diff --git a/src/librustc_typeck/constrained_generic_params.rs b/src/librustc_typeck/constrained_generic_params.rs
index aff3768e35c55..b16aa6ff3b2e4 100644
--- a/src/librustc_typeck/constrained_generic_params.rs
+++ b/src/librustc_typeck/constrained_generic_params.rs
@@ -79,10 +79,18 @@ impl<'tcx> TypeVisitor<'tcx> for ParameterCollector {
     }
 
     fn visit_const(&mut self, c: &'tcx ty::Const<'tcx>) -> bool {
-        if let ty::ConstKind::Param(data) = c.val {
-            self.parameters.push(Parameter::from(data));
+        match c.val {
+            ty::ConstKind::Unevaluated(..) if !self.include_nonconstraining => {
+                // Constant expressions are not injective
+                return c.ty.visit_with(self);
+            }
+            ty::ConstKind::Param(data) => {
+                self.parameters.push(Parameter::from(data));
+            }
+            _ => {}
         }
-        false
+
+        c.super_visit_with(self)
     }
 }
 
diff --git a/src/librustc_typeck/impl_wf_check.rs b/src/librustc_typeck/impl_wf_check.rs
index 0a765a1f9c93c..7ee25bbede2c5 100644
--- a/src/librustc_typeck/impl_wf_check.rs
+++ b/src/librustc_typeck/impl_wf_check.rs
@@ -9,6 +9,8 @@
 //! fixed, but for the moment it's easier to do these checks early.
 
 use crate::constrained_generic_params as cgp;
+use min_specialization::check_min_specialization;
+
 use rustc::ty::query::Providers;
 use rustc::ty::{self, TyCtxt, TypeFoldable};
 use rustc_data_structures::fx::{FxHashMap, FxHashSet};
@@ -16,9 +18,11 @@ use rustc_errors::struct_span_err;
 use rustc_hir as hir;
 use rustc_hir::def_id::DefId;
 use rustc_hir::itemlikevisit::ItemLikeVisitor;
+use rustc_span::Span;
+
 use std::collections::hash_map::Entry::{Occupied, Vacant};
 
-use rustc_span::Span;
+mod min_specialization;
 
 /// Checks that all the type/lifetime parameters on an impl also
 /// appear in the trait ref or self type (or are constrained by a
@@ -60,7 +64,9 @@ pub fn impl_wf_check(tcx: TyCtxt<'_>) {
 }
 
 fn check_mod_impl_wf(tcx: TyCtxt<'_>, module_def_id: DefId) {
-    tcx.hir().visit_item_likes_in_module(module_def_id, &mut ImplWfCheck { tcx });
+    let min_specialization = tcx.features().min_specialization;
+    tcx.hir()
+        .visit_item_likes_in_module(module_def_id, &mut ImplWfCheck { tcx, min_specialization });
 }
 
 pub fn provide(providers: &mut Providers<'_>) {
@@ -69,6 +75,7 @@ pub fn provide(providers: &mut Providers<'_>) {
 
 struct ImplWfCheck<'tcx> {
     tcx: TyCtxt<'tcx>,
+    min_specialization: bool,
 }
 
 impl ItemLikeVisitor<'tcx> for ImplWfCheck<'tcx> {
@@ -77,6 +84,9 @@ impl ItemLikeVisitor<'tcx> for ImplWfCheck<'tcx> {
             let impl_def_id = self.tcx.hir().local_def_id(item.hir_id);
             enforce_impl_params_are_constrained(self.tcx, impl_def_id, items);
             enforce_impl_items_are_distinct(self.tcx, items);
+            if self.min_specialization {
+                check_min_specialization(self.tcx, impl_def_id, item.span);
+            }
         }
     }
 
diff --git a/src/librustc_typeck/impl_wf_check/min_specialization.rs b/src/librustc_typeck/impl_wf_check/min_specialization.rs
new file mode 100644
index 0000000000000..cd52dae5f9ca2
--- /dev/null
+++ b/src/librustc_typeck/impl_wf_check/min_specialization.rs
@@ -0,0 +1,390 @@
+//! # Minimal Specialization
+//!
+//! This module contains the checks for sound specialization used when the
+//! `min_specialization` feature is enabled. This requires that the impl is
+//! *always applicable*.
+//!
+//! If `impl1` specializes `impl2` then `impl1` is always applicable if we know
+//! that all the bounds of `impl2` are satisfied, and all of the bounds of
+//! `impl1` are satisfied for some choice of lifetimes then we know that
+//! `impl1` applies for any choice of lifetimes.
+//!
+//! ## Basic approach
+//!
+//! To enforce this requirement on specializations we take the following
+//! approach:
+//!
+//! 1. Match up the substs for `impl2` so that the implemented trait and
+//!    self-type match those for `impl1`.
+//! 2. Check for any direct use of `'static` in the substs of `impl2`.
+//! 3. Check that all of the generic parameters of `impl1` occur at most once
+//!    in the *unconstrained* substs for `impl2`. A parameter is constrained if
+//!    its value is completely determined by an associated type projection
+//!    predicate.
+//! 4. Check that all predicates on `impl1` also exist on `impl2` (after
+//!    matching substs).
+//!
+//! ## Example
+//!
+//! Suppose we have the following always applicable impl:
+//!
+//! ```rust
+//! impl<T> SpecExtend<T> for std::vec::IntoIter<T> { /* specialized impl */ }
+//! impl<T, I: Iterator<Item=T>> SpecExtend<T> for I { /* default impl */ }
+//! ```
+//!
+//! We get that the subst for `impl2` are `[T, std::vec::IntoIter<T>]`. `T` is
+//! constrained to be `<I as Iterator>::Item`, so we check only
+//! `std::vec::IntoIter<T>` for repeated parameters, which it doesn't have. The
+//! predicates of `impl1` are only `T: Sized`, which is also a predicate of
+//! `impl2`. So this specialization is sound.
+//!
+//! ## Extensions
+//!
+//! Unfortunately not all specializations in the standard library are allowed
+//! by this. So there are two extensions to these rules that allow specializing
+//! on some traits: that is, using them as bounds on the specializing impl,
+//! even when they don't occur in the base impl.
+//!
+//! ### rustc_specialization_trait
+//!
+//! If a trait is always applicable, then it's sound to specialize on it. We
+//! check trait is always applicable in the same way as impls, except that step
+//! 4 is now "all predicates on `impl1` are always applicable". We require that
+//! `specialization` or `min_specialization` is enabled to implement these
+//! traits.
+//!
+//! ### rustc_unsafe_specialization_marker
+//!
+//! There are also some specialization on traits with no methods, including the
+//! stable `FusedIterator` trait. We allow marking marker traits with an
+//! unstable attribute that means we ignore them in point 3 of the checks
+//! above. This is unsound, in the sense that the specialized impl may be used
+//! when it doesn't apply, but we allow it in the short term since it can't
+//! cause use after frees with purely safe code in the same way as specializing
+//! on traits with methods can.
+
+use crate::constrained_generic_params as cgp;
+
+use rustc::middle::region::ScopeTree;
+use rustc::ty::subst::{GenericArg, InternalSubsts, SubstsRef};
+use rustc::ty::trait_def::TraitSpecializationKind;
+use rustc::ty::{self, InstantiatedPredicates, TyCtxt, TypeFoldable};
+use rustc_data_structures::fx::FxHashSet;
+use rustc_hir as hir;
+use rustc_hir::def_id::DefId;
+use rustc_infer::infer::outlives::env::OutlivesEnvironment;
+use rustc_infer::infer::{InferCtxt, SuppressRegionErrors, TyCtxtInferExt};
+use rustc_infer::traits::specialization_graph::Node;
+use rustc_span::Span;
+use rustc_trait_selection::traits::{self, translate_substs, wf};
+
+pub(super) fn check_min_specialization(tcx: TyCtxt<'_>, impl_def_id: DefId, span: Span) {
+    if let Some(node) = parent_specialization_node(tcx, impl_def_id) {
+        tcx.infer_ctxt().enter(|infcx| {
+            check_always_applicable(&infcx, impl_def_id, node, span);
+        });
+    }
+}
+
+fn parent_specialization_node(tcx: TyCtxt<'_>, impl1_def_id: DefId) -> Option<Node> {
+    let trait_ref = tcx.impl_trait_ref(impl1_def_id)?;
+    let trait_def = tcx.trait_def(trait_ref.def_id);
+
+    let impl2_node = trait_def.ancestors(tcx, impl1_def_id).ok()?.nth(1)?;
+
+    let always_applicable_trait =
+        matches!(trait_def.specialization_kind, TraitSpecializationKind::AlwaysApplicable);
+    if impl2_node.is_from_trait() && !always_applicable_trait {
+        // Implementing a normal trait isn't a specialization.
+        return None;
+    }
+    Some(impl2_node)
+}
+
+/// Check that `impl1` is a sound specialization
+fn check_always_applicable(
+    infcx: &InferCtxt<'_, '_>,
+    impl1_def_id: DefId,
+    impl2_node: Node,
+    span: Span,
+) {
+    if let Some((impl1_substs, impl2_substs)) =
+        get_impl_substs(infcx, impl1_def_id, impl2_node, span)
+    {
+        let impl2_def_id = impl2_node.def_id();
+        debug!(
+            "check_always_applicable(\nimpl1_def_id={:?},\nimpl2_def_id={:?},\nimpl2_substs={:?}\n)",
+            impl1_def_id, impl2_def_id, impl2_substs
+        );
+
+        let tcx = infcx.tcx;
+
+        let parent_substs = if impl2_node.is_from_trait() {
+            impl2_substs.to_vec()
+        } else {
+            unconstrained_parent_impl_substs(tcx, impl2_def_id, impl2_substs)
+        };
+
+        check_static_lifetimes(tcx, &parent_substs, span);
+        check_duplicate_params(tcx, impl1_substs, &parent_substs, span);
+
+        check_predicates(tcx, impl1_def_id, impl1_substs, impl2_node, impl2_substs, span);
+    }
+}
+
+/// Given a specializing impl `impl1`, and the base impl `impl2`, returns two
+/// substitutions `(S1, S2)` that equate their trait references. The returned
+/// types are expressed in terms of the generics of `impl1`.
+///
+/// Example
+///
+/// impl<A, B> Foo<A> for B { /* impl2 */ }
+/// impl<C> Foo<Vec<C>> for C { /* impl1 */ }
+///
+/// Would return `S1 = [C]` and `S2 = [Vec<C>, C]`.
+fn get_impl_substs<'tcx>(
+    infcx: &InferCtxt<'_, 'tcx>,
+    impl1_def_id: DefId,
+    impl2_node: Node,
+    span: Span,
+) -> Option<(SubstsRef<'tcx>, SubstsRef<'tcx>)> {
+    let tcx = infcx.tcx;
+    let param_env = tcx.param_env(impl1_def_id);
+
+    let impl1_substs = InternalSubsts::identity_for_item(tcx, impl1_def_id);
+    let impl2_substs = translate_substs(infcx, param_env, impl1_def_id, impl1_substs, impl2_node);
+
+    // Conservatively use an empty `ParamEnv`.
+    let outlives_env = OutlivesEnvironment::new(ty::ParamEnv::empty());
+    infcx.resolve_regions_and_report_errors(
+        impl1_def_id,
+        &ScopeTree::default(),
+        &outlives_env,
+        SuppressRegionErrors::default(),
+    );
+    let impl2_substs = match infcx.fully_resolve(&impl2_substs) {
+        Ok(s) => s,
+        Err(_) => {
+            tcx.sess.struct_span_err(span, "could not resolve substs on overridden impl").emit();
+            return None;
+        }
+    };
+    Some((impl1_substs, impl2_substs))
+}
+
+/// Returns a list of all of the unconstrained subst of the given impl.
+///
+/// For example given the impl:
+///
+/// impl<'a, T, I> ... where &'a I: IntoIterator<Item=&'a T>
+///
+/// This would return the substs corresponding to `['a, I]`, because knowing
+/// `'a` and `I` determines the value of `T`.
+fn unconstrained_parent_impl_substs<'tcx>(
+    tcx: TyCtxt<'tcx>,
+    impl_def_id: DefId,
+    impl_substs: SubstsRef<'tcx>,
+) -> Vec<GenericArg<'tcx>> {
+    let impl_generic_predicates = tcx.predicates_of(impl_def_id);
+    let mut unconstrained_parameters = FxHashSet::default();
+    let mut constrained_params = FxHashSet::default();
+    let impl_trait_ref = tcx.impl_trait_ref(impl_def_id);
+
+    // Unfortunately the functions in `constrained_generic_parameters` don't do
+    // what we want here. We want only a list of constrained parameters while
+    // the functions in `cgp` add the constrained parameters to a list of
+    // unconstrained parameters.
+    for (predicate, _) in impl_generic_predicates.predicates.iter() {
+        if let ty::Predicate::Projection(proj) = predicate {
+            let projection_ty = proj.skip_binder().projection_ty;
+            let projected_ty = proj.skip_binder().ty;
+
+            let unbound_trait_ref = projection_ty.trait_ref(tcx);
+            if Some(unbound_trait_ref) == impl_trait_ref {
+                continue;
+            }
+
+            unconstrained_parameters.extend(cgp::parameters_for(&projection_ty, true));
+
+            for param in cgp::parameters_for(&projected_ty, false) {
+                if !unconstrained_parameters.contains(&param) {
+                    constrained_params.insert(param.0);
+                }
+            }
+
+            unconstrained_parameters.extend(cgp::parameters_for(&projected_ty, true));
+        }
+    }
+
+    impl_substs
+        .iter()
+        .enumerate()
+        .filter(|&(idx, _)| !constrained_params.contains(&(idx as u32)))
+        .map(|(_, arg)| *arg)
+        .collect()
+}
+
+/// Check that parameters of the derived impl don't occur more than once in the
+/// equated substs of the base impl.
+///
+/// For example forbid the following:
+///
+/// impl<A> Tr for A { }
+/// impl<B> Tr for (B, B) { }
+///
+/// Note that only consider the unconstrained parameters of the base impl:
+///
+/// impl<S, I: IntoIterator<Item = S>> Tr<S> for I { }
+/// impl<T> Tr<T> for Vec<T> { }
+///
+/// The substs for the parent impl here are `[T, Vec<T>]`, which repeats `T`,
+/// but `S` is constrained in the parent impl, so `parent_substs` is only
+/// `[Vec<T>]`. This means we allow this impl.
+fn check_duplicate_params<'tcx>(
+    tcx: TyCtxt<'tcx>,
+    impl1_substs: SubstsRef<'tcx>,
+    parent_substs: &Vec<GenericArg<'tcx>>,
+    span: Span,
+) {
+    let mut base_params = cgp::parameters_for(parent_substs, true);
+    base_params.sort_by_key(|param| param.0);
+    if let (_, [duplicate, ..]) = base_params.partition_dedup() {
+        let param = impl1_substs[duplicate.0 as usize];
+        tcx.sess
+            .struct_span_err(span, &format!("specializing impl repeats parameter `{}`", param))
+            .emit();
+    }
+}
+
+/// Check that `'static` lifetimes are not introduced by the specializing impl.
+///
+/// For example forbid the following:
+///
+/// impl<A> Tr for A { }
+/// impl Tr for &'static i32 { }
+fn check_static_lifetimes<'tcx>(
+    tcx: TyCtxt<'tcx>,
+    parent_substs: &Vec<GenericArg<'tcx>>,
+    span: Span,
+) {
+    if tcx.any_free_region_meets(parent_substs, |r| *r == ty::ReStatic) {
+        tcx.sess.struct_span_err(span, &format!("cannot specialize on `'static` lifetime")).emit();
+    }
+}
+
+/// Check whether predicates on the specializing impl (`impl1`) are allowed.
+///
+/// Each predicate `P` must be:
+///
+/// * global (not reference any parameters)
+/// * `T: Tr` predicate where `Tr` is an always-applicable trait
+/// * on the base `impl impl2`
+///     * Currently this check is done using syntactic equality, which is
+///       conservative but generally sufficient.
+fn check_predicates<'tcx>(
+    tcx: TyCtxt<'tcx>,
+    impl1_def_id: DefId,
+    impl1_substs: SubstsRef<'tcx>,
+    impl2_node: Node,
+    impl2_substs: SubstsRef<'tcx>,
+    span: Span,
+) {
+    let impl1_predicates = tcx.predicates_of(impl1_def_id).instantiate(tcx, impl1_substs);
+    let mut impl2_predicates = if impl2_node.is_from_trait() {
+        // Always applicable traits have to be always applicable without any
+        // assumptions.
+        InstantiatedPredicates::empty()
+    } else {
+        tcx.predicates_of(impl2_node.def_id()).instantiate(tcx, impl2_substs)
+    };
+    debug!(
+        "check_always_applicable(\nimpl1_predicates={:?},\nimpl2_predicates={:?}\n)",
+        impl1_predicates, impl2_predicates,
+    );
+
+    // Since impls of always applicable traits don't get to assume anything, we
+    // can also assume their supertraits apply.
+    //
+    // For example, we allow:
+    //
+    // #[rustc_specialization_trait]
+    // trait AlwaysApplicable: Debug { }
+    //
+    // impl<T> Tr for T { }
+    // impl<T: AlwaysApplicable> Tr for T { }
+    //
+    // Specializing on `AlwaysApplicable` allows also specializing on `Debug`
+    // which is sound because we forbid impls like the following
+    //
+    // impl<D: Debug> AlwaysApplicable for D { }
+    let always_applicable_traits: Vec<_> = impl1_predicates
+        .predicates
+        .iter()
+        .filter(|predicate| {
+            matches!(
+                trait_predicate_kind(tcx, predicate),
+                Some(TraitSpecializationKind::AlwaysApplicable)
+            )
+        })
+        .copied()
+        .collect();
+    impl2_predicates.predicates.extend(traits::elaborate_predicates(tcx, always_applicable_traits));
+
+    for predicate in impl1_predicates.predicates {
+        if !impl2_predicates.predicates.contains(&predicate) {
+            check_specialization_on(tcx, &predicate, span)
+        }
+    }
+}
+
+fn check_specialization_on<'tcx>(tcx: TyCtxt<'tcx>, predicate: &ty::Predicate<'tcx>, span: Span) {
+    debug!("can_specialize_on(predicate = {:?})", predicate);
+    match predicate {
+        // Global predicates are either always true or always false, so we
+        // are fine to specialize on.
+        _ if predicate.is_global() => (),
+        // We allow specializing on explicitly marked traits with no associated
+        // items.
+        ty::Predicate::Trait(pred, hir::Constness::NotConst) => {
+            if !matches!(
+                trait_predicate_kind(tcx, predicate),
+                Some(TraitSpecializationKind::Marker)
+            ) {
+                tcx.sess
+                    .struct_span_err(
+                        span,
+                        &format!(
+                            "cannot specialize on trait `{}`",
+                            tcx.def_path_str(pred.def_id()),
+                        ),
+                    )
+                    .emit()
+            }
+        }
+        _ => tcx
+            .sess
+            .struct_span_err(span, &format!("cannot specialize on `{:?}`", predicate))
+            .emit(),
+    }
+}
+
+fn trait_predicate_kind<'tcx>(
+    tcx: TyCtxt<'tcx>,
+    predicate: &ty::Predicate<'tcx>,
+) -> Option<TraitSpecializationKind> {
+    match predicate {
+        ty::Predicate::Trait(pred, hir::Constness::NotConst) => {
+            Some(tcx.trait_def(pred.def_id()).specialization_kind)
+        }
+        ty::Predicate::Trait(_, hir::Constness::Const)
+        | ty::Predicate::RegionOutlives(_)
+        | ty::Predicate::TypeOutlives(_)
+        | ty::Predicate::Projection(_)
+        | ty::Predicate::WellFormed(_)
+        | ty::Predicate::Subtype(_)
+        | ty::Predicate::ObjectSafe(_)
+        | ty::Predicate::ClosureKind(..)
+        | ty::Predicate::ConstEvaluatable(..) => None,
+    }
+}
diff --git a/src/librustc_typeck/lib.rs b/src/librustc_typeck/lib.rs
index 4e7985dd98812..bab354cff7c9c 100644
--- a/src/librustc_typeck/lib.rs
+++ b/src/librustc_typeck/lib.rs
@@ -64,6 +64,7 @@ This API is completely unstable and subject to change.
 #![feature(nll)]
 #![feature(try_blocks)]
 #![feature(never_type)]
+#![feature(slice_partition_dedup)]
 #![recursion_limit = "256"]
 
 #[macro_use]
diff --git a/src/test/ui/specialization/min_specialization/dyn-trait-assoc-types.rs b/src/test/ui/specialization/min_specialization/dyn-trait-assoc-types.rs
new file mode 100644
index 0000000000000..03cab00b0fb96
--- /dev/null
+++ b/src/test/ui/specialization/min_specialization/dyn-trait-assoc-types.rs
@@ -0,0 +1,32 @@
+// Test that associated types in trait objects are not considered to be
+// constrained.
+
+#![feature(min_specialization)]
+
+trait Specializable {
+    fn f();
+}
+
+trait B<T> {
+    type Y;
+}
+
+trait C {
+    type Y;
+}
+
+impl<A: ?Sized> Specializable for A {
+    default fn f() {}
+}
+
+impl<'a, T> Specializable for dyn B<T, Y = T> + 'a {
+    //~^ ERROR specializing impl repeats parameter `T`
+    fn f() {}
+}
+
+impl<'a, T> Specializable for dyn C<Y = (T, T)> + 'a {
+    //~^ ERROR specializing impl repeats parameter `T`
+    fn f() {}
+}
+
+fn main() {}
diff --git a/src/test/ui/specialization/min_specialization/dyn-trait-assoc-types.stderr b/src/test/ui/specialization/min_specialization/dyn-trait-assoc-types.stderr
new file mode 100644
index 0000000000000..6345cee2c3781
--- /dev/null
+++ b/src/test/ui/specialization/min_specialization/dyn-trait-assoc-types.stderr
@@ -0,0 +1,20 @@
+error: specializing impl repeats parameter `T`
+  --> $DIR/dyn-trait-assoc-types.rs:22:1
+   |
+LL | / impl<'a, T> Specializable for dyn B<T, Y = T> + 'a {
+LL | |
+LL | |     fn f() {}
+LL | | }
+   | |_^
+
+error: specializing impl repeats parameter `T`
+  --> $DIR/dyn-trait-assoc-types.rs:27:1
+   |
+LL | / impl<'a, T> Specializable for dyn C<Y = (T, T)> + 'a {
+LL | |
+LL | |     fn f() {}
+LL | | }
+   | |_^
+
+error: aborting due to 2 previous errors
+
diff --git a/src/test/ui/specialization/min_specialization/repeated_projection_type.rs b/src/test/ui/specialization/min_specialization/repeated_projection_type.rs
new file mode 100644
index 0000000000000..f21f39f066981
--- /dev/null
+++ b/src/test/ui/specialization/min_specialization/repeated_projection_type.rs
@@ -0,0 +1,24 @@
+// Test that projection bounds can't be specialized on.
+
+#![feature(min_specialization)]
+
+trait X {
+    fn f();
+}
+trait Id {
+    type This;
+}
+impl<T> Id for T {
+    type This = T;
+}
+
+impl<T: Id> X for T {
+    default fn f() {}
+}
+
+impl<I, V: Id<This = (I,)>> X for V {
+    //~^ ERROR cannot specialize on
+    fn f() {}
+}
+
+fn main() {}
diff --git a/src/test/ui/specialization/min_specialization/repeated_projection_type.stderr b/src/test/ui/specialization/min_specialization/repeated_projection_type.stderr
new file mode 100644
index 0000000000000..7cc4357a704c0
--- /dev/null
+++ b/src/test/ui/specialization/min_specialization/repeated_projection_type.stderr
@@ -0,0 +1,11 @@
+error: cannot specialize on `Binder(ProjectionPredicate(ProjectionTy { substs: [V], item_def_id: DefId(0:6 ~ repeated_projection_type[317d]::Id[0]::This[0]) }, (I,)))`
+  --> $DIR/repeated_projection_type.rs:19:1
+   |
+LL | / impl<I, V: Id<This = (I,)>> X for V {
+LL | |
+LL | |     fn f() {}
+LL | | }
+   | |_^
+
+error: aborting due to previous error
+
diff --git a/src/test/ui/specialization/min_specialization/repeating_lifetimes.rs b/src/test/ui/specialization/min_specialization/repeating_lifetimes.rs
new file mode 100644
index 0000000000000..49bfacec0ae12
--- /dev/null
+++ b/src/test/ui/specialization/min_specialization/repeating_lifetimes.rs
@@ -0,0 +1,19 @@
+// Test that directly specializing on repeated lifetime parameters is not
+// allowed.
+
+#![feature(min_specialization)]
+
+trait X {
+    fn f();
+}
+
+impl<T> X for T {
+    default fn f() {}
+}
+
+impl<'a> X for (&'a u8, &'a u8) {
+    //~^ ERROR specializing impl repeats parameter `'a`
+    fn f() {}
+}
+
+fn main() {}
diff --git a/src/test/ui/specialization/min_specialization/repeating_lifetimes.stderr b/src/test/ui/specialization/min_specialization/repeating_lifetimes.stderr
new file mode 100644
index 0000000000000..ce9309f70122b
--- /dev/null
+++ b/src/test/ui/specialization/min_specialization/repeating_lifetimes.stderr
@@ -0,0 +1,11 @@
+error: specializing impl repeats parameter `'a`
+  --> $DIR/repeating_lifetimes.rs:14:1
+   |
+LL | / impl<'a> X for (&'a u8, &'a u8) {
+LL | |
+LL | |     fn f() {}
+LL | | }
+   | |_^
+
+error: aborting due to previous error
+
diff --git a/src/test/ui/specialization/min_specialization/repeating_param.rs b/src/test/ui/specialization/min_specialization/repeating_param.rs
new file mode 100644
index 0000000000000..5a1c97fd321a9
--- /dev/null
+++ b/src/test/ui/specialization/min_specialization/repeating_param.rs
@@ -0,0 +1,17 @@
+// Test that specializing on two type parameters being equal is not allowed.
+
+#![feature(min_specialization)]
+
+trait X {
+    fn f();
+}
+
+impl<T> X for T {
+    default fn f() {}
+}
+impl<T> X for (T, T) {
+    //~^ ERROR specializing impl repeats parameter `T`
+    fn f() {}
+}
+
+fn main() {}
diff --git a/src/test/ui/specialization/min_specialization/repeating_param.stderr b/src/test/ui/specialization/min_specialization/repeating_param.stderr
new file mode 100644
index 0000000000000..8b4be1c499537
--- /dev/null
+++ b/src/test/ui/specialization/min_specialization/repeating_param.stderr
@@ -0,0 +1,11 @@
+error: specializing impl repeats parameter `T`
+  --> $DIR/repeating_param.rs:12:1
+   |
+LL | / impl<T> X for (T, T) {
+LL | |
+LL | |     fn f() {}
+LL | | }
+   | |_^
+
+error: aborting due to previous error
+
diff --git a/src/test/ui/specialization/min_specialization/spec-iter.rs b/src/test/ui/specialization/min_specialization/spec-iter.rs
new file mode 100644
index 0000000000000..e17e9dd5f133c
--- /dev/null
+++ b/src/test/ui/specialization/min_specialization/spec-iter.rs
@@ -0,0 +1,20 @@
+// Check that we can specialize on a concrete iterator type. This requires us
+// to consider which parameters in the parent impl are constrained.
+
+// check-pass
+
+#![feature(min_specialization)]
+
+trait SpecFromIter<T> {
+    fn f(&self);
+}
+
+impl<'a, T: 'a, I: Iterator<Item = &'a T>> SpecFromIter<T> for I {
+    default fn f(&self) {}
+}
+
+impl<'a, T> SpecFromIter<T> for std::slice::Iter<'a, T> {
+    fn f(&self) {}
+}
+
+fn main() {}
diff --git a/src/test/ui/specialization/min_specialization/spec-reference.rs b/src/test/ui/specialization/min_specialization/spec-reference.rs
new file mode 100644
index 0000000000000..377889e2ccad2
--- /dev/null
+++ b/src/test/ui/specialization/min_specialization/spec-reference.rs
@@ -0,0 +1,19 @@
+// Check that lifetime parameters are allowed in specializing impls.
+
+// check-pass
+
+#![feature(min_specialization)]
+
+trait MySpecTrait {
+    fn f();
+}
+
+impl<T> MySpecTrait for T {
+    default fn f() {}
+}
+
+impl<'a, T: ?Sized> MySpecTrait for &'a T {
+    fn f() {}
+}
+
+fn main() {}
diff --git a/src/test/ui/specialization/min_specialization/specialization_marker.rs b/src/test/ui/specialization/min_specialization/specialization_marker.rs
new file mode 100644
index 0000000000000..93462d02ea578
--- /dev/null
+++ b/src/test/ui/specialization/min_specialization/specialization_marker.rs
@@ -0,0 +1,17 @@
+// Test that `rustc_unsafe_specialization_marker` is only allowed on marker traits.
+
+#![feature(rustc_attrs)]
+
+#[rustc_unsafe_specialization_marker]
+trait SpecMarker {
+    fn f();
+    //~^ ERROR marker traits
+}
+
+#[rustc_unsafe_specialization_marker]
+trait SpecMarker2 {
+    type X;
+    //~^ ERROR marker traits
+}
+
+fn main() {}
diff --git a/src/test/ui/specialization/min_specialization/specialization_marker.stderr b/src/test/ui/specialization/min_specialization/specialization_marker.stderr
new file mode 100644
index 0000000000000..ffeced198211f
--- /dev/null
+++ b/src/test/ui/specialization/min_specialization/specialization_marker.stderr
@@ -0,0 +1,15 @@
+error[E0714]: marker traits cannot have associated items
+  --> $DIR/specialization_marker.rs:7:5
+   |
+LL |     fn f();
+   |     ^^^^^^^
+
+error[E0714]: marker traits cannot have associated items
+  --> $DIR/specialization_marker.rs:13:5
+   |
+LL |     type X;
+   |     ^^^^^^^
+
+error: aborting due to 2 previous errors
+
+For more information about this error, try `rustc --explain E0714`.
diff --git a/src/test/ui/specialization/min_specialization/specialization_trait.rs b/src/test/ui/specialization/min_specialization/specialization_trait.rs
new file mode 100644
index 0000000000000..d597278d29693
--- /dev/null
+++ b/src/test/ui/specialization/min_specialization/specialization_trait.rs
@@ -0,0 +1,26 @@
+// Test that `rustc_specialization_trait` requires always applicable impls.
+
+#![feature(min_specialization)]
+#![feature(rustc_attrs)]
+
+#[rustc_specialization_trait]
+trait SpecMarker {
+    fn f();
+}
+
+impl SpecMarker for &'static u8 {
+    //~^ ERROR cannot specialize
+    fn f() {}
+}
+
+impl<T> SpecMarker for (T, T) {
+    //~^ ERROR specializing impl
+    fn f() {}
+}
+
+impl<T: Clone> SpecMarker for [T] {
+    //~^ ERROR cannot specialize
+    fn f() {}
+}
+
+fn main() {}
diff --git a/src/test/ui/specialization/min_specialization/specialization_trait.stderr b/src/test/ui/specialization/min_specialization/specialization_trait.stderr
new file mode 100644
index 0000000000000..4357d2318fc5d
--- /dev/null
+++ b/src/test/ui/specialization/min_specialization/specialization_trait.stderr
@@ -0,0 +1,29 @@
+error: cannot specialize on `'static` lifetime
+  --> $DIR/specialization_trait.rs:11:1
+   |
+LL | / impl SpecMarker for &'static u8 {
+LL | |
+LL | |     fn f() {}
+LL | | }
+   | |_^
+
+error: specializing impl repeats parameter `T`
+  --> $DIR/specialization_trait.rs:16:1
+   |
+LL | / impl<T> SpecMarker for (T, T) {
+LL | |
+LL | |     fn f() {}
+LL | | }
+   | |_^
+
+error: cannot specialize on trait `std::clone::Clone`
+  --> $DIR/specialization_trait.rs:21:1
+   |
+LL | / impl<T: Clone> SpecMarker for [T] {
+LL | |
+LL | |     fn f() {}
+LL | | }
+   | |_^
+
+error: aborting due to 3 previous errors
+
diff --git a/src/test/ui/specialization/min_specialization/specialize_on_marker.rs b/src/test/ui/specialization/min_specialization/specialize_on_marker.rs
new file mode 100644
index 0000000000000..4219bd13b1816
--- /dev/null
+++ b/src/test/ui/specialization/min_specialization/specialize_on_marker.rs
@@ -0,0 +1,24 @@
+// Test that specializing on a `rustc_unsafe_specialization_marker` trait is
+// allowed.
+
+// check-pass
+
+#![feature(min_specialization)]
+#![feature(rustc_attrs)]
+
+#[rustc_unsafe_specialization_marker]
+trait SpecMarker {}
+
+trait X {
+    fn f();
+}
+
+impl<T> X for T {
+    default fn f() {}
+}
+
+impl<T: SpecMarker> X for T {
+    fn f() {}
+}
+
+fn main() {}
diff --git a/src/test/ui/specialization/min_specialization/specialize_on_spec_trait.rs b/src/test/ui/specialization/min_specialization/specialize_on_spec_trait.rs
new file mode 100644
index 0000000000000..abbab5c23dbb7
--- /dev/null
+++ b/src/test/ui/specialization/min_specialization/specialize_on_spec_trait.rs
@@ -0,0 +1,27 @@
+// Test that specializing on a `rustc_specialization_trait` trait is allowed.
+
+// check-pass
+
+#![feature(min_specialization)]
+#![feature(rustc_attrs)]
+
+#[rustc_specialization_trait]
+trait SpecTrait {
+    fn g(&self);
+}
+
+trait X {
+    fn f(&self);
+}
+
+impl<T> X for T {
+    default fn f(&self) {}
+}
+
+impl<T: SpecTrait> X for T {
+    fn f(&self) {
+        self.g();
+    }
+}
+
+fn main() {}
diff --git a/src/test/ui/specialization/min_specialization/specialize_on_static.rs b/src/test/ui/specialization/min_specialization/specialize_on_static.rs
new file mode 100644
index 0000000000000..dd1b05401e6e7
--- /dev/null
+++ b/src/test/ui/specialization/min_specialization/specialize_on_static.rs
@@ -0,0 +1,18 @@
+// Test that directly specializing on `'static` is not allowed.
+
+#![feature(min_specialization)]
+
+trait X {
+    fn f();
+}
+
+impl<T> X for &'_ T {
+    default fn f() {}
+}
+
+impl X for &'static u8 {
+    //~^ ERROR cannot specialize on `'static` lifetime
+    fn f() {}
+}
+
+fn main() {}
diff --git a/src/test/ui/specialization/min_specialization/specialize_on_static.stderr b/src/test/ui/specialization/min_specialization/specialize_on_static.stderr
new file mode 100644
index 0000000000000..d1809d6dfbb52
--- /dev/null
+++ b/src/test/ui/specialization/min_specialization/specialize_on_static.stderr
@@ -0,0 +1,11 @@
+error: cannot specialize on `'static` lifetime
+  --> $DIR/specialize_on_static.rs:13:1
+   |
+LL | / impl X for &'static u8 {
+LL | |
+LL | |     fn f() {}
+LL | | }
+   | |_^
+
+error: aborting due to previous error
+
diff --git a/src/test/ui/specialization/min_specialization/specialize_on_trait.rs b/src/test/ui/specialization/min_specialization/specialize_on_trait.rs
new file mode 100644
index 0000000000000..0588442c32080
--- /dev/null
+++ b/src/test/ui/specialization/min_specialization/specialize_on_trait.rs
@@ -0,0 +1,20 @@
+// Test that specializing on a trait is not allowed in general.
+
+#![feature(min_specialization)]
+
+trait SpecMarker {}
+
+trait X {
+    fn f();
+}
+
+impl<T> X for T {
+    default fn f() {}
+}
+
+impl<T: SpecMarker> X for T {
+    //~^ ERROR cannot specialize on trait `SpecMarker`
+    fn f() {}
+}
+
+fn main() {}
diff --git a/src/test/ui/specialization/min_specialization/specialize_on_trait.stderr b/src/test/ui/specialization/min_specialization/specialize_on_trait.stderr
new file mode 100644
index 0000000000000..35445fd09b949
--- /dev/null
+++ b/src/test/ui/specialization/min_specialization/specialize_on_trait.stderr
@@ -0,0 +1,11 @@
+error: cannot specialize on trait `SpecMarker`
+  --> $DIR/specialize_on_trait.rs:15:1
+   |
+LL | / impl<T: SpecMarker> X for T {
+LL | |
+LL | |     fn f() {}
+LL | | }
+   | |_^
+
+error: aborting due to previous error
+