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import OAI.LinearAlgebra.MatrixFields.Histories.GroupOrbitTransport
namespace OAI
namespace MatrixAllFields
open scoped BigOperators Topology Polynomial
section
noncomputable section
namespace MatrixMultiplication.JointAmbientDegree
open MatrixMultiplication.Foundation JointPopulation
open scoped BigOperators
attribute [local instance] Classical.propDecidable
variable {H : Type*} [Fintype H] [DecidableEq H]
(counts : H → Shape → ℕ)
def localWord (t : JointCoarseHashing.Triple (Position counts)) (h : H) :
Positions counts h → Shape :=
fun i => (t.1 ⟨h, i⟩, t.2.1 ⟨h, i⟩, t.2.2 ⟨h, i⟩)
omit [Fintype H] [DecidableEq H] in
theorem localWord_injective : Function.Injective (localWord counts) := by
intro t u he
apply Prod.ext
· funext p
exact congrArg (fun w => (w p.1 p.2).1) he
· apply Prod.ext
· funext p
exact congrArg (fun w => (w p.1 p.2).2.1) he
· funext p
exact congrArg (fun w => (w p.1 p.2).2.2) he
def localAllowed (sum : H → ℕ) (h : H) (w : Positions counts h → Shape) : Prop :=
(∀ i, (w i).1.val + (w i).2.1.val + (w i).2.2.val = sum h) ∧
∀ s a, wordPopulation (shapeSide s ∘ w) a =
wordPopulation (sideWord counts s (triple counts (canonicalTarget counts)) h) a
theorem localAllowed_of_mem_ambient (sum : H → ℕ)
(t : JointCoarseHashing.Triple (Position counts))
(ht : t ∈ ambientSet counts sum) (h : H) :
localAllowed counts sum h (localWord counts t h) := by
obtain ⟨hs, hm⟩ := (Finset.mem_filter.mp ht).2
exact ⟨fun i => hs ⟨h, i⟩, fun s a => hm h s a⟩
def sameSide (s : Fin 3) (t e : JointCoarseHashing.Triple (Position counts)) : Prop :=
∀ h, sideWord counts s t h = sideWord counts s e h
def sharedAmbient (sum : H → ℕ) (s : Fin 3)
(e : JointCoarseHashing.Triple (Position counts)) :
Finset (JointCoarseHashing.Triple (Position counts)) :=
(ambientSet counts sum).filter (sameSide counts s · e)
def classBound (s : Fin 3) (e : JointCoarseHashing.Triple (Position counts))
(h : H) (entropyCap : ℝ) : ℝ :=
((Fintype.card (Positions counts h) : ℝ) + 1) ^ Fintype.card Shape *
Real.exp ((Fintype.card (Positions counts h) : ℝ) *
(entropyCap - finiteEntropy (fun a =>
(wordPopulation (sideWord counts s e h) a : ℝ) /
Fintype.card (Positions counts h))) +
((Fintype.card Shape : ℝ) + (Fintype.card (Fin 17) : ℝ) + 2) *
(1 + Real.log (Fintype.card (Positions counts h) + 1 : ℕ)))
def classExponent (s : Fin 3) (e : JointCoarseHashing.Triple (Position counts))
(h : H) (entropyCap : ℝ) : ℝ :=
(Fintype.card (Positions counts h) : ℝ) *
(entropyCap - finiteEntropy (fun a =>
(wordPopulation (sideWord counts s e h) a : ℝ) /
Fintype.card (Positions counts h))) +
((Fintype.card Shape : ℝ) + (Fintype.card (Fin 17) : ℝ) + 2) *
(1 + Real.log (Fintype.card (Positions counts h) + 1 : ℕ)) +
(Fintype.card Shape : ℝ) * Real.log (Fintype.card (Positions counts h) + 1 : ℕ)
omit [Fintype H] [DecidableEq H] in
theorem classBound_eq_exp (s : Fin 3)
(e : JointCoarseHashing.Triple (Position counts)) (h : H) (entropyCap : ℝ) :
classBound counts s e h entropyCap = Real.exp (classExponent counts s e h entropyCap) := by
unfold classBound classExponent
conv_rhs => rw [Real.exp_add, Real.exp_nat_mul, Real.exp_log (by positivity)]
simp only [Nat.cast_add, Nat.cast_one]
exact mul_comm _ _
omit [Fintype H] [DecidableEq H] in
theorem classExponent_target_eq (s : Fin 3) (e : Target counts) (h : H)
(entropyCap : ℝ) :
classExponent counts s (triple counts e) h entropyCap =
classExponent counts s (triple counts (canonicalTarget counts)) h entropyCap := by
simp_rw [classExponent, marginal_population_eq counts e (canonicalTarget counts)]
theorem sharedAmbient_card_le (sum : H → ℕ) (s : Fin 3)
(e : JointCoarseHashing.Triple (Position counts)) (entropyCap : H → ℝ)
(hentropy : ∀ h w, localAllowed counts sum h w →
finiteEntropy (fun a => (wordPopulation w a : ℝ) /
Fintype.card (Positions counts h)) ≤ entropyCap h) :
((sharedAmbient counts sum s e).card : ℝ) ≤
∏ h, classBound counts s e h (entropyCap h) := by
classical
let Allowed := fun h (w : Positions counts h → Shape) =>
localAllowed counts sum h w ∧ ∀ i, shapeSide s (w i) = sideWord counts s e h i
let Local := fun h => {w : Positions counts h → Shape // Allowed h w}
let encode : {t // t ∈ sharedAmbient counts sum s e} → ∀ h, Local h := fun t h =>
⟨localWord counts t.val h, localAllowed_of_mem_ambient counts sum t.val
(Finset.mem_filter.mp t.property).1 h,
fun i => congrFun ((Finset.mem_filter.mp t.property).2 h) i⟩
have hinj : Function.Injective encode := by
intro t u he
apply Subtype.ext
apply localWord_injective counts
funext h
exact congrArg Subtype.val (congrFun he h)
have hcard : (sharedAmbient counts sum s e).card ≤ ∏ h, Fintype.card (Local h) := by
simpa only [Fintype.card_coe, Fintype.card_pi] using
Fintype.card_le_of_injective encode hinj
have hlocal (h : H) : (Fintype.card (Local h) : ℝ) ≤
classBound counts s e h (entropyCap h) := by
exact JointTypeCounts.card_allowed_over_coarse_le (shapeSide s)
(sideWord counts s e h) (Allowed h) (entropyCap h)
(fun w hw => hw.2) (fun w hw => hentropy h w hw.1)
calc
((sharedAmbient counts sum s e).card : ℝ) ≤
∏ h, (Fintype.card (Local h) : ℝ) := by exact_mod_cast hcard
_ ≤ _ := Finset.prod_le_prod₀ (fun _ _ => Nat.cast_nonneg _) (fun history _ => hlocal history)
omit [Fintype H] [DecidableEq H] in
theorem sharesSide_iff (t e : JointCoarseHashing.Triple (Position counts)) :
JointCoarseHashing.SharesSide t e ↔ ∃ s : Fin 3, sameSide counts s t e := by
constructor
· rintro (hx | hy | hz)
· refine ⟨0, fun h => ?_⟩
funext i
change t.1 ⟨h, i⟩ = e.1 ⟨h, i⟩
exact congrFun hx.symm ⟨h, i⟩
· refine ⟨1, fun h => ?_⟩
funext i
change t.2.1 ⟨h, i⟩ = e.2.1 ⟨h, i⟩
exact congrFun hy.symm ⟨h, i⟩
· refine ⟨2, fun h => ?_⟩
funext i
change t.2.2 ⟨h, i⟩ = e.2.2 ⟨h, i⟩
exact congrFun hz.symm ⟨h, i⟩
· rintro ⟨s, hs⟩
fin_cases s
· apply Or.inl
funext p
simpa [sideWord, shapeSide] using (congrFun (hs p.1) p.2).symm
· apply Or.inr ∘ Or.inl
funext p
simpa [sideWord, shapeSide] using (congrFun (hs p.1) p.2).symm
· apply Or.inr ∘ Or.inr
funext p
simpa [sideWord, shapeSide] using (congrFun (hs p.1) p.2).symm
def neighbors (sum : H → ℕ) (e : JointCoarseHashing.Triple (Position counts)) :
Finset (JointCoarseHashing.Triple (Position counts)) :=
(ambientSet counts sum).filter (fun t => t ≠ e ∧ JointCoarseHashing.SharesSide t e)
theorem neighbors_card_le_sum (sum : H → ℕ)
(e : JointCoarseHashing.Triple (Position counts)) :
(neighbors counts sum e).card ≤ ∑ s : Fin 3, (sharedAmbient counts sum s e).card := by
classical
have hsub : neighbors counts sum e ⊆
(ambientSet counts sum).filter (fun t => ∃ s ∈ (Finset.univ : Finset (Fin 3)),
sameSide counts s t e) := by
intro t ht
obtain ⟨ha, _, hs⟩ := Finset.mem_filter.mp ht
obtain ⟨s, hs⟩ := (sharesSide_iff counts t e).mp hs
exact Finset.mem_filter.mpr ⟨ha, s, Finset.mem_univ _, hs⟩
apply (Finset.card_le_card hsub).trans
convert JointLossCounts.card_filter_exists_le (ambientSet counts sum) Finset.univ
(fun side target => sameSide counts side target e) using 1 <;>
simp [sharedAmbient]
theorem neighbors_card_le (sum : H → ℕ)
(e : JointCoarseHashing.Triple (Position counts)) (entropyCap : H → ℝ)
(hentropy : ∀ h w, localAllowed counts sum h w →
finiteEntropy (fun a => (wordPopulation w a : ℝ) /
Fintype.card (Positions counts h)) ≤ entropyCap h) :
((neighbors counts sum e).card : ℝ) ≤
∑ s : Fin 3, ∏ h, classBound counts s e h (entropyCap h) := by
calc
((neighbors counts sum e).card : ℝ) ≤
∑ s : Fin 3, ((sharedAmbient counts sum s e).card : ℝ) := by
exact_mod_cast neighbors_card_le_sum counts sum e
_ ≤ _ := Finset.sum_le_sum fun s _ => sharedAmbient_card_le counts sum s e entropyCap hentropy
theorem neighbors_card_le_three_exp (sum : H → ℕ)
(e : JointCoarseHashing.Triple (Position counts)) (entropyCap : H → ℝ)
(hentropy : ∀ h w, localAllowed counts sum h w →
finiteEntropy (fun a => (wordPopulation w a : ℝ) /
Fintype.card (Positions counts h)) ≤ entropyCap h)
(exponent : ℝ)
(hupper : ∀ s : Fin 3, ∑ h, classExponent counts s e h (entropyCap h) ≤ exponent) :
((neighbors counts sum e).card : ℝ) ≤ 3 * Real.exp exponent := by
have hprod (s : Fin 3) : (∏ h, classBound counts s e h (entropyCap h)) ≤
Real.exp exponent := by
simp_rw [classBound_eq_exp]
rw [← Real.exp_sum]
exact Real.exp_le_exp.mpr (hupper s)
calc
((neighbors counts sum e).card : ℝ) ≤
∑ s : Fin 3, ∏ h, classBound counts s e h (entropyCap h) :=
neighbors_card_le counts sum e entropyCap hentropy
_ ≤ ∑ _s : Fin 3, Real.exp exponent := Finset.sum_le_sum fun s _ => hprod s
_ = _ := by simp
theorem log_neighbors_card_le (sum : H → ℕ)
(e : JointCoarseHashing.Triple (Position counts)) (entropyCap : H → ℝ)
(hentropy : ∀ h w, localAllowed counts sum h w →
finiteEntropy (fun a => (wordPopulation w a : ℝ) /
Fintype.card (Positions counts h)) ≤ entropyCap h)
(exponent : ℝ)
(hupper : ∀ s : Fin 3, ∑ h, classExponent counts s e h (entropyCap h) ≤ exponent)
(hne : 0 < (neighbors counts sum e).card) :
Real.log ((neighbors counts sum e).card : ℝ) ≤ Real.log 3 + exponent := by
have hb := Real.log_le_log (by exact_mod_cast hne :
(0 : ℝ) < (neighbors counts sum e).card)
(neighbors_card_le_three_exp counts sum e entropyCap hentropy exponent hupper)
simpa only [Real.log_mul (by norm_num : (3 : ℝ) ≠ 0) (Real.exp_ne_zero _),
Real.log_exp] using hb
end MatrixMultiplication.JointAmbientDegree
namespace MatrixMultiplication.AllFieldGroupDegrees
open AllFieldHistory AllFieldHistorySupport AllFieldHistoryChildLaws
open AllFieldGroupOrbitData JointPopulation JointCanonicalization
open MatrixMultiplication.Foundation
open scoped BigOperators
attribute [local instance] Classical.propDecidable Classical.decEq
variable {K tick : ℕ}
abbrev Letter (sigma : Placement) (h : ActiveOrder K tick sigma) :=
Fin (activeHalfLength h.val) → Fin 7
abbrev Class (sigma : Placement) := ActiveOrder K tick sigma × Fin 17
abbrev Symbol (sigma : Placement) (c : Class (K := K) (tick := tick) sigma) :=
Statistic c.1.val
def projectedStatistic (sigma : Placement)
(c : Class (K := K) (tick := tick) sigma) : Letter (K := K) (tick := tick) sigma c.1 → Symbol (K := K) (tick := tick) sigma c :=
statistic c.1.val
def classDesignated (right : Bool) (side : Fin 3) (sigma : Placement)
(c : Class (K := K) (tick := tick) sigma) (u : JointPopulation.Shape) : Prop :=
designated right side sigma c.1 u
def classLaw (right : Bool) (side : Fin 3) (sigma : Placement)
(c : Class (K := K) (tick := tick) sigma) (u : JointPopulation.Shape) :
Symbol (K := K) (tick := tick) sigma c → ℝ := compatibilityLaw right side sigma c.1 u
theorem tripleSide_eq_sideWord (allocation : Allocation) (m : ℕ) (sigma : Placement)
(s : Fin 3) (t : JointCoarseHashing.Triple (Pos (K := K) (tick := tick) allocation m sigma))
(h : ActiveOrder K tick sigma) (i : JointPopulation.Positions (Counts (K := K) (tick := tick) allocation m sigma) h) :
tripleSide s t ⟨h, i⟩ = sideWord (Counts (K := K) (tick := tick) allocation m sigma) s t h i := by
fin_cases s <;> rfl
theorem tripleSide_triple (allocation : Allocation) (m : ℕ) (sigma : Placement)
(s : Fin 3) (e : Targets (K := K) (tick := tick) allocation m sigma)
(h : ActiveOrder K tick sigma) (i : JointPopulation.Positions (Counts (K := K) (tick := tick) allocation m sigma) h) :
tripleSide s (triple (Counts (K := K) (tick := tick) allocation m sigma) e) ⟨h, i⟩ =
shapeSide s ((e h).val i) := by
fin_cases s <;> rfl
theorem childWidth_le {ε : ℝ} (hε : 0 ≤ ε) (h : Active K tick) :
AllFieldHistoryMasks.childWidth ε h ≤ ε := by
have hp : 0 ≤ AllFieldHistoryMasks.pairWidth ε h := by
unfold AllFieldHistoryMasks.pairWidth
positivity
calc
AllFieldHistoryMasks.childWidth ε h ≤ AllFieldHistoryMasks.pairWidth ε h :=
div_le_self hp (by norm_num)
_ ≤ ε := div_le_self hε (one_le_pow₀ (by norm_num))
theorem fineCompatible_uniform (allocation : Allocation) (m : ℕ)
{ε χ : ℝ} (hε : 0 ≤ ε) (hχ : ε ≤ χ) (sigma : Placement) (side : Fin 3)
(e : Targets (K := K) (tick := tick) allocation m sigma)
(w : Raw (K := K) (tick := tick) allocation m sigma)
(hw : fineCompatible allocation m ε sigma side e w) :
JointPopulationCompatibility.Compatible (Counts (K := K) (tick := tick) allocation m sigma)
(Letter (K := K) (tick := tick) sigma) (Letter (K := K) (tick := tick) sigma) (Symbol (K := K) (tick := tick) sigma) (Symbol (K := K) (tick := tick) sigma)
(projectedStatistic sigma) (projectedStatistic sigma)
(classDesignated false side sigma) (classDesignated true side sigma)
(classLaw false side sigma) (classLaw true side sigma) χ (sigma side) e w := by
constructor
· intro h u hd a
exact (hw false h u hd a).trans
(mul_le_mul_of_nonneg_left ((childWidth_le hε h.val).trans hχ) (Nat.cast_nonneg _))
· intro h u hd a
exact (hw true h u hd a).trans
(mul_le_mul_of_nonneg_left ((childWidth_le hε h.val).trans hχ) (Nat.cast_nonneg _))
def uniformCompatible (allocation : Allocation) (m : ℕ) (χ : ℝ)
(sigma : Placement) (side : Fin 3)
(e : Targets (K := K) (tick := tick) allocation m sigma)
(w : Raw (K := K) (tick := tick) allocation m sigma) : Prop :=
JointPopulationCompatibility.compatibleWithCoarse (Counts (K := K) (tick := tick) allocation m sigma)
(Letter (K := K) (tick := tick) sigma) (Letter (K := K) (tick := tick) sigma) (Symbol (K := K) (tick := tick) sigma) (Symbol (K := K) (tick := tick) sigma)
(projectedStatistic sigma) (projectedStatistic sigma)
(classDesignated false side sigma) (classDesignated true side sigma)
(classLaw false side sigma) (classLaw true side sigma) χ
(ownWord allocation m sigma) (sigma side) (triple (Counts (K := K) (tick := tick) allocation m sigma) e) w
theorem Compatible_uniform (allocation : Allocation) (m : ℕ)
{ε χ : ℝ} (hε : 0 ≤ ε) (hχ : ε ≤ χ) (sigma : Placement) (side : Fin 3)
(e : Targets (K := K) (tick := tick) allocation m sigma)
(w : Raw (K := K) (tick := tick) allocation m sigma)
(hw : Compatible allocation m ε sigma side e w) :
uniformCompatible allocation m χ sigma side e w := by
apply (JointPopulationCompatibility.compatibleWithCoarse_target
(Counts (K := K) (tick := tick) allocation m sigma) (Letter (K := K) (tick := tick) sigma) (Letter (K := K) (tick := tick) sigma)
(Symbol (K := K) (tick := tick) sigma) (Symbol (K := K) (tick := tick) sigma) (projectedStatistic sigma) (projectedStatistic sigma)
(classDesignated false side sigma) (classDesignated true side sigma)
(classLaw false side sigma) (classLaw true side sigma) χ
(ownWord allocation m sigma) (sigma side) e w).2
refine ⟨?_, fineCompatible_uniform allocation m hε hχ sigma side e w hw.2⟩
intro h i
have hh := congrFun hw.1 ⟨h, i⟩
simpa only [coarse, tripleSide_reorder, tripleSide_triple] using hh
def conditionalCount (allocation : Allocation) (m : ℕ) (χ : ℝ)
(sigma : Placement) (side : Fin 3) (w : Raw (K := K) (tick := tick) allocation m sigma) : ℕ :=
JointPopulationCompatibility.compatibleTargetCount (Counts (K := K) (tick := tick) allocation m sigma)
(Letter (K := K) (tick := tick) sigma) (Letter (K := K) (tick := tick) sigma) (Symbol (K := K) (tick := tick) sigma) (Symbol (K := K) (tick := tick) sigma)
(projectedStatistic sigma) (projectedStatistic sigma)
(classDesignated false side sigma) (classDesignated true side sigma)
(classLaw false side sigma) (classLaw true side sigma) χ
(sigma side) (ownWord allocation m sigma w) w
theorem uniformCompatible_count (allocation : Allocation) (m : ℕ) (χ : ℝ)
(sigma : Placement) (side : Fin 3) (w : Raw (K := K) (tick := tick) allocation m sigma) :
Fintype.card {e : Targets (K := K) (tick := tick) allocation m sigma //
uniformCompatible allocation m χ sigma side e w} =
conditionalCount allocation m χ sigma side w :=
JointPopulationCompatibility.compatibleWithCoarse_target_count
(Counts (K := K) (tick := tick) allocation m sigma) (Letter (K := K) (tick := tick) sigma) (Letter (K := K) (tick := tick) sigma)
(Symbol (K := K) (tick := tick) sigma) (Symbol (K := K) (tick := tick) sigma) (projectedStatistic sigma) (projectedStatistic sigma)
(classDesignated false side sigma) (classDesignated true side sigma)
(classLaw false side sigma) (classLaw true side sigma) χ
(ownWord allocation m sigma) (sigma side) w
theorem competitors_card_le_conditionalCount (allocation : Allocation) (m : ℕ)
{ε χ : ℝ} (hε : 0 ≤ ε) (hχ : ε ≤ χ) (sigma : Placement)
(e : Targets (K := K) (tick := tick) allocation m sigma)
(o : Orbit (K := K) (tick := tick) allocation m sigma) (v : Variable (K := K) (tick := tick) allocation m sigma) :
(competitors allocation m ε sigma e o v).card ≤
conditionalCount allocation m χ sigma v.1 v.2 := by
classical
let candidates : Finset (Targets (K := K) (tick := tick) allocation m sigma) :=
Finset.univ.filter fun f => uniformCompatible allocation m χ sigma v.1 f v.2
have hsub : competitors allocation m ε sigma e o v ⊆
candidates.image (coarse allocation m sigma) := by
intro t ht
obtain ⟨f, hf, hc⟩ := (Finset.mem_filter.mp ht).2.2.2
exact Finset.mem_image.mpr ⟨f, Finset.mem_filter.mpr
⟨Finset.mem_univ _, Compatible_uniform allocation m hε hχ sigma v.1 f v.2 hc⟩, hf⟩
calc
(competitors allocation m ε sigma e o v).card ≤
(candidates.image (coarse allocation m sigma)).card := Finset.card_le_card hsub
_ ≤ candidates.card := Finset.card_image_le
_ = Fintype.card {f : Targets (K := K) (tick := tick) allocation m sigma //
uniformCompatible allocation m χ sigma v.1 f v.2} := by
simp only [candidates, Fintype.card_subtype]
_ = conditionalCount allocation m χ sigma v.1 v.2 :=
uniformCompatible_count allocation m χ sigma v.1 v.2
theorem eligibility_card_le_sharedAmbient (allocation : Allocation) (m : ℕ)
(ε : ℝ) (sigma : Placement)
(e : Targets (K := K) (tick := tick) allocation m sigma) :
((data allocation m ε sigma).eligibilityCompetitors e).card ≤
(JointAmbientDegree.sharedAmbient (Counts (K := K) (tick := tick) allocation m sigma)
(fun h => 2 * activeHalfLength h.val) (sigma 0)
(triple (Counts (K := K) (tick := tick) allocation m sigma) e)).card := by
let a := JointAmbientDegree.sharedAmbient (Counts (K := K) (tick := tick) allocation m sigma)
(fun h => 2 * activeHalfLength h.val) (sigma 0)
(triple (Counts (K := K) (tick := tick) allocation m sigma) e)
have hs : (data allocation m ε sigma).eligibilityCompetitors e ⊆
a.image (reorder sigma) := by
intro f hf
obtain ⟨ha, _, hx⟩ :=
((data allocation m ε sigma).mem_eligibilityCompetitors e f).1 hf
obtain ⟨t, ht, rfl⟩ := Finset.mem_image.mp ha
apply Finset.mem_image.mpr
refine ⟨t, Finset.mem_filter.mpr ⟨ht, ?_⟩, rfl⟩
intro h
funext i
have hh := congrFun hx ⟨h, i⟩
change tripleSide (sigma 0) t ⟨h, i⟩ =
tripleSide (sigma 0) (triple (Counts (K := K) (tick := tick) allocation m sigma) e) ⟨h, i⟩ at hh
simpa only [tripleSide_eq_sideWord] using hh
exact (Finset.card_le_card hs).trans Finset.card_image_le
end MatrixMultiplication.AllFieldGroupDegrees
namespace MatrixMultiplication.JointMarginalSupport
open MatrixMultiplication.Foundation JointPopulation JointAmbientDegree
open scoped BigOperators
attribute [local instance] Classical.propDecidable
variable {H : Type*} [Fintype H] [DecidableEq H]
(counts : H → Shape → ℕ) (sum : H → ℕ)
omit [Fintype H] [DecidableEq H] in
theorem localAllowed_coordinate_witness (h : H) (w : Positions counts h → Shape)
(hw : localAllowed counts sum h w) (s : Fin 3) (i : Positions counts h) :
∃ j : Positions counts h,
shapeSide s ((canonicalTarget counts h).val j) = shapeSide s (w i) := by
have hp : 0 < wordPopulation (shapeSide s ∘ w) (shapeSide s (w i)) := by
unfold wordPopulation
exact Fintype.card_pos_iff.mpr ⟨⟨i, rfl⟩⟩
rw [hw.2 s (shapeSide s (w i))] at hp
obtain ⟨⟨j, hj⟩⟩ := Fintype.card_pos_iff.mp hp
exact ⟨j, hj⟩
omit [Fintype H] [DecidableEq H] in
theorem localAllowed_coordinate_support (h : H) (w : Positions counts h → Shape)
(hw : localAllowed counts sum h w) (allowed : Fin 3 → Fin 17 → Prop)
(hallowed : ∀ u, 0 < counts h u → ∀ s, allowed s (shapeSide s u)) :
∀ i s, allowed s (shapeSide s (w i)) := by
intro i s
obtain ⟨j, hj⟩ := localAllowed_coordinate_witness counts sum h w hw s i
rw [← hj]
exact hallowed ((canonicalTarget counts h).val j)
(symbol_count_pos counts (canonicalTarget counts) h j) s
omit [Fintype H] [DecidableEq H] in
theorem localAllowed_parent_bounds (h : H) (w : Positions counts h → Shape)
(hw : localAllowed counts sum h w) (parent : Fin 3 → ℕ)
(hparent : ∀ u, 0 < counts h u → ∀ s, (shapeSide s u).val ≤ parent s) :
∀ i s, (shapeSide s (w i)).val ≤ parent s :=
localAllowed_coordinate_support counts sum h w hw
(fun s a => a.val ≤ parent s) hparent
omit [Fintype H] [DecidableEq H] in
theorem localAllowed_positive_reference (h : H) (w : Positions counts h → Shape)
(hw : localAllowed counts sum h w) (parent : Fin 3 → ℕ) (p : Shape → ℝ)
(hparent : ∀ u, 0 < counts h u → ∀ s, (shapeSide s u).val ≤ parent s)
(hp : ∀ u : Shape, u.1.val + u.2.1.val + u.2.2.val = sum h →
(∀ s, (shapeSide s u).val ≤ parent s) → 0 < p u) :
∀ i, 0 < p (w i) := by
intro i
exact hp (w i) (hw.1 i) (localAllowed_parent_bounds counts sum h w hw parent hparent i)
end MatrixMultiplication.JointMarginalSupport
end
end
end MatrixAllFields
end OAI
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