awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.Metrizable.Uniformity (hypothesis)
Goal:
X : Type u_1
d : X → X → ℝ≥0
dist_self : ∀ (x : X), d x x = 0
dist_comm : ∀ (x y : X), d x y = d y x
hd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))
this : IsTrans X fun (x y : X) ↦ d x y = 0
x y : X
l : List X
ihn : ∀ m < l.length, ∀ (x y : X) (l : List X) (hn : l.length = m), d x y ≤ 2 * (zipWith d (x :: l) (l ++ [y])).sum
L : List ℝ≥0 := zipWith d (x :: l) (l ++ [y])
hL_len : L.length = l.length + 1
hd₀ : d x y ≠ 0
s : Set ℕ := {m : ℕ | 2 * (take m L).sum ≤ L.sum}
hs₀ : 0 ∈ s
hsne : s.Nonempty
M : ℕ
hMl : M ≤ l.length
hMs : IsGreatest s M
⊢ ∃ (z : X) (z' : X), d x z ≤ L.sum ∧ d z z' ≤ L.sum ∧ d z' y ≤ L.sum
have hM_lt : M < L.length
Tactic: have hM_lt : M < length L := by rwa [hL_len, Nat.lt_succ_iff]
-- Lean 4 proof step from Mathlib.Topology.Metrizable.Uniformity (hypothesis)
Goal:
X : Type u_1
d : X → X → ℝ≥0
dist_self : ∀ (x : X), d x x = 0
dist_comm : ∀ (x y : X), d x y = d y x
hd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))
this : IsTrans X fun (x y : X) ↦ d x y = 0
x y : X
l : List X
ihn : ∀ m < l.length, ∀ (x y : X) (l : List X) (hn : l.length = m), d x y ≤ 2 * (zipWith d (x :: l) (l ++ [y])).sum
L : List ℝ≥0 := zipWith d (x :: l) (l ++ [y])
hL_len : L.length = l.length + 1
hd₀ : d x y ≠ 0
s : Set ℕ := {m : ℕ | 2 * (take m L).sum ≤ L.sum}
hs₀ : 0 ∈ s
hsne : s.Nonempty
M : ℕ
hMl : M ≤ l.length
hMs : IsGreatest s M
hM_lt : M < L.length
⊢ ∃ (z : X) (z' : X), d x z ≤ L.sum ∧ d z z' ≤ L.sum ∧ d z' y ≤ L.sum
have hM_ltx : M < (x :: l).length
Tactic: have hM_ltx : M < length (x::l) := lt_length_left_of_zipWith hM_lt
-- Lean 4 proof step from Mathlib.Topology.Metrizable.Uniformity (hypothesis)
Goal:
X : Type u_1
d : X → X → ℝ≥0
dist_self : ∀ (x : X), d x x = 0
dist_comm : ∀ (x y : X), d x y = d y x
hd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))
this : IsTrans X fun (x y : X) ↦ d x y = 0
x y : X
l : List X
ihn : ∀ m < l.length, ∀ (x y : X) (l : List X) (hn : l.length = m), d x y ≤ 2 * (zipWith d (x :: l) (l ++ [y])).sum
L : List ℝ≥0 := zipWith d (x :: l) (l ++ [y])
hL_len : L.length = l.length + 1
hd₀ : d x y ≠ 0
s : Set ℕ := {m : ℕ | 2 * (take m L).sum ≤ L.sum}
hs₀ : 0 ∈ s
hsne : s.Nonempty
M : ℕ
hMl : M ≤ l.length
hMs : IsGreatest s M
hM_lt : M < L.length
hM_ltx : M < (x :: l).length
⊢ ∃ (z : X) (z' : X), d x z ≤ L.sum ∧ d z z' ≤ L.sum ∧ d z' y ≤ L.sum
have hM_lty : M < (l ++ [y]).length
Tactic: have hM_lty : M < length (l ++ [y]) := lt_length_right_of_zipWith hM_lt
-- Lean 4 proof step from Mathlib.Topology.Metrizable.Uniformity (goal)
Goal:
X : Type u_1
d : X → X → ℝ≥0
dist_self : ∀ (x : X), d x x = 0
dist_comm : ∀ (x y : X), d x y = d y x
hd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))
this : IsTrans X fun (x y : X) ↦ d x y = 0
x y : X
l : List X
ihn : ∀ m < l.length, ∀ (x y : X) (l : List X) (hn : l.length = m), d x y ≤ 2 * (zipWith d (x :: l) (l ++ [y])).sum
L : List ℝ≥0 := zipWith d (x :: l) (l ++ [y])
hL_len : L.length = l.length + 1
hd₀ : d x y ≠ 0
s : Set ℕ := {m : ℕ | 2 * (take m L).sum ≤ L.sum}
hs₀ : 0 ∈ s
hsne : s.Nonempty
M : ℕ
hMl : M ≤ l.length
hMs : IsGreatest s M
hM_lt : M < L.length
hM_ltx : M < (x :: l).length
hM_lty : M < (l ++ [y]).length
⊢ ∃ (z : X) (z' : X), d x z ≤ L.sum ∧ d z z' ≤ L.sum ∧ d z' y ≤ L.sum
have refine_1 : d x (getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) M hM_ltx) ≤ L.sum
Tactic: refine ⟨(x::l)[M], (l ++ [y])[M], ?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Metrizable.Uniformity (goal)
Goal:
X : Type u_1
d : X → X → ℝ≥0
dist_self : ∀ (x : X), d x x = 0
dist_comm : ∀ (x y : X), d x y = d y x
hd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))
this : IsTrans X fun (x y : X) ↦ d x y = 0
x y : X
l : List X
ihn : ∀ m < l.length, ∀ (x y : X) (l : List X) (hn : l.length = m), d x y ≤ 2 * (zipWith d (x :: l) (l ++ [y])).sum
L : List ℝ≥0 := zipWith d (x :: l) (l ++ [y])
hL_len : L.length = l.length + 1
hd₀ : d x y ≠ 0
s : Set ℕ := {m : ℕ | 2 * (take m L).sum ≤ L.sum}
hs₀ : 0 ∈ s
hsne : s.Nonempty
M : ℕ
hMl : M ≤ l.length
hMs : IsGreatest s M
hM_lt : M < L.length
hM_ltx : M < (x :: l).length
hM_lty : M < (l ++ [y]).length
refine_1 : d x (getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) M hM_ltx) ≤ L.sum
⊢ ∃ (z : X) (z' : X), d x z ≤ L.sum ∧ d z z' ≤ L.sum ∧ d z' y ≤ L.sum
have refine_2 : d (getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) M hM_ltx)
    (getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (l ++ [y]) M hM_lty) ≤
  L.sum
Tactic: refine ⟨(x::l)[M], (l ++ [y])[M], ?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Metrizable.Uniformity (goal)
Goal:
X : Type u_1
d : X → X → ℝ≥0
dist_self : ∀ (x : X), d x x = 0
dist_comm : ∀ (x y : X), d x y = d y x
hd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))
this : IsTrans X fun (x y : X) ↦ d x y = 0
x y : X
l : List X
ihn : ∀ m < l.length, ∀ (x y : X) (l : List X) (hn : l.length = m), d x y ≤ 2 * (zipWith d (x :: l) (l ++ [y])).sum
L : List ℝ≥0 := zipWith d (x :: l) (l ++ [y])
hL_len : L.length = l.length + 1
hd₀ : d x y ≠ 0
s : Set ℕ := {m : ℕ | 2 * (take m L).sum ≤ L.sum}
hs₀ : 0 ∈ s
hsne : s.Nonempty
M : ℕ
hMl : M ≤ l.length
hMs : IsGreatest s M
hM_lt : M < L.length
hM_ltx : M < (x :: l).length
hM_lty : M < (l ++ [y]).length
refine_1 : d x (getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) M hM_ltx) ≤ L.sum
refine_2 :
  d (getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) M hM_ltx)
      (getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (l ++ [y]) M hM_lty) ≤
    L.sum
⊢ ∃ (z : X) (z' : X), d x z ≤ L.sum ∧ d z z' ≤ L.sum ∧ d z' y ≤ L.sum
have refine_3 : d (getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (l ++ [y]) M hM_lty) y ≤ L.sum
Tactic: refine ⟨(x::l)[M], (l ++ [y])[M], ?_, ?_, ?_⟩
-- Lean 4 proof step from Mathlib.Topology.Metrizable.Uniformity (goal)
Goal:
X : Type u_1
d : X → X → ℝ≥0
dist_self : ∀ (x : X), d x x = 0
dist_comm : ∀ (x y : X), d x y = d y x
hd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))
this : IsTrans X fun (x y : X) ↦ d x y = 0
x y : X
l : List X
ihn : ∀ m < l.length, ∀ (x y : X) (l : List X) (hn : l.length = m), d x y ≤ 2 * (zipWith d (x :: l) (l ++ [y])).sum
L : List ℝ≥0 := zipWith d (x :: l) (l ++ [y])
hL_len : L.length = l.length + 1
hd₀ : d x y ≠ 0
s : Set ℕ := {m : ℕ | 2 * (take m L).sum ≤ L.sum}
hs₀ : 0 ∈ s
hsne : s.Nonempty
M : ℕ
hMl : M + 1 ≤ l.length
hMs : IsGreatest s (M + 1)
hM_lt : M + 1 < L.length
hM_ltx : M + 1 < (x :: l).length
hM_lty : M + 1 < (l ++ [y]).length
⊢ d x (getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx) ≤ L.sum
have refine_1 : ∀ (hMl : M < l.length) (hMs : IsGreatest s (M + 1)) (hM_lt : M + 1 < L.length) (hM_ltx : M + 1 < (x :: l).length)
  (hM_lty : M + 1 < (l ++ [y]).length),
  d x (getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx) ≤ L.sum
Tactic: rw [Nat.succ_le_iff] at hMl
Cleared hypotheses: hMl, hMs, hM_lt, hM_ltx, hM_lty
-- Lean 4 proof step from Mathlib.Topology.Metrizable.Uniformity (hypothesis)
Goal:
X : Type u_1
d : X → X → ℝ≥0
dist_self : ∀ (x : X), d x x = 0
dist_comm : ∀ (x y : X), d x y = d y x
hd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))
this : IsTrans X fun (x y : X) ↦ d x y = 0
x y : X
l : List X
ihn : ∀ m < l.length, ∀ (x y : X) (l : List X) (hn : l.length = m), d x y ≤ 2 * (zipWith d (x :: l) (l ++ [y])).sum
L : List ℝ≥0 := zipWith d (x :: l) (l ++ [y])
hL_len : L.length = l.length + 1
hd₀ : d x y ≠ 0
s : Set ℕ := {m : ℕ | 2 * (take m L).sum ≤ L.sum}
hs₀ : 0 ∈ s
hsne : s.Nonempty
M : ℕ
hMl : M < l.length
hMs : IsGreatest s (M + 1)
hM_lt : M + 1 < L.length
hM_ltx : M + 1 < (x :: l).length
hM_lty : M + 1 < (l ++ [y]).length
⊢ d x (getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx) ≤ L.sum
have hMl' : (take M l).length = M
Tactic: have hMl' : length (take M l) = M := length_take.trans (min_eq_left hMl.le)
-- Lean 4 proof step from Mathlib.Topology.Metrizable.Uniformity (goal)
Goal:
X : Type u_1
d : X → X → ℝ≥0
dist_self : ∀ (x : X), d x x = 0
dist_comm : ∀ (x y : X), d x y = d y x
hd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))
this : IsTrans X fun (x y : X) ↦ d x y = 0
x y : X
l : List X
ihn : ∀ m < l.length, ∀ (x y : X) (l : List X) (hn : l.length = m), d x y ≤ 2 * (zipWith d (x :: l) (l ++ [y])).sum
L : List ℝ≥0 := zipWith d (x :: l) (l ++ [y])
hL_len : L.length = l.length + 1
hd₀ : d x y ≠ 0
s : Set ℕ := {m : ℕ | 2 * (take m L).sum ≤ L.sum}
hs₀ : 0 ∈ s
hsne : s.Nonempty
M : ℕ
hMl : M < l.length
hMs : IsGreatest s (M + 1)
hM_lt : M + 1 < L.length
hM_ltx : M + 1 < (x :: l).length
hM_lty : M + 1 < (l ++ [y]).length
hMl' : (take M l).length = M
⊢ d x (getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx) ≤ L.sum
have refine_1 : 2 *
    (zipWith d (x :: take M l)
        (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx])).sum ≤
  L.sum
Tactic: refine (ihn _ hMl _ _ _ hMl').trans ?_
-- Lean 4 proof step from Mathlib.Topology.Metrizable.Uniformity (goal)
Goal:
X : Type u_1
d : X → X → ℝ≥0
dist_self : ∀ (x : X), d x x = 0
dist_comm : ∀ (x y : X), d x y = d y x
hd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))
this : IsTrans X fun (x y : X) ↦ d x y = 0
x y : X
l : List X
ihn : ∀ m < l.length, ∀ (x y : X) (l : List X) (hn : l.length = m), d x y ≤ 2 * (zipWith d (x :: l) (l ++ [y])).sum
L : List ℝ≥0 := zipWith d (x :: l) (l ++ [y])
hL_len : L.length = l.length + 1
hd₀ : d x y ≠ 0
s : Set ℕ := {m : ℕ | 2 * (take m L).sum ≤ L.sum}
hs₀ : 0 ∈ s
hsne : s.Nonempty
M : ℕ
hMl : M < l.length
hMs : IsGreatest s (M + 1)
hM_lt : M + 1 < L.length
hM_ltx : M + 1 < (x :: l).length
hM_lty : M + 1 < (l ++ [y]).length
hMl' : (take M l).length = M
⊢ 2 *
      (zipWith d (x :: take M l)
          (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx])).sum ≤
    L.sum
have h : zipWith d (x :: take M l)
    (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
  take (M + 1) L
Tactic: convert! hMs.1.out
-- Lean 4 proof step from Mathlib.Topology.Metrizable.Uniformity (goal)
Goal:
X : Type u_1
d : X → X → ℝ≥0
dist_self : ∀ (x : X), d x x = 0
dist_comm : ∀ (x y : X), d x y = d y x
hd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))
this : IsTrans X fun (x y : X) ↦ d x y = 0
x y : X
l : List X
ihn : ∀ m < l.length, ∀ (x y : X) (l : List X) (hn : l.length = m), d x y ≤ 2 * (zipWith d (x :: l) (l ++ [y])).sum
L : List ℝ≥0 := zipWith d (x :: l) (l ++ [y])
hL_len : L.length = l.length + 1
hd₀ : d x y ≠ 0
s : Set ℕ := {m : ℕ | 2 * (take m L).sum ≤ L.sum}
hs₀ : 0 ∈ s
hsne : s.Nonempty
M : ℕ
hMl : M < l.length
hMs : IsGreatest s (M + 1)
hM_lt : M + 1 < L.length
hM_ltx : M + 1 < (x :: l).length
hM_lty : M + 1 < (l ++ [y]).length
hMl' : (take M l).length = M
⊢ zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    take (M + 1) L
have h : zipWith d (x :: take M l)
    (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
  zipWith d (take (M + 1) (x :: l)) (take (M + 1) (l ++ [y]))
Tactic: rw [take_zipWith, take, take_add_one, getElem?_append_left hMl, getElem?_eq_getElem hMl,
        ← Option.coe_def, Option.toList_some, take_append_of_le_length hMl.le, getElem_cons_succ]
-- Lean 4 proof step from Mathlib.Topology.Metrizable.Uniformity (goal)
Goal:
X : Type u_1
d : X → X → ℝ≥0
dist_self : ∀ (x : X), d x x = 0
dist_comm : ∀ (x y : X), d x y = d y x
hd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))
this : IsTrans X fun (x y : X) ↦ d x y = 0
x y : X
l : List X
ihn : ∀ m < l.length, ∀ (x y : X) (l : List X) (hn : l.length = m), d x y ≤ 2 * (zipWith d (x :: l) (l ++ [y])).sum
L : List ℝ≥0 := zipWith d (x :: l) (l ++ [y])
hL_len : L.length = l.length + 1
hd₀ : d x y ≠ 0
s : Set ℕ := {m : ℕ | 2 * (take m L).sum ≤ L.sum}
hs₀ : 0 ∈ s
hsne : s.Nonempty
M : ℕ
hMl : M < l.length
hMs : IsGreatest s (M + 1)
hM_lt : M + 1 < L.length
hM_ltx : M + 1 < (x :: l).length
hM_lty : M + 1 < (l ++ [y]).length
hMl' : (take M l).length = M
h :
  zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    zipWith d (take (M + 1) (x :: l)) (take (M + 1) (l ++ [y]))
⊢ zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    take (M + 1) L
have h₁ : zipWith d (x :: take M l)
    (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
  zipWith d (x :: take M l) (take (M + 1) (l ++ [y]))
Tactic: rw [take_zipWith, take, take_add_one, getElem?_append_left hMl, getElem?_eq_getElem hMl,
        ← Option.coe_def, Option.toList_some, take_append_of_le_length hMl.le, getElem_cons_succ]
-- Lean 4 proof step from Mathlib.Topology.Metrizable.Uniformity (goal)
Goal:
X : Type u_1
d : X → X → ℝ≥0
dist_self : ∀ (x : X), d x x = 0
dist_comm : ∀ (x y : X), d x y = d y x
hd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))
this : IsTrans X fun (x y : X) ↦ d x y = 0
x y : X
l : List X
ihn : ∀ m < l.length, ∀ (x y : X) (l : List X) (hn : l.length = m), d x y ≤ 2 * (zipWith d (x :: l) (l ++ [y])).sum
L : List ℝ≥0 := zipWith d (x :: l) (l ++ [y])
hL_len : L.length = l.length + 1
hd₀ : d x y ≠ 0
s : Set ℕ := {m : ℕ | 2 * (take m L).sum ≤ L.sum}
hs₀ : 0 ∈ s
hsne : s.Nonempty
M : ℕ
hMl : M < l.length
hMs : IsGreatest s (M + 1)
hM_lt : M + 1 < L.length
hM_ltx : M + 1 < (x :: l).length
hM_lty : M + 1 < (l ++ [y]).length
hMl' : (take M l).length = M
h :
  zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    zipWith d (take (M + 1) (x :: l)) (take (M + 1) (l ++ [y]))
h₁ :
  zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    zipWith d (x :: take M l) (take (M + 1) (l ++ [y]))
⊢ zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    take (M + 1) L
have h₂ : zipWith d (x :: take M l)
    (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
  zipWith d (x :: take M l)
    (take M (l ++ [y]) ++ (getElem? (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (l ++ [y]) M).toList)
Tactic: rw [take_zipWith, take, take_add_one, getElem?_append_left hMl, getElem?_eq_getElem hMl,
        ← Option.coe_def, Option.toList_some, take_append_of_le_length hMl.le, getElem_cons_succ]
-- Lean 4 proof step from Mathlib.Topology.Metrizable.Uniformity (goal)
Goal:
X : Type u_1
d : X → X → ℝ≥0
dist_self : ∀ (x : X), d x x = 0
dist_comm : ∀ (x y : X), d x y = d y x
hd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))
this : IsTrans X fun (x y : X) ↦ d x y = 0
x y : X
l : List X
ihn : ∀ m < l.length, ∀ (x y : X) (l : List X) (hn : l.length = m), d x y ≤ 2 * (zipWith d (x :: l) (l ++ [y])).sum
L : List ℝ≥0 := zipWith d (x :: l) (l ++ [y])
hL_len : L.length = l.length + 1
hd₀ : d x y ≠ 0
s : Set ℕ := {m : ℕ | 2 * (take m L).sum ≤ L.sum}
hs₀ : 0 ∈ s
hsne : s.Nonempty
M : ℕ
hMl : M < l.length
hMs : IsGreatest s (M + 1)
hM_lt : M + 1 < L.length
hM_ltx : M + 1 < (x :: l).length
hM_lty : M + 1 < (l ++ [y]).length
hMl' : (take M l).length = M
h :
  zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    zipWith d (take (M + 1) (x :: l)) (take (M + 1) (l ++ [y]))
h₁ :
  zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    zipWith d (x :: take M l) (take (M + 1) (l ++ [y]))
h₂ :
  zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    zipWith d (x :: take M l)
      (take M (l ++ [y]) ++ (getElem? (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (l ++ [y]) M).toList)
⊢ zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    take (M + 1) L
have h₃ : zipWith d (x :: take M l)
    (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
  zipWith d (x :: take M l)
    (take M (l ++ [y]) ++ (getElem? (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) l M).toList)
Tactic: rw [take_zipWith, take, take_add_one, getElem?_append_left hMl, getElem?_eq_getElem hMl,
        ← Option.coe_def, Option.toList_some, take_append_of_le_length hMl.le, getElem_cons_succ]
-- Lean 4 proof step from Mathlib.Topology.Metrizable.Uniformity (goal)
Goal:
X : Type u_1
d : X → X → ℝ≥0
dist_self : ∀ (x : X), d x x = 0
dist_comm : ∀ (x y : X), d x y = d y x
hd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))
this : IsTrans X fun (x y : X) ↦ d x y = 0
x y : X
l : List X
ihn : ∀ m < l.length, ∀ (x y : X) (l : List X) (hn : l.length = m), d x y ≤ 2 * (zipWith d (x :: l) (l ++ [y])).sum
L : List ℝ≥0 := zipWith d (x :: l) (l ++ [y])
hL_len : L.length = l.length + 1
hd₀ : d x y ≠ 0
s : Set ℕ := {m : ℕ | 2 * (take m L).sum ≤ L.sum}
hs₀ : 0 ∈ s
hsne : s.Nonempty
M : ℕ
hMl : M < l.length
hMs : IsGreatest s (M + 1)
hM_lt : M + 1 < L.length
hM_ltx : M + 1 < (x :: l).length
hM_lty : M + 1 < (l ++ [y]).length
hMl' : (take M l).length = M
h :
  zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    zipWith d (take (M + 1) (x :: l)) (take (M + 1) (l ++ [y]))
h₁ :
  zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    zipWith d (x :: take M l) (take (M + 1) (l ++ [y]))
h₂ :
  zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    zipWith d (x :: take M l)
      (take M (l ++ [y]) ++ (getElem? (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (l ++ [y]) M).toList)
h₃ :
  zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    zipWith d (x :: take M l)
      (take M (l ++ [y]) ++ (getElem? (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) l M).toList)
⊢ zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    take (M + 1) L
have h₄ : zipWith d (x :: take M l)
    (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
  zipWith d (x :: take M l)
    (take M (l ++ [y]) ++ (some (getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) l M hMl)).toList)
Tactic: rw [take_zipWith, take, take_add_one, getElem?_append_left hMl, getElem?_eq_getElem hMl,
        ← Option.coe_def, Option.toList_some, take_append_of_le_length hMl.le, getElem_cons_succ]
-- Lean 4 proof step from Mathlib.Topology.Metrizable.Uniformity (goal)
Goal:
X : Type u_1
d : X → X → ℝ≥0
dist_self : ∀ (x : X), d x x = 0
dist_comm : ∀ (x y : X), d x y = d y x
hd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))
this : IsTrans X fun (x y : X) ↦ d x y = 0
x y : X
l : List X
ihn : ∀ m < l.length, ∀ (x y : X) (l : List X) (hn : l.length = m), d x y ≤ 2 * (zipWith d (x :: l) (l ++ [y])).sum
L : List ℝ≥0 := zipWith d (x :: l) (l ++ [y])
hL_len : L.length = l.length + 1
hd₀ : d x y ≠ 0
s : Set ℕ := {m : ℕ | 2 * (take m L).sum ≤ L.sum}
hs₀ : 0 ∈ s
hsne : s.Nonempty
M : ℕ
hMl : M < l.length
hMs : IsGreatest s (M + 1)
hM_lt : M + 1 < L.length
hM_ltx : M + 1 < (x :: l).length
hM_lty : M + 1 < (l ++ [y]).length
hMl' : (take M l).length = M
h :
  zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    zipWith d (take (M + 1) (x :: l)) (take (M + 1) (l ++ [y]))
h₁ :
  zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    zipWith d (x :: take M l) (take (M + 1) (l ++ [y]))
h₂ :
  zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    zipWith d (x :: take M l)
      (take M (l ++ [y]) ++ (getElem? (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (l ++ [y]) M).toList)
h₃ :
  zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    zipWith d (x :: take M l)
      (take M (l ++ [y]) ++ (getElem? (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) l M).toList)
h₄ :
  zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    zipWith d (x :: take M l)
      (take M (l ++ [y]) ++ (some (getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) l M hMl)).toList)
h₅ :
  zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    zipWith d (x :: take M l)
      (take M (l ++ [y]) ++
        Option.toList ((fun (a : X) ↦ some a) (getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) l M hMl)))
⊢ zipWith d (x :: take M l)
      (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
    take (M + 1) L
have h₆ : zipWith d (x :: take M l)
    (take M l ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) (M + 1) hM_ltx]) =
  zipWith d (x :: take M l)
    (take M (l ++ [y]) ++ [getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) l M hMl])
Tactic: rw [take_zipWith, take, take_add_one, getElem?_append_left hMl, getElem?_eq_getElem hMl,
        ← Option.coe_def, Option.toList_some, take_append_of_le_length hMl.le, getElem_cons_succ]
-- Lean 4 proof step from Mathlib.Topology.Metrizable.Uniformity (goal)
Goal:
X : Type u_1
d : X → X → ℝ≥0
dist_self : ∀ (x : X), d x x = 0
dist_comm : ∀ (x y : X), d x y = d y x
hd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))
this : IsTrans X fun (x y : X) ↦ d x y = 0
x y : X
l : List X
ihn : ∀ m < l.length, ∀ (x y : X) (l : List X) (hn : l.length = m), d x y ≤ 2 * (zipWith d (x :: l) (l ++ [y])).sum
L : List ℝ≥0 := zipWith d (x :: l) (l ++ [y])
hL_len : L.length = l.length + 1
hd₀ : d x y ≠ 0
s : Set ℕ := {m : ℕ | 2 * (take m L).sum ≤ L.sum}
hs₀ : 0 ∈ s
hsne : s.Nonempty
M : ℕ
hMl : M ≤ l.length
hMs : IsGreatest s M
hM_lt : M < L.length
hM_ltx : M < (x :: l).length
hM_lty : M < (l ++ [y]).length
⊢ d x (getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) M hM_ltx) ≤ L.sum
have refine_1 : ∀ (hMl : 0 ≤ l.length) (hMs : IsGreatest s 0) (hM_lt : 0 < L.length) (hM_ltx : 0 < (x :: l).length)
  (hM_lty : 0 < (l ++ [y]).length),
  d x (getElem (valid := fun (as : List X) (i : ℕ) ↦ i < as.length) (x :: l) 0 hM_ltx) ≤ L.sum
Tactic: cases M with
    | zero =>
      simp [dist_self]
    | succ M =>
      rw [Nat.succ_le_iff] at hMl
      have hMl' : length (take M l) = M := length_take.trans (min_eq_left hMl.le)
      refine (ihn _ hMl _ _ _ hMl').trans ?_
      convert! hMs.1.out
      rw [take_zipWith, take, take_add_one, getElem?_append_left hMl, getElem?_eq_getElem hMl,
        ← Option.coe_def, Option.toList_some, take_append_of_le_length hMl.le, getElem_cons_succ]
Cleared hypotheses: M, hMl, hMs, hM_lt, hM_ltx, hM_lty