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/Geometry/Euclidean/Angle/Unoriented/Affine.lean
|
EuclideanGeometry.right_dist_ne_zero_of_angle_eq_pi
|
[] |
[
217,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Mathlib/Topology/Homotopy/Basic.lean
|
ContinuousMap.Homotopic.piMap
|
[] |
[
403,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
400,
11
] |
Mathlib/Order/Filter/Extr.lean
|
IsMinOn.comp_mono
|
[] |
[
343,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
341,
1
] |
Mathlib/FieldTheory/Finite/Basic.lean
|
FiniteField.expand_card
|
[
{
"state_after": "case intro\nK : Type u_1\nR : Type ?u.820624\ninst✝³ : Field K\ninst✝² : Fintype K\np✝ : ℕ\ninst✝¹ : Fact (Nat.Prime p✝)\ninst✝ : Algebra (ZMod p✝) K\nf : K[X]\np : ℕ\nhp : CharP K p\n⊢ ↑(expand K q) f = f ^ q",
"state_before": "K : Type u_1\nR : Type ?u.820624\ninst✝³ : Field K\ninst✝² : Fintype K\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : Algebra (ZMod p) K\nf : K[X]\n⊢ ↑(expand K q) f = f ^ q",
"tactic": "cases' CharP.exists K with p hp"
},
{
"state_after": "case intro\nK : Type u_1\nR : Type ?u.820624\ninst✝³ : Field K\ninst✝² : Fintype K\np✝ : ℕ\ninst✝¹ : Fact (Nat.Prime p✝)\ninst✝ : Algebra (ZMod p✝) K\nf : K[X]\np : ℕ\nhp : CharP K p\nthis : CharP K p := hp\n⊢ ↑(expand K q) f = f ^ q",
"state_before": "case intro\nK : Type u_1\nR : Type ?u.820624\ninst✝³ : Field K\ninst✝² : Fintype K\np✝ : ℕ\ninst✝¹ : Fact (Nat.Prime p✝)\ninst✝ : Algebra (ZMod p✝) K\nf : K[X]\np : ℕ\nhp : CharP K p\n⊢ ↑(expand K q) f = f ^ q",
"tactic": "letI := hp"
},
{
"state_after": "case intro.intro.mk.intro\nK : Type u_1\nR : Type ?u.820624\ninst✝³ : Field K\ninst✝² : Fintype K\np✝ : ℕ\ninst✝¹ : Fact (Nat.Prime p✝)\ninst✝ : Algebra (ZMod p✝) K\nf : K[X]\np : ℕ\nhp✝ : CharP K p\nthis : CharP K p := hp✝\nn : ℕ\nnpos : 0 < n\nhp : Nat.Prime p\nhn : q = p ^ ↑{ val := n, property := npos }\n⊢ ↑(expand K q) f = f ^ q",
"state_before": "case intro\nK : Type u_1\nR : Type ?u.820624\ninst✝³ : Field K\ninst✝² : Fintype K\np✝ : ℕ\ninst✝¹ : Fact (Nat.Prime p✝)\ninst✝ : Algebra (ZMod p✝) K\nf : K[X]\np : ℕ\nhp : CharP K p\nthis : CharP K p := hp\n⊢ ↑(expand K q) f = f ^ q",
"tactic": "rcases FiniteField.card K p with ⟨⟨n, npos⟩, ⟨hp, hn⟩⟩"
},
{
"state_after": "case intro.intro.mk.intro\nK : Type u_1\nR : Type ?u.820624\ninst✝³ : Field K\ninst✝² : Fintype K\np✝ : ℕ\ninst✝¹ : Fact (Nat.Prime p✝)\ninst✝ : Algebra (ZMod p✝) K\nf : K[X]\np : ℕ\nhp✝ : CharP K p\nthis✝ : CharP K p := hp✝\nn : ℕ\nnpos : 0 < n\nhp : Nat.Prime p\nhn : q = p ^ ↑{ val := n, property := npos }\nthis : Fact (Nat.Prime p)\n⊢ ↑(expand K q) f = f ^ q",
"state_before": "case intro.intro.mk.intro\nK : Type u_1\nR : Type ?u.820624\ninst✝³ : Field K\ninst✝² : Fintype K\np✝ : ℕ\ninst✝¹ : Fact (Nat.Prime p✝)\ninst✝ : Algebra (ZMod p✝) K\nf : K[X]\np : ℕ\nhp✝ : CharP K p\nthis : CharP K p := hp✝\nn : ℕ\nnpos : 0 < n\nhp : Nat.Prime p\nhn : q = p ^ ↑{ val := n, property := npos }\n⊢ ↑(expand K q) f = f ^ q",
"tactic": "haveI : Fact p.Prime := ⟨hp⟩"
},
{
"state_after": "case intro.intro.mk.intro\nK : Type u_1\nR : Type ?u.820624\ninst✝³ : Field K\ninst✝² : Fintype K\np✝ : ℕ\ninst✝¹ : Fact (Nat.Prime p✝)\ninst✝ : Algebra (ZMod p✝) K\nf : K[X]\np : ℕ\nhp✝ : CharP K p\nthis✝ : CharP K p := hp✝\nn : ℕ\nnpos : 0 < n\nhp : Nat.Prime p\nhn : q = p ^ n\nthis : Fact (Nat.Prime p)\n⊢ ↑(expand K q) f = f ^ q",
"state_before": "case intro.intro.mk.intro\nK : Type u_1\nR : Type ?u.820624\ninst✝³ : Field K\ninst✝² : Fintype K\np✝ : ℕ\ninst✝¹ : Fact (Nat.Prime p✝)\ninst✝ : Algebra (ZMod p✝) K\nf : K[X]\np : ℕ\nhp✝ : CharP K p\nthis✝ : CharP K p := hp✝\nn : ℕ\nnpos : 0 < n\nhp : Nat.Prime p\nhn : q = p ^ ↑{ val := n, property := npos }\nthis : Fact (Nat.Prime p)\n⊢ ↑(expand K q) f = f ^ q",
"tactic": "dsimp at hn"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.mk.intro\nK : Type u_1\nR : Type ?u.820624\ninst✝³ : Field K\ninst✝² : Fintype K\np✝ : ℕ\ninst✝¹ : Fact (Nat.Prime p✝)\ninst✝ : Algebra (ZMod p✝) K\nf : K[X]\np : ℕ\nhp✝ : CharP K p\nthis✝ : CharP K p := hp✝\nn : ℕ\nnpos : 0 < n\nhp : Nat.Prime p\nhn : q = p ^ n\nthis : Fact (Nat.Prime p)\n⊢ ↑(expand K q) f = f ^ q",
"tactic": "rw [hn, ← map_expand_pow_char, frobenius_pow hn, RingHom.one_def, map_id]"
}
] |
[
301,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
295,
1
] |
Mathlib/MeasureTheory/Integral/FundThmCalculus.lean
|
intervalIntegral.derivWithin_integral_of_tendsto_ae_right
|
[] |
[
926,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
921,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.ConnectedComponent.exists
|
[] |
[
2035,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2033,
11
] |
Mathlib/Topology/Homeomorph.lean
|
Homeomorph.comp_continuousWithinAt_iff
|
[] |
[
431,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
429,
1
] |
Mathlib/Topology/Constructions.lean
|
nhdsWithin_subtype_eq_bot_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.11692\nδ : Type ?u.11695\nε : Type ?u.11698\nζ : Type ?u.11701\ninst✝ : TopologicalSpace α\ns t : Set α\nx : ↑s\n⊢ 𝓝[Subtype.val ⁻¹' t] x = ⊥ ↔ 𝓝[t] ↑x ⊓ 𝓟 s = ⊥",
"tactic": "rw [inf_principal_eq_bot_iff_comap, nhdsWithin, nhdsWithin, comap_inf, comap_principal,\n nhds_induced]"
}
] |
[
235,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
232,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.SameCycle.of_pow
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.204487\nα : Type u_1\nβ : Type ?u.204493\nf g : Perm α\np : α → Prop\nx y z : α\nn : ℕ\nx✝ : SameCycle (f ^ n) x y\nm : ℤ\nh : ↑((f ^ n) ^ m) x = y\n⊢ ↑(f ^ (↑n * m)) x = y",
"tactic": "simp [zpow_mul, h]"
}
] |
[
216,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsBigOWith.prod_left
|
[] |
[
988,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
986,
1
] |
Mathlib/Order/Filter/Pointwise.lean
|
Filter.vsub_bot
|
[] |
[
1103,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1102,
1
] |
Mathlib/CategoryTheory/EqToHom.lean
|
CategoryTheory.Functor.postcomp_map_hEq'
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\nE : Type u₃\ninst✝ : Category E\nF G : C ⥤ D\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\nH : D ⥤ E\nhobj : ∀ (X : C), F.obj X = G.obj X\nhmap : ∀ {X Y : C} (f : X ⟶ Y), HEq (F.map f) (G.map f)\n⊢ HEq ((F ⋙ H).map f) ((G ⋙ H).map f)",
"tactic": "rw [Functor.hext hobj fun _ _ => hmap]"
}
] |
[
256,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
254,
1
] |
Mathlib/Algebra/GCDMonoid/Basic.lean
|
gcd_dvd_gcd_mul_right
|
[] |
[
488,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
487,
1
] |
Mathlib/RingTheory/Localization/Basic.lean
|
IsLocalization.isDomain_localization
|
[] |
[
1263,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1261,
1
] |
Mathlib/Data/Matrix/Block.lean
|
Matrix.fromBlocks_multiply
|
[
{
"state_after": "case a.h\nl : Type u_1\nm : Type u_2\nn : Type u_4\no : Type u_5\np : Type u_6\nq : Type u_7\nm' : o → Type ?u.37702\nn' : o → Type ?u.37707\np' : o → Type ?u.37712\nR : Type ?u.37715\nS : Type ?u.37718\nα : Type u_3\nβ : Type ?u.37724\ninst✝² : Fintype l\ninst✝¹ : Fintype m\ninst✝ : NonUnitalNonAssocSemiring α\nA : Matrix n l α\nB : Matrix n m α\nC : Matrix o l α\nD : Matrix o m α\nA' : Matrix l p α\nB' : Matrix l q α\nC' : Matrix m p α\nD' : Matrix m q α\ni : n ⊕ o\nj : p ⊕ q\n⊢ (fromBlocks A B C D ⬝ fromBlocks A' B' C' D') i j =\n fromBlocks (A ⬝ A' + B ⬝ C') (A ⬝ B' + B ⬝ D') (C ⬝ A' + D ⬝ C') (C ⬝ B' + D ⬝ D') i j",
"state_before": "l : Type u_1\nm : Type u_2\nn : Type u_4\no : Type u_5\np : Type u_6\nq : Type u_7\nm' : o → Type ?u.37702\nn' : o → Type ?u.37707\np' : o → Type ?u.37712\nR : Type ?u.37715\nS : Type ?u.37718\nα : Type u_3\nβ : Type ?u.37724\ninst✝² : Fintype l\ninst✝¹ : Fintype m\ninst✝ : NonUnitalNonAssocSemiring α\nA : Matrix n l α\nB : Matrix n m α\nC : Matrix o l α\nD : Matrix o m α\nA' : Matrix l p α\nB' : Matrix l q α\nC' : Matrix m p α\nD' : Matrix m q α\n⊢ fromBlocks A B C D ⬝ fromBlocks A' B' C' D' =\n fromBlocks (A ⬝ A' + B ⬝ C') (A ⬝ B' + B ⬝ D') (C ⬝ A' + D ⬝ C') (C ⬝ B' + D ⬝ D')",
"tactic": "ext i j"
},
{
"state_after": "no goals",
"state_before": "case a.h\nl : Type u_1\nm : Type u_2\nn : Type u_4\no : Type u_5\np : Type u_6\nq : Type u_7\nm' : o → Type ?u.37702\nn' : o → Type ?u.37707\np' : o → Type ?u.37712\nR : Type ?u.37715\nS : Type ?u.37718\nα : Type u_3\nβ : Type ?u.37724\ninst✝² : Fintype l\ninst✝¹ : Fintype m\ninst✝ : NonUnitalNonAssocSemiring α\nA : Matrix n l α\nB : Matrix n m α\nC : Matrix o l α\nD : Matrix o m α\nA' : Matrix l p α\nB' : Matrix l q α\nC' : Matrix m p α\nD' : Matrix m q α\ni : n ⊕ o\nj : p ⊕ q\n⊢ (fromBlocks A B C D ⬝ fromBlocks A' B' C' D') i j =\n fromBlocks (A ⬝ A' + B ⬝ C') (A ⬝ B' + B ⬝ D') (C ⬝ A' + D ⬝ C') (C ⬝ B' + D ⬝ D') i j",
"tactic": "rcases i with ⟨⟩ <;> rcases j with ⟨⟩ <;> simp only [fromBlocks, mul_apply, of_apply,\n Sum.elim_inr, Fintype.sum_sum_type, Sum.elim_inl, add_apply]"
}
] |
[
257,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
250,
1
] |
Std/Data/List/Basic.lean
|
List.intersperse_eq_intersperseTR
|
[
{
"state_after": "case h.h.h\nα : Type u_1\nsep : α\nl : List α\n⊢ intersperse sep l = intersperseTR sep l",
"state_before": "⊢ @intersperse = @intersperseTR",
"tactic": "funext α sep l"
},
{
"state_after": "case h.h.h\nα : Type u_1\nsep : α\nl : List α\n⊢ intersperse sep l =\n match l with\n | [] => []\n | [x] => [x]\n | x :: y :: xs => x :: sep :: y :: foldr (fun a r => sep :: a :: r) [] xs",
"state_before": "case h.h.h\nα : Type u_1\nsep : α\nl : List α\n⊢ intersperse sep l = intersperseTR sep l",
"tactic": "simp [intersperseTR]"
},
{
"state_after": "no goals",
"state_before": "case h.h.h\nα : Type u_1\nsep : α\nl : List α\n⊢ intersperse sep l =\n match l with\n | [] => []\n | [x] => [x]\n | x :: y :: xs => x :: sep :: y :: foldr (fun a r => sep :: a :: r) [] xs",
"tactic": "match l with\n| [] | [_] => rfl\n| x::y::xs => simp [intersperse]; induction xs generalizing y <;> simp [*]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nsep : α\nl : List α\nhead✝ : α\n⊢ intersperse sep [head✝] =\n match [head✝] with\n | [] => []\n | [x] => [x]\n | x :: y :: xs => x :: sep :: y :: foldr (fun a r => sep :: a :: r) [] xs",
"tactic": "rfl"
},
{
"state_after": "α : Type u_1\nsep : α\nl : List α\nx y : α\nxs : List α\n⊢ intersperse sep (y :: xs) = y :: foldr (fun a r => sep :: a :: r) [] xs",
"state_before": "α : Type u_1\nsep : α\nl : List α\nx y : α\nxs : List α\n⊢ intersperse sep (x :: y :: xs) =\n match x :: y :: xs with\n | [] => []\n | [x] => [x]\n | x :: y :: xs => x :: sep :: y :: foldr (fun a r => sep :: a :: r) [] xs",
"tactic": "simp [intersperse]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nsep : α\nl : List α\nx y : α\nxs : List α\n⊢ intersperse sep (y :: xs) = y :: foldr (fun a r => sep :: a :: r) [] xs",
"tactic": "induction xs generalizing y <;> simp [*]"
}
] |
[
229,
77
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
225,
10
] |
Std/Data/List/Lemmas.lean
|
List.mem_of_mem_drop
|
[] |
[
1751,
89
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1751,
1
] |
Mathlib/Logic/Encodable/Basic.lean
|
Encodable.axiom_of_choice
|
[] |
[
600,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
598,
1
] |
Mathlib/CategoryTheory/Limits/Preserves/Shapes/Terminal.lean
|
CategoryTheory.Limits.PreservesInitial.iso_hom
|
[] |
[
217,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Mathlib/NumberTheory/Padics/PadicNumbers.lean
|
PadicSeq.norm_nonarchimedean_aux
|
[
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ (if hf : f + g ≈ 0 then 0 else padicNorm p (↑(f + g) (stationaryPoint hf))) ≤\n max (if hf : f ≈ 0 then 0 else padicNorm p (↑f (stationaryPoint hf)))\n (if hf : g ≈ 0 then 0 else padicNorm p (↑g (stationaryPoint hf)))",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ norm (f + g) ≤ max (norm f) (norm g)",
"tactic": "unfold norm"
},
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ padicNorm p (↑(f + g) (stationaryPoint hfg)) ≤\n max (padicNorm p (↑f (stationaryPoint hf))) (padicNorm p (↑g (stationaryPoint hg)))",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ (if hf : f + g ≈ 0 then 0 else padicNorm p (↑(f + g) (stationaryPoint hf))) ≤\n max (if hf : f ≈ 0 then 0 else padicNorm p (↑f (stationaryPoint hf)))\n (if hf : g ≈ 0 then 0 else padicNorm p (↑g (stationaryPoint hf)))",
"tactic": "split_ifs"
},
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ padicNorm p (↑(f + g) (max (stationaryPoint hfg) (max ?v2 ?v3))) ≤\n max (padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))))\n (padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg)))))\n\ncase v1\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ ℕ\n\ncase v2\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ ℕ\n\ncase v2\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ ℕ\n\ncase v3\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ ℕ\n\ncase v1\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ ℕ\n\ncase v3\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ ℕ\n\ncase v1\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ ℕ\n\ncase v3\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ ℕ\n\ncase v2\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ ℕ\n\ncase v3\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ ℕ\n\ncase v2\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ ℕ\n\ncase v3\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ ℕ",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ padicNorm p (↑(f + g) (stationaryPoint hfg)) ≤\n max (padicNorm p (↑f (stationaryPoint hf))) (padicNorm p (↑g (stationaryPoint hg)))",
"tactic": "rw [lift_index_left_left hfg, lift_index_left hf, lift_index_right hg]"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ padicNorm p (↑(f + g) (max (stationaryPoint hfg) (max ?v2 ?v3))) ≤\n max (padicNorm p (↑f (max ?v1 (max (stationaryPoint hf) ?v3))))\n (padicNorm p (↑g (max ?v1 (max ?v2 (stationaryPoint hg)))))\n\ncase v1\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ ℕ\n\ncase v2\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ ℕ\n\ncase v2\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ ℕ\n\ncase v3\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ ℕ\n\ncase v1\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ ℕ\n\ncase v3\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ ℕ\n\ncase v1\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ ℕ\n\ncase v3\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ ℕ\n\ncase v2\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ ℕ\n\ncase v3\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ ℕ\n\ncase v2\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ ℕ\n\ncase v3\np : ℕ\nhp : Fact (Nat.Prime p)\nf g : PadicSeq p\nhfg : ¬f + g ≈ 0\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\n⊢ ℕ",
"tactic": "apply padicNorm.nonarchimedean"
}
] |
[
377,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
372,
9
] |
Mathlib/Data/List/Cycle.lean
|
List.next_cons_cons_eq'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nl : List α\nx y z : α\nh : x ∈ y :: z :: l\nhx : x = y\n⊢ next (y :: z :: l) x h = z",
"tactic": "rw [next, nextOr, if_pos hx]"
}
] |
[
159,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/Topology/Algebra/UniformRing.lean
|
UniformSpace.Completion.map_smul_eq_mul_coe
|
[
{
"state_after": "case h\nα : Type ?u.206596\ninst✝¹⁴ : Ring α\ninst✝¹³ : UniformSpace α\ninst✝¹² : TopologicalRing α\ninst✝¹¹ : UniformAddGroup α\nβ : Type u\ninst✝¹⁰ : UniformSpace β\ninst✝⁹ : Ring β\ninst✝⁸ : UniformAddGroup β\ninst✝⁷ : TopologicalRing β\nf : α →+* β\nhf : Continuous ↑f\nA : Type u_1\ninst✝⁶ : Ring A\ninst✝⁵ : UniformSpace A\ninst✝⁴ : UniformAddGroup A\ninst✝³ : TopologicalRing A\nR : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Algebra R A\ninst✝ : UniformContinuousConstSMul R A\nr : R\nx : Completion ((fun x => A) r)\n⊢ Completion.map ((fun x x_1 => x • x_1) r) x = (fun x x_1 => x * x_1) (↑((fun x => A) r) (↑(algebraMap R A) r)) x",
"state_before": "α : Type ?u.206596\ninst✝¹⁴ : Ring α\ninst✝¹³ : UniformSpace α\ninst✝¹² : TopologicalRing α\ninst✝¹¹ : UniformAddGroup α\nβ : Type u\ninst✝¹⁰ : UniformSpace β\ninst✝⁹ : Ring β\ninst✝⁸ : UniformAddGroup β\ninst✝⁷ : TopologicalRing β\nf : α →+* β\nhf : Continuous ↑f\nA : Type u_1\ninst✝⁶ : Ring A\ninst✝⁵ : UniformSpace A\ninst✝⁴ : UniformAddGroup A\ninst✝³ : TopologicalRing A\nR : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Algebra R A\ninst✝ : UniformContinuousConstSMul R A\nr : R\n⊢ Completion.map ((fun x x_1 => x • x_1) r) = (fun x x_1 => x * x_1) (↑((fun x => A) r) (↑(algebraMap R A) r))",
"tactic": "ext x"
},
{
"state_after": "case h.refine'_1\nα : Type ?u.206596\ninst✝¹⁴ : Ring α\ninst✝¹³ : UniformSpace α\ninst✝¹² : TopologicalRing α\ninst✝¹¹ : UniformAddGroup α\nβ : Type u\ninst✝¹⁰ : UniformSpace β\ninst✝⁹ : Ring β\ninst✝⁸ : UniformAddGroup β\ninst✝⁷ : TopologicalRing β\nf : α →+* β\nhf : Continuous ↑f\nA : Type u_1\ninst✝⁶ : Ring A\ninst✝⁵ : UniformSpace A\ninst✝⁴ : UniformAddGroup A\ninst✝³ : TopologicalRing A\nR : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Algebra R A\ninst✝ : UniformContinuousConstSMul R A\nr : R\nx : Completion ((fun x => A) r)\n⊢ IsClosed\n {a |\n Completion.map ((fun x x_1 => x • x_1) r) a = (fun x x_1 => x * x_1) (↑((fun x => A) r) (↑(algebraMap R A) r)) a}\n\ncase h.refine'_2\nα : Type ?u.206596\ninst✝¹⁴ : Ring α\ninst✝¹³ : UniformSpace α\ninst✝¹² : TopologicalRing α\ninst✝¹¹ : UniformAddGroup α\nβ : Type u\ninst✝¹⁰ : UniformSpace β\ninst✝⁹ : Ring β\ninst✝⁸ : UniformAddGroup β\ninst✝⁷ : TopologicalRing β\nf : α →+* β\nhf : Continuous ↑f\nA : Type u_1\ninst✝⁶ : Ring A\ninst✝⁵ : UniformSpace A\ninst✝⁴ : UniformAddGroup A\ninst✝³ : TopologicalRing A\nR : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Algebra R A\ninst✝ : UniformContinuousConstSMul R A\nr : R\nx : Completion ((fun x => A) r)\na : (fun x => A) r\n⊢ Completion.map ((fun x x_1 => x • x_1) r) (↑((fun x => A) r) a) =\n (fun x x_1 => x * x_1) (↑((fun x => A) r) (↑(algebraMap R A) r)) (↑((fun x => A) r) a)",
"state_before": "case h\nα : Type ?u.206596\ninst✝¹⁴ : Ring α\ninst✝¹³ : UniformSpace α\ninst✝¹² : TopologicalRing α\ninst✝¹¹ : UniformAddGroup α\nβ : Type u\ninst✝¹⁰ : UniformSpace β\ninst✝⁹ : Ring β\ninst✝⁸ : UniformAddGroup β\ninst✝⁷ : TopologicalRing β\nf : α →+* β\nhf : Continuous ↑f\nA : Type u_1\ninst✝⁶ : Ring A\ninst✝⁵ : UniformSpace A\ninst✝⁴ : UniformAddGroup A\ninst✝³ : TopologicalRing A\nR : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Algebra R A\ninst✝ : UniformContinuousConstSMul R A\nr : R\nx : Completion ((fun x => A) r)\n⊢ Completion.map ((fun x x_1 => x • x_1) r) x = (fun x x_1 => x * x_1) (↑((fun x => A) r) (↑(algebraMap R A) r)) x",
"tactic": "refine' Completion.induction_on x _ fun a => _"
},
{
"state_after": "no goals",
"state_before": "case h.refine'_1\nα : Type ?u.206596\ninst✝¹⁴ : Ring α\ninst✝¹³ : UniformSpace α\ninst✝¹² : TopologicalRing α\ninst✝¹¹ : UniformAddGroup α\nβ : Type u\ninst✝¹⁰ : UniformSpace β\ninst✝⁹ : Ring β\ninst✝⁸ : UniformAddGroup β\ninst✝⁷ : TopologicalRing β\nf : α →+* β\nhf : Continuous ↑f\nA : Type u_1\ninst✝⁶ : Ring A\ninst✝⁵ : UniformSpace A\ninst✝⁴ : UniformAddGroup A\ninst✝³ : TopologicalRing A\nR : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Algebra R A\ninst✝ : UniformContinuousConstSMul R A\nr : R\nx : Completion ((fun x => A) r)\n⊢ IsClosed\n {a |\n Completion.map ((fun x x_1 => x • x_1) r) a = (fun x x_1 => x * x_1) (↑((fun x => A) r) (↑(algebraMap R A) r)) a}",
"tactic": "exact isClosed_eq Completion.continuous_map (continuous_mul_left _)"
},
{
"state_after": "no goals",
"state_before": "case h.refine'_2\nα : Type ?u.206596\ninst✝¹⁴ : Ring α\ninst✝¹³ : UniformSpace α\ninst✝¹² : TopologicalRing α\ninst✝¹¹ : UniformAddGroup α\nβ : Type u\ninst✝¹⁰ : UniformSpace β\ninst✝⁹ : Ring β\ninst✝⁸ : UniformAddGroup β\ninst✝⁷ : TopologicalRing β\nf : α →+* β\nhf : Continuous ↑f\nA : Type u_1\ninst✝⁶ : Ring A\ninst✝⁵ : UniformSpace A\ninst✝⁴ : UniformAddGroup A\ninst✝³ : TopologicalRing A\nR : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : Algebra R A\ninst✝ : UniformContinuousConstSMul R A\nr : R\nx : Completion ((fun x => A) r)\na : (fun x => A) r\n⊢ Completion.map ((fun x x_1 => x • x_1) r) (↑((fun x => A) r) a) =\n (fun x x_1 => x * x_1) (↑((fun x => A) r) (↑(algebraMap R A) r)) (↑((fun x => A) r) a)",
"tactic": "simp_rw [map_coe (uniformContinuous_const_smul r) a, Algebra.smul_def, coe_mul]"
}
] |
[
201,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
Ordinal.IsFundamentalSequence.zero
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.69006\nr : α → α → Prop\na o : Ordinal\nf✝ : (b : Ordinal) → b < o → Ordinal\nf : (b : Ordinal) → b < 0 → Ordinal\n⊢ 0 ≤ ord (cof 0)",
"tactic": "rw [cof_zero, ord_zero]"
}
] |
[
583,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
582,
11
] |
Mathlib/CategoryTheory/Limits/ConcreteCategory.lean
|
CategoryTheory.Limits.Concrete.isColimit_rep_eq_iff_exists
|
[] |
[
291,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
288,
1
] |
Mathlib/Logic/Equiv/Defs.lean
|
Equiv.symm_symm_apply
|
[] |
[
301,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
301,
24
] |
Mathlib/Algebra/CharP/Two.lean
|
neg_one_eq_one_iff
|
[
{
"state_after": "R : Type u_1\nι : Type ?u.22943\ninst✝¹ : Ring R\ninst✝ : Nontrivial R\nh : -1 = 1\n⊢ ringChar R = 2",
"state_before": "R : Type u_1\nι : Type ?u.22943\ninst✝¹ : Ring R\ninst✝ : Nontrivial R\n⊢ -1 = 1 ↔ ringChar R = 2",
"tactic": "refine' ⟨fun h => _, fun h => @CharTwo.neg_eq _ _ (ringChar.of_eq h) 1⟩"
},
{
"state_after": "R : Type u_1\nι : Type ?u.22943\ninst✝¹ : Ring R\ninst✝ : Nontrivial R\nh✝ : 1 = -1\nh : ↑(1 + 1) = 0\n⊢ ringChar R = 2",
"state_before": "R : Type u_1\nι : Type ?u.22943\ninst✝¹ : Ring R\ninst✝ : Nontrivial R\nh : -1 = 1\n⊢ ringChar R = 2",
"tactic": "rw [eq_comm, ← sub_eq_zero, sub_neg_eq_add, ← Nat.cast_one, ← Nat.cast_add] at h"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nι : Type ?u.22943\ninst✝¹ : Ring R\ninst✝ : Nontrivial R\nh✝ : 1 = -1\nh : ↑(1 + 1) = 0\n⊢ ringChar R = 2",
"tactic": "exact ((Nat.dvd_prime Nat.prime_two).mp (ringChar.dvd h)).resolve_left CharP.ringChar_ne_one"
}
] |
[
134,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/Data/MvPolynomial/Supported.lean
|
MvPolynomial.supported_univ
|
[
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nτ : Type ?u.126223\nR : Type u\nS : Type v\nr : R\ne : ℕ\nn m : σ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\ns t : Set σ\n⊢ supported R Set.univ = ⊤",
"tactic": "simp [Algebra.eq_top_iff, mem_supported]"
}
] |
[
106,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/Topology/DenseEmbedding.lean
|
DenseEmbedding.mk'
|
[] |
[
243,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
240,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
lt_one_of_mul_lt_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.23306\ninst✝² : MulOneClass α\ninst✝¹ : LT α\ninst✝ : ContravariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x < x_1\na b : α\nh : a * b < a\n⊢ ?m.23646 h * b < ?m.23646 h * 1",
"tactic": "simpa only [mul_one]"
}
] |
[
484,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
482,
1
] |
Mathlib/Combinatorics/Young/YoungDiagram.lean
|
YoungDiagram.cells_subset_iff
|
[] |
[
106,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
105,
1
] |
Mathlib/Data/Nat/Cast/Basic.lean
|
map_ofNat
|
[] |
[
271,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
269,
1
] |
Mathlib/Data/List/Sort.lean
|
List.Sorted.of_cons
|
[] |
[
62,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
61,
1
] |
Mathlib/MeasureTheory/Decomposition/Jordan.lean
|
MeasureTheory.SignedMeasure.toJordanDecomposition_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.86482\ninst✝ : MeasurableSpace α\ns : SignedMeasure α\nj : JordanDecomposition α\nh : s = toSignedMeasure j\n⊢ toJordanDecomposition s = j",
"tactic": "rw [h, toJordanDecomposition_toSignedMeasure]"
}
] |
[
495,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
493,
1
] |
Mathlib/CategoryTheory/Limits/HasLimits.lean
|
CategoryTheory.Limits.HasColimit.isoOfNatIso_ι_inv
|
[] |
[
920,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
918,
1
] |
Mathlib/Logic/Embedding/Basic.lean
|
Function.Embedding.equiv_symm_toEmbedding_trans_toEmbedding
|
[
{
"state_after": "case h\nα : Sort u_1\nβ : Sort u_2\ne : α ≃ β\nx✝ : β\n⊢ ↑(Embedding.trans (Equiv.toEmbedding e.symm) (Equiv.toEmbedding e)) x✝ = ↑(Embedding.refl β) x✝",
"state_before": "α : Sort u_1\nβ : Sort u_2\ne : α ≃ β\n⊢ Embedding.trans (Equiv.toEmbedding e.symm) (Equiv.toEmbedding e) = Embedding.refl β",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Sort u_1\nβ : Sort u_2\ne : α ≃ β\nx✝ : β\n⊢ ↑(Embedding.trans (Equiv.toEmbedding e.symm) (Equiv.toEmbedding e)) x✝ = ↑(Embedding.refl β) x✝",
"tactic": "simp"
}
] |
[
160,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
157,
1
] |
Mathlib/Analysis/Normed/Group/Hom.lean
|
NormedAddGroupHom.ker_zero
|
[
{
"state_after": "case h\nV : Type ?u.485691\nW : Type ?u.485694\nV₁ : Type u_1\nV₂ : Type u_2\nV₃ : Type ?u.485703\ninst✝⁴ : SeminormedAddCommGroup V\ninst✝³ : SeminormedAddCommGroup W\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf : NormedAddGroupHom V₁ V₂\ng : NormedAddGroupHom V₂ V₃\nx✝ : V₁\n⊢ x✝ ∈ ker 0 ↔ x✝ ∈ ⊤",
"state_before": "V : Type ?u.485691\nW : Type ?u.485694\nV₁ : Type u_1\nV₂ : Type u_2\nV₃ : Type ?u.485703\ninst✝⁴ : SeminormedAddCommGroup V\ninst✝³ : SeminormedAddCommGroup W\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf : NormedAddGroupHom V₁ V₂\ng : NormedAddGroupHom V₂ V₃\n⊢ ker 0 = ⊤",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nV : Type ?u.485691\nW : Type ?u.485694\nV₁ : Type u_1\nV₂ : Type u_2\nV₃ : Type ?u.485703\ninst✝⁴ : SeminormedAddCommGroup V\ninst✝³ : SeminormedAddCommGroup W\ninst✝² : SeminormedAddCommGroup V₁\ninst✝¹ : SeminormedAddCommGroup V₂\ninst✝ : SeminormedAddCommGroup V₃\nf : NormedAddGroupHom V₁ V₂\ng : NormedAddGroupHom V₂ V₃\nx✝ : V₁\n⊢ x✝ ∈ ker 0 ↔ x✝ ∈ ⊤",
"tactic": "simp [mem_ker]"
}
] |
[
767,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
765,
1
] |
Mathlib/RingTheory/FractionalIdeal.lean
|
FractionalIdeal.coeIdeal_le_coeIdeal'
|
[] |
[
269,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
267,
1
] |
Mathlib/Topology/UniformSpace/Completion.lean
|
CauchyFilter.comp_gen
|
[
{
"state_after": "case hg\nα : Type u\ninst✝² : UniformSpace α\nβ : Type v\nγ : Type w\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\n⊢ Monotone gen\n\ncase hh\nα : Type u\ninst✝² : UniformSpace α\nβ : Type v\nγ : Type w\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\n⊢ Monotone fun s => s ○ s",
"state_before": "α : Type u\ninst✝² : UniformSpace α\nβ : Type v\nγ : Type w\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\n⊢ (Filter.lift' (Filter.lift' (𝓤 α) gen) fun s => s ○ s) = Filter.lift' (𝓤 α) fun s => gen s ○ gen s",
"tactic": "rw [lift'_lift'_assoc]"
},
{
"state_after": "no goals",
"state_before": "case hg\nα : Type u\ninst✝² : UniformSpace α\nβ : Type v\nγ : Type w\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\n⊢ Monotone gen",
"tactic": "exact monotone_gen"
},
{
"state_after": "no goals",
"state_before": "case hh\nα : Type u\ninst✝² : UniformSpace α\nβ : Type v\nγ : Type w\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\n⊢ Monotone fun s => s ○ s",
"tactic": "exact monotone_id.compRel monotone_id"
},
{
"state_after": "case hg\nα : Type u\ninst✝² : UniformSpace α\nβ : Type v\nγ : Type w\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\n⊢ Monotone fun s => s ○ s\n\ncase hh\nα : Type u\ninst✝² : UniformSpace α\nβ : Type v\nγ : Type w\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\n⊢ Monotone gen",
"state_before": "α : Type u\ninst✝² : UniformSpace α\nβ : Type v\nγ : Type w\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\n⊢ (Filter.lift' (𝓤 α) fun s => gen (s ○ s)) = Filter.lift' (Filter.lift' (𝓤 α) fun s => s ○ s) gen",
"tactic": "rw [lift'_lift'_assoc]"
},
{
"state_after": "no goals",
"state_before": "case hg\nα : Type u\ninst✝² : UniformSpace α\nβ : Type v\nγ : Type w\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\n⊢ Monotone fun s => s ○ s",
"tactic": "exact monotone_id.compRel monotone_id"
},
{
"state_after": "no goals",
"state_before": "case hh\nα : Type u\ninst✝² : UniformSpace α\nβ : Type v\nγ : Type w\ninst✝¹ : UniformSpace β\ninst✝ : UniformSpace γ\n⊢ Monotone gen",
"tactic": "exact monotone_gen"
}
] |
[
131,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
9
] |
Mathlib/Data/Prod/Basic.lean
|
Prod.map_mk
|
[] |
[
62,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
61,
1
] |
Mathlib/MeasureTheory/Integral/IntegrableOn.lean
|
MeasureTheory.integrableOn_Lp_of_measure_ne_top
|
[
{
"state_after": "α : Type u_2\nβ : Type ?u.1655667\nE✝ : Type ?u.1655670\nF : Type ?u.1655673\ninst✝² : MeasurableSpace α\ninst✝¹ : NormedAddCommGroup E✝\nf✝ g : α → E✝\ns✝ t : Set α\nμ ν : Measure α\nE : Type u_1\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\ns : Set α\nf : { x // x ∈ Lp E p }\nhp : 1 ≤ p\nhμs : ↑↑μ s ≠ ⊤\n⊢ Memℒp (↑↑f) 1",
"state_before": "α : Type u_2\nβ : Type ?u.1655667\nE✝ : Type ?u.1655670\nF : Type ?u.1655673\ninst✝² : MeasurableSpace α\ninst✝¹ : NormedAddCommGroup E✝\nf✝ g : α → E✝\ns✝ t : Set α\nμ ν : Measure α\nE : Type u_1\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\ns : Set α\nf : { x // x ∈ Lp E p }\nhp : 1 ≤ p\nhμs : ↑↑μ s ≠ ⊤\n⊢ IntegrableOn (↑↑f) s",
"tactic": "refine' memℒp_one_iff_integrable.mp _"
},
{
"state_after": "α : Type u_2\nβ : Type ?u.1655667\nE✝ : Type ?u.1655670\nF : Type ?u.1655673\ninst✝² : MeasurableSpace α\ninst✝¹ : NormedAddCommGroup E✝\nf✝ g : α → E✝\ns✝ t : Set α\nμ ν : Measure α\nE : Type u_1\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\ns : Set α\nf : { x // x ∈ Lp E p }\nhp : 1 ≤ p\nhμs : ↑↑μ s ≠ ⊤\nhμ_restrict_univ : ↑↑(Measure.restrict μ s) univ < ⊤\n⊢ Memℒp (↑↑f) 1",
"state_before": "α : Type u_2\nβ : Type ?u.1655667\nE✝ : Type ?u.1655670\nF : Type ?u.1655673\ninst✝² : MeasurableSpace α\ninst✝¹ : NormedAddCommGroup E✝\nf✝ g : α → E✝\ns✝ t : Set α\nμ ν : Measure α\nE : Type u_1\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\ns : Set α\nf : { x // x ∈ Lp E p }\nhp : 1 ≤ p\nhμs : ↑↑μ s ≠ ⊤\n⊢ Memℒp (↑↑f) 1",
"tactic": "have hμ_restrict_univ : (μ.restrict s) Set.univ < ∞ := by\n simpa only [Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply, lt_top_iff_ne_top]"
},
{
"state_after": "α : Type u_2\nβ : Type ?u.1655667\nE✝ : Type ?u.1655670\nF : Type ?u.1655673\ninst✝² : MeasurableSpace α\ninst✝¹ : NormedAddCommGroup E✝\nf✝ g : α → E✝\ns✝ t : Set α\nμ ν : Measure α\nE : Type u_1\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\ns : Set α\nf : { x // x ∈ Lp E p }\nhp : 1 ≤ p\nhμs : ↑↑μ s ≠ ⊤\nhμ_restrict_univ : ↑↑(Measure.restrict μ s) univ < ⊤\nhμ_finite : IsFiniteMeasure (Measure.restrict μ s)\n⊢ Memℒp (↑↑f) 1",
"state_before": "α : Type u_2\nβ : Type ?u.1655667\nE✝ : Type ?u.1655670\nF : Type ?u.1655673\ninst✝² : MeasurableSpace α\ninst✝¹ : NormedAddCommGroup E✝\nf✝ g : α → E✝\ns✝ t : Set α\nμ ν : Measure α\nE : Type u_1\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\ns : Set α\nf : { x // x ∈ Lp E p }\nhp : 1 ≤ p\nhμs : ↑↑μ s ≠ ⊤\nhμ_restrict_univ : ↑↑(Measure.restrict μ s) univ < ⊤\n⊢ Memℒp (↑↑f) 1",
"tactic": "haveI hμ_finite : IsFiniteMeasure (μ.restrict s) := ⟨hμ_restrict_univ⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.1655667\nE✝ : Type ?u.1655670\nF : Type ?u.1655673\ninst✝² : MeasurableSpace α\ninst✝¹ : NormedAddCommGroup E✝\nf✝ g : α → E✝\ns✝ t : Set α\nμ ν : Measure α\nE : Type u_1\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\ns : Set α\nf : { x // x ∈ Lp E p }\nhp : 1 ≤ p\nhμs : ↑↑μ s ≠ ⊤\nhμ_restrict_univ : ↑↑(Measure.restrict μ s) univ < ⊤\nhμ_finite : IsFiniteMeasure (Measure.restrict μ s)\n⊢ Memℒp (↑↑f) 1",
"tactic": "exact ((Lp.memℒp _).restrict s).memℒp_of_exponent_le hp"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.1655667\nE✝ : Type ?u.1655670\nF : Type ?u.1655673\ninst✝² : MeasurableSpace α\ninst✝¹ : NormedAddCommGroup E✝\nf✝ g : α → E✝\ns✝ t : Set α\nμ ν : Measure α\nE : Type u_1\ninst✝ : NormedAddCommGroup E\np : ℝ≥0∞\ns : Set α\nf : { x // x ∈ Lp E p }\nhp : 1 ≤ p\nhμs : ↑↑μ s ≠ ⊤\n⊢ ↑↑(Measure.restrict μ s) univ < ⊤",
"tactic": "simpa only [Set.univ_inter, MeasurableSet.univ, Measure.restrict_apply, lt_top_iff_ne_top]"
}
] |
[
372,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
366,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.map_monic_eq_zero_iff
|
[
{
"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 : R[X]\ninst✝ : Semiring S\nf : R →+* S\nhp : Monic p\nhfp : map f p = 0\nx : R\n⊢ ↑f x = ↑f x * ↑f (leadingCoeff p)",
"tactic": "simp only [mul_one, hp.leadingCoeff, f.map_one]"
},
{
"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 : R[X]\ninst✝ : Semiring S\nf : R →+* S\nhp : Monic p\nhfp : map f p = 0\nx : R\n⊢ ↑f x * coeff (map f p) (natDegree p) = 0",
"tactic": "simp only [hfp, mul_zero, coeff_zero]"
},
{
"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 : R[X]\ninst✝ : Semiring S\nf : R →+* S\nhp : Monic p\nh : ∀ (x : R), ↑f x = 0\nn : ℕ\n⊢ coeff (map f p) n = coeff 0 n",
"tactic": "simp only [h, coeff_map, coeff_zero]"
}
] |
[
864,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
857,
1
] |
Mathlib/Order/Lattice.lean
|
right_lt_sup
|
[] |
[
214,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
213,
1
] |
Mathlib/LinearAlgebra/BilinearForm.lean
|
BilinForm.IsRefl.groupSMul
|
[] |
[
881,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
879,
11
] |
src/lean/Init/Data/Nat/Basic.lean
|
Nat.zero_lt_one
|
[] |
[
402,
17
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
401,
11
] |
Mathlib/Topology/Sets/Closeds.lean
|
TopologicalSpace.Closeds.coe_bot
|
[] |
[
131,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/Data/Real/Basic.lean
|
Real.sInf_def
|
[] |
[
751,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
750,
1
] |
Mathlib/GroupTheory/Submonoid/Membership.lean
|
Submonoid.mem_iSup_of_mem
|
[
{
"state_after": "M : Type u_2\nA : Type ?u.45573\nB : Type ?u.45576\ninst✝ : MulOneClass M\nι : Sort u_1\nS : ι → Submonoid M\ni : ι\n⊢ S i ≤ iSup S",
"state_before": "M : Type u_2\nA : Type ?u.45573\nB : Type ?u.45576\ninst✝ : MulOneClass M\nι : Sort u_1\nS : ι → Submonoid M\ni : ι\n⊢ ∀ {x : M}, x ∈ S i → x ∈ iSup S",
"tactic": "rw [←SetLike.le_def]"
},
{
"state_after": "no goals",
"state_before": "M : Type u_2\nA : Type ?u.45573\nB : Type ?u.45576\ninst✝ : MulOneClass M\nι : Sort u_1\nS : ι → Submonoid M\ni : ι\n⊢ S i ≤ iSup S",
"tactic": "exact le_iSup _ _"
}
] |
[
260,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
1
] |
Mathlib/Analysis/Convex/Cone/Dual.lean
|
innerDualCone_zero
|
[] |
[
69,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
68,
1
] |
Mathlib/Topology/Algebra/Order/Filter.lean
|
Filter.tendsto_nhds_atTop
|
[] |
[
29,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
28,
11
] |
Mathlib/CategoryTheory/Idempotents/FunctorExtension.lean
|
CategoryTheory.Idempotents.functorExtension₂_comp_whiskeringLeft_toKaroubi
|
[
{
"state_after": "no goals",
"state_before": "C : Type u_1\nD : Type u_2\nE : Type ?u.69364\ninst✝² : Category C\ninst✝¹ : Category D\ninst✝ : Category E\n⊢ functorExtension₂ C D ⋙ (whiskeringLeft C (Karoubi C) (Karoubi D)).obj (toKaroubi C) =\n (whiskeringRight C D (Karoubi D)).obj (toKaroubi D)",
"tactic": "simp only [functorExtension₂, Functor.assoc, functorExtension₁_comp_whiskeringLeft_toKaroubi,\n Functor.comp_id]"
}
] |
[
214,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
210,
1
] |
Mathlib/RingTheory/Ideal/Over.lean
|
Ideal.IntegralClosure.isMaximal_of_isMaximal_comap
|
[] |
[
348,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
346,
1
] |
Mathlib/Algebra/Algebra/Equiv.lean
|
AlgEquiv.arrowCongr_symm
|
[
{
"state_after": "case H.H\nR : Type u\nA₁ : Type v\nA₂ : Type w\nA₃ : Type u₁\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : Semiring A₁\ninst✝⁸ : Semiring A₂\ninst✝⁷ : Semiring A₃\ninst✝⁶ : Algebra R A₁\ninst✝⁵ : Algebra R A₂\ninst✝⁴ : Algebra R A₃\ne : A₁ ≃ₐ[R] A₂\nA₁' : Type u_1\nA₂' : Type u_2\ninst✝³ : Semiring A₁'\ninst✝² : Semiring A₂'\ninst✝¹ : Algebra R A₁'\ninst✝ : Algebra R A₂'\ne₁ : A₁ ≃ₐ[R] A₁'\ne₂ : A₂ ≃ₐ[R] A₂'\nx✝¹ : A₁' →ₐ[R] A₂'\nx✝ : A₁\n⊢ ↑(↑(arrowCongr e₁ e₂).symm x✝¹) x✝ = ↑(↑(arrowCongr (symm e₁) (symm e₂)) x✝¹) x✝",
"state_before": "R : Type u\nA₁ : Type v\nA₂ : Type w\nA₃ : Type u₁\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : Semiring A₁\ninst✝⁸ : Semiring A₂\ninst✝⁷ : Semiring A₃\ninst✝⁶ : Algebra R A₁\ninst✝⁵ : Algebra R A₂\ninst✝⁴ : Algebra R A₃\ne : A₁ ≃ₐ[R] A₂\nA₁' : Type u_1\nA₂' : Type u_2\ninst✝³ : Semiring A₁'\ninst✝² : Semiring A₂'\ninst✝¹ : Algebra R A₁'\ninst✝ : Algebra R A₂'\ne₁ : A₁ ≃ₐ[R] A₁'\ne₂ : A₂ ≃ₐ[R] A₂'\n⊢ (arrowCongr e₁ e₂).symm = arrowCongr (symm e₁) (symm e₂)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case H.H\nR : Type u\nA₁ : Type v\nA₂ : Type w\nA₃ : Type u₁\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : Semiring A₁\ninst✝⁸ : Semiring A₂\ninst✝⁷ : Semiring A₃\ninst✝⁶ : Algebra R A₁\ninst✝⁵ : Algebra R A₂\ninst✝⁴ : Algebra R A₃\ne : A₁ ≃ₐ[R] A₂\nA₁' : Type u_1\nA₂' : Type u_2\ninst✝³ : Semiring A₁'\ninst✝² : Semiring A₂'\ninst✝¹ : Algebra R A₁'\ninst✝ : Algebra R A₂'\ne₁ : A₁ ≃ₐ[R] A₁'\ne₂ : A₂ ≃ₐ[R] A₂'\nx✝¹ : A₁' →ₐ[R] A₂'\nx✝ : A₁\n⊢ ↑(↑(arrowCongr e₁ e₂).symm x✝¹) x✝ = ↑(↑(arrowCongr (symm e₁) (symm e₂)) x✝¹) x✝",
"tactic": "rfl"
}
] |
[
485,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
481,
1
] |
Mathlib/Init/Data/Nat/Lemmas.lean
|
Nat.eq_zero_of_mul_eq_zero
|
[
{
"state_after": "n m : ℕ\n⊢ n * m + m = 0 → succ n = 0 ∨ m = 0",
"state_before": "n m : ℕ\n⊢ succ n * m = 0 → succ n = 0 ∨ m = 0",
"tactic": "rw [succ_mul]"
},
{
"state_after": "n m : ℕ\nh : n * m + m = 0\n⊢ succ n = 0 ∨ m = 0",
"state_before": "n m : ℕ\n⊢ n * m + m = 0 → succ n = 0 ∨ m = 0",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "n m : ℕ\nh : n * m + m = 0\n⊢ succ n = 0 ∨ m = 0",
"tactic": "exact Or.inr (Nat.eq_zero_of_add_eq_zero_left h)"
}
] |
[
22,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
17,
1
] |
Mathlib/CategoryTheory/Adjunction/Basic.lean
|
CategoryTheory.Adjunction.homEquiv_naturality_left_symm
|
[
{
"state_after": "no goals",
"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\nf : X' ⟶ X\ng : X ⟶ G.obj Y\n⊢ ↑(homEquiv adj X' Y).symm (f ≫ g) = F.map f ≫ ↑(homEquiv adj X Y).symm g",
"tactic": "rw [homEquiv_counit, F.map_comp, assoc, adj.homEquiv_counit.symm]"
}
] |
[
153,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.restrict_apply_univ
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.285491\nγ : Type ?u.285494\nδ : Type ?u.285497\nι : Type ?u.285500\nR : Type ?u.285503\nR' : Type ?u.285506\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nμ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t s : Set α\n⊢ ↑↑(restrict μ s) univ = ↑↑μ s",
"tactic": "rw [restrict_apply MeasurableSet.univ, Set.univ_inter]"
}
] |
[
1598,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1597,
1
] |
Mathlib/Algebra/Algebra/Hom.lean
|
AlgHom.algebraMap_eq_apply
|
[] |
[
451,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
449,
1
] |
Mathlib/Algebra/Lie/Nilpotent.lean
|
LieIdeal.map_lowerCentralSeries_le
|
[
{
"state_after": "case zero\nR : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\n⊢ map f (lowerCentralSeries R L L Nat.zero) ≤ lowerCentralSeries R L' L' Nat.zero\n\ncase succ\nR : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nk : ℕ\nih : map f (lowerCentralSeries R L L k) ≤ lowerCentralSeries R L' L' k\n⊢ map f (lowerCentralSeries R L L (Nat.succ k)) ≤ lowerCentralSeries R L' L' (Nat.succ k)",
"state_before": "R : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nk : ℕ\nf : L →ₗ⁅R⁆ L'\n⊢ map f (lowerCentralSeries R L L k) ≤ lowerCentralSeries R L' L' k",
"tactic": "induction' k with k ih"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\n⊢ map f (lowerCentralSeries R L L Nat.zero) ≤ lowerCentralSeries R L' L' Nat.zero",
"tactic": "simp only [Nat.zero_eq, LieModule.lowerCentralSeries_zero, le_top]"
},
{
"state_after": "case succ\nR : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nk : ℕ\nih : map f (lowerCentralSeries R L L k) ≤ lowerCentralSeries R L' L' k\n⊢ map f ⁅⊤, lowerCentralSeries R L L k⁆ ≤ ⁅⊤, lowerCentralSeries R L' L' k⁆",
"state_before": "case succ\nR : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nk : ℕ\nih : map f (lowerCentralSeries R L L k) ≤ lowerCentralSeries R L' L' k\n⊢ map f (lowerCentralSeries R L L (Nat.succ k)) ≤ lowerCentralSeries R L' L' (Nat.succ k)",
"tactic": "simp only [LieModule.lowerCentralSeries_succ]"
},
{
"state_after": "no goals",
"state_before": "case succ\nR : Type u\nL : Type v\nL' : Type w\ninst✝⁴ : CommRing R\ninst✝³ : LieRing L\ninst✝² : LieAlgebra R L\ninst✝¹ : LieRing L'\ninst✝ : LieAlgebra R L'\nf : L →ₗ⁅R⁆ L'\nk : ℕ\nih : map f (lowerCentralSeries R L L k) ≤ lowerCentralSeries R L' L' k\n⊢ map f ⁅⊤, lowerCentralSeries R L L k⁆ ≤ ⁅⊤, lowerCentralSeries R L' L' k⁆",
"tactic": "exact le_trans (LieIdeal.map_bracket_le f) (LieSubmodule.mono_lie _ _ _ _ le_top ih)"
}
] |
[
608,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
603,
1
] |
Mathlib/Order/Hom/Lattice.lean
|
map_compl'
|
[] |
[
289,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
288,
1
] |
Mathlib/Algebra/Ring/BooleanRing.lean
|
BooleanRing.sup_inf_self
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.14060\nγ : Type ?u.14063\ninst✝² : BooleanRing α\ninst✝¹ : BooleanRing β\ninst✝ : BooleanRing γ\na b : α\n⊢ a + a * b + a * (a * b) = a",
"tactic": "rw [← mul_assoc, mul_self, add_assoc, add_self, add_zero]"
}
] |
[
220,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
218,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.isLittleO_neg_right
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.132478\nE : Type u_2\nF : Type ?u.132484\nG : Type ?u.132487\nE' : Type ?u.132490\nF' : Type u_3\nG' : Type ?u.132496\nE'' : Type ?u.132499\nF'' : Type ?u.132502\nG'' : Type ?u.132505\nR : Type ?u.132508\nR' : Type ?u.132511\n𝕜 : Type ?u.132514\n𝕜' : Type ?u.132517\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\n⊢ (∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f fun x => -g' x) ↔ ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f g'",
"state_before": "α : Type u_1\nβ : Type ?u.132478\nE : Type u_2\nF : Type ?u.132484\nG : Type ?u.132487\nE' : Type ?u.132490\nF' : Type u_3\nG' : Type ?u.132496\nE'' : Type ?u.132499\nF'' : Type ?u.132502\nG'' : Type ?u.132505\nR : Type ?u.132508\nR' : Type ?u.132511\n𝕜 : Type ?u.132514\n𝕜' : Type ?u.132517\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\n⊢ (f =o[l] fun x => -g' x) ↔ f =o[l] g'",
"tactic": "simp only [IsLittleO_def]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.132478\nE : Type u_2\nF : Type ?u.132484\nG : Type ?u.132487\nE' : Type ?u.132490\nF' : Type u_3\nG' : Type ?u.132496\nE'' : Type ?u.132499\nF'' : Type ?u.132502\nG'' : Type ?u.132505\nR : Type ?u.132508\nR' : Type ?u.132511\n𝕜 : Type ?u.132514\n𝕜' : Type ?u.132517\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\n⊢ (∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f fun x => -g' x) ↔ ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f g'",
"tactic": "exact forall₂_congr fun _ _ => isBigOWith_neg_right"
}
] |
[
884,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
882,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.toNNReal_sSup
|
[
{
"state_after": "α : Type ?u.841724\nβ : Type ?u.841727\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nι : Sort ?u.841744\nf g : ι → ℝ≥0∞\ns : Set ℝ≥0∞\nhs : ∀ (r : ℝ≥0∞), r ∈ s → r ≠ ⊤\nhf : ∀ (i : { x // x ∈ s }), ↑i ≠ ⊤\n⊢ ENNReal.toNNReal (sSup s) = sSup (ENNReal.toNNReal '' s)",
"state_before": "α : Type ?u.841724\nβ : Type ?u.841727\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nι : Sort ?u.841744\nf g : ι → ℝ≥0∞\ns : Set ℝ≥0∞\nhs : ∀ (r : ℝ≥0∞), r ∈ s → r ≠ ⊤\n⊢ ENNReal.toNNReal (sSup s) = sSup (ENNReal.toNNReal '' s)",
"tactic": "have hf : ∀ i, ((↑) : s → ℝ≥0∞) i ≠ ∞ := fun ⟨r, rs⟩ => hs r rs"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.841724\nβ : Type ?u.841727\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nι : Sort ?u.841744\nf g : ι → ℝ≥0∞\ns : Set ℝ≥0∞\nhs : ∀ (r : ℝ≥0∞), r ∈ s → r ≠ ⊤\nhf : ∀ (i : { x // x ∈ s }), ↑i ≠ ⊤\n⊢ ENNReal.toNNReal (sSup s) = sSup (ENNReal.toNNReal '' s)",
"tactic": "simpa only [← sSup_range, ← image_eq_range, Subtype.range_coe_subtype] using (toNNReal_iSup hf)"
}
] |
[
2377,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2372,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
LowerSet.mem_iInf₂_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.77881\nγ : Type ?u.77884\nι : Sort u_2\nκ : ι → Sort u_3\ninst✝ : LE α\nS : Set (LowerSet α)\ns t : LowerSet α\na : α\nf : (i : ι) → κ i → LowerSet α\n⊢ (a ∈ ⨅ (i : ι) (j : κ i), f i j) ↔ ∀ (i : ι) (j : κ i), a ∈ f i j",
"tactic": "simp_rw [mem_iInf_iff]"
}
] |
[
770,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
769,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.Infinite.ne_real
|
[] |
[
624,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
623,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Ici_subset_Ico_union_Ici
|
[] |
[
1267,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1266,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.OuterMeasure.isCaratheodory_compl
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nm : OuterMeasure α\ns s₁ s₂ : Set α\n⊢ IsCaratheodory m s₁ → IsCaratheodory m (s₁ᶜ)",
"tactic": "simp [IsCaratheodory, diff_eq, add_comm]"
}
] |
[
958,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
957,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.commute_add_int
|
[
{
"state_after": "no goals",
"state_before": "f g : CircleDeg1Lift\nn : ℕ\n⊢ Function.Commute ↑f fun x => x + ↑-[n+1]",
"tactic": "simpa [sub_eq_add_neg] using f.commute_sub_nat (n + 1)"
}
] |
[
349,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
347,
1
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.eval_X
|
[
{
"state_after": "no goals",
"state_before": "K : Type u\ninst✝¹ : Field K\nL : Type u_1\ninst✝ : Field L\nf : K →+* L\na : L\n⊢ eval f a X = a",
"tactic": "simp [eval]"
}
] |
[
1494,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1494,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.isImage_source_target
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.31348\nδ : Type ?u.31351\ne : LocalEquiv α β\ne' : LocalEquiv β γ\nx : α\nhx : x ∈ e.source\n⊢ ↑e x ∈ e.target ↔ x ∈ e.source",
"tactic": "simp [hx]"
}
] |
[
477,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
477,
1
] |
Mathlib/Computability/TuringMachine.lean
|
Turing.ListBlank.exists_cons
|
[] |
[
280,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
278,
1
] |
Mathlib/Order/Filter/Ultrafilter.lean
|
Nat.hyperfilter_le_atTop
|
[] |
[
484,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
483,
1
] |
Mathlib/Algebra/Module/Submodule/Lattice.lean
|
Submodule.disjoint_def
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.170489\nM : Type u_2\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module S M\ninst✝¹ : SMul S R\ninst✝ : IsScalarTower S R M\np✝ q p p' : Submodule R M\n⊢ (∀ (x : M), x ∈ p ∧ x ∈ p' → x ∈ {0}) ↔ ∀ (x : M), x ∈ p → x ∈ p' → x = 0",
"tactic": "simp"
}
] |
[
328,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
327,
1
] |
Mathlib/RingTheory/Polynomial/Opposites.lean
|
Polynomial.support_opRingEquiv
|
[
{
"state_after": "case h\nR : Type u_1\ninst✝ : Semiring R\np : R[X]\n⊢ support (↑(opRingEquiv R) (op p)) = support (unop (op p))",
"state_before": "R : Type u_1\ninst✝ : Semiring R\np : R[X]ᵐᵒᵖ\n⊢ support (↑(opRingEquiv R) p) = support (unop p)",
"tactic": "induction' p using MulOpposite.rec' with p"
},
{
"state_after": "case h.ofFinsupp\nR : Type u_1\ninst✝ : Semiring R\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ support (↑(opRingEquiv R) (op { toFinsupp := toFinsupp✝ })) = support (unop (op { toFinsupp := toFinsupp✝ }))",
"state_before": "case h\nR : Type u_1\ninst✝ : Semiring R\np : R[X]\n⊢ support (↑(opRingEquiv R) (op p)) = support (unop (op p))",
"tactic": "cases p"
},
{
"state_after": "no goals",
"state_before": "case h.ofFinsupp\nR : Type u_1\ninst✝ : Semiring R\ntoFinsupp✝ : AddMonoidAlgebra R ℕ\n⊢ support (↑(opRingEquiv R) (op { toFinsupp := toFinsupp✝ })) = support (unop (op { toFinsupp := toFinsupp✝ }))",
"tactic": "exact Finsupp.support_mapRange_of_injective (map_zero _) _ op_injective"
}
] |
[
109,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
1
] |
Mathlib/Order/CompleteLattice.lean
|
iSup_plift_down
|
[] |
[
672,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
671,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.sin_int_mul_two_pi_sub
|
[] |
[
304,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
303,
1
] |
Std/Data/List/Lemmas.lean
|
List.exists_erase_eq
|
[
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\na : α\nl : List α\nh : a ∈ l\nw✝ : α\nl₁ l₂ : List α\nh₁ : ∀ (b : α), b ∈ l₁ → ¬(a == b) = true\ne : (a == w✝) = true\nh₂ : l = l₁ ++ w✝ :: l₂\nh₃ : eraseP (BEq.beq a) l = l₁ ++ l₂\n⊢ ∃ l₁ l₂, ¬a ∈ l₁ ∧ l = l₁ ++ a :: l₂ ∧ List.erase l a = l₁ ++ l₂",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\na : α\nl : List α\nh : a ∈ l\n⊢ ∃ l₁ l₂, ¬a ∈ l₁ ∧ l = l₁ ++ a :: l₂ ∧ List.erase l a = l₁ ++ l₂",
"tactic": "let ⟨_, l₁, l₂, h₁, e, h₂, h₃⟩ := exists_of_eraseP h (beq_self_eq_true _)"
},
{
"state_after": "α : Type u_1\ninst✝ : DecidableEq α\na : α\nl : List α\nh : a ∈ l\nw✝ : α\nl₁ l₂ : List α\nh₁ : ∀ (b : α), b ∈ l₁ → ¬(a == b) = true\ne : (a == w✝) = true\nh₂ : l = l₁ ++ w✝ :: l₂\nh₃ : eraseP (BEq.beq a) l = l₁ ++ l₂\n⊢ ∃ l₁ l₂, ¬a ∈ l₁ ∧ l = l₁ ++ a :: l₂ ∧ eraseP (fun b => decide (a = b)) l = l₁ ++ l₂",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\na : α\nl : List α\nh : a ∈ l\nw✝ : α\nl₁ l₂ : List α\nh₁ : ∀ (b : α), b ∈ l₁ → ¬(a == b) = true\ne : (a == w✝) = true\nh₂ : l = l₁ ++ w✝ :: l₂\nh₃ : eraseP (BEq.beq a) l = l₁ ++ l₂\n⊢ ∃ l₁ l₂, ¬a ∈ l₁ ∧ l = l₁ ++ a :: l₂ ∧ List.erase l a = l₁ ++ l₂",
"tactic": "rw [erase_eq_eraseP]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\na : α\nl : List α\nh : a ∈ l\nw✝ : α\nl₁ l₂ : List α\nh₁ : ∀ (b : α), b ∈ l₁ → ¬(a == b) = true\ne : (a == w✝) = true\nh₂ : l = l₁ ++ w✝ :: l₂\nh₃ : eraseP (BEq.beq a) l = l₁ ++ l₂\n⊢ ∃ l₁ l₂, ¬a ∈ l₁ ∧ l = l₁ ++ a :: l₂ ∧ eraseP (fun b => decide (a = b)) l = l₁ ++ l₂",
"tactic": "exact ⟨l₁, l₂, fun h => h₁ _ h (beq_self_eq_true _), eq_of_beq e ▸ h₂, h₃⟩"
}
] |
[
1059,
99
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1056,
1
] |
Mathlib/Analysis/Convex/Cone/Basic.lean
|
riesz_extension
|
[
{
"state_after": "case intro.mk.intro.intro.intro\n𝕜 : Type ?u.280700\nE : Type u_1\nF : Type ?u.280706\nG : Type ?u.280709\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : ConvexCone ℝ E\nf : E →ₗ.[ℝ] ℝ\nnonneg : ∀ (x : { x // x ∈ f.domain }), ↑x ∈ s → 0 ≤ ↑f x\ndense : ∀ (y : E), ∃ x, ↑x + y ∈ s\ng : { x // x ∈ ⊤ } →ₗ[ℝ] ℝ\nhfg :\n ∀ ⦃x : { x // x ∈ f.domain }⦄ ⦃y : { x // x ∈ { domain := ⊤, toFun := g }.domain }⦄,\n ↑x = ↑y → ↑f x = ↑{ domain := ⊤, toFun := g } y\nhgs : ∀ (x : { x // x ∈ { domain := ⊤, toFun := g }.domain }), ↑x ∈ s → 0 ≤ ↑{ domain := ⊤, toFun := g } x\n⊢ ∃ g, (∀ (x : { x // x ∈ f.domain }), ↑g ↑x = ↑f x) ∧ ∀ (x : E), x ∈ s → 0 ≤ ↑g x",
"state_before": "𝕜 : Type ?u.280700\nE : Type u_1\nF : Type ?u.280706\nG : Type ?u.280709\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : ConvexCone ℝ E\nf : E →ₗ.[ℝ] ℝ\nnonneg : ∀ (x : { x // x ∈ f.domain }), ↑x ∈ s → 0 ≤ ↑f x\ndense : ∀ (y : E), ∃ x, ↑x + y ∈ s\n⊢ ∃ g, (∀ (x : { x // x ∈ f.domain }), ↑g ↑x = ↑f x) ∧ ∀ (x : E), x ∈ s → 0 ≤ ↑g x",
"tactic": "rcases RieszExtension.exists_top s f nonneg dense\n with ⟨⟨g_dom, g⟩, ⟨-, hfg⟩, rfl : g_dom = ⊤, hgs⟩"
},
{
"state_after": "case intro.mk.intro.intro.intro.refine'_1\n𝕜 : Type ?u.280700\nE : Type u_1\nF : Type ?u.280706\nG : Type ?u.280709\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : ConvexCone ℝ E\nf : E →ₗ.[ℝ] ℝ\nnonneg : ∀ (x : { x // x ∈ f.domain }), ↑x ∈ s → 0 ≤ ↑f x\ndense : ∀ (y : E), ∃ x, ↑x + y ∈ s\ng : { x // x ∈ ⊤ } →ₗ[ℝ] ℝ\nhfg :\n ∀ ⦃x : { x // x ∈ f.domain }⦄ ⦃y : { x // x ∈ { domain := ⊤, toFun := g }.domain }⦄,\n ↑x = ↑y → ↑f x = ↑{ domain := ⊤, toFun := g } y\nhgs : ∀ (x : { x // x ∈ { domain := ⊤, toFun := g }.domain }), ↑x ∈ s → 0 ≤ ↑{ domain := ⊤, toFun := g } x\n⊢ ∀ (x : { x // x ∈ f.domain }), ↑(comp g (LinearMap.codRestrict ⊤ LinearMap.id (_ : E → True))) ↑x = ↑f x\n\ncase intro.mk.intro.intro.intro.refine'_2\n𝕜 : Type ?u.280700\nE : Type u_1\nF : Type ?u.280706\nG : Type ?u.280709\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : ConvexCone ℝ E\nf : E →ₗ.[ℝ] ℝ\nnonneg : ∀ (x : { x // x ∈ f.domain }), ↑x ∈ s → 0 ≤ ↑f x\ndense : ∀ (y : E), ∃ x, ↑x + y ∈ s\ng : { x // x ∈ ⊤ } →ₗ[ℝ] ℝ\nhfg :\n ∀ ⦃x : { x // x ∈ f.domain }⦄ ⦃y : { x // x ∈ { domain := ⊤, toFun := g }.domain }⦄,\n ↑x = ↑y → ↑f x = ↑{ domain := ⊤, toFun := g } y\nhgs : ∀ (x : { x // x ∈ { domain := ⊤, toFun := g }.domain }), ↑x ∈ s → 0 ≤ ↑{ domain := ⊤, toFun := g } x\n⊢ ∀ (x : E), x ∈ s → 0 ≤ ↑(comp g (LinearMap.codRestrict ⊤ LinearMap.id (_ : E → True))) x",
"state_before": "case intro.mk.intro.intro.intro\n𝕜 : Type ?u.280700\nE : Type u_1\nF : Type ?u.280706\nG : Type ?u.280709\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : ConvexCone ℝ E\nf : E →ₗ.[ℝ] ℝ\nnonneg : ∀ (x : { x // x ∈ f.domain }), ↑x ∈ s → 0 ≤ ↑f x\ndense : ∀ (y : E), ∃ x, ↑x + y ∈ s\ng : { x // x ∈ ⊤ } →ₗ[ℝ] ℝ\nhfg :\n ∀ ⦃x : { x // x ∈ f.domain }⦄ ⦃y : { x // x ∈ { domain := ⊤, toFun := g }.domain }⦄,\n ↑x = ↑y → ↑f x = ↑{ domain := ⊤, toFun := g } y\nhgs : ∀ (x : { x // x ∈ { domain := ⊤, toFun := g }.domain }), ↑x ∈ s → 0 ≤ ↑{ domain := ⊤, toFun := g } x\n⊢ ∃ g, (∀ (x : { x // x ∈ f.domain }), ↑g ↑x = ↑f x) ∧ ∀ (x : E), x ∈ s → 0 ≤ ↑g x",
"tactic": "refine' ⟨g.comp (LinearMap.id.codRestrict ⊤ fun _ ↦ trivial), _, _⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.mk.intro.intro.intro.refine'_1\n𝕜 : Type ?u.280700\nE : Type u_1\nF : Type ?u.280706\nG : Type ?u.280709\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : ConvexCone ℝ E\nf : E →ₗ.[ℝ] ℝ\nnonneg : ∀ (x : { x // x ∈ f.domain }), ↑x ∈ s → 0 ≤ ↑f x\ndense : ∀ (y : E), ∃ x, ↑x + y ∈ s\ng : { x // x ∈ ⊤ } →ₗ[ℝ] ℝ\nhfg :\n ∀ ⦃x : { x // x ∈ f.domain }⦄ ⦃y : { x // x ∈ { domain := ⊤, toFun := g }.domain }⦄,\n ↑x = ↑y → ↑f x = ↑{ domain := ⊤, toFun := g } y\nhgs : ∀ (x : { x // x ∈ { domain := ⊤, toFun := g }.domain }), ↑x ∈ s → 0 ≤ ↑{ domain := ⊤, toFun := g } x\n⊢ ∀ (x : { x // x ∈ f.domain }), ↑(comp g (LinearMap.codRestrict ⊤ LinearMap.id (_ : E → True))) ↑x = ↑f x",
"tactic": "exact fun x => (hfg rfl).symm"
},
{
"state_after": "no goals",
"state_before": "case intro.mk.intro.intro.intro.refine'_2\n𝕜 : Type ?u.280700\nE : Type u_1\nF : Type ?u.280706\nG : Type ?u.280709\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : ConvexCone ℝ E\nf : E →ₗ.[ℝ] ℝ\nnonneg : ∀ (x : { x // x ∈ f.domain }), ↑x ∈ s → 0 ≤ ↑f x\ndense : ∀ (y : E), ∃ x, ↑x + y ∈ s\ng : { x // x ∈ ⊤ } →ₗ[ℝ] ℝ\nhfg :\n ∀ ⦃x : { x // x ∈ f.domain }⦄ ⦃y : { x // x ∈ { domain := ⊤, toFun := g }.domain }⦄,\n ↑x = ↑y → ↑f x = ↑{ domain := ⊤, toFun := g } y\nhgs : ∀ (x : { x // x ∈ { domain := ⊤, toFun := g }.domain }), ↑x ∈ s → 0 ≤ ↑{ domain := ⊤, toFun := g } x\n⊢ ∀ (x : E), x ∈ s → 0 ≤ ↑(comp g (LinearMap.codRestrict ⊤ LinearMap.id (_ : E → True))) x",
"tactic": "exact fun x hx => hgs ⟨x, _⟩ hx"
}
] |
[
826,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
818,
1
] |
Mathlib/SetTheory/Cardinal/Ordinal.lean
|
Cardinal.aleph0_mul_mk_eq
|
[] |
[
581,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
580,
1
] |
Mathlib/RingTheory/AdjoinRoot.lean
|
AdjoinRoot.mk_leftInverse
|
[
{
"state_after": "R : Type u\nS : Type v\nK : Type w\ninst✝ : CommRing R\ng : R[X]\nhg : Monic g\nf : AdjoinRoot g\n⊢ ↑(mk g) (↑(modByMonicHom hg) f) = f",
"state_before": "R : Type u\nS : Type v\nK : Type w\ninst✝ : CommRing R\ng : R[X]\nhg : Monic g\n⊢ Function.LeftInverse ↑(mk g) ↑(modByMonicHom hg)",
"tactic": "intro f"
},
{
"state_after": "case ih\nR : Type u\nS : Type v\nK : Type w\ninst✝ : CommRing R\ng : R[X]\nhg : Monic g\np✝ : R[X]\n⊢ ↑(mk g) (↑(modByMonicHom hg) (↑(mk g) p✝)) = ↑(mk g) p✝",
"state_before": "R : Type u\nS : Type v\nK : Type w\ninst✝ : CommRing R\ng : R[X]\nhg : Monic g\nf : AdjoinRoot g\n⊢ ↑(mk g) (↑(modByMonicHom hg) f) = f",
"tactic": "induction f using AdjoinRoot.induction_on"
},
{
"state_after": "case ih\nR : Type u\nS : Type v\nK : Type w\ninst✝ : CommRing R\ng : R[X]\nhg : Monic g\np✝ : R[X]\n⊢ g ∣ g * (p✝ /ₘ g)",
"state_before": "case ih\nR : Type u\nS : Type v\nK : Type w\ninst✝ : CommRing R\ng : R[X]\nhg : Monic g\np✝ : R[X]\n⊢ ↑(mk g) (↑(modByMonicHom hg) (↑(mk g) p✝)) = ↑(mk g) p✝",
"tactic": "rw [modByMonicHom_mk hg, mk_eq_mk, modByMonic_eq_sub_mul_div _ hg, sub_sub_cancel_left,\n dvd_neg]"
},
{
"state_after": "no goals",
"state_before": "case ih\nR : Type u\nS : Type v\nK : Type w\ninst✝ : CommRing R\ng : R[X]\nhg : Monic g\np✝ : R[X]\n⊢ g ∣ g * (p✝ /ₘ g)",
"tactic": "apply dvd_mul_right"
}
] |
[
469,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
464,
1
] |
Mathlib/Data/Nat/Order/Basic.lean
|
Nat.findGreatest_is_greatest
|
[] |
[
673,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
672,
1
] |
Mathlib/GroupTheory/SpecificGroups/Alternating.lean
|
alternatingGroup_eq_sign_ker
|
[] |
[
67,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
LinearMap.sum_apply
|
[] |
[
285,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
1
] |
Mathlib/Data/Rat/Defs.lean
|
Rat.mkRat_one
|
[
{
"state_after": "no goals",
"state_before": "a b c : ℚ\nn : ℤ\n⊢ mkRat n 1 = ↑n",
"tactic": "simp [Rat.mkRat_eq, Rat.divInt_one]"
}
] |
[
259,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
258,
1
] |
Mathlib/CategoryTheory/Limits/Preserves/Shapes/Biproducts.lean
|
CategoryTheory.Functor.mapBiprod_hom
|
[] |
[
403,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
402,
1
] |
Mathlib/Data/Real/ConjugateExponents.lean
|
Real.IsConjugateExponent.one_div_ne_zero
|
[] |
[
71,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Mathlib/Algebra/RingQuot.lean
|
RingQuot.Rel.sub_right
|
[
{
"state_after": "no goals",
"state_before": "R✝ : Type u₁\ninst✝⁴ : Semiring R✝\nS : Type u₂\ninst✝³ : CommSemiring S\nA : Type u₃\ninst✝² : Semiring A\ninst✝¹ : Algebra S A\nR : Type u₁\ninst✝ : Ring R\nr : R → R → Prop\na b c : R\nh : Rel r b c\n⊢ Rel r (a - b) (a - c)",
"tactic": "simp only [sub_eq_add_neg, h.neg.add_right]"
}
] |
[
81,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
80,
1
] |
Mathlib/GroupTheory/Perm/Fin.lean
|
Fin.coe_cycleRange_of_lt
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\ni j : Fin (Nat.succ n)\nh : j < i\n⊢ ↑(↑(cycleRange i) j) = ↑j + 1",
"tactic": "rw [coe_cycleRange_of_le h.le, if_neg h.ne]"
}
] |
[
207,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
Mathlib/RingTheory/ClassGroup.lean
|
ClassGroup.mk0_eq_one_iff
|
[] |
[
386,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
384,
1
] |
Mathlib/LinearAlgebra/SesquilinearForm.lean
|
LinearMap.IsRefl.ker_flip_eq_bot
|
[
{
"state_after": "R : Type u_1\nR₁ : Type u_2\nR₂ : Type ?u.117040\nR₃ : Type ?u.117043\nM : Type ?u.117046\nM₁ : Type u_3\nM₂ : Type ?u.117052\nMₗ₁ : Type ?u.117055\nMₗ₁' : Type ?u.117058\nMₗ₂ : Type ?u.117061\nMₗ₂' : Type ?u.117064\nK : Type ?u.117067\nK₁ : Type ?u.117070\nK₂ : Type ?u.117073\nV : Type ?u.117076\nV₁ : Type ?u.117079\nV₂ : Type ?u.117082\nn : Type ?u.117085\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring R₁\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nI₁ I₂ : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\nH✝ H : IsRefl B\nh : ker B = ⊥\nx✝ : M₁\nhx : ↑(flip B) x✝ = 0\n⊢ ↑B x✝ = 0",
"state_before": "R : Type u_1\nR₁ : Type u_2\nR₂ : Type ?u.117040\nR₃ : Type ?u.117043\nM : Type ?u.117046\nM₁ : Type u_3\nM₂ : Type ?u.117052\nMₗ₁ : Type ?u.117055\nMₗ₁' : Type ?u.117058\nMₗ₂ : Type ?u.117061\nMₗ₂' : Type ?u.117064\nK : Type ?u.117067\nK₁ : Type ?u.117070\nK₂ : Type ?u.117073\nV : Type ?u.117076\nV₁ : Type ?u.117079\nV₂ : Type ?u.117082\nn : Type ?u.117085\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring R₁\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nI₁ I₂ : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\nH✝ H : IsRefl B\nh : ker B = ⊥\n⊢ ker (flip B) = ⊥",
"tactic": "refine' ker_eq_bot'.mpr fun _ hx ↦ ker_eq_bot'.mp h _ _"
},
{
"state_after": "case h\nR : Type u_1\nR₁ : Type u_2\nR₂ : Type ?u.117040\nR₃ : Type ?u.117043\nM : Type ?u.117046\nM₁ : Type u_3\nM₂ : Type ?u.117052\nMₗ₁ : Type ?u.117055\nMₗ₁' : Type ?u.117058\nMₗ₂ : Type ?u.117061\nMₗ₂' : Type ?u.117064\nK : Type ?u.117067\nK₁ : Type ?u.117070\nK₂ : Type ?u.117073\nV : Type ?u.117076\nV₁ : Type ?u.117079\nV₂ : Type ?u.117082\nn : Type ?u.117085\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring R₁\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nI₁ I₂ : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\nH✝ H : IsRefl B\nh : ker B = ⊥\nx✝¹ : M₁\nhx : ↑(flip B) x✝¹ = 0\nx✝ : M₁\n⊢ ↑(↑B x✝¹) x✝ = ↑0 x✝",
"state_before": "R : Type u_1\nR₁ : Type u_2\nR₂ : Type ?u.117040\nR₃ : Type ?u.117043\nM : Type ?u.117046\nM₁ : Type u_3\nM₂ : Type ?u.117052\nMₗ₁ : Type ?u.117055\nMₗ₁' : Type ?u.117058\nMₗ₂ : Type ?u.117061\nMₗ₂' : Type ?u.117064\nK : Type ?u.117067\nK₁ : Type ?u.117070\nK₂ : Type ?u.117073\nV : Type ?u.117076\nV₁ : Type ?u.117079\nV₂ : Type ?u.117082\nn : Type ?u.117085\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring R₁\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nI₁ I₂ : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\nH✝ H : IsRefl B\nh : ker B = ⊥\nx✝ : M₁\nhx : ↑(flip B) x✝ = 0\n⊢ ↑B x✝ = 0",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_1\nR₁ : Type u_2\nR₂ : Type ?u.117040\nR₃ : Type ?u.117043\nM : Type ?u.117046\nM₁ : Type u_3\nM₂ : Type ?u.117052\nMₗ₁ : Type ?u.117055\nMₗ₁' : Type ?u.117058\nMₗ₂ : Type ?u.117061\nMₗ₂' : Type ?u.117064\nK : Type ?u.117067\nK₁ : Type ?u.117070\nK₂ : Type ?u.117073\nV : Type ?u.117076\nV₁ : Type ?u.117079\nV₂ : Type ?u.117082\nn : Type ?u.117085\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring R₁\ninst✝¹ : AddCommMonoid M₁\ninst✝ : Module R₁ M₁\nI₁ I₂ : R₁ →+* R\nB : M₁ →ₛₗ[I₁] M₁ →ₛₗ[I₂] R\nH✝ H : IsRefl B\nh : ker B = ⊥\nx✝¹ : M₁\nhx : ↑(flip B) x✝¹ = 0\nx✝ : M₁\n⊢ ↑(↑B x✝¹) x✝ = ↑0 x✝",
"tactic": "exact H _ _ (LinearMap.congr_fun hx _)"
}
] |
[
199,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Mathlib/RingTheory/IntegralClosure.lean
|
Algebra.isIntegral_trans
|
[] |
[
1017,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1015,
8
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean
|
DifferentiableOn.add_const
|
[] |
[
220,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Deriv.lean
|
HasStrictFDerivAt.cpow
|
[
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf g : E → ℂ\nf' g' : E →L[ℂ] ℂ\nx : E\ns : Set E\nc : ℂ\nhf : HasStrictFDerivAt f f' x\nhg : HasStrictFDerivAt g g' x\nh0 : 0 < (f x).re ∨ (f x).im ≠ 0\n⊢ HasStrictFDerivAt (fun x => f x ^ g x) ((g x * f x ^ (g x - 1)) • f' + (f x ^ g x * Complex.log (f x)) • g') x",
"tactic": "convert (@hasStrictFDerivAt_cpow ((fun x => (f x, g x)) x) h0).comp x (hf.prod hg)"
}
] |
[
85,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Complex.cos_nat_mul_two_pi_sub
|
[] |
[
1256,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1255,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
|
subset_affineSpan
|
[] |
[
554,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
553,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
|
MeasureTheory.exists_measurable_superset_forall_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.14786\nγ : Type ?u.14789\nδ : Type ?u.14792\nι✝ : Type ?u.14795\ninst✝¹ : MeasurableSpace α\nμ✝ μ₁ μ₂ : Measure α\ns✝ s₁ s₂ t : Set α\nι : Sort u_1\ninst✝ : Countable ι\nμ : ι → Measure α\ns : Set α\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ∀ (i : ι), ↑↑(μ i) t = ↑↑(μ i) s",
"tactic": "simpa only [← measure_eq_trim] using\n OuterMeasure.exists_measurable_superset_forall_eq_trim (fun i => (μ i).toOuterMeasure) s"
}
] |
[
218,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
1
] |
Mathlib/Algebra/Module/Equiv.lean
|
LinearEquiv.trans_refl
|
[] |
[
398,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
397,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
OrderIso.map_csInf
|
[] |
[
1346,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1344,
1
] |
Mathlib/Algebra/BigOperators/Finsupp.lean
|
Finsupp.multiset_map_sum
|
[] |
[
566,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
564,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPowerSeries.coeff_zero_mul_X
|
[
{
"state_after": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nφ : MvPowerSeries σ R\ns : σ\nthis : ¬single s 1 ≤ 0\n⊢ ↑(coeff R 0) (φ * X s) = 0",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nφ : MvPowerSeries σ R\ns : σ\n⊢ ↑(coeff R 0) (φ * X s) = 0",
"tactic": "have : ¬single s 1 ≤ 0 := fun h => by simpa using h s"
},
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nφ : MvPowerSeries σ R\ns : σ\nthis : ¬single s 1 ≤ 0\n⊢ ↑(coeff R 0) (φ * X s) = 0",
"tactic": "simp only [X, coeff_mul_monomial, if_neg this]"
},
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nφ : MvPowerSeries σ R\ns : σ\nh : single s 1 ≤ 0\n⊢ False",
"tactic": "simpa using h s"
}
] |
[
461,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
459,
1
] |
Mathlib/CategoryTheory/Whiskering.lean
|
CategoryTheory.isoWhiskerLeft_hom
|
[] |
[
164,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/Order/Zorn.lean
|
zorn_nonempty_Ici₀
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.6442\nr : α → α → Prop\nc : Set α\ninst✝ : Preorder α\na : α\nih :\n ∀ (c : Set α),\n c ⊆ Ici a → IsChain (fun x x_1 => x ≤ x_1) c → ∀ (y : α), y ∈ c → ∃ ub, a ≤ ub ∧ ∀ (z : α), z ∈ c → z ≤ ub\nx : α\nhax : a ≤ x\n⊢ ∀ (c : Set α),\n c ⊆ Ici a → IsChain (fun x x_1 => x ≤ x_1) c → ∀ (y : α), y ∈ c → ∃ ub, ub ∈ Ici a ∧ ∀ (z : α), z ∈ c → z ≤ ub",
"tactic": "simpa using ih"
}
] |
[
152,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
148,
1
] |
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