feat(MultiTapeTM): Nondeterministic multi-tape Turing machines

[barni120400]

Adds nondeterministic multi-tape Turing machines: the transition function is replaced by a transition relation, and a computation path is the list of configurations it passes through together with a proof that they form a chain of steps from the initial configuration to the one it ends at, so List.IsChainFromTo carries the whole validity condition. MultiTapeTM.toNTM embeds the deterministic machine and preserves computation in bounded time and space.

Configurations and the effect of an action move to a shared MultiTape/Configuration.lean. No existing statement changes meaning.

NB: claude was used heavily throughout.

[crei]

OK this is a bit weird. ne_nil is already enforced by isChainFromTo, but you need it for cfg.getLast ne_nil below. Is there a nicer way to do it? Maybe

start : cfgs.head? = some (ntm.initCfg input)
isChain : cfgs.IsChain ntm.Step

[crei]

Another way would be

last : Cfg k Symbol State input
isChainFromTo : cfgs.IsChainFromTo ntm.Step (ntm.initCfg input) last

then you don't need def last below.

[barni120400]

I prefer the second suggestion. It's still a bit weird that we duplicate the last element - it's written both at "last" and it's the last element in the chain.

[crei]

Yet more options:

structure ComputationPath (ntm : MultiTapeNTM k Symbol State) (input : List Symbol) where
  /-- the configurations passed through after the initial one -/
  tail : List (Cfg k Symbol State input)
  isChain : ((ntm.initCfg input) :: tail).IsChain ntm.Step

def ComputationPath.last (p : ntm.ComputationPath input) : Cfg k Symbol State input :=
    ((ntm.initCfg input) :: tail).getLast sorry

or just

def _root_.List.IsComputationPathTo {ntm : MultiTapeNTM k Symbol State} {input : List Symbol}
    (cfgs : List (Cfg k Symbol State input)) (last : Cfg k Symbol State input) :=
  List.IsChainFromTo ntm.Step (ntm.initCfg input) last

[crei]

This looks good as it is, maybe some fine tuning for ComputationPath, but also not needed. In the worst case, we can adjust it if we see it doesn't work well for proofs.

[barni120400]

This looks good as it is, maybe some fine tuning for ComputationPath, but also not needed. In the worst case, we can adjust it if we see it doesn't work well for proofs.

Ok sounds good

[crei]

Suggested change

	def Action.apply (out : Action k Symbol State) (cfg : Cfg k Symbol State input) :
	def Action.apply (action : Action k Symbol State) (cfg : Cfg k Symbol State input) :

[crei]

Suggested change

	lemma workTapePos_apply_le (out : Action k Symbol State)
	lemma workTapePos_apply_le (action : Action k Symbol State)

[crei]

Just noticed that this includes #827, which is probably fine, but please rename all mentions of out / transitionout to action.