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namespace OAI
/-! Active numerical labels in minimal prohibited witnesses. -/
namespace TwoPointCorrelations
open Finset
noncomputable def wordPrimeContribution (h : ℕ) (w : List SignedStep) (p a b : ℕ) : ℤ := by
classical
exact ∑ i ∈ Ico a b, if TuplePrimeAt w p i then wordStepDisplacement h w i else 0
/-- A contiguous relation with a nonzero contribution from this label. -/
def ActivePrime (h : ℕ) (w : List SignedStep) (p : ℕ) : Prop :=
∃ c a b, a ≤ b ∧ b ≤ w.length ∧ (∃ i, TuplePrimeAt w c i) ∧
(c : ℤ) ∣ intervalDisplacement (wordStepDisplacement h w) a b ∧
¬(c : ℤ) ∣ wordPrimeContribution h w p a b
lemma wordPrimeContribution_eq_interval (h : ℕ) (w : List SignedStep) (p a b : ℕ)
(hp : ∀ i ∈ Ico a b, TuplePrimeAt w p i) :
wordPrimeContribution h w p a b = intervalDisplacement (wordStepDisplacement h w) a b := by
classical
unfold wordPrimeContribution intervalDisplacement
apply sum_congr rfl
intro i hi
simp only [hp i hi, ite_true]
lemma wordPrimeContribution_eq_singleton (h : ℕ) (w : List SignedStep) (p a b j : ℕ)
(hj : j ∈ Ico a b) (hp : TuplePrimeAt w p j)
(hunique : ∀ i ∈ Ico a b, TuplePrimeAt w p i → i = j) :
wordPrimeContribution h w p a b = wordStepDisplacement h w j := by
classical
unfold wordPrimeContribution
rw [sum_eq_single j]
· simp only [hp, ite_true]
· intro i hi hne
have hnot : ¬TuplePrimeAt w p i := fun hpi => hne (hunique i hi hpi)
simp only [hnot, ite_false]
· exact fun h => False.elim (h hj)
/-- A label first entering at the last step is active in the original
prohibited suffix, because its contribution is exactly that last step. -/
lemma ForwardProhibited.last_new_active {h s : ℕ} {supply : ℕ → ℕ → Prop}
{w : List SignedStep} (hw : ForwardProhibited h s supply w) {z : ℕ}
(hz : TuplePrimeAt w z (w.length - 1))
(hearlier : ∀ i, i < w.length - 1 → ¬TuplePrimeAt w z i)
(hoff : ∀ p i, (∃ j, TuplePrimeAt w p j) → i < w.length →
¬TuplePrimeAt w p i → ¬(p : ℤ) ∣ wordStepDisplacement h w i) :
ActivePrime h w z := by
rcases hw with ⟨hlen, _, _, _, _, c, hc, hclast, a, ha, halast, hdiv⟩
refine ⟨c, a, w.length, by omega, le_rfl, ⟨0, hc⟩, ?_, ?_⟩
· simpa only [wordSlice, Nat.sub_self, List.take_length, List.length_drop] using
(show (c : ℤ) ∣ intervalDisplacement (wordStepDisplacement h w) a w.length from by
rw [← wordSlice_displacement h w (by omega)]
simpa only [wordSlice, ← List.length_drop, List.take_length] using hdiv)
· rw [wordPrimeContribution_eq_singleton h w z a w.length (w.length - 1)
(by simp only [mem_Ico]; omega) hz]
· exact hoff c (w.length - 1) ⟨0, hc⟩ (by omega) hclast
· intro i hi hzi
have hi' := (mem_Ico.mp hi).2
by_contra hne
exact hearlier i (by omega) hzi
lemma prime_dvd_wordPrimeContribution (h : ℕ) (w : List SignedStep) (p a b : ℕ) :
(p : ℤ) ∣ wordPrimeContribution h w p a b := by
classical
unfold wordPrimeContribution
apply Finset.dvd_sum
intro i _
by_cases hpi : TuplePrimeAt w p i
· simpa only [hpi, ite_true] using (tuplePrime_dvd_step (h := h) hpi)
· simp only [hpi, ite_false, dvd_zero]
lemma ActivePrime.distinct_controller {h p : ℕ} {w : List SignedStep}
(hp : ActivePrime h w p) :
∃ c a b, c ≠ p ∧ a ≤ b ∧ b ≤ w.length ∧ (∃ i, TuplePrimeAt w c i) ∧
(c : ℤ) ∣ intervalDisplacement (wordStepDisplacement h w) a b ∧
¬(c : ℤ) ∣ wordPrimeContribution h w p a b := by
obtain ⟨c, a, b, hab, hb, hc, hd, hn⟩ := hp
refine ⟨c, a, b, ?_, hab, hb, hc, hd, hn⟩
intro heq
subst c
exact hn (prime_dvd_wordPrimeContribution h w p a b)
/-- Intersecting an interval-shaped occurrence set with a prefix gives an
explicit nonempty interval of indices. -/
lemma TuplePrimeIntervals.prefix_occurrences {w : List SignedStep}
(hw : TuplePrimeIntervals w) (p n : ℕ) (hn : n ≤ w.length)
(hp : ∃ j, j < n ∧ TuplePrimeAt w p j) :
∃ a b, a < b ∧ b ≤ n ∧ ∀ i, i < n →
(TuplePrimeAt w p i ↔ a ≤ i ∧ i < b) := by
classical
let S := (range n).filter (TuplePrimeAt w p)
have hS : S.Nonempty := by
obtain ⟨j, hj, hpj⟩ := hp
exact ⟨j, mem_filter.mpr ⟨mem_range.mpr hj, hpj⟩⟩
let a := S.min' hS
let b := S.max' hS + 1
have haS : a ∈ S := min'_mem S hS
have hbS : b - 1 ∈ S := by
simpa only [b, Nat.add_sub_cancel] using max'_mem S hS
have han := (mem_range.mp (mem_filter.mp haS).1)
have hbn := (mem_range.mp (mem_filter.mp hbS).1)
have hab : a < b := by
have h := min'_le S (S.max' hS) (max'_mem S hS)
dsimp [a, b]
omega
refine ⟨a, b, hab, by dsimp [b] at *; omega, ?_⟩
intro i hi
constructor
· intro hpi
have hiS : i ∈ S := mem_filter.mpr ⟨mem_range.mpr hi, hpi⟩
exact ⟨min'_le S i hiS, Nat.lt_succ_of_le (le_max' S i hiS)⟩
· rintro ⟨hai, hib⟩
exact hw p a i (b - 1) hai (by omega) (by omega)
(mem_filter.mp haS).2 (mem_filter.mp hbS).2
lemma wordPrimeContribution_restrict (h : ℕ) (w : List SignedStep) (p n a b : ℕ)
(hbn : b ≤ n)
(hp : ∀ i, i < n → (TuplePrimeAt w p i ↔ a ≤ i ∧ i < b)) :
wordPrimeContribution h w p 0 n = intervalDisplacement (wordStepDisplacement h w) a b := by
classical
unfold wordPrimeContribution intervalDisplacement
calc
_ = ∑ i ∈ Ico a b, if TuplePrimeAt w p i then wordStepDisplacement h w i else 0 := by
symm
apply Finset.sum_subset (Ico_subset_Ico (Nat.zero_le a) hbn)
intro i hi hin
have hnot : ¬TuplePrimeAt w p i := by
intro hpi
exact hin (mem_Ico.mpr ((hp i (mem_Ico.mp hi).2).mp hpi))
simp only [hnot, ite_false]
_ = _ := by
apply sum_congr rfl
intro i hi
simp only [(hp i (lt_of_lt_of_le (mem_Ico.mp hi).2 hbn)).mpr
⟨(mem_Ico.mp hi).1, (mem_Ico.mp hi).2⟩, ite_true]
/-- Minimality forbids a reversed suffix starting at a strictly positive
index, when an internal initial segment is divisible by its final prime. -/
lemma MinimalWord.no_reversed_divisible_interval {h s : ℕ} {supply : ℕ → ℕ → Prop}
{w : List SignedStep} (hw : MinimalWord (ForwardProhibited h s supply) w)
{a c p : ℕ} (ha : 0 < a) (hac : a + 1 < c) (hc : c < w.length)
(hp : TuplePrimeAt w p (w.length - 1)) (hnot : ¬TuplePrimeAt w p a) :
¬(p : ℤ) ∣ intervalDisplacement (wordStepDisplacement h w) a c := by
intro hdiv
have hpro := reversed_interval_forwardProhibited h s supply w hw.1.2.1 hw.1.2.2.1
hw.1.2.2.2.1 hw.1.2.2.2.2.1 hac hc le_rfl hp hnot hdiv
apply hw.2 (reverseWord (wordSlice w a w.length))
(Or.inr (reverseWord_infix (wordSlice_infix w a w.length))) ?_ hpro
rw [reverseWord_length, wordSlice_length w (by omega) le_rfl]
omega
/-- Every label of a minimal prohibited word is active or has a nonzero
prefix contribution modulo an active label used at that departure. -/
theorem MinimalWord.active_dichotomy {h s : ℕ} {supply : ℕ → ℕ → Prop}
{w : List SignedStep} (hw : MinimalWord (ForwardProhibited h s supply) w)
(hnew : ∀ i, i + 1 < w.length →
∃ p, TuplePrimeAt w p (i + 1) ∧ ¬TuplePrimeAt w p i)
(hoff : ∀ p i, (∃ j, TuplePrimeAt w p j) → i < w.length →
¬TuplePrimeAt w p i → ¬(p : ℤ) ∣ wordStepDisplacement h w i)
(y : ℕ) (hy : ∃ j, TuplePrimeAt w y j) :
ActivePrime h w y ∨ ∃ z i, ActivePrime h w z ∧ TuplePrimeAt w z i ∧
¬(z : ℤ) ∣ wordPrimeContribution h w y 0 i := by
classical
have hlen : 3 ≤ w.length := hw.1.1
have hinter : TuplePrimeIntervals w := hw.1.2.2.2.2.1
obtain ⟨z, hzlast', hzprev⟩ := hnew (w.length - 2) (by omega)
have heqlast : w.length - 2 + 1 = w.length - 1 := by omega
have hzlast : TuplePrimeAt w z (w.length - 1) := heqlast ▸ hzlast'
have hzearly : ∀ i, i < w.length - 1 → ¬TuplePrimeAt w z i := by
intro i hi hzi
exact hzprev (hinter z i (w.length - 2) (w.length - 1)
(by omega) (by omega) (by omega) hzi hzlast)
have hzactive := hw.1.last_new_active hzlast hzearly hoff
by_cases hypre : ∃ j, j < w.length - 1 ∧ TuplePrimeAt w y j
· obtain ⟨a, b, hab, hbn, hyinterval⟩ :=
hinter.prefix_occurrences y (w.length - 1) (by omega) hypre
have hyeq := wordPrimeContribution_restrict h w y (w.length - 1) a b hbn hyinterval
by_cases hzd : (z : ℤ) ∣ wordPrimeContribution h w y 0 (w.length - 1)
· have hdiv : (z : ℤ) ∣ intervalDisplacement (wordStepDisplacement h w) a b := by
rwa [hyeq] at hzd
have hab2 : a + 1 < b := by
by_contra hn
have hb : b = a + 1 := by omega
have hd := hoff z a ⟨w.length - 1, hzlast⟩ (by omega) (hzearly a (by omega))
exact hd (by simpa only [hb, intervalDisplacement_singleton] using hdiv)
have ha : a = 0 := by
by_contra hn
exact hw.no_reversed_divisible_interval (by omega) hab2 (by omega)
hzlast (hzearly a (by omega)) hdiv
obtain ⟨z', hz'one, hz'zero⟩ := hnew 0 (by omega)
obtain ⟨a', b', hab', hb'b, hz'interval⟩ :=
hinter.prefix_occurrences z' b (by omega) ⟨1, by omega, hz'one⟩
have ha' : a' = 1 := by
have hle := ((hz'interval 1 (by omega)).mp hz'one).1
have hne : a' ≠ 0 := by
intro heq
have hp := (hz'interval a' (by omega)).mpr ⟨le_rfl, hab'⟩
exact hz'zero (by simpa only [heq] using hp)
omega
have hzeq := wordPrimeContribution_restrict h w z' b a' b' hb'b hz'interval
have hnonzero : ¬(z : ℤ) ∣ wordPrimeContribution h w z' 0 b := by
intro hzero
have hdiv' : (z : ℤ) ∣ intervalDisplacement (wordStepDisplacement h w) 1 b' := by
simpa only [hzeq, ha'] using hzero
have hb' : 2 < b' := by
by_contra hn
have hb'eq : b' = 1 + 1 := by omega
exact (hoff z 1 ⟨w.length - 1, hzlast⟩ (by omega) (hzearly 1 (by omega)))
(by simpa only [hb'eq, intervalDisplacement_singleton] using hdiv')
exact hw.no_reversed_divisible_interval (by omega) hb' (by omega)
hzlast (hzearly 1 (by omega)) hdiv'
have hz'active : ActivePrime h w z' := by
refine ⟨z, 0, b, by omega, by omega, ⟨w.length - 1, hzlast⟩, ?_, hnonzero⟩
simpa only [ha] using hdiv
right
refine ⟨z', 1, hz'active, hz'one, ?_⟩
have hyzero : TuplePrimeAt w y 0 := (hyinterval 0 (by omega)).mpr ⟨by omega, by omega⟩
rw [wordPrimeContribution_eq_singleton h w y 0 1 0 (by simp) hyzero]
· exact hoff z' 0 ⟨1, hz'one⟩ (by omega) hz'zero
· intro i hi _
have := mem_Ico.mp hi
omega
· exact Or.inr ⟨z, w.length - 1, hzactive, hzlast, hzd⟩
· left
have hyearly : ∀ i, i < w.length - 1 → ¬TuplePrimeAt w y i := by
intro i hi hpi
exact hypre ⟨i, hi, hpi⟩
obtain ⟨j, hyj⟩ := hy
have hj := hyj.index_lt
have hjeq : j = w.length - 1 := by
by_contra hn
exact hyearly j (by omega) hyj
exact hw.1.last_new_active (hjeq ▸ hyj) hyearly hoff
/-- Equal-size squarefree tuples that change must also acquire a new prime. -/
lemma adjacent_tuple_new_prime (w : List SignedStep) (J : ℕ)
(hsq : ∀ a ∈ w, Squarefree a.tuple)
(hcard : ∀ a ∈ w, a.tuple.primeFactors.card = J)
(hchain : w.IsChain (fun a b => a.tuple ≠ b.tuple)) :
∀ i, i + 1 < w.length → ∃ p, TuplePrimeAt w p (i + 1) ∧ ¬TuplePrimeAt w p i := by
intro i hi
have hi0 : i < w.length := by omega
have ha := List.getElem_mem hi0
have hb := List.getElem_mem hi
have hneq : w[i].tuple ≠ w[i + 1].tuple := (List.isChain_iff_getElem.mp hchain) i hi
have hsets : w[i + 1].tuple.primeFactors ≠ w[i].tuple.primeFactors := by
intro heq
apply hneq
calc
_ = ∏ p ∈ w[i].tuple.primeFactors, p := (Nat.prod_primeFactors_of_squarefree (hsq _ ha)).symm
_ = ∏ p ∈ w[i + 1].tuple.primeFactors, p := by rw [heq]
_ = _ := Nat.prod_primeFactors_of_squarefree (hsq _ hb)
have hnot : ¬w[i + 1].tuple.primeFactors ⊆ w[i].tuple.primeFactors := by
intro hsub
apply hsets
exact Finset.eq_of_subset_of_card_le hsub (by rw [hcard _ ha, hcard _ hb])
obtain ⟨p, hpb, hpa⟩ := Finset.not_subset.mp hnot
have hp := Nat.mem_primeFactors.mp hpb
refine ⟨p, (tuplePrimeAt_iff_getElem w p (i + 1) hi).mpr ⟨hp.1, hp.2.1⟩, ?_⟩
intro hpi
have hpai := (tuplePrimeAt_iff_getElem w p i hi0).mp hpi
exact hpa (Nat.mem_primeFactors.mpr ⟨hpai.1, hpai.2, (hsq _ ha).ne_zero⟩)
/-- The active-label dichotomy with the manuscript's concrete arithmetic
hypotheses: squarefree whole tuples of fixed prime-factor count, and prime
exclusion from the common multiplier and all padding factors. -/
theorem MinimalWord.active_dichotomy_of_squarefree {h s J : ℕ}
{supply : ℕ → ℕ → Prop} {w : List SignedStep}
(hw : MinimalWord (ForwardProhibited h s supply) w)
(hsq : ∀ a ∈ w, Squarefree a.tuple)
(hcard : ∀ a ∈ w, a.tuple.primeFactors.card = J)
(hsupport : ∀ p j, TuplePrimeAt w p j → ¬p ∣ h ∧ ∀ a ∈ w, ¬p ∣ a.padding)
(y : ℕ) (hy : ∃ j, TuplePrimeAt w y j) :
ActivePrime h w y ∨ ∃ z i, ActivePrime h w z ∧ TuplePrimeAt w z i ∧
¬(z : ℤ) ∣ wordPrimeContribution h w y 0 i := by
apply hw.active_dichotomy (adjacent_tuple_new_prime w J hsq hcard hw.1.2.2.2.1) ?_ y hy
intro p i hseen hi hnot
apply word_prime_support hsupport p i ?_ hi hnot
obtain ⟨j, hj⟩ := hseen
exact ⟨j, hj.index_lt, hj⟩
end TwoPointCorrelations
end OAI
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