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/Analysis/Asymptotics/Asymptotics.lean
|
Asymptotics.isLittleO_iff_nat_mul_le
|
[] |
[
266,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
265,
1
] |
Mathlib/Analysis/Convex/Combination.lean
|
Finset.centroid_eq_centerMass
|
[] |
[
256,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
254,
1
] |
Mathlib/Algebra/AddTorsor.lean
|
Prod.snd_vadd
|
[] |
[
311,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
310,
1
] |
Mathlib/Computability/Ackermann.lean
|
ack_three
|
[
{
"state_after": "case zero\n\n⊢ ack 3 zero = 2 ^ (zero + 3) - 3\n\ncase succ\nn : ℕ\nIH : ack 3 n = 2 ^ (n + 3) - 3\n⊢ ack 3 (succ n) = 2 ^ (succ n + 3) - 3",
"state_before": "n : ℕ\n⊢ ack 3 n = 2 ^ (n + 3) - 3",
"tactic": "induction' n with n IH"
},
{
"state_after": "no goals",
"state_before": "case zero\n\n⊢ ack 3 zero = 2 ^ (zero + 3) - 3",
"tactic": "rfl"
},
{
"state_after": "case succ\nn : ℕ\nIH : ack 3 n = 2 ^ (n + 3) - 3\n⊢ 2 * 3 ≤ 2 * 2 ^ (n + 3)",
"state_before": "case succ\nn : ℕ\nIH : ack 3 n = 2 ^ (n + 3) - 3\n⊢ ack 3 (succ n) = 2 ^ (succ n + 3) - 3",
"tactic": "rw [ack_succ_succ, IH, ack_two, Nat.succ_add, Nat.pow_succ 2 (n + 3), mul_comm _ 2,\n Nat.mul_sub_left_distrib, ← Nat.sub_add_comm, two_mul 3, Nat.add_sub_add_right]"
},
{
"state_after": "case succ\nn : ℕ\nIH : ack 3 n = 2 ^ (n + 3) - 3\nH : 2 * 3 ≤ 2 * 2 ^ 3\n⊢ 2 * 3 ≤ 2 * 2 ^ (n + 3)",
"state_before": "case succ\nn : ℕ\nIH : ack 3 n = 2 ^ (n + 3) - 3\n⊢ 2 * 3 ≤ 2 * 2 ^ (n + 3)",
"tactic": "have H : 2 * 3 ≤ 2 * 2 ^ 3 := by norm_num"
},
{
"state_after": "case succ\nn : ℕ\nIH : ack 3 n = 2 ^ (n + 3) - 3\nH : 2 * 3 ≤ 2 * 2 ^ 3\n⊢ 2 * 2 ^ 3 ≤ 2 * 2 ^ (n + 3)",
"state_before": "case succ\nn : ℕ\nIH : ack 3 n = 2 ^ (n + 3) - 3\nH : 2 * 3 ≤ 2 * 2 ^ 3\n⊢ 2 * 3 ≤ 2 * 2 ^ (n + 3)",
"tactic": "apply H.trans"
},
{
"state_after": "no goals",
"state_before": "case succ\nn : ℕ\nIH : ack 3 n = 2 ^ (n + 3) - 3\nH : 2 * 3 ≤ 2 * 2 ^ 3\n⊢ 2 * 2 ^ 3 ≤ 2 * 2 ^ (n + 3)",
"tactic": "simp [pow_le_pow]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\nIH : ack 3 n = 2 ^ (n + 3) - 3\n⊢ 2 * 3 ≤ 2 * 2 ^ 3",
"tactic": "norm_num"
}
] |
[
106,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
99,
1
] |
Mathlib/NumberTheory/Padics/PadicIntegers.lean
|
PadicInt.p_nonnunit
|
[
{
"state_after": "p : ℕ\nhp : Fact (Nat.Prime p)\nthis : (↑p)⁻¹ < 1\n⊢ ↑p ∈ nonunits ℤ_[p]",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\n⊢ ↑p ∈ nonunits ℤ_[p]",
"tactic": "have : (p : ℝ)⁻¹ < 1 := inv_lt_one <| by exact_mod_cast hp.1.one_lt"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\nthis : (↑p)⁻¹ < 1\n⊢ ↑p ∈ nonunits ℤ_[p]",
"tactic": "rwa [← norm_p, ← mem_nonunits] at this"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Fact (Nat.Prime p)\n⊢ 1 < ↑p",
"tactic": "exact_mod_cast hp.1.one_lt"
}
] |
[
596,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
594,
1
] |
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean
|
Matrix.separatingLeft_toLinearMap₂_iff
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.2748126\nR₁ : Type u_2\nR₂ : Type ?u.2748132\nM✝ : Type ?u.2748135\nM₁ : Type u_3\nM₂ : Type ?u.2748141\nM₁' : Type ?u.2748144\nM₂' : Type ?u.2748147\nn : Type ?u.2748150\nm : Type ?u.2748153\nn' : Type ?u.2748156\nm' : Type ?u.2748159\nι : Type u_1\ninst✝⁴ : CommRing R₁\ninst✝³ : AddCommMonoid M₁\ninst✝² : Module R₁ M₁\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nB : M₁ →ₗ[R₁] M₁ →ₗ[R₁] R₁\nM : Matrix ι ι R₁\nb : Basis ι R₁ M₁\n⊢ SeparatingLeft (↑(toLinearMap₂ b b) M) ↔ Matrix.Nondegenerate M",
"tactic": "rw [← Matrix.separatingLeft_toLinearMap₂'_iff_separatingLeft_toLinearMap₂,\n Matrix.separatingLeft_toLinearMap₂'_iff]"
}
] |
[
712,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
709,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearMap.coe_smul
|
[] |
[
592,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
590,
1
] |
Mathlib/Init/Algebra/Order.lean
|
le_refl
|
[] |
[
55,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/Topology/Order/Hom/Esakia.lean
|
EsakiaHom.comp_id
|
[] |
[
342,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
341,
1
] |
Mathlib/Deprecated/Group.lean
|
IsMonoidHom.map_mul'
|
[] |
[
173,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
172,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean
|
ZFSet.toSet_range
|
[
{
"state_after": "case h\nα : Type u\nf : α → ZFSet\nx✝ : ZFSet\n⊢ x✝ ∈ toSet (range f) ↔ x✝ ∈ Set.range f",
"state_before": "α : Type u\nf : α → ZFSet\n⊢ toSet (range f) = Set.range f",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u\nf : α → ZFSet\nx✝ : ZFSet\n⊢ x✝ ∈ toSet (range f) ↔ x✝ ∈ Set.range f",
"tactic": "simp"
}
] |
[
1282,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1279,
1
] |
Mathlib/MeasureTheory/Integral/IntegralEqImproper.lean
|
MeasureTheory.AECover.integrable_of_integral_bounded_of_nonneg_ae
|
[] |
[
468,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
464,
1
] |
Mathlib/Topology/Instances/ENNReal.lean
|
ENNReal.tsum_mono_subtype
|
[] |
[
996,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
994,
1
] |
Mathlib/RingTheory/DedekindDomain/Ideal.lean
|
FractionalIdeal.spanSingleton_div_self
|
[
{
"state_after": "no goals",
"state_before": "R : Type ?u.45600\nA : Type ?u.45603\nK : Type u_1\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing A\ninst✝⁸ : Field K\ninst✝⁷ : IsDomain A\nR₁ : Type u_2\ninst✝⁶ : CommRing R₁\ninst✝⁵ : IsDomain R₁\ninst✝⁴ : Algebra R₁ K\ninst✝³ : IsFractionRing R₁ K\nI J : FractionalIdeal R₁⁰ K\nK' : Type ?u.46614\ninst✝² : Field K'\ninst✝¹ : Algebra R₁ K'\ninst✝ : IsFractionRing R₁ K'\nx : K\nhx : x ≠ 0\n⊢ spanSingleton R₁⁰ x / spanSingleton R₁⁰ x = 1",
"tactic": "rw [spanSingleton_div_spanSingleton, div_self hx, spanSingleton_one]"
}
] |
[
164,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/Topology/Basic.lean
|
tendsto_atTop_nhds
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\na✝ : α\ns s₁ s₂ t : Set α\np p₁ p₂ : α → Prop\ninst✝² : TopologicalSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\nf : β → α\na : α\n⊢ (∀ (ib : Set α), a ∈ ib ∧ IsOpen ib → ∃ ia, True ∧ ∀ (x : β), x ∈ Ici ia → f x ∈ ib) ↔\n ∀ (U : Set α), a ∈ U → IsOpen U → ∃ N, ∀ (n : β), N ≤ n → f n ∈ U",
"tactic": "simp only [and_imp, exists_prop, true_and_iff, mem_Ici, ge_iff_le]"
}
] |
[
1033,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1030,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Prod.lean
|
hasStrictFDerivAt_pi'
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type ?u.395005\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type ?u.395100\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nG' : Type ?u.395195\ninst✝⁴ : NormedAddCommGroup G'\ninst✝³ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nι : Type u_3\ninst✝² : Fintype ι\nF' : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nφ : (i : ι) → E → F' i\nφ' : (i : ι) → E →L[𝕜] F' i\nΦ : E → (i : ι) → F' i\nΦ' : E →L[𝕜] (i : ι) → F' i\n⊢ ((fun p => Φ p.fst - Φ p.snd - ↑Φ' (p.fst - p.snd)) =o[𝓝 (x, x)] fun p => p.fst - p.snd) ↔\n ∀ (i : ι),\n (fun p => Φ p.fst i - Φ p.snd i - ↑(comp (proj i) Φ') (p.fst - p.snd)) =o[𝓝 (x, x)] fun p => p.fst - p.snd",
"state_before": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type ?u.395005\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type ?u.395100\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nG' : Type ?u.395195\ninst✝⁴ : NormedAddCommGroup G'\ninst✝³ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nι : Type u_3\ninst✝² : Fintype ι\nF' : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nφ : (i : ι) → E → F' i\nφ' : (i : ι) → E →L[𝕜] F' i\nΦ : E → (i : ι) → F' i\nΦ' : E →L[𝕜] (i : ι) → F' i\n⊢ HasStrictFDerivAt Φ Φ' x ↔ ∀ (i : ι), HasStrictFDerivAt (fun x => Φ x i) (comp (proj i) Φ') x",
"tactic": "simp only [HasStrictFDerivAt, ContinuousLinearMap.coe_pi]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝¹¹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nF : Type ?u.395005\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nG : Type ?u.395100\ninst✝⁶ : NormedAddCommGroup G\ninst✝⁵ : NormedSpace 𝕜 G\nG' : Type ?u.395195\ninst✝⁴ : NormedAddCommGroup G'\ninst✝³ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nι : Type u_3\ninst✝² : Fintype ι\nF' : ι → Type u_4\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nφ : (i : ι) → E → F' i\nφ' : (i : ι) → E →L[𝕜] F' i\nΦ : E → (i : ι) → F' i\nΦ' : E →L[𝕜] (i : ι) → F' i\n⊢ ((fun p => Φ p.fst - Φ p.snd - ↑Φ' (p.fst - p.snd)) =o[𝓝 (x, x)] fun p => p.fst - p.snd) ↔\n ∀ (i : ι),\n (fun p => Φ p.fst i - Φ p.snd i - ↑(comp (proj i) Φ') (p.fst - p.snd)) =o[𝓝 (x, x)] fun p => p.fst - p.snd",
"tactic": "exact isLittleO_pi"
}
] |
[
383,
21
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
380,
1
] |
Mathlib/CategoryTheory/Subobject/Lattice.lean
|
CategoryTheory.Subobject.inf_eq_map_pullback'
|
[
{
"state_after": "case h\nC : Type u₁\ninst✝² : Category C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nA : C\nf₁ f₂ : MonoOver A\n⊢ (inf.obj (Quotient.mk'' f₁)).obj (Quotient.mk'' f₂) =\n (map (MonoOver.arrow f₁)).obj ((pullback (MonoOver.arrow f₁)).obj (Quotient.mk'' f₂))",
"state_before": "C : Type u₁\ninst✝² : Category C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nA : C\nf₁ : MonoOver A\nf₂ : Subobject A\n⊢ (inf.obj (Quotient.mk'' f₁)).obj f₂ = (map (MonoOver.arrow f₁)).obj ((pullback (MonoOver.arrow f₁)).obj f₂)",
"tactic": "induction' f₂ using Quotient.inductionOn' with f₂"
},
{
"state_after": "no goals",
"state_before": "case h\nC : Type u₁\ninst✝² : Category C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nA : C\nf₁ f₂ : MonoOver A\n⊢ (inf.obj (Quotient.mk'' f₁)).obj (Quotient.mk'' f₂) =\n (map (MonoOver.arrow f₁)).obj ((pullback (MonoOver.arrow f₁)).obj (Quotient.mk'' f₂))",
"tactic": "rfl"
}
] |
[
467,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
463,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean
|
Finset.summable
|
[] |
[
185,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
183,
11
] |
Mathlib/RingTheory/Valuation/Basic.lean
|
Valuation.map_add_eq_of_lt_left
|
[
{
"state_after": "K : Type ?u.2215047\nF : Type ?u.2215050\nR : Type u_2\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type ?u.2215062\nΓ''₀ : Type ?u.2215065\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx y z : R\nh : ↑v y < ↑v x\n⊢ ↑v (y + x) = ↑v x",
"state_before": "K : Type ?u.2215047\nF : Type ?u.2215050\nR : Type u_2\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type ?u.2215062\nΓ''₀ : Type ?u.2215065\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx y z : R\nh : ↑v y < ↑v x\n⊢ ↑v (x + y) = ↑v x",
"tactic": "rw [add_comm]"
},
{
"state_after": "no goals",
"state_before": "K : Type ?u.2215047\nF : Type ?u.2215050\nR : Type u_2\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type ?u.2215062\nΓ''₀ : Type ?u.2215065\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : Ring R\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nx y z : R\nh : ↑v y < ↑v x\n⊢ ↑v (y + x) = ↑v x",
"tactic": "exact map_add_eq_of_lt_right _ h"
}
] |
[
331,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
330,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Prod.lean
|
fderiv_snd
|
[] |
[
316,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
315,
1
] |
Mathlib/Analysis/Complex/Basic.lean
|
Complex.norm_eq_abs
|
[] |
[
54,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/Algebra/GroupWithZero/Basic.lean
|
mul_self_div_self
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.25591\nM₀ : Type ?u.25594\nG₀ : Type u_1\nM₀' : Type ?u.25600\nG₀' : Type ?u.25603\nF : Type ?u.25606\nF' : Type ?u.25609\ninst✝ : GroupWithZero G₀\na✝ b c a : G₀\n⊢ a * a / a = a",
"tactic": "rw [div_eq_mul_inv, mul_self_mul_inv a]"
}
] |
[
375,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
375,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/WideEqualizers.lean
|
CategoryTheory.Limits.Trident.IsLimit.hom_ext
|
[] |
[
304,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
302,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.of_sub_of
|
[] |
[
333,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
332,
1
] |
Mathlib/Algebra/Star/Subalgebra.lean
|
StarSubalgebra.ext
|
[] |
[
87,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
86,
1
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
|
SimpleGraph.Subgraph.neighborSet_sup
|
[] |
[
498,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
497,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
Real.cosh_le_cosh
|
[] |
[
731,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
730,
1
] |
Mathlib/Algebra/Order/ToIntervalMod.lean
|
toIcoMod_apply_left
|
[
{
"state_after": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b c : α\nn : ℤ\na : α\n⊢ a < a + p ∧ ∃ z, a = a + z • p",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b c : α\nn : ℤ\na : α\n⊢ toIcoMod hp a a = a",
"tactic": "rw [toIcoMod_eq_iff hp, Set.left_mem_Ico]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b c : α\nn : ℤ\na : α\n⊢ a < a + p ∧ ∃ z, a = a + z • p",
"tactic": "exact ⟨lt_add_of_pos_right _ hp, 0, by simp⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrderedAddCommGroup α\nhα : Archimedean α\np : α\nhp : 0 < p\na✝ b c : α\nn : ℤ\na : α\n⊢ a = a + 0 • p",
"tactic": "simp"
}
] |
[
207,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Mathlib/Algebra/Lie/Free.lean
|
FreeLieAlgebra.lift_comp_of
|
[
{
"state_after": "R : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nF : FreeLieAlgebra R X →ₗ⁅R⁆ L\n⊢ ↑(lift R) (↑(lift R).symm F) = F",
"state_before": "R : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nF : FreeLieAlgebra R X →ₗ⁅R⁆ L\n⊢ ↑(lift R) (↑F ∘ of R) = F",
"tactic": "rw [← lift_symm_apply]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nX : Type v\ninst✝² : CommRing R\nL : Type w\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nF : FreeLieAlgebra R X →ₗ⁅R⁆ L\n⊢ ↑(lift R) (↑(lift R).symm F) = F",
"tactic": "exact (lift R).apply_symm_apply F"
}
] |
[
266,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
265,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.inf_lt_iff
|
[] |
[
729,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
728,
11
] |
Mathlib/Topology/Instances/RatLemmas.lean
|
Rat.dense_compl_compact
|
[] |
[
50,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
49,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Subtype.nndist_eq
|
[] |
[
1647,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1646,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.eventually_right_inverse'
|
[] |
[
253,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
251,
1
] |
Mathlib/Topology/Algebra/Order/LiminfLimsup.lean
|
bddBelow_range_of_tendsto_atTop_atTop
|
[] |
[
128,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
1
] |
Mathlib/AlgebraicGeometry/PrimeSpectrum/IsOpenComapC.lean
|
AlgebraicGeometry.Polynomial.imageOfDf_eq_comap_C_compl_zeroLocus
|
[
{
"state_after": "case h\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx : PrimeSpectrum R\n⊢ x ∈ imageOfDf f ↔ x ∈ ↑(PrimeSpectrum.comap C) '' zeroLocus {f}ᶜ",
"state_before": "R : Type u_1\ninst✝ : CommRing R\nf : R[X]\n⊢ imageOfDf f = ↑(PrimeSpectrum.comap C) '' zeroLocus {f}ᶜ",
"tactic": "ext x"
},
{
"state_after": "case h.refine'_1\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx : PrimeSpectrum R\nhx : x ∈ imageOfDf f\n⊢ { asIdeal := Ideal.map C x.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map C x.asIdeal)) } ∈ zeroLocus {f}ᶜ\n\ncase h.refine'_2\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx : PrimeSpectrum R\nhx : x ∈ imageOfDf f\n⊢ ↑(PrimeSpectrum.comap C)\n { asIdeal := Ideal.map C x.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map C x.asIdeal)) } =\n x\n\ncase h.refine'_3\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx : PrimeSpectrum R\n⊢ x ∈ ↑(PrimeSpectrum.comap C) '' zeroLocus {f}ᶜ → x ∈ imageOfDf f",
"state_before": "case h\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx : PrimeSpectrum R\n⊢ x ∈ imageOfDf f ↔ x ∈ ↑(PrimeSpectrum.comap C) '' zeroLocus {f}ᶜ",
"tactic": "refine' ⟨fun hx => ⟨⟨map C x.asIdeal, isPrime_map_C_of_isPrime x.IsPrime⟩, ⟨_, _⟩⟩, _⟩"
},
{
"state_after": "case h.refine'_1\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx : PrimeSpectrum R\nhx : x ∈ imageOfDf f\n⊢ ¬f ∈ ↑{ asIdeal := Ideal.map C x.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map C x.asIdeal)) }.asIdeal",
"state_before": "case h.refine'_1\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx : PrimeSpectrum R\nhx : x ∈ imageOfDf f\n⊢ { asIdeal := Ideal.map C x.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map C x.asIdeal)) } ∈ zeroLocus {f}ᶜ",
"tactic": "rw [mem_compl_iff, mem_zeroLocus, singleton_subset_iff]"
},
{
"state_after": "case h.refine'_1.intro\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx : PrimeSpectrum R\ni : ℕ\nhi : ¬coeff f i ∈ x.asIdeal\n⊢ ¬f ∈ ↑{ asIdeal := Ideal.map C x.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map C x.asIdeal)) }.asIdeal",
"state_before": "case h.refine'_1\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx : PrimeSpectrum R\nhx : x ∈ imageOfDf f\n⊢ ¬f ∈ ↑{ asIdeal := Ideal.map C x.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map C x.asIdeal)) }.asIdeal",
"tactic": "cases' hx with i hi"
},
{
"state_after": "no goals",
"state_before": "case h.refine'_1.intro\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx : PrimeSpectrum R\ni : ℕ\nhi : ¬coeff f i ∈ x.asIdeal\n⊢ ¬f ∈ ↑{ asIdeal := Ideal.map C x.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map C x.asIdeal)) }.asIdeal",
"tactic": "exact fun a => hi (mem_map_C_iff.mp a i)"
},
{
"state_after": "case h.refine'_2.asIdeal.h\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx✝ : PrimeSpectrum R\nhx : x✝ ∈ imageOfDf f\nx : R\n⊢ x ∈\n (↑(PrimeSpectrum.comap C)\n { asIdeal := Ideal.map C x✝.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map C x✝.asIdeal)) }).asIdeal ↔\n x ∈ x✝.asIdeal",
"state_before": "case h.refine'_2\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx : PrimeSpectrum R\nhx : x ∈ imageOfDf f\n⊢ ↑(PrimeSpectrum.comap C)\n { asIdeal := Ideal.map C x.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map C x.asIdeal)) } =\n x",
"tactic": "ext x"
},
{
"state_after": "case h.refine'_2.asIdeal.h\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx✝ : PrimeSpectrum R\nhx : x✝ ∈ imageOfDf f\nx : R\nh :\n x ∈\n (↑(PrimeSpectrum.comap C)\n { asIdeal := Ideal.map C x✝.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map C x✝.asIdeal)) }).asIdeal\n⊢ x ∈ x✝.asIdeal",
"state_before": "case h.refine'_2.asIdeal.h\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx✝ : PrimeSpectrum R\nhx : x✝ ∈ imageOfDf f\nx : R\n⊢ x ∈\n (↑(PrimeSpectrum.comap C)\n { asIdeal := Ideal.map C x✝.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map C x✝.asIdeal)) }).asIdeal ↔\n x ∈ x✝.asIdeal",
"tactic": "refine' ⟨fun h => _, fun h => subset_span (mem_image_of_mem C.1 h)⟩"
},
{
"state_after": "case h.refine'_2.asIdeal.h\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx✝ : PrimeSpectrum R\nhx : x✝ ∈ imageOfDf f\nx : R\nh :\n x ∈\n (↑(PrimeSpectrum.comap C)\n { asIdeal := Ideal.map C x✝.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map C x✝.asIdeal)) }).asIdeal\n⊢ coeff (↑C x) 0 ∈ x✝.asIdeal",
"state_before": "case h.refine'_2.asIdeal.h\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx✝ : PrimeSpectrum R\nhx : x✝ ∈ imageOfDf f\nx : R\nh :\n x ∈\n (↑(PrimeSpectrum.comap C)\n { asIdeal := Ideal.map C x✝.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map C x✝.asIdeal)) }).asIdeal\n⊢ x ∈ x✝.asIdeal",
"tactic": "rw [← @coeff_C_zero R x _]"
},
{
"state_after": "no goals",
"state_before": "case h.refine'_2.asIdeal.h\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx✝ : PrimeSpectrum R\nhx : x✝ ∈ imageOfDf f\nx : R\nh :\n x ∈\n (↑(PrimeSpectrum.comap C)\n { asIdeal := Ideal.map C x✝.asIdeal, IsPrime := (_ : Ideal.IsPrime (Ideal.map C x✝.asIdeal)) }).asIdeal\n⊢ coeff (↑C x) 0 ∈ x✝.asIdeal",
"tactic": "exact mem_map_C_iff.mp h 0"
},
{
"state_after": "case h.refine'_3.intro.intro\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nxli : PrimeSpectrum R[X]\ncomplement : xli ∈ zeroLocus {f}ᶜ\n⊢ ↑(PrimeSpectrum.comap C) xli ∈ imageOfDf f",
"state_before": "case h.refine'_3\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nx : PrimeSpectrum R\n⊢ x ∈ ↑(PrimeSpectrum.comap C) '' zeroLocus {f}ᶜ → x ∈ imageOfDf f",
"tactic": "rintro ⟨xli, complement, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case h.refine'_3.intro.intro\nR : Type u_1\ninst✝ : CommRing R\nf : R[X]\nxli : PrimeSpectrum R[X]\ncomplement : xli ∈ zeroLocus {f}ᶜ\n⊢ ↑(PrimeSpectrum.comap C) xli ∈ imageOfDf f",
"tactic": "exact comap_C_mem_imageOfDf complement"
}
] |
[
69,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
57,
1
] |
Mathlib/Algebra/Algebra/Basic.lean
|
NoZeroSMulDivisors.trans
|
[
{
"state_after": "R✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\nr : R\nm : M\nh : r • m = 0\n⊢ r = 0 ∨ m = 0",
"state_before": "R✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\n⊢ NoZeroSMulDivisors R M",
"tactic": "refine' ⟨fun {r m} h => _⟩"
},
{
"state_after": "R✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\nr : R\nm : M\nh : ↑(algebraMap R A) r • m = 0\n⊢ r = 0 ∨ m = 0",
"state_before": "R✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\nr : R\nm : M\nh : r • m = 0\n⊢ r = 0 ∨ m = 0",
"tactic": "rw [algebra_compatible_smul A r m] at h"
},
{
"state_after": "case inl\nR✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\nr : R\nm : M\nh : ↑(algebraMap R A) r • m = 0\nH : ↑(algebraMap R A) r = 0\n⊢ r = 0 ∨ m = 0\n\ncase inr\nR✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\nr : R\nm : M\nh : ↑(algebraMap R A) r • m = 0\nH : m = 0\n⊢ r = 0 ∨ m = 0",
"state_before": "R✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\nr : R\nm : M\nh : ↑(algebraMap R A) r • m = 0\n⊢ r = 0 ∨ m = 0",
"tactic": "cases' smul_eq_zero.1 h with H H"
},
{
"state_after": "case inl\nR✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\nr : R\nm : M\nh : ↑(algebraMap R A) r • m = 0\nH : ↑(algebraMap R A) r = 0\nthis : Function.Injective ↑(algebraMap R A)\n⊢ r = 0 ∨ m = 0",
"state_before": "case inl\nR✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\nr : R\nm : M\nh : ↑(algebraMap R A) r • m = 0\nH : ↑(algebraMap R A) r = 0\n⊢ r = 0 ∨ m = 0",
"tactic": "have : Function.Injective (algebraMap R A) :=\n NoZeroSMulDivisors.iff_algebraMap_injective.1 inferInstance"
},
{
"state_after": "case inl.h\nR✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\nr : R\nm : M\nh : ↑(algebraMap R A) r • m = 0\nH : ↑(algebraMap R A) r = 0\nthis : Function.Injective ↑(algebraMap R A)\n⊢ r = 0",
"state_before": "case inl\nR✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\nr : R\nm : M\nh : ↑(algebraMap R A) r • m = 0\nH : ↑(algebraMap R A) r = 0\nthis : Function.Injective ↑(algebraMap R A)\n⊢ r = 0 ∨ m = 0",
"tactic": "left"
},
{
"state_after": "no goals",
"state_before": "case inl.h\nR✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\nr : R\nm : M\nh : ↑(algebraMap R A) r • m = 0\nH : ↑(algebraMap R A) r = 0\nthis : Function.Injective ↑(algebraMap R A)\n⊢ r = 0",
"tactic": "exact (injective_iff_map_eq_zero _).1 this _ H"
},
{
"state_after": "case inr.h\nR✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\nr : R\nm : M\nh : ↑(algebraMap R A) r • m = 0\nH : m = 0\n⊢ m = 0",
"state_before": "case inr\nR✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\nr : R\nm : M\nh : ↑(algebraMap R A) r • m = 0\nH : m = 0\n⊢ r = 0 ∨ m = 0",
"tactic": "right"
},
{
"state_after": "no goals",
"state_before": "case inr.h\nR✝ : Type ?u.984855\ninst✝²⁰ : CommSemiring R✝\nA✝ : Type ?u.984861\ninst✝¹⁹ : Semiring A✝\ninst✝¹⁸ : Algebra R✝ A✝\nM✝ : Type ?u.984890\ninst✝¹⁷ : AddCommMonoid M✝\ninst✝¹⁶ : Module A✝ M✝\ninst✝¹⁵ : Module R✝ M✝\ninst✝¹⁴ : IsScalarTower R✝ A✝ M✝\nN : Type ?u.986785\ninst✝¹³ : AddCommMonoid N\ninst✝¹² : Module A✝ N\ninst✝¹¹ : Module R✝ N\ninst✝¹⁰ : IsScalarTower R✝ A✝ N\nR : Type u_1\nA : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : Ring A\ninst✝⁷ : IsDomain A\ninst✝⁶ : Algebra R A\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\ninst✝¹ : NoZeroSMulDivisors R A\ninst✝ : NoZeroSMulDivisors A M\nr : R\nm : M\nh : ↑(algebraMap R A) r • m = 0\nH : m = 0\n⊢ m = 0",
"tactic": "exact H"
}
] |
[
892,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
881,
1
] |
Mathlib/FieldTheory/IntermediateField.lean
|
IntermediateField.sum_mem
|
[] |
[
236,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
234,
11
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.filter_disj_union
|
[] |
[
2794,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2792,
1
] |
Mathlib/MeasureTheory/Function/AEEqOfIntegral.lean
|
MeasureTheory.AEFinStronglyMeasurable.ae_eq_of_forall_set_integral_eq
|
[
{
"state_after": "α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn f s\nhg_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn g s\nhfg_eq : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in s, g x ∂μ\nhf : AEFinStronglyMeasurable f μ\nhg : AEFinStronglyMeasurable g μ\n⊢ f - g =ᵐ[μ] 0",
"state_before": "α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn f s\nhg_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn g s\nhfg_eq : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in s, g x ∂μ\nhf : AEFinStronglyMeasurable f μ\nhg : AEFinStronglyMeasurable g μ\n⊢ f =ᵐ[μ] g",
"tactic": "rw [← sub_ae_eq_zero]"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn f s\nhg_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn g s\nhfg_eq : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in s, g x ∂μ\nhf : AEFinStronglyMeasurable f μ\nhg : AEFinStronglyMeasurable g μ\nhfg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, (f - g) x ∂μ) = 0\n⊢ f - g =ᵐ[μ] 0",
"state_before": "α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn f s\nhg_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn g s\nhfg_eq : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in s, g x ∂μ\nhf : AEFinStronglyMeasurable f μ\nhg : AEFinStronglyMeasurable g μ\n⊢ f - g =ᵐ[μ] 0",
"tactic": "have hfg : ∀ s : Set α, MeasurableSet s → μ s < ∞ → (∫ x in s, (f - g) x ∂μ) = 0 := by\n intro s hs hμs\n rw [integral_sub' (hf_int_finite s hs hμs) (hg_int_finite s hs hμs),\n sub_eq_zero.mpr (hfg_eq s hs hμs)]"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn f s\nhg_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn g s\nhfg_eq : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in s, g x ∂μ\nhf : AEFinStronglyMeasurable f μ\nhg : AEFinStronglyMeasurable g μ\nhfg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, (f - g) x ∂μ) = 0\nhfg_int : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (f - g) s\n⊢ f - g =ᵐ[μ] 0",
"state_before": "α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn f s\nhg_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn g s\nhfg_eq : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in s, g x ∂μ\nhf : AEFinStronglyMeasurable f μ\nhg : AEFinStronglyMeasurable g μ\nhfg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, (f - g) x ∂μ) = 0\n⊢ f - g =ᵐ[μ] 0",
"tactic": "have hfg_int : ∀ s, MeasurableSet s → μ s < ∞ → IntegrableOn (f - g) s μ := fun s hs hμs =>\n (hf_int_finite s hs hμs).sub (hg_int_finite s hs hμs)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn f s\nhg_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn g s\nhfg_eq : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in s, g x ∂μ\nhf : AEFinStronglyMeasurable f μ\nhg : AEFinStronglyMeasurable g μ\nhfg : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, (f - g) x ∂μ) = 0\nhfg_int : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn (f - g) s\n⊢ f - g =ᵐ[μ] 0",
"tactic": "exact (hf.sub hg).ae_eq_zero_of_forall_set_integral_eq_zero hfg_int hfg"
},
{
"state_after": "α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn f s\nhg_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn g s\nhfg_eq : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in s, g x ∂μ\nhf : AEFinStronglyMeasurable f μ\nhg : AEFinStronglyMeasurable g μ\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s < ⊤\n⊢ (∫ (x : α) in s, (f - g) x ∂μ) = 0",
"state_before": "α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ : Measure α\ns t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn f s\nhg_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn g s\nhfg_eq : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in s, g x ∂μ\nhf : AEFinStronglyMeasurable f μ\nhg : AEFinStronglyMeasurable g μ\n⊢ ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, (f - g) x ∂μ) = 0",
"tactic": "intro s hs hμs"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : Type u_2\nm m0 : MeasurableSpace α\nμ : Measure α\ns✝ t : Set α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\np : ℝ≥0∞\nf g : α → E\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn f s\nhg_int_finite : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → IntegrableOn g s\nhfg_eq : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → (∫ (x : α) in s, f x ∂μ) = ∫ (x : α) in s, g x ∂μ\nhf : AEFinStronglyMeasurable f μ\nhg : AEFinStronglyMeasurable g μ\ns : Set α\nhs : MeasurableSet s\nhμs : ↑↑μ s < ⊤\n⊢ (∫ (x : α) in s, (f - g) x ∂μ) = 0",
"tactic": "rw [integral_sub' (hf_int_finite s hs hμs) (hg_int_finite s hs hμs),\n sub_eq_zero.mpr (hfg_eq s hs hμs)]"
}
] |
[
471,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
459,
1
] |
Mathlib/LinearAlgebra/Lagrange.lean
|
Lagrange.basisDivisor_ne_zero_iff
|
[
{
"state_after": "no goals",
"state_before": "F : Type u_1\ninst✝ : Field F\nx y : F\n⊢ basisDivisor x y ≠ 0 ↔ x ≠ y",
"tactic": "rw [Ne.def, basisDivisor_eq_zero_iff]"
}
] |
[
148,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
147,
1
] |
Mathlib/Order/SupIndep.lean
|
CompleteLattice.independent_pair
|
[
{
"state_after": "case mp\nα : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\ni j : ι\nhij : i ≠ j\nhuniv : ∀ (k : ι), k = i ∨ k = j\n⊢ Independent t → Disjoint (t i) (t j)\n\ncase mpr\nα : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\ni j : ι\nhij : i ≠ j\nhuniv : ∀ (k : ι), k = i ∨ k = j\n⊢ Disjoint (t i) (t j) → Independent t",
"state_before": "α : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\ni j : ι\nhij : i ≠ j\nhuniv : ∀ (k : ι), k = i ∨ k = j\n⊢ Independent t ↔ Disjoint (t i) (t j)",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "case mp\nα : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\ni j : ι\nhij : i ≠ j\nhuniv : ∀ (k : ι), k = i ∨ k = j\n⊢ Independent t → Disjoint (t i) (t j)",
"tactic": "exact fun h => h.pairwiseDisjoint hij"
},
{
"state_after": "case mpr\nα : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\ni j : ι\nhij : i ≠ j\nhuniv : ∀ (k : ι), k = i ∨ k = j\nh : Disjoint (t i) (t j)\nk : ι\n⊢ Disjoint (t k) (⨆ (j : ι) (_ : j ≠ k), t j)",
"state_before": "case mpr\nα : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\ni j : ι\nhij : i ≠ j\nhuniv : ∀ (k : ι), k = i ∨ k = j\n⊢ Disjoint (t i) (t j) → Independent t",
"tactic": "rintro h k"
},
{
"state_after": "case mpr.inl\nα : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\nj k : ι\nhij : k ≠ j\nhuniv : ∀ (k_1 : ι), k_1 = k ∨ k_1 = j\nh : Disjoint (t k) (t j)\n⊢ Disjoint (t k) (⨆ (j : ι) (_ : j ≠ k), t j)\n\ncase mpr.inr\nα : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\ni k : ι\nhij : i ≠ k\nhuniv : ∀ (k_1 : ι), k_1 = i ∨ k_1 = k\nh : Disjoint (t i) (t k)\n⊢ Disjoint (t k) (⨆ (j : ι) (_ : j ≠ k), t j)",
"state_before": "case mpr\nα : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\ni j : ι\nhij : i ≠ j\nhuniv : ∀ (k : ι), k = i ∨ k = j\nh : Disjoint (t i) (t j)\nk : ι\n⊢ Disjoint (t k) (⨆ (j : ι) (_ : j ≠ k), t j)",
"tactic": "obtain rfl | rfl := huniv k"
},
{
"state_after": "case mpr.inl\nα : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\nj k : ι\nhij : k ≠ j\nhuniv : ∀ (k_1 : ι), k_1 = k ∨ k_1 = j\nh : Disjoint (t k) (t j)\ni : ι\nhi : i ≠ k\n⊢ t i = t j",
"state_before": "case mpr.inl\nα : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\nj k : ι\nhij : k ≠ j\nhuniv : ∀ (k_1 : ι), k_1 = k ∨ k_1 = j\nh : Disjoint (t k) (t j)\n⊢ Disjoint (t k) (⨆ (j : ι) (_ : j ≠ k), t j)",
"tactic": "refine' h.mono_right (iSup_le fun i => iSup_le fun hi => Eq.le _)"
},
{
"state_after": "no goals",
"state_before": "case mpr.inl\nα : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\nj k : ι\nhij : k ≠ j\nhuniv : ∀ (k_1 : ι), k_1 = k ∨ k_1 = j\nh : Disjoint (t k) (t j)\ni : ι\nhi : i ≠ k\n⊢ t i = t j",
"tactic": "rw [(huniv i).resolve_left hi]"
},
{
"state_after": "case mpr.inr\nα : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\ni k : ι\nhij : i ≠ k\nhuniv : ∀ (k_1 : ι), k_1 = i ∨ k_1 = k\nh : Disjoint (t i) (t k)\nj : ι\nhj : j ≠ k\n⊢ t j = t i",
"state_before": "case mpr.inr\nα : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\ni k : ι\nhij : i ≠ k\nhuniv : ∀ (k_1 : ι), k_1 = i ∨ k_1 = k\nh : Disjoint (t i) (t k)\n⊢ Disjoint (t k) (⨆ (j : ι) (_ : j ≠ k), t j)",
"tactic": "refine' h.symm.mono_right (iSup_le fun j => iSup_le fun hj => Eq.le _)"
},
{
"state_after": "no goals",
"state_before": "case mpr.inr\nα : Type u_2\nβ : Type ?u.54428\nι : Type u_1\nι' : Type ?u.54434\ninst✝ : CompleteLattice α\ns : Set α\nhs : SetIndependent s\nt : ι → α\nht : Independent t\ni k : ι\nhij : i ≠ k\nhuniv : ∀ (k_1 : ι), k_1 = i ∨ k_1 = k\nh : Disjoint (t i) (t k)\nj : ι\nhj : j ≠ k\n⊢ t j = t i",
"tactic": "rw [(huniv j).resolve_right hj]"
}
] |
[
334,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
325,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.Bounded.trans_left
|
[] |
[
958,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
955,
1
] |
Mathlib/Tactic/Ring/Basic.lean
|
Mathlib.Tactic.Ring.atom_pf
|
[
{
"state_after": "no goals",
"state_before": "u : Lean.Level\nR : Type u_1\nα : Q(Type u)\nsα : Q(CommSemiring «$α»)\ninst✝ : CommSemiring R\na : R\n⊢ a = a ^ Nat.rawCast 1 * Nat.rawCast 1 + 0",
"tactic": "simp"
}
] |
[
865,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
865,
1
] |
Mathlib/Data/Polynomial/Eval.lean
|
Polynomial.eval₂_multiset_prod
|
[] |
[
1109,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1107,
1
] |
Mathlib/CategoryTheory/Sites/Adjunction.lean
|
CategoryTheory.Sheaf.adjunctionToTypes_unit_app_val
|
[
{
"state_after": "C : Type u\ninst✝⁹ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁸ : Category D\nE : Type w₂\ninst✝⁷ : Category E\nF : D ⥤ E\nG✝ : E ⥤ D\ninst✝⁶ : (X : C) → (S : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst✝⁵ : ConcreteCategory D\ninst✝⁴ : PreservesLimits (forget D)\ninst✝³ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\nG : Type (max v u) ⥤ D\nadj : G ⊣ forget D\nY : SheafOfTypes J\n⊢ ((Equivalence.toAdjunction (Equivalence.symm (sheafEquivSheafOfTypes J))).unit.app Y).val ≫\n ((adjunction J adj).unit.app ((sheafEquivSheafOfTypes J).inverse.obj Y)).val ≫\n 𝟙 (sheafify J (Y.val ⋙ G) ⋙ forget D) =\n (Adjunction.whiskerRight Cᵒᵖ adj).unit.app Y.val ≫ whiskerRight (toSheafify J (Y.val ⋙ G)) (forget D)",
"state_before": "C : Type u\ninst✝⁹ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁸ : Category D\nE : Type w₂\ninst✝⁷ : Category E\nF : D ⥤ E\nG✝ : E ⥤ D\ninst✝⁶ : (X : C) → (S : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst✝⁵ : ConcreteCategory D\ninst✝⁴ : PreservesLimits (forget D)\ninst✝³ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\nG : Type (max v u) ⥤ D\nadj : G ⊣ forget D\nY : SheafOfTypes J\n⊢ ((adjunctionToTypes J adj).unit.app Y).val =\n (Adjunction.whiskerRight Cᵒᵖ adj).unit.app ((sheafOfTypesToPresheaf J).obj Y) ≫\n whiskerRight\n (toSheafify J (((whiskeringRight Cᵒᵖ (Type (max v u)) D).obj G).obj ((sheafOfTypesToPresheaf J).obj Y)))\n (forget D)",
"tactic": "dsimp [adjunctionToTypes, Adjunction.comp]"
},
{
"state_after": "C : Type u\ninst✝⁹ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁸ : Category D\nE : Type w₂\ninst✝⁷ : Category E\nF : D ⥤ E\nG✝ : E ⥤ D\ninst✝⁶ : (X : C) → (S : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst✝⁵ : ConcreteCategory D\ninst✝⁴ : PreservesLimits (forget D)\ninst✝³ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\nG : Type (max v u) ⥤ D\nadj : G ⊣ forget D\nY : SheafOfTypes J\n⊢ ((Equivalence.toAdjunction (Equivalence.symm (sheafEquivSheafOfTypes J))).unit.app Y).val ≫\n ↑(Adjunction.homEquiv (Adjunction.whiskerRight Cᵒᵖ adj) Y.val (sheafify J (Y.val ⋙ G)))\n (toSheafify J (Y.val ⋙ G)) =\n (Adjunction.whiskerRight Cᵒᵖ adj).unit.app Y.val ≫ whiskerRight (toSheafify J (Y.val ⋙ G)) (forget D)",
"state_before": "C : Type u\ninst✝⁹ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁸ : Category D\nE : Type w₂\ninst✝⁷ : Category E\nF : D ⥤ E\nG✝ : E ⥤ D\ninst✝⁶ : (X : C) → (S : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst✝⁵ : ConcreteCategory D\ninst✝⁴ : PreservesLimits (forget D)\ninst✝³ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\nG : Type (max v u) ⥤ D\nadj : G ⊣ forget D\nY : SheafOfTypes J\n⊢ ((Equivalence.toAdjunction (Equivalence.symm (sheafEquivSheafOfTypes J))).unit.app Y).val ≫\n ((adjunction J adj).unit.app ((sheafEquivSheafOfTypes J).inverse.obj Y)).val ≫\n 𝟙 (sheafify J (Y.val ⋙ G) ⋙ forget D) =\n (Adjunction.whiskerRight Cᵒᵖ adj).unit.app Y.val ≫ whiskerRight (toSheafify J (Y.val ⋙ G)) (forget D)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁹ : Category C\nJ : GrothendieckTopology C\nD : Type w₁\ninst✝⁸ : Category D\nE : Type w₂\ninst✝⁷ : Category E\nF : D ⥤ E\nG✝ : E ⥤ D\ninst✝⁶ : (X : C) → (S : Cover J X) → (P : Cᵒᵖ ⥤ D) → PreservesLimit (MulticospanIndex.multicospan (Cover.index S P)) F\ninst✝⁵ : ConcreteCategory D\ninst✝⁴ : PreservesLimits (forget D)\ninst✝³ : ∀ (P : Cᵒᵖ ⥤ D) (X : C) (S : Cover J X), HasMultiequalizer (Cover.index S P)\ninst✝² : ∀ (X : C), HasColimitsOfShape (Cover J X)ᵒᵖ D\ninst✝¹ : (X : C) → PreservesColimitsOfShape (Cover J X)ᵒᵖ (forget D)\ninst✝ : ReflectsIsomorphisms (forget D)\nG : Type (max v u) ⥤ D\nadj : G ⊣ forget D\nY : SheafOfTypes J\n⊢ ((Equivalence.toAdjunction (Equivalence.symm (sheafEquivSheafOfTypes J))).unit.app Y).val ≫\n ↑(Adjunction.homEquiv (Adjunction.whiskerRight Cᵒᵖ adj) Y.val (sheafify J (Y.val ⋙ G)))\n (toSheafify J (Y.val ⋙ G)) =\n (Adjunction.whiskerRight Cᵒᵖ adj).unit.app Y.val ≫ whiskerRight (toSheafify J (Y.val ⋙ G)) (forget D)",
"tactic": "rfl"
}
] |
[
136,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/Order/WithBot.lean
|
WithTop.lt_iff_exists_coe_btwn
|
[] |
[
1354,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1348,
1
] |
Mathlib/Data/Polynomial/Coeff.lean
|
Polynomial.finset_sum_coeff
|
[] |
[
109,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/Analysis/NormedSpace/Multilinear.lean
|
ContinuousMultilinearMap.mkPiField_apply_one_eq_self
|
[] |
[
902,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
900,
1
] |
Mathlib/Data/Setoid/Partition.lean
|
Finpartition.isPartition_parts
|
[] |
[
316,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
312,
1
] |
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean
|
LinearMap.toMatrix'_toLinearMapₛₗ₂'
|
[] |
[
259,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
257,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.smul_finset_subset_smul_finset_iff
|
[] |
[
1970,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1969,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
div_le_one_of_ge
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.148913\nα : Type u_1\nβ : Type ?u.148919\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\nh : b ≤ a\nhb : b ≤ 0\n⊢ a / b ≤ 1",
"tactic": "simpa only [neg_div_neg_eq] using div_le_one_of_le (neg_le_neg h) (neg_nonneg_of_nonpos hb)"
}
] |
[
753,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
752,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.mul_eq_one_iff
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.557283\nα : Type u_1\nβ : Type ?u.557289\nγ : Type ?u.557292\ninst✝² : DecidableEq α\ninst✝¹ : DecidableEq β\ninst✝ : DivisionMonoid α\ns t : Finset α\n⊢ s * t = 1 ↔ ∃ a b, s = {a} ∧ t = {b} ∧ a * b = 1",
"tactic": "simp_rw [← coe_inj, coe_mul, coe_one, Set.mul_eq_one_iff, coe_singleton]"
}
] |
[
1005,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1004,
11
] |
Mathlib/LinearAlgebra/Lagrange.lean
|
Polynomial.eq_zero_of_degree_lt_of_eval_index_eq_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nι : Type u_1\nv : ι → R\ns : Finset ι\nhvs : Set.InjOn v ↑s\ndegree_f_lt : degree f < ↑(card s)\neval_f : ∀ (i : ι), i ∈ s → eval (v i) f = 0\n⊢ f = 0",
"tactic": "classical\n rw [← card_image_of_injOn hvs] at degree_f_lt\n refine' eq_zero_of_degree_lt_of_eval_finset_eq_zero _ degree_f_lt _\n intro x hx\n rcases mem_image.mp hx with ⟨_, hj, rfl⟩\n exact eval_f _ hj"
},
{
"state_after": "R : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nι : Type u_1\nv : ι → R\ns : Finset ι\nhvs : Set.InjOn v ↑s\ndegree_f_lt : degree f < ↑(card (image v s))\neval_f : ∀ (i : ι), i ∈ s → eval (v i) f = 0\n⊢ f = 0",
"state_before": "R : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nι : Type u_1\nv : ι → R\ns : Finset ι\nhvs : Set.InjOn v ↑s\ndegree_f_lt : degree f < ↑(card s)\neval_f : ∀ (i : ι), i ∈ s → eval (v i) f = 0\n⊢ f = 0",
"tactic": "rw [← card_image_of_injOn hvs] at degree_f_lt"
},
{
"state_after": "R : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nι : Type u_1\nv : ι → R\ns : Finset ι\nhvs : Set.InjOn v ↑s\ndegree_f_lt : degree f < ↑(card (image v s))\neval_f : ∀ (i : ι), i ∈ s → eval (v i) f = 0\n⊢ ∀ (x : R), x ∈ image v s → eval x f = 0",
"state_before": "R : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nι : Type u_1\nv : ι → R\ns : Finset ι\nhvs : Set.InjOn v ↑s\ndegree_f_lt : degree f < ↑(card (image v s))\neval_f : ∀ (i : ι), i ∈ s → eval (v i) f = 0\n⊢ f = 0",
"tactic": "refine' eq_zero_of_degree_lt_of_eval_finset_eq_zero _ degree_f_lt _"
},
{
"state_after": "R : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nι : Type u_1\nv : ι → R\ns : Finset ι\nhvs : Set.InjOn v ↑s\ndegree_f_lt : degree f < ↑(card (image v s))\neval_f : ∀ (i : ι), i ∈ s → eval (v i) f = 0\nx : R\nhx : x ∈ image v s\n⊢ eval x f = 0",
"state_before": "R : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nι : Type u_1\nv : ι → R\ns : Finset ι\nhvs : Set.InjOn v ↑s\ndegree_f_lt : degree f < ↑(card (image v s))\neval_f : ∀ (i : ι), i ∈ s → eval (v i) f = 0\n⊢ ∀ (x : R), x ∈ image v s → eval x f = 0",
"tactic": "intro x hx"
},
{
"state_after": "case intro.intro\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nι : Type u_1\nv : ι → R\ns : Finset ι\nhvs : Set.InjOn v ↑s\ndegree_f_lt : degree f < ↑(card (image v s))\neval_f : ∀ (i : ι), i ∈ s → eval (v i) f = 0\nw✝ : ι\nhj : w✝ ∈ s\nhx : v w✝ ∈ image v s\n⊢ eval (v w✝) f = 0",
"state_before": "R : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nι : Type u_1\nv : ι → R\ns : Finset ι\nhvs : Set.InjOn v ↑s\ndegree_f_lt : degree f < ↑(card (image v s))\neval_f : ∀ (i : ι), i ∈ s → eval (v i) f = 0\nx : R\nhx : x ∈ image v s\n⊢ eval x f = 0",
"tactic": "rcases mem_image.mp hx with ⟨_, hj, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nf g : R[X]\nι : Type u_1\nv : ι → R\ns : Finset ι\nhvs : Set.InjOn v ↑s\ndegree_f_lt : degree f < ↑(card (image v s))\neval_f : ∀ (i : ι), i ∈ s → eval (v i) f = 0\nw✝ : ι\nhj : w✝ ∈ s\nhx : v w✝ ∈ image v s\n⊢ eval (v w✝) f = 0",
"tactic": "exact eval_f _ hj"
}
] |
[
88,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
81,
1
] |
Mathlib/Computability/Language.lean
|
Language.map_id
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.42797\nγ : Type ?u.42800\nl✝ m : Language α\na b x : List α\nl : Language α\n⊢ ↑(map id) l = l",
"tactic": "simp [map]"
}
] |
[
171,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
171,
1
] |
Mathlib/Analysis/Normed/Group/Seminorm.lean
|
NonarchAddGroupNorm.ext
|
[] |
[
924,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
923,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.eq_C_of_degree_eq_zero
|
[] |
[
627,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
626,
1
] |
Mathlib/CategoryTheory/Preadditive/Biproducts.lean
|
CategoryTheory.Limits.HasBinaryBiproduct.of_hasBinaryCoproduct
|
[] |
[
433,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
429,
1
] |
Mathlib/Data/Set/Image.lean
|
Subtype.preimage_coe_compl
|
[] |
[
1477,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1476,
1
] |
Mathlib/Logic/Equiv/Basic.lean
|
Equiv.swap_self
|
[] |
[
1551,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1550,
1
] |
Mathlib/CategoryTheory/Abelian/RightDerived.lean
|
CategoryTheory.NatTrans.rightDerived_id
|
[
{
"state_after": "C : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Functor.Additive F\nn : ℕ\n⊢ 𝟙\n (injectiveResolutions C ⋙\n Functor.mapHomotopyCategory F (ComplexShape.up ℕ) ⋙ HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n) =\n 𝟙 (Functor.rightDerived F n)",
"state_before": "C : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Functor.Additive F\nn : ℕ\n⊢ rightDerived (𝟙 F) n = 𝟙 (Functor.rightDerived F n)",
"tactic": "simp [NatTrans.rightDerived]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁵ : Category C\nD : Type u_2\ninst✝⁴ : Category D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : Functor.Additive F\nn : ℕ\n⊢ 𝟙\n (injectiveResolutions C ⋙\n Functor.mapHomotopyCategory F (ComplexShape.up ℕ) ⋙ HomotopyCategory.homologyFunctor D (ComplexShape.up ℕ) n) =\n 𝟙 (Functor.rightDerived F n)",
"tactic": "rfl"
}
] |
[
142,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
139,
1
] |
Mathlib/Algebra/Module/Basic.lean
|
rat_cast_smul_eq
|
[] |
[
545,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
543,
1
] |
Mathlib/Algebra/Group/Semiconj.lean
|
SemiconjBy.units_inv_symm_left_iff
|
[] |
[
147,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
InnerProductSpace.Core.inner_smul_right
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type ?u.527762\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\nr : 𝕜\n⊢ ↑(starRingEnd 𝕜) (↑(starRingEnd 𝕜) r * inner y x) = r * inner x y",
"state_before": "𝕜 : Type u_1\nE : Type ?u.527762\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\nr : 𝕜\n⊢ inner x (r • y) = r * inner x y",
"tactic": "rw [← inner_conj_symm, inner_smul_left]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type ?u.527762\nF : Type u_2\ninst✝² : IsROrC 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : Core 𝕜 F\nx y : F\nr : 𝕜\n⊢ ↑(starRingEnd 𝕜) (↑(starRingEnd 𝕜) r * inner y x) = r * inner x y",
"tactic": "simp only [conj_conj, inner_conj_symm, RingHom.map_mul]"
}
] |
[
246,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
244,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.AEStronglyMeasurable.nnnorm
|
[] |
[
1471,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1469,
11
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
MeasureTheory.Lp.norm_def
|
[] |
[
263,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
262,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.OuterMeasure.smul_apply
|
[] |
[
317,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
316,
1
] |
Mathlib/Algebra/GroupWithZero/Units/Lemmas.lean
|
div_mul_left
|
[] |
[
120,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Algebra/Ring/BooleanRing.lean
|
mul_self
|
[] |
[
65,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/MeasureTheory/Group/Measure.lean
|
MeasureTheory.forall_measure_preimage_mul_right_iff
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.183132\nG : Type u_1\nH : Type ?u.183138\ninst✝³ : MeasurableSpace G\ninst✝² : MeasurableSpace H\ninst✝¹ : Mul G\nμ✝ : Measure G\ninst✝ : MeasurableMul G\nμ : Measure G\n⊢ (∀ (g : G), Measure.map (fun x => x * g) μ = μ) ↔ IsMulRightInvariant μ",
"tactic": "exact ⟨fun h => ⟨h⟩, fun h => h.1⟩"
},
{
"state_after": "𝕜 : Type ?u.183132\nG : Type u_1\nH : Type ?u.183138\ninst✝³ : MeasurableSpace G\ninst✝² : MeasurableSpace H\ninst✝¹ : Mul G\nμ✝ : Measure G\ninst✝ : MeasurableMul G\nμ : Measure G\n⊢ (∀ (g : G) (A : Set G), MeasurableSet A → ↑↑μ ((fun h => h * g) ⁻¹' A) = ↑↑μ A) ↔\n ∀ (g : G) (s : Set G), MeasurableSet s → ↑↑(Measure.map (fun x => x * g) μ) s = ↑↑μ s",
"state_before": "𝕜 : Type ?u.183132\nG : Type u_1\nH : Type ?u.183138\ninst✝³ : MeasurableSpace G\ninst✝² : MeasurableSpace H\ninst✝¹ : Mul G\nμ✝ : Measure G\ninst✝ : MeasurableMul G\nμ : Measure G\n⊢ (∀ (g : G) (A : Set G), MeasurableSet A → ↑↑μ ((fun h => h * g) ⁻¹' A) = ↑↑μ A) ↔\n ∀ (g : G), Measure.map (fun x => x * g) μ = μ",
"tactic": "simp_rw [Measure.ext_iff]"
},
{
"state_after": "𝕜 : Type ?u.183132\nG : Type u_1\nH : Type ?u.183138\ninst✝³ : MeasurableSpace G\ninst✝² : MeasurableSpace H\ninst✝¹ : Mul G\nμ✝ : Measure G\ninst✝ : MeasurableMul G\nμ : Measure G\ng : G\nA : Set G\nhA : MeasurableSet A\n⊢ ↑↑μ ((fun h => h * g) ⁻¹' A) = ↑↑μ A ↔ ↑↑(Measure.map (fun x => x * g) μ) A = ↑↑μ A",
"state_before": "𝕜 : Type ?u.183132\nG : Type u_1\nH : Type ?u.183138\ninst✝³ : MeasurableSpace G\ninst✝² : MeasurableSpace H\ninst✝¹ : Mul G\nμ✝ : Measure G\ninst✝ : MeasurableMul G\nμ : Measure G\n⊢ (∀ (g : G) (A : Set G), MeasurableSet A → ↑↑μ ((fun h => h * g) ⁻¹' A) = ↑↑μ A) ↔\n ∀ (g : G) (s : Set G), MeasurableSet s → ↑↑(Measure.map (fun x => x * g) μ) s = ↑↑μ s",
"tactic": "refine' forall_congr' fun g => forall_congr' fun A => forall_congr' fun hA => _"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.183132\nG : Type u_1\nH : Type ?u.183138\ninst✝³ : MeasurableSpace G\ninst✝² : MeasurableSpace H\ninst✝¹ : Mul G\nμ✝ : Measure G\ninst✝ : MeasurableMul G\nμ : Measure G\ng : G\nA : Set G\nhA : MeasurableSet A\n⊢ ↑↑μ ((fun h => h * g) ⁻¹' A) = ↑↑μ A ↔ ↑↑(Measure.map (fun x => x * g) μ) A = ↑↑μ A",
"tactic": "rw [map_apply (measurable_mul_const g) hA]"
}
] |
[
201,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
uniformContinuous_def
|
[] |
[
1107,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1105,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean
|
Prod.edist_eq
|
[] |
[
464,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
462,
1
] |
Mathlib/Topology/Bornology/Constructions.lean
|
Bornology.isBounded_pi
|
[
{
"state_after": "case pos\nα : Type ?u.78124\nβ : Type ?u.78127\nι : Type u_2\nπ : ι → Type u_1\ninst✝³ : Fintype ι\ninst✝² : Bornology α\ninst✝¹ : Bornology β\ninst✝ : (i : ι) → Bornology (π i)\ns : Set α\nt : Set β\nS : (i : ι) → Set (π i)\nhne : ∃ i, S i = ∅\n⊢ IsBounded (Set.pi univ S) ↔ (∃ i, S i = ∅) ∨ ∀ (i : ι), IsBounded (S i)\n\ncase neg\nα : Type ?u.78124\nβ : Type ?u.78127\nι : Type u_2\nπ : ι → Type u_1\ninst✝³ : Fintype ι\ninst✝² : Bornology α\ninst✝¹ : Bornology β\ninst✝ : (i : ι) → Bornology (π i)\ns : Set α\nt : Set β\nS : (i : ι) → Set (π i)\nhne : ¬∃ i, S i = ∅\n⊢ IsBounded (Set.pi univ S) ↔ (∃ i, S i = ∅) ∨ ∀ (i : ι), IsBounded (S i)",
"state_before": "α : Type ?u.78124\nβ : Type ?u.78127\nι : Type u_2\nπ : ι → Type u_1\ninst✝³ : Fintype ι\ninst✝² : Bornology α\ninst✝¹ : Bornology β\ninst✝ : (i : ι) → Bornology (π i)\ns : Set α\nt : Set β\nS : (i : ι) → Set (π i)\n⊢ IsBounded (Set.pi univ S) ↔ (∃ i, S i = ∅) ∨ ∀ (i : ι), IsBounded (S i)",
"tactic": "by_cases hne : ∃ i, S i = ∅"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type ?u.78124\nβ : Type ?u.78127\nι : Type u_2\nπ : ι → Type u_1\ninst✝³ : Fintype ι\ninst✝² : Bornology α\ninst✝¹ : Bornology β\ninst✝ : (i : ι) → Bornology (π i)\ns : Set α\nt : Set β\nS : (i : ι) → Set (π i)\nhne : ∃ i, S i = ∅\n⊢ IsBounded (Set.pi univ S) ↔ (∃ i, S i = ∅) ∨ ∀ (i : ι), IsBounded (S i)",
"tactic": "simp [hne, univ_pi_eq_empty_iff.2 hne]"
},
{
"state_after": "case neg\nα : Type ?u.78124\nβ : Type ?u.78127\nι : Type u_2\nπ : ι → Type u_1\ninst✝³ : Fintype ι\ninst✝² : Bornology α\ninst✝¹ : Bornology β\ninst✝ : (i : ι) → Bornology (π i)\ns : Set α\nt : Set β\nS : (i : ι) → Set (π i)\nhne : ¬∃ i, S i = ∅\n⊢ IsBounded (Set.pi univ S) ↔ ∀ (i : ι), IsBounded (S i)",
"state_before": "case neg\nα : Type ?u.78124\nβ : Type ?u.78127\nι : Type u_2\nπ : ι → Type u_1\ninst✝³ : Fintype ι\ninst✝² : Bornology α\ninst✝¹ : Bornology β\ninst✝ : (i : ι) → Bornology (π i)\ns : Set α\nt : Set β\nS : (i : ι) → Set (π i)\nhne : ¬∃ i, S i = ∅\n⊢ IsBounded (Set.pi univ S) ↔ (∃ i, S i = ∅) ∨ ∀ (i : ι), IsBounded (S i)",
"tactic": "simp only [hne, false_or_iff]"
},
{
"state_after": "case neg\nα : Type ?u.78124\nβ : Type ?u.78127\nι : Type u_2\nπ : ι → Type u_1\ninst✝³ : Fintype ι\ninst✝² : Bornology α\ninst✝¹ : Bornology β\ninst✝ : (i : ι) → Bornology (π i)\ns : Set α\nt : Set β\nS : (i : ι) → Set (π i)\nhne : Set.Nonempty (Set.pi univ fun i => S i)\n⊢ IsBounded (Set.pi univ S) ↔ ∀ (i : ι), IsBounded (S i)",
"state_before": "case neg\nα : Type ?u.78124\nβ : Type ?u.78127\nι : Type u_2\nπ : ι → Type u_1\ninst✝³ : Fintype ι\ninst✝² : Bornology α\ninst✝¹ : Bornology β\ninst✝ : (i : ι) → Bornology (π i)\ns : Set α\nt : Set β\nS : (i : ι) → Set (π i)\nhne : ¬∃ i, S i = ∅\n⊢ IsBounded (Set.pi univ S) ↔ ∀ (i : ι), IsBounded (S i)",
"tactic": "simp only [not_exists, ← Ne.def, ← nonempty_iff_ne_empty, ← univ_pi_nonempty_iff] at hne"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type ?u.78124\nβ : Type ?u.78127\nι : Type u_2\nπ : ι → Type u_1\ninst✝³ : Fintype ι\ninst✝² : Bornology α\ninst✝¹ : Bornology β\ninst✝ : (i : ι) → Bornology (π i)\ns : Set α\nt : Set β\nS : (i : ι) → Set (π i)\nhne : Set.Nonempty (Set.pi univ fun i => S i)\n⊢ IsBounded (Set.pi univ S) ↔ ∀ (i : ι), IsBounded (S i)",
"tactic": "exact isBounded_pi_of_nonempty hne"
}
] |
[
126,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/Topology/Algebra/Polynomial.lean
|
Polynomial.exists_forall_norm_le
|
[
{
"state_after": "α : Type ?u.177157\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : IsAbsoluteValue norm\ninst✝ : ProperSpace R\np : R[X]\nhp0 : ¬0 < degree p\n⊢ ∀ (y : R), ‖eval (coeff (↑C (coeff p 0)) 0) (↑C (coeff p 0))‖ ≤ ‖eval y (↑C (coeff p 0))‖",
"state_before": "α : Type ?u.177157\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : IsAbsoluteValue norm\ninst✝ : ProperSpace R\np : R[X]\nhp0 : ¬0 < degree p\n⊢ ∀ (y : R), ‖eval (coeff p 0) p‖ ≤ ‖eval y p‖",
"tactic": "rw [eq_C_of_degree_le_zero (le_of_not_gt hp0)]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.177157\nR : Type u_1\ninst✝² : NormedRing R\ninst✝¹ : IsAbsoluteValue norm\ninst✝ : ProperSpace R\np : R[X]\nhp0 : ¬0 < degree p\n⊢ ∀ (y : R), ‖eval (coeff (↑C (coeff p 0)) 0) (↑C (coeff p 0))‖ ≤ ‖eval y (↑C (coeff p 0))‖",
"tactic": "simp"
}
] |
[
145,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
141,
1
] |
Mathlib/Data/Bitvec/Lemmas.lean
|
Bitvec.ofNat_toNat
|
[
{
"state_after": "case mk\nn : ℕ\nxs : List Bool\nh : List.length xs = n\n⊢ Bitvec.ofNat n (Bitvec.toNat { val := xs, property := h }) = { val := xs, property := h }",
"state_before": "n : ℕ\nv : Bitvec n\n⊢ Bitvec.ofNat n (Bitvec.toNat v) = v",
"tactic": "cases' v with xs h"
},
{
"state_after": "case mk.a\nn : ℕ\nxs : List Bool\nh : List.length xs = n\n⊢ ↑(Bitvec.ofNat n (Bitvec.toNat { val := xs, property := h })) = ↑{ val := xs, property := h }",
"state_before": "case mk\nn : ℕ\nxs : List Bool\nh : List.length xs = n\n⊢ Bitvec.ofNat n (Bitvec.toNat { val := xs, property := h }) = { val := xs, property := h }",
"tactic": "apply Subtype.ext"
},
{
"state_after": "case mk.a\nn : ℕ\nxs : List Bool\nh : List.length xs = n\n⊢ Vector.toList (Bitvec.ofNat n (Bitvec.toNat { val := xs, property := h })) = xs",
"state_before": "case mk.a\nn : ℕ\nxs : List Bool\nh : List.length xs = n\n⊢ ↑(Bitvec.ofNat n (Bitvec.toNat { val := xs, property := h })) = ↑{ val := xs, property := h }",
"tactic": "change Vector.toList _ = xs"
},
{
"state_after": "case mk.a\nn : ℕ\nxs : List Bool\nh : List.length xs = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldl addLsb 0 xs)) = xs",
"state_before": "case mk.a\nn : ℕ\nxs : List Bool\nh : List.length xs = n\n⊢ Vector.toList (Bitvec.ofNat n (Bitvec.toNat { val := xs, property := h })) = xs",
"tactic": "dsimp [Bitvec.toNat, bitsToNat]"
},
{
"state_after": "case mk.a\nn : ℕ\nxs : List Bool\nh : List.length (List.reverse xs) = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldl addLsb 0 xs)) = xs",
"state_before": "case mk.a\nn : ℕ\nxs : List Bool\nh : List.length xs = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldl addLsb 0 xs)) = xs",
"tactic": "rw [← List.length_reverse] at h"
},
{
"state_after": "case mk.a\nn : ℕ\nxs : List Bool\nh : List.length (List.reverse xs) = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 (List.reverse xs))) =\n List.reverse (List.reverse xs)",
"state_before": "case mk.a\nn : ℕ\nxs : List Bool\nh : List.length (List.reverse xs) = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldl addLsb 0 xs)) = xs",
"tactic": "rw [← List.reverse_reverse xs, List.foldl_reverse]"
},
{
"state_after": "case mk.a\nn : ℕ\nxs ys : List Bool\nh : List.length ys = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys)) = List.reverse ys",
"state_before": "case mk.a\nn : ℕ\nxs : List Bool\nh : List.length (List.reverse xs) = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 (List.reverse xs))) =\n List.reverse (List.reverse xs)",
"tactic": "generalize xs.reverse = ys at h⊢"
},
{
"state_after": "case mk.a\nn : ℕ\nys : List Bool\nh : List.length ys = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys)) = List.reverse ys",
"state_before": "case mk.a\nn : ℕ\nxs ys : List Bool\nh : List.length ys = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys)) = List.reverse ys",
"tactic": "clear xs"
},
{
"state_after": "case mk.a.nil\nn✝ : ℕ\nys : List Bool\nh✝ : List.length ys = n✝\nn : ℕ\nh : List.length [] = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 [])) = List.reverse []\n\ncase mk.a.cons\nn✝ : ℕ\nys : List Bool\nh✝ : List.length ys = n✝\nys_head : Bool\nys_tail : List Bool\nys_ih :\n ∀ {n : ℕ},\n List.length ys_tail = n →\n Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\nn : ℕ\nh : List.length (ys_head :: ys_tail) = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 (ys_head :: ys_tail))) =\n List.reverse (ys_head :: ys_tail)",
"state_before": "case mk.a\nn : ℕ\nys : List Bool\nh : List.length ys = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys)) = List.reverse ys",
"tactic": "induction' ys with ys_head ys_tail ys_ih generalizing n"
},
{
"state_after": "case mk.a.nil.refl\nn : ℕ\nys : List Bool\nh : List.length ys = n\n⊢ Vector.toList (Bitvec.ofNat (List.length []) (List.foldr (fun x y => addLsb y x) 0 [])) = List.reverse []",
"state_before": "case mk.a.nil\nn✝ : ℕ\nys : List Bool\nh✝ : List.length ys = n✝\nn : ℕ\nh : List.length [] = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 [])) = List.reverse []",
"tactic": "cases h"
},
{
"state_after": "no goals",
"state_before": "case mk.a.nil.refl\nn : ℕ\nys : List Bool\nh : List.length ys = n\n⊢ Vector.toList (Bitvec.ofNat (List.length []) (List.foldr (fun x y => addLsb y x) 0 [])) = List.reverse []",
"tactic": "simp [Bitvec.ofNat]"
},
{
"state_after": "case mk.a.cons\nn✝ : ℕ\nys : List Bool\nh✝ : List.length ys = n✝\nys_head : Bool\nys_tail : List Bool\nys_ih :\n ∀ {n : ℕ},\n List.length ys_tail = n →\n Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\nn : ℕ\nh : succ (List.length ys_tail) = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 (ys_head :: ys_tail))) =\n List.reverse (ys_head :: ys_tail)",
"state_before": "case mk.a.cons\nn✝ : ℕ\nys : List Bool\nh✝ : List.length ys = n✝\nys_head : Bool\nys_tail : List Bool\nys_ih :\n ∀ {n : ℕ},\n List.length ys_tail = n →\n Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\nn : ℕ\nh : List.length (ys_head :: ys_tail) = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 (ys_head :: ys_tail))) =\n List.reverse (ys_head :: ys_tail)",
"tactic": "simp only [← Nat.succ_eq_add_one, List.length] at h"
},
{
"state_after": "case mk.a.cons\nn : ℕ\nys : List Bool\nh : List.length ys = n\nys_head : Bool\nys_tail : List Bool\nys_ih :\n ∀ {n : ℕ},\n List.length ys_tail = n →\n Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\n⊢ Vector.toList\n (Bitvec.ofNat (succ (List.length ys_tail)) (List.foldr (fun x y => addLsb y x) 0 (ys_head :: ys_tail))) =\n List.reverse (ys_head :: ys_tail)",
"state_before": "case mk.a.cons\nn✝ : ℕ\nys : List Bool\nh✝ : List.length ys = n✝\nys_head : Bool\nys_tail : List Bool\nys_ih :\n ∀ {n : ℕ},\n List.length ys_tail = n →\n Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\nn : ℕ\nh : succ (List.length ys_tail) = n\n⊢ Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 (ys_head :: ys_tail))) =\n List.reverse (ys_head :: ys_tail)",
"tactic": "subst n"
},
{
"state_after": "case mk.a.cons\nn : ℕ\nys : List Bool\nh : List.length ys = n\nys_head : Bool\nys_tail : List Bool\nys_ih :\n ∀ {n : ℕ},\n List.length ys_tail = n →\n Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\n⊢ Vector.toList\n (Bitvec.ofNat (List.length ys_tail) (addLsb (List.foldr (fun x y => addLsb y x) 0 ys_tail) ys_head / 2)) ++\n [decide (addLsb (List.foldr (fun x y => addLsb y x) 0 ys_tail) ys_head % 2 = 1)] =\n List.reverse ys_tail ++ [ys_head]",
"state_before": "case mk.a.cons\nn : ℕ\nys : List Bool\nh : List.length ys = n\nys_head : Bool\nys_tail : List Bool\nys_ih :\n ∀ {n : ℕ},\n List.length ys_tail = n →\n Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\n⊢ Vector.toList\n (Bitvec.ofNat (succ (List.length ys_tail)) (List.foldr (fun x y => addLsb y x) 0 (ys_head :: ys_tail))) =\n List.reverse (ys_head :: ys_tail)",
"tactic": "simp only [Bitvec.ofNat, Vector.toList_cons, Vector.toList_nil, List.reverse_cons,\n Vector.toList_append, List.foldr]"
},
{
"state_after": "case mk.a.cons\nn : ℕ\nys : List Bool\nh : List.length ys = n\nys_head : Bool\nys_tail : List Bool\nys_ih :\n ∀ {n : ℕ},\n List.length ys_tail = n →\n Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\n⊢ Vector.toList (Bitvec.ofNat (List.length ys_tail) (List.foldr (fun x y => addLsb y x) 0 ys_tail)) ++ [ys_head] =\n List.reverse ys_tail ++ [ys_head]",
"state_before": "case mk.a.cons\nn : ℕ\nys : List Bool\nh : List.length ys = n\nys_head : Bool\nys_tail : List Bool\nys_ih :\n ∀ {n : ℕ},\n List.length ys_tail = n →\n Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\n⊢ Vector.toList\n (Bitvec.ofNat (List.length ys_tail) (addLsb (List.foldr (fun x y => addLsb y x) 0 ys_tail) ys_head / 2)) ++\n [decide (addLsb (List.foldr (fun x y => addLsb y x) 0 ys_tail) ys_head % 2 = 1)] =\n List.reverse ys_tail ++ [ys_head]",
"tactic": "erw [addLsb_div_two, decide_addLsb_mod_two]"
},
{
"state_after": "case mk.a.cons.e_a\nn : ℕ\nys : List Bool\nh : List.length ys = n\nys_head : Bool\nys_tail : List Bool\nys_ih :\n ∀ {n : ℕ},\n List.length ys_tail = n →\n Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\n⊢ Vector.toList (Bitvec.ofNat (List.length ys_tail) (List.foldr (fun x y => addLsb y x) 0 ys_tail)) =\n List.reverse ys_tail",
"state_before": "case mk.a.cons\nn : ℕ\nys : List Bool\nh : List.length ys = n\nys_head : Bool\nys_tail : List Bool\nys_ih :\n ∀ {n : ℕ},\n List.length ys_tail = n →\n Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\n⊢ Vector.toList (Bitvec.ofNat (List.length ys_tail) (List.foldr (fun x y => addLsb y x) 0 ys_tail)) ++ [ys_head] =\n List.reverse ys_tail ++ [ys_head]",
"tactic": "congr"
},
{
"state_after": "case mk.a.cons.e_a.h\nn : ℕ\nys : List Bool\nh : List.length ys = n\nys_head : Bool\nys_tail : List Bool\nys_ih :\n ∀ {n : ℕ},\n List.length ys_tail = n →\n Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\n⊢ List.length ys_tail = List.length ys_tail",
"state_before": "case mk.a.cons.e_a\nn : ℕ\nys : List Bool\nh : List.length ys = n\nys_head : Bool\nys_tail : List Bool\nys_ih :\n ∀ {n : ℕ},\n List.length ys_tail = n →\n Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\n⊢ Vector.toList (Bitvec.ofNat (List.length ys_tail) (List.foldr (fun x y => addLsb y x) 0 ys_tail)) =\n List.reverse ys_tail",
"tactic": "apply ys_ih"
},
{
"state_after": "no goals",
"state_before": "case mk.a.cons.e_a.h\nn : ℕ\nys : List Bool\nh : List.length ys = n\nys_head : Bool\nys_tail : List Bool\nys_ih :\n ∀ {n : ℕ},\n List.length ys_tail = n →\n Vector.toList (Bitvec.ofNat n (List.foldr (fun x y => addLsb y x) 0 ys_tail)) = List.reverse ys_tail\n⊢ List.length ys_tail = List.length ys_tail",
"tactic": "rfl"
}
] |
[
147,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
128,
1
] |
Mathlib/Topology/Algebra/Order/Field.lean
|
tendsto_inv_zero_atTop
|
[
{
"state_after": "𝕜 : Type u_1\nα : Type ?u.24190\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\nb : 𝕜\nhb : 1 ≤ b\n⊢ ∀ᶠ (x : 𝕜) in 𝓝[Ioi 0] 0, x⁻¹ ∈ Ici b",
"state_before": "𝕜 : Type u_1\nα : Type ?u.24190\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\n⊢ Tendsto (fun x => x⁻¹) (𝓝[Ioi 0] 0) atTop",
"tactic": "refine' (atTop_basis' 1).tendsto_right_iff.2 fun b hb => _"
},
{
"state_after": "𝕜 : Type u_1\nα : Type ?u.24190\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\nb : 𝕜\nhb : 1 ≤ b\nhb' : 0 < b\n⊢ ∀ᶠ (x : 𝕜) in 𝓝[Ioi 0] 0, x⁻¹ ∈ Ici b",
"state_before": "𝕜 : Type u_1\nα : Type ?u.24190\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\nb : 𝕜\nhb : 1 ≤ b\n⊢ ∀ᶠ (x : 𝕜) in 𝓝[Ioi 0] 0, x⁻¹ ∈ Ici b",
"tactic": "have hb' : 0 < b := by positivity"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nα : Type ?u.24190\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\nb : 𝕜\nhb : 1 ≤ b\nhb' : 0 < b\n⊢ ∀ᶠ (x : 𝕜) in 𝓝[Ioi 0] 0, x⁻¹ ∈ Ici b",
"tactic": "filter_upwards [Ioc_mem_nhdsWithin_Ioi\n ⟨le_rfl, inv_pos.2 hb'⟩]with x hx using(le_inv hx.1 hb').1 hx.2"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nα : Type ?u.24190\ninst✝² : LinearOrderedField 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\nb : 𝕜\nhb : 1 ≤ b\n⊢ 0 < b",
"tactic": "positivity"
}
] |
[
128,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
124,
1
] |
Mathlib/ModelTheory/Syntax.lean
|
FirstOrder.Language.BoundedFormula.sum_elim_comp_relabelAux
|
[
{
"state_after": "case h\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.79266\nP : Type ?u.79269\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.79297\nn m : ℕ\ng : α → β ⊕ Fin n\nv : β → M\nxs : Fin (n + m) → M\nx : α ⊕ Fin m\n⊢ (Sum.elim v xs ∘ relabelAux g m) x = Sum.elim (Sum.elim v (xs ∘ ↑(castAdd m)) ∘ g) (xs ∘ ↑(natAdd n)) x",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.79266\nP : Type ?u.79269\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.79297\nn m : ℕ\ng : α → β ⊕ Fin n\nv : β → M\nxs : Fin (n + m) → M\n⊢ Sum.elim v xs ∘ relabelAux g m = Sum.elim (Sum.elim v (xs ∘ ↑(castAdd m)) ∘ g) (xs ∘ ↑(natAdd n))",
"tactic": "ext x"
},
{
"state_after": "case h.inl\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.79266\nP : Type ?u.79269\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.79297\nn m : ℕ\ng : α → β ⊕ Fin n\nv : β → M\nxs : Fin (n + m) → M\nx : α\n⊢ (Sum.elim v xs ∘ relabelAux g m) (Sum.inl x) =\n Sum.elim (Sum.elim v (xs ∘ ↑(castAdd m)) ∘ g) (xs ∘ ↑(natAdd n)) (Sum.inl x)\n\ncase h.inr\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.79266\nP : Type ?u.79269\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.79297\nn m : ℕ\ng : α → β ⊕ Fin n\nv : β → M\nxs : Fin (n + m) → M\nx : Fin m\n⊢ (Sum.elim v xs ∘ relabelAux g m) (Sum.inr x) =\n Sum.elim (Sum.elim v (xs ∘ ↑(castAdd m)) ∘ g) (xs ∘ ↑(natAdd n)) (Sum.inr x)",
"state_before": "case h\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.79266\nP : Type ?u.79269\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.79297\nn m : ℕ\ng : α → β ⊕ Fin n\nv : β → M\nxs : Fin (n + m) → M\nx : α ⊕ Fin m\n⊢ (Sum.elim v xs ∘ relabelAux g m) x = Sum.elim (Sum.elim v (xs ∘ ↑(castAdd m)) ∘ g) (xs ∘ ↑(natAdd n)) x",
"tactic": "cases' x with x x"
},
{
"state_after": "case h.inl\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.79266\nP : Type ?u.79269\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.79297\nn m : ℕ\ng : α → β ⊕ Fin n\nv : β → M\nxs : Fin (n + m) → M\nx : α\n⊢ Sum.elim v xs (Sum.map id (↑finSumFinEquiv) (↑(Equiv.sumAssoc β (Fin n) (Fin m)) (Sum.inl (g x)))) =\n Sum.elim v (xs ∘ ↑(castAdd m)) (g x)",
"state_before": "case h.inl\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.79266\nP : Type ?u.79269\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.79297\nn m : ℕ\ng : α → β ⊕ Fin n\nv : β → M\nxs : Fin (n + m) → M\nx : α\n⊢ (Sum.elim v xs ∘ relabelAux g m) (Sum.inl x) =\n Sum.elim (Sum.elim v (xs ∘ ↑(castAdd m)) ∘ g) (xs ∘ ↑(natAdd n)) (Sum.inl x)",
"tactic": "simp only [BoundedFormula.relabelAux, Function.comp_apply, Sum.map_inl, Sum.elim_inl]"
},
{
"state_after": "no goals",
"state_before": "case h.inl\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.79266\nP : Type ?u.79269\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.79297\nn m : ℕ\ng : α → β ⊕ Fin n\nv : β → M\nxs : Fin (n + m) → M\nx : α\n⊢ Sum.elim v xs (Sum.map id (↑finSumFinEquiv) (↑(Equiv.sumAssoc β (Fin n) (Fin m)) (Sum.inl (g x)))) =\n Sum.elim v (xs ∘ ↑(castAdd m)) (g x)",
"tactic": "cases' g x with l r <;> simp"
},
{
"state_after": "no goals",
"state_before": "case h.inr\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.79266\nP : Type ?u.79269\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.79297\nn m : ℕ\ng : α → β ⊕ Fin n\nv : β → M\nxs : Fin (n + m) → M\nx : Fin m\n⊢ (Sum.elim v xs ∘ relabelAux g m) (Sum.inr x) =\n Sum.elim (Sum.elim v (xs ∘ ↑(castAdd m)) ∘ g) (xs ∘ ↑(natAdd n)) (Sum.inr x)",
"tactic": "simp [BoundedFormula.relabelAux]"
}
] |
[
582,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
575,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
ContDiffWithinAt.csin
|
[] |
[
434,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
432,
1
] |
Mathlib/Data/Bool/Basic.lean
|
Bool.coe_bool_iff
|
[
{
"state_after": "no goals",
"state_before": "⊢ ∀ {a b : Bool}, (a = true ↔ b = true) ↔ a = b",
"tactic": "decide"
}
] |
[
142,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/CategoryTheory/Monoidal/Preadditive.lean
|
CategoryTheory.monoidalPreadditive_of_faithful
|
[
{
"state_after": "C : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ : W✝ ⟶ X✝\n⊢ f✝ ⊗ 0 = 0",
"state_before": "C : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\n⊢ ∀ {W X Y Z : D} (f : W ⟶ X), f ⊗ 0 = 0",
"tactic": "intros"
},
{
"state_after": "case a\nC : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ : W✝ ⟶ X✝\n⊢ F.map (f✝ ⊗ 0) = F.map 0",
"state_before": "C : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ : W✝ ⟶ X✝\n⊢ f✝ ⊗ 0 = 0",
"tactic": "apply F.toFunctor.map_injective"
},
{
"state_after": "no goals",
"state_before": "case a\nC : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ : W✝ ⟶ X✝\n⊢ F.map (f✝ ⊗ 0) = F.map 0",
"tactic": "simp [F.map_tensor]"
},
{
"state_after": "C : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ : Y✝ ⟶ Z✝\n⊢ 0 ⊗ f✝ = 0",
"state_before": "C : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\n⊢ ∀ {W X Y Z : D} (f : Y ⟶ Z), 0 ⊗ f = 0",
"tactic": "intros"
},
{
"state_after": "case a\nC : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ : Y✝ ⟶ Z✝\n⊢ F.map (0 ⊗ f✝) = F.map 0",
"state_before": "C : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ : Y✝ ⟶ Z✝\n⊢ 0 ⊗ f✝ = 0",
"tactic": "apply F.toFunctor.map_injective"
},
{
"state_after": "no goals",
"state_before": "case a\nC : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ : Y✝ ⟶ Z✝\n⊢ F.map (0 ⊗ f✝) = F.map 0",
"tactic": "simp [F.map_tensor]"
},
{
"state_after": "C : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ : W✝ ⟶ X✝\ng✝ h✝ : Y✝ ⟶ Z✝\n⊢ f✝ ⊗ (g✝ + h✝) = f✝ ⊗ g✝ + f✝ ⊗ h✝",
"state_before": "C : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\n⊢ ∀ {W X Y Z : D} (f : W ⟶ X) (g h : Y ⟶ Z), f ⊗ (g + h) = f ⊗ g + f ⊗ h",
"tactic": "intros"
},
{
"state_after": "case a\nC : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ : W✝ ⟶ X✝\ng✝ h✝ : Y✝ ⟶ Z✝\n⊢ F.map (f✝ ⊗ (g✝ + h✝)) = F.map (f✝ ⊗ g✝ + f✝ ⊗ h✝)",
"state_before": "C : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ : W✝ ⟶ X✝\ng✝ h✝ : Y✝ ⟶ Z✝\n⊢ f✝ ⊗ (g✝ + h✝) = f✝ ⊗ g✝ + f✝ ⊗ h✝",
"tactic": "apply F.toFunctor.map_injective"
},
{
"state_after": "no goals",
"state_before": "case a\nC : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ : W✝ ⟶ X✝\ng✝ h✝ : Y✝ ⟶ Z✝\n⊢ F.map (f✝ ⊗ (g✝ + h✝)) = F.map (f✝ ⊗ g✝ + f✝ ⊗ h✝)",
"tactic": "simp only [F.map_tensor, Functor.map_add, Preadditive.comp_add, Preadditive.add_comp,\n MonoidalPreadditive.tensor_add]"
},
{
"state_after": "C : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ g✝ : W✝ ⟶ X✝\nh✝ : Y✝ ⟶ Z✝\n⊢ (f✝ + g✝) ⊗ h✝ = f✝ ⊗ h✝ + g✝ ⊗ h✝",
"state_before": "C : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\n⊢ ∀ {W X Y Z : D} (f g : W ⟶ X) (h : Y ⟶ Z), (f + g) ⊗ h = f ⊗ h + g ⊗ h",
"tactic": "intros"
},
{
"state_after": "case a\nC : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ g✝ : W✝ ⟶ X✝\nh✝ : Y✝ ⟶ Z✝\n⊢ F.map ((f✝ + g✝) ⊗ h✝) = F.map (f✝ ⊗ h✝ + g✝ ⊗ h✝)",
"state_before": "C : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ g✝ : W✝ ⟶ X✝\nh✝ : Y✝ ⟶ Z✝\n⊢ (f✝ + g✝) ⊗ h✝ = f✝ ⊗ h✝ + g✝ ⊗ h✝",
"tactic": "apply F.toFunctor.map_injective"
},
{
"state_after": "no goals",
"state_before": "case a\nC : Type u_4\ninst✝⁸ : Category C\ninst✝⁷ : Preadditive C\ninst✝⁶ : MonoidalCategory C\ninst✝⁵ : MonoidalPreadditive C\nD : Type u_1\ninst✝⁴ : Category D\ninst✝³ : Preadditive D\ninst✝² : MonoidalCategory D\nF : MonoidalFunctor D C\ninst✝¹ : Faithful F.toFunctor\ninst✝ : Functor.Additive F.toFunctor\nW✝ X✝ Y✝ Z✝ : D\nf✝ g✝ : W✝ ⟶ X✝\nh✝ : Y✝ ⟶ Z✝\n⊢ F.map ((f✝ + g✝) ⊗ h✝) = F.map (f✝ ⊗ h✝ + g✝ ⊗ h✝)",
"tactic": "simp only [F.map_tensor, Functor.map_add, Preadditive.comp_add, Preadditive.add_comp,\n MonoidalPreadditive.add_tensor]"
}
] |
[
93,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
73,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
ConvexOn.le_on_segment'
|
[
{
"state_after": "case h₁.h₁\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.451309\nα : Type ?u.451312\nβ : Type u_3\nι : Type ?u.451318\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nx y : E\nhx : x ∈ s\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ f x ≤ max (f x) (f y)\n\ncase h₂.h₁\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.451309\nα : Type ?u.451312\nβ : Type u_3\nι : Type ?u.451318\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nx y : E\nhx : x ∈ s\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ f y ≤ max (f x) (f y)",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.451309\nα : Type ?u.451312\nβ : Type u_3\nι : Type ?u.451318\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nx y : E\nhx : x ∈ s\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ a • f x + b • f y ≤ a • max (f x) (f y) + b • max (f x) (f y)",
"tactic": "gcongr"
},
{
"state_after": "no goals",
"state_before": "case h₁.h₁\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.451309\nα : Type ?u.451312\nβ : Type u_3\nι : Type ?u.451318\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nx y : E\nhx : x ∈ s\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ f x ≤ max (f x) (f y)",
"tactic": "apply le_max_left"
},
{
"state_after": "no goals",
"state_before": "case h₂.h₁\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.451309\nα : Type ?u.451312\nβ : Type u_3\nι : Type ?u.451318\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedAddCommMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : OrderedSMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nx y : E\nhx : x ∈ s\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ f y ≤ max (f x) (f y)",
"tactic": "apply le_max_right"
}
] |
[
642,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
634,
1
] |
Mathlib/Analysis/InnerProductSpace/l2Space.lean
|
HilbertBasis.orthonormal
|
[
{
"state_after": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.742653\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nb : HilbertBasis ι 𝕜 E\n⊢ ∀ (i j : ι),\n inner (↑(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1))\n (↑(LinearIsometryEquiv.symm b.repr) (lp.single 2 j 1)) =\n if i = j then 1 else 0",
"state_before": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.742653\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nb : HilbertBasis ι 𝕜 E\n⊢ Orthonormal 𝕜 fun i => ↑(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)",
"tactic": "rw [orthonormal_iff_ite]"
},
{
"state_after": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.742653\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nb : HilbertBasis ι 𝕜 E\ni j : ι\n⊢ inner (↑(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) (↑(LinearIsometryEquiv.symm b.repr) (lp.single 2 j 1)) =\n if i = j then 1 else 0",
"state_before": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.742653\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nb : HilbertBasis ι 𝕜 E\n⊢ ∀ (i j : ι),\n inner (↑(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1))\n (↑(LinearIsometryEquiv.symm b.repr) (lp.single 2 j 1)) =\n if i = j then 1 else 0",
"tactic": "intro i j"
},
{
"state_after": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.742653\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nb : HilbertBasis ι 𝕜 E\ni j : ι\n⊢ inner 1 (if h : i = j then (_ : j = i) ▸ 1 else 0) = if i = j then 1 else 0",
"state_before": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.742653\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nb : HilbertBasis ι 𝕜 E\ni j : ι\n⊢ inner (↑(LinearIsometryEquiv.symm b.repr) (lp.single 2 i 1)) (↑(LinearIsometryEquiv.symm b.repr) (lp.single 2 j 1)) =\n if i = j then 1 else 0",
"tactic": "rw [← b.repr.inner_map_map (b i) (b j), b.repr_self, b.repr_self, lp.inner_single_left,\n lp.single_apply]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : IsROrC 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type ?u.742653\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nb : HilbertBasis ι 𝕜 E\ni j : ι\n⊢ inner 1 (if h : i = j then (_ : j = i) ▸ 1 else 0) = if i = j then 1 else 0",
"tactic": "simp"
}
] |
[
453,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
448,
11
] |
Mathlib/Data/MvPolynomial/Equiv.lean
|
MvPolynomial.mapEquiv_trans
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS₁ : Type v\nS₂ : Type w\nS₃ : Type x\nσ : Type u_1\na a' a₁ a₂ : R\ne✝ : ℕ\ns : σ →₀ ℕ\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring S₁\ninst✝¹ : CommSemiring S₂\ninst✝ : CommSemiring S₃\ne : S₁ ≃+* S₂\nf : S₂ ≃+* S₃\np : MvPolynomial σ S₁\n⊢ ↑(RingEquiv.trans (mapEquiv σ e) (mapEquiv σ f)) p = ↑(mapEquiv σ (RingEquiv.trans e f)) p",
"tactic": "simp only [RingEquiv.coe_trans, comp_apply, mapEquiv_apply, RingEquiv.coe_ringHom_trans,\n map_map]"
}
] |
[
123,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Algebra/BigOperators/Finsupp.lean
|
Finsupp.comp_liftAddHom
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_4\nι : Type ?u.518287\nγ : Type ?u.518290\nA : Type ?u.518293\nB : Type ?u.518296\nC : Type ?u.518299\ninst✝⁵ : AddCommMonoid A\ninst✝⁴ : AddCommMonoid B\ninst✝³ : AddCommMonoid C\nt : ι → A → C\nh0 : ∀ (i : ι), t i 0 = 0\nh1 : ∀ (i : ι) (x y : A), t i (x + y) = t i x + t i y\ns : Finset α\nf✝ : α → ι →₀ A\ni : ι\ng✝ : ι →₀ A\nk : ι → A → γ → B\nx : γ\nβ : Type ?u.521429\nM : Type u_1\nM' : Type ?u.521435\nN : Type u_2\nP : Type u_3\nG : Type ?u.521444\nH : Type ?u.521447\nR : Type ?u.521450\nS : Type ?u.521453\ninst✝² : AddCommMonoid M\ninst✝¹ : AddCommMonoid N\ninst✝ : AddCommMonoid P\ng : N →+ P\nf : α → M →+ N\na : α\n⊢ ↑(AddEquiv.symm liftAddHom) (AddMonoidHom.comp g (↑liftAddHom f)) a = AddMonoidHom.comp g (f a)",
"tactic": "rw [liftAddHom_symm_apply, AddMonoidHom.comp_assoc, liftAddHom_comp_single]"
}
] |
[
504,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
498,
1
] |
Mathlib/Algebra/IndicatorFunction.lean
|
Set.prod_mulIndicator_subset_of_eq_one
|
[
{
"state_after": "α : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\n⊢ ∏ i in s, g i (f i) = ∏ x in s, g x (mulIndicator (↑s) f x)\n\nα : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\n⊢ ∀ (x : α), x ∈ t → ¬x ∈ s → g x (mulIndicator (↑s) f x) = 1",
"state_before": "α : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\n⊢ ∏ i in s, g i (f i) = ∏ i in t, g i (mulIndicator (↑s) f i)",
"tactic": "rw [← Finset.prod_subset h _]"
},
{
"state_after": "α : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\n⊢ ∀ (x : α), x ∈ s → g x (f x) = g x (mulIndicator (↑s) f x)",
"state_before": "α : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\n⊢ ∏ i in s, g i (f i) = ∏ x in s, g x (mulIndicator (↑s) f x)",
"tactic": "apply Finset.prod_congr rfl"
},
{
"state_after": "α : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\ni : α\nhi : i ∈ s\n⊢ g i (f i) = g i (mulIndicator (↑s) f i)",
"state_before": "α : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\n⊢ ∀ (x : α), x ∈ s → g x (f x) = g x (mulIndicator (↑s) f x)",
"tactic": "intro i hi"
},
{
"state_after": "case e_a\nα : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\ni : α\nhi : i ∈ s\n⊢ f i = mulIndicator (↑s) f i",
"state_before": "α : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\ni : α\nhi : i ∈ s\n⊢ g i (f i) = g i (mulIndicator (↑s) f i)",
"tactic": "congr"
},
{
"state_after": "case e_a\nα : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\ni : α\nhi : i ∈ s\n⊢ mulIndicator (↑s) f i = f i",
"state_before": "case e_a\nα : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\ni : α\nhi : i ∈ s\n⊢ f i = mulIndicator (↑s) f i",
"tactic": "symm"
},
{
"state_after": "no goals",
"state_before": "case e_a\nα : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\ni : α\nhi : i ∈ s\n⊢ mulIndicator (↑s) f i = f i",
"tactic": "exact mulIndicator_of_mem (α := α) hi f"
},
{
"state_after": "α : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\ni : α\nx✝ : i ∈ t\nhn : ¬i ∈ s\n⊢ g i (mulIndicator (↑s) f i) = 1",
"state_before": "α : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\n⊢ ∀ (x : α), x ∈ t → ¬x ∈ s → g x (mulIndicator (↑s) f x) = 1",
"tactic": "refine' fun i _ hn => _"
},
{
"state_after": "case h.e'_2.h.e'_2\nα : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\ni : α\nx✝ : i ∈ t\nhn : ¬i ∈ s\n⊢ mulIndicator (↑s) f i = 1",
"state_before": "α : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\ni : α\nx✝ : i ∈ t\nhn : ¬i ∈ s\n⊢ g i (mulIndicator (↑s) f i) = 1",
"tactic": "convert hg i"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_2\nα : Type u_2\nβ : Type ?u.100673\nι : Type ?u.100676\nM : Type u_3\nN : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : One N\nf : α → N\ng : α → N → M\ns t : Finset α\nh : s ⊆ t\nhg : ∀ (a : α), g a 1 = 1\ni : α\nx✝ : i ∈ t\nhn : ¬i ∈ s\n⊢ mulIndicator (↑s) f i = 1",
"tactic": "exact mulIndicator_of_not_mem (α := α) hn f"
}
] |
[
597,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
584,
1
] |
Mathlib/Computability/TMToPartrec.lean
|
Turing.PartrecToTM2.tr_init
|
[] |
[
1703,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1701,
1
] |
Mathlib/Data/MvPolynomial/Division.lean
|
MvPolynomial.X_divMonomial
|
[] |
[
172,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
171,
1
] |
Mathlib/Topology/Algebra/UniformRing.lean
|
UniformSpace.Completion.Continuous.mul
|
[] |
[
90,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
Mathlib/Order/Directed.lean
|
directedOn_of_sup_mem
|
[] |
[
129,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
Mathlib/Order/OmegaCompletePartialOrder.lean
|
CompleteLattice.inf_continuous'
|
[] |
[
535,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
533,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean
|
Differentiable.const_smul
|
[] |
[
101,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
100,
1
] |
Mathlib/Order/Filter/Extr.lean
|
IsExtrOn.comp_mapsTo
|
[] |
[
441,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
439,
1
] |
Mathlib/Order/BooleanAlgebra.lean
|
sdiff_lt
|
[
{
"state_after": "α : Type u\nβ : Type ?u.21257\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nhx : y ≤ x\nhy : y ≠ ⊥\nh : x \\ y = x\n⊢ y = ⊥",
"state_before": "α : Type u\nβ : Type ?u.21257\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nhx : y ≤ x\nhy : y ≠ ⊥\n⊢ x \\ y < x",
"tactic": "refine' sdiff_le.lt_of_ne fun h => hy _"
},
{
"state_after": "α : Type u\nβ : Type ?u.21257\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nhx : y ≤ x\nhy : y ≠ ⊥\nh : x ⊓ y = ⊥\n⊢ y = ⊥",
"state_before": "α : Type u\nβ : Type ?u.21257\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nhx : y ≤ x\nhy : y ≠ ⊥\nh : x \\ y = x\n⊢ y = ⊥",
"tactic": "rw [sdiff_eq_self_iff_disjoint', disjoint_iff] at h"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.21257\nw x y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nhx : y ≤ x\nhy : y ≠ ⊥\nh : x ⊓ y = ⊥\n⊢ y = ⊥",
"tactic": "rw [← h, inf_eq_right.mpr hx]"
}
] |
[
324,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
321,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.le_normalClosure
|
[] |
[
2468,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2467,
1
] |
Mathlib/Topology/Algebra/Order/IntermediateValue.lean
|
isConnected_Icc
|
[] |
[
482,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
481,
1
] |
Mathlib/NumberTheory/Divisors.lean
|
Nat.image_snd_divisorsAntidiagonal
|
[
{
"state_after": "n : ℕ\n⊢ image (Prod.snd ∘ ↑(Equiv.toEmbedding (Equiv.prodComm ℕ ℕ))) (divisorsAntidiagonal n) = divisors n",
"state_before": "n : ℕ\n⊢ image Prod.snd (divisorsAntidiagonal n) = divisors n",
"tactic": "rw [← map_swap_divisorsAntidiagonal, map_eq_image, image_image]"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ image (Prod.snd ∘ ↑(Equiv.toEmbedding (Equiv.prodComm ℕ ℕ))) (divisorsAntidiagonal n) = divisors n",
"tactic": "exact image_fst_divisorsAntidiagonal"
}
] |
[
260,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
258,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.map_adj
|
[] |
[
1244,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1242,
1
] |
Mathlib/Combinatorics/SimpleGraph/IncMatrix.lean
|
SimpleGraph.incMatrix_of_mem_incidenceSet
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\nα : Type u_1\nG : SimpleGraph α\ninst✝ : MulZeroOneClass R\na b : α\ne : Sym2 α\nh : e ∈ incidenceSet G a\n⊢ incMatrix R G a e = 1",
"tactic": "rw [incMatrix_apply, Set.indicator_of_mem h, Pi.one_apply]"
}
] |
[
99,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
IsCompact.inter_right
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.6738\nπ : ι → Type ?u.6743\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nhs : IsCompact s\nht : IsClosed t\nf : Filter α\nhnf : NeBot f\nhstf : f ≤ 𝓟 (s ∩ t)\n⊢ ∃ a, a ∈ s ∩ t ∧ ClusterPt a f",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.6738\nπ : ι → Type ?u.6743\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nhs : IsCompact s\nht : IsClosed t\n⊢ IsCompact (s ∩ t)",
"tactic": "intro f hnf hstf"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.6738\nπ : ι → Type ?u.6743\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nhs : IsCompact s\nht : IsClosed t\nf : Filter α\nhnf : NeBot f\nhstf : f ≤ 𝓟 (s ∩ t)\na : α\nhsa : a ∈ s\nha : ClusterPt a f\n⊢ ∃ a, a ∈ s ∩ t ∧ ClusterPt a f",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.6738\nπ : ι → Type ?u.6743\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nhs : IsCompact s\nht : IsClosed t\nf : Filter α\nhnf : NeBot f\nhstf : f ≤ 𝓟 (s ∩ t)\n⊢ ∃ a, a ∈ s ∩ t ∧ ClusterPt a f",
"tactic": "obtain ⟨a, hsa, ha⟩ : ∃ a ∈ s, ClusterPt a f :=\n hs (le_trans hstf (le_principal_iff.2 (inter_subset_left _ _)))"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.6738\nπ : ι → Type ?u.6743\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nhs : IsCompact s\nht : IsClosed t\nf : Filter α\nhnf : NeBot f\nhstf : f ≤ 𝓟 (s ∩ t)\na : α\nhsa : a ∈ s\nha : ClusterPt a f\nthis : a ∈ t\n⊢ ∃ a, a ∈ s ∩ t ∧ ClusterPt a f",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.6738\nπ : ι → Type ?u.6743\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nhs : IsCompact s\nht : IsClosed t\nf : Filter α\nhnf : NeBot f\nhstf : f ≤ 𝓟 (s ∩ t)\na : α\nhsa : a ∈ s\nha : ClusterPt a f\n⊢ ∃ a, a ∈ s ∩ t ∧ ClusterPt a f",
"tactic": "have : a ∈ t := ht.mem_of_nhdsWithin_neBot <|\n ha.mono <| le_trans hstf (le_principal_iff.2 (inter_subset_right _ _))"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.6738\nπ : ι → Type ?u.6743\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nhs : IsCompact s\nht : IsClosed t\nf : Filter α\nhnf : NeBot f\nhstf : f ≤ 𝓟 (s ∩ t)\na : α\nhsa : a ∈ s\nha : ClusterPt a f\nthis : a ∈ t\n⊢ ∃ a, a ∈ s ∩ t ∧ ClusterPt a f",
"tactic": "exact ⟨a, ⟨hsa, this⟩, ha⟩"
}
] |
[
118,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Data/Sign.lean
|
sign_neg
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : Zero α\ninst✝¹ : Preorder α\ninst✝ : DecidableRel fun x x_1 => x < x_1\na : α\nha : a < 0\n⊢ ↑sign a = -1",
"tactic": "rwa [sign_apply, if_neg <| asymm ha, if_pos]"
}
] |
[
328,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
328,
1
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.