| import OAI.Geometry.Convex.GeneralMahler.Middle.Features |
| import OAI.Geometry.Convex.GeneralMahler.Middle.Ratio |
|
|
| namespace OAI |
| /-! Middle-width certificate uses second derivative envelope on the |
| actual endpoint functional. Syntax shares proof of interval and profile estimates. -/ |
| open Set Filter MeasureTheory MeasureTheory.Measure Real |
| namespace GeneralMahler.SCal.Mid |
| open Grid Jet Profile Segment SE Cert Cert.IV Ft |
| inductive Op |
| | src (k:Fin 3) (f:Ft) |
| -- rational constant q/r |
| | num (q r:Int) |
| | add (u v:Op) |
| | sub (u v:Op) |
| | mul (u v:Op) |
| deriving Inhabited |
| namespace Op |
| local notation x " +| " y => Op.add x y |
| noncomputable def eval (k:Fin 3→Ft→ℝ):Op→ℝ |
| | src n feat=> k n feat |
| | num a b=> (a:ℝ)/b |
| | add a b=>eval k a+eval k b |
| | sub a b=>eval k a-eval k b |
| | mul a b=>eval k a*eval k b |
| def boxE (k:Fin 3→Ft→IV):Op→IV |
| | src n feat=> k n feat |
| | num a b=> c a/c b |
| | add a b=>boxE k a+boxE k b |
| | sub a b=>boxE k a-boxE k b |
| | mul a b=>boxE k a*boxE k b |
|
|
| noncomputable def capE (k:Fin 3→Ft→Cap):Op→Cap |
| | src n feat=> k n feat |
| | num a b=> Cap.fl |(a:ℝ)/(b:ℝ)| |
| | add a b=>Cap.plus (capE k a) (capE k b) |
| | sub a b=>Cap.plus (capE k a) (capE k b) |
| | mul a b=>Cap.times (capE k a) (capE k b) |
|
|
| lemma bval {k:Fin 3→Ft→ℝ}{v:Fin 3→Ft→IV}(h:∀ i n,k i n∈ v i n) (x:Op): |
| eval k x∈boxE v x := by |
| induction x with |
| | src=> apply h |
| | num a b=>exact mdiv (mc a) (mc b) |
| | add a b ih hg=>exact madd ih hg |
| | sub a b ih hg=>exact msub ih hg |
| | mul a b ih hg=>exact mmul ih hg |
| lemma valGuard {k:Fin 3→Ft→ℝ→ℝ}{v:Fin 3→Ft→ Cap}(h:∀ i n,Guard (k i n) (v i n))(u:Op): |
| Guard (fun x=> eval (fun i n=> k i n x) u) (capE v u):=by |
| induction u with |
| | src=> apply h |
| | num a b=>exact Guard.const _ _ le_rfl |
| | add a b ih hg=>exact ih.plus hg |
| | sub a b ih hg=>exact ih.minus hg |
| | mul a b ih hg=>exact ih.prod hg |
|
|
| -- endpoints stored 0,1; averages 2 |
| def q (v:Op):Op:=mul v v |
| def ekp:=sub (src 2 Ft.K) (mul (num 1 2) (add (src 0 Ft.K) (src 1 Ft.K))) |
| def gammaOp:Op:= |
| let c:=sub (src 1 Ft.C) (src 0 Ft.C) |
| let k:=sub (src 1 Ft.K) (src 0 Ft.K) |
| mul (mul (num 8 10) (q (num 1000 352))) (add |
| (mul (num 125 100) (q (add k (mul (num 21 1000) c)))) |
| (mul (mul (num 5 1) (q (num 43 1000))) (q c))) |
| def ru2 := q (num 22 100) |
| def sm (p:Fin 3) := add ru2 (src p Ft.M) |
| def dotOp (p:Fin 3) := add (add (mul (src 2 Ft.Q) (src p Ft.Q)) (mul (src 2 Ft.J) (src p Ft.J))) |
| (mul (src 2 Ft.V) (src p Ft.V)) |
| def mb (p:Fin 3) := sub (add (sm 2) (sm p)) (mul (num 2 1) (dotOp p)) |
| def bolt (p q:Fin 3):= sub (sub (src 2 Ft.C) (src q Ft.C)) (mb p) |
| def E:Op := sub (add (add (add ekp gammaOp) |
| (mul (num 10 52) (add (q (bolt 0 1)) (q (bolt 1 0))))) |
| (mul (num 32 1000) (add (mb 0) (mb 1)))) |
| (mul (num 1064 1000) (sub (sm 2) (dotOp 2))) |
| end Op |
| noncomputable section |
| -- choose s=g t profiles |
| def inp (x y:ℝ):Fin 3→Ft→ℝ:= |
| fun n f=> if n=0 then f.g x else if n=1 then f.g y else trav x f.g y |
| def Fval (x y:ℝ):= Op.eval (inp x y) Op.E |
| def caps (i:Fin 3) (f:Ft) := if i=0 then Cap.fl f.cap.a else if i=1 then f.cap else Cap.av f.cap |
| def budgetCap:= Op.capE caps Op.E |
| lemma gv2 (h:NG)(x:ℝ):Guard (Fval x) budgetCap:= by |
| apply Op.valGuard (k:=fun i f y=>inp x y i f) _ _ |
| intro i v |
| have hh:=guarded h v |
| unfold inp caps;split_ifs with hf hg |
| · exact Guard.const _ _ (hh.fa _) |
| · exact hh |
| exact trav_guard hh x |
| lemma Bcost : budgetCap.c ≤ 37:=by |
| have he : (2:Fin 3)≠0:=by decide |
| have hv : (2:Fin 3)≠1:=by decide |
| have ha : (1:Fin 3)≠0:=by decide |
| norm_num [he,hv,ha,budgetCap,Op.E,Op.capE,caps,Cap.plus,Cap.times,Cap.fl,Cap.av, Ft.cap,Ft.cP,Ft.cC, |
| Ft.cJ,Ft.cK,Op.dotOp,Op.sm,Op.ru2,Op.q,Op.mb,Op.bolt,Op.gammaOp,Op.ekp] |
| end |
| end GeneralMahler.SCal.Mid |
|
|
| end OAI |
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