`Compfiles/KolmogorovStreams.lean`

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Project sources: Pinned compfiles documentation, Pinned compfiles corpus index

9 declarations, 0 contain sorry, 0 declared as axioms

Puzzle referenced from this tweet: https://twitter.com/sigfpe/status/1474173467016589323

From the book Out of their Minds: The Lives and Discoveries of 15 Great Computer Scientists by Dennis Shasha and Cathy Lazere.

Problem: Suppose each (finite) word is either "decent" or "indecent". Given an infinite sequence of characters, can you always break it into finite words so that all of them except perhaps the first one belong to the same class?

Imports (4)

Imported by 1 file

Compfiles.lean

def KolmogorovStreams.break_into_words line 38 source
no axioms what this means
Statement uses: Stream', Nat, List
Proof uses: Function.curry, Stream'.corec, Prod, KolmogorovStreams.break_into_words.match_1, Stream'.take, Stream'.head, Prod.mk, Stream'.tail, Stream'.drop
lemma KolmogorovStreams.break_into_words_cons Dropping the first word is equivalent to dropping first_length symbols of the original stream line 52 source
Source documentation
Dropping the first word is equivalent to dropping first_length symbols of the original stream.

standard axioms: Quot.sound, propext what this means
Statement uses: Stream', Nat, Eq, List, Stream'.tail, KolmogorovStreams.break_into_words, Stream'.cons, Stream'.drop
Proof uses: of_eq_true, Stream'.map, Prod, Stream'.take, Stream'.head, Prod.fst, Prod.snd, Stream'.iterate, Prod.mk, Eq.trans, True, congrFun', congrArg, Stream'.tail_iterate, eq_self

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
lemma break_into_words_cons
    (lengths : Stream' ℕ)
    (first_length : ℕ)
    (a : Stream' α) :
    (break_into_words (first_length::lengths) a).tail =
           break_into_words lengths (a.drop first_length) := by
…
```
Mentions: KolmogorovStreams.break_into_words
lemma KolmogorovStreams.break_into_words_closed_form line 60 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Stream', Nat, Eq, List, KolmogorovStreams.break_into_words, Stream'.take, Stream'.drop, Finset.sum, Finset.range
Proof uses: funext, Nat.recAux, Eq.refl, OfNat.ofNat, Stream'.tail, Stream'.head, Eq.mpr, Stream'.cons, id, Eq.rec, Eq.trans, congrArg, Eq.symm, Stream'.eta, KolmogorovStreams.break_into_words_cons, Trans.trans, HAdd.hAdd, congrFun, congr, Stream'.drop_drop and 3 more
10 instances, coercions and auxiliary constants left out.

Tactic trace: 10 steps
Open to load the checked before-and-after goal states.
def KolmogorovStreams.all_prefixes line 80 source
no axioms what this means
Statement uses: List, Stream'
Proof uses: Stream'.All, Stream'.inits
lemma KolmogorovStreams.take_prefix line 82 source
standard axioms: Quot.sound what this means
Statement uses: List, Stream', KolmogorovStreams.all_prefixes, Nat, LT.lt, OfNat.ofNat, Stream'.take
Proof uses: Nat.casesAuxOn, Eq, absurd, lt_irrefl, Eq.symm, HAdd.hAdd, Eq.mp, Stream'.get, Stream'.inits, congrArg, Stream'.get_inits, Eq.refl
8 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma KolmogorovStreams.not_all_prefixes line 92 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: List, Stream', Not, KolmogorovStreams.all_prefixes, Exists, Nat, Stream'.take, Nat.succ
Proof uses: id, Eq.mp, HAdd.hAdd, OfNat.ofNat, Eq.trans, congrArg, forall_congr, Stream'.get, Stream'.inits, Stream'.get_inits
4 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma KolmogorovStreams.exists_break_into_words Documented declaration: exists_break_into_words line 104 source
Source documentation
If from every "good" position of the stream we can carve off a nonempty word satisfying q, landing at another good position, then the stream can be broken into words that all satisfy q.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: List, Nat, Stream', OfNat.ofNat, Exists, And, LT.lt, HAdd.hAdd, Stream'.take, Stream'.drop, Stream'.All, KolmogorovStreams.break_into_words
Proof uses: Subtype, Nat.rec, Subtype.mk, Subtype.property, And.left, And.right, Eq, Finset.sum, Finset.range, Nat.recAux, Eq.refl, Eq.mpr, id, congrArg, Finset.sum_range_succ, Eq.symm, KolmogorovStreams.break_into_words_closed_form, Stream'.get, Exists.intro, And.intro and 2 more
14 instances, coercions and auxiliary constants left out.

Tactic trace: 15 steps
Open to load the checked before-and-after goal states.
```
lemma exists_break_into_words
    (q : List α → Prop)
    (good : ℕ → Prop)
    (a : Stream' α)
    (h0 : good 0)
    (h : ∀ n, good n → ∃ k, 0 < k ∧ good (n + k) ∧ q ((a.drop n).take k)) :
    ∃ lengths : Stream' ℕ,
      lengths.All (0 < ·) ∧ (break_into_words lengths a).All q := by
…
```
Mentions: h, KolmogorovStreams.break_into_words
def KolmogorovStreams.all_same_class line 132 source
no axioms what this means
Statement uses: List, Stream'
Proof uses: Or, Stream'.All, Not
problem KolmogorovStreams.kolmogorov_streams line 138 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: List, Stream', Exists, Nat, And, Stream'.All, LT.lt, OfNat.ofNat, KolmogorovStreams.all_same_class, Stream'.tail, KolmogorovStreams.break_into_words
Proof uses: Not, KolmogorovStreams.all_prefixes, Stream'.drop, HAdd.hAdd, Or, Classical.em, True, Stream'.take, Eq.mpr, id, Eq, congrArg, Stream'.drop_drop, Nat.succ_pos, Nat.succ, KolmogorovStreams.not_all_prefixes, Exists.intro, And.intro, trivial, KolmogorovStreams.exists_break_into_words and 18 more
17 instances, coercions and auxiliary constants left out.

Tactic trace: 41 steps
Open to load the checked before-and-after goal states.