| import OAI.Computability.Logspace.EstimatorMachine |
| import OAI.Computability.Logspace.IterationSchedule |
|
|
| namespace OAI |
|
|
| /-! |
| Choice of simultaneous resource and error parameters for the estimator. |
|
|
| From OpenAI, *Exact Derandomization of Logarithmic Space: L = RL = BPL*. |
| -/ |
|
|
| set_option autoImplicit true |
|
|
| section |
|
|
| namespace ExactDerandomization.Terrain |
| open Analysis Hie Counter Relay Rill Fret Grain Net Est At Buf Stage Finset |
| open Chime Scheme Dome Cone.ConeX _root_.OAI.ExactDerandomization.Sub Swirl |
| open Helix Helix.Age Helix.Mold Helix.Orbit Helix.Foil Helix.Moon |
| open SiteX SiteX.Line |
| open Rain |
|
|
| noncomputable section |
| open scoped Classical |
| variable {q w h:ℕ} {r:Rock q w h} |
|
|
| namespace Bank |
| /- Monetary knobs separated from the syntactic program. |
| Overspending c for canonical tapes scheduled LAST. Width e is |
| small label sizing, independent of c. -/ |
|
|
| def bootC (r:Rock q w h):ℕ:= 50+ Was.Kpre r +clim r.w.p+2*r.w.p.M |
|
|
| abbrev hi (_:MintV)(r:Rock q w h):ℕ:=bootC r+10 |
| abbrev hu (p:MintV)(r:Rock q w h):ℕ:=10*(hi p r+p.vol) |
| abbrev hx (p:MintV)(r:Rock q w h):ℕ:= 10*(hu p r+p.vol) |
| structure Green (r:Rock q w h)(p:MintV) :Prop where |
| money : ∃ t:r.w.Proof, t.T.size^4 ≤ massB p.coin 1∧ t.T.R ≤ massB p.sealR 1 |
| minutes: (massB p.low 1)⁻¹ ≤ delta / 1000 |
| power : 4+Pass.reserve r.w ≤ p.vol |
| honest: 16+p.tokenH ≤ p.sealR |
|
|
| lemma greenUp {r:Rock q w h} (e:r.w.Proof):∃ p:MintV,Green r p:=by |
| have he (c:ℝ):∃ p:ℕ,c ≤ massB p 1 := by |
| obtain ⟨p,h⟩:= exists_nat_gt c |
| exact ⟨p,h.le.trans (swellB (by omega)).le⟩ |
| obtain ⟨k,hk⟩:=he (e.T.size^4) |
| obtain ⟨c,hc⟩:=he e.T.R |
| obtain ⟨w,hw⟩:=he (1000/delta) |
| let v:=4+Pass.reserve r.w |
| let p:MintV:=⟨k,v,c+16+v+20+2*k,w⟩ |
| refine ⟨p,⟨e,hk,le_trans hc (mass_mono (by dsimp only [p]; omega) _)⟩,?_,le_rfl,?_⟩ |
| · apply le_trans (inv_anti₀ (div_pos (by norm_num) dp) hw) |
| exact le_of_eq (by simp) |
| change 16+(v+(20+2*k)) ≤ _ |
| dsimp only [p]; omega |
| |
| structure Turf (p:MintV) (r:Rock q w h) (v:Tags) :Prop where |
| saving : Green r p |
| cost0 : hx p r ≤ v.e |
| coinbook : bootC r ≤ v.bonus |
|
|
| namespace Green |
| variable {p:MintV} (h0:Green r p) |
| include h0 |
| lemma tested : p.fine:=by |
| unfold MintV.fine; have hh:=h0.honest; omega |
| lemma rooms:50+CarryE r.w+r.w.p.K ≤ p.vol:=by |
| have hh:=h0.power; unfold Pass.reserve passport at hh; omega |
| lemma boxesK {d i:ℕ}(hp:1 ≤ d)(he:i<r.w.p.K): |
| lb i ≤ p.vol*d:=by |
| have hm:=h0.rooms; have hh:=lb_le_inc i; nlinarith |
| omit h0 in |
| lemma capacityE (p':Wares)(m i): capacity 2 p'.p.K (Age.L m) i=2^(CarryE p' * gap m i):=by |
| unfold capacity CarryE; change 2^(_*gap m i *4)=2^_ |
| congr 1; ring |
| lemma trunks {d i q0:ℕ}(_:d<Age.L i)(hp:q0< capacity 2 r.w.p.K (Age.L i) d): |
| lb q0 ≤ p.vol*gap i d:=by |
| rw [capacityE] at hp |
| have hk:= gap_pos i d |
| have ht:0 < CarryE r.w:=by unfold CarryE cexpo fexpo; omega |
| exact le_trans (modest hp (Nat.mul_pos ht hk)) (Nat.mul_le_mul_right _ (by have:=h0.rooms; omega)) |
| lemma ray_size{d i:ℕ}(he:1 ≤ d)(hp:i<p.rangeN d):lb i ≤ p.vol*d:= |
| modest hp (Nat.mul_pos (by have :=h0.rooms;omega) he) |
| lemma bill0 {d i j:ℕ}(hd:1 ≤ d) (hp:i<p.rangeN d)(hj:j ≤ bootC r+1): |
| lb (Seek.bin i j) ≤ hu p r*d:=by |
| have h1:lb (enc (i,j)) ≤ 2*lb i+lb j+2:=measure_pair i j |
| have h2:= h0.ray_size hd hp |
| have h3:= lb_le_inc j |
| unfold hu hi Seek.bin |
| nlinarith |
| lemma office {i m j s:ℕ}(hd:i+1=Age.L m) |
| (hj:j < capacity 2 r.w.p.K (Age.L m) i)(hs:lb s ≤ hu p r): |
| lb (Seek.bin j s) ≤ hx p r:= by |
| have h1:lb (enc (j,s)) ≤ 2*lb j+lb s+2:=measure_pair j s |
| have h2:= h0.trunks (by omega) hj |
| have h3 : gap m i=1:=by simp [gap,← hd] |
| rw [h3] at h2; unfold hx Seek.bin; have hx:= h0.rooms; omega |
| end Green |
| |
| end Bank |
| open Bank |
| namespace Stone |
| variable {q w h:ℕ}{u:Line (q:=q) (w:=w) (h:=h)} |
| variable (v:Fabric u) |
|
|
| @[simp] lemma W (a i:ℕ): u.D.W (v.c a i)= Age.weight v.m a i:=u.rolfe .. |
| @[simp] lemma D (a i:ℕ): u.D.D (v.c a i)= slackT v.m a i:=u.rolff .. |
| @[simp] lemma roleR(a i): (uphold (v.c a i)).1=a:=by simp [Fabric.c,rowL,clr,kit] |
| lemma zeroR(a i): 1+u.D.D (v.c a i) ≤ u.D.W (v.c a i):= by |
| have he:=(W v a i); have hp:=(D v a i); have hh:=weights v.m a i; have hq:=pace v.m a i |
| omega |
|
|
| lemma pointing(a b i): pointer (v.c a i) b=v.c b i:=by |
| unfold Fabric.c rowL clr pointer; rw [Wire.eq] |
| lemma alive {a b i o:ℕ}(hi:b=2∨b=4∨b=10∨b=11∨b=15) |
| (he: b=11 ∧ a=9 ∧ o=3 ∨ (a+1=b ∧b≠11∧o=0))(j:ℕ): |
| (coneX u.r u.s.pave).ispath0.f u.x (v.c b i)=true ∧ |
| (coneX u.r u.s.pave).spur.f u.x (v.c b i,j)=post (v.c a i) o j 0:=by |
| change visitX u.x _ = true ∧ send u.x _ = _ |
| unfold visitX send; dsimp only; rw [roleR] |
| refine ⟨decide_eq_true hi,?_⟩ |
| unfold assignT selop; rw [roleR]; simp_rw [Stone.pointing]; congr 1 <;> grind |
| |
| structure Rector (v:Fabric u)(p:MintV):Prop where |
| book:Bank.Turf p u.r u.v |
| chart:u.Chart |
| good:v.Good |
| namespace Rector |
| variable {v:Fabric u}{p:MintV}(h0:Rector v p) |
| include h0 |
| lemma tickets(_ v':ℕ)(hp: v'.succ ≤ Bank.hx p u.r):(lb v') ≤ u.eye.e := |
| (lb_le_inc v').trans (hp.trans h0.book.cost0) |
| lemma wide0 (a i:ℕ):u.smallQ (v.c a i) 0:=by |
| change lb 0 ≤ u.eye.e * u.D.D (v.c a i) |
| rw [lb_zero]; have hx : 1 ≤ u.D.D (v.c a i):=by rw [D]; exact (pace ..).1 |
| have hf : 1 ≤ u.eye.e:=by have h:=h0.book.cost0; change _ ≤ u.eye.e at h; unfold Bank.hx hu hi at h; omega |
| nlinarith |
| lemma aft {a i j:ℕ}(hx:lb j ≤ p.vol*u.D.D (v.c a i)):u.smallQ (v.c a i) j:=by |
| exact le_trans hx (Nat.mul_le_mul_right _ (le_trans (show p.vol ≤ Bank.hx p u.r by unfold Bank.hx;omega) h0.book.cost0)) |
| lemma endUnits (a:ℕ) (hj:a=14∨a=15∨(a=12 ∧ delta0 (bitR v.t))∨(a=16∧redV (bitR v.t))): |
| u.D.Suite (v.c a 0):=by |
| apply Fabric.Good.booked h0.chart h0.good |
| · rw [Age.Period,ite_eq_right (by unfold Brole; omega)]; omega |
| exact ⟨rightR _,Or.inr (by grind)⟩ |
| |
| variable (hc:v.activeR) |
| lemma midUnits (hc:v.activeR) (a i:ℕ)(hj:(1 ≤ a∧a ≤ 4∧i< Age.L v.m)∨i=0∧((9 ≤ a∧a ≤ 11)∨(a=13∧delta0 (bitR v.t)))): |
| u.D.Suite (v.c a i):=by |
| apply Fabric.Good.booked h0.chart h0.good ?_ ⟨rightR _,Or.inl ⟨by grind,(v.workR).1 hc⟩⟩ |
| unfold Age.Period Brole; rcases hj with h|h <;> grind |
| lemma provision {a b i o m:ℕ}(hi:b=2∨b=4∨b=10∨b=11∨b=15) |
| (he: b=11 ∧ a=9 ∧ o=3 ∨ (a+1=b ∧b≠11∧o=0)) |
| (hs:u.D.Suite (v.c a i))(ht:u.D.Suite (v.c b i)) |
| (hh:∀ j:ℕ,j < m → lb j ≤ p.vol*u.D.D (v.c b i)): |
| (CageQ.mk (v.c a i) (v.c b i) o m).Front u:=by |
| have ha: (1 ≤ a∧ a ≤ 17)∧(Age.Brole a↔Age.Brole b):=by unfold Age.Brole;omega |
| have hx:u.D.D (v.c a i)=u.D.D (v.c b i):=by rw [D,D]; simp [slackT,ha.2] |
| refine ⟨hs.stage,ht.stage,?_,?_,hx,fun i hi=> h0.aft (hh i hi), |
| (alive v (i:=i) hi he 0).1,fun j=>(alive v (i:=i) hi he j).2,?_⟩ |
| · have h:= h0.book.cost0; change _ ≤ u.eye.e at h; unfold Bank.hx hu Bank.hi at h; omega |
| · rw [W,W,D]; apply Age.sooner _ (by omega) (by omega) |
| rwa [D,D] at hx |
| intro j hk |
| apply spokenC u hs.stage (Age.cpos _) o j 0 |
| · change lb j ≤ u.eye.e *u.D.D (v.c a i) |
| rw [hx]; exact hk |
| rw [roleR] |
| unfold taps omegaQ colorQ; omega |
| lemma frontI (hc:v.activeR) (i)(he:i < Age.L v.m): (v.α p i).Front u := by |
| apply h0.provision (by decide) (by decide) (h0.midUnits hc 1 i (Or.inl ⟨by decide,by decide,he⟩)) |
| (h0.midUnits hc 2 i (Or.inl ⟨by decide,by decide,he⟩)) |
| intro j hj; rw [D]; exact h0.book.saving.ray_size (gap_pos _ i) hj |
| lemma frontP (hc:v.activeR) (i)(he:i < Age.L v.m): (v.β i).Front u := by |
| apply h0.provision (by decide) (by decide) (h0.midUnits hc 3 i (Or.inl ⟨by decide,by decide,he⟩)) |
| (h0.midUnits hc 4 i (Or.inl ⟨by decide,by decide,he⟩)) |
| intro j hj; rw [D]; exact h0.book.saving.trunks he hj |
| lemma frontT (hc:v.activeR) (b): (v.terminal b).Front u:=by |
| cases b |
| · apply h0.provision (by decide) (by decide) (h0.midUnits hc 9 0 (Or.inr ⟨rfl,Or.inl (by decide)⟩)) |
| (h0.midUnits hc 10 0 (Or.inr ⟨rfl,Or.inl (by decide)⟩)) |
| intro j hj; rw [D]; exact h0.book.saving.boxesK ((pace ..).1) hj |
| apply h0.provision (by decide) (by decide) (h0.midUnits hc 9 0 (Or.inr ⟨rfl,Or.inl (by decide)⟩)) |
| (h0.midUnits hc 11 0 (Or.inr ⟨rfl,Or.inl (by decide)⟩)) |
| intro j hj; rw [D]; exact h0.book.saving.boxesK ((pace ..).1) hj |
| lemma frontE : v.leaveE.Front u:=by |
| apply h0.provision (by decide) (by decide) (h0.endUnits _ (Or.inl rfl)) |
| (h0.endUnits _ (Or.inr (Or.inl rfl))) |
| intro j hj; rw [D]; exact h0.book.saving.boxesK ((pace ..).1) hj |
| end Rector |
| end Stone |
| end |
| end ExactDerandomization.Terrain |
| |
| end |
| |
| end OAI |
| |
