| import OAI.Computability.Logspace.RowEvaluation |
|
|
| namespace OAI |
|
|
| /-! |
| Programs evaluating transitions and the next level of the probability hierarchy. |
|
|
| From OpenAI, *Exact Derandomization of Logarithmic Space: L = RL = BPL*. |
| -/ |
|
|
| set_option autoImplicit true |
|
|
| section |
|
|
| namespace ExactDerandomization.Tiber |
| open Analysis Counter Relay Rill Fret Finset Mint Swirl Grain Net At Est Hie Stage |
| open Helix Helix.Foil Helix.Age Helix.Moon Helix.Night Helix.Mold Helix.Orbit Chime Scheme |
| open Dome Cone ConeX Glass Glass.Rod |
| open Reed Fret.Seats SiteX Terrain Bank Arena Song |
| noncomputable section |
| open scoped Classical |
| variable {q w h:ℕ} {u:Line (q:=q) (w:=w) (h:=h)} |
| namespace Sky |
|
|
| def whichSide (b:Bool) (x:Word)(a:ℕ):ℕ:= if b then (V x a).2.1 else (V x a).2.2 |
| lemma whoK(b):Kn (whichSide b):=by cases b;exact VK.right.right;exact VK.right.left |
| def pre (k o:ℕ) (f:Cub ℕ ℕ)(x:Word)(a:ℕ):ℕ:= |
| hole (pointer (V x a).1 k) (hole o (f.f x a)) |
| abbrev pK(k:ℕ):Kn (fun x a=>pointer (V x a).1 k):= |
| Kn.wash.comp ((Kn.fix k).tuple ((Kn.unpack (α:=Crt)).comp VK.left).right) |
| def seek (k o:ℕ)(f:Cub ℕ ℕ): Sight:= |
| ⟨_,Kn.wash.comp ((pK k).tuple (Kn.wash.comp ((Kn.fix o).tuple f.runs)))⟩ |
| abbrev tag(k o:ℕ)(b:Bool):Cog:=Dome.Bullet.snap (seek k o ⟨_,whoK b⟩) |
|
|
| def rocket (_:Rock q w h)(k : Mode)(b:Bool):Sight:= |
| let v:NoteM:= Helix.Torch.waxy (Helix.Torch.charm k b) |
| ⟨_,Kn.wash.comp ((pK 2).tuple (Kn.wash.comp ((whoK (pivot k)).tuple (Kn.fix v))))⟩ |
| def globe (r:Rock q w h) :Glass.Eye:= ⟨⟨_,r.nKn⟩,fun k b=>Bullet.snap (rocket r k b)⟩ |
| def queryEq (r:Rock q w h) :Cog:= |
| let a:= globe r |
| let u:= Flash.plumb 1 |
| let b:= tag 1 0 |
| let f:=Flash.demand (b false) (a.nexts 4 u Cargo.knock) |
| Flash.demand (b true) f |
|
|
| def rimN (k:Knob)(u:ℕ):Sight:=⟨_,Kn.power u k.bank.runs⟩ |
| def spilt (p:Knob)(i:ℕ):Cub CoinQuery Coin:= |
| ⟨_,Coin.invK ((scaleK (campk p) i).comp Kn.first)⟩ |
|
|
| variable (p:MintV) (r:Rock q w h) (v: Tags) |
| def stadium: Board:= |
| let k:= Mold.knob r v |
| { n:=rimN k p.vol |
| I:=k, kth:=fun b=> if b then ⟨_,Kn.fix r.w.p.K⟩ else rimN k (Rain.CarryE r.w) |
| reserveC:=Brick.solo (campk k) (spilt k (fexpo 2)) p.inputH |
| equal0:=queryEq r |
| pigment:= tag 1 0 true |
| baseC:=Bullet.snap (seek 1 5 ⟨_,Kn.fix 0⟩) |
| pot:=fun b=>tag 1 (if b then 1 else 2) false |
| inp:=p.inputH,lid:=p.sealR |
| lips:= fexpo 2,cup:=p.vol+1 } |
| abbrev wafer(k):= stadium p r k |
| def bodyP (i:ℕ):Cog:= |
| let s:=stadium p r v |
| if i=0 then s.shift else if i=1 ∨ i=2 then s.rateMood (i==1) else s.primary |
| abbrev progP := Studio.arrange (bodyP p r v) 5 |
| abbrev coverC :ℕ:=p.vol+Cambria.Era.tariff r+4 |
|
|
| abbrev baseC :Board:= stadium p r ⟨0,0,0,0,0,0⟩ |
| def festival (w:Board)(c:ℕ):ℕ:=c+w.prompt+w.totalC c |
| def budget0 :ℕ:= |
| let w:=baseC p r; let c:=coverC p r; festival w c |
| namespace Estate |
| open Board Fret.Lyric Cambria.Wear.Paper Fret.Lyric Holds Lu |
| lemma pile (b:Bool)(p q:Branch)(he:p.e=q.e){f g:Lyric}(h:same f g): |
| same (.heap b p f) (.heap b q g):=by |
| obtain ⟨ha,hb,hc⟩:=h |
| simp only [Cambria.Wear.Paper.same,Lyric.weight,Lyric.tail,Lyric.cost,he,ha,hb,hc,Branch.reserve,Branch.foldC]; simp |
| |
| lemma sta {f:Board}{g:Board} |
| (hi:f.inp=g.inp)(hl:f.lid=g.lid)(hc:f.cup=g.cup)(he:f.lips=g.lips): |
| f.prompt = g.prompt ∧ ∀ c,f.totalC c =g.totalC c:=by |
| have h1(b:Bool):same (f.rowB b) (g.rowB b):=by |
| cases b |
| · exact oneself _ |
| unfold rowB; rw [hi]; exact use _ _ _ |
| have h2(b:Bool):same (f.bites b) (g.bites b):=by |
| refine (h1 b).plies (pile _ _ _ hc ?_) _ |
| rw [hi]; exact use _ _ _ |
| have hX {x y:Lyric}(h:same x y):same (Lu.absE x) (Lu.absE y):= |
| h.plies ((oneself _).plies h _) _ |
| have h3:same f.sight g.sight:= (hX (h2 true)).plies (hX (h2 false)) _ |
| have h4(b):same (f.starsR b) (g.starsR b):=by |
| refine (h1 true).plies (pile _ _ _ hc ((use _ _ _).plies ?_ Doop.prod)) _ |
| cases b; exact h3; exact h2 true |
| have h5:same f.riseR g.riseR:=by |
| apply same.plies (oneself _) |
| (same.plies (show same (.use f.inp f.reserveC) (.use g.inp g.reserveC) |
| from by rw [hi];exact use ..) (h4 false) Doop.greater) |
| have h6:same f.shore g.shore:= |
| (oneself _).plies ((oneself _).plies (h3.plies ((oneself _).plies h5 _) _) _) _ |
| have hz:same f.zenith g.zenith:= |
| (oneself _).plies (h5.plies ((h4 true).plies ((oneself _).plies h5 _) _) _) _ |
| have h7(b):same (f.border b) (g.border b):=by cases b; exact oneself _; exact hz |
| have hu:same f.harsh g.harsh := ((oneself _).plies h3 _).plies ((oneself _).plies h5 _) _ |
| have OK{x y u v:Lyric}(h:same x y)(e:same u v) : same (Lu.clippy x u) (Lu.clippy y v) := |
| h.plies (((oneself _).plies h _).plies e _) _ |
| have h8(b):same (f.surf b) (g.surf b):= (h7 b).plies (OK h6 ((h2 b).plies (h7 b) _)) _ |
| have hg:same f.topE g.topE:=by unfold Board.topE; refine ?_; cases f; cases g; dsimp only at hl; subst_vars; exact everF |
| have hr(b):same (f.guardE b) (g.guardE b):=by unfold guardE; rw [hl]; exact (OK hg (h8 b)).stow _ |
| have hy(b):same (f.borderE b) (g.borderE b):=by unfold borderE;rw [hl];exact (h7 b).stow _ |
| have hm(b): f.hills b=g.hills b:=by unfold hills; rw [(hr b).2.1,(hy b).2.1] |
| have hV(c b): f.vibes c b=g.vibes c b:=by unfold vibes; rw [(hr b).2.2,(hy b).2.2] |
| refine ⟨?_,fun c=>?_⟩ |
| · unfold Board.prompt Board.burn Test.extra Test.draw |
| rw [hm,hm,h5.1,h5.2.1,he,hu.1,hu.2.1] |
| unfold Board.totalC tuneC payC Board.bankC Badge.tariff Badge.tariffStart Test.seatC |
| change 4*(16*(16*(4*Oven.price (_ + Test.draw f.riseR 0 +100)+4*Oven.price (_ +Test.draw f.harsh f.lips+100)+_))+_+_+_)=_ |
| simp only [Board.heatp,Board.touchp,Test.arm,Test.draw,h5.1,h5.2.2,hu.1,hu.2.2,he,hV] |
| end Estate |
| lemma booster(a b c d e:ℕ)(h:a=d)(hp:b=e) : c ≤ c+d+e∧a ≤ c+d+e∧b ≤ c+d+e:=by omega |
| lemma swamp (f g:Board)(c:ℕ) (hi:f.inp=g.inp)(hl:f.lid=g.lid)(hc:f.cup=g.cup)(he:f.lips=g.lips): |
| c ≤ festival g c∧f.prompt ≤ festival g c∧f.totalC c ≤ festival g c:=by |
| have ⟨h1,h2⟩:=Estate.sta hi hl hc he |
| rw [h1,h2];unfold festival; omega |
| |
| lemma funded : |
| coverC p r ≤ budget0 p r ∧ (stadium p r v).prompt ≤ budget0 p r ∧ |
| (stadium p r v).totalC (coverC p r) ≤ budget0 p r:= |
| swamp (stadium p r v) (baseC p r) (coverC p r) rfl rfl rfl rfl |
| end Sky |
|
|
| namespace SliceD |
| open Line Line.Fabric Sky |
| structure Fire (u:Line (q:=q) (w:=w) (h:=h)) where |
| p:MintV |
| v:Fabric u |
| i:ℕ |
| x:ℕ |
| namespace Fire |
| variable (v:Fire u) |
| abbrev inc := v.v.α v.p v.i |
| abbrev bud:= v.v.β v.i |
| abbrev frame := stadium v.p u.r u.v |
| abbrev cost := coverC v.p u.r |
| abbrev a:= v.bud.dataN |
| abbrev pond:= Studio.dockA u v.a v.x |
| abbrev d := v.pond.d |
| abbrev μ := gap v.v.m v.i |
| abbrev h0 := u.D.high v.a |
| abbrev H := v.h0 + budget0 v.p u.r * v.d |
| abbrev pan(i j):=film v.pond i j |
| structure Lamp0(v:Fire u) : Prop where |
| he:Terrain.Stone.Rector v.v v.p |
| hz :v.i < Age.L v.v.m |
| ha:v.v.activeR |
| hx:v.x∈u.D.P |
| veil:u.wick0.Veil u.torch v.inc.visN |
| heat: budget0 v.p u.r ≤ u.v.slope |
|
|
| namespace Lamp0 |
| variable {v:Fire u} (h0:v.Lamp0) |
| include h0 |
|
|
| lemma againC : v.inc.Front u:= h0.he.frontI h0.ha v.i h0.hz |
| lemma primal : v.bud.Front u := h0.he.frontP h0.ha v.i h0.hz |
| lemma aimD : u.D.Suite v.a := h0.he.chart.rent h0.primal.pos |
| omit h0 in |
| lemma jump : u.D.D v.inc.dataN=v.d ∧ u.D.W v.inc.visN+v.d ≤ u.D.W v.a ∧ u.D.W v.inc.dataN+v.d ≤ u.D.W v.a:= by |
| change u.D.D (v.v.c 1 v.i)=u.D.D (v.v.c 3 v.i)∧ u.D.W (v.v.c 2 v.i)+u.D.D (v.v.c 3 v.i) ≤ u.D.W (v.v.c 3 v.i)∧ |
| u.D.W (v.v.c 1 v.i)+u.D.D (v.v.c 3 v.i)≤_ |
| rw [Terrain.Stone.D,Terrain.Stone.D,Terrain.Stone.W,Terrain.Stone.W,Terrain.Stone.W] |
| exact ⟨rfl,Age.sooner rfl (by omega) (by omega), Age.sooner rfl (by omega) (by omega)⟩ |
| omit h0 in |
| lemma price :4 ≤ v.cost∧ Bullet.roomS ≤ v.cost∧powder 100≤ v.cost ∧ Cambria.Era.tariff u.r ≤ v.cost:=by |
| have hr:= Cambria.Era.billed u.r |
| unfold cost coverC;omega |
| omit h0 in |
| lemma height0 : v.d=v.μ:= by |
| change u.D.D (v.v.c 3 _) =_ |
| rw [Terrain.Stone.D];rfl |
| lemma stored(i j) : Reed.Lu.Standard (v.pan i j) (campk v.frame.I) v.H:=by |
| have ht:v.H ≤ u.D.H:= Studio.stayedC h0.aimD le_rfl h0.heat |
| exact ⟨Studio.positioned h0.aimD h0.hx i j,spanFilm (Studio.vital (u:=u) v.H ht) i j⟩ |
| omit h0 in |
| lemma urn (i j) (b:Bool)(k): |
| (seek 1 k ⟨_,whoK b⟩).f (v.pan i j).l.x (v.pan i j).info= |
| hole v.inc.dataN (hole k (if b then i else j)):=by |
| change Sky.pre 1 k _ _ _=_ |
| unfold Sky.pre whichSide pan Seats.l film |
| cases b <;> simp only [Reed.V, Seats.info, dec_round] |
| · rw [show pointer v.pond.info 1=v.inc.dataN from Stone.pointing ..] |
| rw [show pointer v.pond.info 1=v.inc.dataN from Stone.pointing ..] |
| |
| omit h0 in |
| lemma quoted(i j)(p:Mode)(b:Bool): |
| (rocket u.r p b).f u.x (v.pan i j).info = |
| hole v.inc.visN (enc ((if pivot p then i else j),Helix.Torch.waxy (Helix.Torch.charm p b))):=by |
| let z:=v.pan i j |
| change hole (pointer (Reed.V u.x z.info).1 2) (enc (whichSide (pivot p) u.x z.info , _))=_ |
| have hp:Reed.V u.x z.info=(v.pond.info,i,j):=dec_round .. |
| rw [show whichSide _ u.x z.info= (if pivot p then i else j) from ?_,hp] |
| · rw [show pointer v.pond.info 2=v.inc.visN from Stone.pointing ..] |
| unfold whichSide;rw [hp] |
| |
| omit h0 in |
| lemma moderate (p:Mode)(b:Bool):u.wick0.Ok (Helix.Torch.waxy (Helix.Torch.charm p b)):=by |
| have hi:=NeZero.pos u.n |
| apply And.intro _ ⟨hi,Nat.zero_le _⟩ |
| exact Helix.Torch.housed u.r u.v u.x (by change Helix.Torch.innate p ≤ 16; cases p<;>decide) (Nat.zero_le _) (Nat.zero_le _) |
| (Nat.zero_le _) (Nat.zero_le _) (Nat.zero_le _) |
| end Lamp0 |
| end Fire |
| end SliceD |
| end |
| end ExactDerandomization.Tiber |
| |
| end |
| |
| section |
| |
| namespace ExactDerandomization.Tiber.SliceD.Fire |
| open Analysis Counter Relay Rill Fret Finset Mint Swirl Grain Net At Est Hie Stage |
| open Helix Helix.Foil Helix.Age Helix.Moon Helix.Night Helix.Mold Helix.Orbit Chime Scheme |
| open Dome Cone ConeX Glass Glass.Rod |
| open Reed Fret.Seats SiteX Terrain Bank Arena Song |
| open Tiber.Sky Line Line.Fabric |
| noncomputable section |
| open scoped Classical |
| variable {q w h:ℕ} {u:Line (q:=q) (w:=w) (h:=h)} |
| variable {v: Fire u} |
| structure Bindq (q:v.v.Yolk v.p v.i v.x):Prop where |
| lamp:v.Lamp0 |
| money:v.v.YFine v.p v.i v.x q |
| tagB:∀ i,q.raw.tag i ≤ Cambria.Era.budget u.r |
|
|
| namespace Bindq |
| variable {f:v.v.Yolk v.p v.i v.x} (hf:Bindq f) |
| include hf |
| lemma fits (i:ℕ)(hi:i<f.raw.m):u.smallQ v.inc.dataN i:= hf.lamp.againC.awake (hf.lamp.againC.size0 _ (by rwa [← hf.money.ns])) |
| lemma trunk(i a b:ℕ):Punct (v.pan a b).head v.cost (f.raw.tag i):=by |
| have he := hf.tagB i; have ht:= lb_le_inc (f.raw.tag i) |
| have hp:4+Cambria.Era.budget u.r ≤ v.cost:=by |
| have hz:=(Lamp0.price (v := v)).2.2.2; unfold Cambria.Era.tariff at hz; omega |
| have hd:1≤ v.d:= hf.lamp.aimD.well.1; change _ +_ ≤ _ *v.d;nlinarith |
| |
| lemma stepC (b:Bool) (i j:ℕ)(hi:(if b then i else j)< f.raw.m): |
| (v.pan i j).rite v.cost (tag 1 0 b) (f.raw.tag (if b then i else j)):=by |
| let n:= if b then i else j |
| have ht:= hf.lamp |
| have hq:=ht.againC |
| have hs:u.smallQ v.inc.dataN n:=hf.fits _ hi |
| have hm:=hf.money.found.next n hs |
| apply Bullet.copy u.D ht.aimD ((by omega:2 ≤ 4).trans (Lamp0.price (v := v)).1) ht.hx (Lamp0.jump (v := v)).2.2 _ (hf.trunk n i j) hm |
| exact (Lamp0.urn (v := v)) i j b 0 |
|
|
| lemma glows (m:Mode) (b:Bool) (i j:ℕ) (hi:(if pivot m then i else j) < f.raw.m) |
| (hv:f.raw.valid (if pivot m then i else j)){c:Pane}(hz:u.D.r.proper c): |
| let x:=v.pan i j; let q:= x.seat (glove c v.x) |
| let e:= f.raw.place (if pivot m then i else j) |
| q.comp 0 v.cost ((globe u.r).visit m b) (0,q.src) (0,glove (pigment u.D.r m b e c) v.x):=by |
| dsimp only |
| let n:=if pivot m then i else j; let c' := Helix.Torch.charm m b |
| let p:= Helix.Torch.waxy c' |
| let e:= f.raw.place n |
| let D:=u.D |
| have hT:= hf.money.found |
| have hs:= hf.lamp; have ht:= (Lamp0.moderate (u := u)) m b |
| have hk:= hT.road n (hf.fits n hi) |
| rw [ite_eq_right hv] at hk |
| have hc := hs.veil _ _ ht v.x _ hk _ hz |
| let f':= pigment D.r m b e c |
| have ho:e ≤ u.n:=Nat.le_of_lt_succ (mem_range.1 (hT.places n hv (hf.fits _ hi))) |
| have hg:u.torch.lit u.x p e c=(false,f'):=by |
| rw [Helix.Torch.litwax u.r u.v u.x (coneX u.r u.s.pave) c' c e ho,Helix.Torch.litmode] |
| rfl |
| have hb:u.wick0.state u.torch p v.x (some (f.raw.tag n,e)) c=(0,glove f' v.x):=by |
| unfold Harp.Wick.state; change (enc (u.torch.lit u.x p e c).1,glove (u.torch.lit u.x p e c).2 v.x)=_; rw [hg];rfl |
| have hp:D.talk (Harp.quoteQ v.inc.visN n p) (glove c v.x) (0,glove f' v.x):= hb ▸ hc |
| have he:D.r.proper f':=by |
| have he: u.D.r.proper (u.torch.lit u.x p e c).2 := (Helix.Torch.life u.r u.v u.x u.prog |
| (coneX u.r u.s.pave) hs.aimD.shape.price).sober p ht _ |
| (mem_range.2 (Nat.lt_succ_of_le ho)) _ hz |
| rwa [hg] at he |
| have hP:= hs.aimD; have hh: 2 ≤ v.cost:= (le_trans (by decide) (Lamp0.price (v := v)).1) |
| exact Bullet.past D hP hh hs.hx (Lamp0.jump (v := v)).2.1 (Lamp0.quoted (v := v) i j m b) 0 0 v.x hs.hx |
| (Arena.Pier.seal0 _ hP hh (q:=(v.pan i j).info)) hz he hp |
|
|
| set_option maxHeartbeats 800000 in |
| lemma equalityP (i j:ℕ)(hi:i < f.raw.m)(hj:j<f.raw.m): |
| (v.pan i j).rite v.cost (queryEq u.r) (Reed.Board.gcat f.raw i j):=by |
| let s:=v.pan i j |
| let R:= hf.lamp.aimD; have hc : 2 ≤ v.cost:= (le_trans (by decide) (Lamp0.price (v := v)).1) |
| let a:=f.raw.tag i; let b:=f.raw.tag j |
| let x:= f.raw.place i; let y:=f.raw.place j |
| let t :ℕ:= if x=y then 1 else 0 |
| have hp: Reed.Board.gcat f.raw i j = Flash.win a (Flash.win b t):= by |
| dsimp only [Reed.Board.gcat,Flash.win,t,a,b,x,y]; unfold Poolx.pals Poolx.joins Poolx.valid; grind |
| rw [hp] |
| apply Flash.wanted u.D R hf.lamp.hx hc a (Flash.win b t) (hf.stepC true i j hi) |
| intro h |
| apply Flash.wanted u.D R hf.lamp.hx hc b t (hf.stepC false i j hj); intro h' |
| rintro n ⟨p,hp,rfl⟩ |
| let r:=globe u.r |
| let q:=postcard s.info 0 |
| let m:Eye.Mantle:=⟨u.D.l,q,⟨u.D.W v.a,0⟩,v.cost,v.d,v.x⟩ |
| have hs:Eye.Twirl r m x y:=by |
| constructor |
| intro e he k b |
| let i':= if pivot k then i else j |
| have hl:i'<f.raw.m:=by unfold i'; split <;> assumption |
| have hv:f.raw.valid i':=by unfold i';split |
| · exact h |
| exact h' |
| have ht:= hf.glows k b i j hl hv he |
| cases hq : pivot k <;> simp only [hq,ite_false,ite_true,Bool.false_eq_true] at * |
| · exact ht |
| exact ht |
| have hx(v:ℕ)(hv:v<f.raw.m)(hm:f.raw.valid v) : |
| f.raw.place v ≤ u.n:= Nat.le_of_lt_succ (mem_range.1 (hf.money.found.places v hm (hf.fits _ hv))) |
| let l:=glove p v.x |
| apply hs.done0 hp (hx i hi h) (hx j hj h') |
| · intro hh |
| change (s.seat l).comp 0 _ _ _ (t,l); rw [show t=1 from ite_eq_left hh] |
| exact Flash.pureQ u.D R hf.lamp.hx hc 1 (Pier.seal1 _ R hc (q:=s.info)) l ⟨p,hp,rfl⟩ |
| intro hh; change (s.seat l).comp 0 _ _ _ (t,l); rw [show t=0 from ite_eq_right hh] |
| exact blankD (R.pit hf.lamp.hx _) hc _ ⟨p,hp,rfl⟩ |
| |
| end Bindq |
| end |
| end ExactDerandomization.Tiber.SliceD.Fire |
| |
| end |
| |
| section |
| |
| namespace ExactDerandomization.Tiber.SliceD.Fire |
| open Analysis Counter Relay Rill Fret Finset Mint Swirl Grain Net At Est Hie Stage |
| open Helix Helix.Foil Helix.Age Helix.Moon Helix.Night Helix.Mold Helix.Orbit Chime Scheme |
| open Dome Cone ConeX Glass Glass.Rod |
| open Reed Fret.Seats SiteX Terrain Bank Arena Song |
| open Tiber.Sky Line Line.Fabric Fret.Lyric |
| noncomputable section |
| open scoped Classical |
| variable {q w h:ℕ} {u:Line (q:=q) (w:=w) (h:=h)} |
| variable {v: Fire u} |
|
|
| lemma bolt {s:Seats}{I:Knob}(H i T:ℕ)(h: Lu.Standard s (campk I) H): |
| let w:= (massB i s.d)⁻¹ |
| Chor (campk I) 1 H s (.solo T (spilt I i)) w:=by |
| intro w |
| let k:=campk I |
| let v:=scaleC k i s.l.x s.info |
| have ht:w=(Coin.inv v).v:=by unfold w v; rw [Coin.inv0]; rw [show (scaleC k i s.l.x s.info).v=massB i s.d from scalen (stable_home h.st) _] |
| rw [ht]; apply Chor.played h.st h.he (fun _ _=>rfl) |
| rw [Coin.inv0,show v.v=massB i s.d from scalen (stable_home h.st) _,abs_inv,abs_of_pos (mass0 ..)] |
| exact le_trans (inv_le_one_of_one_le₀ (mass1 ..)) (mass1 ..) |
| |
| namespace Bindq |
| variable {f:v.v.Yolk v.p v.i v.x}(hf:Bindq f) |
| include hf |
| lemma amp :v.pond.stable v.frame.I ∧ Live v.pond.head v.frame.I v.H:=by |
| let r:= v.frame; let a:=v.a |
| have h:=hf.lamp |
| have he:= h.aimD |
| have hp: v.H ≤ u.D.H:=Studio.stayedC he le_rfl h.heat |
| exact ⟨Mold.knows u.r u.v u.x u.prog a v.x (of_decide_eq_true he.stage) h.hx he.shape.price,Studio.vital _ hp⟩ |
| omit hf in |
| lemma qpres(i): (seek 1 5 ⟨_,Kn.fix 0⟩).f u.x (v.pan i 0).info=hole v.inc.dataN (hole 5 0):= by |
| let a:=V u.x (v.pan i 0).info |
| have hz:a=(v.pond.info,i,0):=dec_round .. |
| change hole (pointer a.1 _) _=_ |
| rw [hz]; exact congrArg (fun a=>hole a (hole 5 0)) (Stone.pointing _ _ _ _) |
| lemma qcall(i): (v.pan i 0).band v.cost v.H v.p.inputH v.frame.baseC f.raw.b:=by |
| have hp:= hf.lamp |
| have hz:= hp.aimD |
| obtain ⟨hd,he,hx⟩:= (Lamp0.jump (v := v)) |
| have hv:=hf.money.baseX |
| rw [show v.v.c 1 v.i=v.inc.dataN from rfl,hd] at hv |
| apply Bullet.clef u.D hz (Lamp0.price (v := v)).2.1 hp.hx hx ((Bindq.qpres (v := v)) i) (Studio.soonerC hz hx hp.heat le_rfl) hv |
| lemma qstamp {b:Bool}(i j:ℕ)(hj:j<f.raw.m): |
| (v.pan i j).band v.cost v.H v.p.inputH (v.frame.pot b) (Reed.sideq (f.raw.chip j) b):=by |
| have hp:=hf.lamp |
| have hz:=hp.aimD |
| obtain ⟨hd,he,hx⟩:=(Lamp0.jump (v := v)) |
| have hv:=(hf.money.hue b).ready j (hf.fits j hj) |
| rw [hd] at hv |
| let k:= if b then 1 else 2 |
| exact Bullet.clef u.D hz (Lamp0.price (v := v)).2.1 hp.hx hx ((Lamp0.urn (v := v)) i j false k) (Studio.soonerC hz hx hp.heat le_rfl) hv |
|
|
| lemma leadX : f.raw.p.K=u.w0.p.K∧ f.raw.p.rho = (massB (fexpo 2) v.d)⁻¹∧ |
| f.raw.p.n=v.v.Snum v.i:=by |
| rw [hf.money.params,(Lamp0.height0 (v := v))];exact ⟨rfl,rfl,rfl⟩ |
| set_option maxRecDepth 6000 in |
| lemma leafC (i j:ℕ) : (v.pan i j).band v.cost v.H v.p.inputH v.frame.reserveC f.raw.p.rho:= by |
| have hp:= hf.lamp.stored i j |
| let k:=fexpo 2 |
| let K:= campk (knob u.r u.v) |
| let q:Lyric:=.solo v.p.inputH (spilt (knob u.r u.v) k) |
| let s:=v.pan i j |
| have he1:= stable_home hp.st |
| let w:= f.raw.p.rho |
| have he:w=(massB k v.d)⁻¹:=hf.leadX.2.1 |
| apply Seats.flex (c:=powder 100) ?_ (show 1 ≤ powder 100 from by unfold powder;omega) |
| (Lamp0.price (v := v)).2.2.1 le_rfl |
| have h :Chor (campk v.frame.I) 1 v.H s q w:=by |
| rw [he]; exact bolt (s:=s) v.H (fexpo 2) _ hp |
| exact Lyric.sing h (by omega) (by unfold q; simp [tail]) |
| |
| lemma sizes0 :let c:=v.frame; let _:=stable_home hf.amp.1 |
| c.n.f u.x v.pond.info=f.raw.m ∧ (c.kth false).f u.x v.pond.info=f.raw.p.n :=by |
| intro c h |
| have hp:v.frame.I.bank.f u.x v.pond.info=2^v.d:=h.2.2 |
| have hh1: ∀ i,(v.frame.I.bank.f u.x v.pond.info)^i=2^(i*v.d):=fun i=> by rw [hp,← pow_mul,Nat.mul_comm] |
| have he: c.n.f u.x v.pond.info=2^(v.p.vol*v.d):=hh1 v.p.vol |
| have hh:(c.kth false).f u.x v.pond.info= 2^(Rain.CarryE u.w0*v.d):=hh1 (Rain.CarryE u.w0) |
| rw [he,hh,hf.money.ns,hf.leadX.2.2,(Lamp0.height0 (v := v))] |
| exact ⟨rfl,(Bank.Green.capacityE u.w0 v.v.m v.i).symm⟩ |
| theorem offeredR : v.frame.Holds v.pond v.cost v.H f.raw:=by |
| have hp:= hf.lamp |
| have hs:= hp.aimD |
| have ha:= hp.he.book.saving |
| have he1:1 ≤ v.d:= hp.aimD.well.1 |
| refine ⟨(Lamp0.price (v := v)).1,hf.money.nice,hf.amp.1,hf.amp.2,hf.sizes0.1,hf.leadX.1.symm, |
| hf.sizes0.2,?_,?_,?_,?_,?_,?_,?_,?_,?_⟩ |
| · rw [hf.money.ns] |
| have hL:v.inc.m = 2^(v.p.vol*v.d):=by rw [(Lamp0.height0 (v := v))];rfl |
| change v.inc.m<2^((v.p.vol+1)*v.d) |
| rw [hL] |
| exact Nat.pow_lt_pow_right (by omega) (by nlinarith) |
| · rw [hf.money.ns]; unfold Punct |
| rw [show v.inc.m=2^(v.p.vol*v.d) from by rw [(Lamp0.height0 (v := v))]; rfl,lb_shift] |
| have hc:v.p.vol+2 ≤ v.cost:=by unfold cost coverC;omega |
| change _ ≤ v.cost*v.d; nlinarith [Nat.mul_le_mul_right v.d hc] |
| · exact le_trans (show (4:ℝ) ≤ 16 by norm_num) (v.p.fluc ha.tested he1) |
| · rw [hf.leadX.2.1]; exact ⟨le_rfl, inv_le_one_of_one_le₀ (mass1 ..)⟩ |
| · exact fun i _=> hf.qcall _ |
| · exact fun i _=> hf.leafC _ _ |
| · intro i hi; exact ⟨hf.stepC true i 0 hi,hf.trunk i i 0⟩ |
| · exact fun i hi j hj=>hf.equalityP _ _ hi hj |
| exact fun i _ j he _ b=> hf.qstamp i j he |
| end Bindq |
| end |
| end ExactDerandomization.Tiber.SliceD.Fire |
| |
| end |
| |
| section |
| |
| namespace ExactDerandomization.Tiber.SliceD.Fire |
| open Analysis Counter Relay Rill Fret Finset Mint Swirl Grain Net At Est Hie Stage |
| open Helix Helix.Foil Helix.Age Helix.Moon Helix.Night Helix.Mold Helix.Orbit Chime Scheme |
| open Dome Cone ConeX Glass Glass.Rod |
| open Reed Fret.Seats SiteX Terrain Bank Arena Song Stem Buf |
| open Tiber.Sky Line Line.Fabric Fret.Lyric Reed.Board |
| noncomputable section |
| open scoped Classical |
| variable {q w h:ℕ} {u:Line (q:=q) (w:=w) (h:=h)} |
| variable {v: Fire u} |
|
|
| namespace Bakin |
| variable (f:v.v.Yolk v.p v.i v.x) |
| abbrev Basket:= Broom v.frame v.pond v.cost f.raw |
| variable (b: Basket f) |
| def trackQ: HistoryC v.v v.p v.x (v.i+1):= ⟨fun i=>if v.i=i then decide (b.hot≠0) else f.hist.kap i⟩ |
| def readP : Stem.Depot := ⟨v.v.S1,b.product,v.bud.m,b.answer,b.spot⟩ |
| def cookP :v.v.Hinge v.p v.i v.x:=⟨trackQ f b,readP f b⟩ |
| structure Receives (b:Basket f) :Prop where |
| knows : Bindq f |
| cap : Reed.Board.Page v.frame v.pond v.cost v.H f.raw b |
| fixed:Studio.Parcel u v.a (progP v.p u.r u.v) |
| namespace Receives |
| variable {f:v.v.Yolk v.p v.i v.x}{b:Basket f}(he:Receives f b) |
| include he |
| omit he in |
| lemma beforeK {i:ℕ}(hi:i ≤ v.i): HistoryC.actual v.v (trackQ f b) i=HistoryC.actual v.v f.hist i:=by |
| unfold HistoryC.actual; induction i with |
| | zero=>rfl |
| | succ i ih=> |
| rw [Run.actual_push,Run.actual_push,ih (by omega),trackQ] |
| dsimp only;rw [ite_eq_right (by omega)] |
| lemma earn0 :HistoryC.actual v.v (trackQ f b) (v.i+1)=b.product :=by |
| have ht:= he.knows.money; have hd:= (Lamp0.height0 (v := v)); let v0:=v.v.marathon v.x |
| change v0.actual _ _ _=_ |
| rw [Run.actual_push] |
| rw [show v0.actual _ (trackQ f b).kap v.i= _ from (beforeK) le_rfl] |
| have hp:v0.pm v.i=f.raw.p:=ht.params.symm |
| have hw:HistoryC.R v.p v.v v.i=v.frame.R v.pond:=by |
| change _=massB _ v.d; rw [hd]; rfl |
| unfold Broom.product trackQ; dsimp only |
| unfold Run.taken |
| have hh:v0.shifted v.i (HistoryC.actual v.v f.hist v.i)=f.raw.raw:=ht.posed.symm |
| rw [hp, hw, hh] |
| by_cases hh':b.hot=0 <;> simp only [hh'] <;> first | rfl| simp ; grind |
| |
| lemma lawful0: HistoryC.Compared v.v (trackQ f b):=by |
| have ht:= he.knows.money |
| constructor |
| · intro i hi h |
| rw [(beforeK) (show i ≤ v.i by omega)] |
| by_cases he0:v.i=i |
| · subst i |
| rw [←ht.params,← ht.posed] |
| have hb:b.hot ≠ 0:=by unfold trackQ at h; simpa using h |
| exact he.cap.fell.lands hb |
| exact ht.compared.cooling _ (by omega) (by unfold trackQ at h;simpa [he0] using h) |
| intro i hi |
| rcases eq_or_lt_of_le (Nat.le_of_lt_succ hi) with hh|hh |
| · rw [hh]; exact ht.cap |
| exact ht.compared.shield _ hh |
| |
| omit he in |
| lemma salary : v.h0+v.frame.prompt*v.d ≤ v.H∧ v.frame.totalC v.cost ≤ budget0 v.p u.r:= |
| let H:=funded v.p u.r u.v |
| ⟨Nat.add_le_add_left (Nat.mul_le_mul_right v.d H.2.1) _,H.2.2⟩ |
|
|
| abbrev coinC(v:Fire u):= budget0 v.p u.r |
| lemma amount :2 ≤ coinC v ∧ coinC v ≤ u.v.slope:= |
| ⟨le_trans (b:=v.cost) ((by omega:2 ≤ 4).trans (Lamp0.price (v := v)).1) |
| (funded _ _ u.v).1,he.knows.lamp.heat⟩ |
|
|
| lemma boot (i:ℕ)(hi:u.smallQ v.a i): |
| u.D.swears (post v.a 0 i 0) v.x (b.answer i):=by |
| have hv:= he.cap |
| have hp: rite (v.pan i 0) (coinC v) (bodyP v.p u.r u.v 0) (b.answer i):= |
| higher (hv.transact i).2 hv.iex (salary).2 |
| exact Studio.echoTicket he.knows.lamp.aimD (by omega:(0:ℕ) ≤ 5) (by omega) hi |
| he.amount.1 he.amount.2 hp he.fixed |
|
|
| lemma corded (i:ℕ)(hi:u.smallQ v.a i)(a:Bool): |
| u.D.Voice v.a v.x (hole (if a then 1 else 2) i) (u.D.high v.a) (v.p.sealR*u.D.D v.a) (b.valueq i a):=by |
| have hv:= he.cap |
| let k:= if a then 1 else 2 |
| have hp:(v.pan i 0).band (coinC v) v.h0 v.p.sealR (bodyP v.p u.r u.v k) (b.valueq i a):=by |
| cases a <;> exact Seats.flex (hv.wave _ _ _ (salary).1) hv.iex (salary).2 le_rfl |
| have hk:k ≤ 5:=by unfold k;split<;>omega |
| exact Studio.scoreTicket he.knows.lamp.aimD hk (by omega) hi he.amount.1 he.amount.2 le_rfl hp he.fixed |
| |
| omit he in |
| lemma bare0: b.product.base=Poolx.Nice.Z f.raw := by unfold Broom.product; split <;> rfl |
| private lemma baseP_band: |
| (v.pan 0 0).band (coinC v) v.h0 v.p.sealR |
| (bodyP v.p u.r u.v 5) (Poolx.Nice.Z f.raw):=by |
| have hp : bodyP v.p u.r u.v 5 = v.frame.primary := by |
| simp only [bodyP, show (5:ℕ) ≠ 0 by decide, show ¬((5:ℕ)=1 ∨ 5=2) by decide, |
| ite_false] |
| rw [hp] |
| exact Seats.flex (he.cap.baseP 0 v.h0 (salary).1) he.cap.iex (salary).2 le_rfl |
|
|
| lemma baseP : |
| u.D.Voice v.a v.x (hole 5 0) (u.D.high v.a) (v.p.sealR*u.D.D v.a) b.product.base:=by |
| rw [(bare0)] |
| exact Studio.scoreTicket (u := u) (a := v.a) (x := v.x) (i := 0) (j := v.p.sealR) |
| (c := coinC v) (o := 5) (m := 5) (H := v.h0) (f := bodyP v.p u.r u.v) |
| he.knows.lamp.aimD (by omega) (by omega) |
| (he.knows.lamp.he.wide0 3 v.i) he.amount.1 he.amount.2 le_rfl he.baseP_band he.fixed |
| lemma taxed(i:ℕ): lb (b.answer i) ≤ Bank.hu v.p u.r*v.μ:=by |
| by_cases ht:b.answer i=0 |
| · rw [ht,lb_zero] |
| have hh:0 < Bank.hu v.p u.r:=by unfold Bank.hu Bank.hi; omega |
| have he:0<v.μ:=Age.gap_pos .. |
| nlinarith |
| have hh:=he.cap.fell |
| obtain ⟨hc,hA,hB,hW⟩:= hh.locale ht |
| conv_lhs=> rw [hB] |
| apply he.knows.lamp.he.book.saving.bill0 (Age.gap_pos ..) _ (he.knows.tagB ..) |
| have hp:= mem_range.1 (hh.there hA).1 |
| exact lt_of_lt_of_le hp (le_of_eq he.knows.money.ns) |
| lemma costed(i:ℕ):u.smallQ v.bud.visN (b.answer i):=by |
| have hp:=he.knows.lamp |
| apply hp.primal.back0 |
| change lb (b.answer i) ≤ u.eye.e*v.d |
| rw [(Lamp0.height0 (v := v))]; exact le_trans (he.taxed i) (Nat.mul_le_mul_right _ (le_trans |
| (show Bank.hu v.p u.r ≤ Bank.hx v.p u.r by unfold Bank.hx;omega) hp.he.book.cost0)) |
| lemma harvest0 (i:ℕ)(hi:b.answer i≠0): |
| i< v.bud.m∧ f.raw.valid (b.who i)∧b.spot i∈v.v.S1:=by |
| have hv:=he.cap.fell |
| obtain ⟨ht,hb,hc,hk⟩:=hv.locale hi |
| obtain ⟨hr,hh,hs⟩:=hv.there hb; have hl:=he.knows.money |
| have hm: f.raw.S=v.v.S1:=by unfold Poolx.S; rw [hl.params];rfl |
| have hz:=hl.nice |
| refine ⟨lt_of_lt_of_le (Broom.Fell.allot.1 hi) ((hv.capacity hz).trans ?_),hh.1, |
| hm ▸ hz.atS _ hr hh.1⟩ |
| exact (le_of_eq he.knows.leadX.2.2) |
| lemma vault : (readP f b).Trust:=by |
| let p:= readP f b |
| have hp:= he.knows; have hf:=he.cap.fell |
| refine ⟨?_,fun i hi=> (he.harvest0 i hi).1,fun i hi=>(he.harvest0 i hi).2.2,?_ ,?_⟩ |
| · change 0 ≤ b.product.base |
| rw [← he.earn0]; exact (v.v.marathon v.x).actual_closed .. |>.1 |
| · intro i j hi hj hk; exact hf.sole hi hj hp.money.nice hk |
| intro i hi hh |
| have hA:4 ≤ v.frame.R v.pond:=hp.offeredR.clampR |
| change b.product.ts i=line b.product.base |
| rw [(bare0)]; apply hf.framed hp.money.nice hA i _ hh |
| unfold Poolx.S |
| rw [hp.money.params]; exact hi |
| |
| lemma pegged(i:ℕ)(hi:u.smallQ v.bud.dataN i): |
| u.eye.Flora u.torch.p v.bud.visN i v.x (if b.answer i=0 then none else some (b.answer i,b.spot i)):=by |
| have hp:=he.knows.lamp; have hz:=he.boot i hi |
| have hW:u.D.swears (post v.bud.dataN v.bud.tagop i 0) v.x (b.answer i):=hz |
| let p:=v.bud |
| by_cases hh:b.answer i=0 |
| · rw [ite_eq_left hh]; apply Flora.abs |
| have hx:=hp.primal.point0 hp.he.chart i (hp.primal.back0 hi) |
| refine ⟨hx,hp.hx,?_⟩ |
| rw [show u.eye.ear u.torch.p _ _ =_ from hp.primal.ear0 i] |
| exact (u.D.identity hW).trans hh |
| rw [ite_eq_right hh] |
| let z:=b.who i; have hv:=he.cap.fell |
| obtain ⟨hX,hL,hT,hO⟩:= he.cap.fell.locale hh |
| have hr:z<f.raw.m:=mem_range.1 (hv.there hL).1 |
| have hk:= he.knows.money.found.caseM z (he.harvest0 i hh).2.1 (he.knows.fits z hr) |
| have hc:p.visN=v.v.c 4 v.i:=rfl |
| apply hp.he.chart.Rope hp.primal i v.x (b.answer i) z v.inc.visN (b.spot i) _ hi (he.costed i) hh |
| hp.hx _ _ _ _ (Fabric.Good.n1 hp.he.chart hp.he.good (he.harvest0 i hh).2.2) hW hk |
| · unfold Mold.offer; dsimp only |
| rw [Stone.roleR, ite_eq_right (by decide:(4:ℕ)≠ 2),ite_eq_left rfl] |
| exact congrArg (fun a=>rope a z) (Stone.pointing ..) |
| · exact Stone.zeroR .. |
| · rw [← hp.primal.grain] |
| exact le_trans (Lamp0.jump (v := v)).2.1 ((Nat.le_add_right ..).trans hp.primal.price) |
| rw [←hp.primal.grain,←hp.againC.grain,(Lamp0.jump (v := v)).1]; change v.d ≤ 2*v.d;omega |
| |
| omit he in |
| lemma plates (i:ℕ)(a):b.valueq i a= Reed.sideq ((readP f b).qpoint i) a:=by |
| unfold Broom.valueq Stem.Depot.qpoint; split |
| · next h=> rw [ite_eq_right (show ¬(readP f b).live _ from by change ¬b.answer i≠0; rw [h]; simp)] |
| rw [←(bare0)]; rfl |
| next h=> rw [ite_eq_left (show (readP f b).live i from h)]; rfl |
| |
| lemma tucked : v.v.HFine v.p v.i v.x (cookP f b):=by |
| have hp:= he.knows |
| refine ⟨he.lawful0,he.earn0.symm,rfl,rfl,he.vault,?_,?_,he.baseP,?_⟩ |
| · refine ⟨fun i hi=>he.boot i hi,fun i _=>he.costed i,?_,fun i hi=>he.pegged i hi⟩ |
| intro i hi |
| by_contra hn; exact Nat.not_lt_of_ge hi (he.harvest0 i hn).1 |
| · intro a; refine ⟨?_⟩; intro i hi |
| have hn:=he.corded i hi a; rw [plates] at hn;exact hn |
| intro h i hk |
| exact (he.harvest0 i hk).2.1 (hp.money.night h _) |
| end Receives |
| |
| theorem beam0 {f:v.v.Yolk v.p v.i v.x} (h:Bindq f)(he:Studio.Parcel u v.a (progP v.p u.r u.v)): |
| ∃ b, v.v.HFine v.p v.i v.x b ∧ ∀ j, lb (b.depot.tag j) ≤ Bank.hu v.p u.r*v.μ:=by |
| have ht:= funded v.p u.r u.v |
| have hp:v.frame.prompt*v.d ≤ v.H:=by |
| unfold H; exact le_trans (Nat.mul_le_mul_right _ ht.2.1) (Nat.le_add_left ..) |
| obtain ⟨b,hv⟩:= Reed.Board.mounted v.frame h.offeredR hp |
| have hq: Receives f b:=⟨h,hv,he⟩ |
| exact ⟨cookP f b,hq.tucked,fun i=>hq.taxed i⟩ |
| end Bakin |
| end |
| end ExactDerandomization.Tiber.SliceD.Fire |
|
|
| end |
|
|
| end OAI |
|
|
