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Mathlib/Combinatorics/Quiver/SingleObj.lean | Quiver.SingleObj.toPrefunctor_comp | []
| [
112,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
110,
1
]
|
Mathlib/CategoryTheory/Limits/Shapes/Types.lean | CategoryTheory.Limits.Types.pullbackIsoPullback_inv_snd | []
| [
606,
44
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
604,
1
]
|
Mathlib/MeasureTheory/Function/LpSeminorm.lean | MeasureTheory.snorm_one_smul_measure | [
{
"state_after": "α : Type u_1\nE : Type ?u.2301795\nF : Type u_2\nG : Type ?u.2301801\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\nc : ℝ≥0∞\n⊢ c ^ ENNReal.toReal (1 / 1) • snorm f 1 μ = c * snorm f 1 μ",
"state_before": "α : Type u_1\nE : Type ?u.2301795\nF : Type u_2\nG : Type ?u.2301801\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\nc : ℝ≥0∞\n⊢ snorm f 1 (c • μ) = c * snorm f 1 μ",
"tactic": "rw [@snorm_smul_measure_of_ne_top _ _ _ μ _ 1 (@ENNReal.coe_ne_top 1) f c]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.2301795\nF : Type u_2\nG : Type ?u.2301801\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\nc : ℝ≥0∞\n⊢ c ^ ENNReal.toReal (1 / 1) • snorm f 1 μ = c * snorm f 1 μ",
"tactic": "simp"
}
]
| [
643,
7
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
641,
1
]
|
Mathlib/CategoryTheory/Limits/Comma.lean | CategoryTheory.CostructuredArrow.epi_iff_epi_left | []
| [
300,
55
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
298,
1
]
|
Mathlib/NumberTheory/Padics/PadicVal.lean | padicValInt_self | []
| [
527,
33
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
526,
1
]
|
Mathlib/MeasureTheory/Measure/OuterMeasure.lean | MeasureTheory.OuterMeasure.biInf_apply' | [
{
"state_after": "no goals",
"state_before": "α : Type u_2\nι : Type u_1\nI : Set ι\nm : ι → OuterMeasure α\ns : Set α\nhs : Set.Nonempty s\n⊢ ↑(⨅ (i : ι) (_ : i ∈ I), m i) s =\n ⨅ (t : ℕ → Set α) (_ : s ⊆ iUnion t), ∑' (n : ℕ), ⨅ (i : ι) (_ : i ∈ I), ↑(m i) (t n)",
"tactic": "simp only [← iInf_subtype'', iInf_apply' _ hs]"
}
]
| [
1219,
49
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1217,
1
]
|
Mathlib/LinearAlgebra/Prod.lean | LinearEquiv.skewProd_apply | []
| [
819,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
818,
1
]
|
Mathlib/Computability/Primrec.lean | Primrec.nat_sqrt | []
| [
1570,
45
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1569,
1
]
|
Mathlib/Algebra/GradedMonoid.lean | SetLike.coe_gOne | []
| [
509,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
507,
1
]
|
Mathlib/RingTheory/PowerSeries/Basic.lean | MvPowerSeries.X_inv | [
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nR : Type ?u.2011299\nk : Type u_2\ninst✝ : Field k\ns : σ\n⊢ (X s)⁻¹ = 0",
"tactic": "rw [inv_eq_zero, constantCoeff_X]"
}
]
| [
1040,
36
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1039,
1
]
|
Mathlib/CategoryTheory/Idempotents/Karoubi.lean | CategoryTheory.Idempotents.Karoubi.eqToHom_f | [
{
"state_after": "C : Type u_1\ninst✝ : Category C\nP : Karoubi C\n⊢ (eqToHom (_ : P = P)).f = P.p ≫ eqToHom (_ : P.X = P.X)",
"state_before": "C : Type u_1\ninst✝ : Category C\nP Q : Karoubi C\nh : P = Q\n⊢ (eqToHom h).f = P.p ≫ eqToHom (_ : P.X = Q.X)",
"tactic": "subst h"
},
{
"state_after": "no goals",
"state_before": "C : Type u_1\ninst✝ : Category C\nP : Karoubi C\n⊢ (eqToHom (_ : P = P)).f = P.p ≫ eqToHom (_ : P.X = P.X)",
"tactic": "simp only [eqToHom_refl, Karoubi.id_eq, comp_id]"
}
]
| [
150,
51
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
147,
1
]
|
Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean | CategoryTheory.Limits.cokernel_not_iso_of_nonzero | [
{
"state_after": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : HasCokernel f\nw : f ≠ 0\nI : IsIso (cokernel.π f)\n⊢ Mono (cokernel.π f)",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : HasCokernel f\nw : f ≠ 0\nI : IsIso (cokernel.π f)\n⊢ Mono (cokernel.π f)",
"tactic": "skip"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ninst✝ : HasCokernel f\nw : f ≠ 0\nI : IsIso (cokernel.π f)\n⊢ Mono (cokernel.π f)",
"tactic": "infer_instance"
}
]
| [
868,
19
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
865,
1
]
|
Mathlib/Data/Nat/Squarefree.lean | Nat.minSqFac_prime | [
{
"state_after": "n d : ℕ\nh : minSqFac n = some d\nthis : MinSqFacProp n (minSqFac n)\n⊢ Prime d",
"state_before": "n d : ℕ\nh : minSqFac n = some d\n⊢ Prime d",
"tactic": "have := minSqFac_has_prop n"
},
{
"state_after": "n d : ℕ\nh : minSqFac n = some d\nthis : MinSqFacProp n (some d)\n⊢ Prime d",
"state_before": "n d : ℕ\nh : minSqFac n = some d\nthis : MinSqFacProp n (minSqFac n)\n⊢ Prime d",
"tactic": "rw [h] at this"
},
{
"state_after": "no goals",
"state_before": "n d : ℕ\nh : minSqFac n = some d\nthis : MinSqFacProp n (some d)\n⊢ Prime d",
"tactic": "exact this.1"
}
]
| [
222,
15
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
219,
1
]
|
Mathlib/Data/Set/Lattice.lean | Set.mapsTo_iInter₂ | []
| [
1460,
45
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1458,
1
]
|
Mathlib/Order/CompleteBooleanAlgebra.lean | sSup_inf_eq | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nκ : ι → Sort ?u.1710\ninst✝ : Frame α\ns t : Set α\na b : α\n⊢ sSup s ⊓ b = ⨆ (a : α) (_ : a ∈ s), a ⊓ b",
"tactic": "simpa only [inf_comm] using @inf_sSup_eq α _ s b"
}
]
| [
98,
51
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
97,
1
]
|
Mathlib/Topology/FiberBundle/Trivialization.lean | Trivialization.continuousAt_of_comp_right | [
{
"state_after": "ι : Type ?u.42663\nB : Type u_2\nF : Type u_3\nE : B → Type ?u.42674\nZ : Type u_4\ninst✝⁴ : TopologicalSpace B\ninst✝³ : TopologicalSpace F\nproj : Z → B\ninst✝² : TopologicalSpace Z\ninst✝¹ : TopologicalSpace (TotalSpace E)\ne✝ : Trivialization F proj\nx : Z\nX : Type u_1\ninst✝ : TopologicalSpace X\nf : Z → X\nz : Z\ne : Trivialization F proj\nhe : proj z ∈ e.baseSet\nhf : ContinuousAt (f ∘ ↑(LocalEquiv.symm e.toLocalEquiv)) (↑e z)\nhez : z ∈ (LocalEquiv.symm e.toLocalEquiv).target\n⊢ ContinuousAt f z",
"state_before": "ι : Type ?u.42663\nB : Type u_2\nF : Type u_3\nE : B → Type ?u.42674\nZ : Type u_4\ninst✝⁴ : TopologicalSpace B\ninst✝³ : TopologicalSpace F\nproj : Z → B\ninst✝² : TopologicalSpace Z\ninst✝¹ : TopologicalSpace (TotalSpace E)\ne✝ : Trivialization F proj\nx : Z\nX : Type u_1\ninst✝ : TopologicalSpace X\nf : Z → X\nz : Z\ne : Trivialization F proj\nhe : proj z ∈ e.baseSet\nhf : ContinuousAt (f ∘ ↑(LocalEquiv.symm e.toLocalEquiv)) (↑e z)\n⊢ ContinuousAt f z",
"tactic": "have hez : z ∈ e.toLocalEquiv.symm.target := by\n rw [LocalEquiv.symm_target, e.mem_source]\n exact he"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.42663\nB : Type u_2\nF : Type u_3\nE : B → Type ?u.42674\nZ : Type u_4\ninst✝⁴ : TopologicalSpace B\ninst✝³ : TopologicalSpace F\nproj : Z → B\ninst✝² : TopologicalSpace Z\ninst✝¹ : TopologicalSpace (TotalSpace E)\ne✝ : Trivialization F proj\nx : Z\nX : Type u_1\ninst✝ : TopologicalSpace X\nf : Z → X\nz : Z\ne : Trivialization F proj\nhe : proj z ∈ e.baseSet\nhf : ContinuousAt (f ∘ ↑(LocalEquiv.symm e.toLocalEquiv)) (↑e z)\nhez : z ∈ (LocalEquiv.symm e.toLocalEquiv).target\n⊢ ContinuousAt f z",
"tactic": "rwa [e.toLocalHomeomorph.symm.continuousAt_iff_continuousAt_comp_right hez,\n LocalHomeomorph.symm_symm]"
},
{
"state_after": "ι : Type ?u.42663\nB : Type u_2\nF : Type u_3\nE : B → Type ?u.42674\nZ : Type u_4\ninst✝⁴ : TopologicalSpace B\ninst✝³ : TopologicalSpace F\nproj : Z → B\ninst✝² : TopologicalSpace Z\ninst✝¹ : TopologicalSpace (TotalSpace E)\ne✝ : Trivialization F proj\nx : Z\nX : Type u_1\ninst✝ : TopologicalSpace X\nf : Z → X\nz : Z\ne : Trivialization F proj\nhe : proj z ∈ e.baseSet\nhf : ContinuousAt (f ∘ ↑(LocalEquiv.symm e.toLocalEquiv)) (↑e z)\n⊢ proj z ∈ e.baseSet",
"state_before": "ι : Type ?u.42663\nB : Type u_2\nF : Type u_3\nE : B → Type ?u.42674\nZ : Type u_4\ninst✝⁴ : TopologicalSpace B\ninst✝³ : TopologicalSpace F\nproj : Z → B\ninst✝² : TopologicalSpace Z\ninst✝¹ : TopologicalSpace (TotalSpace E)\ne✝ : Trivialization F proj\nx : Z\nX : Type u_1\ninst✝ : TopologicalSpace X\nf : Z → X\nz : Z\ne : Trivialization F proj\nhe : proj z ∈ e.baseSet\nhf : ContinuousAt (f ∘ ↑(LocalEquiv.symm e.toLocalEquiv)) (↑e z)\n⊢ z ∈ (LocalEquiv.symm e.toLocalEquiv).target",
"tactic": "rw [LocalEquiv.symm_target, e.mem_source]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.42663\nB : Type u_2\nF : Type u_3\nE : B → Type ?u.42674\nZ : Type u_4\ninst✝⁴ : TopologicalSpace B\ninst✝³ : TopologicalSpace F\nproj : Z → B\ninst✝² : TopologicalSpace Z\ninst✝¹ : TopologicalSpace (TotalSpace E)\ne✝ : Trivialization F proj\nx : Z\nX : Type u_1\ninst✝ : TopologicalSpace X\nf : Z → X\nz : Z\ne : Trivialization F proj\nhe : proj z ∈ e.baseSet\nhf : ContinuousAt (f ∘ ↑(LocalEquiv.symm e.toLocalEquiv)) (↑e z)\n⊢ proj z ∈ e.baseSet",
"tactic": "exact he"
}
]
| [
542,
31
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
535,
1
]
|
Mathlib/MeasureTheory/Integral/Bochner.lean | MeasureTheory.integral_add' | []
| [
869,
21
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
867,
1
]
|
Mathlib/Topology/Category/TopCat/Limits/Products.lean | TopCat.sigmaIsoSigma_hom_ι | [
{
"state_after": "no goals",
"state_before": "J : Type v\ninst✝ : SmallCategory J\nι : Type v\nα : ι → TopCatMax\ni : ι\n⊢ Sigma.ι α i ≫ (sigmaIsoSigma α).hom = sigmaι α i",
"tactic": "simp [sigmaIsoSigma]"
}
]
| [
127,
80
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
126,
1
]
|
Mathlib/Analysis/NormedSpace/Exponential.lean | norm_expSeries_div_summable | []
| [
588,
60
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
586,
1
]
|
Mathlib/MeasureTheory/MeasurableSpace.lean | measurable_snd | []
| [
650,
38
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
648,
1
]
|
Mathlib/RingTheory/Subsemiring/Basic.lean | Subsemiring.closure_iUnion | []
| [
1002,
31
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1001,
1
]
|
Mathlib/Topology/Algebra/Module/Basic.lean | ContinuousLinearMap.coe_copy | []
| [
482,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
481,
1
]
|
Mathlib/Topology/Sheaves/PUnit.lean | TopCat.Presheaf.isSheaf_of_isTerminal_of_indiscrete | [
{
"state_after": "case inl\nC : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\ns : Sieve ⊥\nhs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊥\n⊢ Presieve.IsSheafFor (F ⋙ coyoneda.obj c.op) s.arrows\n\ncase inr\nC : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\nU : TopologicalSpace.Opens ↑X\ns : Sieve U\nhs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) U\nhne : U ≠ ⊥\n⊢ Presieve.IsSheafFor (F ⋙ coyoneda.obj c.op) s.arrows",
"state_before": "C : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\nU : TopologicalSpace.Opens ↑X\ns : Sieve U\nhs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) U\n⊢ Presieve.IsSheafFor (F ⋙ coyoneda.obj c.op) s.arrows",
"tactic": "obtain rfl | hne := eq_or_ne U ⊥"
},
{
"state_after": "case inl\nC : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\ns : Sieve ⊥\nhs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊥\nx✝ : Presieve.FamilyOfElements (F ⋙ coyoneda.obj c.op) s.arrows\na✝ : Presieve.FamilyOfElements.Compatible x✝\n⊢ ∃! t, Presieve.FamilyOfElements.IsAmalgamation x✝ t",
"state_before": "case inl\nC : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\ns : Sieve ⊥\nhs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊥\n⊢ Presieve.IsSheafFor (F ⋙ coyoneda.obj c.op) s.arrows",
"tactic": "intro _ _"
},
{
"state_after": "case inl\nC : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\ns : Sieve ⊥\nhs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊥\nx✝ : Presieve.FamilyOfElements (F ⋙ coyoneda.obj c.op) s.arrows\na✝ : Presieve.FamilyOfElements.Compatible x✝\n⊢ ∃ x, Presieve.FamilyOfElements.IsAmalgamation x✝ x\n\nC : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\ns : Sieve ⊥\nhs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊥\nx✝ : Presieve.FamilyOfElements (F ⋙ coyoneda.obj c.op) s.arrows\na✝ : Presieve.FamilyOfElements.Compatible x✝\n⊢ ∀ (x x_1 : (F ⋙ coyoneda.obj c.op).obj ⊥.op), x = x_1",
"state_before": "case inl\nC : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\ns : Sieve ⊥\nhs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊥\nx✝ : Presieve.FamilyOfElements (F ⋙ coyoneda.obj c.op) s.arrows\na✝ : Presieve.FamilyOfElements.Compatible x✝\n⊢ ∃! t, Presieve.FamilyOfElements.IsAmalgamation x✝ t",
"tactic": "rw [@exists_unique_iff_exists _ ⟨fun _ _ => _⟩]"
},
{
"state_after": "case inl\nC : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\ns : Sieve ⊥\nhs✝ : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊥\nx✝ : Presieve.FamilyOfElements (F ⋙ coyoneda.obj c.op) s.arrows\na✝ : Presieve.FamilyOfElements.Compatible x✝\nU : TopologicalSpace.Opens ↑X\nhU : U ⟶ ⊥\nhs : s.arrows hU\n⊢ IsTerminal (F.obj U.op)",
"state_before": "case inl\nC : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\ns : Sieve ⊥\nhs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊥\nx✝ : Presieve.FamilyOfElements (F ⋙ coyoneda.obj c.op) s.arrows\na✝ : Presieve.FamilyOfElements.Compatible x✝\n⊢ ∃ x, Presieve.FamilyOfElements.IsAmalgamation x✝ x",
"tactic": "refine' ⟨it.from _, fun U hU hs => IsTerminal.hom_ext _ _ _⟩"
},
{
"state_after": "no goals",
"state_before": "case inl\nC : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\ns : Sieve ⊥\nhs✝ : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊥\nx✝ : Presieve.FamilyOfElements (F ⋙ coyoneda.obj c.op) s.arrows\na✝ : Presieve.FamilyOfElements.Compatible x✝\nU : TopologicalSpace.Opens ↑X\nhU : U ⟶ ⊥\nhs : s.arrows hU\n⊢ IsTerminal (F.obj U.op)",
"tactic": "rwa [le_bot_iff.1 hU.le]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\ns : Sieve ⊥\nhs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊥\nx✝ : Presieve.FamilyOfElements (F ⋙ coyoneda.obj c.op) s.arrows\na✝ : Presieve.FamilyOfElements.Compatible x✝\n⊢ ∀ (x x_1 : (F ⋙ coyoneda.obj c.op).obj ⊥.op), x = x_1",
"tactic": "apply it.hom_ext"
},
{
"state_after": "case h.e'_5.h.h.e'_4\nC : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\nU : TopologicalSpace.Opens ↑X\ns : Sieve U\nhs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) U\nhne : U ≠ ⊥\n⊢ s = ⊤",
"state_before": "case inr\nC : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\nU : TopologicalSpace.Opens ↑X\ns : Sieve U\nhs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) U\nhne : U ≠ ⊥\n⊢ Presieve.IsSheafFor (F ⋙ coyoneda.obj c.op) s.arrows",
"tactic": "convert Presieve.isSheafFor_top_sieve (F ⋙ coyoneda.obj (@op C c))"
},
{
"state_after": "case h.e'_5.h.h.e'_4\nC : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\nU : TopologicalSpace.Opens ↑X\ns : Sieve U\nhs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) U\nhne : U ≠ ⊥\n⊢ s.arrows (𝟙 U)",
"state_before": "case h.e'_5.h.h.e'_4\nC : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\nU : TopologicalSpace.Opens ↑X\ns : Sieve U\nhs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) U\nhne : U ≠ ⊥\n⊢ s = ⊤",
"tactic": "rw [← Sieve.id_mem_iff_eq_top]"
},
{
"state_after": "case h.e'_5.h.h.e'_4\nC : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\nU : TopologicalSpace.Opens ↑X\ns : Sieve U\nhs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) U\nhne : U ≠ ⊥\nthis : U = ⊤\n⊢ s.arrows (𝟙 U)",
"state_before": "case h.e'_5.h.h.e'_4\nC : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\nU : TopologicalSpace.Opens ↑X\ns : Sieve U\nhs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) U\nhne : U ≠ ⊥\n⊢ s.arrows (𝟙 U)",
"tactic": "have := (U.eq_bot_or_top hind).resolve_left hne"
},
{
"state_after": "case h.e'_5.h.h.e'_4\nC : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\ns : Sieve ⊤\nhs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊤\nhne : ⊤ ≠ ⊥\n⊢ s.arrows (𝟙 ⊤)",
"state_before": "case h.e'_5.h.h.e'_4\nC : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\nU : TopologicalSpace.Opens ↑X\ns : Sieve U\nhs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) U\nhne : U ≠ ⊥\nthis : U = ⊤\n⊢ s.arrows (𝟙 U)",
"tactic": "subst this"
},
{
"state_after": "case h.e'_5.h.h.e'_4.inl\nC : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\ns : Sieve ⊤\nhs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊤\nhne : ⊤ ≠ ⊥\nhe : IsEmpty ↑X\n⊢ s.arrows (𝟙 ⊤)\n\ncase h.e'_5.h.h.e'_4.inr.intro\nC : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\ns : Sieve ⊤\nhs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊤\nhne : ⊤ ≠ ⊥\nx : ↑X\n⊢ s.arrows (𝟙 ⊤)",
"state_before": "case h.e'_5.h.h.e'_4\nC : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\ns : Sieve ⊤\nhs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊤\nhne : ⊤ ≠ ⊥\n⊢ s.arrows (𝟙 ⊤)",
"tactic": "obtain he | ⟨⟨x⟩⟩ := isEmpty_or_nonempty X"
},
{
"state_after": "case h.e'_5.h.h.e'_4.inr.intro.intro.intro.intro\nC : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\ns : Sieve ⊤\nhs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊤\nhne : ⊤ ≠ ⊥\nx : ↑X\nU : TopologicalSpace.Opens ↑X\nf : U ⟶ ⊤\nhf : s.arrows f\nhm : x ∈ U\n⊢ s.arrows (𝟙 ⊤)",
"state_before": "case h.e'_5.h.h.e'_4.inr.intro\nC : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\ns : Sieve ⊤\nhs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊤\nhne : ⊤ ≠ ⊥\nx : ↑X\n⊢ s.arrows (𝟙 ⊤)",
"tactic": "obtain ⟨U, f, hf, hm⟩ := hs x _root_.trivial"
},
{
"state_after": "case h.e'_5.h.h.e'_4.inr.intro.intro.intro.intro.inl\nC : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\ns : Sieve ⊤\nhs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊤\nhne : ⊤ ≠ ⊥\nx : ↑X\nf : ⊥ ⟶ ⊤\nhf : s.arrows f\nhm : x ∈ ⊥\n⊢ s.arrows (𝟙 ⊤)\n\ncase h.e'_5.h.h.e'_4.inr.intro.intro.intro.intro.inr\nC : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\ns : Sieve ⊤\nhs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊤\nhne : ⊤ ≠ ⊥\nx : ↑X\nf : ⊤ ⟶ ⊤\nhf : s.arrows f\nhm : x ∈ ⊤\n⊢ s.arrows (𝟙 ⊤)",
"state_before": "case h.e'_5.h.h.e'_4.inr.intro.intro.intro.intro\nC : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\ns : Sieve ⊤\nhs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊤\nhne : ⊤ ≠ ⊥\nx : ↑X\nU : TopologicalSpace.Opens ↑X\nf : U ⟶ ⊤\nhf : s.arrows f\nhm : x ∈ U\n⊢ s.arrows (𝟙 ⊤)",
"tactic": "obtain rfl | rfl := U.eq_bot_or_top hind"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h.h.e'_4.inl\nC : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\ns : Sieve ⊤\nhs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊤\nhne : ⊤ ≠ ⊥\nhe : IsEmpty ↑X\n⊢ s.arrows (𝟙 ⊤)",
"tactic": "exact (hne <| SetLike.ext'_iff.2 <| Set.univ_eq_empty_iff.2 he).elim"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h.h.e'_4.inr.intro.intro.intro.intro.inl\nC : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\ns : Sieve ⊤\nhs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊤\nhne : ⊤ ≠ ⊥\nx : ↑X\nf : ⊥ ⟶ ⊤\nhf : s.arrows f\nhm : x ∈ ⊥\n⊢ s.arrows (𝟙 ⊤)",
"tactic": "cases hm"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h.h.e'_4.inr.intro.intro.intro.intro.inr\nC : Type u\ninst✝ : Category C\nX : TopCat\nhind : X.str = ⊤\nF : Presheaf C X\nit : IsTerminal (F.obj ⊥.op)\nc : C\ns : Sieve ⊤\nhs : s ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) ⊤\nhne : ⊤ ≠ ⊥\nx : ↑X\nf : ⊤ ⟶ ⊤\nhf : s.arrows f\nhm : x ∈ ⊤\n⊢ s.arrows (𝟙 ⊤)",
"tactic": "convert hf"
}
]
| [
45,
17
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
28,
1
]
|
Mathlib/Data/Matrix/Basic.lean | Matrix.transposeLinearEquiv_symm | []
| [
2069,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2067,
1
]
|
Mathlib/Data/MvPolynomial/Basic.lean | MvPolynomial.eval₂_map | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S₁\np✝ q : MvPolynomial σ R\nf✝ : R →+* S₁\ninst✝ : CommSemiring S₂\nf : R →+* S₁\ng : σ → S₂\nφ : S₁ →+* S₂\np : MvPolynomial σ R\n⊢ eval₂ φ g (↑(map f) p) = eval₂ (RingHom.comp φ f) g p",
"tactic": "rw [← eval_map, ← eval_map, map_map]"
}
]
| [
1328,
39
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1326,
1
]
|
Mathlib/Computability/Primrec.lean | Nat.Primrec'.prim_iff₂ | [
{
"state_after": "no goals",
"state_before": "f : ℕ → ℕ → ℕ\nh : Primrec fun v => f (Vector.head v) (Vector.head (Vector.tail v))\nv : ℕ × ℕ\n⊢ f (Vector.head (v.fst ::ᵥ v.snd ::ᵥ nil)) (Vector.head (Vector.tail (v.fst ::ᵥ v.snd ::ᵥ nil))) = f v.fst v.snd",
"tactic": "simp"
}
]
| [
1559,
74
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1555,
1
]
|
Mathlib/Data/Nat/Order/Basic.lean | Nat.eq_zero_of_le_div | [
{
"state_after": "m n k l : ℕ\nhn : 2 ≤ n\nh : m ≤ m / n\n⊢ m * n ≤ m",
"state_before": "m n k l : ℕ\nhn : 2 ≤ n\nh : m ≤ m / n\n⊢ n * m ≤ m",
"tactic": "rw [mul_comm]"
},
{
"state_after": "no goals",
"state_before": "m n k l : ℕ\nhn : 2 ≤ n\nh : m ≤ m / n\n⊢ m * n ≤ m",
"tactic": "exact (Nat.le_div_iff_mul_le' (lt_of_lt_of_le (by decide) hn)).1 h"
},
{
"state_after": "no goals",
"state_before": "m n k l : ℕ\nhn : 2 ≤ n\nh : m ≤ m / n\n⊢ 0 < 2",
"tactic": "decide"
}
]
| [
409,
86
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
407,
1
]
|
Mathlib/Analysis/Calculus/FDeriv/Equiv.lean | LinearIsometryEquiv.comp_hasFDerivWithinAt_iff' | []
| [
336,
48
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
334,
1
]
|
Mathlib/Analysis/VonNeumannAlgebra/Basic.lean | VonNeumannAlgebra.coe_commutant | []
| [
135,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
133,
1
]
|
Mathlib/FieldTheory/Separable.lean | isSeparable_tower_bot_of_isSeparable | [
{
"state_after": "F : Type u_1\nK : Type u_3\nE : Type u_2\ninst✝⁶ : Field F\ninst✝⁵ : Field K\ninst✝⁴ : Field E\ninst✝³ : Algebra F K\ninst✝² : Algebra F E\ninst✝¹ : Algebra K E\ninst✝ : IsScalarTower F K E\nh : IsSeparable F E\nx : K\nhs : Separable (minpoly F (↑(algebraMap K E) x))\n⊢ Separable (minpoly F x)",
"state_before": "F : Type u_1\nK : Type u_3\nE : Type u_2\ninst✝⁶ : Field F\ninst✝⁵ : Field K\ninst✝⁴ : Field E\ninst✝³ : Algebra F K\ninst✝² : Algebra F E\ninst✝¹ : Algebra K E\ninst✝ : IsScalarTower F K E\nh : IsSeparable F E\nx : K\n⊢ IsIntegral F x ∧ Separable (minpoly F x)",
"tactic": "refine'\n (isSeparable_iff.1 h (algebraMap K E x)).imp isIntegral_tower_bot_of_isIntegral_field\n fun hs => _"
},
{
"state_after": "case intro\nF : Type u_1\nK : Type u_3\nE : Type u_2\ninst✝⁶ : Field F\ninst✝⁵ : Field K\ninst✝⁴ : Field E\ninst✝³ : Algebra F K\ninst✝² : Algebra F E\ninst✝¹ : Algebra K E\ninst✝ : IsScalarTower F K E\nh : IsSeparable F E\nx : K\nhs : Separable (minpoly F (↑(algebraMap K E) x))\nq : F[X]\nhq : minpoly F (↑(algebraMap K E) x) = minpoly F x * q\n⊢ Separable (minpoly F x)",
"state_before": "F : Type u_1\nK : Type u_3\nE : Type u_2\ninst✝⁶ : Field F\ninst✝⁵ : Field K\ninst✝⁴ : Field E\ninst✝³ : Algebra F K\ninst✝² : Algebra F E\ninst✝¹ : Algebra K E\ninst✝ : IsScalarTower F K E\nh : IsSeparable F E\nx : K\nhs : Separable (minpoly F (↑(algebraMap K E) x))\n⊢ Separable (minpoly F x)",
"tactic": "obtain ⟨q, hq⟩ :=\n minpoly.dvd F x\n ((aeval_algebraMap_eq_zero_iff _ _ _).mp (minpoly.aeval F ((algebraMap K E) x)))"
},
{
"state_after": "case intro\nF : Type u_1\nK : Type u_3\nE : Type u_2\ninst✝⁶ : Field F\ninst✝⁵ : Field K\ninst✝⁴ : Field E\ninst✝³ : Algebra F K\ninst✝² : Algebra F E\ninst✝¹ : Algebra K E\ninst✝ : IsScalarTower F K E\nh : IsSeparable F E\nx : K\nq : F[X]\nhs : Separable (minpoly F x * q)\nhq : minpoly F (↑(algebraMap K E) x) = minpoly F x * q\n⊢ Separable (minpoly F x)",
"state_before": "case intro\nF : Type u_1\nK : Type u_3\nE : Type u_2\ninst✝⁶ : Field F\ninst✝⁵ : Field K\ninst✝⁴ : Field E\ninst✝³ : Algebra F K\ninst✝² : Algebra F E\ninst✝¹ : Algebra K E\ninst✝ : IsScalarTower F K E\nh : IsSeparable F E\nx : K\nhs : Separable (minpoly F (↑(algebraMap K E) x))\nq : F[X]\nhq : minpoly F (↑(algebraMap K E) x) = minpoly F x * q\n⊢ Separable (minpoly F x)",
"tactic": "rw [hq] at hs"
},
{
"state_after": "no goals",
"state_before": "case intro\nF : Type u_1\nK : Type u_3\nE : Type u_2\ninst✝⁶ : Field F\ninst✝⁵ : Field K\ninst✝⁴ : Field E\ninst✝³ : Algebra F K\ninst✝² : Algebra F E\ninst✝¹ : Algebra K E\ninst✝ : IsScalarTower F K E\nh : IsSeparable F E\nx : K\nq : F[X]\nhs : Separable (minpoly F x * q)\nhq : minpoly F (↑(algebraMap K E) x) = minpoly F x * q\n⊢ Separable (minpoly F x)",
"tactic": "exact hs.of_mul_left"
}
]
| [
556,
25
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
547,
1
]
|
Mathlib/Data/Fintype/Pi.lean | Fintype.piFinset_empty | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nδ : α → Type u_2\ninst✝ : Nonempty α\nx✝ : (a : α) → δ a\n⊢ ¬x✝ ∈ piFinset fun x => ∅",
"tactic": "simp"
}
]
| [
62,
46
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
61,
1
]
|
Mathlib/Order/OrdContinuous.lean | LeftOrdContinuous.iterate | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\ng : β → γ\nf✝ : α → β\nf : α → α\nhf : LeftOrdContinuous f\nn : ℕ\n⊢ LeftOrdContinuous (f^[n])",
"tactic": "induction n with\n| zero => exact LeftOrdContinuous.id α\n| succ n ihn => exact ihn.comp hf"
},
{
"state_after": "no goals",
"state_before": "case zero\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\ng : β → γ\nf✝ : α → β\nf : α → α\nhf : LeftOrdContinuous f\n⊢ LeftOrdContinuous (f^[Nat.zero])",
"tactic": "exact LeftOrdContinuous.id α"
},
{
"state_after": "no goals",
"state_before": "case succ\nα : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝² : Preorder α\ninst✝¹ : Preorder β\ninst✝ : Preorder γ\ng : β → γ\nf✝ : α → β\nf : α → α\nhf : LeftOrdContinuous f\nn : ℕ\nihn : LeftOrdContinuous (f^[n])\n⊢ LeftOrdContinuous (f^[Nat.succ n])",
"tactic": "exact ihn.comp hf"
}
]
| [
88,
34
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
84,
11
]
|
Mathlib/Algebra/Order/LatticeGroup.lean | LatticeOrderedCommGroup.pos_mul_neg | [
{
"state_after": "α : Type u\ninst✝² : Lattice α\ninst✝¹ : CommGroup α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na : α\n⊢ abs a = a ⊔ a⁻¹ ⊔ 1",
"state_before": "α : Type u\ninst✝² : Lattice α\ninst✝¹ : CommGroup α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na : α\n⊢ abs a = a⁺ * a⁻",
"tactic": "rw [m_pos_part_def, sup_mul, one_mul, m_neg_part_def, mul_sup, mul_one, mul_inv_self, sup_assoc,\n ← @sup_assoc _ _ a, sup_eq_right.2 le_sup_right]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝² : Lattice α\ninst✝¹ : CommGroup α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na : α\n⊢ abs a = a ⊔ a⁻¹ ⊔ 1",
"tactic": "exact (sup_eq_left.2 <| one_le_abs a).symm"
}
]
| [
382,
45
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
379,
1
]
|
Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | CategoryTheory.Limits.prod.lift_fst_snd | [
{
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},
{
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"tactic": "simp"
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{
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"tactic": "simp"
}
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734,
81
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
733,
1
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Mathlib/CategoryTheory/GlueData.lean | CategoryTheory.GlueData.ι_gluedIso_inv | [
{
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"state_before": "C : Type u₁\ninst✝³ : Category C\nC' : Type u₂\ninst✝² : Category C'\nD : GlueData C\nF : C ⥤ C'\nH : (i j k : D.J) → PreservesLimit (cospan (f D i j) (f D i k)) F\ninst✝¹ : HasMulticoequalizer (diagram D)\ninst✝ : PreservesColimit (MultispanIndex.multispan (diagram D)) F\ni : D.J\n⊢ ι (mapGlueData D F) i ≫ (gluedIso D F).inv = F.map (ι D i)",
"tactic": "rw [Iso.comp_inv_eq, ι_gluedIso_hom]"
}
]
| [
376,
39
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
375,
1
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Mathlib/Algebra/Homology/Exact.lean | CategoryTheory.kernel_comp_cokernel | [
{
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"tactic": "suffices Subobject.arrow (kernelSubobject g) ≫ cokernel.π f = 0 by\n rw [← kernelSubobject_arrow', Category.assoc, this, comp_zero]"
},
{
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"state_before": "V : Type u\ninst✝⁴ : Category V\ninst✝³ : HasImages V\nA B C D : V\nf : A ⟶ B\ng : B ⟶ C\nh✝ : C ⟶ D\ninst✝² : HasZeroMorphisms V\ninst✝¹ : HasEqualizers V\ninst✝ : HasCokernels V\nh : Exact f g\n⊢ Subobject.arrow (kernelSubobject g) ≫ cokernel.π f = 0",
"tactic": "haveI := h.epi"
},
{
"state_after": "V : Type u\ninst✝⁴ : Category V\ninst✝³ : HasImages V\nA B C D : V\nf : A ⟶ B\ng : B ⟶ C\nh✝ : C ⟶ D\ninst✝² : HasZeroMorphisms V\ninst✝¹ : HasEqualizers V\ninst✝ : HasCokernels V\nh : Exact f g\nthis : Epi (imageToKernel f g (_ : f ≫ g = 0))\n⊢ imageToKernel f g (_ : f ≫ g = 0) ≫ Subobject.arrow (kernelSubobject g) ≫ cokernel.π f = 0",
"state_before": "V : Type u\ninst✝⁴ : Category V\ninst✝³ : HasImages V\nA B C D : V\nf : A ⟶ B\ng : B ⟶ C\nh✝ : C ⟶ D\ninst✝² : HasZeroMorphisms V\ninst✝¹ : HasEqualizers V\ninst✝ : HasCokernels V\nh : Exact f g\nthis : Epi (imageToKernel f g (_ : f ≫ g = 0))\n⊢ Subobject.arrow (kernelSubobject g) ≫ cokernel.π f = 0",
"tactic": "apply zero_of_epi_comp (imageToKernel f g h.w) _"
},
{
"state_after": "V : Type u\ninst✝⁴ : Category V\ninst✝³ : HasImages V\nA B C D : V\nf : A ⟶ B\ng : B ⟶ C\nh✝ : C ⟶ D\ninst✝² : HasZeroMorphisms V\ninst✝¹ : HasEqualizers V\ninst✝ : HasCokernels V\nh : Exact f g\nthis : Epi (imageToKernel f g (_ : f ≫ g = 0))\n⊢ image.ι f ≫ cokernel.π f = (imageSubobjectIso f).inv ≫ 0",
"state_before": "V : Type u\ninst✝⁴ : Category V\ninst✝³ : HasImages V\nA B C D : V\nf : A ⟶ B\ng : B ⟶ C\nh✝ : C ⟶ D\ninst✝² : HasZeroMorphisms V\ninst✝¹ : HasEqualizers V\ninst✝ : HasCokernels V\nh : Exact f g\nthis : Epi (imageToKernel f g (_ : f ≫ g = 0))\n⊢ imageToKernel f g (_ : f ≫ g = 0) ≫ Subobject.arrow (kernelSubobject g) ≫ cokernel.π f = 0",
"tactic": "rw [imageToKernel_arrow_assoc, ← imageSubobject_arrow, Category.assoc, ← Iso.eq_inv_comp]"
},
{
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"state_before": "V : Type u\ninst✝⁴ : Category V\ninst✝³ : HasImages V\nA B C D : V\nf : A ⟶ B\ng : B ⟶ C\nh✝ : C ⟶ D\ninst✝² : HasZeroMorphisms V\ninst✝¹ : HasEqualizers V\ninst✝ : HasCokernels V\nh : Exact f g\nthis : Epi (imageToKernel f g (_ : f ≫ g = 0))\n⊢ image.ι f ≫ cokernel.π f = (imageSubobjectIso f).inv ≫ 0",
"tactic": "aesop_cat"
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{
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"state_before": "V : Type u\ninst✝⁴ : Category V\ninst✝³ : HasImages V\nA B C D : V\nf : A ⟶ B\ng : B ⟶ C\nh✝ : C ⟶ D\ninst✝² : HasZeroMorphisms V\ninst✝¹ : HasEqualizers V\ninst✝ : HasCokernels V\nh : Exact f g\nthis : Subobject.arrow (kernelSubobject g) ≫ cokernel.π f = 0\n⊢ kernel.ι g ≫ cokernel.π f = 0",
"tactic": "rw [← kernelSubobject_arrow', Category.assoc, this, comp_zero]"
}
]
| [
280,
12
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
274,
1
]
|
Mathlib/Data/Nat/Bitwise.lean | Nat.lt_lxor'_cases | []
| [
320,
72
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
319,
1
]
|
Mathlib/LinearAlgebra/Dual.lean | LinearMap.dualPairing_nondegenerate | []
| [
1345,
94
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1344,
1
]
|
Mathlib/Order/Concept.lean | Concept.snd_injective | []
| [
210,
84
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
210,
1
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|
Mathlib/Data/Fintype/Basic.lean | Set.ssubset_toFinset | [
{
"state_after": "no goals",
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"tactic": "rw [← Finset.coe_ssubset, coe_toFinset]"
}
]
| [
668,
42
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
667,
1
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|
Mathlib/Data/Real/Basic.lean | Real.ofCauchy_mul | []
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137,
21
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
136,
1
]
|
Mathlib/Data/Set/Semiring.lean | SetSemiring.down_zero | []
| [
111,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
110,
1
]
|
Mathlib/Data/Nat/Pow.lean | Nat.le_self_pow | []
| [
44,
53
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
42,
1
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|
Mathlib/Data/Polynomial/Eval.lean | Polynomial.mem_map_range | []
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931,
19
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
929,
1
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|
Mathlib/Order/Cover.lean | wcovby_of_le_of_le | []
| [
66,
51
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| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
65,
1
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|
Mathlib/Algebra/Lie/IdealOperations.lean | LieIdeal.map_comap_incl | [
{
"state_after": "R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nI₁ I₂ : LieIdeal R L\n⊢ map (incl I₁) (comap (incl I₁) I₂) = LieHom.idealRange (incl I₁) ⊓ I₂",
"state_before": "R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nI₁ I₂ : LieIdeal R L\n⊢ map (incl I₁) (comap (incl I₁) I₂) = I₁ ⊓ I₂",
"tactic": "conv_rhs => rw [← I₁.incl_idealRange]"
},
{
"state_after": "R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nI₁ I₂ : LieIdeal R L\n⊢ LieHom.IsIdealMorphism (incl I₁)",
"state_before": "R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nI₁ I₂ : LieIdeal R L\n⊢ map (incl I₁) (comap (incl I₁) I₂) = LieHom.idealRange (incl I₁) ⊓ I₂",
"tactic": "rw [← map_comap_eq]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nL : Type v\nL' : Type w₂\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nI : LieIdeal R L\nJ : LieIdeal R L'\nI₁ I₂ : LieIdeal R L\n⊢ LieHom.IsIdealMorphism (incl I₁)",
"tactic": "exact I₁.incl_isIdealMorphism"
}
]
| [
303,
32
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
300,
1
]
|
Mathlib/AlgebraicGeometry/LocallyRingedSpace.lean | AlgebraicGeometry.LocallyRingedSpace.basicOpen_zero | [
{
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"state_before": "X✝ : LocallyRingedSpace\nX : LocallyRingedSpace\nU : Opens ↑↑X.toPresheafedSpace\n⊢ RingedSpace.basicOpen (toRingedSpace X) 0 = ⊥",
"tactic": "ext x"
},
{
"state_after": "case h.h\nX✝ : LocallyRingedSpace\nX : LocallyRingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nx : ↑↑(toRingedSpace X).toPresheafedSpace\n⊢ ∀ (x_1 : x ∈ U),\n ¬IsUnit (↑(Presheaf.germ (toRingedSpace X).toPresheafedSpace.presheaf { val := x, property := (_ : x ∈ U) }) 0)",
"state_before": "case h.h\nX✝ : LocallyRingedSpace\nX : LocallyRingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nx : ↑↑(toRingedSpace X).toPresheafedSpace\n⊢ x ∈ ↑(RingedSpace.basicOpen (toRingedSpace X) 0) ↔ x ∈ ↑⊥",
"tactic": "simp only [RingedSpace.basicOpen, Opens.coe_mk, Set.mem_image, Set.mem_setOf_eq, Subtype.exists,\n exists_and_right, exists_eq_right, Opens.coe_bot, Set.mem_empty_iff_false,\n iff_false, not_exists]"
},
{
"state_after": "case h.h\nX✝ : LocallyRingedSpace\nX : LocallyRingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nx : ↑↑(toRingedSpace X).toPresheafedSpace\nhx : x ∈ U\n⊢ ¬IsUnit (↑(Presheaf.germ (toRingedSpace X).toPresheafedSpace.presheaf { val := x, property := (_ : x ∈ U) }) 0)",
"state_before": "case h.h\nX✝ : LocallyRingedSpace\nX : LocallyRingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nx : ↑↑(toRingedSpace X).toPresheafedSpace\n⊢ ∀ (x_1 : x ∈ U),\n ¬IsUnit (↑(Presheaf.germ (toRingedSpace X).toPresheafedSpace.presheaf { val := x, property := (_ : x ∈ U) }) 0)",
"tactic": "intros hx"
},
{
"state_after": "case h.h\nX✝ : LocallyRingedSpace\nX : LocallyRingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nx : ↑↑(toRingedSpace X).toPresheafedSpace\nhx : x ∈ U\n⊢ ¬0 = 1",
"state_before": "case h.h\nX✝ : LocallyRingedSpace\nX : LocallyRingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nx : ↑↑(toRingedSpace X).toPresheafedSpace\nhx : x ∈ U\n⊢ ¬IsUnit (↑(Presheaf.germ (toRingedSpace X).toPresheafedSpace.presheaf { val := x, property := (_ : x ∈ U) }) 0)",
"tactic": "rw [map_zero, isUnit_zero_iff]"
},
{
"state_after": "case h.h\nX✝ : LocallyRingedSpace\nX : LocallyRingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nx : ↑↑(toRingedSpace X).toPresheafedSpace\nhx : x ∈ U\n⊢ 0 ≠ 1",
"state_before": "case h.h\nX✝ : LocallyRingedSpace\nX : LocallyRingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nx : ↑↑(toRingedSpace X).toPresheafedSpace\nhx : x ∈ U\n⊢ ¬0 = 1",
"tactic": "change (0 : X.stalk x) ≠ (1 : X.stalk x)"
},
{
"state_after": "no goals",
"state_before": "case h.h\nX✝ : LocallyRingedSpace\nX : LocallyRingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nx : ↑↑(toRingedSpace X).toPresheafedSpace\nhx : x ∈ U\n⊢ 0 ≠ 1",
"tactic": "exact zero_ne_one"
}
]
| [
333,
20
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
324,
1
]
|
Mathlib/Analysis/NormedSpace/Multilinear.lean | ContinuousMultilinearMap.uncurry0_norm | [
{
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"tactic": "simp"
}
]
| [
1682,
99
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1682,
1
]
|
Mathlib/Algebra/Hom/Ring.lean | NonUnitalRingHom.coe_comp_mulHom | []
| [
276,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
274,
1
]
|
Mathlib/Order/Closure.lean | ClosureOperator.closure_inf_le | []
| [
245,
28
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
243,
1
]
|
Mathlib/Data/Finset/Basic.lean | Finset.union_sdiff_of_subset | []
| [
2068,
27
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2067,
1
]
|
Mathlib/Data/Set/Lattice.lean | Set.surjOn_iUnion | []
| [
1587,
40
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1585,
1
]
|
Mathlib/SetTheory/Game/PGame.lean | LE.le.not_gf | []
| [
414,
11
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
413,
1
]
|
Mathlib/MeasureTheory/Integral/Lebesgue.lean | MeasureTheory.exists_simpleFunc_forall_lintegral_sub_lt_of_pos | [
{
"state_after": "α : Type u_1\nβ : Type ?u.85282\nγ : Type ?u.85285\nδ : Type ?u.85288\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh : (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\n⊢ ∃ φ,\n (∀ (x : α), ↑(↑φ x) ≤ f x) ∧\n ∀ (ψ : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑ψ x) ≤ f x) → SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (ψ - φ)) μ < ε",
"state_before": "α : Type u_1\nβ : Type ?u.85282\nγ : Type ?u.85285\nδ : Type ?u.85288\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh : (∫⁻ (x : α), f x ∂μ) ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\n⊢ ∃ φ,\n (∀ (x : α), ↑(↑φ x) ≤ f x) ∧\n ∀ (ψ : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑ψ x) ≤ f x) → SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (ψ - φ)) μ < ε",
"tactic": "rw [lintegral_eq_nnreal] at h"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.85282\nγ : Type ?u.85285\nδ : Type ?u.85288\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh : (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nthis :\n (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) <\n (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) + ε\n⊢ ∃ φ,\n (∀ (x : α), ↑(↑φ x) ≤ f x) ∧\n ∀ (ψ : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑ψ x) ≤ f x) → SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (ψ - φ)) μ < ε",
"state_before": "α : Type u_1\nβ : Type ?u.85282\nγ : Type ?u.85285\nδ : Type ?u.85288\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh : (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\n⊢ ∃ φ,\n (∀ (x : α), ↑(↑φ x) ≤ f x) ∧\n ∀ (ψ : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑ψ x) ≤ f x) → SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (ψ - φ)) μ < ε",
"tactic": "have := ENNReal.lt_add_right h hε"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.85282\nγ : Type ?u.85285\nδ : Type ?u.85288\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh : (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nthis :\n (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) 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\n⊢ ∃ φ,\n (∀ (x : α), ↑(↑φ x) ≤ f x) ∧\n ∀ (ψ : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑ψ x) ≤ f x) → SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (ψ - φ)) μ < ε",
"state_before": "α : Type u_1\nβ : Type ?u.85282\nγ : Type ?u.85285\nδ : Type ?u.85288\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh : (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nthis :\n (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) <\n (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) + ε\n⊢ ∃ φ,\n (∀ (x : α), ↑(↑φ x) ≤ f x) ∧\n ∀ (ψ : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑ψ x) ≤ f x) → SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (ψ - φ)) μ < ε",
"tactic": "erw [ENNReal.biSup_add] at this <;> [skip; exact ⟨0, fun x => zero_le _⟩]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.85282\nγ : Type ?u.85285\nδ : Type ?u.85288\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh : (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nthis :\n ∃ i h b,\n b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) μ + ε ∧\n ∀ (i : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑i x) ≤ f x) → SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) μ ≤ b\n⊢ ∃ φ,\n (∀ (x : α), ↑(↑φ x) ≤ f x) ∧\n ∀ (ψ : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑ψ x) ≤ f x) → SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (ψ - φ)) μ < ε",
"state_before": "α : Type u_1\nβ : Type ?u.85282\nγ : Type ?u.85285\nδ : Type ?u.85288\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh : (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nthis :\n (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) 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\n⊢ ∃ φ,\n (∀ (x : α), ↑(↑φ x) ≤ f x) ∧\n ∀ (ψ : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑ψ x) ≤ f x) → SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (ψ - φ)) μ < ε",
"tactic": "simp_rw [lt_iSup_iff, iSup_lt_iff, iSup_le_iff] at this"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.85282\nγ : Type ?u.85285\nδ : Type ?u.85288\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh : (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nφ : α →ₛ ℝ≥0\nhle : ∀ (x : α), ↑(↑φ x) ≤ f x\nb : ℝ≥0∞\nhbφ : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ + ε\nhb : ∀ (i : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑i x) ≤ f x) → SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) μ ≤ b\n⊢ ∃ φ,\n (∀ (x : α), ↑(↑φ x) ≤ f x) ∧\n ∀ (ψ : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑ψ x) ≤ f x) → SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (ψ - φ)) μ < ε",
"state_before": "α : Type u_1\nβ : Type ?u.85282\nγ : Type ?u.85285\nδ : Type ?u.85288\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh : (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nthis :\n ∃ i h b,\n b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) μ + ε ∧\n ∀ (i : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑i x) ≤ f x) → SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) μ ≤ b\n⊢ ∃ φ,\n (∀ (x : α), ↑(↑φ x) ≤ f x) ∧\n ∀ (ψ : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑ψ x) ≤ f x) → SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (ψ - φ)) μ < ε",
"tactic": "rcases this with ⟨φ, hle : ∀ x, ↑(φ x) ≤ f x, b, hbφ, hb⟩"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.85282\nγ : Type ?u.85285\nδ : Type ?u.85288\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh : (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nφ : α →ₛ ℝ≥0\nhle : ∀ (x : α), ↑(↑φ x) ≤ f x\nb : ℝ≥0∞\nhbφ : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ + ε\nhb : ∀ (i : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑i x) ≤ f x) → SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) μ ≤ b\nψ : α →ₛ ℝ≥0\nhψ : ∀ (x : α), ↑(↑ψ x) ≤ f x\n⊢ SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (ψ - φ)) μ < ε",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.85282\nγ : Type ?u.85285\nδ : Type ?u.85288\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh : (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nφ : α →ₛ ℝ≥0\nhle : ∀ (x : α), ↑(↑φ x) ≤ f x\nb : ℝ≥0∞\nhbφ : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ + ε\nhb : ∀ (i : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑i x) ≤ f x) → SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) μ ≤ b\n⊢ ∃ φ,\n (∀ (x : α), ↑(↑φ x) ≤ f x) ∧\n ∀ (ψ : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑ψ x) ≤ f x) → SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (ψ - φ)) μ < ε",
"tactic": "refine' ⟨φ, hle, fun ψ hψ => _⟩"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.85282\nγ : Type ?u.85285\nδ : Type ?u.85288\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh : (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nφ : α →ₛ ℝ≥0\nhle : ∀ (x : α), ↑(↑φ x) ≤ f x\nb : ℝ≥0∞\nhbφ : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ + ε\nhb : ∀ (i : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑i x) ≤ f x) → SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) μ ≤ b\nψ : α →ₛ ℝ≥0\nhψ : ∀ (x : α), ↑(↑ψ x) ≤ f x\nthis : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ ≠ ⊤\n⊢ SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (ψ - φ)) μ < ε",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.85282\nγ : Type ?u.85285\nδ : Type ?u.85288\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh : (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nφ : α →ₛ ℝ≥0\nhle : ∀ (x : α), ↑(↑φ x) ≤ f x\nb : ℝ≥0∞\nhbφ : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ + ε\nhb : ∀ (i : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑i x) ≤ f x) → SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) μ ≤ b\nψ : α →ₛ ℝ≥0\nhψ : ∀ (x : α), ↑(↑ψ x) ≤ f x\n⊢ SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (ψ - φ)) μ < ε",
"tactic": "have : (map (↑) φ).lintegral μ ≠ ∞ := ne_top_of_le_ne_top h (by exact le_iSup₂ (α := ℝ≥0∞) φ hle)"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.85282\nγ : Type ?u.85285\nδ : Type ?u.85288\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh : (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nφ : α →ₛ ℝ≥0\nhle : ∀ (x : α), ↑(↑φ x) ≤ f x\nb : ℝ≥0∞\nhbφ : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ + ε\nhb : ∀ (i : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑i x) ≤ f x) → SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) μ ≤ b\nψ : α →ₛ ℝ≥0\nhψ : ∀ (x : α), ↑(↑ψ x) ≤ f x\nthis : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ ≠ ⊤\n⊢ SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (φ + (ψ - φ))) μ <\n SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ + ε",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.85282\nγ : Type ?u.85285\nδ : Type ?u.85288\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh : (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nφ : α →ₛ ℝ≥0\nhle : ∀ (x : α), ↑(↑φ x) ≤ f x\nb : ℝ≥0∞\nhbφ : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ + ε\nhb : ∀ (i : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑i x) ≤ f x) → SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) μ ≤ b\nψ : α →ₛ ℝ≥0\nhψ : ∀ (x : α), ↑(↑ψ x) ≤ f x\nthis : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ ≠ ⊤\n⊢ SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (ψ - φ)) μ < ε",
"tactic": "rw [← ENNReal.add_lt_add_iff_left this, ← add_lintegral, ← SimpleFunc.map_add @ENNReal.coe_add]"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.85282\nγ : Type ?u.85285\nδ : Type ?u.85288\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh : (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nφ : α →ₛ ℝ≥0\nhle : ∀ (x : α), ↑(↑φ x) ≤ f x\nb : ℝ≥0∞\nhbφ : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ + ε\nhb : ∀ (i : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑i x) ≤ f x) → SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) μ ≤ b\nψ : α →ₛ ℝ≥0\nhψ : ∀ (x : α), ↑(↑ψ x) ≤ f x\nthis : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ ≠ ⊤\nx : α\n⊢ ↑(↑(φ + (ψ - φ)) x) ≤ max ↑(↑φ x) ↑(↑ψ x)",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.85282\nγ : Type ?u.85285\nδ : Type ?u.85288\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh : (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nφ : α →ₛ ℝ≥0\nhle : ∀ (x : α), ↑(↑φ x) ≤ f x\nb : ℝ≥0∞\nhbφ : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ + ε\nhb : ∀ (i : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑i x) ≤ f x) → SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) μ ≤ b\nψ : α →ₛ ℝ≥0\nhψ : ∀ (x : α), ↑(↑ψ x) ≤ f x\nthis : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ ≠ ⊤\n⊢ SimpleFunc.lintegral (SimpleFunc.map ENNReal.some (φ + (ψ - φ))) μ <\n SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ + ε",
"tactic": "refine' (hb _ fun x => le_trans _ (max_le (hle x) (hψ x))).trans_lt hbφ"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.85282\nγ : Type ?u.85285\nδ : Type ?u.85288\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh : (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nφ : α →ₛ ℝ≥0\nhle : ∀ (x : α), ↑(↑φ x) ≤ f x\nb : ℝ≥0∞\nhbφ : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ + ε\nhb : ∀ (i : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑i x) ≤ f x) → SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) μ ≤ b\nψ : α →ₛ ℝ≥0\nhψ : ∀ (x : α), ↑(↑ψ x) ≤ f x\nthis : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ ≠ ⊤\nx : α\n⊢ ↑(φ + (ψ - φ)) x ≤ max (↑φ x) (↑ψ x)",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.85282\nγ : Type ?u.85285\nδ : Type ?u.85288\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh : (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nφ : α →ₛ ℝ≥0\nhle : ∀ (x : α), ↑(↑φ x) ≤ f x\nb : ℝ≥0∞\nhbφ : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ + ε\nhb : ∀ (i : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑i x) ≤ f x) → SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) μ ≤ b\nψ : α →ₛ ℝ≥0\nhψ : ∀ (x : α), ↑(↑ψ x) ≤ f x\nthis : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ ≠ ⊤\nx : α\n⊢ ↑(↑(φ + (ψ - φ)) x) ≤ max ↑(↑φ x) ↑(↑ψ x)",
"tactic": "norm_cast"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.85282\nγ : Type ?u.85285\nδ : Type ?u.85288\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh : (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nφ : α →ₛ ℝ≥0\nhle : ∀ (x : α), ↑(↑φ x) ≤ f x\nb : ℝ≥0∞\nhbφ : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ + ε\nhb : ∀ (i : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑i x) ≤ f x) → SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) μ ≤ b\nψ : α →ₛ ℝ≥0\nhψ : ∀ (x : α), ↑(↑ψ x) ≤ f x\nthis : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ ≠ ⊤\nx : α\n⊢ max (↑φ x) (↑ψ x) ≤ max (↑φ x) (↑ψ x)",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.85282\nγ : Type ?u.85285\nδ : Type ?u.85288\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh : (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nφ : α →ₛ ℝ≥0\nhle : ∀ (x : α), ↑(↑φ x) ≤ f x\nb : ℝ≥0∞\nhbφ : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ + ε\nhb : ∀ (i : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑i x) ≤ f x) → SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) μ ≤ b\nψ : α →ₛ ℝ≥0\nhψ : ∀ (x : α), ↑(↑ψ x) ≤ f x\nthis : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ ≠ ⊤\nx : α\n⊢ ↑(φ + (ψ - φ)) x ≤ max (↑φ x) (↑ψ x)",
"tactic": "simp only [add_apply, sub_apply, add_tsub_eq_max]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.85282\nγ : Type ?u.85285\nδ : Type ?u.85288\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh : (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nφ : α →ₛ ℝ≥0\nhle : ∀ (x : α), ↑(↑φ x) ≤ f x\nb : ℝ≥0∞\nhbφ : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ + ε\nhb : ∀ (i : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑i x) ≤ f x) → SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) μ ≤ b\nψ : α →ₛ ℝ≥0\nhψ : ∀ (x : α), ↑(↑ψ x) ≤ f x\nthis : SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ ≠ ⊤\nx : α\n⊢ max (↑φ x) (↑ψ x) ≤ max (↑φ x) (↑ψ x)",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.85282\nγ : Type ?u.85285\nδ : Type ?u.85288\nm : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ≥0∞\nh : (⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ) ≠ ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nφ : α →ₛ ℝ≥0\nhle : ∀ (x : α), ↑(↑φ x) ≤ f x\nb : ℝ≥0∞\nhbφ : b < SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ + ε\nhb : ∀ (i : α →ₛ ℝ≥0), (∀ (x : α), ↑(↑i x) ≤ f x) → SimpleFunc.lintegral (SimpleFunc.map ENNReal.some i) μ ≤ b\nψ : α →ₛ ℝ≥0\nhψ : ∀ (x : α), ↑(↑ψ x) ≤ f x\n⊢ SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ ≤\n ⨆ (φ : α →ₛ ℝ≥0) (_ : ∀ (x : α), ↑(↑φ x) ≤ f x), SimpleFunc.lintegral (SimpleFunc.map ENNReal.some φ) μ",
"tactic": "exact le_iSup₂ (α := ℝ≥0∞) φ hle"
}
]
| [
268,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
252,
1
]
|
Mathlib/Data/QPF/Univariate/Basic.lean | Qpf.Wequiv.symm | [
{
"state_after": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nx y : PFunctor.W (P F)\nh : Wequiv x y\n⊢ Wequiv y x",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nx y : PFunctor.W (P F)\n⊢ Wequiv x y → Wequiv y x",
"tactic": "intro h"
},
{
"state_after": "case ind\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nx y : PFunctor.W (P F)\na✝¹ : (P F).A\nf✝ f'✝ : PFunctor.B (P F) a✝¹ → PFunctor.W (P F)\na✝ : ∀ (x : PFunctor.B (P F) a✝¹), Wequiv (f✝ x) (f'✝ x)\na_ih✝ : ∀ (x : PFunctor.B (P F) a✝¹), Wequiv (f'✝ x) (f✝ x)\n⊢ Wequiv (WType.mk a✝¹ f'✝) (WType.mk a✝¹ f✝)\n\ncase abs\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nx y : PFunctor.W (P F)\na✝¹ : (P F).A\nf✝ : PFunctor.B (P F) a✝¹ → PFunctor.W (P F)\na'✝ : (P F).A\nf'✝ : PFunctor.B (P F) a'✝ → PFunctor.W (P F)\na✝ : Qpf.abs { fst := a✝¹, snd := f✝ } = Qpf.abs { fst := a'✝, snd := f'✝ }\n⊢ Wequiv (WType.mk a'✝ f'✝) (WType.mk a✝¹ f✝)\n\ncase trans\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nx y u✝ v✝ w✝ : PFunctor.W (P F)\na✝¹ : Wequiv u✝ v✝\na✝ : Wequiv v✝ w✝\na_ih✝¹ : Wequiv v✝ u✝\na_ih✝ : Wequiv w✝ v✝\n⊢ Wequiv w✝ u✝",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nx y : PFunctor.W (P F)\nh : Wequiv x y\n⊢ Wequiv y x",
"tactic": "induction h"
},
{
"state_after": "case abs\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nx y : PFunctor.W (P F)\na✝¹ : (P F).A\nf✝ : PFunctor.B (P F) a✝¹ → PFunctor.W (P F)\na'✝ : (P F).A\nf'✝ : PFunctor.B (P F) a'✝ → PFunctor.W (P F)\na✝ : Qpf.abs { fst := a✝¹, snd := f✝ } = Qpf.abs { fst := a'✝, snd := f'✝ }\n⊢ Wequiv (WType.mk a'✝ f'✝) (WType.mk a✝¹ f✝)\n\ncase trans\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nx y u✝ v✝ w✝ : PFunctor.W (P F)\na✝¹ : Wequiv u✝ v✝\na✝ : Wequiv v✝ w✝\na_ih✝¹ : Wequiv v✝ u✝\na_ih✝ : Wequiv w✝ v✝\n⊢ Wequiv w✝ u✝",
"state_before": "case ind\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nx y : PFunctor.W (P F)\na✝¹ : (P F).A\nf✝ f'✝ : PFunctor.B (P F) a✝¹ → PFunctor.W (P F)\na✝ : ∀ (x : PFunctor.B (P F) a✝¹), Wequiv (f✝ x) (f'✝ x)\na_ih✝ : ∀ (x : PFunctor.B (P F) a✝¹), Wequiv (f'✝ x) (f✝ x)\n⊢ Wequiv (WType.mk a✝¹ f'✝) (WType.mk a✝¹ f✝)\n\ncase abs\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nx y : PFunctor.W (P F)\na✝¹ : (P F).A\nf✝ : PFunctor.B (P F) a✝¹ → PFunctor.W (P F)\na'✝ : (P F).A\nf'✝ : PFunctor.B (P F) a'✝ → PFunctor.W (P F)\na✝ : Qpf.abs { fst := a✝¹, snd := f✝ } = Qpf.abs { fst := a'✝, snd := f'✝ }\n⊢ Wequiv (WType.mk a'✝ f'✝) (WType.mk a✝¹ f✝)\n\ncase trans\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nx y u✝ v✝ w✝ : PFunctor.W (P F)\na✝¹ : Wequiv u✝ v✝\na✝ : Wequiv v✝ w✝\na_ih✝¹ : Wequiv v✝ u✝\na_ih✝ : Wequiv w✝ v✝\n⊢ Wequiv w✝ u✝",
"tactic": "case ind a f f' _ ih => exact Wequiv.ind _ _ _ ih"
},
{
"state_after": "case trans\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nx y u✝ v✝ w✝ : PFunctor.W (P F)\na✝¹ : Wequiv u✝ v✝\na✝ : Wequiv v✝ w✝\na_ih✝¹ : Wequiv v✝ u✝\na_ih✝ : Wequiv w✝ v✝\n⊢ Wequiv w✝ u✝",
"state_before": "case abs\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nx y : PFunctor.W (P F)\na✝¹ : (P F).A\nf✝ : PFunctor.B (P F) a✝¹ → PFunctor.W (P F)\na'✝ : (P F).A\nf'✝ : PFunctor.B (P F) a'✝ → PFunctor.W (P F)\na✝ : Qpf.abs { fst := a✝¹, snd := f✝ } = Qpf.abs { fst := a'✝, snd := f'✝ }\n⊢ Wequiv (WType.mk a'✝ f'✝) (WType.mk a✝¹ f✝)\n\ncase trans\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nx y u✝ v✝ w✝ : PFunctor.W (P F)\na✝¹ : Wequiv u✝ v✝\na✝ : Wequiv v✝ w✝\na_ih✝¹ : Wequiv v✝ u✝\na_ih✝ : Wequiv w✝ v✝\n⊢ Wequiv w✝ u✝",
"tactic": "case abs a f a' f' h => exact Wequiv.abs _ _ _ _ h.symm"
},
{
"state_after": "no goals",
"state_before": "case trans\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nx y u✝ v✝ w✝ : PFunctor.W (P F)\na✝¹ : Wequiv u✝ v✝\na✝ : Wequiv v✝ w✝\na_ih✝¹ : Wequiv v✝ u✝\na_ih✝ : Wequiv w✝ v✝\n⊢ Wequiv w✝ u✝",
"tactic": "case trans x y z _ _ ih₁ ih₂ => exact Qpf.Wequiv.trans _ _ _ ih₂ ih₁"
},
{
"state_after": "no goals",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nx y : PFunctor.W (P F)\na : (P F).A\nf f' : PFunctor.B (P F) a → PFunctor.W (P F)\na✝ : ∀ (x : PFunctor.B (P F) a), Wequiv (f x) (f' x)\nih : ∀ (x : PFunctor.B (P F) a), Wequiv (f' x) (f x)\n⊢ Wequiv (WType.mk a f') (WType.mk a f)",
"tactic": "exact Wequiv.ind _ _ _ ih"
},
{
"state_after": "no goals",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nx y : PFunctor.W (P F)\na : (P F).A\nf : PFunctor.B (P F) a → PFunctor.W (P F)\na' : (P F).A\nf' : PFunctor.B (P F) a' → PFunctor.W (P F)\nh : Qpf.abs { fst := a, snd := f } = Qpf.abs { fst := a', snd := f' }\n⊢ Wequiv (WType.mk a' f') (WType.mk a f)",
"tactic": "exact Wequiv.abs _ _ _ _ h.symm"
},
{
"state_after": "no goals",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nx✝ y✝ x y z : PFunctor.W (P F)\na✝¹ : Wequiv x y\na✝ : Wequiv y z\nih₁ : Wequiv y x\nih₂ : Wequiv z y\n⊢ Wequiv z x",
"tactic": "exact Qpf.Wequiv.trans _ _ _ ih₂ ih₁"
}
]
| [
224,
71
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
219,
1
]
|
Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean | ContinuousAffineMap.comp_contLinear | []
| [
92,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
90,
1
]
|
Mathlib/Analysis/SpecialFunctions/Arsinh.lean | Real.arsinh_neg_iff | []
| [
179,
43
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
178,
1
]
|
src/lean/Init/Core.lean | cast_eq | []
| [
554,
6
]
| d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
553,
1
]
|
Mathlib/Analysis/Calculus/FDeriv/Mul.lean | fderiv_smul_const | []
| [
270,
39
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
268,
1
]
|
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean | CategoryTheory.Limits.PullbackCone.mono_of_isLimitMkIdId | [
{
"state_after": "case mk.intro\nC : Type u\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nW X Y Z✝ : C\nf✝ : X ⟶ Z✝\ng : Y ⟶ Z✝\nf : X ⟶ Y\nt : IsLimit (mk (𝟙 X) (𝟙 X) (_ : 𝟙 X ≫ f = 𝟙 X ≫ f))\nZ : C\nval✝ : Z ⟶ (mk (𝟙 X) (𝟙 X) (_ : 𝟙 X ≫ f = 𝟙 X ≫ f)).pt\neq :\n (val✝ ≫ fst (mk (𝟙 X) (𝟙 X) (_ : 𝟙 X ≫ f = 𝟙 X ≫ f))) ≫ f = (val✝ ≫ snd (mk (𝟙 X) (𝟙 X) (_ : 𝟙 X ≫ f = 𝟙 X ≫ f))) ≫ f\n⊢ val✝ ≫ fst (mk (𝟙 X) (𝟙 X) (_ : 𝟙 X ≫ f = 𝟙 X ≫ f)) = val✝ ≫ snd (mk (𝟙 X) (𝟙 X) (_ : 𝟙 X ≫ f = 𝟙 X ≫ f))",
"state_before": "C : Type u\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nW X Y Z✝ : C\nf✝ : X ⟶ Z✝\ng✝ : Y ⟶ Z✝\nf : X ⟶ Y\nt : IsLimit (mk (𝟙 X) (𝟙 X) (_ : 𝟙 X ≫ f = 𝟙 X ≫ f))\nZ : C\ng h : Z ⟶ X\neq : g ≫ f = h ≫ f\n⊢ g = h",
"tactic": "rcases PullbackCone.IsLimit.lift' t _ _ eq with ⟨_, rfl, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mk.intro\nC : Type u\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nW X Y Z✝ : C\nf✝ : X ⟶ Z✝\ng : Y ⟶ Z✝\nf : X ⟶ Y\nt : IsLimit (mk (𝟙 X) (𝟙 X) (_ : 𝟙 X ≫ f = 𝟙 X ≫ f))\nZ : C\nval✝ : Z ⟶ (mk (𝟙 X) (𝟙 X) (_ : 𝟙 X ≫ f = 𝟙 X ≫ f)).pt\neq :\n (val✝ ≫ fst (mk (𝟙 X) (𝟙 X) (_ : 𝟙 X ≫ f = 𝟙 X ≫ f))) ≫ f = (val✝ ≫ snd (mk (𝟙 X) (𝟙 X) (_ : 𝟙 X ≫ f = 𝟙 X ≫ f))) ≫ f\n⊢ val✝ ≫ fst (mk (𝟙 X) (𝟙 X) (_ : 𝟙 X ≫ f = 𝟙 X ≫ f)) = val✝ ≫ snd (mk (𝟙 X) (𝟙 X) (_ : 𝟙 X ≫ f = 𝟙 X ≫ f))",
"tactic": "rfl"
}
]
| [
720,
9
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
716,
1
]
|
Std/Data/Int/Lemmas.lean | Int.one_mul | [
{
"state_after": "no goals",
"state_before": "n : Nat\n⊢ ofNat (1 * n) = ofNat n",
"tactic": "rw [Nat.one_mul]"
},
{
"state_after": "no goals",
"state_before": "n : Nat\n⊢ -[1 * n+1] = -[n+1]",
"tactic": "rw [Nat.one_mul]"
}
]
| [
522,
61
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
520,
19
]
|
Mathlib/Data/Finset/MulAntidiagonal.lean | Finset.mem_mulAntidiagonal | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedCancelCommMonoid α\ns t : Set α\nhs : Set.IsPwo s\nht : Set.IsPwo t\na : α\nu : Set α\nhu : Set.IsPwo u\nx : α × α\n⊢ x ∈ mulAntidiagonal hs ht a ↔ x.fst ∈ s ∧ x.snd ∈ t ∧ x.fst * x.snd = a",
"tactic": "simp only [mulAntidiagonal, Set.Finite.mem_toFinset, Set.mem_mulAntidiagonal]"
}
]
| [
77,
80
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
76,
1
]
|
Mathlib/Algebra/Order/Floor.lean | Nat.sub_one_lt_floor | []
| [
498,
44
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
497,
1
]
|
Std/Data/Int/Lemmas.lean | Int.subNatNat_elim | [
{
"state_after": "m n : Nat\nmotive : Nat → Nat → Int → Prop\nhp : ∀ (i n : Nat), motive (n + i) n ↑i\nhn : ∀ (i m : Nat), motive m (m + i + 1) -[i+1]\n⊢ motive m n\n (match n - m with\n | 0 => ofNat (m - n)\n | succ k => -[k+1])",
"state_before": "m n : Nat\nmotive : Nat → Nat → Int → Prop\nhp : ∀ (i n : Nat), motive (n + i) n ↑i\nhn : ∀ (i m : Nat), motive m (m + i + 1) -[i+1]\n⊢ motive m n (subNatNat m n)",
"tactic": "unfold subNatNat"
},
{
"state_after": "no goals",
"state_before": "m n : Nat\nmotive : Nat → Nat → Int → Prop\nhp : ∀ (i n : Nat), motive (n + i) n ↑i\nhn : ∀ (i m : Nat), motive m (m + i + 1) -[i+1]\n⊢ motive m n\n (match n - m with\n | 0 => ofNat (m - n)\n | succ k => -[k+1])",
"tactic": "match h : n - m with\n| 0 =>\n have ⟨k, h⟩ := Nat.le.dest (Nat.le_of_sub_eq_zero h)\n rw [h.symm, Nat.add_sub_cancel_left]; apply hp\n| succ k =>\n rw [Nat.sub_eq_iff_eq_add (Nat.le_of_lt (Nat.lt_of_sub_eq_succ h))] at h\n rw [h, Nat.add_comm]; apply hn"
},
{
"state_after": "m n : Nat\nmotive : Nat → Nat → Int → Prop\nhp : ∀ (i n : Nat), motive (n + i) n ↑i\nhn : ∀ (i m : Nat), motive m (m + i + 1) -[i+1]\nh✝ : n - m = 0\nk : Nat\nh : n + k = m\n⊢ motive m n\n (match 0 with\n | 0 => ofNat (m - n)\n | succ k => -[k+1])",
"state_before": "m n : Nat\nmotive : Nat → Nat → Int → Prop\nhp : ∀ (i n : Nat), motive (n + i) n ↑i\nhn : ∀ (i m : Nat), motive m (m + i + 1) -[i+1]\nh : n - m = 0\n⊢ motive m n\n (match 0 with\n | 0 => ofNat (m - n)\n | succ k => -[k+1])",
"tactic": "have ⟨k, h⟩ := Nat.le.dest (Nat.le_of_sub_eq_zero h)"
},
{
"state_after": "m n : Nat\nmotive : Nat → Nat → Int → Prop\nhp : ∀ (i n : Nat), motive (n + i) n ↑i\nhn : ∀ (i m : Nat), motive m (m + i + 1) -[i+1]\nh✝ : n - m = 0\nk : Nat\nh : n + k = m\n⊢ motive (n + k) n\n (match 0 with\n | 0 => ofNat k\n | succ k => -[k+1])",
"state_before": "m n : Nat\nmotive : Nat → Nat → Int → Prop\nhp : ∀ (i n : Nat), motive (n + i) n ↑i\nhn : ∀ (i m : Nat), motive m (m + i + 1) -[i+1]\nh✝ : n - m = 0\nk : Nat\nh : n + k = m\n⊢ motive m n\n (match 0 with\n | 0 => ofNat (m - n)\n | succ k => -[k+1])",
"tactic": "rw [h.symm, Nat.add_sub_cancel_left]"
},
{
"state_after": "no goals",
"state_before": "m n : Nat\nmotive : Nat → Nat → Int → Prop\nhp : ∀ (i n : Nat), motive (n + i) n ↑i\nhn : ∀ (i m : Nat), motive m (m + i + 1) -[i+1]\nh✝ : n - m = 0\nk : Nat\nh : n + k = m\n⊢ motive (n + k) n\n (match 0 with\n | 0 => ofNat k\n | succ k => -[k+1])",
"tactic": "apply hp"
},
{
"state_after": "m n : Nat\nmotive : Nat → Nat → Int → Prop\nhp : ∀ (i n : Nat), motive (n + i) n ↑i\nhn : ∀ (i m : Nat), motive m (m + i + 1) -[i+1]\nk : Nat\nh : n = succ k + m\n⊢ motive m n\n (match succ k with\n | 0 => ofNat (m - n)\n | succ k => -[k+1])",
"state_before": "m n : Nat\nmotive : Nat → Nat → Int → Prop\nhp : ∀ (i n : Nat), motive (n + i) n ↑i\nhn : ∀ (i m : Nat), motive m (m + i + 1) -[i+1]\nk : Nat\nh : n - m = succ k\n⊢ motive m n\n (match succ k with\n | 0 => ofNat (m - n)\n | succ k => -[k+1])",
"tactic": "rw [Nat.sub_eq_iff_eq_add (Nat.le_of_lt (Nat.lt_of_sub_eq_succ h))] at h"
},
{
"state_after": "m n : Nat\nmotive : Nat → Nat → Int → Prop\nhp : ∀ (i n : Nat), motive (n + i) n ↑i\nhn : ∀ (i m : Nat), motive m (m + i + 1) -[i+1]\nk : Nat\nh : n = succ k + m\n⊢ motive m (m + succ k)\n (match succ k with\n | 0 => ofNat (m - (m + succ k))\n | succ k => -[k+1])",
"state_before": "m n : Nat\nmotive : Nat → Nat → Int → Prop\nhp : ∀ (i n : Nat), motive (n + i) n ↑i\nhn : ∀ (i m : Nat), motive m (m + i + 1) -[i+1]\nk : Nat\nh : n = succ k + m\n⊢ motive m n\n (match succ k with\n | 0 => ofNat (m - n)\n | succ k => -[k+1])",
"tactic": "rw [h, Nat.add_comm]"
},
{
"state_after": "no goals",
"state_before": "m n : Nat\nmotive : Nat → Nat → Int → Prop\nhp : ∀ (i n : Nat), motive (n + i) n ↑i\nhn : ∀ (i m : Nat), motive m (m + i + 1) -[i+1]\nk : Nat\nh : n = succ k + m\n⊢ motive m (m + succ k)\n (match succ k with\n | 0 => ofNat (m - (m + succ k))\n | succ k => -[k+1])",
"tactic": "apply hn"
}
]
| [
112,
35
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
101,
1
]
|
Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | Cycle.support_formPerm | [
{
"state_after": "case h\nα : Type u_1\ninst✝¹ : DecidableEq α\ns✝¹ s' : Cycle α\ninst✝ : Fintype α\ns✝ : Cycle α\nh✝ : Nodup s✝\nhn✝ : Nontrivial s✝\ns : List α\nh : Nodup (Quot.mk Setoid.r s)\nhn : Nontrivial (Quot.mk Setoid.r s)\n⊢ support (formPerm (Quot.mk Setoid.r s) h) = toFinset (Quot.mk Setoid.r s)",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ns✝ s' : Cycle α\ninst✝ : Fintype α\ns : Cycle α\nh : Nodup s\nhn : Nontrivial s\n⊢ support (formPerm s h) = toFinset s",
"tactic": "induction' s using Quot.inductionOn with s"
},
{
"state_after": "case h\nα : Type u_1\ninst✝¹ : DecidableEq α\ns✝¹ s' : Cycle α\ninst✝ : Fintype α\ns✝ : Cycle α\nh✝ : Nodup s✝\nhn✝ : Nontrivial s✝\ns : List α\nh : Nodup (Quot.mk Setoid.r s)\nhn : Nontrivial (Quot.mk Setoid.r s)\n⊢ ∀ (x : α), s ≠ [x]",
"state_before": "case h\nα : Type u_1\ninst✝¹ : DecidableEq α\ns✝¹ s' : Cycle α\ninst✝ : Fintype α\ns✝ : Cycle α\nh✝ : Nodup s✝\nhn✝ : Nontrivial s✝\ns : List α\nh : Nodup (Quot.mk Setoid.r s)\nhn : Nontrivial (Quot.mk Setoid.r s)\n⊢ support (formPerm (Quot.mk Setoid.r s) h) = toFinset (Quot.mk Setoid.r s)",
"tactic": "refine' support_formPerm_of_nodup s h _"
},
{
"state_after": "case h\nα : Type u_1\ninst✝¹ : DecidableEq α\ns✝ s' : Cycle α\ninst✝ : Fintype α\ns : Cycle α\nh✝ : Nodup s\nhn✝ : Nontrivial s\nx✝ : α\nh : Nodup (Quot.mk Setoid.r [x✝])\nhn : Nontrivial (Quot.mk Setoid.r [x✝])\n⊢ False",
"state_before": "case h\nα : Type u_1\ninst✝¹ : DecidableEq α\ns✝¹ s' : Cycle α\ninst✝ : Fintype α\ns✝ : Cycle α\nh✝ : Nodup s✝\nhn✝ : Nontrivial s✝\ns : List α\nh : Nodup (Quot.mk Setoid.r s)\nhn : Nontrivial (Quot.mk Setoid.r s)\n⊢ ∀ (x : α), s ≠ [x]",
"tactic": "rintro _ rfl"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\ninst✝¹ : DecidableEq α\ns✝ s' : Cycle α\ninst✝ : Fintype α\ns : Cycle α\nh✝ : Nodup s\nhn✝ : Nontrivial s\nx✝ : α\nh : Nodup (Quot.mk Setoid.r [x✝])\nhn : Nontrivial (Quot.mk Setoid.r [x✝])\n⊢ False",
"tactic": "simpa [Nat.succ_le_succ_iff] using length_nontrivial hn"
}
]
| [
174,
58
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
169,
1
]
|
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean | AffineSubspace.bot_ne_top | [
{
"state_after": "k : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ncontra : ⊥ = ⊤\n⊢ False",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\n⊢ ⊥ ≠ ⊤",
"tactic": "intro contra"
},
{
"state_after": "k : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ncontra : ∅ = univ\n⊢ False",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ncontra : ⊥ = ⊤\n⊢ False",
"tactic": "rw [← ext_iff, bot_coe, top_coe] at contra"
},
{
"state_after": "no goals",
"state_before": "k : Type u_2\nV : Type u_3\nP : Type u_1\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\np₁ p₂ : P\ncontra : ∅ = univ\n⊢ False",
"tactic": "exact Set.empty_ne_univ contra"
}
]
| [
774,
33
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
771,
1
]
|
Mathlib/Topology/Connected.lean | IsPreconnected.subset_connectedComponent | []
| [
641,
90
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
640,
1
]
|
Mathlib/Algebra/DirectSum/Decomposition.lean | DirectSum.decompose_symm_zero | []
| [
152,
38
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
151,
1
]
|
Mathlib/GroupTheory/Perm/Cycle/Concrete.lean | Equiv.Perm.formPerm_toList | [
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nf : Perm α\nx : α\nhx : ↑f x = x\n⊢ formPerm (toList f x) = cycleOf f x\n\ncase neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nf : Perm α\nx : α\nhx : ¬↑f x = x\n⊢ formPerm (toList f x) = cycleOf f x",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nf : Perm α\nx : α\n⊢ formPerm (toList f x) = cycleOf f x",
"tactic": "by_cases hx : f x = x"
},
{
"state_after": "case neg.H\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nf : Perm α\nx : α\nhx : ¬↑f x = x\ny : α\n⊢ ↑(formPerm (toList f x)) y = ↑(cycleOf f x) y",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nf : Perm α\nx : α\nhx : ¬↑f x = x\n⊢ formPerm (toList f x) = cycleOf f x",
"tactic": "ext y"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nf : Perm α\nx : α\nhx : ¬↑f x = x\ny : α\nhy : SameCycle f x y\n⊢ ↑(formPerm (toList f x)) y = ↑(cycleOf f x) y\n\ncase neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nf : Perm α\nx : α\nhx : ¬↑f x = x\ny : α\nhy : ¬SameCycle f x y\n⊢ ↑(formPerm (toList f x)) y = ↑(cycleOf f x) y",
"state_before": "case neg.H\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nf : Perm α\nx : α\nhx : ¬↑f x = x\ny : α\n⊢ ↑(formPerm (toList f x)) y = ↑(cycleOf f x) y",
"tactic": "by_cases hy : SameCycle f x y"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nf : Perm α\nx : α\nhx : ↑f x = x\n⊢ formPerm (toList f x) = cycleOf f x",
"tactic": "rw [(cycleOf_eq_one_iff f).mpr hx, toList_eq_nil_iff.mpr (not_mem_support.mpr hx),\n formPerm_nil]"
},
{
"state_after": "case pos.intro.intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nf : Perm α\nx : α\nhx : ¬↑f x = x\nk : ℕ\nw✝ : k < Finset.card (support (cycleOf f x))\nhy : SameCycle f x (↑(f ^ k) x)\n⊢ ↑(formPerm (toList f x)) (↑(f ^ k) x) = ↑(cycleOf f x) (↑(f ^ k) x)",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nf : Perm α\nx : α\nhx : ¬↑f x = x\ny : α\nhy : SameCycle f x y\n⊢ ↑(formPerm (toList f x)) y = ↑(cycleOf f x) y",
"tactic": "obtain ⟨k, _, rfl⟩ := hy.exists_pow_eq_of_mem_support (mem_support.mpr hx)"
},
{
"state_after": "case pos.intro.intro.hy\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nf : Perm α\nx : α\nhx : ¬↑f x = x\nk : ℕ\nw✝ : k < Finset.card (support (cycleOf f x))\nhy : SameCycle f x (↑(f ^ k) x)\n⊢ ↑(f ^ k) x ∈ toList f x",
"state_before": "case pos.intro.intro\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nf : Perm α\nx : α\nhx : ¬↑f x = x\nk : ℕ\nw✝ : k < Finset.card (support (cycleOf f x))\nhy : SameCycle f x (↑(f ^ k) x)\n⊢ ↑(formPerm (toList f x)) (↑(f ^ k) x) = ↑(cycleOf f x) (↑(f ^ k) x)",
"tactic": "rw [cycleOf_apply_apply_pow_self, List.formPerm_apply_mem_eq_next (nodup_toList f x),\n next_toList_eq_apply, pow_succ, mul_apply]"
},
{
"state_after": "case pos.intro.intro.hy\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nf : Perm α\nx : α\nhx : ¬↑f x = x\nk : ℕ\nw✝ : k < Finset.card (support (cycleOf f x))\nhy : SameCycle f x (↑(f ^ k) x)\n⊢ SameCycle f x (↑(f ^ k) x) ∧ x ∈ support f",
"state_before": "case pos.intro.intro.hy\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nf : Perm α\nx : α\nhx : ¬↑f x = x\nk : ℕ\nw✝ : k < Finset.card (support (cycleOf f x))\nhy : SameCycle f x (↑(f ^ k) x)\n⊢ ↑(f ^ k) x ∈ toList f x",
"tactic": "rw [mem_toList_iff]"
},
{
"state_after": "no goals",
"state_before": "case pos.intro.intro.hy\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nf : Perm α\nx : α\nhx : ¬↑f x = x\nk : ℕ\nw✝ : k < Finset.card (support (cycleOf f x))\nhy : SameCycle f x (↑(f ^ k) x)\n⊢ SameCycle f x (↑(f ^ k) x) ∧ x ∈ support f",
"tactic": "exact ⟨⟨k, rfl⟩, mem_support.mpr hx⟩"
},
{
"state_after": "case neg.h\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nf : Perm α\nx : α\nhx : ¬↑f x = x\ny : α\nhy : ¬SameCycle f x y\n⊢ ¬y ∈ toList f x",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nf : Perm α\nx : α\nhx : ¬↑f x = x\ny : α\nhy : ¬SameCycle f x y\n⊢ ↑(formPerm (toList f x)) y = ↑(cycleOf f x) y",
"tactic": "rw [cycleOf_apply_of_not_sameCycle hy, formPerm_apply_of_not_mem]"
},
{
"state_after": "no goals",
"state_before": "case neg.h\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nf : Perm α\nx : α\nhx : ¬↑f x = x\ny : α\nhy : ¬SameCycle f x y\n⊢ ¬y ∈ toList f x",
"tactic": "simp [mem_toList_iff, hy]"
}
]
| [
392,
30
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
380,
1
]
|
Mathlib/Order/FixedPoints.lean | OrderHom.le_nextFixed | []
| [
219,
25
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
218,
1
]
|
Mathlib/Data/Analysis/Filter.lean | Filter.Realizer.top_F | []
| [
192,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
191,
1
]
|
Mathlib/Analysis/Convex/Side.lean | AffineSubspace.wOppSide_of_right_mem | []
| [
273,
35
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
271,
1
]
|
Mathlib/Data/Polynomial/Eval.lean | Polynomial.eval_list_prod | []
| [
1120,
34
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1119,
1
]
|
Mathlib/Analysis/InnerProductSpace/Projection.lean | reflection_symm | []
| [
658,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
657,
1
]
|
Mathlib/Combinatorics/SimpleGraph/Partition.lean | SimpleGraph.Partition.partOfVertex_mem | [
{
"state_after": "case intro\nV : Type u\nG : SimpleGraph V\nP : Partition G\nv : V\nh : Exists.choose (_ : ∃! b x, v ∈ b) ∈ P.parts\n⊢ partOfVertex P v ∈ P.parts",
"state_before": "V : Type u\nG : SimpleGraph V\nP : Partition G\nv : V\n⊢ partOfVertex P v ∈ P.parts",
"tactic": "obtain ⟨h, -⟩ := (P.isPartition.2 v).choose_spec.1"
},
{
"state_after": "no goals",
"state_before": "case intro\nV : Type u\nG : SimpleGraph V\nP : Partition G\nv : V\nh : Exists.choose (_ : ∃! b x, v ∈ b) ∈ P.parts\n⊢ partOfVertex P v ∈ P.parts",
"tactic": "exact h"
}
]
| [
93,
10
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
91,
1
]
|
Mathlib/Order/Hom/CompleteLattice.lean | FrameHom.copy_eq | []
| [
583,
17
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
582,
1
]
|
Mathlib/Data/Polynomial/Inductions.lean | Polynomial.natDegree_ne_zero_induction_on | [
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\np q : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\n⊢ natDegree f = 0 ∨ M f",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\np q : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\n⊢ M f",
"tactic": "suffices f.natDegree = 0 ∨ M f from Or.recOn this (fun h => (f0 h).elim) id"
},
{
"state_after": "case refine_1\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\np q : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\n⊢ ∀ (a : R), natDegree (↑C a) = 0 ∨ M (↑C a)\n\ncase refine_2\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\np q : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\n⊢ ∀ (p q : R[X]), natDegree p = 0 ∨ M p → natDegree q = 0 ∨ M q → natDegree (p + q) = 0 ∨ M (p + q)\n\ncase refine_3\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\np q : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\n⊢ ∀ (n : ℕ) (a : R),\n natDegree (↑C a * X ^ n) = 0 ∨ M (↑C a * X ^ n) → natDegree (↑C a * X ^ (n + 1)) = 0 ∨ M (↑C a * X ^ (n + 1))",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\np q : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\n⊢ natDegree f = 0 ∨ M f",
"tactic": "refine Polynomial.induction_on f ?_ ?_ ?_"
},
{
"state_after": "no goals",
"state_before": "case refine_1\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\np q : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\n⊢ ∀ (a : R), natDegree (↑C a) = 0 ∨ M (↑C a)",
"tactic": "exact fun a => Or.inl (natDegree_C _)"
},
{
"state_after": "case refine_2.inl.inl\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\np q : R[X]\nhp : natDegree p = 0\nhq : natDegree q = 0\n⊢ natDegree (p + q) = 0 ∨ M (p + q)\n\ncase refine_2.inl.inr\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\np q : R[X]\nhp : natDegree p = 0\nhq : M q\n⊢ natDegree (p + q) = 0 ∨ M (p + q)\n\ncase refine_2.inr.inl\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\np q : R[X]\nhp : M p\nhq : natDegree q = 0\n⊢ natDegree (p + q) = 0 ∨ M (p + q)\n\ncase refine_2.inr.inr\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\np q : R[X]\nhp : M p\nhq : M q\n⊢ natDegree (p + q) = 0 ∨ M (p + q)",
"state_before": "case refine_2\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\np q : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\n⊢ ∀ (p q : R[X]), natDegree p = 0 ∨ M p → natDegree q = 0 ∨ M q → natDegree (p + q) = 0 ∨ M (p + q)",
"tactic": "rintro p q (hp | hp) (hq | hq)"
},
{
"state_after": "case refine_2.inl.inl\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\np q : R[X]\nhp : natDegree p = 0\nhq : natDegree q = 0\n⊢ natDegree (p + q) = 0",
"state_before": "case refine_2.inl.inl\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\np q : R[X]\nhp : natDegree p = 0\nhq : natDegree q = 0\n⊢ natDegree (p + q) = 0 ∨ M (p + q)",
"tactic": "refine' Or.inl _"
},
{
"state_after": "no goals",
"state_before": "case refine_2.inl.inl\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\np q : R[X]\nhp : natDegree p = 0\nhq : natDegree q = 0\n⊢ natDegree (p + q) = 0",
"tactic": "rw [eq_C_of_natDegree_eq_zero hp, eq_C_of_natDegree_eq_zero hq, ← C_add, natDegree_C]"
},
{
"state_after": "case refine_2.inl.inr\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\np q : R[X]\nhp : natDegree p = 0\nhq : M q\n⊢ M (p + q)",
"state_before": "case refine_2.inl.inr\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\np q : R[X]\nhp : natDegree p = 0\nhq : M q\n⊢ natDegree (p + q) = 0 ∨ M (p + q)",
"tactic": "refine' Or.inr _"
},
{
"state_after": "case refine_2.inl.inr\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\np q : R[X]\nhp : natDegree p = 0\nhq : M q\n⊢ M (↑C (coeff p 0) + q)",
"state_before": "case refine_2.inl.inr\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\np q : R[X]\nhp : natDegree p = 0\nhq : M q\n⊢ M (p + q)",
"tactic": "rw [eq_C_of_natDegree_eq_zero hp]"
},
{
"state_after": "no goals",
"state_before": "case refine_2.inl.inr\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\np q : R[X]\nhp : natDegree p = 0\nhq : M q\n⊢ M (↑C (coeff p 0) + q)",
"tactic": "exact h_C_add hq"
},
{
"state_after": "case refine_2.inr.inl\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\np q : R[X]\nhp : M p\nhq : natDegree q = 0\n⊢ M (p + q)",
"state_before": "case refine_2.inr.inl\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\np q : R[X]\nhp : M p\nhq : natDegree q = 0\n⊢ natDegree (p + q) = 0 ∨ M (p + q)",
"tactic": "refine' Or.inr _"
},
{
"state_after": "case refine_2.inr.inl\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\np q : R[X]\nhp : M p\nhq : natDegree q = 0\n⊢ M (↑C (coeff q 0) + p)",
"state_before": "case refine_2.inr.inl\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\np q : R[X]\nhp : M p\nhq : natDegree q = 0\n⊢ M (p + q)",
"tactic": "rw [eq_C_of_natDegree_eq_zero hq, add_comm]"
},
{
"state_after": "no goals",
"state_before": "case refine_2.inr.inl\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\np q : R[X]\nhp : M p\nhq : natDegree q = 0\n⊢ M (↑C (coeff q 0) + p)",
"tactic": "exact h_C_add hp"
},
{
"state_after": "no goals",
"state_before": "case refine_2.inr.inr\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\np✝ q✝ : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\np q : R[X]\nhp : M p\nhq : M q\n⊢ natDegree (p + q) = 0 ∨ M (p + q)",
"tactic": "exact Or.inr (h_add hp hq)"
},
{
"state_after": "case refine_3\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na✝¹ b : R\nn✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\nn : ℕ\na : R\na✝ : natDegree (↑C a * X ^ n) = 0 ∨ M (↑C a * X ^ n)\n⊢ natDegree (↑C a * X ^ (n + 1)) = 0 ∨ M (↑C a * X ^ (n + 1))",
"state_before": "case refine_3\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\np q : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\n⊢ ∀ (n : ℕ) (a : R),\n natDegree (↑C a * X ^ n) = 0 ∨ M (↑C a * X ^ n) → natDegree (↑C a * X ^ (n + 1)) = 0 ∨ M (↑C a * X ^ (n + 1))",
"tactic": "intro n a _"
},
{
"state_after": "case pos\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na✝¹ b : R\nn✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\nn : ℕ\na : R\na✝ : natDegree (↑C a * X ^ n) = 0 ∨ M (↑C a * X ^ n)\na0 : a = 0\n⊢ natDegree (↑C a * X ^ (n + 1)) = 0 ∨ M (↑C a * X ^ (n + 1))\n\ncase neg\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na✝¹ b : R\nn✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\nn : ℕ\na : R\na✝ : natDegree (↑C a * X ^ n) = 0 ∨ M (↑C a * X ^ n)\na0 : ¬a = 0\n⊢ natDegree (↑C a * X ^ (n + 1)) = 0 ∨ M (↑C a * X ^ (n + 1))",
"state_before": "case refine_3\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na✝¹ b : R\nn✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\nn : ℕ\na : R\na✝ : natDegree (↑C a * X ^ n) = 0 ∨ M (↑C a * X ^ n)\n⊢ natDegree (↑C a * X ^ (n + 1)) = 0 ∨ M (↑C a * X ^ (n + 1))",
"tactic": "by_cases a0 : a = 0"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na✝¹ b : R\nn✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\nn : ℕ\na : R\na✝ : natDegree (↑C a * X ^ n) = 0 ∨ M (↑C a * X ^ n)\na0 : a = 0\n⊢ natDegree (↑C a * X ^ (n + 1)) = 0 ∨ M (↑C a * X ^ (n + 1))",
"tactic": "exact Or.inl (by rw [a0, C_0, zero_mul, natDegree_zero])"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na✝¹ b : R\nn✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\nn : ℕ\na : R\na✝ : natDegree (↑C a * X ^ n) = 0 ∨ M (↑C a * X ^ n)\na0 : a = 0\n⊢ natDegree (↑C a * X ^ (n + 1)) = 0",
"tactic": "rw [a0, C_0, zero_mul, natDegree_zero]"
},
{
"state_after": "case neg\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na✝¹ b : R\nn✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\nn : ℕ\na : R\na✝ : natDegree (↑C a * X ^ n) = 0 ∨ M (↑C a * X ^ n)\na0 : ¬a = 0\n⊢ M (↑C a * X ^ (n + 1))",
"state_before": "case neg\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na✝¹ b : R\nn✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\nn : ℕ\na : R\na✝ : natDegree (↑C a * X ^ n) = 0 ∨ M (↑C a * X ^ n)\na0 : ¬a = 0\n⊢ natDegree (↑C a * X ^ (n + 1)) = 0 ∨ M (↑C a * X ^ (n + 1))",
"tactic": "refine' Or.inr _"
},
{
"state_after": "case neg\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na✝¹ b : R\nn✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\nn : ℕ\na : R\na✝ : natDegree (↑C a * X ^ n) = 0 ∨ M (↑C a * X ^ n)\na0 : ¬a = 0\n⊢ M (↑(monomial (n + 1)) a)",
"state_before": "case neg\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na✝¹ b : R\nn✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\nn : ℕ\na : R\na✝ : natDegree (↑C a * X ^ n) = 0 ∨ M (↑C a * X ^ n)\na0 : ¬a = 0\n⊢ M (↑C a * X ^ (n + 1))",
"tactic": "rw [C_mul_X_pow_eq_monomial]"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na✝¹ b : R\nn✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nM : R[X] → Prop\nf : R[X]\nf0 : natDegree f ≠ 0\nh_C_add : ∀ {a : R} {p : R[X]}, M p → M (↑C a + p)\nh_add : ∀ {p q : R[X]}, M p → M q → M (p + q)\nh_monomial : ∀ {n : ℕ} {a : R}, a ≠ 0 → n ≠ 0 → M (↑(monomial n) a)\nn : ℕ\na : R\na✝ : natDegree (↑C a * X ^ n) = 0 ∨ M (↑C a * X ^ n)\na0 : ¬a = 0\n⊢ M (↑(monomial (n + 1)) a)",
"tactic": "exact h_monomial a0 n.succ_ne_zero"
}
]
| [
181,
41
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
160,
1
]
|
Mathlib/Order/Atoms.lean | bot_covby_top | []
| [
497,
23
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
496,
1
]
|
Mathlib/Analysis/NormedSpace/AddTorsor.lean | eventually_homothety_mem_of_mem_interior | [
{
"state_after": "α : Type ?u.85021\nV : Type ?u.85024\nP : Type ?u.85027\nW : Type u_3\nQ : Type u_1\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\n⊢ ∃ i, 0 < i ∧ ∀ ⦃x_1 : 𝕜⦄, x_1 ∈ {y | ‖y - 1‖ < i} → ↑(homothety x x_1) y ∈ s",
"state_before": "α : Type ?u.85021\nV : Type ?u.85024\nP : Type ?u.85027\nW : Type u_3\nQ : Type u_1\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\n⊢ ∀ᶠ (δ : 𝕜) in 𝓝 1, ↑(homothety x δ) y ∈ s",
"tactic": "rw [(NormedAddCommGroup.nhds_basis_norm_lt (1 : 𝕜)).eventually_iff]"
},
{
"state_after": "case inl\nα : Type ?u.85021\nV : Type ?u.85024\nP : Type ?u.85027\nW : Type u_3\nQ : Type u_1\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\nh : y = x\n⊢ ∃ i, 0 < i ∧ ∀ ⦃x_1 : 𝕜⦄, x_1 ∈ {y | ‖y - 1‖ < i} → ↑(homothety x x_1) y ∈ s\n\ncase inr\nα : Type ?u.85021\nV : Type ?u.85024\nP : Type ?u.85027\nW : Type u_3\nQ : Type u_1\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\nh : y ≠ x\n⊢ ∃ i, 0 < i ∧ ∀ ⦃x_1 : 𝕜⦄, x_1 ∈ {y | ‖y - 1‖ < i} → ↑(homothety x x_1) y ∈ s",
"state_before": "α : Type ?u.85021\nV : Type ?u.85024\nP : Type ?u.85027\nW : Type u_3\nQ : Type u_1\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\n⊢ ∃ i, 0 < i ∧ ∀ ⦃x_1 : 𝕜⦄, x_1 ∈ {y | ‖y - 1‖ < i} → ↑(homothety x x_1) y ∈ s",
"tactic": "cases' eq_or_ne y x with h h"
},
{
"state_after": "case inr\nα : Type ?u.85021\nV : Type ?u.85024\nP : Type ?u.85027\nW : Type u_3\nQ : Type u_1\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\nh : y ≠ x\nhxy : 0 < ‖y -ᵥ x‖\n⊢ ∃ i, 0 < i ∧ ∀ ⦃x_1 : 𝕜⦄, x_1 ∈ {y | ‖y - 1‖ < i} → ↑(homothety x x_1) y ∈ s",
"state_before": "case inr\nα : Type ?u.85021\nV : Type ?u.85024\nP : Type ?u.85027\nW : Type u_3\nQ : Type u_1\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\nh : y ≠ x\n⊢ ∃ i, 0 < i ∧ ∀ ⦃x_1 : 𝕜⦄, x_1 ∈ {y | ‖y - 1‖ < i} → ↑(homothety x x_1) y ∈ s",
"tactic": "have hxy : 0 < ‖y -ᵥ x‖ := by rwa [norm_pos_iff, vsub_ne_zero]"
},
{
"state_after": "case inr.intro.intro.intro\nα : Type ?u.85021\nV : Type ?u.85024\nP : Type ?u.85027\nW : Type u_3\nQ : Type u_1\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\nh : y ≠ x\nhxy : 0 < ‖y -ᵥ x‖\nu : Set Q\nhu₁ : u ⊆ s\nhu₂ : IsOpen u\nhu₃ : y ∈ u\n⊢ ∃ i, 0 < i ∧ ∀ ⦃x_1 : 𝕜⦄, x_1 ∈ {y | ‖y - 1‖ < i} → ↑(homothety x x_1) y ∈ s",
"state_before": "case inr\nα : Type ?u.85021\nV : Type ?u.85024\nP : Type ?u.85027\nW : Type u_3\nQ : Type u_1\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\nh : y ≠ x\nhxy : 0 < ‖y -ᵥ x‖\n⊢ ∃ i, 0 < i ∧ ∀ ⦃x_1 : 𝕜⦄, x_1 ∈ {y | ‖y - 1‖ < i} → ↑(homothety x x_1) y ∈ s",
"tactic": "obtain ⟨u, hu₁, hu₂, hu₃⟩ := mem_interior.mp hy"
},
{
"state_after": "case inr.intro.intro.intro.intro.intro\nα : Type ?u.85021\nV : Type ?u.85024\nP : Type ?u.85027\nW : Type u_3\nQ : Type u_1\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\nh : y ≠ x\nhxy : 0 < ‖y -ᵥ x‖\nu : Set Q\nhu₁ : u ⊆ s\nhu₂ : IsOpen u\nhu₃ : y ∈ u\nε : ℝ\nhε : ε > 0\nhyε : Metric.ball y ε ⊆ u\n⊢ ∃ i, 0 < i ∧ ∀ ⦃x_1 : 𝕜⦄, x_1 ∈ {y | ‖y - 1‖ < i} → ↑(homothety x x_1) y ∈ s",
"state_before": "case inr.intro.intro.intro\nα : Type ?u.85021\nV : Type ?u.85024\nP : Type ?u.85027\nW : Type u_3\nQ : Type u_1\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\nh : y ≠ x\nhxy : 0 < ‖y -ᵥ x‖\nu : Set Q\nhu₁ : u ⊆ s\nhu₂ : IsOpen u\nhu₃ : y ∈ u\n⊢ ∃ i, 0 < i ∧ ∀ ⦃x_1 : 𝕜⦄, x_1 ∈ {y | ‖y - 1‖ < i} → ↑(homothety x x_1) y ∈ s",
"tactic": "obtain ⟨ε, hε, hyε⟩ := Metric.isOpen_iff.mp hu₂ y hu₃"
},
{
"state_after": "case inr.intro.intro.intro.intro.intro\nα : Type ?u.85021\nV : Type ?u.85024\nP : Type ?u.85027\nW : Type u_3\nQ : Type u_1\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\nh : y ≠ x\nhxy : 0 < ‖y -ᵥ x‖\nu : Set Q\nhu₁ : u ⊆ s\nhu₂ : IsOpen u\nhu₃ : y ∈ u\nε : ℝ\nhε : ε > 0\nhyε : Metric.ball y ε ⊆ u\nδ : 𝕜\nhδ : ‖δ - 1‖ < ε / ‖y -ᵥ x‖\n⊢ ↑(homothety x δ) y ∈ Metric.ball y ε",
"state_before": "case inr.intro.intro.intro.intro.intro\nα : Type ?u.85021\nV : Type ?u.85024\nP : Type ?u.85027\nW : Type u_3\nQ : Type u_1\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\nh : y ≠ x\nhxy : 0 < ‖y -ᵥ x‖\nu : Set Q\nhu₁ : u ⊆ s\nhu₂ : IsOpen u\nhu₃ : y ∈ u\nε : ℝ\nhε : ε > 0\nhyε : Metric.ball y ε ⊆ u\n⊢ ∃ i, 0 < i ∧ ∀ ⦃x_1 : 𝕜⦄, x_1 ∈ {y | ‖y - 1‖ < i} → ↑(homothety x x_1) y ∈ s",
"tactic": "refine' ⟨ε / ‖y -ᵥ x‖, div_pos hε hxy, fun δ (hδ : ‖δ - 1‖ < ε / ‖y -ᵥ x‖) => hu₁ (hyε _)⟩"
},
{
"state_after": "case inr.intro.intro.intro.intro.intro\nα : Type ?u.85021\nV : Type ?u.85024\nP : Type ?u.85027\nW : Type u_3\nQ : Type u_1\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\nh : y ≠ x\nhxy : 0 < ‖y -ᵥ x‖\nu : Set Q\nhu₁ : u ⊆ s\nhu₂ : IsOpen u\nhu₃ : y ∈ u\nε : ℝ\nhε : ε > 0\nhyε : Metric.ball y ε ⊆ u\nδ : 𝕜\nhδ✝ : ‖δ - 1‖ * ‖y -ᵥ x‖ < ε\nhδ : ‖δ • (y -ᵥ x) - (y -ᵥ x)‖ < ε\n⊢ ↑(homothety x δ) y ∈ Metric.ball y ε",
"state_before": "case inr.intro.intro.intro.intro.intro\nα : Type ?u.85021\nV : Type ?u.85024\nP : Type ?u.85027\nW : Type u_3\nQ : Type u_1\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\nh : y ≠ x\nhxy : 0 < ‖y -ᵥ x‖\nu : Set Q\nhu₁ : u ⊆ s\nhu₂ : IsOpen u\nhu₃ : y ∈ u\nε : ℝ\nhε : ε > 0\nhyε : Metric.ball y ε ⊆ u\nδ : 𝕜\nhδ : ‖δ - 1‖ < ε / ‖y -ᵥ x‖\n⊢ ↑(homothety x δ) y ∈ Metric.ball y ε",
"tactic": "rw [lt_div_iff hxy, ← norm_smul, sub_smul, one_smul] at hδ"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.intro.intro.intro.intro\nα : Type ?u.85021\nV : Type ?u.85024\nP : Type ?u.85027\nW : Type u_3\nQ : Type u_1\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\nh : y ≠ x\nhxy : 0 < ‖y -ᵥ x‖\nu : Set Q\nhu₁ : u ⊆ s\nhu₂ : IsOpen u\nhu₃ : y ∈ u\nε : ℝ\nhε : ε > 0\nhyε : Metric.ball y ε ⊆ u\nδ : 𝕜\nhδ✝ : ‖δ - 1‖ * ‖y -ᵥ x‖ < ε\nhδ : ‖δ • (y -ᵥ x) - (y -ᵥ x)‖ < ε\n⊢ ↑(homothety x δ) y ∈ Metric.ball y ε",
"tactic": "rwa [homothety_apply, Metric.mem_ball, dist_eq_norm_vsub W, vadd_vsub_eq_sub_vsub]"
},
{
"state_after": "case inl\nα : Type ?u.85021\nV : Type ?u.85024\nP : Type ?u.85027\nW : Type u_3\nQ : Type u_1\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\nh : y = x\n⊢ 0 < 1 ∧ ∀ ⦃x_1 : 𝕜⦄, x_1 ∈ {y | ‖y - 1‖ < 1} → ↑(homothety x x_1) y ∈ s",
"state_before": "case inl\nα : Type ?u.85021\nV : Type ?u.85024\nP : Type ?u.85027\nW : Type u_3\nQ : Type u_1\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\nh : y = x\n⊢ ∃ i, 0 < i ∧ ∀ ⦃x_1 : 𝕜⦄, x_1 ∈ {y | ‖y - 1‖ < i} → ↑(homothety x x_1) y ∈ s",
"tactic": "use 1"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type ?u.85021\nV : Type ?u.85024\nP : Type ?u.85027\nW : Type u_3\nQ : Type u_1\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\nh : y = x\n⊢ 0 < 1 ∧ ∀ ⦃x_1 : 𝕜⦄, x_1 ∈ {y | ‖y - 1‖ < 1} → ↑(homothety x x_1) y ∈ s",
"tactic": "simp [h.symm, interior_subset hy]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.85021\nV : Type ?u.85024\nP : Type ?u.85027\nW : Type u_3\nQ : Type u_1\ninst✝⁸ : SeminormedAddCommGroup V\ninst✝⁷ : PseudoMetricSpace P\ninst✝⁶ : NormedAddTorsor V P\ninst✝⁵ : NormedAddCommGroup W\ninst✝⁴ : MetricSpace Q\ninst✝³ : NormedAddTorsor W Q\n𝕜 : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 V\ninst✝ : NormedSpace 𝕜 W\nx : Q\ns : Set Q\ny : Q\nhy : y ∈ interior s\nh : y ≠ x\n⊢ 0 < ‖y -ᵥ x‖",
"tactic": "rwa [norm_pos_iff, vsub_ne_zero]"
}
]
| [
250,
85
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
239,
1
]
|
Mathlib/MeasureTheory/Measure/MutuallySingular.lean | MeasureTheory.Measure.MutuallySingular.mk | [
{
"state_after": "α : Type u_1\nm0 : MeasurableSpace α\nμ μ₁ μ₂ ν ν₁ ν₂ : Measure α\ns t : Set α\nhs : ↑↑μ s = 0\nht : ↑↑ν t = 0\nhst : univ ⊆ s ∪ t\n⊢ ↑↑ν (toMeasurable μ sᶜ) = 0",
"state_before": "α : Type u_1\nm0 : MeasurableSpace α\nμ μ₁ μ₂ ν ν₁ ν₂ : Measure α\ns t : Set α\nhs : ↑↑μ s = 0\nht : ↑↑ν t = 0\nhst : univ ⊆ s ∪ t\n⊢ μ ⟂ₘ ν",
"tactic": "use toMeasurable μ s, measurableSet_toMeasurable _ _, (measure_toMeasurable _).trans hs"
},
{
"state_after": "α : Type u_1\nm0 : MeasurableSpace α\nμ μ₁ μ₂ ν ν₁ ν₂ : Measure α\ns t : Set α\nhs : ↑↑μ s = 0\nht : ↑↑ν t = 0\nhst : univ ⊆ s ∪ t\nx : α\nhx : x ∈ toMeasurable μ sᶜ\nhxs : x ∈ s\n⊢ x ∈ toMeasurable μ s",
"state_before": "α : Type u_1\nm0 : MeasurableSpace α\nμ μ₁ μ₂ ν ν₁ ν₂ : Measure α\ns t : Set α\nhs : ↑↑μ s = 0\nht : ↑↑ν t = 0\nhst : univ ⊆ s ∪ t\n⊢ ↑↑ν (toMeasurable μ sᶜ) = 0",
"tactic": "refine' measure_mono_null (fun x hx => (hst trivial).resolve_left fun hxs => hx _) ht"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nm0 : MeasurableSpace α\nμ μ₁ μ₂ ν ν₁ ν₂ : Measure α\ns t : Set α\nhs : ↑↑μ s = 0\nht : ↑↑ν t = 0\nhst : univ ⊆ s ∪ t\nx : α\nhx : x ∈ toMeasurable μ sᶜ\nhxs : x ∈ s\n⊢ x ∈ toMeasurable μ s",
"tactic": "exact subset_toMeasurable _ _ hxs"
}
]
| [
56,
36
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
52,
1
]
|
Mathlib/Data/Set/Intervals/Basic.lean | Set.Ico_diff_left | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.37269\ninst✝ : PartialOrder α\na b c x : α\n⊢ x ∈ Ico a b \\ {a} ↔ x ∈ Ioo a b",
"tactic": "simp [and_right_comm, ← lt_iff_le_and_ne, eq_comm]"
}
]
| [
786,
69
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
785,
1
]
|
Mathlib/LinearAlgebra/AffineSpace/Combination.lean | Finset.weightedVSubVSubWeights_apply_left | [
{
"state_after": "no goals",
"state_before": "k : Type u_2\nV : Type ?u.420383\nP : Type ?u.420386\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nS : AffineSpace V P\nι : Type u_1\ns : Finset ι\nι₂ : Type ?u.421042\ns₂ : Finset ι₂\ninst✝ : DecidableEq ι\ni j : ι\nh : i ≠ j\n⊢ weightedVSubVSubWeights k i j i = 1",
"tactic": "simp [weightedVSubVSubWeights, h]"
}
]
| [
686,
80
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
685,
1
]
|
Mathlib/RingTheory/HahnSeries.lean | HahnSeries.embDomain_smul | [
{
"state_after": "case coeff.h\nΓ : Type u_1\nR : Type u_3\ninst✝⁴ : PartialOrder Γ\ninst✝³ : Semiring R\nV : Type ?u.831652\ninst✝² : AddCommMonoid V\ninst✝¹ : Module R V\nΓ' : Type u_2\ninst✝ : PartialOrder Γ'\nf : Γ ↪o Γ'\nr : R\nx : HahnSeries Γ R\ng : Γ'\n⊢ coeff (embDomain f (r • x)) g = coeff (r • embDomain f x) g",
"state_before": "Γ : Type u_1\nR : Type u_3\ninst✝⁴ : PartialOrder Γ\ninst✝³ : Semiring R\nV : Type ?u.831652\ninst✝² : AddCommMonoid V\ninst✝¹ : Module R V\nΓ' : Type u_2\ninst✝ : PartialOrder Γ'\nf : Γ ↪o Γ'\nr : R\nx : HahnSeries Γ R\n⊢ embDomain f (r • x) = r • embDomain f x",
"tactic": "ext g"
},
{
"state_after": "case pos\nΓ : Type u_1\nR : Type u_3\ninst✝⁴ : PartialOrder Γ\ninst✝³ : Semiring R\nV : Type ?u.831652\ninst✝² : AddCommMonoid V\ninst✝¹ : Module R V\nΓ' : Type u_2\ninst✝ : PartialOrder Γ'\nf : Γ ↪o Γ'\nr : R\nx : HahnSeries Γ R\ng : Γ'\nhg : g ∈ Set.range ↑f\n⊢ coeff (embDomain f (r • x)) g = coeff (r • embDomain f x) g\n\ncase neg\nΓ : Type u_1\nR : Type u_3\ninst✝⁴ : PartialOrder Γ\ninst✝³ : Semiring R\nV : Type ?u.831652\ninst✝² : AddCommMonoid V\ninst✝¹ : Module R V\nΓ' : Type u_2\ninst✝ : PartialOrder Γ'\nf : Γ ↪o Γ'\nr : R\nx : HahnSeries Γ R\ng : Γ'\nhg : ¬g ∈ Set.range ↑f\n⊢ coeff (embDomain f (r • x)) g = coeff (r • embDomain f x) g",
"state_before": "case coeff.h\nΓ : Type u_1\nR : Type u_3\ninst✝⁴ : PartialOrder Γ\ninst✝³ : Semiring R\nV : Type ?u.831652\ninst✝² : AddCommMonoid V\ninst✝¹ : Module R V\nΓ' : Type u_2\ninst✝ : PartialOrder Γ'\nf : Γ ↪o Γ'\nr : R\nx : HahnSeries Γ R\ng : Γ'\n⊢ coeff (embDomain f (r • x)) g = coeff (r • embDomain f x) g",
"tactic": "by_cases hg : g ∈ Set.range f"
},
{
"state_after": "case pos.intro\nΓ : Type u_1\nR : Type u_3\ninst✝⁴ : PartialOrder Γ\ninst✝³ : Semiring R\nV : Type ?u.831652\ninst✝² : AddCommMonoid V\ninst✝¹ : Module R V\nΓ' : Type u_2\ninst✝ : PartialOrder Γ'\nf : Γ ↪o Γ'\nr : R\nx : HahnSeries Γ R\na : Γ\n⊢ coeff (embDomain f (r • x)) (↑f a) = coeff (r • embDomain f x) (↑f a)",
"state_before": "case pos\nΓ : Type u_1\nR : Type u_3\ninst✝⁴ : PartialOrder Γ\ninst✝³ : Semiring R\nV : Type ?u.831652\ninst✝² : AddCommMonoid V\ninst✝¹ : Module R V\nΓ' : Type u_2\ninst✝ : PartialOrder Γ'\nf : Γ ↪o Γ'\nr : R\nx : HahnSeries Γ R\ng : Γ'\nhg : g ∈ Set.range ↑f\n⊢ coeff (embDomain f (r • x)) g = coeff (r • embDomain f x) g",
"tactic": "obtain ⟨a, rfl⟩ := hg"
},
{
"state_after": "no goals",
"state_before": "case pos.intro\nΓ : Type u_1\nR : Type u_3\ninst✝⁴ : PartialOrder Γ\ninst✝³ : Semiring R\nV : Type ?u.831652\ninst✝² : AddCommMonoid V\ninst✝¹ : Module R V\nΓ' : Type u_2\ninst✝ : PartialOrder Γ'\nf : Γ ↪o Γ'\nr : R\nx : HahnSeries Γ R\na : Γ\n⊢ coeff (embDomain f (r • x)) (↑f a) = coeff (r • embDomain f x) (↑f a)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case neg\nΓ : Type u_1\nR : Type u_3\ninst✝⁴ : PartialOrder Γ\ninst✝³ : Semiring R\nV : Type ?u.831652\ninst✝² : AddCommMonoid V\ninst✝¹ : Module R V\nΓ' : Type u_2\ninst✝ : PartialOrder Γ'\nf : Γ ↪o Γ'\nr : R\nx : HahnSeries Γ R\ng : Γ'\nhg : ¬g ∈ Set.range ↑f\n⊢ coeff (embDomain f (r • x)) g = coeff (r • embDomain f x) g",
"tactic": "simp [embDomain_notin_range hg]"
}
]
| [
573,
36
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
567,
1
]
|
Mathlib/Data/Nat/Factorization/Basic.lean | Nat.Ioc_filter_dvd_card_eq_div | [
{
"state_after": "case zero\np : ℕ\n⊢ card (Finset.filter (fun x => p ∣ x) (Ioc 0 zero)) = zero / p\n\ncase succ\np n : ℕ\nIH : card (Finset.filter (fun x => p ∣ x) (Ioc 0 n)) = n / p\n⊢ card (Finset.filter (fun x => p ∣ x) (Ioc 0 (succ n))) = succ n / p",
"state_before": "n p : ℕ\n⊢ card (Finset.filter (fun x => p ∣ x) (Ioc 0 n)) = n / p",
"tactic": "induction' n with n IH"
},
{
"state_after": "no goals",
"state_before": "case succ\np n : ℕ\nIH : card (Finset.filter (fun x => p ∣ x) (Ioc 0 n)) = n / p\nh1 : Ioc 0 (succ n) = insert (succ n) (Ioc 0 n)\n⊢ card (Finset.filter (fun x => p ∣ x) (Ioc 0 (succ n))) = succ n / p",
"tactic": "simp [Nat.succ_div, add_ite, add_zero, h1, filter_insert, apply_ite card, card_insert_eq_ite, IH,\n Finset.mem_filter, mem_Ioc, not_le.2 (lt_add_one n), Nat.succ_eq_add_one]"
},
{
"state_after": "no goals",
"state_before": "case zero\np : ℕ\n⊢ card (Finset.filter (fun x => p ∣ x) (Ioc 0 zero)) = zero / p",
"tactic": "simp"
},
{
"state_after": "case inl\np : ℕ\nIH : card (Finset.filter (fun x => p ∣ x) (Ioc 0 0)) = 0 / p\n⊢ Ioc 0 (succ 0) = insert (succ 0) (Ioc 0 0)\n\ncase inr\np n : ℕ\nIH : card (Finset.filter (fun x => p ∣ x) (Ioc 0 n)) = n / p\nhn : n > 0\n⊢ Ioc 0 (succ n) = insert (succ n) (Ioc 0 n)",
"state_before": "p n : ℕ\nIH : card (Finset.filter (fun x => p ∣ x) (Ioc 0 n)) = n / p\n⊢ Ioc 0 (succ n) = insert (succ n) (Ioc 0 n)",
"tactic": "rcases n.eq_zero_or_pos with (rfl | hn)"
},
{
"state_after": "no goals",
"state_before": "case inr\np n : ℕ\nIH : card (Finset.filter (fun x => p ∣ x) (Ioc 0 n)) = n / p\nhn : n > 0\n⊢ Ioc 0 (succ n) = insert (succ n) (Ioc 0 n)",
"tactic": "simp_rw [← Ico_succ_succ, Ico_insert_right (succ_le_succ hn.le), Ico_succ_right]"
},
{
"state_after": "no goals",
"state_before": "case inl\np : ℕ\nIH : card (Finset.filter (fun x => p ∣ x) (Ioc 0 0)) = 0 / p\n⊢ Ioc 0 (succ 0) = insert (succ 0) (Ioc 0 0)",
"tactic": "simp"
}
]
| [
964,
78
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
955,
1
]
|
Mathlib/Control/Functor/Multivariate.lean | MvFunctor.exists_iff_exists_of_mono | [
{
"state_after": "case mp.intro\nn : ℕ\nα β γ : TypeVec n\nF : TypeVec n → Type v\ninst✝¹ : MvFunctor F\nP✝ : α ⟹ TypeVec.repeat n Prop\nR : α ⊗ α ⟹ TypeVec.repeat n Prop\ninst✝ : LawfulMvFunctor F\nP : F α → Prop\nq : F β → Prop\nf : α ⟹ β\ng : β ⟹ α\nh₀ : f ⊚ g = TypeVec.id\nh₁ : ∀ (u : F α), P u ↔ q (f <$$> u)\nu : F α\nh₂ : P u\n⊢ ∃ u, q u\n\ncase mpr.intro\nn : ℕ\nα β γ : TypeVec n\nF : TypeVec n → Type v\ninst✝¹ : MvFunctor F\nP✝ : α ⟹ TypeVec.repeat n Prop\nR : α ⊗ α ⟹ TypeVec.repeat n Prop\ninst✝ : LawfulMvFunctor F\nP : F α → Prop\nq : F β → Prop\nf : α ⟹ β\ng : β ⟹ α\nh₀ : f ⊚ g = TypeVec.id\nh₁ : ∀ (u : F α), P u ↔ q (f <$$> u)\nu : F β\nh₂ : q u\n⊢ ∃ u, P u",
"state_before": "n : ℕ\nα β γ : TypeVec n\nF : TypeVec n → Type v\ninst✝¹ : MvFunctor F\nP✝ : α ⟹ TypeVec.repeat n Prop\nR : α ⊗ α ⟹ TypeVec.repeat n Prop\ninst✝ : LawfulMvFunctor F\nP : F α → Prop\nq : F β → Prop\nf : α ⟹ β\ng : β ⟹ α\nh₀ : f ⊚ g = TypeVec.id\nh₁ : ∀ (u : F α), P u ↔ q (f <$$> u)\n⊢ (∃ u, P u) ↔ ∃ u, q u",
"tactic": "constructor <;> rintro ⟨u, h₂⟩"
},
{
"state_after": "case mp.intro\nn : ℕ\nα β γ : TypeVec n\nF : TypeVec n → Type v\ninst✝¹ : MvFunctor F\nP✝ : α ⟹ TypeVec.repeat n Prop\nR : α ⊗ α ⟹ TypeVec.repeat n Prop\ninst✝ : LawfulMvFunctor F\nP : F α → Prop\nq : F β → Prop\nf : α ⟹ β\ng : β ⟹ α\nh₀ : f ⊚ g = TypeVec.id\nh₁ : ∀ (u : F α), P u ↔ q (f <$$> u)\nu : F α\nh₂ : P u\n⊢ q (f <$$> u)",
"state_before": "case mp.intro\nn : ℕ\nα β γ : TypeVec n\nF : TypeVec n → Type v\ninst✝¹ : MvFunctor F\nP✝ : α ⟹ TypeVec.repeat n Prop\nR : α ⊗ α ⟹ TypeVec.repeat n Prop\ninst✝ : LawfulMvFunctor F\nP : F α → Prop\nq : F β → Prop\nf : α ⟹ β\ng : β ⟹ α\nh₀ : f ⊚ g = TypeVec.id\nh₁ : ∀ (u : F α), P u ↔ q (f <$$> u)\nu : F α\nh₂ : P u\n⊢ ∃ u, q u",
"tactic": "refine ⟨f <$$> u, ?_⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.intro\nn : ℕ\nα β γ : TypeVec n\nF : TypeVec n → Type v\ninst✝¹ : MvFunctor F\nP✝ : α ⟹ TypeVec.repeat n Prop\nR : α ⊗ α ⟹ TypeVec.repeat n Prop\ninst✝ : LawfulMvFunctor F\nP : F α → Prop\nq : F β → Prop\nf : α ⟹ β\ng : β ⟹ α\nh₀ : f ⊚ g = TypeVec.id\nh₁ : ∀ (u : F α), P u ↔ q (f <$$> u)\nu : F α\nh₂ : P u\n⊢ q (f <$$> u)",
"tactic": "apply (h₁ u).mp h₂"
},
{
"state_after": "case mpr.intro\nn : ℕ\nα β γ : TypeVec n\nF : TypeVec n → Type v\ninst✝¹ : MvFunctor F\nP✝ : α ⟹ TypeVec.repeat n Prop\nR : α ⊗ α ⟹ TypeVec.repeat n Prop\ninst✝ : LawfulMvFunctor F\nP : F α → Prop\nq : F β → Prop\nf : α ⟹ β\ng : β ⟹ α\nh₀ : f ⊚ g = TypeVec.id\nh₁ : ∀ (u : F α), P u ↔ q (f <$$> u)\nu : F β\nh₂ : q u\n⊢ P (g <$$> u)",
"state_before": "case mpr.intro\nn : ℕ\nα β γ : TypeVec n\nF : TypeVec n → Type v\ninst✝¹ : MvFunctor F\nP✝ : α ⟹ TypeVec.repeat n Prop\nR : α ⊗ α ⟹ TypeVec.repeat n Prop\ninst✝ : LawfulMvFunctor F\nP : F α → Prop\nq : F β → Prop\nf : α ⟹ β\ng : β ⟹ α\nh₀ : f ⊚ g = TypeVec.id\nh₁ : ∀ (u : F α), P u ↔ q (f <$$> u)\nu : F β\nh₂ : q u\n⊢ ∃ u, P u",
"tactic": "refine ⟨g <$$> u, ?_⟩"
},
{
"state_after": "n : ℕ\nα β γ : TypeVec n\nF : TypeVec n → Type v\ninst✝¹ : MvFunctor F\nP✝ : α ⟹ TypeVec.repeat n Prop\nR : α ⊗ α ⟹ TypeVec.repeat n Prop\ninst✝ : LawfulMvFunctor F\nP : F α → Prop\nq : F β → Prop\nf : α ⟹ β\ng : β ⟹ α\nh₀ : f ⊚ g = TypeVec.id\nh₁ : ∀ (u : F α), P u ↔ q (f <$$> u)\nu : F β\nh₂ : q u\n⊢ q (f <$$> g <$$> u)",
"state_before": "case mpr.intro\nn : ℕ\nα β γ : TypeVec n\nF : TypeVec n → Type v\ninst✝¹ : MvFunctor F\nP✝ : α ⟹ TypeVec.repeat n Prop\nR : α ⊗ α ⟹ TypeVec.repeat n Prop\ninst✝ : LawfulMvFunctor F\nP : F α → Prop\nq : F β → Prop\nf : α ⟹ β\ng : β ⟹ α\nh₀ : f ⊚ g = TypeVec.id\nh₁ : ∀ (u : F α), P u ↔ q (f <$$> u)\nu : F β\nh₂ : q u\n⊢ P (g <$$> u)",
"tactic": "apply (h₁ _).mpr _"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nα β γ : TypeVec n\nF : TypeVec n → Type v\ninst✝¹ : MvFunctor F\nP✝ : α ⟹ TypeVec.repeat n Prop\nR : α ⊗ α ⟹ TypeVec.repeat n Prop\ninst✝ : LawfulMvFunctor F\nP : F α → Prop\nq : F β → Prop\nf : α ⟹ β\ng : β ⟹ α\nh₀ : f ⊚ g = TypeVec.id\nh₁ : ∀ (u : F α), P u ↔ q (f <$$> u)\nu : F β\nh₂ : q u\n⊢ q (f <$$> g <$$> u)",
"tactic": "simp only [MvFunctor.map_map, h₀, LawfulMvFunctor.id_map, h₂]"
}
]
| [
136,
66
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
126,
1
]
|
Mathlib/Topology/Instances/Matrix.lean | Summable.matrix_blockDiagonal' | []
| [
426,
43
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
424,
1
]
|
Mathlib/Topology/Homeomorph.lean | HasCompactMulSupport.comp_homeomorph | []
| [
370,
44
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
368,
1
]
|
Std/Data/PairingHeap.lean | Std.PairingHeapImp.Heap.size_deleteMin | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nle : α → α → Bool\na : α\ns' s : Heap α\nh : NoSibling s\neq : deleteMin le s = some (a, s')\n⊢ size s = size s' + 1",
"tactic": "cases h with cases eq | node a c => rw [size_combine, size, size]"
},
{
"state_after": "no goals",
"state_before": "case node.refl\nα : Type u_1\nle : α → α → Bool\na : α\nc : Heap α\n⊢ size (node a c nil) = size (combine le c) + 1",
"tactic": "rw [size_combine, size, size]"
}
]
| [
141,
68
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
139,
1
]
|
Mathlib/LinearAlgebra/Dimension.lean | rank_eq_of_injective | []
| [
206,
40
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
204,
1
]
|
Mathlib/Algebra/Order/Field/Basic.lean | inv_lt_of_inv_lt | []
| [
292,
43
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
291,
1
]
|
Mathlib/Data/Real/CauSeqCompletion.lean | CauSeq.Completion.ofRat_intCast | []
| [
144,
6
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
143,
1
]
|
Mathlib/Data/Set/Ncard.lean | Set.ncard_le_ncard_iff_ncard_diff_le_ncard_diff | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.129077\ns t : Set α\na b x y : α\nf : α → β\nhs : autoParam (Set.Finite s) _auto✝\nht : autoParam (Set.Finite t) _auto✝\n⊢ ncard s ≤ ncard t ↔ ncard (s \\ t) ≤ ncard (t \\ s)",
"tactic": "rw [← ncard_inter_add_ncard_diff_eq_ncard s t hs, ← ncard_inter_add_ncard_diff_eq_ncard t s ht,\n inter_comm, add_le_add_iff_left]"
}
]
| [
557,
37
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
554,
1
]
|
Mathlib/Data/Finsupp/Basic.lean | Finsupp.filter_eq_self_iff | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.425326\nγ : Type ?u.425329\nι : Type ?u.425332\nM : Type u_2\nM' : Type ?u.425338\nN : Type ?u.425341\nP : Type ?u.425344\nG : Type ?u.425347\nH : Type ?u.425350\nR : Type ?u.425353\nS : Type ?u.425356\ninst✝ : Zero M\np : α → Prop\nf : α →₀ M\n⊢ filter p f = f ↔ ∀ (x : α), ↑f x ≠ 0 → p x",
"tactic": "simp only [FunLike.ext_iff, filter_eq_indicator, Set.indicator_apply_eq_self, Set.mem_setOf_eq,\n not_imp_comm]"
}
]
| [
910,
18
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
908,
1
]
|
Mathlib/Algebra/Module/Submodule/Basic.lean | Submodule.coe_sub | []
| [
579,
31
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
578,
11
]
|
Std/Data/List/Basic.lean | List.takeD_zero | []
| [
588,
65
]
| e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
588,
9
]
|
Mathlib/Analysis/Complex/RealDeriv.lean | HasStrictDerivAt.real_of_complex | [
{
"state_after": "e : ℂ → ℂ\ne' : ℂ\nz : ℝ\nh : HasStrictDerivAt e e' ↑z\nA : HasStrictFDerivAt ofReal' ofRealClm z\n⊢ HasStrictDerivAt (fun x => (e ↑x).re) e'.re z",
"state_before": "e : ℂ → ℂ\ne' : ℂ\nz : ℝ\nh : HasStrictDerivAt e e' ↑z\n⊢ HasStrictDerivAt (fun x => (e ↑x).re) e'.re z",
"tactic": "have A : HasStrictFDerivAt ((↑) : ℝ → ℂ) ofRealClm z := ofRealClm.hasStrictFDerivAt"
},
{
"state_after": "e : ℂ → ℂ\ne' : ℂ\nz : ℝ\nh : HasStrictDerivAt e e' ↑z\nA : HasStrictFDerivAt ofReal' ofRealClm z\nB : HasStrictFDerivAt e (ContinuousLinearMap.restrictScalars ℝ (ContinuousLinearMap.smulRight 1 e')) (↑ofRealClm z)\n⊢ HasStrictDerivAt (fun x => (e ↑x).re) e'.re z",
"state_before": "e : ℂ → ℂ\ne' : ℂ\nz : ℝ\nh : HasStrictDerivAt e e' ↑z\nA : HasStrictFDerivAt ofReal' ofRealClm z\n⊢ HasStrictDerivAt (fun x => (e ↑x).re) e'.re z",
"tactic": "have B :\n HasStrictFDerivAt e ((ContinuousLinearMap.smulRight 1 e' : ℂ →L[ℂ] ℂ).restrictScalars ℝ)\n (ofRealClm z) :=\n h.hasStrictFDerivAt.restrictScalars ℝ"
},
{
"state_after": "e : ℂ → ℂ\ne' : ℂ\nz : ℝ\nh : HasStrictDerivAt e e' ↑z\nA : HasStrictFDerivAt ofReal' ofRealClm z\nB : HasStrictFDerivAt e (ContinuousLinearMap.restrictScalars ℝ (ContinuousLinearMap.smulRight 1 e')) (↑ofRealClm z)\nC : HasStrictFDerivAt re reClm (e (↑ofRealClm z))\n⊢ HasStrictDerivAt (fun x => (e ↑x).re) e'.re z",
"state_before": "e : ℂ → ℂ\ne' : ℂ\nz : ℝ\nh : HasStrictDerivAt e e' ↑z\nA : HasStrictFDerivAt ofReal' ofRealClm z\nB : HasStrictFDerivAt e (ContinuousLinearMap.restrictScalars ℝ (ContinuousLinearMap.smulRight 1 e')) (↑ofRealClm z)\n⊢ HasStrictDerivAt (fun x => (e ↑x).re) e'.re z",
"tactic": "have C : HasStrictFDerivAt re reClm (e (ofRealClm z)) := reClm.hasStrictFDerivAt"
},
{
"state_after": "case h.e'_7\ne : ℂ → ℂ\ne' : ℂ\nz : ℝ\nh : HasStrictDerivAt e e' ↑z\nA : HasStrictFDerivAt ofReal' ofRealClm z\nB : HasStrictFDerivAt e (ContinuousLinearMap.restrictScalars ℝ (ContinuousLinearMap.smulRight 1 e')) (↑ofRealClm z)\nC : HasStrictFDerivAt re reClm (e (↑ofRealClm z))\n⊢ e'.re =\n ↑(ContinuousLinearMap.comp reClm\n (ContinuousLinearMap.comp (ContinuousLinearMap.restrictScalars ℝ (ContinuousLinearMap.smulRight 1 e'))\n ofRealClm))\n 1",
"state_before": "e : ℂ → ℂ\ne' : ℂ\nz : ℝ\nh : HasStrictDerivAt e e' ↑z\nA : HasStrictFDerivAt ofReal' ofRealClm z\nB : HasStrictFDerivAt e (ContinuousLinearMap.restrictScalars ℝ (ContinuousLinearMap.smulRight 1 e')) (↑ofRealClm z)\nC : HasStrictFDerivAt re reClm (e (↑ofRealClm z))\n⊢ HasStrictDerivAt (fun x => (e ↑x).re) e'.re z",
"tactic": "convert (C.comp z (B.comp z A)).hasStrictDerivAt"
},
{
"state_after": "case h.e'_7\ne : ℂ → ℂ\ne' : ℂ\nz : ℝ\nh : HasStrictDerivAt e e' ↑z\nA : HasStrictFDerivAt ofReal' ofRealClm z\nB : HasStrictFDerivAt e (ContinuousLinearMap.restrictScalars ℝ (ContinuousLinearMap.smulRight 1 e')) (↑ofRealClm z)\nC : HasStrictFDerivAt re reClm (e (↑ofRealClm z))\n⊢ e'.re = ↑reClm (↑(ContinuousLinearMap.restrictScalars ℝ (ContinuousLinearMap.smulRight 1 e')) (↑ofRealClm 1))",
"state_before": "case h.e'_7\ne : ℂ → ℂ\ne' : ℂ\nz : ℝ\nh : HasStrictDerivAt e e' ↑z\nA : HasStrictFDerivAt ofReal' ofRealClm z\nB : HasStrictFDerivAt e (ContinuousLinearMap.restrictScalars ℝ (ContinuousLinearMap.smulRight 1 e')) (↑ofRealClm z)\nC : HasStrictFDerivAt re reClm (e (↑ofRealClm z))\n⊢ e'.re =\n ↑(ContinuousLinearMap.comp reClm\n (ContinuousLinearMap.comp (ContinuousLinearMap.restrictScalars ℝ (ContinuousLinearMap.smulRight 1 e'))\n ofRealClm))\n 1",
"tactic": "rw [ContinuousLinearMap.comp_apply, ContinuousLinearMap.comp_apply]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_7\ne : ℂ → ℂ\ne' : ℂ\nz : ℝ\nh : HasStrictDerivAt e e' ↑z\nA : HasStrictFDerivAt ofReal' ofRealClm z\nB : HasStrictFDerivAt e (ContinuousLinearMap.restrictScalars ℝ (ContinuousLinearMap.smulRight 1 e')) (↑ofRealClm z)\nC : HasStrictFDerivAt re reClm (e (↑ofRealClm z))\n⊢ e'.re = ↑reClm (↑(ContinuousLinearMap.restrictScalars ℝ (ContinuousLinearMap.smulRight 1 e')) (↑ofRealClm 1))",
"tactic": "simp"
}
]
| [
65,
7
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
52,
1
]
|
Mathlib/RingTheory/IsTensorProduct.lean | Algebra.pushoutDesc_left | [
{
"state_after": "R : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝³³ : AddCommMonoid M\ninst✝³² : AddCommMonoid N\ninst✝³¹ : CommRing R\ninst✝³⁰ : CommRing S\ninst✝²⁹ : Algebra R S\ninst✝²⁸ : Module R M\ninst✝²⁷ : Module R N\ninst✝²⁶ : Module S N\ninst✝²⁵ : IsScalarTower R S N\nf✝ : M →ₗ[R] N\nh : IsBaseChange S f✝\nP : Type ?u.1281852\nQ : Type ?u.1281855\ninst✝²⁴ : AddCommMonoid P\ninst✝²³ : Module R P\ninst✝²² : AddCommMonoid Q\ninst✝²¹ : Module S Q\nT : Type ?u.1282170\nO : Type ?u.1282173\ninst✝²⁰ : CommRing T\ninst✝¹⁹ : Algebra R T\ninst✝¹⁸ : Algebra S T\ninst✝¹⁷ : IsScalarTower R S T\ninst✝¹⁶ : AddCommMonoid O\ninst✝¹⁵ : Module R O\ninst✝¹⁴ : Module S O\ninst✝¹³ : Module T O\ninst✝¹² : IsScalarTower S T O\ninst✝¹¹ : IsScalarTower R S O\ninst✝¹⁰ : IsScalarTower R T O\nR' : Type u_2\nS' : Type u_3\ninst✝⁹ : CommRing R'\ninst✝⁸ : CommRing S'\ninst✝⁷ : Algebra R R'\ninst✝⁶ : Algebra S S'\ninst✝⁵ : Algebra R' S'\ninst✝⁴ : Algebra R S'\ninst✝³ : IsScalarTower R R' S'\ninst✝² : IsScalarTower R S S'\nH✝ : IsPushout R S R' S'\nA : Type u_4\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nf : S →ₐ[R] A\ng : R' →ₐ[R] A\nH : ∀ (x : S) (y : R'), ↑f x * ↑g y = ↑g y * ↑f x\nx : S\nthis : Module S A := Module.compHom A ↑f\n⊢ ↑(pushoutDesc S' f g H) (↑(algebraMap S S') x) = ↑f x",
"state_before": "R : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝³³ : AddCommMonoid M\ninst✝³² : AddCommMonoid N\ninst✝³¹ : CommRing R\ninst✝³⁰ : CommRing S\ninst✝²⁹ : Algebra R S\ninst✝²⁸ : Module R M\ninst✝²⁷ : Module R N\ninst✝²⁶ : Module S N\ninst✝²⁵ : IsScalarTower R S N\nf✝ : M →ₗ[R] N\nh : IsBaseChange S f✝\nP : Type ?u.1281852\nQ : Type ?u.1281855\ninst✝²⁴ : AddCommMonoid P\ninst✝²³ : Module R P\ninst✝²² : AddCommMonoid Q\ninst✝²¹ : Module S Q\nT : Type ?u.1282170\nO : Type ?u.1282173\ninst✝²⁰ : CommRing T\ninst✝¹⁹ : Algebra R T\ninst✝¹⁸ : Algebra S T\ninst✝¹⁷ : IsScalarTower R S T\ninst✝¹⁶ : AddCommMonoid O\ninst✝¹⁵ : Module R O\ninst✝¹⁴ : Module S O\ninst✝¹³ : Module T O\ninst✝¹² : IsScalarTower S T O\ninst✝¹¹ : IsScalarTower R S O\ninst✝¹⁰ : IsScalarTower R T O\nR' : Type u_2\nS' : Type u_3\ninst✝⁹ : CommRing R'\ninst✝⁸ : CommRing S'\ninst✝⁷ : Algebra R R'\ninst✝⁶ : Algebra S S'\ninst✝⁵ : Algebra R' S'\ninst✝⁴ : Algebra R S'\ninst✝³ : IsScalarTower R R' S'\ninst✝² : IsScalarTower R S S'\nH✝ : IsPushout R S R' S'\nA : Type u_4\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nf : S →ₐ[R] A\ng : R' →ₐ[R] A\nH : ∀ (x : S) (y : R'), ↑f x * ↑g y = ↑g y * ↑f x\nx : S\n⊢ ↑(pushoutDesc S' f g H) (↑(algebraMap S S') x) = ↑f x",
"tactic": "letI := Module.compHom A f.toRingHom"
},
{
"state_after": "R : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝³³ : AddCommMonoid M\ninst✝³² : AddCommMonoid N\ninst✝³¹ : CommRing R\ninst✝³⁰ : CommRing S\ninst✝²⁹ : Algebra R S\ninst✝²⁸ : Module R M\ninst✝²⁷ : Module R N\ninst✝²⁶ : Module S N\ninst✝²⁵ : IsScalarTower R S N\nf✝ : M →ₗ[R] N\nh : IsBaseChange S f✝\nP : Type ?u.1281852\nQ : Type ?u.1281855\ninst✝²⁴ : AddCommMonoid P\ninst✝²³ : Module R P\ninst✝²² : AddCommMonoid Q\ninst✝²¹ : Module S Q\nT : Type ?u.1282170\nO : Type ?u.1282173\ninst✝²⁰ : CommRing T\ninst✝¹⁹ : Algebra R T\ninst✝¹⁸ : Algebra S T\ninst✝¹⁷ : IsScalarTower R S T\ninst✝¹⁶ : AddCommMonoid O\ninst✝¹⁵ : Module R O\ninst✝¹⁴ : Module S O\ninst✝¹³ : Module T O\ninst✝¹² : IsScalarTower S T O\ninst✝¹¹ : IsScalarTower R S O\ninst✝¹⁰ : IsScalarTower R T O\nR' : Type u_2\nS' : Type u_3\ninst✝⁹ : CommRing R'\ninst✝⁸ : CommRing S'\ninst✝⁷ : Algebra R R'\ninst✝⁶ : Algebra S S'\ninst✝⁵ : Algebra R' S'\ninst✝⁴ : Algebra R S'\ninst✝³ : IsScalarTower R R' S'\ninst✝² : IsScalarTower R S S'\nH✝ : IsPushout R S R' S'\nA : Type u_4\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nf : S →ₐ[R] A\ng : R' →ₐ[R] A\nH : ∀ (x : S) (y : R'), ↑f x * ↑g y = ↑g y * ↑f x\nx : S\nthis✝ : Module S A := Module.compHom A ↑f\nthis : IsScalarTower R S A\n⊢ ↑(pushoutDesc S' f g H) (↑(algebraMap S S') x) = ↑f x",
"state_before": "R : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝³³ : AddCommMonoid M\ninst✝³² : AddCommMonoid N\ninst✝³¹ : CommRing R\ninst✝³⁰ : CommRing S\ninst✝²⁹ : Algebra R S\ninst✝²⁸ : Module R M\ninst✝²⁷ : Module R N\ninst✝²⁶ : Module S N\ninst✝²⁵ : IsScalarTower R S N\nf✝ : M →ₗ[R] N\nh : IsBaseChange S f✝\nP : Type ?u.1281852\nQ : Type ?u.1281855\ninst✝²⁴ : AddCommMonoid P\ninst✝²³ : Module R P\ninst✝²² : AddCommMonoid Q\ninst✝²¹ : Module S Q\nT : Type ?u.1282170\nO : Type ?u.1282173\ninst✝²⁰ : CommRing T\ninst✝¹⁹ : Algebra R T\ninst✝¹⁸ : Algebra S T\ninst✝¹⁷ : IsScalarTower R S T\ninst✝¹⁶ : AddCommMonoid O\ninst✝¹⁵ : Module R O\ninst✝¹⁴ : Module S O\ninst✝¹³ : Module T O\ninst✝¹² : IsScalarTower S T O\ninst✝¹¹ : IsScalarTower R S O\ninst✝¹⁰ : IsScalarTower R T O\nR' : Type u_2\nS' : Type u_3\ninst✝⁹ : CommRing R'\ninst✝⁸ : CommRing S'\ninst✝⁷ : Algebra R R'\ninst✝⁶ : Algebra S S'\ninst✝⁵ : Algebra R' S'\ninst✝⁴ : Algebra R S'\ninst✝³ : IsScalarTower R R' S'\ninst✝² : IsScalarTower R S S'\nH✝ : IsPushout R S R' S'\nA : Type u_4\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nf : S →ₐ[R] A\ng : R' →ₐ[R] A\nH : ∀ (x : S) (y : R'), ↑f x * ↑g y = ↑g y * ↑f x\nx : S\nthis : Module S A := Module.compHom A ↑f\n⊢ ↑(pushoutDesc S' f g H) (↑(algebraMap S S') x) = ↑f x",
"tactic": "haveI : IsScalarTower R S A :=\n { smul_assoc := fun r s a =>\n show f (r • s) * a = r • (f s * a) by rw [f.map_smul, smul_mul_assoc] }"
},
{
"state_after": "R : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝³³ : AddCommMonoid M\ninst✝³² : AddCommMonoid N\ninst✝³¹ : CommRing R\ninst✝³⁰ : CommRing S\ninst✝²⁹ : Algebra R S\ninst✝²⁸ : Module R M\ninst✝²⁷ : Module R N\ninst✝²⁶ : Module S N\ninst✝²⁵ : IsScalarTower R S N\nf✝ : M →ₗ[R] N\nh : IsBaseChange S f✝\nP : Type ?u.1281852\nQ : Type ?u.1281855\ninst✝²⁴ : AddCommMonoid P\ninst✝²³ : Module R P\ninst✝²² : AddCommMonoid Q\ninst✝²¹ : Module S Q\nT : Type ?u.1282170\nO : Type ?u.1282173\ninst✝²⁰ : CommRing T\ninst✝¹⁹ : Algebra R T\ninst✝¹⁸ : Algebra S T\ninst✝¹⁷ : IsScalarTower R S T\ninst✝¹⁶ : AddCommMonoid O\ninst✝¹⁵ : Module R O\ninst✝¹⁴ : Module S O\ninst✝¹³ : Module T O\ninst✝¹² : IsScalarTower S T O\ninst✝¹¹ : IsScalarTower R S O\ninst✝¹⁰ : IsScalarTower R T O\nR' : Type u_2\nS' : Type u_3\ninst✝⁹ : CommRing R'\ninst✝⁸ : CommRing S'\ninst✝⁷ : Algebra R R'\ninst✝⁶ : Algebra S S'\ninst✝⁵ : Algebra R' S'\ninst✝⁴ : Algebra R S'\ninst✝³ : IsScalarTower R R' S'\ninst✝² : IsScalarTower R S S'\nH✝ : IsPushout R S R' S'\nA : Type u_4\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nf : S →ₐ[R] A\ng : R' →ₐ[R] A\nH : ∀ (x : S) (y : R'), ↑f x * ↑g y = ↑g y * ↑f x\nx : S\nthis✝¹ : Module S A := Module.compHom A ↑f\nthis✝ : IsScalarTower R S A\nthis : IsScalarTower S A A\n⊢ ↑(pushoutDesc S' f g H) (↑(algebraMap S S') x) = ↑f x",
"state_before": "R : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝³³ : AddCommMonoid M\ninst✝³² : AddCommMonoid N\ninst✝³¹ : CommRing R\ninst✝³⁰ : CommRing S\ninst✝²⁹ : Algebra R S\ninst✝²⁸ : Module R M\ninst✝²⁷ : Module R N\ninst✝²⁶ : Module S N\ninst✝²⁵ : IsScalarTower R S N\nf✝ : M →ₗ[R] N\nh : IsBaseChange S f✝\nP : Type ?u.1281852\nQ : Type ?u.1281855\ninst✝²⁴ : AddCommMonoid P\ninst✝²³ : Module R P\ninst✝²² : AddCommMonoid Q\ninst✝²¹ : Module S Q\nT : Type ?u.1282170\nO : Type ?u.1282173\ninst✝²⁰ : CommRing T\ninst✝¹⁹ : Algebra R T\ninst✝¹⁸ : Algebra S T\ninst✝¹⁷ : IsScalarTower R S T\ninst✝¹⁶ : AddCommMonoid O\ninst✝¹⁵ : Module R O\ninst✝¹⁴ : Module S O\ninst✝¹³ : Module T O\ninst✝¹² : IsScalarTower S T O\ninst✝¹¹ : IsScalarTower R S O\ninst✝¹⁰ : IsScalarTower R T O\nR' : Type u_2\nS' : Type u_3\ninst✝⁹ : CommRing R'\ninst✝⁸ : CommRing S'\ninst✝⁷ : Algebra R R'\ninst✝⁶ : Algebra S S'\ninst✝⁵ : Algebra R' S'\ninst✝⁴ : Algebra R S'\ninst✝³ : IsScalarTower R R' S'\ninst✝² : IsScalarTower R S S'\nH✝ : IsPushout R S R' S'\nA : Type u_4\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nf : S →ₐ[R] A\ng : R' →ₐ[R] A\nH : ∀ (x : S) (y : R'), ↑f x * ↑g y = ↑g y * ↑f x\nx : S\nthis✝ : Module S A := Module.compHom A ↑f\nthis : IsScalarTower R S A\n⊢ ↑(pushoutDesc S' f g H) (↑(algebraMap S S') x) = ↑f x",
"tactic": "haveI : IsScalarTower S A A := { smul_assoc := fun r a b => mul_assoc _ _ _ }"
},
{
"state_after": "R : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝³³ : AddCommMonoid M\ninst✝³² : AddCommMonoid N\ninst✝³¹ : CommRing R\ninst✝³⁰ : CommRing S\ninst✝²⁹ : Algebra R S\ninst✝²⁸ : Module R M\ninst✝²⁷ : Module R N\ninst✝²⁶ : Module S N\ninst✝²⁵ : IsScalarTower R S N\nf✝ : M →ₗ[R] N\nh : IsBaseChange S f✝\nP : Type ?u.1281852\nQ : Type ?u.1281855\ninst✝²⁴ : AddCommMonoid P\ninst✝²³ : Module R P\ninst✝²² : AddCommMonoid Q\ninst✝²¹ : Module S Q\nT : Type ?u.1282170\nO : Type ?u.1282173\ninst✝²⁰ : CommRing T\ninst✝¹⁹ : Algebra R T\ninst✝¹⁸ : Algebra S T\ninst✝¹⁷ : IsScalarTower R S T\ninst✝¹⁶ : AddCommMonoid O\ninst✝¹⁵ : Module R O\ninst✝¹⁴ : Module S O\ninst✝¹³ : Module T O\ninst✝¹² : IsScalarTower S T O\ninst✝¹¹ : IsScalarTower R S O\ninst✝¹⁰ : IsScalarTower R T O\nR' : Type u_2\nS' : Type u_3\ninst✝⁹ : CommRing R'\ninst✝⁸ : CommRing S'\ninst✝⁷ : Algebra R R'\ninst✝⁶ : Algebra S S'\ninst✝⁵ : Algebra R' S'\ninst✝⁴ : Algebra R S'\ninst✝³ : IsScalarTower R R' S'\ninst✝² : IsScalarTower R S S'\nH✝ : IsPushout R S R' S'\nA : Type u_4\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nf : S →ₐ[R] A\ng : R' →ₐ[R] A\nH : ∀ (x : S) (y : R'), ↑f x * ↑g y = ↑g y * ↑f x\nx : S\nthis✝¹ : Module S A := Module.compHom A ↑f\nthis✝ : IsScalarTower R S A\nthis : IsScalarTower S A A\n⊢ x • 1 = ↑f x",
"state_before": "R : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝³³ : AddCommMonoid M\ninst✝³² : AddCommMonoid N\ninst✝³¹ : CommRing R\ninst✝³⁰ : CommRing S\ninst✝²⁹ : Algebra R S\ninst✝²⁸ : Module R M\ninst✝²⁷ : Module R N\ninst✝²⁶ : Module S N\ninst✝²⁵ : IsScalarTower R S N\nf✝ : M →ₗ[R] N\nh : IsBaseChange S f✝\nP : Type ?u.1281852\nQ : Type ?u.1281855\ninst✝²⁴ : AddCommMonoid P\ninst✝²³ : Module R P\ninst✝²² : AddCommMonoid Q\ninst✝²¹ : Module S Q\nT : Type ?u.1282170\nO : Type ?u.1282173\ninst✝²⁰ : CommRing T\ninst✝¹⁹ : Algebra R T\ninst✝¹⁸ : Algebra S T\ninst✝¹⁷ : IsScalarTower R S T\ninst✝¹⁶ : AddCommMonoid O\ninst✝¹⁵ : Module R O\ninst✝¹⁴ : Module S O\ninst✝¹³ : Module T O\ninst✝¹² : IsScalarTower S T O\ninst✝¹¹ : IsScalarTower R S O\ninst✝¹⁰ : IsScalarTower R T O\nR' : Type u_2\nS' : Type u_3\ninst✝⁹ : CommRing R'\ninst✝⁸ : CommRing S'\ninst✝⁷ : Algebra R R'\ninst✝⁶ : Algebra S S'\ninst✝⁵ : Algebra R' S'\ninst✝⁴ : Algebra R S'\ninst✝³ : IsScalarTower R R' S'\ninst✝² : IsScalarTower R S S'\nH✝ : IsPushout R S R' S'\nA : Type u_4\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nf : S →ₐ[R] A\ng : R' →ₐ[R] A\nH : ∀ (x : S) (y : R'), ↑f x * ↑g y = ↑g y * ↑f x\nx : S\nthis✝¹ : Module S A := Module.compHom A ↑f\nthis✝ : IsScalarTower R S A\nthis : IsScalarTower S A A\n⊢ ↑(pushoutDesc S' f g H) (↑(algebraMap S S') x) = ↑f x",
"tactic": "rw [Algebra.algebraMap_eq_smul_one, pushoutDesc_apply, map_smul, ←\n Algebra.pushoutDesc_apply S' f g H, _root_.map_one]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝³³ : AddCommMonoid M\ninst✝³² : AddCommMonoid N\ninst✝³¹ : CommRing R\ninst✝³⁰ : CommRing S\ninst✝²⁹ : Algebra R S\ninst✝²⁸ : Module R M\ninst✝²⁷ : Module R N\ninst✝²⁶ : Module S N\ninst✝²⁵ : IsScalarTower R S N\nf✝ : M →ₗ[R] N\nh : IsBaseChange S f✝\nP : Type ?u.1281852\nQ : Type ?u.1281855\ninst✝²⁴ : AddCommMonoid P\ninst✝²³ : Module R P\ninst✝²² : AddCommMonoid Q\ninst✝²¹ : Module S Q\nT : Type ?u.1282170\nO : Type ?u.1282173\ninst✝²⁰ : CommRing T\ninst✝¹⁹ : Algebra R T\ninst✝¹⁸ : Algebra S T\ninst✝¹⁷ : IsScalarTower R S T\ninst✝¹⁶ : AddCommMonoid O\ninst✝¹⁵ : Module R O\ninst✝¹⁴ : Module S O\ninst✝¹³ : Module T O\ninst✝¹² : IsScalarTower S T O\ninst✝¹¹ : IsScalarTower R S O\ninst✝¹⁰ : IsScalarTower R T O\nR' : Type u_2\nS' : Type u_3\ninst✝⁹ : CommRing R'\ninst✝⁸ : CommRing S'\ninst✝⁷ : Algebra R R'\ninst✝⁶ : Algebra S S'\ninst✝⁵ : Algebra R' S'\ninst✝⁴ : Algebra R S'\ninst✝³ : IsScalarTower R R' S'\ninst✝² : IsScalarTower R S S'\nH✝ : IsPushout R S R' S'\nA : Type u_4\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nf : S →ₐ[R] A\ng : R' →ₐ[R] A\nH : ∀ (x : S) (y : R'), ↑f x * ↑g y = ↑g y * ↑f x\nx : S\nthis✝¹ : Module S A := Module.compHom A ↑f\nthis✝ : IsScalarTower R S A\nthis : IsScalarTower S A A\n⊢ x • 1 = ↑f x",
"tactic": "exact mul_one (f x)"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nM : Type v₁\nN : Type v₂\nS : Type v₃\ninst✝³³ : AddCommMonoid M\ninst✝³² : AddCommMonoid N\ninst✝³¹ : CommRing R\ninst✝³⁰ : CommRing S\ninst✝²⁹ : Algebra R S\ninst✝²⁸ : Module R M\ninst✝²⁷ : Module R N\ninst✝²⁶ : Module S N\ninst✝²⁵ : IsScalarTower R S N\nf✝ : M →ₗ[R] N\nh : IsBaseChange S f✝\nP : Type ?u.1281852\nQ : Type ?u.1281855\ninst✝²⁴ : AddCommMonoid P\ninst✝²³ : Module R P\ninst✝²² : AddCommMonoid Q\ninst✝²¹ : Module S Q\nT : Type ?u.1282170\nO : Type ?u.1282173\ninst✝²⁰ : CommRing T\ninst✝¹⁹ : Algebra R T\ninst✝¹⁸ : Algebra S T\ninst✝¹⁷ : IsScalarTower R S T\ninst✝¹⁶ : AddCommMonoid O\ninst✝¹⁵ : Module R O\ninst✝¹⁴ : Module S O\ninst✝¹³ : Module T O\ninst✝¹² : IsScalarTower S T O\ninst✝¹¹ : IsScalarTower R S O\ninst✝¹⁰ : IsScalarTower R T O\nR' : Type u_2\nS' : Type u_3\ninst✝⁹ : CommRing R'\ninst✝⁸ : CommRing S'\ninst✝⁷ : Algebra R R'\ninst✝⁶ : Algebra S S'\ninst✝⁵ : Algebra R' S'\ninst✝⁴ : Algebra R S'\ninst✝³ : IsScalarTower R R' S'\ninst✝² : IsScalarTower R S S'\nH✝ : IsPushout R S R' S'\nA : Type u_4\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nf : S →ₐ[R] A\ng : R' →ₐ[R] A\nH : ∀ (x : S) (y : R'), ↑f x * ↑g y = ↑g y * ↑f x\nx : S\nthis : Module S A := Module.compHom A ↑f\nr : R\ns : S\na : A\n⊢ ↑f (r • s) * a = r • (↑f s * a)",
"tactic": "rw [f.map_smul, smul_mul_assoc]"
}
]
| [
494,
22
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
484,
1
]
|
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | SimpleGraph.Path.mapEmbedding_injective | []
| [
1664,
32
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1662,
1
]
|
Mathlib/Analysis/SpecialFunctions/Pow/Asymptotics.lean | isLittleO_exp_neg_mul_rpow_atTop | [
{
"state_after": "case hgf\na : ℝ\nha : 0 < a\nb : ℝ\n⊢ ∀ᶠ (x : ℝ) in atTop, x ^ b = 0 → exp (-a * x) = 0\n\ncase a\na : ℝ\nha : 0 < a\nb : ℝ\n⊢ Tendsto (fun x => exp (-a * x) / x ^ b) atTop (𝓝 0)",
"state_before": "a : ℝ\nha : 0 < a\nb : ℝ\n⊢ (fun x => exp (-a * x)) =o[atTop] fun x => x ^ b",
"tactic": "apply isLittleO_of_tendsto'"
},
{
"state_after": "case hgf\na : ℝ\nha : 0 < a\nb t : ℝ\nht : 0 < t\nh : t ^ b = 0\n⊢ exp (-a * t) = 0",
"state_before": "case hgf\na : ℝ\nha : 0 < a\nb : ℝ\n⊢ ∀ᶠ (x : ℝ) in atTop, x ^ b = 0 → exp (-a * x) = 0",
"tactic": "refine' (eventually_gt_atTop 0).mp (eventually_of_forall fun t ht h => _)"
},
{
"state_after": "case hgf\na : ℝ\nha : 0 < a\nb t : ℝ\nht : 0 < t\nh : t = 0 ∧ b ≠ 0\n⊢ exp (-a * t) = 0",
"state_before": "case hgf\na : ℝ\nha : 0 < a\nb t : ℝ\nht : 0 < t\nh : t ^ b = 0\n⊢ exp (-a * t) = 0",
"tactic": "rw [rpow_eq_zero_iff_of_nonneg ht.le] at h"
},
{
"state_after": "no goals",
"state_before": "case hgf\na : ℝ\nha : 0 < a\nb t : ℝ\nht : 0 < t\nh : t = 0 ∧ b ≠ 0\n⊢ exp (-a * t) = 0",
"tactic": "exact (ht.ne' h.1).elim"
},
{
"state_after": "case a\na : ℝ\nha : 0 < a\nb : ℝ\n⊢ (fun x => exp (a * x) / x ^ (-b))⁻¹ =ᶠ[atTop] fun x => exp (-a * x) / x ^ b",
"state_before": "case a\na : ℝ\nha : 0 < a\nb : ℝ\n⊢ Tendsto (fun x => exp (-a * x) / x ^ b) atTop (𝓝 0)",
"tactic": "refine' (tendsto_exp_mul_div_rpow_atTop (-b) a ha).inv_tendsto_atTop.congr' _"
},
{
"state_after": "case a\na : ℝ\nha : 0 < a\nb t : ℝ\nht : 0 ≤ t\n⊢ (fun x => exp (a * x) / x ^ (-b))⁻¹ t = (fun x => exp (-a * x) / x ^ b) t",
"state_before": "case a\na : ℝ\nha : 0 < a\nb : ℝ\n⊢ (fun x => exp (a * x) / x ^ (-b))⁻¹ =ᶠ[atTop] fun x => exp (-a * x) / x ^ b",
"tactic": "refine' (eventually_ge_atTop 0).mp (eventually_of_forall fun t ht => _)"
},
{
"state_after": "case a\na : ℝ\nha : 0 < a\nb t : ℝ\nht : 0 ≤ t\n⊢ (fun x => exp (a * x) / x ^ (-b))⁻¹ t = exp (-a * t) / t ^ b",
"state_before": "case a\na : ℝ\nha : 0 < a\nb t : ℝ\nht : 0 ≤ t\n⊢ (fun x => exp (a * x) / x ^ (-b))⁻¹ t = (fun x => exp (-a * x) / x ^ b) t",
"tactic": "dsimp only"
},
{
"state_after": "no goals",
"state_before": "case a\na : ℝ\nha : 0 < a\nb t : ℝ\nht : 0 ≤ t\n⊢ (fun x => exp (a * x) / x ^ (-b))⁻¹ t = exp (-a * t) / t ^ b",
"tactic": "rw [Pi.inv_apply, inv_div, ← inv_div_inv, neg_mul, Real.exp_neg, rpow_neg ht, inv_inv]"
}
]
| [
282,
91
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
273,
1
]
|
Mathlib/Topology/Algebra/Equicontinuity.lean | equicontinuous_of_equicontinuousAt_one | [
{
"state_after": "ι : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : UniformSpace M\ninst✝⁴ : Group G\ninst✝³ : Group M\ninst✝² : TopologicalGroup G\ninst✝¹ : UniformGroup M\ninst✝ : MonoidHomClass hom G M\nF : ι → hom\nhf : EquicontinuousAt (FunLike.coe ∘ F) 1\n⊢ Continuous (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F))",
"state_before": "ι : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : UniformSpace M\ninst✝⁴ : Group G\ninst✝³ : Group M\ninst✝² : TopologicalGroup G\ninst✝¹ : UniformGroup M\ninst✝ : MonoidHomClass hom G M\nF : ι → hom\nhf : EquicontinuousAt (FunLike.coe ∘ F) 1\n⊢ Equicontinuous (FunLike.coe ∘ F)",
"tactic": "rw [equicontinuous_iff_continuous]"
},
{
"state_after": "ι : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : UniformSpace M\ninst✝⁴ : Group G\ninst✝³ : Group M\ninst✝² : TopologicalGroup G\ninst✝¹ : UniformGroup M\ninst✝ : MonoidHomClass hom G M\nF : ι → hom\nhf : ContinuousAt (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F)) 1\n⊢ Continuous (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F))",
"state_before": "ι : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : UniformSpace M\ninst✝⁴ : Group G\ninst✝³ : Group M\ninst✝² : TopologicalGroup G\ninst✝¹ : UniformGroup M\ninst✝ : MonoidHomClass hom G M\nF : ι → hom\nhf : EquicontinuousAt (FunLike.coe ∘ F) 1\n⊢ Continuous (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F))",
"tactic": "rw [equicontinuousAt_iff_continuousAt] at hf"
},
{
"state_after": "ι : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : UniformSpace M\ninst✝⁴ : Group G\ninst✝³ : Group M\ninst✝² : TopologicalGroup G\ninst✝¹ : UniformGroup M\ninst✝ : MonoidHomClass hom G M\nF : ι → hom\nhf : ContinuousAt (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F)) 1\nφ : G →* ι →ᵤ M :=\n { toOneHom := { toFun := swap (FunLike.coe ∘ F), map_one' := (_ : swap (FunLike.coe ∘ F) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (a b : G),\n OneHom.toFun { toFun := swap (FunLike.coe ∘ F), map_one' := (_ : swap (FunLike.coe ∘ F) 1 = 1) } (a * b) =\n OneHom.toFun { toFun := swap (FunLike.coe ∘ F), map_one' := (_ : swap (FunLike.coe ∘ F) 1 = 1) } a *\n OneHom.toFun { toFun := swap (FunLike.coe ∘ F), map_one' := (_ : swap (FunLike.coe ∘ F) 1 = 1) } b) }\n⊢ Continuous (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F))",
"state_before": "ι : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : UniformSpace M\ninst✝⁴ : Group G\ninst✝³ : Group M\ninst✝² : TopologicalGroup G\ninst✝¹ : UniformGroup M\ninst✝ : MonoidHomClass hom G M\nF : ι → hom\nhf : ContinuousAt (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F)) 1\n⊢ Continuous (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F))",
"tactic": "let φ : G →* (ι →ᵤ M) :=\n { toFun := swap ((↑) ∘ F)\n map_one' := by dsimp [UniformFun] ; ext ; exact map_one _\n map_mul' := fun a b => by dsimp [UniformFun] ; ext ; exact map_mul _ _ _ }"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : UniformSpace M\ninst✝⁴ : Group G\ninst✝³ : Group M\ninst✝² : TopologicalGroup G\ninst✝¹ : UniformGroup M\ninst✝ : MonoidHomClass hom G M\nF : ι → hom\nhf : ContinuousAt (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F)) 1\nφ : G →* ι →ᵤ M :=\n { toOneHom := { toFun := swap (FunLike.coe ∘ F), map_one' := (_ : swap (FunLike.coe ∘ F) 1 = 1) },\n map_mul' :=\n (_ :\n ∀ (a b : G),\n OneHom.toFun { toFun := swap (FunLike.coe ∘ F), map_one' := (_ : swap (FunLike.coe ∘ F) 1 = 1) } (a * b) =\n OneHom.toFun { toFun := swap (FunLike.coe ∘ F), map_one' := (_ : swap (FunLike.coe ∘ F) 1 = 1) } a *\n OneHom.toFun { toFun := swap (FunLike.coe ∘ F), map_one' := (_ : swap (FunLike.coe ∘ F) 1 = 1) } b) }\n⊢ Continuous (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F))",
"tactic": "exact continuous_of_continuousAt_one φ hf"
},
{
"state_after": "ι : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : UniformSpace M\ninst✝⁴ : Group G\ninst✝³ : Group M\ninst✝² : TopologicalGroup G\ninst✝¹ : UniformGroup M\ninst✝ : MonoidHomClass hom G M\nF : ι → hom\nhf : ContinuousAt (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F)) 1\n⊢ swap (FunLike.coe ∘ F) 1 = 1",
"state_before": "ι : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : UniformSpace M\ninst✝⁴ : Group G\ninst✝³ : Group M\ninst✝² : TopologicalGroup G\ninst✝¹ : UniformGroup M\ninst✝ : MonoidHomClass hom G M\nF : ι → hom\nhf : ContinuousAt (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F)) 1\n⊢ swap (FunLike.coe ∘ F) 1 = 1",
"tactic": "dsimp [UniformFun]"
},
{
"state_after": "case h\nι : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : UniformSpace M\ninst✝⁴ : Group G\ninst✝³ : Group M\ninst✝² : TopologicalGroup G\ninst✝¹ : UniformGroup M\ninst✝ : MonoidHomClass hom G M\nF : ι → hom\nhf : ContinuousAt (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F)) 1\nx✝ : ι\n⊢ swap (FunLike.coe ∘ F) 1 x✝ = OfNat.ofNat 1 x✝",
"state_before": "ι : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : UniformSpace M\ninst✝⁴ : Group G\ninst✝³ : Group M\ninst✝² : TopologicalGroup G\ninst✝¹ : UniformGroup M\ninst✝ : MonoidHomClass hom G M\nF : ι → hom\nhf : ContinuousAt (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F)) 1\n⊢ swap (FunLike.coe ∘ F) 1 = 1",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nι : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : UniformSpace M\ninst✝⁴ : Group G\ninst✝³ : Group M\ninst✝² : TopologicalGroup G\ninst✝¹ : UniformGroup M\ninst✝ : MonoidHomClass hom G M\nF : ι → hom\nhf : ContinuousAt (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F)) 1\nx✝ : ι\n⊢ swap (FunLike.coe ∘ F) 1 x✝ = OfNat.ofNat 1 x✝",
"tactic": "exact map_one _"
},
{
"state_after": "ι : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : UniformSpace M\ninst✝⁴ : Group G\ninst✝³ : Group M\ninst✝² : TopologicalGroup G\ninst✝¹ : UniformGroup M\ninst✝ : MonoidHomClass hom G M\nF : ι → hom\nhf : ContinuousAt (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F)) 1\na b : G\n⊢ swap (FunLike.coe ∘ F) (a * b) = swap (FunLike.coe ∘ F) a * swap (FunLike.coe ∘ F) b",
"state_before": "ι : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : UniformSpace M\ninst✝⁴ : Group G\ninst✝³ : Group M\ninst✝² : TopologicalGroup G\ninst✝¹ : UniformGroup M\ninst✝ : MonoidHomClass hom G M\nF : ι → hom\nhf : ContinuousAt (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F)) 1\na b : G\n⊢ OneHom.toFun { toFun := swap (FunLike.coe ∘ F), map_one' := (_ : swap (FunLike.coe ∘ F) 1 = 1) } (a * b) =\n OneHom.toFun { toFun := swap (FunLike.coe ∘ F), map_one' := (_ : swap (FunLike.coe ∘ F) 1 = 1) } a *\n OneHom.toFun { toFun := swap (FunLike.coe ∘ F), map_one' := (_ : swap (FunLike.coe ∘ F) 1 = 1) } b",
"tactic": "dsimp [UniformFun]"
},
{
"state_after": "case h\nι : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : UniformSpace M\ninst✝⁴ : Group G\ninst✝³ : Group M\ninst✝² : TopologicalGroup G\ninst✝¹ : UniformGroup M\ninst✝ : MonoidHomClass hom G M\nF : ι → hom\nhf : ContinuousAt (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F)) 1\na b : G\nx✝ : ι\n⊢ swap (FunLike.coe ∘ F) (a * b) x✝ = (swap (FunLike.coe ∘ F) a * swap (FunLike.coe ∘ F) b) x✝",
"state_before": "ι : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : UniformSpace M\ninst✝⁴ : Group G\ninst✝³ : Group M\ninst✝² : TopologicalGroup G\ninst✝¹ : UniformGroup M\ninst✝ : MonoidHomClass hom G M\nF : ι → hom\nhf : ContinuousAt (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F)) 1\na b : G\n⊢ swap (FunLike.coe ∘ F) (a * b) = swap (FunLike.coe ∘ F) a * swap (FunLike.coe ∘ F) b",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nι : Type u_1\nG : Type u_2\nM : Type u_3\nhom : Type u_4\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : UniformSpace M\ninst✝⁴ : Group G\ninst✝³ : Group M\ninst✝² : TopologicalGroup G\ninst✝¹ : UniformGroup M\ninst✝ : MonoidHomClass hom G M\nF : ι → hom\nhf : ContinuousAt (↑UniformFun.ofFun ∘ swap (FunLike.coe ∘ F)) 1\na b : G\nx✝ : ι\n⊢ swap (FunLike.coe ∘ F) (a * b) x✝ = (swap (FunLike.coe ∘ F) a * swap (FunLike.coe ∘ F) b) x✝",
"tactic": "exact map_mul _ _ _"
}
]
| [
34,
44
]
| 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
23,
1
]
|
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