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/List/Basic.lean
|
List.zipRight'_nil_cons
|
[] |
[
4085,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
4083,
1
] |
Mathlib/Algebra/Group/Units.lean
|
Units.ext_iff
|
[] |
[
150,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.filter_true_of_mem
|
[] |
[
2698,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2697,
1
] |
Mathlib/GroupTheory/GroupAction/Defs.lean
|
MulAction.surjective_smul
|
[] |
[
162,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/Computability/Primrec.lean
|
PrimrecPred.of_eq
|
[] |
[
512,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
510,
1
] |
Mathlib/Order/Max.lean
|
NoTopOrder.to_noMaxOrder
|
[
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.5028\nβ : Type ?u.5031\nα : Type u_1\ninst✝¹ : LinearOrder α\ninst✝ : NoTopOrder α\na : α\n⊢ ∃ b, a < b",
"tactic": "simpa [not_le] using exists_not_le a"
}
] |
[
150,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
|
aemeasurable_congr
|
[] |
[
708,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
707,
1
] |
Mathlib/RingTheory/RootsOfUnity/Basic.lean
|
IsPrimitiveRoot.orderOf
|
[] |
[
456,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
455,
11
] |
Mathlib/Algebra/QuaternionBasis.lean
|
QuaternionAlgebra.Basis.k_mul_j
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\nA : Type u_1\nB : Type ?u.27887\ninst✝⁴ : CommRing R\ninst✝³ : Ring A\ninst✝² : Ring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nc₁ c₂ : R\nq : Basis A c₁ c₂\n⊢ q.k * q.j = c₂ • q.i",
"tactic": "rw [← i_mul_j, mul_assoc, j_mul_j, mul_smul_comm, mul_one]"
}
] |
[
99,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/Data/Matrix/Block.lean
|
Matrix.blockDiag'_one
|
[] |
[
871,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
869,
1
] |
Mathlib/Order/Hom/Lattice.lean
|
SupHom.coe_comp
|
[] |
[
407,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
406,
1
] |
Mathlib/Algebra/Order/SMul.lean
|
BddBelow.smul_of_nonneg
|
[] |
[
156,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
155,
1
] |
Mathlib/Algebra/GeomSum.lean
|
geom_sum_eq_zero_iff_neg_one
|
[
{
"state_after": "α : Type u\nβ : Type ?u.252702\nn : ℕ\nx : α\ninst✝ : LinearOrderedRing α\nhn : n ≠ 0\nh : ∑ i in range n, x ^ i = 0\n⊢ x = -1 ∧ Even n",
"state_before": "α : Type u\nβ : Type ?u.252702\nn : ℕ\nx : α\ninst✝ : LinearOrderedRing α\nhn : n ≠ 0\n⊢ ∑ i in range n, x ^ i = 0 ↔ x = -1 ∧ Even n",
"tactic": "refine' ⟨fun h => _, @fun ⟨h, hn⟩ => by simp only [h, hn, neg_one_geom_sum, if_true]⟩"
},
{
"state_after": "α : Type u\nβ : Type ?u.252702\nn : ℕ\nx : α\ninst✝ : LinearOrderedRing α\nhn : n ≠ 0\nh : x = -1 → ¬Even n\n⊢ ∑ i in range n, x ^ i ≠ 0",
"state_before": "α : Type u\nβ : Type ?u.252702\nn : ℕ\nx : α\ninst✝ : LinearOrderedRing α\nhn : n ≠ 0\nh : ∑ i in range n, x ^ i = 0\n⊢ x = -1 ∧ Even n",
"tactic": "contrapose! h"
},
{
"state_after": "α : Type u\nβ : Type ?u.252702\nn : ℕ\nx : α\ninst✝ : LinearOrderedRing α\nhn : n ≠ 0\nh : x = -1 → ¬Even n\nhx : x = -1 ∨ x ≠ -1\n⊢ ∑ i in range n, x ^ i ≠ 0",
"state_before": "α : Type u\nβ : Type ?u.252702\nn : ℕ\nx : α\ninst✝ : LinearOrderedRing α\nhn : n ≠ 0\nh : x = -1 → ¬Even n\n⊢ ∑ i in range n, x ^ i ≠ 0",
"tactic": "have hx := eq_or_ne x (-1)"
},
{
"state_after": "case inl\nα : Type u\nβ : Type ?u.252702\nn : ℕ\nx : α\ninst✝ : LinearOrderedRing α\nhn : n ≠ 0\nh : x = -1 → ¬Even n\nhx : x = -1\n⊢ ∑ i in range n, x ^ i ≠ 0\n\ncase inr\nα : Type u\nβ : Type ?u.252702\nn : ℕ\nx : α\ninst✝ : LinearOrderedRing α\nhn : n ≠ 0\nh : x = -1 → ¬Even n\nhx : x ≠ -1\n⊢ ∑ i in range n, x ^ i ≠ 0",
"state_before": "α : Type u\nβ : Type ?u.252702\nn : ℕ\nx : α\ninst✝ : LinearOrderedRing α\nhn : n ≠ 0\nh : x = -1 → ¬Even n\nhx : x = -1 ∨ x ≠ -1\n⊢ ∑ i in range n, x ^ i ≠ 0",
"tactic": "cases' hx with hx hx"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.252702\nn : ℕ\nx : α\ninst✝ : LinearOrderedRing α\nhn✝ : n ≠ 0\nx✝ : x = -1 ∧ Even n\nh : x = -1\nhn : Even n\n⊢ ∑ i in range n, x ^ i = 0",
"tactic": "simp only [h, hn, neg_one_geom_sum, if_true]"
},
{
"state_after": "case inl\nα : Type u\nβ : Type ?u.252702\nn : ℕ\nx : α\ninst✝ : LinearOrderedRing α\nhn : n ≠ 0\nh : x = -1 → ¬Even n\nhx : x = -1\n⊢ (if Even n then 0 else 1) ≠ 0",
"state_before": "case inl\nα : Type u\nβ : Type ?u.252702\nn : ℕ\nx : α\ninst✝ : LinearOrderedRing α\nhn : n ≠ 0\nh : x = -1 → ¬Even n\nhx : x = -1\n⊢ ∑ i in range n, x ^ i ≠ 0",
"tactic": "rw [hx, neg_one_geom_sum]"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u\nβ : Type ?u.252702\nn : ℕ\nx : α\ninst✝ : LinearOrderedRing α\nhn : n ≠ 0\nh : x = -1 → ¬Even n\nhx : x = -1\n⊢ (if Even n then 0 else 1) ≠ 0",
"tactic": "simp only [h hx, ne_eq, ite_eq_left_iff, one_ne_zero, not_forall, exists_prop, and_true]"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u\nβ : Type ?u.252702\nn : ℕ\nx : α\ninst✝ : LinearOrderedRing α\nhn : n ≠ 0\nh : x = -1 → ¬Even n\nhx : x ≠ -1\n⊢ ∑ i in range n, x ^ i ≠ 0",
"tactic": "exact geom_sum_ne_zero hx hn"
}
] |
[
566,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
558,
1
] |
Mathlib/Algebra/Order/Group/Abs.lean
|
lt_of_abs_lt
|
[] |
[
208,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
207,
1
] |
Std/Classes/Order.lean
|
Std.TransCmp.cmp_congr_left
|
[] |
[
83,
17
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
76,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.Finite.pi
|
[
{
"state_after": "case intro\nα : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nδ : Type u_1\ninst✝ : Finite δ\nκ : δ → Type u_2\nt : (d : δ) → Set (κ d)\nht : ∀ (d : δ), Set.Finite (t d)\nval✝ : Fintype δ\n⊢ Set.Finite (Set.pi univ t)",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nδ : Type u_1\ninst✝ : Finite δ\nκ : δ → Type u_2\nt : (d : δ) → Set (κ d)\nht : ∀ (d : δ), Set.Finite (t d)\n⊢ Set.Finite (Set.pi univ t)",
"tactic": "cases _root_.nonempty_fintype δ"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nδ : Type u_1\ninst✝ : Finite δ\nκ : δ → Type u_2\nval✝ : Fintype δ\nt : (i : δ) → Finset (κ i)\n⊢ Set.Finite (Set.pi univ fun i => ↑(t i))",
"state_before": "case intro\nα : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nδ : Type u_1\ninst✝ : Finite δ\nκ : δ → Type u_2\nt : (d : δ) → Set (κ d)\nht : ∀ (d : δ), Set.Finite (t d)\nval✝ : Fintype δ\n⊢ Set.Finite (Set.pi univ t)",
"tactic": "lift t to ∀ d, Finset (κ d) using ht"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nδ : Type u_1\ninst✝ : Finite δ\nκ : δ → Type u_2\nval✝ : Fintype δ\nt : (i : δ) → Finset (κ i)\n⊢ Set.Finite (Set.pi univ fun i => ↑(t i))",
"tactic": "classical\n rw [← Fintype.coe_piFinset]\n apply Finset.finite_toSet"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nδ : Type u_1\ninst✝ : Finite δ\nκ : δ → Type u_2\nval✝ : Fintype δ\nt : (i : δ) → Finset (κ i)\n⊢ Set.Finite ↑(Fintype.piFinset fun i => t i)",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nδ : Type u_1\ninst✝ : Finite δ\nκ : δ → Type u_2\nval✝ : Fintype δ\nt : (i : δ) → Finset (κ i)\n⊢ Set.Finite (Set.pi univ fun i => ↑(t i))",
"tactic": "rw [← Fintype.coe_piFinset]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nι : Sort w\nγ : Type x\nδ : Type u_1\ninst✝ : Finite δ\nκ : δ → Type u_2\nval✝ : Fintype δ\nt : (i : δ) → Finset (κ i)\n⊢ Set.Finite ↑(Fintype.piFinset fun i => t i)",
"tactic": "apply Finset.finite_toSet"
}
] |
[
1010,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1004,
1
] |
Std/Data/Int/DivMod.lean
|
Int.dvd_iff_mod_eq_zero
|
[] |
[
685,
43
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
684,
1
] |
Mathlib/GroupTheory/MonoidLocalization.lean
|
Submonoid.LocalizationMap.mulEquivOfLocalizations_right_inv
|
[] |
[
1440,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1438,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.singleton_inj
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.25432\nγ : Type ?u.25435\na b : α\n⊢ a ::ₘ 0 = b ::ₘ 0 ↔ a = b",
"state_before": "α : Type u_1\nβ : Type ?u.25432\nγ : Type ?u.25435\na b : α\n⊢ {a} = {b} ↔ a = b",
"tactic": "simp_rw [← cons_zero]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.25432\nγ : Type ?u.25435\na b : α\n⊢ a ::ₘ 0 = b ::ₘ 0 ↔ a = b",
"tactic": "exact cons_inj_left _"
}
] |
[
350,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
348,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.mem_Icc_of_Ico
|
[] |
[
667,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
666,
1
] |
Mathlib/Data/List/Basic.lean
|
List.length_erase_add_one
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.425241\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na : α\nl : List α\nh : a ∈ l\n⊢ length (List.erase l a) + 1 = length l",
"tactic": "rw [erase_eq_eraseP, length_eraseP_add_one h (decide_eq_true rfl)]"
}
] |
[
3729,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3727,
9
] |
Mathlib/Data/Set/Ncard.lean
|
Set.exists_intermediate_Set
|
[
{
"state_after": "case inl\nα : Type u_1\nβ : Type ?u.130854\ns t : Set α\na b x y : α\nf : α → β\ni : ℕ\nh₁ : i + ncard s ≤ ncard t\nh₂ : s ⊆ t\nht : Set.Finite t\n⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ ncard r = i + ncard s\n\ncase inr\nα : Type u_1\nβ : Type ?u.130854\ns t : Set α\na b x y : α\nf : α → β\ni : ℕ\nh₁ : i + ncard s ≤ ncard t\nh₂ : s ⊆ t\nht : Set.Infinite t\n⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ ncard r = i + ncard s",
"state_before": "α : Type u_1\nβ : Type ?u.130854\ns t : Set α\na b x y : α\nf : α → β\ni : ℕ\nh₁ : i + ncard s ≤ ncard t\nh₂ : s ⊆ t\n⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ ncard r = i + ncard s",
"tactic": "cases' t.finite_or_infinite with ht ht"
},
{
"state_after": "case inr\nα : Type u_1\nβ : Type ?u.130854\ns t : Set α\na b x y : α\nf : α → β\ni : ℕ\nh₁ : i + ncard s ≤ 0\nh₂ : s ⊆ t\nht : Set.Infinite t\n⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ ncard r = i + ncard s",
"state_before": "case inr\nα : Type u_1\nβ : Type ?u.130854\ns t : Set α\na b x y : α\nf : α → β\ni : ℕ\nh₁ : i + ncard s ≤ ncard t\nh₂ : s ⊆ t\nht : Set.Infinite t\n⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ ncard r = i + ncard s",
"tactic": "rw [ht.ncard] at h₁"
},
{
"state_after": "case inr\nα : Type u_1\nβ : Type ?u.130854\ns t : Set α\na b x y : α\nf : α → β\ni : ℕ\nh₁ : i + ncard s ≤ 0\nh₂ : s ⊆ t\nht : Set.Infinite t\nh₁' : i + ncard s = 0\n⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ ncard r = i + ncard s",
"state_before": "case inr\nα : Type u_1\nβ : Type ?u.130854\ns t : Set α\na b x y : α\nf : α → β\ni : ℕ\nh₁ : i + ncard s ≤ 0\nh₂ : s ⊆ t\nht : Set.Infinite t\n⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ ncard r = i + ncard s",
"tactic": "have h₁' := Nat.eq_zero_of_le_zero h₁"
},
{
"state_after": "case inr\nα : Type u_1\nβ : Type ?u.130854\ns t : Set α\na b x y : α\nf : α → β\ni : ℕ\nh₁ : i + ncard s ≤ 0\nh₂ : s ⊆ t\nht : Set.Infinite t\nh₁' : i = 0 ∧ ncard s = 0\n⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ ncard r = i + ncard s",
"state_before": "case inr\nα : Type u_1\nβ : Type ?u.130854\ns t : Set α\na b x y : α\nf : α → β\ni : ℕ\nh₁ : i + ncard s ≤ 0\nh₂ : s ⊆ t\nht : Set.Infinite t\nh₁' : i + ncard s = 0\n⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ ncard r = i + ncard s",
"tactic": "rw [add_eq_zero_iff] at h₁'"
},
{
"state_after": "case inr\nα : Type u_1\nβ : Type ?u.130854\ns t : Set α\na b x y : α\nf : α → β\ni : ℕ\nh₁ : i + ncard s ≤ 0\nh₂ : s ⊆ t\nht : Set.Infinite t\nh₁' : i = 0 ∧ ncard s = 0\n⊢ ncard t = i + ncard s",
"state_before": "case inr\nα : Type u_1\nβ : Type ?u.130854\ns t : Set α\na b x y : α\nf : α → β\ni : ℕ\nh₁ : i + ncard s ≤ 0\nh₂ : s ⊆ t\nht : Set.Infinite t\nh₁' : i = 0 ∧ ncard s = 0\n⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ ncard r = i + ncard s",
"tactic": "refine' ⟨t, h₂, rfl.subset, _⟩"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\nβ : Type ?u.130854\ns t : Set α\na b x y : α\nf : α → β\ni : ℕ\nh₁ : i + ncard s ≤ 0\nh₂ : s ⊆ t\nht : Set.Infinite t\nh₁' : i = 0 ∧ ncard s = 0\n⊢ ncard t = i + ncard s",
"tactic": "rw [h₁'.2, h₁'.1, ht.ncard, add_zero]"
},
{
"state_after": "case inl\nα : Type u_1\nβ : Type ?u.130854\ns t : Set α\na b x y : α\nf : α → β\ni : ℕ\nh₂ : s ⊆ t\nht : Set.Finite t\nh₁ : i + Finset.card (Finite.toFinset (_ : Set.Finite s)) ≤ ncard t\n⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ ncard r = i + Finset.card (Finite.toFinset (_ : Set.Finite s))",
"state_before": "case inl\nα : Type u_1\nβ : Type ?u.130854\ns t : Set α\na b x y : α\nf : α → β\ni : ℕ\nh₁ : i + ncard s ≤ ncard t\nh₂ : s ⊆ t\nht : Set.Finite t\n⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ ncard r = i + ncard s",
"tactic": "rw [ncard_eq_toFinset_card _ (ht.subset h₂)] at h₁ ⊢"
},
{
"state_after": "case inl\nα : Type u_1\nβ : Type ?u.130854\ns t : Set α\na b x y : α\nf : α → β\ni : ℕ\nh₂ : s ⊆ t\nht : Set.Finite t\nh₁ : i + Finset.card (Finite.toFinset (_ : Set.Finite s)) ≤ Finset.card (Finite.toFinset ht)\n⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ ncard r = i + Finset.card (Finite.toFinset (_ : Set.Finite s))",
"state_before": "case inl\nα : Type u_1\nβ : Type ?u.130854\ns t : Set α\na b x y : α\nf : α → β\ni : ℕ\nh₂ : s ⊆ t\nht : Set.Finite t\nh₁ : i + Finset.card (Finite.toFinset (_ : Set.Finite s)) ≤ ncard t\n⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ ncard r = i + Finset.card (Finite.toFinset (_ : Set.Finite s))",
"tactic": "rw [ncard_eq_toFinset_card t ht] at h₁"
},
{
"state_after": "case inl.intro.intro.intro\nα : Type u_1\nβ : Type ?u.130854\ns t : Set α\na b x y : α\nf : α → β\ni : ℕ\nh₂ : s ⊆ t\nht : Set.Finite t\nh₁ : i + Finset.card (Finite.toFinset (_ : Set.Finite s)) ≤ Finset.card (Finite.toFinset ht)\nr' : Finset α\nhsr' : Finite.toFinset (_ : Set.Finite s) ⊆ r'\nhr't : r' ⊆ Finite.toFinset ht\nhr' : Finset.card r' = i + Finset.card (Finite.toFinset (_ : Set.Finite s))\n⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ ncard r = i + Finset.card (Finite.toFinset (_ : Set.Finite s))",
"state_before": "case inl\nα : Type u_1\nβ : Type ?u.130854\ns t : Set α\na b x y : α\nf : α → β\ni : ℕ\nh₂ : s ⊆ t\nht : Set.Finite t\nh₁ : i + Finset.card (Finite.toFinset (_ : Set.Finite s)) ≤ Finset.card (Finite.toFinset ht)\n⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ ncard r = i + Finset.card (Finite.toFinset (_ : Set.Finite s))",
"tactic": "obtain ⟨r', hsr', hr't, hr'⟩ := Finset.exists_intermediate_set _ h₁ (by simpa)"
},
{
"state_after": "no goals",
"state_before": "case inl.intro.intro.intro\nα : Type u_1\nβ : Type ?u.130854\ns t : Set α\na b x y : α\nf : α → β\ni : ℕ\nh₂ : s ⊆ t\nht : Set.Finite t\nh₁ : i + Finset.card (Finite.toFinset (_ : Set.Finite s)) ≤ Finset.card (Finite.toFinset ht)\nr' : Finset α\nhsr' : Finite.toFinset (_ : Set.Finite s) ⊆ r'\nhr't : r' ⊆ Finite.toFinset ht\nhr' : Finset.card r' = i + Finset.card (Finite.toFinset (_ : Set.Finite s))\n⊢ ∃ r, s ⊆ r ∧ r ⊆ t ∧ ncard r = i + Finset.card (Finite.toFinset (_ : Set.Finite s))",
"tactic": "exact ⟨r', by simpa using hsr', by simpa using hr't, by rw [← hr', ncard_coe_Finset]⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.130854\ns t : Set α\na b x y : α\nf : α → β\ni : ℕ\nh₂ : s ⊆ t\nht : Set.Finite t\nh₁ : i + Finset.card (Finite.toFinset (_ : Set.Finite s)) ≤ Finset.card (Finite.toFinset ht)\n⊢ Finite.toFinset (_ : Set.Finite s) ⊆ Finite.toFinset ht",
"tactic": "simpa"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.130854\ns t : Set α\na b x y : α\nf : α → β\ni : ℕ\nh₂ : s ⊆ t\nht : Set.Finite t\nh₁ : i + Finset.card (Finite.toFinset (_ : Set.Finite s)) ≤ Finset.card (Finite.toFinset ht)\nr' : Finset α\nhsr' : Finite.toFinset (_ : Set.Finite s) ⊆ r'\nhr't : r' ⊆ Finite.toFinset ht\nhr' : Finset.card r' = i + Finset.card (Finite.toFinset (_ : Set.Finite s))\n⊢ s ⊆ ↑r'",
"tactic": "simpa using hsr'"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.130854\ns t : Set α\na b x y : α\nf : α → β\ni : ℕ\nh₂ : s ⊆ t\nht : Set.Finite t\nh₁ : i + Finset.card (Finite.toFinset (_ : Set.Finite s)) ≤ Finset.card (Finite.toFinset ht)\nr' : Finset α\nhsr' : Finite.toFinset (_ : Set.Finite s) ⊆ r'\nhr't : r' ⊆ Finite.toFinset ht\nhr' : Finset.card r' = i + Finset.card (Finite.toFinset (_ : Set.Finite s))\n⊢ ↑r' ⊆ t",
"tactic": "simpa using hr't"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.130854\ns t : Set α\na b x y : α\nf : α → β\ni : ℕ\nh₂ : s ⊆ t\nht : Set.Finite t\nh₁ : i + Finset.card (Finite.toFinset (_ : Set.Finite s)) ≤ Finset.card (Finite.toFinset ht)\nr' : Finset α\nhsr' : Finite.toFinset (_ : Set.Finite s) ⊆ r'\nhr't : r' ⊆ Finite.toFinset ht\nhr' : Finset.card r' = i + Finset.card (Finite.toFinset (_ : Set.Finite s))\n⊢ ncard ↑r' = i + Finset.card (Finite.toFinset (_ : Set.Finite s))",
"tactic": "rw [← hr', ncard_coe_Finset]"
}
] |
[
588,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
577,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
Subsemiring.coe_inf
|
[] |
[
656,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
655,
1
] |
src/lean/Init/Control/Lawful.lean
|
map_eq_pure_bind
|
[
{
"state_after": "no goals",
"state_before": "m : Type u_1 → Type u_2\nα β : Type u_1\ninst✝¹ : Monad m\ninst✝ : LawfulMonad m\nf : α → β\nx : m α\n⊢ f <$> x = do\n let a ← x\n pure (f a)",
"tactic": "rw [← bind_pure_comp]"
}
] |
[
63,
24
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
62,
1
] |
Mathlib/GroupTheory/Perm/Sign.lean
|
Equiv.Perm.perm_inv_mapsTo_iff_mapsTo
|
[] |
[
86,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
1
] |
Mathlib/Algebra/GCDMonoid/Basic.lean
|
dvd_lcm_left
|
[] |
[
716,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
715,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPolynomial.coe_injective
|
[
{
"state_after": "case a\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nφ ψ x y : MvPolynomial σ R\nh : Coe.coe x = Coe.coe y\nm✝ : σ →₀ ℕ\n⊢ coeff m✝ x = coeff m✝ y",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nφ ψ x y : MvPolynomial σ R\nh : Coe.coe x = Coe.coe y\n⊢ x = y",
"tactic": "ext"
},
{
"state_after": "case a\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nφ ψ x y : MvPolynomial σ R\nh : Coe.coe x = Coe.coe y\nm✝ : σ →₀ ℕ\n⊢ ↑(MvPowerSeries.coeff R m✝) ↑x = ↑(MvPowerSeries.coeff R m✝) ↑y",
"state_before": "case a\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nφ ψ x y : MvPolynomial σ R\nh : Coe.coe x = Coe.coe y\nm✝ : σ →₀ ℕ\n⊢ coeff m✝ x = coeff m✝ y",
"tactic": "simp_rw [← coeff_coe]"
},
{
"state_after": "no goals",
"state_before": "case a\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nφ ψ x y : MvPolynomial σ R\nh : Coe.coe x = Coe.coe y\nm✝ : σ →₀ ℕ\n⊢ ↑(MvPowerSeries.coeff R m✝) ↑x = ↑(MvPowerSeries.coeff R m✝) ↑y",
"tactic": "congr"
}
] |
[
1139,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1135,
1
] |
Std/Data/List/Basic.lean
|
List.zipWithLeft_eq_zipWithLeftTR
|
[
{
"state_after": "case h.h.h.h.h.h\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nf : α → Option β → γ\nas : List α\nbs : List β\n⊢ zipWithLeft f as bs = zipWithLeftTR f as bs",
"state_before": "⊢ @zipWithLeft = @zipWithLeftTR",
"tactic": "funext α β γ f as bs"
},
{
"state_after": "case h.h.h.h.h.h\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nf : α → Option β → γ\nas : List α\nbs : List β\n⊢ zipWithLeft f as bs = zipWithLeftTR.go f as bs #[]",
"state_before": "case h.h.h.h.h.h\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nf : α → Option β → γ\nas : List α\nbs : List β\n⊢ zipWithLeft f as bs = zipWithLeftTR f as bs",
"tactic": "simp [zipWithLeftTR]"
},
{
"state_after": "case h.h.h.h.h.h\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nf : α → Option β → γ\nas : List α\nbs : List β\n⊢ zipWithLeft f as bs = zipWithLeftTR.go f as bs #[]",
"state_before": "case h.h.h.h.h.h\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nf : α → Option β → γ\nas : List α\nbs : List β\n⊢ zipWithLeft f as bs = zipWithLeftTR.go f as bs #[]",
"tactic": "let rec go (acc) : ∀ as bs, zipWithLeftTR.go f as bs acc = acc.toList ++ as.zipWithLeft f bs\n| [], bs => by simp [zipWithLeftTR.go]\n| _::_, [] => by simp [zipWithLeftTR.go, Array.foldl_data_eq_map]\n| a::as, b::bs => by simp [zipWithLeftTR.go, go _ as bs]"
},
{
"state_after": "no goals",
"state_before": "case h.h.h.h.h.h\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\nf : α → Option β → γ\nas : List α\nbs : List β\n⊢ zipWithLeft f as bs = zipWithLeftTR.go f as bs #[]",
"tactic": "simp [zipWithLeftTR, go]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type u_2\nγ : Type u_1\nf : α → Option β → γ\nas : List α\nbs✝ : List β\nacc : Array γ\nbs : List β\n⊢ zipWithLeftTR.go f [] bs acc = Array.toList acc ++ zipWithLeft f [] bs",
"tactic": "simp [zipWithLeftTR.go]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type u_2\nγ : Type u_1\nf : α → Option β → γ\nas : List α\nbs : List β\nacc : Array γ\nhead✝ : α\ntail✝ : List α\n⊢ zipWithLeftTR.go f (head✝ :: tail✝) [] acc = Array.toList acc ++ zipWithLeft f (head✝ :: tail✝) []",
"tactic": "simp [zipWithLeftTR.go, Array.foldl_data_eq_map]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type u_2\nγ : Type u_1\nf : α → Option β → γ\nas✝ : List α\nbs✝ : List β\nacc : Array γ\na : α\nas : List α\nb : β\nbs : List β\n⊢ zipWithLeftTR.go f (a :: as) (b :: bs) acc = Array.toList acc ++ zipWithLeft f (a :: as) (b :: bs)",
"tactic": "simp [zipWithLeftTR.go, go _ as bs]"
}
] |
[
1374,
27
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1368,
10
] |
Mathlib/Data/MvPolynomial/Variables.lean
|
MvPolynomial.degrees_sum
|
[
{
"state_after": "case refine'_1\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.34158\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\nι : Type u_1\ninst✝ : DecidableEq σ\ns : Finset ι\nf : ι → MvPolynomial σ R\n⊢ degrees (∑ i in ∅, f i) ≤ Finset.sup ∅ fun i => degrees (f i)\n\ncase refine'_2\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.34158\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\nι : Type u_1\ninst✝ : DecidableEq σ\ns : Finset ι\nf : ι → MvPolynomial σ R\n⊢ ∀ ⦃a : ι⦄ {s : Finset ι},\n ¬a ∈ s →\n (degrees (∑ i in s, f i) ≤ Finset.sup s fun i => degrees (f i)) →\n degrees (∑ i in insert a s, f i) ≤ Finset.sup (insert a s) fun i => degrees (f i)",
"state_before": "R : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.34158\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\nι : Type u_1\ninst✝ : DecidableEq σ\ns : Finset ι\nf : ι → MvPolynomial σ R\n⊢ degrees (∑ i in s, f i) ≤ Finset.sup s fun i => degrees (f i)",
"tactic": "refine' s.induction _ _"
},
{
"state_after": "case refine'_1\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.34158\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\nι : Type u_1\ninst✝ : DecidableEq σ\ns : Finset ι\nf : ι → MvPolynomial σ R\n⊢ 0 ≤ ⊥",
"state_before": "case refine'_1\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.34158\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\nι : Type u_1\ninst✝ : DecidableEq σ\ns : Finset ι\nf : ι → MvPolynomial σ R\n⊢ degrees (∑ i in ∅, f i) ≤ Finset.sup ∅ fun i => degrees (f i)",
"tactic": "simp only [Finset.sum_empty, Finset.sup_empty, degrees_zero]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.34158\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\nι : Type u_1\ninst✝ : DecidableEq σ\ns : Finset ι\nf : ι → MvPolynomial σ R\n⊢ 0 ≤ ⊥",
"tactic": "exact le_rfl"
},
{
"state_after": "case refine'_2\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.34158\nr : R\ne : ℕ\nn m : σ\ns✝¹ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\nι : Type u_1\ninst✝ : DecidableEq σ\ns✝ : Finset ι\nf : ι → MvPolynomial σ R\ni : ι\ns : Finset ι\nhis : ¬i ∈ s\nih : degrees (∑ i in s, f i) ≤ Finset.sup s fun i => degrees (f i)\n⊢ degrees (∑ i in insert i s, f i) ≤ Finset.sup (insert i s) fun i => degrees (f i)",
"state_before": "case refine'_2\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.34158\nr : R\ne : ℕ\nn m : σ\ns✝ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\nι : Type u_1\ninst✝ : DecidableEq σ\ns : Finset ι\nf : ι → MvPolynomial σ R\n⊢ ∀ ⦃a : ι⦄ {s : Finset ι},\n ¬a ∈ s →\n (degrees (∑ i in s, f i) ≤ Finset.sup s fun i => degrees (f i)) →\n degrees (∑ i in insert a s, f i) ≤ Finset.sup (insert a s) fun i => degrees (f i)",
"tactic": "intro i s his ih"
},
{
"state_after": "case refine'_2\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.34158\nr : R\ne : ℕ\nn m : σ\ns✝¹ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\nι : Type u_1\ninst✝ : DecidableEq σ\ns✝ : Finset ι\nf : ι → MvPolynomial σ R\ni : ι\ns : Finset ι\nhis : ¬i ∈ s\nih : degrees (∑ i in s, f i) ≤ Finset.sup s fun i => degrees (f i)\n⊢ degrees (f i + ∑ x in s, f x) ≤ degrees (f i) ⊔ Finset.sup s fun i => degrees (f i)",
"state_before": "case refine'_2\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.34158\nr : R\ne : ℕ\nn m : σ\ns✝¹ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\nι : Type u_1\ninst✝ : DecidableEq σ\ns✝ : Finset ι\nf : ι → MvPolynomial σ R\ni : ι\ns : Finset ι\nhis : ¬i ∈ s\nih : degrees (∑ i in s, f i) ≤ Finset.sup s fun i => degrees (f i)\n⊢ degrees (∑ i in insert i s, f i) ≤ Finset.sup (insert i s) fun i => degrees (f i)",
"tactic": "rw [Finset.sup_insert, Finset.sum_insert his]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nR : Type u\nS : Type v\nσ : Type u_2\nτ : Type ?u.34158\nr : R\ne : ℕ\nn m : σ\ns✝¹ : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\nι : Type u_1\ninst✝ : DecidableEq σ\ns✝ : Finset ι\nf : ι → MvPolynomial σ R\ni : ι\ns : Finset ι\nhis : ¬i ∈ s\nih : degrees (∑ i in s, f i) ≤ Finset.sup s fun i => degrees (f i)\n⊢ degrees (f i + ∑ x in s, f x) ≤ degrees (f i) ⊔ Finset.sup s fun i => degrees (f i)",
"tactic": "exact le_trans (degrees_add _ _) (sup_le_sup_left ih _)"
}
] |
[
159,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Inner.lean
|
MeasureTheory.StronglyMeasurable.inner
|
[] |
[
32,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
29,
11
] |
Mathlib/Topology/Homotopy/HomotopyGroup.lean
|
GenLoop.Homotopic.trans
|
[] |
[
185,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
184,
8
] |
Mathlib/Data/Set/Lattice.lean
|
Set.iUnion_range_eq_iUnion
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.180962\nι : Sort u_3\nι' : Sort ?u.180968\nι₂ : Sort ?u.180971\nκ : ι → Sort ?u.180976\nκ₁ : ι → Sort ?u.180981\nκ₂ : ι → Sort ?u.180986\nκ' : ι' → Sort ?u.180991\nC : ι → Set α\nf : (x : ι) → β → ↑(C x)\nhf : ∀ (x : ι), Surjective (f x)\nx : α\n⊢ (x ∈ ⋃ (y : β), range fun x => ↑(f x y)) ↔ x ∈ ⋃ (x : ι), C x",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.180962\nι : Sort u_3\nι' : Sort ?u.180968\nι₂ : Sort ?u.180971\nκ : ι → Sort ?u.180976\nκ₁ : ι → Sort ?u.180981\nκ₂ : ι → Sort ?u.180986\nκ' : ι' → Sort ?u.180991\nC : ι → Set α\nf : (x : ι) → β → ↑(C x)\nhf : ∀ (x : ι), Surjective (f x)\n⊢ (⋃ (y : β), range fun x => ↑(f x y)) = ⋃ (x : ι), C x",
"tactic": "ext x"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.180962\nι : Sort u_3\nι' : Sort ?u.180968\nι₂ : Sort ?u.180971\nκ : ι → Sort ?u.180976\nκ₁ : ι → Sort ?u.180981\nκ₂ : ι → Sort ?u.180986\nκ' : ι' → Sort ?u.180991\nC : ι → Set α\nf : (x : ι) → β → ↑(C x)\nhf : ∀ (x : ι), Surjective (f x)\nx : α\n⊢ (∃ i, x ∈ range fun x => ↑(f x i)) ↔ ∃ i, x ∈ C i",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.180962\nι : Sort u_3\nι' : Sort ?u.180968\nι₂ : Sort ?u.180971\nκ : ι → Sort ?u.180976\nκ₁ : ι → Sort ?u.180981\nκ₂ : ι → Sort ?u.180986\nκ' : ι' → Sort ?u.180991\nC : ι → Set α\nf : (x : ι) → β → ↑(C x)\nhf : ∀ (x : ι), Surjective (f x)\nx : α\n⊢ (x ∈ ⋃ (y : β), range fun x => ↑(f x y)) ↔ x ∈ ⋃ (x : ι), C x",
"tactic": "rw [mem_iUnion, mem_iUnion]"
},
{
"state_after": "case h.mp\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.180962\nι : Sort u_3\nι' : Sort ?u.180968\nι₂ : Sort ?u.180971\nκ : ι → Sort ?u.180976\nκ₁ : ι → Sort ?u.180981\nκ₂ : ι → Sort ?u.180986\nκ' : ι' → Sort ?u.180991\nC : ι → Set α\nf : (x : ι) → β → ↑(C x)\nhf : ∀ (x : ι), Surjective (f x)\nx : α\n⊢ (∃ i, x ∈ range fun x => ↑(f x i)) → ∃ i, x ∈ C i\n\ncase h.mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.180962\nι : Sort u_3\nι' : Sort ?u.180968\nι₂ : Sort ?u.180971\nκ : ι → Sort ?u.180976\nκ₁ : ι → Sort ?u.180981\nκ₂ : ι → Sort ?u.180986\nκ' : ι' → Sort ?u.180991\nC : ι → Set α\nf : (x : ι) → β → ↑(C x)\nhf : ∀ (x : ι), Surjective (f x)\nx : α\n⊢ (∃ i, x ∈ C i) → ∃ i, x ∈ range fun x => ↑(f x i)",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.180962\nι : Sort u_3\nι' : Sort ?u.180968\nι₂ : Sort ?u.180971\nκ : ι → Sort ?u.180976\nκ₁ : ι → Sort ?u.180981\nκ₂ : ι → Sort ?u.180986\nκ' : ι' → Sort ?u.180991\nC : ι → Set α\nf : (x : ι) → β → ↑(C x)\nhf : ∀ (x : ι), Surjective (f x)\nx : α\n⊢ (∃ i, x ∈ range fun x => ↑(f x i)) ↔ ∃ i, x ∈ C i",
"tactic": "constructor"
},
{
"state_after": "case h.mp.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.180962\nι : Sort u_3\nι' : Sort ?u.180968\nι₂ : Sort ?u.180971\nκ : ι → Sort ?u.180976\nκ₁ : ι → Sort ?u.180981\nκ₂ : ι → Sort ?u.180986\nκ' : ι' → Sort ?u.180991\nC : ι → Set α\nf : (x : ι) → β → ↑(C x)\nhf : ∀ (x : ι), Surjective (f x)\ny : β\ni : ι\n⊢ ∃ i_1, (fun x => ↑(f x y)) i ∈ C i_1",
"state_before": "case h.mp\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.180962\nι : Sort u_3\nι' : Sort ?u.180968\nι₂ : Sort ?u.180971\nκ : ι → Sort ?u.180976\nκ₁ : ι → Sort ?u.180981\nκ₂ : ι → Sort ?u.180986\nκ' : ι' → Sort ?u.180991\nC : ι → Set α\nf : (x : ι) → β → ↑(C x)\nhf : ∀ (x : ι), Surjective (f x)\nx : α\n⊢ (∃ i, x ∈ range fun x => ↑(f x i)) → ∃ i, x ∈ C i",
"tactic": "rintro ⟨y, i, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mp.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.180962\nι : Sort u_3\nι' : Sort ?u.180968\nι₂ : Sort ?u.180971\nκ : ι → Sort ?u.180976\nκ₁ : ι → Sort ?u.180981\nκ₂ : ι → Sort ?u.180986\nκ' : ι' → Sort ?u.180991\nC : ι → Set α\nf : (x : ι) → β → ↑(C x)\nhf : ∀ (x : ι), Surjective (f x)\ny : β\ni : ι\n⊢ ∃ i_1, (fun x => ↑(f x y)) i ∈ C i_1",
"tactic": "exact ⟨i, (f i y).2⟩"
},
{
"state_after": "case h.mpr.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.180962\nι : Sort u_3\nι' : Sort ?u.180968\nι₂ : Sort ?u.180971\nκ : ι → Sort ?u.180976\nκ₁ : ι → Sort ?u.180981\nκ₂ : ι → Sort ?u.180986\nκ' : ι' → Sort ?u.180991\nC : ι → Set α\nf : (x : ι) → β → ↑(C x)\nhf : ∀ (x : ι), Surjective (f x)\nx : α\ni : ι\nhx : x ∈ C i\n⊢ ∃ i, x ∈ range fun x => ↑(f x i)",
"state_before": "case h.mpr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.180962\nι : Sort u_3\nι' : Sort ?u.180968\nι₂ : Sort ?u.180971\nκ : ι → Sort ?u.180976\nκ₁ : ι → Sort ?u.180981\nκ₂ : ι → Sort ?u.180986\nκ' : ι' → Sort ?u.180991\nC : ι → Set α\nf : (x : ι) → β → ↑(C x)\nhf : ∀ (x : ι), Surjective (f x)\nx : α\n⊢ (∃ i, x ∈ C i) → ∃ i, x ∈ range fun x => ↑(f x i)",
"tactic": "rintro ⟨i, hx⟩"
},
{
"state_after": "case h.mpr.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.180962\nι : Sort u_3\nι' : Sort ?u.180968\nι₂ : Sort ?u.180971\nκ : ι → Sort ?u.180976\nκ₁ : ι → Sort ?u.180981\nκ₂ : ι → Sort ?u.180986\nκ' : ι' → Sort ?u.180991\nC : ι → Set α\nf : (x : ι) → β → ↑(C x)\nhf : ∀ (x : ι), Surjective (f x)\nx : α\ni : ι\nhx : x ∈ C i\ny : β\nhy : f i y = { val := x, property := hx }\n⊢ ∃ i, x ∈ range fun x => ↑(f x i)",
"state_before": "case h.mpr.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.180962\nι : Sort u_3\nι' : Sort ?u.180968\nι₂ : Sort ?u.180971\nκ : ι → Sort ?u.180976\nκ₁ : ι → Sort ?u.180981\nκ₂ : ι → Sort ?u.180986\nκ' : ι' → Sort ?u.180991\nC : ι → Set α\nf : (x : ι) → β → ↑(C x)\nhf : ∀ (x : ι), Surjective (f x)\nx : α\ni : ι\nhx : x ∈ C i\n⊢ ∃ i, x ∈ range fun x => ↑(f x i)",
"tactic": "cases' hf i ⟨x, hx⟩ with y hy"
},
{
"state_after": "no goals",
"state_before": "case h.mpr.intro.intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.180962\nι : Sort u_3\nι' : Sort ?u.180968\nι₂ : Sort ?u.180971\nκ : ι → Sort ?u.180976\nκ₁ : ι → Sort ?u.180981\nκ₂ : ι → Sort ?u.180986\nκ' : ι' → Sort ?u.180991\nC : ι → Set α\nf : (x : ι) → β → ↑(C x)\nhf : ∀ (x : ι), Surjective (f x)\nx : α\ni : ι\nhx : x ∈ C i\ny : β\nhy : f i y = { val := x, property := hx }\n⊢ ∃ i, x ∈ range fun x => ↑(f x i)",
"tactic": "exact ⟨y, i, congr_arg Subtype.val hy⟩"
}
] |
[
1394,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1387,
1
] |
Mathlib/Analysis/SpecificLimits/Normed.lean
|
not_summable_of_ratio_test_tendsto_gt_one
|
[
{
"state_after": "α✝ : Type ?u.1387057\nβ : Type ?u.1387060\nι : Type ?u.1387063\nα : Type u_1\ninst✝ : SeminormedAddCommGroup α\nf : ℕ → α\nl : ℝ\nhl : 1 < l\nh : Tendsto (fun n => ‖f (n + 1)‖ / ‖f n‖) atTop (𝓝 l)\nkey : ∀ᶠ (n : ℕ) in atTop, ‖f n‖ ≠ 0\n⊢ ¬Summable f",
"state_before": "α✝ : Type ?u.1387057\nβ : Type ?u.1387060\nι : Type ?u.1387063\nα : Type u_1\ninst✝ : SeminormedAddCommGroup α\nf : ℕ → α\nl : ℝ\nhl : 1 < l\nh : Tendsto (fun n => ‖f (n + 1)‖ / ‖f n‖) atTop (𝓝 l)\n⊢ ¬Summable f",
"tactic": "have key : ∀ᶠ n in atTop, ‖f n‖ ≠ 0 := by\n filter_upwards [eventually_ge_of_tendsto_gt hl h]with _ hn hc\n rw [hc, _root_.div_zero] at hn\n linarith"
},
{
"state_after": "case intro.intro\nα✝ : Type ?u.1387057\nβ : Type ?u.1387060\nι : Type ?u.1387063\nα : Type u_1\ninst✝ : SeminormedAddCommGroup α\nf : ℕ → α\nl : ℝ\nhl : 1 < l\nh : Tendsto (fun n => ‖f (n + 1)‖ / ‖f n‖) atTop (𝓝 l)\nkey : ∀ᶠ (n : ℕ) in atTop, ‖f n‖ ≠ 0\nr : ℝ\nhr₀ : 1 < r\nhr₁ : r < l\n⊢ ¬Summable f",
"state_before": "α✝ : Type ?u.1387057\nβ : Type ?u.1387060\nι : Type ?u.1387063\nα : Type u_1\ninst✝ : SeminormedAddCommGroup α\nf : ℕ → α\nl : ℝ\nhl : 1 < l\nh : Tendsto (fun n => ‖f (n + 1)‖ / ‖f n‖) atTop (𝓝 l)\nkey : ∀ᶠ (n : ℕ) in atTop, ‖f n‖ ≠ 0\n⊢ ¬Summable f",
"tactic": "rcases exists_between hl with ⟨r, hr₀, hr₁⟩"
},
{
"state_after": "case intro.intro\nα✝ : Type ?u.1387057\nβ : Type ?u.1387060\nι : Type ?u.1387063\nα : Type u_1\ninst✝ : SeminormedAddCommGroup α\nf : ℕ → α\nl : ℝ\nhl : 1 < l\nh : Tendsto (fun n => ‖f (n + 1)‖ / ‖f n‖) atTop (𝓝 l)\nkey : ∀ᶠ (n : ℕ) in atTop, ‖f n‖ ≠ 0\nr : ℝ\nhr₀ : 1 < r\nhr₁ : r < l\n⊢ ∀ᶠ (n : ℕ) in atTop, r * ‖f n‖ ≤ ‖f (n + 1)‖",
"state_before": "case intro.intro\nα✝ : Type ?u.1387057\nβ : Type ?u.1387060\nι : Type ?u.1387063\nα : Type u_1\ninst✝ : SeminormedAddCommGroup α\nf : ℕ → α\nl : ℝ\nhl : 1 < l\nh : Tendsto (fun n => ‖f (n + 1)‖ / ‖f n‖) atTop (𝓝 l)\nkey : ∀ᶠ (n : ℕ) in atTop, ‖f n‖ ≠ 0\nr : ℝ\nhr₀ : 1 < r\nhr₁ : r < l\n⊢ ¬Summable f",
"tactic": "refine' not_summable_of_ratio_norm_eventually_ge hr₀ key.frequently _"
},
{
"state_after": "case h\nα✝ : Type ?u.1387057\nβ : Type ?u.1387060\nι : Type ?u.1387063\nα : Type u_1\ninst✝ : SeminormedAddCommGroup α\nf : ℕ → α\nl : ℝ\nhl : 1 < l\nh : Tendsto (fun n => ‖f (n + 1)‖ / ‖f n‖) atTop (𝓝 l)\nkey : ∀ᶠ (n : ℕ) in atTop, ‖f n‖ ≠ 0\nr : ℝ\nhr₀ : 1 < r\nhr₁ : r < l\na✝¹ : ℕ\na✝ : r ≤ ‖f (a✝¹ + 1)‖ / ‖f a✝¹‖\nh₁ : ‖f a✝¹‖ ≠ 0\n⊢ r * ‖f a✝¹‖ ≤ ‖f (a✝¹ + 1)‖",
"state_before": "case intro.intro\nα✝ : Type ?u.1387057\nβ : Type ?u.1387060\nι : Type ?u.1387063\nα : Type u_1\ninst✝ : SeminormedAddCommGroup α\nf : ℕ → α\nl : ℝ\nhl : 1 < l\nh : Tendsto (fun n => ‖f (n + 1)‖ / ‖f n‖) atTop (𝓝 l)\nkey : ∀ᶠ (n : ℕ) in atTop, ‖f n‖ ≠ 0\nr : ℝ\nhr₀ : 1 < r\nhr₁ : r < l\n⊢ ∀ᶠ (n : ℕ) in atTop, r * ‖f n‖ ≤ ‖f (n + 1)‖",
"tactic": "filter_upwards [eventually_ge_of_tendsto_gt hr₁ h, key]with _ _ h₁"
},
{
"state_after": "no goals",
"state_before": "case h\nα✝ : Type ?u.1387057\nβ : Type ?u.1387060\nι : Type ?u.1387063\nα : Type u_1\ninst✝ : SeminormedAddCommGroup α\nf : ℕ → α\nl : ℝ\nhl : 1 < l\nh : Tendsto (fun n => ‖f (n + 1)‖ / ‖f n‖) atTop (𝓝 l)\nkey : ∀ᶠ (n : ℕ) in atTop, ‖f n‖ ≠ 0\nr : ℝ\nhr₀ : 1 < r\nhr₁ : r < l\na✝¹ : ℕ\na✝ : r ≤ ‖f (a✝¹ + 1)‖ / ‖f a✝¹‖\nh₁ : ‖f a✝¹‖ ≠ 0\n⊢ r * ‖f a✝¹‖ ≤ ‖f (a✝¹ + 1)‖",
"tactic": "rwa [← le_div_iff (lt_of_le_of_ne (norm_nonneg _) h₁.symm)]"
},
{
"state_after": "case h\nα✝ : Type ?u.1387057\nβ : Type ?u.1387060\nι : Type ?u.1387063\nα : Type u_1\ninst✝ : SeminormedAddCommGroup α\nf : ℕ → α\nl : ℝ\nhl : 1 < l\nh : Tendsto (fun n => ‖f (n + 1)‖ / ‖f n‖) atTop (𝓝 l)\na✝ : ℕ\nhn : 1 ≤ ‖f (a✝ + 1)‖ / ‖f a✝‖\nhc : ‖f a✝‖ = 0\n⊢ False",
"state_before": "α✝ : Type ?u.1387057\nβ : Type ?u.1387060\nι : Type ?u.1387063\nα : Type u_1\ninst✝ : SeminormedAddCommGroup α\nf : ℕ → α\nl : ℝ\nhl : 1 < l\nh : Tendsto (fun n => ‖f (n + 1)‖ / ‖f n‖) atTop (𝓝 l)\n⊢ ∀ᶠ (n : ℕ) in atTop, ‖f n‖ ≠ 0",
"tactic": "filter_upwards [eventually_ge_of_tendsto_gt hl h]with _ hn hc"
},
{
"state_after": "case h\nα✝ : Type ?u.1387057\nβ : Type ?u.1387060\nι : Type ?u.1387063\nα : Type u_1\ninst✝ : SeminormedAddCommGroup α\nf : ℕ → α\nl : ℝ\nhl : 1 < l\nh : Tendsto (fun n => ‖f (n + 1)‖ / ‖f n‖) atTop (𝓝 l)\na✝ : ℕ\nhn : 1 ≤ 0\nhc : ‖f a✝‖ = 0\n⊢ False",
"state_before": "case h\nα✝ : Type ?u.1387057\nβ : Type ?u.1387060\nι : Type ?u.1387063\nα : Type u_1\ninst✝ : SeminormedAddCommGroup α\nf : ℕ → α\nl : ℝ\nhl : 1 < l\nh : Tendsto (fun n => ‖f (n + 1)‖ / ‖f n‖) atTop (𝓝 l)\na✝ : ℕ\nhn : 1 ≤ ‖f (a✝ + 1)‖ / ‖f a✝‖\nhc : ‖f a✝‖ = 0\n⊢ False",
"tactic": "rw [hc, _root_.div_zero] at hn"
},
{
"state_after": "no goals",
"state_before": "case h\nα✝ : Type ?u.1387057\nβ : Type ?u.1387060\nι : Type ?u.1387063\nα : Type u_1\ninst✝ : SeminormedAddCommGroup α\nf : ℕ → α\nl : ℝ\nhl : 1 < l\nh : Tendsto (fun n => ‖f (n + 1)‖ / ‖f n‖) atTop (𝓝 l)\na✝ : ℕ\nhn : 1 ≤ 0\nhc : ‖f a✝‖ = 0\n⊢ False",
"tactic": "linarith"
}
] |
[
569,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
559,
1
] |
Mathlib/Data/Rat/Cast.lean
|
RingHom.ext_rat
|
[] |
[
487,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
484,
1
] |
Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean
|
SimpleGraph.ComponentCompl.subset_hom
|
[
{
"state_after": "case intro\nV : Type u\nG : SimpleGraph V\nK L L' M : Set V\nh : K ⊆ L\nc : V\ncL : ¬c ∈ L\n⊢ c ∈ ↑(hom h (componentComplMk G cL))",
"state_before": "V : Type u\nG : SimpleGraph V\nK L L' M : Set V\nC : ComponentCompl G L\nh : K ⊆ L\n⊢ ↑C ⊆ ↑(hom h C)",
"tactic": "rintro c ⟨cL, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nV : Type u\nG : SimpleGraph V\nK L L' M : Set V\nh : K ⊆ L\nc : V\ncL : ¬c ∈ L\n⊢ c ∈ ↑(hom h (componentComplMk G cL))",
"tactic": "exact ⟨fun h' => cL (h h'), rfl⟩"
}
] |
[
183,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
181,
1
] |
Mathlib/Algebra/Module/Submodule/Pointwise.lean
|
Submodule.smul_bot'
|
[] |
[
237,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
236,
1
] |
Mathlib/RingTheory/MvPolynomial/WeightedHomogeneous.lean
|
MvPolynomial.weightedTotalDegree_coe
|
[
{
"state_after": "R : Type u_2\nM : Type u_3\ninst✝³ : CommSemiring R\nσ : Type u_1\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nhp : ∃ a, ↑a = weightedTotalDegree' w p\n⊢ weightedTotalDegree' w p = ↑(weightedTotalDegree w p)",
"state_before": "R : Type u_2\nM : Type u_3\ninst✝³ : CommSemiring R\nσ : Type u_1\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nhp : p ≠ 0\n⊢ weightedTotalDegree' w p = ↑(weightedTotalDegree w p)",
"tactic": "rw [Ne.def, ← weightedTotalDegree'_eq_bot_iff w p, ← Ne.def, WithBot.ne_bot_iff_exists] at hp"
},
{
"state_after": "case intro\nR : Type u_2\nM : Type u_3\ninst✝³ : CommSemiring R\nσ : Type u_1\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nm : M\nhm : ↑m = weightedTotalDegree' w p\n⊢ weightedTotalDegree' w p = ↑(weightedTotalDegree w p)",
"state_before": "R : Type u_2\nM : Type u_3\ninst✝³ : CommSemiring R\nσ : Type u_1\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nhp : ∃ a, ↑a = weightedTotalDegree' w p\n⊢ weightedTotalDegree' w p = ↑(weightedTotalDegree w p)",
"tactic": "obtain ⟨m, hm⟩ := hp"
},
{
"state_after": "case intro.a\nR : Type u_2\nM : Type u_3\ninst✝³ : CommSemiring R\nσ : Type u_1\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nm : M\nhm : ↑m = weightedTotalDegree' w p\n⊢ weightedTotalDegree' w p ≤ ↑(weightedTotalDegree w p)\n\ncase intro.a\nR : Type u_2\nM : Type u_3\ninst✝³ : CommSemiring R\nσ : Type u_1\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nm : M\nhm : ↑m = weightedTotalDegree' w p\n⊢ ↑(weightedTotalDegree w p) ≤ weightedTotalDegree' w p",
"state_before": "case intro\nR : Type u_2\nM : Type u_3\ninst✝³ : CommSemiring R\nσ : Type u_1\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nm : M\nhm : ↑m = weightedTotalDegree' w p\n⊢ weightedTotalDegree' w p = ↑(weightedTotalDegree w p)",
"tactic": "apply le_antisymm"
},
{
"state_after": "case intro.a\nR : Type u_2\nM : Type u_3\ninst✝³ : CommSemiring R\nσ : Type u_1\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nm : M\nhm : ↑m = weightedTotalDegree' w p\n⊢ ∀ (b : σ →₀ ℕ), b ∈ support p → ↑(weightedDegree' w) b ≤ sup (support p) fun s => ↑(weightedDegree' w) s",
"state_before": "case intro.a\nR : Type u_2\nM : Type u_3\ninst✝³ : CommSemiring R\nσ : Type u_1\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nm : M\nhm : ↑m = weightedTotalDegree' w p\n⊢ weightedTotalDegree' w p ≤ ↑(weightedTotalDegree w p)",
"tactic": "simp only [weightedTotalDegree, weightedTotalDegree', Finset.sup_le_iff, WithBot.coe_le_coe]"
},
{
"state_after": "case intro.a\nR : Type u_2\nM : Type u_3\ninst✝³ : CommSemiring R\nσ : Type u_1\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nm : M\nhm : ↑m = weightedTotalDegree' w p\nb : σ →₀ ℕ\n⊢ b ∈ support p → ↑(weightedDegree' w) b ≤ sup (support p) fun s => ↑(weightedDegree' w) s",
"state_before": "case intro.a\nR : Type u_2\nM : Type u_3\ninst✝³ : CommSemiring R\nσ : Type u_1\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nm : M\nhm : ↑m = weightedTotalDegree' w p\n⊢ ∀ (b : σ →₀ ℕ), b ∈ support p → ↑(weightedDegree' w) b ≤ sup (support p) fun s => ↑(weightedDegree' w) s",
"tactic": "intro b"
},
{
"state_after": "no goals",
"state_before": "case intro.a\nR : Type u_2\nM : Type u_3\ninst✝³ : CommSemiring R\nσ : Type u_1\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nm : M\nhm : ↑m = weightedTotalDegree' w p\nb : σ →₀ ℕ\n⊢ b ∈ support p → ↑(weightedDegree' w) b ≤ sup (support p) fun s => ↑(weightedDegree' w) s",
"tactic": "exact Finset.le_sup"
},
{
"state_after": "case intro.a\nR : Type u_2\nM : Type u_3\ninst✝³ : CommSemiring R\nσ : Type u_1\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nm : M\nhm : ↑m = weightedTotalDegree' w p\n⊢ ↑(sup (support p) fun s => ↑(weightedDegree' w) s) ≤ weightedTotalDegree' w p",
"state_before": "case intro.a\nR : Type u_2\nM : Type u_3\ninst✝³ : CommSemiring R\nσ : Type u_1\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nm : M\nhm : ↑m = weightedTotalDegree' w p\n⊢ ↑(weightedTotalDegree w p) ≤ weightedTotalDegree' w p",
"tactic": "simp only [weightedTotalDegree]"
},
{
"state_after": "case intro.a\nR : Type u_2\nM : Type u_3\ninst✝³ : CommSemiring R\nσ : Type u_1\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nm : M\nhm : ↑m = weightedTotalDegree' w p\nhm' : weightedTotalDegree' w p ≤ ↑m\n⊢ ↑(sup (support p) fun s => ↑(weightedDegree' w) s) ≤ weightedTotalDegree' w p",
"state_before": "case intro.a\nR : Type u_2\nM : Type u_3\ninst✝³ : CommSemiring R\nσ : Type u_1\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nm : M\nhm : ↑m = weightedTotalDegree' w p\n⊢ ↑(sup (support p) fun s => ↑(weightedDegree' w) s) ≤ weightedTotalDegree' w p",
"tactic": "have hm' : weightedTotalDegree' w p ≤ m := le_of_eq hm.symm"
},
{
"state_after": "case intro.a\nR : Type u_2\nM : Type u_3\ninst✝³ : CommSemiring R\nσ : Type u_1\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nm : M\nhm : ↑m = weightedTotalDegree' w p\nhm' : weightedTotalDegree' w p ≤ ↑m\n⊢ ↑(sup (support p) fun s => ↑(weightedDegree' w) s) ≤ ↑m",
"state_before": "case intro.a\nR : Type u_2\nM : Type u_3\ninst✝³ : CommSemiring R\nσ : Type u_1\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nm : M\nhm : ↑m = weightedTotalDegree' w p\nhm' : weightedTotalDegree' w p ≤ ↑m\n⊢ ↑(sup (support p) fun s => ↑(weightedDegree' w) s) ≤ weightedTotalDegree' w p",
"tactic": "rw [← hm]"
},
{
"state_after": "no goals",
"state_before": "case intro.a\nR : Type u_2\nM : Type u_3\ninst✝³ : CommSemiring R\nσ : Type u_1\ninst✝² : AddCommMonoid M\ninst✝¹ : SemilatticeSup M\ninst✝ : OrderBot M\nw : σ → M\np : MvPolynomial σ R\nm : M\nhm : ↑m = weightedTotalDegree' w p\nhm' : weightedTotalDegree' w p ≤ ↑m\n⊢ ↑(sup (support p) fun s => ↑(weightedDegree' w) s) ≤ ↑m",
"tactic": "simpa [weightedTotalDegree'] using hm'"
}
] |
[
116,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Basic.lean
|
Real.tendsto_log_nhdsWithin_zero
|
[
{
"state_after": "x y : ℝ\n⊢ Tendsto (fun x => log (abs x)) (𝓝[{0}ᶜ] 0) atBot",
"state_before": "x y : ℝ\n⊢ Tendsto log (𝓝[{0}ᶜ] 0) atBot",
"tactic": "rw [← show _ = log from funext log_abs]"
},
{
"state_after": "x y : ℝ\n⊢ Tendsto log (𝓝[Ioi 0] 0) atBot",
"state_before": "x y : ℝ\n⊢ Tendsto (fun x => log (abs x)) (𝓝[{0}ᶜ] 0) atBot",
"tactic": "refine' Tendsto.comp (g := log) _ tendsto_abs_nhdsWithin_zero"
},
{
"state_after": "no goals",
"state_before": "x y : ℝ\n⊢ Tendsto log (𝓝[Ioi 0] 0) atBot",
"tactic": "simpa [← tendsto_comp_exp_atBot] using tendsto_id"
}
] |
[
301,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
298,
1
] |
Mathlib/RingTheory/Localization/Basic.lean
|
IsLocalization.mk'_spec_mk
|
[] |
[
273,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
271,
1
] |
Mathlib/Analysis/Calculus/Deriv/Inv.lean
|
derivWithin_inv'
|
[] |
[
170,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
167,
1
] |
Mathlib/Data/Real/EReal.lean
|
EReal.coe_ennreal_injective
|
[] |
[
485,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
484,
1
] |
Mathlib/Algebra/Algebra/Hom.lean
|
AlgHom.coe_addMonoidHom_injective
|
[] |
[
211,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
210,
1
] |
Mathlib/Data/Seq/Computation.lean
|
Computation.LiftRelAux.ret_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\nC : Computation α → Computation β → Prop\nb : β\nca : Computation α\n⊢ LiftRelAux R C (destruct ca) (Sum.inl b) ↔ ∃ a, a ∈ ca ∧ R a b",
"tactic": "rw [← LiftRelAux.swap, LiftRelAux.ret_left]"
}
] |
[
1282,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1280,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.tendsto_atTop_atBot
|
[] |
[
1281,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1279,
1
] |
Mathlib/Data/Polynomial/Splits.lean
|
Polynomial.splits_of_map_degree_eq_one
|
[
{
"state_after": "F : Type u\nK : Type v\nL : Type w\ninst✝² : CommRing K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nf : K[X]\nhf : degree (map i f) = 1\ng✝ : L[X]\nhg : Irreducible g✝\nx✝ : g✝ ∣ map i f\np : L[X]\nhp : map i f = g✝ * p\nthis : degree (map i f) = degree (g✝ * p)\n⊢ degree g✝ = 1",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝² : CommRing K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nf : K[X]\nhf : degree (map i f) = 1\ng✝ : L[X]\nhg : Irreducible g✝\nx✝ : g✝ ∣ map i f\np : L[X]\nhp : map i f = g✝ * p\n⊢ degree g✝ = 1",
"tactic": "have := congr_arg degree hp"
},
{
"state_after": "F : Type u\nK : Type v\nL : Type w\ninst✝² : CommRing K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nf : K[X]\nhf : degree (map i f) = 1\ng✝ : L[X]\nhg : Irreducible g✝\nx✝ : g✝ ∣ map i f\np : L[X]\nhp : map i f = g✝ * p\nthis : degree g✝ = 1 ∧ degree p = 0\n⊢ degree g✝ = 1",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝² : CommRing K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nf : K[X]\nhf : degree (map i f) = 1\ng✝ : L[X]\nhg : Irreducible g✝\nx✝ : g✝ ∣ map i f\np : L[X]\nhp : map i f = g✝ * p\nthis : degree (map i f) = degree (g✝ * p)\n⊢ degree g✝ = 1",
"tactic": "simp [Nat.WithBot.add_eq_one_iff, hf, @eq_comm (WithBot ℕ) 1,\n mt isUnit_iff_degree_eq_zero.2 hg.1] at this"
},
{
"state_after": "no goals",
"state_before": "F : Type u\nK : Type v\nL : Type w\ninst✝² : CommRing K\ninst✝¹ : Field L\ninst✝ : Field F\ni : K →+* L\nf : K[X]\nhf : degree (map i f) = 1\ng✝ : L[X]\nhg : Irreducible g✝\nx✝ : g✝ ∣ map i f\np : L[X]\nhp : map i f = g✝ * p\nthis : degree g✝ = 1 ∧ degree p = 0\n⊢ degree g✝ = 1",
"tactic": "tauto"
}
] |
[
91,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Mathlib/NumberTheory/Padics/PadicVal.lean
|
padicValNat_self
|
[
{
"state_after": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\n⊢ Part.get (multiplicity p p) (_ : multiplicity.Finite p p) = 1",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\n⊢ padicValNat p p = 1",
"tactic": "rw [padicValNat_def (@Fact.out p.Prime).pos]"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\n⊢ Part.get (multiplicity p p) (_ : multiplicity.Finite p p) = 1",
"tactic": "simp"
}
] |
[
236,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
234,
1
] |
Mathlib/Data/Set/Function.lean
|
Set.EqOn.congr_strictAntiOn
|
[] |
[
287,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
286,
1
] |
Mathlib/Dynamics/PeriodicPts.lean
|
Function.periodicOrbit_eq_nil_of_not_periodic_pt
|
[] |
[
514,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
512,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean
|
Real.tan_add'
|
[] |
[
43,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
40,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.eval₂_comp
|
[
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q : R[X]\nx✝ : R\ninst✝ : CommSemiring S\nf : R →+* S\nx : S\n⊢ eval₂ f x (eval₂ C q (∑ i in range (natDegree p + 1), ↑(monomial i) (coeff p i))) =\n eval₂ f (eval₂ f x q) (∑ i in range (natDegree p + 1), ↑(monomial i) (coeff p i))",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q : R[X]\nx✝ : R\ninst✝ : CommSemiring S\nf : R →+* S\nx : S\n⊢ eval₂ f x (comp p q) = eval₂ f (eval₂ f x q) p",
"tactic": "rw [comp, p.as_sum_range]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np q : R[X]\nx✝ : R\ninst✝ : CommSemiring S\nf : R →+* S\nx : S\n⊢ eval₂ f x (eval₂ C q (∑ i in range (natDegree p + 1), ↑(monomial i) (coeff p i))) =\n eval₂ f (eval₂ f x q) (∑ i in range (natDegree p + 1), ↑(monomial i) (coeff p i))",
"tactic": "simp [eval₂_finset_sum, eval₂_pow]"
}
] |
[
1034,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1033,
1
] |
Mathlib/Data/Set/Intervals/Disjoint.lean
|
IsLUB.biUnion_Iic_eq_Iio
|
[] |
[
223,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
221,
1
] |
Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean
|
NonUnitalRingHom.srangeRestrict_surjective
|
[] |
[
885,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
881,
1
] |
Mathlib/LinearAlgebra/TensorProduct.lean
|
TensorProduct.add_tmul
|
[] |
[
143,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Data/Setoid/Partition.lean
|
Setoid.eqv_classes_of_disjoint_union
|
[
{
"state_after": "α : Type u_1\nc : Set (Set α)\nhu : ⋃₀ c = Set.univ\nH : Set.PairwiseDisjoint c id\na : α\n⊢ a ∈ Set.univ",
"state_before": "α : Type u_1\nc : Set (Set α)\nhu : ⋃₀ c = Set.univ\nH : Set.PairwiseDisjoint c id\na : α\n⊢ a ∈ ⋃₀ c",
"tactic": "rw [hu]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nc : Set (Set α)\nhu : ⋃₀ c = Set.univ\nH : Set.PairwiseDisjoint c id\na : α\n⊢ a ∈ Set.univ",
"tactic": "exact Set.mem_univ a"
}
] |
[
181,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Mathlib/Algebra/AddTorsor.lean
|
Prod.fst_vadd
|
[] |
[
306,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
305,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
natCast_mem
|
[
{
"state_after": "no goals",
"state_before": "S : Type u_1\nR : Type u_2\ninst✝² : AddMonoidWithOne R\ninst✝¹ : SetLike S R\ns : S\ninst✝ : AddSubmonoidWithOneClass S R\nn : ℕ\n⊢ ↑n ∈ s",
"tactic": "induction n <;> simp [zero_mem, add_mem, one_mem, *]"
}
] |
[
44,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
43,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.mapDomain_support
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.187796\nι : Type ?u.187799\nM : Type u_3\nM' : Type ?u.187805\nN : Type ?u.187808\nP : Type ?u.187811\nG : Type ?u.187814\nH : Type ?u.187817\nR : Type ?u.187820\nS : Type ?u.187823\ninst✝¹ : AddCommMonoid M\nv v₁ v₂ : α →₀ M\ninst✝ : DecidableEq β\nf : α → β\ns : α →₀ M\n⊢ (Finset.biUnion s.support fun a => {f a}) ⊆ image f s.support",
"tactic": "rw [Finset.biUnion_singleton]"
}
] |
[
543,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
539,
1
] |
Mathlib/LinearAlgebra/BilinearMap.lean
|
LinearMap.map_zero₂
|
[] |
[
159,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/Data/Multiset/Antidiagonal.lean
|
Multiset.antidiagonal_coe'
|
[] |
[
41,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
40,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Type.lean
|
Equiv.Perm.cycleType_eq'
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\ns : Finset (Perm α)\nh1 : ∀ (f : Perm α), f ∈ s → IsCycle f\nh2 : Set.Pairwise (↑s) Disjoint\nh0 : Finset.noncommProd s id (_ : Set.Pairwise ↑s fun a b => Commute (id a) (id b)) = σ\n⊢ map (Finset.card ∘ support) (cycleFactorsFinset σ).val = map (Finset.card ∘ support) s.val",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\ns : Finset (Perm α)\nh1 : ∀ (f : Perm α), f ∈ s → IsCycle f\nh2 : Set.Pairwise (↑s) Disjoint\nh0 : Finset.noncommProd s id (_ : Set.Pairwise ↑s fun a b => Commute (id a) (id b)) = σ\n⊢ cycleType σ = map (Finset.card ∘ support) s.val",
"tactic": "rw [cycleType_def]"
},
{
"state_after": "case e_s.e_self\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\ns : Finset (Perm α)\nh1 : ∀ (f : Perm α), f ∈ s → IsCycle f\nh2 : Set.Pairwise (↑s) Disjoint\nh0 : Finset.noncommProd s id (_ : Set.Pairwise ↑s fun a b => Commute (id a) (id b)) = σ\n⊢ cycleFactorsFinset σ = s",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\ns : Finset (Perm α)\nh1 : ∀ (f : Perm α), f ∈ s → IsCycle f\nh2 : Set.Pairwise (↑s) Disjoint\nh0 : Finset.noncommProd s id (_ : Set.Pairwise ↑s fun a b => Commute (id a) (id b)) = σ\n⊢ map (Finset.card ∘ support) (cycleFactorsFinset σ).val = map (Finset.card ∘ support) s.val",
"tactic": "congr"
},
{
"state_after": "case e_s.e_self\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\ns : Finset (Perm α)\nh1 : ∀ (f : Perm α), f ∈ s → IsCycle f\nh2 : Set.Pairwise (↑s) Disjoint\nh0 : Finset.noncommProd s id (_ : Set.Pairwise ↑s fun a b => Commute (id a) (id b)) = σ\n⊢ (∀ (f : Perm α), f ∈ s → IsCycle f) ∧\n ∃ h, Finset.noncommProd s id (_ : Set.Pairwise ↑s fun a b => Commute (id a) (id b)) = σ",
"state_before": "case e_s.e_self\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\ns : Finset (Perm α)\nh1 : ∀ (f : Perm α), f ∈ s → IsCycle f\nh2 : Set.Pairwise (↑s) Disjoint\nh0 : Finset.noncommProd s id (_ : Set.Pairwise ↑s fun a b => Commute (id a) (id b)) = σ\n⊢ cycleFactorsFinset σ = s",
"tactic": "rw [cycleFactorsFinset_eq_finset]"
},
{
"state_after": "no goals",
"state_before": "case e_s.e_self\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\ns : Finset (Perm α)\nh1 : ∀ (f : Perm α), f ∈ s → IsCycle f\nh2 : Set.Pairwise (↑s) Disjoint\nh0 : Finset.noncommProd s id (_ : Set.Pairwise ↑s fun a b => Commute (id a) (id b)) = σ\n⊢ (∀ (f : Perm α), f ∈ s → IsCycle f) ∧\n ∃ h, Finset.noncommProd s id (_ : Set.Pairwise ↑s fun a b => Commute (id a) (id b)) = σ",
"tactic": "exact ⟨h1, h2, h0⟩"
}
] |
[
66,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
59,
1
] |
Mathlib/Order/Cover.lean
|
Wcovby.trans_antisymm_rel
|
[] |
[
92,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/Data/List/Basic.lean
|
List.get?_injective
|
[
{
"state_after": "no goals",
"state_before": "case nil\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\ni j : ℕ\nh₀ : i < length []\nh₁ : Nodup []\nh₂ : get? [] i = get? [] j\n⊢ i = j",
"tactic": "cases h₀"
},
{
"state_after": "case cons.zero.zero\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nh₁ : Nodup (x :: xs)\nh₀ : zero < length (x :: xs)\nh₂ : get? (x :: xs) zero = get? (x :: xs) zero\n⊢ zero = zero\n\ncase cons.zero.succ\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nh₁ : Nodup (x :: xs)\nh₀ : zero < length (x :: xs)\nn✝ : ℕ\nh₂ : get? (x :: xs) zero = get? (x :: xs) (succ n✝)\n⊢ zero = succ n✝\n\ncase cons.succ.zero\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nh₁ : Nodup (x :: xs)\nn✝ : ℕ\nh₀ : succ n✝ < length (x :: xs)\nh₂ : get? (x :: xs) (succ n✝) = get? (x :: xs) zero\n⊢ succ n✝ = zero\n\ncase cons.succ.succ\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nh₁ : Nodup (x :: xs)\nn✝¹ : ℕ\nh₀ : succ n✝¹ < length (x :: xs)\nn✝ : ℕ\nh₂ : get? (x :: xs) (succ n✝¹) = get? (x :: xs) (succ n✝)\n⊢ succ n✝¹ = succ n✝",
"state_before": "case cons\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\ni j : ℕ\nh₀ : i < length (x :: xs)\nh₁ : Nodup (x :: xs)\nh₂ : get? (x :: xs) i = get? (x :: xs) j\n⊢ i = j",
"tactic": "cases i <;> cases j"
},
{
"state_after": "case cons.zero.succ\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nh₁ : Nodup (x :: xs)\nh₀ : zero < length (x :: xs)\nn✝ : ℕ\nh₂ : get? (x :: xs) zero = get? (x :: xs) (succ n✝)\n⊢ zero = succ n✝\n\ncase cons.succ.zero\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nh₁ : Nodup (x :: xs)\nn✝ : ℕ\nh₀ : succ n✝ < length (x :: xs)\nh₂ : get? (x :: xs) (succ n✝) = get? (x :: xs) zero\n⊢ succ n✝ = zero\n\ncase cons.succ.succ\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nh₁ : Nodup (x :: xs)\nn✝¹ : ℕ\nh₀ : succ n✝¹ < length (x :: xs)\nn✝ : ℕ\nh₂ : get? (x :: xs) (succ n✝¹) = get? (x :: xs) (succ n✝)\n⊢ succ n✝¹ = succ n✝",
"state_before": "case cons.zero.zero\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nh₁ : Nodup (x :: xs)\nh₀ : zero < length (x :: xs)\nh₂ : get? (x :: xs) zero = get? (x :: xs) zero\n⊢ zero = zero\n\ncase cons.zero.succ\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nh₁ : Nodup (x :: xs)\nh₀ : zero < length (x :: xs)\nn✝ : ℕ\nh₂ : get? (x :: xs) zero = get? (x :: xs) (succ n✝)\n⊢ zero = succ n✝\n\ncase cons.succ.zero\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nh₁ : Nodup (x :: xs)\nn✝ : ℕ\nh₀ : succ n✝ < length (x :: xs)\nh₂ : get? (x :: xs) (succ n✝) = get? (x :: xs) zero\n⊢ succ n✝ = zero\n\ncase cons.succ.succ\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nh₁ : Nodup (x :: xs)\nn✝¹ : ℕ\nh₀ : succ n✝¹ < length (x :: xs)\nn✝ : ℕ\nh₂ : get? (x :: xs) (succ n✝¹) = get? (x :: xs) (succ n✝)\n⊢ succ n✝¹ = succ n✝",
"tactic": "case zero.zero => rfl"
},
{
"state_after": "case cons.zero.succ\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nh₁ : Nodup (x :: xs)\nh₀ : zero < length (x :: xs)\nn✝ : ℕ\nh₂ : get? (x :: xs) zero = get? (x :: xs) (succ n✝)\n⊢ zero = succ n✝\n\ncase cons.succ.zero\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nh₁ : Nodup (x :: xs)\nn✝ : ℕ\nh₀ : succ n✝ < length (x :: xs)\nh₂ : get? (x :: xs) (succ n✝) = get? (x :: xs) zero\n⊢ succ n✝ = zero",
"state_before": "case cons.zero.succ\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nh₁ : Nodup (x :: xs)\nh₀ : zero < length (x :: xs)\nn✝ : ℕ\nh₂ : get? (x :: xs) zero = get? (x :: xs) (succ n✝)\n⊢ zero = succ n✝\n\ncase cons.succ.zero\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nh₁ : Nodup (x :: xs)\nn✝ : ℕ\nh₀ : succ n✝ < length (x :: xs)\nh₂ : get? (x :: xs) (succ n✝) = get? (x :: xs) zero\n⊢ succ n✝ = zero\n\ncase cons.succ.succ\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nh₁ : Nodup (x :: xs)\nn✝¹ : ℕ\nh₀ : succ n✝¹ < length (x :: xs)\nn✝ : ℕ\nh₂ : get? (x :: xs) (succ n✝¹) = get? (x :: xs) (succ n✝)\n⊢ succ n✝¹ = succ n✝",
"tactic": "case succ.succ =>\n congr; cases h₁\n apply tail_ih <;> solve_by_elim [lt_of_succ_lt_succ]"
},
{
"state_after": "case cons.zero.succ.cons\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nh₀ : zero < length (x :: xs)\nn✝ : ℕ\nh₂ : some x = get? xs n✝\nh' : Pairwise (fun x x_1 => x ≠ x_1) xs\nh : ∀ (a' : α), a' ∈ xs → x ≠ a'\n⊢ zero = succ n✝\n\ncase cons.succ.zero.cons\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nn✝ : ℕ\nh₀ : succ n✝ < length (x :: xs)\nh₂ : get? xs n✝ = some x\nh' : Pairwise (fun x x_1 => x ≠ x_1) xs\nh : ∀ (a' : α), a' ∈ xs → x ≠ a'\n⊢ succ n✝ = zero",
"state_before": "case cons.zero.succ\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nh₁ : Nodup (x :: xs)\nh₀ : zero < length (x :: xs)\nn✝ : ℕ\nh₂ : get? (x :: xs) zero = get? (x :: xs) (succ n✝)\n⊢ zero = succ n✝\n\ncase cons.succ.zero\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nh₁ : Nodup (x :: xs)\nn✝ : ℕ\nh₀ : succ n✝ < length (x :: xs)\nh₂ : get? (x :: xs) (succ n✝) = get? (x :: xs) zero\n⊢ succ n✝ = zero",
"tactic": "all_goals ( dsimp at h₂; cases' h₁ with _ _ h h')"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nh₁ : Nodup (x :: xs)\nh₀ : zero < length (x :: xs)\nh₂ : get? (x :: xs) zero = get? (x :: xs) zero\n⊢ zero = zero",
"tactic": "rfl"
},
{
"state_after": "case e_n\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nh₁ : Nodup (x :: xs)\nn✝¹ : ℕ\nh₀ : succ n✝¹ < length (x :: xs)\nn✝ : ℕ\nh₂ : get? (x :: xs) (succ n✝¹) = get? (x :: xs) (succ n✝)\n⊢ n✝¹ = n✝",
"state_before": "ι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nh₁ : Nodup (x :: xs)\nn✝¹ : ℕ\nh₀ : succ n✝¹ < length (x :: xs)\nn✝ : ℕ\nh₂ : get? (x :: xs) (succ n✝¹) = get? (x :: xs) (succ n✝)\n⊢ succ n✝¹ = succ n✝",
"tactic": "congr"
},
{
"state_after": "case e_n.cons\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nn✝¹ : ℕ\nh₀ : succ n✝¹ < length (x :: xs)\nn✝ : ℕ\nh₂ : get? (x :: xs) (succ n✝¹) = get? (x :: xs) (succ n✝)\na✝¹ : Pairwise (fun x x_1 => x ≠ x_1) xs\na✝ : ∀ (a' : α), a' ∈ xs → x ≠ a'\n⊢ n✝¹ = n✝",
"state_before": "case e_n\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nh₁ : Nodup (x :: xs)\nn✝¹ : ℕ\nh₀ : succ n✝¹ < length (x :: xs)\nn✝ : ℕ\nh₂ : get? (x :: xs) (succ n✝¹) = get? (x :: xs) (succ n✝)\n⊢ n✝¹ = n✝",
"tactic": "cases h₁"
},
{
"state_after": "no goals",
"state_before": "case e_n.cons\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nn✝¹ : ℕ\nh₀ : succ n✝¹ < length (x :: xs)\nn✝ : ℕ\nh₂ : get? (x :: xs) (succ n✝¹) = get? (x :: xs) (succ n✝)\na✝¹ : Pairwise (fun x x_1 => x ≠ x_1) xs\na✝ : ∀ (a' : α), a' ∈ xs → x ≠ a'\n⊢ n✝¹ = n✝",
"tactic": "apply tail_ih <;> solve_by_elim [lt_of_succ_lt_succ]"
},
{
"state_after": "case cons.succ.zero.cons\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nn✝ : ℕ\nh₀ : succ n✝ < length (x :: xs)\nh₂ : get? xs n✝ = some x\nh' : Pairwise (fun x x_1 => x ≠ x_1) xs\nh : ∀ (a' : α), a' ∈ xs → x ≠ a'\n⊢ succ n✝ = zero",
"state_before": "case cons.succ.zero\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nh₁ : Nodup (x :: xs)\nn✝ : ℕ\nh₀ : succ n✝ < length (x :: xs)\nh₂ : get? (x :: xs) (succ n✝) = get? (x :: xs) zero\n⊢ succ n✝ = zero",
"tactic": "( dsimp at h₂; cases' h₁ with _ _ h h')"
},
{
"state_after": "case cons.succ.zero\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nh₁ : Nodup (x :: xs)\nn✝ : ℕ\nh₀ : succ n✝ < length (x :: xs)\nh₂ : get? xs n✝ = some x\n⊢ succ n✝ = zero",
"state_before": "case cons.succ.zero\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nh₁ : Nodup (x :: xs)\nn✝ : ℕ\nh₀ : succ n✝ < length (x :: xs)\nh₂ : get? (x :: xs) (succ n✝) = get? (x :: xs) zero\n⊢ succ n✝ = zero",
"tactic": "dsimp at h₂"
},
{
"state_after": "case cons.succ.zero.cons\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nn✝ : ℕ\nh₀ : succ n✝ < length (x :: xs)\nh₂ : get? xs n✝ = some x\nh' : Pairwise (fun x x_1 => x ≠ x_1) xs\nh : ∀ (a' : α), a' ∈ xs → x ≠ a'\n⊢ succ n✝ = zero",
"state_before": "case cons.succ.zero\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nh₁ : Nodup (x :: xs)\nn✝ : ℕ\nh₀ : succ n✝ < length (x :: xs)\nh₂ : get? xs n✝ = some x\n⊢ succ n✝ = zero",
"tactic": "cases' h₁ with _ _ h h'"
},
{
"state_after": "no goals",
"state_before": "case cons.zero.succ.cons\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nh₀ : zero < length (x :: xs)\nn✝ : ℕ\nh₂ : some x = get? xs n✝\nh' : Pairwise (fun x x_1 => x ≠ x_1) xs\nh : ∀ (a' : α), a' ∈ xs → x ≠ a'\n⊢ zero = succ n✝",
"tactic": "cases (h x (mem_iff_get?.mpr ⟨_, h₂.symm⟩) rfl)"
},
{
"state_after": "no goals",
"state_before": "case cons.succ.zero.cons\nι : Type ?u.83779\nα✝ : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α✝\nα : Type u\nx : α\nxs : List α\ntail_ih : ∀ {i j : ℕ}, i < length xs → Nodup xs → get? xs i = get? xs j → i = j\nn✝ : ℕ\nh₀ : succ n✝ < length (x :: xs)\nh₂ : get? xs n✝ = some x\nh' : Pairwise (fun x x_1 => x ≠ x_1) xs\nh : ∀ (a' : α), a' ∈ xs → x ≠ a'\n⊢ succ n✝ = zero",
"tactic": "cases (h x (mem_iff_get?.mpr ⟨_, h₂⟩) rfl)"
}
] |
[
1286,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1274,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
iUnion_Icc_add_int_cast
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : LinearOrderedRing α\ninst✝ : Archimedean α\na : α\n⊢ (⋃ (n : ℤ), Icc (a + ↑n) (a + ↑n + 1)) = univ",
"tactic": "simpa only [zsmul_one, Int.cast_add, Int.cast_one, ← add_assoc] using\n iUnion_Icc_add_zsmul zero_lt_one a"
}
] |
[
1104,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1102,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.closedBall_disjoint_ball
|
[] |
[
532,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
530,
1
] |
Mathlib/RingTheory/HahnSeries.lean
|
HahnSeries.single_mul_single
|
[
{
"state_after": "case coeff.h\nΓ : Type u_1\nR : Type u_2\ninst✝¹ : OrderedCancelAddCommMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\na b : Γ\nr s : R\nx : Γ\n⊢ coeff (↑(single a) r * ↑(single b) s) x = coeff (↑(single (a + b)) (r * s)) x",
"state_before": "Γ : Type u_1\nR : Type u_2\ninst✝¹ : OrderedCancelAddCommMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\na b : Γ\nr s : R\n⊢ ↑(single a) r * ↑(single b) s = ↑(single (a + b)) (r * s)",
"tactic": "ext x"
},
{
"state_after": "case pos\nΓ : Type u_1\nR : Type u_2\ninst✝¹ : OrderedCancelAddCommMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\na b : Γ\nr s : R\nx : Γ\nh : x = a + b\n⊢ coeff (↑(single a) r * ↑(single b) s) x = coeff (↑(single (a + b)) (r * s)) x\n\ncase neg\nΓ : Type u_1\nR : Type u_2\ninst✝¹ : OrderedCancelAddCommMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\na b : Γ\nr s : R\nx : Γ\nh : ¬x = a + b\n⊢ coeff (↑(single a) r * ↑(single b) s) x = coeff (↑(single (a + b)) (r * s)) x",
"state_before": "case coeff.h\nΓ : Type u_1\nR : Type u_2\ninst✝¹ : OrderedCancelAddCommMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\na b : Γ\nr s : R\nx : Γ\n⊢ coeff (↑(single a) r * ↑(single b) s) x = coeff (↑(single (a + b)) (r * s)) x",
"tactic": "by_cases h : x = a + b"
},
{
"state_after": "case pos\nΓ : Type u_1\nR : Type u_2\ninst✝¹ : OrderedCancelAddCommMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\na b : Γ\nr s : R\nx : Γ\nh : x = a + b\n⊢ coeff (↑(single a) r) a * s = coeff (↑(single (a + b)) (r * s)) (a + b)",
"state_before": "case pos\nΓ : Type u_1\nR : Type u_2\ninst✝¹ : OrderedCancelAddCommMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\na b : Γ\nr s : R\nx : Γ\nh : x = a + b\n⊢ coeff (↑(single a) r * ↑(single b) s) x = coeff (↑(single (a + b)) (r * s)) x",
"tactic": "rw [h, mul_single_coeff_add]"
},
{
"state_after": "no goals",
"state_before": "case pos\nΓ : Type u_1\nR : Type u_2\ninst✝¹ : OrderedCancelAddCommMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\na b : Γ\nr s : R\nx : Γ\nh : x = a + b\n⊢ coeff (↑(single a) r) a * s = coeff (↑(single (a + b)) (r * s)) (a + b)",
"tactic": "simp"
},
{
"state_after": "case neg\nΓ : Type u_1\nR : Type u_2\ninst✝¹ : OrderedCancelAddCommMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\na b : Γ\nr s : R\nx : Γ\nh : ¬x = a + b\n⊢ ∀ (x_1 : Γ × Γ),\n x_1 ∈ addAntidiagonal (_ : Set.IsPwo (support (↑(single a) r))) (_ : Set.IsPwo (support (↑(single b) s))) x →\n coeff (↑(single a) r) x_1.fst * coeff (↑(single b) s) x_1.snd = 0",
"state_before": "case neg\nΓ : Type u_1\nR : Type u_2\ninst✝¹ : OrderedCancelAddCommMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\na b : Γ\nr s : R\nx : Γ\nh : ¬x = a + b\n⊢ coeff (↑(single a) r * ↑(single b) s) x = coeff (↑(single (a + b)) (r * s)) x",
"tactic": "rw [single_coeff_of_ne h, mul_coeff, sum_eq_zero]"
},
{
"state_after": "case neg\nΓ : Type u_1\nR : Type u_2\ninst✝¹ : OrderedCancelAddCommMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\na b : Γ\nr s : R\nx : Γ\nh : ¬x = a + b\n⊢ ∀ (x_1 : Γ × Γ),\n x_1.fst ∈ support (↑(single a) r) ∧ x_1.snd ∈ support (↑(single b) s) ∧ x_1.fst + x_1.snd = x →\n coeff (↑(single a) r) x_1.fst * coeff (↑(single b) s) x_1.snd = 0",
"state_before": "case neg\nΓ : Type u_1\nR : Type u_2\ninst✝¹ : OrderedCancelAddCommMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\na b : Γ\nr s : R\nx : Γ\nh : ¬x = a + b\n⊢ ∀ (x_1 : Γ × Γ),\n x_1 ∈ addAntidiagonal (_ : Set.IsPwo (support (↑(single a) r))) (_ : Set.IsPwo (support (↑(single b) s))) x →\n coeff (↑(single a) r) x_1.fst * coeff (↑(single b) s) x_1.snd = 0",
"tactic": "simp_rw [mem_addAntidiagonal]"
},
{
"state_after": "case neg.mk.intro.intro\nΓ : Type u_1\nR : Type u_2\ninst✝¹ : OrderedCancelAddCommMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\na b : Γ\nr s : R\ny z : Γ\nhy : (y, z).fst ∈ support (↑(single a) r)\nhz : (y, z).snd ∈ support (↑(single b) s)\nh : ¬(y, z).fst + (y, z).snd = a + b\n⊢ coeff (↑(single a) r) (y, z).fst * coeff (↑(single b) s) (y, z).snd = 0",
"state_before": "case neg\nΓ : Type u_1\nR : Type u_2\ninst✝¹ : OrderedCancelAddCommMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\na b : Γ\nr s : R\nx : Γ\nh : ¬x = a + b\n⊢ ∀ (x_1 : Γ × Γ),\n x_1.fst ∈ support (↑(single a) r) ∧ x_1.snd ∈ support (↑(single b) s) ∧ x_1.fst + x_1.snd = x →\n coeff (↑(single a) r) x_1.fst * coeff (↑(single b) s) x_1.snd = 0",
"tactic": "rintro ⟨y, z⟩ ⟨hy, hz, rfl⟩"
},
{
"state_after": "case neg.mk.intro.intro\nΓ : Type u_1\nR : Type u_2\ninst✝¹ : OrderedCancelAddCommMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\na b : Γ\nr s : R\ny z : Γ\nhy : (y, z).fst ∈ support (↑(single a) r)\nhz : (y, z).snd ∈ support (↑(single b) s)\nh : ¬a + b = a + b\n⊢ coeff (↑(single a) r) (y, z).fst * coeff (↑(single b) s) (y, z).snd = 0",
"state_before": "case neg.mk.intro.intro\nΓ : Type u_1\nR : Type u_2\ninst✝¹ : OrderedCancelAddCommMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\na b : Γ\nr s : R\ny z : Γ\nhy : (y, z).fst ∈ support (↑(single a) r)\nhz : (y, z).snd ∈ support (↑(single b) s)\nh : ¬(y, z).fst + (y, z).snd = a + b\n⊢ coeff (↑(single a) r) (y, z).fst * coeff (↑(single b) s) (y, z).snd = 0",
"tactic": "rw [eq_of_mem_support_single hy, eq_of_mem_support_single hz] at h"
},
{
"state_after": "no goals",
"state_before": "case neg.mk.intro.intro\nΓ : Type u_1\nR : Type u_2\ninst✝¹ : OrderedCancelAddCommMonoid Γ\ninst✝ : NonUnitalNonAssocSemiring R\na b : Γ\nr s : R\ny z : Γ\nhy : (y, z).fst ∈ support (↑(single a) r)\nhz : (y, z).snd ∈ support (↑(single b) s)\nh : ¬a + b = a + b\n⊢ coeff (↑(single a) r) (y, z).fst * coeff (↑(single b) s) (y, z).snd = 0",
"tactic": "exact (h rfl).elim"
}
] |
[
945,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
935,
1
] |
Mathlib/Algebra/Module/Equiv.lean
|
LinearEquiv.symm_apply_apply
|
[] |
[
383,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
382,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
ContDiffAt.prod
|
[] |
[
540,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
537,
1
] |
Mathlib/Data/List/Cycle.lean
|
Cycle.chain_map
|
[
{
"state_after": "case nil\nα : Type u_2\nβ : Type u_1\nr : α → α → Prop\nf : β → α\ns : Cycle β\n⊢ Chain r (map f (Quotient.mk'' [])) ↔ Chain (fun a b => r (f a) (f b)) (Quotient.mk'' [])\n\ncase cons\nα : Type u_2\nβ : Type u_1\nr : α → α → Prop\nf : β → α\ns : Cycle β\na : β\nl : List β\n⊢ Chain r (map f (Quotient.mk'' (a :: l))) ↔ Chain (fun a b => r (f a) (f b)) (Quotient.mk'' (a :: l))",
"state_before": "α : Type u_2\nβ : Type u_1\nr : α → α → Prop\nf : β → α\ns : Cycle β\nl : List β\n⊢ Chain r (map f (Quotient.mk'' l)) ↔ Chain (fun a b => r (f a) (f b)) (Quotient.mk'' l)",
"tactic": "cases' l with a l"
},
{
"state_after": "case cons\nα : Type u_2\nβ : Type u_1\nr : α → α → Prop\nf : β → α\ns : Cycle β\na : β\nl : List β\n⊢ Chain r (map f (Quotient.mk'' (a :: l))) ↔ Chain (fun a b => r (f a) (f b)) (Quotient.mk'' (a :: l))",
"state_before": "case nil\nα : Type u_2\nβ : Type u_1\nr : α → α → Prop\nf : β → α\ns : Cycle β\n⊢ Chain r (map f (Quotient.mk'' [])) ↔ Chain (fun a b => r (f a) (f b)) (Quotient.mk'' [])\n\ncase cons\nα : Type u_2\nβ : Type u_1\nr : α → α → Prop\nf : β → α\ns : Cycle β\na : β\nl : List β\n⊢ Chain r (map f (Quotient.mk'' (a :: l))) ↔ Chain (fun a b => r (f a) (f b)) (Quotient.mk'' (a :: l))",
"tactic": "rfl"
},
{
"state_after": "case cons\nα : Type u_2\nβ : Type u_1\nr : α → α → Prop\nf : β → α\ns : Cycle β\na : β\nl : List β\n⊢ List.Chain r (f a) (List.map f l ++ [f a]) ↔ List.Chain (fun a b => r (f a) (f b)) a (l ++ [a])",
"state_before": "case cons\nα : Type u_2\nβ : Type u_1\nr : α → α → Prop\nf : β → α\ns : Cycle β\na : β\nl : List β\n⊢ Chain r (map f (Quotient.mk'' (a :: l))) ↔ Chain (fun a b => r (f a) (f b)) (Quotient.mk'' (a :: l))",
"tactic": "dsimp only [Chain, ← mk''_eq_coe, Quotient.liftOn'_mk'', Cycle.map, Quotient.map', Quot.map,\n Quotient.mk'', Quotient.liftOn', Quotient.liftOn, Quot.liftOn_mk, List.map]"
},
{
"state_after": "case cons\nα : Type u_2\nβ : Type u_1\nr : α → α → Prop\nf : β → α\ns : Cycle β\na : β\nl : List β\n⊢ List.Chain (fun a b => r (f a) (f b)) a (concat l a) ↔ List.Chain (fun a b => r (f a) (f b)) a (l ++ [a])",
"state_before": "case cons\nα : Type u_2\nβ : Type u_1\nr : α → α → Prop\nf : β → α\ns : Cycle β\na : β\nl : List β\n⊢ List.Chain r (f a) (List.map f l ++ [f a]) ↔ List.Chain (fun a b => r (f a) (f b)) a (l ++ [a])",
"tactic": "rw [← concat_eq_append, ← List.map_concat, List.chain_map f]"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u_2\nβ : Type u_1\nr : α → α → Prop\nf : β → α\ns : Cycle β\na : β\nl : List β\n⊢ List.Chain (fun a b => r (f a) (f b)) a (concat l a) ↔ List.Chain (fun a b => r (f a) (f b)) a (l ++ [a])",
"tactic": "simp"
}
] |
[
964,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
956,
1
] |
Mathlib/Data/Finset/NAry.lean
|
Finset.card_image₂_singleton_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nα' : Type ?u.51463\nβ : Type u_3\nβ' : Type ?u.51469\nγ : Type u_2\nγ' : Type ?u.51475\nδ : Type ?u.51478\nδ' : Type ?u.51481\nε : Type ?u.51484\nε' : Type ?u.51487\nζ : Type ?u.51490\nζ' : Type ?u.51493\nν : Type ?u.51496\ninst✝⁷ : DecidableEq α'\ninst✝⁶ : DecidableEq β'\ninst✝⁵ : DecidableEq γ\ninst✝⁴ : DecidableEq γ'\ninst✝³ : DecidableEq δ\ninst✝² : DecidableEq δ'\ninst✝¹ : DecidableEq ε\ninst✝ : DecidableEq ε'\nf f' : α → β → γ\ng g' : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\nhf : Injective fun a => f a b\n⊢ card (image₂ f s {b}) = card s",
"tactic": "rw [image₂_singleton_right, card_image_of_injective _ hf]"
}
] |
[
262,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
261,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
discreteTopology_iff_open_singleton_one
|
[] |
[
164,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
1
] |
Mathlib/Algebra/Associated.lean
|
Associates.mk_surjective
|
[] |
[
767,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
766,
1
] |
Mathlib/CategoryTheory/Limits/FullSubcategory.lean
|
CategoryTheory.Limits.ClosedUnderLimitsOfShape.limit
|
[] |
[
51,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
49,
1
] |
Mathlib/Data/Int/SuccPred.lean
|
Int.covby_iff_succ_eq
|
[] |
[
80,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
11
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearMap.copy_eq
|
[] |
[
486,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
485,
1
] |
Mathlib/Order/Concept.lean
|
Concept.swap_swap
|
[] |
[
384,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
383,
1
] |
Mathlib/Algebra/Hom/Equiv/Units/Basic.lean
|
Equiv.divLeft_eq_inv_trans_mulLeft
|
[] |
[
204,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
1
] |
Mathlib/MeasureTheory/Function/SimpleFunc.lean
|
MeasureTheory.SimpleFunc.extend_apply
|
[] |
[
377,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
375,
1
] |
Mathlib/Data/List/Basic.lean
|
List.map₂Right'_cons_cons
|
[] |
[
4027,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
4023,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
continuousWithinAt_inter
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.321718\nδ : Type ?u.321721\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nf : α → β\ns t : Set α\nx : α\nh : t ∈ 𝓝 x\n⊢ ContinuousWithinAt f (s ∩ t) x ↔ ContinuousWithinAt f s x",
"tactic": "simp [ContinuousWithinAt, nhdsWithin_restrict' s h]"
}
] |
[
713,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
711,
1
] |
Mathlib/Algebra/Order/SMul.lean
|
smul_lt_smul_of_pos
|
[] |
[
93,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
11
] |
Mathlib/Combinatorics/SimpleGraph/Clique.lean
|
SimpleGraph.CliqueFree.mono
|
[
{
"state_after": "α : Type u_1\nG H : SimpleGraph α\nm n : ℕ\ns✝ : Finset α\nh : m ≤ n\nhG : CliqueFree G m\ns : Finset α\nhs : IsNClique G n s\n⊢ False",
"state_before": "α : Type u_1\nG H : SimpleGraph α\nm n : ℕ\ns : Finset α\nh : m ≤ n\n⊢ CliqueFree G m → CliqueFree G n",
"tactic": "intro hG s hs"
},
{
"state_after": "case intro.intro\nα : Type u_1\nG H : SimpleGraph α\nm n : ℕ\ns✝ : Finset α\nh : m ≤ n\nhG : CliqueFree G m\ns : Finset α\nhs : IsNClique G n s\nt : Finset α\nhts : t ⊆ s\nht : Finset.card t = m\n⊢ False",
"state_before": "α : Type u_1\nG H : SimpleGraph α\nm n : ℕ\ns✝ : Finset α\nh : m ≤ n\nhG : CliqueFree G m\ns : Finset α\nhs : IsNClique G n s\n⊢ False",
"tactic": "obtain ⟨t, hts, ht⟩ := s.exists_smaller_set _ (h.trans hs.card_eq.ge)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\nG H : SimpleGraph α\nm n : ℕ\ns✝ : Finset α\nh : m ≤ n\nhG : CliqueFree G m\ns : Finset α\nhs : IsNClique G n s\nt : Finset α\nhts : t ⊆ s\nht : Finset.card t = m\n⊢ False",
"tactic": "exact hG _ ⟨hs.clique.subset hts, ht⟩"
}
] |
[
219,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Mathlib/Algebra/DirectLimit.lean
|
Ring.DirectLimit.of_f
|
[] |
[
410,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
409,
1
] |
Mathlib/Algebra/Algebra/Subalgebra/Basic.lean
|
Subalgebra.rangeS_le
|
[] |
[
112,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Combination.lean
|
Finset.weightedVSub_const_smul
|
[] |
[
364,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
362,
1
] |
Mathlib/Logic/Basic.lean
|
or_congr_left'
|
[] |
[
390,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
390,
1
] |
Mathlib/Topology/UniformSpace/UniformEmbedding.lean
|
closedEmbedding_of_spaced_out
|
[
{
"state_after": "α✝ : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : UniformSpace α✝\ninst✝³ : UniformSpace β\ninst✝² : UniformSpace γ\nα : Type u_1\ninst✝¹ : SeparatedSpace β\nf : α → β\ns : Set (β × β)\nhs : s ∈ 𝓤 β\nhf : Pairwise fun x y => ¬(f x, f y) ∈ s\ninst✝ : DiscreteTopology α\n⊢ ClosedEmbedding f",
"state_before": "α✝ : Type u\nβ : Type v\nγ : Type w\ninst✝⁵ : UniformSpace α✝\ninst✝⁴ : UniformSpace β\ninst✝³ : UniformSpace γ\nα : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : DiscreteTopology α\ninst✝ : SeparatedSpace β\nf : α → β\ns : Set (β × β)\nhs : s ∈ 𝓤 β\nhf : Pairwise fun x y => ¬(f x, f y) ∈ s\n⊢ ClosedEmbedding f",
"tactic": "rcases @DiscreteTopology.eq_bot α _ _ with rfl"
},
{
"state_after": "α✝ : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : UniformSpace α✝\ninst✝³ : UniformSpace β\ninst✝² : UniformSpace γ\nα : Type u_1\ninst✝¹ : SeparatedSpace β\nf : α → β\ns : Set (β × β)\nhs : s ∈ 𝓤 β\nhf : Pairwise fun x y => ¬(f x, f y) ∈ s\ninst✝ : DiscreteTopology α\nx✝ : UniformSpace α := ⊥\n⊢ ClosedEmbedding f",
"state_before": "α✝ : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : UniformSpace α✝\ninst✝³ : UniformSpace β\ninst✝² : UniformSpace γ\nα : Type u_1\ninst✝¹ : SeparatedSpace β\nf : α → β\ns : Set (β × β)\nhs : s ∈ 𝓤 β\nhf : Pairwise fun x y => ¬(f x, f y) ∈ s\ninst✝ : DiscreteTopology α\n⊢ ClosedEmbedding f",
"tactic": "let _ : UniformSpace α := ⊥"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : UniformSpace α✝\ninst✝³ : UniformSpace β\ninst✝² : UniformSpace γ\nα : Type u_1\ninst✝¹ : SeparatedSpace β\nf : α → β\ns : Set (β × β)\nhs : s ∈ 𝓤 β\nhf : Pairwise fun x y => ¬(f x, f y) ∈ s\ninst✝ : DiscreteTopology α\nx✝ : UniformSpace α := ⊥\n⊢ ClosedEmbedding f",
"tactic": "exact\n { (uniformEmbedding_of_spaced_out hs hf).embedding with\n closed_range := isClosed_range_of_spaced_out hs hf }"
}
] |
[
243,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Mathlib/Data/Nat/PartENat.lean
|
PartENat.toWithTop_zero'
|
[
{
"state_after": "no goals",
"state_before": "h : Decidable 0.Dom\n⊢ toWithTop 0 = 0",
"tactic": "convert toWithTop_zero"
}
] |
[
570,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
569,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.dart_edge_eq_iff
|
[
{
"state_after": "case mk.mk\nι : Sort ?u.97949\n𝕜 : Type ?u.97952\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne : Sym2 V\np : V × V\nhp : Adj G p.fst p.snd\nq : V × V\nhq : Adj G q.fst q.snd\n⊢ Dart.edge { toProd := p, is_adj := hp } = Dart.edge { toProd := q, is_adj := hq } ↔\n { toProd := p, is_adj := hp } = { toProd := q, is_adj := hq } ∨\n { toProd := p, is_adj := hp } = Dart.symm { toProd := q, is_adj := hq }",
"state_before": "ι : Sort ?u.97949\n𝕜 : Type ?u.97952\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne : Sym2 V\n⊢ ∀ (d₁ d₂ : Dart G), Dart.edge d₁ = Dart.edge d₂ ↔ d₁ = d₂ ∨ d₁ = Dart.symm d₂",
"tactic": "rintro ⟨p, hp⟩ ⟨q, hq⟩"
},
{
"state_after": "no goals",
"state_before": "case mk.mk\nι : Sort ?u.97949\n𝕜 : Type ?u.97952\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v w : V\ne : Sym2 V\np : V × V\nhp : Adj G p.fst p.snd\nq : V × V\nhq : Adj G q.fst q.snd\n⊢ Dart.edge { toProd := p, is_adj := hp } = Dart.edge { toProd := q, is_adj := hq } ↔\n { toProd := p, is_adj := hp } = { toProd := q, is_adj := hq } ∨\n { toProd := p, is_adj := hp } = Dart.symm { toProd := q, is_adj := hq }",
"tactic": "simp [Sym2.mk''_eq_mk''_iff, -Quotient.eq]"
}
] |
[
776,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
774,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
le_one_div
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.79768\nα : Type u_1\nβ : Type ?u.79774\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nha : 0 < a\nhb : 0 < b\n⊢ a ≤ 1 / b ↔ b ≤ 1 / a",
"tactic": "simpa using le_inv ha hb"
}
] |
[
442,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
442,
1
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
Basis.ext_linearIsometry
|
[] |
[
1196,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1194,
1
] |
Mathlib/RingTheory/OreLocalization/OreSet.lean
|
OreLocalization.ore_left_cancel
|
[] |
[
51,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
50,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.monomial_zero_left
|
[] |
[
491,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
490,
1
] |
Mathlib/Analysis/Calculus/Deriv/Linear.lean
|
ContinuousLinearMap.hasDerivAt
|
[] |
[
61,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
11
] |
Mathlib/Data/Real/EReal.lean
|
EReal.abs_def
|
[] |
[
1062,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1062,
1
] |
Mathlib/Topology/FiberBundle/Trivialization.lean
|
Pretrivialization.eqOn
|
[] |
[
128,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
11
] |
Mathlib/Combinatorics/Pigeonhole.lean
|
Finset.exists_card_fiber_le_of_card_le_mul
|
[] |
[
310,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
308,
1
] |
Mathlib/LinearAlgebra/BilinearForm.lean
|
BilinForm.ext_iff
|
[] |
[
165,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/Data/PNat/Factors.lean
|
PrimeMultiset.prod_ofPNatMultiset
|
[
{
"state_after": "v : Multiset ℕ+\nh : ∀ (p : ℕ+), p ∈ v → PNat.Prime p\n⊢ Multiset.prod (toPNatMultiset (ofPNatMultiset v h)) = Multiset.prod v",
"state_before": "v : Multiset ℕ+\nh : ∀ (p : ℕ+), p ∈ v → PNat.Prime p\n⊢ prod (ofPNatMultiset v h) = Multiset.prod v",
"tactic": "dsimp [prod]"
},
{
"state_after": "no goals",
"state_before": "v : Multiset ℕ+\nh : ∀ (p : ℕ+), p ∈ v → PNat.Prime p\n⊢ Multiset.prod (toPNatMultiset (ofPNatMultiset v h)) = Multiset.prod v",
"tactic": "rw [to_ofPNatMultiset]"
}
] |
[
189,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
187,
1
] |
Mathlib/MeasureTheory/Function/EssSup.lean
|
ENNReal.coe_essSup
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.5738175\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\nf✝ : α → ℝ≥0∞\nf : α → ℝ≥0\nhf : IsBoundedUnder (fun x x_1 => x ≤ x_1) (Measure.ae μ) f\nr : ℝ≥0∞\n⊢ (∀ (i : ℝ≥0), (i ∈ fun x => sets (map f (Measure.ae μ)) {x_1 | x_1 ≤ x}) → r ≤ ↑i) ↔\n ∀ (r_1 : ℝ≥0), (∀ᵐ (a : α) ∂μ, f a ≤ r_1) → r ≤ ↑r_1",
"state_before": "α : Type u_1\nβ : Type ?u.5738175\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\nf✝ : α → ℝ≥0∞\nf : α → ℝ≥0\nhf : IsBoundedUnder (fun x x_1 => x ≤ x_1) (Measure.ae μ) f\nr : ℝ≥0∞\n⊢ (r ≤\n ⨅ (a : ℝ≥0) (_ : a ∈ fun x => sets (map f (Measure.ae μ)) {x_1 | (fun x_2 => (fun x x_3 => x ≤ x_3) x_2 x) x_1}),\n ↑a) ↔\n r ≤ essSup (fun x => ↑(f x)) μ",
"tactic": "simp [essSup, limsup, limsSup, eventually_map, ENNReal.forall_ennreal]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.5738175\nm : MeasurableSpace α\nμ ν : MeasureTheory.Measure α\nf✝ : α → ℝ≥0∞\nf : α → ℝ≥0\nhf : IsBoundedUnder (fun x x_1 => x ≤ x_1) (Measure.ae μ) f\nr : ℝ≥0∞\n⊢ (∀ (i : ℝ≥0), (i ∈ fun x => sets (map f (Measure.ae μ)) {x_1 | x_1 ≤ x}) → r ≤ ↑i) ↔\n ∀ (r_1 : ℝ≥0), (∀ᵐ (a : α) ∂μ, f a ≤ r_1) → r ≤ ↑r_1",
"tactic": "rfl"
}
] |
[
341,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
337,
1
] |
Mathlib/CategoryTheory/Limits/ExactFunctor.lean
|
CategoryTheory.RightExactFunctor.forget_obj
|
[] |
[
175,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
174,
1
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.