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Mathlib/Analysis/NormedSpace/Exponential.lean | exp_zsmul | [
{
"state_after": "case intro.inl\n𝕂 : Type u_2\n𝔸 : Type u_1\ninst✝³ : IsROrC 𝕂\ninst✝² : NormedDivisionRing 𝔸\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : CompleteSpace 𝔸\nx : 𝔸\nn : ℕ\n⊢ exp 𝕂 (↑n • x) = exp 𝕂 x ^ ↑n\n\ncase intro.inr\n𝕂 : Type u_2\n𝔸 : Type u_1\ninst✝³ : IsROrC 𝕂\ninst✝² : NormedDivisionRing 𝔸\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : CompleteSpace 𝔸\nx : 𝔸\nn : ℕ\n⊢ exp 𝕂 (-↑n • x) = exp 𝕂 x ^ (-↑n)",
"state_before": "𝕂 : Type u_2\n𝔸 : Type u_1\ninst✝³ : IsROrC 𝕂\ninst✝² : NormedDivisionRing 𝔸\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : CompleteSpace 𝔸\nz : ℤ\nx : 𝔸\n⊢ exp 𝕂 (z • x) = exp 𝕂 x ^ z",
"tactic": "obtain ⟨n, rfl | rfl⟩ := z.eq_nat_or_neg"
},
{
"state_after": "no goals",
"state_before": "case intro.inl\n𝕂 : Type u_2\n𝔸 : Type u_1\ninst✝³ : IsROrC 𝕂\ninst✝² : NormedDivisionRing 𝔸\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : CompleteSpace 𝔸\nx : 𝔸\nn : ℕ\n⊢ exp 𝕂 (↑n • x) = exp 𝕂 x ^ ↑n",
"tactic": "rw [zpow_ofNat, coe_nat_zsmul, exp_nsmul]"
},
{
"state_after": "no goals",
"state_before": "case intro.inr\n𝕂 : Type u_2\n𝔸 : Type u_1\ninst✝³ : IsROrC 𝕂\ninst✝² : NormedDivisionRing 𝔸\ninst✝¹ : NormedAlgebra 𝕂 𝔸\ninst✝ : CompleteSpace 𝔸\nx : 𝔸\nn : ℕ\n⊢ exp 𝕂 (-↑n • x) = exp 𝕂 x ^ (-↑n)",
"tactic": "rw [zpow_neg, zpow_ofNat, neg_smul, exp_neg, coe_nat_zsmul, exp_nsmul]"
}
] | [
610,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
607,
1
] |
Mathlib/Analysis/NormedSpace/Banach.lean | ContinuousLinearMap.nonlinearRightInverseOfSurjective_nnnorm_pos | [
{
"state_after": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf✝ : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nf : E →L[𝕜] F\nhsurj : LinearMap.range f = ⊤\n⊢ 0 < (Classical.choose (_ : ∃ fsymm, 0 < fsymm.nnnorm)).nnnorm",
"state_before": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf✝ : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nf : E →L[𝕜] F\nhsurj : LinearMap.range f = ⊤\n⊢ 0 < (nonlinearRightInverseOfSurjective f hsurj).nnnorm",
"tactic": "rw [nonlinearRightInverseOfSurjective]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nf✝ : E →L[𝕜] F\ninst✝¹ : CompleteSpace F\ninst✝ : CompleteSpace E\nf : E →L[𝕜] F\nhsurj : LinearMap.range f = ⊤\n⊢ 0 < (Classical.choose (_ : ∃ fsymm, 0 < fsymm.nnnorm)).nnnorm",
"tactic": "exact Classical.choose_spec (exists_nonlinearRightInverse_of_surjective f hsurj)"
}
] | [
317,
83
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
314,
1
] |
Mathlib/Analysis/Convex/Combination.lean | convexHull_basis_eq_stdSimplex | [
{
"state_after": "case refine'_1\nR : Type u_1\nE : Type ?u.418534\nF : Type ?u.418537\nι : Type u_2\nι' : Type ?u.418543\nα : Type ?u.418546\ninst✝⁸ : LinearOrderedField R\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : LinearOrderedAddCommGroup α\ninst✝⁴ : Module R E\ninst✝³ : Module R F\ninst✝² : Module R α\ninst✝¹ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\ninst✝ : Fintype ι\nf : ι → R\n⊢ (Set.range fun i j => if i = j then 1 else 0) ⊆ stdSimplex R ι\n\ncase refine'_2\nR : Type u_1\nE : Type ?u.418534\nF : Type ?u.418537\nι : Type u_2\nι' : Type ?u.418543\nα : Type ?u.418546\ninst✝⁸ : LinearOrderedField R\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : LinearOrderedAddCommGroup α\ninst✝⁴ : Module R E\ninst✝³ : Module R F\ninst✝² : Module R α\ninst✝¹ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\ninst✝ : Fintype ι\nf : ι → R\n⊢ stdSimplex R ι ⊆ ↑(convexHull R).toOrderHom (Set.range fun i j => if i = j then 1 else 0)",
"state_before": "R : Type u_1\nE : Type ?u.418534\nF : Type ?u.418537\nι : Type u_2\nι' : Type ?u.418543\nα : Type ?u.418546\ninst✝⁸ : LinearOrderedField R\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : LinearOrderedAddCommGroup α\ninst✝⁴ : Module R E\ninst✝³ : Module R F\ninst✝² : Module R α\ninst✝¹ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\ninst✝ : Fintype ι\nf : ι → R\n⊢ ↑(convexHull R).toOrderHom (Set.range fun i j => if i = j then 1 else 0) = stdSimplex R ι",
"tactic": "refine' Subset.antisymm (convexHull_min _ (convex_stdSimplex R ι)) _"
},
{
"state_after": "case refine'_1.intro\nR : Type u_1\nE : Type ?u.418534\nF : Type ?u.418537\nι : Type u_2\nι' : Type ?u.418543\nα : Type ?u.418546\ninst✝⁸ : LinearOrderedField R\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : LinearOrderedAddCommGroup α\ninst✝⁴ : Module R E\ninst✝³ : Module R F\ninst✝² : Module R α\ninst✝¹ : OrderedSMul R α\ns : Set E\ni✝ j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\ninst✝ : Fintype ι\nf : ι → R\ni : ι\n⊢ (fun i j => if i = j then 1 else 0) i ∈ stdSimplex R ι",
"state_before": "case refine'_1\nR : Type u_1\nE : Type ?u.418534\nF : Type ?u.418537\nι : Type u_2\nι' : Type ?u.418543\nα : Type ?u.418546\ninst✝⁸ : LinearOrderedField R\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : LinearOrderedAddCommGroup α\ninst✝⁴ : Module R E\ninst✝³ : Module R F\ninst✝² : Module R α\ninst✝¹ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\ninst✝ : Fintype ι\nf : ι → R\n⊢ (Set.range fun i j => if i = j then 1 else 0) ⊆ stdSimplex R ι",
"tactic": "rintro _ ⟨i, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro\nR : Type u_1\nE : Type ?u.418534\nF : Type ?u.418537\nι : Type u_2\nι' : Type ?u.418543\nα : Type ?u.418546\ninst✝⁸ : LinearOrderedField R\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : LinearOrderedAddCommGroup α\ninst✝⁴ : Module R E\ninst✝³ : Module R F\ninst✝² : Module R α\ninst✝¹ : OrderedSMul R α\ns : Set E\ni✝ j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\ninst✝ : Fintype ι\nf : ι → R\ni : ι\n⊢ (fun i j => if i = j then 1 else 0) i ∈ stdSimplex R ι",
"tactic": "exact ite_eq_mem_stdSimplex R i"
},
{
"state_after": "case refine'_2.intro\nR : Type u_1\nE : Type ?u.418534\nF : Type ?u.418537\nι : Type u_2\nι' : Type ?u.418543\nα : Type ?u.418546\ninst✝⁸ : LinearOrderedField R\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : LinearOrderedAddCommGroup α\ninst✝⁴ : Module R E\ninst✝³ : Module R F\ninst✝² : Module R α\ninst✝¹ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt : Finset ι\nw✝ : ι → R\nz : ι → E\ninst✝ : Fintype ι\nf w : ι → R\nhw₀ : ∀ (x : ι), 0 ≤ w x\nhw₁ : ∑ x : ι, w x = 1\n⊢ w ∈ ↑(convexHull R).toOrderHom (Set.range fun i j => if i = j then 1 else 0)",
"state_before": "case refine'_2\nR : Type u_1\nE : Type ?u.418534\nF : Type ?u.418537\nι : Type u_2\nι' : Type ?u.418543\nα : Type ?u.418546\ninst✝⁸ : LinearOrderedField R\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : LinearOrderedAddCommGroup α\ninst✝⁴ : Module R E\ninst✝³ : Module R F\ninst✝² : Module R α\ninst✝¹ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt : Finset ι\nw : ι → R\nz : ι → E\ninst✝ : Fintype ι\nf : ι → R\n⊢ stdSimplex R ι ⊆ ↑(convexHull R).toOrderHom (Set.range fun i j => if i = j then 1 else 0)",
"tactic": "rintro w ⟨hw₀, hw₁⟩"
},
{
"state_after": "case refine'_2.intro\nR : Type u_1\nE : Type ?u.418534\nF : Type ?u.418537\nι : Type u_2\nι' : Type ?u.418543\nα : Type ?u.418546\ninst✝⁸ : LinearOrderedField R\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : LinearOrderedAddCommGroup α\ninst✝⁴ : Module R E\ninst✝³ : Module R F\ninst✝² : Module R α\ninst✝¹ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt : Finset ι\nw✝ : ι → R\nz : ι → E\ninst✝ : Fintype ι\nf w : ι → R\nhw₀ : ∀ (x : ι), 0 ≤ w x\nhw₁ : ∑ x : ι, w x = 1\n⊢ (centerMass Finset.univ (fun i => w i) fun i j => if i = j then 1 else 0) ∈\n ↑(convexHull R).toOrderHom (Set.range fun i j => if i = j then 1 else 0)",
"state_before": "case refine'_2.intro\nR : Type u_1\nE : Type ?u.418534\nF : Type ?u.418537\nι : Type u_2\nι' : Type ?u.418543\nα : Type ?u.418546\ninst✝⁸ : LinearOrderedField R\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : LinearOrderedAddCommGroup α\ninst✝⁴ : Module R E\ninst✝³ : Module R F\ninst✝² : Module R α\ninst✝¹ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt : Finset ι\nw✝ : ι → R\nz : ι → E\ninst✝ : Fintype ι\nf w : ι → R\nhw₀ : ∀ (x : ι), 0 ≤ w x\nhw₁ : ∑ x : ι, w x = 1\n⊢ w ∈ ↑(convexHull R).toOrderHom (Set.range fun i j => if i = j then 1 else 0)",
"tactic": "rw [pi_eq_sum_univ w, ← Finset.univ.centerMass_eq_of_sum_1 _ hw₁]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro\nR : Type u_1\nE : Type ?u.418534\nF : Type ?u.418537\nι : Type u_2\nι' : Type ?u.418543\nα : Type ?u.418546\ninst✝⁸ : LinearOrderedField R\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : AddCommGroup F\ninst✝⁵ : LinearOrderedAddCommGroup α\ninst✝⁴ : Module R E\ninst✝³ : Module R F\ninst✝² : Module R α\ninst✝¹ : OrderedSMul R α\ns : Set E\ni j : ι\nc : R\nt : Finset ι\nw✝ : ι → R\nz : ι → E\ninst✝ : Fintype ι\nf w : ι → R\nhw₀ : ∀ (x : ι), 0 ≤ w x\nhw₁ : ∑ x : ι, w x = 1\n⊢ (centerMass Finset.univ (fun i => w i) fun i j => if i = j then 1 else 0) ∈\n ↑(convexHull R).toOrderHom (Set.range fun i j => if i = j then 1 else 0)",
"tactic": "exact Finset.univ.centerMass_mem_convexHull (fun i _ => hw₀ i) (hw₁.symm ▸ zero_lt_one)\n fun i _ => mem_range_self i"
}
] | [
452,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
444,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Bilinear.lean | ContinuousLinearMap.hasFDerivWithinAt_of_bilinear | [] | [
134,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
129,
1
] |
Mathlib/Analysis/NormedSpace/Dual.lean | NormedSpace.dualPairing_apply | [] | [
121,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
120,
1
] |
Mathlib/MeasureTheory/Integral/IntegrableOn.lean | integrableOn_Ici_iff_integrableOn_Ioi | [
{
"state_after": "α : Type u_1\nβ : Type ?u.4570798\nE : Type u_2\nF : Type ?u.4570804\ninst✝⁴ : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : PartialOrder α\ninst✝¹ : MeasurableSingletonClass α\nf : α → E\nμ : MeasureTheory.Measure α\na b : α\ninst✝ : NoAtoms μ\n⊢ 0 ≠ ⊤",
"state_before": "α : Type u_1\nβ : Type ?u.4570798\nE : Type u_2\nF : Type ?u.4570804\ninst✝⁴ : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : PartialOrder α\ninst✝¹ : MeasurableSingletonClass α\nf : α → E\nμ : MeasureTheory.Measure α\na b : α\ninst✝ : NoAtoms μ\n⊢ ↑↑μ {b} ≠ ⊤",
"tactic": "rw [measure_singleton]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.4570798\nE : Type u_2\nF : Type ?u.4570804\ninst✝⁴ : MeasurableSpace α\ninst✝³ : NormedAddCommGroup E\ninst✝² : PartialOrder α\ninst✝¹ : MeasurableSingletonClass α\nf : α → E\nμ : MeasureTheory.Measure α\na b : α\ninst✝ : NoAtoms μ\n⊢ 0 ≠ ⊤",
"tactic": "exact ENNReal.zero_ne_top"
}
] | [
704,
96
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
702,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean | Convex.combo_affine_apply | [
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : Ring 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx y : E\na b : 𝕜\nf : E →ᵃ[𝕜] F\nh : a + b = 1\n⊢ ↑f (b • (y -ᵥ x) + x) = b • (↑f y -ᵥ ↑f x) + ↑f x",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : Ring 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx y : E\na b : 𝕜\nf : E →ᵃ[𝕜] F\nh : a + b = 1\n⊢ ↑f (a • x + b • y) = a • ↑f x + b • ↑f y",
"tactic": "simp only [Convex.combo_eq_smul_sub_add h, ← vsub_eq_sub]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : Ring 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx y : E\na b : 𝕜\nf : E →ᵃ[𝕜] F\nh : a + b = 1\n⊢ ↑f (b • (y -ᵥ x) + x) = b • (↑f y -ᵥ ↑f x) + ↑f x",
"tactic": "exact f.apply_lineMap _ _ _"
}
] | [
873,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
870,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Mul.lean | DifferentiableOn.clm_comp | [] | [
94,
97
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
93,
1
] |
Mathlib/Analysis/Calculus/Deriv/Linear.lean | ContinuousLinearMap.hasStrictDerivAt | [] | [
57,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
56,
11
] |
Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean | MvPolynomial.weightedHomogeneousComponent_isWeightedHomogeneous | [
{
"state_after": "no goals",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\n⊢ IsWeightedHomogeneous w (↑(weightedHomogeneousComponent w n) φ) n",
"tactic": "classical\nintro d hd\ncontrapose! hd\nrw [coeff_weightedHomogeneousComponent, if_neg hd]"
},
{
"state_after": "R : Type u_1\nM : Type u_2\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\nhd : coeff d (↑(weightedHomogeneousComponent w n) φ) ≠ 0\n⊢ ↑(weightedDegree' w) d = n",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\n⊢ IsWeightedHomogeneous w (↑(weightedHomogeneousComponent w n) φ) n",
"tactic": "intro d hd"
},
{
"state_after": "R : Type u_1\nM : Type u_2\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\nhd : ↑(weightedDegree' w) d ≠ n\n⊢ coeff d (↑(weightedHomogeneousComponent w n) φ) = 0",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\nhd : coeff d (↑(weightedHomogeneousComponent w n) φ) ≠ 0\n⊢ ↑(weightedDegree' w) d = n",
"tactic": "contrapose! hd"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nw : σ → M\nn : M\nφ ψ : MvPolynomial σ R\nd : σ →₀ ℕ\nhd : ↑(weightedDegree' w) d ≠ n\n⊢ coeff d (↑(weightedHomogeneousComponent w n) φ) = 0",
"tactic": "rw [coeff_weightedHomogeneousComponent, if_neg hd]"
}
] | [
357,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
352,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean | div_le_div_of_le_left | [
{
"state_after": "ι : Type ?u.62267\nα : Type u_1\nβ : Type ?u.62273\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nha : 0 ≤ a\nhc : 0 < c\nh : c ≤ b\n⊢ a * b⁻¹ ≤ a * c⁻¹",
"state_before": "ι : Type ?u.62267\nα : Type u_1\nβ : Type ?u.62273\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nha : 0 ≤ a\nhc : 0 < c\nh : c ≤ b\n⊢ a / b ≤ a / c",
"tactic": "rw [div_eq_mul_inv, div_eq_mul_inv]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.62267\nα : Type u_1\nβ : Type ?u.62273\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nha : 0 ≤ a\nhc : 0 < c\nh : c ≤ b\n⊢ a * b⁻¹ ≤ a * c⁻¹",
"tactic": "exact mul_le_mul_of_nonneg_left ((inv_le_inv (hc.trans_le h) hc).mpr h) ha"
}
] | [
353,
77
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
351,
1
] |
Mathlib/Order/BooleanAlgebra.lean | disjoint_sdiff_iff_le | [
{
"state_after": "α : Type u\nβ : Type ?u.13662\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nhz : z ≤ y\nhx : x ≤ y\nH : Disjoint z (y \\ x)\n⊢ z ⊔ y \\ x ≤ y",
"state_before": "α : Type u\nβ : Type ?u.13662\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nhz : z ≤ y\nhx : x ≤ y\nH : Disjoint z (y \\ x)\n⊢ z ⊔ y \\ x ≤ x ⊔ y \\ x",
"tactic": "rw [sup_sdiff_cancel_right hx]"
},
{
"state_after": "α : Type u\nβ : Type ?u.13662\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nhz : z ≤ y\nhx : x ≤ y\nH : Disjoint z (y \\ x)\n⊢ z ⊔ y ≤ y",
"state_before": "α : Type u\nβ : Type ?u.13662\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nhz : z ≤ y\nhx : x ≤ y\nH : Disjoint z (y \\ x)\n⊢ z ⊔ y \\ x ≤ y",
"tactic": "refine' le_trans (sup_le_sup_left sdiff_le z) _"
},
{
"state_after": "no goals",
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"tactic": "rw [sup_eq_right.2 hz]"
}
] | [
254,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
247,
1
] |
Mathlib/Data/Set/Intervals/Pi.lean | Set.image_update_uIcc | [
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Lattice (α i)\ninst✝ : DecidableEq ι\nf : (i : ι) → α i\ni : ι\na b : α i\n⊢ Icc (update f i (a ⊓ b)) (update f i (a ⊔ b)) = uIcc (update f i a) (update f i b)",
"tactic": "simp_rw [uIcc, update_sup, update_inf]"
}
] | [
296,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
294,
1
] |
Mathlib/Algebra/Order/Ring/Defs.lean | nonneg_of_mul_nonpos_left | [] | [
1188,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1187,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean | contDiffAt_prod | [] | [
1426,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1424,
1
] |
Mathlib/Data/List/Perm.lean | List.map_append_bind_perm | [
{
"state_after": "no goals",
"state_before": "α : Type uu\nβ : Type vv\nl₁ l₂ l : List α\nf : α → β\ng : α → List β\n⊢ map f l ++ List.bind l g ~ List.bind l fun x => f x :: g x",
"tactic": "simpa [← map_eq_bind] using bind_append_perm l (fun x => [f x]) g"
}
] | [
1103,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1101,
1
] |
Std/Data/Int/Lemmas.lean | Int.mul_le_mul | [] | [
1176,
98
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
1174,
11
] |
Mathlib/GroupTheory/GroupAction/Pi.lean | Function.update_smul | [] | [
251,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
249,
1
] |
Std/Data/Int/DivMod.lean | Int.div_zero | [
{
"state_after": "no goals",
"state_before": "a✝ : Nat\n⊢ ofNat (a✝ / 0) = 0",
"tactic": "simp"
}
] | [
43,
18
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
41,
19
] |
Mathlib/Data/Polynomial/Degree/Lemmas.lean | Polynomial.degree_pos_of_eval₂_root | [] | [
279,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
277,
1
] |
Mathlib/Analysis/NormedSpace/Basic.lean | closure_ball | [
{
"state_after": "α : Type ?u.32233\nβ : Type ?u.32236\nγ : Type ?u.32239\nι : Type ?u.32242\ninst✝⁶ : NormedField α\ninst✝⁵ : SeminormedAddCommGroup β\nE : Type u_1\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : NormedSpace α E\nF : Type ?u.32273\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace α F\ninst✝ : NormedSpace ℝ E\nx : E\nr : ℝ\nhr : r ≠ 0\ny : E\nhy : y ∈ closedBall x r\n⊢ y ∈ closure (ball x r)",
"state_before": "α : Type ?u.32233\nβ : Type ?u.32236\nγ : Type ?u.32239\nι : Type ?u.32242\ninst✝⁶ : NormedField α\ninst✝⁵ : SeminormedAddCommGroup β\nE : Type u_1\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : NormedSpace α E\nF : Type ?u.32273\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace α F\ninst✝ : NormedSpace ℝ E\nx : E\nr : ℝ\nhr : r ≠ 0\n⊢ closure (ball x r) = closedBall x r",
"tactic": "refine' Subset.antisymm closure_ball_subset_closedBall fun y hy => _"
},
{
"state_after": "α : Type ?u.32233\nβ : Type ?u.32236\nγ : Type ?u.32239\nι : Type ?u.32242\ninst✝⁶ : NormedField α\ninst✝⁵ : SeminormedAddCommGroup β\nE : Type u_1\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : NormedSpace α E\nF : Type ?u.32273\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace α F\ninst✝ : NormedSpace ℝ E\nx : E\nr : ℝ\nhr : r ≠ 0\ny : E\nhy : y ∈ closedBall x r\nthis : ContinuousWithinAt (fun c => c • (y - x) + x) (Ico 0 1) 1\n⊢ y ∈ closure (ball x r)",
"state_before": "α : Type ?u.32233\nβ : Type ?u.32236\nγ : Type ?u.32239\nι : Type ?u.32242\ninst✝⁶ : NormedField α\ninst✝⁵ : SeminormedAddCommGroup β\nE : Type u_1\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : NormedSpace α E\nF : Type ?u.32273\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace α F\ninst✝ : NormedSpace ℝ E\nx : E\nr : ℝ\nhr : r ≠ 0\ny : E\nhy : y ∈ closedBall x r\n⊢ y ∈ closure (ball x r)",
"tactic": "have : ContinuousWithinAt (fun c : ℝ => c • (y - x) + x) (Ico 0 1) 1 :=\n ((continuous_id.smul continuous_const).add continuous_const).continuousWithinAt"
},
{
"state_after": "case h.e'_4\nα : Type ?u.32233\nβ : Type ?u.32236\nγ : Type ?u.32239\nι : Type ?u.32242\ninst✝⁶ : NormedField α\ninst✝⁵ : SeminormedAddCommGroup β\nE : Type u_1\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : NormedSpace α E\nF : Type ?u.32273\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace α F\ninst✝ : NormedSpace ℝ E\nx : E\nr : ℝ\nhr : r ≠ 0\ny : E\nhy : y ∈ closedBall x r\nthis : ContinuousWithinAt (fun c => c • (y - x) + x) (Ico 0 1) 1\n⊢ y = 1 • (y - x) + x\n\ncase convert_2\nα : Type ?u.32233\nβ : Type ?u.32236\nγ : Type ?u.32239\nι : Type ?u.32242\ninst✝⁶ : NormedField α\ninst✝⁵ : SeminormedAddCommGroup β\nE : Type u_1\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : NormedSpace α E\nF : Type ?u.32273\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace α F\ninst✝ : NormedSpace ℝ E\nx : E\nr : ℝ\nhr : r ≠ 0\ny : E\nhy : y ∈ closedBall x r\nthis : ContinuousWithinAt (fun c => c • (y - x) + x) (Ico 0 1) 1\n⊢ 1 ∈ closure (Ico 0 1)\n\ncase convert_3\nα : Type ?u.32233\nβ : Type ?u.32236\nγ : Type ?u.32239\nι : Type ?u.32242\ninst✝⁶ : NormedField α\ninst✝⁵ : SeminormedAddCommGroup β\nE : Type u_1\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : NormedSpace α E\nF : Type ?u.32273\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace α F\ninst✝ : NormedSpace ℝ E\nx : E\nr : ℝ\nhr : r ≠ 0\ny : E\nhy : y ∈ closedBall x r\nthis : ContinuousWithinAt (fun c => c • (y - x) + x) (Ico 0 1) 1\n⊢ MapsTo (fun c => c • (y - x) + x) (Ico 0 1) (ball x r)",
"state_before": "α : Type ?u.32233\nβ : Type ?u.32236\nγ : Type ?u.32239\nι : Type ?u.32242\ninst✝⁶ : NormedField α\ninst✝⁵ : SeminormedAddCommGroup β\nE : Type u_1\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : NormedSpace α E\nF : Type ?u.32273\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace α F\ninst✝ : NormedSpace ℝ E\nx : E\nr : ℝ\nhr : r ≠ 0\ny : E\nhy : y ∈ closedBall x r\nthis : ContinuousWithinAt (fun c => c • (y - x) + x) (Ico 0 1) 1\n⊢ y ∈ closure (ball x r)",
"tactic": "convert this.mem_closure _ _"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nα : Type ?u.32233\nβ : Type ?u.32236\nγ : Type ?u.32239\nι : Type ?u.32242\ninst✝⁶ : NormedField α\ninst✝⁵ : SeminormedAddCommGroup β\nE : Type u_1\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : NormedSpace α E\nF : Type ?u.32273\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace α F\ninst✝ : NormedSpace ℝ E\nx : E\nr : ℝ\nhr : r ≠ 0\ny : E\nhy : y ∈ closedBall x r\nthis : ContinuousWithinAt (fun c => c • (y - x) + x) (Ico 0 1) 1\n⊢ y = 1 • (y - x) + x",
"tactic": "rw [one_smul, sub_add_cancel]"
},
{
"state_after": "no goals",
"state_before": "case convert_2\nα : Type ?u.32233\nβ : Type ?u.32236\nγ : Type ?u.32239\nι : Type ?u.32242\ninst✝⁶ : NormedField α\ninst✝⁵ : SeminormedAddCommGroup β\nE : Type u_1\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : NormedSpace α E\nF : Type ?u.32273\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace α F\ninst✝ : NormedSpace ℝ E\nx : E\nr : ℝ\nhr : r ≠ 0\ny : E\nhy : y ∈ closedBall x r\nthis : ContinuousWithinAt (fun c => c • (y - x) + x) (Ico 0 1) 1\n⊢ 1 ∈ closure (Ico 0 1)",
"tactic": "simp [closure_Ico zero_ne_one, zero_le_one]"
},
{
"state_after": "case convert_3.intro\nα : Type ?u.32233\nβ : Type ?u.32236\nγ : Type ?u.32239\nι : Type ?u.32242\ninst✝⁶ : NormedField α\ninst✝⁵ : SeminormedAddCommGroup β\nE : Type u_1\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : NormedSpace α E\nF : Type ?u.32273\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace α F\ninst✝ : NormedSpace ℝ E\nx : E\nr : ℝ\nhr : r ≠ 0\ny : E\nhy : y ∈ closedBall x r\nthis : ContinuousWithinAt (fun c => c • (y - x) + x) (Ico 0 1) 1\nc : ℝ\nhc0 : 0 ≤ c\nhc1 : c < 1\n⊢ (fun c => c • (y - x) + x) c ∈ ball x r",
"state_before": "case convert_3\nα : Type ?u.32233\nβ : Type ?u.32236\nγ : Type ?u.32239\nι : Type ?u.32242\ninst✝⁶ : NormedField α\ninst✝⁵ : SeminormedAddCommGroup β\nE : Type u_1\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : NormedSpace α E\nF : Type ?u.32273\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace α F\ninst✝ : NormedSpace ℝ E\nx : E\nr : ℝ\nhr : r ≠ 0\ny : E\nhy : y ∈ closedBall x r\nthis : ContinuousWithinAt (fun c => c • (y - x) + x) (Ico 0 1) 1\n⊢ MapsTo (fun c => c • (y - x) + x) (Ico 0 1) (ball x r)",
"tactic": "rintro c ⟨hc0, hc1⟩"
},
{
"state_after": "case convert_3.intro\nα : Type ?u.32233\nβ : Type ?u.32236\nγ : Type ?u.32239\nι : Type ?u.32242\ninst✝⁶ : NormedField α\ninst✝⁵ : SeminormedAddCommGroup β\nE : Type u_1\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : NormedSpace α E\nF : Type ?u.32273\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace α F\ninst✝ : NormedSpace ℝ E\nx : E\nr : ℝ\nhr : r ≠ 0\ny : E\nhy : y ∈ closedBall x r\nthis : ContinuousWithinAt (fun c => c • (y - x) + x) (Ico 0 1) 1\nc : ℝ\nhc0 : 0 ≤ c\nhc1 : c < 1\n⊢ ‖y - x‖ * c < r * 1",
"state_before": "case convert_3.intro\nα : Type ?u.32233\nβ : Type ?u.32236\nγ : Type ?u.32239\nι : Type ?u.32242\ninst✝⁶ : NormedField α\ninst✝⁵ : SeminormedAddCommGroup β\nE : Type u_1\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : NormedSpace α E\nF : Type ?u.32273\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace α F\ninst✝ : NormedSpace ℝ E\nx : E\nr : ℝ\nhr : r ≠ 0\ny : E\nhy : y ∈ closedBall x r\nthis : ContinuousWithinAt (fun c => c • (y - x) + x) (Ico 0 1) 1\nc : ℝ\nhc0 : 0 ≤ c\nhc1 : c < 1\n⊢ (fun c => c • (y - x) + x) c ∈ ball x r",
"tactic": "rw [mem_ball, dist_eq_norm, add_sub_cancel, norm_smul, Real.norm_eq_abs, abs_of_nonneg hc0,\n mul_comm, ← mul_one r]"
},
{
"state_after": "case convert_3.intro\nα : Type ?u.32233\nβ : Type ?u.32236\nγ : Type ?u.32239\nι : Type ?u.32242\ninst✝⁶ : NormedField α\ninst✝⁵ : SeminormedAddCommGroup β\nE : Type u_1\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : NormedSpace α E\nF : Type ?u.32273\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace α F\ninst✝ : NormedSpace ℝ E\nx : E\nr : ℝ\nhr : r ≠ 0\ny : E\nhy : ‖y - x‖ ≤ r\nthis : ContinuousWithinAt (fun c => c • (y - x) + x) (Ico 0 1) 1\nc : ℝ\nhc0 : 0 ≤ c\nhc1 : c < 1\n⊢ ‖y - x‖ * c < r * 1",
"state_before": "case convert_3.intro\nα : Type ?u.32233\nβ : Type ?u.32236\nγ : Type ?u.32239\nι : Type ?u.32242\ninst✝⁶ : NormedField α\ninst✝⁵ : SeminormedAddCommGroup β\nE : Type u_1\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : NormedSpace α E\nF : Type ?u.32273\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace α F\ninst✝ : NormedSpace ℝ E\nx : E\nr : ℝ\nhr : r ≠ 0\ny : E\nhy : y ∈ closedBall x r\nthis : ContinuousWithinAt (fun c => c • (y - x) + x) (Ico 0 1) 1\nc : ℝ\nhc0 : 0 ≤ c\nhc1 : c < 1\n⊢ ‖y - x‖ * c < r * 1",
"tactic": "rw [mem_closedBall, dist_eq_norm] at hy"
},
{
"state_after": "case convert_3.intro\nα : Type ?u.32233\nβ : Type ?u.32236\nγ : Type ?u.32239\nι : Type ?u.32242\ninst✝⁶ : NormedField α\ninst✝⁵ : SeminormedAddCommGroup β\nE : Type u_1\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : NormedSpace α E\nF : Type ?u.32273\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace α F\ninst✝ : NormedSpace ℝ E\nx : E\nr : ℝ\ny : E\nhy : ‖y - x‖ ≤ r\nthis : ContinuousWithinAt (fun c => c • (y - x) + x) (Ico 0 1) 1\nc : ℝ\nhc0 : 0 ≤ c\nhc1 : c < 1\nhr : 0 < r\n⊢ ‖y - x‖ * c < r * 1",
"state_before": "case convert_3.intro\nα : Type ?u.32233\nβ : Type ?u.32236\nγ : Type ?u.32239\nι : Type ?u.32242\ninst✝⁶ : NormedField α\ninst✝⁵ : SeminormedAddCommGroup β\nE : Type u_1\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : NormedSpace α E\nF : Type ?u.32273\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace α F\ninst✝ : NormedSpace ℝ E\nx : E\nr : ℝ\nhr : r ≠ 0\ny : E\nhy : ‖y - x‖ ≤ r\nthis : ContinuousWithinAt (fun c => c • (y - x) + x) (Ico 0 1) 1\nc : ℝ\nhc0 : 0 ≤ c\nhc1 : c < 1\n⊢ ‖y - x‖ * c < r * 1",
"tactic": "replace hr : 0 < r := ((norm_nonneg _).trans hy).lt_of_ne hr.symm"
},
{
"state_after": "no goals",
"state_before": "case convert_3.intro\nα : Type ?u.32233\nβ : Type ?u.32236\nγ : Type ?u.32239\nι : Type ?u.32242\ninst✝⁶ : NormedField α\ninst✝⁵ : SeminormedAddCommGroup β\nE : Type u_1\ninst✝⁴ : SeminormedAddCommGroup E\ninst✝³ : NormedSpace α E\nF : Type ?u.32273\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace α F\ninst✝ : NormedSpace ℝ E\nx : E\nr : ℝ\ny : E\nhy : ‖y - x‖ ≤ r\nthis : ContinuousWithinAt (fun c => c • (y - x) + x) (Ico 0 1) 1\nc : ℝ\nhc0 : 0 ≤ c\nhc1 : c < 1\nhr : 0 < r\n⊢ ‖y - x‖ * c < r * 1",
"tactic": "apply mul_lt_mul' <;> assumption"
}
] | [
124,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
111,
1
] |
Mathlib/Order/Directed.lean | Antitone.directed_ge | [] | [
123,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
121,
1
] |
Mathlib/CategoryTheory/DiscreteCategory.lean | CategoryTheory.Discrete.functor_map | [
{
"state_after": "no goals",
"state_before": "α : Type u₁\nC : Type u₂\ninst✝ : Category C\nI : Type u₁\nF : I → C\ni : Discrete I\nf : i ⟶ i\n⊢ (functor F).map f = 𝟙 (F i.as)",
"tactic": "aesop_cat"
}
] | [
190,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
189,
1
] |
Mathlib/Data/Nat/Bits.lean | Nat.binaryRec_eq' | [
{
"state_after": "n✝ : ℕ\nC : ℕ → Sort u_1\nz : C 0\nf : (b : Bool) → (n : ℕ) → C n → C (bit b n)\nb : Bool\nn : ℕ\nh : f false 0 z = z ∨ (n = 0 → b = true)\n⊢ (if n0 : bit b n = 0 then Eq.mpr (_ : C (bit b n) = C 0) z\n else\n let n' := div2 (bit b n);\n let_fun _x := (_ : bit (bodd (bit b n)) (div2 (bit b n)) = bit b n);\n Eq.mpr (_ : C (bit b n) = C (bit (bodd (bit b n)) n')) (f (bodd (bit b n)) n' (binaryRec z f n'))) =\n f b n (binaryRec z f n)",
"state_before": "n✝ : ℕ\nC : ℕ → Sort u_1\nz : C 0\nf : (b : Bool) → (n : ℕ) → C n → C (bit b n)\nb : Bool\nn : ℕ\nh : f false 0 z = z ∨ (n = 0 → b = true)\n⊢ binaryRec z f (bit b n) = f b n (binaryRec z f n)",
"tactic": "rw [binaryRec]"
},
{
"state_after": "case inl\nn✝ : ℕ\nC : ℕ → Sort u_1\nz : C 0\nf : (b : Bool) → (n : ℕ) → C n → C (bit b n)\nb : Bool\nn : ℕ\nh : f false 0 z = z ∨ (n = 0 → b = true)\nh' : bit b n = 0\n⊢ Eq.mpr (_ : C (bit b n) = C 0) z = f b n (binaryRec z f n)\n\ncase inr\nn✝ : ℕ\nC : ℕ → Sort u_1\nz : C 0\nf : (b : Bool) → (n : ℕ) → C n → C (bit b n)\nb : Bool\nn : ℕ\nh : f false 0 z = z ∨ (n = 0 → b = true)\nh' : ¬bit b n = 0\n⊢ (let n' := div2 (bit b n);\n let_fun _x := (_ : bit (bodd (bit b n)) (div2 (bit b n)) = bit b n);\n Eq.mpr (_ : C (bit b n) = C (bit (bodd (bit b n)) n')) (f (bodd (bit b n)) n' (binaryRec z f n'))) =\n f b n (binaryRec z f n)",
"state_before": "n✝ : ℕ\nC : ℕ → Sort u_1\nz : C 0\nf : (b : Bool) → (n : ℕ) → C n → C (bit b n)\nb : Bool\nn : ℕ\nh : f false 0 z = z ∨ (n = 0 → b = true)\n⊢ (if n0 : bit b n = 0 then Eq.mpr (_ : C (bit b n) = C 0) z\n else\n let n' := div2 (bit b n);\n let_fun _x := (_ : bit (bodd (bit b n)) (div2 (bit b n)) = bit b n);\n Eq.mpr (_ : C (bit b n) = C (bit (bodd (bit b n)) n')) (f (bodd (bit b n)) n' (binaryRec z f n'))) =\n f b n (binaryRec z f n)",
"tactic": "split_ifs with h'"
},
{
"state_after": "case inl.intro\nn : ℕ\nC : ℕ → Sort u_1\nz : C 0\nf : (b : Bool) → (n : ℕ) → C n → C (bit b n)\nh : f false 0 z = z ∨ (0 = 0 → false = true)\nh' : bit false 0 = 0\n⊢ Eq.mpr (_ : C (bit false 0) = C 0) z = f false 0 (binaryRec z f 0)",
"state_before": "case inl\nn✝ : ℕ\nC : ℕ → Sort u_1\nz : C 0\nf : (b : Bool) → (n : ℕ) → C n → C (bit b n)\nb : Bool\nn : ℕ\nh : f false 0 z = z ∨ (n = 0 → b = true)\nh' : bit b n = 0\n⊢ Eq.mpr (_ : C (bit b n) = C 0) z = f b n (binaryRec z f n)",
"tactic": "rcases bit_eq_zero_iff.mp h' with ⟨rfl, rfl⟩"
},
{
"state_after": "case inl.intro\nn : ℕ\nC : ℕ → Sort u_1\nz : C 0\nf : (b : Bool) → (n : ℕ) → C n → C (bit b n)\nh : f false 0 z = z ∨ (0 = 0 → false = true)\nh' : bit false 0 = 0\n⊢ Eq.mpr (_ : C (bit false 0) = C 0) z = f false 0 z",
"state_before": "case inl.intro\nn : ℕ\nC : ℕ → Sort u_1\nz : C 0\nf : (b : Bool) → (n : ℕ) → C n → C (bit b n)\nh : f false 0 z = z ∨ (0 = 0 → false = true)\nh' : bit false 0 = 0\n⊢ Eq.mpr (_ : C (bit false 0) = C 0) z = f false 0 (binaryRec z f 0)",
"tactic": "rw [binaryRec_zero]"
},
{
"state_after": "case inl.intro\nn : ℕ\nC : ℕ → Sort u_1\nz : C 0\nf : (b : Bool) → (n : ℕ) → C n → C (bit b n)\nh' : bit false 0 = 0\nh : f false 0 z = z\n⊢ Eq.mpr (_ : C (bit false 0) = C 0) z = f false 0 z",
"state_before": "case inl.intro\nn : ℕ\nC : ℕ → Sort u_1\nz : C 0\nf : (b : Bool) → (n : ℕ) → C n → C (bit b n)\nh : f false 0 z = z ∨ (0 = 0 → false = true)\nh' : bit false 0 = 0\n⊢ Eq.mpr (_ : C (bit false 0) = C 0) z = f false 0 z",
"tactic": "simp only [imp_false, or_false_iff, eq_self_iff_true, not_true] at h"
},
{
"state_after": "no goals",
"state_before": "case inl.intro\nn : ℕ\nC : ℕ → Sort u_1\nz : C 0\nf : (b : Bool) → (n : ℕ) → C n → C (bit b n)\nh' : bit false 0 = 0\nh : f false 0 z = z\n⊢ Eq.mpr (_ : C (bit false 0) = C 0) z = f false 0 z",
"tactic": "exact h.symm"
},
{
"state_after": "case inr\nn✝ : ℕ\nC : ℕ → Sort u_1\nz : C 0\nf : (b : Bool) → (n : ℕ) → C n → C (bit b n)\nb : Bool\nn : ℕ\nh : f false 0 z = z ∨ (n = 0 → b = true)\nh' : ¬bit b n = 0\n⊢ Eq.mpr (_ : C (bit b n) = C (bit (bodd (bit b n)) (div2 (bit b n))))\n (f (bodd (bit b n)) (div2 (bit b n)) (binaryRec z f (div2 (bit b n)))) =\n f b n (binaryRec z f n)",
"state_before": "case inr\nn✝ : ℕ\nC : ℕ → Sort u_1\nz : C 0\nf : (b : Bool) → (n : ℕ) → C n → C (bit b n)\nb : Bool\nn : ℕ\nh : f false 0 z = z ∨ (n = 0 → b = true)\nh' : ¬bit b n = 0\n⊢ (let n' := div2 (bit b n);\n let_fun _x := (_ : bit (bodd (bit b n)) (div2 (bit b n)) = bit b n);\n Eq.mpr (_ : C (bit b n) = C (bit (bodd (bit b n)) n')) (f (bodd (bit b n)) n' (binaryRec z f n'))) =\n f b n (binaryRec z f n)",
"tactic": "dsimp only []"
},
{
"state_after": "case inr\nn✝ : ℕ\nC : ℕ → Sort u_1\nz : C 0\nf : (b : Bool) → (n : ℕ) → C n → C (bit b n)\nb : Bool\nn : ℕ\nh : f false 0 z = z ∨ (n = 0 → b = true)\nh' : ¬bit b n = 0\ne : C (bit b n) = C (bit (bodd (bit b n)) (div2 (bit b n)))\n⊢ Eq.mpr e (f (bodd (bit b n)) (div2 (bit b n)) (binaryRec z f (div2 (bit b n)))) = f b n (binaryRec z f n)",
"state_before": "case inr\nn✝ : ℕ\nC : ℕ → Sort u_1\nz : C 0\nf : (b : Bool) → (n : ℕ) → C n → C (bit b n)\nb : Bool\nn : ℕ\nh : f false 0 z = z ∨ (n = 0 → b = true)\nh' : ¬bit b n = 0\n⊢ Eq.mpr (_ : C (bit b n) = C (bit (bodd (bit b n)) (div2 (bit b n))))\n (f (bodd (bit b n)) (div2 (bit b n)) (binaryRec z f (div2 (bit b n)))) =\n f b n (binaryRec z f n)",
"tactic": "generalize @id (C (bit b n) = C (bit (bodd (bit b n)) (div2 (bit b n))))\n (Eq.symm (bit_decomp (bit b n)) ▸ Eq.refl (C (bit b n))) = e"
},
{
"state_after": "case inr\nn✝ : ℕ\nC : ℕ → Sort u_1\nz : C 0\nf : (b : Bool) → (n : ℕ) → C n → C (bit b n)\nb : Bool\nn : ℕ\nh : f false 0 z = z ∨ (n = 0 → b = true)\nh' : ¬bit b n = 0\n⊢ ∀ (e : C (bit b n) = C (bit (bodd (bit b n)) (div2 (bit b n)))),\n Eq.mpr e (f (bodd (bit b n)) (div2 (bit b n)) (binaryRec z f (div2 (bit b n)))) = f b n (binaryRec z f n)",
"state_before": "case inr\nn✝ : ℕ\nC : ℕ → Sort u_1\nz : C 0\nf : (b : Bool) → (n : ℕ) → C n → C (bit b n)\nb : Bool\nn : ℕ\nh : f false 0 z = z ∨ (n = 0 → b = true)\nh' : ¬bit b n = 0\ne : C (bit b n) = C (bit (bodd (bit b n)) (div2 (bit b n)))\n⊢ Eq.mpr e (f (bodd (bit b n)) (div2 (bit b n)) (binaryRec z f (div2 (bit b n)))) = f b n (binaryRec z f n)",
"tactic": "revert e"
},
{
"state_after": "case inr\nn✝ : ℕ\nC : ℕ → Sort u_1\nz : C 0\nf : (b : Bool) → (n : ℕ) → C n → C (bit b n)\nb : Bool\nn : ℕ\nh : f false 0 z = z ∨ (n = 0 → b = true)\nh' : ¬bit b n = 0\n⊢ ∀ (e : C (bit b n) = C (bit b n)), Eq.mpr e (f b n (binaryRec z f n)) = f b n (binaryRec z f n)",
"state_before": "case inr\nn✝ : ℕ\nC : ℕ → Sort u_1\nz : C 0\nf : (b : Bool) → (n : ℕ) → C n → C (bit b n)\nb : Bool\nn : ℕ\nh : f false 0 z = z ∨ (n = 0 → b = true)\nh' : ¬bit b n = 0\n⊢ ∀ (e : C (bit b n) = C (bit (bodd (bit b n)) (div2 (bit b n)))),\n Eq.mpr e (f (bodd (bit b n)) (div2 (bit b n)) (binaryRec z f (div2 (bit b n)))) = f b n (binaryRec z f n)",
"tactic": "rw [bodd_bit, div2_bit]"
},
{
"state_after": "case inr\nn✝ : ℕ\nC : ℕ → Sort u_1\nz : C 0\nf : (b : Bool) → (n : ℕ) → C n → C (bit b n)\nb : Bool\nn : ℕ\nh : f false 0 z = z ∨ (n = 0 → b = true)\nh' : ¬bit b n = 0\ne✝ : C (bit b n) = C (bit b n)\n⊢ Eq.mpr e✝ (f b n (binaryRec z f n)) = f b n (binaryRec z f n)",
"state_before": "case inr\nn✝ : ℕ\nC : ℕ → Sort u_1\nz : C 0\nf : (b : Bool) → (n : ℕ) → C n → C (bit b n)\nb : Bool\nn : ℕ\nh : f false 0 z = z ∨ (n = 0 → b = true)\nh' : ¬bit b n = 0\n⊢ ∀ (e : C (bit b n) = C (bit b n)), Eq.mpr e (f b n (binaryRec z f n)) = f b n (binaryRec z f n)",
"tactic": "intros"
},
{
"state_after": "no goals",
"state_before": "case inr\nn✝ : ℕ\nC : ℕ → Sort u_1\nz : C 0\nf : (b : Bool) → (n : ℕ) → C n → C (bit b n)\nb : Bool\nn : ℕ\nh : f false 0 z = z ∨ (n = 0 → b = true)\nh' : ¬bit b n = 0\ne✝ : C (bit b n) = C (bit b n)\n⊢ Eq.mpr e✝ (f b n (binaryRec z f n)) = f b n (binaryRec z f n)",
"tactic": "rfl"
}
] | [
185,
8
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
169,
1
] |
Mathlib/Algebra/Module/LinearMap.lean | LinearMap.copy_eq | [] | [
251,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
250,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean | mem_nhds_uniformity_iff_right | [
{
"state_after": "case refine'_1\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.56643\ninst✝ : UniformSpace α\nx : α\ns : Set α\n⊢ s ∈ 𝓝 x → {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α\n\ncase refine'_2\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.56643\ninst✝ : UniformSpace α\nx : α\ns : Set α\nhs : {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α\n⊢ s ∈ 𝓝 x",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.56643\ninst✝ : UniformSpace α\nx : α\ns : Set α\n⊢ s ∈ 𝓝 x ↔ {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α",
"tactic": "refine' ⟨_, fun hs => _⟩"
},
{
"state_after": "case refine'_1\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.56643\ninst✝ : UniformSpace α\nx : α\ns : Set α\n⊢ ∀ (x_1 : Set α),\n x_1 ⊆ s → (∀ (x : α), x ∈ x_1 → {p | p.fst = x → p.snd ∈ x_1} ∈ 𝓤 α) → x ∈ x_1 → {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α",
"state_before": "case refine'_1\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.56643\ninst✝ : UniformSpace α\nx : α\ns : Set α\n⊢ s ∈ 𝓝 x → {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α",
"tactic": "simp only [mem_nhds_iff, isOpen_uniformity, and_imp, exists_imp]"
},
{
"state_after": "case refine'_1\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.56643\ninst✝ : UniformSpace α\nx : α\ns t : Set α\nts : t ⊆ s\nht : ∀ (x : α), x ∈ t → {p | p.fst = x → p.snd ∈ t} ∈ 𝓤 α\nxt : x ∈ t\n⊢ {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α",
"state_before": "case refine'_1\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.56643\ninst✝ : UniformSpace α\nx : α\ns : Set α\n⊢ ∀ (x_1 : Set α),\n x_1 ⊆ s → (∀ (x : α), x ∈ x_1 → {p | p.fst = x → p.snd ∈ x_1} ∈ 𝓤 α) → x ∈ x_1 → {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α",
"tactic": "intro t ts ht xt"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.56643\ninst✝ : UniformSpace α\nx : α\ns t : Set α\nts : t ⊆ s\nht : ∀ (x : α), x ∈ t → {p | p.fst = x → p.snd ∈ t} ∈ 𝓤 α\nxt : x ∈ t\n⊢ {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α",
"tactic": "filter_upwards [ht x xt]using fun y h eq => ts (h eq)"
},
{
"state_after": "case refine'_2.refine'_1\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.56643\ninst✝ : UniformSpace α\nx : α\ns : Set α\nhs : {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α\n⊢ {x | {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α} ⊆ s\n\ncase refine'_2.refine'_2\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.56643\ninst✝ : UniformSpace α\nx : α\ns : Set α\nhs : {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α\n⊢ IsOpen {x | {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α}",
"state_before": "case refine'_2\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.56643\ninst✝ : UniformSpace α\nx : α\ns : Set α\nhs : {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α\n⊢ s ∈ 𝓝 x",
"tactic": "refine' mem_nhds_iff.mpr ⟨{ x | { p : α × α | p.1 = x → p.2 ∈ s } ∈ 𝓤 α }, _, _, hs⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.refine'_1\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.56643\ninst✝ : UniformSpace α\nx : α\ns : Set α\nhs : {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α\n⊢ {x | {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α} ⊆ s",
"tactic": "exact fun y hy => refl_mem_uniformity hy rfl"
},
{
"state_after": "case refine'_2.refine'_2\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.56643\ninst✝ : UniformSpace α\nx : α\ns : Set α\nhs : {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α\ny : α\nhy : y ∈ {x | {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α}\n⊢ {p | p.fst = y → p.snd ∈ {x | {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α}} ∈ 𝓤 α",
"state_before": "case refine'_2.refine'_2\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.56643\ninst✝ : UniformSpace α\nx : α\ns : Set α\nhs : {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α\n⊢ IsOpen {x | {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α}",
"tactic": "refine' isOpen_uniformity.mpr fun y hy => _"
},
{
"state_after": "case refine'_2.refine'_2.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.56643\ninst✝ : UniformSpace α\nx : α\ns : Set α\nhs : {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α\ny : α\nhy : y ∈ {x | {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α}\nt : Set (α × α)\nht : t ∈ 𝓤 α\ntr : t ○ t ⊆ {p | p.fst = y → p.snd ∈ s}\n⊢ {p | p.fst = y → p.snd ∈ {x | {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α}} ∈ 𝓤 α",
"state_before": "case refine'_2.refine'_2\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.56643\ninst✝ : UniformSpace α\nx : α\ns : Set α\nhs : {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α\ny : α\nhy : y ∈ {x | {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α}\n⊢ {p | p.fst = y → p.snd ∈ {x | {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α}} ∈ 𝓤 α",
"tactic": "rcases comp_mem_uniformity_sets hy with ⟨t, ht, tr⟩"
},
{
"state_after": "case h\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.56643\ninst✝ : UniformSpace α\nx : α\ns : Set α\nhs : {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α\ny : α\nhy : y ∈ {x | {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α}\nt : Set (α × α)\nht : t ∈ 𝓤 α\ntr : t ○ t ⊆ {p | p.fst = y → p.snd ∈ s}\n⊢ ∀ (a : α × α), a ∈ t → a.fst = y → {p | p.fst = a.snd → p.snd ∈ s} ∈ 𝓤 α",
"state_before": "case refine'_2.refine'_2.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.56643\ninst✝ : UniformSpace α\nx : α\ns : Set α\nhs : {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α\ny : α\nhy : y ∈ {x | {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α}\nt : Set (α × α)\nht : t ∈ 𝓤 α\ntr : t ○ t ⊆ {p | p.fst = y → p.snd ∈ s}\n⊢ {p | p.fst = y → p.snd ∈ {x | {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α}} ∈ 𝓤 α",
"tactic": "filter_upwards [ht]"
},
{
"state_after": "case h.mk\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.56643\ninst✝ : UniformSpace α\nx : α\ns : Set α\nhs : {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α\nt : Set (α × α)\nht : t ∈ 𝓤 α\na b : α\nhp' : (a, b) ∈ t\nhy : (a, b).fst ∈ {x | {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α}\ntr : t ○ t ⊆ {p | p.fst = (a, b).fst → p.snd ∈ s}\n⊢ {p | p.fst = (a, b).snd → p.snd ∈ s} ∈ 𝓤 α",
"state_before": "case h\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.56643\ninst✝ : UniformSpace α\nx : α\ns : Set α\nhs : {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α\ny : α\nhy : y ∈ {x | {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α}\nt : Set (α × α)\nht : t ∈ 𝓤 α\ntr : t ○ t ⊆ {p | p.fst = y → p.snd ∈ s}\n⊢ ∀ (a : α × α), a ∈ t → a.fst = y → {p | p.fst = a.snd → p.snd ∈ s} ∈ 𝓤 α",
"tactic": "rintro ⟨a, b⟩ hp' rfl"
},
{
"state_after": "case h\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.56643\ninst✝ : UniformSpace α\nx : α\ns : Set α\nhs : {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α\nt : Set (α × α)\nht : t ∈ 𝓤 α\na b : α\nhp' : (a, b) ∈ t\nhy : (a, b).fst ∈ {x | {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α}\ntr : t ○ t ⊆ {p | p.fst = (a, b).fst → p.snd ∈ s}\n⊢ ∀ (a : α × α), a ∈ t → a.fst = b → a.snd ∈ s",
"state_before": "case h.mk\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.56643\ninst✝ : UniformSpace α\nx : α\ns : Set α\nhs : {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α\nt : Set (α × α)\nht : t ∈ 𝓤 α\na b : α\nhp' : (a, b) ∈ t\nhy : (a, b).fst ∈ {x | {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α}\ntr : t ○ t ⊆ {p | p.fst = (a, b).fst → p.snd ∈ s}\n⊢ {p | p.fst = (a, b).snd → p.snd ∈ s} ∈ 𝓤 α",
"tactic": "filter_upwards [ht]"
},
{
"state_after": "case h.mk\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.56643\ninst✝ : UniformSpace α\nx : α\ns : Set α\nhs : {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α\nt : Set (α × α)\nht : t ∈ 𝓤 α\na a' b' : α\nhp'' : (a', b') ∈ t\nhp' : (a, (a', b').fst) ∈ t\nhy : (a, (a', b').fst).fst ∈ {x | {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α}\ntr : t ○ t ⊆ {p | p.fst = (a, (a', b').fst).fst → p.snd ∈ s}\n⊢ (a', b').snd ∈ s",
"state_before": "case h\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.56643\ninst✝ : UniformSpace α\nx : α\ns : Set α\nhs : {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α\nt : Set (α × α)\nht : t ∈ 𝓤 α\na b : α\nhp' : (a, b) ∈ t\nhy : (a, b).fst ∈ {x | {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α}\ntr : t ○ t ⊆ {p | p.fst = (a, b).fst → p.snd ∈ s}\n⊢ ∀ (a : α × α), a ∈ t → a.fst = b → a.snd ∈ s",
"tactic": "rintro ⟨a', b'⟩ hp'' rfl"
},
{
"state_after": "no goals",
"state_before": "case h.mk\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.56643\ninst✝ : UniformSpace α\nx : α\ns : Set α\nhs : {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α\nt : Set (α × α)\nht : t ∈ 𝓤 α\na a' b' : α\nhp'' : (a', b') ∈ t\nhp' : (a, (a', b').fst) ∈ t\nhy : (a, (a', b').fst).fst ∈ {x | {p | p.fst = x → p.snd ∈ s} ∈ 𝓤 α}\ntr : t ○ t ⊆ {p | p.fst = (a, (a', b').fst).fst → p.snd ∈ s}\n⊢ (a', b').snd ∈ s",
"tactic": "exact @tr (a, b') ⟨a', hp', hp''⟩ rfl"
}
] | [
703,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
689,
1
] |
Mathlib/CategoryTheory/Sites/Sheafification.lean | CategoryTheory.Meq.condition | [] | [
72,
8
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
70,
1
] |
Mathlib/LinearAlgebra/Determinant.lean | LinearMap.det_id | [] | [
257,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
256,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean | mul_lt_mul_of_pos_of_pos | [] | [
243,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
241,
1
] |
Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean | FormalMultilinearSeries.ne_iff | [] | [
90,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
89,
11
] |
Mathlib/Data/Sum/Order.lean | OrderIso.sumDualDistrib_symm_inl | [] | [
627,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
626,
1
] |
Mathlib/Algebra/Module/Zlattice.lean | Zspan.repr_ceil_apply | [
{
"state_after": "no goals",
"state_before": "E : Type u_3\nι : Type u_2\nK : Type u_1\ninst✝⁴ : NormedLinearOrderedField K\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace K E\nb : Basis ι K E\ninst✝¹ : FloorRing K\ninst✝ : Fintype ι\nm : E\ni : ι\n⊢ ↑(↑b.repr ↑(ceil b m)) i = ↑⌈↑(↑b.repr m) i⌉",
"tactic": "classical simp only [ceil, zsmul_eq_smul_cast K, b.repr.map_smul, Finsupp.single_apply,\n Finset.sum_apply', Basis.repr_self, Finsupp.smul_single', mul_one, Finset.sum_ite_eq', coe_sum,\n Finset.mem_univ, if_true, coe_smul_of_tower, Basis.restrictScalars_apply, LinearEquiv.map_sum]"
},
{
"state_after": "no goals",
"state_before": "E : Type u_3\nι : Type u_2\nK : Type u_1\ninst✝⁴ : NormedLinearOrderedField K\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace K E\nb : Basis ι K E\ninst✝¹ : FloorRing K\ninst✝ : Fintype ι\nm : E\ni : ι\n⊢ ↑(↑b.repr ↑(ceil b m)) i = ↑⌈↑(↑b.repr m) i⌉",
"tactic": "simp only [ceil, zsmul_eq_smul_cast K, b.repr.map_smul, Finsupp.single_apply,\nFinset.sum_apply', Basis.repr_self, Finsupp.smul_single', mul_one, Finset.sum_ite_eq', coe_sum,\nFinset.mem_univ, if_true, coe_smul_of_tower, Basis.restrictScalars_apply, LinearEquiv.map_sum]"
}
] | [
89,
99
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
86,
1
] |
Mathlib/RingTheory/Derivation/Basic.lean | Derivation.eqOn_adjoin | [
{
"state_after": "no goals",
"state_before": "R : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_1\ninst✝⁴ : CommSemiring A\ninst✝³ : Algebra R A\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : Module A M\ninst✝ : Module R M\nD D1 D2 : Derivation R A M\nr : R\na b : A\ns : Set A\nh : Set.EqOn (↑D1) (↑D2) s\nx✝ : A\nhx✝ : x✝ ∈ ↑(adjoin R s)\nx y : A\nhx : ↑D1 x = ↑D2 x\nhy : ↑D1 y = ↑D2 y\n⊢ ↑D1 (x + y) = ↑D2 (x + y)",
"tactic": "simp only [map_add, *]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_3\ninst✝⁵ : CommSemiring R\nA : Type u_1\ninst✝⁴ : CommSemiring A\ninst✝³ : Algebra R A\nM : Type u_2\ninst✝² : AddCommMonoid M\ninst✝¹ : Module A M\ninst✝ : Module R M\nD D1 D2 : Derivation R A M\nr : R\na b : A\ns : Set A\nh : Set.EqOn (↑D1) (↑D2) s\nx✝ : A\nhx✝ : x✝ ∈ ↑(adjoin R s)\nx y : A\nhx : ↑D1 x = ↑D2 x\nhy : ↑D1 y = ↑D2 y\n⊢ ↑D1 (x * y) = ↑D2 (x * y)",
"tactic": "simp only [leibniz, *]"
}
] | [
171,
92
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
169,
1
] |
Mathlib/Topology/MetricSpace/Completion.lean | UniformSpace.Completion.dist_comm | [
{
"state_after": "case refine'_1\nα : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\nx y : Completion α\n⊢ IsClosed {x | dist x.fst x.snd = dist x.snd x.fst}\n\ncase refine'_2\nα : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\nx y : Completion α\n⊢ ∀ (a b : α), dist (↑α a) (↑α b) = dist (↑α b) (↑α a)",
"state_before": "α : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\nx y : Completion α\n⊢ dist x y = dist y x",
"tactic": "refine' induction_on₂ x y _ _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\nx y : Completion α\n⊢ IsClosed {x | dist x.fst x.snd = dist x.snd x.fst}",
"tactic": "exact isClosed_eq (Completion.continuous_dist continuous_fst continuous_snd)\n (Completion.continuous_dist continuous_snd continuous_fst)"
},
{
"state_after": "case refine'_2\nα : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\nx y : Completion α\na b : α\n⊢ dist (↑α a) (↑α b) = dist (↑α b) (↑α a)",
"state_before": "case refine'_2\nα : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\nx y : Completion α\n⊢ ∀ (a b : α), dist (↑α a) (↑α b) = dist (↑α b) (↑α a)",
"tactic": "intro a b"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\nx y : Completion α\na b : α\n⊢ dist (↑α a) (↑α b) = dist (↑α b) (↑α a)",
"tactic": "rw [Completion.dist_eq, Completion.dist_eq, dist_comm]"
}
] | [
75,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
70,
11
] |
Mathlib/SetTheory/Cardinal/Ordinal.lean | Cardinal.mul_eq_max_of_aleph0_le_left | [
{
"state_after": "case inl\na b : Cardinal\nh : ℵ₀ ≤ a\nh' : b ≠ 0\nhb : ℵ₀ ≤ b\n⊢ a * b = max a b\n\ncase inr\na b : Cardinal\nh : ℵ₀ ≤ a\nh' : b ≠ 0\nhb : b < ℵ₀\n⊢ a * b = max a b",
"state_before": "a b : Cardinal\nh : ℵ₀ ≤ a\nh' : b ≠ 0\n⊢ a * b = max a b",
"tactic": "cases' le_or_lt ℵ₀ b with hb hb"
},
{
"state_after": "case inr\na b : Cardinal\nh : ℵ₀ ≤ a\nh' : b ≠ 0\nhb : b < ℵ₀\n⊢ max a b ≤ a * b",
"state_before": "case inr\na b : Cardinal\nh : ℵ₀ ≤ a\nh' : b ≠ 0\nhb : b < ℵ₀\n⊢ a * b = max a b",
"tactic": "refine' (mul_le_max_of_aleph0_le_left h).antisymm _"
},
{
"state_after": "case inr\na b : Cardinal\nh : ℵ₀ ≤ a\nh' : b ≠ 0\nhb : b < ℵ₀\nthis : b ≤ a\n⊢ max a b ≤ a * b",
"state_before": "case inr\na b : Cardinal\nh : ℵ₀ ≤ a\nh' : b ≠ 0\nhb : b < ℵ₀\n⊢ max a b ≤ a * b",
"tactic": "have : b ≤ a := hb.le.trans h"
},
{
"state_after": "case inr\na b : Cardinal\nh : ℵ₀ ≤ a\nh' : b ≠ 0\nhb : b < ℵ₀\nthis : b ≤ a\n⊢ a ≤ a * b",
"state_before": "case inr\na b : Cardinal\nh : ℵ₀ ≤ a\nh' : b ≠ 0\nhb : b < ℵ₀\nthis : b ≤ a\n⊢ max a b ≤ a * b",
"tactic": "rw [max_eq_left this]"
},
{
"state_after": "case h.e'_3\na b : Cardinal\nh : ℵ₀ ≤ a\nh' : b ≠ 0\nhb : b < ℵ₀\nthis : b ≤ a\n⊢ a = a * 1",
"state_before": "case inr\na b : Cardinal\nh : ℵ₀ ≤ a\nh' : b ≠ 0\nhb : b < ℵ₀\nthis : b ≤ a\n⊢ a ≤ a * b",
"tactic": "convert mul_le_mul_left' (one_le_iff_ne_zero.mpr h') a"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\na b : Cardinal\nh : ℵ₀ ≤ a\nh' : b ≠ 0\nhb : b < ℵ₀\nthis : b ≤ a\n⊢ a = a * 1",
"tactic": "rw [mul_one]"
},
{
"state_after": "no goals",
"state_before": "case inl\na b : Cardinal\nh : ℵ₀ ≤ a\nh' : b ≠ 0\nhb : ℵ₀ ≤ b\n⊢ a * b = max a b",
"tactic": "exact mul_eq_max h hb"
}
] | [
620,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
612,
1
] |
Mathlib/Order/Bounds/Basic.lean | upperBounds_Ico | [] | [
780,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
779,
1
] |
Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean | div_helper | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.13686\nM₀ : Type ?u.13689\nG₀ : Type u_1\nM₀' : Type ?u.13695\nG₀' : Type ?u.13698\nF : Type ?u.13701\nF' : Type ?u.13704\ninst✝¹ : MonoidWithZero M₀\ninst✝ : CommGroupWithZero G₀\na b✝ c d b : G₀\nh : a ≠ 0\n⊢ 1 / (a * b) * a = 1 / b",
"tactic": "rw [div_mul_eq_mul_div, one_mul, div_mul_right _ h]"
}
] | [
202,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
201,
1
] |
Mathlib/Data/ENat/Basic.lean | ENat.toNat_add | [
{
"state_after": "case intro\nm✝ n✝ n : ℕ∞\nhn : n ≠ ⊤\nm : ℕ\n⊢ ↑toNat (↑m + n) = ↑toNat ↑m + ↑toNat n",
"state_before": "m✝ n✝ m n : ℕ∞\nhm : m ≠ ⊤\nhn : n ≠ ⊤\n⊢ ↑toNat (m + n) = ↑toNat m + ↑toNat n",
"tactic": "lift m to ℕ using hm"
},
{
"state_after": "case intro.intro\nm✝ n✝ : ℕ∞\nm n : ℕ\n⊢ ↑toNat (↑m + ↑n) = ↑toNat ↑m + ↑toNat ↑n",
"state_before": "case intro\nm✝ n✝ n : ℕ∞\nhn : n ≠ ⊤\nm : ℕ\n⊢ ↑toNat (↑m + n) = ↑toNat ↑m + ↑toNat n",
"tactic": "lift n to ℕ using hn"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nm✝ n✝ : ℕ∞\nm n : ℕ\n⊢ ↑toNat (↑m + ↑n) = ↑toNat ↑m + ↑toNat ↑n",
"tactic": "rfl"
}
] | [
170,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
167,
1
] |
Mathlib/Analysis/Calculus/FDerivAnalytic.lean | HasFPowerSeriesOnBall.differentiableOn | [] | [
71,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
69,
1
] |
Mathlib/GroupTheory/FreeAbelianGroup.lean | FreeAbelianGroup.mul_def | [] | [
413,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
411,
1
] |
Mathlib/Computability/TuringMachine.lean | Turing.Reaches₀.tail | [] | [
802,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
800,
1
] |
Mathlib/Topology/Constructions.lean | Filter.Eventually.prod_nhds | [] | [
582,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
580,
1
] |
Mathlib/Algebra/Module/LinearMap.lean | IsLinearMap.isLinearMap_smul' | [] | [
701,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
699,
1
] |
Mathlib/Order/Heyting/Hom.lean | BiheytingHom.id_apply | [] | [
560,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
559,
1
] |
Mathlib/Order/WithBot.lean | WithTop.ofDual_lt_ofDual_iff | [] | [
911,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
910,
1
] |
Mathlib/MeasureTheory/Function/SimpleFuncDenseLp.lean | MeasureTheory.SimpleFunc.exists_forall_norm_le | [] | [
294,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
293,
1
] |
Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean | IsBoundedBilinearMap.isBoundedLinearMap_left | [] | [
433,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
431,
1
] |
Mathlib/Topology/Category/TopCat/Limits/Products.lean | TopCat.sigmaIsoSigma_hom_ι_apply | [] | [
133,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
131,
1
] |
Mathlib/MeasureTheory/Measure/VectorMeasure.lean | MeasureTheory.Measure.toENNRealVectorMeasure_zero | [
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.147666\nm : MeasurableSpace α\ni : Set α\n⊢ MeasurableSet i → ↑(toENNRealVectorMeasure 0) i = ↑0 i",
"state_before": "α : Type u_1\nβ : Type ?u.147666\nm : MeasurableSpace α\n⊢ toENNRealVectorMeasure 0 = 0",
"tactic": "ext i"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.147666\nm : MeasurableSpace α\ni : Set α\n⊢ MeasurableSet i → ↑(toENNRealVectorMeasure 0) i = ↑0 i",
"tactic": "simp"
}
] | [
508,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
506,
1
] |
Mathlib/Data/Finset/Image.lean | Finset.map_disjUnion' | [] | [
209,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
207,
1
] |
Mathlib/Topology/Basic.lean | DenseRange.subset_closure_image_preimage_of_isOpen | [
{
"state_after": "α : Type ?u.175524\nβ : Type u_1\nγ : Type ?u.175530\nδ : Type ?u.175533\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nκ : Type u_2\nι : Type ?u.175548\nf : κ → β\ng : β → γ\nhf : DenseRange f\ns : Set β\nhs : IsOpen s\n⊢ s ⊆ closure (s ∩ range f)",
"state_before": "α : Type ?u.175524\nβ : Type u_1\nγ : Type ?u.175530\nδ : Type ?u.175533\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nκ : Type u_2\nι : Type ?u.175548\nf : κ → β\ng : β → γ\nhf : DenseRange f\ns : Set β\nhs : IsOpen s\n⊢ s ⊆ closure (f '' (f ⁻¹' s))",
"tactic": "rw [image_preimage_eq_inter_range]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.175524\nβ : Type u_1\nγ : Type ?u.175530\nδ : Type ?u.175533\ninst✝² : TopologicalSpace α\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\nκ : Type u_2\nι : Type ?u.175548\nf : κ → β\ng : β → γ\nhf : DenseRange f\ns : Set β\nhs : IsOpen s\n⊢ s ⊆ closure (s ∩ range f)",
"tactic": "exact hf.open_subset_closure_inter hs"
}
] | [
1835,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1832,
1
] |
Mathlib/Topology/Basic.lean | mem_closure_iff_nhds' | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns✝ s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\ns : Set α\na : α\n⊢ a ∈ closure s ↔ ∀ (t : Set α), t ∈ 𝓝 a → ∃ y, ↑y ∈ t",
"tactic": "simp only [mem_closure_iff_nhds, Set.inter_nonempty_iff_exists_right, SetCoe.exists, exists_prop]"
}
] | [
1337,
100
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1336,
1
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Mathlib/Algebra/EuclideanDomain/Basic.lean | EuclideanDomain.gcd_mul_lcm | [
{
"state_after": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx y : R\n⊢ gcd x y * (x * y / gcd x y) = x * y",
"state_before": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx y : R\n⊢ gcd x y * lcm x y = x * y",
"tactic": "rw [lcm]"
},
{
"state_after": "case pos\nR : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx y : R\nh : gcd x y = 0\n⊢ gcd x y * (x * y / gcd x y) = x * y\n\ncase neg\nR : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx y : R\nh : ¬gcd x y = 0\n⊢ gcd x y * (x * y / gcd x y) = x * y",
"state_before": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx y : R\n⊢ gcd x y * (x * y / gcd x y) = x * y",
"tactic": "by_cases h : gcd x y = 0"
},
{
"state_after": "case neg.intro.intro.intro\nR : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx y : R\nh : ¬gcd x y = 0\nr : R\nhr : x = gcd x y * r\ns : R\nhs : y = gcd x y * s\n⊢ gcd x y * (x * y / gcd x y) = x * y",
"state_before": "case neg\nR : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx y : R\nh : ¬gcd x y = 0\n⊢ gcd x y * (x * y / gcd x y) = x * y",
"tactic": "rcases gcd_dvd x y with ⟨⟨r, hr⟩, ⟨s, hs⟩⟩"
},
{
"state_after": "case neg.intro.intro.intro\nR : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx y r s : R\nhs : y = gcd x y * s\ng : R\nh : ¬g = 0\nhr : x = g * r\n⊢ g * (x * y / g) = x * y",
"state_before": "case neg.intro.intro.intro\nR : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx y : R\nh : ¬gcd x y = 0\nr : R\nhr : x = gcd x y * r\ns : R\nhs : y = gcd x y * s\n⊢ gcd x y * (x * y / gcd x y) = x * y",
"tactic": "generalize gcd x y = g at h hr⊢"
},
{
"state_after": "case neg.intro.intro.intro\nR : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\ny r s g : R\nh : ¬g = 0\nhs : y = gcd (g * r) y * s\n⊢ g * (g * r * y / g) = g * r * y",
"state_before": "case neg.intro.intro.intro\nR : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx y r s : R\nhs : y = gcd x y * s\ng : R\nh : ¬g = 0\nhr : x = g * r\n⊢ g * (x * y / g) = x * y",
"tactic": "subst hr"
},
{
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"tactic": "rw [mul_assoc, mul_div_cancel_left _ h]"
},
{
"state_after": "case pos\nR : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx y : R\nh : gcd x y = 0\n⊢ 0 = x * y",
"state_before": "case pos\nR : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx y : R\nh : gcd x y = 0\n⊢ gcd x y * (x * y / gcd x y) = x * y",
"tactic": "rw [h, zero_mul]"
},
{
"state_after": "case pos\nR : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx y : R\nh : x = 0 ∧ y = 0\n⊢ 0 = x * y",
"state_before": "case pos\nR : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx y : R\nh : gcd x y = 0\n⊢ 0 = x * y",
"tactic": "rw [EuclideanDomain.gcd_eq_zero_iff] at h"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx y : R\nh : x = 0 ∧ y = 0\n⊢ 0 = x * y",
"tactic": "rw [h.1, zero_mul]"
}
] | [
336,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
329,
1
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Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean | Matrix.Nondegenerate.toLinearMap₂' | [
{
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"tactic": "simpa only [toLinearMap₂'_apply'] using hx y"
}
] | [
693,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
691,
1
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Mathlib/Topology/Sets/Closeds.lean | TopologicalSpace.Closeds.coe_sInf | [] | [
143,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
142,
1
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Mathlib/GroupTheory/MonoidLocalization.lean | Submonoid.LocalizationMap.lift_unique | [
{
"state_after": "case h\nM : Type u_3\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_1\ninst✝¹ : CommMonoid N\nP : Type u_2\ninst✝ : CommMonoid P\nf : LocalizationMap S N\ng : M →* P\nhg : ∀ (y : { x // x ∈ S }), IsUnit (↑g ↑y)\nj : N →* P\nhj : ∀ (x : M), ↑j (↑(toMap f) x) = ↑g x\nx✝ : N\n⊢ ↑(lift f hg) x✝ = ↑j x✝",
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"tactic": "ext"
},
{
"state_after": "case h\nM : Type u_3\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_1\ninst✝¹ : CommMonoid N\nP : Type u_2\ninst✝ : CommMonoid P\nf : LocalizationMap S N\ng : M →* P\nhg : ∀ (y : { x // x ∈ S }), IsUnit (↑g ↑y)\nj : N →* P\nhj : ∀ (x : M), ↑j (↑(toMap f) x) = ↑g x\nx✝ : N\n⊢ ↑j (↑(toMap f) (sec f x✝).fst) = ↑j (↑(toMap f) ↑(sec f x✝).snd * x✝)",
"state_before": "case h\nM : Type u_3\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_1\ninst✝¹ : CommMonoid N\nP : Type u_2\ninst✝ : CommMonoid P\nf : LocalizationMap S N\ng : M →* P\nhg : ∀ (y : { x // x ∈ S }), IsUnit (↑g ↑y)\nj : N →* P\nhj : ∀ (x : M), ↑j (↑(toMap f) x) = ↑g x\nx✝ : N\n⊢ ↑(lift f hg) x✝ = ↑j x✝",
"tactic": "rw [lift_spec, ← hj, ← hj, ← j.map_mul]"
},
{
"state_after": "case h.h\nM : Type u_3\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_1\ninst✝¹ : CommMonoid N\nP : Type u_2\ninst✝ : CommMonoid P\nf : LocalizationMap S N\ng : M →* P\nhg : ∀ (y : { x // x ∈ S }), IsUnit (↑g ↑y)\nj : N →* P\nhj : ∀ (x : M), ↑j (↑(toMap f) x) = ↑g x\nx✝ : N\n⊢ ↑(toMap f) (sec f x✝).fst = ↑(toMap f) ↑(sec f x✝).snd * x✝",
"state_before": "case h\nM : Type u_3\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_1\ninst✝¹ : CommMonoid N\nP : Type u_2\ninst✝ : CommMonoid P\nf : LocalizationMap S N\ng : M →* P\nhg : ∀ (y : { x // x ∈ S }), IsUnit (↑g ↑y)\nj : N →* P\nhj : ∀ (x : M), ↑j (↑(toMap f) x) = ↑g x\nx✝ : N\n⊢ ↑j (↑(toMap f) (sec f x✝).fst) = ↑j (↑(toMap f) ↑(sec f x✝).snd * x✝)",
"tactic": "apply congr_arg"
},
{
"state_after": "no goals",
"state_before": "case h.h\nM : Type u_3\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_1\ninst✝¹ : CommMonoid N\nP : Type u_2\ninst✝ : CommMonoid P\nf : LocalizationMap S N\ng : M →* P\nhg : ∀ (y : { x // x ∈ S }), IsUnit (↑g ↑y)\nj : N →* P\nhj : ∀ (x : M), ↑j (↑(toMap f) x) = ↑g x\nx✝ : N\n⊢ ↑(toMap f) (sec f x✝).fst = ↑(toMap f) ↑(sec f x✝).snd * x✝",
"tactic": "rw [← sec_spec']"
}
] | [
1049,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1045,
1
] |
Mathlib/Data/IsROrC/Basic.lean | IsROrC.conj_smul | [
{
"state_after": "no goals",
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"tactic": "rw [conj_eq_re_sub_im, conj_eq_re_sub_im, smul_re, smul_im, ofReal_mul, ofReal_mul,\n real_smul_eq_coe_mul r (_ - _), mul_sub, mul_assoc]"
}
] | [
391,
56
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
389,
1
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Mathlib/Data/Multiset/LocallyFinite.lean | Multiset.Ico_sub_Ico_right | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : LocallyFiniteOrder α\na✝ b✝ c✝ d a b c : α\n⊢ Ico a b - Ico c b = Ico a (min b c)",
"tactic": "rw [Ico, Ico, Ico, ← Finset.sdiff_val, Finset.Ico_diff_Ico_right]"
}
] | [
280,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
279,
1
] |
Mathlib/Init/Data/Nat/Lemmas.lean | Nat.bit1_succ_eq | [] | [
86,
82
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
85,
11
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Mathlib/Topology/Homotopy/Basic.lean | ContinuousMap.HomotopyWith.coe_toHomotopy | [] | [
470,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
469,
1
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Mathlib/Analysis/Quaternion.lean | Quaternion.continuous_re | [] | [
209,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
208,
1
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Mathlib/MeasureTheory/Integral/Bochner.lean | MeasureTheory.integral_sub | [
{
"state_after": "α : Type u_2\nE : Type u_1\nF : Type ?u.913636\n𝕜 : Type ?u.913639\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : SMulCommClass ℝ 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf g : α → E\nm : MeasurableSpace α\nμ : Measure α\nhf : Integrable f\nhg : Integrable g\n⊢ (if hf : Integrable fun a => f a - g a then ↑L1.integralCLM (Integrable.toL1 (fun a => f a - g a) hf) else 0) =\n (if hf : Integrable fun a => f a then ↑L1.integralCLM (Integrable.toL1 (fun a => f a) hf) else 0) -\n if hf : Integrable fun a => g a then ↑L1.integralCLM (Integrable.toL1 (fun a => g a) hf) else 0",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.913636\n𝕜 : Type ?u.913639\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : SMulCommClass ℝ 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf g : α → E\nm : MeasurableSpace α\nμ : Measure α\nhf : Integrable f\nhg : Integrable g\n⊢ (∫ (a : α), f a - g a ∂μ) = (∫ (a : α), f a ∂μ) - ∫ (a : α), g a ∂μ",
"tactic": "simp only [integral, L1.integral]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.913636\n𝕜 : Type ?u.913639\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : CompleteSpace E\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : SMulCommClass ℝ 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\nf g : α → E\nm : MeasurableSpace α\nμ : Measure α\nhf : Integrable f\nhg : Integrable g\n⊢ (if hf : Integrable fun a => f a - g a then ↑L1.integralCLM (Integrable.toL1 (fun a => f a - g a) hf) else 0) =\n (if hf : Integrable fun a => f a then ↑L1.integralCLM (Integrable.toL1 (fun a => f a) hf) else 0) -\n if hf : Integrable fun a => g a then ↑L1.integralCLM (Integrable.toL1 (fun a => g a) hf) else 0",
"tactic": "exact setToFun_sub (dominatedFinMeasAdditive_weightedSMul μ) hf hg"
}
] | [
890,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
887,
1
] |
Mathlib/Order/Monotone/Basic.lean | StrictMono.maximal_of_maximal_image | [] | [
855,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
853,
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Mathlib/Data/Polynomial/Basic.lean | Polynomial.support_add | [
{
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"state_before": "R : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q : R[X]\n⊢ support (p + q) ⊆ support p ∪ support q",
"tactic": "rcases p with ⟨⟩"
},
{
"state_after": "case ofFinsupp.ofFinsupp\nR : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\ntoFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ support ({ toFinsupp := toFinsupp✝¹ } + { toFinsupp := toFinsupp✝ }) ⊆\n support { toFinsupp := toFinsupp✝¹ } ∪ support { toFinsupp := toFinsupp✝ }",
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"tactic": "rcases q with ⟨⟩"
},
{
"state_after": "case ofFinsupp.ofFinsupp\nR : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\ntoFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ (toFinsupp✝¹ + toFinsupp✝).support ⊆ toFinsupp✝¹.support ∪ toFinsupp✝.support",
"state_before": "case ofFinsupp.ofFinsupp\nR : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\ntoFinsupp✝¹ toFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ support ({ toFinsupp := toFinsupp✝¹ } + { toFinsupp := toFinsupp✝ }) ⊆\n support { toFinsupp := toFinsupp✝¹ } ∪ support { toFinsupp := toFinsupp✝ }",
"tactic": "simp only [← ofFinsupp_add, support]"
},
{
"state_after": "no goals",
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"tactic": "exact support_add"
}
] | [
476,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
473,
1
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Mathlib/Analysis/Asymptotics/Asymptotics.lean | summable_of_isBigO | [] | [
2162,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2159,
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Mathlib/CategoryTheory/Abelian/Images.lean | CategoryTheory.Abelian.coimage_image_factorisation | [
{
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"tactic": "simp [coimageImageComparison]"
}
] | [
129,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
128,
1
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Mathlib/GroupTheory/Subsemigroup/Operations.lean | Subsemigroup.map_surjective_of_surjective | [] | [
465,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
464,
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Mathlib/GroupTheory/SemidirectProduct.lean | SemidirectProduct.hom_ext | [
{
"state_after": "N : Type u_2\nG : Type u_1\nH : Type u_3\ninst✝² : Group N\ninst✝¹ : Group G\ninst✝ : Group H\nφ : G →* MulAut N\nf₁ : N →* H\nf₂ : G →* H\nh :\n ∀ (g : G),\n MonoidHom.comp f₁ (MulEquiv.toMonoidHom (↑φ g)) = MonoidHom.comp (MulEquiv.toMonoidHom (↑MulAut.conj (↑f₂ g))) f₁\nf g : N ⋊[φ] G →* H\nhl : MonoidHom.comp f inl = MonoidHom.comp g inl\nhr : MonoidHom.comp f inr = MonoidHom.comp g inr\n⊢ lift (MonoidHom.comp f inl) (MonoidHom.comp f inr)\n (_ :\n ∀ (x : G),\n MonoidHom.comp (MonoidHom.comp f inl) (MulEquiv.toMonoidHom (↑φ x)) =\n MonoidHom.comp (MulEquiv.toMonoidHom (↑MulAut.conj (↑(MonoidHom.comp f inr) x))) (MonoidHom.comp f inl)) =\n lift (MonoidHom.comp g inl) (MonoidHom.comp g inr)\n (_ :\n ∀ (x : G),\n MonoidHom.comp (MonoidHom.comp g inl) (MulEquiv.toMonoidHom (↑φ x)) =\n MonoidHom.comp (MulEquiv.toMonoidHom (↑MulAut.conj (↑(MonoidHom.comp g inr) x))) (MonoidHom.comp g inl))",
"state_before": "N : Type u_2\nG : Type u_1\nH : Type u_3\ninst✝² : Group N\ninst✝¹ : Group G\ninst✝ : Group H\nφ : G →* MulAut N\nf₁ : N →* H\nf₂ : G →* H\nh :\n ∀ (g : G),\n MonoidHom.comp f₁ (MulEquiv.toMonoidHom (↑φ g)) = MonoidHom.comp (MulEquiv.toMonoidHom (↑MulAut.conj (↑f₂ g))) f₁\nf g : N ⋊[φ] G →* H\nhl : MonoidHom.comp f inl = MonoidHom.comp g inl\nhr : MonoidHom.comp f inr = MonoidHom.comp g inr\n⊢ f = g",
"tactic": "rw [lift_unique f, lift_unique g]"
},
{
"state_after": "no goals",
"state_before": "N : Type u_2\nG : Type u_1\nH : Type u_3\ninst✝² : Group N\ninst✝¹ : Group G\ninst✝ : Group H\nφ : G →* MulAut N\nf₁ : N →* H\nf₂ : G →* H\nh :\n ∀ (g : G),\n MonoidHom.comp f₁ (MulEquiv.toMonoidHom (↑φ g)) = MonoidHom.comp (MulEquiv.toMonoidHom (↑MulAut.conj (↑f₂ g))) f₁\nf g : N ⋊[φ] G →* H\nhl : MonoidHom.comp f inl = MonoidHom.comp g inl\nhr : MonoidHom.comp f inr = MonoidHom.comp g inr\n⊢ lift (MonoidHom.comp f inl) (MonoidHom.comp f inr)\n (_ :\n ∀ (x : G),\n MonoidHom.comp (MonoidHom.comp f inl) (MulEquiv.toMonoidHom (↑φ x)) =\n MonoidHom.comp (MulEquiv.toMonoidHom (↑MulAut.conj (↑(MonoidHom.comp f inr) x))) (MonoidHom.comp f inl)) =\n lift (MonoidHom.comp g inl) (MonoidHom.comp g inr)\n (_ :\n ∀ (x : G),\n MonoidHom.comp (MonoidHom.comp g inl) (MulEquiv.toMonoidHom (↑φ x)) =\n MonoidHom.comp (MulEquiv.toMonoidHom (↑MulAut.conj (↑(MonoidHom.comp g inr) x))) (MonoidHom.comp g inl))",
"tactic": "simp only [*]"
}
] | [
259,
16
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
256,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean | Complex.tendsto_arg_nhdsWithin_im_nonneg_of_re_neg_of_im_zero | [
{
"state_after": "no goals",
"state_before": "x z✝ z : ℂ\nhre : z.re < 0\nhim : z.im = 0\n⊢ Tendsto arg (𝓝[{z | 0 ≤ z.im}] z) (𝓝 π)",
"tactic": "simpa only [arg_eq_pi_iff.2 ⟨hre, him⟩] using\n (continuousWithinAt_arg_of_re_neg_of_im_zero hre him).tendsto"
}
] | [
603,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
600,
1
] |
Mathlib/Topology/Instances/ENNReal.lean | ENNReal.add_biSup' | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.131868\nβ : Type ?u.131871\nγ : Type ?u.131874\na b c d : ℝ≥0∞\nr p✝ q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nι : Sort u_1\np : ι → Prop\nh : ∃ i, p i\nf : ι → ℝ≥0∞\n⊢ (a + ⨆ (i : ι) (_ : p i), f i) = ⨆ (i : ι) (_ : p i), a + f i",
"tactic": "simp only [add_comm a, biSup_add' h]"
}
] | [
576,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
574,
1
] |
Mathlib/LinearAlgebra/Prod.lean | LinearMap.comap_prod_prod | [] | [
480,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
478,
1
] |
Mathlib/Algebra/Bounds.lean | isLUB_inv | [] | [
60,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
59,
1
] |
Mathlib/Order/Filter/Ultrafilter.lean | Ultrafilter.union_mem_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.11570\nf g : Ultrafilter α\ns t : Set α\np q : α → Prop\n⊢ s ∪ t ∈ f ↔ s ∈ f ∨ t ∈ f",
"tactic": "simp only [← mem_coe, ← le_principal_iff, ← sup_principal, le_sup_iff]"
}
] | [
179,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
178,
1
] |
Mathlib/Data/Set/Image.lean | Subtype.preimage_coe_nonempty | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ns t : Set α\n⊢ Set.Nonempty (val ⁻¹' t) ↔ Set.Nonempty (s ∩ t)",
"tactic": "rw [inter_comm, ← image_preimage_coe, nonempty_image_iff]"
}
] | [
1467,
63
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1465,
1
] |
Mathlib/Data/Set/Basic.lean | Set.mem_of_mem_inter_left | [] | [
903,
9
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
902,
1
] |
Mathlib/Algebra/Group/Semiconj.lean | SemiconjBy.units_inv_right_iff | [] | [
131,
37
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
130,
1
] |
Mathlib/Combinatorics/SimpleGraph/AdjMatrix.lean | Matrix.IsAdjMatrix.toGraph_compl_eq | [
{
"state_after": "case Adj.h.h.a\nV : Type u_2\nα : Type u_1\nβ : Type ?u.18379\ninst✝³ : DecidableEq α\ninst✝² : DecidableEq V\nA : Matrix V V α\ninst✝¹ : MulZeroOneClass α\ninst✝ : Nontrivial α\nh : IsAdjMatrix A\nv w : V\n⊢ Adj (toGraph (_ : IsAdjMatrix (Matrix.compl A))) v w ↔ Adj (toGraph hᶜ) v w",
"state_before": "V : Type u_2\nα : Type u_1\nβ : Type ?u.18379\ninst✝³ : DecidableEq α\ninst✝² : DecidableEq V\nA : Matrix V V α\ninst✝¹ : MulZeroOneClass α\ninst✝ : Nontrivial α\nh : IsAdjMatrix A\n⊢ toGraph (_ : IsAdjMatrix (Matrix.compl A)) = toGraph hᶜ",
"tactic": "ext (v w)"
},
{
"state_after": "no goals",
"state_before": "case Adj.h.h.a\nV : Type u_2\nα : Type u_1\nβ : Type ?u.18379\ninst✝³ : DecidableEq α\ninst✝² : DecidableEq V\nA : Matrix V V α\ninst✝¹ : MulZeroOneClass α\ninst✝ : Nontrivial α\nh : IsAdjMatrix A\nv w : V\n⊢ Adj (toGraph (_ : IsAdjMatrix (Matrix.compl A))) v w ↔ Adj (toGraph hᶜ) v w",
"tactic": "cases' h.zero_or_one v w with h h <;> by_cases hvw : v = w <;> simp [Matrix.compl, h, hvw]"
}
] | [
140,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
137,
1
] |
Mathlib/Order/Filter/Archimedean.lean | Filter.Tendsto.const_mul_atTop' | [
{
"state_after": "α : Type u_2\nR : Type u_1\nl : Filter α\nf : α → R\nr : R\ninst✝¹ : LinearOrderedSemiring R\ninst✝ : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\n⊢ ∀ᶠ (a : α) in l, b ≤ r * f a",
"state_before": "α : Type u_2\nR : Type u_1\nl : Filter α\nf : α → R\nr : R\ninst✝¹ : LinearOrderedSemiring R\ninst✝ : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\n⊢ Tendsto (fun x => r * f x) l atTop",
"tactic": "refine' tendsto_atTop.2 fun b => _"
},
{
"state_after": "case intro\nα : Type u_2\nR : Type u_1\nl : Filter α\nf : α → R\nr : R\ninst✝¹ : LinearOrderedSemiring R\ninst✝ : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\nn : ℕ\nhn : 1 ≤ n • r\n⊢ ∀ᶠ (a : α) in l, b ≤ r * f a",
"state_before": "α : Type u_2\nR : Type u_1\nl : Filter α\nf : α → R\nr : R\ninst✝¹ : LinearOrderedSemiring R\ninst✝ : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\n⊢ ∀ᶠ (a : α) in l, b ≤ r * f a",
"tactic": "obtain ⟨n : ℕ, hn : 1 ≤ n • r⟩ := Archimedean.arch 1 hr"
},
{
"state_after": "case intro\nα : Type u_2\nR : Type u_1\nl : Filter α\nf : α → R\nr : R\ninst✝¹ : LinearOrderedSemiring R\ninst✝ : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\nn : ℕ\nhn : 1 ≤ r * ↑n\n⊢ ∀ᶠ (a : α) in l, b ≤ r * f a",
"state_before": "case intro\nα : Type u_2\nR : Type u_1\nl : Filter α\nf : α → R\nr : R\ninst✝¹ : LinearOrderedSemiring R\ninst✝ : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\nn : ℕ\nhn : 1 ≤ n • r\n⊢ ∀ᶠ (a : α) in l, b ≤ r * f a",
"tactic": "rw [nsmul_eq_mul'] at hn"
},
{
"state_after": "case h\nα : Type u_2\nR : Type u_1\nl : Filter α\nf : α → R\nr : R\ninst✝¹ : LinearOrderedSemiring R\ninst✝ : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\nn : ℕ\nhn : 1 ≤ r * ↑n\nx : α\nhx : ↑n * max b 0 ≤ f x\n⊢ b ≤ r * f x",
"state_before": "case intro\nα : Type u_2\nR : Type u_1\nl : Filter α\nf : α → R\nr : R\ninst✝¹ : LinearOrderedSemiring R\ninst✝ : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\nn : ℕ\nhn : 1 ≤ r * ↑n\n⊢ ∀ᶠ (a : α) in l, b ≤ r * f a",
"tactic": "filter_upwards [tendsto_atTop.1 hf (n * max b 0)]with x hx"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_2\nR : Type u_1\nl : Filter α\nf : α → R\nr : R\ninst✝¹ : LinearOrderedSemiring R\ninst✝ : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\nn : ℕ\nhn : 1 ≤ r * ↑n\nx : α\nhx : ↑n * max b 0 ≤ f x\n⊢ b ≤ r * f x",
"tactic": "calc\n b ≤ 1 * max b 0 := by\n { rw [one_mul]\n exact le_max_left _ _ }\n _ ≤ r * n * max b 0 := by gcongr\n _ = r * (n * max b 0) := by rw [mul_assoc]\n _ ≤ r * f x := by gcongr"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nR : Type u_1\nl : Filter α\nf : α → R\nr : R\ninst✝¹ : LinearOrderedSemiring R\ninst✝ : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\nn : ℕ\nhn : 1 ≤ r * ↑n\nx : α\nhx : ↑n * max b 0 ≤ f x\n⊢ 1 * max b 0 ≤ r * ↑n * max b 0",
"tactic": "gcongr"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nR : Type u_1\nl : Filter α\nf : α → R\nr : R\ninst✝¹ : LinearOrderedSemiring R\ninst✝ : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\nn : ℕ\nhn : 1 ≤ r * ↑n\nx : α\nhx : ↑n * max b 0 ≤ f x\n⊢ r * ↑n * max b 0 = r * (↑n * max b 0)",
"tactic": "rw [mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nR : Type u_1\nl : Filter α\nf : α → R\nr : R\ninst✝¹ : LinearOrderedSemiring R\ninst✝ : Archimedean R\nhr : 0 < r\nhf : Tendsto f l atTop\nb : R\nn : ℕ\nhn : 1 ≤ r * ↑n\nx : α\nhx : ↑n * max b 0 ≤ f x\n⊢ r * (↑n * max b 0) ≤ r * f x",
"tactic": "gcongr"
}
] | [
158,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
146,
1
] |
Mathlib/CategoryTheory/Closed/Cartesian.lean | CategoryTheory.initial_mono | [
{
"state_after": "C : Type u\ninst✝³ : Category C\nA B✝ X X' Y Y' Z : C\ninst✝² : HasFiniteProducts C\ninst✝¹ : Exponentiable A\nI B : C\nt : IsInitial I\ninst✝ : CartesianClosed C\nZ✝ : C\ng h : Z✝ ⟶ I\nx✝ : g ≫ IsInitial.to t B = h ≫ IsInitial.to t B\nthis : IsIso g\n⊢ g = h",
"state_before": "C : Type u\ninst✝³ : Category C\nA B✝ X X' Y Y' Z : C\ninst✝² : HasFiniteProducts C\ninst✝¹ : Exponentiable A\nI B : C\nt : IsInitial I\ninst✝ : CartesianClosed C\nZ✝ : C\ng h : Z✝ ⟶ I\nx✝ : g ≫ IsInitial.to t B = h ≫ IsInitial.to t B\n⊢ g = h",
"tactic": "haveI := strict_initial t g"
},
{
"state_after": "C : Type u\ninst✝³ : Category C\nA B✝ X X' Y Y' Z : C\ninst✝² : HasFiniteProducts C\ninst✝¹ : Exponentiable A\nI B : C\nt : IsInitial I\ninst✝ : CartesianClosed C\nZ✝ : C\ng h : Z✝ ⟶ I\nx✝ : g ≫ IsInitial.to t B = h ≫ IsInitial.to t B\nthis✝ : IsIso g\nthis : IsIso h\n⊢ g = h",
"state_before": "C : Type u\ninst✝³ : Category C\nA B✝ X X' Y Y' Z : C\ninst✝² : HasFiniteProducts C\ninst✝¹ : Exponentiable A\nI B : C\nt : IsInitial I\ninst✝ : CartesianClosed C\nZ✝ : C\ng h : Z✝ ⟶ I\nx✝ : g ≫ IsInitial.to t B = h ≫ IsInitial.to t B\nthis : IsIso g\n⊢ g = h",
"tactic": "haveI := strict_initial t h"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\nA B✝ X X' Y Y' Z : C\ninst✝² : HasFiniteProducts C\ninst✝¹ : Exponentiable A\nI B : C\nt : IsInitial I\ninst✝ : CartesianClosed C\nZ✝ : C\ng h : Z✝ ⟶ I\nx✝ : g ≫ IsInitial.to t B = h ≫ IsInitial.to t B\nthis✝ : IsIso g\nthis : IsIso h\n⊢ g = h",
"tactic": "exact eq_of_inv_eq_inv (t.hom_ext _ _)"
}
] | [
390,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
386,
1
] |
Mathlib/Data/List/Forall2.lean | List.forall₂_iff_zip | [
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.40478\nδ : Type ?u.40481\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁ : List α\nl₂ : List β\nh₁ : length l₁ = length l₂\nh₂ : ∀ {a : α} {b : β}, (a, b) ∈ zip l₁ l₂ → R a b\n⊢ Forall₂ R l₁ l₂",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.40478\nδ : Type ?u.40481\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁ : List α\nl₂ : List β\nh : length l₁ = length l₂ ∧ ∀ {a : α} {b : β}, (a, b) ∈ zip l₁ l₂ → R a b\n⊢ Forall₂ R l₁ l₂",
"tactic": "cases' h with h₁ h₂"
},
{
"state_after": "case intro.nil\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.40478\nδ : Type ?u.40481\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁ : List α\nl₂✝ : List β\nh₁✝ : length l₁ = length l₂✝\nh₂✝ : ∀ {a : α} {b : β}, (a, b) ∈ zip l₁ l₂✝ → R a b\nl₂ : List β\nh₁ : length [] = length l₂\nh₂ : ∀ {a : α} {b : β}, (a, b) ∈ zip [] l₂ → R a b\n⊢ Forall₂ R [] l₂\n\ncase intro.cons\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.40478\nδ : Type ?u.40481\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁✝ : List α\nl₂✝ : List β\nh₁✝ : length l₁✝ = length l₂✝\nh₂✝ : ∀ {a : α} {b : β}, (a, b) ∈ zip l₁✝ l₂✝ → R a b\na : α\nl₁ : List α\nIH : ∀ {l₂ : List β}, length l₁ = length l₂ → (∀ {a : α} {b : β}, (a, b) ∈ zip l₁ l₂ → R a b) → Forall₂ R l₁ l₂\nl₂ : List β\nh₁ : length (a :: l₁) = length l₂\nh₂ : ∀ {a_1 : α} {b : β}, (a_1, b) ∈ zip (a :: l₁) l₂ → R a_1 b\n⊢ Forall₂ R (a :: l₁) l₂",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.40478\nδ : Type ?u.40481\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁ : List α\nl₂ : List β\nh₁ : length l₁ = length l₂\nh₂ : ∀ {a : α} {b : β}, (a, b) ∈ zip l₁ l₂ → R a b\n⊢ Forall₂ R l₁ l₂",
"tactic": "induction' l₁ with a l₁ IH generalizing l₂"
},
{
"state_after": "case intro.nil.refl\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.40478\nδ : Type ?u.40481\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁ : List α\nl₂ : List β\nh₁✝ : length l₁ = length l₂\nh₂✝ : ∀ {a : α} {b : β}, (a, b) ∈ zip l₁ l₂ → R a b\nh₁ : length [] = length []\nh₂ : ∀ {a : α} {b : β}, (a, b) ∈ zip [] [] → R a b\n⊢ Forall₂ R [] []",
"state_before": "case intro.nil\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.40478\nδ : Type ?u.40481\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁ : List α\nl₂✝ : List β\nh₁✝ : length l₁ = length l₂✝\nh₂✝ : ∀ {a : α} {b : β}, (a, b) ∈ zip l₁ l₂✝ → R a b\nl₂ : List β\nh₁ : length [] = length l₂\nh₂ : ∀ {a : α} {b : β}, (a, b) ∈ zip [] l₂ → R a b\n⊢ Forall₂ R [] l₂",
"tactic": "cases length_eq_zero.1 h₁.symm"
},
{
"state_after": "no goals",
"state_before": "case intro.nil.refl\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.40478\nδ : Type ?u.40481\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁ : List α\nl₂ : List β\nh₁✝ : length l₁ = length l₂\nh₂✝ : ∀ {a : α} {b : β}, (a, b) ∈ zip l₁ l₂ → R a b\nh₁ : length [] = length []\nh₂ : ∀ {a : α} {b : β}, (a, b) ∈ zip [] [] → R a b\n⊢ Forall₂ R [] []",
"tactic": "constructor"
},
{
"state_after": "case intro.cons.nil\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.40478\nδ : Type ?u.40481\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁✝ : List α\nl₂ : List β\nh₁✝ : length l₁✝ = length l₂\nh₂✝ : ∀ {a : α} {b : β}, (a, b) ∈ zip l₁✝ l₂ → R a b\na : α\nl₁ : List α\nIH : ∀ {l₂ : List β}, length l₁ = length l₂ → (∀ {a : α} {b : β}, (a, b) ∈ zip l₁ l₂ → R a b) → Forall₂ R l₁ l₂\nh₁ : length (a :: l₁) = length []\nh₂ : ∀ {a_1 : α} {b : β}, (a_1, b) ∈ zip (a :: l₁) [] → R a_1 b\n⊢ Forall₂ R (a :: l₁) []\n\ncase intro.cons.cons\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.40478\nδ : Type ?u.40481\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁✝ : List α\nl₂✝ : List β\nh₁✝ : length l₁✝ = length l₂✝\nh₂✝ : ∀ {a : α} {b : β}, (a, b) ∈ zip l₁✝ l₂✝ → R a b\na : α\nl₁ : List α\nIH : ∀ {l₂ : List β}, length l₁ = length l₂ → (∀ {a : α} {b : β}, (a, b) ∈ zip l₁ l₂ → R a b) → Forall₂ R l₁ l₂\nb : β\nl₂ : List β\nh₁ : length (a :: l₁) = length (b :: l₂)\nh₂ : ∀ {a_1 : α} {b_1 : β}, (a_1, b_1) ∈ zip (a :: l₁) (b :: l₂) → R a_1 b_1\n⊢ Forall₂ R (a :: l₁) (b :: l₂)",
"state_before": "case intro.cons\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.40478\nδ : Type ?u.40481\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁✝ : List α\nl₂✝ : List β\nh₁✝ : length l₁✝ = length l₂✝\nh₂✝ : ∀ {a : α} {b : β}, (a, b) ∈ zip l₁✝ l₂✝ → R a b\na : α\nl₁ : List α\nIH : ∀ {l₂ : List β}, length l₁ = length l₂ → (∀ {a : α} {b : β}, (a, b) ∈ zip l₁ l₂ → R a b) → Forall₂ R l₁ l₂\nl₂ : List β\nh₁ : length (a :: l₁) = length l₂\nh₂ : ∀ {a_1 : α} {b : β}, (a_1, b) ∈ zip (a :: l₁) l₂ → R a_1 b\n⊢ Forall₂ R (a :: l₁) l₂",
"tactic": "cases' l₂ with b l₂"
},
{
"state_after": "case intro.cons.cons\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.40478\nδ : Type ?u.40481\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁✝ : List α\nl₂✝ : List β\nh₁✝ : length l₁✝ = length l₂✝\nh₂✝ : ∀ {a : α} {b : β}, (a, b) ∈ zip l₁✝ l₂✝ → R a b\na : α\nl₁ : List α\nIH : ∀ {l₂ : List β}, length l₁ = length l₂ → (∀ {a : α} {b : β}, (a, b) ∈ zip l₁ l₂ → R a b) → Forall₂ R l₁ l₂\nb : β\nl₂ : List β\nh₁ : length (a :: l₁) = length (b :: l₂)\nh₂ : ∀ {a_1 : α} {b_1 : β}, (a_1, b_1) ∈ zip (a :: l₁) (b :: l₂) → R a_1 b_1\n⊢ Forall₂ R (a :: l₁) (b :: l₂)",
"state_before": "case intro.cons.nil\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.40478\nδ : Type ?u.40481\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁✝ : List α\nl₂ : List β\nh₁✝ : length l₁✝ = length l₂\nh₂✝ : ∀ {a : α} {b : β}, (a, b) ∈ zip l₁✝ l₂ → R a b\na : α\nl₁ : List α\nIH : ∀ {l₂ : List β}, length l₁ = length l₂ → (∀ {a : α} {b : β}, (a, b) ∈ zip l₁ l₂ → R a b) → Forall₂ R l₁ l₂\nh₁ : length (a :: l₁) = length []\nh₂ : ∀ {a_1 : α} {b : β}, (a_1, b) ∈ zip (a :: l₁) [] → R a_1 b\n⊢ Forall₂ R (a :: l₁) []\n\ncase intro.cons.cons\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.40478\nδ : Type ?u.40481\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁✝ : List α\nl₂✝ : List β\nh₁✝ : length l₁✝ = length l₂✝\nh₂✝ : ∀ {a : α} {b : β}, (a, b) ∈ zip l₁✝ l₂✝ → R a b\na : α\nl₁ : List α\nIH : ∀ {l₂ : List β}, length l₁ = length l₂ → (∀ {a : α} {b : β}, (a, b) ∈ zip l₁ l₂ → R a b) → Forall₂ R l₁ l₂\nb : β\nl₂ : List β\nh₁ : length (a :: l₁) = length (b :: l₂)\nh₂ : ∀ {a_1 : α} {b_1 : β}, (a_1, b_1) ∈ zip (a :: l₁) (b :: l₂) → R a_1 b_1\n⊢ Forall₂ R (a :: l₁) (b :: l₂)",
"tactic": ". simp at h₁"
},
{
"state_after": "no goals",
"state_before": "case intro.cons.cons\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.40478\nδ : Type ?u.40481\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁✝ : List α\nl₂✝ : List β\nh₁✝ : length l₁✝ = length l₂✝\nh₂✝ : ∀ {a : α} {b : β}, (a, b) ∈ zip l₁✝ l₂✝ → R a b\na : α\nl₁ : List α\nIH : ∀ {l₂ : List β}, length l₁ = length l₂ → (∀ {a : α} {b : β}, (a, b) ∈ zip l₁ l₂ → R a b) → Forall₂ R l₁ l₂\nb : β\nl₂ : List β\nh₁ : length (a :: l₁) = length (b :: l₂)\nh₂ : ∀ {a_1 : α} {b_1 : β}, (a_1, b_1) ∈ zip (a :: l₁) (b :: l₂) → R a_1 b_1\n⊢ Forall₂ R (a :: l₁) (b :: l₂)",
"tactic": ". simp only [length_cons, succ.injEq] at h₁\n exact Forall₂.cons (h₂ <| by simp [zip])\n (IH h₁ <| fun h => h₂ <| by\n simp only [zip, zipWith, find?, mem_cons, Prod.mk.injEq]; right\n simpa [zip] using h)"
},
{
"state_after": "no goals",
"state_before": "case intro.cons.nil\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.40478\nδ : Type ?u.40481\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁✝ : List α\nl₂ : List β\nh₁✝ : length l₁✝ = length l₂\nh₂✝ : ∀ {a : α} {b : β}, (a, b) ∈ zip l₁✝ l₂ → R a b\na : α\nl₁ : List α\nIH : ∀ {l₂ : List β}, length l₁ = length l₂ → (∀ {a : α} {b : β}, (a, b) ∈ zip l₁ l₂ → R a b) → Forall₂ R l₁ l₂\nh₁ : length (a :: l₁) = length []\nh₂ : ∀ {a_1 : α} {b : β}, (a_1, b) ∈ zip (a :: l₁) [] → R a_1 b\n⊢ Forall₂ R (a :: l₁) []",
"tactic": "simp at h₁"
},
{
"state_after": "case intro.cons.cons\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.40478\nδ : Type ?u.40481\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁✝ : List α\nl₂✝ : List β\nh₁✝ : length l₁✝ = length l₂✝\nh₂✝ : ∀ {a : α} {b : β}, (a, b) ∈ zip l₁✝ l₂✝ → R a b\na : α\nl₁ : List α\nIH : ∀ {l₂ : List β}, length l₁ = length l₂ → (∀ {a : α} {b : β}, (a, b) ∈ zip l₁ l₂ → R a b) → Forall₂ R l₁ l₂\nb : β\nl₂ : List β\nh₂ : ∀ {a_1 : α} {b_1 : β}, (a_1, b_1) ∈ zip (a :: l₁) (b :: l₂) → R a_1 b_1\nh₁ : length l₁ = length l₂\n⊢ Forall₂ R (a :: l₁) (b :: l₂)",
"state_before": "case intro.cons.cons\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.40478\nδ : Type ?u.40481\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁✝ : List α\nl₂✝ : List β\nh₁✝ : length l₁✝ = length l₂✝\nh₂✝ : ∀ {a : α} {b : β}, (a, b) ∈ zip l₁✝ l₂✝ → R a b\na : α\nl₁ : List α\nIH : ∀ {l₂ : List β}, length l₁ = length l₂ → (∀ {a : α} {b : β}, (a, b) ∈ zip l₁ l₂ → R a b) → Forall₂ R l₁ l₂\nb : β\nl₂ : List β\nh₁ : length (a :: l₁) = length (b :: l₂)\nh₂ : ∀ {a_1 : α} {b_1 : β}, (a_1, b_1) ∈ zip (a :: l₁) (b :: l₂) → R a_1 b_1\n⊢ Forall₂ R (a :: l₁) (b :: l₂)",
"tactic": "simp only [length_cons, succ.injEq] at h₁"
},
{
"state_after": "no goals",
"state_before": "case intro.cons.cons\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.40478\nδ : Type ?u.40481\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁✝ : List α\nl₂✝ : List β\nh₁✝ : length l₁✝ = length l₂✝\nh₂✝ : ∀ {a : α} {b : β}, (a, b) ∈ zip l₁✝ l₂✝ → R a b\na : α\nl₁ : List α\nIH : ∀ {l₂ : List β}, length l₁ = length l₂ → (∀ {a : α} {b : β}, (a, b) ∈ zip l₁ l₂ → R a b) → Forall₂ R l₁ l₂\nb : β\nl₂ : List β\nh₂ : ∀ {a_1 : α} {b_1 : β}, (a_1, b_1) ∈ zip (a :: l₁) (b :: l₂) → R a_1 b_1\nh₁ : length l₁ = length l₂\n⊢ Forall₂ R (a :: l₁) (b :: l₂)",
"tactic": "exact Forall₂.cons (h₂ <| by simp [zip])\n (IH h₁ <| fun h => h₂ <| by\n simp only [zip, zipWith, find?, mem_cons, Prod.mk.injEq]; right\n simpa [zip] using h)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.40478\nδ : Type ?u.40481\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁✝ : List α\nl₂✝ : List β\nh₁✝ : length l₁✝ = length l₂✝\nh₂✝ : ∀ {a : α} {b : β}, (a, b) ∈ zip l₁✝ l₂✝ → R a b\na : α\nl₁ : List α\nIH : ∀ {l₂ : List β}, length l₁ = length l₂ → (∀ {a : α} {b : β}, (a, b) ∈ zip l₁ l₂ → R a b) → Forall₂ R l₁ l₂\nb : β\nl₂ : List β\nh₂ : ∀ {a_1 : α} {b_1 : β}, (a_1, b_1) ∈ zip (a :: l₁) (b :: l₂) → R a_1 b_1\nh₁ : length l₁ = length l₂\n⊢ (a, b) ∈ zip (a :: l₁) (b :: l₂)",
"tactic": "simp [zip]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.40478\nδ : Type ?u.40481\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁✝ : List α\nl₂✝ : List β\nh₁✝ : length l₁✝ = length l₂✝\nh₂✝ : ∀ {a : α} {b : β}, (a, b) ∈ zip l₁✝ l₂✝ → R a b\na : α\nl₁ : List α\nIH : ∀ {l₂ : List β}, length l₁ = length l₂ → (∀ {a : α} {b : β}, (a, b) ∈ zip l₁ l₂ → R a b) → Forall₂ R l₁ l₂\nb : β\nl₂ : List β\nh₂ : ∀ {a_1 : α} {b_1 : β}, (a_1, b_1) ∈ zip (a :: l₁) (b :: l₂) → R a_1 b_1\nh₁ : length l₁ = length l₂\na✝ : α\nb✝ : β\nh : (a✝, b✝) ∈ zip l₁ l₂\n⊢ a✝ = a ∧ b✝ = b ∨ (a✝, b✝) ∈ zipWith Prod.mk l₁ l₂",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.40478\nδ : Type ?u.40481\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁✝ : List α\nl₂✝ : List β\nh₁✝ : length l₁✝ = length l₂✝\nh₂✝ : ∀ {a : α} {b : β}, (a, b) ∈ zip l₁✝ l₂✝ → R a b\na : α\nl₁ : List α\nIH : ∀ {l₂ : List β}, length l₁ = length l₂ → (∀ {a : α} {b : β}, (a, b) ∈ zip l₁ l₂ → R a b) → Forall₂ R l₁ l₂\nb : β\nl₂ : List β\nh₂ : ∀ {a_1 : α} {b_1 : β}, (a_1, b_1) ∈ zip (a :: l₁) (b :: l₂) → R a_1 b_1\nh₁ : length l₁ = length l₂\na✝ : α\nb✝ : β\nh : (a✝, b✝) ∈ zip l₁ l₂\n⊢ (a✝, b✝) ∈ zip (a :: l₁) (b :: l₂)",
"tactic": "simp only [zip, zipWith, find?, mem_cons, Prod.mk.injEq]"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.40478\nδ : Type ?u.40481\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁✝ : List α\nl₂✝ : List β\nh₁✝ : length l₁✝ = length l₂✝\nh₂✝ : ∀ {a : α} {b : β}, (a, b) ∈ zip l₁✝ l₂✝ → R a b\na : α\nl₁ : List α\nIH : ∀ {l₂ : List β}, length l₁ = length l₂ → (∀ {a : α} {b : β}, (a, b) ∈ zip l₁ l₂ → R a b) → Forall₂ R l₁ l₂\nb : β\nl₂ : List β\nh₂ : ∀ {a_1 : α} {b_1 : β}, (a_1, b_1) ∈ zip (a :: l₁) (b :: l₂) → R a_1 b_1\nh₁ : length l₁ = length l₂\na✝ : α\nb✝ : β\nh : (a✝, b✝) ∈ zip l₁ l₂\n⊢ (a✝, b✝) ∈ zipWith Prod.mk l₁ l₂",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.40478\nδ : Type ?u.40481\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁✝ : List α\nl₂✝ : List β\nh₁✝ : length l₁✝ = length l₂✝\nh₂✝ : ∀ {a : α} {b : β}, (a, b) ∈ zip l₁✝ l₂✝ → R a b\na : α\nl₁ : List α\nIH : ∀ {l₂ : List β}, length l₁ = length l₂ → (∀ {a : α} {b : β}, (a, b) ∈ zip l₁ l₂ → R a b) → Forall₂ R l₁ l₂\nb : β\nl₂ : List β\nh₂ : ∀ {a_1 : α} {b_1 : β}, (a_1, b_1) ∈ zip (a :: l₁) (b :: l₂) → R a_1 b_1\nh₁ : length l₁ = length l₂\na✝ : α\nb✝ : β\nh : (a✝, b✝) ∈ zip l₁ l₂\n⊢ a✝ = a ∧ b✝ = b ∨ (a✝, b✝) ∈ zipWith Prod.mk l₁ l₂",
"tactic": "right"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.40478\nδ : Type ?u.40481\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\nl₁✝ : List α\nl₂✝ : List β\nh₁✝ : length l₁✝ = length l₂✝\nh₂✝ : ∀ {a : α} {b : β}, (a, b) ∈ zip l₁✝ l₂✝ → R a b\na : α\nl₁ : List α\nIH : ∀ {l₂ : List β}, length l₁ = length l₂ → (∀ {a : α} {b : β}, (a, b) ∈ zip l₁ l₂ → R a b) → Forall₂ R l₁ l₂\nb : β\nl₂ : List β\nh₂ : ∀ {a_1 : α} {b_1 : β}, (a_1, b_1) ∈ zip (a :: l₁) (b :: l₂) → R a_1 b_1\nh₁ : length l₁ = length l₂\na✝ : α\nb✝ : β\nh : (a✝, b✝) ∈ zip l₁ l₂\n⊢ (a✝, b✝) ∈ zipWith Prod.mk l₁ l₂",
"tactic": "simpa [zip] using h"
}
] | [
210,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
197,
1
] |
Mathlib/Algebra/Order/Rearrangement.lean | AntivaryOn.sum_smul_lt_sum_smul_comp_perm_iff | [
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝³ : LinearOrderedRing α\ninst✝² : LinearOrderedAddCommGroup β\ninst✝¹ : Module α β\ninst✝ : OrderedSMul α β\ns : Finset ι\nσ : Perm ι\nf : ι → α\ng : ι → β\nhfg : AntivaryOn f g ↑s\nhσ : {x | ↑σ x ≠ x} ⊆ ↑s\n⊢ ∑ i in s, f i • g i < ∑ i in s, f i • g (↑σ i) ↔ ¬AntivaryOn f (g ∘ ↑σ) ↑s",
"tactic": "simp [← hfg.sum_smul_eq_sum_smul_comp_perm_iff hσ, lt_iff_le_and_ne, eq_comm,\n hfg.sum_smul_le_sum_smul_comp_perm hσ]"
}
] | [
214,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
210,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Combination.lean | Finset.sum_centroidWeights_eq_one_of_cast_card_ne_zero | [
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type ?u.465735\nP : Type ?u.465738\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_2\ns : Finset ι\nι₂ : Type ?u.466257\ns₂ : Finset ι₂\nh : ↑(card s) ≠ 0\n⊢ ∑ i in s, centroidWeights k s i = 1",
"tactic": "simp [h]"
}
] | [
808,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
807,
1
] |
Mathlib/Topology/ContinuousFunction/Compact.lean | ContinuousMap.dist_lt_iff_of_nonempty | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nE : Type ?u.76000\ninst✝⁴ : TopologicalSpace α\ninst✝³ : CompactSpace α\ninst✝² : MetricSpace β\ninst✝¹ : NormedAddCommGroup E\nf g : C(α, β)\nC : ℝ\ninst✝ : Nonempty α\n⊢ dist f g < C ↔ ∀ (x : α), dist (↑f x) (↑g x) < C",
"tactic": "simp only [← dist_mkOfCompact, dist_lt_iff_of_nonempty_compact, mkOfCompact_apply]"
}
] | [
148,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
147,
1
] |
Mathlib/Algebra/Hom/Group.lean | MulHom.coe_mk | [] | [
596,
97
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
596,
1
] |
Mathlib/Algebra/Group/Commute.lean | Commute.self_pow | [] | [
194,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
193,
1
] |
Mathlib/Topology/Algebra/GroupWithZero.lean | ContinuousOn.div | [] | [
186,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
185,
1
] |
Mathlib/Analysis/Normed/Group/AddCircle.lean | AddCircle.norm_eq' | [
{
"state_after": "p : ℝ\nhp : 0 < p\nx : ℝ\n⊢ ‖↑x‖ = abs p * abs (p⁻¹ * x - ↑(round (p⁻¹ * x)))",
"state_before": "p : ℝ\nhp : 0 < p\nx : ℝ\n⊢ ‖↑x‖ = p * abs (p⁻¹ * x - ↑(round (p⁻¹ * x)))",
"tactic": "conv_rhs =>\n congr\n rw [← abs_eq_self.mpr hp.le]"
},
{
"state_after": "no goals",
"state_before": "p : ℝ\nhp : 0 < p\nx : ℝ\n⊢ ‖↑x‖ = abs p * abs (p⁻¹ * x - ↑(round (p⁻¹ * x)))",
"tactic": "rw [← abs_mul, mul_sub, mul_inv_cancel_left₀ hp.ne.symm, norm_eq, mul_comm p]"
}
] | [
128,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
124,
1
] |
Std/Data/Nat/Gcd.lean | Nat.coprime.dvd_of_dvd_mul_right | [
{
"state_after": "k n m : Nat\nH1 : coprime k n\nH2 : k ∣ m * n\nt : k ∣ gcd (m * k) (m * n) := dvd_gcd (Nat.dvd_mul_left k m) H2\n⊢ k ∣ m",
"state_before": "k n m : Nat\nH1 : coprime k n\nH2 : k ∣ m * n\n⊢ k ∣ m",
"tactic": "let t := dvd_gcd (Nat.dvd_mul_left k m) H2"
},
{
"state_after": "no goals",
"state_before": "k n m : Nat\nH1 : coprime k n\nH2 : k ∣ m * n\nt : k ∣ gcd (m * k) (m * n) := dvd_gcd (Nat.dvd_mul_left k m) H2\n⊢ k ∣ m",
"tactic": "rwa [gcd_mul_left, H1.gcd_eq_one, Nat.mul_one] at t"
}
] | [
250,
54
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
248,
1
] |
Mathlib/Topology/List.lean | List.tendsto_prod | [
{
"state_after": "case nil\nα : Type u_1\nβ : Type ?u.56124\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : Monoid α\ninst✝ : ContinuousMul α\n⊢ Tendsto prod (𝓝 []) (𝓝 (prod []))\n\ncase cons\nα : Type u_1\nβ : Type ?u.56124\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : Monoid α\ninst✝ : ContinuousMul α\nx : α\nl : List α\nih : Tendsto prod (𝓝 l) (𝓝 (prod l))\n⊢ Tendsto prod (𝓝 (x :: l)) (𝓝 (prod (x :: l)))",
"state_before": "α : Type u_1\nβ : Type ?u.56124\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : Monoid α\ninst✝ : ContinuousMul α\nl : List α\n⊢ Tendsto prod (𝓝 l) (𝓝 (prod l))",
"tactic": "induction' l with x l ih"
},
{
"state_after": "case cons\nα : Type u_1\nβ : Type ?u.56124\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : Monoid α\ninst✝ : ContinuousMul α\nx : α\nl : List α\nih : Tendsto prod (𝓝 l) (𝓝 (prod l))\n⊢ Tendsto (fun p => p.fst * prod p.snd) (𝓝 x ×ˢ 𝓝 l) (𝓝 (x * prod l))",
"state_before": "case cons\nα : Type u_1\nβ : Type ?u.56124\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : Monoid α\ninst✝ : ContinuousMul α\nx : α\nl : List α\nih : Tendsto prod (𝓝 l) (𝓝 (prod l))\n⊢ Tendsto prod (𝓝 (x :: l)) (𝓝 (prod (x :: l)))",
"tactic": "simp_rw [tendsto_cons_iff, prod_cons]"
},
{
"state_after": "case cons\nα : Type u_1\nβ : Type ?u.56124\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : Monoid α\ninst✝ : ContinuousMul α\nx : α\nl : List α\nih : Tendsto prod (𝓝 l) (𝓝 (prod l))\nthis : ContinuousAt (fun p => p.fst * p.snd) (x, prod l)\n⊢ Tendsto (fun p => p.fst * prod p.snd) (𝓝 x ×ˢ 𝓝 l) (𝓝 (x * prod l))",
"state_before": "case cons\nα : Type u_1\nβ : Type ?u.56124\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : Monoid α\ninst✝ : ContinuousMul α\nx : α\nl : List α\nih : Tendsto prod (𝓝 l) (𝓝 (prod l))\n⊢ Tendsto (fun p => p.fst * prod p.snd) (𝓝 x ×ˢ 𝓝 l) (𝓝 (x * prod l))",
"tactic": "have := continuous_iff_continuousAt.mp continuous_mul (x, l.prod)"
},
{
"state_after": "case cons\nα : Type u_1\nβ : Type ?u.56124\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : Monoid α\ninst✝ : ContinuousMul α\nx : α\nl : List α\nih : Tendsto prod (𝓝 l) (𝓝 (prod l))\nthis : Tendsto (fun p => p.fst * p.snd) (𝓝 x ×ˢ 𝓝 (prod l)) (𝓝 ((x, prod l).fst * (x, prod l).snd))\n⊢ Tendsto (fun p => p.fst * prod p.snd) (𝓝 x ×ˢ 𝓝 l) (𝓝 (x * prod l))",
"state_before": "case cons\nα : Type u_1\nβ : Type ?u.56124\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : Monoid α\ninst✝ : ContinuousMul α\nx : α\nl : List α\nih : Tendsto prod (𝓝 l) (𝓝 (prod l))\nthis : ContinuousAt (fun p => p.fst * p.snd) (x, prod l)\n⊢ Tendsto (fun p => p.fst * prod p.snd) (𝓝 x ×ˢ 𝓝 l) (𝓝 (x * prod l))",
"tactic": "rw [ContinuousAt, nhds_prod_eq] at this"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u_1\nβ : Type ?u.56124\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : Monoid α\ninst✝ : ContinuousMul α\nx : α\nl : List α\nih : Tendsto prod (𝓝 l) (𝓝 (prod l))\nthis : Tendsto (fun p => p.fst * p.snd) (𝓝 x ×ˢ 𝓝 (prod l)) (𝓝 ((x, prod l).fst * (x, prod l).snd))\n⊢ Tendsto (fun p => p.fst * prod p.snd) (𝓝 x ×ˢ 𝓝 l) (𝓝 (x * prod l))",
"tactic": "exact this.comp (tendsto_id.prod_map ih)"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\nβ : Type ?u.56124\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : Monoid α\ninst✝ : ContinuousMul α\n⊢ Tendsto prod (𝓝 []) (𝓝 (prod []))",
"tactic": "simp (config := { contextual := true }) [nhds_nil, mem_of_mem_nhds, tendsto_pure_left]"
}
] | [
174,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
167,
1
] |
Mathlib/AlgebraicGeometry/StructureSheaf.lean | AlgebraicGeometry.StructureSheaf.locally_const_basicOpen | [
{
"state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\n⊢ ∃ f g i,\n ↑x ∈ PrimeSpectrum.basicOpen g ∧\n const R f g (PrimeSpectrum.basicOpen g)\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen g → y ∈ PrimeSpectrum.basicOpen g) =\n ↑((structureSheaf R).val.map i.op) s",
"state_before": "R : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\n⊢ ∃ f g i,\n ↑x ∈ PrimeSpectrum.basicOpen g ∧\n const R f g (PrimeSpectrum.basicOpen g)\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen g → y ∈ PrimeSpectrum.basicOpen g) =\n ↑((structureSheaf R).val.map i.op) s",
"tactic": "obtain ⟨V, hxV : x.1 ∈ V.1, iVU, f, g, hVDg : V ≤ PrimeSpectrum.basicOpen g, s_eq⟩ :=\n exists_const R U s x.1 x.2"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\n⊢ ∃ f g i,\n ↑x ∈ PrimeSpectrum.basicOpen g ∧\n const R f g (PrimeSpectrum.basicOpen g)\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen g → y ∈ PrimeSpectrum.basicOpen g) =\n ↑((structureSheaf R).val.map i.op) s",
"state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\n⊢ ∃ f g i,\n ↑x ∈ PrimeSpectrum.basicOpen g ∧\n const R f g (PrimeSpectrum.basicOpen g)\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen g → y ∈ PrimeSpectrum.basicOpen g) =\n ↑((structureSheaf R).val.map i.op) s",
"tactic": "obtain ⟨_, ⟨h, rfl⟩, hxDh, hDhV : PrimeSpectrum.basicOpen h ≤ V⟩ :=\n PrimeSpectrum.isTopologicalBasis_basic_opens.exists_subset_of_mem_open hxV V.2"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nhn : h ^ n ∈ Ideal.span {g}\n⊢ ∃ f g i,\n ↑x ∈ PrimeSpectrum.basicOpen g ∧\n const R f g (PrimeSpectrum.basicOpen g)\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen g → y ∈ PrimeSpectrum.basicOpen g) =\n ↑((structureSheaf R).val.map i.op) s",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\n⊢ ∃ f g i,\n ↑x ∈ PrimeSpectrum.basicOpen g ∧\n const R f g (PrimeSpectrum.basicOpen g)\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen g → y ∈ PrimeSpectrum.basicOpen g) =\n ↑((structureSheaf R).val.map i.op) s",
"tactic": "cases' (PrimeSpectrum.basicOpen_le_basicOpen_iff h g).mp (Set.Subset.trans hDhV hVDg) with n hn"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nhn : h * h ^ n ∈ Ideal.span {g}\n⊢ ∃ f g i,\n ↑x ∈ PrimeSpectrum.basicOpen g ∧\n const R f g (PrimeSpectrum.basicOpen g)\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen g → y ∈ PrimeSpectrum.basicOpen g) =\n ↑((structureSheaf R).val.map i.op) s",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nhn : h ^ n ∈ Ideal.span {g}\n⊢ ∃ f g i,\n ↑x ∈ PrimeSpectrum.basicOpen g ∧\n const R f g (PrimeSpectrum.basicOpen g)\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen g → y ∈ PrimeSpectrum.basicOpen g) =\n ↑((structureSheaf R).val.map i.op) s",
"tactic": "replace hn := Ideal.mul_mem_left (Ideal.span {g}) h hn"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nhn : ∃ a, a * g = h ^ (n + 1)\n⊢ ∃ f g i,\n ↑x ∈ PrimeSpectrum.basicOpen g ∧\n const R f g (PrimeSpectrum.basicOpen g)\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen g → y ∈ PrimeSpectrum.basicOpen g) =\n ↑((structureSheaf R).val.map i.op) s",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nhn : h * h ^ n ∈ Ideal.span {g}\n⊢ ∃ f g i,\n ↑x ∈ PrimeSpectrum.basicOpen g ∧\n const R f g (PrimeSpectrum.basicOpen g)\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen g → y ∈ PrimeSpectrum.basicOpen g) =\n ↑((structureSheaf R).val.map i.op) s",
"tactic": "rw [← pow_succ, Ideal.mem_span_singleton'] at hn"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nc : R\nhc : c * g = h ^ (n + 1)\n⊢ ∃ f g i,\n ↑x ∈ PrimeSpectrum.basicOpen g ∧\n const R f g (PrimeSpectrum.basicOpen g)\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen g → y ∈ PrimeSpectrum.basicOpen g) =\n ↑((structureSheaf R).val.map i.op) s",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nhn : ∃ a, a * g = h ^ (n + 1)\n⊢ ∃ f g i,\n ↑x ∈ PrimeSpectrum.basicOpen g ∧\n const R f g (PrimeSpectrum.basicOpen g)\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen g → y ∈ PrimeSpectrum.basicOpen g) =\n ↑((structureSheaf R).val.map i.op) s",
"tactic": "cases' hn with c hc"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nc : R\nhc : c * g = h ^ (n + 1)\nbasic_opens_eq : PrimeSpectrum.basicOpen (h ^ (n + 1)) = PrimeSpectrum.basicOpen h\n⊢ ∃ f g i,\n ↑x ∈ PrimeSpectrum.basicOpen g ∧\n const R f g (PrimeSpectrum.basicOpen g)\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen g → y ∈ PrimeSpectrum.basicOpen g) =\n ↑((structureSheaf R).val.map i.op) s",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nc : R\nhc : c * g = h ^ (n + 1)\n⊢ ∃ f g i,\n ↑x ∈ PrimeSpectrum.basicOpen g ∧\n const R f g (PrimeSpectrum.basicOpen g)\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen g → y ∈ PrimeSpectrum.basicOpen g) =\n ↑((structureSheaf R).val.map i.op) s",
"tactic": "have basic_opens_eq := PrimeSpectrum.basicOpen_pow h (n + 1) (by linarith)"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nc : R\nhc : c * g = h ^ (n + 1)\nbasic_opens_eq : PrimeSpectrum.basicOpen (h ^ (n + 1)) = PrimeSpectrum.basicOpen h\ni_basic_open : PrimeSpectrum.basicOpen (h ^ (n + 1)) ⟶ V\n⊢ ∃ f g i,\n ↑x ∈ PrimeSpectrum.basicOpen g ∧\n const R f g (PrimeSpectrum.basicOpen g)\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen g → y ∈ PrimeSpectrum.basicOpen g) =\n ↑((structureSheaf R).val.map i.op) s",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nc : R\nhc : c * g = h ^ (n + 1)\nbasic_opens_eq : PrimeSpectrum.basicOpen (h ^ (n + 1)) = PrimeSpectrum.basicOpen h\n⊢ ∃ f g i,\n ↑x ∈ PrimeSpectrum.basicOpen g ∧\n const R f g (PrimeSpectrum.basicOpen g)\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen g → y ∈ PrimeSpectrum.basicOpen g) =\n ↑((structureSheaf R).val.map i.op) s",
"tactic": "have i_basic_open := eqToHom basic_opens_eq ≫ homOfLE hDhV"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nc : R\nhc : c * g = h ^ (n + 1)\nbasic_opens_eq : PrimeSpectrum.basicOpen (h ^ (n + 1)) = PrimeSpectrum.basicOpen h\ni_basic_open : PrimeSpectrum.basicOpen (h ^ (n + 1)) ⟶ V\n⊢ const R (f * c) (h ^ (n + 1)) (PrimeSpectrum.basicOpen (h ^ (n + 1)))\n (_ :\n ∀ (y : ↑(PrimeSpectrum.Top R)),\n y ∈ PrimeSpectrum.basicOpen (h ^ (n + 1)) → y ∈ PrimeSpectrum.basicOpen (h ^ (n + 1))) =\n ↑((structureSheaf R).val.map (i_basic_open ≫ iVU).op) s",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nc : R\nhc : c * g = h ^ (n + 1)\nbasic_opens_eq : PrimeSpectrum.basicOpen (h ^ (n + 1)) = PrimeSpectrum.basicOpen h\ni_basic_open : PrimeSpectrum.basicOpen (h ^ (n + 1)) ⟶ V\n⊢ ∃ f g i,\n ↑x ∈ PrimeSpectrum.basicOpen g ∧\n const R f g (PrimeSpectrum.basicOpen g)\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen g → y ∈ PrimeSpectrum.basicOpen g) =\n ↑((structureSheaf R).val.map i.op) s",
"tactic": "use f * c, h ^ (n + 1), i_basic_open ≫ iVU, (basic_opens_eq.symm.le : _) hxDh"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nc : R\nhc : c * g = h ^ (n + 1)\nbasic_opens_eq : PrimeSpectrum.basicOpen (h ^ (n + 1)) = PrimeSpectrum.basicOpen h\ni_basic_open : PrimeSpectrum.basicOpen (h ^ (n + 1)) ⟶ V\n⊢ const R (f * c) (h ^ (n + 1)) (PrimeSpectrum.basicOpen (h ^ (n + 1)))\n (_ :\n ∀ (y : ↑(PrimeSpectrum.Top R)),\n y ∈ PrimeSpectrum.basicOpen (h ^ (n + 1)) → y ∈ PrimeSpectrum.basicOpen (h ^ (n + 1))) =\n ↑((structureSheaf R).val.map iVU.op ≫ (structureSheaf R).val.map i_basic_open.op) s",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nc : R\nhc : c * g = h ^ (n + 1)\nbasic_opens_eq : PrimeSpectrum.basicOpen (h ^ (n + 1)) = PrimeSpectrum.basicOpen h\ni_basic_open : PrimeSpectrum.basicOpen (h ^ (n + 1)) ⟶ V\n⊢ const R (f * c) (h ^ (n + 1)) (PrimeSpectrum.basicOpen (h ^ (n + 1)))\n (_ :\n ∀ (y : ↑(PrimeSpectrum.Top R)),\n y ∈ PrimeSpectrum.basicOpen (h ^ (n + 1)) → y ∈ PrimeSpectrum.basicOpen (h ^ (n + 1))) =\n ↑((structureSheaf R).val.map (i_basic_open ≫ iVU).op) s",
"tactic": "rw [op_comp, Functor.map_comp]"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nc : R\nhc : c * g = h ^ (n + 1)\nbasic_opens_eq : PrimeSpectrum.basicOpen (h ^ (n + 1)) = PrimeSpectrum.basicOpen h\ni_basic_open : PrimeSpectrum.basicOpen (h ^ (n + 1)) ⟶ V\n⊢ const R (f * c) (h ^ (n + 1)) (PrimeSpectrum.basicOpen (h ^ (n + 1)))\n (_ :\n ∀ (y : ↑(PrimeSpectrum.Top R)),\n y ∈ PrimeSpectrum.basicOpen (h ^ (n + 1)) → y ∈ PrimeSpectrum.basicOpen (h ^ (n + 1))) =\n ↑((structureSheaf R).val.map i_basic_open.op) (↑((structureSheaf R).val.map iVU.op) s)",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nc : R\nhc : c * g = h ^ (n + 1)\nbasic_opens_eq : PrimeSpectrum.basicOpen (h ^ (n + 1)) = PrimeSpectrum.basicOpen h\ni_basic_open : PrimeSpectrum.basicOpen (h ^ (n + 1)) ⟶ V\n⊢ const R (f * c) (h ^ (n + 1)) (PrimeSpectrum.basicOpen (h ^ (n + 1)))\n (_ :\n ∀ (y : ↑(PrimeSpectrum.Top R)),\n y ∈ PrimeSpectrum.basicOpen (h ^ (n + 1)) → y ∈ PrimeSpectrum.basicOpen (h ^ (n + 1))) =\n ↑((structureSheaf R).val.map iVU.op ≫ (structureSheaf R).val.map i_basic_open.op) s",
"tactic": "change const R _ _ _ _ = (structureSheaf R).1.map i_basic_open.op\n ((structureSheaf R).1.map iVU.op s)"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nc : R\nhc : c * g = h ^ (n + 1)\nbasic_opens_eq : PrimeSpectrum.basicOpen (h ^ (n + 1)) = PrimeSpectrum.basicOpen h\ni_basic_open : PrimeSpectrum.basicOpen (h ^ (n + 1)) ⟶ V\n⊢ const R (f * c) (h ^ (n + 1)) (PrimeSpectrum.basicOpen (h ^ (n + 1)))\n (_ :\n ∀ (y : ↑(PrimeSpectrum.Top R)),\n y ∈ PrimeSpectrum.basicOpen (h ^ (n + 1)) → y ∈ PrimeSpectrum.basicOpen (h ^ (n + 1))) =\n const R f g (PrimeSpectrum.basicOpen (h ^ (n + 1)))\n ?intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.hv\n\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.hv\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nc : R\nhc : c * g = h ^ (n + 1)\nbasic_opens_eq : PrimeSpectrum.basicOpen (h ^ (n + 1)) = PrimeSpectrum.basicOpen h\ni_basic_open : PrimeSpectrum.basicOpen (h ^ (n + 1)) ⟶ V\n⊢ ∀ (x : ↑(PrimeSpectrum.Top R)), x ∈ PrimeSpectrum.basicOpen (h ^ (n + 1)) → g ∈ Ideal.primeCompl x.asIdeal",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nc : R\nhc : c * g = h ^ (n + 1)\nbasic_opens_eq : PrimeSpectrum.basicOpen (h ^ (n + 1)) = PrimeSpectrum.basicOpen h\ni_basic_open : PrimeSpectrum.basicOpen (h ^ (n + 1)) ⟶ V\n⊢ const R (f * c) (h ^ (n + 1)) (PrimeSpectrum.basicOpen (h ^ (n + 1)))\n (_ :\n ∀ (y : ↑(PrimeSpectrum.Top R)),\n y ∈ PrimeSpectrum.basicOpen (h ^ (n + 1)) → y ∈ PrimeSpectrum.basicOpen (h ^ (n + 1))) =\n ↑((structureSheaf R).val.map i_basic_open.op) (↑((structureSheaf R).val.map iVU.op) s)",
"tactic": "rw [← s_eq, res_const]"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.hv\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nc : R\nhc : c * g = h ^ (n + 1)\nbasic_opens_eq : PrimeSpectrum.basicOpen (h ^ (n + 1)) = PrimeSpectrum.basicOpen h\ni_basic_open : PrimeSpectrum.basicOpen (h ^ (n + 1)) ⟶ V\n⊢ ∀ (x : ↑(PrimeSpectrum.Top R)), x ∈ PrimeSpectrum.basicOpen (h ^ (n + 1)) → g ∈ Ideal.primeCompl x.asIdeal\n\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nc : R\nhc : c * g = h ^ (n + 1)\nbasic_opens_eq : PrimeSpectrum.basicOpen (h ^ (n + 1)) = PrimeSpectrum.basicOpen h\ni_basic_open : PrimeSpectrum.basicOpen (h ^ (n + 1)) ⟶ V\n⊢ const R (f * c) (h ^ (n + 1)) (PrimeSpectrum.basicOpen (h ^ (n + 1)))\n (_ :\n ∀ (y : ↑(PrimeSpectrum.Top R)),\n y ∈ PrimeSpectrum.basicOpen (h ^ (n + 1)) → y ∈ PrimeSpectrum.basicOpen (h ^ (n + 1))) =\n const R f g (PrimeSpectrum.basicOpen (h ^ (n + 1)))\n ?intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.hv",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nc : R\nhc : c * g = h ^ (n + 1)\nbasic_opens_eq : PrimeSpectrum.basicOpen (h ^ (n + 1)) = PrimeSpectrum.basicOpen h\ni_basic_open : PrimeSpectrum.basicOpen (h ^ (n + 1)) ⟶ V\n⊢ const R (f * c) (h ^ (n + 1)) (PrimeSpectrum.basicOpen (h ^ (n + 1)))\n (_ :\n ∀ (y : ↑(PrimeSpectrum.Top R)),\n y ∈ PrimeSpectrum.basicOpen (h ^ (n + 1)) → y ∈ PrimeSpectrum.basicOpen (h ^ (n + 1))) =\n const R f g (PrimeSpectrum.basicOpen (h ^ (n + 1)))\n ?intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.hv\n\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.hv\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nc : R\nhc : c * g = h ^ (n + 1)\nbasic_opens_eq : PrimeSpectrum.basicOpen (h ^ (n + 1)) = PrimeSpectrum.basicOpen h\ni_basic_open : PrimeSpectrum.basicOpen (h ^ (n + 1)) ⟶ V\n⊢ ∀ (x : ↑(PrimeSpectrum.Top R)), x ∈ PrimeSpectrum.basicOpen (h ^ (n + 1)) → g ∈ Ideal.primeCompl x.asIdeal",
"tactic": "swap"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nc : R\nhc : c * g = h ^ (n + 1)\nbasic_opens_eq : PrimeSpectrum.basicOpen (h ^ (n + 1)) = PrimeSpectrum.basicOpen h\ni_basic_open : PrimeSpectrum.basicOpen (h ^ (n + 1)) ⟶ V\n⊢ f * c * g = f * h ^ (n + 1)",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nc : R\nhc : c * g = h ^ (n + 1)\nbasic_opens_eq : PrimeSpectrum.basicOpen (h ^ (n + 1)) = PrimeSpectrum.basicOpen h\ni_basic_open : PrimeSpectrum.basicOpen (h ^ (n + 1)) ⟶ V\n⊢ const R (f * c) (h ^ (n + 1)) (PrimeSpectrum.basicOpen (h ^ (n + 1)))\n (_ :\n ∀ (y : ↑(PrimeSpectrum.Top R)),\n y ∈ PrimeSpectrum.basicOpen (h ^ (n + 1)) → y ∈ PrimeSpectrum.basicOpen (h ^ (n + 1))) =\n const R f g (PrimeSpectrum.basicOpen (h ^ (n + 1)))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h ^ (n + 1)) → y ∈ ↑(PrimeSpectrum.basicOpen g))",
"tactic": "apply const_ext"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nc : R\nhc : c * g = h ^ (n + 1)\nbasic_opens_eq : PrimeSpectrum.basicOpen (h ^ (n + 1)) = PrimeSpectrum.basicOpen h\ni_basic_open : PrimeSpectrum.basicOpen (h ^ (n + 1)) ⟶ V\n⊢ f * c * g = f * h ^ (n + 1)",
"tactic": "rw [mul_assoc f c g, hc]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nc : R\nhc : c * g = h ^ (n + 1)\n⊢ 0 < n + 1",
"tactic": "linarith"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.hv\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nc : R\nhc : c * g = h ^ (n + 1)\nbasic_opens_eq : PrimeSpectrum.basicOpen (h ^ (n + 1)) = PrimeSpectrum.basicOpen h\ni_basic_open : PrimeSpectrum.basicOpen (h ^ (n + 1)) ⟶ V\ny : ↑(PrimeSpectrum.Top R)\nhy : y ∈ PrimeSpectrum.basicOpen (h ^ (n + 1))\n⊢ g ∈ Ideal.primeCompl y.asIdeal",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.hv\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nc : R\nhc : c * g = h ^ (n + 1)\nbasic_opens_eq : PrimeSpectrum.basicOpen (h ^ (n + 1)) = PrimeSpectrum.basicOpen h\ni_basic_open : PrimeSpectrum.basicOpen (h ^ (n + 1)) ⟶ V\n⊢ ∀ (x : ↑(PrimeSpectrum.Top R)), x ∈ PrimeSpectrum.basicOpen (h ^ (n + 1)) → g ∈ Ideal.primeCompl x.asIdeal",
"tactic": "intro y hy"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.hv\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nc : R\nhc : c * g = h ^ (n + 1)\nbasic_opens_eq : PrimeSpectrum.basicOpen (h ^ (n + 1)) = PrimeSpectrum.basicOpen h\ni_basic_open : PrimeSpectrum.basicOpen (h ^ (n + 1)) ⟶ V\ny : ↑(PrimeSpectrum.Top R)\nhy : y ∈ PrimeSpectrum.basicOpen h\n⊢ g ∈ Ideal.primeCompl y.asIdeal",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.hv\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nc : R\nhc : c * g = h ^ (n + 1)\nbasic_opens_eq : PrimeSpectrum.basicOpen (h ^ (n + 1)) = PrimeSpectrum.basicOpen h\ni_basic_open : PrimeSpectrum.basicOpen (h ^ (n + 1)) ⟶ V\ny : ↑(PrimeSpectrum.Top R)\nhy : y ∈ PrimeSpectrum.basicOpen (h ^ (n + 1))\n⊢ g ∈ Ideal.primeCompl y.asIdeal",
"tactic": "rw [basic_opens_eq] at hy"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.hv\nR : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\ns : ↑((structureSheaf R).val.obj U.op)\nx : { x // x ∈ U }\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑x ∈ V.carrier\niVU : V ⟶ U\nf g : R\nhVDg : V ≤ PrimeSpectrum.basicOpen g\ns_eq : const R f g V hVDg = ↑((structureSheaf R).val.map iVU.op) s\nh : R\nhxDh : ↑x ∈ (fun r => ↑(PrimeSpectrum.basicOpen r)) h\nhDhV : PrimeSpectrum.basicOpen h ≤ V\nn : ℕ\nc : R\nhc : c * g = h ^ (n + 1)\nbasic_opens_eq : PrimeSpectrum.basicOpen (h ^ (n + 1)) = PrimeSpectrum.basicOpen h\ni_basic_open : PrimeSpectrum.basicOpen (h ^ (n + 1)) ⟶ V\ny : ↑(PrimeSpectrum.Top R)\nhy : y ∈ PrimeSpectrum.basicOpen h\n⊢ g ∈ Ideal.primeCompl y.asIdeal",
"tactic": "exact (Set.Subset.trans hDhV hVDg : _) hy"
}
] | [
722,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
685,
1
] |
Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean | MeasureTheory.exists_upperSemicontinuous_le_lintegral_le | [
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nfs : α →ₛ ℝ≥0\nfs_le_f : ∀ (x : α), ↑fs x ≤ f x\nint_fs : (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(↑fs x) ∂μ) + ε / 2\n⊢ ∃ g, (∀ (x : α), g x ≤ f x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\n⊢ ∃ g, (∀ (x : α), g x ≤ f x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε",
"tactic": "obtain ⟨fs, fs_le_f, int_fs⟩ :\n ∃ fs : α →ₛ ℝ≥0, (∀ x, fs x ≤ f x) ∧ (∫⁻ x, f x ∂μ) ≤ (∫⁻ x, fs x ∂μ) + ε / 2 := by\n have aux := ENNReal.lt_add_right int_f (ENNReal.half_pos ε0).ne'\n conv_rhs at aux => rw [lintegral_eq_nnreal (fun x => (f x : ℝ≥0∞)) μ]\n erw [ENNReal.biSup_add] at aux <;> [skip; exact ⟨0, fun x => by simp⟩]\n simp only [lt_iSup_iff] at aux\n rcases aux with ⟨fs, fs_le_f, int_fs⟩\n refine' ⟨fs, fun x => by simpa only [ENNReal.coe_le_coe] using fs_le_f x, _⟩\n convert int_fs.le\n rw [← SimpleFunc.lintegral_eq_lintegral]\n rfl"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nfs : α →ₛ ℝ≥0\nfs_le_f : ∀ (x : α), ↑fs x ≤ f x\nint_fs : (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(↑fs x) ∂μ) + ε / 2\nint_fs_lt_top : (∫⁻ (x : α), ↑(↑fs x) ∂μ) ≠ ⊤\n⊢ ∃ g, (∀ (x : α), g x ≤ f x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε",
"state_before": "case intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nfs : α →ₛ ℝ≥0\nfs_le_f : ∀ (x : α), ↑fs x ≤ f x\nint_fs : (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(↑fs x) ∂μ) + ε / 2\n⊢ ∃ g, (∀ (x : α), g x ≤ f x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε",
"tactic": "have int_fs_lt_top : (∫⁻ x, fs x ∂μ) ≠ ∞ := by\n refine' ne_top_of_le_ne_top int_f (lintegral_mono fun x => _)\n simpa only [ENNReal.coe_le_coe] using fs_le_f x"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nfs : α →ₛ ℝ≥0\nfs_le_f : ∀ (x : α), ↑fs x ≤ f x\nint_fs : (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(↑fs x) ∂μ) + ε / 2\nint_fs_lt_top : (∫⁻ (x : α), ↑(↑fs x) ∂μ) ≠ ⊤\ng : α → ℝ≥0\ng_le_fs : ∀ (x : α), g x ≤ ↑fs x\ngcont : UpperSemicontinuous g\ngint : (∫⁻ (x : α), ↑(↑fs x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε / 2\n⊢ ∃ g, (∀ (x : α), g x ≤ f x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε",
"state_before": "case intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nfs : α →ₛ ℝ≥0\nfs_le_f : ∀ (x : α), ↑fs x ≤ f x\nint_fs : (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(↑fs x) ∂μ) + ε / 2\nint_fs_lt_top : (∫⁻ (x : α), ↑(↑fs x) ∂μ) ≠ ⊤\n⊢ ∃ g, (∀ (x : α), g x ≤ f x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε",
"tactic": "obtain ⟨g, g_le_fs, gcont, gint⟩ :\n ∃ g : α → ℝ≥0,\n (∀ x, g x ≤ fs x) ∧ UpperSemicontinuous g ∧ (∫⁻ x, fs x ∂μ) ≤ (∫⁻ x, g x ∂μ) + ε / 2 :=\n fs.exists_upperSemicontinuous_le_lintegral_le int_fs_lt_top (ENNReal.half_pos ε0).ne'"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nfs : α →ₛ ℝ≥0\nfs_le_f : ∀ (x : α), ↑fs x ≤ f x\nint_fs : (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(↑fs x) ∂μ) + ε / 2\nint_fs_lt_top : (∫⁻ (x : α), ↑(↑fs x) ∂μ) ≠ ⊤\ng : α → ℝ≥0\ng_le_fs : ∀ (x : α), g x ≤ ↑fs x\ngcont : UpperSemicontinuous g\ngint : (∫⁻ (x : α), ↑(↑fs x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε / 2\n⊢ (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nfs : α →ₛ ℝ≥0\nfs_le_f : ∀ (x : α), ↑fs x ≤ f x\nint_fs : (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(↑fs x) ∂μ) + ε / 2\nint_fs_lt_top : (∫⁻ (x : α), ↑(↑fs x) ∂μ) ≠ ⊤\ng : α → ℝ≥0\ng_le_fs : ∀ (x : α), g x ≤ ↑fs x\ngcont : UpperSemicontinuous g\ngint : (∫⁻ (x : α), ↑(↑fs x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε / 2\n⊢ ∃ g, (∀ (x : α), g x ≤ f x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε",
"tactic": "refine' ⟨g, fun x => (g_le_fs x).trans (fs_le_f x), gcont, _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nfs : α →ₛ ℝ≥0\nfs_le_f : ∀ (x : α), ↑fs x ≤ f x\nint_fs : (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(↑fs x) ∂μ) + ε / 2\nint_fs_lt_top : (∫⁻ (x : α), ↑(↑fs x) ∂μ) ≠ ⊤\ng : α → ℝ≥0\ng_le_fs : ∀ (x : α), g x ≤ ↑fs x\ngcont : UpperSemicontinuous g\ngint : (∫⁻ (x : α), ↑(↑fs x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε / 2\n⊢ (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε",
"tactic": "calc\n (∫⁻ x, f x ∂μ) ≤ (∫⁻ x, fs x ∂μ) + ε / 2 := int_fs\n _ ≤ (∫⁻ x, g x ∂μ) + ε / 2 + ε / 2 := (add_le_add gint le_rfl)\n _ = (∫⁻ x, g x ∂μ) + ε := by rw [add_assoc, ENNReal.add_halves]"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\naux : (∫⁻ (x : α), ↑(f x) ∂μ) < (∫⁻ (x : α), ↑(f x) ∂μ) + ε / 2\n⊢ ∃ fs, (∀ (x : α), ↑fs x ≤ f x) ∧ (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(↑fs x) ∂μ) + ε / 2",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\n⊢ ∃ fs, (∀ (x : α), ↑fs x ≤ f x) ∧ (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(↑fs x) ∂μ) + ε / 2",
"tactic": "have aux := ENNReal.lt_add_right int_f (ENNReal.half_pos ε0).ne'"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\naux :\n (∫⁻ (x : α), ↑(f x) ∂μ) <\n (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ ↑(f x)), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) + ε / 2\n⊢ ∃ fs, (∀ (x : α), ↑fs x ≤ f x) ∧ (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(↑fs x) ∂μ) + ε / 2",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\naux : (∫⁻ (x : α), ↑(f x) ∂μ) < (∫⁻ (x : α), ↑(f x) ∂μ) + ε / 2\n⊢ ∃ fs, (∀ (x : α), ↑fs x ≤ f x) ∧ (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(↑fs x) ∂μ) + ε / 2",
"tactic": "conv_rhs at aux => rw [lintegral_eq_nnreal (fun x => (f x : ℝ≥0∞)) μ]"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\naux :\n (∫⁻ (x : α), ↑(f x) ∂μ) <\n ⨆ (i : α →ₛ ℝ≥0) (_ : i ∈ fun φ => ∀ (x : α), ↑(↑φ x) ≤ ↑(f x)),\n SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) μ + ε / 2\n⊢ ∃ fs, (∀ (x : α), ↑fs x ≤ f x) ∧ (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(↑fs x) ∂μ) + ε / 2",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\naux :\n (∫⁻ (x : α), ↑(f x) ∂μ) <\n (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ ↑(f x)), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) + ε / 2\n⊢ ∃ fs, (∀ (x : α), ↑fs x ≤ f x) ∧ (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(↑fs x) ∂μ) + ε / 2",
"tactic": "erw [ENNReal.biSup_add] at aux <;> [skip; exact ⟨0, fun x => by simp⟩]"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\naux : ∃ i i_1, (∫⁻ (x : α), ↑(f x) ∂μ) < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) μ + ε / 2\n⊢ ∃ fs, (∀ (x : α), ↑fs x ≤ f x) ∧ (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(↑fs x) ∂μ) + ε / 2",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\naux :\n (∫⁻ (x : α), ↑(f x) ∂μ) <\n ⨆ (i : α →ₛ ℝ≥0) (_ : i ∈ fun φ => ∀ (x : α), ↑(↑φ x) ≤ ↑(f x)),\n SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) μ + ε / 2\n⊢ ∃ fs, (∀ (x : α), ↑fs x ≤ f x) ∧ (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(↑fs x) ∂μ) + ε / 2",
"tactic": "simp only [lt_iSup_iff] at aux"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nfs : α →ₛ ℝ≥0\nfs_le_f : fs ∈ fun φ => ∀ (x : α), ↑(↑φ x) ≤ ↑(f x)\nint_fs : (∫⁻ (x : α), ↑(f x) ∂μ) < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some fs) μ + ε / 2\n⊢ ∃ fs, (∀ (x : α), ↑fs x ≤ f x) ∧ (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(↑fs x) ∂μ) + ε / 2",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\naux : ∃ i i_1, (∫⁻ (x : α), ↑(f x) ∂μ) < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) μ + ε / 2\n⊢ ∃ fs, (∀ (x : α), ↑fs x ≤ f x) ∧ (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(↑fs x) ∂μ) + ε / 2",
"tactic": "rcases aux with ⟨fs, fs_le_f, int_fs⟩"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nfs : α →ₛ ℝ≥0\nfs_le_f : fs ∈ fun φ => ∀ (x : α), ↑(↑φ x) ≤ ↑(f x)\nint_fs : (∫⁻ (x : α), ↑(f x) ∂μ) < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some fs) μ + ε / 2\n⊢ (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(↑fs x) ∂μ) + ε / 2",
"state_before": "case intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nfs : α →ₛ ℝ≥0\nfs_le_f : fs ∈ fun φ => ∀ (x : α), ↑(↑φ x) ≤ ↑(f x)\nint_fs : (∫⁻ (x : α), ↑(f x) ∂μ) < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some fs) μ + ε / 2\n⊢ ∃ fs, (∀ (x : α), ↑fs x ≤ f x) ∧ (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(↑fs x) ∂μ) + ε / 2",
"tactic": "refine' ⟨fs, fun x => by simpa only [ENNReal.coe_le_coe] using fs_le_f x, _⟩"
},
{
"state_after": "case h.e'_4.h.e'_5\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nfs : α →ₛ ℝ≥0\nfs_le_f : fs ∈ fun φ => ∀ (x : α), ↑(↑φ x) ≤ ↑(f x)\nint_fs : (∫⁻ (x : α), ↑(f x) ∂μ) < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some fs) μ + ε / 2\n⊢ (∫⁻ (x : α), ↑(↑fs x) ∂μ) = SimpleFunc.lintegral (SimpleFunc.map ENNReal.some fs) μ",
"state_before": "case intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nfs : α →ₛ ℝ≥0\nfs_le_f : fs ∈ fun φ => ∀ (x : α), ↑(↑φ x) ≤ ↑(f x)\nint_fs : (∫⁻ (x : α), ↑(f x) ∂μ) < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some fs) μ + ε / 2\n⊢ (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(↑fs x) ∂μ) + ε / 2",
"tactic": "convert int_fs.le"
},
{
"state_after": "case h.e'_4.h.e'_5\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nfs : α →ₛ ℝ≥0\nfs_le_f : fs ∈ fun φ => ∀ (x : α), ↑(↑φ x) ≤ ↑(f x)\nint_fs : (∫⁻ (x : α), ↑(f x) ∂μ) < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some fs) μ + ε / 2\n⊢ (∫⁻ (x : α), ↑(↑fs x) ∂μ) = ∫⁻ (a : α), ↑(SimpleFunc.map ENNReal.some fs) a ∂μ",
"state_before": "case h.e'_4.h.e'_5\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nfs : α →ₛ ℝ≥0\nfs_le_f : fs ∈ fun φ => ∀ (x : α), ↑(↑φ x) ≤ ↑(f x)\nint_fs : (∫⁻ (x : α), ↑(f x) ∂μ) < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some fs) μ + ε / 2\n⊢ (∫⁻ (x : α), ↑(↑fs x) ∂μ) = SimpleFunc.lintegral (SimpleFunc.map ENNReal.some fs) μ",
"tactic": "rw [← SimpleFunc.lintegral_eq_lintegral]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4.h.e'_5\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nfs : α →ₛ ℝ≥0\nfs_le_f : fs ∈ fun φ => ∀ (x : α), ↑(↑φ x) ≤ ↑(f x)\nint_fs : (∫⁻ (x : α), ↑(f x) ∂μ) < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some fs) μ + ε / 2\n⊢ (∫⁻ (x : α), ↑(↑fs x) ∂μ) = ∫⁻ (a : α), ↑(SimpleFunc.map ENNReal.some fs) a ∂μ",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\naux :\n (∫⁻ (x : α), ↑(f x) ∂μ) <\n (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ ↑(f x)), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) + ε / 2\nx : α\n⊢ ↑(↑0 x) ≤ ↑(f x)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nfs : α →ₛ ℝ≥0\nfs_le_f : fs ∈ fun φ => ∀ (x : α), ↑(↑φ x) ≤ ↑(f x)\nint_fs : (∫⁻ (x : α), ↑(f x) ∂μ) < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some fs) μ + ε / 2\nx : α\n⊢ ↑fs x ≤ f x",
"tactic": "simpa only [ENNReal.coe_le_coe] using fs_le_f x"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nfs : α →ₛ ℝ≥0\nfs_le_f : ∀ (x : α), ↑fs x ≤ f x\nint_fs : (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(↑fs x) ∂μ) + ε / 2\nx : α\n⊢ ↑(↑fs x) ≤ ↑(f x)",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nfs : α →ₛ ℝ≥0\nfs_le_f : ∀ (x : α), ↑fs x ≤ f x\nint_fs : (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(↑fs x) ∂μ) + ε / 2\n⊢ (∫⁻ (x : α), ↑(↑fs x) ∂μ) ≠ ⊤",
"tactic": "refine' ne_top_of_le_ne_top int_f (lintegral_mono fun x => _)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nfs : α →ₛ ℝ≥0\nfs_le_f : ∀ (x : α), ↑fs x ≤ f x\nint_fs : (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(↑fs x) ∂μ) + ε / 2\nx : α\n⊢ ↑(↑fs x) ≤ ↑(f x)",
"tactic": "simpa only [ENNReal.coe_le_coe] using fs_le_f x"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α → ℝ≥0\nint_f : (∫⁻ (x : α), ↑(f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nfs : α →ₛ ℝ≥0\nfs_le_f : ∀ (x : α), ↑fs x ≤ f x\nint_fs : (∫⁻ (x : α), ↑(f x) ∂μ) ≤ (∫⁻ (x : α), ↑(↑fs x) ∂μ) + ε / 2\nint_fs_lt_top : (∫⁻ (x : α), ↑(↑fs x) ∂μ) ≠ ⊤\ng : α → ℝ≥0\ng_le_fs : ∀ (x : α), g x ≤ ↑fs x\ngcont : UpperSemicontinuous g\ngint : (∫⁻ (x : α), ↑(↑fs x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε / 2\n⊢ (∫⁻ (x : α), ↑(g x) ∂μ) + ε / 2 + ε / 2 = (∫⁻ (x : α), ↑(g x) ∂μ) + ε",
"tactic": "rw [add_assoc, ENNReal.add_halves]"
}
] | [
422,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
395,
1
] |
Mathlib/Data/Multiset/Basic.lean | Multiset.filterMap_some | [] | [
2169,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2168,
1
] |
Mathlib/Geometry/Manifold/ChartedSpace.lean | ChartedSpace.locallyCompact | [
{
"state_after": "H : Type u\nH' : Type ?u.45087\nM : Type u_1\nM' : Type ?u.45093\nM'' : Type ?u.45096\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : LocallyCompactSpace H\nthis :\n ∀ (x : M),\n HasBasis (𝓝 x) (fun s => s ∈ 𝓝 (↑(chartAt H x) x) ∧ IsCompact s ∧ s ⊆ (chartAt H x).toLocalEquiv.target) fun s =>\n ↑(LocalHomeomorph.symm (chartAt H x)) '' s\n⊢ LocallyCompactSpace M",
"state_before": "H : Type u\nH' : Type ?u.45087\nM : Type u_1\nM' : Type ?u.45093\nM'' : Type ?u.45096\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : LocallyCompactSpace H\n⊢ LocallyCompactSpace M",
"tactic": "have : ∀ x : M, (𝓝 x).HasBasis\n (fun s ↦ s ∈ 𝓝 (chartAt H x x) ∧ IsCompact s ∧ s ⊆ (chartAt H x).target)\n fun s ↦ (chartAt H x).symm '' s := fun x ↦ by\n rw [← (chartAt H x).symm_map_nhds_eq (mem_chart_source H x)]\n exact ((compact_basis_nhds (chartAt H x x)).hasBasis_self_subset\n (chart_target_mem_nhds H x)).map _"
},
{
"state_after": "H : Type u\nH' : Type ?u.45087\nM : Type u_1\nM' : Type ?u.45093\nM'' : Type ?u.45096\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : LocallyCompactSpace H\nthis :\n ∀ (x : M),\n HasBasis (𝓝 x) (fun s => s ∈ 𝓝 (↑(chartAt H x) x) ∧ IsCompact s ∧ s ⊆ (chartAt H x).toLocalEquiv.target) fun s =>\n ↑(LocalHomeomorph.symm (chartAt H x)) '' s\n⊢ ∀ (x : M) (i : Set H),\n i ∈ 𝓝 (↑(chartAt H x) x) ∧ IsCompact i ∧ i ⊆ (chartAt H x).toLocalEquiv.target →\n IsCompact (↑(LocalHomeomorph.symm (chartAt H x)) '' i)",
"state_before": "H : Type u\nH' : Type ?u.45087\nM : Type u_1\nM' : Type ?u.45093\nM'' : Type ?u.45096\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : LocallyCompactSpace H\nthis :\n ∀ (x : M),\n HasBasis (𝓝 x) (fun s => s ∈ 𝓝 (↑(chartAt H x) x) ∧ IsCompact s ∧ s ⊆ (chartAt H x).toLocalEquiv.target) fun s =>\n ↑(LocalHomeomorph.symm (chartAt H x)) '' s\n⊢ LocallyCompactSpace M",
"tactic": "refine locallyCompactSpace_of_hasBasis this ?_"
},
{
"state_after": "case intro.intro\nH : Type u\nH' : Type ?u.45087\nM : Type u_1\nM' : Type ?u.45093\nM'' : Type ?u.45096\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : LocallyCompactSpace H\nthis :\n ∀ (x : M),\n HasBasis (𝓝 x) (fun s => s ∈ 𝓝 (↑(chartAt H x) x) ∧ IsCompact s ∧ s ⊆ (chartAt H x).toLocalEquiv.target) fun s =>\n ↑(LocalHomeomorph.symm (chartAt H x)) '' s\nx : M\ns : Set H\nleft✝ : s ∈ 𝓝 (↑(chartAt H x) x)\nh₂ : IsCompact s\nh₃ : s ⊆ (chartAt H x).toLocalEquiv.target\n⊢ IsCompact (↑(LocalHomeomorph.symm (chartAt H x)) '' s)",
"state_before": "H : Type u\nH' : Type ?u.45087\nM : Type u_1\nM' : Type ?u.45093\nM'' : Type ?u.45096\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : LocallyCompactSpace H\nthis :\n ∀ (x : M),\n HasBasis (𝓝 x) (fun s => s ∈ 𝓝 (↑(chartAt H x) x) ∧ IsCompact s ∧ s ⊆ (chartAt H x).toLocalEquiv.target) fun s =>\n ↑(LocalHomeomorph.symm (chartAt H x)) '' s\n⊢ ∀ (x : M) (i : Set H),\n i ∈ 𝓝 (↑(chartAt H x) x) ∧ IsCompact i ∧ i ⊆ (chartAt H x).toLocalEquiv.target →\n IsCompact (↑(LocalHomeomorph.symm (chartAt H x)) '' i)",
"tactic": "rintro x s ⟨_, h₂, h₃⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nH : Type u\nH' : Type ?u.45087\nM : Type u_1\nM' : Type ?u.45093\nM'' : Type ?u.45096\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : LocallyCompactSpace H\nthis :\n ∀ (x : M),\n HasBasis (𝓝 x) (fun s => s ∈ 𝓝 (↑(chartAt H x) x) ∧ IsCompact s ∧ s ⊆ (chartAt H x).toLocalEquiv.target) fun s =>\n ↑(LocalHomeomorph.symm (chartAt H x)) '' s\nx : M\ns : Set H\nleft✝ : s ∈ 𝓝 (↑(chartAt H x) x)\nh₂ : IsCompact s\nh₃ : s ⊆ (chartAt H x).toLocalEquiv.target\n⊢ IsCompact (↑(LocalHomeomorph.symm (chartAt H x)) '' s)",
"tactic": "exact h₂.image_of_continuousOn ((chartAt H x).continuousOn_symm.mono h₃)"
},
{
"state_after": "H : Type u\nH' : Type ?u.45087\nM : Type u_1\nM' : Type ?u.45093\nM'' : Type ?u.45096\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : LocallyCompactSpace H\nx : M\n⊢ HasBasis (map (↑(LocalHomeomorph.symm (chartAt H x))) (𝓝 (↑(chartAt H x) x)))\n (fun s => s ∈ 𝓝 (↑(chartAt H x) x) ∧ IsCompact s ∧ s ⊆ (chartAt H x).toLocalEquiv.target) fun s =>\n ↑(LocalHomeomorph.symm (chartAt H x)) '' s",
"state_before": "H : Type u\nH' : Type ?u.45087\nM : Type u_1\nM' : Type ?u.45093\nM'' : Type ?u.45096\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : LocallyCompactSpace H\nx : M\n⊢ HasBasis (𝓝 x) (fun s => s ∈ 𝓝 (↑(chartAt H x) x) ∧ IsCompact s ∧ s ⊆ (chartAt H x).toLocalEquiv.target) fun s =>\n ↑(LocalHomeomorph.symm (chartAt H x)) '' s",
"tactic": "rw [← (chartAt H x).symm_map_nhds_eq (mem_chart_source H x)]"
},
{
"state_after": "no goals",
"state_before": "H : Type u\nH' : Type ?u.45087\nM : Type u_1\nM' : Type ?u.45093\nM'' : Type ?u.45096\ninst✝³ : TopologicalSpace H\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\ninst✝ : LocallyCompactSpace H\nx : M\n⊢ HasBasis (map (↑(LocalHomeomorph.symm (chartAt H x))) (𝓝 (↑(chartAt H x) x)))\n (fun s => s ∈ 𝓝 (↑(chartAt H x) x) ∧ IsCompact s ∧ s ⊆ (chartAt H x).toLocalEquiv.target) fun s =>\n ↑(LocalHomeomorph.symm (chartAt H x)) '' s",
"tactic": "exact ((compact_basis_nhds (chartAt H x x)).hasBasis_self_subset\n (chart_target_mem_nhds H x)).map _"
}
] | [
612,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
603,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean | CategoryTheory.Limits.diagonalObjPullbackFstIso_hom_fst_fst | [
{
"state_after": "C : Type u_2\ninst✝⁵ : Category C\nX✝ Y✝ Z✝ : C\ninst✝⁴ : HasPullbacks C\nS T : C\nf✝ : X✝ ⟶ T\ng✝ : Y✝ ⟶ T\ni : T ⟶ S\ninst✝³ : HasPullback i i\ninst✝² : HasPullback f✝ g✝\ninst✝¹ : HasPullback (f✝ ≫ i) (g✝ ≫ i)\ninst✝ :\n HasPullback (diagonal i)\n (map (f✝ ≫ i) (g✝ ≫ i) i i f✝ g✝ (𝟙 S) (_ : (f✝ ≫ i) ≫ 𝟙 S = f✝ ≫ i) (_ : (g✝ ≫ i) ≫ 𝟙 S = g✝ ≫ i))\nX Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\n⊢ (pullbackRightPullbackFstIso f g fst ≪≫\n congrHom (_ : fst ≫ f = snd ≫ g) (_ : g = g) ≪≫\n pullbackAssoc f g g g ≪≫ pullbackSymmetry f (fst ≫ g) ≪≫ congrHom (_ : fst ≫ g = snd ≫ g) (_ : f = f)).hom ≫\n fst ≫ fst =\n fst ≫ snd",
"state_before": "C : Type u_2\ninst✝⁵ : Category C\nX✝ Y✝ Z✝ : C\ninst✝⁴ : HasPullbacks C\nS T : C\nf✝ : X✝ ⟶ T\ng✝ : Y✝ ⟶ T\ni : T ⟶ S\ninst✝³ : HasPullback i i\ninst✝² : HasPullback f✝ g✝\ninst✝¹ : HasPullback (f✝ ≫ i) (g✝ ≫ i)\ninst✝ :\n HasPullback (diagonal i)\n (map (f✝ ≫ i) (g✝ ≫ i) i i f✝ g✝ (𝟙 S) (_ : (f✝ ≫ i) ≫ 𝟙 S = f✝ ≫ i) (_ : (g✝ ≫ i) ≫ 𝟙 S = g✝ ≫ i))\nX Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\n⊢ (diagonalObjPullbackFstIso f g).hom ≫ fst ≫ fst = fst ≫ snd",
"tactic": "delta diagonalObjPullbackFstIso"
},
{
"state_after": "no goals",
"state_before": "C : Type u_2\ninst✝⁵ : Category C\nX✝ Y✝ Z✝ : C\ninst✝⁴ : HasPullbacks C\nS T : C\nf✝ : X✝ ⟶ T\ng✝ : Y✝ ⟶ T\ni : T ⟶ S\ninst✝³ : HasPullback i i\ninst✝² : HasPullback f✝ g✝\ninst✝¹ : HasPullback (f✝ ≫ i) (g✝ ≫ i)\ninst✝ :\n HasPullback (diagonal i)\n (map (f✝ ≫ i) (g✝ ≫ i) i i f✝ g✝ (𝟙 S) (_ : (f✝ ≫ i) ≫ 𝟙 S = f✝ ≫ i) (_ : (g✝ ≫ i) ≫ 𝟙 S = g✝ ≫ i))\nX Y Z : C\nf : X ⟶ Z\ng : Y ⟶ Z\n⊢ (pullbackRightPullbackFstIso f g fst ≪≫\n congrHom (_ : fst ≫ f = snd ≫ g) (_ : g = g) ≪≫\n pullbackAssoc f g g g ≪≫ pullbackSymmetry f (fst ≫ g) ≪≫ congrHom (_ : fst ≫ g = snd ≫ g) (_ : f = f)).hom ≫\n fst ≫ fst =\n fst ≫ snd",
"tactic": "simp"
}
] | [
309,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
305,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean | norm_iteratedFDeriv_clm_apply_const | [
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\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 ?u.4423078\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\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf : E → F →L[𝕜] G\nc : F\nx : E\nN : ℕ∞\nn : ℕ\nhf : ContDiff 𝕜 N f\nhn : ↑n ≤ N\n⊢ ‖iteratedFDerivWithin 𝕜 n (fun y => ↑(f y) c) univ x‖ ≤ ‖c‖ * ‖iteratedFDerivWithin 𝕜 n f univ x‖",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\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 ?u.4423078\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\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf : E → F →L[𝕜] G\nc : F\nx : E\nN : ℕ∞\nn : ℕ\nhf : ContDiff 𝕜 N f\nhn : ↑n ≤ N\n⊢ ‖iteratedFDeriv 𝕜 n (fun y => ↑(f y) c) x‖ ≤ ‖c‖ * ‖iteratedFDeriv 𝕜 n f x‖",
"tactic": "simp only [← iteratedFDerivWithin_univ]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\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 ?u.4423078\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\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf : E → F →L[𝕜] G\nc : F\nx : E\nN : ℕ∞\nn : ℕ\nhf : ContDiff 𝕜 N f\nhn : ↑n ≤ N\n⊢ ‖iteratedFDerivWithin 𝕜 n (fun y => ↑(f y) c) univ x‖ ≤ ‖c‖ * ‖iteratedFDerivWithin 𝕜 n f univ x‖",
"tactic": "exact norm_iteratedFDerivWithin_clm_apply_const hf.contDiffOn uniqueDiffOn_univ\n (Set.mem_univ x) hn"
}
] | [
2755,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2750,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean | FractionalIdeal.map_mul | [
{
"state_after": "R : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nP' : Type u_1\ninst✝³ : CommRing P'\ninst✝² : Algebra R P'\nloc' : IsLocalization S P'\nP'' : Type ?u.565642\ninst✝¹ : CommRing P''\ninst✝ : Algebra R P''\nloc'' : IsLocalization S P''\nI J : FractionalIdeal S P\ng : P →ₐ[R] P'\n⊢ map g { val := ↑I * ↑J, property := (_ : IsFractional S (↑I * ↑J)) } =\n { val := ↑(map g I) * ↑(map g J), property := (_ : IsFractional S (↑(map g I) * ↑(map g J))) }",
"state_before": "R : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nP' : Type u_1\ninst✝³ : CommRing P'\ninst✝² : Algebra R P'\nloc' : IsLocalization S P'\nP'' : Type ?u.565642\ninst✝¹ : CommRing P''\ninst✝ : Algebra R P''\nloc'' : IsLocalization S P''\nI J : FractionalIdeal S P\ng : P →ₐ[R] P'\n⊢ map g (I * J) = map g I * map g J",
"tactic": "simp only [mul_def]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝⁶ : CommRing R\nS : Submonoid R\nP : Type u_3\ninst✝⁵ : CommRing P\ninst✝⁴ : Algebra R P\nloc : IsLocalization S P\nP' : Type u_1\ninst✝³ : CommRing P'\ninst✝² : Algebra R P'\nloc' : IsLocalization S P'\nP'' : Type ?u.565642\ninst✝¹ : CommRing P''\ninst✝ : Algebra R P''\nloc'' : IsLocalization S P''\nI J : FractionalIdeal S P\ng : P →ₐ[R] P'\n⊢ map g { val := ↑I * ↑J, property := (_ : IsFractional S (↑I * ↑J)) } =\n { val := ↑(map g I) * ↑(map g J), property := (_ : IsFractional S (↑(map g I) * ↑(map g J))) }",
"tactic": "exact coeToSubmodule_injective (Submodule.map_mul _ _ _)"
}
] | [
786,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
784,
1
] |
Mathlib/Data/List/Count.lean | List.one_le_count_iff_mem | [] | [
249,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
249,
1
] |
Mathlib/Data/Set/Intervals/SurjOn.lean | surjOn_Ico_of_monotone_surjective | [
{
"state_after": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : PartialOrder β\nf : α → β\nh_mono : Monotone f\nh_surj : Surjective f\na b : α\nhab : a < b\n⊢ SurjOn f (Ico a b) (Ico (f a) (f b))\n\ncase inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : PartialOrder β\nf : α → β\nh_mono : Monotone f\nh_surj : Surjective f\na b : α\nhab : b ≤ a\n⊢ SurjOn f (Ico a b) (Ico (f a) (f b))",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : PartialOrder β\nf : α → β\nh_mono : Monotone f\nh_surj : Surjective f\na b : α\n⊢ SurjOn f (Ico a b) (Ico (f a) (f b))",
"tactic": "obtain hab | hab := lt_or_le a b"
},
{
"state_after": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : PartialOrder β\nf : α → β\nh_mono : Monotone f\nh_surj : Surjective f\na b : α\nhab : a < b\np : β\nhp : p ∈ Ico (f a) (f b)\n⊢ p ∈ f '' Ico a b",
"state_before": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : PartialOrder β\nf : α → β\nh_mono : Monotone f\nh_surj : Surjective f\na b : α\nhab : a < b\n⊢ SurjOn f (Ico a b) (Ico (f a) (f b))",
"tactic": "intro p hp"
},
{
"state_after": "case inl.inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : PartialOrder β\nf : α → β\nh_mono : Monotone f\nh_surj : Surjective f\na b : α\nhab : a < b\nhp : f a ∈ Ico (f a) (f b)\n⊢ f a ∈ f '' Ico a b\n\ncase inl.inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : PartialOrder β\nf : α → β\nh_mono : Monotone f\nh_surj : Surjective f\na b : α\nhab : a < b\np : β\nhp : p ∈ Ico (f a) (f b)\nhp' : p ∈ Ioo (f a) (f b)\n⊢ p ∈ f '' Ico a b",
"state_before": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : PartialOrder β\nf : α → β\nh_mono : Monotone f\nh_surj : Surjective f\na b : α\nhab : a < b\np : β\nhp : p ∈ Ico (f a) (f b)\n⊢ p ∈ f '' Ico a b",
"tactic": "rcases eq_left_or_mem_Ioo_of_mem_Ico hp with (rfl | hp')"
},
{
"state_after": "no goals",
"state_before": "case inl.inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : PartialOrder β\nf : α → β\nh_mono : Monotone f\nh_surj : Surjective f\na b : α\nhab : a < b\nhp : f a ∈ Ico (f a) (f b)\n⊢ f a ∈ f '' Ico a b",
"tactic": "exact mem_image_of_mem f (left_mem_Ico.mpr hab)"
},
{
"state_after": "case inl.inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : PartialOrder β\nf : α → β\nh_mono : Monotone f\nh_surj : Surjective f\na b : α\nhab : a < b\np : β\nhp : p ∈ Ico (f a) (f b)\nhp' : p ∈ Ioo (f a) (f b)\nthis : p ∈ f '' Ioo a b\n⊢ p ∈ f '' Ico a b",
"state_before": "case inl.inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : PartialOrder β\nf : α → β\nh_mono : Monotone f\nh_surj : Surjective f\na b : α\nhab : a < b\np : β\nhp : p ∈ Ico (f a) (f b)\nhp' : p ∈ Ioo (f a) (f b)\n⊢ p ∈ f '' Ico a b",
"tactic": "have := surjOn_Ioo_of_monotone_surjective h_mono h_surj a b hp'"
},
{
"state_after": "no goals",
"state_before": "case inl.inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : PartialOrder β\nf : α → β\nh_mono : Monotone f\nh_surj : Surjective f\na b : α\nhab : a < b\np : β\nhp : p ∈ Ico (f a) (f b)\nhp' : p ∈ Ioo (f a) (f b)\nthis : p ∈ f '' Ioo a b\n⊢ p ∈ f '' Ico a b",
"tactic": "exact image_subset f Ioo_subset_Ico_self this"
},
{
"state_after": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : PartialOrder β\nf : α → β\nh_mono : Monotone f\nh_surj : Surjective f\na b : α\nhab : b ≤ a\n⊢ SurjOn f (Ico a b) ∅",
"state_before": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : PartialOrder β\nf : α → β\nh_mono : Monotone f\nh_surj : Surjective f\na b : α\nhab : b ≤ a\n⊢ SurjOn f (Ico a b) (Ico (f a) (f b))",
"tactic": "rw [Ico_eq_empty (h_mono hab).not_lt]"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : PartialOrder β\nf : α → β\nh_mono : Monotone f\nh_surj : Surjective f\na b : α\nhab : b ≤ a\n⊢ SurjOn f (Ico a b) ∅",
"tactic": "exact surjOn_empty f _"
}
] | [
48,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
39,
1
] |
Mathlib/Combinatorics/SimpleGraph/Prod.lean | SimpleGraph.Walk.ofBoxProdLeft_boxProdLeft | [
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.57688\nG : SimpleGraph α\nH : SimpleGraph β\nb : β\ninst✝¹ : DecidableEq β\ninst✝ : DecidableRel G.Adj\na₁ a₂ x z y : α\nh : Adj G x y\nw : Walk G y z\n⊢ cons (_ : Adj G (x, b).fst (↑(Embedding.toHom (boxProdLeft G H b)) y).fst) (ofBoxProdLeft (Walk.boxProdLeft H b w)) =\n cons' x y z h w\n\ncase hc\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.57688\nG : SimpleGraph α\nH : SimpleGraph β\nb : β\ninst✝¹ : DecidableEq β\ninst✝ : DecidableRel G.Adj\na₁ a₂ x z y : α\nh : Adj G x y\nw : Walk G y z\n⊢ Adj G (x, b).fst (↑(Embedding.toHom (boxProdLeft G H b)) y).fst ∧\n (x, b).snd = (↑(Embedding.toHom (boxProdLeft G H b)) y).snd\n\ncase hc\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.57688\nG : SimpleGraph α\nH : SimpleGraph β\nb : β\ninst✝¹ : DecidableEq β\ninst✝ : DecidableRel G.Adj\na₁ a₂ x z y : α\nh : Adj G x y\nw : Walk G y z\n⊢ Adj G (x, b).fst (↑(Embedding.toHom (boxProdLeft G H b)) y).fst ∧\n (x, b).snd = (↑(Embedding.toHom (boxProdLeft G H b)) y).snd",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.57688\nG : SimpleGraph α\nH : SimpleGraph β\nb : β\ninst✝¹ : DecidableEq β\ninst✝ : DecidableRel G.Adj\na₁ a₂ x z y : α\nh : Adj G x y\nw : Walk G y z\n⊢ ofBoxProdLeft (Walk.boxProdLeft H b (cons' x y z h w)) = cons' x y z h w",
"tactic": "rw [Walk.boxProdLeft, map_cons, ofBoxProdLeft, Or.by_cases, dif_pos, ← Walk.boxProdLeft]"
},
{
"state_after": "case hc\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.57688\nG : SimpleGraph α\nH : SimpleGraph β\nb : β\ninst✝¹ : DecidableEq β\ninst✝ : DecidableRel G.Adj\na₁ a₂ x z y : α\nh : Adj G x y\nw : Walk G y z\n⊢ Adj G (x, b).fst (↑(Embedding.toHom (boxProdLeft G H b)) y).fst ∧\n (x, b).snd = (↑(Embedding.toHom (boxProdLeft G H b)) y).snd\n\ncase hc\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.57688\nG : SimpleGraph α\nH : SimpleGraph β\nb : β\ninst✝¹ : DecidableEq β\ninst✝ : DecidableRel G.Adj\na₁ a₂ x z y : α\nh : Adj G x y\nw : Walk G y z\n⊢ Adj G (x, b).fst (↑(Embedding.toHom (boxProdLeft G H b)) y).fst ∧\n (x, b).snd = (↑(Embedding.toHom (boxProdLeft G H b)) y).snd",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.57688\nG : SimpleGraph α\nH : SimpleGraph β\nb : β\ninst✝¹ : DecidableEq β\ninst✝ : DecidableRel G.Adj\na₁ a₂ x z y : α\nh : Adj G x y\nw : Walk G y z\n⊢ cons (_ : Adj G (x, b).fst (↑(Embedding.toHom (boxProdLeft G H b)) y).fst) (ofBoxProdLeft (Walk.boxProdLeft H b w)) =\n cons' x y z h w\n\ncase hc\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.57688\nG : SimpleGraph α\nH : SimpleGraph β\nb : β\ninst✝¹ : DecidableEq β\ninst✝ : DecidableRel G.Adj\na₁ a₂ x z y : α\nh : Adj G x y\nw : Walk G y z\n⊢ Adj G (x, b).fst (↑(Embedding.toHom (boxProdLeft G H b)) y).fst ∧\n (x, b).snd = (↑(Embedding.toHom (boxProdLeft G H b)) y).snd\n\ncase hc\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.57688\nG : SimpleGraph α\nH : SimpleGraph β\nb : β\ninst✝¹ : DecidableEq β\ninst✝ : DecidableRel G.Adj\na₁ a₂ x z y : α\nh : Adj G x y\nw : Walk G y z\n⊢ Adj G (x, b).fst (↑(Embedding.toHom (boxProdLeft G H b)) y).fst ∧\n (x, b).snd = (↑(Embedding.toHom (boxProdLeft G H b)) y).snd",
"tactic": "simp [ofBoxProdLeft_boxProdLeft]"
},
{
"state_after": "no goals",
"state_before": "case hc\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.57688\nG : SimpleGraph α\nH : SimpleGraph β\nb : β\ninst✝¹ : DecidableEq β\ninst✝ : DecidableRel G.Adj\na₁ a₂ x z y : α\nh : Adj G x y\nw : Walk G y z\n⊢ Adj G (x, b).fst (↑(Embedding.toHom (boxProdLeft G H b)) y).fst ∧\n (x, b).snd = (↑(Embedding.toHom (boxProdLeft G H b)) y).snd\n\ncase hc\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.57688\nG : SimpleGraph α\nH : SimpleGraph β\nb : β\ninst✝¹ : DecidableEq β\ninst✝ : DecidableRel G.Adj\na₁ a₂ x z y : α\nh : Adj G x y\nw : Walk G y z\n⊢ Adj G (x, b).fst (↑(Embedding.toHom (boxProdLeft G H b)) y).fst ∧\n (x, b).snd = (↑(Embedding.toHom (boxProdLeft G H b)) y).snd",
"tactic": "exact ⟨h, rfl⟩"
}
] | [
154,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
148,
1
] |
Mathlib/Data/Int/Parity.lean | Int.odd_add' | [
{
"state_after": "no goals",
"state_before": "m n : ℤ\n⊢ Odd (m + n) ↔ (Odd n ↔ Even m)",
"tactic": "rw [add_comm, odd_add]"
}
] | [
180,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
179,
1
] |
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