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/PEquiv.lean
|
PEquiv.mem_single
|
[] |
[
347,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
346,
1
] |
Mathlib/RingTheory/Congruence.lean
|
RingCon.coe_zsmul
|
[] |
[
271,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
270,
1
] |
Mathlib/Data/Fintype/Card.lean
|
Finset.card_le_univ
|
[] |
[
267,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
266,
1
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
MonoidHom.dfinsupp_prod_apply
|
[] |
[
2263,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2261,
1
] |
Mathlib/Data/Finset/Image.lean
|
Finset.map_inj
|
[] |
[
170,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean
|
Class.mem_irrefl
|
[] |
[
1557,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1556,
1
] |
Mathlib/Data/Multiset/Pi.lean
|
Multiset.Pi.cons_ne
|
[] |
[
49,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
47,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.finite_range_findGreatest
|
[] |
[
1567,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1565,
1
] |
Mathlib/Analysis/Calculus/BumpFunctionInner.lean
|
ContDiffBump.integrable
|
[] |
[
515,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
514,
11
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.tsum_coe_ne_top_iff_summable
|
[
{
"state_after": "α : Type ?u.233118\nβ : Type u_1\nγ : Type ?u.233124\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf✝ g : α → ℝ≥0∞\nf : β → ℝ≥0\nh : (∑' (b : β), ↑(f b)) ≠ ⊤\n⊢ Summable f",
"state_before": "α : Type ?u.233118\nβ : Type u_1\nγ : Type ?u.233124\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf✝ g : α → ℝ≥0∞\nf : β → ℝ≥0\n⊢ (∑' (b : β), ↑(f b)) ≠ ⊤ ↔ Summable f",
"tactic": "refine ⟨fun h => ?_, fun h => ENNReal.coe_tsum h ▸ ENNReal.coe_ne_top⟩"
},
{
"state_after": "case intro\nα : Type ?u.233118\nβ : Type u_1\nγ : Type ?u.233124\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf✝ g : α → ℝ≥0∞\nf : β → ℝ≥0\na : ℝ≥0\nha : ↑a = ∑' (b : β), ↑(f b)\n⊢ Summable f",
"state_before": "α : Type ?u.233118\nβ : Type u_1\nγ : Type ?u.233124\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf✝ g : α → ℝ≥0∞\nf : β → ℝ≥0\nh : (∑' (b : β), ↑(f b)) ≠ ⊤\n⊢ Summable f",
"tactic": "lift ∑' b, (f b : ℝ≥0∞) to ℝ≥0 using h with a ha"
},
{
"state_after": "case intro\nα : Type ?u.233118\nβ : Type u_1\nγ : Type ?u.233124\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf✝ g : α → ℝ≥0∞\nf : β → ℝ≥0\na : ℝ≥0\nha : ↑a = ∑' (b : β), ↑(f b)\n⊢ HasSum (fun a => ↑(f a)) ↑a",
"state_before": "case intro\nα : Type ?u.233118\nβ : Type u_1\nγ : Type ?u.233124\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf✝ g : α → ℝ≥0∞\nf : β → ℝ≥0\na : ℝ≥0\nha : ↑a = ∑' (b : β), ↑(f b)\n⊢ Summable f",
"tactic": "refine' ⟨a, ENNReal.hasSum_coe.1 _⟩"
},
{
"state_after": "case intro\nα : Type ?u.233118\nβ : Type u_1\nγ : Type ?u.233124\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf✝ g : α → ℝ≥0∞\nf : β → ℝ≥0\na : ℝ≥0\nha : ↑a = ∑' (b : β), ↑(f b)\n⊢ HasSum (fun a => ↑(f a)) (∑' (b : β), ↑(f b))",
"state_before": "case intro\nα : Type ?u.233118\nβ : Type u_1\nγ : Type ?u.233124\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf✝ g : α → ℝ≥0∞\nf : β → ℝ≥0\na : ℝ≥0\nha : ↑a = ∑' (b : β), ↑(f b)\n⊢ HasSum (fun a => ↑(f a)) ↑a",
"tactic": "rw [ha]"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type ?u.233118\nβ : Type u_1\nγ : Type ?u.233124\na✝ b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\nf✝ g : α → ℝ≥0∞\nf : β → ℝ≥0\na : ℝ≥0\nha : ↑a = ∑' (b : β), ↑(f b)\n⊢ HasSum (fun a => ↑(f a)) (∑' (b : β), ↑(f b))",
"tactic": "exact ENNReal.summable.hasSum"
}
] |
[
789,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
784,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.zero_le
|
[
{
"state_after": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : FractionalIdeal S P\nx : P\nhx : x ∈ 0\n⊢ x ∈ I",
"state_before": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : FractionalIdeal S P\n⊢ 0 ≤ I",
"tactic": "intro x hx"
},
{
"state_after": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : FractionalIdeal S P\nx : P\nhx : x ∈ 0\n⊢ 0 ∈ I",
"state_before": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : FractionalIdeal S P\nx : P\nhx : x ∈ 0\n⊢ x ∈ I",
"tactic": "rw [(mem_zero_iff _).mp hx]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nloc : IsLocalization S P\nI : FractionalIdeal S P\nx : P\nhx : x ∈ 0\n⊢ 0 ∈ I",
"tactic": "exact zero_mem (I : Submodule R P)"
}
] |
[
400,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
396,
1
] |
Mathlib/Topology/Algebra/ConstMulAction.lean
|
isClosedMap_smul
|
[] |
[
262,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
261,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.OuterMeasure.ofFunction_le
|
[
{
"state_after": "case zero\nα : Type u_1\nm : Set α → ℝ≥0∞\nm_empty : m ∅ = 0\ns : Set α\nf : ℕ → Set α := fun i => Nat.casesOn i s fun x => ∅\n⊢ Nat.zero ≠ 0 → m (f Nat.zero) = 0\n\ncase succ\nα : Type u_1\nm : Set α → ℝ≥0∞\nm_empty : m ∅ = 0\ns : Set α\nf : ℕ → Set α := fun i => Nat.casesOn i s fun x => ∅\ni : ℕ\n⊢ Nat.succ i ≠ 0 → m (f (Nat.succ i)) = 0",
"state_before": "α : Type u_1\nm : Set α → ℝ≥0∞\nm_empty : m ∅ = 0\ns : Set α\nf : ℕ → Set α := fun i => Nat.casesOn i s fun x => ∅\n⊢ ∀ (b' : ℕ), b' ≠ 0 → m (f b') = 0",
"tactic": "rintro (_ | i)"
},
{
"state_after": "case succ\nα : Type u_1\nm : Set α → ℝ≥0∞\nm_empty : m ∅ = 0\ns : Set α\nf : ℕ → Set α := fun i => Nat.casesOn i s fun x => ∅\ni : ℕ\n⊢ Nat.succ i ≠ 0 → m (f (Nat.succ i)) = 0",
"state_before": "case zero\nα : Type u_1\nm : Set α → ℝ≥0∞\nm_empty : m ∅ = 0\ns : Set α\nf : ℕ → Set α := fun i => Nat.casesOn i s fun x => ∅\n⊢ Nat.zero ≠ 0 → m (f Nat.zero) = 0\n\ncase succ\nα : Type u_1\nm : Set α → ℝ≥0∞\nm_empty : m ∅ = 0\ns : Set α\nf : ℕ → Set α := fun i => Nat.casesOn i s fun x => ∅\ni : ℕ\n⊢ Nat.succ i ≠ 0 → m (f (Nat.succ i)) = 0",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case succ\nα : Type u_1\nm : Set α → ℝ≥0∞\nm_empty : m ∅ = 0\ns : Set α\nf : ℕ → Set α := fun i => Nat.casesOn i s fun x => ∅\ni : ℕ\n⊢ Nat.succ i ≠ 0 → m (f (Nat.succ i)) = 0",
"tactic": "simp [m_empty]"
}
] |
[
707,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
703,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Kernels.lean
|
CategoryTheory.Limits.kernelZeroIsoSource_hom
|
[] |
[
334,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
334,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.replicate_le_replicate
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.91808\nγ : Type ?u.91811\na : α\nk n : ℕ\n⊢ replicate k a ≤ replicate n a ↔ List.replicate k a <+ List.replicate n a",
"tactic": "rw [← replicate_le_coe, coe_replicate]"
}
] |
[
969,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
968,
1
] |
Mathlib/Data/Matrix/Basis.lean
|
Matrix.StdBasisMatrix.trace_zero
|
[
{
"state_after": "no goals",
"state_before": "l : Type ?u.27282\nm : Type ?u.27285\nn : Type u_1\nR : Type ?u.27291\nα : Type u_2\ninst✝⁴ : DecidableEq l\ninst✝³ : DecidableEq m\ninst✝² : DecidableEq n\ninst✝¹ : Semiring α\ni j : n\nc : α\ni' j' : n\ninst✝ : Fintype n\nh : j ≠ i\n⊢ trace (stdBasisMatrix i j c) = 0",
"tactic": "simp [trace, -diag_apply, h]"
}
] |
[
167,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
1
] |
Mathlib/Algebra/GCDMonoid/Basic.lean
|
dvd_normalize_iff
|
[] |
[
199,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.star_eq_conjTranspose
|
[] |
[
2343,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2342,
1
] |
Mathlib/Algebra/Group/Units.lean
|
Units.val_one
|
[] |
[
233,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
232,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.iInter_eq_empty_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.129049\nγ : Type ?u.129052\nι : Sort u_2\nι' : Sort ?u.129058\nι₂ : Sort ?u.129061\nκ : ι → Sort ?u.129066\nκ₁ : ι → Sort ?u.129071\nκ₂ : ι → Sort ?u.129076\nκ' : ι' → Sort ?u.129081\nf : ι → Set α\n⊢ (⋂ (i : ι), f i) = ∅ ↔ ∀ (x : α), ∃ i, ¬x ∈ f i",
"tactic": "simp [Set.eq_empty_iff_forall_not_mem]"
}
] |
[
1202,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1201,
1
] |
src/lean/Init/Data/Nat/Linear.lean
|
Nat.Linear.Poly.of_isNonZero
|
[
{
"state_after": "no goals",
"state_before": "ctx : Context\np : Poly\nh : isNonZero [] = true\n⊢ denote ctx [] > 0",
"tactic": "contradiction"
},
{
"state_after": "case inl\nctx : Context\np✝ : Poly\nk : Nat\nv : Var\np : List (Nat × Var)\nhe : (v == fixedVar) = true\nh : k > 0\n⊢ denote ctx p + k * Var.denote ctx v > 0\n\ncase inr\nctx : Context\np✝ : Poly\nk : Nat\nv : Var\np : List (Nat × Var)\nhe : ¬(v == fixedVar) = true\nh : isNonZero p = true\n⊢ denote ctx p + k * Var.denote ctx v > 0",
"state_before": "ctx : Context\np✝ : Poly\nk : Nat\nv : Var\np : List (Nat × Var)\nh : isNonZero ((k, v) :: p) = true\n⊢ denote ctx ((k, v) :: p) > 0",
"tactic": "by_cases he : v == fixedVar <;> simp [he, isNonZero] at h ⊢"
},
{
"state_after": "case inl\nctx : Context\np✝ : Poly\nk : Nat\nv : Var\np : List (Nat × Var)\nhe : (v == fixedVar) = true\nh : k > 0\n⊢ k + denote ctx p > 0",
"state_before": "case inl\nctx : Context\np✝ : Poly\nk : Nat\nv : Var\np : List (Nat × Var)\nhe : (v == fixedVar) = true\nh : k > 0\n⊢ denote ctx p + k * Var.denote ctx v > 0",
"tactic": "simp [eq_of_beq he, Var.denote]"
},
{
"state_after": "case inl.h\nctx : Context\np✝ : Poly\nk : Nat\nv : Var\np : List (Nat × Var)\nhe : (v == fixedVar) = true\nh : k > 0\n⊢ succ 0 ≤ k + denote ctx p",
"state_before": "case inl\nctx : Context\np✝ : Poly\nk : Nat\nv : Var\np : List (Nat × Var)\nhe : (v == fixedVar) = true\nh : k > 0\n⊢ k + denote ctx p > 0",
"tactic": "apply Nat.lt_of_succ_le"
},
{
"state_after": "no goals",
"state_before": "case inl.h\nctx : Context\np✝ : Poly\nk : Nat\nv : Var\np : List (Nat × Var)\nhe : (v == fixedVar) = true\nh : k > 0\n⊢ succ 0 ≤ k + denote ctx p",
"tactic": "exact Nat.le_trans h (Nat.le_add_right ..)"
},
{
"state_after": "case inr\nctx : Context\np✝ : Poly\nk : Nat\nv : Var\np : List (Nat × Var)\nhe : ¬(v == fixedVar) = true\nh : isNonZero p = true\nih : denote ctx p > 0\n⊢ denote ctx p + k * Var.denote ctx v > 0",
"state_before": "case inr\nctx : Context\np✝ : Poly\nk : Nat\nv : Var\np : List (Nat × Var)\nhe : ¬(v == fixedVar) = true\nh : isNonZero p = true\n⊢ denote ctx p + k * Var.denote ctx v > 0",
"tactic": "have ih := of_isNonZero ctx h"
},
{
"state_after": "no goals",
"state_before": "case inr\nctx : Context\np✝ : Poly\nk : Nat\nv : Var\np : List (Nat × Var)\nhe : ¬(v == fixedVar) = true\nh : isNonZero p = true\nih : denote ctx p > 0\n⊢ denote ctx p + k * Var.denote ctx v > 0",
"tactic": "exact Nat.le_trans ih (Nat.le_add_right ..)"
}
] |
[
637,
50
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
630,
1
] |
Mathlib/Data/Finsupp/ToDfinsupp.lean
|
sigmaFinsuppEquivDfinsupp_add
|
[
{
"state_after": "case h.h\nι : Type u_3\nR : Type ?u.76301\nM : Type ?u.76304\nη : ι → Type u_2\nN : Type u_1\ninst✝¹ : Semiring R\ninst✝ : AddZeroClass N\nf g : (i : ι) × η i →₀ N\ni✝ : ι\na✝ : η i✝\n⊢ ↑(↑(↑sigmaFinsuppEquivDfinsupp (f + g)) i✝) a✝ =\n ↑(↑(↑sigmaFinsuppEquivDfinsupp f + ↑sigmaFinsuppEquivDfinsupp g) i✝) a✝",
"state_before": "ι : Type u_3\nR : Type ?u.76301\nM : Type ?u.76304\nη : ι → Type u_2\nN : Type u_1\ninst✝¹ : Semiring R\ninst✝ : AddZeroClass N\nf g : (i : ι) × η i →₀ N\n⊢ ↑sigmaFinsuppEquivDfinsupp (f + g) = ↑sigmaFinsuppEquivDfinsupp f + ↑sigmaFinsuppEquivDfinsupp g",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h.h\nι : Type u_3\nR : Type ?u.76301\nM : Type ?u.76304\nη : ι → Type u_2\nN : Type u_1\ninst✝¹ : Semiring R\ninst✝ : AddZeroClass N\nf g : (i : ι) × η i →₀ N\ni✝ : ι\na✝ : η i✝\n⊢ ↑(↑(↑sigmaFinsuppEquivDfinsupp (f + g)) i✝) a✝ =\n ↑(↑(↑sigmaFinsuppEquivDfinsupp f + ↑sigmaFinsuppEquivDfinsupp g) i✝) a✝",
"tactic": "rfl"
}
] |
[
347,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
343,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.map_equiv_symm
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.284168\nι : Sort x\ne : α ≃ β\nf : Filter β\n⊢ map (↑e) (map (↑e.symm) f) = map (↑e) (comap (↑e) f)",
"tactic": "rw [map_map, e.self_comp_symm, map_id, map_comap_of_surjective e.surjective]"
}
] |
[
2533,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2531,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.monomial_add_erase
|
[
{
"state_after": "case ofFinsupp\nR : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ (↑(monomial n) (coeff { toFinsupp := toFinsupp✝ } n) + erase n { toFinsupp := toFinsupp✝ }).toFinsupp =\n { toFinsupp := toFinsupp✝ }.toFinsupp",
"state_before": "R : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np✝ q p : R[X]\nn : ℕ\n⊢ (↑(monomial n) (coeff p n) + erase n p).toFinsupp = p.toFinsupp",
"tactic": "rcases p with ⟨⟩"
},
{
"state_after": "case ofFinsupp\nR : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ Finsupp.single n\n ((match (motive := R[X] → ℕ → R) { toFinsupp := toFinsupp✝ } with\n | { toFinsupp := p } => ↑p)\n n) +\n Finsupp.erase n { toFinsupp := toFinsupp✝ }.toFinsupp =\n { toFinsupp := toFinsupp✝ }.toFinsupp",
"state_before": "case ofFinsupp\nR : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ (↑(monomial n) (coeff { toFinsupp := toFinsupp✝ } n) + erase n { toFinsupp := toFinsupp✝ }).toFinsupp =\n { toFinsupp := toFinsupp✝ }.toFinsupp",
"tactic": "rw [toFinsupp_add, toFinsupp_monomial, toFinsupp_erase, coeff]"
},
{
"state_after": "no goals",
"state_before": "case ofFinsupp\nR : Type u\na b : R\nm n✝ : ℕ\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ Finsupp.single n\n ((match (motive := R[X] → ℕ → R) { toFinsupp := toFinsupp✝ } with\n | { toFinsupp := p } => ↑p)\n n) +\n Finsupp.erase n { toFinsupp := toFinsupp✝ }.toFinsupp =\n { toFinsupp := toFinsupp✝ }.toFinsupp",
"tactic": "exact Finsupp.single_add_erase _ _"
}
] |
[
1044,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1040,
1
] |
Mathlib/Order/GaloisConnection.lean
|
Nat.galoisConnection_mul_div
|
[] |
[
456,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
455,
1
] |
Mathlib/Logic/Equiv/Defs.lean
|
Equiv.bijective
|
[] |
[
206,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
11
] |
Mathlib/Data/Nat/Parity.lean
|
Nat.even_or_odd'
|
[
{
"state_after": "no goals",
"state_before": "m n✝ n : ℕ\n⊢ ∃ k, n = 2 * k ∨ n = 2 * k + 1",
"tactic": "simpa only [← two_mul, exists_or, Odd, Even] using even_or_odd n"
}
] |
[
83,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
1
] |
Mathlib/Topology/PathConnected.lean
|
Path.target
|
[] |
[
129,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
11
] |
Mathlib/Algebra/Algebra/Basic.lean
|
algebraMap_int_eq
|
[] |
[
782,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
781,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
Subsemiring.mem_mk'
|
[] |
[
285,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
1
] |
Mathlib/Analysis/Complex/UnitDisc/Basic.lean
|
Complex.UnitDisc.conj_neg
|
[] |
[
233,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
232,
1
] |
Mathlib/Data/Matrix/Kronecker.lean
|
Matrix.one_kronecker_one
|
[] |
[
344,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
342,
1
] |
src/lean/Init/Data/Nat/Basic.lean
|
Nat.all_eq_allTR
|
[
{
"state_after": "no goals",
"state_before": "f : Nat → Bool\nn✝ n : Nat\n⊢ (all f n && allTR.loop f (0 + n) 0) = all f (0 + n)",
"tactic": "simp [allTR.loop]"
},
{
"state_after": "f : Nat → Bool\nn✝ m n : Nat\n⊢ (all f n && f n && allTR.loop f (succ (m + n)) m) = all f (succ (m + n))",
"state_before": "f : Nat → Bool\nn✝ m n : Nat\n⊢ (all f n && allTR.loop f (succ m + n) (succ m)) = all f (succ m + n)",
"tactic": "rw [allTR.loop, add_sub_self_left, ← Bool.and_assoc, succ_add]"
},
{
"state_after": "no goals",
"state_before": "f : Nat → Bool\nn✝ m n : Nat\n⊢ (all f n && f n && allTR.loop f (succ (m + n)) m) = all f (succ (m + n))",
"tactic": "exact go m (succ n)"
}
] |
[
758,
16
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
751,
10
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
dist_ne_zero
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nX : Type ?u.549105\nι : Type ?u.549108\ninst✝¹ : PseudoMetricSpace α\nγ : Type w\ninst✝ : MetricSpace γ\nx y : γ\n⊢ dist x y ≠ 0 ↔ x ≠ y",
"tactic": "simpa only [not_iff_not] using dist_eq_zero"
}
] |
[
2860,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2859,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.ite_empty
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns✝ s₁ s₂ t t₁ t₂ u s s' : Set α\n⊢ Set.ite ∅ s s' = s'",
"tactic": "simp [Set.ite]"
}
] |
[
2279,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2279,
1
] |
Mathlib/Combinatorics/Additive/Behrend.lean
|
Behrend.sum_sq_le_of_mem_box
|
[
{
"state_after": "α : Type ?u.502797\nβ : Type ?u.502800\nn d k N : ℕ\nx : Fin n → ℕ\nhx : ∀ (i : Fin n), x i < d\n⊢ ∑ i : Fin n, x i ^ 2 ≤ n * (d - 1) ^ 2",
"state_before": "α : Type ?u.502797\nβ : Type ?u.502800\nn d k N : ℕ\nx : Fin n → ℕ\nhx : x ∈ box n d\n⊢ ∑ i : Fin n, x i ^ 2 ≤ n * (d - 1) ^ 2",
"tactic": "rw [mem_box] at hx"
},
{
"state_after": "α : Type ?u.502797\nβ : Type ?u.502800\nn d k N : ℕ\nx : Fin n → ℕ\nhx : ∀ (i : Fin n), x i < d\nthis : ∀ (i : Fin n), x i ^ 2 ≤ (d - 1) ^ 2\n⊢ ∑ i : Fin n, x i ^ 2 ≤ n * (d - 1) ^ 2",
"state_before": "α : Type ?u.502797\nβ : Type ?u.502800\nn d k N : ℕ\nx : Fin n → ℕ\nhx : ∀ (i : Fin n), x i < d\n⊢ ∑ i : Fin n, x i ^ 2 ≤ n * (d - 1) ^ 2",
"tactic": "have : ∀ i, x i ^ 2 ≤ (d - 1) ^ 2 := fun i =>\n Nat.pow_le_pow_of_le_left (Nat.le_pred_of_lt (hx i)) _"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.502797\nβ : Type ?u.502800\nn d k N : ℕ\nx : Fin n → ℕ\nhx : ∀ (i : Fin n), x i < d\nthis : ∀ (i : Fin n), x i ^ 2 ≤ (d - 1) ^ 2\n⊢ ∑ i : Fin n, x i ^ 2 ≤ n * (d - 1) ^ 2",
"tactic": "exact (sum_le_card_nsmul univ _ _ fun i _ => this i).trans (by rw [card_fin, smul_eq_mul])"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.502797\nβ : Type ?u.502800\nn d k N : ℕ\nx : Fin n → ℕ\nhx : ∀ (i : Fin n), x i < d\nthis : ∀ (i : Fin n), x i ^ 2 ≤ (d - 1) ^ 2\n⊢ card univ • (d - 1) ^ 2 ≤ n * (d - 1) ^ 2",
"tactic": "rw [card_fin, smul_eq_mul]"
}
] |
[
203,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
199,
1
] |
Mathlib/Control/EquivFunctor.lean
|
EquivFunctor.mapEquiv_trans
|
[
{
"state_after": "no goals",
"state_before": "f : Type u₀ → Type u₁\ninst✝ : EquivFunctor f\nα β : Type u₀\ne : α ≃ β\nγ : Type u₀\nab : α ≃ β\nbc : β ≃ γ\nx : f α\n⊢ ↑((mapEquiv f ab).trans (mapEquiv f bc)) x = ↑(mapEquiv f (ab.trans bc)) x",
"tactic": "simp [mapEquiv, map_trans']"
}
] |
[
88,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/ArctanDeriv.lean
|
HasStrictDerivAt.arctan
|
[] |
[
136,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/Data/List/Count.lean
|
List.count_le_count_map
|
[
{
"state_after": "α : Type u_2\nl✝ : List α\ninst✝¹ : DecidableEq α\nβ : Type u_1\ninst✝ : DecidableEq β\nl : List α\nf : α → β\nx : α\n⊢ countp (fun x_1 => x_1 == x) l ≤ countp ((fun x_1 => x_1 == f x) ∘ f) l",
"state_before": "α : Type u_2\nl✝ : List α\ninst✝¹ : DecidableEq α\nβ : Type u_1\ninst✝ : DecidableEq β\nl : List α\nf : α → β\nx : α\n⊢ count x l ≤ count (f x) (map f l)",
"tactic": "rw [count, count, countp_map]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nl✝ : List α\ninst✝¹ : DecidableEq α\nβ : Type u_1\ninst✝ : DecidableEq β\nl : List α\nf : α → β\nx : α\n⊢ countp (fun x_1 => x_1 == x) l ≤ countp ((fun x_1 => x_1 == f x) ∘ f) l",
"tactic": "exact countp_mono_left <| by simp (config := {contextual := true})"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nl✝ : List α\ninst✝¹ : DecidableEq α\nβ : Type u_1\ninst✝ : DecidableEq β\nl : List α\nf : α → β\nx : α\n⊢ ∀ (x_1 : α), x_1 ∈ l → (x_1 == x) = true → ((fun x_2 => x_2 == f x) ∘ f) x_1 = true",
"tactic": "simp (config := {contextual := true})"
}
] |
[
334,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
331,
1
] |
Std/Data/List/Lemmas.lean
|
List.diff_sublist
|
[] |
[
1531,
34
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1525,
1
] |
Mathlib/NumberTheory/Liouville/Basic.lean
|
Liouville.exists_one_le_pow_mul_dist
|
[
{
"state_after": "Z : Type u_1\nN : Type u_2\nR : Type u_3\ninst✝ : PseudoMetricSpace R\nd : N → ℝ\nj : Z → N → R\nf : R → R\nα : R\nε M : ℝ\nd0 : ∀ (a : N), 1 ≤ d a\ne0 : 0 < ε\nB : ∀ ⦃y : R⦄, y ∈ closedBall α ε → dist (f α) (f y) ≤ dist α y * M\nL : ∀ ⦃z : Z⦄ ⦃a : N⦄, j z a ∈ closedBall α ε → 1 ≤ d a * dist (f α) (f (j z a))\nme0 : 0 < max (1 / ε) M\n⊢ ∃ A, 0 < A ∧ ∀ (z : Z) (a : N), 1 ≤ d a * (dist α (j z a) * A)",
"state_before": "Z : Type u_1\nN : Type u_2\nR : Type u_3\ninst✝ : PseudoMetricSpace R\nd : N → ℝ\nj : Z → N → R\nf : R → R\nα : R\nε M : ℝ\nd0 : ∀ (a : N), 1 ≤ d a\ne0 : 0 < ε\nB : ∀ ⦃y : R⦄, y ∈ closedBall α ε → dist (f α) (f y) ≤ dist α y * M\nL : ∀ ⦃z : Z⦄ ⦃a : N⦄, j z a ∈ closedBall α ε → 1 ≤ d a * dist (f α) (f (j z a))\n⊢ ∃ A, 0 < A ∧ ∀ (z : Z) (a : N), 1 ≤ d a * (dist α (j z a) * A)",
"tactic": "have me0 : 0 < max (1 / ε) M := lt_max_iff.mpr (Or.inl (one_div_pos.mpr e0))"
},
{
"state_after": "Z : Type u_1\nN : Type u_2\nR : Type u_3\ninst✝ : PseudoMetricSpace R\nd : N → ℝ\nj : Z → N → R\nf : R → R\nα : R\nε M : ℝ\nd0 : ∀ (a : N), 1 ≤ d a\ne0 : 0 < ε\nB : ∀ ⦃y : R⦄, y ∈ closedBall α ε → dist (f α) (f y) ≤ dist α y * M\nL : ∀ ⦃z : Z⦄ ⦃a : N⦄, j z a ∈ closedBall α ε → 1 ≤ d a * dist (f α) (f (j z a))\nme0 : 0 < max (1 / ε) M\nz : Z\na : N\n⊢ 1 ≤ d a * (dist α (j z a) * max (1 / ε) M)",
"state_before": "Z : Type u_1\nN : Type u_2\nR : Type u_3\ninst✝ : PseudoMetricSpace R\nd : N → ℝ\nj : Z → N → R\nf : R → R\nα : R\nε M : ℝ\nd0 : ∀ (a : N), 1 ≤ d a\ne0 : 0 < ε\nB : ∀ ⦃y : R⦄, y ∈ closedBall α ε → dist (f α) (f y) ≤ dist α y * M\nL : ∀ ⦃z : Z⦄ ⦃a : N⦄, j z a ∈ closedBall α ε → 1 ≤ d a * dist (f α) (f (j z a))\nme0 : 0 < max (1 / ε) M\n⊢ ∃ A, 0 < A ∧ ∀ (z : Z) (a : N), 1 ≤ d a * (dist α (j z a) * A)",
"tactic": "refine' ⟨max (1 / ε) M, me0, fun z a => _⟩"
},
{
"state_after": "case pos\nZ : Type u_1\nN : Type u_2\nR : Type u_3\ninst✝ : PseudoMetricSpace R\nd : N → ℝ\nj : Z → N → R\nf : R → R\nα : R\nε M : ℝ\nd0 : ∀ (a : N), 1 ≤ d a\ne0 : 0 < ε\nB : ∀ ⦃y : R⦄, y ∈ closedBall α ε → dist (f α) (f y) ≤ dist α y * M\nL : ∀ ⦃z : Z⦄ ⦃a : N⦄, j z a ∈ closedBall α ε → 1 ≤ d a * dist (f α) (f (j z a))\nme0 : 0 < max (1 / ε) M\nz : Z\na : N\ndm1 : 1 ≤ dist α (j z a) * max (1 / ε) M\n⊢ 1 ≤ d a * (dist α (j z a) * max (1 / ε) M)\n\ncase neg\nZ : Type u_1\nN : Type u_2\nR : Type u_3\ninst✝ : PseudoMetricSpace R\nd : N → ℝ\nj : Z → N → R\nf : R → R\nα : R\nε M : ℝ\nd0 : ∀ (a : N), 1 ≤ d a\ne0 : 0 < ε\nB : ∀ ⦃y : R⦄, y ∈ closedBall α ε → dist (f α) (f y) ≤ dist α y * M\nL : ∀ ⦃z : Z⦄ ⦃a : N⦄, j z a ∈ closedBall α ε → 1 ≤ d a * dist (f α) (f (j z a))\nme0 : 0 < max (1 / ε) M\nz : Z\na : N\ndm1 : ¬1 ≤ dist α (j z a) * max (1 / ε) M\n⊢ 1 ≤ d a * (dist α (j z a) * max (1 / ε) M)",
"state_before": "Z : Type u_1\nN : Type u_2\nR : Type u_3\ninst✝ : PseudoMetricSpace R\nd : N → ℝ\nj : Z → N → R\nf : R → R\nα : R\nε M : ℝ\nd0 : ∀ (a : N), 1 ≤ d a\ne0 : 0 < ε\nB : ∀ ⦃y : R⦄, y ∈ closedBall α ε → dist (f α) (f y) ≤ dist α y * M\nL : ∀ ⦃z : Z⦄ ⦃a : N⦄, j z a ∈ closedBall α ε → 1 ≤ d a * dist (f α) (f (j z a))\nme0 : 0 < max (1 / ε) M\nz : Z\na : N\n⊢ 1 ≤ d a * (dist α (j z a) * max (1 / ε) M)",
"tactic": "by_cases dm1 : 1 ≤ dist α (j z a) * max (1 / ε) M"
},
{
"state_after": "no goals",
"state_before": "case pos\nZ : Type u_1\nN : Type u_2\nR : Type u_3\ninst✝ : PseudoMetricSpace R\nd : N → ℝ\nj : Z → N → R\nf : R → R\nα : R\nε M : ℝ\nd0 : ∀ (a : N), 1 ≤ d a\ne0 : 0 < ε\nB : ∀ ⦃y : R⦄, y ∈ closedBall α ε → dist (f α) (f y) ≤ dist α y * M\nL : ∀ ⦃z : Z⦄ ⦃a : N⦄, j z a ∈ closedBall α ε → 1 ≤ d a * dist (f α) (f (j z a))\nme0 : 0 < max (1 / ε) M\nz : Z\na : N\ndm1 : 1 ≤ dist α (j z a) * max (1 / ε) M\n⊢ 1 ≤ d a * (dist α (j z a) * max (1 / ε) M)",
"tactic": "exact one_le_mul_of_one_le_of_one_le (d0 a) dm1"
},
{
"state_after": "case neg\nZ : Type u_1\nN : Type u_2\nR : Type u_3\ninst✝ : PseudoMetricSpace R\nd : N → ℝ\nj : Z → N → R\nf : R → R\nα : R\nε M : ℝ\nd0 : ∀ (a : N), 1 ≤ d a\ne0 : 0 < ε\nB : ∀ ⦃y : R⦄, y ∈ closedBall α ε → dist (f α) (f y) ≤ dist α y * M\nL : ∀ ⦃z : Z⦄ ⦃a : N⦄, j z a ∈ closedBall α ε → 1 ≤ d a * dist (f α) (f (j z a))\nme0 : 0 < max (1 / ε) M\nz : Z\na : N\ndm1 : ¬1 ≤ dist α (j z a) * max (1 / ε) M\nthis : j z a ∈ closedBall α ε\n⊢ 1 ≤ d a * (dist α (j z a) * max (1 / ε) M)",
"state_before": "case neg\nZ : Type u_1\nN : Type u_2\nR : Type u_3\ninst✝ : PseudoMetricSpace R\nd : N → ℝ\nj : Z → N → R\nf : R → R\nα : R\nε M : ℝ\nd0 : ∀ (a : N), 1 ≤ d a\ne0 : 0 < ε\nB : ∀ ⦃y : R⦄, y ∈ closedBall α ε → dist (f α) (f y) ≤ dist α y * M\nL : ∀ ⦃z : Z⦄ ⦃a : N⦄, j z a ∈ closedBall α ε → 1 ≤ d a * dist (f α) (f (j z a))\nme0 : 0 < max (1 / ε) M\nz : Z\na : N\ndm1 : ¬1 ≤ dist α (j z a) * max (1 / ε) M\n⊢ 1 ≤ d a * (dist α (j z a) * max (1 / ε) M)",
"tactic": "have : j z a ∈ closedBall α ε := by\n refine' mem_closedBall'.mp (le_trans _ ((one_div_le me0 e0).mpr (le_max_left _ _)))\n exact (le_div_iff me0).mpr (not_le.mp dm1).le"
},
{
"state_after": "case neg\nZ : Type u_1\nN : Type u_2\nR : Type u_3\ninst✝ : PseudoMetricSpace R\nd : N → ℝ\nj : Z → N → R\nf : R → R\nα : R\nε M : ℝ\nd0 : ∀ (a : N), 1 ≤ d a\ne0 : 0 < ε\nB : ∀ ⦃y : R⦄, y ∈ closedBall α ε → dist (f α) (f y) ≤ dist α y * M\nL : ∀ ⦃z : Z⦄ ⦃a : N⦄, j z a ∈ closedBall α ε → 1 ≤ d a * dist (f α) (f (j z a))\nme0 : 0 < max (1 / ε) M\nz : Z\na : N\ndm1 : ¬1 ≤ dist α (j z a) * max (1 / ε) M\nthis : j z a ∈ closedBall α ε\n⊢ d a * dist (f α) (f (j z a)) ≤ d a * (dist α (j z a) * max (1 / ε) M)",
"state_before": "case neg\nZ : Type u_1\nN : Type u_2\nR : Type u_3\ninst✝ : PseudoMetricSpace R\nd : N → ℝ\nj : Z → N → R\nf : R → R\nα : R\nε M : ℝ\nd0 : ∀ (a : N), 1 ≤ d a\ne0 : 0 < ε\nB : ∀ ⦃y : R⦄, y ∈ closedBall α ε → dist (f α) (f y) ≤ dist α y * M\nL : ∀ ⦃z : Z⦄ ⦃a : N⦄, j z a ∈ closedBall α ε → 1 ≤ d a * dist (f α) (f (j z a))\nme0 : 0 < max (1 / ε) M\nz : Z\na : N\ndm1 : ¬1 ≤ dist α (j z a) * max (1 / ε) M\nthis : j z a ∈ closedBall α ε\n⊢ 1 ≤ d a * (dist α (j z a) * max (1 / ε) M)",
"tactic": "refine' (L this).trans _"
},
{
"state_after": "case neg\nZ : Type u_1\nN : Type u_2\nR : Type u_3\ninst✝ : PseudoMetricSpace R\nd : N → ℝ\nj : Z → N → R\nf : R → R\nα : R\nε M : ℝ\nd0 : ∀ (a : N), 1 ≤ d a\ne0 : 0 < ε\nB : ∀ ⦃y : R⦄, y ∈ closedBall α ε → dist (f α) (f y) ≤ dist α y * M\nL : ∀ ⦃z : Z⦄ ⦃a : N⦄, j z a ∈ closedBall α ε → 1 ≤ d a * dist (f α) (f (j z a))\nme0 : 0 < max (1 / ε) M\nz : Z\na : N\ndm1 : ¬1 ≤ dist α (j z a) * max (1 / ε) M\nthis : j z a ∈ closedBall α ε\n⊢ dist α (j z a) * M ≤ dist α (j z a) * max (1 / ε) M",
"state_before": "case neg\nZ : Type u_1\nN : Type u_2\nR : Type u_3\ninst✝ : PseudoMetricSpace R\nd : N → ℝ\nj : Z → N → R\nf : R → R\nα : R\nε M : ℝ\nd0 : ∀ (a : N), 1 ≤ d a\ne0 : 0 < ε\nB : ∀ ⦃y : R⦄, y ∈ closedBall α ε → dist (f α) (f y) ≤ dist α y * M\nL : ∀ ⦃z : Z⦄ ⦃a : N⦄, j z a ∈ closedBall α ε → 1 ≤ d a * dist (f α) (f (j z a))\nme0 : 0 < max (1 / ε) M\nz : Z\na : N\ndm1 : ¬1 ≤ dist α (j z a) * max (1 / ε) M\nthis : j z a ∈ closedBall α ε\n⊢ d a * dist (f α) (f (j z a)) ≤ d a * (dist α (j z a) * max (1 / ε) M)",
"tactic": "refine' mul_le_mul_of_nonneg_left ((B this).trans _) (zero_le_one.trans (d0 a))"
},
{
"state_after": "no goals",
"state_before": "case neg\nZ : Type u_1\nN : Type u_2\nR : Type u_3\ninst✝ : PseudoMetricSpace R\nd : N → ℝ\nj : Z → N → R\nf : R → R\nα : R\nε M : ℝ\nd0 : ∀ (a : N), 1 ≤ d a\ne0 : 0 < ε\nB : ∀ ⦃y : R⦄, y ∈ closedBall α ε → dist (f α) (f y) ≤ dist α y * M\nL : ∀ ⦃z : Z⦄ ⦃a : N⦄, j z a ∈ closedBall α ε → 1 ≤ d a * dist (f α) (f (j z a))\nme0 : 0 < max (1 / ε) M\nz : Z\na : N\ndm1 : ¬1 ≤ dist α (j z a) * max (1 / ε) M\nthis : j z a ∈ closedBall α ε\n⊢ dist α (j z a) * M ≤ dist α (j z a) * max (1 / ε) M",
"tactic": "exact mul_le_mul_of_nonneg_left (le_max_right _ M) dist_nonneg"
},
{
"state_after": "Z : Type u_1\nN : Type u_2\nR : Type u_3\ninst✝ : PseudoMetricSpace R\nd : N → ℝ\nj : Z → N → R\nf : R → R\nα : R\nε M : ℝ\nd0 : ∀ (a : N), 1 ≤ d a\ne0 : 0 < ε\nB : ∀ ⦃y : R⦄, y ∈ closedBall α ε → dist (f α) (f y) ≤ dist α y * M\nL : ∀ ⦃z : Z⦄ ⦃a : N⦄, j z a ∈ closedBall α ε → 1 ≤ d a * dist (f α) (f (j z a))\nme0 : 0 < max (1 / ε) M\nz : Z\na : N\ndm1 : ¬1 ≤ dist α (j z a) * max (1 / ε) M\n⊢ dist α (j z a) ≤ 1 / max (1 / ε) M",
"state_before": "Z : Type u_1\nN : Type u_2\nR : Type u_3\ninst✝ : PseudoMetricSpace R\nd : N → ℝ\nj : Z → N → R\nf : R → R\nα : R\nε M : ℝ\nd0 : ∀ (a : N), 1 ≤ d a\ne0 : 0 < ε\nB : ∀ ⦃y : R⦄, y ∈ closedBall α ε → dist (f α) (f y) ≤ dist α y * M\nL : ∀ ⦃z : Z⦄ ⦃a : N⦄, j z a ∈ closedBall α ε → 1 ≤ d a * dist (f α) (f (j z a))\nme0 : 0 < max (1 / ε) M\nz : Z\na : N\ndm1 : ¬1 ≤ dist α (j z a) * max (1 / ε) M\n⊢ j z a ∈ closedBall α ε",
"tactic": "refine' mem_closedBall'.mp (le_trans _ ((one_div_le me0 e0).mpr (le_max_left _ _)))"
},
{
"state_after": "no goals",
"state_before": "Z : Type u_1\nN : Type u_2\nR : Type u_3\ninst✝ : PseudoMetricSpace R\nd : N → ℝ\nj : Z → N → R\nf : R → R\nα : R\nε M : ℝ\nd0 : ∀ (a : N), 1 ≤ d a\ne0 : 0 < ε\nB : ∀ ⦃y : R⦄, y ∈ closedBall α ε → dist (f α) (f y) ≤ dist α y * M\nL : ∀ ⦃z : Z⦄ ⦃a : N⦄, j z a ∈ closedBall α ε → 1 ≤ d a * dist (f α) (f (j z a))\nme0 : 0 < max (1 / ε) M\nz : Z\na : N\ndm1 : ¬1 ≤ dist α (j z a) * max (1 / ε) M\n⊢ dist α (j z a) ≤ 1 / max (1 / ε) M",
"tactic": "exact (le_div_iff me0).mpr (not_le.mp dm1).le"
}
] |
[
121,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean
|
CategoryTheory.Limits.HasZeroObject.zeroIsoInitial_inv
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁴ : Category C\nD : Type u'\ninst✝³ : Category D\ninst✝² : HasZeroObject C\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasInitial C\n⊢ zeroIsoInitial.inv = 0",
"tactic": "ext"
}
] |
[
324,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
324,
1
] |
Mathlib/Data/Countable/Basic.lean
|
Function.Embedding.countable
|
[] |
[
46,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
45,
11
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
LinearEquiv.coe_isometryOfInner
|
[] |
[
1310,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1309,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.neg_eq_neg_one_mul
|
[
{
"state_after": "n : Nat\n⊢ -↑(succ n) = -[n+1]",
"state_before": "n : Nat\n⊢ -↑(succ n) = -[1 * n+1]",
"tactic": "rw [Nat.one_mul]"
},
{
"state_after": "no goals",
"state_before": "n : Nat\n⊢ -↑(succ n) = -[n+1]",
"tactic": "rfl"
},
{
"state_after": "n : Nat\n⊢ - -[n+1] = ofNat (succ n)",
"state_before": "n : Nat\n⊢ - -[n+1] = ofNat (succ 0 * succ n)",
"tactic": "rw [Nat.one_mul]"
},
{
"state_after": "no goals",
"state_before": "n : Nat\n⊢ - -[n+1] = ofNat (succ n)",
"tactic": "rfl"
}
] |
[
531,
56
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
528,
11
] |
Mathlib/Data/Finsupp/Defs.lean
|
Finsupp.single_eq_pi_single
|
[] |
[
342,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
341,
1
] |
Mathlib/LinearAlgebra/Matrix/ToLin.lean
|
Algebra.leftMulMatrix_eq_repr_mul
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : Ring S\ninst✝² : Algebra R S\nm : Type u_3\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nb : Basis m R S\nx : S\ni j : m\n⊢ ↑(leftMulMatrix b) x i j = ↑(↑b.repr (x * ↑b j)) i",
"tactic": "rw [leftMulMatrix_apply, toMatrix_lmul' b x i j]"
}
] |
[
895,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
892,
1
] |
Mathlib/MeasureTheory/Measure/Portmanteau.lean
|
MeasureTheory.ProbabilityMeasure.tendsto_measure_of_null_frontier_of_tendsto
|
[
{
"state_after": "Ω✝ : Type ?u.40691\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : ProbabilityMeasure Ω\nμs : ι → ProbabilityMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nE : Set Ω\nE_nullbdry : (fun s => ENNReal.toNNReal (↑↑↑μ s)) (frontier E) = 0\nE_nullbdry' : ↑↑↑μ (frontier E) = 0\n⊢ Tendsto (fun i => (fun s => ENNReal.toNNReal (↑↑↑(μs i) s)) E) L (𝓝 ((fun s => ENNReal.toNNReal (↑↑↑μ s)) E))",
"state_before": "Ω✝ : Type ?u.40691\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : ProbabilityMeasure Ω\nμs : ι → ProbabilityMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nE : Set Ω\nE_nullbdry : (fun s => ENNReal.toNNReal (↑↑↑μ s)) (frontier E) = 0\n⊢ Tendsto (fun i => (fun s => ENNReal.toNNReal (↑↑↑(μs i) s)) E) L (𝓝 ((fun s => ENNReal.toNNReal (↑↑↑μ s)) E))",
"tactic": "have E_nullbdry' : (μ : Measure Ω) (frontier E) = 0 := by\n rw [← ProbabilityMeasure.ennreal_coeFn_eq_coeFn_toMeasure, E_nullbdry, ENNReal.coe_zero]"
},
{
"state_after": "Ω✝ : Type ?u.40691\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : ProbabilityMeasure Ω\nμs : ι → ProbabilityMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nE : Set Ω\nE_nullbdry : (fun s => ENNReal.toNNReal (↑↑↑μ s)) (frontier E) = 0\nE_nullbdry' : ↑↑↑μ (frontier E) = 0\nkey : Tendsto (fun i => ↑↑↑(μs i) E) L (𝓝 (↑↑↑μ E))\n⊢ Tendsto (fun i => (fun s => ENNReal.toNNReal (↑↑↑(μs i) s)) E) L (𝓝 ((fun s => ENNReal.toNNReal (↑↑↑μ s)) E))",
"state_before": "Ω✝ : Type ?u.40691\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : ProbabilityMeasure Ω\nμs : ι → ProbabilityMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nE : Set Ω\nE_nullbdry : (fun s => ENNReal.toNNReal (↑↑↑μ s)) (frontier E) = 0\nE_nullbdry' : ↑↑↑μ (frontier E) = 0\n⊢ Tendsto (fun i => (fun s => ENNReal.toNNReal (↑↑↑(μs i) s)) E) L (𝓝 ((fun s => ENNReal.toNNReal (↑↑↑μ s)) E))",
"tactic": "have key := ProbabilityMeasure.tendsto_measure_of_null_frontier_of_tendsto' μs_lim E_nullbdry'"
},
{
"state_after": "no goals",
"state_before": "Ω✝ : Type ?u.40691\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : ProbabilityMeasure Ω\nμs : ι → ProbabilityMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nE : Set Ω\nE_nullbdry : (fun s => ENNReal.toNNReal (↑↑↑μ s)) (frontier E) = 0\nE_nullbdry' : ↑↑↑μ (frontier E) = 0\nkey : Tendsto (fun i => ↑↑↑(μs i) E) L (𝓝 (↑↑↑μ E))\n⊢ Tendsto (fun i => (fun s => ENNReal.toNNReal (↑↑↑(μs i) s)) E) L (𝓝 ((fun s => ENNReal.toNNReal (↑↑↑μ s)) E))",
"tactic": "exact (ENNReal.tendsto_toNNReal (measure_ne_top (↑μ) E)).comp key"
},
{
"state_after": "no goals",
"state_before": "Ω✝ : Type ?u.40691\ninst✝³ : MeasurableSpace Ω✝\nΩ : Type u_1\nι : Type u_2\nL : Filter ι\ninst✝² : MeasurableSpace Ω\ninst✝¹ : PseudoEMetricSpace Ω\ninst✝ : OpensMeasurableSpace Ω\nμ : ProbabilityMeasure Ω\nμs : ι → ProbabilityMeasure Ω\nμs_lim : Tendsto μs L (𝓝 μ)\nE : Set Ω\nE_nullbdry : (fun s => ENNReal.toNNReal (↑↑↑μ s)) (frontier E) = 0\n⊢ ↑↑↑μ (frontier E) = 0",
"tactic": "rw [← ProbabilityMeasure.ennreal_coeFn_eq_coeFn_toMeasure, E_nullbdry, ENNReal.coe_zero]"
}
] |
[
427,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
420,
1
] |
Mathlib/Analysis/SpecialFunctions/Complex/Arg.lean
|
Complex.arg_eq_pi_div_two_iff
|
[
{
"state_after": "case pos\nz : ℂ\nh₀ : z = 0\n⊢ arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im\n\ncase neg\nz : ℂ\nh₀ : ¬z = 0\n⊢ arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im",
"state_before": "z : ℂ\n⊢ arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im",
"tactic": "by_cases h₀ : z = 0"
},
{
"state_after": "case neg.mp\nz : ℂ\nh₀ : ¬z = 0\n⊢ arg z = π / 2 → z.re = 0 ∧ 0 < z.im\n\ncase neg.mpr\nz : ℂ\nh₀ : ¬z = 0\n⊢ z.re = 0 ∧ 0 < z.im → arg z = π / 2",
"state_before": "case neg\nz : ℂ\nh₀ : ¬z = 0\n⊢ arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case pos\nz : ℂ\nh₀ : z = 0\n⊢ arg z = π / 2 ↔ z.re = 0 ∧ 0 < z.im",
"tactic": "simp [h₀, lt_irrefl, Real.pi_div_two_pos.ne]"
},
{
"state_after": "case neg.mp\nz : ℂ\nh₀ : ¬z = 0\nh : arg z = π / 2\n⊢ z.re = 0 ∧ 0 < z.im",
"state_before": "case neg.mp\nz : ℂ\nh₀ : ¬z = 0\n⊢ arg z = π / 2 → z.re = 0 ∧ 0 < z.im",
"tactic": "intro h"
},
{
"state_after": "case neg.mp\nz : ℂ\nh₀ : ¬z = 0\nh : arg z = π / 2\n⊢ (↑(↑abs z) * (cos ↑(π / 2) + sin ↑(π / 2) * I)).re = 0 ∧ 0 < (↑(↑abs z) * (cos ↑(π / 2) + sin ↑(π / 2) * I)).im",
"state_before": "case neg.mp\nz : ℂ\nh₀ : ¬z = 0\nh : arg z = π / 2\n⊢ z.re = 0 ∧ 0 < z.im",
"tactic": "rw [← abs_mul_cos_add_sin_mul_I z, h]"
},
{
"state_after": "no goals",
"state_before": "case neg.mp\nz : ℂ\nh₀ : ¬z = 0\nh : arg z = π / 2\n⊢ (↑(↑abs z) * (cos ↑(π / 2) + sin ↑(π / 2) * I)).re = 0 ∧ 0 < (↑(↑abs z) * (cos ↑(π / 2) + sin ↑(π / 2) * I)).im",
"tactic": "simp [h₀]"
},
{
"state_after": "case neg.mpr.mk\nx y : ℝ\nh₀ : ¬{ re := x, im := y } = 0\n⊢ { re := x, im := y }.re = 0 ∧ 0 < { re := x, im := y }.im → arg { re := x, im := y } = π / 2",
"state_before": "case neg.mpr\nz : ℂ\nh₀ : ¬z = 0\n⊢ z.re = 0 ∧ 0 < z.im → arg z = π / 2",
"tactic": "cases' z with x y"
},
{
"state_after": "case neg.mpr.mk.intro\ny : ℝ\nh₀ : ¬{ re := 0, im := y } = 0\nhy : 0 < y\n⊢ arg { re := 0, im := y } = π / 2",
"state_before": "case neg.mpr.mk\nx y : ℝ\nh₀ : ¬{ re := x, im := y } = 0\n⊢ { re := x, im := y }.re = 0 ∧ 0 < { re := x, im := y }.im → arg { re := x, im := y } = π / 2",
"tactic": "rintro ⟨rfl : x = 0, hy : 0 < y⟩"
},
{
"state_after": "no goals",
"state_before": "case neg.mpr.mk.intro\ny : ℝ\nh₀ : ¬{ re := 0, im := y } = 0\nhy : 0 < y\n⊢ arg { re := 0, im := y } = π / 2",
"tactic": "rw [← arg_I, ← arg_real_mul I hy, ofReal_mul', I_re, I_im, MulZeroClass.mul_zero, mul_one]"
}
] |
[
261,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
253,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
MulOpposite.nndist_unop
|
[] |
[
1667,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1667,
1
] |
Mathlib/Order/Antichain.lean
|
isStrongAntichain_insert
|
[] |
[
331,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
328,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.subNatNat_eq_coe
|
[
{
"state_after": "case hp\nm n : Nat\n⊢ ∀ (i n : Nat), ↑i = ↑(n + i) - ↑n\n\ncase hn\nm n : Nat\n⊢ ∀ (i m : Nat), -[i+1] = ↑m - ↑(m + i + 1)",
"state_before": "m n : Nat\n⊢ subNatNat m n = ↑m - ↑n",
"tactic": "apply subNatNat_elim m n fun m n i => i = m - n"
},
{
"state_after": "case hp\nm n✝ i n : Nat\n⊢ ↑i = ↑(n + i) - ↑n",
"state_before": "case hp\nm n : Nat\n⊢ ∀ (i n : Nat), ↑i = ↑(n + i) - ↑n",
"tactic": "intros i n"
},
{
"state_after": "no goals",
"state_before": "case hp\nm n✝ i n : Nat\n⊢ ↑i = ↑(n + i) - ↑n",
"tactic": "rw [Int.ofNat_add, Int.sub_eq_add_neg, Int.add_assoc, Int.add_left_comm,\n Int.add_right_neg, Int.add_zero]"
},
{
"state_after": "case hn\nm n✝ i n : Nat\n⊢ -[i+1] = ↑n - ↑(n + i + 1)",
"state_before": "case hn\nm n : Nat\n⊢ ∀ (i m : Nat), -[i+1] = ↑m - ↑(m + i + 1)",
"tactic": "intros i n"
},
{
"state_after": "case hn\nm n✝ i n : Nat\n⊢ -↑i + -↑1 = ↑n + -↑n + -↑i + -↑1",
"state_before": "case hn\nm n✝ i n : Nat\n⊢ -[i+1] = ↑n - ↑(n + i + 1)",
"tactic": "simp only [negSucc_coe, ofNat_add, Int.sub_eq_add_neg, Int.neg_add, ← Int.add_assoc]"
},
{
"state_after": "case hn\nm n✝ i n : Nat\n⊢ n ≤ n",
"state_before": "case hn\nm n✝ i n : Nat\n⊢ -↑i + -↑1 = ↑n + -↑n + -↑i + -↑1",
"tactic": "rw [← @Int.sub_eq_add_neg n, ← ofNat_sub, Nat.sub_self, ofNat_zero, Int.zero_add]"
},
{
"state_after": "no goals",
"state_before": "case hn\nm n✝ i n : Nat\n⊢ n ≤ n",
"tactic": "apply Nat.le_refl"
}
] |
[
512,
22
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
504,
11
] |
Mathlib/Algebra/Module/Submodule/Basic.lean
|
Submodule.toAddSubgroup_mono
|
[] |
[
555,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
554,
1
] |
Mathlib/CategoryTheory/Sites/Grothendieck.lean
|
CategoryTheory.GrothendieckTopology.trivial_eq_bot
|
[] |
[
320,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
319,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.map_mul
|
[
{
"state_after": "R : Type u\nS : Type v\nF : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\nr : R\nhri : r ∈ I\ns : R\nhsj : s ∈ J\n⊢ ↑f r * ↑f s ∈ map f I * map f J",
"state_before": "R : Type u\nS : Type v\nF : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\nr : R\nhri : r ∈ I\ns : R\nhsj : s ∈ J\n⊢ ↑f (r * s) ∈ map f I * map f J",
"tactic": "rw [_root_.map_mul]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nF : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\nr : R\nhri : r ∈ I\ns : R\nhsj : s ∈ J\n⊢ ↑f r * ↑f s ∈ map f I * map f J",
"tactic": "exact mul_mem_mul (mem_map_of_mem f hri) (mem_map_of_mem f hsj)"
},
{
"state_after": "R : Type u\nS : Type v\nF : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\ni : S\nx✝¹ : i ∈ ↑f '' ↑I\nr : R\nhri : r ∈ ↑I\nhfri : ↑f r = i\nj : S\nx✝ : j ∈ ↑f '' ↑J\ns : R\nhsj : s ∈ ↑J\nhfsj : ↑f s = j\n⊢ ↑f (r * s) ∈ ↑(map f (I * J))",
"state_before": "R : Type u\nS : Type v\nF : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\ni : S\nx✝¹ : i ∈ ↑f '' ↑I\nr : R\nhri : r ∈ ↑I\nhfri : ↑f r = i\nj : S\nx✝ : j ∈ ↑f '' ↑J\ns : R\nhsj : s ∈ ↑J\nhfsj : ↑f s = j\n⊢ ↑f r * ↑f s ∈ ↑(map f (I * J))",
"tactic": "rw [← _root_.map_mul]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nF : Type u_1\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nrc : RingHomClass F R S\nf : F\nI J : Ideal R\nK L : Ideal S\ni : S\nx✝¹ : i ∈ ↑f '' ↑I\nr : R\nhri : r ∈ ↑I\nhfri : ↑f r = i\nj : S\nx✝ : j ∈ ↑f '' ↑J\ns : R\nhsj : s ∈ ↑J\nhfsj : ↑f s = j\n⊢ ↑f (r * s) ∈ ↑(map f (I * J))",
"tactic": "exact mem_map_of_mem f (mul_mem_mul hri hsj)"
}
] |
[
1790,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1780,
1
] |
Mathlib/Algebra/Homology/HomologicalComplex.lean
|
ChainComplex.next_nat_succ
|
[] |
[
138,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
137,
1
] |
Mathlib/Order/Basic.lean
|
lt_of_eq_of_lt
|
[] |
[
187,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/GroupTheory/SpecificGroups/Dihedral.lean
|
DihedralGroup.one_def
|
[] |
[
102,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/Data/Real/NNReal.lean
|
NNReal.coe_image
|
[] |
[
473,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
471,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Real.expNear_succ
|
[
{
"state_after": "n : ℕ\nx r : ℝ\n⊢ x * x ^ n * ((↑(Nat.factorial n))⁻¹ * (↑n + 1)⁻¹) * r = x ^ n * (↑(Nat.factorial n))⁻¹ * (x * (↑n + 1)⁻¹ * r)",
"state_before": "n : ℕ\nx r : ℝ\n⊢ expNear (n + 1) x r = expNear n x (1 + x / (↑n + 1) * r)",
"tactic": "simp [expNear, range_succ, mul_add, add_left_comm, add_assoc, pow_succ, div_eq_mul_inv,\n mul_inv]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nx r : ℝ\n⊢ x * x ^ n * ((↑(Nat.factorial n))⁻¹ * (↑n + 1)⁻¹) * r = x ^ n * (↑(Nat.factorial n))⁻¹ * (x * (↑n + 1)⁻¹ * r)",
"tactic": "ac_rfl"
}
] |
[
1759,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1756,
1
] |
Mathlib/LinearAlgebra/Dual.lean
|
Basis.toDual_apply_right
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nM : Type v\nK : Type u₁\nV : Type u₂\nι : Type w\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : DecidableEq ι\nb : Basis ι R M\ni : ι\nm : M\n⊢ ↑(↑(toDual b) (↑b i)) m = ↑(↑b.repr m) i",
"tactic": "rw [← b.toDual_total_right, b.total_repr]"
}
] |
[
328,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
327,
1
] |
Mathlib/Analysis/Calculus/Taylor.lean
|
has_deriv_within_taylorWithinEval_at_Icc
|
[] |
[
230,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
224,
1
] |
Mathlib/Algebra/Group/Units.lean
|
Units.mul_left_inj
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\ninst✝ : Monoid α\na✝ b✝ c✝ u a : αˣ\nb c : α\nh : b * ↑a = c * ↑a\n⊢ b = c",
"tactic": "simpa only [mul_inv_cancel_right] using congr_arg (fun x : α => x * ↑(a⁻¹ : αˣ)) h"
}
] |
[
317,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
315,
1
] |
Mathlib/MeasureTheory/MeasurableSpace.lean
|
measurableSet_prod
|
[
{
"state_after": "case inl\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.84459\nδ : Type ?u.84462\nδ' : Type ?u.84465\nι : Sort uι\ns✝ t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ns : Set α\nt : Set β\nh : s ×ˢ t = ∅\n⊢ MeasurableSet (s ×ˢ t) ↔ MeasurableSet s ∧ MeasurableSet t ∨ s = ∅ ∨ t = ∅\n\ncase inr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.84459\nδ : Type ?u.84462\nδ' : Type ?u.84465\nι : Sort uι\ns✝ t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ns : Set α\nt : Set β\nh : Set.Nonempty (s ×ˢ t)\n⊢ MeasurableSet (s ×ˢ t) ↔ MeasurableSet s ∧ MeasurableSet t ∨ s = ∅ ∨ t = ∅",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.84459\nδ : Type ?u.84462\nδ' : Type ?u.84465\nι : Sort uι\ns✝ t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ns : Set α\nt : Set β\n⊢ MeasurableSet (s ×ˢ t) ↔ MeasurableSet s ∧ MeasurableSet t ∨ s = ∅ ∨ t = ∅",
"tactic": "cases' (s ×ˢ t).eq_empty_or_nonempty with h h"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.84459\nδ : Type ?u.84462\nδ' : Type ?u.84465\nι : Sort uι\ns✝ t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ns : Set α\nt : Set β\nh : s ×ˢ t = ∅\n⊢ MeasurableSet (s ×ˢ t) ↔ MeasurableSet s ∧ MeasurableSet t ∨ s = ∅ ∨ t = ∅",
"tactic": "simp [h, prod_eq_empty_iff.mp h]"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.84459\nδ : Type ?u.84462\nδ' : Type ?u.84465\nι : Sort uι\ns✝ t✝ u : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ns : Set α\nt : Set β\nh : Set.Nonempty (s ×ˢ t)\n⊢ MeasurableSet (s ×ˢ t) ↔ MeasurableSet s ∧ MeasurableSet t ∨ s = ∅ ∨ t = ∅",
"tactic": "simp [← not_nonempty_iff_eq_empty, prod_nonempty_iff.mp h, measurableSet_prod_of_nonempty h]"
}
] |
[
738,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
734,
1
] |
Mathlib/Analysis/NormedSpace/Multilinear.lean
|
MultilinearMap.mkContinuous_norm_le
|
[] |
[
768,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
766,
1
] |
Mathlib/Topology/Filter.lean
|
Filter.Tendsto.nhds
|
[] |
[
227,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
11
] |
Mathlib/RingTheory/Ideal/Quotient.lean
|
Ideal.fst_comp_quotientInfEquivQuotientProd
|
[
{
"state_after": "case h\nR : Type u\ninst✝ : CommRing R\nI✝ : Ideal R\na b : R\nS : Type v\nI J : Ideal R\ncoprime : I ⊔ J = ⊤\n⊢ RingHom.comp (RingHom.comp (RingHom.fst (R ⧸ I) (R ⧸ J)) ↑(quotientInfEquivQuotientProd I J coprime))\n (Quotient.mk (I ⊓ J)) =\n RingHom.comp (Quotient.factor (I ⊓ J) I (_ : I ⊓ J ≤ I)) (Quotient.mk (I ⊓ J))",
"state_before": "R : Type u\ninst✝ : CommRing R\nI✝ : Ideal R\na b : R\nS : Type v\nI J : Ideal R\ncoprime : I ⊔ J = ⊤\n⊢ RingHom.comp (RingHom.fst (R ⧸ I) (R ⧸ J)) ↑(quotientInfEquivQuotientProd I J coprime) =\n Quotient.factor (I ⊓ J) I (_ : I ⊓ J ≤ I)",
"tactic": "apply Quotient.ringHom_ext"
},
{
"state_after": "case h.a\nR : Type u\ninst✝ : CommRing R\nI✝ : Ideal R\na b : R\nS : Type v\nI J : Ideal R\ncoprime : I ⊔ J = ⊤\nx✝ : R\n⊢ ↑(RingHom.comp (RingHom.comp (RingHom.fst (R ⧸ I) (R ⧸ J)) ↑(quotientInfEquivQuotientProd I J coprime))\n (Quotient.mk (I ⊓ J)))\n x✝ =\n ↑(RingHom.comp (Quotient.factor (I ⊓ J) I (_ : I ⊓ J ≤ I)) (Quotient.mk (I ⊓ J))) x✝",
"state_before": "case h\nR : Type u\ninst✝ : CommRing R\nI✝ : Ideal R\na b : R\nS : Type v\nI J : Ideal R\ncoprime : I ⊔ J = ⊤\n⊢ RingHom.comp (RingHom.comp (RingHom.fst (R ⧸ I) (R ⧸ J)) ↑(quotientInfEquivQuotientProd I J coprime))\n (Quotient.mk (I ⊓ J)) =\n RingHom.comp (Quotient.factor (I ⊓ J) I (_ : I ⊓ J ≤ I)) (Quotient.mk (I ⊓ J))",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h.a\nR : Type u\ninst✝ : CommRing R\nI✝ : Ideal R\na b : R\nS : Type v\nI J : Ideal R\ncoprime : I ⊔ J = ⊤\nx✝ : R\n⊢ ↑(RingHom.comp (RingHom.comp (RingHom.fst (R ⧸ I) (R ⧸ J)) ↑(quotientInfEquivQuotientProd I J coprime))\n (Quotient.mk (I ⊓ J)))\n x✝ =\n ↑(RingHom.comp (Quotient.factor (I ⊓ J) I (_ : I ⊓ J ≤ I)) (Quotient.mk (I ⊓ J))) x✝",
"tactic": "rfl"
}
] |
[
534,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
530,
1
] |
Mathlib/Init/Data/Fin/Basic.lean
|
Fin.eq_of_veq
|
[
{
"state_after": "case refl\nn iv : ℕ\nilt₁ jlt₁ : iv < n\n⊢ { val := iv, isLt := ilt₁ } = { val := iv, isLt := jlt₁ }",
"state_before": "n iv : ℕ\nilt₁ : iv < n\njv : ℕ\njlt₁ : jv < n\nh : { val := iv, isLt := ilt₁ }.val = { val := jv, isLt := jlt₁ }.val\n⊢ { val := iv, isLt := ilt₁ } = { val := jv, isLt := jlt₁ }",
"tactic": "cases h"
},
{
"state_after": "no goals",
"state_before": "case refl\nn iv : ℕ\nilt₁ jlt₁ : iv < n\n⊢ { val := iv, isLt := ilt₁ } = { val := iv, isLt := jlt₁ }",
"tactic": "rfl"
}
] |
[
15,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
14,
1
] |
Mathlib/Topology/Algebra/OpenSubgroup.lean
|
OpenSubgroup.toOpens_top
|
[] |
[
162,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/Topology/ContinuousFunction/Algebra.lean
|
ContinuousMap.C_apply
|
[] |
[
724,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
723,
1
] |
Mathlib/Data/Polynomial/RingDivision.lean
|
Polynomial.rootSet_mapsTo
|
[
{
"state_after": "R : Type u\nS✝ : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝⁹ : CommRing R\ninst✝⁸ : IsDomain R\np✝ q : R[X]\ninst✝⁷ : CommRing T\np : T[X]\nS : Type u_1\nS' : Type u_2\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra T S\ninst✝³ : CommRing S'\ninst✝² : IsDomain S'\ninst✝¹ : Algebra T S'\ninst✝ : NoZeroSMulDivisors T S'\nf : S →ₐ[T] S'\nh₀ : map (algebraMap T S') p = 0\n⊢ map (algebraMap T S) p = 0",
"state_before": "R : Type u\nS✝ : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝⁹ : CommRing R\ninst✝⁸ : IsDomain R\np✝ q : R[X]\ninst✝⁷ : CommRing T\np : T[X]\nS : Type u_1\nS' : Type u_2\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra T S\ninst✝³ : CommRing S'\ninst✝² : IsDomain S'\ninst✝¹ : Algebra T S'\ninst✝ : NoZeroSMulDivisors T S'\nf : S →ₐ[T] S'\n⊢ Set.MapsTo (↑f) (rootSet p S) (rootSet p S')",
"tactic": "refine' rootSet_maps_to' (fun h₀ => _) f"
},
{
"state_after": "R : Type u\nS✝ : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝⁹ : CommRing R\ninst✝⁸ : IsDomain R\np q : R[X]\ninst✝⁷ : CommRing T\nS : Type u_1\nS' : Type u_2\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra T S\ninst✝³ : CommRing S'\ninst✝² : IsDomain S'\ninst✝¹ : Algebra T S'\ninst✝ : NoZeroSMulDivisors T S'\nf : S →ₐ[T] S'\nh₀ : map (algebraMap T S') 0 = 0\n⊢ map (algebraMap T S) 0 = 0",
"state_before": "R : Type u\nS✝ : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝⁹ : CommRing R\ninst✝⁸ : IsDomain R\np✝ q : R[X]\ninst✝⁷ : CommRing T\np : T[X]\nS : Type u_1\nS' : Type u_2\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra T S\ninst✝³ : CommRing S'\ninst✝² : IsDomain S'\ninst✝¹ : Algebra T S'\ninst✝ : NoZeroSMulDivisors T S'\nf : S →ₐ[T] S'\nh₀ : map (algebraMap T S') p = 0\n⊢ map (algebraMap T S) p = 0",
"tactic": "obtain rfl : p = 0 :=\n map_injective _ (NoZeroSMulDivisors.algebraMap_injective T S') (by rwa [Polynomial.map_zero])"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS✝ : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝⁹ : CommRing R\ninst✝⁸ : IsDomain R\np q : R[X]\ninst✝⁷ : CommRing T\nS : Type u_1\nS' : Type u_2\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra T S\ninst✝³ : CommRing S'\ninst✝² : IsDomain S'\ninst✝¹ : Algebra T S'\ninst✝ : NoZeroSMulDivisors T S'\nf : S →ₐ[T] S'\nh₀ : map (algebraMap T S') 0 = 0\n⊢ map (algebraMap T S) 0 = 0",
"tactic": "exact Polynomial.map_zero _"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS✝ : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝⁹ : CommRing R\ninst✝⁸ : IsDomain R\np✝ q : R[X]\ninst✝⁷ : CommRing T\np : T[X]\nS : Type u_1\nS' : Type u_2\ninst✝⁶ : CommRing S\ninst✝⁵ : IsDomain S\ninst✝⁴ : Algebra T S\ninst✝³ : CommRing S'\ninst✝² : IsDomain S'\ninst✝¹ : Algebra T S'\ninst✝ : NoZeroSMulDivisors T S'\nf : S →ₐ[T] S'\nh₀ : map (algebraMap T S') p = 0\n⊢ map (algebraMap T S') p = map (algebraMap T S') 0",
"tactic": "rwa [Polynomial.map_zero]"
}
] |
[
976,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
970,
1
] |
Mathlib/SetTheory/Ordinal/Exponential.lean
|
Ordinal.opow_zero
|
[
{
"state_after": "case pos\na : Ordinal\nh : a = 0\n⊢ a ^ 0 = 1\n\ncase neg\na : Ordinal\nh : ¬a = 0\n⊢ a ^ 0 = 1",
"state_before": "a : Ordinal\n⊢ a ^ 0 = 1",
"tactic": "by_cases h : a = 0"
},
{
"state_after": "no goals",
"state_before": "case pos\na : Ordinal\nh : a = 0\n⊢ a ^ 0 = 1",
"tactic": "simp only [opow_def, if_pos h, sub_zero]"
},
{
"state_after": "no goals",
"state_before": "case neg\na : Ordinal\nh : ¬a = 0\n⊢ a ^ 0 = 1",
"tactic": "simp only [opow_def, if_neg h, limitRecOn_zero]"
}
] |
[
56,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean
|
Real.continuousAt_rpow
|
[] |
[
260,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
258,
1
] |
Mathlib/Order/Monotone/Monovary.lean
|
antivaryOn_toDual_left
|
[] |
[
279,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
278,
1
] |
Mathlib/CategoryTheory/Subobject/Lattice.lean
|
CategoryTheory.Subobject.nontrivial_of_not_isZero
|
[] |
[
746,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
745,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.sup_add_nat
|
[] |
[
2514,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2513,
1
] |
Mathlib/Order/SymmDiff.lean
|
symmDiff_right_comm
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.72761\nα : Type u_1\nβ : Type ?u.72767\nπ : ι → Type ?u.72772\ninst✝ : GeneralizedBooleanAlgebra α\na b c d : α\n⊢ a ∆ b ∆ c = a ∆ c ∆ b",
"tactic": "simp_rw [symmDiff_assoc, symmDiff_comm]"
}
] |
[
489,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
489,
1
] |
Mathlib/CategoryTheory/Adjunction/Basic.lean
|
CategoryTheory.Adjunction.left_triangle
|
[
{
"state_after": "case w.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX' X : C\nY Y' : D\nx✝ : C\n⊢ (whiskerRight adj.unit F ≫ whiskerLeft F adj.counit).app x✝ = (𝟙 (𝟭 C ⋙ F)).app x✝",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX' X : C\nY Y' : D\n⊢ whiskerRight adj.unit F ≫ whiskerLeft F adj.counit = 𝟙 (𝟭 C ⋙ F)",
"tactic": "ext"
},
{
"state_after": "case w.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX' X : C\nY Y' : D\nx✝ : C\n⊢ F.map (adj.unit.app x✝) ≫ adj.counit.app (F.obj x✝) = 𝟙 (F.obj x✝)",
"state_before": "case w.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX' X : C\nY Y' : D\nx✝ : C\n⊢ (whiskerRight adj.unit F ≫ whiskerLeft F adj.counit).app x✝ = (𝟙 (𝟭 C ⋙ F)).app x✝",
"tactic": "dsimp"
},
{
"state_after": "case w.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX' X : C\nY Y' : D\nx✝ : C\n⊢ adj.unit.app x✝ = adj.unit.app x✝ ≫ G.map (𝟙 (F.obj x✝))",
"state_before": "case w.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX' X : C\nY Y' : D\nx✝ : C\n⊢ F.map (adj.unit.app x✝) ≫ adj.counit.app (F.obj x✝) = 𝟙 (F.obj x✝)",
"tactic": "erw [← adj.homEquiv_counit, Equiv.symm_apply_eq, adj.homEquiv_unit]"
},
{
"state_after": "no goals",
"state_before": "case w.h\nC : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nG : D ⥤ C\nadj : F ⊣ G\nX' X : C\nY Y' : D\nx✝ : C\n⊢ adj.unit.app x✝ = adj.unit.app x✝ ≫ G.map (𝟙 (F.obj x✝))",
"tactic": "simp"
}
] |
[
183,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
real_inner_eq_norm_add_mul_self_sub_norm_mul_self_sub_norm_mul_self_div_two
|
[] |
[
1449,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1446,
1
] |
Mathlib/Order/Hom/CompleteLattice.lean
|
sInfHom.copy_eq
|
[] |
[
429,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
428,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.le_sub_iff_add_le
|
[
{
"state_after": "no goals",
"state_before": "x y k : Nat\nh : k ≤ y\n⊢ x ≤ y - k ↔ x + k ≤ y",
"tactic": "rw [← Nat.add_sub_cancel x k, Nat.sub_le_sub_right_iff h, Nat.add_sub_cancel]"
}
] |
[
465,
80
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
464,
1
] |
Mathlib/Algebra/Algebra/Operations.lean
|
Submodule.comap_op_mul
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM✝ N✝ P Q : Submodule R A\nm n : A\nM N : Submodule R Aᵐᵒᵖ\n⊢ comap (↑(opLinearEquiv R)) (M * N) = comap (↑(opLinearEquiv R)) N * comap (↑(opLinearEquiv R)) M",
"tactic": "simp_rw [comap_equiv_eq_map_symm, map_unop_mul]"
}
] |
[
318,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
314,
1
] |
Mathlib/Logic/Equiv/Basic.lean
|
Equiv.uniqueProd_symm_apply
|
[] |
[
214,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/Data/ZMod/Basic.lean
|
ZMod.le_div_two_iff_lt_neg
|
[
{
"state_after": "n : ℕ\nhn : Fact (n % 2 = 1)\nx : ZMod n\nhx0 : x ≠ 0\nnpos : NeZero n\n⊢ val x ≤ n / 2 ↔ n / 2 < val (-x)",
"state_before": "n : ℕ\nhn : Fact (n % 2 = 1)\nx : ZMod n\nhx0 : x ≠ 0\n⊢ val x ≤ n / 2 ↔ n / 2 < val (-x)",
"tactic": "haveI npos : NeZero n :=\n ⟨by\n rintro rfl\n simp [fact_iff] at hn⟩"
},
{
"state_after": "n : ℕ\nhn : Fact (n % 2 = 1)\nx : ZMod n\nhx0 : x ≠ 0\nnpos : NeZero n\n_hn2 : n / 2 < n\n⊢ val x ≤ n / 2 ↔ n / 2 < val (-x)",
"state_before": "n : ℕ\nhn : Fact (n % 2 = 1)\nx : ZMod n\nhx0 : x ≠ 0\nnpos : NeZero n\n⊢ val x ≤ n / 2 ↔ n / 2 < val (-x)",
"tactic": "have _hn2 : (n : ℕ) / 2 < n :=\n Nat.div_lt_of_lt_mul ((lt_mul_iff_one_lt_left <| NeZero.pos n).2 (by decide))"
},
{
"state_after": "n : ℕ\nhn : Fact (n % 2 = 1)\nx : ZMod n\nhx0 : x ≠ 0\nnpos : NeZero n\n_hn2 : n / 2 < n\nhn2' : n - n / 2 = n / 2 + 1\n⊢ val x ≤ n / 2 ↔ n / 2 < val (-x)",
"state_before": "n : ℕ\nhn : Fact (n % 2 = 1)\nx : ZMod n\nhx0 : x ≠ 0\nnpos : NeZero n\n_hn2 : n / 2 < n\n⊢ val x ≤ n / 2 ↔ n / 2 < val (-x)",
"tactic": "have hn2' : (n : ℕ) - n / 2 = n / 2 + 1 := by\n conv =>\n lhs\n congr\n rw [← Nat.succ_sub_one n, Nat.succ_sub <| NeZero.pos n]\n rw [← Nat.two_mul_odd_div_two hn.1, two_mul, ← Nat.succ_add, add_tsub_cancel_right]"
},
{
"state_after": "n : ℕ\nhn : Fact (n % 2 = 1)\nx : ZMod n\nhx0 : x ≠ 0\nnpos : NeZero n\n_hn2 : n / 2 < n\nhn2' : n - n / 2 = n / 2 + 1\nhxn : n - val x < n\n⊢ val x ≤ n / 2 ↔ n / 2 < val (-x)",
"state_before": "n : ℕ\nhn : Fact (n % 2 = 1)\nx : ZMod n\nhx0 : x ≠ 0\nnpos : NeZero n\n_hn2 : n / 2 < n\nhn2' : n - n / 2 = n / 2 + 1\n⊢ val x ≤ n / 2 ↔ n / 2 < val (-x)",
"tactic": "have hxn : (n : ℕ) - x.val < n := by\n rw [tsub_lt_iff_tsub_lt x.val_le le_rfl, tsub_self]\n rw [← ZMod.nat_cast_zmod_val x] at hx0\n exact Nat.pos_of_ne_zero fun h => by simp [h] at hx0"
},
{
"state_after": "hn : Fact (0 % 2 = 1)\nx : ZMod 0\nhx0 : x ≠ 0\n⊢ False",
"state_before": "n : ℕ\nhn : Fact (n % 2 = 1)\nx : ZMod n\nhx0 : x ≠ 0\n⊢ n ≠ 0",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "hn : Fact (0 % 2 = 1)\nx : ZMod 0\nhx0 : x ≠ 0\n⊢ False",
"tactic": "simp [fact_iff] at hn"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nhn : Fact (n % 2 = 1)\nx : ZMod n\nhx0 : x ≠ 0\nnpos : NeZero n\n⊢ 1 < 2",
"tactic": "decide"
},
{
"state_after": "n : ℕ\nhn : Fact (n % 2 = 1)\nx : ZMod n\nhx0 : x ≠ 0\nnpos : NeZero n\n_hn2 : n / 2 < n\n⊢ Nat.succ (n - Nat.succ 0) - n / 2 = n / 2 + 1",
"state_before": "n : ℕ\nhn : Fact (n % 2 = 1)\nx : ZMod n\nhx0 : x ≠ 0\nnpos : NeZero n\n_hn2 : n / 2 < n\n⊢ n - n / 2 = n / 2 + 1",
"tactic": "conv =>\n lhs\n congr\n rw [← Nat.succ_sub_one n, Nat.succ_sub <| NeZero.pos n]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nhn : Fact (n % 2 = 1)\nx : ZMod n\nhx0 : x ≠ 0\nnpos : NeZero n\n_hn2 : n / 2 < n\n⊢ Nat.succ (n - Nat.succ 0) - n / 2 = n / 2 + 1",
"tactic": "rw [← Nat.two_mul_odd_div_two hn.1, two_mul, ← Nat.succ_add, add_tsub_cancel_right]"
},
{
"state_after": "n : ℕ\nhn : Fact (n % 2 = 1)\nx : ZMod n\nhx0 : x ≠ 0\nnpos : NeZero n\n_hn2 : n / 2 < n\nhn2' : n - n / 2 = n / 2 + 1\n⊢ 0 < val x",
"state_before": "n : ℕ\nhn : Fact (n % 2 = 1)\nx : ZMod n\nhx0 : x ≠ 0\nnpos : NeZero n\n_hn2 : n / 2 < n\nhn2' : n - n / 2 = n / 2 + 1\n⊢ n - val x < n",
"tactic": "rw [tsub_lt_iff_tsub_lt x.val_le le_rfl, tsub_self]"
},
{
"state_after": "n : ℕ\nhn : Fact (n % 2 = 1)\nx : ZMod n\nhx0 : ↑(val x) ≠ 0\nnpos : NeZero n\n_hn2 : n / 2 < n\nhn2' : n - n / 2 = n / 2 + 1\n⊢ 0 < val x",
"state_before": "n : ℕ\nhn : Fact (n % 2 = 1)\nx : ZMod n\nhx0 : x ≠ 0\nnpos : NeZero n\n_hn2 : n / 2 < n\nhn2' : n - n / 2 = n / 2 + 1\n⊢ 0 < val x",
"tactic": "rw [← ZMod.nat_cast_zmod_val x] at hx0"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nhn : Fact (n % 2 = 1)\nx : ZMod n\nhx0 : ↑(val x) ≠ 0\nnpos : NeZero n\n_hn2 : n / 2 < n\nhn2' : n - n / 2 = n / 2 + 1\n⊢ 0 < val x",
"tactic": "exact Nat.pos_of_ne_zero fun h => by simp [h] at hx0"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nhn : Fact (n % 2 = 1)\nx : ZMod n\nhx0 : ↑(val x) ≠ 0\nnpos : NeZero n\n_hn2 : n / 2 < n\nhn2' : n - n / 2 = n / 2 + 1\nh : val x = 0\n⊢ False",
"tactic": "simp [h] at hx0"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nhn : Fact (n % 2 = 1)\nx : ZMod n\nhx0 : x ≠ 0\nnpos : NeZero n\n_hn2 : n / 2 < n\nhn2' : n - n / 2 = n / 2 + 1\nhxn : n - val x < n\n⊢ val x ≤ n / 2 ↔ n / 2 < val (-x)",
"tactic": "conv =>\n rhs\n rw [← Nat.succ_le_iff, Nat.succ_eq_add_one, ← hn2', ← zero_add (-x), ← ZMod.nat_cast_self, ←\n sub_eq_add_neg, ← ZMod.nat_cast_zmod_val x, ← Nat.cast_sub x.val_le, ZMod.val_nat_cast,\n Nat.mod_eq_of_lt hxn, tsub_le_tsub_iff_left x.val_le]"
}
] |
[
814,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
792,
1
] |
Mathlib/Logic/Function/Conjugate.lean
|
Function.Semiconj.comp_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf fab : α → β\nfbc : β → γ\nga ga' : α → α\ngb gb' : β → β\ngc gc' : γ → γ\nhab : Semiconj fab ga gb\nhbc : Semiconj fbc gb gc\nx : α\n⊢ (fbc ∘ fab) (ga x) = gc ((fbc ∘ fab) x)",
"tactic": "simp only [comp_apply, hab.eq, hbc.eq]"
}
] |
[
58,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/Algebra/Order/AbsoluteValue.lean
|
AbsoluteValue.ne_zero_iff
|
[] |
[
123,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
11
] |
Mathlib/Order/Heyting/Hom.lean
|
map_bihimp
|
[
{
"state_after": "no goals",
"state_before": "F : Type u_3\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.33397\nδ : Type ?u.33400\ninst✝² : HeytingAlgebra α\ninst✝¹ : HeytingAlgebra β\ninst✝ : HeytingHomClass F α β\nf : F\na b : α\n⊢ ↑f (a ⇔ b) = ↑f a ⇔ ↑f b",
"tactic": "simp_rw [bihimp, map_inf, map_himp]"
}
] |
[
197,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
1
] |
Mathlib/Algebra/Group/Defs.lean
|
pow_zero
|
[] |
[
630,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
629,
1
] |
Mathlib/Order/Filter/CountableInter.lean
|
EventuallyLE.countable_bInter
|
[
{
"state_after": "ι✝ : Sort ?u.5259\nα : Type u_2\nβ : Type ?u.5265\nl : Filter α\ninst✝ : CountableInterFilter l\nι : Type u_1\nS : Set ι\nhS : Set.Countable S\ns t : (i : ι) → i ∈ S → Set α\nh : ∀ (i : ι) (hi : i ∈ S), s i hi ≤ᶠ[l] t i hi\n⊢ (⋂ (x : ↑S), s ↑x (_ : ↑x ∈ S)) ≤ᶠ[l] ⋂ (x : ↑S), t ↑x (_ : ↑x ∈ S)",
"state_before": "ι✝ : Sort ?u.5259\nα : Type u_2\nβ : Type ?u.5265\nl : Filter α\ninst✝ : CountableInterFilter l\nι : Type u_1\nS : Set ι\nhS : Set.Countable S\ns t : (i : ι) → i ∈ S → Set α\nh : ∀ (i : ι) (hi : i ∈ S), s i hi ≤ᶠ[l] t i hi\n⊢ (⋂ (i : ι) (h : i ∈ S), s i h) ≤ᶠ[l] ⋂ (i : ι) (h : i ∈ S), t i h",
"tactic": "simp only [biInter_eq_iInter]"
},
{
"state_after": "ι✝ : Sort ?u.5259\nα : Type u_2\nβ : Type ?u.5265\nl : Filter α\ninst✝ : CountableInterFilter l\nι : Type u_1\nS : Set ι\nhS : Set.Countable S\ns t : (i : ι) → i ∈ S → Set α\nh : ∀ (i : ι) (hi : i ∈ S), s i hi ≤ᶠ[l] t i hi\nthis : Encodable ↑S\n⊢ (⋂ (x : ↑S), s ↑x (_ : ↑x ∈ S)) ≤ᶠ[l] ⋂ (x : ↑S), t ↑x (_ : ↑x ∈ S)",
"state_before": "ι✝ : Sort ?u.5259\nα : Type u_2\nβ : Type ?u.5265\nl : Filter α\ninst✝ : CountableInterFilter l\nι : Type u_1\nS : Set ι\nhS : Set.Countable S\ns t : (i : ι) → i ∈ S → Set α\nh : ∀ (i : ι) (hi : i ∈ S), s i hi ≤ᶠ[l] t i hi\n⊢ (⋂ (x : ↑S), s ↑x (_ : ↑x ∈ S)) ≤ᶠ[l] ⋂ (x : ↑S), t ↑x (_ : ↑x ∈ S)",
"tactic": "haveI := hS.toEncodable"
},
{
"state_after": "no goals",
"state_before": "ι✝ : Sort ?u.5259\nα : Type u_2\nβ : Type ?u.5265\nl : Filter α\ninst✝ : CountableInterFilter l\nι : Type u_1\nS : Set ι\nhS : Set.Countable S\ns t : (i : ι) → i ∈ S → Set α\nh : ∀ (i : ι) (hi : i ∈ S), s i hi ≤ᶠ[l] t i hi\nthis : Encodable ↑S\n⊢ (⋂ (x : ↑S), s ↑x (_ : ↑x ∈ S)) ≤ᶠ[l] ⋂ (x : ↑S), t ↑x (_ : ↑x ∈ S)",
"tactic": "exact EventuallyLE.countable_iInter fun i => h i i.2"
}
] |
[
120,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Data/List/Infix.lean
|
List.subset_insert
|
[] |
[
498,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
497,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
extChartAt_model_space_eq_id
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nM : Type ?u.234410\nH : Type ?u.234413\nE' : Type ?u.234416\nM' : Type ?u.234419\nH' : Type ?u.234422\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝⁵ : NormedAddCommGroup E'\ninst✝⁴ : NormedSpace 𝕜 E'\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx✝ : M\ns t : Set M\ninst✝¹ : ChartedSpace H M\ninst✝ : ChartedSpace H' M'\nx : E\n⊢ extChartAt 𝓘(𝕜, E) x = LocalEquiv.refl E",
"tactic": "simp only [mfld_simps]"
}
] |
[
1283,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1282,
1
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.restrictScalars_closedBall
|
[] |
[
1098,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1096,
1
] |
Mathlib/LinearAlgebra/Prod.lean
|
Submodule.prod_comap_inr
|
[
{
"state_after": "case h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.303059\nM₆ : Type ?u.303062\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq : Submodule R M₂\nx✝ : M₂\n⊢ x✝ ∈ comap (inr R M M₂) (prod p q) ↔ x✝ ∈ q",
"state_before": "R : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.303059\nM₆ : Type ?u.303062\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq : Submodule R M₂\n⊢ comap (inr R M M₂) (prod p q) = q",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u\nK : Type u'\nM : Type v\nV : Type v'\nM₂ : Type w\nV₂ : Type w'\nM₃ : Type y\nV₃ : Type y'\nM₄ : Type z\nι : Type x\nM₅ : Type ?u.303059\nM₆ : Type ?u.303062\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M\ninst✝ : Module R M₂\np : Submodule R M\nq : Submodule R M₂\nx✝ : M₂\n⊢ x✝ ∈ comap (inr R M M₂) (prod p q) ↔ x✝ ∈ q",
"tactic": "simp"
}
] |
[
575,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
575,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.mk_add_moveLeft_inl
|
[] |
[
1503,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1500,
1
] |
Mathlib/Data/List/Basic.lean
|
List.replicate_add
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.17053\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nm n : ℕ\na : α\n⊢ replicate (m + n) a = replicate m a ++ replicate n a",
"tactic": "induction m <;> simp [*, zero_add, succ_add, replicate]"
}
] |
[
441,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
440,
1
] |
Mathlib/Algebra/FreeMonoid/Basic.lean
|
FreeMonoid.toList_of_mul
|
[] |
[
150,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
IsCompact.compl_mem_sets
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.205\nπ : ι → Type ?u.210\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nhs : IsCompact s\nf : Filter α\nhf : ¬sᶜ ∈ f\n⊢ ∃ a, a ∈ s ∧ ¬sᶜ ∈ 𝓝 a ⊓ f",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.205\nπ : ι → Type ?u.210\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nhs : IsCompact s\nf : Filter α\nhf : ∀ (a : α), a ∈ s → sᶜ ∈ 𝓝 a ⊓ f\n⊢ sᶜ ∈ f",
"tactic": "contrapose! hf"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.205\nπ : ι → Type ?u.210\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nhs : IsCompact s\nf : Filter α\nhf : NeBot (f ⊓ 𝓟 s)\n⊢ ∃ a, a ∈ s ∧ NeBot (𝓝 a ⊓ (f ⊓ 𝓟 s))",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.205\nπ : ι → Type ?u.210\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nhs : IsCompact s\nf : Filter α\nhf : ¬sᶜ ∈ f\n⊢ ∃ a, a ∈ s ∧ ¬sᶜ ∈ 𝓝 a ⊓ f",
"tactic": "simp only [not_mem_iff_inf_principal_compl, compl_compl, inf_assoc] at hf ⊢"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.205\nπ : ι → Type ?u.210\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nhs : IsCompact s\nf : Filter α\nhf : NeBot (f ⊓ 𝓟 s)\n⊢ ∃ a, a ∈ s ∧ NeBot (𝓝 a ⊓ (f ⊓ 𝓟 s))",
"tactic": "exact @hs _ hf inf_le_right"
}
] |
[
81,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
properSpace_of_compact_closedBall_of_le
|
[] |
[
2166,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2163,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.diagonal_transpose
|
[
{
"state_after": "case a.h\nl : Type ?u.49070\nm : Type ?u.49073\nn : Type u_1\no : Type ?u.49079\nm' : o → Type ?u.49084\nn' : o → Type ?u.49089\nR : Type ?u.49092\nS : Type ?u.49095\nα : Type v\nβ : Type w\nγ : Type ?u.49102\ninst✝¹ : DecidableEq n\ninst✝ : Zero α\nv : n → α\ni j : n\n⊢ (diagonal v)ᵀ i j = diagonal v i j",
"state_before": "l : Type ?u.49070\nm : Type ?u.49073\nn : Type u_1\no : Type ?u.49079\nm' : o → Type ?u.49084\nn' : o → Type ?u.49089\nR : Type ?u.49092\nS : Type ?u.49095\nα : Type v\nβ : Type w\nγ : Type ?u.49102\ninst✝¹ : DecidableEq n\ninst✝ : Zero α\nv : n → α\n⊢ (diagonal v)ᵀ = diagonal v",
"tactic": "ext i j"
},
{
"state_after": "case pos\nl : Type ?u.49070\nm : Type ?u.49073\nn : Type u_1\no : Type ?u.49079\nm' : o → Type ?u.49084\nn' : o → Type ?u.49089\nR : Type ?u.49092\nS : Type ?u.49095\nα : Type v\nβ : Type w\nγ : Type ?u.49102\ninst✝¹ : DecidableEq n\ninst✝ : Zero α\nv : n → α\ni j : n\nh : i = j\n⊢ (diagonal v)ᵀ i j = diagonal v i j\n\ncase neg\nl : Type ?u.49070\nm : Type ?u.49073\nn : Type u_1\no : Type ?u.49079\nm' : o → Type ?u.49084\nn' : o → Type ?u.49089\nR : Type ?u.49092\nS : Type ?u.49095\nα : Type v\nβ : Type w\nγ : Type ?u.49102\ninst✝¹ : DecidableEq n\ninst✝ : Zero α\nv : n → α\ni j : n\nh : ¬i = j\n⊢ (diagonal v)ᵀ i j = diagonal v i j",
"state_before": "case a.h\nl : Type ?u.49070\nm : Type ?u.49073\nn : Type u_1\no : Type ?u.49079\nm' : o → Type ?u.49084\nn' : o → Type ?u.49089\nR : Type ?u.49092\nS : Type ?u.49095\nα : Type v\nβ : Type w\nγ : Type ?u.49102\ninst✝¹ : DecidableEq n\ninst✝ : Zero α\nv : n → α\ni j : n\n⊢ (diagonal v)ᵀ i j = diagonal v i j",
"tactic": "by_cases h : i = j"
},
{
"state_after": "no goals",
"state_before": "case pos\nl : Type ?u.49070\nm : Type ?u.49073\nn : Type u_1\no : Type ?u.49079\nm' : o → Type ?u.49084\nn' : o → Type ?u.49089\nR : Type ?u.49092\nS : Type ?u.49095\nα : Type v\nβ : Type w\nγ : Type ?u.49102\ninst✝¹ : DecidableEq n\ninst✝ : Zero α\nv : n → α\ni j : n\nh : i = j\n⊢ (diagonal v)ᵀ i j = diagonal v i j",
"tactic": "simp [h, transpose]"
},
{
"state_after": "no goals",
"state_before": "case neg\nl : Type ?u.49070\nm : Type ?u.49073\nn : Type u_1\no : Type ?u.49079\nm' : o → Type ?u.49084\nn' : o → Type ?u.49089\nR : Type ?u.49092\nS : Type ?u.49095\nα : Type v\nβ : Type w\nγ : Type ?u.49102\ninst✝¹ : DecidableEq n\ninst✝ : Zero α\nv : n → α\ni j : n\nh : ¬i = j\n⊢ (diagonal v)ᵀ i j = diagonal v i j",
"tactic": "simp [h, transpose, diagonal_apply_ne' _ h]"
}
] |
[
469,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
465,
1
] |
Mathlib/Combinatorics/Quiver/Path.lean
|
Quiver.Path.toList_inj
|
[] |
[
200,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
199,
1
] |
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