awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Haar.Extension (goal)
Goal:
A : Type u_1
B : Type u_2
C : Type u_3
inst : Group A
inst_1 : Group B
inst_2 : Group C
inst_3 : TopologicalSpace A
inst_4 : TopologicalSpace B
inst_5 : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst_6 : IsTopologicalGroup A
inst_7 : IsTopologicalGroup B
inst_8 : MeasurableSpace A
inst_9 : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst_10 : IsTopologicalGroup C
inst_11 : LocallyCompactSpace B
inst_12 : MeasurableSpace C
inst_13 : BorelSpace C
μC : Measure C
hμC : μC.IsHaarMeasure
inst_14 : T2Space B
inst_15 : MeasurableSpace B
inst_16 : BorelSpace B
U : Set B
hU : IsOpen U
inst_17 : DiscreteTopology A
h_1 :
  HMul.hMul (α := ENNReal) (β := ENNReal) ((μC : (a : Set C) → ENNReal) Set.univ : ENNReal)
      ((μA : (a : Set A) → ENNReal) {1} : ENNReal) <
    ((H.inducedMeasure μA μC : Measure B) : (a : Set B) → ENNReal) U
ho : LT.lt (α := ENNReal) (0 : ENNReal) ((μA : (a : Set A) → ENNReal) {1} : ENNReal)
ht : LT.lt (α := ENNReal) ((μA : (a : Set A) → ENNReal) {1} : ENNReal) (⊤ : ENNReal)
K : Set B
hKU : K ⊆ U
hK : IsCompact K
f : C(B, ℝ)
hf1 : Set.EqOn (β := ℝ) (⇑f : (a : B) → ℝ) 1 K
hf2 : IsCompact (X := B) (tsupport (α := ℝ) (⇑f : (a : B) → ℝ))
hf3 : tsupport (α := ℝ) (⇑f : (a : B) → ℝ) ⊆ U
hf4 : ∀ (x : B), Membership.mem (α := ℝ) (γ := Set ℝ) (Set.Icc 0 1) ((f : (a : B) → ℝ) x : ℝ)
h :
  HMul.hMul (α := ENNReal) (β := ENNReal) ((μC : (a : Set C) → ENNReal) Set.univ : ENNReal)
      ((μA : (a : Set A) → ENNReal) {1} : ENNReal) <
    ENNReal.ofReal
      (∫ (c : C),
        ((((H.pushforward μA :
                    LinearMap.{0, 0, u_2, u_3} (RingHom.id ℝ) (CompactlySupportedContinuousMap B ℝ)
                      (CompactlySupportedContinuousMap C ℝ)) :
                  (a : CompactlySupportedContinuousMap B ℝ) → CompactlySupportedContinuousMap C ℝ)
                { toContinuousMap := f, hasCompactSupport' := hf2 } :
              CompactlySupportedContinuousMap C ℝ) :
            (a : C) → ℝ)
          c ∂μC)
c : C
hc : Set.InjOn.{u_2, u_3} (β := C) (⇑ψ : (a : B) → C) U
⊢ ((((H.pushforward μA :
                LinearMap.{0, 0, u_2, u_3} (RingHom.id ℝ) (CompactlySupportedContinuousMap B ℝ)
                  (CompactlySupportedContinuousMap C ℝ)) :
              (a : CompactlySupportedContinuousMap B ℝ) → CompactlySupportedContinuousMap C ℝ)
            { toContinuousMap := f, hasCompactSupport' := hf2 } :
          CompactlySupportedContinuousMap C ℝ) :
        (a : C) → ℝ)
      c ≤
    ENNReal.toReal ((μA : (a : Set A) → ENNReal) {1})
have obtain : ∀ (b : B),
  ((((H.pushforward μA :
                LinearMap.{0, 0, u_2, u_3} (RingHom.id ℝ) (CompactlySupportedContinuousMap B ℝ)
                  (CompactlySupportedContinuousMap C ℝ)) :
              (a : CompactlySupportedContinuousMap B ℝ) → CompactlySupportedContinuousMap C ℝ)
            { toContinuousMap := f, hasCompactSupport' := hf2 } :
          CompactlySupportedContinuousMap C ℝ) :
        (a : C) → ℝ)
      ((ψ : (a : B) → C) b : C) ≤
    ENNReal.toReal ((μA : (a : Set A) → ENNReal) {1})
Tactic: obtain ⟨b, rfl⟩ := H.isOpenQuotientMap.surjective c
Cleared hypotheses: c
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Haar.Extension (goal)
Goal:
A : Type u_1
B : Type u_2
C : Type u_3
inst : Group A
inst_1 : Group B
inst_2 : Group C
inst_3 : TopologicalSpace A
inst_4 : TopologicalSpace B
inst_5 : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst_6 : IsTopologicalGroup A
inst_7 : IsTopologicalGroup B
inst_8 : MeasurableSpace A
inst_9 : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst_10 : IsTopologicalGroup C
inst_11 : LocallyCompactSpace B
inst_12 : MeasurableSpace C
inst_13 : BorelSpace C
μC : Measure C
hμC : μC.IsHaarMeasure
inst_14 : T2Space B
inst_15 : MeasurableSpace B
inst_16 : BorelSpace B
U : Set B
hU : IsOpen U
inst_17 : DiscreteTopology A
h_1 :
  HMul.hMul (α := ENNReal) (β := ENNReal) ((μC : (a : Set C) → ENNReal) Set.univ : ENNReal)
      ((μA : (a : Set A) → ENNReal) {1} : ENNReal) <
    ((H.inducedMeasure μA μC : Measure B) : (a : Set B) → ENNReal) U
ho : LT.lt (α := ENNReal) (0 : ENNReal) ((μA : (a : Set A) → ENNReal) {1} : ENNReal)
ht : LT.lt (α := ENNReal) ((μA : (a : Set A) → ENNReal) {1} : ENNReal) (⊤ : ENNReal)
K : Set B
hKU : K ⊆ U
hK : IsCompact K
f : C(B, ℝ)
hf1 : Set.EqOn (β := ℝ) (⇑f : (a : B) → ℝ) 1 K
hf2 : IsCompact (X := B) (tsupport (α := ℝ) (⇑f : (a : B) → ℝ))
hf3 : tsupport (α := ℝ) (⇑f : (a : B) → ℝ) ⊆ U
hf4 : ∀ (x : B), Membership.mem (α := ℝ) (γ := Set ℝ) (Set.Icc 0 1) ((f : (a : B) → ℝ) x : ℝ)
h :
  HMul.hMul (α := ENNReal) (β := ENNReal) ((μC : (a : Set C) → ENNReal) Set.univ : ENNReal)
      ((μA : (a : Set A) → ENNReal) {1} : ENNReal) <
    ENNReal.ofReal
      (∫ (c : C),
        ((((H.pushforward μA :
                    LinearMap.{0, 0, u_2, u_3} (RingHom.id ℝ) (CompactlySupportedContinuousMap B ℝ)
                      (CompactlySupportedContinuousMap C ℝ)) :
                  (a : CompactlySupportedContinuousMap B ℝ) → CompactlySupportedContinuousMap C ℝ)
                { toContinuousMap := f, hasCompactSupport' := hf2 } :
              CompactlySupportedContinuousMap C ℝ) :
            (a : C) → ℝ)
          c ∂μC)
hc : Set.InjOn.{u_2, u_3} (β := C) (⇑ψ : (a : B) → C) U
b : B
⊢ ((((H.pushforward μA :
                LinearMap.{0, 0, u_2, u_3} (RingHom.id ℝ) (CompactlySupportedContinuousMap B ℝ)
                  (CompactlySupportedContinuousMap C ℝ)) :
              (a : CompactlySupportedContinuousMap B ℝ) → CompactlySupportedContinuousMap C ℝ)
            { toContinuousMap := f, hasCompactSupport' := hf2 } :
          CompactlySupportedContinuousMap C ℝ) :
        (a : C) → ℝ)
      ((ψ : (a : B) → C) b : C) ≤
    ENNReal.toReal ((μA : (a : Set A) → ENNReal) {1})
have simp : ∫ (a : A), (f : (a : B) → ℝ) (HMul.hMul (β := B) b ((φ : (a : A) → B) a : B)) ∂μA ≤
  ENNReal.toReal ((μA : (a : Set A) → ENNReal) {1})
Tactic: simp only [pushforward_apply_apply, pullback_def, CompactlySupportedContinuousMap.coe_mk]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Haar.Extension (goal)
Goal:
A : Type u_1
B : Type u_2
C : Type u_3
inst : Group A
inst_1 : Group B
inst_2 : Group C
inst_3 : TopologicalSpace A
inst_4 : TopologicalSpace B
inst_5 : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst_6 : IsTopologicalGroup A
inst_7 : IsTopologicalGroup B
inst_8 : MeasurableSpace A
inst_9 : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst_10 : IsTopologicalGroup C
inst_11 : LocallyCompactSpace B
inst_12 : MeasurableSpace C
inst_13 : BorelSpace C
μC : Measure C
hμC : μC.IsHaarMeasure
inst_14 : T2Space B
inst_15 : MeasurableSpace B
inst_16 : BorelSpace B
U : Set B
hU : IsOpen U
inst_17 : DiscreteTopology A
h_1 :
  HMul.hMul (α := ENNReal) (β := ENNReal) ((μC : (a : Set C) → ENNReal) Set.univ : ENNReal)
      ((μA : (a : Set A) → ENNReal) {1} : ENNReal) <
    ((H.inducedMeasure μA μC : Measure B) : (a : Set B) → ENNReal) U
ho : LT.lt (α := ENNReal) (0 : ENNReal) ((μA : (a : Set A) → ENNReal) {1} : ENNReal)
ht : LT.lt (α := ENNReal) ((μA : (a : Set A) → ENNReal) {1} : ENNReal) (⊤ : ENNReal)
K : Set B
hKU : K ⊆ U
hK : IsCompact K
f : C(B, ℝ)
hf1 : Set.EqOn (β := ℝ) (⇑f : (a : B) → ℝ) 1 K
hf2 : IsCompact (X := B) (tsupport (α := ℝ) (⇑f : (a : B) → ℝ))
hf3 : tsupport (α := ℝ) (⇑f : (a : B) → ℝ) ⊆ U
hf4 : ∀ (x : B), Membership.mem (α := ℝ) (γ := Set ℝ) (Set.Icc 0 1) ((f : (a : B) → ℝ) x : ℝ)
h :
  HMul.hMul (α := ENNReal) (β := ENNReal) ((μC : (a : Set C) → ENNReal) Set.univ : ENNReal)
      ((μA : (a : Set A) → ENNReal) {1} : ENNReal) <
    ENNReal.ofReal
      (∫ (c : C),
        ((((H.pushforward μA :
                    LinearMap.{0, 0, u_2, u_3} (RingHom.id ℝ) (CompactlySupportedContinuousMap B ℝ)
                      (CompactlySupportedContinuousMap C ℝ)) :
                  (a : CompactlySupportedContinuousMap B ℝ) → CompactlySupportedContinuousMap C ℝ)
                { toContinuousMap := f, hasCompactSupport' := hf2 } :
              CompactlySupportedContinuousMap C ℝ) :
            (a : C) → ℝ)
          c ∂μC)
hc : Set.InjOn.{u_2, u_3} (β := C) (⇑ψ : (a : B) → C) U
b : B
⊢ ∫ (a : A), (f : (a : B) → ℝ) (HMul.hMul (β := B) b ((φ : (a : A) → B) a : B)) ∂μA ≤
    ENNReal.toReal ((μA : (a : Set A) → ENNReal) {1})
have rw : ∫ (x : A) in Function.support (M := ℝ) fun (a : A) ↦ (f : (a : B) → ℝ) (HMul.hMul (β := B) b ((φ : (a : A) → B) a : B)),
    (f : (a : B) → ℝ) (HMul.hMul (β := B) b ((φ : (a : A) → B) x : B)) ∂μA ≤
  ENNReal.toReal ((μA : (a : Set A) → ENNReal) {1})
Tactic: rw [← setIntegral_support]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Haar.Extension (hypothesis)
Goal:
A : Type u_1
B : Type u_2
C : Type u_3
inst : Group A
inst_1 : Group B
inst_2 : Group C
inst_3 : TopologicalSpace A
inst_4 : TopologicalSpace B
inst_5 : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst_6 : IsTopologicalGroup A
inst_7 : IsTopologicalGroup B
inst_8 : MeasurableSpace A
inst_9 : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst_10 : IsTopologicalGroup C
inst_11 : LocallyCompactSpace B
inst_12 : MeasurableSpace C
inst_13 : BorelSpace C
μC : Measure C
hμC : μC.IsHaarMeasure
inst_14 : T2Space B
inst_15 : MeasurableSpace B
inst_16 : BorelSpace B
U : Set B
hU : IsOpen U
inst_17 : DiscreteTopology A
h_1 :
  HMul.hMul (α := ENNReal) (β := ENNReal) ((μC : (a : Set C) → ENNReal) Set.univ : ENNReal)
      ((μA : (a : Set A) → ENNReal) {1} : ENNReal) <
    ((H.inducedMeasure μA μC : Measure B) : (a : Set B) → ENNReal) U
ho : LT.lt (α := ENNReal) (0 : ENNReal) ((μA : (a : Set A) → ENNReal) {1} : ENNReal)
ht : LT.lt (α := ENNReal) ((μA : (a : Set A) → ENNReal) {1} : ENNReal) (⊤ : ENNReal)
K : Set B
hKU : K ⊆ U
hK : IsCompact K
f : C(B, ℝ)
hf1 : Set.EqOn (β := ℝ) (⇑f : (a : B) → ℝ) 1 K
hf2 : IsCompact (X := B) (tsupport (α := ℝ) (⇑f : (a : B) → ℝ))
hf3 : tsupport (α := ℝ) (⇑f : (a : B) → ℝ) ⊆ U
hf4 : ∀ (x : B), Membership.mem (α := ℝ) (γ := Set ℝ) (Set.Icc 0 1) ((f : (a : B) → ℝ) x : ℝ)
h :
  HMul.hMul (α := ENNReal) (β := ENNReal) ((μC : (a : Set C) → ENNReal) Set.univ : ENNReal)
      ((μA : (a : Set A) → ENNReal) {1} : ENNReal) <
    ENNReal.ofReal
      (∫ (c : C),
        ((((H.pushforward μA :
                    LinearMap.{0, 0, u_2, u_3} (RingHom.id ℝ) (CompactlySupportedContinuousMap B ℝ)
                      (CompactlySupportedContinuousMap C ℝ)) :
                  (a : CompactlySupportedContinuousMap B ℝ) → CompactlySupportedContinuousMap C ℝ)
                { toContinuousMap := f, hasCompactSupport' := hf2 } :
              CompactlySupportedContinuousMap C ℝ) :
            (a : C) → ℝ)
          c ∂μC)
hc : Set.InjOn.{u_2, u_3} (β := C) (⇑ψ : (a : B) → C) U
b : B
⊢ ∫ (x : A) in
      Function.support (M := ℝ) fun (a : A) ↦ (f : (a : B) → ℝ) (HMul.hMul (β := B) b ((φ : (a : A) → B) a : B)),
      (f : (a : B) → ℝ) (HMul.hMul (β := B) b ((φ : (a : A) → B) x : B)) ∂μA ≤
    ENNReal.toReal ((μA : (a : Set A) → ENNReal) {1})
have key : Set.Subsingleton.{u_1} (α := A)
  (Function.support (M := ℝ) fun (a : A) ↦ (f : (a : B) → ℝ) (HMul.hMul (β := B) b ((φ : (a : A) → B) a : B)))
Tactic: have key : (Function.support fun a ↦ f (b * φ a)).Subsingleton := by
    intro a ha b hb
    simpa [H.isClosedEmbedding.injective.eq_iff] using hc (hf3 (subset_tsupport _ ha))
      (hf3 (subset_tsupport _ hb)) (by simp [H.mulExact.apply_apply_eq_one])
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Haar.Extension (goal)
Goal:
A : Type u_1
B : Type u_2
C : Type u_3
inst : Group A
inst_1 : Group B
inst_2 : Group C
inst_3 : TopologicalSpace A
inst_4 : TopologicalSpace B
inst_5 : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst_6 : IsTopologicalGroup A
inst_7 : IsTopologicalGroup B
inst_8 : MeasurableSpace A
inst_9 : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst_10 : IsTopologicalGroup C
inst_11 : LocallyCompactSpace B
inst_12 : MeasurableSpace C
inst_13 : BorelSpace C
μC : Measure C
hμC : μC.IsHaarMeasure
inst_14 : T2Space B
inst_15 : MeasurableSpace B
inst_16 : BorelSpace B
U : Set B
hU : IsOpen U
inst_17 : DiscreteTopology A
h_1 :
  HMul.hMul (α := ENNReal) (β := ENNReal) ((μC : (a : Set C) → ENNReal) Set.univ : ENNReal)
      ((μA : (a : Set A) → ENNReal) {1} : ENNReal) <
    ((H.inducedMeasure μA μC : Measure B) : (a : Set B) → ENNReal) U
ho : LT.lt (α := ENNReal) (0 : ENNReal) ((μA : (a : Set A) → ENNReal) {1} : ENNReal)
ht : LT.lt (α := ENNReal) ((μA : (a : Set A) → ENNReal) {1} : ENNReal) (⊤ : ENNReal)
K : Set B
hKU : K ⊆ U
hK : IsCompact K
f : C(B, ℝ)
hf1 : Set.EqOn (β := ℝ) (⇑f : (a : B) → ℝ) 1 K
hf2 : IsCompact (X := B) (tsupport (α := ℝ) (⇑f : (a : B) → ℝ))
hf3 : tsupport (α := ℝ) (⇑f : (a : B) → ℝ) ⊆ U
hf4 : ∀ (x : B), Membership.mem (α := ℝ) (γ := Set ℝ) (Set.Icc 0 1) ((f : (a : B) → ℝ) x : ℝ)
h :
  HMul.hMul (α := ENNReal) (β := ENNReal) ((μC : (a : Set C) → ENNReal) Set.univ : ENNReal)
      ((μA : (a : Set A) → ENNReal) {1} : ENNReal) <
    ENNReal.ofReal
      (∫ (c : C),
        ((((H.pushforward μA :
                    LinearMap.{0, 0, u_2, u_3} (RingHom.id ℝ) (CompactlySupportedContinuousMap B ℝ)
                      (CompactlySupportedContinuousMap C ℝ)) :
                  (a : CompactlySupportedContinuousMap B ℝ) → CompactlySupportedContinuousMap C ℝ)
                { toContinuousMap := f, hasCompactSupport' := hf2 } :
              CompactlySupportedContinuousMap C ℝ) :
            (a : C) → ℝ)
          c ∂μC)
hc : Set.InjOn.{u_2, u_3} (β := C) (⇑ψ : (a : B) → C) U
b : B
key :
  Set.Subsingleton.{u_1} (α := A)
    (Function.support (M := ℝ) fun (a : A) ↦ (f : (a : B) → ℝ) (HMul.hMul (β := B) b ((φ : (a : A) → B) a : B)))
⊢ ∫ (x : A) in
      Function.support (M := ℝ) fun (a : A) ↦ (f : (a : B) → ℝ) (HMul.hMul (β := B) b ((φ : (a : A) → B) a : B)),
      (f : (a : B) → ℝ) (HMul.hMul (β := B) b ((φ : (a : A) → B) x : B)) ∂μA ≤
    ENNReal.toReal ((μA : (a : Set A) → ENNReal) {1})
have inl : ∀
  (h :
    Eq.{u_1 + 1} (α := Set A)
      (Function.support (M := ℝ) fun (a : A) ↦ (f : (a : B) → ℝ) (HMul.hMul (β := B) b ((φ : (a : A) → B) a : B))) ∅),
  ∫ (x : A) in
      Function.support (M := ℝ) fun (a : A) ↦ (f : (a : B) → ℝ) (HMul.hMul (β := B) b ((φ : (a : A) → B) a : B)),
      (f : (a : B) → ℝ) (HMul.hMul (β := B) b ((φ : (a : A) → B) x : B)) ∂μA ≤
    ENNReal.toReal ((μA : (a : Set A) → ENNReal) {1})
Tactic: obtain h | ⟨a, ha⟩ := key.eq_empty_or_singleton
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Haar.Extension (goal)
Goal:
A : Type u_1
B : Type u_2
C : Type u_3
inst : Group A
inst_1 : Group B
inst_2 : Group C
inst_3 : TopologicalSpace A
inst_4 : TopologicalSpace B
inst_5 : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst_6 : IsTopologicalGroup A
inst_7 : IsTopologicalGroup B
inst_8 : MeasurableSpace A
inst_9 : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst_10 : IsTopologicalGroup C
inst_11 : LocallyCompactSpace B
inst_12 : MeasurableSpace C
inst_13 : BorelSpace C
μC : Measure C
hμC : μC.IsHaarMeasure
inst_14 : T2Space B
inst_15 : MeasurableSpace B
inst_16 : BorelSpace B
U : Set B
hU : IsOpen U
inst_17 : DiscreteTopology A
h_1 :
  HMul.hMul (α := ENNReal) (β := ENNReal) ((μC : (a : Set C) → ENNReal) Set.univ : ENNReal)
      ((μA : (a : Set A) → ENNReal) {1} : ENNReal) <
    ((H.inducedMeasure μA μC : Measure B) : (a : Set B) → ENNReal) U
ho : LT.lt (α := ENNReal) (0 : ENNReal) ((μA : (a : Set A) → ENNReal) {1} : ENNReal)
ht : LT.lt (α := ENNReal) ((μA : (a : Set A) → ENNReal) {1} : ENNReal) (⊤ : ENNReal)
K : Set B
hKU : K ⊆ U
hK : IsCompact K
f : C(B, ℝ)
hf1 : Set.EqOn (β := ℝ) (⇑f : (a : B) → ℝ) 1 K
hf2 : IsCompact (X := B) (tsupport (α := ℝ) (⇑f : (a : B) → ℝ))
hf3 : tsupport (α := ℝ) (⇑f : (a : B) → ℝ) ⊆ U
hf4 : ∀ (x : B), Membership.mem (α := ℝ) (γ := Set ℝ) (Set.Icc 0 1) ((f : (a : B) → ℝ) x : ℝ)
h :
  HMul.hMul (α := ENNReal) (β := ENNReal) ((μC : (a : Set C) → ENNReal) Set.univ : ENNReal)
      ((μA : (a : Set A) → ENNReal) {1} : ENNReal) <
    ENNReal.ofReal
      (∫ (c : C),
        ((((H.pushforward μA :
                    LinearMap.{0, 0, u_2, u_3} (RingHom.id ℝ) (CompactlySupportedContinuousMap B ℝ)
                      (CompactlySupportedContinuousMap C ℝ)) :
                  (a : CompactlySupportedContinuousMap B ℝ) → CompactlySupportedContinuousMap C ℝ)
                { toContinuousMap := f, hasCompactSupport' := hf2 } :
              CompactlySupportedContinuousMap C ℝ) :
            (a : C) → ℝ)
          c ∂μC)
hc : Set.InjOn.{u_2, u_3} (β := C) (⇑ψ : (a : B) → C) U
b : B
key :
  Set.Subsingleton.{u_1} (α := A)
    (Function.support (M := ℝ) fun (a : A) ↦ (f : (a : B) → ℝ) (HMul.hMul (β := B) b ((φ : (a : A) → B) a : B)))
a : A
ha :
  Eq (α := Set A)
    (Function.support (M := ℝ) fun (a : A) ↦ (f : (a : B) → ℝ) (HMul.hMul (β := B) b ((φ : (a : A) → B) a : B))) {a}
⊢ ∫ (x : A) in
      Function.support (M := ℝ) fun (a : A) ↦ (f : (a : B) → ℝ) (HMul.hMul (β := B) b ((φ : (a : A) → B) a : B)),
      (f : (a : B) → ℝ) (HMul.hMul (β := B) b ((φ : (a : A) → B) x : B)) ∂μA ≤
    ENNReal.toReal ((μA : (a : Set A) → ENNReal) {1})
have inr : LE.le (α := ℝ) ((f : (a : B) → ℝ) (HMul.hMul (β := B) b ((φ : (a : A) → B) a : B)) : ℝ) (1 : ℝ)
Tactic: rw [ha, integral_singleton, real_def, haar_singleton, smul_eq_mul, mul_le_iff_le_one_right]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Haar.Extension (goal)
Goal:
A : Type u_1
B : Type u_2
C : Type u_3
inst : Group A
inst_1 : Group B
inst_2 : Group C
inst_3 : TopologicalSpace A
inst_4 : TopologicalSpace B
inst_5 : TopologicalSpace C
φ : A →* B
ψ : B →* C
H : IsSES φ ψ
inst_6 : IsTopologicalGroup A
inst_7 : IsTopologicalGroup B
inst_8 : MeasurableSpace A
inst_9 : BorelSpace A
μA : Measure A
hμA : μA.IsHaarMeasure
inst_10 : IsTopologicalGroup C
inst_11 : LocallyCompactSpace B
inst_12 : MeasurableSpace C
inst_13 : BorelSpace C
μC : Measure C
hμC : μC.IsHaarMeasure
inst_14 : T2Space B
inst_15 : MeasurableSpace B
inst_16 : BorelSpace B
U : Set B
hU : IsOpen U
inst_17 : DiscreteTopology A
h_1 :
  HMul.hMul (α := ENNReal) (β := ENNReal) ((μC : (a : Set C) → ENNReal) Set.univ : ENNReal)
      ((μA : (a : Set A) → ENNReal) {1} : ENNReal) <
    ((H.inducedMeasure μA μC : Measure B) : (a : Set B) → ENNReal) U
ho : LT.lt (α := ENNReal) (0 : ENNReal) ((μA : (a : Set A) → ENNReal) {1} : ENNReal)
ht : LT.lt (α := ENNReal) ((μA : (a : Set A) → ENNReal) {1} : ENNReal) (⊤ : ENNReal)
K : Set B
hKU : K ⊆ U
hK : IsCompact K
f : C(B, ℝ)
hf1 : Set.EqOn (β := ℝ) (⇑f : (a : B) → ℝ) 1 K
hf2 : IsCompact (X := B) (tsupport (α := ℝ) (⇑f : (a : B) → ℝ))
hf3 : tsupport (α := ℝ) (⇑f : (a : B) → ℝ) ⊆ U
hf4 : ∀ (x : B), Membership.mem (α := ℝ) (γ := Set ℝ) (Set.Icc 0 1) ((f : (a : B) → ℝ) x : ℝ)
h :
  HMul.hMul (α := ENNReal) (β := ENNReal) ((μC : (a : Set C) → ENNReal) Set.univ : ENNReal)
      ((μA : (a : Set A) → ENNReal) {1} : ENNReal) <
    ENNReal.ofReal
      (∫ (c : C),
        ((((H.pushforward μA :
                    LinearMap.{0, 0, u_2, u_3} (RingHom.id ℝ) (CompactlySupportedContinuousMap B ℝ)
                      (CompactlySupportedContinuousMap C ℝ)) :
                  (a : CompactlySupportedContinuousMap B ℝ) → CompactlySupportedContinuousMap C ℝ)
                { toContinuousMap := f, hasCompactSupport' := hf2 } :
              CompactlySupportedContinuousMap C ℝ) :
            (a : C) → ℝ)
          c ∂μC)
hc : Set.InjOn.{u_2, u_3} (β := C) (⇑ψ : (a : B) → C) U
b : B
key :
  Set.Subsingleton.{u_1} (α := A)
    (Function.support (M := ℝ) fun (a : A) ↦ (f : (a : B) → ℝ) (HMul.hMul (β := B) b ((φ : (a : A) → B) a : B)))
a : A
ha :
  Eq (α := Set A)
    (Function.support (M := ℝ) fun (a : A) ↦ (f : (a : B) → ℝ) (HMul.hMul (β := B) b ((φ : (a : A) → B) a : B))) {a}
inr : LE.le (α := ℝ) ((f : (a : B) → ℝ) (HMul.hMul (β := B) b ((φ : (a : A) → B) a : B)) : ℝ) (1 : ℝ)
⊢ ∫ (x : A) in
      Function.support (M := ℝ) fun (a : A) ↦ (f : (a : B) → ℝ) (HMul.hMul (β := B) b ((φ : (a : A) → B) a : B)),
      (f : (a : B) → ℝ) (HMul.hMul (β := B) b ((φ : (a : A) → B) x : B)) ∂μA ≤
    ENNReal.toReal ((μA : (a : Set A) → ENNReal) {1})
have inr₁ : 0 < ENNReal.toReal ((μA : (a : Set A) → ENNReal) {1})
Tactic: rw [ha, integral_singleton, real_def, haar_singleton, smul_eq_mul, mul_le_iff_le_one_right]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace (goal)
Goal:
ι : Type u_1
F : Type u_3
inst : NormedAddCommGroup F
inst_1 : InnerProductSpace ℝ F
inst_2 : MeasurableSpace F
inst_3 : BorelSpace F
inst_4 : Fintype ι
inst_5 : FiniteDimensional ℝ F
n : ℕ
_i : Fact (finrank ℝ F = n)
o : Orientation ℝ F (Fin n)
b : OrthonormalBasis ι ℝ F
⊢ ι ≃ Fin n
have refine : Fintype.card ι = n
Tactic: refine Fintype.equivFinOfCardEq ?_
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace (goal)
Goal:
ι : Type u_1
F : Type u_3
inst : NormedAddCommGroup F
inst_1 : InnerProductSpace ℝ F
inst_2 : MeasurableSpace F
inst_3 : BorelSpace F
inst_4 : Fintype ι
inst_5 : FiniteDimensional ℝ F
n : ℕ
_i : Fact (finrank ℝ F = n)
o : Orientation ℝ F (Fin n)
b : OrthonormalBasis ι ℝ F
⊢ Fintype.card ι = n
have rw : Fintype.card ι = finrank ℝ F
Tactic: rw [← _i.out, finrank_eq_card_basis b.toBasis]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace (goal)
Goal:
ι : Type u_1
F : Type u_3
inst : NormedAddCommGroup F
inst_1 : InnerProductSpace ℝ F
inst_2 : MeasurableSpace F
inst_3 : BorelSpace F
inst_4 : Fintype ι
inst_5 : FiniteDimensional ℝ F
n : ℕ
_i : Fact (finrank ℝ F = n)
o : Orientation ℝ F (Fin n)
b : OrthonormalBasis ι ℝ F
rw : Fintype.card ι = finrank ℝ F
⊢ Fintype.card ι = n
have rw₁ : Fintype.card ι = Fintype.card ι
Tactic: rw [← _i.out, finrank_eq_card_basis b.toBasis]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace (goal)
Goal:
ι : Type u_1
F : Type u_3
inst : NormedAddCommGroup F
inst_1 : InnerProductSpace ℝ F
inst_2 : MeasurableSpace F
inst_3 : BorelSpace F
inst_4 : Fintype ι
inst_5 : FiniteDimensional ℝ F
n : ℕ
_i : Fact (finrank ℝ F = n)
o : Orientation ℝ F (Fin n)
b : OrthonormalBasis ι ℝ F
⊢ Eq (α := ENNReal)
    (((o.volumeForm.measure : Measure F) : (a : Set F) → ENNReal)
        (parallelepiped.{u_1, u_3} (ι := ι) (⇑b : (a : ι) → F)) :
      ENNReal)
    (1 : ENNReal)
have have : ∀ (e : ι ≃ Fin n),
  Eq (α := ENNReal)
    (((o.volumeForm.measure : Measure F) : (a : Set F) → ENNReal)
        (parallelepiped.{u_1, u_3} (ι := ι) (⇑b : (a : ι) → F)) :
      ENNReal)
    (1 : ENNReal)
Tactic: have e : ι ≃ Fin n := by
    refine Fintype.equivFinOfCardEq ?_
    rw [← _i.out, finrank_eq_card_basis b.toBasis]
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace (goal)
Goal:
ι : Type u_1
F : Type u_3
inst : NormedAddCommGroup F
inst_1 : InnerProductSpace ℝ F
inst_2 : MeasurableSpace F
inst_3 : BorelSpace F
inst_4 : Fintype ι
inst_5 : FiniteDimensional ℝ F
n : ℕ
_i : Fact (finrank ℝ F = n)
o : Orientation ℝ F (Fin n)
b : OrthonormalBasis ι ℝ F
e : ι ≃ Fin n
⊢ Eq (α := (a : ι) → F) (⇑b : (a : ι) → F)
    (Function.comp (β := Fin n) (⇑(b.reindex e) : (a : Fin n) → F) (⇑e : (a : ι) → Fin n))
have h : ∀ (x : ι),
  Eq (α := F) ((b : (a : ι) → F) x : F)
    (Function.comp (β := Fin n) (⇑(b.reindex e) : (a : Fin n) → F) (⇑e : (a : ι) → Fin n) x : F)
Tactic: ext x
-- Lean 4 proof step from Mathlib.MeasureTheory.Measure.Haar.InnerProductSpace (hypothesis)
Goal:
ι : Type u_1
F : Type u_3
inst : NormedAddCommGroup F
inst_1 : InnerProductSpace ℝ F
inst_2 : MeasurableSpace F
inst_3 : BorelSpace F
inst_4 : Fintype ι
inst_5 : FiniteDimensional ℝ F
n : ℕ
_i : Fact (finrank ℝ F = n)
o : Orientation ℝ F (Fin n)
b : OrthonormalBasis ι ℝ F
e : ι ≃ Fin n
⊢ Eq (α := ENNReal)
    (((o.volumeForm.measure : Measure F) : (a : Set F) → ENNReal)
        (parallelepiped.{u_1, u_3} (ι := ι) (⇑b : (a : ι) → F)) :
      ENNReal)
    (1 : ENNReal)
have A : Eq (α := (a : ι) → F) (⇑b : (a : ι) → F)
  (Function.comp (β := Fin n) (⇑(b.reindex e) : (a : Fin n) → F) (⇑e : (a : ι) → Fin n))
Tactic: have A : ⇑b = b.reindex e ∘ e := by
    ext x
    simp only [OrthonormalBasis.coe_reindex, Function.comp_apply, Equiv.symm_apply_apply]