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/LinearAlgebra/AdicCompletion.lean
|
IsAdicComplete.subsingleton
|
[] |
[
280,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
279,
11
] |
Mathlib/MeasureTheory/Integral/IntegrableOn.lean
|
integrableOn_Iic_iff_integrableOn_Iio'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.3764222\nE : Type u_2\nF : Type ?u.3764228\ninst✝³ : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : PartialOrder α\ninst✝ : MeasurableSingletonClass α\nf : α → E\nμ : MeasureTheory.Measure α\na b : α\nhb : ↑↑μ {b} ≠ ⊤\n⊢ IntegrableOn f (Iic b) ↔ IntegrableOn f (Iio b)",
"tactic": "rw [← Iio_union_right, integrableOn_union,\n eq_true (integrableOn_singleton_iff.mpr <| Or.inr hb.lt_top), and_true_iff]"
}
] |
[
672,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
669,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.map_sub_int
|
[] |
[
373,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
372,
1
] |
Mathlib/Data/Sum/Basic.lean
|
Sum.inl_injective
|
[] |
[
65,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
|
NNReal.one_rpow
|
[] |
[
77,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
76,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Basic.lean
|
differentiableWithinAt_univ
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.301608\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.301703\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\n⊢ DifferentiableWithinAt 𝕜 f univ x ↔ DifferentiableAt 𝕜 f x",
"tactic": "simp only [DifferentiableWithinAt, hasFDerivWithinAt_univ, DifferentiableAt]"
}
] |
[
581,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
580,
1
] |
Mathlib/NumberTheory/Zsqrtd/Basic.lean
|
Zsqrtd.norm_nat_cast
|
[] |
[
543,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
542,
1
] |
Mathlib/Order/BooleanAlgebra.lean
|
inf_sdiff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.34896\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x ⊓ y ⊓ z ⊔ x \\ z ⊓ y \\ z = (x ⊓ y ⊓ z ⊔ x \\ z) ⊓ (x ⊓ y ⊓ z ⊔ y \\ z)",
"tactic": "rw [sup_inf_left]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.34896\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ (x ⊓ y ⊓ z ⊔ x \\ z) ⊓ (x ⊓ y ⊓ z ⊔ y \\ z) = (x ⊓ y ⊓ (z ⊔ x) ⊔ x \\ z) ⊓ (x ⊓ y ⊓ z ⊔ y \\ z)",
"tactic": "rw [sup_inf_right, sup_sdiff_self_right, inf_sup_right, inf_sdiff_sup_right]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.34896\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ (x ⊓ y ⊓ (z ⊔ x) ⊔ x \\ z) ⊓ (x ⊓ y ⊓ z ⊔ y \\ z) = (y ⊓ (x ⊓ (x ⊔ z)) ⊔ x \\ z) ⊓ (x ⊓ y ⊓ z ⊔ y \\ z)",
"tactic": "ac_rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.34896\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ (y ⊓ (x ⊓ (x ⊔ z)) ⊔ x \\ z) ⊓ (x ⊓ y ⊓ z ⊔ y \\ z) = (y ⊓ x ⊔ x \\ z) ⊓ (x ⊓ y ⊔ y \\ z)",
"tactic": "rw [inf_sup_self, sup_inf_inf_sdiff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.34896\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ (y ⊓ x ⊔ x \\ z) ⊓ (x ⊓ y ⊔ y \\ z) = x ⊓ y ⊔ x \\ z ⊓ y \\ z",
"tactic": "rw [@inf_comm _ _ y, sup_inf_left]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.34896\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x ⊓ y ⊓ z ⊓ (x \\ z ⊓ y \\ z) = x ⊓ y ⊓ (z ⊓ x \\ z) ⊓ y \\ z",
"tactic": "ac_rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.34896\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x ⊓ y ⊓ (z ⊓ x \\ z) ⊓ y \\ z = ⊥",
"tactic": "rw [inf_sdiff_self_right, inf_bot_eq, bot_inf_eq]"
}
] |
[
435,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
423,
1
] |
Mathlib/Data/Nat/Interval.lean
|
Finset.range_eq_Ico
|
[] |
[
93,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Topology/ContinuousFunction/Basic.lean
|
ContinuousMap.comp_id
|
[] |
[
262,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
261,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.prod_coe_symm
|
[] |
[
958,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
956,
1
] |
Mathlib/LinearAlgebra/TensorProduct.lean
|
TensorProduct.curry_apply
|
[] |
[
586,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
585,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.ofReal_re
|
[] |
[
95,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
94,
1
] |
Mathlib/Analysis/NormedSpace/BoundedLinearMaps.lean
|
IsBoundedBilinearMap.isBoundedLinearMap_right
|
[] |
[
438,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
436,
1
] |
Mathlib/CategoryTheory/Bicategory/Basic.lean
|
CategoryTheory.Bicategory.leftUnitor_whiskerRight
|
[
{
"state_after": "no goals",
"state_before": "B : Type u\ninst✝ : Bicategory B\na b c d e : B\nf : a ⟶ b\ng : b ⟶ c\n⊢ (λ_ f).hom ▷ g = (α_ (𝟙 a) f g).hom ≫ (λ_ (f ≫ g)).hom",
"tactic": "rw [← whiskerLeft_iff, whiskerLeft_comp, ← cancel_epi (α_ _ _ _).hom, ←\n cancel_epi ((α_ _ _ _).hom ▷ _), pentagon_assoc, triangle, ← associator_naturality_middle, ←\n comp_whiskerRight_assoc, triangle, associator_naturality_left]"
}
] |
[
431,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
427,
1
] |
Mathlib/Algebra/Order/SMul.lean
|
le_inv_smul_iff
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.113737\n𝕜 : Type u_1\nR : Type ?u.113743\nM : Type u_2\nN : Type ?u.113749\ninst✝⁵ : LinearOrderedSemifield 𝕜\ninst✝⁴ : OrderedAddCommMonoid M\ninst✝³ : OrderedAddCommMonoid N\ninst✝² : MulActionWithZero 𝕜 M\ninst✝¹ : MulActionWithZero 𝕜 N\ninst✝ : OrderedSMul 𝕜 M\ns : Set M\na b : M\nc : 𝕜\nh : 0 < c\n⊢ a ≤ c⁻¹ • b ↔ c • a ≤ b",
"tactic": "rw [← smul_le_smul_iff_of_pos h, smul_inv_smul₀ h.ne']"
}
] |
[
256,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
255,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.smul_eq_mul_diagonal
|
[
{
"state_after": "case a.h\nl : Type ?u.482923\nm : Type u_2\nn : Type u_1\no : Type ?u.482932\nm' : o → Type ?u.482937\nn' : o → Type ?u.482942\nR : Type ?u.482945\nS : Type ?u.482948\nα : Type v\nβ : Type w\nγ : Type ?u.482955\ninst✝² : CommSemiring α\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nM : Matrix m n α\na : α\ni✝ : m\nx✝ : n\n⊢ (a • M) i✝ x✝ = (M ⬝ diagonal fun x => a) i✝ x✝",
"state_before": "l : Type ?u.482923\nm : Type u_2\nn : Type u_1\no : Type ?u.482932\nm' : o → Type ?u.482937\nn' : o → Type ?u.482942\nR : Type ?u.482945\nS : Type ?u.482948\nα : Type v\nβ : Type w\nγ : Type ?u.482955\ninst✝² : CommSemiring α\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nM : Matrix m n α\na : α\n⊢ a • M = M ⬝ diagonal fun x => a",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a.h\nl : Type ?u.482923\nm : Type u_2\nn : Type u_1\no : Type ?u.482932\nm' : o → Type ?u.482937\nn' : o → Type ?u.482942\nR : Type ?u.482945\nS : Type ?u.482948\nα : Type v\nβ : Type w\nγ : Type ?u.482955\ninst✝² : CommSemiring α\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nM : Matrix m n α\na : α\ni✝ : m\nx✝ : n\n⊢ (a • M) i✝ x✝ = (M ⬝ diagonal fun x => a) i✝ x✝",
"tactic": "simp [mul_comm]"
}
] |
[
1284,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1281,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.tendsto_atBot_add_nonpos_right
|
[] |
[
646,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
644,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
ConcaveOn.comp_convexOn
|
[] |
[
151,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/Data/List/Zip.lean
|
List.zip_of_prod
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type u_1\nγ : Type ?u.87473\nδ : Type ?u.87476\nε : Type ?u.87479\nl : List α\nl' : List β\nlp : List (α × β)\nhl : map Prod.fst lp = l\nhr : map Prod.snd lp = l'\n⊢ lp = zip l l'",
"tactic": "rw [← hl, ← hr, ← zip_unzip lp, ← unzip_left, ← unzip_right, zip_unzip, zip_unzip]"
}
] |
[
253,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
251,
1
] |
Mathlib/Data/Nat/PartENat.lean
|
PartENat.ne_top_iff
|
[
{
"state_after": "no goals",
"state_before": "x : PartENat\n⊢ x ≠ ⊤ ↔ ∃ n, x = ↑n",
"tactic": "simpa only [← some_eq_natCast] using Part.ne_none_iff"
}
] |
[
366,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
365,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Mul.lean
|
DifferentiableWithinAt.mul_const
|
[] |
[
425,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
423,
1
] |
Mathlib/Algebra/Ring/Commute.lean
|
Commute.neg_left_iff
|
[] |
[
109,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean
|
Ideal.homogeneousHull.gc
|
[] |
[
615,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
613,
1
] |
Mathlib/Data/List/Rotate.lean
|
List.rotate_perm
|
[
{
"state_after": "α : Type u\nl : List α\nn : ℕ\n⊢ rotate' l n ~ l",
"state_before": "α : Type u\nl : List α\nn : ℕ\n⊢ rotate l n ~ l",
"tactic": "rw [rotate_eq_rotate']"
},
{
"state_after": "case zero\nα : Type u\nl✝ l : List α\n⊢ rotate' l zero ~ l\n\ncase succ\nα : Type u\nl✝ : List α\nn : ℕ\nhn : ∀ (l : List α), rotate' l n ~ l\nl : List α\n⊢ rotate' l (succ n) ~ l",
"state_before": "α : Type u\nl : List α\nn : ℕ\n⊢ rotate' l n ~ l",
"tactic": "induction' n with n hn generalizing l"
},
{
"state_after": "no goals",
"state_before": "case zero\nα : Type u\nl✝ l : List α\n⊢ rotate' l zero ~ l",
"tactic": "simp"
},
{
"state_after": "case succ.nil\nα : Type u\nl : List α\nn : ℕ\nhn : ∀ (l : List α), rotate' l n ~ l\n⊢ rotate' [] (succ n) ~ []\n\ncase succ.cons\nα : Type u\nl : List α\nn : ℕ\nhn : ∀ (l : List α), rotate' l n ~ l\nhd : α\ntl : List α\n⊢ rotate' (hd :: tl) (succ n) ~ hd :: tl",
"state_before": "case succ\nα : Type u\nl✝ : List α\nn : ℕ\nhn : ∀ (l : List α), rotate' l n ~ l\nl : List α\n⊢ rotate' l (succ n) ~ l",
"tactic": "cases' l with hd tl"
},
{
"state_after": "no goals",
"state_before": "case succ.nil\nα : Type u\nl : List α\nn : ℕ\nhn : ∀ (l : List α), rotate' l n ~ l\n⊢ rotate' [] (succ n) ~ []",
"tactic": "simp"
},
{
"state_after": "case succ.cons\nα : Type u\nl : List α\nn : ℕ\nhn : ∀ (l : List α), rotate' l n ~ l\nhd : α\ntl : List α\n⊢ rotate' (tl ++ [hd]) n ~ hd :: tl",
"state_before": "case succ.cons\nα : Type u\nl : List α\nn : ℕ\nhn : ∀ (l : List α), rotate' l n ~ l\nhd : α\ntl : List α\n⊢ rotate' (hd :: tl) (succ n) ~ hd :: tl",
"tactic": "rw [rotate'_cons_succ]"
},
{
"state_after": "no goals",
"state_before": "case succ.cons\nα : Type u\nl : List α\nn : ℕ\nhn : ∀ (l : List α), rotate' l n ~ l\nhd : α\ntl : List α\n⊢ rotate' (tl ++ [hd]) n ~ hd :: tl",
"tactic": "exact (hn _).trans (perm_append_singleton _ _)"
}
] |
[
193,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/Data/Set/Semiring.lean
|
SetSemiring.down_one
|
[] |
[
198,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
1
] |
Mathlib/Order/Lattice.lean
|
inf_right_comm
|
[] |
[
528,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
527,
1
] |
Mathlib/GroupTheory/Coset.lean
|
normal_of_eq_cosets
|
[
{
"state_after": "α : Type u_1\ninst✝ : Group α\ns : Subgroup α\nh : ∀ (g : α), g *l ↑s = ↑s *r g\na : α\nha : a ∈ s\ng : α\n⊢ g * a ∈ g *l ↑s",
"state_before": "α : Type u_1\ninst✝ : Group α\ns : Subgroup α\nh : ∀ (g : α), g *l ↑s = ↑s *r g\na : α\nha : a ∈ s\ng : α\n⊢ g * a * g⁻¹ ∈ ↑s",
"tactic": "rw [← mem_rightCoset_iff, ← h]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : Group α\ns : Subgroup α\nh : ∀ (g : α), g *l ↑s = ↑s *r g\na : α\nha : a ∈ s\ng : α\n⊢ g * a ∈ g *l ↑s",
"tactic": "exact mem_leftCoset g ha"
}
] |
[
261,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
259,
1
] |
Mathlib/GroupTheory/SpecificGroups/Alternating.lean
|
alternatingGroup.nontrivial_of_three_le_card
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh3 : 3 ≤ card α\nthis : Nontrivial α\n⊢ Nontrivial { x // x ∈ alternatingGroup α }",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh3 : 3 ≤ card α\n⊢ Nontrivial { x // x ∈ alternatingGroup α }",
"tactic": "haveI := Fintype.one_lt_card_iff_nontrivial.1 (lt_trans (by decide) h3)"
},
{
"state_after": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh3 : 3 ≤ card α\nthis : Nontrivial α\n⊢ 1 < card { x // x ∈ alternatingGroup α }",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh3 : 3 ≤ card α\nthis : Nontrivial α\n⊢ Nontrivial { x // x ∈ alternatingGroup α }",
"tactic": "rw [← Fintype.one_lt_card_iff_nontrivial]"
},
{
"state_after": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh3 : 3 ≤ card α\nthis : Nontrivial α\n⊢ 2 * 1 < 2 * card { x // x ∈ alternatingGroup α }",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh3 : 3 ≤ card α\nthis : Nontrivial α\n⊢ 1 < card { x // x ∈ alternatingGroup α }",
"tactic": "refine' lt_of_mul_lt_mul_left _ (le_of_lt Nat.prime_two.pos)"
},
{
"state_after": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh3 : 3 ≤ card α\nthis : Nontrivial α\n⊢ Nat.succ (2 * 1) ≤ Nat.factorial (card α)",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh3 : 3 ≤ card α\nthis : Nontrivial α\n⊢ 2 * 1 < 2 * card { x // x ∈ alternatingGroup α }",
"tactic": "rw [two_mul_card_alternatingGroup, card_perm, ← Nat.succ_le_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh3 : 3 ≤ card α\nthis : Nontrivial α\n⊢ Nat.succ (2 * 1) ≤ Nat.factorial (card α)",
"tactic": "exact le_trans h3 (card α).self_le_factorial"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh3 : 3 ≤ card α\n⊢ 1 < 2",
"tactic": "decide"
}
] |
[
224,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
1
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.mul_inv_cancel
|
[
{
"state_after": "K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\np : FractionRing K[X]\nh : { toFractionRing := p } ≠ 0\nthis : p ≠ 0\n⊢ { toFractionRing := p } * { toFractionRing := p }⁻¹ = 1",
"state_before": "K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\np : FractionRing K[X]\nh : { toFractionRing := p } ≠ 0\n⊢ { toFractionRing := p } * { toFractionRing := p }⁻¹ = 1",
"tactic": "have : p ≠ 0 := fun hp => h <| by rw [hp, ofFractionRing_zero]"
},
{
"state_after": "no goals",
"state_before": "K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\np : FractionRing K[X]\nh : { toFractionRing := p } ≠ 0\nthis : p ≠ 0\n⊢ { toFractionRing := p } * { toFractionRing := p }⁻¹ = 1",
"tactic": "simpa only [← ofFractionRing_inv, ← ofFractionRing_mul, ← ofFractionRing_one,\n ofFractionRing.injEq] using _root_.mul_inv_cancel this"
},
{
"state_after": "no goals",
"state_before": "K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\np : FractionRing K[X]\nh : { toFractionRing := p } ≠ 0\nhp : p = 0\n⊢ { toFractionRing := p } = 0",
"tactic": "rw [hp, ofFractionRing_zero]"
}
] |
[
426,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
421,
1
] |
Mathlib/RingTheory/Polynomial/ScaleRoots.lean
|
Polynomial.scaleRoots_aeval_eq_zero_of_aeval_div_eq_zero
|
[] |
[
140,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
136,
1
] |
Mathlib/Data/Num/Lemmas.lean
|
ZNum.cast_to_int
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\n⊢ ↑↑0 = ↑0",
"tactic": "rw [cast_zero, cast_zero, Int.cast_zero]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\np : PosNum\n⊢ ↑↑(pos p) = ↑(pos p)",
"tactic": "rw [cast_pos, cast_pos, PosNum.cast_to_int]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : AddGroupWithOne α\np : PosNum\n⊢ ↑↑(neg p) = ↑(neg p)",
"tactic": "rw [cast_neg, cast_neg, Int.cast_neg, PosNum.cast_to_int]"
}
] |
[
1113,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1110,
1
] |
Mathlib/CategoryTheory/Elements.lean
|
CategoryTheory.CategoryOfElements.fromStructuredArrow_obj
|
[] |
[
164,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
163,
1
] |
Mathlib/Order/Filter/Prod.lean
|
Filter.mem_coprod_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.53677\nδ : Type ?u.53680\nι : Sort ?u.53683\nf✝ : Filter α\ng✝ : Filter β\ns : Set (α × β)\nf : Filter α\ng : Filter β\n⊢ s ∈ Filter.coprod f g ↔ (∃ t₁, t₁ ∈ f ∧ Prod.fst ⁻¹' t₁ ⊆ s) ∧ ∃ t₂, t₂ ∈ g ∧ Prod.snd ⁻¹' t₂ ⊆ s",
"tactic": "simp [Filter.coprod]"
}
] |
[
476,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
474,
1
] |
Mathlib/Data/List/Nodup.lean
|
List.Nodup.product
|
[
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nl l₁ l₂✝ : List α\nr : α → α → Prop\na b : α\nl₂ : List β\nd₁ : Nodup l₁\nd₂ : Nodup l₂\na₁ a₂ : α\nn : a₁ ≠ a₂\nb₁ : β\nleft✝ : b₁ ∈ l₂\nh₁ : (a₁, b₁) ∈ map (Prod.mk a₁) l₂\nh₂ : (a₁, b₁) ∈ map (Prod.mk a₂) l₂\n⊢ False",
"state_before": "α : Type u\nβ : Type v\nl l₁ l₂✝ : List α\nr : α → α → Prop\na b : α\nl₂ : List β\nd₁ : Nodup l₁\nd₂ : Nodup l₂\na₁ a₂ : α\nn : a₁ ≠ a₂\nx : α × β\nh₁ : x ∈ map (Prod.mk a₁) l₂\nh₂ : x ∈ map (Prod.mk a₂) l₂\n⊢ False",
"tactic": "rcases mem_map.1 h₁ with ⟨b₁, _, rfl⟩"
},
{
"state_after": "case intro.intro.intro.intro.refl\nα : Type u\nβ : Type v\nl l₁ l₂✝ : List α\nr : α → α → Prop\na b : α\nl₂ : List β\nd₁ : Nodup l₁\nd₂ : Nodup l₂\na₁ : α\nb₁ : β\nleft✝ : b₁ ∈ l₂\nh₁ : (a₁, b₁) ∈ map (Prod.mk a₁) l₂\nn : a₁ ≠ a₁\nh₂ : (a₁, b₁) ∈ map (Prod.mk a₁) l₂\nmb₂ : b₁ ∈ l₂\n⊢ False",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nl l₁ l₂✝ : List α\nr : α → α → Prop\na b : α\nl₂ : List β\nd₁ : Nodup l₁\nd₂ : Nodup l₂\na₁ a₂ : α\nn : a₁ ≠ a₂\nb₁ : β\nleft✝ : b₁ ∈ l₂\nh₁ : (a₁, b₁) ∈ map (Prod.mk a₁) l₂\nh₂ : (a₁, b₁) ∈ map (Prod.mk a₂) l₂\n⊢ False",
"tactic": "rcases mem_map.1 h₂ with ⟨b₂, mb₂, ⟨⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.refl\nα : Type u\nβ : Type v\nl l₁ l₂✝ : List α\nr : α → α → Prop\na b : α\nl₂ : List β\nd₁ : Nodup l₁\nd₂ : Nodup l₂\na₁ : α\nb₁ : β\nleft✝ : b₁ ∈ l₂\nh₁ : (a₁, b₁) ∈ map (Prod.mk a₁) l₂\nn : a₁ ≠ a₁\nh₂ : (a₁, b₁) ∈ map (Prod.mk a₁) l₂\nmb₂ : b₁ ∈ l₂\n⊢ False",
"tactic": "exact n rfl"
}
] |
[
353,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
346,
11
] |
Mathlib/Data/Set/Basic.lean
|
Set.empty_diff
|
[] |
[
1898,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1897,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.coe_sup_of_nonempty
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.291052\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.291061\nι : Type ?u.291064\nκ : Type ?u.291067\ninst✝¹ : SemilatticeSup α\ninst✝ : OrderBot α\ns : Finset β\nh : Finset.Nonempty s\nf : β → α\n⊢ ↑(sup s f) = sup s (WithBot.some ∘ f)",
"tactic": "simp only [← sup'_eq_sup h, coe_sup' h]"
}
] |
[
1002,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1001,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
|
Real.zero_rpow_eq_iff
|
[
{
"state_after": "case mp\nx a : ℝ\n⊢ 0 ^ x = a → x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1\n\ncase mpr\nx a : ℝ\n⊢ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1 → 0 ^ x = a",
"state_before": "x a : ℝ\n⊢ 0 ^ x = a ↔ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1",
"tactic": "constructor"
},
{
"state_after": "case mp\nx a : ℝ\nhyp : 0 ^ x = a\n⊢ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1",
"state_before": "case mp\nx a : ℝ\n⊢ 0 ^ x = a → x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1",
"tactic": "intro hyp"
},
{
"state_after": "case mp\nx a : ℝ\nhyp : (0 ^ ↑x).re = a\n⊢ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1",
"state_before": "case mp\nx a : ℝ\nhyp : 0 ^ x = a\n⊢ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1",
"tactic": "simp only [rpow_def, Complex.ofReal_zero] at hyp"
},
{
"state_after": "case pos\nx a : ℝ\nhyp : (0 ^ ↑x).re = a\nh : x = 0\n⊢ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1\n\ncase neg\nx a : ℝ\nhyp : (0 ^ ↑x).re = a\nh : ¬x = 0\n⊢ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1",
"state_before": "case mp\nx a : ℝ\nhyp : (0 ^ ↑x).re = a\n⊢ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1",
"tactic": "by_cases x = 0"
},
{
"state_after": "case pos\na : ℝ\nhyp : (0 ^ ↑0).re = a\n⊢ 0 ≠ 0 ∧ a = 0 ∨ 0 = 0 ∧ a = 1",
"state_before": "case pos\nx a : ℝ\nhyp : (0 ^ ↑x).re = a\nh : x = 0\n⊢ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1",
"tactic": "subst h"
},
{
"state_after": "case pos\na : ℝ\nhyp : 1 = a\n⊢ 0 ≠ 0 ∧ a = 0 ∨ 0 = 0 ∧ a = 1",
"state_before": "case pos\na : ℝ\nhyp : (0 ^ ↑0).re = a\n⊢ 0 ≠ 0 ∧ a = 0 ∨ 0 = 0 ∧ a = 1",
"tactic": "simp only [Complex.one_re, Complex.ofReal_zero, Complex.cpow_zero] at hyp"
},
{
"state_after": "no goals",
"state_before": "case pos\na : ℝ\nhyp : 1 = a\n⊢ 0 ≠ 0 ∧ a = 0 ∨ 0 = 0 ∧ a = 1",
"tactic": "exact Or.inr ⟨rfl, hyp.symm⟩"
},
{
"state_after": "case neg\nx a : ℝ\nhyp : 0.re = a\nh : ¬x = 0\n⊢ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1",
"state_before": "case neg\nx a : ℝ\nhyp : (0 ^ ↑x).re = a\nh : ¬x = 0\n⊢ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1",
"tactic": "rw [Complex.zero_cpow (Complex.ofReal_ne_zero.mpr h)] at hyp"
},
{
"state_after": "no goals",
"state_before": "case neg\nx a : ℝ\nhyp : 0.re = a\nh : ¬x = 0\n⊢ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1",
"tactic": "exact Or.inl ⟨h, hyp.symm⟩"
},
{
"state_after": "case mpr.inl.intro\nx : ℝ\nh : x ≠ 0\n⊢ 0 ^ x = 0\n\ncase mpr.inr.intro\n\n⊢ 0 ^ 0 = 1",
"state_before": "case mpr\nx a : ℝ\n⊢ x ≠ 0 ∧ a = 0 ∨ x = 0 ∧ a = 1 → 0 ^ x = a",
"tactic": "rintro (⟨h, rfl⟩ | ⟨rfl, rfl⟩)"
},
{
"state_after": "no goals",
"state_before": "case mpr.inl.intro\nx : ℝ\nh : x ≠ 0\n⊢ 0 ^ x = 0",
"tactic": "exact zero_rpow h"
},
{
"state_after": "no goals",
"state_before": "case mpr.inr.intro\n\n⊢ 0 ^ 0 = 1",
"tactic": "exact rpow_zero _"
}
] |
[
121,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
109,
1
] |
Mathlib/LinearAlgebra/Span.lean
|
Submodule.span_nat_eq_addSubmonoid_closure
|
[
{
"state_after": "R : Type ?u.32732\nR₂ : Type ?u.32735\nK : Type ?u.32738\nM : Type u_1\nM₂ : Type ?u.32744\nV : Type ?u.32747\nS : Type ?u.32750\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns✝ t s : Set M\n⊢ (span ℕ s).toAddSubmonoid ≤ AddSubmonoid.closure s",
"state_before": "R : Type ?u.32732\nR₂ : Type ?u.32735\nK : Type ?u.32738\nM : Type u_1\nM₂ : Type ?u.32744\nV : Type ?u.32747\nS : Type ?u.32750\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns✝ t s : Set M\n⊢ (span ℕ s).toAddSubmonoid = AddSubmonoid.closure s",
"tactic": "refine' Eq.symm (AddSubmonoid.closure_eq_of_le subset_span _)"
},
{
"state_after": "R : Type ?u.32732\nR₂ : Type ?u.32735\nK : Type ?u.32738\nM : Type u_1\nM₂ : Type ?u.32744\nV : Type ?u.32747\nS : Type ?u.32750\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns✝ t s : Set M\n⊢ span ℕ s ≤ ↑(OrderIso.symm (OrderIso.symm AddSubmonoid.toNatSubmodule)) (AddSubmonoid.closure s)",
"state_before": "R : Type ?u.32732\nR₂ : Type ?u.32735\nK : Type ?u.32738\nM : Type u_1\nM₂ : Type ?u.32744\nV : Type ?u.32747\nS : Type ?u.32750\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns✝ t s : Set M\n⊢ (span ℕ s).toAddSubmonoid ≤ AddSubmonoid.closure s",
"tactic": "apply (OrderIso.to_galoisConnection (AddSubmonoid.toNatSubmodule (M := M)).symm).l_le\n (a := span ℕ s) (b := AddSubmonoid.closure s)"
},
{
"state_after": "R : Type ?u.32732\nR₂ : Type ?u.32735\nK : Type ?u.32738\nM : Type u_1\nM₂ : Type ?u.32744\nV : Type ?u.32747\nS : Type ?u.32750\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns✝ t s : Set M\n⊢ s ⊆ ↑(↑(OrderIso.symm (OrderIso.symm AddSubmonoid.toNatSubmodule)) (AddSubmonoid.closure s))",
"state_before": "R : Type ?u.32732\nR₂ : Type ?u.32735\nK : Type ?u.32738\nM : Type u_1\nM₂ : Type ?u.32744\nV : Type ?u.32747\nS : Type ?u.32750\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns✝ t s : Set M\n⊢ span ℕ s ≤ ↑(OrderIso.symm (OrderIso.symm AddSubmonoid.toNatSubmodule)) (AddSubmonoid.closure s)",
"tactic": "rw [span_le]"
},
{
"state_after": "no goals",
"state_before": "R : Type ?u.32732\nR₂ : Type ?u.32735\nK : Type ?u.32738\nM : Type u_1\nM₂ : Type ?u.32744\nV : Type ?u.32747\nS : Type ?u.32750\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nx : M\np p' : Submodule R M\ninst✝² : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝¹ : AddCommMonoid M₂\ninst✝ : Module R₂ M₂\ns✝ t s : Set M\n⊢ s ⊆ ↑(↑(OrderIso.symm (OrderIso.symm AddSubmonoid.toNatSubmodule)) (AddSubmonoid.closure s))",
"tactic": "exact AddSubmonoid.subset_closure"
}
] |
[
189,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.isCycle_def
|
[] |
[
907,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
905,
1
] |
Mathlib/Analysis/InnerProductSpace/Orientation.lean
|
OrthonormalBasis.same_orientation_iff_det_eq_det
|
[
{
"state_after": "case mp\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nne : Nonempty ι\ne f : OrthonormalBasis ι ℝ E\nx : Orientation ℝ E ι\n⊢ Basis.det (OrthonormalBasis.toBasis e) = Basis.det (OrthonormalBasis.toBasis f) →\n Basis.orientation (OrthonormalBasis.toBasis e) = Basis.orientation (OrthonormalBasis.toBasis f)\n\ncase mpr\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nne : Nonempty ι\ne f : OrthonormalBasis ι ℝ E\nx : Orientation ℝ E ι\n⊢ Basis.orientation (OrthonormalBasis.toBasis e) = Basis.orientation (OrthonormalBasis.toBasis f) →\n Basis.det (OrthonormalBasis.toBasis e) = Basis.det (OrthonormalBasis.toBasis f)",
"state_before": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nne : Nonempty ι\ne f : OrthonormalBasis ι ℝ E\nx : Orientation ℝ E ι\n⊢ Basis.det (OrthonormalBasis.toBasis e) = Basis.det (OrthonormalBasis.toBasis f) ↔\n Basis.orientation (OrthonormalBasis.toBasis e) = Basis.orientation (OrthonormalBasis.toBasis f)",
"tactic": "constructor"
},
{
"state_after": "case mp\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nne : Nonempty ι\ne f : OrthonormalBasis ι ℝ E\nx : Orientation ℝ E ι\nh : Basis.det (OrthonormalBasis.toBasis e) = Basis.det (OrthonormalBasis.toBasis f)\n⊢ Basis.orientation (OrthonormalBasis.toBasis e) = Basis.orientation (OrthonormalBasis.toBasis f)",
"state_before": "case mp\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nne : Nonempty ι\ne f : OrthonormalBasis ι ℝ E\nx : Orientation ℝ E ι\n⊢ Basis.det (OrthonormalBasis.toBasis e) = Basis.det (OrthonormalBasis.toBasis f) →\n Basis.orientation (OrthonormalBasis.toBasis e) = Basis.orientation (OrthonormalBasis.toBasis f)",
"tactic": "intro h"
},
{
"state_after": "case mp\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nne : Nonempty ι\ne f : OrthonormalBasis ι ℝ E\nx : Orientation ℝ E ι\nh : Basis.det (OrthonormalBasis.toBasis e) = Basis.det (OrthonormalBasis.toBasis f)\n⊢ rayOfNeZero ℝ (Basis.det (OrthonormalBasis.toBasis e)) (_ : Basis.det (OrthonormalBasis.toBasis e) ≠ 0) =\n rayOfNeZero ℝ (Basis.det (OrthonormalBasis.toBasis f)) (_ : Basis.det (OrthonormalBasis.toBasis f) ≠ 0)",
"state_before": "case mp\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nne : Nonempty ι\ne f : OrthonormalBasis ι ℝ E\nx : Orientation ℝ E ι\nh : Basis.det (OrthonormalBasis.toBasis e) = Basis.det (OrthonormalBasis.toBasis f)\n⊢ Basis.orientation (OrthonormalBasis.toBasis e) = Basis.orientation (OrthonormalBasis.toBasis f)",
"tactic": "dsimp [Basis.orientation]"
},
{
"state_after": "no goals",
"state_before": "case mp\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nne : Nonempty ι\ne f : OrthonormalBasis ι ℝ E\nx : Orientation ℝ E ι\nh : Basis.det (OrthonormalBasis.toBasis e) = Basis.det (OrthonormalBasis.toBasis f)\n⊢ rayOfNeZero ℝ (Basis.det (OrthonormalBasis.toBasis e)) (_ : Basis.det (OrthonormalBasis.toBasis e) ≠ 0) =\n rayOfNeZero ℝ (Basis.det (OrthonormalBasis.toBasis f)) (_ : Basis.det (OrthonormalBasis.toBasis f) ≠ 0)",
"tactic": "congr"
},
{
"state_after": "case mpr\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nne : Nonempty ι\ne f : OrthonormalBasis ι ℝ E\nx : Orientation ℝ E ι\nh : Basis.orientation (OrthonormalBasis.toBasis e) = Basis.orientation (OrthonormalBasis.toBasis f)\n⊢ Basis.det (OrthonormalBasis.toBasis e) = Basis.det (OrthonormalBasis.toBasis f)",
"state_before": "case mpr\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nne : Nonempty ι\ne f : OrthonormalBasis ι ℝ E\nx : Orientation ℝ E ι\n⊢ Basis.orientation (OrthonormalBasis.toBasis e) = Basis.orientation (OrthonormalBasis.toBasis f) →\n Basis.det (OrthonormalBasis.toBasis e) = Basis.det (OrthonormalBasis.toBasis f)",
"tactic": "intro h"
},
{
"state_after": "case mpr\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nne : Nonempty ι\ne f : OrthonormalBasis ι ℝ E\nx : Orientation ℝ E ι\nh : Basis.orientation (OrthonormalBasis.toBasis e) = Basis.orientation (OrthonormalBasis.toBasis f)\n⊢ ↑(Basis.det (OrthonormalBasis.toBasis e)) ↑(OrthonormalBasis.toBasis f) • Basis.det (OrthonormalBasis.toBasis f) =\n Basis.det (OrthonormalBasis.toBasis f)",
"state_before": "case mpr\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nne : Nonempty ι\ne f : OrthonormalBasis ι ℝ E\nx : Orientation ℝ E ι\nh : Basis.orientation (OrthonormalBasis.toBasis e) = Basis.orientation (OrthonormalBasis.toBasis f)\n⊢ Basis.det (OrthonormalBasis.toBasis e) = Basis.det (OrthonormalBasis.toBasis f)",
"tactic": "rw [e.toBasis.det.eq_smul_basis_det f.toBasis]"
},
{
"state_after": "no goals",
"state_before": "case mpr\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace ℝ E\nι : Type u_2\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nne : Nonempty ι\ne f : OrthonormalBasis ι ℝ E\nx : Orientation ℝ E ι\nh : Basis.orientation (OrthonormalBasis.toBasis e) = Basis.orientation (OrthonormalBasis.toBasis f)\n⊢ ↑(Basis.det (OrthonormalBasis.toBasis e)) ↑(OrthonormalBasis.toBasis f) • Basis.det (OrthonormalBasis.toBasis f) =\n Basis.det (OrthonormalBasis.toBasis f)",
"tactic": "simp [e.det_to_matrix_orthonormalBasis_of_same_orientation f h]"
}
] |
[
87,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/Data/MvPolynomial/Variables.lean
|
MvPolynomial.degrees_rename
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nf : σ → τ\nφ : MvPolynomial σ R\n⊢ degrees (↑(rename f) φ) ⊆ Multiset.map f (degrees φ)",
"tactic": "classical\nintro i\nrw [mem_degrees, Multiset.mem_map]\nrintro ⟨d, hd, hi⟩\nobtain ⟨x, rfl, hx⟩ := coeff_rename_ne_zero _ _ _ hd\nsimp only [Finsupp.mapDomain, Finsupp.mem_support_iff] at hi\nrw [sum_apply, Finsupp.sum] at hi\ncontrapose! hi\nrw [Finset.sum_eq_zero]\nintro j hj\nsimp only [exists_prop, mem_degrees] at hi\nspecialize hi j ⟨x, hx, hj⟩\nrw [Finsupp.single_apply, if_neg hi]"
},
{
"state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nf : σ → τ\nφ : MvPolynomial σ R\ni : τ\n⊢ i ∈ degrees (↑(rename f) φ) → i ∈ Multiset.map f (degrees φ)",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nf : σ → τ\nφ : MvPolynomial σ R\n⊢ degrees (↑(rename f) φ) ⊆ Multiset.map f (degrees φ)",
"tactic": "intro i"
},
{
"state_after": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nf : σ → τ\nφ : MvPolynomial σ R\ni : τ\n⊢ (∃ d, coeff d (↑(rename f) φ) ≠ 0 ∧ i ∈ d.support) → ∃ a, a ∈ degrees φ ∧ f a = i",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nf : σ → τ\nφ : MvPolynomial σ R\ni : τ\n⊢ i ∈ degrees (↑(rename f) φ) → i ∈ Multiset.map f (degrees φ)",
"tactic": "rw [mem_degrees, Multiset.mem_map]"
},
{
"state_after": "case intro.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nf : σ → τ\nφ : MvPolynomial σ R\ni : τ\nd : τ →₀ ℕ\nhd : coeff d (↑(rename f) φ) ≠ 0\nhi : i ∈ d.support\n⊢ ∃ a, a ∈ degrees φ ∧ f a = i",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nf : σ → τ\nφ : MvPolynomial σ R\ni : τ\n⊢ (∃ d, coeff d (↑(rename f) φ) ≠ 0 ∧ i ∈ d.support) → ∃ a, a ∈ degrees φ ∧ f a = i",
"tactic": "rintro ⟨d, hd, hi⟩"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nf : σ → τ\nφ : MvPolynomial σ R\ni : τ\nx : σ →₀ ℕ\nhx : coeff x φ ≠ 0\nhd : coeff (Finsupp.mapDomain f x) (↑(rename f) φ) ≠ 0\nhi : i ∈ (Finsupp.mapDomain f x).support\n⊢ ∃ a, a ∈ degrees φ ∧ f a = i",
"state_before": "case intro.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nf : σ → τ\nφ : MvPolynomial σ R\ni : τ\nd : τ →₀ ℕ\nhd : coeff d (↑(rename f) φ) ≠ 0\nhi : i ∈ d.support\n⊢ ∃ a, a ∈ degrees φ ∧ f a = i",
"tactic": "obtain ⟨x, rfl, hx⟩ := coeff_rename_ne_zero _ _ _ hd"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nf : σ → τ\nφ : MvPolynomial σ R\ni : τ\nx : σ →₀ ℕ\nhx : coeff x φ ≠ 0\nhd : coeff (Finsupp.mapDomain f x) (↑(rename f) φ) ≠ 0\nhi : ↑(sum x fun a => Finsupp.single (f a)) i ≠ 0\n⊢ ∃ a, a ∈ degrees φ ∧ f a = i",
"state_before": "case intro.intro.intro.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nf : σ → τ\nφ : MvPolynomial σ R\ni : τ\nx : σ →₀ ℕ\nhx : coeff x φ ≠ 0\nhd : coeff (Finsupp.mapDomain f x) (↑(rename f) φ) ≠ 0\nhi : i ∈ (Finsupp.mapDomain f x).support\n⊢ ∃ a, a ∈ degrees φ ∧ f a = i",
"tactic": "simp only [Finsupp.mapDomain, Finsupp.mem_support_iff] at hi"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nf : σ → τ\nφ : MvPolynomial σ R\ni : τ\nx : σ →₀ ℕ\nhx : coeff x φ ≠ 0\nhd : coeff (Finsupp.mapDomain f x) (↑(rename f) φ) ≠ 0\nhi : ∑ a in x.support, ↑(Finsupp.single (f a) (↑x a)) i ≠ 0\n⊢ ∃ a, a ∈ degrees φ ∧ f a = i",
"state_before": "case intro.intro.intro.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nf : σ → τ\nφ : MvPolynomial σ R\ni : τ\nx : σ →₀ ℕ\nhx : coeff x φ ≠ 0\nhd : coeff (Finsupp.mapDomain f x) (↑(rename f) φ) ≠ 0\nhi : ↑(sum x fun a => Finsupp.single (f a)) i ≠ 0\n⊢ ∃ a, a ∈ degrees φ ∧ f a = i",
"tactic": "rw [sum_apply, Finsupp.sum] at hi"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nf : σ → τ\nφ : MvPolynomial σ R\ni : τ\nx : σ →₀ ℕ\nhx : coeff x φ ≠ 0\nhd : coeff (Finsupp.mapDomain f x) (↑(rename f) φ) ≠ 0\nhi : ∀ (a : σ), a ∈ degrees φ → f a ≠ i\n⊢ ∑ x_1 in x.support, ↑(Finsupp.single (f x_1) (↑x x_1)) i = 0",
"state_before": "case intro.intro.intro.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nf : σ → τ\nφ : MvPolynomial σ R\ni : τ\nx : σ →₀ ℕ\nhx : coeff x φ ≠ 0\nhd : coeff (Finsupp.mapDomain f x) (↑(rename f) φ) ≠ 0\nhi : ∑ a in x.support, ↑(Finsupp.single (f a) (↑x a)) i ≠ 0\n⊢ ∃ a, a ∈ degrees φ ∧ f a = i",
"tactic": "contrapose! hi"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nf : σ → τ\nφ : MvPolynomial σ R\ni : τ\nx : σ →₀ ℕ\nhx : coeff x φ ≠ 0\nhd : coeff (Finsupp.mapDomain f x) (↑(rename f) φ) ≠ 0\nhi : ∀ (a : σ), a ∈ degrees φ → f a ≠ i\n⊢ ∀ (x_1 : σ), x_1 ∈ x.support → ↑(Finsupp.single (f x_1) (↑x x_1)) i = 0",
"state_before": "case intro.intro.intro.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nf : σ → τ\nφ : MvPolynomial σ R\ni : τ\nx : σ →₀ ℕ\nhx : coeff x φ ≠ 0\nhd : coeff (Finsupp.mapDomain f x) (↑(rename f) φ) ≠ 0\nhi : ∀ (a : σ), a ∈ degrees φ → f a ≠ i\n⊢ ∑ x_1 in x.support, ↑(Finsupp.single (f x_1) (↑x x_1)) i = 0",
"tactic": "rw [Finset.sum_eq_zero]"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nf : σ → τ\nφ : MvPolynomial σ R\ni : τ\nx : σ →₀ ℕ\nhx : coeff x φ ≠ 0\nhd : coeff (Finsupp.mapDomain f x) (↑(rename f) φ) ≠ 0\nhi : ∀ (a : σ), a ∈ degrees φ → f a ≠ i\nj : σ\nhj : j ∈ x.support\n⊢ ↑(Finsupp.single (f j) (↑x j)) i = 0",
"state_before": "case intro.intro.intro.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nf : σ → τ\nφ : MvPolynomial σ R\ni : τ\nx : σ →₀ ℕ\nhx : coeff x φ ≠ 0\nhd : coeff (Finsupp.mapDomain f x) (↑(rename f) φ) ≠ 0\nhi : ∀ (a : σ), a ∈ degrees φ → f a ≠ i\n⊢ ∀ (x_1 : σ), x_1 ∈ x.support → ↑(Finsupp.single (f x_1) (↑x x_1)) i = 0",
"tactic": "intro j hj"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nf : σ → τ\nφ : MvPolynomial σ R\ni : τ\nx : σ →₀ ℕ\nhx : coeff x φ ≠ 0\nhd : coeff (Finsupp.mapDomain f x) (↑(rename f) φ) ≠ 0\nj : σ\nhj : j ∈ x.support\nhi : ∀ (a : σ), (∃ d, coeff d φ ≠ 0 ∧ a ∈ d.support) → f a ≠ i\n⊢ ↑(Finsupp.single (f j) (↑x j)) i = 0",
"state_before": "case intro.intro.intro.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nf : σ → τ\nφ : MvPolynomial σ R\ni : τ\nx : σ →₀ ℕ\nhx : coeff x φ ≠ 0\nhd : coeff (Finsupp.mapDomain f x) (↑(rename f) φ) ≠ 0\nhi : ∀ (a : σ), a ∈ degrees φ → f a ≠ i\nj : σ\nhj : j ∈ x.support\n⊢ ↑(Finsupp.single (f j) (↑x j)) i = 0",
"tactic": "simp only [exists_prop, mem_degrees] at hi"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nf : σ → τ\nφ : MvPolynomial σ R\ni : τ\nx : σ →₀ ℕ\nhx : coeff x φ ≠ 0\nhd : coeff (Finsupp.mapDomain f x) (↑(rename f) φ) ≠ 0\nj : σ\nhj : j ∈ x.support\nhi : f j ≠ i\n⊢ ↑(Finsupp.single (f j) (↑x j)) i = 0",
"state_before": "case intro.intro.intro.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nf : σ → τ\nφ : MvPolynomial σ R\ni : τ\nx : σ →₀ ℕ\nhx : coeff x φ ≠ 0\nhd : coeff (Finsupp.mapDomain f x) (↑(rename f) φ) ≠ 0\nj : σ\nhj : j ∈ x.support\nhi : ∀ (a : σ), (∃ d, coeff d φ ≠ 0 ∧ a ∈ d.support) → f a ≠ i\n⊢ ↑(Finsupp.single (f j) (↑x j)) i = 0",
"tactic": "specialize hi j ⟨x, hx, hj⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nR : Type u\nS : Type v\nσ : Type u_1\nτ : Type u_2\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\np q : MvPolynomial σ R\nf : σ → τ\nφ : MvPolynomial σ R\ni : τ\nx : σ →₀ ℕ\nhx : coeff x φ ≠ 0\nhd : coeff (Finsupp.mapDomain f x) (↑(rename f) φ) ≠ 0\nj : σ\nhj : j ∈ x.support\nhi : f j ≠ i\n⊢ ↑(Finsupp.single (f j) (↑x j)) i = 0",
"tactic": "rw [Finsupp.single_apply, if_neg hi]"
}
] |
[
251,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.eventually_nhds'
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.33714\nδ : Type ?u.33717\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ne✝ : LocalHomeomorph α β\ne' : LocalHomeomorph β γ\ne : LocalHomeomorph α β\nx : α\np : α → Prop\nhx : x ∈ e.source\n⊢ (∀ᶠ (x : α) in 𝓝 x, p (↑(LocalHomeomorph.symm e) (↑e x))) ↔ ∀ᶠ (x : α) in 𝓝 x, p x",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.33714\nδ : Type ?u.33717\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ne✝ : LocalHomeomorph α β\ne' : LocalHomeomorph β γ\ne : LocalHomeomorph α β\nx : α\np : α → Prop\nhx : x ∈ e.source\n⊢ (∀ᶠ (y : β) in 𝓝 (↑e x), p (↑(LocalHomeomorph.symm e) y)) ↔ ∀ᶠ (x : α) in 𝓝 x, p x",
"tactic": "rw [e.eventually_nhds _ hx]"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.33714\nδ : Type ?u.33717\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ne✝ : LocalHomeomorph α β\ne' : LocalHomeomorph β γ\ne : LocalHomeomorph α β\nx : α\np : α → Prop\nhx : x ∈ e.source\ny : α\nhy : ↑(LocalHomeomorph.symm e) (↑e y) = y\n⊢ p (↑(LocalHomeomorph.symm e) (↑e y)) ↔ p y",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.33714\nδ : Type ?u.33717\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ne✝ : LocalHomeomorph α β\ne' : LocalHomeomorph β γ\ne : LocalHomeomorph α β\nx : α\np : α → Prop\nhx : x ∈ e.source\n⊢ (∀ᶠ (x : α) in 𝓝 x, p (↑(LocalHomeomorph.symm e) (↑e x))) ↔ ∀ᶠ (x : α) in 𝓝 x, p x",
"tactic": "refine' eventually_congr ((e.eventually_left_inverse hx).mono fun y hy => _)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.33714\nδ : Type ?u.33717\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\ne✝ : LocalHomeomorph α β\ne' : LocalHomeomorph β γ\ne : LocalHomeomorph α β\nx : α\np : α → Prop\nhx : x ∈ e.source\ny : α\nhy : ↑(LocalHomeomorph.symm e) (↑e y) = y\n⊢ p (↑(LocalHomeomorph.symm e) (↑e y)) ↔ p y",
"tactic": "rw [hy]"
}
] |
[
405,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
401,
1
] |
Mathlib/Algebra/Order/Group/Defs.lean
|
StrictMonoOn.inv
|
[] |
[
1330,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1329,
1
] |
Mathlib/Algebra/Group/Basic.lean
|
mul_right_eq_self
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.12442\nβ : Type ?u.12445\nG : Type ?u.12448\nM : Type u\ninst✝ : LeftCancelMonoid M\na b : M\n⊢ a * b = a ↔ a * b = a * 1",
"tactic": "rw [mul_one]"
}
] |
[
184,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
182,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.uniformContinuousOn_iff_le
|
[] |
[
833,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
830,
1
] |
Mathlib/MeasureTheory/MeasurableSpaceDef.lean
|
Set.Countable.measurableSet
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.13908\nγ : Type ?u.13911\nδ : Type ?u.13914\nδ' : Type ?u.13917\nι : Sort ?u.13920\ns✝ t u : Set α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\ns : Set α\nhs : Set.Countable s\n⊢ MeasurableSet (⋃ (x : α) (_ : x ∈ s), {x})",
"state_before": "α : Type u_1\nβ : Type ?u.13908\nγ : Type ?u.13911\nδ : Type ?u.13914\nδ' : Type ?u.13917\nι : Sort ?u.13920\ns✝ t u : Set α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\ns : Set α\nhs : Set.Countable s\n⊢ MeasurableSet s",
"tactic": "rw [← biUnion_of_singleton s]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.13908\nγ : Type ?u.13911\nδ : Type ?u.13914\nδ' : Type ?u.13917\nι : Sort ?u.13920\ns✝ t u : Set α\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\ns : Set α\nhs : Set.Countable s\n⊢ MeasurableSet (⋃ (x : α) (_ : x ∈ s), {x})",
"tactic": "exact .biUnion hs fun b _ => .singleton b"
}
] |
[
320,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
318,
1
] |
Mathlib/RingTheory/Localization/NumDen.lean
|
IsFractionRing.eq_zero_of_num_eq_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.150711\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type ?u.150917\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.151171\ninst✝⁶ : CommRing P\nA : Type u_1\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nx : K\nh : num A x = 0\n⊢ 0 * ↑(algebraMap (?m.152136 h) K) ↑(den (?m.152136 h) x) = ↑(algebraMap (?m.152136 h) K) (num (?m.152136 h) x)",
"tactic": "rw [MulZeroClass.zero_mul, h, RingHom.map_zero]"
}
] |
[
99,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
src/lean/Init/Data/List/Basic.lean
|
List.append_cons
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nas : List α\nb : α\nbs : List α\n⊢ as ++ b :: bs = as ++ b :: nil ++ bs",
"tactic": "induction as with\n| nil => simp\n| cons a as ih => simp [ih]"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u\nβ : Type v\nγ : Type w\nb : α\nbs : List α\n⊢ nil ++ b :: bs = nil ++ b :: nil ++ bs",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u\nβ : Type v\nγ : Type w\nb : α\nbs : List α\na : α\nas : List α\nih : as ++ b :: bs = as ++ b :: nil ++ bs\n⊢ a :: as ++ b :: bs = a :: as ++ b :: nil ++ bs",
"tactic": "simp [ih]"
}
] |
[
111,
30
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
108,
1
] |
Mathlib/Topology/Semicontinuous.lean
|
UpperSemicontinuous.add'
|
[] |
[
928,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
925,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.mapsTo_iUnion₂
|
[] |
[
1434,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1432,
1
] |
Mathlib/Order/Hom/CompleteLattice.lean
|
sInfHom.comp_assoc
|
[] |
[
474,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
472,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.bound_of_isBigO_nat_atTop
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.686737\nβ : Type ?u.686740\nE : Type u_1\nF : Type ?u.686746\nG : Type ?u.686749\nE' : Type ?u.686752\nF' : Type ?u.686755\nG' : Type ?u.686758\nE'' : Type u_2\nF'' : Type ?u.686764\nG'' : Type ?u.686767\nR : Type ?u.686770\nR' : Type ?u.686773\n𝕜 : Type ?u.686776\n𝕜' : Type ?u.686779\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\nf : ℕ → E\ng'' : ℕ → E''\nh : f =O[atTop] g''\n⊢ (fun x => f x) =O[cofinite] fun x => g'' x",
"tactic": "rwa [Nat.cofinite_eq_atTop]"
}
] |
[
2120,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2118,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergence.lean
|
uniformCauchySeqOn_iff_uniformCauchySeqOnFilter
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\n⊢ (∀ (u : Set (β × β)), u ∈ 𝓤 β → ∀ᶠ (m : ι × ι) in p ×ˢ p, ∀ (x : α), x ∈ s → (F m.fst x, F m.snd x) ∈ u) ↔\n ∀ (u : Set (β × β)), u ∈ 𝓤 β → ∀ᶠ (m : (ι × ι) × α) in (p ×ˢ p) ×ˢ 𝓟 s, (F m.fst.fst m.snd, F m.fst.snd m.snd) ∈ u",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\n⊢ UniformCauchySeqOn F p s ↔ UniformCauchySeqOnFilter F p (𝓟 s)",
"tactic": "simp only [UniformCauchySeqOn, UniformCauchySeqOnFilter]"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nu : Set (β × β)\nhu : u ∈ 𝓤 β\n⊢ (∀ᶠ (m : ι × ι) in p ×ˢ p, ∀ (x : α), x ∈ s → (F m.fst x, F m.snd x) ∈ u) ↔\n ∀ᶠ (m : (ι × ι) × α) in (p ×ˢ p) ×ˢ 𝓟 s, (F m.fst.fst m.snd, F m.fst.snd m.snd) ∈ u",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\n⊢ (∀ (u : Set (β × β)), u ∈ 𝓤 β → ∀ᶠ (m : ι × ι) in p ×ˢ p, ∀ (x : α), x ∈ s → (F m.fst x, F m.snd x) ∈ u) ↔\n ∀ (u : Set (β × β)), u ∈ 𝓤 β → ∀ᶠ (m : (ι × ι) × α) in (p ×ˢ p) ×ˢ 𝓟 s, (F m.fst.fst m.snd, F m.fst.snd m.snd) ∈ u",
"tactic": "refine' forall₂_congr fun u hu => _"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nu : Set (β × β)\nhu : u ∈ 𝓤 β\n⊢ (∀ᶠ (m : ι × ι) in p ×ˢ p, ∀ (x : α), x ∈ s → (F m.fst x, F m.snd x) ∈ u) ↔\n ∀ᶠ (m : (ι × ι) × α) in (p ×ˢ p) ×ˢ 𝓟 s, (F m.fst.fst m.snd, F m.fst.snd m.snd) ∈ u",
"tactic": "rw [eventually_prod_principal_iff]"
}
] |
[
414,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
410,
1
] |
src/lean/Init/Data/Nat/Linear.lean
|
Nat.Linear.ExprCnstr.denote_toPoly
|
[
{
"state_after": "case mk\nctx : Context\neq✝ : Bool\nlhs✝ rhs✝ : Expr\n⊢ PolyCnstr.denote ctx (toPoly { eq := eq✝, lhs := lhs✝, rhs := rhs✝ }) =\n denote ctx { eq := eq✝, lhs := lhs✝, rhs := rhs✝ }",
"state_before": "ctx : Context\nc : ExprCnstr\n⊢ PolyCnstr.denote ctx (toPoly c) = denote ctx c",
"tactic": "cases c"
},
{
"state_after": "case mk\nctx : Context\neq : Bool\nlhs rhs : Expr\n⊢ PolyCnstr.denote ctx (toPoly { eq := eq, lhs := lhs, rhs := rhs }) = denote ctx { eq := eq, lhs := lhs, rhs := rhs }",
"state_before": "case mk\nctx : Context\neq✝ : Bool\nlhs✝ rhs✝ : Expr\n⊢ PolyCnstr.denote ctx (toPoly { eq := eq✝, lhs := lhs✝, rhs := rhs✝ }) =\n denote ctx { eq := eq✝, lhs := lhs✝, rhs := rhs✝ }",
"tactic": "rename_i eq lhs rhs"
},
{
"state_after": "case mk\nctx : Context\neq : Bool\nlhs rhs : Expr\n⊢ (bif eq then Poly.denote_eq ctx (Expr.toPoly lhs, Expr.toPoly rhs)\n else Poly.denote_le ctx (Expr.toPoly lhs, Expr.toPoly rhs)) =\n bif eq then Expr.denote ctx lhs = Expr.denote ctx rhs else Expr.denote ctx lhs ≤ Expr.denote ctx rhs",
"state_before": "case mk\nctx : Context\neq : Bool\nlhs rhs : Expr\n⊢ PolyCnstr.denote ctx (toPoly { eq := eq, lhs := lhs, rhs := rhs }) = denote ctx { eq := eq, lhs := lhs, rhs := rhs }",
"tactic": "simp [ExprCnstr.denote, PolyCnstr.denote, ExprCnstr.toPoly]"
},
{
"state_after": "case mk.inl\nctx : Context\neq : Bool\nlhs rhs : Expr\nh : eq = true\n⊢ Poly.denote_eq ctx (Expr.toPoly lhs, Expr.toPoly rhs) = (Expr.denote ctx lhs = Expr.denote ctx rhs)\n\ncase mk.inr\nctx : Context\neq : Bool\nlhs rhs : Expr\nh : ¬eq = true\n⊢ Poly.denote_le ctx (Expr.toPoly lhs, Expr.toPoly rhs) = (Expr.denote ctx lhs ≤ Expr.denote ctx rhs)",
"state_before": "case mk\nctx : Context\neq : Bool\nlhs rhs : Expr\n⊢ (bif eq then Poly.denote_eq ctx (Expr.toPoly lhs, Expr.toPoly rhs)\n else Poly.denote_le ctx (Expr.toPoly lhs, Expr.toPoly rhs)) =\n bif eq then Expr.denote ctx lhs = Expr.denote ctx rhs else Expr.denote ctx lhs ≤ Expr.denote ctx rhs",
"tactic": "by_cases h : eq = true <;> simp [h]"
},
{
"state_after": "no goals",
"state_before": "case mk.inl\nctx : Context\neq : Bool\nlhs rhs : Expr\nh : eq = true\n⊢ Poly.denote_eq ctx (Expr.toPoly lhs, Expr.toPoly rhs) = (Expr.denote ctx lhs = Expr.denote ctx rhs)",
"tactic": "simp [Poly.denote_eq, Expr.toPoly]"
},
{
"state_after": "no goals",
"state_before": "case mk.inr\nctx : Context\neq : Bool\nlhs rhs : Expr\nh : ¬eq = true\n⊢ Poly.denote_le ctx (Expr.toPoly lhs, Expr.toPoly rhs) = (Expr.denote ctx lhs ≤ Expr.denote ctx rhs)",
"tactic": "simp [Poly.denote_le, Expr.toPoly]"
}
] |
[
564,
39
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
559,
1
] |
Mathlib/Topology/Order/Basic.lean
|
nhdsWithin_Iic_neBot
|
[] |
[
259,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
258,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.mk_preimage_of_subset_range_lift
|
[
{
"state_after": "α✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : s ⊆ range f\n⊢ Nonempty (↑s ↪ ↑(f ⁻¹' s))",
"state_before": "α✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : s ⊆ range f\n⊢ lift (#↑s) ≤ lift (#↑(f ⁻¹' s))",
"tactic": "rw [lift_mk_le.{v, u, 0}]"
},
{
"state_after": "case refine'_1\nα✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : s ⊆ range f\n⊢ ↑s → ↑(f ⁻¹' s)\n\ncase refine'_2\nα✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : s ⊆ range f\n⊢ Injective ?refine'_1",
"state_before": "α✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : s ⊆ range f\n⊢ Nonempty (↑s ↪ ↑(f ⁻¹' s))",
"tactic": "refine' ⟨⟨_, _⟩⟩"
},
{
"state_after": "case refine'_2.mk.mk\nα✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : s ⊆ range f\ny : β\nhy : y ∈ s\ny' : β\nhy' : y' ∈ s\n⊢ (fun a =>\n Subtype.casesOn a fun y hy =>\n Subtype.casesOn (subtype_of_exists (_ : y ∈ range f)) fun x property =>\n Eq.ndrec (motive := fun y => y ∈ s → ↑(f ⁻¹' s)) (fun hy => { val := x, property := hy }) property hy)\n { val := y, property := hy } =\n (fun a =>\n Subtype.casesOn a fun y hy =>\n Subtype.casesOn (subtype_of_exists (_ : y ∈ range f)) fun x property =>\n Eq.ndrec (motive := fun y => y ∈ s → ↑(f ⁻¹' s)) (fun hy => { val := x, property := hy }) property hy)\n { val := y', property := hy' } →\n { val := y, property := hy } = { val := y', property := hy' }",
"state_before": "case refine'_2\nα✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : s ⊆ range f\n⊢ Injective fun a =>\n Subtype.casesOn a fun y hy =>\n Subtype.casesOn (subtype_of_exists (_ : y ∈ range f)) fun x property =>\n Eq.ndrec (motive := fun y => y ∈ s → ↑(f ⁻¹' s)) (fun hy => { val := x, property := hy }) property hy",
"tactic": "rintro ⟨y, hy⟩ ⟨y', hy'⟩"
},
{
"state_after": "case refine'_2.mk.mk\nα✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : s ⊆ range f\ny : β\nhy : y ∈ s\ny' : β\nhy' : y' ∈ s\n⊢ Eq.rec (motive := fun x x_1 => x ∈ s → ↑(f ⁻¹' s))\n (fun hy_1 => { val := ↑(subtype_of_exists (_ : y ∈ range f)), property := hy_1 })\n (_ : f ↑(subtype_of_exists (_ : y ∈ range f)) = y) hy =\n Eq.rec (motive := fun x x_1 => x ∈ s → ↑(f ⁻¹' s))\n (fun hy => { val := ↑(subtype_of_exists (_ : y' ∈ range f)), property := hy })\n (_ : f ↑(subtype_of_exists (_ : y' ∈ range f)) = y') hy' →\n { val := y, property := hy } = { val := y', property := hy' }",
"state_before": "case refine'_2.mk.mk\nα✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : s ⊆ range f\ny : β\nhy : y ∈ s\ny' : β\nhy' : y' ∈ s\n⊢ (fun a =>\n Subtype.casesOn a fun y hy =>\n Subtype.casesOn (subtype_of_exists (_ : y ∈ range f)) fun x property =>\n Eq.ndrec (motive := fun y => y ∈ s → ↑(f ⁻¹' s)) (fun hy => { val := x, property := hy }) property hy)\n { val := y, property := hy } =\n (fun a =>\n Subtype.casesOn a fun y hy =>\n Subtype.casesOn (subtype_of_exists (_ : y ∈ range f)) fun x property =>\n Eq.ndrec (motive := fun y => y ∈ s → ↑(f ⁻¹' s)) (fun hy => { val := x, property := hy }) property hy)\n { val := y', property := hy' } →\n { val := y, property := hy } = { val := y', property := hy' }",
"tactic": "dsimp"
},
{
"state_after": "case refine'_2.mk.mk.mk\nα✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : s ⊆ range f\ny' : β\nhy' : y' ∈ s\nx : α\nhy : f x ∈ s\n⊢ Eq.rec (motive := fun x_1 x => x_1 ∈ s → ↑(f ⁻¹' s))\n (fun hy => { val := ↑{ val := x, property := (_ : f x = f x) }, property := hy })\n (_ : f ↑{ val := x, property := (_ : f x = f x) } = f x) hy =\n Eq.rec (motive := fun x x_1 => x ∈ s → ↑(f ⁻¹' s))\n (fun hy => { val := ↑(subtype_of_exists (_ : y' ∈ range f)), property := hy })\n (_ : f ↑(subtype_of_exists (_ : y' ∈ range f)) = y') hy' →\n { val := f x, property := hy } = { val := y', property := hy' }",
"state_before": "case refine'_2.mk.mk\nα✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : s ⊆ range f\ny : β\nhy : y ∈ s\ny' : β\nhy' : y' ∈ s\n⊢ Eq.rec (motive := fun x x_1 => x ∈ s → ↑(f ⁻¹' s))\n (fun hy_1 => { val := ↑(subtype_of_exists (_ : y ∈ range f)), property := hy_1 })\n (_ : f ↑(subtype_of_exists (_ : y ∈ range f)) = y) hy =\n Eq.rec (motive := fun x x_1 => x ∈ s → ↑(f ⁻¹' s))\n (fun hy => { val := ↑(subtype_of_exists (_ : y' ∈ range f)), property := hy })\n (_ : f ↑(subtype_of_exists (_ : y' ∈ range f)) = y') hy' →\n { val := y, property := hy } = { val := y', property := hy' }",
"tactic": "rcases Classical.subtype_of_exists (h hy) with ⟨x, rfl⟩"
},
{
"state_after": "case refine'_2.mk.mk.mk.mk\nα✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : s ⊆ range f\nx : α\nhy : f x ∈ s\nx' : α\nhy' : f x' ∈ s\n⊢ Eq.rec (motive := fun x_1 x => x_1 ∈ s → ↑(f ⁻¹' s))\n (fun hy => { val := ↑{ val := x, property := (_ : f x = f x) }, property := hy })\n (_ : f ↑{ val := x, property := (_ : f x = f x) } = f x) hy =\n Eq.rec (motive := fun x x_1 => x ∈ s → ↑(f ⁻¹' s))\n (fun hy => { val := ↑{ val := x', property := (_ : f x' = f x') }, property := hy })\n (_ : f ↑{ val := x', property := (_ : f x' = f x') } = f x') hy' →\n { val := f x, property := hy } = { val := f x', property := hy' }",
"state_before": "case refine'_2.mk.mk.mk\nα✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : s ⊆ range f\ny' : β\nhy' : y' ∈ s\nx : α\nhy : f x ∈ s\n⊢ Eq.rec (motive := fun x_1 x => x_1 ∈ s → ↑(f ⁻¹' s))\n (fun hy => { val := ↑{ val := x, property := (_ : f x = f x) }, property := hy })\n (_ : f ↑{ val := x, property := (_ : f x = f x) } = f x) hy =\n Eq.rec (motive := fun x x_1 => x ∈ s → ↑(f ⁻¹' s))\n (fun hy => { val := ↑(subtype_of_exists (_ : y' ∈ range f)), property := hy })\n (_ : f ↑(subtype_of_exists (_ : y' ∈ range f)) = y') hy' →\n { val := f x, property := hy } = { val := y', property := hy' }",
"tactic": "rcases Classical.subtype_of_exists (h hy') with ⟨x', rfl⟩"
},
{
"state_after": "case refine'_2.mk.mk.mk.mk\nα✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : s ⊆ range f\nx : α\nhy : f x ∈ s\nx' : α\nhy' : f x' ∈ s\n⊢ x = x' → f x = f x'",
"state_before": "case refine'_2.mk.mk.mk.mk\nα✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : s ⊆ range f\nx : α\nhy : f x ∈ s\nx' : α\nhy' : f x' ∈ s\n⊢ Eq.rec (motive := fun x_1 x => x_1 ∈ s → ↑(f ⁻¹' s))\n (fun hy => { val := ↑{ val := x, property := (_ : f x = f x) }, property := hy })\n (_ : f ↑{ val := x, property := (_ : f x = f x) } = f x) hy =\n Eq.rec (motive := fun x x_1 => x ∈ s → ↑(f ⁻¹' s))\n (fun hy => { val := ↑{ val := x', property := (_ : f x' = f x') }, property := hy })\n (_ : f ↑{ val := x', property := (_ : f x' = f x') } = f x') hy' →\n { val := f x, property := hy } = { val := f x', property := hy' }",
"tactic": "simp"
},
{
"state_after": "case refine'_2.mk.mk.mk.mk\nα✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : s ⊆ range f\nx : α\nhy : f x ∈ s\nx' : α\nhy' : f x' ∈ s\nhxx' : x = x'\n⊢ f x = f x'",
"state_before": "case refine'_2.mk.mk.mk.mk\nα✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : s ⊆ range f\nx : α\nhy : f x ∈ s\nx' : α\nhy' : f x' ∈ s\n⊢ x = x' → f x = f x'",
"tactic": "intro hxx'"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.mk.mk.mk.mk\nα✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : s ⊆ range f\nx : α\nhy : f x ∈ s\nx' : α\nhy' : f x' ∈ s\nhxx' : x = x'\n⊢ f x = f x'",
"tactic": "rw [hxx']"
},
{
"state_after": "case refine'_1.mk\nα✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : s ⊆ range f\ny : β\nhy : y ∈ s\n⊢ ↑(f ⁻¹' s)",
"state_before": "case refine'_1\nα✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : s ⊆ range f\n⊢ ↑s → ↑(f ⁻¹' s)",
"tactic": "rintro ⟨y, hy⟩"
},
{
"state_after": "case refine'_1.mk.mk\nα✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : s ⊆ range f\nx : α\nhy : f x ∈ s\n⊢ ↑(f ⁻¹' s)",
"state_before": "case refine'_1.mk\nα✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : s ⊆ range f\ny : β\nhy : y ∈ s\n⊢ ↑(f ⁻¹' s)",
"tactic": "rcases Classical.subtype_of_exists (h hy) with ⟨x, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.mk.mk\nα✝ β✝ α : Type u\nβ : Type v\nf : α → β\ns : Set β\nh : s ⊆ range f\nx : α\nhy : f x ∈ s\n⊢ ↑(f ⁻¹' s)",
"tactic": "exact ⟨x, hy⟩"
}
] |
[
2182,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2172,
1
] |
Mathlib/RingTheory/Artinian.lean
|
IsArtinian.exists_endomorphism_iterate_ker_sup_range_eq_top
|
[
{
"state_after": "case intro\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw :\n ∀ (m : ℕ),\n n ≤ m →\n ↑(OrderHom.comp (LinearMap.iterateRange f)\n { toFun := fun n => n + 1,\n monotone' := (_ : ∀ (n m : ℕ), n ≤ m → (fun n => n + 1) n ≤ (fun n => n + 1) m) })\n n =\n ↑(OrderHom.comp (LinearMap.iterateRange f)\n { toFun := fun n => n + 1,\n monotone' := (_ : ∀ (n m : ℕ), n ≤ m → (fun n => n + 1) n ≤ (fun n => n + 1) m) })\n m\n⊢ ∃ n, n ≠ 0 ∧ LinearMap.ker (f ^ n) ⊔ LinearMap.range (f ^ n) = ⊤",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\n⊢ ∃ n, n ≠ 0 ∧ LinearMap.ker (f ^ n) ⊔ LinearMap.range (f ^ n) = ⊤",
"tactic": "obtain ⟨n, w⟩ :=\n monotone_stabilizes (f.iterateRange.comp ⟨fun n => n + 1, fun n m w => by linarith⟩)"
},
{
"state_after": "case intro\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw :\n ↑(OrderHom.comp (LinearMap.iterateRange f)\n { toFun := fun n => n + 1, monotone' := (_ : ∀ (n m : ℕ), n ≤ m → (fun n => n + 1) n ≤ (fun n => n + 1) m) })\n n =\n ↑(OrderHom.comp (LinearMap.iterateRange f)\n { toFun := fun n => n + 1, monotone' := (_ : ∀ (n m : ℕ), n ≤ m → (fun n => n + 1) n ≤ (fun n => n + 1) m) })\n (n + 1 + n)\n⊢ ∃ n, n ≠ 0 ∧ LinearMap.ker (f ^ n) ⊔ LinearMap.range (f ^ n) = ⊤",
"state_before": "case intro\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw :\n ∀ (m : ℕ),\n n ≤ m →\n ↑(OrderHom.comp (LinearMap.iterateRange f)\n { toFun := fun n => n + 1,\n monotone' := (_ : ∀ (n m : ℕ), n ≤ m → (fun n => n + 1) n ≤ (fun n => n + 1) m) })\n n =\n ↑(OrderHom.comp (LinearMap.iterateRange f)\n { toFun := fun n => n + 1,\n monotone' := (_ : ∀ (n m : ℕ), n ≤ m → (fun n => n + 1) n ≤ (fun n => n + 1) m) })\n m\n⊢ ∃ n, n ≠ 0 ∧ LinearMap.ker (f ^ n) ⊔ LinearMap.range (f ^ n) = ⊤",
"tactic": "specialize w (n + 1 + n) (by linarith)"
},
{
"state_after": "case intro\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\n⊢ ∃ n, n ≠ 0 ∧ LinearMap.ker (f ^ n) ⊔ LinearMap.range (f ^ n) = ⊤",
"state_before": "case intro\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw :\n ↑(OrderHom.comp (LinearMap.iterateRange f)\n { toFun := fun n => n + 1, monotone' := (_ : ∀ (n m : ℕ), n ≤ m → (fun n => n + 1) n ≤ (fun n => n + 1) m) })\n n =\n ↑(OrderHom.comp (LinearMap.iterateRange f)\n { toFun := fun n => n + 1, monotone' := (_ : ∀ (n m : ℕ), n ≤ m → (fun n => n + 1) n ≤ (fun n => n + 1) m) })\n (n + 1 + n)\n⊢ ∃ n, n ≠ 0 ∧ LinearMap.ker (f ^ n) ⊔ LinearMap.range (f ^ n) = ⊤",
"tactic": "dsimp at w"
},
{
"state_after": "case intro\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\n⊢ LinearMap.ker (f ^ (n + 1)) ⊔ LinearMap.range (f ^ (n + 1)) = ⊤",
"state_before": "case intro\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\n⊢ ∃ n, n ≠ 0 ∧ LinearMap.ker (f ^ n) ⊔ LinearMap.range (f ^ n) = ⊤",
"tactic": "refine' ⟨n + 1, Nat.succ_ne_zero _, _⟩"
},
{
"state_after": "case intro\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\n⊢ ∀ (x : M), ∃ y, y ∈ LinearMap.ker (f ^ (n + 1)) ∧ ∃ z, z ∈ LinearMap.range (f ^ (n + 1)) ∧ y + z = x",
"state_before": "case intro\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\n⊢ LinearMap.ker (f ^ (n + 1)) ⊔ LinearMap.range (f ^ (n + 1)) = ⊤",
"tactic": "simp_rw [eq_top_iff', mem_sup]"
},
{
"state_after": "case intro\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\nx : M\n⊢ ∃ y, y ∈ LinearMap.ker (f ^ (n + 1)) ∧ ∃ z, z ∈ LinearMap.range (f ^ (n + 1)) ∧ y + z = x",
"state_before": "case intro\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\n⊢ ∀ (x : M), ∃ y, y ∈ LinearMap.ker (f ^ (n + 1)) ∧ ∃ z, z ∈ LinearMap.range (f ^ (n + 1)) ∧ y + z = x",
"tactic": "intro x"
},
{
"state_after": "case intro\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\nx : M\nthis : ↑(f ^ (n + 1)) x ∈ LinearMap.range (f ^ (n + 1 + n + 1))\n⊢ ∃ y, y ∈ LinearMap.ker (f ^ (n + 1)) ∧ ∃ z, z ∈ LinearMap.range (f ^ (n + 1)) ∧ y + z = x",
"state_before": "case intro\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\nx : M\n⊢ ∃ y, y ∈ LinearMap.ker (f ^ (n + 1)) ∧ ∃ z, z ∈ LinearMap.range (f ^ (n + 1)) ∧ y + z = x",
"tactic": "have : (f ^ (n + 1)) x ∈ LinearMap.range (f ^ (n + 1 + n + 1)) := by\n rw [← w]\n exact mem_range_self _"
},
{
"state_after": "case intro.intro\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\nx y : M\nhy : ↑(f ^ (n + 1 + n + 1)) y = ↑(f ^ (n + 1)) x\n⊢ ∃ y, y ∈ LinearMap.ker (f ^ (n + 1)) ∧ ∃ z, z ∈ LinearMap.range (f ^ (n + 1)) ∧ y + z = x",
"state_before": "case intro\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\nx : M\nthis : ↑(f ^ (n + 1)) x ∈ LinearMap.range (f ^ (n + 1 + n + 1))\n⊢ ∃ y, y ∈ LinearMap.ker (f ^ (n + 1)) ∧ ∃ z, z ∈ LinearMap.range (f ^ (n + 1)) ∧ y + z = x",
"tactic": "rcases this with ⟨y, hy⟩"
},
{
"state_after": "case intro.intro\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\nx y : M\nhy : ↑(f ^ (n + 1 + n + 1)) y = ↑(f ^ (n + 1)) x\n⊢ x - ↑(f ^ (n + 1)) y ∈ LinearMap.ker (f ^ (n + 1)) ∧\n ∃ z, z ∈ LinearMap.range (f ^ (n + 1)) ∧ x - ↑(f ^ (n + 1)) y + z = x",
"state_before": "case intro.intro\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\nx y : M\nhy : ↑(f ^ (n + 1 + n + 1)) y = ↑(f ^ (n + 1)) x\n⊢ ∃ y, y ∈ LinearMap.ker (f ^ (n + 1)) ∧ ∃ z, z ∈ LinearMap.range (f ^ (n + 1)) ∧ y + z = x",
"tactic": "use x - (f ^ (n + 1)) y"
},
{
"state_after": "case intro.intro.left\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\nx y : M\nhy : ↑(f ^ (n + 1 + n + 1)) y = ↑(f ^ (n + 1)) x\n⊢ x - ↑(f ^ (n + 1)) y ∈ LinearMap.ker (f ^ (n + 1))\n\ncase intro.intro.right\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\nx y : M\nhy : ↑(f ^ (n + 1 + n + 1)) y = ↑(f ^ (n + 1)) x\n⊢ ∃ z, z ∈ LinearMap.range (f ^ (n + 1)) ∧ x - ↑(f ^ (n + 1)) y + z = x",
"state_before": "case intro.intro\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\nx y : M\nhy : ↑(f ^ (n + 1 + n + 1)) y = ↑(f ^ (n + 1)) x\n⊢ x - ↑(f ^ (n + 1)) y ∈ LinearMap.ker (f ^ (n + 1)) ∧\n ∃ z, z ∈ LinearMap.range (f ^ (n + 1)) ∧ x - ↑(f ^ (n + 1)) y + z = x",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn m : ℕ\nw : n ≤ m\n⊢ (fun n => n + 1) n ≤ (fun n => n + 1) m",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw :\n ∀ (m : ℕ),\n n ≤ m →\n ↑(OrderHom.comp (LinearMap.iterateRange f)\n { toFun := fun n => n + 1,\n monotone' := (_ : ∀ (n m : ℕ), n ≤ m → (fun n => n + 1) n ≤ (fun n => n + 1) m) })\n n =\n ↑(OrderHom.comp (LinearMap.iterateRange f)\n { toFun := fun n => n + 1,\n monotone' := (_ : ∀ (n m : ℕ), n ≤ m → (fun n => n + 1) n ≤ (fun n => n + 1) m) })\n m\n⊢ n ≤ n + 1 + n",
"tactic": "linarith"
},
{
"state_after": "R : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\nx : M\n⊢ ↑(f ^ (n + 1)) x ∈ LinearMap.range (f ^ (n + 1))",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\nx : M\n⊢ ↑(f ^ (n + 1)) x ∈ LinearMap.range (f ^ (n + 1 + n + 1))",
"tactic": "rw [← w]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\nx : M\n⊢ ↑(f ^ (n + 1)) x ∈ LinearMap.range (f ^ (n + 1))",
"tactic": "exact mem_range_self _"
},
{
"state_after": "case intro.intro.left\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\nx y : M\nhy : ↑(f ^ (n + 1 + n + 1)) y = ↑(f ^ (n + 1)) x\n⊢ ↑(f ^ (n + 1 + n + 1)) y = ↑(f ^ n * f ^ 1) (↑(f ^ n * f ^ 1) y)",
"state_before": "case intro.intro.left\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\nx y : M\nhy : ↑(f ^ (n + 1 + n + 1)) y = ↑(f ^ (n + 1)) x\n⊢ x - ↑(f ^ (n + 1)) y ∈ LinearMap.ker (f ^ (n + 1))",
"tactic": "rw [LinearMap.mem_ker, LinearMap.map_sub, ← hy, sub_eq_zero, pow_add]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.left\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\nx y : M\nhy : ↑(f ^ (n + 1 + n + 1)) y = ↑(f ^ (n + 1)) x\n⊢ ↑(f ^ (n + 1 + n + 1)) y = ↑(f ^ n * f ^ 1) (↑(f ^ n * f ^ 1) y)",
"tactic": "simp [iterate_add_apply]"
},
{
"state_after": "case intro.intro.right\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\nx y : M\nhy : ↑(f ^ (n + 1 + n + 1)) y = ↑(f ^ (n + 1)) x\n⊢ ↑(f ^ (n + 1)) y ∈ LinearMap.range (f ^ (n + 1)) ∧ x - ↑(f ^ (n + 1)) y + ↑(f ^ (n + 1)) y = x",
"state_before": "case intro.intro.right\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\nx y : M\nhy : ↑(f ^ (n + 1 + n + 1)) y = ↑(f ^ (n + 1)) x\n⊢ ∃ z, z ∈ LinearMap.range (f ^ (n + 1)) ∧ x - ↑(f ^ (n + 1)) y + z = x",
"tactic": "use (f ^ (n + 1)) y"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.right\nR : Type u_1\nM : Type u_2\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsArtinian R M\nf : M →ₗ[R] M\nn : ℕ\nw : LinearMap.range (f ^ (n + 1)) = LinearMap.range (f ^ (n + 1 + n + 1))\nx y : M\nhy : ↑(f ^ (n + 1 + n + 1)) y = ↑(f ^ (n + 1)) x\n⊢ ↑(f ^ (n + 1)) y ∈ LinearMap.range (f ^ (n + 1)) ∧ x - ↑(f ^ (n + 1)) y + ↑(f ^ (n + 1)) y = x",
"tactic": "simp"
}
] |
[
239,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
221,
1
] |
Mathlib/CategoryTheory/Sums/Basic.lean
|
CategoryTheory.Sum.swap_obj_inl
|
[] |
[
122,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/Order/UpperLower/Basic.lean
|
LowerSet.compl_sSup
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.87716\nγ : Type ?u.87719\nι : Sort ?u.87722\nκ : ι → Sort ?u.87727\ninst✝ : LE α\ns t : LowerSet α\na : α\nS : Set (LowerSet α)\n⊢ ↑(compl (sSup S)) = ↑(⨆ (s : LowerSet α) (_ : s ∈ S), compl s)",
"tactic": "simp only [coe_compl, coe_sSup, compl_iUnion₂, UpperSet.coe_iSup₂]"
}
] |
[
909,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
908,
11
] |
Mathlib/Analysis/Convex/Caratheodory.lean
|
Caratheodory.minCardFinsetOfMemConvexHull_nonempty
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nx : E\nhx : x ∈ ↑(convexHull 𝕜).toOrderHom s\n⊢ Set.Nonempty (↑(convexHull 𝕜).toOrderHom ↑(minCardFinsetOfMemConvexHull hx))",
"state_before": "𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nx : E\nhx : x ∈ ↑(convexHull 𝕜).toOrderHom s\n⊢ Finset.Nonempty (minCardFinsetOfMemConvexHull hx)",
"tactic": "rw [← Finset.coe_nonempty, ← @convexHull_nonempty_iff 𝕜]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nx : E\nhx : x ∈ ↑(convexHull 𝕜).toOrderHom s\n⊢ Set.Nonempty (↑(convexHull 𝕜).toOrderHom ↑(minCardFinsetOfMemConvexHull hx))",
"tactic": "exact ⟨x, mem_minCardFinsetOfMemConvexHull hx⟩"
}
] |
[
126,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/Data/Polynomial/Mirror.lean
|
Polynomial.mirror_X
|
[] |
[
65,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
src/lean/Init/Classical.lean
|
Classical.em
|
[
{
"state_after": "no goals",
"state_before": "p : Prop\nU : Prop → Prop := fun x => x = True ∨ p\nV : Prop → Prop := fun x => x = False ∨ p\nexU : ∃ x, U x\nexV : ∃ x, V x\nu : Prop := choose exU\nv : Prop := choose exV\nu_def : U u\nv_def : V v\nhut : u = True\nhvf : v = False\n⊢ u ≠ v",
"tactic": "simp [hvf, hut, true_ne_false]"
},
{
"state_after": "p : Prop\nU : Prop → Prop := fun x => x = True ∨ p\nV : Prop → Prop := fun x => x = False ∨ p\nexU : ∃ x, U x\nexV : ∃ x, V x\nu : Prop := choose exU\nv : Prop := choose exV\nu_def : U u\nv_def : V v\nnot_uv_or_p : u ≠ v ∨ p\nhp : p\nhpred : U = V\n⊢ ∀ (exU exV : ∃ x, V x), choose exU = choose exV",
"state_before": "p : Prop\nU : Prop → Prop := fun x => x = True ∨ p\nV : Prop → Prop := fun x => x = False ∨ p\nexU : ∃ x, U x\nexV : ∃ x, V x\nu : Prop := choose exU\nv : Prop := choose exV\nu_def : U u\nv_def : V v\nnot_uv_or_p : u ≠ v ∨ p\nhp : p\nhpred : U = V\n⊢ ∀ (exU : ∃ x, U x) (exV : ∃ x, V x), choose exU = choose exV",
"tactic": "rw [hpred]"
},
{
"state_after": "p : Prop\nU : Prop → Prop := fun x => x = True ∨ p\nV : Prop → Prop := fun x => x = False ∨ p\nexU : ∃ x, U x\nexV : ∃ x, V x\nu : Prop := choose exU\nv : Prop := choose exV\nu_def : U u\nv_def : V v\nnot_uv_or_p : u ≠ v ∨ p\nhp : p\nhpred : U = V\nexU✝ exV✝ : ∃ x, V x\n⊢ choose exU✝ = choose exV✝",
"state_before": "p : Prop\nU : Prop → Prop := fun x => x = True ∨ p\nV : Prop → Prop := fun x => x = False ∨ p\nexU : ∃ x, U x\nexV : ∃ x, V x\nu : Prop := choose exU\nv : Prop := choose exV\nu_def : U u\nv_def : V v\nnot_uv_or_p : u ≠ v ∨ p\nhp : p\nhpred : U = V\n⊢ ∀ (exU exV : ∃ x, V x), choose exU = choose exV",
"tactic": "intros"
},
{
"state_after": "no goals",
"state_before": "p : Prop\nU : Prop → Prop := fun x => x = True ∨ p\nV : Prop → Prop := fun x => x = False ∨ p\nexU : ∃ x, U x\nexV : ∃ x, V x\nu : Prop := choose exU\nv : Prop := choose exV\nu_def : U u\nv_def : V v\nnot_uv_or_p : u ≠ v ∨ p\nhp : p\nhpred : U = V\nexU✝ exV✝ : ∃ x, V x\n⊢ choose exU✝ = choose exV✝",
"tactic": "rfl"
}
] |
[
57,
27
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
26,
1
] |
Mathlib/Algebra/Hom/Ring.lean
|
RingHom.copy_eq
|
[] |
[
514,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
513,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
WithTop.coe_sSup
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.95088\nγ : Type ?u.95091\nι : Sort ?u.95094\ninst✝ : ConditionallyCompleteLinearOrderBot α\ns : Set α\nhb : BddAbove s\n⊢ ↑(sSup s) = ⨆ (a : α) (_ : a ∈ s), ↑a",
"tactic": "rw [coe_sSup' hb, sSup_image]"
}
] |
[
1226,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1225,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.max_eq_left
|
[
{
"state_after": "a b : Nat\nh : b ≤ a\n⊢ max b a = a",
"state_before": "a b : Nat\nh : b ≤ a\n⊢ max a b = a",
"tactic": "rw [← Nat.max_comm b a]"
},
{
"state_after": "no goals",
"state_before": "a b : Nat\nh : b ≤ a\n⊢ max b a = a",
"tactic": "exact Nat.max_eq_right h"
}
] |
[
214,
52
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
213,
11
] |
Mathlib/Topology/Algebra/Module/FiniteDimension.lean
|
LinearMap.det_toContinuousLinearMap
|
[] |
[
318,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
316,
1
] |
Mathlib/Data/Set/Prod.lean
|
Set.mk_preimage_prod_left_eq_empty
|
[
{
"state_after": "case h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.26517\nδ : Type ?u.26520\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na✝ : α\nb : β\nhb : ¬b ∈ t\na : α\n⊢ a ∈ (fun a => (a, b)) ⁻¹' s ×ˢ t ↔ a ∈ ∅",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.26517\nδ : Type ?u.26520\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na : α\nb : β\nhb : ¬b ∈ t\n⊢ (fun a => (a, b)) ⁻¹' s ×ˢ t = ∅",
"tactic": "ext a"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.26517\nδ : Type ?u.26520\ns s₁ s₂ : Set α\nt t₁ t₂ : Set β\na✝ : α\nb : β\nhb : ¬b ∈ t\na : α\n⊢ a ∈ (fun a => (a, b)) ⁻¹' s ×ˢ t ↔ a ∈ ∅",
"tactic": "simp [hb]"
}
] |
[
239,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
Int.floor_eq_iff
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.139599\nα : Type u_1\nβ : Type ?u.139605\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nz : ℤ\na : α\n⊢ ⌊a⌋ = z ↔ ↑z ≤ a ∧ a < ↑z + 1",
"tactic": "rw [le_antisymm_iff, le_floor, ← Int.lt_add_one_iff, floor_lt, Int.cast_add, Int.cast_one,\n and_comm]"
}
] |
[
781,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
779,
1
] |
Mathlib/Topology/Order/Basic.lean
|
IsLUB.mem_lowerBounds_of_tendsto
|
[] |
[
2065,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2062,
1
] |
Mathlib/Topology/LocalExtr.lean
|
IsLocalMax.on
|
[] |
[
98,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Topology/UniformSpace/Equiv.lean
|
UniformEquiv.image_preimage
|
[] |
[
233,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
232,
1
] |
Mathlib/LinearAlgebra/Quotient.lean
|
Submodule.Quotient.mk_smul
|
[] |
[
144,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
143,
1
] |
Std/Data/List/Lemmas.lean
|
List.singleton_sublist
|
[
{
"state_after": "α : Type u_1\na : α\nl : List α\nh : a ∈ l\n⊢ [a] <+ l",
"state_before": "α : Type u_1\na : α\nl : List α\n⊢ [a] <+ l ↔ a ∈ l",
"tactic": "refine ⟨fun h => h.subset (mem_singleton_self _), fun h => ?_⟩"
},
{
"state_after": "case intro.intro\nα : Type u_1\na : α\nw✝¹ w✝ : List α\nh : a ∈ w✝¹ ++ a :: w✝\n⊢ [a] <+ w✝¹ ++ a :: w✝",
"state_before": "α : Type u_1\na : α\nl : List α\nh : a ∈ l\n⊢ [a] <+ l",
"tactic": "obtain ⟨_, _, rfl⟩ := append_of_mem h"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_1\na : α\nw✝¹ w✝ : List α\nh : a ∈ w✝¹ ++ a :: w✝\n⊢ [a] <+ w✝¹ ++ a :: w✝",
"tactic": "exact ((nil_sublist _).cons₂ _).trans (sublist_append_right ..)"
}
] |
[
426,
66
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
423,
9
] |
Mathlib/Algebra/Category/GroupCat/Colimits.lean
|
AddCommGroupCat.Colimits.cocone_naturality
|
[
{
"state_after": "case w\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ AddCommGroupCat\nj j' : J\nf : j ⟶ j'\nx✝ : ↑(F.obj j)\n⊢ ↑(F.map f ≫ coconeMorphism F j') x✝ = ↑(coconeMorphism F j) x✝",
"state_before": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ AddCommGroupCat\nj j' : J\nf : j ⟶ j'\n⊢ F.map f ≫ coconeMorphism F j' = coconeMorphism F j",
"tactic": "ext"
},
{
"state_after": "case w.a\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ AddCommGroupCat\nj j' : J\nf : j ⟶ j'\nx✝ : ↑(F.obj j)\n⊢ Setoid.r (Prequotient.of j' (↑(F.map f) x✝)) (Prequotient.of j x✝)",
"state_before": "case w\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ AddCommGroupCat\nj j' : J\nf : j ⟶ j'\nx✝ : ↑(F.obj j)\n⊢ ↑(F.map f ≫ coconeMorphism F j') x✝ = ↑(coconeMorphism F j) x✝",
"tactic": "apply Quot.sound"
},
{
"state_after": "no goals",
"state_before": "case w.a\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ AddCommGroupCat\nj j' : J\nf : j ⟶ j'\nx✝ : ↑(F.obj j)\n⊢ Setoid.r (Prequotient.of j' (↑(F.map f) x✝)) (Prequotient.of j x✝)",
"tactic": "apply Relation.map"
}
] |
[
219,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
1
] |
Mathlib/CategoryTheory/Opposites.lean
|
Quiver.Hom.unop_op
|
[] |
[
55,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/Data/Sum/Order.lean
|
WithBot.orderIsoPUnitSumLex_symm_inr
|
[] |
[
766,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
765,
1
] |
Std/Data/String/Lemmas.lean
|
String.join_eq
|
[
{
"state_after": "no goals",
"state_before": "ss : List String\nx✝ : List Char\n⊢ List.foldl (fun x x_1 => x ++ x_1) { data := x✝ } [] = { data := x✝ ++ List.join (List.map data []) }",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "ss✝ : List String\ns : List Char\nss : List String\nx✝ : List Char\n⊢ { data := x✝ ++ s ++ List.join (List.map data ss) } =\n { data := x✝ ++ List.join (List.map data ({ data := s } :: ss)) }",
"tactic": "simp"
}
] |
[
478,
46
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
475,
1
] |
Mathlib/Control/Basic.lean
|
CommApplicative.commutative_map
|
[
{
"state_after": "no goals",
"state_before": "α✝ β✝ γ✝ : Type u\nm✝ : ?m.20189\nm : Type u → Type v\nh : Applicative m\ninst✝ : CommApplicative m\nα β γ : Type u\na : m α\nb : m β\nf : α → β → γ\n⊢ (Seq.seq (f <$> a) fun x => b) = (fun p => f p.fst p.snd) <$> Seq.seq (Prod.mk <$> a) fun x => b",
"tactic": "simp [seq_map_assoc, map_seq, seq_assoc, seq_pure, map_map] <;> rfl"
}
] |
[
275,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
267,
1
] |
Mathlib/Order/SuccPred/Basic.lean
|
Order.Ioo_succ_right_eq_insert
|
[] |
[
512,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
511,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.le_iff_val_le_val
|
[] |
[
290,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
289,
1
] |
Mathlib/Data/Polynomial/EraseLead.lean
|
Polynomial.lt_natDegree_of_mem_eraseLead_support
|
[
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nf : R[X]\na : ℕ\nh : a ≠ natDegree f ∧ a ∈ support f\n⊢ a < natDegree f",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nf : R[X]\na : ℕ\nh : a ∈ support (eraseLead f)\n⊢ a < natDegree f",
"tactic": "rw [eraseLead_support, mem_erase] at h"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nf : R[X]\na : ℕ\nh : a ≠ natDegree f ∧ a ∈ support f\n⊢ a < natDegree f",
"tactic": "exact (le_natDegree_of_mem_supp a h.2).lt_of_ne h.1"
}
] |
[
100,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Order/Disjoint.lean
|
Disjoint.mono_left
|
[] |
[
73,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/Data/Finite/Basic.lean
|
Finite.prod_left
|
[] |
[
77,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
76,
1
] |
Mathlib/RingTheory/MvPolynomial/Ideal.lean
|
MvPolynomial.mem_ideal_span_monomial_image
|
[
{
"state_after": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nx : MvPolynomial σ R\ns : Set (σ →₀ ℕ)\n⊢ (∀ (m : σ →₀ ℕ), m ∈ x.support → ∃ m', m' ∈ s ∧ ∃ d, m = d + m') ↔\n ∀ (xi : σ →₀ ℕ), xi ∈ support x → ∃ si, si ∈ s ∧ si ≤ xi",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nx : MvPolynomial σ R\ns : Set (σ →₀ ℕ)\n⊢ x ∈ Ideal.span ((fun s => ↑(monomial s) 1) '' s) ↔ ∀ (xi : σ →₀ ℕ), xi ∈ support x → ∃ si, si ∈ s ∧ si ≤ xi",
"tactic": "refine' AddMonoidAlgebra.mem_ideal_span_of'_image.trans _"
},
{
"state_after": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nx : MvPolynomial σ R\ns : Set (σ →₀ ℕ)\n⊢ (∀ (m : σ →₀ ℕ), m ∈ x.support → ∃ m', m' ∈ s ∧ ∃ d, m = m' + d) ↔\n ∀ (xi : σ →₀ ℕ), xi ∈ support x → ∃ si, si ∈ s ∧ ∃ c, xi = si + c",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nx : MvPolynomial σ R\ns : Set (σ →₀ ℕ)\n⊢ (∀ (m : σ →₀ ℕ), m ∈ x.support → ∃ m', m' ∈ s ∧ ∃ d, m = d + m') ↔\n ∀ (xi : σ →₀ ℕ), xi ∈ support x → ∃ si, si ∈ s ∧ si ≤ xi",
"tactic": "simp_rw [le_iff_exists_add, add_comm]"
},
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nx : MvPolynomial σ R\ns : Set (σ →₀ ℕ)\n⊢ (∀ (m : σ →₀ ℕ), m ∈ x.support → ∃ m', m' ∈ s ∧ ∃ d, m = m' + d) ↔\n ∀ (xi : σ →₀ ℕ), xi ∈ support x → ∃ si, si ∈ s ∧ ∃ c, xi = si + c",
"tactic": "rfl"
}
] |
[
39,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
35,
1
] |
Mathlib/Data/Polynomial/Lifts.lean
|
Polynomial.X_pow_mem_lifts
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : Semiring R\nS : Type v\ninst✝ : Semiring S\nf✝ f : R →+* S\nn : ℕ\n⊢ ↑(mapRingHom f) (X ^ n) = X ^ n",
"tactic": "simp only [coe_mapRingHom, map_pow, Set.mem_univ, Subsemiring.coe_top, eq_self_iff_true,\n map_X, and_self_iff]"
}
] |
[
110,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/NumberTheory/ArithmeticFunction.lean
|
Nat.ArithmeticFunction.cardDistinctFactors_apply
|
[] |
[
918,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
917,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
WithBot.ciSup_empty
|
[] |
[
136,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/CategoryTheory/Types.lean
|
CategoryTheory.types_id_apply
|
[] |
[
75,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.ae_eq_restrict_biUnion_finset_iff
|
[] |
[
2782,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2780,
1
] |
Mathlib/Analysis/Convex/Normed.lean
|
isConnected_setOf_sameRay
|
[
{
"state_after": "case pos\nι : Type ?u.49013\nE : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns t : Set E\nx : E\nhx : x = 0\n⊢ IsConnected {y | SameRay ℝ x y}\n\ncase neg\nι : Type ?u.49013\nE : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns t : Set E\nx : E\nhx : ¬x = 0\n⊢ IsConnected {y | SameRay ℝ x y}",
"state_before": "ι : Type ?u.49013\nE : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns t : Set E\nx : E\n⊢ IsConnected {y | SameRay ℝ x y}",
"tactic": "by_cases hx : x = 0"
},
{
"state_after": "case neg\nι : Type ?u.49013\nE : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns t : Set E\nx : E\nhx : ¬x = 0\n⊢ IsConnected {y | ∃ r, 0 ≤ r ∧ r • x = y}",
"state_before": "case neg\nι : Type ?u.49013\nE : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns t : Set E\nx : E\nhx : ¬x = 0\n⊢ IsConnected {y | SameRay ℝ x y}",
"tactic": "simp_rw [← exists_nonneg_left_iff_sameRay hx]"
},
{
"state_after": "no goals",
"state_before": "case neg\nι : Type ?u.49013\nE : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns t : Set E\nx : E\nhx : ¬x = 0\n⊢ IsConnected {y | ∃ r, 0 ≤ r ∧ r • x = y}",
"tactic": "exact isConnected_Ici.image _ (continuous_id.smul continuous_const).continuousOn"
},
{
"state_after": "no goals",
"state_before": "case pos\nι : Type ?u.49013\nE : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns t : Set E\nx : E\nhx : x = 0\n⊢ IsConnected {y | SameRay ℝ x y}",
"tactic": "simpa [hx] using isConnected_univ"
}
] |
[
145,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Combinatorics/Composition.lean
|
Composition.index_embedding
|
[
{
"state_after": "n : ℕ\nc : Composition n\ni : Fin (length c)\nj : Fin (blocksFun c i)\n⊢ i = index c (↑(embedding c i) j)",
"state_before": "n : ℕ\nc : Composition n\ni : Fin (length c)\nj : Fin (blocksFun c i)\n⊢ index c (↑(embedding c i) j) = i",
"tactic": "symm"
},
{
"state_after": "n : ℕ\nc : Composition n\ni : Fin (length c)\nj : Fin (blocksFun c i)\n⊢ ↑(embedding c i) j ∈ Set.range ↑(embedding c i)",
"state_before": "n : ℕ\nc : Composition n\ni : Fin (length c)\nj : Fin (blocksFun c i)\n⊢ i = index c (↑(embedding c i) j)",
"tactic": "rw [← mem_range_embedding_iff']"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nc : Composition n\ni : Fin (length c)\nj : Fin (blocksFun c i)\n⊢ ↑(embedding c i) j ∈ Set.range ↑(embedding c i)",
"tactic": "apply Set.mem_range_self"
}
] |
[
438,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
434,
1
] |
Mathlib/Order/Hom/Set.lean
|
OrderIso.range_eq
|
[] |
[
31,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
30,
1
] |
Mathlib/RingTheory/HahnSeries.lean
|
HahnSeries.SummableFamily.coe_ofFinsupp
|
[] |
[
1680,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1679,
1
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
Cubic.degree_of_b_eq_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type ?u.580862\nF : Type ?u.580865\nK : Type ?u.580868\nP Q : Cubic R\na b c d a' b' c' d' : R\ninst✝ : Semiring R\nha : P.a = 0\nhb : P.b = 0\n⊢ degree (toPoly P) ≤ 1",
"tactic": "simpa only [of_b_eq_zero ha hb] using degree_linear_le"
}
] |
[
332,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
331,
1
] |
Mathlib/Topology/Instances/TrivSqZeroExt.lean
|
TrivSqZeroExt.nhds_inl
|
[] |
[
54,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/Topology/Hom/Open.lean
|
ContinuousOpenMap.copy_eq
|
[] |
[
101,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/Data/Real/Basic.lean
|
Real.sSup_nonpos
|
[] |
[
890,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
889,
1
] |
Mathlib/Order/Hom/Basic.lean
|
OrderIso.injective
|
[] |
[
818,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
817,
11
] |
Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean
|
AffineEquiv.injective_pointReflection_left_of_module
|
[
{
"state_after": "no goals",
"state_before": "k : Type u_1\nP₁ : Type u_2\nP₂ : Type ?u.663302\nP₃ : Type ?u.663305\nP₄ : Type ?u.663308\nV₁ : Type u_3\nV₂ : Type ?u.663314\nV₃ : Type ?u.663317\nV₄ : Type ?u.663320\ninst✝¹³ : Ring k\ninst✝¹² : AddCommGroup V₁\ninst✝¹¹ : Module k V₁\ninst✝¹⁰ : AffineSpace V₁ P₁\ninst✝⁹ : AddCommGroup V₂\ninst✝⁸ : Module k V₂\ninst✝⁷ : AffineSpace V₂ P₂\ninst✝⁶ : AddCommGroup V₃\ninst✝⁵ : Module k V₃\ninst✝⁴ : AffineSpace V₃ P₃\ninst✝³ : AddCommGroup V₄\ninst✝² : Module k V₄\ninst✝¹ : AffineSpace V₄ P₄\ninst✝ : Invertible 2\nx y : V₁\nh : bit0 x = bit0 y\n⊢ x = y",
"tactic": "rwa [bit0, bit0, ← two_smul k x, ← two_smul k y,\n (isUnit_of_invertible (2 : k)).smul_left_cancel] at h"
}
] |
[
603,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
599,
1
] |
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