file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Data/List/Lattice.lean
|
List.inter_cons_of_mem
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl l₁✝ l₂ : List α\np : α → Prop\na : α\ninst✝ : DecidableEq α\nl₁ : List α\nh : a ∈ l₂\n⊢ (a :: l₁) ∩ l₂ = a :: l₁ ∩ l₂",
"tactic": "simp only [instInterList, List.inter, filter_cons_of_pos, h]"
}
] |
[
147,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean
|
SimpleGraph.incMatrix_apply_mul_incMatrix_apply_of_not_adj
|
[
{
"state_after": "case h\nR : Type u_2\nα : Type u_1\nG : SimpleGraph α\ninst✝ : MulZeroOneClass R\na b : α\ne : Sym2 α\nhab : a ≠ b\nh : ¬Adj G a b\n⊢ ¬e ∈ incidenceSet G a ∩ incidenceSet G b",
"state_before": "R : Type u_2\nα : Type u_1\nG : SimpleGraph α\ninst✝ : MulZeroOneClass R\na b : α\ne : Sym2 α\nhab : a ≠ b\nh : ¬Adj G a b\n⊢ incMatrix R G a e * incMatrix R G b e = 0",
"tactic": "rw [incMatrix_apply_mul_incMatrix_apply, Set.indicator_of_not_mem]"
},
{
"state_after": "case h\nR : Type u_2\nα : Type u_1\nG : SimpleGraph α\ninst✝ : MulZeroOneClass R\na b : α\ne : Sym2 α\nhab : a ≠ b\nh : ¬Adj G a b\n⊢ ¬e ∈ ∅",
"state_before": "case h\nR : Type u_2\nα : Type u_1\nG : SimpleGraph α\ninst✝ : MulZeroOneClass R\na b : α\ne : Sym2 α\nhab : a ≠ b\nh : ¬Adj G a b\n⊢ ¬e ∈ incidenceSet G a ∩ incidenceSet G b",
"tactic": "rw [G.incidenceSet_inter_incidenceSet_of_not_adj h hab]"
},
{
"state_after": "no goals",
"state_before": "case h\nR : Type u_2\nα : Type u_1\nG : SimpleGraph α\ninst✝ : MulZeroOneClass R\na b : α\ne : Sym2 α\nhab : a ≠ b\nh : ¬Adj G a b\n⊢ ¬e ∈ ∅",
"tactic": "exact Set.not_mem_empty e"
}
] |
[
91,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/RingTheory/WittVector/Defs.lean
|
WittVector.wittZSMul_vars
|
[] |
[
422,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
420,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.multiset_prod_span_singleton
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nι : Type ?u.251146\ninst✝ : CommSemiring R\nI J K L : Ideal R\nm : Multiset R\n⊢ Multiset.prod (Multiset.map (fun x => span {x}) 0) = span {Multiset.prod 0}",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nι : Type ?u.251146\ninst✝ : CommSemiring R\nI J K L : Ideal R\nm✝ : Multiset R\na : R\nm : Multiset R\nih : Multiset.prod (Multiset.map (fun x => span {x}) m) = span {Multiset.prod m}\n⊢ Multiset.prod (Multiset.map (fun x => span {x}) (a ::ₘ m)) = span {Multiset.prod (a ::ₘ m)}",
"tactic": "simp only [Multiset.map_cons, Multiset.prod_cons, ih, ← Ideal.span_singleton_mul_span_singleton]"
}
] |
[
624,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
621,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.zero_toReal
|
[] |
[
239,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
239,
9
] |
Mathlib/Logic/Basic.lean
|
Function.mt
|
[] |
[
184,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
184,
11
] |
Mathlib/Topology/Order/Basic.lean
|
interior_Ico
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : DenselyOrdered α\na✝ b✝ : α\ns : Set α\ninst✝ : NoMinOrder α\na b : α\n⊢ interior (Ico a b) = Ioo a b",
"tactic": "rw [← Ici_inter_Iio, interior_inter, interior_Ici, interior_Iio, Ioi_inter_Iio]"
}
] |
[
2296,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2295,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
MvPowerSeries.eq_mul_inv_iff_mul_eq
|
[
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nR : Type ?u.1886295\nk✝ : Type u_2\ninst✝ : Field k✝\nφ₁ φ₂ φ₃ : MvPowerSeries σ k✝\nh : ↑(constantCoeff σ k✝) φ₃ ≠ 0\nk : φ₁ = φ₂ * φ₃⁻¹\n⊢ φ₁ * φ₃ = φ₂",
"tactic": "simp [k, mul_assoc, MvPowerSeries.inv_mul_cancel _ h]"
},
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nR : Type ?u.1886295\nk✝ : Type u_2\ninst✝ : Field k✝\nφ₁ φ₂ φ₃ : MvPowerSeries σ k✝\nh : ↑(constantCoeff σ k✝) φ₃ ≠ 0\nk : φ₁ * φ₃ = φ₂\n⊢ φ₁ = φ₂ * φ₃⁻¹",
"tactic": "simp [← k, mul_assoc, MvPowerSeries.mul_inv_cancel _ h]"
}
] |
[
997,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
994,
11
] |
Mathlib/Data/Rat/Floor.lean
|
Rat.round_cast
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : ℚ\nH : ↑2 = 2\n⊢ round ↑x = round x",
"state_before": "α : Type u_1\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : ℚ\n⊢ round ↑x = round x",
"tactic": "have H : ((2 : ℚ) : α) = (2 : α) := Rat.cast_coe_nat 2"
},
{
"state_after": "α : Type u_1\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : ℚ\nH : ↑2 = 2\nthis : ↑(x + 1 / 2) = ↑x + 1 / 2\n⊢ round ↑x = round x",
"state_before": "α : Type u_1\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : ℚ\nH : ↑2 = 2\n⊢ round ↑x = round x",
"tactic": "have : ((x + 1 / 2 : ℚ) : α) = x + 1 / 2 := by simp [H]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : ℚ\nH : ↑2 = 2\nthis : ↑(x + 1 / 2) = ↑x + 1 / 2\n⊢ round ↑x = round x",
"tactic": "rw [round_eq, round_eq, ← this, floor_cast]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : ℚ\nH : ↑2 = 2\n⊢ ↑(x + 1 / 2) = ↑x + 1 / 2",
"tactic": "simp [H]"
}
] |
[
87,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Analysis/Normed/Group/AddCircle.lean
|
AddCircle.coe_real_preimage_closedBall_eq_iUnion
|
[
{
"state_after": "case inl\nx ε : ℝ\n⊢ QuotientAddGroup.mk ⁻¹' closedBall (↑x) ε = ⋃ (z : ℤ), closedBall (x + z • 0) ε\n\ncase inr\np x ε : ℝ\nhp : p ≠ 0\n⊢ QuotientAddGroup.mk ⁻¹' closedBall (↑x) ε = ⋃ (z : ℤ), closedBall (x + z • p) ε",
"state_before": "p x ε : ℝ\n⊢ QuotientAddGroup.mk ⁻¹' closedBall (↑x) ε = ⋃ (z : ℤ), closedBall (x + z • p) ε",
"tactic": "rcases eq_or_ne p 0 with (rfl | hp)"
},
{
"state_after": "case inr.h\np x ε : ℝ\nhp : p ≠ 0\ny : ℝ\n⊢ y ∈ QuotientAddGroup.mk ⁻¹' closedBall (↑x) ε ↔ y ∈ ⋃ (z : ℤ), closedBall (x + z • p) ε",
"state_before": "case inr\np x ε : ℝ\nhp : p ≠ 0\n⊢ QuotientAddGroup.mk ⁻¹' closedBall (↑x) ε = ⋃ (z : ℤ), closedBall (x + z • p) ε",
"tactic": "ext y"
},
{
"state_after": "case inr.h\np x ε : ℝ\nhp : p ≠ 0\ny : ℝ\n⊢ abs (y - x - ↑(round (p⁻¹ * (y - x))) * p) ≤ ε ↔ ∃ i, abs (y - x - ↑i * p) ≤ ε",
"state_before": "case inr.h\np x ε : ℝ\nhp : p ≠ 0\ny : ℝ\n⊢ y ∈ QuotientAddGroup.mk ⁻¹' closedBall (↑x) ε ↔ y ∈ ⋃ (z : ℤ), closedBall (x + z • p) ε",
"tactic": "simp only [dist_eq_norm, mem_preimage, mem_closedBall, zsmul_eq_mul, mem_iUnion, Real.norm_eq_abs,\n ← QuotientAddGroup.mk_sub, norm_eq, ← sub_sub]"
},
{
"state_after": "case inr.h\np x ε : ℝ\nhp : p ≠ 0\ny : ℝ\n⊢ (∃ i, abs (y - x - ↑i * p) ≤ ε) → abs (y - x - ↑(round (p⁻¹ * (y - x))) * p) ≤ ε",
"state_before": "case inr.h\np x ε : ℝ\nhp : p ≠ 0\ny : ℝ\n⊢ abs (y - x - ↑(round (p⁻¹ * (y - x))) * p) ≤ ε ↔ ∃ i, abs (y - x - ↑i * p) ≤ ε",
"tactic": "refine' ⟨fun h => ⟨round (p⁻¹ * (y - x)), h⟩, _⟩"
},
{
"state_after": "case inr.h.intro\np x ε : ℝ\nhp : p ≠ 0\ny : ℝ\nn : ℤ\nhn : abs (y - x - ↑n * p) ≤ ε\n⊢ abs (y - x - ↑(round (p⁻¹ * (y - x))) * p) ≤ ε",
"state_before": "case inr.h\np x ε : ℝ\nhp : p ≠ 0\ny : ℝ\n⊢ (∃ i, abs (y - x - ↑i * p) ≤ ε) → abs (y - x - ↑(round (p⁻¹ * (y - x))) * p) ≤ ε",
"tactic": "rintro ⟨n, hn⟩"
},
{
"state_after": "case inr.h.intro\np x ε : ℝ\nhp : p ≠ 0\ny : ℝ\nn : ℤ\nhn : abs (p⁻¹ * (y - x) - ↑n) ≤ abs p⁻¹ * ε\n⊢ abs (p⁻¹ * (y - x) - ↑(round (p⁻¹ * (y - x)))) ≤ abs p⁻¹ * ε",
"state_before": "case inr.h.intro\np x ε : ℝ\nhp : p ≠ 0\ny : ℝ\nn : ℤ\nhn : abs (y - x - ↑n * p) ≤ ε\n⊢ abs (y - x - ↑(round (p⁻¹ * (y - x))) * p) ≤ ε",
"tactic": "rw [← mul_le_mul_left (abs_pos.mpr <| inv_ne_zero hp), ← abs_mul, mul_sub, mul_comm _ p,\n inv_mul_cancel_left₀ hp] at hn⊢"
},
{
"state_after": "no goals",
"state_before": "case inr.h.intro\np x ε : ℝ\nhp : p ≠ 0\ny : ℝ\nn : ℤ\nhn : abs (p⁻¹ * (y - x) - ↑n) ≤ abs p⁻¹ * ε\n⊢ abs (p⁻¹ * (y - x) - ↑(round (p⁻¹ * (y - x)))) ≤ abs p⁻¹ * ε",
"tactic": "exact (round_le (p⁻¹ * (y - x)) n).trans hn"
},
{
"state_after": "no goals",
"state_before": "case inl\nx ε : ℝ\n⊢ QuotientAddGroup.mk ⁻¹' closedBall (↑x) ε = ⋃ (z : ℤ), closedBall (x + z • 0) ε",
"tactic": "simp [iUnion_const]"
}
] |
[
199,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/Analysis/Convex/Between.lean
|
Wbtw.trans_right_left
|
[
{
"state_after": "R : Type u_1\nV : Type u_2\nV' : Type ?u.578788\nP : Type u_3\nP' : Type ?u.578794\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nw x y z : P\nh₁ : Wbtw R z x w\nh₂ : Wbtw R z y x\n⊢ Wbtw R y x w",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.578788\nP : Type u_3\nP' : Type ?u.578794\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nw x y z : P\nh₁ : Wbtw R w x z\nh₂ : Wbtw R x y z\n⊢ Wbtw R w x y",
"tactic": "rw [wbtw_comm] at *"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.578788\nP : Type u_3\nP' : Type ?u.578794\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nw x y z : P\nh₁ : Wbtw R z x w\nh₂ : Wbtw R z y x\n⊢ Wbtw R y x w",
"tactic": "exact h₁.trans_left_right h₂"
}
] |
[
836,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
833,
1
] |
Mathlib/Order/Filter/FilterProduct.lean
|
Filter.Germ.max_def
|
[
{
"state_after": "case inl\nα : Type u\nβ : Type v\nφ : Ultrafilter α\ninst✝ : LinearOrder β\nx y : Germ (↑φ) β\na b : α → β\nh : ↑a ≤ ↑b\n⊢ max ↑a ↑b = map₂ max ↑a ↑b\n\ncase inr\nα : Type u\nβ : Type v\nφ : Ultrafilter α\ninst✝ : LinearOrder β\nx y : Germ (↑φ) β\na b : α → β\nh : ↑b ≤ ↑a\n⊢ max ↑a ↑b = map₂ max ↑a ↑b",
"state_before": "α : Type u\nβ : Type v\nφ : Ultrafilter α\ninst✝ : LinearOrder β\nx y : Germ (↑φ) β\na b : α → β\n⊢ max ↑a ↑b = map₂ max ↑a ↑b",
"tactic": "cases' le_total (a : β*) b with h h"
},
{
"state_after": "case inl\nα : Type u\nβ : Type v\nφ : Ultrafilter α\ninst✝ : LinearOrder β\nx y : Germ (↑φ) β\na b : α → β\nh : ↑a ≤ ↑b\n⊢ b =ᶠ[↑φ] fun x => max (a x) (b x)",
"state_before": "case inl\nα : Type u\nβ : Type v\nφ : Ultrafilter α\ninst✝ : LinearOrder β\nx y : Germ (↑φ) β\na b : α → β\nh : ↑a ≤ ↑b\n⊢ max ↑a ↑b = map₂ max ↑a ↑b",
"tactic": "rw [max_eq_right h, map₂_coe, coe_eq]"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u\nβ : Type v\nφ : Ultrafilter α\ninst✝ : LinearOrder β\nx y : Germ (↑φ) β\na b : α → β\nh : ↑a ≤ ↑b\n⊢ b =ᶠ[↑φ] fun x => max (a x) (b x)",
"tactic": "exact h.mono fun i hi => (max_eq_right hi).symm"
},
{
"state_after": "case inr\nα : Type u\nβ : Type v\nφ : Ultrafilter α\ninst✝ : LinearOrder β\nx y : Germ (↑φ) β\na b : α → β\nh : ↑b ≤ ↑a\n⊢ a =ᶠ[↑φ] fun x => max (a x) (b x)",
"state_before": "case inr\nα : Type u\nβ : Type v\nφ : Ultrafilter α\ninst✝ : LinearOrder β\nx y : Germ (↑φ) β\na b : α → β\nh : ↑b ≤ ↑a\n⊢ max ↑a ↑b = map₂ max ↑a ↑b",
"tactic": "rw [max_eq_left h, map₂_coe, coe_eq]"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u\nβ : Type v\nφ : Ultrafilter α\ninst✝ : LinearOrder β\nx y : Germ (↑φ) β\na b : α → β\nh : ↑b ≤ ↑a\n⊢ a =ᶠ[↑φ] fun x => max (a x) (b x)",
"tactic": "exact h.mono fun i hi => (max_eq_left hi).symm"
}
] |
[
224,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
218,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean
|
ContDiffWithinAt.congr'
|
[] |
[
481,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
479,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
MonoidHom.liftOfRightInverse_comp
|
[] |
[
3366,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3364,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_disj_sum
|
[
{
"state_after": "ι : Type ?u.308476\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α ⊕ γ → β\n⊢ (∏ x in s, f (↑Function.Embedding.inl x)) * ∏ x in t, f (↑Function.Embedding.inr x) =\n (∏ x in s, f (Sum.inl x)) * ∏ x in t, f (Sum.inr x)",
"state_before": "ι : Type ?u.308476\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α ⊕ γ → β\n⊢ ∏ x in disjSum s t, f x = (∏ x in s, f (Sum.inl x)) * ∏ x in t, f (Sum.inr x)",
"tactic": "rw [← map_inl_disjUnion_map_inr, prod_disjUnion, prod_map, prod_map]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.308476\nβ : Type u\nα : Type v\nγ : Type w\ns✝ s₁ s₂ : Finset α\na : α\nf✝ g : α → β\ninst✝ : CommMonoid β\ns : Finset α\nt : Finset γ\nf : α ⊕ γ → β\n⊢ (∏ x in s, f (↑Function.Embedding.inl x)) * ∏ x in t, f (↑Function.Embedding.inr x) =\n (∏ x in s, f (Sum.inl x)) * ∏ x in t, f (Sum.inr x)",
"tactic": "rfl"
}
] |
[
506,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
503,
1
] |
Mathlib/Data/Nat/Basic.lean
|
Nat.one_add_le_iff
|
[
{
"state_after": "no goals",
"state_before": "m n k a b : ℕ\n⊢ 1 + a ≤ b ↔ a < b",
"tactic": "simp only [add_comm, add_one_le_iff]"
}
] |
[
233,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
233,
1
] |
Mathlib/GroupTheory/Submonoid/Pointwise.lean
|
Submonoid.smul_bot
|
[] |
[
254,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
253,
1
] |
Mathlib/Topology/Algebra/Valuation.lean
|
Valued.mem_nhds
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\n_i : Valued R Γ₀\ns : Set R\nx : R\n⊢ s ∈ 𝓝 x ↔ ∃ γ, {y | ↑v (y - x) < ↑γ} ⊆ s",
"tactic": "simp only [← nhds_translation_add_neg x, ← sub_eq_add_neg, preimage_setOf_eq, true_and,\n ((hasBasis_nhds_zero R Γ₀).comap fun y => y - x).mem_iff]"
}
] |
[
142,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
src/lean/Init/Classical.lean
|
Classical.epsilon_spec_aux
|
[] |
[
98,
45
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
97,
1
] |
Mathlib/GroupTheory/GroupAction/Group.lean
|
smul_ne_zero_iff_ne'
|
[] |
[
292,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
291,
1
] |
Mathlib/Data/Rat/Cast.lean
|
Rat.cast_ne_zero
|
[] |
[
215,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
214,
1
] |
Mathlib/Algebra/Module/Equiv.lean
|
LinearEquiv.invFun_eq_symm
|
[] |
[
303,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
302,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.IsSt.neg
|
[] |
[
408,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
407,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.smul_finset_inter₀
|
[] |
[
2061,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2060,
1
] |
Mathlib/Algebra/Order/Sub/Canonical.lean
|
tsub_lt_tsub_right_of_le
|
[] |
[
278,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
277,
1
] |
Mathlib/SetTheory/Ordinal/FixedPoint.lean
|
Ordinal.IsNormal.deriv_fp
|
[] |
[
539,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
538,
1
] |
Mathlib/Combinatorics/Quiver/Symmetric.lean
|
Quiver.symmetrify_reverse
|
[] |
[
126,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Order/CompactlyGenerated.lean
|
DirectedOn.inf_sSup_eq
|
[
{
"state_after": "ι : Sort ?u.89037\nα : Type u_1\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns : Set α\nh : DirectedOn (fun x x_1 => x ≤ x_1) s\n⊢ ∀ (c : α), CompleteLattice.IsCompactElement c → c ≤ a ⊓ sSup s → c ≤ ⨆ (b : α) (_ : b ∈ s), a ⊓ b",
"state_before": "ι : Sort ?u.89037\nα : Type u_1\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns : Set α\nh : DirectedOn (fun x x_1 => x ≤ x_1) s\n⊢ a ⊓ sSup s ≤ ⨆ (b : α) (_ : b ∈ s), a ⊓ b",
"tactic": "rw [le_iff_compact_le_imp]"
},
{
"state_after": "case pos\nι : Sort ?u.89037\nα : Type u_1\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns : Set α\nh : DirectedOn (fun x x_1 => x ≤ x_1) s\nhs : Set.Nonempty s\n⊢ ∀ (c : α), CompleteLattice.IsCompactElement c → c ≤ a ⊓ sSup s → c ≤ ⨆ (b : α) (_ : b ∈ s), a ⊓ b\n\ncase neg\nι : Sort ?u.89037\nα : Type u_1\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns : Set α\nh : DirectedOn (fun x x_1 => x ≤ x_1) s\nhs : ¬Set.Nonempty s\n⊢ ∀ (c : α), CompleteLattice.IsCompactElement c → c ≤ a ⊓ sSup s → c ≤ ⨆ (b : α) (_ : b ∈ s), a ⊓ b",
"state_before": "ι : Sort ?u.89037\nα : Type u_1\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns : Set α\nh : DirectedOn (fun x x_1 => x ≤ x_1) s\n⊢ ∀ (c : α), CompleteLattice.IsCompactElement c → c ≤ a ⊓ sSup s → c ≤ ⨆ (b : α) (_ : b ∈ s), a ⊓ b",
"tactic": "by_cases hs : s.Nonempty"
},
{
"state_after": "case pos\nι : Sort ?u.89037\nα : Type u_1\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns : Set α\nh : DirectedOn (fun x x_1 => x ≤ x_1) s\nhs : Set.Nonempty s\nc : α\nhc : CompleteLattice.IsCompactElement c\nhcinf : c ≤ a ⊓ sSup s\n⊢ c ≤ ⨆ (b : α) (_ : b ∈ s), a ⊓ b",
"state_before": "case pos\nι : Sort ?u.89037\nα : Type u_1\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns : Set α\nh : DirectedOn (fun x x_1 => x ≤ x_1) s\nhs : Set.Nonempty s\n⊢ ∀ (c : α), CompleteLattice.IsCompactElement c → c ≤ a ⊓ sSup s → c ≤ ⨆ (b : α) (_ : b ∈ s), a ⊓ b",
"tactic": "intro c hc hcinf"
},
{
"state_after": "case pos\nι : Sort ?u.89037\nα : Type u_1\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns : Set α\nh : DirectedOn (fun x x_1 => x ≤ x_1) s\nhs : Set.Nonempty s\nc : α\nhc : CompleteLattice.IsCompactElement c\nhcinf : c ≤ a ∧ c ≤ sSup s\n⊢ c ≤ ⨆ (b : α) (_ : b ∈ s), a ⊓ b",
"state_before": "case pos\nι : Sort ?u.89037\nα : Type u_1\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns : Set α\nh : DirectedOn (fun x x_1 => x ≤ x_1) s\nhs : Set.Nonempty s\nc : α\nhc : CompleteLattice.IsCompactElement c\nhcinf : c ≤ a ⊓ sSup s\n⊢ c ≤ ⨆ (b : α) (_ : b ∈ s), a ⊓ b",
"tactic": "rw [le_inf_iff] at hcinf"
},
{
"state_after": "case pos\nι : Sort ?u.89037\nα : Type u_1\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns : Set α\nh : DirectedOn (fun x x_1 => x ≤ x_1) s\nhs : Set.Nonempty s\nc : α\nhc : ∀ (s : Set α), Set.Nonempty s → DirectedOn (fun x x_1 => x ≤ x_1) s → c ≤ sSup s → ∃ x, x ∈ s ∧ c ≤ x\nhcinf : c ≤ a ∧ c ≤ sSup s\n⊢ c ≤ ⨆ (b : α) (_ : b ∈ s), a ⊓ b",
"state_before": "case pos\nι : Sort ?u.89037\nα : Type u_1\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns : Set α\nh : DirectedOn (fun x x_1 => x ≤ x_1) s\nhs : Set.Nonempty s\nc : α\nhc : CompleteLattice.IsCompactElement c\nhcinf : c ≤ a ∧ c ≤ sSup s\n⊢ c ≤ ⨆ (b : α) (_ : b ∈ s), a ⊓ b",
"tactic": "rw [CompleteLattice.isCompactElement_iff_le_of_directed_sSup_le] at hc"
},
{
"state_after": "case pos.intro.intro\nι : Sort ?u.89037\nα : Type u_1\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns : Set α\nh : DirectedOn (fun x x_1 => x ≤ x_1) s\nhs : Set.Nonempty s\nc : α\nhc : ∀ (s : Set α), Set.Nonempty s → DirectedOn (fun x x_1 => x ≤ x_1) s → c ≤ sSup s → ∃ x, x ∈ s ∧ c ≤ x\nhcinf : c ≤ a ∧ c ≤ sSup s\nd : α\nds : d ∈ s\ncd : c ≤ d\n⊢ c ≤ ⨆ (b : α) (_ : b ∈ s), a ⊓ b",
"state_before": "case pos\nι : Sort ?u.89037\nα : Type u_1\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns : Set α\nh : DirectedOn (fun x x_1 => x ≤ x_1) s\nhs : Set.Nonempty s\nc : α\nhc : ∀ (s : Set α), Set.Nonempty s → DirectedOn (fun x x_1 => x ≤ x_1) s → c ≤ sSup s → ∃ x, x ∈ s ∧ c ≤ x\nhcinf : c ≤ a ∧ c ≤ sSup s\n⊢ c ≤ ⨆ (b : α) (_ : b ∈ s), a ⊓ b",
"tactic": "rcases hc s hs h hcinf.2 with ⟨d, ds, cd⟩"
},
{
"state_after": "case pos.intro.intro\nι : Sort ?u.89037\nα : Type u_1\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns : Set α\nh : DirectedOn (fun x x_1 => x ≤ x_1) s\nhs : Set.Nonempty s\nc : α\nhc : ∀ (s : Set α), Set.Nonempty s → DirectedOn (fun x x_1 => x ≤ x_1) s → c ≤ sSup s → ∃ x, x ∈ s ∧ c ≤ x\nhcinf : c ≤ a ∧ c ≤ sSup s\nd : α\nds : d ∈ s\ncd : c ≤ d\n⊢ a ⊓ d ≤ a ⊓ d",
"state_before": "case pos.intro.intro\nι : Sort ?u.89037\nα : Type u_1\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns : Set α\nh : DirectedOn (fun x x_1 => x ≤ x_1) s\nhs : Set.Nonempty s\nc : α\nhc : ∀ (s : Set α), Set.Nonempty s → DirectedOn (fun x x_1 => x ≤ x_1) s → c ≤ sSup s → ∃ x, x ∈ s ∧ c ≤ x\nhcinf : c ≤ a ∧ c ≤ sSup s\nd : α\nds : d ∈ s\ncd : c ≤ d\n⊢ c ≤ ⨆ (b : α) (_ : b ∈ s), a ⊓ b",
"tactic": "refine' (le_inf hcinf.1 cd).trans (le_trans _ (le_iSup₂ d ds))"
},
{
"state_after": "no goals",
"state_before": "case pos.intro.intro\nι : Sort ?u.89037\nα : Type u_1\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns : Set α\nh : DirectedOn (fun x x_1 => x ≤ x_1) s\nhs : Set.Nonempty s\nc : α\nhc : ∀ (s : Set α), Set.Nonempty s → DirectedOn (fun x x_1 => x ≤ x_1) s → c ≤ sSup s → ∃ x, x ∈ s ∧ c ≤ x\nhcinf : c ≤ a ∧ c ≤ sSup s\nd : α\nds : d ∈ s\ncd : c ≤ d\n⊢ a ⊓ d ≤ a ⊓ d",
"tactic": "rfl"
},
{
"state_after": "case neg\nι : Sort ?u.89037\nα : Type u_1\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns : Set α\nh : DirectedOn (fun x x_1 => x ≤ x_1) s\nhs : s = ∅\n⊢ ∀ (c : α), CompleteLattice.IsCompactElement c → c ≤ a ⊓ sSup s → c ≤ ⨆ (b : α) (_ : b ∈ s), a ⊓ b",
"state_before": "case neg\nι : Sort ?u.89037\nα : Type u_1\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns : Set α\nh : DirectedOn (fun x x_1 => x ≤ x_1) s\nhs : ¬Set.Nonempty s\n⊢ ∀ (c : α), CompleteLattice.IsCompactElement c → c ≤ a ⊓ sSup s → c ≤ ⨆ (b : α) (_ : b ∈ s), a ⊓ b",
"tactic": "rw [Set.not_nonempty_iff_eq_empty] at hs"
},
{
"state_after": "no goals",
"state_before": "case neg\nι : Sort ?u.89037\nα : Type u_1\ninst✝² : CompleteLattice α\nf : ι → α\ninst✝¹ : CompleteLattice α\ninst✝ : IsCompactlyGenerated α\na b : α\ns : Set α\nh : DirectedOn (fun x x_1 => x ≤ x_1) s\nhs : s = ∅\n⊢ ∀ (c : α), CompleteLattice.IsCompactElement c → c ≤ a ⊓ sSup s → c ≤ ⨆ (b : α) (_ : b ∈ s), a ⊓ b",
"tactic": "simp [hs]"
}
] |
[
376,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
363,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.Relabelling.mk'_rightMovesEquiv
|
[] |
[
1065,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1063,
1
] |
Mathlib/Algebra/BigOperators/Multiset/Lemmas.lean
|
Multiset.prod_eq_one_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : CanonicallyOrderedMonoid α\nm : Multiset α\nl : List α\n⊢ prod (Quotient.mk (List.isSetoid α) l) = 1 ↔ ∀ (x : α), x ∈ Quotient.mk (List.isSetoid α) l → x = 1",
"tactic": "simpa using List.prod_eq_one_iff l"
}
] |
[
28,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
26,
1
] |
Mathlib/Data/Matrix/Notation.lean
|
Matrix.dotProduct_empty
|
[] |
[
143,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean
|
Set.Ioo_subset_Ioo_right
|
[] |
[
444,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
443,
1
] |
Mathlib/Dynamics/PeriodicPts.lean
|
Function.isPeriodicPt_iff_minimalPeriod_dvd
|
[] |
[
405,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
404,
1
] |
Mathlib/GroupTheory/Complement.lean
|
Subgroup.card_left_transversal
|
[] |
[
282,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
281,
1
] |
Mathlib/CategoryTheory/StructuredArrow.lean
|
CategoryTheory.StructuredArrow.map_comp
|
[
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nS S' S'' : D\nY Y' : C\nT : C ⥤ D\nf : S ⟶ S'\nf' : S' ⟶ S''\nh : StructuredArrow S'' T\n⊢ (map (f ≫ f')).obj (mk h.hom) = (map f).obj ((map f').obj (mk h.hom))",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nS S' S'' : D\nY Y' : C\nT : C ⥤ D\nf : S ⟶ S'\nf' : S' ⟶ S''\nh : StructuredArrow S'' T\n⊢ (map (f ≫ f')).obj h = (map f).obj ((map f').obj h)",
"tactic": "rw [eq_mk h]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nS S' S'' : D\nY Y' : C\nT : C ⥤ D\nf : S ⟶ S'\nf' : S' ⟶ S''\nh : StructuredArrow S'' T\n⊢ (map (f ≫ f')).obj (mk h.hom) = (map f).obj ((map f').obj (mk h.hom))",
"tactic": "simp"
}
] |
[
218,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
1
] |
Mathlib/Topology/Category/Profinite/Basic.lean
|
Profinite.isIso_of_bijective
|
[] |
[
326,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
324,
1
] |
Mathlib/Analysis/ODE/PicardLindelof.lean
|
PicardLindelof.FunSpace.dist_le_of_forall
|
[] |
[
249,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
246,
1
] |
Mathlib/Order/LocallyFinite.lean
|
WithTop.Icc_coe_top
|
[] |
[
1113,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1112,
1
] |
Mathlib/Analysis/NormedSpace/lpSpace.lean
|
lp.coeFn_zero
|
[] |
[
362,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
361,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.coeSubtype
|
[] |
[
796,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
795,
1
] |
Mathlib/Data/Nat/Choose/Central.lean
|
Nat.four_pow_le_two_mul_self_mul_centralBinom
|
[
{
"state_after": "no goals",
"state_before": "x✝ : 0 < 1\n⊢ 4 ^ 1 ≤ 2 * 1 * centralBinom 1",
"tactic": "norm_num [centralBinom, choose]"
},
{
"state_after": "no goals",
"state_before": "x✝ : 0 < 2\n⊢ 4 ^ 2 ≤ 2 * 2 * centralBinom 2",
"tactic": "norm_num [centralBinom, choose]"
},
{
"state_after": "no goals",
"state_before": "x✝ : 0 < 3\n⊢ 4 ^ 3 ≤ 2 * 3 * centralBinom 3",
"tactic": "norm_num [centralBinom, choose]"
},
{
"state_after": "n : ℕ\nx✝ : 0 < n + 4\n⊢ (n + 4) * centralBinom (n + 4) ≤ 2 * ((n + 4) * centralBinom (n + 4))",
"state_before": "n : ℕ\nx✝ : 0 < n + 4\n⊢ (n + 4) * centralBinom (n + 4) ≤ 2 * (n + 4) * centralBinom (n + 4)",
"tactic": "rw [mul_assoc]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nx✝ : 0 < n + 4\n⊢ (n + 4) * centralBinom (n + 4) ≤ 2 * ((n + 4) * centralBinom (n + 4))",
"tactic": "refine' le_mul_of_pos_left zero_lt_two"
}
] |
[
117,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/Algebra/CharP/Basic.lean
|
frobenius_sub
|
[] |
[
438,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
437,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.nhds_zero
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.48158\nβ : Type ?u.48161\nγ : Type ?u.48164\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y z ε ε₁ ε₂ : ℝ≥0∞\ns : Set ℝ≥0∞\n⊢ (⨅ (l : ℝ≥0∞) (_ : ⊥ < l), 𝓟 (Iio l)) = ⨅ (a : ℝ≥0∞) (_ : a ≠ 0), 𝓟 (Iio a)",
"tactic": "simp [pos_iff_ne_zero, Iio]"
}
] |
[
197,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Mathlib/Data/Set/Pairwise/Basic.lean
|
pairwise_subtype_iff_pairwise_set
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.31833\nγ : Type ?u.31836\nι : Type ?u.31839\nι' : Type ?u.31842\nr✝ p q : α → α → Prop\ns : Set α\nr : α → α → Prop\n⊢ (Pairwise fun x y => r ↑x ↑y) ↔ Set.Pairwise s r",
"tactic": "simp only [Pairwise, Set.Pairwise, SetCoe.forall, Ne.def, Subtype.ext_iff, Subtype.coe_mk]"
}
] |
[
226,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
224,
1
] |
Mathlib/Topology/Algebra/UniformGroup.lean
|
TendstoUniformlyOnFilter.div
|
[] |
[
491,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
488,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
topologicalGroup_sInf
|
[] |
[
1794,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1787,
1
] |
Std/Data/RBMap/Alter.lean
|
Std.RBNode.Path.zoom_zoomed₁
|
[
{
"state_after": "α✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt'✝ : RBNode α✝\npath'✝ : Path α✝\ne : zoom cut t path = (t'✝, path'✝)\npath' : Path α✝\nt' : RBNode α✝\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\n⊢ zoom cut (node c✝ l✝ v✝ r✝) path = (t', path') → OnRoot (fun x => cut x = Ordering.eq) t'",
"state_before": "α✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt'✝ : RBNode α✝\npath'✝ : Path α✝\ne✝ : zoom cut t path = (t'✝, path'✝)\npath' : Path α✝\nt' : RBNode α✝\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\ne : zoom cut (node c✝ l✝ v✝ r✝) path = (t', path')\n⊢ OnRoot (fun x => cut x = Ordering.eq) t'",
"tactic": "revert e"
},
{
"state_after": "α✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt'✝ : RBNode α✝\npath'✝ : Path α✝\ne : zoom cut t path = (t'✝, path'✝)\npath' : Path α✝\nt' : RBNode α✝\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\n⊢ (match cut v✝ with\n | Ordering.lt => zoom cut l✝ (left c✝ path v✝ r✝)\n | Ordering.gt => zoom cut r✝ (right c✝ l✝ v✝ path)\n | Ordering.eq => (node c✝ l✝ v✝ r✝, path)) =\n (t', path') →\n OnRoot (fun x => cut x = Ordering.eq) t'",
"state_before": "α✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt'✝ : RBNode α✝\npath'✝ : Path α✝\ne : zoom cut t path = (t'✝, path'✝)\npath' : Path α✝\nt' : RBNode α✝\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\n⊢ zoom cut (node c✝ l✝ v✝ r✝) path = (t', path') → OnRoot (fun x => cut x = Ordering.eq) t'",
"tactic": "unfold zoom"
},
{
"state_after": "case h_1\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt'✝ : RBNode α✝\npath'✝ : Path α✝\ne : zoom cut t path = (t'✝, path'✝)\npath' : Path α✝\nt' : RBNode α✝\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.lt\n⊢ zoom cut l✝ (left c✝ path v✝ r✝) = (t', path') → OnRoot (fun x => cut x = Ordering.eq) t'\n\ncase h_2\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt'✝ : RBNode α✝\npath'✝ : Path α✝\ne : zoom cut t path = (t'✝, path'✝)\npath' : Path α✝\nt' : RBNode α✝\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.gt\n⊢ zoom cut r✝ (right c✝ l✝ v✝ path) = (t', path') → OnRoot (fun x => cut x = Ordering.eq) t'\n\ncase h_3\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt'✝ : RBNode α✝\npath'✝ : Path α✝\ne : zoom cut t path = (t'✝, path'✝)\npath' : Path α✝\nt' : RBNode α✝\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.eq\n⊢ (node c✝ l✝ v✝ r✝, path) = (t', path') → OnRoot (fun x => cut x = Ordering.eq) t'",
"state_before": "α✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt'✝ : RBNode α✝\npath'✝ : Path α✝\ne : zoom cut t path = (t'✝, path'✝)\npath' : Path α✝\nt' : RBNode α✝\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\n⊢ (match cut v✝ with\n | Ordering.lt => zoom cut l✝ (left c✝ path v✝ r✝)\n | Ordering.gt => zoom cut r✝ (right c✝ l✝ v✝ path)\n | Ordering.eq => (node c✝ l✝ v✝ r✝, path)) =\n (t', path') →\n OnRoot (fun x => cut x = Ordering.eq) t'",
"tactic": "split"
},
{
"state_after": "no goals",
"state_before": "case h_1\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt'✝ : RBNode α✝\npath'✝ : Path α✝\ne : zoom cut t path = (t'✝, path'✝)\npath' : Path α✝\nt' : RBNode α✝\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.lt\n⊢ zoom cut l✝ (left c✝ path v✝ r✝) = (t', path') → OnRoot (fun x => cut x = Ordering.eq) t'",
"tactic": "exact zoom_zoomed₁"
},
{
"state_after": "no goals",
"state_before": "case h_2\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt'✝ : RBNode α✝\npath'✝ : Path α✝\ne : zoom cut t path = (t'✝, path'✝)\npath' : Path α✝\nt' : RBNode α✝\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.gt\n⊢ zoom cut r✝ (right c✝ l✝ v✝ path) = (t', path') → OnRoot (fun x => cut x = Ordering.eq) t'",
"tactic": "exact zoom_zoomed₁"
},
{
"state_after": "no goals",
"state_before": "case h_3\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt'✝ : RBNode α✝\npath'✝ : Path α✝\ne : zoom cut t path = (t'✝, path'✝)\npath' : Path α✝\nt' : RBNode α✝\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\nx✝ : Ordering\nheq✝ : cut v✝ = Ordering.eq\n⊢ (node c✝ l✝ v✝ r✝, path) = (t', path') → OnRoot (fun x => cut x = Ordering.eq) t'",
"tactic": "next H => intro e; cases e; exact H"
},
{
"state_after": "α✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt'✝ : RBNode α✝\npath'✝ : Path α✝\ne✝ : zoom cut t path = (t'✝, path'✝)\npath' : Path α✝\nt' : RBNode α✝\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\nx✝ : Ordering\nH : cut v✝ = Ordering.eq\ne : (node c✝ l✝ v✝ r✝, path) = (t', path')\n⊢ OnRoot (fun x => cut x = Ordering.eq) t'",
"state_before": "α✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt'✝ : RBNode α✝\npath'✝ : Path α✝\ne : zoom cut t path = (t'✝, path'✝)\npath' : Path α✝\nt' : RBNode α✝\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\nx✝ : Ordering\nH : cut v✝ = Ordering.eq\n⊢ (node c✝ l✝ v✝ r✝, path) = (t', path') → OnRoot (fun x => cut x = Ordering.eq) t'",
"tactic": "intro e"
},
{
"state_after": "case refl\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt' : RBNode α✝\npath' : Path α✝\ne : zoom cut t path = (t', path')\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\nx✝ : Ordering\nH : cut v✝ = Ordering.eq\n⊢ OnRoot (fun x => cut x = Ordering.eq) (node c✝ l✝ v✝ r✝)",
"state_before": "α✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt'✝ : RBNode α✝\npath'✝ : Path α✝\ne✝ : zoom cut t path = (t'✝, path'✝)\npath' : Path α✝\nt' : RBNode α✝\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\nx✝ : Ordering\nH : cut v✝ = Ordering.eq\ne : (node c✝ l✝ v✝ r✝, path) = (t', path')\n⊢ OnRoot (fun x => cut x = Ordering.eq) t'",
"tactic": "cases e"
},
{
"state_after": "no goals",
"state_before": "case refl\nα✝ : Type u_1\ncut : α✝ → Ordering\nt : RBNode α✝\npath : Path α✝\nt' : RBNode α✝\npath' : Path α✝\ne : zoom cut t path = (t', path')\nc✝ : RBColor\nl✝ : RBNode α✝\nv✝ : α✝\nr✝ : RBNode α✝\nx✝ : Ordering\nH : cut v✝ = Ordering.eq\n⊢ OnRoot (fun x => cut x = Ordering.eq) (node c✝ l✝ v✝ r✝)",
"tactic": "exact H"
}
] |
[
225,
42
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
218,
1
] |
Mathlib/Analysis/NormedSpace/AddTorsorBases.lean
|
IsOpen.affineSpan_eq_top
|
[] |
[
134,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
131,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
IsCompact.exists_open_superset_measure_lt_top'
|
[
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type ?u.3553252\nγ : Type ?u.3553255\nδ : Type ?u.3553258\nι : Type ?u.3553261\nR : Type ?u.3553264\nR' : Type ?u.3553267\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\ns : Set α\nh : IsCompact s\nhμ : ∀ (x : α), x ∈ s → FiniteAtFilter μ (𝓝 x)\n⊢ ∃ U x, IsOpen U ∧ ↑↑μ U < ⊤\n\ncase refine'_2\nα : Type u_1\nβ : Type ?u.3553252\nγ : Type ?u.3553255\nδ : Type ?u.3553258\nι : Type ?u.3553261\nR : Type ?u.3553264\nR' : Type ?u.3553267\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\ns : Set α\nh : IsCompact s\nhμ : ∀ (x : α), x ∈ s → FiniteAtFilter μ (𝓝 x)\n⊢ ∀ ⦃s t : Set α⦄, s ⊆ t → (∃ U x, IsOpen U ∧ ↑↑μ U < ⊤) → ∃ U x, IsOpen U ∧ ↑↑μ U < ⊤\n\ncase refine'_3\nα : Type u_1\nβ : Type ?u.3553252\nγ : Type ?u.3553255\nδ : Type ?u.3553258\nι : Type ?u.3553261\nR : Type ?u.3553264\nR' : Type ?u.3553267\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\ns : Set α\nh : IsCompact s\nhμ : ∀ (x : α), x ∈ s → FiniteAtFilter μ (𝓝 x)\n⊢ ∀ ⦃s t : Set α⦄, (∃ U x, IsOpen U ∧ ↑↑μ U < ⊤) → (∃ U x, IsOpen U ∧ ↑↑μ U < ⊤) → ∃ U x, IsOpen U ∧ ↑↑μ U < ⊤\n\ncase refine'_4\nα : Type u_1\nβ : Type ?u.3553252\nγ : Type ?u.3553255\nδ : Type ?u.3553258\nι : Type ?u.3553261\nR : Type ?u.3553264\nR' : Type ?u.3553267\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\ns : Set α\nh : IsCompact s\nhμ : ∀ (x : α), x ∈ s → FiniteAtFilter μ (𝓝 x)\n⊢ ∀ (x : α), x ∈ s → ∃ t, t ∈ 𝓝[s] x ∧ ∃ U x, IsOpen U ∧ ↑↑μ U < ⊤",
"state_before": "α : Type u_1\nβ : Type ?u.3553252\nγ : Type ?u.3553255\nδ : Type ?u.3553258\nι : Type ?u.3553261\nR : Type ?u.3553264\nR' : Type ?u.3553267\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\ns : Set α\nh : IsCompact s\nhμ : ∀ (x : α), x ∈ s → FiniteAtFilter μ (𝓝 x)\n⊢ ∃ U x, IsOpen U ∧ ↑↑μ U < ⊤",
"tactic": "refine' IsCompact.induction_on h _ _ _ _"
},
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type ?u.3553252\nγ : Type ?u.3553255\nδ : Type ?u.3553258\nι : Type ?u.3553261\nR : Type ?u.3553264\nR' : Type ?u.3553267\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\ns : Set α\nh : IsCompact s\nhμ : ∀ (x : α), x ∈ s → FiniteAtFilter μ (𝓝 x)\n⊢ ∃ x, IsOpen ∅ ∧ ↑↑μ ∅ < ⊤",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type ?u.3553252\nγ : Type ?u.3553255\nδ : Type ?u.3553258\nι : Type ?u.3553261\nR : Type ?u.3553264\nR' : Type ?u.3553267\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\ns : Set α\nh : IsCompact s\nhμ : ∀ (x : α), x ∈ s → FiniteAtFilter μ (𝓝 x)\n⊢ ∃ U x, IsOpen U ∧ ↑↑μ U < ⊤",
"tactic": "use ∅"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type ?u.3553252\nγ : Type ?u.3553255\nδ : Type ?u.3553258\nι : Type ?u.3553261\nR : Type ?u.3553264\nR' : Type ?u.3553267\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\ns : Set α\nh : IsCompact s\nhμ : ∀ (x : α), x ∈ s → FiniteAtFilter μ (𝓝 x)\n⊢ ∃ x, IsOpen ∅ ∧ ↑↑μ ∅ < ⊤",
"tactic": "simp [Superset]"
},
{
"state_after": "case refine'_2.intro.intro.intro\nα : Type u_1\nβ : Type ?u.3553252\nγ : Type ?u.3553255\nδ : Type ?u.3553258\nι : Type ?u.3553261\nR : Type ?u.3553264\nR' : Type ?u.3553267\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\ns✝ : Set α\nh : IsCompact s✝\nhμ : ∀ (x : α), x ∈ s✝ → FiniteAtFilter μ (𝓝 x)\ns t : Set α\nhst : s ⊆ t\nU : Set α\nhtU : U ⊇ t\nhUo : IsOpen U\nhU : ↑↑μ U < ⊤\n⊢ ∃ U x, IsOpen U ∧ ↑↑μ U < ⊤",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type ?u.3553252\nγ : Type ?u.3553255\nδ : Type ?u.3553258\nι : Type ?u.3553261\nR : Type ?u.3553264\nR' : Type ?u.3553267\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\ns : Set α\nh : IsCompact s\nhμ : ∀ (x : α), x ∈ s → FiniteAtFilter μ (𝓝 x)\n⊢ ∀ ⦃s t : Set α⦄, s ⊆ t → (∃ U x, IsOpen U ∧ ↑↑μ U < ⊤) → ∃ U x, IsOpen U ∧ ↑↑μ U < ⊤",
"tactic": "rintro s t hst ⟨U, htU, hUo, hU⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro.intro\nα : Type u_1\nβ : Type ?u.3553252\nγ : Type ?u.3553255\nδ : Type ?u.3553258\nι : Type ?u.3553261\nR : Type ?u.3553264\nR' : Type ?u.3553267\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\ns✝ : Set α\nh : IsCompact s✝\nhμ : ∀ (x : α), x ∈ s✝ → FiniteAtFilter μ (𝓝 x)\ns t : Set α\nhst : s ⊆ t\nU : Set α\nhtU : U ⊇ t\nhUo : IsOpen U\nhU : ↑↑μ U < ⊤\n⊢ ∃ U x, IsOpen U ∧ ↑↑μ U < ⊤",
"tactic": "exact ⟨U, hst.trans htU, hUo, hU⟩"
},
{
"state_after": "case refine'_3.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.3553252\nγ : Type ?u.3553255\nδ : Type ?u.3553258\nι : Type ?u.3553261\nR : Type ?u.3553264\nR' : Type ?u.3553267\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\ns✝ : Set α\nh : IsCompact s✝\nhμ : ∀ (x : α), x ∈ s✝ → FiniteAtFilter μ (𝓝 x)\ns t U : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhU : ↑↑μ U < ⊤\nV : Set α\nhtV : V ⊇ t\nhVo : IsOpen V\nhV : ↑↑μ V < ⊤\n⊢ ∃ U x, IsOpen U ∧ ↑↑μ U < ⊤",
"state_before": "case refine'_3\nα : Type u_1\nβ : Type ?u.3553252\nγ : Type ?u.3553255\nδ : Type ?u.3553258\nι : Type ?u.3553261\nR : Type ?u.3553264\nR' : Type ?u.3553267\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\ns : Set α\nh : IsCompact s\nhμ : ∀ (x : α), x ∈ s → FiniteAtFilter μ (𝓝 x)\n⊢ ∀ ⦃s t : Set α⦄, (∃ U x, IsOpen U ∧ ↑↑μ U < ⊤) → (∃ U x, IsOpen U ∧ ↑↑μ U < ⊤) → ∃ U x, IsOpen U ∧ ↑↑μ U < ⊤",
"tactic": "rintro s t ⟨U, hsU, hUo, hU⟩ ⟨V, htV, hVo, hV⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_3.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.3553252\nγ : Type ?u.3553255\nδ : Type ?u.3553258\nι : Type ?u.3553261\nR : Type ?u.3553264\nR' : Type ?u.3553267\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\ns✝ : Set α\nh : IsCompact s✝\nhμ : ∀ (x : α), x ∈ s✝ → FiniteAtFilter μ (𝓝 x)\ns t U : Set α\nhsU : U ⊇ s\nhUo : IsOpen U\nhU : ↑↑μ U < ⊤\nV : Set α\nhtV : V ⊇ t\nhVo : IsOpen V\nhV : ↑↑μ V < ⊤\n⊢ ∃ U x, IsOpen U ∧ ↑↑μ U < ⊤",
"tactic": "refine'\n ⟨U ∪ V, union_subset_union hsU htV, hUo.union hVo,\n (measure_union_le _ _).trans_lt <| ENNReal.add_lt_top.2 ⟨hU, hV⟩⟩"
},
{
"state_after": "case refine'_4\nα : Type u_1\nβ : Type ?u.3553252\nγ : Type ?u.3553255\nδ : Type ?u.3553258\nι : Type ?u.3553261\nR : Type ?u.3553264\nR' : Type ?u.3553267\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\ns : Set α\nh : IsCompact s\nhμ : ∀ (x : α), x ∈ s → FiniteAtFilter μ (𝓝 x)\nx : α\nhx : x ∈ s\n⊢ ∃ t, t ∈ 𝓝[s] x ∧ ∃ U x, IsOpen U ∧ ↑↑μ U < ⊤",
"state_before": "case refine'_4\nα : Type u_1\nβ : Type ?u.3553252\nγ : Type ?u.3553255\nδ : Type ?u.3553258\nι : Type ?u.3553261\nR : Type ?u.3553264\nR' : Type ?u.3553267\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\ns : Set α\nh : IsCompact s\nhμ : ∀ (x : α), x ∈ s → FiniteAtFilter μ (𝓝 x)\n⊢ ∀ (x : α), x ∈ s → ∃ t, t ∈ 𝓝[s] x ∧ ∃ U x, IsOpen U ∧ ↑↑μ U < ⊤",
"tactic": "intro x hx"
},
{
"state_after": "case refine'_4.intro.intro.intro\nα : Type u_1\nβ : Type ?u.3553252\nγ : Type ?u.3553255\nδ : Type ?u.3553258\nι : Type ?u.3553261\nR : Type ?u.3553264\nR' : Type ?u.3553267\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\ns : Set α\nh : IsCompact s\nhμ : ∀ (x : α), x ∈ s → FiniteAtFilter μ (𝓝 x)\nx : α\nhx✝ : x ∈ s\nU : Set α\nhU : ↑↑μ U < ⊤\nhx : x ∈ U\nhUo : IsOpen U\n⊢ ∃ t, t ∈ 𝓝[s] x ∧ ∃ U x, IsOpen U ∧ ↑↑μ U < ⊤",
"state_before": "case refine'_4\nα : Type u_1\nβ : Type ?u.3553252\nγ : Type ?u.3553255\nδ : Type ?u.3553258\nι : Type ?u.3553261\nR : Type ?u.3553264\nR' : Type ?u.3553267\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\ns : Set α\nh : IsCompact s\nhμ : ∀ (x : α), x ∈ s → FiniteAtFilter μ (𝓝 x)\nx : α\nhx : x ∈ s\n⊢ ∃ t, t ∈ 𝓝[s] x ∧ ∃ U x, IsOpen U ∧ ↑↑μ U < ⊤",
"tactic": "rcases(hμ x hx).exists_mem_basis (nhds_basis_opens _) with ⟨U, ⟨hx, hUo⟩, hU⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_4.intro.intro.intro\nα : Type u_1\nβ : Type ?u.3553252\nγ : Type ?u.3553255\nδ : Type ?u.3553258\nι : Type ?u.3553261\nR : Type ?u.3553264\nR' : Type ?u.3553267\ninst✝¹ : TopologicalSpace α\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\ns : Set α\nh : IsCompact s\nhμ : ∀ (x : α), x ∈ s → FiniteAtFilter μ (𝓝 x)\nx : α\nhx✝ : x ∈ s\nU : Set α\nhU : ↑↑μ U < ⊤\nhx : x ∈ U\nhUo : IsOpen U\n⊢ ∃ t, t ∈ 𝓝[s] x ∧ ∃ U x, IsOpen U ∧ ↑↑μ U < ⊤",
"tactic": "exact ⟨U, nhdsWithin_le_nhds (hUo.mem_nhds hx), U, Subset.rfl, hUo, hU⟩"
}
] |
[
4474,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
4461,
1
] |
Mathlib/Data/Holor.lean
|
Holor.cprankMax_1
|
[
{
"state_after": "α : Type\nd : ℕ\nds ds₁ ds₂ ds₃ : List ℕ\ninst✝¹ : Monoid α\ninst✝ : AddMonoid α\nx : Holor α ds\nh : CPRankMax1 x\nh' : CPRankMax (0 + 1) (x + 0)\n⊢ CPRankMax 1 x",
"state_before": "α : Type\nd : ℕ\nds ds₁ ds₂ ds₃ : List ℕ\ninst✝¹ : Monoid α\ninst✝ : AddMonoid α\nx : Holor α ds\nh : CPRankMax1 x\n⊢ CPRankMax 1 x",
"tactic": "have h' := CPRankMax.succ 0 x 0 h CPRankMax.zero"
},
{
"state_after": "no goals",
"state_before": "α : Type\nd : ℕ\nds ds₁ ds₂ ds₃ : List ℕ\ninst✝¹ : Monoid α\ninst✝ : AddMonoid α\nx : Holor α ds\nh : CPRankMax1 x\nh' : CPRankMax (0 + 1) (x + 0)\n⊢ CPRankMax 1 x",
"tactic": "rwa [zero_add, add_zero] at h'"
}
] |
[
335,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
332,
1
] |
Mathlib/Order/SuccPred/Basic.lean
|
Order.lt_succ
|
[] |
[
332,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
331,
1
] |
Mathlib/Order/JordanHolder.lean
|
CompositionSeries.Equivalent.append
|
[
{
"state_after": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ t₁ t₂ : CompositionSeries X\nhs : top s₁ = bot s₂\nht : top t₁ = bot t₂\nh₁ : Equivalent s₁ t₁\nh₂ : Equivalent s₂ t₂\ne : Fin (s₁.length + s₂.length) ≃ Fin (t₁.length + t₂.length) :=\n Trans.trans (Trans.trans finSumFinEquiv.symm (Equiv.sumCongr (Exists.choose h₁) (Exists.choose h₂))) finSumFinEquiv\ni : Fin (CompositionSeries.append s₁ s₂ hs).length\n⊢ Iso\n (series (CompositionSeries.append s₁ s₂ hs) (↑Fin.castSucc i),\n series (CompositionSeries.append s₁ s₂ hs) (Fin.succ i))\n (series (CompositionSeries.append t₁ t₂ ht) (↑Fin.castSucc (↑e i)),\n series (CompositionSeries.append t₁ t₂ ht) (Fin.succ (↑e i)))",
"state_before": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ t₁ t₂ : CompositionSeries X\nhs : top s₁ = bot s₂\nht : top t₁ = bot t₂\nh₁ : Equivalent s₁ t₁\nh₂ : Equivalent s₂ t₂\ne : Fin (s₁.length + s₂.length) ≃ Fin (t₁.length + t₂.length) :=\n Trans.trans (Trans.trans finSumFinEquiv.symm (Equiv.sumCongr (Exists.choose h₁) (Exists.choose h₂))) finSumFinEquiv\n⊢ ∀ (i : Fin (CompositionSeries.append s₁ s₂ hs).length),\n Iso\n (series (CompositionSeries.append s₁ s₂ hs) (↑Fin.castSucc i),\n series (CompositionSeries.append s₁ s₂ hs) (Fin.succ i))\n (series (CompositionSeries.append t₁ t₂ ht) (↑Fin.castSucc (↑e i)),\n series (CompositionSeries.append t₁ t₂ ht) (Fin.succ (↑e i)))",
"tactic": "intro i"
},
{
"state_after": "case refine'_1\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ t₁ t₂ : CompositionSeries X\nhs : top s₁ = bot s₂\nht : top t₁ = bot t₂\nh₁ : Equivalent s₁ t₁\nh₂ : Equivalent s₂ t₂\ne : Fin (s₁.length + s₂.length) ≃ Fin (t₁.length + t₂.length) :=\n Trans.trans (Trans.trans finSumFinEquiv.symm (Equiv.sumCongr (Exists.choose h₁) (Exists.choose h₂))) finSumFinEquiv\ni : Fin (CompositionSeries.append s₁ s₂ hs).length\n⊢ ∀ (i : Fin s₁.length),\n Iso\n (series (CompositionSeries.append s₁ s₂ hs) (↑Fin.castSucc (↑(Fin.castAdd s₂.length) i)),\n series (CompositionSeries.append s₁ s₂ hs) (Fin.succ (↑(Fin.castAdd s₂.length) i)))\n (series (CompositionSeries.append t₁ t₂ ht) (↑Fin.castSucc (↑e (↑(Fin.castAdd s₂.length) i))),\n series (CompositionSeries.append t₁ t₂ ht) (Fin.succ (↑e (↑(Fin.castAdd s₂.length) i))))\n\ncase refine'_2\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ t₁ t₂ : CompositionSeries X\nhs : top s₁ = bot s₂\nht : top t₁ = bot t₂\nh₁ : Equivalent s₁ t₁\nh₂ : Equivalent s₂ t₂\ne : Fin (s₁.length + s₂.length) ≃ Fin (t₁.length + t₂.length) :=\n Trans.trans (Trans.trans finSumFinEquiv.symm (Equiv.sumCongr (Exists.choose h₁) (Exists.choose h₂))) finSumFinEquiv\ni : Fin (CompositionSeries.append s₁ s₂ hs).length\n⊢ ∀ (i : Fin s₂.length),\n Iso\n (series (CompositionSeries.append s₁ s₂ hs) (↑Fin.castSucc (↑(Fin.natAdd s₁.length) i)),\n series (CompositionSeries.append s₁ s₂ hs) (Fin.succ (↑(Fin.natAdd s₁.length) i)))\n (series (CompositionSeries.append t₁ t₂ ht) (↑Fin.castSucc (↑e (↑(Fin.natAdd s₁.length) i))),\n series (CompositionSeries.append t₁ t₂ ht) (Fin.succ (↑e (↑(Fin.natAdd s₁.length) i))))",
"state_before": "X : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ t₁ t₂ : CompositionSeries X\nhs : top s₁ = bot s₂\nht : top t₁ = bot t₂\nh₁ : Equivalent s₁ t₁\nh₂ : Equivalent s₂ t₂\ne : Fin (s₁.length + s₂.length) ≃ Fin (t₁.length + t₂.length) :=\n Trans.trans (Trans.trans finSumFinEquiv.symm (Equiv.sumCongr (Exists.choose h₁) (Exists.choose h₂))) finSumFinEquiv\ni : Fin (CompositionSeries.append s₁ s₂ hs).length\n⊢ Iso\n (series (CompositionSeries.append s₁ s₂ hs) (↑Fin.castSucc i),\n series (CompositionSeries.append s₁ s₂ hs) (Fin.succ i))\n (series (CompositionSeries.append t₁ t₂ ht) (↑Fin.castSucc (↑e i)),\n series (CompositionSeries.append t₁ t₂ ht) (Fin.succ (↑e i)))",
"tactic": "refine' Fin.addCases _ _ i"
},
{
"state_after": "case refine'_1\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ t₁ t₂ : CompositionSeries X\nhs : top s₁ = bot s₂\nht : top t₁ = bot t₂\nh₁ : Equivalent s₁ t₁\nh₂ : Equivalent s₂ t₂\ne : Fin (s₁.length + s₂.length) ≃ Fin (t₁.length + t₂.length) :=\n Trans.trans (Trans.trans finSumFinEquiv.symm (Equiv.sumCongr (Exists.choose h₁) (Exists.choose h₂))) finSumFinEquiv\ni✝ : Fin (CompositionSeries.append s₁ s₂ hs).length\ni : Fin s₁.length\n⊢ Iso\n (series (CompositionSeries.append s₁ s₂ hs) (↑Fin.castSucc (↑(Fin.castAdd s₂.length) i)),\n series (CompositionSeries.append s₁ s₂ hs) (Fin.succ (↑(Fin.castAdd s₂.length) i)))\n (series (CompositionSeries.append t₁ t₂ ht) (↑Fin.castSucc (↑e (↑(Fin.castAdd s₂.length) i))),\n series (CompositionSeries.append t₁ t₂ ht) (Fin.succ (↑e (↑(Fin.castAdd s₂.length) i))))",
"state_before": "case refine'_1\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ t₁ t₂ : CompositionSeries X\nhs : top s₁ = bot s₂\nht : top t₁ = bot t₂\nh₁ : Equivalent s₁ t₁\nh₂ : Equivalent s₂ t₂\ne : Fin (s₁.length + s₂.length) ≃ Fin (t₁.length + t₂.length) :=\n Trans.trans (Trans.trans finSumFinEquiv.symm (Equiv.sumCongr (Exists.choose h₁) (Exists.choose h₂))) finSumFinEquiv\ni : Fin (CompositionSeries.append s₁ s₂ hs).length\n⊢ ∀ (i : Fin s₁.length),\n Iso\n (series (CompositionSeries.append s₁ s₂ hs) (↑Fin.castSucc (↑(Fin.castAdd s₂.length) i)),\n series (CompositionSeries.append s₁ s₂ hs) (Fin.succ (↑(Fin.castAdd s₂.length) i)))\n (series (CompositionSeries.append t₁ t₂ ht) (↑Fin.castSucc (↑e (↑(Fin.castAdd s₂.length) i))),\n series (CompositionSeries.append t₁ t₂ ht) (Fin.succ (↑e (↑(Fin.castAdd s₂.length) i))))",
"tactic": "intro i"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ t₁ t₂ : CompositionSeries X\nhs : top s₁ = bot s₂\nht : top t₁ = bot t₂\nh₁ : Equivalent s₁ t₁\nh₂ : Equivalent s₂ t₂\ne : Fin (s₁.length + s₂.length) ≃ Fin (t₁.length + t₂.length) :=\n Trans.trans (Trans.trans finSumFinEquiv.symm (Equiv.sumCongr (Exists.choose h₁) (Exists.choose h₂))) finSumFinEquiv\ni✝ : Fin (CompositionSeries.append s₁ s₂ hs).length\ni : Fin s₁.length\n⊢ Iso\n (series (CompositionSeries.append s₁ s₂ hs) (↑Fin.castSucc (↑(Fin.castAdd s₂.length) i)),\n series (CompositionSeries.append s₁ s₂ hs) (Fin.succ (↑(Fin.castAdd s₂.length) i)))\n (series (CompositionSeries.append t₁ t₂ ht) (↑Fin.castSucc (↑e (↑(Fin.castAdd s₂.length) i))),\n series (CompositionSeries.append t₁ t₂ ht) (Fin.succ (↑e (↑(Fin.castAdd s₂.length) i))))",
"tactic": "simpa [top, bot] using h₁.choose_spec i"
},
{
"state_after": "case refine'_2\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ t₁ t₂ : CompositionSeries X\nhs : top s₁ = bot s₂\nht : top t₁ = bot t₂\nh₁ : Equivalent s₁ t₁\nh₂ : Equivalent s₂ t₂\ne : Fin (s₁.length + s₂.length) ≃ Fin (t₁.length + t₂.length) :=\n Trans.trans (Trans.trans finSumFinEquiv.symm (Equiv.sumCongr (Exists.choose h₁) (Exists.choose h₂))) finSumFinEquiv\ni✝ : Fin (CompositionSeries.append s₁ s₂ hs).length\ni : Fin s₂.length\n⊢ Iso\n (series (CompositionSeries.append s₁ s₂ hs) (↑Fin.castSucc (↑(Fin.natAdd s₁.length) i)),\n series (CompositionSeries.append s₁ s₂ hs) (Fin.succ (↑(Fin.natAdd s₁.length) i)))\n (series (CompositionSeries.append t₁ t₂ ht) (↑Fin.castSucc (↑e (↑(Fin.natAdd s₁.length) i))),\n series (CompositionSeries.append t₁ t₂ ht) (Fin.succ (↑e (↑(Fin.natAdd s₁.length) i))))",
"state_before": "case refine'_2\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ t₁ t₂ : CompositionSeries X\nhs : top s₁ = bot s₂\nht : top t₁ = bot t₂\nh₁ : Equivalent s₁ t₁\nh₂ : Equivalent s₂ t₂\ne : Fin (s₁.length + s₂.length) ≃ Fin (t₁.length + t₂.length) :=\n Trans.trans (Trans.trans finSumFinEquiv.symm (Equiv.sumCongr (Exists.choose h₁) (Exists.choose h₂))) finSumFinEquiv\ni : Fin (CompositionSeries.append s₁ s₂ hs).length\n⊢ ∀ (i : Fin s₂.length),\n Iso\n (series (CompositionSeries.append s₁ s₂ hs) (↑Fin.castSucc (↑(Fin.natAdd s₁.length) i)),\n series (CompositionSeries.append s₁ s₂ hs) (Fin.succ (↑(Fin.natAdd s₁.length) i)))\n (series (CompositionSeries.append t₁ t₂ ht) (↑Fin.castSucc (↑e (↑(Fin.natAdd s₁.length) i))),\n series (CompositionSeries.append t₁ t₂ ht) (Fin.succ (↑e (↑(Fin.natAdd s₁.length) i))))",
"tactic": "intro i"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nX : Type u\ninst✝¹ : Lattice X\ninst✝ : JordanHolderLattice X\ns₁ s₂ t₁ t₂ : CompositionSeries X\nhs : top s₁ = bot s₂\nht : top t₁ = bot t₂\nh₁ : Equivalent s₁ t₁\nh₂ : Equivalent s₂ t₂\ne : Fin (s₁.length + s₂.length) ≃ Fin (t₁.length + t₂.length) :=\n Trans.trans (Trans.trans finSumFinEquiv.symm (Equiv.sumCongr (Exists.choose h₁) (Exists.choose h₂))) finSumFinEquiv\ni✝ : Fin (CompositionSeries.append s₁ s₂ hs).length\ni : Fin s₂.length\n⊢ Iso\n (series (CompositionSeries.append s₁ s₂ hs) (↑Fin.castSucc (↑(Fin.natAdd s₁.length) i)),\n series (CompositionSeries.append s₁ s₂ hs) (Fin.succ (↑(Fin.natAdd s₁.length) i)))\n (series (CompositionSeries.append t₁ t₂ ht) (↑Fin.castSucc (↑e (↑(Fin.natAdd s₁.length) i))),\n series (CompositionSeries.append t₁ t₂ ht) (Fin.succ (↑e (↑(Fin.natAdd s₁.length) i))))",
"tactic": "simpa [top, bot] using h₂.choose_spec i"
}
] |
[
660,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
645,
1
] |
Mathlib/LinearAlgebra/TensorAlgebra/Basic.lean
|
TensorAlgebra.hom_ext
|
[
{
"state_after": "R : Type u_2\ninst✝⁴ : CommSemiring R\nM : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_1\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nf g : TensorAlgebra R M →ₐ[R] A\nw : ↑(lift R).symm f = ↑(lift R).symm g\n⊢ f = g",
"state_before": "R : Type u_2\ninst✝⁴ : CommSemiring R\nM : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_1\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nf g : TensorAlgebra R M →ₐ[R] A\nw : LinearMap.comp (AlgHom.toLinearMap f) (ι R) = LinearMap.comp (AlgHom.toLinearMap g) (ι R)\n⊢ f = g",
"tactic": "rw [← lift_symm_apply, ← lift_symm_apply] at w"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝⁴ : CommSemiring R\nM : Type u_3\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nA : Type u_1\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nf g : TensorAlgebra R M →ₐ[R] A\nw : ↑(lift R).symm f = ↑(lift R).symm g\n⊢ f = g",
"tactic": "exact (lift R).symm.injective w"
}
] |
[
162,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
159,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
MeasureTheory.Memℒp.neg_part
|
[] |
[
1100,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1098,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Products.lean
|
CategoryTheory.Limits.Sigma.ι_reindex_inv
|
[
{
"state_after": "no goals",
"state_before": "β : Type w\nC : Type u\ninst✝² : Category C\nγ : Type v\nε : β ≃ γ\nf : γ → C\ninst✝¹ : HasCoproduct f\ninst✝ : HasCoproduct (f ∘ ↑ε)\nb : β\n⊢ ι f (↑ε b) ≫ (reindex ε f).inv = ι (f ∘ ↑ε) b",
"tactic": "simp [Iso.comp_inv_eq]"
}
] |
[
445,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
444,
1
] |
Mathlib/LinearAlgebra/Multilinear/Basic.lean
|
LinearMap.coe_compMultilinearMap
|
[] |
[
790,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
788,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.coe_ne_top
|
[] |
[
283,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
9
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.typein_apply
|
[
{
"state_after": "α✝ : Type ?u.73654\nβ✝ : Type ?u.73657\nγ : Type ?u.73660\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt : γ → γ → Prop\nα β : Type u_1\nr : α → α → Prop\ns : β → β → Prop\ninst✝¹ : IsWellOrder α r\ninst✝ : IsWellOrder β s\nf : r ≼i s\na : α\nx✝ : ↑{b | r b a}\nx : α\nh : x ∈ {b | r b a}\n⊢ ↑f.toRelEmbedding (↑(Subrel.relEmbedding r {b | r b a}) { val := x, property := h }) ∈ {b | s b (↑f a)}",
"state_before": "α✝ : Type ?u.73654\nβ✝ : Type ?u.73657\nγ : Type ?u.73660\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt : γ → γ → Prop\nα β : Type u_1\nr : α → α → Prop\ns : β → β → Prop\ninst✝¹ : IsWellOrder α r\ninst✝ : IsWellOrder β s\nf : r ≼i s\na : α\nx✝ : ↑{b | r b a}\nx : α\nh : x ∈ {b | r b a}\n⊢ ↑(RelEmbedding.trans (Subrel.relEmbedding r {b | r b a}) f.toRelEmbedding) { val := x, property := h } ∈\n {b | s b (↑f a)}",
"tactic": "rw [RelEmbedding.trans_apply]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.73654\nβ✝ : Type ?u.73657\nγ : Type ?u.73660\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt : γ → γ → Prop\nα β : Type u_1\nr : α → α → Prop\ns : β → β → Prop\ninst✝¹ : IsWellOrder α r\ninst✝ : IsWellOrder β s\nf : r ≼i s\na : α\nx✝ : ↑{b | r b a}\nx : α\nh : x ∈ {b | r b a}\n⊢ ↑f.toRelEmbedding (↑(Subrel.relEmbedding r {b | r b a}) { val := x, property := h }) ∈ {b | s b (↑f a)}",
"tactic": "exact f.toRelEmbedding.map_rel_iff.2 h"
},
{
"state_after": "case intro\nα✝ : Type ?u.73654\nβ✝ : Type ?u.73657\nγ : Type ?u.73660\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt : γ → γ → Prop\nα β : Type u_1\nr : α → α → Prop\ns : β → β → Prop\ninst✝¹ : IsWellOrder α r\ninst✝ : IsWellOrder β s\nf : r ≼i s\na✝ : α\nx✝ : ↑{b | s b (↑f a✝)}\na : α\nh : ↑f a ∈ {b | s b (↑f a✝)}\n⊢ ∃ a_1,\n ↑(RelEmbedding.codRestrict {b | s b (↑f a✝)}\n (RelEmbedding.trans (Subrel.relEmbedding r {b | r b a✝}) f.toRelEmbedding)\n (_ :\n ∀ (x : ↑{b | r b a✝}),\n ↑(RelEmbedding.trans (Subrel.relEmbedding r {b | r b a✝}) f.toRelEmbedding) x ∈ {b | s b (↑f a✝)}))\n a_1 =\n { val := ↑f a, property := h }",
"state_before": "α✝ : Type ?u.73654\nβ✝ : Type ?u.73657\nγ : Type ?u.73660\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt : γ → γ → Prop\nα β : Type u_1\nr : α → α → Prop\ns : β → β → Prop\ninst✝¹ : IsWellOrder α r\ninst✝ : IsWellOrder β s\nf : r ≼i s\na : α\nx✝ : ↑{b | s b (↑f a)}\ny : β\nh : y ∈ {b | s b (↑f a)}\n⊢ ∃ a_1,\n ↑(RelEmbedding.codRestrict {b | s b (↑f a)}\n (RelEmbedding.trans (Subrel.relEmbedding r {b | r b a}) f.toRelEmbedding)\n (_ :\n ∀ (x : ↑{b | r b a}),\n ↑(RelEmbedding.trans (Subrel.relEmbedding r {b | r b a}) f.toRelEmbedding) x ∈ {b | s b (↑f a)}))\n a_1 =\n { val := y, property := h }",
"tactic": "rcases f.init h with ⟨a, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nα✝ : Type ?u.73654\nβ✝ : Type ?u.73657\nγ : Type ?u.73660\nr✝ : α✝ → α✝ → Prop\ns✝ : β✝ → β✝ → Prop\nt : γ → γ → Prop\nα β : Type u_1\nr : α → α → Prop\ns : β → β → Prop\ninst✝¹ : IsWellOrder α r\ninst✝ : IsWellOrder β s\nf : r ≼i s\na✝ : α\nx✝ : ↑{b | s b (↑f a✝)}\na : α\nh : ↑f a ∈ {b | s b (↑f a✝)}\n⊢ ∃ a_1,\n ↑(RelEmbedding.codRestrict {b | s b (↑f a✝)}\n (RelEmbedding.trans (Subrel.relEmbedding r {b | r b a✝}) f.toRelEmbedding)\n (_ :\n ∀ (x : ↑{b | r b a✝}),\n ↑(RelEmbedding.trans (Subrel.relEmbedding r {b | r b a✝}) f.toRelEmbedding) x ∈ {b | s b (↑f a✝)}))\n a_1 =\n { val := ↑f a, property := h }",
"tactic": "exact ⟨⟨a, f.toRelEmbedding.map_rel_iff.1 h⟩,\n Subtype.eq <| RelEmbedding.trans_apply _ _ _⟩"
}
] |
[
471,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
461,
1
] |
Mathlib/MeasureTheory/Covering/BesicovitchVectorSpace.lean
|
Besicovitch.SatelliteConfig.centerAndRescale_radius
|
[
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN✝ : ℕ\nτ✝ : ℝ\na✝ : SatelliteConfig E N✝ τ✝\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\n⊢ r (centerAndRescale a) (last N) = 1",
"tactic": "simp [SatelliteConfig.centerAndRescale, inv_mul_cancel (a.rpos _).ne']"
}
] |
[
124,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/Data/Polynomial/Degree/Lemmas.lean
|
Polynomial.degree_pos_of_root
|
[
{
"state_after": "R : Type u\nS : Type v\nι : Type w\na b : R\nm n : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nhp : p ≠ 0\nh : IsRoot p a\nhlt : 0 ≥ degree p\nthis : p = ↑C (coeff p 0)\n⊢ False",
"state_before": "R : Type u\nS : Type v\nι : Type w\na b : R\nm n : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nhp : p ≠ 0\nh : IsRoot p a\nhlt : 0 ≥ degree p\n⊢ False",
"tactic": "have := eq_C_of_degree_le_zero hlt"
},
{
"state_after": "R : Type u\nS : Type v\nι : Type w\na b : R\nm n : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nhp : p ≠ 0\nh : coeff p 0 = 0\nhlt : 0 ≥ degree p\nthis : p = ↑C (coeff p 0)\n⊢ False",
"state_before": "R : Type u\nS : Type v\nι : Type w\na b : R\nm n : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nhp : p ≠ 0\nh : IsRoot p a\nhlt : 0 ≥ degree p\nthis : p = ↑C (coeff p 0)\n⊢ False",
"tactic": "rw [IsRoot, this, eval_C] at h"
},
{
"state_after": "R : Type u\nS : Type v\nι : Type w\na b : R\nm n : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nhp : p ≠ 0\nh : coeff p 0 = 0\nhlt : 0 ≥ degree p\nthis : p = 0\n⊢ False",
"state_before": "R : Type u\nS : Type v\nι : Type w\na b : R\nm n : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nhp : p ≠ 0\nh : coeff p 0 = 0\nhlt : 0 ≥ degree p\nthis : p = ↑C (coeff p 0)\n⊢ False",
"tactic": "simp only [h, RingHom.map_zero] at this"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nι : Type w\na b : R\nm n : ℕ\ninst✝ : Semiring R\np✝ q r p : R[X]\nhp : p ≠ 0\nh : coeff p 0 = 0\nhlt : 0 ≥ degree p\nthis : p = 0\n⊢ False",
"tactic": "exact hp this"
}
] |
[
71,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/RingTheory/Localization/Basic.lean
|
IsLocalization.sec_spec'
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_1\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type ?u.248906\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nz : S\n⊢ ↑(algebraMap R S) (sec M z).fst = ↑(algebraMap R S) ↑(sec M z).snd * z",
"tactic": "rw [mul_comm, sec_spec]"
}
] |
[
206,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
204,
1
] |
Mathlib/RingTheory/Localization/LocalizationLocalization.lean
|
IsLocalization.localization_localization_map_units
|
[
{
"state_after": "case intro.intro\nR : Type u_3\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.58093\ninst✝⁶ : CommRing P\nN : Submonoid S\nT : Type u_2\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsLocalization N T\ny : { x // x ∈ localizationLocalizationSubmodule M N }\ny' : { x // x ∈ N }\nz : { x // x ∈ M }\neq : ↑(algebraMap R S) ↑y = ↑y' * ↑(algebraMap R S) ↑z\n⊢ IsUnit (↑(algebraMap R T) ↑y)",
"state_before": "R : Type u_3\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.58093\ninst✝⁶ : CommRing P\nN : Submonoid S\nT : Type u_2\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsLocalization N T\ny : { x // x ∈ localizationLocalizationSubmodule M N }\n⊢ IsUnit (↑(algebraMap R T) ↑y)",
"tactic": "obtain ⟨y', z, eq⟩ := mem_localizationLocalizationSubmodule.mp y.prop"
},
{
"state_after": "case intro.intro\nR : Type u_3\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.58093\ninst✝⁶ : CommRing P\nN : Submonoid S\nT : Type u_2\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsLocalization N T\ny : { x // x ∈ localizationLocalizationSubmodule M N }\ny' : { x // x ∈ N }\nz : { x // x ∈ M }\neq : ↑(algebraMap R S) ↑y = ↑y' * ↑(algebraMap R S) ↑z\n⊢ IsUnit (↑(algebraMap S T) ↑y') ∧ IsUnit (↑(algebraMap S T) (↑(algebraMap R S) ↑z))",
"state_before": "case intro.intro\nR : Type u_3\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.58093\ninst✝⁶ : CommRing P\nN : Submonoid S\nT : Type u_2\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsLocalization N T\ny : { x // x ∈ localizationLocalizationSubmodule M N }\ny' : { x // x ∈ N }\nz : { x // x ∈ M }\neq : ↑(algebraMap R S) ↑y = ↑y' * ↑(algebraMap R S) ↑z\n⊢ IsUnit (↑(algebraMap R T) ↑y)",
"tactic": "rw [IsScalarTower.algebraMap_apply R S T, eq, RingHom.map_mul, IsUnit.mul_iff]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nR : Type u_3\ninst✝⁹ : CommRing R\nM : Submonoid R\nS : Type u_1\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Type ?u.58093\ninst✝⁶ : CommRing P\nN : Submonoid S\nT : Type u_2\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsLocalization N T\ny : { x // x ∈ localizationLocalizationSubmodule M N }\ny' : { x // x ∈ N }\nz : { x // x ∈ M }\neq : ↑(algebraMap R S) ↑y = ↑y' * ↑(algebraMap R S) ↑z\n⊢ IsUnit (↑(algebraMap S T) ↑y') ∧ IsUnit (↑(algebraMap S T) (↑(algebraMap R S) ↑z))",
"tactic": "exact ⟨IsLocalization.map_units T y', (IsLocalization.map_units _ z).map (algebraMap S T)⟩"
}
] |
[
73,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
69,
1
] |
Mathlib/Data/List/Cycle.lean
|
Cycle.next_reverse_eq_prev'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\ns : Cycle α\nhs : Nodup (reverse s)\nx : α\nhx : x ∈ reverse s\n⊢ next (reverse s) hs x hx = prev s (_ : Nodup s) x (_ : x ∈ s)",
"tactic": "simp [← prev_reverse_eq_next]"
}
] |
[
868,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
866,
1
] |
Mathlib/MeasureTheory/Function/LocallyIntegrable.lean
|
MeasureTheory.locallyIntegrableOn_const
|
[] |
[
215,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
213,
1
] |
Mathlib/Order/BoundedOrder.lean
|
not_isBot_iff_ne_bot
|
[] |
[
354,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
353,
1
] |
Mathlib/Combinatorics/SetFamily/LYM.lean
|
Finset.IsAntichain.sperner
|
[
{
"state_after": "𝕜 : Type ?u.33953\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜\n⊢ ∑ r in Iic (Fintype.card α), ↑(card (𝒜 # r)) / ↑(Nat.choose (Fintype.card α) (Fintype.card α / 2)) ≤ 1",
"state_before": "𝕜 : Type ?u.33953\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜\n⊢ card 𝒜 ≤ Nat.choose (Fintype.card α) (Fintype.card α / 2)",
"tactic": "suffices (∑ r in Iic (Fintype.card α),\n ((𝒜 # r).card : ℚ) / (Fintype.card α).choose (Fintype.card α / 2)) ≤ 1 by\n rw [← sum_div, ← Nat.cast_sum, div_le_one] at this\n simp only [cast_le] at this\n rwa [sum_card_slice] at this\n simp only [cast_pos]\n exact choose_pos (Nat.div_le_self _ _)"
},
{
"state_after": "𝕜 : Type ?u.33953\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜\n⊢ ∑ r in range (succ (Fintype.card α)), ↑(card (𝒜 # r)) / ↑(Nat.choose (Fintype.card α) (Fintype.card α / 2)) ≤ 1",
"state_before": "𝕜 : Type ?u.33953\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜\n⊢ ∑ r in Iic (Fintype.card α), ↑(card (𝒜 # r)) / ↑(Nat.choose (Fintype.card α) (Fintype.card α / 2)) ≤ 1",
"tactic": "rw [Iic_eq_Icc, ← Ico_succ_right, bot_eq_zero, Ico_zero_eq_range]"
},
{
"state_after": "𝕜 : Type ?u.33953\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜\nr : ℕ\nhr : r ∈ range (succ (Fintype.card α))\n⊢ ↑(card (𝒜 # r)) / ↑(Nat.choose (Fintype.card α) (Fintype.card α / 2)) ≤\n ↑(card (𝒜 # r)) / ↑(Nat.choose (Fintype.card α) r)",
"state_before": "𝕜 : Type ?u.33953\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜\n⊢ ∑ r in range (succ (Fintype.card α)), ↑(card (𝒜 # r)) / ↑(Nat.choose (Fintype.card α) (Fintype.card α / 2)) ≤ 1",
"tactic": "refine' (sum_le_sum fun r hr => _).trans (sum_card_slice_div_choose_le_one h𝒜)"
},
{
"state_after": "𝕜 : Type ?u.33953\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜\nr : ℕ\nhr : r < succ (Fintype.card α)\n⊢ ↑(card (𝒜 # r)) / ↑(Nat.choose (Fintype.card α) (Fintype.card α / 2)) ≤\n ↑(card (𝒜 # r)) / ↑(Nat.choose (Fintype.card α) r)",
"state_before": "𝕜 : Type ?u.33953\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜\nr : ℕ\nhr : r ∈ range (succ (Fintype.card α))\n⊢ ↑(card (𝒜 # r)) / ↑(Nat.choose (Fintype.card α) (Fintype.card α / 2)) ≤\n ↑(card (𝒜 # r)) / ↑(Nat.choose (Fintype.card α) r)",
"tactic": "rw [mem_range] at hr"
},
{
"state_after": "case refine'_1\n𝕜 : Type ?u.33953\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜\nr : ℕ\nhr : r < succ (Fintype.card α)\n⊢ 0 ≤ card (𝒜 # r)\n\ncase refine'_2\n𝕜 : Type ?u.33953\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜\nr : ℕ\nhr : r < succ (Fintype.card α)\n⊢ 0 < Nat.choose (Fintype.card α) r\n\ncase refine'_3\n𝕜 : Type ?u.33953\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜\nr : ℕ\nhr : r < succ (Fintype.card α)\n⊢ Nat.choose (Fintype.card α) r ≤ Nat.choose (Fintype.card α) (Fintype.card α / 2)",
"state_before": "𝕜 : Type ?u.33953\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜\nr : ℕ\nhr : r < succ (Fintype.card α)\n⊢ ↑(card (𝒜 # r)) / ↑(Nat.choose (Fintype.card α) (Fintype.card α / 2)) ≤\n ↑(card (𝒜 # r)) / ↑(Nat.choose (Fintype.card α) r)",
"tactic": "refine' div_le_div_of_le_left _ _ _ <;> norm_cast"
},
{
"state_after": "𝕜 : Type ?u.33953\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜\nthis : ↑(∑ x in Iic (Fintype.card α), card (𝒜 # x)) ≤ ↑(Nat.choose (Fintype.card α) (Fintype.card α / 2))\n⊢ card 𝒜 ≤ Nat.choose (Fintype.card α) (Fintype.card α / 2)\n\n𝕜 : Type ?u.33953\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜\nthis : ↑(∑ x in Iic (Fintype.card α), card (𝒜 # x)) / ↑(Nat.choose (Fintype.card α) (Fintype.card α / 2)) ≤ 1\n⊢ 0 < ↑(Nat.choose (Fintype.card α) (Fintype.card α / 2))",
"state_before": "𝕜 : Type ?u.33953\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜\nthis : ∑ r in Iic (Fintype.card α), ↑(card (𝒜 # r)) / ↑(Nat.choose (Fintype.card α) (Fintype.card α / 2)) ≤ 1\n⊢ card 𝒜 ≤ Nat.choose (Fintype.card α) (Fintype.card α / 2)",
"tactic": "rw [← sum_div, ← Nat.cast_sum, div_le_one] at this"
},
{
"state_after": "𝕜 : Type ?u.33953\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜\nthis : ∑ x in Iic (Fintype.card α), card (𝒜 # x) ≤ Nat.choose (Fintype.card α) (Fintype.card α / 2)\n⊢ card 𝒜 ≤ Nat.choose (Fintype.card α) (Fintype.card α / 2)\n\n𝕜 : Type ?u.33953\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜\nthis : ↑(∑ x in Iic (Fintype.card α), card (𝒜 # x)) / ↑(Nat.choose (Fintype.card α) (Fintype.card α / 2)) ≤ 1\n⊢ 0 < ↑(Nat.choose (Fintype.card α) (Fintype.card α / 2))",
"state_before": "𝕜 : Type ?u.33953\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜\nthis : ↑(∑ x in Iic (Fintype.card α), card (𝒜 # x)) ≤ ↑(Nat.choose (Fintype.card α) (Fintype.card α / 2))\n⊢ card 𝒜 ≤ Nat.choose (Fintype.card α) (Fintype.card α / 2)\n\n𝕜 : Type ?u.33953\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜\nthis : ↑(∑ x in Iic (Fintype.card α), card (𝒜 # x)) / ↑(Nat.choose (Fintype.card α) (Fintype.card α / 2)) ≤ 1\n⊢ 0 < ↑(Nat.choose (Fintype.card α) (Fintype.card α / 2))",
"tactic": "simp only [cast_le] at this"
},
{
"state_after": "𝕜 : Type ?u.33953\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜\nthis : ↑(∑ x in Iic (Fintype.card α), card (𝒜 # x)) / ↑(Nat.choose (Fintype.card α) (Fintype.card α / 2)) ≤ 1\n⊢ 0 < ↑(Nat.choose (Fintype.card α) (Fintype.card α / 2))",
"state_before": "𝕜 : Type ?u.33953\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜\nthis : ∑ x in Iic (Fintype.card α), card (𝒜 # x) ≤ Nat.choose (Fintype.card α) (Fintype.card α / 2)\n⊢ card 𝒜 ≤ Nat.choose (Fintype.card α) (Fintype.card α / 2)\n\n𝕜 : Type ?u.33953\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜\nthis : ↑(∑ x in Iic (Fintype.card α), card (𝒜 # x)) / ↑(Nat.choose (Fintype.card α) (Fintype.card α / 2)) ≤ 1\n⊢ 0 < ↑(Nat.choose (Fintype.card α) (Fintype.card α / 2))",
"tactic": "rwa [sum_card_slice] at this"
},
{
"state_after": "𝕜 : Type ?u.33953\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜\nthis : ↑(∑ x in Iic (Fintype.card α), card (𝒜 # x)) / ↑(Nat.choose (Fintype.card α) (Fintype.card α / 2)) ≤ 1\n⊢ 0 < Nat.choose (Fintype.card α) (Fintype.card α / 2)",
"state_before": "𝕜 : Type ?u.33953\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜\nthis : ↑(∑ x in Iic (Fintype.card α), card (𝒜 # x)) / ↑(Nat.choose (Fintype.card α) (Fintype.card α / 2)) ≤ 1\n⊢ 0 < ↑(Nat.choose (Fintype.card α) (Fintype.card α / 2))",
"tactic": "simp only [cast_pos]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.33953\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜\nthis : ↑(∑ x in Iic (Fintype.card α), card (𝒜 # x)) / ↑(Nat.choose (Fintype.card α) (Fintype.card α / 2)) ≤ 1\n⊢ 0 < Nat.choose (Fintype.card α) (Fintype.card α / 2)",
"tactic": "exact choose_pos (Nat.div_le_self _ _)"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\n𝕜 : Type ?u.33953\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜\nr : ℕ\nhr : r < succ (Fintype.card α)\n⊢ 0 ≤ card (𝒜 # r)",
"tactic": "exact Nat.zero_le _"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\n𝕜 : Type ?u.33953\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜\nr : ℕ\nhr : r < succ (Fintype.card α)\n⊢ 0 < Nat.choose (Fintype.card α) r",
"tactic": "exact choose_pos (lt_succ_iff.1 hr)"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\n𝕜 : Type ?u.33953\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\nh𝒜 : IsAntichain (fun x x_1 => x ⊆ x_1) ↑𝒜\nr : ℕ\nhr : r < succ (Fintype.card α)\n⊢ Nat.choose (Fintype.card α) r ≤ Nat.choose (Fintype.card α) (Fintype.card α / 2)",
"tactic": "exact choose_le_middle _ _"
}
] |
[
249,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
232,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.compl_range_some
|
[] |
[
1192,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1191,
1
] |
Mathlib/CategoryTheory/Limits/Types.lean
|
CategoryTheory.Limits.Types.colimitEquivQuot_apply
|
[
{
"state_after": "case a\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ TypeMax\nj : J\nx : F.obj j\n⊢ ↑(colimitEquivQuot F).symm (↑(colimitEquivQuot F) (colimit.ι F j x)) =\n ↑(colimitEquivQuot F).symm (Quot.mk (Quot.Rel F) { fst := j, snd := x })",
"state_before": "J : Type v\ninst✝ : SmallCategory J\nF : J ⥤ TypeMax\nj : J\nx : F.obj j\n⊢ ↑(colimitEquivQuot F) (colimit.ι F j x) = Quot.mk (Quot.Rel F) { fst := j, snd := x }",
"tactic": "apply (colimitEquivQuot F).symm.injective"
},
{
"state_after": "no goals",
"state_before": "case a\nJ : Type v\ninst✝ : SmallCategory J\nF : J ⥤ TypeMax\nj : J\nx : F.obj j\n⊢ ↑(colimitEquivQuot F).symm (↑(colimitEquivQuot F) (colimit.ι F j x)) =\n ↑(colimitEquivQuot F).symm (Quot.mk (Quot.Rel F) { fst := j, snd := x })",
"tactic": "simp"
}
] |
[
304,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
301,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.insert_val
|
[] |
[
1048,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1047,
1
] |
Mathlib/AlgebraicGeometry/PresheafedSpace.lean
|
AlgebraicGeometry.PresheafedSpace.ext
|
[
{
"state_after": "no goals",
"state_before": "C : Type ?u.71852\ninst✝ : Category C\nX Y : PresheafedSpace C\nα β : X ⟶ Y\nw : α.base = β.base\n⊢ Functor.op (Opens.map α.base) = Functor.op (Opens.map β.base)",
"tactic": "rw [w]"
}
] |
[
192,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
190,
1
] |
Mathlib/LinearAlgebra/Span.lean
|
Submodule.prod_bot
|
[
{
"state_after": "case h.mk\nR : Type u_3\nR₂ : Type ?u.251481\nK : Type ?u.251484\nM : Type u_1\nM₂ : Type ?u.251490\nV : Type ?u.251493\nS : Type ?u.251496\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝⁴ : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R₂ M₂\ns t : Set M\nM' : Type u_2\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nq₁ q₁' : Submodule R M'\nx : M\ny : M'\n⊢ (x, y) ∈ prod ⊥ ⊥ ↔ (x, y) ∈ ⊥",
"state_before": "R : Type u_3\nR₂ : Type ?u.251481\nK : Type ?u.251484\nM : Type u_1\nM₂ : Type ?u.251490\nV : Type ?u.251493\nS : Type ?u.251496\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\nx : M\np p' : Submodule R M\ninst✝⁴ : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R₂ M₂\ns t : Set M\nM' : Type u_2\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nq₁ q₁' : Submodule R M'\n⊢ prod ⊥ ⊥ = ⊥",
"tactic": "ext ⟨x, y⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mk\nR : Type u_3\nR₂ : Type ?u.251481\nK : Type ?u.251484\nM : Type u_1\nM₂ : Type ?u.251490\nV : Type ?u.251493\nS : Type ?u.251496\ninst✝⁷ : Semiring R\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\nx✝ : M\np p' : Submodule R M\ninst✝⁴ : Semiring R₂\nσ₁₂ : R →+* R₂\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R₂ M₂\ns t : Set M\nM' : Type u_2\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nq₁ q₁' : Submodule R M'\nx : M\ny : M'\n⊢ (x, y) ∈ prod ⊥ ⊥ ↔ (x, y) ∈ ⊥",
"tactic": "simp [Prod.zero_eq_mk]"
}
] |
[
790,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
790,
1
] |
src/lean/Init/SimpLemmas.lean
|
Bool.not_eq_true
|
[
{
"state_after": "no goals",
"state_before": "b : Bool\n⊢ (¬b = true) = (b = false)",
"tactic": "cases b <;> decide"
}
] |
[
136,
99
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
136,
9
] |
Mathlib/Data/List/Perm.lean
|
List.perm_ext
|
[] |
[
746,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
743,
1
] |
Mathlib/Topology/Sheaves/SheafCondition/Sites.lean
|
TopCat.Presheaf.presieveOfCovering.mem_grothendieckTopology
|
[
{
"state_after": "X : TopCat\nι : Type v\nU : ι → Opens ↑X\nx : ↑X\nhx : x ∈ iSup U\n⊢ ∃ U_1 f, (Sieve.generate (presieveOfCovering U)).arrows f ∧ x ∈ U_1",
"state_before": "X : TopCat\nι : Type v\nU : ι → Opens ↑X\n⊢ Sieve.generate (presieveOfCovering U) ∈ GrothendieckTopology.sieves (Opens.grothendieckTopology ↑X) (iSup U)",
"tactic": "intro x hx"
},
{
"state_after": "case intro\nX : TopCat\nι : Type v\nU : ι → Opens ↑X\nx : ↑X\nhx : x ∈ iSup U\ni : ι\nhxi : x ∈ U i\n⊢ ∃ U_1 f, (Sieve.generate (presieveOfCovering U)).arrows f ∧ x ∈ U_1",
"state_before": "X : TopCat\nι : Type v\nU : ι → Opens ↑X\nx : ↑X\nhx : x ∈ iSup U\n⊢ ∃ U_1 f, (Sieve.generate (presieveOfCovering U)).arrows f ∧ x ∈ U_1",
"tactic": "obtain ⟨i, hxi⟩ := Opens.mem_iSup.mp hx"
},
{
"state_after": "no goals",
"state_before": "case intro\nX : TopCat\nι : Type v\nU : ι → Opens ↑X\nx : ↑X\nhx : x ∈ iSup U\ni : ι\nhxi : x ∈ U i\n⊢ ∃ U_1 f, (Sieve.generate (presieveOfCovering U)).arrows f ∧ x ∈ U_1",
"tactic": "exact ⟨U i, Opens.leSupr U i, ⟨U i, 𝟙 _, Opens.leSupr U i, ⟨i, rfl⟩, Category.id_comp _⟩, hxi⟩"
}
] |
[
110,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
1
] |
Std/Data/String/Lemmas.lean
|
String.utf8GetAux_addChar_right_cancel
|
[] |
[
197,
86
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
196,
1
] |
Mathlib/Topology/Algebra/Order/Floor.lean
|
tendsto_fract_left
|
[] |
[
177,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
174,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean
|
HasFDerivAtFilter.sub
|
[
{
"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.446548\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.446643\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\nhf : HasFDerivAtFilter f f' x L\nhg : HasFDerivAtFilter g g' x L\n⊢ HasFDerivAtFilter (fun x => f x - g x) (f' - g') x L",
"tactic": "simpa only [sub_eq_add_neg] using hf.add hg.neg"
}
] |
[
483,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
481,
1
] |
Mathlib/Analysis/Calculus/TangentCone.lean
|
uniqueDiffWithinAt_Iio
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.188395\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type ?u.188401\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type ?u.188496\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type ?u.188586\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nx y : E\ns t : Set E\na : ℝ\n⊢ Set.Nonempty (interior (Iio a))",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.188395\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type ?u.188401\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type ?u.188496\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type ?u.188586\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace ℝ G\nx y : E\ns t : Set E\na : ℝ\n⊢ a ∈ closure (Iio a)",
"tactic": "simp"
}
] |
[
466,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
465,
1
] |
Mathlib/Algebra/Order/UpperLower.lean
|
IsLowerSet.smul_subset
|
[] |
[
37,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
36,
1
] |
Mathlib/Data/Finsupp/AList.lean
|
AList.lookupFinsupp_apply
|
[
{
"state_after": "α : Type u_1\nM : Type u_2\ninst✝¹ : Zero M\ninst✝ : DecidableEq α\nl : AList fun _x => M\na : α\n⊢ Option.getD (lookup a l) 0 = Option.getD (lookup a l) 0",
"state_before": "α : Type u_1\nM : Type u_2\ninst✝¹ : Zero M\ninst✝ : DecidableEq α\nl : AList fun _x => M\na : α\n⊢ ↑(lookupFinsupp l) a = Option.getD (lookup a l) 0",
"tactic": "simp only [lookupFinsupp, ne_eq, Finsupp.coe_mk]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nM : Type u_2\ninst✝¹ : Zero M\ninst✝ : DecidableEq α\nl : AList fun _x => M\na : α\n⊢ Option.getD (lookup a l) 0 = Option.getD (lookup a l) 0",
"tactic": "congr"
}
] |
[
83,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
80,
1
] |
Mathlib/CategoryTheory/Preadditive/ProjectiveResolution.lean
|
CategoryTheory.ProjectiveResolution.lift_commutes
|
[
{
"state_after": "case h\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroObject C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasEqualizers C\ninst✝ : HasImages C\nY Z : C\nf : Y ⟶ Z\nP : ProjectiveResolution Y\nQ : ProjectiveResolution Z\n⊢ HomologicalComplex.Hom.f (lift f P Q ≫ Q.π) 0 = HomologicalComplex.Hom.f (P.π ≫ (ChainComplex.single₀ C).map f) 0",
"state_before": "C : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroObject C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasEqualizers C\ninst✝ : HasImages C\nY Z : C\nf : Y ⟶ Z\nP : ProjectiveResolution Y\nQ : ProjectiveResolution Z\n⊢ lift f P Q ≫ Q.π = P.π ≫ (ChainComplex.single₀ C).map f",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nC : Type u\ninst✝⁴ : Category C\ninst✝³ : HasZeroObject C\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasEqualizers C\ninst✝ : HasImages C\nY Z : C\nf : Y ⟶ Z\nP : ProjectiveResolution Y\nQ : ProjectiveResolution Z\n⊢ HomologicalComplex.Hom.f (lift f P Q ≫ Q.π) 0 = HomologicalComplex.Hom.f (P.π ≫ (ChainComplex.single₀ C).map f) 0",
"tactic": "simp [lift, liftZero]"
}
] |
[
209,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
207,
1
] |
Mathlib/CategoryTheory/Sites/Sieves.lean
|
CategoryTheory.Sieve.union_apply
|
[] |
[
374,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
373,
1
] |
Mathlib/MeasureTheory/Measure/NullMeasurable.lean
|
Finset.nullMeasurableSet
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.22585\nα : Type u_1\nβ : Type ?u.22591\nγ : Type ?u.22594\nm0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝ : MeasurableSingletonClass (NullMeasurableSpace α)\ns : Finset α\n⊢ NullMeasurableSet ↑s",
"tactic": "apply Finset.measurableSet"
}
] |
[
370,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
369,
11
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.natDegree_monomial_le
|
[
{
"state_after": "R : Type u\nS : Type v\na✝ b c d : R\nn m✝ : ℕ\ninst✝ : Semiring R\np q r : R[X]\na : R\nm : ℕ\n⊢ (if a = 0 then 0 else m) ≤ m",
"state_before": "R : Type u\nS : Type v\na✝ b c d : R\nn m✝ : ℕ\ninst✝ : Semiring R\np q r : R[X]\na : R\nm : ℕ\n⊢ natDegree (↑(monomial m) a) ≤ m",
"tactic": "rw [Polynomial.natDegree_monomial]"
},
{
"state_after": "case inl\nR : Type u\nS : Type v\na✝ b c d : R\nn m✝ : ℕ\ninst✝ : Semiring R\np q r : R[X]\na : R\nm : ℕ\nh✝ : a = 0\n⊢ 0 ≤ m\n\ncase inr\nR : Type u\nS : Type v\na✝ b c d : R\nn m✝ : ℕ\ninst✝ : Semiring R\np q r : R[X]\na : R\nm : ℕ\nh✝ : ¬a = 0\n⊢ m ≤ m",
"state_before": "R : Type u\nS : Type v\na✝ b c d : R\nn m✝ : ℕ\ninst✝ : Semiring R\np q r : R[X]\na : R\nm : ℕ\n⊢ (if a = 0 then 0 else m) ≤ m",
"tactic": "split_ifs"
},
{
"state_after": "no goals",
"state_before": "case inl\nR : Type u\nS : Type v\na✝ b c d : R\nn m✝ : ℕ\ninst✝ : Semiring R\np q r : R[X]\na : R\nm : ℕ\nh✝ : a = 0\n⊢ 0 ≤ m\n\ncase inr\nR : Type u\nS : Type v\na✝ b c d : R\nn m✝ : ℕ\ninst✝ : Semiring R\np q r : R[X]\na : R\nm : ℕ\nh✝ : ¬a = 0\n⊢ m ≤ m",
"tactic": "exacts [Nat.zero_le _, rfl.le]"
}
] |
[
337,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
334,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.prod_map_atBot_eq
|
[] |
[
1421,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1419,
1
] |
Mathlib/Data/PFun.lean
|
PFun.dom_coe
|
[] |
[
153,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.Finite.toFinset_symmDiff
|
[
{
"state_after": "case a\nα : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns t : Set α\na : α\nhs✝ : Set.Finite s\nht✝ : Set.Finite t\ninst✝ : DecidableEq α\nhs : Set.Finite s\nht : Set.Finite t\nh : Set.Finite (s ∆ t)\na✝ : α\n⊢ a✝ ∈ Finite.toFinset h ↔ a✝ ∈ Finite.toFinset hs ∆ Finite.toFinset ht",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns t : Set α\na : α\nhs✝ : Set.Finite s\nht✝ : Set.Finite t\ninst✝ : DecidableEq α\nhs : Set.Finite s\nht : Set.Finite t\nh : Set.Finite (s ∆ t)\n⊢ Finite.toFinset h = Finite.toFinset hs ∆ Finite.toFinset ht",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns t : Set α\na : α\nhs✝ : Set.Finite s\nht✝ : Set.Finite t\ninst✝ : DecidableEq α\nhs : Set.Finite s\nht : Set.Finite t\nh : Set.Finite (s ∆ t)\na✝ : α\n⊢ a✝ ∈ Finite.toFinset h ↔ a✝ ∈ Finite.toFinset hs ∆ Finite.toFinset ht",
"tactic": "simp [mem_symmDiff, Finset.mem_symmDiff]"
}
] |
[
260,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
11
] |
Mathlib/MeasureTheory/MeasurableSpaceDef.lean
|
MeasurableSet.univ
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.3526\nγ : Type ?u.3529\nδ : Type ?u.3532\nδ' : Type ?u.3535\nι : Sort ?u.3538\ns t u : Set α\nm : MeasurableSpace α\n⊢ MeasurableSet (univᶜ)",
"tactic": "simp"
}
] |
[
105,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
11
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
UniformSpace.le_sInf
|
[] |
[
1180,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1178,
11
] |
Mathlib/CategoryTheory/Limits/Opposites.lean
|
CategoryTheory.Limits.hasLimitsOfShape_of_hasColimitsOfShape_op
|
[] |
[
263,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
261,
1
] |
Mathlib/Data/Set/Intervals/ProjIcc.lean
|
Set.projIcc_surjective
|
[] |
[
87,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/Topology/Order.lean
|
induced_iff_nhds_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.44785\nf✝ : α → β\nι : Sort ?u.44792\ntα : TopologicalSpace α\ntβ : TopologicalSpace β\nf : β → α\nh : ∀ (b : β), 𝓝 b = comap f (𝓝 (f b))\nx : β\n⊢ 𝓝 x = 𝓝 x",
"tactic": "rw [h, nhds_induced]"
}
] |
[
847,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
844,
1
] |
Mathlib/Topology/Algebra/Order/LiminfLimsup.lean
|
Filter.Tendsto.limsup_eq
|
[] |
[
201,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
199,
1
] |
Mathlib/NumberTheory/ModularForms/CongruenceSubgroups.lean
|
Gamma_normal
|
[] |
[
73,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
72,
1
] |
Mathlib/Combinatorics/SetFamily/Intersecting.lean
|
Set.Intersecting.is_max_iff_card_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\n⊢ (∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t) ↔ 2 * card s = Fintype.card α",
"tactic": "classical\n refine'\n ⟨fun h => _, fun h t ht hst =>\n Finset.eq_of_subset_of_card_le hst <|\n le_of_mul_le_mul_left (ht.card_le.trans_eq h.symm) two_pos⟩\n suffices s.disjUnion (s.map ⟨compl, compl_injective⟩) hs.disjoint_map_compl = Finset.univ by\n rw [Fintype.card, ← this, two_mul, card_disjUnion, card_map]\n rw [← coe_eq_univ, disjUnion_eq_union, coe_union, coe_map, Function.Embedding.coeFn_mk,\n image_eq_preimage_of_inverse compl_compl compl_compl]\n refine' eq_univ_of_forall fun a => _\n simp_rw [mem_union, mem_preimage]\n by_contra' ha\n refine' s.ne_insert_of_not_mem _ ha.1 (h _ _ <| s.subset_insert _)\n rw [coe_insert]\n refine' hs.insert _ fun b hb hab => ha.2 <| (hs.isUpperSet' h) hab.le_compl_left hb\n rintro rfl\n have := h {⊤} (by rw [coe_singleton]; exact intersecting_singleton.2 top_ne_bot)\n rw [compl_bot] at ha\n rw [coe_eq_empty.1 ((hs.isUpperSet' h).not_top_mem.1 ha.2)] at this\n exact Finset.singleton_ne_empty _ (this <| Finset.empty_subset _).symm"
},
{
"state_after": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\n⊢ 2 * card s = Fintype.card α",
"state_before": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\n⊢ (∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t) ↔ 2 * card s = Fintype.card α",
"tactic": "refine'\n ⟨fun h => _, fun h t ht hst =>\n Finset.eq_of_subset_of_card_le hst <|\n le_of_mul_le_mul_left (ht.card_le.trans_eq h.symm) two_pos⟩"
},
{
"state_after": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\n⊢ disjUnion s (map { toFun := compl, inj' := (_ : Function.Injective compl) } s)\n (_ : Disjoint s (map { toFun := compl, inj' := (_ : Function.Injective compl) } s)) =\n Finset.univ",
"state_before": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\n⊢ 2 * card s = Fintype.card α",
"tactic": "suffices s.disjUnion (s.map ⟨compl, compl_injective⟩) hs.disjoint_map_compl = Finset.univ by\n rw [Fintype.card, ← this, two_mul, card_disjUnion, card_map]"
},
{
"state_after": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\n⊢ ↑s ∪ compl ⁻¹' ↑s = univ",
"state_before": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\n⊢ disjUnion s (map { toFun := compl, inj' := (_ : Function.Injective compl) } s)\n (_ : Disjoint s (map { toFun := compl, inj' := (_ : Function.Injective compl) } s)) =\n Finset.univ",
"tactic": "rw [← coe_eq_univ, disjUnion_eq_union, coe_union, coe_map, Function.Embedding.coeFn_mk,\n image_eq_preimage_of_inverse compl_compl compl_compl]"
},
{
"state_after": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\na : α\n⊢ a ∈ ↑s ∪ compl ⁻¹' ↑s",
"state_before": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\n⊢ ↑s ∪ compl ⁻¹' ↑s = univ",
"tactic": "refine' eq_univ_of_forall fun a => _"
},
{
"state_after": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\na : α\n⊢ a ∈ ↑s ∨ aᶜ ∈ ↑s",
"state_before": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\na : α\n⊢ a ∈ ↑s ∪ compl ⁻¹' ↑s",
"tactic": "simp_rw [mem_union, mem_preimage]"
},
{
"state_after": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\na : α\nha : ¬a ∈ ↑s ∧ ¬aᶜ ∈ ↑s\n⊢ False",
"state_before": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\na : α\n⊢ a ∈ ↑s ∨ aᶜ ∈ ↑s",
"tactic": "by_contra' ha"
},
{
"state_after": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\na : α\nha : ¬a ∈ ↑s ∧ ¬aᶜ ∈ ↑s\n⊢ Intersecting ↑(Insert.insert a s)",
"state_before": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\na : α\nha : ¬a ∈ ↑s ∧ ¬aᶜ ∈ ↑s\n⊢ False",
"tactic": "refine' s.ne_insert_of_not_mem _ ha.1 (h _ _ <| s.subset_insert _)"
},
{
"state_after": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\na : α\nha : ¬a ∈ ↑s ∧ ¬aᶜ ∈ ↑s\n⊢ Intersecting (Insert.insert a ↑s)",
"state_before": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\na : α\nha : ¬a ∈ ↑s ∧ ¬aᶜ ∈ ↑s\n⊢ Intersecting ↑(Insert.insert a s)",
"tactic": "rw [coe_insert]"
},
{
"state_after": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\na : α\nha : ¬a ∈ ↑s ∧ ¬aᶜ ∈ ↑s\n⊢ a ≠ ⊥",
"state_before": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\na : α\nha : ¬a ∈ ↑s ∧ ¬aᶜ ∈ ↑s\n⊢ Intersecting (Insert.insert a ↑s)",
"tactic": "refine' hs.insert _ fun b hb hab => ha.2 <| (hs.isUpperSet' h) hab.le_compl_left hb"
},
{
"state_after": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\nha : ¬⊥ ∈ ↑s ∧ ¬⊥ᶜ ∈ ↑s\n⊢ False",
"state_before": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\na : α\nha : ¬a ∈ ↑s ∧ ¬aᶜ ∈ ↑s\n⊢ a ≠ ⊥",
"tactic": "rintro rfl"
},
{
"state_after": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\nha : ¬⊥ ∈ ↑s ∧ ¬⊥ᶜ ∈ ↑s\nthis : s ⊆ {⊤} → s = {⊤}\n⊢ False",
"state_before": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\nha : ¬⊥ ∈ ↑s ∧ ¬⊥ᶜ ∈ ↑s\n⊢ False",
"tactic": "have := h {⊤} (by rw [coe_singleton]; exact intersecting_singleton.2 top_ne_bot)"
},
{
"state_after": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\nha : ¬⊥ ∈ ↑s ∧ ¬⊤ ∈ ↑s\nthis : s ⊆ {⊤} → s = {⊤}\n⊢ False",
"state_before": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\nha : ¬⊥ ∈ ↑s ∧ ¬⊥ᶜ ∈ ↑s\nthis : s ⊆ {⊤} → s = {⊤}\n⊢ False",
"tactic": "rw [compl_bot] at ha"
},
{
"state_after": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\nha : ¬⊥ ∈ ↑s ∧ ¬⊤ ∈ ↑s\nthis : ∅ ⊆ {⊤} → ∅ = {⊤}\n⊢ False",
"state_before": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\nha : ¬⊥ ∈ ↑s ∧ ¬⊤ ∈ ↑s\nthis : s ⊆ {⊤} → s = {⊤}\n⊢ False",
"tactic": "rw [coe_eq_empty.1 ((hs.isUpperSet' h).not_top_mem.1 ha.2)] at this"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\nha : ¬⊥ ∈ ↑s ∧ ¬⊤ ∈ ↑s\nthis : ∅ ⊆ {⊤} → ∅ = {⊤}\n⊢ False",
"tactic": "exact Finset.singleton_ne_empty _ (this <| Finset.empty_subset _).symm"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\nthis :\n disjUnion s (map { toFun := compl, inj' := (_ : Function.Injective compl) } s)\n (_ : Disjoint s (map { toFun := compl, inj' := (_ : Function.Injective compl) } s)) =\n Finset.univ\n⊢ 2 * card s = Fintype.card α",
"tactic": "rw [Fintype.card, ← this, two_mul, card_disjUnion, card_map]"
},
{
"state_after": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\nha : ¬⊥ ∈ ↑s ∧ ¬⊥ᶜ ∈ ↑s\n⊢ Intersecting {⊤}",
"state_before": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\nha : ¬⊥ ∈ ↑s ∧ ¬⊥ᶜ ∈ ↑s\n⊢ Intersecting ↑{⊤}",
"tactic": "rw [coe_singleton]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : BooleanAlgebra α\ninst✝¹ : Nontrivial α\ninst✝ : Fintype α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\nha : ¬⊥ ∈ ↑s ∧ ¬⊥ᶜ ∈ ↑s\n⊢ Intersecting {⊤}",
"tactic": "exact intersecting_singleton.2 top_ne_bot"
}
] |
[
197,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
176,
1
] |
Mathlib/LinearAlgebra/Basis.lean
|
Basis.extend_apply_self
|
[] |
[
1421,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1419,
1
] |
Mathlib/Topology/UniformSpace/Equiv.lean
|
UniformEquiv.prodComm_symm
|
[] |
[
310,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
309,
1
] |
Mathlib/SetTheory/Cardinal/Continuum.lean
|
Cardinal.aleph0_mul_continuum
|
[] |
[
167,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
166,
1
] |
Mathlib/Data/Dfinsupp/Interval.lean
|
Dfinsupp.mem_rangeIcc_apply_iff
|
[] |
[
125,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/CategoryTheory/EqToHom.lean
|
CategoryTheory.eqToHom_app
|
[
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX : C\n⊢ (eqToHom (_ : F = F)).app X = eqToHom (_ : F.obj X = F.obj X)",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF G : C ⥤ D\nh : F = G\nX : C\n⊢ (eqToHom h).app X = eqToHom (_ : F.obj X = G.obj X)",
"tactic": "subst h"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nF : C ⥤ D\nX : C\n⊢ (eqToHom (_ : F = F)).app X = eqToHom (_ : F.obj X = F.obj X)",
"tactic": "rfl"
}
] |
[
285,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
284,
1
] |
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