awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : EDist α
s : Set α
x : ℝ≥0∞
simp_rw :
  (∀ x_1 ∈ s, ∀ y ∈ s, x_1 ≠ y → x ≤ edist x_1 y) ↔ x ≤ ⨅ (d : (↑s.offDiag : Type u_1)), uncurry edist (↑d : α × α)
⊢ x ≤ s.einfsep ↔ x ≤ ⨅ (d : (↑s.offDiag : Type u_1)), uncurry edist (↑d : α × α)
have simp_rw₁ : (∀ x_1 ∈ s, ∀ y ∈ s, x_1 ≠ y → x ≤ edist x_1 y) ↔ ∀ (i : (↑s.offDiag : Type u_1)), x ≤ uncurry edist (↑i : α × α)
Tactic: simp_rw [le_einfsep_iff, le_iInf_iff, imp_forall_iff, SetCoe.forall, 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 : EDist α
s : Set α
x : ℝ≥0∞
simp_rw :
  (∀ x_1 ∈ s, ∀ y ∈ s, x_1 ≠ y → x ≤ edist x_1 y) ↔ x ≤ ⨅ (d : (↑s.offDiag : Type u_1)), uncurry edist (↑d : α × α)
simp_rw₁ :
  (∀ x_1 ∈ s, ∀ y ∈ s, x_1 ≠ y → x ≤ edist x_1 y) ↔ ∀ (i : (↑s.offDiag : Type u_1)), x ≤ uncurry edist (↑i : α × α)
⊢ x ≤ s.einfsep ↔ x ≤ ⨅ (d : (↑s.offDiag : Type u_1)), uncurry edist (↑d : α × α)
have simp_rw₂ : (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
  ∀ (i : (↑s.offDiag : Type u_1)), x ≤ uncurry edist (↑i : α × α)
Tactic: simp_rw [le_einfsep_iff, le_iInf_iff, imp_forall_iff, SetCoe.forall, 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 : EDist α
s : Set α
x : ℝ≥0∞
simp_rw :
  (∀ x_1 ∈ s, ∀ y ∈ s, x_1 ≠ y → x ≤ edist x_1 y) ↔ x ≤ ⨅ (d : (↑s.offDiag : Type u_1)), uncurry edist (↑d : α × α)
simp_rw₁ :
  (∀ x_1 ∈ s, ∀ y ∈ s, x_1 ≠ y → x ≤ edist x_1 y) ↔ ∀ (i : (↑s.offDiag : Type u_1)), x ≤ uncurry edist (↑i : α × α)
simp_rw₂ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
    ∀ (i : (↑s.offDiag : Type u_1)), x ≤ uncurry edist (↑i : α × α)
⊢ x ≤ s.einfsep ↔ x ≤ ⨅ (d : (↑s.offDiag : Type u_1)), uncurry edist (↑d : α × α)
have simp_rw₃ : (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
  ∀ (x_1 : α × α) (h : x_1 ∈ s.offDiag), x ≤ uncurry edist x_1
Tactic: simp_rw [le_einfsep_iff, le_iInf_iff, imp_forall_iff, SetCoe.forall, 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 : EDist α
s : Set α
x : ℝ≥0∞
simp_rw :
  (∀ x_1 ∈ s, ∀ y ∈ s, x_1 ≠ y → x ≤ edist x_1 y) ↔ x ≤ ⨅ (d : (↑s.offDiag : Type u_1)), uncurry edist (↑d : α × α)
simp_rw₁ :
  (∀ x_1 ∈ s, ∀ y ∈ s, x_1 ≠ y → x ≤ edist x_1 y) ↔ ∀ (i : (↑s.offDiag : Type u_1)), x ≤ uncurry edist (↑i : α × α)
simp_rw₂ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
    ∀ (i : (↑s.offDiag : Type u_1)), x ≤ uncurry edist (↑i : α × α)
simp_rw₃ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
    ∀ (x_1 : α × α) (h : x_1 ∈ s.offDiag), x ≤ uncurry edist x_1
⊢ x ≤ s.einfsep ↔ x ≤ ⨅ (d : (↑s.offDiag : Type u_1)), uncurry edist (↑d : α × α)
have simp_rw₄ : (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
  ∀ (x_1 : α × α) (h : x_1.1 ∈ s ∧ x_1.2 ∈ s ∧ x_1.1 ≠ x_1.2), x ≤ uncurry edist x_1
Tactic: simp_rw [le_einfsep_iff, le_iInf_iff, imp_forall_iff, SetCoe.forall, 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 : EDist α
s : Set α
x : ℝ≥0∞
simp_rw :
  (∀ x_1 ∈ s, ∀ y ∈ s, x_1 ≠ y → x ≤ edist x_1 y) ↔ x ≤ ⨅ (d : (↑s.offDiag : Type u_1)), uncurry edist (↑d : α × α)
simp_rw₁ :
  (∀ x_1 ∈ s, ∀ y ∈ s, x_1 ≠ y → x ≤ edist x_1 y) ↔ ∀ (i : (↑s.offDiag : Type u_1)), x ≤ uncurry edist (↑i : α × α)
simp_rw₂ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
    ∀ (i : (↑s.offDiag : Type u_1)), x ≤ uncurry edist (↑i : α × α)
simp_rw₃ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
    ∀ (x_1 : α × α) (h : x_1 ∈ s.offDiag), x ≤ uncurry edist x_1
simp_rw₄ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
    ∀ (x_1 : α × α) (h : x_1.1 ∈ s ∧ x_1.2 ∈ s ∧ x_1.1 ≠ x_1.2), x ≤ uncurry edist x_1
⊢ x ≤ s.einfsep ↔ x ≤ ⨅ (d : (↑s.offDiag : Type u_1)), uncurry edist (↑d : α × α)
have simp_rw₅ : (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
  ∀ (a b : α) (h : a ∈ s ∧ b ∈ s ∧ a ≠ b), x ≤ uncurry edist (a, b)
Tactic: simp_rw [le_einfsep_iff, le_iInf_iff, imp_forall_iff, SetCoe.forall, 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 : EDist α
s : Set α
x : ℝ≥0∞
simp_rw :
  (∀ x_1 ∈ s, ∀ y ∈ s, x_1 ≠ y → x ≤ edist x_1 y) ↔ x ≤ ⨅ (d : (↑s.offDiag : Type u_1)), uncurry edist (↑d : α × α)
simp_rw₁ :
  (∀ x_1 ∈ s, ∀ y ∈ s, x_1 ≠ y → x ≤ edist x_1 y) ↔ ∀ (i : (↑s.offDiag : Type u_1)), x ≤ uncurry edist (↑i : α × α)
simp_rw₂ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
    ∀ (i : (↑s.offDiag : Type u_1)), x ≤ uncurry edist (↑i : α × α)
simp_rw₃ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
    ∀ (x_1 : α × α) (h : x_1 ∈ s.offDiag), x ≤ uncurry edist x_1
simp_rw₄ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
    ∀ (x_1 : α × α) (h : x_1.1 ∈ s ∧ x_1.2 ∈ s ∧ x_1.1 ≠ x_1.2), x ≤ uncurry edist x_1
simp_rw₅ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
    ∀ (a b : α) (h : a ∈ s ∧ b ∈ s ∧ a ≠ b), x ≤ uncurry edist (a, b)
⊢ x ≤ s.einfsep ↔ x ≤ ⨅ (d : (↑s.offDiag : Type u_1)), uncurry edist (↑d : α × α)
have simp_rw₆ : (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
  ∀ (a b : α) (h : a ∈ s ∧ b ∈ s ∧ a ≠ b), x ≤ edist a b
Tactic: simp_rw [le_einfsep_iff, le_iInf_iff, imp_forall_iff, SetCoe.forall, 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 : EDist α
s : Set α
inst_1 : Fintype (↑s : Type u_1)
⊢ s.einfsep = s.offDiag.toFinset.inf (uncurry edist)
have refine : ∀ (x : ℝ≥0∞), x ≤ s.einfsep ↔ x ≤ s.offDiag.toFinset.inf (uncurry edist)
Tactic: refine eq_of_forall_le_iff fun _ => ?_
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : EDist α
s : Set α
inst_1 : Fintype (↑s : Type u_1)
x : ℝ≥0∞
⊢ x ≤ s.einfsep ↔ x ≤ s.offDiag.toFinset.inf (uncurry edist)
have simp_rw : (∀ x_1 ∈ s, ∀ y ∈ s, x_1 ≠ y → x ≤ edist x_1 y) ↔ x ≤ s.offDiag.toFinset.inf (uncurry edist)
Tactic: simp_rw [le_einfsep_iff, imp_forall_iff, Finset.le_inf_iff, 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 : EDist α
s : Set α
inst_1 : Fintype (↑s : Type u_1)
x : ℝ≥0∞
simp_rw : (∀ x_1 ∈ s, ∀ y ∈ s, x_1 ≠ y → x ≤ edist x_1 y) ↔ x ≤ s.offDiag.toFinset.inf (uncurry edist)
⊢ x ≤ s.einfsep ↔ x ≤ s.offDiag.toFinset.inf (uncurry edist)
have simp_rw₁ : (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔ x ≤ s.offDiag.toFinset.inf (uncurry edist)
Tactic: simp_rw [le_einfsep_iff, imp_forall_iff, Finset.le_inf_iff, 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 : EDist α
s : Set α
inst_1 : Fintype (↑s : Type u_1)
x : ℝ≥0∞
simp_rw : (∀ x_1 ∈ s, ∀ y ∈ s, x_1 ≠ y → x ≤ edist x_1 y) ↔ x ≤ s.offDiag.toFinset.inf (uncurry edist)
simp_rw₁ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔ x ≤ s.offDiag.toFinset.inf (uncurry edist)
⊢ x ≤ s.einfsep ↔ x ≤ s.offDiag.toFinset.inf (uncurry edist)
have simp_rw₂ : (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔ ∀ b ∈ s.offDiag.toFinset, x ≤ uncurry edist b
Tactic: simp_rw [le_einfsep_iff, imp_forall_iff, Finset.le_inf_iff, 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 : EDist α
s : Set α
inst_1 : Fintype (↑s : Type u_1)
x : ℝ≥0∞
simp_rw : (∀ x_1 ∈ s, ∀ y ∈ s, x_1 ≠ y → x ≤ edist x_1 y) ↔ x ≤ s.offDiag.toFinset.inf (uncurry edist)
simp_rw₁ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔ x ≤ s.offDiag.toFinset.inf (uncurry edist)
simp_rw₂ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔ ∀ b ∈ s.offDiag.toFinset, x ≤ uncurry edist b
⊢ x ≤ s.einfsep ↔ x ≤ s.offDiag.toFinset.inf (uncurry edist)
have simp_rw₃ : (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔ ∀ b ∈ s.offDiag, x ≤ uncurry edist b
Tactic: simp_rw [le_einfsep_iff, imp_forall_iff, Finset.le_inf_iff, 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 : EDist α
s : Set α
inst_1 : Fintype (↑s : Type u_1)
x : ℝ≥0∞
simp_rw : (∀ x_1 ∈ s, ∀ y ∈ s, x_1 ≠ y → x ≤ edist x_1 y) ↔ x ≤ s.offDiag.toFinset.inf (uncurry edist)
simp_rw₁ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔ x ≤ s.offDiag.toFinset.inf (uncurry edist)
simp_rw₂ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔ ∀ b ∈ s.offDiag.toFinset, x ≤ uncurry edist b
simp_rw₃ : (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔ ∀ b ∈ s.offDiag, x ≤ uncurry edist b
⊢ x ≤ s.einfsep ↔ x ≤ s.offDiag.toFinset.inf (uncurry edist)
have simp_rw₄ : (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
  ∀ (b : α × α), b.1 ∈ s ∧ b.2 ∈ s ∧ b.1 ≠ b.2 → x ≤ uncurry edist b
Tactic: simp_rw [le_einfsep_iff, imp_forall_iff, Finset.le_inf_iff, 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 : EDist α
s : Set α
inst_1 : Fintype (↑s : Type u_1)
x : ℝ≥0∞
simp_rw : (∀ x_1 ∈ s, ∀ y ∈ s, x_1 ≠ y → x ≤ edist x_1 y) ↔ x ≤ s.offDiag.toFinset.inf (uncurry edist)
simp_rw₁ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔ x ≤ s.offDiag.toFinset.inf (uncurry edist)
simp_rw₂ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔ ∀ b ∈ s.offDiag.toFinset, x ≤ uncurry edist b
simp_rw₃ : (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔ ∀ b ∈ s.offDiag, x ≤ uncurry edist b
simp_rw₄ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
    ∀ (b : α × α), b.1 ∈ s ∧ b.2 ∈ s ∧ b.1 ≠ b.2 → x ≤ uncurry edist b
⊢ x ≤ s.einfsep ↔ x ≤ s.offDiag.toFinset.inf (uncurry edist)
have simp_rw₅ : (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
  ∀ (a b : α), a ∈ s ∧ b ∈ s ∧ a ≠ b → x ≤ uncurry edist (a, b)
Tactic: simp_rw [le_einfsep_iff, imp_forall_iff, Finset.le_inf_iff, 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 : EDist α
s : Set α
inst_1 : Fintype (↑s : Type u_1)
x : ℝ≥0∞
simp_rw : (∀ x_1 ∈ s, ∀ y ∈ s, x_1 ≠ y → x ≤ edist x_1 y) ↔ x ≤ s.offDiag.toFinset.inf (uncurry edist)
simp_rw₁ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔ x ≤ s.offDiag.toFinset.inf (uncurry edist)
simp_rw₂ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔ ∀ b ∈ s.offDiag.toFinset, x ≤ uncurry edist b
simp_rw₃ : (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔ ∀ b ∈ s.offDiag, x ≤ uncurry edist b
simp_rw₄ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
    ∀ (b : α × α), b.1 ∈ s ∧ b.2 ∈ s ∧ b.1 ≠ b.2 → x ≤ uncurry edist b
simp_rw₅ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
    ∀ (a b : α), a ∈ s ∧ b ∈ s ∧ a ≠ b → x ≤ uncurry edist (a, b)
⊢ x ≤ s.einfsep ↔ x ≤ s.offDiag.toFinset.inf (uncurry edist)
have simp_rw₆ : (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
  ∀ (a b : α), a ∈ s ∧ b ∈ s ∧ a ≠ b → x ≤ edist a b
Tactic: simp_rw [le_einfsep_iff, imp_forall_iff, Finset.le_inf_iff, 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 : EDist α
s : Set α
hs : s.Finite
⊢ s.einfsep = ⋯.toFinset (s := s.offDiag).inf (uncurry edist)
have refine : ∀ (x : ℝ≥0∞), x ≤ s.einfsep ↔ x ≤ sorry.toFinset (s := s.offDiag).inf (uncurry edist)
Tactic: refine eq_of_forall_le_iff fun _ => ?_
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : EDist α
s : Set α
hs : s.Finite
x : ℝ≥0∞
⊢ x ≤ s.einfsep ↔ x ≤ ⋯.toFinset (s := s.offDiag).inf (uncurry edist)
have simp_rw : (∀ x_1 ∈ s, ∀ y ∈ s, x_1 ≠ y → x ≤ edist x_1 y) ↔ x ≤ sorry.toFinset (s := s.offDiag).inf (uncurry edist)
Tactic: simp_rw [le_einfsep_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 : EDist α
s : Set α
hs : s.Finite
x : ℝ≥0∞
simp_rw : (∀ x_1 ∈ s, ∀ y ∈ s, x_1 ≠ y → x ≤ edist x_1 y) ↔ x ≤ ⋯.toFinset (s := s.offDiag).inf (uncurry edist)
⊢ x ≤ s.einfsep ↔ x ≤ ⋯.toFinset (s := s.offDiag).inf (uncurry edist)
have simp_rw₁ : (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
  x ≤ sorry.toFinset (s := s.offDiag).inf (uncurry edist)
Tactic: simp_rw [le_einfsep_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 : EDist α
s : Set α
hs : s.Finite
x : ℝ≥0∞
simp_rw : (∀ x_1 ∈ s, ∀ y ∈ s, x_1 ≠ y → x ≤ edist x_1 y) ↔ x ≤ ⋯.toFinset (s := s.offDiag).inf (uncurry edist)
simp_rw₁ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
    x ≤ ⋯.toFinset (s := s.offDiag).inf (uncurry edist)
⊢ x ≤ s.einfsep ↔ x ≤ ⋯.toFinset (s := s.offDiag).inf (uncurry edist)
have simp_rw₂ : (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
  ∀ b ∈ sorry.toFinset (s := s.offDiag), x ≤ uncurry edist b
Tactic: simp_rw [le_einfsep_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 : EDist α
s : Set α
hs : s.Finite
x : ℝ≥0∞
simp_rw : (∀ x_1 ∈ s, ∀ y ∈ s, x_1 ≠ y → x ≤ edist x_1 y) ↔ x ≤ ⋯.toFinset (s := s.offDiag).inf (uncurry edist)
simp_rw₁ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
    x ≤ ⋯.toFinset (s := s.offDiag).inf (uncurry edist)
simp_rw₂ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
    ∀ b ∈ ⋯.toFinset (s := s.offDiag), x ≤ uncurry edist b
⊢ x ≤ s.einfsep ↔ x ≤ ⋯.toFinset (s := s.offDiag).inf (uncurry edist)
have simp_rw₃ : (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔ ∀ b ∈ s.offDiag, x ≤ uncurry edist b
Tactic: simp_rw [le_einfsep_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 : EDist α
s : Set α
hs : s.Finite
x : ℝ≥0∞
simp_rw : (∀ x_1 ∈ s, ∀ y ∈ s, x_1 ≠ y → x ≤ edist x_1 y) ↔ x ≤ ⋯.toFinset (s := s.offDiag).inf (uncurry edist)
simp_rw₁ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
    x ≤ ⋯.toFinset (s := s.offDiag).inf (uncurry edist)
simp_rw₂ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
    ∀ b ∈ ⋯.toFinset (s := s.offDiag), x ≤ uncurry edist b
simp_rw₃ : (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔ ∀ b ∈ s.offDiag, x ≤ uncurry edist b
⊢ x ≤ s.einfsep ↔ x ≤ ⋯.toFinset (s := s.offDiag).inf (uncurry edist)
have simp_rw₄ : (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
  ∀ (b : α × α), b.1 ∈ s ∧ b.2 ∈ s ∧ b.1 ≠ b.2 → x ≤ uncurry edist b
Tactic: simp_rw [le_einfsep_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 : EDist α
s : Set α
hs : s.Finite
x : ℝ≥0∞
simp_rw : (∀ x_1 ∈ s, ∀ y ∈ s, x_1 ≠ y → x ≤ edist x_1 y) ↔ x ≤ ⋯.toFinset (s := s.offDiag).inf (uncurry edist)
simp_rw₁ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
    x ≤ ⋯.toFinset (s := s.offDiag).inf (uncurry edist)
simp_rw₂ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
    ∀ b ∈ ⋯.toFinset (s := s.offDiag), x ≤ uncurry edist b
simp_rw₃ : (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔ ∀ b ∈ s.offDiag, x ≤ uncurry edist b
simp_rw₄ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
    ∀ (b : α × α), b.1 ∈ s ∧ b.2 ∈ s ∧ b.1 ≠ b.2 → x ≤ uncurry edist b
⊢ x ≤ s.einfsep ↔ x ≤ ⋯.toFinset (s := s.offDiag).inf (uncurry edist)
have simp_rw₅ : (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
  ∀ (a b : α), a ∈ s ∧ b ∈ s ∧ a ≠ b → x ≤ uncurry edist (a, b)
Tactic: simp_rw [le_einfsep_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 : EDist α
s : Set α
hs : s.Finite
x : ℝ≥0∞
simp_rw : (∀ x_1 ∈ s, ∀ y ∈ s, x_1 ≠ y → x ≤ edist x_1 y) ↔ x ≤ ⋯.toFinset (s := s.offDiag).inf (uncurry edist)
simp_rw₁ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
    x ≤ ⋯.toFinset (s := s.offDiag).inf (uncurry edist)
simp_rw₂ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
    ∀ b ∈ ⋯.toFinset (s := s.offDiag), x ≤ uncurry edist b
simp_rw₃ : (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔ ∀ b ∈ s.offDiag, x ≤ uncurry edist b
simp_rw₄ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
    ∀ (b : α × α), b.1 ∈ s ∧ b.2 ∈ s ∧ b.1 ≠ b.2 → x ≤ uncurry edist b
simp_rw₅ :
  (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
    ∀ (a b : α), a ∈ s ∧ b ∈ s ∧ a ≠ b → x ≤ uncurry edist (a, b)
⊢ x ≤ s.einfsep ↔ x ≤ ⋯.toFinset (s := s.offDiag).inf (uncurry edist)
have simp_rw₆ : (∀ (x_1 x_2 : α), x_1 ∈ s → x_2 ∈ s → x_1 ≠ x_2 → x ≤ edist x_1 x_2) ↔
  ∀ (a b : α), a ∈ s ∧ b ∈ s ∧ a ≠ b → x ≤ edist a b
Tactic: simp_rw [le_einfsep_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 : EDist α
s : Finset α
⊢ (↑s : Set α).einfsep = s.offDiag.inf (uncurry edist)
have simp_rw : (↑s : Set α).offDiag.toFinset.inf (uncurry edist) = s.offDiag.inf (uncurry edist)
Tactic: simp_rw [einfsep_of_fintype, ← Finset.coe_offDiag, Finset.toFinset_coe]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : EDist α
s : Finset α
simp_rw : (↑s : Set α).offDiag.toFinset.inf (uncurry edist) = s.offDiag.inf (uncurry edist)
⊢ (↑s : Set α).einfsep = s.offDiag.inf (uncurry edist)
have simp_rw₁ : (↑s.offDiag : Set (α × α)).toFinset.inf (uncurry edist) = s.offDiag.inf (uncurry edist)
Tactic: simp_rw [einfsep_of_fintype, ← Finset.coe_offDiag, Finset.toFinset_coe]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : EDist α
s : Finset α
⊢ (↑s : Set α).einfsep = s.offDiag.inf (uncurry edist)
have classical : (↑s : Set α).offDiag.toFinset.inf (uncurry edist) = s.offDiag.inf (uncurry edist)
Tactic: classical simp_rw [einfsep_of_fintype, ← Finset.coe_offDiag, Finset.toFinset_coe]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : EDist α
s : Finset α
classical : (↑s : Set α).offDiag.toFinset.inf (uncurry edist) = s.offDiag.inf (uncurry edist)
⊢ (↑s : Set α).einfsep = s.offDiag.inf (uncurry edist)
have classical₁ : (↑s.offDiag : Set (α × α)).toFinset.inf (uncurry edist) = s.offDiag.inf (uncurry edist)
Tactic: classical simp_rw [einfsep_of_fintype, ← Finset.coe_offDiag, Finset.toFinset_coe]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : EDist α
s : Set α
inst_1 : Finite.{u_1 + 1} (↑s : Type u_1)
hs : s.Nontrivial
⊢ ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.einfsep = edist x y
have intro : ∀ (val : Fintype (↑s : Type u_1)), ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.einfsep = edist x y
Tactic: cases nonempty_fintype s
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : EDist 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.einfsep = edist x y
have intro : ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.offDiag.toFinset.inf (uncurry edist) = edist x y
Tactic: simp_rw [einfsep_of_fintype]
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (goal)
Goal:
α : Type u_1
inst : EDist 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 edist) = edist x y
have intro : ∀ (w : α × α) (hxy : w ∈ s.offDiag.toFinset) (hed : s.offDiag.toFinset.inf (uncurry edist) = uncurry edist w),
  ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.offDiag.toFinset.inf (uncurry edist) = edist x y
Tactic: rcases Finset.exists_mem_eq_inf s.offDiag.toFinset (by simpa) (uncurry edist) with ⟨w, hxy, hed⟩
-- Lean 4 proof step from Mathlib.Topology.MetricSpace.Infsep (hypothesis)
Goal:
α : Type u_1
inst : EDist 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 edist) = uncurry edist w
⊢ ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.offDiag.toFinset.inf (uncurry edist) = edist 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 : EDist α
s : Set α
inst_1 : Finite.{u_1 + 1} (↑s : Type u_1)
hs : s.Nontrivial
⊢ ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.einfsep = edist x y
have intro : ∀ (val : Fintype (↑s : Type u_1)), ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.einfsep = edist x y
Tactic: classical
    cases nonempty_fintype s
    simp_rw [einfsep_of_fintype]
    rcases Finset.exists_mem_eq_inf s.offDiag.toFinset (by simpa) (uncurry edist) 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 : EDist 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.einfsep = edist x y
⊢ ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.einfsep = edist x y
have intro₁ : ∀ (val : Fintype (↑s : Type u_1)), ∃ x ∈ s, ∃ y ∈ s, x ≠ y ∧ s.offDiag.toFinset.inf (uncurry edist) = edist x y
Tactic: classical
    cases nonempty_fintype s
    simp_rw [einfsep_of_fintype]
    rcases Finset.exists_mem_eq_inf s.offDiag.toFinset (by simpa) (uncurry edist) with ⟨w, hxy, hed⟩
    simp_rw [mem_toFinset] at hxy
    exact ⟨w.fst, hxy.1, w.snd, hxy.2.1, hxy.2.2, hed⟩