| import OAI.Combinatorics.GraphThreshold.GraphEncoding | |
| import Mathlib.Tactic | |
| namespace OAI | |
| noncomputable section | |
| namespace LeanBlast.SecondKahnKalai | |
| section BernoulliEndpoints | |
| variable {X : Type*} [Fintype X] [DecidableEq X] | |
| theorem bernoulliWeight_zero (W : Finset X) : | |
| bernoulliWeight 0 W = if W = ∅ then 1 else 0 := by | |
| by_cases hW : W = ∅ | |
| · subst W | |
| simp [bernoulliWeight] | |
| · rw [ite_eq_right hW] | |
| obtain ⟨x, hx⟩ := Finset.nonempty_iff_ne_empty.mpr hW | |
| exact Finset.prod_eq_zero (Finset.mem_univ x) (by simp [hx]) | |
| theorem bernoulliWeight_one (W : Finset X) : | |
| bernoulliWeight 1 W = if W = Finset.univ then 1 else 0 := by | |
| by_cases hW : W = Finset.univ | |
| · subst W | |
| simp [bernoulliWeight] | |
| · rw [ite_eq_right hW] | |
| obtain ⟨x, hx⟩ : ∃ x : X, x ∉ W := by | |
| simpa [Finset.eq_univ_iff_forall] using hW | |
| exact Finset.prod_eq_zero (Finset.mem_univ x) (by simp [hx]) | |
| @[simp] | |
| theorem bernoulliExpectation_zero (f : Finset X → ℝ) : | |
| bernoulliExpectation 0 f = f ∅ := by | |
| simp [bernoulliExpectation, bernoulliWeight_zero] | |
| @[simp] | |
| theorem bernoulliExpectation_one (f : Finset X → ℝ) : | |
| bernoulliExpectation 1 f = f Finset.univ := by | |
| simp [bernoulliExpectation, bernoulliWeight_one] | |
| @[simp] | |
| theorem bernoulliProbability_zero (P : Finset X → Prop) [DecidablePred P] : | |
| bernoulliProbability 0 P = if P ∅ then 1 else 0 := by | |
| classical | |
| unfold bernoulliProbability | |
| rw [Finset.sum_eq_single ∅] | |
| · simp [bernoulliWeight_zero] | |
| · intro W _ hW | |
| simp [bernoulliWeight_zero, hW] | |
| · simp | |
| @[simp] | |
| theorem bernoulliProbability_one (P : Finset X → Prop) [DecidablePred P] : | |
| bernoulliProbability 1 P = if P Finset.univ then 1 else 0 := by | |
| classical | |
| unfold bernoulliProbability | |
| rw [Finset.sum_eq_single Finset.univ] | |
| · simp [bernoulliWeight_one] | |
| · intro W _ hW | |
| simp [bernoulliWeight_one, hW] | |
| · simp | |
| end BernoulliEndpoints | |
| theorem copyCount_iso_target {V W U : Type*} [Fintype V] | |
| (G : SimpleGraph V) {H : SimpleGraph W} {F : SimpleGraph U} | |
| (e : H ≃g F) : G.copyCount H = G.copyCount F := by | |
| classical | |
| unfold SimpleGraph.copyCount | |
| congr 1 | |
| apply Finset.filter_congr | |
| intro J _ | |
| constructor | |
| · rintro ⟨g⟩ | |
| exact ⟨g.comp e.symm⟩ | |
| · rintro ⟨g⟩ | |
| exact ⟨g.comp e⟩ | |
| theorem expectedCopies_iso {V W : Type*} (n : ℕ) (p : ℝ) | |
| {H : SimpleGraph V} {F : SimpleGraph W} (e : H ≃g F) : | |
| expectedCopies n p H = expectedCopies n p F := by | |
| unfold expectedCopies bernoulliExpectation | |
| simp_rw [copyCount_iso_target _ e] | |
| theorem containmentProbability_iso {V W : Type*} (n : ℕ) (p : ℝ) | |
| {H : SimpleGraph V} {F : SimpleGraph W} (e : H ≃g F) : | |
| containmentProbability n p H = containmentProbability n p F := by | |
| unfold containmentProbability | |
| congr 1 | |
| funext K | |
| rw [copyCount_iso_target _ e] | |
| theorem isContained_top_of_card_le {V : Type*} [Fintype V] (n : ℕ) | |
| (H : SimpleGraph V) (hcard : Fintype.card V ≤ n) : | |
| H.IsContained (⊤ : SimpleGraph (Fin n)) := by | |
| obtain ⟨f⟩ := Function.Embedding.nonempty_of_card_le | |
| (show Fintype.card V ≤ Fintype.card (Fin n) by simpa using hcard) | |
| refine ⟨{ toHom := ⟨f, ?_⟩, injective' := f.injective }⟩ | |
| intro a b hab | |
| exact f.injective.ne hab.ne | |
| theorem expectedCopies_one_ge_one {V : Type*} [Fintype V] (n : ℕ) | |
| (H : SimpleGraph V) (hcard : Fintype.card V ≤ n) : | |
| 1 ≤ expectedCopies n 1 H := by | |
| have hp : 0 < (⊤ : SimpleGraph (Fin n)).copyCount H := | |
| SimpleGraph.copyCount_pos.mpr (isContained_top_of_card_le n H hcard) | |
| simpa only [expectedCopies, bernoulliExpectation_one, graphOfEdges_univ, | |
| Nat.one_le_cast] using Nat.succ_le_of_lt hp | |
| theorem containmentProbability_one {V : Type*} [Fintype V] (n : ℕ) | |
| (H : SimpleGraph V) (hcard : Fintype.card V ≤ n) : | |
| containmentProbability n 1 H = 1 := by | |
| classical | |
| have hp : 0 < (⊤ : SimpleGraph (Fin n)).copyCount H := | |
| SimpleGraph.copyCount_pos.mpr (isContained_top_of_card_le n H hcard) | |
| simp only [containmentProbability, bernoulliProbability_one, graphOfEdges_univ, hp, | |
| ite_true] | |
| theorem copyCount_bot_of_edge {V W : Type*} [Fintype V] [Fintype W] | |
| (H : SimpleGraph V) (he : 1 ≤ edgeCount H) : | |
| (⊥ : SimpleGraph W).copyCount H = 0 := by | |
| apply SimpleGraph.copyCount_eq_zero.mpr | |
| rintro ⟨f⟩ | |
| have hbot : H = ⊥ := by | |
| apply SimpleGraph.eq_bot_iff_forall_not_adj.mpr | |
| intro a b hab | |
| exact f.toHom.map_adj hab | |
| have hz := (edgeCount_eq_zero_iff H).mpr hbot | |
| omega | |
| theorem expectedCopies_zero {V : Type*} [Fintype V] (n : ℕ) | |
| (H : SimpleGraph V) (he : 1 ≤ edgeCount H) : | |
| expectedCopies n 0 H = 0 := by | |
| simp [expectedCopies, copyCount_bot_of_edge H he] | |
| theorem containmentProbability_zero {V : Type*} [Fintype V] (n : ℕ) | |
| (H : SimpleGraph V) (he : 1 ≤ edgeCount H) : | |
| containmentProbability n 0 H = 0 := by | |
| classical | |
| simp only [containmentProbability, bernoulliProbability_zero, graphOfEdges_empty, | |
| copyCount_bot_of_edge H he, lt_self_iff_false, ite_false] | |
| end LeanBlast.SecondKahnKalai | |
| end | |
| end OAI | |