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Mathlib/Algebra/RingQuot.lean | RingQuot.smul_quot | [
{
"state_after": "R : Type u₁\ninst✝⁴ : Semiring R\nS : Type u₂\ninst✝³ : CommSemiring S\nA : Type u₃\ninst✝² : Semiring A\ninst✝¹ : Algebra S A\nr : R → R → Prop\ninst✝ : Algebra S R\nn : S\na : R\n⊢ RingQuot.smul r n { toQuot := Quot.mk (Rel r) a } = { toQuot := Quot.mk (Rel r) (n • a) }",
"state_before": "R : Type u₁\ninst✝⁴ : Semiring R\nS : Type u₂\ninst✝³ : CommSemiring S\nA : Type u₃\ninst✝² : Semiring A\ninst✝¹ : Algebra S A\nr : R → R → Prop\ninst✝ : Algebra S R\nn : S\na : R\n⊢ n • { toQuot := Quot.mk (Rel r) a } = { toQuot := Quot.mk (Rel r) (n • a) }",
"tactic": "show smul r _ _ = _"
},
{
"state_after": "R : Type u₁\ninst✝⁴ : Semiring R\nS : Type u₂\ninst✝³ : CommSemiring S\nA : Type u₃\ninst✝² : Semiring A\ninst✝¹ : Algebra S A\nr : R → R → Prop\ninst✝ : Algebra S R\nn : S\na : R\n⊢ (match { toQuot := Quot.mk (Rel r) a } with\n | { toQuot := a } =>\n { toQuot := Quot.map (fun a => n • a) (_ : ∀ ⦃a b : R⦄, Rel r a b → Rel r (n • a) (n • b)) a }) =\n { toQuot := Quot.mk (Rel r) (n • a) }",
"state_before": "R : Type u₁\ninst✝⁴ : Semiring R\nS : Type u₂\ninst✝³ : CommSemiring S\nA : Type u₃\ninst✝² : Semiring A\ninst✝¹ : Algebra S A\nr : R → R → Prop\ninst✝ : Algebra S R\nn : S\na : R\n⊢ RingQuot.smul r n { toQuot := Quot.mk (Rel r) a } = { toQuot := Quot.mk (Rel r) (n • a) }",
"tactic": "rw [smul_def]"
},
{
"state_after": "no goals",
"state_before": "R : Type u₁\ninst✝⁴ : Semiring R\nS : Type u₂\ninst✝³ : CommSemiring S\nA : Type u₃\ninst✝² : Semiring A\ninst✝¹ : Algebra S A\nr : R → R → Prop\ninst✝ : Algebra S R\nn : S\na : R\n⊢ (match { toQuot := Quot.mk (Rel r) a } with\n | { toQuot := a } =>\n { toQuot := Quot.map (fun a => n • a) (_ : ∀ ⦃a b : R⦄, Rel r a b → Rel r (n • a) (n • b)) a }) =\n { toQuot := Quot.mk (Rel r) (n • a) }",
"tactic": "rfl"
}
] | [
271,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
267,
1
] |
Mathlib/ModelTheory/Semantics.lean | FirstOrder.Language.BoundedFormula.realize_top | [
{
"state_after": "no goals",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.31282\nP : Type ?u.31285\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nn l : ℕ\nφ ψ : BoundedFormula L α l\nθ : BoundedFormula L α (Nat.succ l)\nv : α → M\nxs : Fin l → M\n⊢ Realize ⊤ v xs ↔ True",
"tactic": "simp [Top.top]"
}
] | [
282,
90
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
282,
1
] |
Mathlib/Geometry/Euclidean/Sphere/Basic.lean | EuclideanGeometry.Sphere.mk_center_radius | [
{
"state_after": "no goals",
"state_before": "V : Type ?u.1295\nP : Type u_1\ninst✝ : MetricSpace P\ns : Sphere P\n⊢ { center := s.center, radius := s.radius } = s",
"tactic": "ext <;> rfl"
}
] | [
76,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
75,
1
] |
Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | Submodule.isOrtho_self | [] | [
306,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
304,
1
] |
Mathlib/Data/Bitvec/Lemmas.lean | Bitvec.ofFin_le_ofFin_of_le | [
{
"state_after": "n : ℕ\ni j : Fin (2 ^ n)\nh : i ≤ j\n⊢ ↑i ≤ ↑j",
"state_before": "n : ℕ\ni j : Fin (2 ^ n)\nh : i ≤ j\n⊢ Bitvec.toNat (Bitvec.ofNat n ↑i) ≤ Bitvec.toNat (Bitvec.ofNat n ↑j)",
"tactic": "simp only [toNat_ofNat, Nat.mod_eq_of_lt, Fin.is_lt]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\ni j : Fin (2 ^ n)\nh : i ≤ j\n⊢ ↑i ≤ ↑j",
"tactic": "exact h"
}
] | [
164,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
161,
1
] |
Mathlib/Data/Set/Pointwise/Basic.lean | Set.iUnion_mul_left_image | [] | [
478,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
477,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Mul.lean | Differentiable.mul | [] | [
339,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
338,
1
] |
Mathlib/Topology/Algebra/Order/IntermediateValue.lean | IsConnected.Icc_subset | [] | [
229,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
227,
1
] |
Mathlib/RingTheory/DedekindDomain/Ideal.lean | Ideal.dvd_iff_le | [
{
"state_after": "case pos\nR : Type ?u.752313\nA : Type u_1\nK : Type ?u.752319\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : I = ⊥\n⊢ I ∣ J\n\ncase neg\nR : Type ?u.752313\nA : Type u_1\nK : Type ?u.752319\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : ¬I = ⊥\n⊢ I ∣ J",
"state_before": "R : Type ?u.752313\nA : Type u_1\nK : Type ?u.752319\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\n⊢ I ∣ J",
"tactic": "by_cases hI : I = ⊥"
},
{
"state_after": "case neg\nR : Type ?u.752313\nA : Type u_1\nK : Type ?u.752319\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : ¬I = ⊥\nhI' : ↑I ≠ 0\n⊢ I ∣ J",
"state_before": "case neg\nR : Type ?u.752313\nA : Type u_1\nK : Type ?u.752319\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : ¬I = ⊥\n⊢ I ∣ J",
"tactic": "have hI' : (I : FractionalIdeal A⁰ (FractionRing A)) ≠ 0 := coeIdeal_ne_zero.mpr hI"
},
{
"state_after": "case neg\nR : Type ?u.752313\nA : Type u_1\nK : Type ?u.752319\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : ¬I = ⊥\nhI' : ↑I ≠ 0\nthis : (↑I)⁻¹ * ↑J ≤ 1\n⊢ I ∣ J",
"state_before": "case neg\nR : Type ?u.752313\nA : Type u_1\nK : Type ?u.752319\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : ¬I = ⊥\nhI' : ↑I ≠ 0\n⊢ I ∣ J",
"tactic": "have : (I : FractionalIdeal A⁰ (FractionRing A))⁻¹ * J ≤ 1 :=\n le_trans (mul_left_mono (↑I)⁻¹ ((coeIdeal_le_coeIdeal _).mpr h))\n (le_of_eq (inv_mul_cancel hI'))"
},
{
"state_after": "case neg.intro\nR : Type ?u.752313\nA : Type u_1\nK : Type ?u.752319\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : ¬I = ⊥\nhI' : ↑I ≠ 0\nthis : (↑I)⁻¹ * ↑J ≤ 1\nH : Ideal A\nhH : ↑H = (↑I)⁻¹ * ↑J\n⊢ I ∣ J",
"state_before": "case neg\nR : Type ?u.752313\nA : Type u_1\nK : Type ?u.752319\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : ¬I = ⊥\nhI' : ↑I ≠ 0\nthis : (↑I)⁻¹ * ↑J ≤ 1\n⊢ I ∣ J",
"tactic": "obtain ⟨H, hH⟩ := le_one_iff_exists_coeIdeal.mp this"
},
{
"state_after": "case neg.intro\nR : Type ?u.752313\nA : Type u_1\nK : Type ?u.752319\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : ¬I = ⊥\nhI' : ↑I ≠ 0\nthis : (↑I)⁻¹ * ↑J ≤ 1\nH : Ideal A\nhH : ↑H = (↑I)⁻¹ * ↑J\n⊢ J = I * H",
"state_before": "case neg.intro\nR : Type ?u.752313\nA : Type u_1\nK : Type ?u.752319\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : ¬I = ⊥\nhI' : ↑I ≠ 0\nthis : (↑I)⁻¹ * ↑J ≤ 1\nH : Ideal A\nhH : ↑H = (↑I)⁻¹ * ↑J\n⊢ I ∣ J",
"tactic": "use H"
},
{
"state_after": "case neg.intro\nR : Type ?u.752313\nA : Type u_1\nK : Type ?u.752319\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : ¬I = ⊥\nhI' : ↑I ≠ 0\nthis : (↑I)⁻¹ * ↑J ≤ 1\nH : Ideal A\nhH : ↑H = (↑I)⁻¹ * ↑J\n⊢ ↑J = ↑(I * H)",
"state_before": "case neg.intro\nR : Type ?u.752313\nA : Type u_1\nK : Type ?u.752319\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : ¬I = ⊥\nhI' : ↑I ≠ 0\nthis : (↑I)⁻¹ * ↑J ≤ 1\nH : Ideal A\nhH : ↑H = (↑I)⁻¹ * ↑J\n⊢ J = I * H",
"tactic": "refine coeIdeal_injective (show (J : FractionalIdeal A⁰ (FractionRing A)) = ↑(I * H) from ?_)"
},
{
"state_after": "no goals",
"state_before": "case neg.intro\nR : Type ?u.752313\nA : Type u_1\nK : Type ?u.752319\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : ¬I = ⊥\nhI' : ↑I ≠ 0\nthis : (↑I)⁻¹ * ↑J ≤ 1\nH : Ideal A\nhH : ↑H = (↑I)⁻¹ * ↑J\n⊢ ↑J = ↑(I * H)",
"tactic": "rw [coeIdeal_mul, hH, ← mul_assoc, mul_inv_cancel hI', one_mul]"
},
{
"state_after": "case pos\nR : Type ?u.752313\nA : Type u_1\nK : Type ?u.752319\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : I = ⊥\nhJ : J = ⊥\n⊢ I ∣ J",
"state_before": "case pos\nR : Type ?u.752313\nA : Type u_1\nK : Type ?u.752319\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : I = ⊥\n⊢ I ∣ J",
"tactic": "have hJ : J = ⊥ := by rwa [hI, ← eq_bot_iff] at h"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type ?u.752313\nA : Type u_1\nK : Type ?u.752319\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : I = ⊥\nhJ : J = ⊥\n⊢ I ∣ J",
"tactic": "rw [hI, hJ]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.752313\nA : Type u_1\nK : Type ?u.752319\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : IsDedekindDomain A\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nI J : Ideal A\nh : J ≤ I\nhI : I = ⊥\n⊢ J = ⊥",
"tactic": "rwa [hI, ← eq_bot_iff] at h"
}
] | [
655,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
643,
1
] |
Mathlib/Topology/Order/Basic.lean | closure_Ici | [] | [
169,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
168,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | SimpleGraph.singletonSubgraph_le_iff | [
{
"state_after": "ι : Sort ?u.207802\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nv : V\nH : Subgraph G\n⊢ v ∈ H.verts → SimpleGraph.singletonSubgraph G v ≤ H",
"state_before": "ι : Sort ?u.207802\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nv : V\nH : Subgraph G\n⊢ SimpleGraph.singletonSubgraph G v ≤ H ↔ v ∈ H.verts",
"tactic": "refine' ⟨fun h ↦ h.1 (Set.mem_singleton v), _⟩"
},
{
"state_after": "ι : Sort ?u.207802\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nv : V\nH : Subgraph G\nh : v ∈ H.verts\n⊢ SimpleGraph.singletonSubgraph G v ≤ H",
"state_before": "ι : Sort ?u.207802\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nv : V\nH : Subgraph G\n⊢ v ∈ H.verts → SimpleGraph.singletonSubgraph G v ≤ H",
"tactic": "intro h"
},
{
"state_after": "case left\nι : Sort ?u.207802\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nv : V\nH : Subgraph G\nh : v ∈ H.verts\n⊢ (SimpleGraph.singletonSubgraph G v).verts ⊆ H.verts\n\ncase right\nι : Sort ?u.207802\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nv : V\nH : Subgraph G\nh : v ∈ H.verts\n⊢ ∀ ⦃v_1 w : V⦄, Subgraph.Adj (SimpleGraph.singletonSubgraph G v) v_1 w → Subgraph.Adj H v_1 w",
"state_before": "ι : Sort ?u.207802\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nv : V\nH : Subgraph G\nh : v ∈ H.verts\n⊢ SimpleGraph.singletonSubgraph G v ≤ H",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case left\nι : Sort ?u.207802\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nv : V\nH : Subgraph G\nh : v ∈ H.verts\n⊢ (SimpleGraph.singletonSubgraph G v).verts ⊆ H.verts",
"tactic": "rwa [singletonSubgraph_verts, Set.singleton_subset_iff]"
},
{
"state_after": "no goals",
"state_before": "case right\nι : Sort ?u.207802\nV : Type u\nW : Type v\nG : SimpleGraph V\nG' : SimpleGraph W\nv : V\nH : Subgraph G\nh : v ∈ H.verts\n⊢ ∀ ⦃v_1 w : V⦄, Subgraph.Adj (SimpleGraph.singletonSubgraph G v) v_1 w → Subgraph.Adj H v_1 w",
"tactic": "exact fun _ _ ↦ False.elim"
}
] | [
858,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
852,
1
] |
Mathlib/Order/InitialSeg.lean | InitialSeg.trans_apply | [] | [
131,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
130,
1
] |
Std/Data/Int/Lemmas.lean | Int.zero_add | [] | [
228,
95
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
228,
25
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean | deriv_id | [] | [
653,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
652,
1
] |
Mathlib/Order/WellFounded.lean | Acc.induction_bot | [] | [
257,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
255,
1
] |
Std/Data/Nat/Gcd.lean | Nat.gcd_ne_zero_left | [] | [
112,
81
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
112,
1
] |
Mathlib/Data/Nat/Basic.lean | Nat.succ_add_sub_one | [
{
"state_after": "no goals",
"state_before": "m✝ n✝ k n m : ℕ\n⊢ succ n + m - 1 = n + m",
"tactic": "rw [succ_add, succ_sub_one]"
}
] | [
304,
94
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
304,
1
] |
src/lean/Init/SimpLemmas.lean | let_val_congr | [] | [
49,
81
] | d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
48,
1
] |
Mathlib/Data/Set/Function.lean | Set.piecewise_preimage | [
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.89108\nι : Sort ?u.89111\nπ : α → Type ?u.89116\nδ : α → Sort ?u.89121\ns : Set α\nf✝ g✝ : (i : α) → δ i\ninst✝ : (j : α) → Decidable (j ∈ s)\nf g : α → β\nt : Set β\nx : α\n⊢ x ∈ piecewise s f g ⁻¹' t ↔ x ∈ Set.ite s (f ⁻¹' t) (g ⁻¹' t)",
"tactic": "by_cases x ∈ s <;> simp [*, Set.ite]"
}
] | [
1469,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1468,
1
] |
Mathlib/RingTheory/DedekindDomain/Ideal.lean | FractionalIdeal.exists_not_mem_one_of_ne_bot | [
{
"state_after": "R : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\n⊢ ∀ {M : Ideal A}, IsMaximal M → ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1",
"state_before": "R : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\n⊢ ∃ x, x ∈ (↑I)⁻¹ ∧ ¬x ∈ 1",
"tactic": "suffices\n ∀ {M : Ideal A} (_hM : M.IsMaximal),\n ∃ x : K, x ∈ (M⁻¹ : FractionalIdeal A⁰ K) ∧ x ∉ (1 : FractionalIdeal A⁰ K) by\n obtain ⟨M, hM, hIM⟩ : ∃ M : Ideal A, IsMaximal M ∧ I ≤ M := Ideal.exists_le_maximal I hI1\n skip\n have hM0 := (M.bot_lt_of_maximal hNF).ne'\n obtain ⟨x, hxM, hx1⟩ := this hM\n refine ⟨x, inv_anti_mono ?_ ?_ ((coeIdeal_le_coeIdeal _).mpr hIM) hxM, hx1⟩ <;>\n rw [coeIdeal_ne_zero] <;> assumption"
},
{
"state_after": "R : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\n⊢ ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1",
"state_before": "R : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\n⊢ ∀ {M : Ideal A}, IsMaximal M → ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1",
"tactic": "intro M hM"
},
{
"state_after": "R : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\n⊢ ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1",
"state_before": "R : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\n⊢ ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1",
"tactic": "skip"
},
{
"state_after": "case intro.mk\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : { val := a, property := haM } ≠ 0\n⊢ ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1",
"state_before": "R : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\n⊢ ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1",
"tactic": "obtain ⟨⟨a, haM⟩, ha0⟩ := Submodule.nonzero_mem_of_bot_lt (M.bot_lt_of_maximal hNF)"
},
{
"state_after": "case intro.mk\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\n⊢ ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1",
"state_before": "case intro.mk\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : { val := a, property := haM } ≠ 0\n⊢ ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1",
"tactic": "replace ha0 : a ≠ 0 := Subtype.coe_injective.ne ha0"
},
{
"state_after": "case intro.mk\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\n⊢ ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1",
"state_before": "case intro.mk\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\n⊢ ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1",
"tactic": "let J : Ideal A := Ideal.span {a}"
},
{
"state_after": "case intro.mk\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\n⊢ ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1",
"state_before": "case intro.mk\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\n⊢ ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1",
"tactic": "have hJ0 : J ≠ ⊥ := mt Ideal.span_singleton_eq_bot.mp ha0"
},
{
"state_after": "case intro.mk\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\n⊢ ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1",
"state_before": "case intro.mk\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\n⊢ ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1",
"tactic": "have hJM : J ≤ M := Ideal.span_le.mpr (Set.singleton_subset_iff.mpr haM)"
},
{
"state_after": "case intro.mk\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\n⊢ ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1",
"state_before": "case intro.mk\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\n⊢ ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1",
"tactic": "have hM0 : ⊥ < M := M.bot_lt_of_maximal hNF"
},
{
"state_after": "case intro.mk.intro.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\n⊢ ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1",
"state_before": "case intro.mk\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\n⊢ ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1",
"tactic": "obtain ⟨Z, hle, hnle⟩ := exists_multiset_prod_cons_le_and_prod_not_le hNF hJ0 hJM"
},
{
"state_after": "case intro.mk.intro.intro.intro.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\n⊢ ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1",
"state_before": "case intro.mk.intro.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\n⊢ ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1",
"tactic": "obtain ⟨b, hbZ, hbJ⟩ := SetLike.not_le_iff_exists.mp hnle"
},
{
"state_after": "case intro.mk.intro.intro.intro.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n⊢ ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1",
"state_before": "case intro.mk.intro.intro.intro.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\n⊢ ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1",
"tactic": "have hnz_fa : algebraMap A K a ≠ 0 :=\n mt ((injective_iff_map_eq_zero _).mp (IsFractionRing.injective A K) a) ha0"
},
{
"state_after": "case intro.mk.intro.intro.intro.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\n⊢ ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1",
"state_before": "case intro.mk.intro.intro.intro.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n⊢ ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1",
"tactic": "have _hb0 : algebraMap A K b ≠ 0 :=\n mt ((injective_iff_map_eq_zero _).mp (IsFractionRing.injective A K) b) fun h =>\n hbJ <| h.symm ▸ J.zero_mem"
},
{
"state_after": "case intro.mk.intro.intro.intro.intro.refine'_1\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\n⊢ ↑M ≠ 0\n\ncase intro.mk.intro.intro.intro.intro.refine'_2\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\n⊢ ∀ (y : K), y ∈ ↑M → ↑(algebraMap A K) b * (↑(algebraMap A K) a)⁻¹ * y ∈ 1\n\ncase intro.mk.intro.intro.intro.intro.refine'_3\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\n⊢ ¬↑(algebraMap A K) b * (↑(algebraMap A K) a)⁻¹ ∈ 1",
"state_before": "case intro.mk.intro.intro.intro.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\n⊢ ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1",
"tactic": "refine' ⟨algebraMap A K b * (algebraMap A K a)⁻¹, (mem_inv_iff _).mpr _, _⟩"
},
{
"state_after": "case intro.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nthis : ∀ {M : Ideal A}, IsMaximal M → ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1\nM : Ideal A\nhM : IsMaximal M\nhIM : I ≤ M\n⊢ ∃ x, x ∈ (↑I)⁻¹ ∧ ¬x ∈ 1",
"state_before": "R : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nthis : ∀ {M : Ideal A}, IsMaximal M → ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1\n⊢ ∃ x, x ∈ (↑I)⁻¹ ∧ ¬x ∈ 1",
"tactic": "obtain ⟨M, hM, hIM⟩ : ∃ M : Ideal A, IsMaximal M ∧ I ≤ M := Ideal.exists_le_maximal I hI1"
},
{
"state_after": "case intro.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nthis : ∀ {M : Ideal A}, IsMaximal M → ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1\nM : Ideal A\nhM : IsMaximal M\nhIM : I ≤ M\n⊢ ∃ x, x ∈ (↑I)⁻¹ ∧ ¬x ∈ 1",
"state_before": "case intro.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nthis : ∀ {M : Ideal A}, IsMaximal M → ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1\nM : Ideal A\nhM : IsMaximal M\nhIM : I ≤ M\n⊢ ∃ x, x ∈ (↑I)⁻¹ ∧ ¬x ∈ 1",
"tactic": "skip"
},
{
"state_after": "case intro.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nthis : ∀ {M : Ideal A}, IsMaximal M → ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1\nM : Ideal A\nhM : IsMaximal M\nhIM : I ≤ M\nhM0 : M ≠ ⊥\n⊢ ∃ x, x ∈ (↑I)⁻¹ ∧ ¬x ∈ 1",
"state_before": "case intro.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nthis : ∀ {M : Ideal A}, IsMaximal M → ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1\nM : Ideal A\nhM : IsMaximal M\nhIM : I ≤ M\n⊢ ∃ x, x ∈ (↑I)⁻¹ ∧ ¬x ∈ 1",
"tactic": "have hM0 := (M.bot_lt_of_maximal hNF).ne'"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nthis : ∀ {M : Ideal A}, IsMaximal M → ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1\nM : Ideal A\nhM : IsMaximal M\nhIM : I ≤ M\nhM0 : M ≠ ⊥\nx : K\nhxM : x ∈ (↑M)⁻¹\nhx1 : ¬x ∈ 1\n⊢ ∃ x, x ∈ (↑I)⁻¹ ∧ ¬x ∈ 1",
"state_before": "case intro.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nthis : ∀ {M : Ideal A}, IsMaximal M → ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1\nM : Ideal A\nhM : IsMaximal M\nhIM : I ≤ M\nhM0 : M ≠ ⊥\n⊢ ∃ x, x ∈ (↑I)⁻¹ ∧ ¬x ∈ 1",
"tactic": "obtain ⟨x, hxM, hx1⟩ := this hM"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nthis : ∀ {M : Ideal A}, IsMaximal M → ∃ x, x ∈ (↑M)⁻¹ ∧ ¬x ∈ 1\nM : Ideal A\nhM : IsMaximal M\nhIM : I ≤ M\nhM0 : M ≠ ⊥\nx : K\nhxM : x ∈ (↑M)⁻¹\nhx1 : ¬x ∈ 1\n⊢ ∃ x, x ∈ (↑I)⁻¹ ∧ ¬x ∈ 1",
"tactic": "refine ⟨x, inv_anti_mono ?_ ?_ ((coeIdeal_le_coeIdeal _).mpr hIM) hxM, hx1⟩ <;>\n rw [coeIdeal_ne_zero] <;> assumption"
},
{
"state_after": "no goals",
"state_before": "case intro.mk.intro.intro.intro.intro.refine'_1\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\n⊢ ↑M ≠ 0",
"tactic": "exact coeIdeal_ne_zero.mpr hM0.ne'"
},
{
"state_after": "case intro.mk.intro.intro.intro.intro.refine'_2\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\ny₀ : K\nhy₀ : y₀ ∈ ↑M\n⊢ ↑(algebraMap A K) b * (↑(algebraMap A K) a)⁻¹ * y₀ ∈ 1",
"state_before": "case intro.mk.intro.intro.intro.intro.refine'_2\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\n⊢ ∀ (y : K), y ∈ ↑M → ↑(algebraMap A K) b * (↑(algebraMap A K) a)⁻¹ * y ∈ 1",
"tactic": "rintro y₀ hy₀"
},
{
"state_after": "case intro.mk.intro.intro.intro.intro.refine'_2.intro.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\ny : A\nh_Iy : y ∈ M\nhy₀ : ↑(algebraMap A K) y ∈ ↑M\n⊢ ↑(algebraMap A K) b * (↑(algebraMap A K) a)⁻¹ * ↑(algebraMap A K) y ∈ 1",
"state_before": "case intro.mk.intro.intro.intro.intro.refine'_2\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\ny₀ : K\nhy₀ : y₀ ∈ ↑M\n⊢ ↑(algebraMap A K) b * (↑(algebraMap A K) a)⁻¹ * y₀ ∈ 1",
"tactic": "obtain ⟨y, h_Iy, rfl⟩ := (mem_coeIdeal _).mp hy₀"
},
{
"state_after": "case intro.mk.intro.intro.intro.intro.refine'_2.intro.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\ny : A\nh_Iy : y ∈ M\nhy₀ : ↑(algebraMap A K) y ∈ ↑M\n⊢ ↑(algebraMap A K) (y * b) * (↑(algebraMap A K) a)⁻¹ ∈ 1",
"state_before": "case intro.mk.intro.intro.intro.intro.refine'_2.intro.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\ny : A\nh_Iy : y ∈ M\nhy₀ : ↑(algebraMap A K) y ∈ ↑M\n⊢ ↑(algebraMap A K) b * (↑(algebraMap A K) a)⁻¹ * ↑(algebraMap A K) y ∈ 1",
"tactic": "rw [mul_comm, ← mul_assoc, ← RingHom.map_mul]"
},
{
"state_after": "case intro.mk.intro.intro.intro.intro.refine'_2.intro.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\ny : A\nh_Iy : y ∈ M\nhy₀ : ↑(algebraMap A K) y ∈ ↑M\nh_yb : y * b ∈ J\n⊢ ↑(algebraMap A K) (y * b) * (↑(algebraMap A K) a)⁻¹ ∈ 1",
"state_before": "case intro.mk.intro.intro.intro.intro.refine'_2.intro.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\ny : A\nh_Iy : y ∈ M\nhy₀ : ↑(algebraMap A K) y ∈ ↑M\n⊢ ↑(algebraMap A K) (y * b) * (↑(algebraMap A K) a)⁻¹ ∈ 1",
"tactic": "have h_yb : y * b ∈ J := by\n apply hle\n rw [Multiset.prod_cons]\n exact Submodule.smul_mem_smul h_Iy hbZ"
},
{
"state_after": "case intro.mk.intro.intro.intro.intro.refine'_2.intro.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\ny : A\nh_Iy : y ∈ M\nhy₀ : ↑(algebraMap A K) y ∈ ↑M\nh_yb : ∃ a_1, a_1 * a = y * b\n⊢ ↑(algebraMap A K) (y * b) * (↑(algebraMap A K) a)⁻¹ ∈ 1",
"state_before": "case intro.mk.intro.intro.intro.intro.refine'_2.intro.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\ny : A\nh_Iy : y ∈ M\nhy₀ : ↑(algebraMap A K) y ∈ ↑M\nh_yb : y * b ∈ J\n⊢ ↑(algebraMap A K) (y * b) * (↑(algebraMap A K) a)⁻¹ ∈ 1",
"tactic": "rw [Ideal.mem_span_singleton'] at h_yb"
},
{
"state_after": "case intro.mk.intro.intro.intro.intro.refine'_2.intro.intro.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\ny : A\nh_Iy : y ∈ M\nhy₀ : ↑(algebraMap A K) y ∈ ↑M\nc : A\nhc : c * a = y * b\n⊢ ↑(algebraMap A K) (y * b) * (↑(algebraMap A K) a)⁻¹ ∈ 1",
"state_before": "case intro.mk.intro.intro.intro.intro.refine'_2.intro.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\ny : A\nh_Iy : y ∈ M\nhy₀ : ↑(algebraMap A K) y ∈ ↑M\nh_yb : ∃ a_1, a_1 * a = y * b\n⊢ ↑(algebraMap A K) (y * b) * (↑(algebraMap A K) a)⁻¹ ∈ 1",
"tactic": "rcases h_yb with ⟨c, hc⟩"
},
{
"state_after": "case intro.mk.intro.intro.intro.intro.refine'_2.intro.intro.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\ny : A\nh_Iy : y ∈ M\nhy₀ : ↑(algebraMap A K) y ∈ ↑M\nc : A\nhc : c * a = y * b\n⊢ ↑(algebraMap A K) c ∈ 1",
"state_before": "case intro.mk.intro.intro.intro.intro.refine'_2.intro.intro.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\ny : A\nh_Iy : y ∈ M\nhy₀ : ↑(algebraMap A K) y ∈ ↑M\nc : A\nhc : c * a = y * b\n⊢ ↑(algebraMap A K) (y * b) * (↑(algebraMap A K) a)⁻¹ ∈ 1",
"tactic": "rw [← hc, RingHom.map_mul, mul_assoc, mul_inv_cancel hnz_fa, mul_one]"
},
{
"state_after": "no goals",
"state_before": "case intro.mk.intro.intro.intro.intro.refine'_2.intro.intro.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\ny : A\nh_Iy : y ∈ M\nhy₀ : ↑(algebraMap A K) y ∈ ↑M\nc : A\nhc : c * a = y * b\n⊢ ↑(algebraMap A K) c ∈ 1",
"tactic": "apply coe_mem_one"
},
{
"state_after": "case a\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\ny : A\nh_Iy : y ∈ M\nhy₀ : ↑(algebraMap A K) y ∈ ↑M\n⊢ y * b ∈ Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z)",
"state_before": "R : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\ny : A\nh_Iy : y ∈ M\nhy₀ : ↑(algebraMap A K) y ∈ ↑M\n⊢ y * b ∈ J",
"tactic": "apply hle"
},
{
"state_after": "case a\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\ny : A\nh_Iy : y ∈ M\nhy₀ : ↑(algebraMap A K) y ∈ ↑M\n⊢ y * b ∈ M * Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)",
"state_before": "case a\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\ny : A\nh_Iy : y ∈ M\nhy₀ : ↑(algebraMap A K) y ∈ ↑M\n⊢ y * b ∈ Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z)",
"tactic": "rw [Multiset.prod_cons]"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\ny : A\nh_Iy : y ∈ M\nhy₀ : ↑(algebraMap A K) y ∈ ↑M\n⊢ y * b ∈ M * Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)",
"tactic": "exact Submodule.smul_mem_smul h_Iy hbZ"
},
{
"state_after": "case intro.mk.intro.intro.intro.intro.refine'_3\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\n⊢ ¬∃ x', ↑(algebraMap A K) x' = ↑(algebraMap A K) b * (↑(algebraMap A K) a)⁻¹",
"state_before": "case intro.mk.intro.intro.intro.intro.refine'_3\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\n⊢ ¬↑(algebraMap A K) b * (↑(algebraMap A K) a)⁻¹ ∈ 1",
"tactic": "refine' mt (mem_one_iff _).mp _"
},
{
"state_after": "case intro.mk.intro.intro.intro.intro.refine'_3.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\nx' : A\nh₂_abs : ↑(algebraMap A K) x' = ↑(algebraMap A K) b * (↑(algebraMap A K) a)⁻¹\n⊢ False",
"state_before": "case intro.mk.intro.intro.intro.intro.refine'_3\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\n⊢ ¬∃ x', ↑(algebraMap A K) x' = ↑(algebraMap A K) b * (↑(algebraMap A K) a)⁻¹",
"tactic": "rintro ⟨x', h₂_abs⟩"
},
{
"state_after": "case intro.mk.intro.intro.intro.intro.refine'_3.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\nx' : A\nh₂_abs : ↑(algebraMap A K) (x' * a) = ↑(algebraMap A K) b\n⊢ False",
"state_before": "case intro.mk.intro.intro.intro.intro.refine'_3.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\nx' : A\nh₂_abs : ↑(algebraMap A K) x' = ↑(algebraMap A K) b * (↑(algebraMap A K) a)⁻¹\n⊢ False",
"tactic": "rw [← div_eq_mul_inv, eq_div_iff_mul_eq hnz_fa, ← RingHom.map_mul] at h₂_abs"
},
{
"state_after": "case intro.mk.intro.intro.intro.intro.refine'_3.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\nx' : A\nh₂_abs : ↑(algebraMap A K) (x' * a) = ↑(algebraMap A K) b\nthis : b ∈ span {a}\n⊢ False",
"state_before": "case intro.mk.intro.intro.intro.intro.refine'_3.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\nx' : A\nh₂_abs : ↑(algebraMap A K) (x' * a) = ↑(algebraMap A K) b\n⊢ False",
"tactic": "have := Ideal.mem_span_singleton'.mpr ⟨x', IsFractionRing.injective A K h₂_abs⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.mk.intro.intro.intro.intro.refine'_3.intro\nR : Type ?u.159729\nA : Type u_1\nK : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Field K\ninst✝³ : IsDomain A\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nhNF : ¬IsField A\nI : Ideal A\nhI0 : I ≠ ⊥\nhI1 : I ≠ ⊤\nM : Ideal A\nhM : IsMaximal M\na : A\nhaM : a ∈ M\nha0 : a ≠ 0\nJ : Ideal A := span {a}\nhJ0 : J ≠ ⊥\nhJM : J ≤ M\nhM0 : ⊥ < M\nZ : Multiset (PrimeSpectrum A)\nhle : Multiset.prod (M ::ₘ Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nhnle : ¬Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z) ≤ J\nb : A\nhbZ : b ∈ Multiset.prod (Multiset.map PrimeSpectrum.asIdeal Z)\nhbJ : ¬b ∈ J\nhnz_fa : ↑(algebraMap A K) a ≠ 0\n_hb0 : ↑(algebraMap A K) b ≠ 0\nx' : A\nh₂_abs : ↑(algebraMap A K) (x' * a) = ↑(algebraMap A K) b\nthis : b ∈ span {a}\n⊢ False",
"tactic": "contradiction"
}
] | [
451,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
401,
1
] |
Mathlib/CategoryTheory/EqToHom.lean | CategoryTheory.Functor.postcomp_map_hEq | [
{
"state_after": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF G : C ⥤ D\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nH : D ⥤ E\nhx : F.obj X = G.obj X\nhy : F.obj Y = G.obj Y\nhmap : HEq (F.map f) (G.map f)\n⊢ HEq (H.map (F.map f)) (H.map (G.map f))",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF G : C ⥤ D\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nH : D ⥤ E\nhx : F.obj X = G.obj X\nhy : F.obj Y = G.obj Y\nhmap : HEq (F.map f) (G.map f)\n⊢ HEq ((F ⋙ H).map f) ((G ⋙ H).map f)",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF G : C ⥤ D\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nH : D ⥤ E\nhx : F.obj X = G.obj X\nhy : F.obj Y = G.obj Y\nhmap : HEq (F.map f) (G.map f)\n⊢ HEq (H.map (F.map f)) (H.map (G.map f))",
"tactic": "congr"
}
] | [
251,
8
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
248,
1
] |
Mathlib/Probability/Independence/Basic.lean | ProbabilityTheory.iIndepFun.indepFun_prod_range_succ | [] | [
898,
77
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
894,
1
] |
Mathlib/Order/UpperLower/Basic.lean | bddBelow_upperClosure | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.166274\nγ : Type ?u.166277\nι : Sort ?u.166280\nκ : ι → Sort ?u.166285\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns t : Set α\nx : α\n⊢ BddBelow ↑(upperClosure s) ↔ BddBelow s",
"tactic": "simp_rw [BddBelow, lowerBounds_upperClosure]"
}
] | [
1492,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1491,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean | dist_le_Ico_sum_dist | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.11251\nι : Type ?u.11254\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\nm n : ℕ\nh : m ≤ n\n⊢ dist (f m) (f n) ≤ ∑ i in Finset.Ico m n, dist (f i) (f (i + 1))",
"tactic": "induction n, h using Nat.le_induction with\n| base => rw [Finset.Ico_self, Finset.sum_empty, dist_self]\n| succ n hle ihn =>\n calc\n dist (f m) (f (n + 1)) ≤ dist (f m) (f n) + dist (f n) (f (n + 1)) := dist_triangle _ _ _\n _ ≤ (∑ i in Finset.Ico m n, _) + _ := add_le_add ihn le_rfl\n _ = ∑ i in Finset.Ico m (n + 1), _ := by\n { rw [Nat.Ico_succ_right_eq_insert_Ico hle, Finset.sum_insert, add_comm]; simp }"
},
{
"state_after": "no goals",
"state_before": "case base\nα : Type u\nβ : Type v\nX : Type ?u.11251\nι : Type ?u.11254\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\nm n : ℕ\n⊢ dist (f m) (f m) ≤ ∑ i in Finset.Ico m m, dist (f i) (f (i + 1))",
"tactic": "rw [Finset.Ico_self, Finset.sum_empty, dist_self]"
},
{
"state_after": "no goals",
"state_before": "case succ\nα : Type u\nβ : Type v\nX : Type ?u.11251\nι : Type ?u.11254\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\nm n✝ n : ℕ\nhle : m ≤ n\nihn : dist (f m) (f n) ≤ ∑ i in Finset.Ico m n, dist (f i) (f (i + 1))\n⊢ dist (f m) (f (n + 1)) ≤ ∑ i in Finset.Ico m (n + 1), dist (f i) (f (i + 1))",
"tactic": "calc\n dist (f m) (f (n + 1)) ≤ dist (f m) (f n) + dist (f n) (f (n + 1)) := dist_triangle _ _ _\n _ ≤ (∑ i in Finset.Ico m n, _) + _ := add_le_add ihn le_rfl\n _ = ∑ i in Finset.Ico m (n + 1), _ := by\n { rw [Nat.Ico_succ_right_eq_insert_Ico hle, Finset.sum_insert, add_comm]; simp }"
}
] | [
239,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
230,
1
] |
Mathlib/Logic/Basic.lean | ball_congr | [] | [
1035,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1034,
1
] |
Mathlib/RingTheory/Polynomial/Vieta.lean | Multiset.prod_X_sub_X_eq_sum_esymm | [
{
"state_after": "R : Type u_1\ninst✝ : CommRing R\ns : Multiset R\n⊢ prod (map (fun x => X + ↑C (-x)) s) =\n ∑ j in Finset.range (↑card s + 1), (-1) ^ j * (↑C (esymm s j) * X ^ (↑card s - j))",
"state_before": "R : Type u_1\ninst✝ : CommRing R\ns : Multiset R\n⊢ prod (map (fun t => X - ↑C t) s) = ∑ j in Finset.range (↑card s + 1), (-1) ^ j * (↑C (esymm s j) * X ^ (↑card s - j))",
"tactic": "conv_lhs =>\n congr\n congr\n ext x\n rw [sub_eq_add_neg]\n rw [← map_neg C x]"
},
{
"state_after": "case h.e'_2\nR : Type u_1\ninst✝ : CommRing R\ns : Multiset R\n⊢ prod (map (fun x => X + ↑C (-x)) s) = prod (map (fun r => X + ↑C r) (map (fun t => -t) s))\n\ncase h.e'_3\nR : Type u_1\ninst✝ : CommRing R\ns : Multiset R\n⊢ ∑ j in Finset.range (↑card s + 1), (-1) ^ j * (↑C (esymm s j) * X ^ (↑card s - j)) =\n ∑ j in Finset.range (↑card (map (fun t => -t) s) + 1),\n ↑C (esymm (map (fun t => -t) s) j) * X ^ (↑card (map (fun t => -t) s) - j)",
"state_before": "R : Type u_1\ninst✝ : CommRing R\ns : Multiset R\n⊢ prod (map (fun x => X + ↑C (-x)) s) =\n ∑ j in Finset.range (↑card s + 1), (-1) ^ j * (↑C (esymm s j) * X ^ (↑card s - j))",
"tactic": "convert prod_X_add_C_eq_sum_esymm (map (fun t => -t) s) using 1"
},
{
"state_after": "case h.e'_2\nR : Type u_1\ninst✝ : CommRing R\ns : Multiset R\n⊢ prod (map (fun x => X + ↑C (-x)) s) = prod (map ((fun r => X + ↑C r) ∘ fun t => -t) s)",
"state_before": "case h.e'_2\nR : Type u_1\ninst✝ : CommRing R\ns : Multiset R\n⊢ prod (map (fun x => X + ↑C (-x)) s) = prod (map (fun r => X + ↑C r) (map (fun t => -t) s))",
"tactic": "rw [map_map]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2\nR : Type u_1\ninst✝ : CommRing R\ns : Multiset R\n⊢ prod (map (fun x => X + ↑C (-x)) s) = prod (map ((fun r => X + ↑C r) ∘ fun t => -t) s)",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nR : Type u_1\ninst✝ : CommRing R\ns : Multiset R\n⊢ ∑ j in Finset.range (↑card s + 1), (-1) ^ j * (↑C (esymm s j) * X ^ (↑card s - j)) =\n ∑ j in Finset.range (↑card (map (fun t => -t) s) + 1),\n ↑C (esymm (map (fun t => -t) s) j) * X ^ (↑card (map (fun t => -t) s) - j)",
"tactic": "simp only [esymm_neg, card_map, mul_assoc, map_mul, map_pow, map_neg, map_one]"
}
] | [
119,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
107,
1
] |
Mathlib/Analysis/NormedSpace/Star/Basic.lean | starₗᵢ_toContinuousLinearEquiv | [] | [
314,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
312,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | SimpleGraph.Walk.append_assoc | [
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v w x : V\np : Walk G u v\nq : Walk G v w\nr : Walk G w x\n⊢ append p (append q r) = append (append p q) r",
"tactic": "induction p with\n| nil => rfl\n| cons h p' ih =>\n dsimp only [append]\n rw [ih]"
},
{
"state_after": "no goals",
"state_before": "case nil\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v w x : V\nr : Walk G w x\nu✝ : V\nq : Walk G u✝ w\n⊢ append nil (append q r) = append (append nil q) r",
"tactic": "rfl"
},
{
"state_after": "case cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v w x : V\nr : Walk G w x\nu✝ v✝ w✝ : V\nh : Adj G u✝ v✝\np' : Walk G v✝ w✝\nih : ∀ (q : Walk G w✝ w), append p' (append q r) = append (append p' q) r\nq : Walk G w✝ w\n⊢ cons h (append p' (append q r)) = cons h (append (append p' q) r)",
"state_before": "case cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v w x : V\nr : Walk G w x\nu✝ v✝ w✝ : V\nh : Adj G u✝ v✝\np' : Walk G v✝ w✝\nih : ∀ (q : Walk G w✝ w), append p' (append q r) = append (append p' q) r\nq : Walk G w✝ w\n⊢ append (cons h p') (append q r) = append (append (cons h p') q) r",
"tactic": "dsimp only [append]"
},
{
"state_after": "no goals",
"state_before": "case cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v w x : V\nr : Walk G w x\nu✝ v✝ w✝ : V\nh : Adj G u✝ v✝\np' : Walk G v✝ w✝\nih : ∀ (q : Walk G w✝ w), append p' (append q r) = append (append p' q) r\nq : Walk G w✝ w\n⊢ cons h (append p' (append q r)) = cons h (append (append p' q) r)",
"tactic": "rw [ih]"
}
] | [
273,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
267,
1
] |
Mathlib/MeasureTheory/Decomposition/UnsignedHahn.lean | MeasureTheory.hahn_decomposition | [
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"tactic": "let d : Set α → ℝ := fun s => ((μ s).toNNReal : ℝ) - (ν s).toNNReal"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"tactic": "let c : Set ℝ := d '' { s | MeasurableSet s }"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"tactic": "let γ : ℝ := sSup c"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"tactic": "have hμ : ∀ s, μ s ≠ ∞ := measure_ne_top μ"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"tactic": "have hν : ∀ s, ν s ≠ ∞ := measure_ne_top ν"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"tactic": "have to_nnreal_μ : ∀ s, ((μ s).toNNReal : ℝ≥0∞) = μ s := fun s => ENNReal.coe_toNNReal <| hμ _"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"tactic": "have to_nnreal_ν : ∀ s, ((ν s).toNNReal : ℝ≥0∞) = ν s := fun s => ENNReal.coe_toNNReal <| hν _"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"tactic": "have d_split : ∀ s t, MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t) := by\n intro s t _hs ht\n dsimp only\n rw [← measure_inter_add_diff s ht, ← measure_inter_add_diff s ht,\n ENNReal.toNNReal_add (hμ _) (hμ _), ENNReal.toNNReal_add (hν _) (hν _), NNReal.coe_add,\n NNReal.coe_add]\n simp only [sub_eq_add_neg, neg_add]\n abel"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"tactic": "have d_Union :\n ∀ s : ℕ → Set α, Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ n, s n))) := by\n intro s hm\n refine' Tendsto.sub _ _ <;>\n refine' NNReal.tendsto_coe.2 <| (ENNReal.tendsto_toNNReal _).comp <| tendsto_measure_iUnion hm\n exact hμ _\n exact hν _"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"tactic": "have d_Inter :\n ∀ s : ℕ → Set α,\n (∀ n, MeasurableSet (s n)) →\n (∀ n m, n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ n, s n))) := by\n intro s hs hm\n refine' Tendsto.sub _ _ <;>\n refine'\n NNReal.tendsto_coe.2 <|\n (ENNReal.tendsto_toNNReal <| _).comp <| tendsto_measure_iInter hs hm _\n exacts [hμ _, ⟨0, hμ _⟩, hν _, ⟨0, hν _⟩]"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"tactic": "have bdd_c : BddAbove c := by\n use (μ univ).toNNReal\n rintro r ⟨s, _hs, rfl⟩\n refine' le_trans (sub_le_self _ <| NNReal.coe_nonneg _) _\n rw [NNReal.coe_le_coe, ← ENNReal.coe_le_coe, to_nnreal_μ, to_nnreal_μ]\n exact measure_mono (subset_univ _)"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"tactic": "have c_nonempty : c.Nonempty := Nonempty.image _ ⟨_, MeasurableSet.empty⟩"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"tactic": "have d_le_γ : ∀ s, MeasurableSet s → d s ≤ γ := fun s hs => le_csSup bdd_c ⟨s, hs, rfl⟩"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"tactic": "have : ∀ n : ℕ, ∃ s : Set α, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s := by\n intro n\n have : γ - (1 / 2) ^ n < γ := sub_lt_self γ (pow_pos (half_pos zero_lt_one) n)\n rcases exists_lt_of_lt_csSup c_nonempty this with ⟨r, ⟨s, hs, rfl⟩, hlt⟩\n exact ⟨s, hs, hlt⟩"
},
{
"state_after": "case intro\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"tactic": "rcases Classical.axiom_of_choice this with ⟨e, he⟩"
},
{
"state_after": "case intro\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"state_before": "case intro\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"tactic": "change ℕ → Set α at e"
},
{
"state_after": "case intro\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"state_before": "case intro\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"tactic": "have he₁ : ∀ n, MeasurableSet (e n) := fun n => (he n).1"
},
{
"state_after": "case intro\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"state_before": "case intro\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"tactic": "have he₂ : ∀ n, γ - (1 / 2) ^ n < d (e n) := fun n => (he n).2"
},
{
"state_after": "case intro\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"state_before": "case intro\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"tactic": "let f : ℕ → ℕ → Set α := fun n m => (Finset.Ico n (m + 1)).inf e"
},
{
"state_after": "case intro\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"state_before": "case intro\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"tactic": "have hf : ∀ n m, MeasurableSet (f n m) := by\n intro n m\n simp only [Finset.inf_eq_iInf]\n exact MeasurableSet.biInter (to_countable _) fun i _ => he₁ _"
},
{
"state_after": "case intro\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"state_before": "case intro\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"tactic": "have f_subset_f : ∀ {a b c d}, a ≤ b → c ≤ d → f a d ⊆ f b c := by\n intro a b c d hab hcd\n simp_rw [Finset.inf_eq_iInf, Finset.inf_eq_iInf]\n exact biInter_subset_biInter_left (Finset.Ico_subset_Ico hab <| Nat.succ_le_succ hcd)"
},
{
"state_after": "case intro\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"state_before": "case intro\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"tactic": "have f_succ : ∀ n m, n ≤ m → f n (m + 1) = f n m ∩ e (m + 1) := by\n intro n m hnm\n have : n ≤ m + 1 := le_of_lt (Nat.succ_le_succ hnm)\n simp_rw [Nat.Ico_succ_right_eq_insert_Ico this, Finset.inf_insert, Set.inter_comm]\n rfl"
},
{
"state_after": "case intro\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"state_before": "case intro\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"tactic": "let s := ⋃ m, ⋂ n, f m n"
},
{
"state_after": "case intro\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nγ_le_d_s : γ ≤ d s\nhs : MeasurableSet s\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"state_before": "case intro\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nγ_le_d_s : γ ≤ d s\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"tactic": "have hs : MeasurableSet s := MeasurableSet.iUnion fun n => MeasurableSet.iInter fun m => hf _ _"
},
{
"state_after": "case intro.refine'_1\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nγ_le_d_s : γ ≤ d s\nhs : MeasurableSet s\n⊢ ∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t\n\ncase intro.refine'_2\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nγ_le_d_s : γ ≤ d s\nhs : MeasurableSet s\n⊢ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"state_before": "case intro\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nγ_le_d_s : γ ≤ d s\nhs : MeasurableSet s\n⊢ ∃ s,\n MeasurableSet s ∧\n (∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t) ∧ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"tactic": "refine' ⟨s, hs, _, _⟩"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\ns t : Set α\n_hs : MeasurableSet s\nht : MeasurableSet t\n⊢ d s = d (s \\ t) + d (s ∩ t)",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\n⊢ ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)",
"tactic": "intro s t _hs ht"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\ns t : Set α\n_hs : MeasurableSet s\nht : MeasurableSet t\n⊢ ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) =\n ↑(ENNReal.toNNReal (↑↑μ (s \\ t))) - ↑(ENNReal.toNNReal (↑↑ν (s \\ t))) +\n (↑(ENNReal.toNNReal (↑↑μ (s ∩ t))) - ↑(ENNReal.toNNReal (↑↑ν (s ∩ t))))",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\ns t : Set α\n_hs : MeasurableSet s\nht : MeasurableSet t\n⊢ d s = d (s \\ t) + d (s ∩ t)",
"tactic": "dsimp only"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\ns t : Set α\n_hs : MeasurableSet s\nht : MeasurableSet t\n⊢ ↑(ENNReal.toNNReal (↑↑μ (s ∩ t))) + ↑(ENNReal.toNNReal (↑↑μ (s \\ t))) -\n (↑(ENNReal.toNNReal (↑↑ν (s ∩ t))) + ↑(ENNReal.toNNReal (↑↑ν (s \\ t)))) =\n ↑(ENNReal.toNNReal (↑↑μ (s \\ t))) - ↑(ENNReal.toNNReal (↑↑ν (s \\ t))) +\n (↑(ENNReal.toNNReal (↑↑μ (s ∩ t))) - ↑(ENNReal.toNNReal (↑↑ν (s ∩ t))))",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\ns t : Set α\n_hs : MeasurableSet s\nht : MeasurableSet t\n⊢ ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s)) =\n ↑(ENNReal.toNNReal (↑↑μ (s \\ t))) - ↑(ENNReal.toNNReal (↑↑ν (s \\ t))) +\n (↑(ENNReal.toNNReal (↑↑μ (s ∩ t))) - ↑(ENNReal.toNNReal (↑↑ν (s ∩ t))))",
"tactic": "rw [← measure_inter_add_diff s ht, ← measure_inter_add_diff s ht,\n ENNReal.toNNReal_add (hμ _) (hμ _), ENNReal.toNNReal_add (hν _) (hν _), NNReal.coe_add,\n NNReal.coe_add]"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\ns t : Set α\n_hs : MeasurableSet s\nht : MeasurableSet t\n⊢ ↑(ENNReal.toNNReal (↑↑μ (s ∩ t))) + ↑(ENNReal.toNNReal (↑↑μ (s \\ t))) +\n (-↑(ENNReal.toNNReal (↑↑ν (s ∩ t))) + -↑(ENNReal.toNNReal (↑↑ν (s \\ t)))) =\n ↑(ENNReal.toNNReal (↑↑μ (s \\ t))) + -↑(ENNReal.toNNReal (↑↑ν (s \\ t))) +\n (↑(ENNReal.toNNReal (↑↑μ (s ∩ t))) + -↑(ENNReal.toNNReal (↑↑ν (s ∩ t))))",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\ns t : Set α\n_hs : MeasurableSet s\nht : MeasurableSet t\n⊢ ↑(ENNReal.toNNReal (↑↑μ (s ∩ t))) + ↑(ENNReal.toNNReal (↑↑μ (s \\ t))) -\n (↑(ENNReal.toNNReal (↑↑ν (s ∩ t))) + ↑(ENNReal.toNNReal (↑↑ν (s \\ t)))) =\n ↑(ENNReal.toNNReal (↑↑μ (s \\ t))) - ↑(ENNReal.toNNReal (↑↑ν (s \\ t))) +\n (↑(ENNReal.toNNReal (↑↑μ (s ∩ t))) - ↑(ENNReal.toNNReal (↑↑ν (s ∩ t))))",
"tactic": "simp only [sub_eq_add_neg, neg_add]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\ns t : Set α\n_hs : MeasurableSet s\nht : MeasurableSet t\n⊢ ↑(ENNReal.toNNReal (↑↑μ (s ∩ t))) + ↑(ENNReal.toNNReal (↑↑μ (s \\ t))) +\n (-↑(ENNReal.toNNReal (↑↑ν (s ∩ t))) + -↑(ENNReal.toNNReal (↑↑ν (s \\ t)))) =\n ↑(ENNReal.toNNReal (↑↑μ (s \\ t))) + -↑(ENNReal.toNNReal (↑↑ν (s \\ t))) +\n (↑(ENNReal.toNNReal (↑↑μ (s ∩ t))) + -↑(ENNReal.toNNReal (↑↑ν (s ∩ t))))",
"tactic": "abel"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\ns : ℕ → Set α\nhm : Monotone s\n⊢ Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\n⊢ ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))",
"tactic": "intro s hm"
},
{
"state_after": "case refine'_1\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\ns : ℕ → Set α\nhm : Monotone s\n⊢ ↑↑μ (⋃ (n : ℕ), s n) ≠ ⊤\n\ncase refine'_2\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\ns : ℕ → Set α\nhm : Monotone s\n⊢ ↑↑ν (⋃ (n : ℕ), s n) ≠ ⊤",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\ns : ℕ → Set α\nhm : Monotone s\n⊢ Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))",
"tactic": "refine' Tendsto.sub _ _ <;>\n refine' NNReal.tendsto_coe.2 <| (ENNReal.tendsto_toNNReal _).comp <| tendsto_measure_iUnion hm"
},
{
"state_after": "case refine'_2\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\ns : ℕ → Set α\nhm : Monotone s\n⊢ ↑↑ν (⋃ (n : ℕ), s n) ≠ ⊤",
"state_before": "case refine'_1\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\ns : ℕ → Set α\nhm : Monotone s\n⊢ ↑↑μ (⋃ (n : ℕ), s n) ≠ ⊤\n\ncase refine'_2\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\ns : ℕ → Set α\nhm : Monotone s\n⊢ ↑↑ν (⋃ (n : ℕ), s n) ≠ ⊤",
"tactic": "exact hμ _"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\ns : ℕ → Set α\nhm : Monotone s\n⊢ ↑↑ν (⋃ (n : ℕ), s n) ≠ ⊤",
"tactic": "exact hν _"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n\n⊢ Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\n⊢ ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))",
"tactic": "intro s hs hm"
},
{
"state_after": "case refine'_1.refine'_1\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n\n⊢ ↑↑μ (⋂ (n : ℕ), s n) ≠ ⊤\n\ncase refine'_1.refine'_2\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n\n⊢ ∃ i, ↑↑μ (s i) ≠ ⊤\n\ncase refine'_2.refine'_1\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n\n⊢ ↑↑ν (⋂ (n : ℕ), s n) ≠ ⊤\n\ncase refine'_2.refine'_2\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n\n⊢ ∃ i, ↑↑ν (s i) ≠ ⊤",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n\n⊢ Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))",
"tactic": "refine' Tendsto.sub _ _ <;>\n refine'\n NNReal.tendsto_coe.2 <|\n (ENNReal.tendsto_toNNReal <| _).comp <| tendsto_measure_iInter hs hm _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.refine'_1\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n\n⊢ ↑↑μ (⋂ (n : ℕ), s n) ≠ ⊤\n\ncase refine'_1.refine'_2\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n\n⊢ ∃ i, ↑↑μ (s i) ≠ ⊤\n\ncase refine'_2.refine'_1\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n\n⊢ ↑↑ν (⋂ (n : ℕ), s n) ≠ ⊤\n\ncase refine'_2.refine'_2\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\ns : ℕ → Set α\nhs : ∀ (n : ℕ), MeasurableSet (s n)\nhm : ∀ (n m : ℕ), n ≤ m → s m ⊆ s n\n⊢ ∃ i, ↑↑ν (s i) ≠ ⊤",
"tactic": "exacts [hμ _, ⟨0, hμ _⟩, hν _, ⟨0, hν _⟩]"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\n⊢ ↑(ENNReal.toNNReal (↑↑μ univ)) ∈ upperBounds c",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\n⊢ BddAbove c",
"tactic": "use (μ univ).toNNReal"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\ns : Set α\n_hs : s ∈ {s | MeasurableSet s}\n⊢ d s ≤ ↑(ENNReal.toNNReal (↑↑μ univ))",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\n⊢ ↑(ENNReal.toNNReal (↑↑μ univ)) ∈ upperBounds c",
"tactic": "rintro r ⟨s, _hs, rfl⟩"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\ns : Set α\n_hs : s ∈ {s | MeasurableSet s}\n⊢ ↑(ENNReal.toNNReal (↑↑μ s)) ≤ ↑(ENNReal.toNNReal (↑↑μ univ))",
"state_before": "case intro.intro\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\ns : Set α\n_hs : s ∈ {s | MeasurableSet s}\n⊢ d s ≤ ↑(ENNReal.toNNReal (↑↑μ univ))",
"tactic": "refine' le_trans (sub_le_self _ <| NNReal.coe_nonneg _) _"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\ns : Set α\n_hs : s ∈ {s | MeasurableSet s}\n⊢ ↑↑μ s ≤ ↑↑μ univ",
"state_before": "case intro.intro\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\ns : Set α\n_hs : s ∈ {s | MeasurableSet s}\n⊢ ↑(ENNReal.toNNReal (↑↑μ s)) ≤ ↑(ENNReal.toNNReal (↑↑μ univ))",
"tactic": "rw [NNReal.coe_le_coe, ← ENNReal.coe_le_coe, to_nnreal_μ, to_nnreal_μ]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\ns : Set α\n_hs : s ∈ {s | MeasurableSet s}\n⊢ ↑↑μ s ≤ ↑↑μ univ",
"tactic": "exact measure_mono (subset_univ _)"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nn : ℕ\n⊢ ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\n⊢ ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s",
"tactic": "intro n"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nn : ℕ\nthis : γ - (1 / 2) ^ n < γ\n⊢ ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nn : ℕ\n⊢ ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s",
"tactic": "have : γ - (1 / 2) ^ n < γ := sub_lt_self γ (pow_pos (half_pos zero_lt_one) n)"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nn : ℕ\nthis : γ - (1 / 2) ^ n < γ\ns : Set α\nhs : s ∈ {s | MeasurableSet s}\nhlt : γ - (1 / 2) ^ n < d s\n⊢ ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nn : ℕ\nthis : γ - (1 / 2) ^ n < γ\n⊢ ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s",
"tactic": "rcases exists_lt_of_lt_csSup c_nonempty this with ⟨r, ⟨s, hs, rfl⟩, hlt⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nn : ℕ\nthis : γ - (1 / 2) ^ n < γ\ns : Set α\nhs : s ∈ {s | MeasurableSet s}\nhlt : γ - (1 / 2) ^ n < d s\n⊢ ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s",
"tactic": "exact ⟨s, hs, hlt⟩"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nn m : ℕ\n⊢ MeasurableSet (f n m)",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\n⊢ ∀ (n m : ℕ), MeasurableSet (f n m)",
"tactic": "intro n m"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nn m : ℕ\n⊢ MeasurableSet (⨅ (a : ℕ) (_ : a ∈ Finset.Ico n (m + 1)), e a)",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nn m : ℕ\n⊢ MeasurableSet (f n m)",
"tactic": "simp only [Finset.inf_eq_iInf]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nn m : ℕ\n⊢ MeasurableSet (⨅ (a : ℕ) (_ : a ∈ Finset.Ico n (m + 1)), e a)",
"tactic": "exact MeasurableSet.biInter (to_countable _) fun i _ => he₁ _"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd✝ : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc✝ : Set ℝ := d✝ '' {s | MeasurableSet s}\nγ : ℝ := sSup c✝\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d✝ s = d✝ (s \\ t) + d✝ (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d✝ (s n)) atTop (𝓝 (d✝ (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d✝ (s n)) atTop (𝓝 (d✝ (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c✝\nc_nonempty : Set.Nonempty c✝\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d✝ s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d✝ s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d✝ (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d✝ (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\na b c d : ℕ\nhab : a ≤ b\nhcd : c ≤ d\n⊢ f a d ⊆ f b c",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\n⊢ ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c",
"tactic": "intro a b c d hab hcd"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd✝ : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc✝ : Set ℝ := d✝ '' {s | MeasurableSet s}\nγ : ℝ := sSup c✝\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d✝ s = d✝ (s \\ t) + d✝ (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d✝ (s n)) atTop (𝓝 (d✝ (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d✝ (s n)) atTop (𝓝 (d✝ (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c✝\nc_nonempty : Set.Nonempty c✝\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d✝ s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d✝ s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d✝ (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d✝ (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\na b c d : ℕ\nhab : a ≤ b\nhcd : c ≤ d\n⊢ (⨅ (a_1 : ℕ) (_ : a_1 ∈ Finset.Ico a (d + 1)), e a_1) ⊆ ⨅ (a : ℕ) (_ : a ∈ Finset.Ico b (c + 1)), e a",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd✝ : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc✝ : Set ℝ := d✝ '' {s | MeasurableSet s}\nγ : ℝ := sSup c✝\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d✝ s = d✝ (s \\ t) + d✝ (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d✝ (s n)) atTop (𝓝 (d✝ (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d✝ (s n)) atTop (𝓝 (d✝ (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c✝\nc_nonempty : Set.Nonempty c✝\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d✝ s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d✝ s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d✝ (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d✝ (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\na b c d : ℕ\nhab : a ≤ b\nhcd : c ≤ d\n⊢ f a d ⊆ f b c",
"tactic": "simp_rw [Finset.inf_eq_iInf, Finset.inf_eq_iInf]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd✝ : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc✝ : Set ℝ := d✝ '' {s | MeasurableSet s}\nγ : ℝ := sSup c✝\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d✝ s = d✝ (s \\ t) + d✝ (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d✝ (s n)) atTop (𝓝 (d✝ (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d✝ (s n)) atTop (𝓝 (d✝ (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c✝\nc_nonempty : Set.Nonempty c✝\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d✝ s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d✝ s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d✝ (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d✝ (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\na b c d : ℕ\nhab : a ≤ b\nhcd : c ≤ d\n⊢ (⨅ (a_1 : ℕ) (_ : a_1 ∈ Finset.Ico a (d + 1)), e a_1) ⊆ ⨅ (a : ℕ) (_ : a ∈ Finset.Ico b (c + 1)), e a",
"tactic": "exact biInter_subset_biInter_left (Finset.Ico_subset_Ico hab <| Nat.succ_le_succ hcd)"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nn m : ℕ\nhnm : n ≤ m\n⊢ f n (m + 1) = f n m ∩ e (m + 1)",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\n⊢ ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)",
"tactic": "intro n m hnm"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nn m : ℕ\nhnm : n ≤ m\nthis : n ≤ m + 1\n⊢ f n (m + 1) = f n m ∩ e (m + 1)",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nn m : ℕ\nhnm : n ≤ m\n⊢ f n (m + 1) = f n m ∩ e (m + 1)",
"tactic": "have : n ≤ m + 1 := le_of_lt (Nat.succ_le_succ hnm)"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nn m : ℕ\nhnm : n ≤ m\nthis : n ≤ m + 1\n⊢ e (m + 1) ⊓ Finset.inf (Finset.Ico n (m + 1)) e = e (m + 1) ∩ Finset.inf (Finset.Ico n (m + 1)) e",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nn m : ℕ\nhnm : n ≤ m\nthis : n ≤ m + 1\n⊢ f n (m + 1) = f n m ∩ e (m + 1)",
"tactic": "simp_rw [Nat.Ico_succ_right_eq_insert_Ico this, Finset.inf_insert, Set.inter_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nn m : ℕ\nhnm : n ≤ m\nthis : n ≤ m + 1\n⊢ e (m + 1) ⊓ Finset.inf (Finset.Ico n (m + 1)) e = e (m + 1) ∩ Finset.inf (Finset.Ico n (m + 1)) e",
"tactic": "rfl"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn m : ℕ\nh : m ≤ n\n⊢ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\n⊢ ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)",
"tactic": "intro n m h"
},
{
"state_after": "case refine'_1\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn m : ℕ\nh : m ≤ n\n⊢ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ m ≤ d (f m m)\n\ncase refine'_2\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn m : ℕ\nh : m ≤ n\n⊢ ∀ (n : ℕ),\n m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1) ≤ d (f m (n + 1))",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn m : ℕ\nh : m ≤ n\n⊢ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)",
"tactic": "refine' Nat.le_induction _ _ n h"
},
{
"state_after": "case refine'_1\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn m : ℕ\nh : m ≤ n\nthis : γ - (1 / 2) ^ m < d (e m)\n⊢ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ m ≤ d (f m m)",
"state_before": "case refine'_1\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn m : ℕ\nh : m ≤ n\n⊢ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ m ≤ d (f m m)",
"tactic": "have := he₂ m"
},
{
"state_after": "case refine'_1\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn m : ℕ\nh : m ≤ n\nthis : γ - (1 / 2) ^ m < d (e m)\n⊢ sSup ((fun a => ↑(ENNReal.toNNReal (↑↑μ a)) - ↑(ENNReal.toNNReal (↑↑ν a))) '' {s | MeasurableSet s}) -\n 2 * (1 / 2) ^ m +\n (1 / 2) ^ m ≤\n ↑(ENNReal.toNNReal (↑↑μ (e m))) - ↑(ENNReal.toNNReal (↑↑ν (e m)))",
"state_before": "case refine'_1\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn m : ℕ\nh : m ≤ n\nthis : γ - (1 / 2) ^ m < d (e m)\n⊢ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ m ≤ d (f m m)",
"tactic": "simp_rw [Nat.Ico_succ_singleton, Finset.inf_singleton]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn m : ℕ\nh : m ≤ n\nthis : γ - (1 / 2) ^ m < d (e m)\n⊢ sSup ((fun a => ↑(ENNReal.toNNReal (↑↑μ a)) - ↑(ENNReal.toNNReal (↑↑ν a))) '' {s | MeasurableSet s}) -\n 2 * (1 / 2) ^ m +\n (1 / 2) ^ m ≤\n ↑(ENNReal.toNNReal (↑↑μ (e m))) - ↑(ENNReal.toNNReal (↑↑ν (e m)))",
"tactic": "linarith"
},
{
"state_after": "case refine'_2\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn✝ m : ℕ\nh : m ≤ n✝\nn : ℕ\nhmn : m ≤ n\nih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\n⊢ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1) ≤ d (f m (n + 1))",
"state_before": "case refine'_2\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn m : ℕ\nh : m ≤ n\n⊢ ∀ (n : ℕ),\n m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n) → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1) ≤ d (f m (n + 1))",
"tactic": "intro n(hmn : m ≤ n)ih"
},
{
"state_after": "case refine'_2\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn✝ m : ℕ\nh : m ≤ n✝\nn : ℕ\nhmn : m ≤ n\nih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\nthis : γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤ γ + d (f m (n + 1))\n⊢ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1) ≤ d (f m (n + 1))",
"state_before": "case refine'_2\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn✝ m : ℕ\nh : m ≤ n✝\nn : ℕ\nhmn : m ≤ n\nih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\n⊢ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1) ≤ d (f m (n + 1))",
"tactic": "have : γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤ γ + d (f m (n + 1)) := by\n calc\n γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤\n γ + (γ - 2 * (1 / 2) ^ m + ((1 / 2) ^ n - (1 / 2) ^ (n + 1))) := by\n refine' add_le_add_left (add_le_add_left _ _) γ\n simp only [pow_add, pow_one, le_sub_iff_add_le]\n linarith\n _ = γ - (1 / 2) ^ (n + 1) + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n) := by\n simp only [sub_eq_add_neg] ; abel\n _ ≤ d (e (n + 1)) + d (f m n) := (add_le_add (le_of_lt <| he₂ _) ih)\n _ ≤ d (e (n + 1)) + d (f m n \\ e (n + 1)) + d (f m (n + 1)) := by\n rw [f_succ _ _ hmn, d_split (f m n) (e (n + 1)) (hf _ _) (he₁ _), add_assoc]\n _ = d (e (n + 1) ∪ f m n) + d (f m (n + 1)) := by\n rw [d_split (e (n + 1) ∪ f m n) (e (n + 1)), union_diff_left, union_inter_cancel_left]\n abel\n exact (he₁ _).union (hf _ _)\n exact he₁ _\n _ ≤ γ + d (f m (n + 1)) := add_le_add_right (d_le_γ _ <| (he₁ _).union (hf _ _)) _"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn✝ m : ℕ\nh : m ≤ n✝\nn : ℕ\nhmn : m ≤ n\nih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\nthis : γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤ γ + d (f m (n + 1))\n⊢ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1) ≤ d (f m (n + 1))",
"tactic": "exact (add_le_add_iff_left γ).1 this"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn✝ m : ℕ\nh : m ≤ n✝\nn : ℕ\nhmn : m ≤ n\nih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\n⊢ γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤ γ + d (f m (n + 1))",
"tactic": "calc\n γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤\n γ + (γ - 2 * (1 / 2) ^ m + ((1 / 2) ^ n - (1 / 2) ^ (n + 1))) := by\n refine' add_le_add_left (add_le_add_left _ _) γ\n simp only [pow_add, pow_one, le_sub_iff_add_le]\n linarith\n _ = γ - (1 / 2) ^ (n + 1) + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n) := by\n simp only [sub_eq_add_neg] ; abel\n _ ≤ d (e (n + 1)) + d (f m n) := (add_le_add (le_of_lt <| he₂ _) ih)\n _ ≤ d (e (n + 1)) + d (f m n \\ e (n + 1)) + d (f m (n + 1)) := by\n rw [f_succ _ _ hmn, d_split (f m n) (e (n + 1)) (hf _ _) (he₁ _), add_assoc]\n _ = d (e (n + 1) ∪ f m n) + d (f m (n + 1)) := by\n rw [d_split (e (n + 1) ∪ f m n) (e (n + 1)), union_diff_left, union_inter_cancel_left]\n abel\n exact (he₁ _).union (hf _ _)\n exact he₁ _\n _ ≤ γ + d (f m (n + 1)) := add_le_add_right (d_le_γ _ <| (he₁ _).union (hf _ _)) _"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn✝ m : ℕ\nh : m ≤ n✝\nn : ℕ\nhmn : m ≤ n\nih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\n⊢ (1 / 2) ^ (n + 1) ≤ (1 / 2) ^ n - (1 / 2) ^ (n + 1)",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn✝ m : ℕ\nh : m ≤ n✝\nn : ℕ\nhmn : m ≤ n\nih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\n⊢ γ + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ (n + 1)) ≤ γ + (γ - 2 * (1 / 2) ^ m + ((1 / 2) ^ n - (1 / 2) ^ (n + 1)))",
"tactic": "refine' add_le_add_left (add_le_add_left _ _) γ"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn✝ m : ℕ\nh : m ≤ n✝\nn : ℕ\nhmn : m ≤ n\nih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\n⊢ (1 / 2) ^ n * (1 / 2) + (1 / 2) ^ n * (1 / 2) ≤ (1 / 2) ^ n",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn✝ m : ℕ\nh : m ≤ n✝\nn : ℕ\nhmn : m ≤ n\nih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\n⊢ (1 / 2) ^ (n + 1) ≤ (1 / 2) ^ n - (1 / 2) ^ (n + 1)",
"tactic": "simp only [pow_add, pow_one, le_sub_iff_add_le]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn✝ m : ℕ\nh : m ≤ n✝\nn : ℕ\nhmn : m ≤ n\nih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\n⊢ (1 / 2) ^ n * (1 / 2) + (1 / 2) ^ n * (1 / 2) ≤ (1 / 2) ^ n",
"tactic": "linarith"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn✝ m : ℕ\nh : m ≤ n✝\nn : ℕ\nhmn : m ≤ n\nih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\n⊢ sSup ((fun a => ↑(ENNReal.toNNReal (↑↑μ a)) + -↑(ENNReal.toNNReal (↑↑ν a))) '' {s | MeasurableSet s}) +\n (sSup ((fun a => ↑(ENNReal.toNNReal (↑↑μ a)) + -↑(ENNReal.toNNReal (↑↑ν a))) '' {s | MeasurableSet s}) +\n -(2 * (1 / 2) ^ m) +\n ((1 / 2) ^ n + -(1 / 2) ^ (n + 1))) =\n sSup ((fun a => ↑(ENNReal.toNNReal (↑↑μ a)) + -↑(ENNReal.toNNReal (↑↑ν a))) '' {s | MeasurableSet s}) +\n -(1 / 2) ^ (n + 1) +\n (sSup ((fun a => ↑(ENNReal.toNNReal (↑↑μ a)) + -↑(ENNReal.toNNReal (↑↑ν a))) '' {s | MeasurableSet s}) +\n -(2 * (1 / 2) ^ m) +\n (1 / 2) ^ n)",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn✝ m : ℕ\nh : m ≤ n✝\nn : ℕ\nhmn : m ≤ n\nih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\n⊢ γ + (γ - 2 * (1 / 2) ^ m + ((1 / 2) ^ n - (1 / 2) ^ (n + 1))) =\n γ - (1 / 2) ^ (n + 1) + (γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n)",
"tactic": "simp only [sub_eq_add_neg]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn✝ m : ℕ\nh : m ≤ n✝\nn : ℕ\nhmn : m ≤ n\nih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\n⊢ sSup ((fun a => ↑(ENNReal.toNNReal (↑↑μ a)) + -↑(ENNReal.toNNReal (↑↑ν a))) '' {s | MeasurableSet s}) +\n (sSup ((fun a => ↑(ENNReal.toNNReal (↑↑μ a)) + -↑(ENNReal.toNNReal (↑↑ν a))) '' {s | MeasurableSet s}) +\n -(2 * (1 / 2) ^ m) +\n ((1 / 2) ^ n + -(1 / 2) ^ (n + 1))) =\n sSup ((fun a => ↑(ENNReal.toNNReal (↑↑μ a)) + -↑(ENNReal.toNNReal (↑↑ν a))) '' {s | MeasurableSet s}) +\n -(1 / 2) ^ (n + 1) +\n (sSup ((fun a => ↑(ENNReal.toNNReal (↑↑μ a)) + -↑(ENNReal.toNNReal (↑↑ν a))) '' {s | MeasurableSet s}) +\n -(2 * (1 / 2) ^ m) +\n (1 / 2) ^ n)",
"tactic": "abel"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn✝ m : ℕ\nh : m ≤ n✝\nn : ℕ\nhmn : m ≤ n\nih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\n⊢ d (e (n + 1)) + d (f m n) ≤ d (e (n + 1)) + d (f m n \\ e (n + 1)) + d (f m (n + 1))",
"tactic": "rw [f_succ _ _ hmn, d_split (f m n) (e (n + 1)) (hf _ _) (he₁ _), add_assoc]"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn✝ m : ℕ\nh : m ≤ n✝\nn : ℕ\nhmn : m ≤ n\nih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\n⊢ d (e (n + 1)) + d (f m n \\ e (n + 1)) + d (f m (n + 1)) = d (f m n \\ e (n + 1)) + d (e (n + 1)) + d (f m (n + 1))\n\ncase a\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn✝ m : ℕ\nh : m ≤ n✝\nn : ℕ\nhmn : m ≤ n\nih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\n⊢ MeasurableSet (e (n + 1) ∪ f m n)\n\ncase a\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn✝ m : ℕ\nh : m ≤ n✝\nn : ℕ\nhmn : m ≤ n\nih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\n⊢ MeasurableSet (e (n + 1))",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn✝ m : ℕ\nh : m ≤ n✝\nn : ℕ\nhmn : m ≤ n\nih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\n⊢ d (e (n + 1)) + d (f m n \\ e (n + 1)) + d (f m (n + 1)) = d (e (n + 1) ∪ f m n) + d (f m (n + 1))",
"tactic": "rw [d_split (e (n + 1) ∪ f m n) (e (n + 1)), union_diff_left, union_inter_cancel_left]"
},
{
"state_after": "case a\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn✝ m : ℕ\nh : m ≤ n✝\nn : ℕ\nhmn : m ≤ n\nih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\n⊢ MeasurableSet (e (n + 1) ∪ f m n)\n\ncase a\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn✝ m : ℕ\nh : m ≤ n✝\nn : ℕ\nhmn : m ≤ n\nih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\n⊢ MeasurableSet (e (n + 1))",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn✝ m : ℕ\nh : m ≤ n✝\nn : ℕ\nhmn : m ≤ n\nih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\n⊢ d (e (n + 1)) + d (f m n \\ e (n + 1)) + d (f m (n + 1)) = d (f m n \\ e (n + 1)) + d (e (n + 1)) + d (f m (n + 1))\n\ncase a\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn✝ m : ℕ\nh : m ≤ n✝\nn : ℕ\nhmn : m ≤ n\nih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\n⊢ MeasurableSet (e (n + 1) ∪ f m n)\n\ncase a\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn✝ m : ℕ\nh : m ≤ n✝\nn : ℕ\nhmn : m ≤ n\nih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\n⊢ MeasurableSet (e (n + 1))",
"tactic": "abel"
},
{
"state_after": "case a\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn✝ m : ℕ\nh : m ≤ n✝\nn : ℕ\nhmn : m ≤ n\nih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\n⊢ MeasurableSet (e (n + 1))",
"state_before": "case a\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn✝ m : ℕ\nh : m ≤ n✝\nn : ℕ\nhmn : m ≤ n\nih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\n⊢ MeasurableSet (e (n + 1) ∪ f m n)\n\ncase a\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn✝ m : ℕ\nh : m ≤ n✝\nn : ℕ\nhmn : m ≤ n\nih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\n⊢ MeasurableSet (e (n + 1))",
"tactic": "exact (he₁ _).union (hf _ _)"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nn✝ m : ℕ\nh : m ≤ n✝\nn : ℕ\nhmn : m ≤ n\nih : γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\n⊢ MeasurableSet (e (n + 1))",
"tactic": "exact he₁ _"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nhγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)\n⊢ γ ≤ d s",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\n⊢ γ ≤ d s",
"tactic": "have hγ : Tendsto (fun m : ℕ => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ) := by\n suffices Tendsto (fun m : ℕ => γ - 2 * (1 / 2) ^ m) atTop (𝓝 (γ - 2 * 0)) by\n simpa only [MulZeroClass.mul_zero, tsub_zero]\n exact\n tendsto_const_nhds.sub <|\n tendsto_const_nhds.mul <|\n tendsto_pow_atTop_nhds_0_of_lt_1 (le_of_lt <| half_pos <| zero_lt_one)\n (half_lt_self zero_lt_one)"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nhγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)\nhd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))\n⊢ γ ≤ d s",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nhγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)\n⊢ γ ≤ d s",
"tactic": "have hd : Tendsto (fun m => d (⋂ n, f m n)) atTop (𝓝 (d (⋃ m, ⋂ n, f m n))) := by\n refine' d_Union _ _\n exact fun n m hnm =>\n subset_iInter fun i => Subset.trans (iInter_subset (f n) i) <| f_subset_f hnm <| le_rfl"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nhγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)\nhd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))\nm : ℕ\n⊢ γ - 2 * (1 / 2) ^ m ≤ d (⋂ (n : ℕ), f m n)",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nhγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)\nhd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))\n⊢ γ ≤ d s",
"tactic": "refine' le_of_tendsto_of_tendsto' hγ hd fun m => _"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nhγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)\nhd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))\nm : ℕ\nthis : Tendsto (fun n => d (f m n)) atTop (𝓝 (d (⋂ (n : ℕ), f m n)))\nn : ℕ\nhmn : n ≥ m\n⊢ γ - 2 * (1 / 2) ^ m ≤ d (f m n)",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nhγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)\nhd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))\nm : ℕ\nthis : Tendsto (fun n => d (f m n)) atTop (𝓝 (d (⋂ (n : ℕ), f m n)))\n⊢ γ - 2 * (1 / 2) ^ m ≤ d (⋂ (n : ℕ), f m n)",
"tactic": "refine' ge_of_tendsto this (eventually_atTop.2 ⟨m, fun n hmn => _⟩)"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nhγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)\nhd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))\nm : ℕ\nthis : Tendsto (fun n => d (f m n)) atTop (𝓝 (d (⋂ (n : ℕ), f m n)))\nn : ℕ\nhmn : n ≥ m\n⊢ γ - 2 * (1 / 2) ^ m ≤ d (f m n)",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nhγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)\nhd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))\nm : ℕ\nthis : Tendsto (fun n => d (f m n)) atTop (𝓝 (d (⋂ (n : ℕ), f m n)))\nn : ℕ\nhmn : n ≥ m\n⊢ γ - 2 * (1 / 2) ^ m ≤ d (f m n)",
"tactic": "change γ - 2 * (1 / 2) ^ m ≤ d (f m n)"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nhγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)\nhd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))\nm : ℕ\nthis : Tendsto (fun n => d (f m n)) atTop (𝓝 (d (⋂ (n : ℕ), f m n)))\nn : ℕ\nhmn : n ≥ m\n⊢ γ - 2 * (1 / 2) ^ m ≤ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nhγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)\nhd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))\nm : ℕ\nthis : Tendsto (fun n => d (f m n)) atTop (𝓝 (d (⋂ (n : ℕ), f m n)))\nn : ℕ\nhmn : n ≥ m\n⊢ γ - 2 * (1 / 2) ^ m ≤ d (f m n)",
"tactic": "refine' le_trans _ (le_d_f _ _ hmn)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nhγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)\nhd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))\nm : ℕ\nthis : Tendsto (fun n => d (f m n)) atTop (𝓝 (d (⋂ (n : ℕ), f m n)))\nn : ℕ\nhmn : n ≥ m\n⊢ γ - 2 * (1 / 2) ^ m ≤ γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n",
"tactic": "exact le_add_of_le_of_nonneg le_rfl (pow_nonneg (le_of_lt <| half_pos <| zero_lt_one) _)"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\n⊢ Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 (γ - 2 * 0))",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\n⊢ Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)",
"tactic": "suffices Tendsto (fun m : ℕ => γ - 2 * (1 / 2) ^ m) atTop (𝓝 (γ - 2 * 0)) by\n simpa only [MulZeroClass.mul_zero, tsub_zero]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\n⊢ Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 (γ - 2 * 0))",
"tactic": "exact\n tendsto_const_nhds.sub <|\n tendsto_const_nhds.mul <|\n tendsto_pow_atTop_nhds_0_of_lt_1 (le_of_lt <| half_pos <| zero_lt_one)\n (half_lt_self zero_lt_one)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nthis : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 (γ - 2 * 0))\n⊢ Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)",
"tactic": "simpa only [MulZeroClass.mul_zero, tsub_zero]"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nhγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)\n⊢ Monotone fun m => ⋂ (n : ℕ), f m n",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nhγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)\n⊢ Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))",
"tactic": "refine' d_Union _ _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nhγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)\n⊢ Monotone fun m => ⋂ (n : ℕ), f m n",
"tactic": "exact fun n m hnm =>\n subset_iInter fun i => Subset.trans (iInter_subset (f n) i) <| f_subset_f hnm <| le_rfl"
},
{
"state_after": "case refine'_1\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nhγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)\nhd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))\nm : ℕ\n⊢ ∀ (n : ℕ), MeasurableSet (f m n)\n\ncase refine'_2\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nhγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)\nhd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))\nm : ℕ\n⊢ ∀ (n m_1 : ℕ), n ≤ m_1 → f m m_1 ⊆ f m n",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nhγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)\nhd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))\nm : ℕ\n⊢ Tendsto (fun n => d (f m n)) atTop (𝓝 (d (⋂ (n : ℕ), f m n)))",
"tactic": "refine' d_Inter _ _ _"
},
{
"state_after": "case refine'_1\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nhγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)\nhd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))\nm n : ℕ\n⊢ MeasurableSet (f m n)",
"state_before": "case refine'_1\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nhγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)\nhd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))\nm : ℕ\n⊢ ∀ (n : ℕ), MeasurableSet (f m n)",
"tactic": "intro n"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nhγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)\nhd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))\nm n : ℕ\n⊢ MeasurableSet (f m n)",
"tactic": "exact hf _ _"
},
{
"state_after": "case refine'_2\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nhγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)\nhd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))\nm✝ n m : ℕ\nhnm : n ≤ m\n⊢ f m✝ m ⊆ f m✝ n",
"state_before": "case refine'_2\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nhγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)\nhd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))\nm : ℕ\n⊢ ∀ (n m_1 : ℕ), n ≤ m_1 → f m m_1 ⊆ f m n",
"tactic": "intro n m hnm"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nhγ : Tendsto (fun m => γ - 2 * (1 / 2) ^ m) atTop (𝓝 γ)\nhd : Tendsto (fun m => d (⋂ (n : ℕ), f m n)) atTop (𝓝 (d (⋃ (m : ℕ), ⋂ (n : ℕ), f m n)))\nm✝ n m : ℕ\nhnm : n ≤ m\n⊢ f m✝ m ⊆ f m✝ n",
"tactic": "exact f_subset_f le_rfl hnm"
},
{
"state_after": "case intro.refine'_1\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nγ_le_d_s : γ ≤ d s\nhs : MeasurableSet s\nt : Set α\nht : MeasurableSet t\nhts : t ⊆ s\n⊢ ↑↑ν t ≤ ↑↑μ t",
"state_before": "case intro.refine'_1\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nγ_le_d_s : γ ≤ d s\nhs : MeasurableSet s\n⊢ ∀ (t : Set α), MeasurableSet t → t ⊆ s → ↑↑ν t ≤ ↑↑μ t",
"tactic": "intro t ht hts"
},
{
"state_after": "case intro.refine'_1\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nγ_le_d_s : γ ≤ d s\nhs : MeasurableSet s\nt : Set α\nht : MeasurableSet t\nhts : t ⊆ s\nthis : 0 ≤ d t\n⊢ ↑↑ν t ≤ ↑↑μ t",
"state_before": "case intro.refine'_1\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nγ_le_d_s : γ ≤ d s\nhs : MeasurableSet s\nt : Set α\nht : MeasurableSet t\nhts : t ⊆ s\n⊢ ↑↑ν t ≤ ↑↑μ t",
"tactic": "have : 0 ≤ d t :=\n (add_le_add_iff_left γ).1 <|\n calc\n γ + 0 ≤ d s := by rw [add_zero] ; exact γ_le_d_s\n _ = d (s \\ t) + d t := by rw [d_split _ _ hs ht, inter_eq_self_of_subset_right hts]\n _ ≤ γ + d t := add_le_add (d_le_γ _ (hs.diff ht)) le_rfl"
},
{
"state_after": "case intro.refine'_1\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nγ_le_d_s : γ ≤ d s\nhs : MeasurableSet s\nt : Set α\nht : MeasurableSet t\nhts : t ⊆ s\nthis : 0 ≤ d t\n⊢ ↑(ENNReal.toNNReal (↑↑ν t)) ≤ ↑(ENNReal.toNNReal (↑↑μ t))",
"state_before": "case intro.refine'_1\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nγ_le_d_s : γ ≤ d s\nhs : MeasurableSet s\nt : Set α\nht : MeasurableSet t\nhts : t ⊆ s\nthis : 0 ≤ d t\n⊢ ↑↑ν t ≤ ↑↑μ t",
"tactic": "rw [← to_nnreal_μ, ← to_nnreal_ν, ENNReal.coe_le_coe, ← NNReal.coe_le_coe]"
},
{
"state_after": "no goals",
"state_before": "case intro.refine'_1\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nγ_le_d_s : γ ≤ d s\nhs : MeasurableSet s\nt : Set α\nht : MeasurableSet t\nhts : t ⊆ s\nthis : 0 ≤ d t\n⊢ ↑(ENNReal.toNNReal (↑↑ν t)) ≤ ↑(ENNReal.toNNReal (↑↑μ t))",
"tactic": "simpa only [le_sub_iff_add_le, zero_add] using this"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nγ_le_d_s : γ ≤ d s\nhs : MeasurableSet s\nt : Set α\nht : MeasurableSet t\nhts : t ⊆ s\n⊢ γ ≤ d s",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nγ_le_d_s : γ ≤ d s\nhs : MeasurableSet s\nt : Set α\nht : MeasurableSet t\nhts : t ⊆ s\n⊢ γ + 0 ≤ d s",
"tactic": "rw [add_zero]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nγ_le_d_s : γ ≤ d s\nhs : MeasurableSet s\nt : Set α\nht : MeasurableSet t\nhts : t ⊆ s\n⊢ γ ≤ d s",
"tactic": "exact γ_le_d_s"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nγ_le_d_s : γ ≤ d s\nhs : MeasurableSet s\nt : Set α\nht : MeasurableSet t\nhts : t ⊆ s\n⊢ d s = d (s \\ t) + d t",
"tactic": "rw [d_split _ _ hs ht, inter_eq_self_of_subset_right hts]"
},
{
"state_after": "case intro.refine'_2\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nγ_le_d_s : γ ≤ d s\nhs : MeasurableSet s\nt : Set α\nht : MeasurableSet t\nhts : t ⊆ sᶜ\n⊢ ↑↑μ t ≤ ↑↑ν t",
"state_before": "case intro.refine'_2\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nγ_le_d_s : γ ≤ d s\nhs : MeasurableSet s\n⊢ ∀ (t : Set α), MeasurableSet t → t ⊆ sᶜ → ↑↑μ t ≤ ↑↑ν t",
"tactic": "intro t ht hts"
},
{
"state_after": "case intro.refine'_2\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nγ_le_d_s : γ ≤ d s\nhs : MeasurableSet s\nt : Set α\nht : MeasurableSet t\nhts : t ⊆ sᶜ\nthis : d t ≤ 0\n⊢ ↑↑μ t ≤ ↑↑ν t",
"state_before": "case intro.refine'_2\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nγ_le_d_s : γ ≤ d s\nhs : MeasurableSet s\nt : Set α\nht : MeasurableSet t\nhts : t ⊆ sᶜ\n⊢ ↑↑μ t ≤ ↑↑ν t",
"tactic": "have : d t ≤ 0 :=\n (add_le_add_iff_left γ).1 <|\n calc\n γ + d t ≤ d s + d t := add_le_add γ_le_d_s le_rfl\n _ = d (s ∪ t) := by\n rw [d_split _ _ (hs.union ht) ht, union_diff_right, union_inter_cancel_right,\n (subset_compl_iff_disjoint_left.1 hts).sdiff_eq_left]\n _ ≤ γ + 0 := by rw [add_zero] ; exact d_le_γ _ (hs.union ht)"
},
{
"state_after": "case intro.refine'_2\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nγ_le_d_s : γ ≤ d s\nhs : MeasurableSet s\nt : Set α\nht : MeasurableSet t\nhts : t ⊆ sᶜ\nthis : d t ≤ 0\n⊢ ↑(ENNReal.toNNReal (↑↑μ t)) ≤ ↑(ENNReal.toNNReal (↑↑ν t))",
"state_before": "case intro.refine'_2\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nγ_le_d_s : γ ≤ d s\nhs : MeasurableSet s\nt : Set α\nht : MeasurableSet t\nhts : t ⊆ sᶜ\nthis : d t ≤ 0\n⊢ ↑↑μ t ≤ ↑↑ν t",
"tactic": "rw [← to_nnreal_μ, ← to_nnreal_ν, ENNReal.coe_le_coe, ← NNReal.coe_le_coe]"
},
{
"state_after": "no goals",
"state_before": "case intro.refine'_2\nα : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis✝ : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nγ_le_d_s : γ ≤ d s\nhs : MeasurableSet s\nt : Set α\nht : MeasurableSet t\nhts : t ⊆ sᶜ\nthis : d t ≤ 0\n⊢ ↑(ENNReal.toNNReal (↑↑μ t)) ≤ ↑(ENNReal.toNNReal (↑↑ν t))",
"tactic": "simpa only [sub_le_iff_le_add, zero_add] using this"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nγ_le_d_s : γ ≤ d s\nhs : MeasurableSet s\nt : Set α\nht : MeasurableSet t\nhts : t ⊆ sᶜ\n⊢ d s + d t = d (s ∪ t)",
"tactic": "rw [d_split _ _ (hs.union ht) ht, union_diff_right, union_inter_cancel_right,\n (subset_compl_iff_disjoint_left.1 hts).sdiff_eq_left]"
},
{
"state_after": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nγ_le_d_s : γ ≤ d s\nhs : MeasurableSet s\nt : Set α\nht : MeasurableSet t\nhts : t ⊆ sᶜ\n⊢ d (s ∪ t) ≤ γ",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nγ_le_d_s : γ ≤ d s\nhs : MeasurableSet s\nt : Set α\nht : MeasurableSet t\nhts : t ⊆ sᶜ\n⊢ d (s ∪ t) ≤ γ + 0",
"tactic": "rw [add_zero]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s => ↑(ENNReal.toNNReal (↑↑μ s)) - ↑(ENNReal.toNNReal (↑↑ν s))\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), ↑↑μ s ≠ ⊤\nhν : ∀ (s : Set α), ↑↑ν s ≠ ⊤\nto_nnreal_μ : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑μ s)) = ↑↑μ s\nto_nnreal_ν : ∀ (s : Set α), ↑(ENNReal.toNNReal (↑↑ν s)) = ↑↑ν s\nd_split : ∀ (s t : Set α), MeasurableSet s → MeasurableSet t → d s = d (s \\ t) + d (s ∩ t)\nd_Union : ∀ (s : ℕ → Set α), Monotone s → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋃ (n : ℕ), s n)))\nd_Inter :\n ∀ (s : ℕ → Set α),\n (∀ (n : ℕ), MeasurableSet (s n)) →\n (∀ (n m : ℕ), n ≤ m → s m ⊆ s n) → Tendsto (fun n => d (s n)) atTop (𝓝 (d (⋂ (n : ℕ), s n)))\nbdd_c : BddAbove c\nc_nonempty : Set.Nonempty c\nd_le_γ : ∀ (s : Set α), MeasurableSet s → d s ≤ γ\nthis : ∀ (n : ℕ), ∃ s, MeasurableSet s ∧ γ - (1 / 2) ^ n < d s\ne : ℕ → Set α\nhe : ∀ (x : ℕ), MeasurableSet (e x) ∧ γ - (1 / 2) ^ x < d (e x)\nhe₁ : ∀ (n : ℕ), MeasurableSet (e n)\nhe₂ : ∀ (n : ℕ), γ - (1 / 2) ^ n < d (e n)\nf : ℕ → ℕ → Set α := fun n m => Finset.inf (Finset.Ico n (m + 1)) e\nhf : ∀ (n m : ℕ), MeasurableSet (f n m)\nf_subset_f : ∀ {a b c d : ℕ}, a ≤ b → c ≤ d → f a d ⊆ f b c\nf_succ : ∀ (n m : ℕ), n ≤ m → f n (m + 1) = f n m ∩ e (m + 1)\nle_d_f : ∀ (n m : ℕ), m ≤ n → γ - 2 * (1 / 2) ^ m + (1 / 2) ^ n ≤ d (f m n)\ns : Set α := ⋃ (m : ℕ), ⋂ (n : ℕ), f m n\nγ_le_d_s : γ ≤ d s\nhs : MeasurableSet s\nt : Set α\nht : MeasurableSet t\nhts : t ⊆ sᶜ\n⊢ d (s ∪ t) ≤ γ",
"tactic": "exact d_le_γ _ (hs.union ht)"
}
] | [
181,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
39,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean | MeasureTheory.Lp.mul_meas_ge_le_pow_norm | [] | [
1794,
82
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1791,
1
] |
Mathlib/Topology/LocalHomeomorph.lean | LocalHomeomorph.preimage_frontier | [] | [
641,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
639,
1
] |
Mathlib/LinearAlgebra/QuadraticForm/Basic.lean | QuadraticForm.associated_rightInverse | [] | [
821,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
819,
1
] |
Mathlib/Topology/Algebra/Order/Floor.lean | tendsto_ceil_right' | [] | [
152,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
150,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean | PrimeSpectrum.vanishingIdeal_closure | [] | [
475,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
473,
1
] |
Mathlib/Order/Hom/Basic.lean | OrderIso.strictMono | [] | [
1006,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1005,
11
] |
Mathlib/Data/Fintype/Basic.lean | Finset.coe_compl | [] | [
178,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
177,
1
] |
Mathlib/Algebra/Algebra/Tower.lean | Submodule.smul_mem_span_smul_of_mem | [
{
"state_after": "R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid A\ninst✝³ : Module R S\ninst✝² : Module S A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\ns : Set S\nt : Set A\nk : S\nhks : k ∈ span R s\nx : A\nhx : x ∈ t\n⊢ 0 ∈ span R (s • t)",
"state_before": "R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid A\ninst✝³ : Module R S\ninst✝² : Module S A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\ns : Set S\nt : Set A\nk : S\nhks : k ∈ span R s\nx : A\nhx : x ∈ t\n⊢ 0 • x ∈ span R (s • t)",
"tactic": "rw [zero_smul]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid A\ninst✝³ : Module R S\ninst✝² : Module S A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\ns : Set S\nt : Set A\nk : S\nhks : k ∈ span R s\nx : A\nhx : x ∈ t\n⊢ 0 ∈ span R (s • t)",
"tactic": "exact zero_mem _"
},
{
"state_after": "R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid A\ninst✝³ : Module R S\ninst✝² : Module S A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\ns : Set S\nt : Set A\nk : S\nhks : k ∈ span R s\nx : A\nhx : x ∈ t\nc₁ c₂ : S\nih₁ : c₁ • x ∈ span R (s • t)\nih₂ : c₂ • x ∈ span R (s • t)\n⊢ c₁ • x + c₂ • x ∈ span R (s • t)",
"state_before": "R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid A\ninst✝³ : Module R S\ninst✝² : Module S A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\ns : Set S\nt : Set A\nk : S\nhks : k ∈ span R s\nx : A\nhx : x ∈ t\nc₁ c₂ : S\nih₁ : c₁ • x ∈ span R (s • t)\nih₂ : c₂ • x ∈ span R (s • t)\n⊢ (c₁ + c₂) • x ∈ span R (s • t)",
"tactic": "rw [add_smul]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid A\ninst✝³ : Module R S\ninst✝² : Module S A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\ns : Set S\nt : Set A\nk : S\nhks : k ∈ span R s\nx : A\nhx : x ∈ t\nc₁ c₂ : S\nih₁ : c₁ • x ∈ span R (s • t)\nih₂ : c₂ • x ∈ span R (s • t)\n⊢ c₁ • x + c₂ • x ∈ span R (s • t)",
"tactic": "exact add_mem ih₁ ih₂"
},
{
"state_after": "R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid A\ninst✝³ : Module R S\ninst✝² : Module S A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\ns : Set S\nt : Set A\nk : S\nhks : k ∈ span R s\nx : A\nhx : x ∈ t\nb : R\nc : S\nhc : c • x ∈ span R (s • t)\n⊢ b • c • x ∈ span R (s • t)",
"state_before": "R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid A\ninst✝³ : Module R S\ninst✝² : Module S A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\ns : Set S\nt : Set A\nk : S\nhks : k ∈ span R s\nx : A\nhx : x ∈ t\nb : R\nc : S\nhc : c • x ∈ span R (s • t)\n⊢ (b • c) • x ∈ span R (s • t)",
"tactic": "rw [IsScalarTower.smul_assoc]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nA : Type w\nB : Type u₁\nM : Type v₁\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid A\ninst✝³ : Module R S\ninst✝² : Module S A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\ns : Set S\nt : Set A\nk : S\nhks : k ∈ span R s\nx : A\nhx : x ∈ t\nb : R\nc : S\nhc : c • x ∈ span R (s • t)\n⊢ b • c • x ∈ span R (s • t)",
"tactic": "exact smul_mem _ _ hc"
}
] | [
286,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
281,
1
] |
Mathlib/GroupTheory/Submonoid/Operations.lean | Submonoid.map_sup_comap_of_surjective | [] | [
483,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
482,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean | Real.Angle.two_nsmul_neg_pi_div_two | [
{
"state_after": "no goals",
"state_before": "⊢ 2 • ↑(-π / 2) = ↑π",
"tactic": "rw [two_nsmul_coe_div_two, coe_neg, neg_coe_pi]"
}
] | [
152,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
151,
1
] |
Mathlib/Topology/Algebra/GroupCompletion.lean | UniformSpace.Completion.coe_neg | [] | [
89,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
88,
1
] |
Mathlib/Analysis/Convex/Strict.lean | strictConvex_Icc | [] | [
189,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
188,
1
] |
Mathlib/Algebra/Ring/Equiv.lean | RingEquiv.toOpposite_apply | [] | [
405,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
404,
1
] |
Mathlib/Order/Heyting/Basic.lean | disjoint_compl_compl_left_iff | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.163358\nα : Type u_1\nβ : Type ?u.163364\ninst✝ : HeytingAlgebra α\na b c : α\n⊢ Disjoint (aᶜᶜ) b ↔ Disjoint a b",
"tactic": "simp_rw [← le_compl_iff_disjoint_left, compl_compl_compl]"
}
] | [
917,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
916,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean | BoxIntegral.TaggedPrepartition.mem_filter | [] | [
112,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
111,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean | Subalgebra.one_mem | [] | [
129,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
128,
11
] |
Mathlib/Algebra/CubicDiscriminant.lean | Cubic.degree_of_b_ne_zero' | [] | [
328,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
327,
1
] |
Mathlib/SetTheory/Ordinal/Exponential.lean | Ordinal.mod_opow_log_lt_self | [
{
"state_after": "case inl\no : Ordinal\nho : o ≠ 0\n⊢ o % 0 ^ log 0 o < o\n\ncase inr\nb o : Ordinal\nho : o ≠ 0\nhb : b ≠ 0\n⊢ o % b ^ log b o < o",
"state_before": "b o : Ordinal\nho : o ≠ 0\n⊢ o % b ^ log b o < o",
"tactic": "rcases eq_or_ne b 0 with (rfl | hb)"
},
{
"state_after": "no goals",
"state_before": "case inl\no : Ordinal\nho : o ≠ 0\n⊢ o % 0 ^ log 0 o < o",
"tactic": "simpa using Ordinal.pos_iff_ne_zero.2 ho"
},
{
"state_after": "no goals",
"state_before": "case inr\nb o : Ordinal\nho : o ≠ 0\nhb : b ≠ 0\n⊢ o % b ^ log b o < o",
"tactic": "exact (mod_lt _ <| opow_ne_zero _ hb).trans_le (opow_log_le_self _ ho)"
}
] | [
382,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
379,
1
] |
Mathlib/GroupTheory/QuotientGroup.lean | QuotientGroup.mk_div | [] | [
174,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
173,
1
] |
Mathlib/LinearAlgebra/TensorProduct.lean | TensorProduct.lift.tmul | [] | [
481,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
480,
1
] |
Mathlib/Order/Filter/Ultrafilter.lean | Ultrafilter.diff_mem_iff | [] | [
141,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
140,
1
] |
Mathlib/Topology/Separation.lean | normal_separation | [] | [
1711,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1709,
1
] |
Mathlib/Data/Polynomial/Div.lean | Polynomial.divByMonic_eq_zero_iff | [
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝¹ : CommRing R\np q : R[X]\ninst✝ : Nontrivial R\nhq : Monic q\nh : p /ₘ q = 0\nthis : p %ₘ q + q * (p /ₘ q) = p\n⊢ degree p < degree q",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝¹ : CommRing R\np q : R[X]\ninst✝ : Nontrivial R\nhq : Monic q\nh : p /ₘ q = 0\n⊢ degree p < degree q",
"tactic": "have := modByMonic_add_div p hq"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝¹ : CommRing R\np q : R[X]\ninst✝ : Nontrivial R\nhq : Monic q\nh : p /ₘ q = 0\nthis : p %ₘ q + q * (p /ₘ q) = p\n⊢ degree p < degree q",
"tactic": "rwa [h, MulZeroClass.mul_zero, add_zero, modByMonic_eq_self_iff hq] at this"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝¹ : CommRing R\np q : R[X]\ninst✝ : Nontrivial R\nhq : Monic q\nh : degree p < degree q\nthis : ¬degree q ≤ degree p\n⊢ p /ₘ q = 0",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝¹ : CommRing R\np q : R[X]\ninst✝ : Nontrivial R\nhq : Monic q\nh : degree p < degree q\n⊢ p /ₘ q = 0",
"tactic": "have : ¬degree q ≤ degree p := not_le_of_gt h"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝¹ : CommRing R\np q : R[X]\ninst✝ : Nontrivial R\nhq : Monic q\nh : degree p < degree q\nthis : ¬degree q ≤ degree p\n⊢ (if h : Monic q then\n (if h_1 : degree q ≤ degree p ∧ p ≠ 0 then\n let z := ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q);\n let_fun _wf := (_ : degree (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) < degree p);\n let dm := divModByMonicAux (p - z * q) (_ : Monic q);\n (z + dm.fst, dm.snd)\n else (0, p)).fst\n else 0) =\n 0",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝¹ : CommRing R\np q : R[X]\ninst✝ : Nontrivial R\nhq : Monic q\nh : degree p < degree q\nthis : ¬degree q ≤ degree p\n⊢ p /ₘ q = 0",
"tactic": "unfold divByMonic divModByMonicAux"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝¹ : CommRing R\np q : R[X]\ninst✝ : Nontrivial R\nhq : Monic q\nh : degree p < degree q\nthis : ¬degree q ≤ degree p\n⊢ (if h : Monic q then\n (if degree q ≤ degree p ∧ ¬p = 0 then\n (↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) +\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).fst,\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).snd)\n else (0, p)).fst\n else 0) =\n 0",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝¹ : CommRing R\np q : R[X]\ninst✝ : Nontrivial R\nhq : Monic q\nh : degree p < degree q\nthis : ¬degree q ≤ degree p\n⊢ (if h : Monic q then\n (if h_1 : degree q ≤ degree p ∧ p ≠ 0 then\n let z := ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q);\n let_fun _wf := (_ : degree (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) < degree p);\n let dm := divModByMonicAux (p - z * q) (_ : Monic q);\n (z + dm.fst, dm.snd)\n else (0, p)).fst\n else 0) =\n 0",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝¹ : CommRing R\np q : R[X]\ninst✝ : Nontrivial R\nhq : Monic q\nh : degree p < degree q\nthis : ¬degree q ≤ degree p\n⊢ (if h : Monic q then\n (if degree q ≤ degree p ∧ ¬p = 0 then\n (↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) +\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).fst,\n (divModByMonicAux (p - ↑C (leadingCoeff p) * X ^ (natDegree p - natDegree q) * q) (_ : Monic q)).snd)\n else (0, p)).fst\n else 0) =\n 0",
"tactic": "rw [dif_pos hq, if_neg (mt And.left this)]"
}
] | [
262,
91
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
256,
1
] |
Mathlib/Computability/TMToPartrec.lean | Turing.ToPartrec.Code.case_eval | [
{
"state_after": "no goals",
"state_before": "f g : Code\n⊢ eval (case f g) = fun v => Nat.rec (eval f (List.tail v)) (fun y x => eval g (y :: List.tail v)) (List.headI v)",
"tactic": "simp [eval]"
}
] | [
163,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
161,
1
] |
Mathlib/Algebra/Symmetrized.lean | SymAlg.mul_def | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁴ : Add α\ninst✝³ : Mul α\ninst✝² : One α\ninst✝¹ : OfNat α 2\ninst✝ : Invertible 2\na b : αˢʸᵐ\n⊢ a * b = ↑sym (⅟2 * (↑unsym a * ↑unsym b + ↑unsym b * ↑unsym a))",
"tactic": "rfl"
}
] | [
204,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
203,
1
] |
Mathlib/Data/Finsupp/Defs.lean | Finsupp.coe_add | [] | [
975,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
974,
1
] |
Mathlib/Analysis/Convex/Quasiconvex.lean | ConvexOn.quasiconvexOn | [] | [
166,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
165,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean | Finset.Ico_diff_Ico_left | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.155748\nα : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : LocallyFiniteOrder α\na✝ b✝ a b c : α\n⊢ Ico a b \\ Ico a c = Ico (max a c) b",
"tactic": "cases le_total a c with\n| inl h =>\n ext x\n rw [mem_sdiff, mem_Ico, mem_Ico, mem_Ico, max_eq_right h, and_right_comm, not_and, not_lt]\n exact and_congr_left' ⟨fun hx => hx.2 hx.1, fun hx => ⟨h.trans hx, fun _ => hx⟩⟩\n| inr h => rw [Ico_eq_empty_of_le h, sdiff_empty, max_eq_left h]"
},
{
"state_after": "case inl.a\nι : Type ?u.155748\nα : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : LocallyFiniteOrder α\na✝ b✝ a b c : α\nh : a ≤ c\nx : α\n⊢ x ∈ Ico a b \\ Ico a c ↔ x ∈ Ico (max a c) b",
"state_before": "case inl\nι : Type ?u.155748\nα : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : LocallyFiniteOrder α\na✝ b✝ a b c : α\nh : a ≤ c\n⊢ Ico a b \\ Ico a c = Ico (max a c) b",
"tactic": "ext x"
},
{
"state_after": "case inl.a\nι : Type ?u.155748\nα : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : LocallyFiniteOrder α\na✝ b✝ a b c : α\nh : a ≤ c\nx : α\n⊢ (a ≤ x ∧ (a ≤ x → c ≤ x)) ∧ x < b ↔ c ≤ x ∧ x < b",
"state_before": "case inl.a\nι : Type ?u.155748\nα : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : LocallyFiniteOrder α\na✝ b✝ a b c : α\nh : a ≤ c\nx : α\n⊢ x ∈ Ico a b \\ Ico a c ↔ x ∈ Ico (max a c) b",
"tactic": "rw [mem_sdiff, mem_Ico, mem_Ico, mem_Ico, max_eq_right h, and_right_comm, not_and, not_lt]"
},
{
"state_after": "no goals",
"state_before": "case inl.a\nι : Type ?u.155748\nα : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : LocallyFiniteOrder α\na✝ b✝ a b c : α\nh : a ≤ c\nx : α\n⊢ (a ≤ x ∧ (a ≤ x → c ≤ x)) ∧ x < b ↔ c ≤ x ∧ x < b",
"tactic": "exact and_congr_left' ⟨fun hx => hx.2 hx.1, fun hx => ⟨h.trans hx, fun _ => hx⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case inr\nι : Type ?u.155748\nα : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : LocallyFiniteOrder α\na✝ b✝ a b c : α\nh : c ≤ a\n⊢ Ico a b \\ Ico a c = Ico (max a c) b",
"tactic": "rw [Ico_eq_empty_of_le h, sdiff_empty, max_eq_left h]"
}
] | [
821,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
815,
1
] |
Mathlib/Data/List/Count.lean | List.filter_eq' | [] | [
288,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
287,
1
] |
Mathlib/Analysis/SpecialFunctions/Stirling.lean | Stirling.stirlingSeq_pow_four_div_stirlingSeq_pow_two_eq | [
{
"state_after": "n : ℕ\nhn : n ≠ 0\nthis : 4 = 2 * 2\n⊢ stirlingSeq n ^ 4 / stirlingSeq (2 * n) ^ 2 * (↑n / (2 * ↑n + 1)) = Wallis.W n",
"state_before": "n : ℕ\nhn : n ≠ 0\n⊢ stirlingSeq n ^ 4 / stirlingSeq (2 * n) ^ 2 * (↑n / (2 * ↑n + 1)) = Wallis.W n",
"tactic": "have : 4 = 2 * 2 := by rfl"
},
{
"state_after": "n : ℕ\nhn : n ≠ 0\nthis : 4 = 2 * 2\n⊢ ((↑n ! / (Real.sqrt (2 * ↑n) * (↑n / exp 1) ^ n)) ^ 2) ^ 2 /\n (↑(2 * n)! / (Real.sqrt (2 * ↑(2 * n)) * (↑(2 * n) / exp 1) ^ (2 * n))) ^ 2 *\n (↑n / (2 * ↑n + 1)) =\n ↑(2 ^ (4 * n)) * ↑(n ! ^ 4) / (↑((2 * n)! ^ 2) * (2 * ↑n + 1))",
"state_before": "n : ℕ\nhn : n ≠ 0\nthis : 4 = 2 * 2\n⊢ stirlingSeq n ^ 4 / stirlingSeq (2 * n) ^ 2 * (↑n / (2 * ↑n + 1)) = Wallis.W n",
"tactic": "rw [stirlingSeq, this, pow_mul, stirlingSeq, Wallis.W_eq_factorial_ratio]"
},
{
"state_after": "n : ℕ\nhn : n ≠ 0\nthis : 4 = 2 * 2\n⊢ (↑n ! ^ 2) ^ 2 / ((Real.sqrt (2 * ↑n) ^ 2) ^ 2 * ((↑n ^ n / exp 1 ^ n) ^ 2) ^ 2) /\n (↑(2 * n)! ^ 2 / (Real.sqrt (2 * ↑(2 * n)) ^ 2 * (↑(2 * n) ^ (2 * n) / exp 1 ^ (2 * n)) ^ 2)) *\n (↑n / (2 * ↑n + 1)) =\n ↑(2 ^ (4 * n)) * ↑(n ! ^ 4) / (↑((2 * n)! ^ 2) * (2 * ↑n + 1))",
"state_before": "n : ℕ\nhn : n ≠ 0\nthis : 4 = 2 * 2\n⊢ ((↑n ! / (Real.sqrt (2 * ↑n) * (↑n / exp 1) ^ n)) ^ 2) ^ 2 /\n (↑(2 * n)! / (Real.sqrt (2 * ↑(2 * n)) * (↑(2 * n) / exp 1) ^ (2 * n))) ^ 2 *\n (↑n / (2 * ↑n + 1)) =\n ↑(2 ^ (4 * n)) * ↑(n ! ^ 4) / (↑((2 * n)! ^ 2) * (2 * ↑n + 1))",
"tactic": "simp_rw [div_pow, mul_pow]"
},
{
"state_after": "n : ℕ\nhn : n ≠ 0\nthis : 4 = 2 * 2\n⊢ (↑n ! ^ 2) ^ 2 / ((2 * ↑n) ^ 2 * ((↑n ^ n / exp 1 ^ n) ^ 2) ^ 2) /\n (↑(2 * n)! ^ 2 / (2 * ↑(2 * n) * (↑(2 * n) ^ (2 * n) / exp 1 ^ (2 * n)) ^ 2)) *\n (↑n / (2 * ↑n + 1)) =\n ↑(2 ^ (4 * n)) * ↑(n ! ^ 4) / (↑((2 * n)! ^ 2) * (2 * ↑n + 1))\n\nn : ℕ\nhn : n ≠ 0\nthis : 4 = 2 * 2\n⊢ 0 ≤ 2 * ↑(2 * n)\n\nn : ℕ\nhn : n ≠ 0\nthis : 4 = 2 * 2\n⊢ 0 ≤ 2 * ↑n",
"state_before": "n : ℕ\nhn : n ≠ 0\nthis : 4 = 2 * 2\n⊢ (↑n ! ^ 2) ^ 2 / ((Real.sqrt (2 * ↑n) ^ 2) ^ 2 * ((↑n ^ n / exp 1 ^ n) ^ 2) ^ 2) /\n (↑(2 * n)! ^ 2 / (Real.sqrt (2 * ↑(2 * n)) ^ 2 * (↑(2 * n) ^ (2 * n) / exp 1 ^ (2 * n)) ^ 2)) *\n (↑n / (2 * ↑n + 1)) =\n ↑(2 ^ (4 * n)) * ↑(n ! ^ 4) / (↑((2 * n)! ^ 2) * (2 * ↑n + 1))",
"tactic": "rw [sq_sqrt, sq_sqrt]"
},
{
"state_after": "n : ℕ\nhn : n ≠ 0\nthis : 4 = 2 * 2\n⊢ (↑n ! ^ 2) ^ 2 / ((2 * ↑n) ^ 2 * ((↑n ^ n / exp 1 ^ n) ^ 2) ^ 2) /\n (↑(2 * n)! ^ 2 / (2 * ↑(2 * n) * (↑(2 * n) ^ (2 * n) / exp 1 ^ (2 * n)) ^ 2)) *\n (↑n / (2 * ↑n + 1)) =\n ↑(2 ^ (4 * n)) * ↑(n ! ^ 4) / (↑((2 * n)! ^ 2) * (2 * ↑n + 1))",
"state_before": "n : ℕ\nhn : n ≠ 0\nthis : 4 = 2 * 2\n⊢ (↑n ! ^ 2) ^ 2 / ((2 * ↑n) ^ 2 * ((↑n ^ n / exp 1 ^ n) ^ 2) ^ 2) /\n (↑(2 * n)! ^ 2 / (2 * ↑(2 * n) * (↑(2 * n) ^ (2 * n) / exp 1 ^ (2 * n)) ^ 2)) *\n (↑n / (2 * ↑n + 1)) =\n ↑(2 ^ (4 * n)) * ↑(n ! ^ 4) / (↑((2 * n)! ^ 2) * (2 * ↑n + 1))\n\nn : ℕ\nhn : n ≠ 0\nthis : 4 = 2 * 2\n⊢ 0 ≤ 2 * ↑(2 * n)\n\nn : ℕ\nhn : n ≠ 0\nthis : 4 = 2 * 2\n⊢ 0 ≤ 2 * ↑n",
"tactic": "any_goals positivity"
},
{
"state_after": "n : ℕ\nhn : n ≠ 0\nthis✝ : 4 = 2 * 2\nthis : ↑n ≠ 0\n⊢ (↑n ! ^ 2) ^ 2 / ((2 * ↑n) ^ 2 * ((↑n ^ n / exp 1 ^ n) ^ 2) ^ 2) /\n (↑(2 * n)! ^ 2 / (2 * ↑(2 * n) * (↑(2 * n) ^ (2 * n) / exp 1 ^ (2 * n)) ^ 2)) *\n (↑n / (2 * ↑n + 1)) =\n ↑(2 ^ (4 * n)) * ↑(n ! ^ 4) / (↑((2 * n)! ^ 2) * (2 * ↑n + 1))",
"state_before": "n : ℕ\nhn : n ≠ 0\nthis : 4 = 2 * 2\n⊢ (↑n ! ^ 2) ^ 2 / ((2 * ↑n) ^ 2 * ((↑n ^ n / exp 1 ^ n) ^ 2) ^ 2) /\n (↑(2 * n)! ^ 2 / (2 * ↑(2 * n) * (↑(2 * n) ^ (2 * n) / exp 1 ^ (2 * n)) ^ 2)) *\n (↑n / (2 * ↑n + 1)) =\n ↑(2 ^ (4 * n)) * ↑(n ! ^ 4) / (↑((2 * n)! ^ 2) * (2 * ↑n + 1))",
"tactic": "have : (n : ℝ) ≠ 0 := cast_ne_zero.mpr hn"
},
{
"state_after": "n : ℕ\nhn : n ≠ 0\nthis✝¹ : 4 = 2 * 2\nthis✝ : ↑n ≠ 0\nthis : exp 1 ≠ 0\n⊢ (↑n ! ^ 2) ^ 2 / ((2 * ↑n) ^ 2 * ((↑n ^ n / exp 1 ^ n) ^ 2) ^ 2) /\n (↑(2 * n)! ^ 2 / (2 * ↑(2 * n) * (↑(2 * n) ^ (2 * n) / exp 1 ^ (2 * n)) ^ 2)) *\n (↑n / (2 * ↑n + 1)) =\n ↑(2 ^ (4 * n)) * ↑(n ! ^ 4) / (↑((2 * n)! ^ 2) * (2 * ↑n + 1))",
"state_before": "n : ℕ\nhn : n ≠ 0\nthis✝ : 4 = 2 * 2\nthis : ↑n ≠ 0\n⊢ (↑n ! ^ 2) ^ 2 / ((2 * ↑n) ^ 2 * ((↑n ^ n / exp 1 ^ n) ^ 2) ^ 2) /\n (↑(2 * n)! ^ 2 / (2 * ↑(2 * n) * (↑(2 * n) ^ (2 * n) / exp 1 ^ (2 * n)) ^ 2)) *\n (↑n / (2 * ↑n + 1)) =\n ↑(2 ^ (4 * n)) * ↑(n ! ^ 4) / (↑((2 * n)! ^ 2) * (2 * ↑n + 1))",
"tactic": "have : exp 1 ≠ 0 := exp_ne_zero 1"
},
{
"state_after": "n : ℕ\nhn : n ≠ 0\nthis✝² : 4 = 2 * 2\nthis✝¹ : ↑n ≠ 0\nthis✝ : exp 1 ≠ 0\nthis : ↑(2 * n)! ≠ 0\n⊢ (↑n ! ^ 2) ^ 2 / ((2 * ↑n) ^ 2 * ((↑n ^ n / exp 1 ^ n) ^ 2) ^ 2) /\n (↑(2 * n)! ^ 2 / (2 * ↑(2 * n) * (↑(2 * n) ^ (2 * n) / exp 1 ^ (2 * n)) ^ 2)) *\n (↑n / (2 * ↑n + 1)) =\n ↑(2 ^ (4 * n)) * ↑(n ! ^ 4) / (↑((2 * n)! ^ 2) * (2 * ↑n + 1))",
"state_before": "n : ℕ\nhn : n ≠ 0\nthis✝¹ : 4 = 2 * 2\nthis✝ : ↑n ≠ 0\nthis : exp 1 ≠ 0\n⊢ (↑n ! ^ 2) ^ 2 / ((2 * ↑n) ^ 2 * ((↑n ^ n / exp 1 ^ n) ^ 2) ^ 2) /\n (↑(2 * n)! ^ 2 / (2 * ↑(2 * n) * (↑(2 * n) ^ (2 * n) / exp 1 ^ (2 * n)) ^ 2)) *\n (↑n / (2 * ↑n + 1)) =\n ↑(2 ^ (4 * n)) * ↑(n ! ^ 4) / (↑((2 * n)! ^ 2) * (2 * ↑n + 1))",
"tactic": "have : ((2 * n)! : ℝ) ≠ 0 := cast_ne_zero.mpr (factorial_ne_zero (2 * n))"
},
{
"state_after": "n : ℕ\nhn : n ≠ 0\nthis✝³ : 4 = 2 * 2\nthis✝² : ↑n ≠ 0\nthis✝¹ : exp 1 ≠ 0\nthis✝ : ↑(2 * n)! ≠ 0\nthis : 2 * ↑n + 1 ≠ 0\n⊢ (↑n ! ^ 2) ^ 2 / ((2 * ↑n) ^ 2 * ((↑n ^ n / exp 1 ^ n) ^ 2) ^ 2) /\n (↑(2 * n)! ^ 2 / (2 * ↑(2 * n) * (↑(2 * n) ^ (2 * n) / exp 1 ^ (2 * n)) ^ 2)) *\n (↑n / (2 * ↑n + 1)) =\n ↑(2 ^ (4 * n)) * ↑(n ! ^ 4) / (↑((2 * n)! ^ 2) * (2 * ↑n + 1))",
"state_before": "n : ℕ\nhn : n ≠ 0\nthis✝² : 4 = 2 * 2\nthis✝¹ : ↑n ≠ 0\nthis✝ : exp 1 ≠ 0\nthis : ↑(2 * n)! ≠ 0\n⊢ (↑n ! ^ 2) ^ 2 / ((2 * ↑n) ^ 2 * ((↑n ^ n / exp 1 ^ n) ^ 2) ^ 2) /\n (↑(2 * n)! ^ 2 / (2 * ↑(2 * n) * (↑(2 * n) ^ (2 * n) / exp 1 ^ (2 * n)) ^ 2)) *\n (↑n / (2 * ↑n + 1)) =\n ↑(2 ^ (4 * n)) * ↑(n ! ^ 4) / (↑((2 * n)! ^ 2) * (2 * ↑n + 1))",
"tactic": "have : 2 * (n : ℝ) + 1 ≠ 0 := by norm_cast; exact succ_ne_zero (2 * n)"
},
{
"state_after": "n : ℕ\nhn : n ≠ 0\nthis✝³ : 4 = 2 * 2\nthis✝² : ↑n ≠ 0\nthis✝¹ : exp 1 ≠ 0\nthis✝ : ↑(2 * n)! ≠ 0\nthis : 2 * ↑n + 1 ≠ 0\n⊢ (↑n ! ^ 2) ^ 2 * ((exp 1 ^ n) ^ 2) ^ 2 * (2 * (2 * ↑n) * ((2 * ↑n) ^ (2 * n)) ^ 2) * ↑n *\n (↑(2 * n)! ^ 2 * (2 * ↑n + 1)) =\n 2 ^ (4 * n) * ↑n ! ^ 4 *\n ((2 * ↑n) ^ 2 * ((↑n ^ n) ^ 2) ^ 2 * (↑(2 * n)! ^ 2 * (exp 1 ^ (2 * n)) ^ 2) * (2 * ↑n + 1))",
"state_before": "n : ℕ\nhn : n ≠ 0\nthis✝³ : 4 = 2 * 2\nthis✝² : ↑n ≠ 0\nthis✝¹ : exp 1 ≠ 0\nthis✝ : ↑(2 * n)! ≠ 0\nthis : 2 * ↑n + 1 ≠ 0\n⊢ (↑n ! ^ 2) ^ 2 / ((2 * ↑n) ^ 2 * ((↑n ^ n / exp 1 ^ n) ^ 2) ^ 2) /\n (↑(2 * n)! ^ 2 / (2 * ↑(2 * n) * (↑(2 * n) ^ (2 * n) / exp 1 ^ (2 * n)) ^ 2)) *\n (↑n / (2 * ↑n + 1)) =\n ↑(2 ^ (4 * n)) * ↑(n ! ^ 4) / (↑((2 * n)! ^ 2) * (2 * ↑n + 1))",
"tactic": "field_simp"
},
{
"state_after": "n : ℕ\nhn : n ≠ 0\nthis✝³ : 4 = 2 * 2\nthis✝² : ↑n ≠ 0\nthis✝¹ : exp 1 ≠ 0\nthis✝ : ↑(2 * n)! ≠ 0\nthis : 2 * ↑n + 1 ≠ 0\n⊢ (↑n ! ^ 2) ^ 2 * ((exp 1 ^ n) ^ 2) ^ 2 * (2 * (2 * ↑n) * (((2 ^ n) ^ 2) ^ 2 * ((↑n ^ n) ^ 2) ^ 2)) * ↑n *\n (↑(n * 2)! ^ 2 * (2 * ↑n + 1)) =\n (2 ^ n) ^ 4 * ↑n ! ^ 4 *\n (2 ^ 2 * ↑n ^ 2 * ((↑n ^ n) ^ 2) ^ 2 * (↑(n * 2)! ^ 2 * ((exp 1 ^ n) ^ 2) ^ 2) * (2 * ↑n + 1))",
"state_before": "n : ℕ\nhn : n ≠ 0\nthis✝³ : 4 = 2 * 2\nthis✝² : ↑n ≠ 0\nthis✝¹ : exp 1 ≠ 0\nthis✝ : ↑(2 * n)! ≠ 0\nthis : 2 * ↑n + 1 ≠ 0\n⊢ (↑n ! ^ 2) ^ 2 * ((exp 1 ^ n) ^ 2) ^ 2 * (2 * (2 * ↑n) * ((2 * ↑n) ^ (2 * n)) ^ 2) * ↑n *\n (↑(2 * n)! ^ 2 * (2 * ↑n + 1)) =\n 2 ^ (4 * n) * ↑n ! ^ 4 *\n ((2 * ↑n) ^ 2 * ((↑n ^ n) ^ 2) ^ 2 * (↑(2 * n)! ^ 2 * (exp 1 ^ (2 * n)) ^ 2) * (2 * ↑n + 1))",
"tactic": "simp only [mul_pow, mul_comm 2 n, mul_comm 4 n, pow_mul]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nhn : n ≠ 0\nthis✝³ : 4 = 2 * 2\nthis✝² : ↑n ≠ 0\nthis✝¹ : exp 1 ≠ 0\nthis✝ : ↑(2 * n)! ≠ 0\nthis : 2 * ↑n + 1 ≠ 0\n⊢ (↑n ! ^ 2) ^ 2 * ((exp 1 ^ n) ^ 2) ^ 2 * (2 * (2 * ↑n) * (((2 ^ n) ^ 2) ^ 2 * ((↑n ^ n) ^ 2) ^ 2)) * ↑n *\n (↑(n * 2)! ^ 2 * (2 * ↑n + 1)) =\n (2 ^ n) ^ 4 * ↑n ! ^ 4 *\n (2 ^ 2 * ↑n ^ 2 * ((↑n ^ n) ^ 2) ^ 2 * (↑(n * 2)! ^ 2 * ((exp 1 ^ n) ^ 2) ^ 2) * (2 * ↑n + 1))",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nhn : n ≠ 0\n⊢ 4 = 2 * 2",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nhn : n ≠ 0\nthis : 4 = 2 * 2\n⊢ 0 ≤ 2 * ↑n",
"tactic": "positivity"
},
{
"state_after": "n : ℕ\nhn : n ≠ 0\nthis✝² : 4 = 2 * 2\nthis✝¹ : ↑n ≠ 0\nthis✝ : exp 1 ≠ 0\nthis : ↑(2 * n)! ≠ 0\n⊢ ¬2 * n + 1 = 0",
"state_before": "n : ℕ\nhn : n ≠ 0\nthis✝² : 4 = 2 * 2\nthis✝¹ : ↑n ≠ 0\nthis✝ : exp 1 ≠ 0\nthis : ↑(2 * n)! ≠ 0\n⊢ 2 * ↑n + 1 ≠ 0",
"tactic": "norm_cast"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nhn : n ≠ 0\nthis✝² : 4 = 2 * 2\nthis✝¹ : ↑n ≠ 0\nthis✝ : exp 1 ≠ 0\nthis : ↑(2 * n)! ≠ 0\n⊢ ¬2 * n + 1 = 0",
"tactic": "exact succ_ne_zero (2 * n)"
}
] | [
248,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
235,
1
] |
Mathlib/LinearAlgebra/BilinearMap.lean | LinearMap.lsmul_injective | [] | [
419,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
417,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean | Finset.prod_subtype_eq_prod_filter | [
{
"state_after": "ι : Type ?u.379810\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\nf : α → β\np : α → Prop\ninst✝ : DecidablePred p\n⊢ ∏ x in map (Function.Embedding.subtype p) (Finset.subtype p s), f x = ∏ x in filter p s, f x",
"state_before": "ι : Type ?u.379810\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\nf : α → β\np : α → Prop\ninst✝ : DecidablePred p\n⊢ ∏ x in Finset.subtype p s, f ↑x = ∏ x in filter p s, f x",
"tactic": "conv_lhs => erw [← prod_map (s.subtype p) (Function.Embedding.subtype _) f]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.379810\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\nf : α → β\np : α → Prop\ninst✝ : DecidablePred p\n⊢ ∏ x in map (Function.Embedding.subtype p) (Finset.subtype p s), f x = ∏ x in filter p s, f x",
"tactic": "exact prod_congr (subtype_map _) fun x _hx => rfl"
}
] | [
870,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
867,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean | Set.Ici_inj | [] | [
969,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
968,
1
] |
Mathlib/Data/Multiset/LocallyFinite.lean | Multiset.Ioo_eq_zero | [] | [
78,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
77,
1
] |
Mathlib/Topology/DenseEmbedding.lean | DenseInducing.nhdsWithin_neBot | [] | [
132,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
131,
11
] |
Mathlib/FieldTheory/IsAlgClosed/Basic.lean | IsAlgClosure.equivOfEquiv_comp_algebraMap | [] | [
511,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
509,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean | Equiv.Perm.not_isCycle_one | [] | [
295,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
295,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean | Subgroup.eq_top_iff' | [] | [
1086,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1085,
1
] |
Mathlib/Topology/Basic.lean | ClusterPt.mono | [] | [
1128,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1127,
1
] |
Mathlib/Algebra/Group/Commute.lean | Commute.pow_left | [] | [
178,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
177,
1
] |
Mathlib/Data/Finset/Pointwise.lean | Finset.coe_coeMonoidHom | [] | [
836,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
835,
1
] |
Mathlib/Analysis/Calculus/BumpFunctionInner.lean | ContDiffBump.le_one | [] | [
400,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
399,
1
] |
Mathlib/Data/Real/ENNReal.lean | ENNReal.coe_two | [] | [
375,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
375,
1
] |
Mathlib/Data/Fintype/Basic.lean | Function.Injective.right_inv_of_invOfMemRange | [] | [
502,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
501,
1
] |
Mathlib/Algebra/FreeMonoid/Count.lean | FreeMonoid.count_apply | [] | [
85,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
84,
1
] |
Mathlib/Data/Polynomial/Basic.lean | Polynomial.monomial_mul_C | [
{
"state_after": "no goals",
"state_before": "R : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q : R[X]\n⊢ ↑(monomial n) a * ↑C b = ↑(monomial n) (a * b)",
"tactic": "simp only [← monomial_zero_left, monomial_mul_monomial, add_zero]"
}
] | [
546,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
545,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean | MeasureTheory.AEFinStronglyMeasurable.finStronglyMeasurable_mk | [] | [
1854,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1852,
1
] |
Mathlib/Init/Data/Sigma/Basic.lean | PSigma.eq | [] | [
22,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
20,
11
] |
Mathlib/Init/Algebra/Order.lean | lt_or_gt_of_ne | [] | [
360,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
356,
1
] |
Mathlib/Computability/PartrecCode.lean | Nat.Partrec.Code.curry_inj | [
{
"state_after": "no goals",
"state_before": "c₁ c₂ : Code\nn₁ n₂ : ℕ\nh : curry c₁ n₁ = curry c₂ n₂\n⊢ c₁ = c₂",
"tactic": "injection h"
},
{
"state_after": "c₁ c₂ : Code\nn₁ n₂ : ℕ\nh₁ : c₁ = c₂\nh₂ : pair (Code.const n₁) Code.id = pair (Code.const n₂) Code.id\n⊢ n₁ = n₂",
"state_before": "c₁ c₂ : Code\nn₁ n₂ : ℕ\nh : curry c₁ n₁ = curry c₂ n₂\n⊢ n₁ = n₂",
"tactic": "injection h with h₁ h₂"
},
{
"state_after": "c₁ c₂ : Code\nn₁ n₂ : ℕ\nh₁ : c₁ = c₂\nh₃ : Code.const n₁ = Code.const n₂\nh₄ : Code.id = Code.id\n⊢ n₁ = n₂",
"state_before": "c₁ c₂ : Code\nn₁ n₂ : ℕ\nh₁ : c₁ = c₂\nh₂ : pair (Code.const n₁) Code.id = pair (Code.const n₂) Code.id\n⊢ n₁ = n₂",
"tactic": "injection h₂ with h₃ h₄"
},
{
"state_after": "no goals",
"state_before": "c₁ c₂ : Code\nn₁ n₂ : ℕ\nh₁ : c₁ = c₂\nh₃ : Code.const n₁ = Code.const n₂\nh₄ : Code.id = Code.id\n⊢ n₁ = n₂",
"tactic": "exact const_inj h₃"
}
] | [
689,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
685,
1
] |
Mathlib/LinearAlgebra/Span.lean | Submodule.closure_subset_span | [] | [
135,
77
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
134,
1
] |
Mathlib/Data/Matrix/Notation.lean | Matrix.dotProduct_cons | [
{
"state_after": "no goals",
"state_before": "α : Type u\no n m : ℕ\nm' : Type uₘ\nn' : Type uₙ\no' : Type uₒ\na b : ℕ\ninst✝¹ : AddCommMonoid α\ninst✝ : Mul α\nv : Fin (Nat.succ n) → α\nx : α\nw : Fin n → α\n⊢ v ⬝ᵥ vecCons x w = vecHead v * x + vecTail v ⬝ᵥ w",
"tactic": "simp [dotProduct, Fin.sum_univ_succ, vecHead, vecTail]"
}
] | [
155,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
153,
1
] |
Mathlib/SetTheory/Cardinal/Ordinal.lean | Cardinal.extend_function_finite | [
{
"state_after": "α : Type u\nβ : Type v\ninst✝ : Finite α\ns : Set α\nf : ↑s ↪ β\nh : Nonempty (α ≃ β)\n⊢ Nonempty (↑(sᶜ) ≃ ↑(range ↑fᶜ))",
"state_before": "α : Type u\nβ : Type v\ninst✝ : Finite α\ns : Set α\nf : ↑s ↪ β\nh : Nonempty (α ≃ β)\n⊢ ∃ g, ∀ (x : ↑s), ↑g ↑x = ↑f x",
"tactic": "apply extend_function.{v, u} f"
},
{
"state_after": "case intro\nα : Type u\nβ : Type v\ninst✝ : Finite α\ns : Set α\nf : ↑s ↪ β\nh : Nonempty (α ≃ β)\ng : α ≃ β\n⊢ Nonempty (↑(sᶜ) ≃ ↑(range ↑fᶜ))",
"state_before": "α : Type u\nβ : Type v\ninst✝ : Finite α\ns : Set α\nf : ↑s ↪ β\nh : Nonempty (α ≃ β)\n⊢ Nonempty (↑(sᶜ) ≃ ↑(range ↑fᶜ))",
"tactic": "cases' id h with g"
},
{
"state_after": "case intro\nα : Type u\nβ : Type v\ninst✝ : Finite α\ns : Set α\nf : ↑s ↪ β\nh : lift (#α) = lift (#β)\ng : α ≃ β\n⊢ Nonempty (↑(sᶜ) ≃ ↑(range ↑fᶜ))",
"state_before": "case intro\nα : Type u\nβ : Type v\ninst✝ : Finite α\ns : Set α\nf : ↑s ↪ β\nh : Nonempty (α ≃ β)\ng : α ≃ β\n⊢ Nonempty (↑(sᶜ) ≃ ↑(range ↑fᶜ))",
"tactic": "rw [← lift_mk_eq.{u, v, max u v}] at h"
},
{
"state_after": "case intro\nα : Type u\nβ : Type v\ninst✝ : Finite α\ns : Set α\nf : ↑s ↪ β\nh : lift (#α) = lift (#β)\ng : α ≃ β\n⊢ lift (#↑s) = lift (#↑(range ↑f))",
"state_before": "case intro\nα : Type u\nβ : Type v\ninst✝ : Finite α\ns : Set α\nf : ↑s ↪ β\nh : lift (#α) = lift (#β)\ng : α ≃ β\n⊢ Nonempty (↑(sᶜ) ≃ ↑(range ↑fᶜ))",
"tactic": "rw [← lift_mk_eq.{u, v, max u v}, mk_compl_eq_mk_compl_finite_lift.{u, v, max u v} h]"
},
{
"state_after": "case intro\nα : Type u\nβ : Type v\ninst✝ : Finite α\ns : Set α\nf : ↑s ↪ β\nh : lift (#α) = lift (#β)\ng : α ≃ β\n⊢ Injective ↑f",
"state_before": "case intro\nα : Type u\nβ : Type v\ninst✝ : Finite α\ns : Set α\nf : ↑s ↪ β\nh : lift (#α) = lift (#β)\ng : α ≃ β\n⊢ lift (#↑s) = lift (#↑(range ↑f))",
"tactic": "rw [mk_range_eq_lift.{u, v, max u v}]"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u\nβ : Type v\ninst✝ : Finite α\ns : Set α\nf : ↑s ↪ β\nh : lift (#α) = lift (#β)\ng : α ≃ β\n⊢ Injective ↑f",
"tactic": "exact f.2"
}
] | [
1211,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1205,
1
] |
Mathlib/Order/Filter/Partial.lean | Filter.ptendsto_def | [] | [
231,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
229,
1
] |
Mathlib/GroupTheory/FreeGroup.lean | FreeGroup.Red.Step.sublist | [
{
"state_after": "case not\nα : Type u\nL L₃ L₄ L₁✝ L₂✝ : List (α × Bool)\nx✝ : α\nb✝ : Bool\n⊢ L₁✝ ++ L₂✝ <+ L₁✝ ++ (x✝, b✝) :: (x✝, !b✝) :: L₂✝",
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"tactic": "cases H"
},
{
"state_after": "case not\nα : Type u\nL L₃ L₄ L₁✝ L₂✝ : List (α × Bool)\nx✝ : α\nb✝ : Bool\n⊢ L₂✝ <+ (x✝, b✝) :: (x✝, !b✝) :: L₂✝",
"state_before": "case not\nα : Type u\nL L₃ L₄ L₁✝ L₂✝ : List (α × Bool)\nx✝ : α\nb✝ : Bool\n⊢ L₁✝ ++ L₂✝ <+ L₁✝ ++ (x✝, b✝) :: (x✝, !b✝) :: L₂✝",
"tactic": "simp"
},
{
"state_after": "case not.a\nα : Type u\nL L₃ L₄ L₁✝ L₂✝ : List (α × Bool)\nx✝ : α\nb✝ : Bool\n⊢ L₂✝ <+ (x✝, !b✝) :: L₂✝",
"state_before": "case not\nα : Type u\nL L₃ L₄ L₁✝ L₂✝ : List (α × Bool)\nx✝ : α\nb✝ : Bool\n⊢ L₂✝ <+ (x✝, b✝) :: (x✝, !b✝) :: L₂✝",
"tactic": "constructor"
},
{
"state_after": "case not.a.a\nα : Type u\nL L₃ L₄ L₁✝ L₂✝ : List (α × Bool)\nx✝ : α\nb✝ : Bool\n⊢ L₂✝ <+ L₂✝",
"state_before": "case not.a\nα : Type u\nL L₃ L₄ L₁✝ L₂✝ : List (α × Bool)\nx✝ : α\nb✝ : Bool\n⊢ L₂✝ <+ (x✝, !b✝) :: L₂✝",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case not.a.a\nα : Type u\nL L₃ L₄ L₁✝ L₂✝ : List (α × Bool)\nx✝ : α\nb✝ : Bool\n⊢ L₂✝ <+ L₂✝",
"tactic": "rfl"
}
] | [
383,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
382,
1
] |
Mathlib/Data/Finset/NatAntidiagonal.lean | Finset.Nat.antidiagonal.fst_le | [
{
"state_after": "n : ℕ\nkl : ℕ × ℕ\nhlk : kl ∈ antidiagonal n\n⊢ ∃ c, n = kl.fst + c",
"state_before": "n : ℕ\nkl : ℕ × ℕ\nhlk : kl ∈ antidiagonal n\n⊢ kl.fst ≤ n",
"tactic": "rw [le_iff_exists_add]"
},
{
"state_after": "n : ℕ\nkl : ℕ × ℕ\nhlk : kl ∈ antidiagonal n\n⊢ n = kl.fst + kl.snd",
"state_before": "n : ℕ\nkl : ℕ × ℕ\nhlk : kl ∈ antidiagonal n\n⊢ ∃ c, n = kl.fst + c",
"tactic": "use kl.2"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nkl : ℕ × ℕ\nhlk : kl ∈ antidiagonal n\n⊢ n = kl.fst + kl.snd",
"tactic": "rwa [mem_antidiagonal, eq_comm] at hlk"
}
] | [
104,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
101,
1
] |
Mathlib/CategoryTheory/Abelian/Basic.lean | CategoryTheory.Abelian.imageIsoImage_hom_comp_image_ι | [
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nX Y : C\nf : X ⟶ Y\n⊢ (imageIsoImage f).hom ≫ Limits.image.ι f = kernel.ι (cokernel.π f)",
"tactic": "simp only [IsImage.isoExt_hom, IsImage.lift_ι, imageStrongEpiMonoFactorisation_m]"
}
] | [
442,
84
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
441,
1
] |
Mathlib/Algebra/Star/Unitary.lean | unitary.coe_zpow | [
{
"state_after": "case ofNat\nR : Type u_1\ninst✝¹ : GroupWithZero R\ninst✝ : StarSemigroup R\nU : { x // x ∈ unitary R }\na✝ : ℕ\n⊢ ↑(U ^ Int.ofNat a✝) = ↑U ^ Int.ofNat a✝\n\ncase negSucc\nR : Type u_1\ninst✝¹ : GroupWithZero R\ninst✝ : StarSemigroup R\nU : { x // x ∈ unitary R }\na✝ : ℕ\n⊢ ↑(U ^ Int.negSucc a✝) = ↑U ^ Int.negSucc a✝",
"state_before": "R : Type u_1\ninst✝¹ : GroupWithZero R\ninst✝ : StarSemigroup R\nU : { x // x ∈ unitary R }\nz : ℤ\n⊢ ↑(U ^ z) = ↑U ^ z",
"tactic": "induction z"
},
{
"state_after": "no goals",
"state_before": "case ofNat\nR : Type u_1\ninst✝¹ : GroupWithZero R\ninst✝ : StarSemigroup R\nU : { x // x ∈ unitary R }\na✝ : ℕ\n⊢ ↑(U ^ Int.ofNat a✝) = ↑U ^ Int.ofNat a✝",
"tactic": "simp [SubmonoidClass.coe_pow]"
},
{
"state_after": "no goals",
"state_before": "case negSucc\nR : Type u_1\ninst✝¹ : GroupWithZero R\ninst✝ : StarSemigroup R\nU : { x // x ∈ unitary R }\na✝ : ℕ\n⊢ ↑(U ^ Int.negSucc a✝) = ↑U ^ Int.negSucc a✝",
"tactic": "simp [coe_inv]"
}
] | [
182,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
179,
1
] |
Mathlib/Algebra/ContinuedFractions/Translations.lean | GeneralizedContinuedFraction.part_num_eq_s_a | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ng : GeneralizedContinuedFraction α\nn : ℕ\ngp : Pair α\ns_nth_eq : Stream'.Seq.get? g.s n = some gp\n⊢ Stream'.Seq.get? (partialNumerators g) n = some gp.a",
"tactic": "simp [partialNumerators, s_nth_eq]"
}
] | [
60,
84
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
59,
1
] |
Mathlib/Analysis/Seminorm.lean | Seminorm.comp_smul | [
{
"state_after": "no goals",
"state_before": "R : Type ?u.552397\nR' : Type ?u.552400\n𝕜 : Type u_3\n𝕜₂ : Type u_1\n𝕜₃ : Type ?u.552409\n𝕝 : Type ?u.552412\nE : Type u_4\nE₂ : Type u_2\nE₃ : Type ?u.552421\nF : Type ?u.552424\nG : Type ?u.552427\nι : Type ?u.552430\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : SeminormedCommRing 𝕜₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝⁴ : RingHomIsometric σ₁₂\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜₂ E₂\np : Seminorm 𝕜₂ E₂\nf : E →ₛₗ[σ₁₂] E₂\nc : 𝕜₂\nx✝ : E\n⊢ ↑(comp p (c • f)) x✝ = ↑(‖c‖₊ • comp p f) x✝",
"tactic": "rw [comp_apply, smul_apply, LinearMap.smul_apply, map_smul_eq_mul, NNReal.smul_def, coe_nnnorm,\n smul_eq_mul, comp_apply]"
}
] | [
445,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
441,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean | map_extChartAt_nhdsWithin | [] | [
1178,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1176,
1
] |
Mathlib/Data/List/Basic.lean | List.drop_eq_get_cons | [] | [
2157,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2155,
1
] |
Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_finsupp | [
{
"state_after": "R : Type u_2\nM : Type u_1\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\n⊢ (Function.support fun m => ↑(weightedHomogeneousComponent w m) φ) ⊆ (fun d => ↑(weightedDegree' w) d) '' ↑(support φ)",
"state_before": "R : Type u_2\nM : Type u_1\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\n⊢ Set.Finite (Function.support fun m => ↑(weightedHomogeneousComponent w m) φ)",
"tactic": "suffices\n (Function.support fun m => weightedHomogeneousComponent w m φ) ⊆\n (fun d => weightedDegree' w d) '' φ.support by\n exact Finite.subset ((fun d : σ →₀ ℕ => (weightedDegree' w) d) '' ↑(support φ)).toFinite this"
},
{
"state_after": "R : Type u_2\nM : Type u_1\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nm : M\nhm : m ∈ Function.support fun m => ↑(weightedHomogeneousComponent w m) φ\n⊢ m ∈ (fun d => ↑(weightedDegree' w) d) '' ↑(support φ)",
"state_before": "R : Type u_2\nM : Type u_1\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\n⊢ (Function.support fun m => ↑(weightedHomogeneousComponent w m) φ) ⊆ (fun d => ↑(weightedDegree' w) d) '' ↑(support φ)",
"tactic": "intro m hm"
},
{
"state_after": "R : Type u_2\nM : Type u_1\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nm : M\nhm : m ∈ Function.support fun m => ↑(weightedHomogeneousComponent w m) φ\nhm' : ¬m ∈ (fun d => ↑(weightedDegree' w) d) '' ↑(support φ)\n⊢ False",
"state_before": "R : Type u_2\nM : Type u_1\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nm : M\nhm : m ∈ Function.support fun m => ↑(weightedHomogeneousComponent w m) φ\n⊢ m ∈ (fun d => ↑(weightedDegree' w) d) '' ↑(support φ)",
"tactic": "by_contra hm'"
},
{
"state_after": "R : Type u_2\nM : Type u_1\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nm : M\nhm : m ∈ Function.support fun m => ↑(weightedHomogeneousComponent w m) φ\nhm' : ¬m ∈ (fun d => ↑(weightedDegree' w) d) '' ↑(support φ)\n⊢ (fun m => ↑(weightedHomogeneousComponent w m) φ) m = 0",
"state_before": "R : Type u_2\nM : Type u_1\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nm : M\nhm : m ∈ Function.support fun m => ↑(weightedHomogeneousComponent w m) φ\nhm' : ¬m ∈ (fun d => ↑(weightedDegree' w) d) '' ↑(support φ)\n⊢ False",
"tactic": "apply hm"
},
{
"state_after": "R : Type u_2\nM : Type u_1\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nm : M\nhm' : ¬m ∈ (fun d => ↑(weightedDegree' w) d) '' ↑(support φ)\nhm : ¬↑(weightedHomogeneousComponent w m) φ = 0\n⊢ (fun m => ↑(weightedHomogeneousComponent w m) φ) m = 0",
"state_before": "R : Type u_2\nM : Type u_1\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nm : M\nhm : m ∈ Function.support fun m => ↑(weightedHomogeneousComponent w m) φ\nhm' : ¬m ∈ (fun d => ↑(weightedDegree' w) d) '' ↑(support φ)\n⊢ (fun m => ↑(weightedHomogeneousComponent w m) φ) m = 0",
"tactic": "simp only [mem_support, Ne.def] at hm"
},
{
"state_after": "R : Type u_2\nM : Type u_1\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nm : M\nhm : ¬↑(weightedHomogeneousComponent w m) φ = 0\nhm' : ∀ (x : σ →₀ ℕ), x ∈ ↑(support φ) → ¬↑(weightedDegree' w) x = m\n⊢ (fun m => ↑(weightedHomogeneousComponent w m) φ) m = 0",
"state_before": "R : Type u_2\nM : Type u_1\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nm : M\nhm' : ¬m ∈ (fun d => ↑(weightedDegree' w) d) '' ↑(support φ)\nhm : ¬↑(weightedHomogeneousComponent w m) φ = 0\n⊢ (fun m => ↑(weightedHomogeneousComponent w m) φ) m = 0",
"tactic": "simp only [Set.mem_image, not_exists, not_and] at hm'"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nM : Type u_1\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nm : M\nhm : ¬↑(weightedHomogeneousComponent w m) φ = 0\nhm' : ∀ (x : σ →₀ ℕ), x ∈ ↑(support φ) → ¬↑(weightedDegree' w) x = m\n⊢ (fun m => ↑(weightedHomogeneousComponent w m) φ) m = 0",
"tactic": "exact weightedHomogeneousComponent_eq_zero' m φ hm'"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nM : Type u_1\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nthis :\n (Function.support fun m => ↑(weightedHomogeneousComponent w m) φ) ⊆ (fun d => ↑(weightedDegree' w) d) '' ↑(support φ)\n⊢ Set.Finite (Function.support fun m => ↑(weightedHomogeneousComponent w m) φ)",
"tactic": "exact Finite.subset ((fun d : σ →₀ ℕ => (weightedDegree' w) d) '' ↑(support φ)).toFinite this"
}
] | [
399,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
388,
1
] |
Mathlib/GroupTheory/Perm/List.lean | List.formPerm_pair | [] | [
71,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
70,
1
] |
Mathlib/GroupTheory/OrderOfElement.lean | orderOf_inv | [
{
"state_after": "no goals",
"state_before": "G : Type u\nA : Type v\nx✝ y : G\na b : A\nn m : ℕ\ninst✝¹ : Group G\ninst✝ : AddGroup A\ni : ℤ\nx : G\n⊢ orderOf x⁻¹ = orderOf x",
"tactic": "simp [orderOf_eq_orderOf_iff]"
}
] | [
565,
90
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
565,
1
] |
Mathlib/Analysis/Seminorm.lean | Seminorm.ball_zero_absorbs_ball_zero | [
{
"state_after": "case intro.intro\nR : Type ?u.1218120\nR' : Type ?u.1218123\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1218129\n𝕜₃ : Type ?u.1218132\n𝕝 : Type ?u.1218135\nE : Type u_2\nE₂ : Type ?u.1218141\nE₃ : Type ?u.1218144\nF : Type ?u.1218147\nG : Type ?u.1218150\nι : Type ?u.1218153\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np✝ : Seminorm 𝕜 E\nA B : Set E\na : 𝕜\nr✝ : ℝ\nx : E\np : Seminorm 𝕜 E\nr₁ r₂ : ℝ\nhr₁ : 0 < r₁\nr : ℝ\nhr₀ : 0 < r\nhr : r₂ < r * r₁\n⊢ Absorbs 𝕜 (ball p 0 r₁) (ball p 0 r₂)",
"state_before": "R : Type ?u.1218120\nR' : Type ?u.1218123\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1218129\n𝕜₃ : Type ?u.1218132\n𝕝 : Type ?u.1218135\nE : Type u_2\nE₂ : Type ?u.1218141\nE₃ : Type ?u.1218144\nF : Type ?u.1218147\nG : Type ?u.1218150\nι : Type ?u.1218153\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np✝ : Seminorm 𝕜 E\nA B : Set E\na : 𝕜\nr : ℝ\nx : E\np : Seminorm 𝕜 E\nr₁ r₂ : ℝ\nhr₁ : 0 < r₁\n⊢ Absorbs 𝕜 (ball p 0 r₁) (ball p 0 r₂)",
"tactic": "rcases exists_pos_lt_mul hr₁ r₂ with ⟨r, hr₀, hr⟩"
},
{
"state_after": "case intro.intro\nR : Type ?u.1218120\nR' : Type ?u.1218123\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1218129\n𝕜₃ : Type ?u.1218132\n𝕝 : Type ?u.1218135\nE : Type u_2\nE₂ : Type ?u.1218141\nE₃ : Type ?u.1218144\nF : Type ?u.1218147\nG : Type ?u.1218150\nι : Type ?u.1218153\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np✝ : Seminorm 𝕜 E\nA B : Set E\na✝ : 𝕜\nr✝ : ℝ\nx✝ : E\np : Seminorm 𝕜 E\nr₁ r₂ : ℝ\nhr₁ : 0 < r₁\nr : ℝ\nhr₀ : 0 < r\nhr : r₂ < r * r₁\na : 𝕜\nha : r ≤ ‖a‖\nx : E\nhx : x ∈ ball p 0 r₂\n⊢ x ∈ a • ball p 0 r₁",
"state_before": "case intro.intro\nR : Type ?u.1218120\nR' : Type ?u.1218123\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1218129\n𝕜₃ : Type ?u.1218132\n𝕝 : Type ?u.1218135\nE : Type u_2\nE₂ : Type ?u.1218141\nE₃ : Type ?u.1218144\nF : Type ?u.1218147\nG : Type ?u.1218150\nι : Type ?u.1218153\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np✝ : Seminorm 𝕜 E\nA B : Set E\na : 𝕜\nr✝ : ℝ\nx : E\np : Seminorm 𝕜 E\nr₁ r₂ : ℝ\nhr₁ : 0 < r₁\nr : ℝ\nhr₀ : 0 < r\nhr : r₂ < r * r₁\n⊢ Absorbs 𝕜 (ball p 0 r₁) (ball p 0 r₂)",
"tactic": "refine' ⟨r, hr₀, fun a ha x hx => _⟩"
},
{
"state_after": "case intro.intro\nR : Type ?u.1218120\nR' : Type ?u.1218123\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1218129\n𝕜₃ : Type ?u.1218132\n𝕝 : Type ?u.1218135\nE : Type u_2\nE₂ : Type ?u.1218141\nE₃ : Type ?u.1218144\nF : Type ?u.1218147\nG : Type ?u.1218150\nι : Type ?u.1218153\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np✝ : Seminorm 𝕜 E\nA B : Set E\na✝ : 𝕜\nr✝ : ℝ\nx✝ : E\np : Seminorm 𝕜 E\nr₁ r₂ : ℝ\nhr₁ : 0 < r₁\nr : ℝ\nhr₀ : 0 < r\nhr : r₂ < r * r₁\na : 𝕜\nha : r ≤ ‖a‖\nx : E\nhx : x ∈ ball p 0 r₂\n⊢ ↑p x < ‖a‖ * r₁",
"state_before": "case intro.intro\nR : Type ?u.1218120\nR' : Type ?u.1218123\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1218129\n𝕜₃ : Type ?u.1218132\n𝕝 : Type ?u.1218135\nE : Type u_2\nE₂ : Type ?u.1218141\nE₃ : Type ?u.1218144\nF : Type ?u.1218147\nG : Type ?u.1218150\nι : Type ?u.1218153\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np✝ : Seminorm 𝕜 E\nA B : Set E\na✝ : 𝕜\nr✝ : ℝ\nx✝ : E\np : Seminorm 𝕜 E\nr₁ r₂ : ℝ\nhr₁ : 0 < r₁\nr : ℝ\nhr₀ : 0 < r\nhr : r₂ < r * r₁\na : 𝕜\nha : r ≤ ‖a‖\nx : E\nhx : x ∈ ball p 0 r₂\n⊢ x ∈ a • ball p 0 r₁",
"tactic": "rw [smul_ball_zero (norm_pos_iff.1 <| hr₀.trans_le ha), p.mem_ball_zero]"
},
{
"state_after": "case intro.intro\nR : Type ?u.1218120\nR' : Type ?u.1218123\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1218129\n𝕜₃ : Type ?u.1218132\n𝕝 : Type ?u.1218135\nE : Type u_2\nE₂ : Type ?u.1218141\nE₃ : Type ?u.1218144\nF : Type ?u.1218147\nG : Type ?u.1218150\nι : Type ?u.1218153\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np✝ : Seminorm 𝕜 E\nA B : Set E\na✝ : 𝕜\nr✝ : ℝ\nx✝ : E\np : Seminorm 𝕜 E\nr₁ r₂ : ℝ\nhr₁ : 0 < r₁\nr : ℝ\nhr₀ : 0 < r\nhr : r₂ < r * r₁\na : 𝕜\nha : r ≤ ‖a‖\nx : E\nhx : ↑p x < r₂\n⊢ ↑p x < ‖a‖ * r₁",
"state_before": "case intro.intro\nR : Type ?u.1218120\nR' : Type ?u.1218123\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1218129\n𝕜₃ : Type ?u.1218132\n𝕝 : Type ?u.1218135\nE : Type u_2\nE₂ : Type ?u.1218141\nE₃ : Type ?u.1218144\nF : Type ?u.1218147\nG : Type ?u.1218150\nι : Type ?u.1218153\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np✝ : Seminorm 𝕜 E\nA B : Set E\na✝ : 𝕜\nr✝ : ℝ\nx✝ : E\np : Seminorm 𝕜 E\nr₁ r₂ : ℝ\nhr₁ : 0 < r₁\nr : ℝ\nhr₀ : 0 < r\nhr : r₂ < r * r₁\na : 𝕜\nha : r ≤ ‖a‖\nx : E\nhx : x ∈ ball p 0 r₂\n⊢ ↑p x < ‖a‖ * r₁",
"tactic": "rw [p.mem_ball_zero] at hx"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nR : Type ?u.1218120\nR' : Type ?u.1218123\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1218129\n𝕜₃ : Type ?u.1218132\n𝕝 : Type ?u.1218135\nE : Type u_2\nE₂ : Type ?u.1218141\nE₃ : Type ?u.1218144\nF : Type ?u.1218147\nG : Type ?u.1218150\nι : Type ?u.1218153\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np✝ : Seminorm 𝕜 E\nA B : Set E\na✝ : 𝕜\nr✝ : ℝ\nx✝ : E\np : Seminorm 𝕜 E\nr₁ r₂ : ℝ\nhr₁ : 0 < r₁\nr : ℝ\nhr₀ : 0 < r\nhr : r₂ < r * r₁\na : 𝕜\nha : r ≤ ‖a‖\nx : E\nhx : ↑p x < r₂\n⊢ ↑p x < ‖a‖ * r₁",
"tactic": "exact hx.trans (hr.trans_le <| by gcongr)"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.1218120\nR' : Type ?u.1218123\n𝕜 : Type u_1\n𝕜₂ : Type ?u.1218129\n𝕜₃ : Type ?u.1218132\n𝕝 : Type ?u.1218135\nE : Type u_2\nE₂ : Type ?u.1218141\nE₃ : Type ?u.1218144\nF : Type ?u.1218147\nG : Type ?u.1218150\nι : Type ?u.1218153\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np✝ : Seminorm 𝕜 E\nA B : Set E\na✝ : 𝕜\nr✝ : ℝ\nx✝ : E\np : Seminorm 𝕜 E\nr₁ r₂ : ℝ\nhr₁ : 0 < r₁\nr : ℝ\nhr₀ : 0 < r\nhr : r₂ < r * r₁\na : 𝕜\nha : r ≤ ‖a‖\nx : E\nhx : ↑p x < r₂\n⊢ r * r₁ ≤ ‖a‖ * r₁",
"tactic": "gcongr"
}
] | [
979,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
973,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Basic.lean | HasFDerivWithinAt.continuousWithinAt | [] | [
724,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
722,
1
] |
Mathlib/Data/Set/Prod.lean | Set.prod_sdiff_diagonal | [] | [
588,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
587,
1
] |
Mathlib/MeasureTheory/Measure/OpenPos.lean | MeasureTheory.Measure.eqOn_of_ae_eq | [] | [
118,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
113,
1
] |
Std/Data/List/Lemmas.lean | List.erase_of_not_mem | [
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\na b : α\nl : List α\nh : ¬a = b ∧ ¬a ∈ l\n⊢ List.erase (b :: l) a = b :: l",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\na b : α\nl : List α\nh : ¬a ∈ b :: l\n⊢ List.erase (b :: l) a = b :: l",
"tactic": "rw [mem_cons, not_or] at h"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\na b : α\nl : List α\nh : ¬a = b ∧ ¬a ∈ l\n⊢ List.erase (b :: l) a = b :: l",
"tactic": "rw [erase_cons, if_neg (Ne.symm h.1), erase_of_not_mem h.2]"
}
] | [
1054,
64
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
1050,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Mul.lean | HasStrictFDerivAt.clm_comp | [] | [
66,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
63,
1
] |
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