Index: llvm/lib/Transforms/IPO/FunctionSpecialization.cpp
===================================================================
--- llvm/lib/Transforms/IPO/FunctionSpecialization.cpp
+++ llvm/lib/Transforms/IPO/FunctionSpecialization.cpp
@@ -789,12 +789,27 @@
       Solver.trackValueOfGlobalVariable(&G);
   }
 
+  auto &TrackedFuncs = Solver.getArgumentTrackedFunctions();
+  SmallVector<Function *, 16> FuncDecls(TrackedFuncs.begin(),
+                                        TrackedFuncs.end());
+
+  // No tracked functions, so nothing to do: don't run the solver and remove
+  // the ssa_copy intrinsics that may have been introduced.
+  if (TrackedFuncs.empty()) {
+    removeSSACopy(M);
+    return false;
+  }
+
   // Solve for constants.
   auto RunSCCPSolver = [&](auto &WorkList) {
     bool ResolvedUndefs = true;
 
     while (ResolvedUndefs) {
+      // Not running the solver unnecessary is checked in regression test
+      // nothing-to-do.ll, so if this debug message is changed, this regression
+      // test needs updating too.
       LLVM_DEBUG(dbgs() << "FnSpecialization: Running solver\n");
+
       Solver.solve();
       LLVM_DEBUG(dbgs() << "FnSpecialization: Resolving undefs\n");
       ResolvedUndefs = false;
@@ -815,9 +830,6 @@
     }
   };
 
-  auto &TrackedFuncs = Solver.getArgumentTrackedFunctions();
-  SmallVector<Function *, 16> FuncDecls(TrackedFuncs.begin(),
-                                        TrackedFuncs.end());
 #ifndef NDEBUG
   LLVM_DEBUG(dbgs() << "FnSpecialization: Worklist fn decls:\n");
   for (auto *F : FuncDecls)
Index: llvm/test/Transforms/FunctionSpecialization/function-specialization-nothing-todo.ll
===================================================================
--- /dev/null
+++ llvm/test/Transforms/FunctionSpecialization/function-specialization-nothing-todo.ll
@@ -0,0 +1,51 @@
+; REQUIRES: asserts
+; RUN: opt -function-specialization -debug -S < %s 2>&1 | FileCheck %s
+
+; The purpose of this test is to check that we don't run the solver as there's
+; nothing to do here. For a test that doesn't trigger function specialisation,
+; it is intentionally 'big' because we also want to check that the ssa.copy
+; intrinsics that are introduced by the solver are cleaned up if we bail
+; early. Thus, first check the debug messages for the introduction of these
+; intrinsics:
+
+; CHECK: FnSpecialization: Analysing decl: foo
+; CHECK: Found replacement{{.*}} call i32 @llvm.ssa.copy.i32
+; CHECK: Found replacement{{.*}} call i32 @llvm.ssa.copy.i32
+
+; Then, make sure the solver didn't run:
+
+; CHECK-NOT: Running solver
+
+; Finally, check the absence and thus removal of these intrinsics:
+
+; CHECK-LABEL: @foo
+; CHECK-NOT:   call i32 @llvm.ssa.copy.i32
+
+@N = external dso_local global i32, align 4
+@B = external dso_local global i32*, align 8
+@A = external dso_local global i32*, align 8
+
+define dso_local i32 @foo() {
+entry:
+  br label %for.cond
+
+for.cond:
+  %i.0 = phi i32 [ 0, %entry ], [ %inc, %for.body ]
+  %0 = load i32, i32* @N, align 4
+  %cmp = icmp slt i32 %i.0, %0
+  br i1 %cmp, label %for.body, label %for.cond.cleanup
+
+for.cond.cleanup:
+  ret i32 undef
+
+for.body:
+  %1 = load i32*, i32** @B, align 8
+  %idxprom = sext i32 %i.0 to i64
+  %arrayidx = getelementptr inbounds i32, i32* %1, i64 %idxprom
+  %2 = load i32, i32* %arrayidx, align 4
+  %3 = load i32*, i32** @A, align 8
+  %arrayidx2 = getelementptr inbounds i32, i32* %3, i64 %idxprom
+  store i32 %2, i32* %arrayidx2, align 4
+  %inc = add nsw i32 %i.0, 1
+  br label %for.cond
+}