Index: llvm/trunk/include/llvm/Analysis/ScalarEvolution.h
===================================================================
--- llvm/trunk/include/llvm/Analysis/ScalarEvolution.h
+++ llvm/trunk/include/llvm/Analysis/ScalarEvolution.h
@@ -576,6 +576,14 @@
     bool isKnownPredicateWithRanges(ICmpInst::Predicate Pred,
                                     const SCEV *LHS, const SCEV *RHS);
 
+    /// Try to prove the condition described by "LHS Pred RHS" by ruling out
+    /// integer overflow.
+    ///
+    /// For instance, this will return true for "A s< (A + C)<nsw>" if C is
+    /// positive.
+    bool isKnownPredicateViaNoOverflow(ICmpInst::Predicate Pred,
+                                       const SCEV *LHS, const SCEV *RHS);
+
     /// Try to split Pred LHS RHS into logical conjunctions (and's) and try to
     /// prove them individually.
     bool isKnownPredicateViaSplitting(ICmpInst::Predicate Pred, const SCEV *LHS,
Index: llvm/trunk/lib/Analysis/ScalarEvolution.cpp
===================================================================
--- llvm/trunk/lib/Analysis/ScalarEvolution.cpp
+++ llvm/trunk/lib/Analysis/ScalarEvolution.cpp
@@ -7162,6 +7162,60 @@
   return false;
 }
 
+bool ScalarEvolution::isKnownPredicateViaNoOverflow(ICmpInst::Predicate Pred,
+                                                    const SCEV *LHS,
+                                                    const SCEV *RHS) {
+
+  // Match Result to (X + Y)<ExpectedFlags> where Y is a constant integer.
+  // Return Y via OutY.
+  auto MatchBinaryAddToConst =
+      [this](const SCEV *Result, const SCEV *X, APInt &OutY,
+             SCEV::NoWrapFlags ExpectedFlags) {
+    const SCEV *NonConstOp, *ConstOp;
+    SCEV::NoWrapFlags FlagsPresent;
+
+    if (!splitBinaryAdd(Result, ConstOp, NonConstOp, FlagsPresent) ||
+        !isa<SCEVConstant>(ConstOp) || NonConstOp != X)
+      return false;
+
+    OutY = cast<SCEVConstant>(ConstOp)->getValue()->getValue();
+    return (FlagsPresent & ExpectedFlags) != 0;
+  };
+
+  APInt C;
+
+  switch (Pred) {
+  default:
+    break;
+
+  case ICmpInst::ICMP_SGE:
+    std::swap(LHS, RHS);
+  case ICmpInst::ICMP_SLE:
+    // X s<= (X + C)<nsw> if C >= 0
+    if (MatchBinaryAddToConst(RHS, LHS, C, SCEV::FlagNSW) && C.isNonNegative())
+      return true;
+
+    // (X + C)<nsw> s<= X if C <= 0
+    if (MatchBinaryAddToConst(LHS, RHS, C, SCEV::FlagNSW) &&
+        !C.isStrictlyPositive())
+      return true;
+
+  case ICmpInst::ICMP_SGT:
+    std::swap(LHS, RHS);
+  case ICmpInst::ICMP_SLT:
+    // X s< (X + C)<nsw> if C > 0
+    if (MatchBinaryAddToConst(RHS, LHS, C, SCEV::FlagNSW) &&
+        C.isStrictlyPositive())
+      return true;
+
+    // (X + C)<nsw> s< X if C < 0
+    if (MatchBinaryAddToConst(LHS, RHS, C, SCEV::FlagNSW) && C.isNegative())
+      return true;
+  }
+
+  return false;
+}
+
 bool ScalarEvolution::isKnownPredicateViaSplitting(ICmpInst::Predicate Pred,
                                                    const SCEV *LHS,
                                                    const SCEV *RHS) {
@@ -7811,8 +7865,9 @@
   auto IsKnownPredicateFull =
       [this](ICmpInst::Predicate Pred, const SCEV *LHS, const SCEV *RHS) {
     return isKnownPredicateWithRanges(Pred, LHS, RHS) ||
-        IsKnownPredicateViaMinOrMax(*this, Pred, LHS, RHS) ||
-        IsKnownPredicateViaAddRecStart(*this, Pred, LHS, RHS);
+           IsKnownPredicateViaMinOrMax(*this, Pred, LHS, RHS) ||
+           IsKnownPredicateViaAddRecStart(*this, Pred, LHS, RHS) ||
+           isKnownPredicateViaNoOverflow(Pred, LHS, RHS);
   };
 
   switch (Pred) {
Index: llvm/trunk/test/Transforms/IndVarSimplify/eliminate-comparison.ll
===================================================================
--- llvm/trunk/test/Transforms/IndVarSimplify/eliminate-comparison.ll
+++ llvm/trunk/test/Transforms/IndVarSimplify/eliminate-comparison.ll
@@ -477,4 +477,63 @@
   ret void
 }
 
+define void @func_22(i32* %length.ptr) {
+; CHECK-LABEL: @func_22(
+
+; This checks that the backedge condition, (I + 1) < Length - 1 implies
+; (I + 1) < Length
+ entry:
+  %length = load i32, i32* %length.ptr, !range !0
+  %lim = sub i32 %length, 1
+  %entry.cond = icmp sgt i32 %length, 1
+  br i1 %entry.cond, label %loop, label %leave
+
+ loop:
+; CHECK: loop:
+  %iv = phi i32 [ 0, %entry ], [ %iv.inc, %be ]
+  %iv.inc = add i32 %iv, 1
+  %range.check = icmp slt i32 %iv, %length
+  br i1 %range.check, label %be, label %leave
+; CHECK:   br i1 true, label %be, label %leave.loopexit
+; CHECK: be:
+
+ be:
+  call void @side_effect()
+  %be.cond = icmp slt i32 %iv.inc, %lim
+  br i1 %be.cond, label %loop, label %leave
+
+ leave:
+  ret void
+}
+
+define void @func_23(i32* %length.ptr) {
+; CHECK-LABEL: @func_23(
+
+; This checks that the backedge condition, (I + 1) < Length - 1 implies
+; (I + 1) < Length
+ entry:
+  %length = load i32, i32* %length.ptr, !range !0
+  %lim = sub i32 %length, 1
+  %entry.cond = icmp sgt i32 %length, 1
+  br i1 %entry.cond, label %loop, label %leave
+
+ loop:
+; CHECK: loop:
+  %iv = phi i32 [ 0, %entry ], [ %iv.inc, %be ]
+  %iv.inc = add i32 %iv, 1
+  %range.check = icmp sle i32 %iv, %length
+  br i1 %range.check, label %be, label %leave
+; CHECK:   br i1 true, label %be, label %leave.loopexit
+; CHECK: be:
+
+ be:
+  call void @side_effect()
+  %be.cond = icmp sle i32 %iv.inc, %lim
+  br i1 %be.cond, label %loop, label %leave
+
+ leave:
+  ret void
+}
+
+
 !0 = !{i32 0, i32 2147483647}