file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Analysis/NormedSpace/Dual.lean
|
NormedSpace.mem_polar_iff
|
[] |
[
196,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearEquiv.map_sub
|
[] |
[
2279,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2278,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.aeval_prod
|
[] |
[
1561,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1559,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearMap.smulRight_one_one
|
[
{
"state_after": "case h\nR₁ : Type u_1\nR₂ : Type ?u.652857\nR₃ : Type ?u.652860\ninst✝²⁰ : Semiring R₁\ninst✝¹⁹ : Semiring R₂\ninst✝¹⁸ : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\nM₁ : Type ?u.652938\ninst✝¹⁷ : TopologicalSpace M₁\ninst✝¹⁶ : AddCommMonoid M₁\nM'₁ : Type ?u.652947\ninst✝¹⁵ : TopologicalSpace M'₁\ninst✝¹⁴ : AddCommMonoid M'₁\nM₂ : Type u_2\ninst✝¹³ : TopologicalSpace M₂\ninst✝¹² : AddCommMonoid M₂\nM₃ : Type ?u.652965\ninst✝¹¹ : TopologicalSpace M₃\ninst✝¹⁰ : AddCommMonoid M₃\nM₄ : Type ?u.652974\ninst✝⁹ : TopologicalSpace M₄\ninst✝⁸ : AddCommMonoid M₄\ninst✝⁷ : Module R₁ M₁\ninst✝⁶ : Module R₁ M'₁\ninst✝⁵ : Module R₂ M₂\ninst✝⁴ : Module R₃ M₃\ninst✝³ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\nF : Type ?u.653212\ninst✝² : Module R₁ M₂\ninst✝¹ : TopologicalSpace R₁\ninst✝ : ContinuousSMul R₁ M₂\nc : R₁ →L[R₁] M₂\n⊢ ↑(smulRight 1 (↑c 1)) 1 = ↑c 1",
"state_before": "R₁ : Type u_1\nR₂ : Type ?u.652857\nR₃ : Type ?u.652860\ninst✝²⁰ : Semiring R₁\ninst✝¹⁹ : Semiring R₂\ninst✝¹⁸ : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\nM₁ : Type ?u.652938\ninst✝¹⁷ : TopologicalSpace M₁\ninst✝¹⁶ : AddCommMonoid M₁\nM'₁ : Type ?u.652947\ninst✝¹⁵ : TopologicalSpace M'₁\ninst✝¹⁴ : AddCommMonoid M'₁\nM₂ : Type u_2\ninst✝¹³ : TopologicalSpace M₂\ninst✝¹² : AddCommMonoid M₂\nM₃ : Type ?u.652965\ninst✝¹¹ : TopologicalSpace M₃\ninst✝¹⁰ : AddCommMonoid M₃\nM₄ : Type ?u.652974\ninst✝⁹ : TopologicalSpace M₄\ninst✝⁸ : AddCommMonoid M₄\ninst✝⁷ : Module R₁ M₁\ninst✝⁶ : Module R₁ M'₁\ninst✝⁵ : Module R₂ M₂\ninst✝⁴ : Module R₃ M₃\ninst✝³ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\nF : Type ?u.653212\ninst✝² : Module R₁ M₂\ninst✝¹ : TopologicalSpace R₁\ninst✝ : ContinuousSMul R₁ M₂\nc : R₁ →L[R₁] M₂\n⊢ smulRight 1 (↑c 1) = c",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nR₁ : Type u_1\nR₂ : Type ?u.652857\nR₃ : Type ?u.652860\ninst✝²⁰ : Semiring R₁\ninst✝¹⁹ : Semiring R₂\ninst✝¹⁸ : Semiring R₃\nσ₁₂ : R₁ →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R₁ →+* R₃\nM₁ : Type ?u.652938\ninst✝¹⁷ : TopologicalSpace M₁\ninst✝¹⁶ : AddCommMonoid M₁\nM'₁ : Type ?u.652947\ninst✝¹⁵ : TopologicalSpace M'₁\ninst✝¹⁴ : AddCommMonoid M'₁\nM₂ : Type u_2\ninst✝¹³ : TopologicalSpace M₂\ninst✝¹² : AddCommMonoid M₂\nM₃ : Type ?u.652965\ninst✝¹¹ : TopologicalSpace M₃\ninst✝¹⁰ : AddCommMonoid M₃\nM₄ : Type ?u.652974\ninst✝⁹ : TopologicalSpace M₄\ninst✝⁸ : AddCommMonoid M₄\ninst✝⁷ : Module R₁ M₁\ninst✝⁶ : Module R₁ M'₁\ninst✝⁵ : Module R₂ M₂\ninst✝⁴ : Module R₃ M₃\ninst✝³ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\nF : Type ?u.653212\ninst✝² : Module R₁ M₂\ninst✝¹ : TopologicalSpace R₁\ninst✝ : ContinuousSMul R₁ M₂\nc : R₁ →L[R₁] M₂\n⊢ ↑(smulRight 1 (↑c 1)) 1 = ↑c 1",
"tactic": "simp [← ContinuousLinearMap.map_smul_of_tower]"
}
] |
[
1182,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1180,
1
] |
Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean
|
Finset.mul_aux
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA B C : Finset α\nhA : Finset.Nonempty A\nhAB : A ⊆ B\nh : ∀ (A' : Finset α), A' ∈ erase (powerset B) ∅ → ↑(card (A * C)) / ↑(card A) ≤ ↑(card (A' * C)) / ↑(card A')\nA' : Finset α\nhAA' : A' ⊆ A\n⊢ card (A * C) * card A' ≤ card (A' * C) * card A",
"state_before": "α : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA B C : Finset α\nhA : Finset.Nonempty A\nhAB : A ⊆ B\nh : ∀ (A' : Finset α), A' ∈ erase (powerset B) ∅ → ↑(card (A * C)) / ↑(card A) ≤ ↑(card (A' * C)) / ↑(card A')\n⊢ ∀ (A' : Finset α), A' ⊆ A → card (A * C) * card A' ≤ card (A' * C) * card A",
"tactic": "rintro A' hAA'"
},
{
"state_after": "case inl\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA B C : Finset α\nhA : Finset.Nonempty A\nhAB : A ⊆ B\nh : ∀ (A' : Finset α), A' ∈ erase (powerset B) ∅ → ↑(card (A * C)) / ↑(card A) ≤ ↑(card (A' * C)) / ↑(card A')\nhAA' : ∅ ⊆ A\n⊢ card (A * C) * card ∅ ≤ card (∅ * C) * card A\n\ncase inr\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA B C : Finset α\nhA : Finset.Nonempty A\nhAB : A ⊆ B\nh : ∀ (A' : Finset α), A' ∈ erase (powerset B) ∅ → ↑(card (A * C)) / ↑(card A) ≤ ↑(card (A' * C)) / ↑(card A')\nA' : Finset α\nhAA' : A' ⊆ A\nhA' : Finset.Nonempty A'\n⊢ card (A * C) * card A' ≤ card (A' * C) * card A",
"state_before": "α : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA B C : Finset α\nhA : Finset.Nonempty A\nhAB : A ⊆ B\nh : ∀ (A' : Finset α), A' ∈ erase (powerset B) ∅ → ↑(card (A * C)) / ↑(card A) ≤ ↑(card (A' * C)) / ↑(card A')\nA' : Finset α\nhAA' : A' ⊆ A\n⊢ card (A * C) * card A' ≤ card (A' * C) * card A",
"tactic": "obtain rfl | hA' := A'.eq_empty_or_nonempty"
},
{
"state_after": "case inr\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA B C : Finset α\nhA : Finset.Nonempty A\nhAB : A ⊆ B\nh : ∀ (A' : Finset α), A' ∈ erase (powerset B) ∅ → ↑(card (A * C)) / ↑(card A) ≤ ↑(card (A' * C)) / ↑(card A')\nA' : Finset α\nhAA' : A' ⊆ A\nhA' : Finset.Nonempty A'\nhA₀ : 0 < ↑(card A)\n⊢ card (A * C) * card A' ≤ card (A' * C) * card A",
"state_before": "case inr\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA B C : Finset α\nhA : Finset.Nonempty A\nhAB : A ⊆ B\nh : ∀ (A' : Finset α), A' ∈ erase (powerset B) ∅ → ↑(card (A * C)) / ↑(card A) ≤ ↑(card (A' * C)) / ↑(card A')\nA' : Finset α\nhAA' : A' ⊆ A\nhA' : Finset.Nonempty A'\n⊢ card (A * C) * card A' ≤ card (A' * C) * card A",
"tactic": "have hA₀ : (0 : ℚ≥0) < A.card := cast_pos.2 hA.card_pos"
},
{
"state_after": "case inr\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA B C : Finset α\nhA : Finset.Nonempty A\nhAB : A ⊆ B\nh : ∀ (A' : Finset α), A' ∈ erase (powerset B) ∅ → ↑(card (A * C)) / ↑(card A) ≤ ↑(card (A' * C)) / ↑(card A')\nA' : Finset α\nhAA' : A' ⊆ A\nhA' : Finset.Nonempty A'\nhA₀ : 0 < ↑(card A)\nhA₀' : 0 < ↑(card A')\n⊢ card (A * C) * card A' ≤ card (A' * C) * card A",
"state_before": "case inr\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA B C : Finset α\nhA : Finset.Nonempty A\nhAB : A ⊆ B\nh : ∀ (A' : Finset α), A' ∈ erase (powerset B) ∅ → ↑(card (A * C)) / ↑(card A) ≤ ↑(card (A' * C)) / ↑(card A')\nA' : Finset α\nhAA' : A' ⊆ A\nhA' : Finset.Nonempty A'\nhA₀ : 0 < ↑(card A)\n⊢ card (A * C) * card A' ≤ card (A' * C) * card A",
"tactic": "have hA₀' : (0 : ℚ≥0) < A'.card := cast_pos.2 hA'.card_pos"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA B C : Finset α\nhA : Finset.Nonempty A\nhAB : A ⊆ B\nh : ∀ (A' : Finset α), A' ∈ erase (powerset B) ∅ → ↑(card (A * C)) / ↑(card A) ≤ ↑(card (A' * C)) / ↑(card A')\nA' : Finset α\nhAA' : A' ⊆ A\nhA' : Finset.Nonempty A'\nhA₀ : 0 < ↑(card A)\nhA₀' : 0 < ↑(card A')\n⊢ card (A * C) * card A' ≤ card (A' * C) * card A",
"tactic": "exact_mod_cast\n (div_le_div_iff hA₀ hA₀').1\n (h _ <| mem_erase_of_ne_of_mem hA'.ne_empty <| mem_powerset.2 <| hAA'.trans hAB)"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA B C : Finset α\nhA : Finset.Nonempty A\nhAB : A ⊆ B\nh : ∀ (A' : Finset α), A' ∈ erase (powerset B) ∅ → ↑(card (A * C)) / ↑(card A) ≤ ↑(card (A' * C)) / ↑(card A')\nhAA' : ∅ ⊆ A\n⊢ card (A * C) * card ∅ ≤ card (∅ * C) * card A",
"tactic": "simp"
}
] |
[
137,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
9
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.tanh_conj
|
[
{
"state_after": "no goals",
"state_before": "x y : ℂ\n⊢ tanh (↑(starRingEnd ℂ) x) = ↑(starRingEnd ℂ) (tanh x)",
"tactic": "rw [tanh, sinh_conj, cosh_conj, ← map_div₀, tanh]"
}
] |
[
705,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
704,
1
] |
Mathlib/Order/WithBot.lean
|
WithBot.some_le_some
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.7924\nγ : Type ?u.7927\nδ : Type ?u.7930\na b : α\ninst✝ : LE α\n⊢ Option.some a ≤ Option.some b ↔ a ≤ b",
"tactic": "simp [LE.le]"
}
] |
[
203,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
1
] |
Mathlib/Order/CompleteLattice.lean
|
Prod.fst_iInf
|
[] |
[
1858,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1857,
1
] |
Mathlib/MeasureTheory/Decomposition/Lebesgue.lean
|
MeasureTheory.SignedMeasure.singularPart_sub
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.640818\nm : MeasurableSpace α\nμ✝ ν : Measure α\ns✝ t✝ s t : SignedMeasure α\nμ : Measure α\ninst✝¹ : HaveLebesgueDecomposition s μ\ninst✝ : HaveLebesgueDecomposition t μ\n⊢ singularPart (s - t) μ = singularPart s μ - singularPart t μ",
"tactic": "rw [sub_eq_add_neg, sub_eq_add_neg, singularPart_add, singularPart_neg]"
}
] |
[
1127,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1124,
1
] |
Mathlib/Data/Nat/Factorization/Basic.lean
|
Nat.exists_eq_pow_mul_and_not_dvd
|
[] |
[
578,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
573,
1
] |
Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean
|
ProjectiveSpectrum.gc_set
|
[
{
"state_after": "R : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nideal_gc : GaloisConnection Ideal.span SetLike.coe\n⊢ GaloisConnection (fun s => zeroLocus 𝒜 s) fun t => ↑(vanishingIdeal t)",
"state_before": "R : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\n⊢ GaloisConnection (fun s => zeroLocus 𝒜 s) fun t => ↑(vanishingIdeal t)",
"tactic": "have ideal_gc : GaloisConnection Ideal.span _ := (Submodule.gi A _).gc"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\nideal_gc : GaloisConnection Ideal.span SetLike.coe\n⊢ GaloisConnection (fun s => zeroLocus 𝒜 s) fun t => ↑(vanishingIdeal t)",
"tactic": "simpa [zeroLocus_span, Function.comp] using GaloisConnection.compose ideal_gc (gc_ideal 𝒜)"
}
] |
[
145,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordset.pos_size_of_mem
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\nx : α\nt : Ordset α\nh_mem : Ordnode.mem x ↑t = true\n⊢ 0 < size t",
"state_before": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\nx : α\nt : Ordset α\nh_mem : x ∈ t\n⊢ 0 < size t",
"tactic": "simp [Membership.mem, mem] at h_mem"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Preorder α\ninst✝ : DecidableRel fun x x_1 => x ≤ x_1\nx : α\nt : Ordset α\nh_mem : Ordnode.mem x ↑t = true\n⊢ 0 < size t",
"tactic": "apply Ordnode.pos_size_of_mem t.property.sz h_mem"
}
] |
[
1779,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1777,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
upperBounds_empty
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\ninst✝¹ : Preorder α\ninst✝ : Preorder β\ns t : Set α\na b : α\n⊢ upperBounds ∅ = univ",
"tactic": "simp only [upperBounds, eq_univ_iff_forall, mem_setOf_eq, ball_empty_iff, forall_true_iff]"
}
] |
[
865,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
864,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.mul_diagonal
|
[
{
"state_after": "l : Type ?u.272607\nm : Type u_2\nn : Type u_1\no : Type ?u.272616\nm' : o → Type ?u.272621\nn' : o → Type ?u.272626\nR : Type ?u.272629\nS : Type ?u.272632\nα : Type v\nβ : Type w\nγ : Type ?u.272639\ninst✝² : NonUnitalNonAssocSemiring α\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nd : n → α\nM : Matrix m n α\ni : m\nj : n\n⊢ (M ⬝ (diagonal d)ᵀ) i j = M i j * d j",
"state_before": "l : Type ?u.272607\nm : Type u_2\nn : Type u_1\no : Type ?u.272616\nm' : o → Type ?u.272621\nn' : o → Type ?u.272626\nR : Type ?u.272629\nS : Type ?u.272632\nα : Type v\nβ : Type w\nγ : Type ?u.272639\ninst✝² : NonUnitalNonAssocSemiring α\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nd : n → α\nM : Matrix m n α\ni : m\nj : n\n⊢ (M ⬝ diagonal d) i j = M i j * d j",
"tactic": "rw [← diagonal_transpose]"
},
{
"state_after": "no goals",
"state_before": "l : Type ?u.272607\nm : Type u_2\nn : Type u_1\no : Type ?u.272616\nm' : o → Type ?u.272621\nn' : o → Type ?u.272626\nR : Type ?u.272629\nS : Type ?u.272632\nα : Type v\nβ : Type w\nγ : Type ?u.272639\ninst✝² : NonUnitalNonAssocSemiring α\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nd : n → α\nM : Matrix m n α\ni : m\nj : n\n⊢ (M ⬝ (diagonal d)ᵀ) i j = M i j * d j",
"tactic": "apply dotProduct_diagonal"
}
] |
[
1032,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1029,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.toReal_pos_iff
|
[] |
[
2076,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2075,
1
] |
Mathlib/Data/Polynomial/Monic.lean
|
Polynomial.Monic.as_sum
|
[
{
"state_after": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q r : R[X]\nhp : Monic p\n⊢ ↑C (coeff p (natDegree p)) * X ^ natDegree p + ∑ x in range (natDegree p), ↑C (coeff p x) * X ^ x =\n X ^ natDegree p + ∑ i in range (natDegree p), ↑C (coeff p i) * X ^ i",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q r : R[X]\nhp : Monic p\n⊢ p = X ^ natDegree p + ∑ i in range (natDegree p), ↑C (coeff p i) * X ^ i",
"tactic": "conv_lhs => rw [p.as_sum_range_C_mul_X_pow, sum_range_succ_comm]"
},
{
"state_after": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q r : R[X]\nhp : Monic p\n⊢ ↑C (coeff p (natDegree p)) = 1",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q r : R[X]\nhp : Monic p\n⊢ ↑C (coeff p (natDegree p)) * X ^ natDegree p + ∑ x in range (natDegree p), ↑C (coeff p x) * X ^ x =\n X ^ natDegree p + ∑ i in range (natDegree p), ↑C (coeff p i) * X ^ i",
"tactic": "suffices C (p.coeff p.natDegree) = 1 by rw [this, one_mul]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q r : R[X]\nhp : Monic p\n⊢ ↑C (coeff p (natDegree p)) = 1",
"tactic": "exact congr_arg C hp"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np q r : R[X]\nhp : Monic p\nthis : ↑C (coeff p (natDegree p)) = 1\n⊢ ↑C (coeff p (natDegree p)) * X ^ natDegree p + ∑ x in range (natDegree p), ↑C (coeff p x) * X ^ x =\n X ^ natDegree p + ∑ i in range (natDegree p), ↑C (coeff p i) * X ^ i",
"tactic": "rw [this, one_mul]"
}
] |
[
58,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/Probability/ProbabilityMassFunction/Basic.lean
|
Pmf.toOuterMeasure_apply_fintype
|
[] |
[
233,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
232,
1
] |
Mathlib/Topology/UniformSpace/Completion.lean
|
CauchyFilter.mem_uniformity'
|
[
{
"state_after": "α : Type u\ninst✝² : UniformSpace α\nβ : Type v\nγ : Type w\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\ns : Set (CauchyFilter α × CauchyFilter α)\nt : Set (α × α)\n_h : t ∈ 𝓤 α\n⊢ gen t ⊆ s ↔ ∀ (f g : CauchyFilter α), t ∈ ↑f ×ˢ ↑g → (f, g) ∈ s",
"state_before": "α : Type u\ninst✝² : UniformSpace α\nβ : Type v\nγ : Type w\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\ns : Set (CauchyFilter α × CauchyFilter α)\n⊢ s ∈ 𝓤 (CauchyFilter α) ↔ ∃ t, t ∈ 𝓤 α ∧ ∀ (f g : CauchyFilter α), t ∈ ↑f ×ˢ ↑g → (f, g) ∈ s",
"tactic": "refine mem_uniformity.trans (exists_congr (fun t => and_congr_right_iff.mpr (fun _h => ?_)))"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝² : UniformSpace α\nβ : Type v\nγ : Type w\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\ns : Set (CauchyFilter α × CauchyFilter α)\nt : Set (α × α)\n_h : t ∈ 𝓤 α\n⊢ gen t ⊆ s ↔ ∀ (f g : CauchyFilter α), t ∈ ↑f ×ˢ ↑g → (f, g) ∈ s",
"tactic": "exact ⟨fun h _f _g ht => h ht, fun h _p hp => h _ _ hp⟩"
}
] |
[
150,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
147,
1
] |
Mathlib/Analysis/NormedSpace/Dual.lean
|
NormedSpace.eq_iff_forall_dual_eq
|
[
{
"state_after": "𝕜 : Type v\ninst✝² : IsROrC 𝕜\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx y : E\n⊢ (∀ (g : Dual 𝕜 E), ↑g (x - y) = 0) ↔ ∀ (g : Dual 𝕜 E), ↑g x = ↑g y",
"state_before": "𝕜 : Type v\ninst✝² : IsROrC 𝕜\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx y : E\n⊢ x = y ↔ ∀ (g : Dual 𝕜 E), ↑g x = ↑g y",
"tactic": "rw [← sub_eq_zero, eq_zero_iff_forall_dual_eq_zero 𝕜 (x - y)]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type v\ninst✝² : IsROrC 𝕜\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx y : E\n⊢ (∀ (g : Dual 𝕜 E), ↑g (x - y) = 0) ↔ ∀ (g : Dual 𝕜 E), ↑g x = ↑g y",
"tactic": "simp [sub_eq_zero]"
}
] |
[
161,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
1
] |
Mathlib/Algebra/Squarefree.lean
|
Squarefree.gcd_right
|
[] |
[
96,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
95,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean
|
MeasureTheory.Integrable.aemeasurable
|
[] |
[
457,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
455,
1
] |
Mathlib/Topology/Homotopy/Contractible.lean
|
ContinuousMap.Nullhomotopic.comp_right
|
[
{
"state_after": "case intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : C(X, Y)\ng : C(Y, Z)\ny : Y\nhy : Homotopic f (const X y)\n⊢ Nullhomotopic (comp g f)",
"state_before": "X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : C(X, Y)\nhf : Nullhomotopic f\ng : C(Y, Z)\n⊢ Nullhomotopic (comp g f)",
"tactic": "cases' hf with y hy"
},
{
"state_after": "case intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : C(X, Y)\ng : C(Y, Z)\ny : Y\nhy : Homotopic f (const X y)\n⊢ Homotopic (comp g f) (const X (↑g y))",
"state_before": "case intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : C(X, Y)\ng : C(Y, Z)\ny : Y\nhy : Homotopic f (const X y)\n⊢ Nullhomotopic (comp g f)",
"tactic": "use g y"
},
{
"state_after": "no goals",
"state_before": "case intro\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : C(X, Y)\ng : C(Y, Z)\ny : Y\nhy : Homotopic f (const X y)\n⊢ Homotopic (comp g f) (const X (↑g y))",
"tactic": "exact Homotopic.hcomp hy (Homotopic.refl g)"
}
] |
[
40,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
36,
1
] |
Mathlib/Order/Max.lean
|
IsTop.mono
|
[] |
[
304,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
304,
1
] |
Mathlib/Data/Matrix/Kronecker.lean
|
Matrix.mul_kroneckerTMul_mul
|
[] |
[
553,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
551,
1
] |
Mathlib/Order/ModularLattice.lean
|
covby_sup_of_inf_covby_left
|
[] |
[
150,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/Analysis/Convex/Quasiconvex.lean
|
quasiconvexOn_iff_le_max
|
[] |
[
131,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Dynamics/PeriodicPts.lean
|
Function.Commute.minimalPeriod_of_comp_dvd_mul
|
[] |
[
435,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
433,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.mul_subset_iff_left
|
[] |
[
1850,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1849,
1
] |
Mathlib/LinearAlgebra/LinearPMap.lean
|
LinearPMap.exists_of_le
|
[] |
[
225,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
223,
1
] |
Mathlib/Order/Filter/Lift.lean
|
Filter.lift'_mono
|
[] |
[
283,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
282,
1
] |
Mathlib/LinearAlgebra/Dual.lean
|
Submodule.dualPairing_apply
|
[] |
[
1237,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1235,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.nontrivial_iff
|
[] |
[
1226,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1222,
1
] |
Mathlib/Data/PNat/Xgcd.lean
|
PNat.XgcdType.finish_v
|
[
{
"state_after": "u : XgcdType\nhr : r u = 0\nha : r u + ↑(b u) * q u = ↑(a u) := rq_eq u\n⊢ v (finish u) = v u",
"state_before": "u : XgcdType\nhr : r u = 0\n⊢ v (finish u) = v u",
"tactic": "let ha : u.r + u.b * u.q = u.a := u.rq_eq"
},
{
"state_after": "u : XgcdType\nhr : r u = 0\nha : ↑(b u) * q u = ↑(a u)\n⊢ v (finish u) = v u",
"state_before": "u : XgcdType\nhr : r u = 0\nha : r u + ↑(b u) * q u = ↑(a u) := rq_eq u\n⊢ v (finish u) = v u",
"tactic": "rw [hr, zero_add] at ha"
},
{
"state_after": "case h₁\nu : XgcdType\nhr : r u = 0\nha : ↑(b u) * q u = ↑(a u)\n⊢ (v (finish u)).fst = (v u).fst\n\ncase h₂\nu : XgcdType\nhr : r u = 0\nha : ↑(b u) * q u = ↑(a u)\n⊢ (v (finish u)).snd = (v u).snd",
"state_before": "u : XgcdType\nhr : r u = 0\nha : ↑(b u) * q u = ↑(a u)\n⊢ v (finish u) = v u",
"tactic": "ext"
},
{
"state_after": "case h₁\nu : XgcdType\nhr : r u = 0\nha : ↑(b u) * q u = ↑(a u)\n⊢ (u.wp + 1) * ↑(b u) + ((u.wp + 1) * qp u + u.x) * ↑(b u) = ↑(w u) * ↑(a u) + u.x * ↑(b u)",
"state_before": "case h₁\nu : XgcdType\nhr : r u = 0\nha : ↑(b u) * q u = ↑(a u)\n⊢ (v (finish u)).fst = (v u).fst",
"tactic": "change (u.wp + 1) * u.b + ((u.wp + 1) * u.qp + u.x) * u.b = u.w * u.a + u.x * u.b"
},
{
"state_after": "case h₁\nu : XgcdType\nhr : r u = 0\nha : ↑(b u) * q u = ↑(a u)\nthis : u.wp + 1 = ↑(w u)\n⊢ (u.wp + 1) * ↑(b u) + ((u.wp + 1) * qp u + u.x) * ↑(b u) = ↑(w u) * ↑(a u) + u.x * ↑(b u)",
"state_before": "case h₁\nu : XgcdType\nhr : r u = 0\nha : ↑(b u) * q u = ↑(a u)\n⊢ (u.wp + 1) * ↑(b u) + ((u.wp + 1) * qp u + u.x) * ↑(b u) = ↑(w u) * ↑(a u) + u.x * ↑(b u)",
"tactic": "have : u.wp + 1 = u.w := rfl"
},
{
"state_after": "case h₁\nu : XgcdType\nhr : r u = 0\nha : ↑(b u) * q u = ↑(a u)\nthis : u.wp + 1 = ↑(w u)\n⊢ ↑(w u) * ↑(b u) + (↑(w u) * qp u + u.x) * ↑(b u) = ↑(w u) * (↑(b u) * (qp u + 1)) + u.x * ↑(b u)",
"state_before": "case h₁\nu : XgcdType\nhr : r u = 0\nha : ↑(b u) * q u = ↑(a u)\nthis : u.wp + 1 = ↑(w u)\n⊢ (u.wp + 1) * ↑(b u) + ((u.wp + 1) * qp u + u.x) * ↑(b u) = ↑(w u) * ↑(a u) + u.x * ↑(b u)",
"tactic": "rw [this, ← ha, u.qp_eq hr]"
},
{
"state_after": "no goals",
"state_before": "case h₁\nu : XgcdType\nhr : r u = 0\nha : ↑(b u) * q u = ↑(a u)\nthis : u.wp + 1 = ↑(w u)\n⊢ ↑(w u) * ↑(b u) + (↑(w u) * qp u + u.x) * ↑(b u) = ↑(w u) * (↑(b u) * (qp u + 1)) + u.x * ↑(b u)",
"tactic": "ring"
},
{
"state_after": "case h₂\nu : XgcdType\nhr : r u = 0\nha : ↑(b u) * q u = ↑(a u)\n⊢ u.y * ↑(b u) + (u.y * qp u + ↑(z u)) * ↑(b u) = u.y * ↑(a u) + ↑(z u) * ↑(b u)",
"state_before": "case h₂\nu : XgcdType\nhr : r u = 0\nha : ↑(b u) * q u = ↑(a u)\n⊢ (v (finish u)).snd = (v u).snd",
"tactic": "change u.y * u.b + (u.y * u.qp + u.z) * u.b = u.y * u.a + u.z * u.b"
},
{
"state_after": "case h₂\nu : XgcdType\nhr : r u = 0\nha : ↑(b u) * q u = ↑(a u)\n⊢ u.y * ↑(b u) + (u.y * qp u + ↑(z u)) * ↑(b u) = u.y * (↑(b u) * (qp u + 1)) + ↑(z u) * ↑(b u)",
"state_before": "case h₂\nu : XgcdType\nhr : r u = 0\nha : ↑(b u) * q u = ↑(a u)\n⊢ u.y * ↑(b u) + (u.y * qp u + ↑(z u)) * ↑(b u) = u.y * ↑(a u) + ↑(z u) * ↑(b u)",
"tactic": "rw [← ha, u.qp_eq hr]"
},
{
"state_after": "no goals",
"state_before": "case h₂\nu : XgcdType\nhr : r u = 0\nha : ↑(b u) * q u = ↑(a u)\n⊢ u.y * ↑(b u) + (u.y * qp u + ↑(z u)) * ↑(b u) = u.y * (↑(b u) * (qp u + 1)) + ↑(z u) * ↑(b u)",
"tactic": "ring"
}
] |
[
299,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
289,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearMap.ext_ring
|
[] |
[
527,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
526,
1
] |
Mathlib/Data/Part.lean
|
Part.bind_le
|
[
{
"state_after": "case mp\nα✝ : Type ?u.51205\nβ : Type u_1\nγ : Type ?u.51211\nα : Type u_1\nx : Part α\nf : α → Part β\ny : Part β\nh : x >>= f ≤ y\n⊢ ∀ (a : α), a ∈ x → f a ≤ y\n\ncase mpr\nα✝ : Type ?u.51205\nβ : Type u_1\nγ : Type ?u.51211\nα : Type u_1\nx : Part α\nf : α → Part β\ny : Part β\nh : ∀ (a : α), a ∈ x → f a ≤ y\n⊢ x >>= f ≤ y",
"state_before": "α✝ : Type ?u.51205\nβ : Type u_1\nγ : Type ?u.51211\nα : Type u_1\nx : Part α\nf : α → Part β\ny : Part β\n⊢ x >>= f ≤ y ↔ ∀ (a : α), a ∈ x → f a ≤ y",
"tactic": "constructor <;> intro h"
},
{
"state_after": "case mp\nα✝ : Type ?u.51205\nβ : Type u_1\nγ : Type ?u.51211\nα : Type u_1\nx : Part α\nf : α → Part β\ny : Part β\nh : x >>= f ≤ y\na : α\nh' : a ∈ x\nb : β\n⊢ b ∈ f a → b ∈ y",
"state_before": "case mp\nα✝ : Type ?u.51205\nβ : Type u_1\nγ : Type ?u.51211\nα : Type u_1\nx : Part α\nf : α → Part β\ny : Part β\nh : x >>= f ≤ y\n⊢ ∀ (a : α), a ∈ x → f a ≤ y",
"tactic": "intro a h' b"
},
{
"state_after": "case mp\nα✝ : Type ?u.51205\nβ : Type u_1\nγ : Type ?u.51211\nα : Type u_1\nx : Part α\nf : α → Part β\ny : Part β\nh✝ : x >>= f ≤ y\na : α\nh' : a ∈ x\nb : β\nh : b ∈ x >>= f → b ∈ y\n⊢ b ∈ f a → b ∈ y",
"state_before": "case mp\nα✝ : Type ?u.51205\nβ : Type u_1\nγ : Type ?u.51211\nα : Type u_1\nx : Part α\nf : α → Part β\ny : Part β\nh : x >>= f ≤ y\na : α\nh' : a ∈ x\nb : β\n⊢ b ∈ f a → b ∈ y",
"tactic": "have h := h b"
},
{
"state_after": "case mp\nα✝ : Type ?u.51205\nβ : Type u_1\nγ : Type ?u.51211\nα : Type u_1\nx : Part α\nf : α → Part β\ny : Part β\nh✝ : x >>= f ≤ y\na : α\nh' : a ∈ x\nb : β\nh : ∀ (x_1 : α), x_1 ∈ x → b ∈ f x_1 → b ∈ y\n⊢ b ∈ f a → b ∈ y",
"state_before": "case mp\nα✝ : Type ?u.51205\nβ : Type u_1\nγ : Type ?u.51211\nα : Type u_1\nx : Part α\nf : α → Part β\ny : Part β\nh✝ : x >>= f ≤ y\na : α\nh' : a ∈ x\nb : β\nh : b ∈ x >>= f → b ∈ y\n⊢ b ∈ f a → b ∈ y",
"tactic": "simp only [and_imp, exists_prop, bind_eq_bind, mem_bind_iff, exists_imp] at h"
},
{
"state_after": "no goals",
"state_before": "case mp\nα✝ : Type ?u.51205\nβ : Type u_1\nγ : Type ?u.51211\nα : Type u_1\nx : Part α\nf : α → Part β\ny : Part β\nh✝ : x >>= f ≤ y\na : α\nh' : a ∈ x\nb : β\nh : ∀ (x_1 : α), x_1 ∈ x → b ∈ f x_1 → b ∈ y\n⊢ b ∈ f a → b ∈ y",
"tactic": "apply h _ h'"
},
{
"state_after": "case mpr\nα✝ : Type ?u.51205\nβ : Type u_1\nγ : Type ?u.51211\nα : Type u_1\nx : Part α\nf : α → Part β\ny : Part β\nh : ∀ (a : α), a ∈ x → f a ≤ y\nb : β\nh' : b ∈ x >>= f\n⊢ b ∈ y",
"state_before": "case mpr\nα✝ : Type ?u.51205\nβ : Type u_1\nγ : Type ?u.51211\nα : Type u_1\nx : Part α\nf : α → Part β\ny : Part β\nh : ∀ (a : α), a ∈ x → f a ≤ y\n⊢ x >>= f ≤ y",
"tactic": "intro b h'"
},
{
"state_after": "case mpr\nα✝ : Type ?u.51205\nβ : Type u_1\nγ : Type ?u.51211\nα : Type u_1\nx : Part α\nf : α → Part β\ny : Part β\nh : ∀ (a : α), a ∈ x → f a ≤ y\nb : β\nh' : ∃ a, a ∈ x ∧ b ∈ f a\n⊢ b ∈ y",
"state_before": "case mpr\nα✝ : Type ?u.51205\nβ : Type u_1\nγ : Type ?u.51211\nα : Type u_1\nx : Part α\nf : α → Part β\ny : Part β\nh : ∀ (a : α), a ∈ x → f a ≤ y\nb : β\nh' : b ∈ x >>= f\n⊢ b ∈ y",
"tactic": "simp only [exists_prop, bind_eq_bind, mem_bind_iff] at h'"
},
{
"state_after": "case mpr.intro.intro\nα✝ : Type ?u.51205\nβ : Type u_1\nγ : Type ?u.51211\nα : Type u_1\nx : Part α\nf : α → Part β\ny : Part β\nh : ∀ (a : α), a ∈ x → f a ≤ y\nb : β\na : α\nh₀ : a ∈ x\nh₁ : b ∈ f a\n⊢ b ∈ y",
"state_before": "case mpr\nα✝ : Type ?u.51205\nβ : Type u_1\nγ : Type ?u.51211\nα : Type u_1\nx : Part α\nf : α → Part β\ny : Part β\nh : ∀ (a : α), a ∈ x → f a ≤ y\nb : β\nh' : ∃ a, a ∈ x ∧ b ∈ f a\n⊢ b ∈ y",
"tactic": "rcases h' with ⟨a, h₀, h₁⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro\nα✝ : Type ?u.51205\nβ : Type u_1\nγ : Type ?u.51211\nα : Type u_1\nx : Part α\nf : α → Part β\ny : Part β\nh : ∀ (a : α), a ∈ x → f a ≤ y\nb : β\na : α\nh₀ : a ∈ x\nh₁ : b ∈ f a\n⊢ b ∈ y",
"tactic": "apply h _ h₀ _ h₁"
}
] |
[
631,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
621,
1
] |
Mathlib/RingTheory/Ideal/Cotangent.lean
|
Ideal.toCotangent_surjective
|
[] |
[
87,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Std/Data/Rat/Lemmas.lean
|
Rat.divInt_zero
|
[] |
[
126,
61
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
126,
9
] |
Mathlib/Algebra/GroupWithZero/Basic.lean
|
mul_inv_cancel_left₀
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.14933\nM₀ : Type ?u.14936\nG₀ : Type u_1\nM₀' : Type ?u.14942\nG₀' : Type ?u.14945\nF : Type ?u.14948\nF' : Type ?u.14951\ninst✝ : GroupWithZero G₀\na b✝ c g h✝ x : G₀\nh : a ≠ 0\nb : G₀\n⊢ a * a⁻¹ * b = b",
"tactic": "simp [h]"
}
] |
[
249,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
246,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean
|
fderiv_sub
|
[] |
[
524,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
522,
1
] |
Mathlib/CategoryTheory/Abelian/Homology.lean
|
homology.map_eq_desc'_lift_left
|
[
{
"state_after": "no goals",
"state_before": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0",
"tactic": "simp"
},
{
"state_after": "case h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ (kernel.lift g f w ≫\n lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0)) ≫\n ι f' g' w' =\n 0 ≫ ι f' g' w'",
"state_before": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ kernel.lift g f w ≫\n lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0) =\n 0",
"tactic": "ext"
},
{
"state_after": "case h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ f ≫ α.right ≫ cokernel.π f' = 0",
"state_before": "case h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ (kernel.lift g f w ≫\n lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0)) ≫\n ι f' g' w' =\n 0 ≫ ι f' g' w'",
"tactic": "simp only [← h, Category.assoc, zero_comp, lift_ι, kernel.lift_ι_assoc]"
},
{
"state_after": "case h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ α.left ≫ (Arrow.mk f').hom ≫ cokernel.π f' = 0",
"state_before": "case h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ f ≫ α.right ≫ cokernel.π f' = 0",
"tactic": "erw [← reassoc_of% α.w]"
},
{
"state_after": "no goals",
"state_before": "case h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ α.left ≫ (Arrow.mk f').hom ≫ cokernel.π f' = 0",
"tactic": "simp"
},
{
"state_after": "case h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ π' f g w ≫ map w w' α β h =\n π' f g w ≫\n desc' f g w\n (lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0))\n (_ :\n kernel.lift g f w ≫\n lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0) =\n 0)",
"state_before": "A : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ map w w' α β h =\n desc' f g w\n (lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0))\n (_ :\n kernel.lift g f w ≫\n lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0) =\n 0)",
"tactic": "apply homology.hom_from_ext"
},
{
"state_after": "case h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ kernel.map g g' α.right β.right (_ : g ≫ β.right = α.right ≫ g') ≫ π' f' g' w' =\n lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0)",
"state_before": "case h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ π' f g w ≫ map w w' α β h =\n π' f g w ≫\n desc' f g w\n (lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0))\n (_ :\n kernel.lift g f w ≫\n lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0) =\n 0)",
"tactic": "simp only [π'_map, π'_desc']"
},
{
"state_after": "case h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ kernel.map g g' α.right β.right (_ : g ≫ β.right = α.right ≫ g') ≫\n cokernel.π (kernel.lift g' f' w') ≫ (homologyIsoCokernelLift f' g' w').inv =\n kernel.lift (cokernel.desc f' g' w') (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0) ≫\n (homologyIsoKernelDesc f' g' w').inv",
"state_before": "case h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ kernel.map g g' α.right β.right (_ : g ≫ β.right = α.right ≫ g') ≫ π' f' g' w' =\n lift f' g' w' (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0)",
"tactic": "dsimp [π', lift]"
},
{
"state_after": "case h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ (kernel.map g g' α.right β.right (_ : g ≫ β.right = α.right ≫ g') ≫\n cokernel.π (kernel.lift g' f' w') ≫ (homologyIsoCokernelLift f' g' w').inv) ≫\n (homologyIsoKernelDesc f' g' w').hom =\n kernel.lift (cokernel.desc f' g' w') (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0)",
"state_before": "case h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ kernel.map g g' α.right β.right (_ : g ≫ β.right = α.right ≫ g') ≫\n cokernel.π (kernel.lift g' f' w') ≫ (homologyIsoCokernelLift f' g' w').inv =\n kernel.lift (cokernel.desc f' g' w') (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0) ≫\n (homologyIsoKernelDesc f' g' w').inv",
"tactic": "rw [Iso.eq_comp_inv]"
},
{
"state_after": "case h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ (kernel.map g g' α.right β.right (_ : g ≫ β.right = α.right ≫ g') ≫\n cokernel.π (kernel.lift g' f' w') ≫ (homologyIsoCokernelLift f' g' w').inv) ≫\n (homologyIsoCokernelLift f' g' w').hom ≫ Abelian.homologyCToK f' g' w' =\n kernel.lift (cokernel.desc f' g' w') (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0)",
"state_before": "case h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ (kernel.map g g' α.right β.right (_ : g ≫ β.right = α.right ≫ g') ≫\n cokernel.π (kernel.lift g' f' w') ≫ (homologyIsoCokernelLift f' g' w').inv) ≫\n (homologyIsoKernelDesc f' g' w').hom =\n kernel.lift (cokernel.desc f' g' w') (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0)",
"tactic": "dsimp [homologyIsoKernelDesc]"
},
{
"state_after": "case h.h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ ((kernel.map g g' α.right β.right (_ : g ≫ β.right = α.right ≫ g') ≫\n cokernel.π (kernel.lift g' f' w') ≫ (homologyIsoCokernelLift f' g' w').inv) ≫\n (homologyIsoCokernelLift f' g' w').hom ≫ Abelian.homologyCToK f' g' w') ≫\n equalizer.ι (cokernel.desc f' g' w') 0 =\n kernel.lift (cokernel.desc f' g' w') (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0) ≫\n equalizer.ι (cokernel.desc f' g' w') 0",
"state_before": "case h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ (kernel.map g g' α.right β.right (_ : g ≫ β.right = α.right ≫ g') ≫\n cokernel.π (kernel.lift g' f' w') ≫ (homologyIsoCokernelLift f' g' w').inv) ≫\n (homologyIsoCokernelLift f' g' w').hom ≫ Abelian.homologyCToK f' g' w' =\n kernel.lift (cokernel.desc f' g' w') (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0)",
"tactic": "apply Limits.equalizer.hom_ext"
},
{
"state_after": "no goals",
"state_before": "case h.h\nA : Type u\ninst✝¹ : Category A\ninst✝ : Abelian A\nX Y Z : A\nf : X ⟶ Y\ng : Y ⟶ Z\nw : f ≫ g = 0\nX' Y' Z' : A\nf' : X' ⟶ Y'\ng' : Y' ⟶ Z'\nw' : f' ≫ g' = 0\nα : Arrow.mk f ⟶ Arrow.mk f'\nβ : Arrow.mk g ⟶ Arrow.mk g'\nh : α.right = β.left\n⊢ ((kernel.map g g' α.right β.right (_ : g ≫ β.right = α.right ≫ g') ≫\n cokernel.π (kernel.lift g' f' w') ≫ (homologyIsoCokernelLift f' g' w').inv) ≫\n (homologyIsoCokernelLift f' g' w').hom ≫ Abelian.homologyCToK f' g' w') ≫\n equalizer.ι (cokernel.desc f' g' w') 0 =\n kernel.lift (cokernel.desc f' g' w') (kernel.ι g ≫ β.left ≫ cokernel.π f')\n (_ : (kernel.ι g ≫ β.left ≫ cokernel.π f') ≫ cokernel.desc f' g' w' = 0) ≫\n equalizer.ι (cokernel.desc f' g' w') 0",
"tactic": "simp [h]"
}
] |
[
257,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
242,
1
] |
Mathlib/Data/Finmap.lean
|
Finmap.empty_union
|
[
{
"state_after": "α : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\ns₁✝ : Finmap β\ns₁ : AList β\n⊢ ⟦∅⟧ ∪ ⟦s₁⟧ = ⟦s₁⟧",
"state_before": "α : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\ns₁✝ : Finmap β\ns₁ : AList β\n⊢ ∅ ∪ ⟦s₁⟧ = ⟦s₁⟧",
"tactic": "rw [← empty_toFinmap]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\ns₁✝ : Finmap β\ns₁ : AList β\n⊢ ⟦∅⟧ ∪ ⟦s₁⟧ = ⟦s₁⟧",
"tactic": "simp [-empty_toFinmap, AList.toFinmap_eq, union_toFinmap, AList.union_assoc]"
}
] |
[
626,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
623,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
Int.fract_eq_iff
|
[
{
"state_after": "F : Type ?u.161626\nα : Type u_1\nβ : Type ?u.161632\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a b : α\nh : fract a = b\n⊢ 0 ≤ fract a ∧ fract a < 1 ∧ ∃ z, a - fract a = ↑z",
"state_before": "F : Type ?u.161626\nα : Type u_1\nβ : Type ?u.161632\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a b : α\nh : fract a = b\n⊢ 0 ≤ b ∧ b < 1 ∧ ∃ z, a - b = ↑z",
"tactic": "rw [← h]"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.161626\nα : Type u_1\nβ : Type ?u.161632\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a b : α\nh : fract a = b\n⊢ 0 ≤ fract a ∧ fract a < 1 ∧ ∃ z, a - fract a = ↑z",
"tactic": "exact ⟨fract_nonneg _, fract_lt_one _, ⟨⌊a⌋, sub_sub_cancel _ _⟩⟩"
},
{
"state_after": "case intro.intro.intro\nF : Type ?u.161626\nα : Type u_1\nβ : Type ?u.161632\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz✝ : ℤ\na✝ a b : α\nh₀ : 0 ≤ b\nh₁ : b < 1\nz : ℤ\nhz : a - b = ↑z\n⊢ fract a = b",
"state_before": "F : Type ?u.161626\nα : Type u_1\nβ : Type ?u.161632\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na✝ a b : α\n⊢ (0 ≤ b ∧ b < 1 ∧ ∃ z, a - b = ↑z) → fract a = b",
"tactic": "rintro ⟨h₀, h₁, z, hz⟩"
},
{
"state_after": "case intro.intro.intro\nF : Type ?u.161626\nα : Type u_1\nβ : Type ?u.161632\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz✝ : ℤ\na✝ a b : α\nh₀ : 0 ≤ b\nh₁ : b < 1\nz : ℤ\nhz : a - b = ↑z\n⊢ a - b ≤ a ∧ a < a - b + 1",
"state_before": "case intro.intro.intro\nF : Type ?u.161626\nα : Type u_1\nβ : Type ?u.161632\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz✝ : ℤ\na✝ a b : α\nh₀ : 0 ≤ b\nh₁ : b < 1\nz : ℤ\nhz : a - b = ↑z\n⊢ fract a = b",
"tactic": "rw [← self_sub_floor, eq_comm, eq_sub_iff_add_eq, add_comm, ← eq_sub_iff_add_eq, hz,\n Int.cast_inj, floor_eq_iff, ← hz]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nF : Type ?u.161626\nα : Type u_1\nβ : Type ?u.161632\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz✝ : ℤ\na✝ a b : α\nh₀ : 0 ≤ b\nh₁ : b < 1\nz : ℤ\nhz : a - b = ↑z\n⊢ a - b ≤ a ∧ a < a - b + 1",
"tactic": "constructor <;> simpa [sub_eq_add_neg, add_assoc]"
}
] |
[
933,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
925,
1
] |
Mathlib/NumberTheory/PellMatiyasevic.lean
|
Pell.xn_zero
|
[] |
[
133,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.mul_mem_cancel_left
|
[] |
[
621,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
620,
11
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_filter
|
[
{
"state_after": "case hc\nι : Type ?u.337590\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\np : α → Prop\ninst✝ : DecidablePred p\nf : α → β\na : α\nh : a ∈ filter p s\n⊢ p a",
"state_before": "ι : Type ?u.337590\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\np : α → Prop\ninst✝ : DecidablePred p\nf : α → β\na : α\nh : a ∈ filter p s\n⊢ f a = if p a then f a else 1",
"tactic": "rw [if_pos]"
},
{
"state_after": "no goals",
"state_before": "case hc\nι : Type ?u.337590\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\np : α → Prop\ninst✝ : DecidablePred p\nf : α → β\na : α\nh : a ∈ filter p s\n⊢ p a",
"tactic": "simpa using (mem_filter.1 h).2"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.337590\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝¹ : CommMonoid β\np : α → Prop\ninst✝ : DecidablePred p\nf : α → β\nx : α\nhs : x ∈ s\nh : x ∈ s → ¬p x\n⊢ ¬p x",
"tactic": "simpa using h hs"
}
] |
[
782,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
774,
1
] |
Mathlib/Analysis/Convex/Cone/Dual.lean
|
innerDualCone_singleton
|
[] |
[
92,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
90,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
ne_one_of_nnnorm_ne_zero
|
[
{
"state_after": "𝓕 : Type ?u.321774\n𝕜 : Type ?u.321777\nα : Type ?u.321780\nι : Type ?u.321783\nκ : Type ?u.321786\nE : Type u_1\nF : Type ?u.321792\nG : Type ?u.321795\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\n⊢ ‖1‖₊ = 0",
"state_before": "𝓕 : Type ?u.321774\n𝕜 : Type ?u.321777\nα : Type ?u.321780\nι : Type ?u.321783\nκ : Type ?u.321786\nE : Type u_1\nF : Type ?u.321792\nG : Type ?u.321795\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na✝ a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\na : E\n⊢ a = 1 → ‖a‖₊ = 0",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "𝓕 : Type ?u.321774\n𝕜 : Type ?u.321777\nα : Type ?u.321780\nι : Type ?u.321783\nκ : Type ?u.321786\nE : Type u_1\nF : Type ?u.321792\nG : Type ?u.321795\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\n⊢ ‖1‖₊ = 0",
"tactic": "exact nnnorm_one'"
}
] |
[
923,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
920,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.add_div_left
|
[
{
"state_after": "no goals",
"state_before": "x z : Nat\nH : 0 < z\n⊢ (z + x) / z = succ (x / z)",
"tactic": "rw [Nat.add_comm, add_div_right x H]"
}
] |
[
568,
39
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
567,
9
] |
Mathlib/LinearAlgebra/Lagrange.lean
|
Lagrange.eval_nodal_at_node
|
[
{
"state_after": "F : Type u_2\ninst✝ : Field F\nι : Type u_1\ns : Finset ι\nv : ι → F\ni : ι\nr : ι → F\nx : F\nhi : i ∈ s\n⊢ ∃ a, a ∈ s ∧ v i - v a = 0",
"state_before": "F : Type u_2\ninst✝ : Field F\nι : Type u_1\ns : Finset ι\nv : ι → F\ni : ι\nr : ι → F\nx : F\nhi : i ∈ s\n⊢ eval (v i) (nodal s v) = 0",
"tactic": "rw [eval_nodal, prod_eq_zero_iff]"
},
{
"state_after": "no goals",
"state_before": "F : Type u_2\ninst✝ : Field F\nι : Type u_1\ns : Finset ι\nv : ι → F\ni : ι\nr : ι → F\nx : F\nhi : i ∈ s\n⊢ ∃ a, a ∈ s ∧ v i - v a = 0",
"tactic": "exact ⟨i, hi, sub_eq_zero_of_eq rfl⟩"
}
] |
[
510,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
508,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean
|
EuclideanGeometry.angle_neg
|
[
{
"state_after": "no goals",
"state_before": "V : Type u_1\nP : Type ?u.43288\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\nv₁ v₂ v₃ : V\n⊢ ∠ (-v₁) (-v₂) (-v₃) = ∠ v₁ v₂ v₃",
"tactic": "simpa only [zero_sub] using angle_const_sub 0 v₁ v₂ v₃"
}
] |
[
124,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/Order/Filter/Extr.lean
|
IsExtrOn.inter
|
[] |
[
305,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
304,
1
] |
Mathlib/Geometry/Euclidean/Basic.lean
|
EuclideanGeometry.reflection_involutive
|
[] |
[
579,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
577,
1
] |
Mathlib/Order/WellFoundedSet.lean
|
Set.isPwo_iff_exists_monotone_subseq
|
[] |
[
415,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
413,
1
] |
Mathlib/Topology/Basic.lean
|
IsClosed.mem_of_frequently_of_tendsto
|
[] |
[
1449,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1447,
1
] |
Mathlib/Data/ENat/Basic.lean
|
ENat.coe_toNat_eq_self
|
[
{
"state_after": "no goals",
"state_before": "m n : ℕ∞\n⊢ ↑(↑toNat ⊤) = ⊤ ↔ ⊤ ≠ ⊤",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "m n : ℕ∞\nx✝ : ℕ\n⊢ ↑(↑toNat ↑x✝) = ↑x✝ ↔ ↑x✝ ≠ ⊤",
"tactic": "simp [toNat_coe]"
}
] |
[
157,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/Logic/Equiv/TransferInstance.lean
|
Equiv.div_def
|
[] |
[
87,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
1
] |
Mathlib/Analysis/InnerProductSpace/Calculus.lean
|
DifferentiableOn.norm
|
[] |
[
282,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
281,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
Num.ofNat'_one
|
[
{
"state_after": "α : Type ?u.179670\n⊢ (match (motive := Bool → Num → Num) true with\n | true => Num.bit1\n | false => Num.bit0)\n 0 =\n 1",
"state_before": "α : Type ?u.179670\n⊢ ofNat' 1 = 1",
"tactic": "erw [ofNat'_bit true 0, cond, ofNat'_zero]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.179670\n⊢ (match (motive := Bool → Num → Num) true with\n | true => Num.bit1\n | false => Num.bit0)\n 0 =\n 1",
"tactic": "rfl"
}
] |
[
246,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
246,
1
] |
Mathlib/Combinatorics/Derangements/Basic.lean
|
derangements.Equiv.RemoveNone.mem_fiber
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.70902\ninst✝ : DecidableEq α\na : Option α\nf : Perm α\n⊢ f ∈ fiber a ↔ ∃ F, F ∈ derangements (Option α) ∧ ↑F none = a ∧ removeNone F = f",
"tactic": "simp [RemoveNone.fiber, derangements]"
}
] |
[
129,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
1
] |
Mathlib/Order/Filter/Partial.lean
|
Filter.ptendsto_iff_rtendsto
|
[] |
[
236,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
234,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.spanCompIso_hom_app_zero
|
[] |
[
364,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
364,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.tr_reaches₁
|
[
{
"state_after": "case single\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₁ c₁ : σ₁\nac : c₁ ∈ f₁ a₁\n⊢ ∃ b₂, tr c₁ b₂ ∧ Reaches₁ f₂ a₂ b₂\n\ncase tail\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₁ c₁ d₁ : σ₁\na✝ : TransGen (fun a b => b ∈ f₁ a) a₁ c₁\ncd : d₁ ∈ f₁ c₁\nIH : ∃ b₂, tr c₁ b₂ ∧ Reaches₁ f₂ a₂ b₂\n⊢ ∃ b₂, tr d₁ b₂ ∧ Reaches₁ f₂ a₂ b₂",
"state_before": "σ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₁ : σ₁\nab : Reaches₁ f₁ a₁ b₁\n⊢ ∃ b₂, tr b₁ b₂ ∧ Reaches₁ f₂ a₂ b₂",
"tactic": "induction' ab with c₁ ac c₁ d₁ _ cd IH"
},
{
"state_after": "case single\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₁ c₁ : σ₁\nac : c₁ ∈ f₁ a₁\nthis :\n match f₁ a₁ with\n | some b₁ => ∃ b₂, tr b₁ b₂ ∧ Reaches₁ f₂ a₂ b₂\n | none => f₂ a₂ = none\n⊢ ∃ b₂, tr c₁ b₂ ∧ Reaches₁ f₂ a₂ b₂",
"state_before": "case single\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₁ c₁ : σ₁\nac : c₁ ∈ f₁ a₁\n⊢ ∃ b₂, tr c₁ b₂ ∧ Reaches₁ f₂ a₂ b₂",
"tactic": "have := H aa"
},
{
"state_after": "no goals",
"state_before": "case single\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₁ c₁ : σ₁\nac : c₁ ∈ f₁ a₁\nthis :\n match f₁ a₁ with\n | some b₁ => ∃ b₂, tr b₁ b₂ ∧ Reaches₁ f₂ a₂ b₂\n | none => f₂ a₂ = none\n⊢ ∃ b₂, tr c₁ b₂ ∧ Reaches₁ f₂ a₂ b₂",
"tactic": "rwa [show f₁ a₁ = _ from ac] at this"
},
{
"state_after": "case tail.intro.intro\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₁ c₁ d₁ : σ₁\na✝ : TransGen (fun a b => b ∈ f₁ a) a₁ c₁\ncd : d₁ ∈ f₁ c₁\nc₂ : σ₂\ncc : tr c₁ c₂\nac₂ : Reaches₁ f₂ a₂ c₂\n⊢ ∃ b₂, tr d₁ b₂ ∧ Reaches₁ f₂ a₂ b₂",
"state_before": "case tail\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₁ c₁ d₁ : σ₁\na✝ : TransGen (fun a b => b ∈ f₁ a) a₁ c₁\ncd : d₁ ∈ f₁ c₁\nIH : ∃ b₂, tr c₁ b₂ ∧ Reaches₁ f₂ a₂ b₂\n⊢ ∃ b₂, tr d₁ b₂ ∧ Reaches₁ f₂ a₂ b₂",
"tactic": "rcases IH with ⟨c₂, cc, ac₂⟩"
},
{
"state_after": "case tail.intro.intro\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₁ c₁ d₁ : σ₁\na✝ : TransGen (fun a b => b ∈ f₁ a) a₁ c₁\ncd : d₁ ∈ f₁ c₁\nc₂ : σ₂\ncc : tr c₁ c₂\nac₂ : Reaches₁ f₂ a₂ c₂\nthis :\n match f₁ c₁ with\n | some b₁ => ∃ b₂, tr b₁ b₂ ∧ Reaches₁ f₂ c₂ b₂\n | none => f₂ c₂ = none\n⊢ ∃ b₂, tr d₁ b₂ ∧ Reaches₁ f₂ a₂ b₂",
"state_before": "case tail.intro.intro\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₁ c₁ d₁ : σ₁\na✝ : TransGen (fun a b => b ∈ f₁ a) a₁ c₁\ncd : d₁ ∈ f₁ c₁\nc₂ : σ₂\ncc : tr c₁ c₂\nac₂ : Reaches₁ f₂ a₂ c₂\n⊢ ∃ b₂, tr d₁ b₂ ∧ Reaches₁ f₂ a₂ b₂",
"tactic": "have := H cc"
},
{
"state_after": "case tail.intro.intro\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₁ c₁ d₁ : σ₁\na✝ : TransGen (fun a b => b ∈ f₁ a) a₁ c₁\ncd : d₁ ∈ f₁ c₁\nc₂ : σ₂\ncc : tr c₁ c₂\nac₂ : Reaches₁ f₂ a₂ c₂\nthis :\n match some d₁ with\n | some b₁ => ∃ b₂, tr b₁ b₂ ∧ Reaches₁ f₂ c₂ b₂\n | none => f₂ c₂ = none\n⊢ ∃ b₂, tr d₁ b₂ ∧ Reaches₁ f₂ a₂ b₂",
"state_before": "case tail.intro.intro\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₁ c₁ d₁ : σ₁\na✝ : TransGen (fun a b => b ∈ f₁ a) a₁ c₁\ncd : d₁ ∈ f₁ c₁\nc₂ : σ₂\ncc : tr c₁ c₂\nac₂ : Reaches₁ f₂ a₂ c₂\nthis :\n match f₁ c₁ with\n | some b₁ => ∃ b₂, tr b₁ b₂ ∧ Reaches₁ f₂ c₂ b₂\n | none => f₂ c₂ = none\n⊢ ∃ b₂, tr d₁ b₂ ∧ Reaches₁ f₂ a₂ b₂",
"tactic": "rw [show f₁ c₁ = _ from cd] at this"
},
{
"state_after": "case tail.intro.intro.intro.intro\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₁ c₁ d₁ : σ₁\na✝ : TransGen (fun a b => b ∈ f₁ a) a₁ c₁\ncd : d₁ ∈ f₁ c₁\nc₂ : σ₂\ncc : tr c₁ c₂\nac₂ : Reaches₁ f₂ a₂ c₂\nd₂ : σ₂\ndd : tr d₁ d₂\ncd₂ : Reaches₁ f₂ c₂ d₂\n⊢ ∃ b₂, tr d₁ b₂ ∧ Reaches₁ f₂ a₂ b₂",
"state_before": "case tail.intro.intro\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₁ c₁ d₁ : σ₁\na✝ : TransGen (fun a b => b ∈ f₁ a) a₁ c₁\ncd : d₁ ∈ f₁ c₁\nc₂ : σ₂\ncc : tr c₁ c₂\nac₂ : Reaches₁ f₂ a₂ c₂\nthis :\n match some d₁ with\n | some b₁ => ∃ b₂, tr b₁ b₂ ∧ Reaches₁ f₂ c₂ b₂\n | none => f₂ c₂ = none\n⊢ ∃ b₂, tr d₁ b₂ ∧ Reaches₁ f₂ a₂ b₂",
"tactic": "rcases this with ⟨d₂, dd, cd₂⟩"
},
{
"state_after": "no goals",
"state_before": "case tail.intro.intro.intro.intro\nσ₁ : Type u_1\nσ₂ : Type u_2\nf₁ : σ₁ → Option σ₁\nf₂ : σ₂ → Option σ₂\ntr : σ₁ → σ₂ → Prop\nH : Respects f₁ f₂ tr\na₁ : σ₁\na₂ : σ₂\naa : tr a₁ a₂\nb₁ c₁ d₁ : σ₁\na✝ : TransGen (fun a b => b ∈ f₁ a) a₁ c₁\ncd : d₁ ∈ f₁ c₁\nc₂ : σ₂\ncc : tr c₁ c₂\nac₂ : Reaches₁ f₂ a₂ c₂\nd₂ : σ₂\ndd : tr d₁ d₂\ncd₂ : Reaches₁ f₂ c₂ d₂\n⊢ ∃ b₂, tr d₁ b₂ ∧ Reaches₁ f₂ a₂ b₂",
"tactic": "exact ⟨_, dd, ac₂.trans cd₂⟩"
}
] |
[
895,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
886,
1
] |
Mathlib/ModelTheory/Basic.lean
|
FirstOrder.Language.Equiv.refl_apply
|
[
{
"state_after": "L : Language\nL' : Language\nM : Type w\nN : Type w'\ninst✝³ : Structure L M\ninst✝² : Structure L N\nP : Type ?u.257731\ninst✝¹ : Structure L P\nQ : Type ?u.257739\ninst✝ : Structure L Q\nx : M\n⊢ ↑(mk (_root_.Equiv.refl M)) x = x",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type w'\ninst✝³ : Structure L M\ninst✝² : Structure L N\nP : Type ?u.257731\ninst✝¹ : Structure L P\nQ : Type ?u.257739\ninst✝ : Structure L Q\nx : M\n⊢ ↑(refl L M) x = x",
"tactic": "simp [refl]"
},
{
"state_after": "no goals",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type w'\ninst✝³ : Structure L M\ninst✝² : Structure L N\nP : Type ?u.257731\ninst✝¹ : Structure L P\nQ : Type ?u.257739\ninst✝ : Structure L Q\nx : M\n⊢ ↑(mk (_root_.Equiv.refl M)) x = x",
"tactic": "rfl"
}
] |
[
869,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
869,
1
] |
Mathlib/Analysis/InnerProductSpace/Adjoint.lean
|
LinearMap.IsSymmetric.coe_toSelfAdjoint
|
[] |
[
349,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
348,
1
] |
Mathlib/Algebra/Quandle.lean
|
Rack.left_cancel_inv
|
[
{
"state_after": "case mp\nR : Type u_1\ninst✝ : Rack R\nx y y' : R\n⊢ x ◃⁻¹ y = x ◃⁻¹ y' → y = y'\n\ncase mpr\nR : Type u_1\ninst✝ : Rack R\nx y y' : R\n⊢ y = y' → x ◃⁻¹ y = x ◃⁻¹ y'",
"state_before": "R : Type u_1\ninst✝ : Rack R\nx y y' : R\n⊢ x ◃⁻¹ y = x ◃⁻¹ y' ↔ y = y'",
"tactic": "constructor"
},
{
"state_after": "case mpr\nR : Type u_1\ninst✝ : Rack R\nx y y' : R\n⊢ y = y' → x ◃⁻¹ y = x ◃⁻¹ y'",
"state_before": "case mp\nR : Type u_1\ninst✝ : Rack R\nx y y' : R\n⊢ x ◃⁻¹ y = x ◃⁻¹ y' → y = y'\n\ncase mpr\nR : Type u_1\ninst✝ : Rack R\nx y y' : R\n⊢ y = y' → x ◃⁻¹ y = x ◃⁻¹ y'",
"tactic": "apply (act' x).symm.injective"
},
{
"state_after": "case mpr\nR : Type u_1\ninst✝ : Rack R\nx y : R\n⊢ x ◃⁻¹ y = x ◃⁻¹ y",
"state_before": "case mpr\nR : Type u_1\ninst✝ : Rack R\nx y y' : R\n⊢ y = y' → x ◃⁻¹ y = x ◃⁻¹ y'",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "case mpr\nR : Type u_1\ninst✝ : Rack R\nx y : R\n⊢ x ◃⁻¹ y = x ◃⁻¹ y",
"tactic": "rfl"
}
] |
[
239,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
235,
1
] |
Mathlib/CategoryTheory/Subterminal.lean
|
CategoryTheory.IsSubterminal.mono_terminal_from
|
[] |
[
67,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean
|
MeasureTheory.L1.norm_setToL1_le
|
[] |
[
1243,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1242,
1
] |
Mathlib/Data/Nat/Fib.lean
|
Nat.fib_succ_eq_succ_sum
|
[
{
"state_after": "case zero\n\n⊢ fib (zero + 1) = ∑ k in Finset.range zero, fib k + 1\n\ncase succ\nn : ℕ\nih : fib (n + 1) = ∑ k in Finset.range n, fib k + 1\n⊢ fib (succ n + 1) = ∑ k in Finset.range (succ n), fib k + 1",
"state_before": "n : ℕ\n⊢ fib (n + 1) = ∑ k in Finset.range n, fib k + 1",
"tactic": "induction' n with n ih"
},
{
"state_after": "no goals",
"state_before": "case zero\n\n⊢ fib (zero + 1) = ∑ k in Finset.range zero, fib k + 1",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case succ\nn : ℕ\nih : fib (n + 1) = ∑ k in Finset.range n, fib k + 1\n⊢ fib (succ n + 1) = ∑ k in Finset.range (succ n), fib k + 1",
"tactic": "calc\n fib (n + 2) = fib n + fib (n + 1) := fib_add_two\n _ = (fib n + ∑ k in Finset.range n, fib k) + 1 := by rw [ih, add_assoc]\n _ = (∑ k in Finset.range (n + 1), fib k) + 1 := by simp [Finset.range_add_one]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nih : fib (n + 1) = ∑ k in Finset.range n, fib k + 1\n⊢ fib n + fib (n + 1) = fib n + ∑ k in Finset.range n, fib k + 1",
"tactic": "rw [ih, add_assoc]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nih : fib (n + 1) = ∑ k in Finset.range n, fib k + 1\n⊢ fib n + ∑ k in Finset.range n, fib k + 1 = ∑ k in Finset.range (n + 1), fib k + 1",
"tactic": "simp [Finset.range_add_one]"
}
] |
[
305,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
299,
1
] |
Mathlib/Topology/PathConnected.lean
|
JoinedIn.joined_subtype
|
[
{
"state_after": "no goals",
"state_before": "X : Type u_1\nY : Type ?u.563745\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.563760\nF : Set X\nh : JoinedIn F x y\n⊢ Continuous fun t => { val := ↑(somePath h) t, property := (_ : ↑(somePath h) t ∈ F) }",
"tactic": "continuity"
},
{
"state_after": "no goals",
"state_before": "X : Type u_1\nY : Type ?u.563745\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.563760\nF : Set X\nh : JoinedIn F x y\n⊢ ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(somePath h) t, property := (_ : ↑(somePath h) t ∈ F) }) 0 =\n { val := x, property := (_ : x ∈ F) }",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "X : Type u_1\nY : Type ?u.563745\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y z : X\nι : Type ?u.563760\nF : Set X\nh : JoinedIn F x y\n⊢ ContinuousMap.toFun (ContinuousMap.mk fun t => { val := ↑(somePath h) t, property := (_ : ↑(somePath h) t ∈ F) }) 1 =\n { val := y, property := (_ : y ∈ F) }",
"tactic": "simp"
}
] |
[
850,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
845,
1
] |
Mathlib/Data/Real/CauSeq.lean
|
CauSeq.mul_pos
|
[] |
[
699,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
694,
11
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.QuasiMeasurePreserving.id
|
[] |
[
2447,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2446,
11
] |
Mathlib/Order/Grade.lean
|
grade_ne_grade_iff
|
[] |
[
194,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/Order/Cover.lean
|
Wcovby.Ioo_eq
|
[] |
[
109,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/Probability/Independence/Basic.lean
|
ProbabilityTheory.indepSets_piiUnionInter_of_disjoint
|
[
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\n⊢ ↑↑μ (t1 ∩ t2) = ↑↑μ t1 * ↑↑μ t2",
"state_before": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\n⊢ IndepSets (piiUnionInter s S) (piiUnionInter s T)",
"tactic": "rintro t1 t2 ⟨p1, hp1, f1, ht1_m, ht1_eq⟩ ⟨p2, hp2, f2, ht2_m, ht2_eq⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\n⊢ ↑↑μ (t1 ∩ t2) = ↑↑μ t1 * ↑↑μ t2",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\n⊢ ↑↑μ (t1 ∩ t2) = ↑↑μ t1 * ↑↑μ t2",
"tactic": "let g i := ite (i ∈ p1) (f1 i) Set.univ ∩ ite (i ∈ p2) (f2 i) Set.univ"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\nh_P_inter : ↑↑μ (t1 ∩ t2) = ∏ n in p1 ∪ p2, ↑↑μ (g n)\nh_μg : ∀ (n : ι), ↑↑μ (g n) = (if n ∈ p1 then ↑↑μ (f1 n) else 1) * if n ∈ p2 then ↑↑μ (f2 n) else 1\n⊢ ↑↑μ (t1 ∩ t2) = ↑↑μ t1 * ↑↑μ t2",
"tactic": "simp_rw [h_P_inter, h_μg, Finset.prod_mul_distrib,\n Finset.prod_ite_mem (p1 ∪ p2) p1 fun x => μ (f1 x), Finset.union_inter_cancel_left,\n Finset.prod_ite_mem (p1 ∪ p2) p2 fun x => μ (f2 x), Finset.union_inter_cancel_right, ht1_eq, ←\n h_indep p1 ht1_m, ht2_eq, ← h_indep p2 ht2_m]"
},
{
"state_after": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\nhgm : ∀ (i : ι), i ∈ p1 ∪ p2 → g i ∈ s i\nh_p1_inter_p2 :\n ((⋂ (x : ι) (_ : x ∈ p1), f1 x) ∩ ⋂ (x : ι) (_ : x ∈ p2), f2 x) =\n ⋂ (i : ι) (_ : i ∈ p1 ∪ p2), (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\n⊢ ↑↑μ (t1 ∩ t2) = ∏ n in p1 ∪ p2, ↑↑μ (g n)",
"state_before": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\nhgm : ∀ (i : ι), i ∈ p1 ∪ p2 → g i ∈ s i\n⊢ ↑↑μ (t1 ∩ t2) = ∏ n in p1 ∪ p2, ↑↑μ (g n)",
"tactic": "have h_p1_inter_p2 :\n ((⋂ x ∈ p1, f1 x) ∩ ⋂ x ∈ p2, f2 x) =\n ⋂ i ∈ p1 ∪ p2, ite (i ∈ p1) (f1 i) Set.univ ∩ ite (i ∈ p2) (f2 i) Set.univ := by\n ext1 x\n simp only [Set.mem_ite_univ_right, Set.mem_inter_iff, Set.mem_iInter, Finset.mem_union]\n exact\n ⟨fun h i _ => ⟨h.1 i, h.2 i⟩, fun h =>\n ⟨fun i hi => (h i (Or.inl hi)).1 hi, fun i hi => (h i (Or.inr hi)).2 hi⟩⟩"
},
{
"state_after": "no goals",
"state_before": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\nhgm : ∀ (i : ι), i ∈ p1 ∪ p2 → g i ∈ s i\nh_p1_inter_p2 :\n ((⋂ (x : ι) (_ : x ∈ p1), f1 x) ∩ ⋂ (x : ι) (_ : x ∈ p2), f2 x) =\n ⋂ (i : ι) (_ : i ∈ p1 ∪ p2), (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\n⊢ ↑↑μ (t1 ∩ t2) = ∏ n in p1 ∪ p2, ↑↑μ (g n)",
"tactic": "rw [ht1_eq, ht2_eq, h_p1_inter_p2, ← h_indep _ hgm]"
},
{
"state_after": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\ni : ι\nhi_mem_union : i ∈ p1 ∪ p2\n⊢ g i ∈ s i",
"state_before": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\n⊢ ∀ (i : ι), i ∈ p1 ∪ p2 → g i ∈ s i",
"tactic": "intro i hi_mem_union"
},
{
"state_after": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\ni : ι\nhi_mem_union : i ∈ p1 ∨ i ∈ p2\n⊢ g i ∈ s i",
"state_before": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\ni : ι\nhi_mem_union : i ∈ p1 ∪ p2\n⊢ g i ∈ s i",
"tactic": "rw [Finset.mem_union] at hi_mem_union"
},
{
"state_after": "case inl\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\ni : ι\nhi1 : i ∈ p1\n⊢ g i ∈ s i\n\ncase inr\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\ni : ι\nhi2 : i ∈ p2\n⊢ g i ∈ s i",
"state_before": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\ni : ι\nhi_mem_union : i ∈ p1 ∨ i ∈ p2\n⊢ g i ∈ s i",
"tactic": "cases' hi_mem_union with hi1 hi2"
},
{
"state_after": "case inl\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\ni : ι\nhi1 : i ∈ p1\nhi2 : ¬i ∈ p2\n⊢ g i ∈ s i",
"state_before": "case inl\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\ni : ι\nhi1 : i ∈ p1\n⊢ g i ∈ s i",
"tactic": "have hi2 : i ∉ p2 := fun hip2 => Set.disjoint_left.mp hST (hp1 hi1) (hp2 hip2)"
},
{
"state_after": "case inl\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\ni : ι\nhi1 : i ∈ p1\nhi2 : ¬i ∈ p2\n⊢ f1 i ∈ s i",
"state_before": "case inl\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\ni : ι\nhi1 : i ∈ p1\nhi2 : ¬i ∈ p2\n⊢ g i ∈ s i",
"tactic": "simp_rw [if_pos hi1, if_neg hi2, Set.inter_univ]"
},
{
"state_after": "no goals",
"state_before": "case inl\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\ni : ι\nhi1 : i ∈ p1\nhi2 : ¬i ∈ p2\n⊢ f1 i ∈ s i",
"tactic": "exact ht1_m i hi1"
},
{
"state_after": "case inr\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\ni : ι\nhi2 : i ∈ p2\nhi1 : ¬i ∈ p1\n⊢ g i ∈ s i",
"state_before": "case inr\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\ni : ι\nhi2 : i ∈ p2\n⊢ g i ∈ s i",
"tactic": "have hi1 : i ∉ p1 := fun hip1 => Set.disjoint_right.mp hST (hp2 hi2) (hp1 hip1)"
},
{
"state_after": "case inr\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\ni : ι\nhi2 : i ∈ p2\nhi1 : ¬i ∈ p1\n⊢ f2 i ∈ s i",
"state_before": "case inr\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\ni : ι\nhi2 : i ∈ p2\nhi1 : ¬i ∈ p1\n⊢ g i ∈ s i",
"tactic": "simp_rw [if_neg hi1, if_pos hi2, Set.univ_inter]"
},
{
"state_after": "no goals",
"state_before": "case inr\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\ni : ι\nhi2 : i ∈ p2\nhi1 : ¬i ∈ p1\n⊢ f2 i ∈ s i",
"tactic": "exact ht2_m i hi2"
},
{
"state_after": "case h\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\nhgm : ∀ (i : ι), i ∈ p1 ∪ p2 → g i ∈ s i\nx : Ω\n⊢ (x ∈ (⋂ (x : ι) (_ : x ∈ p1), f1 x) ∩ ⋂ (x : ι) (_ : x ∈ p2), f2 x) ↔\n x ∈ ⋂ (i : ι) (_ : i ∈ p1 ∪ p2), (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ",
"state_before": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\nhgm : ∀ (i : ι), i ∈ p1 ∪ p2 → g i ∈ s i\n⊢ ((⋂ (x : ι) (_ : x ∈ p1), f1 x) ∩ ⋂ (x : ι) (_ : x ∈ p2), f2 x) =\n ⋂ (i : ι) (_ : i ∈ p1 ∪ p2), (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ",
"tactic": "ext1 x"
},
{
"state_after": "case h\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\nhgm : ∀ (i : ι), i ∈ p1 ∪ p2 → g i ∈ s i\nx : Ω\n⊢ ((∀ (i : ι), i ∈ p1 → x ∈ f1 i) ∧ ∀ (i : ι), i ∈ p2 → x ∈ f2 i) ↔\n ∀ (i : ι), i ∈ p1 ∨ i ∈ p2 → (i ∈ p1 → x ∈ f1 i) ∧ (i ∈ p2 → x ∈ f2 i)",
"state_before": "case h\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\nhgm : ∀ (i : ι), i ∈ p1 ∪ p2 → g i ∈ s i\nx : Ω\n⊢ (x ∈ (⋂ (x : ι) (_ : x ∈ p1), f1 x) ∩ ⋂ (x : ι) (_ : x ∈ p2), f2 x) ↔\n x ∈ ⋂ (i : ι) (_ : i ∈ p1 ∪ p2), (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ",
"tactic": "simp only [Set.mem_ite_univ_right, Set.mem_inter_iff, Set.mem_iInter, Finset.mem_union]"
},
{
"state_after": "no goals",
"state_before": "case h\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\nhgm : ∀ (i : ι), i ∈ p1 ∪ p2 → g i ∈ s i\nx : Ω\n⊢ ((∀ (i : ι), i ∈ p1 → x ∈ f1 i) ∧ ∀ (i : ι), i ∈ p2 → x ∈ f2 i) ↔\n ∀ (i : ι), i ∈ p1 ∨ i ∈ p2 → (i ∈ p1 → x ∈ f1 i) ∧ (i ∈ p2 → x ∈ f2 i)",
"tactic": "exact\n ⟨fun h i _ => ⟨h.1 i, h.2 i⟩, fun h =>\n ⟨fun i hi => (h i (Or.inl hi)).1 hi, fun i hi => (h i (Or.inr hi)).2 hi⟩⟩"
},
{
"state_after": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\nh_P_inter : ↑↑μ (t1 ∩ t2) = ∏ n in p1 ∪ p2, ↑↑μ (g n)\nn : ι\n⊢ ↑↑μ (g n) = (if n ∈ p1 then ↑↑μ (f1 n) else 1) * if n ∈ p2 then ↑↑μ (f2 n) else 1",
"state_before": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\nh_P_inter : ↑↑μ (t1 ∩ t2) = ∏ n in p1 ∪ p2, ↑↑μ (g n)\n⊢ ∀ (n : ι), ↑↑μ (g n) = (if n ∈ p1 then ↑↑μ (f1 n) else 1) * if n ∈ p2 then ↑↑μ (f2 n) else 1",
"tactic": "intro n"
},
{
"state_after": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\nh_P_inter : ↑↑μ (t1 ∩ t2) = ∏ n in p1 ∪ p2, ↑↑μ (g n)\nn : ι\n⊢ ↑↑μ ((if n ∈ p1 then f1 n else Set.univ) ∩ if n ∈ p2 then f2 n else Set.univ) =\n (if n ∈ p1 then ↑↑μ (f1 n) else 1) * if n ∈ p2 then ↑↑μ (f2 n) else 1",
"state_before": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\nh_P_inter : ↑↑μ (t1 ∩ t2) = ∏ n in p1 ∪ p2, ↑↑μ (g n)\nn : ι\n⊢ ↑↑μ (g n) = (if n ∈ p1 then ↑↑μ (f1 n) else 1) * if n ∈ p2 then ↑↑μ (f2 n) else 1",
"tactic": "dsimp only"
},
{
"state_after": "case inl.inl\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\nh_P_inter : ↑↑μ (t1 ∩ t2) = ∏ n in p1 ∪ p2, ↑↑μ (g n)\nn : ι\nh1 : n ∈ p1\nh2 : n ∈ p2\n⊢ ↑↑μ (f1 n ∩ f2 n) = ↑↑μ (f1 n) * ↑↑μ (f2 n)\n\ncase inl.inr\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\nh_P_inter : ↑↑μ (t1 ∩ t2) = ∏ n in p1 ∪ p2, ↑↑μ (g n)\nn : ι\nh1 : n ∈ p1\nh2 : ¬n ∈ p2\n⊢ ↑↑μ (f1 n ∩ Set.univ) = ↑↑μ (f1 n) * 1\n\ncase inr.inl\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\nh_P_inter : ↑↑μ (t1 ∩ t2) = ∏ n in p1 ∪ p2, ↑↑μ (g n)\nn : ι\nh1 : ¬n ∈ p1\nh✝ : n ∈ p2\n⊢ ↑↑μ (Set.univ ∩ f2 n) = 1 * ↑↑μ (f2 n)\n\ncase inr.inr\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\nh_P_inter : ↑↑μ (t1 ∩ t2) = ∏ n in p1 ∪ p2, ↑↑μ (g n)\nn : ι\nh1 : ¬n ∈ p1\nh✝ : ¬n ∈ p2\n⊢ ↑↑μ (Set.univ ∩ Set.univ) = 1 * 1",
"state_before": "Ω : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\nh_P_inter : ↑↑μ (t1 ∩ t2) = ∏ n in p1 ∪ p2, ↑↑μ (g n)\nn : ι\n⊢ ↑↑μ ((if n ∈ p1 then f1 n else Set.univ) ∩ if n ∈ p2 then f2 n else Set.univ) =\n (if n ∈ p1 then ↑↑μ (f1 n) else 1) * if n ∈ p2 then ↑↑μ (f2 n) else 1",
"tactic": "split_ifs with h1 h2"
},
{
"state_after": "no goals",
"state_before": "case inl.inr\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\nh_P_inter : ↑↑μ (t1 ∩ t2) = ∏ n in p1 ∪ p2, ↑↑μ (g n)\nn : ι\nh1 : n ∈ p1\nh2 : ¬n ∈ p2\n⊢ ↑↑μ (f1 n ∩ Set.univ) = ↑↑μ (f1 n) * 1\n\ncase inr.inl\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\nh_P_inter : ↑↑μ (t1 ∩ t2) = ∏ n in p1 ∪ p2, ↑↑μ (g n)\nn : ι\nh1 : ¬n ∈ p1\nh✝ : n ∈ p2\n⊢ ↑↑μ (Set.univ ∩ f2 n) = 1 * ↑↑μ (f2 n)\n\ncase inr.inr\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\nh_P_inter : ↑↑μ (t1 ∩ t2) = ∏ n in p1 ∪ p2, ↑↑μ (g n)\nn : ι\nh1 : ¬n ∈ p1\nh✝ : ¬n ∈ p2\n⊢ ↑↑μ (Set.univ ∩ Set.univ) = 1 * 1",
"tactic": "all_goals simp only [measure_univ, one_mul, mul_one, Set.inter_univ, Set.univ_inter]"
},
{
"state_after": "no goals",
"state_before": "case inl.inl\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\nh_P_inter : ↑↑μ (t1 ∩ t2) = ∏ n in p1 ∪ p2, ↑↑μ (g n)\nn : ι\nh1 : n ∈ p1\nh2 : n ∈ p2\n⊢ ↑↑μ (f1 n ∩ f2 n) = ↑↑μ (f1 n) * ↑↑μ (f2 n)",
"tactic": "exact absurd rfl (Set.disjoint_iff_forall_ne.mp hST (hp1 h1) (hp2 h2))"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nΩ : Type u_1\nι : Type u_2\nm0 : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\ninst✝ : IsProbabilityMeasure μ\ns : ι → Set (Set Ω)\nS T : Set ι\nh_indep : iIndepSets s\nhST : Disjoint S T\nt1 t2 : Set Ω\np1 : Finset ι\nhp1 : ↑p1 ⊆ S\nf1 : ι → Set Ω\nht1_m : ∀ (x : ι), x ∈ p1 → f1 x ∈ s x\nht1_eq : t1 = ⋂ (x : ι) (_ : x ∈ p1), f1 x\np2 : Finset ι\nhp2 : ↑p2 ⊆ T\nf2 : ι → Set Ω\nht2_m : ∀ (x : ι), x ∈ p2 → f2 x ∈ s x\nht2_eq : t2 = ⋂ (x : ι) (_ : x ∈ p2), f2 x\ng : ι → Set Ω := fun i => (if i ∈ p1 then f1 i else Set.univ) ∩ if i ∈ p2 then f2 i else Set.univ\nh_P_inter : ↑↑μ (t1 ∩ t2) = ∏ n in p1 ∪ p2, ↑↑μ (g n)\nn : ι\nh1 : ¬n ∈ p1\nh✝ : ¬n ∈ p2\n⊢ ↑↑μ (Set.univ ∩ Set.univ) = 1 * 1",
"tactic": "simp only [measure_univ, one_mul, mul_one, Set.inter_univ, Set.univ_inter]"
}
] |
[
425,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
390,
1
] |
Mathlib/Algebra/Ring/BooleanRing.lean
|
BooleanRing.sup_comm
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.12720\nγ : Type ?u.12723\ninst✝² : BooleanRing α\ninst✝¹ : BooleanRing β\ninst✝ : BooleanRing γ\na b : α\n⊢ a + b + a * b = b + a + b * a",
"tactic": "ring"
}
] |
[
200,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
1
] |
src/lean/Init/Data/List/Basic.lean
|
List.cons_getElem_zero
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\na : α\nas : List α\nh : 0 < length (a :: as)\n⊢ getElem (a :: as) 0 h = a",
"tactic": "rfl"
}
] |
[
22,
6
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
21,
9
] |
Mathlib/Data/Polynomial/Monic.lean
|
Polynomial.monic_multiset_prod_of_monic
|
[
{
"state_after": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : CommSemiring R\np : R[X]\nt : Multiset ι\nf : ι → R[X]\n⊢ (∀ (i : ι), i ∈ t → Monic (f i)) → Monic (Multiset.prod (Multiset.map f t))",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : CommSemiring R\np : R[X]\nt : Multiset ι\nf : ι → R[X]\nht : ∀ (i : ι), i ∈ t → Monic (f i)\n⊢ Monic (Multiset.prod (Multiset.map f t))",
"tactic": "revert ht"
},
{
"state_after": "case refine'_1\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : CommSemiring R\np : R[X]\nt : Multiset ι\nf : ι → R[X]\n⊢ (∀ (i : ι), i ∈ 0 → Monic (f i)) → Monic (Multiset.prod (Multiset.map f 0))\n\ncase refine'_2\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : CommSemiring R\np : R[X]\nt : Multiset ι\nf : ι → R[X]\n⊢ ∀ ⦃a : ι⦄ {s : Multiset ι},\n ((∀ (i : ι), i ∈ s → Monic (f i)) → Monic (Multiset.prod (Multiset.map f s))) →\n (∀ (i : ι), i ∈ a ::ₘ s → Monic (f i)) → Monic (Multiset.prod (Multiset.map f (a ::ₘ s)))",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : CommSemiring R\np : R[X]\nt : Multiset ι\nf : ι → R[X]\n⊢ (∀ (i : ι), i ∈ t → Monic (f i)) → Monic (Multiset.prod (Multiset.map f t))",
"tactic": "refine' t.induction_on _ _"
},
{
"state_after": "case refine'_2\nR : Type u\nS : Type v\na✝ b : R\nm n : ℕ\nι : Type y\ninst✝ : CommSemiring R\np : R[X]\nt✝ : Multiset ι\nf : ι → R[X]\na : ι\nt : Multiset ι\nih : (∀ (i : ι), i ∈ t → Monic (f i)) → Monic (Multiset.prod (Multiset.map f t))\nht : ∀ (i : ι), i ∈ a ::ₘ t → Monic (f i)\n⊢ Monic (Multiset.prod (Multiset.map f (a ::ₘ t)))",
"state_before": "case refine'_2\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : CommSemiring R\np : R[X]\nt : Multiset ι\nf : ι → R[X]\n⊢ ∀ ⦃a : ι⦄ {s : Multiset ι},\n ((∀ (i : ι), i ∈ s → Monic (f i)) → Monic (Multiset.prod (Multiset.map f s))) →\n (∀ (i : ι), i ∈ a ::ₘ s → Monic (f i)) → Monic (Multiset.prod (Multiset.map f (a ::ₘ s)))",
"tactic": "intro a t ih ht"
},
{
"state_after": "case refine'_2\nR : Type u\nS : Type v\na✝ b : R\nm n : ℕ\nι : Type y\ninst✝ : CommSemiring R\np : R[X]\nt✝ : Multiset ι\nf : ι → R[X]\na : ι\nt : Multiset ι\nih : (∀ (i : ι), i ∈ t → Monic (f i)) → Monic (Multiset.prod (Multiset.map f t))\nht : ∀ (i : ι), i ∈ a ::ₘ t → Monic (f i)\n⊢ Monic (f a * Multiset.prod (Multiset.map f t))",
"state_before": "case refine'_2\nR : Type u\nS : Type v\na✝ b : R\nm n : ℕ\nι : Type y\ninst✝ : CommSemiring R\np : R[X]\nt✝ : Multiset ι\nf : ι → R[X]\na : ι\nt : Multiset ι\nih : (∀ (i : ι), i ∈ t → Monic (f i)) → Monic (Multiset.prod (Multiset.map f t))\nht : ∀ (i : ι), i ∈ a ::ₘ t → Monic (f i)\n⊢ Monic (Multiset.prod (Multiset.map f (a ::ₘ t)))",
"tactic": "rw [Multiset.map_cons, Multiset.prod_cons]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nR : Type u\nS : Type v\na✝ b : R\nm n : ℕ\nι : Type y\ninst✝ : CommSemiring R\np : R[X]\nt✝ : Multiset ι\nf : ι → R[X]\na : ι\nt : Multiset ι\nih : (∀ (i : ι), i ∈ t → Monic (f i)) → Monic (Multiset.prod (Multiset.map f t))\nht : ∀ (i : ι), i ∈ a ::ₘ t → Monic (f i)\n⊢ Monic (f a * Multiset.prod (Multiset.map f t))",
"tactic": "exact (ht _ (Multiset.mem_cons_self _ _)).mul (ih fun _ hi => ht _ (Multiset.mem_cons_of_mem hi))"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : CommSemiring R\np : R[X]\nt : Multiset ι\nf : ι → R[X]\n⊢ (∀ (i : ι), i ∈ 0 → Monic (f i)) → Monic (Multiset.prod (Multiset.map f 0))",
"tactic": "simp"
}
] |
[
271,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
265,
1
] |
Mathlib/Data/List/AList.lean
|
AList.mem_of_perm
|
[] |
[
102,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/LinearAlgebra/FreeModule/Finite/Rank.lean
|
FiniteDimensional.rank_lt_aleph0
|
[
{
"state_after": "R : Type u\nM : Type v\nN : Type w\ninst✝⁷ : Ring R\ninst✝⁶ : StrongRankCondition R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module.Finite R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : Module.Finite R N\n⊢ (⨆ (ι : { s // LinearIndependent R Subtype.val }), #↑↑ι) < ℵ₀",
"state_before": "R : Type u\nM : Type v\nN : Type w\ninst✝⁷ : Ring R\ninst✝⁶ : StrongRankCondition R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module.Finite R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : Module.Finite R N\n⊢ Module.rank R M < ℵ₀",
"tactic": "simp only [Module.rank_def]"
},
{
"state_after": "R : Type u\nM : Type v\nN : Type w\ninst✝⁷ : Ring R\ninst✝⁶ : StrongRankCondition R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module.Finite R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : Module.Finite R N\nthis : Nontrivial R := nontrivial_of_invariantBasisNumber R\n⊢ (⨆ (ι : { s // LinearIndependent R Subtype.val }), #↑↑ι) < ℵ₀",
"state_before": "R : Type u\nM : Type v\nN : Type w\ninst✝⁷ : Ring R\ninst✝⁶ : StrongRankCondition R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module.Finite R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : Module.Finite R N\n⊢ (⨆ (ι : { s // LinearIndependent R Subtype.val }), #↑↑ι) < ℵ₀",
"tactic": "letI := nontrivial_of_invariantBasisNumber R"
},
{
"state_after": "case intro\nR : Type u\nM : Type v\nN : Type w\ninst✝⁷ : Ring R\ninst✝⁶ : StrongRankCondition R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module.Finite R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : Module.Finite R N\nthis : Nontrivial R := nontrivial_of_invariantBasisNumber R\nS : Finset M\nhS : Submodule.span R ↑S = ⊤\n⊢ (⨆ (ι : { s // LinearIndependent R Subtype.val }), #↑↑ι) < ℵ₀",
"state_before": "R : Type u\nM : Type v\nN : Type w\ninst✝⁷ : Ring R\ninst✝⁶ : StrongRankCondition R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module.Finite R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : Module.Finite R N\nthis : Nontrivial R := nontrivial_of_invariantBasisNumber R\n⊢ (⨆ (ι : { s // LinearIndependent R Subtype.val }), #↑↑ι) < ℵ₀",
"tactic": "obtain ⟨S, hS⟩ := Module.finite_def.mp ‹Module.Finite R M›"
},
{
"state_after": "case intro\nR : Type u\nM : Type v\nN : Type w\ninst✝⁷ : Ring R\ninst✝⁶ : StrongRankCondition R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module.Finite R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : Module.Finite R N\nthis : Nontrivial R := nontrivial_of_invariantBasisNumber R\nS : Finset M\nhS : Submodule.span R ↑S = ⊤\ni : { s // LinearIndependent R Subtype.val }\n⊢ (#↑↑i) ≤ ↑(Finset.card S)",
"state_before": "case intro\nR : Type u\nM : Type v\nN : Type w\ninst✝⁷ : Ring R\ninst✝⁶ : StrongRankCondition R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module.Finite R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : Module.Finite R N\nthis : Nontrivial R := nontrivial_of_invariantBasisNumber R\nS : Finset M\nhS : Submodule.span R ↑S = ⊤\n⊢ (⨆ (ι : { s // LinearIndependent R Subtype.val }), #↑↑ι) < ℵ₀",
"tactic": "refine' (ciSup_le' fun i => _).trans_lt (nat_lt_aleph0 S.card)"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type u\nM : Type v\nN : Type w\ninst✝⁷ : Ring R\ninst✝⁶ : StrongRankCondition R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module.Finite R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : Module.Finite R N\nthis : Nontrivial R := nontrivial_of_invariantBasisNumber R\nS : Finset M\nhS : Submodule.span R ↑S = ⊤\ni : { s // LinearIndependent R Subtype.val }\n⊢ (#↑↑i) ≤ ↑(Finset.card S)",
"tactic": "exact linearIndependent_le_span_finset _ i.prop S hS"
}
] |
[
68,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/Data/Nat/Digits.lean
|
Nat.digits_of_lt
|
[
{
"state_after": "case intro\nn b x : ℕ\nhx : succ x ≠ 0\nhxb : succ x < b\n⊢ digits b (succ x) = [succ x]",
"state_before": "n b x : ℕ\nhx : x ≠ 0\nhxb : x < b\n⊢ digits b x = [x]",
"tactic": "rcases exists_eq_succ_of_ne_zero hx with ⟨x, rfl⟩"
},
{
"state_after": "case intro.intro\nn x : ℕ\nhx : succ x ≠ 0\nb : ℕ\nhxb : succ x < b + succ 1\n⊢ digits (b + succ 1) (succ x) = [succ x]",
"state_before": "case intro\nn b x : ℕ\nhx : succ x ≠ 0\nhxb : succ x < b\n⊢ digits b (succ x) = [succ x]",
"tactic": "rcases exists_eq_add_of_le' ((Nat.le_add_left 1 x).trans_lt hxb) with ⟨b, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nn x : ℕ\nhx : succ x ≠ 0\nb : ℕ\nhxb : succ x < b + succ 1\n⊢ digits (b + succ 1) (succ x) = [succ x]",
"tactic": "rw [digits_add_two_add_one, div_eq_of_lt hxb, digits_zero, mod_eq_of_lt hxb]"
}
] |
[
135,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Mathlib/RingTheory/Localization/Ideal.lean
|
IsLocalization.map_comap
|
[
{
"state_after": "case intro.intro\nR : Type u_2\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_1\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nJ : Ideal S\nx : S\nhJ : x ∈ J\nr : R\ns : { x // x ∈ M }\nhx : mk' S r s = x\n⊢ x ∈ Ideal.map (algebraMap R S) (Ideal.comap (algebraMap R S) J)",
"state_before": "R : Type u_2\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_1\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nJ : Ideal S\nx : S\nhJ : x ∈ J\n⊢ x ∈ Ideal.map (algebraMap R S) (Ideal.comap (algebraMap R S) J)",
"tactic": "obtain ⟨r, s, hx⟩ := mk'_surjective M x"
},
{
"state_after": "case intro.intro\nR : Type u_2\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_1\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nJ : Ideal S\nx : S\nr : R\ns : { x // x ∈ M }\nhJ : mk' S r s ∈ J\nhx : mk' S r s = x\n⊢ mk' S r s ∈ Ideal.map (algebraMap R S) (Ideal.comap (algebraMap R S) J)",
"state_before": "case intro.intro\nR : Type u_2\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_1\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nJ : Ideal S\nx : S\nhJ : x ∈ J\nr : R\ns : { x // x ∈ M }\nhx : mk' S r s = x\n⊢ x ∈ Ideal.map (algebraMap R S) (Ideal.comap (algebraMap R S) J)",
"tactic": "rw [← hx] at hJ⊢"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nR : Type u_2\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_1\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nJ : Ideal S\nx : S\nr : R\ns : { x // x ∈ M }\nhJ : mk' S r s ∈ J\nhx : mk' S r s = x\n⊢ mk' S r s ∈ Ideal.map (algebraMap R S) (Ideal.comap (algebraMap R S) J)",
"tactic": "exact\n Ideal.mul_mem_right _ _\n (Ideal.mem_map_of_mem _\n (show (algebraMap R S) r ∈ J from\n mk'_spec S r s ▸ J.mul_mem_right ((algebraMap R S) s) hJ))"
}
] |
[
78,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.degree_lt_wf
|
[] |
[
58,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/Analysis/Asymptotics/Theta.lean
|
Asymptotics.isTheta_const_mul_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.59587\nE : Type ?u.59590\nF : Type u_3\nG : Type ?u.59596\nE' : Type ?u.59599\nF' : Type ?u.59602\nG' : Type ?u.59605\nE'' : Type ?u.59608\nF'' : Type ?u.59611\nG'' : Type ?u.59614\nR : Type ?u.59617\nR' : Type ?u.59620\n𝕜 : Type u_1\n𝕜' : Type ?u.59626\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''\nl l' : Filter α\nc : 𝕜\nf : α → 𝕜\nhc : c ≠ 0\n⊢ (fun x => c * f x) =Θ[l] g ↔ f =Θ[l] g",
"tactic": "simpa only [← smul_eq_mul] using isTheta_const_smul_left hc"
}
] |
[
301,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
299,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/Basic.lean
|
PrimeSpectrum.zeroLocus_pow
|
[] |
[
381,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
379,
1
] |
Mathlib/Analysis/Normed/Group/Quotient.lean
|
AddSubgroup.normedMk.apply
|
[] |
[
290,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
289,
1
] |
Mathlib/Algebra/Order/Monoid/MinMax.lean
|
max_le_mul_of_one_le
|
[] |
[
153,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/Data/Finset/Image.lean
|
Finset.mapEmbedding_apply
|
[] |
[
179,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.mul_left_comm
|
[
{
"state_after": "no goals",
"state_before": "a b c : Int\n⊢ a * (b * c) = b * (a * c)",
"tactic": "rw [← Int.mul_assoc, ← Int.mul_assoc, Int.mul_comm a]"
}
] |
[
401,
56
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
400,
11
] |
Mathlib/Topology/Algebra/Ring/Basic.lean
|
Subsemiring.topologicalClosure_minimal
|
[] |
[
127,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Algebra/Group/WithOne/Defs.lean
|
WithZero.inv_zero
|
[] |
[
298,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
297,
1
] |
Mathlib/Init/Algebra/Classes.lean
|
refl
|
[] |
[
261,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
260,
1
] |
Mathlib/LinearAlgebra/PiTensorProduct.lean
|
PiTensorProduct.reindex_symm
|
[] |
[
520,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
520,
1
] |
Mathlib/Data/Rat/NNRat.lean
|
NNRat.coe_mono
|
[] |
[
181,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
LocalHomeomorph.isBigO_congr
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nE : Type u_3\ninst✝¹ : Norm E\nF : Type u_4\ninst✝ : Norm F\ne : LocalHomeomorph α β\nb : β\nhb : b ∈ e.target\nf : β → E\ng : β → F\n⊢ (∃ c, IsBigOWith c (𝓝 b) f g) ↔ ∃ c, IsBigOWith c (𝓝 (↑(LocalHomeomorph.symm e) b)) (f ∘ ↑e) (g ∘ ↑e)",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nE : Type u_3\ninst✝¹ : Norm E\nF : Type u_4\ninst✝ : Norm F\ne : LocalHomeomorph α β\nb : β\nhb : b ∈ e.target\nf : β → E\ng : β → F\n⊢ f =O[𝓝 b] g ↔ (f ∘ ↑e) =O[𝓝 (↑(LocalHomeomorph.symm e) b)] (g ∘ ↑e)",
"tactic": "simp only [IsBigO_def]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\nE : Type u_3\ninst✝¹ : Norm E\nF : Type u_4\ninst✝ : Norm F\ne : LocalHomeomorph α β\nb : β\nhb : b ∈ e.target\nf : β → E\ng : β → F\n⊢ (∃ c, IsBigOWith c (𝓝 b) f g) ↔ ∃ c, IsBigOWith c (𝓝 (↑(LocalHomeomorph.symm e) b)) (f ∘ ↑e) (g ∘ ↑e)",
"tactic": "exact exists_congr fun C => e.isBigOWith_congr hb"
}
] |
[
2196,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2193,
1
] |
Mathlib/Data/Polynomial/Monic.lean
|
Polynomial.Monic.mul_natDegree_lt_iff
|
[
{
"state_after": "case pos\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : Monic p\nq : R[X]\nhq : q = 0\n⊢ natDegree (p * q) < natDegree p ↔ p ≠ 1 ∧ q = 0\n\ncase neg\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : Monic p\nq : R[X]\nhq : ¬q = 0\n⊢ natDegree (p * q) < natDegree p ↔ p ≠ 1 ∧ q = 0",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : Monic p\nq : R[X]\n⊢ natDegree (p * q) < natDegree p ↔ p ≠ 1 ∧ q = 0",
"tactic": "by_cases hq : q = 0"
},
{
"state_after": "case pos\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : Monic p\nq : R[X]\nhq : q = 0\n⊢ 0 < natDegree p ↔ natDegree p ≠ 0",
"state_before": "case pos\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : Monic p\nq : R[X]\nhq : q = 0\n⊢ natDegree (p * q) < natDegree p ↔ p ≠ 1 ∧ q = 0",
"tactic": "suffices 0 < p.natDegree ↔ p.natDegree ≠ 0 by simpa [hq, ← h.natDegree_eq_zero_iff_eq_one]"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : Monic p\nq : R[X]\nhq : q = 0\n⊢ 0 < natDegree p ↔ natDegree p ≠ 0",
"tactic": "exact ⟨fun h => h.ne', fun h => lt_of_le_of_ne (Nat.zero_le _) h.symm⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : Monic p\nq : R[X]\nhq : q = 0\nthis : 0 < natDegree p ↔ natDegree p ≠ 0\n⊢ natDegree (p * q) < natDegree p ↔ p ≠ 1 ∧ q = 0",
"tactic": "simpa [hq, ← h.natDegree_eq_zero_iff_eq_one]"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u\nS : Type v\na b : R\nm n : ℕ\nι : Type y\ninst✝ : Semiring R\np : R[X]\nh : Monic p\nq : R[X]\nhq : ¬q = 0\n⊢ natDegree (p * q) < natDegree p ↔ p ≠ 1 ∧ q = 0",
"tactic": "simp [h.natDegree_mul', hq]"
}
] |
[
463,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
458,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.BlankRel.trans
|
[
{
"state_after": "case inl.inl\nΓ : Type u_1\ninst✝ : Inhabited Γ\nl₁ l₂ l₃ : List Γ\nh₁ : BlankExtends l₁ l₂\nh₂ : BlankExtends l₂ l₃\n⊢ BlankRel l₁ l₃\n\ncase inl.inr\nΓ : Type u_1\ninst✝ : Inhabited Γ\nl₁ l₂ l₃ : List Γ\nh₁ : BlankExtends l₁ l₂\nh₂ : BlankExtends l₃ l₂\n⊢ BlankRel l₁ l₃\n\ncase inr.inl\nΓ : Type u_1\ninst✝ : Inhabited Γ\nl₁ l₂ l₃ : List Γ\nh₁ : BlankExtends l₂ l₁\nh₂ : BlankExtends l₂ l₃\n⊢ BlankRel l₁ l₃\n\ncase inr.inr\nΓ : Type u_1\ninst✝ : Inhabited Γ\nl₁ l₂ l₃ : List Γ\nh₁ : BlankExtends l₂ l₁\nh₂ : BlankExtends l₃ l₂\n⊢ BlankRel l₁ l₃",
"state_before": "Γ : Type u_1\ninst✝ : Inhabited Γ\nl₁ l₂ l₃ : List Γ\n⊢ BlankRel l₁ l₂ → BlankRel l₂ l₃ → BlankRel l₁ l₃",
"tactic": "rintro (h₁ | h₁) (h₂ | h₂)"
},
{
"state_after": "no goals",
"state_before": "case inl.inl\nΓ : Type u_1\ninst✝ : Inhabited Γ\nl₁ l₂ l₃ : List Γ\nh₁ : BlankExtends l₁ l₂\nh₂ : BlankExtends l₂ l₃\n⊢ BlankRel l₁ l₃",
"tactic": "exact Or.inl (h₁.trans h₂)"
},
{
"state_after": "case inl.inr.inl\nΓ : Type u_1\ninst✝ : Inhabited Γ\nl₁ l₂ l₃ : List Γ\nh₁ : BlankExtends l₁ l₂\nh₂ : BlankExtends l₃ l₂\nh : List.length l₁ ≤ List.length l₃\n⊢ BlankRel l₁ l₃\n\ncase inl.inr.inr\nΓ : Type u_1\ninst✝ : Inhabited Γ\nl₁ l₂ l₃ : List Γ\nh₁ : BlankExtends l₁ l₂\nh₂ : BlankExtends l₃ l₂\nh : List.length l₃ ≤ List.length l₁\n⊢ BlankRel l₁ l₃",
"state_before": "case inl.inr\nΓ : Type u_1\ninst✝ : Inhabited Γ\nl₁ l₂ l₃ : List Γ\nh₁ : BlankExtends l₁ l₂\nh₂ : BlankExtends l₃ l₂\n⊢ BlankRel l₁ l₃",
"tactic": "cases' le_total l₁.length l₃.length with h h"
},
{
"state_after": "no goals",
"state_before": "case inl.inr.inl\nΓ : Type u_1\ninst✝ : Inhabited Γ\nl₁ l₂ l₃ : List Γ\nh₁ : BlankExtends l₁ l₂\nh₂ : BlankExtends l₃ l₂\nh : List.length l₁ ≤ List.length l₃\n⊢ BlankRel l₁ l₃",
"tactic": "exact Or.inl (h₁.above_of_le h₂ h)"
},
{
"state_after": "no goals",
"state_before": "case inl.inr.inr\nΓ : Type u_1\ninst✝ : Inhabited Γ\nl₁ l₂ l₃ : List Γ\nh₁ : BlankExtends l₁ l₂\nh₂ : BlankExtends l₃ l₂\nh : List.length l₃ ≤ List.length l₁\n⊢ BlankRel l₁ l₃",
"tactic": "exact Or.inr (h₂.above_of_le h₁ h)"
},
{
"state_after": "case inr.inl.inl\nΓ : Type u_1\ninst✝ : Inhabited Γ\nl₁ l₂ l₃ : List Γ\nh₁ : BlankExtends l₂ l₁\nh₂ : BlankExtends l₂ l₃\nh : List.length l₁ ≤ List.length l₃\n⊢ BlankRel l₁ l₃\n\ncase inr.inl.inr\nΓ : Type u_1\ninst✝ : Inhabited Γ\nl₁ l₂ l₃ : List Γ\nh₁ : BlankExtends l₂ l₁\nh₂ : BlankExtends l₂ l₃\nh : List.length l₃ ≤ List.length l₁\n⊢ BlankRel l₁ l₃",
"state_before": "case inr.inl\nΓ : Type u_1\ninst✝ : Inhabited Γ\nl₁ l₂ l₃ : List Γ\nh₁ : BlankExtends l₂ l₁\nh₂ : BlankExtends l₂ l₃\n⊢ BlankRel l₁ l₃",
"tactic": "cases' le_total l₁.length l₃.length with h h"
},
{
"state_after": "no goals",
"state_before": "case inr.inl.inl\nΓ : Type u_1\ninst✝ : Inhabited Γ\nl₁ l₂ l₃ : List Γ\nh₁ : BlankExtends l₂ l₁\nh₂ : BlankExtends l₂ l₃\nh : List.length l₁ ≤ List.length l₃\n⊢ BlankRel l₁ l₃",
"tactic": "exact Or.inl (h₁.below_of_le h₂ h)"
},
{
"state_after": "no goals",
"state_before": "case inr.inl.inr\nΓ : Type u_1\ninst✝ : Inhabited Γ\nl₁ l₂ l₃ : List Γ\nh₁ : BlankExtends l₂ l₁\nh₂ : BlankExtends l₂ l₃\nh : List.length l₃ ≤ List.length l₁\n⊢ BlankRel l₁ l₃",
"tactic": "exact Or.inr (h₂.below_of_le h₁ h)"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nΓ : Type u_1\ninst✝ : Inhabited Γ\nl₁ l₂ l₃ : List Γ\nh₁ : BlankExtends l₂ l₁\nh₂ : BlankExtends l₃ l₂\n⊢ BlankRel l₁ l₃",
"tactic": "exact Or.inr (h₂.trans h₁)"
}
] |
[
145,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.min_empty
|
[] |
[
1248,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1247,
1
] |
Mathlib/Data/PFun.lean
|
PFun.core_restrict
|
[
{
"state_after": "case h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48707\nδ : Type ?u.48710\nε : Type ?u.48713\nι : Type ?u.48716\nf✝ : α →. β\nf : α → β\ns : Set β\nx : α\n⊢ x ∈ core (↑f) s ↔ x ∈ f ⁻¹' s",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.48707\nδ : Type ?u.48710\nε : Type ?u.48713\nι : Type ?u.48716\nf✝ : α →. β\nf : α → β\ns : Set β\n⊢ core (↑f) s = f ⁻¹' s",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.48707\nδ : Type ?u.48710\nε : Type ?u.48713\nι : Type ?u.48716\nf✝ : α →. β\nf : α → β\ns : Set β\nx : α\n⊢ x ∈ core (↑f) s ↔ x ∈ f ⁻¹' s",
"tactic": "simp [core_def]"
}
] |
[
504,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
503,
1
] |
Std/Data/Option/Lemmas.lean
|
Option.mem_unique
|
[] |
[
41,
22
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
40,
1
] |
Mathlib/LinearAlgebra/Matrix/Circulant.lean
|
Matrix.circulant_apply
|
[] |
[
55,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
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