openAI-math-dataset / lean /OAI /Probability /CriticalPercolation /Exploration /PivotalComparison.lean
| import Mathlib | |
| namespace OAI | |
| section | |
| section | |
| open Set MeasureTheory ProbabilityTheory | |
| open Set | |
| open scoped BigOperators | |
| namespace CriticalPercolation.DFS | |
| lemma pivotal_comparison {u v x a s : ℝ} | |
| (hu : 0 < u) (hv : 0 ≤ v) (ha : 0 ≤ a) | |
| (havoid : s*u ≤ x*a) (harris : (u+v)*(u+x) ≤ u) : | |
| (a+s)*(u+v) ≤ a := by | |
| have h1 := mul_le_mul_of_nonneg_right havoid (add_nonneg hu.le hv) | |
| have h2 := mul_le_mul_of_nonneg_left harris ha | |
| nlinarith | |
| lemma tilted_increment {u v a b g : ℝ} | |
| (h0 : 0 < u+v) (h1 : 0 < u+v-b-g) (hc : u*g ≤ v*b) : | |
| a/(u+v-b-g) ≤ (v+a-g)/(u+v-b-g)-v/(u+v) := by | |
| have hid : (v+a-g)/(u+v-b-g)-v/(u+v)-a/(u+v-b-g) = | |
| (v*b-u*g)/((u+v)*(u+v-b-g)) := by | |
| field_simp | |
| ring | |
| have hn : 0 ≤ (v*b-u*g)/((u+v)*(u+v-b-g)) := | |
| div_nonneg (sub_nonneg.mpr hc) (mul_pos h0 h1).le | |
| linarith | |
| lemma converted_variance {p u v x a b g s : ℝ} | |
| (hp : p ∈ Set.Ioo 0 1) (hu : 0 ≤ u) (hv : 0 ≤ v) | |
| (ha : 0 ≤ a) (hb : 0 ≤ b) (hg : 0 ≤ g) (hs : 0 ≤ s) | |
| (hau : a ≤ u) (ha1 : a ≤ u+v-b-g) | |
| (havoid₁ : u*g ≤ v*b) (havoid₂ : s*u ≤ x*a) | |
| (harris : (u+v)*(u+x) ≤ u) : | |
| p*(1-p)*a*(a+s) ≤ | |
| p*(1-p)*(u+v)*(u+v-b-g)/((1-p)*(u+v)+p*(u+v-b-g)) * | |
| ((v+a-g)/(u+v-b-g)-v/(u+v))^2 := by | |
| let U0 := u+v | |
| let U1 := u+v-b-g | |
| let U := (1-p)*U0+p*U1 | |
| let r := (v+a-g)/U1-v/U0 | |
| have hpq : 0 ≤ p*(1-p) := mul_nonneg hp.1.le (sub_nonneg.mpr hp.2.le) | |
| have h0 : 0 ≤ U0 := add_nonneg hu hv | |
| have h1 : 0 ≤ U1 := ha.trans ha1 | |
| have hU : 0 ≤ U := add_nonneg (mul_nonneg (sub_nonneg.mpr hp.2.le) h0) | |
| (mul_nonneg hp.1.le h1) | |
| change p*(1-p)*a*(a+s) ≤ p*(1-p)*U0*U1/U*r^2 | |
| rcases eq_or_lt_of_le ha with hz | ha | |
| · rw [← hz] | |
| simpa only [mul_zero, zero_mul] using | |
| mul_nonneg (div_nonneg (mul_nonneg (mul_nonneg hpq h0) h1) hU) (sq_nonneg r) | |
| · have hup : 0 < u := ha.trans_le hau | |
| have h0p : 0 < U0 := add_pos_of_pos_of_nonneg hup hv | |
| have h1p : 0 < U1 := ha.trans_le ha1 | |
| have hUp : 0 < U := add_pos | |
| (mul_pos (sub_pos.mpr hp.2) h0p) (mul_pos hp.1 h1p) | |
| have h10 : U1 ≤ U0 := by dsimp [U1,U0]; linarith | |
| have hU0 : U ≤ U0 := by | |
| dsimp [U] | |
| have ht := mul_le_mul_of_nonneg_left h10 hp.1.le | |
| nlinarith | |
| have hprod : U*U1 ≤ U0^2 := by | |
| have hh := mul_le_mul hU0 h10 h1 h0 | |
| nlinarith | |
| have hcmp : (a+s)*U0 ≤ a := pivotal_comparison hup hv ha.le havoid₂ harris | |
| have hr : a/U1 ≤ r := tilted_increment h0p h1p havoid₁ | |
| have hr0 : 0 ≤ r := (div_nonneg ha.le h1).trans hr | |
| have hsqr : (a/U1)^2 ≤ r^2 := by | |
| exact pow_le_pow_left₀ (div_nonneg ha.le h1) hr 2 | |
| have hab : 0 ≤ a*(a+s) := mul_nonneg ha.le (add_nonneg ha.le hs) | |
| have hbase : a*(a+s) ≤ U0*a^2/(U*U1) := by | |
| apply (le_div_iff₀ (mul_pos hUp h1p)).mpr | |
| calc | |
| _ ≤ (a*(a+s))*U0^2 := mul_le_mul_of_nonneg_left hprod hab | |
| _ ≤ U0*a^2 := by | |
| have hh := mul_le_mul_of_nonneg_left hcmp (mul_nonneg ha.le h0) | |
| nlinarith | |
| calc | |
| _ = (p*(1-p))*(a*(a+s)) := by ring | |
| _ ≤ (p*(1-p))*(U0*a^2/(U*U1)) := mul_le_mul_of_nonneg_left hbase hpq | |
| _ = (p*(1-p)*U0*U1/U)*(a/U1)^2 := by field_simp | |
| _ ≤ _ := mul_le_mul_of_nonneg_left hsqr | |
| (div_nonneg (mul_nonneg (mul_nonneg hpq h0) h1) hU) | |
| end CriticalPercolation.DFS | |
| end | |
| end | |
| end OAI | |