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/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.coe_two_pi
|
[
{
"state_after": "no goals",
"state_before": "⊢ 2 * π - 0 = 2 * π * ↑1",
"tactic": "rw [sub_zero, Int.cast_one, mul_one]"
}
] |
[
130,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/NumberTheory/Liouville/LiouvilleWith.lean
|
LiouvilleWith.rat_mul
|
[] |
[
149,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.le_of_forall_lt
|
[] |
[
617,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
615,
1
] |
Mathlib/Data/Part.lean
|
Part.right_dom_of_div_dom
|
[] |
[
768,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
768,
1
] |
Mathlib/RingTheory/AdjoinRoot.lean
|
AdjoinRoot.isIntegral_root
|
[] |
[
544,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
543,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
ModelWithCorners.image_eq
|
[
{
"state_after": "case refine'_1\n𝕜 : Type u_3\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\nH : Type u_1\ninst✝ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\ns : Set H\n⊢ s ⊆ I.source\n\ncase refine'_2\n𝕜 : Type u_3\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\nH : Type u_1\ninst✝ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\ns : Set H\n⊢ I.target ∩ ↑(LocalEquiv.symm I.toLocalEquiv) ⁻¹' s = ↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I",
"state_before": "𝕜 : Type u_3\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\nH : Type u_1\ninst✝ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\ns : Set H\n⊢ ↑I '' s = ↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I",
"tactic": "refine' (I.toLocalEquiv.image_eq_target_inter_inv_preimage _).trans _"
},
{
"state_after": "case refine'_1\n𝕜 : Type u_3\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\nH : Type u_1\ninst✝ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\ns : Set H\n⊢ s ⊆ univ",
"state_before": "case refine'_1\n𝕜 : Type u_3\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\nH : Type u_1\ninst✝ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\ns : Set H\n⊢ s ⊆ I.source",
"tactic": "rw [I.source_eq]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\n𝕜 : Type u_3\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\nH : Type u_1\ninst✝ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\ns : Set H\n⊢ s ⊆ univ",
"tactic": "exact subset_univ _"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\n𝕜 : Type u_3\ninst✝³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\nH : Type u_1\ninst✝ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\ns : Set H\n⊢ I.target ∩ ↑(LocalEquiv.symm I.toLocalEquiv) ⁻¹' s = ↑(ModelWithCorners.symm I) ⁻¹' s ∩ range ↑I",
"tactic": "rw [inter_comm, I.target_eq, I.toLocalEquiv_coe_symm]"
}
] |
[
294,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
291,
11
] |
Mathlib/AlgebraicTopology/SplitSimplicialObject.lean
|
SimplicialObject.Splitting.IndexSet.eqId_iff_mono
|
[
{
"state_after": "case mp\nC : Type ?u.5440\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\n⊢ EqId A → Mono (e A)\n\ncase mpr\nC : Type ?u.5440\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\n⊢ Mono (e A) → EqId A",
"state_before": "C : Type ?u.5440\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\n⊢ EqId A ↔ Mono (e A)",
"tactic": "constructor"
},
{
"state_after": "case mp\nC : Type ?u.5440\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\nh : EqId A\n⊢ Mono (e A)",
"state_before": "case mp\nC : Type ?u.5440\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\n⊢ EqId A → Mono (e A)",
"tactic": "intro h"
},
{
"state_after": "case mp\nC : Type ?u.5440\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\nh : A = id Δ\n⊢ Mono (e A)",
"state_before": "case mp\nC : Type ?u.5440\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\nh : EqId A\n⊢ Mono (e A)",
"tactic": "dsimp at h"
},
{
"state_after": "case mp\nC : Type ?u.5440\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\n⊢ Mono (e (id Δ))",
"state_before": "case mp\nC : Type ?u.5440\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\nh : A = id Δ\n⊢ Mono (e A)",
"tactic": "subst h"
},
{
"state_after": "case mp\nC : Type ?u.5440\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\n⊢ Mono (𝟙 Δ.unop)",
"state_before": "case mp\nC : Type ?u.5440\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\n⊢ Mono (e (id Δ))",
"tactic": "dsimp only [id, e]"
},
{
"state_after": "no goals",
"state_before": "case mp\nC : Type ?u.5440\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\n⊢ Mono (𝟙 Δ.unop)",
"tactic": "infer_instance"
},
{
"state_after": "case mpr\nC : Type ?u.5440\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\nh : Mono (e A)\n⊢ EqId A",
"state_before": "case mpr\nC : Type ?u.5440\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\n⊢ Mono (e A) → EqId A",
"tactic": "intro h"
},
{
"state_after": "case mpr\nC : Type ?u.5440\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\nh : Mono (e A)\n⊢ len Δ.unop ≤ len A.fst.unop",
"state_before": "case mpr\nC : Type ?u.5440\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\nh : Mono (e A)\n⊢ EqId A",
"tactic": "rw [eqId_iff_len_le]"
},
{
"state_after": "no goals",
"state_before": "case mpr\nC : Type ?u.5440\ninst✝ : Category C\nΔ : SimplexCategoryᵒᵖ\nA : IndexSet Δ\nh : Mono (e A)\n⊢ len Δ.unop ≤ len A.fst.unop",
"tactic": "exact len_le_of_mono h"
}
] |
[
175,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
166,
1
] |
Mathlib/Logic/Equiv/Basic.lean
|
Equiv.sumCompl_apply_inl
|
[] |
[
521,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
519,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
intervalIntegral.norm_integral_le_of_norm_le_const
|
[] |
[
576,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
574,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.hasFiniteIntegral_norm_iff
|
[] |
[
257,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
255,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.val_one
|
[
{
"state_after": "n : ℕ\ninst✝ : Fact (1 < n)\n⊢ 1 % n = 1",
"state_before": "n : ℕ\ninst✝ : Fact (1 < n)\n⊢ val 1 = 1",
"tactic": "rw [val_one_eq_one_mod]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\ninst✝ : Fact (1 < n)\n⊢ 1 % n = 1",
"tactic": "exact Nat.mod_eq_of_lt Fact.out"
}
] |
[
594,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
592,
1
] |
Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
|
intervalIntegral.integral_deriv_comp_smul_deriv
|
[] |
[
1450,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1445,
1
] |
Mathlib/Order/SuccPred/Basic.lean
|
Order.le_pred_iff_isMin
|
[] |
[
617,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
616,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.strictMonoOn_tan
|
[] |
[
990,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
989,
1
] |
Mathlib/Algebra/Order/Sub/Canonical.lean
|
AddLECancellable.tsub_lt_iff_left
|
[
{
"state_after": "α : Type u_1\ninst✝⁵ : AddCommSemigroup α\ninst✝⁴ : PartialOrder α\ninst✝³ : ExistsAddOfLE α\ninst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nhb : AddLECancellable b\nhba : b ≤ a\n⊢ a < b + c → a - b < c",
"state_before": "α : Type u_1\ninst✝⁵ : AddCommSemigroup α\ninst✝⁴ : PartialOrder α\ninst✝³ : ExistsAddOfLE α\ninst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nhb : AddLECancellable b\nhba : b ≤ a\n⊢ a - b < c ↔ a < b + c",
"tactic": "refine' ⟨hb.lt_add_of_tsub_lt_left, _⟩"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : AddCommSemigroup α\ninst✝⁴ : PartialOrder α\ninst✝³ : ExistsAddOfLE α\ninst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nhb : AddLECancellable b\nhba : b ≤ a\nh : a < b + c\n⊢ a - b < c",
"state_before": "α : Type u_1\ninst✝⁵ : AddCommSemigroup α\ninst✝⁴ : PartialOrder α\ninst✝³ : ExistsAddOfLE α\ninst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nhb : AddLECancellable b\nhba : b ≤ a\n⊢ a < b + c → a - b < c",
"tactic": "intro h"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : AddCommSemigroup α\ninst✝⁴ : PartialOrder α\ninst✝³ : ExistsAddOfLE α\ninst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nhb : AddLECancellable b\nhba : b ≤ a\nh : a < b + c\n⊢ a - b ≠ c",
"state_before": "α : Type u_1\ninst✝⁵ : AddCommSemigroup α\ninst✝⁴ : PartialOrder α\ninst✝³ : ExistsAddOfLE α\ninst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nhb : AddLECancellable b\nhba : b ≤ a\nh : a < b + c\n⊢ a - b < c",
"tactic": "refine' (tsub_le_iff_left.mpr h.le).lt_of_ne _"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : AddCommSemigroup α\ninst✝⁴ : PartialOrder α\ninst✝³ : ExistsAddOfLE α\ninst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b d : α\nhb : AddLECancellable b\nhba : b ≤ a\nh : a < b + (a - b)\n⊢ False",
"state_before": "α : Type u_1\ninst✝⁵ : AddCommSemigroup α\ninst✝⁴ : PartialOrder α\ninst✝³ : ExistsAddOfLE α\ninst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nhb : AddLECancellable b\nhba : b ≤ a\nh : a < b + c\n⊢ a - b ≠ c",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁵ : AddCommSemigroup α\ninst✝⁴ : PartialOrder α\ninst✝³ : ExistsAddOfLE α\ninst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b d : α\nhb : AddLECancellable b\nhba : b ≤ a\nh : a < b + (a - b)\n⊢ False",
"tactic": "exact h.ne' (add_tsub_cancel_of_le hba)"
}
] |
[
123,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
11
] |
Mathlib/Analysis/NormedSpace/Exponential.lean
|
exp_eq_tsum_div
|
[] |
[
185,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
184,
1
] |
Mathlib/Order/Cover.lean
|
Wcovby.snd
|
[] |
[
514,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
513,
1
] |
Mathlib/LinearAlgebra/Matrix/BilinearForm.lean
|
Matrix.toBilin_comp
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.1830938\nM✝ : Type ?u.1830941\ninst✝²⁰ : Semiring R\ninst✝¹⁹ : AddCommMonoid M✝\ninst✝¹⁸ : Module R M✝\nR₁ : Type ?u.1830977\nM₁ : Type ?u.1830980\ninst✝¹⁷ : Ring R₁\ninst✝¹⁶ : AddCommGroup M₁\ninst✝¹⁵ : Module R₁ M₁\nR₂ : Type u_2\nM₂ : Type u_5\ninst✝¹⁴ : CommSemiring R₂\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : Module R₂ M₂\nR₃ : Type ?u.1831779\nM₃ : Type ?u.1831782\ninst✝¹¹ : CommRing R₃\ninst✝¹⁰ : AddCommGroup M₃\ninst✝⁹ : Module R₃ M₃\nV : Type ?u.1832370\nK : Type ?u.1832373\ninst✝⁸ : Field K\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\nB : BilinForm R M✝\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_1\no : Type u_3\ninst✝⁵ : Fintype n\ninst✝⁴ : Fintype o\ninst✝³ : DecidableEq n\nb : Basis n R₂ M₂\nM₂' : Type u_4\ninst✝² : AddCommMonoid M₂'\ninst✝¹ : Module R₂ M₂'\nc : Basis o R₂ M₂'\ninst✝ : DecidableEq o\nM : Matrix n n R₂\nP Q : Matrix n o R₂\n⊢ ↑(BilinForm.toMatrix c) (BilinForm.comp (↑(toBilin b) M) (↑(toLin c b) P) (↑(toLin c b) Q)) =\n ↑(BilinForm.toMatrix c) (↑(toBilin c) (Pᵀ ⬝ M ⬝ Q))",
"tactic": "simp only [BilinForm.toMatrix_comp b c, BilinForm.toMatrix_toBilin, toMatrix_toLin]"
}
] |
[
423,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
420,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.comp_familyOfBFamily'
|
[] |
[
1233,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1230,
1
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
EsakiaHom.coe_id_continuousOrderHom
|
[] |
[
306,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
306,
1
] |
Mathlib/Data/Nat/Choose/Multinomial.lean
|
Nat.succ_mul_binomial
|
[
{
"state_after": "α : Type u_1\ns : Finset α\nf : α → ℕ\na b : α\nn : ℕ\ninst✝ : DecidableEq α\nh : a ≠ b\n⊢ succ (f a + f b) * choose (f a + f b) (f a) = choose (succ (f a) + f b) (succ (f a)) * succ (f a)",
"state_before": "α : Type u_1\ns : Finset α\nf : α → ℕ\na b : α\nn : ℕ\ninst✝ : DecidableEq α\nh : a ≠ b\n⊢ succ (f a + f b) * multinomial {a, b} f = succ (f a) * multinomial {a, b} (Function.update f a (succ (f a)))",
"tactic": "rw [binomial_eq_choose _ h, binomial_eq_choose _ h, mul_comm (f a).succ, Function.update_same,\n Function.update_noteq (ne_comm.mp h)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ns : Finset α\nf : α → ℕ\na b : α\nn : ℕ\ninst✝ : DecidableEq α\nh : a ≠ b\n⊢ succ (f a + f b) * choose (f a + f b) (f a) = choose (succ (f a) + f b) (succ (f a)) * succ (f a)",
"tactic": "rw [succ_mul_choose_eq (f a + f b) (f a), succ_add (f a) (f b)]"
}
] |
[
144,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
139,
1
] |
Mathlib/Topology/ContinuousFunction/CocompactMap.lean
|
CocompactMap.comp_apply
|
[] |
[
156,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/Data/Set/Basic.lean
|
Disjoint.inter_right
|
[] |
[
2969,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2968,
1
] |
Mathlib/AlgebraicGeometry/StructureSheaf.lean
|
AlgebraicGeometry.StructureSheaf.toOpen_germ
|
[
{
"state_after": "R : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\nx : { x // x ∈ U }\n⊢ toOpen R ⊤ ≫ germ (structureSheaf R).val ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑⊤) }) x) = toStalk R ↑x",
"state_before": "R : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\nx : { x // x ∈ U }\n⊢ toOpen R U ≫ germ (Sheaf.presheaf (structureSheaf R)) x = toStalk R ↑x",
"tactic": "rw [← toOpen_res R ⊤ U (homOfLE le_top : U ⟶ ⊤), Category.assoc, Presheaf.germ_res]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : CommRing R\nU : Opens ↑(PrimeSpectrum.Top R)\nx : { x // x ∈ U }\n⊢ toOpen R ⊤ ≫ germ (structureSheaf R).val ((fun x => { val := ↑x, property := (_ : ↑x ∈ ↑⊤) }) x) = toStalk R ↑x",
"tactic": "rfl"
}
] |
[
450,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
448,
1
] |
Mathlib/Data/Finite/Card.lean
|
Finite.card_sum
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.5063\ninst✝¹ : Finite α\ninst✝ : Finite β\nthis : Fintype α\n⊢ Nat.card (α ⊕ β) = Nat.card α + Nat.card β",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.5063\ninst✝¹ : Finite α\ninst✝ : Finite β\n⊢ Nat.card (α ⊕ β) = Nat.card α + Nat.card β",
"tactic": "haveI := Fintype.ofFinite α"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.5063\ninst✝¹ : Finite α\ninst✝ : Finite β\nthis✝ : Fintype α\nthis : Fintype β\n⊢ Nat.card (α ⊕ β) = Nat.card α + Nat.card β",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.5063\ninst✝¹ : Finite α\ninst✝ : Finite β\nthis : Fintype α\n⊢ Nat.card (α ⊕ β) = Nat.card α + Nat.card β",
"tactic": "haveI := Fintype.ofFinite β"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.5063\ninst✝¹ : Finite α\ninst✝ : Finite β\nthis✝ : Fintype α\nthis : Fintype β\n⊢ Nat.card (α ⊕ β) = Nat.card α + Nat.card β",
"tactic": "simp only [Nat.card_eq_fintype_card, Fintype.card_sum]"
}
] |
[
172,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/RingTheory/FinitePresentation.lean
|
RingHom.FinitePresentation.of_finiteType
|
[] |
[
448,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
447,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.map_injective_of_injective
|
[
{
"state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf✝ : G →g G'\nf' : G' →g G''\nu✝ v✝ u' v' : V\np✝ : Walk G u✝ v✝\nf : G →g G'\nhinj : Injective ↑f\nu v : V\np p' : Walk G u v\nh : Walk.map f p = Walk.map f p'\n⊢ p = p'",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf✝ : G →g G'\nf' : G' →g G''\nu✝ v✝ u' v' : V\np : Walk G u✝ v✝\nf : G →g G'\nhinj : Injective ↑f\nu v : V\n⊢ Injective (Walk.map f)",
"tactic": "intro p p' h"
},
{
"state_after": "case nil.nil\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf✝ : G →g G'\nf' : G' →g G''\nu✝¹ v✝ u' v' : V\np : Walk G u✝¹ v✝\nf : G →g G'\nhinj : Injective ↑f\nu v u✝ : V\nh : Walk.map f nil = Walk.map f nil\n⊢ nil = nil\n\ncase nil.cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf✝ : G →g G'\nf' : G' →g G''\nu✝¹ v✝¹ u' v' : V\np : Walk G u✝¹ v✝¹\nf : G →g G'\nhinj : Injective ↑f\nu v u✝ v✝ : V\nh✝ : Adj G u✝ v✝\np✝ : Walk G v✝ u✝\nh : Walk.map f nil = Walk.map f (cons h✝ p✝)\n⊢ nil = cons h✝ p✝",
"state_before": "case nil\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf✝ : G →g G'\nf' : G' →g G''\nu✝¹ v✝ u' v' : V\np : Walk G u✝¹ v✝\nf : G →g G'\nhinj : Injective ↑f\nu v u✝ : V\np' : Walk G u✝ u✝\nh : Walk.map f nil = Walk.map f p'\n⊢ nil = p'",
"tactic": "cases p'"
},
{
"state_after": "no goals",
"state_before": "case nil.nil\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf✝ : G →g G'\nf' : G' →g G''\nu✝¹ v✝ u' v' : V\np : Walk G u✝¹ v✝\nf : G →g G'\nhinj : Injective ↑f\nu v u✝ : V\nh : Walk.map f nil = Walk.map f nil\n⊢ nil = nil",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case nil.cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf✝ : G →g G'\nf' : G' →g G''\nu✝¹ v✝¹ u' v' : V\np : Walk G u✝¹ v✝¹\nf : G →g G'\nhinj : Injective ↑f\nu v u✝ v✝ : V\nh✝ : Adj G u✝ v✝\np✝ : Walk G v✝ u✝\nh : Walk.map f nil = Walk.map f (cons h✝ p✝)\n⊢ nil = cons h✝ p✝",
"tactic": "simp at h"
},
{
"state_after": "no goals",
"state_before": "case cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf✝ : G →g G'\nf' : G' →g G''\nu✝¹ v✝¹ u' v' : V\np : Walk G u✝¹ v✝¹\nf : G →g G'\nhinj : Injective ↑f\nu v u✝ v✝ w✝ : V\nh✝ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nih : ∀ ⦃p' : Walk G v✝ w✝⦄, Walk.map f p✝ = Walk.map f p' → p✝ = p'\np' : Walk G u✝ w✝\nh : Walk.map f (cons h✝ p✝) = Walk.map f p'\n⊢ cons h✝ p✝ = p'",
"tactic": "cases p' with\n| nil => simp at h\n| cons _ _ =>\n simp only [map_cons, cons.injEq] at h\n cases hinj h.1\n simp only [cons.injEq, heq_iff_eq, true_and_iff]\n apply ih\n simpa using h.2"
},
{
"state_after": "no goals",
"state_before": "case cons.nil\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf✝ : G →g G'\nf' : G' →g G''\nu✝¹ v✝¹ u' v' : V\np : Walk G u✝¹ v✝¹\nf : G →g G'\nhinj : Injective ↑f\nu v u✝ v✝ : V\nh✝ : Adj G u✝ v✝\np✝ : Walk G v✝ u✝\nih : ∀ ⦃p' : Walk G v✝ u✝⦄, Walk.map f p✝ = Walk.map f p' → p✝ = p'\nh : Walk.map f (cons h✝ p✝) = Walk.map f nil\n⊢ cons h✝ p✝ = nil",
"tactic": "simp at h"
},
{
"state_after": "case cons.cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf✝ : G →g G'\nf' : G' →g G''\nu✝¹ v✝² u' v' : V\np : Walk G u✝¹ v✝²\nf : G →g G'\nhinj : Injective ↑f\nu v u✝ v✝¹ w✝ : V\nh✝¹ : Adj G u✝ v✝¹\np✝¹ : Walk G v✝¹ w✝\nih : ∀ ⦃p' : Walk G v✝¹ w✝⦄, Walk.map f p✝¹ = Walk.map f p' → p✝¹ = p'\nv✝ : V\nh✝ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nh : ↑f v✝¹ = ↑f v✝ ∧ HEq (Walk.map f p✝¹) (Walk.map f p✝)\n⊢ cons h✝¹ p✝¹ = cons h✝ p✝",
"state_before": "case cons.cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf✝ : G →g G'\nf' : G' →g G''\nu✝¹ v✝² u' v' : V\np : Walk G u✝¹ v✝²\nf : G →g G'\nhinj : Injective ↑f\nu v u✝ v✝¹ w✝ : V\nh✝¹ : Adj G u✝ v✝¹\np✝¹ : Walk G v✝¹ w✝\nih : ∀ ⦃p' : Walk G v✝¹ w✝⦄, Walk.map f p✝¹ = Walk.map f p' → p✝¹ = p'\nv✝ : V\nh✝ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nh : Walk.map f (cons h✝¹ p✝¹) = Walk.map f (cons h✝ p✝)\n⊢ cons h✝¹ p✝¹ = cons h✝ p✝",
"tactic": "simp only [map_cons, cons.injEq] at h"
},
{
"state_after": "case cons.cons.refl\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf✝ : G →g G'\nf' : G' →g G''\nu✝¹ v✝¹ u' v' : V\np : Walk G u✝¹ v✝¹\nf : G →g G'\nhinj : Injective ↑f\nu v u✝ v✝ w✝ : V\nh✝¹ : Adj G u✝ v✝\np✝¹ : Walk G v✝ w✝\nih : ∀ ⦃p' : Walk G v✝ w✝⦄, Walk.map f p✝¹ = Walk.map f p' → p✝¹ = p'\nh✝ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nh : ↑f v✝ = ↑f v✝ ∧ HEq (Walk.map f p✝¹) (Walk.map f p✝)\n⊢ cons h✝¹ p✝¹ = cons h✝ p✝",
"state_before": "case cons.cons\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf✝ : G →g G'\nf' : G' →g G''\nu✝¹ v✝² u' v' : V\np : Walk G u✝¹ v✝²\nf : G →g G'\nhinj : Injective ↑f\nu v u✝ v✝¹ w✝ : V\nh✝¹ : Adj G u✝ v✝¹\np✝¹ : Walk G v✝¹ w✝\nih : ∀ ⦃p' : Walk G v✝¹ w✝⦄, Walk.map f p✝¹ = Walk.map f p' → p✝¹ = p'\nv✝ : V\nh✝ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nh : ↑f v✝¹ = ↑f v✝ ∧ HEq (Walk.map f p✝¹) (Walk.map f p✝)\n⊢ cons h✝¹ p✝¹ = cons h✝ p✝",
"tactic": "cases hinj h.1"
},
{
"state_after": "case cons.cons.refl\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf✝ : G →g G'\nf' : G' →g G''\nu✝¹ v✝¹ u' v' : V\np : Walk G u✝¹ v✝¹\nf : G →g G'\nhinj : Injective ↑f\nu v u✝ v✝ w✝ : V\nh✝¹ : Adj G u✝ v✝\np✝¹ : Walk G v✝ w✝\nih : ∀ ⦃p' : Walk G v✝ w✝⦄, Walk.map f p✝¹ = Walk.map f p' → p✝¹ = p'\nh✝ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nh : ↑f v✝ = ↑f v✝ ∧ HEq (Walk.map f p✝¹) (Walk.map f p✝)\n⊢ p✝¹ = p✝",
"state_before": "case cons.cons.refl\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf✝ : G →g G'\nf' : G' →g G''\nu✝¹ v✝¹ u' v' : V\np : Walk G u✝¹ v✝¹\nf : G →g G'\nhinj : Injective ↑f\nu v u✝ v✝ w✝ : V\nh✝¹ : Adj G u✝ v✝\np✝¹ : Walk G v✝ w✝\nih : ∀ ⦃p' : Walk G v✝ w✝⦄, Walk.map f p✝¹ = Walk.map f p' → p✝¹ = p'\nh✝ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nh : ↑f v✝ = ↑f v✝ ∧ HEq (Walk.map f p✝¹) (Walk.map f p✝)\n⊢ cons h✝¹ p✝¹ = cons h✝ p✝",
"tactic": "simp only [cons.injEq, heq_iff_eq, true_and_iff]"
},
{
"state_after": "case cons.cons.refl.h\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf✝ : G →g G'\nf' : G' →g G''\nu✝¹ v✝¹ u' v' : V\np : Walk G u✝¹ v✝¹\nf : G →g G'\nhinj : Injective ↑f\nu v u✝ v✝ w✝ : V\nh✝¹ : Adj G u✝ v✝\np✝¹ : Walk G v✝ w✝\nih : ∀ ⦃p' : Walk G v✝ w✝⦄, Walk.map f p✝¹ = Walk.map f p' → p✝¹ = p'\nh✝ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nh : ↑f v✝ = ↑f v✝ ∧ HEq (Walk.map f p✝¹) (Walk.map f p✝)\n⊢ Walk.map f p✝¹ = Walk.map f p✝",
"state_before": "case cons.cons.refl\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf✝ : G →g G'\nf' : G' →g G''\nu✝¹ v✝¹ u' v' : V\np : Walk G u✝¹ v✝¹\nf : G →g G'\nhinj : Injective ↑f\nu v u✝ v✝ w✝ : V\nh✝¹ : Adj G u✝ v✝\np✝¹ : Walk G v✝ w✝\nih : ∀ ⦃p' : Walk G v✝ w✝⦄, Walk.map f p✝¹ = Walk.map f p' → p✝¹ = p'\nh✝ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nh : ↑f v✝ = ↑f v✝ ∧ HEq (Walk.map f p✝¹) (Walk.map f p✝)\n⊢ p✝¹ = p✝",
"tactic": "apply ih"
},
{
"state_after": "no goals",
"state_before": "case cons.cons.refl.h\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf✝ : G →g G'\nf' : G' →g G''\nu✝¹ v✝¹ u' v' : V\np : Walk G u✝¹ v✝¹\nf : G →g G'\nhinj : Injective ↑f\nu v u✝ v✝ w✝ : V\nh✝¹ : Adj G u✝ v✝\np✝¹ : Walk G v✝ w✝\nih : ∀ ⦃p' : Walk G v✝ w✝⦄, Walk.map f p✝¹ = Walk.map f p' → p✝¹ = p'\nh✝ : Adj G u✝ v✝\np✝ : Walk G v✝ w✝\nh : ↑f v✝ = ↑f v✝ ∧ HEq (Walk.map f p✝¹) (Walk.map f p✝)\n⊢ Walk.map f p✝¹ = Walk.map f p✝",
"tactic": "simpa using h.2"
}
] |
[
1597,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1581,
1
] |
Mathlib/RingTheory/IntegralClosure.lean
|
isField_of_isIntegral_of_isField'
|
[
{
"state_after": "R✝ : Type ?u.1647177\nA : Type ?u.1647180\nB : Type ?u.1647183\nS✝ : Type ?u.1647186\nT : Type ?u.1647189\ninst✝¹² : CommRing R✝\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : CommRing S✝\ninst✝⁸ : CommRing T\ninst✝⁷ : Algebra A B\ninst✝⁶ : Algebra R✝ B\nf : R✝ →+* S✝\ng : S✝ →+* T\ninst✝⁵ : Algebra R✝ A\ninst✝⁴ : IsScalarTower R✝ A B\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : IsDomain S\ninst✝ : Algebra R S\nH : Algebra.IsIntegral R S\nhR : IsField R\nthis : Field R := IsField.toField hR\n⊢ IsField S",
"state_before": "R✝ : Type ?u.1647177\nA : Type ?u.1647180\nB : Type ?u.1647183\nS✝ : Type ?u.1647186\nT : Type ?u.1647189\ninst✝¹² : CommRing R✝\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : CommRing S✝\ninst✝⁸ : CommRing T\ninst✝⁷ : Algebra A B\ninst✝⁶ : Algebra R✝ B\nf : R✝ →+* S✝\ng : S✝ →+* T\ninst✝⁵ : Algebra R✝ A\ninst✝⁴ : IsScalarTower R✝ A B\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : IsDomain S\ninst✝ : Algebra R S\nH : Algebra.IsIntegral R S\nhR : IsField R\n⊢ IsField S",
"tactic": "letI := hR.toField"
},
{
"state_after": "R✝ : Type ?u.1647177\nA : Type ?u.1647180\nB : Type ?u.1647183\nS✝ : Type ?u.1647186\nT : Type ?u.1647189\ninst✝¹² : CommRing R✝\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : CommRing S✝\ninst✝⁸ : CommRing T\ninst✝⁷ : Algebra A B\ninst✝⁶ : Algebra R✝ B\nf : R✝ →+* S✝\ng : S✝ →+* T\ninst✝⁵ : Algebra R✝ A\ninst✝⁴ : IsScalarTower R✝ A B\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : IsDomain S\ninst✝ : Algebra R S\nH : Algebra.IsIntegral R S\nhR : IsField R\nthis : Field R := IsField.toField hR\nx : S\nhx : x ≠ 0\n⊢ ∃ b, x * b = 1",
"state_before": "R✝ : Type ?u.1647177\nA : Type ?u.1647180\nB : Type ?u.1647183\nS✝ : Type ?u.1647186\nT : Type ?u.1647189\ninst✝¹² : CommRing R✝\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : CommRing S✝\ninst✝⁸ : CommRing T\ninst✝⁷ : Algebra A B\ninst✝⁶ : Algebra R✝ B\nf : R✝ →+* S✝\ng : S✝ →+* T\ninst✝⁵ : Algebra R✝ A\ninst✝⁴ : IsScalarTower R✝ A B\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : IsDomain S\ninst✝ : Algebra R S\nH : Algebra.IsIntegral R S\nhR : IsField R\nthis : Field R := IsField.toField hR\n⊢ IsField S",
"tactic": "refine' ⟨⟨0, 1, zero_ne_one⟩, mul_comm, fun {x} hx => _⟩"
},
{
"state_after": "R✝ : Type ?u.1647177\nA✝ : Type ?u.1647180\nB : Type ?u.1647183\nS✝ : Type ?u.1647186\nT : Type ?u.1647189\ninst✝¹² : CommRing R✝\ninst✝¹¹ : CommRing A✝\ninst✝¹⁰ : CommRing B\ninst✝⁹ : CommRing S✝\ninst✝⁸ : CommRing T\ninst✝⁷ : Algebra A✝ B\ninst✝⁶ : Algebra R✝ B\nf : R✝ →+* S✝\ng : S✝ →+* T\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : IsScalarTower R✝ A✝ B\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : IsDomain S\ninst✝ : Algebra R S\nH : Algebra.IsIntegral R S\nhR : IsField R\nthis : Field R := IsField.toField hR\nx : S\nhx : x ≠ 0\nA : Subalgebra R S := adjoin R {x}\n⊢ ∃ b, x * b = 1",
"state_before": "R✝ : Type ?u.1647177\nA : Type ?u.1647180\nB : Type ?u.1647183\nS✝ : Type ?u.1647186\nT : Type ?u.1647189\ninst✝¹² : CommRing R✝\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : CommRing S✝\ninst✝⁸ : CommRing T\ninst✝⁷ : Algebra A B\ninst✝⁶ : Algebra R✝ B\nf : R✝ →+* S✝\ng : S✝ →+* T\ninst✝⁵ : Algebra R✝ A\ninst✝⁴ : IsScalarTower R✝ A B\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : IsDomain S\ninst✝ : Algebra R S\nH : Algebra.IsIntegral R S\nhR : IsField R\nthis : Field R := IsField.toField hR\nx : S\nhx : x ≠ 0\n⊢ ∃ b, x * b = 1",
"tactic": "let A := Algebra.adjoin R ({x} : Set S)"
},
{
"state_after": "R✝ : Type ?u.1647177\nA✝ : Type ?u.1647180\nB : Type ?u.1647183\nS✝ : Type ?u.1647186\nT : Type ?u.1647189\ninst✝¹² : CommRing R✝\ninst✝¹¹ : CommRing A✝\ninst✝¹⁰ : CommRing B\ninst✝⁹ : CommRing S✝\ninst✝⁸ : CommRing T\ninst✝⁷ : Algebra A✝ B\ninst✝⁶ : Algebra R✝ B\nf : R✝ →+* S✝\ng : S✝ →+* T\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : IsScalarTower R✝ A✝ B\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : IsDomain S\ninst✝ : Algebra R S\nH : Algebra.IsIntegral R S\nhR : IsField R\nthis✝ : Field R := IsField.toField hR\nx : S\nhx : x ≠ 0\nA : Subalgebra R S := adjoin R {x}\nthis : IsNoetherian R { x // x ∈ A }\n⊢ ∃ b, x * b = 1",
"state_before": "R✝ : Type ?u.1647177\nA✝ : Type ?u.1647180\nB : Type ?u.1647183\nS✝ : Type ?u.1647186\nT : Type ?u.1647189\ninst✝¹² : CommRing R✝\ninst✝¹¹ : CommRing A✝\ninst✝¹⁰ : CommRing B\ninst✝⁹ : CommRing S✝\ninst✝⁸ : CommRing T\ninst✝⁷ : Algebra A✝ B\ninst✝⁶ : Algebra R✝ B\nf : R✝ →+* S✝\ng : S✝ →+* T\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : IsScalarTower R✝ A✝ B\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : IsDomain S\ninst✝ : Algebra R S\nH : Algebra.IsIntegral R S\nhR : IsField R\nthis : Field R := IsField.toField hR\nx : S\nhx : x ≠ 0\nA : Subalgebra R S := adjoin R {x}\n⊢ ∃ b, x * b = 1",
"tactic": "haveI : IsNoetherian R A :=\n isNoetherian_of_fg_of_noetherian (Subalgebra.toSubmodule A)\n (FG_adjoin_singleton_of_integral x (H x))"
},
{
"state_after": "R✝ : Type ?u.1647177\nA✝ : Type ?u.1647180\nB : Type ?u.1647183\nS✝ : Type ?u.1647186\nT : Type ?u.1647189\ninst✝¹² : CommRing R✝\ninst✝¹¹ : CommRing A✝\ninst✝¹⁰ : CommRing B\ninst✝⁹ : CommRing S✝\ninst✝⁸ : CommRing T\ninst✝⁷ : Algebra A✝ B\ninst✝⁶ : Algebra R✝ B\nf : R✝ →+* S✝\ng : S✝ →+* T\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : IsScalarTower R✝ A✝ B\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : IsDomain S\ninst✝ : Algebra R S\nH : Algebra.IsIntegral R S\nhR : IsField R\nthis✝¹ : Field R := IsField.toField hR\nx : S\nhx : x ≠ 0\nA : Subalgebra R S := adjoin R {x}\nthis✝ : IsNoetherian R { x // x ∈ A }\nthis : Module.Finite R { x // x ∈ A }\n⊢ ∃ b, x * b = 1",
"state_before": "R✝ : Type ?u.1647177\nA✝ : Type ?u.1647180\nB : Type ?u.1647183\nS✝ : Type ?u.1647186\nT : Type ?u.1647189\ninst✝¹² : CommRing R✝\ninst✝¹¹ : CommRing A✝\ninst✝¹⁰ : CommRing B\ninst✝⁹ : CommRing S✝\ninst✝⁸ : CommRing T\ninst✝⁷ : Algebra A✝ B\ninst✝⁶ : Algebra R✝ B\nf : R✝ →+* S✝\ng : S✝ →+* T\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : IsScalarTower R✝ A✝ B\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : IsDomain S\ninst✝ : Algebra R S\nH : Algebra.IsIntegral R S\nhR : IsField R\nthis✝ : Field R := IsField.toField hR\nx : S\nhx : x ≠ 0\nA : Subalgebra R S := adjoin R {x}\nthis : IsNoetherian R { x // x ∈ A }\n⊢ ∃ b, x * b = 1",
"tactic": "haveI : Module.Finite R A := Module.IsNoetherian.finite R A"
},
{
"state_after": "case intro\nR✝ : Type ?u.1647177\nA✝ : Type ?u.1647180\nB : Type ?u.1647183\nS✝ : Type ?u.1647186\nT : Type ?u.1647189\ninst✝¹² : CommRing R✝\ninst✝¹¹ : CommRing A✝\ninst✝¹⁰ : CommRing B\ninst✝⁹ : CommRing S✝\ninst✝⁸ : CommRing T\ninst✝⁷ : Algebra A✝ B\ninst✝⁶ : Algebra R✝ B\nf : R✝ →+* S✝\ng : S✝ →+* T\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : IsScalarTower R✝ A✝ B\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : IsDomain S\ninst✝ : Algebra R S\nH : Algebra.IsIntegral R S\nhR : IsField R\nthis✝¹ : Field R := IsField.toField hR\nx : S\nhx : x ≠ 0\nA : Subalgebra R S := adjoin R {x}\nthis✝ : IsNoetherian R { x // x ∈ A }\nthis : Module.Finite R { x // x ∈ A }\ny : { x // x ∈ A }\nhy : ↑(LinearMap.mulLeft R { val := x, property := (_ : x ∈ ↑(adjoin R {x})) }) y = 1\n⊢ ∃ b, x * b = 1",
"state_before": "R✝ : Type ?u.1647177\nA✝ : Type ?u.1647180\nB : Type ?u.1647183\nS✝ : Type ?u.1647186\nT : Type ?u.1647189\ninst✝¹² : CommRing R✝\ninst✝¹¹ : CommRing A✝\ninst✝¹⁰ : CommRing B\ninst✝⁹ : CommRing S✝\ninst✝⁸ : CommRing T\ninst✝⁷ : Algebra A✝ B\ninst✝⁶ : Algebra R✝ B\nf : R✝ →+* S✝\ng : S✝ →+* T\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : IsScalarTower R✝ A✝ B\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : IsDomain S\ninst✝ : Algebra R S\nH : Algebra.IsIntegral R S\nhR : IsField R\nthis✝¹ : Field R := IsField.toField hR\nx : S\nhx : x ≠ 0\nA : Subalgebra R S := adjoin R {x}\nthis✝ : IsNoetherian R { x // x ∈ A }\nthis : Module.Finite R { x // x ∈ A }\n⊢ ∃ b, x * b = 1",
"tactic": "obtain ⟨y, hy⟩ :=\n LinearMap.surjective_of_injective\n (@LinearMap.mulLeft_injective R A _ _ _ _ ⟨x, subset_adjoin (Set.mem_singleton x)⟩ fun h =>\n hx (Subtype.ext_iff.mp h))\n 1"
},
{
"state_after": "no goals",
"state_before": "case intro\nR✝ : Type ?u.1647177\nA✝ : Type ?u.1647180\nB : Type ?u.1647183\nS✝ : Type ?u.1647186\nT : Type ?u.1647189\ninst✝¹² : CommRing R✝\ninst✝¹¹ : CommRing A✝\ninst✝¹⁰ : CommRing B\ninst✝⁹ : CommRing S✝\ninst✝⁸ : CommRing T\ninst✝⁷ : Algebra A✝ B\ninst✝⁶ : Algebra R✝ B\nf : R✝ →+* S✝\ng : S✝ →+* T\ninst✝⁵ : Algebra R✝ A✝\ninst✝⁴ : IsScalarTower R✝ A✝ B\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : IsDomain S\ninst✝ : Algebra R S\nH : Algebra.IsIntegral R S\nhR : IsField R\nthis✝¹ : Field R := IsField.toField hR\nx : S\nhx : x ≠ 0\nA : Subalgebra R S := adjoin R {x}\nthis✝ : IsNoetherian R { x // x ∈ A }\nthis : Module.Finite R { x // x ∈ A }\ny : { x // x ∈ A }\nhy : ↑(LinearMap.mulLeft R { val := x, property := (_ : x ∈ ↑(adjoin R {x})) }) y = 1\n⊢ ∃ b, x * b = 1",
"tactic": "exact ⟨y, Subtype.ext_iff.mp hy⟩"
}
] |
[
1159,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1145,
1
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.le_sup_toSubalgebra
|
[] |
[
1176,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1175,
1
] |
Std/Data/Array/Lemmas.lean
|
Array.singleton_def
|
[] |
[
35,
66
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
35,
9
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.eventually_inf_principal
|
[] |
[
1242,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1240,
1
] |
Mathlib/NumberTheory/LegendreSymbol/MulCharacter.lean
|
MulChar.mul_apply
|
[] |
[
308,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
307,
1
] |
Mathlib/Order/Filter/Ultrafilter.lean
|
Filter.isAtom_pure
|
[] |
[
413,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
412,
1
] |
Mathlib/RingTheory/AdjoinRoot.lean
|
AdjoinRoot.quotMapCMapSpanMkEquivQuotMapCQuotMapSpanMk_mk
|
[] |
[
771,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
768,
1
] |
Mathlib/CategoryTheory/Subobject/Lattice.lean
|
CategoryTheory.Subobject.bot_factors_iff_zero
|
[
{
"state_after": "case intro\nC : Type u₁\ninst✝³ : Category C\nX Y Z : C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nA B : C\nh : A ⟶ ⊥.obj.left\n⊢ h ≫ MonoOver.arrow ⊥ = 0",
"state_before": "C : Type u₁\ninst✝³ : Category C\nX Y Z : C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nA B : C\nf : A ⟶ B\n⊢ Factors ⊥ f → f = 0",
"tactic": "rintro ⟨h, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nC : Type u₁\ninst✝³ : Category C\nX Y Z : C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nA B : C\nh : A ⟶ ⊥.obj.left\n⊢ h ≫ MonoOver.arrow ⊥ = 0",
"tactic": "simp only [MonoOver.bot_arrow_eq_zero, Functor.id_obj, Functor.const_obj_obj,\n MonoOver.bot_left, comp_zero]"
},
{
"state_after": "C : Type u₁\ninst✝³ : Category C\nX Y Z : C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nA B : C\n⊢ Factors ⊥ 0",
"state_before": "C : Type u₁\ninst✝³ : Category C\nX Y Z : C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nA B : C\nf : A ⟶ B\n⊢ f = 0 → Factors ⊥ f",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝³ : Category C\nX Y Z : C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nA B : C\n⊢ Factors ⊥ 0",
"tactic": "exact ⟨0, by simp⟩"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝³ : Category C\nX Y Z : C\nD : Type u₂\ninst✝² : Category D\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nA B : C\n⊢ 0 ≫ MonoOver.arrow ⊥ = 0",
"tactic": "simp"
}
] |
[
353,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
346,
1
] |
Mathlib/Algebra/Quaternion.lean
|
QuaternionAlgebra.im_imI
|
[] |
[
120,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Algebra/Order/Rearrangement.lean
|
Monovary.sum_comp_perm_mul_eq_sum_mul_iff
|
[] |
[
488,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
486,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.aevalTower_comp_toAlgHom
|
[] |
[
1629,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1627,
1
] |
Mathlib/Topology/Algebra/Order/IntermediateValue.lean
|
isConnected_Iic
|
[] |
[
470,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
469,
1
] |
Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean
|
NonUnitalSubsemiring.center_eq_top
|
[] |
[
499,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
498,
1
] |
Mathlib/Order/CompleteLattice.lean
|
iSup_pair
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nβ₂ : Type ?u.141744\nγ : Type ?u.141747\nι : Sort ?u.141750\nι' : Sort ?u.141753\nκ : ι → Sort ?u.141758\nκ' : ι' → Sort ?u.141763\ninst✝ : CompleteLattice α\nf✝ g s t : ι → α\na✝ b✝ : α\nf : β → α\na b : β\n⊢ (⨆ (x : β) (_ : x ∈ {a, b}), f x) = f a ⊔ f b",
"tactic": "rw [iSup_insert, iSup_singleton]"
}
] |
[
1466,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1465,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
Set.Nonempty.bddBelow_upperBounds
|
[] |
[
270,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
269,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Module.lean
|
ContinuousLinearEquiv.hasSum'
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_5\nR : Type u_1\nR₂ : Type u_2\nM : Type u_3\nM₂ : Type u_4\ninst✝⁹ : Semiring R\ninst✝⁸ : Semiring R₂\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommMonoid M₂\ninst✝⁴ : Module R₂ M₂\ninst✝³ : TopologicalSpace M\ninst✝² : TopologicalSpace M₂\nσ : R →+* R₂\nσ' : R₂ →+* R\ninst✝¹ : RingHomInvPair σ σ'\ninst✝ : RingHomInvPair σ' σ\nf : ι → M\ne : M ≃SL[σ] M₂\nx : M\n⊢ HasSum (fun b => ↑e (f b)) (↑e x) ↔ HasSum f x",
"tactic": "rw [e.hasSum, ContinuousLinearEquiv.symm_apply_apply]"
}
] |
[
80,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
11
] |
Mathlib/SetTheory/Lists.lean
|
Lists'.of_toList
|
[
{
"state_after": "α : Type u_1\nb : Bool\nh : true = b\nl : Lists' α b\n⊢ Lists' α b",
"state_before": "α : Type u_1\nb : Bool\nh : true = b\nl : Lists' α b\n⊢ Lists' α true",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nb : Bool\nh : true = b\nl : Lists' α b\n⊢ Lists' α b",
"tactic": "exact l"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nb : Bool\nh : true = b\nl : Lists' α b\n⊢ let l' := Eq.mpr (_ : Lists' α true = Lists' α b) l;\n ofList (toList l') = l'",
"tactic": "induction l with\n| atom => cases h\n| nil => simp\n| cons' b a _ IH =>\n intro l'\n simpa [cons] using IH rfl"
},
{
"state_after": "no goals",
"state_before": "case atom\nα : Type u_1\nb : Bool\na✝ : α\nh : true = false\n⊢ let l' := Eq.mpr (_ : Lists' α true = Lists' α false) (atom a✝);\n ofList (toList l') = l'",
"tactic": "cases h"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\nb : Bool\nh : true = true\n⊢ let l' := Eq.mpr (_ : Lists' α true = Lists' α true) nil;\n ofList (toList l') = l'",
"tactic": "simp"
},
{
"state_after": "case cons'\nα : Type u_1\nb✝¹ b✝ : Bool\nb : Lists' α b✝\na : Lists' α true\na_ih✝ :\n ∀ (h : true = b✝),\n let l' := Eq.mpr (_ : Lists' α true = Lists' α b✝) b;\n ofList (toList l') = l'\nIH :\n ∀ (h : true = true),\n let l' := Eq.mpr (_ : Lists' α true = Lists' α true) a;\n ofList (toList l') = l'\nh : true = true\nl' : Lists' α true := Eq.mpr (_ : Lists' α true = Lists' α true) (cons' b a)\n⊢ ofList (toList l') = l'",
"state_before": "case cons'\nα : Type u_1\nb✝¹ b✝ : Bool\nb : Lists' α b✝\na : Lists' α true\na_ih✝ :\n ∀ (h : true = b✝),\n let l' := Eq.mpr (_ : Lists' α true = Lists' α b✝) b;\n ofList (toList l') = l'\nIH :\n ∀ (h : true = true),\n let l' := Eq.mpr (_ : Lists' α true = Lists' α true) a;\n ofList (toList l') = l'\nh : true = true\n⊢ let l' := Eq.mpr (_ : Lists' α true = Lists' α true) (cons' b a);\n ofList (toList l') = l'",
"tactic": "intro l'"
},
{
"state_after": "no goals",
"state_before": "case cons'\nα : Type u_1\nb✝¹ b✝ : Bool\nb : Lists' α b✝\na : Lists' α true\na_ih✝ :\n ∀ (h : true = b✝),\n let l' := Eq.mpr (_ : Lists' α true = Lists' α b✝) b;\n ofList (toList l') = l'\nIH :\n ∀ (h : true = true),\n let l' := Eq.mpr (_ : Lists' α true = Lists' α true) a;\n ofList (toList l') = l'\nh : true = true\nl' : Lists' α true := Eq.mpr (_ : Lists' α true = Lists' α true) (cons' b a)\n⊢ ofList (toList l') = l'",
"tactic": "simpa [cons] using IH rfl"
}
] |
[
122,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/MeasureTheory/Function/SpecialFunctions/IsROrC.lean
|
Measurable.im
|
[] |
[
56,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/Analysis/Convex/Cone/Basic.lean
|
ConvexCone.blunt_iff_not_pointed
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.62541\nG : Type ?u.62544\ninst✝² : OrderedSemiring 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\nS✝ S : ConvexCone 𝕜 E\n⊢ Blunt S ↔ ¬Pointed S",
"tactic": "rw [pointed_iff_not_blunt, Classical.not_not]"
}
] |
[
360,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
359,
1
] |
Mathlib/Analysis/Complex/CauchyIntegral.lean
|
Complex.integral_boundary_rect_eq_zero_of_continuousOn_of_differentiableOn
|
[] |
[
280,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
272,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.bit0_re
|
[] |
[
207,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Additive.lean
|
BoxIntegral.BoxAdditiveMap.coe_mk
|
[] |
[
73,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/ModelTheory/ElementaryMaps.lean
|
FirstOrder.Language.ElementaryEmbedding.map_fun
|
[
{
"state_after": "L : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.308452\nQ : Type ?u.308455\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nφ : M ↪ₑ[L] N\nn : ℕ\nf : Functions L n\nx : Fin n → M\nh :\n Formula.Realize (Formula.graph f) (↑φ ∘ Fin.cons (funMap f x) x) ↔\n Formula.Realize (Formula.graph f) (Fin.cons (funMap f x) x)\n⊢ ↑φ (funMap f x) = funMap f (↑φ ∘ x)",
"state_before": "L : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.308452\nQ : Type ?u.308455\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nφ : M ↪ₑ[L] N\nn : ℕ\nf : Functions L n\nx : Fin n → M\n⊢ ↑φ (funMap f x) = funMap f (↑φ ∘ x)",
"tactic": "have h := φ.map_formula (Formula.graph f) (Fin.cons (funMap f x) x)"
},
{
"state_after": "L : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.308452\nQ : Type ?u.308455\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nφ : M ↪ₑ[L] N\nn : ℕ\nf : Functions L n\nx : Fin n → M\nh : funMap f (↑φ ∘ x) = ↑φ (funMap f x) ↔ funMap f x = funMap f x\n⊢ ↑φ (funMap f x) = funMap f (↑φ ∘ x)",
"state_before": "L : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.308452\nQ : Type ?u.308455\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nφ : M ↪ₑ[L] N\nn : ℕ\nf : Functions L n\nx : Fin n → M\nh :\n Formula.Realize (Formula.graph f) (↑φ ∘ Fin.cons (funMap f x) x) ↔\n Formula.Realize (Formula.graph f) (Fin.cons (funMap f x) x)\n⊢ ↑φ (funMap f x) = funMap f (↑φ ∘ x)",
"tactic": "rw [Formula.realize_graph, Fin.comp_cons, Formula.realize_graph] at h"
},
{
"state_after": "no goals",
"state_before": "L : Language\nM : Type u_3\nN : Type u_4\nP : Type ?u.308452\nQ : Type ?u.308455\ninst✝³ : Structure L M\ninst✝² : Structure L N\ninst✝¹ : Structure L P\ninst✝ : Structure L Q\nφ : M ↪ₑ[L] N\nn : ℕ\nf : Functions L n\nx : Fin n → M\nh : funMap f (↑φ ∘ x) = ↑φ (funMap f x) ↔ funMap f x = funMap f x\n⊢ ↑φ (funMap f x) = funMap f (↑φ ∘ x)",
"tactic": "rw [eq_comm, h]"
}
] |
[
146,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Data/MvPolynomial/Polynomial.lean
|
MvPolynomial.polynomial_eval_eval₂
|
[
{
"state_after": "case h_C\nR : Type u_1\nS : Type u_2\nσ : Type u_3\nx : S\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* Polynomial S\ng : σ → Polynomial S\np : MvPolynomial σ R\n⊢ ∀ (a : R),\n Polynomial.eval x (eval₂ f g (↑C a)) =\n eval₂ (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) (↑C a)\n\ncase h_add\nR : Type u_1\nS : Type u_2\nσ : Type u_3\nx : S\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* Polynomial S\ng : σ → Polynomial S\np : MvPolynomial σ R\n⊢ ∀ (p q : MvPolynomial σ R),\n Polynomial.eval x (eval₂ f g p) =\n eval₂ (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) p →\n Polynomial.eval x (eval₂ f g q) =\n eval₂ (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) q →\n Polynomial.eval x (eval₂ f g (p + q)) =\n eval₂ (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) (p + q)\n\ncase h_X\nR : Type u_1\nS : Type u_2\nσ : Type u_3\nx : S\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* Polynomial S\ng : σ → Polynomial S\np : MvPolynomial σ R\n⊢ ∀ (p : MvPolynomial σ R) (n : σ),\n Polynomial.eval x (eval₂ f g p) =\n eval₂ (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) p →\n Polynomial.eval x (eval₂ f g (p * X n)) =\n eval₂ (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) (p * X n)",
"state_before": "R : Type u_1\nS : Type u_2\nσ : Type u_3\nx : S\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* Polynomial S\ng : σ → Polynomial S\np : MvPolynomial σ R\n⊢ Polynomial.eval x (eval₂ f g p) =\n eval₂ (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) p",
"tactic": "apply induction_on p"
},
{
"state_after": "no goals",
"state_before": "case h_C\nR : Type u_1\nS : Type u_2\nσ : Type u_3\nx : S\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* Polynomial S\ng : σ → Polynomial S\np : MvPolynomial σ R\n⊢ ∀ (a : R),\n Polynomial.eval x (eval₂ f g (↑C a)) =\n eval₂ (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) (↑C a)",
"tactic": "simp"
},
{
"state_after": "case h_add\nR : Type u_1\nS : Type u_2\nσ : Type u_3\nx : S\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* Polynomial S\ng : σ → Polynomial S\np✝ p q : MvPolynomial σ R\nhp :\n Polynomial.eval x (eval₂ f g p) =\n eval₂ (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) p\nhq :\n Polynomial.eval x (eval₂ f g q) =\n eval₂ (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) q\n⊢ Polynomial.eval x (eval₂ f g (p + q)) =\n eval₂ (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) (p + q)",
"state_before": "case h_add\nR : Type u_1\nS : Type u_2\nσ : Type u_3\nx : S\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* Polynomial S\ng : σ → Polynomial S\np : MvPolynomial σ R\n⊢ ∀ (p q : MvPolynomial σ R),\n Polynomial.eval x (eval₂ f g p) =\n eval₂ (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) p →\n Polynomial.eval x (eval₂ f g q) =\n eval₂ (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) q →\n Polynomial.eval x (eval₂ f g (p + q)) =\n eval₂ (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) (p + q)",
"tactic": "intro p q hp hq"
},
{
"state_after": "no goals",
"state_before": "case h_add\nR : Type u_1\nS : Type u_2\nσ : Type u_3\nx : S\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* Polynomial S\ng : σ → Polynomial S\np✝ p q : MvPolynomial σ R\nhp :\n Polynomial.eval x (eval₂ f g p) =\n eval₂ (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) p\nhq :\n Polynomial.eval x (eval₂ f g q) =\n eval₂ (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) q\n⊢ Polynomial.eval x (eval₂ f g (p + q)) =\n eval₂ (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) (p + q)",
"tactic": "simp [hp, hq]"
},
{
"state_after": "case h_X\nR : Type u_1\nS : Type u_2\nσ : Type u_3\nx : S\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* Polynomial S\ng : σ → Polynomial S\np✝ p : MvPolynomial σ R\nn : σ\nhp :\n Polynomial.eval x (eval₂ f g p) =\n eval₂ (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) p\n⊢ Polynomial.eval x (eval₂ f g (p * X n)) =\n eval₂ (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) (p * X n)",
"state_before": "case h_X\nR : Type u_1\nS : Type u_2\nσ : Type u_3\nx : S\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* Polynomial S\ng : σ → Polynomial S\np : MvPolynomial σ R\n⊢ ∀ (p : MvPolynomial σ R) (n : σ),\n Polynomial.eval x (eval₂ f g p) =\n eval₂ (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) p →\n Polynomial.eval x (eval₂ f g (p * X n)) =\n eval₂ (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) (p * X n)",
"tactic": "intro p n hp"
},
{
"state_after": "no goals",
"state_before": "case h_X\nR : Type u_1\nS : Type u_2\nσ : Type u_3\nx : S\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nf : R →+* Polynomial S\ng : σ → Polynomial S\np✝ p : MvPolynomial σ R\nn : σ\nhp :\n Polynomial.eval x (eval₂ f g p) =\n eval₂ (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) p\n⊢ Polynomial.eval x (eval₂ f g (p * X n)) =\n eval₂ (RingHom.comp (Polynomial.evalRingHom x) f) (fun s => Polynomial.eval x (g s)) (p * X n)",
"tactic": "simp [hp]"
}
] |
[
29,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
20,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.half_lt_self
|
[
{
"state_after": "case intro\nα : Type ?u.326127\nβ : Type ?u.326130\nb c d : ℝ≥0∞\nr p q a : ℝ≥0\nhz : ↑a ≠ 0\n⊢ ↑a / 2 < ↑a",
"state_before": "α : Type ?u.326127\nβ : Type ?u.326130\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nhz : a ≠ 0\nht : a ≠ ⊤\n⊢ a / 2 < a",
"tactic": "lift a to ℝ≥0 using ht"
},
{
"state_after": "case intro\nα : Type ?u.326127\nβ : Type ?u.326130\nb c d : ℝ≥0∞\nr p q a : ℝ≥0\nhz : a ≠ 0\n⊢ ↑a / 2 < ↑a",
"state_before": "case intro\nα : Type ?u.326127\nβ : Type ?u.326130\nb c d : ℝ≥0∞\nr p q a : ℝ≥0\nhz : ↑a ≠ 0\n⊢ ↑a / 2 < ↑a",
"tactic": "rw [coe_ne_zero] at hz"
},
{
"state_after": "case intro\nα : Type ?u.326127\nβ : Type ?u.326130\nb c d : ℝ≥0∞\nr p q a : ℝ≥0\nhz : a ≠ 0\n⊢ a / 2 < a\n\ncase intro\nα : Type ?u.326127\nβ : Type ?u.326130\nb c d : ℝ≥0∞\nr p q a : ℝ≥0\nhz : a ≠ 0\n⊢ 2 ≠ 0",
"state_before": "case intro\nα : Type ?u.326127\nβ : Type ?u.326130\nb c d : ℝ≥0∞\nr p q a : ℝ≥0\nhz : a ≠ 0\n⊢ ↑a / 2 < ↑a",
"tactic": "rw [← coe_two, ← coe_div, coe_lt_coe]"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type ?u.326127\nβ : Type ?u.326130\nb c d : ℝ≥0∞\nr p q a : ℝ≥0\nhz : a ≠ 0\n⊢ a / 2 < a\n\ncase intro\nα : Type ?u.326127\nβ : Type ?u.326130\nb c d : ℝ≥0∞\nr p q a : ℝ≥0\nhz : a ≠ 0\n⊢ 2 ≠ 0",
"tactic": "exacts [NNReal.half_lt_self hz, two_ne_zero' _]"
}
] |
[
1755,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1751,
11
] |
Mathlib/Analysis/LocallyConvex/Polar.lean
|
LinearMap.polar_zero
|
[
{
"state_after": "𝕜 : Type u_2\nE : Type u_3\nF : Type u_1\ninst✝⁴ : NormedCommRing 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\ny : F\nx : E\nhx : x ∈ {0}\n⊢ ‖↑(↑B x) y‖ ≤ 1",
"state_before": "𝕜 : Type u_2\nE : Type u_3\nF : Type u_1\ninst✝⁴ : NormedCommRing 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\n⊢ polar B {0} = Set.univ",
"tactic": "refine' Set.eq_univ_iff_forall.mpr fun y x hx => _"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_3\nF : Type u_1\ninst✝⁴ : NormedCommRing 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\ny : F\nx : E\nhx : x ∈ {0}\n⊢ 0 ≤ 1",
"state_before": "𝕜 : Type u_2\nE : Type u_3\nF : Type u_1\ninst✝⁴ : NormedCommRing 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\ny : F\nx : E\nhx : x ∈ {0}\n⊢ ‖↑(↑B x) y‖ ≤ 1",
"tactic": "rw [Set.mem_singleton_iff.mp hx, map_zero, LinearMap.zero_apply, norm_zero]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_3\nF : Type u_1\ninst✝⁴ : NormedCommRing 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\ny : F\nx : E\nhx : x ∈ {0}\n⊢ 0 ≤ 1",
"tactic": "exact zero_le_one"
}
] |
[
113,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/Analysis/Complex/AbsMax.lean
|
Complex.eqOn_closedBall_of_isMaxOn_norm
|
[] |
[
330,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
327,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.dotProduct_sub
|
[
{
"state_after": "no goals",
"state_before": "l : Type ?u.181483\nm : Type u_1\nn : Type ?u.181489\no : Type ?u.181492\nm' : o → Type ?u.181497\nn' : o → Type ?u.181502\nR : Type ?u.181505\nS : Type ?u.181508\nα : Type v\nβ : Type w\nγ : Type ?u.181515\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : NonUnitalNonAssocRing α\nu v w : m → α\n⊢ u ⬝ᵥ (v - w) = u ⬝ᵥ v - u ⬝ᵥ w",
"tactic": "simp [sub_eq_add_neg]"
}
] |
[
874,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
874,
1
] |
Mathlib/Order/SymmDiff.lean
|
symmDiff_symmDiff_right
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.69972\nα : Type u_1\nβ : Type ?u.69978\nπ : ι → Type ?u.69983\ninst✝ : GeneralizedBooleanAlgebra α\na b c d : α\n⊢ a \\ (b ⊔ c) ⊔ a ⊓ b ⊓ c ⊔ (b \\ (c ⊔ a) ⊔ c \\ (b ⊔ a)) = a \\ (b ⊔ c) ⊔ b \\ (a ⊔ c) ⊔ c \\ (a ⊔ b) ⊔ a ⊓ b ⊓ c",
"tactic": "ac_rfl"
}
] |
[
474,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
468,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
|
AffineMap.lineMap_same_apply
|
[
{
"state_after": "k : Type u_2\nV1 : Type u_3\nP1 : Type u_1\nV2 : Type ?u.434982\nP2 : Type ?u.434985\nV3 : Type ?u.434988\nP3 : Type ?u.434991\nV4 : Type ?u.434994\nP4 : Type ?u.434997\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P2\ninst✝⁵ : AddCommGroup V3\ninst✝⁴ : Module k V3\ninst✝³ : AffineSpace V3 P3\ninst✝² : AddCommGroup V4\ninst✝¹ : Module k V4\ninst✝ : AffineSpace V4 P4\np : P1\nc : k\nthis : AddAction V1 P1 := inferInstance\n⊢ ↑(lineMap p p) c = p",
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},
{
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"tactic": "simp [(lineMap_apply), (vsub_self)]"
}
] |
[
550,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
547,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.Tendsto.atBot_mul_neg_const
|
[] |
[
1236,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1234,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
|
MeasureTheory.union_ae_eq_univ_of_ae_eq_univ_left
|
[
{
"state_after": "case h.e'_5\nα : Type u_1\nβ : Type ?u.112649\nγ : Type ?u.112652\nδ : Type ?u.112655\nι : Type ?u.112658\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t : Set α\nh : s =ᵐ[μ] univ\n⊢ univ = univ ∪ t",
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"tactic": "convert ae_eq_set_union h (ae_eq_refl t)"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5\nα : Type u_1\nβ : Type ?u.112649\nγ : Type ?u.112652\nδ : Type ?u.112655\nι : Type ?u.112658\ninst✝ : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns s₁ s₂ t : Set α\nh : s =ᵐ[μ] univ\n⊢ univ = univ ∪ t",
"tactic": "rw [univ_union]"
}
] |
[
513,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
511,
1
] |
Mathlib/GroupTheory/Abelianization.lean
|
Abelianization.hom_ext
|
[] |
[
169,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
OrthogonalFamily.independent
|
[
{
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"tactic": "classical!"
},
{
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"tactic": "apply CompleteLattice.independent_of_dfinsupp_lsum_injective"
},
{
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"tactic": "refine LinearMap.ker_eq_bot.mp ?_"
},
{
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"tactic": "rw [Submodule.eq_bot_iff]"
},
{
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},
{
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"tactic": "rw [LinearMap.mem_ker] at hv"
},
{
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"tactic": "ext i"
},
{
"state_after": "case h.h.a\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3754417\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_3\ndec_ι : DecidableEq ι\nG : ι → Type ?u.3754480\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV✝ : (i : ι) → G i →ₗᵢ[𝕜] E\nhV✝ : OrthogonalFamily 𝕜 G V✝\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nV : ι → Submodule 𝕜 E\nhV : OrthogonalFamily 𝕜 (fun i => { x // x ∈ V i }) fun i => Submodule.subtypeₗᵢ (V i)\nem✝ : (a : Prop) → Decidable a\nv : Π₀ (i : ι), { x // x ∈ V i }\nhv : ↑(↑(Dfinsupp.lsum ℕ) fun i => Submodule.subtype (V i)) v = 0\ni : ι\n⊢ inner ↑(↑v i) ↑(↑v i) = 0",
"state_before": "case h.h.a\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3754417\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_3\ndec_ι : DecidableEq ι\nG : ι → Type ?u.3754480\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV✝ : (i : ι) → G i →ₗᵢ[𝕜] E\nhV✝ : OrthogonalFamily 𝕜 G V✝\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nV : ι → Submodule 𝕜 E\nhV : OrthogonalFamily 𝕜 (fun i => { x // x ∈ V i }) fun i => Submodule.subtypeₗᵢ (V i)\nem✝ : (a : Prop) → Decidable a\nv : Π₀ (i : ι), { x // x ∈ V i }\nhv : ↑(↑(Dfinsupp.lsum ℕ) fun i => Submodule.subtype (V i)) v = 0\ni : ι\n⊢ ↑(↑v i) = ↑(↑0 i)",
"tactic": "suffices ⟪(v i : E), v i⟫ = 0 by simpa only [inner_self_eq_zero] using this"
},
{
"state_after": "no goals",
"state_before": "case h.h.a\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3754417\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_3\ndec_ι : DecidableEq ι\nG : ι → Type ?u.3754480\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV✝ : (i : ι) → G i →ₗᵢ[𝕜] E\nhV✝ : OrthogonalFamily 𝕜 G V✝\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nV : ι → Submodule 𝕜 E\nhV : OrthogonalFamily 𝕜 (fun i => { x // x ∈ V i }) fun i => Submodule.subtypeₗᵢ (V i)\nem✝ : (a : Prop) → Decidable a\nv : Π₀ (i : ι), { x // x ∈ V i }\nhv : ↑(↑(Dfinsupp.lsum ℕ) fun i => Submodule.subtype (V i)) v = 0\ni : ι\n⊢ inner ↑(↑v i) ↑(↑v i) = 0",
"tactic": "calc\n ⟪(v i : E), v i⟫ = ⟪(v i : E), Dfinsupp.lsum ℕ (fun i => (V i).subtype) v⟫ := by\n simpa only [Dfinsupp.sumAddHom_apply, Dfinsupp.lsum_apply_apply] using\n (hV.inner_right_dfinsupp v i (v i)).symm\n _ = 0 := by simp only [hv, inner_zero_right]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3754417\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_3\ndec_ι : DecidableEq ι\nG : ι → Type ?u.3754480\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV✝ : (i : ι) → G i →ₗᵢ[𝕜] E\nhV✝ : OrthogonalFamily 𝕜 G V✝\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nV : ι → Submodule 𝕜 E\nhV : OrthogonalFamily 𝕜 (fun i => { x // x ∈ V i }) fun i => Submodule.subtypeₗᵢ (V i)\nem✝ : (a : Prop) → Decidable a\nv : Π₀ (i : ι), { x // x ∈ V i }\nhv : ↑(↑(Dfinsupp.lsum ℕ) fun i => Submodule.subtype (V i)) v = 0\ni : ι\nthis : inner ↑(↑v i) ↑(↑v i) = 0\n⊢ ↑(↑v i) = ↑(↑0 i)",
"tactic": "simpa only [inner_self_eq_zero] using this"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3754417\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_3\ndec_ι : DecidableEq ι\nG : ι → Type ?u.3754480\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV✝ : (i : ι) → G i →ₗᵢ[𝕜] E\nhV✝ : OrthogonalFamily 𝕜 G V✝\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nV : ι → Submodule 𝕜 E\nhV : OrthogonalFamily 𝕜 (fun i => { x // x ∈ V i }) fun i => Submodule.subtypeₗᵢ (V i)\nem✝ : (a : Prop) → Decidable a\nv : Π₀ (i : ι), { x // x ∈ V i }\nhv : ↑(↑(Dfinsupp.lsum ℕ) fun i => Submodule.subtype (V i)) v = 0\ni : ι\n⊢ inner ↑(↑v i) ↑(↑v i) = inner (↑(↑v i)) (↑(↑(Dfinsupp.lsum ℕ) fun i => Submodule.subtype (V i)) v)",
"tactic": "simpa only [Dfinsupp.sumAddHom_apply, Dfinsupp.lsum_apply_apply] using\n (hV.inner_right_dfinsupp v i (v i)).symm"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.3754417\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nι : Type u_3\ndec_ι : DecidableEq ι\nG : ι → Type ?u.3754480\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nV✝ : (i : ι) → G i →ₗᵢ[𝕜] E\nhV✝ : OrthogonalFamily 𝕜 G V✝\ndec_V : (i : ι) → (x : G i) → Decidable (x ≠ 0)\nV : ι → Submodule 𝕜 E\nhV : OrthogonalFamily 𝕜 (fun i => { x // x ∈ V i }) fun i => Submodule.subtypeₗᵢ (V i)\nem✝ : (a : Prop) → Decidable a\nv : Π₀ (i : ι), { x // x ∈ V i }\nhv : ↑(↑(Dfinsupp.lsum ℕ) fun i => Submodule.subtype (V i)) v = 0\ni : ι\n⊢ inner (↑(↑v i)) (↑(↑(Dfinsupp.lsum ℕ) fun i => Submodule.subtype (V i)) v) = 0",
"tactic": "simp only [hv, inner_zero_right]"
}
] |
[
2172,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2157,
1
] |
Mathlib/Algebra/Hom/Equiv/Basic.lean
|
MulEquiv.piCongrRight_refl
|
[] |
[
662,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
661,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.one_add_of_omega_le
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.135821\nβ : Type ?u.135824\nγ : Type ?u.135827\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nh : ω ≤ o\n⊢ 1 + o = o",
"tactic": "rw [← Ordinal.add_sub_cancel_of_le h, ← add_assoc, one_add_omega]"
}
] |
[
632,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
631,
1
] |
Mathlib/Data/Set/Function.lean
|
Function.Semiconj.mapsTo_preimage
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.113229\nι : Sort ?u.113232\nπ : α → Type ?u.113237\nfa : α → α\nfb : β → β\nf : α → β\ng : β → γ\ns✝ t✝ : Set α\nh : Semiconj f fa fb\ns t : Set β\nhb : MapsTo fb s t\nx : α\nhx : x ∈ f ⁻¹' s\n⊢ fa x ∈ f ⁻¹' t",
"tactic": "simp only [mem_preimage, h x, hb hx]"
}
] |
[
1665,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1664,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean
|
EuclideanGeometry.dist_div_sin_angle_of_angle_eq_pi_div_two
|
[
{
"state_after": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : inner (p₂ -ᵥ p₃) (p₁ -ᵥ p₂) = 0\nh0 : p₁ ≠ p₂ ∨ p₃ = p₂\n⊢ dist p₁ p₂ / Real.sin (∠ p₂ p₃ p₁) = dist p₁ p₃",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : ∠ p₁ p₂ p₃ = π / 2\nh0 : p₁ ≠ p₂ ∨ p₃ = p₂\n⊢ dist p₁ p₂ / Real.sin (∠ p₂ p₃ p₁) = dist p₁ p₃",
"tactic": "rw [angle, ← inner_eq_zero_iff_angle_eq_pi_div_two, real_inner_comm, ← neg_eq_zero, ←\n inner_neg_left, neg_vsub_eq_vsub_rev] at h"
},
{
"state_after": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : inner (p₂ -ᵥ p₃) (p₁ -ᵥ p₂) = 0\nh0 : p₂ -ᵥ p₃ = 0 ∨ p₁ -ᵥ p₂ ≠ 0\n⊢ dist p₁ p₂ / Real.sin (∠ p₂ p₃ p₁) = dist p₁ p₃",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : inner (p₂ -ᵥ p₃) (p₁ -ᵥ p₂) = 0\nh0 : p₁ ≠ p₂ ∨ p₃ = p₂\n⊢ dist p₁ p₂ / Real.sin (∠ p₂ p₃ p₁) = dist p₁ p₃",
"tactic": "rw [eq_comm, ← @vsub_ne_zero V, ← @vsub_eq_zero_iff_eq V, or_comm] at h0"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ p₃ : P\nh : inner (p₂ -ᵥ p₃) (p₁ -ᵥ p₂) = 0\nh0 : p₂ -ᵥ p₃ = 0 ∨ p₁ -ᵥ p₂ ≠ 0\n⊢ dist p₁ p₂ / Real.sin (∠ p₂ p₃ p₁) = dist p₁ p₃",
"tactic": "rw [angle, dist_eq_norm_vsub V p₁ p₂, dist_eq_norm_vsub V p₁ p₃, ← vsub_add_vsub_cancel p₁ p₂ p₃,\n add_comm, norm_div_sin_angle_add_of_inner_eq_zero h h0]"
}
] |
[
511,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
505,
1
] |
Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean
|
lt_iff_exists_mul
|
[
{
"state_after": "α : Type u\ninst✝¹ : CanonicallyOrderedMonoid α\na b c d : α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\n⊢ (∃ x, b = a * x ∧ a ≠ b) ↔ ∃ c, c > 1 ∧ b = a * c",
"state_before": "α : Type u\ninst✝¹ : CanonicallyOrderedMonoid α\na b c d : α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\n⊢ a < b ↔ ∃ c, c > 1 ∧ b = a * c",
"tactic": "rw [lt_iff_le_and_ne, le_iff_exists_mul, ←exists_and_right]"
},
{
"state_after": "case h\nα : Type u\ninst✝¹ : CanonicallyOrderedMonoid α\na b c d : α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\n⊢ ∀ (a_1 : α), b = a * a_1 ∧ a ≠ b ↔ a_1 > 1 ∧ b = a * a_1",
"state_before": "α : Type u\ninst✝¹ : CanonicallyOrderedMonoid α\na b c d : α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\n⊢ (∃ x, b = a * x ∧ a ≠ b) ↔ ∃ c, c > 1 ∧ b = a * c",
"tactic": "apply exists_congr"
},
{
"state_after": "case h\nα : Type u\ninst✝¹ : CanonicallyOrderedMonoid α\na b c✝ d : α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\nc : α\n⊢ b = a * c ∧ a ≠ b ↔ c > 1 ∧ b = a * c",
"state_before": "case h\nα : Type u\ninst✝¹ : CanonicallyOrderedMonoid α\na b c d : α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\n⊢ ∀ (a_1 : α), b = a * a_1 ∧ a ≠ b ↔ a_1 > 1 ∧ b = a * a_1",
"tactic": "intro c"
},
{
"state_after": "case h\nα : Type u\ninst✝¹ : CanonicallyOrderedMonoid α\na b c✝ d : α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\nc : α\n⊢ b = a * c → (a ≠ b ↔ 1 < c)",
"state_before": "case h\nα : Type u\ninst✝¹ : CanonicallyOrderedMonoid α\na b c✝ d : α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\nc : α\n⊢ b = a * c ∧ a ≠ b ↔ c > 1 ∧ b = a * c",
"tactic": "rw [and_comm, and_congr_left_iff, gt_iff_lt]"
},
{
"state_after": "case h\nα : Type u\ninst✝¹ : CanonicallyOrderedMonoid α\na c✝ d : α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\nc : α\n⊢ a ≠ a * c ↔ 1 < c",
"state_before": "case h\nα : Type u\ninst✝¹ : CanonicallyOrderedMonoid α\na b c✝ d : α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\nc : α\n⊢ b = a * c → (a ≠ b ↔ 1 < c)",
"tactic": "rintro rfl"
},
{
"state_after": "case h.mp\nα : Type u\ninst✝¹ : CanonicallyOrderedMonoid α\na c✝ d : α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\nc : α\n⊢ a ≠ a * c → 1 < c\n\ncase h.mpr\nα : Type u\ninst✝¹ : CanonicallyOrderedMonoid α\na c✝ d : α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\nc : α\n⊢ 1 < c → a ≠ a * c",
"state_before": "case h\nα : Type u\ninst✝¹ : CanonicallyOrderedMonoid α\na c✝ d : α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\nc : α\n⊢ a ≠ a * c ↔ 1 < c",
"tactic": "constructor"
},
{
"state_after": "case h.mp\nα : Type u\ninst✝¹ : CanonicallyOrderedMonoid α\na c✝ d : α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\nc : α\n⊢ a ≠ a * c → c ≠ 1",
"state_before": "case h.mp\nα : Type u\ninst✝¹ : CanonicallyOrderedMonoid α\na c✝ d : α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\nc : α\n⊢ a ≠ a * c → 1 < c",
"tactic": "rw [one_lt_iff_ne_one]"
},
{
"state_after": "case h.mp.h₁\nα : Type u\ninst✝¹ : CanonicallyOrderedMonoid α\na c✝ d : α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\nc : α\n⊢ c = 1 → a = a * c",
"state_before": "case h.mp\nα : Type u\ninst✝¹ : CanonicallyOrderedMonoid α\na c✝ d : α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\nc : α\n⊢ a ≠ a * c → c ≠ 1",
"tactic": "apply mt"
},
{
"state_after": "case h.mp.h₁\nα : Type u\ninst✝¹ : CanonicallyOrderedMonoid α\na c d : α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\n⊢ a = a * 1",
"state_before": "case h.mp.h₁\nα : Type u\ninst✝¹ : CanonicallyOrderedMonoid α\na c✝ d : α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\nc : α\n⊢ c = 1 → a = a * c",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "case h.mp.h₁\nα : Type u\ninst✝¹ : CanonicallyOrderedMonoid α\na c d : α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\n⊢ a = a * 1",
"tactic": "rw [mul_one]"
},
{
"state_after": "case h.mpr\nα : Type u\ninst✝¹ : CanonicallyOrderedMonoid α\na c✝ d : α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\nc : α\n⊢ 1 < c → a < a * c",
"state_before": "case h.mpr\nα : Type u\ninst✝¹ : CanonicallyOrderedMonoid α\na c✝ d : α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\nc : α\n⊢ 1 < c → a ≠ a * c",
"tactic": "rw [← (self_le_mul_right a c).lt_iff_ne]"
},
{
"state_after": "no goals",
"state_before": "case h.mpr\nα : Type u\ninst✝¹ : CanonicallyOrderedMonoid α\na c✝ d : α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\nc : α\n⊢ 1 < c → a < a * c",
"tactic": "apply lt_mul_of_one_lt_right'"
}
] |
[
300,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
288,
1
] |
Mathlib/Init/Algebra/Order.lean
|
lt_of_le_of_ne
|
[] |
[
201,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
200,
1
] |
Mathlib/Data/List/Chain.lean
|
List.chain'_append_cons_cons
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nR r : α → α → Prop\nl l₁✝ l₂✝ : List α\na b✝ b c : α\nl₁ l₂ : List α\n⊢ Chain' R (l₁ ++ b :: c :: l₂) ↔ Chain' R (l₁ ++ [b]) ∧ R b c ∧ Chain' R (c :: l₂)",
"tactic": "rw [chain'_split, chain'_cons]"
}
] |
[
224,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
222,
1
] |
Mathlib/Analysis/Calculus/Deriv/Basic.lean
|
deriv_mem_iff
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type w\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf✝ f₀ f₁ g : 𝕜 → F\nf' f₀' f₁' g' : F\nx✝ : 𝕜\ns✝ t : Set 𝕜\nL L₁ L₂ : Filter 𝕜\nf : 𝕜 → F\ns : Set F\nx : 𝕜\n⊢ deriv f x ∈ s ↔ DifferentiableAt 𝕜 f x ∧ deriv f x ∈ s ∨ ¬DifferentiableAt 𝕜 f x ∧ 0 ∈ s",
"tactic": "by_cases hx : DifferentiableAt 𝕜 f x <;> simp [deriv_zero_of_not_differentiableAt, *]"
}
] |
[
531,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
528,
1
] |
Mathlib/GroupTheory/Complement.lean
|
Subgroup.mem_transferSet
|
[] |
[
620,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
619,
1
] |
Mathlib/Data/TypeVec.lean
|
TypeVec.subtypeVal_diagSub
|
[
{
"state_after": "case a.h\nn : ℕ\nα : TypeVec n\ni : Fin2 n\nx : α i\n⊢ (subtypeVal (repeatEq α) ⊚ diagSub) i x = prod.diag i x",
"state_before": "n : ℕ\nα : TypeVec n\n⊢ subtypeVal (repeatEq α) ⊚ diagSub = prod.diag",
"tactic": "ext i x"
},
{
"state_after": "case a.h.fz\nn : ℕ\nα✝ : TypeVec n\ni : Fin2 n\nx✝ : α✝ i\nn✝ : ℕ\nα : TypeVec (Nat.succ n✝)\nx : α Fin2.fz\n⊢ (subtypeVal (repeatEq α) ⊚ diagSub) Fin2.fz x = prod.diag Fin2.fz x\n\ncase a.h.fs\nn : ℕ\nα✝ : TypeVec n\ni : Fin2 n\nx✝ : α✝ i\nn✝ : ℕ\na✝ : Fin2 n✝\ni_ih : ∀ {α : TypeVec n✝} (x : α a✝), (subtypeVal (repeatEq α) ⊚ diagSub) a✝ x = prod.diag a✝ x\nα : TypeVec (Nat.succ n✝)\nx : α (Fin2.fs a✝)\n⊢ (subtypeVal (repeatEq α) ⊚ diagSub) (Fin2.fs a✝) x = prod.diag (Fin2.fs a✝) x",
"state_before": "case a.h\nn : ℕ\nα : TypeVec n\ni : Fin2 n\nx : α i\n⊢ (subtypeVal (repeatEq α) ⊚ diagSub) i x = prod.diag i x",
"tactic": "induction' i with _ _ _ i_ih"
},
{
"state_after": "case a.h.fs\nn : ℕ\nα✝ : TypeVec n\ni : Fin2 n\nx✝ : α✝ i\nn✝ : ℕ\na✝ : Fin2 n✝\ni_ih : ∀ {α : TypeVec n✝} (x : α a✝), (subtypeVal (repeatEq α) ⊚ diagSub) a✝ x = prod.diag a✝ x\nα : TypeVec (Nat.succ n✝)\nx : α (Fin2.fs a✝)\n⊢ (subtypeVal (repeatEq α) ⊚ diagSub) (Fin2.fs a✝) x = prod.diag (Fin2.fs a✝) x",
"state_before": "case a.h.fz\nn : ℕ\nα✝ : TypeVec n\ni : Fin2 n\nx✝ : α✝ i\nn✝ : ℕ\nα : TypeVec (Nat.succ n✝)\nx : α Fin2.fz\n⊢ (subtypeVal (repeatEq α) ⊚ diagSub) Fin2.fz x = prod.diag Fin2.fz x\n\ncase a.h.fs\nn : ℕ\nα✝ : TypeVec n\ni : Fin2 n\nx✝ : α✝ i\nn✝ : ℕ\na✝ : Fin2 n✝\ni_ih : ∀ {α : TypeVec n✝} (x : α a✝), (subtypeVal (repeatEq α) ⊚ diagSub) a✝ x = prod.diag a✝ x\nα : TypeVec (Nat.succ n✝)\nx : α (Fin2.fs a✝)\n⊢ (subtypeVal (repeatEq α) ⊚ diagSub) (Fin2.fs a✝) x = prod.diag (Fin2.fs a✝) x",
"tactic": ". simp [comp, diagSub, subtypeVal, prod.diag]"
},
{
"state_after": "no goals",
"state_before": "case a.h.fs\nn : ℕ\nα✝ : TypeVec n\ni : Fin2 n\nx✝ : α✝ i\nn✝ : ℕ\na✝ : Fin2 n✝\ni_ih : ∀ {α : TypeVec n✝} (x : α a✝), (subtypeVal (repeatEq α) ⊚ diagSub) a✝ x = prod.diag a✝ x\nα : TypeVec (Nat.succ n✝)\nx : α (Fin2.fs a✝)\n⊢ (subtypeVal (repeatEq α) ⊚ diagSub) (Fin2.fs a✝) x = prod.diag (Fin2.fs a✝) x",
"tactic": ". simp [prod.diag]\n simp [comp, diagSub, subtypeVal, prod.diag] at *\n apply @i_ih (drop _)"
},
{
"state_after": "no goals",
"state_before": "case a.h.fz\nn : ℕ\nα✝ : TypeVec n\ni : Fin2 n\nx✝ : α✝ i\nn✝ : ℕ\nα : TypeVec (Nat.succ n✝)\nx : α Fin2.fz\n⊢ (subtypeVal (repeatEq α) ⊚ diagSub) Fin2.fz x = prod.diag Fin2.fz x",
"tactic": "simp [comp, diagSub, subtypeVal, prod.diag]"
},
{
"state_after": "case a.h.fs\nn : ℕ\nα✝ : TypeVec n\ni : Fin2 n\nx✝ : α✝ i\nn✝ : ℕ\na✝ : Fin2 n✝\ni_ih : ∀ {α : TypeVec n✝} (x : α a✝), (subtypeVal (repeatEq α) ⊚ diagSub) a✝ x = prod.diag a✝ x\nα : TypeVec (Nat.succ n✝)\nx : α (Fin2.fs a✝)\n⊢ (subtypeVal (repeatEq α) ⊚ diagSub) (Fin2.fs a✝) x = prod.diag a✝ x",
"state_before": "case a.h.fs\nn : ℕ\nα✝ : TypeVec n\ni : Fin2 n\nx✝ : α✝ i\nn✝ : ℕ\na✝ : Fin2 n✝\ni_ih : ∀ {α : TypeVec n✝} (x : α a✝), (subtypeVal (repeatEq α) ⊚ diagSub) a✝ x = prod.diag a✝ x\nα : TypeVec (Nat.succ n✝)\nx : α (Fin2.fs a✝)\n⊢ (subtypeVal (repeatEq α) ⊚ diagSub) (Fin2.fs a✝) x = prod.diag (Fin2.fs a✝) x",
"tactic": "simp [prod.diag]"
},
{
"state_after": "case a.h.fs\nn : ℕ\nα✝ : TypeVec n\ni : Fin2 n\nx✝ : α✝ i\nn✝ : ℕ\na✝ : Fin2 n✝\ni_ih : ∀ {α : TypeVec n✝} (x : α a✝), subtypeVal (repeatEq α) a✝ (diagSub a✝ x) = prod.diag a✝ x\nα : TypeVec (Nat.succ n✝)\nx : α (Fin2.fs a✝)\n⊢ subtypeVal (dropFun (repeatEq α)) a✝ (diagSub a✝ x) = prod.diag a✝ x",
"state_before": "case a.h.fs\nn : ℕ\nα✝ : TypeVec n\ni : Fin2 n\nx✝ : α✝ i\nn✝ : ℕ\na✝ : Fin2 n✝\ni_ih : ∀ {α : TypeVec n✝} (x : α a✝), (subtypeVal (repeatEq α) ⊚ diagSub) a✝ x = prod.diag a✝ x\nα : TypeVec (Nat.succ n✝)\nx : α (Fin2.fs a✝)\n⊢ (subtypeVal (repeatEq α) ⊚ diagSub) (Fin2.fs a✝) x = prod.diag a✝ x",
"tactic": "simp [comp, diagSub, subtypeVal, prod.diag] at *"
},
{
"state_after": "no goals",
"state_before": "case a.h.fs\nn : ℕ\nα✝ : TypeVec n\ni : Fin2 n\nx✝ : α✝ i\nn✝ : ℕ\na✝ : Fin2 n✝\ni_ih : ∀ {α : TypeVec n✝} (x : α a✝), subtypeVal (repeatEq α) a✝ (diagSub a✝ x) = prod.diag a✝ x\nα : TypeVec (Nat.succ n✝)\nx : α (Fin2.fs a✝)\n⊢ subtypeVal (dropFun (repeatEq α)) a✝ (diagSub a✝ x) = prod.diag a✝ x",
"tactic": "apply @i_ih (drop _)"
}
] |
[
790,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
784,
1
] |
Mathlib/Data/ENat/Basic.lean
|
ENat.succ_def
|
[
{
"state_after": "no goals",
"state_before": "m✝ n m : ℕ∞\n⊢ Order.succ m = m + 1",
"tactic": "cases m <;> rfl"
}
] |
[
185,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
185,
1
] |
Mathlib/Algebra/GroupWithZero/Power.lean
|
zpow_ne_zero
|
[] |
[
181,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.reClm_coe
|
[] |
[
890,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
889,
1
] |
Mathlib/Algebra/Homology/Additive.lean
|
HomologicalComplex.singleMapHomologicalComplex_hom_app_ne
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nV : Type u\ninst✝⁶ : Category V\ninst✝⁵ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf g : C ⟶ D\nh✝ k : D ⟶ E\ni✝ : ι\ninst✝⁴ : HasZeroObject V\nW : Type u_3\ninst✝³ : Category W\ninst✝² : Preadditive W\ninst✝¹ : HasZeroObject W\nF : V ⥤ W\ninst✝ : Functor.Additive F\ni j : ι\nh : i ≠ j\nX : V\n⊢ Hom.f ((singleMapHomologicalComplex F c j).hom.app X) i = 0",
"tactic": "simp [singleMapHomologicalComplex, h]"
}
] |
[
324,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
322,
1
] |
Mathlib/Order/JordanHolder.lean
|
CompositionSeries.inj
|
[] |
[
184,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
11
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
Real.sinh_injective
|
[] |
[
686,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
685,
1
] |
Mathlib/Data/List/Sigma.lean
|
List.kerase_sublist
|
[] |
[
419,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
418,
1
] |
Mathlib/Topology/MetricSpace/Infsep.lean
|
Set.einfsep_ne_top
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.10722\ninst✝ : EDist α\nx y : α\ns t : Set α\n⊢ einfsep s ≠ ⊤ ↔ ∃ x x_1 y x_2 _hxy, edist x y ≠ ⊤",
"tactic": "simp_rw [← lt_top_iff_ne_top, einfsep_lt_top]"
}
] |
[
83,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/CategoryTheory/FullSubcategory.lean
|
CategoryTheory.fullSubcategoryInclusion_map_lift_map
|
[] |
[
199,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Mathlib/Topology/Order.lean
|
TopologicalSpace.nhds_mkOfNhds_filterBasis
|
[
{
"state_after": "case h₀.intro.intro\nα : Type u\nB : α → FilterBasis α\na : α\nh₀ : ∀ (x : α) (n : Set α), n ∈ B x → x ∈ n\nh₁ : ∀ (x : α) (n : Set α), n ∈ B x → ∃ n₁, n₁ ∈ B x ∧ n₁ ⊆ n ∧ ∀ (x' : α), x' ∈ n₁ → ∃ n₂, n₂ ∈ B x' ∧ n₂ ⊆ n\nx : α\nn : Set α\nhn : n ∈ (fun x => FilterBasis.filter (B x)) x\nm : Set α\nhm₁ : m ∈ B x\nhm₂ : m ⊆ n\n⊢ n ∈ pure x\n\ncase h₁.intro.intro\nα : Type u\nB : α → FilterBasis α\na : α\nh₀ : ∀ (x : α) (n : Set α), n ∈ B x → x ∈ n\nh₁ : ∀ (x : α) (n : Set α), n ∈ B x → ∃ n₁, n₁ ∈ B x ∧ n₁ ⊆ n ∧ ∀ (x' : α), x' ∈ n₁ → ∃ n₂, n₂ ∈ B x' ∧ n₂ ⊆ n\nx : α\nn : Set α\nhn : n ∈ FilterBasis.filter (B x)\nm : Set α\nhm₁ : m ∈ B x\nhm₂ : m ⊆ n\n⊢ ∃ t, t ∈ FilterBasis.filter (B x) ∧ t ⊆ n ∧ ∀ (a' : α), a' ∈ t → n ∈ FilterBasis.filter (B a')",
"state_before": "α : Type u\nB : α → FilterBasis α\na : α\nh₀ : ∀ (x : α) (n : Set α), n ∈ B x → x ∈ n\nh₁ : ∀ (x : α) (n : Set α), n ∈ B x → ∃ n₁, n₁ ∈ B x ∧ n₁ ⊆ n ∧ ∀ (x' : α), x' ∈ n₁ → ∃ n₂, n₂ ∈ B x' ∧ n₂ ⊆ n\n⊢ 𝓝 a = FilterBasis.filter (B a)",
"tactic": "rw [TopologicalSpace.nhds_mkOfNhds] <;> intro x n hn <;>\n obtain ⟨m, hm₁, hm₂⟩ := (B x).mem_filter_iff.mp hn"
},
{
"state_after": "no goals",
"state_before": "case h₀.intro.intro\nα : Type u\nB : α → FilterBasis α\na : α\nh₀ : ∀ (x : α) (n : Set α), n ∈ B x → x ∈ n\nh₁ : ∀ (x : α) (n : Set α), n ∈ B x → ∃ n₁, n₁ ∈ B x ∧ n₁ ⊆ n ∧ ∀ (x' : α), x' ∈ n₁ → ∃ n₂, n₂ ∈ B x' ∧ n₂ ⊆ n\nx : α\nn : Set α\nhn : n ∈ (fun x => FilterBasis.filter (B x)) x\nm : Set α\nhm₁ : m ∈ B x\nhm₂ : m ⊆ n\n⊢ n ∈ pure x",
"tactic": "exact hm₂ (h₀ _ _ hm₁)"
},
{
"state_after": "case h₁.intro.intro.intro.intro.intro\nα : Type u\nB : α → FilterBasis α\na : α\nh₀ : ∀ (x : α) (n : Set α), n ∈ B x → x ∈ n\nh₁ : ∀ (x : α) (n : Set α), n ∈ B x → ∃ n₁, n₁ ∈ B x ∧ n₁ ⊆ n ∧ ∀ (x' : α), x' ∈ n₁ → ∃ n₂, n₂ ∈ B x' ∧ n₂ ⊆ n\nx : α\nn : Set α\nhn : n ∈ FilterBasis.filter (B x)\nm : Set α\nhm₁ : m ∈ B x\nhm₂ : m ⊆ n\nn₁ : Set α\nhn₁ : n₁ ∈ B x\nhn₂ : n₁ ⊆ m\nhn₃ : ∀ (x' : α), x' ∈ n₁ → ∃ n₂, n₂ ∈ B x' ∧ n₂ ⊆ m\n⊢ ∃ t, t ∈ FilterBasis.filter (B x) ∧ t ⊆ n ∧ ∀ (a' : α), a' ∈ t → n ∈ FilterBasis.filter (B a')",
"state_before": "case h₁.intro.intro\nα : Type u\nB : α → FilterBasis α\na : α\nh₀ : ∀ (x : α) (n : Set α), n ∈ B x → x ∈ n\nh₁ : ∀ (x : α) (n : Set α), n ∈ B x → ∃ n₁, n₁ ∈ B x ∧ n₁ ⊆ n ∧ ∀ (x' : α), x' ∈ n₁ → ∃ n₂, n₂ ∈ B x' ∧ n₂ ⊆ n\nx : α\nn : Set α\nhn : n ∈ FilterBasis.filter (B x)\nm : Set α\nhm₁ : m ∈ B x\nhm₂ : m ⊆ n\n⊢ ∃ t, t ∈ FilterBasis.filter (B x) ∧ t ⊆ n ∧ ∀ (a' : α), a' ∈ t → n ∈ FilterBasis.filter (B a')",
"tactic": "obtain ⟨n₁, hn₁, hn₂, hn₃⟩ := h₁ x m hm₁"
},
{
"state_after": "case h₁.intro.intro.intro.intro.intro\nα : Type u\nB : α → FilterBasis α\na : α\nh₀ : ∀ (x : α) (n : Set α), n ∈ B x → x ∈ n\nh₁ : ∀ (x : α) (n : Set α), n ∈ B x → ∃ n₁, n₁ ∈ B x ∧ n₁ ⊆ n ∧ ∀ (x' : α), x' ∈ n₁ → ∃ n₂, n₂ ∈ B x' ∧ n₂ ⊆ n\nx : α\nn : Set α\nhn : n ∈ FilterBasis.filter (B x)\nm : Set α\nhm₁ : m ∈ B x\nhm₂ : m ⊆ n\nn₁ : Set α\nhn₁ : n₁ ∈ B x\nhn₂ : n₁ ⊆ m\nhn₃ : ∀ (x' : α), x' ∈ n₁ → ∃ n₂, n₂ ∈ B x' ∧ n₂ ⊆ m\nx' : α\nhx' : x' ∈ n₁\n⊢ n ∈ FilterBasis.filter (B x')",
"state_before": "case h₁.intro.intro.intro.intro.intro\nα : Type u\nB : α → FilterBasis α\na : α\nh₀ : ∀ (x : α) (n : Set α), n ∈ B x → x ∈ n\nh₁ : ∀ (x : α) (n : Set α), n ∈ B x → ∃ n₁, n₁ ∈ B x ∧ n₁ ⊆ n ∧ ∀ (x' : α), x' ∈ n₁ → ∃ n₂, n₂ ∈ B x' ∧ n₂ ⊆ n\nx : α\nn : Set α\nhn : n ∈ FilterBasis.filter (B x)\nm : Set α\nhm₁ : m ∈ B x\nhm₂ : m ⊆ n\nn₁ : Set α\nhn₁ : n₁ ∈ B x\nhn₂ : n₁ ⊆ m\nhn₃ : ∀ (x' : α), x' ∈ n₁ → ∃ n₂, n₂ ∈ B x' ∧ n₂ ⊆ m\n⊢ ∃ t, t ∈ FilterBasis.filter (B x) ∧ t ⊆ n ∧ ∀ (a' : α), a' ∈ t → n ∈ FilterBasis.filter (B a')",
"tactic": "refine'\n ⟨n₁, (B x).mem_filter_of_mem hn₁, hn₂.trans hm₂, fun x' hx' => (B x').mem_filter_iff.mp _⟩"
},
{
"state_after": "case h₁.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nB : α → FilterBasis α\na : α\nh₀ : ∀ (x : α) (n : Set α), n ∈ B x → x ∈ n\nh₁ : ∀ (x : α) (n : Set α), n ∈ B x → ∃ n₁, n₁ ∈ B x ∧ n₁ ⊆ n ∧ ∀ (x' : α), x' ∈ n₁ → ∃ n₂, n₂ ∈ B x' ∧ n₂ ⊆ n\nx : α\nn : Set α\nhn : n ∈ FilterBasis.filter (B x)\nm : Set α\nhm₁ : m ∈ B x\nhm₂ : m ⊆ n\nn₁ : Set α\nhn₁ : n₁ ∈ B x\nhn₂ : n₁ ⊆ m\nhn₃ : ∀ (x' : α), x' ∈ n₁ → ∃ n₂, n₂ ∈ B x' ∧ n₂ ⊆ m\nx' : α\nhx' : x' ∈ n₁\nn₂ : Set α\nhn₄ : n₂ ∈ B x'\nhn₅ : n₂ ⊆ m\n⊢ n ∈ FilterBasis.filter (B x')",
"state_before": "case h₁.intro.intro.intro.intro.intro\nα : Type u\nB : α → FilterBasis α\na : α\nh₀ : ∀ (x : α) (n : Set α), n ∈ B x → x ∈ n\nh₁ : ∀ (x : α) (n : Set α), n ∈ B x → ∃ n₁, n₁ ∈ B x ∧ n₁ ⊆ n ∧ ∀ (x' : α), x' ∈ n₁ → ∃ n₂, n₂ ∈ B x' ∧ n₂ ⊆ n\nx : α\nn : Set α\nhn : n ∈ FilterBasis.filter (B x)\nm : Set α\nhm₁ : m ∈ B x\nhm₂ : m ⊆ n\nn₁ : Set α\nhn₁ : n₁ ∈ B x\nhn₂ : n₁ ⊆ m\nhn₃ : ∀ (x' : α), x' ∈ n₁ → ∃ n₂, n₂ ∈ B x' ∧ n₂ ⊆ m\nx' : α\nhx' : x' ∈ n₁\n⊢ n ∈ FilterBasis.filter (B x')",
"tactic": "obtain ⟨n₂, hn₄, hn₅⟩ := hn₃ x' hx'"
},
{
"state_after": "no goals",
"state_before": "case h₁.intro.intro.intro.intro.intro.intro.intro\nα : Type u\nB : α → FilterBasis α\na : α\nh₀ : ∀ (x : α) (n : Set α), n ∈ B x → x ∈ n\nh₁ : ∀ (x : α) (n : Set α), n ∈ B x → ∃ n₁, n₁ ∈ B x ∧ n₁ ⊆ n ∧ ∀ (x' : α), x' ∈ n₁ → ∃ n₂, n₂ ∈ B x' ∧ n₂ ⊆ n\nx : α\nn : Set α\nhn : n ∈ FilterBasis.filter (B x)\nm : Set α\nhm₁ : m ∈ B x\nhm₂ : m ⊆ n\nn₁ : Set α\nhn₁ : n₁ ∈ B x\nhn₂ : n₁ ⊆ m\nhn₃ : ∀ (x' : α), x' ∈ n₁ → ∃ n₂, n₂ ∈ B x' ∧ n₂ ⊆ m\nx' : α\nhx' : x' ∈ n₁\nn₂ : Set α\nhn₄ : n₂ ∈ B x'\nhn₅ : n₂ ⊆ m\n⊢ n ∈ FilterBasis.filter (B x')",
"tactic": "exact ⟨n₂, hn₄, hn₅.trans hm₂⟩"
}
] |
[
150,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Mathlib/Init/Data/Nat/Lemmas.lean
|
Nat.bit0_ne_one
|
[] |
[
98,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
11
] |
Mathlib/Data/Polynomial/Laurent.lean
|
Polynomial.toLaurent_apply
|
[] |
[
109,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/Algebra/CharP/Two.lean
|
CharTwo.sub_eq_add'
|
[] |
[
80,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/Geometry/Manifold/ChartedSpace.lean
|
StructureGroupoid.subset_maximalAtlas
|
[] |
[
916,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
915,
1
] |
Mathlib/CategoryTheory/Types.lean
|
CategoryTheory.homOfElement_eq_iff
|
[
{
"state_after": "no goals",
"state_before": "X : Type u\nx y : X\n⊢ x = y → homOfElement x = homOfElement y",
"tactic": "aesop"
}
] |
[
236,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
235,
1
] |
Mathlib/Dynamics/Flow.lean
|
IsFwInvariant.isInvariant
|
[] |
[
71,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/Algebra/Order/Monoid/WithTop.lean
|
WithTop.map_add
|
[
{
"state_after": "case top\nα : Type u\nβ : Type v\ninst✝² : Add α\na b✝ c d : WithTop α\nx y : α\nF : Type u_1\ninst✝¹ : Add β\ninst✝ : AddHomClass F α β\nf : F\nb : WithTop α\n⊢ map (↑f) (⊤ + b) = map ↑f ⊤ + map (↑f) b\n\ncase coe\nα : Type u\nβ : Type v\ninst✝² : Add α\na b✝ c d : WithTop α\nx y : α\nF : Type u_1\ninst✝¹ : Add β\ninst✝ : AddHomClass F α β\nf : F\nb : WithTop α\na✝ : α\n⊢ map (↑f) (↑a✝ + b) = map ↑f ↑a✝ + map (↑f) b",
"state_before": "α : Type u\nβ : Type v\ninst✝² : Add α\na✝ b✝ c d : WithTop α\nx y : α\nF : Type u_1\ninst✝¹ : Add β\ninst✝ : AddHomClass F α β\nf : F\na b : WithTop α\n⊢ map (↑f) (a + b) = map (↑f) a + map (↑f) b",
"tactic": "induction a using WithTop.recTopCoe"
},
{
"state_after": "no goals",
"state_before": "case top\nα : Type u\nβ : Type v\ninst✝² : Add α\na b✝ c d : WithTop α\nx y : α\nF : Type u_1\ninst✝¹ : Add β\ninst✝ : AddHomClass F α β\nf : F\nb : WithTop α\n⊢ map (↑f) (⊤ + b) = map ↑f ⊤ + map (↑f) b",
"tactic": "exact (top_add _).symm"
},
{
"state_after": "case coe.top\nα : Type u\nβ : Type v\ninst✝² : Add α\na b c d : WithTop α\nx y : α\nF : Type u_1\ninst✝¹ : Add β\ninst✝ : AddHomClass F α β\nf : F\na✝ : α\n⊢ map (↑f) (↑a✝ + ⊤) = map ↑f ↑a✝ + map ↑f ⊤\n\ncase coe.coe\nα : Type u\nβ : Type v\ninst✝² : Add α\na b c d : WithTop α\nx y : α\nF : Type u_1\ninst✝¹ : Add β\ninst✝ : AddHomClass F α β\nf : F\na✝¹ a✝ : α\n⊢ map (↑f) (↑a✝¹ + ↑a✝) = map ↑f ↑a✝¹ + map ↑f ↑a✝",
"state_before": "case coe\nα : Type u\nβ : Type v\ninst✝² : Add α\na b✝ c d : WithTop α\nx y : α\nF : Type u_1\ninst✝¹ : Add β\ninst✝ : AddHomClass F α β\nf : F\nb : WithTop α\na✝ : α\n⊢ map (↑f) (↑a✝ + b) = map ↑f ↑a✝ + map (↑f) b",
"tactic": "induction b using WithTop.recTopCoe"
},
{
"state_after": "no goals",
"state_before": "case coe.top\nα : Type u\nβ : Type v\ninst✝² : Add α\na b c d : WithTop α\nx y : α\nF : Type u_1\ninst✝¹ : Add β\ninst✝ : AddHomClass F α β\nf : F\na✝ : α\n⊢ map (↑f) (↑a✝ + ⊤) = map ↑f ↑a✝ + map ↑f ⊤",
"tactic": "exact (add_top _).symm"
},
{
"state_after": "case coe.coe\nα : Type u\nβ : Type v\ninst✝² : Add α\na b c d : WithTop α\nx y : α\nF : Type u_1\ninst✝¹ : Add β\ninst✝ : AddHomClass F α β\nf : F\na✝¹ a✝ : α\n⊢ map ↑f ↑(a✝¹ + a✝) = ↑(↑f (a✝¹ + a✝))",
"state_before": "case coe.coe\nα : Type u\nβ : Type v\ninst✝² : Add α\na b c d : WithTop α\nx y : α\nF : Type u_1\ninst✝¹ : Add β\ninst✝ : AddHomClass F α β\nf : F\na✝¹ a✝ : α\n⊢ map (↑f) (↑a✝¹ + ↑a✝) = map ↑f ↑a✝¹ + map ↑f ↑a✝",
"tactic": "rw [map_coe, map_coe, ← coe_add, ← coe_add, ← map_add]"
},
{
"state_after": "no goals",
"state_before": "case coe.coe\nα : Type u\nβ : Type v\ninst✝² : Add α\na b c d : WithTop α\nx y : α\nF : Type u_1\ninst✝¹ : Add β\ninst✝ : AddHomClass F α β\nf : F\na✝¹ a✝ : α\n⊢ map ↑f ↑(a✝¹ + a✝) = ↑(↑f (a✝¹ + a✝))",
"tactic": "rfl"
}
] |
[
301,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
294,
11
] |
Mathlib/MeasureTheory/Constructions/Prod/Basic.lean
|
Measurable.lintegral_prod_right'
|
[
{
"state_after": "α : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nm :\n ∀ {α : Type ?u.3611327} {β : Type ?u.3611326} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α},\n Measurable (Prod.mk x)\n⊢ ∀ {f : α × β → ℝ≥0∞}, Measurable f → Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν",
"state_before": "α : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\n⊢ ∀ {f : α × β → ℝ≥0∞}, Measurable f → Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν",
"tactic": "have m := @measurable_prod_mk_left"
},
{
"state_after": "case refine'_1\nα : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nm :\n ∀ {α : Type ?u.3611327} {β : Type ?u.3611326} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α},\n Measurable (Prod.mk x)\n⊢ ∀ (c : ℝ≥0∞) ⦃s : Set (α × β)⦄,\n MeasurableSet s → (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) (Set.indicator s fun x => c)\n\ncase refine'_2\nα : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nm :\n ∀ {α : Type ?u.3611327} {β : Type ?u.3611326} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α},\n Measurable (Prod.mk x)\n⊢ ∀ ⦃f g : α × β → ℝ≥0∞⦄,\n Disjoint (support f) (support g) →\n Measurable f →\n Measurable g →\n (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) f →\n (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) g →\n (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) (f + g)\n\ncase refine'_3\nα : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nm :\n ∀ {α : Type ?u.3611327} {β : Type ?u.3611326} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α},\n Measurable (Prod.mk x)\n⊢ ∀ ⦃f : ℕ → α × β → ℝ≥0∞⦄,\n (∀ (n : ℕ), Measurable (f n)) →\n Monotone f →\n (∀ (n : ℕ), (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) (f n)) →\n (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) fun x => ⨆ (n : ℕ), f n x",
"state_before": "α : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nm :\n ∀ {α : Type ?u.3611327} {β : Type ?u.3611326} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α},\n Measurable (Prod.mk x)\n⊢ ∀ {f : α × β → ℝ≥0∞}, Measurable f → Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν",
"tactic": "refine' Measurable.ennreal_induction (P := fun f => Measurable fun (x : α) => ∫⁻ y, f (x, y) ∂ν)\n _ _ _"
},
{
"state_after": "case refine'_1\nα : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nm :\n ∀ {α : Type ?u.3611327} {β : Type ?u.3611326} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α},\n Measurable (Prod.mk x)\nc : ℝ≥0∞\ns : Set (α × β)\nhs : MeasurableSet s\n⊢ Measurable fun x => ∫⁻ (y : β), Set.indicator s (fun x => c) (x, y) ∂ν",
"state_before": "case refine'_1\nα : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nm :\n ∀ {α : Type ?u.3611327} {β : Type ?u.3611326} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α},\n Measurable (Prod.mk x)\n⊢ ∀ (c : ℝ≥0∞) ⦃s : Set (α × β)⦄,\n MeasurableSet s → (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) (Set.indicator s fun x => c)",
"tactic": "intro c s hs"
},
{
"state_after": "case refine'_1\nα : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nm :\n ∀ {α : Type ?u.3611327} {β : Type ?u.3611326} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α},\n Measurable (Prod.mk x)\nc : ℝ≥0∞\ns : Set (α × β)\nhs : MeasurableSet s\n⊢ Measurable fun x => ∫⁻ (y : β), Set.indicator (Prod.mk x ⁻¹' s) ((fun x => c) ∘ Prod.mk x) y ∂ν",
"state_before": "case refine'_1\nα : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nm :\n ∀ {α : Type ?u.3611327} {β : Type ?u.3611326} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α},\n Measurable (Prod.mk x)\nc : ℝ≥0∞\ns : Set (α × β)\nhs : MeasurableSet s\n⊢ Measurable fun x => ∫⁻ (y : β), Set.indicator s (fun x => c) (x, y) ∂ν",
"tactic": "simp only [← indicator_comp_right]"
},
{
"state_after": "case refine'_1\nα : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nm :\n ∀ {α : Type ?u.3611327} {β : Type ?u.3611326} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α},\n Measurable (Prod.mk x)\nc : ℝ≥0∞\ns : Set (α × β)\nhs : MeasurableSet s\n⊢ Measurable fun x => c * ↑↑ν (Prod.mk x ⁻¹' s)",
"state_before": "case refine'_1\nα : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nm :\n ∀ {α : Type ?u.3611327} {β : Type ?u.3611326} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α},\n Measurable (Prod.mk x)\nc : ℝ≥0∞\ns : Set (α × β)\nhs : MeasurableSet s\n⊢ Measurable fun x => ∫⁻ (y : β), Set.indicator (Prod.mk x ⁻¹' s) ((fun x => c) ∘ Prod.mk x) y ∂ν",
"tactic": "suffices Measurable fun x => c * ν (Prod.mk x ⁻¹' s) by simpa [lintegral_indicator _ (m hs)]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nm :\n ∀ {α : Type ?u.3611327} {β : Type ?u.3611326} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α},\n Measurable (Prod.mk x)\nc : ℝ≥0∞\ns : Set (α × β)\nhs : MeasurableSet s\n⊢ Measurable fun x => c * ↑↑ν (Prod.mk x ⁻¹' s)",
"tactic": "exact (measurable_measure_prod_mk_left hs).const_mul _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nm :\n ∀ {α : Type ?u.3611327} {β : Type ?u.3611326} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α},\n Measurable (Prod.mk x)\nc : ℝ≥0∞\ns : Set (α × β)\nhs : MeasurableSet s\nthis : Measurable fun x => c * ↑↑ν (Prod.mk x ⁻¹' s)\n⊢ Measurable fun x => ∫⁻ (y : β), Set.indicator (Prod.mk x ⁻¹' s) ((fun x => c) ∘ Prod.mk x) y ∂ν",
"tactic": "simpa [lintegral_indicator _ (m hs)]"
},
{
"state_after": "case refine'_2\nα : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nm :\n ∀ {α : Type ?u.3611327} {β : Type ?u.3611326} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α},\n Measurable (Prod.mk x)\nf g : α × β → ℝ≥0∞\nhf : Measurable f\nh2f : Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν\nh2g : Measurable fun x => ∫⁻ (y : β), g (x, y) ∂ν\n⊢ Measurable fun x => ∫⁻ (y : β), (f + g) (x, y) ∂ν",
"state_before": "case refine'_2\nα : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nm :\n ∀ {α : Type ?u.3611327} {β : Type ?u.3611326} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α},\n Measurable (Prod.mk x)\n⊢ ∀ ⦃f g : α × β → ℝ≥0∞⦄,\n Disjoint (support f) (support g) →\n Measurable f →\n Measurable g →\n (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) f →\n (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) g →\n (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) (f + g)",
"tactic": "rintro f g - hf - h2f h2g"
},
{
"state_after": "case refine'_2\nα : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nm :\n ∀ {α : Type ?u.3611327} {β : Type ?u.3611326} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α},\n Measurable (Prod.mk x)\nf g : α × β → ℝ≥0∞\nhf : Measurable f\nh2f : Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν\nh2g : Measurable fun x => ∫⁻ (y : β), g (x, y) ∂ν\n⊢ Measurable fun x => ∫⁻ (y : β), f (x, y) + g (x, y) ∂ν",
"state_before": "case refine'_2\nα : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nm :\n ∀ {α : Type ?u.3611327} {β : Type ?u.3611326} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α},\n Measurable (Prod.mk x)\nf g : α × β → ℝ≥0∞\nhf : Measurable f\nh2f : Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν\nh2g : Measurable fun x => ∫⁻ (y : β), g (x, y) ∂ν\n⊢ Measurable fun x => ∫⁻ (y : β), (f + g) (x, y) ∂ν",
"tactic": "simp only [Pi.add_apply]"
},
{
"state_after": "case refine'_2\nα : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nm : ∀ {α : Type u_2} {β : Type u_1} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x)\nf g : α × β → ℝ≥0∞\nhf : Measurable f\nh2f : Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν\nh2g : Measurable fun x => ∫⁻ (y : β), g (x, y) ∂ν\n⊢ Measurable fun x => (∫⁻ (a : β), (f ∘ Prod.mk x) a ∂ν) + ∫⁻ (a : β), g (x, a) ∂ν",
"state_before": "case refine'_2\nα : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nm :\n ∀ {α : Type ?u.3611327} {β : Type ?u.3611326} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α},\n Measurable (Prod.mk x)\nf g : α × β → ℝ≥0∞\nhf : Measurable f\nh2f : Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν\nh2g : Measurable fun x => ∫⁻ (y : β), g (x, y) ∂ν\n⊢ Measurable fun x => ∫⁻ (y : β), f (x, y) + g (x, y) ∂ν",
"tactic": "conv => enter [1, x]; erw [lintegral_add_left (hf.comp m)]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nm : ∀ {α : Type u_2} {β : Type u_1} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x)\nf g : α × β → ℝ≥0∞\nhf : Measurable f\nh2f : Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν\nh2g : Measurable fun x => ∫⁻ (y : β), g (x, y) ∂ν\n⊢ Measurable fun x => (∫⁻ (a : β), (f ∘ Prod.mk x) a ∂ν) + ∫⁻ (a : β), g (x, a) ∂ν",
"tactic": "exact h2f.add h2g"
},
{
"state_after": "case refine'_3\nα : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nm : ∀ {α : Type u_2} {β : Type u_1} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x)\nf : ℕ → α × β → ℝ≥0∞\nhf : ∀ (n : ℕ), Measurable (f n)\nh2f : Monotone f\nh3f : ∀ (n : ℕ), (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) (f n)\n⊢ Measurable fun x => ∫⁻ (y : β), (fun x => ⨆ (n : ℕ), f n x) (x, y) ∂ν",
"state_before": "case refine'_3\nα : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nm : ∀ {α : Type u_2} {β : Type u_1} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x)\n⊢ ∀ ⦃f : ℕ → α × β → ℝ≥0∞⦄,\n (∀ (n : ℕ), Measurable (f n)) →\n Monotone f →\n (∀ (n : ℕ), (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) (f n)) →\n (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) fun x => ⨆ (n : ℕ), f n x",
"tactic": "intro f hf h2f h3f"
},
{
"state_after": "case refine'_3\nα : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nm : ∀ {α : Type u_2} {β : Type u_1} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x)\nf : ℕ → α × β → ℝ≥0∞\nhf : ∀ (n : ℕ), Measurable (f n)\nh2f : Monotone f\nh3f : ∀ (n : ℕ), (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) (f n)\nthis : Measurable fun b => ⨆ (i : ℕ), ∫⁻ (y : β), f i (b, y) ∂ν\n⊢ Measurable fun x => ∫⁻ (y : β), (fun x => ⨆ (n : ℕ), f n x) (x, y) ∂ν",
"state_before": "case refine'_3\nα : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nm : ∀ {α : Type u_2} {β : Type u_1} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x)\nf : ℕ → α × β → ℝ≥0∞\nhf : ∀ (n : ℕ), Measurable (f n)\nh2f : Monotone f\nh3f : ∀ (n : ℕ), (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) (f n)\n⊢ Measurable fun x => ∫⁻ (y : β), (fun x => ⨆ (n : ℕ), f n x) (x, y) ∂ν",
"tactic": "have := measurable_iSup h3f"
},
{
"state_after": "case refine'_3\nα : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nm : ∀ {α : Type u_2} {β : Type u_1} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x)\nf : ℕ → α × β → ℝ≥0∞\nhf : ∀ (n : ℕ), Measurable (f n)\nh2f : Monotone f\nh3f : ∀ (n : ℕ), (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) (f n)\nthis✝ : Measurable fun b => ⨆ (i : ℕ), ∫⁻ (y : β), f i (b, y) ∂ν\nthis : ∀ (x : α), Monotone fun n y => f n (x, y)\n⊢ Measurable fun x => ∫⁻ (y : β), (fun x => ⨆ (n : ℕ), f n x) (x, y) ∂ν",
"state_before": "case refine'_3\nα : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nm : ∀ {α : Type u_2} {β : Type u_1} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x)\nf : ℕ → α × β → ℝ≥0∞\nhf : ∀ (n : ℕ), Measurable (f n)\nh2f : Monotone f\nh3f : ∀ (n : ℕ), (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) (f n)\nthis : Measurable fun b => ⨆ (i : ℕ), ∫⁻ (y : β), f i (b, y) ∂ν\n⊢ Measurable fun x => ∫⁻ (y : β), (fun x => ⨆ (n : ℕ), f n x) (x, y) ∂ν",
"tactic": "have : ∀ x, Monotone fun n y => f n (x, y) := fun x i j hij y => h2f hij (x, y)"
},
{
"state_after": "case refine'_3\nα : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nm : ∀ {α : Type u_2} {β : Type u_1} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x)\nf : ℕ → α × β → ℝ≥0∞\nhf : ∀ (n : ℕ), Measurable (f n)\nh2f : Monotone f\nh3f : ∀ (n : ℕ), (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) (f n)\nthis✝ : Measurable fun b => ⨆ (i : ℕ), ∫⁻ (y : β), f i (b, y) ∂ν\nthis : ∀ (x : α), Monotone fun n y => f n (x, y)\n⊢ Measurable fun x => ⨆ (n : ℕ), ∫⁻ (a : β), (f n ∘ Prod.mk x) a ∂ν",
"state_before": "case refine'_3\nα : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nm : ∀ {α : Type u_2} {β : Type u_1} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x)\nf : ℕ → α × β → ℝ≥0∞\nhf : ∀ (n : ℕ), Measurable (f n)\nh2f : Monotone f\nh3f : ∀ (n : ℕ), (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) (f n)\nthis✝ : Measurable fun b => ⨆ (i : ℕ), ∫⁻ (y : β), f i (b, y) ∂ν\nthis : ∀ (x : α), Monotone fun n y => f n (x, y)\n⊢ Measurable fun x => ∫⁻ (y : β), (fun x => ⨆ (n : ℕ), f n x) (x, y) ∂ν",
"tactic": "conv => enter [1, x]; erw [lintegral_iSup (fun n => (hf n).comp m) (this x)]"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nα : Type u_2\nα' : Type ?u.3342539\nβ : Type u_1\nβ' : Type ?u.3342545\nγ : Type ?u.3342548\nE : Type ?u.3342551\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ μ' : MeasureTheory.Measure α\nν ν' : MeasureTheory.Measure β\nτ : MeasureTheory.Measure γ\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SigmaFinite ν\nm : ∀ {α : Type u_2} {β : Type u_1} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x)\nf : ℕ → α × β → ℝ≥0∞\nhf : ∀ (n : ℕ), Measurable (f n)\nh2f : Monotone f\nh3f : ∀ (n : ℕ), (fun f => Measurable fun x => ∫⁻ (y : β), f (x, y) ∂ν) (f n)\nthis✝ : Measurable fun b => ⨆ (i : ℕ), ∫⁻ (y : β), f i (b, y) ∂ν\nthis : ∀ (x : α), Monotone fun n y => f n (x, y)\n⊢ Measurable fun x => ⨆ (n : ℕ), ∫⁻ (a : β), (f n ∘ Prod.mk x) a ∂ν",
"tactic": "assumption"
}
] |
[
261,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
244,
1
] |
Mathlib/Algebra/CharP/Basic.lean
|
CharP.neg_one_pow_char
|
[
{
"state_after": "R : Type u_1\ninst✝² : CommRing R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (Nat.Prime p)\n⊢ (-1) ^ p + 1 = 0",
"state_before": "R : Type u_1\ninst✝² : CommRing R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (Nat.Prime p)\n⊢ (-1) ^ p = -1",
"tactic": "rw [eq_neg_iff_add_eq_zero]"
},
{
"state_after": "R : Type u_1\ninst✝² : CommRing R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (Nat.Prime p)\n⊢ (-1) ^ p + 1 ^ p = 0",
"state_before": "R : Type u_1\ninst✝² : CommRing R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (Nat.Prime p)\n⊢ (-1) ^ p + 1 = 0",
"tactic": "nth_rw 2 [← one_pow p]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝² : CommRing R\np : ℕ\ninst✝¹ : CharP R p\ninst✝ : Fact (Nat.Prime p)\n⊢ (-1) ^ p + 1 ^ p = 0",
"tactic": "rw [← add_pow_char, add_left_neg, zero_pow (Fact.out (p := Nat.Prime p)).pos]"
}
] |
[
315,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
311,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
IntervalIntegrable.trans
|
[] |
[
154,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
1
] |
Mathlib/MeasureTheory/Function/UniformIntegrable.lean
|
MeasureTheory.UniformIntegrable.aeStronglyMeasurable
|
[] |
[
83,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
11
] |
Mathlib/Order/Hom/Bounded.lean
|
BoundedOrderHom.coe_id
|
[] |
[
630,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
629,
1
] |
Mathlib/Analysis/BoxIntegral/Basic.lean
|
BoxIntegral.Integrable.of_neg
|
[] |
[
288,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
287,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean
|
ZFSet.omega_zero
|
[] |
[
948,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
947,
1
] |
Mathlib/NumberTheory/Padics/PadicVal.lean
|
padicValInt.mul
|
[
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\na b : ℤ\nha : a ≠ 0\nhb : b ≠ 0\n⊢ padicValNat p (Int.natAbs (a * b)) = padicValNat p (Int.natAbs a) + padicValNat p (Int.natAbs b)",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\na b : ℤ\nha : a ≠ 0\nhb : b ≠ 0\n⊢ padicValInt p (a * b) = padicValInt p a + padicValInt p b",
"tactic": "simp_rw [padicValInt]"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\na b : ℤ\nha : a ≠ 0\nhb : b ≠ 0\n⊢ padicValNat p (Int.natAbs (a * b)) = padicValNat p (Int.natAbs a) + padicValNat p (Int.natAbs b)",
"tactic": "rw [Int.natAbs_mul, padicValNat.mul] <;> rwa [Int.natAbs_ne_zero]"
}
] |
[
533,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
530,
1
] |
Mathlib/Order/JordanHolder.lean
|
CompositionSeries.append_castAdd_aux
|
[
{
"state_after": "case mk\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\nα : Type u_1\nm n : ℕ\na : Fin (Nat.succ m) → α\nb : Fin (Nat.succ n) → α\nval✝ : ℕ\nisLt✝ : val✝ < m\n⊢ Matrix.vecAppend (_ : Nat.succ (m + n) = m + Nat.succ n) (a ∘ ↑Fin.castSucc) b\n (↑Fin.castSucc (↑(Fin.castAdd n) { val := val✝, isLt := isLt✝ })) =\n a (↑Fin.castSucc { val := val✝, isLt := isLt✝ })",
"state_before": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\nα : Type u_1\nm n : ℕ\na : Fin (Nat.succ m) → α\nb : Fin (Nat.succ n) → α\ni : Fin m\n⊢ Matrix.vecAppend (_ : Nat.succ (m + n) = m + Nat.succ n) (a ∘ ↑Fin.castSucc) b (↑Fin.castSucc (↑(Fin.castAdd n) i)) =\n a (↑Fin.castSucc i)",
"tactic": "cases i"
},
{
"state_after": "no goals",
"state_before": "case mk\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\nα : Type u_1\nm n : ℕ\na : Fin (Nat.succ m) → α\nb : Fin (Nat.succ n) → α\nval✝ : ℕ\nisLt✝ : val✝ < m\n⊢ Matrix.vecAppend (_ : Nat.succ (m + n) = m + Nat.succ n) (a ∘ ↑Fin.castSucc) b\n (↑Fin.castSucc (↑(Fin.castAdd n) { val := val✝, isLt := isLt✝ })) =\n a (↑Fin.castSucc { val := val✝, isLt := isLt✝ })",
"tactic": "simp [Matrix.vecAppend_eq_ite, *]"
}
] |
[
462,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
457,
1
] |
Mathlib/Data/Set/Function.lean
|
StrictAntiOn.strictAnti
|
[] |
[
332,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
330,
11
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.eval₂_smul
|
[
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np✝ q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nx✝ : S\ng : R →+* S\np : R[X]\nx : S\ns : R\nA : natDegree p < Nat.succ (natDegree p)\n⊢ eval₂ g x (s • p) = ↑g s * eval₂ g x p",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np✝ q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nx✝ : S\ng : R →+* S\np : R[X]\nx : S\ns : R\n⊢ eval₂ g x (s • p) = ↑g s * eval₂ g x p",
"tactic": "have A : p.natDegree < p.natDegree.succ := Nat.lt_succ_self _"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np✝ q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nx✝ : S\ng : R →+* S\np : R[X]\nx : S\ns : R\nA : natDegree p < Nat.succ (natDegree p)\nB : natDegree (s • p) < Nat.succ (natDegree p)\n⊢ eval₂ g x (s • p) = ↑g s * eval₂ g x p",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np✝ q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nx✝ : S\ng : R →+* S\np : R[X]\nx : S\ns : R\nA : natDegree p < Nat.succ (natDegree p)\n⊢ eval₂ g x (s • p) = ↑g s * eval₂ g x p",
"tactic": "have B : (s • p).natDegree < p.natDegree.succ := (natDegree_smul_le _ _).trans_lt A"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np✝ q r : R[X]\ninst✝ : Semiring S\nf : R →+* S\nx✝ : S\ng : R →+* S\np : R[X]\nx : S\ns : R\nA : natDegree p < Nat.succ (natDegree p)\nB : natDegree (s • p) < Nat.succ (natDegree p)\n⊢ eval₂ g x (s • p) = ↑g s * eval₂ g x p",
"tactic": "rw [eval₂_eq_sum, eval₂_eq_sum, sum_over_range' _ _ _ A, sum_over_range' _ _ _ B] <;>\n simp [mul_sum, mul_assoc]"
}
] |
[
119,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/Order/Filter/Bases.lean
|
Filter.hasBasis_iInf_principal
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.62085\nγ : Type ?u.62088\nι : Sort u_2\nι' : Sort ?u.62094\nl l' : Filter α\np : ι → Prop\ns✝ : ι → Set α\nt✝ : Set α\ni : ι\np' : ι' → Prop\ns' : ι' → Set α\ni' : ι'\ns : ι → Set α\nh : Directed (fun x x_1 => x ≥ x_1) s\ninst✝ : Nonempty ι\nt : Set α\n⊢ (t ∈ ⨅ (i : ι), 𝓟 (s i)) ↔ ∃ i, True ∧ s i ⊆ t",
"tactic": "simpa only [true_and] using mem_iInf_of_directed (h.mono_comp monotone_principal.dual) t"
}
] |
[
764,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
761,
1
] |
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