Two Craigavon men, jailed for murdering Constable Stephen Carroll in 2009, this week began an appeal against their convictions.
Brendan McConville and John Paul Wootton were found guilty of murder last year.
Constable Carroll, 48, was shot dead after he responded to a 999 emergency call in Craigavon in March 2009.
At the time the dissident republican group, the Continuity IRA, claimed it was responsible for the shooting.
McConville, 42, of Glenholme Avenue in Craigavon is serving at least a 25 year sentence.
While Wootton, 22, of Collingdale, Lurgan, received a minimum 14-year term.
Dressed in dark suits, both men were led handcuffed into a packed Court of Appeal in Belfast for their bid to have their convictions quashed.
Their family, friends and supporters - including Gerry Conlon, one of the Guildford Four - gathered in the public gallery, a few feet away from the murdered officer’s widow Kate Carroll.
McConville’s legal team opened the hearing with a scathing assessment of the account given by a man referred to only as Witness M who claimed to have seen their client in the area around the time of the killing.
Lord Chief Justice Sir Declan Morgan, Lord Justice Higgins and Lord Justice Coghlin were told the prosecution case depended entirely on circumstantial evidence based on this alleged sighting.
Barry Macdonald QC said: “The appellant contends that Witness M was a dishonest, demonstrably unreliable witness whose account of what he saw on the night in question bordered on the farcical.
“We will be seeking to adduce fresh evidence to the effect that Witness M was known to his own family as a Walter Mitty character.
“He had the habit of making up stories and was such a compulsive liar that, in the words of the new witness, you couldn’t believe anything he said.”
The barrister continued: “Witness M lives in a world of his own and would often say things that simply weren’t true.”
As McConville took notes at various points in the hearing, the court heard he maintains he had nothing to do with the murder.
It was set out how Constable Carroll was shot twice in the head as he sat in his car.
The murder weapon, an AK47, was recovered from under an oil tank during subsequent police searches.
Witness M only phoned police to make his claims 11 months later, the court heard.
Mr Macdonald added: “He had been drinking until the early hours of the morning and he had drunk so much that one could tell at the other end of the telephone line that he was under the influence of alcohol.”
The accuracy of Witness M’s claims about being threatened and warned to “keep your mouth shut” was also called into question.
Mr Macdonald argued that different timings given for this alleged approach should have “set alarm bells ringing” among police.
The barrister then challenged his account on when he claims to have made a connection between what he saw and the shooting.
He set out how Witness M at one stage recalled going home and discussing it with his partner.
“Once again that ought to have alerted the police officer and indeed, we say, the trial judge, to the fact that this man was not a reliable historian, to put it mildly,” the lawyer said.
The hearing continues.