file_path
stringlengths 11
79
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stringclasses 4
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Mathlib/LinearAlgebra/Basic.lean
|
LinearMap.pi_apply_eq_sum_univ
|
[
{
"state_after": "R : Type u_2\nR₁ : Type ?u.256942\nR₂ : Type ?u.256945\nR₃ : Type ?u.256948\nR₄ : Type ?u.256951\nS : Type ?u.256954\nK : Type ?u.256957\nK₂ : Type ?u.256960\nM : Type u_3\nM' : Type ?u.256966\nM₁ : Type ?u.256969\nM₂ : Type ?u.256972\nM₃ : Type ?u.256975\nM₄ : Type ?u.256978\nN : Type ?u.256981\nN₂ : Type ?u.256984\nι : Type u_1\nV : Type ?u.256990\nV₂ : Type ?u.256993\ninst✝¹⁹ : Semiring R\ninst✝¹⁸ : Semiring R₂\ninst✝¹⁷ : Semiring R₃\ninst✝¹⁶ : Semiring R₄\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid M₃\ninst✝¹¹ : AddCommMonoid M₄\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R₂ M₂\ninst✝⁷ : Module R₃ M₃\ninst✝⁶ : Module R₄ M₄\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₃₄ : R₃ →+* R₄\nσ₁₃ : R →+* R₃\nσ₂₄ : R₂ →+* R₄\nσ₁₄ : R →+* R₄\ninst✝⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁴ : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄\ninst✝³ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄\ninst✝² : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄\nf✝ : M →ₛₗ[σ₁₂] M₂\ng : M₂ →ₛₗ[σ₂₃] M₃\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nf : (ι → R) →ₗ[R] M\nx : ι → R\n⊢ ∑ i : ι, ↑f (x i • fun j => if i = j then 1 else 0) = ∑ i : ι, x i • ↑f fun j => if i = j then 1 else 0",
"state_before": "R : Type u_2\nR₁ : Type ?u.256942\nR₂ : Type ?u.256945\nR₃ : Type ?u.256948\nR₄ : Type ?u.256951\nS : Type ?u.256954\nK : Type ?u.256957\nK₂ : Type ?u.256960\nM : Type u_3\nM' : Type ?u.256966\nM₁ : Type ?u.256969\nM₂ : Type ?u.256972\nM₃ : Type ?u.256975\nM₄ : Type ?u.256978\nN : Type ?u.256981\nN₂ : Type ?u.256984\nι : Type u_1\nV : Type ?u.256990\nV₂ : Type ?u.256993\ninst✝¹⁹ : Semiring R\ninst✝¹⁸ : Semiring R₂\ninst✝¹⁷ : Semiring R₃\ninst✝¹⁶ : Semiring R₄\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid M₃\ninst✝¹¹ : AddCommMonoid M₄\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R₂ M₂\ninst✝⁷ : Module R₃ M₃\ninst✝⁶ : Module R₄ M₄\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₃₄ : R₃ →+* R₄\nσ₁₃ : R →+* R₃\nσ₂₄ : R₂ →+* R₄\nσ₁₄ : R →+* R₄\ninst✝⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁴ : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄\ninst✝³ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄\ninst✝² : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄\nf✝ : M →ₛₗ[σ₁₂] M₂\ng : M₂ →ₛₗ[σ₂₃] M₃\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nf : (ι → R) →ₗ[R] M\nx : ι → R\n⊢ ↑f x = ∑ i : ι, x i • ↑f fun j => if i = j then 1 else 0",
"tactic": "conv_lhs => rw [pi_eq_sum_univ x, f.map_sum]"
},
{
"state_after": "R : Type u_2\nR₁ : Type ?u.256942\nR₂ : Type ?u.256945\nR₃ : Type ?u.256948\nR₄ : Type ?u.256951\nS : Type ?u.256954\nK : Type ?u.256957\nK₂ : Type ?u.256960\nM : Type u_3\nM' : Type ?u.256966\nM₁ : Type ?u.256969\nM₂ : Type ?u.256972\nM₃ : Type ?u.256975\nM₄ : Type ?u.256978\nN : Type ?u.256981\nN₂ : Type ?u.256984\nι : Type u_1\nV : Type ?u.256990\nV₂ : Type ?u.256993\ninst✝¹⁹ : Semiring R\ninst✝¹⁸ : Semiring R₂\ninst✝¹⁷ : Semiring R₃\ninst✝¹⁶ : Semiring R₄\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid M₃\ninst✝¹¹ : AddCommMonoid M₄\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R₂ M₂\ninst✝⁷ : Module R₃ M₃\ninst✝⁶ : Module R₄ M₄\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₃₄ : R₃ →+* R₄\nσ₁₃ : R →+* R₃\nσ₂₄ : R₂ →+* R₄\nσ₁₄ : R →+* R₄\ninst✝⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁴ : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄\ninst✝³ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄\ninst✝² : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄\nf✝ : M →ₛₗ[σ₁₂] M₂\ng : M₂ →ₛₗ[σ₂₃] M₃\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nf : (ι → R) →ₗ[R] M\nx : ι → R\nx✝¹ : ι\nx✝ : x✝¹ ∈ Finset.univ\n⊢ ↑f (x x✝¹ • fun j => if x✝¹ = j then 1 else 0) = x x✝¹ • ↑f fun j => if x✝¹ = j then 1 else 0",
"state_before": "R : Type u_2\nR₁ : Type ?u.256942\nR₂ : Type ?u.256945\nR₃ : Type ?u.256948\nR₄ : Type ?u.256951\nS : Type ?u.256954\nK : Type ?u.256957\nK₂ : Type ?u.256960\nM : Type u_3\nM' : Type ?u.256966\nM₁ : Type ?u.256969\nM₂ : Type ?u.256972\nM₃ : Type ?u.256975\nM₄ : Type ?u.256978\nN : Type ?u.256981\nN₂ : Type ?u.256984\nι : Type u_1\nV : Type ?u.256990\nV₂ : Type ?u.256993\ninst✝¹⁹ : Semiring R\ninst✝¹⁸ : Semiring R₂\ninst✝¹⁷ : Semiring R₃\ninst✝¹⁶ : Semiring R₄\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid M₃\ninst✝¹¹ : AddCommMonoid M₄\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R₂ M₂\ninst✝⁷ : Module R₃ M₃\ninst✝⁶ : Module R₄ M₄\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₃₄ : R₃ →+* R₄\nσ₁₃ : R →+* R₃\nσ₂₄ : R₂ →+* R₄\nσ₁₄ : R →+* R₄\ninst✝⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁴ : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄\ninst✝³ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄\ninst✝² : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄\nf✝ : M →ₛₗ[σ₁₂] M₂\ng : M₂ →ₛₗ[σ₂₃] M₃\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nf : (ι → R) →ₗ[R] M\nx : ι → R\n⊢ ∑ i : ι, ↑f (x i • fun j => if i = j then 1 else 0) = ∑ i : ι, x i • ↑f fun j => if i = j then 1 else 0",
"tactic": "refine Finset.sum_congr rfl (fun _ _ => ?_)"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nR₁ : Type ?u.256942\nR₂ : Type ?u.256945\nR₃ : Type ?u.256948\nR₄ : Type ?u.256951\nS : Type ?u.256954\nK : Type ?u.256957\nK₂ : Type ?u.256960\nM : Type u_3\nM' : Type ?u.256966\nM₁ : Type ?u.256969\nM₂ : Type ?u.256972\nM₃ : Type ?u.256975\nM₄ : Type ?u.256978\nN : Type ?u.256981\nN₂ : Type ?u.256984\nι : Type u_1\nV : Type ?u.256990\nV₂ : Type ?u.256993\ninst✝¹⁹ : Semiring R\ninst✝¹⁸ : Semiring R₂\ninst✝¹⁷ : Semiring R₃\ninst✝¹⁶ : Semiring R₄\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : AddCommMonoid M₁\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : AddCommMonoid M₃\ninst✝¹¹ : AddCommMonoid M₄\ninst✝¹⁰ : Module R M\ninst✝⁹ : Module R M₁\ninst✝⁸ : Module R₂ M₂\ninst✝⁷ : Module R₃ M₃\ninst✝⁶ : Module R₄ M₄\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₃₄ : R₃ →+* R₄\nσ₁₃ : R →+* R₃\nσ₂₄ : R₂ →+* R₄\nσ₁₄ : R →+* R₄\ninst✝⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁴ : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄\ninst✝³ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄\ninst✝² : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄\nf✝ : M →ₛₗ[σ₁₂] M₂\ng : M₂ →ₛₗ[σ₂₃] M₃\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nf : (ι → R) →ₗ[R] M\nx : ι → R\nx✝¹ : ι\nx✝ : x✝¹ ∈ Finset.univ\n⊢ ↑f (x x✝¹ • fun j => if x✝¹ = j then 1 else 0) = x x✝¹ • ↑f fun j => if x✝¹ = j then 1 else 0",
"tactic": "rw [map_smul]"
}
] |
[
433,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
429,
1
] |
Mathlib/Algebra/Order/Ring/Defs.lean
|
mul_pos_iff
|
[] |
[
1054,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1052,
1
] |
Mathlib/Algebra/Order/Sub/Canonical.lean
|
tsub_eq_iff_eq_add_of_le
|
[] |
[
213,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/Algebra/MonoidAlgebra/Division.lean
|
AddMonoidAlgebra.of'_mul_divOf
|
[
{
"state_after": "k : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCancelCommMonoid G\na : G\nx : AddMonoidAlgebra k G\nx✝ : G\n⊢ ↑(of' k G a * x /ᵒᶠ a) x✝ = ↑x x✝",
"state_before": "k : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCancelCommMonoid G\na : G\nx : AddMonoidAlgebra k G\n⊢ of' k G a * x /ᵒᶠ a = x",
"tactic": "refine Finsupp.ext fun _ => ?_"
},
{
"state_after": "case H\nk : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCancelCommMonoid G\na : G\nx : AddMonoidAlgebra k G\nx✝ : G\n⊢ ∀ (a_1 : G), a + a_1 = a + x✝ ↔ a_1 = x✝",
"state_before": "k : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCancelCommMonoid G\na : G\nx : AddMonoidAlgebra k G\nx✝ : G\n⊢ ↑(of' k G a * x /ᵒᶠ a) x✝ = ↑x x✝",
"tactic": "rw [AddMonoidAlgebra.divOf_apply, of'_apply, single_mul_apply_aux, one_mul]"
},
{
"state_after": "case H\nk : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCancelCommMonoid G\na : G\nx : AddMonoidAlgebra k G\nx✝ c : G\n⊢ a + c = a + x✝ ↔ c = x✝",
"state_before": "case H\nk : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCancelCommMonoid G\na : G\nx : AddMonoidAlgebra k G\nx✝ : G\n⊢ ∀ (a_1 : G), a + a_1 = a + x✝ ↔ a_1 = x✝",
"tactic": "intro c"
},
{
"state_after": "no goals",
"state_before": "case H\nk : Type u_1\nG : Type u_2\ninst✝¹ : Semiring k\ninst✝ : AddCancelCommMonoid G\na : G\nx : AddMonoidAlgebra k G\nx✝ c : G\n⊢ a + c = a + x✝ ↔ c = x✝",
"tactic": "exact add_right_inj _"
}
] |
[
113,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
109,
1
] |
Mathlib/Data/Set/Pointwise/SMul.lean
|
Set.smul_inter_ne_empty_iff
|
[
{
"state_after": "F : Type ?u.137905\nα : Type u_1\nβ : Type ?u.137911\nγ : Type ?u.137914\ninst✝¹ : Group α\ninst✝ : MulAction α β\ns✝ t✝ A B : Set β\na : α\nx✝ : β\ns t : Set α\nx : α\n⊢ Set.Nonempty (x • s ∩ t) ↔ ∃ a b, (a ∈ t ∧ b ∈ s) ∧ a * b⁻¹ = x",
"state_before": "F : Type ?u.137905\nα : Type u_1\nβ : Type ?u.137911\nγ : Type ?u.137914\ninst✝¹ : Group α\ninst✝ : MulAction α β\ns✝ t✝ A B : Set β\na : α\nx✝ : β\ns t : Set α\nx : α\n⊢ x • s ∩ t ≠ ∅ ↔ ∃ a b, (a ∈ t ∧ b ∈ s) ∧ a * b⁻¹ = x",
"tactic": "rw [← nonempty_iff_ne_empty]"
},
{
"state_after": "case mp\nF : Type ?u.137905\nα : Type u_1\nβ : Type ?u.137911\nγ : Type ?u.137914\ninst✝¹ : Group α\ninst✝ : MulAction α β\ns✝ t✝ A B : Set β\na : α\nx✝ : β\ns t : Set α\nx : α\n⊢ Set.Nonempty (x • s ∩ t) → ∃ a b, (a ∈ t ∧ b ∈ s) ∧ a * b⁻¹ = x\n\ncase mpr\nF : Type ?u.137905\nα : Type u_1\nβ : Type ?u.137911\nγ : Type ?u.137914\ninst✝¹ : Group α\ninst✝ : MulAction α β\ns✝ t✝ A B : Set β\na : α\nx✝ : β\ns t : Set α\nx : α\n⊢ (∃ a b, (a ∈ t ∧ b ∈ s) ∧ a * b⁻¹ = x) → Set.Nonempty (x • s ∩ t)",
"state_before": "F : Type ?u.137905\nα : Type u_1\nβ : Type ?u.137911\nγ : Type ?u.137914\ninst✝¹ : Group α\ninst✝ : MulAction α β\ns✝ t✝ A B : Set β\na : α\nx✝ : β\ns t : Set α\nx : α\n⊢ Set.Nonempty (x • s ∩ t) ↔ ∃ a b, (a ∈ t ∧ b ∈ s) ∧ a * b⁻¹ = x",
"tactic": "constructor"
},
{
"state_after": "case mp.intro.intro\nF : Type ?u.137905\nα : Type u_1\nβ : Type ?u.137911\nγ : Type ?u.137914\ninst✝¹ : Group α\ninst✝ : MulAction α β\ns✝ t✝ A B : Set β\na✝ : α\nx✝ : β\ns t : Set α\nx a : α\nh : a ∈ x • s\nha : a ∈ t\n⊢ ∃ a b, (a ∈ t ∧ b ∈ s) ∧ a * b⁻¹ = x",
"state_before": "case mp\nF : Type ?u.137905\nα : Type u_1\nβ : Type ?u.137911\nγ : Type ?u.137914\ninst✝¹ : Group α\ninst✝ : MulAction α β\ns✝ t✝ A B : Set β\na : α\nx✝ : β\ns t : Set α\nx : α\n⊢ Set.Nonempty (x • s ∩ t) → ∃ a b, (a ∈ t ∧ b ∈ s) ∧ a * b⁻¹ = x",
"tactic": "rintro ⟨a, h, ha⟩"
},
{
"state_after": "case mp.intro.intro.intro.intro\nF : Type ?u.137905\nα : Type u_1\nβ : Type ?u.137911\nγ : Type ?u.137914\ninst✝¹ : Group α\ninst✝ : MulAction α β\ns✝ t✝ A B : Set β\na : α\nx✝ : β\ns t : Set α\nx b : α\nhb : b ∈ s\nh : x • b ∈ x • s\nha : x • b ∈ t\n⊢ ∃ a b, (a ∈ t ∧ b ∈ s) ∧ a * b⁻¹ = x",
"state_before": "case mp.intro.intro\nF : Type ?u.137905\nα : Type u_1\nβ : Type ?u.137911\nγ : Type ?u.137914\ninst✝¹ : Group α\ninst✝ : MulAction α β\ns✝ t✝ A B : Set β\na✝ : α\nx✝ : β\ns t : Set α\nx a : α\nh : a ∈ x • s\nha : a ∈ t\n⊢ ∃ a b, (a ∈ t ∧ b ∈ s) ∧ a * b⁻¹ = x",
"tactic": "obtain ⟨b, hb, rfl⟩ := mem_smul_set.mp h"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro\nF : Type ?u.137905\nα : Type u_1\nβ : Type ?u.137911\nγ : Type ?u.137914\ninst✝¹ : Group α\ninst✝ : MulAction α β\ns✝ t✝ A B : Set β\na : α\nx✝ : β\ns t : Set α\nx b : α\nhb : b ∈ s\nh : x • b ∈ x • s\nha : x • b ∈ t\n⊢ ∃ a b, (a ∈ t ∧ b ∈ s) ∧ a * b⁻¹ = x",
"tactic": "exact ⟨x • b, b, ⟨ha, hb⟩, by simp⟩"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.137905\nα : Type u_1\nβ : Type ?u.137911\nγ : Type ?u.137914\ninst✝¹ : Group α\ninst✝ : MulAction α β\ns✝ t✝ A B : Set β\na : α\nx✝ : β\ns t : Set α\nx b : α\nhb : b ∈ s\nh : x • b ∈ x • s\nha : x • b ∈ t\n⊢ x • b * b⁻¹ = x",
"tactic": "simp"
},
{
"state_after": "case mpr.intro.intro.intro.intro\nF : Type ?u.137905\nα : Type u_1\nβ : Type ?u.137911\nγ : Type ?u.137914\ninst✝¹ : Group α\ninst✝ : MulAction α β\ns✝ t✝ A B : Set β\na✝ : α\nx : β\ns t : Set α\na b : α\nha : a ∈ t\nhb : b ∈ s\n⊢ Set.Nonempty ((a * b⁻¹) • s ∩ t)",
"state_before": "case mpr\nF : Type ?u.137905\nα : Type u_1\nβ : Type ?u.137911\nγ : Type ?u.137914\ninst✝¹ : Group α\ninst✝ : MulAction α β\ns✝ t✝ A B : Set β\na : α\nx✝ : β\ns t : Set α\nx : α\n⊢ (∃ a b, (a ∈ t ∧ b ∈ s) ∧ a * b⁻¹ = x) → Set.Nonempty (x • s ∩ t)",
"tactic": "rintro ⟨a, b, ⟨ha, hb⟩, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.intro.intro\nF : Type ?u.137905\nα : Type u_1\nβ : Type ?u.137911\nγ : Type ?u.137914\ninst✝¹ : Group α\ninst✝ : MulAction α β\ns✝ t✝ A B : Set β\na✝ : α\nx : β\ns t : Set α\na b : α\nha : a ∈ t\nhb : b ∈ s\n⊢ Set.Nonempty ((a * b⁻¹) • s ∩ t)",
"tactic": "exact ⟨a, mem_inter (mem_smul_set.mpr ⟨b, hb, by simp⟩) ha⟩"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.137905\nα : Type u_1\nβ : Type ?u.137911\nγ : Type ?u.137914\ninst✝¹ : Group α\ninst✝ : MulAction α β\ns✝ t✝ A B : Set β\na✝ : α\nx : β\ns t : Set α\na b : α\nha : a ∈ t\nhb : b ∈ s\n⊢ (a * b⁻¹) • b = a",
"tactic": "simp"
}
] |
[
969,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
961,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.rat_smul_eq_C_mul
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\na✝ b : R\nm n : ℕ\ninst✝ : DivisionRing R\na : ℚ\nf : R[X]\n⊢ a • f = ↑C ↑a * f",
"tactic": "rw [← Rat.smul_one_eq_coe, ← Polynomial.smul_C, C_1, smul_one_mul]"
}
] |
[
1216,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1215,
1
] |
Mathlib/Analysis/InnerProductSpace/Projection.lean
|
orthogonalProjection_singleton
|
[
{
"state_after": "case pos\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.606270\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ K }\nv w : E\nhv : v = 0\n⊢ ↑(↑(orthogonalProjection (Submodule.span 𝕜 {v})) w) = (inner v w / ↑(‖v‖ ^ 2)) • v\n\ncase neg\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.606270\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ K }\nv w : E\nhv : ¬v = 0\n⊢ ↑(↑(orthogonalProjection (Submodule.span 𝕜 {v})) w) = (inner v w / ↑(‖v‖ ^ 2)) • v",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.606270\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ K }\nv w : E\n⊢ ↑(↑(orthogonalProjection (Submodule.span 𝕜 {v})) w) = (inner v w / ↑(‖v‖ ^ 2)) • v",
"tactic": "by_cases hv : v = 0"
},
{
"state_after": "case neg\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.606270\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ K }\nv w : E\nhv : ¬v = 0\nhv' : ‖v‖ ≠ 0\n⊢ ↑(↑(orthogonalProjection (Submodule.span 𝕜 {v})) w) = (inner v w / ↑(‖v‖ ^ 2)) • v",
"state_before": "case neg\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.606270\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ K }\nv w : E\nhv : ¬v = 0\n⊢ ↑(↑(orthogonalProjection (Submodule.span 𝕜 {v})) w) = (inner v w / ↑(‖v‖ ^ 2)) • v",
"tactic": "have hv' : ‖v‖ ≠ 0 := ne_of_gt (norm_pos_iff.mpr hv)"
},
{
"state_after": "case neg\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.606270\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ K }\nv w : E\nhv : ¬v = 0\nhv' : ‖v‖ ≠ 0\nkey :\n ((↑(‖v‖ ^ 2))⁻¹ * ↑(‖v‖ ^ 2)) • ↑(↑(orthogonalProjection (Submodule.span 𝕜 {v})) w) = ((↑(‖v‖ ^ 2))⁻¹ * inner v w) • v\n⊢ ↑(↑(orthogonalProjection (Submodule.span 𝕜 {v})) w) = (inner v w / ↑(‖v‖ ^ 2)) • v",
"state_before": "case neg\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.606270\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ K }\nv w : E\nhv : ¬v = 0\nhv' : ‖v‖ ≠ 0\n⊢ ↑(↑(orthogonalProjection (Submodule.span 𝕜 {v})) w) = (inner v w / ↑(‖v‖ ^ 2)) • v",
"tactic": "have key :\n (((‖v‖ ^ 2 : ℝ) : 𝕜)⁻¹ * ((‖v‖ ^ 2 : ℝ) : 𝕜)) • ((orthogonalProjection (𝕜 ∙ v) w) : E) =\n (((‖v‖ ^ 2 : ℝ) : 𝕜)⁻¹ * ⟪v, w⟫) • v :=\n by simp [mul_smul, smul_orthogonalProjection_singleton 𝕜 w, -ofReal_pow]"
},
{
"state_after": "no goals",
"state_before": "case neg\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.606270\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ K }\nv w : E\nhv : ¬v = 0\nhv' : ‖v‖ ≠ 0\nkey :\n ((↑(‖v‖ ^ 2))⁻¹ * ↑(‖v‖ ^ 2)) • ↑(↑(orthogonalProjection (Submodule.span 𝕜 {v})) w) = ((↑(‖v‖ ^ 2))⁻¹ * inner v w) • v\n⊢ ↑(↑(orthogonalProjection (Submodule.span 𝕜 {v})) w) = (inner v w / ↑(‖v‖ ^ 2)) • v",
"tactic": "convert key using 1 <;> field_simp [hv']"
},
{
"state_after": "case pos\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.606270\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ K }\nv w : E\nhv : v = 0\n⊢ ↑(↑(orthogonalProjection ⊥) w) = (inner 0 w / ↑(‖0‖ ^ 2)) • 0",
"state_before": "case pos\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.606270\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ K }\nv w : E\nhv : v = 0\n⊢ ↑(↑(orthogonalProjection (Submodule.span 𝕜 {v})) w) = (inner v w / ↑(‖v‖ ^ 2)) • v",
"tactic": "rw [hv, eq_orthogonalProjection_of_eq_submodule (Submodule.span_zero_singleton 𝕜)]"
},
{
"state_after": "no goals",
"state_before": "case pos\n𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.606270\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ K }\nv w : E\nhv : v = 0\n⊢ ↑(↑(orthogonalProjection ⊥) w) = (inner 0 w / ↑(‖0‖ ^ 2)) • 0",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.606270\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace { x // x ∈ K }\nv w : E\nhv : ¬v = 0\nhv' : ‖v‖ ≠ 0\n⊢ ((↑(‖v‖ ^ 2))⁻¹ * ↑(‖v‖ ^ 2)) • ↑(↑(orthogonalProjection (Submodule.span 𝕜 {v})) w) = ((↑(‖v‖ ^ 2))⁻¹ * inner v w) • v",
"tactic": "simp [mul_smul, smul_orthogonalProjection_singleton 𝕜 w, -ofReal_pow]"
}
] |
[
601,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
591,
1
] |
Mathlib/Algebra/Order/LatticeGroup.lean
|
LatticeOrderedCommGroup.pos_div_neg
|
[
{
"state_after": "α : Type u\ninst✝² : Lattice α\ninst✝¹ : CommGroup α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na : α\n⊢ a = a⁺ / a⁻",
"state_before": "α : Type u\ninst✝² : Lattice α\ninst✝¹ : CommGroup α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na : α\n⊢ a⁺ / a⁻ = a",
"tactic": "symm"
},
{
"state_after": "α : Type u\ninst✝² : Lattice α\ninst✝¹ : CommGroup α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na : α\n⊢ a = a⁺ * (a⁻)⁻¹",
"state_before": "α : Type u\ninst✝² : Lattice α\ninst✝¹ : CommGroup α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na : α\n⊢ a = a⁺ / a⁻",
"tactic": "rw [div_eq_mul_inv]"
},
{
"state_after": "case h\nα : Type u\ninst✝² : Lattice α\ninst✝¹ : CommGroup α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na : α\n⊢ a * a⁻ = a⁺",
"state_before": "α : Type u\ninst✝² : Lattice α\ninst✝¹ : CommGroup α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na : α\n⊢ a = a⁺ * (a⁻)⁻¹",
"tactic": "apply eq_mul_inv_of_mul_eq"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u\ninst✝² : Lattice α\ninst✝¹ : CommGroup α\ninst✝ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\na : α\n⊢ a * a⁻ = a⁺",
"tactic": "rw [m_neg_part_def, mul_sup, mul_one, mul_right_inv, sup_comm, m_pos_part_def]"
}
] |
[
291,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
287,
1
] |
Mathlib/Data/Polynomial/Mirror.lean
|
Polynomial.mirror_smul
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : Ring R\np q : R[X]\ninst✝ : NoZeroDivisors R\na : R\n⊢ mirror (a • p) = a • mirror p",
"tactic": "rw [← C_mul', ← C_mul', mirror_mul_of_domain, mirror_C]"
}
] |
[
213,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/Topology/Order/Basic.lean
|
Filter.tendsto_nhds_max_right
|
[
{
"state_after": "case intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderClosedTopology α\nf g : β → α\nl : Filter β\na : α\nh : Tendsto f l (𝓝[Ioi a] a)\nh₁ : Tendsto f l (𝓝 a)\nh₂ : ∀ᶠ (i : β) in l, f i ∈ Ioi a\n⊢ Tendsto (fun i => max a (f i)) l (𝓝[Ioi a] a)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderClosedTopology α\nf g : β → α\nl : Filter β\na : α\nh : Tendsto f l (𝓝[Ioi a] a)\n⊢ Tendsto (fun i => max a (f i)) l (𝓝[Ioi a] a)",
"tactic": "obtain ⟨h₁ : Tendsto f l (𝓝 a), h₂ : ∀ᶠ i in l, f i ∈ Ioi a⟩ := tendsto_nhdsWithin_iff.mp h"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u\nβ : Type v\nγ : Type w\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderClosedTopology α\nf g : β → α\nl : Filter β\na : α\nh : Tendsto f l (𝓝[Ioi a] a)\nh₁ : Tendsto f l (𝓝 a)\nh₂ : ∀ᶠ (i : β) in l, f i ∈ Ioi a\n⊢ Tendsto (fun i => max a (f i)) l (𝓝[Ioi a] a)",
"tactic": "exact tendsto_nhdsWithin_iff.mpr ⟨h₁.max_right, h₂.mono fun i hi => lt_max_of_lt_right hi⟩"
}
] |
[
726,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
723,
1
] |
Mathlib/LinearAlgebra/Finsupp.lean
|
Finsupp.sigmaFinsuppLEquivPiFinsupp_apply
|
[] |
[
988,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
986,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsLittleO.sub
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.227373\nE : Type ?u.227376\nF : Type u_3\nG : Type ?u.227382\nE' : Type u_2\nF' : Type ?u.227388\nG' : Type ?u.227391\nE'' : Type ?u.227394\nF'' : Type ?u.227397\nG'' : Type ?u.227400\nR : Type ?u.227403\nR' : Type ?u.227406\n𝕜 : Type ?u.227409\n𝕜' : Type ?u.227412\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nf₁ f₂ : α → E'\ng₁ g₂ : α → F'\nh₁ : f₁ =o[l] g\nh₂ : f₂ =o[l] g\n⊢ (fun x => f₁ x - f₂ x) =o[l] g",
"tactic": "simpa only [sub_eq_add_neg] using h₁.add h₂.neg_left"
}
] |
[
1119,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1118,
1
] |
Mathlib/Data/Rat/BigOperators.lean
|
Rat.cast_prod
|
[] |
[
60,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/Algebra/MonoidAlgebra/Basic.lean
|
AddMonoidAlgebra.nat_cast_def
|
[] |
[
1373,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1372,
1
] |
Mathlib/RingTheory/DedekindDomain/Ideal.lean
|
Ideal.strictAnti_pow
|
[] |
[
736,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
733,
1
] |
Mathlib/MeasureTheory/Covering/VitaliFamily.lean
|
VitaliFamily.FineSubfamilyOn.index_countable
|
[] |
[
160,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Comp.lean
|
HasFDerivAt.comp
|
[] |
[
115,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/Data/Set/Function.lean
|
Set.maps_image_to
|
[] |
[
515,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
513,
1
] |
Mathlib/RingTheory/Ideal/QuotientOperations.lean
|
Ideal.Quotient.liftₐ_apply
|
[] |
[
259,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
1
] |
Mathlib/Logic/Basic.lean
|
dec_em'
|
[] |
[
192,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
192,
1
] |
Mathlib/Topology/Order/Basic.lean
|
Monotone.map_iInf_of_continuousAt'
|
[] |
[
2695,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2693,
1
] |
Mathlib/RingTheory/PowerBasis.lean
|
PowerBasis.equivOfRoot_map
|
[
{
"state_after": "case h\nR : Type ?u.705600\nS : Type u_2\nT : Type ?u.705606\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : Ring S\ninst✝⁹ : Algebra R S\nA : Type u_1\nB : Type ?u.705912\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : IsDomain B\ninst✝⁵ : Algebra A B\nK : Type ?u.706334\ninst✝⁴ : Field K\nS' : Type u_3\ninst✝³ : CommRing S'\ninst✝² : Algebra R S'\ninst✝¹ : Algebra A S\ninst✝ : Algebra A S'\npb : PowerBasis A S\ne : S ≃ₐ[A] S'\nh₁ : ↑(aeval pb.gen) (minpoly A (map pb e).gen) = 0\nh₂ : ↑(aeval (map pb e).gen) (minpoly A pb.gen) = 0\nx : S\n⊢ ↑(equivOfRoot pb (map pb e) h₁ h₂) x = ↑e x",
"state_before": "R : Type ?u.705600\nS : Type u_2\nT : Type ?u.705606\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : Ring S\ninst✝⁹ : Algebra R S\nA : Type u_1\nB : Type ?u.705912\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : IsDomain B\ninst✝⁵ : Algebra A B\nK : Type ?u.706334\ninst✝⁴ : Field K\nS' : Type u_3\ninst✝³ : CommRing S'\ninst✝² : Algebra R S'\ninst✝¹ : Algebra A S\ninst✝ : Algebra A S'\npb : PowerBasis A S\ne : S ≃ₐ[A] S'\nh₁ : ↑(aeval pb.gen) (minpoly A (map pb e).gen) = 0\nh₂ : ↑(aeval (map pb e).gen) (minpoly A pb.gen) = 0\n⊢ equivOfRoot pb (map pb e) h₁ h₂ = e",
"tactic": "ext x"
},
{
"state_after": "case h.intro\nR : Type ?u.705600\nS : Type u_2\nT : Type ?u.705606\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : Ring S\ninst✝⁹ : Algebra R S\nA : Type u_1\nB : Type ?u.705912\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : IsDomain B\ninst✝⁵ : Algebra A B\nK : Type ?u.706334\ninst✝⁴ : Field K\nS' : Type u_3\ninst✝³ : CommRing S'\ninst✝² : Algebra R S'\ninst✝¹ : Algebra A S\ninst✝ : Algebra A S'\npb : PowerBasis A S\ne : S ≃ₐ[A] S'\nh₁ : ↑(aeval pb.gen) (minpoly A (map pb e).gen) = 0\nh₂ : ↑(aeval (map pb e).gen) (minpoly A pb.gen) = 0\nf : A[X]\n⊢ ↑(equivOfRoot pb (map pb e) h₁ h₂) (↑(aeval pb.gen) f) = ↑e (↑(aeval pb.gen) f)",
"state_before": "case h\nR : Type ?u.705600\nS : Type u_2\nT : Type ?u.705606\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : Ring S\ninst✝⁹ : Algebra R S\nA : Type u_1\nB : Type ?u.705912\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : IsDomain B\ninst✝⁵ : Algebra A B\nK : Type ?u.706334\ninst✝⁴ : Field K\nS' : Type u_3\ninst✝³ : CommRing S'\ninst✝² : Algebra R S'\ninst✝¹ : Algebra A S\ninst✝ : Algebra A S'\npb : PowerBasis A S\ne : S ≃ₐ[A] S'\nh₁ : ↑(aeval pb.gen) (minpoly A (map pb e).gen) = 0\nh₂ : ↑(aeval (map pb e).gen) (minpoly A pb.gen) = 0\nx : S\n⊢ ↑(equivOfRoot pb (map pb e) h₁ h₂) x = ↑e x",
"tactic": "obtain ⟨f, rfl⟩ := pb.exists_eq_aeval' x"
},
{
"state_after": "no goals",
"state_before": "case h.intro\nR : Type ?u.705600\nS : Type u_2\nT : Type ?u.705606\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : Ring S\ninst✝⁹ : Algebra R S\nA : Type u_1\nB : Type ?u.705912\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : IsDomain B\ninst✝⁵ : Algebra A B\nK : Type ?u.706334\ninst✝⁴ : Field K\nS' : Type u_3\ninst✝³ : CommRing S'\ninst✝² : Algebra R S'\ninst✝¹ : Algebra A S\ninst✝ : Algebra A S'\npb : PowerBasis A S\ne : S ≃ₐ[A] S'\nh₁ : ↑(aeval pb.gen) (minpoly A (map pb e).gen) = 0\nh₂ : ↑(aeval (map pb e).gen) (minpoly A pb.gen) = 0\nf : A[X]\n⊢ ↑(equivOfRoot pb (map pb e) h₁ h₂) (↑(aeval pb.gen) f) = ↑e (↑(aeval pb.gen) f)",
"tactic": "simp [aeval_algEquiv]"
}
] |
[
490,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
486,
1
] |
Mathlib/MeasureTheory/Integral/CircleIntegral.lean
|
deriv_circleMap_eq_zero_iff
|
[
{
"state_after": "no goals",
"state_before": "E : Type ?u.25080\ninst✝ : NormedAddCommGroup E\nc : ℂ\nR θ : ℝ\n⊢ deriv (circleMap c R) θ = 0 ↔ R = 0",
"tactic": "simp [I_ne_zero]"
}
] |
[
203,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.mem_image_elim_on
|
[] |
[
249,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
247,
1
] |
Mathlib/Topology/Separation.lean
|
minimal_nonempty_open_eq_singleton
|
[] |
[
297,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
295,
1
] |
Mathlib/Data/UnionFind.lean
|
UFModel.Models.empty
|
[] |
[
136,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
136,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.Valid'.balanceL
|
[
{
"state_after": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\nr : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nhr : Valid' (↑x) r o₂\nH : (∃ l', Raised l' (size l) ∧ BalancedSz l' (size r)) ∨ ∃ r', Raised (size r) r' ∧ BalancedSz (size l) r'\n⊢ Valid' o₁ (Ordnode.balance' l x r) o₂",
"state_before": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\nr : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nhr : Valid' (↑x) r o₂\nH : (∃ l', Raised l' (size l) ∧ BalancedSz l' (size r)) ∨ ∃ r', Raised (size r) r' ∧ BalancedSz (size l) r'\n⊢ Valid' o₁ (Ordnode.balanceL l x r) o₂",
"tactic": "rw [balanceL_eq_balance' hl.3 hr.3 hl.2 hr.2 H]"
},
{
"state_after": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\nr : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nhr : Valid' (↑x) r o₂\nH : (∃ l', Raised l' (size l) ∧ BalancedSz l' (size r)) ∨ ∃ r', Raised (size r) r' ∧ BalancedSz (size l) r'\n⊢ ∃ l' r', BalancedSz l' r' ∧ (Nat.dist (size l) l' ≤ 1 ∧ size r = r' ∨ Nat.dist (size r) r' ≤ 1 ∧ size l = l')",
"state_before": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\nr : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nhr : Valid' (↑x) r o₂\nH : (∃ l', Raised l' (size l) ∧ BalancedSz l' (size r)) ∨ ∃ r', Raised (size r) r' ∧ BalancedSz (size l) r'\n⊢ Valid' o₁ (Ordnode.balance' l x r) o₂",
"tactic": "refine' hl.balance' hr _"
},
{
"state_after": "case inl.intro.intro\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\nr : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nhr : Valid' (↑x) r o₂\nl' : ℕ\ne : Raised l' (size l)\nH : BalancedSz l' (size r)\n⊢ ∃ l' r', BalancedSz l' r' ∧ (Nat.dist (size l) l' ≤ 1 ∧ size r = r' ∨ Nat.dist (size r) r' ≤ 1 ∧ size l = l')\n\ncase inr.intro.intro\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\nr : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nhr : Valid' (↑x) r o₂\nr' : ℕ\ne : Raised (size r) r'\nH : BalancedSz (size l) r'\n⊢ ∃ l' r', BalancedSz l' r' ∧ (Nat.dist (size l) l' ≤ 1 ∧ size r = r' ∨ Nat.dist (size r) r' ≤ 1 ∧ size l = l')",
"state_before": "α : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\nr : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nhr : Valid' (↑x) r o₂\nH : (∃ l', Raised l' (size l) ∧ BalancedSz l' (size r)) ∨ ∃ r', Raised (size r) r' ∧ BalancedSz (size l) r'\n⊢ ∃ l' r', BalancedSz l' r' ∧ (Nat.dist (size l) l' ≤ 1 ∧ size r = r' ∨ Nat.dist (size r) r' ≤ 1 ∧ size l = l')",
"tactic": "rcases H with (⟨l', e, H⟩ | ⟨r', e, H⟩)"
},
{
"state_after": "no goals",
"state_before": "case inl.intro.intro\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\nr : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nhr : Valid' (↑x) r o₂\nl' : ℕ\ne : Raised l' (size l)\nH : BalancedSz l' (size r)\n⊢ ∃ l' r', BalancedSz l' r' ∧ (Nat.dist (size l) l' ≤ 1 ∧ size r = r' ∨ Nat.dist (size r) r' ≤ 1 ∧ size l = l')",
"tactic": "exact ⟨_, _, H, Or.inl ⟨e.dist_le', rfl⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.intro.intro\nα : Type u_1\ninst✝ : Preorder α\nl : Ordnode α\nx : α\nr : Ordnode α\no₁ : WithBot α\no₂ : WithTop α\nhl : Valid' o₁ l ↑x\nhr : Valid' (↑x) r o₂\nr' : ℕ\ne : Raised (size r) r'\nH : BalancedSz (size l) r'\n⊢ ∃ l' r', BalancedSz l' r' ∧ (Nat.dist (size l) l' ≤ 1 ∧ size r = r' ∨ Nat.dist (size r) r' ≤ 1 ∧ size l = l')",
"tactic": "exact ⟨_, _, H, Or.inr ⟨e.dist_le, rfl⟩⟩"
}
] |
[
1370,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1362,
1
] |
Mathlib/Topology/Inseparable.lean
|
inseparable_of_nhdsWithin_eq
|
[] |
[
312,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
311,
1
] |
Mathlib/Data/Sym/Sym2.lean
|
Sym2.mem_map
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.38357\nf : α → β\nb : β\nx y : α\n⊢ b ∈ map f (Quotient.mk (Rel.setoid α) (x, y)) ↔ ∃ a, a ∈ Quotient.mk (Rel.setoid α) (x, y) ∧ f a = b",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.38357\nf : α → β\nb : β\nz : Sym2 α\n⊢ b ∈ map f z ↔ ∃ a, a ∈ z ∧ f a = b",
"tactic": "induction' z using Sym2.ind with x y"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.38357\nf : α → β\nb : β\nx y : α\n⊢ b = f x ∨ b = f y ↔ ∃ a, (a = x ∨ a = y) ∧ f a = b",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.38357\nf : α → β\nb : β\nx y : α\n⊢ b ∈ map f (Quotient.mk (Rel.setoid α) (x, y)) ↔ ∃ a, a ∈ Quotient.mk (Rel.setoid α) (x, y) ∧ f a = b",
"tactic": "simp only [map, Quotient.map_mk, Prod.map_mk, mem_iff]"
},
{
"state_after": "case h.mp\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.38357\nf : α → β\nb : β\nx y : α\n⊢ b = f x ∨ b = f y → ∃ a, (a = x ∨ a = y) ∧ f a = b\n\ncase h.mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.38357\nf : α → β\nb : β\nx y : α\n⊢ (∃ a, (a = x ∨ a = y) ∧ f a = b) → b = f x ∨ b = f y",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.38357\nf : α → β\nb : β\nx y : α\n⊢ b = f x ∨ b = f y ↔ ∃ a, (a = x ∨ a = y) ∧ f a = b",
"tactic": "constructor"
},
{
"state_after": "case h.mp.inl\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.38357\nf : α → β\nx y : α\n⊢ ∃ a, (a = x ∨ a = y) ∧ f a = f x\n\ncase h.mp.inr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.38357\nf : α → β\nx y : α\n⊢ ∃ a, (a = x ∨ a = y) ∧ f a = f y",
"state_before": "case h.mp\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.38357\nf : α → β\nb : β\nx y : α\n⊢ b = f x ∨ b = f y → ∃ a, (a = x ∨ a = y) ∧ f a = b",
"tactic": "rintro (rfl | rfl)"
},
{
"state_after": "no goals",
"state_before": "case h.mp.inl\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.38357\nf : α → β\nx y : α\n⊢ ∃ a, (a = x ∨ a = y) ∧ f a = f x",
"tactic": "exact ⟨x, by simp⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.38357\nf : α → β\nx y : α\n⊢ (x = x ∨ x = y) ∧ f x = f x",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case h.mp.inr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.38357\nf : α → β\nx y : α\n⊢ ∃ a, (a = x ∨ a = y) ∧ f a = f y",
"tactic": "exact ⟨y, by simp⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.38357\nf : α → β\nx y : α\n⊢ (y = x ∨ y = y) ∧ f y = f y",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case h.mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.38357\nf : α → β\nb : β\nx y : α\n⊢ (∃ a, (a = x ∨ a = y) ∧ f a = b) → b = f x ∨ b = f y",
"tactic": "rintro ⟨w, rfl | rfl, rfl⟩ <;> simp"
}
] |
[
402,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
395,
1
] |
Mathlib/Algebra/BigOperators/Order.lean
|
Finset.one_le_prod'
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : Type ?u.16280\nβ : Type ?u.16283\nM : Type ?u.16286\nN : Type u_2\nG : Type ?u.16292\nk : Type ?u.16295\nR : Type ?u.16298\ninst✝¹ : CommMonoid M\ninst✝ : OrderedCommMonoid N\nf g : ι → N\ns t : Finset ι\nh : ∀ (i : ι), i ∈ s → 1 ≤ f i\n⊢ 1 ≤ ∏ i in s, 1",
"tactic": "rw [prod_const_one]"
}
] |
[
143,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
MulOpposite.norm_op
|
[] |
[
2610,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2609,
1
] |
src/lean/Init/Classical.lean
|
Classical.exists_true_of_nonempty
|
[] |
[
60,
24
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
59,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean
|
BoundedContinuousFunction.coe_neg
|
[] |
[
948,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
948,
1
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.intDegree_one
|
[
{
"state_after": "no goals",
"state_before": "K : Type u\ninst✝ : Field K\n⊢ intDegree 1 = 0",
"tactic": "rw [intDegree, num_one, denom_one, sub_self]"
}
] |
[
1577,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1576,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
|
SimpleGraph.Subgraph.verts_iSup
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : Subgraph G\na b : V\nf : ι → Subgraph G\n⊢ (⨆ (i : ι), f i).verts = ⋃ (i : ι), (f i).verts",
"tactic": "simp [iSup]"
}
] |
[
433,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
433,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_indicator_const_comp
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.1365626\nδ : Type ?u.1365629\nm : MeasurableSpace α\nμ ν : Measure α\nmβ : MeasurableSpace β\nf : α → β\ns : Set β\nhf : Measurable f\nhs : MeasurableSet s\nc : ℝ≥0∞\n⊢ (∫⁻ (a : α), indicator s (fun x => c) (f a) ∂μ) = c * ↑↑μ (f ⁻¹' s)",
"tactic": "erw [lintegral_comp (measurable_const.indicator hs) hf, lintegral_indicator_const hs,\n Measure.map_apply hf hs]"
}
] |
[
1305,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1301,
1
] |
Mathlib/Data/Nat/Order/Lemmas.lean
|
Nat.mod_div_self
|
[
{
"state_after": "case zero\na b m✝ n k m : ℕ\n⊢ m % zero / zero = 0\n\ncase succ\na b m✝ n k m n✝ : ℕ\n⊢ m % succ n✝ / succ n✝ = 0",
"state_before": "a b m✝ n✝ k m n : ℕ\n⊢ m % n / n = 0",
"tactic": "cases n"
},
{
"state_after": "no goals",
"state_before": "case zero\na b m✝ n k m : ℕ\n⊢ m % zero / zero = 0",
"tactic": "exact (m % 0).div_zero"
},
{
"state_after": "no goals",
"state_before": "case succ\na b m✝ n k m n✝ : ℕ\n⊢ m % succ n✝ / succ n✝ = 0",
"tactic": "case succ n => exact Nat.div_eq_zero (m.mod_lt n.succ_pos)"
},
{
"state_after": "no goals",
"state_before": "a b m✝ n✝ k m n : ℕ\n⊢ m % succ n / succ n = 0",
"tactic": "exact Nat.div_eq_zero (m.mod_lt n.succ_pos)"
}
] |
[
229,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
226,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
ContDiffOn.ccosh
|
[] |
[
497,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
495,
1
] |
Mathlib/CategoryTheory/Limits/Opposites.lean
|
CategoryTheory.Limits.hasCoproductsOfShape_of_opposite
|
[] |
[
345,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
342,
1
] |
Std/Data/Nat/Gcd.lean
|
Nat.lcm_zero_right
|
[
{
"state_after": "no goals",
"state_before": "m : Nat\n⊢ lcm m 0 = 0",
"tactic": "simp [lcm]"
}
] |
[
187,
72
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
187,
9
] |
Mathlib/Algebra/Module/Submodule/Lattice.lean
|
AddSubmonoid.toNatSubmodule_symm
|
[] |
[
358,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
356,
1
] |
Mathlib/NumberTheory/Zsqrtd/Basic.lean
|
Zsqrtd.add_def
|
[] |
[
117,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/RingTheory/Localization/FractionRing.lean
|
IsFractionRing.lift_algebraMap
|
[] |
[
213,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/Geometry/Euclidean/Sphere/Basic.lean
|
EuclideanGeometry.cospherical_pair
|
[
{
"state_after": "case inl\nV : Type ?u.9686\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₂ p : P\n⊢ dist p (midpoint ℝ p p₂) = ‖2‖⁻¹ * dist p p₂\n\ncase inr.refl\nV : Type ?u.9686\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ : P\n⊢ dist p₂ (midpoint ℝ p₁ p₂) = ‖2‖⁻¹ * dist p₁ p₂",
"state_before": "V : Type ?u.9686\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ : P\n⊢ ∀ (p : P), p ∈ {p₁, p₂} → dist p (midpoint ℝ p₁ p₂) = ‖2‖⁻¹ * dist p₁ p₂",
"tactic": "rintro p (rfl | rfl | _)"
},
{
"state_after": "no goals",
"state_before": "case inl\nV : Type ?u.9686\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₂ p : P\n⊢ dist p (midpoint ℝ p p₂) = ‖2‖⁻¹ * dist p p₂",
"tactic": "rw [dist_comm, dist_midpoint_left (𝕜 := ℝ)]"
},
{
"state_after": "no goals",
"state_before": "case inr.refl\nV : Type ?u.9686\nP : Type u_1\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\np₁ p₂ : P\n⊢ dist p₂ (midpoint ℝ p₁ p₂) = ‖2‖⁻¹ * dist p₁ p₂",
"tactic": "rw [dist_comm, dist_midpoint_right (𝕜 := ℝ)]"
}
] |
[
205,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
201,
1
] |
Mathlib/GroupTheory/Congruence.lean
|
Con.inv
|
[
{
"state_after": "no goals",
"state_before": "M : Type u_1\nN : Type ?u.96860\nP : Type ?u.96863\ninst✝² : Group M\ninst✝¹ : Group N\ninst✝ : Group P\nc : Con M\nx y : M\nh : ↑c x y\n⊢ ↑c x⁻¹ y⁻¹",
"tactic": "simpa using c.symm (c.mul (c.mul (c.refl x⁻¹) h) (c.refl y⁻¹))"
}
] |
[
1225,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1224,
11
] |
Mathlib/FieldTheory/Subfield.lean
|
Subfield.zsmul_mem
|
[] |
[
325,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
324,
11
] |
Mathlib/LinearAlgebra/Matrix/Charpoly/Eigs.lean
|
Matrix.trace_eq_sum_roots_charpoly
|
[] |
[
90,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
89,
1
] |
Mathlib/Data/Vector/Basic.lean
|
Vector.get_tail_succ
|
[
{
"state_after": "n : ℕ\nα : Type u_1\na : α\nl : List α\ne : List.length (a :: l) = Nat.succ n\ni : ℕ\nh : i < n\n⊢ List.get (toList (tail { val := a :: l, property := e }))\n { val := i, isLt := (_ : i < List.length (toList (tail { val := a :: l, property := e }))) } =\n List.get l { val := i, isLt := (_ : i < List.length l) }",
"state_before": "n : ℕ\nα : Type u_1\na : α\nl : List α\ne : List.length (a :: l) = Nat.succ n\ni : ℕ\nh : i < n\n⊢ get (tail { val := a :: l, property := e }) { val := i, isLt := h } =\n get { val := a :: l, property := e } (Fin.succ { val := i, isLt := h })",
"tactic": "simp [get_eq_get]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nα : Type u_1\na : α\nl : List α\ne : List.length (a :: l) = Nat.succ n\ni : ℕ\nh : i < n\n⊢ List.get (toList (tail { val := a :: l, property := e }))\n { val := i, isLt := (_ : i < List.length (toList (tail { val := a :: l, property := e }))) } =\n List.get l { val := i, isLt := (_ : i < List.length l) }",
"tactic": "rfl"
}
] |
[
171,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Mathlib/Data/List/Perm.lean
|
List.nil_subperm
|
[
{
"state_after": "no goals",
"state_before": "α : Type uu\nβ : Type vv\nl₁ l₂ l : List α\n⊢ [] <+ l",
"tactic": "simp"
}
] |
[
404,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
403,
1
] |
Mathlib/RingTheory/TensorProduct.lean
|
Algebra.TensorProduct.includeLeft_comp_algebraMap
|
[
{
"state_after": "case a\nR✝ : Type u\ninst✝¹⁵ : CommSemiring R✝\nA : Type v₁\ninst✝¹⁴ : Semiring A\ninst✝¹³ : Algebra R✝ A\nB : Type v₂\ninst✝¹² : Semiring B\ninst✝¹¹ : Algebra R✝ B\nS✝ : Type ?u.805787\ninst✝¹⁰ : CommSemiring S✝\ninst✝⁹ : Algebra R✝ S✝\ninst✝⁸ : Algebra S✝ A\ninst✝⁷ : IsScalarTower R✝ S✝ A\nC : Type v₃\ninst✝⁶ : Semiring C\ninst✝⁵ : Algebra R✝ C\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nx✝ : R\n⊢ ↑(RingHom.comp (↑includeLeft) (algebraMap R S)) x✝ = ↑(RingHom.comp (↑includeRight) (algebraMap R T)) x✝",
"state_before": "R✝ : Type u\ninst✝¹⁵ : CommSemiring R✝\nA : Type v₁\ninst✝¹⁴ : Semiring A\ninst✝¹³ : Algebra R✝ A\nB : Type v₂\ninst✝¹² : Semiring B\ninst✝¹¹ : Algebra R✝ B\nS✝ : Type ?u.805787\ninst✝¹⁰ : CommSemiring S✝\ninst✝⁹ : Algebra R✝ S✝\ninst✝⁸ : Algebra S✝ A\ninst✝⁷ : IsScalarTower R✝ S✝ A\nC : Type v₃\ninst✝⁶ : Semiring C\ninst✝⁵ : Algebra R✝ C\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\n⊢ RingHom.comp (↑includeLeft) (algebraMap R S) = RingHom.comp (↑includeRight) (algebraMap R T)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nR✝ : Type u\ninst✝¹⁵ : CommSemiring R✝\nA : Type v₁\ninst✝¹⁴ : Semiring A\ninst✝¹³ : Algebra R✝ A\nB : Type v₂\ninst✝¹² : Semiring B\ninst✝¹¹ : Algebra R✝ B\nS✝ : Type ?u.805787\ninst✝¹⁰ : CommSemiring S✝\ninst✝⁹ : Algebra R✝ S✝\ninst✝⁸ : Algebra S✝ A\ninst✝⁷ : IsScalarTower R✝ S✝ A\nC : Type v₃\ninst✝⁶ : Semiring C\ninst✝⁵ : Algebra R✝ C\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nx✝ : R\n⊢ ↑(RingHom.comp (↑includeLeft) (algebraMap R S)) x✝ = ↑(RingHom.comp (↑includeRight) (algebraMap R T)) x✝",
"tactic": "simp"
}
] |
[
573,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
568,
1
] |
Mathlib/Algebra/BigOperators/Finsupp.lean
|
Finsupp.sum_ite_self_eq'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nι : Type ?u.74791\nγ : Type ?u.74794\nA : Type ?u.74797\nB : Type ?u.74800\nC : Type ?u.74803\ninst✝⁷ : AddCommMonoid A\ninst✝⁶ : AddCommMonoid B\ninst✝⁵ : AddCommMonoid C\nt : ι → A → C\nh0 : ∀ (i : ι), t i 0 = 0\nh1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y\ns : Finset α\nf✝ : α → ι →₀ A\ni : ι\ng : ι →₀ A\nk : ι → A → γ → B\nx : γ\nβ : Type ?u.77933\nM : Type ?u.77936\nM' : Type ?u.77939\nN✝ : Type ?u.77942\nP : Type ?u.77945\nG : Type ?u.77948\nH : Type ?u.77951\nR : Type ?u.77954\nS : Type ?u.77957\ninst✝⁴ : Zero M\ninst✝³ : Zero M'\ninst✝² : CommMonoid N✝\ninst✝¹ : DecidableEq α\nN : Type u_2\ninst✝ : AddCommMonoid N\nf : α →₀ N\na : α\n⊢ (sum f fun x v => if x = a then v else 0) = ↑f a",
"tactic": "classical\n convert f.sum_ite_eq' a fun _ => id\n simp [ite_eq_right_iff.2 Eq.symm]"
},
{
"state_after": "case h.e'_3\nα : Type u_1\nι : Type ?u.74791\nγ : Type ?u.74794\nA : Type ?u.74797\nB : Type ?u.74800\nC : Type ?u.74803\ninst✝⁷ : AddCommMonoid A\ninst✝⁶ : AddCommMonoid B\ninst✝⁵ : AddCommMonoid C\nt : ι → A → C\nh0 : ∀ (i : ι), t i 0 = 0\nh1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y\ns : Finset α\nf✝ : α → ι →₀ A\ni : ι\ng : ι →₀ A\nk : ι → A → γ → B\nx : γ\nβ : Type ?u.77933\nM : Type ?u.77936\nM' : Type ?u.77939\nN✝ : Type ?u.77942\nP : Type ?u.77945\nG : Type ?u.77948\nH : Type ?u.77951\nR : Type ?u.77954\nS : Type ?u.77957\ninst✝⁴ : Zero M\ninst✝³ : Zero M'\ninst✝² : CommMonoid N✝\ninst✝¹ : DecidableEq α\nN : Type u_2\ninst✝ : AddCommMonoid N\nf : α →₀ N\na : α\n⊢ ↑f a = if a ∈ f.support then id (↑f a) else 0",
"state_before": "α : Type u_1\nι : Type ?u.74791\nγ : Type ?u.74794\nA : Type ?u.74797\nB : Type ?u.74800\nC : Type ?u.74803\ninst✝⁷ : AddCommMonoid A\ninst✝⁶ : AddCommMonoid B\ninst✝⁵ : AddCommMonoid C\nt : ι → A → C\nh0 : ∀ (i : ι), t i 0 = 0\nh1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y\ns : Finset α\nf✝ : α → ι →₀ A\ni : ι\ng : ι →₀ A\nk : ι → A → γ → B\nx : γ\nβ : Type ?u.77933\nM : Type ?u.77936\nM' : Type ?u.77939\nN✝ : Type ?u.77942\nP : Type ?u.77945\nG : Type ?u.77948\nH : Type ?u.77951\nR : Type ?u.77954\nS : Type ?u.77957\ninst✝⁴ : Zero M\ninst✝³ : Zero M'\ninst✝² : CommMonoid N✝\ninst✝¹ : DecidableEq α\nN : Type u_2\ninst✝ : AddCommMonoid N\nf : α →₀ N\na : α\n⊢ (sum f fun x v => if x = a then v else 0) = ↑f a",
"tactic": "convert f.sum_ite_eq' a fun _ => id"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nα : Type u_1\nι : Type ?u.74791\nγ : Type ?u.74794\nA : Type ?u.74797\nB : Type ?u.74800\nC : Type ?u.74803\ninst✝⁷ : AddCommMonoid A\ninst✝⁶ : AddCommMonoid B\ninst✝⁵ : AddCommMonoid C\nt : ι → A → C\nh0 : ∀ (i : ι), t i 0 = 0\nh1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y\ns : Finset α\nf✝ : α → ι →₀ A\ni : ι\ng : ι →₀ A\nk : ι → A → γ → B\nx : γ\nβ : Type ?u.77933\nM : Type ?u.77936\nM' : Type ?u.77939\nN✝ : Type ?u.77942\nP : Type ?u.77945\nG : Type ?u.77948\nH : Type ?u.77951\nR : Type ?u.77954\nS : Type ?u.77957\ninst✝⁴ : Zero M\ninst✝³ : Zero M'\ninst✝² : CommMonoid N✝\ninst✝¹ : DecidableEq α\nN : Type u_2\ninst✝ : AddCommMonoid N\nf : α →₀ N\na : α\n⊢ ↑f a = if a ∈ f.support then id (↑f a) else 0",
"tactic": "simp [ite_eq_right_iff.2 Eq.symm]"
}
] |
[
151,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
147,
1
] |
Mathlib/NumberTheory/Liouville/Measure.lean
|
ae_not_liouvilleWith
|
[
{
"state_after": "no goals",
"state_before": "⊢ ∀ᵐ (x : ℝ), ∀ (p : ℝ), p > 2 → ¬LiouvilleWith p x",
"tactic": "simpa only [ae_iff, not_forall, Classical.not_not, setOf_exists] using\n volume_iUnion_setOf_liouvilleWith"
}
] |
[
116,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
Submodule.ker_subtypeL
|
[] |
[
1059,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1058,
1
] |
Mathlib/Probability/ProbabilityMassFunction/Constructions.lean
|
Pmf.support_map
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.4250\nf : α → β\np : Pmf α\nb✝ b : β\n⊢ b ∈ support (map f p) ↔ b ∈ f '' support p",
"tactic": "simp [map, @eq_comm β b]"
}
] |
[
58,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/MeasureTheory/Measure/VectorMeasure.lean
|
MeasureTheory.VectorMeasure.restrict_le_restrict_iUnion
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\n⊢ ↑v a ≤ ↑w a",
"state_before": "α : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\n⊢ restrict v (⋃ (n : ℕ), f n) ≤ restrict w (⋃ (n : ℕ), f n)",
"tactic": "refine' restrict_le_restrict_of_subset_le v w fun a ha₁ ha₂ => _"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\n⊢ ↑v a ≤ ↑w a",
"state_before": "α : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\n⊢ ↑v a ≤ ↑w a",
"tactic": "have ha₃ : (⋃ n, a ∩ disjointed f n) = a := by\n rwa [← Set.inter_iUnion, iUnion_disjointed, Set.inter_eq_left_iff_subset]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\n⊢ ↑v a ≤ ↑w a",
"state_before": "α : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\n⊢ ↑v a ≤ ↑w a",
"tactic": "have ha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n) :=\n (disjoint_disjointed _).mono fun i j => Disjoint.mono inf_le_right inf_le_right"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\n⊢ (∑' (i : ℕ), ↑v (a ∩ disjointed f i)) ≤ ∑' (i : ℕ), ↑w (a ∩ disjointed f i)\n\nα : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\n⊢ ∀ (i : ℕ), MeasurableSet (a ∩ disjointed f i)\n\nα : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\n⊢ ∀ (i : ℕ), MeasurableSet (a ∩ disjointed f i)",
"state_before": "α : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\n⊢ ↑v a ≤ ↑w a",
"tactic": "rw [← ha₃, v.of_disjoint_iUnion_nat _ ha₄, w.of_disjoint_iUnion_nat _ ha₄]"
},
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\nn : ℕ\n⊢ MeasurableSet (a ∩ disjointed f n)\n\ncase refine'_2\nα : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\nn : ℕ\n⊢ a ∩ disjointed f n ⊆ f n\n\ncase refine'_3\nα : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\n⊢ Summable fun i => ↑v (a ∩ disjointed f i)\n\ncase refine'_4\nα : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\n⊢ Summable fun i => ↑w (a ∩ disjointed f i)\n\nα : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\n⊢ ∀ (i : ℕ), MeasurableSet (a ∩ disjointed f i)\n\nα : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\n⊢ ∀ (i : ℕ), MeasurableSet (a ∩ disjointed f i)",
"state_before": "α : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\n⊢ (∑' (i : ℕ), ↑v (a ∩ disjointed f i)) ≤ ∑' (i : ℕ), ↑w (a ∩ disjointed f i)\n\nα : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\n⊢ ∀ (i : ℕ), MeasurableSet (a ∩ disjointed f i)\n\nα : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\n⊢ ∀ (i : ℕ), MeasurableSet (a ∩ disjointed f i)",
"tactic": "refine' tsum_le_tsum (fun n => (restrict_le_restrict_iff v w (hf₁ n)).1 (hf₂ n) _ _) _ _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\n⊢ (⋃ (n : ℕ), a ∩ disjointed f n) = a",
"tactic": "rwa [← Set.inter_iUnion, iUnion_disjointed, Set.inter_eq_left_iff_subset]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\nn : ℕ\n⊢ MeasurableSet (a ∩ disjointed f n)",
"tactic": "exact ha₁.inter (MeasurableSet.disjointed hf₁ n)"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\nn : ℕ\n⊢ a ∩ disjointed f n ⊆ f n",
"tactic": "exact Set.Subset.trans (Set.inter_subset_right _ _) (disjointed_subset _ _)"
},
{
"state_after": "case refine'_3.refine'_1\nα : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\nn : ℕ\n⊢ MeasurableSet (a ∩ disjointed f n)\n\ncase refine'_3.refine'_2\nα : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\n⊢ Pairwise (Disjoint on fun i => a ∩ disjointed f i)",
"state_before": "case refine'_3\nα : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\n⊢ Summable fun i => ↑v (a ∩ disjointed f i)",
"tactic": "refine' (v.m_iUnion (fun n => _) _).summable"
},
{
"state_after": "no goals",
"state_before": "case refine'_3.refine'_1\nα : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\nn : ℕ\n⊢ MeasurableSet (a ∩ disjointed f n)",
"tactic": "exact ha₁.inter (MeasurableSet.disjointed hf₁ n)"
},
{
"state_after": "no goals",
"state_before": "case refine'_3.refine'_2\nα : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\n⊢ Pairwise (Disjoint on fun i => a ∩ disjointed f i)",
"tactic": "exact (disjoint_disjointed _).mono fun i j => Disjoint.mono inf_le_right inf_le_right"
},
{
"state_after": "case refine'_4.refine'_1\nα : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\nn : ℕ\n⊢ MeasurableSet (a ∩ disjointed f n)\n\ncase refine'_4.refine'_2\nα : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\n⊢ Pairwise (Disjoint on fun i => a ∩ disjointed f i)",
"state_before": "case refine'_4\nα : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\n⊢ Summable fun i => ↑w (a ∩ disjointed f i)",
"tactic": "refine' (w.m_iUnion (fun n => _) _).summable"
},
{
"state_after": "no goals",
"state_before": "case refine'_4.refine'_1\nα : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\nn : ℕ\n⊢ MeasurableSet (a ∩ disjointed f n)",
"tactic": "exact ha₁.inter (MeasurableSet.disjointed hf₁ n)"
},
{
"state_after": "no goals",
"state_before": "case refine'_4.refine'_2\nα : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\n⊢ Pairwise (Disjoint on fun i => a ∩ disjointed f i)",
"tactic": "exact (disjoint_disjointed _).mono fun i j => Disjoint.mono inf_le_right inf_le_right"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\nn : ℕ\n⊢ MeasurableSet (a ∩ disjointed f n)",
"state_before": "α : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\n⊢ ∀ (i : ℕ), MeasurableSet (a ∩ disjointed f i)",
"tactic": "intro n"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\nn : ℕ\n⊢ MeasurableSet (a ∩ disjointed f n)",
"tactic": "exact ha₁.inter (MeasurableSet.disjointed hf₁ n)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.555378\nm : MeasurableSpace α\nM : Type u_2\ninst✝² : TopologicalSpace M\ninst✝¹ : OrderedAddCommMonoid M\ninst✝ : OrderClosedTopology M\nv w : VectorMeasure α M\ni j : Set α\nf : ℕ → Set α\nhf₁ : ∀ (n : ℕ), MeasurableSet (f n)\nhf₂ : ∀ (n : ℕ), restrict v (f n) ≤ restrict w (f n)\na : Set α\nha₁ : MeasurableSet a\nha₂ : a ⊆ ⋃ (n : ℕ), f n\nha₃ : (⋃ (n : ℕ), a ∩ disjointed f n) = a\nha₄ : Pairwise (Disjoint on fun n => a ∩ disjointed f n)\n⊢ ∀ (i : ℕ), MeasurableSet (a ∩ disjointed f i)",
"tactic": "exact fun n => ha₁.inter (MeasurableSet.disjointed hf₁ n)"
}
] |
[
965,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
946,
1
] |
Mathlib/Topology/Constructions.lean
|
frontier_univ_prod_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.61888\nδ : Type ?u.61891\nε : Type ?u.61894\nζ : Type ?u.61897\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace γ\ninst✝² : TopologicalSpace δ\ninst✝¹ : TopologicalSpace ε\ninst✝ : TopologicalSpace ζ\ns : Set β\n⊢ frontier (univ ×ˢ s) = univ ×ˢ frontier s",
"tactic": "simp [frontier_prod_eq]"
}
] |
[
775,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
774,
1
] |
Mathlib/Topology/Algebra/Order/LiminfLimsup.lean
|
limsInf_nhds
|
[] |
[
173,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean
|
CategoryTheory.Limits.biproduct.lift_π
|
[] |
[
439,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
438,
1
] |
Mathlib/Algebra/Algebra/Operations.lean
|
Submodule.prod_span_singleton
|
[
{
"state_after": "no goals",
"state_before": "ι✝ : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nm n : A\nι : Type u_1\ns : Finset ι\nx : ι → A\n⊢ ∏ i in s, span R {x i} = span R {∏ i in s, x i}",
"tactic": "rw [prod_span, Set.finset_prod_singleton]"
}
] |
[
619,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
617,
1
] |
Mathlib/MeasureTheory/MeasurableSpaceDef.lean
|
MeasurableSpace.generateFrom_induction
|
[
{
"state_after": "case basic\nα : Type u_1\nβ : Type ?u.16280\nγ : Type ?u.16283\nδ : Type ?u.16286\nδ' : Type ?u.16289\nι : Sort ?u.16292\ns✝ t u : Set α\np : Set α → Prop\nC : Set (Set α)\nhC : ∀ (t : Set α), t ∈ C → p t\nh_empty : p ∅\nh_compl : ∀ (t : Set α), p t → p (tᶜ)\nh_Union : ∀ (f : ℕ → Set α), (∀ (n : ℕ), p (f n)) → p (⋃ (i : ℕ), f i)\ns u✝ : Set α\na✝ : u✝ ∈ C\n⊢ p u✝\n\ncase empty\nα : Type u_1\nβ : Type ?u.16280\nγ : Type ?u.16283\nδ : Type ?u.16286\nδ' : Type ?u.16289\nι : Sort ?u.16292\ns✝ t u : Set α\np : Set α → Prop\nC : Set (Set α)\nhC : ∀ (t : Set α), t ∈ C → p t\nh_empty : p ∅\nh_compl : ∀ (t : Set α), p t → p (tᶜ)\nh_Union : ∀ (f : ℕ → Set α), (∀ (n : ℕ), p (f n)) → p (⋃ (i : ℕ), f i)\ns : Set α\n⊢ p ∅\n\ncase compl\nα : Type u_1\nβ : Type ?u.16280\nγ : Type ?u.16283\nδ : Type ?u.16286\nδ' : Type ?u.16289\nι : Sort ?u.16292\ns✝ t u : Set α\np : Set α → Prop\nC : Set (Set α)\nhC : ∀ (t : Set α), t ∈ C → p t\nh_empty : p ∅\nh_compl : ∀ (t : Set α), p t → p (tᶜ)\nh_Union : ∀ (f : ℕ → Set α), (∀ (n : ℕ), p (f n)) → p (⋃ (i : ℕ), f i)\ns t✝ : Set α\na✝ : GenerateMeasurable C t✝\na_ih✝ : p t✝\n⊢ p (t✝ᶜ)\n\ncase iUnion\nα : Type u_1\nβ : Type ?u.16280\nγ : Type ?u.16283\nδ : Type ?u.16286\nδ' : Type ?u.16289\nι : Sort ?u.16292\ns✝ t u : Set α\np : Set α → Prop\nC : Set (Set α)\nhC : ∀ (t : Set α), t ∈ C → p t\nh_empty : p ∅\nh_compl : ∀ (t : Set α), p t → p (tᶜ)\nh_Union : ∀ (f : ℕ → Set α), (∀ (n : ℕ), p (f n)) → p (⋃ (i : ℕ), f i)\ns : Set α\nf✝ : ℕ → Set α\na✝ : ∀ (n : ℕ), GenerateMeasurable C (f✝ n)\na_ih✝ : ∀ (n : ℕ), p (f✝ n)\n⊢ p (⋃ (i : ℕ), f✝ i)",
"state_before": "α : Type u_1\nβ : Type ?u.16280\nγ : Type ?u.16283\nδ : Type ?u.16286\nδ' : Type ?u.16289\nι : Sort ?u.16292\ns✝ t u : Set α\np : Set α → Prop\nC : Set (Set α)\nhC : ∀ (t : Set α), t ∈ C → p t\nh_empty : p ∅\nh_compl : ∀ (t : Set α), p t → p (tᶜ)\nh_Union : ∀ (f : ℕ → Set α), (∀ (n : ℕ), p (f n)) → p (⋃ (i : ℕ), f i)\ns : Set α\nhs : MeasurableSet s\n⊢ p s",
"tactic": "induction hs"
},
{
"state_after": "no goals",
"state_before": "case basic\nα : Type u_1\nβ : Type ?u.16280\nγ : Type ?u.16283\nδ : Type ?u.16286\nδ' : Type ?u.16289\nι : Sort ?u.16292\ns✝ t u : Set α\np : Set α → Prop\nC : Set (Set α)\nhC : ∀ (t : Set α), t ∈ C → p t\nh_empty : p ∅\nh_compl : ∀ (t : Set α), p t → p (tᶜ)\nh_Union : ∀ (f : ℕ → Set α), (∀ (n : ℕ), p (f n)) → p (⋃ (i : ℕ), f i)\ns u✝ : Set α\na✝ : u✝ ∈ C\n⊢ p u✝\n\ncase empty\nα : Type u_1\nβ : Type ?u.16280\nγ : Type ?u.16283\nδ : Type ?u.16286\nδ' : Type ?u.16289\nι : Sort ?u.16292\ns✝ t u : Set α\np : Set α → Prop\nC : Set (Set α)\nhC : ∀ (t : Set α), t ∈ C → p t\nh_empty : p ∅\nh_compl : ∀ (t : Set α), p t → p (tᶜ)\nh_Union : ∀ (f : ℕ → Set α), (∀ (n : ℕ), p (f n)) → p (⋃ (i : ℕ), f i)\ns : Set α\n⊢ p ∅\n\ncase compl\nα : Type u_1\nβ : Type ?u.16280\nγ : Type ?u.16283\nδ : Type ?u.16286\nδ' : Type ?u.16289\nι : Sort ?u.16292\ns✝ t u : Set α\np : Set α → Prop\nC : Set (Set α)\nhC : ∀ (t : Set α), t ∈ C → p t\nh_empty : p ∅\nh_compl : ∀ (t : Set α), p t → p (tᶜ)\nh_Union : ∀ (f : ℕ → Set α), (∀ (n : ℕ), p (f n)) → p (⋃ (i : ℕ), f i)\ns t✝ : Set α\na✝ : GenerateMeasurable C t✝\na_ih✝ : p t✝\n⊢ p (t✝ᶜ)\n\ncase iUnion\nα : Type u_1\nβ : Type ?u.16280\nγ : Type ?u.16283\nδ : Type ?u.16286\nδ' : Type ?u.16289\nι : Sort ?u.16292\ns✝ t u : Set α\np : Set α → Prop\nC : Set (Set α)\nhC : ∀ (t : Set α), t ∈ C → p t\nh_empty : p ∅\nh_compl : ∀ (t : Set α), p t → p (tᶜ)\nh_Union : ∀ (f : ℕ → Set α), (∀ (n : ℕ), p (f n)) → p (⋃ (i : ℕ), f i)\ns : Set α\nf✝ : ℕ → Set α\na✝ : ∀ (n : ℕ), GenerateMeasurable C (f✝ n)\na_ih✝ : ∀ (n : ℕ), p (f✝ n)\n⊢ p (⋃ (i : ℕ), f✝ i)",
"tactic": "exacts [hC _ ‹_›, h_empty, h_compl _ ‹_›, h_Union ‹_› ‹_›]"
}
] |
[
385,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
380,
1
] |
Mathlib/RepresentationTheory/Basic.lean
|
Representation.dual_apply
|
[] |
[
453,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
452,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
Nat.ceil_pos
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.53526\nα : Type u_1\nβ : Type ?u.53532\ninst✝¹ : LinearOrderedSemiring α\ninst✝ : FloorSemiring α\na : α\nn : ℕ\n⊢ 0 < ⌈a⌉₊ ↔ 0 < a",
"tactic": "rw [lt_ceil, cast_zero]"
}
] |
[
336,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
336,
1
] |
Mathlib/Algebra/Order/Hom/Ring.lean
|
OrderRingHom.comp_apply
|
[] |
[
314,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
313,
1
] |
Mathlib/Order/Filter/Ultrafilter.lean
|
Filter.forall_neBot_le_iff
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type ?u.53069\nf : Filter α\ns : Set α\na : α\ng : Filter α\np : Filter α → Prop\nhp : Monotone p\n⊢ (∀ (f : Ultrafilter α), ↑f ≤ g → p ↑f) → ∀ (f : Filter α), NeBot f → f ≤ g → p f",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.53069\nf : Filter α\ns : Set α\na : α\ng : Filter α\np : Filter α → Prop\nhp : Monotone p\n⊢ (∀ (f : Filter α), NeBot f → f ≤ g → p f) ↔ ∀ (f : Ultrafilter α), ↑f ≤ g → p ↑f",
"tactic": "refine' ⟨fun H f hf => H f f.neBot hf, _⟩"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type ?u.53069\nf✝ : Filter α\ns : Set α\na : α\ng : Filter α\np : Filter α → Prop\nhp : Monotone p\nH : ∀ (f : Ultrafilter α), ↑f ≤ g → p ↑f\nf : Filter α\nhf : NeBot f\nhfg : f ≤ g\n⊢ p f",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.53069\nf : Filter α\ns : Set α\na : α\ng : Filter α\np : Filter α → Prop\nhp : Monotone p\n⊢ (∀ (f : Ultrafilter α), ↑f ≤ g → p ↑f) → ∀ (f : Filter α), NeBot f → f ≤ g → p f",
"tactic": "intro H f hf hfg"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.53069\nf✝ : Filter α\ns : Set α\na : α\ng : Filter α\np : Filter α → Prop\nhp : Monotone p\nH : ∀ (f : Ultrafilter α), ↑f ≤ g → p ↑f\nf : Filter α\nhf : NeBot f\nhfg : f ≤ g\n⊢ p f",
"tactic": "exact hp (of_le f) (H _ ((of_le f).trans hfg))"
}
] |
[
465,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
461,
1
] |
Mathlib/Algebra/Module/Equiv.lean
|
LinearEquiv.self_trans_symm
|
[
{
"state_after": "case h\nR : Type ?u.216670\nR₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type ?u.216679\nk : Type ?u.216682\nS : Type ?u.216685\nM : Type ?u.216688\nM₁ : Type u_3\nM₂ : Type u_4\nM₃ : Type ?u.216697\nN₁ : Type ?u.216700\nN₂ : Type ?u.216703\nN₃ : Type ?u.216706\nN₄ : Type ?u.216709\nι : Type ?u.216712\nM₄ : Type ?u.216715\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring S\ninst✝¹⁵ : Semiring R₁\ninst✝¹⁴ : Semiring R₂\ninst✝¹³ : Semiring R₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid M₁\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid M₄\ninst✝⁷ : AddCommMonoid N₁\ninst✝⁶ : AddCommMonoid N₂\nmodule_M : Module R M\nmodule_S_M₂ : Module S M₂\nσ : R →+* S\nσ' : S →+* R\nre₁ : RingHomInvPair σ σ'\nre₂ : RingHomInvPair σ' σ\ne e' : M ≃ₛₗ[σ] M₂\nmodule_M₁ : Module R₁ M₁\nmodule_M₂ : Module R₂ M₂\nmodule_M₃ : Module R₃ M₃\nmodule_N₁ : Module R₁ N₁\nmodule_N₂ : Module R₁ N₂\nσ₁₂ : R₁ →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\nσ₂₁ : R₂ →+* R₁\nσ₃₂ : R₃ →+* R₂\nσ₃₁ : R₃ →+* R₁\ninst✝⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁴ : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁\nre₁₂ : RingHomInvPair σ₁₂ σ₂₁\nre₂₃ : RingHomInvPair σ₂₃ σ₃₂\ninst✝³ : RingHomInvPair σ₁₃ σ₃₁\nre₂₁ : RingHomInvPair σ₂₁ σ₁₂\nre₃₂ : RingHomInvPair σ₃₂ σ₂₃\ninst✝² : RingHomInvPair σ₃₁ σ₁₃\ne₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂\ne₂₃ : M₂ ≃ₛₗ[σ₂₃] M₃\ninst✝¹ : RingHomCompTriple σ₂₁ σ₁₃ σ₂₃\ninst✝ : RingHomCompTriple σ₃₁ σ₁₂ σ₃₂\nf : M₁ ≃ₛₗ[σ₁₂] M₂\nx : M₁\n⊢ ↑(trans f (symm f)) x = ↑(refl R₁ M₁) x",
"state_before": "R : Type ?u.216670\nR₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type ?u.216679\nk : Type ?u.216682\nS : Type ?u.216685\nM : Type ?u.216688\nM₁ : Type u_3\nM₂ : Type u_4\nM₃ : Type ?u.216697\nN₁ : Type ?u.216700\nN₂ : Type ?u.216703\nN₃ : Type ?u.216706\nN₄ : Type ?u.216709\nι : Type ?u.216712\nM₄ : Type ?u.216715\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring S\ninst✝¹⁵ : Semiring R₁\ninst✝¹⁴ : Semiring R₂\ninst✝¹³ : Semiring R₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid M₁\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid M₄\ninst✝⁷ : AddCommMonoid N₁\ninst✝⁶ : AddCommMonoid N₂\nmodule_M : Module R M\nmodule_S_M₂ : Module S M₂\nσ : R →+* S\nσ' : S →+* R\nre₁ : RingHomInvPair σ σ'\nre₂ : RingHomInvPair σ' σ\ne e' : M ≃ₛₗ[σ] M₂\nmodule_M₁ : Module R₁ M₁\nmodule_M₂ : Module R₂ M₂\nmodule_M₃ : Module R₃ M₃\nmodule_N₁ : Module R₁ N₁\nmodule_N₂ : Module R₁ N₂\nσ₁₂ : R₁ →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\nσ₂₁ : R₂ →+* R₁\nσ₃₂ : R₃ →+* R₂\nσ₃₁ : R₃ →+* R₁\ninst✝⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁴ : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁\nre₁₂ : RingHomInvPair σ₁₂ σ₂₁\nre₂₃ : RingHomInvPair σ₂₃ σ₃₂\ninst✝³ : RingHomInvPair σ₁₃ σ₃₁\nre₂₁ : RingHomInvPair σ₂₁ σ₁₂\nre₃₂ : RingHomInvPair σ₃₂ σ₂₃\ninst✝² : RingHomInvPair σ₃₁ σ₁₃\ne₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂\ne₂₃ : M₂ ≃ₛₗ[σ₂₃] M₃\ninst✝¹ : RingHomCompTriple σ₂₁ σ₁₃ σ₂₃\ninst✝ : RingHomCompTriple σ₃₁ σ₁₂ σ₃₂\nf : M₁ ≃ₛₗ[σ₁₂] M₂\n⊢ trans f (symm f) = refl R₁ M₁",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type ?u.216670\nR₁ : Type u_1\nR₂ : Type u_2\nR₃ : Type ?u.216679\nk : Type ?u.216682\nS : Type ?u.216685\nM : Type ?u.216688\nM₁ : Type u_3\nM₂ : Type u_4\nM₃ : Type ?u.216697\nN₁ : Type ?u.216700\nN₂ : Type ?u.216703\nN₃ : Type ?u.216706\nN₄ : Type ?u.216709\nι : Type ?u.216712\nM₄ : Type ?u.216715\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring S\ninst✝¹⁵ : Semiring R₁\ninst✝¹⁴ : Semiring R₂\ninst✝¹³ : Semiring R₃\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : AddCommMonoid M₁\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : AddCommMonoid M₃\ninst✝⁸ : AddCommMonoid M₄\ninst✝⁷ : AddCommMonoid N₁\ninst✝⁶ : AddCommMonoid N₂\nmodule_M : Module R M\nmodule_S_M₂ : Module S M₂\nσ : R →+* S\nσ' : S →+* R\nre₁ : RingHomInvPair σ σ'\nre₂ : RingHomInvPair σ' σ\ne e' : M ≃ₛₗ[σ] M₂\nmodule_M₁ : Module R₁ M₁\nmodule_M₂ : Module R₂ M₂\nmodule_M₃ : Module R₃ M₃\nmodule_N₁ : Module R₁ N₁\nmodule_N₂ : Module R₁ N₂\nσ₁₂ : R₁ →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\nσ₂₁ : R₂ →+* R₁\nσ₃₂ : R₃ →+* R₂\nσ₃₁ : R₃ →+* R₁\ninst✝⁵ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝⁴ : RingHomCompTriple σ₃₂ σ₂₁ σ₃₁\nre₁₂ : RingHomInvPair σ₁₂ σ₂₁\nre₂₃ : RingHomInvPair σ₂₃ σ₃₂\ninst✝³ : RingHomInvPair σ₁₃ σ₃₁\nre₂₁ : RingHomInvPair σ₂₁ σ₁₂\nre₃₂ : RingHomInvPair σ₃₂ σ₂₃\ninst✝² : RingHomInvPair σ₃₁ σ₁₃\ne₁₂ : M₁ ≃ₛₗ[σ₁₂] M₂\ne₂₃ : M₂ ≃ₛₗ[σ₂₃] M₃\ninst✝¹ : RingHomCompTriple σ₂₁ σ₁₃ σ₂₃\ninst✝ : RingHomCompTriple σ₃₁ σ₁₂ σ₃₂\nf : M₁ ≃ₛₗ[σ₁₂] M₂\nx : M₁\n⊢ ↑(trans f (symm f)) x = ↑(refl R₁ M₁) x",
"tactic": "simp"
}
] |
[
468,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
466,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.integral_mono_ae
|
[
{
"state_after": "α : Type u_1\nE : Type ?u.1219372\nF : Type ?u.1219375\n𝕜 : Type ?u.1219378\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g✝ : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1222069\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf g : α → ℝ\nhf : Integrable f\nhg : Integrable g\nh : f ≤ᵐ[μ] g\n⊢ (if hf : Integrable fun a => f a then ↑L1.integralCLM (Integrable.toL1 (fun a => f a) hf) else 0) ≤\n if hf : Integrable fun a => g a then ↑L1.integralCLM (Integrable.toL1 (fun a => g a) hf) else 0",
"state_before": "α : Type u_1\nE : Type ?u.1219372\nF : Type ?u.1219375\n𝕜 : Type ?u.1219378\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g✝ : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1222069\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf g : α → ℝ\nhf : Integrable f\nhg : Integrable g\nh : f ≤ᵐ[μ] g\n⊢ (∫ (a : α), f a ∂μ) ≤ ∫ (a : α), g a ∂μ",
"tactic": "simp only [integral, L1.integral]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.1219372\nF : Type ?u.1219375\n𝕜 : Type ?u.1219378\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : CompleteSpace E\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℝ 𝕜 E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : CompleteSpace F\nf✝ g✝ : α → E\nm : MeasurableSpace α\nμ : Measure α\nX : Type ?u.1222069\ninst✝¹ : TopologicalSpace X\ninst✝ : FirstCountableTopology X\nf g : α → ℝ\nhf : Integrable f\nhg : Integrable g\nh : f ≤ᵐ[μ] g\n⊢ (if hf : Integrable fun a => f a then ↑L1.integralCLM (Integrable.toL1 (fun a => f a) hf) else 0) ≤\n if hf : Integrable fun a => g a then ↑L1.integralCLM (Integrable.toL1 (fun a => g a) hf) else 0",
"tactic": "exact setToFun_mono (dominatedFinMeasAdditive_weightedSMul μ)\n (fun s _ _ => weightedSMul_nonneg s) hf hg h"
}
] |
[
1299,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1295,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
ContDiff.iterate_deriv'
|
[] |
[
2180,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2177,
1
] |
Mathlib/Topology/Bornology/Constructions.lean
|
Bornology.isBounded_prod_of_nonempty
|
[] |
[
84,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
1
] |
Mathlib/Probability/Independence/Basic.lean
|
ProbabilityTheory.IndepFun.comp
|
[
{
"state_after": "case intro.intro.intro.intro\nΩ : Type u_5\nι : Type ?u.3311612\nβ : Type u_1\nβ' : Type u_2\nγ : Type u_3\nγ' : Type u_4\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf : Ω → β\ng : Ω → β'\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\nmγ : MeasurableSpace γ\nmγ' : MeasurableSpace γ'\nφ : β → γ\nψ : β' → γ'\nhfg : IndepFun f g\nhφ : Measurable φ\nhψ : Measurable ψ\nA : Set γ\nhA : MeasurableSet A\nB : Set γ'\nhB : MeasurableSet B\n⊢ ↑↑μ (φ ∘ f ⁻¹' A ∩ ψ ∘ g ⁻¹' B) = ↑↑μ (φ ∘ f ⁻¹' A) * ↑↑μ (ψ ∘ g ⁻¹' B)",
"state_before": "Ω : Type u_5\nι : Type ?u.3311612\nβ : Type u_1\nβ' : Type u_2\nγ : Type u_3\nγ' : Type u_4\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf : Ω → β\ng : Ω → β'\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\nmγ : MeasurableSpace γ\nmγ' : MeasurableSpace γ'\nφ : β → γ\nψ : β' → γ'\nhfg : IndepFun f g\nhφ : Measurable φ\nhψ : Measurable ψ\n⊢ IndepFun (φ ∘ f) (ψ ∘ g)",
"tactic": "rintro _ _ ⟨A, hA, rfl⟩ ⟨B, hB, rfl⟩"
},
{
"state_after": "case intro.intro.intro.intro.a\nΩ : Type u_5\nι : Type ?u.3311612\nβ : Type u_1\nβ' : Type u_2\nγ : Type u_3\nγ' : Type u_4\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf : Ω → β\ng : Ω → β'\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\nmγ : MeasurableSpace γ\nmγ' : MeasurableSpace γ'\nφ : β → γ\nψ : β' → γ'\nhfg : IndepFun f g\nhφ : Measurable φ\nhψ : Measurable ψ\nA : Set γ\nhA : MeasurableSet A\nB : Set γ'\nhB : MeasurableSet B\n⊢ φ ∘ f ⁻¹' A ∈ {s | MeasurableSet s}\n\ncase intro.intro.intro.intro.a\nΩ : Type u_5\nι : Type ?u.3311612\nβ : Type u_1\nβ' : Type u_2\nγ : Type u_3\nγ' : Type u_4\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf : Ω → β\ng : Ω → β'\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\nmγ : MeasurableSpace γ\nmγ' : MeasurableSpace γ'\nφ : β → γ\nψ : β' → γ'\nhfg : IndepFun f g\nhφ : Measurable φ\nhψ : Measurable ψ\nA : Set γ\nhA : MeasurableSet A\nB : Set γ'\nhB : MeasurableSet B\n⊢ ψ ∘ g ⁻¹' B ∈ {s | MeasurableSet s}",
"state_before": "case intro.intro.intro.intro\nΩ : Type u_5\nι : Type ?u.3311612\nβ : Type u_1\nβ' : Type u_2\nγ : Type u_3\nγ' : Type u_4\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf : Ω → β\ng : Ω → β'\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\nmγ : MeasurableSpace γ\nmγ' : MeasurableSpace γ'\nφ : β → γ\nψ : β' → γ'\nhfg : IndepFun f g\nhφ : Measurable φ\nhψ : Measurable ψ\nA : Set γ\nhA : MeasurableSet A\nB : Set γ'\nhB : MeasurableSet B\n⊢ ↑↑μ (φ ∘ f ⁻¹' A ∩ ψ ∘ g ⁻¹' B) = ↑↑μ (φ ∘ f ⁻¹' A) * ↑↑μ (ψ ∘ g ⁻¹' B)",
"tactic": "apply hfg"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.a\nΩ : Type u_5\nι : Type ?u.3311612\nβ : Type u_1\nβ' : Type u_2\nγ : Type u_3\nγ' : Type u_4\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf : Ω → β\ng : Ω → β'\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\nmγ : MeasurableSpace γ\nmγ' : MeasurableSpace γ'\nφ : β → γ\nψ : β' → γ'\nhfg : IndepFun f g\nhφ : Measurable φ\nhψ : Measurable ψ\nA : Set γ\nhA : MeasurableSet A\nB : Set γ'\nhB : MeasurableSet B\n⊢ φ ∘ f ⁻¹' A ∈ {s | MeasurableSet s}",
"tactic": "exact ⟨φ ⁻¹' A, hφ hA, Set.preimage_comp.symm⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.a\nΩ : Type u_5\nι : Type ?u.3311612\nβ : Type u_1\nβ' : Type u_2\nγ : Type u_3\nγ' : Type u_4\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf : Ω → β\ng : Ω → β'\nmβ : MeasurableSpace β\nmβ' : MeasurableSpace β'\nmγ : MeasurableSpace γ\nmγ' : MeasurableSpace γ'\nφ : β → γ\nψ : β' → γ'\nhfg : IndepFun f g\nhφ : Measurable φ\nhψ : Measurable ψ\nA : Set γ\nhA : MeasurableSet A\nB : Set γ'\nhB : MeasurableSet B\n⊢ ψ ∘ g ⁻¹' B ∈ {s | MeasurableSet s}",
"tactic": "exact ⟨ψ ⁻¹' B, hψ hB, Set.preimage_comp.symm⟩"
}
] |
[
729,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
722,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
Uniform.tendsto_nhds_right
|
[
{
"state_after": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.196733\ninst✝ : UniformSpace α\nf : Filter β\nu : β → α\na : α\n⊢ Tendsto (Prod.mk a ∘ u) f (𝓤 α) ↔ Tendsto (fun x => (a, u x)) f (𝓤 α)",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.196733\ninst✝ : UniformSpace α\nf : Filter β\nu : β → α\na : α\n⊢ Tendsto u f (𝓝 a) ↔ Tendsto (fun x => (a, u x)) f (𝓤 α)",
"tactic": "rw [nhds_eq_comap_uniformity, tendsto_comap_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort ?u.196733\ninst✝ : UniformSpace α\nf : Filter β\nu : β → α\na : α\n⊢ Tendsto (Prod.mk a ∘ u) f (𝓤 α) ↔ Tendsto (fun x => (a, u x)) f (𝓤 α)",
"tactic": "rfl"
}
] |
[
1915,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1913,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean
|
CategoryTheory.Limits.epi_of_target_iso_zero
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝³ : Category C\nD : Type u'\ninst✝² : Category D\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nX Y : C\nf : X ⟶ Y\ni : Y ≅ 0\nZ : C\ng h : Y ⟶ Z\nx✝ : f ≫ g = f ≫ h\n⊢ g = h",
"tactic": "rw [zero_of_source_iso_zero g i, zero_of_source_iso_zero h i]"
}
] |
[
408,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
407,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Diagonal.lean
|
CategoryTheory.Limits.pullback.diagonal_fst
|
[] |
[
50,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
49,
1
] |
Mathlib/Analysis/Analytic/Basic.lean
|
HasFPowerSeriesOnBall.neg
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.532806\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f pf x r\n⊢ r ≤ FormalMultilinearSeries.radius pf",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.532806\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f pf x r\n⊢ r ≤ FormalMultilinearSeries.radius (-pf)",
"tactic": "rw [pf.radius_neg]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type ?u.532806\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf g : E → F\np pf pg : FormalMultilinearSeries 𝕜 E F\nx : E\nr r' : ℝ≥0∞\nhf : HasFPowerSeriesOnBall f pf x r\n⊢ r ≤ FormalMultilinearSeries.radius pf",
"tactic": "exact hf.r_le"
}
] |
[
562,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
556,
1
] |
Mathlib/Order/Hom/Lattice.lean
|
SupHom.symm_dual_id
|
[] |
[
1389,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1388,
1
] |
Mathlib/Data/List/Basic.lean
|
List.get_drop
|
[
{
"state_after": "ι : Type ?u.209291\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ L : List α\ni j : ℕ\nh : i + j < length L\nA : i < length L\n⊢ j < length (drop i L)",
"state_before": "ι : Type ?u.209291\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ L : List α\ni j : ℕ\nh : i + j < length L\n⊢ j < length (drop i L)",
"tactic": "have A : i < L.length := lt_of_le_of_lt (Nat.le.intro rfl) h"
},
{
"state_after": "ι : Type ?u.209291\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ L : List α\ni j : ℕ\nh : i + j < length (take i L ++ drop i L)\nA : i < length L\n⊢ j < length (drop i L)",
"state_before": "ι : Type ?u.209291\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ L : List α\ni j : ℕ\nh : i + j < length L\nA : i < length L\n⊢ j < length (drop i L)",
"tactic": "rw [(take_append_drop i L).symm] at h"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.209291\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ L : List α\ni j : ℕ\nh : i + j < length (take i L ++ drop i L)\nA : i < length L\n⊢ j < length (drop i L)",
"tactic": "simpa only [le_of_lt A, min_eq_left, add_lt_add_iff_left, length_take,\n length_append] using h"
},
{
"state_after": "ι : Type ?u.209291\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ L : List α\ni j : ℕ\nh : i + j < length L\n⊢ nthLe L (i + j) h = nthLe (drop i L) j (_ : j < length (drop i L))",
"state_before": "ι : Type ?u.209291\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ L : List α\ni j : ℕ\nh : i + j < length L\n⊢ get L { val := i + j, isLt := h } = get (drop i L) { val := j, isLt := (_ : j < length (drop i L)) }",
"tactic": "rw [← nthLe_eq, ← nthLe_eq]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.209291\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ L : List α\ni j : ℕ\nh : i + j < length L\n⊢ nthLe L (i + j) h = nthLe (drop i L) j (_ : j < length (drop i L))",
"tactic": "rw [nthLe_of_eq (take_append_drop i L).symm h, nthLe_append_right] <;>\nsimp [min_eq_left (show i ≤ length L from le_trans (by simp) (le_of_lt h))]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.209291\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ L : List α\ni j : ℕ\nh : i + j < length L\n⊢ i ≤ i + j",
"tactic": "simp"
}
] |
[
2211,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2203,
1
] |
Mathlib/Topology/Algebra/ContinuousAffineMap.lean
|
ContinuousLinearMap.toContinuousAffineMap_map_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nV : Type u_2\nW : Type u_3\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : TopologicalSpace V\ninst✝² : AddCommGroup W\ninst✝¹ : Module R W\ninst✝ : TopologicalSpace W\nf : V →L[R] W\n⊢ ↑(toContinuousAffineMap f) 0 = 0",
"tactic": "simp"
}
] |
[
287,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
287,
1
] |
Mathlib/Data/Pi/Algebra.lean
|
Pi.comp_one
|
[] |
[
75,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.ofReal_bit0
|
[] |
[
215,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
214,
1
] |
Mathlib/Data/Vector/Basic.lean
|
Vector.singleton_tail
|
[] |
[
188,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
187,
1
] |
Mathlib/Data/List/Chain.lean
|
List.chain'_iff_get
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na b : α\nR : α → α → Prop\n⊢ Chain' R []",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na b : α\nR : α → α → Prop\nx✝ : ℕ\nh : x✝ < length [] - 1\n⊢ R (get [] { val := x✝, isLt := (_ : x✝ < length []) }) (get [] { val := x✝ + 1, isLt := (_ : succ x✝ < length []) })",
"tactic": "simp at h"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b : α\nR : α → α → Prop\na : α\n⊢ Chain' R [a]",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b : α\nR : α → α → Prop\na : α\nx✝ : ℕ\nh : x✝ < length [a] - 1\n⊢ R (get [a] { val := x✝, isLt := (_ : x✝ < length [a]) })\n (get [a] { val := x✝ + 1, isLt := (_ : succ x✝ < length [a]) })",
"tactic": "simp at h"
},
{
"state_after": "α : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b✝ : α\nR : α → α → Prop\na b : α\nt : List α\n⊢ (R a b ∧\n ∀ (i : ℕ) (h : i < length (b :: t) - 1),\n R (get (b :: t) { val := i, isLt := (_ : i < length (b :: t)) })\n (get (b :: t) { val := i + 1, isLt := (_ : succ i < length (b :: t)) })) ↔\n (∀ (h : 0 < length (a :: b :: t) - 1),\n R (get (a :: b :: t) { val := 0, isLt := (_ : 0 < length (a :: b :: t)) })\n (get (a :: b :: t) { val := 0 + 1, isLt := (_ : succ 0 < length (a :: b :: t)) })) ∧\n ∀ (n : ℕ) (h : n + 1 < length (a :: b :: t) - 1),\n R (get (a :: b :: t) { val := n + 1, isLt := (_ : n + 1 < length (a :: b :: t)) })\n (get (a :: b :: t) { val := n + 1 + 1, isLt := (_ : succ (n + 1) < length (a :: b :: t)) })",
"state_before": "α : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b✝ : α\nR : α → α → Prop\na b : α\nt : List α\n⊢ Chain' R (a :: b :: t) ↔\n ∀ (i : ℕ) (h : i < length (a :: b :: t) - 1),\n R (get (a :: b :: t) { val := i, isLt := (_ : i < length (a :: b :: t)) })\n (get (a :: b :: t) { val := i + 1, isLt := (_ : succ i < length (a :: b :: t)) })",
"tactic": "rw [← and_forall_succ, chain'_cons, chain'_iff_get]"
},
{
"state_after": "α : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b✝ : α\nR : α → α → Prop\na b : α\nt : List α\n⊢ R a b →\n ((∀ (i : ℕ) (h : i < length (b :: t) - 1),\n R (get (b :: t) { val := i, isLt := (_ : i < length (b :: t)) })\n (get t { val := i, isLt := (_ : i < length t) })) ↔\n ∀ (n : ℕ) (h : n + 1 < length (a :: b :: t) - 1),\n R (get (b :: t) { val := n, isLt := (_ : n < length (b :: t)) })\n (get t { val := n, isLt := (_ : n < length t) }))",
"state_before": "α : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b✝ : α\nR : α → α → Prop\na b : α\nt : List α\n⊢ (R a b ∧\n ∀ (i : ℕ) (h : i < length (b :: t) - 1),\n R (get (b :: t) { val := i, isLt := (_ : i < length (b :: t)) })\n (get (b :: t) { val := i + 1, isLt := (_ : succ i < length (b :: t)) })) ↔\n (∀ (h : 0 < length (a :: b :: t) - 1),\n R (get (a :: b :: t) { val := 0, isLt := (_ : 0 < length (a :: b :: t)) })\n (get (a :: b :: t) { val := 0 + 1, isLt := (_ : succ 0 < length (a :: b :: t)) })) ∧\n ∀ (n : ℕ) (h : n + 1 < length (a :: b :: t) - 1),\n R (get (a :: b :: t) { val := n + 1, isLt := (_ : n + 1 < length (a :: b :: t)) })\n (get (a :: b :: t) { val := n + 1 + 1, isLt := (_ : succ (n + 1) < length (a :: b :: t)) })",
"tactic": "simp"
},
{
"state_after": "α : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b✝ : α\nR : α → α → Prop\na b : α\nt : List α\n⊢ R a b →\n ((∀ (i : ℕ) (h : i < length t),\n R (get (b :: t) { val := i, isLt := (_ : i < succ (length t)) })\n (get t { val := i, isLt := (_ : succ (i + 0) ≤ length t) })) ↔\n ∀ (n : ℕ) (h : n + 1 < length t + 1),\n R (get (b :: t) { val := n, isLt := (_ : succ (n + 0) ≤ length (b :: t)) })\n (get t { val := n, isLt := (_ : succ (n + 0) ≤ length t) }))",
"state_before": "α : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b✝ : α\nR : α → α → Prop\na b : α\nt : List α\n⊢ R a b →\n ((∀ (i : ℕ) (h : i < length (b :: t) - 1),\n R (get (b :: t) { val := i, isLt := (_ : i < length (b :: t)) })\n (get t { val := i, isLt := (_ : i < length t) })) ↔\n ∀ (n : ℕ) (h : n + 1 < length (a :: b :: t) - 1),\n R (get (b :: t) { val := n, isLt := (_ : n < length (b :: t)) })\n (get t { val := n, isLt := (_ : n < length t) }))",
"tactic": "dsimp [succ_sub_one]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nR✝ r : α → α → Prop\nl l₁ l₂ : List α\na✝ b✝ : α\nR : α → α → Prop\na b : α\nt : List α\n⊢ R a b →\n ((∀ (i : ℕ) (h : i < length t),\n R (get (b :: t) { val := i, isLt := (_ : i < succ (length t)) })\n (get t { val := i, isLt := (_ : succ (i + 0) ≤ length t) })) ↔\n ∀ (n : ℕ) (h : n + 1 < length t + 1),\n R (get (b :: t) { val := n, isLt := (_ : succ (n + 0) ≤ length (b :: t)) })\n (get t { val := n, isLt := (_ : succ (n + 0) ≤ length t) }))",
"tactic": "exact fun _ => ⟨fun h i hi => h i (Nat.lt_of_succ_lt_succ hi),\n fun h i hi => h i (Nat.succ_lt_succ hi)⟩"
}
] |
[
361,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
351,
1
] |
Mathlib/CategoryTheory/Limits/IsLimit.lean
|
CategoryTheory.Limits.IsLimit.OfNatIso.coneOfHom_homOfCone
|
[
{
"state_after": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt : Cone F\nX : C\nh : yoneda.obj X ⋙ uliftFunctor ≅ Functor.cones F\ns : Cone F\n⊢ { pt := s.pt, π := h.hom.app s.pt.op { down := (h.inv.app s.pt.op s.π).down } } = s",
"state_before": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt : Cone F\nX : C\nh : yoneda.obj X ⋙ uliftFunctor ≅ Functor.cones F\ns : Cone F\n⊢ coneOfHom h (homOfCone h s) = s",
"tactic": "dsimp [coneOfHom, homOfCone]"
},
{
"state_after": "no goals",
"state_before": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt : Cone F\nX : C\nh : yoneda.obj X ⋙ uliftFunctor ≅ Functor.cones F\ns : Cone F\n⊢ { pt := s.pt, π := h.hom.app s.pt.op { down := (h.inv.app s.pt.op s.π).down } } = s",
"tactic": "match s with\n| .mk s_pt s_π =>\n congr; dsimp\n convert congrFun (congrFun (congrArg NatTrans.app h.inv_hom_id) (op s_pt)) s_π using 1"
},
{
"state_after": "case e_π\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt : Cone F\nX : C\nh : yoneda.obj X ⋙ uliftFunctor ≅ Functor.cones F\ns : Cone F\ns_pt : C\ns_π : (const J).obj s_pt ⟶ F\n⊢ h.hom.app { pt := s_pt, π := s_π }.pt.op\n { down := (h.inv.app { pt := s_pt, π := s_π }.pt.op { pt := s_pt, π := s_π }.π).down } =\n s_π",
"state_before": "J : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt : Cone F\nX : C\nh : yoneda.obj X ⋙ uliftFunctor ≅ Functor.cones F\ns : Cone F\ns_pt : C\ns_π : (const J).obj s_pt ⟶ F\n⊢ { pt := { pt := s_pt, π := s_π }.pt,\n π :=\n h.hom.app { pt := s_pt, π := s_π }.pt.op\n { down := (h.inv.app { pt := s_pt, π := s_π }.pt.op { pt := s_pt, π := s_π }.π).down } } =\n { pt := s_pt, π := s_π }",
"tactic": "congr"
},
{
"state_after": "case e_π\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt : Cone F\nX : C\nh : yoneda.obj X ⋙ uliftFunctor ≅ Functor.cones F\ns : Cone F\ns_pt : C\ns_π : (const J).obj s_pt ⟶ F\n⊢ h.hom.app s_pt.op { down := (h.inv.app s_pt.op s_π).down } = s_π",
"state_before": "case e_π\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt : Cone F\nX : C\nh : yoneda.obj X ⋙ uliftFunctor ≅ Functor.cones F\ns : Cone F\ns_pt : C\ns_π : (const J).obj s_pt ⟶ F\n⊢ h.hom.app { pt := s_pt, π := s_π }.pt.op\n { down := (h.inv.app { pt := s_pt, π := s_π }.pt.op { pt := s_pt, π := s_π }.π).down } =\n s_π",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "case e_π\nJ : Type u₁\ninst✝² : Category J\nK : Type u₂\ninst✝¹ : Category K\nC : Type u₃\ninst✝ : Category C\nF : J ⥤ C\nt : Cone F\nX : C\nh : yoneda.obj X ⋙ uliftFunctor ≅ Functor.cones F\ns : Cone F\ns_pt : C\ns_π : (const J).obj s_pt ⟶ F\n⊢ h.hom.app s_pt.op { down := (h.inv.app s_pt.op s_π).down } = s_π",
"tactic": "convert congrFun (congrFun (congrArg NatTrans.app h.inv_hom_id) (op s_pt)) s_π using 1"
}
] |
[
493,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
488,
1
] |
Mathlib/Data/Sum/Basic.lean
|
Sum.inr_ne_inl
|
[] |
[
168,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.cos_add_pi
|
[] |
[
316,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
315,
1
] |
Mathlib/Data/Int/Bitwise.lean
|
Int.bit0_ne_bit1
|
[
{
"state_after": "no goals",
"state_before": "m n : ℤ\n⊢ ¬bodd (bit0 m) = bodd (bit1 n)",
"tactic": "simp"
}
] |
[
184,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
1
] |
Mathlib/Data/Real/Irrational.lean
|
Irrational.nat_mul
|
[] |
[
386,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
385,
1
] |
Mathlib/Order/Interval.lean
|
NonemptyInterval.coe_ssubset_coe
|
[] |
[
259,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
258,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
Commute.cast_nat_mul_left
|
[] |
[
1094,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1093,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Ioo_eq_empty_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.28100\ninst✝¹ : Preorder α\na a₁ a₂ b b₁ b₂ c x : α\ninst✝ : DenselyOrdered α\n⊢ Ioo a b = ∅ ↔ ¬a < b",
"tactic": "rw [← not_nonempty_iff_eq_empty, not_iff_not, nonempty_Ioo]"
}
] |
[
695,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
694,
1
] |
Mathlib/LinearAlgebra/Projection.lean
|
LinearMap.equivProdOfSurjectiveOfIsCompl_apply
|
[] |
[
339,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
337,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
toIcoMod_add_right_eq_add
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b✝ c✝ : α\nn : ℤ\na b c : α\n⊢ toIcoMod hp a (b + c) = toIcoMod hp (a - c) b + c",
"tactic": "simp_rw [toIcoMod, toIcoDiv_sub_eq_toIcoDiv_add', sub_add_eq_add_sub]"
}
] |
[
541,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
540,
1
] |
Std/Data/List/Init/Lemmas.lean
|
List.bind_id
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl : List (List α)\n⊢ List.bind l id = join l",
"tactic": "simp [List.bind]"
}
] |
[
109,
93
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
109,
9
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.exists_succAbove_eq
|
[
{
"state_after": "case inl\nn m : ℕ\nx y : Fin (n + 1)\nh : x ≠ y\nhlt : x < y\n⊢ ∃ z, ↑(succAbove y) z = x\n\ncase inr\nn m : ℕ\nx y : Fin (n + 1)\nh : x ≠ y\nhlt : y < x\n⊢ ∃ z, ↑(succAbove y) z = x",
"state_before": "n m : ℕ\nx y : Fin (n + 1)\nh : x ≠ y\n⊢ ∃ z, ↑(succAbove y) z = x",
"tactic": "cases' h.lt_or_lt with hlt hlt"
},
{
"state_after": "no goals",
"state_before": "case inl\nn m : ℕ\nx y : Fin (n + 1)\nh : x ≠ y\nhlt : x < y\n⊢ ∃ z, ↑(succAbove y) z = x\n\ncase inr\nn m : ℕ\nx y : Fin (n + 1)\nh : x ≠ y\nhlt : y < x\n⊢ ∃ z, ↑(succAbove y) z = x",
"tactic": "exacts [⟨_, succAbove_castLT hlt⟩, ⟨_, succAbove_pred hlt⟩]"
}
] |
[
2157,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2155,
1
] |
Mathlib/Data/List/TFAE.lean
|
List.tfae_cons_cons
|
[] |
[
52,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
51,
1
] |
Mathlib/Data/Polynomial/Degree/TrailingDegree.lean
|
Polynomial.natTrailingDegree_X_le
|
[] |
[
297,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
296,
1
] |
Mathlib/Algebra/Homology/Augment.lean
|
ChainComplex.augment_d_succ_succ
|
[
{
"state_after": "no goals",
"state_before": "V : Type u\ninst✝¹ : Category V\ninst✝ : HasZeroMorphisms V\nC : ChainComplex V ℕ\nX : V\nf : HomologicalComplex.X C 0 ⟶ X\nw : d C 1 0 ≫ f = 0\ni j : ℕ\n⊢ d (augment C f w) (i + 1) (j + 1) = d C i j",
"tactic": "cases i <;> rfl"
}
] |
[
96,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/RingTheory/Multiplicity.lean
|
multiplicity.pow_dvd_iff_le_multiplicity
|
[] |
[
151,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/Data/Seq/Seq.lean
|
Stream'.Seq.enum_cons
|
[
{
"state_after": "case h.zero\nα : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nx : α\n⊢ get? (enum (cons x s)) Nat.zero = get? (cons (0, x) (map (Prod.map Nat.succ id) (enum s))) Nat.zero\n\ncase h.succ\nα : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nx : α\nn✝ : ℕ\n⊢ get? (enum (cons x s)) (Nat.succ n✝) = get? (cons (0, x) (map (Prod.map Nat.succ id) (enum s))) (Nat.succ n✝)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nx : α\n⊢ enum (cons x s) = cons (0, x) (map (Prod.map Nat.succ id) (enum s))",
"tactic": "ext ⟨n⟩ : 1"
},
{
"state_after": "no goals",
"state_before": "case h.zero\nα : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nx : α\n⊢ get? (enum (cons x s)) Nat.zero = get? (cons (0, x) (map (Prod.map Nat.succ id) (enum s))) Nat.zero",
"tactic": "simp"
},
{
"state_after": "case h.succ\nα : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nx : α\nn✝ : ℕ\n⊢ Option.map (Prod.mk (Nat.succ n✝)) (get? s n✝) = Option.map (Prod.map Nat.succ id ∘ Prod.mk n✝) (get? s n✝)",
"state_before": "case h.succ\nα : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nx : α\nn✝ : ℕ\n⊢ get? (enum (cons x s)) (Nat.succ n✝) = get? (cons (0, x) (map (Prod.map Nat.succ id) (enum s))) (Nat.succ n✝)",
"tactic": "simp only [get?_enum, get?_cons_succ, map_get?, Option.map_map]"
},
{
"state_after": "no goals",
"state_before": "case h.succ\nα : Type u\nβ : Type v\nγ : Type w\ns : Seq α\nx : α\nn✝ : ℕ\n⊢ Option.map (Prod.mk (Nat.succ n✝)) (get? s n✝) = Option.map (Prod.map Nat.succ id ∘ Prod.mk n✝) (get? s n✝)",
"tactic": "congr"
}
] |
[
899,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
894,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Filter.lean
|
BoxIntegral.IntegrationParams.toFilteriUnion_congr
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\ninst✝ : Fintype ι\nI✝ J : Box ι\nc c₁ c₂ : ℝ≥0\nr r₁ r₂ : (ι → ℝ) → ↑(Set.Ioi 0)\nπ π₁✝ π₂✝ : TaggedPrepartition I✝\nl✝ l₁ l₂ : IntegrationParams\nI : Box ι\nl : IntegrationParams\nπ₁ π₂ : Prepartition I\nh : Prepartition.iUnion π₁ = Prepartition.iUnion π₂\n⊢ toFilteriUnion l I π₁ = toFilteriUnion l I π₂",
"tactic": "simp only [toFilteriUnion, toFilterDistortioniUnion, h]"
}
] |
[
467,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
465,
1
] |
Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean
|
Set.ordConnectedProj_mem_ordConnectedComponent
|
[] |
[
114,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/GroupTheory/Submonoid/Membership.lean
|
Submonoid.coe_powers
|
[] |
[
437,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
436,
1
] |
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