openAI-math-dataset / lean /OAI /Combinatorics /HadwigerCounterexample /CompatibleOptionRealization.lean
| import OAI.Combinatorics.HadwigerCounterexample.PairedEventBounds | |
| namespace OAI | |
| /-! # Compatible Option Realization -/ | |
| noncomputable section | |
| namespace HadwigerCounterexample | |
| open Finset Matrix Function Module LinearMap | |
| open scoped BigOperators | |
| variable {p:Shape} {n N M K s0:ℕ} | |
| namespace StatusAnalysis | |
| variable {q:Format} {d e: Rates} {b:Bool} | |
| abbrev Stalls (p:Shape) := Bool→Finset (Label p.b) | |
| def booths (p:Shape) (K:ℕ) (s: Plan p n) (H:Stalls p) := | |
| (∀ i,(H i).card≤frameBills p.g K) ∧ | |
| ∀ i j, protectedBy p (sPin s j i) (qorth K) (H i) | |
| theorem buildBooths (K:ℕ) (s:Plan p n) (h:planRank s≤K) | |
| (hj:(K+1)*qorth K≤p.j+1) : | |
| ∃ H,booths p K s H := by | |
| classical | |
| choose H hH hG using fun i : Bool=> pin_protection s i K (qorth K) h hj | |
| exact ⟨H,hH,hG⟩ | |
| theorem shelter (K :ℕ) (S:Pin p n N) (t : Quill p q d b) (z : Tagging p (Messenger q)) (H:Stalls p) | |
| (h:booths p K S.sp H) (hp : Injective t.sel) | |
| (ht:∀ i:Messenger q,t.sel i∉H (parcel b i)) | |
| (hg : ∀ i,1≤q i ∧ q i≤7) : t.penSpace z S K := by | |
| intro x i | |
| let v:=(t.pen z).draw x | |
| refine ⟨H i,h.2 i,fun j hj=>?_,?_,hp.injOn⟩ | |
| · have he : parcel b j=i := Finset.mem_filter.mp hj|>.2 | |
| rw [←he]; exact ht j | |
| exact (Volley.here_card v (shots_ok b v rfl hg) i).trans (by unfold qorth; omega) | |
| theorem sizedCharts (K:ℕ) (w:LeafMixture p n N) (t:Station p n N K w) (ht:t.bank) | |
| (i:w.T) (h:planRank (w.pin i.val).sp≤K) : | |
| ∃ d:Stakes K,charted (t.cAxes i (toRates d)) (w.pin i.val).sp := by | |
| let s:=(w.pin i.val).sp | |
| let d (i:Nodes):= dim (effective s i) | |
| have hd i : d i < K^2+1 := Nat.lt_of_le_of_lt | |
| ((effective_dim s i).trans (by gcongr)) (Nat.lt_succ_self _) | |
| exact ⟨fun i=>⟨d i,hd i⟩,ht.1 i⟩ | |
| structure Fabric (p:Shape) (q:Format) (d:Rates) (b:Bool) where | |
| sel : Messenger q → Label p.b | |
| flav : Messenger q → Flavor p.g | |
| amp : Amounts p q d b | |
| namespace Fabric | |
| variable (w:Fabric p q d b) (t:Tagging p (Messenger q)) | |
| def hang (s:Script p (Messenger q)) : Quill p q d b := | |
| ⟨w.sel,w.flav,w.amp,s⟩ | |
| def up (w:Fabric p q d b) (t:Tagging p (Messenger q)) (z: Messenger q→F) := | |
| (w.hang 0).moral t z | |
| end Fabric | |
| variable {i j:Bool} {L:ℕ} | |
| set_option maxHeartbeats 1200000 in | |
| theorem optionVia (q:Format) (d e : Rates) (s t:Pin p n N) (T:Cross p n) (K:ℕ) | |
| (ha: tableInj s.sp t.sp T) (hb : tableInj t.sp s.sp (reciprocal p T)) | |
| (p':Tagging p (Messenger q)) (x:Charts p n d) (y:Charts p n e) | |
| (hx:charted x s.sp) (hy:charted y t.sp) | |
| (f:Fabric p q d true) (g:Fabric p q e false) | |
| (hs h:Stalls p) (hm:booths p K s.sp hs) (hu : booths p K t.sp h) | |
| (hf:∀ i,f.sel i∉ hs (parcel true i)) (hg:∀ i,g.sel i∉ h (parcel false i)) | |
| (z:Messenger q→F) (hF:f.up p' z)(hG:g.up p' z) | |
| (hj : 1≤p.J) (h2: 2≤p.g) (h₀:∀ i,1≤q i∧q i≤7) | |
| (he:speak _ _ (sendSum z f.amp g.amp)=(hailing p x y T,0)) : | |
| ∃ (a b:Script p (Messenger q)),Typus.chances ⟨p',z,f.hang a,g.hang b⟩ x y s t T K := by | |
| obtain ⟨hQ,hR,hL,hS⟩:= (spoken x y T z f.amp g.amp).mp he | |
| obtain ⟨a,ha'⟩:= proclaim_script true f.amp g.amp p' y (reciprocal p T) hR (fun i=>(h₀ i).1) | |
| hS h2 hj | |
| have hh : rhyme g.amp f.amp := fun i=> (add_comm ..).trans (hR i) | |
| obtain ⟨b,hb'⟩:= proclaim_script false g.amp f.amp p' x T hh (fun i=>(h₀ i).1) | |
| hL h2 hj | |
| exact ⟨a,b, ⟨h₀,hF,hG,hQ,hR,ha',hb'⟩,⟨hx,shelter K s (f.hang a) p' hs hm hF.1 hf h₀,hL,ha⟩, | |
| ⟨hy,shelter K t (g.hang b) p' h hu hG.1 hg h₀,hS,hb⟩⟩ | |
| def berthTicket (a b:Pin p n N) (T: Cross p n) | |
| (x:UnitType p n N × UnitType p n N) := | |
| tableInj a.sp b.sp T ∧ tableInj b.sp a.sp (reciprocal p T) | |
| ∧ fitsTable a.sp b.sp T (actual p x.1 x.2) | |
| theorem issueTicket (x y:UnitType p n N) (w : LeafMixture p n N) (t:Station p n N K w) | |
| (i j:w.T) (ha:apart w.T) | |
| (hx:x∈i.val) (hy:y∈j.val) (ht:t.bank) | |
| (he : secureAt p w.rooting (x,y)) : | |
| ∃ k, berthTicket (w.pin i.val) (w.pin j.val) (t.mails i j k) (x,y) := by | |
| obtain ⟨k,hk⟩:=ht.2 i j (actual p x y) | |
| obtain ⟨h1,h2⟩:=he | |
| let a:= w.pin i.val; let b:= w.pin j.val | |
| have hh : w.rooting x=a.sp:= congrArg Pin.sp (LeafMixture.poller_eq w ha i.property hx) | |
| have hg : w.rooting y=b.sp:= congrArg Pin.sp (LeafMixture.poller_eq w ha j.property hy) | |
| change tableInj (w.rooting x) (w.rooting y) (actual p x y) at h1 | |
| change tableInj (w.rooting y) (w.rooting x) (actual p y x) at h2 | |
| rw [hh,hg] at h1 h2 | |
| rw [← reciprocal_actual] at h2 | |
| exact ⟨k, fits_inj hk h1,fits_inj (fits_recip hk) h2,hk⟩ | |
| theorem berth_flags (a b:Pin p n N) (T:Cross p n) {x:UnitType p n N × UnitType p n N} | |
| (h:berthTicket a b T x) (ha:a.agree x.1) (hb:b.agree x.2) : | |
| tableFlag a b T x.1 ∧ tableFlag b a (reciprocal p T) x.2 := by | |
| have hg:=fits_recip h.2.2; rw [reciprocal_actual] at hg | |
| exact ⟨fits_flag T _ a b x.1 h.2.2 (flag_actual _ _ hb), | |
| fits_flag _ _ b a x.2 hg (flag_actual _ _ ha)⟩ | |
| def cardTone (m:BMark p n N) (x:UnitType p n N) (T:Cross p n) (j:Bool) : F := | |
| ∑ i:Bool,contract p T i j (m x i) | |
| theorem actualTone (m:BMark p n N) (a b : UnitType p n N) (j:Bool) : | |
| cardTone m a (actual p a b) j = tone p b (unary j) (delta m a) := by | |
| rw [lone] | |
| unfold cardTone delta | |
| rw [map_add,Fintype.sum_bool] | |
| simp_rw [contract_actual]; rfl | |
| theorem fitsTone {w v: Plan p n} {T U:Cross p n} | |
| (m:BMark p n N) (x:UnitType p n N) | |
| (h:fitsTable w v T U) (hx:∀ i,m x i∈effective w i) (j) : | |
| cardTone m x T j=cardTone m x U j := | |
| Finset.sum_congr rfl (fun i _=>fits_contract h i j _ (hx i)) | |
| end StatusAnalysis | |
| end HadwigerCounterexample | |
| end | |
| end OAI | |