| import OAI.Computability.Logspace.CompressionEvaluation |
| import OAI.Computability.Logspace.DiagonalEvaluation |
| import OAI.Computability.Logspace.TerminalEvaluation |
| import OAI.Computability.Logspace.TransitionEvaluation |
|
|
| namespace OAI |
|
|
| /-! |
| Simultaneous correctness of one uniform estimator program on all inputs. |
|
|
| From OpenAI, *Exact Derandomization of Logarithmic Space: L = RL = BPL*. |
| -/ |
|
|
| set_option autoImplicit true |
|
|
| section |
|
|
| namespace ExactDerandomization.Umbra |
| open Analysis Relay Counter Rill Fret At Hie |
| open Arena Amber Song Fret.Lyric Reed Lu Est Net |
| open Grain Terrain Bank Helix Helix.Age Helix.Mold Helix.Foil Helix.Moon Helix.Orbit Chime Dome Scheme Finset Stage |
| open SiteX Line Fret.Seats |
| open Rain Spring |
| open Vessel Gamut |
| noncomputable section |
| open scoped Classical |
| variable {q w h:ℕ} |
| namespace Codex |
| variable (p:MintV)(r:Rock q w h)(v:Tags) |
| def atlas : Pose:= Lily.poseX r |
| abbrev mdi:= BevelP p r v where |
| BevelP:= Chron.Bevel.rentG |
| abbrev boxI0:=Lily.prog r v p (mdi p r v) |
| abbrev boxI1:=Tiber.Sky.progP p r v |
| abbrev boxT:= Aurora.Fort.baseTc r p v |
| abbrev boxE:= Gyre.Term.fullD r v p |
| abbrev capT:=Line.Fabric.capprog r v (atlas r) |
| abbrev rankT1:=Line.Fabric.Stand.score0 (Pond.reapT r.w) r v (atlas r) |
| abbrev rankE1:=Line.Fabric.Stand.score0 (Pond.reapE r.w) r v (atlas r) |
| abbrev VI := Sun.scheduleR p (knob r v) (PrintV.shell r p v) |
| abbrev VF := Ma.promptV p r v |
|
|
| def produce (k:ℕ):Cog := |
| match k with |
| | 1=>boxI0 p r v |
| | 3=>boxI1 p r v |
| | 9=>boxT p r v |
| | 12=>capT r v |
| | 13=>rankT1 r v |
| | 14=>boxE p r v |
| | 16=>rankE1 r v |
| | 5=>VI p r v |
| | 17=>VF p r v |
| | _=> Cargo.knock |
| abbrev pages :Stash:=⟨atlas r,produce p r v⟩ |
|
|
| abbrev hcap := v.A*(2*v.A+40) |
| abbrev compass (s:Fabric.Stand)(v:Tags):ℕ:=powder (v.A*(2*v.A+40)+6*(s.k+50)) |
| abbrev coins3:= GoodF.treasury p r |
| abbrev debit3:= Sun.priceC r p (coins3 p r) |
|
|
| structure Paying (p:MintV)(r:Rock q w h)(v:Tags) :Prop where |
| h1:Cambria.Wear.Paper.budgetD p r ≤ v.slope |
| h2:Chron.Bevel.median p r≤v.slope |
| h3:Tiber.Sky.budget0 p r≤v.slope |
| h4:Aurora.Fort.budget4 p r≤v.slope |
| h5:hcap v ≤ v.slope |
| h6:compass (Pond.reapT r.w) v≤v.slope |
| h7:compass (Pond.reapE r.w) v≤ v.slope |
| h8:Gyre.Term.Budget0.cost r v p ≤ v.slope |
| h9:debit3 p r+ coins3 p r ≤ v.slope |
| he:Ma.render p≤ v.slope |
| def sump(p:MintV)(r:Rock q w h)(v:Tags):ℕ:= |
| Cambria.Wear.Paper.budgetD p r+ Chron.Bevel.median p r+Tiber.Sky.budget0 p r+ |
| Aurora.Fort.budget4 p r+ hcap v + compass (Pond.reapT r.w) v+compass (Pond.reapE r.w) v+ |
| Gyre.Term.Budget0.cost r v p+ (debit3 p r+coins3 p r)+Ma.render p |
| lemma liable (ht: sump p r v ≤ v.slope):Paying p r v:=by |
| unfold sump at ht |
| constructor <;>omega |
| |
| structure Fortune (r:Rock q w h)(p:MintV)(v:Tags)(κ:ℕ):Prop where |
| funds:Paying p r v |
| t0:Turf p r v |
| coast0:max 6 (10*v.e) ≤ v.slope |
| earth0:100 ≤ κ∧ lb capD ≤ κ∧100*v.e ≤ κ |
| boom:1000*κ^2 ≤ v.A |
| city:v.A ≤ v.slope |
| cargo:r.w.p.K ≤ κ |
|
|
| lemma enacted (t:ℕ) (e:r.w.Proof):∃ p v κ, Fortune r p v κ ∧ v.extra=t:=by |
| obtain ⟨p,hp⟩:= Bank.greenUp (r:=r) e |
| let x:=hx p r |
| let B:=bootC r |
| let κ:=100 + lb capD +100*x+r.w.p.K |
| let A:=1000*κ^2 |
| let f(s):Tags:=⟨x,A,s,t,B,0⟩ |
| let b:=max 6 (10*x) |
| let k:=sump p r (f 0) + A+b |
| have ht:sump p r (f k)=sump p r (f 0):=rfl |
| have hh:sump p r (f k)≤k:=by rw [ht]; exact le_trans (Nat.le_add_right ..) (Nat.le_add_right ..) |
| have ha:A ≤ k:= le_trans (Nat.le_add_left ..) (Nat.le_add_right ..) |
| exact ⟨p,f k,κ,⟨liable _ _ _ hh,⟨hp,le_rfl,le_rfl⟩,(show b ≤ k from Nat.le_add_left ..), |
| ⟨(by unfold κ;omega),(by unfold κ;omega),by change 100*x≤κ; unfold κ;omega⟩,le_rfl,ha,(by unfold κ;omega)⟩,rfl⟩ |
| end Codex |
| |
| variable {u:Line (q:=q) (w:=w) (h:=h)} |
|
|
| structure TrustQ(u:Line (q:=q) (w:=w) (h:=h)) (p:MintV)(κ:ℕ):Prop where |
| plan:Codex.Fortune u.r p u.v κ |
| page:u.s=Codex.pages p u.r u.v |
| chart:u.Chart |
| namespace TrustQ |
| variable {p:MintV}{k:ℕ}(he:TrustQ u p k) |
| include he |
| lemma owns(v:Fabric u)(hp:v.Good):Stone.Rector v p:=⟨he.plan.t0,he.chart,hp⟩ |
| lemma ownsV(v:OrbitV u)(hp:v.estate):v.Counsel p:=⟨he.chart,hp,he.plan.t0⟩ |
| lemma grants(v:Fabric u):v.Mantle k:=by |
| have hh:=he.plan |
| refine ⟨he.chart,hh.earth0.1,hh.earth0.2.1,hh.earth0.2.2,hh.boom,hh.city,?_⟩ |
| intro i hi; exact le_trans (le_trans (lb_le_inc i) (Nat.succ_le_of_lt hi)) hh.cargo |
| |
| lemma scripted(i):u.s.scripted i=Codex.produce p u.r u.v i:=by rw [he.page] |
| lemma stock:u.s.pave=Codex.atlas u.r:=by rw [he.page] |
| lemma capdraw:u.s.scripted 12=Line.Fabric.capprog u.r u.v u.s.pave:=by rw [he.scripted,he.stock]; rfl |
| |
| lemma pearls {a i:ℕ}(hp:u.D.active a)(hm:a≠0)(hk:(uphold a).1=i) |
| (hz:i≠0)(he1:∀ k,k=2∨k=4∨k=10∨k=11∨k=15→i≠k): |
| Studio.Parcel u a (Codex.produce p u.r u.v i):= by |
| have hx:=he.chart |
| apply Studio.stationed |
| all_goals first|exact hp|exact hx|exact hm|skip |
| · rw [hk]; exact ⟨hz,he1⟩ |
| rw [hk]; exact he.scripted i |
| |
| lemma pearlR (v: Fabric u)(i j:ℕ)(h:u.D.active (v.c i j))(hz:i≠0)(he1:∀ k,k=2∨k=4∨k=10∨k=11∨k=15→i≠k): |
| Studio.Parcel u (v.c i j) (Codex.produce p u.r u.v i):= |
| he.pearls h (Age.cpos _) (Stone.roleR ..) hz he1 |
| lemma pearlV (v: OrbitV u)(i j:ℕ)(h:u.D.active (v.channel i j))(hz:i≠0)(he1:∀ k,k=2∨k=4∨k=10∨k=11∨k=15→i≠k): |
| Studio.Parcel u (v.channel i j) (Codex.produce p u.r u.v i):= |
| he.pearls h (Age.cpos _) (show (uphold (v.channel i j)).1=i from by simp [uphold,OrbitV.channel,vecL,clr,Foil.kit]) hz he1 |
|
|
| lemma insures (v:Fabric u)(h:v.Good): Pond.Books p v k:=by |
| have H:=he.plan.funds |
| refine ⟨he.owns v h,he.grants v,H.h6,H.h7,?_,?_⟩ |
| · rw [he.scripted,he.stock]; rfl |
| rw [he.scripted,he.stock];rfl |
| end TrustQ |
| |
| end |
| end ExactDerandomization.Umbra |
| |
| end |
| |
| section |
| |
| namespace ExactDerandomization.Umbra |
| open Analysis Relay Counter Rill Fret At Hie |
| open Arena Amber Song Fret.Lyric Reed Lu Est Net |
| open Grain Terrain Bank Helix Helix.Age Helix.Mold Helix.Foil Helix.Moon Helix.Orbit Chime Dome Scheme Finset Stage |
| open SiteX Line Fret.Seats |
| open Vessel Gamut |
| noncomputable section |
| open scoped Classical |
| variable {q w h:ℕ}{u:Line (q:=q) (w:=w) (h:=h)} |
|
|
| namespace Elect |
| variable (p:MintV) (v:Fabric u) |
| abbrev Birth (i:ℕ) :Prop :=∀ x∈u.D.P,∃ a:v.Yolk p i x,v.YFine p i x a∧ |
| ∀ j,a.raw.tag j ≤ Cambria.Era.budget u.r |
| abbrev Mark (i:ℕ):Prop:=∀ x∈u.D.P,∃ a:v.Hinge p i x,v.HFine p i x a∧ |
| ∀ j,lb (a.depot.tag j) ≤ Bank.hu p u.r * gap v.m i |
| abbrev CeilingR (p:MintV)(v:Fabric u):Prop:= |
| ∃ f:Bool → Producer, ∃ tb, v.Tgood p f ∧ v.Profession (f false) tb |
| /-- Specific contract per version, Good row assumed.Pulse.EI at any m scoreboard. |
| If inactive potential, no TC or T services. End serves zeros. -/ |
| structure Covenant (u:Line (q:=q) (w:=w) (h:=h))(p:MintV)(a:ℕ):Prop where |
| θ:∀ v:Fabric u,v.Good→v.activeR→∀ i<Age.L v.m,v.c 1 i=a→ Birth p v i |
| φ:∀ v:Fabric u,v.Good→v.activeR→∀ i<Age.L v.m,v.c 3 i=a→ Mark p v i |
| T0:∀ v:Fabric u,v.Good→v.activeR→v.c 9 0=a→∃ f,v.Tgood p f |
| ψ:∀ v:Fabric u,v.Good→v.activeR→v.govern=true → v.c 12 0=a→ CeilingR p v |
| ω:∀ v:Fabric u,v.Good→v.activeR→v.govern=true →v.c 13 0=a→ Pond.TracedR v |
| R0:∀ v:Fabric u,v.Good→v.c 14 0=a→∃ f,v.Efine p f |
| IN:∀ v:Fabric u,v.Good→ v.t.main=true→ (∃ k,v.t.tag=.score k) →v.c 16 0=a→Pond.TracedE p v |
| ζ:∀ v:OrbitV u,v.estate→v.works→∀ i<Age.L v.m,v.channel 5 i=a→∀ x∈u.D.P,v.Earn p i x |
| ε:∀ v:OrbitV u,v.estate→v.channel 17 0=a→v.FEnd |
|
|
| structure FullV (u:Line (q:=q) (w:=w) (h:=h))(p:MintV)(i:ℕ):Prop where |
| old:Covenant u p i |
| wick:u.wick0.Veil u.torch i |
| ear:u.D.Ears0 i |
| def Promised (u:Line (q:=q) (w:=w) (h:=h))(p:MintV)(i:ℕ):Prop:= |
| ∀ j,u.D.active j→u.D.W j<u.D.W i→FullV u p j |
|
|
| lemma glacial (v:Fabric u): delta0 (bitR v.t) ↔ v.govern=true:=by |
| unfold Line.Fabric.govern bitR delta0 |
| cases v.t.tag |
| all_goals simp [shaped,Kind.narrow] |
| all_goals split <;> simp_all |
| lemma ruddy(v:Fabric u) {k}(h:v.t.main=true)(hp:v.t.tag=.score k):redV (bitR v.t):=by |
| unfold redV bitR; rw [h,hp]; exact ⟨rfl,rfl⟩ |
| lemma haze (v:OrbitV u)(a i:ℕ):(uphold (v.channel a i)).1=a:=by |
| simp [uphold,OrbitV.channel,vecL,clr,Foil.kit] |
|
|
| namespace Covenant |
| variable {p:MintV} {k:ℕ}(h0:TrustQ u p k) |
| include h0 |
| omit h0 in |
| lemma mute {a:ℕ}(_:u.D.active a)(hk:a≠0)(j:ℕ)(hQ:(uphold a).1=j) |
| (hh:¬ (j=1∨j=3∨j=9∨j=12∨j=14)) : |
| u.D.Ears0 a:=Shift.free u hk hQ (fun i t=>by unfold taps colorQ omegaQ;omega) (by omega) |
| |
| lemma fandom (w:Fabric u)(i j:ℕ)(h:u.D.active (w.c i j))(hp:Covenant u p (w.c i j)): |
| u.D.Ears0 (w.c i j):=by |
| have hs:=h0.chart |
| have hf:= hs.rent h |
| obtain ⟨h1,h2,h3,h4⟩:= ((Line.Chart.courseR (u := u)) w.dye).1 h |
| have hz:w.Good:= ⟨h1,h4⟩ |
| have hc : i=1∨i=3∨i=9 → w.activeR:=by |
| intro he |
| apply w.workR.2 |
| rcases h3.2 with hk|hk; exact hk.2 |
| omega |
| have hh:i=1∨i=3→j<Age.L w.m:=by |
| intro hi; rwa [Period,ite_eq_left (by unfold Brole;omega)] at h2 |
| have hh':i≥9→j=0:=by |
| intro hi; rw [Period,ite_eq_right (by unfold Brole;omega)] at h2; omega |
| by_cases h:i=1 |
| · subst i; apply Shift.earsTag u w 1 j (by omega) |
| intro x hx i hi |
| obtain ⟨b,hb,_⟩:=hp.θ w hz (hc (by omega)) j (hh (by omega)) rfl x hx |
| exact ⟨_,hb.found.next i hi⟩ |
| by_cases h':i=3 |
| · subst i; apply Shift.earsTag u w 3 j (by omega) |
| intro x hx i hi |
| obtain ⟨b,hb,_⟩:=hp.φ w hz (hc (by omega)) j (hh (by omega)) rfl x hx |
| exact ⟨_,hb.found.next i hi⟩ |
| by_cases hT:i=9 |
| · have ht:=hh' (show 9 ≤ i from by omega) |
| subst i; subst j |
| obtain ⟨f,he⟩:=hp.T0 w hz (hc (by omega)) rfl |
| exact Shift.terminal u w p f he |
| by_cases hR:i=12 |
| · have ht:=hh' (show 9 ≤ i from by omega) |
| subst i;subst j |
| have hk:w.govern=true:=by |
| apply (glacial w).1 |
| have hm:=h3.2 |
| unfold Bob at hm; grind |
| apply Shift.court u w hf |
| intro i hi; by_cases hQ:w.activeR |
| · obtain ⟨f,tb,h1,h2⟩:=hp.ψ w hz hQ hk rfl |
| have hc:1 ≤ clim u.w0.p:=by unfold clim; omega |
| exact ⟨_,(h2.guide 0 (mem_range.2 (Nat.zero_lt_succ _)) i hi 1 Nat.zero_lt_one hc).2.1⟩ |
| exact ⟨_,Line.Fabric.desert12 i hz hQ hf hs h0.capdraw⟩ |
| by_cases hU:i=14 |
| · have ht:=hh' (show 9 ≤ i from by omega) |
| subst i;subst j |
| obtain ⟨f,he⟩:=hp.R0 w hz rfl |
| exact Shift.eventE u w p f he |
| apply mute ‹u.D.active (w.c i j)› (Age.cpos _) i (Stone.roleR ..) |
| omega |
|
|
| lemma orchard {a:ℕ}(h:Covenant u p a)(he:u.D.active a):u.D.Ears0 a:=by |
| by_cases hk:a=0 |
| · subst a; exact Shift.basic u h0.chart |
| rcases (Line.Chart.parse (u := u)) he hk with ⟨P,m,i,j,w,rfl⟩|⟨P,m,i,j,w,rfl⟩ |
| · exact fandom h0 (⟨P,m,w⟩:Fabric u) i j he h |
| let f:OrbitV u:=⟨m,P,w⟩ |
| apply mute he hk i (haze f ..) |
| have h4:= (((Line.Chart.courseV (u := u)) w).1 he).2.2.1.2; omega |
| end Covenant |
| |
| namespace Promised |
| variable {p:MintV}{i:ℕ}(hg:Promised u p i) |
| include hg |
| lemma seeking:u.D.Heard i:=fun _ he hi _=> (hg _ he hi).ear |
| lemma sooner {b}(hi:u.D.active b)(hp:u.D.W b+u.D.D i ≤ u.D.W i)(he:u.D.Suite i):FullV u p b:= |
| hg b hi (by have := he.well.1; omega) |
| end Promised |
| end Elect |
| |
| end |
| end ExactDerandomization.Umbra |
| |
| end |
| |
| section |
| |
| namespace ExactDerandomization.Umbra |
| open Analysis Relay Counter Rill Fret At Hie |
| open Arena Amber Song Fret.Lyric Reed Lu Est Net |
| open Grain Terrain Bank Helix Helix.Age Helix.Mold Helix.Foil Helix.Moon Helix.Orbit Chime Dome Scheme Finset Stage |
| open SiteX Line Fret.Seats |
| open Vessel Gamut Spring |
| noncomputable section |
| open scoped Classical |
| variable {q w h:ℕ}{u:Line (q:=q) (w:=w) (h:=h)} |
| open Elect |
| namespace TrustQ |
| variable {p:MintV}{k:ℕ}(hQ:TrustQ u p k) |
| include hQ |
|
|
| lemma stara (v:Fabric u) (i:ℕ) (hp:v.Good)(hi:v.activeR)(hj:i<Age.L v.m) |
| (hc:Promised u p (v.c 1 i)):Birth p v i:=by |
| let e:Chron.Moon u:=⟨p,v,i,Codex.mdi p u.r u.v⟩ |
| have ht:=hQ.owns v hp |
| have hs:= (ht.midUnits hi 1 i (Or.inl ⟨by omega,by omega,hj⟩)) |
| have he:e.Lo:= ⟨ht,hi,hj,hQ.stock.symm,hQ.plan.funds.h1,hQ.pearlR v 1 i hs.stage (by omega) |
| (by intro j;omega), |
| fun b h hh=>(hc.sooner h hh hs).wick⟩ |
| have hS:e.Spire:=⟨he,rfl,hQ.plan.funds.h2⟩ |
| have hf:Chron.Moon.Weather.Crude e:=by |
| refine ⟨he,?_,?_⟩ |
| · intro a h hk |
| have hl:u.D.W (a.c 14 0)+u.D.D (v.c 1 i) ≤ u.D.W (v.c 1 i):=by rw [Stone.W,Stone.D,Stone.W];exact Age.renewal hk |
| have ha:= (hQ.owns a h).endUnits 14 (Or.inl rfl) |
| exact (hc.sooner ha.stage hl hs).old.R0 a h rfl |
| intro t m w hl hy hk |
| let x:OrbitV u:=⟨m,t,w⟩ |
| have hf:x.estate:=⟨hl,hy⟩ |
| have ha: u.D.W (x.channel 17 0)+u.D.D (v.c 1 i) ≤ u.D.W (v.c 1 i):=by |
| rw [show u.D.W (x.channel 17 0)=Age.weight m 17 0 from u.rolfe ..,Stone.D,Stone.W] |
| exact Age.renewal hk |
| exact (hc.sooner (hQ.ownsV x hf).unitV.stage ha hs).old.ε x hf rfl |
| apply e.nursery hf hS |
| intro x hx j he1 |
| have hh:i=j+1:=he1 |
| have hl:j+1<Age.L v.m:= hh ▸hj |
| have hm:u.D.W (v.c 3 j)+u.D.D (v.c 1 i) ≤ u.D.W (v.c 1 i):=by |
| rw [Stone.W,Stone.D,Stone.W,hh];exact Age.oldpay hl (by decide) (by decide) |
| have hk:= (hc.sooner (ht.midUnits hi 3 j (Or.inl ⟨by decide,by decide,by omega⟩)).stage hm hs).old |
| obtain ⟨a,ha,hf⟩:=hk.φ v hp hi j (by omega) rfl x hx |
| exact ⟨a,ha⟩ |
|
|
| lemma starb (v:Fabric u)(i:ℕ)(hp:v.Good)(hi:v.activeR)(hj:i<Age.L v.m) |
| (hc:Promised u p (v.c 3 i)):Mark p v i:=by |
| intro x hx |
| let s:Tiber.SliceD.Fire u:=⟨p,v,i,x⟩ |
| have ht:=hQ.owns v hp |
| have h2(k:ℕ)(he:k=1∨k=2∨k=3):= ht.midUnits hi k i (Or.inl ⟨by omega,by omega,hj⟩) |
| have hs:=h2 3 (by omega) |
| have h1(k:ℕ)(hh:k=1∨k=2):FullV u p (v.c k i):=by |
| apply hc.sooner (h2 k (by omega)).stage ?_ hs |
| rw [Stone.W,Stone.D,Stone.W] |
| apply Age.sooner ?_ (by omega) (by omega) |
| rcases hh with hh|hh <;> rw [hh] <;> rfl |
| obtain ⟨f,hf,he⟩:=(h1 1 (by omega)).old.θ v hp hi i hj rfl x hx |
| apply Tiber.SliceD.Fire.Bakin.beam0 (v:=s) (f:=f) |
| · exact ⟨⟨ht,hj,hi,hx,(h1 2 (by omega)).wick,hQ.plan.funds.h3⟩,hf,he⟩ |
| exact hQ.pearlR v 3 i hs.stage (by omega) (by intro i;omega) |
| |
| lemma stard (v:Fabric u)(hp:v.Good)(hi:v.activeR) |
| (hc:Promised u p (v.c 9 0)):∃ f,v.Tgood p f:=by |
| let s:Aurora.Grape.Dew u:=⟨p,v⟩ |
| have ht:=hQ.owns v hp |
| have hs:= ht.midUnits hi 9 0 (Or.inr ⟨rfl,Or.inl (by omega)⟩) |
| have he:s.Ready:=⟨ht,hi,hQ.plan.funds.h4⟩ |
| let j:=s.g |
| have hm:j<Age.L v.m:=by have hh:j+1=Age.L v.m:=(Aurora.Grape.Dew.Ready.now (e := s)); omega |
| apply Aurora.Grape.Dew.Supply.givenE he (hQ.pearlR v 9 0 hs.stage (by omega) (by intro i; omega)) |
| intro x hx |
| have hn :u.D.W (v.c 3 j)+u.D.D (v.c 9 0) ≤ u.D.W (v.c 9 0):=by |
| rw [show u.D.D (v.c 9 0)=1 from (Aurora.Grape.Dew.Ready.last (e := s))]; exact (Aurora.Grape.Dew.Ready.toward (e := s)).2.2 |
| have hl:= hc.sooner (ht.midUnits hi 3 j (Or.inl ⟨by decide,by decide,hm⟩)).stage hn hs |
| obtain ⟨a,h1,hA⟩:= hl.old.φ v hp hi j hm rfl x hx |
| refine ⟨a,he,hx,h1,fun i=>?_⟩ |
| have hh:= hA i; rwa [show gap v.m j=1 from (by unfold gap; rw [show Age.L v.m=j+1 from (Aurora.Grape.Dew.Ready.now (e := s)).symm]; simp),mul_one] at hh |
| |
| end TrustQ |
| |
| end |
| end ExactDerandomization.Umbra |
| |
| end |
| |
| section |
| |
| namespace ExactDerandomization.Umbra |
| open Analysis Relay Counter Rill Fret At Hie |
| open Arena Amber Song Fret.Lyric Reed Lu Est Net |
| open Grain Terrain Bank Helix Helix.Age Helix.Mold Helix.Foil Helix.Moon Helix.Orbit Chime Dome Scheme Finset Stage |
| open SiteX Line Fret.Seats |
| open Vessel Gamut |
| noncomputable section |
| open scoped Classical |
| variable {q w h:ℕ}{u:Line (q:=q) (w:=w) (h:=h)} |
| open Elect |
| namespace TrustQ |
| variable {p:MintV}{k:ℕ}(hr:TrustQ u p k) |
| include hr |
|
|
| omit hr in |
| lemma promiseE {v:Fabric u}{i j:ℕ}(h:Promised u p (v.c j 0))(hi:5 < i)(he:i<j)(hp:j ≤ 64) |
| (h1:u.D.active (v.c i 0)): |
| FullV u p (v.c i 0):= h (v.c i 0) h1 (v.parentL (by unfold Brole;omega) he hp) |
| lemma earlierT (v:Fabric u)(j:ℕ)(h:Promised u p (v.c j 0))(he: v.Good)(hu :v.activeR) |
| (hj:9<j)(hk:j ≤ 64):∃ f,v.Tgood p f := |
| (TrustQ.promiseE h (show 5<(9:ℕ) by omega) hj hk |
| ((hr.owns v he).midUnits hu 9 0 (Or.inr ⟨rfl,Or.inl (by omega)⟩)).stage).old.T0 v he hu rfl |
|
|
| lemma crownC (v:Fabric u)(he:v.Good)(hp:v.govern=true): u.D.Suite (v.c 12 0):= |
| (hr.owns v he).endUnits _ (Or.inr (Or.inr (Or.inl ⟨rfl,(glacial v).2 hp⟩))) |
|
|
| lemma deputed (v:Fabric u)(h:Promised u p (v.c 12 0))(he: v.Good)(hu:v.activeR)(hp:v.govern=true): |
| CeilingR p v:=by |
| obtain ⟨f,hf⟩:=hr.earlierT v 12 h he hu (by omega) (by omega) |
| have hs:=hr.crownC v he hp |
| obtain ⟨g,hg⟩:= Line.Fabric.tribunal (hf.secure false) (hr.grants v) he hs h.seeking hu hr.plan.funds.h5 hr.capdraw |
| exact ⟨f,g,hf,hg⟩ |
|
|
| lemma starR (v:Fabric u)(h:Promised u p (v.c 13 0))(he:v.Good)(hu:v.activeR)(hp:v.govern=true): |
| Pond.TracedR v:=by |
| obtain ⟨f,hf⟩:=hr.earlierT v 13 h he hu (by omega) (by omega) |
| have hq:=hr.insures v he |
| exact hq.echoR hf (hq.iris.midUnits hu _ _ (Or.inr ⟨rfl,Or.inr ⟨rfl,(glacial v).2 hp⟩⟩)) h.seeking |
|
|
| lemma staE (v:Fabric u)(h: Promised u p (v.c 16 0))(he:v.Good) |
| (hk:v.t.main = true) (hh:∃ k,v.t.tag=.score k): |
| Pond.TracedE p v:=by |
| obtain ⟨i,hi⟩:=hh |
| have hq:=hr.insures v he |
| obtain ⟨f,hf⟩:= ((TrustQ.promiseE h (show 5< (14:ℕ) by omega) (by omega) (by omega) |
| (hq.iris.endUnits _ (Or.inl rfl)).stage).old.R0 v he rfl) |
| exact hq.echoE hf (hq.iris.endUnits _ (Or.inr (Or.inr (Or.inr ⟨rfl,ruddy v hk hi⟩)))) h.seeking |
|
|
| open Gyre Gyre.OrbitR |
| abbrev pr(t:MintV)(v:Fabric u): Prism u:=⟨v,t⟩ |
| lemma upK (v:Fabric u)(he:v.Good): (pr p v).Ace:=by |
| have hm:= (hr.owns v he) |
| have hq:=hr.plan.funds.h8 |
| have hp:= (Gyre.Term.Budget0.fareE u.r u.v p 0).1 |
| refine ⟨⟨hm,hp.trans hq⟩,hq,?_⟩ |
| exact hr.pearlR v 14 0 (hm.endUnits 14 (Or.inl rfl)).stage (by omega) (by intro i;omega) |
| lemma sovereign(v:Fabric u)(he:v.Good) (hu:v.activeR)(h:Promised u p (v.c 14 0)): |
| ∃ f tb,(pr p v).Enabled f tb:=by |
| have hh:=hr.upK v he |
| cases hp:v.govern |
| · obtain ⟨f,hf⟩:=hr.earlierT v 14 h he hu (by omega) (by omega) |
| refine ⟨f,fun _ _=>0,hh.he,hu,hf,fun ht=>?_⟩ |
| change v.govern=true at ht; simp [hp] at ht |
| obtain ⟨f,g,h1,h2⟩:= ((TrustQ.promiseE h (show 5<(12:ℕ) by omega) (by omega) (by omega) |
| (hr.crownC v he hp).stage).old.ψ v he hu hp rfl) |
| exact ⟨f,g,hh.he,hu,h1,fun _=>h2⟩ |
| open Gyre.OrbitR.Prism |
|
|
| omit hr in |
| lemma hstep(v:Fabric u)(h:v.t.ready u.plotM v.m) : |
| v.t.steps u.plotM ≤ u.w0.p.T:=by |
| have hm:=h.2.2.2.2.2 |
| unfold Quest.steps |
| cases hq : v.t.tag <;> rw [hq] at hm <;> first|exact Nat.zero_le _| skip |
| next b i=> |
| have hp:i< u.w0.p.T:=hm.2 |
| cases b <;> first | exact le_rfl | exact hp |
| lemma umbrella (v:Fabric u)(he:v.Good)(hu:v.activeR)(h:Promised u p (v.c 14 0)) |
| (j:ℕ) (hh:j<heightR (pr p v)) : |
| let e:=pr p v |
| ∀ y∈u.D.P, u.D.Voice (e.stationE j) y (hole 1 0) (u.D.high (e.stationE j)) 0 (e.paintE j y):=by |
| intro e y hy |
| have hm:0<v.m:=by |
| have hq:=hu; unfold Fabric.activeR Net.onQ at hq |
| by_contra hn; simp [Nat.eq_zero_of_not_pos hn] at hq |
| let f:OrbitV u:=⟨e.vale,e.euro j,e.across⟩ |
| have hx:f.m+1=v.m:=by change v.m-1+1=_; omega |
| have hz:e.heightR≤v.t.steps u.plotM:=by |
| unfold heightR;split;exact le_rfl;exact Nat.zero_le _ |
| have hT:j<u.w0.p.T:= lt_of_lt_of_le hh (hz.trans (TrustQ.hstep v he.2)) |
| have hF:f.estate:=by |
| have hL:=he.1;obtain ⟨h1,h2,h3,h4,h5,h6⟩:=he.2 |
| exact ⟨by omega,h1,le_rfl,by change v.t.i+f.m≤u.plotM.roof;omega,rfl,hT⟩ |
| have hJ:=hr.ownsV f hF |
| have ht:= (hr.owns v he).endUnits _ (Or.inl (show (14:ℕ)=14 from rfl)) |
| have hk:u.D.W (f.channel 17 0)+u.D.D (v.c 14 0) ≤ u.D.W (v.c 14 0):=by |
| rw [Vessel.OrbitV.Counsel.costZ,Stone.D,Stone.W]; exact Age.renewal hx.le |
| have hv := (h.sooner hJ.unitV.stage hk ht).old.ε f hF rfl |
| rw [paintE,ite_eq_left (show j<e.heightR from hh)] |
| have hh:e.s.roll e.vale= forecast u.plotM e.vale:=vault _ _ (show e.vale<e.s.m from by change f.m<v.m; omega) |
| rw [hh]; exact hv y hy |
| |
| lemma terrarium (v:Fabric u)(he:v.Good)(h:Promised u p (v.c 14 0)): |
| ∃ f,v.Efine p f:=by |
| let e:=pr p v |
| have hx:= hr.upK v he |
| have ht:=hx.he.season |
| by_cases hu:v.activeR |
| · obtain ⟨f,g,hf⟩:=hr.sovereign v he hu h |
| have hh : Court e f g:=by |
| refine ⟨hf,hx.hC,?_ ,?_ ,fun _ j hj=>hr.umbrella v he hu h j hj⟩ |
| · apply (TrustQ.promiseE h (show 5<(10:ℕ) by omega) (by omega) (by omega) |
| (ht.midUnits hu _ _ (Or.inr ⟨rfl,Or.inl (by omega)⟩)).stage).wick |
| intro hk |
| have ha:= (ht.midUnits hu 13 0 (Or.inr ⟨rfl,Or.inr ⟨rfl,(glacial v).2 hk⟩⟩)) |
| exact ((TrustQ.promiseE h (show 5<(13:ℕ) by omega) (by omega) (by omega) ha.stage).old.ω v he hu hk rfl) |
| exact ⟨_,hh.plants hx⟩ |
| exact ⟨_,hx.water hu hr.capdraw⟩ |
|
|
| end TrustQ |
| end |
| end ExactDerandomization.Umbra |
|
|
| end |
|
|
| section |
|
|
| namespace ExactDerandomization.Umbra |
| open Analysis Relay Counter Rill Fret At Hie |
| open Arena Amber Song Fret.Lyric Reed Lu Est Net |
| open Grain Terrain Bank Helix Helix.Age Helix.Mold Helix.Foil Helix.Moon Helix.Orbit Chime Dome Scheme Finset Stage |
| open SiteX Line Fret.Seats |
| open Vessel Gamut |
| noncomputable section |
| open scoped Classical |
| variable {q w h:ℕ}{u:Line (q:=q) (w:=w) (h:=h)} |
| open Elect |
| namespace TrustQ |
| variable {p:MintV}{k:ℕ}(ht:TrustQ u p k) |
| include ht |
| open Ferry Basket |
| abbrev bread(p:MintV)(v:OrbitV u)(i:ℕ):Basket.Sack u:= ⟨v,p,i,Codex.coins3 p u.r, PrintV.shell u.r p u.v⟩ |
| lemma sunken(v:OrbitV u) (i:ℕ)(hi:i<Age.L v.m)(hv:v.estate)(he:v.works): |
| Ferry.TorchE (bread p v i):=by |
| let s:= bread p v i |
| have hs:= ht.ownsV v hv |
| have hx:=hs.pred0 hi he |
| have hz:s.wallet+s.c ≤ u.v.slope:= ht.plan.funds.h9 |
| refine ⟨⟨hs,he,hi,le_trans (Nat.le_add_right ..) hz,?_,ht.pearlV v 5 i hx.stage (by omega) |
| (by intro i;omega)⟩,rfl,hz,rfl⟩ |
| have hh:=GoodF.estimated p u.r u.v; have hl:GoodF.hour p u.r≤ s.c:=hh.2.2.1 |
| have hM:=Cambria.Era.billed u.r; have hM0:Cambria.Era.tariff u.r ≤ s.c:=le_trans (Nat.le_add_right ..) hl |
| exact ⟨hM.2.2.trans hM0,hM.1.trans hM0,hM.2.1.trans hM0⟩ |
|
|
| lemma relicV(v:OrbitV u)(i:ℕ)(hi:i<Age.L v.m)(hv:v.estate)(he:v.works) |
| (hh:Promised u p (v.channel 5 i)):Ferry.Wells (bread p v i):=by |
| have hb:=ht.sunken v i hi hv he |
| have hp:= hb.may.hc |
| have hc:= hp.pred0 hi he |
| refine ⟨hb,?_,?_,?_⟩ |
| · intro a ha hV |
| have hj:=ht.ownsV a ha |
| have hs:FullV u p (a.channel 17 0):=by |
| apply hh.sooner hj.unitV.stage _ hc |
| rw [show u.D.D (v.channel 5 i)=gap v.m i from Vessel.OrbitV.Counsel.heightZ .., |
| show u.D.W (v.channel 5 i)=Age.weight v.m 5 i from Vessel.OrbitV.Counsel.costZ ..,Vessel.OrbitV.Counsel.costZ] |
| exact Age.renewal (show a.m+slackT v.m 5 i ≤ v.m from hV) |
| exact hs.old.ε a ha rfl |
| · intro a ha hV |
| have hj:=(ht.owns a ha).endUnits _ (Or.inl (show (14:ℕ)=14 from rfl)) |
| exact (hh.sooner hj.stage ((Vessel.OrbitV.Counsel.foreign (v := v)) a hV _ _) hc).old.R0 a ha rfl |
| intro a ha hV ht' ho |
| obtain ⟨j,hj⟩:=ho |
| have hx:= (ht.owns a ha).endUnits 16 (Or.inr (Or.inr (Or.inr ⟨rfl,ruddy a ht' hj⟩))) |
| exact (hh.sooner hx.stage ((Vessel.OrbitV.Counsel.foreign (v := v)) a hV _ _) hc).old.IN a ha ht' ⟨j,hj⟩ rfl |
|
|
| lemma treasuryV (v:OrbitV u) (i:ℕ)(hi:i<Age.L v.m)(hv:v.estate)(he:v.works) |
| (hh:Promised u p (v.channel 5 i)) (x:ℕ)(hx:x∈u.D.P): |
| v.Earn p i x:=by |
| have hb:= ht.relicV v i hi hv he hh |
| have hp:=ht.ownsV v hv |
| apply hb.cockpit x hx |
| intro j hj |
| change i=j+1 at hj; have hk:j<Age.L v.m:=by omega |
| have hm:u.D.W (v.channel 5 j)+u.D.D (v.channel 5 i) ≤ u.D.W (v.channel 5 i):=by |
| rw [Vessel.OrbitV.Counsel.costZ,Vessel.OrbitV.Counsel.heightZ,Vessel.OrbitV.Counsel.costZ,hj] |
| exact Age.oldpay (by omega) (by decide) (by decide) |
| have ha:=hh.sooner (hp.pred0 hk he).stage hm (hp.pred0 hi he) |
| exact ha.old.ζ v hv he j hk rfl x hx |
|
|
| lemma auraV (v:OrbitV u)(hv:v.estate)(h:Promised u p (v.channel 17 0)): |
| v.FEnd:=by |
| have hs:=ht.ownsV v hv |
| have hx:=hs.unitV |
| apply hs.knownEnd ht.plan.funds.he (ht.pearlV v 17 0 hx.stage (by omega) |
| (by intro i;omega)) |
| intro hb j hj |
| have hi:j<Age.L v.m:=by omega |
| have hm:u.D.W (v.channel 5 j)<u.D.W (v.channel 17 0):=by |
| rw [Vessel.OrbitV.Counsel.costZ,Vessel.OrbitV.Counsel.costZ] |
| exact (Age.endbirth hj (by decide) (by decide) (by decide)).2 |
| exact (h _ (hs.pred0 hi hb).stage hm).old.ζ v hv hb j hi rfl |
|
|
| lemma earned {a:ℕ}(h:Promised u p a): |
| Covenant u p a := by |
| refine ⟨fun v hp he i hi hm=> ht.stara v i hp he hi (hm.symm ▸ h), |
| fun v hp he i hi hm=> ht.starb v i hp he hi (hm.symm ▸ h), |
| fun v hp he hm=> ht.stard v hp he (hm.symm ▸ h), |
| fun v hp he hQ hm=> ht.deputed v (hm.symm ▸ h) hp he hQ, |
| fun v hp he hQ hm=> ht.starR v (hm.symm ▸ h) hp he hQ, |
| fun v hp hm=> ht.terrarium v hp (hm.symm ▸ h), |
| fun v hp hx he hm=> ht.staE v (hm.symm ▸ h) hp hx he, |
| fun v he hp i hj hm=> ht.treasuryV v i hj he hp (hm.symm ▸ h), |
| fun v hp hm=> ht.auraV v hp (hm.symm ▸ h)⟩ |
| /-- full version closure. -/ |
| theorem coverage (u0:ℕ) (hx:u.D.active u0): |
| FullV u p u0:=by |
| have hh (j:ℕ):∀ i,u.D.W i=j→ u.D.active i→FullV u p i:=by |
| induction j using Nat.strong_induction_on with |
| | h j ih=> |
| intro i hi he |
| have hp:Promised u p i:= fun b hb hx=> ih _ (hi ▸hx) b rfl hb |
| have hg:=ht.earned hp |
| have hm:= ht.chart.chamber he hp.seeking fun b hb hx _=> (hp b hb hx).wick |
| exact ⟨hg,hm,Covenant.orchard ht hg he⟩ |
| exact hh _ u0 rfl hx |
|
|
| end TrustQ |
| end |
| end ExactDerandomization.Umbra |
|
|
| end |
|
|
| section |
|
|
| namespace ExactDerandomization.Umbra |
| open Analysis Relay Counter Rill Fret At Hie |
| open Arena Amber Song Fret.Lyric Reed Lu Est Net |
| open Grain Terrain Bank Helix Helix.Age Helix.Mold Helix.Foil Helix.Moon Helix.Night Helix.Orbit Chime Dome Scheme Finset Stage |
| open SiteX Line Fret.Seats |
| open Vessel Gamut |
| noncomputable section |
| open scoped Classical |
| variable {q w h:ℕ} |
| open Elect |
| namespace Scope |
| variable (p:MintV) (r:Rock q w h) (v:Tags) |
| abbrev selfS := Codex.pages p r v |
| abbrev Dscope(x:Word):Line (q:=q) (w:=w) (h:=h):=⟨r,v,selfS p r v,x⟩ |
| abbrev prog0 :Cog:=Mold.woven r v (selfS p r v) |
| def cosmos(x:Word)(i:ℕ):ℕ:= |
| enc (Foil.kit 17 (r.w.top (r.B x),0) (bitV (prizep (r.B x) (r.B x))) i) |
| abbrev request (x:Word)(i t:ℕ):ℕ:=post (cosmos r x i) 1 0 t |
| abbrev U (x:Word):ℕ:= Night.fRock r v x+ (2:ℕ)^charge v.slope (planets r x) (Moon.T r v x) |
| abbrev shield(x:Word):ℕ:= Dome.glove Dome.clearG (r.camp x) |
| def study(t:ℕ)(x:Word)(i:ℕ) : Grid.Job:=((U r v x,request r x i t),(0,shield r x)) |
| def chop0(f:PT Grid.Job Site)(x:Word)(i t:ℕ):ℤ:=dec (f x (study r v t x i)).1 |
| def dustF(f:PT Grid.Job Site)(j:ℕ)(x:Word)(i:ℕ):ℤ := |
| ∑ t∈range j,(chop0 r v f x i t)*(2:ℤ)^(6-t) |
| abbrev solve(f:PT Grid.Job Site)(x:Word)(i:ℕ):ℕ:= (dustF r v f 7 x i).toNat |
|
|
| lemma cosmosK:Kn (cosmos r):=by |
| have hp:= r.BKn (α:=ℕ) |
| let s:=(hp.tuple (hp.tuple (Kn.fix ((0,0):Rill.History)))).tuple (Kn.fix ((5,0,false):ChoiceX)) |
| exact Kn.wash.comp ((Kn.fix _).tuple (((topK _ hp).tuple (Kn.fix _)).tuple (s.tuple Kn.identity))) |
| lemma UK {α}[Wire α]:Kn (fun x (_:α)=>U r v x):=by |
| have hf:Kn (fun x (_:α)=>(2:ℕ)^charge v.slope (planets r x) (T r v x)):=by |
| convert ((stationK r v (α:=α)).power v.slope).times (upclock r (v.slope*v.slope)) using 1 |
| ext x a; rw [← pow_mul,← pow_add]; congr 1; unfold charge; ring |
| exact (Night.fRockK r v).add hf |
| lemma studyK(t): Kn (study r v t):=by |
| let a := Kn.wash.comp ((cosmosK r).tuple (Kn.fix (hole 1 0))) |
| let q:=Kn.wash.comp (a.tuple (Kn.fix t)) |
| have hp:= Kn.wash.comp ((r.campK (α:=ℕ)).tuple (Kn.fix clearG)) |
| exact ((UK r v).tuple q).tuple ((Kn.fix 0).tuple hp) |
| lemma shake {f:PT Grid.Job Site}(hf:Kn f):Kn (solve r v f):=by |
| have hs(j:ℕ):Kn (dustF r v f j):=by |
| induction j with |
| | zero=>have hh:dustF r v f 0=fun _ _=>0:=by ext; simp [dustF] |
| exact hh ▸ Kn.fix _ |
| | succ j ih=> |
| have he:= (Kn.unpack (α:=ℤ)).comp (hf.comp (studyK r v j)).left |
| convert ih.plusZ (he.prodZ (Kn.fix ((2:ℤ)^(6-j)))) using 1 |
| ext x a;unfold dustF; rw [sum_range_succ]; rfl |
| exact (hs 7).topNat |
| |
| structure Sane (r:Rock q w h)(p:MintV)(v:Tags)(k:ℕ) : Prop where |
| funds:Codex.Fortune r p v k |
| stake:6 ≤ v.extra |
| proofM :∀ x, (Dscope p r v x).Chart |
|
|
| abbrev Page(p:MintV)(r:Rock q w h)(v:Tags)(x:Word)(s:Pool (r.n x)):OrbitV (Dscope p r v x):= |
| ⟨r.w.top (r.B x),prizep (r.B x) (r.B x),s⟩ |
| namespace Sane |
| variable {r:Rock q w h} {p:MintV}{v:Tags}{k:ℕ} (he:Sane r p v k) |
| include he |
| omit he in |
| lemma stableQ (x:Word) (s:Pool (r.n x)): |
| let u:=Dscope p r v x |
| let P:=Page p r v x s |
| P.m=u.limitM ∧ P.estate:=by |
| intro u P |
| exact ⟨rfl,le_rfl,(r.matter0 x).reachesB⟩ |
| lemma swept(x:Word): TrustQ (Dscope p r v x) p k:=⟨he.funds,rfl,he.proofM x⟩ |
| omit he in |
| lemma date0 (x:Word)(s:Pool (r.n x)): |
| cosmos r x (r.printid x s)=(Page p r v x s).channel 17 0:=rfl |
| abbrev stalkC (r:Rock q w h)(x:Word)(s:Pool (r.n x)):ℕ:= cosmos r x (r.printid x s) |
| lemma suiteV(x:Word)(s:Pool (r.n x)): |
| (Dscope p r v x).D.Suite (stalkC r x s) := |
| ((he.swept x).ownsV (Page p r v x s) ((stableQ (r := r) (p := p) (v := v)) x s).2).unitV |
|
|
| lemma calend (x:Word)(s:Pool (r.n x)): |
| 6 ≤ (Dscope p r v x).D.high (stalkC r x s):=by |
| let u:= Dscope p r v x |
| let a:= stalkC r x s; let D:=u.D |
| have hz := he.suiteV x s |
| have hm :D.W a ≤ planets r x:= by |
| have ht:=he.proofM x |
| exact Orbit.licensedW _ ((Line.Chart.auth (u := u) a).1 hz.stage) |
| have hs:= he.stake |
| change 6 ≤ T r v x-v.slope*D.W a |
| have hh:=Nat.mul_le_mul_left v.slope hm |
| rw [T]; rw [Nat.mul_add]; omega |
| |
| lemma money0 (x:Word)(s:Pool (r.n x)) (i:ℕ)(hi:i ≤ 6): |
| let u:= Dscope p r v x; let Q:= request r x (r.printid x s) i |
| (2:ℕ)^(u.D.F.meter Q) ≤ U r v x ∧ u.D.F.limit ≤ U r v x:=by |
| intro u Q |
| have hs:=he.proofM x; let a:=stalkC r x s |
| have h0:=he.suiteV x s |
| have hm:u.D.W a ≤ planets r x:= Orbit.licensedW _ ((Line.Chart.auth (u := u) a).1 h0.stage) |
| have ht:i ≤ T r v x:=cap_time r v x u.prog a i (hi.trans (he.calend ..)) |
| refine ⟨le_trans (b:=(2:ℕ)^charge v.slope (planets r x) (T r v x)) ?_ (Nat.le_add_left ..), |
| Nat.le_add_right ..⟩ |
| apply Nat.pow_le_pow_right (by omega) |
| rw [h0.shape.rateC] |
| rw [show u.D.l.book Q=⟨u.D.W a,i⟩ from h0.shape.bookB _ _ _] |
| unfold charge; dsimp only; change v.slope*i+v.slope*v.slope*u.D.W a ≤_; gcongr |
| |
| def Agent(f:PT Grid.Job Site)(r:Rock q w h)(p:MintV)(v:Tags):Prop:= |
| ∀ {x U F q a b}, let c:=prog0 p r v; |
| (2:ℕ)^(Fare.meter F q) ≤ U → Fare.limit F ≤ U → Boots x F c q c a b → |
| f x ((U,q),a)=b |
| lemma bells0 (x:Word)(s:Pool (r.n x))(hS:r.P.Space){ |
| f:PT Grid.Job Site}(hf:Agent f r p v): |
| let m:=r.matter0 x |
| let Z:=m.oracleV s m.i |
| |(dustF r v f 7 x (r.printid x s):ℝ)- (2:ℝ)^6*Z| ≤ 2:=by |
| intro m Z |
| let u:= Dscope p r v x; let a:=stalkC r x s; let D:=u.D |
| let P:=Page p r v x s |
| have hv:=he.suiteV x s |
| have hs:=he.proofM x |
| have hZ : m.i ∈ u.D.P:=by |
| have hh:=hs.sc |
| have hq:= hh.ids (r.B x) le_rfl (m.origin hS) |
| exact mem_range.2 ((mem_range.1 hq).trans (by change r.n x<r.n x+1; omega)) |
| have hk:P.FEnd:= ((he.swept x).coverage a hv.stage).old.ε P ((stableQ (r := r) (p := p) (v := v)) x s).2 rfl |
| obtain ⟨w,hb,hP⟩:= hk m.i hZ |
| have ht:Checked w Z (u.D.high a):=by |
| change Checked w (Z/(2:ℝ)^0) (u.D.high a) at hb; simpa only [pow_zero,div_one] using hb |
| have h6 :6 ≤ u.D.high a := he.calend x s |
| have hh:dustF r v f 7 x (r.printid x s)=w 6:=by |
| unfold dustF; rw [←sumsign w 6]; apply sum_congr rfl |
| intro i hi; have hi0:i≤6:=Nat.le_of_lt_succ (mem_range.1 hi) |
| have heE:= hP i (hi0.trans h6) Dome.clearG u.D.cleared |
| have hv:= he.money0 x s i hi0 |
| have hp:=hf hv.1 hv.2 heE |
| have hl: f x (study r v i x (r.printid x s))=(enc (dsign w i), Dome.glove Dome.clearG m.i):=by |
| unfold study shield; rw [r.camps x]; exact hp |
| unfold chop0; rw [hl]; dsimp only; rw [dec_round] |
| rw [hh]; exact ht.2 _ h6 |
| end Sane |
| |
| end Scope |
| end |
| end ExactDerandomization.Umbra |
| |
| end |
| |
| end OAI |
| |
