file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Data/Nat/Factors.lean
|
Nat.le_of_mem_factors
|
[
{
"state_after": "case inl\np : ℕ\nh : p ∈ factors 0\n⊢ p ≤ 0\n\ncase inr\nn p : ℕ\nh : p ∈ factors n\nhn : n > 0\n⊢ p ≤ n",
"state_before": "n p : ℕ\nh : p ∈ factors n\n⊢ p ≤ n",
"tactic": "rcases n.eq_zero_or_pos with (rfl | hn)"
},
{
"state_after": "case inl\np : ℕ\nh : p ∈ []\n⊢ p ≤ 0",
"state_before": "case inl\np : ℕ\nh : p ∈ factors 0\n⊢ p ≤ 0",
"tactic": "rw [factors_zero] at h"
},
{
"state_after": "no goals",
"state_before": "case inl\np : ℕ\nh : p ∈ []\n⊢ p ≤ 0",
"tactic": "cases h"
},
{
"state_after": "no goals",
"state_before": "case inr\nn p : ℕ\nh : p ∈ factors n\nhn : n > 0\n⊢ p ≤ n",
"tactic": "exact le_of_dvd hn (dvd_of_mem_factors h)"
}
] |
[
166,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
Subsemiring.comap_top
|
[] |
[
1034,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1033,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.lean
|
derivWithin_arctan
|
[] |
[
151,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.bddAbove_iff
|
[
{
"state_after": "R : Type ?u.762821\nR' : Type ?u.762824\n𝕜 : Type u_2\n𝕜₂ : Type ?u.762830\n𝕜₃ : Type ?u.762833\n𝕝 : Type ?u.762836\nE : Type u_1\nE₂ : Type ?u.762842\nE₃ : Type ?u.762845\nF : Type ?u.762848\nG : Type ?u.762851\nι : Type ?u.762854\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np✝ q : Seminorm 𝕜 E\nx✝ : E\ns : Set (Seminorm 𝕜 E)\nH : BddAbove (FunLike.coe '' s)\np : Seminorm 𝕜 E\nhp : p ∈ s\nx : E\n⊢ ↑p x ≤ ↑(sSup s) x",
"state_before": "R : Type ?u.762821\nR' : Type ?u.762824\n𝕜 : Type u_2\n𝕜₂ : Type ?u.762830\n𝕜₃ : Type ?u.762833\n𝕝 : Type ?u.762836\nE : Type u_1\nE₂ : Type ?u.762842\nE₃ : Type ?u.762845\nF : Type ?u.762848\nG : Type ?u.762851\nι : Type ?u.762854\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np✝ q : Seminorm 𝕜 E\nx✝ : E\ns : Set (Seminorm 𝕜 E)\nH : BddAbove (FunLike.coe '' s)\np : Seminorm 𝕜 E\nhp : p ∈ s\nx : E\n⊢ (fun f => ↑f) p x ≤ (fun f => ↑f) (sSup s) x",
"tactic": "dsimp"
},
{
"state_after": "R : Type ?u.762821\nR' : Type ?u.762824\n𝕜 : Type u_2\n𝕜₂ : Type ?u.762830\n𝕜₃ : Type ?u.762833\n𝕝 : Type ?u.762836\nE : Type u_1\nE₂ : Type ?u.762842\nE₃ : Type ?u.762845\nF : Type ?u.762848\nG : Type ?u.762851\nι : Type ?u.762854\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np✝ q : Seminorm 𝕜 E\nx✝ : E\ns : Set (Seminorm 𝕜 E)\nH : BddAbove (FunLike.coe '' s)\np : Seminorm 𝕜 E\nhp : p ∈ s\nx : E\n⊢ ↑p x ≤ ⨆ (i : ↑s), ↑↑i x",
"state_before": "R : Type ?u.762821\nR' : Type ?u.762824\n𝕜 : Type u_2\n𝕜₂ : Type ?u.762830\n𝕜₃ : Type ?u.762833\n𝕝 : Type ?u.762836\nE : Type u_1\nE₂ : Type ?u.762842\nE₃ : Type ?u.762845\nF : Type ?u.762848\nG : Type ?u.762851\nι : Type ?u.762854\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np✝ q : Seminorm 𝕜 E\nx✝ : E\ns : Set (Seminorm 𝕜 E)\nH : BddAbove (FunLike.coe '' s)\np : Seminorm 𝕜 E\nhp : p ∈ s\nx : E\n⊢ ↑p x ≤ ↑(sSup s) x",
"tactic": "rw [Seminorm.coe_sSup_eq' H, iSup_apply]"
},
{
"state_after": "case intro\nR : Type ?u.762821\nR' : Type ?u.762824\n𝕜 : Type u_2\n𝕜₂ : Type ?u.762830\n𝕜₃ : Type ?u.762833\n𝕝 : Type ?u.762836\nE : Type u_1\nE₂ : Type ?u.762842\nE₃ : Type ?u.762845\nF : Type ?u.762848\nG : Type ?u.762851\nι : Type ?u.762854\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np✝ q✝ : Seminorm 𝕜 E\nx✝ : E\ns : Set (Seminorm 𝕜 E)\np : Seminorm 𝕜 E\nhp : p ∈ s\nx : E\nq : E → ℝ\nhq : q ∈ upperBounds (FunLike.coe '' s)\n⊢ ↑p x ≤ ⨆ (i : ↑s), ↑↑i x",
"state_before": "R : Type ?u.762821\nR' : Type ?u.762824\n𝕜 : Type u_2\n𝕜₂ : Type ?u.762830\n𝕜₃ : Type ?u.762833\n𝕝 : Type ?u.762836\nE : Type u_1\nE₂ : Type ?u.762842\nE₃ : Type ?u.762845\nF : Type ?u.762848\nG : Type ?u.762851\nι : Type ?u.762854\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np✝ q : Seminorm 𝕜 E\nx✝ : E\ns : Set (Seminorm 𝕜 E)\nH : BddAbove (FunLike.coe '' s)\np : Seminorm 𝕜 E\nhp : p ∈ s\nx : E\n⊢ ↑p x ≤ ⨆ (i : ↑s), ↑↑i x",
"tactic": "rcases H with ⟨q, hq⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type ?u.762821\nR' : Type ?u.762824\n𝕜 : Type u_2\n𝕜₂ : Type ?u.762830\n𝕜₃ : Type ?u.762833\n𝕝 : Type ?u.762836\nE : Type u_1\nE₂ : Type ?u.762842\nE₃ : Type ?u.762845\nF : Type ?u.762848\nG : Type ?u.762851\nι : Type ?u.762854\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np✝ q✝ : Seminorm 𝕜 E\nx✝ : E\ns : Set (Seminorm 𝕜 E)\np : Seminorm 𝕜 E\nhp : p ∈ s\nx : E\nq : E → ℝ\nhq : q ∈ upperBounds (FunLike.coe '' s)\n⊢ ↑p x ≤ ⨆ (i : ↑s), ↑↑i x",
"tactic": "exact\n le_ciSup ⟨q x, forall_range_iff.mpr fun i : s => hq (mem_image_of_mem _ i.2) x⟩ ⟨p, hp⟩"
}
] |
[
580,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
572,
11
] |
Mathlib/LinearAlgebra/FiniteDimensional.lean
|
LinearMap.finrank_range_add_finrank_ker
|
[
{
"state_after": "K : Type u\nV : Type v\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nV₂ : Type v'\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : FiniteDimensional K V\nf : V →ₗ[K] V₂\n⊢ finrank K (V ⧸ ker f) + finrank K { x // x ∈ ker f } = finrank K V",
"state_before": "K : Type u\nV : Type v\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nV₂ : Type v'\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : FiniteDimensional K V\nf : V →ₗ[K] V₂\n⊢ finrank K { x // x ∈ range f } + finrank K { x // x ∈ ker f } = finrank K V",
"tactic": "rw [← f.quotKerEquivRange.finrank_eq]"
},
{
"state_after": "no goals",
"state_before": "K : Type u\nV : Type v\ninst✝⁵ : DivisionRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nV₂ : Type v'\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : FiniteDimensional K V\nf : V →ₗ[K] V₂\n⊢ finrank K (V ⧸ ker f) + finrank K { x // x ∈ ker f } = finrank K V",
"tactic": "exact Submodule.finrank_quotient_add_finrank _"
}
] |
[
958,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
955,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.centralizer_eq_top_iff_subset
|
[] |
[
879,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
878,
1
] |
Mathlib/Analysis/LocallyConvex/Bounded.lean
|
Bornology.IsVonNBounded.subset
|
[] |
[
93,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/CategoryTheory/Sites/Sieves.lean
|
CategoryTheory.Sieve.functorPullback_id
|
[
{
"state_after": "case h\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS R✝ : Sieve X\nE : Type u₃\ninst✝ : Category E\nG : D ⥤ E\nR : Sieve X\nY✝ : C\nf✝ : Y✝ ⟶ X\n⊢ (functorPullback (𝟭 C) R).arrows f✝ ↔ R.arrows f✝",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS R✝ : Sieve X\nE : Type u₃\ninst✝ : Category E\nG : D ⥤ E\nR : Sieve X\n⊢ functorPullback (𝟭 C) R = R",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nC : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nF : C ⥤ D\nX Y Z : C\nf : Y ⟶ X\nS R✝ : Sieve X\nE : Type u₃\ninst✝ : Category E\nG : D ⥤ E\nR : Sieve X\nY✝ : C\nf✝ : Y✝ ⟶ X\n⊢ (functorPullback (𝟭 C) R).arrows f✝ ↔ R.arrows f✝",
"tactic": "rfl"
}
] |
[
606,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
604,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/StrongEpi.lean
|
CategoryTheory.StrongEpi.of_arrow_iso
|
[
{
"state_after": "C : Type u\ninst✝ : Category C\nP Q R : C\nf✝ : P ⟶ Q\ng✝ : Q ⟶ R\nA B A' B' : C\nf : A ⟶ B\ng : A' ⟶ B'\ne : Arrow.mk f ≅ Arrow.mk g\nh : StrongEpi f\n⊢ Epi (e.inv.left ≫ f ≫ e.hom.right)",
"state_before": "C : Type u\ninst✝ : Category C\nP Q R : C\nf✝ : P ⟶ Q\ng✝ : Q ⟶ R\nA B A' B' : C\nf : A ⟶ B\ng : A' ⟶ B'\ne : Arrow.mk f ≅ Arrow.mk g\nh : StrongEpi f\n⊢ Epi g",
"tactic": "rw [Arrow.iso_w' e]"
},
{
"state_after": "C : Type u\ninst✝ : Category C\nP Q R : C\nf✝ : P ⟶ Q\ng✝ : Q ⟶ R\nA B A' B' : C\nf : A ⟶ B\ng : A' ⟶ B'\ne : Arrow.mk f ≅ Arrow.mk g\nh : StrongEpi f\nthis : Epi (f ≫ e.hom.right)\n⊢ Epi (e.inv.left ≫ f ≫ e.hom.right)",
"state_before": "C : Type u\ninst✝ : Category C\nP Q R : C\nf✝ : P ⟶ Q\ng✝ : Q ⟶ R\nA B A' B' : C\nf : A ⟶ B\ng : A' ⟶ B'\ne : Arrow.mk f ≅ Arrow.mk g\nh : StrongEpi f\n⊢ Epi (e.inv.left ≫ f ≫ e.hom.right)",
"tactic": "haveI := epi_comp f e.hom.right"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nP Q R : C\nf✝ : P ⟶ Q\ng✝ : Q ⟶ R\nA B A' B' : C\nf : A ⟶ B\ng : A' ⟶ B'\ne : Arrow.mk f ≅ Arrow.mk g\nh : StrongEpi f\nthis : Epi (f ≫ e.hom.right)\n⊢ Epi (e.inv.left ≫ f ≫ e.hom.right)",
"tactic": "apply epi_comp"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\nP Q R : C\nf✝ : P ⟶ Q\ng✝ : Q ⟶ R\nA B A' B' : C\nf : A ⟶ B\ng : A' ⟶ B'\ne : Arrow.mk f ≅ Arrow.mk g\nh : StrongEpi f\nX Y : C\nz : X ⟶ Y\ninst✝ : Mono z\n⊢ HasLiftingProperty g z",
"state_before": "C : Type u\ninst✝ : Category C\nP Q R : C\nf✝ : P ⟶ Q\ng✝ : Q ⟶ R\nA B A' B' : C\nf : A ⟶ B\ng : A' ⟶ B'\ne : Arrow.mk f ≅ Arrow.mk g\nh : StrongEpi f\nX Y : C\nz : X ⟶ Y\n⊢ ∀ [inst : Mono z], HasLiftingProperty g z",
"tactic": "intro"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\nP Q R : C\nf✝ : P ⟶ Q\ng✝ : Q ⟶ R\nA B A' B' : C\nf : A ⟶ B\ng : A' ⟶ B'\ne : Arrow.mk f ≅ Arrow.mk g\nh : StrongEpi f\nX Y : C\nz : X ⟶ Y\ninst✝ : Mono z\n⊢ HasLiftingProperty g z",
"tactic": "apply HasLiftingProperty.of_arrow_iso_left e z"
}
] |
[
162,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/Topology/Inseparable.lean
|
Inseparable.map
|
[] |
[
378,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
377,
1
] |
Mathlib/Data/Real/EReal.lean
|
EReal.coe_real_ereal_eq_coe_toNNReal_sub_coe_toNNReal
|
[
{
"state_after": "case inl\nx : ℝ\nh : 0 ≤ x\n⊢ ↑x = ↑↑(Real.toNNReal x) - ↑↑(Real.toNNReal (-x))\n\ncase inr\nx : ℝ\nh : x ≤ 0\n⊢ ↑x = ↑↑(Real.toNNReal x) - ↑↑(Real.toNNReal (-x))",
"state_before": "x : ℝ\n⊢ ↑x = ↑↑(Real.toNNReal x) - ↑↑(Real.toNNReal (-x))",
"tactic": "rcases le_total 0 x with (h | h)"
},
{
"state_after": "case inl.intro\nx : ℝ≥0\n⊢ ↑↑x = ↑↑(Real.toNNReal ↑x) - ↑↑(Real.toNNReal (-↑x))",
"state_before": "case inl\nx : ℝ\nh : 0 ≤ x\n⊢ ↑x = ↑↑(Real.toNNReal x) - ↑↑(Real.toNNReal (-x))",
"tactic": "lift x to ℝ≥0 using h"
},
{
"state_after": "case inl.intro\nx : ℝ≥0\n⊢ ↑↑x = ↑↑x",
"state_before": "case inl.intro\nx : ℝ≥0\n⊢ ↑↑x = ↑↑(Real.toNNReal ↑x) - ↑↑(Real.toNNReal (-↑x))",
"tactic": "rw [Real.toNNReal_of_nonpos (neg_nonpos.mpr x.coe_nonneg), Real.toNNReal_coe, ENNReal.coe_zero,\n coe_ennreal_zero, sub_zero]"
},
{
"state_after": "no goals",
"state_before": "case inl.intro\nx : ℝ≥0\n⊢ ↑↑x = ↑↑x",
"tactic": "rfl"
},
{
"state_after": "case inr.hr\nx : ℝ\nh : x ≤ 0\n⊢ 0 ≤ -x",
"state_before": "case inr\nx : ℝ\nh : x ≤ 0\n⊢ ↑x = ↑↑(Real.toNNReal x) - ↑↑(Real.toNNReal (-x))",
"tactic": "rw [Real.toNNReal_of_nonpos h, ENNReal.coe_zero, coe_ennreal_zero, coe_nnreal_eq_coe_real,\n Real.coe_toNNReal, zero_sub, coe_neg, neg_neg]"
},
{
"state_after": "no goals",
"state_before": "case inr.hr\nx : ℝ\nh : x ≤ 0\n⊢ 0 ≤ -x",
"tactic": "exact neg_nonneg.2 h"
}
] |
[
882,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
873,
1
] |
Mathlib/Data/Polynomial/RingDivision.lean
|
Polynomial.Monic.comp_X_sub_C
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nhp : Monic p\nr : R\n⊢ Monic (Polynomial.comp p (X - ↑C r))",
"tactic": "simpa using hp.comp_X_add_C (-r)"
}
] |
[
840,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
839,
1
] |
Mathlib/Order/SuccPred/Limit.lean
|
Order.isPredLimitRecOn_limit
|
[] |
[
380,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
378,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.subset_iInter
|
[] |
[
295,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
294,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Combination.lean
|
Finset.attach_affineCombination_coe
|
[
{
"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\nι : Type ?u.279611\ns✝ : Finset ι\nι₂ : Type ?u.279617\ns₂ : Finset ι₂\ns : Finset P\nw : P → k\n⊢ ↑(affineCombination k (attach s) Subtype.val) (w ∘ Subtype.val) = ↑(affineCombination k s id) w",
"tactic": "classical rw [attach_affineCombination_of_injective s w ((↑) : s → P) Subtype.coe_injective,\n univ_eq_attach, attach_image_val]"
},
{
"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\nι : Type ?u.279611\ns✝ : Finset ι\nι₂ : Type ?u.279617\ns₂ : Finset ι₂\ns : Finset P\nw : P → k\n⊢ ↑(affineCombination k (attach s) Subtype.val) (w ∘ Subtype.val) = ↑(affineCombination k s id) w",
"tactic": "rw [attach_affineCombination_of_injective s w ((↑) : s → P) Subtype.coe_injective,\nuniv_eq_attach, attach_image_val]"
}
] |
[
460,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
457,
1
] |
Mathlib/GroupTheory/Subsemigroup/Operations.lean
|
Subsemigroup.mem_map_of_mem
|
[] |
[
244,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
243,
1
] |
Mathlib/RingTheory/WittVector/StructurePolynomial.lean
|
constantCoeff_wittStructureInt_zero
|
[
{
"state_after": "p : ℕ\nR : Type ?u.1978909\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\ninj : Function.Injective ↑(Int.castRingHom ℚ)\n⊢ ↑constantCoeff (wittStructureInt p Φ 0) = ↑constantCoeff Φ",
"state_before": "p : ℕ\nR : Type ?u.1978909\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\n⊢ ↑constantCoeff (wittStructureInt p Φ 0) = ↑constantCoeff Φ",
"tactic": "have inj : Function.Injective (Int.castRingHom ℚ) := by intro m n; exact Int.cast_inj.mp"
},
{
"state_after": "case a\np : ℕ\nR : Type ?u.1978909\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\ninj : Function.Injective ↑(Int.castRingHom ℚ)\n⊢ ↑(Int.castRingHom ℚ) (↑constantCoeff (wittStructureInt p Φ 0)) = ↑(Int.castRingHom ℚ) (↑constantCoeff Φ)",
"state_before": "p : ℕ\nR : Type ?u.1978909\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\ninj : Function.Injective ↑(Int.castRingHom ℚ)\n⊢ ↑constantCoeff (wittStructureInt p Φ 0) = ↑constantCoeff Φ",
"tactic": "apply inj"
},
{
"state_after": "no goals",
"state_before": "case a\np : ℕ\nR : Type ?u.1978909\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\ninj : Function.Injective ↑(Int.castRingHom ℚ)\n⊢ ↑(Int.castRingHom ℚ) (↑constantCoeff (wittStructureInt p Φ 0)) = ↑(Int.castRingHom ℚ) (↑constantCoeff Φ)",
"tactic": "rw [← constantCoeff_map, map_wittStructureInt, constantCoeff_wittStructureRat_zero,\n constantCoeff_map]"
},
{
"state_after": "p : ℕ\nR : Type ?u.1978909\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nm n : ℤ\n⊢ ↑(Int.castRingHom ℚ) m = ↑(Int.castRingHom ℚ) n → m = n",
"state_before": "p : ℕ\nR : Type ?u.1978909\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\n⊢ Function.Injective ↑(Int.castRingHom ℚ)",
"tactic": "intro m n"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nR : Type ?u.1978909\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℤ\nm n : ℤ\n⊢ ↑(Int.castRingHom ℚ) m = ↑(Int.castRingHom ℚ) n → m = n",
"tactic": "exact Int.cast_inj.mp"
}
] |
[
379,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
374,
1
] |
Mathlib/Algebra/Order/Rearrangement.lean
|
Antivary.sum_mul_eq_sum_mul_comp_perm_iff
|
[] |
[
511,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
509,
1
] |
Mathlib/Data/MvPolynomial/Variables.lean
|
MvPolynomial.totalDegree_mul
|
[
{
"state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.449883\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q a b : MvPolynomial σ R\nn : σ →₀ ℕ\nhn : n ∈ support (a * b)\nthis : n ∈ Finset.biUnion a.support fun a₁ => Finset.biUnion b.support fun a₂ => {a₁ + a₂}\n⊢ (sum n fun x e => e) ≤ totalDegree a + totalDegree b",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.449883\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q a b : MvPolynomial σ R\nn : σ →₀ ℕ\nhn : n ∈ support (a * b)\n⊢ (sum n fun x e => e) ≤ totalDegree a + totalDegree b",
"tactic": "have := AddMonoidAlgebra.support_mul a b hn"
},
{
"state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.449883\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q a b : MvPolynomial σ R\nn : σ →₀ ℕ\nhn : n ∈ support (a * b)\nthis : ∃ a_1, a_1 ∈ a.support ∧ ∃ a, a ∈ b.support ∧ n = a_1 + a\n⊢ (sum n fun x e => e) ≤ totalDegree a + totalDegree b",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.449883\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q a b : MvPolynomial σ R\nn : σ →₀ ℕ\nhn : n ∈ support (a * b)\nthis : n ∈ Finset.biUnion a.support fun a₁ => Finset.biUnion b.support fun a₂ => {a₁ + a₂}\n⊢ (sum n fun x e => e) ≤ totalDegree a + totalDegree b",
"tactic": "simp only [Finset.mem_biUnion, Finset.mem_singleton] at this"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.449883\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q a b : MvPolynomial σ R\na₁ : σ →₀ ℕ\nh₁ : a₁ ∈ a.support\na₂ : σ →₀ ℕ\nh₂ : a₂ ∈ b.support\nhn : a₁ + a₂ ∈ support (a * b)\n⊢ (sum (a₁ + a₂) fun x e => e) ≤ totalDegree a + totalDegree b",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.449883\nr : R\ne : ℕ\nn✝ m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q a b : MvPolynomial σ R\nn : σ →₀ ℕ\nhn : n ∈ support (a * b)\nthis : ∃ a_1, a_1 ∈ a.support ∧ ∃ a, a ∈ b.support ∧ n = a_1 + a\n⊢ (sum n fun x e => e) ≤ totalDegree a + totalDegree b",
"tactic": "rcases this with ⟨a₁, h₁, a₂, h₂, rfl⟩"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.449883\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q a b : MvPolynomial σ R\na₁ : σ →₀ ℕ\nh₁ : a₁ ∈ a.support\na₂ : σ →₀ ℕ\nh₂ : a₂ ∈ b.support\nhn : a₁ + a₂ ∈ support (a * b)\n⊢ ((sum a₁ fun x e => e) + sum a₂ fun x e => e) ≤ totalDegree a + totalDegree b\n\ncase intro.intro.intro.intro.h_zero\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.449883\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q a b : MvPolynomial σ R\na₁ : σ →₀ ℕ\nh₁ : a₁ ∈ a.support\na₂ : σ →₀ ℕ\nh₂ : a₂ ∈ b.support\nhn : a₁ + a₂ ∈ support (a * b)\n⊢ σ → 0 = 0\n\ncase intro.intro.intro.intro.h_add\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.449883\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q a b : MvPolynomial σ R\na₁ : σ →₀ ℕ\nh₁ : a₁ ∈ a.support\na₂ : σ →₀ ℕ\nh₂ : a₂ ∈ b.support\nhn : a₁ + a₂ ∈ support (a * b)\n⊢ σ → ∀ (b₁ b₂ : ℕ), b₁ + b₂ = b₁ + b₂",
"state_before": "case intro.intro.intro.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.449883\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q a b : MvPolynomial σ R\na₁ : σ →₀ ℕ\nh₁ : a₁ ∈ a.support\na₂ : σ →₀ ℕ\nh₂ : a₂ ∈ b.support\nhn : a₁ + a₂ ∈ support (a * b)\n⊢ (sum (a₁ + a₂) fun x e => e) ≤ totalDegree a + totalDegree b",
"tactic": "rw [Finsupp.sum_add_index']"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.449883\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q a b : MvPolynomial σ R\na₁ : σ →₀ ℕ\nh₁ : a₁ ∈ a.support\na₂ : σ →₀ ℕ\nh₂ : a₂ ∈ b.support\nhn : a₁ + a₂ ∈ support (a * b)\n⊢ ((sum a₁ fun x e => e) + sum a₂ fun x e => e) ≤\n (Finset.sup (support a) fun s => sum s fun x e => e) + Finset.sup (support b) fun s => sum s fun x e => e",
"state_before": "case intro.intro.intro.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.449883\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q a b : MvPolynomial σ R\na₁ : σ →₀ ℕ\nh₁ : a₁ ∈ a.support\na₂ : σ →₀ ℕ\nh₂ : a₂ ∈ b.support\nhn : a₁ + a₂ ∈ support (a * b)\n⊢ ((sum a₁ fun x e => e) + sum a₂ fun x e => e) ≤ totalDegree a + totalDegree b",
"tactic": "dsimp [totalDegree]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.449883\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q a b : MvPolynomial σ R\na₁ : σ →₀ ℕ\nh₁ : a₁ ∈ a.support\na₂ : σ →₀ ℕ\nh₂ : a₂ ∈ b.support\nhn : a₁ + a₂ ∈ support (a * b)\n⊢ ((sum a₁ fun x e => e) + sum a₂ fun x e => e) ≤\n (Finset.sup (support a) fun s => sum s fun x e => e) + Finset.sup (support b) fun s => sum s fun x e => e",
"tactic": "exact add_le_add (le_totalDegree h₁) (le_totalDegree h₂)"
},
{
"state_after": "case intro.intro.intro.intro.h_zero\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.449883\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q a b : MvPolynomial σ R\na₁ : σ →₀ ℕ\nh₁ : a₁ ∈ a.support\na₂ : σ →₀ ℕ\nh₂ : a₂ ∈ b.support\nhn : a₁ + a₂ ∈ support (a * b)\na✝ : σ\n⊢ 0 = 0",
"state_before": "case intro.intro.intro.intro.h_zero\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.449883\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q a b : MvPolynomial σ R\na₁ : σ →₀ ℕ\nh₁ : a₁ ∈ a.support\na₂ : σ →₀ ℕ\nh₂ : a₂ ∈ b.support\nhn : a₁ + a₂ ∈ support (a * b)\n⊢ σ → 0 = 0",
"tactic": "intro _"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.h_zero\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.449883\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q a b : MvPolynomial σ R\na₁ : σ →₀ ℕ\nh₁ : a₁ ∈ a.support\na₂ : σ →₀ ℕ\nh₂ : a₂ ∈ b.support\nhn : a₁ + a₂ ∈ support (a * b)\na✝ : σ\n⊢ 0 = 0",
"tactic": "rfl"
},
{
"state_after": "case intro.intro.intro.intro.h_add\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.449883\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q a b : MvPolynomial σ R\na₁ : σ →₀ ℕ\nh₁ : a₁ ∈ a.support\na₂ : σ →₀ ℕ\nh₂ : a₂ ∈ b.support\nhn : a₁ + a₂ ∈ support (a * b)\na✝ : σ\nb₁ b₂ : ℕ\n⊢ b₁ + b₂ = b₁ + b₂",
"state_before": "case intro.intro.intro.intro.h_add\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.449883\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q a b : MvPolynomial σ R\na₁ : σ →₀ ℕ\nh₁ : a₁ ∈ a.support\na₂ : σ →₀ ℕ\nh₂ : a₂ ∈ b.support\nhn : a₁ + a₂ ∈ support (a * b)\n⊢ σ → ∀ (b₁ b₂ : ℕ), b₁ + b₂ = b₁ + b₂",
"tactic": "intro _ b₁ b₂"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.h_add\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.449883\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q a b : MvPolynomial σ R\na₁ : σ →₀ ℕ\nh₁ : a₁ ∈ a.support\na₂ : σ →₀ ℕ\nh₂ : a₂ ∈ b.support\nhn : a₁ + a₂ ∈ support (a * b)\na✝ : σ\nb₁ b₂ : ℕ\n⊢ b₁ + b₂ = b₁ + b₂",
"tactic": "rfl"
}
] |
[
696,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
683,
1
] |
Mathlib/Data/Polynomial/RingDivision.lean
|
Polynomial.rootSet_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS✝ : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\np q : R[X]\ninst✝³ : CommRing T\nS : Type u_1\ninst✝² : CommRing S\ninst✝¹ : IsDomain S\ninst✝ : Algebra T S\n⊢ rootSet 0 S = ∅",
"tactic": "rw [← C_0, rootSet_C]"
}
] |
[
905,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
904,
1
] |
Mathlib/NumberTheory/Multiplicity.lean
|
sq_dvd_add_pow_sub_sub
|
[
{
"state_after": "case zero\nR : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x✝ y p x : R\n⊢ p ^ 2 ∣ (x + p) ^ Nat.zero - x ^ (Nat.zero - 1) * p * ↑Nat.zero - x ^ Nat.zero\n\ncase succ\nR : Type u_1\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y p x : R\nn : ℕ\n⊢ p ^ 2 ∣ (x + p) ^ Nat.succ n - x ^ (Nat.succ n - 1) * p * ↑(Nat.succ n) - x ^ Nat.succ n",
"state_before": "R : Type u_1\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y p x : R\nn : ℕ\n⊢ p ^ 2 ∣ (x + p) ^ n - x ^ (n - 1) * p * ↑n - x ^ n",
"tactic": "cases' n with n n"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u_1\nn : ℕ\ninst✝ : CommRing R\na b x✝ y p x : R\n⊢ p ^ 2 ∣ (x + p) ^ Nat.zero - x ^ (Nat.zero - 1) * p * ↑Nat.zero - x ^ Nat.zero",
"tactic": "simp only [pow_zero, Nat.cast_zero, sub_zero, sub_self, dvd_zero, Nat.zero_eq, mul_zero]"
},
{
"state_after": "case succ\nR : Type u_1\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y p x : R\nn : ℕ\n⊢ p ^ 2 ∣\n ∑ m in range n, x ^ m * p ^ (n + 1 - m) * ↑(Nat.choose (n + 1) m) + x ^ n * p * (↑n + 1) + x ^ (n + 1) -\n x ^ n * p * (↑n + 1) -\n x ^ (n + 1)",
"state_before": "case succ\nR : Type u_1\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y p x : R\nn : ℕ\n⊢ p ^ 2 ∣ (x + p) ^ Nat.succ n - x ^ (Nat.succ n - 1) * p * ↑(Nat.succ n) - x ^ Nat.succ n",
"tactic": "simp only [Nat.succ_sub_succ_eq_sub, tsub_zero, Nat.cast_succ, add_pow, Finset.sum_range_succ,\n Nat.choose_self, Nat.succ_sub _, tsub_self, pow_one, Nat.choose_succ_self_right, pow_zero,\n mul_one, Nat.cast_zero, zero_add, Nat.succ_eq_add_one, add_tsub_cancel_left]"
},
{
"state_after": "case succ\nR : Type u_1\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y p x : R\nn : ℕ\n⊢ p ^ 2 ∣ ∑ i in range n, x ^ i * p ^ (n + 1 - i) * ↑(Nat.choose (n + 1) i)",
"state_before": "case succ\nR : Type u_1\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y p x : R\nn : ℕ\n⊢ p ^ 2 ∣\n ∑ m in range n, x ^ m * p ^ (n + 1 - m) * ↑(Nat.choose (n + 1) m) + x ^ n * p * (↑n + 1) + x ^ (n + 1) -\n x ^ n * p * (↑n + 1) -\n x ^ (n + 1)",
"tactic": "suffices p ^ 2 ∣ ∑ i : ℕ in range n, x ^ i * p ^ (n + 1 - i) * ↑((n + 1).choose i) by\n convert this; abel"
},
{
"state_after": "case h.e'_4\nR : Type u_1\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y p x : R\nn : ℕ\nthis : p ^ 2 ∣ ∑ i in range n, x ^ i * p ^ (n + 1 - i) * ↑(Nat.choose (n + 1) i)\n⊢ ∑ m in range n, x ^ m * p ^ (n + 1 - m) * ↑(Nat.choose (n + 1) m) + x ^ n * p * (↑n + 1) + x ^ (n + 1) -\n x ^ n * p * (↑n + 1) -\n x ^ (n + 1) =\n ∑ i in range n, x ^ i * p ^ (n + 1 - i) * ↑(Nat.choose (n + 1) i)",
"state_before": "R : Type u_1\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y p x : R\nn : ℕ\nthis : p ^ 2 ∣ ∑ i in range n, x ^ i * p ^ (n + 1 - i) * ↑(Nat.choose (n + 1) i)\n⊢ p ^ 2 ∣\n ∑ m in range n, x ^ m * p ^ (n + 1 - m) * ↑(Nat.choose (n + 1) m) + x ^ n * p * (↑n + 1) + x ^ (n + 1) -\n x ^ n * p * (↑n + 1) -\n x ^ (n + 1)",
"tactic": "convert this"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nR : Type u_1\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y p x : R\nn : ℕ\nthis : p ^ 2 ∣ ∑ i in range n, x ^ i * p ^ (n + 1 - i) * ↑(Nat.choose (n + 1) i)\n⊢ ∑ m in range n, x ^ m * p ^ (n + 1 - m) * ↑(Nat.choose (n + 1) m) + x ^ n * p * (↑n + 1) + x ^ (n + 1) -\n x ^ n * p * (↑n + 1) -\n x ^ (n + 1) =\n ∑ i in range n, x ^ i * p ^ (n + 1 - i) * ↑(Nat.choose (n + 1) i)",
"tactic": "abel"
},
{
"state_after": "case succ.h\nR : Type u_1\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y p x : R\nn : ℕ\n⊢ ∀ (x_1 : ℕ), x_1 ∈ range n → p ^ 2 ∣ x ^ x_1 * p ^ (n + 1 - x_1) * ↑(Nat.choose (n + 1) x_1)",
"state_before": "case succ\nR : Type u_1\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y p x : R\nn : ℕ\n⊢ p ^ 2 ∣ ∑ i in range n, x ^ i * p ^ (n + 1 - i) * ↑(Nat.choose (n + 1) i)",
"tactic": "apply Finset.dvd_sum"
},
{
"state_after": "case succ.h\nR : Type u_1\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ p x : R\nn y : ℕ\nhy : y ∈ range n\n⊢ p ^ 2 ∣ x ^ y * p ^ (n + 1 - y) * ↑(Nat.choose (n + 1) y)",
"state_before": "case succ.h\nR : Type u_1\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y p x : R\nn : ℕ\n⊢ ∀ (x_1 : ℕ), x_1 ∈ range n → p ^ 2 ∣ x ^ x_1 * p ^ (n + 1 - x_1) * ↑(Nat.choose (n + 1) x_1)",
"tactic": "intro y hy"
},
{
"state_after": "no goals",
"state_before": "case succ.h\nR : Type u_1\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ p x : R\nn y : ℕ\nhy : y ∈ range n\n⊢ p ^ 2 ∣ x ^ y * p ^ (n + 1 - y) * ↑(Nat.choose (n + 1) y)",
"tactic": "calc\n p ^ 2 ∣ p ^ (n + 1 - y) :=\n pow_dvd_pow p (le_tsub_of_add_le_left (by linarith [Finset.mem_range.mp hy]))\n _ ∣ x ^ y * p ^ (n + 1 - y) * ↑((n + 1).choose y) :=\n dvd_mul_of_dvd_left (dvd_mul_left _ _) _"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nn✝ : ℕ\ninst✝ : CommRing R\na b x✝ y✝ p x : R\nn y : ℕ\nhy : y ∈ range n\n⊢ y + 2 ≤ n + 1",
"tactic": "linarith [Finset.mem_range.mp hy]"
}
] |
[
77,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/StrictInitial.lean
|
CategoryTheory.Limits.IsTerminal.subsingleton_to
|
[] |
[
202,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
201,
1
] |
Mathlib/AlgebraicTopology/SplitSimplicialObject.lean
|
SimplicialObject.Splitting.hom_ext
|
[
{
"state_after": "case h\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nf g : X ⟶ Y\nh : ∀ (n : ℕ), φ s f n = φ s g n\nΔ : SimplexCategoryᵒᵖ\n⊢ f.app Δ = g.app Δ",
"state_before": "C : Type u_2\ninst✝¹ : Category C\ninst✝ : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nf g : X ⟶ Y\nh : ∀ (n : ℕ), φ s f n = φ s g n\n⊢ f = g",
"tactic": "ext Δ"
},
{
"state_after": "case h.h\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nf g : X ⟶ Y\nh : ∀ (n : ℕ), φ s f n = φ s g n\nΔ : SimplexCategoryᵒᵖ\n⊢ ∀ (A : IndexSet Δ), ιSummand s A ≫ f.app Δ = ιSummand s A ≫ g.app Δ",
"state_before": "case h\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nf g : X ⟶ Y\nh : ∀ (n : ℕ), φ s f n = φ s g n\nΔ : SimplexCategoryᵒᵖ\n⊢ f.app Δ = g.app Δ",
"tactic": "apply s.hom_ext'"
},
{
"state_after": "case h.h\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nf g : X ⟶ Y\nh : ∀ (n : ℕ), φ s f n = φ s g n\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\n⊢ ιSummand s A ≫ f.app Δ = ιSummand s A ≫ g.app Δ",
"state_before": "case h.h\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nf g : X ⟶ Y\nh : ∀ (n : ℕ), φ s f n = φ s g n\nΔ : SimplexCategoryᵒᵖ\n⊢ ∀ (A : IndexSet Δ), ιSummand s A ≫ f.app Δ = ιSummand s A ≫ g.app Δ",
"tactic": "intro A"
},
{
"state_after": "case h.h.mk\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nf g : X ⟶ Y\nh : ∀ (n : ℕ), φ s f n = φ s g n\nΔ✝ : SimplexCategoryᵒᵖ\nA✝ : IndexSet Δ✝\nΔ : SimplexCategory\nA : IndexSet { unop := Δ }\n⊢ ιSummand s A ≫ f.app { unop := Δ } = ιSummand s A ≫ g.app { unop := Δ }",
"state_before": "case h.h\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nf g : X ⟶ Y\nh : ∀ (n : ℕ), φ s f n = φ s g n\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\n⊢ ιSummand s A ≫ f.app Δ = ιSummand s A ≫ g.app Δ",
"tactic": "induction' Δ using Opposite.rec with Δ"
},
{
"state_after": "case h.h.mk.h\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nf g : X ⟶ Y\nh : ∀ (n : ℕ), φ s f n = φ s g n\nΔ✝ : SimplexCategoryᵒᵖ\nA✝¹ : IndexSet Δ✝\nΔ : SimplexCategory\nA✝ : IndexSet { unop := Δ }\nn : ℕ\nA : IndexSet { unop := [n] }\n⊢ ιSummand s A ≫ f.app { unop := [n] } = ιSummand s A ≫ g.app { unop := [n] }",
"state_before": "case h.h.mk\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nf g : X ⟶ Y\nh : ∀ (n : ℕ), φ s f n = φ s g n\nΔ✝ : SimplexCategoryᵒᵖ\nA✝ : IndexSet Δ✝\nΔ : SimplexCategory\nA : IndexSet { unop := Δ }\n⊢ ιSummand s A ≫ f.app { unop := Δ } = ιSummand s A ≫ g.app { unop := Δ }",
"tactic": "induction' Δ using SimplexCategory.rec with n"
},
{
"state_after": "case h.h.mk.h\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nf g : X ⟶ Y\nh : ∀ (n : ℕ), φ s f n = φ s g n\nΔ✝ : SimplexCategoryᵒᵖ\nA✝¹ : IndexSet Δ✝\nΔ : SimplexCategory\nA✝ : IndexSet { unop := Δ }\nn : ℕ\nA : IndexSet { unop := [n] }\n⊢ ιSummand s A ≫ f.app { unop := [n] } = ιSummand s A ≫ g.app { unop := [n] }",
"state_before": "case h.h.mk.h\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nf g : X ⟶ Y\nh : ∀ (n : ℕ), φ s f n = φ s g n\nΔ✝ : SimplexCategoryᵒᵖ\nA✝¹ : IndexSet Δ✝\nΔ : SimplexCategory\nA✝ : IndexSet { unop := Δ }\nn : ℕ\nA : IndexSet { unop := [n] }\n⊢ ιSummand s A ≫ f.app { unop := [n] } = ιSummand s A ≫ g.app { unop := [n] }",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "case h.h.mk.h\nC : Type u_2\ninst✝¹ : Category C\ninst✝ : HasFiniteCoproducts C\nX Y : SimplicialObject C\ns : Splitting X\nf g : X ⟶ Y\nh : ∀ (n : ℕ), φ s f n = φ s g n\nΔ✝ : SimplexCategoryᵒᵖ\nA✝¹ : IndexSet Δ✝\nΔ : SimplexCategory\nA✝ : IndexSet { unop := Δ }\nn : ℕ\nA : IndexSet { unop := [n] }\n⊢ ιSummand s A ≫ f.app { unop := [n] } = ιSummand s A ≫ g.app { unop := [n] }",
"tactic": "simp only [s.ιSummand_comp_app, h]"
}
] |
[
316,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
309,
1
] |
Mathlib/Data/Real/NNReal.lean
|
NNReal.coe_div
|
[] |
[
200,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
199,
11
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean
|
Real.mapsTo_sin_Ioo
|
[
{
"state_after": "no goals",
"state_before": "x : ℝ\nh : x ∈ Ioo (-(π / 2)) (π / 2)\n⊢ sin x ∈ Ioo (-1) 1",
"tactic": "rwa [mem_Ioo, ← arcsin_lt_pi_div_two, ← neg_pi_div_two_lt_arcsin, arcsin_sin h.1.le h.2.le]"
}
] |
[
281,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
280,
1
] |
Mathlib/Topology/Compactification/OnePoint.lean
|
OnePoint.tendsto_nhds_infty
|
[
{
"state_after": "no goals",
"state_before": "X : Type u_2\ninst✝ : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\nα : Type u_1\nf : OnePoint X → α\nl : Filter α\n⊢ Tendsto f (pure ∞) l ∧ Tendsto (f ∘ some) (coclosedCompact X) l ↔\n ∀ (s : Set α), s ∈ l → f ∞ ∈ s ∧ ∃ t, IsClosed t ∧ IsCompact t ∧ MapsTo (f ∘ some) (tᶜ) s",
"tactic": "simp only [tendsto_pure_left, hasBasis_coclosedCompact.tendsto_left_iff, forall_and,\n and_assoc, exists_prop]"
}
] |
[
370,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
365,
1
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.adjoin_simple_eq_bot_iff
|
[
{
"state_after": "no goals",
"state_before": "F : Type u_2\ninst✝² : Field F\nE : Type u_1\ninst✝¹ : Field E\ninst✝ : Algebra F E\nα : E\nS : Set E\n⊢ F⟮α⟯ = ⊥ ↔ α ∈ ⊥",
"tactic": "simp"
}
] |
[
682,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
681,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
Subalgebra.sub_mem
|
[] |
[
195,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
11
] |
Mathlib/Algebra/TrivSqZeroExt.lean
|
TrivSqZeroExt.liftAux_comp_inrHom
|
[] |
[
827,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
825,
1
] |
Mathlib/LinearAlgebra/Matrix/ToLin.lean
|
Matrix.toLinearMapRight'_mul
|
[] |
[
152,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.blsub_eq_lsub
|
[] |
[
1794,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1792,
1
] |
Mathlib/Algebra/BigOperators/Associated.lean
|
Associates.prod_mk
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.33213\nγ : Type ?u.33216\nδ : Type ?u.33219\ninst✝ : CommMonoid α\np : Multiset α\n⊢ Multiset.prod (Multiset.map Associates.mk 0) = Associates.mk (Multiset.prod 0)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.33213\nγ : Type ?u.33216\nδ : Type ?u.33219\ninst✝ : CommMonoid α\np : Multiset α\na : α\ns : Multiset α\nih : Multiset.prod (Multiset.map Associates.mk s) = Associates.mk (Multiset.prod s)\n⊢ Multiset.prod (Multiset.map Associates.mk (a ::ₘ s)) = Associates.mk (Multiset.prod (a ::ₘ s))",
"tactic": "simp [ih, Associates.mk_mul_mk]"
}
] |
[
113,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Analysis/NormedSpace/Extend.lean
|
ContinuousLinearMap.norm_extendTo𝕜'_bound
|
[
{
"state_after": "𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nF : Type u_1\ninst✝³ : SeminormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : IsScalarTower ℝ 𝕜 F\nfr : F →L[ℝ] ℝ\nx : F\nlm : F →ₗ[𝕜] 𝕜 := LinearMap.extendTo𝕜' ↑fr\n⊢ ‖↑lm x‖ ≤ ‖fr‖ * ‖x‖",
"state_before": "𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nF : Type u_1\ninst✝³ : SeminormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : IsScalarTower ℝ 𝕜 F\nfr : F →L[ℝ] ℝ\nx : F\n⊢ ‖↑(LinearMap.extendTo𝕜' ↑fr) x‖ ≤ ‖fr‖ * ‖x‖",
"tactic": "set lm : F →ₗ[𝕜] 𝕜 := fr.toLinearMap.extendTo𝕜'"
},
{
"state_after": "case pos\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nF : Type u_1\ninst✝³ : SeminormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : IsScalarTower ℝ 𝕜 F\nfr : F →L[ℝ] ℝ\nx : F\nlm : F →ₗ[𝕜] 𝕜 := LinearMap.extendTo𝕜' ↑fr\nh : ↑lm x = 0\n⊢ ‖↑lm x‖ ≤ ‖fr‖ * ‖x‖\n\ncase neg\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nF : Type u_1\ninst✝³ : SeminormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : IsScalarTower ℝ 𝕜 F\nfr : F →L[ℝ] ℝ\nx : F\nlm : F →ₗ[𝕜] 𝕜 := LinearMap.extendTo𝕜' ↑fr\nh : ¬↑lm x = 0\n⊢ ‖↑lm x‖ ≤ ‖fr‖ * ‖x‖",
"state_before": "𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nF : Type u_1\ninst✝³ : SeminormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : IsScalarTower ℝ 𝕜 F\nfr : F →L[ℝ] ℝ\nx : F\nlm : F →ₗ[𝕜] 𝕜 := LinearMap.extendTo𝕜' ↑fr\n⊢ ‖↑lm x‖ ≤ ‖fr‖ * ‖x‖",
"tactic": "by_cases h : lm x = 0"
},
{
"state_after": "case neg\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nF : Type u_1\ninst✝³ : SeminormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : IsScalarTower ℝ 𝕜 F\nfr : F →L[ℝ] ℝ\nx : F\nlm : F →ₗ[𝕜] 𝕜 := LinearMap.extendTo𝕜' ↑fr\nh : ¬↑lm x = 0\n⊢ ‖↑lm x‖ ^ 2 ≤ ‖↑lm x‖ * (‖fr‖ * ‖x‖)",
"state_before": "case neg\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nF : Type u_1\ninst✝³ : SeminormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : IsScalarTower ℝ 𝕜 F\nfr : F →L[ℝ] ℝ\nx : F\nlm : F →ₗ[𝕜] 𝕜 := LinearMap.extendTo𝕜' ↑fr\nh : ¬↑lm x = 0\n⊢ ‖↑lm x‖ ≤ ‖fr‖ * ‖x‖",
"tactic": "rw [← mul_le_mul_left (norm_pos_iff.2 h), ← sq]"
},
{
"state_after": "no goals",
"state_before": "case neg\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nF : Type u_1\ninst✝³ : SeminormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : IsScalarTower ℝ 𝕜 F\nfr : F →L[ℝ] ℝ\nx : F\nlm : F →ₗ[𝕜] 𝕜 := LinearMap.extendTo𝕜' ↑fr\nh : ¬↑lm x = 0\n⊢ ‖↑lm x‖ ^ 2 ≤ ‖↑lm x‖ * (‖fr‖ * ‖x‖)",
"tactic": "calc\n ‖lm x‖ ^ 2 = fr (conj (lm x : 𝕜) • x) := fr.toLinearMap.norm_extendTo𝕜'_apply_sq x\n _ ≤ ‖fr (conj (lm x : 𝕜) • x)‖ := (le_abs_self _)\n _ ≤ ‖fr‖ * ‖conj (lm x : 𝕜) • x‖ := (le_op_norm _ _)\n _ = ‖(lm x : 𝕜)‖ * (‖fr‖ * ‖x‖) := by rw [norm_smul, norm_conj, mul_left_comm]"
},
{
"state_after": "case pos\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nF : Type u_1\ninst✝³ : SeminormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : IsScalarTower ℝ 𝕜 F\nfr : F →L[ℝ] ℝ\nx : F\nlm : F →ₗ[𝕜] 𝕜 := LinearMap.extendTo𝕜' ↑fr\nh : ↑lm x = 0\n⊢ 0 ≤ ‖fr‖ * ‖x‖",
"state_before": "case pos\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nF : Type u_1\ninst✝³ : SeminormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : IsScalarTower ℝ 𝕜 F\nfr : F →L[ℝ] ℝ\nx : F\nlm : F →ₗ[𝕜] 𝕜 := LinearMap.extendTo𝕜' ↑fr\nh : ↑lm x = 0\n⊢ ‖↑lm x‖ ≤ ‖fr‖ * ‖x‖",
"tactic": "rw [h, norm_zero]"
},
{
"state_after": "no goals",
"state_before": "case pos\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nF : Type u_1\ninst✝³ : SeminormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : IsScalarTower ℝ 𝕜 F\nfr : F →L[ℝ] ℝ\nx : F\nlm : F →ₗ[𝕜] 𝕜 := LinearMap.extendTo𝕜' ↑fr\nh : ↑lm x = 0\n⊢ 0 ≤ ‖fr‖ * ‖x‖",
"tactic": "apply mul_nonneg <;> exact norm_nonneg _"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nF : Type u_1\ninst✝³ : SeminormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : IsScalarTower ℝ 𝕜 F\nfr : F →L[ℝ] ℝ\nx : F\nlm : F →ₗ[𝕜] 𝕜 := LinearMap.extendTo𝕜' ↑fr\nh : ¬↑lm x = 0\n⊢ ‖fr‖ * ‖↑(starRingEnd ((fun x => 𝕜) x)) (↑lm x) • x‖ = ‖↑lm x‖ * (‖fr‖ * ‖x‖)",
"tactic": "rw [norm_smul, norm_conj, mul_left_comm]"
}
] |
[
123,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.coeSubtype
|
[] |
[
501,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
500,
1
] |
Mathlib/Data/Seq/WSeq.lean
|
Stream'.WSeq.think_congr
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\n⊢ LiftRel (fun x x_1 => x = x_1) (think s) (think t)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\n⊢ think s ~ʷ think t",
"tactic": "unfold Equiv"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\n⊢ LiftRel (fun x x_1 => x = x_1) s t",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\n⊢ LiftRel (fun x x_1 => x = x_1) (think s) (think t)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns t : WSeq α\nh : s ~ʷ t\n⊢ LiftRel (fun x x_1 => x = x_1) s t",
"tactic": "exact h"
}
] |
[
1124,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1123,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.finsuppTotal_apply
|
[
{
"state_after": "ι : Type u_1\nM : Type u_3\ninst✝² : AddCommGroup M\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : Module R M\nI : Ideal R\nv : ι → M\nhv : Submodule.span R (Set.range v) = ⊤\nf : ι →₀ { x // x ∈ I }\n⊢ ↑(Finsupp.total ι M R v) (Finsupp.mapRange Subtype.val (_ : ↑(Submodule.subtype I) 0 = 0) f) =\n Finsupp.sum f fun i x => ↑x • v i",
"state_before": "ι : Type u_1\nM : Type u_3\ninst✝² : AddCommGroup M\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : Module R M\nI : Ideal R\nv : ι → M\nhv : Submodule.span R (Set.range v) = ⊤\nf : ι →₀ { x // x ∈ I }\n⊢ ↑(finsuppTotal ι M I v) f = Finsupp.sum f fun i x => ↑x • v i",
"tactic": "dsimp [finsuppTotal]"
},
{
"state_after": "ι : Type u_1\nM : Type u_3\ninst✝² : AddCommGroup M\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : Module R M\nI : Ideal R\nv : ι → M\nhv : Submodule.span R (Set.range v) = ⊤\nf : ι →₀ { x // x ∈ I }\n⊢ ∀ (a : ι), 0 • v a = 0",
"state_before": "ι : Type u_1\nM : Type u_3\ninst✝² : AddCommGroup M\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : Module R M\nI : Ideal R\nv : ι → M\nhv : Submodule.span R (Set.range v) = ⊤\nf : ι →₀ { x // x ∈ I }\n⊢ ↑(Finsupp.total ι M R v) (Finsupp.mapRange Subtype.val (_ : ↑(Submodule.subtype I) 0 = 0) f) =\n Finsupp.sum f fun i x => ↑x • v i",
"tactic": "rw [Finsupp.total_apply, Finsupp.sum_mapRange_index]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nM : Type u_3\ninst✝² : AddCommGroup M\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : Module R M\nI : Ideal R\nv : ι → M\nhv : Submodule.span R (Set.range v) = ⊤\nf : ι →₀ { x // x ∈ I }\n⊢ ∀ (a : ι), 0 • v a = 0",
"tactic": "exact fun _ => zero_smul _ _"
}
] |
[
1903,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1899,
1
] |
Mathlib/Algebra/Order/LatticeGroup.lean
|
LatticeOrderedCommGroup.pos_inf_neg_eq_one
|
[
{
"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⊢ a⁺ ⊓ a⁻ = 1",
"tactic": "rw [← mul_right_inj (a⁻)⁻¹, mul_inf, mul_one, mul_left_inv, mul_comm, ← div_eq_mul_inv,\n pos_div_neg, neg_eq_inv_inf_one, inv_inv]"
}
] |
[
300,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
298,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
Ordinal.aleph0_le_cof
|
[
{
"state_after": "case inl\nα : Type ?u.100268\nr : α → α → Prop\n⊢ ℵ₀ ≤ cof 0 ↔ IsLimit 0\n\ncase inr.inl.intro\nα : Type ?u.100268\nr : α → α → Prop\no : Ordinal\n⊢ ℵ₀ ≤ cof (succ o) ↔ IsLimit (succ o)\n\ncase inr.inr\nα : Type ?u.100268\nr : α → α → Prop\no : Ordinal\nl : IsLimit o\n⊢ ℵ₀ ≤ cof o ↔ IsLimit o",
"state_before": "α : Type ?u.100268\nr : α → α → Prop\no : Ordinal\n⊢ ℵ₀ ≤ cof o ↔ IsLimit o",
"tactic": "rcases zero_or_succ_or_limit o with (rfl | ⟨o, rfl⟩ | l)"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type ?u.100268\nr : α → α → Prop\n⊢ ℵ₀ ≤ cof 0 ↔ IsLimit 0",
"tactic": "simp [not_zero_isLimit, Cardinal.aleph0_ne_zero]"
},
{
"state_after": "no goals",
"state_before": "case inr.inl.intro\nα : Type ?u.100268\nr : α → α → Prop\no : Ordinal\n⊢ ℵ₀ ≤ cof (succ o) ↔ IsLimit (succ o)",
"tactic": "simp [not_succ_isLimit, Cardinal.one_lt_aleph0]"
},
{
"state_after": "case inr.inr\nα : Type ?u.100268\nr : α → α → Prop\no : Ordinal\nl : IsLimit o\n⊢ ℵ₀ ≤ cof o",
"state_before": "case inr.inr\nα : Type ?u.100268\nr : α → α → Prop\no : Ordinal\nl : IsLimit o\n⊢ ℵ₀ ≤ cof o ↔ IsLimit o",
"tactic": "simp [l]"
},
{
"state_after": "case inr.inr\nα : Type ?u.100268\nr : α → α → Prop\no : Ordinal\nl : IsLimit o\nh : cof o < ℵ₀\n⊢ False",
"state_before": "case inr.inr\nα : Type ?u.100268\nr : α → α → Prop\no : Ordinal\nl : IsLimit o\n⊢ ℵ₀ ≤ cof o",
"tactic": "refine' le_of_not_lt fun h => _"
},
{
"state_after": "case inr.inr.intro\nα : Type ?u.100268\nr : α → α → Prop\no : Ordinal\nl : IsLimit o\nh : cof o < ℵ₀\nn : ℕ\ne : cof o = ↑n\n⊢ False",
"state_before": "case inr.inr\nα : Type ?u.100268\nr : α → α → Prop\no : Ordinal\nl : IsLimit o\nh : cof o < ℵ₀\n⊢ False",
"tactic": "cases' Cardinal.lt_aleph0.1 h with n e"
},
{
"state_after": "case inr.inr.intro\nα : Type ?u.100268\nr : α → α → Prop\no : Ordinal\nl : IsLimit o\nh : cof o < ℵ₀\nn : ℕ\ne : cof o = ↑n\nthis : cof (ord (cof o)) = cof o\n⊢ False",
"state_before": "case inr.inr.intro\nα : Type ?u.100268\nr : α → α → Prop\no : Ordinal\nl : IsLimit o\nh : cof o < ℵ₀\nn : ℕ\ne : cof o = ↑n\n⊢ False",
"tactic": "have := cof_cof o"
},
{
"state_after": "case inr.inr.intro\nα : Type ?u.100268\nr : α → α → Prop\no : Ordinal\nl : IsLimit o\nh : cof o < ℵ₀\nn : ℕ\ne : cof o = ↑n\nthis : cof ↑n = ↑n\n⊢ False",
"state_before": "case inr.inr.intro\nα : Type ?u.100268\nr : α → α → Prop\no : Ordinal\nl : IsLimit o\nh : cof o < ℵ₀\nn : ℕ\ne : cof o = ↑n\nthis : cof (ord (cof o)) = cof o\n⊢ False",
"tactic": "rw [e, ord_nat] at this"
},
{
"state_after": "case inr.inr.intro.zero\nα : Type ?u.100268\nr : α → α → Prop\no : Ordinal\nl : IsLimit o\nh : cof o < ℵ₀\ne : cof o = ↑Nat.zero\nthis : cof ↑Nat.zero = ↑Nat.zero\n⊢ False\n\ncase inr.inr.intro.succ\nα : Type ?u.100268\nr : α → α → Prop\no : Ordinal\nl : IsLimit o\nh : cof o < ℵ₀\nn✝ : ℕ\ne : cof o = ↑(Nat.succ n✝)\nthis : cof ↑(Nat.succ n✝) = ↑(Nat.succ n✝)\n⊢ False",
"state_before": "case inr.inr.intro\nα : Type ?u.100268\nr : α → α → Prop\no : Ordinal\nl : IsLimit o\nh : cof o < ℵ₀\nn : ℕ\ne : cof o = ↑n\nthis : cof ↑n = ↑n\n⊢ False",
"tactic": "cases n"
},
{
"state_after": "case inr.inr.intro.zero\nα : Type ?u.100268\nr : α → α → Prop\no : Ordinal\nl : IsLimit o\nh : cof o < ℵ₀\nthis : cof ↑Nat.zero = ↑Nat.zero\ne : o = 0\n⊢ False",
"state_before": "case inr.inr.intro.zero\nα : Type ?u.100268\nr : α → α → Prop\no : Ordinal\nl : IsLimit o\nh : cof o < ℵ₀\ne : cof o = ↑Nat.zero\nthis : cof ↑Nat.zero = ↑Nat.zero\n⊢ False",
"tactic": "simp at e"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.intro.zero\nα : Type ?u.100268\nr : α → α → Prop\no : Ordinal\nl : IsLimit o\nh : cof o < ℵ₀\nthis : cof ↑Nat.zero = ↑Nat.zero\ne : o = 0\n⊢ False",
"tactic": "simp [e, not_zero_isLimit] at l"
},
{
"state_after": "case inr.inr.intro.succ\nα : Type ?u.100268\nr : α → α → Prop\no : Ordinal\nl : IsLimit o\nh : cof o < ℵ₀\nn✝ : ℕ\ne : cof o = ↑(Nat.succ n✝)\nthis : 1 = ↑(Nat.succ n✝)\n⊢ False",
"state_before": "case inr.inr.intro.succ\nα : Type ?u.100268\nr : α → α → Prop\no : Ordinal\nl : IsLimit o\nh : cof o < ℵ₀\nn✝ : ℕ\ne : cof o = ↑(Nat.succ n✝)\nthis : cof ↑(Nat.succ n✝) = ↑(Nat.succ n✝)\n⊢ False",
"tactic": "rw [nat_cast_succ, cof_succ] at this"
},
{
"state_after": "case inr.inr.intro.succ\nα : Type ?u.100268\nr : α → α → Prop\no : Ordinal\nl : IsLimit o\nh : cof o < ℵ₀\nn✝ : ℕ\ne : ∃ a, o = succ a\nthis : 1 = ↑(Nat.succ n✝)\n⊢ False",
"state_before": "case inr.inr.intro.succ\nα : Type ?u.100268\nr : α → α → Prop\no : Ordinal\nl : IsLimit o\nh : cof o < ℵ₀\nn✝ : ℕ\ne : cof o = ↑(Nat.succ n✝)\nthis : 1 = ↑(Nat.succ n✝)\n⊢ False",
"tactic": "rw [← this, cof_eq_one_iff_is_succ] at e"
},
{
"state_after": "case inr.inr.intro.succ.intro\nα : Type ?u.100268\nr : α → α → Prop\nn✝ : ℕ\nthis : 1 = ↑(Nat.succ n✝)\na : Ordinal\nl : IsLimit (succ a)\nh : cof (succ a) < ℵ₀\n⊢ False",
"state_before": "case inr.inr.intro.succ\nα : Type ?u.100268\nr : α → α → Prop\no : Ordinal\nl : IsLimit o\nh : cof o < ℵ₀\nn✝ : ℕ\ne : ∃ a, o = succ a\nthis : 1 = ↑(Nat.succ n✝)\n⊢ False",
"tactic": "rcases e with ⟨a, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.intro.succ.intro\nα : Type ?u.100268\nr : α → α → Prop\nn✝ : ℕ\nthis : 1 = ↑(Nat.succ n✝)\na : Ordinal\nl : IsLimit (succ a)\nh : cof (succ a) < ℵ₀\n⊢ False",
"tactic": "exact not_succ_isLimit _ l"
}
] |
[
721,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
706,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.meas_snormEssSup_lt
|
[] |
[
895,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
894,
1
] |
Mathlib/Analysis/NormedSpace/lpSpace.lean
|
lp.norm_zero
|
[
{
"state_after": "case inl\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\n⊢ ‖0‖ = 0\n\ncase inr.inl\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\n⊢ ‖0‖ = 0\n\ncase inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\n⊢ ‖0‖ = 0",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\n⊢ ‖0‖ = 0",
"tactic": "rcases p.trichotomy with (rfl | rfl | hp)"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\n⊢ ‖0‖ = 0",
"tactic": "simp [lp.norm_eq_card_dsupport]"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nα : Type u_1\nE : α → Type u_2\nq : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\n⊢ ‖0‖ = 0",
"tactic": "simp [lp.norm_eq_ciSup]"
},
{
"state_after": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\n⊢ (∑' (i : α), ‖↑0 i‖ ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) = 0",
"state_before": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\n⊢ ‖0‖ = 0",
"tactic": "rw [lp.norm_eq_tsum_rpow hp]"
},
{
"state_after": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\nhp' : 1 / ENNReal.toReal p ≠ 0\n⊢ (∑' (i : α), ‖↑0 i‖ ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) = 0",
"state_before": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\n⊢ (∑' (i : α), ‖↑0 i‖ ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) = 0",
"tactic": "have hp' : 1 / p.toReal ≠ 0 := one_div_ne_zero hp.ne'"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nα : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\nhp' : 1 / ENNReal.toReal p ≠ 0\n⊢ (∑' (i : α), ‖↑0 i‖ ^ ENNReal.toReal p) ^ (1 / ENNReal.toReal p) = 0",
"tactic": "simpa [Real.zero_rpow hp.ne'] using Real.zero_rpow hp'"
}
] |
[
453,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
447,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.cos_three_mul
|
[
{
"state_after": "x y : ℂ\nh1 : x + 2 * x = 3 * x\n⊢ cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x",
"state_before": "x y : ℂ\n⊢ cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x",
"tactic": "have h1 : x + 2 * x = 3 * x := by ring"
},
{
"state_after": "x y : ℂ\nh1 : x + 2 * x = 3 * x\n⊢ cos x * cos (2 * x) - sin x * sin (2 * x) = 4 * cos x ^ 3 - 3 * cos x",
"state_before": "x y : ℂ\nh1 : x + 2 * x = 3 * x\n⊢ cos (3 * x) = 4 * cos x ^ 3 - 3 * cos x",
"tactic": "rw [← h1, cos_add x (2 * x)]"
},
{
"state_after": "x y : ℂ\nh1 : x + 2 * x = 3 * x\n⊢ cos x * (2 * (cos x * cos x)) - cos x - sin x * (2 * sin x * cos x) = 4 * cos x ^ 3 - 3 * cos x",
"state_before": "x y : ℂ\nh1 : x + 2 * x = 3 * x\n⊢ cos x * cos (2 * x) - sin x * sin (2 * x) = 4 * cos x ^ 3 - 3 * cos x",
"tactic": "simp only [cos_two_mul, sin_two_mul, mul_add, mul_sub, mul_one, sq]"
},
{
"state_after": "x y : ℂ\nh1 : x + 2 * x = 3 * x\nh2 : 4 * cos x ^ 3 = 2 * cos x * cos x * cos x + 2 * cos x * cos x ^ 2\n⊢ cos x * (2 * (cos x * cos x)) - cos x - sin x * (2 * sin x * cos x) = 4 * cos x ^ 3 - 3 * cos x",
"state_before": "x y : ℂ\nh1 : x + 2 * x = 3 * x\n⊢ cos x * (2 * (cos x * cos x)) - cos x - sin x * (2 * sin x * cos x) = 4 * cos x ^ 3 - 3 * cos x",
"tactic": "have h2 : 4 * cos x ^ 3 = 2 * cos x * cos x * cos x + 2 * cos x * cos x ^ 2 := by ring"
},
{
"state_after": "x y : ℂ\nh1 : x + 2 * x = 3 * x\nh2 : 4 * cos x ^ 3 = 2 * cos x * cos x * cos x + 2 * cos x * cos x ^ 2\n⊢ cos x * (2 * (cos x * cos x)) - cos x - sin x * (2 * sin x * cos x) =\n 2 * cos x * cos x * cos x + 2 * cos x * (1 - sin x ^ 2) - 3 * cos x",
"state_before": "x y : ℂ\nh1 : x + 2 * x = 3 * x\nh2 : 4 * cos x ^ 3 = 2 * cos x * cos x * cos x + 2 * cos x * cos x ^ 2\n⊢ cos x * (2 * (cos x * cos x)) - cos x - sin x * (2 * sin x * cos x) = 4 * cos x ^ 3 - 3 * cos x",
"tactic": "rw [h2, cos_sq']"
},
{
"state_after": "no goals",
"state_before": "x y : ℂ\nh1 : x + 2 * x = 3 * x\nh2 : 4 * cos x ^ 3 = 2 * cos x * cos x * cos x + 2 * cos x * cos x ^ 2\n⊢ cos x * (2 * (cos x * cos x)) - cos x - sin x * (2 * sin x * cos x) =\n 2 * cos x * cos x * cos x + 2 * cos x * (1 - sin x ^ 2) - 3 * cos x",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "x y : ℂ\n⊢ x + 2 * x = 3 * x",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "x y : ℂ\nh1 : x + 2 * x = 3 * x\n⊢ 4 * cos x ^ 3 = 2 * cos x * cos x * cos x + 2 * cos x * cos x ^ 2",
"tactic": "ring"
}
] |
[
1068,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1062,
1
] |
Mathlib/Data/Prod/TProd.lean
|
List.TProd.elim_of_ne
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u\nα : ι → Type v\ni j : ι\nl : List ι\nf : (i : ι) → α i\ninst✝ : DecidableEq ι\nhj : j ∈ i :: l\nhji : j ≠ i\nv : TProd α (i :: l)\n⊢ TProd.elim v hj = TProd.elim v.snd (_ : j ∈ l)",
"tactic": "simp [TProd.elim, hji]"
}
] |
[
98,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.iInter_inter_distrib
|
[] |
[
536,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
534,
1
] |
Mathlib/Data/PNat/Defs.lean
|
PNat.coe_toPNat'
|
[] |
[
173,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
1
] |
Mathlib/Analysis/Normed/MulAction.lean
|
nndist_smul_le
|
[] |
[
46,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
45,
1
] |
Mathlib/Algebra/MonoidAlgebra/Basic.lean
|
MonoidAlgebra.coe_algebraMap
|
[] |
[
825,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
823,
1
] |
Mathlib/Analysis/NormedSpace/Multilinear.lean
|
ContinuousMultilinearMap.norm_mkPiAlgebraFin_zero
|
[
{
"state_after": "case refine'_1\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁶ : Fintype ι\ninst✝¹⁵ : Fintype ι'\ninst✝¹⁴ : NontriviallyNormedField 𝕜\ninst✝¹³ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹² : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁹ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nA : Type u_1\ninst✝¹ : NormedRing A\ninst✝ : NormedAlgebra 𝕜 A\n⊢ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A‖ ≤ ‖1‖\n\ncase refine'_2\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁶ : Fintype ι\ninst✝¹⁵ : Fintype ι'\ninst✝¹⁴ : NontriviallyNormedField 𝕜\ninst✝¹³ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹² : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁹ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nA : Type u_1\ninst✝¹ : NormedRing A\ninst✝ : NormedAlgebra 𝕜 A\n⊢ ‖1‖ ≤ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A‖",
"state_before": "𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁶ : Fintype ι\ninst✝¹⁵ : Fintype ι'\ninst✝¹⁴ : NontriviallyNormedField 𝕜\ninst✝¹³ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹² : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁹ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nA : Type u_1\ninst✝¹ : NormedRing A\ninst✝ : NormedAlgebra 𝕜 A\n⊢ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A‖ = ‖1‖",
"tactic": "refine' le_antisymm _ _"
},
{
"state_after": "case refine'_1\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁶ : Fintype ι\ninst✝¹⁵ : Fintype ι'\ninst✝¹⁴ : NontriviallyNormedField 𝕜\ninst✝¹³ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹² : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁹ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nA : Type u_1\ninst✝¹ : NormedRing A\ninst✝ : NormedAlgebra 𝕜 A\nthis :\n ∀ (f : ContinuousMultilinearMap 𝕜 (fun x => A) A), (∀ (m : Fin 0 → A), ‖↑f m‖ ≤ ‖1‖ * ∏ i : Fin 0, ‖m i‖) → ‖f‖ ≤ ‖1‖\n⊢ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A‖ ≤ ‖1‖",
"state_before": "case refine'_1\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁶ : Fintype ι\ninst✝¹⁵ : Fintype ι'\ninst✝¹⁴ : NontriviallyNormedField 𝕜\ninst✝¹³ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹² : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁹ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nA : Type u_1\ninst✝¹ : NormedRing A\ninst✝ : NormedAlgebra 𝕜 A\n⊢ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A‖ ≤ ‖1‖",
"tactic": "have := fun f =>\n @op_norm_le_bound 𝕜 (Fin 0) (fun _ => A) A _ _ _ _ _ _ f _ (norm_nonneg (1 : A))"
},
{
"state_after": "case refine'_1\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁶ : Fintype ι\ninst✝¹⁵ : Fintype ι'\ninst✝¹⁴ : NontriviallyNormedField 𝕜\ninst✝¹³ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹² : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁹ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nA : Type u_1\ninst✝¹ : NormedRing A\ninst✝ : NormedAlgebra 𝕜 A\nthis :\n ∀ (f : ContinuousMultilinearMap 𝕜 (fun x => A) A), (∀ (m : Fin 0 → A), ‖↑f m‖ ≤ ‖1‖ * ∏ i : Fin 0, ‖m i‖) → ‖f‖ ≤ ‖1‖\n⊢ ∀ (m : Fin 0 → A), ‖↑(ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A) m‖ ≤ ‖1‖ * ∏ i : Fin 0, ‖m i‖",
"state_before": "case refine'_1\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁶ : Fintype ι\ninst✝¹⁵ : Fintype ι'\ninst✝¹⁴ : NontriviallyNormedField 𝕜\ninst✝¹³ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹² : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁹ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nA : Type u_1\ninst✝¹ : NormedRing A\ninst✝ : NormedAlgebra 𝕜 A\nthis :\n ∀ (f : ContinuousMultilinearMap 𝕜 (fun x => A) A), (∀ (m : Fin 0 → A), ‖↑f m‖ ≤ ‖1‖ * ∏ i : Fin 0, ‖m i‖) → ‖f‖ ≤ ‖1‖\n⊢ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A‖ ≤ ‖1‖",
"tactic": "refine' this _ _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁶ : Fintype ι\ninst✝¹⁵ : Fintype ι'\ninst✝¹⁴ : NontriviallyNormedField 𝕜\ninst✝¹³ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹² : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁹ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nA : Type u_1\ninst✝¹ : NormedRing A\ninst✝ : NormedAlgebra 𝕜 A\nthis :\n ∀ (f : ContinuousMultilinearMap 𝕜 (fun x => A) A), (∀ (m : Fin 0 → A), ‖↑f m‖ ≤ ‖1‖ * ∏ i : Fin 0, ‖m i‖) → ‖f‖ ≤ ‖1‖\n⊢ ∀ (m : Fin 0 → A), ‖↑(ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A) m‖ ≤ ‖1‖ * ∏ i : Fin 0, ‖m i‖",
"tactic": "simp"
},
{
"state_after": "case h.e'_3\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁶ : Fintype ι\ninst✝¹⁵ : Fintype ι'\ninst✝¹⁴ : NontriviallyNormedField 𝕜\ninst✝¹³ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹² : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁹ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nA : Type u_1\ninst✝¹ : NormedRing A\ninst✝ : NormedAlgebra 𝕜 A\n⊢ ‖1‖ = ‖↑(ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A) fun x => 1‖ / ∏ i : Fin 0, ‖1‖",
"state_before": "case refine'_2\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁶ : Fintype ι\ninst✝¹⁵ : Fintype ι'\ninst✝¹⁴ : NontriviallyNormedField 𝕜\ninst✝¹³ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹² : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁹ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nA : Type u_1\ninst✝¹ : NormedRing A\ninst✝ : NormedAlgebra 𝕜 A\n⊢ ‖1‖ ≤ ‖ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A‖",
"tactic": "convert ratio_le_op_norm (ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A) fun _ => (1 : A)"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁶ : Fintype ι\ninst✝¹⁵ : Fintype ι'\ninst✝¹⁴ : NontriviallyNormedField 𝕜\ninst✝¹³ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹² : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹¹ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹⁰ : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝⁹ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝⁸ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁷ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁶ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝⁵ : NormedAddCommGroup G\ninst✝⁴ : NormedSpace 𝕜 G\ninst✝³ : NormedAddCommGroup G'\ninst✝² : NormedSpace 𝕜 G'\nA : Type u_1\ninst✝¹ : NormedRing A\ninst✝ : NormedAlgebra 𝕜 A\n⊢ ‖1‖ = ‖↑(ContinuousMultilinearMap.mkPiAlgebraFin 𝕜 0 A) fun x => 1‖ / ∏ i : Fin 0, ‖1‖",
"tactic": "simp"
}
] |
[
865,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
858,
1
] |
Mathlib/Algebra/Divisibility/Basic.lean
|
pow_dvd_pow_of_dvd
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : CommMonoid α\na✝ b✝ a b : α\nh : a ∣ b\n⊢ a ^ 0 ∣ b ^ 0",
"tactic": "rw [pow_zero, pow_zero]"
},
{
"state_after": "α : Type u_1\ninst✝ : CommMonoid α\na✝ b✝ a b : α\nh : a ∣ b\nn : ℕ\n⊢ a * a ^ n ∣ b * b ^ n",
"state_before": "α : Type u_1\ninst✝ : CommMonoid α\na✝ b✝ a b : α\nh : a ∣ b\nn : ℕ\n⊢ a ^ (n + 1) ∣ b ^ (n + 1)",
"tactic": "rw [pow_succ, pow_succ]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : CommMonoid α\na✝ b✝ a b : α\nh : a ∣ b\nn : ℕ\n⊢ a * a ^ n ∣ b * b ^ n",
"tactic": "exact mul_dvd_mul h (pow_dvd_pow_of_dvd h n)"
}
] |
[
205,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
201,
1
] |
Mathlib/Algebra/AlgebraicCard.lean
|
Algebraic.cardinal_mk_lift_le_max
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : IsDomain A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroSMulDivisors R A\n⊢ lift (max (#R) ℵ₀) * ℵ₀ ≤ max (lift (#R)) ℵ₀",
"tactic": "simp"
}
] |
[
63,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/CategoryTheory/Limits/Types.lean
|
CategoryTheory.Limits.Types.limit_ext
|
[
{
"state_after": "case a\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ TypeMax\nx y : limit F\nw : ∀ (j : J), limit.π F j x = limit.π F j y\n⊢ ↑(limitEquivSections F) x = ↑(limitEquivSections F) y",
"state_before": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ TypeMax\nx y : limit F\nw : ∀ (j : J), limit.π F j x = limit.π F j y\n⊢ x = y",
"tactic": "apply (limitEquivSections.{v, u} F).injective"
},
{
"state_after": "case a.a.h\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ TypeMax\nx y : limit F\nw : ∀ (j : J), limit.π F j x = limit.π F j y\nj : J\n⊢ ↑(↑(limitEquivSections F) x) j = ↑(↑(limitEquivSections F) y) j",
"state_before": "case a\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ TypeMax\nx y : limit F\nw : ∀ (j : J), limit.π F j x = limit.π F j y\n⊢ ↑(limitEquivSections F) x = ↑(limitEquivSections F) y",
"tactic": "ext j"
},
{
"state_after": "no goals",
"state_before": "case a.a.h\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ TypeMax\nx y : limit F\nw : ∀ (j : J), limit.π F j x = limit.π F j y\nj : J\n⊢ ↑(↑(limitEquivSections F) x) j = ↑(↑(limitEquivSections F) y) j",
"tactic": "simp [w j]"
}
] |
[
165,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Std/Data/RBMap/Lemmas.lean
|
Std.RBNode.Any_def
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\np : α → Prop\nt : RBNode α\n⊢ Any p t ↔ ∃ x, x ∈ t ∧ p x",
"tactic": "induction t <;> simp [or_and_right, exists_or, *]"
}
] |
[
104,
52
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
103,
1
] |
Mathlib/Algebra/GCDMonoid/Basic.lean
|
gcd_eq_of_dvd_sub_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : NormalizedGCDMonoid α\na b c : α\nh : a ∣ b - c\n⊢ gcd b a = gcd c a",
"tactic": "rw [gcd_comm _ a, gcd_comm _ a, gcd_eq_of_dvd_sub_right h]"
}
] |
[
998,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
997,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
|
ContDiff.rpow_const_of_ne
|
[] |
[
536,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
534,
1
] |
Mathlib/Logic/Equiv/Set.lean
|
Equiv.Set.sumCompl_symm_apply_of_mem
|
[
{
"state_after": "α✝ : Sort u\nβ : Sort v\nγ : Sort w\nα : Type u\ns : Set α\ninst✝ : DecidablePred fun x => x ∈ s\nx : α\nhx : x ∈ s\nthis : ↑{ val := x, property := (_ : x ∈ s ∨ x ∈ sᶜ) } ∈ s\n⊢ ↑(Set.sumCompl s).symm x = Sum.inl { val := x, property := hx }",
"state_before": "α✝ : Sort u\nβ : Sort v\nγ : Sort w\nα : Type u\ns : Set α\ninst✝ : DecidablePred fun x => x ∈ s\nx : α\nhx : x ∈ s\n⊢ ↑(Set.sumCompl s).symm x = Sum.inl { val := x, property := hx }",
"tactic": "have : ((⟨x, Or.inl hx⟩ : (s ∪ sᶜ : Set α)) : α) ∈ s := hx"
},
{
"state_after": "α✝ : Sort u\nβ : Sort v\nγ : Sort w\nα : Type u\ns : Set α\ninst✝ : DecidablePred fun x => x ∈ s\nx : α\nhx : x ∈ s\nthis : ↑{ val := x, property := (_ : x ∈ s ∨ x ∈ sᶜ) } ∈ s\n⊢ ↑(Trans.trans (Trans.trans (Set.union (_ : s ∩ sᶜ ⊆ ∅)).symm (Set.ofEq (_ : s ∪ sᶜ = univ))) (Set.univ α)).symm x =\n Sum.inl { val := x, property := hx }",
"state_before": "α✝ : Sort u\nβ : Sort v\nγ : Sort w\nα : Type u\ns : Set α\ninst✝ : DecidablePred fun x => x ∈ s\nx : α\nhx : x ∈ s\nthis : ↑{ val := x, property := (_ : x ∈ s ∨ x ∈ sᶜ) } ∈ s\n⊢ ↑(Set.sumCompl s).symm x = Sum.inl { val := x, property := hx }",
"tactic": "rw [Equiv.Set.sumCompl]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Sort u\nβ : Sort v\nγ : Sort w\nα : Type u\ns : Set α\ninst✝ : DecidablePred fun x => x ∈ s\nx : α\nhx : x ∈ s\nthis : ↑{ val := x, property := (_ : x ∈ s ∨ x ∈ sᶜ) } ∈ s\n⊢ ↑(Trans.trans (Trans.trans (Set.union (_ : s ∩ sᶜ ⊆ ∅)).symm (Set.ofEq (_ : s ∪ sᶜ = univ))) (Set.univ α)).symm x =\n Sum.inl { val := x, property := hx }",
"tactic": "simpa using Set.union_apply_left (by simp) this"
},
{
"state_after": "no goals",
"state_before": "α✝ : Sort u\nβ : Sort v\nγ : Sort w\nα : Type u\ns : Set α\ninst✝ : DecidablePred fun x => x ∈ s\nx : α\nhx : x ∈ s\nthis : ↑{ val := x, property := (_ : x ∈ s ∨ x ∈ sᶜ) } ∈ s\n⊢ s ∩ sᶜ ⊆ ∅",
"tactic": "simp"
}
] |
[
335,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
331,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.sinh_sq
|
[
{
"state_after": "x y : ℂ\n⊢ sinh x ^ 2 = cosh x ^ 2 - (cosh x ^ 2 - sinh x ^ 2)",
"state_before": "x y : ℂ\n⊢ sinh x ^ 2 = cosh x ^ 2 - 1",
"tactic": "rw [← cosh_sq_sub_sinh_sq x]"
},
{
"state_after": "no goals",
"state_before": "x y : ℂ\n⊢ sinh x ^ 2 = cosh x ^ 2 - (cosh x ^ 2 - sinh x ^ 2)",
"tactic": "ring"
}
] |
[
766,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
764,
1
] |
Mathlib/Data/Fintype/BigOperators.lean
|
Fintype.prod_sum_type
|
[] |
[
297,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
295,
1
] |
Mathlib/Combinatorics/Additive/SalemSpencer.lean
|
mulRothNumber_spec
|
[
{
"state_after": "F : Type ?u.131282\nα : Type u_1\nβ : Type ?u.131288\n𝕜 : Type ?u.131291\nE : Type ?u.131294\ninst✝³ : DecidableEq α\ninst✝² : Monoid α\ninst✝¹ : DecidableEq β\ninst✝ : Monoid β\ns t : Finset α\n⊢ MulSalemSpencer ∅",
"state_before": "F : Type ?u.131282\nα : Type u_1\nβ : Type ?u.131288\n𝕜 : Type ?u.131291\nE : Type ?u.131294\ninst✝³ : DecidableEq α\ninst✝² : Monoid α\ninst✝¹ : DecidableEq β\ninst✝ : Monoid β\ns t : Finset α\n⊢ MulSalemSpencer ↑∅",
"tactic": "norm_cast"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.131282\nα : Type u_1\nβ : Type ?u.131288\n𝕜 : Type ?u.131291\nE : Type ?u.131294\ninst✝³ : DecidableEq α\ninst✝² : Monoid α\ninst✝¹ : DecidableEq β\ninst✝ : Monoid β\ns t : Finset α\n⊢ MulSalemSpencer ∅",
"tactic": "exact mulSalemSpencer_empty"
}
] |
[
346,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
342,
1
] |
Std/Classes/Order.lean
|
Std.TransCmp.gt_asymm
|
[] |
[
62,
46
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
61,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.map_id'
|
[] |
[
1862,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1861,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.mul_lt_top
|
[] |
[
584,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
584,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
ContDiffOn.sin
|
[] |
[
1044,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1043,
1
] |
Mathlib/Algebra/Quaternion.lean
|
QuaternionAlgebra.zero_im
|
[] |
[
182,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
182,
9
] |
Mathlib/Computability/Primrec.lean
|
Nat.Primrec.const
|
[] |
[
99,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Order/Grade.lean
|
IsMin.grade
|
[] |
[
133,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
11
] |
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
|
Real.rpow_neg
|
[
{
"state_after": "x✝ y✝ z x : ℝ\nhx : 0 ≤ x\ny : ℝ\n⊢ (if x = 0 then if -y = 0 then 1 else 0 else exp (log x * -y)) =\n (if x = 0 then if y = 0 then 1 else 0 else exp (log x * y))⁻¹",
"state_before": "x✝ y✝ z x : ℝ\nhx : 0 ≤ x\ny : ℝ\n⊢ x ^ (-y) = (x ^ y)⁻¹",
"tactic": "simp only [rpow_def_of_nonneg hx]"
},
{
"state_after": "no goals",
"state_before": "x✝ y✝ z x : ℝ\nhx : 0 ≤ x\ny : ℝ\n⊢ (if x = 0 then if -y = 0 then 1 else 0 else exp (log x * -y)) =\n (if x = 0 then if y = 0 then 1 else 0 else exp (log x * y))⁻¹",
"tactic": "split_ifs <;> simp_all [exp_neg]"
}
] |
[
225,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
224,
1
] |
Mathlib/NumberTheory/Multiplicity.lean
|
Int.two_pow_sub_pow
|
[
{
"state_after": "R : Type ?u.865616\nn✝ : ℕ\nx y : ℤ\nn : ℕ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhn : Even n\nhy : Odd y\n⊢ multiplicity 2 (x ^ n - y ^ n) + 1 = multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 ↑n",
"state_before": "R : Type ?u.865616\nn✝ : ℕ\nx y : ℤ\nn : ℕ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhn : Even n\n⊢ multiplicity 2 (x ^ n - y ^ n) + 1 = multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 ↑n",
"tactic": "have hy : Odd y := by\n rw [← even_iff_two_dvd, ← Int.odd_iff_not_even] at hx\n replace hxy := (@even_neg _ _ (x - y)).mpr (even_iff_two_dvd.mpr hxy)\n convert Even.add_odd hxy hx\n abel"
},
{
"state_after": "case intro\nR : Type ?u.865616\nn✝ : ℕ\nx y : ℤ\nn : ℕ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhd : n = d + d\n⊢ multiplicity 2 (x ^ n - y ^ n) + 1 = multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 ↑n",
"state_before": "R : Type ?u.865616\nn✝ : ℕ\nx y : ℤ\nn : ℕ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhn : Even n\nhy : Odd y\n⊢ multiplicity 2 (x ^ n - y ^ n) + 1 = multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 ↑n",
"tactic": "cases' hn with d hd"
},
{
"state_after": "case intro\nR : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\n⊢ multiplicity 2 (x ^ (d + d) - y ^ (d + d)) + 1 =\n multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 ↑(d + d)",
"state_before": "case intro\nR : Type ?u.865616\nn✝ : ℕ\nx y : ℤ\nn : ℕ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhd : n = d + d\n⊢ multiplicity 2 (x ^ n - y ^ n) + 1 = multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 ↑n",
"tactic": "subst hd"
},
{
"state_after": "case intro\nR : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\n⊢ multiplicity 2 ((x ^ 2) ^ d - (y ^ 2) ^ d) + 1 =\n multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 ↑(2 * d)",
"state_before": "case intro\nR : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\n⊢ multiplicity 2 (x ^ (d + d) - y ^ (d + d)) + 1 =\n multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 ↑(d + d)",
"tactic": "simp only [← two_mul, pow_mul]"
},
{
"state_after": "case intro\nR : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhxy4 : 4 ∣ x ^ 2 - y ^ 2\n⊢ multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 ↑d + 1 =\n multiplicity 2 (x + y) + multiplicity 2 (x - y) + (multiplicity 2 ↑2 + multiplicity 2 ↑d)\n\nR : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhxy4 : 4 ∣ x ^ 2 - y ^ 2\n⊢ ¬2 ∣ x ^ 2",
"state_before": "case intro\nR : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhxy4 : 4 ∣ x ^ 2 - y ^ 2\n⊢ multiplicity 2 ((x ^ 2) ^ d - (y ^ 2) ^ d) + 1 =\n multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 ↑(2 * d)",
"tactic": "rw [Int.two_pow_sub_pow' d hxy4 _, sq_sub_sq, ← Int.ofNat_mul_out, multiplicity.mul Int.prime_two,\n multiplicity.mul Int.prime_two]"
},
{
"state_after": "case intro\nR : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhxy4 : 4 ∣ x ^ 2 - y ^ 2\n⊢ multiplicity 2 ↑2 = 1\n\nR : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhxy4 : 4 ∣ x ^ 2 - y ^ 2\n⊢ ¬2 ∣ x ^ 2",
"state_before": "case intro\nR : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhxy4 : 4 ∣ x ^ 2 - y ^ 2\n⊢ multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 ↑d + 1 =\n multiplicity 2 (x + y) + multiplicity 2 (x - y) + (multiplicity 2 ↑2 + multiplicity 2 ↑d)\n\nR : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhxy4 : 4 ∣ x ^ 2 - y ^ 2\n⊢ ¬2 ∣ x ^ 2",
"tactic": "suffices multiplicity (2 : ℤ) ↑(2 : ℕ) = 1 by rw [this, add_comm (1 : PartENat), ← add_assoc]"
},
{
"state_after": "R : Type ?u.865616\nn✝ : ℕ\nx y : ℤ\nn : ℕ\nhxy : 2 ∣ x - y\nhx : Odd x\nhn : Even n\n⊢ Odd y",
"state_before": "R : Type ?u.865616\nn✝ : ℕ\nx y : ℤ\nn : ℕ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhn : Even n\n⊢ Odd y",
"tactic": "rw [← even_iff_two_dvd, ← Int.odd_iff_not_even] at hx"
},
{
"state_after": "R : Type ?u.865616\nn✝ : ℕ\nx y : ℤ\nn : ℕ\nhx : Odd x\nhn : Even n\nhxy : Even (-(x - y))\n⊢ Odd y",
"state_before": "R : Type ?u.865616\nn✝ : ℕ\nx y : ℤ\nn : ℕ\nhxy : 2 ∣ x - y\nhx : Odd x\nhn : Even n\n⊢ Odd y",
"tactic": "replace hxy := (@even_neg _ _ (x - y)).mpr (even_iff_two_dvd.mpr hxy)"
},
{
"state_after": "case h.e'_3\nR : Type ?u.865616\nn✝ : ℕ\nx y : ℤ\nn : ℕ\nhx : Odd x\nhn : Even n\nhxy : Even (-(x - y))\n⊢ y = -(x - y) + x",
"state_before": "R : Type ?u.865616\nn✝ : ℕ\nx y : ℤ\nn : ℕ\nhx : Odd x\nhn : Even n\nhxy : Even (-(x - y))\n⊢ Odd y",
"tactic": "convert Even.add_odd hxy hx"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nR : Type ?u.865616\nn✝ : ℕ\nx y : ℤ\nn : ℕ\nhx : Odd x\nhn : Even n\nhxy : Even (-(x - y))\n⊢ y = -(x - y) + x",
"tactic": "abel"
},
{
"state_after": "R : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\n⊢ (1 - 1) % 4 = 0\n\nR : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\n⊢ Odd x",
"state_before": "R : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\n⊢ 4 ∣ x ^ 2 - y ^ 2",
"tactic": "rw [Int.dvd_iff_emod_eq_zero, Int.sub_emod, Int.sq_mod_four_eq_one_of_odd _,\n Int.sq_mod_four_eq_one_of_odd hy]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\n⊢ (1 - 1) % 4 = 0",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\n⊢ Odd x",
"tactic": "simp only [Int.odd_iff_not_even, even_iff_two_dvd, hx, not_false_iff]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhxy4 : 4 ∣ x ^ 2 - y ^ 2\nthis : multiplicity 2 ↑2 = 1\n⊢ multiplicity 2 (x + y) + multiplicity 2 (x - y) + multiplicity 2 ↑d + 1 =\n multiplicity 2 (x + y) + multiplicity 2 (x - y) + (multiplicity 2 ↑2 + multiplicity 2 ↑d)",
"tactic": "rw [this, add_comm (1 : PartENat), ← add_assoc]"
},
{
"state_after": "case intro\nR : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhxy4 : 4 ∣ x ^ 2 - y ^ 2\n⊢ multiplicity 2 2 = 1",
"state_before": "case intro\nR : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhxy4 : 4 ∣ x ^ 2 - y ^ 2\n⊢ multiplicity 2 ↑2 = 1",
"tactic": "norm_cast"
},
{
"state_after": "R : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhxy4 : 4 ∣ x ^ 2 - y ^ 2\n⊢ ¬IsUnit 2\n\nR : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhxy4 : 4 ∣ x ^ 2 - y ^ 2\n⊢ 2 ≠ 0",
"state_before": "case intro\nR : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhxy4 : 4 ∣ x ^ 2 - y ^ 2\n⊢ multiplicity 2 2 = 1",
"tactic": "rw [multiplicity.multiplicity_self _ _]"
},
{
"state_after": "case hp\nR : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhxy4 : 4 ∣ x ^ 2 - y ^ 2\n⊢ Prime 2",
"state_before": "R : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhxy4 : 4 ∣ x ^ 2 - y ^ 2\n⊢ ¬IsUnit 2",
"tactic": "apply Prime.not_unit"
},
{
"state_after": "no goals",
"state_before": "case hp\nR : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhxy4 : 4 ∣ x ^ 2 - y ^ 2\n⊢ Prime 2",
"tactic": "simp only [← Nat.prime_iff, Nat.prime_two]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhxy4 : 4 ∣ x ^ 2 - y ^ 2\n⊢ 2 ≠ 0",
"tactic": "exact two_ne_zero"
},
{
"state_after": "R : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhxy4 : 4 ∣ x ^ 2 - y ^ 2\n⊢ Odd (x ^ 2)",
"state_before": "R : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhxy4 : 4 ∣ x ^ 2 - y ^ 2\n⊢ ¬2 ∣ x ^ 2",
"tactic": "rw [← even_iff_two_dvd, ← Int.odd_iff_not_even]"
},
{
"state_after": "case hm\nR : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhxy4 : 4 ∣ x ^ 2 - y ^ 2\n⊢ Odd x",
"state_before": "R : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhxy4 : 4 ∣ x ^ 2 - y ^ 2\n⊢ Odd (x ^ 2)",
"tactic": "apply Odd.pow"
},
{
"state_after": "no goals",
"state_before": "case hm\nR : Type ?u.865616\nn : ℕ\nx y : ℤ\nhxy : 2 ∣ x - y\nhx : ¬2 ∣ x\nhy : Odd y\nd : ℕ\nhxy4 : 4 ∣ x ^ 2 - y ^ 2\n⊢ Odd x",
"tactic": "simp only [Int.odd_iff_not_even, even_iff_two_dvd, hx, not_false_iff]"
}
] |
[
360,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
334,
1
] |
Mathlib/GroupTheory/Submonoid/Pointwise.lean
|
Set.IsPwo.submonoid_closure
|
[
{
"state_after": "α : Type u_1\nG : Type ?u.364481\nM : Type ?u.364484\nR : Type ?u.364487\nA : Type ?u.364490\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : OrderedCancelCommMonoid α\ns : Set α\nhpos : ∀ (x : α), x ∈ s → 1 ≤ x\nh : IsPwo s\n⊢ IsPwo (List.prod '' {l | ∀ (x : α), x ∈ l → x ∈ s})",
"state_before": "α : Type u_1\nG : Type ?u.364481\nM : Type ?u.364484\nR : Type ?u.364487\nA : Type ?u.364490\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : OrderedCancelCommMonoid α\ns : Set α\nhpos : ∀ (x : α), x ∈ s → 1 ≤ x\nh : IsPwo s\n⊢ IsPwo ↑(Submonoid.closure s)",
"tactic": "rw [Submonoid.closure_eq_image_prod]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nG : Type ?u.364481\nM : Type ?u.364484\nR : Type ?u.364487\nA : Type ?u.364490\ninst✝² : Monoid M\ninst✝¹ : AddMonoid A\ninst✝ : OrderedCancelCommMonoid α\ns : Set α\nhpos : ∀ (x : α), x ∈ s → 1 ≤ x\nh : IsPwo s\n⊢ ∀ (a₁ : List α),\n a₁ ∈ {l | ∀ (x : α), x ∈ l → x ∈ s} →\n ∀ (a₂ : List α),\n a₂ ∈ {l | ∀ (x : α), x ∈ l → x ∈ s} →\n List.SublistForall₂ (fun x x_1 => x ≤ x_1) a₁ a₂ → List.prod a₁ ≤ List.prod a₂",
"tactic": "exact fun l1 _ l2 hl2 h12 => h12.prod_le_prod' fun x hx => hpos x <| hl2 x hx"
}
] |
[
708,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
704,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.closedBall_zero
|
[] |
[
2897,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2897,
9
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.ball_subset
|
[
{
"state_after": "α : Type u\nβ : Type v\nX : Type ?u.43836\nι : Type ?u.43839\ninst✝ : PseudoMetricSpace α\nx y z✝ : α\nδ ε ε₁ ε₂ : ℝ\ns : Set α\nh : dist x y ≤ ε₂ - ε₁\nz : α\nzx : z ∈ ball x ε₁\n⊢ z ∈ ball y (ε₁ + (ε₂ - ε₁))",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.43836\nι : Type ?u.43839\ninst✝ : PseudoMetricSpace α\nx y z✝ : α\nδ ε ε₁ ε₂ : ℝ\ns : Set α\nh : dist x y ≤ ε₂ - ε₁\nz : α\nzx : z ∈ ball x ε₁\n⊢ z ∈ ball y ε₂",
"tactic": "rw [← add_sub_cancel'_right ε₁ ε₂]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.43836\nι : Type ?u.43839\ninst✝ : PseudoMetricSpace α\nx y z✝ : α\nδ ε ε₁ ε₂ : ℝ\ns : Set α\nh : dist x y ≤ ε₂ - ε₁\nz : α\nzx : z ∈ ball x ε₁\n⊢ z ∈ ball y (ε₁ + (ε₂ - ε₁))",
"tactic": "exact lt_of_le_of_lt (dist_triangle z x y) (add_lt_add_of_lt_of_le zx h)"
}
] |
[
662,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
660,
1
] |
Mathlib/InformationTheory/Hamming.lean
|
hammingDist_eq_zero
|
[
{
"state_after": "α : Type ?u.9836\nι : Type u_1\nβ : ι → Type u_2\ninst✝² : Fintype ι\ninst✝¹ : (i : ι) → DecidableEq (β i)\nγ : ι → Type ?u.9868\ninst✝ : (i : ι) → DecidableEq (γ i)\nx y : (i : ι) → β i\nH : x = y\n⊢ hammingDist y y = 0",
"state_before": "α : Type ?u.9836\nι : Type u_1\nβ : ι → Type u_2\ninst✝² : Fintype ι\ninst✝¹ : (i : ι) → DecidableEq (β i)\nγ : ι → Type ?u.9868\ninst✝ : (i : ι) → DecidableEq (γ i)\nx y : (i : ι) → β i\nH : x = y\n⊢ hammingDist x y = 0",
"tactic": "rw [H]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.9836\nι : Type u_1\nβ : ι → Type u_2\ninst✝² : Fintype ι\ninst✝¹ : (i : ι) → DecidableEq (β i)\nγ : ι → Type ?u.9868\ninst✝ : (i : ι) → DecidableEq (γ i)\nx y : (i : ι) → β i\nH : x = y\n⊢ hammingDist y y = 0",
"tactic": "exact hammingDist_self _"
}
] |
[
105,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
1
] |
Mathlib/Analysis/Convex/SpecificFunctions/Basic.lean
|
one_add_mul_self_le_rpow_one_add
|
[
{
"state_after": "case inl\ns : ℝ\nhs : -1 ≤ s\nhp : 1 ≤ 1\n⊢ 1 + 1 * s ≤ (1 + s) ^ 1\n\ncase inr\ns : ℝ\nhs : -1 ≤ s\np : ℝ\nhp✝ : 1 ≤ p\nhp : 1 < p\n⊢ 1 + p * s ≤ (1 + s) ^ p",
"state_before": "s : ℝ\nhs : -1 ≤ s\np : ℝ\nhp : 1 ≤ p\n⊢ 1 + p * s ≤ (1 + s) ^ p",
"tactic": "rcases eq_or_lt_of_le hp with (rfl | hp)"
},
{
"state_after": "case pos\ns : ℝ\nhs : -1 ≤ s\np : ℝ\nhp✝ : 1 ≤ p\nhp : 1 < p\nhs' : s = 0\n⊢ 1 + p * s ≤ (1 + s) ^ p\n\ncase neg\ns : ℝ\nhs : -1 ≤ s\np : ℝ\nhp✝ : 1 ≤ p\nhp : 1 < p\nhs' : ¬s = 0\n⊢ 1 + p * s ≤ (1 + s) ^ p",
"state_before": "case inr\ns : ℝ\nhs : -1 ≤ s\np : ℝ\nhp✝ : 1 ≤ p\nhp : 1 < p\n⊢ 1 + p * s ≤ (1 + s) ^ p",
"tactic": "by_cases hs' : s = 0"
},
{
"state_after": "no goals",
"state_before": "case neg\ns : ℝ\nhs : -1 ≤ s\np : ℝ\nhp✝ : 1 ≤ p\nhp : 1 < p\nhs' : ¬s = 0\n⊢ 1 + p * s ≤ (1 + s) ^ p",
"tactic": "exact (one_add_mul_self_lt_rpow_one_add hs hs' hp).le"
},
{
"state_after": "no goals",
"state_before": "case inl\ns : ℝ\nhs : -1 ≤ s\nhp : 1 ≤ 1\n⊢ 1 + 1 * s ≤ (1 + s) ^ 1",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case pos\ns : ℝ\nhs : -1 ≤ s\np : ℝ\nhp✝ : 1 ≤ p\nhp : 1 < p\nhs' : s = 0\n⊢ 1 + p * s ≤ (1 + s) ^ p",
"tactic": "simp [hs']"
}
] |
[
231,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/Algebra/Order/Sub/Canonical.lean
|
tsub_pos_of_lt
|
[] |
[
357,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
356,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.image_inter_nonempty_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.52785\nι : Sort ?u.52788\nι' : Sort ?u.52791\nf✝ : α → β\ns✝ t✝ : Set α\nf : α → β\ns : Set α\nt : Set β\n⊢ Set.Nonempty (f '' s ∩ t) ↔ Set.Nonempty (s ∩ f ⁻¹' t)",
"tactic": "rw [← image_inter_preimage, nonempty_image_iff]"
}
] |
[
522,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
520,
1
] |
Mathlib/Data/Set/MulAntidiagonal.lean
|
Set.swap_mem_mulAntidiagonal_aux
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : CommSemigroup α\ns t : Set α\na : α\nx : α × α\n⊢ x.snd ∈ s ∧ x.fst ∈ t ∧ x.snd * x.fst = a ↔ x ∈ mulAntidiagonal t s a",
"tactic": "simp [mul_comm, and_left_comm]"
}
] |
[
66,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
1
] |
Mathlib/Analysis/InnerProductSpace/l2Space.lean
|
IsHilbertSum.mk
|
[
{
"state_after": "ι : Type u_4\n𝕜 : Type u_2\ninst✝⁵ : IsROrC 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type u_1\ninst✝² : (i : ι) → NormedAddCommGroup (G i)\ninst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nF : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace (G i)\nhVortho : OrthogonalFamily 𝕜 G V\nhVtotal : ⊤ ≤ topologicalClosure (⨆ (i : ι), LinearMap.range (V i).toLinearMap)\n⊢ Function.Surjective ↑(OrthogonalFamily.linearIsometry hVortho).toLinearMap",
"state_before": "ι : Type u_4\n𝕜 : Type u_2\ninst✝⁵ : IsROrC 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type u_1\ninst✝² : (i : ι) → NormedAddCommGroup (G i)\ninst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nF : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace (G i)\nhVortho : OrthogonalFamily 𝕜 G V\nhVtotal : ⊤ ≤ topologicalClosure (⨆ (i : ι), LinearMap.range (V i).toLinearMap)\n⊢ Function.Surjective ↑(OrthogonalFamily.linearIsometry hVortho)",
"tactic": "rw [← LinearIsometry.coe_toLinearMap]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_4\n𝕜 : Type u_2\ninst✝⁵ : IsROrC 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type u_1\ninst✝² : (i : ι) → NormedAddCommGroup (G i)\ninst✝¹ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nF : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace (G i)\nhVortho : OrthogonalFamily 𝕜 G V\nhVtotal : ⊤ ≤ topologicalClosure (⨆ (i : ι), LinearMap.range (V i).toLinearMap)\n⊢ Function.Surjective ↑(OrthogonalFamily.linearIsometry hVortho).toLinearMap",
"tactic": "exact LinearMap.range_eq_top.mp\n (eq_top_iff.mpr <| hVtotal.trans_eq hVortho.range_linearIsometry.symm)"
}
] |
[
313,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
306,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
MeasurableSet.measurableSet_bliminf
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.910929\nγ : Type ?u.910932\nδ : Type ?u.910935\nδ' : Type ?u.910938\nι : Sort uι\ns✝ t u : Set α\ninst✝ : MeasurableSpace α\ns : ℕ → Set α\np : ℕ → Prop\nh : ∀ (n : ℕ), p n → MeasurableSet (s n)\n⊢ MeasurableSet (⋃ (i : ℕ), ⋂ (j : ℕ) (_ : p j ∧ i ≤ j), s j)",
"state_before": "α : Type u_1\nβ : Type ?u.910929\nγ : Type ?u.910932\nδ : Type ?u.910935\nδ' : Type ?u.910938\nι : Sort uι\ns✝ t u : Set α\ninst✝ : MeasurableSpace α\ns : ℕ → Set α\np : ℕ → Prop\nh : ∀ (n : ℕ), p n → MeasurableSet (s n)\n⊢ MeasurableSet (bliminf s atTop p)",
"tactic": "simp only [Filter.bliminf_eq_iSup_biInf_of_nat, iInf_eq_iInter, iSup_eq_iUnion]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.910929\nγ : Type ?u.910932\nδ : Type ?u.910935\nδ' : Type ?u.910938\nι : Sort uι\ns✝ t u : Set α\ninst✝ : MeasurableSpace α\ns : ℕ → Set α\np : ℕ → Prop\nh : ∀ (n : ℕ), p n → MeasurableSet (s n)\n⊢ MeasurableSet (⋃ (i : ℕ), ⋂ (j : ℕ) (_ : p j ∧ i ≤ j), s j)",
"tactic": "exact .iUnion fun n => .iInter fun m => .iInter fun hm => h m hm.1"
}
] |
[
1901,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1898,
1
] |
Mathlib/Data/Set/Intervals/OrderIso.lean
|
OrderIso.image_Ioi
|
[] |
[
86,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/Data/Real/EReal.lean
|
EReal.toReal_top
|
[] |
[
262,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
261,
1
] |
Mathlib/Topology/Order/Basic.lean
|
pi_Icc_mem_nhds'
|
[] |
[
1481,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1480,
1
] |
Mathlib/Data/Polynomial/RingDivision.lean
|
Polynomial.card_roots
|
[
{
"state_after": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nhp0 : p ≠ 0\n⊢ ↑(↑Multiset.card\n (if h : p = 0 then ∅\n else\n Classical.choose (_ : ∃ s, ↑(↑Multiset.card s) ≤ degree p ∧ ∀ (a : R), count a s = rootMultiplicity a p))) ≤\n degree p",
"state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nhp0 : p ≠ 0\n⊢ ↑(↑Multiset.card (roots p)) ≤ degree p",
"tactic": "unfold roots"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nhp0 : p ≠ 0\n⊢ ↑(↑Multiset.card\n (Classical.choose (_ : ∃ s, ↑(↑Multiset.card s) ≤ degree p ∧ ∀ (a : R), count a s = rootMultiplicity a p))) ≤\n degree p",
"state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nhp0 : p ≠ 0\n⊢ ↑(↑Multiset.card\n (if h : p = 0 then ∅\n else\n Classical.choose (_ : ∃ s, ↑(↑Multiset.card s) ≤ degree p ∧ ∀ (a : R), count a s = rootMultiplicity a p))) ≤\n degree p",
"tactic": "rw [dif_neg hp0]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nhp0 : p ≠ 0\n⊢ ↑(↑Multiset.card\n (Classical.choose (_ : ∃ s, ↑(↑Multiset.card s) ≤ degree p ∧ ∀ (a : R), count a s = rootMultiplicity a p))) ≤\n degree p",
"tactic": "exact (Classical.choose_spec (exists_multiset_roots hp0)).1"
}
] |
[
541,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
538,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.ae_eq_dirac'
|
[] |
[
3022,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3020,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.neg_ofNat_zero
|
[] |
[
39,
69
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
39,
15
] |
Mathlib/Analysis/SpecialFunctions/Log/Basic.lean
|
Real.exp_log_of_neg
|
[
{
"state_after": "x y : ℝ\nhx : x < 0\n⊢ abs x = -x",
"state_before": "x y : ℝ\nhx : x < 0\n⊢ exp (log x) = -x",
"tactic": "rw [exp_log_eq_abs (ne_of_lt hx)]"
},
{
"state_after": "no goals",
"state_before": "x y : ℝ\nhx : x < 0\n⊢ abs x = -x",
"tactic": "exact abs_of_neg hx"
}
] |
[
69,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/RingTheory/Noetherian.lean
|
isNoetherian_of_linearEquiv
|
[] |
[
136,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Mathlib/Analysis/Calculus/Series.lean
|
contDiff_tsum
|
[
{
"state_after": "α : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\n⊢ (∀ (m : ℕ), ↑m ≤ N → Continuous fun x => iteratedFDeriv 𝕜 m (fun x => ∑' (i : α), f i x) x) ∧\n ∀ (m : ℕ), ↑m < N → Differentiable 𝕜 fun x => iteratedFDeriv 𝕜 m (fun x => ∑' (i : α), f i x) x",
"state_before": "α : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\n⊢ ContDiff 𝕜 N fun x => ∑' (i : α), f i x",
"tactic": "rw [contDiff_iff_continuous_differentiable]"
},
{
"state_after": "case left\nα : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\n⊢ ∀ (m : ℕ), ↑m ≤ N → Continuous fun x => iteratedFDeriv 𝕜 m (fun x => ∑' (i : α), f i x) x\n\ncase right\nα : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\n⊢ ∀ (m : ℕ), ↑m < N → Differentiable 𝕜 fun x => iteratedFDeriv 𝕜 m (fun x => ∑' (i : α), f i x) x",
"state_before": "α : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\n⊢ (∀ (m : ℕ), ↑m ≤ N → Continuous fun x => iteratedFDeriv 𝕜 m (fun x => ∑' (i : α), f i x) x) ∧\n ∀ (m : ℕ), ↑m < N → Differentiable 𝕜 fun x => iteratedFDeriv 𝕜 m (fun x => ∑' (i : α), f i x) x",
"tactic": "constructor"
},
{
"state_after": "case left\nα : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\n⊢ Continuous fun x => iteratedFDeriv 𝕜 m (fun x => ∑' (i : α), f i x) x",
"state_before": "case left\nα : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\n⊢ ∀ (m : ℕ), ↑m ≤ N → Continuous fun x => iteratedFDeriv 𝕜 m (fun x => ∑' (i : α), f i x) x",
"tactic": "intro m hm"
},
{
"state_after": "case left\nα : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\n⊢ Continuous fun x => (fun x => ∑' (n : α), iteratedFDeriv 𝕜 m (f n) x) x",
"state_before": "case left\nα : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\n⊢ Continuous fun x => iteratedFDeriv 𝕜 m (fun x => ∑' (i : α), f i x) x",
"tactic": "rw [iteratedFDeriv_tsum hf hv h'f hm]"
},
{
"state_after": "case left.refine'_1\nα : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\n⊢ ∀ (i : α), Continuous fun x => iteratedFDeriv 𝕜 m (f i) x\n\ncase left.refine'_2\nα : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\n⊢ ∀ (n : α) (x : E), ‖iteratedFDeriv 𝕜 m (f n) x‖ ≤ v m n",
"state_before": "case left\nα : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\n⊢ Continuous fun x => (fun x => ∑' (n : α), iteratedFDeriv 𝕜 m (f n) x) x",
"tactic": "refine' continuous_tsum _ (hv m hm) _"
},
{
"state_after": "case left.refine'_1\nα : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\ni : α\n⊢ Continuous fun x => iteratedFDeriv 𝕜 m (f i) x",
"state_before": "case left.refine'_1\nα : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\n⊢ ∀ (i : α), Continuous fun x => iteratedFDeriv 𝕜 m (f i) x",
"tactic": "intro i"
},
{
"state_after": "no goals",
"state_before": "case left.refine'_1\nα : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\ni : α\n⊢ Continuous fun x => iteratedFDeriv 𝕜 m (f i) x",
"tactic": "exact ContDiff.continuous_iteratedFDeriv hm (hf i)"
},
{
"state_after": "case left.refine'_2\nα : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x✝ : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\nn : α\nx : E\n⊢ ‖iteratedFDeriv 𝕜 m (f n) x‖ ≤ v m n",
"state_before": "case left.refine'_2\nα : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\n⊢ ∀ (n : α) (x : E), ‖iteratedFDeriv 𝕜 m (f n) x‖ ≤ v m n",
"tactic": "intro n x"
},
{
"state_after": "no goals",
"state_before": "case left.refine'_2\nα : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x✝ : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m ≤ N\nn : α\nx : E\n⊢ ‖iteratedFDeriv 𝕜 m (f n) x‖ ≤ v m n",
"tactic": "exact h'f _ _ _ hm"
},
{
"state_after": "case right\nα : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m < N\n⊢ Differentiable 𝕜 fun x => iteratedFDeriv 𝕜 m (fun x => ∑' (i : α), f i x) x",
"state_before": "case right\nα : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\n⊢ ∀ (m : ℕ), ↑m < N → Differentiable 𝕜 fun x => iteratedFDeriv 𝕜 m (fun x => ∑' (i : α), f i x) x",
"tactic": "intro m hm"
},
{
"state_after": "case right\nα : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m < N\nh'm : ↑(m + 1) ≤ N\n⊢ Differentiable 𝕜 fun x => iteratedFDeriv 𝕜 m (fun x => ∑' (i : α), f i x) x",
"state_before": "case right\nα : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m < N\n⊢ Differentiable 𝕜 fun x => iteratedFDeriv 𝕜 m (fun x => ∑' (i : α), f i x) x",
"tactic": "have h'm : ((m + 1 : ℕ) : ℕ∞) ≤ N := by\n simpa only [ENat.coe_add, Nat.cast_withBot, ENat.coe_one] using ENat.add_one_le_of_lt hm"
},
{
"state_after": "case right\nα : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m < N\nh'm : ↑(m + 1) ≤ N\n⊢ Differentiable 𝕜 fun x => (fun x => ∑' (n : α), iteratedFDeriv 𝕜 m (f n) x) x",
"state_before": "case right\nα : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m < N\nh'm : ↑(m + 1) ≤ N\n⊢ Differentiable 𝕜 fun x => iteratedFDeriv 𝕜 m (fun x => ∑' (i : α), f i x) x",
"tactic": "rw [iteratedFDeriv_tsum hf hv h'f hm.le]"
},
{
"state_after": "case right\nα : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m < N\nh'm : ↑(m + 1) ≤ N\nA : ∀ (n : α) (x : E), HasFDerivAt (iteratedFDeriv 𝕜 m (f n)) (fderiv 𝕜 (iteratedFDeriv 𝕜 m (f n)) x) x\n⊢ Differentiable 𝕜 fun x => (fun x => ∑' (n : α), iteratedFDeriv 𝕜 m (f n) x) x",
"state_before": "case right\nα : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m < N\nh'm : ↑(m + 1) ≤ N\n⊢ Differentiable 𝕜 fun x => (fun x => ∑' (n : α), iteratedFDeriv 𝕜 m (f n) x) x",
"tactic": "have A :\n ∀ n x, HasFDerivAt (iteratedFDeriv 𝕜 m (f n)) (fderiv 𝕜 (iteratedFDeriv 𝕜 m (f n)) x) x :=\n fun n x => (ContDiff.differentiable_iteratedFDeriv hm (hf n)).differentiableAt.hasFDerivAt"
},
{
"state_after": "case right\nα : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x✝ : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m < N\nh'm : ↑(m + 1) ≤ N\nA : ∀ (n : α) (x : E), HasFDerivAt (iteratedFDeriv 𝕜 m (f n)) (fderiv 𝕜 (iteratedFDeriv 𝕜 m (f n)) x) x\nn : α\nx : E\n⊢ ‖fderiv 𝕜 (iteratedFDeriv 𝕜 m (f n)) x‖ ≤ v (m + 1) n",
"state_before": "case right\nα : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m < N\nh'm : ↑(m + 1) ≤ N\nA : ∀ (n : α) (x : E), HasFDerivAt (iteratedFDeriv 𝕜 m (f n)) (fderiv 𝕜 (iteratedFDeriv 𝕜 m (f n)) x) x\n⊢ Differentiable 𝕜 fun x => (fun x => ∑' (n : α), iteratedFDeriv 𝕜 m (f n) x) x",
"tactic": "refine differentiable_tsum (hv _ h'm) A fun n x => ?_"
},
{
"state_after": "case right\nα : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x✝ : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m < N\nh'm : ↑(m + 1) ≤ N\nA : ∀ (n : α) (x : E), HasFDerivAt (iteratedFDeriv 𝕜 m (f n)) (fderiv 𝕜 (iteratedFDeriv 𝕜 m (f n)) x) x\nn : α\nx : E\n⊢ ‖iteratedFDeriv 𝕜 (m + 1) (f n) x‖ ≤ v (m + 1) n",
"state_before": "case right\nα : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x✝ : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m < N\nh'm : ↑(m + 1) ≤ N\nA : ∀ (n : α) (x : E), HasFDerivAt (iteratedFDeriv 𝕜 m (f n)) (fderiv 𝕜 (iteratedFDeriv 𝕜 m (f n)) x) x\nn : α\nx : E\n⊢ ‖fderiv 𝕜 (iteratedFDeriv 𝕜 m (f n)) x‖ ≤ v (m + 1) n",
"tactic": "rw [fderiv_iteratedFDeriv, comp_apply, LinearIsometryEquiv.norm_map]"
},
{
"state_after": "no goals",
"state_before": "case right\nα : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x✝ : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m < N\nh'm : ↑(m + 1) ≤ N\nA : ∀ (n : α) (x : E), HasFDerivAt (iteratedFDeriv 𝕜 m (f n)) (fderiv 𝕜 (iteratedFDeriv 𝕜 m (f n)) x) x\nn : α\nx : E\n⊢ ‖iteratedFDeriv 𝕜 (m + 1) (f n) x‖ ≤ v (m + 1) n",
"tactic": "exact h'f _ _ _ h'm"
},
{
"state_after": "no goals",
"state_before": "α : Type u_4\nβ : Type ?u.121726\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : CompleteSpace F\nu : α → ℝ\ninst✝ : NormedSpace 𝕜 F\nf : α → E → F\nf' : α → E → E →L[𝕜] F\nv : ℕ → α → ℝ\ns : Set E\nx₀ x : E\nN : ℕ∞\nhf : ∀ (i : α), ContDiff 𝕜 N (f i)\nhv : ∀ (k : ℕ), ↑k ≤ N → Summable (v k)\nh'f : ∀ (k : ℕ) (i : α) (x : E), ↑k ≤ N → ‖iteratedFDeriv 𝕜 k (f i) x‖ ≤ v k i\nm : ℕ\nhm : ↑m < N\n⊢ ↑(m + 1) ≤ N",
"tactic": "simpa only [ENat.coe_add, Nat.cast_withBot, ENat.coe_one] using ENat.add_one_le_of_lt hm"
}
] |
[
252,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
231,
1
] |
Mathlib/Analysis/Convex/Extreme.lean
|
isExtreme_biInter
|
[
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nF✝ : Type ?u.3511\nι : Type ?u.3514\nπ : ι → Type ?u.3519\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\nA B C : Set E\nx : E\nF : Set (Set E)\nhF : Set.Nonempty F\nhA : ∀ (B : Set E), B ∈ F → IsExtreme 𝕜 A B\nthis : Nonempty ↑F\n⊢ IsExtreme 𝕜 A (⋂ (B : Set E) (_ : B ∈ F), B)",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF✝ : Type ?u.3511\nι : Type ?u.3514\nπ : ι → Type ?u.3519\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\nA B C : Set E\nx : E\nF : Set (Set E)\nhF : Set.Nonempty F\nhA : ∀ (B : Set E), B ∈ F → IsExtreme 𝕜 A B\n⊢ IsExtreme 𝕜 A (⋂ (B : Set E) (_ : B ∈ F), B)",
"tactic": "haveI := hF.to_subtype"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF✝ : Type ?u.3511\nι : Type ?u.3514\nπ : ι → Type ?u.3519\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\nA B C : Set E\nx : E\nF : Set (Set E)\nhF : Set.Nonempty F\nhA : ∀ (B : Set E), B ∈ F → IsExtreme 𝕜 A B\nthis : Nonempty ↑F\n⊢ IsExtreme 𝕜 A (⋂ (B : Set E) (_ : B ∈ F), B)",
"tactic": "simpa only [iInter_subtype] using isExtreme_iInter fun i : F ↦ hA _ i.2"
}
] |
[
125,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/Topology/UniformSpace/CompactConvergence.lean
|
ContinuousMap.compactConvNhd_filter_isBasis
|
[
{
"state_after": "case mk.mk.intro.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nK₁ : Set α\nV₁ : Set (β × β)\nK₂ : Set α\nV₂ : Set (β × β)\nhK₁ : IsCompact (K₁, V₁).fst\nhV₁ : (K₁, V₁).snd ∈ 𝓤 β\nhK₂ : IsCompact (K₂, V₂).fst\nhV₂ : (K₂, V₂).snd ∈ 𝓤 β\n⊢ ∃ k,\n (IsCompact k.fst ∧ k.snd ∈ 𝓤 β) ∧\n compactConvNhd k.fst k.snd f ⊆\n compactConvNhd (K₁, V₁).fst (K₁, V₁).snd f ∩ compactConvNhd (K₂, V₂).fst (K₂, V₂).snd f",
"state_before": "α : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\n⊢ ∀ {i j : Set α × Set (β × β)},\n IsCompact i.fst ∧ i.snd ∈ 𝓤 β →\n IsCompact j.fst ∧ j.snd ∈ 𝓤 β →\n ∃ k,\n (IsCompact k.fst ∧ k.snd ∈ 𝓤 β) ∧\n compactConvNhd k.fst k.snd f ⊆ compactConvNhd i.fst i.snd f ∩ compactConvNhd j.fst j.snd f",
"tactic": "rintro ⟨K₁, V₁⟩ ⟨K₂, V₂⟩ ⟨hK₁, hV₁⟩ ⟨hK₂, hV₂⟩"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.intro.intro\nα : Type u₁\nβ : Type u₂\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nK : Set α\nV : Set (β × β)\nf : C(α, β)\nK₁ : Set α\nV₁ : Set (β × β)\nK₂ : Set α\nV₂ : Set (β × β)\nhK₁ : IsCompact (K₁, V₁).fst\nhV₁ : (K₁, V₁).snd ∈ 𝓤 β\nhK₂ : IsCompact (K₂, V₂).fst\nhV₂ : (K₂, V₂).snd ∈ 𝓤 β\n⊢ ∃ k,\n (IsCompact k.fst ∧ k.snd ∈ 𝓤 β) ∧\n compactConvNhd k.fst k.snd f ⊆\n compactConvNhd (K₁, V₁).fst (K₁, V₁).snd f ∩ compactConvNhd (K₂, V₂).fst (K₂, V₂).snd f",
"tactic": "exact\n ⟨⟨K₁ ∪ K₂, V₁ ∩ V₂⟩, ⟨hK₁.union hK₂, Filter.inter_mem hV₁ hV₂⟩,\n compactConvNhd_subset_inter f K₁ K₂ V₁ V₂⟩"
}
] |
[
150,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Data/Set/Sups.lean
|
Set.iUnion_image_inf_right
|
[] |
[
338,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
337,
1
] |
Mathlib/Data/List/Chain.lean
|
List.Chain'.append
|
[] |
[
298,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
296,
1
] |
src/lean/Init/SimpLemmas.lean
|
not_false_eq_true
|
[] |
[
100,
75
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
100,
9
] |
Mathlib/Algebra/Ring/Commute.lean
|
Commute.neg_left
|
[] |
[
104,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/Algebra/Regular/SMul.lean
|
IsSMulRegular.of_mul
|
[
{
"state_after": "R : Type u_1\nS : Type ?u.5473\nM : Type u_2\na b : R\ns : S\ninst✝⁵ : SMul R M\ninst✝⁴ : SMul R S\ninst✝³ : SMul S M\ninst✝² : IsScalarTower R S M\ninst✝¹ : Mul R\ninst✝ : IsScalarTower R R M\nab : IsSMulRegular M (a • b)\n⊢ IsSMulRegular M b",
"state_before": "R : Type u_1\nS : Type ?u.5473\nM : Type u_2\na b : R\ns : S\ninst✝⁵ : SMul R M\ninst✝⁴ : SMul R S\ninst✝³ : SMul S M\ninst✝² : IsScalarTower R S M\ninst✝¹ : Mul R\ninst✝ : IsScalarTower R R M\nab : IsSMulRegular M (a * b)\n⊢ IsSMulRegular M b",
"tactic": "rw [← smul_eq_mul] at ab"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.5473\nM : Type u_2\na b : R\ns : S\ninst✝⁵ : SMul R M\ninst✝⁴ : SMul R S\ninst✝³ : SMul S M\ninst✝² : IsScalarTower R S M\ninst✝¹ : Mul R\ninst✝ : IsScalarTower R R M\nab : IsSMulRegular M (a • b)\n⊢ IsSMulRegular M b",
"tactic": "exact ab.of_smul _"
}
] |
[
108,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/RingTheory/Int/Basic.lean
|
Int.eq_of_associated_of_nonneg
|
[] |
[
133,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/Data/Sym/Basic.lean
|
Sym.of_vector_cons
|
[
{
"state_after": "case mk\nα : Type u_1\nβ : Type ?u.13471\nn n' m : ℕ\ns : Sym α n\na✝ b a : α\nval✝ : List α\nproperty✝ : List.length val✝ = n\n⊢ ofVector (a ::ᵥ { val := val✝, property := property✝ }) = a ::ₛ ofVector { val := val✝, property := property✝ }",
"state_before": "α : Type u_1\nβ : Type ?u.13471\nn n' m : ℕ\ns : Sym α n\na✝ b a : α\nv : Vector α n\n⊢ ofVector (a ::ᵥ v) = a ::ₛ ofVector v",
"tactic": "cases v"
},
{
"state_after": "no goals",
"state_before": "case mk\nα : Type u_1\nβ : Type ?u.13471\nn n' m : ℕ\ns : Sym α n\na✝ b a : α\nval✝ : List α\nproperty✝ : List.length val✝ = n\n⊢ ofVector (a ::ᵥ { val := val✝, property := property✝ }) = a ::ₛ ofVector { val := val✝, property := property✝ }",
"tactic": "rfl"
}
] |
[
157,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/Order/OmegaCompletePartialOrder.lean
|
Part.ωSup_eq_some
|
[] |
[
337,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
332,
1
] |
Mathlib/Analysis/NormedSpace/PiLp.lean
|
PiLp.equiv_smul
|
[] |
[
812,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
811,
1
] |
Mathlib/Analysis/Convex/Star.lean
|
StarConvex.segment_subset
|
[] |
[
86,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/Algebra/Polynomial/BigOperators.lean
|
Polynomial.leadingCoeff_prod
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nι : Type w\ns : Finset ι\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\nf : ι → R[X]\nt : Multiset R[X]\n⊢ leadingCoeff (∏ i in s, f i) = ∏ i in s, leadingCoeff (f i)",
"tactic": "simpa using leadingCoeff_multiset_prod (s.1.map f)"
}
] |
[
365,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
364,
1
] |
Mathlib/Data/Semiquot.lean
|
Semiquot.mem_univ
|
[] |
[
265,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
264,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.not_mem_commonNeighbors_right
|
[] |
[
1029,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1028,
1
] |
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