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a44267b | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 | import OAI.NumberTheory.DukePrimeDegree.Entropy.FirstRootConditionalProduct
namespace OAI
/-!
# A concrete first adjacent-root product step
The first adjacent root admits a neutral contracting diagonal direction.
Starting the Gaussian chain with this root gives actual local charts to
which the proved conditional-product theorem applies.
-/
noncomputable section
open Matrix Set MeasureTheory Filter
open scoped Topology BigOperators
namespace DukePrimeDegree
open PrimeDegreePackets
theorem exists_complete_elimination_chain_from_initial {n : ℕ}
(P : Finset (Fin n × Fin n)) (D : Finset (Fin n))
(hP : InitialRootPattern P) (hD : PivotsFromPattern P D) :
∃ c : EliminationChain P D, c.finalEntries = allPositiveRoots n := by
suffices ∀ m : ℕ, ∀ (P : Finset (Fin n × Fin n)) (D : Finset (Fin n)),
(allPositiveRoots n \ P).card = m → InitialRootPattern P → PivotsFromPattern P D →
∃ c : EliminationChain P D, c.finalEntries = allPositiveRoots n by
exact this _ P D rfl hP hD
intro m
induction m using Nat.strong_induction_on with
| h m ih =>
intro P D hcard hP hD
by_cases hempty : allPositiveRoots n \ P = ∅
· refine ⟨.done, ?_⟩
change P = allPositiveRoots n
apply Finset.Subset.antisymm
· exact fun p hp => (hP p hp).1
· exact Finset.sdiff_eq_empty_iff_subset.mp hempty
· obtain ⟨p, hp, hmin⟩ := Finset.exists_min_image (allPositiveRoots n \ P)
(fun q => toLex q) (Finset.nonempty_iff_ne_empty.mpr hempty)
have hab : p.1 < p.2 := (Finset.mem_filter.mp (Finset.mem_sdiff.mp hp).1).2
have hnot : p ∉ P := (Finset.mem_sdiff.mp hp).2
have hless : (allPositiveRoots n \ insert p P).card < m := by
rw [Finset.sdiff_insert, ← hcard]
exact Finset.card_erase_lt_of_mem hp
obtain ⟨c, hc⟩ := ih _ hless (insert p P) (insert p.1 D) rfl
(initialRootPattern_insert_min hP p hp hmin) (pivotsFromPattern_insert hD p.1 p.2)
exact ⟨.step p.1 p.2 (ne_of_lt hab)
(initialRootPattern_stable hP hab hnot)
(initialRootPattern_pivot_notMem hP hD hab hnot) c, hc⟩
lemma adjacent_first_initial (d : ℕ) :
InitialRootPattern ({((0 : Fin (d + 2)), (1 : Fin (d + 2)))} :
Finset (Fin (d + 2) × Fin (d + 2))) := by
intro p hp
have hp' : p = (0, 1) := Finset.mem_singleton.mp hp
subst p
refine ⟨Finset.mem_filter.mpr ⟨Finset.mem_univ _, by change (0 : Fin (d + 2)) < 1; norm_num⟩, ?_⟩
intro q hq hlt
have hq' := (Finset.mem_filter.mp hq).2
rcases Prod.Lex.toLex_lt_toLex.mp hlt with hr | ⟨hr, hc⟩
· exfalso
change q.1 < (0 : Fin (d + 2)) at hr
exact (Fin.not_lt_zero q.1) hr
· exfalso
have hv := congrArg Fin.val hr
fin_omega
lemma adjacent_first_pivots (d : ℕ) :
PivotsFromPattern ({((0 : Fin (d + 2)), (1 : Fin (d + 2)))} :
Finset (Fin (d + 2) × Fin (d + 2))) {0} := by
intro a ha
have ha' : a = 0 := Finset.mem_singleton.mp ha
subst a
exact ⟨1, Finset.mem_singleton_self _⟩
def adjacentFirstTail (d : ℕ) :
EliminationChain ({((0 : Fin (d + 2)), (1 : Fin (d + 2)))} :
Finset (Fin (d + 2) × Fin (d + 2))) {0} :=
(exists_complete_elimination_chain_from_initial _ _
(adjacent_first_initial d) (adjacent_first_pivots d)).choose
lemma adjacentFirstTail_final (d : ℕ) :
(adjacentFirstTail d).finalEntries = allPositiveRoots (d + 2) :=
(exists_complete_elimination_chain_from_initial _ _
(adjacent_first_initial d) (adjacent_first_pivots d)).choose_spec
lemma exists_adjacent_first_weights (d : ℕ) :
∃ ell : Fin (d + 2) → ℝ, (∑ i, ell i) = 0 ∧ ell 0 = ell 1 ∧
∀ p ∈ allPositiveRoots (d + 2), p ≠ (0, 1) → ell p.1 < ell p.2 := by
refine ⟨centerWeights (d + 1) (midpointRootRaw 0 1), centerWeights_sum _ _, ?_, ?_⟩
· unfold centerWeights
rw [midpointRootRaw_equal]
· intro p hp hne
apply sub_neg.mp
rw [centerWeights_sub]
apply sub_neg.mpr
apply midpointRootRaw_strict (by norm_num : (0 : Fin (d + 2)) < 1) (Finset.mem_filter.mp hp).2
· have hp' := (Finset.mem_filter.mp hp).2
have hnat : p.1.val + 1 ≤ p.2.val := by fin_omega
have hreal : (p.1.val : ℝ) + 1 ≤ p.2.val := by exact_mod_cast hnat
norm_num
linarith
· intro h
exact hne (Prod.ext h.1 h.2)
end DukePrimeDegree
end
end OAI
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