A jury convicted Eric Clemmons of capital murder and first-degree assault. The Circuit Court of the City of St. Louis sentenced Clemmons to life imprisonment, with eligibility for parole in fifty years, for the murder conviction, and a concurrent sentence of ten years for the assault conviction. The convictions were affirmed by the Missouri Court of Appeals. State v. Clemmons, 682 S.W.2d 843 (Mo.App.1984). Pursuant to Missouri Supreme Court Rule 29.15, Clemmons then sought post-conviction relief. His motion for post-conviction relief was denied by the Circuit Court. The Missouri Court of Appeals again affirmed. Clemmons v. State, 795 S.W.2d 414 (Mo.App.1990), cert. denied, 500 U.S. 907, 111 S.Ct. 1689, 114 L.Ed.2d 83 (1991). Clemmons then filed a 28 U.S.C. § 2254 petition for habeas corpus relief. The District Court1 denied the petition, and Clem-mons now appeals.
*683Petitioner raised a number of issues in his original habeas petition. The magistrate judge’s2 report and recommendation held that some of the claims were procedurally barred because of Clemmons’s failure to raise them at the appropriate stages of the Missouri state-court proceedings. The magistrate judge recommended denying the remaining claims on their merits. After Clemmons objected to the magistrate judge’s report, the magistrate judge reconsidered some of the issues raised in the petition. Prior to the District Court’s ruling, but after the magistrate judge’s second report and recommendation, Clem-mons filed a motion to amend his habeas petition, seeking to add an additional claim. The District Court denied both the habeas petition, following the magistrate judge’s recommendations, and the motion to amend. Clemmons raises five issues in his appeal, abandoning some of the issues brought before the District Court. We affirm.
I.
In the early morning hours of August 14, 1982, Clemmons and some friends were standing on a street corner in St. Louis when they heard Clemmons’s brother, Stanley Barnes, yell from up the street that someone was trying to rob him. Clemmons and his friends approached Barnes and found him swinging a pipe at Lindsey Washington. On their way up the street to reach Barnes, Clemmons and his friends passed Todd Weems running in the opposite direction. When Clemmons reached Barnes, he pulled him off Washington and began hitting Washington himself. During his trial, Clemmons testified that Barnes then told him that Weems had been with Washington and that Weems had robbed Barnes as well. Clemmons then chased Weems. Clemmons testified that Weems swung- a board at him, and that Clemmons then found a pipe on the ground, which he swung at Weems in self-defense. Clemmons stated that he did not have the pipe when he first approached Weems. Testimony from the prosecution’s witnesses — the friends who had been with Clemmons on the street corner — indicated that Barnes also attacked Weems, although the testimony differed as to who reached Weems first, and who hit him first.
Weems died after the attack. Washington survived. Weems suffered bruises and abrasions on the left side of his face; abrasions on the bridge of his nose, above his right eye, and on his mouth; and lacerations on his ear. Weems also suffered blows to the back of his head. The medical examiner, Dr. Case, testified at Clem-mons’s trial that the cause of death was a closed-head injury.
II.
Clemmons’s first claim is that he received ineffective assistance of counsel because his trial counsel failed to elicit Weems’s exact cause of death from the medical examiner. The District Court denied this argument — claim VIII in the District Court’s opinion — on its merits. The standard for claims of ineffective assistance of counsel is well established. See Strickland v. Washington, 466 U.S. 668, 104 S.Ct. 2052, 80 L.Ed.2d 674 (1984). A petitioner must show that counsel’s performance “fell below an objective standard of reasonableness.” Strickland, 466 U.S. at 688. The petitioner must also show that there was a reasonable probability of a different outcome if counsel had performed properly. Id. at 687.
Clemmons’s argument focuses on the difference in testimony provided by the medical examiner at Clemmons’s trial and at Stanley Barnes’s trial. During Clem-mons’s trial, Dr. Case testified that some of the marks on Weems’s face were proba*684bly the result of being struck with a linear instrument, such as a pipe (Trial Tr. 289-90; Index to Exhibits vi.) In addition to the injuries Weems suffered on his face, he was also struck on the head. According to Dr. Case’s testimony, the blows to Weems’s head were likely the fatal blows (Trial Tr. 295-96.) When asked by the prosecution whether the injuries to Weems’s head “could ... have been consistent with an object such as a pipe that caused the injuries” on Weems’s face, Dr. Case responded “Yes” (Trial Tr. 309.) Clemmons’s attorney did not question Dr. Case about this testimony. When questioned about the blows to Weems’s head during Barnes’s trial, Dr. Case stated her opinion that “the blows to the head, to the scalp area were not made by the same linear instrument that caused the injuries” to Weems’s face (Barnes Trial Tr. 44.) She also said that the three blows to Weems’s face would not, by themselves, have been lethal.
Clemmons argues that Dr. Case’s testimony differed at the two trials, and that Clemmons’s trial counsel should have questioned Dr. Case further about the cause of the blows to the head. Clemmons must show a reasonable probability that this evidence would change the outcome of the trial. Clemmons has failed to do this. In the first place, we do not know what Dr. Case would have said if Clemmons’s counsel had pressed her on the matter. She might have repeated with even greater force that the fatal blows were “consistent with ... a pipe.” The fact that Dr. Case (arguably) testified differently at the later Barnes trial may mean only that she had had a chance to reflect and had somewhat modified her opinion. When questioned repeatedly about the subject, witnesses will sometimes give answers with different emphases, and do so entirely innocently. In the second place, Dr. Case’s testimony at the later trial was not necessarily inconsistent with what she had said at Clem-mons’s trial. When she testified at Barnes’s trial, she did not entirely rule out a pipe or similar object as the instrument that inflicted the injuries that caused death (Barnes Trial Tr. 39.) Finally, the jury could still have convicted Clemmons, on an accomplice theory, even if it thought someone else struck the fatal blows. On the whole, our confidence in the outcome of the trial is not undermined. Clemmons has not borne the burden of showing prejudice, even if counsel’s performance fell below an acceptable level.
III.
Clemmons’s second claim is essentially a replay of his first one. He argues that the State wrongfully withheld from him Dr. Case’s actual opinion — -as shown by her testimony at the subsequent trial of Stanley Barnes — that the fatal blows were probably not struck by the pipe that caused the injuries to the victim’s face. We cannot agree with the characterization of this claim as one of withholding material exculpatory evidence. There is no showing that the State knew, in advance of the Barnes trial, exactly what Dr. Case’s testimony at that trial was going to be. In our view, it is much more likely that the expert’s testimony, as is often the case, simply came out in a slightly different way on the two occasions.
It is certainly true that the prosecution has a duty to disclose evidence favorable to a defendant. See Kyles v. Whitley, 514 U.S. 419, 433, 115 S.Ct. 1555, 131 L.Ed.2d 490 (1995). However, even if there is a withholding of exculpatory evidence, constitutional error occurs only “if there is a reasonable probability that, had the evidence been disclosed to the defense, the result of the proceeding would have been different.” United States v. Bagley, 473 U.S. 667, 682, 105 S.Ct. 3375, 87 L.Ed.2d 481 (1985), quoted with approval in Kyles, supra, 514 U.S. at 433-34. The Bagley materiality standard is the same as the standard for Strickland prejudice, and we have already explained, in part II of this opinion, why we believe Clemmons has not met this standard.
*685IV.
Clemmons’s third claim is that the trial court erred in failing to give a proper self-defense instruction to the jury. Federal habeas relief is not available simply because of a faulty jury instruction. See Estelle v. McGuire, 502 U.S. 62, 71-72, 112 S.Ct. 475, 116 L.Ed.2d 385 (1991). For a petitioner to obtain relief based on a faulty jury instruction, the instruction must “so infect[ ] the entire trial that the resulting conviction violates due process.” Cupp v. Naughten, 414 U.S. 141, 147, 94 S.Ct. 396, 38 L.Ed.2d 368 (1973).
Clemmons is entitled to a self-defense instruction if there is evidence to support his theory. See United States v. Solem, 646 F.2d 322, 328 (8th Cir.1980). A self-defense instruction was given to the jury, based on the Missouri model jury instructions, but it did not contain the paragraph from the model instructions addressing the lawful use of deadly force. Instead, the jury instruction was based on the section of the model instructions to be used when there is no evidence supporting the lawful use of deadly force by the defendant. See MAI-CR 2d 2.41.1(2)(2) (1979). The problem, Clemmons argues, is that the instruction did not leave the jury room to. acquit him if it found that he acted reasonably in using deadly force. Assuming for the sake of argument that Clem-mons is correct, this is no more than an error of state law. This is not a case where the defense of self-defense was withheld completely from the jury. The argument, rather, is that the contours of the defense, under Missouri law, were not explained properly. We do not think that this sort of claim qualifies for habeas relief, nor do we believe, on the present record, that the absence of a correct (again assuming the validity of Clemmons’s point) self-defense instruction rendered this trial fundamentally unfair.
V.
Clemmons’s fourth claim is that his right to confront the witnesses against him was violated when the trial court admitted Lindsey Washington’s medical reports into evidence without requiring the prosecution to show that the treating physician was unavailable. (This claim, even if valid, appears to call in question only the conviction on Count II, the assault against Mr. Washington.) The District Court ruled that this claim was procedurally barred. Clemmons initially raised this claim in his Rule 29.15 motion for post-conviction relief. The Missouri Court of Appeals noted that allegations of trial error cannot be reviewed in post-conviction relief proceedings, and that Clemmons should have raised this argument in his direct appeal. See Clemmons v. State, 795 S.W.2d 414, 417 (Mo.App.1990). Clemmons’s trial counsel did object to the admission of the medical records during the trial, and the issue could have been raised in Clemmons’s direct appeal. Id.
Ineffectiveness of direct-appeal counsel can constitute cause for purposes of avoiding this procedural bar. This argument itself needs to have been raised properly in the state courts, and Clem-mons asserts that it was raised in the motion to recall the mandate. He appears to be correct about this point, but he still has to show “actual prejudice resulting from the claimed constitutional violation,” Stanley v. Jones, 973 F.2d 680, 682 (8th Cir.1992), in order to avoid the procedural bar. We do not believe that such prejudice has been shown. It seems to us reasonably probable that Clemmons’s point would have been rejected on its merits even if direct-appeal counsel had properly presented it. The medical records in question were admitted under the business-records exception to the hearsay rule. Brief for Appellant 36-37. This is an exception in its own right, and does not require that any witness be shown to have been unavailable. See Ohio v. Roberts, 448 U.S. 56, 66 & n. 8, 100 S.Ct. 2531, 65 L.Ed.2d 597 (1980) (specifically endorsing the business-records exception).
*686VI.
Clemmons’s final claim is that the District Court erred in denying his motion for leave to amend his habeas petition. Clemmons filed a motion with the District Court to amend his habeas petition after the magistrate judge made his second report and recommendation. Clemmons’s amended petition included a claim that the capital-murder instruction given to the jury was defective.
The rule governing habeas petitions allows amendments as provided in the applicable rules of civil procedure. See 28 U.S.C. § 2242. Federal Rule of Civil Procedure 15 provides that leave to amend “shall be freely given when justice so requires.” Fed.R.Civ.P. 15. The decision to grant or deny a motion to amend is within the discretion of the District Court. See Foman v. Davis, 371 U.S. 178, 83 S.Ct. 227, 9 L.Ed.2d 222 (1962). As the District Court observed, Clemmons had had plenty of opportunity to assert this claim earlier. He waited until after the magistrate judge’s second report and recommendation. The case had already been pending for some time. We see no abuse of discretion in the District Court’s unwillingness to allow the injection of a new issue at this late stage.
In addition, we agree with the District Court that the new claim, even if leave to assert it had been granted, would not have succeeded on the merits. Clemmons argues that the jury instruction on the capital-murder count used in his case did not require a finding of cool deliberation, which is an essential element of capital murder. We disagree. Under the instruction, the jury had to find either that defendant took the life of Todd Weems and reflected upon the matter coolly and fully before doing so, or that the defendant “aided or encouraged other persons in causing the death of Todd Weems and reflected upon this matter coolly and fully....” We see no error in this instruction. In State v. Bell, 854 S.W.2d 612, 614 (Mo. App.1993), the instruction was differently worded and would have allowed the jury to find Bell guilty of capital murder without finding cool deliberation on his part. That is not true here.
For all of these reasons, the judgment is affirmed.
. The Hon. John F. Nangle, United States District Judge for the Eastern District of Missouri.
. The Hon. Terry I. Adelxnan, Üniled States Magistrate Judge for the Eastern District of Missouri.