Datasets:

roseking
/

openAI-math-dataset

import Mathlib
import OAI.Computability.VertexCover.Games.Basic
import OAI.Computability.VertexCover.Games.Stochastic
import OAI.Computability.VertexCover.Repetition.Factorization

namespace OAI

section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
section
                                                                                          
section

namespace UniqueGames.Foundations.Repetition

open scoped BigOperators
open Games
noncomputable section

variable {X Y : Type*} [Fintype X] [Fintype Y]

def normalizeOr (w : X → ℝ) (hw : ∀ x, 0 ≤ w x)
    (fallback : FiniteDistribution X) : FiniteDistribution X :=
  if h : 0 < ∑ x, w x then
    { weight := normalizedWeight w
      nonnegative := (normalizedWeight_isProbability w hw h).1
      normalized := (normalizedWeight_isProbability w hw h).2 }
  else fallback

theorem sum_mul_normalizeOr (w : X → ℝ) (hw : ∀ x, 0 ≤ w x)
    (fallback : FiniteDistribution X) (x : X) :
    (∑ z, w z) * (normalizeOr w hw fallback).weight x = w x := by
  classical
  have hs : 0 ≤ ∑ z, w z := Finset.sum_nonneg (fun z _ => hw z)
  by_cases h : 0 < ∑ z, w z
  · simp only [normalizeOr, dite_eq_left h, normalizedWeight]
    field_simp
  · have hz : ∑ z, w z = 0 := le_antisymm (le_of_not_gt h) hs
    have hx : w x = 0 := by
      apply le_antisymm _ (hw x)
      have hl := Finset.single_le_sum (fun z _ => hw z) (Finset.mem_univ x)
      simpa only [hz] using hl
    simp [normalizeOr, hz, hx]

theorem local_completion_recombination
    (left : X → ℝ) (right : Y → ℝ)
    (hl : ∀ x, 0 ≤ left x) (hr : ∀ y, 0 ≤ right y)
    (defaultLeft : FiniteDistribution X) (defaultRight : FiniteDistribution Y)
    («c» : ℝ) (x : X) (y : Y) :
    («c» * (∑ z, left z) * (∑ z, right z)) *
      ((normalizeOr left hl defaultLeft).weight x *
        (normalizeOr right hr defaultRight).weight y) = «c» * left x * right y := by
  calc
    _ = «c» * ((∑ z, left z) * (normalizeOr left hl defaultLeft).weight x) *
        ((∑ z, right z) * (normalizeOr right hr defaultRight).weight y) := by ring
    _ = _ := by rw [sum_mul_normalizeOr, sum_mul_normalizeOr]

theorem factorized_completion_row
    (left : X → ℝ) (right : Y → ℝ)
    (hl : ∀ x, 0 ≤ left x) (hr : ∀ y, 0 ≤ right y)
    (defaultLeft : FiniteDistribution X) (defaultRight : FiniteDistribution Y)
    («c» : ℝ) (xy : X × Y) :
    (∑ z : X × Y, «c» * left z.1 * right z.2) *
      ((normalizeOr left hl defaultLeft).product
        (normalizeOr right hr defaultRight)).weight xy = «c» * left xy.1 * right xy.2 := by
  have hm : (∑ z : X × Y, «c» * left z.1 * right z.2) =
      «c» * (∑ z, left z) * (∑ z, right z) := by
    simp_rw [mul_assoc, ← Finset.mul_sum]
    rw [productWeight_mass]
  rw [hm]
  exact local_completion_recombination left right hl hr defaultLeft defaultRight «c» xy.1 xy.2

end
end UniqueGames.Foundations.Repetition

end


end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end
end

end OAI

	import Mathlib
	import OAI.Computability.VertexCover.Games.Basic
	import OAI.Computability.VertexCover.Games.Stochastic
	import OAI.Computability.VertexCover.Repetition.Factorization

	namespace OAI

	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section
	section

	section

	namespace UniqueGames.Foundations.Repetition

	open scoped BigOperators
	open Games
	noncomputable section

	variable {X Y : Type*} [Fintype X] [Fintype Y]

	def normalizeOr (w : X → ℝ) (hw : ∀ x, 0 ≤ w x)
	(fallback : FiniteDistribution X) : FiniteDistribution X :=
	if h : 0 < ∑ x, w x then
	{ weight := normalizedWeight w
	nonnegative := (normalizedWeight_isProbability w hw h).1
	normalized := (normalizedWeight_isProbability w hw h).2 }
	else fallback

	theorem sum_mul_normalizeOr (w : X → ℝ) (hw : ∀ x, 0 ≤ w x)
	(fallback : FiniteDistribution X) (x : X) :
	(∑ z, w z) * (normalizeOr w hw fallback).weight x = w x := by
	classical
	have hs : 0 ≤ ∑ z, w z := Finset.sum_nonneg (fun z _ => hw z)
	by_cases h : 0 < ∑ z, w z
	· simp only [normalizeOr, dite_eq_left h, normalizedWeight]
	field_simp
	· have hz : ∑ z, w z = 0 := le_antisymm (le_of_not_gt h) hs
	have hx : w x = 0 := by
	apply le_antisymm _ (hw x)
	have hl := Finset.single_le_sum (fun z _ => hw z) (Finset.mem_univ x)
	simpa only [hz] using hl
	simp [normalizeOr, hz, hx]

	theorem local_completion_recombination
	(left : X → ℝ) (right : Y → ℝ)
	(hl : ∀ x, 0 ≤ left x) (hr : ∀ y, 0 ≤ right y)
	(defaultLeft : FiniteDistribution X) (defaultRight : FiniteDistribution Y)
	(«c» : ℝ) (x : X) (y : Y) :
	(«c» * (∑ z, left z) * (∑ z, right z)) *
	((normalizeOr left hl defaultLeft).weight x *
	(normalizeOr right hr defaultRight).weight y) = «c» * left x * right y := by
	calc
	_ = «c» * ((∑ z, left z) * (normalizeOr left hl defaultLeft).weight x) *
	((∑ z, right z) * (normalizeOr right hr defaultRight).weight y) := by ring
	_ = _ := by rw [sum_mul_normalizeOr, sum_mul_normalizeOr]

	theorem factorized_completion_row
	(left : X → ℝ) (right : Y → ℝ)
	(hl : ∀ x, 0 ≤ left x) (hr : ∀ y, 0 ≤ right y)
	(defaultLeft : FiniteDistribution X) (defaultRight : FiniteDistribution Y)
	(«c» : ℝ) (xy : X × Y) :
	(∑ z : X × Y, «c» * left z.1 * right z.2) *
	((normalizeOr left hl defaultLeft).product
	(normalizeOr right hr defaultRight)).weight xy = «c» * left xy.1 * right xy.2 := by
	have hm : (∑ z : X × Y, «c» * left z.1 * right z.2) =
	«c» * (∑ z, left z) * (∑ z, right z) := by
	simp_rw [mul_assoc, ← Finset.mul_sum]
	rw [productWeight_mass]
	rw [hm]
	exact local_completion_recombination left right hl hr defaultLeft defaultRight «c» xy.1 xy.2

	end
	end UniqueGames.Foundations.Repetition

	end


	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end
	end

	end OAI