| import OAI.Probability.ClassicalON.PinnedCylinder |
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| namespace OAI |
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| universe uE uV |
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| noncomputable section |
| open scoped BigOperators |
| namespace ClassicalON |
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| abbrev SignField := ZMod 2 |
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| variable {V : Type uV} {E : Type uE} |
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| def bondSubspace (left right : E → V) (η : E → Bool) : Submodule SignField (V → SignField) where |
| carrier := {s | ∀ e,η e=true → s (left e)=s (right e)} |
| zero_mem' := by simp |
| add_mem' := by intro f g hf hg e he; simp only [Pi.add_apply,hf e he,hg e he] |
| smul_mem' := by intro c f hf e he; simp only [Pi.smul_apply,hf e he] |
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| @[simp] theorem mem_bondSubspace (left right : E → V) (η : E → Bool) (s : V → SignField) : |
| s∈bondSubspace left right η ↔ ∀ e,η e=true → s (left e)=s (right e) := Iff.rfl |
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| theorem bondSubspace_antitone (left right : E → V) : Antitone (bondSubspace left right) := by |
| intro η ξ h s hs e he |
| apply hs e |
| have hh := h e |
| rw [he] at hh |
| exact le_antisymm (show ξ e ≤ true from Bool.le_true _) hh |
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| theorem bondSubspace_sup (left right : E → V) (η ξ : E → Bool) : |
| bondSubspace left right (η⊔ξ)=bondSubspace left right η⊓bondSubspace left right ξ := by |
| apply le_antisymm |
| · exact le_inf (bondSubspace_antitone left right le_sup_left) (bondSubspace_antitone left right le_sup_right) |
| · intro s hs e he |
| have ht : η e=true ∨ ξ e=true := by simpa using he |
| rcases ht with ht | ht |
| · exact hs.1 e ht |
| · exact hs.2 e ht |
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| variable [Fintype V] |
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| def bondRank (left right : E → V) (η : E → Bool) : ℕ := |
| Module.finrank SignField (bondSubspace left right η) |
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| theorem bondRank_supermodular (left right : E → V) (η ξ : E → Bool) : |
| bondRank left right η+bondRank left right ξ≤ |
| bondRank left right (η⊓ξ)+bondRank left right (η⊔ξ) := by |
| unfold bondRank |
| rw [bondSubspace_sup,← Submodule.finrank_sup_add_finrank_inf_eq] |
| apply Nat.add_le_add_right |
| apply Submodule.finrank_mono |
| exact sup_le (bondSubspace_antitone left right inf_le_left) (bondSubspace_antitone left right inf_le_right) |
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| def clusterWeight (left right : E → V) (η : E → Bool) : ℝ := |
| 2^bondRank left right η |
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| omit [Fintype V] in |
| theorem clusterWeight_pos (left right : E → V) (η : E → Bool) : 0<clusterWeight left right η := by |
| unfold clusterWeight; positivity |
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| theorem clusterWeight_supermodular (left right : E → V) (η ξ : E → Bool) : |
| clusterWeight left right η*clusterWeight left right ξ≤ |
| clusterWeight left right (η⊓ξ)*clusterWeight left right (η⊔ξ) := by |
| unfold clusterWeight |
| rw [← pow_add,← pow_add] |
| exact pow_le_pow_right₀ (by norm_num) (bondRank_supermodular left right η ξ) |
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| end ClassicalON |
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| end |
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| end OAI |
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