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Mathlib/Algebra/Order/ToIntervalMod.lean | toIcoMod_zero_one | [
{
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"tactic": "simp [toIcoMod_eq_add_fract_mul]"
}
] | [
1037,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1036,
1
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Mathlib/ModelTheory/Semantics.lean | FirstOrder.Language.BoundedFormula.realize_bdEqual | [] | [
278,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
276,
1
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Mathlib/FieldTheory/IntermediateField.lean | IntermediateField.isIntegral_iff | [
{
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"tactic": "rw [← isAlgebraic_iff_isIntegral, isAlgebraic_iff, isAlgebraic_iff_isIntegral]"
}
] | [
734,
81
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
733,
1
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Mathlib/Data/Int/Log.lean | Int.clog_inv | [
{
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"state_before": "R : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nr : R\n⊢ clog b r⁻¹ = -log b r",
"tactic": "cases' lt_or_le 0 r with hrp hrp"
},
{
"state_after": "case inl.inl\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nr : R\nhrp : 0 < r\nhr : 1 ≤ r\n⊢ clog b r⁻¹ = -log b r\n\ncase inl.inr\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nr : R\nhrp : 0 < r\nhr : r ≤ 1\n⊢ clog b r⁻¹ = -log b r",
"state_before": "case inl\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nr : R\nhrp : 0 < r\n⊢ clog b r⁻¹ = -log b r",
"tactic": "obtain hr | hr := le_total 1 r"
},
{
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"tactic": "rw [clog_of_right_le_one _ (inv_le_one hr), log_of_one_le_right _ hr, inv_inv]"
},
{
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"tactic": "rw [clog_of_one_le_right _ (one_le_inv hrp hr), log_of_right_le_one _ hr, neg_neg]"
},
{
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"tactic": "rw [clog_of_right_le_zero _ (inv_nonpos.mpr hrp), log_of_right_le_zero _ hrp, neg_zero]"
}
] | [
218,
92
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
213,
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Mathlib/NumberTheory/Padics/Hensel.lean | newton_seq_dist_tendsto | [] | [
406,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
402,
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Mathlib/CategoryTheory/Limits/Shapes/Images.lean | CategoryTheory.Limits.image.factor_map | [
{
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"tactic": "simp"
}
] | [
800,
10
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799,
1
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Mathlib/RingTheory/Ideal/Cotangent.lean | Ideal.cotangentEquivIdeal_symm_apply | [
{
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"tactic": "apply I.cotangentEquivIdeal.injective"
},
{
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"state_before": "case a\nR : Type u\nS : Type v\nS' : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra S R\ninst✝³ : CommSemiring S'\ninst✝² : Algebra S' R\ninst✝¹ : Algebra S S'\ninst✝ : IsScalarTower S S' R\nI : Ideal R\nx : R\nhx : x ∈ I\n⊢ ↑(cotangentEquivIdeal I)\n (↑(LinearEquiv.symm (cotangentEquivIdeal I))\n { val := ↑(Submodule.mkQ (I ^ 2)) x,\n property := (_ : ↑(Submodule.mkQ (I ^ 2)) x ∈ Submodule.map (Submodule.mkQ (I ^ 2)) I) }) =\n ↑(cotangentEquivIdeal I) (↑(toCotangent I) { val := x, property := hx })",
"tactic": "rw [I.cotangentEquivIdeal.apply_symm_apply]"
},
{
"state_after": "case a.a\nR : Type u\nS : Type v\nS' : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra S R\ninst✝³ : CommSemiring S'\ninst✝² : Algebra S' R\ninst✝¹ : Algebra S S'\ninst✝ : IsScalarTower S S' R\nI : Ideal R\nx : R\nhx : x ∈ I\n⊢ ↑{ val := ↑(Submodule.mkQ (I ^ 2)) x,\n property := (_ : ↑(Submodule.mkQ (I ^ 2)) x ∈ Submodule.map (Submodule.mkQ (I ^ 2)) I) } =\n ↑(↑(cotangentEquivIdeal I) (↑(toCotangent I) { val := x, property := hx }))",
"state_before": "case a\nR : Type u\nS : Type v\nS' : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra S R\ninst✝³ : CommSemiring S'\ninst✝² : Algebra S' R\ninst✝¹ : Algebra S S'\ninst✝ : IsScalarTower S S' R\nI : Ideal R\nx : R\nhx : x ∈ I\n⊢ { val := ↑(Submodule.mkQ (I ^ 2)) x,\n property := (_ : ↑(Submodule.mkQ (I ^ 2)) x ∈ Submodule.map (Submodule.mkQ (I ^ 2)) I) } =\n ↑(cotangentEquivIdeal I) (↑(toCotangent I) { val := x, property := hx })",
"tactic": "ext"
},
{
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"tactic": "rfl"
}
] | [
175,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
169,
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Mathlib/Algebra/Order/Floor.lean | Int.preimage_ceil_singleton | [] | [
1234,
27
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1233,
1
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Mathlib/Data/Finset/Lattice.lean | Finset.inf_inf | [] | [
362,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
361,
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Mathlib/FieldTheory/Finite/Polynomial.lean | MvPolynomial.frobenius_zmod | [
{
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"state_before": "σ : Type u_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : MvPolynomial σ (ZMod p)\n⊢ ↑(frobenius (MvPolynomial σ (ZMod p)) p) f = ↑(expand p) f",
"tactic": "apply induction_on f"
},
{
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"tactic": "intro a"
},
{
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"tactic": "rw [expand_C, frobenius_def, ← C_pow, ZMod.pow_card]"
},
{
"state_after": "case h_add\nσ : Type u_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : MvPolynomial σ (ZMod p)\n⊢ ∀ (p_1 q : MvPolynomial σ (ZMod p)),\n ↑(frobenius (MvPolynomial σ (ZMod p)) p) p_1 = ↑(expand p) p_1 →\n ↑(frobenius (MvPolynomial σ (ZMod p)) p) q = ↑(expand p) q →\n ↑(frobenius (MvPolynomial σ (ZMod p)) p) p_1 + ↑(frobenius (MvPolynomial σ (ZMod p)) p) q =\n ↑(expand p) p_1 + ↑(expand p) q",
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"tactic": "simp only [AlgHom.map_add, RingHom.map_add]"
},
{
"state_after": "case h_add\nσ : Type u_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf p✝ q✝ : MvPolynomial σ (ZMod p)\nhf : ↑(frobenius (MvPolynomial σ (ZMod p)) p) p✝ = ↑(expand p) p✝\nhg : ↑(frobenius (MvPolynomial σ (ZMod p)) p) q✝ = ↑(expand p) q✝\n⊢ ↑(frobenius (MvPolynomial σ (ZMod p)) p) p✝ + ↑(frobenius (MvPolynomial σ (ZMod p)) p) q✝ =\n ↑(expand p) p✝ + ↑(expand p) q✝",
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"tactic": "intro _ _ hf hg"
},
{
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"tactic": "rw [hf, hg]"
},
{
"state_after": "case h_X\nσ : Type u_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : MvPolynomial σ (ZMod p)\n⊢ ∀ (p_1 : MvPolynomial σ (ZMod p)) (n : σ),\n ↑(frobenius (MvPolynomial σ (ZMod p)) p) p_1 = ↑(expand p) p_1 →\n ↑(frobenius (MvPolynomial σ (ZMod p)) p) p_1 * ↑(frobenius (MvPolynomial σ (ZMod p)) p) (X n) =\n ↑(expand p) p_1 * X n ^ p",
"state_before": "case h_X\nσ : Type u_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : MvPolynomial σ (ZMod p)\n⊢ ∀ (p_1 : MvPolynomial σ (ZMod p)) (n : σ),\n ↑(frobenius (MvPolynomial σ (ZMod p)) p) p_1 = ↑(expand p) p_1 →\n ↑(frobenius (MvPolynomial σ (ZMod p)) p) (p_1 * X n) = ↑(expand p) (p_1 * X n)",
"tactic": "simp only [expand_X, RingHom.map_mul, AlgHom.map_mul]"
},
{
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"tactic": "intro _ _ hf"
},
{
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"tactic": "rw [hf, frobenius_def]"
}
] | [
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] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/Data/Fintype/Card.lean | Infinite.natEmbeddingAux_injective | [
{
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{
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"tactic": "letI := Classical.decEq α"
},
{
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"tactic": "rcases lt_trichotomy m n with hmn | rfl | hnm"
},
{
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"state_before": "case inl\nα✝ : Type ?u.80507\nβ : Type ?u.80510\nγ : Type ?u.80513\nα : Type u_1\ninst✝ : Infinite α\nm n : ℕ\nh : Infinite.natEmbeddingAux α m = Infinite.natEmbeddingAux α n\nthis : DecidableEq α := Classical.decEq α\nhmn : m < n\n⊢ m = n",
"tactic": "apply natEmbeddingAux_injective_aux α m n h hmn"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nα✝ : Type ?u.80507\nβ : Type ?u.80510\nγ : Type ?u.80513\nα : Type u_1\ninst✝ : Infinite α\nm : ℕ\nthis : DecidableEq α := Classical.decEq α\nh : Infinite.natEmbeddingAux α m = Infinite.natEmbeddingAux α m\n⊢ m = m",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nα✝ : Type ?u.80507\nβ : Type ?u.80510\nγ : Type ?u.80513\nα : Type u_1\ninst✝ : Infinite α\nm n : ℕ\nh : Infinite.natEmbeddingAux α m = Infinite.natEmbeddingAux α n\nthis : DecidableEq α := Classical.decEq α\nhnm : n < m\n⊢ m = n",
"tactic": "apply (natEmbeddingAux_injective_aux α n m h.symm hnm).symm"
}
] | [
1101,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1094,
9
] |
Mathlib/Analysis/NormedSpace/Basic.lean | frontier_sphere' | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.310915\nβ : Type ?u.310918\nγ : Type ?u.310921\nι : Type ?u.310924\ninst✝⁶ : NormedField α\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace α E\nF : Type ?u.311017\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace α F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Nontrivial E\nx : E\nr : ℝ\n⊢ frontier (sphere x r) = sphere x r",
"tactic": "rw [isClosed_sphere.frontier_eq, interior_sphere' x, diff_empty]"
}
] | [
403,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
401,
1
] |
Mathlib/LinearAlgebra/Matrix/Block.lean | Matrix.blockTriangular_transpose_iff | [] | [
84,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
82,
11
] |
Mathlib/CategoryTheory/Products/Bifunctor.lean | CategoryTheory.Bifunctor.map_comp_id | [
{
"state_after": "no goals",
"state_before": "C : Type u₁\nD : Type u₂\nE : Type u₃\ninst✝² : Category C\ninst✝¹ : Category D\ninst✝ : Category E\nF : C × D ⥤ E\nX Y Z : C\nW : D\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ F.map (f ≫ g, 𝟙 W) = F.map (f, 𝟙 W) ≫ F.map (g, 𝟙 W)",
"tactic": "rw [← Functor.map_comp, prod_comp, Category.comp_id]"
}
] | [
45,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
42,
1
] |
Mathlib/MeasureTheory/Function/L2Space.lean | MeasureTheory.L2.integral_inner_eq_sq_snorm | [
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.50966\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp E 2 }\n⊢ (∫ (a : α), ↑‖↑↑f a‖ ^ 2 ∂μ) = ↑(ENNReal.toReal (∫⁻ (a : α), ↑‖↑↑f a‖₊ ^ 2 ∂μ))",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.50966\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp E 2 }\n⊢ (∫ (a : α), inner (↑↑f a) (↑↑f a) ∂μ) = ↑(ENNReal.toReal (∫⁻ (a : α), ↑‖↑↑f a‖₊ ^ 2 ∂μ))",
"tactic": "simp_rw [inner_self_eq_norm_sq_to_K]"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.50966\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp E 2 }\n⊢ (∫ (a : α), ‖↑↑f a‖ ^ 2 ∂μ) = ENNReal.toReal (∫⁻ (a : α), ↑(‖↑↑f a‖₊ ^ 2) ∂μ)",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.50966\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp E 2 }\n⊢ (∫ (a : α), ↑‖↑↑f a‖ ^ 2 ∂μ) = ↑(ENNReal.toReal (∫⁻ (a : α), ↑‖↑↑f a‖₊ ^ 2 ∂μ))",
"tactic": "norm_cast"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nF : Type ?u.50966\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp E 2 }\n⊢ ENNReal.toReal (∫⁻ (a : α), ENNReal.ofReal (‖↑↑f a‖ ^ 2) ∂μ) = ENNReal.toReal (∫⁻ (a : α), ↑(‖↑↑f a‖₊ ^ 2) ∂μ)\n\ncase hf\nα : Type u_1\nE : Type u_2\nF : Type ?u.50966\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp E 2 }\n⊢ 0 ≤ᵐ[μ] fun a => ‖↑↑f a‖ ^ 2\n\ncase hfm\nα : Type u_1\nE : Type u_2\nF : Type ?u.50966\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp E 2 }\n⊢ AEStronglyMeasurable (fun a => ‖↑↑f a‖ ^ 2) μ",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.50966\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp E 2 }\n⊢ (∫ (a : α), ‖↑↑f a‖ ^ 2 ∂μ) = ENNReal.toReal (∫⁻ (a : α), ↑(‖↑↑f a‖₊ ^ 2) ∂μ)",
"tactic": "rw [integral_eq_lintegral_of_nonneg_ae]"
},
{
"state_after": "case hf\nα : Type u_1\nE : Type u_2\nF : Type ?u.50966\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp E 2 }\n⊢ 0 ≤ᵐ[μ] fun a => ‖↑↑f a‖ ^ 2\n\ncase hfm\nα : Type u_1\nE : Type u_2\nF : Type ?u.50966\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp E 2 }\n⊢ AEStronglyMeasurable (fun a => ‖↑↑f a‖ ^ 2) μ\n\nα : Type u_1\nE : Type u_2\nF : Type ?u.50966\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp E 2 }\n⊢ ENNReal.toReal (∫⁻ (a : α), ENNReal.ofReal (‖↑↑f a‖ ^ 2) ∂μ) = ENNReal.toReal (∫⁻ (a : α), ↑(‖↑↑f a‖₊ ^ 2) ∂μ)",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.50966\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp E 2 }\n⊢ ENNReal.toReal (∫⁻ (a : α), ENNReal.ofReal (‖↑↑f a‖ ^ 2) ∂μ) = ENNReal.toReal (∫⁻ (a : α), ↑(‖↑↑f a‖₊ ^ 2) ∂μ)\n\ncase hf\nα : Type u_1\nE : Type u_2\nF : Type ?u.50966\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp E 2 }\n⊢ 0 ≤ᵐ[μ] fun a => ‖↑↑f a‖ ^ 2\n\ncase hfm\nα : Type u_1\nE : Type u_2\nF : Type ?u.50966\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp E 2 }\n⊢ AEStronglyMeasurable (fun a => ‖↑↑f a‖ ^ 2) μ",
"tactic": "rotate_left"
},
{
"state_after": "case e_a.e_f\nα : Type u_1\nE : Type u_2\nF : Type ?u.50966\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp E 2 }\n⊢ (fun a => ENNReal.ofReal (‖↑↑f a‖ ^ 2)) = fun a => ↑(‖↑↑f a‖₊ ^ 2)",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.50966\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp E 2 }\n⊢ ENNReal.toReal (∫⁻ (a : α), ENNReal.ofReal (‖↑↑f a‖ ^ 2) ∂μ) = ENNReal.toReal (∫⁻ (a : α), ↑(‖↑↑f a‖₊ ^ 2) ∂μ)",
"tactic": "congr"
},
{
"state_after": "case e_a.e_f.h\nα : Type u_1\nE : Type u_2\nF : Type ?u.50966\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp E 2 }\nx : α\n⊢ ENNReal.ofReal (‖↑↑f x‖ ^ 2) = ↑(‖↑↑f x‖₊ ^ 2)",
"state_before": "case e_a.e_f\nα : Type u_1\nE : Type u_2\nF : Type ?u.50966\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp E 2 }\n⊢ (fun a => ENNReal.ofReal (‖↑↑f a‖ ^ 2)) = fun a => ↑(‖↑↑f a‖₊ ^ 2)",
"tactic": "ext1 x"
},
{
"state_after": "case e_a.e_f.h\nα : Type u_1\nE : Type u_2\nF : Type ?u.50966\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp E 2 }\nx : α\nh_two : 2 = ↑2\n⊢ ENNReal.ofReal (‖↑↑f x‖ ^ 2) = ↑(‖↑↑f x‖₊ ^ 2)",
"state_before": "case e_a.e_f.h\nα : Type u_1\nE : Type u_2\nF : Type ?u.50966\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp E 2 }\nx : α\n⊢ ENNReal.ofReal (‖↑↑f x‖ ^ 2) = ↑(‖↑↑f x‖₊ ^ 2)",
"tactic": "have h_two : (2 : ℝ) = ((2 : ℕ) : ℝ) := by simp"
},
{
"state_after": "case e_a.e_f.h\nα : Type u_1\nE : Type u_2\nF : Type ?u.50966\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp E 2 }\nx : α\nh_two : 2 = ↑2\n⊢ ↑‖↑↑f x‖₊ ^ 2 = ↑(‖↑↑f x‖₊ ^ 2)",
"state_before": "case e_a.e_f.h\nα : Type u_1\nE : Type u_2\nF : Type ?u.50966\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp E 2 }\nx : α\nh_two : 2 = ↑2\n⊢ ENNReal.ofReal (‖↑↑f x‖ ^ 2) = ↑(‖↑↑f x‖₊ ^ 2)",
"tactic": "rw [← Real.rpow_nat_cast _ 2, ← h_two, ←\n ENNReal.ofReal_rpow_of_nonneg (norm_nonneg _) zero_le_two, ofReal_norm_eq_coe_nnnorm]"
},
{
"state_after": "no goals",
"state_before": "case e_a.e_f.h\nα : Type u_1\nE : Type u_2\nF : Type ?u.50966\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp E 2 }\nx : α\nh_two : 2 = ↑2\n⊢ ↑‖↑↑f x‖₊ ^ 2 = ↑(‖↑↑f x‖₊ ^ 2)",
"tactic": "norm_cast"
},
{
"state_after": "no goals",
"state_before": "case hf\nα : Type u_1\nE : Type u_2\nF : Type ?u.50966\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp E 2 }\n⊢ 0 ≤ᵐ[μ] fun a => ‖↑↑f a‖ ^ 2",
"tactic": "exact Filter.eventually_of_forall fun x => sq_nonneg _"
},
{
"state_after": "no goals",
"state_before": "case hfm\nα : Type u_1\nE : Type u_2\nF : Type ?u.50966\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp E 2 }\n⊢ AEStronglyMeasurable (fun a => ‖↑↑f a‖ ^ 2) μ",
"tactic": "exact ((Lp.aestronglyMeasurable f).norm.aemeasurable.pow_const _).aestronglyMeasurable"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.50966\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : NormedAddCommGroup F\nf : { x // x ∈ Lp E 2 }\nx : α\n⊢ 2 = ↑2",
"tactic": "simp"
}
] | [
173,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
160,
1
] |
Mathlib/Topology/Separation.lean | RegularSpace.ofLift'_closure | [] | [
1506,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1505,
1
] |
Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean | NonUnitalSubsemiring.toSubsemigroup_injective | [] | [
148,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
146,
1
] |
Mathlib/Topology/MetricSpace/Holder.lean | HolderWith.comp | [] | [
206,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
204,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Pointwise.lean | Subalgebra.mul_toSubmodule_le | [
{
"state_after": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ ∀ (m : A), m ∈ ↑toSubmodule S → ∀ (n : A), n ∈ ↑toSubmodule T → m * n ∈ ↑toSubmodule (S ⊔ T)",
"state_before": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ ↑toSubmodule S * ↑toSubmodule T ≤ ↑toSubmodule (S ⊔ T)",
"tactic": "rw [Submodule.mul_le]"
},
{
"state_after": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ ↑toSubmodule S\nz : A\nhz : z ∈ ↑toSubmodule T\n⊢ y * z ∈ ↑toSubmodule (S ⊔ T)",
"state_before": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\n⊢ ∀ (m : A), m ∈ ↑toSubmodule S → ∀ (n : A), n ∈ ↑toSubmodule T → m * n ∈ ↑toSubmodule (S ⊔ T)",
"tactic": "intro y hy z hz"
},
{
"state_after": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ ↑toSubmodule S\nz : A\nhz : z ∈ ↑toSubmodule T\n⊢ y * z ∈ S ⊔ T",
"state_before": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ ↑toSubmodule S\nz : A\nhz : z ∈ ↑toSubmodule T\n⊢ y * z ∈ ↑toSubmodule (S ⊔ T)",
"tactic": "show y * z ∈ S ⊔ T"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nA : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Subalgebra R A\ny : A\nhy : y ∈ ↑toSubmodule S\nz : A\nhz : z ∈ ↑toSubmodule T\n⊢ y * z ∈ S ⊔ T",
"tactic": "exact mul_mem (Algebra.mem_sup_left hy) (Algebra.mem_sup_right hz)"
}
] | [
35,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
30,
1
] |
Mathlib/Data/Finsupp/WellFounded.lean | Finsupp.Lex.wellFounded | [] | [
49,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
48,
1
] |
Mathlib/Data/Nat/Log.lean | Nat.log_eq_of_pow_le_of_lt_pow | [
{
"state_after": "case inl\nb n : ℕ\nh₁ : b ^ 0 ≤ n\nh₂ : n < b ^ (0 + 1)\n⊢ log b n = 0\n\ncase inr\nb m n : ℕ\nh₁ : b ^ m ≤ n\nh₂ : n < b ^ (m + 1)\nhm : m ≠ 0\n⊢ log b n = m",
"state_before": "b m n : ℕ\nh₁ : b ^ m ≤ n\nh₂ : n < b ^ (m + 1)\n⊢ log b n = m",
"tactic": "rcases eq_or_ne m 0 with (rfl | hm)"
},
{
"state_after": "case inl\nb n : ℕ\nh₁ : b ^ 0 ≤ n\nh₂ : n < b\n⊢ log b n = 0",
"state_before": "case inl\nb n : ℕ\nh₁ : b ^ 0 ≤ n\nh₂ : n < b ^ (0 + 1)\n⊢ log b n = 0",
"tactic": "rw [pow_one] at h₂"
},
{
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"tactic": "exact log_of_lt h₂"
},
{
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"tactic": "exact (log_eq_iff (Or.inl hm)).2 ⟨h₁, h₂⟩"
}
] | [
157,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
152,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean | HasFTaylorSeriesUpToOn.shift_of_succ | [
{
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"tactic": "constructor"
},
{
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"tactic": "intro x _"
},
{
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"state_before": "case zero_eq\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nH : HasFTaylorSeriesUpToOn (↑(n + 1)) f p s\nx : E\na✝ : x ∈ s\n⊢ ContinuousMultilinearMap.uncurry0 (FormalMultilinearSeries.shift (p x) 0) =\n ↑(continuousMultilinearCurryFin1 𝕜 E F) (p x 1)",
"tactic": "rfl"
},
{
"state_after": "case fderivWithin\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝ n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nH : HasFTaylorSeriesUpToOn (↑(n + 1)) f p s\nm : ℕ\nhm : ↑m < ↑n\nx : E\nhx : x ∈ s\n⊢ HasFDerivWithinAt (fun x => FormalMultilinearSeries.shift (p x) m)\n (ContinuousMultilinearMap.curryLeft (FormalMultilinearSeries.shift (p x) (Nat.succ m))) s x",
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"tactic": "intro m (hm : (m : ℕ∞) < n) x (hx : x ∈ s)"
},
{
"state_after": "case A\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝ n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nH : HasFTaylorSeriesUpToOn (↑(n + 1)) f p s\nm : ℕ\nhm : ↑m < ↑n\nx : E\nhx : x ∈ s\n⊢ ↑(Nat.succ m) < ↑(Nat.succ n)\n\ncase fderivWithin\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝ n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nH : HasFTaylorSeriesUpToOn (↑(n + 1)) f p s\nm : ℕ\nhm : ↑m < ↑n\nx : E\nhx : x ∈ s\nA : ↑(Nat.succ m) < ↑(Nat.succ n)\n⊢ HasFDerivWithinAt (fun x => FormalMultilinearSeries.shift (p x) m)\n (ContinuousMultilinearMap.curryLeft (FormalMultilinearSeries.shift (p x) (Nat.succ m))) s x",
"state_before": "case fderivWithin\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝ n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nH : HasFTaylorSeriesUpToOn (↑(n + 1)) f p s\nm : ℕ\nhm : ↑m < ↑n\nx : E\nhx : x ∈ s\n⊢ HasFDerivWithinAt (fun x => FormalMultilinearSeries.shift (p x) m)\n (ContinuousMultilinearMap.curryLeft (FormalMultilinearSeries.shift (p x) (Nat.succ m))) s x",
"tactic": "have A : (m.succ : ℕ∞) < n.succ"
},
{
"state_after": "case fderivWithin\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝ n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nH : HasFTaylorSeriesUpToOn (↑(n + 1)) f p s\nm : ℕ\nhm : ↑m < ↑n\nx : E\nhx : x ∈ s\nA : ↑(Nat.succ m) < ↑(Nat.succ n)\n⊢ HasFDerivWithinAt (fun x => p x (Nat.succ m))\n (ContinuousLinearMap.comp\n (↑(ContinuousLinearEquiv.mk\n (LinearIsometryEquiv.symm\n (LinearIsometryEquiv.symm (continuousMultilinearCurryRightEquiv' 𝕜 m E F))).toLinearEquiv))\n (ContinuousMultilinearMap.curryLeft (ContinuousMultilinearMap.curryRight (p x (Nat.succ (Nat.succ m))))))\n s x",
"state_before": "case fderivWithin\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝ n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nH : HasFTaylorSeriesUpToOn (↑(n + 1)) f p s\nm : ℕ\nhm : ↑m < ↑n\nx : E\nhx : x ∈ s\nA : ↑(Nat.succ m) < ↑(Nat.succ n)\n⊢ HasFDerivWithinAt\n (↑(LinearIsometryEquiv.symm (continuousMultilinearCurryRightEquiv' 𝕜 m E F)) ∘ fun x => p x (Nat.succ m))\n (ContinuousMultilinearMap.curryLeft (ContinuousMultilinearMap.curryRight (p x (Nat.succ (Nat.succ m))))) s x",
"tactic": "rw [((continuousMultilinearCurryRightEquiv' 𝕜 m E F).symm).comp_hasFDerivWithinAt_iff']"
},
{
"state_after": "case h.e'_10.h.h\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝ n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nH : HasFTaylorSeriesUpToOn (↑(n + 1)) f p s\nm : ℕ\nhm : ↑m < ↑n\nx : E\nhx : x ∈ s\nA : ↑(Nat.succ m) < ↑(Nat.succ n)\ne_7✝ : ContinuousMultilinearMap.normedAddCommGroup' = ContinuousMultilinearMap.normedAddCommGroup\n⊢ ContinuousLinearMap.comp\n (↑(ContinuousLinearEquiv.mk\n (LinearIsometryEquiv.symm\n (LinearIsometryEquiv.symm (continuousMultilinearCurryRightEquiv' 𝕜 m E F))).toLinearEquiv))\n (ContinuousMultilinearMap.curryLeft (ContinuousMultilinearMap.curryRight (p x (Nat.succ (Nat.succ m))))) =\n ContinuousMultilinearMap.curryLeft (p x (Nat.succ (Nat.succ m)))",
"state_before": "case fderivWithin\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝ n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nH : HasFTaylorSeriesUpToOn (↑(n + 1)) f p s\nm : ℕ\nhm : ↑m < ↑n\nx : E\nhx : x ∈ s\nA : ↑(Nat.succ m) < ↑(Nat.succ n)\n⊢ HasFDerivWithinAt (fun x => p x (Nat.succ m))\n (ContinuousLinearMap.comp\n (↑(ContinuousLinearEquiv.mk\n (LinearIsometryEquiv.symm\n (LinearIsometryEquiv.symm (continuousMultilinearCurryRightEquiv' 𝕜 m E F))).toLinearEquiv))\n (ContinuousMultilinearMap.curryLeft (ContinuousMultilinearMap.curryRight (p x (Nat.succ (Nat.succ m))))))\n s x",
"tactic": "convert H.fderivWithin _ A x hx"
},
{
"state_after": "case h.e'_10.h.h.h.H\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝ n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nH : HasFTaylorSeriesUpToOn (↑(n + 1)) f p s\nm : ℕ\nhm : ↑m < ↑n\nx : E\nhx : x ∈ s\nA : ↑(Nat.succ m) < ↑(Nat.succ n)\ne_7✝ : ContinuousMultilinearMap.normedAddCommGroup' = ContinuousMultilinearMap.normedAddCommGroup\ny : E\nv : Fin (Nat.succ m) → E\n⊢ ↑(↑(ContinuousLinearMap.comp\n (↑(ContinuousLinearEquiv.mk\n (LinearIsometryEquiv.symm\n (LinearIsometryEquiv.symm (continuousMultilinearCurryRightEquiv' 𝕜 m E F))).toLinearEquiv))\n (ContinuousMultilinearMap.curryLeft (ContinuousMultilinearMap.curryRight (p x (Nat.succ (Nat.succ m))))))\n y)\n v =\n ↑(↑(ContinuousMultilinearMap.curryLeft (p x (Nat.succ (Nat.succ m)))) y) v",
"state_before": "case h.e'_10.h.h\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm✝ n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nH : HasFTaylorSeriesUpToOn (↑(n + 1)) f p s\nm : ℕ\nhm : ↑m < ↑n\nx : E\nhx : x ∈ s\nA : ↑(Nat.succ m) < ↑(Nat.succ n)\ne_7✝ : ContinuousMultilinearMap.normedAddCommGroup' = ContinuousMultilinearMap.normedAddCommGroup\n⊢ ContinuousLinearMap.comp\n (↑(ContinuousLinearEquiv.mk\n (LinearIsometryEquiv.symm\n (LinearIsometryEquiv.symm (continuousMultilinearCurryRightEquiv' 𝕜 m E F))).toLinearEquiv))\n (ContinuousMultilinearMap.curryLeft (ContinuousMultilinearMap.curryRight (p x (Nat.succ (Nat.succ m))))) =\n ContinuousMultilinearMap.curryLeft (p x (Nat.succ (Nat.succ m)))",
"tactic": "ext y v"
},
{
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{
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},
{
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},
{
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{
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},
{
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361,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
338,
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Mathlib/Data/Multiset/Basic.lean | Multiset.disjoint_iff_ne | [
{
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2913,
37
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2912,
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Mathlib/Data/List/Perm.lean | List.Perm.nodup_iff | [] | [
1068,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1067,
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Mathlib/Topology/Constructions.lean | Continuous.snd' | [] | [
374,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
373,
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Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean | CategoryTheory.Limits.pullbackSymmetry_inv_comp_snd | [
{
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1539,
90
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1538,
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Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean | map_div₀ | [] | [
246,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
245,
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Mathlib/Data/Dfinsupp/Basic.lean | Dfinsupp.support_eq_empty | [
{
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{
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1147,
98
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1146,
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Mathlib/RingTheory/Ideal/LocalRing.lean | LocalRing.ResidueField.map_comp_residue | [] | [
431,
6
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429,
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Mathlib/SetTheory/Ordinal/FixedPoint.lean | Ordinal.nfpBFamily_le_apply | [
{
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"tactic": "rw [← not_iff_not]"
},
{
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"tactic": "push_neg"
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{
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"tactic": "exact apply_lt_nfpBFamily_iff.{u, v} ho H"
}
] | [
314,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/Analysis/InnerProductSpace/GramSchmidtOrtho.lean | gramSchmidt_ne_zero | [] | [
234,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
232,
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Mathlib/Data/Set/Intervals/Basic.lean | Set.Iic_union_Ioc_eq_Iic | [] | [
1405,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1403,
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Mathlib/Data/Fin/Basic.lean | Fin.cases_zero | [
{
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] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1736,
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Mathlib/Analysis/Calculus/Deriv/Comp.lean | HasStrictDerivAt.iterate | [
{
"state_after": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nE : Type w\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nf✝ f₀ f₁ g : 𝕜 → F\nf'✝ f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\n𝕜' : Type ?u.192344\ninst✝³ : NontriviallyNormedField 𝕜'\ninst✝² : NormedAlgebra 𝕜 𝕜'\ninst✝¹ : NormedSpace 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\ns' t' : Set 𝕜'\nh : 𝕜 → 𝕜'\nh₁ : 𝕜 → 𝕜\nh₂ : 𝕜' → 𝕜'\nh' h₂' : 𝕜'\nh₁' : 𝕜\ng₁ : 𝕜' → F\ng₁' : F\nL' : Filter 𝕜'\nf : 𝕜 → 𝕜\nf' : 𝕜\nhf : HasStrictDerivAt f f' x\nhx : f x = x\nn : ℕ\nthis : HasStrictFDerivAt (f^[n]) (smulRight 1 f' ^ n) x\n⊢ HasStrictDerivAt (f^[n]) (f' ^ n) x",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nE : Type w\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nf✝ f₀ f₁ g : 𝕜 → F\nf'✝ f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\n𝕜' : Type ?u.192344\ninst✝³ : NontriviallyNormedField 𝕜'\ninst✝² : NormedAlgebra 𝕜 𝕜'\ninst✝¹ : NormedSpace 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\ns' t' : Set 𝕜'\nh : 𝕜 → 𝕜'\nh₁ : 𝕜 → 𝕜\nh₂ : 𝕜' → 𝕜'\nh' h₂' : 𝕜'\nh₁' : 𝕜\ng₁ : 𝕜' → F\ng₁' : F\nL' : Filter 𝕜'\nf : 𝕜 → 𝕜\nf' : 𝕜\nhf : HasStrictDerivAt f f' x\nhx : f x = x\nn : ℕ\n⊢ HasStrictDerivAt (f^[n]) (f' ^ n) x",
"tactic": "have := hf.iterate hx n"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : NormedSpace 𝕜 F\nE : Type w\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nf✝ f₀ f₁ g : 𝕜 → F\nf'✝ f₀' f₁' g' : F\nx : 𝕜\ns t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\n𝕜' : Type ?u.192344\ninst✝³ : NontriviallyNormedField 𝕜'\ninst✝² : NormedAlgebra 𝕜 𝕜'\ninst✝¹ : NormedSpace 𝕜' F\ninst✝ : IsScalarTower 𝕜 𝕜' F\ns' t' : Set 𝕜'\nh : 𝕜 → 𝕜'\nh₁ : 𝕜 → 𝕜\nh₂ : 𝕜' → 𝕜'\nh' h₂' : 𝕜'\nh₁' : 𝕜\ng₁ : 𝕜' → F\ng₁' : F\nL' : Filter 𝕜'\nf : 𝕜 → 𝕜\nf' : 𝕜\nhf : HasStrictDerivAt f f' x\nhx : f x = x\nn : ℕ\nthis : HasStrictFDerivAt (f^[n]) (smulRight 1 f' ^ n) x\n⊢ HasStrictDerivAt (f^[n]) (f' ^ n) x",
"tactic": "rwa [ContinuousLinearMap.smulRight_one_pow] at this"
}
] | [
226,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
222,
19
] |
Mathlib/Topology/Inseparable.lean | SeparationQuotient.comap_mk_nhdsSet | [
{
"state_after": "no goals",
"state_before": "X : Type u_1\nY : Type ?u.79044\nZ : Type ?u.79047\nα : Type ?u.79050\nι : Type ?u.79053\nπ : ι → Type ?u.79058\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : TopologicalSpace Z\ninst✝ : (i : ι) → TopologicalSpace (π i)\nx y z : X\ns : Set X\nf : X → Y\nt : Set (SeparationQuotient X)\n⊢ comap mk (𝓝ˢ t) = 𝓝ˢ (mk ⁻¹' t)",
"tactic": "conv_lhs => rw [← image_preimage_eq t surjective_mk, comap_mk_nhdsSet_image]"
}
] | [
495,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
494,
1
] |
Mathlib/Data/Nat/Prime.lean | Nat.Prime.dvd_mul_of_dvd_ne | [] | [
702,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
700,
1
] |
Mathlib/Data/Fin/Tuple/NatAntidiagonal.lean | Multiset.Nat.antidiagonalTuple_one | [] | [
216,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
215,
1
] |
Mathlib/Data/Set/Basic.lean | Set.sep_inter | [] | [
1469,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1468,
1
] |
Mathlib/Data/Nat/Order/Basic.lean | Nat.div_eq_sub_mod_div | [
{
"state_after": "case pos\nm n k l : ℕ\nn0 : n = 0\n⊢ m / n = (m - m % n) / n\n\ncase neg\nm n k l : ℕ\nn0 : ¬n = 0\n⊢ m / n = (m - m % n) / n",
"state_before": "m n k l : ℕ\n⊢ m / n = (m - m % n) / n",
"tactic": "by_cases n0 : n = 0"
},
{
"state_after": "no goals",
"state_before": "case pos\nm n k l : ℕ\nn0 : n = 0\n⊢ m / n = (m - m % n) / n",
"tactic": "rw [n0, Nat.div_zero, Nat.div_zero]"
},
{
"state_after": "case neg\nm n k l : ℕ\nn0 : ¬n = 0\nthis : m - m % n = n * (m / n)\n⊢ m / n = (m - m % n) / n",
"state_before": "case neg\nm n k l : ℕ\nn0 : ¬n = 0\n⊢ m / n = (m - m % n) / n",
"tactic": "have : m - m % n = n * (m / n) := by\n rw [tsub_eq_iff_eq_add_of_le (Nat.mod_le _ _), add_comm, mod_add_div]"
},
{
"state_after": "no goals",
"state_before": "case neg\nm n k l : ℕ\nn0 : ¬n = 0\nthis : m - m % n = n * (m / n)\n⊢ m / n = (m - m % n) / n",
"tactic": "rw [this, mul_div_right _ (Nat.pos_of_ne_zero n0)]"
},
{
"state_after": "no goals",
"state_before": "m n k l : ℕ\nn0 : ¬n = 0\n⊢ m - m % n = n * (m / n)",
"tactic": "rw [tsub_eq_iff_eq_add_of_le (Nat.mod_le _ _), add_comm, mod_add_div]"
}
] | [
558,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
553,
1
] |
Mathlib/Data/TypeVec.lean | TypeVec.casesCons_append1 | [] | [
333,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
330,
11
] |
Mathlib/Data/Int/Dvd/Basic.lean | Int.coe_nat_dvd_right | [
{
"state_after": "no goals",
"state_before": "n : ℕ\nz : ℤ\n⊢ z ∣ ↑n ↔ natAbs z ∣ n",
"tactic": "rcases natAbs_eq z with (eq | eq) <;> rw [eq] <;> simp [←coe_nat_dvd]"
}
] | [
42,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
41,
1
] |
Mathlib/GroupTheory/Nilpotent.lean | nilpotencyClass_prod | [
{
"state_after": "G : Type ?u.332663\ninst✝⁵ : Group G\nH : Subgroup G\ninst✝⁴ : Normal H\nG₁ : Type u_1\nG₂ : Type u_2\ninst✝³ : Group G₁\ninst✝² : Group G₂\ninst✝¹ : Group.IsNilpotent G₁\ninst✝ : Group.IsNilpotent G₂\nk : ℕ\n⊢ nilpotencyClass (G₁ × G₂) ≤ k ↔ max (nilpotencyClass G₁) (nilpotencyClass G₂) ≤ k",
"state_before": "G : Type ?u.332663\ninst✝⁵ : Group G\nH : Subgroup G\ninst✝⁴ : Normal H\nG₁ : Type u_1\nG₂ : Type u_2\ninst✝³ : Group G₁\ninst✝² : Group G₂\ninst✝¹ : Group.IsNilpotent G₁\ninst✝ : Group.IsNilpotent G₂\n⊢ nilpotencyClass (G₁ × G₂) = max (nilpotencyClass G₁) (nilpotencyClass G₂)",
"tactic": "refine' eq_of_forall_ge_iff fun k => _"
},
{
"state_after": "no goals",
"state_before": "G : Type ?u.332663\ninst✝⁵ : Group G\nH : Subgroup G\ninst✝⁴ : Normal H\nG₁ : Type u_1\nG₂ : Type u_2\ninst✝³ : Group G₁\ninst✝² : Group G₂\ninst✝¹ : Group.IsNilpotent G₁\ninst✝ : Group.IsNilpotent G₂\nk : ℕ\n⊢ nilpotencyClass (G₁ × G₂) ≤ k ↔ max (nilpotencyClass G₁) (nilpotencyClass G₂) ≤ k",
"tactic": "simp only [max_le_iff, ← lowerCentralSeries_eq_bot_iff_nilpotencyClass_le,\n lowerCentralSeries_prod, prod_eq_bot_iff]"
}
] | [
742,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
737,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean | ULift.norm_def | [] | [
2106,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2105,
1
] |
Mathlib/Data/Int/Div.lean | Int.eq_zero_of_dvd_of_nonneg_of_lt | [] | [
43,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
42,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean | Cardinal.sum_nat_eq_add_sum_succ | [
{
"state_after": "α β : Type u\nf : ℕ → Cardinal\n⊢ (#Quotient.out (f 0) ⊕ (n : ℕ) × Quotient.out (f (n + 1))) = f 0 + sum fun i => f (i + 1)",
"state_before": "α β : Type u\nf : ℕ → Cardinal\n⊢ sum f = f 0 + sum fun i => f (i + 1)",
"tactic": "refine' (Equiv.sigmaNatSucc fun i => Quotient.out (f i)).cardinal_eq.trans _"
},
{
"state_after": "no goals",
"state_before": "α β : Type u\nf : ℕ → Cardinal\n⊢ (#Quotient.out (f 0) ⊕ (n : ℕ) × Quotient.out (f (n + 1))) = f 0 + sum fun i => f (i + 1)",
"tactic": "simp only [mk_sum, mk_out, lift_id, mk_sigma]"
}
] | [
1005,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1002,
1
] |
Mathlib/MeasureTheory/Group/Action.lean | MeasureTheory.vadd_ae_eq_self_of_mem_zmultiples | [
{
"state_after": "G✝ : Type ?u.114459\nM : Type ?u.114462\nα : Type u_2\ns : Set α\nm : MeasurableSpace α\ninst✝⁹ : Group G✝\ninst✝⁸ : MulAction G✝ α\ninst✝⁷ : MeasurableSpace G✝\ninst✝⁶ : MeasurableSMul G✝ α\nc : G✝\nμ : Measure α\ninst✝⁵ : SMulInvariantMeasure G✝ α μ\nG : Type u_1\ninst✝⁴ : MeasurableSpace G\ninst✝³ : AddGroup G\ninst✝² : AddAction G α\ninst✝¹ : VAddInvariantMeasure G α μ\ninst✝ : MeasurableVAdd G α\nx y : G\nhs : x +ᵥ s =ᶠ[ae μ] s\nhy : y ∈ AddSubgroup.zmultiples x\nthis : MeasurableSpace (Multiplicative G) := inferInstanceAs (MeasurableSpace G)\n⊢ y +ᵥ s =ᶠ[ae μ] s",
"state_before": "G✝ : Type ?u.114459\nM : Type ?u.114462\nα : Type u_2\ns : Set α\nm : MeasurableSpace α\ninst✝⁹ : Group G✝\ninst✝⁸ : MulAction G✝ α\ninst✝⁷ : MeasurableSpace G✝\ninst✝⁶ : MeasurableSMul G✝ α\nc : G✝\nμ : Measure α\ninst✝⁵ : SMulInvariantMeasure G✝ α μ\nG : Type u_1\ninst✝⁴ : MeasurableSpace G\ninst✝³ : AddGroup G\ninst✝² : AddAction G α\ninst✝¹ : VAddInvariantMeasure G α μ\ninst✝ : MeasurableVAdd G α\nx y : G\nhs : x +ᵥ s =ᶠ[ae μ] s\nhy : y ∈ AddSubgroup.zmultiples x\n⊢ y +ᵥ s =ᶠ[ae μ] s",
"tactic": "letI : MeasurableSpace (Multiplicative G) := (inferInstanceAs (MeasurableSpace G))"
},
{
"state_after": "G✝ : Type ?u.114459\nM : Type ?u.114462\nα : Type u_2\ns : Set α\nm : MeasurableSpace α\ninst✝⁹ : Group G✝\ninst✝⁸ : MulAction G✝ α\ninst✝⁷ : MeasurableSpace G✝\ninst✝⁶ : MeasurableSMul G✝ α\nc : G✝\nμ : Measure α\ninst✝⁵ : SMulInvariantMeasure G✝ α μ\nG : Type u_1\ninst✝⁴ : MeasurableSpace G\ninst✝³ : AddGroup G\ninst✝² : AddAction G α\ninst✝¹ : VAddInvariantMeasure G α μ\ninst✝ : MeasurableVAdd G α\nx y : G\nhs : x +ᵥ s =ᶠ[ae μ] s\nhy : y ∈ AddSubgroup.zmultiples x\nthis✝ : MeasurableSpace (Multiplicative G) := inferInstanceAs (MeasurableSpace G)\nthis : SMulInvariantMeasure (Multiplicative G) α μ :=\n { measure_preimage_smul := fun g => VAddInvariantMeasure.measure_preimage_vadd (↑Multiplicative.toAdd g) }\n⊢ y +ᵥ s =ᶠ[ae μ] s",
"state_before": "G✝ : Type ?u.114459\nM : Type ?u.114462\nα : Type u_2\ns : Set α\nm : MeasurableSpace α\ninst✝⁹ : Group G✝\ninst✝⁸ : MulAction G✝ α\ninst✝⁷ : MeasurableSpace G✝\ninst✝⁶ : MeasurableSMul G✝ α\nc : G✝\nμ : Measure α\ninst✝⁵ : SMulInvariantMeasure G✝ α μ\nG : Type u_1\ninst✝⁴ : MeasurableSpace G\ninst✝³ : AddGroup G\ninst✝² : AddAction G α\ninst✝¹ : VAddInvariantMeasure G α μ\ninst✝ : MeasurableVAdd G α\nx y : G\nhs : x +ᵥ s =ᶠ[ae μ] s\nhy : y ∈ AddSubgroup.zmultiples x\nthis : MeasurableSpace (Multiplicative G) := inferInstanceAs (MeasurableSpace G)\n⊢ y +ᵥ s =ᶠ[ae μ] s",
"tactic": "letI : SMulInvariantMeasure (Multiplicative G) α μ :=\n ⟨fun g => VAddInvariantMeasure.measure_preimage_vadd (Multiplicative.toAdd g)⟩"
},
{
"state_after": "G✝ : Type ?u.114459\nM : Type ?u.114462\nα : Type u_2\ns : Set α\nm : MeasurableSpace α\ninst✝⁹ : Group G✝\ninst✝⁸ : MulAction G✝ α\ninst✝⁷ : MeasurableSpace G✝\ninst✝⁶ : MeasurableSMul G✝ α\nc : G✝\nμ : Measure α\ninst✝⁵ : SMulInvariantMeasure G✝ α μ\nG : Type u_1\ninst✝⁴ : MeasurableSpace G\ninst✝³ : AddGroup G\ninst✝² : AddAction G α\ninst✝¹ : VAddInvariantMeasure G α μ\ninst✝ : MeasurableVAdd G α\nx y : G\nhs : x +ᵥ s =ᶠ[ae μ] s\nhy : y ∈ AddSubgroup.zmultiples x\nthis✝¹ : MeasurableSpace (Multiplicative G) := inferInstanceAs (MeasurableSpace G)\nthis✝ : SMulInvariantMeasure (Multiplicative G) α μ :=\n { measure_preimage_smul := fun g => VAddInvariantMeasure.measure_preimage_vadd (↑Multiplicative.toAdd g) }\nthis : MeasurableSMul (Multiplicative G) α :=\n { measurable_const_smul := fun g => measurable_const_vadd (↑Multiplicative.toAdd g),\n measurable_smul_const := fun a => measurable_vadd_const a }\n⊢ y +ᵥ s =ᶠ[ae μ] s",
"state_before": "G✝ : Type ?u.114459\nM : Type ?u.114462\nα : Type u_2\ns : Set α\nm : MeasurableSpace α\ninst✝⁹ : Group G✝\ninst✝⁸ : MulAction G✝ α\ninst✝⁷ : MeasurableSpace G✝\ninst✝⁶ : MeasurableSMul G✝ α\nc : G✝\nμ : Measure α\ninst✝⁵ : SMulInvariantMeasure G✝ α μ\nG : Type u_1\ninst✝⁴ : MeasurableSpace G\ninst✝³ : AddGroup G\ninst✝² : AddAction G α\ninst✝¹ : VAddInvariantMeasure G α μ\ninst✝ : MeasurableVAdd G α\nx y : G\nhs : x +ᵥ s =ᶠ[ae μ] s\nhy : y ∈ AddSubgroup.zmultiples x\nthis✝ : MeasurableSpace (Multiplicative G) := inferInstanceAs (MeasurableSpace G)\nthis : SMulInvariantMeasure (Multiplicative G) α μ :=\n { measure_preimage_smul := fun g => VAddInvariantMeasure.measure_preimage_vadd (↑Multiplicative.toAdd g) }\n⊢ y +ᵥ s =ᶠ[ae μ] s",
"tactic": "letI : MeasurableSMul (Multiplicative G) α :=\n { measurable_const_smul := fun g => measurable_const_vadd (Multiplicative.toAdd g)\n measurable_smul_const := fun a =>\n @measurable_vadd_const (Multiplicative G) α (inferInstanceAs (VAdd G α)) _ _\n (inferInstanceAs (MeasurableVAdd G α)) a }"
},
{
"state_after": "no goals",
"state_before": "G✝ : Type ?u.114459\nM : Type ?u.114462\nα : Type u_2\ns : Set α\nm : MeasurableSpace α\ninst✝⁹ : Group G✝\ninst✝⁸ : MulAction G✝ α\ninst✝⁷ : MeasurableSpace G✝\ninst✝⁶ : MeasurableSMul G✝ α\nc : G✝\nμ : Measure α\ninst✝⁵ : SMulInvariantMeasure G✝ α μ\nG : Type u_1\ninst✝⁴ : MeasurableSpace G\ninst✝³ : AddGroup G\ninst✝² : AddAction G α\ninst✝¹ : VAddInvariantMeasure G α μ\ninst✝ : MeasurableVAdd G α\nx y : G\nhs : x +ᵥ s =ᶠ[ae μ] s\nhy : y ∈ AddSubgroup.zmultiples x\nthis✝¹ : MeasurableSpace (Multiplicative G) := inferInstanceAs (MeasurableSpace G)\nthis✝ : SMulInvariantMeasure (Multiplicative G) α μ :=\n { measure_preimage_smul := fun g => VAddInvariantMeasure.measure_preimage_vadd (↑Multiplicative.toAdd g) }\nthis : MeasurableSMul (Multiplicative G) α :=\n { measurable_const_smul := fun g => measurable_const_vadd (↑Multiplicative.toAdd g),\n measurable_smul_const := fun a => measurable_vadd_const a }\n⊢ y +ᵥ s =ᶠ[ae μ] s",
"tactic": "exact @smul_ae_eq_self_of_mem_zpowers (Multiplicative G) α _ _ _ _ _ _ _ _ _ _ hs hy"
}
] | [
289,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
277,
1
] |
Mathlib/Algebra/Parity.lean | even_two_mul | [] | [
291,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
290,
1
] |
Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean | Finset.card_div_mul_le_card_mul_mul_card_div | [
{
"state_after": "α : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\n⊢ card (A * C⁻¹) * card B ≤ card (A * B) * card (B * C⁻¹)",
"state_before": "α : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\n⊢ card (A / C) * card B ≤ card (A * B) * card (B / C)",
"tactic": "rw [div_eq_mul_inv, div_eq_mul_inv]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\n⊢ card (A * C⁻¹) * card B ≤ card (A * B) * card (B * C⁻¹)",
"tactic": "exact card_mul_mul_card_le_card_mul_mul_card_mul _ _ _"
}
] | [
175,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
172,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean | BoundedContinuousFunction.dist_le_iff_of_nonempty | [] | [
189,
77
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
187,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean | MeasureTheory.ae_le_set | [
{
"state_after": "α : Type u_1\nβ : Type ?u.101868\nγ : Type ?u.101871\nδ : Type ?u.101874\nι : Type ?u.101877\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t : Set α\n⊢ ↑↑μ {a | a ∈ s ∧ ¬a ∈ t} = 0 ↔ ↑↑μ (s \\ t) = 0",
"state_before": "α : Type u_1\nβ : Type ?u.101868\nγ : Type ?u.101871\nδ : Type ?u.101874\nι : Type ?u.101877\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t : Set α\n⊢ (∀ᵐ (x : α) ∂μ, x ∈ s → x ∈ t) ↔ ↑↑μ (s \\ t) = 0",
"tactic": "simp [ae_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.101868\nγ : Type ?u.101871\nδ : Type ?u.101874\nι : Type ?u.101877\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t : Set α\n⊢ ↑↑μ {a | a ∈ s ∧ ¬a ∈ t} = 0 ↔ ↑↑μ (s \\ t) = 0",
"tactic": "rfl"
}
] | [
455,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
452,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | intervalIntegral.integral_smul | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.12999603\n𝕜✝ : Type ?u.12999606\nE : Type u_2\nF : Type ?u.12999612\nA : Type ?u.12999615\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : CompleteSpace E\ninst✝³ : NormedSpace ℝ E\na b : ℝ\nf✝ g : ℝ → E\nμ : MeasureTheory.Measure ℝ\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : SMulCommClass ℝ 𝕜 E\nr : 𝕜\nf : ℝ → E\n⊢ (∫ (x : ℝ) in a..b, r • f x ∂μ) = r • ∫ (x : ℝ) in a..b, f x ∂μ",
"tactic": "simp only [intervalIntegral, integral_smul, smul_sub]"
}
] | [
607,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
604,
8
] |
Mathlib/RingTheory/SimpleModule.lean | is_semisimple_iff_top_eq_sSup_simples | [
{
"state_after": "R : Type u_2\ninst✝⁴ : Ring R\nM : Type u_1\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nm : Submodule R M\nN : Type ?u.47667\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\na✝ : IsSemisimpleModule R M\n⊢ sSup {m | IsSimpleModule R { x // x ∈ m }} = ⊤",
"state_before": "R : Type u_2\ninst✝⁴ : Ring R\nM : Type u_1\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nm : Submodule R M\nN : Type ?u.47667\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\n⊢ IsSemisimpleModule R M → sSup {m | IsSimpleModule R { x // x ∈ m }} = ⊤",
"tactic": "intro"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝⁴ : Ring R\nM : Type u_1\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nm : Submodule R M\nN : Type ?u.47667\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\na✝ : IsSemisimpleModule R M\n⊢ sSup {m | IsSimpleModule R { x // x ∈ m }} = ⊤",
"tactic": "exact IsSemisimpleModule.sSup_simples_eq_top"
}
] | [
127,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
123,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | SimpleGraph.Walk.copy_rfl_rfl | [] | [
138,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
138,
1
] |
Mathlib/Order/Filter/Bases.lean | Filter.hasBasis_generate | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.9944\nγ : Type ?u.9947\nι : Sort ?u.9950\nι' : Sort ?u.9953\nl l' : Filter α\np : ι → Prop\ns✝ : ι → Set α\nt : Set α\ni : ι\np' : ι' → Prop\ns' : ι' → Set α\ni' : ι'\ns : Set (Set α)\nU : Set α\n⊢ U ∈ generate s ↔ ∃ i, (Set.Finite i ∧ i ⊆ s) ∧ ⋂₀ i ⊆ U",
"tactic": "simp only [mem_generate_iff, exists_prop, and_assoc, and_left_comm]"
}
] | [
245,
84
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
243,
1
] |
Mathlib/Topology/Instances/TrivSqZeroExt.lean | TrivSqZeroExt.hasSum_inr | [] | [
156,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
154,
1
] |
Mathlib/Algebra/GCDMonoid/Basic.lean | gcd_one_right' | [] | [
422,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
421,
1
] |
Mathlib/Probability/Independence/ZeroOne.lean | ProbabilityTheory.indep_iSup_directed_limsup | [
{
"state_after": "case refine'_1\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\nα : Type u_3\np : Set ι → Prop\nf : Filter ι\nns : α → Set ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nhf : ∀ (t : Set ι), p t → tᶜ ∈ f\nhns : Directed (fun x x_1 => x ≤ x_1) ns\nhnsp : ∀ (a : α), p (ns a)\n⊢ ∀ (i : α), Indep (⨆ (n : ι) (_ : n ∈ ns i), s n) (limsup s f)\n\ncase refine'_2\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\nα : Type u_3\np : Set ι → Prop\nf : Filter ι\nns : α → Set ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nhf : ∀ (t : Set ι), p t → tᶜ ∈ f\nhns : Directed (fun x x_1 => x ≤ x_1) ns\nhnsp : ∀ (a : α), p (ns a)\n⊢ ∀ (i : α), (⨆ (n : ι) (_ : n ∈ ns i), s n) ≤ m0\n\ncase refine'_3\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\nα : Type u_3\np : Set ι → Prop\nf : Filter ι\nns : α → Set ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nhf : ∀ (t : Set ι), p t → tᶜ ∈ f\nhns : Directed (fun x x_1 => x ≤ x_1) ns\nhnsp : ∀ (a : α), p (ns a)\n⊢ limsup s f ≤ m0\n\ncase refine'_4\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\nα : Type u_3\np : Set ι → Prop\nf : Filter ι\nns : α → Set ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nhf : ∀ (t : Set ι), p t → tᶜ ∈ f\nhns : Directed (fun x x_1 => x ≤ x_1) ns\nhnsp : ∀ (a : α), p (ns a)\n⊢ Directed (fun x x_1 => x ≤ x_1) fun a => ⨆ (n : ι) (_ : n ∈ ns a), s n",
"state_before": "Ω : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\nα : Type u_3\np : Set ι → Prop\nf : Filter ι\nns : α → Set ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nhf : ∀ (t : Set ι), p t → tᶜ ∈ f\nhns : Directed (fun x x_1 => x ≤ x_1) ns\nhnsp : ∀ (a : α), p (ns a)\n⊢ Indep (⨆ (a : α) (n : ι) (_ : n ∈ ns a), s n) (limsup s f)",
"tactic": "refine' indep_iSup_of_directed_le _ _ _ _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\nα : Type u_3\np : Set ι → Prop\nf : Filter ι\nns : α → Set ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nhf : ∀ (t : Set ι), p t → tᶜ ∈ f\nhns : Directed (fun x x_1 => x ≤ x_1) ns\nhnsp : ∀ (a : α), p (ns a)\n⊢ ∀ (i : α), Indep (⨆ (n : ι) (_ : n ∈ ns i), s n) (limsup s f)",
"tactic": "exact fun a => indep_biSup_limsup h_le h_indep hf (hnsp a)"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\nα : Type u_3\np : Set ι → Prop\nf : Filter ι\nns : α → Set ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nhf : ∀ (t : Set ι), p t → tᶜ ∈ f\nhns : Directed (fun x x_1 => x ≤ x_1) ns\nhnsp : ∀ (a : α), p (ns a)\n⊢ ∀ (i : α), (⨆ (n : ι) (_ : n ∈ ns i), s n) ≤ m0",
"tactic": "exact fun a => iSup₂_le fun n _ => h_le n"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\nα : Type u_3\np : Set ι → Prop\nf : Filter ι\nns : α → Set ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nhf : ∀ (t : Set ι), p t → tᶜ ∈ f\nhns : Directed (fun x x_1 => x ≤ x_1) ns\nhnsp : ∀ (a : α), p (ns a)\n⊢ limsup s f ≤ m0",
"tactic": "exact limsup_le_iSup.trans (iSup_le h_le)"
},
{
"state_after": "case refine'_4\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\nα : Type u_3\np : Set ι → Prop\nf : Filter ι\nns : α → Set ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nhf : ∀ (t : Set ι), p t → tᶜ ∈ f\nhns : Directed (fun x x_1 => x ≤ x_1) ns\nhnsp : ∀ (a : α), p (ns a)\na b : α\n⊢ ∃ z,\n (fun x x_1 => x ≤ x_1) ((fun a => ⨆ (n : ι) (_ : n ∈ ns a), s n) a) ((fun a => ⨆ (n : ι) (_ : n ∈ ns a), s n) z) ∧\n (fun x x_1 => x ≤ x_1) ((fun a => ⨆ (n : ι) (_ : n ∈ ns a), s n) b) ((fun a => ⨆ (n : ι) (_ : n ∈ ns a), s n) z)",
"state_before": "case refine'_4\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\nα : Type u_3\np : Set ι → Prop\nf : Filter ι\nns : α → Set ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nhf : ∀ (t : Set ι), p t → tᶜ ∈ f\nhns : Directed (fun x x_1 => x ≤ x_1) ns\nhnsp : ∀ (a : α), p (ns a)\n⊢ Directed (fun x x_1 => x ≤ x_1) fun a => ⨆ (n : ι) (_ : n ∈ ns a), s n",
"tactic": "intro a b"
},
{
"state_after": "case refine'_4.intro\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\nα : Type u_3\np : Set ι → Prop\nf : Filter ι\nns : α → Set ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nhf : ∀ (t : Set ι), p t → tᶜ ∈ f\nhns : Directed (fun x x_1 => x ≤ x_1) ns\nhnsp : ∀ (a : α), p (ns a)\na b c : α\nhc : (fun x x_1 => x ≤ x_1) (ns a) (ns c) ∧ (fun x x_1 => x ≤ x_1) (ns b) (ns c)\n⊢ ∃ z,\n (fun x x_1 => x ≤ x_1) ((fun a => ⨆ (n : ι) (_ : n ∈ ns a), s n) a) ((fun a => ⨆ (n : ι) (_ : n ∈ ns a), s n) z) ∧\n (fun x x_1 => x ≤ x_1) ((fun a => ⨆ (n : ι) (_ : n ∈ ns a), s n) b) ((fun a => ⨆ (n : ι) (_ : n ∈ ns a), s n) z)",
"state_before": "case refine'_4\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\nα : Type u_3\np : Set ι → Prop\nf : Filter ι\nns : α → Set ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nhf : ∀ (t : Set ι), p t → tᶜ ∈ f\nhns : Directed (fun x x_1 => x ≤ x_1) ns\nhnsp : ∀ (a : α), p (ns a)\na b : α\n⊢ ∃ z,\n (fun x x_1 => x ≤ x_1) ((fun a => ⨆ (n : ι) (_ : n ∈ ns a), s n) a) ((fun a => ⨆ (n : ι) (_ : n ∈ ns a), s n) z) ∧\n (fun x x_1 => x ≤ x_1) ((fun a => ⨆ (n : ι) (_ : n ∈ ns a), s n) b) ((fun a => ⨆ (n : ι) (_ : n ∈ ns a), s n) z)",
"tactic": "obtain ⟨c, hc⟩ := hns a b"
},
{
"state_after": "case refine'_4.intro.refine'_1\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\nα : Type u_3\np : Set ι → Prop\nf : Filter ι\nns : α → Set ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nhf : ∀ (t : Set ι), p t → tᶜ ∈ f\nhns : Directed (fun x x_1 => x ≤ x_1) ns\nhnsp : ∀ (a : α), p (ns a)\na b c : α\nhc : (fun x x_1 => x ≤ x_1) (ns a) (ns c) ∧ (fun x x_1 => x ≤ x_1) (ns b) (ns c)\nn : ι\nhn : n ∈ ns a\n⊢ n ∈ ns c\n\ncase refine'_4.intro.refine'_2\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\nα : Type u_3\np : Set ι → Prop\nf : Filter ι\nns : α → Set ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nhf : ∀ (t : Set ι), p t → tᶜ ∈ f\nhns : Directed (fun x x_1 => x ≤ x_1) ns\nhnsp : ∀ (a : α), p (ns a)\na b c : α\nhc : (fun x x_1 => x ≤ x_1) (ns a) (ns c) ∧ (fun x x_1 => x ≤ x_1) (ns b) (ns c)\nn : ι\nhn : n ∈ ns b\n⊢ n ∈ ns c",
"state_before": "case refine'_4.intro\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\nα : Type u_3\np : Set ι → Prop\nf : Filter ι\nns : α → Set ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nhf : ∀ (t : Set ι), p t → tᶜ ∈ f\nhns : Directed (fun x x_1 => x ≤ x_1) ns\nhnsp : ∀ (a : α), p (ns a)\na b c : α\nhc : (fun x x_1 => x ≤ x_1) (ns a) (ns c) ∧ (fun x x_1 => x ≤ x_1) (ns b) (ns c)\n⊢ ∃ z,\n (fun x x_1 => x ≤ x_1) ((fun a => ⨆ (n : ι) (_ : n ∈ ns a), s n) a) ((fun a => ⨆ (n : ι) (_ : n ∈ ns a), s n) z) ∧\n (fun x x_1 => x ≤ x_1) ((fun a => ⨆ (n : ι) (_ : n ∈ ns a), s n) b) ((fun a => ⨆ (n : ι) (_ : n ∈ ns a), s n) z)",
"tactic": "refine' ⟨c, _, _⟩ <;> refine' iSup_mono fun n => iSup_mono' fun hn => ⟨_, le_rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_4.intro.refine'_1\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\nα : Type u_3\np : Set ι → Prop\nf : Filter ι\nns : α → Set ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nhf : ∀ (t : Set ι), p t → tᶜ ∈ f\nhns : Directed (fun x x_1 => x ≤ x_1) ns\nhnsp : ∀ (a : α), p (ns a)\na b c : α\nhc : (fun x x_1 => x ≤ x_1) (ns a) (ns c) ∧ (fun x x_1 => x ≤ x_1) (ns b) (ns c)\nn : ι\nhn : n ∈ ns a\n⊢ n ∈ ns c",
"tactic": "exact hc.1 hn"
},
{
"state_after": "no goals",
"state_before": "case refine'_4.intro.refine'_2\nΩ : Type u_1\nι : Type u_2\nm m0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → MeasurableSpace Ω\nα : Type u_3\np : Set ι → Prop\nf : Filter ι\nns : α → Set ι\nh_le : ∀ (n : ι), s n ≤ m0\nh_indep : iIndep s\nhf : ∀ (t : Set ι), p t → tᶜ ∈ f\nhns : Directed (fun x x_1 => x ≤ x_1) ns\nhnsp : ∀ (a : α), p (ns a)\na b c : α\nhc : (fun x x_1 => x ≤ x_1) (ns a) (ns c) ∧ (fun x x_1 => x ≤ x_1) (ns b) (ns c)\nn : ι\nhn : n ∈ ns b\n⊢ n ∈ ns c",
"tactic": "exact hc.2 hn"
}
] | [
96,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
85,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Comp.lean | DifferentiableWithinAt.iterate | [] | [
263,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
261,
11
] |
Mathlib/CategoryTheory/Subterminal.lean | CategoryTheory.IsSubterminal.mono_isTerminal_from | [] | [
59,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
57,
1
] |
Mathlib/Data/List/Rotate.lean | List.rotate_eq_rotate | [] | [
336,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
335,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean | ModelWithCorners.right_inv | [] | [
284,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
283,
11
] |
Mathlib/AlgebraicTopology/SimplexCategory.lean | SimplexCategory.eq_δ_of_mono | [
{
"state_after": "case intro.intro\nn : ℕ\nθ : [n] ⟶ [n + 1]\ninst✝ : Mono θ\ni : Fin (n + 2)\nθ' : [n] ⟶ [n]\nh : θ = θ' ≫ δ i\n⊢ ∃ i, θ = δ i",
"state_before": "n : ℕ\nθ : [n] ⟶ [n + 1]\ninst✝ : Mono θ\n⊢ ∃ i, θ = δ i",
"tactic": "rcases eq_comp_δ_of_not_surjective θ (by\n by_contra h\n simpa using le_of_epi (epi_iff_surjective.mpr h)) with ⟨i, θ', h⟩"
},
{
"state_after": "case intro.intro\nn : ℕ\nθ : [n] ⟶ [n + 1]\ninst✝ : Mono θ\ni : Fin (n + 2)\nθ' : [n] ⟶ [n]\nh : θ = θ' ≫ δ i\n⊢ θ = δ i",
"state_before": "case intro.intro\nn : ℕ\nθ : [n] ⟶ [n + 1]\ninst✝ : Mono θ\ni : Fin (n + 2)\nθ' : [n] ⟶ [n]\nh : θ = θ' ≫ δ i\n⊢ ∃ i, θ = δ i",
"tactic": "use i"
},
{
"state_after": "case intro.intro\nn : ℕ\nθ : [n] ⟶ [n + 1]\ninst✝ : Mono θ\ni : Fin (n + 2)\nθ' : [n] ⟶ [n]\nh : θ = θ' ≫ δ i\nthis : Mono (θ' ≫ δ i)\n⊢ θ = δ i",
"state_before": "case intro.intro\nn : ℕ\nθ : [n] ⟶ [n + 1]\ninst✝ : Mono θ\ni : Fin (n + 2)\nθ' : [n] ⟶ [n]\nh : θ = θ' ≫ δ i\n⊢ θ = δ i",
"tactic": "haveI : Mono (θ' ≫ δ i) := by\n rw [← h]\n infer_instance"
},
{
"state_after": "case intro.intro\nn : ℕ\nθ : [n] ⟶ [n + 1]\ninst✝ : Mono θ\ni : Fin (n + 2)\nθ' : [n] ⟶ [n]\nh : θ = θ' ≫ δ i\nthis✝ : Mono (θ' ≫ δ i)\nthis : Mono θ'\n⊢ θ = δ i",
"state_before": "case intro.intro\nn : ℕ\nθ : [n] ⟶ [n + 1]\ninst✝ : Mono θ\ni : Fin (n + 2)\nθ' : [n] ⟶ [n]\nh : θ = θ' ≫ δ i\nthis : Mono (θ' ≫ δ i)\n⊢ θ = δ i",
"tactic": "haveI := CategoryTheory.mono_of_mono θ' (δ i)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nn : ℕ\nθ : [n] ⟶ [n + 1]\ninst✝ : Mono θ\ni : Fin (n + 2)\nθ' : [n] ⟶ [n]\nh : θ = θ' ≫ δ i\nthis✝ : Mono (θ' ≫ δ i)\nthis : Mono θ'\n⊢ θ = δ i",
"tactic": "rw [h, eq_id_of_mono θ', Category.id_comp]"
},
{
"state_after": "n : ℕ\nθ : [n] ⟶ [n + 1]\ninst✝ : Mono θ\nh : Function.Surjective ↑(Hom.toOrderHom θ)\n⊢ False",
"state_before": "n : ℕ\nθ : [n] ⟶ [n + 1]\ninst✝ : Mono θ\n⊢ ¬Function.Surjective ↑(Hom.toOrderHom θ)",
"tactic": "by_contra h"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nθ : [n] ⟶ [n + 1]\ninst✝ : Mono θ\nh : Function.Surjective ↑(Hom.toOrderHom θ)\n⊢ False",
"tactic": "simpa using le_of_epi (epi_iff_surjective.mpr h)"
},
{
"state_after": "n : ℕ\nθ : [n] ⟶ [n + 1]\ninst✝ : Mono θ\ni : Fin (n + 2)\nθ' : [n] ⟶ [n]\nh : θ = θ' ≫ δ i\n⊢ Mono θ",
"state_before": "n : ℕ\nθ : [n] ⟶ [n + 1]\ninst✝ : Mono θ\ni : Fin (n + 2)\nθ' : [n] ⟶ [n]\nh : θ = θ' ≫ δ i\n⊢ Mono (θ' ≫ δ i)",
"tactic": "rw [← h]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nθ : [n] ⟶ [n + 1]\ninst✝ : Mono θ\ni : Fin (n + 2)\nθ' : [n] ⟶ [n]\nh : θ = θ' ≫ δ i\n⊢ Mono θ",
"tactic": "infer_instance"
}
] | [
755,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
746,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean | Metric.mem_thickening_iff_infEdist_lt | [] | [
908,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
907,
1
] |
Mathlib/RingTheory/Int/Basic.lean | Int.zmultiples_natAbs | [] | [
389,
89
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
386,
1
] |
Mathlib/RingTheory/PolynomialAlgebra.lean | polyEquivTensor_symm_apply_tmul | [] | [
212,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
210,
1
] |
Mathlib/Algebra/Module/LocalizedModule.lean | LocalizedModule.induction_on₂ | [
{
"state_after": "case mk.mk.mk.mk\nR : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nβ : LocalizedModule S M → LocalizedModule S M → Prop\nh : ∀ (m m' : M) (s s' : { x // x ∈ S }), β (mk m s) (mk m' s')\nx✝ : LocalizedModule S M\nm : M\ns : { x // x ∈ S }\ny✝ : LocalizedModule S M\nm' : M\ns' : { x // x ∈ S }\n⊢ β (Quot.mk Setoid.r (m, s)) (Quot.mk Setoid.r (m', s'))",
"state_before": "R : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nβ : LocalizedModule S M → LocalizedModule S M → Prop\nh : ∀ (m m' : M) (s s' : { x // x ∈ S }), β (mk m s) (mk m' s')\n⊢ ∀ (x y : LocalizedModule S M), β x y",
"tactic": "rintro ⟨⟨m, s⟩⟩ ⟨⟨m', s'⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.mk.mk\nR : Type u\ninst✝² : CommSemiring R\nS : Submonoid R\nM : Type v\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nβ : LocalizedModule S M → LocalizedModule S M → Prop\nh : ∀ (m m' : M) (s s' : { x // x ∈ S }), β (mk m s) (mk m' s')\nx✝ : LocalizedModule S M\nm : M\ns : { x // x ∈ S }\ny✝ : LocalizedModule S M\nm' : M\ns' : { x // x ∈ S }\n⊢ β (Quot.mk Setoid.r (m, s)) (Quot.mk Setoid.r (m', s'))",
"tactic": "exact h m m' s s'"
}
] | [
113,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
110,
1
] |
Mathlib/Topology/Maps.lean | IsOpenMap.to_quotientMap | [] | [
405,
95
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
402,
1
] |
Mathlib/Algebra/BigOperators/Intervals.lean | Finset.prod_Ioc_consecutive | [
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nm n k : ℕ\nhmn : m ≤ n\nhnk : n ≤ k\n⊢ Disjoint (Ioc m n) (Ioc n k)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nm n k : ℕ\nhmn : m ≤ n\nhnk : n ≤ k\n⊢ (∏ i in Ioc m n, f i) * ∏ i in Ioc n k, f i = ∏ i in Ioc m k, f i",
"tactic": "rw [← Ioc_union_Ioc_eq_Ioc hmn hnk, prod_union]"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nm n k : ℕ\nhmn : m ≤ n\nhnk : n ≤ k\n⊢ ∀ (x : ℕ), x ∈ Ioc m n → x ∈ Ioc n k → False",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nm n k : ℕ\nhmn : m ≤ n\nhnk : n ≤ k\n⊢ Disjoint (Ioc m n) (Ioc n k)",
"tactic": "apply disjoint_left.2 fun x hx h'x => _"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nm n k : ℕ\nhmn : m ≤ n\nhnk : n ≤ k\nx : ℕ\nhx : x ∈ Ioc m n\nh'x : x ∈ Ioc n k\n⊢ False",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nm n k : ℕ\nhmn : m ≤ n\nhnk : n ≤ k\n⊢ ∀ (x : ℕ), x ∈ Ioc m n → x ∈ Ioc n k → False",
"tactic": "intros x hx h'x"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns₂ s₁ s : Finset α\na : α\ng f✝ : α → β\ninst✝ : CommMonoid β\nf : ℕ → β\nm n k : ℕ\nhmn : m ≤ n\nhnk : n ≤ k\nx : ℕ\nhx : x ∈ Ioc m n\nh'x : x ∈ Ioc n k\n⊢ False",
"tactic": "exact lt_irrefl _ ((mem_Ioc.1 h'x).1.trans_le (mem_Ioc.1 hx).2)"
}
] | [
80,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
75,
1
] |
Mathlib/Order/Filter/CountableInter.lean | countable_iInter_mem | [] | [
54,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
53,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean | RingHom.ker_eq_comap_bot | [] | [
2009,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2008,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean | AffineMap.lineMap_apply_zero | [
{
"state_after": "k : Type u_2\nV1 : Type u_3\nP1 : Type u_1\nV2 : Type ?u.445314\nP2 : Type ?u.445317\nV3 : Type ?u.445320\nP3 : Type ?u.445323\nV4 : Type ?u.445326\nP4 : Type ?u.445329\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\np₀ p₁ : P1\nthis : AddAction V1 P1 := inferInstance\n⊢ ↑(lineMap p₀ p₁) 0 = p₀",
"state_before": "k : Type u_2\nV1 : Type u_3\nP1 : Type u_1\nV2 : Type ?u.445314\nP2 : Type ?u.445317\nV3 : Type ?u.445320\nP3 : Type ?u.445323\nV4 : Type ?u.445326\nP4 : Type ?u.445329\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\np₀ p₁ : P1\n⊢ ↑(lineMap p₀ p₁) 0 = p₀",
"tactic": "letI : AddAction V1 P1 := inferInstance"
},
{
"state_after": "no goals",
"state_before": "k : Type u_2\nV1 : Type u_3\nP1 : Type u_1\nV2 : Type ?u.445314\nP2 : Type ?u.445317\nV3 : Type ?u.445320\nP3 : Type ?u.445323\nV4 : Type ?u.445326\nP4 : Type ?u.445329\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\np₀ p₁ : P1\nthis : AddAction V1 P1 := inferInstance\n⊢ ↑(lineMap p₀ p₁) 0 = p₀",
"tactic": "simp [(lineMap_apply)]"
}
] | [
562,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
559,
1
] |
Mathlib/Algebra/Homology/HomologicalComplex.lean | CochainComplex.of_x | [] | [
900,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
899,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean | CircleDeg1Lift.translationNumber_eq_of_tendsto₀ | [] | [
660,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
656,
1
] |
Mathlib/Data/Seq/WSeq.lean | Stream'.WSeq.map_ret | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nf : α → β\na : α\n⊢ map f (ret a) = ret (f a)",
"tactic": "simp [ret]"
}
] | [
1418,
77
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1418,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean | CategoryTheory.Limits.IsZero.of_epi_zero | [
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroMorphisms C\nX Y : C\ninst✝ : Epi 0\n⊢ 0 ≫ 𝟙 Y = 0 ≫ 0",
"tactic": "simp"
}
] | [
202,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
201,
1
] |
Mathlib/Data/Finsupp/NeLocus.lean | Finsupp.not_mem_neLocus | [] | [
51,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
50,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean | NNReal.dist_eq | [] | [
1682,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1682,
1
] |
Mathlib/Algebra/Order/Group/Defs.lean | inv_lt_iff_one_lt_mul | [
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝² : Group α\ninst✝¹ : LT 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 < b * a ↔ 1 < b * a",
"tactic": "rw [inv_mul_self]"
}
] | [
297,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
296,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean | Real.arcsin_zero | [] | [
116,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
115,
1
] |
Mathlib/Topology/Order/Hom/Basic.lean | ContinuousOrderHom.toFun_eq_coe | [] | [
114,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
114,
1
] |
Mathlib/LinearAlgebra/LinearPMap.lean | Submodule.valFromGraph_mem | [] | [
949,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
946,
1
] |
Mathlib/Order/Hom/Bounded.lean | BoundedOrderHom.cancel_left | [
{
"state_after": "no goals",
"state_before": "F : Type ?u.76609\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.76621\ninst✝⁷ : Preorder α\ninst✝⁶ : Preorder β\ninst✝⁵ : Preorder γ\ninst✝⁴ : Preorder δ\ninst✝³ : BoundedOrder α\ninst✝² : BoundedOrder β\ninst✝¹ : BoundedOrder γ\ninst✝ : BoundedOrder δ\ng : BoundedOrderHom β γ\nf₁ f₂ : BoundedOrderHom α β\nhg : Injective ↑g\nh : comp g f₁ = comp g f₂\na : α\n⊢ ↑g (↑f₁ a) = ↑g (↑f₂ a)",
"tactic": "rw [← BoundedOrderHom.comp_apply, h, BoundedOrderHom.comp_apply]"
}
] | [
701,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
696,
1
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean | Ordinal.foldr_le_nfpBFamily | [] | [
269,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
266,
1
] |
Std/Data/String/Lemmas.lean | Substring.Valid.valid | [
{
"state_after": "no goals",
"state_before": "l m r : List Char\n⊢ l ++ (m ++ r) =\n { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },\n stopPos := { byteIdx := utf8Len l + utf8Len m } }.str.data ∧\n { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },\n stopPos := { byteIdx := utf8Len l + utf8Len m } }.startPos.byteIdx =\n utf8Len l",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "l m r : List Char\n⊢ l ++ m ++ r =\n { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },\n stopPos := { byteIdx := utf8Len l + utf8Len m } }.str.data ∧\n { str := { data := l ++ m ++ r }, startPos := { byteIdx := utf8Len l },\n stopPos := { byteIdx := utf8Len l + utf8Len m } }.stopPos.byteIdx =\n utf8Len (l ++ m)",
"tactic": "simp"
}
] | [
970,
79
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
969,
1
] |
Mathlib/SetTheory/Ordinal/NaturalOps.lean | NatOrdinal.toOrdinal_max | [] | [
125,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
124,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Base.lean | Real.logb_prod | [
{
"state_after": "case empty\nb x y : ℝ\nα : Type u_1\ns : Finset α\nf : α → ℝ\nhf✝ : ∀ (x : α), x ∈ s → f x ≠ 0\nhf : ∀ (x : α), x ∈ ∅ → f x ≠ 0\n⊢ logb b (∏ i in ∅, f i) = ∑ i in ∅, logb b (f i)\n\ncase insert\nb x y : ℝ\nα : Type u_1\ns✝ : Finset α\nf : α → ℝ\nhf✝ : ∀ (x : α), x ∈ s✝ → f x ≠ 0\na : α\ns : Finset α\nha : ¬a ∈ s\nih : (∀ (x : α), x ∈ s → f x ≠ 0) → logb b (∏ i in s, f i) = ∑ i in s, logb b (f i)\nhf : ∀ (x : α), x ∈ insert a s → f x ≠ 0\n⊢ logb b (∏ i in insert a s, f i) = ∑ i in insert a s, logb b (f i)",
"state_before": "b x y : ℝ\nα : Type u_1\ns : Finset α\nf : α → ℝ\nhf : ∀ (x : α), x ∈ s → f x ≠ 0\n⊢ logb b (∏ i in s, f i) = ∑ i in s, logb b (f i)",
"tactic": "induction' s using Finset.induction_on with a s ha ih"
},
{
"state_after": "case insert\nb x y : ℝ\nα : Type u_1\ns✝ : Finset α\nf : α → ℝ\nhf✝ : ∀ (x : α), x ∈ s✝ → f x ≠ 0\na : α\ns : Finset α\nha : ¬a ∈ s\nih : (∀ (x : α), x ∈ s → f x ≠ 0) → logb b (∏ i in s, f i) = ∑ i in s, logb b (f i)\nhf : f a ≠ 0 ∧ ∀ (a : α), a ∈ s → f a ≠ 0\n⊢ logb b (∏ i in insert a s, f i) = ∑ i in insert a s, logb b (f i)",
"state_before": "case insert\nb x y : ℝ\nα : Type u_1\ns✝ : Finset α\nf : α → ℝ\nhf✝ : ∀ (x : α), x ∈ s✝ → f x ≠ 0\na : α\ns : Finset α\nha : ¬a ∈ s\nih : (∀ (x : α), x ∈ s → f x ≠ 0) → logb b (∏ i in s, f i) = ∑ i in s, logb b (f i)\nhf : ∀ (x : α), x ∈ insert a s → f x ≠ 0\n⊢ logb b (∏ i in insert a s, f i) = ∑ i in insert a s, logb b (f i)",
"tactic": "simp only [Finset.mem_insert, forall_eq_or_imp] at hf"
},
{
"state_after": "no goals",
"state_before": "case insert\nb x y : ℝ\nα : Type u_1\ns✝ : Finset α\nf : α → ℝ\nhf✝ : ∀ (x : α), x ∈ s✝ → f x ≠ 0\na : α\ns : Finset α\nha : ¬a ∈ s\nih : (∀ (x : α), x ∈ s → f x ≠ 0) → logb b (∏ i in s, f i) = ∑ i in s, logb b (f i)\nhf : f a ≠ 0 ∧ ∀ (a : α), a ∈ s → f a ≠ 0\n⊢ logb b (∏ i in insert a s, f i) = ∑ i in insert a s, logb b (f i)",
"tactic": "simp [ha, ih hf.2, logb_mul hf.1 (Finset.prod_ne_zero_iff.2 hf.2)]"
},
{
"state_after": "no goals",
"state_before": "case empty\nb x y : ℝ\nα : Type u_1\ns : Finset α\nf : α → ℝ\nhf✝ : ∀ (x : α), x ∈ s → f x ≠ 0\nhf : ∀ (x : α), x ∈ ∅ → f x ≠ 0\n⊢ logb b (∏ i in ∅, f i) = ∑ i in ∅, logb b (f i)",
"tactic": "simp"
}
] | [
397,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
391,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean | CategoryTheory.Limits.Cofork.π_ofπ | [] | [
388,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
387,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean | Asymptotics.isBigO_norm_norm | [] | [
826,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
825,
1
] |
Mathlib/Algebra/Hom/Centroid.lean | CentroidHom.coe_sub | [] | [
483,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
482,
1
] |
Mathlib/CategoryTheory/Whiskering.lean | CategoryTheory.Functor.assoc | [] | [
276,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
275,
11
] |
Mathlib/Topology/Homeomorph.lean | Homeomorph.preimage_closure | [] | [
344,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
343,
1
] |
Mathlib/Data/Int/Parity.lean | Int.odd_mul | [
{
"state_after": "no goals",
"state_before": "m n : ℤ\n⊢ Odd (m * n) ↔ Odd m ∧ Odd n",
"tactic": "simp [not_or, parity_simps]"
}
] | [
146,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
146,
1
] |
Mathlib/Order/UpperLower/Basic.lean | LowerSet.coe_iInf | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.71521\nγ : Type ?u.71524\nι : Sort u_2\nκ : ι → Sort ?u.71532\ninst✝ : LE α\nS : Set (LowerSet α)\ns t : LowerSet α\na : α\nf : ι → LowerSet α\n⊢ ↑(⨅ (i : ι), f i) = ⋂ (i : ι), ↑(f i)",
"tactic": "simp_rw [iInf, coe_sInf, mem_range, iInter_exists, iInter_iInter_eq']"
}
] | [
708,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
707,
1
] |
Mathlib/Data/Matrix/Block.lean | Matrix.blockDiag_neg | [] | [
595,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
594,
1
] |
Mathlib/Logic/Embedding/Basic.lean | Equiv.embeddingCongr_trans | [] | [
438,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
434,
1
] |
src/lean/Init/Data/Nat/Div.lean | Nat.div.inductionOn | [
{
"state_after": "case a\nmotive : Nat → Nat → Sort u\nx y : Nat\nind : (x y : Nat) → 0 < y ∧ y ≤ x → motive (x - y) y → motive x y\nbase : (x y : Nat) → ¬(0 < y ∧ y ≤ x) → motive x y\nh : 0 < y ∧ y ≤ x\n⊢ 0 < y ∧ y ≤ x",
"state_before": "motive : Nat → Nat → Sort u\nx y : Nat\nind : (x y : Nat) → 0 < y ∧ y ≤ x → motive (x - y) y → motive x y\nbase : (x y : Nat) → ¬(0 < y ∧ y ≤ x) → motive x y\nh : 0 < y ∧ y ≤ x\n⊢ (invImage\n (fun a =>\n PSigma.casesOn a fun x snd => PSigma.casesOn snd fun y snd => PSigma.casesOn snd fun ind snd => sizeOf x)\n instWellFoundedRelation).1\n { fst := x - y, snd := { fst := y, snd := { fst := ind, snd := base } } }\n { fst := x, snd := { fst := y, snd := { fst := ind, snd := base } } }",
"tactic": "apply div_rec_lemma"
},
{
"state_after": "no goals",
"state_before": "case a\nmotive : Nat → Nat → Sort u\nx y : Nat\nind : (x y : Nat) → 0 < y ∧ y ≤ x → motive (x - y) y → motive x y\nbase : (x y : Nat) → ¬(0 < y ∧ y ≤ x) → motive x y\nh : 0 < y ∧ y ≤ x\n⊢ 0 < y ∧ y ≤ x",
"tactic": "assumption"
}
] | [
40,
46
] | d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
30,
1
] |
Mathlib/Order/Filter/Basic.lean | Filter.eventuallyEq_set | [] | [
1446,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1445,
1
] |
Mathlib/Data/Finset/NoncommProd.lean | Multiset.noncommFold_empty | [] | [
97,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
96,
1
] |
Std/Data/String/Lemmas.lean | String.Pos.mk_le_mk | [] | [
141,
81
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
141,
9
] |
Mathlib/Computability/TMToPartrec.lean | Turing.PartrecToTM2.contSupp_fix | [
{
"state_after": "no goals",
"state_before": "f : Code\nk : Cont'\n⊢ contSupp (Cont'.fix f k) = codeSupp f (Cont'.fix f k)",
"tactic": "simp (config := { contextual := true }) [codeSupp, codeSupp', contSupp, Finset.union_assoc,\n Finset.subset_iff]"
}
] | [
1886,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1884,
1
] |
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