awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Module: Mathlib.Topology.EMetricSpace.PairReduction
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
E : Type u_2
inst_2 : PseudoEMetricSpace E
ha : 1 < a
f : T → E
s : T
hs : s ∈ J
rw :
  ∀ (i : { t : ↥J // edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) t ≤ c }),
    edist (f (↑(Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) : T)) (f (↑(↑i : ↥J) : T)) ≤
      HMul.hMul (α := ℝ≥0∞) 2 (⨆ (p : ↥(pairSet J a c)), edist (f (↑p : T × T).1) (f (↑p : T × T).2))
⊢ ⨆ (t : { t : ↥J // edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) t ≤ c }),
      edist (f (↑(Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) : T)) (f (↑(↑t : ↥J) : T)) ≤
    HMul.hMul (α := ℝ≥0∞) 2 (⨆ (p : ↥(pairSet J a c)), edist (f (↑p : T × T).1) (f (↑p : T × T).2))

Tactic:
rw [iSup_le_iff]

Premises used: iSup_le_iff
-- Module: Mathlib.Topology.EMetricSpace.PairReduction
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
E : Type u_2
inst_2 : PseudoEMetricSpace E
ha : 1 < a
f : T → E
s : T
hs : s ∈ J
t : T
ht : t ∈ J
hst : edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) ≤ c
⊢ J.Nonempty

Tactic:
have hJ : J.Nonempty := ⟨s, hs⟩
-- Module: Mathlib.Topology.EMetricSpace.PairReduction
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
E : Type u_2
inst_2 : PseudoEMetricSpace E
ha : 1 < a
f : T → E
s : T
hs : s ∈ J
t : T
ht : t ∈ J
hst : edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) ≤ c
hJ : J.Nonempty
P : (l : ℕ) → Prop := fun (l : ℕ) ↦ s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset
l : ℕ := Nat.findGreatest P (#J - 1)
apply : P 0
⊢ P l

Tactic:
apply Nat.findGreatest_spec zero_le

Premises used: zero_le, Nat.findGreatest_spec
-- Module: Mathlib.Topology.EMetricSpace.PairReduction
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
E : Type u_2
inst_2 : PseudoEMetricSpace E
ha : 1 < a
f : T → E
s : T
hs : s ∈ J
t : T
ht : t ∈ J
hst : edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) ≤ c
hJ : J.Nonempty
P : (l : ℕ) → Prop := fun (l : ℕ) ↦ s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset
l : ℕ := Nat.findGreatest P (#J - 1)
⊢ P 0

Tactic:
simpa [P, finset_logSizeBallSeq_zero] using ⟨hs, ht⟩

Premises used: PairReduction.finset_logSizeBallSeq_zero
-- Module: Mathlib.Topology.EMetricSpace.PairReduction
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
E : Type u_2
inst_2 : PseudoEMetricSpace E
ha : 1 < a
f : T → E
s : T
hs : s ∈ J
t : T
ht : t ∈ J
hst : edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) ≤ c
hJ : J.Nonempty
P : (l : ℕ) → Prop := fun (l : ℕ) ↦ s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset
l : ℕ := Nat.findGreatest P (#J - 1)
⊢ s ∈ (logSizeBallSeq J hJ a c l).finset

Tactic:
obtain ⟨hsV, htV⟩ : P l := by
    apply Nat.findGreatest_spec zero_le
    simpa [P, finset_logSizeBallSeq_zero] using ⟨hs, ht⟩
-- Module: Mathlib.Topology.EMetricSpace.PairReduction
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
E : Type u_2
inst_2 : PseudoEMetricSpace E
ha : 1 < a
f : T → E
s : T
hs : s ∈ J
t : T
ht : t ∈ J
hst : edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) ≤ c
hJ : J.Nonempty
P : (l : ℕ) → Prop := fun (l : ℕ) ↦ s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset
l : ℕ := Nat.findGreatest P (#J - 1)
hsV : s ∈ (logSizeBallSeq J hJ a c l).finset
⊢ t ∈ (logSizeBallSeq J hJ a c l).finset

Tactic:
obtain ⟨hsV, htV⟩ : P l := by
    apply Nat.findGreatest_spec zero_le
    simpa [P, finset_logSizeBallSeq_zero] using ⟨hs, ht⟩
-- Module: Mathlib.Topology.EMetricSpace.PairReduction
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
E : Type u_2
inst_2 : PseudoEMetricSpace E
ha : 1 < a
f : T → E
s : T
hs : s ∈ J
t : T
ht : t ∈ J
hst : edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) ≤ c
hJ : J.Nonempty
P : (l : ℕ) → Prop := fun (l : ℕ) ↦ s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset
l : ℕ := Nat.findGreatest P (#J - 1)
hsV : s ∈ (logSizeBallSeq J hJ a c l).finset
htV : t ∈ (logSizeBallSeq J hJ a c l).finset
this :
  ∀ (s : T) (hs : s ∈ J) (t : T) (ht : t ∈ J)
    (hst : edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) ≤ c),
    let P : (l : ℕ) → Prop := fun (l : ℕ) ↦
      s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset;
    let l := Nat.findGreatest P (#J - 1);
    ∀ (hsV : s ∈ (logSizeBallSeq J hJ a c l).finset) (htV : t ∈ (logSizeBallSeq J hJ a c l).finset)
      (h : s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset),
      edist (f (↑(Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) : T))
          (f
            (↑(↑(Subtype.mk (p := fun (t : ↥J) ↦ edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) t ≤ c)
                      (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) hst) :
                  ↥J) :
              T)) ≤
        HMul.hMul (α := ℝ≥0∞) 2 (⨆ (p : ↥(pairSet J a c)), edist (f (↑p : T × T).1) (f (↑p : T × T).2))
h : ¬s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
by_contra : ∀ (hl : ¬l < #J - 1), False
⊢ l < #J - 1

Tactic:
by_contra hl
-- Module: Mathlib.Topology.EMetricSpace.PairReduction
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
E : Type u_2
inst_2 : PseudoEMetricSpace E
ha : 1 < a
f : T → E
s : T
hs : s ∈ J
t : T
ht : t ∈ J
hst : edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) ≤ c
hJ : J.Nonempty
P : (l : ℕ) → Prop := fun (l : ℕ) ↦ s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset
l : ℕ := Nat.findGreatest P (#J - 1)
hsV : s ∈ (logSizeBallSeq J hJ a c l).finset
htV : t ∈ (logSizeBallSeq J hJ a c l).finset
this :
  ∀ (s : T) (hs : s ∈ J) (t : T) (ht : t ∈ J)
    (hst : edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) ≤ c),
    let P : (l : ℕ) → Prop := fun (l : ℕ) ↦
      s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset;
    let l := Nat.findGreatest P (#J - 1);
    ∀ (hsV : s ∈ (logSizeBallSeq J hJ a c l).finset) (htV : t ∈ (logSizeBallSeq J hJ a c l).finset)
      (h : s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset),
      edist (f (↑(Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) : T))
          (f
            (↑(↑(Subtype.mk (p := fun (t : ↥J) ↦ edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) t ≤ c)
                      (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) hst) :
                  ↥J) :
              T)) ≤
        HMul.hMul (α := ℝ≥0∞) 2 (⨆ (p : ↥(pairSet J a c)), edist (f (↑p : T × T).1) (f (↑p : T × T).2))
h : ¬s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
hl : ¬l < #J - 1
⊢ #J ≤ l + 1

Tactic:
simp only [not_lt, tsub_le_iff_right] at hl

Premises used: tsub_le_iff_right, not_lt
-- Module: Mathlib.Topology.EMetricSpace.PairReduction
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
E : Type u_2
inst_2 : PseudoEMetricSpace E
ha : 1 < a
f : T → E
s : T
hs : s ∈ J
t : T
ht : t ∈ J
hst : edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) ≤ c
hJ : J.Nonempty
P : (l : ℕ) → Prop := fun (l : ℕ) ↦ s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset
l : ℕ := Nat.findGreatest P (#J - 1)
hsV : s ∈ (logSizeBallSeq J hJ a c l).finset
htV : t ∈ (logSizeBallSeq J hJ a c l).finset
this :
  ∀ (s : T) (hs : s ∈ J) (t : T) (ht : t ∈ J)
    (hst : edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) ≤ c),
    let P : (l : ℕ) → Prop := fun (l : ℕ) ↦
      s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset;
    let l := Nat.findGreatest P (#J - 1);
    ∀ (hsV : s ∈ (logSizeBallSeq J hJ a c l).finset) (htV : t ∈ (logSizeBallSeq J hJ a c l).finset)
      (h : s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset),
      edist (f (↑(Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) : T))
          (f
            (↑(↑(Subtype.mk (p := fun (t : ↥J) ↦ edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) t ≤ c)
                      (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) hst) :
                  ↥J) :
              T)) ≤
        HMul.hMul (α := ℝ≥0∞) 2 (⨆ (p : ↥(pairSet J a c)), edist (f (↑p : T × T).1) (f (↑p : T × T).2))
h : ¬s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
hl : #J ≤ l + 1
⊢ l + 1 = #J

Tactic:
Nat.le_antisymm_iff.mpr ⟨?_, hl⟩

Premises used: Iff.mpr, And.intro, Nat.le_antisymm_iff
-- Module: Mathlib.Topology.EMetricSpace.PairReduction
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
E : Type u_2
inst_2 : PseudoEMetricSpace E
ha : 1 < a
f : T → E
s : T
hs : s ∈ J
t : T
ht : t ∈ J
hst : edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) ≤ c
hJ : J.Nonempty
P : (l : ℕ) → Prop := fun (l : ℕ) ↦ s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset
l : ℕ := Nat.findGreatest P (#J - 1)
hsV : s ∈ (logSizeBallSeq J hJ a c l).finset
htV : t ∈ (logSizeBallSeq J hJ a c l).finset
this :
  ∀ (s : T) (hs : s ∈ J) (t : T) (ht : t ∈ J)
    (hst : edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) ≤ c),
    let P : (l : ℕ) → Prop := fun (l : ℕ) ↦
      s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset;
    let l := Nat.findGreatest P (#J - 1);
    ∀ (hsV : s ∈ (logSizeBallSeq J hJ a c l).finset) (htV : t ∈ (logSizeBallSeq J hJ a c l).finset)
      (h : s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset),
      edist (f (↑(Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) : T))
          (f
            (↑(↑(Subtype.mk (p := fun (t : ↥J) ↦ edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) t ≤ c)
                      (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) hst) :
                  ↥J) :
              T)) ≤
        HMul.hMul (α := ℝ≥0∞) 2 (⨆ (p : ↥(pairSet J a c)), edist (f (↑p : T × T).1) (f (↑p : T × T).2))
h : ¬s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
hl : #J ≤ l + 1
refine : l + 1 ≤ #J
⊢ l + 1 = #J

Tactic:
refine Nat.le_antisymm_iff.mpr ⟨?_, hl⟩

Premises used: Iff.mpr, And.intro, Nat.le_antisymm_iff
-- Module: Mathlib.Topology.EMetricSpace.PairReduction
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
E : Type u_2
inst_2 : PseudoEMetricSpace E
ha : 1 < a
f : T → E
s : T
hs : s ∈ J
t : T
ht : t ∈ J
hst : edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) ≤ c
hJ : J.Nonempty
P : (l : ℕ) → Prop := fun (l : ℕ) ↦ s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset
l : ℕ := Nat.findGreatest P (#J - 1)
hsV : s ∈ (logSizeBallSeq J hJ a c l).finset
htV : t ∈ (logSizeBallSeq J hJ a c l).finset
this :
  ∀ (s : T) (hs : s ∈ J) (t : T) (ht : t ∈ J)
    (hst : edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) ≤ c),
    let P : (l : ℕ) → Prop := fun (l : ℕ) ↦
      s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset;
    let l := Nat.findGreatest P (#J - 1);
    ∀ (hsV : s ∈ (logSizeBallSeq J hJ a c l).finset) (htV : t ∈ (logSizeBallSeq J hJ a c l).finset)
      (h : s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset),
      edist (f (↑(Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) : T))
          (f
            (↑(↑(Subtype.mk (p := fun (t : ↥J) ↦ edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) t ≤ c)
                      (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) hst) :
                  ↥J) :
              T)) ≤
        HMul.hMul (α := ℝ≥0∞) 2 (⨆ (p : ↥(pairSet J a c)), edist (f (↑p : T × T).1) (f (↑p : T × T).2))
h : ¬s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
hl : #J ≤ l + 1
dsimp : Nat.findGreatest P (#J - 1) + 1 ≤ #J
⊢ l + 1 ≤ #J

Tactic:
dsimp [l]
-- Module: Mathlib.Topology.EMetricSpace.PairReduction
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
E : Type u_2
inst_2 : PseudoEMetricSpace E
ha : 1 < a
f : T → E
s : T
hs : s ∈ J
t : T
ht : t ∈ J
hst : edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) ≤ c
hJ : J.Nonempty
P : (l : ℕ) → Prop := fun (l : ℕ) ↦ s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset
l : ℕ := Nat.findGreatest P (#J - 1)
hsV : s ∈ (logSizeBallSeq J hJ a c l).finset
htV : t ∈ (logSizeBallSeq J hJ a c l).finset
this :
  ∀ (s : T) (hs : s ∈ J) (t : T) (ht : t ∈ J)
    (hst : edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) ≤ c),
    let P : (l : ℕ) → Prop := fun (l : ℕ) ↦
      s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset;
    let l := Nat.findGreatest P (#J - 1);
    ∀ (hsV : s ∈ (logSizeBallSeq J hJ a c l).finset) (htV : t ∈ (logSizeBallSeq J hJ a c l).finset)
      (h : s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset),
      edist (f (↑(Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) : T))
          (f
            (↑(↑(Subtype.mk (p := fun (t : ↥J) ↦ edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) t ≤ c)
                      (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) hst) :
                  ↥J) :
              T)) ≤
        HMul.hMul (α := ℝ≥0∞) 2 (⨆ (p : ↥(pairSet J a c)), edist (f (↑p : T × T).1) (f (↑p : T × T).2))
h : ¬s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
hl : #J ≤ l + 1
grw : #J - 1 + 1 - 1 + 1 ≤ #J - 1 + 1 - 1 + 1
⊢ Nat.findGreatest P (#J - 1) + 1 ≤ #J

Tactic:
grw [← Nat.sub_add_cancel <| Order.one_le_iff_pos.mpr (Finset.card_pos.mpr hJ),
            Nat.findGreatest_le]

Premises used: Iff.mpr, Nat.sub_add_cancel, Finset.card_pos, Nat.findGreatest_le, Order.one_le_iff_pos
-- Module: Mathlib.Topology.EMetricSpace.PairReduction
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
E : Type u_2
inst_2 : PseudoEMetricSpace E
ha : 1 < a
f : T → E
s : T
hs : s ∈ J
t : T
ht : t ∈ J
hst : edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) ≤ c
hJ : J.Nonempty
P : (l : ℕ) → Prop := fun (l : ℕ) ↦ s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset
l : ℕ := Nat.findGreatest P (#J - 1)
hsV : s ∈ (logSizeBallSeq J hJ a c l).finset
htV : t ∈ (logSizeBallSeq J hJ a c l).finset
this :
  ∀ (s : T) (hs : s ∈ J) (t : T) (ht : t ∈ J)
    (hst : edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) ≤ c),
    let P : (l : ℕ) → Prop := fun (l : ℕ) ↦
      s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset;
    let l := Nat.findGreatest P (#J - 1);
    ∀ (hsV : s ∈ (logSizeBallSeq J hJ a c l).finset) (htV : t ∈ (logSizeBallSeq J hJ a c l).finset)
      (h : s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset),
      edist (f (↑(Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) : T))
          (f
            (↑(↑(Subtype.mk (p := fun (t : ↥J) ↦ edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) t ≤ c)
                      (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) hst) :
                  ↥J) :
              T)) ≤
        HMul.hMul (α := ℝ≥0∞) 2 (⨆ (p : ↥(pairSet J a c)), edist (f (↑p : T × T).1) (f (↑p : T × T).2))
h : ¬s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
hl : #J ≤ l + 1
⊢ #J - 1 + 1 - 1 + 1 ≤ #J - 1 + 1 - 1 + 1

Tactic:
rfl
-- Module: Mathlib.Topology.EMetricSpace.PairReduction
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
E : Type u_2
inst_2 : PseudoEMetricSpace E
ha : 1 < a
f : T → E
s : T
hs : s ∈ J
t : T
ht : t ∈ J
hst : edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) ≤ c
hJ : J.Nonempty
P : (l : ℕ) → Prop := fun (l : ℕ) ↦ s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset
l : ℕ := Nat.findGreatest P (#J - 1)
hsV : s ∈ (logSizeBallSeq J hJ a c l).finset
htV : t ∈ (logSizeBallSeq J hJ a c l).finset
this :
  ∀ (s : T) (hs : s ∈ J) (t : T) (ht : t ∈ J)
    (hst : edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) ≤ c),
    let P : (l : ℕ) → Prop := fun (l : ℕ) ↦
      s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset;
    let l := Nat.findGreatest P (#J - 1);
    ∀ (hsV : s ∈ (logSizeBallSeq J hJ a c l).finset) (htV : t ∈ (logSizeBallSeq J hJ a c l).finset)
      (h : s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset),
      edist (f (↑(Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) : T))
          (f
            (↑(↑(Subtype.mk (p := fun (t : ↥J) ↦ edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) t ≤ c)
                      (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) hst) :
                  ↥J) :
              T)) ≤
        HMul.hMul (α := ℝ≥0∞) 2 (⨆ (p : ↥(pairSet J a c)), edist (f (↑p : T × T).1) (f (↑p : T × T).2))
h : ¬s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
hl : #J ≤ l + 1
⊢ l + 1 = #J

Tactic:
have hlJ : l + 1 = #J := by
          refine Nat.le_antisymm_iff.mpr ⟨?_, hl⟩
          dsimp [l]
          grw [← Nat.sub_add_cancel <| Order.one_le_iff_pos.mpr (Finset.card_pos.mpr hJ),
            Nat.findGreatest_le]
          rfl
-- Module: Mathlib.Topology.EMetricSpace.PairReduction
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
E : Type u_2
inst_2 : PseudoEMetricSpace E
ha : 1 < a
f : T → E
s : T
hs : s ∈ J
t : T
ht : t ∈ J
hst : edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) ≤ c
hJ : J.Nonempty
P : (l : ℕ) → Prop := fun (l : ℕ) ↦ s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset
l : ℕ := Nat.findGreatest P (#J - 1)
hsV : s ∈ (logSizeBallSeq J hJ a c l).finset
htV : t ∈ (logSizeBallSeq J hJ a c l).finset
this :
  ∀ (s : T) (hs : s ∈ J) (t : T) (ht : t ∈ J)
    (hst : edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) ≤ c),
    let P : (l : ℕ) → Prop := fun (l : ℕ) ↦
      s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset;
    let l := Nat.findGreatest P (#J - 1);
    ∀ (hsV : s ∈ (logSizeBallSeq J hJ a c l).finset) (htV : t ∈ (logSizeBallSeq J hJ a c l).finset)
      (h : s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset),
      edist (f (↑(Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) : T))
          (f
            (↑(↑(Subtype.mk (p := fun (t : ↥J) ↦ edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) t ≤ c)
                      (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) hst) :
                  ↥J) :
              T)) ≤
        HMul.hMul (α := ℝ≥0∞) 2 (⨆ (p : ↥(pairSet J a c)), edist (f (↑p : T × T).1) (f (↑p : T × T).2))
h : ¬s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
hl : #J ≤ l + 1
hlJ : l + 1 = #J
apply : s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
⊢ False

Tactic:
apply h
-- Module: Mathlib.Topology.EMetricSpace.PairReduction
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
E : Type u_2
inst_2 : PseudoEMetricSpace E
ha : 1 < a
f : T → E
s : T
hs : s ∈ J
t : T
ht : t ∈ J
hst : edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) ≤ c
hJ : J.Nonempty
P : (l : ℕ) → Prop := fun (l : ℕ) ↦ s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset
l : ℕ := Nat.findGreatest P (#J - 1)
hsV : s ∈ (logSizeBallSeq J hJ a c l).finset
htV : t ∈ (logSizeBallSeq J hJ a c l).finset
this :
  ∀ (s : T) (hs : s ∈ J) (t : T) (ht : t ∈ J)
    (hst : edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) ≤ c),
    let P : (l : ℕ) → Prop := fun (l : ℕ) ↦
      s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset;
    let l := Nat.findGreatest P (#J - 1);
    ∀ (hsV : s ∈ (logSizeBallSeq J hJ a c l).finset) (htV : t ∈ (logSizeBallSeq J hJ a c l).finset)
      (h : s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset),
      edist (f (↑(Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) : T))
          (f
            (↑(↑(Subtype.mk (p := fun (t : ↥J) ↦ edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) t ≤ c)
                      (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) hst) :
                  ↥J) :
              T)) ≤
        HMul.hMul (α := ℝ≥0∞) 2 (⨆ (p : ↥(pairSet J a c)), edist (f (↑p : T × T).1) (f (↑p : T × T).2))
h : ¬s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
hl : #J ≤ l + 1
hlJ : l + 1 = #J
h_emp : (logSizeBallSeq J hJ a c (l + 1)).finset = ∅
⊢ s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset

Tactic:
simp [h_emp]
-- Module: Mathlib.Topology.EMetricSpace.PairReduction
Goal:
T : Type u_1
inst : PseudoEMetricSpace T
a c : ℝ≥0∞
J : Finset T
inst_1 : DecidableEq T
E : Type u_2
inst_2 : PseudoEMetricSpace E
ha : 1 < a
f : T → E
s : T
hs : s ∈ J
t : T
ht : t ∈ J
hst : edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) ≤ c
hJ : J.Nonempty
P : (l : ℕ) → Prop := fun (l : ℕ) ↦ s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset
l : ℕ := Nat.findGreatest P (#J - 1)
hsV : s ∈ (logSizeBallSeq J hJ a c l).finset
htV : t ∈ (logSizeBallSeq J hJ a c l).finset
this :
  ∀ (s : T) (hs : s ∈ J) (t : T) (ht : t ∈ J)
    (hst : edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) ≤ c),
    let P : (l : ℕ) → Prop := fun (l : ℕ) ↦
      s ∈ (logSizeBallSeq J hJ a c l).finset ∧ t ∈ (logSizeBallSeq J hJ a c l).finset;
    let l := Nat.findGreatest P (#J - 1);
    ∀ (hsV : s ∈ (logSizeBallSeq J hJ a c l).finset) (htV : t ∈ (logSizeBallSeq J hJ a c l).finset)
      (h : s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset),
      edist (f (↑(Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) : T))
          (f
            (↑(↑(Subtype.mk (p := fun (t : ↥J) ↦ edist (Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) t ≤ c)
                      (Subtype.mk (p := fun (x : T) ↦ x ∈ J) t ht) hst) :
                  ↥J) :
              T)) ≤
        HMul.hMul (α := ℝ≥0∞) 2 (⨆ (p : ↥(pairSet J a c)), edist (f (↑p : T × T).1) (f (↑p : T × T).2))
h : ¬s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
hl : #J ≤ l + 1
hlJ : l + 1 = #J
suffices : (logSizeBallSeq J hJ a c (l + 1)).finset = ∅
⊢ s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset

Tactic:
suffices h_emp : (logSizeBallSeq J hJ a c (l + 1)).finset = ∅ from by simp [h_emp]