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/Topology/Sets/Closeds.lean
|
TopologicalSpace.Closeds.coe_eq_empty
|
[] |
[
136,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.inter_nonempty_iff_exists_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns s₁ s₂ t t₁ t₂ u : Set α\n⊢ Set.Nonempty (s ∩ t) ↔ ∃ x, x ∈ s ∧ x ∈ t",
"tactic": "simp_rw [inter_nonempty]"
}
] |
[
524,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
523,
1
] |
Mathlib/Algebra/GroupWithZero/Basic.lean
|
right_eq_mul₀
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.12533\nM₀ : Type u_1\nG₀ : Type ?u.12539\nM₀' : Type ?u.12542\nG₀' : Type ?u.12545\nF : Type ?u.12548\nF' : Type ?u.12551\ninst✝ : CancelMonoidWithZero M₀\na b c : M₀\nhb : b ≠ 0\n⊢ b = a * b ↔ a = 1",
"tactic": "rw [eq_comm, mul_eq_right₀ hb]"
}
] |
[
216,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.le.intro_sub
|
[
{
"state_after": "a b : Int\nn : Nat\nh : b - a = ↑n\n⊢ NonNeg ↑n",
"state_before": "a b : Int\nn : Nat\nh : b - a = ↑n\n⊢ a ≤ b",
"tactic": "simp [le_def, h]"
},
{
"state_after": "no goals",
"state_before": "a b : Int\nn : Nat\nh : b - a = ↑n\n⊢ NonNeg ↑n",
"tactic": "constructor"
}
] |
[
558,
32
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
557,
1
] |
Mathlib/Analysis/Convex/Side.lean
|
AffineSubspace.wOppSide_lineMap_left
|
[] |
[
366,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
364,
1
] |
Mathlib/Algebra/Invertible.lean
|
mul_left_eq_iff_eq_invOf_mul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nc a b : α\ninst✝¹ : Monoid α\ninst✝ : Invertible c\n⊢ c * a = b ↔ a = ⅟c * b",
"tactic": "rw [← mul_left_inj_of_invertible (c := ⅟c), invOf_mul_self_assoc]"
}
] |
[
313,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
311,
1
] |
Mathlib/Topology/Paracompact.lean
|
Homeomorph.paracompactSpace_iff
|
[] |
[
131,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/Order/Disjoint.lean
|
codisjoint_iff
|
[] |
[
313,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
312,
1
] |
Mathlib/Computability/RegularExpressions.lean
|
RegularExpression.matches'_add
|
[] |
[
139,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Std/Data/List/Lemmas.lean
|
List.append_of_mem
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\na : α\nl : List α\nb : α\nas✝ : List α\nh : Mem a as✝\ns t : List α\nh' : as✝ = s ++ a :: t\n⊢ b :: as✝ = b :: s ++ a :: t",
"tactic": "rw [h', cons_append]"
}
] |
[
88,
87
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
86,
1
] |
Mathlib/Algebra/Quaternion.lean
|
Cardinal.mk_univ_quaternion
|
[] |
[
1469,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1468,
1
] |
Mathlib/Algebra/GroupWithZero/Basic.lean
|
mul_right_eq_self₀
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.9281\nM₀ : Type u_1\nG₀ : Type ?u.9287\nM₀' : Type ?u.9290\nG₀' : Type ?u.9293\nF : Type ?u.9296\nF' : Type ?u.9299\ninst✝ : CancelMonoidWithZero M₀\na b c : M₀\n⊢ a * b = a ↔ a * b = a * 1",
"tactic": "rw [mul_one]"
}
] |
[
191,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Images.lean
|
CategoryTheory.Limits.IsImage.isoExt_inv_m
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX Y : C\nf : X ⟶ Y\nF F' : MonoFactorisation f\nhF : IsImage F\nhF' : IsImage F'\n⊢ (isoExt hF hF').inv ≫ F.m = F'.m",
"tactic": "simp"
}
] |
[
226,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
226,
1
] |
Mathlib/Algebra/Homology/Homology.lean
|
cyclesMap_arrow
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nV : Type u\ninst✝² : Category V\ninst✝¹ : HasZeroMorphisms V\nc : ComplexShape ι\nC : HomologicalComplex V c\ninst✝ : HasKernels V\nC₁ C₂ C₃ : HomologicalComplex V c\nf✝ f : C₁ ⟶ C₂\ni : ι\n⊢ cyclesMap f i ≫ Subobject.arrow (cycles C₂ i) = Subobject.arrow (cycles C₁ i) ≫ Hom.f f i",
"tactic": "simp"
}
] |
[
210,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
209,
1
] |
Mathlib/Data/Finsupp/Basic.lean
|
Finsupp.equivMapDomain_zero
|
[
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.144676\nι : Type ?u.144679\nM : Type u_3\nM' : Type ?u.144685\nN : Type ?u.144688\nP : Type ?u.144691\nG : Type ?u.144694\nH : Type ?u.144697\nR : Type ?u.144700\nS : Type ?u.144703\ninst✝ : Zero M\nf : α ≃ β\na✝ : β\n⊢ ↑(equivMapDomain f 0) a✝ = ↑0 a✝",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.144676\nι : Type ?u.144679\nM : Type u_3\nM' : Type ?u.144685\nN : Type ?u.144688\nP : Type ?u.144691\nG : Type ?u.144694\nH : Type ?u.144697\nR : Type ?u.144700\nS : Type ?u.144703\ninst✝ : Zero M\nf : α ≃ β\n⊢ equivMapDomain f 0 = 0",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.144676\nι : Type ?u.144679\nM : Type u_3\nM' : Type ?u.144685\nN : Type ?u.144688\nP : Type ?u.144691\nG : Type ?u.144694\nH : Type ?u.144697\nR : Type ?u.144700\nS : Type ?u.144703\ninst✝ : Zero M\nf : α ≃ β\na✝ : β\n⊢ ↑(equivMapDomain f 0) a✝ = ↑0 a✝",
"tactic": "simp only [equivMapDomain_apply, coe_zero, Pi.zero_apply]"
}
] |
[
354,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
353,
1
] |
Mathlib/Combinatorics/Additive/Behrend.lean
|
Behrend.sum_lt
|
[] |
[
213,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/Data/Set/Intervals/OrdConnectedComponent.lean
|
Set.subset_ordConnectedComponent
|
[] |
[
51,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
50,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
nnnorm_ofAdd
|
[] |
[
2250,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2249,
1
] |
Mathlib/Dynamics/Circle/RotationNumber/TranslationNumber.lean
|
CircleDeg1Lift.transnumAuxSeq_zero
|
[
{
"state_after": "no goals",
"state_before": "f g : CircleDeg1Lift\n⊢ transnumAuxSeq f 0 = ↑f 0",
"tactic": "simp [transnumAuxSeq]"
}
] |
[
668,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
668,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.join_le
|
[] |
[
1796,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1795,
1
] |
Mathlib/Analysis/Calculus/Deriv/ZPow.lean
|
iter_deriv_inv
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nE : Type v\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx✝ : 𝕜\ns : Set 𝕜\nm : ℤ\nk : ℕ\nx : 𝕜\n⊢ (deriv^[k]) Inv.inv x = (∏ i in Finset.range k, (-1 - ↑i)) * x ^ (-1 - ↑k)",
"tactic": "simpa only [zpow_neg_one, Int.cast_neg, Int.cast_one] using iter_deriv_zpow (-1) x k"
}
] |
[
146,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
144,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Mul.lean
|
fderiv_const_mul
|
[] |
[
504,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
502,
1
] |
Mathlib/Data/List/BigOperators/Basic.lean
|
List.headI_mul_tail_prod_of_ne_nil
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.69762\nα : Type ?u.69765\nM : Type u_1\nN : Type ?u.69771\nP : Type ?u.69774\nM₀ : Type ?u.69777\nG : Type ?u.69780\nR : Type ?u.69783\ninst✝³ : Monoid M\ninst✝² : Monoid N\ninst✝¹ : Monoid P\nl✝ l₁ l₂ : List M\na : M\ninst✝ : Inhabited M\nl : List M\nh : l ≠ []\n⊢ headI l * prod (tail l) = prod l",
"tactic": "cases l <;> [contradiction; simp]"
}
] |
[
241,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
240,
1
] |
Std/Data/Int/Lemmas.lean
|
Int.default_eq_zero
|
[] |
[
23,
61
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
23,
9
] |
Mathlib/Algebra/Associated.lean
|
Associated.isUnit_iff
|
[] |
[
633,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
632,
1
] |
Mathlib/Data/Polynomial/Monic.lean
|
Polynomial.Monic.pow
|
[
{
"state_after": "R : Type u\nS : Type v\na b : R\nm n✝ : ℕ\nι : Type y\ninst✝ : Semiring R\np q r : R[X]\nhp : Monic p\nn : ℕ\n⊢ Monic (p * p ^ n)",
"state_before": "R : Type u\nS : Type v\na b : R\nm n✝ : ℕ\nι : Type y\ninst✝ : Semiring R\np q r : R[X]\nhp : Monic p\nn : ℕ\n⊢ Monic (p ^ (n + 1))",
"tactic": "rw [pow_succ]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nm n✝ : ℕ\nι : Type y\ninst✝ : Semiring R\np q r : R[X]\nhp : Monic p\nn : ℕ\n⊢ Monic (p * p ^ n)",
"tactic": "exact hp.mul (Monic.pow hp n)"
}
] |
[
129,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Data/Polynomial/Derivative.lean
|
Polynomial.derivative_X_sq
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Semiring R\n⊢ ↑derivative (X ^ 2) = ↑C 2 * X",
"tactic": "rw [derivative_X_pow, Nat.cast_two, pow_one]"
}
] |
[
121,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
120,
1
] |
Mathlib/Algebra/Lie/DirectSum.lean
|
DirectSum.lieAlgebra_ext
|
[] |
[
160,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.two_nsmul_coe_pi
|
[
{
"state_after": "no goals",
"state_before": "⊢ 2 • ↑π = 0",
"tactic": "simp [← coe_nat_mul_eq_nsmul]"
}
] |
[
165,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
1
] |
Mathlib/Topology/MetricSpace/Gluing.lean
|
Metric.glueDist_triangle_inl_inr_inl
|
[
{
"state_after": "X : Type u\nY : Type v\nZ : Type w\ninst✝² : MetricSpace X\ninst✝¹ : MetricSpace Y\nΦ✝ : Z → X\nΨ✝ : Z → Y\nε✝ : ℝ\ninst✝ : Nonempty Z\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nH : ∀ (p q : Z), abs (dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)) ≤ 2 * ε\nx : X\ny : Y\nz : X\n⊢ dist x z ≤ (⨅ (p : Z), dist x (Φ p) + dist y (Ψ p)) + ((⨅ (p : Z), dist z (Φ p) + dist y (Ψ p)) + (ε + ε))",
"state_before": "X : Type u\nY : Type v\nZ : Type w\ninst✝² : MetricSpace X\ninst✝¹ : MetricSpace Y\nΦ✝ : Z → X\nΨ✝ : Z → Y\nε✝ : ℝ\ninst✝ : Nonempty Z\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nH : ∀ (p q : Z), abs (dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)) ≤ 2 * ε\nx : X\ny : Y\nz : X\n⊢ glueDist Φ Ψ ε (Sum.inl x) (Sum.inl z) ≤\n glueDist Φ Ψ ε (Sum.inl x) (Sum.inr y) + glueDist Φ Ψ ε (Sum.inr y) (Sum.inl z)",
"tactic": "simp_rw [glueDist, add_add_add_comm _ ε, add_assoc]"
},
{
"state_after": "X : Type u\nY : Type v\nZ : Type w\ninst✝² : MetricSpace X\ninst✝¹ : MetricSpace Y\nΦ✝ : Z → X\nΨ✝ : Z → Y\nε✝ : ℝ\ninst✝ : Nonempty Z\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nH : ∀ (p q : Z), abs (dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)) ≤ 2 * ε\nx : X\ny : Y\nz : X\np : Z\n⊢ dist x z ≤ dist x (Φ p) + dist y (Ψ p) + ((⨅ (p : Z), dist z (Φ p) + dist y (Ψ p)) + (ε + ε))",
"state_before": "X : Type u\nY : Type v\nZ : Type w\ninst✝² : MetricSpace X\ninst✝¹ : MetricSpace Y\nΦ✝ : Z → X\nΨ✝ : Z → Y\nε✝ : ℝ\ninst✝ : Nonempty Z\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nH : ∀ (p q : Z), abs (dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)) ≤ 2 * ε\nx : X\ny : Y\nz : X\n⊢ dist x z ≤ (⨅ (p : Z), dist x (Φ p) + dist y (Ψ p)) + ((⨅ (p : Z), dist z (Φ p) + dist y (Ψ p)) + (ε + ε))",
"tactic": "refine le_ciInf_add fun p => ?_"
},
{
"state_after": "X : Type u\nY : Type v\nZ : Type w\ninst✝² : MetricSpace X\ninst✝¹ : MetricSpace Y\nΦ✝ : Z → X\nΨ✝ : Z → Y\nε✝ : ℝ\ninst✝ : Nonempty Z\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nH : ∀ (p q : Z), abs (dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)) ≤ 2 * ε\nx : X\ny : Y\nz : X\np : Z\n⊢ dist x z ≤ (⨅ (p : Z), dist z (Φ p) + dist y (Ψ p)) + (dist x (Φ p) + (dist y (Ψ p) + 2 * ε))",
"state_before": "X : Type u\nY : Type v\nZ : Type w\ninst✝² : MetricSpace X\ninst✝¹ : MetricSpace Y\nΦ✝ : Z → X\nΨ✝ : Z → Y\nε✝ : ℝ\ninst✝ : Nonempty Z\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nH : ∀ (p q : Z), abs (dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)) ≤ 2 * ε\nx : X\ny : Y\nz : X\np : Z\n⊢ dist x z ≤ dist x (Φ p) + dist y (Ψ p) + ((⨅ (p : Z), dist z (Φ p) + dist y (Ψ p)) + (ε + ε))",
"tactic": "rw [add_left_comm, add_assoc, ← two_mul]"
},
{
"state_after": "X : Type u\nY : Type v\nZ : Type w\ninst✝² : MetricSpace X\ninst✝¹ : MetricSpace Y\nΦ✝ : Z → X\nΨ✝ : Z → Y\nε✝ : ℝ\ninst✝ : Nonempty Z\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nH : ∀ (p q : Z), abs (dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)) ≤ 2 * ε\nx : X\ny : Y\nz : X\np q : Z\n⊢ dist x z ≤ dist z (Φ q) + dist y (Ψ q) + (dist x (Φ p) + (dist y (Ψ p) + 2 * ε))",
"state_before": "X : Type u\nY : Type v\nZ : Type w\ninst✝² : MetricSpace X\ninst✝¹ : MetricSpace Y\nΦ✝ : Z → X\nΨ✝ : Z → Y\nε✝ : ℝ\ninst✝ : Nonempty Z\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nH : ∀ (p q : Z), abs (dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)) ≤ 2 * ε\nx : X\ny : Y\nz : X\np : Z\n⊢ dist x z ≤ (⨅ (p : Z), dist z (Φ p) + dist y (Ψ p)) + (dist x (Φ p) + (dist y (Ψ p) + 2 * ε))",
"tactic": "refine le_ciInf_add fun q => ?_"
},
{
"state_after": "X : Type u\nY : Type v\nZ : Type w\ninst✝² : MetricSpace X\ninst✝¹ : MetricSpace Y\nΦ✝ : Z → X\nΨ✝ : Z → Y\nε✝ : ℝ\ninst✝ : Nonempty Z\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nH : ∀ (p q : Z), abs (dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)) ≤ 2 * ε\nx : X\ny : Y\nz : X\np q : Z\n⊢ dist x z ≤ dist (Φ q) z + dist y (Ψ q) + (dist x (Φ p) + (dist y (Ψ p) + 2 * ε))",
"state_before": "X : Type u\nY : Type v\nZ : Type w\ninst✝² : MetricSpace X\ninst✝¹ : MetricSpace Y\nΦ✝ : Z → X\nΨ✝ : Z → Y\nε✝ : ℝ\ninst✝ : Nonempty Z\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nH : ∀ (p q : Z), abs (dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)) ≤ 2 * ε\nx : X\ny : Y\nz : X\np q : Z\n⊢ dist x z ≤ dist z (Φ q) + dist y (Ψ q) + (dist x (Φ p) + (dist y (Ψ p) + 2 * ε))",
"tactic": "rw [dist_comm z]"
},
{
"state_after": "no goals",
"state_before": "X : Type u\nY : Type v\nZ : Type w\ninst✝² : MetricSpace X\ninst✝¹ : MetricSpace Y\nΦ✝ : Z → X\nΨ✝ : Z → Y\nε✝ : ℝ\ninst✝ : Nonempty Z\nΦ : Z → X\nΨ : Z → Y\nε : ℝ\nH : ∀ (p q : Z), abs (dist (Φ p) (Φ q) - dist (Ψ p) (Ψ q)) ≤ 2 * ε\nx : X\ny : Y\nz : X\np q : Z\n⊢ dist x z ≤ dist (Φ q) z + dist y (Ψ q) + (dist x (Φ p) + (dist y (Ψ p) + 2 * ε))",
"tactic": "linarith [dist_triangle4 x (Φ p) (Φ q) z, dist_triangle_left (Ψ p) (Ψ q) y, (abs_le.1 (H p q)).2]"
}
] |
[
135,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
9
] |
Mathlib/Data/Polynomial/FieldDivision.lean
|
Polynomial.coeff_inv_units
|
[
{
"state_after": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn✝ : ℕ\ninst✝ : Field R\np q : R[X]\nu : R[X]ˣ\nn : ℕ\n⊢ (1 / if n = 0 then coeff (↑u) 0 else 0) = if n = 0 then coeff (↑u⁻¹) 0 else 0",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn✝ : ℕ\ninst✝ : Field R\np q : R[X]\nu : R[X]ˣ\nn : ℕ\n⊢ (coeff (↑u) n)⁻¹ = coeff (↑u⁻¹) n",
"tactic": "rw [eq_C_of_degree_eq_zero (degree_coe_units u), eq_C_of_degree_eq_zero (degree_coe_units u⁻¹),\n coeff_C, coeff_C, inv_eq_one_div]"
},
{
"state_after": "case inl\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn✝ : ℕ\ninst✝ : Field R\np q : R[X]\nu : R[X]ˣ\nn : ℕ\nh✝ : n = 0\n⊢ 1 / coeff (↑u) 0 = coeff (↑u⁻¹) 0\n\ncase inr\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn✝ : ℕ\ninst✝ : Field R\np q : R[X]\nu : R[X]ˣ\nn : ℕ\nh✝ : ¬n = 0\n⊢ 1 / 0 = 0",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn✝ : ℕ\ninst✝ : Field R\np q : R[X]\nu : R[X]ˣ\nn : ℕ\n⊢ (1 / if n = 0 then coeff (↑u) 0 else 0) = if n = 0 then coeff (↑u⁻¹) 0 else 0",
"tactic": "split_ifs"
},
{
"state_after": "case inl\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn✝ : ℕ\ninst✝ : Field R\np q : R[X]\nu : R[X]ˣ\nn : ℕ\nh✝ : n = 0\n⊢ eval 0 ↑1 = 1",
"state_before": "case inl\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn✝ : ℕ\ninst✝ : Field R\np q : R[X]\nu : R[X]ˣ\nn : ℕ\nh✝ : n = 0\n⊢ 1 / coeff (↑u) 0 = coeff (↑u⁻¹) 0",
"tactic": "rw [div_eq_iff_mul_eq (coeff_coe_units_zero_ne_zero u), coeff_zero_eq_eval_zero,\n coeff_zero_eq_eval_zero, ← eval_mul, ← Units.val_mul, inv_mul_self]"
},
{
"state_after": "no goals",
"state_before": "case inl\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn✝ : ℕ\ninst✝ : Field R\np q : R[X]\nu : R[X]ˣ\nn : ℕ\nh✝ : n = 0\n⊢ eval 0 ↑1 = 1",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case inr\nR : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn✝ : ℕ\ninst✝ : Field R\np q : R[X]\nu : R[X]ˣ\nn : ℕ\nh✝ : ¬n = 0\n⊢ 1 / 0 = 0",
"tactic": "simp"
}
] |
[
411,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
404,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.add_eq_union_iff_disjoint
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.504974\nγ : Type ?u.504977\ninst✝ : DecidableEq α\ns t : Multiset α\n⊢ s + t = s ∪ t ↔ Disjoint s t",
"tactic": "simp_rw [← inter_eq_zero_iff_disjoint, ext, count_add, count_union, count_inter, count_zero,\n Nat.min_eq_zero_iff, Nat.add_eq_max_iff]"
}
] |
[
2987,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2984,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.StronglyMeasurable.measurableSet_eq_fun
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.224536\nγ : Type ?u.224539\nι : Type ?u.224542\ninst✝² : Countable ι\nf✝ g✝ : α → β\nm : MeasurableSpace α\nE : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : MetrizableSpace E\nf g : α → E\nhf : StronglyMeasurable f\nhg : StronglyMeasurable g\nthis✝¹ : MeasurableSpace (E × E) := borel (E × E)\nthis✝ : BorelSpace (E × E)\n⊢ MeasurableSet {x | f x = g x}",
"state_before": "α : Type u_1\nβ : Type ?u.224536\nγ : Type ?u.224539\nι : Type ?u.224542\ninst✝² : Countable ι\nf✝ g✝ : α → β\nm : MeasurableSpace α\nE : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : MetrizableSpace E\nf g : α → E\nhf : StronglyMeasurable f\nhg : StronglyMeasurable g\n⊢ MeasurableSet {x | f x = g x}",
"tactic": "borelize (E × E)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.224536\nγ : Type ?u.224539\nι : Type ?u.224542\ninst✝² : Countable ι\nf✝ g✝ : α → β\nm : MeasurableSpace α\nE : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : MetrizableSpace E\nf g : α → E\nhf : StronglyMeasurable f\nhg : StronglyMeasurable g\nthis✝¹ : MeasurableSpace (E × E) := borel (E × E)\nthis✝ : BorelSpace (E × E)\n⊢ MeasurableSet {x | f x = g x}",
"tactic": "exact (hf.prod_mk hg).measurable isClosed_diagonal.measurableSet"
}
] |
[
870,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
866,
1
] |
Mathlib/MeasureTheory/PiSystem.lean
|
MeasurableSpace.DynkinSystem.has_diff
|
[
{
"state_after": "α : Type u_1\nd : DynkinSystem α\ns₁ s₂ : Set α\nh₁ : Has d s₁\nh₂ : Has d s₂\nh : s₂ ⊆ s₁\n⊢ Has d ((s₁ \\ s₂)ᶜ)",
"state_before": "α : Type u_1\nd : DynkinSystem α\ns₁ s₂ : Set α\nh₁ : Has d s₁\nh₂ : Has d s₂\nh : s₂ ⊆ s₁\n⊢ Has d (s₁ \\ s₂)",
"tactic": "apply d.has_compl_iff.1"
},
{
"state_after": "α : Type u_1\nd : DynkinSystem α\ns₁ s₂ : Set α\nh₁ : Has d s₁\nh₂ : Has d s₂\nh : s₂ ⊆ s₁\n⊢ Has d (s₁ᶜ ∪ s₂)",
"state_before": "α : Type u_1\nd : DynkinSystem α\ns₁ s₂ : Set α\nh₁ : Has d s₁\nh₂ : Has d s₂\nh : s₂ ⊆ s₁\n⊢ Has d ((s₁ \\ s₂)ᶜ)",
"tactic": "simp [diff_eq, compl_inter]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nd : DynkinSystem α\ns₁ s₂ : Set α\nh₁ : Has d s₁\nh₂ : Has d s₂\nh : s₂ ⊆ s₁\n⊢ Has d (s₁ᶜ ∪ s₂)",
"tactic": "exact d.has_union (d.has_compl h₁) h₂ (disjoint_compl_left.mono_right h)"
}
] |
[
594,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
590,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.IsLittleO.add_isBigOWith
|
[] |
[
1101,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1099,
1
] |
Mathlib/CategoryTheory/Generator.lean
|
CategoryTheory.IsCodetecting.isCoseparating
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : HasCoequalizers C\n𝒢 : Set C\n⊢ IsCodetecting 𝒢 → IsCoseparating 𝒢",
"tactic": "simpa only [← isSeparating_op_iff, ← isDetecting_op_iff] using IsDetecting.isSeparating"
}
] |
[
163,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
161,
1
] |
Mathlib/Data/Fintype/Card.lean
|
Fintype.bijective_iff_injective_and_card
|
[] |
[
669,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
666,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/InverseDeriv.lean
|
Real.differentiableWithinAt_arcsin_Ici
|
[
{
"state_after": "x : ℝ\n⊢ DifferentiableWithinAt ℝ arcsin (Ici x) x → x ≠ -1",
"state_before": "x : ℝ\n⊢ DifferentiableWithinAt ℝ arcsin (Ici x) x ↔ x ≠ -1",
"tactic": "refine' ⟨_, fun h => (hasDerivWithinAt_arcsin_Ici h).differentiableWithinAt⟩"
},
{
"state_after": "h : DifferentiableWithinAt ℝ arcsin (Ici (-1)) (-1)\n⊢ False",
"state_before": "x : ℝ\n⊢ DifferentiableWithinAt ℝ arcsin (Ici x) x → x ≠ -1",
"tactic": "rintro h rfl"
},
{
"state_after": "h : DifferentiableWithinAt ℝ arcsin (Ici (-1)) (-1)\nthis : sin ∘ arcsin =ᶠ[𝓝[Ici (-1)] (-1)] id\n⊢ False",
"state_before": "h : DifferentiableWithinAt ℝ arcsin (Ici (-1)) (-1)\n⊢ False",
"tactic": "have : sin ∘ arcsin =ᶠ[𝓝[≥] (-1 : ℝ)] id := by\n filter_upwards [Icc_mem_nhdsWithin_Ici ⟨le_rfl, neg_lt_self (zero_lt_one' ℝ)⟩] with x using\n sin_arcsin'"
},
{
"state_after": "h : DifferentiableWithinAt ℝ arcsin (Ici (-1)) (-1)\nthis✝ : sin ∘ arcsin =ᶠ[𝓝[Ici (-1)] (-1)] id\nthis : HasDerivWithinAt id (cos (arcsin (-1)) * derivWithin arcsin (Ici (-1)) (-1)) (Ici (-1)) (-1)\n⊢ False",
"state_before": "h : DifferentiableWithinAt ℝ arcsin (Ici (-1)) (-1)\nthis : sin ∘ arcsin =ᶠ[𝓝[Ici (-1)] (-1)] id\n⊢ False",
"tactic": "have := h.hasDerivWithinAt.sin.congr_of_eventuallyEq this.symm (by simp)"
},
{
"state_after": "no goals",
"state_before": "h : DifferentiableWithinAt ℝ arcsin (Ici (-1)) (-1)\nthis✝ : sin ∘ arcsin =ᶠ[𝓝[Ici (-1)] (-1)] id\nthis : HasDerivWithinAt id (cos (arcsin (-1)) * derivWithin arcsin (Ici (-1)) (-1)) (Ici (-1)) (-1)\n⊢ False",
"tactic": "simpa using (uniqueDiffOn_Ici _ _ left_mem_Ici).eq_deriv _ this (hasDerivWithinAt_id _ _)"
},
{
"state_after": "no goals",
"state_before": "h : DifferentiableWithinAt ℝ arcsin (Ici (-1)) (-1)\n⊢ sin ∘ arcsin =ᶠ[𝓝[Ici (-1)] (-1)] id",
"tactic": "filter_upwards [Icc_mem_nhdsWithin_Ici ⟨le_rfl, neg_lt_self (zero_lt_one' ℝ)⟩] with x using\n sin_arcsin'"
},
{
"state_after": "no goals",
"state_before": "h : DifferentiableWithinAt ℝ arcsin (Ici (-1)) (-1)\nthis : sin ∘ arcsin =ᶠ[𝓝[Ici (-1)] (-1)] id\n⊢ id (-1) = sin (arcsin (-1))",
"tactic": "simp"
}
] |
[
93,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/Logic/Function/Iterate.lean
|
Function.iterate_commute
|
[] |
[
224,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
223,
1
] |
Mathlib/Data/List/Cycle.lean
|
List.pmap_prev_eq_rotate_length_sub_one
|
[
{
"state_after": "case hl\nα : Type u_1\ninst✝ : DecidableEq α\nl : List α\nx : α\nh : Nodup l\n⊢ length (pmap (prev l) l (_ : ∀ (x : α), x ∈ l → x ∈ l)) = length (rotate l (length l - 1))\n\ncase h\nα : Type u_1\ninst✝ : DecidableEq α\nl : List α\nx : α\nh : Nodup l\n⊢ ∀ (n : ℕ) (h₁ : n < length (pmap (prev l) l (_ : ∀ (x : α), x ∈ l → x ∈ l)))\n (h₂ : n < length (rotate l (length l - 1))),\n nthLe (pmap (prev l) l (_ : ∀ (x : α), x ∈ l → x ∈ l)) n h₁ = nthLe (rotate l (length l - 1)) n h₂",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\nl : List α\nx : α\nh : Nodup l\n⊢ pmap (prev l) l (_ : ∀ (x : α), x ∈ l → x ∈ l) = rotate l (length l - 1)",
"tactic": "apply List.ext_nthLe"
},
{
"state_after": "no goals",
"state_before": "case hl\nα : Type u_1\ninst✝ : DecidableEq α\nl : List α\nx : α\nh : Nodup l\n⊢ length (pmap (prev l) l (_ : ∀ (x : α), x ∈ l → x ∈ l)) = length (rotate l (length l - 1))",
"tactic": "simp"
},
{
"state_after": "case h\nα : Type u_1\ninst✝ : DecidableEq α\nl : List α\nx : α\nh : Nodup l\nn : ℕ\nhn : n < length (pmap (prev l) l (_ : ∀ (x : α), x ∈ l → x ∈ l))\nhn' : n < length (rotate l (length l - 1))\n⊢ nthLe (pmap (prev l) l (_ : ∀ (x : α), x ∈ l → x ∈ l)) n hn = nthLe (rotate l (length l - 1)) n hn'",
"state_before": "case h\nα : Type u_1\ninst✝ : DecidableEq α\nl : List α\nx : α\nh : Nodup l\n⊢ ∀ (n : ℕ) (h₁ : n < length (pmap (prev l) l (_ : ∀ (x : α), x ∈ l → x ∈ l)))\n (h₂ : n < length (rotate l (length l - 1))),\n nthLe (pmap (prev l) l (_ : ∀ (x : α), x ∈ l → x ∈ l)) n h₁ = nthLe (rotate l (length l - 1)) n h₂",
"tactic": "intro n hn hn'"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\ninst✝ : DecidableEq α\nl : List α\nx : α\nh : Nodup l\nn : ℕ\nhn : n < length (pmap (prev l) l (_ : ∀ (x : α), x ∈ l → x ∈ l))\nhn' : n < length (rotate l (length l - 1))\n⊢ nthLe (pmap (prev l) l (_ : ∀ (x : α), x ∈ l → x ∈ l)) n hn = nthLe (rotate l (length l - 1)) n hn'",
"tactic": "rw [nthLe_rotate, nthLe_pmap, prev_nthLe _ h]"
}
] |
[
372,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
367,
1
] |
Mathlib/GroupTheory/Subsemigroup/Basic.lean
|
Subsemigroup.closure_induction₂
|
[] |
[
384,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
379,
1
] |
Std/Data/HashMap/WF.lean
|
Std.HashMap.Imp.Bucket.ext
|
[] |
[
19,
35
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
18,
18
] |
Mathlib/Algebra/Order/Group/Abs.lean
|
sub_le_of_abs_sub_le_right
|
[] |
[
297,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
296,
1
] |
Mathlib/Combinatorics/SimpleGraph/Ends/Defs.lean
|
SimpleGraph.ComponentCompl.not_mem_of_mem
|
[] |
[
131,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
130,
1
] |
Mathlib/GroupTheory/GroupAction/ConjAct.lean
|
ConjAct.card
|
[] |
[
63,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/LinearAlgebra/Matrix/Block.lean
|
Matrix.blockTriangular_zero
|
[] |
[
88,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.one_out_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.186066\nβ : Type ?u.186069\nγ : Type ?u.186072\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nx : (Quotient.out 1).α\n⊢ 0 < type fun x x_1 => x < x_1",
"tactic": "simp"
}
] |
[
1127,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1126,
1
] |
Mathlib/Data/List/Basic.lean
|
List.comp_map
|
[] |
[
1850,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1849,
1
] |
Mathlib/MeasureTheory/Group/Arithmetic.lean
|
aemeasurable_inv_iff
|
[
{
"state_after": "no goals",
"state_before": "G✝ : Type ?u.2473457\nα : Type u_2\ninst✝⁵ : Inv G✝\ninst✝⁴ : MeasurableSpace G✝\ninst✝³ : MeasurableInv G✝\nm : MeasurableSpace α\nf✝ : α → G✝\nμ : MeasureTheory.Measure α\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : MeasurableSpace G\ninst✝ : MeasurableInv G\nf : α → G\nh : AEMeasurable fun x => (f x)⁻¹\n⊢ AEMeasurable f",
"tactic": "simpa only [inv_inv] using h.inv"
}
] |
[
479,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
477,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
|
AffineSubspace.direction_sup
|
[
{
"state_after": "case refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ s1\nhp2 : p2 ∈ s2\n⊢ direction (s1 ⊔ s2) ≤ direction s1 ⊔ direction s2 ⊔ Submodule.span k {p2 -ᵥ p1}\n\ncase refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ s1\nhp2 : p2 ∈ s2\n⊢ direction s1 ⊔ direction s2 ⊔ Submodule.span k {p2 -ᵥ p1} ≤ direction (s1 ⊔ s2)",
"state_before": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ s1\nhp2 : p2 ∈ s2\n⊢ direction (s1 ⊔ s2) = direction s1 ⊔ direction s2 ⊔ Submodule.span k {p2 -ᵥ p1}",
"tactic": "refine' le_antisymm _ _"
},
{
"state_after": "case refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ s1\nhp2 : p2 ∈ s2\n⊢ direction (affineSpan k (↑s1 ∪ ↑s2)) ≤ direction s1 ⊔ direction s2 ⊔ Submodule.span k {p2 -ᵥ p1}",
"state_before": "case refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ s1\nhp2 : p2 ∈ s2\n⊢ direction (s1 ⊔ s2) ≤ direction s1 ⊔ direction s2 ⊔ Submodule.span k {p2 -ᵥ p1}",
"tactic": "change (affineSpan k ((s1 : Set P) ∪ s2)).direction ≤ _"
},
{
"state_after": "case refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ ↑s1\nhp2 : p2 ∈ s2\n⊢ direction (affineSpan k (↑s1 ∪ ↑s2)) ≤ direction s1 ⊔ direction s2 ⊔ Submodule.span k {p2 -ᵥ p1}",
"state_before": "case refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ s1\nhp2 : p2 ∈ s2\n⊢ direction (affineSpan k (↑s1 ∪ ↑s2)) ≤ direction s1 ⊔ direction s2 ⊔ Submodule.span k {p2 -ᵥ p1}",
"tactic": "rw [← mem_coe] at hp1"
},
{
"state_after": "case refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ ↑s1\nhp2 : p2 ∈ s2\n⊢ (fun x => x -ᵥ p1) '' (↑s1 ∪ ↑s2) ⊆ ↑(direction s1 ⊔ direction s2 ⊔ Submodule.span k {p2 -ᵥ p1})",
"state_before": "case refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ ↑s1\nhp2 : p2 ∈ s2\n⊢ direction (affineSpan k (↑s1 ∪ ↑s2)) ≤ direction s1 ⊔ direction s2 ⊔ Submodule.span k {p2 -ᵥ p1}",
"tactic": "rw [direction_affineSpan, vectorSpan_eq_span_vsub_set_right k (Set.mem_union_left _ hp1),\n Submodule.span_le]"
},
{
"state_after": "case refine'_1.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ ↑s1\nhp2 : p2 ∈ s2\np3 : P\nhp3 : p3 ∈ ↑s1 ∪ ↑s2\n⊢ (fun x => x -ᵥ p1) p3 ∈ ↑(direction s1 ⊔ direction s2 ⊔ Submodule.span k {p2 -ᵥ p1})",
"state_before": "case refine'_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ ↑s1\nhp2 : p2 ∈ s2\n⊢ (fun x => x -ᵥ p1) '' (↑s1 ∪ ↑s2) ⊆ ↑(direction s1 ⊔ direction s2 ⊔ Submodule.span k {p2 -ᵥ p1})",
"tactic": "rintro v ⟨p3, hp3, rfl⟩"
},
{
"state_after": "case refine'_1.intro.intro.inl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ ↑s1\nhp2 : p2 ∈ s2\np3 : P\nhp3 : p3 ∈ ↑s1\n⊢ (fun x => x -ᵥ p1) p3 ∈ ↑(direction s1 ⊔ direction s2 ⊔ Submodule.span k {p2 -ᵥ p1})\n\ncase refine'_1.intro.intro.inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ ↑s1\nhp2 : p2 ∈ s2\np3 : P\nhp3 : p3 ∈ ↑s2\n⊢ (fun x => x -ᵥ p1) p3 ∈ ↑(direction s1 ⊔ direction s2 ⊔ Submodule.span k {p2 -ᵥ p1})",
"state_before": "case refine'_1.intro.intro\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ ↑s1\nhp2 : p2 ∈ s2\np3 : P\nhp3 : p3 ∈ ↑s1 ∪ ↑s2\n⊢ (fun x => x -ᵥ p1) p3 ∈ ↑(direction s1 ⊔ direction s2 ⊔ Submodule.span k {p2 -ᵥ p1})",
"tactic": "cases' hp3 with hp3 hp3"
},
{
"state_after": "case refine'_1.intro.intro.inl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ ↑s1\nhp2 : p2 ∈ s2\np3 : P\nhp3 : p3 ∈ ↑s1\n⊢ ∃ y, y ∈ direction s2 ⊔ Submodule.span k {p2 -ᵥ p1} ∧ ∃ z, z ∈ direction s1 ∧ y + z = (fun x => x -ᵥ p1) p3",
"state_before": "case refine'_1.intro.intro.inl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ ↑s1\nhp2 : p2 ∈ s2\np3 : P\nhp3 : p3 ∈ ↑s1\n⊢ (fun x => x -ᵥ p1) p3 ∈ ↑(direction s1 ⊔ direction s2 ⊔ Submodule.span k {p2 -ᵥ p1})",
"tactic": "rw [sup_assoc, sup_comm, SetLike.mem_coe, Submodule.mem_sup]"
},
{
"state_after": "case refine'_1.intro.intro.inl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ ↑s1\nhp2 : p2 ∈ s2\np3 : P\nhp3 : p3 ∈ ↑s1\n⊢ 0 + (p3 -ᵥ p1) = (fun x => x -ᵥ p1) p3",
"state_before": "case refine'_1.intro.intro.inl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ ↑s1\nhp2 : p2 ∈ s2\np3 : P\nhp3 : p3 ∈ ↑s1\n⊢ ∃ y, y ∈ direction s2 ⊔ Submodule.span k {p2 -ᵥ p1} ∧ ∃ z, z ∈ direction s1 ∧ y + z = (fun x => x -ᵥ p1) p3",
"tactic": "use 0, Submodule.zero_mem _, p3 -ᵥ p1, vsub_mem_direction hp3 hp1"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.intro.inl\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ ↑s1\nhp2 : p2 ∈ s2\np3 : P\nhp3 : p3 ∈ ↑s1\n⊢ 0 + (p3 -ᵥ p1) = (fun x => x -ᵥ p1) p3",
"tactic": "rw [zero_add]"
},
{
"state_after": "case refine'_1.intro.intro.inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ ↑s1\nhp2 : p2 ∈ s2\np3 : P\nhp3 : p3 ∈ ↑s2\n⊢ ∃ y, y ∈ direction s1 ∧ ∃ z, z ∈ direction s2 ⊔ Submodule.span k {p2 -ᵥ p1} ∧ y + z = (fun x => x -ᵥ p1) p3",
"state_before": "case refine'_1.intro.intro.inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ ↑s1\nhp2 : p2 ∈ s2\np3 : P\nhp3 : p3 ∈ ↑s2\n⊢ (fun x => x -ᵥ p1) p3 ∈ ↑(direction s1 ⊔ direction s2 ⊔ Submodule.span k {p2 -ᵥ p1})",
"tactic": "rw [sup_assoc, SetLike.mem_coe, Submodule.mem_sup]"
},
{
"state_after": "case refine'_1.intro.intro.inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ ↑s1\nhp2 : p2 ∈ s2\np3 : P\nhp3 : p3 ∈ ↑s2\n⊢ p3 -ᵥ p1 ∈ direction s2 ⊔ Submodule.span k {p2 -ᵥ p1} ∧ 0 + (p3 -ᵥ p1) = (fun x => x -ᵥ p1) p3",
"state_before": "case refine'_1.intro.intro.inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ ↑s1\nhp2 : p2 ∈ s2\np3 : P\nhp3 : p3 ∈ ↑s2\n⊢ ∃ y, y ∈ direction s1 ∧ ∃ z, z ∈ direction s2 ⊔ Submodule.span k {p2 -ᵥ p1} ∧ y + z = (fun x => x -ᵥ p1) p3",
"tactic": "use 0, Submodule.zero_mem _, p3 -ᵥ p1"
},
{
"state_after": "case refine'_1.intro.intro.inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ ↑s1\nhp2 : p2 ∈ s2\np3 : P\nhp3 : p3 ∈ ↑s2\n⊢ p3 -ᵥ p1 = (fun x => x -ᵥ p1) p3 ∧ p3 -ᵥ p1 ∈ direction s2 ⊔ Submodule.span k {p2 -ᵥ p1}",
"state_before": "case refine'_1.intro.intro.inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ ↑s1\nhp2 : p2 ∈ s2\np3 : P\nhp3 : p3 ∈ ↑s2\n⊢ p3 -ᵥ p1 ∈ direction s2 ⊔ Submodule.span k {p2 -ᵥ p1} ∧ 0 + (p3 -ᵥ p1) = (fun x => x -ᵥ p1) p3",
"tactic": "rw [and_comm, zero_add]"
},
{
"state_after": "case refine'_1.intro.intro.inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ ↑s1\nhp2 : p2 ∈ s2\np3 : P\nhp3 : p3 ∈ ↑s2\n⊢ p3 -ᵥ p1 ∈ direction s2 ⊔ Submodule.span k {p2 -ᵥ p1}",
"state_before": "case refine'_1.intro.intro.inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ ↑s1\nhp2 : p2 ∈ s2\np3 : P\nhp3 : p3 ∈ ↑s2\n⊢ p3 -ᵥ p1 = (fun x => x -ᵥ p1) p3 ∧ p3 -ᵥ p1 ∈ direction s2 ⊔ Submodule.span k {p2 -ᵥ p1}",
"tactic": "use rfl"
},
{
"state_after": "case refine'_1.intro.intro.inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ ↑s1\nhp2 : p2 ∈ s2\np3 : P\nhp3 : p3 ∈ ↑s2\n⊢ ∃ y, y ∈ direction s2 ∧ ∃ z, z ∈ Submodule.span k {p2 -ᵥ p1} ∧ y + z = p3 -ᵥ p2 + (p2 -ᵥ p1)",
"state_before": "case refine'_1.intro.intro.inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ ↑s1\nhp2 : p2 ∈ s2\np3 : P\nhp3 : p3 ∈ ↑s2\n⊢ p3 -ᵥ p1 ∈ direction s2 ⊔ Submodule.span k {p2 -ᵥ p1}",
"tactic": "rw [← vsub_add_vsub_cancel p3 p2 p1, Submodule.mem_sup]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.intro.inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ ↑s1\nhp2 : p2 ∈ s2\np3 : P\nhp3 : p3 ∈ ↑s2\n⊢ ∃ y, y ∈ direction s2 ∧ ∃ z, z ∈ Submodule.span k {p2 -ᵥ p1} ∧ y + z = p3 -ᵥ p2 + (p2 -ᵥ p1)",
"tactic": "use p3 -ᵥ p2, vsub_mem_direction hp3 hp2, p2 -ᵥ p1, Submodule.mem_span_singleton_self _"
},
{
"state_after": "case refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ s1\nhp2 : p2 ∈ s2\n⊢ Submodule.span k {p2 -ᵥ p1} ≤ direction (s1 ⊔ s2)",
"state_before": "case refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ s1\nhp2 : p2 ∈ s2\n⊢ direction s1 ⊔ direction s2 ⊔ Submodule.span k {p2 -ᵥ p1} ≤ direction (s1 ⊔ s2)",
"tactic": "refine' sup_le (sup_direction_le _ _) _"
},
{
"state_after": "case refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ s1\nhp2 : p2 ∈ s2\n⊢ Submodule.span k {p2 -ᵥ p1} ≤ Submodule.span k (↑(s1 ⊔ s2) -ᵥ ↑(s1 ⊔ s2))",
"state_before": "case refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ s1\nhp2 : p2 ∈ s2\n⊢ Submodule.span k {p2 -ᵥ p1} ≤ direction (s1 ⊔ s2)",
"tactic": "rw [direction_eq_vectorSpan, vectorSpan_def]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns1 s2 : AffineSubspace k P\np1 p2 : P\nhp1 : p1 ∈ s1\nhp2 : p2 ∈ s2\n⊢ Submodule.span k {p2 -ᵥ p1} ≤ Submodule.span k (↑(s1 ⊔ s2) -ᵥ ↑(s1 ⊔ s2))",
"tactic": "exact\n sInf_le_sInf fun p hp =>\n Set.Subset.trans\n (Set.singleton_subset_iff.2\n (vsub_mem_vsub (mem_spanPoints k p2 _ (Set.mem_union_right _ hp2))\n (mem_spanPoints k p1 _ (Set.mem_union_left _ hp1))))\n hp"
}
] |
[
1440,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1414,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.diff_singleton_subset_iff
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns✝ s₁ s₂ t✝ t₁ t₂ u : Set α\nx : α\ns t : Set α\n⊢ s \\ {x} ⊆ t ↔ s ⊆ {x} ∪ t",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns✝ s₁ s₂ t✝ t₁ t₂ u : Set α\nx : α\ns t : Set α\n⊢ s \\ {x} ⊆ t ↔ s ⊆ insert x t",
"tactic": "rw [← union_singleton, union_comm]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\na b : α\ns✝ s₁ s₂ t✝ t₁ t₂ u : Set α\nx : α\ns t : Set α\n⊢ s \\ {x} ⊆ t ↔ s ⊆ {x} ∪ t",
"tactic": "apply diff_subset_iff"
}
] |
[
1939,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1937,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
Nat.cast_floor_eq_int_floor
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.309882\nα : Type u_1\nβ : Type ?u.309888\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\na : α\nha : 0 ≤ a\n⊢ ↑⌊a⌋₊ = ⌊a⌋",
"tactic": "rw [← Int.floor_toNat, Int.toNat_of_nonneg (Int.floor_nonneg.2 ha)]"
}
] |
[
1603,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1602,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
Filter.Tendsto.nnnorm'
|
[] |
[
1194,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1193,
1
] |
Mathlib/Algebra/Ring/Units.lean
|
Units.val_neg
|
[] |
[
40,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
39,
11
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.coe_sub
|
[] |
[
98,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Std/Data/List/Lemmas.lean
|
List.eraseP_of_forall_not
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\np : α → Bool\nl : List α\nh : ∀ (a : α), a ∈ l → ¬p a = true\n⊢ eraseP p l = l",
"tactic": "induction l with\n| nil => rfl\n| cons _ _ ih => simp [h _ (.head ..), ih (forall_mem_cons.1 h).2]"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\np : α → Bool\nh : ∀ (a : α), a ∈ [] → ¬p a = true\n⊢ eraseP p [] = []",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u_1\np : α → Bool\nhead✝ : α\ntail✝ : List α\nih : (∀ (a : α), a ∈ tail✝ → ¬p a = true) → eraseP p tail✝ = tail✝\nh : ∀ (a : α), a ∈ head✝ :: tail✝ → ¬p a = true\n⊢ eraseP p (head✝ :: tail✝) = head✝ :: tail✝",
"tactic": "simp [h _ (.head ..), ih (forall_mem_cons.1 h).2]"
}
] |
[
940,
69
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
937,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.coeff_monomial
|
[
{
"state_after": "R : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q : R[X]\n⊢ coeff { toFinsupp := Finsupp.single n a } m = if n = m then a else 0",
"state_before": "R : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q : R[X]\n⊢ coeff (↑(monomial n) a) m = if n = m then a else 0",
"tactic": "rw [← ofFinsupp_single]"
},
{
"state_after": "R : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q : R[X]\n⊢ ↑(Finsupp.single n a) m = if n = m then a else 0",
"state_before": "R : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q : R[X]\n⊢ coeff { toFinsupp := Finsupp.single n a } m = if n = m then a else 0",
"tactic": "simp only [coeff, LinearMap.coe_mk]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\na b : R\nm n : ℕ\ninst✝ : Semiring R\np q : R[X]\n⊢ ↑(Finsupp.single n a) m = if n = m then a else 0",
"tactic": "rw [Finsupp.single_apply]"
}
] |
[
675,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
671,
1
] |
Mathlib/Topology/MetricSpace/Contracting.lean
|
ContractingWith.one_sub_K_pos
|
[] |
[
268,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
267,
1
] |
Mathlib/Logic/Nonempty.lean
|
nonempty_sum
|
[] |
[
83,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
continuousAt_extChartAt'
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\nE : Type u_2\nM : Type u_1\nH : Type u_4\nE' : Type ?u.192175\nM' : Type ?u.192178\nH' : Type ?u.192181\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\ninst✝⁷ : TopologicalSpace H\ninst✝⁶ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝⁵ : NormedAddCommGroup E'\ninst✝⁴ : NormedSpace 𝕜 E'\ninst✝³ : TopologicalSpace H'\ninst✝² : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx : M\ns t : Set M\ninst✝¹ : ChartedSpace H M\ninst✝ : ChartedSpace H' M'\nx' : M\nh : x' ∈ (extChartAt I x).source\n⊢ x' ∈ (chartAt H x).toLocalEquiv.source",
"tactic": "rwa [← extChartAt_source I]"
}
] |
[
1091,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1089,
1
] |
Mathlib/SetTheory/Cardinal/Cofinality.lean
|
Cardinal.sup_lt_ord_of_isRegular
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.158454\nr : α → α → Prop\nι : Type (max u_1 u_2)\nf : ι → Ordinal\nc : Cardinal\nhc : IsRegular c\nhι : (#ι) < c\n⊢ (#ι) < Ordinal.cof (ord c)",
"tactic": "rwa [hc.cof_eq]"
}
] |
[
1078,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1076,
1
] |
Mathlib/LinearAlgebra/Span.lean
|
Submodule.span_coe_eq_restrictScalars
|
[] |
[
95,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
ENNReal.measurable_of_measurable_nnreal
|
[] |
[
1824,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1821,
1
] |
Mathlib/Analysis/Analytic/Basic.lean
|
HasFPowerSeriesAt.eventually_hasSum_sub
|
[] |
[
503,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
500,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.toReal_le_pi
|
[
{
"state_after": "case h\nx✝ : ℝ\n⊢ toReal ↑x✝ ≤ π",
"state_before": "θ : Angle\n⊢ toReal θ ≤ π",
"tactic": "induction θ using Real.Angle.induction_on"
},
{
"state_after": "case h.e'_4\nx✝ : ℝ\n⊢ π = -π + 2 * π",
"state_before": "case h\nx✝ : ℝ\n⊢ toReal ↑x✝ ≤ π",
"tactic": "convert toIocMod_le_right two_pi_pos _ _"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nx✝ : ℝ\n⊢ π = -π + 2 * π",
"tactic": "ring"
}
] |
[
567,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
564,
1
] |
Mathlib/Algebra/Order/Ring/Defs.lean
|
lt_of_mul_lt_mul_of_nonpos_right
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.200981\ninst✝ : LinearOrderedRing α\na b c : α\nh : a * c < b * c\nhc : c ≤ 0\n⊢ b * -c < a * -c",
"tactic": "rwa [mul_neg, mul_neg, neg_lt_neg_iff]"
}
] |
[
1155,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1154,
1
] |
Mathlib/Data/List/Basic.lean
|
List.nth_take_of_succ
|
[] |
[
2031,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2030,
1
] |
Mathlib/ModelTheory/Definability.lean
|
Set.Definable.image_comp
|
[
{
"state_after": "case intro\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝ : Fintype α\n⊢ Definable A L ((fun g => g ∘ f) '' s)",
"state_before": "M : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\n⊢ Definable A L ((fun g => g ∘ f) '' s)",
"tactic": "cases nonempty_fintype α"
},
{
"state_after": "case intro.intro\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\n⊢ Definable A L ((fun g => g ∘ f) '' s)",
"state_before": "case intro\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝ : Fintype α\n⊢ Definable A L ((fun g => g ∘ f) '' s)",
"tactic": "cases nonempty_fintype β"
},
{
"state_after": "case intro.intro\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\n⊢ Definable A L ((fun g => g ∘ f) '' s)",
"state_before": "case intro.intro\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\n⊢ Definable A L ((fun g => g ∘ f) '' s)",
"tactic": "have h :=\n (((h.image_comp_equiv (Equiv.Set.sumCompl (range f))).image_comp_equiv\n (Equiv.sumCongr (_root_.Equiv.refl _)\n (Fintype.equivFin _).symm)).image_comp_sum_inl_fin\n _).preimage_comp\n (rangeSplitting f)"
},
{
"state_after": "case intro.intro\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : Definable A L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}\n⊢ Definable A L ((fun g => g ∘ f) '' s)",
"state_before": "case intro.intro\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\n⊢ Definable A L ((fun g => g ∘ f) '' s)",
"tactic": "have h' :\n A.Definable L { x : α → M | ∀ a, x a = x (rangeSplitting f (rangeFactorization f a)) } := by\n have h' : ∀ a,\n A.Definable L { x : α → M | x a = x (rangeSplitting f (rangeFactorization f a)) } := by\n refine' fun a => ⟨(var a).equal (var (rangeSplitting f (rangeFactorization f a))), ext _⟩\n simp\n refine' (congr rfl (ext _)).mp (definable_finset_biInter h' Finset.univ)\n simp"
},
{
"state_after": "case intro.intro\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : Definable A L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}\nx : α → M\n⊢ x ∈\n (fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))) ∩\n {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))} ↔\n x ∈ (fun g => g ∘ f) '' s",
"state_before": "case intro.intro\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : Definable A L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}\n⊢ Definable A L ((fun g => g ∘ f) '' s)",
"tactic": "refine' (congr rfl (ext fun x => _)).mp (h.inter h')"
},
{
"state_after": "case intro.intro\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : Definable A L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}\nx : α → M\n⊢ ((∃ a,\n a ∈ s ∧\n ((a ∘ ↑(Equiv.Set.sumCompl (range f))) ∘\n ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ∘\n Sum.inl =\n x ∘ rangeSplitting f) ∧\n ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))) ↔\n ∃ x_1, x_1 ∈ s ∧ x_1 ∘ f = x",
"state_before": "case intro.intro\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : Definable A L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}\nx : α → M\n⊢ x ∈\n (fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))) ∩\n {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))} ↔\n x ∈ (fun g => g ∘ f) '' s",
"tactic": "simp only [Equiv.coe_trans, mem_inter_iff, mem_preimage, mem_image, exists_exists_and_eq_and,\n mem_setOf_eq]"
},
{
"state_after": "case intro.intro.mp\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : Definable A L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}\nx : α → M\n⊢ ((∃ a,\n a ∈ s ∧\n ((a ∘ ↑(Equiv.Set.sumCompl (range f))) ∘\n ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ∘\n Sum.inl =\n x ∘ rangeSplitting f) ∧\n ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))) →\n ∃ x_1, x_1 ∈ s ∧ x_1 ∘ f = x\n\ncase intro.intro.mpr\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : Definable A L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}\nx : α → M\n⊢ (∃ x_1, x_1 ∈ s ∧ x_1 ∘ f = x) →\n (∃ a,\n a ∈ s ∧\n ((a ∘ ↑(Equiv.Set.sumCompl (range f))) ∘\n ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ∘\n Sum.inl =\n x ∘ rangeSplitting f) ∧\n ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))",
"state_before": "case intro.intro\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : Definable A L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}\nx : α → M\n⊢ ((∃ a,\n a ∈ s ∧\n ((a ∘ ↑(Equiv.Set.sumCompl (range f))) ∘\n ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ∘\n Sum.inl =\n x ∘ rangeSplitting f) ∧\n ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))) ↔\n ∃ x_1, x_1 ∈ s ∧ x_1 ∘ f = x",
"tactic": "constructor"
},
{
"state_after": "M : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : ∀ (a : α), Definable A L {x | x a = x (rangeSplitting f (rangeFactorization f a))}\n⊢ Definable A L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}",
"state_before": "M : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\n⊢ Definable A L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}",
"tactic": "have h' : ∀ a,\n A.Definable L { x : α → M | x a = x (rangeSplitting f (rangeFactorization f a)) } := by\n refine' fun a => ⟨(var a).equal (var (rangeSplitting f (rangeFactorization f a))), ext _⟩\n simp"
},
{
"state_after": "M : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : ∀ (a : α), Definable A L {x | x a = x (rangeSplitting f (rangeFactorization f a))}\n⊢ ∀ (x : α → M),\n (x ∈ ⋂ (i : α) (_ : i ∈ Finset.univ), {x | x i = x (rangeSplitting f (rangeFactorization f i))}) ↔\n x ∈ {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}",
"state_before": "M : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : ∀ (a : α), Definable A L {x | x a = x (rangeSplitting f (rangeFactorization f a))}\n⊢ Definable A L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}",
"tactic": "refine' (congr rfl (ext _)).mp (definable_finset_biInter h' Finset.univ)"
},
{
"state_after": "no goals",
"state_before": "M : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : ∀ (a : α), Definable A L {x | x a = x (rangeSplitting f (rangeFactorization f a))}\n⊢ ∀ (x : α → M),\n (x ∈ ⋂ (i : α) (_ : i ∈ Finset.univ), {x | x i = x (rangeSplitting f (rangeFactorization f i))}) ↔\n x ∈ {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}",
"tactic": "simp"
},
{
"state_after": "M : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\na : α\n⊢ ∀ (x : α → M),\n x ∈ {x | x a = x (rangeSplitting f (rangeFactorization f a))} ↔\n x ∈ setOf (Formula.Realize (Term.equal (var a) (var (rangeSplitting f (rangeFactorization f a)))))",
"state_before": "M : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\n⊢ ∀ (a : α), Definable A L {x | x a = x (rangeSplitting f (rangeFactorization f a))}",
"tactic": "refine' fun a => ⟨(var a).equal (var (rangeSplitting f (rangeFactorization f a))), ext _⟩"
},
{
"state_after": "no goals",
"state_before": "M : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\na : α\n⊢ ∀ (x : α → M),\n x ∈ {x | x a = x (rangeSplitting f (rangeFactorization f a))} ↔\n x ∈ setOf (Formula.Realize (Term.equal (var a) (var (rangeSplitting f (rangeFactorization f a)))))",
"tactic": "simp"
},
{
"state_after": "case intro.intro.mp.intro.intro.intro\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : Definable A L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}\nx : α → M\nhx : ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))\ny : β → M\nys : y ∈ s\nhy :\n ((y ∘ ↑(Equiv.Set.sumCompl (range f))) ∘\n ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ∘\n Sum.inl =\n x ∘ rangeSplitting f\n⊢ ∃ x_1, x_1 ∈ s ∧ x_1 ∘ f = x",
"state_before": "case intro.intro.mp\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : Definable A L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}\nx : α → M\n⊢ ((∃ a,\n a ∈ s ∧\n ((a ∘ ↑(Equiv.Set.sumCompl (range f))) ∘\n ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ∘\n Sum.inl =\n x ∘ rangeSplitting f) ∧\n ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))) →\n ∃ x_1, x_1 ∈ s ∧ x_1 ∘ f = x",
"tactic": "rintro ⟨⟨y, ys, hy⟩, hx⟩"
},
{
"state_after": "case intro.intro.mp.intro.intro.intro\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : Definable A L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}\nx : α → M\nhx : ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))\ny : β → M\nys : y ∈ s\nhy :\n ((y ∘ ↑(Equiv.Set.sumCompl (range f))) ∘\n ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ∘\n Sum.inl =\n x ∘ rangeSplitting f\n⊢ y ∘ f = x",
"state_before": "case intro.intro.mp.intro.intro.intro\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : Definable A L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}\nx : α → M\nhx : ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))\ny : β → M\nys : y ∈ s\nhy :\n ((y ∘ ↑(Equiv.Set.sumCompl (range f))) ∘\n ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ∘\n Sum.inl =\n x ∘ rangeSplitting f\n⊢ ∃ x_1, x_1 ∈ s ∧ x_1 ∘ f = x",
"tactic": "refine' ⟨y, ys, _⟩"
},
{
"state_after": "case intro.intro.mp.intro.intro.intro.h\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : Definable A L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}\nx : α → M\nhx : ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))\ny : β → M\nys : y ∈ s\nhy :\n ((y ∘ ↑(Equiv.Set.sumCompl (range f))) ∘\n ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ∘\n Sum.inl =\n x ∘ rangeSplitting f\na : α\n⊢ (y ∘ f) a = x a",
"state_before": "case intro.intro.mp.intro.intro.intro\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : Definable A L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}\nx : α → M\nhx : ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))\ny : β → M\nys : y ∈ s\nhy :\n ((y ∘ ↑(Equiv.Set.sumCompl (range f))) ∘\n ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ∘\n Sum.inl =\n x ∘ rangeSplitting f\n⊢ y ∘ f = x",
"tactic": "ext a"
},
{
"state_after": "case intro.intro.mp.intro.intro.intro.h\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : Definable A L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}\nx : α → M\nhx : ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))\ny : β → M\nys : y ∈ s\nhy :\n ((y ∘ ↑(Equiv.Set.sumCompl (range f))) ∘\n ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ∘\n Sum.inl =\n x ∘ rangeSplitting f\na : α\n⊢ (y ∘ f) a =\n (((y ∘ ↑(Equiv.Set.sumCompl (range f))) ∘\n ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ∘\n Sum.inl)\n (rangeFactorization f a)",
"state_before": "case intro.intro.mp.intro.intro.intro.h\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : Definable A L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}\nx : α → M\nhx : ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))\ny : β → M\nys : y ∈ s\nhy :\n ((y ∘ ↑(Equiv.Set.sumCompl (range f))) ∘\n ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ∘\n Sum.inl =\n x ∘ rangeSplitting f\na : α\n⊢ (y ∘ f) a = x a",
"tactic": "rw [hx a, ← Function.comp_apply (f := x), ← hy]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.mp.intro.intro.intro.h\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : Definable A L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}\nx : α → M\nhx : ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))\ny : β → M\nys : y ∈ s\nhy :\n ((y ∘ ↑(Equiv.Set.sumCompl (range f))) ∘\n ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ∘\n Sum.inl =\n x ∘ rangeSplitting f\na : α\n⊢ (y ∘ f) a =\n (((y ∘ ↑(Equiv.Set.sumCompl (range f))) ∘\n ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ∘\n Sum.inl)\n (rangeFactorization f a)",
"tactic": "simp"
},
{
"state_after": "case intro.intro.mpr.intro.intro\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : Definable A L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}\ny : β → M\nys : y ∈ s\n⊢ (∃ a,\n a ∈ s ∧\n ((a ∘ ↑(Equiv.Set.sumCompl (range f))) ∘\n ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ∘\n Sum.inl =\n (y ∘ f) ∘ rangeSplitting f) ∧\n ∀ (a : α), (y ∘ f) a = (y ∘ f) (rangeSplitting f (rangeFactorization f a))",
"state_before": "case intro.intro.mpr\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : Definable A L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}\nx : α → M\n⊢ (∃ x_1, x_1 ∈ s ∧ x_1 ∘ f = x) →\n (∃ a,\n a ∈ s ∧\n ((a ∘ ↑(Equiv.Set.sumCompl (range f))) ∘\n ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ∘\n Sum.inl =\n x ∘ rangeSplitting f) ∧\n ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))",
"tactic": "rintro ⟨y, ys, rfl⟩"
},
{
"state_after": "case intro.intro.mpr.intro.intro.refine'_1\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : Definable A L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}\ny : β → M\nys : y ∈ s\n⊢ ((y ∘ ↑(Equiv.Set.sumCompl (range f))) ∘\n ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ∘\n Sum.inl =\n (y ∘ f) ∘ rangeSplitting f\n\ncase intro.intro.mpr.intro.intro.refine'_2\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : Definable A L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}\ny : β → M\nys : y ∈ s\na : α\n⊢ (y ∘ f) a = (y ∘ f) (rangeSplitting f (rangeFactorization f a))",
"state_before": "case intro.intro.mpr.intro.intro\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : Definable A L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}\ny : β → M\nys : y ∈ s\n⊢ (∃ a,\n a ∈ s ∧\n ((a ∘ ↑(Equiv.Set.sumCompl (range f))) ∘\n ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ∘\n Sum.inl =\n (y ∘ f) ∘ rangeSplitting f) ∧\n ∀ (a : α), (y ∘ f) a = (y ∘ f) (rangeSplitting f (rangeFactorization f a))",
"tactic": "refine' ⟨⟨y, ys, _⟩, fun a => _⟩"
},
{
"state_after": "case intro.intro.mpr.intro.intro.refine'_1.h\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : Definable A L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}\ny : β → M\nys : y ∈ s\nx✝ : ↑(range f)\n⊢ (((y ∘ ↑(Equiv.Set.sumCompl (range f))) ∘\n ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ∘\n Sum.inl)\n x✝ =\n ((y ∘ f) ∘ rangeSplitting f) x✝",
"state_before": "case intro.intro.mpr.intro.intro.refine'_1\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : Definable A L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}\ny : β → M\nys : y ∈ s\n⊢ ((y ∘ ↑(Equiv.Set.sumCompl (range f))) ∘\n ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ∘\n Sum.inl =\n (y ∘ f) ∘ rangeSplitting f",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.mpr.intro.intro.refine'_1.h\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : Definable A L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}\ny : β → M\nys : y ∈ s\nx✝ : ↑(range f)\n⊢ (((y ∘ ↑(Equiv.Set.sumCompl (range f))) ∘\n ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ∘\n Sum.inl)\n x✝ =\n ((y ∘ f) ∘ rangeSplitting f) x✝",
"tactic": "simp [Set.apply_rangeSplitting f]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.mpr.intro.intro.refine'_2\nM : Type w\nA : Set M\nL : Language\ninst✝² : Structure L M\nα : Type u₁\nβ : Type u_1\nB : Set M\ns✝ : Set (α → M)\ns : Set (β → M)\nh✝ : Definable A L s\nf : α → β\ninst✝¹ : Finite α\ninst✝ : Finite β\nval✝¹ : Fintype α\nval✝ : Fintype β\nh :\n Definable A L\n ((fun g => g ∘ rangeSplitting f) ⁻¹'\n ((fun g => g ∘ Sum.inl) ''\n ((fun g => g ∘ ↑(Equiv.sumCongr (Equiv.refl ↑(range f)) (Fintype.equivFin ↑(range fᶜ)).symm)) ''\n ((fun g => g ∘ ↑(Equiv.Set.sumCompl (range f))) '' s))))\nh' : Definable A L {x | ∀ (a : α), x a = x (rangeSplitting f (rangeFactorization f a))}\ny : β → M\nys : y ∈ s\na : α\n⊢ (y ∘ f) a = (y ∘ f) (rangeSplitting f (rangeFactorization f a))",
"tactic": "rw [Function.comp_apply, Function.comp_apply, apply_rangeSplitting f,\n rangeFactorization_coe]"
}
] |
[
240,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
207,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
SeminormedGroup.uniformCauchySeqOnFilter_iff_tendstoUniformlyOnFilter_one
|
[
{
"state_after": "case refine'_1\n𝓕 : Type ?u.504291\n𝕜 : Type ?u.504294\nα : Type ?u.504297\nι : Type u_1\nκ : Type u_2\nE : Type ?u.504306\nF : Type ?u.504309\nG : Type u_3\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nf : ι → κ → G\nl : Filter ι\nl' : Filter κ\nhf : UniformCauchySeqOnFilter f l l'\nu : Set (G × G)\nhu : u ∈ 𝓤 G\n⊢ ∀ᶠ (n : (ι × ι) × κ) in (l ×ˢ l) ×ˢ l', (OfNat.ofNat 1 n.snd, (fun n z => f n.fst z / f n.snd z) n.fst n.snd) ∈ u\n\ncase refine'_2\n𝓕 : Type ?u.504291\n𝕜 : Type ?u.504294\nα : Type ?u.504297\nι : Type u_1\nκ : Type u_2\nE : Type ?u.504306\nF : Type ?u.504309\nG : Type u_3\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nf : ι → κ → G\nl : Filter ι\nl' : Filter κ\nhf : TendstoUniformlyOnFilter (fun n z => f n.fst z / f n.snd z) 1 (l ×ˢ l) l'\nu : Set (G × G)\nhu : u ∈ 𝓤 G\n⊢ ∀ᶠ (m : (ι × ι) × κ) in (l ×ˢ l) ×ˢ l', (f m.fst.fst m.snd, f m.fst.snd m.snd) ∈ u",
"state_before": "𝓕 : Type ?u.504291\n𝕜 : Type ?u.504294\nα : Type ?u.504297\nι : Type u_1\nκ : Type u_2\nE : Type ?u.504306\nF : Type ?u.504309\nG : Type u_3\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nf : ι → κ → G\nl : Filter ι\nl' : Filter κ\n⊢ UniformCauchySeqOnFilter f l l' ↔ TendstoUniformlyOnFilter (fun n z => f n.fst z / f n.snd z) 1 (l ×ˢ l) l'",
"tactic": "refine' ⟨fun hf u hu => _, fun hf u hu => _⟩"
},
{
"state_after": "case refine'_1.intro.intro\n𝓕 : Type ?u.504291\n𝕜 : Type ?u.504294\nα : Type ?u.504297\nι : Type u_1\nκ : Type u_2\nE : Type ?u.504306\nF : Type ?u.504309\nG : Type u_3\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nf : ι → κ → G\nl : Filter ι\nl' : Filter κ\nhf : UniformCauchySeqOnFilter f l l'\nu : Set (G × G)\nhu : u ∈ 𝓤 G\nε : ℝ\nhε : 0 < ε\nH : ∀ (a b : G), (a, b) ∈ {p | dist p.fst p.snd < ε} → (a, b) ∈ u\n⊢ ∀ᶠ (n : (ι × ι) × κ) in (l ×ˢ l) ×ˢ l', (OfNat.ofNat 1 n.snd, (fun n z => f n.fst z / f n.snd z) n.fst n.snd) ∈ u",
"state_before": "case refine'_1\n𝓕 : Type ?u.504291\n𝕜 : Type ?u.504294\nα : Type ?u.504297\nι : Type u_1\nκ : Type u_2\nE : Type ?u.504306\nF : Type ?u.504309\nG : Type u_3\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nf : ι → κ → G\nl : Filter ι\nl' : Filter κ\nhf : UniformCauchySeqOnFilter f l l'\nu : Set (G × G)\nhu : u ∈ 𝓤 G\n⊢ ∀ᶠ (n : (ι × ι) × κ) in (l ×ˢ l) ×ˢ l', (OfNat.ofNat 1 n.snd, (fun n z => f n.fst z / f n.snd z) n.fst n.snd) ∈ u",
"tactic": "obtain ⟨ε, hε, H⟩ := uniformity_basis_dist.mem_uniformity_iff.mp hu"
},
{
"state_after": "case refine'_1.intro.intro\n𝓕 : Type ?u.504291\n𝕜 : Type ?u.504294\nα : Type ?u.504297\nι : Type u_1\nκ : Type u_2\nE : Type ?u.504306\nF : Type ?u.504309\nG : Type u_3\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nf : ι → κ → G\nl : Filter ι\nl' : Filter κ\nhf : UniformCauchySeqOnFilter f l l'\nu : Set (G × G)\nhu : u ∈ 𝓤 G\nε : ℝ\nhε : 0 < ε\nH : ∀ (a b : G), (a, b) ∈ {p | dist p.fst p.snd < ε} → (a, b) ∈ u\nx : (ι × ι) × κ\nhx : (f x.fst.fst x.snd, f x.fst.snd x.snd) ∈ {p | dist p.fst p.snd < ε}\n⊢ (1, f x.fst.fst x.snd / f x.fst.snd x.snd) ∈ {p | dist p.fst p.snd < ε}",
"state_before": "case refine'_1.intro.intro\n𝓕 : Type ?u.504291\n𝕜 : Type ?u.504294\nα : Type ?u.504297\nι : Type u_1\nκ : Type u_2\nE : Type ?u.504306\nF : Type ?u.504309\nG : Type u_3\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nf : ι → κ → G\nl : Filter ι\nl' : Filter κ\nhf : UniformCauchySeqOnFilter f l l'\nu : Set (G × G)\nhu : u ∈ 𝓤 G\nε : ℝ\nhε : 0 < ε\nH : ∀ (a b : G), (a, b) ∈ {p | dist p.fst p.snd < ε} → (a, b) ∈ u\n⊢ ∀ᶠ (n : (ι × ι) × κ) in (l ×ˢ l) ×ˢ l', (OfNat.ofNat 1 n.snd, (fun n z => f n.fst z / f n.snd z) n.fst n.snd) ∈ u",
"tactic": "refine'\n (hf { p : G × G | dist p.fst p.snd < ε } <| dist_mem_uniformity hε).mono fun x hx =>\n H 1 (f x.fst.fst x.snd / f x.fst.snd x.snd) _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.intro\n𝓕 : Type ?u.504291\n𝕜 : Type ?u.504294\nα : Type ?u.504297\nι : Type u_1\nκ : Type u_2\nE : Type ?u.504306\nF : Type ?u.504309\nG : Type u_3\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nf : ι → κ → G\nl : Filter ι\nl' : Filter κ\nhf : UniformCauchySeqOnFilter f l l'\nu : Set (G × G)\nhu : u ∈ 𝓤 G\nε : ℝ\nhε : 0 < ε\nH : ∀ (a b : G), (a, b) ∈ {p | dist p.fst p.snd < ε} → (a, b) ∈ u\nx : (ι × ι) × κ\nhx : (f x.fst.fst x.snd, f x.fst.snd x.snd) ∈ {p | dist p.fst p.snd < ε}\n⊢ (1, f x.fst.fst x.snd / f x.fst.snd x.snd) ∈ {p | dist p.fst p.snd < ε}",
"tactic": "simpa [dist_eq_norm_div, norm_div_rev] using hx"
},
{
"state_after": "case refine'_2.intro.intro\n𝓕 : Type ?u.504291\n𝕜 : Type ?u.504294\nα : Type ?u.504297\nι : Type u_1\nκ : Type u_2\nE : Type ?u.504306\nF : Type ?u.504309\nG : Type u_3\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nf : ι → κ → G\nl : Filter ι\nl' : Filter κ\nhf : TendstoUniformlyOnFilter (fun n z => f n.fst z / f n.snd z) 1 (l ×ˢ l) l'\nu : Set (G × G)\nhu : u ∈ 𝓤 G\nε : ℝ\nhε : 0 < ε\nH : ∀ (a b : G), (a, b) ∈ {p | dist p.fst p.snd < ε} → (a, b) ∈ u\n⊢ ∀ᶠ (m : (ι × ι) × κ) in (l ×ˢ l) ×ˢ l', (f m.fst.fst m.snd, f m.fst.snd m.snd) ∈ u",
"state_before": "case refine'_2\n𝓕 : Type ?u.504291\n𝕜 : Type ?u.504294\nα : Type ?u.504297\nι : Type u_1\nκ : Type u_2\nE : Type ?u.504306\nF : Type ?u.504309\nG : Type u_3\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nf : ι → κ → G\nl : Filter ι\nl' : Filter κ\nhf : TendstoUniformlyOnFilter (fun n z => f n.fst z / f n.snd z) 1 (l ×ˢ l) l'\nu : Set (G × G)\nhu : u ∈ 𝓤 G\n⊢ ∀ᶠ (m : (ι × ι) × κ) in (l ×ˢ l) ×ˢ l', (f m.fst.fst m.snd, f m.fst.snd m.snd) ∈ u",
"tactic": "obtain ⟨ε, hε, H⟩ := uniformity_basis_dist.mem_uniformity_iff.mp hu"
},
{
"state_after": "case refine'_2.intro.intro\n𝓕 : Type ?u.504291\n𝕜 : Type ?u.504294\nα : Type ?u.504297\nι : Type u_1\nκ : Type u_2\nE : Type ?u.504306\nF : Type ?u.504309\nG : Type u_3\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nf : ι → κ → G\nl : Filter ι\nl' : Filter κ\nhf : TendstoUniformlyOnFilter (fun n z => f n.fst z / f n.snd z) 1 (l ×ˢ l) l'\nu : Set (G × G)\nhu : u ∈ 𝓤 G\nε : ℝ\nhε : 0 < ε\nH : ∀ (a b : G), (a, b) ∈ {p | dist p.fst p.snd < ε} → (a, b) ∈ u\nx : (ι × ι) × κ\nhx : (OfNat.ofNat 1 x.snd, (fun n z => f n.fst z / f n.snd z) x.fst x.snd) ∈ {p | dist p.fst p.snd < ε}\n⊢ (f x.fst.fst x.snd, f x.fst.snd x.snd) ∈ {p | dist p.fst p.snd < ε}",
"state_before": "case refine'_2.intro.intro\n𝓕 : Type ?u.504291\n𝕜 : Type ?u.504294\nα : Type ?u.504297\nι : Type u_1\nκ : Type u_2\nE : Type ?u.504306\nF : Type ?u.504309\nG : Type u_3\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nf : ι → κ → G\nl : Filter ι\nl' : Filter κ\nhf : TendstoUniformlyOnFilter (fun n z => f n.fst z / f n.snd z) 1 (l ×ˢ l) l'\nu : Set (G × G)\nhu : u ∈ 𝓤 G\nε : ℝ\nhε : 0 < ε\nH : ∀ (a b : G), (a, b) ∈ {p | dist p.fst p.snd < ε} → (a, b) ∈ u\n⊢ ∀ᶠ (m : (ι × ι) × κ) in (l ×ˢ l) ×ˢ l', (f m.fst.fst m.snd, f m.fst.snd m.snd) ∈ u",
"tactic": "refine'\n (hf { p : G × G | dist p.fst p.snd < ε } <| dist_mem_uniformity hε).mono fun x hx =>\n H (f x.fst.fst x.snd) (f x.fst.snd x.snd) _"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro\n𝓕 : Type ?u.504291\n𝕜 : Type ?u.504294\nα : Type ?u.504297\nι : Type u_1\nκ : Type u_2\nE : Type ?u.504306\nF : Type ?u.504309\nG : Type u_3\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nf : ι → κ → G\nl : Filter ι\nl' : Filter κ\nhf : TendstoUniformlyOnFilter (fun n z => f n.fst z / f n.snd z) 1 (l ×ˢ l) l'\nu : Set (G × G)\nhu : u ∈ 𝓤 G\nε : ℝ\nhε : 0 < ε\nH : ∀ (a b : G), (a, b) ∈ {p | dist p.fst p.snd < ε} → (a, b) ∈ u\nx : (ι × ι) × κ\nhx : (OfNat.ofNat 1 x.snd, (fun n z => f n.fst z / f n.snd z) x.fst x.snd) ∈ {p | dist p.fst p.snd < ε}\n⊢ (f x.fst.fst x.snd, f x.fst.snd x.snd) ∈ {p | dist p.fst p.snd < ε}",
"tactic": "simpa [dist_eq_norm_div, norm_div_rev] using hx"
}
] |
[
1313,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1299,
1
] |
Mathlib/Algebra/QuadraticDiscriminant.lean
|
discrim_le_zero
|
[
{
"state_after": "K : Type u_1\ninst✝ : LinearOrderedField K\na b c : K\nh : ∀ (x : K), 0 ≤ a * x * x + b * x + c\n⊢ b * b - 4 * a * c ≤ 0",
"state_before": "K : Type u_1\ninst✝ : LinearOrderedField K\na b c : K\nh : ∀ (x : K), 0 ≤ a * x * x + b * x + c\n⊢ discrim a b c ≤ 0",
"tactic": "rw [discrim, sq]"
},
{
"state_after": "case inl\nK : Type u_1\ninst✝ : LinearOrderedField K\na b c : K\nh : ∀ (x : K), 0 ≤ a * x * x + b * x + c\nha : a < 0\n⊢ b * b - 4 * a * c ≤ 0\n\ncase inr.inl\nK : Type u_1\ninst✝ : LinearOrderedField K\nb c : K\nh : ∀ (x : K), 0 ≤ 0 * x * x + b * x + c\n⊢ b * b - 4 * 0 * c ≤ 0\n\ncase inr.inr\nK : Type u_1\ninst✝ : LinearOrderedField K\na b c : K\nh : ∀ (x : K), 0 ≤ a * x * x + b * x + c\nha : 0 < a\n⊢ b * b - 4 * a * c ≤ 0",
"state_before": "K : Type u_1\ninst✝ : LinearOrderedField K\na b c : K\nh : ∀ (x : K), 0 ≤ a * x * x + b * x + c\n⊢ b * b - 4 * a * c ≤ 0",
"tactic": "obtain ha | rfl | ha : a < 0 ∨ a = 0 ∨ 0 < a := lt_trichotomy a 0"
},
{
"state_after": "case inl\nK : Type u_1\ninst✝ : LinearOrderedField K\na b c : K\nh : ∀ (x : K), 0 ≤ a * x * x + b * x + c\nha : a < 0\nthis : Tendsto (fun x => (a * x + b) * x + c) atTop atBot\n⊢ b * b - 4 * a * c ≤ 0",
"state_before": "case inl\nK : Type u_1\ninst✝ : LinearOrderedField K\na b c : K\nh : ∀ (x : K), 0 ≤ a * x * x + b * x + c\nha : a < 0\n⊢ b * b - 4 * a * c ≤ 0",
"tactic": "have : Tendsto (fun x => (a * x + b) * x + c) atTop atBot :=\n tendsto_atBot_add_const_right _ c\n ((tendsto_atBot_add_const_right _ b (tendsto_id.neg_const_mul_atTop ha)).atBot_mul_atTop\n tendsto_id)"
},
{
"state_after": "case inl.intro\nK : Type u_1\ninst✝ : LinearOrderedField K\na b c : K\nh : ∀ (x : K), 0 ≤ a * x * x + b * x + c\nha : a < 0\nthis : Tendsto (fun x => (a * x + b) * x + c) atTop atBot\nx : K\nhx : (a * x + b) * x + c < 0\n⊢ b * b - 4 * a * c ≤ 0",
"state_before": "case inl\nK : Type u_1\ninst✝ : LinearOrderedField K\na b c : K\nh : ∀ (x : K), 0 ≤ a * x * x + b * x + c\nha : a < 0\nthis : Tendsto (fun x => (a * x + b) * x + c) atTop atBot\n⊢ b * b - 4 * a * c ≤ 0",
"tactic": "rcases(this.eventually (eventually_lt_atBot 0)).exists with ⟨x, hx⟩"
},
{
"state_after": "no goals",
"state_before": "case inl.intro\nK : Type u_1\ninst✝ : LinearOrderedField K\na b c : K\nh : ∀ (x : K), 0 ≤ a * x * x + b * x + c\nha : a < 0\nthis : Tendsto (fun x => (a * x + b) * x + c) atTop atBot\nx : K\nhx : (a * x + b) * x + c < 0\n⊢ b * b - 4 * a * c ≤ 0",
"tactic": "exact False.elim ((h x).not_lt <| by rwa [← add_mul])"
},
{
"state_after": "no goals",
"state_before": "K : Type u_1\ninst✝ : LinearOrderedField K\na b c : K\nh : ∀ (x : K), 0 ≤ a * x * x + b * x + c\nha : a < 0\nthis : Tendsto (fun x => (a * x + b) * x + c) atTop atBot\nx : K\nhx : (a * x + b) * x + c < 0\n⊢ a * x * x + b * x + c < 0",
"tactic": "rwa [← add_mul]"
},
{
"state_after": "case inr.inl.inl\nK : Type u_1\ninst✝ : LinearOrderedField K\nc : K\nh : ∀ (x : K), 0 ≤ 0 * x * x + 0 * x + c\n⊢ 0 * 0 - 4 * 0 * c ≤ 0\n\ncase inr.inl.inr\nK : Type u_1\ninst✝ : LinearOrderedField K\nb c : K\nh : ∀ (x : K), 0 ≤ 0 * x * x + b * x + c\nhb : b ≠ 0\n⊢ b * b - 4 * 0 * c ≤ 0",
"state_before": "case inr.inl\nK : Type u_1\ninst✝ : LinearOrderedField K\nb c : K\nh : ∀ (x : K), 0 ≤ 0 * x * x + b * x + c\n⊢ b * b - 4 * 0 * c ≤ 0",
"tactic": "rcases eq_or_ne b 0 with (rfl | hb)"
},
{
"state_after": "no goals",
"state_before": "case inr.inl.inl\nK : Type u_1\ninst✝ : LinearOrderedField K\nc : K\nh : ∀ (x : K), 0 ≤ 0 * x * x + 0 * x + c\n⊢ 0 * 0 - 4 * 0 * c ≤ 0",
"tactic": "simp"
},
{
"state_after": "case inr.inl.inr\nK : Type u_1\ninst✝ : LinearOrderedField K\nb c : K\nh : ∀ (x : K), 0 ≤ 0 * x * x + b * x + c\nhb : b ≠ 0\nthis : 0 ≤ 0 * ((-c - 1) / b) * ((-c - 1) / b) + b * ((-c - 1) / b) + c\n⊢ b * b - 4 * 0 * c ≤ 0",
"state_before": "case inr.inl.inr\nK : Type u_1\ninst✝ : LinearOrderedField K\nb c : K\nh : ∀ (x : K), 0 ≤ 0 * x * x + b * x + c\nhb : b ≠ 0\n⊢ b * b - 4 * 0 * c ≤ 0",
"tactic": "have := h ((-c - 1) / b)"
},
{
"state_after": "case inr.inl.inr\nK : Type u_1\ninst✝ : LinearOrderedField K\nb c : K\nh : ∀ (x : K), 0 ≤ 0 * x * x + b * x + c\nhb : b ≠ 0\nthis : 0 ≤ 0 * ((-c - 1) / b) * ((-c - 1) / b) + (-c - 1) + c\n⊢ b * b - 4 * 0 * c ≤ 0",
"state_before": "case inr.inl.inr\nK : Type u_1\ninst✝ : LinearOrderedField K\nb c : K\nh : ∀ (x : K), 0 ≤ 0 * x * x + b * x + c\nhb : b ≠ 0\nthis : 0 ≤ 0 * ((-c - 1) / b) * ((-c - 1) / b) + b * ((-c - 1) / b) + c\n⊢ b * b - 4 * 0 * c ≤ 0",
"tactic": "rw [mul_div_cancel' _ hb] at this"
},
{
"state_after": "no goals",
"state_before": "case inr.inl.inr\nK : Type u_1\ninst✝ : LinearOrderedField K\nb c : K\nh : ∀ (x : K), 0 ≤ 0 * x * x + b * x + c\nhb : b ≠ 0\nthis : 0 ≤ 0 * ((-c - 1) / b) * ((-c - 1) / b) + (-c - 1) + c\n⊢ b * b - 4 * 0 * c ≤ 0",
"tactic": "linarith"
},
{
"state_after": "case inr.inr\nK : Type u_1\ninst✝ : LinearOrderedField K\na b c : K\nh : ∀ (x : K), 0 ≤ a * x * x + b * x + c\nha : 0 < a\nha' : 0 ≤ 4 * a\n⊢ b * b - 4 * a * c ≤ 0",
"state_before": "case inr.inr\nK : Type u_1\ninst✝ : LinearOrderedField K\na b c : K\nh : ∀ (x : K), 0 ≤ a * x * x + b * x + c\nha : 0 < a\n⊢ b * b - 4 * a * c ≤ 0",
"tactic": "have ha' : 0 ≤ 4 * a := mul_nonneg zero_le_four ha.le"
},
{
"state_after": "case h.e'_3\nK : Type u_1\ninst✝ : LinearOrderedField K\na b c : K\nh : ∀ (x : K), 0 ≤ a * x * x + b * x + c\nha : 0 < a\nha' : 0 ≤ 4 * a\n⊢ b * b - 4 * a * c = -(4 * a * (a * (-b / (2 * a)) * (-b / (2 * a)) + b * (-b / (2 * a)) + c))",
"state_before": "case inr.inr\nK : Type u_1\ninst✝ : LinearOrderedField K\na b c : K\nh : ∀ (x : K), 0 ≤ a * x * x + b * x + c\nha : 0 < a\nha' : 0 ≤ 4 * a\n⊢ b * b - 4 * a * c ≤ 0",
"tactic": "convert neg_nonpos.2 (mul_nonneg ha' (h (-b / (2 * a)))) using 1"
},
{
"state_after": "case h.e'_3\nK : Type u_1\ninst✝ : LinearOrderedField K\na b c : K\nh : ∀ (x : K), 0 ≤ a * x * x + b * x + c\nha : 0 < a\nha' : 0 ≤ 4 * a\n⊢ (b * b - 4 * a * c) * (2 * a * (2 * a) * (2 * a)) =\n -(4 * a * (a * b * b * (2 * a) + -(b * b * (2 * a * (2 * a))) + c * (2 * a * (2 * a) * (2 * a))))",
"state_before": "case h.e'_3\nK : Type u_1\ninst✝ : LinearOrderedField K\na b c : K\nh : ∀ (x : K), 0 ≤ a * x * x + b * x + c\nha : 0 < a\nha' : 0 ≤ 4 * a\n⊢ b * b - 4 * a * c = -(4 * a * (a * (-b / (2 * a)) * (-b / (2 * a)) + b * (-b / (2 * a)) + c))",
"tactic": "field_simp [ha.ne']"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nK : Type u_1\ninst✝ : LinearOrderedField K\na b c : K\nh : ∀ (x : K), 0 ≤ a * x * x + b * x + c\nha : 0 < a\nha' : 0 ≤ 4 * a\n⊢ (b * b - 4 * a * c) * (2 * a * (2 * a) * (2 * a)) =\n -(4 * a * (a * b * b * (2 * a) + -(b * b * (2 * a * (2 * a))) + c * (2 * a * (2 * a) * (2 * a))))",
"tactic": "ring"
}
] |
[
141,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
contDiffOn_succ_iff_deriv_of_open
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.3014886\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf₂ : 𝕜 → F\ns₂ : Set 𝕜\nn : ℕ\nhs : IsOpen s₂\n⊢ DifferentiableOn 𝕜 f₂ s₂ ∧ ContDiffOn 𝕜 (↑n) (derivWithin f₂ s₂) s₂ ↔\n DifferentiableOn 𝕜 f₂ s₂ ∧ ContDiffOn 𝕜 (↑n) (deriv f₂) s₂",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.3014886\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf₂ : 𝕜 → F\ns₂ : Set 𝕜\nn : ℕ\nhs : IsOpen s₂\n⊢ ContDiffOn 𝕜 (↑(n + 1)) f₂ s₂ ↔ DifferentiableOn 𝕜 f₂ s₂ ∧ ContDiffOn 𝕜 (↑n) (deriv f₂) s₂",
"tactic": "rw [contDiffOn_succ_iff_derivWithin hs.uniqueDiffOn]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝⁹ : NormedAddCommGroup D\ninst✝⁸ : NormedSpace 𝕜 D\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type ?u.3014886\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nf₂ : 𝕜 → F\ns₂ : Set 𝕜\nn : ℕ\nhs : IsOpen s₂\n⊢ DifferentiableOn 𝕜 f₂ s₂ ∧ ContDiffOn 𝕜 (↑n) (derivWithin f₂ s₂) s₂ ↔\n DifferentiableOn 𝕜 f₂ s₂ ∧ ContDiffOn 𝕜 (↑n) (deriv f₂) s₂",
"tactic": "exact Iff.rfl.and (contDiffOn_congr fun _ => derivWithin_of_open hs)"
}
] |
[
2094,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2091,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.normalClosure_subset_iff
|
[] |
[
2494,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2493,
1
] |
Mathlib/Topology/Order.lean
|
TopologicalSpace.nhds_generateFrom
|
[
{
"state_after": "α : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\n⊢ 𝓝 a = ⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s",
"state_before": "α : Type u\ng : Set (Set α)\na : α\n⊢ 𝓝 a = ⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s",
"tactic": "letI := generateFrom g"
},
{
"state_after": "α : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ IsOpen s}), 𝓟 s) = ⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s",
"state_before": "α : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\n⊢ 𝓝 a = ⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s",
"tactic": "rw [nhds_def]"
},
{
"state_after": "α : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ ⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ IsOpen s}), 𝓟 s",
"state_before": "α : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ IsOpen s}), 𝓟 s) = ⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s",
"tactic": "refine le_antisymm (biInf_mono fun s ⟨as, sg⟩ => ⟨as, .basic _ sg⟩) ?_"
},
{
"state_after": "α : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns : Set α\nx✝ : s ∈ {s | a ∈ s ∧ IsOpen s}\nha : a ∈ s\nhs : IsOpen s\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 s",
"state_before": "α : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ ⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ IsOpen s}), 𝓟 s",
"tactic": "refine le_iInf₂ fun s ⟨ha, hs⟩ => ?_"
},
{
"state_after": "α : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns : Set α\nha : a ∈ s\nhs : IsOpen s\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 s",
"state_before": "α : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns : Set α\nx✝ : s ∈ {s | a ∈ s ∧ IsOpen s}\nha : a ∈ s\nhs : IsOpen s\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 s",
"tactic": "clear ‹s ∈ { s | a ∈ s ∧ IsOpen s }›"
},
{
"state_after": "case basic\nα : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns s✝ : Set α\na✝ : s✝ ∈ g\nha : a ∈ s✝\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 s✝\n\ncase univ\nα : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns : Set α\nha : a ∈ univ\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 univ\n\ncase inter\nα : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns s✝ t✝ : Set α\na✝¹ : GenerateOpen g s✝\na✝ : GenerateOpen g t✝\na_ih✝¹ : a ∈ s✝ → (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 s✝\na_ih✝ : a ∈ t✝ → (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 t✝\nha : a ∈ s✝ ∩ t✝\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 (s✝ ∩ t✝)\n\ncase sUnion\nα : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns : Set α\nS✝ : Set (Set α)\na✝ : ∀ (s : Set α), s ∈ S✝ → GenerateOpen g s\na_ih✝ : ∀ (s : Set α), s ∈ S✝ → a ∈ s → (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 s\nha : a ∈ ⋃₀ S✝\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 (⋃₀ S✝)",
"state_before": "α : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns : Set α\nha : a ∈ s\nhs : IsOpen s\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 s",
"tactic": "induction hs"
},
{
"state_after": "case univ\nα : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns : Set α\nha : a ∈ univ\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 univ\n\ncase inter\nα : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns s✝ t✝ : Set α\na✝¹ : GenerateOpen g s✝\na✝ : GenerateOpen g t✝\na_ih✝¹ : a ∈ s✝ → (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 s✝\na_ih✝ : a ∈ t✝ → (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 t✝\nha : a ∈ s✝ ∩ t✝\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 (s✝ ∩ t✝)\n\ncase sUnion\nα : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns : Set α\nS✝ : Set (Set α)\na✝ : ∀ (s : Set α), s ∈ S✝ → GenerateOpen g s\na_ih✝ : ∀ (s : Set α), s ∈ S✝ → a ∈ s → (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 s\nha : a ∈ ⋃₀ S✝\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 (⋃₀ S✝)",
"state_before": "case basic\nα : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns s✝ : Set α\na✝ : s✝ ∈ g\nha : a ∈ s✝\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 s✝\n\ncase univ\nα : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns : Set α\nha : a ∈ univ\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 univ\n\ncase inter\nα : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns s✝ t✝ : Set α\na✝¹ : GenerateOpen g s✝\na✝ : GenerateOpen g t✝\na_ih✝¹ : a ∈ s✝ → (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 s✝\na_ih✝ : a ∈ t✝ → (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 t✝\nha : a ∈ s✝ ∩ t✝\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 (s✝ ∩ t✝)\n\ncase sUnion\nα : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns : Set α\nS✝ : Set (Set α)\na✝ : ∀ (s : Set α), s ∈ S✝ → GenerateOpen g s\na_ih✝ : ∀ (s : Set α), s ∈ S✝ → a ∈ s → (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 s\nha : a ∈ ⋃₀ S✝\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 (⋃₀ S✝)",
"tactic": "case basic hs => exact iInf₂_le _ ⟨ha, hs⟩"
},
{
"state_after": "case inter\nα : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns s✝ t✝ : Set α\na✝¹ : GenerateOpen g s✝\na✝ : GenerateOpen g t✝\na_ih✝¹ : a ∈ s✝ → (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 s✝\na_ih✝ : a ∈ t✝ → (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 t✝\nha : a ∈ s✝ ∩ t✝\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 (s✝ ∩ t✝)\n\ncase sUnion\nα : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns : Set α\nS✝ : Set (Set α)\na✝ : ∀ (s : Set α), s ∈ S✝ → GenerateOpen g s\na_ih✝ : ∀ (s : Set α), s ∈ S✝ → a ∈ s → (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 s\nha : a ∈ ⋃₀ S✝\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 (⋃₀ S✝)",
"state_before": "case univ\nα : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns : Set α\nha : a ∈ univ\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 univ\n\ncase inter\nα : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns s✝ t✝ : Set α\na✝¹ : GenerateOpen g s✝\na✝ : GenerateOpen g t✝\na_ih✝¹ : a ∈ s✝ → (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 s✝\na_ih✝ : a ∈ t✝ → (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 t✝\nha : a ∈ s✝ ∩ t✝\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 (s✝ ∩ t✝)\n\ncase sUnion\nα : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns : Set α\nS✝ : Set (Set α)\na✝ : ∀ (s : Set α), s ∈ S✝ → GenerateOpen g s\na_ih✝ : ∀ (s : Set α), s ∈ S✝ → a ∈ s → (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 s\nha : a ∈ ⋃₀ S✝\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 (⋃₀ S✝)",
"tactic": "case univ => exact le_top.trans_eq principal_univ.symm"
},
{
"state_after": "case sUnion\nα : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns : Set α\nS✝ : Set (Set α)\na✝ : ∀ (s : Set α), s ∈ S✝ → GenerateOpen g s\na_ih✝ : ∀ (s : Set α), s ∈ S✝ → a ∈ s → (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 s\nha : a ∈ ⋃₀ S✝\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 (⋃₀ S✝)",
"state_before": "case inter\nα : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns s✝ t✝ : Set α\na✝¹ : GenerateOpen g s✝\na✝ : GenerateOpen g t✝\na_ih✝¹ : a ∈ s✝ → (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 s✝\na_ih✝ : a ∈ t✝ → (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 t✝\nha : a ∈ s✝ ∩ t✝\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 (s✝ ∩ t✝)\n\ncase sUnion\nα : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns : Set α\nS✝ : Set (Set α)\na✝ : ∀ (s : Set α), s ∈ S✝ → GenerateOpen g s\na_ih✝ : ∀ (s : Set α), s ∈ S✝ → a ∈ s → (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 s\nha : a ∈ ⋃₀ S✝\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 (⋃₀ S✝)",
"tactic": "case inter hs ht => exact (le_inf (hs ha.1) (ht ha.2)).trans_eq inf_principal"
},
{
"state_after": "no goals",
"state_before": "case sUnion\nα : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns : Set α\nS✝ : Set (Set α)\na✝ : ∀ (s : Set α), s ∈ S✝ → GenerateOpen g s\na_ih✝ : ∀ (s : Set α), s ∈ S✝ → a ∈ s → (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 s\nha : a ∈ ⋃₀ S✝\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 (⋃₀ S✝)",
"tactic": "case sUnion _S hS =>\n let ⟨t, htS, hat⟩ := ha\n exact (hS t htS hat).trans (principal_mono.2 <| subset_sUnion_of_mem htS)"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns s✝ : Set α\nhs : s✝ ∈ g\nha : a ∈ s✝\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 s✝",
"tactic": "exact iInf₂_le _ ⟨ha, hs⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns : Set α\nha : a ∈ univ\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 univ",
"tactic": "exact le_top.trans_eq principal_univ.symm"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns s✝ t✝ : Set α\na✝¹ : GenerateOpen g s✝\na✝ : GenerateOpen g t✝\nhs : a ∈ s✝ → (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 s✝\nht : a ∈ t✝ → (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 t✝\nha : a ∈ s✝ ∩ t✝\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 (s✝ ∩ t✝)",
"tactic": "exact (le_inf (hs ha.1) (ht ha.2)).trans_eq inf_principal"
},
{
"state_after": "α : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns : Set α\nS✝ : Set (Set α)\n_S : ∀ (s : Set α), s ∈ S✝ → GenerateOpen g s\nhS : ∀ (s : Set α), s ∈ S✝ → a ∈ s → (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 s\nha : a ∈ ⋃₀ S✝\nt : Set α\nhtS : t ∈ S✝\nhat : a ∈ t\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 (⋃₀ S✝)",
"state_before": "α : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns : Set α\nS✝ : Set (Set α)\n_S : ∀ (s : Set α), s ∈ S✝ → GenerateOpen g s\nhS : ∀ (s : Set α), s ∈ S✝ → a ∈ s → (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 s\nha : a ∈ ⋃₀ S✝\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 (⋃₀ S✝)",
"tactic": "let ⟨t, htS, hat⟩ := ha"
},
{
"state_after": "no goals",
"state_before": "α : Type u\ng : Set (Set α)\na : α\nthis : TopologicalSpace α := generateFrom g\ns : Set α\nS✝ : Set (Set α)\n_S : ∀ (s : Set α), s ∈ S✝ → GenerateOpen g s\nhS : ∀ (s : Set α), s ∈ S✝ → a ∈ s → (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 s\nha : a ∈ ⋃₀ S✝\nt : Set α\nhtS : t ∈ S✝\nhat : a ∈ t\n⊢ (⨅ (s : Set α) (_ : s ∈ {s | a ∈ s ∧ s ∈ g}), 𝓟 s) ≤ 𝓟 (⋃₀ S✝)",
"tactic": "exact (hS t htS hat).trans (principal_mono.2 <| subset_sUnion_of_mem htS)"
}
] |
[
93,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/Data/List/Basic.lean
|
List.map_const'
|
[] |
[
1888,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1887,
9
] |
Mathlib/Order/LiminfLimsup.lean
|
Filter.inf_limsup
|
[] |
[
1010,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1009,
1
] |
Mathlib/RingTheory/Coprime/Basic.lean
|
IsCoprime.mul_add_right_left_iff
|
[] |
[
344,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
343,
1
] |
Mathlib/Data/Setoid/Basic.lean
|
Setoid.ker_def
|
[] |
[
118,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
117,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
|
SimpleGraph.Subgraph.incidenceSet_subset
|
[] |
[
266,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
265,
1
] |
Mathlib/Data/Rat/Floor.lean
|
Rat.floor_cast
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : LinearOrderedField α\ninst✝ : FloorRing α\nx : ℚ\n⊢ ↑⌊x⌋ ≤ ↑x ∧ ↑x < ↑⌊x⌋ + 1",
"tactic": "exact_mod_cast floor_eq_iff.1 (Eq.refl ⌊x⌋)"
}
] |
[
74,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/Algebra/BigOperators/Finprod.lean
|
finprod_prod_comm
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nι : Type ?u.386315\nG : Type ?u.386318\nM : Type u_3\nN : Type ?u.386324\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf✝ g : α → M\na b : α\ns✝ t : Set α\ns : Finset β\nf : α → β → M\nh : ∀ (b : β), b ∈ s → Set.Finite (mulSupport fun a => f a b)\nhU :\n (mulSupport fun a => ∏ b in s, f a b) ⊆\n ↑(Finite.toFinset (_ : Set.Finite (⋃ (i : β) (_ : i ∈ ↑s), mulSupport fun a => f a i)))\n⊢ (∏ᶠ (a : α), ∏ b in s, f a b) = ∏ b in s, ∏ᶠ (a : α), f a b",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.386315\nG : Type ?u.386318\nM : Type u_3\nN : Type ?u.386324\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf✝ g : α → M\na b : α\ns✝ t : Set α\ns : Finset β\nf : α → β → M\nh : ∀ (b : β), b ∈ s → Set.Finite (mulSupport fun a => f a b)\n⊢ (∏ᶠ (a : α), ∏ b in s, f a b) = ∏ b in s, ∏ᶠ (a : α), f a b",
"tactic": "have hU :\n (mulSupport fun a => ∏ b in s, f a b) ⊆\n (s.finite_toSet.biUnion fun b hb => h b (Finset.mem_coe.1 hb)).toFinset := by\n rw [Finite.coe_toFinset]\n intro x hx\n simp only [exists_prop, mem_iUnion, Ne.def, mem_mulSupport, Finset.mem_coe]\n contrapose! hx\n rw [mem_mulSupport, not_not, Finset.prod_congr rfl hx, Finset.prod_const_one]"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nι : Type ?u.386315\nG : Type ?u.386318\nM : Type u_3\nN : Type ?u.386324\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf✝ g : α → M\na b : α\ns✝ t : Set α\ns : Finset β\nf : α → β → M\nh : ∀ (b : β), b ∈ s → Set.Finite (mulSupport fun a => f a b)\nhU :\n (mulSupport fun a => ∏ b in s, f a b) ⊆\n ↑(Finite.toFinset (_ : Set.Finite (⋃ (i : β) (_ : i ∈ ↑s), mulSupport fun a => f a i)))\n⊢ ∏ y in s, ∏ x in Finite.toFinset (_ : Set.Finite (⋃ (i : β) (_ : i ∈ ↑s), mulSupport fun a => f a i)), f x y =\n ∏ b in s, ∏ᶠ (a : α), f a b",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.386315\nG : Type ?u.386318\nM : Type u_3\nN : Type ?u.386324\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf✝ g : α → M\na b : α\ns✝ t : Set α\ns : Finset β\nf : α → β → M\nh : ∀ (b : β), b ∈ s → Set.Finite (mulSupport fun a => f a b)\nhU :\n (mulSupport fun a => ∏ b in s, f a b) ⊆\n ↑(Finite.toFinset (_ : Set.Finite (⋃ (i : β) (_ : i ∈ ↑s), mulSupport fun a => f a i)))\n⊢ (∏ᶠ (a : α), ∏ b in s, f a b) = ∏ b in s, ∏ᶠ (a : α), f a b",
"tactic": "rw [finprod_eq_prod_of_mulSupport_subset _ hU, Finset.prod_comm]"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nι : Type ?u.386315\nG : Type ?u.386318\nM : Type u_3\nN : Type ?u.386324\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf✝ g : α → M\na b✝ : α\ns✝ t : Set α\ns : Finset β\nf : α → β → M\nh : ∀ (b : β), b ∈ s → Set.Finite (mulSupport fun a => f a b)\nhU :\n (mulSupport fun a => ∏ b in s, f a b) ⊆\n ↑(Finite.toFinset (_ : Set.Finite (⋃ (i : β) (_ : i ∈ ↑s), mulSupport fun a => f a i)))\nb : β\nhb : b ∈ s\n⊢ (mulSupport fun x => f x b) ⊆ ↑(Finite.toFinset (_ : Set.Finite (⋃ (i : β) (_ : i ∈ ↑s), mulSupport fun a => f a i)))",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.386315\nG : Type ?u.386318\nM : Type u_3\nN : Type ?u.386324\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf✝ g : α → M\na b : α\ns✝ t : Set α\ns : Finset β\nf : α → β → M\nh : ∀ (b : β), b ∈ s → Set.Finite (mulSupport fun a => f a b)\nhU :\n (mulSupport fun a => ∏ b in s, f a b) ⊆\n ↑(Finite.toFinset (_ : Set.Finite (⋃ (i : β) (_ : i ∈ ↑s), mulSupport fun a => f a i)))\n⊢ ∏ y in s, ∏ x in Finite.toFinset (_ : Set.Finite (⋃ (i : β) (_ : i ∈ ↑s), mulSupport fun a => f a i)), f x y =\n ∏ b in s, ∏ᶠ (a : α), f a b",
"tactic": "refine' Finset.prod_congr rfl fun b hb => (finprod_eq_prod_of_mulSupport_subset _ _).symm"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nι : Type ?u.386315\nG : Type ?u.386318\nM : Type u_3\nN : Type ?u.386324\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf✝ g : α → M\na✝ b✝ : α\ns✝ t : Set α\ns : Finset β\nf : α → β → M\nh : ∀ (b : β), b ∈ s → Set.Finite (mulSupport fun a => f a b)\nhU :\n (mulSupport fun a => ∏ b in s, f a b) ⊆\n ↑(Finite.toFinset (_ : Set.Finite (⋃ (i : β) (_ : i ∈ ↑s), mulSupport fun a => f a i)))\nb : β\nhb : b ∈ s\na : α\nha : a ∈ mulSupport fun x => f x b\n⊢ a ∈ ↑(Finite.toFinset (_ : Set.Finite (⋃ (i : β) (_ : i ∈ ↑s), mulSupport fun a => f a i)))",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.386315\nG : Type ?u.386318\nM : Type u_3\nN : Type ?u.386324\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf✝ g : α → M\na b✝ : α\ns✝ t : Set α\ns : Finset β\nf : α → β → M\nh : ∀ (b : β), b ∈ s → Set.Finite (mulSupport fun a => f a b)\nhU :\n (mulSupport fun a => ∏ b in s, f a b) ⊆\n ↑(Finite.toFinset (_ : Set.Finite (⋃ (i : β) (_ : i ∈ ↑s), mulSupport fun a => f a i)))\nb : β\nhb : b ∈ s\n⊢ (mulSupport fun x => f x b) ⊆ ↑(Finite.toFinset (_ : Set.Finite (⋃ (i : β) (_ : i ∈ ↑s), mulSupport fun a => f a i)))",
"tactic": "intro a ha"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nι : Type ?u.386315\nG : Type ?u.386318\nM : Type u_3\nN : Type ?u.386324\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf✝ g : α → M\na✝ b✝ : α\ns✝ t : Set α\ns : Finset β\nf : α → β → M\nh : ∀ (b : β), b ∈ s → Set.Finite (mulSupport fun a => f a b)\nhU :\n (mulSupport fun a => ∏ b in s, f a b) ⊆\n ↑(Finite.toFinset (_ : Set.Finite (⋃ (i : β) (_ : i ∈ ↑s), mulSupport fun a => f a i)))\nb : β\nhb : b ∈ s\na : α\nha : a ∈ mulSupport fun x => f x b\n⊢ ∃ i i_1, a ∈ mulSupport fun a => f a i",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.386315\nG : Type ?u.386318\nM : Type u_3\nN : Type ?u.386324\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf✝ g : α → M\na✝ b✝ : α\ns✝ t : Set α\ns : Finset β\nf : α → β → M\nh : ∀ (b : β), b ∈ s → Set.Finite (mulSupport fun a => f a b)\nhU :\n (mulSupport fun a => ∏ b in s, f a b) ⊆\n ↑(Finite.toFinset (_ : Set.Finite (⋃ (i : β) (_ : i ∈ ↑s), mulSupport fun a => f a i)))\nb : β\nhb : b ∈ s\na : α\nha : a ∈ mulSupport fun x => f x b\n⊢ a ∈ ↑(Finite.toFinset (_ : Set.Finite (⋃ (i : β) (_ : i ∈ ↑s), mulSupport fun a => f a i)))",
"tactic": "simp only [Finite.coe_toFinset, mem_iUnion]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.386315\nG : Type ?u.386318\nM : Type u_3\nN : Type ?u.386324\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf✝ g : α → M\na✝ b✝ : α\ns✝ t : Set α\ns : Finset β\nf : α → β → M\nh : ∀ (b : β), b ∈ s → Set.Finite (mulSupport fun a => f a b)\nhU :\n (mulSupport fun a => ∏ b in s, f a b) ⊆\n ↑(Finite.toFinset (_ : Set.Finite (⋃ (i : β) (_ : i ∈ ↑s), mulSupport fun a => f a i)))\nb : β\nhb : b ∈ s\na : α\nha : a ∈ mulSupport fun x => f x b\n⊢ ∃ i i_1, a ∈ mulSupport fun a => f a i",
"tactic": "exact ⟨b, hb, ha⟩"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nι : Type ?u.386315\nG : Type ?u.386318\nM : Type u_3\nN : Type ?u.386324\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf✝ g : α → M\na b : α\ns✝ t : Set α\ns : Finset β\nf : α → β → M\nh : ∀ (b : β), b ∈ s → Set.Finite (mulSupport fun a => f a b)\n⊢ (mulSupport fun a => ∏ b in s, f a b) ⊆ ⋃ (i : β) (_ : i ∈ ↑s), mulSupport fun a => f a i",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.386315\nG : Type ?u.386318\nM : Type u_3\nN : Type ?u.386324\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf✝ g : α → M\na b : α\ns✝ t : Set α\ns : Finset β\nf : α → β → M\nh : ∀ (b : β), b ∈ s → Set.Finite (mulSupport fun a => f a b)\n⊢ (mulSupport fun a => ∏ b in s, f a b) ⊆\n ↑(Finite.toFinset (_ : Set.Finite (⋃ (i : β) (_ : i ∈ ↑s), mulSupport fun a => f a i)))",
"tactic": "rw [Finite.coe_toFinset]"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nι : Type ?u.386315\nG : Type ?u.386318\nM : Type u_3\nN : Type ?u.386324\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf✝ g : α → M\na b : α\ns✝ t : Set α\ns : Finset β\nf : α → β → M\nh : ∀ (b : β), b ∈ s → Set.Finite (mulSupport fun a => f a b)\nx : α\nhx : x ∈ mulSupport fun a => ∏ b in s, f a b\n⊢ x ∈ ⋃ (i : β) (_ : i ∈ ↑s), mulSupport fun a => f a i",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.386315\nG : Type ?u.386318\nM : Type u_3\nN : Type ?u.386324\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf✝ g : α → M\na b : α\ns✝ t : Set α\ns : Finset β\nf : α → β → M\nh : ∀ (b : β), b ∈ s → Set.Finite (mulSupport fun a => f a b)\n⊢ (mulSupport fun a => ∏ b in s, f a b) ⊆ ⋃ (i : β) (_ : i ∈ ↑s), mulSupport fun a => f a i",
"tactic": "intro x hx"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nι : Type ?u.386315\nG : Type ?u.386318\nM : Type u_3\nN : Type ?u.386324\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf✝ g : α → M\na b : α\ns✝ t : Set α\ns : Finset β\nf : α → β → M\nh : ∀ (b : β), b ∈ s → Set.Finite (mulSupport fun a => f a b)\nx : α\nhx : x ∈ mulSupport fun a => ∏ b in s, f a b\n⊢ ∃ i, i ∈ s ∧ ¬f x i = 1",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.386315\nG : Type ?u.386318\nM : Type u_3\nN : Type ?u.386324\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf✝ g : α → M\na b : α\ns✝ t : Set α\ns : Finset β\nf : α → β → M\nh : ∀ (b : β), b ∈ s → Set.Finite (mulSupport fun a => f a b)\nx : α\nhx : x ∈ mulSupport fun a => ∏ b in s, f a b\n⊢ x ∈ ⋃ (i : β) (_ : i ∈ ↑s), mulSupport fun a => f a i",
"tactic": "simp only [exists_prop, mem_iUnion, Ne.def, mem_mulSupport, Finset.mem_coe]"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nι : Type ?u.386315\nG : Type ?u.386318\nM : Type u_3\nN : Type ?u.386324\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf✝ g : α → M\na b : α\ns✝ t : Set α\ns : Finset β\nf : α → β → M\nh : ∀ (b : β), b ∈ s → Set.Finite (mulSupport fun a => f a b)\nx : α\nhx : ∀ (i : β), i ∈ s → f x i = 1\n⊢ ¬x ∈ mulSupport fun a => ∏ b in s, f a b",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.386315\nG : Type ?u.386318\nM : Type u_3\nN : Type ?u.386324\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf✝ g : α → M\na b : α\ns✝ t : Set α\ns : Finset β\nf : α → β → M\nh : ∀ (b : β), b ∈ s → Set.Finite (mulSupport fun a => f a b)\nx : α\nhx : x ∈ mulSupport fun a => ∏ b in s, f a b\n⊢ ∃ i, i ∈ s ∧ ¬f x i = 1",
"tactic": "contrapose! hx"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nι : Type ?u.386315\nG : Type ?u.386318\nM : Type u_3\nN : Type ?u.386324\ninst✝¹ : CommMonoid M\ninst✝ : CommMonoid N\nf✝ g : α → M\na b : α\ns✝ t : Set α\ns : Finset β\nf : α → β → M\nh : ∀ (b : β), b ∈ s → Set.Finite (mulSupport fun a => f a b)\nx : α\nhx : ∀ (i : β), i ∈ s → f x i = 1\n⊢ ¬x ∈ mulSupport fun a => ∏ b in s, f a b",
"tactic": "rw [mem_mulSupport, not_not, Finset.prod_congr rfl hx, Finset.prod_const_one]"
}
] |
[
1180,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1165,
1
] |
Mathlib/Analysis/InnerProductSpace/Projection.lean
|
Submodule.orthogonal_eq_bot_iff
|
[
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.794030\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace ↑↑K\n⊢ Kᗮ = ⊥ → K = ⊤",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.794030\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace ↑↑K\n⊢ Kᗮ = ⊥ ↔ K = ⊤",
"tactic": "refine' ⟨_, fun h => by rw [h, Submodule.top_orthogonal_eq_bot]⟩"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.794030\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace ↑↑K\nh : Kᗮ = ⊥\n⊢ K = ⊤",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.794030\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace ↑↑K\n⊢ Kᗮ = ⊥ → K = ⊤",
"tactic": "intro h"
},
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.794030\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace ↑↑K\nh : Kᗮ = ⊥\nthis : K ⊔ Kᗮ = ⊤\n⊢ K = ⊤",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.794030\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace ↑↑K\nh : Kᗮ = ⊥\n⊢ K = ⊤",
"tactic": "have : K ⊔ Kᗮ = ⊤ := Submodule.sup_orthogonal_of_completeSpace"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.794030\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace ↑↑K\nh : Kᗮ = ⊥\nthis : K ⊔ Kᗮ = ⊤\n⊢ K = ⊤",
"tactic": "rwa [h, sup_comm, bot_sup_eq] at this"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.794030\ninst✝⁵ : IsROrC 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\ninst✝ : CompleteSpace ↑↑K\nh : K = ⊤\n⊢ Kᗮ = ⊥",
"tactic": "rw [h, Submodule.top_orthogonal_eq_bot]"
}
] |
[
802,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
798,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.insert_none_range_some
|
[] |
[
1208,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1207,
1
] |
Mathlib/SetTheory/Lists.lean
|
Lists'.to_ofList
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl : List (Lists α)\n⊢ toList (ofList l) = l",
"tactic": "induction l <;> simp [*]"
}
] |
[
101,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
|
vsub_mem_vectorSpan
|
[] |
[
101,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
99,
1
] |
Mathlib/Analysis/Calculus/MeanValue.lean
|
strictAnti_of_deriv_neg
|
[] |
[
951,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
946,
1
] |
Mathlib/GroupTheory/OrderOfElement.lean
|
pow_inj_iff_of_orderOf_eq_zero
|
[
{
"state_after": "no goals",
"state_before": "G : Type u\nA : Type v\nx y : G\na b : A\nn✝ m✝ : ℕ\ninst✝ : LeftCancelMonoid G\nh : orderOf x = 0\nn m : ℕ\n⊢ x ^ n = x ^ m ↔ n = m",
"tactic": "rw [pow_eq_pow_iff_modEq, h, modEq_zero_iff]"
}
] |
[
508,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
507,
1
] |
Mathlib/Geometry/Euclidean/Sphere/Power.lean
|
EuclideanGeometry.mul_dist_eq_mul_dist_of_cospherical_of_angle_eq_zero
|
[
{
"state_after": "case intro.intro.intro\nV : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\nP : Type u_1\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhab : a ≠ b\nhcd : c ≠ d\nhapb : ∠ a p b = 0\nhcpd : ∠ c p d = 0\nk₁ : ℝ\nhab₁ : b -ᵥ p = k₁ • (a -ᵥ p)\n⊢ dist a p * dist b p = dist c p * dist d p",
"state_before": "V : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\nP : Type u_1\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhab : a ≠ b\nhcd : c ≠ d\nhapb : ∠ a p b = 0\nhcpd : ∠ c p d = 0\n⊢ dist a p * dist b p = dist c p * dist d p",
"tactic": "obtain ⟨-, k₁, -, hab₁⟩ := angle_eq_zero_iff.mp hapb"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nV : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\nP : Type u_1\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhab : a ≠ b\nhcd : c ≠ d\nhapb : ∠ a p b = 0\nhcpd : ∠ c p d = 0\nk₁ : ℝ\nhab₁ : b -ᵥ p = k₁ • (a -ᵥ p)\nk₂ : ℝ\nhcd₁ : d -ᵥ p = k₂ • (c -ᵥ p)\n⊢ dist a p * dist b p = dist c p * dist d p",
"state_before": "case intro.intro.intro\nV : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\nP : Type u_1\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhab : a ≠ b\nhcd : c ≠ d\nhapb : ∠ a p b = 0\nhcpd : ∠ c p d = 0\nk₁ : ℝ\nhab₁ : b -ᵥ p = k₁ • (a -ᵥ p)\n⊢ dist a p * dist b p = dist c p * dist d p",
"tactic": "obtain ⟨-, k₂, -, hcd₁⟩ := angle_eq_zero_iff.mp hcpd"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.refine'_1\nV : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\nP : Type u_1\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c d p : P\nk₁ k₂ : ℝ\nh : Cospherical {a, b, c, d}\nhab : a ≠ b\nhcd : c ≠ d\nhapb : ∠ a p b = 0\nhcpd : ∠ c p d = 0\nhab₁ : b -ᵥ p = a -ᵥ p\nhcd₁ : d -ᵥ p = k₂ • (c -ᵥ p)\nhnot : k₁ = 1\n⊢ False\n\ncase intro.intro.intro.intro.intro.intro.refine'_2\nV : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\nP : Type u_1\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c d p : P\nk₁ k₂ : ℝ\nh : Cospherical {a, b, c, d}\nhab : a ≠ b\nhcd : c ≠ d\nhapb : ∠ a p b = 0\nhcpd : ∠ c p d = 0\nhab₁ : b -ᵥ p = k₁ • (a -ᵥ p)\nhcd₁ : d -ᵥ p = c -ᵥ p\nhnot : k₂ = 1\n⊢ False",
"state_before": "case intro.intro.intro.intro.intro.intro\nV : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\nP : Type u_1\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c d p : P\nh : Cospherical {a, b, c, d}\nhab : a ≠ b\nhcd : c ≠ d\nhapb : ∠ a p b = 0\nhcpd : ∠ c p d = 0\nk₁ : ℝ\nhab₁ : b -ᵥ p = k₁ • (a -ᵥ p)\nk₂ : ℝ\nhcd₁ : d -ᵥ p = k₂ • (c -ᵥ p)\n⊢ dist a p * dist b p = dist c p * dist d p",
"tactic": "refine' mul_dist_eq_mul_dist_of_cospherical h ⟨k₁, _, hab₁⟩ ⟨k₂, _, hcd₁⟩ <;> by_contra hnot <;>\n simp_all only [Classical.not_not, one_smul]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.refine'_1\nV : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\nP : Type u_1\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c d p : P\nk₁ k₂ : ℝ\nh : Cospherical {a, b, c, d}\nhab : a ≠ b\nhcd : c ≠ d\nhapb : ∠ a p b = 0\nhcpd : ∠ c p d = 0\nhab₁ : b -ᵥ p = a -ᵥ p\nhcd₁ : d -ᵥ p = k₂ • (c -ᵥ p)\nhnot : k₁ = 1\n⊢ False\n\ncase intro.intro.intro.intro.intro.intro.refine'_2\nV : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : InnerProductSpace ℝ V\nP : Type u_1\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\na b c d p : P\nk₁ k₂ : ℝ\nh : Cospherical {a, b, c, d}\nhab : a ≠ b\nhcd : c ≠ d\nhapb : ∠ a p b = 0\nhcpd : ∠ c p d = 0\nhab₁ : b -ᵥ p = k₁ • (a -ᵥ p)\nhcd₁ : d -ᵥ p = c -ᵥ p\nhnot : k₂ = 1\n⊢ False",
"tactic": "exacts [hab (vsub_left_cancel hab₁).symm, hcd (vsub_left_cancel hcd₁).symm]"
}
] |
[
132,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Algebra/Star/Basic.lean
|
star_pow
|
[] |
[
212,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
210,
1
] |
src/lean/Lean/Data/PersistentHashMap.lean
|
Lean.PersistentHashMap.size_push
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nks : Array α\nvs : Array β\nh : Array.size ks = Array.size vs\nk : α\nv : β\n⊢ Array.size (Array.push ks k) = Array.size (Array.push vs v)",
"tactic": "simp [h]"
}
] |
[
71,
11
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
70,
9
] |
Mathlib/Order/Filter/Partial.lean
|
Filter.tendsto_iff_ptendsto
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nl₁ : Filter α\nl₂ : Filter β\ns : Set α\nf : α → β\n⊢ Tendsto f (l₁ ⊓ 𝓟 s) l₂ ↔ PTendsto (PFun.res f s) l₁ l₂",
"tactic": "simp only [Tendsto, PTendsto, pmap_res]"
}
] |
[
248,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
246,
1
] |
Mathlib/Algebra/Star/Subalgebra.lean
|
StarSubalgebra.mem_toSubalgebra
|
[] |
[
92,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/Order/Hom/Lattice.lean
|
BoundedLatticeHom.id_comp
|
[] |
[
1345,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1345,
9
] |
Mathlib/Data/QPF/Multivariate/Basic.lean
|
MvQPF.suppPreservation_iff_liftpPreservation
|
[
{
"state_after": "case mp\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : SuppPreservation\n⊢ LiftPPreservation\n\ncase mpr\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : LiftPPreservation\n⊢ SuppPreservation",
"state_before": "n : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\n⊢ SuppPreservation ↔ LiftPPreservation",
"tactic": "constructor <;> intro h"
},
{
"state_after": "case mp.mk\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : SuppPreservation\nα : TypeVec n\np : ⦃i : Fin2 n⦄ → α i → Prop\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\n⊢ LiftP p (abs { fst := a, snd := f }) ↔ LiftP p { fst := a, snd := f }",
"state_before": "case mp\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : SuppPreservation\n⊢ LiftPPreservation",
"tactic": "rintro α p ⟨a, f⟩"
},
{
"state_after": "case mp.mk\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : SuppPreservation\nα : TypeVec n\np : ⦃i : Fin2 n⦄ → α i → Prop\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\nh' : SuppPreservation\n⊢ LiftP p (abs { fst := a, snd := f }) ↔ LiftP p { fst := a, snd := f }",
"state_before": "case mp.mk\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : SuppPreservation\nα : TypeVec n\np : ⦃i : Fin2 n⦄ → α i → Prop\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\n⊢ LiftP p (abs { fst := a, snd := f }) ↔ LiftP p { fst := a, snd := f }",
"tactic": "have h' := h"
},
{
"state_after": "case mp.mk\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : SuppPreservation\nα : TypeVec n\np : ⦃i : Fin2 n⦄ → α i → Prop\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\nh' : IsUniform\n⊢ LiftP p (abs { fst := a, snd := f }) ↔ LiftP p { fst := a, snd := f }",
"state_before": "case mp.mk\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : SuppPreservation\nα : TypeVec n\np : ⦃i : Fin2 n⦄ → α i → Prop\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\nh' : SuppPreservation\n⊢ LiftP p (abs { fst := a, snd := f }) ↔ LiftP p { fst := a, snd := f }",
"tactic": "rw [suppPreservation_iff_isUniform] at h'"
},
{
"state_after": "case mp.mk\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh :\n ∀ ⦃α : TypeVec n⦄ (x : MvPFunctor.Obj (P F) α),\n (fun i => {y | ∀ ⦃P : (i : Fin2 n) → α i → Prop⦄, LiftP P (abs x) → P i y}) = fun i =>\n {y | ∀ ⦃P : (i : Fin2 n) → α i → Prop⦄, LiftP P x → P i y}\nα : TypeVec n\np : ⦃i : Fin2 n⦄ → α i → Prop\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\nh' : IsUniform\n⊢ LiftP p (abs { fst := a, snd := f }) ↔ LiftP p { fst := a, snd := f }",
"state_before": "case mp.mk\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : SuppPreservation\nα : TypeVec n\np : ⦃i : Fin2 n⦄ → α i → Prop\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\nh' : IsUniform\n⊢ LiftP p (abs { fst := a, snd := f }) ↔ LiftP p { fst := a, snd := f }",
"tactic": "dsimp only [SuppPreservation, supp] at h"
},
{
"state_after": "case mp.mk\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh :\n ∀ ⦃α : TypeVec n⦄ (x : MvPFunctor.Obj (P F) α),\n (fun i => {y | ∀ ⦃P : (i : Fin2 n) → α i → Prop⦄, LiftP P (abs x) → P i y}) = fun i =>\n {y | ∀ ⦃P : (i : Fin2 n) → α i → Prop⦄, LiftP P x → P i y}\nα : TypeVec n\np : ⦃i : Fin2 n⦄ → α i → Prop\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\nh' : IsUniform\n⊢ (∀ (i : Fin2 n) (u : α i) (x : MvPFunctor.B (P F) a i), f i x = u → p u) ↔\n ∀ (i : Fin2 n) (x : MvPFunctor.B (P F) a i), p (f i x)",
"state_before": "case mp.mk\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh :\n ∀ ⦃α : TypeVec n⦄ (x : MvPFunctor.Obj (P F) α),\n (fun i => {y | ∀ ⦃P : (i : Fin2 n) → α i → Prop⦄, LiftP P (abs x) → P i y}) = fun i =>\n {y | ∀ ⦃P : (i : Fin2 n) → α i → Prop⦄, LiftP P x → P i y}\nα : TypeVec n\np : ⦃i : Fin2 n⦄ → α i → Prop\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\nh' : IsUniform\n⊢ LiftP p (abs { fst := a, snd := f }) ↔ LiftP p { fst := a, snd := f }",
"tactic": "simp only [liftP_iff_of_isUniform, supp_eq_of_isUniform, MvPFunctor.liftP_iff', h',\n image_univ, mem_range, exists_imp]"
},
{
"state_after": "no goals",
"state_before": "case mp.mk\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh :\n ∀ ⦃α : TypeVec n⦄ (x : MvPFunctor.Obj (P F) α),\n (fun i => {y | ∀ ⦃P : (i : Fin2 n) → α i → Prop⦄, LiftP P (abs x) → P i y}) = fun i =>\n {y | ∀ ⦃P : (i : Fin2 n) → α i → Prop⦄, LiftP P x → P i y}\nα : TypeVec n\np : ⦃i : Fin2 n⦄ → α i → Prop\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\nh' : IsUniform\n⊢ (∀ (i : Fin2 n) (u : α i) (x : MvPFunctor.B (P F) a i), f i x = u → p u) ↔\n ∀ (i : Fin2 n) (x : MvPFunctor.B (P F) a i), p (f i x)",
"tactic": "constructor <;> intros <;> subst_vars <;> solve_by_elim"
},
{
"state_after": "case mpr.mk\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : LiftPPreservation\nα : TypeVec n\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\n⊢ supp (abs { fst := a, snd := f }) = supp { fst := a, snd := f }",
"state_before": "case mpr\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : LiftPPreservation\n⊢ SuppPreservation",
"tactic": "rintro α ⟨a, f⟩"
},
{
"state_after": "case mpr.mk\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : ∀ ⦃α : TypeVec n⦄ (p : ⦃i : Fin2 n⦄ → α i → Prop) (x : MvPFunctor.Obj (P F) α), LiftP p (abs x) ↔ LiftP p x\nα : TypeVec n\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\n⊢ supp (abs { fst := a, snd := f }) = supp { fst := a, snd := f }",
"state_before": "case mpr.mk\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : LiftPPreservation\nα : TypeVec n\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\n⊢ supp (abs { fst := a, snd := f }) = supp { fst := a, snd := f }",
"tactic": "simp only [LiftPPreservation] at h"
},
{
"state_after": "case mpr.mk.h.h\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : ∀ ⦃α : TypeVec n⦄ (p : ⦃i : Fin2 n⦄ → α i → Prop) (x : MvPFunctor.Obj (P F) α), LiftP p (abs x) ↔ LiftP p x\nα : TypeVec n\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\nx✝¹ : Fin2 n\nx✝ : α x✝¹\n⊢ x✝ ∈ supp (abs { fst := a, snd := f }) x✝¹ ↔ x✝ ∈ supp { fst := a, snd := f } x✝¹",
"state_before": "case mpr.mk\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : ∀ ⦃α : TypeVec n⦄ (p : ⦃i : Fin2 n⦄ → α i → Prop) (x : MvPFunctor.Obj (P F) α), LiftP p (abs x) ↔ LiftP p x\nα : TypeVec n\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\n⊢ supp (abs { fst := a, snd := f }) = supp { fst := a, snd := f }",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case mpr.mk.h.h\nn : ℕ\nF : TypeVec n → Type u_1\ninst✝ : MvFunctor F\nq : MvQPF F\nh : ∀ ⦃α : TypeVec n⦄ (p : ⦃i : Fin2 n⦄ → α i → Prop) (x : MvPFunctor.Obj (P F) α), LiftP p (abs x) ↔ LiftP p x\nα : TypeVec n\na : (P F).A\nf : MvPFunctor.B (P F) a ⟹ α\nx✝¹ : Fin2 n\nx✝ : α x✝¹\n⊢ x✝ ∈ supp (abs { fst := a, snd := f }) x✝¹ ↔ x✝ ∈ supp { fst := a, snd := f } x✝¹",
"tactic": "simp only [supp, h, mem_setOf_eq]"
}
] |
[
282,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
270,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.div_mem_comm_iff
|
[] |
[
602,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
601,
11
] |
Mathlib/Data/Matrix/Block.lean
|
Matrix.blockDiagonal'_eq_blockDiagonal
|
[] |
[
640,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
638,
1
] |
Mathlib/Data/Nat/Digits.lean
|
Nat.ofDigits_lt_base_pow_length
|
[
{
"state_after": "case succ.succ\nn : ℕ\nl : List ℕ\nb : ℕ\nhl : ∀ (x : ℕ), x ∈ l → x < succ (succ b)\n⊢ ofDigits (succ (succ b)) l < succ (succ b) ^ List.length l",
"state_before": "n b : ℕ\nl : List ℕ\nhb : 1 < b\nhl : ∀ (x : ℕ), x ∈ l → x < b\n⊢ ofDigits b l < b ^ List.length l",
"tactic": "rcases b with (_ | _ | b) <;> try simp_all"
},
{
"state_after": "no goals",
"state_before": "case succ.succ\nn : ℕ\nl : List ℕ\nb : ℕ\nhl : ∀ (x : ℕ), x ∈ l → x < succ (succ b)\n⊢ ofDigits (succ (succ b)) l < succ (succ b) ^ List.length l",
"tactic": "exact ofDigits_lt_base_pow_length' hl"
},
{
"state_after": "case succ.succ\nn : ℕ\nl : List ℕ\nb : ℕ\nhl : ∀ (x : ℕ), x ∈ l → x < succ (succ b)\n⊢ ofDigits (succ (succ b)) l < succ (succ b) ^ List.length l",
"state_before": "case succ.succ\nn : ℕ\nl : List ℕ\nb : ℕ\nhb : 1 < succ (succ b)\nhl : ∀ (x : ℕ), x ∈ l → x < succ (succ b)\n⊢ ofDigits (succ (succ b)) l < succ (succ b) ^ List.length l",
"tactic": "simp_all"
}
] |
[
412,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
409,
1
] |
Mathlib/RingTheory/GradedAlgebra/HomogeneousLocalization.lean
|
HomogeneousLocalization.NumDenSameDeg.num_one
|
[] |
[
134,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.insert_eq
|
[] |
[
1294,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1293,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.sup_biUnion
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.39437\nα : Type u_3\nβ : Type u_1\nγ : Type u_2\nι : Type ?u.39449\nκ : Type ?u.39452\ninst✝² : SemilatticeSup α\ninst✝¹ : OrderBot α\ns✝ s₁ s₂ : Finset β\nf g : β → α\na : α\ninst✝ : DecidableEq β\ns : Finset γ\nt : γ → Finset β\nc : α\n⊢ sup (Finset.biUnion s t) f ≤ c ↔ (sup s fun x => sup (t x) f) ≤ c",
"tactic": "simp [@forall_swap _ β]"
}
] |
[
131,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean
|
ZFSet.ext
|
[] |
[
797,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
796,
1
] |
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