awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
s : Set α
inst_1 : Decidable s.Nontrivial
hsf : s.Finite
hs : s.Nontrivial
x : ℝ
pos : (∀ x_1 ∈ s, ∀ y ∈ s, x_1 ≠ y → x ≤ dist x_1 y) ↔ x ≤ ⋯.toFinset (s := s.offDiag).inf' ⋯ (uncurry dist)
pos₁ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ dist x_1 x_2) ↔
    x ≤ ⋯.toFinset (s := s.offDiag).inf' ⋯ (uncurry dist)
pos₂ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ dist x_1 x_2) ↔
    ∀ b ∈ ⋯.toFinset (s := s.offDiag), x ≤ uncurry dist b
pos₃ : (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ dist x_1 x_2) ↔ ∀ b ∈ s.offDiag, x ≤ uncurry dist b
⊢ x ≤ s.infsep ↔ x ≤ ⋯.toFinset (s := s.offDiag).inf' ⋯ (uncurry dist)
have pos₄ : (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ dist x_1 x_2) ↔
  ∀ (b : α × α), b.1 ∈ s ∧ b.2 ∈ s ∧ b.1 ≠ b.2 → x ≤ uncurry dist b
Tactic: simp_rw [hs.le_infsep_iff, imp_forall_iff, Finset.le_inf'_iff, Finite.mem_toFinset,
      mem_offDiag, Prod.forall, uncurry_apply_pair, and_imp]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
s : Set α
inst_1 : Decidable s.Nontrivial
hsf : s.Finite
hs : s.Nontrivial
x : ℝ
pos : (∀ x_1 ∈ s, ∀ y ∈ s, x_1 ≠ y → x ≤ dist x_1 y) ↔ x ≤ ⋯.toFinset (s := s.offDiag).inf' ⋯ (uncurry dist)
pos₁ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ dist x_1 x_2) ↔
    x ≤ ⋯.toFinset (s := s.offDiag).inf' ⋯ (uncurry dist)
pos₂ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ dist x_1 x_2) ↔
    ∀ b ∈ ⋯.toFinset (s := s.offDiag), x ≤ uncurry dist b
pos₃ : (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ dist x_1 x_2) ↔ ∀ b ∈ s.offDiag, x ≤ uncurry dist b
pos₄ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ dist x_1 x_2) ↔
    ∀ (b : α × α), b.1 ∈ s ∧ b.2 ∈ s ∧ b.1 ≠ b.2 → x ≤ uncurry dist b
⊢ x ≤ s.infsep ↔ x ≤ ⋯.toFinset (s := s.offDiag).inf' ⋯ (uncurry dist)
have pos₅ : (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ dist x_1 x_2) ↔
  ∀ (a b : α), a ∈ s ∧ b ∈ s ∧ a ≠ b → x ≤ uncurry dist (a, b)
Tactic: simp_rw [hs.le_infsep_iff, imp_forall_iff, Finset.le_inf'_iff, Finite.mem_toFinset,
      mem_offDiag, Prod.forall, uncurry_apply_pair, and_imp]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
s : Set α
inst_1 : Decidable s.Nontrivial
hsf : s.Finite
hs : s.Nontrivial
x : ℝ
pos : (∀ x_1 ∈ s, ∀ y ∈ s, x_1 ≠ y → x ≤ dist x_1 y) ↔ x ≤ ⋯.toFinset (s := s.offDiag).inf' ⋯ (uncurry dist)
pos₁ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ dist x_1 x_2) ↔
    x ≤ ⋯.toFinset (s := s.offDiag).inf' ⋯ (uncurry dist)
pos₂ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ dist x_1 x_2) ↔
    ∀ b ∈ ⋯.toFinset (s := s.offDiag), x ≤ uncurry dist b
pos₃ : (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ dist x_1 x_2) ↔ ∀ b ∈ s.offDiag, x ≤ uncurry dist b
pos₄ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ dist x_1 x_2) ↔
    ∀ (b : α × α), b.1 ∈ s ∧ b.2 ∈ s ∧ b.1 ≠ b.2 → x ≤ uncurry dist b
pos₅ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ dist x_1 x_2) ↔
    ∀ (a b : α), a ∈ s ∧ b ∈ s ∧ a ≠ b → x ≤ uncurry dist (a, b)
⊢ x ≤ s.infsep ↔ x ≤ ⋯.toFinset (s := s.offDiag).inf' ⋯ (uncurry dist)
have pos₆ : (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ dist x_1 x_2) ↔ ∀ (a b : α), a ∈ s ∧ b ∈ s ∧ a ≠ b → x ≤ dist a b
Tactic: simp_rw [hs.le_infsep_iff, imp_forall_iff, Finset.le_inf'_iff, Finite.mem_toFinset,
      mem_offDiag, Prod.forall, uncurry_apply_pair, and_imp]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (hypothesis)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
s : Set α
inst_1 : Decidable s.Nontrivial
hsf : s.Finite
hs : ¬s.Nontrivial
⊢ s.infsep = 0
have hs₁ : s.Subsingleton
Tactic: rw [not_nontrivial_iff] at hs
Cleared hypotheses: hs
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
s : Set α
hsf : s.Finite
hs : s.Nontrivial
⊢ s.infsep = ⋯.toFinset (s := s.offDiag).inf' ⋯ (uncurry dist)
have simp_rw : (if hs : s.Nontrivial then sorry.toFinset (s := s.offDiag).inf' sorry (uncurry dist) else 0) =
  sorry.toFinset (s := s.offDiag).inf' sorry (uncurry dist)
Tactic: simp_rw [hsf.infsep, dif_pos hs]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
s : Set α
hsf : s.Finite
hs : s.Nontrivial
⊢ s.infsep = ⋯.toFinset (s := s.offDiag).inf' ⋯ (uncurry dist)
have classical : (if hs : s.Nontrivial then sorry.toFinset (s := s.offDiag).inf' sorry (uncurry dist) else 0) =
  sorry.toFinset (s := s.offDiag).inf' sorry (uncurry dist)
Tactic: classical simp_rw [hsf.infsep, dif_pos hs]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
s : Finset α
⊢ (↑s : Set α).Nontrivial ↔ s.offDiag.Nonempty
have rw : (↑s : Set α).offDiag.Nonempty ↔ s.offDiag.Nonempty
Tactic: rw [← Set.offDiag_nonempty, ← Finset.coe_offDiag, Finset.coe_nonempty]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
s : Finset α
rw : (↑s : Set α).offDiag.Nonempty ↔ s.offDiag.Nonempty
⊢ (↑s : Set α).Nontrivial ↔ s.offDiag.Nonempty
have rw₁ : (↑s.offDiag : Set (α × α)).Nonempty ↔ s.offDiag.Nonempty
Tactic: rw [← Set.offDiag_nonempty, ← Finset.coe_offDiag, Finset.coe_nonempty]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
s : Finset α
rw : (↑s : Set α).offDiag.Nonempty ↔ s.offDiag.Nonempty
rw₁ : (↑s.offDiag : Set (α × α)).Nonempty ↔ s.offDiag.Nonempty
⊢ (↑s : Set α).Nontrivial ↔ s.offDiag.Nonempty
have rw₂ : s.offDiag.Nonempty ↔ s.offDiag.Nonempty
Tactic: rw [← Set.offDiag_nonempty, ← Finset.coe_offDiag, Finset.coe_nonempty]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (hypothesis)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
s : Finset α
⊢ (↑s : Set α).infsep = if hs : s.offDiag.Nonempty then s.offDiag.inf' hs (uncurry dist) else 0
have H : (↑s : Set α).Nontrivial ↔ s.offDiag.Nonempty
Tactic: have H : (s : Set α).Nontrivial ↔ s.offDiag.Nonempty := by
    rw [← Set.offDiag_nonempty, ← Finset.coe_offDiag, Finset.coe_nonempty]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
s : Finset α
H : (↑s : Set α).Nontrivial ↔ s.offDiag.Nonempty
⊢ (↑s : Set α).infsep = if hs : s.offDiag.Nonempty then s.offDiag.inf' hs (uncurry dist) else 0
have pos : ∀ (hs : s.offDiag.Nonempty), (↑s : Set α).infsep = s.offDiag.inf' hs (uncurry dist)
Tactic: split_ifs with hs
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
s : Finset α
H : (↑s : Set α).Nontrivial ↔ s.offDiag.Nonempty
pos : ∀ (hs : s.offDiag.Nonempty), (↑s : Set α).infsep = s.offDiag.inf' hs (uncurry dist)
⊢ (↑s : Set α).infsep = if hs : s.offDiag.Nonempty then s.offDiag.inf' hs (uncurry dist) else 0
have neg : ∀ (hs : ¬s.offDiag.Nonempty), (↑s : Set α).infsep = 0
Tactic: split_ifs with hs
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
s : Finset α
H : (↑s : Set α).Nontrivial ↔ s.offDiag.Nonempty
hs : s.offDiag.Nonempty
⊢ (↑s : Set α).infsep = s.offDiag.inf' hs (uncurry dist)
have pos : (↑s : Set α).offDiag.toFinset.inf' sorry (uncurry dist) = s.offDiag.inf' hs (uncurry dist)
Tactic: simp_rw [(H.mpr hs).infsep_of_fintype, ← Finset.coe_offDiag, Finset.toFinset_coe]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
s : Finset α
H : (↑s : Set α).Nontrivial ↔ s.offDiag.Nonempty
hs : s.offDiag.Nonempty
pos : (↑s : Set α).offDiag.toFinset.inf' ⋯ (uncurry dist) = s.offDiag.inf' hs (uncurry dist)
⊢ (↑s : Set α).infsep = s.offDiag.inf' hs (uncurry dist)
have pos₁ : ((↑s.offDiag : Set (α × α)).toFinset.inf' sorry fun (x : α × α) ↦ uncurry dist x) = s.offDiag.inf' hs (uncurry dist)
Tactic: simp_rw [(H.mpr hs).infsep_of_fintype, ← Finset.coe_offDiag, Finset.toFinset_coe]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
s : Finset α
H : (↑s : Set α).Nontrivial ↔ s.offDiag.Nonempty
hs : s.offDiag.Nonempty
⊢ (↑s : Set α).infsep = s.offDiag.inf' hs (uncurry dist)
have pos : (↑s : Set α).offDiag.toFinset.inf' sorry (uncurry dist) = s.offDiag.inf' hs (uncurry dist)
Tactic: classical simp_rw [(H.mpr hs).infsep_of_fintype, ← Finset.coe_offDiag, Finset.toFinset_coe]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
s : Finset α
H : (↑s : Set α).Nontrivial ↔ s.offDiag.Nonempty
hs : s.offDiag.Nonempty
pos : (↑s : Set α).offDiag.toFinset.inf' ⋯ (uncurry dist) = s.offDiag.inf' hs (uncurry dist)
⊢ (↑s : Set α).infsep = s.offDiag.inf' hs (uncurry dist)
have pos₁ : ((↑s.offDiag : Set (α × α)).toFinset.inf' sorry fun (x : α × α) ↦ uncurry dist x) = s.offDiag.inf' hs (uncurry dist)
Tactic: classical simp_rw [(H.mpr hs).infsep_of_fintype, ← Finset.coe_offDiag, Finset.toFinset_coe]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
s : Finset α
hs : s.offDiag.Nonempty
⊢ (↑s : Set α).infsep = s.offDiag.inf' hs (uncurry dist)
have rw : (if hs : s.offDiag.Nonempty then s.offDiag.inf' hs (uncurry dist) else 0) = s.offDiag.inf' hs (uncurry dist)
Tactic: rw [Finset.coe_infsep, dif_pos hs]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
s : Finset α
hs : s.offDiag.Nonempty
rw : (if hs : s.offDiag.Nonempty then s.offDiag.inf' hs (uncurry dist) else 0) = s.offDiag.inf' hs (uncurry dist)
⊢ (↑s : Set α).infsep = s.offDiag.inf' hs (uncurry dist)
have rw₁ : s.offDiag.inf' hs (uncurry dist) = s.offDiag.inf' hs (uncurry dist)
Tactic: rw [Finset.coe_infsep, dif_pos hs]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (hypothesis)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
s : Finset α
hs : s.offDiag = ∅
⊢ (↑s : Set α).infsep = 0
have hs₁ : ¬s.offDiag.Nonempty
Tactic: rw [← Finset.not_nonempty_iff_eq_empty] at hs
Cleared hypotheses: hs
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
s : Finset α
hs : ¬s.offDiag.Nonempty
⊢ (↑s : Set α).infsep = 0
have rw : Eq (α := ℝ) (if hs : s.offDiag.Nonempty then s.offDiag.inf' hs (uncurry dist) else 0) 0
Tactic: rw [Finset.coe_infsep, dif_neg hs]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
s : Finset α
hs : ¬s.offDiag.Nonempty
rw : Eq (α := ℝ) (if hs : s.offDiag.Nonempty then s.offDiag.inf' hs (uncurry dist) else 0) 0
⊢ (↑s : Set α).infsep = 0
have rw₁ : Eq (α := ℝ) 0 0
Tactic: rw [Finset.coe_infsep, dif_neg hs]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
s : Set α
inst_1 : Finite.{u_1 + 1} (↑s : Type u_1)
hs : s.Nontrivial
⊢ ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.infsep = dist x y
have intro : ∀ (val : Fintype (↑s : Type u_1)), ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.infsep = dist x y
Tactic: cases nonempty_fintype s
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
s : Set α
inst_1 : Finite.{u_1 + 1} (↑s : Type u_1)
hs : s.Nontrivial
val : Fintype (↑s : Type u_1)
⊢ ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.infsep = dist x y
have intro : ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.offDiag.toFinset.inf' sorry (uncurry dist) = dist x y
Tactic: simp_rw [hs.infsep_of_fintype]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
s : Set α
inst_1 : Finite.{u_1 + 1} (↑s : Type u_1)
hs : s.Nontrivial
val : Fintype (↑s : Type u_1)
⊢ ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.offDiag.toFinset.inf' ⋯ (uncurry dist) = dist x y
have intro : ∀ (w : α × α) (hxy : w ∈ s.offDiag.toFinset) (hed : s.offDiag.toFinset.inf' sorry (uncurry dist) = uncurry dist w),
  ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.offDiag.toFinset.inf' sorry (uncurry dist) = dist x y
Tactic: rcases Finset.exists_mem_eq_inf' (s := s.offDiag.toFinset) (by simpa) (uncurry dist) with
      ⟨w, hxy, hed⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (hypothesis)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
s : Set α
inst_1 : Finite.{u_1 + 1} (↑s : Type u_1)
hs : s.Nontrivial
val : Fintype (↑s : Type u_1)
w : α × α
hxy : w ∈ s.offDiag.toFinset
hed : s.offDiag.toFinset.inf' ⋯ (uncurry dist) = uncurry dist w
⊢ ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.offDiag.toFinset.inf' ⋯ (uncurry dist) = dist x y
have hxy₁ : w ∈ s.offDiag
Tactic: simp_rw [mem_toFinset] at hxy
Cleared hypotheses: hxy
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
s : Set α
inst_1 : Finite.{u_1 + 1} (↑s : Type u_1)
hs : s.Nontrivial
⊢ ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.infsep = dist x y
have intro : ∀ (val : Fintype (↑s : Type u_1)), ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.infsep = dist x y
Tactic: classical
    cases nonempty_fintype s
    simp_rw [hs.infsep_of_fintype]
    rcases Finset.exists_mem_eq_inf' (s := s.offDiag.toFinset) (by simpa) (uncurry dist) with
      ⟨w, hxy, hed⟩
    simp_rw [mem_toFinset] at hxy
    exact ⟨w.fst, hxy.1, w.snd, hxy.2.1, hxy.2.2, hed⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
s : Set α
inst_1 : Finite.{u_1 + 1} (↑s : Type u_1)
hs : s.Nontrivial
intro : ∀ (val : Fintype (↑s : Type u_1)), ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.infsep = dist x y
⊢ ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.infsep = dist x y
have intro₁ : ∀ (val : Fintype (↑s : Type u_1)), ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.offDiag.toFinset.inf' sorry (uncurry dist) = dist x y
Tactic: classical
    cases nonempty_fintype s
    simp_rw [hs.infsep_of_fintype]
    rcases Finset.exists_mem_eq_inf' (s := s.offDiag.toFinset) (by simpa) (uncurry dist) with
      ⟨w, hxy, hed⟩
    simp_rw [mem_toFinset] at hxy
    exact ⟨w.fst, hxy.1, w.snd, hxy.2.1, hxy.2.2, hed⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
s : Set α
inst_1 : Finite.{u_1 + 1} (↑s : Type u_1)
hs : s.Nontrivial
intro : ∀ (val : Fintype (↑s : Type u_1)), ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.infsep = dist x y
intro₁ :
  ∀ (val : Fintype (↑s : Type u_1)), ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.offDiag.toFinset.inf' ⋯ (uncurry dist) = dist x y
⊢ ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.infsep = dist x y
have intro₂ : ∀ (val : Fintype (↑s : Type u_1)) (w : α × α) (hxy : w ∈ s.offDiag.toFinset)
  (hed : s.offDiag.toFinset.inf' sorry (uncurry dist) = uncurry dist w),
  ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.offDiag.toFinset.inf' sorry (uncurry dist) = dist x y
Tactic: classical
    cases nonempty_fintype s
    simp_rw [hs.infsep_of_fintype]
    rcases Finset.exists_mem_eq_inf' (s := s.offDiag.toFinset) (by simpa) (uncurry dist) with
      ⟨w, hxy, hed⟩
    simp_rw [mem_toFinset] at hxy
    exact ⟨w.fst, hxy.1, w.snd, hxy.2.1, hxy.2.2, hed⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : PseudoMetricSpace α
s : Set α
inst_1 : Finite.{u_1 + 1} (↑s : Type u_1)
hs : s.Nontrivial
intro : ∀ (val : Fintype (↑s : Type u_1)), ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.infsep = dist x y
intro₁ :
  ∀ (val : Fintype (↑s : Type u_1)), ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.offDiag.toFinset.inf' ⋯ (uncurry dist) = dist x y
intro₂ :
  ∀ (val : Fintype (↑s : Type u_1)) (w : α × α) (hxy : w ∈ s.offDiag.toFinset)
    (hed : s.offDiag.toFinset.inf' ⋯ (uncurry dist) = uncurry dist w),
    ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.offDiag.toFinset.inf' ⋯ (uncurry dist) = dist x y
⊢ ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.infsep = dist x y
have intro₃ : ∀ (val : Fintype (↑s : Type u_1)) (w : α × α) (hed : s.offDiag.toFinset.inf' sorry (uncurry dist) = uncurry dist w)
  (hxy : w ∈ s.offDiag), ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.offDiag.toFinset.inf' sorry (uncurry dist) = dist x y
Tactic: classical
    cases nonempty_fintype s
    simp_rw [hs.infsep_of_fintype]
    rcases Finset.exists_mem_eq_inf' (s := s.offDiag.toFinset) (by simpa) (uncurry dist) with
      ⟨w, hxy, hed⟩
    simp_rw [mem_toFinset] at hxy
    exact ⟨w.fst, hxy.1, w.snd, hxy.2.1, hxy.2.2, hed⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : MetricSpace α
s : Set α
inst_1 : Finite.{u_1 + 1} (↑s : Type u_1)
⊢ s.infsep = 0 ↔ s.Subsingleton
have rw : s.einfsep = 0 → s.Subsingleton
Tactic: rw [infsep_zero, einfsep_eq_top_iff, or_iff_right_iff_imp]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : MetricSpace α
s : Set α
inst_1 : Finite.{u_1 + 1} (↑s : Type u_1)
⊢ 0 < s.infsep ↔ s.Nontrivial
have rw : s.Nontrivial → 0 < s.einfsep
Tactic: rw [infsep_pos, einfsep_lt_top_iff, and_iff_right_iff_imp]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.IsometricSMul (goal)
Goal:
G : Type v
inst : Group G
inst_1 : PseudoEMetricSpace G
inst_2 : IsIsometricSMul G G
inst_3 : IsIsometricSMul Gᵐᵒᵖ G
a b : G
⊢ edist a⁻¹ b⁻¹ = edist a b
have rw : edist (a * a⁻¹) (a * b⁻¹) = edist a b
Tactic: rw [← edist_mul_left a, ← edist_mul_right _ _ b, mul_inv_cancel, one_mul, inv_mul_cancel_right,
    edist_comm]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.IsometricSMul (goal)
Goal:
G : Type v
inst : Group G
inst_1 : PseudoEMetricSpace G
inst_2 : IsIsometricSMul G G
inst_3 : IsIsometricSMul Gᵐᵒᵖ G
a b : G
rw : edist (a * a⁻¹) (a * b⁻¹) = edist a b
⊢ edist a⁻¹ b⁻¹ = edist a b
have rw₁ : edist (a * a⁻¹ * b) (a * b⁻¹ * b) = edist a b
Tactic: rw [← edist_mul_left a, ← edist_mul_right _ _ b, mul_inv_cancel, one_mul, inv_mul_cancel_right,
    edist_comm]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.IsometricSMul (goal)
Goal:
G : Type v
inst : Group G
inst_1 : PseudoEMetricSpace G
inst_2 : IsIsometricSMul G G
inst_3 : IsIsometricSMul Gᵐᵒᵖ G
a b : G
rw : edist (a * a⁻¹) (a * b⁻¹) = edist a b
rw₁ : edist (a * a⁻¹ * b) (a * b⁻¹ * b) = edist a b
⊢ edist a⁻¹ b⁻¹ = edist a b
have rw₂ : edist (1 * b) (a * b⁻¹ * b) = edist a b
Tactic: rw [← edist_mul_left a, ← edist_mul_right _ _ b, mul_inv_cancel, one_mul, inv_mul_cancel_right,
    edist_comm]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.IsometricSMul (goal)
Goal:
G : Type v
inst : Group G
inst_1 : PseudoEMetricSpace G
inst_2 : IsIsometricSMul G G
inst_3 : IsIsometricSMul Gᵐᵒᵖ G
a b : G
rw : edist (a * a⁻¹) (a * b⁻¹) = edist a b
rw₁ : edist (a * a⁻¹ * b) (a * b⁻¹ * b) = edist a b
rw₂ : edist (1 * b) (a * b⁻¹ * b) = edist a b
⊢ edist a⁻¹ b⁻¹ = edist a b
have rw₃ : edist b (a * b⁻¹ * b) = edist a b
Tactic: rw [← edist_mul_left a, ← edist_mul_right _ _ b, mul_inv_cancel, one_mul, inv_mul_cancel_right,
    edist_comm]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.IsometricSMul (goal)
Goal:
G : Type v
inst : Group G
inst_1 : PseudoEMetricSpace G
inst_2 : IsIsometricSMul G G
inst_3 : IsIsometricSMul Gᵐᵒᵖ G
a b : G
rw : edist (a * a⁻¹) (a * b⁻¹) = edist a b
rw₁ : edist (a * a⁻¹ * b) (a * b⁻¹ * b) = edist a b
rw₂ : edist (1 * b) (a * b⁻¹ * b) = edist a b
rw₃ : edist b (a * b⁻¹ * b) = edist a b
⊢ edist a⁻¹ b⁻¹ = edist a b
have rw₄ : edist b a = edist a b
Tactic: rw [← edist_mul_left a, ← edist_mul_right _ _ b, mul_inv_cancel, one_mul, inv_mul_cancel_right,
    edist_comm]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.IsometricSMul (goal)
Goal:
G : Type v
inst : Group G
inst_1 : PseudoEMetricSpace G
inst_2 : IsIsometricSMul G G
inst_3 : IsIsometricSMul Gᵐᵒᵖ G
a b : G
rw : edist (a * a⁻¹) (a * b⁻¹) = edist a b
rw₁ : edist (a * a⁻¹ * b) (a * b⁻¹ * b) = edist a b
rw₂ : edist (1 * b) (a * b⁻¹ * b) = edist a b
rw₃ : edist b (a * b⁻¹ * b) = edist a b
rw₄ : edist b a = edist a b
⊢ edist a⁻¹ b⁻¹ = edist a b
have rw₅ : edist a b = edist a b
Tactic: rw [← edist_mul_left a, ← edist_mul_right _ _ b, mul_inv_cancel, one_mul, inv_mul_cancel_right,
    edist_comm]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.IsometricSMul (goal)
Goal:
G : Type v
inst : Group G
inst_1 : PseudoEMetricSpace G
inst_2 : IsIsometricSMul G G
inst_3 : IsIsometricSMul Gᵐᵒᵖ G
x y : G
⊢ edist x⁻¹ y = edist x y⁻¹
have rw : edist x⁻¹⁻¹ y⁻¹ = edist x y⁻¹
Tactic: rw [← edist_inv_inv, inv_inv]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.IsometricSMul (goal)
Goal:
G : Type v
inst : Group G
inst_1 : PseudoEMetricSpace G
inst_2 : IsIsometricSMul G G
inst_3 : IsIsometricSMul Gᵐᵒᵖ G
x y : G
rw : edist x⁻¹⁻¹ y⁻¹ = edist x y⁻¹
⊢ edist x⁻¹ y = edist x y⁻¹
have rw₁ : edist x y⁻¹ = edist x y⁻¹
Tactic: rw [← edist_inv_inv, inv_inv]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.IsometricSMul (goal)
Goal:
G : Type v
inst : Group G
inst_1 : PseudoEMetricSpace G
inst_2 : IsIsometricSMul G G
inst_3 : IsIsometricSMul Gᵐᵒᵖ G
a b c : G
⊢ edist (a / b) (a / c) = edist b c
have rw : edist (a * b⁻¹) (a / c) = edist b c
Tactic: rw [div_eq_mul_inv, div_eq_mul_inv, edist_mul_left, edist_inv_inv]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.IsometricSMul (goal)
Goal:
G : Type v
inst : Group G
inst_1 : PseudoEMetricSpace G
inst_2 : IsIsometricSMul G G
inst_3 : IsIsometricSMul Gᵐᵒᵖ G
a b c : G
rw : edist (a * b⁻¹) (a / c) = edist b c
⊢ edist (a / b) (a / c) = edist b c
have rw₁ : edist (a * b⁻¹) (a * c⁻¹) = edist b c
Tactic: rw [div_eq_mul_inv, div_eq_mul_inv, edist_mul_left, edist_inv_inv]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.IsometricSMul (goal)
Goal:
G : Type v
inst : Group G
inst_1 : PseudoEMetricSpace G
inst_2 : IsIsometricSMul G G
inst_3 : IsIsometricSMul Gᵐᵒᵖ G
a b c : G
rw : edist (a * b⁻¹) (a / c) = edist b c
rw₁ : edist (a * b⁻¹) (a * c⁻¹) = edist b c
⊢ edist (a / b) (a / c) = edist b c
have rw₂ : edist b⁻¹ c⁻¹ = edist b c
Tactic: rw [div_eq_mul_inv, div_eq_mul_inv, edist_mul_left, edist_inv_inv]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.IsometricSMul (goal)
Goal:
G : Type v
inst : Group G
inst_1 : PseudoEMetricSpace G
inst_2 : IsIsometricSMul G G
inst_3 : IsIsometricSMul Gᵐᵒᵖ G
a b c : G
rw : edist (a * b⁻¹) (a / c) = edist b c
rw₁ : edist (a * b⁻¹) (a * c⁻¹) = edist b c
rw₂ : edist b⁻¹ c⁻¹ = edist b c
⊢ edist (a / b) (a / c) = edist b c
have rw₃ : edist b c = edist b c
Tactic: rw [div_eq_mul_inv, div_eq_mul_inv, edist_mul_left, edist_inv_inv]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.IsometricSMul (goal)
Goal:
G : Type v
X : Type w
inst : PseudoEMetricSpace X
inst_1 : Group G
inst_2 : MulAction G X
inst_3 : IsIsometricSMul G X
c : G
x : X
r : ℝ≥0∞
⊢ (fun (x : X) ↦ c • x) ⁻¹' eball x r = eball (c⁻¹ • x) r
have rw : c⁻¹ • eball x r = eball (c⁻¹ • x) r
Tactic: rw [preimage_smul, smul_eball]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.IsometricSMul (goal)
Goal:
G : Type v
X : Type w
inst : PseudoEMetricSpace X
inst_1 : Group G
inst_2 : MulAction G X
inst_3 : IsIsometricSMul G X
c : G
x : X
r : ℝ≥0∞
rw : c⁻¹ • eball x r = eball (c⁻¹ • x) r
⊢ (fun (x : X) ↦ c • x) ⁻¹' eball x r = eball (c⁻¹ • x) r
have rw₁ : eball (c⁻¹ • x) r = eball (c⁻¹ • x) r
Tactic: rw [preimage_smul, smul_eball]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.IsometricSMul (goal)
Goal:
G : Type v
X : Type w
inst : PseudoEMetricSpace X
inst_1 : Group G
inst_2 : MulAction G X
inst_3 : IsIsometricSMul G X
c : G
x : X
r : ℝ≥0∞
⊢ (fun (x : X) ↦ c • x) ⁻¹' closedEBall x r = closedEBall (c⁻¹ • x) r
have rw : c⁻¹ • closedEBall x r = closedEBall (c⁻¹ • x) r
Tactic: rw [preimage_smul, smul_closedEBall]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.IsometricSMul (goal)
Goal:
G : Type v
X : Type w
inst : PseudoEMetricSpace X
inst_1 : Group G
inst_2 : MulAction G X
inst_3 : IsIsometricSMul G X
c : G
x : X
r : ℝ≥0∞
rw : c⁻¹ • closedEBall x r = closedEBall (c⁻¹ • x) r
⊢ (fun (x : X) ↦ c • x) ⁻¹' closedEBall x r = closedEBall (c⁻¹ • x) r
have rw₁ : closedEBall (c⁻¹ • x) r = closedEBall (c⁻¹ • x) r
Tactic: rw [preimage_smul, smul_closedEBall]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.IsometricSMul (goal)
Goal:
G : Type v
inst : Group G
inst_1 : PseudoEMetricSpace G
inst_2 : IsIsometricSMul Gᵐᵒᵖ G
a b : G
r : ℝ≥0∞
⊢ (fun (x : G) ↦ x * a) ⁻¹' eball b r = eball (b / a) r
have rw : (fun (x : G) ↦ x * a) ⁻¹' eball b r = eball (b * a⁻¹) r
Tactic: rw [div_eq_mul_inv]