file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Data/Matrix/Block.lean
|
Matrix.blockDiagonal'_sub
|
[] |
[
751,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
749,
1
] |
Mathlib/MeasureTheory/Measure/NullMeasurable.lean
|
MeasureTheory.NullMeasurableSet.iUnion
|
[] |
[
151,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
11
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
nndist_eq_zero
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.551874\nι : Type ?u.551877\ninst✝¹ : PseudoMetricSpace α\nγ : Type w\ninst✝ : MetricSpace γ\nx y : γ\n⊢ nndist x y = 0 ↔ x = y",
"tactic": "simp only [← NNReal.eq_iff, ← dist_nndist, imp_self, NNReal.coe_zero, dist_eq_zero]"
}
] |
[
2885,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2884,
1
] |
Mathlib/Data/Set/Pairwise/Basic.lean
|
Set.pairwise_singleton
|
[] |
[
93,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Data/Nat/Lattice.lean
|
Set.biInter_lt_succ'
|
[] |
[
218,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean
|
ProjectiveSpectrum.isClosed_zeroLocus
|
[
{
"state_after": "R : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\ns : Set A\n⊢ ∃ s_1, zeroLocus 𝒜 s = zeroLocus 𝒜 s_1",
"state_before": "R : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\ns : Set A\n⊢ IsClosed (zeroLocus 𝒜 s)",
"tactic": "rw [isClosed_iff_zeroLocus]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\n𝒜 : ℕ → Submodule R A\ninst✝ : GradedAlgebra 𝒜\ns : Set A\n⊢ ∃ s_1, zeroLocus 𝒜 s = zeroLocus 𝒜 s_1",
"tactic": "exact ⟨s, rfl⟩"
}
] |
[
360,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
358,
1
] |
Mathlib/MeasureTheory/Function/Egorov.lean
|
MeasureTheory.tendstoUniformlyOn_of_ae_tendsto
|
[] |
[
212,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
203,
1
] |
Mathlib/Data/Nat/Prime.lean
|
Nat.Prime.mul_eq_prime_sq_iff
|
[
{
"state_after": "x y p : ℕ\nhp : Prime p\nhx : x ≠ 1\nhy : y ≠ 1\nh : x * y = p ^ 2\n⊢ x = p ∧ y = p",
"state_before": "x y p : ℕ\nhp : Prime p\nhx : x ≠ 1\nhy : y ≠ 1\n⊢ x * y = p ^ 2 ↔ x = p ∧ y = p",
"tactic": "refine' ⟨fun h => _, fun ⟨h₁, h₂⟩ => h₁.symm ▸ h₂.symm ▸ (sq _).symm⟩"
},
{
"state_after": "x y p : ℕ\nhp : Prime p\nhx : x ≠ 1\nhy : y ≠ 1\nh : x * y = p ^ 2\npdvdxy : p ∣ x * y\n⊢ x = p ∧ y = p",
"state_before": "x y p : ℕ\nhp : Prime p\nhx : x ≠ 1\nhy : y ≠ 1\nh : x * y = p ^ 2\n⊢ x = p ∧ y = p",
"tactic": "have pdvdxy : p ∣ x * y := by rw [h]; simp [sq]"
},
{
"state_after": "x y p : ℕ\nhp : Prime p\nhx : x ≠ 1\nhy : y ≠ 1\nh : x * y = p ^ 2\npdvdxy : p ∣ x * y\n⊢ ∀ (x' y' : ℕ), x' ≠ 1 → y' ≠ 1 → x' * y' = p ^ 2 → p ∣ x' → x' = p ∧ y' = p",
"state_before": "x y p : ℕ\nhp : Prime p\nhx : x ≠ 1\nhy : y ≠ 1\nh : x * y = p ^ 2\npdvdxy : p ∣ x * y\n⊢ x = p ∧ y = p",
"tactic": "suffices ∀ x' y' : ℕ, x' ≠ 1 → y' ≠ 1 → x' * y' = p ^ 2 → p ∣ x' → x' = p ∧ y' = p by\n obtain hx | hy := hp.dvd_mul.1 pdvdxy <;>\n [skip; rw [And.comm]] <;>\n [skip; rw [mul_comm] at h pdvdxy] <;>\n apply this <;>\n assumption"
},
{
"state_after": "case intro\nx✝ y✝ p : ℕ\nhp : Prime p\nhx✝ : x✝ ≠ 1\nhy✝ : y✝ ≠ 1\nh✝ : x✝ * y✝ = p ^ 2\npdvdxy : p ∣ x✝ * y✝\nx y : ℕ\nhx : x ≠ 1\nhy : y ≠ 1\nh : x * y = p ^ 2\na : ℕ\nha : x = p * a\n⊢ x = p ∧ y = p",
"state_before": "x y p : ℕ\nhp : Prime p\nhx : x ≠ 1\nhy : y ≠ 1\nh : x * y = p ^ 2\npdvdxy : p ∣ x * y\n⊢ ∀ (x' y' : ℕ), x' ≠ 1 → y' ≠ 1 → x' * y' = p ^ 2 → p ∣ x' → x' = p ∧ y' = p",
"tactic": "rintro x y hx hy h ⟨a, ha⟩"
},
{
"state_after": "case intro\nx✝ y✝ p : ℕ\nhp : Prime p\nhx✝ : x✝ ≠ 1\nhy✝ : y✝ ≠ 1\nh✝ : x✝ * y✝ = p ^ 2\npdvdxy : p ∣ x✝ * y✝\nx y : ℕ\nhx : x ≠ 1\nhy : y ≠ 1\nh : x * y = p ^ 2\na : ℕ\nha : x = p * a\nthis : a ∣ p\n⊢ x = p ∧ y = p",
"state_before": "case intro\nx✝ y✝ p : ℕ\nhp : Prime p\nhx✝ : x✝ ≠ 1\nhy✝ : y✝ ≠ 1\nh✝ : x✝ * y✝ = p ^ 2\npdvdxy : p ∣ x✝ * y✝\nx y : ℕ\nhx : x ≠ 1\nhy : y ≠ 1\nh : x * y = p ^ 2\na : ℕ\nha : x = p * a\n⊢ x = p ∧ y = p",
"tactic": "have : a ∣ p := ⟨y, by rwa [ha, sq, mul_assoc, mul_right_inj' hp.ne_zero, eq_comm] at h⟩"
},
{
"state_after": "case intro.inl\nx✝ y✝ p : ℕ\nhp : Prime p\nhx✝ : x✝ ≠ 1\nhy✝ : y✝ ≠ 1\nh✝ : x✝ * y✝ = p ^ 2\npdvdxy : p ∣ x✝ * y✝\nx y : ℕ\nhx : x ≠ 1\nhy : y ≠ 1\nh : x * y = p ^ 2\na : ℕ\nha : x = p * a\nthis : a ∣ p\nha1 : a = 1\n⊢ x = p ∧ y = p\n\ncase intro.inr\nx✝ y✝ p : ℕ\nhp : Prime p\nhx✝ : x✝ ≠ 1\nhy✝ : y✝ ≠ 1\nh✝ : x✝ * y✝ = p ^ 2\npdvdxy : p ∣ x✝ * y✝\nx y : ℕ\nhx : x ≠ 1\nhy : y ≠ 1\nh : x * y = p ^ 2\na : ℕ\nha : x = p * a\nthis : a ∣ p\nhap : a = p\n⊢ x = p ∧ y = p",
"state_before": "case intro\nx✝ y✝ p : ℕ\nhp : Prime p\nhx✝ : x✝ ≠ 1\nhy✝ : y✝ ≠ 1\nh✝ : x✝ * y✝ = p ^ 2\npdvdxy : p ∣ x✝ * y✝\nx y : ℕ\nhx : x ≠ 1\nhy : y ≠ 1\nh : x * y = p ^ 2\na : ℕ\nha : x = p * a\nthis : a ∣ p\n⊢ x = p ∧ y = p",
"tactic": "obtain ha1 | hap := (Nat.dvd_prime hp).mp ‹a ∣ p›"
},
{
"state_after": "x y p : ℕ\nhp : Prime p\nhx : x ≠ 1\nhy : y ≠ 1\nh : x * y = p ^ 2\n⊢ p ∣ p ^ 2",
"state_before": "x y p : ℕ\nhp : Prime p\nhx : x ≠ 1\nhy : y ≠ 1\nh : x * y = p ^ 2\n⊢ p ∣ x * y",
"tactic": "rw [h]"
},
{
"state_after": "no goals",
"state_before": "x y p : ℕ\nhp : Prime p\nhx : x ≠ 1\nhy : y ≠ 1\nh : x * y = p ^ 2\n⊢ p ∣ p ^ 2",
"tactic": "simp [sq]"
},
{
"state_after": "no goals",
"state_before": "x y p : ℕ\nhp : Prime p\nhx : x ≠ 1\nhy : y ≠ 1\nh : x * y = p ^ 2\npdvdxy : p ∣ x * y\nthis : ∀ (x' y' : ℕ), x' ≠ 1 → y' ≠ 1 → x' * y' = p ^ 2 → p ∣ x' → x' = p ∧ y' = p\n⊢ x = p ∧ y = p",
"tactic": "obtain hx | hy := hp.dvd_mul.1 pdvdxy <;>\n [skip; rw [And.comm]] <;>\n [skip; rw [mul_comm] at h pdvdxy] <;>\n apply this <;>\n assumption"
},
{
"state_after": "no goals",
"state_before": "x✝ y✝ p : ℕ\nhp : Prime p\nhx✝ : x✝ ≠ 1\nhy✝ : y✝ ≠ 1\nh✝ : x✝ * y✝ = p ^ 2\npdvdxy : p ∣ x✝ * y✝\nx y : ℕ\nhx : x ≠ 1\nhy : y ≠ 1\nh : x * y = p ^ 2\na : ℕ\nha : x = p * a\n⊢ p = a * y",
"tactic": "rwa [ha, sq, mul_assoc, mul_right_inj' hp.ne_zero, eq_comm] at h"
},
{
"state_after": "case intro.inl\nx✝ y✝ p : ℕ\nhp : Prime p\nhx✝ : x✝ ≠ 1\nhy✝ : y✝ ≠ 1\nh✝ : x✝ * y✝ = p ^ 2\npdvdxy : p ∣ x✝ * y✝\nx y : ℕ\nhx : x ≠ 1\nhy : y ≠ 1\nh : x * y = p ^ 2\nha : x = p * 1\nthis : 1 ∣ p\n⊢ x = p ∧ y = p",
"state_before": "case intro.inl\nx✝ y✝ p : ℕ\nhp : Prime p\nhx✝ : x✝ ≠ 1\nhy✝ : y✝ ≠ 1\nh✝ : x✝ * y✝ = p ^ 2\npdvdxy : p ∣ x✝ * y✝\nx y : ℕ\nhx : x ≠ 1\nhy : y ≠ 1\nh : x * y = p ^ 2\na : ℕ\nha : x = p * a\nthis : a ∣ p\nha1 : a = 1\n⊢ x = p ∧ y = p",
"tactic": "subst ha1"
},
{
"state_after": "case intro.inl\nx✝ y✝ p : ℕ\nhp : Prime p\nhx✝ : x✝ ≠ 1\nhy✝ : y✝ ≠ 1\nh✝ : x✝ * y✝ = p ^ 2\npdvdxy : p ∣ x✝ * y✝\nx y : ℕ\nhx : x ≠ 1\nhy : y ≠ 1\nh : x * y = p ^ 2\nha : x = p\nthis : 1 ∣ p\n⊢ x = p ∧ y = p",
"state_before": "case intro.inl\nx✝ y✝ p : ℕ\nhp : Prime p\nhx✝ : x✝ ≠ 1\nhy✝ : y✝ ≠ 1\nh✝ : x✝ * y✝ = p ^ 2\npdvdxy : p ∣ x✝ * y✝\nx y : ℕ\nhx : x ≠ 1\nhy : y ≠ 1\nh : x * y = p ^ 2\nha : x = p * 1\nthis : 1 ∣ p\n⊢ x = p ∧ y = p",
"tactic": "rw [mul_one] at ha"
},
{
"state_after": "case intro.inl\nx✝ y✝ : ℕ\nhx✝ : x✝ ≠ 1\nhy✝ : y✝ ≠ 1\nx y : ℕ\nhx : x ≠ 1\nhy : y ≠ 1\nhp : Prime x\nh✝ : x✝ * y✝ = x ^ 2\npdvdxy : x ∣ x✝ * y✝\nh : x * y = x ^ 2\nthis : 1 ∣ x\n⊢ x = x ∧ y = x",
"state_before": "case intro.inl\nx✝ y✝ p : ℕ\nhp : Prime p\nhx✝ : x✝ ≠ 1\nhy✝ : y✝ ≠ 1\nh✝ : x✝ * y✝ = p ^ 2\npdvdxy : p ∣ x✝ * y✝\nx y : ℕ\nhx : x ≠ 1\nhy : y ≠ 1\nh : x * y = p ^ 2\nha : x = p\nthis : 1 ∣ p\n⊢ x = p ∧ y = p",
"tactic": "subst ha"
},
{
"state_after": "case intro.inl\nx✝ y✝ : ℕ\nhx✝ : x✝ ≠ 1\nhy✝ : y✝ ≠ 1\nx y : ℕ\nhx : x ≠ 1\nhy : y ≠ 1\nhp : Prime x\nh✝ : x✝ * y✝ = x ^ 2\npdvdxy : x ∣ x✝ * y✝\nthis : 1 ∣ x\nh : y = x\n⊢ x = x ∧ y = x",
"state_before": "case intro.inl\nx✝ y✝ : ℕ\nhx✝ : x✝ ≠ 1\nhy✝ : y✝ ≠ 1\nx y : ℕ\nhx : x ≠ 1\nhy : y ≠ 1\nhp : Prime x\nh✝ : x✝ * y✝ = x ^ 2\npdvdxy : x ∣ x✝ * y✝\nh : x * y = x ^ 2\nthis : 1 ∣ x\n⊢ x = x ∧ y = x",
"tactic": "simp only [sq, mul_right_inj' hp.ne_zero] at h"
},
{
"state_after": "case intro.inl\nx y✝ : ℕ\nhx✝ : x ≠ 1\nhy✝ : y✝ ≠ 1\ny : ℕ\nhy hx : y ≠ 1\nhp : Prime y\nh : x * y✝ = y ^ 2\npdvdxy : y ∣ x * y✝\nthis : 1 ∣ y\n⊢ y = y ∧ y = y",
"state_before": "case intro.inl\nx✝ y✝ : ℕ\nhx✝ : x✝ ≠ 1\nhy✝ : y✝ ≠ 1\nx y : ℕ\nhx : x ≠ 1\nhy : y ≠ 1\nhp : Prime x\nh✝ : x✝ * y✝ = x ^ 2\npdvdxy : x ∣ x✝ * y✝\nthis : 1 ∣ x\nh : y = x\n⊢ x = x ∧ y = x",
"tactic": "subst h"
},
{
"state_after": "no goals",
"state_before": "case intro.inl\nx y✝ : ℕ\nhx✝ : x ≠ 1\nhy✝ : y✝ ≠ 1\ny : ℕ\nhy hx : y ≠ 1\nhp : Prime y\nh : x * y✝ = y ^ 2\npdvdxy : y ∣ x * y✝\nthis : 1 ∣ y\n⊢ y = y ∧ y = y",
"tactic": "exact ⟨rfl, rfl⟩"
},
{
"state_after": "case intro.inr\nx✝ y✝ p : ℕ\nhp : Prime p\nhx✝ : x✝ ≠ 1\nhy✝ : y✝ ≠ 1\nh✝ : x✝ * y✝ = p ^ 2\npdvdxy : p ∣ x✝ * y✝\nx y : ℕ\nhx : x ≠ 1\nhy : y ≠ 1\nh : x * y = p ^ 2\na : ℕ\nha : x = p * a\nthis : a ∣ p\nhap : a = p\n⊢ y = 1",
"state_before": "case intro.inr\nx✝ y✝ p : ℕ\nhp : Prime p\nhx✝ : x✝ ≠ 1\nhy✝ : y✝ ≠ 1\nh✝ : x✝ * y✝ = p ^ 2\npdvdxy : p ∣ x✝ * y✝\nx y : ℕ\nhx : x ≠ 1\nhy : y ≠ 1\nh : x * y = p ^ 2\na : ℕ\nha : x = p * a\nthis : a ∣ p\nhap : a = p\n⊢ x = p ∧ y = p",
"tactic": "refine' (hy ?_).elim"
},
{
"state_after": "case intro.inr\nx✝ y✝ : ℕ\nhx✝ : x✝ ≠ 1\nhy✝ : y✝ ≠ 1\nx y : ℕ\nhx : x ≠ 1\nhy : y ≠ 1\na : ℕ\nhp : Prime a\nh✝ : x✝ * y✝ = a ^ 2\npdvdxy : a ∣ x✝ * y✝\nh : x * y = a ^ 2\nha : x = a * a\nthis : a ∣ a\n⊢ y = 1",
"state_before": "case intro.inr\nx✝ y✝ p : ℕ\nhp : Prime p\nhx✝ : x✝ ≠ 1\nhy✝ : y✝ ≠ 1\nh✝ : x✝ * y✝ = p ^ 2\npdvdxy : p ∣ x✝ * y✝\nx y : ℕ\nhx : x ≠ 1\nhy : y ≠ 1\nh : x * y = p ^ 2\na : ℕ\nha : x = p * a\nthis : a ∣ p\nhap : a = p\n⊢ y = 1",
"tactic": "subst hap"
},
{
"state_after": "case intro.inr\nx y✝ : ℕ\nhx✝ : x ≠ 1\nhy✝ : y✝ ≠ 1\ny : ℕ\nhy : y ≠ 1\na : ℕ\nhp : Prime a\nh✝ : x * y✝ = a ^ 2\npdvdxy : a ∣ x * y✝\nthis : a ∣ a\nhx : a * a ≠ 1\nh : a * a * y = a ^ 2\n⊢ y = 1",
"state_before": "case intro.inr\nx✝ y✝ : ℕ\nhx✝ : x✝ ≠ 1\nhy✝ : y✝ ≠ 1\nx y : ℕ\nhx : x ≠ 1\nhy : y ≠ 1\na : ℕ\nhp : Prime a\nh✝ : x✝ * y✝ = a ^ 2\npdvdxy : a ∣ x✝ * y✝\nh : x * y = a ^ 2\nha : x = a * a\nthis : a ∣ a\n⊢ y = 1",
"tactic": "subst ha"
},
{
"state_after": "case intro.inr\nx y✝ : ℕ\nhx✝ : x ≠ 1\nhy✝ : y✝ ≠ 1\ny : ℕ\nhy : y ≠ 1\na : ℕ\nhp : Prime a\nh✝ : x * y✝ = a ^ 2\npdvdxy : a ∣ x * y✝\nthis : a ∣ a\nhx : a * a ≠ 1\nh : y = 1\n⊢ y = 1",
"state_before": "case intro.inr\nx y✝ : ℕ\nhx✝ : x ≠ 1\nhy✝ : y✝ ≠ 1\ny : ℕ\nhy : y ≠ 1\na : ℕ\nhp : Prime a\nh✝ : x * y✝ = a ^ 2\npdvdxy : a ∣ x * y✝\nthis : a ∣ a\nhx : a * a ≠ 1\nh : a * a * y = a ^ 2\n⊢ y = 1",
"tactic": "rw [sq, Nat.mul_right_eq_self_iff (Nat.mul_pos hp.pos hp.pos : 0 < a * a)] at h"
},
{
"state_after": "no goals",
"state_before": "case intro.inr\nx y✝ : ℕ\nhx✝ : x ≠ 1\nhy✝ : y✝ ≠ 1\ny : ℕ\nhy : y ≠ 1\na : ℕ\nhp : Prime a\nh✝ : x * y✝ = a ^ 2\npdvdxy : a ∣ x * y✝\nthis : a ∣ a\nhx : a * a ≠ 1\nh : y = 1\n⊢ y = 1",
"tactic": "exact h"
}
] |
[
658,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
634,
1
] |
Std/Data/Int/DivMod.lean
|
Int.ediv_one
|
[] |
[
131,
48
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
129,
9
] |
Mathlib/Analysis/Convex/Cone/Basic.lean
|
ConvexCone.comap_comap
|
[] |
[
303,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
301,
1
] |
Mathlib/Analysis/NormedSpace/Exponential.lean
|
exp_add_of_commute
|
[] |
[
463,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
461,
1
] |
Mathlib/RingTheory/FinitePresentation.lean
|
Algebra.FinitePresentation.self
|
[] |
[
118,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
116,
1
] |
Mathlib/CategoryTheory/Closed/Monoidal.lean
|
CategoryTheory.MonoidalClosed.pre_comm_ihom_map
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁵ : Category C\ninst✝⁴ : MonoidalCategory C\nA B X✝ X' Y✝ Y' Z✝ : C\ninst✝³ : Closed A\ninst✝² : Closed B\nW X Y Z : C\ninst✝¹ : Closed W\ninst✝ : Closed X\nf : W ⟶ X\ng : Y ⟶ Z\n⊢ (pre f).app Y ≫ (ihom W).map g = (ihom X).map g ≫ (pre f).app Z",
"tactic": "simp"
}
] |
[
288,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
287,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
ZNum.cmp_to_int
|
[
{
"state_after": "α : Type ?u.968134\na b : PosNum\nthis : Ordering.casesOn (PosNum.cmp a b) (↑a < ↑b) (a = b) (↑b < ↑a)\n⊢ Ordering.casesOn (cmp (pos a) (pos b)) (↑(pos a) < ↑(pos b)) (pos a = pos b) (↑(pos b) < ↑(pos a))",
"state_before": "α : Type ?u.968134\na b : PosNum\n⊢ Ordering.casesOn (cmp (pos a) (pos b)) (↑(pos a) < ↑(pos b)) (pos a = pos b) (↑(pos b) < ↑(pos a))",
"tactic": "have := PosNum.cmp_to_nat a b"
},
{
"state_after": "α : Type ?u.968134\na b : PosNum\n⊢ Ordering.casesOn (PosNum.cmp a b) (↑a < ↑b) (a = b) (↑b < ↑a) →\n Ordering.casesOn (cmp (pos a) (pos b)) (↑(pos a) < ↑(pos b)) (pos a = pos b) (↑(pos b) < ↑(pos a))",
"state_before": "α : Type ?u.968134\na b : PosNum\nthis : Ordering.casesOn (PosNum.cmp a b) (↑a < ↑b) (a = b) (↑b < ↑a)\n⊢ Ordering.casesOn (cmp (pos a) (pos b)) (↑(pos a) < ↑(pos b)) (pos a = pos b) (↑(pos b) < ↑(pos a))",
"tactic": "revert this"
},
{
"state_after": "α : Type ?u.968134\na b : PosNum\n⊢ Ordering.rec (↑a < ↑b) (a = b) (↑b < ↑a) (PosNum.cmp a b) →\n Ordering.rec (↑a < ↑b) (pos a = pos b) (↑b < ↑a) (PosNum.cmp a b)",
"state_before": "α : Type ?u.968134\na b : PosNum\n⊢ Ordering.casesOn (PosNum.cmp a b) (↑a < ↑b) (a = b) (↑b < ↑a) →\n Ordering.casesOn (cmp (pos a) (pos b)) (↑(pos a) < ↑(pos b)) (pos a = pos b) (↑(pos b) < ↑(pos a))",
"tactic": "dsimp [cmp]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.968134\na b : PosNum\n⊢ Ordering.rec (↑a < ↑b) (a = b) (↑b < ↑a) (PosNum.cmp a b) →\n Ordering.rec (↑a < ↑b) (pos a = pos b) (↑b < ↑a) (PosNum.cmp a b)",
"tactic": "cases PosNum.cmp a b <;> dsimp <;> [simp; exact congr_arg pos; simp [GT.gt]]"
},
{
"state_after": "α : Type ?u.968134\na b : PosNum\nthis : Ordering.casesOn (PosNum.cmp b a) (↑b < ↑a) (b = a) (↑a < ↑b)\n⊢ Ordering.casesOn (cmp (neg a) (neg b)) (↑(neg a) < ↑(neg b)) (neg a = neg b) (↑(neg b) < ↑(neg a))",
"state_before": "α : Type ?u.968134\na b : PosNum\n⊢ Ordering.casesOn (cmp (neg a) (neg b)) (↑(neg a) < ↑(neg b)) (neg a = neg b) (↑(neg b) < ↑(neg a))",
"tactic": "have := PosNum.cmp_to_nat b a"
},
{
"state_after": "α : Type ?u.968134\na b : PosNum\n⊢ Ordering.casesOn (PosNum.cmp b a) (↑b < ↑a) (b = a) (↑a < ↑b) →\n Ordering.casesOn (cmp (neg a) (neg b)) (↑(neg a) < ↑(neg b)) (neg a = neg b) (↑(neg b) < ↑(neg a))",
"state_before": "α : Type ?u.968134\na b : PosNum\nthis : Ordering.casesOn (PosNum.cmp b a) (↑b < ↑a) (b = a) (↑a < ↑b)\n⊢ Ordering.casesOn (cmp (neg a) (neg b)) (↑(neg a) < ↑(neg b)) (neg a = neg b) (↑(neg b) < ↑(neg a))",
"tactic": "revert this"
},
{
"state_after": "α : Type ?u.968134\na b : PosNum\n⊢ Ordering.rec (↑b < ↑a) (b = a) (↑a < ↑b) (PosNum.cmp b a) →\n Ordering.rec (-↑a < -↑b) (neg a = neg b) (-↑b < -↑a) (PosNum.cmp b a)",
"state_before": "α : Type ?u.968134\na b : PosNum\n⊢ Ordering.casesOn (PosNum.cmp b a) (↑b < ↑a) (b = a) (↑a < ↑b) →\n Ordering.casesOn (cmp (neg a) (neg b)) (↑(neg a) < ↑(neg b)) (neg a = neg b) (↑(neg b) < ↑(neg a))",
"tactic": "dsimp [cmp]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.968134\na b : PosNum\n⊢ Ordering.rec (↑b < ↑a) (b = a) (↑a < ↑b) (PosNum.cmp b a) →\n Ordering.rec (-↑a < -↑b) (neg a = neg b) (-↑b < -↑a) (PosNum.cmp b a)",
"tactic": "cases PosNum.cmp b a <;> dsimp <;> [simp; simp (config := { contextual := true }); simp [GT.gt]]"
}
] |
[
1389,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1376,
1
] |
Mathlib/Topology/UniformSpace/Cauchy.lean
|
Cauchy.ultrafilter_of
|
[
{
"state_after": "α : Type u\nβ : Type v\ninst✝ : UniformSpace α\nl : Filter α\nh : Cauchy l\nthis : NeBot l\n⊢ Cauchy ↑(Ultrafilter.of l)",
"state_before": "α : Type u\nβ : Type v\ninst✝ : UniformSpace α\nl : Filter α\nh : Cauchy l\n⊢ Cauchy ↑(Ultrafilter.of l)",
"tactic": "haveI := h.1"
},
{
"state_after": "α : Type u\nβ : Type v\ninst✝ : UniformSpace α\nl : Filter α\nh : Cauchy l\nthis✝ : NeBot l\nthis : ↑(Ultrafilter.of l) ≤ l\n⊢ Cauchy ↑(Ultrafilter.of l)",
"state_before": "α : Type u\nβ : Type v\ninst✝ : UniformSpace α\nl : Filter α\nh : Cauchy l\nthis : NeBot l\n⊢ Cauchy ↑(Ultrafilter.of l)",
"tactic": "have := Ultrafilter.of_le l"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝ : UniformSpace α\nl : Filter α\nh : Cauchy l\nthis✝ : NeBot l\nthis : ↑(Ultrafilter.of l) ≤ l\n⊢ Cauchy ↑(Ultrafilter.of l)",
"tactic": "exact ⟨Ultrafilter.neBot _, (Filter.prod_mono this this).trans h.2⟩"
}
] |
[
64,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/Order/LocallyFinite.lean
|
Finset.map_subtype_embedding_Ici
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.223222\ninst✝² : Preorder α\np : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : LocallyFiniteOrderTop α\na : Subtype p\nhp : ∀ ⦃a x : α⦄, a ≤ x → p a → p x\n⊢ map (Embedding.subtype p) (Finset.subtype p (Ici ↑a)) = Ici ↑a",
"state_before": "α : Type u_1\nβ : Type ?u.223222\ninst✝² : Preorder α\np : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : LocallyFiniteOrderTop α\na : Subtype p\nhp : ∀ ⦃a x : α⦄, a ≤ x → p a → p x\n⊢ map (Embedding.subtype p) (Ici a) = Ici ↑a",
"tactic": "rw [subtype_Ici_eq]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.223222\ninst✝² : Preorder α\np : α → Prop\ninst✝¹ : DecidablePred p\ninst✝ : LocallyFiniteOrderTop α\na : Subtype p\nhp : ∀ ⦃a x : α⦄, a ≤ x → p a → p x\n⊢ map (Embedding.subtype p) (Finset.subtype p (Ici ↑a)) = Ici ↑a",
"tactic": "exact Finset.subtype_map_of_mem fun x hx => hp (mem_Ici.1 hx) a.prop"
}
] |
[
1331,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1329,
1
] |
Mathlib/Data/List/Perm.lean
|
List.Perm.bagInter
|
[] |
[
858,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
856,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.singleton_mul
|
[] |
[
404,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
403,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec₂.ofNat_iff
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.68744\nβ✝ : Type ?u.68747\nσ✝ : Type ?u.68750\ninst✝⁵ : Primcodable α✝\ninst✝⁴ : Primcodable β✝\ninst✝³ : Primcodable σ✝\nα : Type u_1\nβ : Type u_2\nσ : Type u_3\ninst✝² : Denumerable α\ninst✝¹ : Denumerable β\ninst✝ : Primcodable σ\nf : α → β → σ\n⊢ (Primrec fun n => f (ofNat (α × β) n).fst (ofNat (α × β) n).snd) ↔\n Primrec (Nat.unpaired fun m n => f (ofNat α m) (ofNat β n))",
"tactic": "simp"
}
] |
[
458,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
456,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.image_div_prod
|
[] |
[
547,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
546,
1
] |
Mathlib/Algebra/GroupPower/Ring.lean
|
sq_eq_one_iff
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.125015\nM : Type ?u.125018\ninst✝¹ : Ring R\na b : R\ninst✝ : NoZeroDivisors R\n⊢ a ^ 2 = 1 ↔ a = 1 ∨ a = -1",
"tactic": "rw [← (Commute.one_right a).sq_eq_sq_iff_eq_or_eq_neg, one_pow]"
}
] |
[
263,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
262,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
intervalIntegral.integral_comp_div
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.14843380\n𝕜 : Type ?u.14843383\nE : Type u_1\nF : Type ?u.14843389\nA : Type ?u.14843392\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na b c d : ℝ\nf : ℝ → E\nhc : c ≠ 0\n⊢ (∫ (x : ℝ) in a..b, f (x / c)) = c • ∫ (x : ℝ) in a / c..b / c, f x",
"tactic": "simpa only [inv_inv] using integral_comp_mul_right f (inv_ne_zero hc)"
}
] |
[
727,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
726,
1
] |
Mathlib/Analysis/Normed/Group/HomCompletion.lean
|
NormedAddGroupHom.extension_def
|
[] |
[
217,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
1
] |
Mathlib/Algebra/MonoidAlgebra/Degree.lean
|
AddMonoidAlgebra.sup_support_pow_le
|
[
{
"state_after": "R : Type u_3\nA : Type u_2\nT : Type ?u.47893\nB : Type u_1\nι : Type ?u.47899\ninst✝¹¹ : SemilatticeSup B\ninst✝¹⁰ : OrderBot B\ninst✝⁹ : SemilatticeInf T\ninst✝⁸ : OrderTop T\ninst✝⁷ : Semiring R\ninst✝⁶ : AddMonoid A\ninst✝⁵ : AddMonoid B\ninst✝⁴ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝³ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝² : AddMonoid T\ninst✝¹ : CovariantClass T T (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass T T (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ndegb : A → B\ndegt : A → T\ndegb0 : degb 0 ≤ 0\ndegbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b\nn : ℕ\nf : AddMonoidAlgebra R A\n⊢ Finset.sup (List.prod (List.replicate n f)).support degb ≤ List.sum (List.replicate n (Finset.sup f.support degb))",
"state_before": "R : Type u_3\nA : Type u_2\nT : Type ?u.47893\nB : Type u_1\nι : Type ?u.47899\ninst✝¹¹ : SemilatticeSup B\ninst✝¹⁰ : OrderBot B\ninst✝⁹ : SemilatticeInf T\ninst✝⁸ : OrderTop T\ninst✝⁷ : Semiring R\ninst✝⁶ : AddMonoid A\ninst✝⁵ : AddMonoid B\ninst✝⁴ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝³ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝² : AddMonoid T\ninst✝¹ : CovariantClass T T (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass T T (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ndegb : A → B\ndegt : A → T\ndegb0 : degb 0 ≤ 0\ndegbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b\nn : ℕ\nf : AddMonoidAlgebra R A\n⊢ Finset.sup (f ^ n).support degb ≤ n • Finset.sup f.support degb",
"tactic": "rw [← List.prod_replicate, ← List.sum_replicate]"
},
{
"state_after": "R : Type u_3\nA : Type u_2\nT : Type ?u.47893\nB : Type u_1\nι : Type ?u.47899\ninst✝¹¹ : SemilatticeSup B\ninst✝¹⁰ : OrderBot B\ninst✝⁹ : SemilatticeInf T\ninst✝⁸ : OrderTop T\ninst✝⁷ : Semiring R\ninst✝⁶ : AddMonoid A\ninst✝⁵ : AddMonoid B\ninst✝⁴ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝³ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝² : AddMonoid T\ninst✝¹ : CovariantClass T T (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass T T (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ndegb : A → B\ndegt : A → T\ndegb0 : degb 0 ≤ 0\ndegbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b\nn : ℕ\nf : AddMonoidAlgebra R A\n⊢ List.sum (List.map (fun f => Finset.sup f.support degb) (List.replicate n f)) =\n List.sum (List.replicate n (Finset.sup f.support degb))",
"state_before": "R : Type u_3\nA : Type u_2\nT : Type ?u.47893\nB : Type u_1\nι : Type ?u.47899\ninst✝¹¹ : SemilatticeSup B\ninst✝¹⁰ : OrderBot B\ninst✝⁹ : SemilatticeInf T\ninst✝⁸ : OrderTop T\ninst✝⁷ : Semiring R\ninst✝⁶ : AddMonoid A\ninst✝⁵ : AddMonoid B\ninst✝⁴ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝³ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝² : AddMonoid T\ninst✝¹ : CovariantClass T T (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass T T (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ndegb : A → B\ndegt : A → T\ndegb0 : degb 0 ≤ 0\ndegbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b\nn : ℕ\nf : AddMonoidAlgebra R A\n⊢ Finset.sup (List.prod (List.replicate n f)).support degb ≤ List.sum (List.replicate n (Finset.sup f.support degb))",
"tactic": "refine' (sup_support_list_prod_le degb0 degbm _).trans_eq _"
},
{
"state_after": "no goals",
"state_before": "R : Type u_3\nA : Type u_2\nT : Type ?u.47893\nB : Type u_1\nι : Type ?u.47899\ninst✝¹¹ : SemilatticeSup B\ninst✝¹⁰ : OrderBot B\ninst✝⁹ : SemilatticeInf T\ninst✝⁸ : OrderTop T\ninst✝⁷ : Semiring R\ninst✝⁶ : AddMonoid A\ninst✝⁵ : AddMonoid B\ninst✝⁴ : CovariantClass B B (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝³ : CovariantClass B B (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝² : AddMonoid T\ninst✝¹ : CovariantClass T T (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass T T (Function.swap fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ndegb : A → B\ndegt : A → T\ndegb0 : degb 0 ≤ 0\ndegbm : ∀ (a b : A), degb (a + b) ≤ degb a + degb b\nn : ℕ\nf : AddMonoidAlgebra R A\n⊢ List.sum (List.map (fun f => Finset.sup f.support degb) (List.replicate n f)) =\n List.sum (List.replicate n (Finset.sup f.support degb))",
"tactic": "rw [List.map_replicate]"
}
] |
[
122,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/Algebra/Homology/Augment.lean
|
ChainComplex.augmentTruncate_inv_f_zero
|
[] |
[
183,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
181,
1
] |
Mathlib/CategoryTheory/Limits/Types.lean
|
CategoryTheory.Limits.Types.FilteredColimit.rel_eq_eqvGen_quot_rel
|
[
{
"state_after": "case h.mk.h.mk.a\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ TypeMax\ninst✝ : IsFilteredOrEmpty J\nj : J\nx : F.obj j\nj' : J\ny : F.obj j'\n⊢ FilteredColimit.Rel F { fst := j, snd := x } { fst := j', snd := y } ↔\n EqvGen (Quot.Rel F) { fst := j, snd := x } { fst := j', snd := y }",
"state_before": "J : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ TypeMax\ninst✝ : IsFilteredOrEmpty J\n⊢ FilteredColimit.Rel F = EqvGen (Quot.Rel F)",
"tactic": "ext ⟨j, x⟩ ⟨j', y⟩"
},
{
"state_after": "case h.mk.h.mk.a.mp\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ TypeMax\ninst✝ : IsFilteredOrEmpty J\nj : J\nx : F.obj j\nj' : J\ny : F.obj j'\n⊢ FilteredColimit.Rel F { fst := j, snd := x } { fst := j', snd := y } →\n EqvGen (Quot.Rel F) { fst := j, snd := x } { fst := j', snd := y }\n\ncase h.mk.h.mk.a.mpr\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ TypeMax\ninst✝ : IsFilteredOrEmpty J\nj : J\nx : F.obj j\nj' : J\ny : F.obj j'\n⊢ EqvGen (Quot.Rel F) { fst := j, snd := x } { fst := j', snd := y } →\n FilteredColimit.Rel F { fst := j, snd := x } { fst := j', snd := y }",
"state_before": "case h.mk.h.mk.a\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ TypeMax\ninst✝ : IsFilteredOrEmpty J\nj : J\nx : F.obj j\nj' : J\ny : F.obj j'\n⊢ FilteredColimit.Rel F { fst := j, snd := x } { fst := j', snd := y } ↔\n EqvGen (Quot.Rel F) { fst := j, snd := x } { fst := j', snd := y }",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case h.mk.h.mk.a.mp\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ TypeMax\ninst✝ : IsFilteredOrEmpty J\nj : J\nx : F.obj j\nj' : J\ny : F.obj j'\n⊢ FilteredColimit.Rel F { fst := j, snd := x } { fst := j', snd := y } →\n EqvGen (Quot.Rel F) { fst := j, snd := x } { fst := j', snd := y }",
"tactic": "apply eqvGen_quot_rel_of_rel"
},
{
"state_after": "case h.mk.h.mk.a.mpr\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ TypeMax\ninst✝ : IsFilteredOrEmpty J\nj : J\nx : F.obj j\nj' : J\ny : F.obj j'\n⊢ EqvGen (Quot.Rel F) { fst := j, snd := x } { fst := j', snd := y } →\n EqvGen (FilteredColimit.Rel F) { fst := j, snd := x } { fst := j', snd := y }",
"state_before": "case h.mk.h.mk.a.mpr\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ TypeMax\ninst✝ : IsFilteredOrEmpty J\nj : J\nx : F.obj j\nj' : J\ny : F.obj j'\n⊢ EqvGen (Quot.Rel F) { fst := j, snd := x } { fst := j', snd := y } →\n FilteredColimit.Rel F { fst := j, snd := x } { fst := j', snd := y }",
"tactic": "rw [← (FilteredColimit.rel_equiv F).eqvGen_iff]"
},
{
"state_after": "no goals",
"state_before": "case h.mk.h.mk.a.mpr\nJ : Type v\ninst✝¹ : SmallCategory J\nF : J ⥤ TypeMax\ninst✝ : IsFilteredOrEmpty J\nj : J\nx : F.obj j\nj' : J\ny : F.obj j'\n⊢ EqvGen (Quot.Rel F) { fst := j, snd := x } { fst := j', snd := y } →\n EqvGen (FilteredColimit.Rel F) { fst := j, snd := x } { fst := j', snd := y }",
"tactic": "exact EqvGen.mono (rel_of_quot_rel F)"
}
] |
[
470,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
464,
11
] |
Mathlib/Data/List/Chain.lean
|
List.chain'_nil
|
[] |
[
194,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
193,
1
] |
Std/Data/List/Lemmas.lean
|
List.getLastD_cons
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\na b : α\nl : List α\n⊢ getLastD (b :: l) a = getLastD l b",
"tactic": "cases l <;> rfl"
}
] |
[
462,
98
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
462,
9
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
nndist_triangle
|
[] |
[
390,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
389,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.toReal_ofNat
|
[] |
[
710,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
708,
9
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.dart_fst_mem_support_of_mem_darts
|
[
{
"state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v v✝ : V\nh : Adj G u v✝\np' : Walk G v✝ v\nd : Dart G\nhd : d = { toProd := (u, v✝), is_adj := h } ∨ d ∈ darts p'\n⊢ d.fst = u ∨ d.fst ∈ support p'",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v v✝ : V\nh : Adj G u v✝\np' : Walk G v✝ v\nd : Dart G\nhd : d ∈ darts (cons h p')\n⊢ d.fst ∈ support (cons h p')",
"tactic": "simp only [support_cons, darts_cons, List.mem_cons] at hd ⊢"
},
{
"state_after": "case inl\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v v✝ : V\nh : Adj G u v✝\np' : Walk G v✝ v\n⊢ { toProd := (u, v✝), is_adj := h }.toProd.fst = u ∨ { toProd := (u, v✝), is_adj := h }.toProd.fst ∈ support p'\n\ncase inr\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v v✝ : V\nh : Adj G u v✝\np' : Walk G v✝ v\nd : Dart G\nhd : d ∈ darts p'\n⊢ d.fst = u ∨ d.fst ∈ support p'",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v v✝ : V\nh : Adj G u v✝\np' : Walk G v✝ v\nd : Dart G\nhd : d = { toProd := (u, v✝), is_adj := h } ∨ d ∈ darts p'\n⊢ d.fst = u ∨ d.fst ∈ support p'",
"tactic": "rcases hd with (rfl | hd)"
},
{
"state_after": "no goals",
"state_before": "case inl\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v v✝ : V\nh : Adj G u v✝\np' : Walk G v✝ v\n⊢ { toProd := (u, v✝), is_adj := h }.toProd.fst = u ∨ { toProd := (u, v✝), is_adj := h }.toProd.fst ∈ support p'",
"tactic": "exact Or.inl rfl"
},
{
"state_after": "no goals",
"state_before": "case inr\nV : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nu v v✝ : V\nh : Adj G u v✝\np' : Walk G v✝ v\nd : Dart G\nhd : d ∈ darts p'\n⊢ d.fst = u ∨ d.fst ∈ support p'",
"tactic": "exact Or.inr (dart_fst_mem_support_of_mem_darts _ hd)"
}
] |
[
790,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
784,
1
] |
Mathlib/Data/Multiset/Powerset.lean
|
Multiset.powersetAux'_nil
|
[] |
[
55,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/MeasureTheory/Function/LocallyIntegrable.lean
|
MeasureTheory.IntegrableOn.mul_continuousOn_of_subset
|
[
{
"state_after": "case intro\nX : Type u_1\nY : Type ?u.2476922\nE : Type ?u.2476925\nR : Type u_2\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\nA K : Set X\ninst✝¹ : NormedRing R\ninst✝ : SecondCountableTopologyEither X R\ng g' : X → R\nhg : IntegrableOn g A\nhg' : ContinuousOn g' K\nhA : MeasurableSet A\nhK : IsCompact K\nhAK : A ⊆ K\nC : ℝ\nhC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C\n⊢ IntegrableOn (fun x => g x * g' x) A",
"state_before": "X : Type u_1\nY : Type ?u.2476922\nE : Type ?u.2476925\nR : Type u_2\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\nA K : Set X\ninst✝¹ : NormedRing R\ninst✝ : SecondCountableTopologyEither X R\ng g' : X → R\nhg : IntegrableOn g A\nhg' : ContinuousOn g' K\nhA : MeasurableSet A\nhK : IsCompact K\nhAK : A ⊆ K\n⊢ IntegrableOn (fun x => g x * g' x) A",
"tactic": "rcases IsCompact.exists_bound_of_continuousOn hK hg' with ⟨C, hC⟩"
},
{
"state_after": "case intro\nX : Type u_1\nY : Type ?u.2476922\nE : Type ?u.2476925\nR : Type u_2\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\nA K : Set X\ninst✝¹ : NormedRing R\ninst✝ : SecondCountableTopologyEither X R\ng g' : X → R\nhg : Memℒp g 1\nhg' : ContinuousOn g' K\nhA : MeasurableSet A\nhK : IsCompact K\nhAK : A ⊆ K\nC : ℝ\nhC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C\n⊢ Memℒp (fun x => g x * g' x) 1",
"state_before": "case intro\nX : Type u_1\nY : Type ?u.2476922\nE : Type ?u.2476925\nR : Type u_2\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\nA K : Set X\ninst✝¹ : NormedRing R\ninst✝ : SecondCountableTopologyEither X R\ng g' : X → R\nhg : IntegrableOn g A\nhg' : ContinuousOn g' K\nhA : MeasurableSet A\nhK : IsCompact K\nhAK : A ⊆ K\nC : ℝ\nhC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C\n⊢ IntegrableOn (fun x => g x * g' x) A",
"tactic": "rw [IntegrableOn, ← memℒp_one_iff_integrable] at hg⊢"
},
{
"state_after": "case intro\nX : Type u_1\nY : Type ?u.2476922\nE : Type ?u.2476925\nR : Type u_2\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\nA K : Set X\ninst✝¹ : NormedRing R\ninst✝ : SecondCountableTopologyEither X R\ng g' : X → R\nhg : Memℒp g 1\nhg' : ContinuousOn g' K\nhA : MeasurableSet A\nhK : IsCompact K\nhAK : A ⊆ K\nC : ℝ\nhC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C\nthis : ∀ᵐ (x : X) ∂Measure.restrict μ A, ‖g x * g' x‖ ≤ C * ‖g x‖\n⊢ Memℒp (fun x => g x * g' x) 1",
"state_before": "case intro\nX : Type u_1\nY : Type ?u.2476922\nE : Type ?u.2476925\nR : Type u_2\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\nA K : Set X\ninst✝¹ : NormedRing R\ninst✝ : SecondCountableTopologyEither X R\ng g' : X → R\nhg : Memℒp g 1\nhg' : ContinuousOn g' K\nhA : MeasurableSet A\nhK : IsCompact K\nhAK : A ⊆ K\nC : ℝ\nhC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C\n⊢ Memℒp (fun x => g x * g' x) 1",
"tactic": "have : ∀ᵐ x ∂μ.restrict A, ‖g x * g' x‖ ≤ C * ‖g x‖ := by\n filter_upwards [ae_restrict_mem hA]with x hx\n refine' (norm_mul_le _ _).trans _\n rw [mul_comm]\n apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _)"
},
{
"state_after": "no goals",
"state_before": "case intro\nX : Type u_1\nY : Type ?u.2476922\nE : Type ?u.2476925\nR : Type u_2\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\nA K : Set X\ninst✝¹ : NormedRing R\ninst✝ : SecondCountableTopologyEither X R\ng g' : X → R\nhg : Memℒp g 1\nhg' : ContinuousOn g' K\nhA : MeasurableSet A\nhK : IsCompact K\nhAK : A ⊆ K\nC : ℝ\nhC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C\nthis : ∀ᵐ (x : X) ∂Measure.restrict μ A, ‖g x * g' x‖ ≤ C * ‖g x‖\n⊢ Memℒp (fun x => g x * g' x) 1",
"tactic": "exact\n Memℒp.of_le_mul hg (hg.aestronglyMeasurable.mul <| (hg'.mono hAK).aestronglyMeasurable hA) this"
},
{
"state_after": "case h\nX : Type u_1\nY : Type ?u.2476922\nE : Type ?u.2476925\nR : Type u_2\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\nA K : Set X\ninst✝¹ : NormedRing R\ninst✝ : SecondCountableTopologyEither X R\ng g' : X → R\nhg : Memℒp g 1\nhg' : ContinuousOn g' K\nhA : MeasurableSet A\nhK : IsCompact K\nhAK : A ⊆ K\nC : ℝ\nhC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C\nx : X\nhx : x ∈ A\n⊢ ‖g x * g' x‖ ≤ C * ‖g x‖",
"state_before": "X : Type u_1\nY : Type ?u.2476922\nE : Type ?u.2476925\nR : Type u_2\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\nA K : Set X\ninst✝¹ : NormedRing R\ninst✝ : SecondCountableTopologyEither X R\ng g' : X → R\nhg : Memℒp g 1\nhg' : ContinuousOn g' K\nhA : MeasurableSet A\nhK : IsCompact K\nhAK : A ⊆ K\nC : ℝ\nhC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C\n⊢ ∀ᵐ (x : X) ∂Measure.restrict μ A, ‖g x * g' x‖ ≤ C * ‖g x‖",
"tactic": "filter_upwards [ae_restrict_mem hA]with x hx"
},
{
"state_after": "case h\nX : Type u_1\nY : Type ?u.2476922\nE : Type ?u.2476925\nR : Type u_2\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\nA K : Set X\ninst✝¹ : NormedRing R\ninst✝ : SecondCountableTopologyEither X R\ng g' : X → R\nhg : Memℒp g 1\nhg' : ContinuousOn g' K\nhA : MeasurableSet A\nhK : IsCompact K\nhAK : A ⊆ K\nC : ℝ\nhC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C\nx : X\nhx : x ∈ A\n⊢ ‖g x‖ * ‖g' x‖ ≤ C * ‖g x‖",
"state_before": "case h\nX : Type u_1\nY : Type ?u.2476922\nE : Type ?u.2476925\nR : Type u_2\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\nA K : Set X\ninst✝¹ : NormedRing R\ninst✝ : SecondCountableTopologyEither X R\ng g' : X → R\nhg : Memℒp g 1\nhg' : ContinuousOn g' K\nhA : MeasurableSet A\nhK : IsCompact K\nhAK : A ⊆ K\nC : ℝ\nhC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C\nx : X\nhx : x ∈ A\n⊢ ‖g x * g' x‖ ≤ C * ‖g x‖",
"tactic": "refine' (norm_mul_le _ _).trans _"
},
{
"state_after": "case h\nX : Type u_1\nY : Type ?u.2476922\nE : Type ?u.2476925\nR : Type u_2\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\nA K : Set X\ninst✝¹ : NormedRing R\ninst✝ : SecondCountableTopologyEither X R\ng g' : X → R\nhg : Memℒp g 1\nhg' : ContinuousOn g' K\nhA : MeasurableSet A\nhK : IsCompact K\nhAK : A ⊆ K\nC : ℝ\nhC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C\nx : X\nhx : x ∈ A\n⊢ ‖g' x‖ * ‖g x‖ ≤ C * ‖g x‖",
"state_before": "case h\nX : Type u_1\nY : Type ?u.2476922\nE : Type ?u.2476925\nR : Type u_2\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\nA K : Set X\ninst✝¹ : NormedRing R\ninst✝ : SecondCountableTopologyEither X R\ng g' : X → R\nhg : Memℒp g 1\nhg' : ContinuousOn g' K\nhA : MeasurableSet A\nhK : IsCompact K\nhAK : A ⊆ K\nC : ℝ\nhC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C\nx : X\nhx : x ∈ A\n⊢ ‖g x‖ * ‖g' x‖ ≤ C * ‖g x‖",
"tactic": "rw [mul_comm]"
},
{
"state_after": "no goals",
"state_before": "case h\nX : Type u_1\nY : Type ?u.2476922\nE : Type ?u.2476925\nR : Type u_2\ninst✝⁷ : MeasurableSpace X\ninst✝⁶ : TopologicalSpace X\ninst✝⁵ : MeasurableSpace Y\ninst✝⁴ : TopologicalSpace Y\ninst✝³ : NormedAddCommGroup E\nf : X → E\nμ : Measure X\ns : Set X\ninst✝² : OpensMeasurableSpace X\nA K : Set X\ninst✝¹ : NormedRing R\ninst✝ : SecondCountableTopologyEither X R\ng g' : X → R\nhg : Memℒp g 1\nhg' : ContinuousOn g' K\nhA : MeasurableSet A\nhK : IsCompact K\nhAK : A ⊆ K\nC : ℝ\nhC : ∀ (x : X), x ∈ K → ‖g' x‖ ≤ C\nx : X\nhx : x ∈ A\n⊢ ‖g' x‖ * ‖g x‖ ≤ C * ‖g x‖",
"tactic": "apply mul_le_mul_of_nonneg_right (hC x (hAK hx)) (norm_nonneg _)"
}
] |
[
396,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
385,
1
] |
Mathlib/Data/Nat/Order/Basic.lean
|
Nat.eq_one_of_mul_eq_one_right
|
[] |
[
155,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
Set.disjoint_toFinset
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.86017\nγ : Type ?u.86020\ns t : Set α\ninst✝¹ : Fintype ↑s\ninst✝ : Fintype ↑t\n⊢ Disjoint (toFinset s) (toFinset t) ↔ Disjoint s t",
"tactic": "simp only [← disjoint_coe, coe_toFinset]"
}
] |
[
689,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
688,
1
] |
Mathlib/NumberTheory/Divisors.lean
|
Nat.divisors_subset_properDivisors
|
[
{
"state_after": "n m : ℕ\nhzero : n ≠ 0\nh : m ∣ n\nhdiff : m ≠ n\n⊢ ∀ ⦃x : ℕ⦄, x ∈ divisors m → x ∈ properDivisors n",
"state_before": "n m : ℕ\nhzero : n ≠ 0\nh : m ∣ n\nhdiff : m ≠ n\n⊢ divisors m ⊆ properDivisors n",
"tactic": "apply Finset.subset_iff.2"
},
{
"state_after": "n m : ℕ\nhzero : n ≠ 0\nh : m ∣ n\nhdiff : m ≠ n\nx : ℕ\nhx : x ∈ divisors m\n⊢ x ∈ properDivisors n",
"state_before": "n m : ℕ\nhzero : n ≠ 0\nh : m ∣ n\nhdiff : m ≠ n\n⊢ ∀ ⦃x : ℕ⦄, x ∈ divisors m → x ∈ properDivisors n",
"tactic": "intro x hx"
},
{
"state_after": "no goals",
"state_before": "n m : ℕ\nhzero : n ≠ 0\nh : m ∣ n\nhdiff : m ≠ n\nx : ℕ\nhx : x ∈ divisors m\n⊢ x ∈ properDivisors n",
"tactic": "exact\n Nat.mem_properDivisors.2\n ⟨(Nat.mem_divisors.1 hx).1.trans h,\n lt_of_le_of_lt (divisor_le hx)\n (lt_of_le_of_ne (divisor_le (Nat.mem_divisors.2 ⟨h, hzero⟩)) hdiff)⟩"
}
] |
[
157,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_filter_of_ne
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.334765\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝¹ : CommMonoid β\np : α → Prop\ninst✝ : DecidablePred p\nhp : ∀ (x : α), x ∈ s → f x ≠ 1 → p x\nx : α\n⊢ x ∈ s → ¬x ∈ filter p s → f x = 1",
"tactic": "classical\n rw [not_imp_comm, mem_filter]\n exact fun h₁ h₂ => ⟨h₁, by simpa using hp _ h₁ h₂⟩"
},
{
"state_after": "ι : Type ?u.334765\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝¹ : CommMonoid β\np : α → Prop\ninst✝ : DecidablePred p\nhp : ∀ (x : α), x ∈ s → f x ≠ 1 → p x\nx : α\n⊢ x ∈ s → ¬f x = 1 → x ∈ s ∧ p x",
"state_before": "ι : Type ?u.334765\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝¹ : CommMonoid β\np : α → Prop\ninst✝ : DecidablePred p\nhp : ∀ (x : α), x ∈ s → f x ≠ 1 → p x\nx : α\n⊢ x ∈ s → ¬x ∈ filter p s → f x = 1",
"tactic": "rw [not_imp_comm, mem_filter]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.334765\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝¹ : CommMonoid β\np : α → Prop\ninst✝ : DecidablePred p\nhp : ∀ (x : α), x ∈ s → f x ≠ 1 → p x\nx : α\n⊢ x ∈ s → ¬f x = 1 → x ∈ s ∧ p x",
"tactic": "exact fun h₁ h₂ => ⟨h₁, by simpa using hp _ h₁ h₂⟩"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.334765\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝¹ : CommMonoid β\np : α → Prop\ninst✝ : DecidablePred p\nhp : ∀ (x : α), x ∈ s → f x ≠ 1 → p x\nx : α\nh₁ : x ∈ s\nh₂ : ¬f x = 1\n⊢ p x",
"tactic": "simpa using hp _ h₁ h₂"
}
] |
[
760,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
755,
1
] |
Mathlib/Init/Data/Fin/Basic.lean
|
Fin.veq_of_eq
|
[] |
[
18,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
17,
1
] |
Mathlib/LinearAlgebra/Matrix/Basis.lean
|
basis_toMatrix_mul_linearMap_toMatrix_mul_basis_toMatrix
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nι' : Type u_2\nκ : Type u_3\nκ' : Type u_5\nR : Type u_4\nM : Type u_7\ninst✝¹³ : CommSemiring R\ninst✝¹² : AddCommMonoid M\ninst✝¹¹ : Module R M\nR₂ : Type ?u.410998\nM₂ : Type ?u.411001\ninst✝¹⁰ : CommRing R₂\ninst✝⁹ : AddCommGroup M₂\ninst✝⁸ : Module R₂ M₂\ne : Basis ι R M\nv : ι' → M\ni : ι\nj : ι'\nN : Type u_6\ninst✝⁷ : AddCommMonoid N\ninst✝⁶ : Module R N\nb : Basis ι R M\nb' : Basis ι' R M\nc : Basis κ R N\nc' : Basis κ' R N\nf : M →ₗ[R] N\ninst✝⁵ : Fintype ι'\ninst✝⁴ : Fintype κ\ninst✝³ : Fintype κ'\ninst✝² : Fintype ι\ninst✝¹ : DecidableEq ι\ninst✝ : DecidableEq ι'\n⊢ Basis.toMatrix c ↑c' ⬝ ↑(toMatrix b' c') f ⬝ Basis.toMatrix b' ↑b = ↑(toMatrix b c) f",
"tactic": "rw [basis_toMatrix_mul_linearMap_toMatrix, linearMap_toMatrix_mul_basis_toMatrix]"
}
] |
[
200,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
1
] |
Mathlib/Topology/Separation.lean
|
Set.Finite.isClosed
|
[
{
"state_after": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns : Set α\nhs : Set.Finite s\n⊢ IsClosed (⋃ (x : α) (_ : x ∈ s), {x})",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns : Set α\nhs : Set.Finite s\n⊢ IsClosed s",
"tactic": "rw [← biUnion_of_singleton s]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\ninst✝¹ : TopologicalSpace α\ninst✝ : T1Space α\ns : Set α\nhs : Set.Finite s\n⊢ IsClosed (⋃ (x : α) (_ : x ∈ s), {x})",
"tactic": "exact isClosed_biUnion hs fun i _ => isClosed_singleton"
}
] |
[
416,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
414,
11
] |
Mathlib/LinearAlgebra/StdBasis.lean
|
Pi.linearIndependent_stdBasis
|
[
{
"state_after": "R : Type u_1\nη : Type u_3\nιs : η → Type u_4\nMs : η → Type u_2\ninst✝³ : Ring R\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module R (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent R (v i)\nhs' : ∀ (j : η), LinearIndependent R fun i => ↑(stdBasis R Ms j) (v j i)\n⊢ LinearIndependent R fun ji => ↑(stdBasis R Ms ji.fst) (v ji.fst ji.snd)",
"state_before": "R : Type u_1\nη : Type u_3\nιs : η → Type u_4\nMs : η → Type u_2\ninst✝³ : Ring R\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module R (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent R (v i)\n⊢ LinearIndependent R fun ji => ↑(stdBasis R Ms ji.fst) (v ji.fst ji.snd)",
"tactic": "have hs' : ∀ j : η, LinearIndependent R fun i : ιs j => stdBasis R Ms j (v j i) := by\n intro j\n exact (hs j).map' _ (ker_stdBasis _ _ _)"
},
{
"state_after": "R : Type u_1\nη : Type u_3\nιs : η → Type u_4\nMs : η → Type u_2\ninst✝³ : Ring R\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module R (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent R (v i)\nhs' : ∀ (j : η), LinearIndependent R fun i => ↑(stdBasis R Ms j) (v j i)\n⊢ ∀ (i : η) (t : Set η),\n Set.Finite t →\n ¬i ∈ t →\n Disjoint (span R (Set.range fun i_1 => ↑(stdBasis R Ms i) (v i i_1)))\n (⨆ (i : η) (_ : i ∈ t), span R (Set.range fun i_1 => ↑(stdBasis R Ms i) (v i i_1)))",
"state_before": "R : Type u_1\nη : Type u_3\nιs : η → Type u_4\nMs : η → Type u_2\ninst✝³ : Ring R\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module R (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent R (v i)\nhs' : ∀ (j : η), LinearIndependent R fun i => ↑(stdBasis R Ms j) (v j i)\n⊢ LinearIndependent R fun ji => ↑(stdBasis R Ms ji.fst) (v ji.fst ji.snd)",
"tactic": "apply linearIndependent_iUnion_finite hs'"
},
{
"state_after": "R : Type u_1\nη : Type u_3\nιs : η → Type u_4\nMs : η → Type u_2\ninst✝³ : Ring R\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module R (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent R (v i)\nj : η\n⊢ LinearIndependent R fun i => ↑(stdBasis R Ms j) (v j i)",
"state_before": "R : Type u_1\nη : Type u_3\nιs : η → Type u_4\nMs : η → Type u_2\ninst✝³ : Ring R\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module R (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent R (v i)\n⊢ ∀ (j : η), LinearIndependent R fun i => ↑(stdBasis R Ms j) (v j i)",
"tactic": "intro j"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nη : Type u_3\nιs : η → Type u_4\nMs : η → Type u_2\ninst✝³ : Ring R\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module R (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent R (v i)\nj : η\n⊢ LinearIndependent R fun i => ↑(stdBasis R Ms j) (v j i)",
"tactic": "exact (hs j).map' _ (ker_stdBasis _ _ _)"
},
{
"state_after": "R : Type u_1\nη : Type u_3\nιs : η → Type u_4\nMs : η → Type u_2\ninst✝³ : Ring R\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module R (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent R (v i)\nhs' : ∀ (j : η), LinearIndependent R fun i => ↑(stdBasis R Ms j) (v j i)\nj : η\nJ : Set η\na✝ : Set.Finite J\nhiJ : ¬j ∈ J\n⊢ Disjoint (span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)))\n (⨆ (i : η) (_ : i ∈ J), span R (Set.range fun i_1 => ↑(stdBasis R Ms i) (v i i_1)))",
"state_before": "R : Type u_1\nη : Type u_3\nιs : η → Type u_4\nMs : η → Type u_2\ninst✝³ : Ring R\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module R (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent R (v i)\nhs' : ∀ (j : η), LinearIndependent R fun i => ↑(stdBasis R Ms j) (v j i)\n⊢ ∀ (i : η) (t : Set η),\n Set.Finite t →\n ¬i ∈ t →\n Disjoint (span R (Set.range fun i_1 => ↑(stdBasis R Ms i) (v i i_1)))\n (⨆ (i : η) (_ : i ∈ t), span R (Set.range fun i_1 => ↑(stdBasis R Ms i) (v i i_1)))",
"tactic": "intro j J _ hiJ"
},
{
"state_after": "R : Type u_1\nη : Type u_3\nιs : η → Type u_4\nMs : η → Type u_2\ninst✝³ : Ring R\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module R (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent R (v i)\nhs' : ∀ (j : η), LinearIndependent R fun i => ↑(stdBasis R Ms j) (v j i)\nj : η\nJ : Set η\na✝ : Set.Finite J\nhiJ : ¬j ∈ J\n⊢ Disjoint (span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)))\n (⨆ (i : η) (_ : i ∈ J), span R (Set.range fun i_1 => ↑(stdBasis R Ms i) (v i i_1)))",
"state_before": "R : Type u_1\nη : Type u_3\nιs : η → Type u_4\nMs : η → Type u_2\ninst✝³ : Ring R\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module R (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent R (v i)\nhs' : ∀ (j : η), LinearIndependent R fun i => ↑(stdBasis R Ms j) (v j i)\nj : η\nJ : Set η\na✝ : Set.Finite J\nhiJ : ¬j ∈ J\n⊢ Disjoint (span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)))\n (⨆ (i : η) (_ : i ∈ J), span R (Set.range fun i_1 => ↑(stdBasis R Ms i) (v i i_1)))",
"tactic": "simp only"
},
{
"state_after": "R : Type u_1\nη : Type u_3\nιs : η → Type u_4\nMs : η → Type u_2\ninst✝³ : Ring R\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module R (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent R (v i)\nhs' : ∀ (j : η), LinearIndependent R fun i => ↑(stdBasis R Ms j) (v j i)\nj : η\nJ : Set η\na✝ : Set.Finite J\nhiJ : ¬j ∈ J\nh₀ : ∀ (j : η), span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)) ≤ LinearMap.range (stdBasis R Ms j)\n⊢ Disjoint (span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)))\n (⨆ (i : η) (_ : i ∈ J), span R (Set.range fun i_1 => ↑(stdBasis R Ms i) (v i i_1)))",
"state_before": "R : Type u_1\nη : Type u_3\nιs : η → Type u_4\nMs : η → Type u_2\ninst✝³ : Ring R\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module R (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent R (v i)\nhs' : ∀ (j : η), LinearIndependent R fun i => ↑(stdBasis R Ms j) (v j i)\nj : η\nJ : Set η\na✝ : Set.Finite J\nhiJ : ¬j ∈ J\n⊢ Disjoint (span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)))\n (⨆ (i : η) (_ : i ∈ J), span R (Set.range fun i_1 => ↑(stdBasis R Ms i) (v i i_1)))",
"tactic": "have h₀ :\n ∀ j, span R (range fun i : ιs j => stdBasis R Ms j (v j i)) ≤\n LinearMap.range (stdBasis R Ms j) := by\n intro j\n rw [span_le, LinearMap.range_coe]\n apply range_comp_subset_range"
},
{
"state_after": "R : Type u_1\nη : Type u_3\nιs : η → Type u_4\nMs : η → Type u_2\ninst✝³ : Ring R\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module R (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent R (v i)\nhs' : ∀ (j : η), LinearIndependent R fun i => ↑(stdBasis R Ms j) (v j i)\nj : η\nJ : Set η\na✝ : Set.Finite J\nhiJ : ¬j ∈ J\nh₀ : ∀ (j : η), span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)) ≤ LinearMap.range (stdBasis R Ms j)\nh₁ : span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)) ≤ ⨆ (i : η) (_ : i ∈ {j}), LinearMap.range (stdBasis R Ms i)\n⊢ Disjoint (span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)))\n (⨆ (i : η) (_ : i ∈ J), span R (Set.range fun i_1 => ↑(stdBasis R Ms i) (v i i_1)))",
"state_before": "R : Type u_1\nη : Type u_3\nιs : η → Type u_4\nMs : η → Type u_2\ninst✝³ : Ring R\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module R (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent R (v i)\nhs' : ∀ (j : η), LinearIndependent R fun i => ↑(stdBasis R Ms j) (v j i)\nj : η\nJ : Set η\na✝ : Set.Finite J\nhiJ : ¬j ∈ J\nh₀ : ∀ (j : η), span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)) ≤ LinearMap.range (stdBasis R Ms j)\n⊢ Disjoint (span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)))\n (⨆ (i : η) (_ : i ∈ J), span R (Set.range fun i_1 => ↑(stdBasis R Ms i) (v i i_1)))",
"tactic": "have h₁ :\n span R (range fun i : ιs j => stdBasis R Ms j (v j i)) ≤\n ⨆ i ∈ ({j} : Set _), LinearMap.range (stdBasis R Ms i) := by\n rw [@iSup_singleton _ _ _ fun i => LinearMap.range (stdBasis R (Ms) i)]\n apply h₀"
},
{
"state_after": "R : Type u_1\nη : Type u_3\nιs : η → Type u_4\nMs : η → Type u_2\ninst✝³ : Ring R\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module R (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent R (v i)\nhs' : ∀ (j : η), LinearIndependent R fun i => ↑(stdBasis R Ms j) (v j i)\nj : η\nJ : Set η\na✝ : Set.Finite J\nhiJ : ¬j ∈ J\nh₀ : ∀ (j : η), span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)) ≤ LinearMap.range (stdBasis R Ms j)\nh₁ : span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)) ≤ ⨆ (i : η) (_ : i ∈ {j}), LinearMap.range (stdBasis R Ms i)\nh₂ :\n (⨆ (j : η) (_ : j ∈ J), span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i))) ≤\n ⨆ (j : η) (_ : j ∈ J), LinearMap.range (stdBasis R (fun j => Ms j) j)\n⊢ Disjoint (span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)))\n (⨆ (i : η) (_ : i ∈ J), span R (Set.range fun i_1 => ↑(stdBasis R Ms i) (v i i_1)))",
"state_before": "R : Type u_1\nη : Type u_3\nιs : η → Type u_4\nMs : η → Type u_2\ninst✝³ : Ring R\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module R (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent R (v i)\nhs' : ∀ (j : η), LinearIndependent R fun i => ↑(stdBasis R Ms j) (v j i)\nj : η\nJ : Set η\na✝ : Set.Finite J\nhiJ : ¬j ∈ J\nh₀ : ∀ (j : η), span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)) ≤ LinearMap.range (stdBasis R Ms j)\nh₁ : span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)) ≤ ⨆ (i : η) (_ : i ∈ {j}), LinearMap.range (stdBasis R Ms i)\n⊢ Disjoint (span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)))\n (⨆ (i : η) (_ : i ∈ J), span R (Set.range fun i_1 => ↑(stdBasis R Ms i) (v i i_1)))",
"tactic": "have h₂ :\n (⨆ j ∈ J, span R (range fun i : ιs j => stdBasis R Ms j (v j i))) ≤\n ⨆ j ∈ J, LinearMap.range (stdBasis R (fun j : η => Ms j) j) :=\n iSup₂_mono fun i _ => h₀ i"
},
{
"state_after": "R : Type u_1\nη : Type u_3\nιs : η → Type u_4\nMs : η → Type u_2\ninst✝³ : Ring R\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module R (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent R (v i)\nhs' : ∀ (j : η), LinearIndependent R fun i => ↑(stdBasis R Ms j) (v j i)\nj : η\nJ : Set η\na✝ : Set.Finite J\nhiJ : ¬j ∈ J\nh₀ : ∀ (j : η), span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)) ≤ LinearMap.range (stdBasis R Ms j)\nh₁ : span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)) ≤ ⨆ (i : η) (_ : i ∈ {j}), LinearMap.range (stdBasis R Ms i)\nh₂ :\n (⨆ (j : η) (_ : j ∈ J), span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i))) ≤\n ⨆ (j : η) (_ : j ∈ J), LinearMap.range (stdBasis R (fun j => Ms j) j)\nh₃ : Disjoint (fun i => i ∈ {j}) J\n⊢ Disjoint (span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)))\n (⨆ (i : η) (_ : i ∈ J), span R (Set.range fun i_1 => ↑(stdBasis R Ms i) (v i i_1)))",
"state_before": "R : Type u_1\nη : Type u_3\nιs : η → Type u_4\nMs : η → Type u_2\ninst✝³ : Ring R\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module R (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent R (v i)\nhs' : ∀ (j : η), LinearIndependent R fun i => ↑(stdBasis R Ms j) (v j i)\nj : η\nJ : Set η\na✝ : Set.Finite J\nhiJ : ¬j ∈ J\nh₀ : ∀ (j : η), span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)) ≤ LinearMap.range (stdBasis R Ms j)\nh₁ : span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)) ≤ ⨆ (i : η) (_ : i ∈ {j}), LinearMap.range (stdBasis R Ms i)\nh₂ :\n (⨆ (j : η) (_ : j ∈ J), span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i))) ≤\n ⨆ (j : η) (_ : j ∈ J), LinearMap.range (stdBasis R (fun j => Ms j) j)\n⊢ Disjoint (span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)))\n (⨆ (i : η) (_ : i ∈ J), span R (Set.range fun i_1 => ↑(stdBasis R Ms i) (v i i_1)))",
"tactic": "have h₃ : Disjoint (fun i : η => i ∈ ({j} : Set _)) J := by\n convert Set.disjoint_singleton_left.2 hiJ using 0"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nη : Type u_3\nιs : η → Type u_4\nMs : η → Type u_2\ninst✝³ : Ring R\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module R (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent R (v i)\nhs' : ∀ (j : η), LinearIndependent R fun i => ↑(stdBasis R Ms j) (v j i)\nj : η\nJ : Set η\na✝ : Set.Finite J\nhiJ : ¬j ∈ J\nh₀ : ∀ (j : η), span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)) ≤ LinearMap.range (stdBasis R Ms j)\nh₁ : span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)) ≤ ⨆ (i : η) (_ : i ∈ {j}), LinearMap.range (stdBasis R Ms i)\nh₂ :\n (⨆ (j : η) (_ : j ∈ J), span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i))) ≤\n ⨆ (j : η) (_ : j ∈ J), LinearMap.range (stdBasis R (fun j => Ms j) j)\nh₃ : Disjoint (fun i => i ∈ {j}) J\n⊢ Disjoint (span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)))\n (⨆ (i : η) (_ : i ∈ J), span R (Set.range fun i_1 => ↑(stdBasis R Ms i) (v i i_1)))",
"tactic": "exact (disjoint_stdBasis_stdBasis _ _ _ _ h₃).mono h₁ h₂"
},
{
"state_after": "R : Type u_1\nη : Type u_3\nιs : η → Type u_4\nMs : η → Type u_2\ninst✝³ : Ring R\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module R (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent R (v i)\nhs' : ∀ (j : η), LinearIndependent R fun i => ↑(stdBasis R Ms j) (v j i)\nj✝ : η\nJ : Set η\na✝ : Set.Finite J\nhiJ : ¬j✝ ∈ J\nj : η\n⊢ span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)) ≤ LinearMap.range (stdBasis R Ms j)",
"state_before": "R : Type u_1\nη : Type u_3\nιs : η → Type u_4\nMs : η → Type u_2\ninst✝³ : Ring R\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module R (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent R (v i)\nhs' : ∀ (j : η), LinearIndependent R fun i => ↑(stdBasis R Ms j) (v j i)\nj : η\nJ : Set η\na✝ : Set.Finite J\nhiJ : ¬j ∈ J\n⊢ ∀ (j : η), span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)) ≤ LinearMap.range (stdBasis R Ms j)",
"tactic": "intro j"
},
{
"state_after": "R : Type u_1\nη : Type u_3\nιs : η → Type u_4\nMs : η → Type u_2\ninst✝³ : Ring R\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module R (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent R (v i)\nhs' : ∀ (j : η), LinearIndependent R fun i => ↑(stdBasis R Ms j) (v j i)\nj✝ : η\nJ : Set η\na✝ : Set.Finite J\nhiJ : ¬j✝ ∈ J\nj : η\n⊢ (Set.range fun i => ↑(stdBasis R Ms j) (v j i)) ⊆ Set.range ↑(stdBasis R Ms j)",
"state_before": "R : Type u_1\nη : Type u_3\nιs : η → Type u_4\nMs : η → Type u_2\ninst✝³ : Ring R\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module R (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent R (v i)\nhs' : ∀ (j : η), LinearIndependent R fun i => ↑(stdBasis R Ms j) (v j i)\nj✝ : η\nJ : Set η\na✝ : Set.Finite J\nhiJ : ¬j✝ ∈ J\nj : η\n⊢ span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)) ≤ LinearMap.range (stdBasis R Ms j)",
"tactic": "rw [span_le, LinearMap.range_coe]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nη : Type u_3\nιs : η → Type u_4\nMs : η → Type u_2\ninst✝³ : Ring R\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module R (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent R (v i)\nhs' : ∀ (j : η), LinearIndependent R fun i => ↑(stdBasis R Ms j) (v j i)\nj✝ : η\nJ : Set η\na✝ : Set.Finite J\nhiJ : ¬j✝ ∈ J\nj : η\n⊢ (Set.range fun i => ↑(stdBasis R Ms j) (v j i)) ⊆ Set.range ↑(stdBasis R Ms j)",
"tactic": "apply range_comp_subset_range"
},
{
"state_after": "R : Type u_1\nη : Type u_3\nιs : η → Type u_4\nMs : η → Type u_2\ninst✝³ : Ring R\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module R (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent R (v i)\nhs' : ∀ (j : η), LinearIndependent R fun i => ↑(stdBasis R Ms j) (v j i)\nj : η\nJ : Set η\na✝ : Set.Finite J\nhiJ : ¬j ∈ J\nh₀ : ∀ (j : η), span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)) ≤ LinearMap.range (stdBasis R Ms j)\n⊢ span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)) ≤ LinearMap.range (stdBasis R Ms j)",
"state_before": "R : Type u_1\nη : Type u_3\nιs : η → Type u_4\nMs : η → Type u_2\ninst✝³ : Ring R\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module R (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent R (v i)\nhs' : ∀ (j : η), LinearIndependent R fun i => ↑(stdBasis R Ms j) (v j i)\nj : η\nJ : Set η\na✝ : Set.Finite J\nhiJ : ¬j ∈ J\nh₀ : ∀ (j : η), span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)) ≤ LinearMap.range (stdBasis R Ms j)\n⊢ span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)) ≤ ⨆ (i : η) (_ : i ∈ {j}), LinearMap.range (stdBasis R Ms i)",
"tactic": "rw [@iSup_singleton _ _ _ fun i => LinearMap.range (stdBasis R (Ms) i)]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nη : Type u_3\nιs : η → Type u_4\nMs : η → Type u_2\ninst✝³ : Ring R\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module R (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent R (v i)\nhs' : ∀ (j : η), LinearIndependent R fun i => ↑(stdBasis R Ms j) (v j i)\nj : η\nJ : Set η\na✝ : Set.Finite J\nhiJ : ¬j ∈ J\nh₀ : ∀ (j : η), span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)) ≤ LinearMap.range (stdBasis R Ms j)\n⊢ span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)) ≤ LinearMap.range (stdBasis R Ms j)",
"tactic": "apply h₀"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nη : Type u_3\nιs : η → Type u_4\nMs : η → Type u_2\ninst✝³ : Ring R\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module R (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent R (v i)\nhs' : ∀ (j : η), LinearIndependent R fun i => ↑(stdBasis R Ms j) (v j i)\nj : η\nJ : Set η\na✝ : Set.Finite J\nhiJ : ¬j ∈ J\nh₀ : ∀ (j : η), span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)) ≤ LinearMap.range (stdBasis R Ms j)\nh₁ : span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i)) ≤ ⨆ (i : η) (_ : i ∈ {j}), LinearMap.range (stdBasis R Ms i)\nh₂ :\n (⨆ (j : η) (_ : j ∈ J), span R (Set.range fun i => ↑(stdBasis R Ms j) (v j i))) ≤\n ⨆ (j : η) (_ : j ∈ J), LinearMap.range (stdBasis R (fun j => Ms j) j)\n⊢ Disjoint (fun i => i ∈ {j}) J",
"tactic": "convert Set.disjoint_singleton_left.2 hiJ using 0"
}
] |
[
206,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
hasSum_iff_hasSum_of_ne_zero_bij
|
[] |
[
250,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
242,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Basic.lean
|
Real.log_nonneg
|
[] |
[
192,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
191,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.prod_span_singleton
|
[] |
[
617,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
615,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.support_monomial'
|
[
{
"state_after": "R : Type u\na✝ b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\na : R\n⊢ (match { toFinsupp := Finsupp.single n a } with\n | { toFinsupp := p } => p.support) ⊆\n {n}",
"state_before": "R : Type u\na✝ b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\na : R\n⊢ support (↑(monomial n) a) ⊆ {n}",
"tactic": "rw [← ofFinsupp_single, support]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\na✝ b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\na : R\n⊢ (match { toFinsupp := Finsupp.single n a } with\n | { toFinsupp := p } => p.support) ⊆\n {n}",
"tactic": "exact Finsupp.support_single_subset"
}
] |
[
841,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
839,
1
] |
Mathlib/Deprecated/Subfield.lean
|
Field.subset_closure
|
[] |
[
133,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
1
] |
Mathlib/Order/Minimal.lean
|
minimals_union
|
[] |
[
157,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/GroupTheory/FreeAbelianGroup.lean
|
FreeAbelianGroup.map_comp_apply
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nf : α → β\ng : β → γ\nx : FreeAbelianGroup α\n⊢ ↑(AddMonoidHom.comp (map g) (map f)) x = ↑(map g) (↑(map f) x)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nf : α → β\ng : β → γ\nx : FreeAbelianGroup α\n⊢ ↑(map (g ∘ f)) x = ↑(map g) (↑(map f) x)",
"tactic": "rw [map_comp]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nf : α → β\ng : β → γ\nx : FreeAbelianGroup α\n⊢ ↑(AddMonoidHom.comp (map g) (map f)) x = ↑(map g) (↑(map f) x)",
"tactic": "rfl"
}
] |
[
391,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
388,
1
] |
Mathlib/Topology/Homotopy/Basic.lean
|
ContinuousMap.Homotopic.symm
|
[] |
[
366,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
365,
1
] |
Mathlib/Probability/Independence/Basic.lean
|
ProbabilityTheory.indepSet_empty_left
|
[] |
[
179,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Mathlib/Analysis/Calculus/Deriv/Slope.lean
|
HasDerivWithinAt.liminf_right_norm_slope_le
|
[] |
[
163,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/Topology/ContinuousFunction/Bounded.lean
|
BoundedContinuousFunction.nnnorm_coe_le_nnnorm
|
[] |
[
1007,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1006,
1
] |
Mathlib/GroupTheory/GroupAction/Quotient.lean
|
MulAction.injective_ofQuotientStabilizer
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² inst✝¹ : Group α\ninst✝ : MulAction α β\nx : β\ny₁ y₂ : α ⧸ stabilizer α x\ng₁ g₂ : α\nH : g₁ • x = g₂ • x\n⊢ g₁⁻¹ * g₂ ∈ stabilizer α x",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² inst✝¹ : Group α\ninst✝ : MulAction α β\nx : β\ny₁ y₂ : α ⧸ stabilizer α x\ng₁ g₂ : α\nH : g₁ • x = g₂ • x\n⊢ Setoid.r g₁ g₂",
"tactic": "rw [leftRel_apply]"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² inst✝¹ : Group α\ninst✝ : MulAction α β\nx : β\ny₁ y₂ : α ⧸ stabilizer α x\ng₁ g₂ : α\nH : g₁ • x = g₂ • x\n⊢ (g₁⁻¹ * g₂) • x = x",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² inst✝¹ : Group α\ninst✝ : MulAction α β\nx : β\ny₁ y₂ : α ⧸ stabilizer α x\ng₁ g₂ : α\nH : g₁ • x = g₂ • x\n⊢ g₁⁻¹ * g₂ ∈ stabilizer α x",
"tactic": "show (g₁⁻¹ * g₂) • x = x"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² inst✝¹ : Group α\ninst✝ : MulAction α β\nx : β\ny₁ y₂ : α ⧸ stabilizer α x\ng₁ g₂ : α\nH : g₁ • x = g₂ • x\n⊢ (g₁⁻¹ * g₂) • x = x",
"tactic": "rw [mul_smul, ← H, inv_smul_smul]"
}
] |
[
190,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
184,
1
] |
Mathlib/Algebra/Bounds.lean
|
bddBelow_inv
|
[] |
[
42,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
41,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Independent.lean
|
sign_eq_of_affineCombination_mem_affineSpan_single_lineMap
|
[
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : LinearOrderedRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 1\ni₁ i₂ i₃ : ι\nh₁ : i₁ ∈ s\nh₂ : i₂ ∈ s\nh₃ : i₃ ∈ s\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\nc : k\nhc0 : 0 < c\nhc1 : c < 1\nhs : ↑(Finset.affineCombination k s p) w ∈ affineSpan k {p i₁, ↑(AffineMap.lineMap (p i₂) (p i₃)) c}\n⊢ ↑SignType.sign (w i₂) = ↑SignType.sign (w i₃)",
"tactic": "classical\n rw [← s.affineCombination_affineCombinationSingleWeights k p h₁, ←\n s.affineCombination_affineCombinationLineMapWeights p h₂ h₃ c] at hs\n refine'\n sign_eq_of_affineCombination_mem_affineSpan_pair h hw\n (s.sum_affineCombinationSingleWeights k h₁)\n (s.sum_affineCombinationLineMapWeights h₂ h₃ c) hs h₂ h₃\n (Finset.affineCombinationSingleWeights_apply_of_ne k h₁₂.symm)\n (Finset.affineCombinationSingleWeights_apply_of_ne k h₁₃.symm) _\n rw [Finset.affineCombinationLineMapWeights_apply_left h₂₃,\n Finset.affineCombinationLineMapWeights_apply_right h₂₃]\n simp_all only [sub_pos, sign_pos]"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : LinearOrderedRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 1\ni₁ i₂ i₃ : ι\nh₁ : i₁ ∈ s\nh₂ : i₂ ∈ s\nh₃ : i₃ ∈ s\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\nc : k\nhc0 : 0 < c\nhc1 : c < 1\nhs :\n ↑(Finset.affineCombination k s p) w ∈\n affineSpan k\n {↑(Finset.affineCombination k s p) (Finset.affineCombinationSingleWeights k i₁),\n ↑(Finset.affineCombination k s p) (Finset.affineCombinationLineMapWeights i₂ i₃ c)}\n⊢ ↑SignType.sign (w i₂) = ↑SignType.sign (w i₃)",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : LinearOrderedRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 1\ni₁ i₂ i₃ : ι\nh₁ : i₁ ∈ s\nh₂ : i₂ ∈ s\nh₃ : i₃ ∈ s\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\nc : k\nhc0 : 0 < c\nhc1 : c < 1\nhs : ↑(Finset.affineCombination k s p) w ∈ affineSpan k {p i₁, ↑(AffineMap.lineMap (p i₂) (p i₃)) c}\n⊢ ↑SignType.sign (w i₂) = ↑SignType.sign (w i₃)",
"tactic": "rw [← s.affineCombination_affineCombinationSingleWeights k p h₁, ←\n s.affineCombination_affineCombinationLineMapWeights p h₂ h₃ c] at hs"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : LinearOrderedRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 1\ni₁ i₂ i₃ : ι\nh₁ : i₁ ∈ s\nh₂ : i₂ ∈ s\nh₃ : i₃ ∈ s\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\nc : k\nhc0 : 0 < c\nhc1 : c < 1\nhs :\n ↑(Finset.affineCombination k s p) w ∈\n affineSpan k\n {↑(Finset.affineCombination k s p) (Finset.affineCombinationSingleWeights k i₁),\n ↑(Finset.affineCombination k s p) (Finset.affineCombinationLineMapWeights i₂ i₃ c)}\n⊢ ↑SignType.sign (Finset.affineCombinationLineMapWeights i₂ i₃ c i₂) =\n ↑SignType.sign (Finset.affineCombinationLineMapWeights i₂ i₃ c i₃)",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : LinearOrderedRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 1\ni₁ i₂ i₃ : ι\nh₁ : i₁ ∈ s\nh₂ : i₂ ∈ s\nh₃ : i₃ ∈ s\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\nc : k\nhc0 : 0 < c\nhc1 : c < 1\nhs :\n ↑(Finset.affineCombination k s p) w ∈\n affineSpan k\n {↑(Finset.affineCombination k s p) (Finset.affineCombinationSingleWeights k i₁),\n ↑(Finset.affineCombination k s p) (Finset.affineCombinationLineMapWeights i₂ i₃ c)}\n⊢ ↑SignType.sign (w i₂) = ↑SignType.sign (w i₃)",
"tactic": "refine'\n sign_eq_of_affineCombination_mem_affineSpan_pair h hw\n (s.sum_affineCombinationSingleWeights k h₁)\n (s.sum_affineCombinationLineMapWeights h₂ h₃ c) hs h₂ h₃\n (Finset.affineCombinationSingleWeights_apply_of_ne k h₁₂.symm)\n (Finset.affineCombinationSingleWeights_apply_of_ne k h₁₃.symm) _"
},
{
"state_after": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : LinearOrderedRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 1\ni₁ i₂ i₃ : ι\nh₁ : i₁ ∈ s\nh₂ : i₂ ∈ s\nh₃ : i₃ ∈ s\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\nc : k\nhc0 : 0 < c\nhc1 : c < 1\nhs :\n ↑(Finset.affineCombination k s p) w ∈\n affineSpan k\n {↑(Finset.affineCombination k s p) (Finset.affineCombinationSingleWeights k i₁),\n ↑(Finset.affineCombination k s p) (Finset.affineCombinationLineMapWeights i₂ i₃ c)}\n⊢ ↑SignType.sign (1 - c) = ↑SignType.sign c",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : LinearOrderedRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 1\ni₁ i₂ i₃ : ι\nh₁ : i₁ ∈ s\nh₂ : i₂ ∈ s\nh₃ : i₃ ∈ s\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\nc : k\nhc0 : 0 < c\nhc1 : c < 1\nhs :\n ↑(Finset.affineCombination k s p) w ∈\n affineSpan k\n {↑(Finset.affineCombination k s p) (Finset.affineCombinationSingleWeights k i₁),\n ↑(Finset.affineCombination k s p) (Finset.affineCombinationLineMapWeights i₂ i₃ c)}\n⊢ ↑SignType.sign (Finset.affineCombinationLineMapWeights i₂ i₃ c i₂) =\n ↑SignType.sign (Finset.affineCombinationLineMapWeights i₂ i₃ c i₃)",
"tactic": "rw [Finset.affineCombinationLineMapWeights_apply_left h₂₃,\n Finset.affineCombinationLineMapWeights_apply_right h₂₃]"
},
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : LinearOrderedRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\nh : AffineIndependent k p\nw : ι → k\ns : Finset ι\nhw : ∑ i in s, w i = 1\ni₁ i₂ i₃ : ι\nh₁ : i₁ ∈ s\nh₂ : i₂ ∈ s\nh₃ : i₃ ∈ s\nh₁₂ : i₁ ≠ i₂\nh₁₃ : i₁ ≠ i₃\nh₂₃ : i₂ ≠ i₃\nc : k\nhc0 : 0 < c\nhc1 : c < 1\nhs :\n ↑(Finset.affineCombination k s p) w ∈\n affineSpan k\n {↑(Finset.affineCombination k s p) (Finset.affineCombinationSingleWeights k i₁),\n ↑(Finset.affineCombination k s p) (Finset.affineCombinationLineMapWeights i₂ i₃ c)}\n⊢ ↑SignType.sign (1 - c) = ↑SignType.sign c",
"tactic": "simp_all only [sub_pos, sign_pos]"
}
] |
[
768,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
751,
1
] |
Mathlib/RingTheory/DedekindDomain/Ideal.lean
|
IsDedekindDomain.exists_forall_sub_mem_ideal
|
[
{
"state_after": "case intro\nR : Type u_2\nA : Type ?u.1320972\nK : Type ?u.1320975\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns : Finset ι\nP : ι → Ideal R\ne : ι → ℕ\nprime : ∀ (i : ι), i ∈ s → Prime (P i)\ncoprime : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → P i ≠ P j\nx : { x // x ∈ s } → R\ny : R\nhy :\n ∀ (i : ι) (hi : i ∈ s),\n ↑(Ideal.Quotient.mk (P i ^ e i)) y =\n ↑(Ideal.Quotient.mk (P ↑{ val := i, property := hi } ^ e ↑{ val := i, property := hi }))\n (x { val := i, property := hi })\n⊢ ∃ y, ∀ (i : ι) (hi : i ∈ s), y - x { val := i, property := hi } ∈ P i ^ e i",
"state_before": "R : Type u_2\nA : Type ?u.1320972\nK : Type ?u.1320975\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns : Finset ι\nP : ι → Ideal R\ne : ι → ℕ\nprime : ∀ (i : ι), i ∈ s → Prime (P i)\ncoprime : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → P i ≠ P j\nx : { x // x ∈ s } → R\n⊢ ∃ y, ∀ (i : ι) (hi : i ∈ s), y - x { val := i, property := hi } ∈ P i ^ e i",
"tactic": "obtain ⟨y, hy⟩ :=\n IsDedekindDomain.exists_representative_mod_finset P e prime coprime fun i =>\n Ideal.Quotient.mk _ (x i)"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type u_2\nA : Type ?u.1320972\nK : Type ?u.1320975\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : IsDomain A\ninst✝¹ : IsDomain R\ninst✝ : IsDedekindDomain R\nι : Type u_1\ns : Finset ι\nP : ι → Ideal R\ne : ι → ℕ\nprime : ∀ (i : ι), i ∈ s → Prime (P i)\ncoprime : ∀ (i : ι), i ∈ s → ∀ (j : ι), j ∈ s → i ≠ j → P i ≠ P j\nx : { x // x ∈ s } → R\ny : R\nhy :\n ∀ (i : ι) (hi : i ∈ s),\n ↑(Ideal.Quotient.mk (P i ^ e i)) y =\n ↑(Ideal.Quotient.mk (P ↑{ val := i, property := hi } ^ e ↑{ val := i, property := hi }))\n (x { val := i, property := hi })\n⊢ ∃ y, ∀ (i : ι) (hi : i ∈ s), y - x { val := i, property := hi } ∈ P i ^ e i",
"tactic": "exact ⟨y, fun i hi => Ideal.Quotient.eq.mp (hy i hi)⟩"
}
] |
[
1396,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1389,
1
] |
Mathlib/LinearAlgebra/Span.lean
|
Submodule.nontrivial_span_singleton
|
[
{
"state_after": "R : Type u_2\nR₂ : Type ?u.114155\nK : Type ?u.114158\nM : Type u_1\nM₂ : Type ?u.114164\nV : Type ?u.114167\nS : Type ?u.114170\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nx : M\nh : x ≠ 0\n⊢ 0 ≠ { val := x, property := (_ : x ∈ span R {x}) }",
"state_before": "R : Type u_2\nR₂ : Type ?u.114155\nK : Type ?u.114158\nM : Type u_1\nM₂ : Type ?u.114164\nV : Type ?u.114167\nS : Type ?u.114170\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nx : M\nh : x ≠ 0\n⊢ ∃ x_1 y, x_1 ≠ y",
"tactic": "use 0, ⟨x, Submodule.mem_span_singleton_self x⟩"
},
{
"state_after": "R : Type u_2\nR₂ : Type ?u.114155\nK : Type ?u.114158\nM : Type u_1\nM₂ : Type ?u.114164\nV : Type ?u.114167\nS : Type ?u.114170\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nx : M\nh : x ≠ 0\nH : 0 = { val := x, property := (_ : x ∈ span R {x}) }\n⊢ False",
"state_before": "R : Type u_2\nR₂ : Type ?u.114155\nK : Type ?u.114158\nM : Type u_1\nM₂ : Type ?u.114164\nV : Type ?u.114167\nS : Type ?u.114170\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nx : M\nh : x ≠ 0\n⊢ 0 ≠ { val := x, property := (_ : x ∈ span R {x}) }",
"tactic": "intro H"
},
{
"state_after": "R : Type u_2\nR₂ : Type ?u.114155\nK : Type ?u.114158\nM : Type u_1\nM₂ : Type ?u.114164\nV : Type ?u.114167\nS : Type ?u.114170\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nx : M\nh : x ≠ 0\nH : x = 0\n⊢ False",
"state_before": "R : Type u_2\nR₂ : Type ?u.114155\nK : Type ?u.114158\nM : Type u_1\nM₂ : Type ?u.114164\nV : Type ?u.114167\nS : Type ?u.114170\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nx : M\nh : x ≠ 0\nH : 0 = { val := x, property := (_ : x ∈ span R {x}) }\n⊢ False",
"tactic": "rw [eq_comm, Submodule.mk_eq_zero] at H"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nR₂ : Type ?u.114155\nK : Type ?u.114158\nM : Type u_1\nM₂ : Type ?u.114164\nV : Type ?u.114167\nS : Type ?u.114170\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns t : Set M\nx : M\nh : x ≠ 0\nH : x = 0\n⊢ False",
"tactic": "exact h H"
}
] |
[
398,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
393,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean
|
BoxIntegral.TaggedPrepartition.disjUnion_boxes
|
[] |
[
358,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
357,
1
] |
Mathlib/GroupTheory/Perm/Basic.lean
|
Equiv.Perm.sigmaCongrRight_inv
|
[] |
[
249,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
247,
1
] |
Mathlib/GroupTheory/OrderOfElement.lean
|
orderOf_ofAdd_eq_addOrderOf
|
[] |
[
147,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/Analysis/LocallyConvex/BalancedCoreHull.lean
|
mem_balancedCoreAux_iff
|
[] |
[
108,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/Logic/Relation.lean
|
Relation.refl_trans_gen_idem
|
[] |
[
558,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
557,
1
] |
Mathlib/Algebra/Order/Sub/Canonical.lean
|
AddLECancellable.lt_tsub_iff_left_of_le
|
[
{
"state_after": "α : Type u_1\ninst✝⁵ : AddCommSemigroup α\ninst✝⁴ : PartialOrder α\ninst✝³ : ExistsAddOfLE α\ninst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nhc : AddLECancellable c\nh : c ≤ b\n⊢ a < b - c ↔ a + c < b",
"state_before": "α : Type u_1\ninst✝⁵ : AddCommSemigroup α\ninst✝⁴ : PartialOrder α\ninst✝³ : ExistsAddOfLE α\ninst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nhc : AddLECancellable c\nh : c ≤ b\n⊢ a < b - c ↔ c + a < b",
"tactic": "rw [add_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁵ : AddCommSemigroup α\ninst✝⁴ : PartialOrder α\ninst✝³ : ExistsAddOfLE α\ninst✝² : CovariantClass α α (fun x x_1 => x + x_1) fun x x_1 => x ≤ x_1\ninst✝¹ : Sub α\ninst✝ : OrderedSub α\na b c d : α\nhc : AddLECancellable c\nh : c ≤ b\n⊢ a < b - c ↔ a + c < b",
"tactic": "exact hc.lt_tsub_iff_right_of_le h"
}
] |
[
152,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
11
] |
Std/Data/Int/Lemmas.lean
|
Int.add_le_add_right
|
[] |
[
758,
64
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
757,
11
] |
Mathlib/RingTheory/TensorProduct.lean
|
Algebra.TensorProduct.algEquivOfLinearEquivTensorProduct_apply
|
[] |
[
710,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
708,
1
] |
Mathlib/Order/PrimeIdeal.lean
|
Order.Ideal.isPrime_of_mem_or_compl_mem
|
[
{
"state_after": "P : Type u_1\ninst✝¹ : BooleanAlgebra P\nx : P\nI : Ideal P\ninst✝ : IsProper I\nh : ∀ {x : P}, x ∈ I ∨ xᶜ ∈ I\n⊢ ∀ {x y : P}, x ⊓ y ∈ I → ¬x ∈ I → y ∈ I",
"state_before": "P : Type u_1\ninst✝¹ : BooleanAlgebra P\nx : P\nI : Ideal P\ninst✝ : IsProper I\nh : ∀ {x : P}, x ∈ I ∨ xᶜ ∈ I\n⊢ IsPrime I",
"tactic": "simp only [isPrime_iff_mem_or_mem, or_iff_not_imp_left]"
},
{
"state_after": "P : Type u_1\ninst✝¹ : BooleanAlgebra P\nx✝ : P\nI : Ideal P\ninst✝ : IsProper I\nh : ∀ {x : P}, x ∈ I ∨ xᶜ ∈ I\nx y : P\nhxy : x ⊓ y ∈ I\nhxI : ¬x ∈ I\n⊢ y ∈ I",
"state_before": "P : Type u_1\ninst✝¹ : BooleanAlgebra P\nx : P\nI : Ideal P\ninst✝ : IsProper I\nh : ∀ {x : P}, x ∈ I ∨ xᶜ ∈ I\n⊢ ∀ {x y : P}, x ⊓ y ∈ I → ¬x ∈ I → y ∈ I",
"tactic": "intro x y hxy hxI"
},
{
"state_after": "P : Type u_1\ninst✝¹ : BooleanAlgebra P\nx✝ : P\nI : Ideal P\ninst✝ : IsProper I\nh : ∀ {x : P}, x ∈ I ∨ xᶜ ∈ I\nx y : P\nhxy : x ⊓ y ∈ I\nhxI : ¬x ∈ I\nhxcI : xᶜ ∈ I\n⊢ y ∈ I",
"state_before": "P : Type u_1\ninst✝¹ : BooleanAlgebra P\nx✝ : P\nI : Ideal P\ninst✝ : IsProper I\nh : ∀ {x : P}, x ∈ I ∨ xᶜ ∈ I\nx y : P\nhxy : x ⊓ y ∈ I\nhxI : ¬x ∈ I\n⊢ y ∈ I",
"tactic": "have hxcI : xᶜ ∈ I := h.resolve_left hxI"
},
{
"state_after": "P : Type u_1\ninst✝¹ : BooleanAlgebra P\nx✝ : P\nI : Ideal P\ninst✝ : IsProper I\nh : ∀ {x : P}, x ∈ I ∨ xᶜ ∈ I\nx y : P\nhxy : x ⊓ y ∈ I\nhxI : ¬x ∈ I\nhxcI : xᶜ ∈ I\nass : x ⊓ y ⊔ y ⊓ xᶜ ∈ I\n⊢ y ∈ I",
"state_before": "P : Type u_1\ninst✝¹ : BooleanAlgebra P\nx✝ : P\nI : Ideal P\ninst✝ : IsProper I\nh : ∀ {x : P}, x ∈ I ∨ xᶜ ∈ I\nx y : P\nhxy : x ⊓ y ∈ I\nhxI : ¬x ∈ I\nhxcI : xᶜ ∈ I\n⊢ y ∈ I",
"tactic": "have ass : x ⊓ y ⊔ y ⊓ xᶜ ∈ I := sup_mem hxy (I.lower inf_le_right hxcI)"
},
{
"state_after": "no goals",
"state_before": "P : Type u_1\ninst✝¹ : BooleanAlgebra P\nx✝ : P\nI : Ideal P\ninst✝ : IsProper I\nh : ∀ {x : P}, x ∈ I ∨ xᶜ ∈ I\nx y : P\nhxy : x ⊓ y ∈ I\nhxI : ¬x ∈ I\nhxcI : xᶜ ∈ I\nass : x ⊓ y ⊔ y ⊓ xᶜ ∈ I\n⊢ y ∈ I",
"tactic": "rwa [inf_comm, sup_inf_inf_compl] at ass"
}
] |
[
193,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.extend_apply'
|
[] |
[
383,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
381,
1
] |
Mathlib/Data/Nat/Choose/Factorization.lean
|
Nat.le_two_mul_of_factorization_centralBinom_pos
|
[] |
[
129,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/Logic/Nontrivial.lean
|
exists_ne
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.349\ninst✝ : Nontrivial α\nx : α\nthis : DecidableEq α := Classical.decEq α\n⊢ ∃ y, y ≠ x",
"state_before": "α : Type u_1\nβ : Type ?u.349\ninst✝ : Nontrivial α\nx : α\n⊢ ∃ y, y ≠ x",
"tactic": "letI := Classical.decEq α"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.349\ninst✝ : Nontrivial α\nx : α\nthis : DecidableEq α := Classical.decEq α\n⊢ ∃ y, y ≠ x",
"tactic": "exact Decidable.exists_ne x"
}
] |
[
57,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/Topology/DenseEmbedding.lean
|
DenseInducing.extend_unique
|
[] |
[
193,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
191,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.mem_of_superset
|
[] |
[
154,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
1
] |
Mathlib/LinearAlgebra/Matrix/BilinearForm.lean
|
BilinForm.toMatrix_apply
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.1078977\nM : Type ?u.1078980\ninst✝¹⁷ : Semiring R\ninst✝¹⁶ : AddCommMonoid M\ninst✝¹⁵ : Module R M\nR₁ : Type ?u.1079016\nM₁ : Type ?u.1079019\ninst✝¹⁴ : Ring R₁\ninst✝¹³ : AddCommGroup M₁\ninst✝¹² : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_2\ninst✝¹¹ : CommSemiring R₂\ninst✝¹⁰ : AddCommMonoid M₂\ninst✝⁹ : Module R₂ M₂\nR₃ : Type ?u.1079818\nM₃ : Type ?u.1079821\ninst✝⁸ : CommRing R₃\ninst✝⁷ : AddCommGroup M₃\ninst✝⁶ : Module R₃ M₃\nV : Type ?u.1080409\nK : Type ?u.1080412\ninst✝⁵ : Field K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_3\no : Type ?u.1081629\ninst✝² : Fintype n\ninst✝¹ : Fintype o\ninst✝ : DecidableEq n\nb : Basis n R₂ M₂\nB : BilinForm R₂ M₂\ni j : n\n⊢ ↑(toMatrix b) B i j = bilin B (↑b i) (↑b j)",
"tactic": "rw [BilinForm.toMatrix, LinearEquiv.trans_apply, BilinForm.toMatrix'_apply, congr_apply,\n b.equivFun_symm_stdBasis, b.equivFun_symm_stdBasis]"
}
] |
[
308,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
305,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.tendsto_measure_iUnion
|
[
{
"state_after": "α : Type u_2\nβ : Type ?u.56609\nγ : Type ?u.56612\nδ : Type ?u.56615\nι : Type u_1\nR : Type ?u.56621\nR' : Type ?u.56624\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\ninst✝¹ : SemilatticeSup ι\ninst✝ : Countable ι\ns : ι → Set α\nhm : Monotone s\n⊢ Tendsto (↑↑μ ∘ s) atTop (𝓝 (⨆ (i : ι), ↑↑μ (s i)))",
"state_before": "α : Type u_2\nβ : Type ?u.56609\nγ : Type ?u.56612\nδ : Type ?u.56615\nι : Type u_1\nR : Type ?u.56621\nR' : Type ?u.56624\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\ninst✝¹ : SemilatticeSup ι\ninst✝ : Countable ι\ns : ι → Set α\nhm : Monotone s\n⊢ Tendsto (↑↑μ ∘ s) atTop (𝓝 (↑↑μ (⋃ (n : ι), s n)))",
"tactic": "rw [measure_iUnion_eq_iSup (directed_of_sup hm)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.56609\nγ : Type ?u.56612\nδ : Type ?u.56615\nι : Type u_1\nR : Type ?u.56621\nR' : Type ?u.56624\nm : MeasurableSpace α\nμ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\ninst✝¹ : SemilatticeSup ι\ninst✝ : Countable ι\ns : ι → Set α\nhm : Monotone s\n⊢ Tendsto (↑↑μ ∘ s) atTop (𝓝 (⨆ (i : ι), ↑↑μ (s i)))",
"tactic": "exact tendsto_atTop_iSup fun n m hnm => measure_mono <| hm hnm"
}
] |
[
521,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
518,
1
] |
Mathlib/CategoryTheory/Sites/Sieves.lean
|
CategoryTheory.Sieve.pullback_eq_top_of_mem
|
[] |
[
486,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
485,
1
] |
Mathlib/GroupTheory/Perm/Support.lean
|
Equiv.Perm.support_prod_le
|
[
{
"state_after": "case nil\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\n⊢ support (List.prod []) ≤ List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support [])\n\ncase cons\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g hd : Perm α\ntl : List (Perm α)\nhl : support (List.prod tl) ≤ List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support tl)\n⊢ support (List.prod (hd :: tl)) ≤ List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support (hd :: tl))",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\nl : List (Perm α)\n⊢ support (List.prod l) ≤ List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support l)",
"tactic": "induction' l with hd tl hl"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g : Perm α\n⊢ support (List.prod []) ≤ List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support [])",
"tactic": "simp"
},
{
"state_after": "case cons\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g hd : Perm α\ntl : List (Perm α)\nhl : support (List.prod tl) ≤ List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support tl)\n⊢ support (hd * List.prod tl) ≤ support hd ⊔ List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support tl)",
"state_before": "case cons\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g hd : Perm α\ntl : List (Perm α)\nhl : support (List.prod tl) ≤ List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support tl)\n⊢ support (List.prod (hd :: tl)) ≤ List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support (hd :: tl))",
"tactic": "rw [List.prod_cons, List.map_cons, List.foldr_cons]"
},
{
"state_after": "case cons\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g hd : Perm α\ntl : List (Perm α)\nhl : support (List.prod tl) ≤ List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support tl)\n⊢ support hd ⊔ support (List.prod tl) ≤ support hd ⊔ List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support tl)",
"state_before": "case cons\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g hd : Perm α\ntl : List (Perm α)\nhl : support (List.prod tl) ≤ List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support tl)\n⊢ support (hd * List.prod tl) ≤ support hd ⊔ List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support tl)",
"tactic": "refine' (support_mul_le hd tl.prod).trans _"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nf g hd : Perm α\ntl : List (Perm α)\nhl : support (List.prod tl) ≤ List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support tl)\n⊢ support hd ⊔ support (List.prod tl) ≤ support hd ⊔ List.foldr (fun x x_1 => x ⊔ x_1) ⊥ (List.map support tl)",
"tactic": "exact sup_le_sup le_rfl hl"
}
] |
[
425,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
420,
1
] |
Mathlib/Topology/Basic.lean
|
interior_singleton
|
[] |
[
1321,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1320,
1
] |
Mathlib/Order/FixedPoints.lean
|
OrderHom.le_lfp
|
[] |
[
69,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
68,
1
] |
Mathlib/CategoryTheory/Category/KleisliCat.lean
|
CategoryTheory.KleisliCat.comp_def
|
[] |
[
68,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
Submodule.comapSubtypeEquivOfLe_apply_coe
|
[] |
[
2472,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2469,
1
] |
Mathlib/Logic/Function/Basic.lean
|
Function.injective_of_subsingleton
|
[] |
[
152,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
1
] |
Mathlib/GroupTheory/Submonoid/Basic.lean
|
Submonoid.closure_induction
|
[] |
[
450,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
448,
1
] |
Mathlib/Data/Multiset/Bind.lean
|
Multiset.bind_zero
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.11661\nδ : Type ?u.11664\na : α\ns t : Multiset α\nf g : α → Multiset β\n⊢ (bind s fun x => 0) = 0",
"tactic": "simp [bind, join, nsmul_zero]"
}
] |
[
123,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
123,
1
] |
Mathlib/Algebra/Tropical/Basic.lean
|
Tropical.trop_mul_def
|
[] |
[
397,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
396,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.eventually_iff_exists_mem
|
[] |
[
1117,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1115,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.eventually_atBot
|
[] |
[
182,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Mathlib/Order/Filter/Prod.lean
|
Filter.prod_comm
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.14817\nδ : Type ?u.14820\nι : Sort ?u.14823\ns : Set α\nt : Set β\nf : Filter α\ng : Filter β\n⊢ Prod.swap <$> (g ×ˢ f) = map (fun p => (p.snd, p.fst)) (g ×ˢ f)",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.14817\nδ : Type ?u.14820\nι : Sort ?u.14823\ns : Set α\nt : Set β\nf : Filter α\ng : Filter β\n⊢ f ×ˢ g = map (fun p => (p.snd, p.fst)) (g ×ˢ f)",
"tactic": "rw [prod_comm', ← map_swap_eq_comap_swap]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.14817\nδ : Type ?u.14820\nι : Sort ?u.14823\ns : Set α\nt : Set β\nf : Filter α\ng : Filter β\n⊢ Prod.swap <$> (g ×ˢ f) = map (fun p => (p.snd, p.fst)) (g ×ˢ f)",
"tactic": "rfl"
}
] |
[
256,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
254,
1
] |
Mathlib/NumberTheory/PellMatiyasevic.lean
|
Pell.pell_eqz
|
[] |
[
239,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
238,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.Infinite.exists_subset_card_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns : Set α\nhs : Set.Infinite s\nn : ℕ\n⊢ ↑(Finset.map (Embedding.subtype fun x => x ∈ s) (Finset.map (natEmbedding s hs) (Finset.range n))) ⊆ s ∧\n Finset.card (Finset.map (Embedding.subtype fun x => x ∈ s) (Finset.map (natEmbedding s hs) (Finset.range n))) = n",
"tactic": "simp"
}
] |
[
1296,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1294,
1
] |
Mathlib/Computability/TMToPartrec.lean
|
Turing.PartrecToTM2.contSupp_cons₂
|
[] |
[
1876,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1874,
1
] |
Std/Data/String/Lemmas.lean
|
String.Pos.zero_addChar_byteIdx
|
[
{
"state_after": "no goals",
"state_before": "c : Char\n⊢ (0 + c).byteIdx = csize c",
"tactic": "simp only [addChar_byteIdx, byteIdx_zero, Nat.zero_add]"
}
] |
[
110,
58
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
109,
1
] |
Mathlib/Order/WithBot.lean
|
WithTop.some_eq_coe
|
[] |
[
601,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
600,
1
] |
Mathlib/CategoryTheory/Idempotents/Karoubi.lean
|
CategoryTheory.Idempotents.Karoubi.decompId_i_naturality
|
[
{
"state_after": "no goals",
"state_before": "C : Type ?u.50202\ninst✝ : Category C\nP Q : Karoubi C\nf : P ⟶ Q\n⊢ mk P.X (𝟙 P.X) ⟶ mk Q.X (𝟙 Q.X)",
"tactic": "exact Hom.mk f.f (by simp)"
},
{
"state_after": "no goals",
"state_before": "C : Type ?u.50202\ninst✝ : Category C\nP Q : Karoubi C\nf : P ⟶ Q\n⊢ f.f = (mk P.X (𝟙 P.X)).p ≫ f.f ≫ (mk Q.X (𝟙 Q.X)).p",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "C : Type u_1\ninst✝ : Category C\nP Q : Karoubi C\nf : P ⟶ Q\n⊢ f ≫ decompId_i Q = decompId_i P ≫ Hom.mk f.f",
"tactic": "aesop_cat"
}
] |
[
297,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
295,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.ListBlank.nth_modifyNth
|
[
{
"state_after": "case zero\nΓ : Type u_1\ninst✝ : Inhabited Γ\nf : Γ → Γ\ni✝ : ℕ\nL✝ : ListBlank Γ\ni : ℕ\nL : ListBlank Γ\n⊢ nth (modifyNth f Nat.zero L) i = if i = Nat.zero then f (nth L i) else nth L i\n\ncase succ\nΓ : Type u_1\ninst✝ : Inhabited Γ\nf : Γ → Γ\ni✝ : ℕ\nL✝ : ListBlank Γ\nn : ℕ\nIH : ∀ (i : ℕ) (L : ListBlank Γ), nth (modifyNth f n L) i = if i = n then f (nth L i) else nth L i\ni : ℕ\nL : ListBlank Γ\n⊢ nth (modifyNth f (Nat.succ n) L) i = if i = Nat.succ n then f (nth L i) else nth L i",
"state_before": "Γ : Type u_1\ninst✝ : Inhabited Γ\nf : Γ → Γ\nn i : ℕ\nL : ListBlank Γ\n⊢ nth (modifyNth f n L) i = if i = n then f (nth L i) else nth L i",
"tactic": "induction' n with n IH generalizing i L"
},
{
"state_after": "no goals",
"state_before": "case zero\nΓ : Type u_1\ninst✝ : Inhabited Γ\nf : Γ → Γ\ni✝ : ℕ\nL✝ : ListBlank Γ\ni : ℕ\nL : ListBlank Γ\n⊢ nth (modifyNth f Nat.zero L) i = if i = Nat.zero then f (nth L i) else nth L i",
"tactic": "cases i <;> simp only [ListBlank.nth_zero, if_true, ListBlank.head_cons, ListBlank.modifyNth,\n ListBlank.nth_succ, if_false, ListBlank.tail_cons, Nat.zero_eq]"
},
{
"state_after": "case succ.zero\nΓ : Type u_1\ninst✝ : Inhabited Γ\nf : Γ → Γ\ni : ℕ\nL✝ : ListBlank Γ\nn : ℕ\nIH : ∀ (i : ℕ) (L : ListBlank Γ), nth (modifyNth f n L) i = if i = n then f (nth L i) else nth L i\nL : ListBlank Γ\n⊢ nth (modifyNth f (Nat.succ n) L) Nat.zero = if Nat.zero = Nat.succ n then f (nth L Nat.zero) else nth L Nat.zero\n\ncase succ.succ\nΓ : Type u_1\ninst✝ : Inhabited Γ\nf : Γ → Γ\ni : ℕ\nL✝ : ListBlank Γ\nn : ℕ\nIH : ∀ (i : ℕ) (L : ListBlank Γ), nth (modifyNth f n L) i = if i = n then f (nth L i) else nth L i\nL : ListBlank Γ\nn✝ : ℕ\n⊢ nth (modifyNth f (Nat.succ n) L) (Nat.succ n✝) =\n if Nat.succ n✝ = Nat.succ n then f (nth L (Nat.succ n✝)) else nth L (Nat.succ n✝)",
"state_before": "case succ\nΓ : Type u_1\ninst✝ : Inhabited Γ\nf : Γ → Γ\ni✝ : ℕ\nL✝ : ListBlank Γ\nn : ℕ\nIH : ∀ (i : ℕ) (L : ListBlank Γ), nth (modifyNth f n L) i = if i = n then f (nth L i) else nth L i\ni : ℕ\nL : ListBlank Γ\n⊢ nth (modifyNth f (Nat.succ n) L) i = if i = Nat.succ n then f (nth L i) else nth L i",
"tactic": "cases i"
},
{
"state_after": "case succ.zero\nΓ : Type u_1\ninst✝ : Inhabited Γ\nf : Γ → Γ\ni : ℕ\nL✝ : ListBlank Γ\nn : ℕ\nIH : ∀ (i : ℕ) (L : ListBlank Γ), nth (modifyNth f n L) i = if i = n then f (nth L i) else nth L i\nL : ListBlank Γ\n⊢ nth (modifyNth f (Nat.succ n) L) Nat.zero = nth L Nat.zero",
"state_before": "case succ.zero\nΓ : Type u_1\ninst✝ : Inhabited Γ\nf : Γ → Γ\ni : ℕ\nL✝ : ListBlank Γ\nn : ℕ\nIH : ∀ (i : ℕ) (L : ListBlank Γ), nth (modifyNth f n L) i = if i = n then f (nth L i) else nth L i\nL : ListBlank Γ\n⊢ nth (modifyNth f (Nat.succ n) L) Nat.zero = if Nat.zero = Nat.succ n then f (nth L Nat.zero) else nth L Nat.zero",
"tactic": "rw [if_neg (Nat.succ_ne_zero _).symm]"
},
{
"state_after": "no goals",
"state_before": "case succ.zero\nΓ : Type u_1\ninst✝ : Inhabited Γ\nf : Γ → Γ\ni : ℕ\nL✝ : ListBlank Γ\nn : ℕ\nIH : ∀ (i : ℕ) (L : ListBlank Γ), nth (modifyNth f n L) i = if i = n then f (nth L i) else nth L i\nL : ListBlank Γ\n⊢ nth (modifyNth f (Nat.succ n) L) Nat.zero = nth L Nat.zero",
"tactic": "simp only [ListBlank.nth_zero, ListBlank.head_cons, ListBlank.modifyNth, Nat.zero_eq]"
},
{
"state_after": "no goals",
"state_before": "case succ.succ\nΓ : Type u_1\ninst✝ : Inhabited Γ\nf : Γ → Γ\ni : ℕ\nL✝ : ListBlank Γ\nn : ℕ\nIH : ∀ (i : ℕ) (L : ListBlank Γ), nth (modifyNth f n L) i = if i = n then f (nth L i) else nth L i\nL : ListBlank Γ\nn✝ : ℕ\n⊢ nth (modifyNth f (Nat.succ n) L) (Nat.succ n✝) =\n if Nat.succ n✝ = Nat.succ n then f (nth L (Nat.succ n✝)) else nth L (Nat.succ n✝)",
"tactic": "simp only [IH, ListBlank.modifyNth, ListBlank.nth_succ, ListBlank.tail_cons, Nat.succ.injEq]"
}
] |
[
348,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
340,
1
] |
Mathlib/Init/CcLemmas.lean
|
or_eq_of_eq_true_right
|
[] |
[
40,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
39,
1
] |
Mathlib/Algebra/Ring/Divisibility.lean
|
dvd_neg
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.2595\ninst✝¹ : Semigroup α\ninst✝ : HasDistribNeg α\na b c : α\n⊢ (∃ b_1, b = a * b_1) ↔ a ∣ b",
"state_before": "α : Type u_1\nβ : Type ?u.2595\ninst✝¹ : Semigroup α\ninst✝ : HasDistribNeg α\na b c : α\n⊢ (∃ b_1, -b = a * ↑(Equiv.neg α).symm b_1) ↔ a ∣ b",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.2595\ninst✝¹ : Semigroup α\ninst✝ : HasDistribNeg α\na b c : α\n⊢ (∃ b_1, b = a * b_1) ↔ a ∣ b",
"tactic": "rfl"
}
] |
[
61,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/RingTheory/AlgebraicIndependent.lean
|
AlgebraicIndependent.map'
|
[] |
[
160,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
Nat.preimage_Iio
|
[
{
"state_after": "case h\nF : Type ?u.75344\nα : Type u_1\nβ : Type ?u.75350\ninst✝¹ : LinearOrderedSemiring α\ninst✝ : FloorSemiring α\na✝ : α\nn : ℕ\na : α\nx✝ : ℕ\n⊢ x✝ ∈ Nat.cast ⁻¹' Iio a ↔ x✝ ∈ Iio ⌈a⌉₊",
"state_before": "F : Type ?u.75344\nα : Type u_1\nβ : Type ?u.75350\ninst✝¹ : LinearOrderedSemiring α\ninst✝ : FloorSemiring α\na✝ : α\nn : ℕ\na : α\n⊢ Nat.cast ⁻¹' Iio a = Iio ⌈a⌉₊",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nF : Type ?u.75344\nα : Type u_1\nβ : Type ?u.75350\ninst✝¹ : LinearOrderedSemiring α\ninst✝ : FloorSemiring α\na✝ : α\nn : ℕ\na : α\nx✝ : ℕ\n⊢ x✝ ∈ Nat.cast ⁻¹' Iio a ↔ x✝ ∈ Iio ⌈a⌉₊",
"tactic": "simp [lt_ceil]"
}
] |
[
428,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
426,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.sinh_two_mul
|
[
{
"state_after": "x y : ℂ\n⊢ sinh x * cosh x + cosh x * sinh x = 2 * sinh x * cosh x",
"state_before": "x y : ℂ\n⊢ sinh (2 * x) = 2 * sinh x * cosh x",
"tactic": "rw [two_mul, sinh_add]"
},
{
"state_after": "no goals",
"state_before": "x y : ℂ\n⊢ sinh x * cosh x + cosh x * sinh x = 2 * sinh x * cosh x",
"tactic": "ring"
}
] |
[
774,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
772,
1
] |
Mathlib/Data/Erased.lean
|
Erased.out_mk
|
[
{
"state_after": "α : Sort u_1\na : α\nh : ∃ a_1, (fun b => a_1 = b) = (mk a).fst := (mk a).snd\n⊢ out (mk a) = a",
"state_before": "α : Sort u_1\na : α\n⊢ out (mk a) = a",
"tactic": "let h := (mk a).2"
},
{
"state_after": "α : Sort u_1\na : α\nh : ∃ a_1, (fun b => a_1 = b) = (mk a).fst := (mk a).snd\n⊢ Classical.choose h = a",
"state_before": "α : Sort u_1\na : α\nh : ∃ a_1, (fun b => a_1 = b) = (mk a).fst := (mk a).snd\n⊢ out (mk a) = a",
"tactic": "show Classical.choose h = a"
},
{
"state_after": "α : Sort u_1\na : α\nh : ∃ a_1, (fun b => a_1 = b) = (mk a).fst := (mk a).snd\nthis : (fun b => Classical.choose h = b) = (mk a).fst\n⊢ Classical.choose h = a",
"state_before": "α : Sort u_1\na : α\nh : ∃ a_1, (fun b => a_1 = b) = (mk a).fst := (mk a).snd\n⊢ Classical.choose h = a",
"tactic": "have := Classical.choose_spec h"
},
{
"state_after": "no goals",
"state_before": "α : Sort u_1\na : α\nh : ∃ a_1, (fun b => a_1 = b) = (mk a).fst := (mk a).snd\nthis : (fun b => Classical.choose h = b) = (mk a).fst\n⊢ Classical.choose h = a",
"tactic": "exact cast (congr_fun this a).symm rfl"
}
] |
[
63,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/Data/Finset/Sups.lean
|
Finset.forall_infs_iff
|
[] |
[
283,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
282,
1
] |
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