import Erdos1135.ND.Band.A5Sweep /-! # A5 Full Strict Tube This leaf freezes the literal finite natural-index tube for the strict A5 affine selector. It deliberately defines the full tube before any physical band interval is imposed. The radius is `ceil W - 1`, so integral widths use the source's strict endpoint convention. -/ namespace Erdos1135 namespace ND /-- The largest natural absolute value strictly below the real tube width. -/ noncomputable def ndA5TubeRadius (W : ℝ) : ℕ := Nat.ceil W - 1 /-- The strict integer radius loses less than one from a positive real width. This is the scalar form of the `ceil W - 1` convention. -/ theorem ndA5TubeWidth_sub_one_le_radius {W : ℝ} (hW : 0 < W) : W - 1 ≤ (ndA5TubeRadius W : ℝ) := by have hceil : 0 < Nat.ceil W := Nat.ceil_pos.mpr hW have hradius : ndA5TubeRadius W + 1 = Nat.ceil W := by unfold ndA5TubeRadius omega have hceilLower : W ≤ (Nat.ceil W : ℝ) := Nat.le_ceil W rw [← hradius] at hceilLower push_cast at hceilLower linarith /-- Above width two, the strict integer radius retains at least half of the real width. -/ theorem ndA5TubeWidth_div_two_le_radius {W : ℝ} (hW : 2 ≤ W) : W / 2 ≤ (ndA5TubeRadius W : ℝ) := by calc W / 2 ≤ W - 1 := by linarith _ ≤ (ndA5TubeRadius W : ℝ) := ndA5TubeWidth_sub_one_le_radius (by linarith) /-- A finite upper cap for all natural indices in the strict affine tube. -/ noncomputable def ndA5TubeUpperCap (A W : ℝ) : ℕ := (Int.floor ((A + (ndA5TubeRadius W : ℝ) + 1) / ndA5PhaseDelta)).toNat /-- The natural lower endpoint obtained from the strict lower real bound. Negative signed endpoints are clipped to zero by `Int.toNat`. -/ noncomputable def ndA5TubeLo (A W : ℝ) : ℕ := (Int.floor ((A - (ndA5TubeRadius W : ℝ)) / ndA5PhaseDelta) + 1).toNat /-- The natural upper endpoint obtained from the weak upper real bound. -/ noncomputable def ndA5TubeHi (A W : ℝ) : ℕ := ndA5TubeUpperCap A W /-- The full finite natural-index tube for the strict affine selector. -/ noncomputable def ndA5FullTube (A W : ℝ) : Finset ℕ := (Finset.range (ndA5TubeUpperCap A W + 1)).filter (fun nu => |(ndA5StrictAffineSweep A nu : ℝ)| < W) private theorem intCast_abs_lt_iff_natAbs_le_tubeRadius {W : ℝ} (hW : 0 < W) (d : ℤ) : |(d : ℝ)| < W ↔ Int.natAbs d ≤ ndA5TubeRadius W := by have hceil : 0 < Nat.ceil W := Nat.ceil_pos.mpr hW have hradius : ndA5TubeRadius W + 1 = Nat.ceil W := by unfold ndA5TubeRadius omega calc |(d : ℝ)| < W ↔ (Int.natAbs d : ℝ) < W := by rw [Nat.cast_natAbs] norm_cast _ ↔ Int.natAbs d < Nat.ceil W := Nat.lt_ceil.symm _ ↔ Int.natAbs d ≤ ndA5TubeRadius W := by omega private theorem natAbs_le_iff_signed_bounds (d : ℤ) (r : ℕ) : Int.natAbs d ≤ r ↔ -(r : ℤ) ≤ d ∧ d ≤ (r : ℤ) := by rw [← Int.ofNat_le, Int.natCast_natAbs] exact abs_le /-- The strict integer tube is exactly the asymmetric pair of affine real bounds dictated by `floor x + 1`. -/ theorem ndA5StrictAffineSweep_abs_lt_iff_affine {W : ℝ} (hW : 0 < W) (A : ℝ) (nu : ℕ) : |(ndA5StrictAffineSweep A nu : ℝ)| < W ↔ (A - (ndA5TubeRadius W : ℝ)) / ndA5PhaseDelta < (nu : ℝ) ∧ (nu : ℝ) ≤ (A + (ndA5TubeRadius W : ℝ) + 1) / ndA5PhaseDelta := by rw [intCast_abs_lt_iff_natAbs_le_tubeRadius hW] rw [natAbs_le_iff_signed_bounds] unfold ndA5StrictAffineSweep constructor · rintro ⟨hlower, hupper⟩ have hlowerFloor : -((ndA5TubeRadius W : ℤ)) - 1 ≤ Int.floor (A - ndA5PhaseDelta * (nu : ℝ)) := by omega have hupperFloor : Int.floor (A - ndA5PhaseDelta * (nu : ℝ)) < (ndA5TubeRadius W : ℤ) := by omega have hlowerReal := Int.le_floor.mp hlowerFloor have hupperReal := Int.floor_lt.mp hupperFloor constructor · apply (div_lt_iff₀ ndA5PhaseDelta_mem_Ioo.1).2 push_cast at hupperReal ⊢ linarith · apply (le_div_iff₀ ndA5PhaseDelta_mem_Ioo.1).2 push_cast at hlowerReal ⊢ linarith · rintro ⟨hlower, hupper⟩ have hupperReal : A - ndA5PhaseDelta * (nu : ℝ) < (ndA5TubeRadius W : ℤ) := by have h := (div_lt_iff₀ ndA5PhaseDelta_mem_Ioo.1).mp hlower push_cast at h ⊢ linarith have hlowerReal : (-((ndA5TubeRadius W : ℤ)) - 1 : ℤ) ≤ A - ndA5PhaseDelta * (nu : ℝ) := by have h := (le_div_iff₀ ndA5PhaseDelta_mem_Ioo.1).mp hupper push_cast at h ⊢ linarith have hupperFloor : Int.floor (A - ndA5PhaseDelta * (nu : ℝ)) < (ndA5TubeRadius W : ℤ) := Int.floor_lt.mpr hupperReal have hlowerFloor : -((ndA5TubeRadius W : ℤ)) - 1 ≤ Int.floor (A - ndA5PhaseDelta * (nu : ℝ)) := Int.le_floor.mpr hlowerReal constructor <;> omega /-- The explicit range cap is total but does not remove any point satisfying the strict tube predicate. -/ theorem mem_ndA5FullTube_iff {W : ℝ} (hW : 0 < W) (A : ℝ) (nu : ℕ) : nu ∈ ndA5FullTube A W ↔ |(ndA5StrictAffineSweep A nu : ℝ)| < W := by constructor · intro h exact (Finset.mem_filter.mp h).2 · intro htube have hbounds := (ndA5StrictAffineSweep_abs_lt_iff_affine hW A nu).mp htube have hnuFloor : (nu : ℤ) ≤ Int.floor ((A + (ndA5TubeRadius W : ℝ) + 1) / ndA5PhaseDelta) := Int.le_floor.mpr hbounds.2 have hfloor : 0 ≤ Int.floor ((A + (ndA5TubeRadius W : ℝ) + 1) / ndA5PhaseDelta) := by omega have hnuCap : nu ≤ ndA5TubeUpperCap A W := by unfold ndA5TubeUpperCap exact (Int.le_toNat hfloor).mpr hnuFloor apply Finset.mem_filter.mpr exact ⟨Finset.mem_range.mpr (Nat.lt_succ_of_le hnuCap), htube⟩ /-- Every positive affine phase has a natural point in a strict tube whose width is larger than the unit strict-selector witness. -/ theorem ndA5FullTube_nonempty_of_phase_pos {A W : ℝ} (hA : 0 < A) (hW : 1 < W) : (ndA5FullTube A W).Nonempty := by have hWpos : 0 < W := by linarith let q := Nat.floor (A / ndA5PhaseDelta) have hcenterNonneg : 0 ≤ A / ndA5PhaseDelta := (div_pos hA ndA5PhaseDelta_mem_Ioo.1).le have hqLower : (q : ℝ) ≤ A / ndA5PhaseDelta := by dsimp [q] exact Nat.floor_le hcenterNonneg have hqUpper : A / ndA5PhaseDelta < (q : ℝ) + 1 := by dsimp [q] simpa using Nat.lt_floor_add_one (A / ndA5PhaseDelta) have hqLowerMul : ndA5PhaseDelta * (q : ℝ) ≤ A := by simpa [mul_comm] using (le_div_iff₀ ndA5PhaseDelta_mem_Ioo.1).mp hqLower have hqUpperMul : A < ndA5PhaseDelta * ((q : ℝ) + 1) := by simpa [mul_comm] using (div_lt_iff₀ ndA5PhaseDelta_mem_Ioo.1).mp hqUpper have hremainderLower : 0 ≤ A - ndA5PhaseDelta * (q : ℝ) := by linarith have hremainderUpper : A - ndA5PhaseDelta * (q : ℝ) < ndA5PhaseDelta := by nlinarith have hfloor : Int.floor (A - ndA5PhaseDelta * (q : ℝ)) = 0 := by rw [Int.floor_eq_iff] constructor · simpa using hremainderLower · simpa using hremainderUpper.trans ndA5PhaseDelta_mem_Ioo.2 have hsweep : ndA5StrictAffineSweep A q = 1 := by unfold ndA5StrictAffineSweep rw [hfloor] omega refine ⟨q, (mem_ndA5FullTube_iff hWpos A q).mpr ?_⟩ rw [hsweep] norm_num exact hW /-- A nonempty full strict tube is the exact natural interval between the strict lower and weak upper affine endpoints. -/ theorem ndA5FullTube_eq_Icc {A W : ℝ} (hW : 0 < W) (hne : (ndA5FullTube A W).Nonempty) : ndA5FullTube A W = Finset.Icc (ndA5TubeLo A W) (ndA5TubeHi A W) := by obtain ⟨nu₀, hnu₀⟩ := hne have hbounds₀ := (ndA5StrictAffineSweep_abs_lt_iff_affine hW A nu₀).mp ((mem_ndA5FullTube_iff hW A nu₀).mp hnu₀) have hnu₀Floor : (nu₀ : ℤ) ≤ Int.floor ((A + (ndA5TubeRadius W : ℝ) + 1) / ndA5PhaseDelta) := Int.le_floor.mpr hbounds₀.2 have hfloor : 0 ≤ Int.floor ((A + (ndA5TubeRadius W : ℝ) + 1) / ndA5PhaseDelta) := by omega ext nu rw [mem_ndA5FullTube_iff hW] rw [Finset.mem_Icc] constructor · intro htube have hbounds := (ndA5StrictAffineSweep_abs_lt_iff_affine hW A nu).mp htube constructor · unfold ndA5TubeLo apply Int.toNat_le.mpr have hfloorLower : Int.floor ((A - (ndA5TubeRadius W : ℝ)) / ndA5PhaseDelta) < (nu : ℤ) := Int.floor_lt.mpr hbounds.1 omega · unfold ndA5TubeHi ndA5TubeUpperCap apply (Int.le_toNat hfloor).mpr exact Int.le_floor.mpr hbounds.2 · rintro ⟨hlower, hupper⟩ apply (ndA5StrictAffineSweep_abs_lt_iff_affine hW A nu).mpr constructor · unfold ndA5TubeLo at hlower have hlowerZ := Int.toNat_le.mp hlower apply Int.floor_lt.mp omega · unfold ndA5TubeHi ndA5TubeUpperCap at hupper have hupperZ := (Int.le_toNat hfloor).mp hupper exact Int.le_floor.mp hupperZ private theorem card_ndA5FullTube_eq_intFloor_sub_intFloor {A W : ℝ} (hW : 0 < W) (hne : (ndA5FullTube A W).Nonempty) (hlo : 0 < ndA5TubeLo A W) : ((ndA5FullTube A W).card : ℝ) = (Int.floor ((A + (ndA5TubeRadius W : ℝ) + 1) / ndA5PhaseDelta) : ℝ) - (Int.floor ((A - (ndA5TubeRadius W : ℝ)) / ndA5PhaseDelta) : ℝ) := by have hEq := ndA5FullTube_eq_Icc hW hne have hlohi : ndA5TubeLo A W ≤ ndA5TubeHi A W := by rw [← Finset.nonempty_Icc] rw [← hEq] exact hne have hloFloorNonneg : 0 ≤ Int.floor ((A - (ndA5TubeRadius W : ℝ)) / ndA5PhaseDelta) + 1 := by unfold ndA5TubeLo at hlo exact (Int.pos_iff_toNat_pos.mpr hlo).le have hloInt : (ndA5TubeLo A W : ℤ) = Int.floor ((A - (ndA5TubeRadius W : ℝ)) / ndA5PhaseDelta) + 1 := by unfold ndA5TubeLo exact Int.toNat_of_nonneg hloFloorNonneg have hhiPos : 0 < ndA5TubeHi A W := hlo.trans_le hlohi have hhiFloorNonneg : 0 ≤ Int.floor ((A + (ndA5TubeRadius W : ℝ) + 1) / ndA5PhaseDelta) := by unfold ndA5TubeHi ndA5TubeUpperCap at hhiPos exact (Int.pos_iff_toNat_pos.mpr hhiPos).le have hhiInt : (ndA5TubeHi A W : ℤ) = Int.floor ((A + (ndA5TubeRadius W : ℝ) + 1) / ndA5PhaseDelta) := by unfold ndA5TubeHi ndA5TubeUpperCap exact Int.toNat_of_nonneg hhiFloorNonneg have hcardInt : ((ndA5FullTube A W).card : ℤ) = Int.floor ((A + (ndA5TubeRadius W : ℝ) + 1) / ndA5PhaseDelta) - Int.floor ((A - (ndA5TubeRadius W : ℝ)) / ndA5PhaseDelta) := by rw [hEq, Nat.card_Icc] rw [Int.ofNat_sub (by omega : ndA5TubeLo A W ≤ ndA5TubeHi A W + 1)] push_cast rw [hloInt, hhiInt] ring norm_cast private theorem abs_intFloor_sub_intFloor_sub_sub_le_one (upper lower : ℝ) : |((Int.floor upper : ℝ) - (Int.floor lower : ℝ)) - (upper - lower)| ≤ 1 := by have hUpperFloor := Int.floor_le upper have hUpperCeil := Int.lt_floor_add_one upper have hLowerFloor := Int.floor_le lower have hLowerCeil := Int.lt_floor_add_one lower push_cast at hUpperFloor hUpperCeil hLowerFloor hLowerCeil rw [abs_le] constructor <;> linarith /-- The cardinality of an unclipped full strict tube has the frozen v10 `K_I = 2 + 1 / delta` error. The positive lower endpoint is load-bearing: without it, natural-index clipping can lose order `W` points. -/ theorem ndA5FullTube_card_error {A W : ℝ} (hW : 0 < W) (hne : (ndA5FullTube A W).Nonempty) (hlo : 0 < ndA5TubeLo A W) : |((ndA5FullTube A W).card : ℝ) - 2 * W / ndA5PhaseDelta| ≤ 2 + 1 / ndA5PhaseDelta := by have hcard := card_ndA5FullTube_eq_intFloor_sub_intFloor hW hne hlo have hceil : 0 < Nat.ceil W := Nat.ceil_pos.mpr hW have hradius : ndA5TubeRadius W + 1 = Nat.ceil W := by unfold ndA5TubeRadius omega have hradiusLt : (ndA5TubeRadius W : ℝ) < W := by have h := Nat.ceil_lt_add_one hW.le rw [← hradius] at h push_cast at h linarith have hWLeRadiusAddOne : W ≤ (ndA5TubeRadius W : ℝ) + 1 := by have h := Nat.le_ceil W rw [← hradius] at h push_cast at h linarith have hradiusError : |(2 * (ndA5TubeRadius W : ℝ) + 1) - 2 * W| ≤ 1 := by rw [abs_le] constructor <;> linarith have hdelta : 0 < ndA5PhaseDelta := ndA5PhaseDelta_mem_Ioo.1 have hwidthError : |(2 * (ndA5TubeRadius W : ℝ) + 1) / ndA5PhaseDelta - 2 * W / ndA5PhaseDelta| ≤ 1 / ndA5PhaseDelta := by rw [← sub_div, abs_div, abs_of_pos hdelta] exact (div_le_div_iff_of_pos_right hdelta).mpr hradiusError let lower : ℝ := (A - (ndA5TubeRadius W : ℝ)) / ndA5PhaseDelta let upper : ℝ := (A + (ndA5TubeRadius W : ℝ) + 1) / ndA5PhaseDelta have hspan : upper - lower = (2 * (ndA5TubeRadius W : ℝ) + 1) / ndA5PhaseDelta := by dsimp [upper, lower] ring have hfloorError : |((Int.floor upper : ℝ) - (Int.floor lower : ℝ)) - (upper - lower)| ≤ 1 := abs_intFloor_sub_intFloor_sub_sub_le_one upper lower have hspanError : |(upper - lower) - 2 * W / ndA5PhaseDelta| ≤ 1 / ndA5PhaseDelta := by rw [hspan] exact hwidthError rw [hcard] change |((Int.floor upper : ℝ) - (Int.floor lower : ℝ)) - 2 * W / ndA5PhaseDelta| ≤ 2 + 1 / ndA5PhaseDelta calc |((Int.floor upper : ℝ) - (Int.floor lower : ℝ)) - 2 * W / ndA5PhaseDelta| = |(((Int.floor upper : ℝ) - (Int.floor lower : ℝ)) - (upper - lower)) + ((upper - lower) - 2 * W / ndA5PhaseDelta)| := by congr 1 ring _ ≤ |((Int.floor upper : ℝ) - (Int.floor lower : ℝ)) - (upper - lower)| + |(upper - lower) - 2 * W / ndA5PhaseDelta| := abs_add_le _ _ _ ≤ 1 + 1 / ndA5PhaseDelta := add_le_add hfloorError hspanError _ ≤ 2 + 1 / ndA5PhaseDelta := by linarith /-- If the nonempty full tube is not clipped at zero on the left, its two natural endpoints attain the exact signed radius values. -/ theorem ndA5FullTube_endpoint_crossings {A W : ℝ} (hW : 0 < W) (hne : (ndA5FullTube A W).Nonempty) (hlo : 0 < ndA5TubeLo A W) : ndA5StrictAffineSweep A (ndA5TubeLo A W) = (ndA5TubeRadius W : ℤ) ∧ ndA5StrictAffineSweep A (ndA5TubeHi A W) = -(ndA5TubeRadius W : ℤ) := by have hEq := ndA5FullTube_eq_Icc hW hne have hlohi : ndA5TubeLo A W ≤ ndA5TubeHi A W := by rw [← Finset.nonempty_Icc] rw [← hEq] exact hne have hloMem : ndA5TubeLo A W ∈ ndA5FullTube A W := by rw [hEq] exact Finset.left_mem_Icc.mpr hlohi have hhiMem : ndA5TubeHi A W ∈ ndA5FullTube A W := by rw [hEq] exact Finset.right_mem_Icc.mpr hlohi have hsigned (nu : ℕ) (hnu : nu ∈ ndA5FullTube A W) : -(ndA5TubeRadius W : ℤ) ≤ ndA5StrictAffineSweep A nu ∧ ndA5StrictAffineSweep A nu ≤ (ndA5TubeRadius W : ℤ) := by apply (natAbs_le_iff_signed_bounds _ _).mp apply (intCast_abs_lt_iff_natAbs_le_tubeRadius hW _).mp exact (mem_ndA5FullTube_iff hW A nu).mp hnu have hloBounds := hsigned (ndA5TubeLo A W) hloMem have hhiBounds := hsigned (ndA5TubeHi A W) hhiMem let p := ndA5TubeLo A W - 1 have hpSucc : p + 1 = ndA5TubeLo A W := by dsimp [p] omega have hpNot : p ∉ ndA5FullTube A W := by intro hp rw [hEq] at hp have hpLower := (Finset.mem_Icc.mp hp).1 dsimp [p] at hpLower omega have hpOutside : ndA5StrictAffineSweep A p < -(ndA5TubeRadius W : ℤ) ∨ (ndA5TubeRadius W : ℤ) < ndA5StrictAffineSweep A p := by by_contra h push Not at h apply hpNot apply (mem_ndA5FullTube_iff hW A p).mpr apply (intCast_abs_lt_iff_natAbs_le_tubeRadius hW _).mpr apply (natAbs_le_iff_signed_bounds _ _).mpr exact h have hpStep := ndA5StrictAffineSweep_succ A p rw [hpSucc] at hpStep have hloExact : ndA5StrictAffineSweep A (ndA5TubeLo A W) = (ndA5TubeRadius W : ℤ) := by rcases hpStep with hpStep | hpStep <;> omega let q := ndA5TubeHi A W + 1 have hqNot : q ∉ ndA5FullTube A W := by intro hq rw [hEq] at hq have hqUpper := (Finset.mem_Icc.mp hq).2 dsimp [q] at hqUpper omega have hqOutside : ndA5StrictAffineSweep A q < -(ndA5TubeRadius W : ℤ) ∨ (ndA5TubeRadius W : ℤ) < ndA5StrictAffineSweep A q := by by_contra h push Not at h apply hqNot apply (mem_ndA5FullTube_iff hW A q).mpr apply (intCast_abs_lt_iff_natAbs_le_tubeRadius hW _).mpr apply (natAbs_le_iff_signed_bounds _ _).mpr exact h have hhiStep := ndA5StrictAffineSweep_succ A (ndA5TubeHi A W) change ndA5StrictAffineSweep A q = ndA5StrictAffineSweep A (ndA5TubeHi A W) ∨ ndA5StrictAffineSweep A q = ndA5StrictAffineSweep A (ndA5TubeHi A W) - 1 at hhiStep have hhiExact : ndA5StrictAffineSweep A (ndA5TubeHi A W) = -(ndA5TubeRadius W : ℤ) := by rcases hhiStep with hhiStep | hhiStep <;> omega exact ⟨hloExact, hhiExact⟩ end ND end Erdos1135