| import OAI.Combinatorics.HadwigerCounterexample.FiniteOptionLabels |
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| namespace OAI |
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| /-! # Binary Status Comparison -/ |
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| noncomputable section |
| namespace HadwigerCounterexample |
| open Finset Matrix Filter Function Submodule Module LinearMap |
| open scoped BigOperators |
| variable {A B J H : Type} [Fintype A] [Fintype B] [Fintype J] |
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|
| theorem disagreement_bound (w : Wgt A) (hw : ∀ x,0 ≤ w x) (f g h:A→ H) : |
| pr w (fun i=> f i≠ h i) ≤ pr w (fun i=>f i≠ g i) + pr w (fun i=>g i≠h i) := |
| let p (i:A) := f i≠g i∨g i≠h i |
| ((pr_mono hw fun i h=>show p i from Classical.byContradiction (fun hi => h (Eq.trans |
| (not_not.mp (mt Or.inl hi)) (not_not.mp (mt Or.inr hi))))).trans (pr_or_le hw)) |
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| def majority (w : Wgt A) (f : A→F) := |
| if 1/2≤ pr w (fun i=>f i=0) then (0:F) else 1 |
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|
| theorem majority_bound (w : Wgt A) (hw:IsLaw w) (f : A→F) : |
| 1/2 ≤ pr w (fun i=> f i= majority w f) := by |
| classical |
| unfold majority |
| split |
| next h => exact h |
| next h => |
| have hi i : f i=1 ↔ ¬f i=0 := |
| ⟨fun hh=>by rw [hh]; exact one_ne_zero,fun hh=>(bit_cases _).resolve_left hh⟩ |
| simp_rw [hi,pr_not,hw.total] |
| exact le_of_lt (by linarith only [h]) |
|
|
| theorem majority_charge (p:Wgt A) (q:Wgt B) (hp:IsLaw p) (hq:IsLaw q) |
| (f:A→F) (g:B→F) : |
| pr p (fun i=> f i≠majority q g)≤2 * pr (pairW p q) (fun i=> f i.1≠g i.2) := by |
| classical |
| let Q i:= g i=majority q g; let P i:= f i≠majority q g |
| let v:= pairW p q |
| have hm (z:A×B) (h:P z.1∧Q z.2): f z.1≠g z.2 := h.2 ▸ h.1 |
| have hh : pr p P*pr q Q ≤ pr v (fun i=>f i.1≠g i.2) := |
| (pairW_pr ..).symm.le.trans (pr_mono (pairW_law hp hq).nz hm) |
| have hx:=majority_bound q hq g |
| have hh' := mul_le_mul_of_nonneg_left hx (pr_nonneg (p:=P) hp.nz) |
| change _* (1/2) ≤ _*pr q Q at hh' |
| change pr p P≤2*pr v _ |
| linarith only [hh,hh'] |
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|
| theorem reflect_cost (p:Wgt A) (q:Wgt B) (hp:IsLaw p) (hq:IsLaw q) |
| (f:A→H) (g:B→H) (k:H) : |
| pr q (fun i=>g i≠k)≤ pr (pairW p q) (fun i=>f i.1≠g i.2)+pr p (fun i=>f i≠k) := by |
| let w:=pairW p q |
| let t i:= g i≠k; let s i:= f i≠k |
| have hi : pr q t=pr w (fun i=> t i.2) := by simpa [hp.total] using |
| (pairW_pr p q (fun _=>True) t).symm |
| have hj : pr p s=pr w (fun i=>s i.1) := by simpa [hq.total] using |
| (pairW_pr p q s (fun _=>True)).symm |
| let P (i:A×B):= f i.1 ≠ g i.2 |
| rw [hi,hj] |
| have hh (i:A×B) (h:t i.2): P i∨ s i.1 := by |
| by_cases he:P i |
| · exact Or.inl he |
| exact Or.inr (fun hc=> h ((not_not.mp he).symm.trans hc)) |
| exact (pr_mono (pairW_law hp hq).nz hh).trans (pr_or_le (pairW_law hp hq).nz) |
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|
| namespace StatusAnalysis |
| namespace DualAnalysis |
| variable {p:Shape} {d e: Rates} {r K:ℕ} |
| abbrev Pulsar (p:Shape) (r:ℕ):= Tag p.g → F → Fin r→F |
| variable (u:Terrain p r) (s:Gazette p r d) (t:Gazette p r e) |
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| def shot (i:Nodes) (x:Appearance p) (a:RhymeSet (d i)) (f:Pulsar p r) := |
| avg (u x.l x.q) (fun j=>pr (s i x j) (fun v=> a.soot x v ≠f x.l x.q j)) |
| def praise (u:Terrain p r) (s:Gazette p r d) (i:Nodes) |
| (A:Finset (Label p.b)) (a:RhymeSet (d i)) (f:Pulsar p r) (ε:ℝ) := |
| ∀ x:Appearance p,x.s∉A →x.apt → shot u s i x a f≤ε |
| def counsel (i:Grid) (A B:Finset (Label p.b)) |
| (a : RhymeSet (d i.1)) (b : RhymeSet (e i.2)) (ε :ℝ) := |
| ∀ x:Zodiac p,x.common → x.s true∉A →x.s false∉ B →charge u x s t i a b≤ε |
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| def amazes (p:Shape) (z:Label p.b) (l: Tag p.g) (q:F):Appearance p:=⟨l,z,Flavor.full,q⟩ |
| def amends (p:Shape) (x y:Appearance p): Zodiac p := |
| {l:=x.l |
| q:=x.q |
| s i:=if i then x.s else y.s |
| f i:=if i then x.f else y.f} |
| theorem amendsB (x y: Appearance p)(h:x.l=y.l)(ht:x.q=y.q): |
| (amends p x y).aspect false=y := by cases x; cases y; simp_all [amends,Zodiac.aspect] |
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| theorem splitCounsel (i:Grid) (A B:Finset (Label p.b)) |
| (a : RhymeSet (d i.1)) (b : RhymeSet (e i.2)) (ε :ℝ) |
| (h:counsel u s t i A B a b ε) (x y:Appearance p) |
| (hx:x.s∉A) (hy:y.s∉B) (he:x.apt) (hf:y.apt) (hg:x.l=y.l) (hk:x.q=y.q) : |
| avg (u x.l x.q) (fun j=>pr (pairW (s i.1 x j) (t i.2 y j)) (fun k=>a.soot x k.1 ≠ |
| b.soot y k.2))≤ε := by |
| classical |
| let Z:=amends p x y |
| have hQ : Z.aspect false=y := amendsB _ _ hg hk |
| have hP : Z.aspect true=x := rfl |
| have hm : Z.common := by intro i;cases i; rw [hQ];exact hf; rw [hP];exact he |
| have hi:= h Z hm hx hy |
| unfold charge circlet pr couplePage at hi |
| rw [mixW_avg,hQ,hP] at hi |
| exact hi |
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| set_option maxHeartbeats 800000 in |
| theorem paeans {β δ:ℝ} (hu:origami p r u δ)(hs:reports p r u s β δ)(ht:reports p r u t β δ) |
| (i:Grid) (A B:Finset (Label p.b)) (hx:A.card<Fintype.card (Label p.b)) |
| (hy:B.card<Fintype.card (Label p.b)) (hg:3≤p.g) |
| (a : RhymeSet (d i.1)) (b : RhymeSet (e i.2)) (ε:ℝ) (he:0≤ε) |
| (h:counsel u s t i A B a b ε) : |
| ∃ f:Pulsar p r, praise u s i.1 A a f (3*ε) ∧ praise u t i.2 B b f (3*ε) := by |
| classical |
| obtain ⟨x,hx⟩:=any_label A hx |
| obtain ⟨y,hy⟩:=any_label B hy |
| let Z:= amazes p y |
| let f l q j:= majority (t i.2 (Z l q) j) (b.soot (Z l q)) |
| have hl:praise u s i.1 A a f (2*ε) := by |
| intro z hz hhq |
| let w:= Z z.l z.q |
| have ht' := generic_apt p hg w rfl |
| have hh := splitCounsel u s t i A B a b ε h z w hz hy hhq ht' rfl rfl |
| refine le_trans (avg_mono (hu.1 z.l z.q).nz (fun j=> majority_charge (s i.1 z j) (t i.2 w j) |
| (hs.law _ _ hhq _) (ht.law _ _ ht' _) (a.soot z) (b.soot w))) ?_ |
| rw [avg_mul'] |
| exact mul_le_mul_of_nonneg_left hh (by norm_num) |
| refine ⟨f,fun w hw hx=>(hl _ hw hx).trans (by linarith only [he]), fun w hw hv=>?_⟩ |
| let z:=amazes p x w.l w.q |
| have hf : z.apt:=generic_apt _ hg _ rfl |
| have hx0:= splitCounsel u s t i A B a b ε h z w hx hw hf hv rfl rfl |
| have hl0:=hl z hx hf |
| let W:=u w.l w.q |
| have heq j:= reflect_cost (s i.1 z j) (t i.2 w j) (hs.law _ _ hf j) (ht.law _ _ hv j) |
| (a.soot z) (b.soot w) (f w.l w.q j) |
| apply (avg_mono (hu.1 w.l w.q).nz heq).trans |
| rw [avg_add]; exact (add_le_add hx0 hl0).trans_eq (by ring) |
| end DualAnalysis |
| end StatusAnalysis |
| end HadwigerCounterexample |
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| end |
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| end OAI |
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