| import OAI.Computability.RandomMean.StateGrowth |
|
|
| namespace OAI |
|
|
| section |
| namespace MeanPayoff.TreeDevice.Certified.RawControl |
| open BinaryData Code |
| local infixl:80 " ≫ₚ " => Code.comp |
| inductive JobLabel where |
| | solve | ask | trials | update | centers | orient | orientTrials | orientPlus | tighten | reweight |
| noncomputable def JobLabel.code : JobLabel → Code Input State |
| | .solve => solveCase |
| | .ask => askCase |
| | .trials => trialsCase |
| | .update => updateCase |
| | .centers => centersCase |
| | .orient => orientCase |
| | .orientTrials => orientTrialsCase |
| | .orientPlus => orientPlusCase |
| | .tighten => tightenCase |
| | .reweight => reweightCase |
| inductive FrameLabel where |
| | trialsHead | vectorCons | median | centerFirst | centerSecond | centersHead | orientLoop | orientTail | orientHead | orientCons | combine | tightenFirst | tightenSecond | floor | ceiling | solveTight | solveOriented | reweightHead | solveReweighted | capHigh | capLow | includeInner |
| noncomputable def FrameLabel.code : FrameLabel → Code Input State |
| | .trialsHead => trialsHeadCase |
| | .vectorCons => vectorConsCase |
| | .median => medianCase |
| | .centerFirst => centerFirstCase |
| | .centerSecond => centerSecondCase |
| | .centersHead => centersHeadCase |
| | .orientLoop => orientLoopCase |
| | .orientTail => orientTailCase |
| | .orientHead => orientHeadCase |
| | .orientCons => orientConsCase |
| | .combine => combineCase |
| | .tightenFirst => tightenFirstCase |
| | .tightenSecond => tightenSecondCase |
| | .floor => boxPassCase |
| | .ceiling => boxPassCase |
| | .solveTight => solveTightCase |
| | .solveOriented => solveOrientedCase |
| | .reweightHead => reweightHeadCase |
| | .solveReweighted => solveReweightedCase |
| | .capHigh => capHighCase |
| | .capLow => capLowCase |
| | .includeInner => includeInnerCase |
| theorem JobLabel.small (j : JobLabel) {n H M E D : ℕ} {x : Input} |
| (hc : (current x).Small n H M) (hk : StackSmall n H M (stack x)) |
| (hs : x.1.2.2.1.length≤H) (hj : x.1.2.2.2.1.length≤H) |
| (he : RawBasic.ArenaBounded x.1.1 E) (hd : x.1.2.2.2.2.length*(E+1)+20≤D) : |
| StateSmall n H (M+D) (JobLabel.code j x) := by |
| have hmd : M≤M+D := by omega |
| have hd3 : M+3≤M+D := by omega |
| have hd14 : M+14≤M+D := by omega |
| have hdB : M+x.1.2.2.2.2.length*(E+1)≤M+D := by omega |
| have hc' := hc.mono hmd |
| have hk' := hk.mono hmd |
| cases j with |
| | solve => |
| simp only [JobLabel.code,solveCase,branch_eval] |
| split |
| · refine ⟨?_,hk'⟩ |
| exact ⟨zeroBudget_small _ _,zeroBox_small _ _,Nat.zero_le _,RawBox.Small.smallVector hc'.b hc'.p, |
| Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| · split |
| · refine ⟨?_,hk'⟩ |
| exact ⟨zeroBudget_small _ _,zeroBox_small _ _,Nat.zero_le _,(basic_small _ hc.b he).mono hdB, |
| Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| · refine ⟨?_,StackSmall.cons hc' hk'⟩ |
| exact ⟨hc'.p,hc'.b,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| | ask => |
| refine ⟨?_,StackSmall.cons hc' hk'⟩ |
| exact ⟨hc'.p,hc'.b,hj,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| | trials => |
| simp only [JobLabel.code,trialsCase,branch_eval] |
| split |
| · exact ⟨Pack.zero_small _ _ _,hk'⟩ |
| · refine ⟨hc',StackSmall.cons ?_ hk'⟩ |
| exact ⟨hc'.p,hc'.b,(by change (current x).t.tail.length≤H; rw [List.length_tail]; have := hc.t; omega),Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| | update => |
| refine ⟨?_,StackSmall.cons hc' hk'⟩ |
| exact ⟨hc.p.reduceK.mono hd14,(narrow_small _ _ hc.b hc.q).mono hd3,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| | centers => |
| simp only [JobLabel.code,centersCase,branch_eval] |
| split |
| · refine ⟨?_,hk'⟩ |
| refine ⟨zeroBudget_small _ _,zeroBox_small _ _,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,?_,zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| intro xs hx |
| change xs∈[(current x).q] at hx |
| rcases List.mem_singleton.mp hx with rfl |
| exact hc'.q |
| · refine ⟨hc',StackSmall.cons ?_ hk'⟩ |
| exact ⟨hc'.p,hc'.b,(by change (current x).t.tail.length≤H; rw [List.length_tail]; have := hc.t; omega),hc'.q,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| | orient => |
| simp only [JobLabel.code,orientCase,branch_eval] |
| split |
| · refine ⟨?_,hk'⟩ |
| exact ⟨zeroBudget_small _ _,zeroBox_small _ _,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),RawOrientation.base_small _ _ ((basic_small _ hc.b he).mono hdB),(by intro v hv; cases hv)⟩ |
| · refine ⟨?_,StackSmall.cons hc' hk'⟩ |
| exact ⟨hc'.p,hc'.b,hs,hc'.b.midpoint,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| | orientTrials => |
| simp only [JobLabel.code,orientTrialsCase,branch_eval] |
| split |
| · exact ⟨Pack.zero_small _ _ _,hk'⟩ |
| · refine ⟨hc',StackSmall.cons ?_ hk'⟩ |
| exact ⟨hc'.p,hc'.b,(by change (current x).t.tail.length≤H; rw [List.length_tail]; have := hc.t; omega),Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| | orientPlus => |
| refine ⟨?_,StackSmall.cons hc' hk'⟩ |
| exact ⟨hc'.p,hc'.b,hj,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| | tighten => |
| refine ⟨?_,StackSmall.cons hc' hk'⟩ |
| exact ⟨hc.p.reduceK.mono hd14,hc'.b,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| | reweight => |
| simp only [JobLabel.code,reweightCase,branch_eval] |
| split |
| · exact ⟨Pack.zero_small _ _ _,hk'⟩ |
| · refine ⟨?_,StackSmall.cons ?_ hk'⟩ |
| · refine ⟨?_,hc'.b,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| simp only [call,makeCell,pair_eval,pack_eval,branch_eval] |
| split |
| · exact (hc.p.reweightM _).mono hd14 |
| · exact (hc.p.reweightK _).mono hd14 |
| · refine ⟨hc'.p,hc'.b,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),?_,(by intro v hv; cases hv)⟩ |
| intro xs hx |
| exact hc.o xs (List.mem_of_mem_tail hx) |
| theorem FrameLabel.small (f : FrameLabel) {n H M D : ℕ} {x : Input} |
| (hc : (current x).Small n H M) (hf : (caller x).Small n H M) (hk : StackSmall n H M (rest x)) |
| (hd : 20≤D) : StateSmall n H (M+D) (FrameLabel.code f x) := by |
| have hmd : M≤M+D := by omega |
| have hd3 : M+3≤M+D := by omega |
| have hd6 : M+3+3≤M+D := by omega |
| have hd14 : M+14≤M+D := by omega |
| have hc' := hc.mono hmd |
| have hf' := hf.mono hmd |
| have hk' := hk.mono hmd |
| cases f with |
| | trialsHead => |
| refine ⟨?_,StackSmall.cons ?_ hk'⟩ |
| · exact ⟨hf'.p,hf'.b,hf'.t,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| · exact ⟨zeroBudget_small _ _,zeroBox_small _ _,Nat.zero_le _,hc'.q,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| | vectorCons => |
| refine ⟨?_,hk'⟩ |
| refine ⟨zeroBudget_small _ _,zeroBox_small _ _,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,?_,zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| intro xs hx |
| change xs∈(caller x).q::(current x).vs at hx |
| rcases List.mem_cons.mp hx with rfl | hx |
| · exact hf'.q |
| · exact hc'.vs xs hx |
| | median => |
| refine ⟨?_,hk'⟩ |
| exact ⟨zeroBudget_small _ _,zeroBox_small _ _,Nat.zero_le _,RawMedian.boxed_small hf'.b hc'.vs,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| | centerFirst => |
| refine ⟨?_,StackSmall.cons ?_ hk'⟩ |
| · exact ⟨hf.p.reduceM.mono hd14,(narrow_small _ _ hf.b hf.q).mono hd3,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| · exact ⟨zeroBudget_small _ _,hf'.b,Nat.zero_le _,Array.Small.nil _ _,hc'.q,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| | centerSecond => |
| refine ⟨?_,hk'⟩ |
| exact ⟨zeroBudget_small _ _,zeroBox_small _ _,Nat.zero_le _,hf'.b.averageClipped hf'.r hc'.q,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| | centersHead => |
| refine ⟨?_,StackSmall.cons ?_ hk'⟩ |
| · exact ⟨hf'.p,hf'.b,hf'.t,hc'.q,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| · exact ⟨zeroBudget_small _ _,zeroBox_small _ _,Nat.zero_le _,hf'.q,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| | orientLoop => |
| have hp := lastPositive_small hf.b hc.vs |
| have hn := lastNegative_small hf.b hc.vs |
| simp only [FrameLabel.code,orientLoopCase,branch_eval] |
| split |
| · refine ⟨?_,hk'⟩ |
| exact ⟨zeroBudget_small _ _,zeroBox_small _ _,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),RawOrientation.trueComplement_small hp,(by intro v hv; cases hv)⟩ |
| · split |
| · refine ⟨?_,hk'⟩ |
| exact ⟨zeroBudget_small _ _,zeroBox_small _ _,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),RawOrientation.falseComplement_small hn,(by intro v hv; cases hv)⟩ |
| · refine ⟨?_,StackSmall.cons ?_ hk'⟩ |
| · exact ⟨hf'.p,(wide_small _ _ hf.b (centerAt_small _ hf.b hc.vs)).mono hd3,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| · refine ⟨zeroBudget_small _ _,zeroBox_small _ _,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),?_,(by intro v hv; cases hv)⟩ |
| intro bs hbs |
| change bs∈[lastPositive x,lastNegative x] at hbs |
| simp only [List.mem_cons,List.not_mem_nil,or_false] at hbs |
| rcases hbs with rfl | rfl <;> assumption |
| | orientTail => |
| refine ⟨?_,hk'⟩ |
| refine ⟨zeroBudget_small _ _,zeroBox_small _ _,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),?_,(by intro v hv; cases hv)⟩ |
| exact RawOrientation.extend_small hc.o (mask_head hf.o) (mask_head (fun bs hbs => hf.o bs (List.mem_of_mem_tail hbs))) |
| | orientHead => |
| refine ⟨?_,StackSmall.cons ?_ hk'⟩ |
| · exact ⟨hf'.p,hf'.b,hf'.t,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| · exact ⟨zeroBudget_small _ _,zeroBox_small _ _,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),hc.o,(by intro v hv; cases hv)⟩ |
| | orientCons => |
| refine ⟨?_,hk'⟩ |
| refine ⟨zeroBudget_small _ _,zeroBox_small _ _,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,?_⟩ |
| intro v hv |
| change v∈(caller x).o::(current x).os at hv |
| rcases List.mem_cons.mp hv with rfl | hv |
| · exact hf.o |
| · exact hc.os v hv |
| | combine => |
| refine ⟨?_,hk'⟩ |
| exact ⟨zeroBudget_small _ _,zeroBox_small _ _,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),RawOrientation.combine_small hc.os,(by intro v hv; cases hv)⟩ |
| | tightenFirst => |
| simp only [FrameLabel.code,tightenFirstCase,branch_eval] |
| split |
| · refine ⟨?_,StackSmall.cons ?_ hk'⟩ |
| · exact ⟨hf'.p,hf'.b.withFloor hc'.q,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| · exact ⟨zeroBudget_small _ _,hf'.b,Nat.zero_le _,hc'.q,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| · refine ⟨?_,StackSmall.cons hf' hk'⟩ |
| exact ⟨hf.p.reduceM.mono hd14,hf'.b,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| | tightenSecond => |
| simp only [FrameLabel.code,tightenSecondCase,branch_eval] |
| split |
| · refine ⟨?_,StackSmall.cons ?_ hk'⟩ |
| · exact ⟨hf'.p,hf'.b.withCeiling hc'.q,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| · exact ⟨zeroBudget_small _ _,hf'.b,Nat.zero_le _,hc'.q,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| · refine ⟨?_,hk'⟩ |
| exact ⟨zeroBudget_small _ _,hf'.b,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| | floor | ceiling => |
| refine ⟨?_,hk'⟩ |
| exact ⟨zeroBudget_small _ _,hc'.b,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| | solveTight => |
| refine ⟨?_,StackSmall.cons ?_ (StackSmall.cons (Pack.zero_small _ _ _) hk')⟩ |
| · exact ⟨hf'.p,hc'.b,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| · exact ⟨hf'.p,hc'.b,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| | solveOriented => |
| refine ⟨?_,StackSmall.cons ?_ hk'⟩ |
| · exact ⟨hf'.p,hf'.b,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),hc.o,(by intro v hv; cases hv)⟩ |
| · exact ⟨hf'.p,hf'.b,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),(by intro v hv; cases hv),(by intro v hv; cases hv)⟩ |
| | reweightHead => |
| refine ⟨?_,StackSmall.cons ?_ hk'⟩ |
| · exact ⟨hf'.p,hf'.b,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),hf.o,(by intro v hv; cases hv)⟩ |
| · exact ⟨zeroBudget_small _ _,zeroBox_small _ _,Nat.zero_le _,hc'.q,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| | solveReweighted => |
| have hw := capWidth_small (x:=x) (caller ≫ₚ b) hf.b |
| simp only [FrameLabel.code,solveReweightedCase,branch_eval] |
| split |
| · refine ⟨?_,StackSmall.cons ?_ hk'⟩ |
| · exact ⟨hf'.p,((hf.b.mono (by omega : M≤M+3)).capHigh hw).mono hd6,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| · exact ⟨zeroBudget_small _ _,hf'.b,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,hc'.vs,zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| · refine ⟨?_,StackSmall.cons ?_ hk'⟩ |
| · exact ⟨hf'.p,((hf.b.mono (by omega : M≤M+3)).capLow hw).mono (by omega),Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| · exact ⟨zeroBudget_small _ _,hf'.b,Nat.zero_le _,Array.Small.nil _ _,Array.Small.nil _ _,hc'.vs,zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| | capHigh => |
| refine ⟨?_,hk'⟩ |
| exact ⟨zeroBudget_small _ _,zeroBox_small _ _,Nat.zero_le _,hc'.q.foldMin hf'.vs,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| | capLow => |
| refine ⟨?_,hk'⟩ |
| exact ⟨zeroBudget_small _ _,zeroBox_small _ _,Nat.zero_le _,hc'.q.foldMax hf'.vs,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| | includeInner => |
| refine ⟨?_,hk'⟩ |
| exact ⟨zeroBudget_small _ _,zeroBox_small _ _,Nat.zero_le _,hc'.q,Array.Small.nil _ _,(by intro v hv; cases hv),zeroOrient_small _,(by intro v hv; cases hv)⟩ |
| theorem JobLabel.mass (j : JobLabel) (x : Input) : |
| stateMass (JobLabel.code j x)≤ stateMass (state x)+3 ∧ (JobLabel.code j x).2.length≤(stack x).length+2 := by |
| cases j <;> |
| simp only [JobLabel.code,solveCase,askCase,trialsCase,updateCase,centersCase,orientCase,orientTrialsCase, |
| orientPlusCase,tightenCase,reweightCase,branch_eval] <;> |
| (try split_ifs) <;> |
| simp only [stateMass,ret,call,save,push,makeCell,pair_eval,comp_eval,pack_eval,const_eval, |
| cons_eval,tail_eval,stackMass_cons,Pack.mass,p_eval,b_eval,t_eval,q_eval,o_eval, List.length_nil,List.length_cons,RawOrientation.listMass_nil, RawOrientation.base_length,List.length_tail] <;> |
| dsimp only [state,stack,current,Code.comp_eval,Code.fst_eval,Code.snd_eval] <;> omega |
| theorem FrameLabel.mass (f : FrameLabel) (x : Input) : |
| stateMass (FrameLabel.code f x)≤(current x).mass+(caller x).mass+stackMass (rest x)+3 ∧ |
| (FrameLabel.code f x).2.length≤(rest x).length+2 := by |
| cases f <;> |
| simp only [FrameLabel.code,trialsHeadCase,vectorConsCase,medianCase,centerFirstCase,centerSecondCase, |
| centersHeadCase,orientLoopCase,orientTailCase,orientHeadCase,orientConsCase,combineCase, |
| tightenFirstCase,tightenSecondCase,boxPassCase,solveTightCase,solveOrientedCase,reweightHeadCase, |
| solveReweightedCase,capHighCase,capLowCase,includeInnerCase,branch_eval] <;> |
| (try split_ifs) <;> |
| simp only [stateMass,ret,call,push,makeCell,pair_eval,comp_eval,pack_eval,const_eval, |
| cons_eval,tail_eval,stackMass_cons,Pack.mass,p_eval,b_eval,t_eval,q_eval,r_eval,vs_eval,o_eval,os_eval, |
| List.length_nil,List.length_cons,RawOrientation.listMass_nil,RawOrientation.listMass_cons, |
| RawOrientation.extend_length,RawOrientation.combine_length, RawOrientation.trueComplement,RawOrientation.falseComplement,RawOrientation.singleton,Pack.zero] <;> |
| simp only [Code.comp_eval,Code.fst_eval,Code.snd_eval, current,caller,stack,rest,state,Code.head_eval,Code.tail_eval] <;> omega |
| end MeanPayoff.TreeDevice.Certified.RawControl |
|
|
| end |
|
|
| section |
| namespace MeanPayoff.TreeDevice.Certified |
| open BinaryData Code |
| namespace Code |
| theorem select_spec {α β : Type} [Encoding α] [Encoding β] (tag : Code α Count) (cases : List (Code α β)) |
| (fallback : Code α β) (x : α) (P : β→Prop) (hc : ∀ c∈cases, P (c x)) (hf : P (fallback x)) : |
| P (select tag cases fallback x) := by |
| induction cases generalizing tag with |
| | nil => exact hf |
| | cons c cs ih => |
| simp only [select,branch_eval] |
| split |
| · exact hc c (by simp) |
| · exact ih _ (fun d hd => hc d (by simp [hd])) |
| end Code |
| namespace RawControl |
| theorem jobs_spec (x : Input) (P : State→Prop) (h : ∀ j : JobLabel, P (j.code x)) : |
| ∀ c ∈ [solveCase,askCase,trialsCase,updateCase,centersCase,orientCase,orientTrialsCase,orientPlusCase,tightenCase,reweightCase], P (c x) := by |
| intro c hc |
| simp only [List.mem_cons,List.not_mem_nil,or_false] at hc |
| rcases hc with rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl |
| · exact h .solve |
| · exact h .ask |
| · exact h .trials |
| · exact h .update |
| · exact h .centers |
| · exact h .orient |
| · exact h .orientTrials |
| · exact h .orientPlus |
| · exact h .tighten |
| · exact h .reweight |
| theorem frames_spec (x : Input) (P : State→Prop) (h : ∀ j : FrameLabel, P (j.code x)) : |
| ∀ c ∈ [trialsHeadCase,vectorConsCase,medianCase,centerFirstCase,centerSecondCase,centersHeadCase,orientLoopCase,orientTailCase,orientHeadCase,orientConsCase,combineCase,tightenFirstCase,tightenSecondCase,boxPassCase,boxPassCase,solveTightCase,solveOrientedCase,reweightHeadCase,solveReweightedCase,capHighCase,capLowCase,includeInnerCase], P (c x) := by |
| intro c hc |
| simp only [List.mem_cons,List.not_mem_nil,or_false] at hc |
| rcases hc with rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl | rfl |
| · exact h .trialsHead |
| · exact h .vectorCons |
| · exact h .median |
| · exact h .centerFirst |
| · exact h .centerSecond |
| · exact h .centersHead |
| · exact h .orientLoop |
| · exact h .orientTail |
| · exact h .orientHead |
| · exact h .orientCons |
| · exact h .combine |
| · exact h .tightenFirst |
| · exact h .tightenSecond |
| · exact h .floor |
| · exact h .ceiling |
| · exact h .solveTight |
| · exact h .solveOriented |
| · exact h .reweightHead |
| · exact h .solveReweighted |
| · exact h .capHigh |
| · exact h .capLow |
| · exact h .includeInner |
| theorem caller_mass (x : Input) : (caller x).mass+stackMass (rest x)=stackMass (stack x) := by |
| change ((stack x).headD ([],Pack.zero)).2.mass+stackMass (stack x).tail=stackMass (stack x) |
| cases stack x <;> rfl |
| theorem dispatch_small {n H M E D : ℕ} {x : Input} |
| (hc : (current x).Small n H M) (hk : StackSmall n H M (stack x)) |
| (hs : x.1.2.2.1.length≤H) (hj : x.1.2.2.2.1.length≤H) |
| (he : RawBasic.ArenaBounded x.1.1 E) (hd : x.1.2.2.2.2.length*(E+1)+20≤D) : |
| StateSmall n H (M+D) (dispatch x) := by |
| have hmd : M≤M+D := by omega |
| have h0 : StateSmall n H (M+D) (state x) := ⟨hc.mono hmd,hk.mono hmd⟩ |
| apply Code.select_spec |
| · exact jobs_spec x _ (fun j => j.small hc hk hs hj he hd) |
| · simp only [resume,branch_eval] |
| split |
| · exact h0 |
| · apply Code.select_spec |
| · exact frames_spec x _ (fun f => f.small hc hk.head hk.tail (by omega)) |
| · exact h0 |
| theorem dispatch_mass (x : Input) : |
| stateMass (dispatch x)≤ stateMass (state x)+3 ∧ (dispatch x).2.length≤(stack x).length+2 := by |
| have h0 : stateMass (state x)≤ stateMass (state x)+3 ∧ (state x).2.length≤(stack x).length+2 := by |
| constructor <;> exact Nat.le_add_right _ _ |
| refine Code.select_spec _ _ _ x (fun z => stateMass z ≤ stateMass (state x)+3 ∧ z.2.length≤(stack x).length+2) ?_ ?_ |
| · exact jobs_spec x (fun z => stateMass z ≤ stateMass (state x)+3 ∧ z.2.length≤(stack x).length+2) (fun j => j.mass x) |
| · simp only [resume,branch_eval] |
| split |
| · exact h0 |
| · refine Code.select_spec _ _ _ x (fun z => stateMass z ≤ stateMass (state x)+3 ∧ z.2.length≤(stack x).length+2) ?_ ?_ |
| · apply frames_spec x (fun z => stateMass z ≤ stateMass (state x)+3 ∧ z.2.length≤(stack x).length+2) |
| intro f |
| have h := f.mass x |
| have hm := caller_mass x |
| have ht : (rest x).length≤(stack x).length := by |
| change (stack x).tail.length≤(stack x).length |
| rw [List.length_tail]; omega |
| change _≤(current x).mass+stackMass (stack x)+3 ∧ _≤_ at ⊢ |
| omega |
| · exact h0 |
| end RawControl |
| end MeanPayoff.TreeDevice.Certified |
|
|
| end |
|
|
| end OAI |
|
|
