openAI-math-dataset / lean /OAI /Combinatorics /HadwigerCounterexample /AdaptiveProjectionBounds.lean
| import OAI.Combinatorics.HadwigerCounterexample.KeyProjectionBounds | |
| namespace OAI | |
| /-! # Adaptive Projection Bounds -/ | |
| noncomputable section | |
| namespace HadwigerCounterexample | |
| open Filter Finset Matrix | |
| open scoped BigOperators Topology | |
| open Template | |
| namespace StatusAnalysis | |
| variable {p:Shape} {n N K M r z s0:ℕ} {q:Format} {d e:Rates} {b:Bool} | |
| private theorem avg_side_bounds {A : Type} [Fintype A] (w : Wgt A) (hw:IsLaw w) | |
| (b:Bool) (f : Wgt A) (F:A×A→ℝ) (h:∀ i,F i≤f (onSide i b)) : | |
| avg (pairW w w) F≤avg w f := by | |
| apply (avg_mono (pairW_law hw hw).nz h).trans | |
| cases b | |
| · rw [pairW_avg] | |
| simp [onSide,hw.avg_const] | |
| rw [pairW_avg]; simp only [onSide,ite_true,hw.avg_const]; rfl | |
| namespace Quill | |
| variable (w:Quill p q d b) | |
| (m:Mixer p n) (t:Tagging p (Messenger q)) (φ:Messenger q→F) | |
| (u:LeafMixture p n N) (Y:Silhouette p N r) | |
| set_option maxHeartbeats 1200000 in | |
| theorem oneTrestle (omega_:Wgt u.T) (hv:IsLaw omega_) (s:u.T→ Charts p n d) | |
| (μ:u.T→Wgt (UnitType p n N)) (hμ:∀ i,IsLaw (μ i)) | |
| (P : Keys (p:=p) t N→ Fin z) (F:u.T×u.T → UnitType p n N→Prop) | |
| (hF : ∀ i o,F i o→PreparationParameters.frameSide p m o) | |
| (hw:∀ i,IsLaw (w.policies t φ n i)) (hν:IsLaw (lawNu p N t φ)) | |
| (ε v A B:ℝ) (he:0≤ε) | |
| (ha:∀ i:Fin z, | |
| pr (mixW omega_ μ) (w.flunk t φ Y (fun x=>ind (P x=i)) m v ε) ≤ A) | |
| (hh: pr (mixW omega_ μ) (fun x=>¬ w.matchesArea t φ Y x v)≤B) : | |
| let L (i:u.T×u.T):=onSide i b | |
| avg (pairW omega_ omega_) (fun i=>varOne (mapW (w.displayMass t φ (μ (L i)) (F i) m (s (L i)) Y) P) | |
| (mapW (w.sales t φ (μ (L i)) (F i) m (s (L i))) P)) ≤ z*(ε+2*(A+B)) := by | |
| classical | |
| intro L | |
| let Q (i:u.T×u.T) (x:Fin z):= | |
| |mapW (w.displayMass t φ (μ (L i)) (F i) m (s (L i)) Y) P x- | |
| mapW (w.sales t φ (μ (L i)) (F i) m (s (L i))) P x| | |
| let V:= pairW omega_ omega_ | |
| have h (x : Fin z) : | |
| avg V (fun i=>Q i x) ≤ ε+ 2*(A+B) := by | |
| let H k:= P k=x | |
| let R:= w.flunk t φ Y (fun i=>ind (H i)) m v ε | |
| let Z (i:UnitType p n N):= ¬ w.matchesArea t φ Y i v | |
| let S o:=R o∨ Z o | |
| have heq (i): Q i x≤ ε+2*pr (μ (L i)) S := by | |
| refine w.display_test t φ (μ (L i)) (F i) m (s (L i)) Y H (hμ _) hw hν S he | |
| (fun y hy hz=> ?_) | |
| by_cases hf :Z y | |
| · exact Or.inr hf | |
| exact Or.inl ⟨hF _ _ hy,not_not.mp hf,s (L i),hz⟩ | |
| let U:=mixW omega_ μ | |
| have hu : IsLaw U:=mixW_law hv hμ | |
| have hm (o) : avg omega_ (fun i=>pr (μ i) o)= pr U o := (mixW_avg ..).symm | |
| apply (avg_side_bounds omega_ hv b (fun i=>ε+2*pr (μ i) S) _ heq).trans | |
| rw [avg_add,avg_mul',hm,hv.avg_const] | |
| exact add_le_add_right (mul_le_mul_of_nonneg_left ((pr_or_le hu.nz).trans | |
| (add_le_add (ha _) hh)) (by norm_num)) _ | |
| change avg V (fun x=> ∑ i,Q x i) ≤ _ | |
| rw [avg_sum]; exact (Finset.sum_le_sum fun i _=> h i).trans_eq (by simp; ring) | |
| end Quill | |
| open Station PreparationParameters | |
| namespace Station | |
| variable {w:LeafMixture p n N} (t:Station p n N K w) | |
| (x:Typus p q d e) (m:Mixer p n) (k: Fin (2^deckMax p K)) | |
| (μ:Wgt (UnitType p n N)) (Y:Silhouette p N r) | |
| theorem gauzed_area (Y:Silhouette p N r) (w : Quill p q d b) | |
| (t: Tagging p (Messenger q)) (u : Messenger q → F) | |
| (h:w.moral t u) (δ:ℝ) | |
| {K : ℕ} {m:Mixer p n} {ε v:ℝ} (he:ε≤v) | |
| (o : UnitType p n N) (hc:gauze p ε δ Y m K o) : | |
| w.matchesArea t u Y o v := by | |
| intro i; have heq := w.allure_apt t u h i | |
| exact ((hc _ (w.allure t u i)).1 heq).trans he | |
| set_option maxHeartbeats 1200000 in | |
| theorem projectSum (P : Keys (p:=p) x.t N→Fin z) | |
| (h :apart w.T) (hm:IsLaw μ) (hw :∀ i∈w.T,0<partMass μ i) (hs : 0<w.amount μ) | |
| (ha : ∀ i,IsLaw (x.qA.policies x.t x.z n i)) (hb:∀ i,IsLaw (x.qB.policies x.t x.z n i)) | |
| (hν : IsLaw (lawNu p N x.t x.z)) (ε v v' τ B δ : ℝ) (he:0≤ε) | |
| (hf : x.permission) | |
| (ht : τ ≤ v ∧ τ≤v') | |
| (hh : pr (w.sap μ) (fun i=>¬gauze p τ δ Y m K i)≤B) | |
| (hx : ∀ i:Fin z,pr (w.sap μ) (x.qA.flunk x.t x.z Y (fun y=>ind (P y=i)) m v ε)≤B) | |
| (hy : ∀ i:Fin z,pr (w.sap μ) (x.qB.flunk x.t x.z Y (fun y=>ind (P y=i)) m v' ε)≤B) : | |
| t.due x m μ k Y P≤2*(z*(ε+2*(B+B))) := by | |
| classical | |
| obtain ⟨hp,hq⟩:=rebuild hm.nz h hw hs | |
| let A (i:w.T):=partW μ i.val | |
| let u:=portionW μ w.T | |
| have hQ : w.sap μ=mixW u A :=hq | |
| let V:=pairW u u; have hA i:IsLaw (A i):=on_law hm.nz (hw i.val i.property) | |
| have hL : avg V (fun i=> taxLeft t x m μ k Y P i.1 i.2) ≤ z*(ε+2*(B+B)) := by | |
| refine x.qA.oneTrestle m x.t x.z w Y u hp (fun i=>t.cAxes i d) A hA P | |
| (fun i=> t.frontPage x m i.1 i.2 k) (fun i o h => h.2.2.1) ha hν ε v B B he | |
| (fun i=>by rw [←hQ]; exact hx i) ?_ | |
| rw [←hQ]; apply (pr_mono (LeafMixture.sapLaw hs hm).nz ?_).trans hh | |
| exact fun i hi h=> hi (gauzed_area _ _ _ _ hf.2.1 δ ht.1 i h) | |
| have hR : avg V (fun i=> taxRight t x m μ k Y P i.1 i.2) ≤ z*(ε+2*(B+B)) := by | |
| refine x.qB.oneTrestle m x.t x.z w Y u hp (fun i=>t.cAxes i e) A hA P | |
| (fun i=> t.backPage x m i.1 i.2 k) (fun i o h => h.2.2.1) hb hν ε v' B B he | |
| (fun i=>by rw [←hQ]; exact hy i) ?_ | |
| rw [←hQ]; apply (pr_mono (LeafMixture.sapLaw hs hm).nz ?_).trans hh | |
| exact fun i hi h=>hi (gauzed_area _ _ _ _ hf.2.2.1 δ ht.2 i h) | |
| unfold due; rw [avg_add,two_mul] | |
| exact add_le_add hL hR | |
| end Station | |
| end StatusAnalysis | |
| end HadwigerCounterexample | |
| end | |
| end OAI | |