| import OAI.MathematicalPhysics.AlternatingFlow.BaseNames |
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| namespace OAI |
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| section VelocityNamesDevelopment |
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| open scoped BigOperators Topology ContDiff |
| namespace AlternatingNS.Effective |
| attribute [local instance] Arithmetic.rationalCoding |
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| variable {A : Type*} [Primcodable A] |
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| lemma JetBound.curl_named {f : A → Space → ℝ} (hf : JetBound f) |
| (hn : Named (fun z : A × RationalPoint => f z.1 (rationalPoint z.2).2)) |
| (i j : A → Fin 3) (hi : Computable i) (hj : Computable j) : |
| Named (fun z : (A × Fin 3) × RationalPoint => |
| Spatial.curl (i z.1.1) (j z.1.1) (f z.1.1) (rationalPoint z.2).2 z.1.2) := by |
| have hp : Computable (fun z : (A × Fin 3) × RationalPoint => (z.1.1,z.2)) := |
| (Computable.fst.comp Computable.fst).pair Computable.snd |
| have hdj := (hf.spatial_d_named hn j hj).comp hp |
| have hdi := (hf.spatial_d_named hn i hi).comp hp |
| have hqi : Computable (fun z : (A × Fin 3) × RationalPoint => decide (z.1.2 = i z.1.1)) := Primrec.eq.decide.to_comp.comp (Computable.snd.comp Computable.fst) |
| (hi.comp (Computable.fst.comp Computable.fst)) |
| have hqj : Computable (fun z : (A × Fin 3) × RationalPoint => decide (z.1.2 = j z.1.1)) := Primrec.eq.decide.to_comp.comp (Computable.snd.comp Computable.fst) |
| (hj.comp (Computable.fst.comp Computable.fst)) |
| exact ((Named.ite _ hqi hdj (Named.const 0)).sub |
| (Named.ite _ hqj hdi (Named.const 0))).congr (by |
| intro z; simp [Spatial.curl, Spatial.e_apply]) |
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| lemma loading_named : Named (fun z : ((Machine × List ℕ) × Fin 3) × RationalPoint => |
| Construction.loading z.1.1.1 z.1.1.2 (rationalPoint z.2).2 z.1.2) := by |
| exact loadPotential_bound.curl_named loadPotential_named (fun _ => 1) (fun _ => 0) |
| (Computable.const 1) (Computable.const 0) |
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| lemma slot_named : Named (fun z : (((Machine × ℕ) × ℕ) × Fin 3) × RationalPoint => |
| Construction.slot z.1.1.1.1 z.1.1.1.2 z.1.1.2 (rationalPoint z.2).2 z.1.2) := by |
| exact slotPotential_bound.curl_named slotPotential_named |
| (fun p => Construction.receiverIndex p.2) (fun _ => 0) |
| (receiver_computable.comp Computable.snd) (Computable.const 0) |
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| lemma piece_named : Named (fun z : (((Machine × List ℕ) × ℕ) × Fin 3) × RationalPoint => |
| Construction.piece z.1.1.1.1 z.1.1.1.2 z.1.1.2 (rationalPoint z.2).2 z.1.2) := by |
| let P := (((Machine × List ℕ) × ℕ) × Fin 3) × RationalPoint |
| have hM : Computable (fun z : P => z.1.1.1.1) := |
| Computable.fst.comp (Computable.fst.comp (Computable.fst.comp Computable.fst)) |
| have hw : Computable (fun z : P => z.1.1.1.2) := |
| Computable.snd.comp (Computable.fst.comp (Computable.fst.comp Computable.fst)) |
| have hn : Computable (fun z : P => z.1.1.2) := |
| Computable.snd.comp (Computable.fst.comp Computable.fst) |
| have hi : Computable (fun z : P => z.1.2) := Computable.snd.comp Computable.fst |
| have hload := loading_named.comp (((hM.pair hw).pair hi).pair Computable.snd) |
| have hslot := slot_named.comp ((((hM.pair (Computable.list_length.comp hw)).pair |
| (Primrec.nat_sub.to_comp.comp hn (Computable.const 1))).pair hi).pair Computable.snd) |
| exact (Named.ite (fun z : P => z.1.1.2 = 0) |
| (Primrec.eq.decide.to_comp.comp hn (Computable.const 0)) hload hslot).congr |
| (by rintro ⟨⟨⟨⟨M,w⟩,n⟩,i⟩,q⟩; cases n <;> rfl) |
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| lemma clock_bound : JetBound (fun _ : Unit => Profiles.clock) := by |
| have hclock := transition_scaled.affine (fun _ _ => zero_le_one) (fun _ _ => 2) |
| (fun _ _ => -1/2) (fun _ => 2) (Computable.const 2) (by intros; norm_num) |
| obtain ⟨hs,B,hB,h⟩ := hclock |
| refine ⟨fun _ => Profiles.clock_smooth, B, hB, fun p r x => ?_⟩ |
| have he : (fun t => Real.smoothTransition (2 * t + -1/2)) = Profiles.clock := by |
| funext t; unfold Profiles.clock; congr 1; ring |
| simpa only [one_pow, mul_one, he] using h p r 0 x |
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| lemma pulse_rational_named : Named (fun q : ℚ => Profiles.pulse (q : ℝ)) := by |
| have hf := clock_bound.comp_unit (fun _ => ContinuousLinearMap.fst ℝ ℝ Space) (fun _ => norm_fst) |
| have hn : Named (fun z : Unit × RationalPoint => Profiles.clock (rationalPoint z.2).1) := by |
| have ht := Named.rational (Computable.fst.comp (Computable.snd : Computable (Prod.snd : Unit × RationalPoint → _))) |
| exact (((Named.const 2).mul ht).sub (Named.const (1/2))).transition.congr (by |
| intro z; simp [Profiles.clock, rationalPoint]) |
| have hd := hf.named_directional hn (fun _ => 0) (Computable.const 0) |
| have hdc := hd.comp ((Computable.const ()).pair (Computable.id.pair |
| ((Computable.const (0:ℚ)).pair ((Computable.const (0:ℚ)).pair (Computable.const (0:ℚ)))))) |
| apply hdc.congr |
| intro q |
| change fderiv ℝ (Profiles.clock ∘ Prod.fst) ((q:ℝ),_) (1,0) = deriv Profiles.clock (q:ℝ) |
| rw [fderiv_comp _ (Profiles.clock_smooth.differentiable (by simp) _) differentiableAt_fst, fderiv_fst] |
| rfl |
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| lemma velocity_named : Named (fun z : ((Machine × List ℕ) × Fin 3) × RationalPoint => |
| Construction.velocity z.1.1.1 z.1.1.2 (rationalPoint z.2).1 (rationalPoint z.2).2 z.1.2) := by |
| let P := ((Machine × List ℕ) × Fin 3) × RationalPoint |
| have hq : Computable (fun z : P => z.2.1) := Computable.fst.comp Computable.snd |
| let N : P → ℕ := fun z => z.2.1.num.natAbs+1 |
| have hN : Computable N := Computable.succ.comp (Arithmetic.int_abs.to_comp.comp (Arithmetic.rat_num.to_comp.comp hq)) |
| have hterm : Named (fun z : P × ℕ => Profiles.pulse ((rationalPoint z.1.2).1 - z.2) * |
| Construction.piece z.1.1.1.1 z.1.1.1.2 z.2 (rationalPoint z.1.2).2 z.1.1.2) := by |
| have hh := pulse_rational_named.comp (Arithmetic.rat_sub.to_comp.comp |
| (hq.comp Computable.fst) (Arithmetic.rat_natCast.to_comp.comp Computable.snd)) |
| have hp := piece_named.comp |
| ((((Computable.fst.comp (Computable.fst.comp Computable.fst)).pair Computable.snd).pair |
| (Computable.snd.comp (Computable.fst.comp Computable.fst))).pair |
| (Computable.snd.comp Computable.fst)) |
| exact (hh.mul hp).congr (by intro z; simp [rationalPoint]) |
| apply (hterm.sum_range N hN).congr |
| intro z |
| have ht : (rationalPoint z.2).1 ≤ (N z : ℝ) := by |
| have hr := le_trans (le_abs_self (z.2.1 : ℝ)) (abs_rat_cast_le_num z.2.1) |
| dsimp [N, rationalPoint]; push_cast; linarith |
| rw [Construction.velocity, TimeGlue.eq_sum _ Profiles.pulse_zero_left _ _ _ (N z) ht] |
| simp [WithLp.ofLp_sum, Finset.sum_apply] |
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| end AlternatingNS.Effective |
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| end VelocityNamesDevelopment |
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| end OAI |
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