Index: lib/Analysis/ScalarEvolution.cpp
===================================================================
--- lib/Analysis/ScalarEvolution.cpp
+++ lib/Analysis/ScalarEvolution.cpp
@@ -6948,6 +6948,92 @@
                                      getNotSCEV(FoundLHS));
 }
 
+
+/// If Expr computes ~A assign `A' to NotOf and return true, else
+/// return false.
+static bool MatchNotExpr(const SCEV *Expr, const SCEV *&NotOf) {
+  const SCEVAddExpr *Add = dyn_cast<SCEVAddExpr>(Expr);
+  if (!Add || Add->getNumOperands() != 2) return false;
+
+  const SCEVConstant *AddLHS = dyn_cast<SCEVConstant>(Add->getOperand(0));
+  if (!(AddLHS && AddLHS->getValue()->getValue().isAllOnesValue()))
+    return false;
+
+  const SCEVMulExpr *AddRHS = dyn_cast<SCEVMulExpr>(Add->getOperand(1));
+  if (!AddRHS || AddRHS->getNumOperands() != 2) return false;
+
+  const SCEVConstant *MulLHS = dyn_cast<SCEVConstant>(AddRHS->getOperand(0));
+  if (!(MulLHS && MulLHS->getValue()->getValue().isAllOnesValue()))
+    return false;
+
+  NotOf = AddRHS->getOperand(1);
+  return true;
+}
+
+
+/// Is MaybeMaxExpr an SMax or UMax of Candidate and some other values?
+template<typename MaxExprType>
+static bool IsMaxConsistingOf(ScalarEvolution &SE,
+                              const SCEV *MaybeMaxExpr,
+                              const SCEV *Candidate) {
+  const MaxExprType *MaxExpr = dyn_cast<MaxExprType>(MaybeMaxExpr);
+  if (!MaxExpr) return false;
+
+  for (const SCEV *Op : MaxExpr->operands())
+    if (SE.getMinusSCEV(Candidate, Op)->isZero())
+      return true;
+
+  return false;
+}
+
+
+/// Is MaybeMinExpr an SMin or UMin of Candidate and some other values?
+template<typename MaxExprType>
+static bool IsMinConsistingOf(ScalarEvolution &SE,
+                              const SCEV *MaybeMinExpr,
+                              const SCEV *Candidate) {
+  const SCEV *MaybeMaxExpr = nullptr;
+  if (!MatchNotExpr(MaybeMinExpr, MaybeMaxExpr))
+    return false;
+
+  return IsMaxConsistingOf<MaxExprType>(
+    SE, MaybeMaxExpr, SE.getNotSCEV(Candidate));
+}
+
+
+/// Is LHS `Pred` RHS true on the virtue of LHS or RHS being a Min or Max
+/// expression?
+static bool IsKnownPredicateViaMinOrMax(ScalarEvolution &SE,
+                                        ICmpInst::Predicate Pred,
+                                        const SCEV *LHS, const SCEV *RHS) {
+  switch (Pred) {
+  default:
+    return false;
+
+  case ICmpInst::ICMP_SGE:
+    std::swap(LHS, RHS);
+    // fall through
+  case ICmpInst::ICMP_SLE:
+    return
+      // min(A, ...) <= A
+      IsMinConsistingOf<SCEVSMaxExpr>(SE, LHS, RHS) ||
+      // A <= max(A, ...)
+      IsMaxConsistingOf<SCEVSMaxExpr>(SE, RHS, LHS);
+
+  case ICmpInst::ICMP_UGE:
+    std::swap(LHS, RHS);
+    // fall through
+  case ICmpInst::ICMP_ULE:
+    return
+      // min(A, ...) <= A
+      IsMinConsistingOf<SCEVUMaxExpr>(SE, LHS, RHS) ||
+      // A <= max(A, ...)
+      IsMaxConsistingOf<SCEVUMaxExpr>(SE, RHS, LHS);
+  }
+
+  llvm_unreachable("covered switch fell through?!");
+}
+
 /// isImpliedCondOperandsHelper - Test whether the condition described by
 /// Pred, LHS, and RHS is true whenever the condition described by Pred,
 /// FoundLHS, and FoundRHS is true.
@@ -6956,6 +7042,12 @@
                                              const SCEV *LHS, const SCEV *RHS,
                                              const SCEV *FoundLHS,
                                              const SCEV *FoundRHS) {
+  auto IsKnownPredicateFull =
+      [this](ICmpInst::Predicate Pred, const SCEV *LHS, const SCEV *RHS) {
+    return isKnownPredicateWithRanges(Pred, LHS, RHS) ||
+        IsKnownPredicateViaMinOrMax(*this, Pred, LHS, RHS);
+  };
+
   switch (Pred) {
   default: llvm_unreachable("Unexpected ICmpInst::Predicate value!");
   case ICmpInst::ICMP_EQ:
@@ -6965,26 +7057,26 @@
     break;
   case ICmpInst::ICMP_SLT:
   case ICmpInst::ICMP_SLE:
-    if (isKnownPredicateWithRanges(ICmpInst::ICMP_SLE, LHS, FoundLHS) &&
-        isKnownPredicateWithRanges(ICmpInst::ICMP_SGE, RHS, FoundRHS))
+    if (IsKnownPredicateFull(ICmpInst::ICMP_SLE, LHS, FoundLHS) &&
+        IsKnownPredicateFull(ICmpInst::ICMP_SGE, RHS, FoundRHS))
       return true;
     break;
   case ICmpInst::ICMP_SGT:
   case ICmpInst::ICMP_SGE:
-    if (isKnownPredicateWithRanges(ICmpInst::ICMP_SGE, LHS, FoundLHS) &&
-        isKnownPredicateWithRanges(ICmpInst::ICMP_SLE, RHS, FoundRHS))
+    if (IsKnownPredicateFull(ICmpInst::ICMP_SGE, LHS, FoundLHS) &&
+        IsKnownPredicateFull(ICmpInst::ICMP_SLE, RHS, FoundRHS))
       return true;
     break;
   case ICmpInst::ICMP_ULT:
   case ICmpInst::ICMP_ULE:
-    if (isKnownPredicateWithRanges(ICmpInst::ICMP_ULE, LHS, FoundLHS) &&
-        isKnownPredicateWithRanges(ICmpInst::ICMP_UGE, RHS, FoundRHS))
+    if (IsKnownPredicateFull(ICmpInst::ICMP_ULE, LHS, FoundLHS) &&
+        IsKnownPredicateFull(ICmpInst::ICMP_UGE, RHS, FoundRHS))
       return true;
     break;
   case ICmpInst::ICMP_UGT:
   case ICmpInst::ICMP_UGE:
-    if (isKnownPredicateWithRanges(ICmpInst::ICMP_UGE, LHS, FoundLHS) &&
-        isKnownPredicateWithRanges(ICmpInst::ICMP_ULE, RHS, FoundRHS))
+    if (IsKnownPredicateFull(ICmpInst::ICMP_UGE, LHS, FoundLHS) &&
+        IsKnownPredicateFull(ICmpInst::ICMP_ULE, RHS, FoundRHS))
       return true;
     break;
   }
Index: test/Transforms/IndVarSimplify/backedge-on-min-max.ll
===================================================================
--- /dev/null
+++ test/Transforms/IndVarSimplify/backedge-on-min-max.ll
@@ -0,0 +1,453 @@
+; RUN: opt < %s -indvars -S | FileCheck %s
+
+;; --- signed ---
+
+define void @min.signed.1(i32* %a, i32 %a_len, i32 %n) {
+; CHECK-LABEL: @min.signed.1
+ entry:
+  %smin.cmp = icmp slt i32 %a_len, %n
+  %smin = select i1 %smin.cmp, i32 %a_len, i32 %n
+  %entry.cond = icmp slt i32 0, %smin
+  br i1 %entry.cond, label %loop, label %exit
+
+ loop:
+  %idx = phi i32 [ 0, %entry ], [ %idx.inc, %latch ]
+  %idx.inc = add i32 %idx, 1
+  %in.bounds = icmp slt i32 %idx, %a_len
+  br i1 %in.bounds, label %ok, label %latch
+; CHECK: br i1 true, label %ok, label %latch
+
+ ok:
+  %addr = getelementptr i32* %a, i32 %idx
+  store i32 %idx, i32* %addr
+  br label %latch
+
+ latch:
+  %be.cond = icmp slt i32 %idx.inc, %smin
+  br i1 %be.cond, label %loop, label %exit
+
+ exit:
+  ret void
+}
+
+define void @min.signed.2(i32* %a, i32 %a_len, i32 %n) {
+; CHECK-LABEL: @min.signed.2
+ entry:
+  %smin.cmp = icmp slt i32 %a_len, %n
+  %smin = select i1 %smin.cmp, i32 %a_len, i32 %n
+  %entry.cond = icmp slt i32 0, %smin
+  br i1 %entry.cond, label %loop, label %exit
+
+ loop:
+  %idx = phi i32 [ 0, %entry ], [ %idx.inc, %latch ]
+  %idx.inc = add i32 %idx, 1
+  %in.bounds = icmp sgt i32 %a_len, %idx
+  br i1 %in.bounds, label %ok, label %latch
+; CHECK: br i1 true, label %ok, label %latch
+
+ ok:
+  %addr = getelementptr i32* %a, i32 %idx
+  store i32 %idx, i32* %addr
+  br label %latch
+
+ latch:
+  %be.cond = icmp slt i32 %idx.inc, %smin
+  br i1 %be.cond, label %loop, label %exit
+
+ exit:
+  ret void
+}
+
+define void @min.signed.3(i32* %a, i32 %n) {
+; CHECK-LABEL: @min.signed.3
+ entry:
+  %smin.cmp = icmp slt i32 42, %n
+  %smin = select i1 %smin.cmp, i32 42, i32 %n
+  %entry.cond = icmp slt i32 0, %smin
+  br i1 %entry.cond, label %loop, label %exit
+
+ loop:
+  %idx = phi i32 [ 0, %entry ], [ %idx.inc, %latch ]
+  %idx.inc = add i32 %idx, 1
+  %in.bounds = icmp slt i32 %idx, 42
+  br i1 %in.bounds, label %ok, label %latch
+; CHECK: br i1 true, label %ok, label %latch
+
+ ok:
+  %addr = getelementptr i32* %a, i32 %idx
+  store i32 %idx, i32* %addr
+  br label %latch
+
+ latch:
+  %be.cond = icmp slt i32 %idx.inc, %smin
+  br i1 %be.cond, label %loop, label %exit
+
+ exit:
+  ret void
+}
+
+define void @min.signed.4(i32* %a, i32 %n) {
+; CHECK-LABEL: @min.signed.4
+ entry:
+  %smin.cmp = icmp slt i32 42, %n
+  %smin = select i1 %smin.cmp, i32 42, i32 %n
+  %entry.cond = icmp slt i32 0, %smin
+  br i1 %entry.cond, label %loop, label %exit
+
+ loop:
+  %idx = phi i32 [ 0, %entry ], [ %idx.inc, %latch ]
+  %idx.inc = add i32 %idx, 1
+  %in.bounds = icmp sgt i32 42, %idx
+  br i1 %in.bounds, label %ok, label %latch
+; CHECK: br i1 true, label %ok, label %latch
+
+ ok:
+  %addr = getelementptr i32* %a, i32 %idx
+  store i32 %idx, i32* %addr
+  br label %latch
+
+ latch:
+  %be.cond = icmp slt i32 %idx.inc, %smin
+  br i1 %be.cond, label %loop, label %exit
+
+ exit:
+  ret void
+}
+
+define void @max.signed.1(i32* %a, i32 %a_len, i32 %n) {
+; CHECK-LABEL: @max.signed.1
+ entry:
+  %smax.cmp = icmp sgt i32 %a_len, %n
+  %smax = select i1 %smax.cmp, i32 %a_len, i32 %n
+  %entry.cond = icmp sgt i32 0, %smax
+  br i1 %entry.cond, label %loop, label %exit
+
+ loop:
+  %idx = phi i32 [ 0, %entry ], [ %idx.inc, %latch ]
+  %idx.inc = add i32 %idx, 1
+  %in.bounds = icmp sgt i32 %idx, %a_len
+  br i1 %in.bounds, label %ok, label %latch
+; CHECK: br i1 true, label %ok, label %latch
+
+ ok:
+  %addr = getelementptr i32* %a, i32 %idx
+  store i32 %idx, i32* %addr
+  br label %latch
+
+ latch:
+  %be.cond = icmp sgt i32 %idx.inc, %smax
+  br i1 %be.cond, label %loop, label %exit
+
+ exit:
+  ret void
+}
+
+define void @max.signed.2(i32* %a, i32 %a_len, i32 %n) {
+; CHECK-LABEL: @max.signed.2
+ entry:
+  %smax.cmp = icmp sgt i32 %a_len, %n
+  %smax = select i1 %smax.cmp, i32 %a_len, i32 %n
+  %entry.cond = icmp sgt i32 0, %smax
+  br i1 %entry.cond, label %loop, label %exit
+
+ loop:
+  %idx = phi i32 [ 0, %entry ], [ %idx.inc, %latch ]
+  %idx.inc = add i32 %idx, 1
+  %in.bounds = icmp slt i32 %a_len, %idx
+  br i1 %in.bounds, label %ok, label %latch
+; CHECK: br i1 true, label %ok, label %latch
+
+ ok:
+  %addr = getelementptr i32* %a, i32 %idx
+  store i32 %idx, i32* %addr
+  br label %latch
+
+ latch:
+  %be.cond = icmp sgt i32 %idx.inc, %smax
+  br i1 %be.cond, label %loop, label %exit
+
+ exit:
+  ret void
+}
+
+define void @max.signed.3(i32* %a, i32 %n, i32 %init) {
+; CHECK-LABEL: @max.signed.3
+ entry:
+  %smax.cmp = icmp sgt i32 42, %n
+  %smax = select i1 %smax.cmp, i32 42, i32 %n
+  %entry.cond = icmp sgt i32 %init, %smax
+  br i1 %entry.cond, label %loop, label %exit
+
+ loop:
+  %idx = phi i32 [ %init, %entry ], [ %idx.inc, %latch ]
+  %idx.inc = add i32 %idx, 1
+  %in.bounds = icmp sgt i32 %idx, 42
+  br i1 %in.bounds, label %ok, label %latch
+; CHECK: br i1 true, label %ok, label %latch
+
+ ok:
+  %addr = getelementptr i32* %a, i32 %idx
+  store i32 %idx, i32* %addr
+  br label %latch
+
+ latch:
+  %be.cond = icmp sgt i32 %idx.inc, %smax
+  br i1 %be.cond, label %loop, label %exit
+
+ exit:
+  ret void
+}
+
+define void @max.signed.4(i32* %a, i32 %n, i32 %init) {
+; CHECK-LABEL: @max.signed.4
+ entry:
+  %smax.cmp = icmp sgt i32 42, %n
+  %smax = select i1 %smax.cmp, i32 42, i32 %n
+  %entry.cond = icmp sgt i32 %init, %smax
+  br i1 %entry.cond, label %loop, label %exit
+
+ loop:
+  %idx = phi i32 [ %init, %entry ], [ %idx.inc, %latch ]
+  %idx.inc = add i32 %idx, 1
+  %in.bounds = icmp slt i32 42, %idx
+  br i1 %in.bounds, label %ok, label %latch
+; CHECK: br i1 true, label %ok, label %latch
+
+ ok:
+  %addr = getelementptr i32* %a, i32 %idx
+  store i32 %idx, i32* %addr
+  br label %latch
+
+ latch:
+  %be.cond = icmp sgt i32 %idx.inc, %smax
+  br i1 %be.cond, label %loop, label %exit
+
+ exit:
+  ret void
+}
+
+;; --- unsigned ---
+
+define void @min.unsigned.1(i32* %a, i32 %a_len, i32 %n) {
+; CHECK-LABEL: @min.unsigned.1
+ entry:
+  %umin.cmp = icmp ult i32 %a_len, %n
+  %umin = select i1 %umin.cmp, i32 %a_len, i32 %n
+  %entry.cond = icmp ult i32 5, %umin
+  br i1 %entry.cond, label %loop, label %exit
+
+ loop:
+  %idx = phi i32 [ 5, %entry ], [ %idx.inc, %latch ]
+  %idx.inc = add i32 %idx, 1
+  %in.bounds = icmp ult i32 %idx, %a_len
+  br i1 %in.bounds, label %ok, label %latch
+; CHECK: br i1 true, label %ok, label %latch
+
+ ok:
+  %addr = getelementptr i32* %a, i32 %idx
+  store i32 %idx, i32* %addr
+  br label %latch
+
+ latch:
+  %be.cond = icmp ult i32 %idx.inc, %umin
+  br i1 %be.cond, label %loop, label %exit
+
+ exit:
+  ret void
+}
+
+define void @min.unsigned.2(i32* %a, i32 %a_len, i32 %n) {
+; CHECK-LABEL: @min.unsigned.2
+ entry:
+  %umin.cmp = icmp ult i32 %a_len, %n
+  %umin = select i1 %umin.cmp, i32 %a_len, i32 %n
+  %entry.cond = icmp ult i32 5, %umin
+  br i1 %entry.cond, label %loop, label %exit
+
+ loop:
+  %idx = phi i32 [ 5, %entry ], [ %idx.inc, %latch ]
+  %idx.inc = add i32 %idx, 1
+  %in.bounds = icmp ugt i32 %a_len, %idx
+  br i1 %in.bounds, label %ok, label %latch
+; CHECK: br i1 true, label %ok, label %latch
+
+ ok:
+  %addr = getelementptr i32* %a, i32 %idx
+  store i32 %idx, i32* %addr
+  br label %latch
+
+ latch:
+  %be.cond = icmp ult i32 %idx.inc, %umin
+  br i1 %be.cond, label %loop, label %exit
+
+ exit:
+  ret void
+}
+
+define void @min.unsigned.3(i32* %a, i32 %n) {
+; CHECK-LABEL: @min.unsigned.3
+ entry:
+  %umin.cmp = icmp ult i32 42, %n
+  %umin = select i1 %umin.cmp, i32 42, i32 %n
+  %entry.cond = icmp ult i32 5, %umin
+  br i1 %entry.cond, label %loop, label %exit
+
+ loop:
+  %idx = phi i32 [ 5, %entry ], [ %idx.inc, %latch ]
+  %idx.inc = add i32 %idx, 1
+  %in.bounds = icmp ult i32 %idx, 42
+  br i1 %in.bounds, label %ok, label %latch
+; CHECK: br i1 true, label %ok, label %latch
+
+ ok:
+  %addr = getelementptr i32* %a, i32 %idx
+  store i32 %idx, i32* %addr
+  br label %latch
+
+ latch:
+  %be.cond = icmp ult i32 %idx.inc, %umin
+  br i1 %be.cond, label %loop, label %exit
+
+ exit:
+  ret void
+}
+
+define void @min.unsigned.4(i32* %a, i32 %n) {
+; CHECK-LABEL: @min.unsigned.4
+ entry:
+  %umin.cmp = icmp ult i32 42, %n
+  %umin = select i1 %umin.cmp, i32 42, i32 %n
+  %entry.cond = icmp ult i32 5, %umin
+  br i1 %entry.cond, label %loop, label %exit
+
+ loop:
+  %idx = phi i32 [ 5, %entry ], [ %idx.inc, %latch ]
+  %idx.inc = add i32 %idx, 1
+  %in.bounds = icmp ugt i32 42, %idx
+  br i1 %in.bounds, label %ok, label %latch
+; CHECK: br i1 true, label %ok, label %latch
+
+ ok:
+  %addr = getelementptr i32* %a, i32 %idx
+  store i32 %idx, i32* %addr
+  br label %latch
+
+ latch:
+  %be.cond = icmp ult i32 %idx.inc, %umin
+  br i1 %be.cond, label %loop, label %exit
+
+ exit:
+  ret void
+}
+
+define void @max.unsigned.1(i32* %a, i32 %a_len, i32 %n) {
+; CHECK-LABEL: @max.unsigned.1
+ entry:
+  %umax.cmp = icmp ugt i32 %a_len, %n
+  %umax = select i1 %umax.cmp, i32 %a_len, i32 %n
+  %entry.cond = icmp ugt i32 5, %umax
+  br i1 %entry.cond, label %loop, label %exit
+
+ loop:
+  %idx = phi i32 [ 5, %entry ], [ %idx.inc, %latch ]
+  %idx.inc = add i32 %idx, 1
+  %in.bounds = icmp ugt i32 %idx, %a_len
+  br i1 %in.bounds, label %ok, label %latch
+; CHECK: br i1 true, label %ok, label %latch
+
+ ok:
+  %addr = getelementptr i32* %a, i32 %idx
+  store i32 %idx, i32* %addr
+  br label %latch
+
+ latch:
+  %be.cond = icmp ugt i32 %idx.inc, %umax
+  br i1 %be.cond, label %loop, label %exit
+
+ exit:
+  ret void
+}
+
+define void @max.unsigned.2(i32* %a, i32 %a_len, i32 %n) {
+; CHECK-LABEL: @max.unsigned.2
+ entry:
+  %umax.cmp = icmp ugt i32 %a_len, %n
+  %umax = select i1 %umax.cmp, i32 %a_len, i32 %n
+  %entry.cond = icmp ugt i32 5, %umax
+  br i1 %entry.cond, label %loop, label %exit
+
+ loop:
+  %idx = phi i32 [ 5, %entry ], [ %idx.inc, %latch ]
+  %idx.inc = add i32 %idx, 1
+  %in.bounds = icmp ult i32 %a_len, %idx
+  br i1 %in.bounds, label %ok, label %latch
+; CHECK: br i1 true, label %ok, label %latch
+
+ ok:
+  %addr = getelementptr i32* %a, i32 %idx
+  store i32 %idx, i32* %addr
+  br label %latch
+
+ latch:
+  %be.cond = icmp ugt i32 %idx.inc, %umax
+  br i1 %be.cond, label %loop, label %exit
+
+ exit:
+  ret void
+}
+
+define void @max.unsigned.3(i32* %a, i32 %n, i32 %init) {
+; CHECK-LABEL: @max.unsigned.3
+ entry:
+  %umax.cmp = icmp ugt i32 42, %n
+  %umax = select i1 %umax.cmp, i32 42, i32 %n
+  %entry.cond = icmp ugt i32 %init, %umax
+  br i1 %entry.cond, label %loop, label %exit
+
+ loop:
+  %idx = phi i32 [ %init, %entry ], [ %idx.inc, %latch ]
+  %idx.inc = add i32 %idx, 1
+  %in.bounds = icmp ugt i32 %idx, 42
+  br i1 %in.bounds, label %ok, label %latch
+; CHECK: br i1 true, label %ok, label %latch
+
+ ok:
+  %addr = getelementptr i32* %a, i32 %idx
+  store i32 %idx, i32* %addr
+  br label %latch
+
+ latch:
+  %be.cond = icmp ugt i32 %idx.inc, %umax
+  br i1 %be.cond, label %loop, label %exit
+
+ exit:
+  ret void
+}
+
+define void @max.unsigned.4(i32* %a, i32 %n, i32 %init) {
+; CHECK-LABEL: @max.unsigned.4
+ entry:
+  %umax.cmp = icmp ugt i32 42, %n
+  %umax = select i1 %umax.cmp, i32 42, i32 %n
+  %entry.cond = icmp ugt i32 %init, %umax
+  br i1 %entry.cond, label %loop, label %exit
+
+ loop:
+  %idx = phi i32 [ %init, %entry ], [ %idx.inc, %latch ]
+  %idx.inc = add i32 %idx, 1
+  %in.bounds = icmp ult i32 42, %idx
+  br i1 %in.bounds, label %ok, label %latch
+; CHECK: br i1 true, label %ok, label %latch
+
+ ok:
+  %addr = getelementptr i32* %a, i32 %idx
+  store i32 %idx, i32* %addr
+  br label %latch
+
+ latch:
+  %be.cond = icmp ugt i32 %idx.inc, %umax
+  br i1 %be.cond, label %loop, label %exit
+
+ exit:
+  ret void
+}