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namespace OAI
section
namespace BinaryMatching.Algorithm
noncomputable section
open Finset
def Piece.evalReal (p : Piece) (t : ℝ) : ℝ := (p.slope:ℝ)*t+(p.constant:ℝ)
def selectedReal (k : ℕ) (t : ℝ) : Piece := by
classical
exact ((pieces k).argmin (fun p => p.evalReal t)).getD ⟨k,0⟩
def psiReal (k : ℕ) (t : ℝ) : ℝ := (selectedReal k t).evalReal t
def dyadic (j : ℕ) : ℝ := (2:ℝ)⁻¹^j
lemma pieces_nonempty (k : ℕ) : pieces k ≠ [] := by simp [pieces]
lemma selectedPiece_mem_argmin (k : ℕ) (t : ℚ) :
selectedPiece k t ∈ (pieces k).argmin (fun p => p.eval t) := by
unfold selectedPiece
cases he : (pieces k).argmin (fun p => p.eval t) with
| none => exact (pieces_nonempty k (List.argmin_eq_none.mp he)).elim
| some p => simp
lemma selectedReal_mem_argmin (k : ℕ) (t : ℝ) :
selectedReal k t ∈ (pieces k).argmin (fun p => p.evalReal t) := by
classical
unfold selectedReal
cases he : (pieces k).argmin (fun p => p.evalReal t) with
| none => exact (pieces_nonempty k (List.argmin_eq_none.mp he)).elim
| some p => simp
lemma psiReal_le_piece (k : ℕ) (t : ℝ) (p : Piece) (hp : p∈pieces k) :
psiReal k t ≤ p.evalReal t :=
List.le_of_mem_argmin hp (selectedReal_mem_argmin k t)
lemma le_psiReal (k : ℕ) (t b : ℝ) (h : ∀ p∈pieces k,b≤p.evalReal t) :
b ≤ psiReal k t := h _ (List.argmin_mem (selectedReal_mem_argmin k t))
lemma realPsi_cast (k : ℕ) (t : ℚ) : psiReal k (t:ℝ)=(psi k t:ℝ) := by
have cast_eval (p : Piece) : p.evalReal (t:ℝ)=(p.eval t:ℝ) := by
simp [Piece.eval,Piece.evalReal]
apply le_antisymm
· have h := psiReal_le_piece k (t:ℝ) _ (List.argmin_mem (selectedPiece_mem_argmin k t))
simpa only [cast_eval,psi] using h
· apply le_psiReal
intro p hp
rw [cast_eval]
exact_mod_cast (List.le_of_mem_argmin hp (selectedPiece_mem_argmin k t))
lemma psiReal_concave (k : ℕ) : ConcaveOn ℝ Set.univ (psiReal k) := by
refine ⟨convex_univ,?_⟩
intro x _ y _ a b ha hb hab
apply le_psiReal
intro p hp
have hx := psiReal_le_piece k x p hp
have hy := psiReal_le_piece k y p hp
simp only [smul_eq_mul]
calc
a*psiReal k x+b*psiReal k y ≤ a*p.evalReal x+b*p.evalReal y := by gcongr
_ = p.evalReal (a*x+b*y) := by
simp only [Piece.evalReal]
have hh := congrArg (fun t => t*(p.constant:ℝ)) hab
nlinarith [hh]
lemma mem_pieces (k : ℕ) (p : Piece) :
p∈pieces k ↔ p=⟨k,0⟩ ∨ ∃ j<k,p=⟨(j:ℤ)-1,(2:ℚ)⁻¹^j⟩ := by
simp only [pieces,List.mem_append,List.mem_map,List.mem_range,List.mem_singleton]
aesop
lemma first_mem_pieces (k : ℕ) : (⟨k,0⟩:Piece)∈pieces k := (mem_pieces k _).2 (Or.inl rfl)
lemma chord_mem_pieces (k j : ℕ) (hj : j<k) :
(⟨(j:ℤ)-1,(2:ℚ)⁻¹^j⟩:Piece)∈pieces k := (mem_pieces k _).2 (Or.inr ⟨j,hj,rfl⟩)
lemma chord_eval (j : ℕ) (t : ℝ) :
(⟨(j:ℤ)-1,(2:ℚ)⁻¹^j⟩:Piece).evalReal t=((j:ℝ)-1)*t+dyadic j := by
simp [Piece.evalReal,dyadic]
lemma first_eval (k : ℕ) (t : ℝ) : (⟨k,0⟩:Piece).evalReal t=(k:ℝ)*t := by
simp [Piece.evalReal]
lemma dyadic_pos (j : ℕ) : 0<dyadic j := by unfold dyadic; positivity
lemma dyadic_nonneg (j : ℕ) : 0≤dyadic j := (dyadic_pos j).le
lemma dyadic_succ (j : ℕ) : dyadic (j+1)=dyadic j/2 := by simp [dyadic,pow_succ,div_eq_mul_inv]
lemma dyadic_zero : dyadic 0=1 := by simp [dyadic]
lemma dyadic_mul_pow (j : ℕ) : dyadic j*(2:ℝ)^j=1 := by
rw [dyadic,←mul_pow]; norm_num
lemma dyadic_scale (j l : ℕ) (h : l≤j) : dyadic l=(2:ℝ)^(j-l)*dyadic j := by
have hj : j=l+(j-l) := by omega
conv_rhs => rw [hj,dyadic,pow_add]
rw [Nat.add_sub_cancel_left]
change (2:ℝ)⁻¹^l=(2:ℝ)^(j-l)*((2:ℝ)⁻¹^l*(2:ℝ)⁻¹^(j-l))
calc
_ = (2:ℝ)⁻¹^l*1 := (mul_one _).symm
_ = _ := by rw [←dyadic_mul_pow (j-l)]; unfold dyadic; ring
lemma chord_above_knot (l j : ℕ) :
(j:ℝ)*dyadic j ≤ ((l:ℝ)-1)*dyadic j+dyadic l := by
rcases lt_trichotomy l j with h|h|h
· have hp : 1+(j-l:ℕ) ≤ (2:ℕ)^(j-l) := by
have hh := one_add_mul_le_pow (show (-2:ℝ)≤1 by norm_num) (j-l)
norm_num at hh
exact_mod_cast hh
have hpR : (1:ℝ)+(j:ℝ)-(l:ℝ) ≤ (2:ℝ)^(j-l) := by
have := Nat.cast_le (α:=ℝ).mpr hp
push_cast [Nat.cast_sub h.le] at this
linarith
rw [dyadic_scale j l h.le]
nlinarith [mul_le_mul_of_nonneg_right hpR (dyadic_nonneg j)]
· subst l; ring_nf; rfl
· have hh : (j:ℝ)+1≤l := by exact_mod_cast h
nlinarith [dyadic_nonneg j,dyadic_nonneg l]
lemma psiReal_knot (k j : ℕ) (hj : j≤k) : psiReal k (dyadic j)=(j:ℝ)*dyadic j := by
apply le_antisymm
· rcases lt_or_eq_of_le hj with h|h
· have hh := psiReal_le_piece k (dyadic j) _ (chord_mem_pieces k j h)
rw [chord_eval] at hh
nlinarith
· subst j
simpa only [first_eval] using psiReal_le_piece k (dyadic k) _ (first_mem_pieces k)
· apply le_psiReal
intro p hp
rcases (mem_pieces k p).1 hp with rfl|⟨l,_,rfl⟩
· rw [first_eval]
exact mul_le_mul_of_nonneg_right (by exact_mod_cast hj) (dyadic_nonneg j)
· rw [chord_eval]; exact chord_above_knot l j
lemma psiReal_zero (k : ℕ) : psiReal k 0=0 := by
apply le_antisymm
· simpa only [first_eval,mul_zero] using psiReal_le_piece k 0 _ (first_mem_pieces k)
· apply le_psiReal
intro p hp
rcases (mem_pieces k p).1 hp with rfl|⟨j,_,rfl⟩
· simp [first_eval]
· simpa only [chord_eval,mul_zero,zero_add] using dyadic_nonneg j
end
end BinaryMatching.Algorithm
end
end OAI
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