Mr Bush was speaking on a visit to the Indonesian island of Bali
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US President George W Bush has insisted good progress is being made on North Korea, despite Pyongyang's dismissal of US efforts as "laughable".
He was speaking after meeting Indonesian President Megawati Sukarnoputri and Islamic leaders on a lightning visit to Bali.
Mr Bush said there were valuable discussions on the nuclear crisis at a regional security summit in Bangkok, where the US suggested Pyongyang might be offered some kind of informal security guarantee.
But North Korea said in a radio broadcast on Tuesday that it would settle for nothing less than a formal non-aggression treaty it has been demanding for months.
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BUSH IN ASIA
Bali agenda:
US support for Indonesia's fight against terrorism
Meeting with Islamic and other religious leaders
Explain US foreign policy, especially over Iraq
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Pyongyang, which has admitted to a nuclear weapons programme, is convinced that it could be the victim of a US-led attack, despite Washington's assurances to the contrary.
During Mr Bush's three-hour stop in Bali - the site of last year's bomb attacks that killed 202 people - he talked with President Megawati and met Indonesian religious leaders.
A US official said President Bush wanted to try to correct the perceptions of Muslims that his foreign policy was anti-Islamic.
"Americans hold a deep respect for the Islamic faith, which is professed by a growing number of my own citizens," Mr Bush said.
"We know that Islam is fully compatible with liberty and tolerance and progress because we see the proof in your country and in our own."
He said that he was pleased that during his meeting with religious leaders there was a general recognition of the importance of religious tolerance.
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HAVE YOUR SAY
Bush is a fundamentalist Christian, thus he can only see Islam in its own fundamentalism
Diren Yardimli, Istanbul, Turkey
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The US, Mr Bush said, would provide increased funding for Indonesia to support that tradition through education.
The talks took place amid intense security, as anti-American sentiment runs at its highest in Indonesia - home to the world's largest Muslim population.
More than 5,000 security personnel were deployed around the island.
North Korea's dismissal
Mr Bush was in Bali as part of a six-nation tour of Asia, which included his participation in an Asia-Pacific Economic Co-operation (Apec) summit in Bangkok on Monday and Tuesday.
There, the US outlined new proposals to try and resolve the standoff over North Korea's nuclear programme, as Pyongyang continued to ratchet up the tensions with the reported test-firing of at least one short-range missile.
Pyongyang said on Tuesday that Washington's offer was "not worth considering".
"We have demanded that the United States drop its hostile policy toward the (North) and sign a bilateral nonaggression treaty with us. We have not demanded some kind of security guarantee," state radio said.
President Bush this week ruled out a nonaggression treaty, which would require Congressional approval and could tie Washington's hands in a conflict.