awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
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
⊢ False
have apply : s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
Tactic: apply h
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
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
⊢ s ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
have suffices : (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]
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
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
⊢ (logSizeBallSeq J hJ a c (l + 1)).finset = ∅
have rw : #(logSizeBallSeq J hJ a c #J).finset ≤ #J - #J
Tactic: rw [← Finset.card_eq_zero, ← Nat.le_zero, ← Nat.sub_self #J, hlJ]
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (hypothesis)
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
⊢ t ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
have hl : l < #J - 1
Tactic: have hl : l < #J - 1 := by
        by_contra hl
        simp only [not_lt, tsub_le_iff_right] at hl
        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
        apply h
        suffices h_emp : (logSizeBallSeq J hJ a c (l + 1)).finset = ∅ from by simp [h_emp]
        rw [← Finset.card_eq_zero, ← Nat.le_zero, ← Nat.sub_self #J, hlJ]
        apply card_finset_logSizeBallSeq_le
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (hypothesis)
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
⊢ t ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
have h₁ : s ∈ (logSizeBallSeq J hJ a c (l + 1)).finset
Tactic: simp only [Decidable.not_not] at h
Cleared hypotheses: h
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (hypothesis)
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))
hl : l < #J - 1
h : s ∈ (logSizeBallSeq J hJ a c (l + 1)).finset
⊢ t ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
have hP : ¬P (l + 1)
Tactic: have hP := Nat.findGreatest_is_greatest (lt_add_one l) (Nat.add_one_le_of_lt hl)
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (hypothesis)
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
⊢ 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))
have h' : t ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
Tactic: have h' : t ∉ (logSizeBallSeq J hJ a c (l + 1)).finset := by
      have hl : l < #J - 1 := by
        by_contra hl
        simp only [not_lt, tsub_le_iff_right] at hl
        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
        apply h
        suffices h_emp : (logSizeBallSeq J hJ a c (l + 1)).finset = ∅ from by simp [h_emp]
        rw [← Finset.card_eq_zero, ← Nat.le_zero, ← Nat.sub_self #J, hlJ]
        apply card_finset_logSizeBallSeq_le
      simp only [Decidable.not_not] at h
      have hP := Nat.findGreatest_is_greatest (lt_add_one l) (Nat.add_one_le_of_lt hl)
      simpa [P, h] using hP
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
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
h' : t ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
⊢ edist t s ≤ c
have rw : edist s t ≤ c
Tactic: rw [edist_comm]
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (hypothesis)
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
h' : t ∉ (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))
have hts : edist t s ≤ c
Tactic: have hts : edist t s ≤ c := by rw [edist_comm]; exact hst
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
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
h' : t ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
hts : edist t s ≤ c
⊢ 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))
have inr : edist
    (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))
    (f (↑(Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) : 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 [edist_comm]
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (goal)
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
h' : t ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
hts : edist t s ≤ c
⊢ P = fun (l : ℕ) ↦ t ∈ (logSizeBallSeq J hJ a c l).finset ∧ s ∈ (logSizeBallSeq J hJ a c l).finset
have h₁ : ∀ (x : ℕ), P x ↔ t ∈ (logSizeBallSeq J hJ a c x).finset ∧ s ∈ (logSizeBallSeq J hJ a c x).finset
Tactic: ext
-- Lean 4 proof step from Mathlib.Topology.EMetricSpace.PairReduction (hypothesis)
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
h' : t ∉ (logSizeBallSeq J hJ a c (l + 1)).finset
hts : edist t s ≤ c
⊢ edist
      (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))
      (f (↑(Subtype.mk (p := fun (x : T) ↦ x ∈ J) s hs) : T)) ≤
    HMul.hMul (α := ℝ≥0∞) 2 (⨆ (p : ↥(pairSet J a c)), edist (f (↑p : T × T).1) (f (↑p : T × T).2))
have hP : P = fun (l : ℕ) ↦ t ∈ (logSizeBallSeq J hJ a c l).finset ∧ s ∈ (logSizeBallSeq J hJ a c l).finset
Tactic: have hP : P = (fun l ↦
      t ∈ (logSizeBallSeq J hJ a c l).finset ∧ s ∈ (logSizeBallSeq J hJ a c l).finset) := by
        ext; simp [P, and_comm]