[Merged by Bors] - feat(computability/tm_to_partrec): prove finiteness #6955
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| any_goals { | ||
| cases finset.insert_subset.1 HS₁ with h₁ h₂, | ||
| id { have h₃ := h₂ W } <|> try { simp [finset.subset_iff] at h₂ } }, | ||
| { exact supports_insert.2 ⟨⟨λ _, h₃, λ _, h₁⟩, q_ih H₁ h₂⟩ }, -- move |
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Should any_goals maybe preserve tags?
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Perhaps, but I'm not planning on doing anything about that in this PR.
| states in `code_supp c k`, so this is a finite state machine. Even though the underlying type of | ||
| state labels `Λ'` is infinite, for a given partial recursive function `c` and continuation `k`, | ||
| only finitely many states are accessed, corresponding roughly to subterms of `c`. -/ | ||
| theorem tr_supports (c k) : @TM2.supports _ _ _ _ _ ⟨tr_normal c k⟩ tr (code_supp c k) := |
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The non-standard inhabited instance is a bit ugly here.
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The TM2 functions assume that there is a single initial state given by the default on the type. For this proof we need a family of TMs with different starting points, so we have to override the inhabited instance in the statement.
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The proof in this file was incomplete, and I finally found the time to finish it. `tm_to_partrec.lean` constructs a turing machine that can simulate a given partial recursive function. Previously, the functional correctness of this machine is proven, but it uses an infinite state space so it is not a priori obvious that it is in fact a true (finite) turing machine, which is important for the Church-Turing theorem. This PR adds a proof that the machine is in fact finite.
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This was already looking good the last time. bors d+ |
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The proof in this file was incomplete, and I finally found the time to finish it. `tm_to_partrec.lean` constructs a turing machine that can simulate a given partial recursive function. Previously, the functional correctness of this machine was proven, but it uses an infinite state space so it is not a priori obvious that it is in fact a true (finite) turing machine, which is important for the Church-Turing theorem. This PR adds a proof that the machine is in fact finite.
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The proof in this file was incomplete, and I finally found the time to finish it. `tm_to_partrec.lean` constructs a turing machine that can simulate a given partial recursive function. Previously, the functional correctness of this machine was proven, but it uses an infinite state space so it is not a priori obvious that it is in fact a true (finite) turing machine, which is important for the Church-Turing theorem. This PR adds a proof that the machine is in fact finite.
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The proof in this file was incomplete, and I finally found the time to
finish it.
tm_to_partrec.leanconstructs a turing machine that cansimulate a given partial recursive function. Previously, the functional
correctness of this machine was proven, but it uses an infinite state
space so it is not a priori obvious that it is in fact a true (finite)
turing machine, which is important for the Church-Turing theorem. This
PR adds a proof that the machine is in fact finite.