| import Mathlib |
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| namespace OAI |
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| /-! Closed gradient subspaces of connected oriented multigraphs. -/ |
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| open scoped lp |
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| noncomputable section |
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| namespace SingerFour |
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| structure OrientedGraph (V E : Type*) where |
| tail : E → V |
| head : E → V |
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| namespace OrientedGraph |
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| variable {V E : Type*} (G : OrientedGraph V E) |
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| inductive Walk : V → V → Type _ |
| | nil (v : V) : Walk v v |
| | forward {v : V} (e : E) (p : Walk v (G.tail e)) : Walk v (G.head e) |
| | backward {v : V} (e : E) (p : Walk v (G.head e)) : Walk v (G.tail e) |
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| def Connected : Prop := ∀ v w : V, Nonempty (G.Walk v w) |
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| def gradients : Submodule ℝ (ℓ²(E, ℝ)) where |
| carrier := {c | ∃ f : V → ℝ, ∀ e, c e = f (G.head e) - f (G.tail e)} |
| zero_mem' := ⟨0, by intro e; simp⟩ |
| add_mem' := by |
| rintro c d ⟨f, hf⟩ ⟨g, hg⟩ |
| refine ⟨f + g, ?_⟩ |
| intro e |
| change c e + d e = (f (G.head e) + g (G.head e)) - |
| (f (G.tail e) + g (G.tail e)) |
| rw [hf e, hg e] |
| ring |
| smul_mem' := by |
| rintro r c ⟨f, hf⟩ |
| refine ⟨r • f, ?_⟩ |
| intro e |
| change r * c e = r * f (G.head e) - r * f (G.tail e) |
| rw [hf e, mul_sub] |
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|
| @[simp] |
| theorem mem_gradients (c : ℓ²(E, ℝ)) : |
| c ∈ G.gradients ↔ ∃ f : V → ℝ, ∀ e, c e = f (G.head e) - f (G.tail e) := Iff.rfl |
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| def Walk.integral {G : OrientedGraph V E} {v w : V} : G.Walk v w → (ℓ²(E, ℝ) →L[ℝ] ℝ) |
| | .nil _ => 0 |
| | .forward e p => p.integral + lp.evalCLM ℝ (fun _ : E => ℝ) 2 e |
| | .backward e p => p.integral - lp.evalCLM ℝ (fun _ : E => ℝ) 2 e |
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| theorem Walk.integral_gradient {G : OrientedGraph V E} {c : ℓ²(E, ℝ)} {f : V → ℝ} |
| (hf : ∀ e, c e = f (G.head e) - f (G.tail e)) {v w : V} |
| (p : G.Walk v w) : p.integral c = f w - f v := by |
| induction p with |
| | nil => simp [Walk.integral] |
| | forward e p ih => |
| change p.integral c + c e = f (G.head e) - f v |
| rw [ih, hf e] |
| ring |
| | backward e p ih => |
| change p.integral c - c e = f (G.tail e) - f v |
| rw [ih, hf e] |
| ring |
|
|
| theorem eq_of_zero_differences (hG : G.Connected) {f : V → ℝ} |
| (hf : ∀ e, f (G.head e) = f (G.tail e)) (v w : V) : f v = f w := by |
| have hgrad : ∀ e, (0 : ℓ²(E, ℝ)) e = f (G.head e) - f (G.tail e) := by |
| intro e |
| simp [hf e] |
| have h := (Classical.choice (hG v w)).integral_gradient hgrad |
| simpa only [map_zero, eq_comm, sub_eq_zero] using h.symm |
|
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| theorem potential_unique (hG : G.Connected) (o : V) {f g : V → ℝ} |
| (hedges : ∀ e, f (G.head e) - f (G.tail e) = g (G.head e) - g (G.tail e)) |
| (ho : f o = g o) : f = g := by |
| funext v |
| have hzero : ∀ e, (f - g) (G.head e) = (f - g) (G.tail e) := by |
| intro e |
| have := hedges e |
| dsimp |
| linarith |
| have h := G.eq_of_zero_differences hG hzero o v |
| dsimp at h |
| linarith |
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| theorem l2_eq_zero_of_constant [Infinite V] (f : ℓ²(V, ℝ)) {r : ℝ} |
| (hf : ∀ v, f v = r) : f = 0 := by |
| have hs : Summable (fun v : V => ‖r‖ ^ (2 : ℕ)) := by |
| simpa only [hf, ENNReal.toReal_ofNat, Real.rpow_two] using |
| (lp.memℓp f).summable (by norm_num : 0 < (2 : ENNReal).toReal) |
| have hr : r = 0 := by |
| have h := (summable_const_iff (β := V) (‖r‖ ^ (2 : ℕ))).mp hs |
| simpa using h |
| apply lp.ext |
| funext v |
| simpa only [lp.coeFn_zero, Pi.zero_apply, hr] using hf v |
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| theorem l2_eq_zero_of_zero_differences [Infinite V] (hG : G.Connected) |
| (f : ℓ²(V, ℝ)) (hf : ∀ e, f (G.head e) = f (G.tail e)) : f = 0 := by |
| let o : V := Classical.choice (inferInstance : Nonempty V) |
| exact l2_eq_zero_of_constant f (fun v => G.eq_of_zero_differences hG hf v o) |
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| noncomputable def normalizedPotential (hG : G.Connected) (o v : V) : |
| ℓ²(E, ℝ) →L[ℝ] ℝ := |
| (Classical.choice (hG o v)).integral |
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| theorem normalizedPotential_gradient (hG : G.Connected) (o v : V) |
| {c : ℓ²(E, ℝ)} {f : V → ℝ} |
| (hf : ∀ e, c e = f (G.head e) - f (G.tail e)) : |
| G.normalizedPotential hG o v c = f v - f o := |
| Walk.integral_gradient hf _ |
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| theorem mem_gradients_iff_normalized (hG : G.Connected) (o : V) (c : ℓ²(E, ℝ)) : |
| c ∈ G.gradients ↔ ∀ e, |
| c e = G.normalizedPotential hG o (G.head e) c - |
| G.normalizedPotential hG o (G.tail e) c := by |
| constructor |
| · rintro ⟨f, hf⟩ e |
| rw [G.normalizedPotential_gradient hG o (G.head e) hf, |
| G.normalizedPotential_gradient hG o (G.tail e) hf, hf e] |
| ring |
| · intro hc |
| exact ⟨fun v => G.normalizedPotential hG o v c, hc⟩ |
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|
| theorem isClosed_gradients (hG : G.Connected) (o : V) : |
| IsClosed (G.gradients : Set (ℓ²(E, ℝ))) := by |
| have hset : (G.gradients : Set (ℓ²(E, ℝ))) = |
| ⋂ e : E, {c : ℓ²(E, ℝ) | c e = |
| G.normalizedPotential hG o (G.head e) c - |
| G.normalizedPotential hG o (G.tail e) c} := by |
| ext c |
| simp only [SetLike.mem_coe, Set.mem_iInter, Set.mem_ofPred, |
| G.mem_gradients_iff_normalized hG o c] |
| rw [hset] |
| apply isClosed_iInter |
| intro e |
| exact isClosed_eq (lp.evalCLM ℝ (fun _ : E => ℝ) 2 e).continuous |
| ((G.normalizedPotential hG o (G.head e)).continuous.sub |
| (G.normalizedPotential hG o (G.tail e)).continuous) |
|
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| theorem completeSpace_gradients (hG : G.Connected) (o : V) : |
| CompleteSpace G.gradients := |
| (G.isClosed_gradients hG o).completeSpace_coe |
|
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| def finiteSupportGradients : Submodule ℝ (ℓ²(E, ℝ)) where |
| carrier := {c | ∃ f : V →₀ ℝ, ∀ e, c e = f (G.head e) - f (G.tail e)} |
| zero_mem' := ⟨0, by intro e; simp⟩ |
| add_mem' := by |
| rintro c d ⟨f, hf⟩ ⟨g, hg⟩ |
| refine ⟨f + g, ?_⟩ |
| intro e |
| change c e + d e = (f (G.head e) + g (G.head e)) - |
| (f (G.tail e) + g (G.tail e)) |
| rw [hf e, hg e] |
| ring |
| smul_mem' := by |
| rintro r c ⟨f, hf⟩ |
| refine ⟨r • f, ?_⟩ |
| intro e |
| change r * c e = r * f (G.head e) - r * f (G.tail e) |
| rw [hf e, mul_sub] |
|
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| def LocallyFinite : Prop := |
| ∀ v : V, Set.Finite {e : E | G.tail e = v ∨ G.head e = v} |
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| theorem hasFiniteSupport_differential (hG : G.LocallyFinite) (f : V →₀ ℝ) : |
| (Function.support (fun e => f (G.head e) - f (G.tail e))).Finite := by |
| classical |
| have hfinite : (⋃ v ∈ (f.support : Set V), |
| {e : E | G.tail e = v ∨ G.head e = v}).Finite := |
| f.support.finite_toSet.biUnion (fun v _ => hG v) |
| apply hfinite.subset |
| intro e he |
| change f (G.head e) - f (G.tail e) ≠ 0 at he |
| have hside : G.head e ∈ f.support ∨ G.tail e ∈ f.support := by |
| by_contra h |
| push Not at h |
| simp [Finsupp.notMem_support_iff.mp h.1, Finsupp.notMem_support_iff.mp h.2] at he |
| rcases hside with hh | ht |
| · exact Set.mem_iUnion₂.mpr ⟨G.head e, hh, Or.inr rfl⟩ |
| · exact Set.mem_iUnion₂.mpr ⟨G.tail e, ht, Or.inl rfl⟩ |
|
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| def finiteSupportDifferential (hG : G.LocallyFinite) : |
| (V →₀ ℝ) →ₗ[ℝ] ℓ²(E, ℝ) where |
| toFun f := ⟨fun e => f (G.head e) - f (G.tail e), |
| (memℓp_zero (G.hasFiniteSupport_differential hG f)).of_exponent_ge (by positivity)⟩ |
| map_add' f g := by |
| apply lp.ext |
| funext e |
| change f (G.head e) + g (G.head e) - (f (G.tail e) + g (G.tail e)) = |
| (f (G.head e) - f (G.tail e)) + (g (G.head e) - g (G.tail e)) |
| ring |
| map_smul' r f := by |
| apply lp.ext |
| funext e |
| exact mul_sub r (f (G.head e)) (f (G.tail e)) |>.symm |
|
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| @[simp] |
| theorem finiteSupportDifferential_apply (hG : G.LocallyFinite) (f : V →₀ ℝ) (e : E) : |
| G.finiteSupportDifferential hG f e = f (G.head e) - f (G.tail e) := rfl |
|
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| theorem range_finiteSupportDifferential (hG : G.LocallyFinite) : |
| LinearMap.range (G.finiteSupportDifferential hG) = G.finiteSupportGradients := by |
| ext c |
| constructor |
| · rintro ⟨f, rfl⟩ |
| exact ⟨f, fun e => rfl⟩ |
| · rintro ⟨f, hf⟩ |
| refine ⟨f, ?_⟩ |
| apply lp.ext |
| funext e |
| exact (hf e).symm |
|
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| def zeroBoundaryGradients : Submodule ℝ (ℓ²(E, ℝ)) := |
| G.finiteSupportGradients.topologicalClosure |
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| theorem finiteSupportGradients_le : G.finiteSupportGradients ≤ G.gradients := by |
| rintro c ⟨f, hf⟩ |
| exact ⟨f, hf⟩ |
|
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| theorem zeroBoundaryGradients_le (hG : G.Connected) (o : V) : |
| G.zeroBoundaryGradients ≤ G.gradients := |
| G.finiteSupportGradients.topologicalClosure_minimal G.finiteSupportGradients_le |
| (G.isClosed_gradients hG o) |
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| end OrientedGraph |
| end SingerFour |
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| end |
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| end OAI |
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