprop, f : (_42 : (eprop A) -> (eprop P) ) , f欧0 : (_43 : (eprop B) -> (eprop P) ) , o : (eprop ( (or A) B) ) ] ( ( ( ( ( (or_introl_case_5 A) B) P) f) f欧0) o) --> ( (or_introl A) B) .
+or_intror_case_5 : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_44 : (eprop A) -> (eprop P) ) -> (f欧0 : (_45 : (eprop B) -> (eprop P) ) -> (o : (eprop ( (or A) B) ) -> (_46 : (eprop B) -> (eprop ( (or A) B) ) ) ) ) ) ) ) ) .
+[A : Uprop, B : Uprop, P : Uprop, f : (_47 : (eprop A) -> (eprop P) ) , f欧0 : (_48 : (eprop B) -> (eprop P) ) , o : (eprop ( (or A) B) ) ] ( ( ( ( ( (or_intror_case_5 A) B) P) f) f欧0) o) --> ( (or_intror A) B) .
+[A : Uprop, B : Uprop, P : Uprop, f : (_37 : (eprop A) -> (eprop P) ) , f欧0 : (_38 : (eprop B) -> (eprop P) ) , o : (eprop ( (or A) B) ) , var_6 : (eprop A) ] ( ( ( ( ( ( (case_5 A) B) P) f) f欧0) o) ( ( ( ( ( ( (or_introl_case_5 A) B) P) f) f欧0) o) var_6) ) --> (f var_6) .
+[A : Uprop, B : Uprop, P : Uprop, f : (_37 : (eprop A) -> (eprop P) ) , f欧0 : (_38 : (eprop B) -> (eprop P) ) , o : (eprop ( (or A) B) ) , var_7 : (eprop B) ] ( ( ( ( ( ( (case_5 A) B) P) f) f欧0) o) ( ( ( ( ( ( (or_intror_case_5 A) B) P) f) f欧0) o) var_7) ) --> (f欧0 var_7) .
+or_ind : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_51 : (eprop A) -> (eprop P) ) -> (f欧0 : (_52 : (eprop B) -> (eprop P) ) -> (o : (eprop ( (or A) B) ) -> (eprop P) ) ) ) ) ) ) .
+[] or_ind --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dotprop) => (f : (eprop ( (dotpipp A) (_50 : (eprop A) => P) ) ) => (f欧0 : (eprop ( (dotpipp B) (_49 : (eprop B) => P) ) ) => (o : (eprop ( (or A) B) ) => ( ( ( ( ( ( (case_5 A) B) P) f) f欧0) o) o) ) ) ) ) ) ) .
+iff : (A : Uprop -> (B : Uprop -> Uprop) ) .
+[] iff --> (A : (etype dotprop) => (B : (etype dotprop) => ( (and ( (dotpipp A) (_53 : (eprop A) => B) ) ) ( (dotpipp B) (_54 : (eprop B) => A) ) ) ) ) .
+iff_refl : (A : Uprop -> (eprop ( (iff A) A) ) ) .
+[] iff_refl --> (A : (etype dotprop) => ( ( ( (conj ( (dotpipp A) (_55 : (eprop A) => A) ) ) ( (dotpipp A) (_56 : (eprop A) => A) ) ) (H : (eprop A) => H) ) (H : (eprop A) => H) ) ) .
+case_6 : (A : Uprop -> (B : Uprop -> (C : Uprop -> (H : (eprop ( (iff A) B) ) -> (_60 : (eprop ( (and ( (dotpipp A) (_57 : (eprop A) => B) ) ) ( (dotpipp B) (_58 : (eprop B) => A) ) ) ) -> (_59 : (eprop ( (iff B) C) ) -> (eprop ( (iff A) C) ) ) ) ) ) ) ) .
+conj_case_6 : (A : Uprop -> (B : Uprop -> (C : Uprop -> (H : (eprop ( (iff A) B) ) -> (_68 : (_63 : (eprop A) -> (eprop B) ) -> (_67 : (_64 : (eprop B) -> (eprop A) ) -> (eprop ( (and ( (dotpipp A) (_65 : (eprop A) => B) ) ) ( (dotpipp B) (_66 : (eprop B) => A) ) ) ) ) ) ) ) ) ) .
+[A : Uprop, B : Uprop, C : Uprop, H : (eprop ( (iff A) B) ) ] ( ( ( (conj_case_6 A) B) C) H) --> ( (conj ( (dotpipp A) (_61 : (eprop A) => B) ) ) ( (dotpipp B) (_62 : (eprop B) => A) ) ) .
+case_7 : (A : Uprop -> (B : Uprop -> (C : Uprop -> (H : (eprop ( (iff A) B) ) -> (H1 : (_71 : (eprop A) -> (eprop B) ) -> (H2 : (_72 : (eprop B) -> (eprop A) ) -> (H欧0 : (eprop ( (iff B) C) ) -> (_75 : (eprop ( (and ( (dotpipp B) (_73 : (eprop B) => C) ) ) ( (dotpipp C) (_74 : (eprop C) => B) ) ) ) -> (eprop ( (iff A) C) ) ) ) ) ) ) ) ) ) .
+conj_case_7 : (A : Uprop -> (B : Uprop -> (C : Uprop -> (H : (eprop ( (iff A) B) ) -> (H1 : (_80 : (eprop A) -> (eprop B) ) -> (H2 : (_81 : (eprop B) -> (eprop A) ) -> (H欧0 : (eprop ( (iff B) C) ) -> (_87 : (_82 : (eprop B) -> (eprop C) ) -> (_86 : (_83 : (eprop C) -> (eprop B) ) -> (eprop ( (and ( (dotpipp B) (_84 : (eprop B) => C) ) ) ( (dotpipp C) (_85 : (eprop C) => B) ) ) ) ) ) ) ) ) ) ) ) ) .
+[A : Uprop, B : Uprop, C : Uprop, H : (eprop ( (iff A) B) ) , H1 : (_88 : (eprop A) -> (eprop B) ) , H2 : (_89 : (eprop B) -> (eprop A) ) , H欧0 : (eprop ( (iff B) C) ) ] ( ( ( ( ( ( (conj_case_7 A) B) C) H) H1) H2) H欧0) --> ( (conj ( (dotpipp B) (_78 : (eprop B) => C) ) ) ( (dotpipp C) (_79 : (eprop C) => B) ) ) .
+[A : Uprop, B : Uprop, C : Uprop, H : (eprop ( (iff A) B) ) , H1 : (_76 : (eprop A) -> (eprop B) ) , H2 : (_77 : (eprop B) -> (eprop A) ) , H欧0 : (eprop ( (iff B) C) ) , var_10 : (_90 : (eprop B) -> (eprop C) ) , var_11 : (_91 : (eprop C) -> (eprop B) ) ] ( ( ( ( ( ( ( (case_7 A) B) C) H) H1) H2) H欧0) ( ( ( ( ( ( ( ( (conj_case_7 A) B) C) H) H1) H2) H欧0) var_10) var_11) ) --> ( ( (H3 : (eprop ( (dotpipp B) (_95 : (eprop B) => C) ) ) => (H4 : (eprop ( (dotpipp C) (_94 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp A) (_92 : (eprop A) => C) ) ) ( (dotpipp C) (_93 : (eprop C) => A) ) ) (H欧1 : (eprop A) => (H3 (H1 H欧1) ) ) ) (H欧1 : (eprop C) => (H2 (H1 (H2 (H4 H欧1) ) ) ) ) ) ) ) var_10) var_11) .
+[A : Uprop, B : Uprop, C : Uprop, H : (eprop ( (iff A) B) ) , var_8 : (_69 : (eprop A) -> (eprop B) ) , var_9 : (_70 : (eprop B) -> (eprop A) ) ] ( ( ( ( (case_6 A) B) C) H) ( ( ( ( ( (conj_case_6 A) B) C) H) var_8) var_9) ) --> ( ( (H1 : (eprop ( (dotpipp A) (_97 : (eprop A) => B) ) ) => (H2 : (eprop ( (dotpipp B) (_96 : (eprop B) => A) ) ) => (H欧0 : (eprop ( (iff B) C) ) => ( ( ( ( ( ( ( (case_7 A) B) C) H) H1) H2) H欧0) H欧0) ) ) ) var_8) var_9) .
+iff_trans : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_99 : (eprop ( (iff A) B) ) -> (_98 : (eprop ( (iff B) C) ) -> (eprop ( (iff A) C) ) ) ) ) ) ) .
+[] iff_trans --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff A) B) ) => ( ( ( ( (case_6 A) B) C) H) H) ) ) ) ) .
+case_8 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (iff A) B) ) -> (_102 : (eprop ( (and ( (dotpipp A) (_100 : (eprop A) => B) ) ) ( (dotpipp B) (_101 : (eprop B) => A) ) ) ) -> (eprop ( (iff B) A) ) ) ) ) ) .
+conj_case_8 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (iff A) B) ) -> (_110 : (_105 : (eprop A) -> (eprop B) ) -> (_109 : (_106 : (eprop B) -> (eprop A) ) -> (eprop ( (and ( (dotpipp A) (_107 : (eprop A) => B) ) ) ( (dotpipp B) (_108 : (eprop B) => A) ) ) ) ) ) ) ) ) .
+[A : Uprop, B : Uprop, H : (eprop ( (iff A) B) ) ] ( ( (conj_case_8 A) B) H) --> ( (conj ( (dotpipp A) (_103 : (eprop A) => B) ) ) ( (dotpipp B) (_104 : (eprop B) => A) ) ) .
+[A : Uprop, B : Uprop, H : (eprop ( (iff A) B) ) , var_12 : (_111 : (eprop A) -> (eprop B) ) , var_13 : (_112 : (eprop B) -> (eprop A) ) ] ( ( ( (case_8 A) B) H) ( ( ( ( (conj_case_8 A) B) H) var_12) var_13) ) --> ( ( (H1 : (eprop ( (dotpipp A) (_116 : (eprop A) => B) ) ) => (H2 : (eprop ( (dotpipp B) (_115 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp B) (_113 : (eprop B) => A) ) ) ( (dotpipp A) (_114 : (eprop A) => B) ) ) H2) H1) ) ) var_12) var_13) .
+iff_sym : (A : Uprop -> (B : Uprop -> (_117 : (eprop ( (iff A) B) ) -> (eprop ( (iff B) A) ) ) ) ) .
+[] iff_sym --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (iff A) B) ) => ( ( ( (case_8 A) B) H) H) ) ) ) .
+case_9 : (A : Uprop -> (H : (eprop ( (iff A) False) ) -> (_128 : (eprop ( (and ( (dotpipp A) (_125 : (eprop A) => False) ) ) ( (dotpipp False) (_126 : (eprop False) => A) ) ) ) -> (_127 : (eprop A) -> (eprop False) ) ) ) ) .
+conj_case_9 : (A : Uprop -> (H : (eprop ( (iff A) False) ) -> (_136 : (_131 : (eprop A) -> (eprop False) ) -> (_135 : (_132 : (eprop False) -> (eprop A) ) -> (eprop ( (and ( (dotpipp A) (_133 : (eprop A) => False) ) ) ( (dotpipp False) (_134 : (eprop False) => A) ) ) ) ) ) ) ) .
+[A : Uprop, H : (eprop ( (iff A) False) ) ] ( (conj_case_9 A) H) --> ( (conj ( (dotpipp A) (_129 : (eprop A) => False) ) ) ( (dotpipp False) (_130 : (eprop False) => A) ) ) .
+[A : Uprop, H : (eprop ( (iff A) False) ) , var_14 : (_137 : (eprop A) -> (eprop False) ) , var_15 : (_138 : (eprop False) -> (eprop A) ) ] ( ( (case_9 A) H) ( ( ( (conj_case_9 A) H) var_14) var_15) ) --> ( ( (H欧0 : (eprop ( (dotpipp A) (_140 : (eprop A) => False) ) ) => (H0 : (eprop ( (dotpipp False) (_139 : (eprop False) => A) ) ) => H欧0) ) var_14) var_15) .
+neg_false : (A : Uprop -> (eprop ( (iff (not A) ) ( (iff A) False) ) ) ) .
+[] neg_false --> (A : (etype dotprop) => ( ( ( (conj ( (dotpipp ( (dotpipp A) (_118 : (eprop A) => False) ) ) (_119 : (eprop ( (dotpipp A) (_118 : (eprop A) => False) ) ) => ( (iff A) False) ) ) ) ( (dotpipp ( (iff A) False) ) (_121 : (eprop ( (iff A) False) ) => ( (dotpipp A) (_120 : (eprop A) => False) ) ) ) ) (H : (eprop ( (dotpipp A) (_124 : (eprop A) => False) ) ) => ( ( ( (conj ( (dotpipp A) (_122 : (eprop A) => False) ) ) ( (dotpipp False) (_123 : (eprop False) => A) ) ) H) (H1 : (eprop False) => ( (False_ind A) H1) ) ) ) ) (H : (eprop ( (iff A) False) ) => ( ( (case_9 A) H) H) ) ) ) .
+and_cancel_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_168 : (_165 : (eprop B) -> (eprop A) ) -> (_167 : (_166 : (eprop C) -> (eprop A) ) -> (eprop ( (iff ( (iff ( (and A) B) ) ( (and A) C) ) ) ( (iff B) C) ) ) ) ) ) ) ) .
+[] and_cancel_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_164 : (eprop B) => A) ) ) => (H0 : (eprop ( (dotpipp C) (_163 : (eprop C) => A) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (and A) B) ) ( (and A) C) ) ) (_141 : (eprop ( (iff ( (and A) B) ) ( (and A) C) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_142 : (eprop ( (iff B) C) ) => ( (iff ( (and A) B) ) ( (and A) C) ) ) ) ) (H1 : (eprop ( (iff ( (and A) B) ) ( (and A) C) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (and A) B) ) (_143 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) ( (dotpipp ( (and A) C) ) (_144 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (and A) B) ) (_156 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) => (H3 : (eprop ( (dotpipp ( (and A) C) ) (_155 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) => ( (H1欧0 : (eprop ( (dotpipp A) (_154 : (eprop A) => ( (dotpipp B) (_153 : (eprop B) => ( (and A) C) ) ) ) ) ) => ( (H2欧0 : (eprop ( (dotpipp A) (_152 : (eprop A) => ( (dotpipp C) (_151 : (eprop C) => ( (and A) B) ) ) ) ) ) => ( ( ( (conj ( (dotpipp B) (_145 : (eprop B) => C) ) ) ( (dotpipp C) (_146 : (eprop C) => B) ) ) (H3欧0 : (eprop B) => ( (H4 : (eprop A) => ( (H欧0 : (eprop ( (dotpipp B) (_148 : (eprop B) => ( (and A) C) ) ) ) => ( (H1欧1 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) C) (H欧1 : (eprop A) => (H5 : (eprop C) => ( (H1欧2 : (eprop ( (dotpipp C) (_147 : (eprop C) => ( (and A) B) ) ) ) => ( (H2欧1 : (eprop A) => ( (H0欧0 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) C) (H1欧3 : (eprop A) => (H6 : (eprop B) => H5) ) ) H0欧0) ) (H1欧2 H5) ) ) (H0 H5) ) ) (H2欧0 H4) ) ) ) ) H1欧1) ) (H欧0 H3欧0) ) ) (H1欧0 H4) ) ) (H H3欧0) ) ) ) (H3欧0 : (eprop C) => ( (H4 : (eprop A) => ( (H0欧0 : (eprop ( (dotpipp B) (_150 : (eprop B) => ( (and A) C) ) ) ) => ( (H1欧1 : (eprop ( (dotpipp C) (_149 : (eprop C) => ( (and A) B) ) ) ) => ( (H2欧1 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) B) (H1欧2 : (eprop A) => (H5 : (eprop B) => ( (H2欧2 : (eprop A) => ( (H欧0 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) B) (H0欧1 : (eprop A) => (H6 : (eprop C) => H5) ) ) H欧0) ) (H0欧0 H5) ) ) (H H5) ) ) ) ) H2欧1) ) (H1欧1 H3欧0) ) ) (H2欧0 H4) ) ) (H1欧0 H4) ) ) (H0 H3欧0) ) ) ) ) (H2欧0 : (eprop A) => (H4 : (eprop C) => (H3 ( ( ( (conj A) C) H2欧0) H4) ) ) ) ) ) (H1欧0 : (eprop A) => (H4 : (eprop B) => (H2 ( ( ( (conj A) B) H1欧0) H4) ) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_157 : (eprop B) => C) ) ) ( (dotpipp C) (_158 : (eprop C) => B) ) ) ( (iff ( (and A) B) ) ( (and A) C) ) ) (H2 : (eprop ( (dotpipp B) (_162 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_161 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and A) B) ) (_159 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) ( (dotpipp ( (and A) C) ) (_160 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) (H1欧0 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) ( (and A) C) ) (H4 : (eprop A) => (H5 : (eprop B) => ( (H1欧1 : (eprop A) => ( (H欧0 : (eprop C) => ( (H2欧0 : (eprop A) => ( (H0欧0 : (eprop B) => ( ( ( (conj A) C) H2欧0) H欧0) ) (H3 H欧0) ) ) (H0 H欧0) ) ) (H2 H5) ) ) (H H5) ) ) ) ) H1欧0) ) ) (H1欧0 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) ( (and A) B) ) (H4 : (eprop A) => (H5 : (eprop C) => ( (H1欧1 : (eprop A) => ( (H0欧0 : (eprop B) => ( (H3欧0 : (eprop A) => ( (H欧0 : (eprop C) => ( ( ( (conj A) B) H3欧0) H0欧0) ) (H2 H0欧0) ) ) (H H0欧0) ) ) (H3 H5) ) ) (H0 H5) ) ) ) ) H1欧0) ) ) ) ) ) H1) ) ) ) ) ) ) ) .
+and_cancel_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_196 : (_193 : (eprop B) -> (eprop A) ) -> (_195 : (_194 : (eprop C) -> (eprop A) ) -> (eprop ( (iff ( (iff ( (and B) A) ) ( (and C) A) ) ) ( (iff B) C) ) ) ) ) ) ) ) .
+[] and_cancel_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_192 : (eprop B) => A) ) ) => (H0 : (eprop ( (dotpipp C) (_191 : (eprop C) => A) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (and B) A) ) ( (and C) A) ) ) (_169 : (eprop ( (iff ( (and B) A) ) ( (and C) A) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_170 : (eprop ( (iff B) C) ) => ( (iff ( (and B) A) ) ( (and C) A) ) ) ) ) (H1 : (eprop ( (iff ( (and B) A) ) ( (and C) A) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (and B) A) ) (_171 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) ( (dotpipp ( (and C) A) ) (_172 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (and B) A) ) (_184 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) => (H3 : (eprop ( (dotpipp ( (and C) A) ) (_183 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) => ( (H1欧0 : (eprop ( (dotpipp B) (_182 : (eprop B) => ( (dotpipp A) (_181 : (eprop A) => ( (and C) A) ) ) ) ) ) => ( (H2欧0 : (eprop ( (dotpipp C) (_180 : (eprop C) => ( (dotpipp A) (_179 : (eprop A) => ( (and B) A) ) ) ) ) ) => ( ( ( (conj ( (dotpipp B) (_173 : (eprop B) => C) ) ) ( (dotpipp C) (_174 : (eprop C) => B) ) ) (H3欧0 : (eprop B) => ( (H4 : (eprop A) => ( (H欧0 : (eprop ( (dotpipp A) (_176 : (eprop A) => ( (and C) A) ) ) ) => ( (H1欧1 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) C) (H欧1 : (eprop C) => (H5 : (eprop A) => ( (H1欧2 : (eprop A) => ( (H0欧0 : (eprop ( (dotpipp A) (_175 : (eprop A) => ( (and B) A) ) ) ) => ( (H2欧1 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) C) (H0欧1 : (eprop B) => (H6 : (eprop A) => H欧1) ) ) H2欧1) ) (H0欧0 H4) ) ) (H2欧0 H欧1) ) ) (H0 H欧1) ) ) ) ) H1欧1) ) (H欧0 H4) ) ) (H1欧0 H3欧0) ) ) (H H3欧0) ) ) ) (H3欧0 : (eprop C) => ( (H4 : (eprop A) => ( (H0欧0 : (eprop ( (dotpipp A) (_178 : (eprop A) => ( (and B) A) ) ) ) => ( (H2欧1 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) B) (H0欧1 : (eprop B) => (H5 : (eprop A) => ( (H2欧2 : (eprop A) => ( (H欧0 : (eprop ( (dotpipp A) (_177 : (eprop A) => ( (and C) A) ) ) ) => ( (H1欧1 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) B) (H欧1 : (eprop C) => (H6 : (eprop A) => H0欧1) ) ) H1欧1) ) (H欧0 H4) ) ) (H1欧0 H0欧1) ) ) (H H0欧1) ) ) ) ) H2欧1) ) (H0欧0 H4) ) ) (H2欧0 H3欧0) ) ) (H0 H3欧0) ) ) ) ) (H2欧0 : (eprop C) => (H4 : (eprop A) => (H3 ( ( ( (conj C) A) H2欧0) H4) ) ) ) ) ) (H1欧0 : (eprop B) => (H4 : (eprop A) => (H2 ( ( ( (conj B) A) H1欧0) H4) ) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_185 : (eprop B) => C) ) ) ( (dotpipp C) (_186 : (eprop C) => B) ) ) ( (iff ( (and B) A) ) ( (and C) A) ) ) (H2 : (eprop ( (dotpipp B) (_190 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_189 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and B) A) ) (_187 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) ( (dotpipp ( (and C) A) ) (_188 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) (H1欧0 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) ( (and C) A) ) (H4 : (eprop B) => (H5 : (eprop A) => ( (H1欧1 : (eprop A) => ( (H欧0 : (eprop C) => ( (H2欧0 : (eprop A) => ( (H0欧0 : (eprop B) => ( ( ( (conj C) A) H欧0) H2欧0) ) (H3 H欧0) ) ) (H0 H欧0) ) ) (H2 H4) ) ) (H H4) ) ) ) ) H1欧0) ) ) (H1欧0 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) ( (and B) A) ) (H4 : (eprop C) => (H5 : (eprop A) => ( (H1欧1 : (eprop A) => ( (H0欧0 : (eprop B) => ( (H3欧0 : (eprop A) => ( (H欧0 : (eprop C) => ( ( ( (conj B) A) H0欧0) H3欧0) ) (H2 H0欧0) ) ) (H H0欧0) ) ) (H3 H4) ) ) (H0 H4) ) ) ) ) H1欧0) ) ) ) ) ) H1) ) ) ) ) ) ) ) .
+or_cancel_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_220 : (_217 : (eprop B) -> (eprop (not A) ) ) -> (_219 : (_218 : (eprop C) -> (eprop (not A) ) ) -> (eprop ( (iff ( (iff ( (or A) B) ) ( (or A) C) ) ) ( (iff B) C) ) ) ) ) ) ) ) .
+[] or_cancel_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_216 : (eprop B) => (not A) ) ) ) => (H0 : (eprop ( (dotpipp C) (_215 : (eprop C) => (not A) ) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (or A) B) ) ( (or A) C) ) ) (_197 : (eprop ( (iff ( (or A) B) ) ( (or A) C) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_198 : (eprop ( (iff B) C) ) => ( (iff ( (or A) B) ) ( (or A) C) ) ) ) ) (H1 : (eprop ( (iff ( (or A) B) ) ( (or A) C) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (or A) B) ) (_199 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) ( (dotpipp ( (or A) C) ) (_200 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (or A) B) ) (_208 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) => (H3 : (eprop ( (dotpipp ( (or A) C) ) (_207 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) => ( (H1欧0 : (eprop ( (dotpipp A) (_206 : (eprop A) => ( (or A) C) ) ) ) => ( (H4 : (eprop ( (dotpipp B) (_205 : (eprop B) => ( (or A) C) ) ) ) => ( (H2欧0 : (eprop ( (dotpipp A) (_204 : (eprop A) => ( (or A) B) ) ) ) => ( (H5 : (eprop ( (dotpipp C) (_203 : (eprop C) => ( (or A) B) ) ) ) => ( ( ( (conj ( (dotpipp B) (_201 : (eprop B) => C) ) ) ( (dotpipp C) (_202 : (eprop C) => B) ) ) (H3欧0 : (eprop B) => ( (H6 : (eprop (not A) ) => ( (H欧0 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) C) (H4欧0 : (eprop A) => ( (H欧1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) C) (H1欧1 : (eprop A) => ( (H欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H2欧1 : (eprop A) => ( (H欧3 : (eprop False) => ( (False_ind C) H欧3) ) (H6 H4欧0) ) ) ) (H2欧1 : (eprop B) => ( (H欧3 : (eprop False) => ( (False_ind C) H欧3) ) (H6 H4欧0) ) ) ) H欧2) ) (H2欧0 H4欧0) ) ) ) (H1欧1 : (eprop C) => ( (H欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H2欧1 : (eprop A) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H0欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H5欧0 : (eprop A) => ( (False_ind C) H0欧0) ) ) (H5欧0 : (eprop B) => ( (False_ind C) H0欧0) ) ) H6欧1) ) (H5 H1欧1) ) ) (H6欧0 H4欧0) ) ) (H0 H1欧1) ) ) (H6 H4欧0) ) ) ) (H2欧1 : (eprop B) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H0欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H5欧0 : (eprop A) => ( (False_ind C) H0欧0) ) ) (H5欧0 : (eprop B) => ( (False_ind C) H0欧0) ) ) H6欧1) ) (H5 H1欧1) ) ) (H6欧0 H4欧0) ) ) (H0 H1欧1) ) ) (H6 H4欧0) ) ) ) H欧2) ) (H2欧0 H4欧0) ) ) ) H欧1) ) (H1欧0 H4欧0) ) ) ) (H4欧0 : (eprop C) => ( (H欧1 : (eprop (not A) ) => ( (H0欧0 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H5欧0 : (eprop A) => ( (H0欧1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) C) (H1欧1 : (eprop A) => ( (H0欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H2欧1 : (eprop A) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H5欧0) ) ) (H6 H5欧0) ) ) ) (H2欧1 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H5欧0) ) ) (H6 H5欧0) ) ) ) H0欧2) ) (H2欧0 H5欧0) ) ) ) (H1欧1 : (eprop C) => ( (H0欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H2欧1 : (eprop A) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H5欧0) ) ) (H6 H5欧0) ) ) ) (H2欧1 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H5欧0) ) ) (H6 H5欧0) ) ) ) H0欧2) ) (H2欧0 H5欧0) ) ) ) H0欧1) ) (H1欧0 H5欧0) ) ) ) (H5欧0 : (eprop B) => H4欧0) ) H0欧0) ) (H5 H4欧0) ) ) (H0 H4欧0) ) ) ) H欧0) ) (H4 H3欧0) ) ) (H H3欧0) ) ) ) (H3欧0 : (eprop C) => ( (H6 : (eprop (not A) ) => ( (H0欧0 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H5欧0 : (eprop A) => ( (H0欧1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H1欧1 : (eprop A) => ( (H0欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H2欧1 : (eprop A) => ( (H0欧3 : (eprop False) => ( (False_ind B) H0欧3) ) (H6 H5欧0) ) ) ) (H2欧1 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H4欧0 : (eprop A) => ( (False_ind B) H欧0) ) ) (H4欧0 : (eprop C) => ( (False_ind B) H欧0) ) ) H6欧1) ) (H4 H2欧1) ) ) (H6欧0 H5欧0) ) ) (H H2欧1) ) ) (H6 H5欧0) ) ) ) H0欧2) ) (H2欧0 H5欧0) ) ) ) (H1欧1 : (eprop C) => ( (H0欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H2欧1 : (eprop A) => ( (H0欧3 : (eprop False) => ( (False_ind B) H0欧3) ) (H6 H5欧0) ) ) ) (H2欧1 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H4欧0 : (eprop A) => ( (False_ind B) H欧0) ) ) (H4欧0 : (eprop C) => ( (False_ind B) H欧0) ) ) H6欧1) ) (H4 H2欧1) ) ) (H6欧0 H5欧0) ) ) (H H2欧1) ) ) (H6 H5欧0) ) ) ) H0欧2) ) (H2欧0 H5欧0) ) ) ) H0欧1) ) (H1欧0 H5欧0) ) ) ) (H5欧0 : (eprop B) => ( (H0欧1 : (eprop (not A) ) => ( (H欧0 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H4欧0 : (eprop A) => ( (H欧1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H1欧1 : (eprop A) => ( (H欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H2欧1 : (eprop A) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H4欧0) ) ) (H6 H4欧0) ) ) ) (H2欧1 : (eprop B) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H4欧0) ) ) (H6 H4欧0) ) ) ) H欧2) ) (H2欧0 H4欧0) ) ) ) (H1欧1 : (eprop C) => ( (H欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H2欧1 : (eprop A) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H4欧0) ) ) (H6 H4欧0) ) ) ) (H2欧1 : (eprop B) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H4欧0) ) ) (H6 H4欧0) ) ) ) H欧2) ) (H2欧0 H4欧0) ) ) ) H欧1) ) (H1欧0 H4欧0) ) ) ) (H4欧0 : (eprop C) => H5欧0) ) H欧0) ) (H4 H5欧0) ) ) (H H5欧0) ) ) ) H0欧0) ) (H5 H3欧0) ) ) (H0 H3欧0) ) ) ) ) (H5 : (eprop C) => (H3 ( ( (or_intror A) C) H5) ) ) ) ) (H2欧0 : (eprop A) => (H3 ( ( (or_introl A) C) H2欧0) ) ) ) ) (H4 : (eprop B) => (H2 ( ( (or_intror A) B) H4) ) ) ) ) (H1欧0 : (eprop A) => (H2 ( ( (or_introl A) B) H1欧0) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_209 : (eprop B) => C) ) ) ( (dotpipp C) (_210 : (eprop C) => B) ) ) ( (iff ( (or A) B) ) ( (or A) C) ) ) (H2 : (eprop ( (dotpipp B) (_214 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_213 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or A) B) ) (_211 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) ( (dotpipp ( (or A) C) ) (_212 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) (H1欧0 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) ( (or A) C) ) (H4 : (eprop A) => ( ( (or_introl A) C) H4) ) ) (H4 : (eprop B) => ( (H1欧1 : (eprop (not A) ) => ( (H欧0 : (eprop C) => ( (H2欧0 : (eprop (not A) ) => ( (H0欧0 : (eprop B) => ( ( (or_intror A) C) H欧0) ) (H3 H欧0) ) ) (H0 H欧0) ) ) (H2 H4) ) ) (H H4) ) ) ) H1欧0) ) ) (H1欧0 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) ( (or A) B) ) (H4 : (eprop A) => ( ( (or_introl A) B) H4) ) ) (H4 : (eprop C) => ( (H1欧1 : (eprop (not A) ) => ( (H0欧0 : (eprop B) => ( (H3欧0 : (eprop (not A) ) => ( (H欧0 : (eprop C) => ( ( (or_intror A) B) H0欧0) ) (H2 H0欧0) ) ) (H H0欧0) ) ) (H3 H4) ) ) (H0 H4) ) ) ) H1欧0) ) ) ) ) ) H1) ) ) ) ) ) ) ) .
+or_cancel_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_244 : (_241 : (eprop B) -> (eprop (not A) ) ) -> (_243 : (_242 : (eprop C) -> (eprop (not A) ) ) -> (eprop ( (iff ( (iff ( (or B) A) ) ( (or C) A) ) ) ( (iff B) C) ) ) ) ) ) ) ) .
+[] or_cancel_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_240 : (eprop B) => (not A) ) ) ) => (H0 : (eprop ( (dotpipp C) (_239 : (eprop C) => (not A) ) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (or B) A) ) ( (or C) A) ) ) (_221 : (eprop ( (iff ( (or B) A) ) ( (or C) A) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_222 : (eprop ( (iff B) C) ) => ( (iff ( (or B) A) ) ( (or C) A) ) ) ) ) (H1 : (eprop ( (iff ( (or B) A) ) ( (or C) A) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (or B) A) ) (_223 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) ( (dotpipp ( (or C) A) ) (_224 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (or B) A) ) (_232 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) => (H3 : (eprop ( (dotpipp ( (or C) A) ) (_231 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) => ( (H1欧0 : (eprop ( (dotpipp B) (_230 : (eprop B) => ( (or C) A) ) ) ) => ( (H4 : (eprop ( (dotpipp A) (_229 : (eprop A) => ( (or C) A) ) ) ) => ( (H2欧0 : (eprop ( (dotpipp C) (_228 : (eprop C) => ( (or B) A) ) ) ) => ( (H5 : (eprop ( (dotpipp A) (_227 : (eprop A) => ( (or B) A) ) ) ) => ( ( ( (conj ( (dotpipp B) (_225 : (eprop B) => C) ) ) ( (dotpipp C) (_226 : (eprop C) => B) ) ) (H3欧0 : (eprop B) => ( (H6 : (eprop (not A) ) => ( (H欧0 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) C) (H1欧1 : (eprop C) => ( (H欧1 : (eprop (not A) ) => ( (H0欧0 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H2欧1 : (eprop B) => H1欧1) ) (H2欧1 : (eprop A) => ( (H0欧1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) C) (H4欧0 : (eprop C) => ( (H0欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H5欧0 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H2欧1) ) ) (H6 H2欧1) ) ) ) (H5欧0 : (eprop A) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H2欧1) ) ) (H6 H2欧1) ) ) ) H0欧2) ) (H5 H2欧1) ) ) ) (H4欧0 : (eprop A) => ( (H0欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H5欧0 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H2欧1) ) ) (H6 H2欧1) ) ) ) (H5欧0 : (eprop A) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H2欧1) ) ) (H6 H2欧1) ) ) ) H0欧2) ) (H5 H2欧1) ) ) ) H0欧1) ) (H4 H2欧1) ) ) ) H0欧0) ) (H2欧0 H1欧1) ) ) (H0 H1欧1) ) ) ) (H1欧1 : (eprop A) => ( (H欧1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) C) (H4欧0 : (eprop C) => ( (H欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H5欧0 : (eprop B) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H0欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H2欧1 : (eprop B) => ( (False_ind C) H0欧0) ) ) (H2欧1 : (eprop A) => ( (False_ind C) H0欧0) ) ) H6欧1) ) (H2欧0 H4欧0) ) ) (H6欧0 H1欧1) ) ) (H0 H4欧0) ) ) (H6 H1欧1) ) ) ) (H5欧0 : (eprop A) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H0欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H2欧1 : (eprop B) => ( (False_ind C) H0欧0) ) ) (H2欧1 : (eprop A) => ( (False_ind C) H0欧0) ) ) H6欧1) ) (H2欧0 H4欧0) ) ) (H6欧0 H1欧1) ) ) (H0 H4欧0) ) ) (H6 H1欧1) ) ) ) H欧2) ) (H5 H1欧1) ) ) ) (H4欧0 : (eprop A) => ( (H欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H5欧0 : (eprop B) => ( (H欧3 : (eprop False) => ( (False_ind C) H欧3) ) (H6 H1欧1) ) ) ) (H5欧0 : (eprop A) => ( (H欧3 : (eprop False) => ( (False_ind C) H欧3) ) (H6 H1欧1) ) ) ) H欧2) ) (H5 H1欧1) ) ) ) H欧1) ) (H4 H1欧1) ) ) ) H欧0) ) (H1欧0 H3欧0) ) ) (H H3欧0) ) ) ) (H3欧0 : (eprop C) => ( (H6 : (eprop (not A) ) => ( (H0欧0 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H2欧1 : (eprop B) => ( (H0欧1 : (eprop (not A) ) => ( (H欧0 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H1欧1 : (eprop C) => H2欧1) ) (H1欧1 : (eprop A) => ( (H欧1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H4欧0 : (eprop C) => ( (H欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H5欧0 : (eprop B) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H1欧1) ) ) (H6 H1欧1) ) ) ) (H5欧0 : (eprop A) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H1欧1) ) ) (H6 H1欧1) ) ) ) H欧2) ) (H5 H1欧1) ) ) ) (H4欧0 : (eprop A) => ( (H欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H5欧0 : (eprop B) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H1欧1) ) ) (H6 H1欧1) ) ) ) (H5欧0 : (eprop A) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H1欧1) ) ) (H6 H1欧1) ) ) ) H欧2) ) (H5 H1欧1) ) ) ) H欧1) ) (H4 H1欧1) ) ) ) H欧0) ) (H1欧0 H2欧1) ) ) (H H2欧1) ) ) ) (H2欧1 : (eprop A) => ( (H0欧1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H4欧0 : (eprop C) => ( (H0欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H5欧0 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H1欧1 : (eprop C) => ( (False_ind B) H欧0) ) ) (H1欧1 : (eprop A) => ( (False_ind B) H欧0) ) ) H6欧1) ) (H1欧0 H5欧0) ) ) (H6欧0 H2欧1) ) ) (H H5欧0) ) ) (H6 H2欧1) ) ) ) (H5欧0 : (eprop A) => ( (H0欧3 : (eprop False) => ( (False_ind B) H0欧3) ) (H6 H2欧1) ) ) ) H0欧2) ) (H5 H2欧1) ) ) ) (H4欧0 : (eprop A) => ( (H0欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H5欧0 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H1欧1 : (eprop C) => ( (False_ind B) H欧0) ) ) (H1欧1 : (eprop A) => ( (False_ind B) H欧0) ) ) H6欧1) ) (H1欧0 H5欧0) ) ) (H6欧0 H2欧1) ) ) (H H5欧0) ) ) (H6 H2欧1) ) ) ) (H5欧0 : (eprop A) => ( (H0欧3 : (eprop False) => ( (False_ind B) H0欧3) ) (H6 H2欧1) ) ) ) H0欧2) ) (H5 H2欧1) ) ) ) H0欧1) ) (H4 H2欧1) ) ) ) H0欧0) ) (H2欧0 H3欧0) ) ) (H0 H3欧0) ) ) ) ) (H5 : (eprop A) => (H3 ( ( (or_intror C) A) H5) ) ) ) ) (H2欧0 : (eprop C) => (H3 ( ( (or_introl C) A) H2欧0) ) ) ) ) (H4 : (eprop A) => (H2 ( ( (or_intror B) A) H4) ) ) ) ) (H1欧0 : (eprop B) => (H2 ( ( (or_introl B) A) H1欧0) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_233 : (eprop B) => C) ) ) ( (dotpipp C) (_234 : (eprop C) => B) ) ) ( (iff ( (or B) A) ) ( (or C) A) ) ) (H2 : (eprop ( (dotpipp B) (_238 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_237 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or B) A) ) (_235 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) ( (dotpipp ( (or C) A) ) (_236 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) (H1欧0 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) ( (or C) A) ) (H4 : (eprop B) => ( (H1欧1 : (eprop (not A) ) => ( (H欧0 : (eprop C) => ( (H2欧0 : (eprop (not A) ) => ( (H0欧0 : (eprop B) => ( ( (or_introl C) A) H欧0) ) (H3 H欧0) ) ) (H0 H欧0) ) ) (H2 H4) ) ) (H H4) ) ) ) (H4 : (eprop A) => ( ( (or_intror C) A) H4) ) ) H1欧0) ) ) (H1欧0 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) ( (or B) A) ) (H4 : (eprop C) => ( (H1欧1 : (eprop (not A) ) => ( (H0欧0 : (eprop B) => ( (H3欧0 : (eprop (not A) ) => ( (H欧0 : (eprop C) => ( ( (or_introl B) A) H0欧0) ) (H2 H0欧0) ) ) (H H0欧0) ) ) (H3 H4) ) ) (H0 H4) ) ) ) (H4 : (eprop A) => ( ( (or_intror B) A) H4) ) ) H1欧0) ) ) ) ) ) H1) ) ) ) ) ) ) ) .
+and_iff_compat_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_251 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (and A) B) ) ( (and A) C) ) ) ) ) ) ) .
+[] and_iff_compat_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_245 : (eprop B) => C) ) ) ( (dotpipp C) (_246 : (eprop C) => B) ) ) ( (iff ( (and A) B) ) ( (and A) C) ) ) (H0 : (eprop ( (dotpipp B) (_250 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_249 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and A) B) ) (_247 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) ( (dotpipp ( (and A) C) ) (_248 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) (H欧0 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) ( (and A) C) ) (H2 : (eprop A) => (H3 : (eprop B) => ( (H欧1 : (eprop C) => ( (H0欧0 : (eprop B) => ( ( ( (conj A) C) H2) H欧1) ) (H1 H欧1) ) ) (H0 H3) ) ) ) ) H欧0) ) ) (H欧0 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) ( (and A) B) ) (H2 : (eprop A) => (H3 : (eprop C) => ( (H欧1 : (eprop B) => ( (H1欧0 : (eprop C) => ( ( ( (conj A) B) H2) H欧1) ) (H0 H欧1) ) ) (H1 H3) ) ) ) ) H欧0) ) ) ) ) ) H) ) ) ) ) .
+and_iff_compat_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_258 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (and B) A) ) ( (and C) A) ) ) ) ) ) ) .
+[] and_iff_compat_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_252 : (eprop B) => C) ) ) ( (dotpipp C) (_253 : (eprop C) => B) ) ) ( (iff ( (and B) A) ) ( (and C) A) ) ) (H0 : (eprop ( (dotpipp B) (_257 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_256 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and B) A) ) (_254 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) ( (dotpipp ( (and C) A) ) (_255 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) (H欧0 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) ( (and C) A) ) (H2 : (eprop B) => (H3 : (eprop A) => ( (H欧1 : (eprop C) => ( (H0欧0 : (eprop B) => ( ( ( (conj C) A) H欧1) H3) ) (H1 H欧1) ) ) (H0 H2) ) ) ) ) H欧0) ) ) (H欧0 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) ( (and B) A) ) (H2 : (eprop C) => (H3 : (eprop A) => ( (H欧1 : (eprop B) => ( (H1欧0 : (eprop C) => ( ( ( (conj B) A) H欧1) H3) ) (H0 H欧1) ) ) (H1 H2) ) ) ) ) H欧0) ) ) ) ) ) H) ) ) ) ) .
+or_iff_compat_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_265 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (or A) B) ) ( (or A) C) ) ) ) ) ) ) .
+[] or_iff_compat_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_259 : (eprop B) => C) ) ) ( (dotpipp C) (_260 : (eprop C) => B) ) ) ( (iff ( (or A) B) ) ( (or A) C) ) ) (H0 : (eprop ( (dotpipp B) (_264 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_263 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or A) B) ) (_261 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) ( (dotpipp ( (or A) C) ) (_262 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) (H欧0 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) ( (or A) C) ) (H2 : (eprop A) => ( ( (or_introl A) C) H2) ) ) (H2 : (eprop B) => ( (H欧1 : (eprop C) => ( (H0欧0 : (eprop B) => ( ( (or_intror A) C) H欧1) ) (H1 H欧1) ) ) (H0 H2) ) ) ) H欧0) ) ) (H欧0 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) ( (or A) B) ) (H2 : (eprop A) => ( ( (or_introl A) B) H2) ) ) (H2 : (eprop C) => ( (H欧1 : (eprop B) => ( (H1欧0 : (eprop C) => ( ( (or_intror A) B) H欧1) ) (H0 H欧1) ) ) (H1 H2) ) ) ) H欧0) ) ) ) ) ) H) ) ) ) ) .
+or_iff_compat_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_272 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (or B) A) ) ( (or C) A) ) ) ) ) ) ) .
+[] or_iff_compat_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_266 : (eprop B) => C) ) ) ( (dotpipp C) (_267 : (eprop C) => B) ) ) ( (iff ( (or B) A) ) ( (or C) A) ) ) (H0 : (eprop ( (dotpipp B) (_271 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_270 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or B) A) ) (_268 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) ( (dotpipp ( (or C) A) ) (_269 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) (H欧0 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) ( (or C) A) ) (H2 : (eprop B) => ( (H欧1 : (eprop C) => ( (H0欧0 : (eprop B) => ( ( (or_introl C) A) H欧1) ) (H1 H欧1) ) ) (H0 H2) ) ) ) (H2 : (eprop A) => ( ( (or_intror C) A) H2) ) ) H欧0) ) ) (H欧0 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) ( (or B) A) ) (H2 : (eprop C) => ( (H欧1 : (eprop B) => ( (H1欧0 : (eprop C) => ( ( (or_introl B) A) H欧1) ) (H0 H欧1) ) ) (H1 H2) ) ) ) (H2 : (eprop A) => ( ( (or_intror B) A) H2) ) ) H欧0) ) ) ) ) ) H) ) ) ) ) .
+case_10 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (iff A) B) ) -> (_277 : (eprop ( (and ( (dotpipp A) (_273 : (eprop A) => B) ) ) ( (dotpipp B) (_274 : (eprop B) => A) ) ) ) -> (eprop ( (and ( (dotpipp A) (_275 : (eprop A) => B) ) ) ( (dotpipp B) (_276 : (eprop B) => A) ) ) ) ) ) ) ) .
+conj_case_10 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (iff A) B) ) -> (_285 : (_280 : (eprop A) -> (eprop B) ) -> (_284 : (_281 : (eprop B) -> (eprop A) ) -> (eprop ( (and ( (dotpipp A) (_282 : (eprop A) => B) ) ) ( (dotpipp B) (_283 : (eprop B) => A) ) ) ) ) ) ) ) ) .
+[A : Uprop, B : Uprop, H : (eprop ( (iff A) B) ) ] ( ( (conj_case_10 A) B) H) --> ( (conj ( (dotpipp A) (_278 : (eprop A) => B) ) ) ( (dotpipp B) (_279 : (eprop B) => A) ) ) .
+[A : Uprop, B : Uprop, H : (eprop ( (iff A) B) ) , var_16 : (_286 : (eprop A) -> (eprop B) ) , var_17 : (_287 : (eprop B) -> (eprop A) ) ] ( ( ( (case_10 A) B) H) ( ( ( ( (conj_case_10 A) B) H) var_16) var_17) ) --> ( ( (H欧0 : (eprop ( (dotpipp A) (_291 : (eprop A) => B) ) ) => (H0 : (eprop ( (dotpipp B) (_290 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp A) (_288 : (eprop A) => B) ) ) ( (dotpipp B) (_289 : (eprop B) => A) ) ) H欧0) H0) ) ) var_16) var_17) .
+iff_and : (A : Uprop -> (B : Uprop -> (_294 : (eprop ( (iff A) B) ) -> (eprop ( (and ( (dotpipp A) (_292 : (eprop A) => B) ) ) ( (dotpipp B) (_293 : (eprop B) => A) ) ) ) ) ) ) .
+[] iff_and --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (iff A) B) ) => ( ( ( (case_10 A) B) H) H) ) ) ) .
+iff_to_and : (A : Uprop -> (B : Uprop -> (eprop ( (iff ( (iff A) B) ) ( (and ( (dotpipp A) (_317 : (eprop A) => B) ) ) ( (dotpipp B) (_318 : (eprop B) => A) ) ) ) ) ) ) .
+[] iff_to_and --> (A : (etype dotprop) => (B : (etype dotprop) => ( ( ( (conj ( (dotpipp ( (iff A) B) ) (_297 : (eprop ( (iff A) B) ) => ( (and ( (dotpipp A) (_295 : (eprop A) => B) ) ) ( (dotpipp B) (_296 : (eprop B) => A) ) ) ) ) ) ( (dotpipp ( (and ( (dotpipp A) (_298 : (eprop A) => B) ) ) ( (dotpipp B) (_299 : (eprop B) => A) ) ) ) (_300 : (eprop ( (and ( (dotpipp A) (_298 : (eprop A) => B) ) ) ( (dotpipp B) (_299 : (eprop B) => A) ) ) ) => ( (iff A) B) ) ) ) (H : (eprop ( (iff A) B) ) => ( ( ( ( (and_ind ( (dotpipp A) (_301 : (eprop A) => B) ) ) ( (dotpipp B) (_302 : (eprop B) => A) ) ) ( (and ( (dotpipp A) (_303 : (eprop A) => B) ) ) ( (dotpipp B) (_304 : (eprop B) => A) ) ) ) (H0 : (eprop ( (dotpipp A) (_308 : (eprop A) => B) ) ) => (H1 : (eprop ( (dotpipp B) (_307 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp A) (_305 : (eprop A) => B) ) ) ( (dotpipp B) (_306 : (eprop B) => A) ) ) (H欧0 : (eprop A) => ( (H2 : (eprop B) => ( (H0欧0 : (eprop A) => H2) (H1 H2) ) ) (H0 H欧0) ) ) ) (H欧0 : (eprop B) => ( (H2 : (eprop A) => ( (H1欧0 : (eprop B) => H2) (H0 H2) ) ) (H1 H欧0) ) ) ) ) ) ) H) ) ) (H : (eprop ( (and ( (dotpipp A) (_315 : (eprop A) => B) ) ) ( (dotpipp B) (_316 : (eprop B) => A) ) ) ) => ( ( ( ( (and_ind ( (dotpipp A) (_309 : (eprop A) => B) ) ) ( (dotpipp B) (_310 : (eprop B) => A) ) ) ( (iff A) B) ) (H0 : (eprop ( (dotpipp A) (_314 : (eprop A) => B) ) ) => (H1 : (eprop ( (dotpipp B) (_313 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp A) (_311 : (eprop A) => B) ) ) ( (dotpipp B) (_312 : (eprop B) => A) ) ) (H欧0 : (eprop A) => ( (H2 : (eprop B) => ( (H0欧0 : (eprop A) => H2) (H1 H2) ) ) (H0 H欧0) ) ) ) (H欧0 : (eprop B) => ( (H2 : (eprop A) => ( (H1欧0 : (eprop B) => H2) (H0 H2) ) ) (H1 H欧0) ) ) ) ) ) ) H) ) ) ) ) .
+IF_then_else : (P : Uprop -> (Q : Uprop -> (R : Uprop -> Uprop) ) ) .
+[] IF_then_else --> (P : (etype dotprop) => (Q : (etype dotprop) => (R : (etype dotprop) => ( (or ( (and P) Q) ) ( (and (not P) ) R) ) ) ) ) .
+ex : (A : Utype -> (P : (_319 : (etype A) -> Uprop) -> Uprop) ) .
+ex_intro : (A : Utype -> (P : (_320 : (etype A) -> Uprop) -> (x : (etype A) -> (_321 : (eprop (P x) ) -> (eprop ( (ex A) P) ) ) ) ) ) .
+case_11 : (A : Utype -> (P : (_322 : (etype A) -> Uprop) -> (P欧0 : Uprop -> (f : (x : (etype A) -> (_323 : (eprop (P x) ) -> (eprop P欧0) ) ) -> (e : (eprop ( (ex A) P) ) -> (_324 : (eprop ( (ex A) P) ) -> (eprop P欧0) ) ) ) ) ) ) .
+ex_intro_case_11 : (A : Utype -> (P : (_327 : (etype A) -> Uprop) -> (P欧0 : Uprop -> (f : (x : (etype A) -> (_328 : (eprop (P x) ) -> (eprop P欧0) ) ) -> (e : (eprop ( (ex A) P) ) -> (x : (etype A) -> (_329 : (eprop (P x) ) -> (eprop ( (ex A) P) ) ) ) ) ) ) ) ) .
+[A : Utype, P : (_330 : (etype A) -> Uprop) , P欧0 : Uprop, f : (x : (etype A) -> (_331 : (eprop (P x) ) -> (eprop P欧0) ) ) , e : (eprop ( (ex A) P) ) ] ( ( ( ( (ex_intro_case_11 A) P) P欧0) f) e) --> ( (ex_intro A) P) .
+[A : Utype, P : (_325 : (etype A) -> Uprop) , P欧0 : Uprop, f : (x : (etype A) -> (_326 : (eprop (P x) ) -> (eprop P欧0) ) ) , e : (eprop ( (ex A) P) ) , var_18 : (etype A) , var_19 : (eprop (P var_18) ) ] ( ( ( ( ( (case_11 A) P) P欧0) f) e) ( ( ( ( ( ( (ex_intro_case_11 A) P) P欧0) f) e) var_18) var_19) ) --> ( (f var_18) var_19) .
+ex_ind : (A : Utype -> (P : (_334 : (etype A) -> Uprop) -> (P欧0 : Uprop -> (f : (x : (etype A) -> (_335 : (eprop (P x) ) -> (eprop P欧0) ) ) -> (e : (eprop ( (ex A) P) ) -> (eprop P欧0) ) ) ) ) ) .
+[] ex_ind --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_333 : (etype A) => dotprop) ) ) => (P欧0 : (etype dotprop) => (f : (eprop ( (dotpitp A) (x : (etype A) => ( (dotpipp (P x) ) (_332 : (eprop (P x) ) => P欧0) ) ) ) ) => (e : (eprop ( (ex A) P) ) => ( ( ( ( ( (case_11 A) P) P欧0) f) e) e) ) ) ) ) ) .
+ex2 : (A : Utype -> (P : (_336 : (etype A) -> Uprop) -> (Q : (_337 : (etype A) -> Uprop) -> Uprop) ) ) .
+ex_intro2 : (A : Utype -> (P : (_338 : (etype A) -> Uprop) -> (Q : (_339 : (etype A) -> Uprop) -> (x : (etype A) -> (_341 : (eprop (P x) ) -> (_340 : (eprop (Q x) ) -> (eprop ( ( (ex2 A) P) Q) ) ) ) ) ) ) ) .
+case_12 : (A : Utype -> (P : (_342 : (etype A) -> Uprop) -> (Q : (_343 : (etype A) -> Uprop) -> (P欧0 : Uprop -> (f : (x : (etype A) -> (_345 : (eprop (P x) ) -> (_344 : (eprop (Q x) ) -> (eprop P欧0) ) ) ) -> (e : (eprop ( ( (ex2 A) P) Q) ) -> (_346 : (eprop ( ( (ex2 A) P) Q) ) -> (eprop P欧0) ) ) ) ) ) ) ) .
+ex_intro2_case_12 : (A : Utype -> (P : (_351 : (etype A) -> Uprop) -> (Q : (_352 : (etype A) -> Uprop) -> (P欧0 : Uprop -> (f : (x : (etype A) -> (_354 : (eprop (P x) ) -> (_353 : (eprop (Q x) ) -> (eprop P欧0) ) ) ) -> (e : (eprop ( ( (ex2 A) P) Q) ) -> (x : (etype A) -> (_356 : (eprop (P x) ) -> (_355 : (eprop (Q x) ) -> (eprop ( ( (ex2 A) P) Q) ) ) ) ) ) ) ) ) ) ) .
+[A : Utype, P : (_357 : (etype A) -> Uprop) , Q : (_358 : (etype A) -> Uprop) , P欧0 : Uprop, f : (x : (etype A) -> (_360 : (eprop (P x) ) -> (_359 : (eprop (Q x) ) -> (eprop P欧0) ) ) ) , e : (eprop ( ( (ex2 A) P) Q) ) ] ( ( ( ( ( (ex_intro2_case_12 A) P) Q) P欧0) f) e) --> ( ( (ex_intro2 A) P) Q) .
+[A : Utype, P : (_347 : (etype A) -> Uprop) , Q : (_348 : (etype A) -> Uprop) , P欧0 : Uprop, f : (x : (etype A) -> (_350 : (eprop (P x) ) -> (_349 : (eprop (Q x) ) -> (eprop P欧0) ) ) ) , e : (eprop ( ( (ex2 A) P) Q) ) , var_20 : (etype A) , var_21 : (eprop (P var_20) ) , var_22 : (eprop (Q var_20) ) ] ( ( ( ( ( ( (case_12 A) P) Q) P欧0) f) e) ( ( ( ( ( ( ( ( (ex_intro2_case_12 A) P) Q) P欧0) f) e) var_20) var_21) var_22) ) --> ( ( (f var_20) var_21) var_22) .
+ex2_ind : (A : Utype -> (P : (_365 : (etype A) -> Uprop) -> (Q : (_366 : (etype A) -> Uprop) -> (P欧0 : Uprop -> (f : (x : (etype A) -> (_368 : (eprop (P x) ) -> (_367 : (eprop (Q x) ) -> (eprop P欧0) ) ) ) -> (e : (eprop ( ( (ex2 A) P) Q) ) -> (eprop P欧0) ) ) ) ) ) ) .
+[] ex2_ind --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_364 : (etype A) => dotprop) ) ) => (Q : (etype ( (dotpitt A) (_363 : (etype A) => dotprop) ) ) => (P欧0 : (etype dotprop) => (f : (eprop ( (dotpitp A) (x : (etype A) => ( (dotpipp (P x) ) (_362 : (eprop (P x) ) => ( (dotpipp (Q x) ) (_361 : (eprop (Q x) ) => P欧0) ) ) ) ) ) ) => (e : (eprop ( ( (ex2 A) P) Q) ) => ( ( ( ( ( ( (case_12 A) P) Q) P欧0) f) e) e) ) ) ) ) ) ) .
+all : (A : Utype -> (P : (_370 : (etype A) -> Uprop) -> Uprop) ) .
+[] all --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_369 : (etype A) => dotprop) ) ) => ( (dotpitp A) (x : (etype A) => (P x) ) ) ) ) .
+inst : (A : Utype -> (P : (_372 : (etype A) -> Uprop) -> (x : (etype A) -> (_373 : (eprop ( (all A) (x欧0 : (etype A) => (P x欧0) ) ) ) -> (eprop (P x) ) ) ) ) ) .
+[] inst --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_371 : (etype A) => dotprop) ) ) => (x : (etype A) => (H : (eprop ( (dotpitp A) (x欧0 : (etype A) => (P x欧0) ) ) ) => (H x) ) ) ) ) .
+gen : (A : Utype -> (P : (_376 : (etype A) -> Uprop) -> (B : Uprop -> (f : (y : (etype A) -> (_377 : (eprop B) -> (eprop (P y) ) ) ) -> (_378 : (eprop B) -> (eprop ( (all A) P) ) ) ) ) ) ) .
+[] gen --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_375 : (etype A) => dotprop) ) ) => (B : (etype dotprop) => (f : (eprop ( (dotpitp A) (y : (etype A) => ( (dotpipp B) (_374 : (eprop B) => (P y) ) ) ) ) ) => (H : (eprop B) => (x : (etype A) => ( (f x) H) ) ) ) ) ) ) .
+eq : (A : Utype -> (x : (etype A) -> (_379 : (etype A) -> Uprop) ) ) .
+refl_equal : (A : Utype -> (x : (etype A) -> (eprop ( ( (eq A) x) x) ) ) ) .
+case_13 : (A : Utype -> (x : (etype A) -> (P : (_380 : (etype A) -> Utype) -> (f : (etype (P x) ) -> (y : (etype A) -> (e : (eprop ( ( (eq A) x) y) ) -> (y欧0 : (etype A) -> (_381 : (eprop ( ( (eq A) x) y欧0) ) -> (etype (P y欧0) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_13 : (A : Utype -> (x : (etype A) -> (P : (_383 : (etype A) -> Utype) -> (f : (etype (P x) ) -> (y : (etype A) -> (e : (eprop ( ( (eq A) x) y) ) -> (eprop ( ( (eq A) x) x) ) ) ) ) ) ) ) .
+[A : Utype, x : (etype A) , P : (_384 : (etype A) -> Utype) , f : (etype (P x) ) , y : (etype A) , e : (eprop ( ( (eq A) x) y) ) ] ( ( ( ( ( (refl_equal_case_13 A) x) P) f) y) e) --> ( (refl_equal A) x) .
+[A : Utype, x : (etype A) , P : (_382 : (etype A) -> Utype) , f : (etype (P x) ) , y : (etype A) , e : (eprop ( ( (eq A) x) y) ) ] ( ( ( ( ( ( ( (case_13 A) x) P) f) y) e) x) ( ( ( ( ( (refl_equal_case_13 A) x) P) f) y) e) ) --> f.
+eq_rect : (A : Utype -> (x : (etype A) -> (P : (_386 : (etype A) -> Utype) -> (f : (etype (P x) ) -> (y : (etype A) -> (e : (eprop ( ( (eq A) x) y) ) -> (etype (P y) ) ) ) ) ) ) ) .
+[] eq_rect --> (A : (etype dottype) => (x : (etype A) => (P : (etype ( (dotpitt A) (_385 : (etype A) => dottype) ) ) => (f : (etype (P x) ) => (y : (etype A) => (e : (eprop ( ( (eq A) x) y) ) => ( ( ( ( ( ( ( (case_13 A) x) P) f) y) e) y) e) ) ) ) ) ) ) .
+eq_ind : (A : Utype -> (x : (etype A) -> (P : (_388 : (etype A) -> Uprop) -> (f : (eprop (P x) ) -> (y : (etype A) -> (e : (eprop ( ( (eq A) x) y) ) -> (eprop (P y) ) ) ) ) ) ) ) .
+[] eq_ind --> (A : (etype dottype) => (x : (etype A) => (P : (etype ( (dotpitt A) (_387 : (etype A) => dotprop) ) ) => ( ( (eq_rect A) x) P) ) ) ) .
+eq_rec : (A : Utype -> (x : (etype A) -> (P : (_390 : (etype A) -> Uset) -> (f : (eset (P x) ) -> (y : (etype A) -> (e : (eprop ( ( (eq A) x) y) ) -> (eset (P y) ) ) ) ) ) ) ) .
+[] eq_rec --> (A : (etype dottype) => (x : (etype A) => (P : (etype ( (dotpitt A) (_389 : (etype A) => dotset) ) ) => ( ( (eq_rect A) x) P) ) ) ) .
+case_14 : (A : Uprop -> (C : Uprop -> (h1 : (eprop A) -> (h2 : (_391 : (eprop A) -> (eprop False) ) -> (f : (eprop False) -> (_392 : (eprop False) -> (eprop C) ) ) ) ) ) ) .
+absurd : (A : Uprop -> (C : Uprop -> (_396 : (eprop A) -> (_395 : (eprop (not A) ) -> (eprop C) ) ) ) ) .
+[] absurd --> (A : (etype dotprop) => (C : (etype dotprop) => (h1 : (eprop A) => (h2 : (eprop ( (dotpipp A) (_394 : (eprop A) => False) ) ) => ( (f : (eprop False) => ( ( ( ( ( (case_14 A) C) h1) h2) f) f) ) (h2 h1) ) ) ) ) ) .
+case_15 : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (H : (eprop ( ( (eq A) x) y) ) -> (y欧0 : (etype A) -> (_397 : (eprop ( ( (eq A) x) y欧0) ) -> (eprop ( ( (eq A) y欧0) x) ) ) ) ) ) ) ) .
+refl_equal_case_15 : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (H : (eprop ( ( (eq A) x) y) ) -> (eprop ( ( (eq A) x) x) ) ) ) ) ) .
+[A : Utype, x : (etype A) , y : (etype A) , H : (eprop ( ( (eq A) x) y) ) ] ( ( ( (refl_equal_case_15 A) x) y) H) --> ( (refl_equal A) x) .
+[A : Utype, x : (etype A) , y : (etype A) , H : (eprop ( ( (eq A) x) y) ) ] ( ( ( ( ( (case_15 A) x) y) H) x) ( ( ( (refl_equal_case_15 A) x) y) H) ) --> ( (refl_equal A) x) .
+sym_eq : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (_398 : (eprop ( ( (eq A) x) y) ) -> (eprop ( ( (eq A) y) x) ) ) ) ) ) .
+[] sym_eq --> (A : (etype dottype) => (x : (etype A) => (y : (etype A) => (H : (eprop ( ( (eq A) x) y) ) => ( ( ( ( ( (case_15 A) x) y) H) y) H) ) ) ) ) .
+case_16 : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (z : (etype A) -> (H : (eprop ( ( (eq A) x) y) ) -> (H0 : (eprop ( ( (eq A) y) z) ) -> (y欧0 : (etype A) -> (_399 : (eprop ( ( (eq A) y) y欧0) ) -> (eprop ( ( (eq A) x) y欧0) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_16 : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (z : (etype A) -> (H : (eprop ( ( (eq A) x) y) ) -> (H0 : (eprop ( ( (eq A) y) z) ) -> (eprop ( ( (eq A) y) y) ) ) ) ) ) ) ) .
+[A : Utype, x : (etype A) , y : (etype A) , z : (etype A) , H : (eprop ( ( (eq A) x) y) ) , H0 : (eprop ( ( (eq A) y) z) ) ] ( ( ( ( ( (refl_equal_case_16 A) x) y) z) H) H0) --> ( (refl_equal A) y) .
+[A : Utype, x : (etype A) , y : (etype A) , z : (etype A) , H : (eprop ( ( (eq A) x) y) ) , H0 : (eprop ( ( (eq A) y) z) ) ] ( ( ( ( ( ( ( (case_16 A) x) y) z) H) H0) y) ( ( ( ( ( (refl_equal_case_16 A) x) y) z) H) H0) ) --> H.
+trans_eq : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (z : (etype A) -> (_401 : (eprop ( ( (eq A) x) y) ) -> (_400 : (eprop ( ( (eq A) y) z) ) -> (eprop ( ( (eq A) x) z) ) ) ) ) ) ) ) .
+[] trans_eq --> (A : (etype dottype) => (x : (etype A) => (y : (etype A) => (z : (etype A) => (H : (eprop ( ( (eq A) x) y) ) => (H0 : (eprop ( ( (eq A) y) z) ) => ( ( ( ( ( ( ( (case_16 A) x) y) z) H) H0) z) H0) ) ) ) ) ) ) .
+case_17 : (A : Utype -> (B : Utype -> (f : (_402 : (etype A) -> (etype B) ) -> (x : (etype A) -> (y : (etype A) -> (H : (eprop ( ( (eq A) x) y) ) -> (y欧0 : (etype A) -> (_403 : (eprop ( ( (eq A) x) y欧0) ) -> (eprop ( ( (eq B) (f x) ) (f y欧0) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_17 : (A : Utype -> (B : Utype -> (f : (_405 : (etype A) -> (etype B) ) -> (x : (etype A) -> (y : (etype A) -> (H : (eprop ( ( (eq A) x) y) ) -> (eprop ( ( (eq A) x) x) ) ) ) ) ) ) ) .
+[A : Utype, B : Utype, f : (_406 : (etype A) -> (etype B) ) , x : (etype A) , y : (etype A) , H : (eprop ( ( (eq A) x) y) ) ] ( ( ( ( ( (refl_equal_case_17 A) B) f) x) y) H) --> ( (refl_equal A) x) .
+[A : Utype, B : Utype, f : (_404 : (etype A) -> (etype B) ) , x : (etype A) , y : (etype A) , H : (eprop ( ( (eq A) x) y) ) ] ( ( ( ( ( ( ( (case_17 A) B) f) x) y) H) x) ( ( ( ( ( (refl_equal_case_17 A) B) f) x) y) H) ) --> ( (refl_equal B) (f x) ) .
+f_equal : (A : Utype -> (B : Utype -> (f : (_408 : (etype A) -> (etype B) ) -> (x : (etype A) -> (y : (etype A) -> (_409 : (eprop ( ( (eq A) x) y) ) -> (eprop ( ( (eq B) (f x) ) (f y) ) ) ) ) ) ) ) ) .
+[] f_equal --> (A : (etype dottype) => (B : (etype dottype) => (f : (etype ( (dotpitt A) (_407 : (etype A) => B) ) ) => (x : (etype A) => (y : (etype A) => (H : (eprop ( ( (eq A) x) y) ) => ( ( ( ( ( ( ( (case_17 A) B) f) x) y) H) y) H) ) ) ) ) ) ) .
+case_18 : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (h1 : (eprop (not ( ( (eq A) x) y) ) ) -> (h2 : (eprop ( ( (eq A) y) x) ) -> (y欧0 : (etype A) -> (_410 : (eprop ( ( (eq A) y) y欧0) ) -> (h1欧0 : (eprop (not ( ( (eq A) y欧0) y) ) ) -> (eprop ( ( (eq A) y欧0) y) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_18 : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (h1 : (eprop (not ( ( (eq A) x) y) ) ) -> (h2 : (eprop ( ( (eq A) y) x) ) -> (eprop ( ( (eq A) y) y) ) ) ) ) ) ) .
+[A : Utype, x : (etype A) , y : (etype A) , h1 : (eprop (not ( ( (eq A) x) y) ) ) , h2 : (eprop ( ( (eq A) y) x) ) ] ( ( ( ( (refl_equal_case_18 A) x) y) h1) h2) --> ( (refl_equal A) y) .
+[A : Utype, x : (etype A) , y : (etype A) , h1 : (eprop (not ( ( (eq A) x) y) ) ) , h2 : (eprop ( ( (eq A) y) x) ) ] ( ( ( ( ( ( (case_18 A) x) y) h1) h2) y) ( ( ( ( (refl_equal_case_18 A) x) y) h1) h2) ) --> (h1欧0 : (eprop (not ( ( (eq A) y) y) ) ) => ( (refl_equal A) y) ) .
+sym_not_eq : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (_411 : (eprop (not ( ( (eq A) x) y) ) ) -> (eprop (not ( ( (eq A) y) x) ) ) ) ) ) ) .
+[] sym_not_eq --> (A : (etype dottype) => (x : (etype A) => (y : (etype A) => (h1 : (eprop (not ( ( (eq A) x) y) ) ) => (h2 : (eprop ( ( (eq A) y) x) ) => (h1 ( ( ( ( ( ( ( (case_18 A) x) y) h1) h2) x) h2) h1) ) ) ) ) ) ) .
+sym_equal : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (_412 : (eprop ( ( (eq A) x) y) ) -> (eprop ( ( (eq A) y) x) ) ) ) ) ) .
+[] sym_equal --> (A : (etype dottype) => (x : (etype A) => (y : (etype A) => ( ( (sym_eq A) x) y) ) ) ) .
+sym_not_equal : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (_413 : (eprop (not ( ( (eq A) x) y) ) ) -> (eprop (not ( ( (eq A) y) x) ) ) ) ) ) ) .
+[] sym_not_equal --> (A : (etype dottype) => (x : (etype A) => (y : (etype A) => ( ( (sym_not_eq A) x) y) ) ) ) .
+trans_equal : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (z : (etype A) -> (_415 : (eprop ( ( (eq A) x) y) ) -> (_414 : (eprop ( ( (eq A) y) z) ) -> (eprop ( ( (eq A) x) z) ) ) ) ) ) ) ) .
+[] trans_equal --> (A : (etype dottype) => (x : (etype A) => (y : (etype A) => (z : (etype A) => ( ( ( (trans_eq A) x) y) z) ) ) ) ) .
+eq_ind_r : (A : Utype -> (x : (etype A) -> (P : (_417 : (etype A) -> Uprop) -> (_419 : (eprop (P x) ) -> (y : (etype A) -> (_418 : (eprop ( ( (eq A) y) x) ) -> (eprop (P y) ) ) ) ) ) ) ) .
+[] eq_ind_r --> (A : (etype dottype) => (x : (etype A) => (P : (etype ( (dotpitt A) (_416 : (etype A) => dotprop) ) ) => (H : (eprop (P x) ) => (y : (etype A) => (H0 : (eprop ( ( (eq A) y) x) ) => ( ( ( ( ( (eq_ind A) x) (y欧0 : (etype A) => (P y欧0) ) ) H) y) ( ( ( (sym_eq A) y) x) H0) ) ) ) ) ) ) ) .
+eq_rec_r : (A : Utype -> (x : (etype A) -> (P : (_421 : (etype A) -> Uset) -> (_423 : (eset (P x) ) -> (y : (etype A) -> (_422 : (eprop ( ( (eq A) y) x) ) -> (eset (P y) ) ) ) ) ) ) ) .
+[] eq_rec_r --> (A : (etype dottype) => (x : (etype A) => (P : (etype ( (dotpitt A) (_420 : (etype A) => dotset) ) ) => (H : (eset (P x) ) => (y : (etype A) => (H0 : (eprop ( ( (eq A) y) x) ) => ( ( ( ( ( (eq_rec A) x) (y欧0 : (etype A) => (P y欧0) ) ) H) y) ( ( ( (sym_eq A) y) x) H0) ) ) ) ) ) ) ) .
+eq_rect_r : (A : Utype -> (x : (etype A) -> (P : (_425 : (etype A) -> Utype) -> (_427 : (etype (P x) ) -> (y : (etype A) -> (_426 : (eprop ( ( (eq A) y) x) ) -> (etype (P y) ) ) ) ) ) ) ) .
+[] eq_rect_r --> (A : (etype dottype) => (x : (etype A) => (P : (etype ( (dotpitt A) (_424 : (etype A) => dottype) ) ) => (H : (etype (P x) ) => (y : (etype A) => (H0 : (eprop ( ( (eq A) y) x) ) => ( ( ( ( ( (eq_rect A) x) (y欧0 : (etype A) => (P y欧0) ) ) H) y) ( ( ( (sym_eq A) y) x) H0) ) ) ) ) ) ) ) .
+case_19 : (A1 : Utype -> (A2 : Utype -> (B : Utype -> (f : (_429 : (etype A1) -> (_428 : (etype A2) -> (etype B) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (y : (etype A1) -> (_431 : (eprop ( ( (eq A1) x1) y) ) -> (_430 : (eprop ( ( (eq A2) x2) y2) ) -> (eprop ( ( (eq B) ( (f x1) x2) ) ( (f y) y2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_19 : (A1 : Utype -> (A2 : Utype -> (B : Utype -> (f : (_435 : (etype A1) -> (_434 : (etype A2) -> (etype B) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (eprop ( ( (eq A1) x1) x1) ) ) ) ) ) ) ) ) ) ) .
+[A1 : Utype, A2 : Utype, B : Utype, f : (_437 : (etype A1) -> (_436 : (etype A2) -> (etype B) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( (refl_equal_case_19 A1) A2) B) f) x1) y1) x2) y2) H) --> ( (refl_equal A1) x1) .
+case_20 : (A1 : Utype -> (A2 : Utype -> (B : Utype -> (f : (_439 : (etype A1) -> (_438 : (etype A2) -> (etype B) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (y : (etype A2) -> (_440 : (eprop ( ( (eq A2) x2) y) ) -> (eprop ( ( (eq B) ( (f x1) x2) ) ( (f x1) y) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_20 : (A1 : Utype -> (A2 : Utype -> (B : Utype -> (f : (_444 : (etype A1) -> (_443 : (etype A2) -> (etype B) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (eprop ( ( (eq A2) x2) x2) ) ) ) ) ) ) ) ) ) ) ) .
+[A1 : Utype, A2 : Utype, B : Utype, f : (_446 : (etype A1) -> (_445 : (etype A2) -> (etype B) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( (refl_equal_case_20 A1) A2) B) f) x1) y1) x2) y2) H) H欧0) --> ( (refl_equal A2) x2) .
+[A1 : Utype, A2 : Utype, B : Utype, f : (_442 : (etype A1) -> (_441 : (etype A2) -> (etype B) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( ( ( (case_20 A1) A2) B) f) x1) y1) x2) y2) H) H欧0) x2) ( ( ( ( ( ( ( ( ( (refl_equal_case_20 A1) A2) B) f) x1) y1) x2) y2) H) H欧0) ) --> ( (refl_equal B) ( (f x1) x2) ) .
+[A1 : Utype, A2 : Utype, B : Utype, f : (_433 : (etype A1) -> (_432 : (etype A2) -> (etype B) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( ( ( (case_19 A1) A2) B) f) x1) y1) x2) y2) H) x1) ( ( ( ( ( ( ( ( (refl_equal_case_19 A1) A2) B) f) x1) y1) x2) y2) H) ) --> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) => ( ( ( ( ( ( ( ( ( ( ( (case_20 A1) A2) B) f) x1) y1) x2) y2) H) H欧0) y2) H欧0) ) .
+f_equal2 : (A1 : Utype -> (A2 : Utype -> (B : Utype -> (f : (_450 : (etype A1) -> (_449 : (etype A2) -> (etype B) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (_452 : (eprop ( ( (eq A1) x1) y1) ) -> (_451 : (eprop ( ( (eq A2) x2) y2) ) -> (eprop ( ( (eq B) ( (f x1) x2) ) ( (f y1) y2) ) ) ) ) ) ) ) ) ) ) ) ) .
+[] f_equal2 --> (A1 : (etype dottype) => (A2 : (etype dottype) => (B : (etype dottype) => (f : (etype ( (dotpitt A1) (_448 : (etype A1) => ( (dotpitt A2) (_447 : (etype A2) => B) ) ) ) ) => (x1 : (etype A1) => (y1 : (etype A1) => (x2 : (etype A2) => (y2 : (etype A2) => (H : (eprop ( ( (eq A1) x1) y1) ) => ( ( ( ( ( ( ( ( ( ( (case_19 A1) A2) B) f) x1) y1) x2) y2) H) y1) H) ) ) ) ) ) ) ) ) ) .
+case_21 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (B : Utype -> (f : (_455 : (etype A1) -> (_454 : (etype A2) -> (_453 : (etype A3) -> (etype B) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (y : (etype A1) -> (_458 : (eprop ( ( (eq A1) x1) y) ) -> (_457 : (eprop ( ( (eq A2) x2) y2) ) -> (_456 : (eprop ( ( (eq A3) x3) y3) ) -> (eprop ( ( (eq B) ( ( (f x1) x2) x3) ) ( ( (f y) y2) y3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_21 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (B : Utype -> (f : (_464 : (etype A1) -> (_463 : (etype A2) -> (_462 : (etype A3) -> (etype B) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (eprop ( ( (eq A1) x1) x1) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, B : Utype, f : (_467 : (etype A1) -> (_466 : (etype A2) -> (_465 : (etype A3) -> (etype B) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_21 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) --> ( (refl_equal A1) x1) .
+case_22 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (B : Utype -> (f : (_470 : (etype A1) -> (_469 : (etype A2) -> (_468 : (etype A3) -> (etype B) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (y : (etype A2) -> (_472 : (eprop ( ( (eq A2) x2) y) ) -> (_471 : (eprop ( ( (eq A3) x3) y3) ) -> (eprop ( ( (eq B) ( ( (f x1) x2) x3) ) ( ( (f x1) y) y3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_22 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (B : Utype -> (f : (_478 : (etype A1) -> (_477 : (etype A2) -> (_476 : (etype A3) -> (etype B) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (eprop ( ( (eq A2) x2) x2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, B : Utype, f : (_481 : (etype A1) -> (_480 : (etype A2) -> (_479 : (etype A3) -> (etype B) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_22 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) --> ( (refl_equal A2) x2) .
+case_23 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (B : Utype -> (f : (_484 : (etype A1) -> (_483 : (etype A2) -> (_482 : (etype A3) -> (etype B) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (y : (etype A3) -> (_485 : (eprop ( ( (eq A3) x3) y) ) -> (eprop ( ( (eq B) ( ( (f x1) x2) x3) ) ( ( (f x1) x2) y) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_23 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (B : Utype -> (f : (_491 : (etype A1) -> (_490 : (etype A2) -> (_489 : (etype A3) -> (etype B) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (eprop ( ( (eq A3) x3) x3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, B : Utype, f : (_494 : (etype A1) -> (_493 : (etype A2) -> (_492 : (etype A3) -> (etype B) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_23 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) H欧1) --> ( (refl_equal A3) x3) .
+[A1 : Utype, A2 : Utype, A3 : Utype, B : Utype, f : (_488 : (etype A1) -> (_487 : (etype A2) -> (_486 : (etype A3) -> (etype B) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_23 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) H欧1) x3) ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_23 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) H欧1) ) --> ( (refl_equal B) ( ( (f x1) x2) x3) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, B : Utype, f : (_475 : (etype A1) -> (_474 : (etype A2) -> (_473 : (etype A3) -> (etype B) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_22 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) x2) ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_22 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) ) --> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_23 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) H欧1) y3) H欧1) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, B : Utype, f : (_461 : (etype A1) -> (_460 : (etype A2) -> (_459 : (etype A3) -> (etype B) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( (case_21 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) x1) ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_21 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) ) --> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_22 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) y2) H欧0) ) .
+f_equal3 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (B : Utype -> (f : (_500 : (etype A1) -> (_499 : (etype A2) -> (_498 : (etype A3) -> (etype B) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (_503 : (eprop ( ( (eq A1) x1) y1) ) -> (_502 : (eprop ( ( (eq A2) x2) y2) ) -> (_501 : (eprop ( ( (eq A3) x3) y3) ) -> (eprop ( ( (eq B) ( ( (f x1) x2) x3) ) ( ( (f y1) y2) y3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[] f_equal3 --> (A1 : (etype dottype) => (A2 : (etype dottype) => (A3 : (etype dottype) => (B : (etype dottype) => (f : (etype ( (dotpitt A1) (_497 : (etype A1) => ( (dotpitt A2) (_496 : (etype A2) => ( (dotpitt A3) (_495 : (etype A3) => B) ) ) ) ) ) ) => (x1 : (etype A1) => (y1 : (etype A1) => (x2 : (etype A2) => (y2 : (etype A2) => (x3 : (etype A3) => (y3 : (etype A3) => (H : (eprop ( ( (eq A1) x1) y1) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( (case_21 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) y1) H) ) ) ) ) ) ) ) ) ) ) ) ) .
+case_24 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_507 : (etype A1) -> (_506 : (etype A2) -> (_505 : (etype A3) -> (_504 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (y : (etype A1) -> (_511 : (eprop ( ( (eq A1) x1) y) ) -> (_510 : (eprop ( ( (eq A2) x2) y2) ) -> (_509 : (eprop ( ( (eq A3) x3) y3) ) -> (_508 : (eprop ( ( (eq A4) x4) y4) ) -> (eprop ( ( (eq B) ( ( ( (f x1) x2) x3) x4) ) ( ( ( (f y) y2) y3) y4) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_24 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_519 : (etype A1) -> (_518 : (etype A2) -> (_517 : (etype A3) -> (_516 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (eprop ( ( (eq A1) x1) x1) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_523 : (etype A1) -> (_522 : (etype A2) -> (_521 : (etype A3) -> (_520 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_24 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) --> ( (refl_equal A1) x1) .
+case_25 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_527 : (etype A1) -> (_526 : (etype A2) -> (_525 : (etype A3) -> (_524 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (y : (etype A2) -> (_530 : (eprop ( ( (eq A2) x2) y) ) -> (_529 : (eprop ( ( (eq A3) x3) y3) ) -> (_528 : (eprop ( ( (eq A4) x4) y4) ) -> (eprop ( ( (eq B) ( ( ( (f x1) x2) x3) x4) ) ( ( ( (f x1) y) y3) y4) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_25 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_538 : (etype A1) -> (_537 : (etype A2) -> (_536 : (etype A3) -> (_535 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (eprop ( ( (eq A2) x2) x2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_542 : (etype A1) -> (_541 : (etype A2) -> (_540 : (etype A3) -> (_539 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_25 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) --> ( (refl_equal A2) x2) .
+case_26 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_546 : (etype A1) -> (_545 : (etype A2) -> (_544 : (etype A3) -> (_543 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (y : (etype A3) -> (_548 : (eprop ( ( (eq A3) x3) y) ) -> (_547 : (eprop ( ( (eq A4) x4) y4) ) -> (eprop ( ( (eq B) ( ( ( (f x1) x2) x3) x4) ) ( ( ( (f x1) x2) y) y4) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_26 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_556 : (etype A1) -> (_555 : (etype A2) -> (_554 : (etype A3) -> (_553 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (eprop ( ( (eq A3) x3) x3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_560 : (etype A1) -> (_559 : (etype A2) -> (_558 : (etype A3) -> (_557 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_26 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) --> ( (refl_equal A3) x3) .
+case_27 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_564 : (etype A1) -> (_563 : (etype A2) -> (_562 : (etype A3) -> (_561 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) -> (y : (etype A4) -> (_565 : (eprop ( ( (eq A4) x4) y) ) -> (eprop ( ( (eq B) ( ( ( (f x1) x2) x3) x4) ) ( ( ( (f x1) x2) x3) y) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_27 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_573 : (etype A1) -> (_572 : (etype A2) -> (_571 : (etype A3) -> (_570 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) -> (eprop ( ( (eq A4) x4) x4) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_577 : (etype A1) -> (_576 : (etype A2) -> (_575 : (etype A3) -> (_574 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) , H欧2 : (eprop ( ( (eq A4) x4) y4) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_27 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) H欧2) --> ( (refl_equal A4) x4) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_569 : (etype A1) -> (_568 : (etype A2) -> (_567 : (etype A3) -> (_566 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) , H欧2 : (eprop ( ( (eq A4) x4) y4) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_27 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) H欧2) x4) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_27 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) H欧2) ) --> ( (refl_equal B) ( ( ( (f x1) x2) x3) x4) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_552 : (etype A1) -> (_551 : (etype A2) -> (_550 : (etype A3) -> (_549 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_26 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) x3) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_26 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) ) --> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_27 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) H欧2) y4) H欧2) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_534 : (etype A1) -> (_533 : (etype A2) -> (_532 : (etype A3) -> (_531 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_25 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) x2) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_25 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) ) --> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_26 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) y3) H欧1) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_515 : (etype A1) -> (_514 : (etype A2) -> (_513 : (etype A3) -> (_512 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_24 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) x1) ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_24 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) ) --> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_25 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) y2) H欧0) ) .
+f_equal4 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_585 : (etype A1) -> (_584 : (etype A2) -> (_583 : (etype A3) -> (_582 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (_589 : (eprop ( ( (eq A1) x1) y1) ) -> (_588 : (eprop ( ( (eq A2) x2) y2) ) -> (_587 : (eprop ( ( (eq A3) x3) y3) ) -> (_586 : (eprop ( ( (eq A4) x4) y4) ) -> (eprop ( ( (eq B) ( ( ( (f x1) x2) x3) x4) ) ( ( ( (f y1) y2) y3) y4) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[] f_equal4 --> (A1 : (etype dottype) => (A2 : (etype dottype) => (A3 : (etype dottype) => (A4 : (etype dottype) => (B : (etype dottype) => (f : (etype ( (dotpitt A1) (_581 : (etype A1) => ( (dotpitt A2) (_580 : (etype A2) => ( (dotpitt A3) (_579 : (etype A3) => ( (dotpitt A4) (_578 : (etype A4) => B) ) ) ) ) ) ) ) ) => (x1 : (etype A1) => (y1 : (etype A1) => (x2 : (etype A2) => (y2 : (etype A2) => (x3 : (etype A3) => (y3 : (etype A3) => (x4 : (etype A4) => (y4 : (etype A4) => (H : (eprop ( ( (eq A1) x1) y1) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_24 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) y1) H) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+case_28 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_594 : (etype A1) -> (_593 : (etype A2) -> (_592 : (etype A3) -> (_591 : (etype A4) -> (_590 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (y : (etype A1) -> (_599 : (eprop ( ( (eq A1) x1) y) ) -> (_598 : (eprop ( ( (eq A2) x2) y2) ) -> (_597 : (eprop ( ( (eq A3) x3) y3) ) -> (_596 : (eprop ( ( (eq A4) x4) y4) ) -> (_595 : (eprop ( ( (eq A5) x5) y5) ) -> (eprop ( ( (eq B) ( ( ( ( (f x1) x2) x3) x4) x5) ) ( ( ( ( (f y) y2) y3) y4) y5) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_28 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_609 : (etype A1) -> (_608 : (etype A2) -> (_607 : (etype A3) -> (_606 : (etype A4) -> (_605 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (eprop ( ( (eq A1) x1) x1) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_614 : (etype A1) -> (_613 : (etype A2) -> (_612 : (etype A3) -> (_611 : (etype A4) -> (_610 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_28 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) --> ( (refl_equal A1) x1) .
+case_29 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_619 : (etype A1) -> (_618 : (etype A2) -> (_617 : (etype A3) -> (_616 : (etype A4) -> (_615 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (y : (etype A2) -> (_623 : (eprop ( ( (eq A2) x2) y) ) -> (_622 : (eprop ( ( (eq A3) x3) y3) ) -> (_621 : (eprop ( ( (eq A4) x4) y4) ) -> (_620 : (eprop ( ( (eq A5) x5) y5) ) -> (eprop ( ( (eq B) ( ( ( ( (f x1) x2) x3) x4) x5) ) ( ( ( ( (f x1) y) y3) y4) y5) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_29 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_633 : (etype A1) -> (_632 : (etype A2) -> (_631 : (etype A3) -> (_630 : (etype A4) -> (_629 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (eprop ( ( (eq A2) x2) x2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_638 : (etype A1) -> (_637 : (etype A2) -> (_636 : (etype A3) -> (_635 : (etype A4) -> (_634 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_29 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) --> ( (refl_equal A2) x2) .
+case_30 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_643 : (etype A1) -> (_642 : (etype A2) -> (_641 : (etype A3) -> (_640 : (etype A4) -> (_639 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (y : (etype A3) -> (_646 : (eprop ( ( (eq A3) x3) y) ) -> (_645 : (eprop ( ( (eq A4) x4) y4) ) -> (_644 : (eprop ( ( (eq A5) x5) y5) ) -> (eprop ( ( (eq B) ( ( ( ( (f x1) x2) x3) x4) x5) ) ( ( ( ( (f x1) x2) y) y4) y5) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_30 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_656 : (etype A1) -> (_655 : (etype A2) -> (_654 : (etype A3) -> (_653 : (etype A4) -> (_652 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (eprop ( ( (eq A3) x3) x3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_661 : (etype A1) -> (_660 : (etype A2) -> (_659 : (etype A3) -> (_658 : (etype A4) -> (_657 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_30 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) --> ( (refl_equal A3) x3) .
+case_31 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_666 : (etype A1) -> (_665 : (etype A2) -> (_664 : (etype A3) -> (_663 : (etype A4) -> (_662 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) -> (y : (etype A4) -> (_668 : (eprop ( ( (eq A4) x4) y) ) -> (_667 : (eprop ( ( (eq A5) x5) y5) ) -> (eprop ( ( (eq B) ( ( ( ( (f x1) x2) x3) x4) x5) ) ( ( ( ( (f x1) x2) x3) y) y5) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_31 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_678 : (etype A1) -> (_677 : (etype A2) -> (_676 : (etype A3) -> (_675 : (etype A4) -> (_674 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) -> (eprop ( ( (eq A4) x4) x4) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_683 : (etype A1) -> (_682 : (etype A2) -> (_681 : (etype A3) -> (_680 : (etype A4) -> (_679 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) , H欧2 : (eprop ( ( (eq A4) x4) y4) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_31 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) --> ( (refl_equal A4) x4) .
+case_32 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_688 : (etype A1) -> (_687 : (etype A2) -> (_686 : (etype A3) -> (_685 : (etype A4) -> (_684 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) -> (H欧3 : (eprop ( ( (eq A5) x5) y5) ) -> (y : (etype A5) -> (_689 : (eprop ( ( (eq A5) x5) y) ) -> (eprop ( ( (eq B) ( ( ( ( (f x1) x2) x3) x4) x5) ) ( ( ( ( (f x1) x2) x3) x4) y) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_32 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_699 : (etype A1) -> (_698 : (etype A2) -> (_697 : (etype A3) -> (_696 : (etype A4) -> (_695 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) -> (H欧3 : (eprop ( ( (eq A5) x5) y5) ) -> (eprop ( ( (eq A5) x5) x5) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_704 : (etype A1) -> (_703 : (etype A2) -> (_702 : (etype A3) -> (_701 : (etype A4) -> (_700 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) , H欧2 : (eprop ( ( (eq A4) x4) y4) ) , H欧3 : (eprop ( ( (eq A5) x5) y5) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_32 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) H欧3) --> ( (refl_equal A5) x5) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_694 : (etype A1) -> (_693 : (etype A2) -> (_692 : (etype A3) -> (_691 : (etype A4) -> (_690 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) , H欧2 : (eprop ( ( (eq A4) x4) y4) ) , H欧3 : (eprop ( ( (eq A5) x5) y5) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_32 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) H欧3) x5) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_32 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) H欧3) ) --> ( (refl_equal B) ( ( ( ( (f x1) x2) x3) x4) x5) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_673 : (etype A1) -> (_672 : (etype A2) -> (_671 : (etype A3) -> (_670 : (etype A4) -> (_669 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) , H欧2 : (eprop ( ( (eq A4) x4) y4) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_31 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) x4) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_31 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) ) --> (H欧3 : (eprop ( ( (eq A5) x5) y5) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_32 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) H欧3) y5) H欧3) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_651 : (etype A1) -> (_650 : (etype A2) -> (_649 : (etype A3) -> (_648 : (etype A4) -> (_647 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_30 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) x3) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_30 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) ) --> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_31 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) y4) H欧2) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_628 : (etype A1) -> (_627 : (etype A2) -> (_626 : (etype A3) -> (_625 : (etype A4) -> (_624 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_29 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) x2) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_29 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) ) --> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_30 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) y3) H欧1) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_604 : (etype A1) -> (_603 : (etype A2) -> (_602 : (etype A3) -> (_601 : (etype A4) -> (_600 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_28 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) x1) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_28 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) ) --> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_29 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) y2) H欧0) ) .
+f_equal5 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_714 : (etype A1) -> (_713 : (etype A2) -> (_712 : (etype A3) -> (_711 : (etype A4) -> (_710 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (_719 : (eprop ( ( (eq A1) x1) y1) ) -> (_718 : (eprop ( ( (eq A2) x2) y2) ) -> (_717 : (eprop ( ( (eq A3) x3) y3) ) -> (_716 : (eprop ( ( (eq A4) x4) y4) ) -> (_715 : (eprop ( ( (eq A5) x5) y5) ) -> (eprop ( ( (eq B) ( ( ( ( (f x1) x2) x3) x4) x5) ) ( ( ( ( (f y1) y2) y3) y4) y5) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[] f_equal5 --> (A1 : (etype dottype) => (A2 : (etype dottype) => (A3 : (etype dottype) => (A4 : (etype dottype) => (A5 : (etype dottype) => (B : (etype dottype) => (f : (etype ( (dotpitt A1) (_709 : (etype A1) => ( (dotpitt A2) (_708 : (etype A2) => ( (dotpitt A3) (_707 : (etype A3) => ( (dotpitt A4) (_706 : (etype A4) => ( (dotpitt A5) (_705 : (etype A5) => B) ) ) ) ) ) ) ) ) ) ) => (x1 : (etype A1) => (y1 : (etype A1) => (x2 : (etype A2) => (y2 : (etype A2) => (x3 : (etype A3) => (y3 : (etype A3) => (x4 : (etype A4) => (y4 : (etype A4) => (x5 : (etype A5) => (y5 : (etype A5) => (H : (eprop ( ( (eq A1) x1) y1) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_28 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) y1) H) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+subrelation : (A : Utype -> (B : Utype -> (R : (_726 : (etype A) -> (_725 : (etype B) -> Uprop) ) -> (R' : (_728 : (etype A) -> (_727 : (etype B) -> Uprop) ) -> Uprop) ) ) ) .
+[] subrelation --> (A : (etype dottype) => (B : (etype dottype) => (R : (etype ( (dotpitt A) (_724 : (etype A) => ( (dotpitt B) (_723 : (etype B) => dotprop) ) ) ) ) => (R' : (etype ( (dotpitt A) (_722 : (etype A) => ( (dotpitt B) (_721 : (etype B) => dotprop) ) ) ) ) => ( (dotpitp A) (x : (etype A) => ( (dotpitp B) (y : (etype B) => ( (dotpipp ( (R x) y) ) (_720 : (eprop ( (R x) y) ) => ( (R' x) y) ) ) ) ) ) ) ) ) ) ) .
+unique : (A : Utype -> (P : (_731 : (etype A) -> Uprop) -> (x : (etype A) -> Uprop) ) ) .
+[] unique --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_730 : (etype A) => dotprop) ) ) => (x : (etype A) => ( (and (P x) ) ( (dotpitp A) (x' : (etype A) => ( (dotpipp (P x') ) (_729 : (eprop (P x') ) => ( ( (eq A) x) x') ) ) ) ) ) ) ) ) .
+uniqueness : (A : Utype -> (P : (_735 : (etype A) -> Uprop) -> Uprop) ) .
+[] uniqueness --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_734 : (etype A) => dotprop) ) ) => ( (dotpitp A) (x : (etype A) => ( (dotpitp A) (y : (etype A) => ( (dotpipp (P x) ) (_733 : (eprop (P x) ) => ( (dotpipp (P y) ) (_732 : (eprop (P y) ) => ( ( (eq A) x) y) ) ) ) ) ) ) ) ) ) ) .
+case_33 : (A : Utype -> (P : (_738 : (etype A) -> Uprop) -> (H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) -> (_739 : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) -> (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) ) ) ) ) .
+conj_case_33 : (A : Utype -> (P : (_741 : (etype A) -> Uprop) -> (H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) -> (_743 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (_742 : (eprop ( (uniqueness A) P) ) -> (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) ) ) ) ) ) .
+[A : Utype, P : (_744 : (etype A) -> Uprop) , H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) ] ( ( (conj_case_33 A) P) H) --> ( (conj ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) .
+case_34 : (A : Utype -> (P : (_745 : (etype A) -> Uprop) -> (H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) -> (H欧0 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (_747 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (_746 : (eprop ( (uniqueness A) P) ) -> (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) ) ) ) ) ) ) .
+ex_intro_case_34 : (A : Utype -> (P : (_749 : (etype A) -> Uprop) -> (H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) -> (H欧0 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (x : (etype A) -> (_750 : (eprop ( (x欧0 : (etype A) => (P x欧0) ) x) ) -> (eprop ( (ex A) (x欧0 : (etype A) => (P x欧0) ) ) ) ) ) ) ) ) ) .
+[A : Utype, P : (_751 : (etype A) -> Uprop) , H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) , H欧0 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) ] ( ( ( (ex_intro_case_34 A) P) H) H欧0) --> ( (ex_intro A) (x : (etype A) => (P x) ) ) .
+[A : Utype, P : (_748 : (etype A) -> Uprop) , H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) , H欧0 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) , var_25 : (etype A) , var_26 : (eprop ( (x : (etype A) => (P x) ) var_25) ) ] ( ( ( ( (case_34 A) P) H) H欧0) ( ( ( ( ( (ex_intro_case_34 A) P) H) H欧0) var_25) var_26) ) --> ( ( (x : (etype A) => (Hx : (eprop (P x) ) => (Huni : (eprop ( (uniqueness A) P) ) => ( ( ( (ex_intro A) ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) ) x) ( ( ( (conj (P x) ) ( (dotpitp A) (x' : (etype A) => ( (dotpipp (P x') ) (_752 : (eprop (P x') ) => ( ( (eq A) x) x') ) ) ) ) ) Hx) (x' : (etype A) => (H欧1 : (eprop (P x') ) => ( ( ( (Huni x) x') Hx) H欧1) ) ) ) ) ) ) ) var_25) var_26) .
+[A : Utype, P : (_740 : (etype A) -> Uprop) , H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) , var_23 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) , var_24 : (eprop ( (uniqueness A) P) ) ] ( ( ( (case_33 A) P) H) ( ( ( ( (conj_case_33 A) P) H) var_23) var_24) ) --> ( ( (H欧0 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) => ( ( ( ( (case_34 A) P) H) H欧0) H欧0) ) var_23) var_24) .
+case_35 : (A : Utype -> (P : (_753 : (etype A) -> Uprop) -> (H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) -> (_754 : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) -> (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) ) ) ) ) .
+ex_intro_case_35 : (A : Utype -> (P : (_756 : (etype A) -> Uprop) -> (H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) -> (x : (etype A) -> (_757 : (eprop ( ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) x) ) -> (eprop ( (ex A) ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) ) ) ) ) ) ) ) .
+[A : Utype, P : (_758 : (etype A) -> Uprop) , H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) ] ( ( (ex_intro_case_35 A) P) H) --> ( (ex_intro A) ( (unique A) (x : (etype A) => (P x) ) ) ) .
+case_36 : (A : Utype -> (P : (_759 : (etype A) -> Uprop) -> (H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) -> (x : (etype A) -> (H欧0 : (eprop ( ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) x) ) -> (_761 : (eprop ( (and (P x) ) ( (dotpitp A) (x' : (etype A) => ( (dotpipp (P x') ) (_760 : (eprop (P x') ) => ( ( (eq A) x) x') ) ) ) ) ) ) -> (eprop ( (and ( (ex A) (x欧0 : (etype A) => (P x欧0) ) ) ) ( (uniqueness A) P) ) ) ) ) ) ) ) ) .
+conj_case_36 : (A : Utype -> (P : (_764 : (etype A) -> Uprop) -> (H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) -> (x : (etype A) -> (H欧0 : (eprop ( ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) x) ) -> (_768 : (eprop (P x) ) -> (_767 : (x' : (etype A) -> (_765 : (eprop (P x') ) -> (eprop ( ( (eq A) x) x') ) ) ) -> (eprop ( (and (P x) ) ( (dotpitp A) (x' : (etype A) => ( (dotpipp (P x') ) (_766 : (eprop (P x') ) => ( ( (eq A) x) x') ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[A : Utype, P : (_769 : (etype A) -> Uprop) , H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) , x : (etype A) , H欧0 : (eprop ( ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) x) ) ] ( ( ( ( (conj_case_36 A) P) H) x) H欧0) --> ( (conj (P x) ) ( (dotpitp A) (x' : (etype A) => ( (dotpipp (P x') ) (_763 : (eprop (P x') ) => ( ( (eq A) x) x') ) ) ) ) ) .
+[A : Utype, P : (_762 : (etype A) -> Uprop) , H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) , x : (etype A) , H欧0 : (eprop ( ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) x) ) , var_29 : (eprop (P x) ) , var_30 : (x' : (etype A) -> (_770 : (eprop (P x') ) -> (eprop ( ( (eq A) x) x') ) ) ) ] ( ( ( ( ( (case_36 A) P) H) x) H欧0) ( ( ( ( ( ( (conj_case_36 A) P) H) x) H欧0) var_29) var_30) ) --> ( ( (Hx : (eprop (P x) ) => (Huni : (eprop ( (dotpitp A) (x' : (etype A) => ( (dotpipp (P x') ) (_771 : (eprop (P x') ) => ( ( (eq A) x) x') ) ) ) ) ) => ( ( ( (conj ( (ex A) (x欧0 : (etype A) => (P x欧0) ) ) ) ( (uniqueness A) P) ) ( ( ( (ex_intro A) (x欧0 : (etype A) => (P x欧0) ) ) x) Hx) ) (x' : (etype A) => (x'' : (etype A) => (Hx' : (eprop (P x') ) => (Hx'' : (eprop (P x'') ) => ( ( ( ( ( (trans_eq A) x') x) x'') ( ( ( (sym_eq A) x) x') ( (Huni x') Hx') ) ) ( (Huni x'') Hx'') ) ) ) ) ) ) ) ) var_29) var_30) .
+[A : Utype, P : (_755 : (etype A) -> Uprop) , H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) , var_27 : (etype A) , var_28 : (eprop ( ( (unique A) (x : (etype A) => (P x) ) ) var_27) ) ] ( ( ( (case_35 A) P) H) ( ( ( ( (ex_intro_case_35 A) P) H) var_27) var_28) ) --> ( ( (x : (etype A) => (H欧0 : (eprop ( ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) x) ) => ( ( ( ( ( (case_36 A) P) H) x) H欧0) H欧0) ) ) var_27) var_28) .
+unique_existence : (A : Utype -> (P : (_773 : (etype A) -> Uprop) -> (eprop ( (iff ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) ) ) ) .
+[] unique_existence --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_772 : (etype A) => dotprop) ) ) => ( ( ( (conj ( (dotpipp ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) (_736 : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) => ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) ) ) ( (dotpipp ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) (_737 : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) => ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) ) ) (H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) => ( ( ( (case_33 A) P) H) H) ) ) (H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) => ( ( ( (case_35 A) P) H) H) ) ) ) ) .
+inhabited : (A : Utype -> Uprop) .
+inhabits : (A : Utype -> (_774 : (etype A) -> (eprop (inhabited A) ) ) ) .
+case_37 : (A : Utype -> (P : Uprop -> (f : (_775 : (etype A) -> (eprop P) ) -> (i : (eprop (inhabited A) ) -> (_776 : (eprop (inhabited A) ) -> (eprop P) ) ) ) ) ) .
+inhabits_case_37 : (A : Utype -> (P : Uprop -> (f : (_778 : (etype A) -> (eprop P) ) -> (i : (eprop (inhabited A) ) -> (_779 : (etype A) -> (eprop (inhabited A) ) ) ) ) ) ) .
+[A : Utype, P : Uprop, f : (_780 : (etype A) -> (eprop P) ) , i : (eprop (inhabited A) ) ] ( ( ( (inhabits_case_37 A) P) f) i) --> (inhabits A) .
+[A : Utype, P : Uprop, f : (_777 : (etype A) -> (eprop P) ) , i : (eprop (inhabited A) ) , var_31 : (etype A) ] ( ( ( ( (case_37 A) P) f) i) ( ( ( ( (inhabits_case_37 A) P) f) i) var_31) ) --> (f var_31) .
+inhabited_ind : (A : Utype -> (P : Uprop -> (f : (_782 : (etype A) -> (eprop P) ) -> (i : (eprop (inhabited A) ) -> (eprop P) ) ) ) ) .
+[] inhabited_ind --> (A : (etype dottype) => (P : (etype dotprop) => (f : (eprop ( (dotpitp A) (_781 : (etype A) => P) ) ) => (i : (eprop (inhabited A) ) => ( ( ( ( (case_37 A) P) f) i) i) ) ) ) ) .
+case_38 : (A : Utype -> (P : (_783 : (etype A) -> Uprop) -> (H : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (_784 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (eprop (inhabited A) ) ) ) ) ) .
+ex_intro_case_38 : (A : Utype -> (P : (_786 : (etype A) -> Uprop) -> (H : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (x : (etype A) -> (_787 : (eprop ( (x欧0 : (etype A) => (P x欧0) ) x) ) -> (eprop ( (ex A) (x欧0 : (etype A) => (P x欧0) ) ) ) ) ) ) ) ) .
+[A : Utype, P : (_788 : (etype A) -> Uprop) , H : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) ] ( ( (ex_intro_case_38 A) P) H) --> ( (ex_intro A) (x : (etype A) => (P x) ) ) .
+[A : Utype, P : (_785 : (etype A) -> Uprop) , H : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) , var_32 : (etype A) , var_33 : (eprop ( (x : (etype A) => (P x) ) var_32) ) ] ( ( ( (case_38 A) P) H) ( ( ( ( (ex_intro_case_38 A) P) H) var_32) var_33) ) --> ( ( (x : (etype A) => (H欧0 : (eprop (P x) ) => ( (inhabits A) x) ) ) var_32) var_33) .
+exists_inhabited : (A : Utype -> (P : (_790 : (etype A) -> Uprop) -> (_791 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (eprop (inhabited A) ) ) ) ) .
+[] exists_inhabited --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_789 : (etype A) => dotprop) ) ) => (H : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) => ( ( ( (case_38 A) P) H) H) ) ) ) .
+eq_stepl : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (z : (etype A) -> (_793 : (eprop ( ( (eq A) x) y) ) -> (_792 : (eprop ( ( (eq A) x) z) ) -> (eprop ( ( (eq A) z) y) ) ) ) ) ) ) ) .
+[] eq_stepl --> (A : (etype dottype) => (x : (etype A) => (y : (etype A) => (z : (etype A) => (H1 : (eprop ( ( (eq A) x) y) ) => (H2 : (eprop ( ( (eq A) x) z) ) => ( ( ( ( ( (eq_ind A) x) (z欧0 : (etype A) => ( ( (eq A) z欧0) y) ) ) H1) z) H2) ) ) ) ) ) ) .
+iff_stepl : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_805 : (eprop ( (iff A) B) ) -> (_804 : (eprop ( (iff A) C) ) -> (eprop ( (iff C) B) ) ) ) ) ) ) .
+[] iff_stepl --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff A) B) ) => (H0 : (eprop ( (iff A) C) ) => ( ( ( ( (and_ind ( (dotpipp A) (_794 : (eprop A) => B) ) ) ( (dotpipp B) (_795 : (eprop B) => A) ) ) ( (iff C) B) ) (H1 : (eprop ( (dotpipp A) (_803 : (eprop A) => B) ) ) => (H2 : (eprop ( (dotpipp B) (_802 : (eprop B) => A) ) ) => ( ( ( ( (and_ind ( (dotpipp A) (_796 : (eprop A) => C) ) ) ( (dotpipp C) (_797 : (eprop C) => A) ) ) ( (iff C) B) ) (H欧0 : (eprop ( (dotpipp A) (_801 : (eprop A) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_800 : (eprop C) => A) ) ) => ( ( ( (conj ( (dotpipp C) (_798 : (eprop C) => B) ) ) ( (dotpipp B) (_799 : (eprop B) => C) ) ) (H0欧0 : (eprop C) => ( (H4 : (eprop A) => ( (H3欧0 : (eprop B) => ( (H1欧0 : (eprop C) => ( (H欧1 : (eprop A) => H3欧0) (H2 H3欧0) ) ) (H欧0 H4) ) ) (H1 H4) ) ) (H3 H0欧0) ) ) ) (H0欧0 : (eprop B) => ( (H4 : (eprop A) => ( (H2欧0 : (eprop B) => ( (H1欧0 : (eprop C) => ( (H欧1 : (eprop A) => H1欧0) (H3 H1欧0) ) ) (H欧0 H4) ) ) (H1 H4) ) ) (H2 H0欧0) ) ) ) ) ) ) H0) ) ) ) H) ) ) ) ) ) .
+;Finished module Logic
diff --git a/t/Logicavecprelude.eu b/t/Logicavecprelude.eu
new file mode 100644
--- /dev/null
+++ b/t/Logicavecprelude.eu
@@ -0,0 +1,156 @@
+Uset : Type.
+Uprop : Type.
+Utype : Type.
+
+eprop : x : Uprop -> Type.
+eset : x : Uset -> Type.
+etype : x : Utype -> Type.
+
+dotset : Utype.
+dotprop : Utype.
+
+; /!\ type : type /!\, should use universes
+dottype : Utype.
+
+; /!\ subtyping in coq, should be unidirectional /!\
+[] Uprop --> Utype.
+[] Uset --> Utype.
+
+dotpipp : x : Uprop -> y : (eprop x -> Uprop) -> Uprop.
+dotpips : x : Uprop -> y : (eprop x -> Uset) -> Uset.
+dotpipt : x : Uprop -> y : (eprop x -> Utype) -> Utype.
+dotpisp : x : Uset -> y : (eset x -> Uprop) -> Uprop.
+dotpitp : x : Utype -> y : (etype x -> Uprop) -> Uprop.
+dotpist : x : Uset -> y : (eset x -> Utype) -> Utype.
+dotpits : x : Utype -> y : (etype x -> Uset) -> Uset.
+dotpiss : x : Uset -> y : (eset x -> Uset) -> Uset.
+dotpitt : x : Utype -> y : (etype x -> Utype) -> Utype.
+
+
+[x:Uprop, y : eprop x -> Uprop]
+ eprop (dotpipp x y) --> w : eprop x -> eprop (y w).
+
+[x:Uset, y : eset x -> Uprop]
+ eprop (dotpisp x y) --> w : eset x -> eprop (y w).
+
+[x:Utype, y : etype x -> Uprop]
+ eprop (dotpitp x y) --> w : etype x -> eprop (y w).
+
+; /!\
+[P : Uprop] eprop P --> etype P.
+
+[x:Uprop, y : eprop x -> Uset]
+ eset (dotpips x y) --> w : eprop x -> eset (y w).
+
+[x:Utype, y : etype x -> Uset]
+ eset (dotpits x y) --> w : etype x -> eset (y w).
+
+[x:Uset, y : eset x -> Uset]
+ eset (dotpiss x y) --> w : eset x -> eset (y w).
+
+; /!\
+[P : Uset] eset P --> etype P.
+
+[x:Uset, y : eset x -> Utype]
+ etype (dotpist x y) --> w : eset x -> etype (y w).
+
+[x:Utype, y : etype x -> Utype]
+ etype (dotpitt x y) --> w : etype x -> etype (y w).
+
+[x:Uprop, y : eprop x -> Utype]
+ etype (dotpipt x y) --> w : eprop x -> etype (y w).
+
+
+[] (etype dotset) --> Uset.
+[] (etype dotprop) --> Uprop.
+; /!\
+[] (etype dottype) --> Utype.
+
+; end of Coq1univ
+
+True : Uprop.
+I : (eprop True) .
+case_0 : (P : Utype -> (f : (etype P) -> (t : (eprop True) -> (_0 : (eprop True) -> (etype P) ) ) ) ) .
+[P : Utype, f : (etype P) , t : (eprop True) ] ( ( ( (case_0 P) f) t) I) --> f.
+True_rect : (P : Utype -> (f : (etype P) -> (t : (eprop True) -> (etype P) ) ) ) .
+[] True_rect --> (P : (etype dottype) => (f : (etype P) => (t : (eprop True) => ( ( ( (case_0 P) f) t) t) ) ) ) .
+True_ind : (P : Uprop -> (f : (eprop P) -> (t : (eprop True) -> (eprop P) ) ) ) .
+[] True_ind --> (P : (etype dotprop) => (True_rect P) ) .
+True_rec : (P : Uset -> (f : (eset P) -> (t : (eprop True) -> (eset P) ) ) ) .
+[] True_rec --> (P : (etype dotset) => (True_rect P) ) .
+False : Uprop.
+case_1 : (P : Utype -> (f : (eprop False) -> (_1 : (eprop False) -> (etype P) ) ) ) .
+False_rect : (P : Utype -> (f : (eprop False) -> (etype P) ) ) .
+[] False_rect --> (P : (etype dottype) => (f : (eprop False) => ( ( (case_1 P) f) f) ) ) .
+False_ind : (P : Uprop -> (f : (eprop False) -> (eprop P) ) ) .
+[] False_ind --> (P : (etype dotprop) => (False_rect P) ) .
+False_rec : (P : Uset -> (f : (eprop False) -> (eset P) ) ) .
+[] False_rec --> (P : (etype dotset) => (False_rect P) ) .
+not : (A : Uprop -> Uprop) .
+[] not --> (A : (etype dotprop) => ( (dotpipp A) (_2 : (eprop A) => False) ) ) .
+and : (A : Uprop -> (B : Uprop -> Uprop) ) .
+conj : (A : Uprop -> (B : Uprop -> (_4 : (eprop A) -> (_3 : (eprop B) -> (eprop ( (and A) B) ) ) ) ) ) .
+case_2 : (A : Uprop -> (B : Uprop -> (P : Utype -> (f : (_6 : (eprop A) -> (_5 : (eprop B) -> (etype P) ) ) -> (a : (eprop ( (and A) B) ) -> (_7 : (eprop ( (and A) B) ) -> (etype P) ) ) ) ) ) ) .
+[A : Uprop, B : Uprop, P : Utype, f : (_9 : (eprop A) -> (_8 : (eprop B) -> (etype P) ) ) , a : (eprop ( (and A) B) ) , var_0 : (eprop A) , var_1 : (eprop B) ] ( ( ( ( ( (case_2 A) B) P) f) a) ( ( ( (conj A) B) var_0) var_1) ) --> ( (f var_0) var_1) .
+and_rect : (A : Uprop -> (B : Uprop -> (P : Utype -> (f : (_13 : (eprop A) -> (_12 : (eprop B) -> (etype P) ) ) -> (a : (eprop ( (and A) B) ) -> (etype P) ) ) ) ) ) .
+[] and_rect --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dottype) => (f : (etype ( (dotpipt A) (_11 : (eprop A) => ( (dotpipt B) (_10 : (eprop B) => P) ) ) ) ) => (a : (eprop ( (and A) B) ) => ( ( ( ( ( (case_2 A) B) P) f) a) a) ) ) ) ) ) .
+and_ind : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_15 : (eprop A) -> (_14 : (eprop B) -> (eprop P) ) ) -> (a : (eprop ( (and A) B) ) -> (eprop P) ) ) ) ) ) .
+[] and_ind --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dotprop) => ( ( (and_rect A) B) P) ) ) ) .
+and_rec : (A : Uprop -> (B : Uprop -> (P : Uset -> (f : (_17 : (eprop A) -> (_16 : (eprop B) -> (eset P) ) ) -> (a : (eprop ( (and A) B) ) -> (eset P) ) ) ) ) ) .
+[] and_rec --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dotset) => ( ( (and_rect A) B) P) ) ) ) .
+case_3 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (and A) B) ) -> (_18 : (eprop ( (and A) B) ) -> (eprop A) ) ) ) ) .
+[A : Uprop, B : Uprop, H : (eprop ( (and A) B) ) , var_2 : (eprop A) , var_3 : (eprop B) ] ( ( ( (case_3 A) B) H) ( ( ( (conj A) B) var_2) var_3) ) --> ( ( (H0 : (eprop A) => (H0 : (eprop B) => H0) ) var_2) var_3) .
+proj1 : (A : Uprop -> (B : Uprop -> (_19 : (eprop ( (and A) B) ) -> (eprop A) ) ) ) .
+[] proj1 --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (and A) B) ) => ( ( ( (case_3 A) B) H) H) ) ) ) .
+case_4 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (and A) B) ) -> (_20 : (eprop ( (and A) B) ) -> (eprop B) ) ) ) ) .
+[A : Uprop, B : Uprop, H : (eprop ( (and A) B) ) , var_4 : (eprop A) , var_5 : (eprop B) ] ( ( ( (case_4 A) B) H) ( ( ( (conj A) B) var_4) var_5) ) --> ( ( (H0 : (eprop A) => (H0 : (eprop B) => H0) ) var_4) var_5) .
+proj2 : (A : Uprop -> (B : Uprop -> (_21 : (eprop ( (and A) B) ) -> (eprop B) ) ) ) .
+[] proj2 --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (and A) B) ) => ( ( ( (case_4 A) B) H) H) ) ) ) .
+or : (A : Uprop -> (B : Uprop -> Uprop) ) .
+or_introl : (A : Uprop -> (B : Uprop -> (_22 : (eprop A) -> (eprop ( (or A) B) ) ) ) ) .
+or_intror : (A : Uprop -> (B : Uprop -> (_23 : (eprop B) -> (eprop ( (or A) B) ) ) ) ) .
+case_5 : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_24 : (eprop A) -> (eprop P) ) -> (f0 : (_25 : (eprop B) -> (eprop P) ) -> (o : (eprop ( (or A) B) ) -> (_26 : (eprop ( (or A) B) ) -> (eprop P) ) ) ) ) ) ) ) .
+[A : Uprop, B : Uprop, P : Uprop, f : (_27 : (eprop A) -> (eprop P) ) , f0 : (_28 : (eprop B) -> (eprop P) ) , o : (eprop ( (or A) B) ) , var_6 : (eprop A) ] ( ( ( ( ( ( (case_5 A) B) P) f) f0) o) ( ( (or_introl A) B) var_6) ) --> (f var_6) .
+[A : Uprop, B : Uprop, P : Uprop, f : (_27 : (eprop A) -> (eprop P) ) , f0 : (_28 : (eprop B) -> (eprop P) ) , o : (eprop ( (or A) B) ) , var_7 : (eprop B) ] ( ( ( ( ( ( (case_5 A) B) P) f) f0) o) ( ( (or_intror A) B) var_7) ) --> (f0 var_7) .
+or_ind : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_31 : (eprop A) -> (eprop P) ) -> (f0 : (_32 : (eprop B) -> (eprop P) ) -> (o : (eprop ( (or A) B) ) -> (eprop P) ) ) ) ) ) ) .
+[] or_ind --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dotprop) => (f : (eprop ( (dotpipp A) (_30 : (eprop A) => P) ) ) => (f0 : (eprop ( (dotpipp B) (_29 : (eprop B) => P) ) ) => (o : (eprop ( (or A) B) ) => ( ( ( ( ( ( (case_5 A) B) P) f) f0) o) o) ) ) ) ) ) ) .
+iff : (A : Uprop -> (B : Uprop -> Uprop) ) .
+[] iff --> (A : (etype dotprop) => (B : (etype dotprop) => ( (and ( (dotpipp A) (_33 : (eprop A) => B) ) ) ( (dotpipp B) (_34 : (eprop B) => A) ) ) ) ) .
+iff_refl : (A : Uprop -> (eprop ( (iff A) A) ) ) .
+[] iff_refl --> (A : (etype dotprop) => ( ( ( (conj ( (dotpipp A) (_35 : (eprop A) => A) ) ) ( (dotpipp A) (_36 : (eprop A) => A) ) ) (H : (eprop A) => H) ) (H : (eprop A) => H) ) ) .
+case_6 : (A : Uprop -> (B : Uprop -> (C : Uprop -> (H : (eprop ( (iff A) B) ) -> (_40 : (eprop ( (and ( (dotpipp A) (_37 : (eprop A) => B) ) ) ( (dotpipp B) (_38 : (eprop B) => A) ) ) ) -> (_39 : (eprop ( (iff B) C) ) -> (eprop ( (iff A) C) ) ) ) ) ) ) ) .
+case_7 : (A : Uprop -> (B : Uprop -> (C : Uprop -> (H : (eprop ( (iff A) B) ) -> (H1 : (_41 : (eprop A) -> (eprop B) ) -> (H2 : (_42 : (eprop B) -> (eprop A) ) -> (H0 : (eprop ( (iff B) C) ) -> (_45 : (eprop ( (and ( (dotpipp B) (_43 : (eprop B) => C) ) ) ( (dotpipp C) (_44 : (eprop C) => B) ) ) ) -> (eprop ( (iff A) C) ) ) ) ) ) ) ) ) ) .
+[A : Uprop, B : Uprop, C : Uprop, H : (eprop ( (iff A) B) ) , H1 : (_46 : (eprop A) -> (eprop B) ) , H2 : (_47 : (eprop B) -> (eprop A) ) , H0 : (eprop ( (iff B) C) ) , var_10 : (eprop B) , var_11 : (eprop C) ] ( ( ( ( ( ( ( (case_7 A) B) C) H) H1) H2) H0) ( ( ( (conj B) C) var_10) var_11) ) --> ( ( (H3 : (eprop ( (dotpipp B) (_51 : (eprop B) => C) ) ) => (H4 : (eprop ( (dotpipp C) (_50 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp A) (_48 : (eprop A) => C) ) ) ( (dotpipp C) (_49 : (eprop C) => A) ) ) (H1 : (eprop A) => (H3 (H1 H1) ) ) ) (H1 : (eprop C) => (H2 (H1 (H2 (H4 H1) ) ) ) ) ) ) ) var_10) var_11) .
+[A : Uprop, B : Uprop, C : Uprop, H : (eprop ( (iff A) B) ) , var_8 : (eprop A) , var_9 : (eprop B) ] ( ( ( ( (case_6 A) B) C) H) ( ( ( (conj A) B) var_8) var_9) ) --> ( ( (H1 : (eprop ( (dotpipp A) (_53 : (eprop A) => B) ) ) => (H2 : (eprop ( (dotpipp B) (_52 : (eprop B) => A) ) ) => (H0 : (eprop ( (iff B) C) ) => ( ( ( ( ( ( ( (case_7 A) B) C) H) H1) H2) H0) H0) ) ) ) var_8) var_9) .
+iff_trans : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_55 : (eprop ( (iff A) B) ) -> (_54 : (eprop ( (iff B) C) ) -> (eprop ( (iff A) C) ) ) ) ) ) ) .
+[] iff_trans --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff A) B) ) => ( ( ( ( (case_6 A) B) C) H) H) ) ) ) ) .
+case_8 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (iff A) B) ) -> (_58 : (eprop ( (and ( (dotpipp A) (_56 : (eprop A) => B) ) ) ( (dotpipp B) (_57 : (eprop B) => A) ) ) ) -> (eprop ( (iff B) A) ) ) ) ) ) .
+[A : Uprop, B : Uprop, H : (eprop ( (iff A) B) ) , var_12 : (eprop A) , var_13 : (eprop B) ] ( ( ( (case_8 A) B) H) ( ( ( (conj A) B) var_12) var_13) ) --> ( ( (H1 : (eprop ( (dotpipp A) (_62 : (eprop A) => B) ) ) => (H2 : (eprop ( (dotpipp B) (_61 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp B) (_59 : (eprop B) => A) ) ) ( (dotpipp A) (_60 : (eprop A) => B) ) ) H2) H1) ) ) var_12) var_13) .
+iff_sym : (A : Uprop -> (B : Uprop -> (_63 : (eprop ( (iff A) B) ) -> (eprop ( (iff B) A) ) ) ) ) .
+[] iff_sym --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (iff A) B) ) => ( ( ( (case_8 A) B) H) H) ) ) ) .
+case_9 : (A : Uprop -> (H : (eprop ( (iff A) False) ) -> (_74 : (eprop ( (and ( (dotpipp A) (_71 : (eprop A) => False) ) ) ( (dotpipp False) (_72 : (eprop False) => A) ) ) ) -> (_73 : (eprop A) -> (eprop False) ) ) ) ) .
+[A : Uprop, H : (eprop ( (iff A) False) ) , var_14 : (eprop A) , var_15 : (eprop False) ] ( ( (case_9 A) H) ( ( ( (conj A) False) var_14) var_15) ) --> ( ( (H0 : (eprop ( (dotpipp A) (_76 : (eprop A) => False) ) ) => (H0 : (eprop ( (dotpipp False) (_75 : (eprop False) => A) ) ) => H0) ) var_14) var_15) .
+neg_false : (A : Uprop -> (eprop ( (iff (not A) ) ( (iff A) False) ) ) ) .
+[] neg_false --> (A : (etype dotprop) => ( ( ( (conj ( (dotpipp ( (dotpipp A) (_64 : (eprop A) => False) ) ) (_65 : (eprop ( (dotpipp A) (_64 : (eprop A) => False) ) ) => ( (iff A) False) ) ) ) ( (dotpipp ( (iff A) False) ) (_67 : (eprop ( (iff A) False) ) => ( (dotpipp A) (_66 : (eprop A) => False) ) ) ) ) (H : (eprop ( (dotpipp A) (_70 : (eprop A) => False) ) ) => ( ( ( (conj ( (dotpipp A) (_68 : (eprop A) => False) ) ) ( (dotpipp False) (_69 : (eprop False) => A) ) ) H) (H1 : (eprop False) => ( (False_ind A) H1) ) ) ) ) (H : (eprop ( (iff A) False) ) => ( ( (case_9 A) H) H) ) ) ) .
+and_cancel_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_104 : (_101 : (eprop B) -> (eprop A) ) -> (_103 : (_102 : (eprop C) -> (eprop A) ) -> (eprop ( (iff ( (iff ( (and A) B) ) ( (and A) C) ) ) ( (iff B) C) ) ) ) ) ) ) ) .
+[] and_cancel_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_100 : (eprop B) => A) ) ) => (H0 : (eprop ( (dotpipp C) (_99 : (eprop C) => A) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (and A) B) ) ( (and A) C) ) ) (_77 : (eprop ( (iff ( (and A) B) ) ( (and A) C) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_78 : (eprop ( (iff B) C) ) => ( (iff ( (and A) B) ) ( (and A) C) ) ) ) ) (H1 : (eprop ( (iff ( (and A) B) ) ( (and A) C) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (and A) B) ) (_79 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) ( (dotpipp ( (and A) C) ) (_80 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (and A) B) ) (_92 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) => (H3 : (eprop ( (dotpipp ( (and A) C) ) (_91 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) => ( (H10 : (eprop ( (dotpipp A) (_90 : (eprop A) => ( (dotpipp B) (_89 : (eprop B) => ( (and A) C) ) ) ) ) ) => ( (H20 : (eprop ( (dotpipp A) (_88 : (eprop A) => ( (dotpipp C) (_87 : (eprop C) => ( (and A) B) ) ) ) ) ) => ( ( ( (conj ( (dotpipp B) (_81 : (eprop B) => C) ) ) ( (dotpipp C) (_82 : (eprop C) => B) ) ) (H30 : (eprop B) => ( (H4 : (eprop A) => ( (H0 : (eprop ( (dotpipp B) (_84 : (eprop B) => ( (and A) C) ) ) ) => ( (H11 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) C) (H1 : (eprop A) => (H5 : (eprop C) => ( (H12 : (eprop ( (dotpipp C) (_83 : (eprop C) => ( (and A) B) ) ) ) => ( (H21 : (eprop A) => ( (H00 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) C) (H13 : (eprop A) => (H6 : (eprop B) => H5) ) ) H00) ) (H12 H5) ) ) (H0 H5) ) ) (H20 H4) ) ) ) ) H11) ) (H0 H30) ) ) (H10 H4) ) ) (H H30) ) ) ) (H30 : (eprop C) => ( (H4 : (eprop A) => ( (H00 : (eprop ( (dotpipp B) (_86 : (eprop B) => ( (and A) C) ) ) ) => ( (H11 : (eprop ( (dotpipp C) (_85 : (eprop C) => ( (and A) B) ) ) ) => ( (H21 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) B) (H12 : (eprop A) => (H5 : (eprop B) => ( (H22 : (eprop A) => ( (H0 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) B) (H01 : (eprop A) => (H6 : (eprop C) => H5) ) ) H0) ) (H00 H5) ) ) (H H5) ) ) ) ) H21) ) (H11 H30) ) ) (H20 H4) ) ) (H10 H4) ) ) (H0 H30) ) ) ) ) (H20 : (eprop A) => (H4 : (eprop C) => (H3 ( ( ( (conj A) C) H20) H4) ) ) ) ) ) (H10 : (eprop A) => (H4 : (eprop B) => (H2 ( ( ( (conj A) B) H10) H4) ) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_93 : (eprop B) => C) ) ) ( (dotpipp C) (_94 : (eprop C) => B) ) ) ( (iff ( (and A) B) ) ( (and A) C) ) ) (H2 : (eprop ( (dotpipp B) (_98 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_97 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and A) B) ) (_95 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) ( (dotpipp ( (and A) C) ) (_96 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) (H10 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) ( (and A) C) ) (H4 : (eprop A) => (H5 : (eprop B) => ( (H11 : (eprop A) => ( (H0 : (eprop C) => ( (H20 : (eprop A) => ( (H00 : (eprop B) => ( ( ( (conj A) C) H20) H0) ) (H3 H0) ) ) (H0 H0) ) ) (H2 H5) ) ) (H H5) ) ) ) ) H10) ) ) (H10 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) ( (and A) B) ) (H4 : (eprop A) => (H5 : (eprop C) => ( (H11 : (eprop A) => ( (H00 : (eprop B) => ( (H30 : (eprop A) => ( (H0 : (eprop C) => ( ( ( (conj A) B) H30) H00) ) (H2 H00) ) ) (H H00) ) ) (H3 H5) ) ) (H0 H5) ) ) ) ) H10) ) ) ) ) ) H1) ) ) ) ) ) ) ) .
+and_cancel_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_132 : (_129 : (eprop B) -> (eprop A) ) -> (_131 : (_130 : (eprop C) -> (eprop A) ) -> (eprop ( (iff ( (iff ( (and B) A) ) ( (and C) A) ) ) ( (iff B) C) ) ) ) ) ) ) ) .
+[] and_cancel_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_128 : (eprop B) => A) ) ) => (H0 : (eprop ( (dotpipp C) (_127 : (eprop C) => A) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (and B) A) ) ( (and C) A) ) ) (_105 : (eprop ( (iff ( (and B) A) ) ( (and C) A) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_106 : (eprop ( (iff B) C) ) => ( (iff ( (and B) A) ) ( (and C) A) ) ) ) ) (H1 : (eprop ( (iff ( (and B) A) ) ( (and C) A) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (and B) A) ) (_107 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) ( (dotpipp ( (and C) A) ) (_108 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (and B) A) ) (_120 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) => (H3 : (eprop ( (dotpipp ( (and C) A) ) (_119 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) => ( (H10 : (eprop ( (dotpipp B) (_118 : (eprop B) => ( (dotpipp A) (_117 : (eprop A) => ( (and C) A) ) ) ) ) ) => ( (H20 : (eprop ( (dotpipp C) (_116 : (eprop C) => ( (dotpipp A) (_115 : (eprop A) => ( (and B) A) ) ) ) ) ) => ( ( ( (conj ( (dotpipp B) (_109 : (eprop B) => C) ) ) ( (dotpipp C) (_110 : (eprop C) => B) ) ) (H30 : (eprop B) => ( (H4 : (eprop A) => ( (H0 : (eprop ( (dotpipp A) (_112 : (eprop A) => ( (and C) A) ) ) ) => ( (H11 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) C) (H1 : (eprop C) => (H5 : (eprop A) => ( (H12 : (eprop A) => ( (H00 : (eprop ( (dotpipp A) (_111 : (eprop A) => ( (and B) A) ) ) ) => ( (H21 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) C) (H01 : (eprop B) => (H6 : (eprop A) => H1) ) ) H21) ) (H00 H4) ) ) (H20 H1) ) ) (H0 H1) ) ) ) ) H11) ) (H0 H4) ) ) (H10 H30) ) ) (H H30) ) ) ) (H30 : (eprop C) => ( (H4 : (eprop A) => ( (H00 : (eprop ( (dotpipp A) (_114 : (eprop A) => ( (and B) A) ) ) ) => ( (H21 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) B) (H01 : (eprop B) => (H5 : (eprop A) => ( (H22 : (eprop A) => ( (H0 : (eprop ( (dotpipp A) (_113 : (eprop A) => ( (and C) A) ) ) ) => ( (H11 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) B) (H1 : (eprop C) => (H6 : (eprop A) => H01) ) ) H11) ) (H0 H4) ) ) (H10 H01) ) ) (H H01) ) ) ) ) H21) ) (H00 H4) ) ) (H20 H30) ) ) (H0 H30) ) ) ) ) (H20 : (eprop C) => (H4 : (eprop A) => (H3 ( ( ( (conj C) A) H20) H4) ) ) ) ) ) (H10 : (eprop B) => (H4 : (eprop A) => (H2 ( ( ( (conj B) A) H10) H4) ) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_121 : (eprop B) => C) ) ) ( (dotpipp C) (_122 : (eprop C) => B) ) ) ( (iff ( (and B) A) ) ( (and C) A) ) ) (H2 : (eprop ( (dotpipp B) (_126 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_125 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and B) A) ) (_123 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) ( (dotpipp ( (and C) A) ) (_124 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) (H10 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) ( (and C) A) ) (H4 : (eprop B) => (H5 : (eprop A) => ( (H11 : (eprop A) => ( (H0 : (eprop C) => ( (H20 : (eprop A) => ( (H00 : (eprop B) => ( ( ( (conj C) A) H0) H20) ) (H3 H0) ) ) (H0 H0) ) ) (H2 H4) ) ) (H H4) ) ) ) ) H10) ) ) (H10 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) ( (and B) A) ) (H4 : (eprop C) => (H5 : (eprop A) => ( (H11 : (eprop A) => ( (H00 : (eprop B) => ( (H30 : (eprop A) => ( (H0 : (eprop C) => ( ( ( (conj B) A) H00) H30) ) (H2 H00) ) ) (H H00) ) ) (H3 H4) ) ) (H0 H4) ) ) ) ) H10) ) ) ) ) ) H1) ) ) ) ) ) ) ) .
+or_cancel_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_156 : (_153 : (eprop B) -> (eprop (not A) ) ) -> (_155 : (_154 : (eprop C) -> (eprop (not A) ) ) -> (eprop ( (iff ( (iff ( (or A) B) ) ( (or A) C) ) ) ( (iff B) C) ) ) ) ) ) ) ) .
+[] or_cancel_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_152 : (eprop B) => (not A) ) ) ) => (H0 : (eprop ( (dotpipp C) (_151 : (eprop C) => (not A) ) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (or A) B) ) ( (or A) C) ) ) (_133 : (eprop ( (iff ( (or A) B) ) ( (or A) C) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_134 : (eprop ( (iff B) C) ) => ( (iff ( (or A) B) ) ( (or A) C) ) ) ) ) (H1 : (eprop ( (iff ( (or A) B) ) ( (or A) C) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (or A) B) ) (_135 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) ( (dotpipp ( (or A) C) ) (_136 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (or A) B) ) (_144 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) => (H3 : (eprop ( (dotpipp ( (or A) C) ) (_143 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) => ( (H10 : (eprop ( (dotpipp A) (_142 : (eprop A) => ( (or A) C) ) ) ) => ( (H4 : (eprop ( (dotpipp B) (_141 : (eprop B) => ( (or A) C) ) ) ) => ( (H20 : (eprop ( (dotpipp A) (_140 : (eprop A) => ( (or A) B) ) ) ) => ( (H5 : (eprop ( (dotpipp C) (_139 : (eprop C) => ( (or A) B) ) ) ) => ( ( ( (conj ( (dotpipp B) (_137 : (eprop B) => C) ) ) ( (dotpipp C) (_138 : (eprop C) => B) ) ) (H30 : (eprop B) => ( (H6 : (eprop (not A) ) => ( (H0 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) C) (H40 : (eprop A) => ( (H1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) C) (H11 : (eprop A) => ( (H2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H21 : (eprop A) => ( (H3 : (eprop False) => ( (False_ind C) H3) ) (H6 H40) ) ) ) (H21 : (eprop B) => ( (H3 : (eprop False) => ( (False_ind C) H3) ) (H6 H40) ) ) ) H2) ) (H20 H40) ) ) ) (H11 : (eprop C) => ( (H2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H21 : (eprop A) => ( (H3 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H00 : (eprop False) => ( (H61 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H50 : (eprop A) => ( (False_ind C) H00) ) ) (H50 : (eprop B) => ( (False_ind C) H00) ) ) H61) ) (H5 H11) ) ) (H60 H40) ) ) (H0 H11) ) ) (H6 H40) ) ) ) (H21 : (eprop B) => ( (H3 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H00 : (eprop False) => ( (H61 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H50 : (eprop A) => ( (False_ind C) H00) ) ) (H50 : (eprop B) => ( (False_ind C) H00) ) ) H61) ) (H5 H11) ) ) (H60 H40) ) ) (H0 H11) ) ) (H6 H40) ) ) ) H2) ) (H20 H40) ) ) ) H1) ) (H10 H40) ) ) ) (H40 : (eprop C) => ( (H1 : (eprop (not A) ) => ( (H00 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H50 : (eprop A) => ( (H01 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) C) (H11 : (eprop A) => ( (H02 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H21 : (eprop A) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H50) ) ) (H6 H50) ) ) ) (H21 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H50) ) ) (H6 H50) ) ) ) H02) ) (H20 H50) ) ) ) (H11 : (eprop C) => ( (H02 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H21 : (eprop A) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H50) ) ) (H6 H50) ) ) ) (H21 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H50) ) ) (H6 H50) ) ) ) H02) ) (H20 H50) ) ) ) H01) ) (H10 H50) ) ) ) (H50 : (eprop B) => H40) ) H00) ) (H5 H40) ) ) (H0 H40) ) ) ) H0) ) (H4 H30) ) ) (H H30) ) ) ) (H30 : (eprop C) => ( (H6 : (eprop (not A) ) => ( (H00 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H50 : (eprop A) => ( (H01 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H11 : (eprop A) => ( (H02 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H21 : (eprop A) => ( (H03 : (eprop False) => ( (False_ind B) H03) ) (H6 H50) ) ) ) (H21 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H0 : (eprop False) => ( (H61 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H40 : (eprop A) => ( (False_ind B) H0) ) ) (H40 : (eprop C) => ( (False_ind B) H0) ) ) H61) ) (H4 H21) ) ) (H60 H50) ) ) (H H21) ) ) (H6 H50) ) ) ) H02) ) (H20 H50) ) ) ) (H11 : (eprop C) => ( (H02 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H21 : (eprop A) => ( (H03 : (eprop False) => ( (False_ind B) H03) ) (H6 H50) ) ) ) (H21 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H0 : (eprop False) => ( (H61 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H40 : (eprop A) => ( (False_ind B) H0) ) ) (H40 : (eprop C) => ( (False_ind B) H0) ) ) H61) ) (H4 H21) ) ) (H60 H50) ) ) (H H21) ) ) (H6 H50) ) ) ) H02) ) (H20 H50) ) ) ) H01) ) (H10 H50) ) ) ) (H50 : (eprop B) => ( (H01 : (eprop (not A) ) => ( (H0 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H40 : (eprop A) => ( (H1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H11 : (eprop A) => ( (H2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H21 : (eprop A) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H40) ) ) (H6 H40) ) ) ) (H21 : (eprop B) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H40) ) ) (H6 H40) ) ) ) H2) ) (H20 H40) ) ) ) (H11 : (eprop C) => ( (H2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H21 : (eprop A) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H40) ) ) (H6 H40) ) ) ) (H21 : (eprop B) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H40) ) ) (H6 H40) ) ) ) H2) ) (H20 H40) ) ) ) H1) ) (H10 H40) ) ) ) (H40 : (eprop C) => H50) ) H0) ) (H4 H50) ) ) (H H50) ) ) ) H00) ) (H5 H30) ) ) (H0 H30) ) ) ) ) (H5 : (eprop C) => (H3 ( ( (or_intror A) C) H5) ) ) ) ) (H20 : (eprop A) => (H3 ( ( (or_introl A) C) H20) ) ) ) ) (H4 : (eprop B) => (H2 ( ( (or_intror A) B) H4) ) ) ) ) (H10 : (eprop A) => (H2 ( ( (or_introl A) B) H10) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_145 : (eprop B) => C) ) ) ( (dotpipp C) (_146 : (eprop C) => B) ) ) ( (iff ( (or A) B) ) ( (or A) C) ) ) (H2 : (eprop ( (dotpipp B) (_150 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_149 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or A) B) ) (_147 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) ( (dotpipp ( (or A) C) ) (_148 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) (H10 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) ( (or A) C) ) (H4 : (eprop A) => ( ( (or_introl A) C) H4) ) ) (H4 : (eprop B) => ( (H11 : (eprop (not A) ) => ( (H0 : (eprop C) => ( (H20 : (eprop (not A) ) => ( (H00 : (eprop B) => ( ( (or_intror A) C) H0) ) (H3 H0) ) ) (H0 H0) ) ) (H2 H4) ) ) (H H4) ) ) ) H10) ) ) (H10 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) ( (or A) B) ) (H4 : (eprop A) => ( ( (or_introl A) B) H4) ) ) (H4 : (eprop C) => ( (H11 : (eprop (not A) ) => ( (H00 : (eprop B) => ( (H30 : (eprop (not A) ) => ( (H0 : (eprop C) => ( ( (or_intror A) B) H00) ) (H2 H00) ) ) (H H00) ) ) (H3 H4) ) ) (H0 H4) ) ) ) H10) ) ) ) ) ) H1) ) ) ) ) ) ) ) .
+or_cancel_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_180 : (_177 : (eprop B) -> (eprop (not A) ) ) -> (_179 : (_178 : (eprop C) -> (eprop (not A) ) ) -> (eprop ( (iff ( (iff ( (or B) A) ) ( (or C) A) ) ) ( (iff B) C) ) ) ) ) ) ) ) .
+[] or_cancel_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_176 : (eprop B) => (not A) ) ) ) => (H0 : (eprop ( (dotpipp C) (_175 : (eprop C) => (not A) ) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (or B) A) ) ( (or C) A) ) ) (_157 : (eprop ( (iff ( (or B) A) ) ( (or C) A) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_158 : (eprop ( (iff B) C) ) => ( (iff ( (or B) A) ) ( (or C) A) ) ) ) ) (H1 : (eprop ( (iff ( (or B) A) ) ( (or C) A) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (or B) A) ) (_159 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) ( (dotpipp ( (or C) A) ) (_160 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (or B) A) ) (_168 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) => (H3 : (eprop ( (dotpipp ( (or C) A) ) (_167 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) => ( (H10 : (eprop ( (dotpipp B) (_166 : (eprop B) => ( (or C) A) ) ) ) => ( (H4 : (eprop ( (dotpipp A) (_165 : (eprop A) => ( (or C) A) ) ) ) => ( (H20 : (eprop ( (dotpipp C) (_164 : (eprop C) => ( (or B) A) ) ) ) => ( (H5 : (eprop ( (dotpipp A) (_163 : (eprop A) => ( (or B) A) ) ) ) => ( ( ( (conj ( (dotpipp B) (_161 : (eprop B) => C) ) ) ( (dotpipp C) (_162 : (eprop C) => B) ) ) (H30 : (eprop B) => ( (H6 : (eprop (not A) ) => ( (H0 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) C) (H11 : (eprop C) => ( (H1 : (eprop (not A) ) => ( (H00 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H21 : (eprop B) => H11) ) (H21 : (eprop A) => ( (H01 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) C) (H40 : (eprop C) => ( (H02 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H50 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H21) ) ) (H6 H21) ) ) ) (H50 : (eprop A) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H21) ) ) (H6 H21) ) ) ) H02) ) (H5 H21) ) ) ) (H40 : (eprop A) => ( (H02 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H50 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H21) ) ) (H6 H21) ) ) ) (H50 : (eprop A) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H21) ) ) (H6 H21) ) ) ) H02) ) (H5 H21) ) ) ) H01) ) (H4 H21) ) ) ) H00) ) (H20 H11) ) ) (H0 H11) ) ) ) (H11 : (eprop A) => ( (H1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) C) (H40 : (eprop C) => ( (H2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H50 : (eprop B) => ( (H3 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H00 : (eprop False) => ( (H61 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H21 : (eprop B) => ( (False_ind C) H00) ) ) (H21 : (eprop A) => ( (False_ind C) H00) ) ) H61) ) (H20 H40) ) ) (H60 H11) ) ) (H0 H40) ) ) (H6 H11) ) ) ) (H50 : (eprop A) => ( (H3 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H00 : (eprop False) => ( (H61 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H21 : (eprop B) => ( (False_ind C) H00) ) ) (H21 : (eprop A) => ( (False_ind C) H00) ) ) H61) ) (H20 H40) ) ) (H60 H11) ) ) (H0 H40) ) ) (H6 H11) ) ) ) H2) ) (H5 H11) ) ) ) (H40 : (eprop A) => ( (H2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H50 : (eprop B) => ( (H3 : (eprop False) => ( (False_ind C) H3) ) (H6 H11) ) ) ) (H50 : (eprop A) => ( (H3 : (eprop False) => ( (False_ind C) H3) ) (H6 H11) ) ) ) H2) ) (H5 H11) ) ) ) H1) ) (H4 H11) ) ) ) H0) ) (H10 H30) ) ) (H H30) ) ) ) (H30 : (eprop C) => ( (H6 : (eprop (not A) ) => ( (H00 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H21 : (eprop B) => ( (H01 : (eprop (not A) ) => ( (H0 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H11 : (eprop C) => H21) ) (H11 : (eprop A) => ( (H1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H40 : (eprop C) => ( (H2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H50 : (eprop B) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H11) ) ) (H6 H11) ) ) ) (H50 : (eprop A) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H11) ) ) (H6 H11) ) ) ) H2) ) (H5 H11) ) ) ) (H40 : (eprop A) => ( (H2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H50 : (eprop B) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H11) ) ) (H6 H11) ) ) ) (H50 : (eprop A) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H11) ) ) (H6 H11) ) ) ) H2) ) (H5 H11) ) ) ) H1) ) (H4 H11) ) ) ) H0) ) (H10 H21) ) ) (H H21) ) ) ) (H21 : (eprop A) => ( (H01 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H40 : (eprop C) => ( (H02 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H50 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H0 : (eprop False) => ( (H61 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H11 : (eprop C) => ( (False_ind B) H0) ) ) (H11 : (eprop A) => ( (False_ind B) H0) ) ) H61) ) (H10 H50) ) ) (H60 H21) ) ) (H H50) ) ) (H6 H21) ) ) ) (H50 : (eprop A) => ( (H03 : (eprop False) => ( (False_ind B) H03) ) (H6 H21) ) ) ) H02) ) (H5 H21) ) ) ) (H40 : (eprop A) => ( (H02 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H50 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H0 : (eprop False) => ( (H61 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H11 : (eprop C) => ( (False_ind B) H0) ) ) (H11 : (eprop A) => ( (False_ind B) H0) ) ) H61) ) (H10 H50) ) ) (H60 H21) ) ) (H H50) ) ) (H6 H21) ) ) ) (H50 : (eprop A) => ( (H03 : (eprop False) => ( (False_ind B) H03) ) (H6 H21) ) ) ) H02) ) (H5 H21) ) ) ) H01) ) (H4 H21) ) ) ) H00) ) (H20 H30) ) ) (H0 H30) ) ) ) ) (H5 : (eprop A) => (H3 ( ( (or_intror C) A) H5) ) ) ) ) (H20 : (eprop C) => (H3 ( ( (or_introl C) A) H20) ) ) ) ) (H4 : (eprop A) => (H2 ( ( (or_intror B) A) H4) ) ) ) ) (H10 : (eprop B) => (H2 ( ( (or_introl B) A) H10) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_169 : (eprop B) => C) ) ) ( (dotpipp C) (_170 : (eprop C) => B) ) ) ( (iff ( (or B) A) ) ( (or C) A) ) ) (H2 : (eprop ( (dotpipp B) (_174 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_173 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or B) A) ) (_171 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) ( (dotpipp ( (or C) A) ) (_172 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) (H10 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) ( (or C) A) ) (H4 : (eprop B) => ( (H11 : (eprop (not A) ) => ( (H0 : (eprop C) => ( (H20 : (eprop (not A) ) => ( (H00 : (eprop B) => ( ( (or_introl C) A) H0) ) (H3 H0) ) ) (H0 H0) ) ) (H2 H4) ) ) (H H4) ) ) ) (H4 : (eprop A) => ( ( (or_intror C) A) H4) ) ) H10) ) ) (H10 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) ( (or B) A) ) (H4 : (eprop C) => ( (H11 : (eprop (not A) ) => ( (H00 : (eprop B) => ( (H30 : (eprop (not A) ) => ( (H0 : (eprop C) => ( ( (or_introl B) A) H00) ) (H2 H00) ) ) (H H00) ) ) (H3 H4) ) ) (H0 H4) ) ) ) (H4 : (eprop A) => ( ( (or_intror B) A) H4) ) ) H10) ) ) ) ) ) H1) ) ) ) ) ) ) ) .
+and_iff_compat_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_187 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (and A) B) ) ( (and A) C) ) ) ) ) ) ) .
+[] and_iff_compat_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_181 : (eprop B) => C) ) ) ( (dotpipp C) (_182 : (eprop C) => B) ) ) ( (iff ( (and A) B) ) ( (and A) C) ) ) (H0 : (eprop ( (dotpipp B) (_186 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_185 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and A) B) ) (_183 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) ( (dotpipp ( (and A) C) ) (_184 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) (H0 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) ( (and A) C) ) (H2 : (eprop A) => (H3 : (eprop B) => ( (H1 : (eprop C) => ( (H00 : (eprop B) => ( ( ( (conj A) C) H2) H1) ) (H1 H1) ) ) (H0 H3) ) ) ) ) H0) ) ) (H0 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) ( (and A) B) ) (H2 : (eprop A) => (H3 : (eprop C) => ( (H1 : (eprop B) => ( (H10 : (eprop C) => ( ( ( (conj A) B) H2) H1) ) (H0 H1) ) ) (H1 H3) ) ) ) ) H0) ) ) ) ) ) H) ) ) ) ) .
+and_iff_compat_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_194 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (and B) A) ) ( (and C) A) ) ) ) ) ) ) .
+[] and_iff_compat_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_188 : (eprop B) => C) ) ) ( (dotpipp C) (_189 : (eprop C) => B) ) ) ( (iff ( (and B) A) ) ( (and C) A) ) ) (H0 : (eprop ( (dotpipp B) (_193 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_192 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and B) A) ) (_190 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) ( (dotpipp ( (and C) A) ) (_191 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) (H0 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) ( (and C) A) ) (H2 : (eprop B) => (H3 : (eprop A) => ( (H1 : (eprop C) => ( (H00 : (eprop B) => ( ( ( (conj C) A) H1) H3) ) (H1 H1) ) ) (H0 H2) ) ) ) ) H0) ) ) (H0 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) ( (and B) A) ) (H2 : (eprop C) => (H3 : (eprop A) => ( (H1 : (eprop B) => ( (H10 : (eprop C) => ( ( ( (conj B) A) H1) H3) ) (H0 H1) ) ) (H1 H2) ) ) ) ) H0) ) ) ) ) ) H) ) ) ) ) .
+or_iff_compat_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_201 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (or A) B) ) ( (or A) C) ) ) ) ) ) ) .
+[] or_iff_compat_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_195 : (eprop B) => C) ) ) ( (dotpipp C) (_196 : (eprop C) => B) ) ) ( (iff ( (or A) B) ) ( (or A) C) ) ) (H0 : (eprop ( (dotpipp B) (_200 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_199 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or A) B) ) (_197 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) ( (dotpipp ( (or A) C) ) (_198 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) (H0 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) ( (or A) C) ) (H2 : (eprop A) => ( ( (or_introl A) C) H2) ) ) (H2 : (eprop B) => ( (H1 : (eprop C) => ( (H00 : (eprop B) => ( ( (or_intror A) C) H1) ) (H1 H1) ) ) (H0 H2) ) ) ) H0) ) ) (H0 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) ( (or A) B) ) (H2 : (eprop A) => ( ( (or_introl A) B) H2) ) ) (H2 : (eprop C) => ( (H1 : (eprop B) => ( (H10 : (eprop C) => ( ( (or_intror A) B) H1) ) (H0 H1) ) ) (H1 H2) ) ) ) H0) ) ) ) ) ) H) ) ) ) ) .
+or_iff_compat_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_208 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (or B) A) ) ( (or C) A) ) ) ) ) ) ) .
+[] or_iff_compat_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_202 : (eprop B) => C) ) ) ( (dotpipp C) (_203 : (eprop C) => B) ) ) ( (iff ( (or B) A) ) ( (or C) A) ) ) (H0 : (eprop ( (dotpipp B) (_207 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_206 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or B) A) ) (_204 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) ( (dotpipp ( (or C) A) ) (_205 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) (H0 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) ( (or C) A) ) (H2 : (eprop B) => ( (H1 : (eprop C) => ( (H00 : (eprop B) => ( ( (or_introl C) A) H1) ) (H1 H1) ) ) (H0 H2) ) ) ) (H2 : (eprop A) => ( ( (or_intror C) A) H2) ) ) H0) ) ) (H0 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) ( (or B) A) ) (H2 : (eprop C) => ( (H1 : (eprop B) => ( (H10 : (eprop C) => ( ( (or_introl B) A) H1) ) (H0 H1) ) ) (H1 H2) ) ) ) (H2 : (eprop A) => ( ( (or_intror B) A) H2) ) ) H0) ) ) ) ) ) H) ) ) ) ) .
+case_10 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (iff A) B) ) -> (_213 : (eprop ( (and ( (dotpipp A) (_209 : (eprop A) => B) ) ) ( (dotpipp B) (_210 : (eprop B) => A) ) ) ) -> (eprop ( (and ( (dotpipp A) (_211 : (eprop A) => B) ) ) ( (dotpipp B) (_212 : (eprop B) => A) ) ) ) ) ) ) ) .
+[A : Uprop, B : Uprop, H : (eprop ( (iff A) B) ) , var_16 : (eprop A) , var_17 : (eprop B) ] ( ( ( (case_10 A) B) H) ( ( ( (conj A) B) var_16) var_17) ) --> ( ( (H0 : (eprop ( (dotpipp A) (_217 : (eprop A) => B) ) ) => (H0 : (eprop ( (dotpipp B) (_216 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp A) (_214 : (eprop A) => B) ) ) ( (dotpipp B) (_215 : (eprop B) => A) ) ) H0) H0) ) ) var_16) var_17) .
+iff_and : (A : Uprop -> (B : Uprop -> (_220 : (eprop ( (iff A) B) ) -> (eprop ( (and ( (dotpipp A) (_218 : (eprop A) => B) ) ) ( (dotpipp B) (_219 : (eprop B) => A) ) ) ) ) ) ) .
+[] iff_and --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (iff A) B) ) => ( ( ( (case_10 A) B) H) H) ) ) ) .
+iff_to_and : (A : Uprop -> (B : Uprop -> (eprop ( (iff ( (iff A) B) ) ( (and ( (dotpipp A) (_243 : (eprop A) => B) ) ) ( (dotpipp B) (_244 : (eprop B) => A) ) ) ) ) ) ) .
+[] iff_to_and --> (A : (etype dotprop) => (B : (etype dotprop) => ( ( ( (conj ( (dotpipp ( (iff A) B) ) (_223 : (eprop ( (iff A) B) ) => ( (and ( (dotpipp A) (_221 : (eprop A) => B) ) ) ( (dotpipp B) (_222 : (eprop B) => A) ) ) ) ) ) ( (dotpipp ( (and ( (dotpipp A) (_224 : (eprop A) => B) ) ) ( (dotpipp B) (_225 : (eprop B) => A) ) ) ) (_226 : (eprop ( (and ( (dotpipp A) (_224 : (eprop A) => B) ) ) ( (dotpipp B) (_225 : (eprop B) => A) ) ) ) => ( (iff A) B) ) ) ) (H : (eprop ( (iff A) B) ) => ( ( ( ( (and_ind ( (dotpipp A) (_227 : (eprop A) => B) ) ) ( (dotpipp B) (_228 : (eprop B) => A) ) ) ( (and ( (dotpipp A) (_229 : (eprop A) => B) ) ) ( (dotpipp B) (_230 : (eprop B) => A) ) ) ) (H0 : (eprop ( (dotpipp A) (_234 : (eprop A) => B) ) ) => (H1 : (eprop ( (dotpipp B) (_233 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp A) (_231 : (eprop A) => B) ) ) ( (dotpipp B) (_232 : (eprop B) => A) ) ) (H0 : (eprop A) => ( (H2 : (eprop B) => ( (H00 : (eprop A) => H2) (H1 H2) ) ) (H0 H0) ) ) ) (H0 : (eprop B) => ( (H2 : (eprop A) => ( (H10 : (eprop B) => H2) (H0 H2) ) ) (H1 H0) ) ) ) ) ) ) H) ) ) (H : (eprop ( (and ( (dotpipp A) (_241 : (eprop A) => B) ) ) ( (dotpipp B) (_242 : (eprop B) => A) ) ) ) => ( ( ( ( (and_ind ( (dotpipp A) (_235 : (eprop A) => B) ) ) ( (dotpipp B) (_236 : (eprop B) => A) ) ) ( (iff A) B) ) (H0 : (eprop ( (dotpipp A) (_240 : (eprop A) => B) ) ) => (H1 : (eprop ( (dotpipp B) (_239 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp A) (_237 : (eprop A) => B) ) ) ( (dotpipp B) (_238 : (eprop B) => A) ) ) (H0 : (eprop A) => ( (H2 : (eprop B) => ( (H00 : (eprop A) => H2) (H1 H2) ) ) (H0 H0) ) ) ) (H0 : (eprop B) => ( (H2 : (eprop A) => ( (H10 : (eprop B) => H2) (H0 H2) ) ) (H1 H0) ) ) ) ) ) ) H) ) ) ) ) .
diff --git a/t/bug.eu b/t/bug.eu
new file mode 100644
--- /dev/null
+++ b/t/bug.eu
@@ -0,0 +1,5 @@
+nat : Type.
+
+x : nat.
+
+y : x.
diff --git a/t/coc.eu b/t/coc.eu
new file mode 100644
--- /dev/null
+++ b/t/coc.eu
@@ -0,0 +1,28 @@
+Utype : Type.
+
+Ukind : Type.
+
+etype : Utype -> Type.
+
+ekind : Ukind -> Type.
+
+dottype : Ukind.
+
+dotpi1 : x : Utype -> y : (etype x -> Utype) -> Utype.
+dotpi2 : x : Utype -> y : (etype x -> Ukind) -> Ukind.
+dotpi3 : x : Ukind -> y : (ekind x -> Utype) -> Utype.
+dotpi4 : x : Ukind -> y : (ekind x -> Ukind) -> Ukind.
+
+[x:Utype, y : etype x -> Utype]
+ etype (dotpi1 x y) --> w : etype x -> etype (y w).
+[x:Ukind, y : ekind x -> Utype]
+ etype (dotpi3 x y) --> w : ekind x -> etype (y w).
+
+[] ekind dottype --> Utype.
+[x:Utype, y : etype x -> Ukind]
+ ekind (dotpi2 x y) --> w : etype x -> ekind (y w).
+[x:Ukind, y : ekind x -> Ukind]
+ ekind (dotpi4 x y) --> w : ekind x -> ekind (y w).
+
+a : x : Utype -> y : etype x -> etype x.
+[] a --> x : Utype => y : etype x => y.
diff --git a/t/conj.eu b/t/conj.eu
new file mode 100644
--- /dev/null
+++ b/t/conj.eu
@@ -0,0 +1,3 @@
+o : Type.
+conj : o -> o -> Type.
+[x : o] conj x x --> conj x x.
diff --git a/t/coqlogicprel.eu b/t/coqlogicprel.eu
new file mode 100644
--- /dev/null
+++ b/t/coqlogicprel.eu
@@ -0,0 +1,156 @@
+Uset : Type.
+Uprop : Type.
+Utype : Type.
+
+eprop : x : Uprop -> Type.
+eset : x : Uset -> Type.
+etype : x : Utype -> Type.
+
+dotset : Utype.
+dotprop : Utype.
+
+; /!\ type : type /!\, should use universes
+dottype : Utype.
+
+; /!\ subtyping in coq, should be unidirectional /!\
+[] Uprop --> Utype.
+[] Uset --> Utype.
+
+dotpipp : x : Uprop -> y : (eprop x -> Uprop) -> Uprop.
+dotpips : x : Uprop -> y : (eprop x -> Uset) -> Uset.
+dotpipt : x : Uprop -> y : (eprop x -> Utype) -> Utype.
+dotpisp : x : Uset -> y : (eset x -> Uprop) -> Uprop.
+dotpitp : x : Utype -> y : (etype x -> Uprop) -> Uprop.
+dotpist : x : Uset -> y : (eset x -> Utype) -> Utype.
+dotpits : x : Utype -> y : (etype x -> Uset) -> Uset.
+dotpiss : x : Uset -> y : (eset x -> Uset) -> Uset.
+dotpitt : x : Utype -> y : (etype x -> Utype) -> Utype.
+
+
+[x:Uprop, y : eprop x -> Uprop]
+ eprop (dotpipp x y) --> w : eprop x -> eprop (y w).
+
+[x:Uset, y : eset x -> Uprop]
+ eprop (dotpisp x y) --> w : eset x -> eprop (y w).
+
+[x:Utype, y : etype x -> Uprop]
+ eprop (dotpitp x y) --> w : etype x -> eprop (y w).
+
+; /!\
+[P : Uprop] eprop P --> etype P.
+
+[x:Uprop, y : eprop x -> Uset]
+ eset (dotpips x y) --> w : eprop x -> eset (y w).
+
+[x:Utype, y : etype x -> Uset]
+ eset (dotpits x y) --> w : etype x -> eset (y w).
+
+[x:Uset, y : eset x -> Uset]
+ eset (dotpiss x y) --> w : eset x -> eset (y w).
+
+; /!\
+[P : Uset] eset P --> etype P.
+
+[x:Uset, y : eset x -> Utype]
+ etype (dotpist x y) --> w : eset x -> etype (y w).
+
+[x:Utype, y : etype x -> Utype]
+ etype (dotpitt x y) --> w : etype x -> etype (y w).
+
+[x:Uprop, y : eprop x -> Utype]
+ etype (dotpipt x y) --> w : eprop x -> etype (y w).
+
+
+[] (etype dotset) --> Uset.
+[] (etype dotprop) --> Uprop.
+; /!\
+[] (etype dottype) --> Utype.
+
+; end of Coq1univ
+
+True : Uprop.
+I : (eprop True) .
+case_0 : (P : Utype -> (f : (etype P) -> (t : (eprop True) -> (_0 : (eprop True) -> (etype P) ) ) ) ) .
+[P : Utype, f : (etype P) , t : (eprop True) ] ( ( ( (case_0 P) f) t) I) --> f.
+True_rect : (P : Utype -> (f : (etype P) -> (t : (eprop True) -> (etype P) ) ) ) .
+[] True_rect --> (P : (etype dottype) => (f : (etype P) => (t : (eprop True) => ( ( ( (case_0 P) f) t) t) ) ) ) .
+True_ind : (P : Uprop -> (f : (eprop P) -> (t : (eprop True) -> (eprop P) ) ) ) .
+[] True_ind --> (P : (etype dotprop) => (True_rect P) ) .
+True_rec : (P : Uset -> (f : (eset P) -> (t : (eprop True) -> (eset P) ) ) ) .
+[] True_rec --> (P : (etype dotset) => (True_rect P) ) .
+False : Uprop.
+case_1 : (P : Utype -> (f : (eprop False) -> (_1 : (eprop False) -> (etype P) ) ) ) .
+False_rect : (P : Utype -> (f : (eprop False) -> (etype P) ) ) .
+[] False_rect --> (P : (etype dottype) => (f : (eprop False) => ( ( (case_1 P) f) f) ) ) .
+False_ind : (P : Uprop -> (f : (eprop False) -> (eprop P) ) ) .
+[] False_ind --> (P : (etype dotprop) => (False_rect P) ) .
+False_rec : (P : Uset -> (f : (eprop False) -> (eset P) ) ) .
+[] False_rec --> (P : (etype dotset) => (False_rect P) ) .
+not : (A : Uprop -> Uprop) .
+[] not --> (A : (etype dotprop) => ( (dotpipp A) (_2 : (eprop A) => False) ) ) .
+and : (A : Uprop -> (B : Uprop -> Uprop) ) .
+conj : (A : Uprop -> (B : Uprop -> (_4 : (eprop A) -> (_3 : (eprop B) -> (eprop ( (and A) B) ) ) ) ) ) .
+case_2 : (A : Uprop -> (B : Uprop -> (P : Utype -> (f : (_6 : (eprop A) -> (_5 : (eprop B) -> (etype P) ) ) -> (a : (eprop ( (and A) B) ) -> (_7 : (eprop ( (and A) B) ) -> (etype P) ) ) ) ) ) ) .
+[A : Uprop, B : Uprop, P : Utype, f : (_9 : (eprop A) -> (_8 : (eprop B) -> (etype P) ) ) , a : (eprop ( (and A) B) ) , var_0 : (eprop A) , var_1 : (eprop B) ] ( ( ( ( ( (case_2 A) B) P) f) a) ( ( ( (conj A) B) var_0) var_1) ) --> ( (f var_0) var_1) .
+and_rect : (A : Uprop -> (B : Uprop -> (P : Utype -> (f : (_13 : (eprop A) -> (_12 : (eprop B) -> (etype P) ) ) -> (a : (eprop ( (and A) B) ) -> (etype P) ) ) ) ) ) .
+[] and_rect --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dottype) => (f : (etype ( (dotpipt A) (_11 : (eprop A) => ( (dotpipt B) (_10 : (eprop B) => P) ) ) ) ) => (a : (eprop ( (and A) B) ) => ( ( ( ( ( (case_2 A) B) P) f) a) a) ) ) ) ) ) .
+and_ind : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_15 : (eprop A) -> (_14 : (eprop B) -> (eprop P) ) ) -> (a : (eprop ( (and A) B) ) -> (eprop P) ) ) ) ) ) .
+[] and_ind --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dotprop) => ( ( (and_rect A) B) P) ) ) ) .
+and_rec : (A : Uprop -> (B : Uprop -> (P : Uset -> (f : (_17 : (eprop A) -> (_16 : (eprop B) -> (eset P) ) ) -> (a : (eprop ( (and A) B) ) -> (eset P) ) ) ) ) ) .
+[] and_rec --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dotset) => ( ( (and_rect A) B) P) ) ) ) .
+case_3 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (and A) B) ) -> (_18 : (eprop ( (and A) B) ) -> (eprop A) ) ) ) ) .
+[A : Uprop, B : Uprop, H : (eprop ( (and A) B) ) , var_2 : (eprop A) , var_3 : (eprop B) ] ( ( ( (case_3 A) B) H) ( ( ( (conj A) B) var_2) var_3) ) --> ( ( (H0 : (eprop A) => (H0 : (eprop B) => H0) ) var_2) var_3) .
+proj1 : (A : Uprop -> (B : Uprop -> (_19 : (eprop ( (and A) B) ) -> (eprop A) ) ) ) .
+[] proj1 --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (and A) B) ) => ( ( ( (case_3 A) B) H) H) ) ) ) .
+case_4 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (and A) B) ) -> (_20 : (eprop ( (and A) B) ) -> (eprop B) ) ) ) ) .
+[A : Uprop, B : Uprop, H : (eprop ( (and A) B) ) , var_4 : (eprop A) , var_5 : (eprop B) ] ( ( ( (case_4 A) B) H) ( ( ( (conj A) B) var_4) var_5) ) --> ( ( (H0 : (eprop A) => (H0 : (eprop B) => H0) ) var_4) var_5) .
+proj2 : (A : Uprop -> (B : Uprop -> (_21 : (eprop ( (and A) B) ) -> (eprop B) ) ) ) .
+[] proj2 --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (and A) B) ) => ( ( ( (case_4 A) B) H) H) ) ) ) .
+or : (A : Uprop -> (B : Uprop -> Uprop) ) .
+or_introl : (A : Uprop -> (B : Uprop -> (_22 : (eprop A) -> (eprop ( (or A) B) ) ) ) ) .
+or_intror : (A : Uprop -> (B : Uprop -> (_23 : (eprop B) -> (eprop ( (or A) B) ) ) ) ) .
+case_5 : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_24 : (eprop A) -> (eprop P) ) -> (f0 : (_25 : (eprop B) -> (eprop P) ) -> (o : (eprop ( (or A) B) ) -> (_26 : (eprop ( (or A) B) ) -> (eprop P) ) ) ) ) ) ) ) .
+[A : Uprop, B : Uprop, P : Uprop, f : (_27 : (eprop A) -> (eprop P) ) , f0 : (_28 : (eprop B) -> (eprop P) ) , o : (eprop ( (or A) B) ) , var_6 : (eprop A) ] ( ( ( ( ( ( (case_5 A) B) P) f) f0) o) ( ( (or_introl A) B) var_6) ) --> (f var_6) .
+[A : Uprop, B : Uprop, P : Uprop, f : (_27 : (eprop A) -> (eprop P) ) , f0 : (_28 : (eprop B) -> (eprop P) ) , o : (eprop ( (or A) B) ) , var_7 : (eprop B) ] ( ( ( ( ( ( (case_5 A) B) P) f) f0) o) ( ( (or_intror A) B) var_7) ) --> (f0 var_7) .
+or_ind : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_31 : (eprop A) -> (eprop P) ) -> (f0 : (_32 : (eprop B) -> (eprop P) ) -> (o : (eprop ( (or A) B) ) -> (eprop P) ) ) ) ) ) ) .
+[] or_ind --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dotprop) => (f : (eprop ( (dotpipp A) (_30 : (eprop A) => P) ) ) => (f0 : (eprop ( (dotpipp B) (_29 : (eprop B) => P) ) ) => (o : (eprop ( (or A) B) ) => ( ( ( ( ( ( (case_5 A) B) P) f) f0) o) o) ) ) ) ) ) ) .
+iff : (A : Uprop -> (B : Uprop -> Uprop) ) .
+[] iff --> (A : (etype dotprop) => (B : (etype dotprop) => ( (and ( (dotpipp A) (_33 : (eprop A) => B) ) ) ( (dotpipp B) (_34 : (eprop B) => A) ) ) ) ) .
+iff_refl : (A : Uprop -> (eprop ( (iff A) A) ) ) .
+[] iff_refl --> (A : (etype dotprop) => ( ( ( (conj ( (dotpipp A) (_35 : (eprop A) => A) ) ) ( (dotpipp A) (_36 : (eprop A) => A) ) ) (H : (eprop A) => H) ) (H : (eprop A) => H) ) ) .
+case_6 : (A : Uprop -> (B : Uprop -> (C : Uprop -> (H : (eprop ( (iff A) B) ) -> (_40 : (eprop ( (and ( (dotpipp A) (_37 : (eprop A) => B) ) ) ( (dotpipp B) (_38 : (eprop B) => A) ) ) ) -> (_39 : (eprop ( (iff B) C) ) -> (eprop ( (iff A) C) ) ) ) ) ) ) ) .
+case_7 : (A : Uprop -> (B : Uprop -> (C : Uprop -> (H : (eprop ( (iff A) B) ) -> (H1 : (_41 : (eprop A) -> (eprop B) ) -> (H2 : (_42 : (eprop B) -> (eprop A) ) -> (H0 : (eprop ( (iff B) C) ) -> (_45 : (eprop ( (and ( (dotpipp B) (_43 : (eprop B) => C) ) ) ( (dotpipp C) (_44 : (eprop C) => B) ) ) ) -> (eprop ( (iff A) C) ) ) ) ) ) ) ) ) ) .
+[A : Uprop, B : Uprop, C : Uprop, H : (eprop ( (iff A) B) ) , H1 : (_46 : (eprop A) -> (eprop B) ) , H2 : (_47 : (eprop B) -> (eprop A) ) , H0 : (eprop ( (iff B) C) ) , var_10 : (eprop B) , var_11 : (eprop C) ] ( ( ( ( ( ( ( (case_7 A) B) C) H) H1) H2) H0) ( ( ( (conj B) C) var_10) var_11) ) --> ( ( (H3 : (eprop ( (dotpipp B) (_51 : (eprop B) => C) ) ) => (H4 : (eprop ( (dotpipp C) (_50 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp A) (_48 : (eprop A) => C) ) ) ( (dotpipp C) (_49 : (eprop C) => A) ) ) (H1 : (eprop A) => (H3 (H1 H1) ) ) ) (H1 : (eprop C) => (H2 (H1 (H2 (H4 H1) ) ) ) ) ) ) ) var_10) var_11) .
+[A : Uprop, B : Uprop, C : Uprop, H : (eprop ( (iff A) B) ) , var_8 : (eprop A) , var_9 : (eprop B) ] ( ( ( ( (case_6 A) B) C) H) ( ( ( (conj A) B) var_8) var_9) ) --> ( ( (H1 : (eprop ( (dotpipp A) (_53 : (eprop A) => B) ) ) => (H2 : (eprop ( (dotpipp B) (_52 : (eprop B) => A) ) ) => (H0 : (eprop ( (iff B) C) ) => ( ( ( ( ( ( ( (case_7 A) B) C) H) H1) H2) H0) H0) ) ) ) var_8) var_9) .
+iff_trans : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_55 : (eprop ( (iff A) B) ) -> (_54 : (eprop ( (iff B) C) ) -> (eprop ( (iff A) C) ) ) ) ) ) ) .
+[] iff_trans --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff A) B) ) => ( ( ( ( (case_6 A) B) C) H) H) ) ) ) ) .
+case_8 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (iff A) B) ) -> (_58 : (eprop ( (and ( (dotpipp A) (_56 : (eprop A) => B) ) ) ( (dotpipp B) (_57 : (eprop B) => A) ) ) ) -> (eprop ( (iff B) A) ) ) ) ) ) .
+[A : Uprop, B : Uprop, H : (eprop ( (iff A) B) ) , var_12 : (eprop A) , var_13 : (eprop B) ] ( ( ( (case_8 A) B) H) ( ( ( (conj A) B) var_12) var_13) ) --> ( ( (H1 : (eprop ( (dotpipp A) (_62 : (eprop A) => B) ) ) => (H2 : (eprop ( (dotpipp B) (_61 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp B) (_59 : (eprop B) => A) ) ) ( (dotpipp A) (_60 : (eprop A) => B) ) ) H2) H1) ) ) var_12) var_13) .
+iff_sym : (A : Uprop -> (B : Uprop -> (_63 : (eprop ( (iff A) B) ) -> (eprop ( (iff B) A) ) ) ) ) .
+[] iff_sym --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (iff A) B) ) => ( ( ( (case_8 A) B) H) H) ) ) ) .
+case_9 : (A : Uprop -> (H : (eprop ( (iff A) False) ) -> (_74 : (eprop ( (and ( (dotpipp A) (_71 : (eprop A) => False) ) ) ( (dotpipp False) (_72 : (eprop False) => A) ) ) ) -> (_73 : (eprop A) -> (eprop False) ) ) ) ) .
+[A : Uprop, H : (eprop ( (iff A) False) ) , var_14 : (eprop A) , var_15 : (eprop False) ] ( ( (case_9 A) H) ( ( ( (conj A) False) var_14) var_15) ) --> ( ( (H0 : (eprop ( (dotpipp A) (_76 : (eprop A) => False) ) ) => (H0 : (eprop ( (dotpipp False) (_75 : (eprop False) => A) ) ) => H0) ) var_14) var_15) .
+neg_false : (A : Uprop -> (eprop ( (iff (not A) ) ( (iff A) False) ) ) ) .
+[] neg_false --> (A : (etype dotprop) => ( ( ( (conj ( (dotpipp ( (dotpipp A) (_64 : (eprop A) => False) ) ) (_65 : (eprop ( (dotpipp A) (_64 : (eprop A) => False) ) ) => ( (iff A) False) ) ) ) ( (dotpipp ( (iff A) False) ) (_67 : (eprop ( (iff A) False) ) => ( (dotpipp A) (_66 : (eprop A) => False) ) ) ) ) (H : (eprop ( (dotpipp A) (_70 : (eprop A) => False) ) ) => ( ( ( (conj ( (dotpipp A) (_68 : (eprop A) => False) ) ) ( (dotpipp False) (_69 : (eprop False) => A) ) ) H) (H1 : (eprop False) => ( (False_ind A) H1) ) ) ) ) (H : (eprop ( (iff A) False) ) => ( ( (case_9 A) H) H) ) ) ) .
+and_cancel_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_104 : (_101 : (eprop B) -> (eprop A) ) -> (_103 : (_102 : (eprop C) -> (eprop A) ) -> (eprop ( (iff ( (iff ( (and A) B) ) ( (and A) C) ) ) ( (iff B) C) ) ) ) ) ) ) ) .
+[] and_cancel_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_100 : (eprop B) => A) ) ) => (H0 : (eprop ( (dotpipp C) (_99 : (eprop C) => A) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (and A) B) ) ( (and A) C) ) ) (_77 : (eprop ( (iff ( (and A) B) ) ( (and A) C) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_78 : (eprop ( (iff B) C) ) => ( (iff ( (and A) B) ) ( (and A) C) ) ) ) ) (H1 : (eprop ( (iff ( (and A) B) ) ( (and A) C) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (and A) B) ) (_79 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) ( (dotpipp ( (and A) C) ) (_80 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (and A) B) ) (_92 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) => (H3 : (eprop ( (dotpipp ( (and A) C) ) (_91 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) => ( (H10 : (eprop ( (dotpipp A) (_90 : (eprop A) => ( (dotpipp B) (_89 : (eprop B) => ( (and A) C) ) ) ) ) ) => ( (H20 : (eprop ( (dotpipp A) (_88 : (eprop A) => ( (dotpipp C) (_87 : (eprop C) => ( (and A) B) ) ) ) ) ) => ( ( ( (conj ( (dotpipp B) (_81 : (eprop B) => C) ) ) ( (dotpipp C) (_82 : (eprop C) => B) ) ) (H30 : (eprop B) => ( (H4 : (eprop A) => ( (H0 : (eprop ( (dotpipp B) (_84 : (eprop B) => ( (and A) C) ) ) ) => ( (H11 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) C) (H1 : (eprop A) => (H5 : (eprop C) => ( (H12 : (eprop ( (dotpipp C) (_83 : (eprop C) => ( (and A) B) ) ) ) => ( (H21 : (eprop A) => ( (H00 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) C) (H13 : (eprop A) => (H6 : (eprop B) => H5) ) ) H00) ) (H12 H5) ) ) (H0 H5) ) ) (H20 H4) ) ) ) ) H11) ) (H0 H30) ) ) (H10 H4) ) ) (H H30) ) ) ) (H30 : (eprop C) => ( (H4 : (eprop A) => ( (H00 : (eprop ( (dotpipp B) (_86 : (eprop B) => ( (and A) C) ) ) ) => ( (H11 : (eprop ( (dotpipp C) (_85 : (eprop C) => ( (and A) B) ) ) ) => ( (H21 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) B) (H12 : (eprop A) => (H5 : (eprop B) => ( (H22 : (eprop A) => ( (H0 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) B) (H01 : (eprop A) => (H6 : (eprop C) => H5) ) ) H0) ) (H00 H5) ) ) (H H5) ) ) ) ) H21) ) (H11 H30) ) ) (H20 H4) ) ) (H10 H4) ) ) (H0 H30) ) ) ) ) (H20 : (eprop A) => (H4 : (eprop C) => (H3 ( ( ( (conj A) C) H20) H4) ) ) ) ) ) (H10 : (eprop A) => (H4 : (eprop B) => (H2 ( ( ( (conj A) B) H10) H4) ) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_93 : (eprop B) => C) ) ) ( (dotpipp C) (_94 : (eprop C) => B) ) ) ( (iff ( (and A) B) ) ( (and A) C) ) ) (H2 : (eprop ( (dotpipp B) (_98 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_97 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and A) B) ) (_95 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) ( (dotpipp ( (and A) C) ) (_96 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) (H10 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) ( (and A) C) ) (H4 : (eprop A) => (H5 : (eprop B) => ( (H11 : (eprop A) => ( (H0 : (eprop C) => ( (H20 : (eprop A) => ( (H00 : (eprop B) => ( ( ( (conj A) C) H20) H0) ) (H3 H0) ) ) (H0 H0) ) ) (H2 H5) ) ) (H H5) ) ) ) ) H10) ) ) (H10 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) ( (and A) B) ) (H4 : (eprop A) => (H5 : (eprop C) => ( (H11 : (eprop A) => ( (H00 : (eprop B) => ( (H30 : (eprop A) => ( (H0 : (eprop C) => ( ( ( (conj A) B) H30) H00) ) (H2 H00) ) ) (H H00) ) ) (H3 H5) ) ) (H0 H5) ) ) ) ) H10) ) ) ) ) ) H1) ) ) ) ) ) ) ) .
+and_cancel_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_132 : (_129 : (eprop B) -> (eprop A) ) -> (_131 : (_130 : (eprop C) -> (eprop A) ) -> (eprop ( (iff ( (iff ( (and B) A) ) ( (and C) A) ) ) ( (iff B) C) ) ) ) ) ) ) ) .
+[] and_cancel_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_128 : (eprop B) => A) ) ) => (H0 : (eprop ( (dotpipp C) (_127 : (eprop C) => A) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (and B) A) ) ( (and C) A) ) ) (_105 : (eprop ( (iff ( (and B) A) ) ( (and C) A) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_106 : (eprop ( (iff B) C) ) => ( (iff ( (and B) A) ) ( (and C) A) ) ) ) ) (H1 : (eprop ( (iff ( (and B) A) ) ( (and C) A) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (and B) A) ) (_107 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) ( (dotpipp ( (and C) A) ) (_108 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (and B) A) ) (_120 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) => (H3 : (eprop ( (dotpipp ( (and C) A) ) (_119 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) => ( (H10 : (eprop ( (dotpipp B) (_118 : (eprop B) => ( (dotpipp A) (_117 : (eprop A) => ( (and C) A) ) ) ) ) ) => ( (H20 : (eprop ( (dotpipp C) (_116 : (eprop C) => ( (dotpipp A) (_115 : (eprop A) => ( (and B) A) ) ) ) ) ) => ( ( ( (conj ( (dotpipp B) (_109 : (eprop B) => C) ) ) ( (dotpipp C) (_110 : (eprop C) => B) ) ) (H30 : (eprop B) => ( (H4 : (eprop A) => ( (H0 : (eprop ( (dotpipp A) (_112 : (eprop A) => ( (and C) A) ) ) ) => ( (H11 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) C) (H1 : (eprop C) => (H5 : (eprop A) => ( (H12 : (eprop A) => ( (H00 : (eprop ( (dotpipp A) (_111 : (eprop A) => ( (and B) A) ) ) ) => ( (H21 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) C) (H01 : (eprop B) => (H6 : (eprop A) => H1) ) ) H21) ) (H00 H4) ) ) (H20 H1) ) ) (H0 H1) ) ) ) ) H11) ) (H0 H4) ) ) (H10 H30) ) ) (H H30) ) ) ) (H30 : (eprop C) => ( (H4 : (eprop A) => ( (H00 : (eprop ( (dotpipp A) (_114 : (eprop A) => ( (and B) A) ) ) ) => ( (H21 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) B) (H01 : (eprop B) => (H5 : (eprop A) => ( (H22 : (eprop A) => ( (H0 : (eprop ( (dotpipp A) (_113 : (eprop A) => ( (and C) A) ) ) ) => ( (H11 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) B) (H1 : (eprop C) => (H6 : (eprop A) => H01) ) ) H11) ) (H0 H4) ) ) (H10 H01) ) ) (H H01) ) ) ) ) H21) ) (H00 H4) ) ) (H20 H30) ) ) (H0 H30) ) ) ) ) (H20 : (eprop C) => (H4 : (eprop A) => (H3 ( ( ( (conj C) A) H20) H4) ) ) ) ) ) (H10 : (eprop B) => (H4 : (eprop A) => (H2 ( ( ( (conj B) A) H10) H4) ) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_121 : (eprop B) => C) ) ) ( (dotpipp C) (_122 : (eprop C) => B) ) ) ( (iff ( (and B) A) ) ( (and C) A) ) ) (H2 : (eprop ( (dotpipp B) (_126 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_125 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and B) A) ) (_123 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) ( (dotpipp ( (and C) A) ) (_124 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) (H10 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) ( (and C) A) ) (H4 : (eprop B) => (H5 : (eprop A) => ( (H11 : (eprop A) => ( (H0 : (eprop C) => ( (H20 : (eprop A) => ( (H00 : (eprop B) => ( ( ( (conj C) A) H0) H20) ) (H3 H0) ) ) (H0 H0) ) ) (H2 H4) ) ) (H H4) ) ) ) ) H10) ) ) (H10 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) ( (and B) A) ) (H4 : (eprop C) => (H5 : (eprop A) => ( (H11 : (eprop A) => ( (H00 : (eprop B) => ( (H30 : (eprop A) => ( (H0 : (eprop C) => ( ( ( (conj B) A) H00) H30) ) (H2 H00) ) ) (H H00) ) ) (H3 H4) ) ) (H0 H4) ) ) ) ) H10) ) ) ) ) ) H1) ) ) ) ) ) ) ) .
+or_cancel_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_156 : (_153 : (eprop B) -> (eprop (not A) ) ) -> (_155 : (_154 : (eprop C) -> (eprop (not A) ) ) -> (eprop ( (iff ( (iff ( (or A) B) ) ( (or A) C) ) ) ( (iff B) C) ) ) ) ) ) ) ) .
+[] or_cancel_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_152 : (eprop B) => (not A) ) ) ) => (H0 : (eprop ( (dotpipp C) (_151 : (eprop C) => (not A) ) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (or A) B) ) ( (or A) C) ) ) (_133 : (eprop ( (iff ( (or A) B) ) ( (or A) C) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_134 : (eprop ( (iff B) C) ) => ( (iff ( (or A) B) ) ( (or A) C) ) ) ) ) (H1 : (eprop ( (iff ( (or A) B) ) ( (or A) C) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (or A) B) ) (_135 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) ( (dotpipp ( (or A) C) ) (_136 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (or A) B) ) (_144 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) => (H3 : (eprop ( (dotpipp ( (or A) C) ) (_143 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) => ( (H10 : (eprop ( (dotpipp A) (_142 : (eprop A) => ( (or A) C) ) ) ) => ( (H4 : (eprop ( (dotpipp B) (_141 : (eprop B) => ( (or A) C) ) ) ) => ( (H20 : (eprop ( (dotpipp A) (_140 : (eprop A) => ( (or A) B) ) ) ) => ( (H5 : (eprop ( (dotpipp C) (_139 : (eprop C) => ( (or A) B) ) ) ) => ( ( ( (conj ( (dotpipp B) (_137 : (eprop B) => C) ) ) ( (dotpipp C) (_138 : (eprop C) => B) ) ) (H30 : (eprop B) => ( (H6 : (eprop (not A) ) => ( (H0 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) C) (H40 : (eprop A) => ( (H1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) C) (H11 : (eprop A) => ( (H2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H21 : (eprop A) => ( (H3 : (eprop False) => ( (False_ind C) H3) ) (H6 H40) ) ) ) (H21 : (eprop B) => ( (H3 : (eprop False) => ( (False_ind C) H3) ) (H6 H40) ) ) ) H2) ) (H20 H40) ) ) ) (H11 : (eprop C) => ( (H2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H21 : (eprop A) => ( (H3 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H00 : (eprop False) => ( (H61 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H50 : (eprop A) => ( (False_ind C) H00) ) ) (H50 : (eprop B) => ( (False_ind C) H00) ) ) H61) ) (H5 H11) ) ) (H60 H40) ) ) (H0 H11) ) ) (H6 H40) ) ) ) (H21 : (eprop B) => ( (H3 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H00 : (eprop False) => ( (H61 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H50 : (eprop A) => ( (False_ind C) H00) ) ) (H50 : (eprop B) => ( (False_ind C) H00) ) ) H61) ) (H5 H11) ) ) (H60 H40) ) ) (H0 H11) ) ) (H6 H40) ) ) ) H2) ) (H20 H40) ) ) ) H1) ) (H10 H40) ) ) ) (H40 : (eprop C) => ( (H1 : (eprop (not A) ) => ( (H00 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H50 : (eprop A) => ( (H01 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) C) (H11 : (eprop A) => ( (H02 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H21 : (eprop A) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H50) ) ) (H6 H50) ) ) ) (H21 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H50) ) ) (H6 H50) ) ) ) H02) ) (H20 H50) ) ) ) (H11 : (eprop C) => ( (H02 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H21 : (eprop A) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H50) ) ) (H6 H50) ) ) ) (H21 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H50) ) ) (H6 H50) ) ) ) H02) ) (H20 H50) ) ) ) H01) ) (H10 H50) ) ) ) (H50 : (eprop B) => H40) ) H00) ) (H5 H40) ) ) (H0 H40) ) ) ) H0) ) (H4 H30) ) ) (H H30) ) ) ) (H30 : (eprop C) => ( (H6 : (eprop (not A) ) => ( (H00 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H50 : (eprop A) => ( (H01 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H11 : (eprop A) => ( (H02 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H21 : (eprop A) => ( (H03 : (eprop False) => ( (False_ind B) H03) ) (H6 H50) ) ) ) (H21 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H0 : (eprop False) => ( (H61 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H40 : (eprop A) => ( (False_ind B) H0) ) ) (H40 : (eprop C) => ( (False_ind B) H0) ) ) H61) ) (H4 H21) ) ) (H60 H50) ) ) (H H21) ) ) (H6 H50) ) ) ) H02) ) (H20 H50) ) ) ) (H11 : (eprop C) => ( (H02 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H21 : (eprop A) => ( (H03 : (eprop False) => ( (False_ind B) H03) ) (H6 H50) ) ) ) (H21 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H0 : (eprop False) => ( (H61 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H40 : (eprop A) => ( (False_ind B) H0) ) ) (H40 : (eprop C) => ( (False_ind B) H0) ) ) H61) ) (H4 H21) ) ) (H60 H50) ) ) (H H21) ) ) (H6 H50) ) ) ) H02) ) (H20 H50) ) ) ) H01) ) (H10 H50) ) ) ) (H50 : (eprop B) => ( (H01 : (eprop (not A) ) => ( (H0 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H40 : (eprop A) => ( (H1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H11 : (eprop A) => ( (H2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H21 : (eprop A) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H40) ) ) (H6 H40) ) ) ) (H21 : (eprop B) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H40) ) ) (H6 H40) ) ) ) H2) ) (H20 H40) ) ) ) (H11 : (eprop C) => ( (H2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H21 : (eprop A) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H40) ) ) (H6 H40) ) ) ) (H21 : (eprop B) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H40) ) ) (H6 H40) ) ) ) H2) ) (H20 H40) ) ) ) H1) ) (H10 H40) ) ) ) (H40 : (eprop C) => H50) ) H0) ) (H4 H50) ) ) (H H50) ) ) ) H00) ) (H5 H30) ) ) (H0 H30) ) ) ) ) (H5 : (eprop C) => (H3 ( ( (or_intror A) C) H5) ) ) ) ) (H20 : (eprop A) => (H3 ( ( (or_introl A) C) H20) ) ) ) ) (H4 : (eprop B) => (H2 ( ( (or_intror A) B) H4) ) ) ) ) (H10 : (eprop A) => (H2 ( ( (or_introl A) B) H10) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_145 : (eprop B) => C) ) ) ( (dotpipp C) (_146 : (eprop C) => B) ) ) ( (iff ( (or A) B) ) ( (or A) C) ) ) (H2 : (eprop ( (dotpipp B) (_150 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_149 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or A) B) ) (_147 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) ( (dotpipp ( (or A) C) ) (_148 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) (H10 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) ( (or A) C) ) (H4 : (eprop A) => ( ( (or_introl A) C) H4) ) ) (H4 : (eprop B) => ( (H11 : (eprop (not A) ) => ( (H0 : (eprop C) => ( (H20 : (eprop (not A) ) => ( (H00 : (eprop B) => ( ( (or_intror A) C) H0) ) (H3 H0) ) ) (H0 H0) ) ) (H2 H4) ) ) (H H4) ) ) ) H10) ) ) (H10 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) ( (or A) B) ) (H4 : (eprop A) => ( ( (or_introl A) B) H4) ) ) (H4 : (eprop C) => ( (H11 : (eprop (not A) ) => ( (H00 : (eprop B) => ( (H30 : (eprop (not A) ) => ( (H0 : (eprop C) => ( ( (or_intror A) B) H00) ) (H2 H00) ) ) (H H00) ) ) (H3 H4) ) ) (H0 H4) ) ) ) H10) ) ) ) ) ) H1) ) ) ) ) ) ) ) .
+or_cancel_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_180 : (_177 : (eprop B) -> (eprop (not A) ) ) -> (_179 : (_178 : (eprop C) -> (eprop (not A) ) ) -> (eprop ( (iff ( (iff ( (or B) A) ) ( (or C) A) ) ) ( (iff B) C) ) ) ) ) ) ) ) .
+[] or_cancel_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_176 : (eprop B) => (not A) ) ) ) => (H0 : (eprop ( (dotpipp C) (_175 : (eprop C) => (not A) ) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (or B) A) ) ( (or C) A) ) ) (_157 : (eprop ( (iff ( (or B) A) ) ( (or C) A) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_158 : (eprop ( (iff B) C) ) => ( (iff ( (or B) A) ) ( (or C) A) ) ) ) ) (H1 : (eprop ( (iff ( (or B) A) ) ( (or C) A) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (or B) A) ) (_159 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) ( (dotpipp ( (or C) A) ) (_160 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (or B) A) ) (_168 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) => (H3 : (eprop ( (dotpipp ( (or C) A) ) (_167 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) => ( (H10 : (eprop ( (dotpipp B) (_166 : (eprop B) => ( (or C) A) ) ) ) => ( (H4 : (eprop ( (dotpipp A) (_165 : (eprop A) => ( (or C) A) ) ) ) => ( (H20 : (eprop ( (dotpipp C) (_164 : (eprop C) => ( (or B) A) ) ) ) => ( (H5 : (eprop ( (dotpipp A) (_163 : (eprop A) => ( (or B) A) ) ) ) => ( ( ( (conj ( (dotpipp B) (_161 : (eprop B) => C) ) ) ( (dotpipp C) (_162 : (eprop C) => B) ) ) (H30 : (eprop B) => ( (H6 : (eprop (not A) ) => ( (H0 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) C) (H11 : (eprop C) => ( (H1 : (eprop (not A) ) => ( (H00 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H21 : (eprop B) => H11) ) (H21 : (eprop A) => ( (H01 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) C) (H40 : (eprop C) => ( (H02 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H50 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H21) ) ) (H6 H21) ) ) ) (H50 : (eprop A) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H21) ) ) (H6 H21) ) ) ) H02) ) (H5 H21) ) ) ) (H40 : (eprop A) => ( (H02 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H50 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H21) ) ) (H6 H21) ) ) ) (H50 : (eprop A) => ( (H03 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind C) H60) ) (H1 H21) ) ) (H6 H21) ) ) ) H02) ) (H5 H21) ) ) ) H01) ) (H4 H21) ) ) ) H00) ) (H20 H11) ) ) (H0 H11) ) ) ) (H11 : (eprop A) => ( (H1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) C) (H40 : (eprop C) => ( (H2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H50 : (eprop B) => ( (H3 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H00 : (eprop False) => ( (H61 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H21 : (eprop B) => ( (False_ind C) H00) ) ) (H21 : (eprop A) => ( (False_ind C) H00) ) ) H61) ) (H20 H40) ) ) (H60 H11) ) ) (H0 H40) ) ) (H6 H11) ) ) ) (H50 : (eprop A) => ( (H3 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H00 : (eprop False) => ( (H61 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H21 : (eprop B) => ( (False_ind C) H00) ) ) (H21 : (eprop A) => ( (False_ind C) H00) ) ) H61) ) (H20 H40) ) ) (H60 H11) ) ) (H0 H40) ) ) (H6 H11) ) ) ) H2) ) (H5 H11) ) ) ) (H40 : (eprop A) => ( (H2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H50 : (eprop B) => ( (H3 : (eprop False) => ( (False_ind C) H3) ) (H6 H11) ) ) ) (H50 : (eprop A) => ( (H3 : (eprop False) => ( (False_ind C) H3) ) (H6 H11) ) ) ) H2) ) (H5 H11) ) ) ) H1) ) (H4 H11) ) ) ) H0) ) (H10 H30) ) ) (H H30) ) ) ) (H30 : (eprop C) => ( (H6 : (eprop (not A) ) => ( (H00 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H21 : (eprop B) => ( (H01 : (eprop (not A) ) => ( (H0 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H11 : (eprop C) => H21) ) (H11 : (eprop A) => ( (H1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H40 : (eprop C) => ( (H2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H50 : (eprop B) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H11) ) ) (H6 H11) ) ) ) (H50 : (eprop A) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H11) ) ) (H6 H11) ) ) ) H2) ) (H5 H11) ) ) ) (H40 : (eprop A) => ( (H2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H50 : (eprop B) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H11) ) ) (H6 H11) ) ) ) (H50 : (eprop A) => ( (H3 : (eprop False) => ( (H60 : (eprop False) => ( (False_ind B) H60) ) (H01 H11) ) ) (H6 H11) ) ) ) H2) ) (H5 H11) ) ) ) H1) ) (H4 H11) ) ) ) H0) ) (H10 H21) ) ) (H H21) ) ) ) (H21 : (eprop A) => ( (H01 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H40 : (eprop C) => ( (H02 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H50 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H0 : (eprop False) => ( (H61 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H11 : (eprop C) => ( (False_ind B) H0) ) ) (H11 : (eprop A) => ( (False_ind B) H0) ) ) H61) ) (H10 H50) ) ) (H60 H21) ) ) (H H50) ) ) (H6 H21) ) ) ) (H50 : (eprop A) => ( (H03 : (eprop False) => ( (False_ind B) H03) ) (H6 H21) ) ) ) H02) ) (H5 H21) ) ) ) (H40 : (eprop A) => ( (H02 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H50 : (eprop B) => ( (H03 : (eprop False) => ( (H60 : (eprop (not A) ) => ( (H0 : (eprop False) => ( (H61 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H11 : (eprop C) => ( (False_ind B) H0) ) ) (H11 : (eprop A) => ( (False_ind B) H0) ) ) H61) ) (H10 H50) ) ) (H60 H21) ) ) (H H50) ) ) (H6 H21) ) ) ) (H50 : (eprop A) => ( (H03 : (eprop False) => ( (False_ind B) H03) ) (H6 H21) ) ) ) H02) ) (H5 H21) ) ) ) H01) ) (H4 H21) ) ) ) H00) ) (H20 H30) ) ) (H0 H30) ) ) ) ) (H5 : (eprop A) => (H3 ( ( (or_intror C) A) H5) ) ) ) ) (H20 : (eprop C) => (H3 ( ( (or_introl C) A) H20) ) ) ) ) (H4 : (eprop A) => (H2 ( ( (or_intror B) A) H4) ) ) ) ) (H10 : (eprop B) => (H2 ( ( (or_introl B) A) H10) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_169 : (eprop B) => C) ) ) ( (dotpipp C) (_170 : (eprop C) => B) ) ) ( (iff ( (or B) A) ) ( (or C) A) ) ) (H2 : (eprop ( (dotpipp B) (_174 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_173 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or B) A) ) (_171 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) ( (dotpipp ( (or C) A) ) (_172 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) (H10 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) ( (or C) A) ) (H4 : (eprop B) => ( (H11 : (eprop (not A) ) => ( (H0 : (eprop C) => ( (H20 : (eprop (not A) ) => ( (H00 : (eprop B) => ( ( (or_introl C) A) H0) ) (H3 H0) ) ) (H0 H0) ) ) (H2 H4) ) ) (H H4) ) ) ) (H4 : (eprop A) => ( ( (or_intror C) A) H4) ) ) H10) ) ) (H10 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) ( (or B) A) ) (H4 : (eprop C) => ( (H11 : (eprop (not A) ) => ( (H00 : (eprop B) => ( (H30 : (eprop (not A) ) => ( (H0 : (eprop C) => ( ( (or_introl B) A) H00) ) (H2 H00) ) ) (H H00) ) ) (H3 H4) ) ) (H0 H4) ) ) ) (H4 : (eprop A) => ( ( (or_intror B) A) H4) ) ) H10) ) ) ) ) ) H1) ) ) ) ) ) ) ) .
+and_iff_compat_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_187 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (and A) B) ) ( (and A) C) ) ) ) ) ) ) .
+[] and_iff_compat_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_181 : (eprop B) => C) ) ) ( (dotpipp C) (_182 : (eprop C) => B) ) ) ( (iff ( (and A) B) ) ( (and A) C) ) ) (H0 : (eprop ( (dotpipp B) (_186 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_185 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and A) B) ) (_183 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) ( (dotpipp ( (and A) C) ) (_184 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) (H0 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) ( (and A) C) ) (H2 : (eprop A) => (H3 : (eprop B) => ( (H1 : (eprop C) => ( (H00 : (eprop B) => ( ( ( (conj A) C) H2) H1) ) (H1 H1) ) ) (H0 H3) ) ) ) ) H0) ) ) (H0 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) ( (and A) B) ) (H2 : (eprop A) => (H3 : (eprop C) => ( (H1 : (eprop B) => ( (H10 : (eprop C) => ( ( ( (conj A) B) H2) H1) ) (H0 H1) ) ) (H1 H3) ) ) ) ) H0) ) ) ) ) ) H) ) ) ) ) .
+and_iff_compat_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_194 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (and B) A) ) ( (and C) A) ) ) ) ) ) ) .
+[] and_iff_compat_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_188 : (eprop B) => C) ) ) ( (dotpipp C) (_189 : (eprop C) => B) ) ) ( (iff ( (and B) A) ) ( (and C) A) ) ) (H0 : (eprop ( (dotpipp B) (_193 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_192 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and B) A) ) (_190 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) ( (dotpipp ( (and C) A) ) (_191 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) (H0 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) ( (and C) A) ) (H2 : (eprop B) => (H3 : (eprop A) => ( (H1 : (eprop C) => ( (H00 : (eprop B) => ( ( ( (conj C) A) H1) H3) ) (H1 H1) ) ) (H0 H2) ) ) ) ) H0) ) ) (H0 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) ( (and B) A) ) (H2 : (eprop C) => (H3 : (eprop A) => ( (H1 : (eprop B) => ( (H10 : (eprop C) => ( ( ( (conj B) A) H1) H3) ) (H0 H1) ) ) (H1 H2) ) ) ) ) H0) ) ) ) ) ) H) ) ) ) ) .
+or_iff_compat_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_201 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (or A) B) ) ( (or A) C) ) ) ) ) ) ) .
+[] or_iff_compat_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_195 : (eprop B) => C) ) ) ( (dotpipp C) (_196 : (eprop C) => B) ) ) ( (iff ( (or A) B) ) ( (or A) C) ) ) (H0 : (eprop ( (dotpipp B) (_200 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_199 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or A) B) ) (_197 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) ( (dotpipp ( (or A) C) ) (_198 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) (H0 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) ( (or A) C) ) (H2 : (eprop A) => ( ( (or_introl A) C) H2) ) ) (H2 : (eprop B) => ( (H1 : (eprop C) => ( (H00 : (eprop B) => ( ( (or_intror A) C) H1) ) (H1 H1) ) ) (H0 H2) ) ) ) H0) ) ) (H0 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) ( (or A) B) ) (H2 : (eprop A) => ( ( (or_introl A) B) H2) ) ) (H2 : (eprop C) => ( (H1 : (eprop B) => ( (H10 : (eprop C) => ( ( (or_intror A) B) H1) ) (H0 H1) ) ) (H1 H2) ) ) ) H0) ) ) ) ) ) H) ) ) ) ) .
+or_iff_compat_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_208 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (or B) A) ) ( (or C) A) ) ) ) ) ) ) .
+[] or_iff_compat_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_202 : (eprop B) => C) ) ) ( (dotpipp C) (_203 : (eprop C) => B) ) ) ( (iff ( (or B) A) ) ( (or C) A) ) ) (H0 : (eprop ( (dotpipp B) (_207 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_206 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or B) A) ) (_204 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) ( (dotpipp ( (or C) A) ) (_205 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) (H0 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) ( (or C) A) ) (H2 : (eprop B) => ( (H1 : (eprop C) => ( (H00 : (eprop B) => ( ( (or_introl C) A) H1) ) (H1 H1) ) ) (H0 H2) ) ) ) (H2 : (eprop A) => ( ( (or_intror C) A) H2) ) ) H0) ) ) (H0 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) ( (or B) A) ) (H2 : (eprop C) => ( (H1 : (eprop B) => ( (H10 : (eprop C) => ( ( (or_introl B) A) H1) ) (H0 H1) ) ) (H1 H2) ) ) ) (H2 : (eprop A) => ( ( (or_intror B) A) H2) ) ) H0) ) ) ) ) ) H) ) ) ) ) .
+case_10 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (iff A) B) ) -> (_213 : (eprop ( (and ( (dotpipp A) (_209 : (eprop A) => B) ) ) ( (dotpipp B) (_210 : (eprop B) => A) ) ) ) -> (eprop ( (and ( (dotpipp A) (_211 : (eprop A) => B) ) ) ( (dotpipp B) (_212 : (eprop B) => A) ) ) ) ) ) ) ) .
+[A : Uprop, B : Uprop, H : (eprop ( (iff A) B) ) , var_16 : (eprop A) , var_17 : (eprop B) ] ( ( ( (case_10 A) B) H) ( ( ( (conj A) B) var_16) var_17) ) --> ( ( (H0 : (eprop ( (dotpipp A) (_217 : (eprop A) => B) ) ) => (H0 : (eprop ( (dotpipp B) (_216 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp A) (_214 : (eprop A) => B) ) ) ( (dotpipp B) (_215 : (eprop B) => A) ) ) H0) H0) ) ) var_16) var_17) .
+iff_and : (A : Uprop -> (B : Uprop -> (_220 : (eprop ( (iff A) B) ) -> (eprop ( (and ( (dotpipp A) (_218 : (eprop A) => B) ) ) ( (dotpipp B) (_219 : (eprop B) => A) ) ) ) ) ) ) .
+[] iff_and --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (iff A) B) ) => ( ( ( (case_10 A) B) H) H) ) ) ) .
+iff_to_and : (A : Uprop -> (B : Uprop -> (eprop ( (iff ( (iff A) B) ) ( (and ( (dotpipp A) (_243 : (eprop A) => B) ) ) ( (dotpipp B) (_244 : (eprop B) => A) ) ) ) ) ) ) .
+[] iff_to_and --> (A : (etype dotprop) => (B : (etype dotprop) => ( ( ( (conj ( (dotpipp ( (iff A) B) ) (_223 : (eprop ( (iff A) B) ) => ( (and ( (dotpipp A) (_221 : (eprop A) => B) ) ) ( (dotpipp B) (_222 : (eprop B) => A) ) ) ) ) ) ( (dotpipp ( (and ( (dotpipp A) (_224 : (eprop A) => B) ) ) ( (dotpipp B) (_225 : (eprop B) => A) ) ) ) (_226 : (eprop ( (and ( (dotpipp A) (_224 : (eprop A) => B) ) ) ( (dotpipp B) (_225 : (eprop B) => A) ) ) ) => ( (iff A) B) ) ) ) (H : (eprop ( (iff A) B) ) => ( ( ( ( (and_ind ( (dotpipp A) (_227 : (eprop A) => B) ) ) ( (dotpipp B) (_228 : (eprop B) => A) ) ) ( (and ( (dotpipp A) (_229 : (eprop A) => B) ) ) ( (dotpipp B) (_230 : (eprop B) => A) ) ) ) (H0 : (eprop ( (dotpipp A) (_234 : (eprop A) => B) ) ) => (H1 : (eprop ( (dotpipp B) (_233 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp A) (_231 : (eprop A) => B) ) ) ( (dotpipp B) (_232 : (eprop B) => A) ) ) (H0 : (eprop A) => ( (H2 : (eprop B) => ( (H00 : (eprop A) => H2) (H1 H2) ) ) (H0 H0) ) ) ) (H0 : (eprop B) => ( (H2 : (eprop A) => ( (H10 : (eprop B) => H2) (H0 H2) ) ) (H1 H0) ) ) ) ) ) ) H) ) ) (H : (eprop ( (and ( (dotpipp A) (_241 : (eprop A) => B) ) ) ( (dotpipp B) (_242 : (eprop B) => A) ) ) ) => ( ( ( ( (and_ind ( (dotpipp A) (_235 : (eprop A) => B) ) ) ( (dotpipp B) (_236 : (eprop B) => A) ) ) ( (iff A) B) ) (H0 : (eprop ( (dotpipp A) (_240 : (eprop A) => B) ) ) => (H1 : (eprop ( (dotpipp B) (_239 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp A) (_237 : (eprop A) => B) ) ) ( (dotpipp B) (_238 : (eprop B) => A) ) ) (H0 : (eprop A) => ( (H2 : (eprop B) => ( (H00 : (eprop A) => H2) (H1 H2) ) ) (H0 H0) ) ) ) (H0 : (eprop B) => ( (H2 : (eprop A) => ( (H10 : (eprop B) => H2) (H0 H2) ) ) (H1 H0) ) ) ) ) ) ) H) ) ) ) ) .
diff --git a/t/delta1.eu b/t/delta1.eu
new file mode 100644
--- /dev/null
+++ b/t/delta1.eu
@@ -0,0 +1,2 @@
+delta : a : Type -> (b : Type -> b -> b) -> a -> a.
+[] delta --> a : Type => x : (Type -> Type) => x (a -> a) (x a).
diff --git a/t/delta2.eu b/t/delta2.eu
new file mode 100644
--- /dev/null
+++ b/t/delta2.eu
@@ -0,0 +1,7 @@
+;; Same as delta1.eu but with d2 declared of type 'delta delta', which of
+;; course is ill-typed.
+
+delta : a : Type -> (b : Type -> b -> b) -> a -> a.
+[] delta --> a : Type => x : (Type -> Type) => x (a -> a) (x a).
+
+d2 : delta delta.
diff --git a/t/exemple.eu b/t/exemple.eu
new file mode 100644
--- /dev/null
+++ b/t/exemple.eu
@@ -0,0 +1,9 @@
+T : Type.
+
+U : Type.
+
+[] U --> T -> T.
+
+app : f : (T -> T) -> T -> T.
+
+[] app --> f : U => x : T => f x.
diff --git a/t/f.eu b/t/f.eu
new file mode 100644
--- /dev/null
+++ b/t/f.eu
@@ -0,0 +1,17 @@
+Utype : Type.
+Ukind : Type.
+etype : Utype -> Type.
+ekind : Ukind -> Type.
+dottype : Ukind.
+dotpi1 : x : Utype -> (etype x -> Utype) -> Utype.
+dotpi3 : x : Ukind -> (ekind x -> Utype) -> Utype.
+[] ekind dottype --> Utype.
+[x:Utype, y : etype x -> Utype]
+ etype ((dotpi1 x) y) --> w : etype x -> etype (y w).
+
+[x:Ukind, y : ekind x -> Utype]
+ etype ((dotpi3 x) y) --> w : ekind x -> etype (y w).
+
+a : x : Utype -> etype x -> etype x.
+
+[] a --> x : Utype => y : etype x => y.
diff --git a/t/gros.eu b/t/gros.eu
new file mode 100644
--- /dev/null
+++ b/t/gros.eu
@@ -0,0 +1,360 @@
+Uset : Type.
+Uprop : Type.
+Utype : Type.
+
+eprop : x : Uprop -> Type.
+eset : x : Uset -> Type.
+etype : x : Utype -> Type.
+
+dotset : Utype.
+dotprop : Utype.
+
+; /!\ type : type /!\, should use universes
+dottype : Utype.
+
+; /!\ subtyping in coq, should be unidirectional /!\
+[] Uprop --> Utype.
+[] Uset --> Utype.
+
+dotpipp : x : Uprop -> y : (eprop x -> Uprop) -> Uprop.
+dotpips : x : Uprop -> y : (eprop x -> Uset) -> Uset.
+dotpipt : x : Uprop -> y : (eprop x -> Utype) -> Utype.
+dotpisp : x : Uset -> y : (eset x -> Uprop) -> Uprop.
+dotpitp : x : Utype -> y : (etype x -> Uprop) -> Uprop.
+dotpist : x : Uset -> y : (eset x -> Utype) -> Utype.
+dotpits : x : Utype -> y : (etype x -> Uset) -> Uset.
+dotpiss : x : Uset -> y : (eset x -> Uset) -> Uset.
+dotpitt : x : Utype -> y : (etype x -> Utype) -> Utype.
+
+
+[x:Uprop, y : eprop x -> Uprop]
+ eprop (dotpipp x y) --> w : eprop x -> eprop (y w).
+
+[x:Uset, y : eset x -> Uprop]
+ eprop (dotpisp x y) --> w : eset x -> eprop (y w).
+
+[x:Utype, y : etype x -> Uprop]
+ eprop (dotpitp x y) --> w : etype x -> eprop (y w).
+
+; /!\
+[P : Uprop] eprop P --> etype P.
+
+[x:Uprop, y : eprop x -> Uset]
+ eset (dotpips x y) --> w : eprop x -> eset (y w).
+
+[x:Utype, y : etype x -> Uset]
+ eset (dotpits x y) --> w : etype x -> eset (y w).
+
+[x:Uset, y : eset x -> Uset]
+ eset (dotpiss x y) --> w : eset x -> eset (y w).
+
+; /!\
+[P : Uset] eset P --> etype P.
+
+[x:Uset, y : eset x -> Utype]
+ etype (dotpist x y) --> w : eset x -> etype (y w).
+
+[x:Utype, y : etype x -> Utype]
+ etype (dotpitt x y) --> w : etype x -> etype (y w).
+
+[x:Uprop, y : eprop x -> Utype]
+ etype (dotpipt x y) --> w : eprop x -> etype (y w).
+
+
+[] (etype dotset) --> Uset.
+[] (etype dotprop) --> Uprop.
+; /!\
+[] (etype dottype) --> Utype.
+
+; end of Coq1univ
+
+True : Uprop.
+I : (eprop True) .
+case_0 : (P : Utype -> (f : (etype P) -> (t : (eprop True) -> (_0 : (eprop True) -> (etype P) ) ) ) ) .
+I_case_0 : (P : Utype -> (f : (etype P) -> (t : (eprop True) -> (eprop True) ) ) ) .
+[P : Utype, f : (etype P) , t : (eprop True) ] ( ( (I_case_0 P) f) t) --> I.
+[P : Utype, f : (etype P) , t : (eprop True) ] ( ( ( (case_0 P) f) t) ( ( (I_case_0 P) f) t) ) --> f.
+True_rect : (P : Utype -> (f : (etype P) -> (t : (eprop True) -> (etype P) ) ) ) .
+[] True_rect --> (P : (etype dottype) => (f : (etype P) => (t : (eprop True) => ( ( ( (case_0 P) f) t) t) ) ) ) .
+True_ind : (P : Uprop -> (f : (eprop P) -> (t : (eprop True) -> (eprop P) ) ) ) .
+[] True_ind --> (P : (etype dotprop) => (True_rect P) ) .
+True_rec : (P : Uset -> (f : (eset P) -> (t : (eprop True) -> (eset P) ) ) ) .
+[] True_rec --> (P : (etype dotset) => (True_rect P) ) .
+False : Uprop.
+case_1 : (P : Utype -> (f : (eprop False) -> (_1 : (eprop False) -> (etype P) ) ) ) .
+False_rect : (P : Utype -> (f : (eprop False) -> (etype P) ) ) .
+[] False_rect --> (P : (etype dottype) => (f : (eprop False) => ( ( (case_1 P) f) f) ) ) .
+False_ind : (P : Uprop -> (f : (eprop False) -> (eprop P) ) ) .
+[] False_ind --> (P : (etype dotprop) => (False_rect P) ) .
+False_rec : (P : Uset -> (f : (eprop False) -> (eset P) ) ) .
+[] False_rec --> (P : (etype dotset) => (False_rect P) ) .
+not : (A : Uprop -> Uprop) .
+[] not --> (A : (etype dotprop) => ( (dotpipp A) (_2 : (eprop A) => False) ) ) .
+and : (A : Uprop -> (B : Uprop -> Uprop) ) .
+conj : (A : Uprop -> (B : Uprop -> (_4 : (eprop A) -> (_3 : (eprop B) -> (eprop ( (and A) B) ) ) ) ) ) .
+case_2 : (A : Uprop -> (B : Uprop -> (P : Utype -> (f : (_6 : (eprop A) -> (_5 : (eprop B) -> (etype P) ) ) -> (a : (eprop ( (and A) B) ) -> (_7 : (eprop ( (and A) B) ) -> (etype P) ) ) ) ) ) ) .
+conj_case_2 : (A : Uprop -> (B : Uprop -> (P : Utype -> (f : (_11 : (eprop A) -> (_10 : (eprop B) -> (etype P) ) ) -> (a : (eprop ( (and A) B) ) -> (_13 : (eprop A) -> (_12 : (eprop B) -> (eprop ( (and A) B) ) ) ) ) ) ) ) ) .
+[A : Uprop, B : Uprop, P : Utype, f : (_15 : (eprop A) -> (_14 : (eprop B) -> (etype P) ) ) , a : (eprop ( (and A) B) ) ] ( ( ( ( (conj_case_2 A) B) P) f) a) --> ( (conj A) B) .
+[A : Uprop, B : Uprop, P : Utype, f : (_9 : (eprop A) -> (_8 : (eprop B) -> (etype P) ) ) , a : (eprop ( (and A) B) ) , var_0 : (eprop A) , var_1 : (eprop B) ] ( ( ( ( ( (case_2 A) B) P) f) a) ( ( ( ( ( ( (conj_case_2 A) B) P) f) a) var_0) var_1) ) --> ( (f var_0) var_1) .
+and_rect : (A : Uprop -> (B : Uprop -> (P : Utype -> (f : (_19 : (eprop A) -> (_18 : (eprop B) -> (etype P) ) ) -> (a : (eprop ( (and A) B) ) -> (etype P) ) ) ) ) ) .
+[] and_rect --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dottype) => (f : (etype ( (dotpipt A) (_17 : (eprop A) => ( (dotpipt B) (_16 : (eprop B) => P) ) ) ) ) => (a : (eprop ( (and A) B) ) => ( ( ( ( ( (case_2 A) B) P) f) a) a) ) ) ) ) ) .
+and_ind : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_21 : (eprop A) -> (_20 : (eprop B) -> (eprop P) ) ) -> (a : (eprop ( (and A) B) ) -> (eprop P) ) ) ) ) ) .
+[] and_ind --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dotprop) => ( ( (and_rect A) B) P) ) ) ) .
+and_rec : (A : Uprop -> (B : Uprop -> (P : Uset -> (f : (_23 : (eprop A) -> (_22 : (eprop B) -> (eset P) ) ) -> (a : (eprop ( (and A) B) ) -> (eset P) ) ) ) ) ) .
+[] and_rec --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dotset) => ( ( (and_rect A) B) P) ) ) ) .
+case_3 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (and A) B) ) -> (_24 : (eprop ( (and A) B) ) -> (eprop A) ) ) ) ) .
+conj_case_3 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (and A) B) ) -> (_26 : (eprop A) -> (_25 : (eprop B) -> (eprop ( (and A) B) ) ) ) ) ) ) .
+[A : Uprop, B : Uprop, H : (eprop ( (and A) B) ) ] ( ( (conj_case_3 A) B) H) --> ( (conj A) B) .
+[A : Uprop, B : Uprop, H : (eprop ( (and A) B) ) , var_2 : (eprop A) , var_3 : (eprop B) ] ( ( ( (case_3 A) B) H) ( ( ( ( (conj_case_3 A) B) H) var_2) var_3) ) --> ( ( (H欧0 : (eprop A) => (H0 : (eprop B) => H欧0) ) var_2) var_3) .
+proj1 : (A : Uprop -> (B : Uprop -> (_27 : (eprop ( (and A) B) ) -> (eprop A) ) ) ) .
+[] proj1 --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (and A) B) ) => ( ( ( (case_3 A) B) H) H) ) ) ) .
+case_4 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (and A) B) ) -> (_28 : (eprop ( (and A) B) ) -> (eprop B) ) ) ) ) .
+conj_case_4 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (and A) B) ) -> (_30 : (eprop A) -> (_29 : (eprop B) -> (eprop ( (and A) B) ) ) ) ) ) ) .
+[A : Uprop, B : Uprop, H : (eprop ( (and A) B) ) ] ( ( (conj_case_4 A) B) H) --> ( (conj A) B) .
+[A : Uprop, B : Uprop, H : (eprop ( (and A) B) ) , var_4 : (eprop A) , var_5 : (eprop B) ] ( ( ( (case_4 A) B) H) ( ( ( ( (conj_case_4 A) B) H) var_4) var_5) ) --> ( ( (H欧0 : (eprop A) => (H0 : (eprop B) => H0) ) var_4) var_5) .
+proj2 : (A : Uprop -> (B : Uprop -> (_31 : (eprop ( (and A) B) ) -> (eprop B) ) ) ) .
+[] proj2 --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (and A) B) ) => ( ( ( (case_4 A) B) H) H) ) ) ) .
+or : (A : Uprop -> (B : Uprop -> Uprop) ) .
+or_introl : (A : Uprop -> (B : Uprop -> (_32 : (eprop A) -> (eprop ( (or A) B) ) ) ) ) .
+or_intror : (A : Uprop -> (B : Uprop -> (_33 : (eprop B) -> (eprop ( (or A) B) ) ) ) ) .
+case_5 : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_34 : (eprop A) -> (eprop P) ) -> (f欧0 : (_35 : (eprop B) -> (eprop P) ) -> (o : (eprop ( (or A) B) ) -> (_36 : (eprop ( (or A) B) ) -> (eprop P) ) ) ) ) ) ) ) .
+or_introl_case_5 : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_39 : (eprop A) -> (eprop P) ) -> (f欧0 : (_40 : (eprop B) -> (eprop P) ) -> (o : (eprop ( (or A) B) ) -> (_41 : (eprop A) -> (eprop ( (or A) B) ) ) ) ) ) ) ) ) .
+[A : Uprop, B : Uprop, P : Uprop, f : (_42 : (eprop A) -> (eprop P) ) , f欧0 : (_43 : (eprop B) -> (eprop P) ) , o : (eprop ( (or A) B) ) ] ( ( ( ( ( (or_introl_case_5 A) B) P) f) f欧0) o) --> ( (or_introl A) B) .
+or_intror_case_5 : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_44 : (eprop A) -> (eprop P) ) -> (f欧0 : (_45 : (eprop B) -> (eprop P) ) -> (o : (eprop ( (or A) B) ) -> (_46 : (eprop B) -> (eprop ( (or A) B) ) ) ) ) ) ) ) ) .
+[A : Uprop, B : Uprop, P : Uprop, f : (_47 : (eprop A) -> (eprop P) ) , f欧0 : (_48 : (eprop B) -> (eprop P) ) , o : (eprop ( (or A) B) ) ] ( ( ( ( ( (or_intror_case_5 A) B) P) f) f欧0) o) --> ( (or_intror A) B) .
+[A : Uprop, B : Uprop, P : Uprop, f : (_37 : (eprop A) -> (eprop P) ) , f欧0 : (_38 : (eprop B) -> (eprop P) ) , o : (eprop ( (or A) B) ) , var_6 : (eprop A) ] ( ( ( ( ( ( (case_5 A) B) P) f) f欧0) o) ( ( ( ( ( ( (or_introl_case_5 A) B) P) f) f欧0) o) var_6) ) --> (f var_6) .
+[A : Uprop, B : Uprop, P : Uprop, f : (_37 : (eprop A) -> (eprop P) ) , f欧0 : (_38 : (eprop B) -> (eprop P) ) , o : (eprop ( (or A) B) ) , var_7 : (eprop B) ] ( ( ( ( ( ( (case_5 A) B) P) f) f欧0) o) ( ( ( ( ( ( (or_intror_case_5 A) B) P) f) f欧0) o) var_7) ) --> (f欧0 var_7) .
+or_ind : (A : Uprop -> (B : Uprop -> (P : Uprop -> (f : (_51 : (eprop A) -> (eprop P) ) -> (f欧0 : (_52 : (eprop B) -> (eprop P) ) -> (o : (eprop ( (or A) B) ) -> (eprop P) ) ) ) ) ) ) .
+[] or_ind --> (A : (etype dotprop) => (B : (etype dotprop) => (P : (etype dotprop) => (f : (eprop ( (dotpipp A) (_50 : (eprop A) => P) ) ) => (f欧0 : (eprop ( (dotpipp B) (_49 : (eprop B) => P) ) ) => (o : (eprop ( (or A) B) ) => ( ( ( ( ( ( (case_5 A) B) P) f) f欧0) o) o) ) ) ) ) ) ) .
+iff : (A : Uprop -> (B : Uprop -> Uprop) ) .
+[] iff --> (A : (etype dotprop) => (B : (etype dotprop) => ( (and ( (dotpipp A) (_53 : (eprop A) => B) ) ) ( (dotpipp B) (_54 : (eprop B) => A) ) ) ) ) .
+iff_refl : (A : Uprop -> (eprop ( (iff A) A) ) ) .
+[] iff_refl --> (A : (etype dotprop) => ( ( ( (conj ( (dotpipp A) (_55 : (eprop A) => A) ) ) ( (dotpipp A) (_56 : (eprop A) => A) ) ) (H : (eprop A) => H) ) (H : (eprop A) => H) ) ) .
+case_6 : (A : Uprop -> (B : Uprop -> (C : Uprop -> (H : (eprop ( (iff A) B) ) -> (_60 : (eprop ( (and ( (dotpipp A) (_57 : (eprop A) => B) ) ) ( (dotpipp B) (_58 : (eprop B) => A) ) ) ) -> (_59 : (eprop ( (iff B) C) ) -> (eprop ( (iff A) C) ) ) ) ) ) ) ) .
+conj_case_6 : (A : Uprop -> (B : Uprop -> (C : Uprop -> (H : (eprop ( (iff A) B) ) -> (_68 : (_63 : (eprop A) -> (eprop B) ) -> (_67 : (_64 : (eprop B) -> (eprop A) ) -> (eprop ( (and ( (dotpipp A) (_65 : (eprop A) => B) ) ) ( (dotpipp B) (_66 : (eprop B) => A) ) ) ) ) ) ) ) ) ) .
+[A : Uprop, B : Uprop, C : Uprop, H : (eprop ( (iff A) B) ) ] ( ( ( (conj_case_6 A) B) C) H) --> ( (conj ( (dotpipp A) (_61 : (eprop A) => B) ) ) ( (dotpipp B) (_62 : (eprop B) => A) ) ) .
+case_7 : (A : Uprop -> (B : Uprop -> (C : Uprop -> (H : (eprop ( (iff A) B) ) -> (H1 : (_71 : (eprop A) -> (eprop B) ) -> (H2 : (_72 : (eprop B) -> (eprop A) ) -> (H欧0 : (eprop ( (iff B) C) ) -> (_75 : (eprop ( (and ( (dotpipp B) (_73 : (eprop B) => C) ) ) ( (dotpipp C) (_74 : (eprop C) => B) ) ) ) -> (eprop ( (iff A) C) ) ) ) ) ) ) ) ) ) .
+conj_case_7 : (A : Uprop -> (B : Uprop -> (C : Uprop -> (H : (eprop ( (iff A) B) ) -> (H1 : (_80 : (eprop A) -> (eprop B) ) -> (H2 : (_81 : (eprop B) -> (eprop A) ) -> (H欧0 : (eprop ( (iff B) C) ) -> (_87 : (_82 : (eprop B) -> (eprop C) ) -> (_86 : (_83 : (eprop C) -> (eprop B) ) -> (eprop ( (and ( (dotpipp B) (_84 : (eprop B) => C) ) ) ( (dotpipp C) (_85 : (eprop C) => B) ) ) ) ) ) ) ) ) ) ) ) ) .
+[A : Uprop, B : Uprop, C : Uprop, H : (eprop ( (iff A) B) ) , H1 : (_88 : (eprop A) -> (eprop B) ) , H2 : (_89 : (eprop B) -> (eprop A) ) , H欧0 : (eprop ( (iff B) C) ) ] ( ( ( ( ( ( (conj_case_7 A) B) C) H) H1) H2) H欧0) --> ( (conj ( (dotpipp B) (_78 : (eprop B) => C) ) ) ( (dotpipp C) (_79 : (eprop C) => B) ) ) .
+[A : Uprop, B : Uprop, C : Uprop, H : (eprop ( (iff A) B) ) , H1 : (_76 : (eprop A) -> (eprop B) ) , H2 : (_77 : (eprop B) -> (eprop A) ) , H欧0 : (eprop ( (iff B) C) ) , var_10 : (_90 : (eprop B) -> (eprop C) ) , var_11 : (_91 : (eprop C) -> (eprop B) ) ] ( ( ( ( ( ( ( (case_7 A) B) C) H) H1) H2) H欧0) ( ( ( ( ( ( ( ( (conj_case_7 A) B) C) H) H1) H2) H欧0) var_10) var_11) ) --> ( ( (H3 : (eprop ( (dotpipp B) (_95 : (eprop B) => C) ) ) => (H4 : (eprop ( (dotpipp C) (_94 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp A) (_92 : (eprop A) => C) ) ) ( (dotpipp C) (_93 : (eprop C) => A) ) ) (H欧1 : (eprop A) => (H3 (H1 H欧1) ) ) ) (H欧1 : (eprop C) => (H2 (H1 (H2 (H4 H欧1) ) ) ) ) ) ) ) var_10) var_11) .
+[A : Uprop, B : Uprop, C : Uprop, H : (eprop ( (iff A) B) ) , var_8 : (_69 : (eprop A) -> (eprop B) ) , var_9 : (_70 : (eprop B) -> (eprop A) ) ] ( ( ( ( (case_6 A) B) C) H) ( ( ( ( ( (conj_case_6 A) B) C) H) var_8) var_9) ) --> ( ( (H1 : (eprop ( (dotpipp A) (_97 : (eprop A) => B) ) ) => (H2 : (eprop ( (dotpipp B) (_96 : (eprop B) => A) ) ) => (H欧0 : (eprop ( (iff B) C) ) => ( ( ( ( ( ( ( (case_7 A) B) C) H) H1) H2) H欧0) H欧0) ) ) ) var_8) var_9) .
+iff_trans : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_99 : (eprop ( (iff A) B) ) -> (_98 : (eprop ( (iff B) C) ) -> (eprop ( (iff A) C) ) ) ) ) ) ) .
+[] iff_trans --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff A) B) ) => ( ( ( ( (case_6 A) B) C) H) H) ) ) ) ) .
+case_8 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (iff A) B) ) -> (_102 : (eprop ( (and ( (dotpipp A) (_100 : (eprop A) => B) ) ) ( (dotpipp B) (_101 : (eprop B) => A) ) ) ) -> (eprop ( (iff B) A) ) ) ) ) ) .
+conj_case_8 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (iff A) B) ) -> (_110 : (_105 : (eprop A) -> (eprop B) ) -> (_109 : (_106 : (eprop B) -> (eprop A) ) -> (eprop ( (and ( (dotpipp A) (_107 : (eprop A) => B) ) ) ( (dotpipp B) (_108 : (eprop B) => A) ) ) ) ) ) ) ) ) .
+[A : Uprop, B : Uprop, H : (eprop ( (iff A) B) ) ] ( ( (conj_case_8 A) B) H) --> ( (conj ( (dotpipp A) (_103 : (eprop A) => B) ) ) ( (dotpipp B) (_104 : (eprop B) => A) ) ) .
+[A : Uprop, B : Uprop, H : (eprop ( (iff A) B) ) , var_12 : (_111 : (eprop A) -> (eprop B) ) , var_13 : (_112 : (eprop B) -> (eprop A) ) ] ( ( ( (case_8 A) B) H) ( ( ( ( (conj_case_8 A) B) H) var_12) var_13) ) --> ( ( (H1 : (eprop ( (dotpipp A) (_116 : (eprop A) => B) ) ) => (H2 : (eprop ( (dotpipp B) (_115 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp B) (_113 : (eprop B) => A) ) ) ( (dotpipp A) (_114 : (eprop A) => B) ) ) H2) H1) ) ) var_12) var_13) .
+iff_sym : (A : Uprop -> (B : Uprop -> (_117 : (eprop ( (iff A) B) ) -> (eprop ( (iff B) A) ) ) ) ) .
+[] iff_sym --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (iff A) B) ) => ( ( ( (case_8 A) B) H) H) ) ) ) .
+case_9 : (A : Uprop -> (H : (eprop ( (iff A) False) ) -> (_128 : (eprop ( (and ( (dotpipp A) (_125 : (eprop A) => False) ) ) ( (dotpipp False) (_126 : (eprop False) => A) ) ) ) -> (_127 : (eprop A) -> (eprop False) ) ) ) ) .
+conj_case_9 : (A : Uprop -> (H : (eprop ( (iff A) False) ) -> (_136 : (_131 : (eprop A) -> (eprop False) ) -> (_135 : (_132 : (eprop False) -> (eprop A) ) -> (eprop ( (and ( (dotpipp A) (_133 : (eprop A) => False) ) ) ( (dotpipp False) (_134 : (eprop False) => A) ) ) ) ) ) ) ) .
+[A : Uprop, H : (eprop ( (iff A) False) ) ] ( (conj_case_9 A) H) --> ( (conj ( (dotpipp A) (_129 : (eprop A) => False) ) ) ( (dotpipp False) (_130 : (eprop False) => A) ) ) .
+[A : Uprop, H : (eprop ( (iff A) False) ) , var_14 : (_137 : (eprop A) -> (eprop False) ) , var_15 : (_138 : (eprop False) -> (eprop A) ) ] ( ( (case_9 A) H) ( ( ( (conj_case_9 A) H) var_14) var_15) ) --> ( ( (H欧0 : (eprop ( (dotpipp A) (_140 : (eprop A) => False) ) ) => (H0 : (eprop ( (dotpipp False) (_139 : (eprop False) => A) ) ) => H欧0) ) var_14) var_15) .
+neg_false : (A : Uprop -> (eprop ( (iff (not A) ) ( (iff A) False) ) ) ) .
+[] neg_false --> (A : (etype dotprop) => ( ( ( (conj ( (dotpipp ( (dotpipp A) (_118 : (eprop A) => False) ) ) (_119 : (eprop ( (dotpipp A) (_118 : (eprop A) => False) ) ) => ( (iff A) False) ) ) ) ( (dotpipp ( (iff A) False) ) (_121 : (eprop ( (iff A) False) ) => ( (dotpipp A) (_120 : (eprop A) => False) ) ) ) ) (H : (eprop ( (dotpipp A) (_124 : (eprop A) => False) ) ) => ( ( ( (conj ( (dotpipp A) (_122 : (eprop A) => False) ) ) ( (dotpipp False) (_123 : (eprop False) => A) ) ) H) (H1 : (eprop False) => ( (False_ind A) H1) ) ) ) ) (H : (eprop ( (iff A) False) ) => ( ( (case_9 A) H) H) ) ) ) .
+and_cancel_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_168 : (_165 : (eprop B) -> (eprop A) ) -> (_167 : (_166 : (eprop C) -> (eprop A) ) -> (eprop ( (iff ( (iff ( (and A) B) ) ( (and A) C) ) ) ( (iff B) C) ) ) ) ) ) ) ) .
+[] and_cancel_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_164 : (eprop B) => A) ) ) => (H0 : (eprop ( (dotpipp C) (_163 : (eprop C) => A) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (and A) B) ) ( (and A) C) ) ) (_141 : (eprop ( (iff ( (and A) B) ) ( (and A) C) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_142 : (eprop ( (iff B) C) ) => ( (iff ( (and A) B) ) ( (and A) C) ) ) ) ) (H1 : (eprop ( (iff ( (and A) B) ) ( (and A) C) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (and A) B) ) (_143 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) ( (dotpipp ( (and A) C) ) (_144 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (and A) B) ) (_156 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) => (H3 : (eprop ( (dotpipp ( (and A) C) ) (_155 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) => ( (H1欧0 : (eprop ( (dotpipp A) (_154 : (eprop A) => ( (dotpipp B) (_153 : (eprop B) => ( (and A) C) ) ) ) ) ) => ( (H2欧0 : (eprop ( (dotpipp A) (_152 : (eprop A) => ( (dotpipp C) (_151 : (eprop C) => ( (and A) B) ) ) ) ) ) => ( ( ( (conj ( (dotpipp B) (_145 : (eprop B) => C) ) ) ( (dotpipp C) (_146 : (eprop C) => B) ) ) (H3欧0 : (eprop B) => ( (H4 : (eprop A) => ( (H欧0 : (eprop ( (dotpipp B) (_148 : (eprop B) => ( (and A) C) ) ) ) => ( (H1欧1 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) C) (H欧1 : (eprop A) => (H5 : (eprop C) => ( (H1欧2 : (eprop ( (dotpipp C) (_147 : (eprop C) => ( (and A) B) ) ) ) => ( (H2欧1 : (eprop A) => ( (H0欧0 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) C) (H1欧3 : (eprop A) => (H6 : (eprop B) => H5) ) ) H0欧0) ) (H1欧2 H5) ) ) (H0 H5) ) ) (H2欧0 H4) ) ) ) ) H1欧1) ) (H欧0 H3欧0) ) ) (H1欧0 H4) ) ) (H H3欧0) ) ) ) (H3欧0 : (eprop C) => ( (H4 : (eprop A) => ( (H0欧0 : (eprop ( (dotpipp B) (_150 : (eprop B) => ( (and A) C) ) ) ) => ( (H1欧1 : (eprop ( (dotpipp C) (_149 : (eprop C) => ( (and A) B) ) ) ) => ( (H2欧1 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) B) (H1欧2 : (eprop A) => (H5 : (eprop B) => ( (H2欧2 : (eprop A) => ( (H欧0 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) B) (H0欧1 : (eprop A) => (H6 : (eprop C) => H5) ) ) H欧0) ) (H0欧0 H5) ) ) (H H5) ) ) ) ) H2欧1) ) (H1欧1 H3欧0) ) ) (H2欧0 H4) ) ) (H1欧0 H4) ) ) (H0 H3欧0) ) ) ) ) (H2欧0 : (eprop A) => (H4 : (eprop C) => (H3 ( ( ( (conj A) C) H2欧0) H4) ) ) ) ) ) (H1欧0 : (eprop A) => (H4 : (eprop B) => (H2 ( ( ( (conj A) B) H1欧0) H4) ) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_157 : (eprop B) => C) ) ) ( (dotpipp C) (_158 : (eprop C) => B) ) ) ( (iff ( (and A) B) ) ( (and A) C) ) ) (H2 : (eprop ( (dotpipp B) (_162 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_161 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and A) B) ) (_159 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) ( (dotpipp ( (and A) C) ) (_160 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) (H1欧0 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) ( (and A) C) ) (H4 : (eprop A) => (H5 : (eprop B) => ( (H1欧1 : (eprop A) => ( (H欧0 : (eprop C) => ( (H2欧0 : (eprop A) => ( (H0欧0 : (eprop B) => ( ( ( (conj A) C) H2欧0) H欧0) ) (H3 H欧0) ) ) (H0 H欧0) ) ) (H2 H5) ) ) (H H5) ) ) ) ) H1欧0) ) ) (H1欧0 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) ( (and A) B) ) (H4 : (eprop A) => (H5 : (eprop C) => ( (H1欧1 : (eprop A) => ( (H0欧0 : (eprop B) => ( (H3欧0 : (eprop A) => ( (H欧0 : (eprop C) => ( ( ( (conj A) B) H3欧0) H0欧0) ) (H2 H0欧0) ) ) (H H0欧0) ) ) (H3 H5) ) ) (H0 H5) ) ) ) ) H1欧0) ) ) ) ) ) H1) ) ) ) ) ) ) ) .
+and_cancel_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_196 : (_193 : (eprop B) -> (eprop A) ) -> (_195 : (_194 : (eprop C) -> (eprop A) ) -> (eprop ( (iff ( (iff ( (and B) A) ) ( (and C) A) ) ) ( (iff B) C) ) ) ) ) ) ) ) .
+[] and_cancel_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_192 : (eprop B) => A) ) ) => (H0 : (eprop ( (dotpipp C) (_191 : (eprop C) => A) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (and B) A) ) ( (and C) A) ) ) (_169 : (eprop ( (iff ( (and B) A) ) ( (and C) A) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_170 : (eprop ( (iff B) C) ) => ( (iff ( (and B) A) ) ( (and C) A) ) ) ) ) (H1 : (eprop ( (iff ( (and B) A) ) ( (and C) A) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (and B) A) ) (_171 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) ( (dotpipp ( (and C) A) ) (_172 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (and B) A) ) (_184 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) => (H3 : (eprop ( (dotpipp ( (and C) A) ) (_183 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) => ( (H1欧0 : (eprop ( (dotpipp B) (_182 : (eprop B) => ( (dotpipp A) (_181 : (eprop A) => ( (and C) A) ) ) ) ) ) => ( (H2欧0 : (eprop ( (dotpipp C) (_180 : (eprop C) => ( (dotpipp A) (_179 : (eprop A) => ( (and B) A) ) ) ) ) ) => ( ( ( (conj ( (dotpipp B) (_173 : (eprop B) => C) ) ) ( (dotpipp C) (_174 : (eprop C) => B) ) ) (H3欧0 : (eprop B) => ( (H4 : (eprop A) => ( (H欧0 : (eprop ( (dotpipp A) (_176 : (eprop A) => ( (and C) A) ) ) ) => ( (H1欧1 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) C) (H欧1 : (eprop C) => (H5 : (eprop A) => ( (H1欧2 : (eprop A) => ( (H0欧0 : (eprop ( (dotpipp A) (_175 : (eprop A) => ( (and B) A) ) ) ) => ( (H2欧1 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) C) (H0欧1 : (eprop B) => (H6 : (eprop A) => H欧1) ) ) H2欧1) ) (H0欧0 H4) ) ) (H2欧0 H欧1) ) ) (H0 H欧1) ) ) ) ) H1欧1) ) (H欧0 H4) ) ) (H1欧0 H3欧0) ) ) (H H3欧0) ) ) ) (H3欧0 : (eprop C) => ( (H4 : (eprop A) => ( (H0欧0 : (eprop ( (dotpipp A) (_178 : (eprop A) => ( (and B) A) ) ) ) => ( (H2欧1 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) B) (H0欧1 : (eprop B) => (H5 : (eprop A) => ( (H2欧2 : (eprop A) => ( (H欧0 : (eprop ( (dotpipp A) (_177 : (eprop A) => ( (and C) A) ) ) ) => ( (H1欧1 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) B) (H欧1 : (eprop C) => (H6 : (eprop A) => H0欧1) ) ) H1欧1) ) (H欧0 H4) ) ) (H1欧0 H0欧1) ) ) (H H0欧1) ) ) ) ) H2欧1) ) (H0欧0 H4) ) ) (H2欧0 H3欧0) ) ) (H0 H3欧0) ) ) ) ) (H2欧0 : (eprop C) => (H4 : (eprop A) => (H3 ( ( ( (conj C) A) H2欧0) H4) ) ) ) ) ) (H1欧0 : (eprop B) => (H4 : (eprop A) => (H2 ( ( ( (conj B) A) H1欧0) H4) ) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_185 : (eprop B) => C) ) ) ( (dotpipp C) (_186 : (eprop C) => B) ) ) ( (iff ( (and B) A) ) ( (and C) A) ) ) (H2 : (eprop ( (dotpipp B) (_190 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_189 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and B) A) ) (_187 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) ( (dotpipp ( (and C) A) ) (_188 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) (H1欧0 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) ( (and C) A) ) (H4 : (eprop B) => (H5 : (eprop A) => ( (H1欧1 : (eprop A) => ( (H欧0 : (eprop C) => ( (H2欧0 : (eprop A) => ( (H0欧0 : (eprop B) => ( ( ( (conj C) A) H欧0) H2欧0) ) (H3 H欧0) ) ) (H0 H欧0) ) ) (H2 H4) ) ) (H H4) ) ) ) ) H1欧0) ) ) (H1欧0 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) ( (and B) A) ) (H4 : (eprop C) => (H5 : (eprop A) => ( (H1欧1 : (eprop A) => ( (H0欧0 : (eprop B) => ( (H3欧0 : (eprop A) => ( (H欧0 : (eprop C) => ( ( ( (conj B) A) H0欧0) H3欧0) ) (H2 H0欧0) ) ) (H H0欧0) ) ) (H3 H4) ) ) (H0 H4) ) ) ) ) H1欧0) ) ) ) ) ) H1) ) ) ) ) ) ) ) .
+or_cancel_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_220 : (_217 : (eprop B) -> (eprop (not A) ) ) -> (_219 : (_218 : (eprop C) -> (eprop (not A) ) ) -> (eprop ( (iff ( (iff ( (or A) B) ) ( (or A) C) ) ) ( (iff B) C) ) ) ) ) ) ) ) .
+[] or_cancel_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_216 : (eprop B) => (not A) ) ) ) => (H0 : (eprop ( (dotpipp C) (_215 : (eprop C) => (not A) ) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (or A) B) ) ( (or A) C) ) ) (_197 : (eprop ( (iff ( (or A) B) ) ( (or A) C) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_198 : (eprop ( (iff B) C) ) => ( (iff ( (or A) B) ) ( (or A) C) ) ) ) ) (H1 : (eprop ( (iff ( (or A) B) ) ( (or A) C) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (or A) B) ) (_199 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) ( (dotpipp ( (or A) C) ) (_200 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (or A) B) ) (_208 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) => (H3 : (eprop ( (dotpipp ( (or A) C) ) (_207 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) => ( (H1欧0 : (eprop ( (dotpipp A) (_206 : (eprop A) => ( (or A) C) ) ) ) => ( (H4 : (eprop ( (dotpipp B) (_205 : (eprop B) => ( (or A) C) ) ) ) => ( (H2欧0 : (eprop ( (dotpipp A) (_204 : (eprop A) => ( (or A) B) ) ) ) => ( (H5 : (eprop ( (dotpipp C) (_203 : (eprop C) => ( (or A) B) ) ) ) => ( ( ( (conj ( (dotpipp B) (_201 : (eprop B) => C) ) ) ( (dotpipp C) (_202 : (eprop C) => B) ) ) (H3欧0 : (eprop B) => ( (H6 : (eprop (not A) ) => ( (H欧0 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) C) (H4欧0 : (eprop A) => ( (H欧1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) C) (H1欧1 : (eprop A) => ( (H欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H2欧1 : (eprop A) => ( (H欧3 : (eprop False) => ( (False_ind C) H欧3) ) (H6 H4欧0) ) ) ) (H2欧1 : (eprop B) => ( (H欧3 : (eprop False) => ( (False_ind C) H欧3) ) (H6 H4欧0) ) ) ) H欧2) ) (H2欧0 H4欧0) ) ) ) (H1欧1 : (eprop C) => ( (H欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H2欧1 : (eprop A) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H0欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H5欧0 : (eprop A) => ( (False_ind C) H0欧0) ) ) (H5欧0 : (eprop B) => ( (False_ind C) H0欧0) ) ) H6欧1) ) (H5 H1欧1) ) ) (H6欧0 H4欧0) ) ) (H0 H1欧1) ) ) (H6 H4欧0) ) ) ) (H2欧1 : (eprop B) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H0欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H5欧0 : (eprop A) => ( (False_ind C) H0欧0) ) ) (H5欧0 : (eprop B) => ( (False_ind C) H0欧0) ) ) H6欧1) ) (H5 H1欧1) ) ) (H6欧0 H4欧0) ) ) (H0 H1欧1) ) ) (H6 H4欧0) ) ) ) H欧2) ) (H2欧0 H4欧0) ) ) ) H欧1) ) (H1欧0 H4欧0) ) ) ) (H4欧0 : (eprop C) => ( (H欧1 : (eprop (not A) ) => ( (H0欧0 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H5欧0 : (eprop A) => ( (H0欧1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) C) (H1欧1 : (eprop A) => ( (H0欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H2欧1 : (eprop A) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H5欧0) ) ) (H6 H5欧0) ) ) ) (H2欧1 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H5欧0) ) ) (H6 H5欧0) ) ) ) H0欧2) ) (H2欧0 H5欧0) ) ) ) (H1欧1 : (eprop C) => ( (H0欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) C) (H2欧1 : (eprop A) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H5欧0) ) ) (H6 H5欧0) ) ) ) (H2欧1 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H5欧0) ) ) (H6 H5欧0) ) ) ) H0欧2) ) (H2欧0 H5欧0) ) ) ) H0欧1) ) (H1欧0 H5欧0) ) ) ) (H5欧0 : (eprop B) => H4欧0) ) H0欧0) ) (H5 H4欧0) ) ) (H0 H4欧0) ) ) ) H欧0) ) (H4 H3欧0) ) ) (H H3欧0) ) ) ) (H3欧0 : (eprop C) => ( (H6 : (eprop (not A) ) => ( (H0欧0 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H5欧0 : (eprop A) => ( (H0欧1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H1欧1 : (eprop A) => ( (H0欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H2欧1 : (eprop A) => ( (H0欧3 : (eprop False) => ( (False_ind B) H0欧3) ) (H6 H5欧0) ) ) ) (H2欧1 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H4欧0 : (eprop A) => ( (False_ind B) H欧0) ) ) (H4欧0 : (eprop C) => ( (False_ind B) H欧0) ) ) H6欧1) ) (H4 H2欧1) ) ) (H6欧0 H5欧0) ) ) (H H2欧1) ) ) (H6 H5欧0) ) ) ) H0欧2) ) (H2欧0 H5欧0) ) ) ) (H1欧1 : (eprop C) => ( (H0欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H2欧1 : (eprop A) => ( (H0欧3 : (eprop False) => ( (False_ind B) H0欧3) ) (H6 H5欧0) ) ) ) (H2欧1 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H4欧0 : (eprop A) => ( (False_ind B) H欧0) ) ) (H4欧0 : (eprop C) => ( (False_ind B) H欧0) ) ) H6欧1) ) (H4 H2欧1) ) ) (H6欧0 H5欧0) ) ) (H H2欧1) ) ) (H6 H5欧0) ) ) ) H0欧2) ) (H2欧0 H5欧0) ) ) ) H0欧1) ) (H1欧0 H5欧0) ) ) ) (H5欧0 : (eprop B) => ( (H0欧1 : (eprop (not A) ) => ( (H欧0 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H4欧0 : (eprop A) => ( (H欧1 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) B) (H1欧1 : (eprop A) => ( (H欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H2欧1 : (eprop A) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H4欧0) ) ) (H6 H4欧0) ) ) ) (H2欧1 : (eprop B) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H4欧0) ) ) (H6 H4欧0) ) ) ) H欧2) ) (H2欧0 H4欧0) ) ) ) (H1欧1 : (eprop C) => ( (H欧2 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) B) (H2欧1 : (eprop A) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H4欧0) ) ) (H6 H4欧0) ) ) ) (H2欧1 : (eprop B) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H4欧0) ) ) (H6 H4欧0) ) ) ) H欧2) ) (H2欧0 H4欧0) ) ) ) H欧1) ) (H1欧0 H4欧0) ) ) ) (H4欧0 : (eprop C) => H5欧0) ) H欧0) ) (H4 H5欧0) ) ) (H H5欧0) ) ) ) H0欧0) ) (H5 H3欧0) ) ) (H0 H3欧0) ) ) ) ) (H5 : (eprop C) => (H3 ( ( (or_intror A) C) H5) ) ) ) ) (H2欧0 : (eprop A) => (H3 ( ( (or_introl A) C) H2欧0) ) ) ) ) (H4 : (eprop B) => (H2 ( ( (or_intror A) B) H4) ) ) ) ) (H1欧0 : (eprop A) => (H2 ( ( (or_introl A) B) H1欧0) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_209 : (eprop B) => C) ) ) ( (dotpipp C) (_210 : (eprop C) => B) ) ) ( (iff ( (or A) B) ) ( (or A) C) ) ) (H2 : (eprop ( (dotpipp B) (_214 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_213 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or A) B) ) (_211 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) ( (dotpipp ( (or A) C) ) (_212 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) (H1欧0 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) ( (or A) C) ) (H4 : (eprop A) => ( ( (or_introl A) C) H4) ) ) (H4 : (eprop B) => ( (H1欧1 : (eprop (not A) ) => ( (H欧0 : (eprop C) => ( (H2欧0 : (eprop (not A) ) => ( (H0欧0 : (eprop B) => ( ( (or_intror A) C) H欧0) ) (H3 H欧0) ) ) (H0 H欧0) ) ) (H2 H4) ) ) (H H4) ) ) ) H1欧0) ) ) (H1欧0 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) ( (or A) B) ) (H4 : (eprop A) => ( ( (or_introl A) B) H4) ) ) (H4 : (eprop C) => ( (H1欧1 : (eprop (not A) ) => ( (H0欧0 : (eprop B) => ( (H3欧0 : (eprop (not A) ) => ( (H欧0 : (eprop C) => ( ( (or_intror A) B) H0欧0) ) (H2 H0欧0) ) ) (H H0欧0) ) ) (H3 H4) ) ) (H0 H4) ) ) ) H1欧0) ) ) ) ) ) H1) ) ) ) ) ) ) ) .
+or_cancel_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_244 : (_241 : (eprop B) -> (eprop (not A) ) ) -> (_243 : (_242 : (eprop C) -> (eprop (not A) ) ) -> (eprop ( (iff ( (iff ( (or B) A) ) ( (or C) A) ) ) ( (iff B) C) ) ) ) ) ) ) ) .
+[] or_cancel_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (dotpipp B) (_240 : (eprop B) => (not A) ) ) ) => (H0 : (eprop ( (dotpipp C) (_239 : (eprop C) => (not A) ) ) ) => ( ( ( (conj ( (dotpipp ( (iff ( (or B) A) ) ( (or C) A) ) ) (_221 : (eprop ( (iff ( (or B) A) ) ( (or C) A) ) ) => ( (iff B) C) ) ) ) ( (dotpipp ( (iff B) C) ) (_222 : (eprop ( (iff B) C) ) => ( (iff ( (or B) A) ) ( (or C) A) ) ) ) ) (H1 : (eprop ( (iff ( (or B) A) ) ( (or C) A) ) ) => ( ( ( ( (and_ind ( (dotpipp ( (or B) A) ) (_223 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) ( (dotpipp ( (or C) A) ) (_224 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) ( (iff B) C) ) (H2 : (eprop ( (dotpipp ( (or B) A) ) (_232 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) => (H3 : (eprop ( (dotpipp ( (or C) A) ) (_231 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) => ( (H1欧0 : (eprop ( (dotpipp B) (_230 : (eprop B) => ( (or C) A) ) ) ) => ( (H4 : (eprop ( (dotpipp A) (_229 : (eprop A) => ( (or C) A) ) ) ) => ( (H2欧0 : (eprop ( (dotpipp C) (_228 : (eprop C) => ( (or B) A) ) ) ) => ( (H5 : (eprop ( (dotpipp A) (_227 : (eprop A) => ( (or B) A) ) ) ) => ( ( ( (conj ( (dotpipp B) (_225 : (eprop B) => C) ) ) ( (dotpipp C) (_226 : (eprop C) => B) ) ) (H3欧0 : (eprop B) => ( (H6 : (eprop (not A) ) => ( (H欧0 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) C) (H1欧1 : (eprop C) => ( (H欧1 : (eprop (not A) ) => ( (H0欧0 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H2欧1 : (eprop B) => H1欧1) ) (H2欧1 : (eprop A) => ( (H0欧1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) C) (H4欧0 : (eprop C) => ( (H0欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H5欧0 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H2欧1) ) ) (H6 H2欧1) ) ) ) (H5欧0 : (eprop A) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H2欧1) ) ) (H6 H2欧1) ) ) ) H0欧2) ) (H5 H2欧1) ) ) ) (H4欧0 : (eprop A) => ( (H0欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H5欧0 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H2欧1) ) ) (H6 H2欧1) ) ) ) (H5欧0 : (eprop A) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind C) H6欧0) ) (H欧1 H2欧1) ) ) (H6 H2欧1) ) ) ) H0欧2) ) (H5 H2欧1) ) ) ) H0欧1) ) (H4 H2欧1) ) ) ) H0欧0) ) (H2欧0 H1欧1) ) ) (H0 H1欧1) ) ) ) (H1欧1 : (eprop A) => ( (H欧1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) C) (H4欧0 : (eprop C) => ( (H欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H5欧0 : (eprop B) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H0欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H2欧1 : (eprop B) => ( (False_ind C) H0欧0) ) ) (H2欧1 : (eprop A) => ( (False_ind C) H0欧0) ) ) H6欧1) ) (H2欧0 H4欧0) ) ) (H6欧0 H1欧1) ) ) (H0 H4欧0) ) ) (H6 H1欧1) ) ) ) (H5欧0 : (eprop A) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H0欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H2欧1 : (eprop B) => ( (False_ind C) H0欧0) ) ) (H2欧1 : (eprop A) => ( (False_ind C) H0欧0) ) ) H6欧1) ) (H2欧0 H4欧0) ) ) (H6欧0 H1欧1) ) ) (H0 H4欧0) ) ) (H6 H1欧1) ) ) ) H欧2) ) (H5 H1欧1) ) ) ) (H4欧0 : (eprop A) => ( (H欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) C) (H5欧0 : (eprop B) => ( (H欧3 : (eprop False) => ( (False_ind C) H欧3) ) (H6 H1欧1) ) ) ) (H5欧0 : (eprop A) => ( (H欧3 : (eprop False) => ( (False_ind C) H欧3) ) (H6 H1欧1) ) ) ) H欧2) ) (H5 H1欧1) ) ) ) H欧1) ) (H4 H1欧1) ) ) ) H欧0) ) (H1欧0 H3欧0) ) ) (H H3欧0) ) ) ) (H3欧0 : (eprop C) => ( (H6 : (eprop (not A) ) => ( (H0欧0 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H2欧1 : (eprop B) => ( (H0欧1 : (eprop (not A) ) => ( (H欧0 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H1欧1 : (eprop C) => H2欧1) ) (H1欧1 : (eprop A) => ( (H欧1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H4欧0 : (eprop C) => ( (H欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H5欧0 : (eprop B) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H1欧1) ) ) (H6 H1欧1) ) ) ) (H5欧0 : (eprop A) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H1欧1) ) ) (H6 H1欧1) ) ) ) H欧2) ) (H5 H1欧1) ) ) ) (H4欧0 : (eprop A) => ( (H欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H5欧0 : (eprop B) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H1欧1) ) ) (H6 H1欧1) ) ) ) (H5欧0 : (eprop A) => ( (H欧3 : (eprop False) => ( (H6欧0 : (eprop False) => ( (False_ind B) H6欧0) ) (H0欧1 H1欧1) ) ) (H6 H1欧1) ) ) ) H欧2) ) (H5 H1欧1) ) ) ) H欧1) ) (H4 H1欧1) ) ) ) H欧0) ) (H1欧0 H2欧1) ) ) (H H2欧1) ) ) ) (H2欧1 : (eprop A) => ( (H0欧1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H4欧0 : (eprop C) => ( (H0欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H5欧0 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H1欧1 : (eprop C) => ( (False_ind B) H欧0) ) ) (H1欧1 : (eprop A) => ( (False_ind B) H欧0) ) ) H6欧1) ) (H1欧0 H5欧0) ) ) (H6欧0 H2欧1) ) ) (H H5欧0) ) ) (H6 H2欧1) ) ) ) (H5欧0 : (eprop A) => ( (H0欧3 : (eprop False) => ( (False_ind B) H0欧3) ) (H6 H2欧1) ) ) ) H0欧2) ) (H5 H2欧1) ) ) ) (H4欧0 : (eprop A) => ( (H0欧2 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) B) (H5欧0 : (eprop B) => ( (H0欧3 : (eprop False) => ( (H6欧0 : (eprop (not A) ) => ( (H欧0 : (eprop False) => ( (H6欧1 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) B) (H1欧1 : (eprop C) => ( (False_ind B) H欧0) ) ) (H1欧1 : (eprop A) => ( (False_ind B) H欧0) ) ) H6欧1) ) (H1欧0 H5欧0) ) ) (H6欧0 H2欧1) ) ) (H H5欧0) ) ) (H6 H2欧1) ) ) ) (H5欧0 : (eprop A) => ( (H0欧3 : (eprop False) => ( (False_ind B) H0欧3) ) (H6 H2欧1) ) ) ) H0欧2) ) (H5 H2欧1) ) ) ) H0欧1) ) (H4 H2欧1) ) ) ) H0欧0) ) (H2欧0 H3欧0) ) ) (H0 H3欧0) ) ) ) ) (H5 : (eprop A) => (H3 ( ( (or_intror C) A) H5) ) ) ) ) (H2欧0 : (eprop C) => (H3 ( ( (or_introl C) A) H2欧0) ) ) ) ) (H4 : (eprop A) => (H2 ( ( (or_intror B) A) H4) ) ) ) ) (H1欧0 : (eprop B) => (H2 ( ( (or_introl B) A) H1欧0) ) ) ) ) ) ) H1) ) ) (H1 : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_233 : (eprop B) => C) ) ) ( (dotpipp C) (_234 : (eprop C) => B) ) ) ( (iff ( (or B) A) ) ( (or C) A) ) ) (H2 : (eprop ( (dotpipp B) (_238 : (eprop B) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_237 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or B) A) ) (_235 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) ( (dotpipp ( (or C) A) ) (_236 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) (H1欧0 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) ( (or C) A) ) (H4 : (eprop B) => ( (H1欧1 : (eprop (not A) ) => ( (H欧0 : (eprop C) => ( (H2欧0 : (eprop (not A) ) => ( (H0欧0 : (eprop B) => ( ( (or_introl C) A) H欧0) ) (H3 H欧0) ) ) (H0 H欧0) ) ) (H2 H4) ) ) (H H4) ) ) ) (H4 : (eprop A) => ( ( (or_intror C) A) H4) ) ) H1欧0) ) ) (H1欧0 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) ( (or B) A) ) (H4 : (eprop C) => ( (H1欧1 : (eprop (not A) ) => ( (H0欧0 : (eprop B) => ( (H3欧0 : (eprop (not A) ) => ( (H欧0 : (eprop C) => ( ( (or_introl B) A) H0欧0) ) (H2 H0欧0) ) ) (H H0欧0) ) ) (H3 H4) ) ) (H0 H4) ) ) ) (H4 : (eprop A) => ( ( (or_intror B) A) H4) ) ) H1欧0) ) ) ) ) ) H1) ) ) ) ) ) ) ) .
+and_iff_compat_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_251 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (and A) B) ) ( (and A) C) ) ) ) ) ) ) .
+[] and_iff_compat_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_245 : (eprop B) => C) ) ) ( (dotpipp C) (_246 : (eprop C) => B) ) ) ( (iff ( (and A) B) ) ( (and A) C) ) ) (H0 : (eprop ( (dotpipp B) (_250 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_249 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and A) B) ) (_247 : (eprop ( (and A) B) ) => ( (and A) C) ) ) ) ( (dotpipp ( (and A) C) ) (_248 : (eprop ( (and A) C) ) => ( (and A) B) ) ) ) (H欧0 : (eprop ( (and A) B) ) => ( ( ( ( (and_ind A) B) ( (and A) C) ) (H2 : (eprop A) => (H3 : (eprop B) => ( (H欧1 : (eprop C) => ( (H0欧0 : (eprop B) => ( ( ( (conj A) C) H2) H欧1) ) (H1 H欧1) ) ) (H0 H3) ) ) ) ) H欧0) ) ) (H欧0 : (eprop ( (and A) C) ) => ( ( ( ( (and_ind A) C) ( (and A) B) ) (H2 : (eprop A) => (H3 : (eprop C) => ( (H欧1 : (eprop B) => ( (H1欧0 : (eprop C) => ( ( ( (conj A) B) H2) H欧1) ) (H0 H欧1) ) ) (H1 H3) ) ) ) ) H欧0) ) ) ) ) ) H) ) ) ) ) .
+and_iff_compat_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_258 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (and B) A) ) ( (and C) A) ) ) ) ) ) ) .
+[] and_iff_compat_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_252 : (eprop B) => C) ) ) ( (dotpipp C) (_253 : (eprop C) => B) ) ) ( (iff ( (and B) A) ) ( (and C) A) ) ) (H0 : (eprop ( (dotpipp B) (_257 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_256 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (and B) A) ) (_254 : (eprop ( (and B) A) ) => ( (and C) A) ) ) ) ( (dotpipp ( (and C) A) ) (_255 : (eprop ( (and C) A) ) => ( (and B) A) ) ) ) (H欧0 : (eprop ( (and B) A) ) => ( ( ( ( (and_ind B) A) ( (and C) A) ) (H2 : (eprop B) => (H3 : (eprop A) => ( (H欧1 : (eprop C) => ( (H0欧0 : (eprop B) => ( ( ( (conj C) A) H欧1) H3) ) (H1 H欧1) ) ) (H0 H2) ) ) ) ) H欧0) ) ) (H欧0 : (eprop ( (and C) A) ) => ( ( ( ( (and_ind C) A) ( (and B) A) ) (H2 : (eprop C) => (H3 : (eprop A) => ( (H欧1 : (eprop B) => ( (H1欧0 : (eprop C) => ( ( ( (conj B) A) H欧1) H3) ) (H0 H欧1) ) ) (H1 H2) ) ) ) ) H欧0) ) ) ) ) ) H) ) ) ) ) .
+or_iff_compat_l : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_265 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (or A) B) ) ( (or A) C) ) ) ) ) ) ) .
+[] or_iff_compat_l --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_259 : (eprop B) => C) ) ) ( (dotpipp C) (_260 : (eprop C) => B) ) ) ( (iff ( (or A) B) ) ( (or A) C) ) ) (H0 : (eprop ( (dotpipp B) (_264 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_263 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or A) B) ) (_261 : (eprop ( (or A) B) ) => ( (or A) C) ) ) ) ( (dotpipp ( (or A) C) ) (_262 : (eprop ( (or A) C) ) => ( (or A) B) ) ) ) (H欧0 : (eprop ( (or A) B) ) => ( ( ( ( ( (or_ind A) B) ( (or A) C) ) (H2 : (eprop A) => ( ( (or_introl A) C) H2) ) ) (H2 : (eprop B) => ( (H欧1 : (eprop C) => ( (H0欧0 : (eprop B) => ( ( (or_intror A) C) H欧1) ) (H1 H欧1) ) ) (H0 H2) ) ) ) H欧0) ) ) (H欧0 : (eprop ( (or A) C) ) => ( ( ( ( ( (or_ind A) C) ( (or A) B) ) (H2 : (eprop A) => ( ( (or_introl A) B) H2) ) ) (H2 : (eprop C) => ( (H欧1 : (eprop B) => ( (H1欧0 : (eprop C) => ( ( (or_intror A) B) H欧1) ) (H0 H欧1) ) ) (H1 H2) ) ) ) H欧0) ) ) ) ) ) H) ) ) ) ) .
+or_iff_compat_r : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_272 : (eprop ( (iff B) C) ) -> (eprop ( (iff ( (or B) A) ) ( (or C) A) ) ) ) ) ) ) .
+[] or_iff_compat_r --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff B) C) ) => ( ( ( ( (and_ind ( (dotpipp B) (_266 : (eprop B) => C) ) ) ( (dotpipp C) (_267 : (eprop C) => B) ) ) ( (iff ( (or B) A) ) ( (or C) A) ) ) (H0 : (eprop ( (dotpipp B) (_271 : (eprop B) => C) ) ) => (H1 : (eprop ( (dotpipp C) (_270 : (eprop C) => B) ) ) => ( ( ( (conj ( (dotpipp ( (or B) A) ) (_268 : (eprop ( (or B) A) ) => ( (or C) A) ) ) ) ( (dotpipp ( (or C) A) ) (_269 : (eprop ( (or C) A) ) => ( (or B) A) ) ) ) (H欧0 : (eprop ( (or B) A) ) => ( ( ( ( ( (or_ind B) A) ( (or C) A) ) (H2 : (eprop B) => ( (H欧1 : (eprop C) => ( (H0欧0 : (eprop B) => ( ( (or_introl C) A) H欧1) ) (H1 H欧1) ) ) (H0 H2) ) ) ) (H2 : (eprop A) => ( ( (or_intror C) A) H2) ) ) H欧0) ) ) (H欧0 : (eprop ( (or C) A) ) => ( ( ( ( ( (or_ind C) A) ( (or B) A) ) (H2 : (eprop C) => ( (H欧1 : (eprop B) => ( (H1欧0 : (eprop C) => ( ( (or_introl B) A) H欧1) ) (H0 H欧1) ) ) (H1 H2) ) ) ) (H2 : (eprop A) => ( ( (or_intror B) A) H2) ) ) H欧0) ) ) ) ) ) H) ) ) ) ) .
+case_10 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (iff A) B) ) -> (_277 : (eprop ( (and ( (dotpipp A) (_273 : (eprop A) => B) ) ) ( (dotpipp B) (_274 : (eprop B) => A) ) ) ) -> (eprop ( (and ( (dotpipp A) (_275 : (eprop A) => B) ) ) ( (dotpipp B) (_276 : (eprop B) => A) ) ) ) ) ) ) ) .
+conj_case_10 : (A : Uprop -> (B : Uprop -> (H : (eprop ( (iff A) B) ) -> (_285 : (_280 : (eprop A) -> (eprop B) ) -> (_284 : (_281 : (eprop B) -> (eprop A) ) -> (eprop ( (and ( (dotpipp A) (_282 : (eprop A) => B) ) ) ( (dotpipp B) (_283 : (eprop B) => A) ) ) ) ) ) ) ) ) .
+[A : Uprop, B : Uprop, H : (eprop ( (iff A) B) ) ] ( ( (conj_case_10 A) B) H) --> ( (conj ( (dotpipp A) (_278 : (eprop A) => B) ) ) ( (dotpipp B) (_279 : (eprop B) => A) ) ) .
+[A : Uprop, B : Uprop, H : (eprop ( (iff A) B) ) , var_16 : (_286 : (eprop A) -> (eprop B) ) , var_17 : (_287 : (eprop B) -> (eprop A) ) ] ( ( ( (case_10 A) B) H) ( ( ( ( (conj_case_10 A) B) H) var_16) var_17) ) --> ( ( (H欧0 : (eprop ( (dotpipp A) (_291 : (eprop A) => B) ) ) => (H0 : (eprop ( (dotpipp B) (_290 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp A) (_288 : (eprop A) => B) ) ) ( (dotpipp B) (_289 : (eprop B) => A) ) ) H欧0) H0) ) ) var_16) var_17) .
+iff_and : (A : Uprop -> (B : Uprop -> (_294 : (eprop ( (iff A) B) ) -> (eprop ( (and ( (dotpipp A) (_292 : (eprop A) => B) ) ) ( (dotpipp B) (_293 : (eprop B) => A) ) ) ) ) ) ) .
+[] iff_and --> (A : (etype dotprop) => (B : (etype dotprop) => (H : (eprop ( (iff A) B) ) => ( ( ( (case_10 A) B) H) H) ) ) ) .
+iff_to_and : (A : Uprop -> (B : Uprop -> (eprop ( (iff ( (iff A) B) ) ( (and ( (dotpipp A) (_317 : (eprop A) => B) ) ) ( (dotpipp B) (_318 : (eprop B) => A) ) ) ) ) ) ) .
+[] iff_to_and --> (A : (etype dotprop) => (B : (etype dotprop) => ( ( ( (conj ( (dotpipp ( (iff A) B) ) (_297 : (eprop ( (iff A) B) ) => ( (and ( (dotpipp A) (_295 : (eprop A) => B) ) ) ( (dotpipp B) (_296 : (eprop B) => A) ) ) ) ) ) ( (dotpipp ( (and ( (dotpipp A) (_298 : (eprop A) => B) ) ) ( (dotpipp B) (_299 : (eprop B) => A) ) ) ) (_300 : (eprop ( (and ( (dotpipp A) (_298 : (eprop A) => B) ) ) ( (dotpipp B) (_299 : (eprop B) => A) ) ) ) => ( (iff A) B) ) ) ) (H : (eprop ( (iff A) B) ) => ( ( ( ( (and_ind ( (dotpipp A) (_301 : (eprop A) => B) ) ) ( (dotpipp B) (_302 : (eprop B) => A) ) ) ( (and ( (dotpipp A) (_303 : (eprop A) => B) ) ) ( (dotpipp B) (_304 : (eprop B) => A) ) ) ) (H0 : (eprop ( (dotpipp A) (_308 : (eprop A) => B) ) ) => (H1 : (eprop ( (dotpipp B) (_307 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp A) (_305 : (eprop A) => B) ) ) ( (dotpipp B) (_306 : (eprop B) => A) ) ) (H欧0 : (eprop A) => ( (H2 : (eprop B) => ( (H0欧0 : (eprop A) => H2) (H1 H2) ) ) (H0 H欧0) ) ) ) (H欧0 : (eprop B) => ( (H2 : (eprop A) => ( (H1欧0 : (eprop B) => H2) (H0 H2) ) ) (H1 H欧0) ) ) ) ) ) ) H) ) ) (H : (eprop ( (and ( (dotpipp A) (_315 : (eprop A) => B) ) ) ( (dotpipp B) (_316 : (eprop B) => A) ) ) ) => ( ( ( ( (and_ind ( (dotpipp A) (_309 : (eprop A) => B) ) ) ( (dotpipp B) (_310 : (eprop B) => A) ) ) ( (iff A) B) ) (H0 : (eprop ( (dotpipp A) (_314 : (eprop A) => B) ) ) => (H1 : (eprop ( (dotpipp B) (_313 : (eprop B) => A) ) ) => ( ( ( (conj ( (dotpipp A) (_311 : (eprop A) => B) ) ) ( (dotpipp B) (_312 : (eprop B) => A) ) ) (H欧0 : (eprop A) => ( (H2 : (eprop B) => ( (H0欧0 : (eprop A) => H2) (H1 H2) ) ) (H0 H欧0) ) ) ) (H欧0 : (eprop B) => ( (H2 : (eprop A) => ( (H1欧0 : (eprop B) => H2) (H0 H2) ) ) (H1 H欧0) ) ) ) ) ) ) H) ) ) ) ) .
+IF_then_else : (P : Uprop -> (Q : Uprop -> (R : Uprop -> Uprop) ) ) .
+[] IF_then_else --> (P : (etype dotprop) => (Q : (etype dotprop) => (R : (etype dotprop) => ( (or ( (and P) Q) ) ( (and (not P) ) R) ) ) ) ) .
+ex : (A : Utype -> (P : (_319 : (etype A) -> Uprop) -> Uprop) ) .
+ex_intro : (A : Utype -> (P : (_320 : (etype A) -> Uprop) -> (x : (etype A) -> (_321 : (eprop (P x) ) -> (eprop ( (ex A) P) ) ) ) ) ) .
+case_11 : (A : Utype -> (P : (_322 : (etype A) -> Uprop) -> (P欧0 : Uprop -> (f : (x : (etype A) -> (_323 : (eprop (P x) ) -> (eprop P欧0) ) ) -> (e : (eprop ( (ex A) P) ) -> (_324 : (eprop ( (ex A) P) ) -> (eprop P欧0) ) ) ) ) ) ) .
+ex_intro_case_11 : (A : Utype -> (P : (_327 : (etype A) -> Uprop) -> (P欧0 : Uprop -> (f : (x : (etype A) -> (_328 : (eprop (P x) ) -> (eprop P欧0) ) ) -> (e : (eprop ( (ex A) P) ) -> (x : (etype A) -> (_329 : (eprop (P x) ) -> (eprop ( (ex A) P) ) ) ) ) ) ) ) ) .
+[A : Utype, P : (_330 : (etype A) -> Uprop) , P欧0 : Uprop, f : (x : (etype A) -> (_331 : (eprop (P x) ) -> (eprop P欧0) ) ) , e : (eprop ( (ex A) P) ) ] ( ( ( ( (ex_intro_case_11 A) P) P欧0) f) e) --> ( (ex_intro A) P) .
+[A : Utype, P : (_325 : (etype A) -> Uprop) , P欧0 : Uprop, f : (x : (etype A) -> (_326 : (eprop (P x) ) -> (eprop P欧0) ) ) , e : (eprop ( (ex A) P) ) , var_18 : (etype A) , var_19 : (eprop (P var_18) ) ] ( ( ( ( ( (case_11 A) P) P欧0) f) e) ( ( ( ( ( ( (ex_intro_case_11 A) P) P欧0) f) e) var_18) var_19) ) --> ( (f var_18) var_19) .
+ex_ind : (A : Utype -> (P : (_334 : (etype A) -> Uprop) -> (P欧0 : Uprop -> (f : (x : (etype A) -> (_335 : (eprop (P x) ) -> (eprop P欧0) ) ) -> (e : (eprop ( (ex A) P) ) -> (eprop P欧0) ) ) ) ) ) .
+[] ex_ind --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_333 : (etype A) => dotprop) ) ) => (P欧0 : (etype dotprop) => (f : (eprop ( (dotpitp A) (x : (etype A) => ( (dotpipp (P x) ) (_332 : (eprop (P x) ) => P欧0) ) ) ) ) => (e : (eprop ( (ex A) P) ) => ( ( ( ( ( (case_11 A) P) P欧0) f) e) e) ) ) ) ) ) .
+ex2 : (A : Utype -> (P : (_336 : (etype A) -> Uprop) -> (Q : (_337 : (etype A) -> Uprop) -> Uprop) ) ) .
+ex_intro2 : (A : Utype -> (P : (_338 : (etype A) -> Uprop) -> (Q : (_339 : (etype A) -> Uprop) -> (x : (etype A) -> (_341 : (eprop (P x) ) -> (_340 : (eprop (Q x) ) -> (eprop ( ( (ex2 A) P) Q) ) ) ) ) ) ) ) .
+case_12 : (A : Utype -> (P : (_342 : (etype A) -> Uprop) -> (Q : (_343 : (etype A) -> Uprop) -> (P欧0 : Uprop -> (f : (x : (etype A) -> (_345 : (eprop (P x) ) -> (_344 : (eprop (Q x) ) -> (eprop P欧0) ) ) ) -> (e : (eprop ( ( (ex2 A) P) Q) ) -> (_346 : (eprop ( ( (ex2 A) P) Q) ) -> (eprop P欧0) ) ) ) ) ) ) ) .
+ex_intro2_case_12 : (A : Utype -> (P : (_351 : (etype A) -> Uprop) -> (Q : (_352 : (etype A) -> Uprop) -> (P欧0 : Uprop -> (f : (x : (etype A) -> (_354 : (eprop (P x) ) -> (_353 : (eprop (Q x) ) -> (eprop P欧0) ) ) ) -> (e : (eprop ( ( (ex2 A) P) Q) ) -> (x : (etype A) -> (_356 : (eprop (P x) ) -> (_355 : (eprop (Q x) ) -> (eprop ( ( (ex2 A) P) Q) ) ) ) ) ) ) ) ) ) ) .
+[A : Utype, P : (_357 : (etype A) -> Uprop) , Q : (_358 : (etype A) -> Uprop) , P欧0 : Uprop, f : (x : (etype A) -> (_360 : (eprop (P x) ) -> (_359 : (eprop (Q x) ) -> (eprop P欧0) ) ) ) , e : (eprop ( ( (ex2 A) P) Q) ) ] ( ( ( ( ( (ex_intro2_case_12 A) P) Q) P欧0) f) e) --> ( ( (ex_intro2 A) P) Q) .
+[A : Utype, P : (_347 : (etype A) -> Uprop) , Q : (_348 : (etype A) -> Uprop) , P欧0 : Uprop, f : (x : (etype A) -> (_350 : (eprop (P x) ) -> (_349 : (eprop (Q x) ) -> (eprop P欧0) ) ) ) , e : (eprop ( ( (ex2 A) P) Q) ) , var_20 : (etype A) , var_21 : (eprop (P var_20) ) , var_22 : (eprop (Q var_20) ) ] ( ( ( ( ( ( (case_12 A) P) Q) P欧0) f) e) ( ( ( ( ( ( ( ( (ex_intro2_case_12 A) P) Q) P欧0) f) e) var_20) var_21) var_22) ) --> ( ( (f var_20) var_21) var_22) .
+ex2_ind : (A : Utype -> (P : (_365 : (etype A) -> Uprop) -> (Q : (_366 : (etype A) -> Uprop) -> (P欧0 : Uprop -> (f : (x : (etype A) -> (_368 : (eprop (P x) ) -> (_367 : (eprop (Q x) ) -> (eprop P欧0) ) ) ) -> (e : (eprop ( ( (ex2 A) P) Q) ) -> (eprop P欧0) ) ) ) ) ) ) .
+[] ex2_ind --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_364 : (etype A) => dotprop) ) ) => (Q : (etype ( (dotpitt A) (_363 : (etype A) => dotprop) ) ) => (P欧0 : (etype dotprop) => (f : (eprop ( (dotpitp A) (x : (etype A) => ( (dotpipp (P x) ) (_362 : (eprop (P x) ) => ( (dotpipp (Q x) ) (_361 : (eprop (Q x) ) => P欧0) ) ) ) ) ) ) => (e : (eprop ( ( (ex2 A) P) Q) ) => ( ( ( ( ( ( (case_12 A) P) Q) P欧0) f) e) e) ) ) ) ) ) ) .
+all : (A : Utype -> (P : (_370 : (etype A) -> Uprop) -> Uprop) ) .
+[] all --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_369 : (etype A) => dotprop) ) ) => ( (dotpitp A) (x : (etype A) => (P x) ) ) ) ) .
+inst : (A : Utype -> (P : (_372 : (etype A) -> Uprop) -> (x : (etype A) -> (_373 : (eprop ( (all A) (x欧0 : (etype A) => (P x欧0) ) ) ) -> (eprop (P x) ) ) ) ) ) .
+[] inst --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_371 : (etype A) => dotprop) ) ) => (x : (etype A) => (H : (eprop ( (dotpitp A) (x欧0 : (etype A) => (P x欧0) ) ) ) => (H x) ) ) ) ) .
+gen : (A : Utype -> (P : (_376 : (etype A) -> Uprop) -> (B : Uprop -> (f : (y : (etype A) -> (_377 : (eprop B) -> (eprop (P y) ) ) ) -> (_378 : (eprop B) -> (eprop ( (all A) P) ) ) ) ) ) ) .
+[] gen --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_375 : (etype A) => dotprop) ) ) => (B : (etype dotprop) => (f : (eprop ( (dotpitp A) (y : (etype A) => ( (dotpipp B) (_374 : (eprop B) => (P y) ) ) ) ) ) => (H : (eprop B) => (x : (etype A) => ( (f x) H) ) ) ) ) ) ) .
+eq : (A : Utype -> (x : (etype A) -> (_379 : (etype A) -> Uprop) ) ) .
+refl_equal : (A : Utype -> (x : (etype A) -> (eprop ( ( (eq A) x) x) ) ) ) .
+case_13 : (A : Utype -> (x : (etype A) -> (P : (_380 : (etype A) -> Utype) -> (f : (etype (P x) ) -> (y : (etype A) -> (e : (eprop ( ( (eq A) x) y) ) -> (y欧0 : (etype A) -> (_381 : (eprop ( ( (eq A) x) y欧0) ) -> (etype (P y欧0) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_13 : (A : Utype -> (x : (etype A) -> (P : (_383 : (etype A) -> Utype) -> (f : (etype (P x) ) -> (y : (etype A) -> (e : (eprop ( ( (eq A) x) y) ) -> (eprop ( ( (eq A) x) x) ) ) ) ) ) ) ) .
+[A : Utype, x : (etype A) , P : (_384 : (etype A) -> Utype) , f : (etype (P x) ) , y : (etype A) , e : (eprop ( ( (eq A) x) y) ) ] ( ( ( ( ( (refl_equal_case_13 A) x) P) f) y) e) --> ( (refl_equal A) x) .
+[A : Utype, x : (etype A) , P : (_382 : (etype A) -> Utype) , f : (etype (P x) ) , y : (etype A) , e : (eprop ( ( (eq A) x) y) ) ] ( ( ( ( ( ( ( (case_13 A) x) P) f) y) e) x) ( ( ( ( ( (refl_equal_case_13 A) x) P) f) y) e) ) --> f.
+eq_rect : (A : Utype -> (x : (etype A) -> (P : (_386 : (etype A) -> Utype) -> (f : (etype (P x) ) -> (y : (etype A) -> (e : (eprop ( ( (eq A) x) y) ) -> (etype (P y) ) ) ) ) ) ) ) .
+[] eq_rect --> (A : (etype dottype) => (x : (etype A) => (P : (etype ( (dotpitt A) (_385 : (etype A) => dottype) ) ) => (f : (etype (P x) ) => (y : (etype A) => (e : (eprop ( ( (eq A) x) y) ) => ( ( ( ( ( ( ( (case_13 A) x) P) f) y) e) y) e) ) ) ) ) ) ) .
+eq_ind : (A : Utype -> (x : (etype A) -> (P : (_388 : (etype A) -> Uprop) -> (f : (eprop (P x) ) -> (y : (etype A) -> (e : (eprop ( ( (eq A) x) y) ) -> (eprop (P y) ) ) ) ) ) ) ) .
+[] eq_ind --> (A : (etype dottype) => (x : (etype A) => (P : (etype ( (dotpitt A) (_387 : (etype A) => dotprop) ) ) => ( ( (eq_rect A) x) P) ) ) ) .
+eq_rec : (A : Utype -> (x : (etype A) -> (P : (_390 : (etype A) -> Uset) -> (f : (eset (P x) ) -> (y : (etype A) -> (e : (eprop ( ( (eq A) x) y) ) -> (eset (P y) ) ) ) ) ) ) ) .
+[] eq_rec --> (A : (etype dottype) => (x : (etype A) => (P : (etype ( (dotpitt A) (_389 : (etype A) => dotset) ) ) => ( ( (eq_rect A) x) P) ) ) ) .
+case_14 : (A : Uprop -> (C : Uprop -> (h1 : (eprop A) -> (h2 : (_391 : (eprop A) -> (eprop False) ) -> (f : (eprop False) -> (_392 : (eprop False) -> (eprop C) ) ) ) ) ) ) .
+absurd : (A : Uprop -> (C : Uprop -> (_396 : (eprop A) -> (_395 : (eprop (not A) ) -> (eprop C) ) ) ) ) .
+[] absurd --> (A : (etype dotprop) => (C : (etype dotprop) => (h1 : (eprop A) => (h2 : (eprop ( (dotpipp A) (_394 : (eprop A) => False) ) ) => ( (f : (eprop False) => ( ( ( ( ( (case_14 A) C) h1) h2) f) f) ) (h2 h1) ) ) ) ) ) .
+case_15 : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (H : (eprop ( ( (eq A) x) y) ) -> (y欧0 : (etype A) -> (_397 : (eprop ( ( (eq A) x) y欧0) ) -> (eprop ( ( (eq A) y欧0) x) ) ) ) ) ) ) ) .
+refl_equal_case_15 : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (H : (eprop ( ( (eq A) x) y) ) -> (eprop ( ( (eq A) x) x) ) ) ) ) ) .
+[A : Utype, x : (etype A) , y : (etype A) , H : (eprop ( ( (eq A) x) y) ) ] ( ( ( (refl_equal_case_15 A) x) y) H) --> ( (refl_equal A) x) .
+[A : Utype, x : (etype A) , y : (etype A) , H : (eprop ( ( (eq A) x) y) ) ] ( ( ( ( ( (case_15 A) x) y) H) x) ( ( ( (refl_equal_case_15 A) x) y) H) ) --> ( (refl_equal A) x) .
+sym_eq : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (_398 : (eprop ( ( (eq A) x) y) ) -> (eprop ( ( (eq A) y) x) ) ) ) ) ) .
+[] sym_eq --> (A : (etype dottype) => (x : (etype A) => (y : (etype A) => (H : (eprop ( ( (eq A) x) y) ) => ( ( ( ( ( (case_15 A) x) y) H) y) H) ) ) ) ) .
+case_16 : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (z : (etype A) -> (H : (eprop ( ( (eq A) x) y) ) -> (H0 : (eprop ( ( (eq A) y) z) ) -> (y欧0 : (etype A) -> (_399 : (eprop ( ( (eq A) y) y欧0) ) -> (eprop ( ( (eq A) x) y欧0) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_16 : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (z : (etype A) -> (H : (eprop ( ( (eq A) x) y) ) -> (H0 : (eprop ( ( (eq A) y) z) ) -> (eprop ( ( (eq A) y) y) ) ) ) ) ) ) ) .
+[A : Utype, x : (etype A) , y : (etype A) , z : (etype A) , H : (eprop ( ( (eq A) x) y) ) , H0 : (eprop ( ( (eq A) y) z) ) ] ( ( ( ( ( (refl_equal_case_16 A) x) y) z) H) H0) --> ( (refl_equal A) y) .
+[A : Utype, x : (etype A) , y : (etype A) , z : (etype A) , H : (eprop ( ( (eq A) x) y) ) , H0 : (eprop ( ( (eq A) y) z) ) ] ( ( ( ( ( ( ( (case_16 A) x) y) z) H) H0) y) ( ( ( ( ( (refl_equal_case_16 A) x) y) z) H) H0) ) --> H.
+trans_eq : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (z : (etype A) -> (_401 : (eprop ( ( (eq A) x) y) ) -> (_400 : (eprop ( ( (eq A) y) z) ) -> (eprop ( ( (eq A) x) z) ) ) ) ) ) ) ) .
+[] trans_eq --> (A : (etype dottype) => (x : (etype A) => (y : (etype A) => (z : (etype A) => (H : (eprop ( ( (eq A) x) y) ) => (H0 : (eprop ( ( (eq A) y) z) ) => ( ( ( ( ( ( ( (case_16 A) x) y) z) H) H0) z) H0) ) ) ) ) ) ) .
+case_17 : (A : Utype -> (B : Utype -> (f : (_402 : (etype A) -> (etype B) ) -> (x : (etype A) -> (y : (etype A) -> (H : (eprop ( ( (eq A) x) y) ) -> (y欧0 : (etype A) -> (_403 : (eprop ( ( (eq A) x) y欧0) ) -> (eprop ( ( (eq B) (f x) ) (f y欧0) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_17 : (A : Utype -> (B : Utype -> (f : (_405 : (etype A) -> (etype B) ) -> (x : (etype A) -> (y : (etype A) -> (H : (eprop ( ( (eq A) x) y) ) -> (eprop ( ( (eq A) x) x) ) ) ) ) ) ) ) .
+[A : Utype, B : Utype, f : (_406 : (etype A) -> (etype B) ) , x : (etype A) , y : (etype A) , H : (eprop ( ( (eq A) x) y) ) ] ( ( ( ( ( (refl_equal_case_17 A) B) f) x) y) H) --> ( (refl_equal A) x) .
+[A : Utype, B : Utype, f : (_404 : (etype A) -> (etype B) ) , x : (etype A) , y : (etype A) , H : (eprop ( ( (eq A) x) y) ) ] ( ( ( ( ( ( ( (case_17 A) B) f) x) y) H) x) ( ( ( ( ( (refl_equal_case_17 A) B) f) x) y) H) ) --> ( (refl_equal B) (f x) ) .
+f_equal : (A : Utype -> (B : Utype -> (f : (_408 : (etype A) -> (etype B) ) -> (x : (etype A) -> (y : (etype A) -> (_409 : (eprop ( ( (eq A) x) y) ) -> (eprop ( ( (eq B) (f x) ) (f y) ) ) ) ) ) ) ) ) .
+[] f_equal --> (A : (etype dottype) => (B : (etype dottype) => (f : (etype ( (dotpitt A) (_407 : (etype A) => B) ) ) => (x : (etype A) => (y : (etype A) => (H : (eprop ( ( (eq A) x) y) ) => ( ( ( ( ( ( ( (case_17 A) B) f) x) y) H) y) H) ) ) ) ) ) ) .
+case_18 : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (h1 : (eprop (not ( ( (eq A) x) y) ) ) -> (h2 : (eprop ( ( (eq A) y) x) ) -> (y欧0 : (etype A) -> (_410 : (eprop ( ( (eq A) y) y欧0) ) -> (h1欧0 : (eprop (not ( ( (eq A) y欧0) y) ) ) -> (eprop ( ( (eq A) y欧0) y) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_18 : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (h1 : (eprop (not ( ( (eq A) x) y) ) ) -> (h2 : (eprop ( ( (eq A) y) x) ) -> (eprop ( ( (eq A) y) y) ) ) ) ) ) ) .
+[A : Utype, x : (etype A) , y : (etype A) , h1 : (eprop (not ( ( (eq A) x) y) ) ) , h2 : (eprop ( ( (eq A) y) x) ) ] ( ( ( ( (refl_equal_case_18 A) x) y) h1) h2) --> ( (refl_equal A) y) .
+[A : Utype, x : (etype A) , y : (etype A) , h1 : (eprop (not ( ( (eq A) x) y) ) ) , h2 : (eprop ( ( (eq A) y) x) ) ] ( ( ( ( ( ( (case_18 A) x) y) h1) h2) y) ( ( ( ( (refl_equal_case_18 A) x) y) h1) h2) ) --> (h1欧0 : (eprop (not ( ( (eq A) y) y) ) ) => ( (refl_equal A) y) ) .
+sym_not_eq : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (_411 : (eprop (not ( ( (eq A) x) y) ) ) -> (eprop (not ( ( (eq A) y) x) ) ) ) ) ) ) .
+[] sym_not_eq --> (A : (etype dottype) => (x : (etype A) => (y : (etype A) => (h1 : (eprop (not ( ( (eq A) x) y) ) ) => (h2 : (eprop ( ( (eq A) y) x) ) => (h1 ( ( ( ( ( ( ( (case_18 A) x) y) h1) h2) x) h2) h1) ) ) ) ) ) ) .
+sym_equal : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (_412 : (eprop ( ( (eq A) x) y) ) -> (eprop ( ( (eq A) y) x) ) ) ) ) ) .
+[] sym_equal --> (A : (etype dottype) => (x : (etype A) => (y : (etype A) => ( ( (sym_eq A) x) y) ) ) ) .
+sym_not_equal : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (_413 : (eprop (not ( ( (eq A) x) y) ) ) -> (eprop (not ( ( (eq A) y) x) ) ) ) ) ) ) .
+[] sym_not_equal --> (A : (etype dottype) => (x : (etype A) => (y : (etype A) => ( ( (sym_not_eq A) x) y) ) ) ) .
+trans_equal : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (z : (etype A) -> (_415 : (eprop ( ( (eq A) x) y) ) -> (_414 : (eprop ( ( (eq A) y) z) ) -> (eprop ( ( (eq A) x) z) ) ) ) ) ) ) ) .
+[] trans_equal --> (A : (etype dottype) => (x : (etype A) => (y : (etype A) => (z : (etype A) => ( ( ( (trans_eq A) x) y) z) ) ) ) ) .
+eq_ind_r : (A : Utype -> (x : (etype A) -> (P : (_417 : (etype A) -> Uprop) -> (_419 : (eprop (P x) ) -> (y : (etype A) -> (_418 : (eprop ( ( (eq A) y) x) ) -> (eprop (P y) ) ) ) ) ) ) ) .
+[] eq_ind_r --> (A : (etype dottype) => (x : (etype A) => (P : (etype ( (dotpitt A) (_416 : (etype A) => dotprop) ) ) => (H : (eprop (P x) ) => (y : (etype A) => (H0 : (eprop ( ( (eq A) y) x) ) => ( ( ( ( ( (eq_ind A) x) (y欧0 : (etype A) => (P y欧0) ) ) H) y) ( ( ( (sym_eq A) y) x) H0) ) ) ) ) ) ) ) .
+eq_rec_r : (A : Utype -> (x : (etype A) -> (P : (_421 : (etype A) -> Uset) -> (_423 : (eset (P x) ) -> (y : (etype A) -> (_422 : (eprop ( ( (eq A) y) x) ) -> (eset (P y) ) ) ) ) ) ) ) .
+[] eq_rec_r --> (A : (etype dottype) => (x : (etype A) => (P : (etype ( (dotpitt A) (_420 : (etype A) => dotset) ) ) => (H : (eset (P x) ) => (y : (etype A) => (H0 : (eprop ( ( (eq A) y) x) ) => ( ( ( ( ( (eq_rec A) x) (y欧0 : (etype A) => (P y欧0) ) ) H) y) ( ( ( (sym_eq A) y) x) H0) ) ) ) ) ) ) ) .
+eq_rect_r : (A : Utype -> (x : (etype A) -> (P : (_425 : (etype A) -> Utype) -> (_427 : (etype (P x) ) -> (y : (etype A) -> (_426 : (eprop ( ( (eq A) y) x) ) -> (etype (P y) ) ) ) ) ) ) ) .
+[] eq_rect_r --> (A : (etype dottype) => (x : (etype A) => (P : (etype ( (dotpitt A) (_424 : (etype A) => dottype) ) ) => (H : (etype (P x) ) => (y : (etype A) => (H0 : (eprop ( ( (eq A) y) x) ) => ( ( ( ( ( (eq_rect A) x) (y欧0 : (etype A) => (P y欧0) ) ) H) y) ( ( ( (sym_eq A) y) x) H0) ) ) ) ) ) ) ) .
+case_19 : (A1 : Utype -> (A2 : Utype -> (B : Utype -> (f : (_429 : (etype A1) -> (_428 : (etype A2) -> (etype B) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (y : (etype A1) -> (_431 : (eprop ( ( (eq A1) x1) y) ) -> (_430 : (eprop ( ( (eq A2) x2) y2) ) -> (eprop ( ( (eq B) ( (f x1) x2) ) ( (f y) y2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_19 : (A1 : Utype -> (A2 : Utype -> (B : Utype -> (f : (_435 : (etype A1) -> (_434 : (etype A2) -> (etype B) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (eprop ( ( (eq A1) x1) x1) ) ) ) ) ) ) ) ) ) ) .
+[A1 : Utype, A2 : Utype, B : Utype, f : (_437 : (etype A1) -> (_436 : (etype A2) -> (etype B) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( (refl_equal_case_19 A1) A2) B) f) x1) y1) x2) y2) H) --> ( (refl_equal A1) x1) .
+case_20 : (A1 : Utype -> (A2 : Utype -> (B : Utype -> (f : (_439 : (etype A1) -> (_438 : (etype A2) -> (etype B) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (y : (etype A2) -> (_440 : (eprop ( ( (eq A2) x2) y) ) -> (eprop ( ( (eq B) ( (f x1) x2) ) ( (f x1) y) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_20 : (A1 : Utype -> (A2 : Utype -> (B : Utype -> (f : (_444 : (etype A1) -> (_443 : (etype A2) -> (etype B) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (eprop ( ( (eq A2) x2) x2) ) ) ) ) ) ) ) ) ) ) ) .
+[A1 : Utype, A2 : Utype, B : Utype, f : (_446 : (etype A1) -> (_445 : (etype A2) -> (etype B) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( (refl_equal_case_20 A1) A2) B) f) x1) y1) x2) y2) H) H欧0) --> ( (refl_equal A2) x2) .
+[A1 : Utype, A2 : Utype, B : Utype, f : (_442 : (etype A1) -> (_441 : (etype A2) -> (etype B) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( ( ( (case_20 A1) A2) B) f) x1) y1) x2) y2) H) H欧0) x2) ( ( ( ( ( ( ( ( ( (refl_equal_case_20 A1) A2) B) f) x1) y1) x2) y2) H) H欧0) ) --> ( (refl_equal B) ( (f x1) x2) ) .
+[A1 : Utype, A2 : Utype, B : Utype, f : (_433 : (etype A1) -> (_432 : (etype A2) -> (etype B) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( ( ( (case_19 A1) A2) B) f) x1) y1) x2) y2) H) x1) ( ( ( ( ( ( ( ( (refl_equal_case_19 A1) A2) B) f) x1) y1) x2) y2) H) ) --> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) => ( ( ( ( ( ( ( ( ( ( ( (case_20 A1) A2) B) f) x1) y1) x2) y2) H) H欧0) y2) H欧0) ) .
+f_equal2 : (A1 : Utype -> (A2 : Utype -> (B : Utype -> (f : (_450 : (etype A1) -> (_449 : (etype A2) -> (etype B) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (_452 : (eprop ( ( (eq A1) x1) y1) ) -> (_451 : (eprop ( ( (eq A2) x2) y2) ) -> (eprop ( ( (eq B) ( (f x1) x2) ) ( (f y1) y2) ) ) ) ) ) ) ) ) ) ) ) ) .
+[] f_equal2 --> (A1 : (etype dottype) => (A2 : (etype dottype) => (B : (etype dottype) => (f : (etype ( (dotpitt A1) (_448 : (etype A1) => ( (dotpitt A2) (_447 : (etype A2) => B) ) ) ) ) => (x1 : (etype A1) => (y1 : (etype A1) => (x2 : (etype A2) => (y2 : (etype A2) => (H : (eprop ( ( (eq A1) x1) y1) ) => ( ( ( ( ( ( ( ( ( ( (case_19 A1) A2) B) f) x1) y1) x2) y2) H) y1) H) ) ) ) ) ) ) ) ) ) .
+case_21 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (B : Utype -> (f : (_455 : (etype A1) -> (_454 : (etype A2) -> (_453 : (etype A3) -> (etype B) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (y : (etype A1) -> (_458 : (eprop ( ( (eq A1) x1) y) ) -> (_457 : (eprop ( ( (eq A2) x2) y2) ) -> (_456 : (eprop ( ( (eq A3) x3) y3) ) -> (eprop ( ( (eq B) ( ( (f x1) x2) x3) ) ( ( (f y) y2) y3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_21 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (B : Utype -> (f : (_464 : (etype A1) -> (_463 : (etype A2) -> (_462 : (etype A3) -> (etype B) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (eprop ( ( (eq A1) x1) x1) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, B : Utype, f : (_467 : (etype A1) -> (_466 : (etype A2) -> (_465 : (etype A3) -> (etype B) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_21 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) --> ( (refl_equal A1) x1) .
+case_22 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (B : Utype -> (f : (_470 : (etype A1) -> (_469 : (etype A2) -> (_468 : (etype A3) -> (etype B) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (y : (etype A2) -> (_472 : (eprop ( ( (eq A2) x2) y) ) -> (_471 : (eprop ( ( (eq A3) x3) y3) ) -> (eprop ( ( (eq B) ( ( (f x1) x2) x3) ) ( ( (f x1) y) y3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_22 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (B : Utype -> (f : (_478 : (etype A1) -> (_477 : (etype A2) -> (_476 : (etype A3) -> (etype B) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (eprop ( ( (eq A2) x2) x2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, B : Utype, f : (_481 : (etype A1) -> (_480 : (etype A2) -> (_479 : (etype A3) -> (etype B) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_22 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) --> ( (refl_equal A2) x2) .
+case_23 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (B : Utype -> (f : (_484 : (etype A1) -> (_483 : (etype A2) -> (_482 : (etype A3) -> (etype B) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (y : (etype A3) -> (_485 : (eprop ( ( (eq A3) x3) y) ) -> (eprop ( ( (eq B) ( ( (f x1) x2) x3) ) ( ( (f x1) x2) y) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_23 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (B : Utype -> (f : (_491 : (etype A1) -> (_490 : (etype A2) -> (_489 : (etype A3) -> (etype B) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (eprop ( ( (eq A3) x3) x3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, B : Utype, f : (_494 : (etype A1) -> (_493 : (etype A2) -> (_492 : (etype A3) -> (etype B) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_23 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) H欧1) --> ( (refl_equal A3) x3) .
+[A1 : Utype, A2 : Utype, A3 : Utype, B : Utype, f : (_488 : (etype A1) -> (_487 : (etype A2) -> (_486 : (etype A3) -> (etype B) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_23 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) H欧1) x3) ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_23 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) H欧1) ) --> ( (refl_equal B) ( ( (f x1) x2) x3) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, B : Utype, f : (_475 : (etype A1) -> (_474 : (etype A2) -> (_473 : (etype A3) -> (etype B) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_22 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) x2) ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_22 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) ) --> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_23 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) H欧1) y3) H欧1) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, B : Utype, f : (_461 : (etype A1) -> (_460 : (etype A2) -> (_459 : (etype A3) -> (etype B) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( (case_21 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) x1) ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_21 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) ) --> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_22 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) H欧0) y2) H欧0) ) .
+f_equal3 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (B : Utype -> (f : (_500 : (etype A1) -> (_499 : (etype A2) -> (_498 : (etype A3) -> (etype B) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (_503 : (eprop ( ( (eq A1) x1) y1) ) -> (_502 : (eprop ( ( (eq A2) x2) y2) ) -> (_501 : (eprop ( ( (eq A3) x3) y3) ) -> (eprop ( ( (eq B) ( ( (f x1) x2) x3) ) ( ( (f y1) y2) y3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[] f_equal3 --> (A1 : (etype dottype) => (A2 : (etype dottype) => (A3 : (etype dottype) => (B : (etype dottype) => (f : (etype ( (dotpitt A1) (_497 : (etype A1) => ( (dotpitt A2) (_496 : (etype A2) => ( (dotpitt A3) (_495 : (etype A3) => B) ) ) ) ) ) ) => (x1 : (etype A1) => (y1 : (etype A1) => (x2 : (etype A2) => (y2 : (etype A2) => (x3 : (etype A3) => (y3 : (etype A3) => (H : (eprop ( ( (eq A1) x1) y1) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( (case_21 A1) A2) A3) B) f) x1) y1) x2) y2) x3) y3) H) y1) H) ) ) ) ) ) ) ) ) ) ) ) ) .
+case_24 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_507 : (etype A1) -> (_506 : (etype A2) -> (_505 : (etype A3) -> (_504 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (y : (etype A1) -> (_511 : (eprop ( ( (eq A1) x1) y) ) -> (_510 : (eprop ( ( (eq A2) x2) y2) ) -> (_509 : (eprop ( ( (eq A3) x3) y3) ) -> (_508 : (eprop ( ( (eq A4) x4) y4) ) -> (eprop ( ( (eq B) ( ( ( (f x1) x2) x3) x4) ) ( ( ( (f y) y2) y3) y4) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_24 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_519 : (etype A1) -> (_518 : (etype A2) -> (_517 : (etype A3) -> (_516 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (eprop ( ( (eq A1) x1) x1) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_523 : (etype A1) -> (_522 : (etype A2) -> (_521 : (etype A3) -> (_520 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_24 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) --> ( (refl_equal A1) x1) .
+case_25 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_527 : (etype A1) -> (_526 : (etype A2) -> (_525 : (etype A3) -> (_524 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (y : (etype A2) -> (_530 : (eprop ( ( (eq A2) x2) y) ) -> (_529 : (eprop ( ( (eq A3) x3) y3) ) -> (_528 : (eprop ( ( (eq A4) x4) y4) ) -> (eprop ( ( (eq B) ( ( ( (f x1) x2) x3) x4) ) ( ( ( (f x1) y) y3) y4) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_25 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_538 : (etype A1) -> (_537 : (etype A2) -> (_536 : (etype A3) -> (_535 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (eprop ( ( (eq A2) x2) x2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_542 : (etype A1) -> (_541 : (etype A2) -> (_540 : (etype A3) -> (_539 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_25 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) --> ( (refl_equal A2) x2) .
+case_26 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_546 : (etype A1) -> (_545 : (etype A2) -> (_544 : (etype A3) -> (_543 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (y : (etype A3) -> (_548 : (eprop ( ( (eq A3) x3) y) ) -> (_547 : (eprop ( ( (eq A4) x4) y4) ) -> (eprop ( ( (eq B) ( ( ( (f x1) x2) x3) x4) ) ( ( ( (f x1) x2) y) y4) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_26 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_556 : (etype A1) -> (_555 : (etype A2) -> (_554 : (etype A3) -> (_553 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (eprop ( ( (eq A3) x3) x3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_560 : (etype A1) -> (_559 : (etype A2) -> (_558 : (etype A3) -> (_557 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_26 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) --> ( (refl_equal A3) x3) .
+case_27 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_564 : (etype A1) -> (_563 : (etype A2) -> (_562 : (etype A3) -> (_561 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) -> (y : (etype A4) -> (_565 : (eprop ( ( (eq A4) x4) y) ) -> (eprop ( ( (eq B) ( ( ( (f x1) x2) x3) x4) ) ( ( ( (f x1) x2) x3) y) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_27 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_573 : (etype A1) -> (_572 : (etype A2) -> (_571 : (etype A3) -> (_570 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) -> (eprop ( ( (eq A4) x4) x4) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_577 : (etype A1) -> (_576 : (etype A2) -> (_575 : (etype A3) -> (_574 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) , H欧2 : (eprop ( ( (eq A4) x4) y4) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_27 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) H欧2) --> ( (refl_equal A4) x4) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_569 : (etype A1) -> (_568 : (etype A2) -> (_567 : (etype A3) -> (_566 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) , H欧2 : (eprop ( ( (eq A4) x4) y4) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_27 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) H欧2) x4) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_27 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) H欧2) ) --> ( (refl_equal B) ( ( ( (f x1) x2) x3) x4) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_552 : (etype A1) -> (_551 : (etype A2) -> (_550 : (etype A3) -> (_549 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_26 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) x3) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_26 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) ) --> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_27 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) H欧2) y4) H欧2) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_534 : (etype A1) -> (_533 : (etype A2) -> (_532 : (etype A3) -> (_531 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_25 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) x2) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_25 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) ) --> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_26 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) H欧1) y3) H欧1) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, B : Utype, f : (_515 : (etype A1) -> (_514 : (etype A2) -> (_513 : (etype A3) -> (_512 : (etype A4) -> (etype B) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_24 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) x1) ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_24 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) ) --> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_25 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) H欧0) y2) H欧0) ) .
+f_equal4 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (B : Utype -> (f : (_585 : (etype A1) -> (_584 : (etype A2) -> (_583 : (etype A3) -> (_582 : (etype A4) -> (etype B) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (_589 : (eprop ( ( (eq A1) x1) y1) ) -> (_588 : (eprop ( ( (eq A2) x2) y2) ) -> (_587 : (eprop ( ( (eq A3) x3) y3) ) -> (_586 : (eprop ( ( (eq A4) x4) y4) ) -> (eprop ( ( (eq B) ( ( ( (f x1) x2) x3) x4) ) ( ( ( (f y1) y2) y3) y4) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[] f_equal4 --> (A1 : (etype dottype) => (A2 : (etype dottype) => (A3 : (etype dottype) => (A4 : (etype dottype) => (B : (etype dottype) => (f : (etype ( (dotpitt A1) (_581 : (etype A1) => ( (dotpitt A2) (_580 : (etype A2) => ( (dotpitt A3) (_579 : (etype A3) => ( (dotpitt A4) (_578 : (etype A4) => B) ) ) ) ) ) ) ) ) => (x1 : (etype A1) => (y1 : (etype A1) => (x2 : (etype A2) => (y2 : (etype A2) => (x3 : (etype A3) => (y3 : (etype A3) => (x4 : (etype A4) => (y4 : (etype A4) => (H : (eprop ( ( (eq A1) x1) y1) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_24 A1) A2) A3) A4) B) f) x1) y1) x2) y2) x3) y3) x4) y4) H) y1) H) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+case_28 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_594 : (etype A1) -> (_593 : (etype A2) -> (_592 : (etype A3) -> (_591 : (etype A4) -> (_590 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (y : (etype A1) -> (_599 : (eprop ( ( (eq A1) x1) y) ) -> (_598 : (eprop ( ( (eq A2) x2) y2) ) -> (_597 : (eprop ( ( (eq A3) x3) y3) ) -> (_596 : (eprop ( ( (eq A4) x4) y4) ) -> (_595 : (eprop ( ( (eq A5) x5) y5) ) -> (eprop ( ( (eq B) ( ( ( ( (f x1) x2) x3) x4) x5) ) ( ( ( ( (f y) y2) y3) y4) y5) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_28 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_609 : (etype A1) -> (_608 : (etype A2) -> (_607 : (etype A3) -> (_606 : (etype A4) -> (_605 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (eprop ( ( (eq A1) x1) x1) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_614 : (etype A1) -> (_613 : (etype A2) -> (_612 : (etype A3) -> (_611 : (etype A4) -> (_610 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_28 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) --> ( (refl_equal A1) x1) .
+case_29 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_619 : (etype A1) -> (_618 : (etype A2) -> (_617 : (etype A3) -> (_616 : (etype A4) -> (_615 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (y : (etype A2) -> (_623 : (eprop ( ( (eq A2) x2) y) ) -> (_622 : (eprop ( ( (eq A3) x3) y3) ) -> (_621 : (eprop ( ( (eq A4) x4) y4) ) -> (_620 : (eprop ( ( (eq A5) x5) y5) ) -> (eprop ( ( (eq B) ( ( ( ( (f x1) x2) x3) x4) x5) ) ( ( ( ( (f x1) y) y3) y4) y5) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_29 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_633 : (etype A1) -> (_632 : (etype A2) -> (_631 : (etype A3) -> (_630 : (etype A4) -> (_629 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (eprop ( ( (eq A2) x2) x2) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_638 : (etype A1) -> (_637 : (etype A2) -> (_636 : (etype A3) -> (_635 : (etype A4) -> (_634 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_29 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) --> ( (refl_equal A2) x2) .
+case_30 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_643 : (etype A1) -> (_642 : (etype A2) -> (_641 : (etype A3) -> (_640 : (etype A4) -> (_639 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (y : (etype A3) -> (_646 : (eprop ( ( (eq A3) x3) y) ) -> (_645 : (eprop ( ( (eq A4) x4) y4) ) -> (_644 : (eprop ( ( (eq A5) x5) y5) ) -> (eprop ( ( (eq B) ( ( ( ( (f x1) x2) x3) x4) x5) ) ( ( ( ( (f x1) x2) y) y4) y5) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_30 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_656 : (etype A1) -> (_655 : (etype A2) -> (_654 : (etype A3) -> (_653 : (etype A4) -> (_652 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (eprop ( ( (eq A3) x3) x3) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_661 : (etype A1) -> (_660 : (etype A2) -> (_659 : (etype A3) -> (_658 : (etype A4) -> (_657 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_30 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) --> ( (refl_equal A3) x3) .
+case_31 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_666 : (etype A1) -> (_665 : (etype A2) -> (_664 : (etype A3) -> (_663 : (etype A4) -> (_662 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) -> (y : (etype A4) -> (_668 : (eprop ( ( (eq A4) x4) y) ) -> (_667 : (eprop ( ( (eq A5) x5) y5) ) -> (eprop ( ( (eq B) ( ( ( ( (f x1) x2) x3) x4) x5) ) ( ( ( ( (f x1) x2) x3) y) y5) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_31 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_678 : (etype A1) -> (_677 : (etype A2) -> (_676 : (etype A3) -> (_675 : (etype A4) -> (_674 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) -> (eprop ( ( (eq A4) x4) x4) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_683 : (etype A1) -> (_682 : (etype A2) -> (_681 : (etype A3) -> (_680 : (etype A4) -> (_679 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) , H欧2 : (eprop ( ( (eq A4) x4) y4) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_31 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) --> ( (refl_equal A4) x4) .
+case_32 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_688 : (etype A1) -> (_687 : (etype A2) -> (_686 : (etype A3) -> (_685 : (etype A4) -> (_684 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) -> (H欧3 : (eprop ( ( (eq A5) x5) y5) ) -> (y : (etype A5) -> (_689 : (eprop ( ( (eq A5) x5) y) ) -> (eprop ( ( (eq B) ( ( ( ( (f x1) x2) x3) x4) x5) ) ( ( ( ( (f x1) x2) x3) x4) y) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+refl_equal_case_32 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_699 : (etype A1) -> (_698 : (etype A2) -> (_697 : (etype A3) -> (_696 : (etype A4) -> (_695 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (H : (eprop ( ( (eq A1) x1) y1) ) -> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) -> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) -> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) -> (H欧3 : (eprop ( ( (eq A5) x5) y5) ) -> (eprop ( ( (eq A5) x5) x5) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_704 : (etype A1) -> (_703 : (etype A2) -> (_702 : (etype A3) -> (_701 : (etype A4) -> (_700 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) , H欧2 : (eprop ( ( (eq A4) x4) y4) ) , H欧3 : (eprop ( ( (eq A5) x5) y5) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_32 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) H欧3) --> ( (refl_equal A5) x5) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_694 : (etype A1) -> (_693 : (etype A2) -> (_692 : (etype A3) -> (_691 : (etype A4) -> (_690 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) , H欧2 : (eprop ( ( (eq A4) x4) y4) ) , H欧3 : (eprop ( ( (eq A5) x5) y5) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_32 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) H欧3) x5) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_32 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) H欧3) ) --> ( (refl_equal B) ( ( ( ( (f x1) x2) x3) x4) x5) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_673 : (etype A1) -> (_672 : (etype A2) -> (_671 : (etype A3) -> (_670 : (etype A4) -> (_669 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) , H欧2 : (eprop ( ( (eq A4) x4) y4) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_31 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) x4) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_31 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) ) --> (H欧3 : (eprop ( ( (eq A5) x5) y5) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_32 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) H欧3) y5) H欧3) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_651 : (etype A1) -> (_650 : (etype A2) -> (_649 : (etype A3) -> (_648 : (etype A4) -> (_647 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) , H欧1 : (eprop ( ( (eq A3) x3) y3) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_30 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) x3) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_30 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) ) --> (H欧2 : (eprop ( ( (eq A4) x4) y4) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_31 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) H欧2) y4) H欧2) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_628 : (etype A1) -> (_627 : (etype A2) -> (_626 : (etype A3) -> (_625 : (etype A4) -> (_624 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) , H欧0 : (eprop ( ( (eq A2) x2) y2) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_29 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) x2) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_29 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) ) --> (H欧1 : (eprop ( ( (eq A3) x3) y3) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_30 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) H欧1) y3) H欧1) ) .
+[A1 : Utype, A2 : Utype, A3 : Utype, A4 : Utype, A5 : Utype, B : Utype, f : (_604 : (etype A1) -> (_603 : (etype A2) -> (_602 : (etype A3) -> (_601 : (etype A4) -> (_600 : (etype A5) -> (etype B) ) ) ) ) ) , x1 : (etype A1) , y1 : (etype A1) , x2 : (etype A2) , y2 : (etype A2) , x3 : (etype A3) , y3 : (etype A3) , x4 : (etype A4) , y4 : (etype A4) , x5 : (etype A5) , y5 : (etype A5) , H : (eprop ( ( (eq A1) x1) y1) ) ] ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_28 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) x1) ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (refl_equal_case_28 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) ) --> (H欧0 : (eprop ( ( (eq A2) x2) y2) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_29 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) H欧0) y2) H欧0) ) .
+f_equal5 : (A1 : Utype -> (A2 : Utype -> (A3 : Utype -> (A4 : Utype -> (A5 : Utype -> (B : Utype -> (f : (_714 : (etype A1) -> (_713 : (etype A2) -> (_712 : (etype A3) -> (_711 : (etype A4) -> (_710 : (etype A5) -> (etype B) ) ) ) ) ) -> (x1 : (etype A1) -> (y1 : (etype A1) -> (x2 : (etype A2) -> (y2 : (etype A2) -> (x3 : (etype A3) -> (y3 : (etype A3) -> (x4 : (etype A4) -> (y4 : (etype A4) -> (x5 : (etype A5) -> (y5 : (etype A5) -> (_719 : (eprop ( ( (eq A1) x1) y1) ) -> (_718 : (eprop ( ( (eq A2) x2) y2) ) -> (_717 : (eprop ( ( (eq A3) x3) y3) ) -> (_716 : (eprop ( ( (eq A4) x4) y4) ) -> (_715 : (eprop ( ( (eq A5) x5) y5) ) -> (eprop ( ( (eq B) ( ( ( ( (f x1) x2) x3) x4) x5) ) ( ( ( ( (f y1) y2) y3) y4) y5) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[] f_equal5 --> (A1 : (etype dottype) => (A2 : (etype dottype) => (A3 : (etype dottype) => (A4 : (etype dottype) => (A5 : (etype dottype) => (B : (etype dottype) => (f : (etype ( (dotpitt A1) (_709 : (etype A1) => ( (dotpitt A2) (_708 : (etype A2) => ( (dotpitt A3) (_707 : (etype A3) => ( (dotpitt A4) (_706 : (etype A4) => ( (dotpitt A5) (_705 : (etype A5) => B) ) ) ) ) ) ) ) ) ) ) => (x1 : (etype A1) => (y1 : (etype A1) => (x2 : (etype A2) => (y2 : (etype A2) => (x3 : (etype A3) => (y3 : (etype A3) => (x4 : (etype A4) => (y4 : (etype A4) => (x5 : (etype A5) => (y5 : (etype A5) => (H : (eprop ( ( (eq A1) x1) y1) ) => ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( (case_28 A1) A2) A3) A4) A5) B) f) x1) y1) x2) y2) x3) y3) x4) y4) x5) y5) H) y1) H) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) .
+subrelation : (A : Utype -> (B : Utype -> (R : (_726 : (etype A) -> (_725 : (etype B) -> Uprop) ) -> (R' : (_728 : (etype A) -> (_727 : (etype B) -> Uprop) ) -> Uprop) ) ) ) .
+[] subrelation --> (A : (etype dottype) => (B : (etype dottype) => (R : (etype ( (dotpitt A) (_724 : (etype A) => ( (dotpitt B) (_723 : (etype B) => dotprop) ) ) ) ) => (R' : (etype ( (dotpitt A) (_722 : (etype A) => ( (dotpitt B) (_721 : (etype B) => dotprop) ) ) ) ) => ( (dotpitp A) (x : (etype A) => ( (dotpitp B) (y : (etype B) => ( (dotpipp ( (R x) y) ) (_720 : (eprop ( (R x) y) ) => ( (R' x) y) ) ) ) ) ) ) ) ) ) ) .
+unique : (A : Utype -> (P : (_731 : (etype A) -> Uprop) -> (x : (etype A) -> Uprop) ) ) .
+[] unique --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_730 : (etype A) => dotprop) ) ) => (x : (etype A) => ( (and (P x) ) ( (dotpitp A) (x' : (etype A) => ( (dotpipp (P x') ) (_729 : (eprop (P x') ) => ( ( (eq A) x) x') ) ) ) ) ) ) ) ) .
+uniqueness : (A : Utype -> (P : (_735 : (etype A) -> Uprop) -> Uprop) ) .
+[] uniqueness --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_734 : (etype A) => dotprop) ) ) => ( (dotpitp A) (x : (etype A) => ( (dotpitp A) (y : (etype A) => ( (dotpipp (P x) ) (_733 : (eprop (P x) ) => ( (dotpipp (P y) ) (_732 : (eprop (P y) ) => ( ( (eq A) x) y) ) ) ) ) ) ) ) ) ) ) .
+case_33 : (A : Utype -> (P : (_738 : (etype A) -> Uprop) -> (H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) -> (_739 : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) -> (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) ) ) ) ) .
+conj_case_33 : (A : Utype -> (P : (_741 : (etype A) -> Uprop) -> (H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) -> (_743 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (_742 : (eprop ( (uniqueness A) P) ) -> (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) ) ) ) ) ) .
+[A : Utype, P : (_744 : (etype A) -> Uprop) , H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) ] ( ( (conj_case_33 A) P) H) --> ( (conj ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) .
+case_34 : (A : Utype -> (P : (_745 : (etype A) -> Uprop) -> (H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) -> (H欧0 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (_747 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (_746 : (eprop ( (uniqueness A) P) ) -> (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) ) ) ) ) ) ) .
+ex_intro_case_34 : (A : Utype -> (P : (_749 : (etype A) -> Uprop) -> (H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) -> (H欧0 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (x : (etype A) -> (_750 : (eprop ( (x欧0 : (etype A) => (P x欧0) ) x) ) -> (eprop ( (ex A) (x欧0 : (etype A) => (P x欧0) ) ) ) ) ) ) ) ) ) .
+[A : Utype, P : (_751 : (etype A) -> Uprop) , H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) , H欧0 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) ] ( ( ( (ex_intro_case_34 A) P) H) H欧0) --> ( (ex_intro A) (x : (etype A) => (P x) ) ) .
+[A : Utype, P : (_748 : (etype A) -> Uprop) , H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) , H欧0 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) , var_25 : (etype A) , var_26 : (eprop ( (x : (etype A) => (P x) ) var_25) ) ] ( ( ( ( (case_34 A) P) H) H欧0) ( ( ( ( ( (ex_intro_case_34 A) P) H) H欧0) var_25) var_26) ) --> ( ( (x : (etype A) => (Hx : (eprop (P x) ) => (Huni : (eprop ( (uniqueness A) P) ) => ( ( ( (ex_intro A) ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) ) x) ( ( ( (conj (P x) ) ( (dotpitp A) (x' : (etype A) => ( (dotpipp (P x') ) (_752 : (eprop (P x') ) => ( ( (eq A) x) x') ) ) ) ) ) Hx) (x' : (etype A) => (H欧1 : (eprop (P x') ) => ( ( ( (Huni x) x') Hx) H欧1) ) ) ) ) ) ) ) var_25) var_26) .
+[A : Utype, P : (_740 : (etype A) -> Uprop) , H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) , var_23 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) , var_24 : (eprop ( (uniqueness A) P) ) ] ( ( ( (case_33 A) P) H) ( ( ( ( (conj_case_33 A) P) H) var_23) var_24) ) --> ( ( (H欧0 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) => ( ( ( ( (case_34 A) P) H) H欧0) H欧0) ) var_23) var_24) .
+case_35 : (A : Utype -> (P : (_753 : (etype A) -> Uprop) -> (H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) -> (_754 : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) -> (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) ) ) ) ) .
+ex_intro_case_35 : (A : Utype -> (P : (_756 : (etype A) -> Uprop) -> (H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) -> (x : (etype A) -> (_757 : (eprop ( ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) x) ) -> (eprop ( (ex A) ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) ) ) ) ) ) ) ) .
+[A : Utype, P : (_758 : (etype A) -> Uprop) , H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) ] ( ( (ex_intro_case_35 A) P) H) --> ( (ex_intro A) ( (unique A) (x : (etype A) => (P x) ) ) ) .
+case_36 : (A : Utype -> (P : (_759 : (etype A) -> Uprop) -> (H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) -> (x : (etype A) -> (H欧0 : (eprop ( ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) x) ) -> (_761 : (eprop ( (and (P x) ) ( (dotpitp A) (x' : (etype A) => ( (dotpipp (P x') ) (_760 : (eprop (P x') ) => ( ( (eq A) x) x') ) ) ) ) ) ) -> (eprop ( (and ( (ex A) (x欧0 : (etype A) => (P x欧0) ) ) ) ( (uniqueness A) P) ) ) ) ) ) ) ) ) .
+conj_case_36 : (A : Utype -> (P : (_764 : (etype A) -> Uprop) -> (H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) -> (x : (etype A) -> (H欧0 : (eprop ( ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) x) ) -> (_768 : (eprop (P x) ) -> (_767 : (x' : (etype A) -> (_765 : (eprop (P x') ) -> (eprop ( ( (eq A) x) x') ) ) ) -> (eprop ( (and (P x) ) ( (dotpitp A) (x' : (etype A) => ( (dotpipp (P x') ) (_766 : (eprop (P x') ) => ( ( (eq A) x) x') ) ) ) ) ) ) ) ) ) ) ) ) ) .
+[A : Utype, P : (_769 : (etype A) -> Uprop) , H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) , x : (etype A) , H欧0 : (eprop ( ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) x) ) ] ( ( ( ( (conj_case_36 A) P) H) x) H欧0) --> ( (conj (P x) ) ( (dotpitp A) (x' : (etype A) => ( (dotpipp (P x') ) (_763 : (eprop (P x') ) => ( ( (eq A) x) x') ) ) ) ) ) .
+[A : Utype, P : (_762 : (etype A) -> Uprop) , H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) , x : (etype A) , H欧0 : (eprop ( ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) x) ) , var_29 : (eprop (P x) ) , var_30 : (x' : (etype A) -> (_770 : (eprop (P x') ) -> (eprop ( ( (eq A) x) x') ) ) ) ] ( ( ( ( ( (case_36 A) P) H) x) H欧0) ( ( ( ( ( ( (conj_case_36 A) P) H) x) H欧0) var_29) var_30) ) --> ( ( (Hx : (eprop (P x) ) => (Huni : (eprop ( (dotpitp A) (x' : (etype A) => ( (dotpipp (P x') ) (_771 : (eprop (P x') ) => ( ( (eq A) x) x') ) ) ) ) ) => ( ( ( (conj ( (ex A) (x欧0 : (etype A) => (P x欧0) ) ) ) ( (uniqueness A) P) ) ( ( ( (ex_intro A) (x欧0 : (etype A) => (P x欧0) ) ) x) Hx) ) (x' : (etype A) => (x'' : (etype A) => (Hx' : (eprop (P x') ) => (Hx'' : (eprop (P x'') ) => ( ( ( ( ( (trans_eq A) x') x) x'') ( ( ( (sym_eq A) x) x') ( (Huni x') Hx') ) ) ( (Huni x'') Hx'') ) ) ) ) ) ) ) ) var_29) var_30) .
+[A : Utype, P : (_755 : (etype A) -> Uprop) , H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) , var_27 : (etype A) , var_28 : (eprop ( ( (unique A) (x : (etype A) => (P x) ) ) var_27) ) ] ( ( ( (case_35 A) P) H) ( ( ( ( (ex_intro_case_35 A) P) H) var_27) var_28) ) --> ( ( (x : (etype A) => (H欧0 : (eprop ( ( (unique A) (x欧0 : (etype A) => (P x欧0) ) ) x) ) => ( ( ( ( ( (case_36 A) P) H) x) H欧0) H欧0) ) ) var_27) var_28) .
+unique_existence : (A : Utype -> (P : (_773 : (etype A) -> Uprop) -> (eprop ( (iff ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) ) ) ) .
+[] unique_existence --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_772 : (etype A) => dotprop) ) ) => ( ( ( (conj ( (dotpipp ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) (_736 : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) => ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) ) ) ( (dotpipp ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) (_737 : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) => ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) ) ) (H : (eprop ( (and ( (ex A) (x : (etype A) => (P x) ) ) ) ( (uniqueness A) P) ) ) => ( ( ( (case_33 A) P) H) H) ) ) (H : (eprop ( (ex A) ( (unique A) (x : (etype A) => (P x) ) ) ) ) => ( ( ( (case_35 A) P) H) H) ) ) ) ) .
+inhabited : (A : Utype -> Uprop) .
+inhabits : (A : Utype -> (_774 : (etype A) -> (eprop (inhabited A) ) ) ) .
+case_37 : (A : Utype -> (P : Uprop -> (f : (_775 : (etype A) -> (eprop P) ) -> (i : (eprop (inhabited A) ) -> (_776 : (eprop (inhabited A) ) -> (eprop P) ) ) ) ) ) .
+inhabits_case_37 : (A : Utype -> (P : Uprop -> (f : (_778 : (etype A) -> (eprop P) ) -> (i : (eprop (inhabited A) ) -> (_779 : (etype A) -> (eprop (inhabited A) ) ) ) ) ) ) .
+[A : Utype, P : Uprop, f : (_780 : (etype A) -> (eprop P) ) , i : (eprop (inhabited A) ) ] ( ( ( (inhabits_case_37 A) P) f) i) --> (inhabits A) .
+[A : Utype, P : Uprop, f : (_777 : (etype A) -> (eprop P) ) , i : (eprop (inhabited A) ) , var_31 : (etype A) ] ( ( ( ( (case_37 A) P) f) i) ( ( ( ( (inhabits_case_37 A) P) f) i) var_31) ) --> (f var_31) .
+inhabited_ind : (A : Utype -> (P : Uprop -> (f : (_782 : (etype A) -> (eprop P) ) -> (i : (eprop (inhabited A) ) -> (eprop P) ) ) ) ) .
+[] inhabited_ind --> (A : (etype dottype) => (P : (etype dotprop) => (f : (eprop ( (dotpitp A) (_781 : (etype A) => P) ) ) => (i : (eprop (inhabited A) ) => ( ( ( ( (case_37 A) P) f) i) i) ) ) ) ) .
+case_38 : (A : Utype -> (P : (_783 : (etype A) -> Uprop) -> (H : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (_784 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (eprop (inhabited A) ) ) ) ) ) .
+ex_intro_case_38 : (A : Utype -> (P : (_786 : (etype A) -> Uprop) -> (H : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (x : (etype A) -> (_787 : (eprop ( (x欧0 : (etype A) => (P x欧0) ) x) ) -> (eprop ( (ex A) (x欧0 : (etype A) => (P x欧0) ) ) ) ) ) ) ) ) .
+[A : Utype, P : (_788 : (etype A) -> Uprop) , H : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) ] ( ( (ex_intro_case_38 A) P) H) --> ( (ex_intro A) (x : (etype A) => (P x) ) ) .
+[A : Utype, P : (_785 : (etype A) -> Uprop) , H : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) , var_32 : (etype A) , var_33 : (eprop ( (x : (etype A) => (P x) ) var_32) ) ] ( ( ( (case_38 A) P) H) ( ( ( ( (ex_intro_case_38 A) P) H) var_32) var_33) ) --> ( ( (x : (etype A) => (H欧0 : (eprop (P x) ) => ( (inhabits A) x) ) ) var_32) var_33) .
+exists_inhabited : (A : Utype -> (P : (_790 : (etype A) -> Uprop) -> (_791 : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) -> (eprop (inhabited A) ) ) ) ) .
+[] exists_inhabited --> (A : (etype dottype) => (P : (etype ( (dotpitt A) (_789 : (etype A) => dotprop) ) ) => (H : (eprop ( (ex A) (x : (etype A) => (P x) ) ) ) => ( ( ( (case_38 A) P) H) H) ) ) ) .
+eq_stepl : (A : Utype -> (x : (etype A) -> (y : (etype A) -> (z : (etype A) -> (_793 : (eprop ( ( (eq A) x) y) ) -> (_792 : (eprop ( ( (eq A) x) z) ) -> (eprop ( ( (eq A) z) y) ) ) ) ) ) ) ) .
+[] eq_stepl --> (A : (etype dottype) => (x : (etype A) => (y : (etype A) => (z : (etype A) => (H1 : (eprop ( ( (eq A) x) y) ) => (H2 : (eprop ( ( (eq A) x) z) ) => ( ( ( ( ( (eq_ind A) x) (z欧0 : (etype A) => ( ( (eq A) z欧0) y) ) ) H1) z) H2) ) ) ) ) ) ) .
+iff_stepl : (A : Uprop -> (B : Uprop -> (C : Uprop -> (_805 : (eprop ( (iff A) B) ) -> (_804 : (eprop ( (iff A) C) ) -> (eprop ( (iff C) B) ) ) ) ) ) ) .
+[] iff_stepl --> (A : (etype dotprop) => (B : (etype dotprop) => (C : (etype dotprop) => (H : (eprop ( (iff A) B) ) => (H0 : (eprop ( (iff A) C) ) => ( ( ( ( (and_ind ( (dotpipp A) (_794 : (eprop A) => B) ) ) ( (dotpipp B) (_795 : (eprop B) => A) ) ) ( (iff C) B) ) (H1 : (eprop ( (dotpipp A) (_803 : (eprop A) => B) ) ) => (H2 : (eprop ( (dotpipp B) (_802 : (eprop B) => A) ) ) => ( ( ( ( (and_ind ( (dotpipp A) (_796 : (eprop A) => C) ) ) ( (dotpipp C) (_797 : (eprop C) => A) ) ) ( (iff C) B) ) (H欧0 : (eprop ( (dotpipp A) (_801 : (eprop A) => C) ) ) => (H3 : (eprop ( (dotpipp C) (_800 : (eprop C) => A) ) ) => ( ( ( (conj ( (dotpipp C) (_798 : (eprop C) => B) ) ) ( (dotpipp B) (_799 : (eprop B) => C) ) ) (H0欧0 : (eprop C) => ( (H4 : (eprop A) => ( (H3欧0 : (eprop B) => ( (H1欧0 : (eprop C) => ( (H欧1 : (eprop A) => H3欧0) (H2 H3欧0) ) ) (H欧0 H4) ) ) (H1 H4) ) ) (H3 H0欧0) ) ) ) (H0欧0 : (eprop B) => ( (H4 : (eprop A) => ( (H2欧0 : (eprop B) => ( (H1欧0 : (eprop C) => ( (H欧1 : (eprop A) => H1欧0) (H3 H1欧0) ) ) (H欧0 H4) ) ) (H1 H4) ) ) (H2 H0欧0) ) ) ) ) ) ) H0) ) ) ) H) ) ) ) ) ) .
+;Finished module Logic
diff --git a/t/linearity.eu b/t/linearity.eu
new file mode 100644
--- /dev/null
+++ b/t/linearity.eu
@@ -0,0 +1,3 @@
+Nat: Type.
+Q : (Nat -> Nat) -> Nat -> Nat -> Type.
+[a:Nat, f : Nat -> Nat] Q f a (f a) --> Nat.
diff --git a/t/logic.eu b/t/logic.eu
new file mode 100644
--- /dev/null
+++ b/t/logic.eu
@@ -0,0 +1,49 @@
+False : coc.Utype.
+
+True : coc.Utype.
+
+I : coc.etype True.
+
+eq : t : coc.Utype -> coc.etype t -> coc.etype t -> Type.
+
+eq_ : t : coc.Utype -> coc.etype t -> coc.etype t -> coc.Utype.
+
+[ t : coc.Utype
+, x : coc.etype t
+, y : coc.etype t ]
+eq t x y --> coc.etype (eq_ t x y).
+
+
+refl_equal : t : coc.Utype -> x : coc.etype t -> eq t x x.
+
+eq_rec : t : coc.Utype
+ -> x : coc.etype t
+ -> p : (coc.etype t -> coc.Utype)
+ -> g : coc.etype (p x)
+ -> y : coc.etype t
+ -> h : eq t x y
+ -> coc.etype (p y).
+
+[ t : coc.Utype
+, x : coc.etype t
+, p : coc.etype t -> coc.Utype
+, f : coc.etype (p x) ]
+eq_rec t x p f x (refl_equal t x) --> f.
+
+f_equal
+ : A : coc.Utype
+ -> B : coc.Utype
+ -> f : (coc.etype A -> coc.etype B)
+ -> x : coc.etype A
+ -> y : coc.etype A
+ -> H : eq A x y
+ -> eq B (f x) (f y).
+
+[] f_equal -->
+ A : coc.Utype
+ => B : coc.Utype
+ => f : (coc.etype A -> coc.etype B)
+ => x : coc.etype A
+ => y : coc.etype A
+ => H : eq A x y
+ => eq_rec A x (z : coc.etype A => eq_ B (f x) (f z)) (refl_equal B (f x)) y H.
diff --git a/t/loop.eu b/t/loop.eu
new file mode 100644
--- /dev/null
+++ b/t/loop.eu
@@ -0,0 +1,4 @@
+A : Type.
+[] A --> A -> A.
+t : A.
+[] t --> x : A => x.
diff --git a/t/nat.eu b/t/nat.eu
new file mode 100644
--- /dev/null
+++ b/t/nat.eu
@@ -0,0 +1,17 @@
+nat : Type.
+
+0 : nat.
+
+S : nat -> nat.
+
+1 : nat.
+
+[] 1 --> (S 0).
+
+plus : nat -> nat -> nat.
+[x : nat] plus x 0 --> x.
+[x : nat] plus 0 x --> x.
+[x : nat, y : nat] plus x (S y) --> S (plus x y).
+[x : nat, y : nat] plus (S x) y --> S (plus x y).
+
+
diff --git a/t/peano.eu b/t/peano.eu
new file mode 100644
--- /dev/null
+++ b/t/peano.eu
@@ -0,0 +1,66 @@
+nat : Type.
+
+nat_ : coc.Utype.
+
+[] nat --> coc.etype nat_.
+
+0 : nat.
+
+S : nat -> nat.
+
+nat_rec : t : coc.Utype
+ -> coc.etype t
+ -> (nat -> coc.etype t -> coc.etype t)
+ -> nat
+ -> coc.etype t.
+
+[ t : coc.Utype
+, a : coc.etype t
+, f : nat -> coc.etype t -> coc.etype t
+] nat_rec t a f 0 --> a.
+
+[ t : coc.Utype
+, a : coc.etype t
+, f : nat -> coc.etype t -> coc.etype t
+, n : nat
+] nat_rec t a f (S n) --> f n (nat_rec t a f (S n)).
+
+plus : nat -> nat -> nat.
+
+[] plus --> x : nat => y : nat => nat_rec nat_ 0 (x : nat => y : nat => y) x.
+
+plus2 : nat -> nat -> nat.
+
+[x : nat] plus2 x 0 --> x.
+[x : nat] plus2 0 x --> x.
+[x : nat, y : nat] plus2 x (S y) --> S (plus2 x y).
+[x : nat, y : nat] plus2 (S x) y --> S (plus2 x y).
+
+eq_S : x : nat
+ -> y : nat
+ -> logic.eq nat_ x y
+ -> logic.eq nat_ (S x) (S y).
+
+[] eq_S --> logic.f_equal nat_ nat_ S.
+
+eq_S2 : coc.etype (coc.dotpi1 nat_ (x : nat
+ => coc.dotpi1 nat_ (y : nat
+ => coc.dotpi1 (logic.eq_ nat_ x y) (h : logic.eq nat_ x y
+ => logic.eq_ nat_ (S x) (S y))))).
+
+[] eq_S2 --> eq_S.
+
+
+pred : nat -> nat.
+
+[] pred --> nat_rec nat_ 0 (x:nat => nat => x).
+
+pred2 : nat -> nat.
+
+[] pred2 0 --> 0.
+
+[x : nat] pred2 (S x) --> x.
+
+pred_Sn : n : nat -> logic.eq nat_ n (pred (S n)).
+
+[] pred_Sn --> n : nat => logic.refl_equal nat_ n.
diff --git a/t/plus.eu b/t/plus.eu
new file mode 100644
--- /dev/null
+++ b/t/plus.eu
@@ -0,0 +1,7 @@
+P : nat.nat -> Type.
+
+y : P (nat.S nat.0).
+
+w : P (nat.S nat.0).
+
+[] w --> (x : P (nat.plus nat.0 (nat.S nat.0)) => x) y.
diff --git a/t/sigma.eu b/t/sigma.eu
new file mode 100644
--- /dev/null
+++ b/t/sigma.eu
@@ -0,0 +1,41 @@
+o : Type.
+eps : o -> Type.
+
+sigma_ : A : o -> (eps A -> o) -> o.
+exist_ : A : o -> P : (eps A -> o) -> x : eps A -> eps (P x) -> eps (sigma_ A P).
+
+fst : A : o -> P : (eps A -> o) -> eps (sigma_ A P) -> eps A.
+[A : o, P : eps A -> o, w : eps A, pi : P w]
+fst _ _ (eps (exist_ _ _ w pi)) --> w.
+
+snd : A : o -> P : (eps A -> o) -> s : eps (sigma_ A P) -> eps (P (fst A P s)).
+[A : o, P : eps A -> o, w : eps A, pi : P w]
+snd _ _ (eps (exist_ _ _ w pi)) --> pi.
+
+
+;; test
+
+nat : Type.
+nat_ : o.
+
+O : nat.
+S : nat -> nat.
+
+plus : nat -> nat -> nat.
+[n:nat,m:nat] plus (S n) m --> S (plus n m).
+[n:nat,m:nat] plus O m --> m.
+
+eq : nat -> nat -> Type.
+[n:nat,m:nat] eq (S n) (S m) --> eq n m.
+ax: eq O O.
+
+eq_ : nat -> nat -> o.
+
+[x:nat,y:nat] eps (eq_ x y) --> eq x y.
+[] eps nat_ --> nat.
+
+thm : n:nat -> eps (sigma_ nat_ (m:nat => eq_ (plus (S O) n) m)).
+[] thm --> n:nat => exist_ nat_ (m:nat => eq_ (plus (S O) n) m) (S n) ax.
+
+verif : eq (fst nat_ (m:nat => eq_ (plus (S O) O) m) (thm O)) (S O).
+[] verif --> ax.
diff --git a/t/sigma2.eu b/t/sigma2.eu
new file mode 100644
--- /dev/null
+++ b/t/sigma2.eu
@@ -0,0 +1,31 @@
+nat : Type.
+0 : nat.
+S : nat -> nat.
+
+plus:nat -> nat -> nat.
+[x:nat] plus x 0 --> x.
+[x:nat] (plus 0) x --> x.
+[x:nat, y:nat] plus x (S y) --> S (plus x y).
+[x:nat, y:nat] (plus (S x)) y --> S (plus x y).
+
+eqnat : nat -> nat -> Type.
+ax: eqnat 0 0.
+[n:nat, m:nat] eqnat (S n) (S m) --> eqnat n m.
+
+o : Type.
+eps : o -> Type.
+
+_nat : o.
+[] (eps _nat) --> nat.
+
+_eqnat : nat -> nat -> o.
+[n:nat,m:nat] eps (_eqnat n m) --> eqnat n m.
+
+[x:o] eps x --> (eps x).
+
+sigma : a:o -> (eps a -> o) -> Type.
+
+
+th : Type.
+[] th --> sigma _nat (n:nat => _nat).
+
diff --git a/t/stt1.eu b/t/stt1.eu
new file mode 100644
--- /dev/null
+++ b/t/stt1.eu
@@ -0,0 +1,57 @@
+; Simple Type Theory as a theory in predicate logic.
+;
+; Use reverse polish notation for names, eg:
+; * i -> i becomes iia,
+; * i -> i -> i becomes iiiaa
+; * (i -> i) -> i becomes iiaia
+
+o : Type.
+eps : o -> Type.
+
+i : Type.
+iia : Type.
+iiiaa : Type.
+iiiiaaa : Type.
+iiaia : Type.
+iiiaaiiaiiaaa : Type.
+iiaiiaa : Type.
+
+ooa : Type.
+oooaa : Type.
+ioaoa : Type.
+iiaoaoa : Type.
+iiiaaoaoa : Type.
+
+O : i.
+S : iia.
+
+ap_iia : iia -> i -> i.
+ap_iiiaa : iiiaa -> i -> iia.
+ap_iiiaaiiaiiaaa : iiiaaiiaiiaaa -> iiiaa -> iiaiiaa.
+ap_iiaiiaa : iiaiiaa -> iia -> iia.
+
+ap_ooa : ooa -> o -> o.
+ap_oooaa : oooaa -> o -> ooa.
+
+one : i.
+[] one --> ap_iia S O.
+
+imp : oooaa.
+forall_i : ioaoa.
+forall_iia : iiaoaoa.
+forall_iiiaa : iiiaaoaoa.
+
+; S and K combinators.
+S_iiiaaiiaiiaaa : iiiaaiiaiiaaa.
+K_iiiaa : iiiaa.
+
+[ x : iiiaa
+, y : iia
+, z : i ]
+ap_iia (ap_iiaiiaa (ap_iiiaaiiaiiaaa S_iiiaaiiaiiaaa x) y) z --> ap_iia (ap_iiiaa x z) (ap_iia y z).
+[ x : i
+, y : i ]
+ap_iia (ap_iiiaa K_iiiaa x) y --> x.
+[ x : o
+, y : o ]
+eps (ap_ooa (ap_oooaa imp x) y) --> eps x -> eps y.
diff --git a/t/test1.eu b/t/test1.eu
new file mode 100644
--- /dev/null
+++ b/t/test1.eu
@@ -0,0 +1,8 @@
+nat : Type.
+0 : nat.
+S : nat -> nat.
+a : Type.
+vec : nat -> Type.
+vec' : n : nat -> vec n.
+nil : vec 0.
+cons : n : nat -> a -> vec n -> vec (S n).
diff --git a/t/testcomplet.eu b/t/testcomplet.eu
new file mode 100644
--- /dev/null
+++ b/t/testcomplet.eu
@@ -0,0 +1,60 @@
+Uset : Type.
+Uprop : Type.
+Utype : Type.
+
+eprop : x : Uprop -> Type.
+eset : x : Uset -> Type.
+etype : x : Utype -> Type.
+
+dotset : Utype.
+dotprop : Utype.
+
+dotpipp : x : Uprop -> y : (eprop x -> Uprop) -> Uprop.
+dotpips : x : Uprop -> y : (eprop x -> Uset) -> Uset.
+dotpipt : x : Uprop -> y : (eprop x -> Utype) -> Utype.
+dotpisp : x : Uset -> y : (eset x -> Uprop) -> Uprop.
+dotpitp : x : Utype -> y : (etype x -> Uprop) -> Uprop.
+dotpist : x : Uset -> y : (eset x -> Utype) -> Utype.
+dotpits : x : Utype -> y : (etype x -> Uset) -> Uset.
+dotpiss : x : Uset -> y : (eset x -> Uset) -> Uset.
+dotpitt : x : Utype -> y : (etype x -> Utype) -> Utype.
+
+
+[x:Uprop, y : eprop x -> Uprop]
+ eprop (dotpipp x y) --> w : eprop x -> eprop (y w).
+
+[x:Uset, y : eset x -> Uprop]
+ eprop (dotpisp x y) --> w : eset x -> eprop (y w).
+
+[x:Utype, y : etype x -> Uprop]
+ eprop (dotpitp x y) --> w : etype x -> eprop (y w).
+
+[x:Uprop, y : eprop x -> Uset]
+ eset (dotpips x y) --> w : eprop x -> eset (y w).
+
+[x:Utype, y : etype x -> Uset]
+ eset (dotpits x y) --> w : etype x -> eset (y w).
+
+[x:Uset, y : eset x -> Uset]
+ eset (dotpiss x y) --> w : eset x -> eset (y w).
+
+[x:Uset, y : eset x -> Utype]
+ etype (dotpist x y) --> w : eset x -> etype (y w).
+
+[x:Utype, y : etype x -> Utype]
+ etype (dotpitt x y) --> w : etype x -> etype (y w).
+
+[x:Uprop, y : eprop x -> Utype]
+ etype (dotpipt x y) --> w : eprop x -> etype (y w).
+
+
+[] (etype dotset) --> Uset.
+[] (etype dotprop) --> Uprop.
+simple : (P : Uprop -> (_ : (eprop P) -> (eprop P) ) ) .
+[] simple --> (P : (etype dotprop) => (H : (eprop P) => H) ) .
+K : (P : Uprop -> (Q : Uprop -> (_ : (eprop P) -> (_ : (eprop Q) -> (eprop P) ) ) ) ) .
+[] K --> (P : (etype dotprop) => (Q : (etype dotprop) => (H : (eprop P) => (H0 : (eprop Q) => ( (simple P) H) ) ) ) ) .
+S : (P : Uprop -> (Q : Uprop -> (R : Uprop -> (_ : (_ : (eprop P) -> (_ : (eprop Q) -> (eprop R) ) ) -> (_ : (_ : (eprop P) -> (eprop Q) ) -> (_ : (eprop P) -> (eprop R) ) ) ) ) ) ) .
+[] S --> (P : (etype dotprop) => (Q : (etype dotprop) => (R : (etype dotprop) => (H : (eprop ( (dotpipp P) (_ : (eprop P) => ( (dotpipp Q) (_ : (eprop Q) => R) ) ) ) ) => (H0 : (eprop ( (dotpipp P) (_ : (eprop P) => Q) ) ) => (H1 : (eprop P) => ( (H H1) (H0 H1) ) ) ) ) ) ) ) .
+I : (P : Uprop -> (_ : (eprop P) -> (eprop P) ) ) .
+[] I --> (P : (etype dotprop) => ( ( ( ( (S P) ( (dotpipp P) (_ : (eprop P) => P) ) ) P) ( (K P) ( (dotpipp P) (_ : (eprop P) => P) ) ) ) ( (K P) P) ) ) .
diff --git a/t/vec.eu b/t/vec.eu
new file mode 100644
--- /dev/null
+++ b/t/vec.eu
@@ -0,0 +1,21 @@
+nat : Type.
+O : nat.
+S : nat -> nat.
+
+o : Type.
+
+nat_ : nat -> o.
+
+eps : o -> Type.
+[n : nat] eps (nat_ n) --> Nat n.
+
+Nat : nat -> Type.
+[n : nat] Nat n --> P : (nat -> o) -> eps (P O)
+ -> (m : nat -> (Nat m) -> eps (P m) -> eps (P (S m)))
+ -> eps (P n).
+
+one : Nat (S O).
+[p : nat -> o, z : eps (p O), s : m : nat -> (Nat m) -> eps (P m) -> eps (P (S m))]
+one p z s --> s O z.
+
+;; suc : n : nat -> Nat n -> Nat (S n).