file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/LinearAlgebra/Matrix/GeneralLinearGroup.lean
|
Matrix.GLPos.coe_neg
|
[] |
[
230,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
229,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.L1.SimpleFunc.integral_add
|
[] |
[
542,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
541,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Deriv.lean
|
Real.differentiableAt_log_iff
|
[] |
[
67,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/Data/MvPolynomial/Variables.lean
|
MvPolynomial.vars_map_of_injective
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nσ : Type u_1\nτ : Type ?u.221880\nr : R\ne : ℕ\nn m : σ\ns : σ →₀ ℕ\ninst✝¹ : CommSemiring R\np q : MvPolynomial σ R\ninst✝ : CommSemiring S\nf : R →+* S\nhf : Injective ↑f\n⊢ vars (↑(map f) p) = vars p",
"tactic": "simp [vars, degrees_map_of_injective _ hf]"
}
] |
[
462,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
461,
1
] |
src/lean/Init/Core.lean
|
Subsingleton.helim
|
[
{
"state_after": "a✝ b✝ c d : Prop\nα : Sort u\nh₁ : Subsingleton α\na b : α\n⊢ HEq a b",
"state_before": "a✝ b✝ c d : Prop\nα β : Sort u\nh₁ : Subsingleton α\nh₂ : α = β\na : α\nb : β\n⊢ HEq a b",
"tactic": "subst h₂"
},
{
"state_after": "case h\na✝ b✝ c d : Prop\nα : Sort u\nh₁ : Subsingleton α\na b : α\n⊢ a = b",
"state_before": "a✝ b✝ c d : Prop\nα : Sort u\nh₁ : Subsingleton α\na b : α\n⊢ HEq a b",
"tactic": "apply heq_of_eq"
},
{
"state_after": "no goals",
"state_before": "case h\na✝ b✝ c d : Prop\nα : Sort u\nh₁ : Subsingleton α\na b : α\n⊢ a = b",
"tactic": "apply Subsingleton.elim"
}
] |
[
879,
26
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
876,
11
] |
Mathlib/Analysis/NormedSpace/LinearIsometry.lean
|
LinearIsometryEquiv.self_comp_symm
|
[] |
[
843,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
842,
1
] |
Mathlib/Analysis/SpecificLimits/Basic.lean
|
edist_le_of_edist_le_geometric_of_tendsto
|
[
{
"state_after": "case h.e'_4\nα : Type u_1\nβ : Type ?u.431910\nι : Type ?u.431913\ninst✝ : PseudoEMetricSpace α\nr C : ℝ≥0∞\nhr : r < 1\nhC : C ≠ ⊤\nf : ℕ → α\nhu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C * r ^ n\na : α\nha : Tendsto f atTop (𝓝 a)\nn : ℕ\n⊢ C * r ^ n / (1 - r) = ∑' (m : ℕ), C * r ^ (n + m)",
"state_before": "α : Type u_1\nβ : Type ?u.431910\nι : Type ?u.431913\ninst✝ : PseudoEMetricSpace α\nr C : ℝ≥0∞\nhr : r < 1\nhC : C ≠ ⊤\nf : ℕ → α\nhu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C * r ^ n\na : α\nha : Tendsto f atTop (𝓝 a)\nn : ℕ\n⊢ edist (f n) a ≤ C * r ^ n / (1 - r)",
"tactic": "convert edist_le_tsum_of_edist_le_of_tendsto _ hu ha _"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nα : Type u_1\nβ : Type ?u.431910\nι : Type ?u.431913\ninst✝ : PseudoEMetricSpace α\nr C : ℝ≥0∞\nhr : r < 1\nhC : C ≠ ⊤\nf : ℕ → α\nhu : ∀ (n : ℕ), edist (f n) (f (n + 1)) ≤ C * r ^ n\na : α\nha : Tendsto f atTop (𝓝 a)\nn : ℕ\n⊢ C * r ^ n / (1 - r) = ∑' (m : ℕ), C * r ^ (n + m)",
"tactic": "simp only [pow_add, ENNReal.tsum_mul_left, ENNReal.tsum_geometric, div_eq_mul_inv, mul_assoc]"
}
] |
[
325,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
322,
1
] |
Mathlib/FieldTheory/RatFunc.lean
|
RatFunc.mk_def_of_mem
|
[
{
"state_after": "no goals",
"state_before": "K : Type u\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\np q : K[X]\nhq : q ∈ K[X]⁰\n⊢ RatFunc.mk p q = { toFractionRing := IsLocalization.mk' (FractionRing K[X]) p { val := q, property := hq } }",
"tactic": "simp only [← mk_coe_def]"
}
] |
[
216,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
213,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean
|
SimpleGraph.dartOfNeighborSet_injective
|
[
{
"state_after": "ι : Sort ?u.99591\n𝕜 : Type ?u.99594\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w : V\ne : Sym2 V\nv : V\ne₁ e₂ : ↑(neighborSet G v)\nh' : (v, ↑e₁) = (v, ↑e₂)\n⊢ ↑e₁ = ↑e₂",
"state_before": "ι : Sort ?u.99591\n𝕜 : Type ?u.99594\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w : V\ne : Sym2 V\nv : V\ne₁ e₂ : ↑(neighborSet G v)\nh : dartOfNeighborSet G v e₁ = dartOfNeighborSet G v e₂\n⊢ ↑e₁ = ↑e₂",
"tactic": "injection h with h'"
},
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.99591\n𝕜 : Type ?u.99594\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w : V\ne : Sym2 V\nv : V\ne₁ e₂ : ↑(neighborSet G v)\nh' : (v, ↑e₁) = (v, ↑e₂)\n⊢ ↑e₁ = ↑e₂",
"tactic": "convert congr_arg Prod.snd h'"
}
] |
[
813,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
809,
1
] |
Mathlib/Order/InitialSeg.lean
|
PrincipalSeg.ofIsEmpty_top
|
[] |
[
422,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
420,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
Metric.disjoint_closedBall_of_lt_infDist
|
[] |
[
551,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
549,
1
] |
Mathlib/Data/MvPolynomial/Derivation.lean
|
MvPolynomial.derivation_eq_of_forall_mem_vars
|
[] |
[
91,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
89,
1
] |
Mathlib/Dynamics/PeriodicPts.lean
|
Function.periodicOrbit_eq_nil_iff_not_periodic_pt
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.30100\nf fa : α → α\nfb : β → β\nx y : α\nm n : ℕ\n⊢ minimalPeriod f x = 0 ↔ ¬x ∈ periodicPts f",
"state_before": "α : Type u_1\nβ : Type ?u.30100\nf fa : α → α\nfb : β → β\nx y : α\nm n : ℕ\n⊢ periodicOrbit f x = Cycle.nil ↔ ¬x ∈ periodicPts f",
"tactic": "simp [periodicOrbit]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.30100\nf fa : α → α\nfb : β → β\nx y : α\nm n : ℕ\n⊢ minimalPeriod f x = 0 ↔ ¬x ∈ periodicPts f",
"tactic": "exact minimalPeriod_eq_zero_iff_nmem_periodicPts"
}
] |
[
509,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
506,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean
|
CategoryTheory.Limits.coprodComparison_natural
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁵ : Category C\nX Y : C\nD : Type u₂\ninst✝⁴ : Category D\nF : C ⥤ D\nA A' B B' : C\ninst✝³ : HasBinaryCoproduct A B\ninst✝² : HasBinaryCoproduct A' B'\ninst✝¹ : HasBinaryCoproduct (F.obj A) (F.obj B)\ninst✝ : HasBinaryCoproduct (F.obj A') (F.obj B')\nf : A ⟶ A'\ng : B ⟶ B'\n⊢ coprodComparison F A B ≫ F.map (coprod.map f g) = coprod.map (F.map f) (F.map g) ≫ coprodComparison F A' B'",
"tactic": "rw [coprodComparison, coprodComparison, coprod.map_desc, ← F.map_comp, ← F.map_comp,\n coprod.desc_comp, ← F.map_comp, coprod.inl_map, ← F.map_comp, coprod.inr_map]"
}
] |
[
1359,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1355,
1
] |
Mathlib/Data/Polynomial/AlgebraMap.lean
|
Polynomial.aevalTower_id
|
[
{
"state_after": "case h.h\nR : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1573756\nB' : Type ?u.1573759\na b : R\nn : ℕ\ninst✝¹⁰ : CommSemiring A'\ninst✝⁹ : Semiring B'\ninst✝⁸ : CommSemiring R\np q : R[X]\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\nB : Type ?u.1573972\ninst✝⁵ : Semiring B\ninst✝⁴ : Algebra R B\nx : A\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Algebra S A'\ninst✝ : Algebra S B'\ng : R →ₐ[S] A'\ny : A'\ns : S\n⊢ ↑(aevalTower (AlgHom.id S S) s) X = ↑(aeval s) X",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1573756\nB' : Type ?u.1573759\na b : R\nn : ℕ\ninst✝¹⁰ : CommSemiring A'\ninst✝⁹ : Semiring B'\ninst✝⁸ : CommSemiring R\np q : R[X]\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\nB : Type ?u.1573972\ninst✝⁵ : Semiring B\ninst✝⁴ : Algebra R B\nx : A\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Algebra S A'\ninst✝ : Algebra S B'\ng : R →ₐ[S] A'\ny : A'\n⊢ aevalTower (AlgHom.id S S) = aeval",
"tactic": "ext s"
},
{
"state_after": "no goals",
"state_before": "case h.h\nR : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1573756\nB' : Type ?u.1573759\na b : R\nn : ℕ\ninst✝¹⁰ : CommSemiring A'\ninst✝⁹ : Semiring B'\ninst✝⁸ : CommSemiring R\np q : R[X]\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\nB : Type ?u.1573972\ninst✝⁵ : Semiring B\ninst✝⁴ : Algebra R B\nx : A\ninst✝³ : CommSemiring S\ninst✝² : Algebra S R\ninst✝¹ : Algebra S A'\ninst✝ : Algebra S B'\ng : R →ₐ[S] A'\ny : A'\ns : S\n⊢ ↑(aevalTower (AlgHom.id S S) s) X = ↑(aeval s) X",
"tactic": "simp only [eval_X, aevalTower_X, coe_aeval_eq_eval]"
}
] |
[
436,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
434,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
|
BoxIntegral.Prepartition.mem_boxes
|
[] |
[
70,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/Data/List/Count.lean
|
List.count_concat
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nl✝ : List α\ninst✝ : DecidableEq α\na : α\nl : List α\n⊢ count a (concat l a) = succ (count a l)",
"tactic": "simp"
}
] |
[
240,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
240,
1
] |
Mathlib/Probability/Kernel/Basic.lean
|
ProbabilityTheory.kernel.ext_iff
|
[] |
[
186,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
186,
1
] |
Mathlib/GroupTheory/Submonoid/Basic.lean
|
MonoidHom.eqLocusM_same
|
[] |
[
594,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
593,
1
] |
Mathlib/Data/List/Perm.lean
|
List.Perm.bind_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type uu\nβ : Type vv\nl₁ l₂ l : List α\nf g : α → List β\nh : ∀ (a : α), a ∈ l → f a ~ g a\n⊢ Forall₂ (fun x x_1 => x ~ x_1) (List.map f l) (List.map g l)",
"tactic": "rwa [List.forall₂_map_right_iff, List.forall₂_map_left_iff, List.forall₂_same]"
}
] |
[
1090,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1087,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.measureUnivNNReal_zero
|
[] |
[
3102,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3101,
1
] |
Mathlib/Probability/ProbabilityMassFunction/Constructions.lean
|
Pmf.monad_seq_eq_seq
|
[] |
[
122,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/Analysis/SpecificLimits/Basic.lean
|
tendsto_nat_floor_mul_div_atTop
|
[
{
"state_after": "α : Type ?u.500878\nβ : Type ?u.500881\nι : Type ?u.500884\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a - x⁻¹) atTop (𝓝 (a - 0))\n⊢ Tendsto (fun x => ↑⌊a * x⌋₊ / x) atTop (𝓝 a)",
"state_before": "α : Type ?u.500878\nβ : Type ?u.500881\nι : Type ?u.500884\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\n⊢ Tendsto (fun x => ↑⌊a * x⌋₊ / x) atTop (𝓝 a)",
"tactic": "have A : Tendsto (fun x : R => a - x⁻¹) atTop (𝓝 (a - 0)) :=\n tendsto_const_nhds.sub tendsto_inv_atTop_zero"
},
{
"state_after": "α : Type ?u.500878\nβ : Type ?u.500881\nι : Type ?u.500884\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a - x⁻¹) atTop (𝓝 a)\n⊢ Tendsto (fun x => ↑⌊a * x⌋₊ / x) atTop (𝓝 a)",
"state_before": "α : Type ?u.500878\nβ : Type ?u.500881\nι : Type ?u.500884\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a - x⁻¹) atTop (𝓝 (a - 0))\n⊢ Tendsto (fun x => ↑⌊a * x⌋₊ / x) atTop (𝓝 a)",
"tactic": "rw [sub_zero] at A"
},
{
"state_after": "case hgf\nα : Type ?u.500878\nβ : Type ?u.500881\nι : Type ?u.500884\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a - x⁻¹) atTop (𝓝 a)\n⊢ ∀ᶠ (b : R) in atTop, a - b⁻¹ ≤ ↑⌊a * b⌋₊ / b\n\ncase hfh\nα : Type ?u.500878\nβ : Type ?u.500881\nι : Type ?u.500884\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a - x⁻¹) atTop (𝓝 a)\n⊢ ∀ᶠ (b : R) in atTop, ↑⌊a * b⌋₊ / b ≤ a",
"state_before": "α : Type ?u.500878\nβ : Type ?u.500881\nι : Type ?u.500884\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a - x⁻¹) atTop (𝓝 a)\n⊢ Tendsto (fun x => ↑⌊a * x⌋₊ / x) atTop (𝓝 a)",
"tactic": "apply tendsto_of_tendsto_of_tendsto_of_le_of_le' A tendsto_const_nhds"
},
{
"state_after": "case hgf\nα : Type ?u.500878\nβ : Type ?u.500881\nι : Type ?u.500884\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a - x⁻¹) atTop (𝓝 a)\nx : R\nhx : x ≥ 1\n⊢ a - x⁻¹ ≤ ↑⌊a * x⌋₊ / x",
"state_before": "case hgf\nα : Type ?u.500878\nβ : Type ?u.500881\nι : Type ?u.500884\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a - x⁻¹) atTop (𝓝 a)\n⊢ ∀ᶠ (b : R) in atTop, a - b⁻¹ ≤ ↑⌊a * b⌋₊ / b",
"tactic": "refine' eventually_atTop.2 ⟨1, fun x hx => _⟩"
},
{
"state_after": "case hgf\nα : Type ?u.500878\nβ : Type ?u.500881\nι : Type ?u.500884\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a - x⁻¹) atTop (𝓝 a)\nx : R\nhx : x ≥ 1\n⊢ a * x - 1 ≤ ↑⌊a * x⌋₊",
"state_before": "case hgf\nα : Type ?u.500878\nβ : Type ?u.500881\nι : Type ?u.500884\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a - x⁻¹) atTop (𝓝 a)\nx : R\nhx : x ≥ 1\n⊢ a - x⁻¹ ≤ ↑⌊a * x⌋₊ / x",
"tactic": "simp only [le_div_iff (zero_lt_one.trans_le hx), _root_.sub_mul,\n inv_mul_cancel (zero_lt_one.trans_le hx).ne']"
},
{
"state_after": "case hgf\nα : Type ?u.500878\nβ : Type ?u.500881\nι : Type ?u.500884\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a - x⁻¹) atTop (𝓝 a)\nx : R\nhx : x ≥ 1\nthis : a * x < ↑⌊a * x⌋₊ + 1\n⊢ a * x - 1 ≤ ↑⌊a * x⌋₊",
"state_before": "case hgf\nα : Type ?u.500878\nβ : Type ?u.500881\nι : Type ?u.500884\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a - x⁻¹) atTop (𝓝 a)\nx : R\nhx : x ≥ 1\n⊢ a * x - 1 ≤ ↑⌊a * x⌋₊",
"tactic": "have := Nat.lt_floor_add_one (a * x)"
},
{
"state_after": "no goals",
"state_before": "case hgf\nα : Type ?u.500878\nβ : Type ?u.500881\nι : Type ?u.500884\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a - x⁻¹) atTop (𝓝 a)\nx : R\nhx : x ≥ 1\nthis : a * x < ↑⌊a * x⌋₊ + 1\n⊢ a * x - 1 ≤ ↑⌊a * x⌋₊",
"tactic": "linarith"
},
{
"state_after": "case hfh\nα : Type ?u.500878\nβ : Type ?u.500881\nι : Type ?u.500884\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a - x⁻¹) atTop (𝓝 a)\nx : R\nhx : x ≥ 1\n⊢ ↑⌊a * x⌋₊ / x ≤ a",
"state_before": "case hfh\nα : Type ?u.500878\nβ : Type ?u.500881\nι : Type ?u.500884\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a - x⁻¹) atTop (𝓝 a)\n⊢ ∀ᶠ (b : R) in atTop, ↑⌊a * b⌋₊ / b ≤ a",
"tactic": "refine' eventually_atTop.2 ⟨1, fun x hx => _⟩"
},
{
"state_after": "case hfh\nα : Type ?u.500878\nβ : Type ?u.500881\nι : Type ?u.500884\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a - x⁻¹) atTop (𝓝 a)\nx : R\nhx : x ≥ 1\n⊢ ↑⌊a * x⌋₊ ≤ a * x",
"state_before": "case hfh\nα : Type ?u.500878\nβ : Type ?u.500881\nι : Type ?u.500884\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a - x⁻¹) atTop (𝓝 a)\nx : R\nhx : x ≥ 1\n⊢ ↑⌊a * x⌋₊ / x ≤ a",
"tactic": "rw [div_le_iff (zero_lt_one.trans_le hx)]"
},
{
"state_after": "no goals",
"state_before": "case hfh\nα : Type ?u.500878\nβ : Type ?u.500881\nι : Type ?u.500884\nR : Type u_1\ninst✝³ : TopologicalSpace R\ninst✝² : LinearOrderedField R\ninst✝¹ : OrderTopology R\ninst✝ : FloorRing R\na : R\nha : 0 ≤ a\nA : Tendsto (fun x => a - x⁻¹) atTop (𝓝 a)\nx : R\nhx : x ≥ 1\n⊢ ↑⌊a * x⌋₊ ≤ a * x",
"tactic": "simp [Nat.floor_le (mul_nonneg ha (zero_le_one.trans hx))]"
}
] |
[
592,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
579,
1
] |
Mathlib/Algebra/Order/Pointwise.lean
|
csSup_inv
|
[
{
"state_after": "α : Type u_1\ninst✝³ : ConditionallyCompleteLattice α\ninst✝² : Group α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ns t : Set α\nhs₀ : Set.Nonempty s\nhs₁ : BddBelow s\n⊢ sSup (Inv.inv '' s) = (sInf s)⁻¹",
"state_before": "α : Type u_1\ninst✝³ : ConditionallyCompleteLattice α\ninst✝² : Group α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ns t : Set α\nhs₀ : Set.Nonempty s\nhs₁ : BddBelow s\n⊢ sSup s⁻¹ = (sInf s)⁻¹",
"tactic": "rw [← image_inv]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : ConditionallyCompleteLattice α\ninst✝² : Group α\ninst✝¹ : CovariantClass α α (fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ninst✝ : CovariantClass α α (swap fun x x_1 => x * x_1) fun x x_1 => x ≤ x_1\ns t : Set α\nhs₀ : Set.Nonempty s\nhs₁ : BddBelow s\n⊢ sSup (Inv.inv '' s) = (sInf s)⁻¹",
"tactic": "exact ((OrderIso.inv α).map_csInf' hs₀ hs₁).symm"
}
] |
[
135,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
133,
1
] |
Mathlib/Combinatorics/Pigeonhole.lean
|
Fintype.exists_lt_card_fiber_of_mul_lt_card
|
[] |
[
393,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
391,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.cospanExt_inv_app_left
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nX Y Z X' Y' Z' : C\niX : X ≅ X'\niY : Y ≅ Y'\niZ : Z ≅ Z'\nf : X ⟶ Z\ng : Y ⟶ Z\nf' : X' ⟶ Z'\ng' : Y' ⟶ Z'\nwf : iX.hom ≫ f' = f ≫ iZ.hom\nwg : iY.hom ≫ g' = g ≫ iZ.hom\n⊢ (cospanExt iX iY iZ wf wg).inv.app WalkingCospan.left = iX.inv",
"tactic": "dsimp [cospanExt]"
}
] |
[
431,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
430,
1
] |
Mathlib/Data/Real/Basic.lean
|
Real.ciSup_empty
|
[
{
"state_after": "x y : ℝ\nα : Sort u_1\ninst✝ : IsEmpty α\nf : α → ℝ\n⊢ sSup (Set.range fun i => f i) = 0",
"state_before": "x y : ℝ\nα : Sort u_1\ninst✝ : IsEmpty α\nf : α → ℝ\n⊢ (⨆ (i : α), f i) = 0",
"tactic": "dsimp [iSup]"
},
{
"state_after": "case h.e'_2.h.e'_3\nx y : ℝ\nα : Sort u_1\ninst✝ : IsEmpty α\nf : α → ℝ\n⊢ (Set.range fun i => f i) = ∅",
"state_before": "x y : ℝ\nα : Sort u_1\ninst✝ : IsEmpty α\nf : α → ℝ\n⊢ sSup (Set.range fun i => f i) = 0",
"tactic": "convert Real.sSup_empty"
},
{
"state_after": "case h.e'_2.h.e'_3\nx y : ℝ\nα : Sort u_1\ninst✝ : IsEmpty α\nf : α → ℝ\n⊢ IsEmpty α",
"state_before": "case h.e'_2.h.e'_3\nx y : ℝ\nα : Sort u_1\ninst✝ : IsEmpty α\nf : α → ℝ\n⊢ (Set.range fun i => f i) = ∅",
"tactic": "rw [Set.range_eq_empty_iff]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h.e'_3\nx y : ℝ\nα : Sort u_1\ninst✝ : IsEmpty α\nf : α → ℝ\n⊢ IsEmpty α",
"tactic": "infer_instance"
}
] |
[
806,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
802,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
|
CategoryTheory.Limits.Fork.IsLimit.homIso_natural
|
[] |
[
566,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
563,
1
] |
Mathlib/SetTheory/Ordinal/Exponential.lean
|
Ordinal.log_of_not_one_lt_left
|
[
{
"state_after": "no goals",
"state_before": "b : Ordinal\nh : ¬1 < b\nx : Ordinal\n⊢ log b x = 0",
"tactic": "simp only [log, dif_neg h]"
}
] |
[
271,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
270,
1
] |
Mathlib/Order/PropInstances.lean
|
Prop.isCompl_iff
|
[
{
"state_after": "P Q : Prop\n⊢ ¬(P ∧ Q) ∧ (P ∨ Q) ↔ (¬P ↔ Q)",
"state_before": "P Q : Prop\n⊢ IsCompl P Q ↔ ¬(P ↔ Q)",
"tactic": "rw [_root_.isCompl_iff, Prop.disjoint_iff, Prop.codisjoint_iff, not_iff]"
},
{
"state_after": "no goals",
"state_before": "P Q : Prop\n⊢ ¬(P ∧ Q) ∧ (P ∨ Q) ↔ (¬P ↔ Q)",
"tactic": "by_cases P <;> by_cases Q <;> simp [*]"
}
] |
[
113,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/Data/Sym/Card.lean
|
Sym2.card_subtype_diag
|
[
{
"state_after": "case h.e'_2\nα : Type u_1\nβ : Type ?u.80631\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n⊢ Fintype.card { a // IsDiag a } = Finset.card (image Quotient.mk' (Finset.diag univ))",
"state_before": "α : Type u_1\nβ : Type ?u.80631\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n⊢ Fintype.card { a // IsDiag a } = Fintype.card α",
"tactic": "convert card_image_diag (univ : Finset α)"
},
{
"state_after": "case h.e'_2\nα : Type u_1\nβ : Type ?u.80631\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n⊢ Fintype.card { a // IsDiag a } = Finset.card (image (fun a => Quotient.mk'' a) (Finset.diag univ))",
"state_before": "case h.e'_2\nα : Type u_1\nβ : Type ?u.80631\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n⊢ Fintype.card { a // IsDiag a } = Finset.card (image Quotient.mk' (Finset.diag univ))",
"tactic": "simp_rw [Quotient.mk', ← Quotient.mk''_eq_mk]"
},
{
"state_after": "case h.e'_2.H\nα : Type u_1\nβ : Type ?u.80631\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n⊢ ∀ (x : Sym2 α), x ∈ filter IsDiag (image Quotient.mk'' (univ ×ˢ univ)) ↔ IsDiag x",
"state_before": "case h.e'_2\nα : Type u_1\nβ : Type ?u.80631\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n⊢ Fintype.card { a // IsDiag a } = Finset.card (image (fun a => Quotient.mk'' a) (Finset.diag univ))",
"tactic": "rw [Fintype.card_of_subtype, ← filter_image_quotient_mk''_isDiag]"
},
{
"state_after": "case h.e'_2.H\nα : Type u_1\nβ : Type ?u.80631\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nx : Sym2 α\n⊢ x ∈ filter IsDiag (image Quotient.mk'' (univ ×ˢ univ)) ↔ IsDiag x",
"state_before": "case h.e'_2.H\nα : Type u_1\nβ : Type ?u.80631\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n⊢ ∀ (x : Sym2 α), x ∈ filter IsDiag (image Quotient.mk'' (univ ×ˢ univ)) ↔ IsDiag x",
"tactic": "rintro x"
},
{
"state_after": "case h.e'_2.H\nα : Type u_1\nβ : Type ?u.80631\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nx : Sym2 α\n⊢ (∃ a, a ∈ univ ∧ Quotient.mk'' a = x) ∧ IsDiag x ↔ IsDiag x",
"state_before": "case h.e'_2.H\nα : Type u_1\nβ : Type ?u.80631\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nx : Sym2 α\n⊢ x ∈ filter IsDiag (image Quotient.mk'' (univ ×ˢ univ)) ↔ IsDiag x",
"tactic": "rw [mem_filter, univ_product_univ, mem_image]"
},
{
"state_after": "case h.e'_2.H.intro\nα : Type u_1\nβ : Type ?u.80631\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nx : Sym2 α\na : α × α\nha : Quotient.mk (Rel.setoid α) a = x\n⊢ (∃ a, a ∈ univ ∧ Quotient.mk'' a = x) ∧ IsDiag x ↔ IsDiag x",
"state_before": "case h.e'_2.H\nα : Type u_1\nβ : Type ?u.80631\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nx : Sym2 α\n⊢ (∃ a, a ∈ univ ∧ Quotient.mk'' a = x) ∧ IsDiag x ↔ IsDiag x",
"tactic": "obtain ⟨a, ha⟩ := Quotient.exists_rep x"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.H.intro\nα : Type u_1\nβ : Type ?u.80631\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nx : Sym2 α\na : α × α\nha : Quotient.mk (Rel.setoid α) a = x\n⊢ (∃ a, a ∈ univ ∧ Quotient.mk'' a = x) ∧ IsDiag x ↔ IsDiag x",
"tactic": "exact and_iff_right ⟨a, mem_univ _, ha⟩"
}
] |
[
182,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
1
] |
Mathlib/Analysis/Calculus/ContDiff.lean
|
hasFTaylorSeriesUpToOn_pi
|
[
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹³ : NormedAddCommGroup D\ninst✝¹² : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nX : Type ?u.1182357\ninst✝⁵ : NormedAddCommGroup X\ninst✝⁴ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nι : Type u_2\nι' : Type ?u.1185838\ninst✝³ : Fintype ι\ninst✝² : Fintype ι'\nF' : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nφ : (i : ι) → E → F' i\np' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)\nΦ : E → (i : ι) → F' i\nP' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)\npr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj\n⊢ HasFTaylorSeriesUpToOn n (fun x i => φ i x) (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s ↔\n ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s",
"state_before": "𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹³ : NormedAddCommGroup D\ninst✝¹² : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nX : Type ?u.1182357\ninst✝⁵ : NormedAddCommGroup X\ninst✝⁴ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nι : Type u_2\nι' : Type ?u.1185838\ninst✝³ : Fintype ι\ninst✝² : Fintype ι'\nF' : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nφ : (i : ι) → E → F' i\np' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)\nΦ : E → (i : ι) → F' i\nP' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)\n⊢ HasFTaylorSeriesUpToOn n (fun x i => φ i x) (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s ↔\n ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s",
"tactic": "set pr := @ContinuousLinearMap.proj 𝕜 _ ι F' _ _ _"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹³ : NormedAddCommGroup D\ninst✝¹² : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nX : Type ?u.1182357\ninst✝⁵ : NormedAddCommGroup X\ninst✝⁴ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nι : Type u_2\nι' : Type ?u.1185838\ninst✝³ : Fintype ι\ninst✝² : Fintype ι'\nF' : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nφ : (i : ι) → E → F' i\np' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)\nΦ : E → (i : ι) → F' i\nP' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)\npr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj\nthis : (m : ℕ) → (i : ι) → NormedSpace 𝕜 (ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) := fun m i => inferInstance\n⊢ HasFTaylorSeriesUpToOn n (fun x i => φ i x) (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s ↔\n ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s",
"state_before": "𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹³ : NormedAddCommGroup D\ninst✝¹² : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nX : Type ?u.1182357\ninst✝⁵ : NormedAddCommGroup X\ninst✝⁴ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nι : Type u_2\nι' : Type ?u.1185838\ninst✝³ : Fintype ι\ninst✝² : Fintype ι'\nF' : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nφ : (i : ι) → E → F' i\np' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)\nΦ : E → (i : ι) → F' i\nP' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)\npr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj\n⊢ HasFTaylorSeriesUpToOn n (fun x i => φ i x) (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s ↔\n ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s",
"tactic": "letI : ∀ (m : ℕ) (i : ι), NormedSpace 𝕜 (E[×m]→L[𝕜] F' i) := fun m i => inferInstance"
},
{
"state_after": "𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹³ : NormedAddCommGroup D\ninst✝¹² : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nX : Type ?u.1182357\ninst✝⁵ : NormedAddCommGroup X\ninst✝⁴ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nι : Type u_2\nι' : Type ?u.1185838\ninst✝³ : Fintype ι\ninst✝² : Fintype ι'\nF' : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nφ : (i : ι) → E → F' i\np' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)\nΦ : E → (i : ι) → F' i\nP' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)\npr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj\nthis : (m : ℕ) → (i : ι) → NormedSpace 𝕜 (ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : ℕ) →\n ((i : ι) → ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) ≃ₗᵢ[𝕜]\n ContinuousMultilinearMap 𝕜 (fun i => E) ((i : ι) → F' i) :=\n fun m => ContinuousMultilinearMap.piₗᵢ 𝕜 fun i => E\n⊢ HasFTaylorSeriesUpToOn n (fun x i => φ i x) (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s ↔\n ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s",
"state_before": "𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹³ : NormedAddCommGroup D\ninst✝¹² : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nX : Type ?u.1182357\ninst✝⁵ : NormedAddCommGroup X\ninst✝⁴ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nι : Type u_2\nι' : Type ?u.1185838\ninst✝³ : Fintype ι\ninst✝² : Fintype ι'\nF' : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nφ : (i : ι) → E → F' i\np' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)\nΦ : E → (i : ι) → F' i\nP' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)\npr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj\nthis : (m : ℕ) → (i : ι) → NormedSpace 𝕜 (ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) := fun m i => inferInstance\n⊢ HasFTaylorSeriesUpToOn n (fun x i => φ i x) (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s ↔\n ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s",
"tactic": "set L : ∀ m : ℕ, (∀ i, E[×m]→L[𝕜] F' i) ≃ₗᵢ[𝕜] E[×m]→L[𝕜] ∀ i, F' i := fun m =>\n ContinuousMultilinearMap.piₗᵢ _ _"
},
{
"state_after": "case refine'_1\n𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹³ : NormedAddCommGroup D\ninst✝¹² : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nX : Type ?u.1182357\ninst✝⁵ : NormedAddCommGroup X\ninst✝⁴ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nι : Type u_2\nι' : Type ?u.1185838\ninst✝³ : Fintype ι\ninst✝² : Fintype ι'\nF' : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nφ : (i : ι) → E → F' i\np' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)\nΦ : E → (i : ι) → F' i\nP' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)\npr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj\nthis : (m : ℕ) → (i : ι) → NormedSpace 𝕜 (ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : ℕ) →\n ((i : ι) → ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) ≃ₗᵢ[𝕜]\n ContinuousMultilinearMap 𝕜 (fun i => E) ((i : ι) → F' i) :=\n fun m => ContinuousMultilinearMap.piₗᵢ 𝕜 fun i => E\nh : HasFTaylorSeriesUpToOn n (fun x i => φ i x) (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s\ni : ι\n⊢ HasFTaylorSeriesUpToOn n (φ i) (p' i) s\n\ncase refine'_2\n𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹³ : NormedAddCommGroup D\ninst✝¹² : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nX : Type ?u.1182357\ninst✝⁵ : NormedAddCommGroup X\ninst✝⁴ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nι : Type u_2\nι' : Type ?u.1185838\ninst✝³ : Fintype ι\ninst✝² : Fintype ι'\nF' : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nφ : (i : ι) → E → F' i\np' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)\nΦ : E → (i : ι) → F' i\nP' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)\npr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj\nthis : (m : ℕ) → (i : ι) → NormedSpace 𝕜 (ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : ℕ) →\n ((i : ι) → ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) ≃ₗᵢ[𝕜]\n ContinuousMultilinearMap 𝕜 (fun i => E) ((i : ι) → F' i) :=\n fun m => ContinuousMultilinearMap.piₗᵢ 𝕜 fun i => E\nh : ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s\nx : E\nhx : x ∈ s\n⊢ ContinuousMultilinearMap.uncurry0 (ContinuousMultilinearMap.pi fun i => p' i x 0) = fun i => φ i x\n\ncase refine'_3\n𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹³ : NormedAddCommGroup D\ninst✝¹² : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nX : Type ?u.1182357\ninst✝⁵ : NormedAddCommGroup X\ninst✝⁴ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nι : Type u_2\nι' : Type ?u.1185838\ninst✝³ : Fintype ι\ninst✝² : Fintype ι'\nF' : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nφ : (i : ι) → E → F' i\np' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)\nΦ : E → (i : ι) → F' i\nP' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)\npr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj\nthis : (m : ℕ) → (i : ι) → NormedSpace 𝕜 (ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : ℕ) →\n ((i : ι) → ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) ≃ₗᵢ[𝕜]\n ContinuousMultilinearMap 𝕜 (fun i => E) ((i : ι) → F' i) :=\n fun m => ContinuousMultilinearMap.piₗᵢ 𝕜 fun i => E\nh : ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s\n⊢ ∀ (m : ℕ),\n ↑m < n →\n ∀ (x : E),\n x ∈ s →\n HasFDerivWithinAt (fun x => ContinuousMultilinearMap.pi fun i => p' i x m)\n (ContinuousMultilinearMap.curryLeft (ContinuousMultilinearMap.pi fun i => p' i x (Nat.succ m))) s x\n\ncase refine'_4\n𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹³ : NormedAddCommGroup D\ninst✝¹² : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nX : Type ?u.1182357\ninst✝⁵ : NormedAddCommGroup X\ninst✝⁴ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nι : Type u_2\nι' : Type ?u.1185838\ninst✝³ : Fintype ι\ninst✝² : Fintype ι'\nF' : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nφ : (i : ι) → E → F' i\np' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)\nΦ : E → (i : ι) → F' i\nP' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)\npr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj\nthis : (m : ℕ) → (i : ι) → NormedSpace 𝕜 (ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : ℕ) →\n ((i : ι) → ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) ≃ₗᵢ[𝕜]\n ContinuousMultilinearMap 𝕜 (fun i => E) ((i : ι) → F' i) :=\n fun m => ContinuousMultilinearMap.piₗᵢ 𝕜 fun i => E\nh : ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s\n⊢ ∀ (m : ℕ), ↑m ≤ n → ContinuousOn (fun x => ContinuousMultilinearMap.pi fun i => p' i x m) s",
"state_before": "𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹³ : NormedAddCommGroup D\ninst✝¹² : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nX : Type ?u.1182357\ninst✝⁵ : NormedAddCommGroup X\ninst✝⁴ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nι : Type u_2\nι' : Type ?u.1185838\ninst✝³ : Fintype ι\ninst✝² : Fintype ι'\nF' : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nφ : (i : ι) → E → F' i\np' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)\nΦ : E → (i : ι) → F' i\nP' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)\npr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj\nthis : (m : ℕ) → (i : ι) → NormedSpace 𝕜 (ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : ℕ) →\n ((i : ι) → ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) ≃ₗᵢ[𝕜]\n ContinuousMultilinearMap 𝕜 (fun i => E) ((i : ι) → F' i) :=\n fun m => ContinuousMultilinearMap.piₗᵢ 𝕜 fun i => E\n⊢ HasFTaylorSeriesUpToOn n (fun x i => φ i x) (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s ↔\n ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s",
"tactic": "refine' ⟨fun h i => _, fun h => ⟨fun x hx => _, _, _⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\n𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹³ : NormedAddCommGroup D\ninst✝¹² : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nX : Type ?u.1182357\ninst✝⁵ : NormedAddCommGroup X\ninst✝⁴ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nι : Type u_2\nι' : Type ?u.1185838\ninst✝³ : Fintype ι\ninst✝² : Fintype ι'\nF' : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nφ : (i : ι) → E → F' i\np' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)\nΦ : E → (i : ι) → F' i\nP' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)\npr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj\nthis : (m : ℕ) → (i : ι) → NormedSpace 𝕜 (ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : ℕ) →\n ((i : ι) → ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) ≃ₗᵢ[𝕜]\n ContinuousMultilinearMap 𝕜 (fun i => E) ((i : ι) → F' i) :=\n fun m => ContinuousMultilinearMap.piₗᵢ 𝕜 fun i => E\nh : HasFTaylorSeriesUpToOn n (fun x i => φ i x) (fun x m => ContinuousMultilinearMap.pi fun i => p' i x m) s\ni : ι\n⊢ HasFTaylorSeriesUpToOn n (φ i) (p' i) s",
"tactic": "convert h.continuousLinearMap_comp (pr i)"
},
{
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{
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"state_before": "case refine'_2.h\n𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹³ : NormedAddCommGroup D\ninst✝¹² : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nX : Type ?u.1182357\ninst✝⁵ : NormedAddCommGroup X\ninst✝⁴ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nι : Type u_2\nι' : Type ?u.1185838\ninst✝³ : Fintype ι\ninst✝² : Fintype ι'\nF' : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nφ : (i : ι) → E → F' i\np' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)\nΦ : E → (i : ι) → F' i\nP' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)\npr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj\nthis : (m : ℕ) → (i : ι) → NormedSpace 𝕜 (ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : ℕ) →\n ((i : ι) → ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) ≃ₗᵢ[𝕜]\n ContinuousMultilinearMap 𝕜 (fun i => E) ((i : ι) → F' i) :=\n fun m => ContinuousMultilinearMap.piₗᵢ 𝕜 fun i => E\nh : ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s\nx : E\nhx : x ∈ s\ni : ι\n⊢ ContinuousMultilinearMap.uncurry0 (ContinuousMultilinearMap.pi fun i => p' i x 0) i = φ i x",
"tactic": "exact (h i).zero_eq x hx"
},
{
"state_after": "case refine'_3\n𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹³ : NormedAddCommGroup D\ninst✝¹² : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nX : Type ?u.1182357\ninst✝⁵ : NormedAddCommGroup X\ninst✝⁴ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nι : Type u_2\nι' : Type ?u.1185838\ninst✝³ : Fintype ι\ninst✝² : Fintype ι'\nF' : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nφ : (i : ι) → E → F' i\np' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)\nΦ : E → (i : ι) → F' i\nP' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)\npr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj\nthis : (m : ℕ) → (i : ι) → NormedSpace 𝕜 (ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : ℕ) →\n ((i : ι) → ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) ≃ₗᵢ[𝕜]\n ContinuousMultilinearMap 𝕜 (fun i => E) ((i : ι) → F' i) :=\n fun m => ContinuousMultilinearMap.piₗᵢ 𝕜 fun i => E\nh : ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s\nm : ℕ\nhm : ↑m < n\nx : E\nhx : x ∈ s\n⊢ HasFDerivWithinAt (fun x => ContinuousMultilinearMap.pi fun i => p' i x m)\n (ContinuousMultilinearMap.curryLeft (ContinuousMultilinearMap.pi fun i => p' i x (Nat.succ m))) s x",
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"tactic": "intro m hm x hx"
},
{
"state_after": "case refine'_3\n𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹³ : NormedAddCommGroup D\ninst✝¹² : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nX : Type ?u.1182357\ninst✝⁵ : NormedAddCommGroup X\ninst✝⁴ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nι : Type u_2\nι' : Type ?u.1185838\ninst✝³ : Fintype ι\ninst✝² : Fintype ι'\nF' : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nφ : (i : ι) → E → F' i\np' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)\nΦ : E → (i : ι) → F' i\nP' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)\npr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj\nthis✝ : (m : ℕ) → (i : ι) → NormedSpace 𝕜 (ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : ℕ) →\n ((i : ι) → ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) ≃ₗᵢ[𝕜]\n ContinuousMultilinearMap 𝕜 (fun i => E) ((i : ι) → F' i) :=\n fun m => ContinuousMultilinearMap.piₗᵢ 𝕜 fun i => E\nh : ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s\nm : ℕ\nhm : ↑m < n\nx : E\nhx : x ∈ s\nthis :\n HasFDerivWithinAt (fun x i => p' i x m)\n (ContinuousLinearMap.pi fun i => ContinuousMultilinearMap.curryLeft (p' i x (Nat.succ m))) s x\n⊢ HasFDerivWithinAt (fun x => ContinuousMultilinearMap.pi fun i => p' i x m)\n (ContinuousMultilinearMap.curryLeft (ContinuousMultilinearMap.pi fun i => p' i x (Nat.succ m))) s x",
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"tactic": "have := hasFDerivWithinAt_pi.2 fun i => (h i).fderivWithin m hm x hx"
},
{
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"state_before": "case refine'_3\n𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹³ : NormedAddCommGroup D\ninst✝¹² : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nX : Type ?u.1182357\ninst✝⁵ : NormedAddCommGroup X\ninst✝⁴ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nι : Type u_2\nι' : Type ?u.1185838\ninst✝³ : Fintype ι\ninst✝² : Fintype ι'\nF' : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nφ : (i : ι) → E → F' i\np' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)\nΦ : E → (i : ι) → F' i\nP' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)\npr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj\nthis✝ : (m : ℕ) → (i : ι) → NormedSpace 𝕜 (ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : ℕ) →\n ((i : ι) → ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) ≃ₗᵢ[𝕜]\n ContinuousMultilinearMap 𝕜 (fun i => E) ((i : ι) → F' i) :=\n fun m => ContinuousMultilinearMap.piₗᵢ 𝕜 fun i => E\nh : ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s\nm : ℕ\nhm : ↑m < n\nx : E\nhx : x ∈ s\nthis :\n HasFDerivWithinAt (fun x i => p' i x m)\n (ContinuousLinearMap.pi fun i => ContinuousMultilinearMap.curryLeft (p' i x (Nat.succ m))) s x\n⊢ HasFDerivWithinAt (fun x => ContinuousMultilinearMap.pi fun i => p' i x m)\n (ContinuousMultilinearMap.curryLeft (ContinuousMultilinearMap.pi fun i => p' i x (Nat.succ m))) s x",
"tactic": "convert(L m).hasFDerivAt.comp_hasFDerivWithinAt x this"
},
{
"state_after": "case refine'_4\n𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹³ : NormedAddCommGroup D\ninst✝¹² : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nX : Type ?u.1182357\ninst✝⁵ : NormedAddCommGroup X\ninst✝⁴ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nι : Type u_2\nι' : Type ?u.1185838\ninst✝³ : Fintype ι\ninst✝² : Fintype ι'\nF' : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nφ : (i : ι) → E → F' i\np' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)\nΦ : E → (i : ι) → F' i\nP' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)\npr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj\nthis : (m : ℕ) → (i : ι) → NormedSpace 𝕜 (ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : ℕ) →\n ((i : ι) → ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) ≃ₗᵢ[𝕜]\n ContinuousMultilinearMap 𝕜 (fun i => E) ((i : ι) → F' i) :=\n fun m => ContinuousMultilinearMap.piₗᵢ 𝕜 fun i => E\nh : ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s\nm : ℕ\nhm : ↑m ≤ n\n⊢ ContinuousOn (fun x => ContinuousMultilinearMap.pi fun i => p' i x m) s",
"state_before": "case refine'_4\n𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹³ : NormedAddCommGroup D\ninst✝¹² : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nX : Type ?u.1182357\ninst✝⁵ : NormedAddCommGroup X\ninst✝⁴ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nι : Type u_2\nι' : Type ?u.1185838\ninst✝³ : Fintype ι\ninst✝² : Fintype ι'\nF' : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nφ : (i : ι) → E → F' i\np' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)\nΦ : E → (i : ι) → F' i\nP' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)\npr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj\nthis : (m : ℕ) → (i : ι) → NormedSpace 𝕜 (ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : ℕ) →\n ((i : ι) → ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) ≃ₗᵢ[𝕜]\n ContinuousMultilinearMap 𝕜 (fun i => E) ((i : ι) → F' i) :=\n fun m => ContinuousMultilinearMap.piₗᵢ 𝕜 fun i => E\nh : ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s\n⊢ ∀ (m : ℕ), ↑m ≤ n → ContinuousOn (fun x => ContinuousMultilinearMap.pi fun i => p' i x m) s",
"tactic": "intro m hm"
},
{
"state_after": "case refine'_4\n𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹³ : NormedAddCommGroup D\ninst✝¹² : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nX : Type ?u.1182357\ninst✝⁵ : NormedAddCommGroup X\ninst✝⁴ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nι : Type u_2\nι' : Type ?u.1185838\ninst✝³ : Fintype ι\ninst✝² : Fintype ι'\nF' : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nφ : (i : ι) → E → F' i\np' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)\nΦ : E → (i : ι) → F' i\nP' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)\npr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj\nthis✝ : (m : ℕ) → (i : ι) → NormedSpace 𝕜 (ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : ℕ) →\n ((i : ι) → ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) ≃ₗᵢ[𝕜]\n ContinuousMultilinearMap 𝕜 (fun i => E) ((i : ι) → F' i) :=\n fun m => ContinuousMultilinearMap.piₗᵢ 𝕜 fun i => E\nh : ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s\nm : ℕ\nhm : ↑m ≤ n\nthis : ContinuousOn (fun y i => p' i y m) s\n⊢ ContinuousOn (fun x => ContinuousMultilinearMap.pi fun i => p' i x m) s",
"state_before": "case refine'_4\n𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹³ : NormedAddCommGroup D\ninst✝¹² : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nX : Type ?u.1182357\ninst✝⁵ : NormedAddCommGroup X\ninst✝⁴ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nι : Type u_2\nι' : Type ?u.1185838\ninst✝³ : Fintype ι\ninst✝² : Fintype ι'\nF' : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nφ : (i : ι) → E → F' i\np' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)\nΦ : E → (i : ι) → F' i\nP' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)\npr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj\nthis : (m : ℕ) → (i : ι) → NormedSpace 𝕜 (ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : ℕ) →\n ((i : ι) → ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) ≃ₗᵢ[𝕜]\n ContinuousMultilinearMap 𝕜 (fun i => E) ((i : ι) → F' i) :=\n fun m => ContinuousMultilinearMap.piₗᵢ 𝕜 fun i => E\nh : ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s\nm : ℕ\nhm : ↑m ≤ n\n⊢ ContinuousOn (fun x => ContinuousMultilinearMap.pi fun i => p' i x m) s",
"tactic": "have := continuousOn_pi.2 fun i => (h i).cont m hm"
},
{
"state_after": "no goals",
"state_before": "case refine'_4\n𝕜 : Type u_1\ninst✝¹⁴ : NontriviallyNormedField 𝕜\nD : Type uD\ninst✝¹³ : NormedAddCommGroup D\ninst✝¹² : NormedSpace 𝕜 D\nE : Type uE\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁹ : NormedAddCommGroup F\ninst✝⁸ : NormedSpace 𝕜 F\nG : Type uG\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\nX : Type ?u.1182357\ninst✝⁵ : NormedAddCommGroup X\ninst✝⁴ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nb : E × F → G\nm✝ n : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nι : Type u_2\nι' : Type ?u.1185838\ninst✝³ : Fintype ι\ninst✝² : Fintype ι'\nF' : ι → Type u_3\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nφ : (i : ι) → E → F' i\np' : (i : ι) → E → FormalMultilinearSeries 𝕜 E (F' i)\nΦ : E → (i : ι) → F' i\nP' : E → FormalMultilinearSeries 𝕜 E ((i : ι) → F' i)\npr : (i : ι) → ((i : ι) → F' i) →L[𝕜] F' i := ContinuousLinearMap.proj\nthis✝ : (m : ℕ) → (i : ι) → NormedSpace 𝕜 (ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) := fun m i => inferInstance\nL : (m : ℕ) →\n ((i : ι) → ContinuousMultilinearMap 𝕜 (fun i => E) (F' i)) ≃ₗᵢ[𝕜]\n ContinuousMultilinearMap 𝕜 (fun i => E) ((i : ι) → F' i) :=\n fun m => ContinuousMultilinearMap.piₗᵢ 𝕜 fun i => E\nh : ∀ (i : ι), HasFTaylorSeriesUpToOn n (φ i) (p' i) s\nm : ℕ\nhm : ↑m ≤ n\nthis : ContinuousOn (fun y i => p' i y m) s\n⊢ ContinuousOn (fun x => ContinuousMultilinearMap.pi fun i => p' i x m) s",
"tactic": "convert(L m).continuous.comp_continuousOn this"
}
] |
[
1134,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1117,
1
] |
Mathlib/Computability/Partrec.lean
|
Nat.Partrec.ppred
|
[
{
"state_after": "case zero\nthis : Primrec₂ fun n m => if n = Nat.succ m then 0 else 1\n⊢ (Nat.rfind fun n => Part.some false) = Part.none\n\ncase succ\nthis : Primrec₂ fun n m => if n = Nat.succ m then 0 else 1\nn✝ : ℕ\n⊢ (Nat.rfind fun n => Part.some (decide (¬n✝ = n → False))) = Part.some n✝",
"state_before": "this : Primrec₂ fun n m => if n = Nat.succ m then 0 else 1\nn : ℕ\n⊢ (Nat.rfind fun n_1 =>\n (fun m => decide (m = 0)) <$> ↑(unpaired fun n m => if n = Nat.succ m then 0 else 1) (Nat.pair n n_1)) =\n ↑(Nat.ppred n)",
"tactic": "cases n <;> simp"
},
{
"state_after": "no goals",
"state_before": "case zero\nthis : Primrec₂ fun n m => if n = Nat.succ m then 0 else 1\n⊢ (Nat.rfind fun n => Part.some false) = Part.none",
"tactic": "exact\n eq_none_iff.2 fun a ⟨⟨m, h, _⟩, _⟩ => by\n simp [show 0 ≠ m.succ by intro h; injection h] at h"
},
{
"state_after": "no goals",
"state_before": "this : Primrec₂ fun n m => if n = Nat.succ m then 0 else 1\na : ℕ\nx✝ : a ∈ Nat.rfind fun n => Part.some false\nm : ℕ\nh : true ∈ (fun n => Part.some false) m\nright✝ : ∀ (k : ℕ), k < m → ((fun n => Part.some false) k).Dom\nh✝ :\n Part.get (Nat.rfind fun n => Part.some false)\n (_ : ∃ n, true ∈ (fun n => Part.some false) n ∧ ∀ (k : ℕ), k < n → ((fun n => Part.some false) k).Dom) =\n a\n⊢ False",
"tactic": "simp [show 0 ≠ m.succ by intro h; injection h] at h"
},
{
"state_after": "this : Primrec₂ fun n m => if n = Nat.succ m then 0 else 1\na : ℕ\nx✝ : a ∈ Nat.rfind fun n => Part.some false\nm : ℕ\nh✝¹ : true ∈ (fun n => Part.some false) m\nright✝ : ∀ (k : ℕ), k < m → ((fun n => Part.some false) k).Dom\nh✝ :\n Part.get (Nat.rfind fun n => Part.some false)\n (_ : ∃ n, true ∈ (fun n => Part.some false) n ∧ ∀ (k : ℕ), k < n → ((fun n => Part.some false) k).Dom) =\n a\nh : 0 = Nat.succ m\n⊢ False",
"state_before": "this : Primrec₂ fun n m => if n = Nat.succ m then 0 else 1\na : ℕ\nx✝ : a ∈ Nat.rfind fun n => Part.some false\nm : ℕ\nh : true ∈ (fun n => Part.some false) m\nright✝ : ∀ (k : ℕ), k < m → ((fun n => Part.some false) k).Dom\nh✝ :\n Part.get (Nat.rfind fun n => Part.some false)\n (_ : ∃ n, true ∈ (fun n => Part.some false) n ∧ ∀ (k : ℕ), k < n → ((fun n => Part.some false) k).Dom) =\n a\n⊢ 0 ≠ Nat.succ m",
"tactic": "intro h"
},
{
"state_after": "no goals",
"state_before": "this : Primrec₂ fun n m => if n = Nat.succ m then 0 else 1\na : ℕ\nx✝ : a ∈ Nat.rfind fun n => Part.some false\nm : ℕ\nh✝¹ : true ∈ (fun n => Part.some false) m\nright✝ : ∀ (k : ℕ), k < m → ((fun n => Part.some false) k).Dom\nh✝ :\n Part.get (Nat.rfind fun n => Part.some false)\n (_ : ∃ n, true ∈ (fun n => Part.some false) n ∧ ∀ (k : ℕ), k < n → ((fun n => Part.some false) k).Dom) =\n a\nh : 0 = Nat.succ m\n⊢ False",
"tactic": "injection h"
},
{
"state_after": "case succ\nthis : Primrec₂ fun n m => if n = Nat.succ m then 0 else 1\nn✝ : ℕ\n⊢ n✝ ∈ Nat.rfind fun n => Part.some (decide (¬n✝ = n → False))",
"state_before": "case succ\nthis : Primrec₂ fun n m => if n = Nat.succ m then 0 else 1\nn✝ : ℕ\n⊢ (Nat.rfind fun n => Part.some (decide (¬n✝ = n → False))) = Part.some n✝",
"tactic": "refine' eq_some_iff.2 _"
},
{
"state_after": "case succ\nthis : Primrec₂ fun n m => if n = Nat.succ m then 0 else 1\nn✝ : ℕ\n⊢ ∀ {m : ℕ}, m < n✝ → ¬(¬n✝ = m → False)",
"state_before": "case succ\nthis : Primrec₂ fun n m => if n = Nat.succ m then 0 else 1\nn✝ : ℕ\n⊢ n✝ ∈ Nat.rfind fun n => Part.some (decide (¬n✝ = n → False))",
"tactic": "simp"
},
{
"state_after": "case succ\nthis : Primrec₂ fun n m => if n = Nat.succ m then 0 else 1\nn✝ m : ℕ\nh : m < n✝\n⊢ ¬(¬n✝ = m → False)",
"state_before": "case succ\nthis : Primrec₂ fun n m => if n = Nat.succ m then 0 else 1\nn✝ : ℕ\n⊢ ∀ {m : ℕ}, m < n✝ → ¬(¬n✝ = m → False)",
"tactic": "intro m h"
},
{
"state_after": "no goals",
"state_before": "case succ\nthis : Primrec₂ fun n m => if n = Nat.succ m then 0 else 1\nn✝ m : ℕ\nh : m < n✝\n⊢ ¬(¬n✝ = m → False)",
"tactic": "simp [ne_of_gt h]"
}
] |
[
232,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
218,
1
] |
Mathlib/Data/MvPolynomial/Rename.lean
|
MvPolynomial.support_rename_of_injective
|
[
{
"state_after": "σ : Type u_1\nτ : Type u_3\nα : Type ?u.1038465\nR : Type u_2\nS : Type ?u.1038471\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\np : MvPolynomial σ R\nf : σ → τ\ninst✝ : DecidableEq τ\nh : Injective f\n⊢ support (Finsupp.mapDomain (Finsupp.mapDomain f) p) = Finset.image (Finsupp.mapDomain f) (support p)",
"state_before": "σ : Type u_1\nτ : Type u_3\nα : Type ?u.1038465\nR : Type u_2\nS : Type ?u.1038471\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\np : MvPolynomial σ R\nf : σ → τ\ninst✝ : DecidableEq τ\nh : Injective f\n⊢ support (↑(rename f) p) = Finset.image (Finsupp.mapDomain f) (support p)",
"tactic": "rw [rename_eq]"
},
{
"state_after": "no goals",
"state_before": "σ : Type u_1\nτ : Type u_3\nα : Type ?u.1038465\nR : Type u_2\nS : Type ?u.1038471\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\np : MvPolynomial σ R\nf : σ → τ\ninst✝ : DecidableEq τ\nh : Injective f\n⊢ support (Finsupp.mapDomain (Finsupp.mapDomain f) p) = Finset.image (Finsupp.mapDomain f) (support p)",
"tactic": "exact Finsupp.mapDomain_support_of_injective (mapDomain_injective h) _"
}
] |
[
347,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
343,
1
] |
Mathlib/LinearAlgebra/LinearIndependent.lean
|
LinearIndependent.sum_type
|
[] |
[
686,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
682,
1
] |
Mathlib/Data/Ordmap/Ordset.lean
|
Ordnode.merge_nil_right
|
[] |
[
645,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
644,
1
] |
Mathlib/Data/Finset/Pointwise.lean
|
Finset.preimage_mul_right_singleton
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.673903\nα : Type u_1\nβ : Type ?u.673909\nγ : Type ?u.673912\ninst✝ : Group α\ns t : Finset α\na b : α\n⊢ preimage {b} (fun x => x * a) (_ : Set.InjOn (fun x => x * a) ((fun x => x * a) ⁻¹' ↑{b})) = {b * a⁻¹}",
"tactic": "classical rw [← image_mul_right', image_singleton]"
},
{
"state_after": "no goals",
"state_before": "F : Type ?u.673903\nα : Type u_1\nβ : Type ?u.673909\nγ : Type ?u.673912\ninst✝ : Group α\ns t : Finset α\na b : α\n⊢ preimage {b} (fun x => x * a) (_ : Set.InjOn (fun x => x * a) ((fun x => x * a) ⁻¹' ↑{b})) = {b * a⁻¹}",
"tactic": "rw [← image_mul_right', image_singleton]"
}
] |
[
1221,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1219,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Multiequalizer.lean
|
CategoryTheory.Limits.Multicofork.fst_app_right
|
[
{
"state_after": "C : Type u\ninst✝ : Category C\nI : MultispanIndex C\nK : Multicofork I\na : I.L\n⊢ (MultispanIndex.multispan I).map (WalkingMultispan.Hom.fst a) ≫\n K.ι.app (WalkingMultispan.right (MultispanIndex.fstFrom I a)) =\n MultispanIndex.fst I a ≫ π K (MultispanIndex.fstFrom I a)",
"state_before": "C : Type u\ninst✝ : Category C\nI : MultispanIndex C\nK : Multicofork I\na : I.L\n⊢ K.ι.app (WalkingMultispan.left a) = MultispanIndex.fst I a ≫ π K (MultispanIndex.fstFrom I a)",
"tactic": "rw [← K.w (WalkingMultispan.Hom.fst a)]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nI : MultispanIndex C\nK : Multicofork I\na : I.L\n⊢ (MultispanIndex.multispan I).map (WalkingMultispan.Hom.fst a) ≫\n K.ι.app (WalkingMultispan.right (MultispanIndex.fstFrom I a)) =\n MultispanIndex.fst I a ≫ π K (MultispanIndex.fstFrom I a)",
"tactic": "rfl"
}
] |
[
548,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
546,
1
] |
Mathlib/Data/Set/Pairwise/Basic.lean
|
Set.pairwise_union_of_symmetric
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.9555\nγ : Type ?u.9558\nι : Type ?u.9561\nι' : Type ?u.9564\nr p q : α → α → Prop\nf g : ι → α\ns t u : Set α\na b : α\nhr : Symmetric r\n⊢ (Set.Pairwise s r ∧ Set.Pairwise t r ∧ ∀ (a : α), a ∈ s → ∀ (b : α), b ∈ t → a ≠ b → r a b ∧ r b a) ↔\n Set.Pairwise s r ∧ Set.Pairwise t r ∧ ∀ (a : α), a ∈ s → ∀ (b : α), b ∈ t → a ≠ b → r a b",
"tactic": "simp only [hr.iff, and_self_iff]"
}
] |
[
151,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
149,
1
] |
Mathlib/Algebra/BigOperators/Fin.lean
|
finFunctionFinEquiv_single
|
[
{
"state_after": "α : Type ?u.147970\nβ : Type ?u.147973\nm n : ℕ\ninst✝ : NeZero m\ni : Fin n\nj : Fin m\n⊢ ∀ (x : Fin n), x ≠ i → ↑(Pi.single i j x) * m ^ ↑x = 0",
"state_before": "α : Type ?u.147970\nβ : Type ?u.147973\nm n : ℕ\ninst✝ : NeZero m\ni : Fin n\nj : Fin m\n⊢ ↑(↑finFunctionFinEquiv (Pi.single i j)) = ↑j * m ^ ↑i",
"tactic": "rw [finFunctionFinEquiv_apply, Fintype.sum_eq_single i, Pi.single_eq_same]"
},
{
"state_after": "α : Type ?u.147970\nβ : Type ?u.147973\nm n : ℕ\ninst✝ : NeZero m\ni : Fin n\nj : Fin m\nx : Fin n\nhx : x ≠ i\n⊢ ↑(Pi.single i j x) * m ^ ↑x = 0",
"state_before": "α : Type ?u.147970\nβ : Type ?u.147973\nm n : ℕ\ninst✝ : NeZero m\ni : Fin n\nj : Fin m\n⊢ ∀ (x : Fin n), x ≠ i → ↑(Pi.single i j x) * m ^ ↑x = 0",
"tactic": "rintro x hx"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.147970\nβ : Type ?u.147973\nm n : ℕ\ninst✝ : NeZero m\ni : Fin n\nj : Fin m\nx : Fin n\nhx : x ≠ i\n⊢ ↑(Pi.single i j x) * m ^ ↑x = 0",
"tactic": "rw [Pi.single_eq_of_ne hx, Fin.val_zero, MulZeroClass.zero_mul]"
}
] |
[
358,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
354,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.isLimit_aleph0
|
[] |
[
1448,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1447,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean
|
PowerSeries.coeff_mul_of_lt_order
|
[
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\nn : ℕ\nh : ↑n < order ψ\nthis : ↑(coeff R n) (φ * ψ) = ∑ p in Finset.Nat.antidiagonal n, 0\n⊢ ↑(coeff R n) (φ * ψ) = 0\n\ncase this\nR : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\nn : ℕ\nh : ↑n < order ψ\n⊢ ↑(coeff R n) (φ * ψ) = ∑ p in Finset.Nat.antidiagonal n, 0",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\nn : ℕ\nh : ↑n < order ψ\n⊢ ↑(coeff R n) (φ * ψ) = 0",
"tactic": "suffices : coeff R n (φ * ψ) = ∑ p in Finset.Nat.antidiagonal n, 0"
},
{
"state_after": "case this\nR : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\nn : ℕ\nh : ↑n < order ψ\n⊢ ↑(coeff R n) (φ * ψ) = ∑ p in Finset.Nat.antidiagonal n, 0",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\nn : ℕ\nh : ↑n < order ψ\nthis : ↑(coeff R n) (φ * ψ) = ∑ p in Finset.Nat.antidiagonal n, 0\n⊢ ↑(coeff R n) (φ * ψ) = 0\n\ncase this\nR : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\nn : ℕ\nh : ↑n < order ψ\n⊢ ↑(coeff R n) (φ * ψ) = ∑ p in Finset.Nat.antidiagonal n, 0",
"tactic": "rw [this, Finset.sum_const_zero]"
},
{
"state_after": "case this\nR : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\nn : ℕ\nh : ↑n < order ψ\n⊢ ∑ p in Finset.Nat.antidiagonal n, ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = ∑ p in Finset.Nat.antidiagonal n, 0",
"state_before": "case this\nR : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\nn : ℕ\nh : ↑n < order ψ\n⊢ ↑(coeff R n) (φ * ψ) = ∑ p in Finset.Nat.antidiagonal n, 0",
"tactic": "rw [coeff_mul]"
},
{
"state_after": "case this\nR : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\nn : ℕ\nh : ↑n < order ψ\n⊢ ∀ (x : ℕ × ℕ), x ∈ Finset.Nat.antidiagonal n → ↑(coeff R x.fst) φ * ↑(coeff R x.snd) ψ = 0",
"state_before": "case this\nR : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\nn : ℕ\nh : ↑n < order ψ\n⊢ ∑ p in Finset.Nat.antidiagonal n, ↑(coeff R p.fst) φ * ↑(coeff R p.snd) ψ = ∑ p in Finset.Nat.antidiagonal n, 0",
"tactic": "apply Finset.sum_congr rfl"
},
{
"state_after": "case this\nR : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\nn : ℕ\nh : ↑n < order ψ\nx : ℕ × ℕ\nhx : x ∈ Finset.Nat.antidiagonal n\n⊢ ↑(coeff R x.fst) φ * ↑(coeff R x.snd) ψ = 0",
"state_before": "case this\nR : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\nn : ℕ\nh : ↑n < order ψ\n⊢ ∀ (x : ℕ × ℕ), x ∈ Finset.Nat.antidiagonal n → ↑(coeff R x.fst) φ * ↑(coeff R x.snd) ψ = 0",
"tactic": "intro x hx"
},
{
"state_after": "case this\nR : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\nn : ℕ\nh : ↑n < order ψ\nx : ℕ × ℕ\nhx : x ∈ Finset.Nat.antidiagonal n\n⊢ ↑x.snd ≤ ↑n",
"state_before": "case this\nR : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\nn : ℕ\nh : ↑n < order ψ\nx : ℕ × ℕ\nhx : x ∈ Finset.Nat.antidiagonal n\n⊢ ↑(coeff R x.fst) φ * ↑(coeff R x.snd) ψ = 0",
"tactic": "refine' mul_eq_zero_of_right (coeff R x.fst φ) (coeff_of_lt_order x.snd (lt_of_le_of_lt _ h))"
},
{
"state_after": "case this\nR : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\nn : ℕ\nh : ↑n < order ψ\nx : ℕ × ℕ\nhx : x.fst + x.snd = n\n⊢ ↑x.snd ≤ ↑n",
"state_before": "case this\nR : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\nn : ℕ\nh : ↑n < order ψ\nx : ℕ × ℕ\nhx : x ∈ Finset.Nat.antidiagonal n\n⊢ ↑x.snd ≤ ↑n",
"tactic": "rw [Finset.Nat.mem_antidiagonal] at hx"
},
{
"state_after": "case this\nR : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\nn : ℕ\nh : ↑n < order ψ\nx : ℕ × ℕ\nhx : x.fst + x.snd = n\n⊢ x.snd ≤ n",
"state_before": "case this\nR : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\nn : ℕ\nh : ↑n < order ψ\nx : ℕ × ℕ\nhx : x.fst + x.snd = n\n⊢ ↑x.snd ≤ ↑n",
"tactic": "norm_cast"
},
{
"state_after": "no goals",
"state_before": "case this\nR : Type u_1\ninst✝ : Semiring R\nφ✝ φ ψ : PowerSeries R\nn : ℕ\nh : ↑n < order ψ\nx : ℕ × ℕ\nhx : x.fst + x.snd = n\n⊢ x.snd ≤ n",
"tactic": "linarith"
}
] |
[
2421,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2411,
1
] |
Mathlib/RingTheory/Localization/Away/Basic.lean
|
selfZpow_neg_mul
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\ninst✝³ : CommRing R\nx : R\nB : Type u_1\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : IsLocalization.Away x B\nd : ℤ\n⊢ selfZpow x B (-d) * selfZpow x B d = 1",
"tactic": "rw [mul_comm, selfZpow_mul_neg x B d]"
}
] |
[
291,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
290,
1
] |
Mathlib/Data/Set/Prod.lean
|
Set.union_pi
|
[
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nα : ι → Type u_2\nβ : ι → Type ?u.144565\ns s₁ s₂ : Set ι\nt t₁ t₂ : (i : ι) → Set (α i)\ni : ι\n⊢ pi (s₁ ∪ s₂) t = pi s₁ t ∩ pi s₂ t",
"tactic": "simp [pi, or_imp, forall_and, setOf_and]"
}
] |
[
762,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
761,
1
] |
Mathlib/CategoryTheory/Simple.lean
|
CategoryTheory.isIso_of_mono_of_nonzero
|
[] |
[
61,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
60,
1
] |
Mathlib/RingTheory/IntegralClosure.lean
|
IsIntegral.prod
|
[] |
[
657,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
655,
1
] |
Mathlib/Topology/Compactification/OnePoint.lean
|
OnePoint.comap_coe_nhds_infty
|
[
{
"state_after": "no goals",
"state_before": "X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set (OnePoint X)\nt : Set X\n⊢ comap some (𝓝 ∞) = coclosedCompact X",
"tactic": "simp [nhds_infty_eq, comap_sup, comap_map coe_injective]"
}
] |
[
346,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
345,
1
] |
Mathlib/Data/Bitvec/Lemmas.lean
|
Bitvec.addLsb_eq_twice_add_one
|
[
{
"state_after": "no goals",
"state_before": "x : ℕ\nb : Bool\n⊢ addLsb x b = 2 * x + bif b then 1 else 0",
"tactic": "simp [addLsb, two_mul]"
}
] |
[
86,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.sin_lt_sin_of_lt_of_le_pi_div_two
|
[
{
"state_after": "x y : ℝ\nhx₁ : -(π / 2) ≤ x\nhy₂ : y ≤ π / 2\nhxy : x < y\n⊢ cos (-(x - π / 2)) < cos (-(y - π / 2))",
"state_before": "x y : ℝ\nhx₁ : -(π / 2) ≤ x\nhy₂ : y ≤ π / 2\nhxy : x < y\n⊢ sin x < sin y",
"tactic": "rw [← cos_sub_pi_div_two, ← cos_sub_pi_div_two, ← cos_neg (x - _), ← cos_neg (y - _)]"
},
{
"state_after": "no goals",
"state_before": "x y : ℝ\nhx₁ : -(π / 2) ≤ x\nhy₂ : y ≤ π / 2\nhxy : x < y\n⊢ cos (-(x - π / 2)) < cos (-(y - π / 2))",
"tactic": "apply cos_lt_cos_of_nonneg_of_le_pi <;> linarith"
}
] |
[
598,
51
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
595,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
|
ENNReal.one_le_rpow
|
[
{
"state_after": "case none\nz : ℝ\nhz : 0 < z\nhx : 1 ≤ none\n⊢ 1 ≤ none ^ z\n\ncase some\nz : ℝ\nhz : 0 < z\nval✝ : ℝ≥0\nhx : 1 ≤ Option.some val✝\n⊢ 1 ≤ Option.some val✝ ^ z",
"state_before": "x : ℝ≥0∞\nz : ℝ\nhx : 1 ≤ x\nhz : 0 < z\n⊢ 1 ≤ x ^ z",
"tactic": "cases x"
},
{
"state_after": "no goals",
"state_before": "case none\nz : ℝ\nhz : 0 < z\nhx : 1 ≤ none\n⊢ 1 ≤ none ^ z",
"tactic": "simp [top_rpow_of_pos hz]"
},
{
"state_after": "case some\nz : ℝ\nhz : 0 < z\nval✝ : ℝ≥0\nhx : 1 ≤ val✝\n⊢ 1 ≤ Option.some val✝ ^ z",
"state_before": "case some\nz : ℝ\nhz : 0 < z\nval✝ : ℝ≥0\nhx : 1 ≤ Option.some val✝\n⊢ 1 ≤ Option.some val✝ ^ z",
"tactic": "simp only [one_le_coe_iff, some_eq_coe] at hx"
},
{
"state_after": "no goals",
"state_before": "case some\nz : ℝ\nhz : 0 < z\nval✝ : ℝ≥0\nhx : 1 ≤ val✝\n⊢ 1 ≤ Option.some val✝ ^ z",
"tactic": "simp [coe_rpow_of_nonneg _ (le_of_lt hz), NNReal.one_le_rpow hx (le_of_lt hz)]"
}
] |
[
704,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
700,
1
] |
Mathlib/CategoryTheory/Generator.lean
|
CategoryTheory.IsCoseparating.isCodetecting
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : Balanced C\n𝒢 : Set C\n⊢ IsCoseparating 𝒢 → IsCodetecting 𝒢",
"tactic": "simpa only [← isDetecting_op_iff, ← isSeparating_op_iff] using IsSeparating.isDetecting"
}
] |
[
186,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
184,
1
] |
Mathlib/Computability/TMToPartrec.lean
|
Turing.ToPartrec.Code.cons_eval
|
[
{
"state_after": "no goals",
"state_before": "f fs : Code\n⊢ eval (cons f fs) = fun v => do\n let n ← eval f v\n let ns ← eval fs v\n pure (List.headI n :: ns)",
"tactic": "simp [eval]"
}
] |
[
155,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/Order/Hom/Lattice.lean
|
LatticeHom.symm_dual_comp
|
[] |
[
1546,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1543,
1
] |
Mathlib/LinearAlgebra/TensorProduct.lean
|
TensorProduct.tmul_neg
|
[] |
[
1248,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1247,
1
] |
Mathlib/MeasureTheory/Integral/MeanInequalities.lean
|
ENNReal.lintegral_Lp_add_le
|
[
{
"state_after": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)",
"tactic": "have hp_pos : 0 < p := lt_of_lt_of_le zero_lt_one hp1"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : (∫⁻ (a : α), f a ^ p ∂μ) = ⊤\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)\n\ncase neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)",
"tactic": "by_cases hf_top : (∫⁻ a, f a ^ p ∂μ) = ⊤"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : (∫⁻ (a : α), g a ^ p ∂μ) = ⊤\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)\n\ncase neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : ¬(∫⁻ (a : α), g a ^ p ∂μ) = ⊤\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)",
"state_before": "case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)",
"tactic": "by_cases hg_top : (∫⁻ a, g a ^ p ∂μ) = ⊤"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : ¬(∫⁻ (a : α), g a ^ p ∂μ) = ⊤\nh1 : p = 1\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)\n\ncase neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : ¬(∫⁻ (a : α), g a ^ p ∂μ) = ⊤\nh1 : ¬p = 1\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)",
"state_before": "case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : ¬(∫⁻ (a : α), g a ^ p ∂μ) = ⊤\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)",
"tactic": "by_cases h1 : p = 1"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : ¬(∫⁻ (a : α), g a ^ p ∂μ) = ⊤\nh1 : ¬p = 1\nhp1_lt : 1 < p\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)",
"state_before": "case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : ¬(∫⁻ (a : α), g a ^ p ∂μ) = ⊤\nh1 : ¬p = 1\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)",
"tactic": "have hp1_lt : 1 < p := by\n refine' lt_of_le_of_ne hp1 _\n symm\n exact h1"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : ¬(∫⁻ (a : α), g a ^ p ∂μ) = ⊤\nh1 : ¬p = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)",
"state_before": "case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : ¬(∫⁻ (a : α), g a ^ p ∂μ) = ⊤\nh1 : ¬p = 1\nhp1_lt : 1 < p\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)",
"tactic": "have hpq := Real.isConjugateExponent_conjugateExponent hp1_lt"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : ¬(∫⁻ (a : α), g a ^ p ∂μ) = ⊤\nh1 : ¬p = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : (∫⁻ (a : α), (f + g) a ^ p ∂μ) = 0\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)\n\ncase neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : ¬(∫⁻ (a : α), g a ^ p ∂μ) = ⊤\nh1 : ¬p = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : ¬(∫⁻ (a : α), (f + g) a ^ p ∂μ) = 0\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)",
"state_before": "case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : ¬(∫⁻ (a : α), g a ^ p ∂μ) = ⊤\nh1 : ¬p = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)",
"tactic": "by_cases h0 : (∫⁻ a, (f + g) a ^ p ∂μ) = 0"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : ¬(∫⁻ (a : α), g a ^ p ∂μ) = ⊤\nh1 : ¬p = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : ¬(∫⁻ (a : α), (f + g) a ^ p ∂μ) = 0\nhtop : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ ⊤\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)",
"state_before": "case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : ¬(∫⁻ (a : α), g a ^ p ∂μ) = ⊤\nh1 : ¬p = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : ¬(∫⁻ (a : α), (f + g) a ^ p ∂μ) = 0\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)",
"tactic": "have htop : (∫⁻ a, (f + g) a ^ p ∂μ) ≠ ⊤ := by\n rw [← Ne.def] at hf_top hg_top\n rw [← lt_top_iff_ne_top] at hf_top hg_top⊢\n exact lintegral_rpow_add_lt_top_of_lintegral_rpow_lt_top hf hf_top hg_top hp1"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : ¬(∫⁻ (a : α), g a ^ p ∂μ) = ⊤\nh1 : ¬p = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : ¬(∫⁻ (a : α), (f + g) a ^ p ∂μ) = 0\nhtop : (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ ⊤\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)",
"tactic": "exact lintegral_Lp_add_le_aux hpq hf hf_top hg hg_top h0 htop"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : (∫⁻ (a : α), f a ^ p ∂μ) = ⊤\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)",
"tactic": "simp [hf_top, hp_pos]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : (∫⁻ (a : α), g a ^ p ∂μ) = ⊤\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)",
"tactic": "simp [hg_top, hp_pos]"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : ¬(∫⁻ (a : α), g a ^ p ∂μ) = ⊤\nh1 : p = 1\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) = (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)",
"state_before": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : ¬(∫⁻ (a : α), g a ^ p ∂μ) = ⊤\nh1 : p = 1\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)",
"tactic": "refine' le_of_eq _"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : ¬(∫⁻ (a : α), g a ^ p ∂μ) = ⊤\nh1 : p = 1\n⊢ (∫⁻ (a : α), (f + g) a ∂μ) = (∫⁻ (a : α), f a ∂μ) + ∫⁻ (a : α), g a ∂μ",
"state_before": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : ¬(∫⁻ (a : α), g a ^ p ∂μ) = ⊤\nh1 : p = 1\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) = (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)",
"tactic": "simp_rw [h1, one_div_one, ENNReal.rpow_one]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : ¬(∫⁻ (a : α), g a ^ p ∂μ) = ⊤\nh1 : p = 1\n⊢ (∫⁻ (a : α), (f + g) a ∂μ) = (∫⁻ (a : α), f a ∂μ) + ∫⁻ (a : α), g a ∂μ",
"tactic": "exact lintegral_add_left' hf _"
},
{
"state_after": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : ¬(∫⁻ (a : α), g a ^ p ∂μ) = ⊤\nh1 : ¬p = 1\n⊢ 1 ≠ p",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : ¬(∫⁻ (a : α), g a ^ p ∂μ) = ⊤\nh1 : ¬p = 1\n⊢ 1 < p",
"tactic": "refine' lt_of_le_of_ne hp1 _"
},
{
"state_after": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : ¬(∫⁻ (a : α), g a ^ p ∂μ) = ⊤\nh1 : ¬p = 1\n⊢ p ≠ 1",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : ¬(∫⁻ (a : α), g a ^ p ∂μ) = ⊤\nh1 : ¬p = 1\n⊢ 1 ≠ p",
"tactic": "symm"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : ¬(∫⁻ (a : α), g a ^ p ∂μ) = ⊤\nh1 : ¬p = 1\n⊢ p ≠ 1",
"tactic": "exact h1"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : ¬(∫⁻ (a : α), g a ^ p ∂μ) = ⊤\nh1 : ¬p = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : (∫⁻ (a : α), (f + g) a ^ p ∂μ) = 0\n⊢ 0 ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)",
"state_before": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : ¬(∫⁻ (a : α), g a ^ p ∂μ) = ⊤\nh1 : ¬p = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : (∫⁻ (a : α), (f + g) a ^ p ∂μ) = 0\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)",
"tactic": "rw [h0, @ENNReal.zero_rpow_of_pos (1 / p) (by simp [lt_of_lt_of_le zero_lt_one hp1])]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : ¬(∫⁻ (a : α), g a ^ p ∂μ) = ⊤\nh1 : ¬p = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : (∫⁻ (a : α), (f + g) a ^ p ∂μ) = 0\n⊢ 0 ≤ (∫⁻ (a : α), f a ^ p ∂μ) ^ (1 / p) + (∫⁻ (a : α), g a ^ p ∂μ) ^ (1 / p)",
"tactic": "exact zero_le _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : ¬(∫⁻ (a : α), g a ^ p ∂μ) = ⊤\nh1 : ¬p = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : (∫⁻ (a : α), (f + g) a ^ p ∂μ) = 0\n⊢ 0 < 1 / p",
"tactic": "simp [lt_of_lt_of_le zero_lt_one hp1]"
},
{
"state_after": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : (∫⁻ (a : α), f a ^ p ∂μ) ≠ ⊤\nhg_top : (∫⁻ (a : α), g a ^ p ∂μ) ≠ ⊤\nh1 : ¬p = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : ¬(∫⁻ (a : α), (f + g) a ^ p ∂μ) = 0\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ ⊤",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : ¬(∫⁻ (a : α), f a ^ p ∂μ) = ⊤\nhg_top : ¬(∫⁻ (a : α), g a ^ p ∂μ) = ⊤\nh1 : ¬p = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : ¬(∫⁻ (a : α), (f + g) a ^ p ∂μ) = 0\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ ⊤",
"tactic": "rw [← Ne.def] at hf_top hg_top"
},
{
"state_after": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : (∫⁻ (a : α), f a ^ p ∂μ) < ⊤\nhg_top : (∫⁻ (a : α), g a ^ p ∂μ) < ⊤\nh1 : ¬p = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : ¬(∫⁻ (a : α), (f + g) a ^ p ∂μ) = 0\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) < ⊤",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : (∫⁻ (a : α), f a ^ p ∂μ) ≠ ⊤\nhg_top : (∫⁻ (a : α), g a ^ p ∂μ) ≠ ⊤\nh1 : ¬p = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : ¬(∫⁻ (a : α), (f + g) a ^ p ∂μ) = 0\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) ≠ ⊤",
"tactic": "rw [← lt_top_iff_ne_top] at hf_top hg_top⊢"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : MeasureTheory.Measure α\np : ℝ\nf g : α → ℝ≥0∞\nhf : AEMeasurable f\nhg : AEMeasurable g\nhp1 : 1 ≤ p\nhp_pos : 0 < p\nhf_top : (∫⁻ (a : α), f a ^ p ∂μ) < ⊤\nhg_top : (∫⁻ (a : α), g a ^ p ∂μ) < ⊤\nh1 : ¬p = 1\nhp1_lt : 1 < p\nhpq : Real.IsConjugateExponent p (Real.conjugateExponent p)\nh0 : ¬(∫⁻ (a : α), (f + g) a ^ p ∂μ) = 0\n⊢ (∫⁻ (a : α), (f + g) a ^ p ∂μ) < ⊤",
"tactic": "exact lintegral_rpow_add_lt_top_of_lintegral_rpow_lt_top hf hf_top hg_top hp1"
}
] |
[
374,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
349,
1
] |
Mathlib/Combinatorics/Quiver/Path.lean
|
Quiver.Path.nil_comp
|
[
{
"state_after": "no goals",
"state_before": "V : Type u\ninst✝ : Quiver V\na✝¹ b c d a x✝ b✝ : V\np : Path a b✝\na✝ : b✝ ⟶ x✝\n⊢ comp nil (cons p a✝) = cons p a✝",
"tactic": "rw [comp_cons, nil_comp p]"
}
] |
[
111,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
109,
1
] |
Mathlib/LinearAlgebra/Finrank.lean
|
FiniteDimensional.nontrivial_of_finrank_eq_succ
|
[
{
"state_after": "K : Type u\nV : Type v\ninst✝⁶ : Ring K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nV₂ : Type v'\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : Nontrivial K\ninst✝ : NoZeroSMulDivisors K V\nn : ℕ\nhn : finrank K V = Nat.succ n\n⊢ 0 < Nat.succ n",
"state_before": "K : Type u\nV : Type v\ninst✝⁶ : Ring K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nV₂ : Type v'\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : Nontrivial K\ninst✝ : NoZeroSMulDivisors K V\nn : ℕ\nhn : finrank K V = Nat.succ n\n⊢ 0 < finrank (?m.96464 hn) V",
"tactic": "rw [hn]"
},
{
"state_after": "no goals",
"state_before": "K : Type u\nV : Type v\ninst✝⁶ : Ring K\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module K V\nV₂ : Type v'\ninst✝³ : AddCommGroup V₂\ninst✝² : Module K V₂\ninst✝¹ : Nontrivial K\ninst✝ : NoZeroSMulDivisors K V\nn : ℕ\nhn : finrank K V = Nat.succ n\n⊢ 0 < Nat.succ n",
"tactic": "exact n.succ_pos"
}
] |
[
109,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/Topology/Sets/Closeds.lean
|
TopologicalSpace.Clopens.coe_top
|
[] |
[
332,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
332,
9
] |
Mathlib/NumberTheory/Padics/PadicIntegers.lean
|
PadicInt.ext
|
[] |
[
79,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/Data/Seq/Computation.lean
|
Computation.liftRel_pure
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\na : α\nb : β\n⊢ (∃ b_1, b_1 ∈ pure b ∧ R a b_1) ↔ R a b",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\na : α\nb : β\n⊢ LiftRel R (pure a) (pure b) ↔ R a b",
"tactic": "rw [liftRel_pure_left]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\na : α\nb : β\n⊢ (∃ b_1, b_1 ∈ pure b ∧ R a b_1) ↔ R a b",
"tactic": "exact ⟨fun ⟨b', mb', ab'⟩ => by rwa [eq_of_pure_mem mb'] at ab', fun ab => ⟨_, ret_mem _, ab⟩⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nR : α → β → Prop\na : α\nb : β\nx✝ : ∃ b_1, b_1 ∈ pure b ∧ R a b_1\nb' : β\nmb' : b' ∈ pure b\nab' : R a b'\n⊢ R a b",
"tactic": "rwa [eq_of_pure_mem mb'] at ab'"
}
] |
[
1197,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1194,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.inf_eq_inter
|
[] |
[
1814,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1813,
1
] |
Mathlib/Analysis/Calculus/IteratedDeriv.lean
|
norm_iteratedFDerivWithin_eq_norm_iteratedDerivWithin
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nF : Type u_1\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nE : Type ?u.24509\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn : ℕ\nf : 𝕜 → F\ns : Set 𝕜\nx : 𝕜\n⊢ ‖iteratedFDerivWithin 𝕜 n f s x‖ = ‖iteratedDerivWithin n f s x‖",
"tactic": "rw [iteratedDerivWithin_eq_equiv_comp, Function.comp_apply, LinearIsometryEquiv.norm_map]"
}
] |
[
114,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Analysis/Seminorm.lean
|
Seminorm.coe_smul
|
[] |
[
176,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
174,
1
] |
Mathlib/LinearAlgebra/SymplecticGroup.lean
|
SymplecticGroup.inv_eq_symplectic_inv
|
[
{
"state_after": "no goals",
"state_before": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA✝ A : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ∈ symplecticGroup l R\n⊢ (-J l R) ⬝ Aᵀ ⬝ J l R ⬝ A = 1",
"tactic": "simp only [Matrix.neg_mul, inv_left_mul_aux hA]"
}
] |
[
211,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
209,
1
] |
Mathlib/Data/List/Basic.lean
|
List.doubleton_eq
|
[
{
"state_after": "ι : Type ?u.7883\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\nx y : α\nh : x ≠ y\n⊢ ¬x ∈ {y}",
"state_before": "ι : Type ?u.7883\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\nx y : α\nh : x ≠ y\n⊢ {x, y} = [x, y]",
"tactic": "rw [insert_neg, singleton_eq]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.7883\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\nx y : α\nh : x ≠ y\n⊢ ¬x ∈ {y}",
"tactic": "rwa [singleton_eq, mem_singleton]"
}
] |
[
264,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
262,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.OuterMeasure.add_apply
|
[] |
[
290,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
289,
1
] |
Mathlib/GroupTheory/Subsemigroup/Operations.lean
|
Subsemigroup.map_iInf_comap_of_surjective
|
[] |
[
485,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
483,
1
] |
Mathlib/Analysis/SpecialFunctions/Integrals.lean
|
intervalIntegral.inv_mul_integral_comp_div_sub
|
[] |
[
303,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
301,
1
] |
Mathlib/LinearAlgebra/Matrix/SesquilinearForm.lean
|
Matrix.toLinearMap₂'_comp
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_3\nR₁ : Type ?u.1177421\nR₂ : Type ?u.1177424\nM✝ : Type ?u.1177427\nM₁ : Type ?u.1177430\nM₂ : Type ?u.1177433\nM₁' : Type ?u.1177436\nM₂' : Type ?u.1177439\nn : Type u_1\nm : Type u_2\nn' : Type u_4\nm' : Type u_5\nι : Type ?u.1177454\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing R₁\ninst✝⁸ : CommRing R₂\ninst✝⁷ : Fintype n\ninst✝⁶ : Fintype m\ninst✝⁵ : DecidableEq n\ninst✝⁴ : DecidableEq m\nσ₁ : R₁ →+* R\nσ₂ : R₂ →+* R\ninst✝³ : Fintype n'\ninst✝² : Fintype m'\ninst✝¹ : DecidableEq n'\ninst✝ : DecidableEq m'\nM : Matrix n m R\nP : Matrix n n' R\nQ : Matrix m m' R\n⊢ ↑toMatrix₂' (compl₁₂ (↑toLinearMap₂' M) (↑toLin' P) (↑toLin' Q)) = ↑toMatrix₂' (↑toLinearMap₂' (Pᵀ ⬝ M ⬝ Q))",
"tactic": "simp"
}
] |
[
337,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
335,
1
] |
Mathlib/Analysis/Convex/Independent.lean
|
ConvexIndependent.mono
|
[] |
[
112,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
11
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.sin_add_int_mul_two_pi
|
[] |
[
284,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
283,
1
] |
Mathlib/AlgebraicGeometry/ProjectiveSpectrum/Topology.lean
|
ProjectiveSpectrum.homogeneousIdeal_le_vanishingIdeal_zeroLocus
|
[] |
[
172,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
170,
1
] |
Mathlib/Combinatorics/SetFamily/LYM.lean
|
Finset.slice_union_shadow_falling_succ
|
[
{
"state_after": "case a\n𝕜 : Type ?u.21156\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : DecidableEq α\nk : ℕ\n𝒜 : Finset (Finset α)\ns✝ s : Finset α\n⊢ s ∈ 𝒜 # k ∪ (∂ ) (falling (k + 1) 𝒜) ↔ s ∈ falling k 𝒜",
"state_before": "𝕜 : Type ?u.21156\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : DecidableEq α\nk : ℕ\n𝒜 : Finset (Finset α)\ns : Finset α\n⊢ 𝒜 # k ∪ (∂ ) (falling (k + 1) 𝒜) = falling k 𝒜",
"tactic": "ext s"
},
{
"state_after": "case a\n𝕜 : Type ?u.21156\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : DecidableEq α\nk : ℕ\n𝒜 : Finset (Finset α)\ns✝ s : Finset α\n⊢ (s ∈ 𝒜 ∧ card s = k ∨ ∃ t, ((∃ t_1, t_1 ∈ 𝒜 ∧ t ⊆ t_1) ∧ card t = k + 1) ∧ ∃ a, a ∈ t ∧ erase t a = s) ↔\n (∃ t, t ∈ 𝒜 ∧ s ⊆ t) ∧ card s = k",
"state_before": "case a\n𝕜 : Type ?u.21156\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : DecidableEq α\nk : ℕ\n𝒜 : Finset (Finset α)\ns✝ s : Finset α\n⊢ s ∈ 𝒜 # k ∪ (∂ ) (falling (k + 1) 𝒜) ↔ s ∈ falling k 𝒜",
"tactic": "simp_rw [mem_union, mem_slice, mem_shadow_iff, mem_falling]"
},
{
"state_after": "case a.mp\n𝕜 : Type ?u.21156\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : DecidableEq α\nk : ℕ\n𝒜 : Finset (Finset α)\ns✝ s : Finset α\n⊢ (s ∈ 𝒜 ∧ card s = k ∨ ∃ t, ((∃ t_1, t_1 ∈ 𝒜 ∧ t ⊆ t_1) ∧ card t = k + 1) ∧ ∃ a, a ∈ t ∧ erase t a = s) →\n (∃ t, t ∈ 𝒜 ∧ s ⊆ t) ∧ card s = k\n\ncase a.mpr\n𝕜 : Type ?u.21156\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : DecidableEq α\nk : ℕ\n𝒜 : Finset (Finset α)\ns✝ s : Finset α\n⊢ (∃ t, t ∈ 𝒜 ∧ s ⊆ t) ∧ card s = k →\n s ∈ 𝒜 ∧ card s = k ∨ ∃ t, ((∃ t_1, t_1 ∈ 𝒜 ∧ t ⊆ t_1) ∧ card t = k + 1) ∧ ∃ a, a ∈ t ∧ erase t a = s",
"state_before": "case a\n𝕜 : Type ?u.21156\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : DecidableEq α\nk : ℕ\n𝒜 : Finset (Finset α)\ns✝ s : Finset α\n⊢ (s ∈ 𝒜 ∧ card s = k ∨ ∃ t, ((∃ t_1, t_1 ∈ 𝒜 ∧ t ⊆ t_1) ∧ card t = k + 1) ∧ ∃ a, a ∈ t ∧ erase t a = s) ↔\n (∃ t, t ∈ 𝒜 ∧ s ⊆ t) ∧ card s = k",
"tactic": "constructor"
},
{
"state_after": "case a.mp.inl\n𝕜 : Type ?u.21156\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : DecidableEq α\nk : ℕ\n𝒜 : Finset (Finset α)\ns✝ s : Finset α\nh : s ∈ 𝒜 ∧ card s = k\n⊢ (∃ t, t ∈ 𝒜 ∧ s ⊆ t) ∧ card s = k\n\ncase a.mp.inr.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type ?u.21156\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : DecidableEq α\nk : ℕ\n𝒜 : Finset (Finset α)\ns✝ s : Finset α\nhs : card s = k + 1\nt : Finset α\nht : t ∈ 𝒜\nhst : s ⊆ t\na : α\nha : a ∈ s\n⊢ (∃ t, t ∈ 𝒜 ∧ erase s a ⊆ t) ∧ card (erase s a) = k",
"state_before": "case a.mp\n𝕜 : Type ?u.21156\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : DecidableEq α\nk : ℕ\n𝒜 : Finset (Finset α)\ns✝ s : Finset α\n⊢ (s ∈ 𝒜 ∧ card s = k ∨ ∃ t, ((∃ t_1, t_1 ∈ 𝒜 ∧ t ⊆ t_1) ∧ card t = k + 1) ∧ ∃ a, a ∈ t ∧ erase t a = s) →\n (∃ t, t ∈ 𝒜 ∧ s ⊆ t) ∧ card s = k",
"tactic": "rintro (h | ⟨s, ⟨⟨t, ht, hst⟩, hs⟩, a, ha, rfl⟩)"
},
{
"state_after": "case a.mp.inr.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type ?u.21156\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : DecidableEq α\nk : ℕ\n𝒜 : Finset (Finset α)\ns✝ s : Finset α\nhs : card s = k + 1\nt : Finset α\nht : t ∈ 𝒜\nhst : s ⊆ t\na : α\nha : a ∈ s\n⊢ card (erase s a) = k",
"state_before": "case a.mp.inr.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type ?u.21156\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : DecidableEq α\nk : ℕ\n𝒜 : Finset (Finset α)\ns✝ s : Finset α\nhs : card s = k + 1\nt : Finset α\nht : t ∈ 𝒜\nhst : s ⊆ t\na : α\nha : a ∈ s\n⊢ (∃ t, t ∈ 𝒜 ∧ erase s a ⊆ t) ∧ card (erase s a) = k",
"tactic": "refine' ⟨⟨t, ht, (erase_subset _ _).trans hst⟩, _⟩"
},
{
"state_after": "case a.mp.inr.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type ?u.21156\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : DecidableEq α\nk : ℕ\n𝒜 : Finset (Finset α)\ns✝ s : Finset α\nhs : card s = k + 1\nt : Finset α\nht : t ∈ 𝒜\nhst : s ⊆ t\na : α\nha : a ∈ s\n⊢ k + 1 - 1 = k",
"state_before": "case a.mp.inr.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type ?u.21156\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : DecidableEq α\nk : ℕ\n𝒜 : Finset (Finset α)\ns✝ s : Finset α\nhs : card s = k + 1\nt : Finset α\nht : t ∈ 𝒜\nhst : s ⊆ t\na : α\nha : a ∈ s\n⊢ card (erase s a) = k",
"tactic": "rw [card_erase_of_mem ha, hs]"
},
{
"state_after": "no goals",
"state_before": "case a.mp.inr.intro.intro.intro.intro.intro.intro.intro\n𝕜 : Type ?u.21156\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : DecidableEq α\nk : ℕ\n𝒜 : Finset (Finset α)\ns✝ s : Finset α\nhs : card s = k + 1\nt : Finset α\nht : t ∈ 𝒜\nhst : s ⊆ t\na : α\nha : a ∈ s\n⊢ k + 1 - 1 = k",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case a.mp.inl\n𝕜 : Type ?u.21156\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : DecidableEq α\nk : ℕ\n𝒜 : Finset (Finset α)\ns✝ s : Finset α\nh : s ∈ 𝒜 ∧ card s = k\n⊢ (∃ t, t ∈ 𝒜 ∧ s ⊆ t) ∧ card s = k",
"tactic": "exact ⟨⟨s, h.1, Subset.refl _⟩, h.2⟩"
},
{
"state_after": "case a.mpr.intro.intro.intro\n𝕜 : Type ?u.21156\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : DecidableEq α\nk : ℕ\n𝒜 : Finset (Finset α)\ns✝ s : Finset α\nhs : card s = k\nt : Finset α\nht : t ∈ 𝒜\nhst : s ⊆ t\n⊢ s ∈ 𝒜 ∧ card s = k ∨ ∃ t, ((∃ t_1, t_1 ∈ 𝒜 ∧ t ⊆ t_1) ∧ card t = k + 1) ∧ ∃ a, a ∈ t ∧ erase t a = s",
"state_before": "case a.mpr\n𝕜 : Type ?u.21156\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : DecidableEq α\nk : ℕ\n𝒜 : Finset (Finset α)\ns✝ s : Finset α\n⊢ (∃ t, t ∈ 𝒜 ∧ s ⊆ t) ∧ card s = k →\n s ∈ 𝒜 ∧ card s = k ∨ ∃ t, ((∃ t_1, t_1 ∈ 𝒜 ∧ t ⊆ t_1) ∧ card t = k + 1) ∧ ∃ a, a ∈ t ∧ erase t a = s",
"tactic": "rintro ⟨⟨t, ht, hst⟩, hs⟩"
},
{
"state_after": "case pos\n𝕜 : Type ?u.21156\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : DecidableEq α\nk : ℕ\n𝒜 : Finset (Finset α)\ns✝ s : Finset α\nhs : card s = k\nt : Finset α\nht : t ∈ 𝒜\nhst : s ⊆ t\nh : s ∈ 𝒜\n⊢ s ∈ 𝒜 ∧ card s = k ∨ ∃ t, ((∃ t_1, t_1 ∈ 𝒜 ∧ t ⊆ t_1) ∧ card t = k + 1) ∧ ∃ a, a ∈ t ∧ erase t a = s\n\ncase neg\n𝕜 : Type ?u.21156\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : DecidableEq α\nk : ℕ\n𝒜 : Finset (Finset α)\ns✝ s : Finset α\nhs : card s = k\nt : Finset α\nht : t ∈ 𝒜\nhst : s ⊆ t\nh : ¬s ∈ 𝒜\n⊢ s ∈ 𝒜 ∧ card s = k ∨ ∃ t, ((∃ t_1, t_1 ∈ 𝒜 ∧ t ⊆ t_1) ∧ card t = k + 1) ∧ ∃ a, a ∈ t ∧ erase t a = s",
"state_before": "case a.mpr.intro.intro.intro\n𝕜 : Type ?u.21156\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : DecidableEq α\nk : ℕ\n𝒜 : Finset (Finset α)\ns✝ s : Finset α\nhs : card s = k\nt : Finset α\nht : t ∈ 𝒜\nhst : s ⊆ t\n⊢ s ∈ 𝒜 ∧ card s = k ∨ ∃ t, ((∃ t_1, t_1 ∈ 𝒜 ∧ t ⊆ t_1) ∧ card t = k + 1) ∧ ∃ a, a ∈ t ∧ erase t a = s",
"tactic": "by_cases h : s ∈ 𝒜"
},
{
"state_after": "case neg.intro.intro\n𝕜 : Type ?u.21156\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : DecidableEq α\nk : ℕ\n𝒜 : Finset (Finset α)\ns✝ s : Finset α\nhs : card s = k\nt : Finset α\nht : t ∈ 𝒜\nhst✝ : s ⊆ t\nh : ¬s ∈ 𝒜\na : α\nha : ¬a ∈ s\nhst : insert a s ⊆ t\n⊢ s ∈ 𝒜 ∧ card s = k ∨ ∃ t, ((∃ t_1, t_1 ∈ 𝒜 ∧ t ⊆ t_1) ∧ card t = k + 1) ∧ ∃ a, a ∈ t ∧ erase t a = s",
"state_before": "case neg\n𝕜 : Type ?u.21156\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : DecidableEq α\nk : ℕ\n𝒜 : Finset (Finset α)\ns✝ s : Finset α\nhs : card s = k\nt : Finset α\nht : t ∈ 𝒜\nhst : s ⊆ t\nh : ¬s ∈ 𝒜\n⊢ s ∈ 𝒜 ∧ card s = k ∨ ∃ t, ((∃ t_1, t_1 ∈ 𝒜 ∧ t ⊆ t_1) ∧ card t = k + 1) ∧ ∃ a, a ∈ t ∧ erase t a = s",
"tactic": "obtain ⟨a, ha, hst⟩ := ssubset_iff.1 (ssubset_of_subset_of_ne hst (ht.ne_of_not_mem h).symm)"
},
{
"state_after": "case neg.intro.intro\n𝕜 : Type ?u.21156\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : DecidableEq α\nk : ℕ\n𝒜 : Finset (Finset α)\ns✝ s : Finset α\nhs : card s = k\nt : Finset α\nht : t ∈ 𝒜\nhst✝ : s ⊆ t\nh : ¬s ∈ 𝒜\na : α\nha : ¬a ∈ s\nhst : insert a s ⊆ t\n⊢ card (insert a s) = k + 1",
"state_before": "case neg.intro.intro\n𝕜 : Type ?u.21156\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : DecidableEq α\nk : ℕ\n𝒜 : Finset (Finset α)\ns✝ s : Finset α\nhs : card s = k\nt : Finset α\nht : t ∈ 𝒜\nhst✝ : s ⊆ t\nh : ¬s ∈ 𝒜\na : α\nha : ¬a ∈ s\nhst : insert a s ⊆ t\n⊢ s ∈ 𝒜 ∧ card s = k ∨ ∃ t, ((∃ t_1, t_1 ∈ 𝒜 ∧ t ⊆ t_1) ∧ card t = k + 1) ∧ ∃ a, a ∈ t ∧ erase t a = s",
"tactic": "refine' Or.inr ⟨insert a s, ⟨⟨t, ht, hst⟩, _⟩, a, mem_insert_self _ _, erase_insert ha⟩"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro\n𝕜 : Type ?u.21156\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : DecidableEq α\nk : ℕ\n𝒜 : Finset (Finset α)\ns✝ s : Finset α\nhs : card s = k\nt : Finset α\nht : t ∈ 𝒜\nhst✝ : s ⊆ t\nh : ¬s ∈ 𝒜\na : α\nha : ¬a ∈ s\nhst : insert a s ⊆ t\n⊢ card (insert a s) = k + 1",
"tactic": "rw [card_insert_of_not_mem ha, hs]"
},
{
"state_after": "no goals",
"state_before": "case pos\n𝕜 : Type ?u.21156\nα : Type u_1\ninst✝¹ : LinearOrderedField 𝕜\ninst✝ : DecidableEq α\nk : ℕ\n𝒜 : Finset (Finset α)\ns✝ s : Finset α\nhs : card s = k\nt : Finset α\nht : t ∈ 𝒜\nhst : s ⊆ t\nh : s ∈ 𝒜\n⊢ s ∈ 𝒜 ∧ card s = k ∨ ∃ t, ((∃ t_1, t_1 ∈ 𝒜 ∧ t ⊆ t_1) ∧ card t = k + 1) ∧ ∃ a, a ∈ t ∧ erase t a = s",
"tactic": "exact Or.inl ⟨h, hs⟩"
}
] |
[
166,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/Analysis/Normed/MulAction.lean
|
BoundedSMul.of_norm_smul_le
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝² : SeminormedRing α\ninst✝¹ : SeminormedAddCommGroup β\ninst✝ : Module α β\nh : ∀ (r : α) (x : β), ‖r • x‖ ≤ ‖r‖ * ‖x‖\na : α\nb₁ b₂ : β\n⊢ dist (a • b₁) (a • b₂) ≤ dist a 0 * dist b₁ b₂",
"tactic": "simpa [smul_sub, dist_eq_norm] using h a (b₁ - b₂)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\ninst✝² : SeminormedRing α\ninst✝¹ : SeminormedAddCommGroup β\ninst✝ : Module α β\nh : ∀ (r : α) (x : β), ‖r • x‖ ≤ ‖r‖ * ‖x‖\na₁ a₂ : α\nb : β\n⊢ dist (a₁ • b) (a₂ • b) ≤ dist a₁ a₂ * dist b 0",
"tactic": "simpa [sub_smul, dist_eq_norm] using h (a₁ - a₂) b"
}
] |
[
81,
94
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
78,
1
] |
Mathlib/Analysis/Convex/Function.lean
|
ConvexOn.lt_right_of_left_lt'
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.592639\nα : Type ?u.592642\nβ : Type u_3\nι : Type ?u.592648\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedCancelAddCommMonoid β\ninst✝² : Module 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\na b : 𝕜\nhx : x ∈ s\nhy : y ∈ s\nha : 0 < a\nhb : 0 < b\nhab : b + a = 1\nhfx : f x < f (b • y + a • x)\n⊢ f (b • y + a • x) < f y",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.592639\nα : Type ?u.592642\nβ : Type u_3\nι : Type ?u.592648\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedCancelAddCommMonoid β\ninst✝² : Module 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\na b : 𝕜\nhx : x ∈ s\nhy : y ∈ s\nha : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhfx : f x < f (a • x + b • y)\n⊢ f (a • x + b • y) < f y",
"tactic": "rw [add_comm] at hab hfx⊢"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.592639\nα : Type ?u.592642\nβ : Type u_3\nι : Type ?u.592648\ninst✝⁶ : OrderedSemiring 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : LinearOrderedCancelAddCommMonoid β\ninst✝² : Module 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\na b : 𝕜\nhx : x ∈ s\nhy : y ∈ s\nha : 0 < a\nhb : 0 < b\nhab : b + a = 1\nhfx : f x < f (b • y + a • x)\n⊢ f (b • y + a • x) < f y",
"tactic": "exact hf.lt_left_of_right_lt' hy hx hb ha hab hfx"
}
] |
[
794,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
790,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.val_injective
|
[] |
[
105,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
104,
1
] |
Mathlib/Data/Finmap.lean
|
Finmap.mem_singleton
|
[
{
"state_after": "α : Type u\nβ : α → Type v\nx y : α\nb : β y\n⊢ x ∈ ⟦AList.singleton y b⟧ ↔ x = y",
"state_before": "α : Type u\nβ : α → Type v\nx y : α\nb : β y\n⊢ x ∈ singleton y b ↔ x = y",
"tactic": "simp only [singleton]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : α → Type v\nx y : α\nb : β y\n⊢ x ∈ ⟦AList.singleton y b⟧ ↔ x = y",
"tactic": "erw [mem_cons, mem_nil_iff, or_false_iff]"
}
] |
[
251,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
250,
1
] |
Mathlib/Analysis/LocallyConvex/WithSeminorms.lean
|
Inducing.withSeminorms
|
[
{
"state_after": "𝕜 : Type u_5\n𝕜₂ : Type u_2\n𝕝 : Type ?u.841276\n𝕝₂ : Type ?u.841279\nE : Type u_4\nF : Type u_3\nG : Type ?u.841288\nι : Type u_1\nι' : Type ?u.841294\ninst✝⁹ : NormedField 𝕜\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : NormedField 𝕜₂\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝³ : RingHomIsometric σ₁₂\ninst✝² : TopologicalSpace F\ninst✝¹ : TopologicalAddGroup F\nhι : Nonempty ι\nq : SeminormFamily 𝕜₂ F ι\nhq : WithSeminorms q\ninst✝ : TopologicalSpace E\nf : E →ₛₗ[σ₁₂] F\nhf : Inducing ↑f\n⊢ WithSeminorms (SeminormFamily.comp q f)",
"state_before": "𝕜 : Type u_5\n𝕜₂ : Type u_2\n𝕝 : Type ?u.841276\n𝕝₂ : Type ?u.841279\nE : Type u_4\nF : Type u_3\nG : Type ?u.841288\nι : Type u_1\nι' : Type ?u.841294\ninst✝⁹ : NormedField 𝕜\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : NormedField 𝕜₂\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝³ : RingHomIsometric σ₁₂\ninst✝² : TopologicalSpace F\ninst✝¹ : TopologicalAddGroup F\nhι : Nonempty ι\nq : SeminormFamily 𝕜₂ F ι\nhq : WithSeminorms q\ninst✝ : TopologicalSpace E\nf : E →ₛₗ[σ₁₂] F\nhf : Inducing ↑f\n⊢ WithSeminorms (SeminormFamily.comp q f)",
"tactic": "rw [hf.induced]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_5\n𝕜₂ : Type u_2\n𝕝 : Type ?u.841276\n𝕝₂ : Type ?u.841279\nE : Type u_4\nF : Type u_3\nG : Type ?u.841288\nι : Type u_1\nι' : Type ?u.841294\ninst✝⁹ : NormedField 𝕜\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : NormedField 𝕜₂\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝³ : RingHomIsometric σ₁₂\ninst✝² : TopologicalSpace F\ninst✝¹ : TopologicalAddGroup F\nhι : Nonempty ι\nq : SeminormFamily 𝕜₂ F ι\nhq : WithSeminorms q\ninst✝ : TopologicalSpace E\nf : E →ₛₗ[σ₁₂] F\nhf : Inducing ↑f\n⊢ WithSeminorms (SeminormFamily.comp q f)",
"tactic": "exact f.withSeminorms_induced hq"
}
] |
[
749,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
746,
1
] |
Mathlib/Analysis/MeanInequalitiesPow.lean
|
ENNReal.rpow_add_le_mul_rpow_add_rpow
|
[
{
"state_after": "case inl\nι : Type u\ns : Finset ι\nz₁ z₂ : ℝ≥0∞\nhp : 1 ≤ 1\n⊢ (z₁ + z₂) ^ 1 ≤ 2 ^ (1 - 1) * (z₁ ^ 1 + z₂ ^ 1)\n\ncase inr\nι : Type u\ns : Finset ι\nz₁ z₂ : ℝ≥0∞\np : ℝ\nhp : 1 ≤ p\nh'p : 1 < p\n⊢ (z₁ + z₂) ^ p ≤ 2 ^ (p - 1) * (z₁ ^ p + z₂ ^ p)",
"state_before": "ι : Type u\ns : Finset ι\nz₁ z₂ : ℝ≥0∞\np : ℝ\nhp : 1 ≤ p\n⊢ (z₁ + z₂) ^ p ≤ 2 ^ (p - 1) * (z₁ ^ p + z₂ ^ p)",
"tactic": "rcases eq_or_lt_of_le hp with (rfl | h'p)"
},
{
"state_after": "case h.e'_3\nι : Type u\ns : Finset ι\nz₁ z₂ : ℝ≥0∞\np : ℝ\nhp : 1 ≤ p\nh'p : 1 < p\n⊢ (z₁ + z₂) ^ p = (1 / 2 * (2 * z₁) + 1 / 2 * (2 * z₂)) ^ p\n\ncase h.e'_4\nι : Type u\ns : Finset ι\nz₁ z₂ : ℝ≥0∞\np : ℝ\nhp : 1 ≤ p\nh'p : 1 < p\n⊢ 2 ^ (p - 1) * (z₁ ^ p + z₂ ^ p) = 1 / 2 * (2 * z₁) ^ p + 1 / 2 * (2 * z₂) ^ p",
"state_before": "case inr\nι : Type u\ns : Finset ι\nz₁ z₂ : ℝ≥0∞\np : ℝ\nhp : 1 ≤ p\nh'p : 1 < p\n⊢ (z₁ + z₂) ^ p ≤ 2 ^ (p - 1) * (z₁ ^ p + z₂ ^ p)",
"tactic": "convert rpow_arith_mean_le_arith_mean2_rpow (1 / 2) (1 / 2) (2 * z₁) (2 * z₂)\n (ENNReal.add_halves 1) hp using 1"
},
{
"state_after": "no goals",
"state_before": "case inl\nι : Type u\ns : Finset ι\nz₁ z₂ : ℝ≥0∞\nhp : 1 ≤ 1\n⊢ (z₁ + z₂) ^ 1 ≤ 2 ^ (1 - 1) * (z₁ ^ 1 + z₂ ^ 1)",
"tactic": "simp only [rpow_one, sub_self, rpow_zero, one_mul, le_refl]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nι : Type u\ns : Finset ι\nz₁ z₂ : ℝ≥0∞\np : ℝ\nhp : 1 ≤ p\nh'p : 1 < p\n⊢ (z₁ + z₂) ^ p = (1 / 2 * (2 * z₁) + 1 / 2 * (2 * z₂)) ^ p",
"tactic": "simp [← mul_assoc, ENNReal.inv_mul_cancel two_ne_zero two_ne_top]"
},
{
"state_after": "case h.e'_4\nι : Type u\ns : Finset ι\nz₁ z₂ : ℝ≥0∞\np : ℝ\nhp : 1 ≤ p\nh'p : 1 < p\nx✝ : p - 1 ≠ 0\n⊢ 2 ^ (p - 1) * (z₁ ^ p + z₂ ^ p) = 1 / 2 * (2 * z₁) ^ p + 1 / 2 * (2 * z₂) ^ p",
"state_before": "case h.e'_4\nι : Type u\ns : Finset ι\nz₁ z₂ : ℝ≥0∞\np : ℝ\nhp : 1 ≤ p\nh'p : 1 < p\n⊢ 2 ^ (p - 1) * (z₁ ^ p + z₂ ^ p) = 1 / 2 * (2 * z₁) ^ p + 1 / 2 * (2 * z₂) ^ p",
"tactic": "have _ : p - 1 ≠ 0 := ne_of_gt (sub_pos.2 h'p)"
},
{
"state_after": "case h.e'_4\nι : Type u\ns : Finset ι\nz₁ z₂ : ℝ≥0∞\np : ℝ\nhp : 1 ≤ p\nh'p : 1 < p\nx✝ : p - 1 ≠ 0\n⊢ 2⁻¹ * 2 ^ p * (z₁ ^ p + z₂ ^ p) = 2⁻¹ * (2 ^ p * z₁ ^ p) + 2⁻¹ * (2 ^ p * z₂ ^ p)",
"state_before": "case h.e'_4\nι : Type u\ns : Finset ι\nz₁ z₂ : ℝ≥0∞\np : ℝ\nhp : 1 ≤ p\nh'p : 1 < p\nx✝ : p - 1 ≠ 0\n⊢ 2 ^ (p - 1) * (z₁ ^ p + z₂ ^ p) = 1 / 2 * (2 * z₁) ^ p + 1 / 2 * (2 * z₂) ^ p",
"tactic": "simp only [mul_rpow_of_nonneg _ _ (zero_le_one.trans hp), rpow_sub _ _ two_ne_zero two_ne_top,\n div_eq_inv_mul, rpow_one, mul_one]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nι : Type u\ns : Finset ι\nz₁ z₂ : ℝ≥0∞\np : ℝ\nhp : 1 ≤ p\nh'p : 1 < p\nx✝ : p - 1 ≠ 0\n⊢ 2⁻¹ * 2 ^ p * (z₁ ^ p + z₂ ^ p) = 2⁻¹ * (2 ^ p * z₁ ^ p) + 2⁻¹ * (2 ^ p * z₂ ^ p)",
"tactic": "ring"
}
] |
[
305,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
295,
1
] |
Mathlib/MeasureTheory/Integral/Bochner.lean
|
MeasureTheory.integral_sum_measure
|
[] |
[
1469,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1467,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
|
MeasureTheory.ae_eq_set_inter
|
[] |
[
503,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
501,
1
] |
Mathlib/GroupTheory/GroupAction/Group.lean
|
smul_eq_zero_iff_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : Group α\ninst✝¹ : AddMonoid β\ninst✝ : DistribMulAction α β\na : α\nx : β\nh : a • x = 0\n⊢ x = 0",
"tactic": "rw [← inv_smul_smul a x, h, smul_zero]"
}
] |
[
274,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
273,
1
] |
Mathlib/Data/Finset/Preimage.lean
|
Finset.image_preimage
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ninst✝¹ : DecidableEq β\nf : α → β\ns : Finset β\ninst✝ : (x : β) → Decidable (x ∈ Set.range f)\nhf : InjOn f (f ⁻¹' ↑s)\n⊢ {x | x ∈ ↑s ∧ x ∈ Set.range f} = {x | x ∈ s ∧ x ∈ Set.range f}",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ninst✝¹ : DecidableEq β\nf : α → β\ns : Finset β\ninst✝ : (x : β) → Decidable (x ∈ Set.range f)\nhf : InjOn f (f ⁻¹' ↑s)\n⊢ ↑(image f (preimage s f hf)) = ↑(filter (fun x => x ∈ Set.range f) s)",
"tactic": "simp only [coe_image, coe_preimage, coe_filter, Set.image_preimage_eq_inter_range,\n ← Set.sep_mem_eq]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ninst✝¹ : DecidableEq β\nf : α → β\ns : Finset β\ninst✝ : (x : β) → Decidable (x ∈ Set.range f)\nhf : InjOn f (f ⁻¹' ↑s)\n⊢ {x | x ∈ ↑s ∧ x ∈ Set.range f} = {x | x ∈ s ∧ x ∈ Set.range f}",
"tactic": "rfl"
}
] |
[
101,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/Analysis/SpecificLimits/Basic.lean
|
NNReal.tendsto_pow_atTop_nhds_0_of_lt_1
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.41397\nβ : Type ?u.41400\nι : Type ?u.41403\nr : ℝ≥0\nhr : r < 1\n⊢ Tendsto (fun a => ↑(r ^ a)) atTop (𝓝 ↑0)",
"tactic": "simp only [NNReal.coe_pow, NNReal.coe_zero,\n _root_.tendsto_pow_atTop_nhds_0_of_lt_1 r.coe_nonneg hr]"
}
] |
[
166,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/Data/Real/NNReal.lean
|
NNReal.coe_sub
|
[
{
"state_after": "no goals",
"state_before": "r₁ r₂ : ℝ≥0\nh : r₂ ≤ r₁\n⊢ ↑r₂ ≤ ↑r₁ - 0",
"tactic": "simp [show (r₂ : ℝ) ≤ r₁ from h]"
}
] |
[
211,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
210,
11
] |
Mathlib/GroupTheory/Transfer.lean
|
MonoidHom.transfer_eq_pow
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\nA : Type u_2\ninst✝¹ : CommGroup A\nϕ : { x // x ∈ H } →* A\nT : ↑(leftTransversals ↑H)\ninst✝ : FiniteIndex H\ng : G\nkey : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g₀⁻¹ * g ^ k * g₀ = g ^ k\n⊢ ↑(transfer ϕ) g = ↑ϕ { val := g ^ index H, property := (_ : g ^ index H ∈ H) }",
"tactic": "classical\n letI := H.fintypeQuotientOfFiniteIndex\n change ∀ (k g₀) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H), ↑(⟨g₀⁻¹ * g ^ k * g₀, hk⟩ : H) = g ^ k at key\n rw [transfer_eq_prod_quotient_orbitRel_zpowers_quot, ← Finset.prod_to_list]\n refine' (List.prod_map_hom _ _ _).trans _ refine' congrArg ϕ (Subtype.coe_injective _)\n simp only rw [H.coe_mk, ← (zpowers g).coe_mk g (mem_zpowers g), ← (zpowers g).coe_pow,\n index_eq_card, Fintype.card_congr (selfEquivSigmaOrbits (zpowers g) (G ⧸ H)),\n Fintype.card_sigma, ← Finset.prod_pow_eq_pow_sum, ← Finset.prod_to_list]\n simp only [Subgroup.val_list_prod, List.map_map, ← minimalPeriod_eq_card]\n congr\n funext\n apply key"
},
{
"state_after": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\nA : Type u_2\ninst✝¹ : CommGroup A\nϕ : { x // x ∈ H } →* A\nT : ↑(leftTransversals ↑H)\ninst✝ : FiniteIndex H\ng : G\nkey : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g₀⁻¹ * g ^ k * g₀ = g ^ k\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex H\n⊢ ↑(transfer ϕ) g = ↑ϕ { val := g ^ index H, property := (_ : g ^ index H ∈ H) }",
"state_before": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\nA : Type u_2\ninst✝¹ : CommGroup A\nϕ : { x // x ∈ H } →* A\nT : ↑(leftTransversals ↑H)\ninst✝ : FiniteIndex H\ng : G\nkey : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g₀⁻¹ * g ^ k * g₀ = g ^ k\n⊢ ↑(transfer ϕ) g = ↑ϕ { val := g ^ index H, property := (_ : g ^ index H ∈ H) }",
"tactic": "letI := H.fintypeQuotientOfFiniteIndex"
},
{
"state_after": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\nA : Type u_2\ninst✝¹ : CommGroup A\nϕ : { x // x ∈ H } →* A\nT : ↑(leftTransversals ↑H)\ninst✝ : FiniteIndex H\ng : G\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex H\nkey : ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H), ↑{ val := g₀⁻¹ * g ^ k * g₀, property := hk } = g ^ k\n⊢ ↑(transfer ϕ) g = ↑ϕ { val := g ^ index H, property := (_ : g ^ index H ∈ H) }",
"state_before": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\nA : Type u_2\ninst✝¹ : CommGroup A\nϕ : { x // x ∈ H } →* A\nT : ↑(leftTransversals ↑H)\ninst✝ : FiniteIndex H\ng : G\nkey : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g₀⁻¹ * g ^ k * g₀ = g ^ k\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex H\n⊢ ↑(transfer ϕ) g = ↑ϕ { val := g ^ index H, property := (_ : g ^ index H ∈ H) }",
"tactic": "change ∀ (k g₀) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H), ↑(⟨g₀⁻¹ * g ^ k * g₀, hk⟩ : H) = g ^ k at key"
},
{
"state_after": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\nA : Type u_2\ninst✝¹ : CommGroup A\nϕ : { x // x ∈ H } →* A\nT : ↑(leftTransversals ↑H)\ninst✝ : FiniteIndex H\ng : G\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex H\nkey : ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H), ↑{ val := g₀⁻¹ * g ^ k * g₀, property := hk } = g ^ k\n⊢ List.prod\n (List.map\n (fun q =>\n ↑ϕ\n {\n val :=\n (Quotient.out' (Quotient.out' q))⁻¹ *\n g ^ Function.minimalPeriod ((fun x x_1 => x • x_1) g) (Quotient.out' q) *\n Quotient.out' (Quotient.out' q),\n property :=\n (_ :\n (Quotient.out' (Quotient.out' q))⁻¹ *\n g ^ Function.minimalPeriod ((fun x x_1 => x • x_1) g) (Quotient.out' q) *\n Quotient.out' (Quotient.out' q) ∈\n H) })\n (Finset.toList Finset.univ)) =\n ↑ϕ { val := g ^ index H, property := (_ : g ^ index H ∈ H) }",
"state_before": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\nA : Type u_2\ninst✝¹ : CommGroup A\nϕ : { x // x ∈ H } →* A\nT : ↑(leftTransversals ↑H)\ninst✝ : FiniteIndex H\ng : G\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex H\nkey : ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H), ↑{ val := g₀⁻¹ * g ^ k * g₀, property := hk } = g ^ k\n⊢ ↑(transfer ϕ) g = ↑ϕ { val := g ^ index H, property := (_ : g ^ index H ∈ H) }",
"tactic": "rw [transfer_eq_prod_quotient_orbitRel_zpowers_quot, ← Finset.prod_to_list]"
},
{
"state_after": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\nA : Type u_2\ninst✝¹ : CommGroup A\nϕ : { x // x ∈ H } →* A\nT : ↑(leftTransversals ↑H)\ninst✝ : FiniteIndex H\ng : G\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex H\nkey : ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H), ↑{ val := g₀⁻¹ * g ^ k * g₀, property := hk } = g ^ k\n⊢ ↑ϕ\n (List.prod\n (List.map\n (fun q =>\n {\n val :=\n (Quotient.out' (Quotient.out' q))⁻¹ *\n g ^ Function.minimalPeriod ((fun x x_1 => x • x_1) g) (Quotient.out' q) *\n Quotient.out' (Quotient.out' q),\n property :=\n (_ :\n (Quotient.out' (Quotient.out' q))⁻¹ *\n g ^ Function.minimalPeriod ((fun x x_1 => x • x_1) g) (Quotient.out' q) *\n Quotient.out' (Quotient.out' q) ∈\n H) })\n (Finset.toList Finset.univ))) =\n ↑ϕ { val := g ^ index H, property := (_ : g ^ index H ∈ H) }",
"state_before": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\nA : Type u_2\ninst✝¹ : CommGroup A\nϕ : { x // x ∈ H } →* A\nT : ↑(leftTransversals ↑H)\ninst✝ : FiniteIndex H\ng : G\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex H\nkey : ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H), ↑{ val := g₀⁻¹ * g ^ k * g₀, property := hk } = g ^ k\n⊢ List.prod\n (List.map\n (fun q =>\n ↑ϕ\n {\n val :=\n (Quotient.out' (Quotient.out' q))⁻¹ *\n g ^ Function.minimalPeriod ((fun x x_1 => x • x_1) g) (Quotient.out' q) *\n Quotient.out' (Quotient.out' q),\n property :=\n (_ :\n (Quotient.out' (Quotient.out' q))⁻¹ *\n g ^ Function.minimalPeriod ((fun x x_1 => x • x_1) g) (Quotient.out' q) *\n Quotient.out' (Quotient.out' q) ∈\n H) })\n (Finset.toList Finset.univ)) =\n ↑ϕ { val := g ^ index H, property := (_ : g ^ index H ∈ H) }",
"tactic": "refine' (List.prod_map_hom _ _ _).trans _"
},
{
"state_after": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\nA : Type u_2\ninst✝¹ : CommGroup A\nϕ : { x // x ∈ H } →* A\nT : ↑(leftTransversals ↑H)\ninst✝ : FiniteIndex H\ng : G\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex H\nkey : ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H), ↑{ val := g₀⁻¹ * g ^ k * g₀, property := hk } = g ^ k\n⊢ (fun a => ↑a)\n (List.prod\n (List.map\n (fun q =>\n {\n val :=\n (Quotient.out' (Quotient.out' q))⁻¹ *\n g ^ Function.minimalPeriod ((fun x x_1 => x • x_1) g) (Quotient.out' q) *\n Quotient.out' (Quotient.out' q),\n property :=\n (_ :\n (Quotient.out' (Quotient.out' q))⁻¹ *\n g ^ Function.minimalPeriod ((fun x x_1 => x • x_1) g) (Quotient.out' q) *\n Quotient.out' (Quotient.out' q) ∈\n H) })\n (Finset.toList Finset.univ))) =\n (fun a => ↑a) { val := g ^ index H, property := (_ : g ^ index H ∈ H) }",
"state_before": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\nA : Type u_2\ninst✝¹ : CommGroup A\nϕ : { x // x ∈ H } →* A\nT : ↑(leftTransversals ↑H)\ninst✝ : FiniteIndex H\ng : G\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex H\nkey : ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H), ↑{ val := g₀⁻¹ * g ^ k * g₀, property := hk } = g ^ k\n⊢ ↑ϕ\n (List.prod\n (List.map\n (fun q =>\n {\n val :=\n (Quotient.out' (Quotient.out' q))⁻¹ *\n g ^ Function.minimalPeriod ((fun x x_1 => x • x_1) g) (Quotient.out' q) *\n Quotient.out' (Quotient.out' q),\n property :=\n (_ :\n (Quotient.out' (Quotient.out' q))⁻¹ *\n g ^ Function.minimalPeriod ((fun x x_1 => x • x_1) g) (Quotient.out' q) *\n Quotient.out' (Quotient.out' q) ∈\n H) })\n (Finset.toList Finset.univ))) =\n ↑ϕ { val := g ^ index H, property := (_ : g ^ index H ∈ H) }",
"tactic": "refine' congrArg ϕ (Subtype.coe_injective _)"
},
{
"state_after": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\nA : Type u_2\ninst✝¹ : CommGroup A\nϕ : { x // x ∈ H } →* A\nT : ↑(leftTransversals ↑H)\ninst✝ : FiniteIndex H\ng : G\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex H\nkey : ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H), ↑{ val := g₀⁻¹ * g ^ k * g₀, property := hk } = g ^ k\n⊢ ↑(List.prod\n (List.map\n (fun q =>\n {\n val :=\n (Quotient.out' (Quotient.out' q))⁻¹ * g ^ Function.minimalPeriod (fun x => g • x) (Quotient.out' q) *\n Quotient.out' (Quotient.out' q),\n property :=\n (_ :\n (Quotient.out' (Quotient.out' q))⁻¹ *\n g ^ Function.minimalPeriod ((fun x x_1 => x • x_1) g) (Quotient.out' q) *\n Quotient.out' (Quotient.out' q) ∈\n H) })\n (Finset.toList Finset.univ))) =\n g ^ index H",
"state_before": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\nA : Type u_2\ninst✝¹ : CommGroup A\nϕ : { x // x ∈ H } →* A\nT : ↑(leftTransversals ↑H)\ninst✝ : FiniteIndex H\ng : G\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex H\nkey : ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H), ↑{ val := g₀⁻¹ * g ^ k * g₀, property := hk } = g ^ k\n⊢ (fun a => ↑a)\n (List.prod\n (List.map\n (fun q =>\n {\n val :=\n (Quotient.out' (Quotient.out' q))⁻¹ *\n g ^ Function.minimalPeriod ((fun x x_1 => x • x_1) g) (Quotient.out' q) *\n Quotient.out' (Quotient.out' q),\n property :=\n (_ :\n (Quotient.out' (Quotient.out' q))⁻¹ *\n g ^ Function.minimalPeriod ((fun x x_1 => x • x_1) g) (Quotient.out' q) *\n Quotient.out' (Quotient.out' q) ∈\n H) })\n (Finset.toList Finset.univ))) =\n (fun a => ↑a) { val := g ^ index H, property := (_ : g ^ index H ∈ H) }",
"tactic": "simp only"
},
{
"state_after": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\nA : Type u_2\ninst✝¹ : CommGroup A\nϕ : { x // x ∈ H } →* A\nT : ↑(leftTransversals ↑H)\ninst✝ : FiniteIndex H\ng : G\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex H\nkey : ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H), ↑{ val := g₀⁻¹ * g ^ k * g₀, property := hk } = g ^ k\n⊢ ↑(List.prod\n (List.map\n (fun q =>\n {\n val :=\n (Quotient.out' (Quotient.out' q))⁻¹ *\n ↑{ val := g, property := (_ : g ∈ zpowers g) } ^\n Function.minimalPeriod (fun x => ↑{ val := g, property := (_ : g ∈ zpowers g) } • x)\n (Quotient.out' q) *\n Quotient.out' (Quotient.out' q),\n property :=\n (_ :\n (Quotient.out' (Quotient.out' q))⁻¹ *\n ↑{ val := g, property := (_ : g ∈ zpowers g) } ^\n Function.minimalPeriod ((fun x x_1 => x • x_1) ↑{ val := g, property := (_ : g ∈ zpowers g) })\n (Quotient.out' q) *\n Quotient.out' (Quotient.out' q) ∈\n H) })\n (Finset.toList Finset.univ))) =\n ↑(List.prod\n (List.map\n (fun i =>\n { val := g, property := (_ : g ∈ zpowers g) } ^\n Fintype.card ↑(orbit { x // x ∈ zpowers g } (Quotient.out' i)))\n (Finset.toList Finset.univ)))",
"state_before": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\nA : Type u_2\ninst✝¹ : CommGroup A\nϕ : { x // x ∈ H } →* A\nT : ↑(leftTransversals ↑H)\ninst✝ : FiniteIndex H\ng : G\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex H\nkey : ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H), ↑{ val := g₀⁻¹ * g ^ k * g₀, property := hk } = g ^ k\n⊢ ↑(List.prod\n (List.map\n (fun q =>\n {\n val :=\n (Quotient.out' (Quotient.out' q))⁻¹ * g ^ Function.minimalPeriod (fun x => g • x) (Quotient.out' q) *\n Quotient.out' (Quotient.out' q),\n property :=\n (_ :\n (Quotient.out' (Quotient.out' q))⁻¹ *\n g ^ Function.minimalPeriod ((fun x x_1 => x • x_1) g) (Quotient.out' q) *\n Quotient.out' (Quotient.out' q) ∈\n H) })\n (Finset.toList Finset.univ))) =\n g ^ index H",
"tactic": "rw [H.coe_mk, ← (zpowers g).coe_mk g (mem_zpowers g), ← (zpowers g).coe_pow,\n index_eq_card, Fintype.card_congr (selfEquivSigmaOrbits (zpowers g) (G ⧸ H)),\n Fintype.card_sigma, ← Finset.prod_pow_eq_pow_sum, ← Finset.prod_to_list]"
},
{
"state_after": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\nA : Type u_2\ninst✝¹ : CommGroup A\nϕ : { x // x ∈ H } →* A\nT : ↑(leftTransversals ↑H)\ninst✝ : FiniteIndex H\ng : G\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex H\nkey : ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H), ↑{ val := g₀⁻¹ * g ^ k * g₀, property := hk } = g ^ k\n⊢ List.prod\n (List.map\n (Subtype.val ∘ fun q =>\n {\n val :=\n (Quotient.out' (Quotient.out' q))⁻¹ * g ^ Function.minimalPeriod (fun x => g • x) (Quotient.out' q) *\n Quotient.out' (Quotient.out' q),\n property :=\n (_ :\n (Quotient.out' (Quotient.out' q))⁻¹ *\n ↑{ val := g, property := (_ : g ∈ zpowers g) } ^\n Function.minimalPeriod ((fun x x_1 => x • x_1) ↑{ val := g, property := (_ : g ∈ zpowers g) })\n (Quotient.out' q) *\n Quotient.out' (Quotient.out' q) ∈\n H) })\n (Finset.toList Finset.univ)) =\n List.prod\n (List.map\n (Subtype.val ∘ fun i =>\n { val := g, property := (_ : g ∈ zpowers g) } ^ Function.minimalPeriod (fun x => g • x) (Quotient.out' i))\n (Finset.toList Finset.univ))",
"state_before": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\nA : Type u_2\ninst✝¹ : CommGroup A\nϕ : { x // x ∈ H } →* A\nT : ↑(leftTransversals ↑H)\ninst✝ : FiniteIndex H\ng : G\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex H\nkey : ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H), ↑{ val := g₀⁻¹ * g ^ k * g₀, property := hk } = g ^ k\n⊢ ↑(List.prod\n (List.map\n (fun q =>\n {\n val :=\n (Quotient.out' (Quotient.out' q))⁻¹ *\n ↑{ val := g, property := (_ : g ∈ zpowers g) } ^\n Function.minimalPeriod (fun x => ↑{ val := g, property := (_ : g ∈ zpowers g) } • x)\n (Quotient.out' q) *\n Quotient.out' (Quotient.out' q),\n property :=\n (_ :\n (Quotient.out' (Quotient.out' q))⁻¹ *\n ↑{ val := g, property := (_ : g ∈ zpowers g) } ^\n Function.minimalPeriod ((fun x x_1 => x • x_1) ↑{ val := g, property := (_ : g ∈ zpowers g) })\n (Quotient.out' q) *\n Quotient.out' (Quotient.out' q) ∈\n H) })\n (Finset.toList Finset.univ))) =\n ↑(List.prod\n (List.map\n (fun i =>\n { val := g, property := (_ : g ∈ zpowers g) } ^\n Fintype.card ↑(orbit { x // x ∈ zpowers g } (Quotient.out' i)))\n (Finset.toList Finset.univ)))",
"tactic": "simp only [Subgroup.val_list_prod, List.map_map, ← minimalPeriod_eq_card]"
},
{
"state_after": "case e_a.e_f\nG : Type u_1\ninst✝² : Group G\nH : Subgroup G\nA : Type u_2\ninst✝¹ : CommGroup A\nϕ : { x // x ∈ H } →* A\nT : ↑(leftTransversals ↑H)\ninst✝ : FiniteIndex H\ng : G\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex H\nkey : ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H), ↑{ val := g₀⁻¹ * g ^ k * g₀, property := hk } = g ^ k\n⊢ (Subtype.val ∘ fun q =>\n {\n val :=\n (Quotient.out' (Quotient.out' q))⁻¹ * g ^ Function.minimalPeriod (fun x => g • x) (Quotient.out' q) *\n Quotient.out' (Quotient.out' q),\n property :=\n (_ :\n (Quotient.out' (Quotient.out' q))⁻¹ *\n ↑{ val := g, property := (_ : g ∈ zpowers g) } ^\n Function.minimalPeriod ((fun x x_1 => x • x_1) ↑{ val := g, property := (_ : g ∈ zpowers g) })\n (Quotient.out' q) *\n Quotient.out' (Quotient.out' q) ∈\n H) }) =\n Subtype.val ∘ fun i =>\n { val := g, property := (_ : g ∈ zpowers g) } ^ Function.minimalPeriod (fun x => g • x) (Quotient.out' i)",
"state_before": "G : Type u_1\ninst✝² : Group G\nH : Subgroup G\nA : Type u_2\ninst✝¹ : CommGroup A\nϕ : { x // x ∈ H } →* A\nT : ↑(leftTransversals ↑H)\ninst✝ : FiniteIndex H\ng : G\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex H\nkey : ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H), ↑{ val := g₀⁻¹ * g ^ k * g₀, property := hk } = g ^ k\n⊢ List.prod\n (List.map\n (Subtype.val ∘ fun q =>\n {\n val :=\n (Quotient.out' (Quotient.out' q))⁻¹ * g ^ Function.minimalPeriod (fun x => g • x) (Quotient.out' q) *\n Quotient.out' (Quotient.out' q),\n property :=\n (_ :\n (Quotient.out' (Quotient.out' q))⁻¹ *\n ↑{ val := g, property := (_ : g ∈ zpowers g) } ^\n Function.minimalPeriod ((fun x x_1 => x • x_1) ↑{ val := g, property := (_ : g ∈ zpowers g) })\n (Quotient.out' q) *\n Quotient.out' (Quotient.out' q) ∈\n H) })\n (Finset.toList Finset.univ)) =\n List.prod\n (List.map\n (Subtype.val ∘ fun i =>\n { val := g, property := (_ : g ∈ zpowers g) } ^ Function.minimalPeriod (fun x => g • x) (Quotient.out' i))\n (Finset.toList Finset.univ))",
"tactic": "congr"
},
{
"state_after": "case e_a.e_f.h\nG : Type u_1\ninst✝² : Group G\nH : Subgroup G\nA : Type u_2\ninst✝¹ : CommGroup A\nϕ : { x // x ∈ H } →* A\nT : ↑(leftTransversals ↑H)\ninst✝ : FiniteIndex H\ng : G\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex H\nkey : ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H), ↑{ val := g₀⁻¹ * g ^ k * g₀, property := hk } = g ^ k\nx✝ : Quotient (orbitRel { x // x ∈ zpowers g } (G ⧸ H))\n⊢ (Subtype.val ∘ fun q =>\n {\n val :=\n (Quotient.out' (Quotient.out' q))⁻¹ * g ^ Function.minimalPeriod (fun x => g • x) (Quotient.out' q) *\n Quotient.out' (Quotient.out' q),\n property :=\n (_ :\n (Quotient.out' (Quotient.out' q))⁻¹ *\n ↑{ val := g, property := (_ : g ∈ zpowers g) } ^\n Function.minimalPeriod ((fun x x_1 => x • x_1) ↑{ val := g, property := (_ : g ∈ zpowers g) })\n (Quotient.out' q) *\n Quotient.out' (Quotient.out' q) ∈\n H) })\n x✝ =\n (Subtype.val ∘ fun i =>\n { val := g, property := (_ : g ∈ zpowers g) } ^ Function.minimalPeriod (fun x => g • x) (Quotient.out' i))\n x✝",
"state_before": "case e_a.e_f\nG : Type u_1\ninst✝² : Group G\nH : Subgroup G\nA : Type u_2\ninst✝¹ : CommGroup A\nϕ : { x // x ∈ H } →* A\nT : ↑(leftTransversals ↑H)\ninst✝ : FiniteIndex H\ng : G\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex H\nkey : ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H), ↑{ val := g₀⁻¹ * g ^ k * g₀, property := hk } = g ^ k\n⊢ (Subtype.val ∘ fun q =>\n {\n val :=\n (Quotient.out' (Quotient.out' q))⁻¹ * g ^ Function.minimalPeriod (fun x => g • x) (Quotient.out' q) *\n Quotient.out' (Quotient.out' q),\n property :=\n (_ :\n (Quotient.out' (Quotient.out' q))⁻¹ *\n ↑{ val := g, property := (_ : g ∈ zpowers g) } ^\n Function.minimalPeriod ((fun x x_1 => x • x_1) ↑{ val := g, property := (_ : g ∈ zpowers g) })\n (Quotient.out' q) *\n Quotient.out' (Quotient.out' q) ∈\n H) }) =\n Subtype.val ∘ fun i =>\n { val := g, property := (_ : g ∈ zpowers g) } ^ Function.minimalPeriod (fun x => g • x) (Quotient.out' i)",
"tactic": "funext"
},
{
"state_after": "no goals",
"state_before": "case e_a.e_f.h\nG : Type u_1\ninst✝² : Group G\nH : Subgroup G\nA : Type u_2\ninst✝¹ : CommGroup A\nϕ : { x // x ∈ H } →* A\nT : ↑(leftTransversals ↑H)\ninst✝ : FiniteIndex H\ng : G\nthis : Fintype (G ⧸ H) := fintypeQuotientOfFiniteIndex H\nkey : ∀ (k : ℕ) (g₀ : G) (hk : g₀⁻¹ * g ^ k * g₀ ∈ H), ↑{ val := g₀⁻¹ * g ^ k * g₀, property := hk } = g ^ k\nx✝ : Quotient (orbitRel { x // x ∈ zpowers g } (G ⧸ H))\n⊢ (Subtype.val ∘ fun q =>\n {\n val :=\n (Quotient.out' (Quotient.out' q))⁻¹ * g ^ Function.minimalPeriod (fun x => g • x) (Quotient.out' q) *\n Quotient.out' (Quotient.out' q),\n property :=\n (_ :\n (Quotient.out' (Quotient.out' q))⁻¹ *\n ↑{ val := g, property := (_ : g ∈ zpowers g) } ^\n Function.minimalPeriod ((fun x x_1 => x • x_1) ↑{ val := g, property := (_ : g ∈ zpowers g) })\n (Quotient.out' q) *\n Quotient.out' (Quotient.out' q) ∈\n H) })\n x✝ =\n (Subtype.val ∘ fun i =>\n { val := g, property := (_ : g ∈ zpowers g) } ^ Function.minimalPeriod (fun x => g • x) (Quotient.out' i))\n x✝",
"tactic": "apply key"
}
] |
[
189,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
173,
1
] |
Mathlib/GroupTheory/MonoidLocalization.lean
|
Submonoid.LocalizationMap.mk'_eq_iff_eq_mul
|
[
{
"state_after": "no goals",
"state_before": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type ?u.800895\ninst✝ : CommMonoid P\nf : LocalizationMap S N\nx : M\ny : { x // x ∈ S }\nz : N\n⊢ mk' f x y = z ↔ ↑(toMap f) x = z * ↑(toMap f) ↑y",
"tactic": "rw [eq_comm, eq_mk'_iff_mul_eq, eq_comm]"
}
] |
[
760,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
759,
1
] |
Mathlib/Logic/Equiv/LocalEquiv.lean
|
LocalEquiv.rightInvOn
|
[] |
[
244,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
244,
11
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.lift_is_succ
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.87265\nβ : Type ?u.87268\nγ : Type ?u.87271\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nx✝ : ∃ a, lift o = succ a\na : Ordinal\nh : lift o = succ a\nb : Ordinal\ne : lift b = a\n⊢ lift o = lift (succ b)",
"tactic": "rw [h, ← e, lift_succ]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.87265\nβ : Type ?u.87268\nγ : Type ?u.87271\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nx✝ : ∃ a, o = succ a\na : Ordinal\nh : o = succ a\n⊢ lift o = succ (lift a)",
"tactic": "simp only [h, lift_succ]"
}
] |
[
227,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
223,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.inter_union_compl
|
[] |
[
1877,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1876,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean
|
Class.mem_sUnion
|
[
{
"state_after": "case mp\nx y : Class\n⊢ y ∈ ⋃₀ x → ∃ z, z ∈ x ∧ y ∈ z\n\ncase mpr\nx y : Class\n⊢ (∃ z, z ∈ x ∧ y ∈ z) → y ∈ ⋃₀ x",
"state_before": "x y : Class\n⊢ y ∈ ⋃₀ x ↔ ∃ z, z ∈ x ∧ y ∈ z",
"tactic": "constructor"
},
{
"state_after": "case mp.intro.intro.intro.intro\nx : Class\nw : ZFSet\nz : Set ZFSet\nhzx : z ∈ classToCong x\nhwz : w ∈ z\n⊢ ∃ z, z ∈ x ∧ ↑w ∈ z",
"state_before": "case mp\nx y : Class\n⊢ y ∈ ⋃₀ x → ∃ z, z ∈ x ∧ y ∈ z",
"tactic": "rintro ⟨w, rfl, z, hzx, hwz⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro\nx : Class\nw : ZFSet\nz : Set ZFSet\nhzx : z ∈ classToCong x\nhwz : w ∈ z\n⊢ ∃ z, z ∈ x ∧ ↑w ∈ z",
"tactic": "exact ⟨z, hzx, coe_mem.2 hwz⟩"
},
{
"state_after": "case mpr.intro.intro.intro.intro\nx w : Class\nhwx : w ∈ x\nz : ZFSet\nhwz : w z\n⊢ ↑z ∈ ⋃₀ x",
"state_before": "case mpr\nx y : Class\n⊢ (∃ z, z ∈ x ∧ y ∈ z) → y ∈ ⋃₀ x",
"tactic": "rintro ⟨w, hwx, z, rfl, hwz⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.intro.intro\nx w : Class\nhwx : w ∈ x\nz : ZFSet\nhwz : w z\n⊢ ↑z ∈ ⋃₀ x",
"tactic": "exact ⟨z, rfl, w, hwx, hwz⟩"
}
] |
[
1697,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1692,
1
] |
Mathlib/GroupTheory/FreeGroup.lean
|
FreeGroup.Red.append_append
|
[] |
[
288,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
287,
1
] |
Mathlib/RingTheory/HahnSeries.lean
|
HahnSeries.zero_coeff
|
[] |
[
122,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
121,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.diag_submatrix
|
[] |
[
2468,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2467,
1
] |
Mathlib/Data/Set/Basic.lean
|
SetCoe.exists'
|
[] |
[
198,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Mathlib/Analysis/NormedSpace/FiniteDimension.lean
|
AffineSubspace.closed_of_finiteDimensional
|
[] |
[
390,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
387,
1
] |
Mathlib/CategoryTheory/Monoidal/Opposite.lean
|
CategoryTheory.MonoidalOpposite.op_injective
|
[] |
[
51,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
50,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec.nat_add
|
[] |
[
669,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
668,
1
] |
Mathlib/Data/Set/Intervals/Group.lean
|
Set.sub_mem_Icc_iff_left
|
[] |
[
104,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/Topology/UniformSpace/UniformConvergence.lean
|
TendstoUniformlyOn.congr
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nF' : ι → α → β\nhf : TendstoUniformlyOnFilter F f p (𝓟 s)\nhff' : ∀ᶠ (n : ι) in p, EqOn (F n) (F' n) s\n⊢ TendstoUniformlyOnFilter F' f p (𝓟 s)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nF' : ι → α → β\nhf : TendstoUniformlyOn F f p s\nhff' : ∀ᶠ (n : ι) in p, EqOn (F n) (F' n) s\n⊢ TendstoUniformlyOn F' f p s",
"tactic": "rw [tendstoUniformlyOn_iff_tendstoUniformlyOnFilter] at hf⊢"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nF' : ι → α → β\nhf : TendstoUniformlyOnFilter F f p (𝓟 s)\nhff' : ∀ᶠ (n : ι) in p, EqOn (F n) (F' n) s\n⊢ ∀ᶠ (n : ι × α) in p ×ˢ 𝓟 s, F n.fst n.snd = F' n.fst n.snd",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nF' : ι → α → β\nhf : TendstoUniformlyOnFilter F f p (𝓟 s)\nhff' : ∀ᶠ (n : ι) in p, EqOn (F n) (F' n) s\n⊢ TendstoUniformlyOnFilter F' f p (𝓟 s)",
"tactic": "refine' hf.congr _"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nF' : ι → α → β\nhf : TendstoUniformlyOnFilter F f p (𝓟 s)\nhff' : {x | EqOn (F x) (F' x) s} ∈ p\n⊢ {x | F x.fst x.snd = F' x.fst x.snd} ∈ p ×ˢ 𝓟 s",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nF' : ι → α → β\nhf : TendstoUniformlyOnFilter F f p (𝓟 s)\nhff' : ∀ᶠ (n : ι) in p, EqOn (F n) (F' n) s\n⊢ ∀ᶠ (n : ι × α) in p ×ˢ 𝓟 s, F n.fst n.snd = F' n.fst n.snd",
"tactic": "rw [eventually_iff] at hff'⊢"
},
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nF' : ι → α → β\nhf : TendstoUniformlyOnFilter F f p (𝓟 s)\nhff' : {x | ∀ ⦃x_1 : α⦄, x_1 ∈ s → F x x_1 = F' x x_1} ∈ p\n⊢ {x | F x.fst x.snd = F' x.fst x.snd} ∈ p ×ˢ 𝓟 s",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nF' : ι → α → β\nhf : TendstoUniformlyOnFilter F f p (𝓟 s)\nhff' : {x | EqOn (F x) (F' x) s} ∈ p\n⊢ {x | F x.fst x.snd = F' x.fst x.snd} ∈ p ×ˢ 𝓟 s",
"tactic": "simp only [Set.EqOn] at hff'"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Type x\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns s' : Set α\nx : α\np : Filter ι\np' : Filter α\ng : ι → α\nF' : ι → α → β\nhf : TendstoUniformlyOnFilter F f p (𝓟 s)\nhff' : {x | ∀ ⦃x_1 : α⦄, x_1 ∈ s → F x x_1 = F' x x_1} ∈ p\n⊢ {x | F x.fst x.snd = F' x.fst x.snd} ∈ p ×ˢ 𝓟 s",
"tactic": "simp only [mem_prod_principal, hff', mem_setOf_eq]"
}
] |
[
221,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
215,
1
] |
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