| import OAI.Analysis.FourierLLogL.Paths.Bridge |
| /-! |
|
|
| open Function Set Filter |
| open scoped Topology BigOperators ComplexConjugate |
| noncomputable section |
|
|
| namespace OAI |
|
|
| namespace FourierLLogL.Dyadic.Procedure.Paths |
| open Law Node Real Period Compress.Params Thermal Fund Scene Fork Road Event Journal Wire Pulse Thread Capture Chart Snapshot |
| variable {M j J : ℕ} {s q : Scene M} {f : Grid M → ℝ} |
| open scoped Classical |
|
|
|
|
| namespace Beam |
|
|
| /-- Camera anchored at a jumping post. -/ |
| structure Lamp (M : ℕ) where |
| u : Grid M |
| width : ℝ |
| W : Paths.Vector M |
| site : Scene M |
|
|
| namespace Lamp |
| def glint (e : Lamp M) (z : Grid M) := |
| glow (e.W z) (letter e.site e.u z) (e.width*matter e.site z) |
| def first (e : Lamp M) (z : Grid M) := |
| e.glint z 0 (factor e.site e.u z) (e.width*shell e.site z) |
| def second (e : Lamp M) (r : Bool) (q : Scene M) (z : Grid M) := |
| e.glint z ((curl r:ℂ)*factor e.site e.u z) (factor q e.u z) (e.width*shell q z) |
| def start (e : Lamp M) := ∑ z, first e z |
| def delayed (e : Lamp M) (r : Bool) (q : Scene M) := ∑ z, second e r q z |
|
|
| /-- update of accumulated same-parity output. -/ |
| def carry (e : Lamp M) (r t : Bool) (z : Grid M) := |
| e.W z + (curl t:ℂ)*((curl r:ℂ)*factor e.site e.u z) |
|
|
| lemma flash_cell (e : Lamp M) (h : Climb e.site) (r t : Bool) (q : Scene M) |
| (hq : Ideal (e.site.smaller r) q) (ha : 0 < e.width) (he : e.width ≤ 1) |
| (z : Grid M) : |
| let u := e.u |
| let a := e.width |
| let s := e.site |
| let k := matter (s.smaller r) z-matter q z |
| let C := 20*(pierce s z+pierce q z) + 400*((k+dew u s z+dew u q z)/a) |
| prism (q.smaller t) u (e.carry r t) a z ≤ |
| prism s u e.W a z + curl r * e.first z + curl t * e.second r q z + 2*k+4*C := by |
| intro u a s k C |
| let w := s.smaller r; let y := q.smaller t |
| let R := prism s u e.W a z |
| let f (s) := letter s u z; let m (s) := matter s z |
| let P (s) := factor s u z; let D (s : Scene M) := a*shell s z |
| let X := curl r; let Y := curl t; let A := e.W z |
| let F (x : Scene M) := a*m x |
| let kill : ℂ := f q-f w; let lost : ℝ := F q-F w |
| let g₀ := (↑X:ℂ)*P s; let g₁ := (↑Y:ℂ)*P q |
| let gain := (↑Y:ℂ)*g₀; let plusf := g₀+kill+g₁ |
| let plusm := X*D s + lost + Y*D q |
| have hk : ‖kill‖ ≤ k := by unfold kill; rw [norm_sub_rev]; exact matter_drain hq u z |
| have hl : lost= -(a*k) := by unfold lost k F w m; ring |
| have hp := (norm_nonneg kill).trans hk |
| have hm : |lost|=a*k := by rw [hl,abs_neg,abs_of_nonneg (mul_nonneg ha.le hp)] |
| have hs : |lost| ≤ k := by rw [hm]; nlinarith |
| have hA : f y=f s+plusf := by |
| have h1 : f w=f s+g₀ := letter_jump h r u z |
| have h2 : f y=f q+g₁ := letter_jump hq.climb t u z |
| rw [h2]; unfold plusf kill; rw [h1]; ring |
| have hB : F y=F s+plusm := by |
| have h1 : m w=m s+X*shell s z := matter_jump h r z |
| have h2 : m y=m q+Y*shell q z := matter_jump hq.climb t z |
| unfold F plusm lost D; unfold F; rw [h1,h2]; ring |
| have h0 : 0 ≤ R := ray_pos _ _ _ |
| have hC : 0 ≤ C := by |
| have h1 := pierce_nonneg h.fit z |
| have h2 := pierce_nonneg hq.climb.fit z |
| have h3 : 0 ≤ k+dew u s z+dew u q z := add_nonneg |
| (add_nonneg hp (dew_nonneg u h z)) (dew_nonneg u hq.climb z) |
| exact add_nonneg (mul_nonneg (by norm_num) (add_nonneg h1 h2)) |
| (mul_nonneg (by norm_num) (div_nonneg h3 ha.le)) |
| have hx : |X|=1 := curl_unit _; have hy : |Y|=1 := curl_unit _ |
| have HG : content gain plusf plusm ≤ R*(4*C) := by |
| have HH := contest g₀ kill g₁ (X*D s) lost (Y*D q) |
| have Heq : ‖g₀‖=‖P s‖ := by unfold g₀; rw [norm_scaleC,hx,one_mul] |
| have Hf : ‖gain‖=‖g₀‖ := by unfold gain; rw [norm_scaleC,hy,one_mul] |
| have Hf' : ‖g₁‖=‖P q‖ := by unfold g₁; rw [norm_scaleC,hy,one_mul] |
| have HV : ‖P s‖^2+‖P q‖^2+D s^2+D q^2+ |
| ‖kill‖^2+(a*k)^2 ≤ R*C := by |
| have H := extra_joints h u z r hq a ha he e.W |
| unfold kill; rw [norm_sub_rev]; exact H |
| rw [show content gain plusf plusm=‖gain‖^2+(‖plusf‖^2+plusm^2) by unfold content; ring,Hf,Heq] |
| change ‖plusf‖^2 + plusm^2 ≤ _ at HH |
| rw [Heq,Hf',mul_pow X (D s),mul_pow Y (D q),show X^2=1 from curl_sq _,show Y^2=1 from curl_sq _,hl] at HH |
| nlinarith [sq_nonneg (D s),sq_nonneg (D q),sq_nonneg ‖P q‖,sq_nonneg ‖kill‖,sq_nonneg (a*k)] |
| let glow' := glow A (f s) (F s) |
| have HL : glow' gain plusf plusm = |
| X*e.first z+glow' 0 kill lost+Y*e.second r q z := by |
| unfold gain plusf plusm g₀ g₁ first second glint X Y glow' |
| let F' := factor e.site e.u z |
| let a' := curl r; let k' := curl t |
|
|
| let L := e.glint z |
| change L ((k':ℂ)*((a':ℂ)*F')) ((a':ℂ)*F'+kill+(k':ℂ)*P q) |
| (a'*D s+lost+k'*D q)=a'*L 0 F' (D s)+L 0 kill lost+k'*L ((a':ℂ)*F') (P q) (D q) |
| have h1 (v w p i : ℂ) (u y : ℝ) : |
| L (v+p) (w+i) (u+y)=L v w u+L p i y := glow_add _ _ _ _ _ _ _ _ _ |
| have h2 (v : ℝ) (y : ℝ) (q w : ℂ) : |
| L ((v:ℂ)*q) ((v:ℂ)*w) (v*y)=v*L q w y := glow_scale _ _ _ _ _ _ _ |
| rw [← h2,← h2,← h1,← h1] |
| congr 1; ring |
| have Hpill : glow' 0 kill lost ≤ 2*k := by |
| apply le_trans (le_abs_self _); apply le_trans (glow_limit _ _ _ _ _ _) |
| rw [norm_zero]; linarith |
| change ray (A+gain) (f y) (F y) ≤ ray A (f s) (F s)+X*e.first z+Y*e.second r q z+2*k+4*C |
| rw [hA,hB] |
| apply (ray_shift A (f s) gain plusf (F s) plusm _ (by positivity) HG).trans |
| change ray A _ _+glow' gain plusf plusm+_ ≤ _ |
| rw [HL]; linarith |
| |
| lemma flash (e : Lamp M) (h : Climb e.site) (r : Bool) (b : Beat M) |
| (hb : b.pre=e.site.smaller r) (hq : Ideal b.pre b.post) |
| (ha : 0 < e.width) (he : e.width ≤ 1) : |
| let s := e.site |
| let u := e.u |
| let a := e.width |
| let q := b.post |
| let t := b.bit |
| let k := killed b |
| let C := 20*(fax s+fax q)+400*((k+shower u s+shower u q)/a) |
| watt (winds b) u (carry e r t) a ≤ watt s u e.W a + curl r *start e + |
| curl t *delayed e r q + 2*k + 4*C := by |
| intro s u a q t k C |
| let K z := matter (s.smaller r) z - matter q z |
| have hK : k=∑ z,K z := by |
| unfold K k; rw [hq.kill_eq]; rw [← matter_sum hq.fit, ← matter_sum hq.climb.fit, |
| Finset.sum_sub_distrib] |
| rw [hb] |
| let B z := 20*(pierce s z+pierce q z)+400*((K z+dew u s z+dew u q z)/a) |
| have hC : C=∑ z,B z := by |
| unfold C B fax shower |
| rw [hK] |
| rw [Finset.sum_add_distrib,←Finset.mul_sum,←Finset.mul_sum,← Finset.sum_div]; simp_rw [Finset.sum_add_distrib] |
| rfl |
| rw [hK,hC,watt,watt,start,delayed, Finset.mul_sum, Finset.mul_sum, Finset.mul_sum,Finset.mul_sum, |
| ← Finset.sum_add_distrib, ← Finset.sum_add_distrib,← Finset.sum_add_distrib,← Finset.sum_add_distrib] |
| apply Finset.sum_le_sum; intro i _ |
| exact flash_cell e h r t q (hb ▸ hq) ha he i |
| |
| def lit (e : Lamp M) (s : Scene M) := (energy s e.u + e.width^2)*fax s + shower e.u s |
|
|
| lemma start_cost (e : Lamp M) (h : Climb e.site) (ha : 0 ≤ e.width) : |
| start e^2 ≤ 100*lit e e.site := by |
| let a := e.width; let W := e.W; let s := e.site; let u := e.u |
| let x := ∑ z,‖factor s u z‖; let y := ∑ z, |shell s z| |
| have H : |start e| ≤ x + a*y := by |
| apply le_trans (Finset.abs_sum_le_sum_abs _ _) |
| unfold x y; rw [Finset.mul_sum,← Finset.sum_add_distrib] |
| apply Finset.sum_le_sum; intro i _ |
| apply (glow_limit _ _ _ _ _ _).trans |
| rw [norm_zero,zero_add,abs_mul,abs_of_nonneg ha] |
| have hX : x^2 ≤ _ := factor_bill _ h u |
| have hY : y^2 ≤ _ := shell_bill _ h |
| have hT : (start e)^2 ≤ (x+a*y)^2 := by |
| rw [←sq_abs]; exact pow_le_pow_left₀ (abs_nonneg _) H _ |
| have hV : (a*y)^2 ≤ a^2*(2*fax s) := by rw [mul_pow]; gcongr |
| change _ ≤ 100*((energy s u+a^2)*fax s+shower u s) |
| nlinarith [mul_nonneg (energy_small h.fit u).1 (fax_floor h).1,shower_nonneg (u:=u) h,sq_nonneg (x-a*y), |
| mul_nonneg (sq_nonneg a) (fax_floor h).1] |
|
|
| lemma delayed_cost (e : Lamp M) (h : Climb e.site) {q : Scene M} (hq : Climb q) |
| (ha : 0 ≤ e.width) (r : Bool) : |
| delayed e r q^2 ≤ 100*(lit e e.site+lit e q) := by |
| let a := e.width; let W := e.W; let s := e.site; let u := e.u |
| let x (s : Scene M) := ∑ z,‖factor s u z‖ |
| let y := ∑ z, |shell q z| |
| have H : |delayed e r q| ≤ x s+x q + a*y := by |
| apply le_trans (Finset.abs_sum_le_sum_abs _ _) |
| unfold x y; rw [Finset.mul_sum,← Finset.sum_add_distrib,← Finset.sum_add_distrib] |
| apply Finset.sum_le_sum; intro i _ |
| apply (glow_limit _ _ _ _ _ _).trans |
| rw [norm_scaleC,curl_unit,one_mul,abs_mul,abs_of_nonneg ha] |
| have hX₁ : x s^2 ≤ _ := factor_bill _ h u |
| have hX₂ : x q^2 ≤ _ := factor_bill _ hq u |
| have hY : y^2 ≤ _ := shell_bill _ hq |
| have hT : delayed e r q^2 ≤ (x s+x q+a*y)^2 := by |
| rw [←sq_abs]; exact pow_le_pow_left₀ (abs_nonneg _) H _ |
| have hV : (a*y)^2 ≤ a^2*(2*fax q) := by rw [mul_pow]; gcongr |
| let A (s : Scene M) := (energy s u+a^2)*fax s+shower u s |
| have HI (w : Scene M) (hw : Climb w) : 0 ≤ energy w u*fax w ∧ 0 ≤ a^2*fax w := |
| ⟨mul_nonneg (energy_small hw.fit u).1 (fax_floor hw).1,mul_nonneg (sq_nonneg a) (fax_floor hw).1⟩ |
| have h1 := HI s h; have h2 := HI q hq |
| change _ ≤ 100*((energy s u+a^2)*fax s+shower u s+((energy q u+a^2)*fax q+shower u q)) |
| nlinarith [shower_nonneg (u:=u) h, shower_nonneg (u:=u) hq,sq_nonneg (x s-x q),sq_nonneg (x q-a*y),sq_nonneg (x s-a*y)] |
|
|
| end Lamp |
| end Beam |
|
|
|
|
| end FourierLLogL.Dyadic.Procedure.Paths |
|
|
| end OAI |
|
|
