TODO #
noncomputable def
logSizeRadius
{T : Type u_1}
[PseudoEMetricSpace T]
(t : T)
(V : Finset T)
(a c : ENNReal)
:
Equations
- logSizeRadius t V a c = if h : 1 < a then Nat.find ⋯ else 0
Instances For
theorem
one_le_logSizeRadius
{T : Type u_1}
[PseudoEMetricSpace T]
{a c : ENNReal}
{V : Finset T}
{t : T}
(ha : 1 < a)
:
theorem
card_le_logSizeRadius_le
{T : Type u_1}
[PseudoEMetricSpace T]
{a c : ENNReal}
{V : Finset T}
{t : T}
(ha : 1 < a)
:
theorem
card_le_logSizeRadius_ge
{T : Type u_1}
[PseudoEMetricSpace T]
{a c : ENNReal}
{V : Finset T}
{t : T}
(ha : 1 < a)
(ht : t ∈ V)
:
noncomputable def
logSizeBallStruct.ball
{T : Type u_1}
[PseudoEMetricSpace T]
(struct : logSizeBallStruct T)
(c : ENNReal)
:
Finset T
Instances For
noncomputable def
logSizeBallSeq
{T : Type u_1}
[PseudoEMetricSpace T]
(J : Finset T)
(hJ : J.Nonempty)
(a c : ENNReal)
:
ℕ → logSizeBallStruct T
Equations
- One or more equations did not get rendered due to their size.
Instances For
theorem
finset_logSizeBallSeq_zero
{T : Type u_1}
[PseudoEMetricSpace T]
{a c : ENNReal}
{J : Finset T}
(hJ : J.Nonempty)
:
theorem
point_logSizeBallSeq_zero
{T : Type u_1}
[PseudoEMetricSpace T]
{a c : ENNReal}
{J : Finset T}
(hJ : J.Nonempty)
:
theorem
radius_logSizeBallSeq_zero
{T : Type u_1}
[PseudoEMetricSpace T]
{a c : ENNReal}
{J : Finset T}
(hJ : J.Nonempty)
:
theorem
finset_logSizeBallSeq_add_one
{T : Type u_1}
[PseudoEMetricSpace T]
{a c : ENNReal}
{J : Finset T}
(hJ : J.Nonempty)
(i : ℕ)
:
(logSizeBallSeq J hJ a c (i + 1)).finset = (logSizeBallSeq J hJ a c i).finset \ (logSizeBallSeq J hJ a c i).smallBall c
theorem
point_logSizeBallSeq_add_one
{T : Type u_1}
[PseudoEMetricSpace T]
{a c : ENNReal}
{J : Finset T}
(hJ : J.Nonempty)
(i : ℕ)
:
(logSizeBallSeq J hJ a c (i + 1)).point = if hV' : (logSizeBallSeq J hJ a c (i + 1)).finset.Nonempty then Exists.choose hV'
else (logSizeBallSeq J hJ a c i).point
theorem
radius_logSizeBallSeq_add_one
{T : Type u_1}
[PseudoEMetricSpace T]
{a c : ENNReal}
{J : Finset T}
(hJ : J.Nonempty)
(i : ℕ)
:
(logSizeBallSeq J hJ a c (i + 1)).radius = logSizeRadius (logSizeBallSeq J hJ a c (i + 1)).point (logSizeBallSeq J hJ a c (i + 1)).finset a c
theorem
finset_logSizeBallSeq_add_one_subset
{T : Type u_1}
[PseudoEMetricSpace T]
{a c : ENNReal}
{J : Finset T}
(hJ : J.Nonempty)
(i : ℕ)
:
theorem
point_mem_finset_logSizeBallSeq
{T : Type u_1}
[PseudoEMetricSpace T]
{a c : ENNReal}
{J : Finset T}
(hJ : J.Nonempty)
(i : ℕ)
(h : (logSizeBallSeq J hJ a c i).finset.Nonempty)
:
theorem
point_notMem_finset_logSizeBallSeq_add_one
{T : Type u_1}
[PseudoEMetricSpace T]
{a c : ENNReal}
{J : Finset T}
(hJ : J.Nonempty)
(i : ℕ)
(h : (logSizeBallSeq J hJ a c i).finset.Nonempty)
:
(logSizeBallSeq J hJ a c i).point ∉ (logSizeBallSeq J hJ a c (i + 1)).finset
theorem
card_finset_logSizeBallSeq_add_one_lt
{T : Type u_1}
[PseudoEMetricSpace T]
{a c : ENNReal}
{J : Finset T}
(hJ : J.Nonempty)
(i : ℕ)
(h : (logSizeBallSeq J hJ a c i).finset.Nonempty)
:
theorem
card_finset_logSizeBallSeq_le
{T : Type u_1}
[PseudoEMetricSpace T]
{a c : ENNReal}
{J : Finset T}
(hJ : J.Nonempty)
(i : ℕ)
:
theorem
card_finset_logSizeBallSeq_card_eq_zero
{T : Type u_1}
[PseudoEMetricSpace T]
{a c : ENNReal}
{J : Finset T}
(hJ : J.Nonempty)
:
theorem
disjoint_smallBall_logSizeBallSeq
{T : Type u_1}
[PseudoEMetricSpace T]
{a c : ENNReal}
{J : Finset T}
(hJ : J.Nonempty)
(i j : ℕ)
(hij : i ≠ j)
:
Disjoint ((logSizeBallSeq J hJ a c i).smallBall c) ((logSizeBallSeq J hJ a c j).smallBall c)
noncomputable def
pairSetSeq
{T : Type u_1}
[PseudoEMetricSpace T]
(J : Finset T)
(a c : ENNReal)
(n : ℕ)
:
Equations
- pairSetSeq J a c n = if hJ : J.Nonempty then {(logSizeBallSeq J hJ a c n).point}.product ((logSizeBallSeq J hJ a c n).ball c) else ∅
Instances For
Equations
- pairSet J a c = (Finset.range J.card).biUnion (pairSetSeq J a c)
Instances For
theorem
pair_reduction
{T : Type u_1}
[PseudoEMetricSpace T]
{a c : ENNReal}
{n : ℕ}
(J : Finset T)
(hJ : ↑J.card ≤ a ^ n)
(ha : 1 < a)
(hc : c ≠ 0)
(E : Type u_2)
[PseudoEMetricSpace E]
: