| import Mathlib |
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| namespace OAI |
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| noncomputable section |
| open scoped NNReal |
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| namespace Problem367 |
| namespace CavityPair |
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| variable {ψ : ℝ → ℝ} {θ : ℝ≥0} |
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| def step (ψ : ℝ → ℝ) (H K x : ℝ) : ℝ := H - ψ (K - ψ x) |
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| theorem contracting_step (hψ : LipschitzWith θ ψ) (hθ : θ < 1) (H K : ℝ) : |
| ContractingWith (θ * θ) (step ψ H K) := by |
| refine ⟨?_, LipschitzWith.of_dist_le_mul ?_⟩ |
| · have hnonneg : (0 : ℝ) ≤ θ := θ.coe_nonneg |
| have hlt : (θ : ℝ) < 1 := hθ |
| change (θ : ℝ) * θ < 1 |
| nlinarith |
| · intro x y |
| simp only [step, dist_sub_left, NNReal.coe_mul] |
| calc |
| dist (ψ (K - ψ x)) (ψ (K - ψ y)) ≤ |
| θ * dist (K - ψ x) (K - ψ y) := hψ.dist_le_mul _ _ |
| _ = θ * dist (ψ x) (ψ y) := by rw [dist_sub_left] |
| _ ≤ θ * (θ * dist x y) := |
| mul_le_mul_of_nonneg_left (hψ.dist_le_mul x y) θ.coe_nonneg |
| _ = θ * θ * dist x y := by ring |
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| theorem existsUnique (hψ : LipschitzWith θ ψ) (hθ : θ < 1) (H K : ℝ) : |
| ∃! p : ℝ × ℝ, p.1 + ψ p.2 = H ∧ p.2 + ψ p.1 = K := by |
| let F := step ψ H K |
| have hF : ContractingWith (θ * θ) F := contracting_step hψ hθ H K |
| let x := hF.fixedPoint F |
| have hx : F x = x := hF.fixedPoint_isFixedPt |
| refine ⟨(x, K - ψ x), ?_, ?_⟩ |
| · change x + ψ (K - ψ x) = H ∧ K - ψ x + ψ x = K |
| dsimp [F, step] at hx |
| constructor <;> linarith |
| · intro p hp |
| have hpF : F p.1 = p.1 := by |
| dsimp [F, step] |
| rw [show K - ψ p.1 = p.2 by linarith [hp.2]] |
| linarith [hp.1] |
| have hp1 : p.1 = x := hF.fixedPoint_unique hpF |
| apply Prod.ext hp1 |
| change p.2 = K - ψ x |
| rw [← hp1] |
| linarith [hp.2] |
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| theorem stability_fst (hψ : LipschitzWith θ ψ) (hθ : θ < 1) |
| {x y x' y' H K H' K' : ℝ} |
| (hx : x + ψ y = H) (hy : y + ψ x = K) |
| (hx' : x' + ψ y' = H') (hy' : y' + ψ x' = K') : |
| dist x x' ≤ (dist H H' + θ * dist K K') / (1 - (θ : ℝ)^2) := by |
| have hθ0 : (0 : ℝ) ≤ θ := θ.coe_nonneg |
| have hθ1 : (θ : ℝ) < 1 := hθ |
| have hden : 0 < 1 - (θ : ℝ)^2 := by nlinarith |
| have hX : dist x x' ≤ dist H H' + θ * dist y y' := by |
| calc |
| dist x x' = dist (H - ψ y) (H' - ψ y') := by |
| congr 1 <;> linarith |
| _ ≤ dist H H' + dist (ψ y) (ψ y') := dist_sub_sub_le _ _ _ _ |
| _ ≤ dist H H' + θ * dist y y' := by |
| gcongr |
| exact hψ.dist_le_mul _ _ |
| have hY : dist y y' ≤ dist K K' + θ * dist x x' := by |
| calc |
| dist y y' = dist (K - ψ x) (K' - ψ x') := by |
| congr 1 <;> linarith |
| _ ≤ dist K K' + dist (ψ x) (ψ x') := dist_sub_sub_le _ _ _ _ |
| _ ≤ dist K K' + θ * dist x x' := by |
| gcongr |
| exact hψ.dist_le_mul _ _ |
| apply (le_div_iff₀ hden).2 |
| have hY' := mul_le_mul_of_nonneg_left hY hθ0 |
| nlinarith |
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| theorem stability (hψ : LipschitzWith θ ψ) (hθ : θ < 1) |
| {p p' z z' : ℝ × ℝ} |
| (hp : p.1 + ψ p.2 = z.1 ∧ p.2 + ψ p.1 = z.2) |
| (hp' : p'.1 + ψ p'.2 = z'.1 ∧ p'.2 + ψ p'.1 = z'.2) : |
| dist p.1 p'.1 ≤ (dist z.1 z'.1 + θ * dist z.2 z'.2) / (1 - (θ : ℝ)^2) ∧ |
| dist p.2 p'.2 ≤ (dist z.2 z'.2 + θ * dist z.1 z'.1) / (1 - (θ : ℝ)^2) := |
| ⟨stability_fst hψ hθ hp.1 hp.2 hp'.1 hp'.2, |
| stability_fst hψ hθ hp.2 hp.1 hp'.2 hp'.1⟩ |
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| def solve (hψ : LipschitzWith θ ψ) (hθ : θ < 1) (z : ℝ × ℝ) : ℝ × ℝ := |
| Classical.choose (existsUnique hψ hθ z.1 z.2) |
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| theorem solve_spec (hψ : LipschitzWith θ ψ) (hθ : θ < 1) (z : ℝ × ℝ) : |
| (solve hψ hθ z).1 + ψ (solve hψ hθ z).2 = z.1 ∧ |
| (solve hψ hθ z).2 + ψ (solve hψ hθ z).1 = z.2 := |
| (Classical.choose_spec (existsUnique hψ hθ z.1 z.2)).1 |
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| theorem solve_eq (hψ : LipschitzWith θ ψ) (hθ : θ < 1) {z p : ℝ × ℝ} |
| (hp : p.1 + ψ p.2 = z.1 ∧ p.2 + ψ p.1 = z.2) : |
| solve hψ hθ z = p := |
| ((Classical.choose_spec (existsUnique hψ hθ z.1 z.2)).2 p hp).symm |
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| theorem lipschitzWith_solve (hψ : LipschitzWith θ ψ) (hθ : θ < 1) : |
| LipschitzWith (((1 + (θ : ℝ)) / (1 - (θ : ℝ)^2)).toNNReal) |
| (solve hψ hθ) := by |
| apply LipschitzWith.of_dist_le' |
| intro z z' |
| have hθ0 : (0 : ℝ) ≤ θ := θ.coe_nonneg |
| have hθ1 : (θ : ℝ) < 1 := hθ |
| have hden : 0 < 1 - (θ : ℝ)^2 := by nlinarith |
| have h := stability hψ hθ (solve_spec hψ hθ z) (solve_spec hψ hθ z') |
| have hfst : dist z.1 z'.1 ≤ dist z z' := by |
| rw [Prod.dist_eq] |
| exact le_max_left _ _ |
| have hsnd : dist z.2 z'.2 ≤ dist z z' := by |
| rw [Prod.dist_eq] |
| exact le_max_right _ _ |
| rw [Prod.dist_eq] |
| apply max_le |
| · apply h.1.trans |
| calc |
| (dist z.1 z'.1 + θ * dist z.2 z'.2) / (1 - (θ : ℝ)^2) ≤ |
| (dist z z' + θ * dist z z') / (1 - (θ : ℝ)^2) := by |
| gcongr |
| _ = (1 + (θ : ℝ)) / (1 - (θ : ℝ)^2) * dist z z' := by ring |
| · apply h.2.trans |
| calc |
| (dist z.2 z'.2 + θ * dist z.1 z'.1) / (1 - (θ : ℝ)^2) ≤ |
| (dist z z' + θ * dist z z') / (1 - (θ : ℝ)^2) := by |
| gcongr |
| _ = (1 + (θ : ℝ)) / (1 - (θ : ℝ)^2) * dist z z' := by ring |
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| theorem continuous_solve (hψ : LipschitzWith θ ψ) (hθ : θ < 1) : |
| Continuous (solve hψ hθ) := |
| (lipschitzWith_solve hψ hθ).continuous |
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| theorem measurable_solve (hψ : LipschitzWith θ ψ) (hθ : θ < 1) : |
| Measurable (solve hψ hθ) := |
| (continuous_solve hψ hθ).measurable |
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| theorem solve_swap (hψ : LipschitzWith θ ψ) (hθ : θ < 1) (z : ℝ × ℝ) : |
| solve hψ hθ z.swap = (solve hψ hθ z).swap := by |
| apply solve_eq hψ hθ |
| exact ⟨(solve_spec hψ hθ z).2, (solve_spec hψ hθ z).1⟩ |
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| theorem continuous_solve_comp {T : Type*} [TopologicalSpace T] |
| (hψ : LipschitzWith θ ψ) (hθ : θ < 1) |
| {H K : T → ℝ} (hH : Continuous H) (hK : Continuous K) : |
| Continuous (fun t => solve hψ hθ (H t, K t)) := |
| (continuous_solve hψ hθ).comp (hH.prodMk hK) |
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| end CavityPair |
| end Problem367 |
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| end |
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| end OAI |
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