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/Order/SuccPred/Basic.lean
|
Order.le_succ
|
[] |
[
217,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.coe_bot
|
[] |
[
379,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
378,
1
] |
Mathlib/Topology/Semicontinuous.lean
|
UpperSemicontinuous.upperSemicontinuousAt
|
[] |
[
700,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
698,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Order.lean
|
tsum_le_tsum
|
[] |
[
109,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/Data/Set/NAry.lean
|
Set.image2_union_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nα' : Type ?u.14061\nβ : Type u_3\nβ' : Type ?u.14067\nγ : Type u_1\nγ' : Type ?u.14073\nδ : Type ?u.14076\nδ' : Type ?u.14079\nε : Type ?u.14082\nε' : Type ?u.14085\nζ : Type ?u.14088\nζ' : Type ?u.14091\nν : Type ?u.14094\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Set α\nt t' : Set β\nu u' : Set γ\nv : Set δ\na a' : α\nb b' : β\nc c' : γ\nd d' : δ\n⊢ image2 f s (t ∪ t') = image2 f s t ∪ image2 f s t'",
"tactic": "rw [← image2_swap, image2_union_left, image2_swap f, image2_swap f]"
}
] |
[
136,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Mathlib/Analysis/NormedSpace/Pointwise.lean
|
ball_add_closedBall
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.933372\nE : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nx y z : E\nδ ε : ℝ\nhε : 0 < ε\nhδ : 0 ≤ δ\na b : E\n⊢ Metric.ball a ε + Metric.closedBall b δ = Metric.ball (a + b) (ε + δ)",
"tactic": "rw [ball_add, thickening_closedBall hε hδ b, Metric.vadd_ball, vadd_eq_add]"
}
] |
[
360,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
358,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean
|
Complex.zero_cpow
|
[
{
"state_after": "no goals",
"state_before": "x : ℂ\nh : x ≠ 0\n⊢ 0 ^ x = 0",
"tactic": "simp [cpow_def, *]"
}
] |
[
58,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/Algebra/Category/ModuleCat/Monoidal/Basic.lean
|
ModuleCat.MonoidalCategory.associator_hom_apply
|
[] |
[
249,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
247,
1
] |
Mathlib/Algebra/GroupPower/Lemmas.lean
|
Int.cast_mul_eq_zsmul_cast
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : AddCommGroupWithOne α\nm : ℤ\n⊢ ∀ (n : ℤ), ↑(0 * n) = 0 • ↑n",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : AddCommGroupWithOne α\nm k : ℤ\nx✝ : 0 ≤ k\nih : ∀ (n : ℤ), ↑(k * n) = k • ↑n\nn : ℤ\n⊢ ↑((k + 1) * n) = (k + 1) • ↑n",
"tactic": "simp [add_mul, add_zsmul, ih]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝ : AddCommGroupWithOne α\nm k : ℤ\nx✝ : k ≤ 0\nih : ∀ (n : ℤ), ↑(k * n) = k • ↑n\nn : ℤ\n⊢ ↑((k - 1) * n) = (k - 1) • ↑n",
"tactic": "simp [sub_mul, sub_zsmul, ih, ← sub_eq_add_neg]"
}
] |
[
587,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
583,
1
] |
Mathlib/Topology/ContinuousFunction/Basic.lean
|
ContinuousMap.id_comp
|
[] |
[
257,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
256,
1
] |
Mathlib/MeasureTheory/Constructions/Prod/Integral.lean
|
MeasureTheory.integrable_swap_iff
|
[] |
[
234,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
232,
1
] |
Mathlib/NumberTheory/Padics/PadicNumbers.lean
|
Padic.norm_lt_pow_iff_norm_le_pow_sub_one
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ℚ_[p]\nn : ℤ\n⊢ ‖x‖ < ↑p ^ n ↔ ‖x‖ ≤ ↑p ^ (n - 1)",
"tactic": "rw [norm_le_pow_iff_norm_lt_pow_add_one, sub_add_cancel]"
}
] |
[
1179,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1177,
1
] |
Mathlib/RingTheory/Ideal/QuotientOperations.lean
|
DoubleQuot.ker_quotQuotMk
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : CommRing R\nI J : Ideal R\n⊢ RingHom.ker (quotQuotMk I J) = I ⊔ J",
"tactic": "rw [RingHom.ker_eq_comap_bot, quotQuotMk, ← comap_comap, ← RingHom.ker, mk_ker,\n comap_map_of_surjective (Ideal.Quotient.mk I) Quotient.mk_surjective, ← RingHom.ker, mk_ker,\n sup_comm]"
}
] |
[
548,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
545,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
LinearMap.submodule_pow_eq_zero_of_pow_eq_zero
|
[
{
"state_after": "case h.a\nR : Type u_1\nR₁ : Type ?u.155563\nR₂ : Type ?u.155566\nR₃ : Type ?u.155569\nR₄ : Type ?u.155572\nS : Type ?u.155575\nK : Type ?u.155578\nK₂ : Type ?u.155581\nM : Type u_2\nM' : Type ?u.155587\nM₁ : Type ?u.155590\nM₂ : Type ?u.155593\nM₃ : Type ?u.155596\nM₄ : Type ?u.155599\nN✝ : Type ?u.155602\nN₂ : Type ?u.155605\nι : Type ?u.155608\nV : Type ?u.155611\nV₂ : Type ?u.155614\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring R₂\ninst✝¹⁵ : Semiring R₃\ninst✝¹⁴ : Semiring R₄\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid M₁\ninst✝¹¹ : AddCommMonoid M₂\ninst✝¹⁰ : AddCommMonoid M₃\ninst✝⁹ : AddCommMonoid M₄\ninst✝⁸ : Module R M\ninst✝⁷ : Module R M₁\ninst✝⁶ : Module R₂ M₂\ninst✝⁵ : Module R₃ M₃\ninst✝⁴ : Module R₄ M₄\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₃₄ : R₃ →+* R₄\nσ₁₃ : R →+* R₃\nσ₂₄ : R₂ →+* R₄\nσ₁₄ : R →+* R₄\ninst✝³ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝² : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄\ninst✝¹ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄\ninst✝ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄\nf : M →ₛₗ[σ₁₂] M₂\ng✝ : M₂ →ₛₗ[σ₂₃] M₃\nN : Submodule R M\ng : Module.End R { x // x ∈ N }\nG : Module.End R M\nh : comp G (Submodule.subtype N) = comp (Submodule.subtype N) g\nk : ℕ\nhG : G ^ k = 0\nm : { x // x ∈ N }\n⊢ ↑(↑(g ^ k) m) = ↑(↑0 m)",
"state_before": "R : Type u_1\nR₁ : Type ?u.155563\nR₂ : Type ?u.155566\nR₃ : Type ?u.155569\nR₄ : Type ?u.155572\nS : Type ?u.155575\nK : Type ?u.155578\nK₂ : Type ?u.155581\nM : Type u_2\nM' : Type ?u.155587\nM₁ : Type ?u.155590\nM₂ : Type ?u.155593\nM₃ : Type ?u.155596\nM₄ : Type ?u.155599\nN✝ : Type ?u.155602\nN₂ : Type ?u.155605\nι : Type ?u.155608\nV : Type ?u.155611\nV₂ : Type ?u.155614\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring R₂\ninst✝¹⁵ : Semiring R₃\ninst✝¹⁴ : Semiring R₄\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid M₁\ninst✝¹¹ : AddCommMonoid M₂\ninst✝¹⁰ : AddCommMonoid M₃\ninst✝⁹ : AddCommMonoid M₄\ninst✝⁸ : Module R M\ninst✝⁷ : Module R M₁\ninst✝⁶ : Module R₂ M₂\ninst✝⁵ : Module R₃ M₃\ninst✝⁴ : Module R₄ M₄\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₃₄ : R₃ →+* R₄\nσ₁₃ : R →+* R₃\nσ₂₄ : R₂ →+* R₄\nσ₁₄ : R →+* R₄\ninst✝³ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝² : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄\ninst✝¹ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄\ninst✝ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄\nf : M →ₛₗ[σ₁₂] M₂\ng✝ : M₂ →ₛₗ[σ₂₃] M₃\nN : Submodule R M\ng : Module.End R { x // x ∈ N }\nG : Module.End R M\nh : comp G (Submodule.subtype N) = comp (Submodule.subtype N) g\nk : ℕ\nhG : G ^ k = 0\n⊢ g ^ k = 0",
"tactic": "ext m"
},
{
"state_after": "case h.a\nR : Type u_1\nR₁ : Type ?u.155563\nR₂ : Type ?u.155566\nR₃ : Type ?u.155569\nR₄ : Type ?u.155572\nS : Type ?u.155575\nK : Type ?u.155578\nK₂ : Type ?u.155581\nM : Type u_2\nM' : Type ?u.155587\nM₁ : Type ?u.155590\nM₂ : Type ?u.155593\nM₃ : Type ?u.155596\nM₄ : Type ?u.155599\nN✝ : Type ?u.155602\nN₂ : Type ?u.155605\nι : Type ?u.155608\nV : Type ?u.155611\nV₂ : Type ?u.155614\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring R₂\ninst✝¹⁵ : Semiring R₃\ninst✝¹⁴ : Semiring R₄\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid M₁\ninst✝¹¹ : AddCommMonoid M₂\ninst✝¹⁰ : AddCommMonoid M₃\ninst✝⁹ : AddCommMonoid M₄\ninst✝⁸ : Module R M\ninst✝⁷ : Module R M₁\ninst✝⁶ : Module R₂ M₂\ninst✝⁵ : Module R₃ M₃\ninst✝⁴ : Module R₄ M₄\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₃₄ : R₃ →+* R₄\nσ₁₃ : R →+* R₃\nσ₂₄ : R₂ →+* R₄\nσ₁₄ : R →+* R₄\ninst✝³ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝² : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄\ninst✝¹ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄\ninst✝ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄\nf : M →ₛₗ[σ₁₂] M₂\ng✝ : M₂ →ₛₗ[σ₂₃] M₃\nN : Submodule R M\ng : Module.End R { x // x ∈ N }\nG : Module.End R M\nh : comp G (Submodule.subtype N) = comp (Submodule.subtype N) g\nk : ℕ\nhG : G ^ k = 0\nm : { x // x ∈ N }\nhg : ↑(comp (Submodule.subtype N) (g ^ k)) m = 0\n⊢ ↑(↑(g ^ k) m) = ↑(↑0 m)",
"state_before": "case h.a\nR : Type u_1\nR₁ : Type ?u.155563\nR₂ : Type ?u.155566\nR₃ : Type ?u.155569\nR₄ : Type ?u.155572\nS : Type ?u.155575\nK : Type ?u.155578\nK₂ : Type ?u.155581\nM : Type u_2\nM' : Type ?u.155587\nM₁ : Type ?u.155590\nM₂ : Type ?u.155593\nM₃ : Type ?u.155596\nM₄ : Type ?u.155599\nN✝ : Type ?u.155602\nN₂ : Type ?u.155605\nι : Type ?u.155608\nV : Type ?u.155611\nV₂ : Type ?u.155614\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring R₂\ninst✝¹⁵ : Semiring R₃\ninst✝¹⁴ : Semiring R₄\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid M₁\ninst✝¹¹ : AddCommMonoid M₂\ninst✝¹⁰ : AddCommMonoid M₃\ninst✝⁹ : AddCommMonoid M₄\ninst✝⁸ : Module R M\ninst✝⁷ : Module R M₁\ninst✝⁶ : Module R₂ M₂\ninst✝⁵ : Module R₃ M₃\ninst✝⁴ : Module R₄ M₄\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₃₄ : R₃ →+* R₄\nσ₁₃ : R →+* R₃\nσ₂₄ : R₂ →+* R₄\nσ₁₄ : R →+* R₄\ninst✝³ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝² : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄\ninst✝¹ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄\ninst✝ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄\nf : M →ₛₗ[σ₁₂] M₂\ng✝ : M₂ →ₛₗ[σ₂₃] M₃\nN : Submodule R M\ng : Module.End R { x // x ∈ N }\nG : Module.End R M\nh : comp G (Submodule.subtype N) = comp (Submodule.subtype N) g\nk : ℕ\nhG : G ^ k = 0\nm : { x // x ∈ N }\n⊢ ↑(↑(g ^ k) m) = ↑(↑0 m)",
"tactic": "have hg : N.subtype.comp (g ^ k) m = 0 := by\n rw [← commute_pow_left_of_commute h, hG, zero_comp, zero_apply]"
},
{
"state_after": "no goals",
"state_before": "case h.a\nR : Type u_1\nR₁ : Type ?u.155563\nR₂ : Type ?u.155566\nR₃ : Type ?u.155569\nR₄ : Type ?u.155572\nS : Type ?u.155575\nK : Type ?u.155578\nK₂ : Type ?u.155581\nM : Type u_2\nM' : Type ?u.155587\nM₁ : Type ?u.155590\nM₂ : Type ?u.155593\nM₃ : Type ?u.155596\nM₄ : Type ?u.155599\nN✝ : Type ?u.155602\nN₂ : Type ?u.155605\nι : Type ?u.155608\nV : Type ?u.155611\nV₂ : Type ?u.155614\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring R₂\ninst✝¹⁵ : Semiring R₃\ninst✝¹⁴ : Semiring R₄\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid M₁\ninst✝¹¹ : AddCommMonoid M₂\ninst✝¹⁰ : AddCommMonoid M₃\ninst✝⁹ : AddCommMonoid M₄\ninst✝⁸ : Module R M\ninst✝⁷ : Module R M₁\ninst✝⁶ : Module R₂ M₂\ninst✝⁵ : Module R₃ M₃\ninst✝⁴ : Module R₄ M₄\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₃₄ : R₃ →+* R₄\nσ₁₃ : R →+* R₃\nσ₂₄ : R₂ →+* R₄\nσ₁₄ : R →+* R₄\ninst✝³ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝² : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄\ninst✝¹ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄\ninst✝ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄\nf : M →ₛₗ[σ₁₂] M₂\ng✝ : M₂ →ₛₗ[σ₂₃] M₃\nN : Submodule R M\ng : Module.End R { x // x ∈ N }\nG : Module.End R M\nh : comp G (Submodule.subtype N) = comp (Submodule.subtype N) g\nk : ℕ\nhG : G ^ k = 0\nm : { x // x ∈ N }\nhg : ↑(comp (Submodule.subtype N) (g ^ k)) m = 0\n⊢ ↑(↑(g ^ k) m) = ↑(↑0 m)",
"tactic": "simpa using hg"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nR₁ : Type ?u.155563\nR₂ : Type ?u.155566\nR₃ : Type ?u.155569\nR₄ : Type ?u.155572\nS : Type ?u.155575\nK : Type ?u.155578\nK₂ : Type ?u.155581\nM : Type u_2\nM' : Type ?u.155587\nM₁ : Type ?u.155590\nM₂ : Type ?u.155593\nM₃ : Type ?u.155596\nM₄ : Type ?u.155599\nN✝ : Type ?u.155602\nN₂ : Type ?u.155605\nι : Type ?u.155608\nV : Type ?u.155611\nV₂ : Type ?u.155614\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : Semiring R₂\ninst✝¹⁵ : Semiring R₃\ninst✝¹⁴ : Semiring R₄\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : AddCommMonoid M₁\ninst✝¹¹ : AddCommMonoid M₂\ninst✝¹⁰ : AddCommMonoid M₃\ninst✝⁹ : AddCommMonoid M₄\ninst✝⁸ : Module R M\ninst✝⁷ : Module R M₁\ninst✝⁶ : Module R₂ M₂\ninst✝⁵ : Module R₃ M₃\ninst✝⁴ : Module R₄ M₄\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₃₄ : R₃ →+* R₄\nσ₁₃ : R →+* R₃\nσ₂₄ : R₂ →+* R₄\nσ₁₄ : R →+* R₄\ninst✝³ : RingHomCompTriple σ₁₂ σ₂₃ σ₁₃\ninst✝² : RingHomCompTriple σ₂₃ σ₃₄ σ₂₄\ninst✝¹ : RingHomCompTriple σ₁₃ σ₃₄ σ₁₄\ninst✝ : RingHomCompTriple σ₁₂ σ₂₄ σ₁₄\nf : M →ₛₗ[σ₁₂] M₂\ng✝ : M₂ →ₛₗ[σ₂₃] M₃\nN : Submodule R M\ng : Module.End R { x // x ∈ N }\nG : Module.End R M\nh : comp G (Submodule.subtype N) = comp (Submodule.subtype N) g\nk : ℕ\nhG : G ^ k = 0\nm : { x // x ∈ N }\n⊢ ↑(comp (Submodule.subtype N) (g ^ k)) m = 0",
"tactic": "rw [← commute_pow_left_of_commute h, hG, zero_comp, zero_apply]"
}
] |
[
354,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
347,
1
] |
Mathlib/Algebra/BigOperators/Order.lean
|
Finset.pow_card_le_prod
|
[] |
[
226,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/Order/BoundedOrder.lean
|
not_lt_top_iff
|
[] |
[
180,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
179,
1
] |
Mathlib/Analysis/BoxIntegral/Box/Basic.lean
|
BoxIntegral.Box.face_mono
|
[] |
[
409,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
407,
1
] |
Mathlib/Topology/FiberBundle/Basic.lean
|
FiberBundleCore.localTrivAt_def
|
[] |
[
630,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
629,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
isGLB_Ici
|
[] |
[
523,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
522,
1
] |
Mathlib/MeasureTheory/Function/AEEqFun.lean
|
ContinuousMap.coeFn_toAEEqFun
|
[] |
[
928,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
927,
1
] |
Mathlib/ModelTheory/Basic.lean
|
FirstOrder.Language.toEquiv_equiv_empty
|
[] |
[
1006,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1005,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.le_degree_of_ne_zero
|
[] |
[
171,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/Data/Prod/Basic.lean
|
Prod.exists'
|
[] |
[
47,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
46,
1
] |
Mathlib/Algebra/Group/Units.lean
|
Units.inv_mul_eq_one
|
[] |
[
385,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
384,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Mul.lean
|
fderivWithin_mul
|
[] |
[
374,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
370,
1
] |
Mathlib/Computability/Partrec.lean
|
Computable.option_some_iff
|
[] |
[
634,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
633,
1
] |
Mathlib/FieldTheory/PerfectClosure.lean
|
PerfectClosure.mk_add_mk
|
[] |
[
289,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
286,
1
] |
Mathlib/Data/Setoid/Basic.lean
|
Setoid.sup_def
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.7769\nr s : Setoid α\n⊢ (EqvGen.Setoid fun x y => Rel r x y ∨ Rel s x y) = EqvGen.Setoid (Rel r ⊔ Rel s)",
"state_before": "α : Type u_1\nβ : Type ?u.7769\nr s : Setoid α\n⊢ r ⊔ s = EqvGen.Setoid (Rel r ⊔ Rel s)",
"tactic": "rw [sup_eq_eqvGen]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.7769\nr s : Setoid α\n⊢ (EqvGen.Setoid fun x y => Rel r x y ∨ Rel s x y) = EqvGen.Setoid (Rel r ⊔ Rel s)",
"tactic": "rfl"
}
] |
[
225,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
224,
1
] |
Mathlib/Algebra/Algebra/Basic.lean
|
algebraMap.coe_ratCast
|
[] |
[
253,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
252,
1
] |
Mathlib/Topology/Order/Basic.lean
|
tendsto_nhds_bot_mono'
|
[] |
[
1180,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1178,
1
] |
Mathlib/Geometry/Euclidean/Sphere/Basic.lean
|
EuclideanGeometry.Sphere.cospherical
|
[] |
[
172,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
171,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.eq_neg_comm
|
[] |
[
312,
47
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
311,
11
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean
|
MeasureTheory.snormEssSup_smul_measure
|
[
{
"state_after": "α : Type u_1\nE : Type ?u.2067351\nF : Type u_2\nG : Type ?u.2067357\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\nc : ℝ≥0∞\nhc : c ≠ 0\n⊢ essSup (fun x => ↑‖f x‖₊) (c • μ) = essSup (fun x => ↑‖f x‖₊) μ",
"state_before": "α : Type u_1\nE : Type ?u.2067351\nF : Type u_2\nG : Type ?u.2067357\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\nc : ℝ≥0∞\nhc : c ≠ 0\n⊢ snormEssSup f (c • μ) = snormEssSup f μ",
"tactic": "simp_rw [snormEssSup]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type ?u.2067351\nF : Type u_2\nG : Type ?u.2067357\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf : α → F\nc : ℝ≥0∞\nhc : c ≠ 0\n⊢ essSup (fun x => ↑‖f x‖₊) (c • μ) = essSup (fun x => ↑‖f x‖₊) μ",
"tactic": "exact essSup_smul_measure hc"
}
] |
[
612,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
609,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
ContinuousLinearMap.op_nnnorm_comp_le
|
[] |
[
462,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
461,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.lift_mul
|
[] |
[
616,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
614,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.mem_cofinite
|
[] |
[
2638,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2637,
1
] |
Mathlib/MeasureTheory/Measure/ProbabilityMeasure.lean
|
MeasureTheory.ProbabilityMeasure.toWeakDualBCNN_continuous
|
[] |
[
249,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
248,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Prod.lean
|
differentiableOn_snd
|
[] |
[
307,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
306,
1
] |
Mathlib/Topology/Order/Basic.lean
|
nhds_order_unbounded
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : TopologicalSpace α\ninst✝ : Preorder α\nt : OrderTopology α\na : α\nhu : ∃ u, a < u\nhl : ∃ l, l < a\n⊢ ((⨅ (b : α) (_ : b ∈ Iio a), 𝓟 (Ioi b)) ⊓ ⨅ (b : α) (_ : b ∈ Ioi a), 𝓟 (Iio b)) =\n (⨅ (i : α) (_ : i < a), 𝓟 (Ioi i)) ⊓ ⨅ (i : α) (_ : a < i), 𝓟 (Iio i)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : TopologicalSpace α\ninst✝ : Preorder α\nt : OrderTopology α\na : α\nhu : ∃ u, a < u\nhl : ∃ l, l < a\n⊢ 𝓝 a = ⨅ (l : α) (_ : l < a) (u : α) (_ : a < u), 𝓟 (Ioo l u)",
"tactic": "simp only [nhds_eq_order, ← inf_biInf, ← biInf_inf, *, ← inf_principal, ← Ioi_inter_Iio]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : TopologicalSpace α\ninst✝ : Preorder α\nt : OrderTopology α\na : α\nhu : ∃ u, a < u\nhl : ∃ l, l < a\n⊢ ((⨅ (b : α) (_ : b ∈ Iio a), 𝓟 (Ioi b)) ⊓ ⨅ (b : α) (_ : b ∈ Ioi a), 𝓟 (Iio b)) =\n (⨅ (i : α) (_ : i < a), 𝓟 (Ioi i)) ⊓ ⨅ (i : α) (_ : a < i), 𝓟 (Iio i)",
"tactic": "rfl"
}
] |
[
978,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
976,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.infinite_iff
|
[
{
"state_after": "no goals",
"state_before": "α✝ β α : Type u\n⊢ Infinite α ↔ ℵ₀ ≤ (#α)",
"tactic": "rw [← not_lt, lt_aleph0_iff_finite, not_finite_iff_infinite]"
}
] |
[
1606,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1605,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.predAbove_right_monotone
|
[
{
"state_after": "n m : ℕ\np : Fin n\na b : Fin (n + 1)\nH : a ≤ b\n⊢ (if h : ↑castSucc p < a then pred a (_ : a ≠ 0) else castLT a (_ : ↑a < n)) ≤\n if h : ↑castSucc p < b then pred b (_ : b ≠ 0) else castLT b (_ : ↑b < n)",
"state_before": "n m : ℕ\np : Fin n\na b : Fin (n + 1)\nH : a ≤ b\n⊢ predAbove p a ≤ predAbove p b",
"tactic": "dsimp [predAbove]"
},
{
"state_after": "case inl.inl\nn m : ℕ\np : Fin n\na b : Fin (n + 1)\nH : a ≤ b\nha : ↑castSucc p < a\nhb : ↑castSucc p < b\n⊢ pred a (_ : a ≠ 0) ≤ pred b (_ : b ≠ 0)\n\ncase inl.inr\nn m : ℕ\np : Fin n\na b : Fin (n + 1)\nH : a ≤ b\nha : ↑castSucc p < a\nhb : ¬↑castSucc p < b\n⊢ pred a (_ : a ≠ 0) ≤ castLT b (_ : ↑b < n)\n\ncase inr.inl\nn m : ℕ\np : Fin n\na b : Fin (n + 1)\nH : a ≤ b\nha : ¬↑castSucc p < a\nhb : ↑castSucc p < b\n⊢ castLT a (_ : ↑a < n) ≤ pred b (_ : b ≠ 0)\n\ncase inr.inr\nn m : ℕ\np : Fin n\na b : Fin (n + 1)\nH : a ≤ b\nha : ¬↑castSucc p < a\nhb : ¬↑castSucc p < b\n⊢ castLT a (_ : ↑a < n) ≤ castLT b (_ : ↑b < n)",
"state_before": "n m : ℕ\np : Fin n\na b : Fin (n + 1)\nH : a ≤ b\n⊢ (if h : ↑castSucc p < a then pred a (_ : a ≠ 0) else castLT a (_ : ↑a < n)) ≤\n if h : ↑castSucc p < b then pred b (_ : b ≠ 0) else castLT b (_ : ↑b < n)",
"tactic": "split_ifs with ha hb hb"
},
{
"state_after": "case inl.inl\nn m : ℕ\np : Fin n\na b : Fin (n + 1)\nH : a ≤ b\nha : ↑castSucc p < a\nhb : ↑castSucc p < b\n⊢ ↑a - 1 ≤ ↑b - 1\n\ncase inl.inr\nn m : ℕ\np : Fin n\na b : Fin (n + 1)\nH : a ≤ b\nha : ↑castSucc p < a\nhb : ¬↑castSucc p < b\n⊢ ↑a - 1 ≤ ↑(castLT b (_ : ↑b < n))\n\ncase inr.inl\nn m : ℕ\np : Fin n\na b : Fin (n + 1)\nH : a ≤ b\nha : ¬↑castSucc p < a\nhb : ↑castSucc p < b\n⊢ ↑(castLT a (_ : ↑a < n)) ≤ ↑b - 1\n\ncase inr.inr\nn m : ℕ\np : Fin n\na b : Fin (n + 1)\nH : a ≤ b\nha : ¬↑castSucc p < a\nhb : ¬↑castSucc p < b\n⊢ ↑(castLT a (_ : ↑a < n)) ≤ ↑(castLT b (_ : ↑b < n))",
"state_before": "case inl.inl\nn m : ℕ\np : Fin n\na b : Fin (n + 1)\nH : a ≤ b\nha : ↑castSucc p < a\nhb : ↑castSucc p < b\n⊢ pred a (_ : a ≠ 0) ≤ pred b (_ : b ≠ 0)\n\ncase inl.inr\nn m : ℕ\np : Fin n\na b : Fin (n + 1)\nH : a ≤ b\nha : ↑castSucc p < a\nhb : ¬↑castSucc p < b\n⊢ pred a (_ : a ≠ 0) ≤ castLT b (_ : ↑b < n)\n\ncase inr.inl\nn m : ℕ\np : Fin n\na b : Fin (n + 1)\nH : a ≤ b\nha : ¬↑castSucc p < a\nhb : ↑castSucc p < b\n⊢ castLT a (_ : ↑a < n) ≤ pred b (_ : b ≠ 0)\n\ncase inr.inr\nn m : ℕ\np : Fin n\na b : Fin (n + 1)\nH : a ≤ b\nha : ¬↑castSucc p < a\nhb : ¬↑castSucc p < b\n⊢ castLT a (_ : ↑a < n) ≤ castLT b (_ : ↑b < n)",
"tactic": "all_goals simp only [le_iff_val_le_val, coe_pred]"
},
{
"state_after": "case inr.inr\nn m : ℕ\np : Fin n\na b : Fin (n + 1)\nH : a ≤ b\nha : ¬↑castSucc p < a\nhb : ¬↑castSucc p < b\n⊢ ↑(castLT a (_ : ↑a < n)) ≤ ↑(castLT b (_ : ↑b < n))",
"state_before": "case inr.inr\nn m : ℕ\np : Fin n\na b : Fin (n + 1)\nH : a ≤ b\nha : ¬↑castSucc p < a\nhb : ¬↑castSucc p < b\n⊢ castLT a (_ : ↑a < n) ≤ castLT b (_ : ↑b < n)",
"tactic": "simp only [le_iff_val_le_val, coe_pred]"
},
{
"state_after": "no goals",
"state_before": "case inl.inl\nn m : ℕ\np : Fin n\na b : Fin (n + 1)\nH : a ≤ b\nha : ↑castSucc p < a\nhb : ↑castSucc p < b\n⊢ ↑a - 1 ≤ ↑b - 1",
"tactic": "exact pred_le_pred H"
},
{
"state_after": "no goals",
"state_before": "case inl.inr\nn m : ℕ\np : Fin n\na b : Fin (n + 1)\nH : a ≤ b\nha : ↑castSucc p < a\nhb : ¬↑castSucc p < b\n⊢ ↑a - 1 ≤ ↑(castLT b (_ : ↑b < n))",
"tactic": "calc\n _ ≤ _ := Nat.pred_le _\n _ ≤ _ := H"
},
{
"state_after": "case inr.inl\nn m : ℕ\np : Fin n\na b : Fin (n + 1)\nH : a ≤ b\nha✝ : ¬↑castSucc p < a\nhb : ↑castSucc p < b\nha : a ≤ ↑castSucc p\n⊢ ↑(castLT a (_ : ↑a < n)) ≤ ↑b - 1",
"state_before": "case inr.inl\nn m : ℕ\np : Fin n\na b : Fin (n + 1)\nH : a ≤ b\nha : ¬↑castSucc p < a\nhb : ↑castSucc p < b\n⊢ ↑(castLT a (_ : ↑a < n)) ≤ ↑b - 1",
"tactic": "simp at ha"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nn m : ℕ\np : Fin n\na b : Fin (n + 1)\nH : a ≤ b\nha✝ : ¬↑castSucc p < a\nhb : ↑castSucc p < b\nha : a ≤ ↑castSucc p\n⊢ ↑(castLT a (_ : ↑a < n)) ≤ ↑b - 1",
"tactic": "exact le_pred_of_lt (lt_of_le_of_lt ha hb)"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nn m : ℕ\np : Fin n\na b : Fin (n + 1)\nH : a ≤ b\nha : ¬↑castSucc p < a\nhb : ¬↑castSucc p < b\n⊢ ↑(castLT a (_ : ↑a < n)) ≤ ↑(castLT b (_ : ↑b < n))",
"tactic": "exact H"
}
] |
[
2277,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2267,
1
] |
Mathlib/NumberTheory/FermatPsp.lean
|
FermatPsp.coprime_of_probablePrime
|
[
{
"state_after": "case pos\nn b : ℕ\nh : ProbablePrime n b\nh₁ : 1 ≤ n\nh₂ : 1 ≤ b\nh₃ : 2 ≤ n\n⊢ Nat.coprime n b\n\ncase neg\nn b : ℕ\nh : ProbablePrime n b\nh₁ : 1 ≤ n\nh₂ : 1 ≤ b\nh₃ : ¬2 ≤ n\n⊢ Nat.coprime n b",
"state_before": "n b : ℕ\nh : ProbablePrime n b\nh₁ : 1 ≤ n\nh₂ : 1 ≤ b\n⊢ Nat.coprime n b",
"tactic": "by_cases h₃ : 2 ≤ n"
},
{
"state_after": "case pos.H\nn b : ℕ\nh : ProbablePrime n b\nh₁ : 1 ≤ n\nh₂ : 1 ≤ b\nh₃ : 2 ≤ n\n⊢ ∀ (k : ℕ), Nat.Prime k → k ∣ n → ¬k ∣ b",
"state_before": "case pos\nn b : ℕ\nh : ProbablePrime n b\nh₁ : 1 ≤ n\nh₂ : 1 ≤ b\nh₃ : 2 ≤ n\n⊢ Nat.coprime n b",
"tactic": "apply Nat.coprime_of_dvd"
},
{
"state_after": "case pos.H.intro.intro\nk : ℕ\nhk : Nat.Prime k\nm : ℕ\nh₁ : 1 ≤ k * m\nh₃ : 2 ≤ k * m\nj : ℕ\nh₂ : 1 ≤ k * j\nh : ProbablePrime (k * m) (k * j)\n⊢ False",
"state_before": "case pos.H\nn b : ℕ\nh : ProbablePrime n b\nh₁ : 1 ≤ n\nh₂ : 1 ≤ b\nh₃ : 2 ≤ n\n⊢ ∀ (k : ℕ), Nat.Prime k → k ∣ n → ¬k ∣ b",
"tactic": "rintro k hk ⟨m, rfl⟩ ⟨j, rfl⟩"
},
{
"state_after": "case pos.H.intro.intro\nk : ℕ\nhk : Nat.Prime k\nm : ℕ\nh₁ : 1 ≤ k * m\nh₃ : 2 ≤ k * m\nj : ℕ\nh₂ : 1 ≤ k * j\nh : ProbablePrime (k * m) (k * j)\n⊢ k ∣ 1",
"state_before": "case pos.H.intro.intro\nk : ℕ\nhk : Nat.Prime k\nm : ℕ\nh₁ : 1 ≤ k * m\nh₃ : 2 ≤ k * m\nj : ℕ\nh₂ : 1 ≤ k * j\nh : ProbablePrime (k * m) (k * j)\n⊢ False",
"tactic": "apply Nat.Prime.not_dvd_one hk"
},
{
"state_after": "case pos.H.intro.intro\nk : ℕ\nhk : Nat.Prime k\nm : ℕ\nh₁ : 1 ≤ k * m\nh₃ : 2 ≤ k * m\nj : ℕ\nh₂ : 1 ≤ k * j\nh : k ∣ (k * j) ^ (k * m - 1) - 1\n⊢ k ∣ 1",
"state_before": "case pos.H.intro.intro\nk : ℕ\nhk : Nat.Prime k\nm : ℕ\nh₁ : 1 ≤ k * m\nh₃ : 2 ≤ k * m\nj : ℕ\nh₂ : 1 ≤ k * j\nh : ProbablePrime (k * m) (k * j)\n⊢ k ∣ 1",
"tactic": "replace h := dvd_of_mul_right_dvd h"
},
{
"state_after": "case pos.H.intro.intro\nk : ℕ\nhk : Nat.Prime k\nm : ℕ\nh₁ : 1 ≤ k * m\nh₃ : 2 ≤ k * m\nj : ℕ\nh₂ : 1 ≤ k * j\nh : k ∣ (k * j) ^ (k * m - 1) - 1\n⊢ k ∣ (k * j) ^ (k * m - 1)",
"state_before": "case pos.H.intro.intro\nk : ℕ\nhk : Nat.Prime k\nm : ℕ\nh₁ : 1 ≤ k * m\nh₃ : 2 ≤ k * m\nj : ℕ\nh₂ : 1 ≤ k * j\nh : k ∣ (k * j) ^ (k * m - 1) - 1\n⊢ k ∣ 1",
"tactic": "rw [Nat.dvd_add_iff_right h, Nat.sub_add_cancel (Nat.one_le_pow _ _ h₂)]"
},
{
"state_after": "case pos.H.intro.intro\nk : ℕ\nhk : Nat.Prime k\nm : ℕ\nh₁ : 1 ≤ k * m\nh₃ : 2 ≤ k * m\nj : ℕ\nh₂ : 1 ≤ k * j\nh : k ∣ (k * j) ^ (k * m - 1) - 1\n⊢ k * m - 1 ≠ 0",
"state_before": "case pos.H.intro.intro\nk : ℕ\nhk : Nat.Prime k\nm : ℕ\nh₁ : 1 ≤ k * m\nh₃ : 2 ≤ k * m\nj : ℕ\nh₂ : 1 ≤ k * j\nh : k ∣ (k * j) ^ (k * m - 1) - 1\n⊢ k ∣ (k * j) ^ (k * m - 1)",
"tactic": "refine' dvd_of_mul_right_dvd (dvd_pow_self (k * j) _)"
},
{
"state_after": "no goals",
"state_before": "case pos.H.intro.intro\nk : ℕ\nhk : Nat.Prime k\nm : ℕ\nh₁ : 1 ≤ k * m\nh₃ : 2 ≤ k * m\nj : ℕ\nh₂ : 1 ≤ k * j\nh : k ∣ (k * j) ^ (k * m - 1) - 1\n⊢ k * m - 1 ≠ 0",
"tactic": "linarith [tsub_pos_of_lt (one_lt_two.trans_le h₃)]"
},
{
"state_after": "case neg\nn b : ℕ\nh : ProbablePrime n b\nh₁ : 1 ≤ n\nh₂ : 1 ≤ b\nh₃ : ¬2 ≤ n\n⊢ Nat.coprime 1 b",
"state_before": "case neg\nn b : ℕ\nh : ProbablePrime n b\nh₁ : 1 ≤ n\nh₂ : 1 ≤ b\nh₃ : ¬2 ≤ n\n⊢ Nat.coprime n b",
"tactic": "rw [show n = 1 by linarith]"
},
{
"state_after": "no goals",
"state_before": "case neg\nn b : ℕ\nh : ProbablePrime n b\nh₁ : 1 ≤ n\nh₂ : 1 ≤ b\nh₃ : ¬2 ≤ n\n⊢ Nat.coprime 1 b",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "n b : ℕ\nh : ProbablePrime n b\nh₁ : 1 ≤ n\nh₂ : 1 ≤ b\nh₃ : ¬2 ≤ n\n⊢ n = 1",
"tactic": "linarith"
}
] |
[
104,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
80,
1
] |
Mathlib/CategoryTheory/Generator.lean
|
CategoryTheory.IsDetector.def
|
[] |
[
495,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
493,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.multiset_prod_le_inf
|
[
{
"state_after": "case refine'_1\nR : Type u\nι : Type ?u.260476\ninst✝ : CommSemiring R\nI J K L : Ideal R\ns : Multiset (Ideal R)\n⊢ Multiset.prod 0 ≤ Multiset.inf 0\n\ncase refine'_2\nR : Type u\nι : Type ?u.260476\ninst✝ : CommSemiring R\nI J K L : Ideal R\ns : Multiset (Ideal R)\n⊢ ∀ ⦃a : Ideal R⦄ {s : Multiset (Ideal R)},\n Multiset.prod s ≤ Multiset.inf s → Multiset.prod (a ::ₘ s) ≤ Multiset.inf (a ::ₘ s)",
"state_before": "R : Type u\nι : Type ?u.260476\ninst✝ : CommSemiring R\nI J K L : Ideal R\ns : Multiset (Ideal R)\n⊢ Multiset.prod s ≤ Multiset.inf s",
"tactic": "refine' s.induction_on _ _"
},
{
"state_after": "case refine'_2\nR : Type u\nι : Type ?u.260476\ninst✝ : CommSemiring R\nI J K L : Ideal R\ns✝ : Multiset (Ideal R)\na : Ideal R\ns : Multiset (Ideal R)\nih : Multiset.prod s ≤ Multiset.inf s\n⊢ Multiset.prod (a ::ₘ s) ≤ Multiset.inf (a ::ₘ s)",
"state_before": "case refine'_2\nR : Type u\nι : Type ?u.260476\ninst✝ : CommSemiring R\nI J K L : Ideal R\ns : Multiset (Ideal R)\n⊢ ∀ ⦃a : Ideal R⦄ {s : Multiset (Ideal R)},\n Multiset.prod s ≤ Multiset.inf s → Multiset.prod (a ::ₘ s) ≤ Multiset.inf (a ::ₘ s)",
"tactic": "intro a s ih"
},
{
"state_after": "case refine'_2\nR : Type u\nι : Type ?u.260476\ninst✝ : CommSemiring R\nI J K L : Ideal R\ns✝ : Multiset (Ideal R)\na : Ideal R\ns : Multiset (Ideal R)\nih : Multiset.prod s ≤ Multiset.inf s\n⊢ a * Multiset.prod s ≤ a ⊓ Multiset.inf s",
"state_before": "case refine'_2\nR : Type u\nι : Type ?u.260476\ninst✝ : CommSemiring R\nI J K L : Ideal R\ns✝ : Multiset (Ideal R)\na : Ideal R\ns : Multiset (Ideal R)\nih : Multiset.prod s ≤ Multiset.inf s\n⊢ Multiset.prod (a ::ₘ s) ≤ Multiset.inf (a ::ₘ s)",
"tactic": "rw [Multiset.prod_cons, Multiset.inf_cons]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nR : Type u\nι : Type ?u.260476\ninst✝ : CommSemiring R\nI J K L : Ideal R\ns✝ : Multiset (Ideal R)\na : Ideal R\ns : Multiset (Ideal R)\nih : Multiset.prod s ≤ Multiset.inf s\n⊢ a * Multiset.prod s ≤ a ⊓ Multiset.inf s",
"tactic": "exact le_trans mul_le_inf (inf_le_inf le_rfl ih)"
},
{
"state_after": "case refine'_1\nR : Type u\nι : Type ?u.260476\ninst✝ : CommSemiring R\nI J K L : Ideal R\ns : Multiset (Ideal R)\n⊢ Multiset.prod 0 ≤ ⊤",
"state_before": "case refine'_1\nR : Type u\nι : Type ?u.260476\ninst✝ : CommSemiring R\nI J K L : Ideal R\ns : Multiset (Ideal R)\n⊢ Multiset.prod 0 ≤ Multiset.inf 0",
"tactic": "rw [Multiset.inf_zero]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nR : Type u\nι : Type ?u.260476\ninst✝ : CommSemiring R\nI J K L : Ideal R\ns : Multiset (Ideal R)\n⊢ Multiset.prod 0 ≤ ⊤",
"tactic": "exact le_top"
}
] |
[
665,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
658,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
Monotone.const_mul'
|
[] |
[
1289,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1288,
1
] |
Mathlib/Data/Set/Basic.lean
|
Disjoint.inter_eq
|
[] |
[
1526,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1525,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
LinearMap.restrict_apply
|
[] |
[
238,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
236,
1
] |
Mathlib/ModelTheory/Basic.lean
|
FirstOrder.Language.Hom.ext_iff
|
[] |
[
530,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
529,
1
] |
Mathlib/Logic/Equiv/Basic.lean
|
Equiv.sumCongr_trans
|
[
{
"state_after": "case H\nα₁ : Type u_1\nβ₁ : Type u_2\nα₂ : Type u_3\nβ₂ : Type u_4\nγ₁ : Type u_5\nγ₂ : Type u_6\ne : α₁ ≃ β₁\nf : α₂ ≃ β₂\ng : β₁ ≃ γ₁\nh : β₂ ≃ γ₂\ni : α₁ ⊕ α₂\n⊢ ↑((sumCongr e f).trans (sumCongr g h)) i = ↑(sumCongr (e.trans g) (f.trans h)) i",
"state_before": "α₁ : Type u_1\nβ₁ : Type u_2\nα₂ : Type u_3\nβ₂ : Type u_4\nγ₁ : Type u_5\nγ₂ : Type u_6\ne : α₁ ≃ β₁\nf : α₂ ≃ β₂\ng : β₁ ≃ γ₁\nh : β₂ ≃ γ₂\n⊢ (sumCongr e f).trans (sumCongr g h) = sumCongr (e.trans g) (f.trans h)",
"tactic": "ext i"
},
{
"state_after": "no goals",
"state_before": "case H\nα₁ : Type u_1\nβ₁ : Type u_2\nα₂ : Type u_3\nβ₂ : Type u_4\nγ₁ : Type u_5\nγ₂ : Type u_6\ne : α₁ ≃ β₁\nf : α₂ ≃ β₂\ng : β₁ ≃ γ₁\nh : β₂ ≃ γ₂\ni : α₁ ⊕ α₂\n⊢ ↑((sumCongr e f).trans (sumCongr g h)) i = ↑(sumCongr (e.trans g) (f.trans h)) i",
"tactic": "cases i <;> rfl"
}
] |
[
282,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
279,
1
] |
Mathlib/Data/List/Cycle.lean
|
Cycle.reverse_coe
|
[] |
[
553,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
552,
1
] |
Mathlib/Data/List/Count.lean
|
List.count_replicate
|
[
{
"state_after": "case inl\nα : Type u_1\nl : List α\ninst✝ : DecidableEq α\na b : α\nn : ℕ\nh✝ : a = b\n⊢ count a (replicate n b) = n\n\ncase inr\nα : Type u_1\nl : List α\ninst✝ : DecidableEq α\na b : α\nn : ℕ\nh✝ : ¬a = b\n⊢ count a (replicate n b) = 0",
"state_before": "α : Type u_1\nl : List α\ninst✝ : DecidableEq α\na b : α\nn : ℕ\n⊢ count a (replicate n b) = if a = b then n else 0",
"tactic": "split"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nl : List α\ninst✝ : DecidableEq α\na b : α\nn : ℕ\nh✝ : a = b\n⊢ count a (replicate n b) = n\n\ncase inr\nα : Type u_1\nl : List α\ninst✝ : DecidableEq α\na b : α\nn : ℕ\nh✝ : ¬a = b\n⊢ count a (replicate n b) = 0",
"tactic": "exacts [‹a = b› ▸ count_replicate_self _ _, count_eq_zero.2 <| mt eq_of_mem_replicate ‹a ≠ b›]"
}
] |
[
279,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
277,
1
] |
Mathlib/Data/Seq/Seq.lean
|
Stream'.Seq.get?_zip
|
[] |
[
604,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
602,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.L1.integral_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nF : Type ?u.411128\n𝕜 : Type ?u.411131\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\nm : MeasurableSpace α\nμ : Measure α\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : SMulCommClass ℝ 𝕜 E\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace E\nf : { x // x ∈ Lp E 1 }\n⊢ integral f = ↑integralCLM f",
"tactic": "simp only [integral]"
}
] |
[
680,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
679,
1
] |
Mathlib/Data/List/Basic.lean
|
List.foldl_fixed
|
[] |
[
2431,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2430,
1
] |
Mathlib/Algebra/Order/Monoid/Canonical/Defs.lean
|
min_one
|
[] |
[
384,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
383,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.StronglyMeasurable.nnnorm
|
[] |
[
829,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
827,
11
] |
Mathlib/Order/Filter/Extr.lean
|
Filter.EventuallyEq.isMaxFilter_iff
|
[] |
[
652,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
650,
1
] |
Mathlib/Data/PFun.lean
|
PFun.dom_comp
|
[
{
"state_after": "case h\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.54232\nε : Type ?u.54235\nι : Type ?u.54238\nf✝ : α →. β\nf : β →. γ\ng : α →. β\nx✝ : α\n⊢ x✝ ∈ Dom (comp f g) ↔ x✝ ∈ preimage g (Dom f)",
"state_before": "α : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.54232\nε : Type ?u.54235\nι : Type ?u.54238\nf✝ : α →. β\nf : β →. γ\ng : α →. β\n⊢ Dom (comp f g) = preimage g (Dom f)",
"tactic": "ext"
},
{
"state_after": "case h\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.54232\nε : Type ?u.54235\nι : Type ?u.54238\nf✝ : α →. β\nf : β →. γ\ng : α →. β\nx✝ : α\n⊢ (∃ y a, a ∈ g x✝ ∧ y ∈ f a) ↔ ∃ y x, x ∈ f y ∧ y ∈ g x✝",
"state_before": "case h\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.54232\nε : Type ?u.54235\nι : Type ?u.54238\nf✝ : α →. β\nf : β →. γ\ng : α →. β\nx✝ : α\n⊢ x✝ ∈ Dom (comp f g) ↔ x✝ ∈ preimage g (Dom f)",
"tactic": "simp_rw [mem_preimage, mem_dom, comp_apply, Part.mem_bind_iff, ← exists_and_right]"
},
{
"state_after": "case h\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.54232\nε : Type ?u.54235\nι : Type ?u.54238\nf✝ : α →. β\nf : β →. γ\ng : α →. β\nx✝ : α\n⊢ (∃ b a, b ∈ g x✝ ∧ a ∈ f b) ↔ ∃ y x, x ∈ f y ∧ y ∈ g x✝",
"state_before": "case h\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.54232\nε : Type ?u.54235\nι : Type ?u.54238\nf✝ : α →. β\nf : β →. γ\ng : α →. β\nx✝ : α\n⊢ (∃ y a, a ∈ g x✝ ∧ y ∈ f a) ↔ ∃ y x, x ∈ f y ∧ y ∈ g x✝",
"tactic": "rw [exists_comm]"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nδ : Type ?u.54232\nε : Type ?u.54235\nι : Type ?u.54238\nf✝ : α →. β\nf : β →. γ\ng : α →. β\nx✝ : α\n⊢ (∃ b a, b ∈ g x✝ ∧ a ∈ f b) ↔ ∃ y x, x ∈ f y ∧ y ∈ g x✝",
"tactic": "simp_rw [and_comm]"
}
] |
[
601,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
597,
1
] |
Mathlib/Algebra/Category/ModuleCat/Projective.lean
|
IsProjective.iff_projective
|
[
{
"state_after": "case refine'_1\nR : Type u\ninst✝² : Ring R\nP : Type (max u v)\ninst✝¹ : AddCommGroup P\ninst✝ : Module R P\nh : Module.Projective R P\n⊢ Projective (of R P)\n\ncase refine'_2\nR : Type u\ninst✝² : Ring R\nP : Type (max u v)\ninst✝¹ : AddCommGroup P\ninst✝ : Module R P\nh : Projective (of R P)\n⊢ Module.Projective R P",
"state_before": "R : Type u\ninst✝² : Ring R\nP : Type (max u v)\ninst✝¹ : AddCommGroup P\ninst✝ : Module R P\n⊢ Module.Projective R P ↔ Projective (of R P)",
"tactic": "refine' ⟨fun h => _, fun h => _⟩"
},
{
"state_after": "case refine'_1\nR : Type u\ninst✝² : Ring R\nP : Type (max u v)\ninst✝¹ : AddCommGroup P\ninst✝ : Module R P\nh : Module.Projective R P\nthis : Module.Projective R ↑(of R P) := h\n⊢ Projective (of R P)",
"state_before": "case refine'_1\nR : Type u\ninst✝² : Ring R\nP : Type (max u v)\ninst✝¹ : AddCommGroup P\ninst✝ : Module R P\nh : Module.Projective R P\n⊢ Projective (of R P)",
"tactic": "letI : Module.Projective R (ModuleCat.of R P) := h"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nR : Type u\ninst✝² : Ring R\nP : Type (max u v)\ninst✝¹ : AddCommGroup P\ninst✝ : Module R P\nh : Module.Projective R P\nthis : Module.Projective R ↑(of R P) := h\n⊢ Projective (of R P)",
"tactic": "exact ⟨fun E X epi => Module.projective_lifting_property _ _\n ((ModuleCat.epi_iff_surjective _).mp epi)⟩"
},
{
"state_after": "case refine'_2\nR : Type u\ninst✝² : Ring R\nP : Type (max u v)\ninst✝¹ : AddCommGroup P\ninst✝ : Module R P\nh : Projective (of R P)\n⊢ ∀ {M : Type (max v u)} {N : Type (max u v)} [inst : AddCommGroup M] [inst_1 : AddCommGroup N] [inst_2 : Module R M]\n [inst_3 : Module R N] (f : M →ₗ[R] N) (g : P →ₗ[R] N), Function.Surjective ↑f → ∃ h, comp f h = g",
"state_before": "case refine'_2\nR : Type u\ninst✝² : Ring R\nP : Type (max u v)\ninst✝¹ : AddCommGroup P\ninst✝ : Module R P\nh : Projective (of R P)\n⊢ Module.Projective R P",
"tactic": "refine' Module.Projective.of_lifting_property.{u,v} _"
},
{
"state_after": "case refine'_2\nR : Type u\ninst✝² : Ring R\nP : Type (max u v)\ninst✝¹ : AddCommGroup P\ninst✝ : Module R P\nh : Projective (of R P)\nE : Type (max v u)\nX : Type (max u v)\nmE : AddCommGroup E\nmX : AddCommGroup X\nsE : Module R E\nsX : Module R X\nf : E →ₗ[R] X\ng : P →ₗ[R] X\ns : Function.Surjective ↑f\n⊢ ∃ h, comp f h = g",
"state_before": "case refine'_2\nR : Type u\ninst✝² : Ring R\nP : Type (max u v)\ninst✝¹ : AddCommGroup P\ninst✝ : Module R P\nh : Projective (of R P)\n⊢ ∀ {M : Type (max v u)} {N : Type (max u v)} [inst : AddCommGroup M] [inst_1 : AddCommGroup N] [inst_2 : Module R M]\n [inst_3 : Module R N] (f : M →ₗ[R] N) (g : P →ₗ[R] N), Function.Surjective ↑f → ∃ h, comp f h = g",
"tactic": "intro E X mE mX sE sX f g s"
},
{
"state_after": "case refine'_2\nR : Type u\ninst✝² : Ring R\nP : Type (max u v)\ninst✝¹ : AddCommGroup P\ninst✝ : Module R P\nh : Projective (of R P)\nE : Type (max v u)\nX : Type (max u v)\nmE : AddCommGroup E\nmX : AddCommGroup X\nsE : Module R E\nsX : Module R X\nf : E →ₗ[R] X\ng : P →ₗ[R] X\ns : Function.Surjective ↑f\nthis : Epi (↟f)\n⊢ ∃ h, comp f h = g",
"state_before": "case refine'_2\nR : Type u\ninst✝² : Ring R\nP : Type (max u v)\ninst✝¹ : AddCommGroup P\ninst✝ : Module R P\nh : Projective (of R P)\nE : Type (max v u)\nX : Type (max u v)\nmE : AddCommGroup E\nmX : AddCommGroup X\nsE : Module R E\nsX : Module R X\nf : E →ₗ[R] X\ng : P →ₗ[R] X\ns : Function.Surjective ↑f\n⊢ ∃ h, comp f h = g",
"tactic": "haveI : Epi (↟f) := (ModuleCat.epi_iff_surjective (↟f)).mpr s"
},
{
"state_after": "case refine'_2\nR : Type u\ninst✝² : Ring R\nP : Type (max u v)\ninst✝¹ : AddCommGroup P\ninst✝ : Module R P\nh : Projective (of R P)\nE : Type (max v u)\nX : Type (max u v)\nmE : AddCommGroup E\nmX : AddCommGroup X\nsE : Module R E\nsX : Module R X\nf : E →ₗ[R] X\ng : P →ₗ[R] X\ns : Function.Surjective ↑f\nthis✝ : Epi (↟f)\nthis : Projective (of R P) := h\n⊢ ∃ h, comp f h = g",
"state_before": "case refine'_2\nR : Type u\ninst✝² : Ring R\nP : Type (max u v)\ninst✝¹ : AddCommGroup P\ninst✝ : Module R P\nh : Projective (of R P)\nE : Type (max v u)\nX : Type (max u v)\nmE : AddCommGroup E\nmX : AddCommGroup X\nsE : Module R E\nsX : Module R X\nf : E →ₗ[R] X\ng : P →ₗ[R] X\ns : Function.Surjective ↑f\nthis : Epi (↟f)\n⊢ ∃ h, comp f h = g",
"tactic": "letI : Projective (ModuleCat.of R P) := h"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nR : Type u\ninst✝² : Ring R\nP : Type (max u v)\ninst✝¹ : AddCommGroup P\ninst✝ : Module R P\nh : Projective (of R P)\nE : Type (max v u)\nX : Type (max u v)\nmE : AddCommGroup E\nmX : AddCommGroup X\nsE : Module R E\nsX : Module R X\nf : E →ₗ[R] X\ng : P →ₗ[R] X\ns : Function.Surjective ↑f\nthis✝ : Epi (↟f)\nthis : Projective (of R P) := h\n⊢ ∃ h, comp f h = g",
"tactic": "exact ⟨Projective.factorThru (↟g) (↟f), Projective.factorThru_comp (↟g) (↟f)⟩"
}
] |
[
44,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
34,
1
] |
Mathlib/Topology/Sheaves/Stalks.lean
|
TopCat.Presheaf.stalkPushforward.comp
|
[
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasColimits C\nX Y Z : TopCat\nℱ : Presheaf C X\nf : X ⟶ Y\ng : Y ⟶ Z\nx : ↑X\n⊢ stalkPushforward C (f ≫ g) ℱ x = stalkPushforward C g (f _* ℱ) ((forget TopCat).map f x) ≫ stalkPushforward C f ℱ x",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasColimits C\nX Y Z : TopCat\nℱ : Presheaf C X\nf : X ⟶ Y\ng : Y ⟶ Z\nx : ↑X\n⊢ stalkPushforward C (f ≫ g) ℱ x = stalkPushforward C g (f _* ℱ) ((forget TopCat).map f x) ≫ stalkPushforward C f ℱ x",
"tactic": "change (_ : colimit _ ⟶ _) = (_ : colimit _ ⟶ _)"
},
{
"state_after": "case w\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasColimits C\nX Y Z : TopCat\nℱ : Presheaf C X\nf : X ⟶ Y\ng : Y ⟶ Z\nx : ↑X\nU : (OpenNhds ((forget TopCat).map (f ≫ g) x))ᵒᵖ\n⊢ colimit.ι\n (((whiskeringLeft (OpenNhds ((forget TopCat).map (f ≫ g) x))ᵒᵖ (Opens ↑Z)ᵒᵖ C).obj\n (Functor.op (OpenNhds.inclusion ((forget TopCat).map (f ≫ g) x)))).obj\n ((f ≫ g) _* ℱ))\n U ≫\n stalkPushforward C (f ≫ g) ℱ x =\n colimit.ι\n (((whiskeringLeft (OpenNhds ((forget TopCat).map (f ≫ g) x))ᵒᵖ (Opens ↑Z)ᵒᵖ C).obj\n (Functor.op (OpenNhds.inclusion ((forget TopCat).map (f ≫ g) x)))).obj\n ((f ≫ g) _* ℱ))\n U ≫\n stalkPushforward C g (f _* ℱ) ((forget TopCat).map f x) ≫ stalkPushforward C f ℱ x",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : HasColimits C\nX Y Z : TopCat\nℱ : Presheaf C X\nf : X ⟶ Y\ng : Y ⟶ Z\nx : ↑X\n⊢ stalkPushforward C (f ≫ g) ℱ x = stalkPushforward C g (f _* ℱ) ((forget TopCat).map f x) ≫ stalkPushforward C f ℱ x",
"tactic": "ext U"
},
{
"state_after": "case w.mk.mk.mk\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasColimits C\nX Y Z : TopCat\nℱ : Presheaf C X\nf : X ⟶ Y\ng : Y ⟶ Z\nx : ↑X\ncarrier✝ : Set ↑Z\nis_open'✝ : IsOpen carrier✝\nproperty✝ : (forget TopCat).map (f ≫ g) x ∈ { carrier := carrier✝, is_open' := is_open'✝ }\n⊢ colimit.ι\n (((whiskeringLeft (OpenNhds ((forget TopCat).map (f ≫ g) x))ᵒᵖ (Opens ↑Z)ᵒᵖ C).obj\n (Functor.op (OpenNhds.inclusion ((forget TopCat).map (f ≫ g) x)))).obj\n ((f ≫ g) _* ℱ))\n { unop := { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ } } ≫\n stalkPushforward C (f ≫ g) ℱ x =\n colimit.ι\n (((whiskeringLeft (OpenNhds ((forget TopCat).map (f ≫ g) x))ᵒᵖ (Opens ↑Z)ᵒᵖ C).obj\n (Functor.op (OpenNhds.inclusion ((forget TopCat).map (f ≫ g) x)))).obj\n ((f ≫ g) _* ℱ))\n { unop := { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ } } ≫\n stalkPushforward C g (f _* ℱ) ((forget TopCat).map f x) ≫ stalkPushforward C f ℱ x",
"state_before": "case w\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasColimits C\nX Y Z : TopCat\nℱ : Presheaf C X\nf : X ⟶ Y\ng : Y ⟶ Z\nx : ↑X\nU : (OpenNhds ((forget TopCat).map (f ≫ g) x))ᵒᵖ\n⊢ colimit.ι\n (((whiskeringLeft (OpenNhds ((forget TopCat).map (f ≫ g) x))ᵒᵖ (Opens ↑Z)ᵒᵖ C).obj\n (Functor.op (OpenNhds.inclusion ((forget TopCat).map (f ≫ g) x)))).obj\n ((f ≫ g) _* ℱ))\n U ≫\n stalkPushforward C (f ≫ g) ℱ x =\n colimit.ι\n (((whiskeringLeft (OpenNhds ((forget TopCat).map (f ≫ g) x))ᵒᵖ (Opens ↑Z)ᵒᵖ C).obj\n (Functor.op (OpenNhds.inclusion ((forget TopCat).map (f ≫ g) x)))).obj\n ((f ≫ g) _* ℱ))\n U ≫\n stalkPushforward C g (f _* ℱ) ((forget TopCat).map f x) ≫ stalkPushforward C f ℱ x",
"tactic": "rcases U with ⟨⟨_, _⟩, _⟩"
},
{
"state_after": "case w.mk.mk.mk\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasColimits C\nX Y Z : TopCat\nℱ : Presheaf C X\nf : X ⟶ Y\ng : Y ⟶ Z\nx : ↑X\ncarrier✝ : Set ↑Z\nis_open'✝ : IsOpen carrier✝\nproperty✝ : (forget TopCat).map (f ≫ g) x ∈ { carrier := carrier✝, is_open' := is_open'✝ }\n⊢ colimit.ι\n (((whiskeringLeft (OpenNhds ((forget TopCat).map (f ≫ g) x))ᵒᵖ (Opens ↑Z)ᵒᵖ C).obj\n (Functor.op (OpenNhds.inclusion ((forget TopCat).map (f ≫ g) x)))).obj\n ((f ≫ g) _* ℱ))\n { unop := { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ } } ≫\n stalkPushforward C (f ≫ g) ℱ x =\n colimit.ι\n (((whiskeringLeft (OpenNhds ((forget TopCat).map (f ≫ g) x))ᵒᵖ (Opens ↑Z)ᵒᵖ C).obj\n (Functor.op (OpenNhds.inclusion ((forget TopCat).map (f ≫ g) x)))).obj\n ((f ≫ g) _* ℱ))\n { unop := { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ } } ≫\n stalkPushforward C g (f _* ℱ) ((forget TopCat).map f x) ≫ stalkPushforward C f ℱ x",
"state_before": "case w.mk.mk.mk\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasColimits C\nX Y Z : TopCat\nℱ : Presheaf C X\nf : X ⟶ Y\ng : Y ⟶ Z\nx : ↑X\ncarrier✝ : Set ↑Z\nis_open'✝ : IsOpen carrier✝\nproperty✝ : (forget TopCat).map (f ≫ g) x ∈ { carrier := carrier✝, is_open' := is_open'✝ }\n⊢ colimit.ι\n (((whiskeringLeft (OpenNhds ((forget TopCat).map (f ≫ g) x))ᵒᵖ (Opens ↑Z)ᵒᵖ C).obj\n (Functor.op (OpenNhds.inclusion ((forget TopCat).map (f ≫ g) x)))).obj\n ((f ≫ g) _* ℱ))\n { unop := { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ } } ≫\n stalkPushforward C (f ≫ g) ℱ x =\n colimit.ι\n (((whiskeringLeft (OpenNhds ((forget TopCat).map (f ≫ g) x))ᵒᵖ (Opens ↑Z)ᵒᵖ C).obj\n (Functor.op (OpenNhds.inclusion ((forget TopCat).map (f ≫ g) x)))).obj\n ((f ≫ g) _* ℱ))\n { unop := { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ } } ≫\n stalkPushforward C g (f _* ℱ) ((forget TopCat).map f x) ≫ stalkPushforward C f ℱ x",
"tactic": "simp only [colimit.ι_map_assoc, colimit.ι_pre_assoc, whiskerRight_app, Category.assoc]"
},
{
"state_after": "no goals",
"state_before": "case w.mk.mk.mk\nC : Type u\ninst✝¹ : Category C\ninst✝ : HasColimits C\nX Y Z : TopCat\nℱ : Presheaf C X\nf : X ⟶ Y\ng : Y ⟶ Z\nx : ↑X\ncarrier✝ : Set ↑Z\nis_open'✝ : IsOpen carrier✝\nproperty✝ : (forget TopCat).map (f ≫ g) x ∈ { carrier := carrier✝, is_open' := is_open'✝ }\n⊢ colimit.ι\n (((whiskeringLeft (OpenNhds ((forget TopCat).map (f ≫ g) x))ᵒᵖ (Opens ↑Z)ᵒᵖ C).obj\n (Functor.op (OpenNhds.inclusion ((forget TopCat).map (f ≫ g) x)))).obj\n ((f ≫ g) _* ℱ))\n { unop := { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ } } ≫\n stalkPushforward C (f ≫ g) ℱ x =\n colimit.ι\n (((whiskeringLeft (OpenNhds ((forget TopCat).map (f ≫ g) x))ᵒᵖ (Opens ↑Z)ᵒᵖ C).obj\n (Functor.op (OpenNhds.inclusion ((forget TopCat).map (f ≫ g) x)))).obj\n ((f ≫ g) _* ℱ))\n { unop := { obj := { carrier := carrier✝, is_open' := is_open'✝ }, property := property✝ } } ≫\n stalkPushforward C g (f _* ℱ) ((forget TopCat).map f x) ≫ stalkPushforward C f ℱ x",
"tactic": "simp [stalkFunctor, stalkPushforward]"
}
] |
[
200,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Mathlib/Data/Nat/Prime.lean
|
Nat.dvd_prime_pow
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ\npp : Prime p\nm i : ℕ\n⊢ i ∣ p ^ m ↔ ∃ k, k ≤ m ∧ i = p ^ k",
"tactic": "simp_rw [_root_.dvd_prime_pow (prime_iff.mp pp) m, associated_eq_eq]"
}
] |
[
697,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
696,
1
] |
Mathlib/Data/Nat/Dist.lean
|
Nat.dist_eq_zero
|
[
{
"state_after": "no goals",
"state_before": "n m : ℕ\nh : n = m\n⊢ dist n m = 0",
"tactic": "rw [h, dist_self]"
}
] |
[
46,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
46,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean
|
InnerProductGeometry.sin_angle_add_of_inner_eq_zero
|
[
{
"state_after": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x y = 0\nh0 : x ≠ 0 ∨ y ≠ 0\n⊢ ‖y‖ ≤ ‖x + y‖",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x y = 0\nh0 : x ≠ 0 ∨ y ≠ 0\n⊢ Real.sin (angle x (x + y)) = ‖y‖ / ‖x + y‖",
"tactic": "rw [angle_add_eq_arcsin_of_inner_eq_zero h h0,\n Real.sin_arcsin (le_trans (by norm_num) (div_nonneg (norm_nonneg _) (norm_nonneg _)))\n (div_le_one_of_le _ (norm_nonneg _))]"
},
{
"state_after": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x y = 0\nh0 : x ≠ 0 ∨ y ≠ 0\n⊢ ‖y‖ * ‖y‖ ≤ ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x y = 0\nh0 : x ≠ 0 ∨ y ≠ 0\n⊢ ‖y‖ ≤ ‖x + y‖",
"tactic": "rw [mul_self_le_mul_self_iff (norm_nonneg _) (norm_nonneg _),\n norm_add_sq_eq_norm_sq_add_norm_sq_real h]"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x y = 0\nh0 : x ≠ 0 ∨ y ≠ 0\n⊢ ‖y‖ * ‖y‖ ≤ ‖x‖ * ‖x‖ + ‖y‖ * ‖y‖",
"tactic": "exact le_add_of_nonneg_left (mul_self_nonneg _)"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x y = 0\nh0 : x ≠ 0 ∨ y ≠ 0\n⊢ -1 ≤ 0",
"tactic": "norm_num"
}
] |
[
157,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/Data/Set/Ncard.lean
|
Set.eq_insert_of_ncard_eq_succ
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.160028\ns t : Set α\na b x y : α\nf : α → β\nn : ℕ\nh : ncard s = n + 1\nhsf : Set.Finite s\n⊢ ∃ a t, ¬a ∈ t ∧ insert a t = s ∧ ncard t = n",
"state_before": "α : Type u_1\nβ : Type ?u.160028\ns t : Set α\na b x y : α\nf : α → β\nn : ℕ\nh : ncard s = n + 1\n⊢ ∃ a t, ¬a ∈ t ∧ insert a t = s ∧ ncard t = n",
"tactic": "have hsf := Finite_of_ncard_pos (n.zero_lt_succ.trans_eq h.symm)"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.160028\ns t : Set α\na b x y : α\nf : α → β\nn : ℕ\nhsf : Set.Finite s\nh✝ : Finset.card (Finite.toFinset hsf) = n + 1\nh : ∃ a t, ¬a ∈ t ∧ insert a t = Finite.toFinset hsf ∧ Finset.card t = n\n⊢ ∃ a t, ¬a ∈ t ∧ insert a t = s ∧ ncard t = n",
"state_before": "α : Type u_1\nβ : Type ?u.160028\ns t : Set α\na b x y : α\nf : α → β\nn : ℕ\nh : ncard s = n + 1\nhsf : Set.Finite s\n⊢ ∃ a t, ¬a ∈ t ∧ insert a t = s ∧ ncard t = n",
"tactic": "rw [ncard_eq_toFinset_card _ hsf, Finset.card_eq_succ] at h"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.160028\ns t✝ : Set α\na✝ b x y : α\nf : α → β\nhsf : Set.Finite s\na : α\nt : Finset α\nhat : ¬a ∈ t\nhts : insert a t = Finite.toFinset hsf\nh : Finset.card (Finite.toFinset hsf) = Finset.card t + 1\n⊢ ∃ a t_1, ¬a ∈ t_1 ∧ insert a t_1 = s ∧ ncard t_1 = Finset.card t",
"state_before": "α : Type u_1\nβ : Type ?u.160028\ns t : Set α\na b x y : α\nf : α → β\nn : ℕ\nhsf : Set.Finite s\nh✝ : Finset.card (Finite.toFinset hsf) = n + 1\nh : ∃ a t, ¬a ∈ t ∧ insert a t = Finite.toFinset hsf ∧ Finset.card t = n\n⊢ ∃ a t, ¬a ∈ t ∧ insert a t = s ∧ ncard t = n",
"tactic": "obtain ⟨a, t, hat, hts, rfl⟩ := h"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.160028\ns t✝ : Set α\na✝ b x y : α\nf : α → β\nhsf : Set.Finite s\na : α\nt : Finset α\nhat : ¬a ∈ t\nh : Finset.card (Finite.toFinset hsf) = Finset.card t + 1\nhts : ∀ (a_1 : α), a_1 = a ∨ a_1 ∈ t ↔ a_1 ∈ s\n⊢ ∃ a t_1, ¬a ∈ t_1 ∧ insert a t_1 = s ∧ ncard t_1 = Finset.card t",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.160028\ns t✝ : Set α\na✝ b x y : α\nf : α → β\nhsf : Set.Finite s\na : α\nt : Finset α\nhat : ¬a ∈ t\nhts : insert a t = Finite.toFinset hsf\nh : Finset.card (Finite.toFinset hsf) = Finset.card t + 1\n⊢ ∃ a t_1, ¬a ∈ t_1 ∧ insert a t_1 = s ∧ ncard t_1 = Finset.card t",
"tactic": "simp only [Finset.ext_iff, Finset.mem_insert, Finite.mem_toFinset] at hts"
},
{
"state_after": "case intro.intro.intro.intro.refine'_1\nα : Type u_1\nβ : Type ?u.160028\ns t✝ : Set α\na✝ b x y : α\nf : α → β\nhsf : Set.Finite s\na : α\nt : Finset α\nhat : ¬a ∈ t\nh : Finset.card (Finite.toFinset hsf) = Finset.card t + 1\nhts : ∀ (a_1 : α), a_1 = a ∨ a_1 ∈ t ↔ a_1 ∈ s\n⊢ insert a ↑t = s\n\ncase intro.intro.intro.intro.refine'_2\nα : Type u_1\nβ : Type ?u.160028\ns t✝ : Set α\na✝ b x y : α\nf : α → β\nhsf : Set.Finite s\na : α\nt : Finset α\nhat : ¬a ∈ t\nh : Finset.card (Finite.toFinset hsf) = Finset.card t + 1\nhts : ∀ (a_1 : α), a_1 = a ∨ a_1 ∈ t ↔ a_1 ∈ s\n⊢ ncard ↑t = Finset.card t",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.160028\ns t✝ : Set α\na✝ b x y : α\nf : α → β\nhsf : Set.Finite s\na : α\nt : Finset α\nhat : ¬a ∈ t\nh : Finset.card (Finite.toFinset hsf) = Finset.card t + 1\nhts : ∀ (a_1 : α), a_1 = a ∨ a_1 ∈ t ↔ a_1 ∈ s\n⊢ ∃ a t_1, ¬a ∈ t_1 ∧ insert a t_1 = s ∧ ncard t_1 = Finset.card t",
"tactic": "refine' ⟨a, t, hat, _, _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.refine'_2\nα : Type u_1\nβ : Type ?u.160028\ns t✝ : Set α\na✝ b x y : α\nf : α → β\nhsf : Set.Finite s\na : α\nt : Finset α\nhat : ¬a ∈ t\nh : Finset.card (Finite.toFinset hsf) = Finset.card t + 1\nhts : ∀ (a_1 : α), a_1 = a ∨ a_1 ∈ t ↔ a_1 ∈ s\n⊢ ncard ↑t = Finset.card t",
"tactic": "simp"
},
{
"state_after": "case intro.intro.intro.intro.refine'_1\nα : Type u_1\nβ : Type ?u.160028\ns t✝ : Set α\na✝ b x y : α\nf : α → β\nhsf : Set.Finite s\na : α\nt : Finset α\nhat : ¬a ∈ t\nh : Finset.card (Finite.toFinset hsf) = Finset.card t + 1\nhts : ∀ (a_1 : α), a_1 = a ∨ a_1 ∈ t ↔ a_1 ∈ s\n⊢ ∀ (x : α), x = a ∨ x ∈ t ↔ x ∈ s",
"state_before": "case intro.intro.intro.intro.refine'_1\nα : Type u_1\nβ : Type ?u.160028\ns t✝ : Set α\na✝ b x y : α\nf : α → β\nhsf : Set.Finite s\na : α\nt : Finset α\nhat : ¬a ∈ t\nh : Finset.card (Finite.toFinset hsf) = Finset.card t + 1\nhts : ∀ (a_1 : α), a_1 = a ∨ a_1 ∈ t ↔ a_1 ∈ s\n⊢ insert a ↑t = s",
"tactic": "simp only [Finset.mem_coe, ext_iff, mem_insert_iff]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.refine'_1\nα : Type u_1\nβ : Type ?u.160028\ns t✝ : Set α\na✝ b x y : α\nf : α → β\nhsf : Set.Finite s\na : α\nt : Finset α\nhat : ¬a ∈ t\nh : Finset.card (Finite.toFinset hsf) = Finset.card t + 1\nhts : ∀ (a_1 : α), a_1 = a ∨ a_1 ∈ t ↔ a_1 ∈ s\n⊢ ∀ (x : α), x = a ∨ x ∈ t ↔ x ∈ s",
"tactic": "tauto"
}
] |
[
735,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
724,
1
] |
Mathlib/Data/Multiset/Lattice.lean
|
Multiset.inf_add
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : SemilatticeInf α\ninst✝ : OrderTop α\ns₁ s₂ : Multiset α\n⊢ inf (s₁ + s₂) = fold (fun x x_1 => x ⊓ x_1) (⊤ ⊓ ⊤) (s₁ + s₂)",
"tactic": "simp [inf]"
}
] |
[
142,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/CategoryTheory/Abelian/Pseudoelements.lean
|
CategoryTheory.Abelian.Pseudoelement.pseudo_pullback
|
[
{
"state_after": "case intro.intro.intro.intro.intro\nC : Type u\ninst✝² : Category C\ninst✝¹ : Abelian C\ninst✝ : HasPullbacks C\nP Q R : C\nf : P ⟶ R\ng : Q ⟶ R\np : Pseudoelement P\nq : Pseudoelement Q\nx : Over P\ny : Over Q\nh : pseudoApply f (Quotient.mk (setoid P) x) = pseudoApply g (Quotient.mk (setoid Q) y)\nZ : C\na : Z ⟶ ((fun g => app f g) x).left\nb : Z ⟶ ((fun g_1 => app g g_1) y).left\nea : Epi a\neb : Epi b\ncomm : a ≫ ((fun g => app f g) x).hom = b ≫ ((fun g_1 => app g g_1) y).hom\n⊢ ∃ s, pseudoApply pullback.fst s = Quotient.mk (setoid P) x ∧ pseudoApply pullback.snd s = Quotient.mk (setoid Q) y",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : Abelian C\ninst✝ : HasPullbacks C\nP Q R : C\nf : P ⟶ R\ng : Q ⟶ R\np : Pseudoelement P\nq : Pseudoelement Q\nx : Over P\ny : Over Q\nh : pseudoApply f (Quotient.mk (setoid P) x) = pseudoApply g (Quotient.mk (setoid Q) y)\n⊢ ∃ s, pseudoApply pullback.fst s = Quotient.mk (setoid P) x ∧ pseudoApply pullback.snd s = Quotient.mk (setoid Q) y",
"tactic": "obtain ⟨Z, a, b, ea, eb, comm⟩ := Quotient.exact h"
},
{
"state_after": "case intro.intro.intro.intro.intro.mk.intro\nC : Type u\ninst✝² : Category C\ninst✝¹ : Abelian C\ninst✝ : HasPullbacks C\nP Q R : C\nf : P ⟶ R\ng : Q ⟶ R\np : Pseudoelement P\nq : Pseudoelement Q\nx : Over P\ny : Over Q\nh : pseudoApply f (Quotient.mk (setoid P) x) = pseudoApply g (Quotient.mk (setoid Q) y)\nZ : C\na : Z ⟶ ((fun g => app f g) x).left\nb : Z ⟶ ((fun g_1 => app g g_1) y).left\nea : Epi a\neb : Epi b\ncomm : a ≫ ((fun g => app f g) x).hom = b ≫ ((fun g_1 => app g g_1) y).hom\nl : Z ⟶ pullback f g\nhl₁ : l ≫ pullback.fst = a ≫ x.hom\nhl₂ : l ≫ pullback.snd = b ≫ y.hom\n⊢ ∃ s, pseudoApply pullback.fst s = Quotient.mk (setoid P) x ∧ pseudoApply pullback.snd s = Quotient.mk (setoid Q) y",
"state_before": "case intro.intro.intro.intro.intro\nC : Type u\ninst✝² : Category C\ninst✝¹ : Abelian C\ninst✝ : HasPullbacks C\nP Q R : C\nf : P ⟶ R\ng : Q ⟶ R\np : Pseudoelement P\nq : Pseudoelement Q\nx : Over P\ny : Over Q\nh : pseudoApply f (Quotient.mk (setoid P) x) = pseudoApply g (Quotient.mk (setoid Q) y)\nZ : C\na : Z ⟶ ((fun g => app f g) x).left\nb : Z ⟶ ((fun g_1 => app g g_1) y).left\nea : Epi a\neb : Epi b\ncomm : a ≫ ((fun g => app f g) x).hom = b ≫ ((fun g_1 => app g g_1) y).hom\n⊢ ∃ s, pseudoApply pullback.fst s = Quotient.mk (setoid P) x ∧ pseudoApply pullback.snd s = Quotient.mk (setoid Q) y",
"tactic": "obtain ⟨l, hl₁, hl₂⟩ := @pullback.lift' _ _ _ _ _ _ f g _ (a ≫ x.hom) (b ≫ y.hom) (by\n simp only [Category.assoc]\n exact comm)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.mk.intro\nC : Type u\ninst✝² : Category C\ninst✝¹ : Abelian C\ninst✝ : HasPullbacks C\nP Q R : C\nf : P ⟶ R\ng : Q ⟶ R\np : Pseudoelement P\nq : Pseudoelement Q\nx : Over P\ny : Over Q\nh : pseudoApply f (Quotient.mk (setoid P) x) = pseudoApply g (Quotient.mk (setoid Q) y)\nZ : C\na : Z ⟶ ((fun g => app f g) x).left\nb : Z ⟶ ((fun g_1 => app g g_1) y).left\nea : Epi a\neb : Epi b\ncomm : a ≫ ((fun g => app f g) x).hom = b ≫ ((fun g_1 => app g g_1) y).hom\nl : Z ⟶ pullback f g\nhl₁ : l ≫ pullback.fst = a ≫ x.hom\nhl₂ : l ≫ pullback.snd = b ≫ y.hom\n⊢ ∃ s, pseudoApply pullback.fst s = Quotient.mk (setoid P) x ∧ pseudoApply pullback.snd s = Quotient.mk (setoid Q) y",
"tactic": "exact ⟨l, ⟨Quotient.sound ⟨Z, 𝟙 Z, a, inferInstance, ea, by rwa [Category.id_comp]⟩,\n Quotient.sound ⟨Z, 𝟙 Z, b, inferInstance, eb, by rwa [Category.id_comp]⟩⟩⟩"
},
{
"state_after": "C : Type u\ninst✝² : Category C\ninst✝¹ : Abelian C\ninst✝ : HasPullbacks C\nP Q R : C\nf : P ⟶ R\ng : Q ⟶ R\np : Pseudoelement P\nq : Pseudoelement Q\nx : Over P\ny : Over Q\nh : pseudoApply f (Quotient.mk (setoid P) x) = pseudoApply g (Quotient.mk (setoid Q) y)\nZ : C\na : Z ⟶ ((fun g => app f g) x).left\nb : Z ⟶ ((fun g_1 => app g g_1) y).left\nea : Epi a\neb : Epi b\ncomm : a ≫ ((fun g => app f g) x).hom = b ≫ ((fun g_1 => app g g_1) y).hom\n⊢ a ≫ x.hom ≫ f = b ≫ y.hom ≫ g",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : Abelian C\ninst✝ : HasPullbacks C\nP Q R : C\nf : P ⟶ R\ng : Q ⟶ R\np : Pseudoelement P\nq : Pseudoelement Q\nx : Over P\ny : Over Q\nh : pseudoApply f (Quotient.mk (setoid P) x) = pseudoApply g (Quotient.mk (setoid Q) y)\nZ : C\na : Z ⟶ ((fun g => app f g) x).left\nb : Z ⟶ ((fun g_1 => app g g_1) y).left\nea : Epi a\neb : Epi b\ncomm : a ≫ ((fun g => app f g) x).hom = b ≫ ((fun g_1 => app g g_1) y).hom\n⊢ (a ≫ x.hom) ≫ f = (b ≫ y.hom) ≫ g",
"tactic": "simp only [Category.assoc]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : Abelian C\ninst✝ : HasPullbacks C\nP Q R : C\nf : P ⟶ R\ng : Q ⟶ R\np : Pseudoelement P\nq : Pseudoelement Q\nx : Over P\ny : Over Q\nh : pseudoApply f (Quotient.mk (setoid P) x) = pseudoApply g (Quotient.mk (setoid Q) y)\nZ : C\na : Z ⟶ ((fun g => app f g) x).left\nb : Z ⟶ ((fun g_1 => app g g_1) y).left\nea : Epi a\neb : Epi b\ncomm : a ≫ ((fun g => app f g) x).hom = b ≫ ((fun g_1 => app g g_1) y).hom\n⊢ a ≫ x.hom ≫ f = b ≫ y.hom ≫ g",
"tactic": "exact comm"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : Abelian C\ninst✝ : HasPullbacks C\nP Q R : C\nf : P ⟶ R\ng : Q ⟶ R\np : Pseudoelement P\nq : Pseudoelement Q\nx : Over P\ny : Over Q\nh : pseudoApply f (Quotient.mk (setoid P) x) = pseudoApply g (Quotient.mk (setoid Q) y)\nZ : C\na : Z ⟶ ((fun g => app f g) x).left\nb : Z ⟶ ((fun g_1 => app g g_1) y).left\nea : Epi a\neb : Epi b\ncomm : a ≫ ((fun g => app f g) x).hom = b ≫ ((fun g_1 => app g g_1) y).hom\nl : Z ⟶ pullback f g\nhl₁ : l ≫ pullback.fst = a ≫ x.hom\nhl₂ : l ≫ pullback.snd = b ≫ y.hom\n⊢ 𝟙 Z ≫ ((fun g_1 => app pullback.fst g_1) (Over.mk l)).hom = a ≫ x.hom",
"tactic": "rwa [Category.id_comp]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝² : Category C\ninst✝¹ : Abelian C\ninst✝ : HasPullbacks C\nP Q R : C\nf : P ⟶ R\ng : Q ⟶ R\np : Pseudoelement P\nq : Pseudoelement Q\nx : Over P\ny : Over Q\nh : pseudoApply f (Quotient.mk (setoid P) x) = pseudoApply g (Quotient.mk (setoid Q) y)\nZ : C\na : Z ⟶ ((fun g => app f g) x).left\nb : Z ⟶ ((fun g_1 => app g g_1) y).left\nea : Epi a\neb : Epi b\ncomm : a ≫ ((fun g => app f g) x).hom = b ≫ ((fun g_1 => app g g_1) y).hom\nl : Z ⟶ pullback f g\nhl₁ : l ≫ pullback.fst = a ≫ x.hom\nhl₂ : l ≫ pullback.snd = b ≫ y.hom\n⊢ 𝟙 Z ≫ ((fun g_1 => app pullback.snd g_1) (Over.mk l)).hom = b ≫ y.hom",
"tactic": "rwa [Category.id_comp]"
}
] |
[
475,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
466,
1
] |
Mathlib/Data/List/BigOperators/Basic.lean
|
List.alternatingProd_singleton
|
[] |
[
630,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
629,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
ENNReal.measurable_of_measurable_nnreal_nnreal
|
[] |
[
1851,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1846,
1
] |
Mathlib/CategoryTheory/Sites/SheafOfTypes.lean
|
CategoryTheory.Presieve.isSheaf_iso
|
[] |
[
754,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
753,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean
|
CategoryTheory.Limits.BinaryCofan.mk_inl
|
[] |
[
331,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
330,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
MonotoneOn.map_bddAbove
|
[] |
[
1186,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1185,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean
|
fderiv_const_sub
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_2\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.579415\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.579510\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nc : F\n⊢ fderiv 𝕜 (fun y => c - f y) x = -fderiv 𝕜 f x",
"tactic": "simp only [← fderivWithin_univ, fderivWithin_const_sub uniqueDiffWithinAt_univ]"
}
] |
[
663,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
662,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/Log.lean
|
Complex.ofReal_log
|
[
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : 0 ≤ x\n⊢ (↑(Real.log x)).re = (log ↑x).re",
"tactic": "rw [log_re, ofReal_re, abs_of_nonneg hx]"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\nhx : 0 ≤ x\n⊢ (↑(Real.log x)).im = (log ↑x).im",
"tactic": "rw [ofReal_im, log_im, arg_ofReal_of_nonneg hx]"
}
] |
[
75,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/LinearAlgebra/Prod.lean
|
LinearMap.prodMap_apply
|
[] |
[
319,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
318,
1
] |
Mathlib/RingTheory/WittVector/Basic.lean
|
WittVector.ghostComponent_apply
|
[] |
[
314,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
313,
1
] |
Mathlib/RingTheory/Int/Basic.lean
|
Int.span_natAbs
|
[
{
"state_after": "a : ℤ\n⊢ Associated (↑(natAbs a)) a",
"state_before": "a : ℤ\n⊢ Ideal.span {↑(natAbs a)} = Ideal.span {a}",
"tactic": "rw [Ideal.span_singleton_eq_span_singleton]"
},
{
"state_after": "no goals",
"state_before": "a : ℤ\n⊢ Associated (↑(natAbs a)) a",
"tactic": "exact (associated_natAbs _).symm"
}
] |
[
394,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
392,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
|
Measurable.aemeasurable
|
[] |
[
680,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
679,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
inv_div_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.24338\nG : Type ?u.24341\ninst✝ : DivisionMonoid α\na b c : α\n⊢ a⁻¹ / b = (b * a)⁻¹",
"tactic": "simp"
}
] |
[
409,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
409,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.iInter_ge_eq_iInter_nat_add
|
[] |
[
2202,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2201,
1
] |
Mathlib/Topology/Algebra/ContinuousMonoidHom.lean
|
ContinuousMonoidHom.closedEmbedding_toContinuousMap
|
[
{
"state_after": "F : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\n⊢ Set.range toContinuousMap =\n ({f | ↑f '' {1} ⊆ {1}ᶜ} ∪\n ⋃ (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (_ : IsOpen U) (_ : IsOpen V) (_ : IsOpen W) (_ :\n Disjoint (U * V) W), {f | ↑f '' {x} ⊆ U} ∩ {f | ↑f '' {y} ⊆ V} ∩ {f | ↑f '' {x * y} ⊆ W})ᶜ",
"state_before": "F : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\n⊢ IsOpen (Set.range toContinuousMapᶜ)",
"tactic": "suffices\n Set.range (toContinuousMap : ContinuousMonoidHom A B → C(A, B)) =\n ({ f | f '' {1} ⊆ {1}ᶜ } ∪\n ⋃ (x) (y) (U) (V) (W) (_ : IsOpen U) (_ : IsOpen V) (_ : IsOpen W) (_ :\n Disjoint (U * V) W),\n { f | f '' {x} ⊆ U } ∩ { f | f '' {y} ⊆ V } ∩ { f | f '' {x * y} ⊆ W } :\n Set C(A , B))ᶜ by\n rw [this, compl_compl]\n refine' (ContinuousMap.isOpen_gen isCompact_singleton isOpen_compl_singleton).union _\n repeat' apply isOpen_iUnion; intro\n repeat' apply IsOpen.inter\n all_goals apply ContinuousMap.isOpen_gen isCompact_singleton; assumption"
},
{
"state_after": "F : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\n⊢ ∀ (x : C(A, B)),\n x ∈ Set.range toContinuousMap ↔\n ¬x ∈ {f | ¬↑f 1 ∈ {1}} ∧\n ∀ (i i_1 : A) (i_2 i_3 i_4 : Set B),\n IsOpen i_2 →\n IsOpen i_3 →\n IsOpen i_4 →\n Disjoint (i_2 * i_3) i_4 → ¬x ∈ {f | ↑f i ∈ i_2} ∩ {f | ↑f i_1 ∈ i_3} ∩ {f | ↑f (i * i_1) ∈ i_4}",
"state_before": "F : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\n⊢ Set.range toContinuousMap =\n ({f | ↑f '' {1} ⊆ {1}ᶜ} ∪\n ⋃ (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (_ : IsOpen U) (_ : IsOpen V) (_ : IsOpen W) (_ :\n Disjoint (U * V) W), {f | ↑f '' {x} ⊆ U} ∩ {f | ↑f '' {y} ⊆ V} ∩ {f | ↑f '' {x * y} ⊆ W})ᶜ",
"tactic": "simp_rw [Set.compl_union, Set.compl_iUnion, Set.image_singleton, Set.singleton_subset_iff,\n Set.ext_iff, Set.mem_inter_iff, Set.mem_iInter, Set.mem_compl_iff]"
},
{
"state_after": "case refine'_1\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nf : C(A, B)\n⊢ f ∈ Set.range toContinuousMap →\n ¬f ∈ {f | ¬↑f 1 ∈ {1}} ∧\n ∀ (i i_1 : A) (i_2 i_3 i_4 : Set B),\n IsOpen i_2 →\n IsOpen i_3 →\n IsOpen i_4 →\n Disjoint (i_2 * i_3) i_4 → ¬f ∈ {f | ↑f i ∈ i_2} ∩ {f | ↑f i_1 ∈ i_3} ∩ {f | ↑f (i * i_1) ∈ i_4}\n\ncase refine'_2\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nf : C(A, B)\n⊢ (¬f ∈ {f | ¬↑f 1 ∈ {1}} ∧\n ∀ (i i_1 : A) (i_2 i_3 i_4 : Set B),\n IsOpen i_2 →\n IsOpen i_3 →\n IsOpen i_4 →\n Disjoint (i_2 * i_3) i_4 → ¬f ∈ {f | ↑f i ∈ i_2} ∩ {f | ↑f i_1 ∈ i_3} ∩ {f | ↑f (i * i_1) ∈ i_4}) →\n f ∈ Set.range toContinuousMap",
"state_before": "F : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\n⊢ ∀ (x : C(A, B)),\n x ∈ Set.range toContinuousMap ↔\n ¬x ∈ {f | ¬↑f 1 ∈ {1}} ∧\n ∀ (i i_1 : A) (i_2 i_3 i_4 : Set B),\n IsOpen i_2 →\n IsOpen i_3 →\n IsOpen i_4 →\n Disjoint (i_2 * i_3) i_4 → ¬x ∈ {f | ↑f i ∈ i_2} ∩ {f | ↑f i_1 ∈ i_3} ∩ {f | ↑f (i * i_1) ∈ i_4}",
"tactic": "refine' fun f => ⟨_, _⟩"
},
{
"state_after": "F : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nthis :\n Set.range toContinuousMap =\n ({f | ↑f '' {1} ⊆ {1}ᶜ} ∪\n ⋃ (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (_ : IsOpen U) (_ : IsOpen V) (_ : IsOpen W) (_ :\n Disjoint (U * V) W), {f | ↑f '' {x} ⊆ U} ∩ {f | ↑f '' {y} ⊆ V} ∩ {f | ↑f '' {x * y} ⊆ W})ᶜ\n⊢ IsOpen\n ({f | ↑f '' {1} ⊆ {1}ᶜ} ∪\n ⋃ (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (_ : IsOpen U) (_ : IsOpen V) (_ : IsOpen W) (_ :\n Disjoint (U * V) W), {f | ↑f '' {x} ⊆ U} ∩ {f | ↑f '' {y} ⊆ V} ∩ {f | ↑f '' {x * y} ⊆ W})",
"state_before": "F : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nthis :\n Set.range toContinuousMap =\n ({f | ↑f '' {1} ⊆ {1}ᶜ} ∪\n ⋃ (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (_ : IsOpen U) (_ : IsOpen V) (_ : IsOpen W) (_ :\n Disjoint (U * V) W), {f | ↑f '' {x} ⊆ U} ∩ {f | ↑f '' {y} ⊆ V} ∩ {f | ↑f '' {x * y} ⊆ W})ᶜ\n⊢ IsOpen (Set.range toContinuousMapᶜ)",
"tactic": "rw [this, compl_compl]"
},
{
"state_after": "F : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nthis :\n Set.range toContinuousMap =\n ({f | ↑f '' {1} ⊆ {1}ᶜ} ∪\n ⋃ (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (_ : IsOpen U) (_ : IsOpen V) (_ : IsOpen W) (_ :\n Disjoint (U * V) W), {f | ↑f '' {x} ⊆ U} ∩ {f | ↑f '' {y} ⊆ V} ∩ {f | ↑f '' {x * y} ⊆ W})ᶜ\n⊢ IsOpen\n (⋃ (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (_ : IsOpen U) (_ : IsOpen V) (_ : IsOpen W) (_ :\n Disjoint (U * V) W), {f | ↑f '' {x} ⊆ U} ∩ {f | ↑f '' {y} ⊆ V} ∩ {f | ↑f '' {x * y} ⊆ W})",
"state_before": "F : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nthis :\n Set.range toContinuousMap =\n ({f | ↑f '' {1} ⊆ {1}ᶜ} ∪\n ⋃ (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (_ : IsOpen U) (_ : IsOpen V) (_ : IsOpen W) (_ :\n Disjoint (U * V) W), {f | ↑f '' {x} ⊆ U} ∩ {f | ↑f '' {y} ⊆ V} ∩ {f | ↑f '' {x * y} ⊆ W})ᶜ\n⊢ IsOpen\n ({f | ↑f '' {1} ⊆ {1}ᶜ} ∪\n ⋃ (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (_ : IsOpen U) (_ : IsOpen V) (_ : IsOpen W) (_ :\n Disjoint (U * V) W), {f | ↑f '' {x} ⊆ U} ∩ {f | ↑f '' {y} ⊆ V} ∩ {f | ↑f '' {x * y} ⊆ W})",
"tactic": "refine' (ContinuousMap.isOpen_gen isCompact_singleton isOpen_compl_singleton).union _"
},
{
"state_after": "case h.h.h.h.h.h.h.h.h\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nthis :\n Set.range toContinuousMap =\n ({f | ↑f '' {1} ⊆ {1}ᶜ} ∪\n ⋃ (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (_ : IsOpen U) (_ : IsOpen V) (_ : IsOpen W) (_ :\n Disjoint (U * V) W), {f | ↑f '' {x} ⊆ U} ∩ {f | ↑f '' {y} ⊆ V} ∩ {f | ↑f '' {x * y} ⊆ W})ᶜ\ni✝⁸ i✝⁷ : A\ni✝⁶ i✝⁵ i✝⁴ : Set B\ni✝³ : IsOpen i✝⁶\ni✝² : IsOpen i✝⁵\ni✝¹ : IsOpen i✝⁴\ni✝ : Disjoint (i✝⁶ * i✝⁵) i✝⁴\n⊢ IsOpen ({f | ↑f '' {i✝⁸} ⊆ i✝⁶} ∩ {f | ↑f '' {i✝⁷} ⊆ i✝⁵} ∩ {f | ↑f '' {i✝⁸ * i✝⁷} ⊆ i✝⁴})",
"state_before": "F : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nthis :\n Set.range toContinuousMap =\n ({f | ↑f '' {1} ⊆ {1}ᶜ} ∪\n ⋃ (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (_ : IsOpen U) (_ : IsOpen V) (_ : IsOpen W) (_ :\n Disjoint (U * V) W), {f | ↑f '' {x} ⊆ U} ∩ {f | ↑f '' {y} ⊆ V} ∩ {f | ↑f '' {x * y} ⊆ W})ᶜ\n⊢ IsOpen\n (⋃ (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (_ : IsOpen U) (_ : IsOpen V) (_ : IsOpen W) (_ :\n Disjoint (U * V) W), {f | ↑f '' {x} ⊆ U} ∩ {f | ↑f '' {y} ⊆ V} ∩ {f | ↑f '' {x * y} ⊆ W})",
"tactic": "repeat' apply isOpen_iUnion; intro"
},
{
"state_after": "case h.h.h.h.h.h.h.h.h.h₁.h₁\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nthis :\n Set.range toContinuousMap =\n ({f | ↑f '' {1} ⊆ {1}ᶜ} ∪\n ⋃ (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (_ : IsOpen U) (_ : IsOpen V) (_ : IsOpen W) (_ :\n Disjoint (U * V) W), {f | ↑f '' {x} ⊆ U} ∩ {f | ↑f '' {y} ⊆ V} ∩ {f | ↑f '' {x * y} ⊆ W})ᶜ\ni✝⁸ i✝⁷ : A\ni✝⁶ i✝⁵ i✝⁴ : Set B\ni✝³ : IsOpen i✝⁶\ni✝² : IsOpen i✝⁵\ni✝¹ : IsOpen i✝⁴\ni✝ : Disjoint (i✝⁶ * i✝⁵) i✝⁴\n⊢ IsOpen {f | ↑f '' {i✝⁸} ⊆ i✝⁶}\n\ncase h.h.h.h.h.h.h.h.h.h₁.h₂\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nthis :\n Set.range toContinuousMap =\n ({f | ↑f '' {1} ⊆ {1}ᶜ} ∪\n ⋃ (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (_ : IsOpen U) (_ : IsOpen V) (_ : IsOpen W) (_ :\n Disjoint (U * V) W), {f | ↑f '' {x} ⊆ U} ∩ {f | ↑f '' {y} ⊆ V} ∩ {f | ↑f '' {x * y} ⊆ W})ᶜ\ni✝⁸ i✝⁷ : A\ni✝⁶ i✝⁵ i✝⁴ : Set B\ni✝³ : IsOpen i✝⁶\ni✝² : IsOpen i✝⁵\ni✝¹ : IsOpen i✝⁴\ni✝ : Disjoint (i✝⁶ * i✝⁵) i✝⁴\n⊢ IsOpen {f | ↑f '' {i✝⁷} ⊆ i✝⁵}\n\ncase h.h.h.h.h.h.h.h.h.h₂\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nthis :\n Set.range toContinuousMap =\n ({f | ↑f '' {1} ⊆ {1}ᶜ} ∪\n ⋃ (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (_ : IsOpen U) (_ : IsOpen V) (_ : IsOpen W) (_ :\n Disjoint (U * V) W), {f | ↑f '' {x} ⊆ U} ∩ {f | ↑f '' {y} ⊆ V} ∩ {f | ↑f '' {x * y} ⊆ W})ᶜ\ni✝⁸ i✝⁷ : A\ni✝⁶ i✝⁵ i✝⁴ : Set B\ni✝³ : IsOpen i✝⁶\ni✝² : IsOpen i✝⁵\ni✝¹ : IsOpen i✝⁴\ni✝ : Disjoint (i✝⁶ * i✝⁵) i✝⁴\n⊢ IsOpen {f | ↑f '' {i✝⁸ * i✝⁷} ⊆ i✝⁴}",
"state_before": "case h.h.h.h.h.h.h.h.h\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nthis :\n Set.range toContinuousMap =\n ({f | ↑f '' {1} ⊆ {1}ᶜ} ∪\n ⋃ (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (_ : IsOpen U) (_ : IsOpen V) (_ : IsOpen W) (_ :\n Disjoint (U * V) W), {f | ↑f '' {x} ⊆ U} ∩ {f | ↑f '' {y} ⊆ V} ∩ {f | ↑f '' {x * y} ⊆ W})ᶜ\ni✝⁸ i✝⁷ : A\ni✝⁶ i✝⁵ i✝⁴ : Set B\ni✝³ : IsOpen i✝⁶\ni✝² : IsOpen i✝⁵\ni✝¹ : IsOpen i✝⁴\ni✝ : Disjoint (i✝⁶ * i✝⁵) i✝⁴\n⊢ IsOpen ({f | ↑f '' {i✝⁸} ⊆ i✝⁶} ∩ {f | ↑f '' {i✝⁷} ⊆ i✝⁵} ∩ {f | ↑f '' {i✝⁸ * i✝⁷} ⊆ i✝⁴})",
"tactic": "repeat' apply IsOpen.inter"
},
{
"state_after": "no goals",
"state_before": "case h.h.h.h.h.h.h.h.h.h₁.h₁\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nthis :\n Set.range toContinuousMap =\n ({f | ↑f '' {1} ⊆ {1}ᶜ} ∪\n ⋃ (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (_ : IsOpen U) (_ : IsOpen V) (_ : IsOpen W) (_ :\n Disjoint (U * V) W), {f | ↑f '' {x} ⊆ U} ∩ {f | ↑f '' {y} ⊆ V} ∩ {f | ↑f '' {x * y} ⊆ W})ᶜ\ni✝⁸ i✝⁷ : A\ni✝⁶ i✝⁵ i✝⁴ : Set B\ni✝³ : IsOpen i✝⁶\ni✝² : IsOpen i✝⁵\ni✝¹ : IsOpen i✝⁴\ni✝ : Disjoint (i✝⁶ * i✝⁵) i✝⁴\n⊢ IsOpen {f | ↑f '' {i✝⁸} ⊆ i✝⁶}\n\ncase h.h.h.h.h.h.h.h.h.h₁.h₂\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nthis :\n Set.range toContinuousMap =\n ({f | ↑f '' {1} ⊆ {1}ᶜ} ∪\n ⋃ (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (_ : IsOpen U) (_ : IsOpen V) (_ : IsOpen W) (_ :\n Disjoint (U * V) W), {f | ↑f '' {x} ⊆ U} ∩ {f | ↑f '' {y} ⊆ V} ∩ {f | ↑f '' {x * y} ⊆ W})ᶜ\ni✝⁸ i✝⁷ : A\ni✝⁶ i✝⁵ i✝⁴ : Set B\ni✝³ : IsOpen i✝⁶\ni✝² : IsOpen i✝⁵\ni✝¹ : IsOpen i✝⁴\ni✝ : Disjoint (i✝⁶ * i✝⁵) i✝⁴\n⊢ IsOpen {f | ↑f '' {i✝⁷} ⊆ i✝⁵}\n\ncase h.h.h.h.h.h.h.h.h.h₂\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nthis :\n Set.range toContinuousMap =\n ({f | ↑f '' {1} ⊆ {1}ᶜ} ∪\n ⋃ (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (_ : IsOpen U) (_ : IsOpen V) (_ : IsOpen W) (_ :\n Disjoint (U * V) W), {f | ↑f '' {x} ⊆ U} ∩ {f | ↑f '' {y} ⊆ V} ∩ {f | ↑f '' {x * y} ⊆ W})ᶜ\ni✝⁸ i✝⁷ : A\ni✝⁶ i✝⁵ i✝⁴ : Set B\ni✝³ : IsOpen i✝⁶\ni✝² : IsOpen i✝⁵\ni✝¹ : IsOpen i✝⁴\ni✝ : Disjoint (i✝⁶ * i✝⁵) i✝⁴\n⊢ IsOpen {f | ↑f '' {i✝⁸ * i✝⁷} ⊆ i✝⁴}",
"tactic": "all_goals apply ContinuousMap.isOpen_gen isCompact_singleton; assumption"
},
{
"state_after": "case h.h.h.h.h.h.h.h.h\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nthis :\n Set.range toContinuousMap =\n ({f | ↑f '' {1} ⊆ {1}ᶜ} ∪\n ⋃ (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (_ : IsOpen U) (_ : IsOpen V) (_ : IsOpen W) (_ :\n Disjoint (U * V) W), {f | ↑f '' {x} ⊆ U} ∩ {f | ↑f '' {y} ⊆ V} ∩ {f | ↑f '' {x * y} ⊆ W})ᶜ\ni✝⁸ i✝⁷ : A\ni✝⁶ i✝⁵ i✝⁴ : Set B\ni✝³ : IsOpen i✝⁶\ni✝² : IsOpen i✝⁵\ni✝¹ : IsOpen i✝⁴\ni✝ : Disjoint (i✝⁶ * i✝⁵) i✝⁴\n⊢ IsOpen ({f | ↑f '' {i✝⁸} ⊆ i✝⁶} ∩ {f | ↑f '' {i✝⁷} ⊆ i✝⁵} ∩ {f | ↑f '' {i✝⁸ * i✝⁷} ⊆ i✝⁴})",
"state_before": "case h.h.h.h.h.h.h.h.h\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nthis :\n Set.range toContinuousMap =\n ({f | ↑f '' {1} ⊆ {1}ᶜ} ∪\n ⋃ (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (_ : IsOpen U) (_ : IsOpen V) (_ : IsOpen W) (_ :\n Disjoint (U * V) W), {f | ↑f '' {x} ⊆ U} ∩ {f | ↑f '' {y} ⊆ V} ∩ {f | ↑f '' {x * y} ⊆ W})ᶜ\ni✝⁸ i✝⁷ : A\ni✝⁶ i✝⁵ i✝⁴ : Set B\ni✝³ : IsOpen i✝⁶\ni✝² : IsOpen i✝⁵\ni✝¹ : IsOpen i✝⁴\ni✝ : Disjoint (i✝⁶ * i✝⁵) i✝⁴\n⊢ IsOpen ({f | ↑f '' {i✝⁸} ⊆ i✝⁶} ∩ {f | ↑f '' {i✝⁷} ⊆ i✝⁵} ∩ {f | ↑f '' {i✝⁸ * i✝⁷} ⊆ i✝⁴})",
"tactic": "apply isOpen_iUnion"
},
{
"state_after": "case h.h.h.h.h.h.h.h.h\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nthis :\n Set.range toContinuousMap =\n ({f | ↑f '' {1} ⊆ {1}ᶜ} ∪\n ⋃ (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (_ : IsOpen U) (_ : IsOpen V) (_ : IsOpen W) (_ :\n Disjoint (U * V) W), {f | ↑f '' {x} ⊆ U} ∩ {f | ↑f '' {y} ⊆ V} ∩ {f | ↑f '' {x * y} ⊆ W})ᶜ\ni✝⁸ i✝⁷ : A\ni✝⁶ i✝⁵ i✝⁴ : Set B\ni✝³ : IsOpen i✝⁶\ni✝² : IsOpen i✝⁵\ni✝¹ : IsOpen i✝⁴\ni✝ : Disjoint (i✝⁶ * i✝⁵) i✝⁴\n⊢ IsOpen ({f | ↑f '' {i✝⁸} ⊆ i✝⁶} ∩ {f | ↑f '' {i✝⁷} ⊆ i✝⁵} ∩ {f | ↑f '' {i✝⁸ * i✝⁷} ⊆ i✝⁴})",
"state_before": "case h.h.h.h.h.h.h.h.h\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nthis :\n Set.range toContinuousMap =\n ({f | ↑f '' {1} ⊆ {1}ᶜ} ∪\n ⋃ (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (_ : IsOpen U) (_ : IsOpen V) (_ : IsOpen W) (_ :\n Disjoint (U * V) W), {f | ↑f '' {x} ⊆ U} ∩ {f | ↑f '' {y} ⊆ V} ∩ {f | ↑f '' {x * y} ⊆ W})ᶜ\ni✝⁷ i✝⁶ : A\ni✝⁵ i✝⁴ i✝³ : Set B\ni✝² : IsOpen i✝⁵\ni✝¹ : IsOpen i✝⁴\ni✝ : IsOpen i✝³\n⊢ Disjoint (i✝⁵ * i✝⁴) i✝³ → IsOpen ({f | ↑f '' {i✝⁷} ⊆ i✝⁵} ∩ {f | ↑f '' {i✝⁶} ⊆ i✝⁴} ∩ {f | ↑f '' {i✝⁷ * i✝⁶} ⊆ i✝³})",
"tactic": "intro"
},
{
"state_after": "case h.h.h.h.h.h.h.h.h.h₂\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nthis :\n Set.range toContinuousMap =\n ({f | ↑f '' {1} ⊆ {1}ᶜ} ∪\n ⋃ (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (_ : IsOpen U) (_ : IsOpen V) (_ : IsOpen W) (_ :\n Disjoint (U * V) W), {f | ↑f '' {x} ⊆ U} ∩ {f | ↑f '' {y} ⊆ V} ∩ {f | ↑f '' {x * y} ⊆ W})ᶜ\ni✝⁸ i✝⁷ : A\ni✝⁶ i✝⁵ i✝⁴ : Set B\ni✝³ : IsOpen i✝⁶\ni✝² : IsOpen i✝⁵\ni✝¹ : IsOpen i✝⁴\ni✝ : Disjoint (i✝⁶ * i✝⁵) i✝⁴\n⊢ IsOpen {f | ↑f '' {i✝⁸ * i✝⁷} ⊆ i✝⁴}",
"state_before": "case h.h.h.h.h.h.h.h.h.h₂\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nthis :\n Set.range toContinuousMap =\n ({f | ↑f '' {1} ⊆ {1}ᶜ} ∪\n ⋃ (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (_ : IsOpen U) (_ : IsOpen V) (_ : IsOpen W) (_ :\n Disjoint (U * V) W), {f | ↑f '' {x} ⊆ U} ∩ {f | ↑f '' {y} ⊆ V} ∩ {f | ↑f '' {x * y} ⊆ W})ᶜ\ni✝⁸ i✝⁷ : A\ni✝⁶ i✝⁵ i✝⁴ : Set B\ni✝³ : IsOpen i✝⁶\ni✝² : IsOpen i✝⁵\ni✝¹ : IsOpen i✝⁴\ni✝ : Disjoint (i✝⁶ * i✝⁵) i✝⁴\n⊢ IsOpen {f | ↑f '' {i✝⁸ * i✝⁷} ⊆ i✝⁴}",
"tactic": "apply IsOpen.inter"
},
{
"state_after": "case h.h.h.h.h.h.h.h.h.h₂\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nthis :\n Set.range toContinuousMap =\n ({f | ↑f '' {1} ⊆ {1}ᶜ} ∪\n ⋃ (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (_ : IsOpen U) (_ : IsOpen V) (_ : IsOpen W) (_ :\n Disjoint (U * V) W), {f | ↑f '' {x} ⊆ U} ∩ {f | ↑f '' {y} ⊆ V} ∩ {f | ↑f '' {x * y} ⊆ W})ᶜ\ni✝⁸ i✝⁷ : A\ni✝⁶ i✝⁵ i✝⁴ : Set B\ni✝³ : IsOpen i✝⁶\ni✝² : IsOpen i✝⁵\ni✝¹ : IsOpen i✝⁴\ni✝ : Disjoint (i✝⁶ * i✝⁵) i✝⁴\n⊢ IsOpen i✝⁴",
"state_before": "case h.h.h.h.h.h.h.h.h.h₂\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nthis :\n Set.range toContinuousMap =\n ({f | ↑f '' {1} ⊆ {1}ᶜ} ∪\n ⋃ (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (_ : IsOpen U) (_ : IsOpen V) (_ : IsOpen W) (_ :\n Disjoint (U * V) W), {f | ↑f '' {x} ⊆ U} ∩ {f | ↑f '' {y} ⊆ V} ∩ {f | ↑f '' {x * y} ⊆ W})ᶜ\ni✝⁸ i✝⁷ : A\ni✝⁶ i✝⁵ i✝⁴ : Set B\ni✝³ : IsOpen i✝⁶\ni✝² : IsOpen i✝⁵\ni✝¹ : IsOpen i✝⁴\ni✝ : Disjoint (i✝⁶ * i✝⁵) i✝⁴\n⊢ IsOpen {f | ↑f '' {i✝⁸ * i✝⁷} ⊆ i✝⁴}",
"tactic": "apply ContinuousMap.isOpen_gen isCompact_singleton"
},
{
"state_after": "no goals",
"state_before": "case h.h.h.h.h.h.h.h.h.h₂\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nthis :\n Set.range toContinuousMap =\n ({f | ↑f '' {1} ⊆ {1}ᶜ} ∪\n ⋃ (x : A) (y : A) (U : Set B) (V : Set B) (W : Set B) (_ : IsOpen U) (_ : IsOpen V) (_ : IsOpen W) (_ :\n Disjoint (U * V) W), {f | ↑f '' {x} ⊆ U} ∩ {f | ↑f '' {y} ⊆ V} ∩ {f | ↑f '' {x * y} ⊆ W})ᶜ\ni✝⁸ i✝⁷ : A\ni✝⁶ i✝⁵ i✝⁴ : Set B\ni✝³ : IsOpen i✝⁶\ni✝² : IsOpen i✝⁵\ni✝¹ : IsOpen i✝⁴\ni✝ : Disjoint (i✝⁶ * i✝⁵) i✝⁴\n⊢ IsOpen i✝⁴",
"tactic": "assumption"
},
{
"state_after": "case refine'_1.intro\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nf : ContinuousMonoidHom A B\n⊢ ¬toContinuousMap f ∈ {f | ¬↑f 1 ∈ {1}} ∧\n ∀ (i i_1 : A) (i_2 i_3 i_4 : Set B),\n IsOpen i_2 →\n IsOpen i_3 →\n IsOpen i_4 →\n Disjoint (i_2 * i_3) i_4 →\n ¬toContinuousMap f ∈ {f | ↑f i ∈ i_2} ∩ {f | ↑f i_1 ∈ i_3} ∩ {f | ↑f (i * i_1) ∈ i_4}",
"state_before": "case refine'_1\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nf : C(A, B)\n⊢ f ∈ Set.range toContinuousMap →\n ¬f ∈ {f | ¬↑f 1 ∈ {1}} ∧\n ∀ (i i_1 : A) (i_2 i_3 i_4 : Set B),\n IsOpen i_2 →\n IsOpen i_3 →\n IsOpen i_4 →\n Disjoint (i_2 * i_3) i_4 → ¬f ∈ {f | ↑f i ∈ i_2} ∩ {f | ↑f i_1 ∈ i_3} ∩ {f | ↑f (i * i_1) ∈ i_4}",
"tactic": "rintro ⟨f, rfl⟩"
},
{
"state_after": "case refine'_2.intro\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nf : C(A, B)\nhf1 : ¬f ∈ {f | ¬↑f 1 ∈ {1}}\nhf2 :\n ∀ (i i_1 : A) (i_2 i_3 i_4 : Set B),\n IsOpen i_2 →\n IsOpen i_3 →\n IsOpen i_4 → Disjoint (i_2 * i_3) i_4 → ¬f ∈ {f | ↑f i ∈ i_2} ∩ {f | ↑f i_1 ∈ i_3} ∩ {f | ↑f (i * i_1) ∈ i_4}\n⊢ f ∈ Set.range toContinuousMap",
"state_before": "case refine'_2\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nf : C(A, B)\n⊢ (¬f ∈ {f | ¬↑f 1 ∈ {1}} ∧\n ∀ (i i_1 : A) (i_2 i_3 i_4 : Set B),\n IsOpen i_2 →\n IsOpen i_3 →\n IsOpen i_4 →\n Disjoint (i_2 * i_3) i_4 → ¬f ∈ {f | ↑f i ∈ i_2} ∩ {f | ↑f i_1 ∈ i_3} ∩ {f | ↑f (i * i_1) ∈ i_4}) →\n f ∈ Set.range toContinuousMap",
"tactic": "rintro ⟨hf1, hf2⟩"
},
{
"state_after": "case refine'_2.intro\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nf : C(A, B)\nhf1 : ¬f ∈ {f | ¬↑f 1 ∈ {1}}\nhf2 :\n ∀ (i i_1 : A) (i_2 i_3 i_4 : Set B),\n IsOpen i_2 →\n IsOpen i_3 →\n IsOpen i_4 → Disjoint (i_2 * i_3) i_4 → ¬f ∈ {f | ↑f i ∈ i_2} ∩ {f | ↑f i_1 ∈ i_3} ∩ {f | ↑f (i * i_1) ∈ i_4}\n⊢ ∀ (x y : A), ↑f (x * y) = ↑f x * ↑f y",
"state_before": "case refine'_2.intro\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nf : C(A, B)\nhf1 : ¬f ∈ {f | ¬↑f 1 ∈ {1}}\nhf2 :\n ∀ (i i_1 : A) (i_2 i_3 i_4 : Set B),\n IsOpen i_2 →\n IsOpen i_3 →\n IsOpen i_4 → Disjoint (i_2 * i_3) i_4 → ¬f ∈ {f | ↑f i ∈ i_2} ∩ {f | ↑f i_1 ∈ i_3} ∩ {f | ↑f (i * i_1) ∈ i_4}\n⊢ f ∈ Set.range toContinuousMap",
"tactic": "suffices ∀ x y, f (x * y) = f x * f y by\n refine'\n ⟨({ f with\n map_one' := of_not_not hf1\n map_mul' := this } :\n ContinuousMonoidHom A B),\n ContinuousMap.ext fun _ => rfl⟩"
},
{
"state_after": "case refine'_2.intro\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nf : C(A, B)\nhf1 : ¬f ∈ {f | ¬↑f 1 ∈ {1}}\nhf2 :\n ∀ (i i_1 : A) (i_2 i_3 i_4 : Set B),\n IsOpen i_2 →\n IsOpen i_3 →\n IsOpen i_4 → Disjoint (i_2 * i_3) i_4 → ¬f ∈ {f | ↑f i ∈ i_2} ∩ {f | ↑f i_1 ∈ i_3} ∩ {f | ↑f (i * i_1) ∈ i_4}\nx y : A\n⊢ ↑f (x * y) = ↑f x * ↑f y",
"state_before": "case refine'_2.intro\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nf : C(A, B)\nhf1 : ¬f ∈ {f | ¬↑f 1 ∈ {1}}\nhf2 :\n ∀ (i i_1 : A) (i_2 i_3 i_4 : Set B),\n IsOpen i_2 →\n IsOpen i_3 →\n IsOpen i_4 → Disjoint (i_2 * i_3) i_4 → ¬f ∈ {f | ↑f i ∈ i_2} ∩ {f | ↑f i_1 ∈ i_3} ∩ {f | ↑f (i * i_1) ∈ i_4}\n⊢ ∀ (x y : A), ↑f (x * y) = ↑f x * ↑f y",
"tactic": "intro x y"
},
{
"state_after": "case refine'_2.intro\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nf : C(A, B)\nhf1 : ¬f ∈ {f | ¬↑f 1 ∈ {1}}\nx y : A\nhf2 : ↑f (x * y) ≠ ↑f x * ↑f y\n⊢ ∃ i i_1 i_2 i_3 i_4,\n IsOpen i_2 ∧\n IsOpen i_3 ∧\n IsOpen i_4 ∧ Disjoint (i_2 * i_3) i_4 ∧ f ∈ {f | ↑f i ∈ i_2} ∩ {f | ↑f i_1 ∈ i_3} ∩ {f | ↑f (i * i_1) ∈ i_4}",
"state_before": "case refine'_2.intro\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nf : C(A, B)\nhf1 : ¬f ∈ {f | ¬↑f 1 ∈ {1}}\nhf2 :\n ∀ (i i_1 : A) (i_2 i_3 i_4 : Set B),\n IsOpen i_2 →\n IsOpen i_3 →\n IsOpen i_4 → Disjoint (i_2 * i_3) i_4 → ¬f ∈ {f | ↑f i ∈ i_2} ∩ {f | ↑f i_1 ∈ i_3} ∩ {f | ↑f (i * i_1) ∈ i_4}\nx y : A\n⊢ ↑f (x * y) = ↑f x * ↑f y",
"tactic": "contrapose! hf2"
},
{
"state_after": "case refine'_2.intro.intro.intro.intro.intro.intro.intro\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nf : C(A, B)\nhf1 : ¬f ∈ {f | ¬↑f 1 ∈ {1}}\nx y : A\nhf2 : ↑f (x * y) ≠ ↑f x * ↑f y\nUV W : Set B\nhUV : IsOpen UV\nhW : IsOpen W\nhfUV : ↑f x * ↑f y ∈ UV\nhfW : ↑f (x * y) ∈ W\nh : Disjoint UV W\n⊢ ∃ i i_1 i_2 i_3 i_4,\n IsOpen i_2 ∧\n IsOpen i_3 ∧\n IsOpen i_4 ∧ Disjoint (i_2 * i_3) i_4 ∧ f ∈ {f | ↑f i ∈ i_2} ∩ {f | ↑f i_1 ∈ i_3} ∩ {f | ↑f (i * i_1) ∈ i_4}",
"state_before": "case refine'_2.intro\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nf : C(A, B)\nhf1 : ¬f ∈ {f | ¬↑f 1 ∈ {1}}\nx y : A\nhf2 : ↑f (x * y) ≠ ↑f x * ↑f y\n⊢ ∃ i i_1 i_2 i_3 i_4,\n IsOpen i_2 ∧\n IsOpen i_3 ∧\n IsOpen i_4 ∧ Disjoint (i_2 * i_3) i_4 ∧ f ∈ {f | ↑f i ∈ i_2} ∩ {f | ↑f i_1 ∈ i_3} ∩ {f | ↑f (i * i_1) ∈ i_4}",
"tactic": "obtain ⟨UV, W, hUV, hW, hfUV, hfW, h⟩ := t2_separation hf2.symm"
},
{
"state_after": "case refine'_2.intro.intro.intro.intro.intro.intro.intro\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nf : C(A, B)\nhf1 : ¬f ∈ {f | ¬↑f 1 ∈ {1}}\nx y : A\nhf2 : ↑f (x * y) ≠ ↑f x * ↑f y\nUV W : Set B\nhUV : IsOpen UV\nhW : IsOpen W\nhfUV : ↑f x * ↑f y ∈ UV\nhfW : ↑f (x * y) ∈ W\nh : Disjoint UV W\nhB : Continuous fun p => p.fst * p.snd\n⊢ ∃ i i_1 i_2 i_3 i_4,\n IsOpen i_2 ∧\n IsOpen i_3 ∧\n IsOpen i_4 ∧ Disjoint (i_2 * i_3) i_4 ∧ f ∈ {f | ↑f i ∈ i_2} ∩ {f | ↑f i_1 ∈ i_3} ∩ {f | ↑f (i * i_1) ∈ i_4}",
"state_before": "case refine'_2.intro.intro.intro.intro.intro.intro.intro\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nf : C(A, B)\nhf1 : ¬f ∈ {f | ¬↑f 1 ∈ {1}}\nx y : A\nhf2 : ↑f (x * y) ≠ ↑f x * ↑f y\nUV W : Set B\nhUV : IsOpen UV\nhW : IsOpen W\nhfUV : ↑f x * ↑f y ∈ UV\nhfW : ↑f (x * y) ∈ W\nh : Disjoint UV W\n⊢ ∃ i i_1 i_2 i_3 i_4,\n IsOpen i_2 ∧\n IsOpen i_3 ∧\n IsOpen i_4 ∧ Disjoint (i_2 * i_3) i_4 ∧ f ∈ {f | ↑f i ∈ i_2} ∩ {f | ↑f i_1 ∈ i_3} ∩ {f | ↑f (i * i_1) ∈ i_4}",
"tactic": "have hB := @continuous_mul B _ _ _"
},
{
"state_after": "case refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nf : C(A, B)\nhf1 : ¬f ∈ {f | ¬↑f 1 ∈ {1}}\nx y : A\nhf2 : ↑f (x * y) ≠ ↑f x * ↑f y\nUV W : Set B\nhUV : IsOpen UV\nhW : IsOpen W\nhfUV : ↑f x * ↑f y ∈ UV\nhfW : ↑f (x * y) ∈ W\nh : Disjoint UV W\nhB : Continuous fun p => p.fst * p.snd\nU V : Set B\nhU : IsOpen U\nhV : IsOpen V\nhfU : ↑f x ∈ U\nhfV : ↑f y ∈ V\nh' : U ×ˢ V ⊆ (fun p => p.fst * p.snd) ⁻¹' UV\n⊢ ∃ i i_1 i_2 i_3 i_4,\n IsOpen i_2 ∧\n IsOpen i_3 ∧\n IsOpen i_4 ∧ Disjoint (i_2 * i_3) i_4 ∧ f ∈ {f | ↑f i ∈ i_2} ∩ {f | ↑f i_1 ∈ i_3} ∩ {f | ↑f (i * i_1) ∈ i_4}",
"state_before": "case refine'_2.intro.intro.intro.intro.intro.intro.intro\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nf : C(A, B)\nhf1 : ¬f ∈ {f | ¬↑f 1 ∈ {1}}\nx y : A\nhf2 : ↑f (x * y) ≠ ↑f x * ↑f y\nUV W : Set B\nhUV : IsOpen UV\nhW : IsOpen W\nhfUV : ↑f x * ↑f y ∈ UV\nhfW : ↑f (x * y) ∈ W\nh : Disjoint UV W\nhB : Continuous fun p => p.fst * p.snd\n⊢ ∃ i i_1 i_2 i_3 i_4,\n IsOpen i_2 ∧\n IsOpen i_3 ∧\n IsOpen i_4 ∧ Disjoint (i_2 * i_3) i_4 ∧ f ∈ {f | ↑f i ∈ i_2} ∩ {f | ↑f i_1 ∈ i_3} ∩ {f | ↑f (i * i_1) ∈ i_4}",
"tactic": "obtain ⟨U, V, hU, hV, hfU, hfV, h'⟩ :=\n isOpen_prod_iff.mp (hUV.preimage hB) (f x) (f y) hfUV"
},
{
"state_after": "case refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nf : C(A, B)\nhf1 : ¬f ∈ {f | ¬↑f 1 ∈ {1}}\nx y : A\nhf2 : ↑f (x * y) ≠ ↑f x * ↑f y\nUV W : Set B\nhUV : IsOpen UV\nhW : IsOpen W\nhfUV : ↑f x * ↑f y ∈ UV\nhfW : ↑f (x * y) ∈ W\nh : Disjoint UV W\nhB : Continuous fun p => p.fst * p.snd\nU V : Set B\nhU : IsOpen U\nhV : IsOpen V\nhfU : ↑f x ∈ U\nhfV : ↑f y ∈ V\nh' : U ×ˢ V ⊆ (fun p => p.fst * p.snd) ⁻¹' UV\n⊢ U * V ≤ UV",
"state_before": "case refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nf : C(A, B)\nhf1 : ¬f ∈ {f | ¬↑f 1 ∈ {1}}\nx y : A\nhf2 : ↑f (x * y) ≠ ↑f x * ↑f y\nUV W : Set B\nhUV : IsOpen UV\nhW : IsOpen W\nhfUV : ↑f x * ↑f y ∈ UV\nhfW : ↑f (x * y) ∈ W\nh : Disjoint UV W\nhB : Continuous fun p => p.fst * p.snd\nU V : Set B\nhU : IsOpen U\nhV : IsOpen V\nhfU : ↑f x ∈ U\nhfV : ↑f y ∈ V\nh' : U ×ˢ V ⊆ (fun p => p.fst * p.snd) ⁻¹' UV\n⊢ ∃ i i_1 i_2 i_3 i_4,\n IsOpen i_2 ∧\n IsOpen i_3 ∧\n IsOpen i_4 ∧ Disjoint (i_2 * i_3) i_4 ∧ f ∈ {f | ↑f i ∈ i_2} ∩ {f | ↑f i_1 ∈ i_3} ∩ {f | ↑f (i * i_1) ∈ i_4}",
"tactic": "refine' ⟨x, y, U, V, W, hU, hV, hW, h.mono_left _, ⟨hfU, hfV⟩, hfW⟩"
},
{
"state_after": "case refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nf : C(A, B)\nhf1 : ¬f ∈ {f | ¬↑f 1 ∈ {1}}\nx✝ y✝ : A\nhf2 : ↑f (x✝ * y✝) ≠ ↑f x✝ * ↑f y✝\nUV W : Set B\nhUV : IsOpen UV\nhW : IsOpen W\nhfUV : ↑f x✝ * ↑f y✝ ∈ UV\nhfW : ↑f (x✝ * y✝) ∈ W\nh : Disjoint UV W\nhB : Continuous fun p => p.fst * p.snd\nU V : Set B\nhU : IsOpen U\nhV : IsOpen V\nhfU : ↑f x✝ ∈ U\nhfV : ↑f y✝ ∈ V\nh' : U ×ˢ V ⊆ (fun p => p.fst * p.snd) ⁻¹' UV\nx y : B\nhx : (x, y).fst ∈ U\nhy : (x, y).snd ∈ V\n⊢ (fun x x_1 => x * x_1) x y ∈ UV",
"state_before": "case refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nf : C(A, B)\nhf1 : ¬f ∈ {f | ¬↑f 1 ∈ {1}}\nx y : A\nhf2 : ↑f (x * y) ≠ ↑f x * ↑f y\nUV W : Set B\nhUV : IsOpen UV\nhW : IsOpen W\nhfUV : ↑f x * ↑f y ∈ UV\nhfW : ↑f (x * y) ∈ W\nh : Disjoint UV W\nhB : Continuous fun p => p.fst * p.snd\nU V : Set B\nhU : IsOpen U\nhV : IsOpen V\nhfU : ↑f x ∈ U\nhfV : ↑f y ∈ V\nh' : U ×ˢ V ⊆ (fun p => p.fst * p.snd) ⁻¹' UV\n⊢ U * V ≤ UV",
"tactic": "rintro _ ⟨x, y, hx : (x, y).1 ∈ U, hy : (x, y).2 ∈ V, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nF : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nf : C(A, B)\nhf1 : ¬f ∈ {f | ¬↑f 1 ∈ {1}}\nx✝ y✝ : A\nhf2 : ↑f (x✝ * y✝) ≠ ↑f x✝ * ↑f y✝\nUV W : Set B\nhUV : IsOpen UV\nhW : IsOpen W\nhfUV : ↑f x✝ * ↑f y✝ ∈ UV\nhfW : ↑f (x✝ * y✝) ∈ W\nh : Disjoint UV W\nhB : Continuous fun p => p.fst * p.snd\nU V : Set B\nhU : IsOpen U\nhV : IsOpen V\nhfU : ↑f x✝ ∈ U\nhfV : ↑f y✝ ∈ V\nh' : U ×ˢ V ⊆ (fun p => p.fst * p.snd) ⁻¹' UV\nx y : B\nhx : (x, y).fst ∈ U\nhy : (x, y).snd ∈ V\n⊢ (fun x x_1 => x * x_1) x y ∈ UV",
"tactic": "exact h' ⟨hx, hy⟩"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.142376\nA : Type u_2\nB : Type u_1\nC : Type ?u.142385\nD : Type ?u.142388\nE : Type ?u.142391\ninst✝¹² : Monoid A\ninst✝¹¹ : Monoid B\ninst✝¹⁰ : Monoid C\ninst✝⁹ : Monoid D\ninst✝⁸ : CommGroup E\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace C\ninst✝⁴ : TopologicalSpace D\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalGroup E\ninst✝¹ : ContinuousMul B\ninst✝ : T2Space B\nf : C(A, B)\nhf1 : ¬f ∈ {f | ¬↑f 1 ∈ {1}}\nhf2 :\n ∀ (i i_1 : A) (i_2 i_3 i_4 : Set B),\n IsOpen i_2 →\n IsOpen i_3 →\n IsOpen i_4 → Disjoint (i_2 * i_3) i_4 → ¬f ∈ {f | ↑f i ∈ i_2} ∩ {f | ↑f i_1 ∈ i_3} ∩ {f | ↑f (i * i_1) ∈ i_4}\nthis : ∀ (x y : A), ↑f (x * y) = ↑f x * ↑f y\n⊢ f ∈ Set.range toContinuousMap",
"tactic": "refine'\n ⟨({ f with\n map_one' := of_not_not hf1\n map_mul' := this } :\n ContinuousMonoidHom A B),\n ContinuousMap.ext fun _ => rfl⟩"
}
] |
[
332,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
293,
1
] |
Mathlib/Data/List/Rotate.lean
|
List.rotate_reverse
|
[
{
"state_after": "α : Type u\nl : List α\nn : ℕ\n⊢ rotate (reverse (reverse (reverse l))) n =\n reverse (rotate (reverse (reverse l)) (length (reverse (reverse l)) - n % length (reverse (reverse l))))",
"state_before": "α : Type u\nl : List α\nn : ℕ\n⊢ rotate (reverse l) n = reverse (rotate l (length l - n % length l))",
"tactic": "rw [← reverse_reverse l]"
},
{
"state_after": "α : Type u\nl : List α\nn : ℕ\n⊢ reverse l = rotate (reverse l) (length l - (length l - n % length l) % length l + (length l - n % length l))",
"state_before": "α : Type u\nl : List α\nn : ℕ\n⊢ rotate (reverse (reverse (reverse l))) n =\n reverse (rotate (reverse (reverse l)) (length (reverse (reverse l)) - n % length (reverse (reverse l))))",
"tactic": "simp_rw [reverse_rotate, reverse_reverse, rotate_eq_iff, rotate_rotate, length_rotate,\n length_reverse]"
},
{
"state_after": "α : Type u\nl : List α\nn : ℕ\n⊢ reverse l =\n rotate (reverse l)\n (length (reverse l) - (length (reverse l) - n % length (reverse l)) % length (reverse l) +\n (length (reverse l) - n % length (reverse l)))",
"state_before": "α : Type u\nl : List α\nn : ℕ\n⊢ reverse l = rotate (reverse l) (length l - (length l - n % length l) % length l + (length l - n % length l))",
"tactic": "rw [← length_reverse l]"
},
{
"state_after": "α : Type u\nl : List α\nn : ℕ\nk : ℕ := n % length (reverse l)\n⊢ reverse l =\n rotate (reverse l)\n (length (reverse l) - (length (reverse l) - n % length (reverse l)) % length (reverse l) +\n (length (reverse l) - n % length (reverse l)))",
"state_before": "α : Type u\nl : List α\nn : ℕ\n⊢ reverse l =\n rotate (reverse l)\n (length (reverse l) - (length (reverse l) - n % length (reverse l)) % length (reverse l) +\n (length (reverse l) - n % length (reverse l)))",
"tactic": "let k := n % l.reverse.length"
},
{
"state_after": "case zero\nα : Type u\nl : List α\nn : ℕ\nk : ℕ := n % length (reverse l)\nhk' : k = zero\n⊢ reverse l =\n rotate (reverse l)\n (length (reverse l) - (length (reverse l) - n % length (reverse l)) % length (reverse l) +\n (length (reverse l) - n % length (reverse l)))\n\ncase succ\nα : Type u\nl : List α\nn : ℕ\nk : ℕ := n % length (reverse l)\nk' : ℕ\nhk' : k = succ k'\n⊢ reverse l =\n rotate (reverse l)\n (length (reverse l) - (length (reverse l) - n % length (reverse l)) % length (reverse l) +\n (length (reverse l) - n % length (reverse l)))",
"state_before": "α : Type u\nl : List α\nn : ℕ\nk : ℕ := n % length (reverse l)\n⊢ reverse l =\n rotate (reverse l)\n (length (reverse l) - (length (reverse l) - n % length (reverse l)) % length (reverse l) +\n (length (reverse l) - n % length (reverse l)))",
"tactic": "cases' hk' : k with k'"
},
{
"state_after": "no goals",
"state_before": "case zero\nα : Type u\nl : List α\nn : ℕ\nk : ℕ := n % length (reverse l)\nhk' : k = zero\n⊢ reverse l =\n rotate (reverse l)\n (length (reverse l) - (length (reverse l) - n % length (reverse l)) % length (reverse l) +\n (length (reverse l) - n % length (reverse l)))",
"tactic": "simp_all! [length_reverse, ← rotate_rotate]"
},
{
"state_after": "case succ.nil\nα : Type u\nn k' : ℕ\nk : ℕ := n % length (reverse [])\nhk' : k = succ k'\n⊢ reverse [] =\n rotate (reverse [])\n (length (reverse []) - (length (reverse []) - n % length (reverse [])) % length (reverse []) +\n (length (reverse []) - n % length (reverse [])))\n\ncase succ.cons\nα : Type u\nn k' : ℕ\nx : α\nl : List α\nk : ℕ := n % length (reverse (x :: l))\nhk' : k = succ k'\n⊢ reverse (x :: l) =\n rotate (reverse (x :: l))\n (length (reverse (x :: l)) -\n (length (reverse (x :: l)) - n % length (reverse (x :: l))) % length (reverse (x :: l)) +\n (length (reverse (x :: l)) - n % length (reverse (x :: l))))",
"state_before": "case succ\nα : Type u\nl : List α\nn : ℕ\nk : ℕ := n % length (reverse l)\nk' : ℕ\nhk' : k = succ k'\n⊢ reverse l =\n rotate (reverse l)\n (length (reverse l) - (length (reverse l) - n % length (reverse l)) % length (reverse l) +\n (length (reverse l) - n % length (reverse l)))",
"tactic": "cases' l with x l"
},
{
"state_after": "no goals",
"state_before": "case succ.nil\nα : Type u\nn k' : ℕ\nk : ℕ := n % length (reverse [])\nhk' : k = succ k'\n⊢ reverse [] =\n rotate (reverse [])\n (length (reverse []) - (length (reverse []) - n % length (reverse [])) % length (reverse []) +\n (length (reverse []) - n % length (reverse [])))",
"tactic": "simp"
},
{
"state_after": "case succ.cons\nα : Type u\nn k' : ℕ\nx : α\nl : List α\nk : ℕ := n % length (reverse (x :: l))\nhk' : k = succ k'\n⊢ length (reverse (x :: l)) - n % length (reverse (x :: l)) ≤ length (reverse (x :: l))\n\ncase succ.cons\nα : Type u\nn k' : ℕ\nx : α\nl : List α\nk : ℕ := n % length (reverse (x :: l))\nhk' : k = succ k'\n⊢ length (reverse (x :: l)) - n % length (reverse (x :: l)) < length (reverse (x :: l))",
"state_before": "case succ.cons\nα : Type u\nn k' : ℕ\nx : α\nl : List α\nk : ℕ := n % length (reverse (x :: l))\nhk' : k = succ k'\n⊢ reverse (x :: l) =\n rotate (reverse (x :: l))\n (length (reverse (x :: l)) -\n (length (reverse (x :: l)) - n % length (reverse (x :: l))) % length (reverse (x :: l)) +\n (length (reverse (x :: l)) - n % length (reverse (x :: l))))",
"tactic": "rw [Nat.mod_eq_of_lt, tsub_add_cancel_of_le, rotate_length]"
},
{
"state_after": "no goals",
"state_before": "case succ.cons\nα : Type u\nn k' : ℕ\nx : α\nl : List α\nk : ℕ := n % length (reverse (x :: l))\nhk' : k = succ k'\n⊢ length (reverse (x :: l)) - n % length (reverse (x :: l)) ≤ length (reverse (x :: l))",
"tactic": "exact tsub_le_self"
},
{
"state_after": "no goals",
"state_before": "case succ.cons\nα : Type u\nn k' : ℕ\nx : α\nl : List α\nk : ℕ := n % length (reverse (x :: l))\nhk' : k = succ k'\n⊢ length (reverse (x :: l)) - n % length (reverse (x :: l)) < length (reverse (x :: l))",
"tactic": "exact tsub_lt_self (by simp) (by simp_all!)"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nn k' : ℕ\nx : α\nl : List α\nk : ℕ := n % length (reverse (x :: l))\nhk' : k = succ k'\n⊢ 0 < length (reverse (x :: l))",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nn k' : ℕ\nx : α\nl : List α\nk : ℕ := n % length (reverse (x :: l))\nhk' : k = succ k'\n⊢ 0 < n % length (reverse (x :: l))",
"tactic": "simp_all!"
}
] |
[
384,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
371,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.mem_or_mem_of_mem_union
|
[] |
[
746,
4
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
745,
1
] |
Mathlib/Order/JordanHolder.lean
|
JordanHolderLattice.isMaximal_of_eq_inf
|
[
{
"state_after": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\nx b a y : X\nha : x ⊓ y = a\nhxy : x ≠ y\nhxb : IsMaximal x b\nhyb : IsMaximal y b\nhb : x ⊔ y = b\n⊢ IsMaximal a y",
"state_before": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\nx b a y : X\nha : x ⊓ y = a\nhxy : x ≠ y\nhxb : IsMaximal x b\nhyb : IsMaximal y b\n⊢ IsMaximal a y",
"tactic": "have hb : x ⊔ y = b := sup_eq_of_isMaximal hxb hyb hxy"
},
{
"state_after": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\nx y : X\nhxy : x ≠ y\nhxb : IsMaximal x (x ⊔ y)\nhyb : IsMaximal y (x ⊔ y)\n⊢ IsMaximal (x ⊓ y) y",
"state_before": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\nx b a y : X\nha : x ⊓ y = a\nhxy : x ≠ y\nhxb : IsMaximal x b\nhyb : IsMaximal y b\nhb : x ⊔ y = b\n⊢ IsMaximal a y",
"tactic": "substs a b"
},
{
"state_after": "no goals",
"state_before": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\nx y : X\nhxy : x ≠ y\nhxb : IsMaximal x (x ⊔ y)\nhyb : IsMaximal y (x ⊔ y)\n⊢ IsMaximal (x ⊓ y) y",
"tactic": "exact isMaximal_inf_right_of_isMaximal_sup hxb hyb"
}
] |
[
111,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.monotoneOn_singleton
|
[] |
[
2694,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2693,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean
|
MvPolynomial.X_injective
|
[] |
[
290,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
289,
1
] |
Mathlib/Analysis/InnerProductSpace/Orthogonal.lean
|
Submodule.IsOrtho.mono
|
[] |
[
300,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
298,
1
] |
Mathlib/MeasureTheory/Function/LocallyIntegrable.lean
|
MeasureTheory.IntegrableOn.smul_continuousOn
|
[
{
"state_after": "X : Type u_1\nY : Type ?u.2491139\nE : Type u_2\nR : Type ?u.2491145\ninst✝⁹ : MeasurableSpace X\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : MeasurableSpace Y\ninst✝⁶ : TopologicalSpace Y\ninst✝⁵ : NormedAddCommGroup E\nf✝ : X → E\nμ : Measure X\ns : Set X\ninst✝⁴ : OpensMeasurableSpace X\nA K : Set X\n𝕜 : Type u_3\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : T2Space X\ninst✝ : SecondCountableTopologyEither X E\nf : X → 𝕜\nhf : IntegrableOn f K\ng : X → E\nhg : ContinuousOn g K\nhK : IsCompact K\n⊢ Integrable fun a => ‖f a • g a‖\n\nX : Type u_1\nY : Type ?u.2491139\nE : Type u_2\nR : Type ?u.2491145\ninst✝⁹ : MeasurableSpace X\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : MeasurableSpace Y\ninst✝⁶ : TopologicalSpace Y\ninst✝⁵ : NormedAddCommGroup E\nf✝ : X → E\nμ : Measure X\ns : Set X\ninst✝⁴ : OpensMeasurableSpace X\nA K : Set X\n𝕜 : Type u_3\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : T2Space X\ninst✝ : SecondCountableTopologyEither X E\nf : X → 𝕜\nhf : IntegrableOn f K\ng : X → E\nhg : ContinuousOn g K\nhK : IsCompact K\n⊢ AEStronglyMeasurable (fun x => f x • g x) (Measure.restrict μ K)",
"state_before": "X : Type u_1\nY : Type ?u.2491139\nE : Type u_2\nR : Type ?u.2491145\ninst✝⁹ : MeasurableSpace X\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : MeasurableSpace Y\ninst✝⁶ : TopologicalSpace Y\ninst✝⁵ : NormedAddCommGroup E\nf✝ : X → E\nμ : Measure X\ns : Set X\ninst✝⁴ : OpensMeasurableSpace X\nA K : Set X\n𝕜 : Type u_3\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : T2Space X\ninst✝ : SecondCountableTopologyEither X E\nf : X → 𝕜\nhf : IntegrableOn f K\ng : X → E\nhg : ContinuousOn g K\nhK : IsCompact K\n⊢ IntegrableOn (fun x => f x • g x) K",
"tactic": "rw [IntegrableOn, ← integrable_norm_iff]"
},
{
"state_after": "X : Type u_1\nY : Type ?u.2491139\nE : Type u_2\nR : Type ?u.2491145\ninst✝⁹ : MeasurableSpace X\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : MeasurableSpace Y\ninst✝⁶ : TopologicalSpace Y\ninst✝⁵ : NormedAddCommGroup E\nf✝ : X → E\nμ : Measure X\ns : Set X\ninst✝⁴ : OpensMeasurableSpace X\nA K : Set X\n𝕜 : Type u_3\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : T2Space X\ninst✝ : SecondCountableTopologyEither X E\nf : X → 𝕜\nhf : IntegrableOn f K\ng : X → E\nhg : ContinuousOn g K\nhK : IsCompact K\n⊢ Integrable fun a => ‖f a‖ * ‖g a‖",
"state_before": "X : Type u_1\nY : Type ?u.2491139\nE : Type u_2\nR : Type ?u.2491145\ninst✝⁹ : MeasurableSpace X\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : MeasurableSpace Y\ninst✝⁶ : TopologicalSpace Y\ninst✝⁵ : NormedAddCommGroup E\nf✝ : X → E\nμ : Measure X\ns : Set X\ninst✝⁴ : OpensMeasurableSpace X\nA K : Set X\n𝕜 : Type u_3\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : T2Space X\ninst✝ : SecondCountableTopologyEither X E\nf : X → 𝕜\nhf : IntegrableOn f K\ng : X → E\nhg : ContinuousOn g K\nhK : IsCompact K\n⊢ Integrable fun a => ‖f a • g a‖",
"tactic": "simp_rw [norm_smul]"
},
{
"state_after": "X : Type u_1\nY : Type ?u.2491139\nE : Type u_2\nR : Type ?u.2491145\ninst✝⁹ : MeasurableSpace X\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : MeasurableSpace Y\ninst✝⁶ : TopologicalSpace Y\ninst✝⁵ : NormedAddCommGroup E\nf✝ : X → E\nμ : Measure X\ns : Set X\ninst✝⁴ : OpensMeasurableSpace X\nA K : Set X\n𝕜 : Type u_3\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : T2Space X\ninst✝ : SecondCountableTopologyEither X E\nf : X → 𝕜\nhf : IntegrableOn f K\ng : X → E\nhg : ContinuousOn g K\nhK : IsCompact K\n⊢ ContinuousOn (fun a => ‖g a‖) K",
"state_before": "X : Type u_1\nY : Type ?u.2491139\nE : Type u_2\nR : Type ?u.2491145\ninst✝⁹ : MeasurableSpace X\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : MeasurableSpace Y\ninst✝⁶ : TopologicalSpace Y\ninst✝⁵ : NormedAddCommGroup E\nf✝ : X → E\nμ : Measure X\ns : Set X\ninst✝⁴ : OpensMeasurableSpace X\nA K : Set X\n𝕜 : Type u_3\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : T2Space X\ninst✝ : SecondCountableTopologyEither X E\nf : X → 𝕜\nhf : IntegrableOn f K\ng : X → E\nhg : ContinuousOn g K\nhK : IsCompact K\n⊢ Integrable fun a => ‖f a‖ * ‖g a‖",
"tactic": "refine' IntegrableOn.mul_continuousOn hf.norm _ hK"
},
{
"state_after": "no goals",
"state_before": "X : Type u_1\nY : Type ?u.2491139\nE : Type u_2\nR : Type ?u.2491145\ninst✝⁹ : MeasurableSpace X\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : MeasurableSpace Y\ninst✝⁶ : TopologicalSpace Y\ninst✝⁵ : NormedAddCommGroup E\nf✝ : X → E\nμ : Measure X\ns : Set X\ninst✝⁴ : OpensMeasurableSpace X\nA K : Set X\n𝕜 : Type u_3\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : T2Space X\ninst✝ : SecondCountableTopologyEither X E\nf : X → 𝕜\nhf : IntegrableOn f K\ng : X → E\nhg : ContinuousOn g K\nhK : IsCompact K\n⊢ ContinuousOn (fun a => ‖g a‖) K",
"tactic": "exact continuous_norm.comp_continuousOn hg"
},
{
"state_after": "no goals",
"state_before": "X : Type u_1\nY : Type ?u.2491139\nE : Type u_2\nR : Type ?u.2491145\ninst✝⁹ : MeasurableSpace X\ninst✝⁸ : TopologicalSpace X\ninst✝⁷ : MeasurableSpace Y\ninst✝⁶ : TopologicalSpace Y\ninst✝⁵ : NormedAddCommGroup E\nf✝ : X → E\nμ : Measure X\ns : Set X\ninst✝⁴ : OpensMeasurableSpace X\nA K : Set X\n𝕜 : Type u_3\ninst✝³ : NormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : T2Space X\ninst✝ : SecondCountableTopologyEither X E\nf : X → 𝕜\nhf : IntegrableOn f K\ng : X → E\nhg : ContinuousOn g K\nhK : IsCompact K\n⊢ AEStronglyMeasurable (fun x => f x • g x) (Measure.restrict μ K)",
"tactic": "exact hf.1.smul (hg.aestronglyMeasurable hK.measurableSet)"
}
] |
[
445,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
438,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
disjoint_compl_left
|
[] |
[
844,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
843,
1
] |
Mathlib/Order/Hom/CompleteLattice.lean
|
FrameHom.id_apply
|
[] |
[
605,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
604,
1
] |
Mathlib/Geometry/Euclidean/Sphere/Basic.lean
|
EuclideanGeometry.Concyclic.subset
|
[] |
[
217,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Mathlib/Data/List/AList.lean
|
AList.ext_iff
|
[] |
[
72,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.val_mul'
|
[] |
[
83,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
1
] |
Mathlib/LinearAlgebra/FiniteDimensional.lean
|
Subalgebra.rank_eq_one_iff
|
[] |
[
1410,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1408,
1
] |
Mathlib/Topology/Category/Compactum.lean
|
compactumToCompHaus.faithful
|
[
{
"state_after": "X✝ Y✝ : Compactum\na₁✝ a₂✝ : X✝ ⟶ Y✝\nh : compactumToCompHaus.map a₁✝ = compactumToCompHaus.map a₂✝\n⊢ a₁✝ = a₂✝",
"state_before": "⊢ ∀ {X Y : Compactum}, Function.Injective compactumToCompHaus.map",
"tactic": "intro _ _ _ _ h"
},
{
"state_after": "case f\nX✝ Y✝ : Compactum\na₁✝ a₂✝ : X✝ ⟶ Y✝\nh : compactumToCompHaus.map a₁✝ = compactumToCompHaus.map a₂✝\n⊢ a₁✝.f = a₂✝.f",
"state_before": "X✝ Y✝ : Compactum\na₁✝ a₂✝ : X✝ ⟶ Y✝\nh : compactumToCompHaus.map a₁✝ = compactumToCompHaus.map a₂✝\n⊢ a₁✝ = a₂✝",
"tactic": "apply Monad.Algebra.Hom.ext"
},
{
"state_after": "no goals",
"state_before": "case f\nX✝ Y✝ : Compactum\na₁✝ a₂✝ : X✝ ⟶ Y✝\nh : compactumToCompHaus.map a₁✝ = compactumToCompHaus.map a₂✝\n⊢ a₁✝.f = a₂✝.f",
"tactic": "apply congrArg (fun f => f.toFun) h"
}
] |
[
462,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
456,
1
] |
Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean
|
isBoundedBilinearMap_smul
|
[] |
[
444,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
441,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.limsup_lintegral_le
|
[
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type ?u.996569\nγ : Type ?u.996572\nδ : Type ?u.996575\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\ng : α → ℝ≥0∞\nhf_meas : ∀ (n : ℕ), Measurable (f n)\nh_bound : ∀ (n : ℕ), f n ≤ᵐ[μ] g\nh_fin : (∫⁻ (a : α), g a ∂μ) ≠ ⊤\n⊢ ∀ (n : ℕ), Measurable fun a => ⨆ (i : ℕ) (_ : i ≥ n), f i a\n\ncase refine'_2\nα : Type u_1\nβ : Type ?u.996569\nγ : Type ?u.996572\nδ : Type ?u.996575\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\ng : α → ℝ≥0∞\nhf_meas : ∀ (n : ℕ), Measurable (f n)\nh_bound : ∀ (n : ℕ), f n ≤ᵐ[μ] g\nh_fin : (∫⁻ (a : α), g a ∂μ) ≠ ⊤\n⊢ Antitone fun n a => ⨆ (i : ℕ) (_ : i ≥ n), f i a\n\ncase refine'_3\nα : Type u_1\nβ : Type ?u.996569\nγ : Type ?u.996572\nδ : Type ?u.996575\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\ng : α → ℝ≥0∞\nhf_meas : ∀ (n : ℕ), Measurable (f n)\nh_bound : ∀ (n : ℕ), f n ≤ᵐ[μ] g\nh_fin : (∫⁻ (a : α), g a ∂μ) ≠ ⊤\n⊢ (∫⁻ (a : α), ⨆ (i : ℕ) (_ : i ≥ 0), f i a ∂μ) ≠ ⊤",
"state_before": "α : Type u_1\nβ : Type ?u.996569\nγ : Type ?u.996572\nδ : Type ?u.996575\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\ng : α → ℝ≥0∞\nhf_meas : ∀ (n : ℕ), Measurable (f n)\nh_bound : ∀ (n : ℕ), f n ≤ᵐ[μ] g\nh_fin : (∫⁻ (a : α), g a ∂μ) ≠ ⊤\n⊢ (⨅ (n : ℕ), ∫⁻ (a : α), ⨆ (i : ℕ) (_ : i ≥ n), f i a ∂μ) = ∫⁻ (a : α), ⨅ (n : ℕ), ⨆ (i : ℕ) (_ : i ≥ n), f i a ∂μ",
"tactic": "refine' (lintegral_iInf _ _ _).symm"
},
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type ?u.996569\nγ : Type ?u.996572\nδ : Type ?u.996575\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\ng : α → ℝ≥0∞\nhf_meas : ∀ (n : ℕ), Measurable (f n)\nh_bound : ∀ (n : ℕ), f n ≤ᵐ[μ] g\nh_fin : (∫⁻ (a : α), g a ∂μ) ≠ ⊤\nn : ℕ\n⊢ Measurable fun a => ⨆ (i : ℕ) (_ : i ≥ n), f i a",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type ?u.996569\nγ : Type ?u.996572\nδ : Type ?u.996575\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\ng : α → ℝ≥0∞\nhf_meas : ∀ (n : ℕ), Measurable (f n)\nh_bound : ∀ (n : ℕ), f n ≤ᵐ[μ] g\nh_fin : (∫⁻ (a : α), g a ∂μ) ≠ ⊤\n⊢ ∀ (n : ℕ), Measurable fun a => ⨆ (i : ℕ) (_ : i ≥ n), f i a",
"tactic": "intro n"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type ?u.996569\nγ : Type ?u.996572\nδ : Type ?u.996575\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\ng : α → ℝ≥0∞\nhf_meas : ∀ (n : ℕ), Measurable (f n)\nh_bound : ∀ (n : ℕ), f n ≤ᵐ[μ] g\nh_fin : (∫⁻ (a : α), g a ∂μ) ≠ ⊤\nn : ℕ\n⊢ Measurable fun a => ⨆ (i : ℕ) (_ : i ≥ n), f i a",
"tactic": "exact measurable_biSup _ (to_countable _) hf_meas"
},
{
"state_after": "case refine'_2\nα : Type u_1\nβ : Type ?u.996569\nγ : Type ?u.996572\nδ : Type ?u.996575\nm✝ : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\ng : α → ℝ≥0∞\nhf_meas : ∀ (n : ℕ), Measurable (f n)\nh_bound : ∀ (n : ℕ), f n ≤ᵐ[μ] g\nh_fin : (∫⁻ (a : α), g a ∂μ) ≠ ⊤\nn m : ℕ\nhnm : n ≤ m\na : α\n⊢ (fun n a => ⨆ (i : ℕ) (_ : i ≥ n), f i a) m a ≤ (fun n a => ⨆ (i : ℕ) (_ : i ≥ n), f i a) n a",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type ?u.996569\nγ : Type ?u.996572\nδ : Type ?u.996575\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\ng : α → ℝ≥0∞\nhf_meas : ∀ (n : ℕ), Measurable (f n)\nh_bound : ∀ (n : ℕ), f n ≤ᵐ[μ] g\nh_fin : (∫⁻ (a : α), g a ∂μ) ≠ ⊤\n⊢ Antitone fun n a => ⨆ (i : ℕ) (_ : i ≥ n), f i a",
"tactic": "intro n m hnm a"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type ?u.996569\nγ : Type ?u.996572\nδ : Type ?u.996575\nm✝ : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\ng : α → ℝ≥0∞\nhf_meas : ∀ (n : ℕ), Measurable (f n)\nh_bound : ∀ (n : ℕ), f n ≤ᵐ[μ] g\nh_fin : (∫⁻ (a : α), g a ∂μ) ≠ ⊤\nn m : ℕ\nhnm : n ≤ m\na : α\n⊢ (fun n a => ⨆ (i : ℕ) (_ : i ≥ n), f i a) m a ≤ (fun n a => ⨆ (i : ℕ) (_ : i ≥ n), f i a) n a",
"tactic": "exact iSup_le_iSup_of_subset fun i hi => le_trans hnm hi"
},
{
"state_after": "case refine'_3\nα : Type u_1\nβ : Type ?u.996569\nγ : Type ?u.996572\nδ : Type ?u.996575\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\ng : α → ℝ≥0∞\nhf_meas : ∀ (n : ℕ), Measurable (f n)\nh_bound : ∀ (n : ℕ), f n ≤ᵐ[μ] g\nh_fin : (∫⁻ (a : α), g a ∂μ) ≠ ⊤\n⊢ ∀ᵐ (a : α) ∂μ, (⨆ (i : ℕ) (_ : i ≥ 0), f i a) ≤ g a",
"state_before": "case refine'_3\nα : Type u_1\nβ : Type ?u.996569\nγ : Type ?u.996572\nδ : Type ?u.996575\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\ng : α → ℝ≥0∞\nhf_meas : ∀ (n : ℕ), Measurable (f n)\nh_bound : ∀ (n : ℕ), f n ≤ᵐ[μ] g\nh_fin : (∫⁻ (a : α), g a ∂μ) ≠ ⊤\n⊢ (∫⁻ (a : α), ⨆ (i : ℕ) (_ : i ≥ 0), f i a ∂μ) ≠ ⊤",
"tactic": "refine' ne_top_of_le_ne_top h_fin (lintegral_mono_ae _)"
},
{
"state_after": "case refine'_3\nα : Type u_1\nβ : Type ?u.996569\nγ : Type ?u.996572\nδ : Type ?u.996575\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\ng : α → ℝ≥0∞\nhf_meas : ∀ (n : ℕ), Measurable (f n)\nh_bound : ∀ (n : ℕ), f n ≤ᵐ[μ] g\nh_fin : (∫⁻ (a : α), g a ∂μ) ≠ ⊤\nn : α\nhn : ∀ (i : ℕ), f i n ≤ g n\n⊢ (⨆ (i : ℕ) (_ : i ≥ 0), f i n) ≤ g n",
"state_before": "case refine'_3\nα : Type u_1\nβ : Type ?u.996569\nγ : Type ?u.996572\nδ : Type ?u.996575\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\ng : α → ℝ≥0∞\nhf_meas : ∀ (n : ℕ), Measurable (f n)\nh_bound : ∀ (n : ℕ), f n ≤ᵐ[μ] g\nh_fin : (∫⁻ (a : α), g a ∂μ) ≠ ⊤\n⊢ ∀ᵐ (a : α) ∂μ, (⨆ (i : ℕ) (_ : i ≥ 0), f i a) ≤ g a",
"tactic": "refine' (ae_all_iff.2 h_bound).mono fun n hn => _"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nα : Type u_1\nβ : Type ?u.996569\nγ : Type ?u.996572\nδ : Type ?u.996575\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\ng : α → ℝ≥0∞\nhf_meas : ∀ (n : ℕ), Measurable (f n)\nh_bound : ∀ (n : ℕ), f n ≤ᵐ[μ] g\nh_fin : (∫⁻ (a : α), g a ∂μ) ≠ ⊤\nn : α\nhn : ∀ (i : ℕ), f i n ≤ g n\n⊢ (⨆ (i : ℕ) (_ : i ≥ 0), f i n) ≤ g n",
"tactic": "exact iSup_le fun i => iSup_le fun _ => hn i"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.996569\nγ : Type ?u.996572\nδ : Type ?u.996575\nm : MeasurableSpace α\nμ ν : Measure α\nf : ℕ → α → ℝ≥0∞\ng : α → ℝ≥0∞\nhf_meas : ∀ (n : ℕ), Measurable (f n)\nh_bound : ∀ (n : ℕ), f n ≤ᵐ[μ] g\nh_fin : (∫⁻ (a : α), g a ∂μ) ≠ ⊤\n⊢ (∫⁻ (a : α), ⨅ (n : ℕ), ⨆ (i : ℕ) (_ : i ≥ n), f i a ∂μ) = ∫⁻ (a : α), limsup (fun n => f n a) atTop ∂μ",
"tactic": "simp only [limsup_eq_iInf_iSup_of_nat]"
}
] |
[
1038,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1022,
1
] |
Mathlib/Topology/MetricSpace/Lipschitz.lean
|
lipschitzOnWith_iff_restrict
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nK : ℝ≥0\nf : α → β\ns : Set α\n⊢ LipschitzOnWith K f s ↔ LipschitzWith K (restrict s f)",
"tactic": "simp only [LipschitzOnWith, LipschitzWith, SetCoe.forall', restrict, Subtype.edist_eq]"
}
] |
[
104,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
1
] |
Mathlib/Order/CompactlyGenerated.lean
|
sSup_compact_eq_top
|
[
{
"state_after": "ι : Sort ?u.87370\nα : Type u_1\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns : Set α\nx : α\n⊢ x ∈ {a | CompleteLattice.IsCompactElement a} ↔ x ∈ {c | CompleteLattice.IsCompactElement c ∧ c ≤ ⊤}",
"state_before": "ι : Sort ?u.87370\nα : Type u_1\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns : Set α\n⊢ sSup {a | CompleteLattice.IsCompactElement a} = ⊤",
"tactic": "refine' Eq.trans (congr rfl (Set.ext fun x => _)) (sSup_compact_le_eq ⊤)"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.87370\nα : Type u_1\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns : Set α\nx : α\n⊢ x ∈ {a | CompleteLattice.IsCompactElement a} ↔ x ∈ {c | CompleteLattice.IsCompactElement c ∧ c ≤ ⊤}",
"tactic": "exact (and_iff_left le_top).symm"
}
] |
[
352,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
350,
1
] |
Mathlib/Data/Fintype/Perm.lean
|
Fintype.card_equiv
|
[] |
[
171,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/Algebra/Order/Rearrangement.lean
|
AntivaryOn.sum_smul_le_sum_smul_comp_perm
|
[] |
[
195,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
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