ruc-ai4math/mathlib_handler_benchmark_410 huggingface dataset:
-- SetTheory.PGame.Impartial.equiv_or_fuzzy_zero
theorem equiv_or_fuzzy_zero : (G ≈ 0) ∨ G ‖ 0 
Arguments: (G : SetTheory.PGame) [G.Impartial]
Goal: G ≈ 0 ∨ G.Fuzzy 0
-- SetTheory.PGame.Impartial.not_equiv_zero_iff
@[simp]
theorem not_equiv_zero_iff : ¬(G ≈ 0) ↔ G ‖ 0 
Arguments: (G : SetTheory.PGame) [G.Impartial]
Goal: ¬G ≈ 0 ↔ G.Fuzzy 0
-- SetTheory.PGame.Impartial.not_fuzzy_zero_iff
@[simp]
theorem not_fuzzy_zero_iff : ¬G ‖ 0 ↔ (G ≈ 0) 
Arguments: (G : SetTheory.PGame) [G.Impartial]
Goal: ¬G.Fuzzy 0 ↔ G ≈ 0
-- SetTheory.PGame.Impartial.add_self
theorem add_self : G + G ≈ 0 
Arguments: (G : SetTheory.PGame) [G.Impartial]
Goal: G + G ≈ 0
-- SetTheory.PGame.Impartial.mk'_add_self
@[simp]
theorem mk'_add_self : (⟦G⟧ : Quotient setoid) + ⟦G⟧ = 0 
Arguments: (G : SetTheory.PGame) [G.Impartial]
Goal: ⟦G⟧ + ⟦G⟧ = 0
-- SetTheory.PGame.Impartial.equiv_iff_add_equiv_zero
theorem equiv_iff_add_equiv_zero (H : PGame) : (H ≈ G) ↔ (H + G ≈ 0) 
Arguments: (G : SetTheory.PGame) [G.Impartial] (H : SetTheory.PGame)
Goal: H ≈ G ↔ H + G ≈ 0
-- SetTheory.PGame.Impartial.equiv_iff_add_equiv_zero'
theorem equiv_iff_add_equiv_zero' (H : PGame) : (G ≈ H) ↔ (G + H ≈ 0) 
Arguments: (G : SetTheory.PGame) [G.Impartial] (H : SetTheory.PGame)
Goal: G ≈ H ↔ G + H ≈ 0
-- SetTheory.PGame.Impartial.le_zero_iff
theorem le_zero_iff {G : PGame} [G.Impartial] : G ≤ 0 ↔ 0 ≤ G 
Arguments: {G : SetTheory.PGame} [G.Impartial]
Goal: G ≤ 0 ↔ 0 ≤ G
-- SetTheory.PGame.Impartial.lf_zero_iff
theorem lf_zero_iff {G : PGame} [G.Impartial] : G ⧏ 0 ↔ 0 ⧏ G 
Arguments: {G : SetTheory.PGame} [G.Impartial]
Goal: G.LF 0 ↔ SetTheory.PGame.LF 0 G
-- SetTheory.PGame.Impartial.equiv_zero_iff_le
theorem equiv_zero_iff_le : (G ≈ 0) ↔ G ≤ 0 
Arguments: (G : SetTheory.PGame) [G.Impartial]
Goal: G ≈ 0 ↔ G ≤ 0
-- SetTheory.PGame.Impartial.fuzzy_zero_iff_lf
theorem fuzzy_zero_iff_lf : G ‖ 0 ↔ G ⧏ 0 
Arguments: (G : SetTheory.PGame) [G.Impartial]
Goal: G.Fuzzy 0 ↔ G.LF 0
-- SetTheory.PGame.Impartial.equiv_zero_iff_ge
theorem equiv_zero_iff_ge : (G ≈ 0) ↔ 0 ≤ G 
Arguments: (G : SetTheory.PGame) [G.Impartial]
Goal: G ≈ 0 ↔ 0 ≤ G
-- SetTheory.PGame.Impartial.fuzzy_zero_iff_gf
theorem fuzzy_zero_iff_gf : G ‖ 0 ↔ 0 ⧏ G 
Arguments: (G : SetTheory.PGame) [G.Impartial]
Goal: G.Fuzzy 0 ↔ SetTheory.PGame.LF 0 G
-- SetTheory.PGame.Impartial.forall_leftMoves_fuzzy_iff_equiv_zero
theorem forall_leftMoves_fuzzy_iff_equiv_zero : (∀ i, G.moveLeft i ‖ 0) ↔ (G ≈ 0) 
Arguments: (G : SetTheory.PGame) [G.Impartial]
Goal: (∀ (i : G.LeftMoves), (G.moveLeft i).Fuzzy 0) ↔ G ≈ 0
-- SetTheory.PGame.Impartial.forall_rightMoves_fuzzy_iff_equiv_zero
theorem forall_rightMoves_fuzzy_iff_equiv_zero : (∀ j, G.moveRight j ‖ 0) ↔ (G ≈ 0) 
Arguments: (G : SetTheory.PGame) [G.Impartial]
Goal: (∀ (j : G.RightMoves), (G.moveRight j).Fuzzy 0) ↔ G ≈ 0
-- SetTheory.PGame.Impartial.exists_left_move_equiv_iff_fuzzy_zero
theorem exists_left_move_equiv_iff_fuzzy_zero : (∃ i, G.moveLeft i ≈ 0) ↔ G ‖ 0 
Arguments: (G : SetTheory.PGame) [G.Impartial]
Goal: (∃ i, G.moveLeft i ≈ 0) ↔ G.Fuzzy 0
-- SetTheory.PGame.Impartial.exists_right_move_equiv_iff_fuzzy_zero
theorem exists_right_move_equiv_iff_fuzzy_zero : (∃ j, G.moveRight j ≈ 0) ↔ G ‖ 0 
Arguments: (G : SetTheory.PGame) [G.Impartial]
Goal: (∃ j, G.moveRight j ≈ 0) ↔ G.Fuzzy 0
-- SetTheory.PGame.subsingleton_short_example
theorem subsingleton_short_example : ∀ x : PGame, Subsingleton (Short x)
  
Arguments: (x : SetTheory.PGame)
Goal: Subsingleton x.Short
-- SetTheory.PGame.short_birthday
theorem short_birthday (x : PGame.{u}) : [Short x] → x.birthday < Ordinal.omega 
Arguments: (x : SetTheory.PGame) [x.Short]
Goal: x.birthday < Ordinal.omega
-- SetTheory.PGame.State.right_bound
class State (S : Type u) where
  turnBound : S → ℕ
  l : S → Finset S
  r : S → Finset S
  left_bound : ∀ {s t : S}, t ∈ l s → turnBound t < turnBound s
  right_bound : ∀ {s t : S}, t ∈ r s → turnBound t < turnBound s
Arguments: {S : Type u} [self : SetTheory.PGame.State S] {s t : S}
Goal: t ∈ SetTheory.PGame.State.r s → SetTheory.PGame.State.turnBound t < SetTheory.PGame.State.turnBound s
-- SetTheory.PGame.State.left_bound
class State (S : Type u) where
  turnBound : S → ℕ
  l : S → Finset S
  r : S → Finset S
  left_bound : ∀ {s t : S}, t ∈ l s → turnBound t < turnBound s
  right_bound : ∀ {s t : S}, t ∈ r s → turnBound t < turnBound s
Arguments: {S : Type u} [self : SetTheory.PGame.State S] {s t : S}
Goal: t ∈ SetTheory.PGame.State.l s → SetTheory.PGame.State.turnBound t < SetTheory.PGame.State.turnBound s
-- SetTheory.PGame.turnBound_ne_zero_of_left_move
theorem turnBound_ne_zero_of_left_move {s t : S} (m : t ∈ l s) : turnBound s ≠ 0 
Arguments: {S : Type u} [SetTheory.PGame.State S] {s t : S} (m : t ∈ SetTheory.PGame.State.l s)
Goal: SetTheory.PGame.State.turnBound s ≠ 0
-- SetTheory.PGame.turnBound_ne_zero_of_right_move
theorem turnBound_ne_zero_of_right_move {s t : S} (m : t ∈ r s) : turnBound s ≠ 0 
Arguments: {S : Type u} [SetTheory.PGame.State S] {s t : S} (m : t ∈ SetTheory.PGame.State.r s)
Goal: SetTheory.PGame.State.turnBound s ≠ 0
-- SetTheory.PGame.turnBound_of_left
theorem turnBound_of_left {s t : S} (m : t ∈ l s) (n : ℕ) (h : turnBound s ≤ n + 1) :
    turnBound t ≤ n 
Arguments: {S : Type u} [SetTheory.PGame.State S] {s t : S} (m : t ∈ SetTheory.PGame.State.l s) (n : ℕ) (h : SetTheory.PGame.State.turnBound s ≤ n + 1)
Goal: SetTheory.PGame.State.turnBound t ≤ n
-- SetTheory.PGame.turnBound_of_right
theorem turnBound_of_right {s t : S} (m : t ∈ r s) (n : ℕ) (h : turnBound s ≤ n + 1) :
    turnBound t ≤ n 
Arguments: {S : Type u} [SetTheory.PGame.State S] {s t : S} (m : t ∈ SetTheory.PGame.State.r s) (n : ℕ) (h : SetTheory.PGame.State.turnBound s ≤ n + 1)
Goal: SetTheory.PGame.State.turnBound t ≤ n
-- PhragmenLindelof.isBigO_sub_exp_exp
theorem isBigO_sub_exp_exp {a : ℝ} {f g : ℂ → E} {l : Filter ℂ} {u : ℂ → ℝ}
    (hBf : ∃ c < a, ∃ B, f =O[l] fun z => expR (B * expR (c * |u z|)))
    (hBg : ∃ c < a, ∃ B, g =O[l] fun z => expR (B * expR (c * |u z|))) :
    ∃ c < a, ∃ B, (f - g) =O[l] fun z => expR (B * expR (c * |u z|)) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] {a : ℝ} {f g : ℂ → E} {l : Filter ℂ} {u : ℂ → ℝ} (hBf : ∃ c < a, ∃ B, f =O[l] fun z => expR (B * expR (c * |u z|))) (hBg : ∃ c < a, ∃ B, g =O[l] fun z => expR (B * expR (c * |u z|)))
Goal: ∃ c < a, ∃ B, (f - g) =O[l] fun z => expR (B * expR (c * |u z|))
-- PhragmenLindelof.isBigO_sub_exp_rpow
theorem isBigO_sub_exp_rpow {a : ℝ} {f g : ℂ → E} {l : Filter ℂ}
    (hBf : ∃ c < a, ∃ B, f =O[cobounded ℂ ⊓ l] fun z => expR (B * abs z ^ c))
    (hBg : ∃ c < a, ∃ B, g =O[cobounded ℂ ⊓ l] fun z => expR (B * abs z ^ c)) :
    ∃ c < a, ∃ B, (f - g) =O[cobounded ℂ ⊓ l] fun z => expR (B * abs z ^ c) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] {a : ℝ} {f g : ℂ → E} {l : Filter ℂ} (hBf : ∃ c < a, ∃ B, f =O[Bornology.cobounded ℂ ⊓ l] fun z => expR (B * Complex.abs z ^ c)) (hBg : ∃ c < a, ∃ B, g =O[Bornology.cobounded ℂ ⊓ l] fun z => expR (B * Complex.abs z ^ c))
Goal: ∃ c < a, ∃ B, (f - g) =O[Bornology.cobounded ℂ ⊓ l] fun z => expR (B * Complex.abs z ^ c)
-- PhragmenLindelof.horizontal_strip
theorem horizontal_strip (hfd : DiffContOnCl ℂ f (im ⁻¹' Ioo a b))
    (hB : ∃ c < π / (b - a), ∃ B, f =O[comap (_root_.abs ∘ re) atTop ⊓ 𝓟 (im ⁻¹' Ioo a b)]
      fun z ↦ expR (B * expR (c * |z.re|)))
    (hle_a : ∀ z : ℂ, im z = a → ‖f z‖ ≤ C) (hle_b : ∀ z, im z = b → ‖f z‖ ≤ C) (hza : a ≤ im z)
    (hzb : im z ≤ b) : ‖f z‖ ≤ C 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {a b C : ℝ} {f : ℂ → E} {z : ℂ} (hfd : DiffContOnCl ℂ f (Complex.im ⁻¹' Set.Ioo a b)) (hB :
  ∃ c < π / (b - a),
    ∃ B,
      f =O[Filter.comap (abs ∘ Complex.re) Filter.atTop ⊓ 𝓟 (Complex.im ⁻¹' Set.Ioo a b)] fun z =>
        expR (B * expR (c * |z.re|))) (hle_a : ∀ (z : ℂ), z.im = a → ‖f z‖ ≤ C) (hle_b : ∀ (z : ℂ), z.im = b → ‖f z‖ ≤ C) (hza : a ≤ z.im) (hzb : z.im ≤ b)
Goal: ‖f z‖ ≤ C
-- PhragmenLindelof.eq_zero_on_horizontal_strip
theorem eq_zero_on_horizontal_strip (hd : DiffContOnCl ℂ f (im ⁻¹' Ioo a b))
    (hB : ∃ c < π / (b - a), ∃ B, f =O[comap (_root_.abs ∘ re) atTop ⊓ 𝓟 (im ⁻¹' Ioo a b)]
      fun z ↦ expR (B * expR (c * |z.re|)))
    (ha : ∀ z : ℂ, z.im = a → f z = 0) (hb : ∀ z : ℂ, z.im = b → f z = 0) :
    EqOn f 0 (im ⁻¹' Icc a b) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {a b : ℝ} {f : ℂ → E} (hd : DiffContOnCl ℂ f (Complex.im ⁻¹' Set.Ioo a b)) (hB :
  ∃ c < π / (b - a),
    ∃ B,
      f =O[Filter.comap (abs ∘ Complex.re) Filter.atTop ⊓ 𝓟 (Complex.im ⁻¹' Set.Ioo a b)] fun z =>
        expR (B * expR (c * |z.re|))) (ha : ∀ (z : ℂ), z.im = a → f z = 0) (hb : ∀ (z : ℂ), z.im = b → f z = 0)
Goal: Set.EqOn f 0 (Complex.im ⁻¹' Set.Icc a b)
-- PhragmenLindelof.eqOn_horizontal_strip
theorem eqOn_horizontal_strip {g : ℂ → E} (hdf : DiffContOnCl ℂ f (im ⁻¹' Ioo a b))
    (hBf : ∃ c < π / (b - a), ∃ B, f =O[comap (_root_.abs ∘ re) atTop ⊓ 𝓟 (im ⁻¹' Ioo a b)]
      fun z ↦ expR (B * expR (c * |z.re|)))
    (hdg : DiffContOnCl ℂ g (im ⁻¹' Ioo a b))
    (hBg : ∃ c < π / (b - a), ∃ B, g =O[comap (_root_.abs ∘ re) atTop ⊓ 𝓟 (im ⁻¹' Ioo a b)]
      fun z ↦ expR (B * expR (c * |z.re|)))
    (ha : ∀ z : ℂ, z.im = a → f z = g z) (hb : ∀ z : ℂ, z.im = b → f z = g z) :
    EqOn f g (im ⁻¹' Icc a b) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {a b : ℝ} {f g : ℂ → E} (hdf : DiffContOnCl ℂ f (Complex.im ⁻¹' Set.Ioo a b)) (hBf :
  ∃ c < π / (b - a),
    ∃ B,
      f =O[Filter.comap (abs ∘ Complex.re) Filter.atTop ⊓ 𝓟 (Complex.im ⁻¹' Set.Ioo a b)] fun z =>
        expR (B * expR (c * |z.re|))) (hdg : DiffContOnCl ℂ g (Complex.im ⁻¹' Set.Ioo a b)) (hBg :
  ∃ c < π / (b - a),
    ∃ B,
      g =O[Filter.comap (abs ∘ Complex.re) Filter.atTop ⊓ 𝓟 (Complex.im ⁻¹' Set.Ioo a b)] fun z =>
        expR (B * expR (c * |z.re|))) (ha : ∀ (z : ℂ), z.im = a → f z = g z) (hb : ∀ (z : ℂ), z.im = b → f z = g z)
Goal: Set.EqOn f g (Complex.im ⁻¹' Set.Icc a b)
-- PhragmenLindelof.vertical_strip
theorem vertical_strip (hfd : DiffContOnCl ℂ f (re ⁻¹' Ioo a b))
    (hB : ∃ c < π / (b - a), ∃ B, f =O[comap (_root_.abs ∘ im) atTop ⊓ 𝓟 (re ⁻¹' Ioo a b)]
      fun z ↦ expR (B * expR (c * |z.im|)))
    (hle_a : ∀ z : ℂ, re z = a → ‖f z‖ ≤ C) (hle_b : ∀ z, re z = b → ‖f z‖ ≤ C) (hza : a ≤ re z)
    (hzb : re z ≤ b) : ‖f z‖ ≤ C 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {a b C : ℝ} {f : ℂ → E} {z : ℂ} (hfd : DiffContOnCl ℂ f (Complex.re ⁻¹' Set.Ioo a b)) (hB :
  ∃ c < π / (b - a),
    ∃ B,
      f =O[Filter.comap (abs ∘ Complex.im) Filter.atTop ⊓ 𝓟 (Complex.re ⁻¹' Set.Ioo a b)] fun z =>
        expR (B * expR (c * |z.im|))) (hle_a : ∀ (z : ℂ), z.re = a → ‖f z‖ ≤ C) (hle_b : ∀ (z : ℂ), z.re = b → ‖f z‖ ≤ C) (hza : a ≤ z.re) (hzb : z.re ≤ b)
Goal: ‖f z‖ ≤ C
-- PhragmenLindelof.eq_zero_on_vertical_strip
theorem eq_zero_on_vertical_strip (hd : DiffContOnCl ℂ f (re ⁻¹' Ioo a b))
    (hB : ∃ c < π / (b - a), ∃ B, f =O[comap (_root_.abs ∘ im) atTop ⊓ 𝓟 (re ⁻¹' Ioo a b)]
      fun z ↦ expR (B * expR (c * |z.im|)))
    (ha : ∀ z : ℂ, re z = a → f z = 0) (hb : ∀ z : ℂ, re z = b → f z = 0) :
    EqOn f 0 (re ⁻¹' Icc a b) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {a b : ℝ} {f : ℂ → E} (hd : DiffContOnCl ℂ f (Complex.re ⁻¹' Set.Ioo a b)) (hB :
  ∃ c < π / (b - a),
    ∃ B,
      f =O[Filter.comap (abs ∘ Complex.im) Filter.atTop ⊓ 𝓟 (Complex.re ⁻¹' Set.Ioo a b)] fun z =>
        expR (B * expR (c * |z.im|))) (ha : ∀ (z : ℂ), z.re = a → f z = 0) (hb : ∀ (z : ℂ), z.re = b → f z = 0)
Goal: Set.EqOn f 0 (Complex.re ⁻¹' Set.Icc a b)
-- PhragmenLindelof.eqOn_vertical_strip
theorem eqOn_vertical_strip {g : ℂ → E} (hdf : DiffContOnCl ℂ f (re ⁻¹' Ioo a b))
    (hBf : ∃ c < π / (b - a), ∃ B, f =O[comap (_root_.abs ∘ im) atTop ⊓ 𝓟 (re ⁻¹' Ioo a b)]
      fun z ↦ expR (B * expR (c * |z.im|)))
    (hdg : DiffContOnCl ℂ g (re ⁻¹' Ioo a b))
    (hBg : ∃ c < π / (b - a), ∃ B, g =O[comap (_root_.abs ∘ im) atTop ⊓ 𝓟 (re ⁻¹' Ioo a b)]
      fun z ↦ expR (B * expR (c * |z.im|)))
    (ha : ∀ z : ℂ, re z = a → f z = g z) (hb : ∀ z : ℂ, re z = b → f z = g z) :
    EqOn f g (re ⁻¹' Icc a b) 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {a b : ℝ} {f g : ℂ → E} (hdf : DiffContOnCl ℂ f (Complex.re ⁻¹' Set.Ioo a b)) (hBf :
  ∃ c < π / (b - a),
    ∃ B,
      f =O[Filter.comap (abs ∘ Complex.im) Filter.atTop ⊓ 𝓟 (Complex.re ⁻¹' Set.Ioo a b)] fun z =>
        expR (B * expR (c * |z.im|))) (hdg : DiffContOnCl ℂ g (Complex.re ⁻¹' Set.Ioo a b)) (hBg :
  ∃ c < π / (b - a),
    ∃ B,
      g =O[Filter.comap (abs ∘ Complex.im) Filter.atTop ⊓ 𝓟 (Complex.re ⁻¹' Set.Ioo a b)] fun z =>
        expR (B * expR (c * |z.im|))) (ha : ∀ (z : ℂ), z.re = a → f z = g z) (hb : ∀ (z : ℂ), z.re = b → f z = g z)
Goal: Set.EqOn f g (Complex.re ⁻¹' Set.Icc a b)
-- PhragmenLindelof.quadrant_I
nonrec theorem quadrant_I (hd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Ioi 0))
    (hB : ∃ c < (2 : ℝ), ∃ B,
      f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Ioi 0)] fun z => expR (B * abs z ^ c))
    (hre : ∀ x : ℝ, 0 ≤ x → ‖f x‖ ≤ C) (him : ∀ x : ℝ, 0 ≤ x → ‖f (x * I)‖ ≤ C) (hz_re : 0 ≤ z.re)
    (hz_im : 0 ≤ z.im) : ‖f z‖ ≤ C 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {C : ℝ} {f : ℂ → E} {z : ℂ} (hd : DiffContOnCl ℂ f (Set.Ioi 0 ×ℂ Set.Ioi 0)) (hB : ∃ c < 2, ∃ B, f =O[Bornology.cobounded ℂ ⊓ 𝓟 (Set.Ioi 0 ×ℂ Set.Ioi 0)] fun z => expR (B * Complex.abs z ^ c)) (hre : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C) (him : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * Complex.I)‖ ≤ C) (hz_re : 0 ≤ z.re) (hz_im : 0 ≤ z.im)
Goal: ‖f z‖ ≤ C
-- PhragmenLindelof.eq_zero_on_quadrant_I
theorem eq_zero_on_quadrant_I (hd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Ioi 0))
    (hB : ∃ c < (2 : ℝ), ∃ B,
      f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Ioi 0)] fun z => expR (B * abs z ^ c))
    (hre : ∀ x : ℝ, 0 ≤ x → f x = 0) (him : ∀ x : ℝ, 0 ≤ x → f (x * I) = 0) :
    EqOn f 0 {z | 0 ≤ z.re ∧ 0 ≤ z.im} 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} (hd : DiffContOnCl ℂ f (Set.Ioi 0 ×ℂ Set.Ioi 0)) (hB : ∃ c < 2, ∃ B, f =O[Bornology.cobounded ℂ ⊓ 𝓟 (Set.Ioi 0 ×ℂ Set.Ioi 0)] fun z => expR (B * Complex.abs z ^ c)) (hre : ∀ (x : ℝ), 0 ≤ x → f ↑x = 0) (him : ∀ (x : ℝ), 0 ≤ x → f (↑x * Complex.I) = 0)
Goal: Set.EqOn f 0 {z | 0 ≤ z.re ∧ 0 ≤ z.im}
-- PhragmenLindelof.eqOn_quadrant_I
theorem eqOn_quadrant_I (hdf : DiffContOnCl ℂ f (Ioi 0 ×ℂ Ioi 0))
    (hBf : ∃ c < (2 : ℝ), ∃ B,
      f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Ioi 0)] fun z => expR (B * abs z ^ c))
    (hdg : DiffContOnCl ℂ g (Ioi 0 ×ℂ Ioi 0))
    (hBg : ∃ c < (2 : ℝ), ∃ B,
      g =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Ioi 0)] fun z => expR (B * abs z ^ c))
    (hre : ∀ x : ℝ, 0 ≤ x → f x = g x) (him : ∀ x : ℝ, 0 ≤ x → f (x * I) = g (x * I)) :
    EqOn f g {z | 0 ≤ z.re ∧ 0 ≤ z.im} 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f g : ℂ → E} (hdf : DiffContOnCl ℂ f (Set.Ioi 0 ×ℂ Set.Ioi 0)) (hBf : ∃ c < 2, ∃ B, f =O[Bornology.cobounded ℂ ⊓ 𝓟 (Set.Ioi 0 ×ℂ Set.Ioi 0)] fun z => expR (B * Complex.abs z ^ c)) (hdg : DiffContOnCl ℂ g (Set.Ioi 0 ×ℂ Set.Ioi 0)) (hBg : ∃ c < 2, ∃ B, g =O[Bornology.cobounded ℂ ⊓ 𝓟 (Set.Ioi 0 ×ℂ Set.Ioi 0)] fun z => expR (B * Complex.abs z ^ c)) (hre : ∀ (x : ℝ), 0 ≤ x → f ↑x = g ↑x) (him : ∀ (x : ℝ), 0 ≤ x → f (↑x * Complex.I) = g (↑x * Complex.I))
Goal: Set.EqOn f g {z | 0 ≤ z.re ∧ 0 ≤ z.im}
-- PhragmenLindelof.quadrant_II
theorem quadrant_II (hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Ioi 0))
    (hB : ∃ c < (2 : ℝ), ∃ B,
      f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)] fun z => expR (B * abs z ^ c))
    (hre : ∀ x : ℝ, x ≤ 0 → ‖f x‖ ≤ C) (him : ∀ x : ℝ, 0 ≤ x → ‖f (x * I)‖ ≤ C) (hz_re : z.re ≤ 0)
    (hz_im : 0 ≤ z.im) : ‖f z‖ ≤ C 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {C : ℝ} {f : ℂ → E} {z : ℂ} (hd : DiffContOnCl ℂ f (Set.Iio 0 ×ℂ Set.Ioi 0)) (hB : ∃ c < 2, ∃ B, f =O[Bornology.cobounded ℂ ⊓ 𝓟 (Set.Iio 0 ×ℂ Set.Ioi 0)] fun z => expR (B * Complex.abs z ^ c)) (hre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C) (him : ∀ (x : ℝ), 0 ≤ x → ‖f (↑x * Complex.I)‖ ≤ C) (hz_re : z.re ≤ 0) (hz_im : 0 ≤ z.im)
Goal: ‖f z‖ ≤ C
-- PhragmenLindelof.eq_zero_on_quadrant_II
theorem eq_zero_on_quadrant_II (hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Ioi 0))
    (hB : ∃ c < (2 : ℝ), ∃ B,
      f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)] fun z => expR (B * abs z ^ c))
    (hre : ∀ x : ℝ, x ≤ 0 → f x = 0) (him : ∀ x : ℝ, 0 ≤ x → f (x * I) = 0) :
    EqOn f 0 {z | z.re ≤ 0 ∧ 0 ≤ z.im} 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} (hd : DiffContOnCl ℂ f (Set.Iio 0 ×ℂ Set.Ioi 0)) (hB : ∃ c < 2, ∃ B, f =O[Bornology.cobounded ℂ ⊓ 𝓟 (Set.Iio 0 ×ℂ Set.Ioi 0)] fun z => expR (B * Complex.abs z ^ c)) (hre : ∀ x ≤ 0, f ↑x = 0) (him : ∀ (x : ℝ), 0 ≤ x → f (↑x * Complex.I) = 0)
Goal: Set.EqOn f 0 {z | z.re ≤ 0 ∧ 0 ≤ z.im}
-- PhragmenLindelof.eqOn_quadrant_II
theorem eqOn_quadrant_II (hdf : DiffContOnCl ℂ f (Iio 0 ×ℂ Ioi 0))
    (hBf : ∃ c < (2 : ℝ), ∃ B,
      f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)] fun z => expR (B * abs z ^ c))
    (hdg : DiffContOnCl ℂ g (Iio 0 ×ℂ Ioi 0))
    (hBg : ∃ c < (2 : ℝ), ∃ B,
      g =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Ioi 0)] fun z => expR (B * abs z ^ c))
    (hre : ∀ x : ℝ, x ≤ 0 → f x = g x) (him : ∀ x : ℝ, 0 ≤ x → f (x * I) = g (x * I)) :
    EqOn f g {z | z.re ≤ 0 ∧ 0 ≤ z.im} 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f g : ℂ → E} (hdf : DiffContOnCl ℂ f (Set.Iio 0 ×ℂ Set.Ioi 0)) (hBf : ∃ c < 2, ∃ B, f =O[Bornology.cobounded ℂ ⊓ 𝓟 (Set.Iio 0 ×ℂ Set.Ioi 0)] fun z => expR (B * Complex.abs z ^ c)) (hdg : DiffContOnCl ℂ g (Set.Iio 0 ×ℂ Set.Ioi 0)) (hBg : ∃ c < 2, ∃ B, g =O[Bornology.cobounded ℂ ⊓ 𝓟 (Set.Iio 0 ×ℂ Set.Ioi 0)] fun z => expR (B * Complex.abs z ^ c)) (hre : ∀ x ≤ 0, f ↑x = g ↑x) (him : ∀ (x : ℝ), 0 ≤ x → f (↑x * Complex.I) = g (↑x * Complex.I))
Goal: Set.EqOn f g {z | z.re ≤ 0 ∧ 0 ≤ z.im}
-- PhragmenLindelof.quadrant_III
theorem quadrant_III (hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0))
    (hB : ∃ c < (2 : ℝ), ∃ B,
      f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z => expR (B * abs z ^ c))
    (hre : ∀ x : ℝ, x ≤ 0 → ‖f x‖ ≤ C) (him : ∀ x : ℝ, x ≤ 0 → ‖f (x * I)‖ ≤ C) (hz_re : z.re ≤ 0)
    (hz_im : z.im ≤ 0) : ‖f z‖ ≤ C 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {C : ℝ} {f : ℂ → E} {z : ℂ} (hd : DiffContOnCl ℂ f (Set.Iio 0 ×ℂ Set.Iio 0)) (hB : ∃ c < 2, ∃ B, f =O[Bornology.cobounded ℂ ⊓ 𝓟 (Set.Iio 0 ×ℂ Set.Iio 0)] fun z => expR (B * Complex.abs z ^ c)) (hre : ∀ x ≤ 0, ‖f ↑x‖ ≤ C) (him : ∀ x ≤ 0, ‖f (↑x * Complex.I)‖ ≤ C) (hz_re : z.re ≤ 0) (hz_im : z.im ≤ 0)
Goal: ‖f z‖ ≤ C
-- PhragmenLindelof.eq_zero_on_quadrant_III
theorem eq_zero_on_quadrant_III (hd : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0))
    (hB : ∃ c < (2 : ℝ), ∃ B,
      f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z => expR (B * abs z ^ c))
    (hre : ∀ x : ℝ, x ≤ 0 → f x = 0) (him : ∀ x : ℝ, x ≤ 0 → f (x * I) = 0) :
    EqOn f 0 {z | z.re ≤ 0 ∧ z.im ≤ 0} 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} (hd : DiffContOnCl ℂ f (Set.Iio 0 ×ℂ Set.Iio 0)) (hB : ∃ c < 2, ∃ B, f =O[Bornology.cobounded ℂ ⊓ 𝓟 (Set.Iio 0 ×ℂ Set.Iio 0)] fun z => expR (B * Complex.abs z ^ c)) (hre : ∀ x ≤ 0, f ↑x = 0) (him : ∀ x ≤ 0, f (↑x * Complex.I) = 0)
Goal: Set.EqOn f 0 {z | z.re ≤ 0 ∧ z.im ≤ 0}
-- PhragmenLindelof.eqOn_quadrant_III
theorem eqOn_quadrant_III (hdf : DiffContOnCl ℂ f (Iio 0 ×ℂ Iio 0))
    (hBf : ∃ c < (2 : ℝ), ∃ B,
      f =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z => expR (B * abs z ^ c))
    (hdg : DiffContOnCl ℂ g (Iio 0 ×ℂ Iio 0))
    (hBg : ∃ c < (2 : ℝ), ∃ B,
      g =O[cobounded ℂ ⊓ 𝓟 (Iio 0 ×ℂ Iio 0)] fun z => expR (B * abs z ^ c))
    (hre : ∀ x : ℝ, x ≤ 0 → f x = g x) (him : ∀ x : ℝ, x ≤ 0 → f (x * I) = g (x * I)) :
    EqOn f g {z | z.re ≤ 0 ∧ z.im ≤ 0} 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f g : ℂ → E} (hdf : DiffContOnCl ℂ f (Set.Iio 0 ×ℂ Set.Iio 0)) (hBf : ∃ c < 2, ∃ B, f =O[Bornology.cobounded ℂ ⊓ 𝓟 (Set.Iio 0 ×ℂ Set.Iio 0)] fun z => expR (B * Complex.abs z ^ c)) (hdg : DiffContOnCl ℂ g (Set.Iio 0 ×ℂ Set.Iio 0)) (hBg : ∃ c < 2, ∃ B, g =O[Bornology.cobounded ℂ ⊓ 𝓟 (Set.Iio 0 ×ℂ Set.Iio 0)] fun z => expR (B * Complex.abs z ^ c)) (hre : ∀ x ≤ 0, f ↑x = g ↑x) (him : ∀ x ≤ 0, f (↑x * Complex.I) = g (↑x * Complex.I))
Goal: Set.EqOn f g {z | z.re ≤ 0 ∧ z.im ≤ 0}
-- PhragmenLindelof.quadrant_IV
theorem quadrant_IV (hd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Iio 0))
    (hB : ∃ c < (2 : ℝ), ∃ B,
      f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Iio 0)] fun z => expR (B * abs z ^ c))
    (hre : ∀ x : ℝ, 0 ≤ x → ‖f x‖ ≤ C) (him : ∀ x : ℝ, x ≤ 0 → ‖f (x * I)‖ ≤ C) (hz_re : 0 ≤ z.re)
    (hz_im : z.im ≤ 0) : ‖f z‖ ≤ C 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {C : ℝ} {f : ℂ → E} {z : ℂ} (hd : DiffContOnCl ℂ f (Set.Ioi 0 ×ℂ Set.Iio 0)) (hB : ∃ c < 2, ∃ B, f =O[Bornology.cobounded ℂ ⊓ 𝓟 (Set.Ioi 0 ×ℂ Set.Iio 0)] fun z => expR (B * Complex.abs z ^ c)) (hre : ∀ (x : ℝ), 0 ≤ x → ‖f ↑x‖ ≤ C) (him : ∀ x ≤ 0, ‖f (↑x * Complex.I)‖ ≤ C) (hz_re : 0 ≤ z.re) (hz_im : z.im ≤ 0)
Goal: ‖f z‖ ≤ C
-- PhragmenLindelof.eq_zero_on_quadrant_IV
theorem eq_zero_on_quadrant_IV (hd : DiffContOnCl ℂ f (Ioi 0 ×ℂ Iio 0))
    (hB : ∃ c < (2 : ℝ), ∃ B,
      f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Iio 0)] fun z => expR (B * abs z ^ c))
    (hre : ∀ x : ℝ, 0 ≤ x → f x = 0) (him : ∀ x : ℝ, x ≤ 0 → f (x * I) = 0) :
    EqOn f 0 {z | 0 ≤ z.re ∧ z.im ≤ 0} 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f : ℂ → E} (hd : DiffContOnCl ℂ f (Set.Ioi 0 ×ℂ Set.Iio 0)) (hB : ∃ c < 2, ∃ B, f =O[Bornology.cobounded ℂ ⊓ 𝓟 (Set.Ioi 0 ×ℂ Set.Iio 0)] fun z => expR (B * Complex.abs z ^ c)) (hre : ∀ (x : ℝ), 0 ≤ x → f ↑x = 0) (him : ∀ x ≤ 0, f (↑x * Complex.I) = 0)
Goal: Set.EqOn f 0 {z | 0 ≤ z.re ∧ z.im ≤ 0}
-- PhragmenLindelof.eqOn_quadrant_IV
theorem eqOn_quadrant_IV (hdf : DiffContOnCl ℂ f (Ioi 0 ×ℂ Iio 0))
    (hBf : ∃ c < (2 : ℝ), ∃ B,
      f =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Iio 0)] fun z => expR (B * abs z ^ c))
    (hdg : DiffContOnCl ℂ g (Ioi 0 ×ℂ Iio 0))
    (hBg : ∃ c < (2 : ℝ), ∃ B,
      g =O[cobounded ℂ ⊓ 𝓟 (Ioi 0 ×ℂ Iio 0)] fun z => expR (B * abs z ^ c))
    (hre : ∀ x : ℝ, 0 ≤ x → f x = g x) (him : ∀ x : ℝ, x ≤ 0 → f (x * I) = g (x * I)) :
    EqOn f g {z | 0 ≤ z.re ∧ z.im ≤ 0} 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {f g : ℂ → E} (hdf : DiffContOnCl ℂ f (Set.Ioi 0 ×ℂ Set.Iio 0)) (hBf : ∃ c < 2, ∃ B, f =O[Bornology.cobounded ℂ ⊓ 𝓟 (Set.Ioi 0 ×ℂ Set.Iio 0)] fun z => expR (B * Complex.abs z ^ c)) (hdg : DiffContOnCl ℂ g (Set.Ioi 0 ×ℂ Set.Iio 0)) (hBg : ∃ c < 2, ∃ B, g =O[Bornology.cobounded ℂ ⊓ 𝓟 (Set.Ioi 0 ×ℂ Set.Iio 0)] fun z => expR (B * Complex.abs z ^ c)) (hre : ∀ (x : ℝ), 0 ≤ x → f ↑x = g ↑x) (him : ∀ x ≤ 0, f (↑x * Complex.I) = g (↑x * Complex.I))
Goal: Set.EqOn f g {z | 0 ≤ z.re ∧ z.im ≤ 0}
-- PhragmenLindelof.right_half_plane_of_tendsto_zero_on_real
theorem right_half_plane_of_tendsto_zero_on_real (hd : DiffContOnCl ℂ f {z | 0 < z.re})
    (hexp : ∃ c < (2 : ℝ), ∃ B,
      f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * abs z ^ c))
    (hre : Tendsto (fun x : ℝ => f x) atTop (𝓝 0)) (him : ∀ x : ℝ, ‖f (x * I)‖ ≤ C)
    (hz : 0 ≤ z.re) : ‖f z‖ ≤ C 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {C : ℝ} {f : ℂ → E} {z : ℂ} (hd : DiffContOnCl ℂ f {z | 0 < z.re}) (hexp : ∃ c < 2, ∃ B, f =O[Bornology.cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * Complex.abs z ^ c)) (hre : Filter.Tendsto (fun x => f ↑x) Filter.atTop (𝓝 0)) (him : ∀ (x : ℝ), ‖f (↑x * Complex.I)‖ ≤ C) (hz : 0 ≤ z.re)
Goal: ‖f z‖ ≤ C
-- PhragmenLindelof.right_half_plane_of_bounded_on_real
theorem right_half_plane_of_bounded_on_real (hd : DiffContOnCl ℂ f {z | 0 < z.re})
    (hexp : ∃ c < (2 : ℝ), ∃ B,
      f =O[cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * abs z ^ c))
    (hre : IsBoundedUnder (· ≤ ·) atTop fun x : ℝ => ‖f x‖) (him : ∀ x : ℝ, ‖f (x * I)‖ ≤ C)
    (hz : 0 ≤ z.re) : ‖f z‖ ≤ C 
Arguments: {E : Type u_1} [NormedAddCommGroup E] [NormedSpace ℂ E] {C : ℝ} {f : ℂ → E} {z : ℂ} (hd : DiffContOnCl ℂ f {z | 0 < z.re}) (hexp : ∃ c < 2, ∃ B, f =O[Bornology.cobounded ℂ ⊓ 𝓟 {z | 0 < z.re}] fun z => expR (B * Complex.abs z ^ c)) (hre : Filter.IsBoundedUnder (fun x x_1 => x ≤ x_1) Filter.atTop fun x => ‖f ↑x‖) (him : ∀ (x : ℝ), ‖f (↑x * Complex.I)‖ ≤ C) (hz : 0 ≤ z.re)
Goal: ‖f z‖ ≤ C