file_path
stringlengths 11
79
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stringlengths 2
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list | end
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stringclasses 4
values | start
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Mathlib/Algebra/IndicatorFunction.lean | Set.mulIndicator_comp_right | [
{
"state_after": "α : Type u_1\nβ : Type u_3\nι : Type ?u.20670\nM : Type u_2\nN : Type ?u.20676\ninst✝¹ : One M\ninst✝ : One N\ns✝ t : Set α\nf✝ g✝ : α → M\na : α\ns : Set α\nf : β → α\ng : α → M\nx : β\n⊢ (if x ∈ f ⁻¹' s then g (f x) else 1) = if f x ∈ s then g (f x) else 1",
"state_before": "α : Type u_1\nβ : Type u_3\nι : Type ?u.20670\nM : Type u_2\nN : Type ?u.20676\ninst✝¹ : One M\ninst✝ : One N\ns✝ t : Set α\nf✝ g✝ : α → M\na : α\ns : Set α\nf : β → α\ng : α → M\nx : β\n⊢ mulIndicator (f ⁻¹' s) (g ∘ f) x = mulIndicator s g (f x)",
"tactic": "simp only [mulIndicator, Function.comp]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_3\nι : Type ?u.20670\nM : Type u_2\nN : Type ?u.20676\ninst✝¹ : One M\ninst✝ : One N\ns✝ t : Set α\nf✝ g✝ : α → M\na : α\ns : Set α\nf : β → α\ng : α → M\nx : β\n⊢ (if x ∈ f ⁻¹' s then g (f x) else 1) = if f x ∈ s then g (f x) else 1",
"tactic": "split_ifs with h h' h'' <;> first | rfl | contradiction"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nα : Type u_1\nβ : Type u_3\nι : Type ?u.20670\nM : Type u_2\nN : Type ?u.20676\ninst✝¹ : One M\ninst✝ : One N\ns✝ t : Set α\nf✝ g✝ : α → M\na : α\ns : Set α\nf : β → α\ng : α → M\nx : β\nh : ¬x ∈ f ⁻¹' s\nh'' : ¬f x ∈ s\n⊢ 1 = 1",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nα : Type u_1\nβ : Type u_3\nι : Type ?u.20670\nM : Type u_2\nN : Type ?u.20676\ninst✝¹ : One M\ninst✝ : One N\ns✝ t : Set α\nf✝ g✝ : α → M\na : α\ns : Set α\nf : β → α\ng : α → M\nx : β\nh : ¬x ∈ f ⁻¹' s\nh'' : f x ∈ s\n⊢ 1 = g (f x)",
"tactic": "contradiction"
}
]
| [
263,
58
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
260,
1
]
|
Mathlib/Topology/MetricSpace/Lipschitz.lean | LipschitzWith.ediam_image_le | [
{
"state_after": "case h\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nK : ℝ≥0\nf : α → β\nx y : α\nr : ℝ≥0∞\nhf : LipschitzWith K f\ns : Set α\n⊢ ∀ (x : β), x ∈ f '' s → ∀ (y : β), y ∈ f '' s → edist x y ≤ ↑K * diam s",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nK : ℝ≥0\nf : α → β\nx y : α\nr : ℝ≥0∞\nhf : LipschitzWith K f\ns : Set α\n⊢ diam (f '' s) ≤ ↑K * diam s",
"tactic": "apply EMetric.diam_le"
},
{
"state_after": "case h.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nK : ℝ≥0\nf : α → β\nx✝ y✝ : α\nr : ℝ≥0∞\nhf : LipschitzWith K f\ns : Set α\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\n⊢ edist (f x) (f y) ≤ ↑K * diam s",
"state_before": "case h\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nK : ℝ≥0\nf : α → β\nx y : α\nr : ℝ≥0∞\nhf : LipschitzWith K f\ns : Set α\n⊢ ∀ (x : β), x ∈ f '' s → ∀ (y : β), y ∈ f '' s → edist x y ≤ ↑K * diam s",
"tactic": "rintro _ ⟨x, hx, rfl⟩ _ ⟨y, hy, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case h.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nK : ℝ≥0\nf : α → β\nx✝ y✝ : α\nr : ℝ≥0∞\nhf : LipschitzWith K f\ns : Set α\nx : α\nhx : x ∈ s\ny : α\nhy : y ∈ s\n⊢ edist (f x) (f y) ≤ ↑K * diam s",
"tactic": "exact hf.edist_le_mul_of_le (EMetric.edist_le_diam_of_mem hx hy)"
}
]
| [
178,
67
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
174,
1
]
|
Mathlib/Algebra/Algebra/Operations.lean | Submodule.pow_mem_pow | []
| [
416,
43
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
415,
1
]
|
Mathlib/MeasureTheory/Measure/MeasureSpace.lean | MeasureTheory.measure_biUnion₀ | [
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.8654\nδ : Type ?u.8657\nι : Type ?u.8660\nR : Type ?u.8663\nR' : Type ?u.8666\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\ns : Set β\nf : β → Set α\nhs : Set.Countable s\nhd : Set.Pairwise s (AEDisjoint μ on f)\nh : ∀ (b : β), b ∈ s → NullMeasurableSet (f b)\nthis : Encodable ↑s\n⊢ ↑↑μ (⋃ (b : β) (_ : b ∈ s), f b) = ∑' (p : ↑s), ↑↑μ (f ↑p)",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.8654\nδ : Type ?u.8657\nι : Type ?u.8660\nR : Type ?u.8663\nR' : Type ?u.8666\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\ns : Set β\nf : β → Set α\nhs : Set.Countable s\nhd : Set.Pairwise s (AEDisjoint μ on f)\nh : ∀ (b : β), b ∈ s → NullMeasurableSet (f b)\n⊢ ↑↑μ (⋃ (b : β) (_ : b ∈ s), f b) = ∑' (p : ↑s), ↑↑μ (f ↑p)",
"tactic": "haveI := hs.toEncodable"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.8654\nδ : Type ?u.8657\nι : Type ?u.8660\nR : Type ?u.8663\nR' : Type ?u.8666\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\ns : Set β\nf : β → Set α\nhs : Set.Countable s\nhd : Set.Pairwise s (AEDisjoint μ on f)\nh : ∀ (b : β), b ∈ s → NullMeasurableSet (f b)\nthis : Encodable ↑s\n⊢ ↑↑μ (⋃ (x : ↑s), f ↑x) = ∑' (p : ↑s), ↑↑μ (f ↑p)",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.8654\nδ : Type ?u.8657\nι : Type ?u.8660\nR : Type ?u.8663\nR' : Type ?u.8666\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\ns : Set β\nf : β → Set α\nhs : Set.Countable s\nhd : Set.Pairwise s (AEDisjoint μ on f)\nh : ∀ (b : β), b ∈ s → NullMeasurableSet (f b)\nthis : Encodable ↑s\n⊢ ↑↑μ (⋃ (b : β) (_ : b ∈ s), f b) = ∑' (p : ↑s), ↑↑μ (f ↑p)",
"tactic": "rw [biUnion_eq_iUnion]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.8654\nδ : Type ?u.8657\nι : Type ?u.8660\nR : Type ?u.8663\nR' : Type ?u.8666\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\ns : Set β\nf : β → Set α\nhs : Set.Countable s\nhd : Set.Pairwise s (AEDisjoint μ on f)\nh : ∀ (b : β), b ∈ s → NullMeasurableSet (f b)\nthis : Encodable ↑s\n⊢ ↑↑μ (⋃ (x : ↑s), f ↑x) = ∑' (p : ↑s), ↑↑μ (f ↑p)",
"tactic": "exact measure_iUnion₀ (hd.on_injective Subtype.coe_injective fun x => x.2) fun x => h x x.2"
}
]
| [
165,
94
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
160,
1
]
|
Mathlib/Topology/Support.lean | hasCompactMulSupport_iff_eventuallyEq | []
| [
176,
98
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
169,
1
]
|
Mathlib/Analysis/Calculus/ContDiff.lean | ContDiffWithinAt.mul | []
| [
1387,
50
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1385,
1
]
|
Mathlib/Topology/Semicontinuous.lean | ContinuousAt.comp_upperSemicontinuousWithinAt_antitone | []
| [
862,
90
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
859,
1
]
|
Mathlib/Analysis/Calculus/FDeriv/Prod.lean | hasStrictFDerivAt_pi | []
| [
390,
24
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
387,
1
]
|
Mathlib/Algebra/Order/AbsoluteValue.lean | AbsoluteValue.sub_le | [
{
"state_after": "no goals",
"state_before": "R : Type u_2\nS : Type u_1\ninst✝¹ : Ring R\ninst✝ : OrderedSemiring S\nabv : AbsoluteValue R S\na b c : R\n⊢ ↑abv (a - c) ≤ ↑abv (a - b) + ↑abv (b - c)",
"tactic": "simpa [sub_eq_add_neg, add_assoc] using abv.add_le (a - b) (b - c)"
}
]
| [
156,
69
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
155,
11
]
|
Mathlib/Logic/Basic.lean | Imp.swap | []
| [
272,
75
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
272,
1
]
|
Mathlib/CategoryTheory/Limits/Creates.lean | CategoryTheory.hasLimit_of_created | []
| [
163,
40
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
159,
1
]
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean | MeasureTheory.lintegral_dirac' | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.1688682\nγ : Type ?u.1688685\nδ : Type ?u.1688688\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝ : MeasurableSpace α\na : α\nf : α → ℝ≥0∞\nhf : Measurable f\n⊢ (∫⁻ (a : α), f a ∂dirac a) = f a",
"tactic": "simp [lintegral_congr_ae (ae_eq_dirac' hf)]"
}
]
| [
1371,
49
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1370,
1
]
|
Mathlib/Algebra/Order/Monoid/WithTop.lean | WithBot.add_lt_add_of_le_of_lt | []
| [
717,
65
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
714,
11
]
|
Mathlib/Topology/Order/Basic.lean | nhds_eq_order | [
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : TopologicalSpace α\ninst✝ : Preorder α\nt : OrderTopology α\na : α\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ {s | ∃ a, s = Ioi a ∨ s = Iio a}}), 𝓟 s) =\n (⨅ (b : α) (_ : b ∈ Iio a), 𝓟 (Ioi b)) ⊓ ⨅ (b : α) (_ : b ∈ Ioi a), 𝓟 (Iio b)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : TopologicalSpace α\ninst✝ : Preorder α\nt : OrderTopology α\na : α\n⊢ 𝓝 a = (⨅ (b : α) (_ : b ∈ Iio a), 𝓟 (Ioi b)) ⊓ ⨅ (b : α) (_ : b ∈ Ioi a), 𝓟 (Iio b)",
"tactic": "rw [t.topology_eq_generate_intervals, nhds_generateFrom]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : TopologicalSpace α\ninst✝ : Preorder α\nt : OrderTopology α\na : α\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ {s | ∃ a, s = Ioi a ∨ s = Iio a}}), 𝓟 s) =\n (⨅ (b : α) (_ : b ∈ Iio a), 𝓟 (Ioi b)) ⊓ ⨅ (b : α) (_ : b ∈ Ioi a), 𝓟 (Iio b)",
"tactic": "simp_rw [mem_setOf_eq, @and_comm (a ∈ _), exists_or, or_and_right, iInf_or, iInf_and, iInf_exists,\n iInf_inf_eq, iInf_comm (ι := Set α), iInf_iInf_eq_left, mem_Ioi, mem_Iio]"
}
]
| [
930,
78
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
927,
1
]
|
Mathlib/Data/Set/Intervals/Basic.lean | Set.Icc_subset_Iio_iff | []
| [
574,
64
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
573,
1
]
|
Mathlib/Algebra/Lie/Submodule.lean | LieIdeal.coe_toSubalgebra | []
| [
251,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
250,
1
]
|
Mathlib/Topology/Algebra/Monoid.lean | Filter.Tendsto.mul_const | []
| [
139,
27
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
137,
1
]
|
Mathlib/LinearAlgebra/Dual.lean | Module.dualProdDualEquivDual_apply | []
| [
178,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
176,
1
]
|
Mathlib/Data/Set/Function.lean | Set.surjective_iff_surjOn_univ | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.44646\nι : Sort ?u.44649\nπ : α → Type ?u.44654\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\np : Set γ\nf f₁ f₂ f₃ : α → β\ng g₁ g₂ : β → γ\nf' f₁' f₂' : β → α\ng' : γ → β\na : α\nb : β\n⊢ Surjective f ↔ SurjOn f univ univ",
"tactic": "simp [Surjective, SurjOn, subset_def]"
}
]
| [
852,
40
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
851,
1
]
|
Mathlib/Data/Int/Log.lean | Int.zpow_le_iff_le_log | []
| [
184,
72
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
182,
1
]
|
Mathlib/Data/List/OfFn.lean | List.ofFn_succ' | [
{
"state_after": "case zero\nα : Type u\nn : ℕ\nf✝ : Fin (succ n) → α\nf : Fin (succ zero) → α\n⊢ ofFn f = concat (ofFn fun i => f (↑Fin.castSucc i)) (f (Fin.last zero))\n\ncase succ\nα : Type u\nn✝ : ℕ\nf✝ : Fin (succ n✝) → α\nn : ℕ\nIH : ∀ (f : Fin (succ n) → α), ofFn f = concat (ofFn fun i => f (↑Fin.castSucc i)) (f (Fin.last n))\nf : Fin (succ (succ n)) → α\n⊢ ofFn f = concat (ofFn fun i => f (↑Fin.castSucc i)) (f (Fin.last (succ n)))",
"state_before": "α : Type u\nn : ℕ\nf : Fin (succ n) → α\n⊢ ofFn f = concat (ofFn fun i => f (↑Fin.castSucc i)) (f (Fin.last n))",
"tactic": "induction' n with n IH"
},
{
"state_after": "case zero\nα : Type u\nn : ℕ\nf✝ : Fin (succ n) → α\nf : Fin (succ zero) → α\n⊢ [f 0] = [f (Fin.last zero)]",
"state_before": "case zero\nα : Type u\nn : ℕ\nf✝ : Fin (succ n) → α\nf : Fin (succ zero) → α\n⊢ ofFn f = concat (ofFn fun i => f (↑Fin.castSucc i)) (f (Fin.last zero))",
"tactic": "rw [ofFn_zero, concat_nil, ofFn_succ, ofFn_zero]"
},
{
"state_after": "no goals",
"state_before": "case zero\nα : Type u\nn : ℕ\nf✝ : Fin (succ n) → α\nf : Fin (succ zero) → α\n⊢ [f 0] = [f (Fin.last zero)]",
"tactic": "rfl"
},
{
"state_after": "case succ\nα : Type u\nn✝ : ℕ\nf✝ : Fin (succ n✝) → α\nn : ℕ\nIH : ∀ (f : Fin (succ n) → α), ofFn f = concat (ofFn fun i => f (↑Fin.castSucc i)) (f (Fin.last n))\nf : Fin (succ (succ n)) → α\n⊢ f 0 :: concat (ofFn fun i => f (Fin.succ (↑Fin.castSucc i))) (f (Fin.succ (Fin.last n))) =\n f 0 :: concat (ofFn fun i => f (↑Fin.castSucc (Fin.succ i))) (f (Fin.last (succ n)))",
"state_before": "case succ\nα : Type u\nn✝ : ℕ\nf✝ : Fin (succ n✝) → α\nn : ℕ\nIH : ∀ (f : Fin (succ n) → α), ofFn f = concat (ofFn fun i => f (↑Fin.castSucc i)) (f (Fin.last n))\nf : Fin (succ (succ n)) → α\n⊢ ofFn f = concat (ofFn fun i => f (↑Fin.castSucc i)) (f (Fin.last (succ n)))",
"tactic": "rw [ofFn_succ, IH, ofFn_succ, concat_cons, Fin.castSucc_zero]"
},
{
"state_after": "no goals",
"state_before": "case succ\nα : Type u\nn✝ : ℕ\nf✝ : Fin (succ n✝) → α\nn : ℕ\nIH : ∀ (f : Fin (succ n) → α), ofFn f = concat (ofFn fun i => f (↑Fin.castSucc i)) (f (Fin.last n))\nf : Fin (succ (succ n)) → α\n⊢ f 0 :: concat (ofFn fun i => f (Fin.succ (↑Fin.castSucc i))) (f (Fin.succ (Fin.last n))) =\n f 0 :: concat (ofFn fun i => f (↑Fin.castSucc (Fin.succ i))) (f (Fin.last (succ n)))",
"tactic": "congr"
}
]
| [
130,
10
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
124,
1
]
|
Mathlib/RingTheory/Subring/Basic.lean | RingHom.eqOn_set_closure | []
| [
1237,
51
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1235,
1
]
|
Mathlib/Order/Bounds/Basic.lean | OrderBot.lowerBounds_univ | []
| [
822,
37
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
820,
1
]
|
Mathlib/CategoryTheory/Limits/Shapes/Types.lean | CategoryTheory.Limits.Types.coequalizer_preimage_image_eq_of_preimage_eq | [
{
"state_after": "X Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : f ⁻¹' U = g ⁻¹' U\nlem : ∀ (x y : Y), CoequalizerRel f g x y → (x ∈ U ↔ y ∈ U)\n⊢ π ⁻¹' (π '' U) = U",
"state_before": "X Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : f ⁻¹' U = g ⁻¹' U\n⊢ π ⁻¹' (π '' U) = U",
"tactic": "have lem : ∀ x y, CoequalizerRel f g x y → (x ∈ U ↔ y ∈ U) := by\n rintro _ _ ⟨x⟩\n change x ∈ f ⁻¹' U ↔ x ∈ g ⁻¹' U\n rw [H]"
},
{
"state_after": "X Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : f ⁻¹' U = g ⁻¹' U\nlem : ∀ (x y : Y), CoequalizerRel f g x y → (x ∈ U ↔ y ∈ U)\neqv : _root_.Equivalence fun x y => x ∈ U ↔ y ∈ U\n⊢ π ⁻¹' (π '' U) = U",
"state_before": "X Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : f ⁻¹' U = g ⁻¹' U\nlem : ∀ (x y : Y), CoequalizerRel f g x y → (x ∈ U ↔ y ∈ U)\n⊢ π ⁻¹' (π '' U) = U",
"tactic": "have eqv : _root_.Equivalence fun x y => x ∈ U ↔ y ∈ U :=\n { refl := by tauto\n symm := by tauto\n trans := by tauto }"
},
{
"state_after": "case h\nX Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : f ⁻¹' U = g ⁻¹' U\nlem : ∀ (x y : Y), CoequalizerRel f g x y → (x ∈ U ↔ y ∈ U)\neqv : _root_.Equivalence fun x y => x ∈ U ↔ y ∈ U\nx✝ : Y\n⊢ x✝ ∈ π ⁻¹' (π '' U) ↔ x✝ ∈ U",
"state_before": "X Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : f ⁻¹' U = g ⁻¹' U\nlem : ∀ (x y : Y), CoequalizerRel f g x y → (x ∈ U ↔ y ∈ U)\neqv : _root_.Equivalence fun x y => x ∈ U ↔ y ∈ U\n⊢ π ⁻¹' (π '' U) = U",
"tactic": "ext"
},
{
"state_after": "case h.mp\nX Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : f ⁻¹' U = g ⁻¹' U\nlem : ∀ (x y : Y), CoequalizerRel f g x y → (x ∈ U ↔ y ∈ U)\neqv : _root_.Equivalence fun x y => x ∈ U ↔ y ∈ U\nx✝ : Y\n⊢ x✝ ∈ π ⁻¹' (π '' U) → x✝ ∈ U\n\ncase h.mpr\nX Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : f ⁻¹' U = g ⁻¹' U\nlem : ∀ (x y : Y), CoequalizerRel f g x y → (x ∈ U ↔ y ∈ U)\neqv : _root_.Equivalence fun x y => x ∈ U ↔ y ∈ U\nx✝ : Y\n⊢ x✝ ∈ U → x✝ ∈ π ⁻¹' (π '' U)",
"state_before": "case h\nX Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : f ⁻¹' U = g ⁻¹' U\nlem : ∀ (x y : Y), CoequalizerRel f g x y → (x ∈ U ↔ y ∈ U)\neqv : _root_.Equivalence fun x y => x ∈ U ↔ y ∈ U\nx✝ : Y\n⊢ x✝ ∈ π ⁻¹' (π '' U) ↔ x✝ ∈ U",
"tactic": "constructor"
},
{
"state_after": "case Rel\nX Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : f ⁻¹' U = g ⁻¹' U\nx : X\n⊢ f x ∈ U ↔ g x ∈ U",
"state_before": "X Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : f ⁻¹' U = g ⁻¹' U\n⊢ ∀ (x y : Y), CoequalizerRel f g x y → (x ∈ U ↔ y ∈ U)",
"tactic": "rintro _ _ ⟨x⟩"
},
{
"state_after": "case Rel\nX Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : f ⁻¹' U = g ⁻¹' U\nx : X\n⊢ x ∈ f ⁻¹' U ↔ x ∈ g ⁻¹' U",
"state_before": "case Rel\nX Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : f ⁻¹' U = g ⁻¹' U\nx : X\n⊢ f x ∈ U ↔ g x ∈ U",
"tactic": "change x ∈ f ⁻¹' U ↔ x ∈ g ⁻¹' U"
},
{
"state_after": "no goals",
"state_before": "case Rel\nX Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : f ⁻¹' U = g ⁻¹' U\nx : X\n⊢ x ∈ f ⁻¹' U ↔ x ∈ g ⁻¹' U",
"tactic": "rw [H]"
},
{
"state_after": "no goals",
"state_before": "X Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : f ⁻¹' U = g ⁻¹' U\nlem : ∀ (x y : Y), CoequalizerRel f g x y → (x ∈ U ↔ y ∈ U)\n⊢ ∀ (x : Y), x ∈ U ↔ x ∈ U",
"tactic": "tauto"
},
{
"state_after": "no goals",
"state_before": "X Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : f ⁻¹' U = g ⁻¹' U\nlem : ∀ (x y : Y), CoequalizerRel f g x y → (x ∈ U ↔ y ∈ U)\n⊢ ∀ {x y : Y}, (x ∈ U ↔ y ∈ U) → (y ∈ U ↔ x ∈ U)",
"tactic": "tauto"
},
{
"state_after": "no goals",
"state_before": "X Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : f ⁻¹' U = g ⁻¹' U\nlem : ∀ (x y : Y), CoequalizerRel f g x y → (x ∈ U ↔ y ∈ U)\n⊢ ∀ {x y z : Y}, (x ∈ U ↔ y ∈ U) → (y ∈ U ↔ z ∈ U) → (x ∈ U ↔ z ∈ U)",
"tactic": "tauto"
},
{
"state_after": "case h.mp\nX Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : f ⁻¹' U = g ⁻¹' U\nlem : ∀ (x y : Y), CoequalizerRel f g x y → (x ∈ U ↔ y ∈ U)\neqv : _root_.Equivalence fun x y => x ∈ U ↔ y ∈ U\nx✝ : Y\n⊢ x✝ ∈\n ((coequalizerColimit f g).cocone.ι.app WalkingParallelPair.one ≫\n (IsColimit.coconePointUniqueUpToIso h (coequalizerColimit f g).isColimit).inv) ⁻¹'\n (((coequalizerColimit f g).cocone.ι.app WalkingParallelPair.one ≫\n (IsColimit.coconePointUniqueUpToIso h (coequalizerColimit f g).isColimit).inv) ''\n U) →\n x✝ ∈ U",
"state_before": "case h.mp\nX Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : f ⁻¹' U = g ⁻¹' U\nlem : ∀ (x y : Y), CoequalizerRel f g x y → (x ∈ U ↔ y ∈ U)\neqv : _root_.Equivalence fun x y => x ∈ U ↔ y ∈ U\nx✝ : Y\n⊢ x✝ ∈ π ⁻¹' (π '' U) → x✝ ∈ U",
"tactic": "rw [←\n show _ = π from\n h.comp_coconePointUniqueUpToIso_inv (coequalizerColimit f g).2\n WalkingParallelPair.one]"
},
{
"state_after": "case h.mp.intro.intro\nX Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : f ⁻¹' U = g ⁻¹' U\nlem : ∀ (x y : Y), CoequalizerRel f g x y → (x ∈ U ↔ y ∈ U)\neqv : _root_.Equivalence fun x y => x ∈ U ↔ y ∈ U\nx✝ y : Y\nhy : y ∈ U\ne' :\n ((coequalizerColimit f g).cocone.ι.app WalkingParallelPair.one ≫\n (IsColimit.coconePointUniqueUpToIso h (coequalizerColimit f g).isColimit).inv)\n y =\n ((coequalizerColimit f g).cocone.ι.app WalkingParallelPair.one ≫\n (IsColimit.coconePointUniqueUpToIso h (coequalizerColimit f g).isColimit).inv)\n x✝\n⊢ x✝ ∈ U",
"state_before": "case h.mp\nX Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : f ⁻¹' U = g ⁻¹' U\nlem : ∀ (x y : Y), CoequalizerRel f g x y → (x ∈ U ↔ y ∈ U)\neqv : _root_.Equivalence fun x y => x ∈ U ↔ y ∈ U\nx✝ : Y\n⊢ x✝ ∈\n ((coequalizerColimit f g).cocone.ι.app WalkingParallelPair.one ≫\n (IsColimit.coconePointUniqueUpToIso h (coequalizerColimit f g).isColimit).inv) ⁻¹'\n (((coequalizerColimit f g).cocone.ι.app WalkingParallelPair.one ≫\n (IsColimit.coconePointUniqueUpToIso h (coequalizerColimit f g).isColimit).inv) ''\n U) →\n x✝ ∈ U",
"tactic": "rintro ⟨y, hy, e'⟩"
},
{
"state_after": "case h.mp.intro.intro\nX Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : f ⁻¹' U = g ⁻¹' U\nlem : ∀ (x y : Y), CoequalizerRel f g x y → (x ∈ U ↔ y ∈ U)\neqv : _root_.Equivalence fun x y => x ∈ U ↔ y ∈ U\nx✝ y : Y\nhy : y ∈ U\ne' :\n (IsColimit.coconePointUniqueUpToIso h (coequalizerColimit f g).isColimit).inv\n (Cofork.π (coequalizerColimit f g).cocone y) =\n (IsColimit.coconePointUniqueUpToIso h (coequalizerColimit f g).isColimit).inv\n (Cofork.π (coequalizerColimit f g).cocone x✝)\n⊢ x✝ ∈ U",
"state_before": "case h.mp.intro.intro\nX Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : f ⁻¹' U = g ⁻¹' U\nlem : ∀ (x y : Y), CoequalizerRel f g x y → (x ∈ U ↔ y ∈ U)\neqv : _root_.Equivalence fun x y => x ∈ U ↔ y ∈ U\nx✝ y : Y\nhy : y ∈ U\ne' :\n ((coequalizerColimit f g).cocone.ι.app WalkingParallelPair.one ≫\n (IsColimit.coconePointUniqueUpToIso h (coequalizerColimit f g).isColimit).inv)\n y =\n ((coequalizerColimit f g).cocone.ι.app WalkingParallelPair.one ≫\n (IsColimit.coconePointUniqueUpToIso h (coequalizerColimit f g).isColimit).inv)\n x✝\n⊢ x✝ ∈ U",
"tactic": "dsimp at e'"
},
{
"state_after": "case h.mp.intro.intro\nX Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : f ⁻¹' U = g ⁻¹' U\nlem : ∀ (x y : Y), CoequalizerRel f g x y → (x ∈ U ↔ y ∈ U)\neqv : _root_.Equivalence fun x y => x ∈ U ↔ y ∈ U\nx✝ y : Y\nhy : y ∈ U\ne' : Cofork.π (coequalizerColimit f g).cocone y = Cofork.π (coequalizerColimit f g).cocone x✝\n⊢ x✝ ∈ U",
"state_before": "case h.mp.intro.intro\nX Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : f ⁻¹' U = g ⁻¹' U\nlem : ∀ (x y : Y), CoequalizerRel f g x y → (x ∈ U ↔ y ∈ U)\neqv : _root_.Equivalence fun x y => x ∈ U ↔ y ∈ U\nx✝ y : Y\nhy : y ∈ U\ne' :\n (IsColimit.coconePointUniqueUpToIso h (coequalizerColimit f g).isColimit).inv\n (Cofork.π (coequalizerColimit f g).cocone y) =\n (IsColimit.coconePointUniqueUpToIso h (coequalizerColimit f g).isColimit).inv\n (Cofork.π (coequalizerColimit f g).cocone x✝)\n⊢ x✝ ∈ U",
"tactic": "replace e' :=\n (mono_iff_injective\n (h.coconePointUniqueUpToIso (coequalizerColimit f g).isColimit).inv).mp\n inferInstance e'"
},
{
"state_after": "no goals",
"state_before": "case h.mp.intro.intro\nX Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : f ⁻¹' U = g ⁻¹' U\nlem : ∀ (x y : Y), CoequalizerRel f g x y → (x ∈ U ↔ y ∈ U)\neqv : _root_.Equivalence fun x y => x ∈ U ↔ y ∈ U\nx✝ y : Y\nhy : y ∈ U\ne' : Cofork.π (coequalizerColimit f g).cocone y = Cofork.π (coequalizerColimit f g).cocone x✝\n⊢ x✝ ∈ U",
"tactic": "exact (eqv.eqvGen_iff.mp (EqvGen.mono lem (Quot.exact _ e'))).mp hy"
},
{
"state_after": "no goals",
"state_before": "case h.mpr\nX Y Z : Type u\nf g : X ⟶ Y\nπ : Y ⟶ Z\ne : f ≫ π = g ≫ π\nh : IsColimit (Cofork.ofπ π e)\nU : Set Y\nH : f ⁻¹' U = g ⁻¹' U\nlem : ∀ (x y : Y), CoequalizerRel f g x y → (x ∈ U ↔ y ∈ U)\neqv : _root_.Equivalence fun x y => x ∈ U ↔ y ∈ U\nx✝ : Y\n⊢ x✝ ∈ U → x✝ ∈ π ⁻¹' (π '' U)",
"tactic": "exact fun hx => ⟨_, hx, rfl⟩"
}
]
| [
503,
33
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
479,
1
]
|
Mathlib/Order/Disjoint.lean | Codisjoint.sup_left | []
| [
339,
26
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
338,
1
]
|
Mathlib/Analysis/InnerProductSpace/Basic.lean | LinearMap.isometryOfInner_toLinearMap | []
| [
1300,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1298,
1
]
|
Mathlib/Data/Set/Pointwise/SMul.lean | Set.iUnion_op_smul_set | []
| [
426,
23
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
425,
1
]
|
Mathlib/Analysis/Calculus/FDeriv/Add.lean | HasFDerivAtFilter.const_smul | []
| [
73,
63
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
71,
1
]
|
Mathlib/Init/Data/Nat/Bitwise.lean | Nat.div2_succ | [
{
"state_after": "n : ℕ\n⊢ (match boddDiv2 n with\n | (false, m) => (true, m)\n | (true, m) => (false, succ m)).snd =\n bif (boddDiv2 n).fst then succ (boddDiv2 n).snd else (boddDiv2 n).snd",
"state_before": "n : ℕ\n⊢ div2 (succ n) = bif bodd n then succ (div2 n) else div2 n",
"tactic": "simp only [bodd, boddDiv2, div2]"
},
{
"state_after": "case mk\nn : ℕ\nfst : Bool\nsnd : ℕ\n⊢ (match (fst, snd) with\n | (false, m) => (true, m)\n | (true, m) => (false, succ m)).snd =\n bif (fst, snd).fst then succ (fst, snd).snd else (fst, snd).snd",
"state_before": "n : ℕ\n⊢ (match boddDiv2 n with\n | (false, m) => (true, m)\n | (true, m) => (false, succ m)).snd =\n bif (boddDiv2 n).fst then succ (boddDiv2 n).snd else (boddDiv2 n).snd",
"tactic": "cases' boddDiv2 n with fst snd"
},
{
"state_after": "case mk.false\nn snd : ℕ\n⊢ (match (false, snd) with\n | (false, m) => (true, m)\n | (true, m) => (false, succ m)).snd =\n bif (false, snd).fst then succ (false, snd).snd else (false, snd).snd\n\ncase mk.true\nn snd : ℕ\n⊢ (match (true, snd) with\n | (false, m) => (true, m)\n | (true, m) => (false, succ m)).snd =\n bif (true, snd).fst then succ (true, snd).snd else (true, snd).snd",
"state_before": "case mk\nn : ℕ\nfst : Bool\nsnd : ℕ\n⊢ (match (fst, snd) with\n | (false, m) => (true, m)\n | (true, m) => (false, succ m)).snd =\n bif (fst, snd).fst then succ (fst, snd).snd else (fst, snd).snd",
"tactic": "cases fst"
},
{
"state_after": "case mk.true\nn snd : ℕ\n⊢ (match (true, snd) with\n | (false, m) => (true, m)\n | (true, m) => (false, succ m)).snd =\n bif (true, snd).fst then succ (true, snd).snd else (true, snd).snd",
"state_before": "case mk.false\nn snd : ℕ\n⊢ (match (false, snd) with\n | (false, m) => (true, m)\n | (true, m) => (false, succ m)).snd =\n bif (false, snd).fst then succ (false, snd).snd else (false, snd).snd\n\ncase mk.true\nn snd : ℕ\n⊢ (match (true, snd) with\n | (false, m) => (true, m)\n | (true, m) => (false, succ m)).snd =\n bif (true, snd).fst then succ (true, snd).snd else (true, snd).snd",
"tactic": "case mk.false =>\n simp"
},
{
"state_after": "no goals",
"state_before": "case mk.true\nn snd : ℕ\n⊢ (match (true, snd) with\n | (false, m) => (true, m)\n | (true, m) => (false, succ m)).snd =\n bif (true, snd).fst then succ (true, snd).snd else (true, snd).snd",
"tactic": "case mk.true =>\n simp"
},
{
"state_after": "no goals",
"state_before": "n snd : ℕ\n⊢ (match (false, snd) with\n | (false, m) => (true, m)\n | (true, m) => (false, succ m)).snd =\n bif (false, snd).fst then succ (false, snd).snd else (false, snd).snd",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "n snd : ℕ\n⊢ (match (true, snd) with\n | (false, m) => (true, m)\n | (true, m) => (false, succ m)).snd =\n bif (true, snd).fst then succ (true, snd).snd else (true, snd).snd",
"tactic": "simp"
}
]
| [
125,
9
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
118,
1
]
|
Mathlib/Topology/DiscreteQuotient.lean | DiscreteQuotient.isClosed_preimage | []
| [
150,
28
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
149,
1
]
|
Mathlib/Data/Matrix/Reflection.lean | Matrix.dotProductᵣ_eq | [
{
"state_after": "no goals",
"state_before": "l m✝ n : ℕ\nα : Type u_1\nβ : Type ?u.26440\ninst✝¹ : Mul α\ninst✝ : AddCommMonoid α\nm : ℕ\na b : Fin m → α\n⊢ dotProductᵣ a b = a ⬝ᵥ b",
"tactic": "simp_rw [dotProductᵣ, dotProduct, FinVec.sum_eq, FinVec.seq_eq, FinVec.map_eq,\n Function.comp_apply]"
}
]
| [
138,
27
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
135,
1
]
|
Mathlib/Data/Int/Bitwise.lean | Int.shiftr_coe_nat | [
{
"state_after": "no goals",
"state_before": "m n : ℕ\n⊢ shiftr ↑m ↑n = ↑(Nat.shiftr m n)",
"tactic": "cases n <;> rfl"
}
]
| [
387,
85
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
387,
1
]
|
Mathlib/Data/Set/Intervals/Monotone.lean | Antitone.Iio | []
| [
97,
32
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
96,
11
]
|
Mathlib/Analysis/Asymptotics/Theta.lean | Asymptotics.isTheta_sup | []
| [
200,
75
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
199,
1
]
|
Mathlib/Data/Nat/Bitwise.lean | Nat.lxor'_self | [
{
"state_after": "no goals",
"state_before": "n i : ℕ\n⊢ testBit (lxor' n n) i = false",
"tactic": "simp"
}
]
| [
239,
44
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
238,
1
]
|
Mathlib/Analysis/SpecialFunctions/Exp.lean | ContinuousOn.cexp | []
| [
111,
28
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
110,
1
]
|
Mathlib/Data/Finset/NAry.lean | Finset.image₂_insert_right | [
{
"state_after": "α : Type u_3\nα' : Type ?u.38450\nβ : Type u_1\nβ' : Type ?u.38456\nγ : Type u_2\nγ' : Type ?u.38462\nδ : Type ?u.38465\nδ' : Type ?u.38468\nε : Type ?u.38471\nε' : Type ?u.38474\nζ : Type ?u.38477\nζ' : Type ?u.38480\nν : Type ?u.38483\ninst✝⁸ : DecidableEq α'\ninst✝⁷ : DecidableEq β'\ninst✝⁶ : DecidableEq γ\ninst✝⁵ : DecidableEq γ'\ninst✝⁴ : DecidableEq δ\ninst✝³ : DecidableEq δ'\ninst✝² : DecidableEq ε\ninst✝¹ : DecidableEq ε'\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\ninst✝ : DecidableEq β\n⊢ image2 f (↑s) (insert b ↑t) = (fun a => f a b) '' ↑s ∪ image2 f ↑s ↑t",
"state_before": "α : Type u_3\nα' : Type ?u.38450\nβ : Type u_1\nβ' : Type ?u.38456\nγ : Type u_2\nγ' : Type ?u.38462\nδ : Type ?u.38465\nδ' : Type ?u.38468\nε : Type ?u.38471\nε' : Type ?u.38474\nζ : Type ?u.38477\nζ' : Type ?u.38480\nν : Type ?u.38483\ninst✝⁸ : DecidableEq α'\ninst✝⁷ : DecidableEq β'\ninst✝⁶ : DecidableEq γ\ninst✝⁵ : DecidableEq γ'\ninst✝⁴ : DecidableEq δ\ninst✝³ : DecidableEq δ'\ninst✝² : DecidableEq ε\ninst✝¹ : DecidableEq ε'\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\ninst✝ : DecidableEq β\n⊢ ↑(image₂ f s (insert b t)) = ↑(image (fun a => f a b) s ∪ image₂ f s t)",
"tactic": "push_cast"
},
{
"state_after": "no goals",
"state_before": "α : Type u_3\nα' : Type ?u.38450\nβ : Type u_1\nβ' : Type ?u.38456\nγ : Type u_2\nγ' : Type ?u.38462\nδ : Type ?u.38465\nδ' : Type ?u.38468\nε : Type ?u.38471\nε' : Type ?u.38474\nζ : Type ?u.38477\nζ' : Type ?u.38480\nν : Type ?u.38483\ninst✝⁸ : DecidableEq α'\ninst✝⁷ : DecidableEq β'\ninst✝⁶ : DecidableEq γ\ninst✝⁵ : DecidableEq γ'\ninst✝⁴ : DecidableEq δ\ninst✝³ : DecidableEq δ'\ninst✝² : DecidableEq ε\ninst✝¹ : DecidableEq ε'\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\ninst✝ : DecidableEq β\n⊢ image2 f (↑s) (insert b ↑t) = (fun a => f a b) '' ↑s ∪ image2 f ↑s ↑t",
"tactic": "exact image2_insert_right"
}
]
| [
195,
30
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
191,
1
]
|
Mathlib/Algebra/Hom/Equiv/Basic.lean | MulEquiv.self_trans_symm | []
| [
458,
37
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
457,
1
]
|
Mathlib/Order/MinMax.lean | min_le_of_right_le | []
| [
104,
21
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
103,
1
]
|
Mathlib/Topology/DiscreteQuotient.lean | DiscreteQuotient.isOpen_preimage | []
| [
146,
28
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
145,
1
]
|
Mathlib/MeasureTheory/Constructions/Polish.lean | MeasurableSet.analyticSet | [
{
"state_after": "case intro.intro.intro.intro\nα✝ : Type ?u.62930\ninst✝³ : TopologicalSpace α✝\nι : Type ?u.62936\nα : Type u_1\nt : TopologicalSpace α\ninst✝² : PolishSpace α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\ns : Set α\nhs : MeasurableSet s\nt' : TopologicalSpace α\nt't : t' ≤ t\nt'_polish : PolishSpace α\ns_closed : IsClosed s\nright✝ : IsOpen s\n⊢ AnalyticSet s",
"state_before": "α✝ : Type ?u.62930\ninst✝³ : TopologicalSpace α✝\nι : Type ?u.62936\nα : Type u_1\nt : TopologicalSpace α\ninst✝² : PolishSpace α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\ns : Set α\nhs : MeasurableSet s\n⊢ AnalyticSet s",
"tactic": "obtain ⟨t', t't, t'_polish, s_closed, _⟩ :\n ∃ t' : TopologicalSpace α, t' ≤ t ∧ @PolishSpace α t' ∧ IsClosed[t'] s ∧ IsOpen[t'] s :=\n hs.isClopenable"
},
{
"state_after": "case intro.intro.intro.intro\nα✝ : Type ?u.62930\ninst✝³ : TopologicalSpace α✝\nι : Type ?u.62936\nα : Type u_1\nt : TopologicalSpace α\ninst✝² : PolishSpace α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\ns : Set α\nhs : MeasurableSet s\nt' : TopologicalSpace α\nt't : t' ≤ t\nt'_polish : PolishSpace α\ns_closed : IsClosed s\nright✝ : IsOpen s\nA : AnalyticSet s\n⊢ AnalyticSet s",
"state_before": "case intro.intro.intro.intro\nα✝ : Type ?u.62930\ninst✝³ : TopologicalSpace α✝\nι : Type ?u.62936\nα : Type u_1\nt : TopologicalSpace α\ninst✝² : PolishSpace α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\ns : Set α\nhs : MeasurableSet s\nt' : TopologicalSpace α\nt't : t' ≤ t\nt'_polish : PolishSpace α\ns_closed : IsClosed s\nright✝ : IsOpen s\n⊢ AnalyticSet s",
"tactic": "have A := @IsClosed.analyticSet α t' t'_polish s s_closed"
},
{
"state_after": "case h.e'_3\nα✝ : Type ?u.62930\ninst✝³ : TopologicalSpace α✝\nι : Type ?u.62936\nα : Type u_1\nt : TopologicalSpace α\ninst✝² : PolishSpace α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\ns : Set α\nhs : MeasurableSet s\nt' : TopologicalSpace α\nt't : t' ≤ t\nt'_polish : PolishSpace α\ns_closed : IsClosed s\nright✝ : IsOpen s\nA : AnalyticSet s\n⊢ s = id '' s",
"state_before": "case intro.intro.intro.intro\nα✝ : Type ?u.62930\ninst✝³ : TopologicalSpace α✝\nι : Type ?u.62936\nα : Type u_1\nt : TopologicalSpace α\ninst✝² : PolishSpace α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\ns : Set α\nhs : MeasurableSet s\nt' : TopologicalSpace α\nt't : t' ≤ t\nt'_polish : PolishSpace α\ns_closed : IsClosed s\nright✝ : IsOpen s\nA : AnalyticSet s\n⊢ AnalyticSet s",
"tactic": "convert @AnalyticSet.image_of_continuous α t' α t s A id (continuous_id_of_le t't)"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nα✝ : Type ?u.62930\ninst✝³ : TopologicalSpace α✝\nι : Type ?u.62936\nα : Type u_1\nt : TopologicalSpace α\ninst✝² : PolishSpace α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\ns : Set α\nhs : MeasurableSet s\nt' : TopologicalSpace α\nt't : t' ≤ t\nt'_polish : PolishSpace α\ns_closed : IsClosed s\nright✝ : IsOpen s\nA : AnalyticSet s\n⊢ s = id '' s",
"tactic": "simp only [id.def, image_id']"
}
]
| [
240,
32
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
230,
1
]
|
Mathlib/SetTheory/ZFC/Basic.lean | PSet.toSet_empty | [
{
"state_after": "no goals",
"state_before": "⊢ toSet ∅ = ∅",
"tactic": "simp [toSet]"
}
]
| [
408,
53
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
408,
1
]
|
Std/Data/Int/Lemmas.lean | Int.natAbs_sign_of_nonzero | [
{
"state_after": "no goals",
"state_before": "z : Int\nhz : z ≠ 0\n⊢ natAbs (sign z) = 1",
"tactic": "rw [Int.natAbs_sign, if_neg hz]"
}
]
| [
205,
34
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
204,
1
]
|
Mathlib/Order/LiminfLimsup.lean | Filter.sup_limsup | [
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.164326\nι : Type ?u.164329\ninst✝¹ : CompleteDistribLattice α\nf : Filter β\np q : β → Prop\nu : β → α\ninst✝ : NeBot f\na : α\n⊢ (⨅ (i : Set β) (_ : i ∈ f), a ⊔ ⨆ (a : β) (_ : a ∈ i), u a) =\n ⨅ (s : Set β) (_ : s ∈ f), (⨆ (x : β) (_ : x ∈ s), a) ⊔ ⨆ (x : β) (_ : x ∈ s), u x",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.164326\nι : Type ?u.164329\ninst✝¹ : CompleteDistribLattice α\nf : Filter β\np q : β → Prop\nu : β → α\ninst✝ : NeBot f\na : α\n⊢ a ⊔ limsup u f = limsup (fun x => a ⊔ u x) f",
"tactic": "simp only [limsup_eq_iInf_iSup, iSup_sup_eq, sup_iInf₂_eq]"
},
{
"state_after": "case e_s\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.164326\nι : Type ?u.164329\ninst✝¹ : CompleteDistribLattice α\nf : Filter β\np q : β → Prop\nu : β → α\ninst✝ : NeBot f\na : α\n⊢ (fun i => ⨅ (_ : i ∈ f), a ⊔ ⨆ (a : β) (_ : a ∈ i), u a) = fun s =>\n ⨅ (_ : s ∈ f), (⨆ (x : β) (_ : x ∈ s), a) ⊔ ⨆ (x : β) (_ : x ∈ s), u x",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.164326\nι : Type ?u.164329\ninst✝¹ : CompleteDistribLattice α\nf : Filter β\np q : β → Prop\nu : β → α\ninst✝ : NeBot f\na : α\n⊢ (⨅ (i : Set β) (_ : i ∈ f), a ⊔ ⨆ (a : β) (_ : a ∈ i), u a) =\n ⨅ (s : Set β) (_ : s ∈ f), (⨆ (x : β) (_ : x ∈ s), a) ⊔ ⨆ (x : β) (_ : x ∈ s), u x",
"tactic": "congr"
},
{
"state_after": "case e_s.h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.164326\nι : Type ?u.164329\ninst✝¹ : CompleteDistribLattice α\nf : Filter β\np q : β → Prop\nu : β → α\ninst✝ : NeBot f\na : α\ns : Set β\n⊢ (⨅ (_ : s ∈ f), a ⊔ ⨆ (a : β) (_ : a ∈ s), u a) =\n ⨅ (_ : s ∈ f), (⨆ (x : β) (_ : x ∈ s), a) ⊔ ⨆ (x : β) (_ : x ∈ s), u x",
"state_before": "case e_s\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.164326\nι : Type ?u.164329\ninst✝¹ : CompleteDistribLattice α\nf : Filter β\np q : β → Prop\nu : β → α\ninst✝ : NeBot f\na : α\n⊢ (fun i => ⨅ (_ : i ∈ f), a ⊔ ⨆ (a : β) (_ : a ∈ i), u a) = fun s =>\n ⨅ (_ : s ∈ f), (⨆ (x : β) (_ : x ∈ s), a) ⊔ ⨆ (x : β) (_ : x ∈ s), u x",
"tactic": "ext s"
},
{
"state_after": "case e_s.h.e_s\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.164326\nι : Type ?u.164329\ninst✝¹ : CompleteDistribLattice α\nf : Filter β\np q : β → Prop\nu : β → α\ninst✝ : NeBot f\na : α\ns : Set β\n⊢ (fun j => a ⊔ ⨆ (a : β) (_ : a ∈ s), u a) = fun x => (⨆ (x : β) (_ : x ∈ s), a) ⊔ ⨆ (x : β) (_ : x ∈ s), u x",
"state_before": "case e_s.h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.164326\nι : Type ?u.164329\ninst✝¹ : CompleteDistribLattice α\nf : Filter β\np q : β → Prop\nu : β → α\ninst✝ : NeBot f\na : α\ns : Set β\n⊢ (⨅ (_ : s ∈ f), a ⊔ ⨆ (a : β) (_ : a ∈ s), u a) =\n ⨅ (_ : s ∈ f), (⨆ (x : β) (_ : x ∈ s), a) ⊔ ⨆ (x : β) (_ : x ∈ s), u x",
"tactic": "congr"
},
{
"state_after": "case e_s.h.e_s.h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.164326\nι : Type ?u.164329\ninst✝¹ : CompleteDistribLattice α\nf : Filter β\np q : β → Prop\nu : β → α\ninst✝ : NeBot f\na : α\ns : Set β\nhs : s ∈ f\n⊢ (a ⊔ ⨆ (a : β) (_ : a ∈ s), u a) = (⨆ (x : β) (_ : x ∈ s), a) ⊔ ⨆ (x : β) (_ : x ∈ s), u x",
"state_before": "case e_s.h.e_s\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.164326\nι : Type ?u.164329\ninst✝¹ : CompleteDistribLattice α\nf : Filter β\np q : β → Prop\nu : β → α\ninst✝ : NeBot f\na : α\ns : Set β\n⊢ (fun j => a ⊔ ⨆ (a : β) (_ : a ∈ s), u a) = fun x => (⨆ (x : β) (_ : x ∈ s), a) ⊔ ⨆ (x : β) (_ : x ∈ s), u x",
"tactic": "ext hs"
},
{
"state_after": "case e_s.h.e_s.h.e_a\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.164326\nι : Type ?u.164329\ninst✝¹ : CompleteDistribLattice α\nf : Filter β\np q : β → Prop\nu : β → α\ninst✝ : NeBot f\na : α\ns : Set β\nhs : s ∈ f\n⊢ a = ⨆ (x : β) (_ : x ∈ s), a",
"state_before": "case e_s.h.e_s.h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.164326\nι : Type ?u.164329\ninst✝¹ : CompleteDistribLattice α\nf : Filter β\np q : β → Prop\nu : β → α\ninst✝ : NeBot f\na : α\ns : Set β\nhs : s ∈ f\n⊢ (a ⊔ ⨆ (a : β) (_ : a ∈ s), u a) = (⨆ (x : β) (_ : x ∈ s), a) ⊔ ⨆ (x : β) (_ : x ∈ s), u x",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case e_s.h.e_s.h.e_a\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.164326\nι : Type ?u.164329\ninst✝¹ : CompleteDistribLattice α\nf : Filter β\np q : β → Prop\nu : β → α\ninst✝ : NeBot f\na : α\ns : Set β\nhs : s ∈ f\n⊢ a = ⨆ (x : β) (_ : x ∈ s), a",
"tactic": "exact (biSup_const (nonempty_of_mem hs)).symm"
}
]
| [
996,
48
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
993,
1
]
|
Mathlib/Order/Lattice.lean | lt_sup_of_lt_right | []
| [
165,
26
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
164,
1
]
|
Mathlib/Data/List/Forall2.lean | List.forall₂_map_right_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type u_2\nδ : Type ?u.21466\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nf : γ → β\nx✝ : List α\n⊢ Forall₂ R x✝ (map f []) ↔ Forall₂ (fun a c => R a (f c)) x✝ []",
"tactic": "simp only [map, forall₂_nil_right_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_3\nγ : Type u_2\nδ : Type ?u.21466\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nf : γ → β\nx✝ : List α\nb : γ\nu : List γ\n⊢ Forall₂ R x✝ (map f (b :: u)) ↔ Forall₂ (fun a c => R a (f c)) x✝ (b :: u)",
"tactic": "simp only [map, forall₂_cons_right_iff, forall₂_map_right_iff]"
}
]
| [
134,
83
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
131,
1
]
|
Mathlib/Init/Logic.lean | ExistsUnique.unique | []
| [
251,
56
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
249,
1
]
|
Mathlib/Data/Fintype/BigOperators.lean | Finset.prod_univ_sum | [
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_3\nγ : Type ?u.45448\ninst✝³ : DecidableEq α\ninst✝² : Fintype α\ninst✝¹ : CommSemiring β\nδ : α → Type u_1\ninst✝ : (a : α) → DecidableEq (δ a)\nt : (a : α) → Finset (δ a)\nf : (a : α) → δ a → β\n⊢ ∏ a : α, ∑ b in t a, f a b = ∑ p in Fintype.piFinset t, ∏ x : α, f x (p x)",
"tactic": "simp only [Finset.prod_attach_univ, prod_sum, Finset.sum_univ_pi]"
}
]
| [
194,
68
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
191,
1
]
|
Mathlib/NumberTheory/Liouville/Basic.lean | Liouville.irrational | [
{
"state_after": "case intro.mk'\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville ↑(Rat.mk' a b)\n⊢ False",
"state_before": "x : ℝ\nh : Liouville x\n⊢ Irrational x",
"tactic": "rintro ⟨⟨a, b, bN0, cop⟩, rfl⟩"
},
{
"state_after": "case intro.mk'\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\n⊢ False",
"state_before": "case intro.mk'\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville ↑(Rat.mk' a b)\n⊢ False",
"tactic": "rw [Rat.cast_mk', ← div_eq_mul_inv] at h"
},
{
"state_after": "case intro.mk'.intro.intro.intro.intro\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\np q : ℤ\nq1 : 1 < q\na0 : ↑a / ↑b ≠ ↑p / ↑q\na1 : abs (↑a / ↑b - ↑p / ↑q) < 1 / ↑q ^ (b + 1)\n⊢ False",
"state_before": "case intro.mk'\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\n⊢ False",
"tactic": "rcases h (b + 1) with ⟨p, q, q1, a0, a1⟩"
},
{
"state_after": "case intro.mk'.intro.intro.intro.intro\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\np q : ℤ\nq1 : 1 < q\na0 : ↑a / ↑b ≠ ↑p / ↑q\na1 : abs (↑a / ↑b - ↑p / ↑q) < 1 / ↑q ^ (b + 1)\nqR0 : 0 < ↑q\n⊢ False",
"state_before": "case intro.mk'.intro.intro.intro.intro\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\np q : ℤ\nq1 : 1 < q\na0 : ↑a / ↑b ≠ ↑p / ↑q\na1 : abs (↑a / ↑b - ↑p / ↑q) < 1 / ↑q ^ (b + 1)\n⊢ False",
"tactic": "have qR0 : (0 : ℝ) < q := Int.cast_pos.mpr (zero_lt_one.trans q1)"
},
{
"state_after": "case intro.mk'.intro.intro.intro.intro\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\np q : ℤ\nq1 : 1 < q\na0 : ↑a / ↑b ≠ ↑p / ↑q\na1 : abs (↑a / ↑b - ↑p / ↑q) < 1 / ↑q ^ (b + 1)\nqR0 : 0 < ↑q\nb0 : ↑b ≠ 0\n⊢ False",
"state_before": "case intro.mk'.intro.intro.intro.intro\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\np q : ℤ\nq1 : 1 < q\na0 : ↑a / ↑b ≠ ↑p / ↑q\na1 : abs (↑a / ↑b - ↑p / ↑q) < 1 / ↑q ^ (b + 1)\nqR0 : 0 < ↑q\n⊢ False",
"tactic": "have b0 : (b : ℝ) ≠ 0 := Nat.cast_ne_zero.mpr bN0"
},
{
"state_after": "case intro.mk'.intro.intro.intro.intro\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\np q : ℤ\nq1 : 1 < q\na0 : ↑a / ↑b ≠ ↑p / ↑q\na1 : abs (↑a / ↑b - ↑p / ↑q) < 1 / ↑q ^ (b + 1)\nqR0 : 0 < ↑q\nb0 : ↑b ≠ 0\nbq0 : 0 < ↑b * ↑q\n⊢ False",
"state_before": "case intro.mk'.intro.intro.intro.intro\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\np q : ℤ\nq1 : 1 < q\na0 : ↑a / ↑b ≠ ↑p / ↑q\na1 : abs (↑a / ↑b - ↑p / ↑q) < 1 / ↑q ^ (b + 1)\nqR0 : 0 < ↑q\nb0 : ↑b ≠ 0\n⊢ False",
"tactic": "have bq0 : (0 : ℝ) < b * q := mul_pos (Nat.cast_pos.mpr bN0.bot_lt) qR0"
},
{
"state_after": "case a1\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\np q : ℤ\nq1 : 1 < q\na0 : ↑a / ↑b ≠ ↑p / ↑q\na1 : abs (↑a / ↑b - ↑p / ↑q) < 1 / ↑q ^ (b + 1)\nqR0 : 0 < ↑q\nb0 : ↑b ≠ 0\nbq0 : 0 < ↑b * ↑q\n⊢ abs (a * q - ↑b * p) * q ^ (b + 1) < ↑b * q\n\ncase intro.mk'.intro.intro.intro.intro\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\np q : ℤ\nq1 : 1 < q\na0 : ↑a / ↑b ≠ ↑p / ↑q\nqR0 : 0 < ↑q\nb0 : ↑b ≠ 0\nbq0 : 0 < ↑b * ↑q\na1 : abs (a * q - ↑b * p) * q ^ (b + 1) < ↑b * q\n⊢ False",
"state_before": "case intro.mk'.intro.intro.intro.intro\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\np q : ℤ\nq1 : 1 < q\na0 : ↑a / ↑b ≠ ↑p / ↑q\na1 : abs (↑a / ↑b - ↑p / ↑q) < 1 / ↑q ^ (b + 1)\nqR0 : 0 < ↑q\nb0 : ↑b ≠ 0\nbq0 : 0 < ↑b * ↑q\n⊢ False",
"tactic": "replace a1 : |a * q - b * p| * q ^ (b + 1) < b * q"
},
{
"state_after": "case a0\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\np q : ℤ\nq1 : 1 < q\na0 : ↑a / ↑b ≠ ↑p / ↑q\nqR0 : 0 < ↑q\nb0 : ↑b ≠ 0\nbq0 : 0 < ↑b * ↑q\na1 : abs (a * q - ↑b * p) * q ^ (b + 1) < ↑b * q\n⊢ a * q - ↑b * p ≠ 0\n\ncase intro.mk'.intro.intro.intro.intro\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\np q : ℤ\nq1 : 1 < q\nqR0 : 0 < ↑q\nb0 : ↑b ≠ 0\nbq0 : 0 < ↑b * ↑q\na1 : abs (a * q - ↑b * p) * q ^ (b + 1) < ↑b * q\na0 : a * q - ↑b * p ≠ 0\n⊢ False",
"state_before": "case intro.mk'.intro.intro.intro.intro\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\np q : ℤ\nq1 : 1 < q\na0 : ↑a / ↑b ≠ ↑p / ↑q\nqR0 : 0 < ↑q\nb0 : ↑b ≠ 0\nbq0 : 0 < ↑b * ↑q\na1 : abs (a * q - ↑b * p) * q ^ (b + 1) < ↑b * q\n⊢ False",
"tactic": "replace a0 : a * q - ↑b * p ≠ 0"
},
{
"state_after": "case intro.mk'.intro.intro.intro.intro.intro\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\np : ℤ\nb0 : ↑b ≠ 0\nq : ℕ\nq1 : 1 < ↑q\nqR0 : 0 < ↑↑q\nbq0 : 0 < ↑b * ↑↑q\na1 : abs (a * ↑q - ↑b * p) * ↑q ^ (b + 1) < ↑b * ↑q\na0 : a * ↑q - ↑b * p ≠ 0\n⊢ False",
"state_before": "case intro.mk'.intro.intro.intro.intro\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\np q : ℤ\nq1 : 1 < q\nqR0 : 0 < ↑q\nb0 : ↑b ≠ 0\nbq0 : 0 < ↑b * ↑q\na1 : abs (a * q - ↑b * p) * q ^ (b + 1) < ↑b * q\na0 : a * q - ↑b * p ≠ 0\n⊢ False",
"tactic": "lift q to ℕ using (zero_lt_one.trans q1).le"
},
{
"state_after": "case intro.mk'.intro.intro.intro.intro.intro\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\np : ℤ\nb0 : ↑b ≠ 0\nq : ℕ\nq1 : 1 < ↑q\nqR0 : 0 < ↑↑q\nbq0 : 0 < ↑b * ↑↑q\na1 : abs (a * ↑q - ↑b * p) * ↑q ^ (b + 1) < ↑b * ↑q\na0 : a * ↑q - ↑b * p ≠ 0\nap : 0 < abs (a * ↑q - ↑b * p)\n⊢ False",
"state_before": "case intro.mk'.intro.intro.intro.intro.intro\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\np : ℤ\nb0 : ↑b ≠ 0\nq : ℕ\nq1 : 1 < ↑q\nqR0 : 0 < ↑↑q\nbq0 : 0 < ↑b * ↑↑q\na1 : abs (a * ↑q - ↑b * p) * ↑q ^ (b + 1) < ↑b * ↑q\na0 : a * ↑q - ↑b * p ≠ 0\n⊢ False",
"tactic": "have ap : 0 < |a * ↑q - ↑b * p| := abs_pos.mpr a0"
},
{
"state_after": "case intro.mk'.intro.intro.intro.intro.intro.intro\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\np : ℤ\nb0 : ↑b ≠ 0\nq : ℕ\nq1 : 1 < ↑q\nqR0 : 0 < ↑↑q\nbq0 : 0 < ↑b * ↑↑q\na0 : a * ↑q - ↑b * p ≠ 0\ne : ℕ\nhe : ↑e = abs (a * ↑q - ↑b * p)\na1✝ a1 : ↑e * ↑q ^ (b + 1) < ↑b * ↑q\nap✝ ap : 0 < ↑e\n⊢ False",
"state_before": "case intro.mk'.intro.intro.intro.intro.intro\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\np : ℤ\nb0 : ↑b ≠ 0\nq : ℕ\nq1 : 1 < ↑q\nqR0 : 0 < ↑↑q\nbq0 : 0 < ↑b * ↑↑q\na1 : abs (a * ↑q - ↑b * p) * ↑q ^ (b + 1) < ↑b * ↑q\na0 : a * ↑q - ↑b * p ≠ 0\nap : 0 < abs (a * ↑q - ↑b * p)\n⊢ False",
"tactic": "lift |a * ↑q - ↑b * p| to ℕ using abs_nonneg (a * ↑q - ↑b * p) with e he"
},
{
"state_after": "case intro.mk'.intro.intro.intro.intro.intro.intro\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\np : ℤ\nb0 : ↑b ≠ 0\nq : ℕ\nq1 : 1 < ↑q\nqR0 : 0 < ↑↑q\nbq0 : 0 < ↑b * ↑↑q\na0 : a * ↑q - ↑b * p ≠ 0\ne : ℕ\nhe : ↑e = abs (a * ↑q - ↑b * p)\na1✝ : ↑e * ↑q ^ (b + 1) < ↑b * ↑q\na1 : e * q ^ (b + 1) < b * q\nap✝ ap : 0 < ↑e\n⊢ False",
"state_before": "case intro.mk'.intro.intro.intro.intro.intro.intro\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\np : ℤ\nb0 : ↑b ≠ 0\nq : ℕ\nq1 : 1 < ↑q\nqR0 : 0 < ↑↑q\nbq0 : 0 < ↑b * ↑↑q\na0 : a * ↑q - ↑b * p ≠ 0\ne : ℕ\nhe : ↑e = abs (a * ↑q - ↑b * p)\na1✝ a1 : ↑e * ↑q ^ (b + 1) < ↑b * ↑q\nap✝ ap : 0 < ↑e\n⊢ False",
"tactic": "rw [← Int.ofNat_mul, ← Int.coe_nat_pow, ← Int.ofNat_mul, Int.ofNat_lt] at a1"
},
{
"state_after": "no goals",
"state_before": "case intro.mk'.intro.intro.intro.intro.intro.intro\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\np : ℤ\nb0 : ↑b ≠ 0\nq : ℕ\nq1 : 1 < ↑q\nqR0 : 0 < ↑↑q\nbq0 : 0 < ↑b * ↑↑q\na0 : a * ↑q - ↑b * p ≠ 0\ne : ℕ\nhe : ↑e = abs (a * ↑q - ↑b * p)\na1✝ : ↑e * ↑q ^ (b + 1) < ↑b * ↑q\na1 : e * q ^ (b + 1) < b * q\nap✝ ap : 0 < ↑e\n⊢ False",
"tactic": "exact not_le.mpr a1 (Nat.mul_lt_mul_pow_succ (Int.coe_nat_pos.mp ap) (Int.ofNat_lt.mp q1)).le"
},
{
"state_after": "case a1\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\np q : ℤ\nq1 : 1 < q\na0 : ↑a / ↑b ≠ ↑p / ↑q\na1 : abs (↑a * ↑q - ↑b * ↑p) * ↑q ^ (b + 1) < ↑b * ↑q\nqR0 : 0 < ↑q\nb0 : ↑b ≠ 0\nbq0 : 0 < ↑b * ↑q\n⊢ abs (a * q - ↑b * p) * q ^ (b + 1) < ↑b * q",
"state_before": "case a1\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\np q : ℤ\nq1 : 1 < q\na0 : ↑a / ↑b ≠ ↑p / ↑q\na1 : abs (↑a / ↑b - ↑p / ↑q) < 1 / ↑q ^ (b + 1)\nqR0 : 0 < ↑q\nb0 : ↑b ≠ 0\nbq0 : 0 < ↑b * ↑q\n⊢ abs (a * q - ↑b * p) * q ^ (b + 1) < ↑b * q",
"tactic": "rw [div_sub_div _ _ b0 qR0.ne', abs_div, div_lt_div_iff (abs_pos.mpr bq0.ne') (pow_pos qR0 _),\n abs_of_pos bq0, one_mul] at a1"
},
{
"state_after": "no goals",
"state_before": "case a1\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\np q : ℤ\nq1 : 1 < q\na0 : ↑a / ↑b ≠ ↑p / ↑q\na1 : abs (↑a * ↑q - ↑b * ↑p) * ↑q ^ (b + 1) < ↑b * ↑q\nqR0 : 0 < ↑q\nb0 : ↑b ≠ 0\nbq0 : 0 < ↑b * ↑q\n⊢ abs (a * q - ↑b * p) * q ^ (b + 1) < ↑b * q",
"tactic": "exact_mod_cast a1"
},
{
"state_after": "case a0\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\np q : ℤ\nq1 : 1 < q\na0✝ : ¬↑a * ↑q = ↑b * ↑p\na0 : ¬↑a * ↑q - ↑b * ↑p = 0\nqR0 : 0 < ↑q\nb0 : ↑b ≠ 0\nbq0 : 0 < ↑b * ↑q\na1 : abs (a * q - ↑b * p) * q ^ (b + 1) < ↑b * q\n⊢ a * q - ↑b * p ≠ 0",
"state_before": "case a0\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\np q : ℤ\nq1 : 1 < q\na0 : ↑a / ↑b ≠ ↑p / ↑q\nqR0 : 0 < ↑q\nb0 : ↑b ≠ 0\nbq0 : 0 < ↑b * ↑q\na1 : abs (a * q - ↑b * p) * q ^ (b + 1) < ↑b * q\n⊢ a * q - ↑b * p ≠ 0",
"tactic": "rw [Ne.def, div_eq_div_iff b0 qR0.ne', mul_comm (p : ℝ), ← sub_eq_zero] at a0"
},
{
"state_after": "no goals",
"state_before": "case a0\na : ℤ\nb : ℕ\nbN0 : b ≠ 0\ncop : Nat.coprime (Int.natAbs a) b\nh : Liouville (↑a / ↑b)\np q : ℤ\nq1 : 1 < q\na0✝ : ¬↑a * ↑q = ↑b * ↑p\na0 : ¬↑a * ↑q - ↑b * ↑p = 0\nqR0 : 0 < ↑q\nb0 : ↑b ≠ 0\nbq0 : 0 < ↑b * ↑q\na1 : abs (a * q - ↑b * p) * q ^ (b + 1) < ↑b * q\n⊢ a * q - ↑b * p ≠ 0",
"tactic": "exact_mod_cast a0"
}
]
| [
72,
96
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
40,
11
]
|
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean | CategoryTheory.Limits.cospan_right | []
| [
212,
52
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
211,
1
]
|
Mathlib/LinearAlgebra/Matrix/ToLin.lean | Algebra.leftMulMatrix_apply | []
| [
889,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
888,
1
]
|
Mathlib/CategoryTheory/Monoidal/CoherenceLemmas.lean | CategoryTheory.MonoidalCategory.triangle_assoc_comp_right_inv | [
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : MonoidalCategory C\nX Y : C\n⊢ ((ρ_ X).inv ⊗ 𝟙 Y) ≫ (α_ X (𝟙_ C) Y).hom = 𝟙 X ⊗ (λ_ Y).inv",
"tactic": "coherence"
}
]
| [
68,
12
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
66,
1
]
|
Mathlib/Algebra/Order/Group/Defs.lean | Right.inv_le_one_iff | [
{
"state_after": "α : Type u\ninst✝² : Group α\ninst✝¹ : LE α\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c : α\n⊢ a⁻¹ * a ≤ 1 * a ↔ 1 ≤ a",
"state_before": "α : Type u\ninst✝² : Group α\ninst✝¹ : LE α\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c : α\n⊢ a⁻¹ ≤ 1 ↔ 1 ≤ a",
"tactic": "rw [← mul_le_mul_iff_right a]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝² : Group α\ninst✝¹ : LE α\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na b c : α\n⊢ a⁻¹ * a ≤ 1 * a ↔ 1 ≤ a",
"tactic": "simp"
}
]
| [
220,
7
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
218,
1
]
|
Mathlib/InformationTheory/Hamming.lean | swap_hammingDist | [
{
"state_after": "case h.h\nα : Type ?u.6729\nι : Type u_1\nβ : ι → Type u_2\ninst✝² : Fintype ι\ninst✝¹ : (i : ι) → DecidableEq (β i)\nγ : ι → Type ?u.6761\ninst✝ : (i : ι) → DecidableEq (γ i)\nx y : (i : ι) → β i\n⊢ swap hammingDist x y = hammingDist x y",
"state_before": "α : Type ?u.6729\nι : Type u_1\nβ : ι → Type u_2\ninst✝² : Fintype ι\ninst✝¹ : (i : ι) → DecidableEq (β i)\nγ : ι → Type ?u.6761\ninst✝ : (i : ι) → DecidableEq (γ i)\n⊢ swap hammingDist = hammingDist",
"tactic": "funext x y"
},
{
"state_after": "no goals",
"state_before": "case h.h\nα : Type ?u.6729\nι : Type u_1\nβ : ι → Type u_2\ninst✝² : Fintype ι\ninst✝¹ : (i : ι) → DecidableEq (β i)\nγ : ι → Type ?u.6761\ninst✝ : (i : ι) → DecidableEq (γ i)\nx y : (i : ι) → β i\n⊢ swap hammingDist x y = hammingDist x y",
"tactic": "exact hammingDist_comm _ _"
}
]
| [
91,
29
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
89,
1
]
|
Mathlib/Order/LocallyFinite.lean | WithTop.Ioc_coe_coe | []
| [
1133,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1132,
1
]
|
Mathlib/GroupTheory/Perm/Cycle/Basic.lean | Equiv.Perm.IsCycleOn.apply_ne | [
{
"state_after": "case intro.intro\nι : Type ?u.1651424\nα : Type u_1\nβ : Type ?u.1651430\nf g : Perm α\ns t : Set α\na b✝ x y : α\nhf : IsCycleOn f s\nhs : Set.Nontrivial s\nha : a ∈ s\nb : α\nhb : b ∈ s\nhba : b ≠ a\n⊢ ↑f a ≠ a",
"state_before": "ι : Type ?u.1651424\nα : Type u_1\nβ : Type ?u.1651430\nf g : Perm α\ns t : Set α\na b x y : α\nhf : IsCycleOn f s\nhs : Set.Nontrivial s\nha : a ∈ s\n⊢ ↑f a ≠ a",
"tactic": "obtain ⟨b, hb, hba⟩ := hs.exists_ne a"
},
{
"state_after": "case intro.intro.intro\nι : Type ?u.1651424\nα : Type u_1\nβ : Type ?u.1651430\nf g : Perm α\ns t : Set α\na b x y : α\nhf : IsCycleOn f s\nhs : Set.Nontrivial s\nha : a ∈ s\nn : ℤ\nhb : ↑(f ^ n) a ∈ s\nhba : ↑(f ^ n) a ≠ a\n⊢ ↑f a ≠ a",
"state_before": "case intro.intro\nι : Type ?u.1651424\nα : Type u_1\nβ : Type ?u.1651430\nf g : Perm α\ns t : Set α\na b✝ x y : α\nhf : IsCycleOn f s\nhs : Set.Nontrivial s\nha : a ∈ s\nb : α\nhb : b ∈ s\nhba : b ≠ a\n⊢ ↑f a ≠ a",
"tactic": "obtain ⟨n, rfl⟩ := hf.2 ha hb"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nι : Type ?u.1651424\nα : Type u_1\nβ : Type ?u.1651430\nf g : Perm α\ns t : Set α\na b x y : α\nhf : IsCycleOn f s\nhs : Set.Nontrivial s\nha : a ∈ s\nn : ℤ\nhb : ↑(f ^ n) a ∈ s\nhba : ↑(f ^ n) a ≠ a\n⊢ ↑f a ≠ a",
"tactic": "exact fun h => hba (IsFixedPt.perm_zpow h n)"
}
]
| [
809,
47
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
805,
11
]
|
Mathlib/Algebra/Star/StarAlgHom.lean | StarAlgHom.coe_prod | []
| [
626,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
625,
1
]
|
Mathlib/Algebra/BigOperators/Basic.lean | Multiset.add_eq_union_left_of_le | [
{
"state_after": "ι : Type ?u.902858\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : DecidableEq α\nx y z : Multiset α\nh : y ≤ x\n⊢ z + x = z ∪ y ↔ z + x = z ∪ x ∧ x = y",
"state_before": "ι : Type ?u.902858\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : DecidableEq α\nx y z : Multiset α\nh : y ≤ x\n⊢ z + x = z ∪ y ↔ Disjoint z x ∧ x = y",
"tactic": "rw [← add_eq_union_iff_disjoint]"
},
{
"state_after": "case mp\nι : Type ?u.902858\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : DecidableEq α\nx y z : Multiset α\nh : y ≤ x\n⊢ z + x = z ∪ y → z + x = z ∪ x ∧ x = y\n\ncase mpr\nι : Type ?u.902858\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : DecidableEq α\nx y z : Multiset α\nh : y ≤ x\n⊢ z + x = z ∪ x ∧ x = y → z + x = z ∪ y",
"state_before": "ι : Type ?u.902858\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : DecidableEq α\nx y z : Multiset α\nh : y ≤ x\n⊢ z + x = z ∪ y ↔ z + x = z ∪ x ∧ x = y",
"tactic": "constructor"
},
{
"state_after": "case mp\nι : Type ?u.902858\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : DecidableEq α\nx y z : Multiset α\nh : y ≤ x\nh0 : z + x = z ∪ y\n⊢ z + x = z ∪ x ∧ x = y",
"state_before": "case mp\nι : Type ?u.902858\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : DecidableEq α\nx y z : Multiset α\nh : y ≤ x\n⊢ z + x = z ∪ y → z + x = z ∪ x ∧ x = y",
"tactic": "intro h0"
},
{
"state_after": "case mp\nι : Type ?u.902858\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : DecidableEq α\nx y z : Multiset α\nh : y ≤ x\nh0 : z + x = z ∪ y\n⊢ x = y\n\ncase mp\nι : Type ?u.902858\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : DecidableEq α\nx y z : Multiset α\nh : y ≤ x\nh0 : z + x = z ∪ y\n⊢ x = y → z + x = z ∪ x",
"state_before": "case mp\nι : Type ?u.902858\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : DecidableEq α\nx y z : Multiset α\nh : y ≤ x\nh0 : z + x = z ∪ y\n⊢ z + x = z ∪ x ∧ x = y",
"tactic": "rw [and_iff_right_of_imp]"
},
{
"state_after": "no goals",
"state_before": "case mp\nι : Type ?u.902858\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : DecidableEq α\nx y z : Multiset α\nh : y ≤ x\nh0 : z + x = z ∪ y\n⊢ x = y",
"tactic": "exact (le_of_add_le_add_left <| h0.trans_le <| union_le_add z y).antisymm h"
},
{
"state_after": "case mp\nι : Type ?u.902858\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : DecidableEq α\nx z : Multiset α\nh : x ≤ x\nh0 : z + x = z ∪ x\n⊢ z + x = z ∪ x",
"state_before": "case mp\nι : Type ?u.902858\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : DecidableEq α\nx y z : Multiset α\nh : y ≤ x\nh0 : z + x = z ∪ y\n⊢ x = y → z + x = z ∪ x",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "case mp\nι : Type ?u.902858\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : DecidableEq α\nx z : Multiset α\nh : x ≤ x\nh0 : z + x = z ∪ x\n⊢ z + x = z ∪ x",
"tactic": "exact h0"
},
{
"state_after": "case mpr.intro\nι : Type ?u.902858\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : DecidableEq α\nx z : Multiset α\nh0 : z + x = z ∪ x\nh : x ≤ x\n⊢ z + x = z ∪ x",
"state_before": "case mpr\nι : Type ?u.902858\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : DecidableEq α\nx y z : Multiset α\nh : y ≤ x\n⊢ z + x = z ∪ x ∧ x = y → z + x = z ∪ y",
"tactic": "rintro ⟨h0, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro\nι : Type ?u.902858\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : DecidableEq α\nx z : Multiset α\nh0 : z + x = z ∪ x\nh : x ≤ x\n⊢ z + x = z ∪ x",
"tactic": "exact h0"
}
]
| [
2102,
13
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2092,
1
]
|
Mathlib/Data/Set/Lattice.lean | Set.Iic_iInf | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.295745\nγ : Type ?u.295748\nι : Sort u_2\nι' : Sort ?u.295754\nι₂ : Sort ?u.295757\nκ : ι → Sort ?u.295762\nκ₁ : ι → Sort ?u.295767\nκ₂ : ι → Sort ?u.295772\nκ' : ι' → Sort ?u.295777\ninst✝ : CompleteLattice α\nf : ι → α\nx✝ : α\n⊢ x✝ ∈ Iic (⨅ (i : ι), f i) ↔ x✝ ∈ ⋂ (i : ι), Iic (f i)",
"tactic": "simp only [mem_Iic, le_iInf_iff, mem_iInter]"
}
]
| [
2127,
63
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2126,
1
]
|
Std/Data/RBMap/Lemmas.lean | Std.RBNode.depthUB_le_two_depthLB | []
| [
65,
30
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
63,
1
]
|
Mathlib/RingTheory/Subsemiring/Basic.lean | Subsemiring.nsmul_mem | []
| [
474,
17
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
473,
11
]
|
Mathlib/SetTheory/Cardinal/Basic.lean | Cardinal.mk_punit | []
| [
1929,
18
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1928,
1
]
|
Mathlib/Data/Set/Prod.lean | Set.prod_eq | []
| [
51,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
50,
1
]
|
Mathlib/Topology/Separation.lean | Embedding.t3Space | []
| [
1663,
51
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1660,
11
]
|
Mathlib/Data/List/AList.lean | AList.keys_replace | []
| [
197,
22
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
196,
1
]
|
Mathlib/Algebra/Order/Ring/WithTop.lean | WithBot.bot_lt_mul | []
| [
262,
38
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
261,
1
]
|
Mathlib/MeasureTheory/Function/LpSeminorm.lean | MeasureTheory.coe_nnnorm_ae_le_snormEssSup | []
| [
730,
48
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
728,
1
]
|
Mathlib/CategoryTheory/Monoidal/Mon_.lean | Mon_.Mon_tensor_mul_assoc | [
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N : Mon_ C\n⊢ ((tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (M.X ⊗ N.X)) ≫ ((M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X))) ≫\n tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =\n (α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫\n (𝟙 (M.X ⊗ N.X) ≫ 𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫\n tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N : Mon_ C\n⊢ (tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X)) ≫\n tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =\n (α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫\n (𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫\n tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)",
"tactic": "rw [← Category.id_comp (𝟙 (M.X ⊗ N.X)), tensor_comp]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N : Mon_ C\n⊢ (tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (M.X ⊗ N.X)) ≫\n (tensor_μ C (((M.X, N.X) ⊗ (M.X, N.X)).fst, ((M.X, N.X) ⊗ (M.X, N.X)).snd) (M.X, N.X) ≫\n ((M.mul ⊗ 𝟙 M.X) ⊗ N.mul ⊗ 𝟙 N.X)) ≫\n (M.mul ⊗ N.mul) =\n (α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫\n (𝟙 (M.X ⊗ N.X) ≫ 𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫\n tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N : Mon_ C\n⊢ ((tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (M.X ⊗ N.X)) ≫ ((M.mul ⊗ N.mul) ⊗ 𝟙 (M.X ⊗ N.X))) ≫\n tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul) =\n (α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫\n (𝟙 (M.X ⊗ N.X) ≫ 𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫\n tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)",
"tactic": "slice_lhs 2 3 => rw [← tensor_id, tensor_μ_natural]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N : Mon_ C\n⊢ (tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (M.X ⊗ N.X)) ≫\n tensor_μ C (((M.X, N.X) ⊗ (M.X, N.X)).fst, ((M.X, N.X) ⊗ (M.X, N.X)).snd) (M.X, N.X) ≫\n ((α_ M.X M.X M.X).hom ⊗ (α_ N.X N.X N.X).hom) ≫ ((𝟙 M.X ⊗ M.mul) ⊗ 𝟙 N.X ⊗ N.mul) ≫ (M.mul ⊗ N.mul) =\n (α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫\n (𝟙 (M.X ⊗ N.X) ≫ 𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫\n tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N : Mon_ C\n⊢ (tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (M.X ⊗ N.X)) ≫\n (tensor_μ C (((M.X, N.X) ⊗ (M.X, N.X)).fst, ((M.X, N.X) ⊗ (M.X, N.X)).snd) (M.X, N.X) ≫\n ((M.mul ⊗ 𝟙 M.X) ⊗ N.mul ⊗ 𝟙 N.X)) ≫\n (M.mul ⊗ N.mul) =\n (α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫\n (𝟙 (M.X ⊗ N.X) ≫ 𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫\n tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)",
"tactic": "slice_lhs 3 4 => rw [← tensor_comp, mul_assoc M, mul_assoc N, tensor_comp, tensor_comp]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N : Mon_ C\n⊢ (((α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫\n (𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X)) ≫ tensor_μ C (M.X, N.X) (M.X ⊗ M.X, N.X ⊗ N.X)) ≫\n ((𝟙 M.X ⊗ M.mul) ⊗ 𝟙 N.X ⊗ N.mul)) ≫\n (M.mul ⊗ N.mul) =\n (α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫\n (𝟙 (M.X ⊗ N.X) ≫ 𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫\n tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N : Mon_ C\n⊢ (tensor_μ C (M.X, N.X) (M.X, N.X) ⊗ 𝟙 (M.X ⊗ N.X)) ≫\n tensor_μ C (((M.X, N.X) ⊗ (M.X, N.X)).fst, ((M.X, N.X) ⊗ (M.X, N.X)).snd) (M.X, N.X) ≫\n ((α_ M.X M.X M.X).hom ⊗ (α_ N.X N.X N.X).hom) ≫ ((𝟙 M.X ⊗ M.mul) ⊗ 𝟙 N.X ⊗ N.mul) ≫ (M.mul ⊗ N.mul) =\n (α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫\n (𝟙 (M.X ⊗ N.X) ≫ 𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫\n tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)",
"tactic": "slice_lhs 1 3 => dsimp; rw [tensor_associativity]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N : Mon_ C\n⊢ (α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫\n (𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X)) ≫\n (((𝟙 M.X ⊗ 𝟙 N.X) ⊗ M.mul ⊗ N.mul) ≫ tensor_μ C (M.X, N.X) (M.X, N.X)) ≫ (M.mul ⊗ N.mul) =\n (α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫\n (𝟙 (M.X ⊗ N.X) ≫ 𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫\n tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N : Mon_ C\n⊢ (((α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫\n (𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X)) ≫ tensor_μ C (M.X, N.X) (M.X ⊗ M.X, N.X ⊗ N.X)) ≫\n ((𝟙 M.X ⊗ M.mul) ⊗ 𝟙 N.X ⊗ N.mul)) ≫\n (M.mul ⊗ N.mul) =\n (α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫\n (𝟙 (M.X ⊗ N.X) ≫ 𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫\n tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)",
"tactic": "slice_lhs 3 4 => rw [← tensor_μ_natural]"
},
{
"state_after": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N : Mon_ C\n⊢ (α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫\n ((𝟙 (M.X ⊗ N.X) ≫ 𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫\n tensor_μ C (M.X, N.X) (M.X, N.X)) ≫\n (M.mul ⊗ N.mul) =\n (α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫\n (𝟙 (M.X ⊗ N.X) ≫ 𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫\n tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N : Mon_ C\n⊢ (α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫\n (𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X)) ≫\n (((𝟙 M.X ⊗ 𝟙 N.X) ⊗ M.mul ⊗ N.mul) ≫ tensor_μ C (M.X, N.X) (M.X, N.X)) ≫ (M.mul ⊗ N.mul) =\n (α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫\n (𝟙 (M.X ⊗ N.X) ≫ 𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫\n tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)",
"tactic": "slice_lhs 2 3 => rw [← tensor_comp, tensor_id]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\ninst✝¹ : MonoidalCategory C\ninst✝ : BraidedCategory C\nM N : Mon_ C\n⊢ (α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫\n ((𝟙 (M.X ⊗ N.X) ≫ 𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫\n tensor_μ C (M.X, N.X) (M.X, N.X)) ≫\n (M.mul ⊗ N.mul) =\n (α_ (M.X ⊗ N.X) (M.X ⊗ N.X) (M.X ⊗ N.X)).hom ≫\n (𝟙 (M.X ⊗ N.X) ≫ 𝟙 (M.X ⊗ N.X) ⊗ tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)) ≫\n tensor_μ C (M.X, N.X) (M.X, N.X) ≫ (M.mul ⊗ N.mul)",
"tactic": "simp only [Category.assoc]"
}
]
| [
445,
29
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
432,
1
]
|
Mathlib/Data/Int/Order/Basic.lean | Int.sign_add_eq_of_sign_eq | [
{
"state_after": "this : 1 ≠ -1\n⊢ ∀ {m n : ℤ}, sign m = sign n → sign (m + n) = sign n",
"state_before": "⊢ ∀ {m n : ℤ}, sign m = sign n → sign (m + n) = sign n",
"tactic": "have : (1 : ℤ) ≠ -1 := by decide"
},
{
"state_after": "case ofNat.succ.ofNat.succ\nthis : 1 ≠ -1\nm n : ℕ\n⊢ sign (↑m + 1 + (↑n + 1)) = 1",
"state_before": "this : 1 ≠ -1\n⊢ ∀ {m n : ℤ}, sign m = sign n → sign (m + n) = sign n",
"tactic": "rintro ((_ | m) | m) ((_ | n) | n) <;> simp [this, this.symm]"
},
{
"state_after": "case ofNat.succ.ofNat.succ\nthis : 1 ≠ -1\nm n : ℕ\n⊢ 0 < ↑m + 1 + (↑n + 1)",
"state_before": "case ofNat.succ.ofNat.succ\nthis : 1 ≠ -1\nm n : ℕ\n⊢ sign (↑m + 1 + (↑n + 1)) = 1",
"tactic": "rw [Int.sign_eq_one_iff_pos]"
},
{
"state_after": "no goals",
"state_before": "⊢ 1 ≠ -1",
"tactic": "decide"
},
{
"state_after": "no goals",
"state_before": "case ofNat.succ.ofNat.succ.hb\nthis : 1 ≠ -1\nm n : ℕ\n⊢ 0 < ↑n + 1",
"tactic": "exact zero_lt_one.trans_le (le_add_of_nonneg_left <| coe_nat_nonneg _)"
}
]
| [
97,
97
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
93,
1
]
|
Std/Data/List/Init/Lemmas.lean | List.length_eq_zero | []
| [
50,
47
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
49,
1
]
|
Mathlib/Data/Setoid/Partition.lean | Setoid.mkClasses_classes | []
| [
195,
73
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
192,
1
]
|
Mathlib/Algebra/Order/Ring/Defs.lean | cmp_mul_pos_left | []
| [
961,
49
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
960,
1
]
|
Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean | CategoryTheory.IsPullback.paste_vert_iff | []
| [
544,
51
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
540,
1
]
|
Std/Data/List/Lemmas.lean | List.get_cons_zero | []
| [
496,
95
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
496,
9
]
|
Mathlib/MeasureTheory/Measure/MeasureSpace.lean | MeasureTheory.Ioo_ae_eq_Icc' | []
| [
2988,
48
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2987,
1
]
|
Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean | MeasureTheory.exists_lt_lowerSemicontinuous_lintegral_ge | [
{
"state_after": "α : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : WeaklyRegular μ\ninst✝ : SigmaFinite μ\nf : α → ℝ≥0\nfmeas : Measurable f\nε : ℝ≥0∞\nε0 : ε ≠ 0\nthis : ε / 2 ≠ 0\n⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ (∫⁻ (x : α), g x ∂μ) ≤ (∫⁻ (x : α), ↑(f x) ∂μ) + ε",
"state_before": "α : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : WeaklyRegular μ\ninst✝ : SigmaFinite μ\nf : α → ℝ≥0\nfmeas : Measurable f\nε : ℝ≥0∞\nε0 : ε ≠ 0\n⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ (∫⁻ (x : α), g x ∂μ) ≤ (∫⁻ (x : α), ↑(f x) ∂μ) + ε",
"tactic": "have : ε / 2 ≠ 0 := (ENNReal.half_pos ε0).ne'"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : WeaklyRegular μ\ninst✝ : SigmaFinite μ\nf : α → ℝ≥0\nfmeas : Measurable f\nε : ℝ≥0∞\nε0 : ε ≠ 0\nthis : ε / 2 ≠ 0\nw : α → ℝ≥0\nwpos : ∀ (x : α), 0 < w x\nwmeas : Measurable w\nwint : (∫⁻ (x : α), ↑(w x) ∂μ) < ε / 2\n⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ (∫⁻ (x : α), g x ∂μ) ≤ (∫⁻ (x : α), ↑(f x) ∂μ) + ε",
"state_before": "α : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : WeaklyRegular μ\ninst✝ : SigmaFinite μ\nf : α → ℝ≥0\nfmeas : Measurable f\nε : ℝ≥0∞\nε0 : ε ≠ 0\nthis : ε / 2 ≠ 0\n⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ (∫⁻ (x : α), g x ∂μ) ≤ (∫⁻ (x : α), ↑(f x) ∂μ) + ε",
"tactic": "rcases exists_pos_lintegral_lt_of_sigmaFinite μ this with ⟨w, wpos, wmeas, wint⟩"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : WeaklyRegular μ\ninst✝ : SigmaFinite μ\nf : α → ℝ≥0\nfmeas : Measurable f\nε : ℝ≥0∞\nε0 : ε ≠ 0\nthis : ε / 2 ≠ 0\nw : α → ℝ≥0\nwpos : ∀ (x : α), 0 < w x\nwmeas : Measurable w\nwint : (∫⁻ (x : α), ↑(w x) ∂μ) < ε / 2\nf' : α → ℝ≥0∞ := fun x => ↑(f x + w x)\n⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ (∫⁻ (x : α), g x ∂μ) ≤ (∫⁻ (x : α), ↑(f x) ∂μ) + ε",
"state_before": "case intro.intro.intro\nα : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : WeaklyRegular μ\ninst✝ : SigmaFinite μ\nf : α → ℝ≥0\nfmeas : Measurable f\nε : ℝ≥0∞\nε0 : ε ≠ 0\nthis : ε / 2 ≠ 0\nw : α → ℝ≥0\nwpos : ∀ (x : α), 0 < w x\nwmeas : Measurable w\nwint : (∫⁻ (x : α), ↑(w x) ∂μ) < ε / 2\n⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ (∫⁻ (x : α), g x ∂μ) ≤ (∫⁻ (x : α), ↑(f x) ∂μ) + ε",
"tactic": "let f' x := ((f x + w x : ℝ≥0) : ℝ≥0∞)"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : WeaklyRegular μ\ninst✝ : SigmaFinite μ\nf : α → ℝ≥0\nfmeas : Measurable f\nε : ℝ≥0∞\nε0 : ε ≠ 0\nthis : ε / 2 ≠ 0\nw : α → ℝ≥0\nwpos : ∀ (x : α), 0 < w x\nwmeas : Measurable w\nwint : (∫⁻ (x : α), ↑(w x) ∂μ) < ε / 2\nf' : α → ℝ≥0∞ := fun x => ↑(f x + w x)\ng : α → ℝ≥0∞\nle_g : ∀ (x : α), f' x ≤ g x\ngcont : LowerSemicontinuous g\ngint : (∫⁻ (x : α), g x ∂μ) ≤ (∫⁻ (x : α), f' x ∂μ) + ε / 2\n⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ (∫⁻ (x : α), g x ∂μ) ≤ (∫⁻ (x : α), ↑(f x) ∂μ) + ε",
"state_before": "case intro.intro.intro\nα : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : WeaklyRegular μ\ninst✝ : SigmaFinite μ\nf : α → ℝ≥0\nfmeas : Measurable f\nε : ℝ≥0∞\nε0 : ε ≠ 0\nthis : ε / 2 ≠ 0\nw : α → ℝ≥0\nwpos : ∀ (x : α), 0 < w x\nwmeas : Measurable w\nwint : (∫⁻ (x : α), ↑(w x) ∂μ) < ε / 2\nf' : α → ℝ≥0∞ := fun x => ↑(f x + w x)\n⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ (∫⁻ (x : α), g x ∂μ) ≤ (∫⁻ (x : α), ↑(f x) ∂μ) + ε",
"tactic": "rcases exists_le_lowerSemicontinuous_lintegral_ge μ f' (fmeas.add wmeas).coe_nnreal_ennreal\n this with\n ⟨g, le_g, gcont, gint⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.refine'_1\nα : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : WeaklyRegular μ\ninst✝ : SigmaFinite μ\nf : α → ℝ≥0\nfmeas : Measurable f\nε : ℝ≥0∞\nε0 : ε ≠ 0\nthis : ε / 2 ≠ 0\nw : α → ℝ≥0\nwpos : ∀ (x : α), 0 < w x\nwmeas : Measurable w\nwint : (∫⁻ (x : α), ↑(w x) ∂μ) < ε / 2\nf' : α → ℝ≥0∞ := fun x => ↑(f x + w x)\ng : α → ℝ≥0∞\nle_g : ∀ (x : α), f' x ≤ g x\ngcont : LowerSemicontinuous g\ngint : (∫⁻ (x : α), g x ∂μ) ≤ (∫⁻ (x : α), f' x ∂μ) + ε / 2\nx : α\n⊢ ↑(f x) < g x\n\ncase intro.intro.intro.intro.intro.intro.refine'_2\nα : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : WeaklyRegular μ\ninst✝ : SigmaFinite μ\nf : α → ℝ≥0\nfmeas : Measurable f\nε : ℝ≥0∞\nε0 : ε ≠ 0\nthis : ε / 2 ≠ 0\nw : α → ℝ≥0\nwpos : ∀ (x : α), 0 < w x\nwmeas : Measurable w\nwint : (∫⁻ (x : α), ↑(w x) ∂μ) < ε / 2\nf' : α → ℝ≥0∞ := fun x => ↑(f x + w x)\ng : α → ℝ≥0∞\nle_g : ∀ (x : α), f' x ≤ g x\ngcont : LowerSemicontinuous g\ngint : (∫⁻ (x : α), g x ∂μ) ≤ (∫⁻ (x : α), f' x ∂μ) + ε / 2\n⊢ (∫⁻ (x : α), g x ∂μ) ≤ (∫⁻ (x : α), ↑(f x) ∂μ) + ε",
"state_before": "case intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : WeaklyRegular μ\ninst✝ : SigmaFinite μ\nf : α → ℝ≥0\nfmeas : Measurable f\nε : ℝ≥0∞\nε0 : ε ≠ 0\nthis : ε / 2 ≠ 0\nw : α → ℝ≥0\nwpos : ∀ (x : α), 0 < w x\nwmeas : Measurable w\nwint : (∫⁻ (x : α), ↑(w x) ∂μ) < ε / 2\nf' : α → ℝ≥0∞ := fun x => ↑(f x + w x)\ng : α → ℝ≥0∞\nle_g : ∀ (x : α), f' x ≤ g x\ngcont : LowerSemicontinuous g\ngint : (∫⁻ (x : α), g x ∂μ) ≤ (∫⁻ (x : α), f' x ∂μ) + ε / 2\n⊢ ∃ g, (∀ (x : α), ↑(f x) < g x) ∧ LowerSemicontinuous g ∧ (∫⁻ (x : α), g x ∂μ) ≤ (∫⁻ (x : α), ↑(f x) ∂μ) + ε",
"tactic": "refine' ⟨g, fun x => _, gcont, _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.refine'_1\nα : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : WeaklyRegular μ\ninst✝ : SigmaFinite μ\nf : α → ℝ≥0\nfmeas : Measurable f\nε : ℝ≥0∞\nε0 : ε ≠ 0\nthis : ε / 2 ≠ 0\nw : α → ℝ≥0\nwpos : ∀ (x : α), 0 < w x\nwmeas : Measurable w\nwint : (∫⁻ (x : α), ↑(w x) ∂μ) < ε / 2\nf' : α → ℝ≥0∞ := fun x => ↑(f x + w x)\ng : α → ℝ≥0∞\nle_g : ∀ (x : α), f' x ≤ g x\ngcont : LowerSemicontinuous g\ngint : (∫⁻ (x : α), g x ∂μ) ≤ (∫⁻ (x : α), f' x ∂μ) + ε / 2\nx : α\n⊢ ↑(f x) < g x",
"tactic": "calc\n (f x : ℝ≥0∞) < f' x := by\n simpa only [← ENNReal.coe_lt_coe, add_zero] using add_lt_add_left (wpos x) (f x)\n _ ≤ g x := le_g x"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : WeaklyRegular μ\ninst✝ : SigmaFinite μ\nf : α → ℝ≥0\nfmeas : Measurable f\nε : ℝ≥0∞\nε0 : ε ≠ 0\nthis : ε / 2 ≠ 0\nw : α → ℝ≥0\nwpos : ∀ (x : α), 0 < w x\nwmeas : Measurable w\nwint : (∫⁻ (x : α), ↑(w x) ∂μ) < ε / 2\nf' : α → ℝ≥0∞ := fun x => ↑(f x + w x)\ng : α → ℝ≥0∞\nle_g : ∀ (x : α), f' x ≤ g x\ngcont : LowerSemicontinuous g\ngint : (∫⁻ (x : α), g x ∂μ) ≤ (∫⁻ (x : α), f' x ∂μ) + ε / 2\nx : α\n⊢ ↑(f x) < f' x",
"tactic": "simpa only [← ENNReal.coe_lt_coe, add_zero] using add_lt_add_left (wpos x) (f x)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.refine'_2\nα : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : WeaklyRegular μ\ninst✝ : SigmaFinite μ\nf : α → ℝ≥0\nfmeas : Measurable f\nε : ℝ≥0∞\nε0 : ε ≠ 0\nthis : ε / 2 ≠ 0\nw : α → ℝ≥0\nwpos : ∀ (x : α), 0 < w x\nwmeas : Measurable w\nwint : (∫⁻ (x : α), ↑(w x) ∂μ) < ε / 2\nf' : α → ℝ≥0∞ := fun x => ↑(f x + w x)\ng : α → ℝ≥0∞\nle_g : ∀ (x : α), f' x ≤ g x\ngcont : LowerSemicontinuous g\ngint : (∫⁻ (x : α), g x ∂μ) ≤ (∫⁻ (x : α), f' x ∂μ) + ε / 2\n⊢ (∫⁻ (x : α), g x ∂μ) ≤ (∫⁻ (x : α), ↑(f x) ∂μ) + ε",
"tactic": "calc\n (∫⁻ x : α, g x ∂μ) ≤ (∫⁻ x : α, f x + w x ∂μ) + ε / 2 := gint\n _ = ((∫⁻ x : α, f x ∂μ) + ∫⁻ x : α, w x ∂μ) + ε / 2 := by\n rw [lintegral_add_right _ wmeas.coe_nnreal_ennreal]\n _ ≤ (∫⁻ x : α, f x ∂μ) + ε / 2 + ε / 2 := (add_le_add_right (add_le_add_left wint.le _) _)\n _ = (∫⁻ x : α, f x ∂μ) + ε := by rw [add_assoc, ENNReal.add_halves]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : WeaklyRegular μ\ninst✝ : SigmaFinite μ\nf : α → ℝ≥0\nfmeas : Measurable f\nε : ℝ≥0∞\nε0 : ε ≠ 0\nthis : ε / 2 ≠ 0\nw : α → ℝ≥0\nwpos : ∀ (x : α), 0 < w x\nwmeas : Measurable w\nwint : (∫⁻ (x : α), ↑(w x) ∂μ) < ε / 2\nf' : α → ℝ≥0∞ := fun x => ↑(f x + w x)\ng : α → ℝ≥0∞\nle_g : ∀ (x : α), f' x ≤ g x\ngcont : LowerSemicontinuous g\ngint : (∫⁻ (x : α), g x ∂μ) ≤ (∫⁻ (x : α), f' x ∂μ) + ε / 2\n⊢ (∫⁻ (x : α), ↑(f x) + ↑(w x) ∂μ) + ε / 2 = ((∫⁻ (x : α), ↑(f x) ∂μ) + ∫⁻ (x : α), ↑(w x) ∂μ) + ε / 2",
"tactic": "rw [lintegral_add_right _ wmeas.coe_nnreal_ennreal]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : WeaklyRegular μ\ninst✝ : SigmaFinite μ\nf : α → ℝ≥0\nfmeas : Measurable f\nε : ℝ≥0∞\nε0 : ε ≠ 0\nthis : ε / 2 ≠ 0\nw : α → ℝ≥0\nwpos : ∀ (x : α), 0 < w x\nwmeas : Measurable w\nwint : (∫⁻ (x : α), ↑(w x) ∂μ) < ε / 2\nf' : α → ℝ≥0∞ := fun x => ↑(f x + w x)\ng : α → ℝ≥0∞\nle_g : ∀ (x : α), f' x ≤ g x\ngcont : LowerSemicontinuous g\ngint : (∫⁻ (x : α), g x ∂μ) ≤ (∫⁻ (x : α), f' x ∂μ) + ε / 2\n⊢ (∫⁻ (x : α), ↑(f x) ∂μ) + ε / 2 + ε / 2 = (∫⁻ (x : α), ↑(f x) ∂μ) + ε",
"tactic": "rw [add_assoc, ENNReal.add_halves]"
}
]
| [
228,
74
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
208,
1
]
|
Mathlib/Analysis/Asymptotics/Asymptotics.lean | Asymptotics.IsLittleO.trans_isBigOWith | [
{
"state_after": "α : Type u_1\nβ : Type ?u.85733\nE : Type u_2\nF : Type u_3\nG : Type u_4\nE' : Type ?u.85745\nF' : Type ?u.85748\nG' : Type ?u.85751\nE'' : Type ?u.85754\nF'' : Type ?u.85757\nG'' : Type ?u.85760\nR : Type ?u.85763\nR' : Type ?u.85766\n𝕜 : Type ?u.85769\n𝕜' : Type ?u.85772\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nhgk : IsBigOWith c l g k\nhc : 0 < c\nhfg : ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f g\n⊢ ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f k",
"state_before": "α : Type u_1\nβ : Type ?u.85733\nE : Type u_2\nF : Type u_3\nG : Type u_4\nE' : Type ?u.85745\nF' : Type ?u.85748\nG' : Type ?u.85751\nE'' : Type ?u.85754\nF'' : Type ?u.85757\nG'' : Type ?u.85760\nR : Type ?u.85763\nR' : Type ?u.85766\n𝕜 : Type ?u.85769\n𝕜' : Type ?u.85772\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nhfg : f =o[l] g\nhgk : IsBigOWith c l g k\nhc : 0 < c\n⊢ f =o[l] k",
"tactic": "simp only [IsLittleO_def] at *"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.85733\nE : Type u_2\nF : Type u_3\nG : Type u_4\nE' : Type ?u.85745\nF' : Type ?u.85748\nG' : Type ?u.85751\nE'' : Type ?u.85754\nF'' : Type ?u.85757\nG'' : Type ?u.85760\nR : Type ?u.85763\nR' : Type ?u.85766\n𝕜 : Type ?u.85769\n𝕜' : Type ?u.85772\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c'✝ c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nhgk : IsBigOWith c l g k\nhc : 0 < c\nhfg : ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f g\nc' : ℝ\nc'pos : 0 < c'\n⊢ IsBigOWith c' l f k",
"state_before": "α : Type u_1\nβ : Type ?u.85733\nE : Type u_2\nF : Type u_3\nG : Type u_4\nE' : Type ?u.85745\nF' : Type ?u.85748\nG' : Type ?u.85751\nE'' : Type ?u.85754\nF'' : Type ?u.85757\nG'' : Type ?u.85760\nR : Type ?u.85763\nR' : Type ?u.85766\n𝕜 : Type ?u.85769\n𝕜' : Type ?u.85772\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nhgk : IsBigOWith c l g k\nhc : 0 < c\nhfg : ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f g\n⊢ ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f k",
"tactic": "intro c' c'pos"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.85733\nE : Type u_2\nF : Type u_3\nG : Type u_4\nE' : Type ?u.85745\nF' : Type ?u.85748\nG' : Type ?u.85751\nE'' : Type ?u.85754\nF'' : Type ?u.85757\nG'' : Type ?u.85760\nR : Type ?u.85763\nR' : Type ?u.85766\n𝕜 : Type ?u.85769\n𝕜' : Type ?u.85772\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c'✝ c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nhgk : IsBigOWith c l g k\nhc : 0 < c\nhfg : ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f g\nc' : ℝ\nc'pos : 0 < c'\nthis : 0 < c' / c\n⊢ IsBigOWith c' l f k",
"state_before": "α : Type u_1\nβ : Type ?u.85733\nE : Type u_2\nF : Type u_3\nG : Type u_4\nE' : Type ?u.85745\nF' : Type ?u.85748\nG' : Type ?u.85751\nE'' : Type ?u.85754\nF'' : Type ?u.85757\nG'' : Type ?u.85760\nR : Type ?u.85763\nR' : Type ?u.85766\n𝕜 : Type ?u.85769\n𝕜' : Type ?u.85772\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c'✝ c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nhgk : IsBigOWith c l g k\nhc : 0 < c\nhfg : ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f g\nc' : ℝ\nc'pos : 0 < c'\n⊢ IsBigOWith c' l f k",
"tactic": "have : 0 < c' / c := div_pos c'pos hc"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.85733\nE : Type u_2\nF : Type u_3\nG : Type u_4\nE' : Type ?u.85745\nF' : Type ?u.85748\nG' : Type ?u.85751\nE'' : Type ?u.85754\nF'' : Type ?u.85757\nG'' : Type ?u.85760\nR : Type ?u.85763\nR' : Type ?u.85766\n𝕜 : Type ?u.85769\n𝕜' : Type ?u.85772\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c'✝ c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nhgk : IsBigOWith c l g k\nhc : 0 < c\nhfg : ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f g\nc' : ℝ\nc'pos : 0 < c'\nthis : 0 < c' / c\n⊢ IsBigOWith c' l f k",
"tactic": "exact ((hfg this).trans hgk this.le).congr_const (div_mul_cancel _ hc.ne')"
}
]
| [
486,
77
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
481,
1
]
|
Mathlib/RingTheory/PowerSeries/Basic.lean | MvPowerSeries.inv_eq_zero | [
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nR : Type ?u.1713985\nk : Type u_2\ninst✝ : Field k\nφ : MvPowerSeries σ k\nh : φ⁻¹ = 0\n⊢ ↑(constantCoeff σ k) φ = 0",
"tactic": "simpa using congr_arg (constantCoeff σ k) h"
},
{
"state_after": "σ : Type u_1\nR : Type ?u.1713985\nk : Type u_2\ninst✝ : Field k\nφ : MvPowerSeries σ k\nh : ↑(constantCoeff σ k) φ = 0\nn : σ →₀ ℕ\n⊢ (if n = 0 then (↑(constantCoeff σ k) φ)⁻¹\n else\n -(↑(constantCoeff σ k) φ)⁻¹ *\n ∑ x in antidiagonal n, if x.snd < n then ↑(coeff k x.fst) φ * ↑(coeff k x.snd) φ⁻¹ else 0) =\n ↑(coeff k n) 0",
"state_before": "σ : Type u_1\nR : Type ?u.1713985\nk : Type u_2\ninst✝ : Field k\nφ : MvPowerSeries σ k\nh : ↑(constantCoeff σ k) φ = 0\nn : σ →₀ ℕ\n⊢ ↑(coeff k n) φ⁻¹ = ↑(coeff k n) 0",
"tactic": "rw [coeff_inv]"
},
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nR : Type ?u.1713985\nk : Type u_2\ninst✝ : Field k\nφ : MvPowerSeries σ k\nh : ↑(constantCoeff σ k) φ = 0\nn : σ →₀ ℕ\n⊢ (if n = 0 then (↑(constantCoeff σ k) φ)⁻¹\n else\n -(↑(constantCoeff σ k) φ)⁻¹ *\n ∑ x in antidiagonal n, if x.snd < n then ↑(coeff k x.fst) φ * ↑(coeff k x.snd) φ⁻¹ else 0) =\n ↑(coeff k n) 0",
"tactic": "split_ifs <;>\n simp only [h, map_zero, zero_mul, inv_zero, neg_zero]"
}
]
| [
960,
63
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
955,
1
]
|
Mathlib/Topology/Category/TopCat/Opens.lean | TopologicalSpace.Opens.op_map_id_obj | [
{
"state_after": "no goals",
"state_before": "X Y Z : TopCat\nU : (Opens ↑X)ᵒᵖ\n⊢ (Functor.op (map (𝟙 X))).obj U = U",
"tactic": "simp"
}
]
| [
182,
78
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
182,
1
]
|
Mathlib/Analysis/NormedSpace/FiniteDimension.lean | lipschitzExtensionConstant_pos | [
{
"state_after": "𝕜 : Type u\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nF : Type w\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁸ : AddCommGroup F'\ninst✝⁷ : Module 𝕜 F'\ninst✝⁶ : TopologicalSpace F'\ninst✝⁵ : TopologicalAddGroup F'\ninst✝⁴ : ContinuousSMul 𝕜 F'\ninst✝³ : CompleteSpace 𝕜\nE' : Type u_1\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : FiniteDimensional ℝ E'\n⊢ 0 <\n let A := LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace ℝ E'));\n max (‖↑(ContinuousLinearEquiv.symm A)‖₊ * ‖↑A‖₊) 1",
"state_before": "𝕜 : Type u\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nF : Type w\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁸ : AddCommGroup F'\ninst✝⁷ : Module 𝕜 F'\ninst✝⁶ : TopologicalSpace F'\ninst✝⁵ : TopologicalAddGroup F'\ninst✝⁴ : ContinuousSMul 𝕜 F'\ninst✝³ : CompleteSpace 𝕜\nE' : Type u_1\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : FiniteDimensional ℝ E'\n⊢ 0 < lipschitzExtensionConstant E'",
"tactic": "rw [lipschitzExtensionConstant]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nF : Type w\ninst✝¹⁰ : NormedAddCommGroup F\ninst✝⁹ : NormedSpace 𝕜 F\nF' : Type x\ninst✝⁸ : AddCommGroup F'\ninst✝⁷ : Module 𝕜 F'\ninst✝⁶ : TopologicalSpace F'\ninst✝⁵ : TopologicalAddGroup F'\ninst✝⁴ : ContinuousSMul 𝕜 F'\ninst✝³ : CompleteSpace 𝕜\nE' : Type u_1\ninst✝² : NormedAddCommGroup E'\ninst✝¹ : NormedSpace ℝ E'\ninst✝ : FiniteDimensional ℝ E'\n⊢ 0 <\n let A := LinearEquiv.toContinuousLinearEquiv (Basis.equivFun (Basis.ofVectorSpace ℝ E'));\n max (‖↑(ContinuousLinearEquiv.symm A)‖₊ * ‖↑A‖₊) 1",
"tactic": "exact zero_lt_one.trans_le (le_max_right _ _)"
}
]
| [
195,
48
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
192,
1
]
|
Mathlib/GroupTheory/Torsion.lean | ExponentExists.isTorsion | [
{
"state_after": "case intro.intro\nG : Type u_1\nH : Type ?u.170289\ninst✝¹ : Group G\nN : Subgroup G\ninst✝ : Group H\ng : G\nn : ℕ\nnpos : 0 < n\nhn : ∀ (g : G), g ^ n = 1\n⊢ IsOfFinOrder g",
"state_before": "G : Type u_1\nH : Type ?u.170289\ninst✝¹ : Group G\nN : Subgroup G\ninst✝ : Group H\nh : ExponentExists G\ng : G\n⊢ IsOfFinOrder g",
"tactic": "obtain ⟨n, npos, hn⟩ := h"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nG : Type u_1\nH : Type ?u.170289\ninst✝¹ : Group G\nN : Subgroup G\ninst✝ : Group H\ng : G\nn : ℕ\nnpos : 0 < n\nhn : ∀ (g : G), g ^ n = 1\n⊢ IsOfFinOrder g",
"tactic": "exact (isOfFinOrder_iff_pow_eq_one g).mpr ⟨n, npos, hn g⟩"
}
]
| [
137,
60
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
135,
1
]
|
Mathlib/CategoryTheory/Sites/SheafOfTypes.lean | CategoryTheory.Presieve.FamilyOfElements.Compatible.restrict | []
| [
178,
69
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
176,
1
]
|
Mathlib/MeasureTheory/Measure/MeasureSpace.lean | MeasureTheory.Measure.countable_meas_pos_of_disjoint_iUnion | [
{
"state_after": "α : Type u_2\nβ : Type ?u.742871\nγ : Type ?u.742874\nδ : Type ?u.742877\nι✝ : Type ?u.742880\nR : Type ?u.742883\nR' : Type ?u.742886\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nι : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nAs : ι → Set α\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nobs : {i | 0 < ↑↑μ (As i)} ⊆ ⋃ (n : ℕ), {i | 0 < ↑↑μ (As i ∩ spanningSets μ n)}\n⊢ Set.Countable {i | 0 < ↑↑μ (As i)}",
"state_before": "α : Type u_2\nβ : Type ?u.742871\nγ : Type ?u.742874\nδ : Type ?u.742877\nι✝ : Type ?u.742880\nR : Type ?u.742883\nR' : Type ?u.742886\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nι : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nAs : ι → Set α\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\n⊢ Set.Countable {i | 0 < ↑↑μ (As i)}",
"tactic": "have obs : { i : ι | 0 < μ (As i) } ⊆ ⋃ n, { i : ι | 0 < μ (As i ∩ spanningSets μ n) } := by\n intro i i_in_nonzeroes\n by_contra con\n simp only [mem_iUnion, mem_setOf_eq, not_exists, not_lt, nonpos_iff_eq_zero] at *\n simp [(forall_measure_inter_spanningSets_eq_zero _).mp con] at i_in_nonzeroes"
},
{
"state_after": "α : Type u_2\nβ : Type ?u.742871\nγ : Type ?u.742874\nδ : Type ?u.742877\nι✝ : Type ?u.742880\nR : Type ?u.742883\nR' : Type ?u.742886\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nι : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nAs : ι → Set α\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nobs : {i | 0 < ↑↑μ (As i)} ⊆ ⋃ (n : ℕ), {i | 0 < ↑↑μ (As i ∩ spanningSets μ n)}\n⊢ Set.Countable (⋃ (n : ℕ), {i | 0 < ↑↑μ (As i ∩ spanningSets μ n)})",
"state_before": "α : Type u_2\nβ : Type ?u.742871\nγ : Type ?u.742874\nδ : Type ?u.742877\nι✝ : Type ?u.742880\nR : Type ?u.742883\nR' : Type ?u.742886\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nι : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nAs : ι → Set α\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nobs : {i | 0 < ↑↑μ (As i)} ⊆ ⋃ (n : ℕ), {i | 0 < ↑↑μ (As i ∩ spanningSets μ n)}\n⊢ Set.Countable {i | 0 < ↑↑μ (As i)}",
"tactic": "apply Countable.mono obs"
},
{
"state_after": "case refine'_1\nα : Type u_2\nβ : Type ?u.742871\nγ : Type ?u.742874\nδ : Type ?u.742877\nι✝ : Type ?u.742880\nR : Type ?u.742883\nR' : Type ?u.742886\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nι : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nAs : ι → Set α\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nobs : {i | 0 < ↑↑μ (As i)} ⊆ ⋃ (n : ℕ), {i | 0 < ↑↑μ (As i ∩ spanningSets μ n)}\nn : ℕ\n⊢ ∀ (i : ι), MeasurableSet (As i ∩ spanningSets μ n)\n\ncase refine'_2\nα : Type u_2\nβ : Type ?u.742871\nγ : Type ?u.742874\nδ : Type ?u.742877\nι✝ : Type ?u.742880\nR : Type ?u.742883\nR' : Type ?u.742886\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nι : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nAs : ι → Set α\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nobs : {i | 0 < ↑↑μ (As i)} ⊆ ⋃ (n : ℕ), {i | 0 < ↑↑μ (As i ∩ spanningSets μ n)}\nn : ℕ\n⊢ Pairwise (Disjoint on fun i => As i ∩ spanningSets μ n)\n\ncase refine'_3\nα : Type u_2\nβ : Type ?u.742871\nγ : Type ?u.742874\nδ : Type ?u.742877\nι✝ : Type ?u.742880\nR : Type ?u.742883\nR' : Type ?u.742886\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nι : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nAs : ι → Set α\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nobs : {i | 0 < ↑↑μ (As i)} ⊆ ⋃ (n : ℕ), {i | 0 < ↑↑μ (As i ∩ spanningSets μ n)}\nn : ℕ\n⊢ ↑↑μ (⋃ (i : ι), As i ∩ spanningSets μ n) ≠ ⊤",
"state_before": "α : Type u_2\nβ : Type ?u.742871\nγ : Type ?u.742874\nδ : Type ?u.742877\nι✝ : Type ?u.742880\nR : Type ?u.742883\nR' : Type ?u.742886\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nι : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nAs : ι → Set α\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nobs : {i | 0 < ↑↑μ (As i)} ⊆ ⋃ (n : ℕ), {i | 0 < ↑↑μ (As i ∩ spanningSets μ n)}\n⊢ Set.Countable (⋃ (n : ℕ), {i | 0 < ↑↑μ (As i ∩ spanningSets μ n)})",
"tactic": "refine' countable_iUnion fun n => countable_meas_pos_of_disjoint_of_meas_iUnion_ne_top μ _ _ _"
},
{
"state_after": "α : Type u_2\nβ : Type ?u.742871\nγ : Type ?u.742874\nδ : Type ?u.742877\nι✝ : Type ?u.742880\nR : Type ?u.742883\nR' : Type ?u.742886\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nι : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nAs : ι → Set α\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\ni : ι\ni_in_nonzeroes : i ∈ {i | 0 < ↑↑μ (As i)}\n⊢ i ∈ ⋃ (n : ℕ), {i | 0 < ↑↑μ (As i ∩ spanningSets μ n)}",
"state_before": "α : Type u_2\nβ : Type ?u.742871\nγ : Type ?u.742874\nδ : Type ?u.742877\nι✝ : Type ?u.742880\nR : Type ?u.742883\nR' : Type ?u.742886\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nι : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nAs : ι → Set α\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\n⊢ {i | 0 < ↑↑μ (As i)} ⊆ ⋃ (n : ℕ), {i | 0 < ↑↑μ (As i ∩ spanningSets μ n)}",
"tactic": "intro i i_in_nonzeroes"
},
{
"state_after": "α : Type u_2\nβ : Type ?u.742871\nγ : Type ?u.742874\nδ : Type ?u.742877\nι✝ : Type ?u.742880\nR : Type ?u.742883\nR' : Type ?u.742886\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nι : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nAs : ι → Set α\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\ni : ι\ni_in_nonzeroes : i ∈ {i | 0 < ↑↑μ (As i)}\ncon : ¬i ∈ ⋃ (n : ℕ), {i | 0 < ↑↑μ (As i ∩ spanningSets μ n)}\n⊢ False",
"state_before": "α : Type u_2\nβ : Type ?u.742871\nγ : Type ?u.742874\nδ : Type ?u.742877\nι✝ : Type ?u.742880\nR : Type ?u.742883\nR' : Type ?u.742886\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nι : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nAs : ι → Set α\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\ni : ι\ni_in_nonzeroes : i ∈ {i | 0 < ↑↑μ (As i)}\n⊢ i ∈ ⋃ (n : ℕ), {i | 0 < ↑↑μ (As i ∩ spanningSets μ n)}",
"tactic": "by_contra con"
},
{
"state_after": "α : Type u_2\nβ : Type ?u.742871\nγ : Type ?u.742874\nδ : Type ?u.742877\nι✝ : Type ?u.742880\nR : Type ?u.742883\nR' : Type ?u.742886\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nι : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nAs : ι → Set α\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\ni : ι\ni_in_nonzeroes : 0 < ↑↑μ (As i)\ncon : ∀ (x : ℕ), ↑↑μ (As i ∩ spanningSets μ x) = 0\n⊢ False",
"state_before": "α : Type u_2\nβ : Type ?u.742871\nγ : Type ?u.742874\nδ : Type ?u.742877\nι✝ : Type ?u.742880\nR : Type ?u.742883\nR' : Type ?u.742886\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nι : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nAs : ι → Set α\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\ni : ι\ni_in_nonzeroes : i ∈ {i | 0 < ↑↑μ (As i)}\ncon : ¬i ∈ ⋃ (n : ℕ), {i | 0 < ↑↑μ (As i ∩ spanningSets μ n)}\n⊢ False",
"tactic": "simp only [mem_iUnion, mem_setOf_eq, not_exists, not_lt, nonpos_iff_eq_zero] at *"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.742871\nγ : Type ?u.742874\nδ : Type ?u.742877\nι✝ : Type ?u.742880\nR : Type ?u.742883\nR' : Type ?u.742886\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nι : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nAs : ι → Set α\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\ni : ι\ni_in_nonzeroes : 0 < ↑↑μ (As i)\ncon : ∀ (x : ℕ), ↑↑μ (As i ∩ spanningSets μ x) = 0\n⊢ False",
"tactic": "simp [(forall_measure_inter_spanningSets_eq_zero _).mp con] at i_in_nonzeroes"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_2\nβ : Type ?u.742871\nγ : Type ?u.742874\nδ : Type ?u.742877\nι✝ : Type ?u.742880\nR : Type ?u.742883\nR' : Type ?u.742886\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nι : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nAs : ι → Set α\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nobs : {i | 0 < ↑↑μ (As i)} ⊆ ⋃ (n : ℕ), {i | 0 < ↑↑μ (As i ∩ spanningSets μ n)}\nn : ℕ\n⊢ ∀ (i : ι), MeasurableSet (As i ∩ spanningSets μ n)",
"tactic": "exact fun i => MeasurableSet.inter (As_mble i) (measurable_spanningSets μ n)"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u_2\nβ : Type ?u.742871\nγ : Type ?u.742874\nδ : Type ?u.742877\nι✝ : Type ?u.742880\nR : Type ?u.742883\nR' : Type ?u.742886\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nι : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nAs : ι → Set α\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nobs : {i | 0 < ↑↑μ (As i)} ⊆ ⋃ (n : ℕ), {i | 0 < ↑↑μ (As i ∩ spanningSets μ n)}\nn : ℕ\n⊢ Pairwise (Disjoint on fun i => As i ∩ spanningSets μ n)",
"tactic": "exact fun i j i_ne_j b hbi hbj =>\n As_disj i_ne_j (hbi.trans (inter_subset_left _ _)) (hbj.trans (inter_subset_left _ _))"
},
{
"state_after": "case refine'_3\nα : Type u_2\nβ : Type ?u.742871\nγ : Type ?u.742874\nδ : Type ?u.742877\nι✝ : Type ?u.742880\nR : Type ?u.742883\nR' : Type ?u.742886\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nι : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nAs : ι → Set α\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nobs : {i | 0 < ↑↑μ (As i)} ⊆ ⋃ (n : ℕ), {i | 0 < ↑↑μ (As i ∩ spanningSets μ n)}\nn : ℕ\n⊢ (⋃ (i : ι), As i ∩ spanningSets μ n) ⊆ spanningSets μ n",
"state_before": "case refine'_3\nα : Type u_2\nβ : Type ?u.742871\nγ : Type ?u.742874\nδ : Type ?u.742877\nι✝ : Type ?u.742880\nR : Type ?u.742883\nR' : Type ?u.742886\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nι : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nAs : ι → Set α\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nobs : {i | 0 < ↑↑μ (As i)} ⊆ ⋃ (n : ℕ), {i | 0 < ↑↑μ (As i ∩ spanningSets μ n)}\nn : ℕ\n⊢ ↑↑μ (⋃ (i : ι), As i ∩ spanningSets μ n) ≠ ⊤",
"tactic": "refine' (lt_of_le_of_lt (measure_mono _) (measure_spanningSets_lt_top μ n)).ne"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nα : Type u_2\nβ : Type ?u.742871\nγ : Type ?u.742874\nδ : Type ?u.742877\nι✝ : Type ?u.742880\nR : Type ?u.742883\nR' : Type ?u.742886\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns s' t : Set α\nι : Type u_1\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\nAs : ι → Set α\nAs_mble : ∀ (i : ι), MeasurableSet (As i)\nAs_disj : Pairwise (Disjoint on As)\nobs : {i | 0 < ↑↑μ (As i)} ⊆ ⋃ (n : ℕ), {i | 0 < ↑↑μ (As i ∩ spanningSets μ n)}\nn : ℕ\n⊢ (⋃ (i : ι), As i ∩ spanningSets μ n) ⊆ spanningSets μ n",
"tactic": "exact iUnion_subset fun i => inter_subset_right _ _"
}
]
| [
3622,
56
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
3608,
1
]
|
Mathlib/MeasureTheory/Measure/OuterMeasure.lean | MeasureTheory.OuterMeasure.iUnion_nat_of_monotone_of_tsum_ne_top | [
{
"state_after": "α : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\n⊢ Tendsto (fun k => ↑m ((⋃ (n : ℕ), s n) \\ s k)) atTop (𝓝 0)",
"state_before": "α : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\n⊢ ↑m (⋃ (n : ℕ), s n) = ⨆ (n : ℕ), ↑m (s n)",
"tactic": "refine' m.iUnion_of_tendsto_zero atTop _"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\nn : ℕ\n⊢ ↑m ((⋃ (n : ℕ), s n) \\ s n) ≤ ∑' (k : ℕ), ↑m (s (k + n + 1) \\ s (k + n))",
"state_before": "α : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\n⊢ Tendsto (fun k => ↑m ((⋃ (n : ℕ), s n) \\ s k)) atTop (𝓝 0)",
"tactic": "refine' tendsto_nhds_bot_mono' (ENNReal.tendsto_sum_nat_add _ h0) fun n => _"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\nn : ℕ\n⊢ (⋃ (n : ℕ), s n) \\ s n ⊆ ⋃ (i : ℕ), s (i + n + 1) \\ s (i + n)",
"state_before": "α : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\nn : ℕ\n⊢ ↑m ((⋃ (n : ℕ), s n) \\ s n) ≤ ∑' (k : ℕ), ↑m (s (k + n + 1) \\ s (k + n))",
"tactic": "refine' (m.mono _).trans (m.iUnion _)"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\nn : ℕ\nh' : Monotone s\n⊢ (⋃ (n : ℕ), s n) \\ s n ⊆ ⋃ (i : ℕ), s (i + n + 1) \\ s (i + n)",
"state_before": "α : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\nn : ℕ\n⊢ (⋃ (n : ℕ), s n) \\ s n ⊆ ⋃ (i : ℕ), s (i + n + 1) \\ s (i + n)",
"tactic": "have h' : Monotone s := @monotone_nat_of_le_succ (Set α) _ _ h_mono"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\nn : ℕ\nh' : Monotone s\n⊢ ∀ (i : ℕ), s i ⊆ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \\ s (i + n)",
"state_before": "α : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\nn : ℕ\nh' : Monotone s\n⊢ (⋃ (n : ℕ), s n) \\ s n ⊆ ⋃ (i : ℕ), s (i + n + 1) \\ s (i + n)",
"tactic": "simp only [diff_subset_iff, iUnion_subset_iff]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\nn : ℕ\nh' : Monotone s\ni : ℕ\nx : α\nhx : x ∈ s i\n⊢ x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \\ s (i + n)",
"state_before": "α : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\nn : ℕ\nh' : Monotone s\n⊢ ∀ (i : ℕ), s i ⊆ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \\ s (i + n)",
"tactic": "intro i x hx"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\nn : ℕ\nh' : Monotone s\ni : ℕ\nx : α\nhx : x ∈ s i\nthis : ∃ i, x ∈ s i\n⊢ x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \\ s (i + n)",
"state_before": "α : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\nn : ℕ\nh' : Monotone s\ni : ℕ\nx : α\nhx : x ∈ s i\n⊢ x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \\ s (i + n)",
"tactic": "have : ∃i, x ∈ s i := by exists i"
},
{
"state_after": "case mk.intro\nα : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\nn : ℕ\nh' : Monotone s\ni : ℕ\nx : α\nhx : x ∈ s i\nthis : ∃ i, x ∈ s i\nj : ℕ\nhj : x ∈ s j\nhlt : ∀ (m : ℕ), m < j → ¬x ∈ s m\n⊢ x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \\ s (i + n)",
"state_before": "α : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\nn : ℕ\nh' : Monotone s\ni : ℕ\nx : α\nhx : x ∈ s i\nthis : ∃ i, x ∈ s i\n⊢ x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \\ s (i + n)",
"tactic": "rcases Nat.findX this with ⟨j, hj, hlt⟩"
},
{
"state_after": "case mk.intro\nα : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\nn : ℕ\nh' : Monotone s\nx : α\nthis : ∃ i, x ∈ s i\nj : ℕ\nhj : x ∈ s j\nhlt : ∀ (m : ℕ), m < j → ¬x ∈ s m\n⊢ x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \\ s (i + n)",
"state_before": "case mk.intro\nα : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\nn : ℕ\nh' : Monotone s\ni : ℕ\nx : α\nhx : x ∈ s i\nthis : ∃ i, x ∈ s i\nj : ℕ\nhj : x ∈ s j\nhlt : ∀ (m : ℕ), m < j → ¬x ∈ s m\n⊢ x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \\ s (i + n)",
"tactic": "clear hx i"
},
{
"state_after": "case mk.intro.inl\nα : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\nn : ℕ\nh' : Monotone s\nx : α\nthis : ∃ i, x ∈ s i\nj : ℕ\nhj : x ∈ s j\nhlt : ∀ (m : ℕ), m < j → ¬x ∈ s m\nhjn : j ≤ n\n⊢ x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \\ s (i + n)\n\ncase mk.intro.inr\nα : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\nn : ℕ\nh' : Monotone s\nx : α\nthis : ∃ i, x ∈ s i\nj : ℕ\nhj : x ∈ s j\nhlt : ∀ (m : ℕ), m < j → ¬x ∈ s m\nhnj : n < j\n⊢ x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \\ s (i + n)",
"state_before": "case mk.intro\nα : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\nn : ℕ\nh' : Monotone s\nx : α\nthis : ∃ i, x ∈ s i\nj : ℕ\nhj : x ∈ s j\nhlt : ∀ (m : ℕ), m < j → ¬x ∈ s m\n⊢ x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \\ s (i + n)",
"tactic": "cases' le_or_lt j n with hjn hnj"
},
{
"state_after": "case mk.intro.inr\nα : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\nn : ℕ\nh' : Monotone s\nx : α\nthis✝ : ∃ i, x ∈ s i\nj : ℕ\nhj : x ∈ s j\nhlt : ∀ (m : ℕ), m < j → ¬x ∈ s m\nhnj : n < j\nthis : j - (n + 1) + n + 1 = j\n⊢ x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \\ s (i + n)",
"state_before": "case mk.intro.inr\nα : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\nn : ℕ\nh' : Monotone s\nx : α\nthis : ∃ i, x ∈ s i\nj : ℕ\nhj : x ∈ s j\nhlt : ∀ (m : ℕ), m < j → ¬x ∈ s m\nhnj : n < j\n⊢ x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \\ s (i + n)",
"tactic": "have : j - (n + 1) + n + 1 = j := by rw [add_assoc, tsub_add_cancel_of_le hnj.nat_succ_le]"
},
{
"state_after": "case mk.intro.inr.refine'_1\nα : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\nn : ℕ\nh' : Monotone s\nx : α\nthis✝ : ∃ i, x ∈ s i\nj : ℕ\nhj : x ∈ s j\nhlt : ∀ (m : ℕ), m < j → ¬x ∈ s m\nhnj : n < j\nthis : j - (n + 1) + n + 1 = j\n⊢ x ∈ s (j - (n + 1) + n + 1)\n\ncase mk.intro.inr.refine'_2\nα : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\nn : ℕ\nh' : Monotone s\nx : α\nthis✝ : ∃ i, x ∈ s i\nj : ℕ\nhj : x ∈ s j\nhlt : ∀ (m : ℕ), m < j → ¬x ∈ s m\nhnj : n < j\nthis : j - (n + 1) + n + 1 = j\n⊢ j - (n + 1) + n < j",
"state_before": "case mk.intro.inr\nα : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\nn : ℕ\nh' : Monotone s\nx : α\nthis✝ : ∃ i, x ∈ s i\nj : ℕ\nhj : x ∈ s j\nhlt : ∀ (m : ℕ), m < j → ¬x ∈ s m\nhnj : n < j\nthis : j - (n + 1) + n + 1 = j\n⊢ x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \\ s (i + n)",
"tactic": "refine' Or.inr (mem_iUnion.2 ⟨j - (n + 1), _, hlt _ _⟩)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\nn : ℕ\nh' : Monotone s\ni : ℕ\nx : α\nhx : x ∈ s i\n⊢ ∃ i, x ∈ s i",
"tactic": "exists i"
},
{
"state_after": "no goals",
"state_before": "case mk.intro.inl\nα : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\nn : ℕ\nh' : Monotone s\nx : α\nthis : ∃ i, x ∈ s i\nj : ℕ\nhj : x ∈ s j\nhlt : ∀ (m : ℕ), m < j → ¬x ∈ s m\nhjn : j ≤ n\n⊢ x ∈ s n ∪ ⋃ (i : ℕ), s (i + n + 1) \\ s (i + n)",
"tactic": "exact Or.inl (h' hjn hj)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\nn : ℕ\nh' : Monotone s\nx : α\nthis : ∃ i, x ∈ s i\nj : ℕ\nhj : x ∈ s j\nhlt : ∀ (m : ℕ), m < j → ¬x ∈ s m\nhnj : n < j\n⊢ j - (n + 1) + n + 1 = j",
"tactic": "rw [add_assoc, tsub_add_cancel_of_le hnj.nat_succ_le]"
},
{
"state_after": "no goals",
"state_before": "case mk.intro.inr.refine'_1\nα : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\nn : ℕ\nh' : Monotone s\nx : α\nthis✝ : ∃ i, x ∈ s i\nj : ℕ\nhj : x ∈ s j\nhlt : ∀ (m : ℕ), m < j → ¬x ∈ s m\nhnj : n < j\nthis : j - (n + 1) + n + 1 = j\n⊢ x ∈ s (j - (n + 1) + n + 1)",
"tactic": "rwa [this]"
},
{
"state_after": "no goals",
"state_before": "case mk.intro.inr.refine'_2\nα : Type u_1\nβ : Type ?u.15791\nR : Type ?u.15794\nR' : Type ?u.15797\nms : Set (OuterMeasure α)\nm✝ m : OuterMeasure α\ns : ℕ → Set α\nh_mono : ∀ (n : ℕ), s n ⊆ s (n + 1)\nh0 : (∑' (k : ℕ), ↑m (s (k + 1) \\ s k)) ≠ ⊤\ninst✝ : (i : ℕ) → DecidablePred fun x => x ∈ s i\nn : ℕ\nh' : Monotone s\nx : α\nthis✝ : ∃ i, x ∈ s i\nj : ℕ\nhj : x ∈ s j\nhlt : ∀ (m : ℕ), m < j → ¬x ∈ s m\nhnj : n < j\nthis : j - (n + 1) + n + 1 = j\n⊢ j - (n + 1) + n < j",
"tactic": "rw [← Nat.succ_le_iff, Nat.succ_eq_add_one, this]"
}
]
| [
217,
54
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
199,
1
]
|
Mathlib/MeasureTheory/Function/LpSeminorm.lean | MeasureTheory.snorm_zero' | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.923765\nF : Type u_2\nG : Type ?u.923771\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\n⊢ snorm (fun x => 0) p μ = 0",
"tactic": "convert snorm_zero (F := F)"
}
]
| [
214,
93
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
214,
1
]
|
Mathlib/RingTheory/MvPolynomial/Symmetric.lean | MvPolynomial.esymm_eq_sum_subtype | []
| [
183,
55
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
181,
1
]
|
Mathlib/Algebra/Order/Group/Defs.lean | div_lt_div_right' | []
| [
885,
31
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
884,
1
]
|
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean | BoxIntegral.Prepartition.IsPartition.restrict | [
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nh : IsPartition π\nhJ : J ≤ I\n⊢ Prepartition.iUnion (restrict π J) = ↑J",
"tactic": "simp [h.iUnion_eq, hJ]"
}
]
| [
773,
59
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
772,
11
]
|
Mathlib/Combinatorics/SetFamily/Compression/UV.lean | UV.aux | [
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu a : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ ∅ ∧ IsCompressed (erase u x) (erase ∅ y) 𝒜\n⊢ u = ∅",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu v a : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ v ∧ IsCompressed (erase u x) (erase v y) 𝒜\n⊢ v = ∅ → u = ∅",
"tactic": "rintro rfl"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu a✝ : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ ∅ ∧ IsCompressed (erase u x) (erase ∅ y) 𝒜\na : α\nha : a ∈ u\n⊢ False",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu a : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ ∅ ∧ IsCompressed (erase u x) (erase ∅ y) 𝒜\n⊢ u = ∅",
"tactic": "refine' eq_empty_of_forall_not_mem fun a ha => _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 : Finset (Finset α)\nu a✝ : Finset α\nhuv : ∀ (x : α), x ∈ u → ∃ y, y ∈ ∅ ∧ IsCompressed (erase u x) (erase ∅ y) 𝒜\na : α\nha : a ∈ u\n⊢ False",
"tactic": "obtain ⟨_, ⟨⟩, -⟩ := huv a ha"
}
]
| [
319,
32
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
315,
9
]
|
Mathlib/Analysis/Complex/Basic.lean | Complex.continuous_re | []
| [
262,
19
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
261,
1
]
|
Mathlib/Algebra/Star/Subalgebra.lean | StarSubalgebra.coe_sInf | []
| [
671,
13
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
670,
1
]
|
Mathlib/SetTheory/Ordinal/Arithmetic.lean | Ordinal.IsNormal.sup | [
{
"state_after": "α : Type ?u.284505\nβ : Type ?u.284508\nγ : Type ?u.284511\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal → Ordinal\nH : IsNormal f\nι : Type u\ng : ι → Ordinal\ninst✝ : Nonempty ι\na : Ordinal\n⊢ f (Ordinal.sup g) ≤ a ↔ ∀ (i : ι), (f ∘ g) i ≤ a",
"state_before": "α : Type ?u.284505\nβ : Type ?u.284508\nγ : Type ?u.284511\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal → Ordinal\nH : IsNormal f\nι : Type u\ng : ι → Ordinal\ninst✝ : Nonempty ι\na : Ordinal\n⊢ f (Ordinal.sup g) ≤ a ↔ Ordinal.sup (f ∘ g) ≤ a",
"tactic": "rw [sup_le_iff]"
},
{
"state_after": "α : Type ?u.284505\nβ : Type ?u.284508\nγ : Type ?u.284511\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal → Ordinal\nH : IsNormal f\nι : Type u\ng : ι → Ordinal\ninst✝ : Nonempty ι\na : Ordinal\n⊢ f (Ordinal.sup g) ≤ a ↔ ∀ (i : ι), f (g i) ≤ a",
"state_before": "α : Type ?u.284505\nβ : Type ?u.284508\nγ : Type ?u.284511\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal → Ordinal\nH : IsNormal f\nι : Type u\ng : ι → Ordinal\ninst✝ : Nonempty ι\na : Ordinal\n⊢ f (Ordinal.sup g) ≤ a ↔ ∀ (i : ι), (f ∘ g) i ≤ a",
"tactic": "simp only [comp]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.284505\nβ : Type ?u.284508\nγ : Type ?u.284511\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nf : Ordinal → Ordinal\nH : IsNormal f\nι : Type u\ng : ι → Ordinal\ninst✝ : Nonempty ι\na : Ordinal\n⊢ f (Ordinal.sup g) ≤ a ↔ ∀ (i : ι), f (g i) ≤ a",
"tactic": "rw [H.le_set' Set.univ Set.univ_nonempty g] <;>\nsimp [sup_le_iff]"
}
]
| [
1306,
24
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1302,
1
]
|
Mathlib/Analysis/Convex/Side.lean | AffineSubspace.wOppSide_smul_vsub_vadd_left | [
{
"state_after": "R : Type u_1\nV : Type u_2\nV' : Type ?u.178003\nP : Type u_3\nP' : Type ?u.178009\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\np₁ p₂ x : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nt : R\nht : t ≤ 0\n⊢ SameRay R (t • (x -ᵥ p₁) +ᵥ p₂ -ᵥ p₂) (p₁ -ᵥ x)",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.178003\nP : Type u_3\nP' : Type ?u.178009\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\np₁ p₂ x : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nt : R\nht : t ≤ 0\n⊢ WOppSide s (t • (x -ᵥ p₁) +ᵥ p₂) x",
"tactic": "refine' ⟨p₂, hp₂, p₁, hp₁, _⟩"
},
{
"state_after": "R : Type u_1\nV : Type u_2\nV' : Type ?u.178003\nP : Type u_3\nP' : Type ?u.178009\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\np₁ p₂ x : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nt : R\nht : t ≤ 0\n⊢ SameRay R (-t • (p₁ -ᵥ x)) (p₁ -ᵥ x)",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.178003\nP : Type u_3\nP' : Type ?u.178009\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\np₁ p₂ x : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nt : R\nht : t ≤ 0\n⊢ SameRay R (t • (x -ᵥ p₁) +ᵥ p₂ -ᵥ p₂) (p₁ -ᵥ x)",
"tactic": "rw [vadd_vsub, ← neg_neg t, neg_smul, ← smul_neg, neg_vsub_eq_vsub_rev]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.178003\nP : Type u_3\nP' : Type ?u.178009\ninst✝⁶ : StrictOrderedCommRing R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\np₁ p₂ x : P\nhp₁ : p₁ ∈ s\nhp₂ : p₂ ∈ s\nt : R\nht : t ≤ 0\n⊢ SameRay R (-t • (p₁ -ᵥ x)) (p₁ -ᵥ x)",
"tactic": "exact SameRay.sameRay_nonneg_smul_left _ (neg_nonneg.2 ht)"
}
]
| [
356,
61
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
352,
1
]
|
Std/Logic.lean | and_iff_left_of_imp | []
| [
199,
35
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
198,
1
]
|
Mathlib/Analysis/Calculus/FDeriv/Star.lean | HasStrictFDerivAt.star | []
| [
41,
74
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
39,
1
]
|
Mathlib/Data/Finset/Image.lean | Finset.fin_map | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.137971\nβ : Type ?u.137974\nγ : Type ?u.137977\nn : ℕ\ns : Finset ℕ\n⊢ map Fin.valEmbedding (Finset.fin n s) = filter (fun x => x < n) s",
"tactic": "simp [Finset.fin, Finset.map_map]"
}
]
| [
742,
36
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
741,
1
]
|
Mathlib/Analysis/Convex/Normed.lean | convexHull_diam | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.41107\nE : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns✝ t s : Set E\n⊢ diam (↑(convexHull ℝ).toOrderHom s) = diam s",
"tactic": "simp only [Metric.diam, convexHull_ediam]"
}
]
| [
117,
44
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
116,
1
]
|
Mathlib/Order/Filter/Lift.lean | Filter.comap_eq_lift' | []
| [
414,
62
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
413,
1
]
|
Mathlib/Algebra/Order/Hom/Ring.lean | OrderRingIso.toOrderIso_eq_coe | []
| [
415,
19
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
414,
1
]
|
Mathlib/Data/Finsupp/Basic.lean | Finsupp.comapSMul_apply | [
{
"state_after": "α : Type u_1\nβ : Type ?u.593102\nγ : Type ?u.593105\nι : Type ?u.593108\nM : Type u_2\nM' : Type ?u.593114\nN : Type ?u.593117\nP : Type ?u.593120\nG : Type u_3\nH : Type ?u.593126\nR : Type ?u.593129\nS : Type ?u.593132\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : AddCommMonoid M\ng : G\nf : α →₀ M\na : α\n⊢ ↑(g • f) (g • g⁻¹ • a) = ↑f (g⁻¹ • a)",
"state_before": "α : Type u_1\nβ : Type ?u.593102\nγ : Type ?u.593105\nι : Type ?u.593108\nM : Type u_2\nM' : Type ?u.593114\nN : Type ?u.593117\nP : Type ?u.593120\nG : Type u_3\nH : Type ?u.593126\nR : Type ?u.593129\nS : Type ?u.593132\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : AddCommMonoid M\ng : G\nf : α →₀ M\na : α\n⊢ ↑(g • f) a = ↑f (g⁻¹ • a)",
"tactic": "conv_lhs => rw [← smul_inv_smul g a]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.593102\nγ : Type ?u.593105\nι : Type ?u.593108\nM : Type u_2\nM' : Type ?u.593114\nN : Type ?u.593117\nP : Type ?u.593120\nG : Type u_3\nH : Type ?u.593126\nR : Type ?u.593129\nS : Type ?u.593132\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : AddCommMonoid M\ng : G\nf : α →₀ M\na : α\n⊢ ↑(g • f) (g • g⁻¹ • a) = ↑f (g⁻¹ • a)",
"tactic": "exact mapDomain_apply (MulAction.injective g) _ (g⁻¹ • a)"
}
]
| [
1477,
60
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1475,
1
]
|
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