diff --git a/.gitignore b/.gitignore
--- a/.gitignore
+++ b/.gitignore
@@ -11,3 +11,21 @@
 *.hi
 *~
 *#
+.stack-work/
+cabal-dev
+*.chi
+*.chs.h
+*.dyn_o
+*.dyn_hi
+.hpc
+.hsenv
+.cabal-sandbox/
+cabal.sandbox.config
+*.prof
+*.aux
+*.hp
+*.eventlog
+cabal.project.local
+cabal.project.local~
+.HTF/
+.ghc.environment.*
diff --git a/.travis.yml b/.travis.yml
--- a/.travis.yml
+++ b/.travis.yml
@@ -1,61 +1,146 @@
-# NB: don't set `language: haskell` here
+# This Travis job script has been generated by a script via
+#
+#   runghc make_travis_yml_2.hs '-o' '.travis.yml' '--irc-channel=irc.freenode.org#haskell-lens' '--no-no-tests-no-bench' '--no-installed' 'cabal.project'
+#
+# For more information, see https://github.com/hvr/multi-ghc-travis
+#
+language: c
+sudo: false
 
-# See also https://github.com/hvr/multi-ghc-travis for more information
-env:
- # we have to use CABALVER=1.16 for GHC<7.6 as well, as there's
- # no package for earlier cabal versions in the PPA
- - GHCVER=7.4.2 CABALVER=1.16
- - GHCVER=7.6.3 CABALVER=1.16
- - GHCVER=7.8.2 CABALVER=1.18
- # NOTE: we can't use Cabal 1.20 yet due to
- #  https://github.com/haskell/cabal/issues/1806
- - GHCVER=head  CABALVER=1.20
+git:
+  submodules: false  # whether to recursively clone submodules
 
+notifications:
+  irc:
+    channels:
+      - "irc.freenode.org#haskell-lens"
+    skip_join: true
+    template:
+      - "\x0313eq\x03/\x0306%{branch}\x03 \x0314%{commit}\x03 %{build_url} %{message}"
+
+cache:
+  directories:
+    - $HOME/.cabal/packages
+    - $HOME/.cabal/store
+
+before_cache:
+  - rm -fv $HOME/.cabal/packages/hackage.haskell.org/build-reports.log
+  # remove files that are regenerated by 'cabal update'
+  - rm -fv $HOME/.cabal/packages/hackage.haskell.org/00-index.*
+  - rm -fv $HOME/.cabal/packages/hackage.haskell.org/*.json
+  - rm -fv $HOME/.cabal/packages/hackage.haskell.org/01-index.cache
+  - rm -fv $HOME/.cabal/packages/hackage.haskell.org/01-index.tar
+  - rm -fv $HOME/.cabal/packages/hackage.haskell.org/01-index.tar.idx
+
+  - rm -rfv $HOME/.cabal/packages/head.hackage
+
 matrix:
+  include:
+    - compiler: "ghc-7.0.4"
+    # env: TEST=--disable-tests BENCH=--disable-benchmarks
+      addons: {apt: {packages: [*apt_packages,cabal-install-2.0,ghc-7.0.4], sources: [hvr-ghc]}}
+    - compiler: "ghc-7.2.2"
+    # env: TEST=--disable-tests BENCH=--disable-benchmarks
+      addons: {apt: {packages: [*apt_packages,cabal-install-2.0,ghc-7.2.2], sources: [hvr-ghc]}}
+    - compiler: "ghc-7.4.2"
+    # env: TEST=--disable-tests BENCH=--disable-benchmarks
+      addons: {apt: {packages: [*apt_packages,cabal-install-2.0,ghc-7.4.2], sources: [hvr-ghc]}}
+    - compiler: "ghc-7.6.3"
+    # env: TEST=--disable-tests BENCH=--disable-benchmarks
+      addons: {apt: {packages: [*apt_packages,cabal-install-2.0,ghc-7.6.3], sources: [hvr-ghc]}}
+    - compiler: "ghc-7.8.4"
+    # env: TEST=--disable-tests BENCH=--disable-benchmarks
+      addons: {apt: {packages: [*apt_packages,cabal-install-2.0,ghc-7.8.4], sources: [hvr-ghc]}}
+    - compiler: "ghc-7.10.3"
+    # env: TEST=--disable-tests BENCH=--disable-benchmarks
+      addons: {apt: {packages: [*apt_packages,cabal-install-2.0,ghc-7.10.3], sources: [hvr-ghc]}}
+    - compiler: "ghc-8.0.2"
+    # env: TEST=--disable-tests BENCH=--disable-benchmarks
+      addons: {apt: {packages: [*apt_packages,cabal-install-2.0,ghc-8.0.2], sources: [hvr-ghc]}}
+    - compiler: "ghc-8.2.2"
+    # env: TEST=--disable-tests BENCH=--disable-benchmarks
+      addons: {apt: {packages: [*apt_packages,cabal-install-2.0,ghc-8.2.2], sources: [hvr-ghc]}}
+    - compiler: "ghc-8.4.1"
+      env: GHCHEAD=true
+      addons: {apt: {packages: [*apt_packages,cabal-install-head,ghc-8.4.1], sources: [hvr-ghc]}}
+    - compiler: "ghc-head"
+      env: GHCHEAD=true
+      addons: {apt: {packages: [*apt_packages,cabal-install-head,ghc-head], sources: [hvr-ghc]}}
+
   allow_failures:
-   - env: GHCVER=head  CABALVER=1.20
+    - compiler: "ghc-7.0.4"
+    - compiler: "ghc-7.2.2"
+    - compiler: "ghc-8.4.1"
+    - compiler: "ghc-head"
 
-# Note: the distinction between `before_install` and `install` is not
-#       important.
 before_install:
- - travis_retry sudo add-apt-repository -y ppa:hvr/ghc
- - travis_retry sudo apt-get update
- - travis_retry sudo apt-get install cabal-install-$CABALVER ghc-$GHCVER
- - export PATH=/opt/ghc/$GHCVER/bin:/opt/cabal/$CABALVER/bin:$PATH
- - cabal --version
+  - HC=${CC}
+  - HCPKG=${HC/ghc/ghc-pkg}
+  - unset CC
+  - ROOTDIR=$(pwd)
+  - mkdir -p $HOME/.local/bin
+  - "PATH=/opt/ghc/bin:/opt/ghc-ppa-tools/bin:$HOME/local/bin:$PATH"
+  - HCNUMVER=$(( $(${HC} --numeric-version|sed -E 's/([0-9]+)\.([0-9]+)\.([0-9]+).*/\1 * 10000 + \2 * 100 + \3/') ))
+  - echo $HCNUMVER
 
 install:
- - travis_retry cabal update
- - cabal install --only-dependencies
+  - cabal --version
+  - echo "$(${HC} --version) [$(${HC} --print-project-git-commit-id 2> /dev/null || echo '?')]"
+  - BENCH=${BENCH---enable-benchmarks}
+  - TEST=${TEST---enable-tests}
+  - HADDOCK=${HADDOCK-true}
+  - INSTALLED=${INSTALLED-true}
+  - GHCHEAD=${GHCHEAD-false}
+  - travis_retry cabal update -v
+  - "sed -i.bak 's/^jobs:/-- jobs:/' ${HOME}/.cabal/config"
+  - rm -fv cabal.project cabal.project.local
+  # Overlay Hackage Package Index for GHC HEAD: https://github.com/hvr/head.hackage
+  - |
+    if $GHCHEAD; then
+      sed -i.bak 's/-- allow-newer:.*/allow-newer: *:base, *:template-haskell, *:ghc, *:Cabal/' ${HOME}/.cabal/config
 
-# Here starts the actual work to be performed for the package under
-# test; any command which exits with a non-zero exit code causes the
-# build to fail.
+      echo 'repository head.hackage'                                                        >> ${HOME}/.cabal/config
+      echo '   url: http://head.hackage.haskell.org/'                                       >> ${HOME}/.cabal/config
+      echo '   secure: True'                                                                >> ${HOME}/.cabal/config
+      echo '   root-keys: 07c59cb65787dedfaef5bd5f987ceb5f7e5ebf88b904bbd4c5cbdeb2ff71b740' >> ${HOME}/.cabal/config
+      echo '              2e8555dde16ebd8df076f1a8ef13b8f14c66bad8eafefd7d9e37d0ed711821fb' >> ${HOME}/.cabal/config
+      echo '              8f79fd2389ab2967354407ec852cbe73f2e8635793ac446d09461ffb99527f6e' >> ${HOME}/.cabal/config
+      echo '   key-threshold: 3'                                                            >> ${HOME}/.cabal.config
+
+      cabal new-update head.hackage -v
+    fi
+  - grep -Ev -- '^\s*--' ${HOME}/.cabal/config | grep -Ev '^\s*$'
+  - "printf 'packages: \".\"\\n' > cabal.project"
+  - cat cabal.project
+  - if [ -f "./configure.ac" ]; then
+      (cd "." && autoreconf -i);
+    fi
+  - rm -f cabal.project.freeze
+  - cabal new-build -w ${HC} ${TEST} ${BENCH} --project-file="cabal.project" --dep -j2 all
+  - rm -rf "."/.ghc.environment.* "."/dist
+  - DISTDIR=$(mktemp -d /tmp/dist-test.XXXX)
+
+# Here starts the actual work to be performed for the package under test;
+# any command which exits with a non-zero exit code causes the build to fail.
 script:
- # -v2 provides useful information for debugging
- - cabal configure -v2
+  # test that source-distributions can be generated
+  - (cd "." && cabal sdist)
+  - mv "."/dist/eq-*.tar.gz ${DISTDIR}/
+  - cd ${DISTDIR} || false
+  - find . -maxdepth 1 -name '*.tar.gz' -exec tar -xvf '{}' \;
+  - "printf 'packages: eq-*/*.cabal\\n' > cabal.project"
+  - cat cabal.project
 
- # this builds all libraries and executables
- # (including tests/benchmarks)
- - cabal build
 
- # tests that a source-distribution can be generated
- - cabal sdist
+  # build & run tests, build benchmarks
+  - cabal new-build -w ${HC} ${TEST} ${BENCH} all
 
- # check that the generated source-distribution can be built & installed
- - export SRC_TGZ=$(cabal info . | awk '{print $2 ".tar.gz";exit}') ;
-   cd dist/;
-   if [ -f "$SRC_TGZ" ]; then
-      cabal install --force-reinstalls "$SRC_TGZ";
-   else
-      echo "expected '$SRC_TGZ' not found";
-      exit 1;
-   fi
+  # cabal check
+  - (cd eq-* && cabal check)
 
-notifications:
-  irc:
-    channels:
-      - "irc.freenode.org#haskell-lens"
-    skip_join: true
-    template:
-      - "\x0313eq\x03/\x0306%{branch}\x03 \x0314%{commit}\x03 %{build_url} %{message}"
+  # haddock
+  - rm -rf ./dist-newstyle
+  - if $HADDOCK; then cabal new-haddock -w ${HC} ${TEST} ${BENCH} all; else echo "Skipping haddock generation";fi
+
+# REGENDATA ["-o",".travis.yml","--irc-channel=irc.freenode.org#haskell-lens","--no-no-tests-no-bench","--no-installed","cabal.project"]
+# EOF
diff --git a/CHANGELOG.markdown b/CHANGELOG.markdown
--- a/CHANGELOG.markdown
+++ b/CHANGELOG.markdown
@@ -1,3 +1,13 @@
+4.1
+---
+* Add `TestEquality` and `TestCoercion` instances for `(:=)`.
+* Add `fromLeibniz` and `toLeibniz` functions for converting between `(:~:)`
+  (from `Data.Type.Equality`) and `(:=)`.
+* Add a `reprLeibniz` function to convert `(:=)` to a `Coercion`
+  (i.e., representational equality).
+* Make `(:=)` a newtype.
+* We can remove the `Trustworthy` claim and infer as `Safe` on modern GHCs.
+
 4.0.2
 -----
 * Made := kind polymorphic.
diff --git a/README.markdown b/README.markdown
--- a/README.markdown
+++ b/README.markdown
@@ -1,7 +1,7 @@
 eq
 ==
 
-[![Build Status](https://secure.travis-ci.org/ekmett/eq.png?branch=master)](http://travis-ci.org/ekmett/eq)
+[![Hackage](https://img.shields.io/hackage/v/eq.svg)](https://hackage.haskell.org/package/eq) [![Build Status](https://secure.travis-ci.org/ekmett/eq.png?branch=master)](http://travis-ci.org/ekmett/eq)
 
 This package provides a data type that witnesses equality between two types using Leibnizian equality.
 
diff --git a/eq.cabal b/eq.cabal
--- a/eq.cabal
+++ b/eq.cabal
@@ -1,6 +1,6 @@
 name:          eq
 category:      Type System
-version:       4.0.4
+version:       4.1
 license:       BSD3
 cabal-version: >= 1.6
 license-file:  LICENSE
@@ -11,8 +11,17 @@
 bug-reports:   http://github.com/ekmett/eq/issues
 copyright:     Copyright (C) 2011-2013 Edward A. Kmett
 synopsis:      Leibnizian equality
-description:   Leibnizian equality
+description:   Leibnizian equality.
 build-type:    Simple
+tested-with:   GHC == 7.0.4
+             , GHC == 7.2.2
+             , GHC == 7.4.2
+             , GHC == 7.6.3
+             , GHC == 7.8.4
+             , GHC == 7.10.3
+             , GHC == 8.0.2
+             , GHC == 8.2.2
+             , GHC == 8.4.1
 
 extra-source-files:
   .ghci
@@ -39,7 +48,7 @@
   exposed-modules:
     Data.Eq.Type
 
-  if impl(ghc >= 7.0) && impl(ghc < 7.6)
+  if impl(ghc >= 7.0)
     extensions: TypeFamilies
     cpp-options: -DLANGUAGE_TypeFamilies
 
diff --git a/src/Data/Eq/Type.hs b/src/Data/Eq/Type.hs
--- a/src/Data/Eq/Type.hs
+++ b/src/Data/Eq/Type.hs
@@ -1,13 +1,15 @@
 {-# LANGUAGE CPP, Rank2Types, TypeOperators #-}
-#if defined(__GLASGOW_HASKELL__) && __GLASGOW_HASKELL__ >= 702
-{-# LANGUAGE Trustworthy #-}
-#endif
 #if defined(__GLASGOW_HASKELL__) && __GLASGOW_HASKELL__ >= 706
 {-# LANGUAGE PolyKinds #-}
 #endif
 #if defined(__GLASGOW_HASKELL__) && __GLASGOW_HASKELL__ >= 707
 {-# LANGUAGE RoleAnnotations #-}
 #endif
+#if defined(__GLASGOW_HASKELL__) && MIN_VERSION_base(4,7,0)
+#define HAS_DATA_TYPE_EQUALITY 1
+{-# LANGUAGE GADTs #-}
+{-# LANGUAGE ScopedTypeVariables #-}
+#endif
 
 -----------------------------------------------------------------------------
 -- |
@@ -20,7 +22,7 @@
 -- Portability :  rank2 types, type operators, (optional) type families
 --
 -- Leibnizian equality. Injectivity in the presence of type families
--- is provided by a generalization of a trick by Oleg Kiselyv posted here:
+-- is provided by a generalization of a trick by Oleg Kiselyov posted here:
 --
 -- <http://www.haskell.org/pipermail/haskell-cafe/2010-May/077177.html>
 ----------------------------------------------------------------------------
@@ -44,18 +46,33 @@
   , lower2
   , lower3
 #endif
+#ifdef HAS_DATA_TYPE_EQUALITY
+  -- * 'Eq.:~:' equivalence
+  -- | "Data.Type.Equality" GADT definition is equivalent in power
+  , fromLeibniz
+  , toLeibniz
+
+  -- * 'Co.Coercion' conversion
+  -- | Leibnizian equality can be converted to representational equality
+  , reprLeibniz
+#endif
   ) where
 
-import Prelude (flip)
+import Prelude (Maybe(..), flip)
 import Control.Category
 import Data.Semigroupoid
 import Data.Groupoid
 
+#ifdef HAS_DATA_TYPE_EQUALITY
+import qualified Data.Type.Coercion as Co
+import qualified Data.Type.Equality as Eq
+#endif
+
 infixl 4 :=
 
 -- | Leibnizian equality states that two things are equal if you can
--- substite one for the other in all contexts
-data a := b = Refl { subst :: forall c. c a -> c b }
+-- substitute one for the other in all contexts
+newtype a := b = Refl { subst :: forall c. c a -> c b }
 #if defined(__GLASGOW_HASKELL__) && __GLASGOW_HASKELL__ >= 707
 type role (:=) nominal nominal
 #endif
@@ -110,24 +127,44 @@
 lift3' ab cd ef = lift3 ab `subst` lift2 cd `subst` lift ef
 
 #ifdef LANGUAGE_TypeFamilies
-type family Inj f
+type family Inj (f :: *) :: *
 type instance Inj (f a) = a
 newtype Lower a b = Lower { unlower :: Inj a := Inj b }
 -- | Type constructors are injective, so you can lower equality through any type constructor
-lower :: f a := f b -> a := b
+lower :: forall (a :: *) (b :: *) (f :: * -> *). f a := f b -> a := b
 lower eq = unlower (subst eq (Lower refl :: Lower (f a) (f a)))
 
-type family Inj2 f
-type instance Inj2 (f a b) = a
+type family Inj2 (f :: *) :: *
+type instance Inj2 (f a (b :: *)) = a
 newtype Lower2 a b = Lower2 { unlower2 :: Inj2 a := Inj2 b }
 -- | ... in any position
-lower2 :: f a c := f b c -> a := b
+lower2 :: forall (a :: *) (b :: *) (c :: *) (f :: * -> * -> *).
+    f a c := f b c -> a := b
 lower2 eq = unlower2 (subst eq (Lower2 refl :: Lower2 (f a c) (f a c)))
 
-type family Inj3 f
-type instance Inj3 (f a b c) = a
+type family Inj3 (f :: *) :: *
+type instance Inj3 (f a (b :: *) (c :: *)) = a
 newtype Lower3 a b = Lower3 { unlower3 :: Inj3 a := Inj3 b }
-lower3 :: f a c d := f b c d -> a := b
+-- | But unfortunately these definitions aren't polykinded. Everything is just a star.
+lower3 :: forall (a :: *) (b :: *) (c :: *) (d :: *) (f :: * -> * -> * -> *).
+    f a c d := f b c d -> a := b
 lower3 eq = unlower3 (subst eq (Lower3 refl :: Lower3 (f a c d) (f a c d)))
 
+#endif
+
+#ifdef HAS_DATA_TYPE_EQUALITY
+fromLeibniz :: a := b -> a Eq.:~: b
+fromLeibniz a = subst a Eq.Refl
+
+toLeibniz :: a Eq.:~: b -> a := b
+toLeibniz Eq.Refl = refl
+
+instance Eq.TestEquality ((:=) a) where
+  testEquality fa fb = Just (fromLeibniz (trans (symm fa) fb))
+
+reprLeibniz :: a := b -> Co.Coercion a b
+reprLeibniz a = subst a Co.Coercion
+
+instance Co.TestCoercion ((:=) a) where
+  testCoercion fa fb = Just (reprLeibniz (trans (symm fa) fb))
 #endif
