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a2ee19a | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 | import Mathlib
import OAI.Probability.Perceptron.Variational.DepthBlock
namespace OAI
noncomputable section
open MeasureTheory ProbabilityTheory Set Filter
open scoped Classical ENNReal NNReal BigOperators Topology
namespace SphericalPerceptronFreeEnergy
lemma depth_block_row_threshold (k r d : ℕ)
(μ : Measure (FiniteOverlap (Fin (k+1)))) (A : FiniteBlock (Fin (k+1)) r)
(i j : Fin r) :
μ.real {R | finiteBlock r R = A ∧ d ≤ (R i j).val} =
μ.real {R | finiteBlock r R = A} * (if d ≤ (A i j).val then 1 else 0) := by
have he (R : FiniteOverlap (Fin (k+1))) (hR : finiteBlock r R = A) : R i j = A i j :=
congrFun (congrFun hR i) j
by_cases hd : d ≤ (A i j).val
· rw [ite_eq_left hd,mul_one]
congr 1
ext R
constructor
· exact And.left
· intro hR
exact ⟨hR,by rw [he R hR]; exact hd⟩
· rw [ite_eq_right hd,mul_zero]
have hEmpty : {R : FiniteOverlap (Fin (k+1)) | finiteBlock r R = A ∧ d ≤ (R i j).val} = ∅ := by
ext R
simp only [Set.mem_ofPred_eq,Set.mem_empty_iff_false,iff_false,not_and]
intro hR hD
exact hd (by simpa only [he R hR] using hD)
rw [hEmpty,measureReal_empty]
lemma indexedDepthLaw_block_zero_of_diagonal (k r : ℕ) (z : Fin k → ℝ)
(A : FiniteBlock (Fin (k+1)) r) (i : Fin r) (hA : A i i ≠ Fin.last k) :
(indexedDepthLaw k z : Measure (FiniteOverlap (Fin (k+1)))).real
{R | finiteBlock r R = A} = 0 := by
convert measureReal_empty (μ := (indexedDepthLaw k z : Measure (FiniteOverlap (Fin (k+1))))) using 1
apply measureReal_congr
filter_upwards [indexedDepthLaw_geometry k z] with R hR
change (finiteBlock r R = A) = False
apply propext
simp only [iff_false]
intro hB
have he : R i i = A i i := congrFun (congrFun hB i) i
exact hA (he.symm.trans (hR.2.1 i))
lemma indexedDepthLaw_focused_formula_real (k : ℕ) (z : Fin k → ℝ)
(hz : StrictMono z) (hz0 : ∀ i, 0 < z i) (hz1 : ∀ i, z i < 1)
(r d : ℕ) (hd : d ≤ k) (A : FiniteBlock (Fin (k+1)) r) (i : Fin r)
(hA : A i i = Fin.last k) :
(indexedDepthLaw k z : Measure (FiniteOverlap (Fin (k+1)))).real
{R | finiteBlock r R = A ∧ d ≤ (R i r).val} =
(indexedDepthLaw k z : Measure (FiniteOverlap (Fin (k+1)))).real {R | finiteBlock r R = A} *
(((∑ j, if d ≤ (A i j).val then (1:ℝ) else 0) -
(if h : d = 0 then 0 else z ⟨d-1,by omega⟩)) / r) := by
have hb : (if h : d = 0 then (0:ℝ) else z ⟨d-1,by omega⟩) ≤ 1 := by
split_ifs
· norm_num
· exact (hz1 _).le
have hc : (1:ℝ) ≤ ∑ j, if d ≤ (A i j).val then (1:ℝ) else 0 := by
have he := Finset.single_le_sum (s := Finset.univ)
(f := fun j : Fin r => if d ≤ (A i j).val then (1:ℝ) else 0)
(fun j _ => by split_ifs <;> norm_num) (Finset.mem_univ i)
simpa only [hA,Fin.val_last,ite_eq_left hd] using he
have hn : 0 ≤ ((∑ j, if d ≤ (A i j).val then (1:ℝ) else 0) -
(if h : d = 0 then (0:ℝ) else z ⟨d-1,by omega⟩)) / r :=
div_nonneg (sub_nonneg.mpr (hb.trans hc)) (Nat.cast_nonneg _)
have he := congrArg ENNReal.toReal (indexedDepthLaw_focused_formula k z hz hz0 hz1 r d hd A i)
simpa only [measureReal_def,ENNReal.toReal_mul,ENNReal.toReal_ofReal hn] using he
lemma indexedDepthLaw_pair_tail_real (k : ℕ) (z : Fin k → ℝ)
(hz : StrictMono z) (hz0 : ∀ i, 0 < z i) (hz1 : ∀ i, z i < 1)
(d : ℕ) (hd : d ≤ k) :
(indexedDepthLaw k z : Measure (FiniteOverlap (Fin (k+1)))).real {R | d ≤ (R 0 1).val} =
1 - (if h : d = 0 then 0 else z ⟨d-1,by omega⟩) := by
have hb : (if h : d = 0 then (0:ℝ) else z ⟨d-1,by omega⟩) ≤ 1 := by
split_ifs
· norm_num
· exact (hz1 _).le
have he := congrArg ENNReal.toReal (indexedDepthLaw_pair_tail k z hz hz0 hz1 d hd)
simpa only [measureReal_def,ENNReal.toReal_ofReal (sub_nonneg.mpr hb)] using he
theorem indexedDepthLaw_finiteGG (k : ℕ) (z : Fin k → ℝ)
(hz : StrictMono z) (hz0 : ∀ i, 0 < z i) (hz1 : ∀ i, z i < 1) :
FiniteGG (indexedDepthLaw k z : Measure (FiniteOverlap (Fin (k+1)))) := by
apply finiteGG_of_depth_thresholds
intro r _hr i A d hd
let μ := (indexedDepthLaw k z : Measure (FiniteOverlap (Fin (k+1))))
change μ.real {R | finiteBlock r R = A ∧ d ≤ (R i r).val} =
μ.real {R | finiteBlock r R = A} * μ.real {R | d ≤ (R 0 1).val} / r +
(∑ j ∈ Finset.univ.erase i, μ.real {R | finiteBlock r R = A ∧ d ≤ (R i j).val}) / r
by_cases hdk : d ≤ k
· by_cases hA : A i i = Fin.last k
· have he := indexedDepthLaw_focused_formula_real k z hz hz0 hz1 r d hdk A i hA
have hp := indexedDepthLaw_pair_tail_real k z hz hz0 hz1 d hdk
change μ.real _ = μ.real _ * _ at he
change μ.real _ = _ at hp
rw [he,hp]
simp_rw [depth_block_row_threshold k r d μ A i]
rw [← Finset.mul_sum,Finset.sum_erase_eq_sub (Finset.mem_univ i)]
simp only [hA,Fin.val_last,ite_eq_left hdk]
ring
· have hB := indexedDepthLaw_block_zero_of_diagonal k r z A i hA
change μ.real _ = 0 at hB
have hF : μ.real {R | finiteBlock r R = A ∧ d ≤ (R i r).val} = 0 :=
measureReal_mono_null (fun _ h => h.1) hB
rw [hF,hB]
simp_rw [depth_block_row_threshold k r d μ A i,hB,zero_mul]
simp
· have he : d = k+1 := by omega
subst d
have hfalse (l : Fin (k+1)) : ¬ k+1 ≤ l.val := Nat.not_le.mpr l.isLt
simp only [hfalse,and_false]
simp
end SphericalPerceptronFreeEnergy
end
end OAI
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