file_path
stringlengths 11
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Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean
|
MeasureTheory.Measure.add_haar_ball_mul_of_pos
|
[
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.1974470\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ : Set E\nx : E\nr : ℝ\nhr : 0 < r\ns : ℝ\nthis : ball 0 (r * s) = r • ball 0 s\n⊢ ↑↑μ (ball x (r * s)) = ENNReal.ofReal (r ^ finrank ℝ E) * ↑↑μ (ball 0 s)",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.1974470\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ : Set E\nx : E\nr : ℝ\nhr : 0 < r\ns : ℝ\n⊢ ↑↑μ (ball x (r * s)) = ENNReal.ofReal (r ^ finrank ℝ E) * ↑↑μ (ball 0 s)",
"tactic": "have : ball (0 : E) (r * s) = r • ball (0 : E) s := by\n simp only [_root_.smul_ball hr.ne' (0 : E) s, Real.norm_eq_abs, abs_of_nonneg hr.le, smul_zero]"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.1974470\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ : Set E\nx : E\nr : ℝ\nhr : 0 < r\ns : ℝ\nthis : ball 0 (r * s) = r • ball 0 s\n⊢ ↑↑μ (ball x (r * s)) = ENNReal.ofReal (r ^ finrank ℝ E) * ↑↑μ (ball 0 s)",
"tactic": "simp only [this, add_haar_smul, abs_of_nonneg hr.le, add_haar_ball_center, abs_pow]"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.1974470\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ : Set E\nx : E\nr : ℝ\nhr : 0 < r\ns : ℝ\n⊢ ball 0 (r * s) = r • ball 0 s",
"tactic": "simp only [_root_.smul_ball hr.ne' (0 : E) s, Real.norm_eq_abs, abs_of_nonneg hr.le, smul_zero]"
}
] |
[
420,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
416,
1
] |
Mathlib/Data/Int/Cast/Lemmas.lean
|
Int.cast_pos
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.18775\nι : Type ?u.18778\nα : Type u_1\nβ : Type ?u.18784\ninst✝¹ : OrderedRing α\ninst✝ : Nontrivial α\nn : ℤ\n⊢ 0 < ↑n ↔ 0 < n",
"tactic": "rw [← cast_zero, cast_lt]"
}
] |
[
151,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
Measurable.infNndist
|
[] |
[
1551,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1549,
1
] |
Mathlib/Data/IsROrC/Basic.lean
|
IsROrC.one_re
|
[
{
"state_after": "no goals",
"state_before": "K : Type u_1\nE : Type ?u.1627867\ninst✝ : IsROrC K\n⊢ ↑re 1 = 1",
"tactic": "rw [← ofReal_one, ofReal_re]"
}
] |
[
160,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
1
] |
Mathlib/MeasureTheory/Function/UniformIntegrable.lean
|
MeasureTheory.unifIntegrable_zero_meas
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nm : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup β\ninst✝ : MeasurableSpace α\np : ℝ≥0∞\nf : ι → α → β\nε : ℝ\nx✝² : 0 < ε\ni : ι\ns : Set α\nx✝¹ : MeasurableSet s\nx✝ : ↑↑0 s ≤ ENNReal.ofReal 1\n⊢ snorm (Set.indicator s (f i)) p 0 ≤ ENNReal.ofReal ε",
"tactic": "simp"
}
] |
[
153,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
1
] |
Mathlib/RingTheory/PrincipalIdealDomain.lean
|
PrincipalIdealRing.associates_irreducible_iff_prime
|
[] |
[
271,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
270,
1
] |
Mathlib/Topology/Sets/Closeds.lean
|
TopologicalSpace.Closeds.coe_nonempty
|
[] |
[
139,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.coe_replicate
|
[] |
[
888,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
888,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean
|
ContinuousLinearMap.op_nnnorm_le_of_lipschitz
|
[] |
[
435,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
433,
1
] |
Mathlib/Topology/Order/Basic.lean
|
induced_orderTopology'
|
[] |
[
1046,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1041,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/ContinuousLinearMap.lean
|
ContinuousLinearMap.measurable_comp
|
[] |
[
39,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
37,
1
] |
Mathlib/Data/Real/ConjugateExponents.lean
|
Real.IsConjugateExponent.one_div_pos
|
[] |
[
65,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
65,
1
] |
Mathlib/MeasureTheory/Function/LpSpace.lean
|
MeasureTheory.Memℒp.toLp_congr
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nE : Type u_1\nF : Type ?u.347109\nG : Type ?u.347112\nm m0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nf g : α → E\nhf : Memℒp f p\nhg : Memℒp g p\nhfg : f =ᵐ[μ] g\n⊢ toLp f hf = toLp g hg",
"tactic": "simp [toLp, hfg]"
}
] |
[
126,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/Topology/ContinuousOn.lean
|
nhdsWithin_restrict
|
[] |
[
198,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
196,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.mul_div_le
|
[
{
"state_after": "no goals",
"state_before": "m n : Nat\n⊢ n * (m / n) ≤ m",
"tactic": "match n, Nat.eq_zero_or_pos n with\n| _, Or.inl rfl => rw [Nat.zero_mul]; exact m.zero_le\n| n, Or.inr h => rw [Nat.mul_comm, ← Nat.le_div_iff_mul_le h]; exact Nat.le_refl _"
},
{
"state_after": "m n : Nat\n⊢ 0 ≤ m",
"state_before": "m n : Nat\n⊢ 0 * (m / 0) ≤ m",
"tactic": "rw [Nat.zero_mul]"
},
{
"state_after": "no goals",
"state_before": "m n : Nat\n⊢ 0 ≤ m",
"tactic": "exact m.zero_le"
},
{
"state_after": "m n✝ n : Nat\nh : n > 0\n⊢ m / n ≤ m / n",
"state_before": "m n✝ n : Nat\nh : n > 0\n⊢ n * (m / n) ≤ m",
"tactic": "rw [Nat.mul_comm, ← Nat.le_div_iff_mul_le h]"
},
{
"state_after": "no goals",
"state_before": "m n✝ n : Nat\nh : n > 0\n⊢ m / n ≤ m / n",
"tactic": "exact Nat.le_refl _"
}
] |
[
647,
85
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
644,
1
] |
Mathlib/Data/Finset/NAry.lean
|
Finset.image_image₂_antidistrib_right
|
[
{
"state_after": "α : Type u_3\nα' : Type u_5\nβ : Type u_4\nβ' : Type ?u.95037\nγ : Type u_2\nγ' : Type ?u.95043\nδ : Type u_1\nδ' : Type ?u.95049\nε : Type ?u.95052\nε' : Type ?u.95055\nζ : Type ?u.95058\nζ' : Type ?u.95061\nν : Type ?u.95064\ninst✝⁷ : DecidableEq α'\ninst✝⁶ : DecidableEq β'\ninst✝⁵ : DecidableEq γ\ninst✝⁴ : DecidableEq γ'\ninst✝³ : DecidableEq δ\ninst✝² : DecidableEq δ'\ninst✝¹ : DecidableEq ε\ninst✝ : DecidableEq ε'\nf f'✝ : α → β → γ\ng✝ g'✝ : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\ng : γ → δ\nf' : β → α' → δ\ng' : α → α'\nh_antidistrib : ∀ (a : α) (b : β), g (f a b) = f' b (g' a)\n⊢ g '' image2 f ↑s ↑t = image2 f' (↑t) (g' '' ↑s)",
"state_before": "α : Type u_3\nα' : Type u_5\nβ : Type u_4\nβ' : Type ?u.95037\nγ : Type u_2\nγ' : Type ?u.95043\nδ : Type u_1\nδ' : Type ?u.95049\nε : Type ?u.95052\nε' : Type ?u.95055\nζ : Type ?u.95058\nζ' : Type ?u.95061\nν : Type ?u.95064\ninst✝⁷ : DecidableEq α'\ninst✝⁶ : DecidableEq β'\ninst✝⁵ : DecidableEq γ\ninst✝⁴ : DecidableEq γ'\ninst✝³ : DecidableEq δ\ninst✝² : DecidableEq δ'\ninst✝¹ : DecidableEq ε\ninst✝ : DecidableEq ε'\nf f'✝ : α → β → γ\ng✝ g'✝ : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\ng : γ → δ\nf' : β → α' → δ\ng' : α → α'\nh_antidistrib : ∀ (a : α) (b : β), g (f a b) = f' b (g' a)\n⊢ ↑(image g (image₂ f s t)) = ↑(image₂ f' t (image g' s))",
"tactic": "push_cast"
},
{
"state_after": "no goals",
"state_before": "α : Type u_3\nα' : Type u_5\nβ : Type u_4\nβ' : Type ?u.95037\nγ : Type u_2\nγ' : Type ?u.95043\nδ : Type u_1\nδ' : Type ?u.95049\nε : Type ?u.95052\nε' : Type ?u.95055\nζ : Type ?u.95058\nζ' : Type ?u.95061\nν : Type ?u.95064\ninst✝⁷ : DecidableEq α'\ninst✝⁶ : DecidableEq β'\ninst✝⁵ : DecidableEq γ\ninst✝⁴ : DecidableEq γ'\ninst✝³ : DecidableEq δ\ninst✝² : DecidableEq δ'\ninst✝¹ : DecidableEq ε\ninst✝ : DecidableEq ε'\nf f'✝ : α → β → γ\ng✝ g'✝ : α → β → γ → δ\ns s' : Finset α\nt t' : Finset β\nu u' : Finset γ\na a' : α\nb b' : β\nc : γ\ng : γ → δ\nf' : β → α' → δ\ng' : α → α'\nh_antidistrib : ∀ (a : α) (b : β), g (f a b) = f' b (g' a)\n⊢ g '' image2 f ↑s ↑t = image2 f' (↑t) (g' '' ↑s)",
"tactic": "exact image_image2_antidistrib_right h_antidistrib"
}
] |
[
491,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
486,
1
] |
Mathlib/RingTheory/Valuation/Basic.lean
|
Valuation.map_sum_le
|
[
{
"state_after": "K : Type ?u.625793\nF : Type ?u.625796\nR : Type u_3\ninst✝⁴ : DivisionRing K\nΓ₀ : Type u_2\nΓ'₀ : Type ?u.625808\nΓ''₀ : Type ?u.625811\ninst✝³ : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommMonoidWithZero Γ₀\ninst✝ : LinearOrderedCommMonoidWithZero Γ'₀\nv : Valuation R Γ₀\nx y z : R\nι : Type u_1\ns✝ : Finset ι\nf : ι → R\ng : Γ₀\nhf✝ : ∀ (i : ι), i ∈ s✝ → ↑v (f i) ≤ g\na : ι\ns : Finset ι\nhas : ¬a ∈ s\nih : (∀ (i : ι), i ∈ s → ↑v (f i) ≤ g) → ↑v (∑ i in s, f i) ≤ g\nhf : ∀ (i : ι), i ∈ insert a s → ↑v (f i) ≤ g\n⊢ ↑v (∑ i in insert a s, f i) ≤ g",
"state_before": "K : Type ?u.625793\nF : Type ?u.625796\nR : Type u_3\ninst✝⁴ : DivisionRing K\nΓ₀ : Type u_2\nΓ'₀ : Type ?u.625808\nΓ''₀ : Type ?u.625811\ninst✝³ : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommMonoidWithZero Γ₀\ninst✝ : LinearOrderedCommMonoidWithZero Γ'₀\nv : Valuation R Γ₀\nx y z : R\nι : Type u_1\ns : Finset ι\nf : ι → R\ng : Γ₀\nhf : ∀ (i : ι), i ∈ s → ↑v (f i) ≤ g\n⊢ ↑v (∑ i in s, f i) ≤ g",
"tactic": "refine'\n Finset.induction_on s (fun _ => v.map_zero ▸ zero_le')\n (fun a s has ih hf => _) hf"
},
{
"state_after": "K : Type ?u.625793\nF : Type ?u.625796\nR : Type u_3\ninst✝⁴ : DivisionRing K\nΓ₀ : Type u_2\nΓ'₀ : Type ?u.625808\nΓ''₀ : Type ?u.625811\ninst✝³ : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommMonoidWithZero Γ₀\ninst✝ : LinearOrderedCommMonoidWithZero Γ'₀\nv : Valuation R Γ₀\nx y z : R\nι : Type u_1\ns✝ : Finset ι\nf : ι → R\ng : Γ₀\nhf✝ : ∀ (i : ι), i ∈ s✝ → ↑v (f i) ≤ g\na : ι\ns : Finset ι\nhas : ¬a ∈ s\nih : (∀ (i : ι), i ∈ s → ↑v (f i) ≤ g) → ↑v (∑ i in s, f i) ≤ g\nhf : ↑v (f a) ≤ g ∧ ∀ (x : ι), x ∈ s → ↑v (f x) ≤ g\n⊢ ↑v (∑ i in insert a s, f i) ≤ g",
"state_before": "K : Type ?u.625793\nF : Type ?u.625796\nR : Type u_3\ninst✝⁴ : DivisionRing K\nΓ₀ : Type u_2\nΓ'₀ : Type ?u.625808\nΓ''₀ : Type ?u.625811\ninst✝³ : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommMonoidWithZero Γ₀\ninst✝ : LinearOrderedCommMonoidWithZero Γ'₀\nv : Valuation R Γ₀\nx y z : R\nι : Type u_1\ns✝ : Finset ι\nf : ι → R\ng : Γ₀\nhf✝ : ∀ (i : ι), i ∈ s✝ → ↑v (f i) ≤ g\na : ι\ns : Finset ι\nhas : ¬a ∈ s\nih : (∀ (i : ι), i ∈ s → ↑v (f i) ≤ g) → ↑v (∑ i in s, f i) ≤ g\nhf : ∀ (i : ι), i ∈ insert a s → ↑v (f i) ≤ g\n⊢ ↑v (∑ i in insert a s, f i) ≤ g",
"tactic": "rw [Finset.forall_mem_insert] at hf"
},
{
"state_after": "K : Type ?u.625793\nF : Type ?u.625796\nR : Type u_3\ninst✝⁴ : DivisionRing K\nΓ₀ : Type u_2\nΓ'₀ : Type ?u.625808\nΓ''₀ : Type ?u.625811\ninst✝³ : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommMonoidWithZero Γ₀\ninst✝ : LinearOrderedCommMonoidWithZero Γ'₀\nv : Valuation R Γ₀\nx y z : R\nι : Type u_1\ns✝ : Finset ι\nf : ι → R\ng : Γ₀\nhf✝ : ∀ (i : ι), i ∈ s✝ → ↑v (f i) ≤ g\na : ι\ns : Finset ι\nhas : ¬a ∈ s\nih : (∀ (i : ι), i ∈ s → ↑v (f i) ≤ g) → ↑v (∑ i in s, f i) ≤ g\nhf : ↑v (f a) ≤ g ∧ ∀ (x : ι), x ∈ s → ↑v (f x) ≤ g\n⊢ ↑v (f a + ∑ x in s, f x) ≤ g",
"state_before": "K : Type ?u.625793\nF : Type ?u.625796\nR : Type u_3\ninst✝⁴ : DivisionRing K\nΓ₀ : Type u_2\nΓ'₀ : Type ?u.625808\nΓ''₀ : Type ?u.625811\ninst✝³ : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommMonoidWithZero Γ₀\ninst✝ : LinearOrderedCommMonoidWithZero Γ'₀\nv : Valuation R Γ₀\nx y z : R\nι : Type u_1\ns✝ : Finset ι\nf : ι → R\ng : Γ₀\nhf✝ : ∀ (i : ι), i ∈ s✝ → ↑v (f i) ≤ g\na : ι\ns : Finset ι\nhas : ¬a ∈ s\nih : (∀ (i : ι), i ∈ s → ↑v (f i) ≤ g) → ↑v (∑ i in s, f i) ≤ g\nhf : ↑v (f a) ≤ g ∧ ∀ (x : ι), x ∈ s → ↑v (f x) ≤ g\n⊢ ↑v (∑ i in insert a s, f i) ≤ g",
"tactic": "rw [Finset.sum_insert has]"
},
{
"state_after": "no goals",
"state_before": "K : Type ?u.625793\nF : Type ?u.625796\nR : Type u_3\ninst✝⁴ : DivisionRing K\nΓ₀ : Type u_2\nΓ'₀ : Type ?u.625808\nΓ''₀ : Type ?u.625811\ninst✝³ : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommMonoidWithZero Γ₀\ninst✝ : LinearOrderedCommMonoidWithZero Γ'₀\nv : Valuation R Γ₀\nx y z : R\nι : Type u_1\ns✝ : Finset ι\nf : ι → R\ng : Γ₀\nhf✝ : ∀ (i : ι), i ∈ s✝ → ↑v (f i) ≤ g\na : ι\ns : Finset ι\nhas : ¬a ∈ s\nih : (∀ (i : ι), i ∈ s → ↑v (f i) ≤ g) → ↑v (∑ i in s, f i) ≤ g\nhf : ↑v (f a) ≤ g ∧ ∀ (x : ι), x ∈ s → ↑v (f x) ≤ g\n⊢ ↑v (f a + ∑ x in s, f x) ≤ g",
"tactic": "exact v.map_add_le hf.1 (ih hf.2)"
}
] |
[
196,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
190,
1
] |
Mathlib/Analysis/Complex/ReImTopology.lean
|
Metric.Bounded.reProdIm
|
[] |
[
217,
60
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Mathlib/FieldTheory/Subfield.lean
|
RingHom.eqOn_field_closure
|
[] |
[
882,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
880,
1
] |
Mathlib/Tactic/Sat/FromLRAT.lean
|
Sat.Valuation.mk_implies
|
[
{
"state_after": "no goals",
"state_before": "as ps : List Prop\np : Prop\nas₁ : List Prop\n⊢ as = List.reverseAux as₁ ps → implies (mk as) p ps (List.length as₁) → p",
"tactic": "induction ps generalizing as₁ with\n| nil => exact fun _ ↦ id\n| cons a as ih =>\n refine fun e H ↦ @ih (a::as₁) e (H ?_)\n subst e; clear ih H\n suffices ∀ n n', n' = List.length as₁ + n →\n ∀ bs, mk (as₁.reverseAux bs) n' ↔ mk bs n from this 0 _ rfl (a::as)\n induction as₁ with simp\n | cons b as₁ ih => exact fun n bs ↦ ih (n+1) _ (Nat.succ_add ..) _"
},
{
"state_after": "no goals",
"state_before": "case nil\nas : List Prop\np : Prop\nas₁ : List Prop\n⊢ as = List.reverseAux as₁ [] → implies (mk as) p [] (List.length as₁) → p",
"tactic": "exact fun _ ↦ id"
},
{
"state_after": "case cons\nas✝ : List Prop\np a : Prop\nas : List Prop\nih : ∀ (as₁ : List Prop), as✝ = List.reverseAux as₁ as → implies (mk as✝) p as (List.length as₁) → p\nas₁ : List Prop\ne : as✝ = List.reverseAux as₁ (a :: as)\nH : implies (mk as✝) p (a :: as) (List.length as₁)\n⊢ mk as✝ (List.length as₁) ↔ a",
"state_before": "case cons\nas✝ : List Prop\np a : Prop\nas : List Prop\nih : ∀ (as₁ : List Prop), as✝ = List.reverseAux as₁ as → implies (mk as✝) p as (List.length as₁) → p\nas₁ : List Prop\n⊢ as✝ = List.reverseAux as₁ (a :: as) → implies (mk as✝) p (a :: as) (List.length as₁) → p",
"tactic": "refine fun e H ↦ @ih (a::as₁) e (H ?_)"
},
{
"state_after": "case cons\np a : Prop\nas as₁ : List Prop\nih :\n ∀ (as₁_1 : List Prop),\n List.reverseAux as₁ (a :: as) = List.reverseAux as₁_1 as →\n implies (mk (List.reverseAux as₁ (a :: as))) p as (List.length as₁_1) → p\nH : implies (mk (List.reverseAux as₁ (a :: as))) p (a :: as) (List.length as₁)\n⊢ mk (List.reverseAux as₁ (a :: as)) (List.length as₁) ↔ a",
"state_before": "case cons\nas✝ : List Prop\np a : Prop\nas : List Prop\nih : ∀ (as₁ : List Prop), as✝ = List.reverseAux as₁ as → implies (mk as✝) p as (List.length as₁) → p\nas₁ : List Prop\ne : as✝ = List.reverseAux as₁ (a :: as)\nH : implies (mk as✝) p (a :: as) (List.length as₁)\n⊢ mk as✝ (List.length as₁) ↔ a",
"tactic": "subst e"
},
{
"state_after": "case cons\np a : Prop\nas as₁ : List Prop\n⊢ mk (List.reverseAux as₁ (a :: as)) (List.length as₁) ↔ a",
"state_before": "case cons\np a : Prop\nas as₁ : List Prop\nih :\n ∀ (as₁_1 : List Prop),\n List.reverseAux as₁ (a :: as) = List.reverseAux as₁_1 as →\n implies (mk (List.reverseAux as₁ (a :: as))) p as (List.length as₁_1) → p\nH : implies (mk (List.reverseAux as₁ (a :: as))) p (a :: as) (List.length as₁)\n⊢ mk (List.reverseAux as₁ (a :: as)) (List.length as₁) ↔ a",
"tactic": "clear ih H"
},
{
"state_after": "case cons\np a : Prop\nas as₁ : List Prop\n⊢ ∀ (n n' : ℕ), n' = List.length as₁ + n → ∀ (bs : List Prop), mk (List.reverseAux as₁ bs) n' ↔ mk bs n",
"state_before": "case cons\np a : Prop\nas as₁ : List Prop\n⊢ mk (List.reverseAux as₁ (a :: as)) (List.length as₁) ↔ a",
"tactic": "suffices ∀ n n', n' = List.length as₁ + n →\n ∀ bs, mk (as₁.reverseAux bs) n' ↔ mk bs n from this 0 _ rfl (a::as)"
},
{
"state_after": "no goals",
"state_before": "case cons\np a : Prop\nas as₁ : List Prop\n⊢ ∀ (n n' : ℕ), n' = List.length as₁ + n → ∀ (bs : List Prop), mk (List.reverseAux as₁ bs) n' ↔ mk bs n",
"tactic": "induction as₁ with simp\n| cons b as₁ ih => exact fun n bs ↦ ih (n+1) _ (Nat.succ_add ..) _"
},
{
"state_after": "no goals",
"state_before": "case cons.cons\np a : Prop\nas : List Prop\nb : Prop\nas₁ : List Prop\nih : ∀ (n n' : ℕ), n' = List.length as₁ + n → ∀ (bs : List Prop), mk (List.reverseAux as₁ bs) n' ↔ mk bs n\n⊢ ∀ (n : ℕ) (bs : List Prop), mk (List.reverseAux as₁ (b :: bs)) (Nat.succ (List.length as₁) + n) ↔ mk bs n",
"tactic": "exact fun n bs ↦ ih (n+1) _ (Nat.succ_add ..) _"
}
] |
[
164,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/LinearAlgebra/Matrix/Hermitian.lean
|
Matrix.isHermitian_transpose_mul_self
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_3\nβ : Type ?u.28730\nm : Type u_1\nn : Type u_2\nA✝ : Matrix n n α\ninst✝⁴ : NonUnitalSemiring α\ninst✝³ : StarRing α\ninst✝² : NonUnitalSemiring β\ninst✝¹ : StarRing β\ninst✝ : Fintype m\nA : Matrix m n α\n⊢ IsHermitian (Aᴴ ⬝ A)",
"tactic": "rw [IsHermitian, conjTranspose_mul, conjTranspose_conjTranspose]"
}
] |
[
206,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.singleton_subset_singleton
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.35449\nγ : Type ?u.35452\ns : Finset α\na b : α\n⊢ {a} ⊆ {b} ↔ a = b",
"tactic": "simp"
}
] |
[
784,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
784,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.aleph0_le
|
[
{
"state_after": "case intro\nα β : Type u\nn : ℕ\nh : ∀ (n_1 : ℕ), ↑n_1 ≤ ↑n\nhn : ↑n < ℵ₀\n⊢ False",
"state_before": "α β : Type u\nc : Cardinal\nh : ∀ (n : ℕ), ↑n ≤ c\nhn : c < ℵ₀\n⊢ False",
"tactic": "rcases lt_aleph0.1 hn with ⟨n, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nα β : Type u\nn : ℕ\nh : ∀ (n_1 : ℕ), ↑n_1 ≤ ↑n\nhn : ↑n < ℵ₀\n⊢ False",
"tactic": "exact (Nat.lt_succ_self _).not_le (natCast_le.1 (h (n + 1)))"
}
] |
[
1437,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1433,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/InverseDeriv.lean
|
Real.contDiffOn_arccos
|
[] |
[
188,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
187,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
|
Real.rpow_div_two_eq_sqrt
|
[
{
"state_after": "z x✝ y x r : ℝ\nhx : 0 ≤ x\n⊢ x ^ (r / 2) = x ^ (1 / 2 * r)",
"state_before": "z x✝ y x r : ℝ\nhx : 0 ≤ x\n⊢ x ^ (r / 2) = sqrt x ^ r",
"tactic": "rw [sqrt_eq_rpow, ← rpow_mul hx]"
},
{
"state_after": "case e_a\nz x✝ y x r : ℝ\nhx : 0 ≤ x\n⊢ r / 2 = 1 / 2 * r",
"state_before": "z x✝ y x r : ℝ\nhx : 0 ≤ x\n⊢ x ^ (r / 2) = x ^ (1 / 2 * r)",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case e_a\nz x✝ y x r : ℝ\nhx : 0 ≤ x\n⊢ r / 2 = 1 / 2 * r",
"tactic": "ring"
}
] |
[
663,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
660,
1
] |
Mathlib/Data/Set/Sigma.lean
|
Set.fst_image_sigma
|
[] |
[
248,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
245,
1
] |
Mathlib/Computability/PartrecCode.lean
|
Nat.Partrec.Code.const_prim
|
[
{
"state_after": "n : ℕ\n⊢ ((fun b => comp succ b)^[n]) zero = Code.const n",
"state_before": "n : ℕ\n⊢ ((fun b => comp succ (n, b).snd)^[id n]) zero = Code.const n",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ ((fun b => comp succ b)^[n]) zero = Code.const n",
"tactic": "induction n <;>\nsimp [*, Code.const, Function.iterate_succ', -Function.iterate_succ]"
}
] |
[
677,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
673,
1
] |
Mathlib/CategoryTheory/Limits/Constructions/Equalizers.lean
|
CategoryTheory.Limits.hasCoequalizers_of_hasPushouts_and_binary_coproducts
|
[] |
[
201,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
195,
1
] |
Mathlib/Algebra/Associated.lean
|
Associates.Prime.le_or_le
|
[] |
[
1010,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1008,
1
] |
Std/Data/List/Lemmas.lean
|
List.ext
|
[
{
"state_after": "α : Type u_1\na : α\nl₁ : List α\na' : α\nl₂ : List α\nh : ∀ (n : Nat), get? (a :: l₁) n = get? (a' :: l₂) n\nh0 : some a = some a'\n⊢ a :: l₁ = a' :: l₂",
"state_before": "α : Type u_1\na : α\nl₁ : List α\na' : α\nl₂ : List α\nh : ∀ (n : Nat), get? (a :: l₁) n = get? (a' :: l₂) n\n⊢ a :: l₁ = a' :: l₂",
"tactic": "have h0 : some a = some a' := h 0"
},
{
"state_after": "α : Type u_1\na : α\nl₁ : List α\na' : α\nl₂ : List α\nh : ∀ (n : Nat), get? (a :: l₁) n = get? (a' :: l₂) n\naa : a = a'\n⊢ a :: l₁ = a' :: l₂",
"state_before": "α : Type u_1\na : α\nl₁ : List α\na' : α\nl₂ : List α\nh : ∀ (n : Nat), get? (a :: l₁) n = get? (a' :: l₂) n\nh0 : some a = some a'\n⊢ a :: l₁ = a' :: l₂",
"tactic": "injection h0 with aa"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\na : α\nl₁ : List α\na' : α\nl₂ : List α\nh : ∀ (n : Nat), get? (a :: l₁) n = get? (a' :: l₂) n\naa : a = a'\n⊢ a :: l₁ = a' :: l₂",
"tactic": "simp only [aa, ext fun n => h (n+1)]"
}
] |
[
655,
63
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
649,
8
] |
Mathlib/Data/Finset/Slice.lean
|
Finset.sized_slice
|
[] |
[
152,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
152,
1
] |
Mathlib/Order/WellFoundedSet.lean
|
Set.Finite.isPwo
|
[] |
[
444,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
444,
11
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.degree_C_mul_X_pow
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nn : ℕ\nha : a ≠ 0\n⊢ degree (↑C a * X ^ n) = ↑n",
"tactic": "rw [C_mul_X_pow_eq_monomial, degree_monomial n ha]"
}
] |
[
295,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
294,
1
] |
Mathlib/Algebra/Order/Group/Abs.lean
|
abs_le'
|
[] |
[
53,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Add.lean
|
differentiableWithinAt_sub_const_iff
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type ?u.497137\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nG' : Type ?u.497232\ninst✝¹ : NormedAddCommGroup G'\ninst✝ : NormedSpace 𝕜 G'\nf f₀ f₁ g : E → F\nf' f₀' f₁' g' e : E →L[𝕜] F\nx : E\ns t : Set E\nL L₁ L₂ : Filter E\nc : F\n⊢ DifferentiableWithinAt 𝕜 (fun y => f y - c) s x ↔ DifferentiableWithinAt 𝕜 f s x",
"tactic": "simp only [sub_eq_add_neg, differentiableWithinAt_add_const_iff]"
}
] |
[
555,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
553,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.SubgroupNormal.mem_comm
|
[
{
"state_after": "G : Type u_1\nG' : Type ?u.714616\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.714625\ninst✝¹ : AddGroup A\nN : Type ?u.714631\ninst✝ : Group N\nH K : Subgroup G\nhK : H ≤ K\nhN : Normal (subgroupOf H K)\na b : G\nhb : b ∈ K\nh : a * b ∈ H\nthis : b * (a * b) * b⁻¹ ∈ H\n⊢ b * a ∈ H",
"state_before": "G : Type u_1\nG' : Type ?u.714616\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.714625\ninst✝¹ : AddGroup A\nN : Type ?u.714631\ninst✝ : Group N\nH K : Subgroup G\nhK : H ≤ K\nhN : Normal (subgroupOf H K)\na b : G\nhb : b ∈ K\nh : a * b ∈ H\n⊢ b * a ∈ H",
"tactic": "have := (normal_subgroupOf_iff hK).mp hN (a * b) b h hb"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\nG' : Type ?u.714616\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.714625\ninst✝¹ : AddGroup A\nN : Type ?u.714631\ninst✝ : Group N\nH K : Subgroup G\nhK : H ≤ K\nhN : Normal (subgroupOf H K)\na b : G\nhb : b ∈ K\nh : a * b ∈ H\nthis : b * (a * b) * b⁻¹ ∈ H\n⊢ b * a ∈ H",
"tactic": "rwa [mul_assoc, mul_assoc, mul_right_inv, mul_one] at this"
}
] |
[
3609,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3606,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.le_lift_iff
|
[] |
[
1128,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1123,
1
] |
Mathlib/Analysis/NormedSpace/lpSpace.lean
|
lp.infty_coeFn_mul
|
[] |
[
800,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
799,
1
] |
Mathlib/CategoryTheory/Monoidal/Preadditive.lean
|
CategoryTheory.leftDistributor_inv
|
[
{
"state_after": "case w\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalPreadditive C\ninst✝¹ : HasFiniteBiproducts C\nJ : Type\ninst✝ : Fintype J\nX : C\nf : J → C\nj✝ : J\n⊢ biproduct.ι (fun j => X ⊗ f j) j✝ ≫ (leftDistributor X f).inv =\n biproduct.ι (fun j => X ⊗ f j) j✝ ≫ ∑ j : J, biproduct.π (fun j => X ⊗ f j) j ≫ (𝟙 X ⊗ biproduct.ι f j)",
"state_before": "C : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalPreadditive C\ninst✝¹ : HasFiniteBiproducts C\nJ : Type\ninst✝ : Fintype J\nX : C\nf : J → C\n⊢ (leftDistributor X f).inv = ∑ j : J, biproduct.π (fun j => X ⊗ f j) j ≫ (𝟙 X ⊗ biproduct.ι f j)",
"tactic": "ext"
},
{
"state_after": "case w\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalPreadditive C\ninst✝¹ : HasFiniteBiproducts C\nJ : Type\ninst✝ : Fintype J\nX : C\nf : J → C\nj✝ : J\n⊢ (biproduct.ι (fun j => X ⊗ f j) j✝ ≫ biproduct.desc fun j => 𝟙 X ⊗ biproduct.ι f j) =\n biproduct.ι (fun j => X ⊗ f j) j✝ ≫ ∑ j : J, biproduct.π (fun j => X ⊗ f j) j ≫ (𝟙 X ⊗ biproduct.ι f j)",
"state_before": "case w\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalPreadditive C\ninst✝¹ : HasFiniteBiproducts C\nJ : Type\ninst✝ : Fintype J\nX : C\nf : J → C\nj✝ : J\n⊢ biproduct.ι (fun j => X ⊗ f j) j✝ ≫ (leftDistributor X f).inv =\n biproduct.ι (fun j => X ⊗ f j) j✝ ≫ ∑ j : J, biproduct.π (fun j => X ⊗ f j) j ≫ (𝟙 X ⊗ biproduct.ι f j)",
"tactic": "dsimp [leftDistributor, Functor.mapBiproduct, Functor.mapBicone]"
},
{
"state_after": "no goals",
"state_before": "case w\nC : Type u_2\ninst✝⁵ : Category C\ninst✝⁴ : Preadditive C\ninst✝³ : MonoidalCategory C\ninst✝² : MonoidalPreadditive C\ninst✝¹ : HasFiniteBiproducts C\nJ : Type\ninst✝ : Fintype J\nX : C\nf : J → C\nj✝ : J\n⊢ (biproduct.ι (fun j => X ⊗ f j) j✝ ≫ biproduct.desc fun j => 𝟙 X ⊗ biproduct.ι f j) =\n biproduct.ι (fun j => X ⊗ f j) j✝ ≫ ∑ j : J, biproduct.π (fun j => X ⊗ f j) j ≫ (𝟙 X ⊗ biproduct.ι f j)",
"tactic": "simp only [Preadditive.comp_sum, biproduct.ι_π_assoc, dite_comp, zero_comp,\n Finset.sum_dite_eq, Finset.mem_univ, ite_true, eqToHom_refl, Category.id_comp,\n biproduct.ι_desc]"
}
] |
[
162,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/Computability/Primrec.lean
|
Primrec.sum_inr
|
[] |
[
1001,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1000,
1
] |
Mathlib/RingTheory/Localization/Basic.lean
|
IsLocalization.ringEquivOfRingEquiv_mk'
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁷ : CommSemiring R\nM : Submonoid R\nS : Type u_4\ninst✝⁶ : CommSemiring S\ninst✝⁵ : Algebra R S\nP : Type u_2\ninst✝⁴ : CommSemiring P\ninst✝³ : IsLocalization M S\ng : R →+* P\nhg : ∀ (y : { x // x ∈ M }), IsUnit (↑g ↑y)\nT : Submonoid P\nQ : Type u_3\ninst✝² : CommSemiring Q\nhy : M ≤ Submonoid.comap g T\ninst✝¹ : Algebra P Q\ninst✝ : IsLocalization T Q\nj : R ≃+* P\nH : Submonoid.map (RingEquiv.toMonoidHom j) M = T\nx : R\ny : { x // x ∈ M }\n⊢ ↑(ringEquivOfRingEquiv S Q j H) (mk' S x y) = mk' Q (↑j x) { val := ↑j ↑y, property := (_ : ↑j ↑y ∈ T) }",
"tactic": "simp [map_mk']"
}
] |
[
710,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
707,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
inv_pow_le_inv_pow_of_le
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.126922\nα : Type u_1\nβ : Type ?u.126928\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm✝ n✝ : ℤ\na1 : 1 ≤ a\nm n : ℕ\nmn : m ≤ n\n⊢ (a ^ n)⁻¹ ≤ (a ^ m)⁻¹",
"tactic": "convert one_div_pow_le_one_div_pow_of_le a1 mn using 1 <;> simp"
}
] |
[
640,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
639,
1
] |
Mathlib/GroupTheory/PGroup.lean
|
IsPGroup.card_center_eq_prime_pow
|
[
{
"state_after": "p : ℕ\nG : Type u_1\ninst✝³ : Group G\ninst✝² : Fintype G\ninst✝¹ : Fact (Nat.Prime p)\nn : ℕ\nhGpn : card G = p ^ n\nhn : 0 < n\ninst✝ : Fintype { x // x ∈ center G }\nhcG : IsPGroup p { x // x ∈ center G }\n⊢ ∃ k, k > 0 ∧ card { x // x ∈ center G } = p ^ k",
"state_before": "p : ℕ\nG : Type u_1\ninst✝³ : Group G\ninst✝² : Fintype G\ninst✝¹ : Fact (Nat.Prime p)\nn : ℕ\nhGpn : card G = p ^ n\nhn : 0 < n\ninst✝ : Fintype { x // x ∈ center G }\n⊢ ∃ k, k > 0 ∧ card { x // x ∈ center G } = p ^ k",
"tactic": "have hcG := to_subgroup (of_card hGpn) (center G)"
},
{
"state_after": "p : ℕ\nG : Type u_1\ninst✝³ : Group G\ninst✝² : Fintype G\ninst✝¹ : Fact (Nat.Prime p)\nn : ℕ\nhGpn : card G = p ^ n\nhn : 0 < n\ninst✝ : Fintype { x // x ∈ center G }\nhcG : IsPGroup p { x // x ∈ center G }\nx✝ : ∃ n, card { x // x ∈ center G } = p ^ n\n⊢ ∃ k, k > 0 ∧ card { x // x ∈ center G } = p ^ k",
"state_before": "p : ℕ\nG : Type u_1\ninst✝³ : Group G\ninst✝² : Fintype G\ninst✝¹ : Fact (Nat.Prime p)\nn : ℕ\nhGpn : card G = p ^ n\nhn : 0 < n\ninst✝ : Fintype { x // x ∈ center G }\nhcG : IsPGroup p { x // x ∈ center G }\n⊢ ∃ k, k > 0 ∧ card { x // x ∈ center G } = p ^ k",
"tactic": "rcases iff_card.1 hcG with _"
},
{
"state_after": "p : ℕ\nG : Type u_1\ninst✝³ : Group G\ninst✝² : Fintype G\ninst✝¹ : Fact (Nat.Prime p)\nn : ℕ\nhGpn : card G = p ^ n\nhn : 0 < n\ninst✝ : Fintype { x // x ∈ center G }\nhcG : IsPGroup p { x // x ∈ center G }\nx✝ : ∃ n, card { x // x ∈ center G } = p ^ n\nthis : Nontrivial G\n⊢ ∃ k, k > 0 ∧ card { x // x ∈ center G } = p ^ k",
"state_before": "p : ℕ\nG : Type u_1\ninst✝³ : Group G\ninst✝² : Fintype G\ninst✝¹ : Fact (Nat.Prime p)\nn : ℕ\nhGpn : card G = p ^ n\nhn : 0 < n\ninst✝ : Fintype { x // x ∈ center G }\nhcG : IsPGroup p { x // x ∈ center G }\nx✝ : ∃ n, card { x // x ∈ center G } = p ^ n\n⊢ ∃ k, k > 0 ∧ card { x // x ∈ center G } = p ^ k",
"tactic": "haveI : Nontrivial G := (nontrivial_iff_card <| of_card hGpn).2 ⟨n, hn, hGpn⟩"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nG : Type u_1\ninst✝³ : Group G\ninst✝² : Fintype G\ninst✝¹ : Fact (Nat.Prime p)\nn : ℕ\nhGpn : card G = p ^ n\nhn : 0 < n\ninst✝ : Fintype { x // x ∈ center G }\nhcG : IsPGroup p { x // x ∈ center G }\nx✝ : ∃ n, card { x // x ∈ center G } = p ^ n\nthis : Nontrivial G\n⊢ ∃ k, k > 0 ∧ card { x // x ∈ center G } = p ^ k",
"tactic": "exact (nontrivial_iff_card hcG).mp (center_nontrivial (of_card hGpn))"
}
] |
[
379,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
374,
1
] |
Mathlib/NumberTheory/Zsqrtd/Basic.lean
|
Zsqrtd.eq_of_smul_eq_smul_left
|
[
{
"state_after": "d a : ℤ\nb c : ℤ√d\nha : a ≠ 0\nh : (↑a * b).re = (↑a * c).re ∧ (↑a * b).im = (↑a * c).im\n⊢ b.re = c.re ∧ b.im = c.im",
"state_before": "d a : ℤ\nb c : ℤ√d\nha : a ≠ 0\nh : ↑a * b = ↑a * c\n⊢ b = c",
"tactic": "rw [ext] at h⊢"
},
{
"state_after": "no goals",
"state_before": "d a : ℤ\nb c : ℤ√d\nha : a ≠ 0\nh : (↑a * b).re = (↑a * c).re ∧ (↑a * b).im = (↑a * c).im\n⊢ b.re = c.re ∧ b.im = c.im",
"tactic": "apply And.imp _ _ h <;> simpa only [smul_re, smul_im] using mul_left_cancel₀ ha"
}
] |
[
384,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
381,
11
] |
Mathlib/LinearAlgebra/Prod.lean
|
LinearMap.ker_snd
|
[] |
[
161,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
1
] |
Mathlib/Data/List/Basic.lean
|
List.takeD_left'
|
[
{
"state_after": "ι : Type ?u.228301\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁✝ l₂✝ l₁ l₂ : List α\nn : ℕ\na : α\nh : length l₁ = n\n⊢ takeD (length l₁) (l₁ ++ l₂) a = l₁",
"state_before": "ι : Type ?u.228301\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁✝ l₂✝ l₁ l₂ : List α\nn : ℕ\na : α\nh : length l₁ = n\n⊢ takeD n (l₁ ++ l₂) a = l₁",
"tactic": "rw [← h]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.228301\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁✝ l₂✝ l₁ l₂ : List α\nn : ℕ\na : α\nh : length l₁ = n\n⊢ takeD (length l₁) (l₁ ++ l₂) a = l₁",
"tactic": "apply takeD_left"
}
] |
[
2371,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2370,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.map_eq_map_iff
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\nG' : Type ?u.590543\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.590552\ninst✝¹ : AddGroup A\nN : Type u_2\ninst✝ : Group N\nf✝ f : G →* N\nH K : Subgroup G\n⊢ map f H = map f K ↔ H ⊔ ker f = K ⊔ ker f",
"tactic": "simp only [le_antisymm_iff, map_le_map_iff']"
}
] |
[
3090,
97
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3089,
1
] |
Mathlib/RingTheory/HahnSeries.lean
|
HahnSeries.sub_coeff'
|
[
{
"state_after": "case h\nΓ : Type u_1\nR : Type u_2\ninst✝¹ : PartialOrder Γ\ninst✝ : AddGroup R\nx y : HahnSeries Γ R\nx✝ : Γ\n⊢ coeff (x - y) x✝ = (x.coeff - y.coeff) x✝",
"state_before": "Γ : Type u_1\nR : Type u_2\ninst✝¹ : PartialOrder Γ\ninst✝ : AddGroup R\nx y : HahnSeries Γ R\n⊢ (x - y).coeff = x.coeff - y.coeff",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nΓ : Type u_1\nR : Type u_2\ninst✝¹ : PartialOrder Γ\ninst✝ : AddGroup R\nx y : HahnSeries Γ R\nx✝ : Γ\n⊢ coeff (x - y) x✝ = (x.coeff - y.coeff) x✝",
"tactic": "simp [sub_eq_add_neg]"
}
] |
[
470,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
468,
1
] |
Mathlib/LinearAlgebra/Quotient.lean
|
Submodule.comap_map_mkQ
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_2\nM : Type u_1\nr : R\nx y : M\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\np p' : Submodule R M\nR₂ : Type ?u.237240\nM₂ : Type ?u.237243\ninst✝² : Ring R₂\ninst✝¹ : AddCommGroup M₂\ninst✝ : Module R₂ M₂\nτ₁₂ : R →+* R₂\n⊢ comap (mkQ p) (map (mkQ p) p') = p ⊔ p'",
"tactic": "simp [comap_map_eq, sup_comm]"
}
] |
[
394,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
394,
1
] |
Mathlib/Algebra/Star/Pointwise.lean
|
Set.star_singleton
|
[
{
"state_after": "case h\nα : Type ?u.11501\ns t : Set α\na : α\nβ : Type u_1\ninst✝ : InvolutiveStar β\nx y : β\n⊢ y ∈ {x}⋆ ↔ y ∈ {x⋆}",
"state_before": "α : Type ?u.11501\ns t : Set α\na : α\nβ : Type u_1\ninst✝ : InvolutiveStar β\nx : β\n⊢ {x}⋆ = {x⋆}",
"tactic": "ext1 y"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type ?u.11501\ns t : Set α\na : α\nβ : Type u_1\ninst✝ : InvolutiveStar β\nx y : β\n⊢ y ∈ {x}⋆ ↔ y ∈ {x⋆}",
"tactic": "rw [mem_star, mem_singleton_iff, mem_singleton_iff, star_eq_iff_star_eq, eq_comm]"
}
] |
[
120,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/Topology/Algebra/Order/Floor.lean
|
tendsto_ceil_atBot
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.2623\nγ : Type ?u.2626\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nb : ℤ\n⊢ b - 1 ≤ b",
"state_before": "α : Type u_1\nβ : Type ?u.2623\nγ : Type ?u.2626\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nb : ℤ\n⊢ ⌈↑(b - 1)⌉ ≤ b",
"tactic": "rw [ceil_intCast]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.2623\nγ : Type ?u.2626\ninst✝¹ : LinearOrderedRing α\ninst✝ : FloorRing α\nb : ℤ\n⊢ b - 1 ≤ b",
"tactic": "exact (sub_one_lt _).le"
}
] |
[
51,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
49,
1
] |
Mathlib/Data/Sign.lean
|
sign_eq_neg_one_iff
|
[
{
"state_after": "α : Type u_1\ninst✝² : Zero α\ninst✝¹ : Preorder α\ninst✝ : DecidableRel fun x x_1 => x < x_1\na : α\nh : ↑sign a = -1\n⊢ a < 0",
"state_before": "α : Type u_1\ninst✝² : Zero α\ninst✝¹ : Preorder α\ninst✝ : DecidableRel fun x x_1 => x < x_1\na : α\n⊢ ↑sign a = -1 ↔ a < 0",
"tactic": "refine' ⟨fun h => _, fun h => sign_neg h⟩"
},
{
"state_after": "α : Type u_1\ninst✝² : Zero α\ninst✝¹ : Preorder α\ninst✝ : DecidableRel fun x x_1 => x < x_1\na : α\nh : (if 0 < a then 1 else if a < 0 then -1 else 0) = -1\n⊢ a < 0",
"state_before": "α : Type u_1\ninst✝² : Zero α\ninst✝¹ : Preorder α\ninst✝ : DecidableRel fun x x_1 => x < x_1\na : α\nh : ↑sign a = -1\n⊢ a < 0",
"tactic": "rw [sign_apply] at h"
},
{
"state_after": "case inr.inl\nα : Type u_1\ninst✝² : Zero α\ninst✝¹ : Preorder α\ninst✝ : DecidableRel fun x x_1 => x < x_1\na : α\nh✝¹ : ¬0 < a\nh✝ : a < 0\nh : -1 = -1\n⊢ a < 0",
"state_before": "α : Type u_1\ninst✝² : Zero α\ninst✝¹ : Preorder α\ninst✝ : DecidableRel fun x x_1 => x < x_1\na : α\nh : (if 0 < a then 1 else if a < 0 then -1 else 0) = -1\n⊢ a < 0",
"tactic": "split_ifs at h"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nα : Type u_1\ninst✝² : Zero α\ninst✝¹ : Preorder α\ninst✝ : DecidableRel fun x x_1 => x < x_1\na : α\nh✝¹ : ¬0 < a\nh✝ : a < 0\nh : -1 = -1\n⊢ a < 0",
"tactic": "assumption"
}
] |
[
342,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
338,
1
] |
Mathlib/Order/WithBot.lean
|
WithBot.coe_max
|
[] |
[
490,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
489,
1
] |
Mathlib/Analysis/SpecialFunctions/Integrals.lean
|
integral_mul_rpow_one_add_sq
|
[
{
"state_after": "a b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s\n⊢ (∫ (x : ℝ) in a..b, x * (1 + x ^ 2) ^ t) =\n (1 + b ^ 2) ^ (t + 1) / (2 * (t + 1)) - (1 + a ^ 2) ^ (t + 1) / (2 * (t + 1))",
"state_before": "a b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\n⊢ (∫ (x : ℝ) in a..b, x * (1 + x ^ 2) ^ t) =\n (1 + b ^ 2) ^ (t + 1) / (2 * (t + 1)) - (1 + a ^ 2) ^ (t + 1) / (2 * (t + 1))",
"tactic": "have : ∀ x s : ℝ, (((↑1 + x ^ 2) ^ s : ℝ) : ℂ) = (1 + (x : ℂ) ^ 2) ^ (s:ℂ) := by\n intro x s\n norm_cast\n rw [ofReal_cpow, ofReal_add, ofReal_pow, ofReal_one]\n exact add_nonneg zero_le_one (sq_nonneg x)"
},
{
"state_after": "a b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s\n⊢ ↑(∫ (x : ℝ) in a..b, x * (1 + x ^ 2) ^ t) =\n ↑((1 + b ^ 2) ^ (t + 1) / (2 * (t + 1)) - (1 + a ^ 2) ^ (t + 1) / (2 * (t + 1)))",
"state_before": "a b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s\n⊢ (∫ (x : ℝ) in a..b, x * (1 + x ^ 2) ^ t) =\n (1 + b ^ 2) ^ (t + 1) / (2 * (t + 1)) - (1 + a ^ 2) ^ (t + 1) / (2 * (t + 1))",
"tactic": "rw [← ofReal_inj]"
},
{
"state_after": "case h.e'_2\na b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s\n⊢ ↑(∫ (x : ℝ) in a..b, x * (1 + x ^ 2) ^ t) = ∫ (x : ℝ) in ?convert_1..?convert_2, ↑x * (1 + ↑x ^ 2) ^ ↑t\n\ncase h.e'_3\na b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s\n⊢ ↑((1 + b ^ 2) ^ (t + 1) / (2 * (t + 1)) - (1 + a ^ 2) ^ (t + 1) / (2 * (t + 1))) =\n (1 + ↑?convert_2 ^ 2) ^ (↑t + 1) / (↑2 * (↑t + 1)) - (1 + ↑?convert_1 ^ 2) ^ (↑t + 1) / (2 * (↑t + 1))\n\ncase convert_1\na b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s\n⊢ ℝ\n\ncase convert_2\na b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s\n⊢ ℝ\n\ncase convert_3\na b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s\n⊢ ↑t ≠ -1",
"state_before": "a b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s\n⊢ ↑(∫ (x : ℝ) in a..b, x * (1 + x ^ 2) ^ t) =\n ↑((1 + b ^ 2) ^ (t + 1) / (2 * (t + 1)) - (1 + a ^ 2) ^ (t + 1) / (2 * (t + 1)))",
"tactic": "convert integral_mul_cpow_one_add_sq (_ : (t : ℂ) ≠ -1)"
},
{
"state_after": "a b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nx s : ℝ\n⊢ ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s",
"state_before": "a b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\n⊢ ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s",
"tactic": "intro x s"
},
{
"state_after": "a b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nx s : ℝ\n⊢ ↑((1 + x ^ 2) ^ s) = ↑(1 + x ^ 2) ^ ↑s",
"state_before": "a b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nx s : ℝ\n⊢ ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s",
"tactic": "norm_cast"
},
{
"state_after": "case hx\na b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nx s : ℝ\n⊢ 0 ≤ 1 + x ^ 2",
"state_before": "a b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nx s : ℝ\n⊢ ↑((1 + x ^ 2) ^ s) = ↑(1 + x ^ 2) ^ ↑s",
"tactic": "rw [ofReal_cpow, ofReal_add, ofReal_pow, ofReal_one]"
},
{
"state_after": "no goals",
"state_before": "case hx\na b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nx s : ℝ\n⊢ 0 ≤ 1 + x ^ 2",
"tactic": "exact add_nonneg zero_le_one (sq_nonneg x)"
},
{
"state_after": "case h.e'_2\na b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s\n⊢ (∫ (x : ℝ) in a..b, ↑(x * (1 + x ^ 2) ^ t)) = ∫ (x : ℝ) in ?convert_1..?convert_2, ↑x * (1 + ↑x ^ 2) ^ ↑t\n\ncase convert_1\na b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s\n⊢ ℝ\n\ncase convert_2\na b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s\n⊢ ℝ",
"state_before": "case h.e'_2\na b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s\n⊢ ↑(∫ (x : ℝ) in a..b, x * (1 + x ^ 2) ^ t) = ∫ (x : ℝ) in ?convert_1..?convert_2, ↑x * (1 + ↑x ^ 2) ^ ↑t",
"tactic": "rw [← intervalIntegral.integral_ofReal]"
},
{
"state_after": "case h.e'_2.e_f.h\na b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s\nx : ℝ\n⊢ ↑(x * (1 + x ^ 2) ^ t) = ↑x * (1 + ↑x ^ 2) ^ ↑t",
"state_before": "case h.e'_2\na b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s\n⊢ (∫ (x : ℝ) in a..b, ↑(x * (1 + x ^ 2) ^ t)) = ∫ (x : ℝ) in ?convert_1..?convert_2, ↑x * (1 + ↑x ^ 2) ^ ↑t\n\ncase convert_1\na b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s\n⊢ ℝ\n\ncase convert_2\na b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s\n⊢ ℝ",
"tactic": "congr with x : 1"
},
{
"state_after": "case h.e'_2.e_f.h\na b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s\nx : ℝ\n⊢ ↑x * (↑1 + ↑x ^ 2) ^ ↑t = ↑x * (1 + ↑x ^ 2) ^ ↑t",
"state_before": "case h.e'_2.e_f.h\na b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s\nx : ℝ\n⊢ ↑(x * (1 + x ^ 2) ^ t) = ↑x * (1 + ↑x ^ 2) ^ ↑t",
"tactic": "rw [ofReal_mul, this x t]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.e_f.h\na b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s\nx : ℝ\n⊢ ↑x * (↑1 + ↑x ^ 2) ^ ↑t = ↑x * (1 + ↑x ^ 2) ^ ↑t",
"tactic": "norm_cast"
},
{
"state_after": "case h.e'_3\na b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s\n⊢ (↑1 + ↑b ^ 2) ^ ↑(t + 1) / ↑(2 * (t + 1)) - (↑1 + ↑a ^ 2) ^ ↑(t + 1) / ↑(2 * (t + 1)) =\n (1 + ↑b ^ 2) ^ (↑t + 1) / (↑2 * (↑t + 1)) - (1 + ↑a ^ 2) ^ (↑t + 1) / (2 * (↑t + 1))",
"state_before": "case h.e'_3\na b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s\n⊢ ↑((1 + b ^ 2) ^ (t + 1) / (2 * (t + 1)) - (1 + a ^ 2) ^ (t + 1) / (2 * (t + 1))) =\n (1 + ↑b ^ 2) ^ (↑t + 1) / (↑2 * (↑t + 1)) - (1 + ↑a ^ 2) ^ (↑t + 1) / (2 * (↑t + 1))",
"tactic": "simp_rw [ofReal_sub, ofReal_div, this a (t + 1), this b (t + 1)]"
},
{
"state_after": "case h.e'_3\na b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s\n⊢ (1 + ↑b ^ 2) ^ (↑t + 1) / (2 * (↑t + 1)) - (1 + ↑a ^ 2) ^ (↑t + 1) / (2 * (↑t + 1)) =\n (1 + ↑b ^ 2) ^ (↑t + 1) / (2 * (↑t + 1)) - (1 + ↑a ^ 2) ^ (↑t + 1) / (2 * (↑t + 1))",
"state_before": "case h.e'_3\na b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s\n⊢ (↑1 + ↑b ^ 2) ^ ↑(t + 1) / ↑(2 * (t + 1)) - (↑1 + ↑a ^ 2) ^ ↑(t + 1) / ↑(2 * (t + 1)) =\n (1 + ↑b ^ 2) ^ (↑t + 1) / (↑2 * (↑t + 1)) - (1 + ↑a ^ 2) ^ (↑t + 1) / (2 * (↑t + 1))",
"tactic": "push_cast"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\na b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s\n⊢ (1 + ↑b ^ 2) ^ (↑t + 1) / (2 * (↑t + 1)) - (1 + ↑a ^ 2) ^ (↑t + 1) / (2 * (↑t + 1)) =\n (1 + ↑b ^ 2) ^ (↑t + 1) / (2 * (↑t + 1)) - (1 + ↑a ^ 2) ^ (↑t + 1) / (2 * (↑t + 1))",
"tactic": "rfl"
},
{
"state_after": "case convert_3\na b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s\n⊢ ¬t = -1",
"state_before": "case convert_3\na b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s\n⊢ ↑t ≠ -1",
"tactic": "rw [← ofReal_one, ← ofReal_neg, Ne.def, ofReal_inj]"
},
{
"state_after": "no goals",
"state_before": "case convert_3\na b : ℝ\nn : ℕ\nt : ℝ\nht : t ≠ -1\nthis : ∀ (x s : ℝ), ↑((1 + x ^ 2) ^ s) = (↑1 + ↑x ^ 2) ^ ↑s\n⊢ ¬t = -1",
"tactic": "exact ht"
}
] |
[
609,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
592,
1
] |
Mathlib/Order/Filter/AtTopBot.lean
|
Filter.exists_le_of_tendsto_atTop
|
[
{
"state_after": "ι : Type ?u.65107\nι' : Type ?u.65110\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.65119\ninst✝¹ : SemilatticeSup α\ninst✝ : Preorder β\nu : α → β\nh : Tendsto u atTop atTop\na : α\nb : β\nthis : Nonempty α\n⊢ ∃ a', a' ≥ a ∧ b ≤ u a'",
"state_before": "ι : Type ?u.65107\nι' : Type ?u.65110\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.65119\ninst✝¹ : SemilatticeSup α\ninst✝ : Preorder β\nu : α → β\nh : Tendsto u atTop atTop\na : α\nb : β\n⊢ ∃ a', a' ≥ a ∧ b ≤ u a'",
"tactic": "have : Nonempty α := ⟨a⟩"
},
{
"state_after": "ι : Type ?u.65107\nι' : Type ?u.65110\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.65119\ninst✝¹ : SemilatticeSup α\ninst✝ : Preorder β\nu : α → β\nh : Tendsto u atTop atTop\na : α\nb : β\nthis✝ : Nonempty α\nthis : ∀ᶠ (x : α) in atTop, a ≤ x ∧ b ≤ u x\n⊢ ∃ a', a' ≥ a ∧ b ≤ u a'",
"state_before": "ι : Type ?u.65107\nι' : Type ?u.65110\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.65119\ninst✝¹ : SemilatticeSup α\ninst✝ : Preorder β\nu : α → β\nh : Tendsto u atTop atTop\na : α\nb : β\nthis : Nonempty α\n⊢ ∃ a', a' ≥ a ∧ b ≤ u a'",
"tactic": "have : ∀ᶠ x in atTop, a ≤ x ∧ b ≤ u x :=\n (eventually_ge_atTop a).and (h.eventually <| eventually_ge_atTop b)"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.65107\nι' : Type ?u.65110\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.65119\ninst✝¹ : SemilatticeSup α\ninst✝ : Preorder β\nu : α → β\nh : Tendsto u atTop atTop\na : α\nb : β\nthis✝ : Nonempty α\nthis : ∀ᶠ (x : α) in atTop, a ≤ x ∧ b ≤ u x\n⊢ ∃ a', a' ≥ a ∧ b ≤ u a'",
"tactic": "exact this.exists"
}
] |
[
517,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
512,
1
] |
Mathlib/Data/Analysis/Filter.lean
|
Filter.Realizer.bot_σ
|
[] |
[
203,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
1
] |
Mathlib/Order/Ideal.lean
|
Order.cofinal_meets_idealOfCofinals
|
[] |
[
601,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
600,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.isSt_iff_abs_sub_lt_delta
|
[
{
"state_after": "no goals",
"state_before": "x : ℝ*\nr : ℝ\n⊢ IsSt x r ↔ ∀ (δ : ℝ), 0 < δ → abs (x - ↑r) < ↑δ",
"tactic": "simp only [abs_sub_lt_iff, sub_lt_iff_lt_add, IsSt, and_comm, add_comm]"
}
] |
[
388,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
387,
1
] |
Mathlib/Algebra/Homology/Homotopy.lean
|
Homotopy.dNext_zero_chainComplex
|
[
{
"state_after": "ι : Type ?u.313500\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni : ι\nP Q : ChainComplex V ℕ\nf : (i j : ℕ) → X P i ⟶ X Q j\n⊢ d P 0 (ComplexShape.next (ComplexShape.down ℕ) 0) ≫ f (ComplexShape.next (ComplexShape.down ℕ) 0) 0 = 0",
"state_before": "ι : Type ?u.313500\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni : ι\nP Q : ChainComplex V ℕ\nf : (i j : ℕ) → X P i ⟶ X Q j\n⊢ ↑(dNext 0) f = 0",
"tactic": "dsimp [dNext]"
},
{
"state_after": "case a\nι : Type ?u.313500\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni : ι\nP Q : ChainComplex V ℕ\nf : (i j : ℕ) → X P i ⟶ X Q j\n⊢ ¬ComplexShape.Rel (ComplexShape.down ℕ) 0 (ComplexShape.next (ComplexShape.down ℕ) 0)",
"state_before": "ι : Type ?u.313500\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni : ι\nP Q : ChainComplex V ℕ\nf : (i j : ℕ) → X P i ⟶ X Q j\n⊢ d P 0 (ComplexShape.next (ComplexShape.down ℕ) 0) ≫ f (ComplexShape.next (ComplexShape.down ℕ) 0) 0 = 0",
"tactic": "rw [P.shape, zero_comp]"
},
{
"state_after": "case a\nι : Type ?u.313500\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni : ι\nP Q : ChainComplex V ℕ\nf : (i j : ℕ) → X P i ⟶ X Q j\n⊢ ¬ComplexShape.Rel (ComplexShape.down ℕ) 0 0",
"state_before": "case a\nι : Type ?u.313500\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni : ι\nP Q : ChainComplex V ℕ\nf : (i j : ℕ) → X P i ⟶ X Q j\n⊢ ¬ComplexShape.Rel (ComplexShape.down ℕ) 0 (ComplexShape.next (ComplexShape.down ℕ) 0)",
"tactic": "rw [ChainComplex.next_nat_zero]"
},
{
"state_after": "case a\nι : Type ?u.313500\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni : ι\nP Q : ChainComplex V ℕ\nf : (i j : ℕ) → X P i ⟶ X Q j\n⊢ ¬0 + 1 = 0",
"state_before": "case a\nι : Type ?u.313500\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni : ι\nP Q : ChainComplex V ℕ\nf : (i j : ℕ) → X P i ⟶ X Q j\n⊢ ¬ComplexShape.Rel (ComplexShape.down ℕ) 0 0",
"tactic": "dsimp"
},
{
"state_after": "no goals",
"state_before": "case a\nι : Type ?u.313500\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni : ι\nP Q : ChainComplex V ℕ\nf : (i j : ℕ) → X P i ⟶ X Q j\n⊢ ¬0 + 1 = 0",
"tactic": "decide"
}
] |
[
481,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
478,
1
] |
Mathlib/MeasureTheory/Function/ContinuousMapDense.lean
|
MeasureTheory.Memℒp.exists_boundedContinuous_snorm_sub_le
|
[
{
"state_after": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε : ℝ≥0∞\nhε : ε ≠ 0\nH : ∃ g, snorm (f - g) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)\n⊢ ∃ g, snorm (f - ↑g) p μ ≤ ε ∧ Memℒp (↑g) p\n\ncase H\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε : ℝ≥0∞\nhε : ε ≠ 0\n⊢ ∃ g, snorm (f - g) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)",
"state_before": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε : ℝ≥0∞\nhε : ε ≠ 0\n⊢ ∃ g, snorm (f - ↑g) p μ ≤ ε ∧ Memℒp (↑g) p",
"tactic": "suffices H :\n ∃ g : α → E, snorm (f - g) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p μ ∧ Metric.Bounded (range g)"
},
{
"state_after": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε : ℝ≥0∞\nhε : ε ≠ 0\n⊢ ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s →\n ↑↑μ s < ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)\n\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε : ℝ≥0∞\nhε : ε ≠ 0\n⊢ ∀ (f g : α → E),\n Continuous f ∧ Memℒp f p ∧ Metric.Bounded (range f) →\n Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g) →\n Continuous (f + g) ∧ Memℒp (f + g) p ∧ Metric.Bounded (range (f + g))\n\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε : ℝ≥0∞\nhε : ε ≠ 0\n⊢ ∀ (f : α → E), Continuous f ∧ Memℒp f p ∧ Metric.Bounded (range f) → AEStronglyMeasurable f μ",
"state_before": "case H\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε : ℝ≥0∞\nhε : ε ≠ 0\n⊢ ∃ g, snorm (f - g) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)",
"tactic": "apply hf.induction_dense hp _ _ _ _ hε"
},
{
"state_after": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε : ℝ≥0∞\nhε : ε ≠ 0\n⊢ ∀ (f g : α → E),\n Continuous f ∧ Memℒp f p ∧ Metric.Bounded (range f) →\n Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g) →\n Continuous (f + g) ∧ Memℒp (f + g) p ∧ Metric.Bounded (range (f + g))\n\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε : ℝ≥0∞\nhε : ε ≠ 0\n⊢ ∀ (f : α → E), Continuous f ∧ Memℒp f p ∧ Metric.Bounded (range f) → AEStronglyMeasurable f μ\n\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε : ℝ≥0∞\nhε : ε ≠ 0\n⊢ ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s →\n ↑↑μ s < ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)",
"state_before": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε : ℝ≥0∞\nhε : ε ≠ 0\n⊢ ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s →\n ↑↑μ s < ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)\n\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε : ℝ≥0∞\nhε : ε ≠ 0\n⊢ ∀ (f g : α → E),\n Continuous f ∧ Memℒp f p ∧ Metric.Bounded (range f) →\n Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g) →\n Continuous (f + g) ∧ Memℒp (f + g) p ∧ Metric.Bounded (range (f + g))\n\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε : ℝ≥0∞\nhε : ε ≠ 0\n⊢ ∀ (f : α → E), Continuous f ∧ Memℒp f p ∧ Metric.Bounded (range f) → AEStronglyMeasurable f μ",
"tactic": "rotate_left"
},
{
"state_after": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\n⊢ ∃ g, snorm (g - Set.indicator t fun x => c) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)",
"state_before": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε : ℝ≥0∞\nhε : ε ≠ 0\n⊢ ∀ (c : E) ⦃s : Set α⦄,\n MeasurableSet s →\n ↑↑μ s < ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, snorm (g - Set.indicator s fun x => c) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)",
"tactic": "intro c t ht htμ ε hε"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\n⊢ ∃ g, snorm (g - Set.indicator t fun x => c) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)",
"state_before": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\n⊢ ∃ g, snorm (g - Set.indicator t fun x => c) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)",
"tactic": "rcases exists_Lp_half E μ p hε with ⟨δ, δpos, hδ⟩"
},
{
"state_after": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\n⊢ ∃ η, 0 < η ∧ ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun _x => c) p μ ≤ δ\n\ncase intro.intro.intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\nη : ℝ≥0\nηpos : 0 < η\nhη : ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun _x => c) p μ ≤ δ\n⊢ ∃ g, snorm (g - Set.indicator t fun x => c) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)",
"state_before": "case intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\n⊢ ∃ g, snorm (g - Set.indicator t fun x => c) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)",
"tactic": "obtain ⟨η, ηpos, hη⟩ :\n ∃ η : ℝ≥0, 0 < η ∧ ∀ s : Set α, μ s ≤ η → snorm (s.indicator fun _x => c) p μ ≤ δ"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\nη : ℝ≥0\nηpos : 0 < η\nhη : ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun _x => c) p μ ≤ δ\n⊢ ∃ g, snorm (g - Set.indicator t fun x => c) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)",
"state_before": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\n⊢ ∃ η, 0 < η ∧ ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun _x => c) p μ ≤ δ\n\ncase intro.intro.intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\nη : ℝ≥0\nηpos : 0 < η\nhη : ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun _x => c) p μ ≤ δ\n⊢ ∃ g, snorm (g - Set.indicator t fun x => c) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)",
"tactic": "exact exists_snorm_indicator_le hp c δpos.ne'"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\nη : ℝ≥0\nηpos : 0 < η\nhη : ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun _x => c) p μ ≤ δ\nhη_pos' : 0 < ↑η\n⊢ ∃ g, snorm (g - Set.indicator t fun x => c) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\nη : ℝ≥0\nηpos : 0 < η\nhη : ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun _x => c) p μ ≤ δ\n⊢ ∃ g, snorm (g - Set.indicator t fun x => c) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)",
"tactic": "have hη_pos' : (0 : ℝ≥0∞) < η := ENNReal.coe_pos.2 ηpos"
},
{
"state_after": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\nη : ℝ≥0\nηpos : 0 < η\nhη : ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun _x => c) p μ ≤ δ\nhη_pos' : 0 < ↑η\n⊢ ∃ s, s ⊆ t ∧ IsClosed s ∧ ↑↑μ (t \\ s) < ↑η\n\ncase intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\nη : ℝ≥0\nηpos : 0 < η\nhη : ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun _x => c) p μ ≤ δ\nhη_pos' : 0 < ↑η\ns : Set α\nst : s ⊆ t\ns_closed : IsClosed s\nμs : ↑↑μ (t \\ s) < ↑η\n⊢ ∃ g, snorm (g - Set.indicator t fun x => c) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\nη : ℝ≥0\nηpos : 0 < η\nhη : ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun _x => c) p μ ≤ δ\nhη_pos' : 0 < ↑η\n⊢ ∃ g, snorm (g - Set.indicator t fun x => c) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)",
"tactic": "obtain ⟨s, st, s_closed, μs⟩ : ∃ s, s ⊆ t ∧ IsClosed s ∧ μ (t \\ s) < η"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\nη : ℝ≥0\nηpos : 0 < η\nhη : ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun _x => c) p μ ≤ δ\nhη_pos' : 0 < ↑η\ns : Set α\nst : s ⊆ t\ns_closed : IsClosed s\nμs : ↑↑μ (t \\ s) < ↑η\n⊢ ∃ g, snorm (g - Set.indicator t fun x => c) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)",
"state_before": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\nη : ℝ≥0\nηpos : 0 < η\nhη : ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun _x => c) p μ ≤ δ\nhη_pos' : 0 < ↑η\n⊢ ∃ s, s ⊆ t ∧ IsClosed s ∧ ↑↑μ (t \\ s) < ↑η\n\ncase intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\nη : ℝ≥0\nηpos : 0 < η\nhη : ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun _x => c) p μ ≤ δ\nhη_pos' : 0 < ↑η\ns : Set α\nst : s ⊆ t\ns_closed : IsClosed s\nμs : ↑↑μ (t \\ s) < ↑η\n⊢ ∃ g, snorm (g - Set.indicator t fun x => c) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)",
"tactic": "exact ht.exists_isClosed_diff_lt htμ.ne hη_pos'.ne'"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\nη : ℝ≥0\nηpos : 0 < η\nhη : ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun _x => c) p μ ≤ δ\nhη_pos' : 0 < ↑η\ns : Set α\nst : s ⊆ t\ns_closed : IsClosed s\nμs : ↑↑μ (t \\ s) < ↑η\nhsμ : ↑↑μ s < ⊤\n⊢ ∃ g, snorm (g - Set.indicator t fun x => c) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)",
"state_before": "case intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\nη : ℝ≥0\nηpos : 0 < η\nhη : ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun _x => c) p μ ≤ δ\nhη_pos' : 0 < ↑η\ns : Set α\nst : s ⊆ t\ns_closed : IsClosed s\nμs : ↑↑μ (t \\ s) < ↑η\n⊢ ∃ g, snorm (g - Set.indicator t fun x => c) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)",
"tactic": "have hsμ : μ s < ∞ := (measure_mono st).trans_lt htμ"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\nη : ℝ≥0\nηpos : 0 < η\nhη : ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun _x => c) p μ ≤ δ\nhη_pos' : 0 < ↑η\ns : Set α\nst : s ⊆ t\ns_closed : IsClosed s\nμs : ↑↑μ (t \\ s) < ↑η\nhsμ : ↑↑μ s < ⊤\nI1 : snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p μ ≤ δ\n⊢ ∃ g, snorm (g - Set.indicator t fun x => c) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)",
"state_before": "case intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\nη : ℝ≥0\nηpos : 0 < η\nhη : ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun _x => c) p μ ≤ δ\nhη_pos' : 0 < ↑η\ns : Set α\nst : s ⊆ t\ns_closed : IsClosed s\nμs : ↑↑μ (t \\ s) < ↑η\nhsμ : ↑↑μ s < ⊤\n⊢ ∃ g, snorm (g - Set.indicator t fun x => c) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)",
"tactic": "have I1 : snorm ((s.indicator fun _y => c) - t.indicator fun _y => c) p μ ≤ δ := by\n rw [← snorm_neg, neg_sub, ← indicator_diff st]\n exact hη _ μs.le"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf✝ : α → E\nhf : Memℒp f✝ p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\nη : ℝ≥0\nηpos : 0 < η\nhη : ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun _x => c) p μ ≤ δ\nhη_pos' : 0 < ↑η\ns : Set α\nst : s ⊆ t\ns_closed : IsClosed s\nμs : ↑↑μ (t \\ s) < ↑η\nhsμ : ↑↑μ s < ⊤\nI1 : snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p μ ≤ δ\nf : α → E\nf_cont : Continuous f\nI2 : snorm (fun x => f x - Set.indicator s (fun _y => c) x) p μ ≤ δ\nf_bound : ∀ (x : α), ‖f x‖ ≤ ‖c‖\nf_mem : Memℒp f p\n⊢ ∃ g, snorm (g - Set.indicator t fun x => c) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)",
"state_before": "case intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\nη : ℝ≥0\nηpos : 0 < η\nhη : ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun _x => c) p μ ≤ δ\nhη_pos' : 0 < ↑η\ns : Set α\nst : s ⊆ t\ns_closed : IsClosed s\nμs : ↑↑μ (t \\ s) < ↑η\nhsμ : ↑↑μ s < ⊤\nI1 : snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p μ ≤ δ\n⊢ ∃ g, snorm (g - Set.indicator t fun x => c) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)",
"tactic": "rcases exists_continuous_snorm_sub_le_of_closed hp s_closed isOpen_univ (subset_univ _) hsμ.ne c\n δpos.ne' with\n ⟨f, f_cont, I2, f_bound, -, f_mem⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf✝ : α → E\nhf : Memℒp f✝ p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\nη : ℝ≥0\nηpos : 0 < η\nhη : ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun _x => c) p μ ≤ δ\nhη_pos' : 0 < ↑η\ns : Set α\nst : s ⊆ t\ns_closed : IsClosed s\nμs : ↑↑μ (t \\ s) < ↑η\nhsμ : ↑↑μ s < ⊤\nI1 : snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p μ ≤ δ\nf : α → E\nf_cont : Continuous f\nI2 : snorm (fun x => f x - Set.indicator s (fun _y => c) x) p μ ≤ δ\nf_bound : ∀ (x : α), ‖f x‖ ≤ ‖c‖\nf_mem : Memℒp f p\nI3 : snorm (f - Set.indicator t fun _y => c) p μ ≤ ε\n⊢ ∃ g, snorm (g - Set.indicator t fun x => c) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf✝ : α → E\nhf : Memℒp f✝ p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\nη : ℝ≥0\nηpos : 0 < η\nhη : ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun _x => c) p μ ≤ δ\nhη_pos' : 0 < ↑η\ns : Set α\nst : s ⊆ t\ns_closed : IsClosed s\nμs : ↑↑μ (t \\ s) < ↑η\nhsμ : ↑↑μ s < ⊤\nI1 : snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p μ ≤ δ\nf : α → E\nf_cont : Continuous f\nI2 : snorm (fun x => f x - Set.indicator s (fun _y => c) x) p μ ≤ δ\nf_bound : ∀ (x : α), ‖f x‖ ≤ ‖c‖\nf_mem : Memℒp f p\n⊢ ∃ g, snorm (g - Set.indicator t fun x => c) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)",
"tactic": "have I3 : snorm (f - t.indicator fun _y => c) p μ ≤ ε := by\n convert\n (hδ _ _\n (f_mem.aestronglyMeasurable.sub\n (aestronglyMeasurable_const.indicator s_closed.measurableSet))\n ((aestronglyMeasurable_const.indicator s_closed.measurableSet).sub\n (aestronglyMeasurable_const.indicator ht))\n I2 I1).le using 2\n simp only [sub_add_sub_cancel]"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf✝ : α → E\nhf : Memℒp f✝ p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\nη : ℝ≥0\nηpos : 0 < η\nhη : ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun _x => c) p μ ≤ δ\nhη_pos' : 0 < ↑η\ns : Set α\nst : s ⊆ t\ns_closed : IsClosed s\nμs : ↑↑μ (t \\ s) < ↑η\nhsμ : ↑↑μ s < ⊤\nI1 : snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p μ ≤ δ\nf : α → E\nf_cont : Continuous f\nI2 : snorm (fun x => f x - Set.indicator s (fun _y => c) x) p μ ≤ δ\nf_bound : ∀ (x : α), ‖f x‖ ≤ ‖c‖\nf_mem : Memℒp f p\nI3 : snorm (f - Set.indicator t fun _y => c) p μ ≤ ε\n⊢ Metric.Bounded (range f)",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf✝ : α → E\nhf : Memℒp f✝ p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\nη : ℝ≥0\nηpos : 0 < η\nhη : ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun _x => c) p μ ≤ δ\nhη_pos' : 0 < ↑η\ns : Set α\nst : s ⊆ t\ns_closed : IsClosed s\nμs : ↑↑μ (t \\ s) < ↑η\nhsμ : ↑↑μ s < ⊤\nI1 : snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p μ ≤ δ\nf : α → E\nf_cont : Continuous f\nI2 : snorm (fun x => f x - Set.indicator s (fun _y => c) x) p μ ≤ δ\nf_bound : ∀ (x : α), ‖f x‖ ≤ ‖c‖\nf_mem : Memℒp f p\nI3 : snorm (f - Set.indicator t fun _y => c) p μ ≤ ε\n⊢ ∃ g, snorm (g - Set.indicator t fun x => c) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)",
"tactic": "refine' ⟨f, I3, f_cont, f_mem, _⟩"
},
{
"state_after": "no goals",
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"tactic": "exact (BoundedContinuousFunction.ofNormedAddCommGroup f f_cont _ f_bound).bounded_range"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε : ℝ≥0∞\nhε : ε ≠ 0\ng : α → E\nhg : snorm (f - g) p μ ≤ ε\ng_cont : Continuous g\ng_mem : Memℒp g p\ng_bd : Metric.Bounded (range g)\n⊢ ∃ g, snorm (f - ↑g) p μ ≤ ε ∧ Memℒp (↑g) p",
"state_before": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε : ℝ≥0∞\nhε : ε ≠ 0\nH : ∃ g, snorm (f - g) p μ ≤ ε ∧ Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g)\n⊢ ∃ g, snorm (f - ↑g) p μ ≤ ε ∧ Memℒp (↑g) p",
"tactic": "rcases H with ⟨g, hg, g_cont, g_mem, g_bd⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε : ℝ≥0∞\nhε : ε ≠ 0\ng : α → E\nhg : snorm (f - g) p μ ≤ ε\ng_cont : Continuous g\ng_mem : Memℒp g p\ng_bd : Metric.Bounded (range g)\n⊢ ∃ g, snorm (f - ↑g) p μ ≤ ε ∧ Memℒp (↑g) p",
"tactic": "exact ⟨⟨⟨g, g_cont⟩, Metric.bounded_range_iff.1 g_bd⟩, hg, g_mem⟩"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf✝ : α → E\nhf : Memℒp f✝ p\nε : ℝ≥0∞\nhε : ε ≠ 0\nf g : α → E\nf_cont : Continuous f\nf_mem : Memℒp f p\nf_bd : Metric.Bounded (range f)\ng_cont : Continuous g\ng_mem : Memℒp g p\ng_bd : Metric.Bounded (range g)\n⊢ Continuous (f + g) ∧ Memℒp (f + g) p ∧ Metric.Bounded (range (f + g))",
"state_before": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε : ℝ≥0∞\nhε : ε ≠ 0\n⊢ ∀ (f g : α → E),\n Continuous f ∧ Memℒp f p ∧ Metric.Bounded (range f) →\n Continuous g ∧ Memℒp g p ∧ Metric.Bounded (range g) →\n Continuous (f + g) ∧ Memℒp (f + g) p ∧ Metric.Bounded (range (f + g))",
"tactic": "rintro f g ⟨f_cont, f_mem, f_bd⟩ ⟨g_cont, g_mem, g_bd⟩"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf✝ : α → E\nhf : Memℒp f✝ p\nε : ℝ≥0∞\nhε : ε ≠ 0\nf g : α → E\nf_cont : Continuous f\nf_mem : Memℒp f p\nf_bd : Metric.Bounded (range f)\ng_cont : Continuous g\ng_mem : Memℒp g p\ng_bd : Metric.Bounded (range g)\n⊢ Metric.Bounded (range (f + g))",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf✝ : α → E\nhf : Memℒp f✝ p\nε : ℝ≥0∞\nhε : ε ≠ 0\nf g : α → E\nf_cont : Continuous f\nf_mem : Memℒp f p\nf_bd : Metric.Bounded (range f)\ng_cont : Continuous g\ng_mem : Memℒp g p\ng_bd : Metric.Bounded (range g)\n⊢ Continuous (f + g) ∧ Memℒp (f + g) p ∧ Metric.Bounded (range (f + g))",
"tactic": "refine' ⟨f_cont.add g_cont, f_mem.add g_mem, _⟩"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf✝ : α → E\nhf : Memℒp f✝ p\nε : ℝ≥0∞\nhε : ε ≠ 0\nf g : α → E\nf_cont : Continuous f\nf_mem : Memℒp f p\nf_bd : Metric.Bounded (range f)\ng_cont : Continuous g\ng_mem : Memℒp g p\ng_bd : Metric.Bounded (range g)\nf' : α →ᵇ E :=\n { toContinuousMap := ContinuousMap.mk f,\n map_bounded' :=\n (_ :\n ∃ C,\n ∀ (x y : α),\n dist (ContinuousMap.toFun (ContinuousMap.mk f) x) (ContinuousMap.toFun (ContinuousMap.mk f) y) ≤ C) }\n⊢ Metric.Bounded (range (f + g))",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf✝ : α → E\nhf : Memℒp f✝ p\nε : ℝ≥0∞\nhε : ε ≠ 0\nf g : α → E\nf_cont : Continuous f\nf_mem : Memℒp f p\nf_bd : Metric.Bounded (range f)\ng_cont : Continuous g\ng_mem : Memℒp g p\ng_bd : Metric.Bounded (range g)\n⊢ Metric.Bounded (range (f + g))",
"tactic": "let f' : α →ᵇ E := ⟨⟨f, f_cont⟩, Metric.bounded_range_iff.1 f_bd⟩"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf✝ : α → E\nhf : Memℒp f✝ p\nε : ℝ≥0∞\nhε : ε ≠ 0\nf g : α → E\nf_cont : Continuous f\nf_mem : Memℒp f p\nf_bd : Metric.Bounded (range f)\ng_cont : Continuous g\ng_mem : Memℒp g p\ng_bd : Metric.Bounded (range g)\nf' : α →ᵇ E :=\n { toContinuousMap := ContinuousMap.mk f,\n map_bounded' :=\n (_ :\n ∃ C,\n ∀ (x y : α),\n dist (ContinuousMap.toFun (ContinuousMap.mk f) x) (ContinuousMap.toFun (ContinuousMap.mk f) y) ≤ C) }\ng' : α →ᵇ E :=\n { toContinuousMap := ContinuousMap.mk g,\n map_bounded' :=\n (_ :\n ∃ C,\n ∀ (x y : α),\n dist (ContinuousMap.toFun (ContinuousMap.mk g) x) (ContinuousMap.toFun (ContinuousMap.mk g) y) ≤ C) }\n⊢ Metric.Bounded (range (f + g))",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf✝ : α → E\nhf : Memℒp f✝ p\nε : ℝ≥0∞\nhε : ε ≠ 0\nf g : α → E\nf_cont : Continuous f\nf_mem : Memℒp f p\nf_bd : Metric.Bounded (range f)\ng_cont : Continuous g\ng_mem : Memℒp g p\ng_bd : Metric.Bounded (range g)\nf' : α →ᵇ E :=\n { toContinuousMap := ContinuousMap.mk f,\n map_bounded' :=\n (_ :\n ∃ C,\n ∀ (x y : α),\n dist (ContinuousMap.toFun (ContinuousMap.mk f) x) (ContinuousMap.toFun (ContinuousMap.mk f) y) ≤ C) }\n⊢ Metric.Bounded (range (f + g))",
"tactic": "let g' : α →ᵇ E := ⟨⟨g, g_cont⟩, Metric.bounded_range_iff.1 g_bd⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf✝ : α → E\nhf : Memℒp f✝ p\nε : ℝ≥0∞\nhε : ε ≠ 0\nf g : α → E\nf_cont : Continuous f\nf_mem : Memℒp f p\nf_bd : Metric.Bounded (range f)\ng_cont : Continuous g\ng_mem : Memℒp g p\ng_bd : Metric.Bounded (range g)\nf' : α →ᵇ E :=\n { toContinuousMap := ContinuousMap.mk f,\n map_bounded' :=\n (_ :\n ∃ C,\n ∀ (x y : α),\n dist (ContinuousMap.toFun (ContinuousMap.mk f) x) (ContinuousMap.toFun (ContinuousMap.mk f) y) ≤ C) }\ng' : α →ᵇ E :=\n { toContinuousMap := ContinuousMap.mk g,\n map_bounded' :=\n (_ :\n ∃ C,\n ∀ (x y : α),\n dist (ContinuousMap.toFun (ContinuousMap.mk g) x) (ContinuousMap.toFun (ContinuousMap.mk g) y) ≤ C) }\n⊢ Metric.Bounded (range (f + g))",
"tactic": "exact (f' + g').bounded_range"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε : ℝ≥0∞\nhε : ε ≠ 0\n⊢ ∀ (f : α → E), Continuous f ∧ Memℒp f p ∧ Metric.Bounded (range f) → AEStronglyMeasurable f μ",
"tactic": "exact fun f ⟨_, h, _⟩ => h.aestronglyMeasurable"
},
{
"state_after": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\nη : ℝ≥0\nηpos : 0 < η\nhη : ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun _x => c) p μ ≤ δ\nhη_pos' : 0 < ↑η\ns : Set α\nst : s ⊆ t\ns_closed : IsClosed s\nμs : ↑↑μ (t \\ s) < ↑η\nhsμ : ↑↑μ s < ⊤\n⊢ snorm (Set.indicator (t \\ s) fun _y => c) p μ ≤ δ",
"state_before": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\nη : ℝ≥0\nηpos : 0 < η\nhη : ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun _x => c) p μ ≤ δ\nhη_pos' : 0 < ↑η\ns : Set α\nst : s ⊆ t\ns_closed : IsClosed s\nμs : ↑↑μ (t \\ s) < ↑η\nhsμ : ↑↑μ s < ⊤\n⊢ snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p μ ≤ δ",
"tactic": "rw [← snorm_neg, neg_sub, ← indicator_diff st]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf : α → E\nhf : Memℒp f p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\nη : ℝ≥0\nηpos : 0 < η\nhη : ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun _x => c) p μ ≤ δ\nhη_pos' : 0 < ↑η\ns : Set α\nst : s ⊆ t\ns_closed : IsClosed s\nμs : ↑↑μ (t \\ s) < ↑η\nhsμ : ↑↑μ s < ⊤\n⊢ snorm (Set.indicator (t \\ s) fun _y => c) p μ ≤ δ",
"tactic": "exact hη _ μs.le"
},
{
"state_after": "case h.e'_3.h.e'_5\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf✝ : α → E\nhf : Memℒp f✝ p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\nη : ℝ≥0\nηpos : 0 < η\nhη : ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun _x => c) p μ ≤ δ\nhη_pos' : 0 < ↑η\ns : Set α\nst : s ⊆ t\ns_closed : IsClosed s\nμs : ↑↑μ (t \\ s) < ↑η\nhsμ : ↑↑μ s < ⊤\nI1 : snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p μ ≤ δ\nf : α → E\nf_cont : Continuous f\nI2 : snorm (fun x => f x - Set.indicator s (fun _y => c) x) p μ ≤ δ\nf_bound : ∀ (x : α), ‖f x‖ ≤ ‖c‖\nf_mem : Memℒp f p\n⊢ (f - Set.indicator t fun _y => c) =\n (f - Set.indicator s fun x => c) + ((Set.indicator s fun x => c) - Set.indicator t fun x => c)",
"state_before": "α : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf✝ : α → E\nhf : Memℒp f✝ p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\nη : ℝ≥0\nηpos : 0 < η\nhη : ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun _x => c) p μ ≤ δ\nhη_pos' : 0 < ↑η\ns : Set α\nst : s ⊆ t\ns_closed : IsClosed s\nμs : ↑↑μ (t \\ s) < ↑η\nhsμ : ↑↑μ s < ⊤\nI1 : snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p μ ≤ δ\nf : α → E\nf_cont : Continuous f\nI2 : snorm (fun x => f x - Set.indicator s (fun _y => c) x) p μ ≤ δ\nf_bound : ∀ (x : α), ‖f x‖ ≤ ‖c‖\nf_mem : Memℒp f p\n⊢ snorm (f - Set.indicator t fun _y => c) p μ ≤ ε",
"tactic": "convert\n (hδ _ _\n (f_mem.aestronglyMeasurable.sub\n (aestronglyMeasurable_const.indicator s_closed.measurableSet))\n ((aestronglyMeasurable_const.indicator s_closed.measurableSet).sub\n (aestronglyMeasurable_const.indicator ht))\n I2 I1).le using 2"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.e'_5\nα : Type u_1\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : NormalSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : Measure.WeaklyRegular μ\nhp : p ≠ ⊤\nf✝ : α → E\nhf : Memℒp f✝ p\nε✝ : ℝ≥0∞\nhε✝ : ε✝ ≠ 0\nc : E\nt : Set α\nht : MeasurableSet t\nhtμ : ↑↑μ t < ⊤\nε : ℝ≥0∞\nhε : ε ≠ 0\nδ : ℝ≥0∞\nδpos : 0 < δ\nhδ :\n ∀ (f g : α → E),\n AEStronglyMeasurable f μ → AEStronglyMeasurable g μ → snorm f p μ ≤ δ → snorm g p μ ≤ δ → snorm (f + g) p μ < ε\nη : ℝ≥0\nηpos : 0 < η\nhη : ∀ (s : Set α), ↑↑μ s ≤ ↑η → snorm (Set.indicator s fun _x => c) p μ ≤ δ\nhη_pos' : 0 < ↑η\ns : Set α\nst : s ⊆ t\ns_closed : IsClosed s\nμs : ↑↑μ (t \\ s) < ↑η\nhsμ : ↑↑μ s < ⊤\nI1 : snorm ((Set.indicator s fun _y => c) - Set.indicator t fun _y => c) p μ ≤ δ\nf : α → E\nf_cont : Continuous f\nI2 : snorm (fun x => f x - Set.indicator s (fun _y => c) x) p μ ≤ δ\nf_bound : ∀ (x : α), ‖f x‖ ≤ ‖c‖\nf_mem : Memℒp f p\n⊢ (f - Set.indicator t fun _y => c) =\n (f - Set.indicator s fun x => c) + ((Set.indicator s fun x => c) - Set.indicator t fun x => c)",
"tactic": "simp only [sub_add_sub_cancel]"
}
] |
[
287,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
240,
1
] |
Mathlib/Algebra/Hom/NonUnitalAlg.lean
|
NonUnitalAlgHom.zero_apply
|
[] |
[
267,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
266,
1
] |
Mathlib/Data/ULift.lean
|
PLift.forall
|
[] |
[
71,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
70,
1
] |
Mathlib/Topology/Semicontinuous.lean
|
lowerSemicontinuousWithinAt_univ_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : TopologicalSpace α\nβ : Type u_2\ninst✝ : Preorder β\nf g : α → β\nx : α\ns t : Set α\ny z : β\n⊢ LowerSemicontinuousWithinAt f univ x ↔ LowerSemicontinuousAt f x",
"tactic": "simp [LowerSemicontinuousWithinAt, LowerSemicontinuousAt, nhdsWithin_univ]"
}
] |
[
141,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
139,
1
] |
Mathlib/Order/Filter/Bases.lean
|
Filter.HasBasis.inf'
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.29550\nγ : Type ?u.29553\nι : Sort u_2\nι' : Sort u_3\nl l' : Filter α\np : ι → Prop\ns : ι → Set α\nt✝ : Set α\ni : ι\np' : ι' → Prop\ns' : ι' → Set α\ni' : ι'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\nt : Set α\n⊢ t ∈ l ⊓ l' ↔ ∃ i, (p i.fst ∧ p' i.snd) ∧ s i.fst ∩ s' i.snd ⊆ t",
"state_before": "α : Type u_1\nβ : Type ?u.29550\nγ : Type ?u.29553\nι : Sort u_2\nι' : Sort u_3\nl l' : Filter α\np : ι → Prop\ns : ι → Set α\nt : Set α\ni : ι\np' : ι' → Prop\ns' : ι' → Set α\ni' : ι'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\n⊢ ∀ (t : Set α), t ∈ l ⊓ l' ↔ ∃ i, (p i.fst ∧ p' i.snd) ∧ s i.fst ∩ s' i.snd ⊆ t",
"tactic": "intro t"
},
{
"state_after": "case mp\nα : Type u_1\nβ : Type ?u.29550\nγ : Type ?u.29553\nι : Sort u_2\nι' : Sort u_3\nl l' : Filter α\np : ι → Prop\ns : ι → Set α\nt✝ : Set α\ni : ι\np' : ι' → Prop\ns' : ι' → Set α\ni' : ι'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\nt : Set α\n⊢ t ∈ l ⊓ l' → ∃ i, (p i.fst ∧ p' i.snd) ∧ s i.fst ∩ s' i.snd ⊆ t\n\ncase mpr\nα : Type u_1\nβ : Type ?u.29550\nγ : Type ?u.29553\nι : Sort u_2\nι' : Sort u_3\nl l' : Filter α\np : ι → Prop\ns : ι → Set α\nt✝ : Set α\ni : ι\np' : ι' → Prop\ns' : ι' → Set α\ni' : ι'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\nt : Set α\n⊢ (∃ i, (p i.fst ∧ p' i.snd) ∧ s i.fst ∩ s' i.snd ⊆ t) → t ∈ l ⊓ l'",
"state_before": "α : Type u_1\nβ : Type ?u.29550\nγ : Type ?u.29553\nι : Sort u_2\nι' : Sort u_3\nl l' : Filter α\np : ι → Prop\ns : ι → Set α\nt✝ : Set α\ni : ι\np' : ι' → Prop\ns' : ι' → Set α\ni' : ι'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\nt : Set α\n⊢ t ∈ l ⊓ l' ↔ ∃ i, (p i.fst ∧ p' i.snd) ∧ s i.fst ∩ s' i.snd ⊆ t",
"tactic": "constructor"
},
{
"state_after": "case mp\nα : Type u_1\nβ : Type ?u.29550\nγ : Type ?u.29553\nι : Sort u_2\nι' : Sort u_3\nl l' : Filter α\np : ι → Prop\ns : ι → Set α\nt✝ : Set α\ni : ι\np' : ι' → Prop\ns' : ι' → Set α\ni' : ι'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\nt : Set α\n⊢ (∃ t₁, (∃ i, p i ∧ s i ⊆ t₁) ∧ ∃ t₂, (∃ i, p' i ∧ s' i ⊆ t₂) ∧ t = t₁ ∩ t₂) →\n ∃ i, (p i.fst ∧ p' i.snd) ∧ s i.fst ∩ s' i.snd ⊆ t",
"state_before": "case mp\nα : Type u_1\nβ : Type ?u.29550\nγ : Type ?u.29553\nι : Sort u_2\nι' : Sort u_3\nl l' : Filter α\np : ι → Prop\ns : ι → Set α\nt✝ : Set α\ni : ι\np' : ι' → Prop\ns' : ι' → Set α\ni' : ι'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\nt : Set α\n⊢ t ∈ l ⊓ l' → ∃ i, (p i.fst ∧ p' i.snd) ∧ s i.fst ∩ s' i.snd ⊆ t",
"tactic": "simp only [mem_inf_iff, hl.mem_iff, hl'.mem_iff]"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.29550\nγ : Type ?u.29553\nι : Sort u_2\nι' : Sort u_3\nl l' : Filter α\np : ι → Prop\ns : ι → Set α\nt✝ : Set α\ni✝ : ι\np' : ι' → Prop\ns' : ι' → Set α\ni'✝ : ι'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\nt : Set α\ni : ι\nhi : p i\nht : s i ⊆ t\nt' : Set α\ni' : ι'\nhi' : p' i'\nht' : s' i' ⊆ t'\n⊢ ∃ i, (p i.fst ∧ p' i.snd) ∧ s i.fst ∩ s' i.snd ⊆ t ∩ t'",
"state_before": "case mp\nα : Type u_1\nβ : Type ?u.29550\nγ : Type ?u.29553\nι : Sort u_2\nι' : Sort u_3\nl l' : Filter α\np : ι → Prop\ns : ι → Set α\nt✝ : Set α\ni : ι\np' : ι' → Prop\ns' : ι' → Set α\ni' : ι'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\nt : Set α\n⊢ (∃ t₁, (∃ i, p i ∧ s i ⊆ t₁) ∧ ∃ t₂, (∃ i, p' i ∧ s' i ⊆ t₂) ∧ t = t₁ ∩ t₂) →\n ∃ i, (p i.fst ∧ p' i.snd) ∧ s i.fst ∩ s' i.snd ⊆ t",
"tactic": "rintro ⟨t, ⟨i, hi, ht⟩, t', ⟨i', hi', ht'⟩, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.29550\nγ : Type ?u.29553\nι : Sort u_2\nι' : Sort u_3\nl l' : Filter α\np : ι → Prop\ns : ι → Set α\nt✝ : Set α\ni✝ : ι\np' : ι' → Prop\ns' : ι' → Set α\ni'✝ : ι'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\nt : Set α\ni : ι\nhi : p i\nht : s i ⊆ t\nt' : Set α\ni' : ι'\nhi' : p' i'\nht' : s' i' ⊆ t'\n⊢ ∃ i, (p i.fst ∧ p' i.snd) ∧ s i.fst ∩ s' i.snd ⊆ t ∩ t'",
"tactic": "exact ⟨⟨i, i'⟩, ⟨hi, hi'⟩, inter_subset_inter ht ht'⟩"
},
{
"state_after": "case mpr.intro.mk.intro.intro\nα : Type u_1\nβ : Type ?u.29550\nγ : Type ?u.29553\nι : Sort u_2\nι' : Sort u_3\nl l' : Filter α\np : ι → Prop\ns : ι → Set α\nt✝ : Set α\ni✝ : ι\np' : ι' → Prop\ns' : ι' → Set α\ni'✝ : ι'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\nt : Set α\ni : ι\ni' : ι'\nH : s { fst := i, snd := i' }.fst ∩ s' { fst := i, snd := i' }.snd ⊆ t\nhi : p { fst := i, snd := i' }.fst\nhi' : p' { fst := i, snd := i' }.snd\n⊢ t ∈ l ⊓ l'",
"state_before": "case mpr\nα : Type u_1\nβ : Type ?u.29550\nγ : Type ?u.29553\nι : Sort u_2\nι' : Sort u_3\nl l' : Filter α\np : ι → Prop\ns : ι → Set α\nt✝ : Set α\ni : ι\np' : ι' → Prop\ns' : ι' → Set α\ni' : ι'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\nt : Set α\n⊢ (∃ i, (p i.fst ∧ p' i.snd) ∧ s i.fst ∩ s' i.snd ⊆ t) → t ∈ l ⊓ l'",
"tactic": "rintro ⟨⟨i, i'⟩, ⟨hi, hi'⟩, H⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.mk.intro.intro\nα : Type u_1\nβ : Type ?u.29550\nγ : Type ?u.29553\nι : Sort u_2\nι' : Sort u_3\nl l' : Filter α\np : ι → Prop\ns : ι → Set α\nt✝ : Set α\ni✝ : ι\np' : ι' → Prop\ns' : ι' → Set α\ni'✝ : ι'\nhl : HasBasis l p s\nhl' : HasBasis l' p' s'\nt : Set α\ni : ι\ni' : ι'\nH : s { fst := i, snd := i' }.fst ∩ s' { fst := i, snd := i' }.snd ⊆ t\nhi : p { fst := i, snd := i' }.fst\nhi' : p' { fst := i, snd := i' }.snd\n⊢ t ∈ l ⊓ l'",
"tactic": "exact mem_inf_of_inter (hl.mem_of_mem hi) (hl'.mem_of_mem hi') H"
}
] |
[
490,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
481,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineSubspace.lean
|
AffineSubspace.map_direction
|
[
{
"state_after": "no goals",
"state_before": "k : Type u_1\nV₁ : Type u_2\nP₁ : Type u_3\nV₂ : Type u_4\nP₂ : Type u_5\nV₃ : Type ?u.566022\nP₃ : Type ?u.566025\ninst✝⁹ : Ring k\ninst✝⁸ : AddCommGroup V₁\ninst✝⁷ : Module k V₁\ninst✝⁶ : AffineSpace V₁ P₁\ninst✝⁵ : AddCommGroup V₂\ninst✝⁴ : Module k V₂\ninst✝³ : AffineSpace V₂ P₂\ninst✝² : AddCommGroup V₃\ninst✝¹ : Module k V₃\ninst✝ : AffineSpace V₃ P₃\nf : P₁ →ᵃ[k] P₂\ns : AffineSubspace k P₁\n⊢ direction (map f s) = Submodule.map f.linear (direction s)",
"tactic": "rw [direction_eq_vectorSpan, direction_eq_vectorSpan, coe_map,\nAffineMap.vectorSpan_image_eq_submodule_map]"
}
] |
[
1559,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1557,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.preimage_const_of_mem
|
[] |
[
135,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/Data/List/Basic.lean
|
List.reverse_foldl
|
[
{
"state_after": "ι : Type ?u.233879\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\n⊢ reverse (foldr (fun h t => h :: t) [] (reverse l)) = l",
"state_before": "ι : Type ?u.233879\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\n⊢ reverse (foldl (fun t h => h :: t) [] l) = l",
"tactic": "rw [← foldr_reverse]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.233879\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ l : List α\n⊢ reverse (foldr (fun h t => h :: t) [] (reverse l)) = l",
"tactic": "simp only [foldr_self_append, append_nil, reverse_reverse]"
}
] |
[
2465,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2464,
1
] |
Mathlib/Computability/Ackermann.lean
|
one_lt_ack_succ_left
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ 1 < ack (0 + 1) n",
"tactic": "simp"
},
{
"state_after": "m : ℕ\n⊢ 1 < ack (m + 1) 1",
"state_before": "m : ℕ\n⊢ 1 < ack (m + 1 + 1) 0",
"tactic": "rw [ack_succ_zero]"
},
{
"state_after": "no goals",
"state_before": "m : ℕ\n⊢ 1 < ack (m + 1) 1",
"tactic": "apply one_lt_ack_succ_left"
},
{
"state_after": "m n : ℕ\n⊢ 1 < ack (m + 1) (ack (m + 1 + 1) n)",
"state_before": "m n : ℕ\n⊢ 1 < ack (m + 1 + 1) (n + 1)",
"tactic": "rw [ack_succ_succ]"
},
{
"state_after": "no goals",
"state_before": "m n : ℕ\n⊢ 1 < ack (m + 1) (ack (m + 1 + 1) n)",
"tactic": "apply one_lt_ack_succ_left"
}
] |
[
126,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
119,
1
] |
Mathlib/Analysis/Convex/Side.lean
|
AffineSubspace.WSameSide.trans
|
[
{
"state_after": "case intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.277529\nP : Type u_3\nP' : Type ?u.277535\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhyz : WSameSide s y z\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (y -ᵥ p₂)\n⊢ WSameSide s x z",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.277529\nP : Type u_3\nP' : Type ?u.277535\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhxy : WSameSide s x y\nhyz : WSameSide s y z\nhy : ¬y ∈ s\n⊢ WSameSide s x z",
"tactic": "rcases hxy with ⟨p₁, hp₁, p₂, hp₂, hxy⟩"
},
{
"state_after": "case intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.277529\nP : Type u_3\nP' : Type ?u.277535\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhyz : ∃ p₂_1, p₂_1 ∈ s ∧ SameRay R (y -ᵥ p₂) (z -ᵥ p₂_1)\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (y -ᵥ p₂)\n⊢ WSameSide s x z",
"state_before": "case intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.277529\nP : Type u_3\nP' : Type ?u.277535\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhyz : WSameSide s y z\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (y -ᵥ p₂)\n⊢ WSameSide s x z",
"tactic": "rw [wSameSide_iff_exists_left hp₂, or_iff_right hy] at hyz"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.277529\nP : Type u_3\nP' : Type ?u.277535\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (y -ᵥ p₂)\np₃ : P\nhp₃ : p₃ ∈ s\nhyz : SameRay R (y -ᵥ p₂) (z -ᵥ p₃)\n⊢ WSameSide s x z",
"state_before": "case intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.277529\nP : Type u_3\nP' : Type ?u.277535\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhyz : ∃ p₂_1, p₂_1 ∈ s ∧ SameRay R (y -ᵥ p₂) (z -ᵥ p₂_1)\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (y -ᵥ p₂)\n⊢ WSameSide s x z",
"tactic": "rcases hyz with ⟨p₃, hp₃, hyz⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.277529\nP : Type u_3\nP' : Type ?u.277535\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (y -ᵥ p₂)\np₃ : P\nhp₃ : p₃ ∈ s\nhyz : SameRay R (y -ᵥ p₂) (z -ᵥ p₃)\n⊢ y -ᵥ p₂ = 0 → x -ᵥ p₁ = 0 ∨ z -ᵥ p₃ = 0",
"state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.277529\nP : Type u_3\nP' : Type ?u.277535\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (y -ᵥ p₂)\np₃ : P\nhp₃ : p₃ ∈ s\nhyz : SameRay R (y -ᵥ p₂) (z -ᵥ p₃)\n⊢ WSameSide s x z",
"tactic": "refine' ⟨p₁, hp₁, p₃, hp₃, hxy.trans hyz _⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.277529\nP : Type u_3\nP' : Type ?u.277535\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (y -ᵥ p₂)\np₃ : P\nhp₃ : p₃ ∈ s\nhyz : SameRay R (y -ᵥ p₂) (z -ᵥ p₃)\nh : y -ᵥ p₂ = 0\n⊢ False",
"state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.277529\nP : Type u_3\nP' : Type ?u.277535\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (y -ᵥ p₂)\np₃ : P\nhp₃ : p₃ ∈ s\nhyz : SameRay R (y -ᵥ p₂) (z -ᵥ p₃)\n⊢ y -ᵥ p₂ = 0 → x -ᵥ p₁ = 0 ∨ z -ᵥ p₃ = 0",
"tactic": "refine' fun h => False.elim _"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.277529\nP : Type u_3\nP' : Type ?u.277535\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (y -ᵥ p₂)\np₃ : P\nhp₃ : p₃ ∈ s\nhyz : SameRay R (y -ᵥ p₂) (z -ᵥ p₃)\nh : y = p₂\n⊢ False",
"state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.277529\nP : Type u_3\nP' : Type ?u.277535\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (y -ᵥ p₂)\np₃ : P\nhp₃ : p₃ ∈ s\nhyz : SameRay R (y -ᵥ p₂) (z -ᵥ p₃)\nh : y -ᵥ p₂ = 0\n⊢ False",
"tactic": "rw [vsub_eq_zero_iff_eq] at h"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u_1\nV : Type u_2\nV' : Type ?u.277529\nP : Type u_3\nP' : Type ?u.277535\ninst✝⁶ : LinearOrderedField R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : AddCommGroup V'\ninst✝¹ : Module R V'\ninst✝ : AddTorsor V' P'\ns : AffineSubspace R P\nx y z : P\nhy : ¬y ∈ s\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhxy : SameRay R (x -ᵥ p₁) (y -ᵥ p₂)\np₃ : P\nhp₃ : p₃ ∈ s\nhyz : SameRay R (y -ᵥ p₂) (z -ᵥ p₃)\nh : y = p₂\n⊢ False",
"tactic": "exact hy (h.symm ▸ hp₂)"
}
] |
[
533,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
525,
1
] |
Mathlib/Data/Set/Image.lean
|
Set.image_univ
|
[
{
"state_after": "case h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.67325\nι : Sort ?u.67328\nι' : Sort ?u.67331\nf✝ : ι → α\ns t : Set α\nf : α → β\nx✝ : β\n⊢ x✝ ∈ f '' univ ↔ x✝ ∈ range f",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.67325\nι : Sort ?u.67328\nι' : Sort ?u.67331\nf✝ : ι → α\ns t : Set α\nf : α → β\n⊢ f '' univ = range f",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.67325\nι : Sort ?u.67328\nι' : Sort ?u.67331\nf✝ : ι → α\ns t : Set α\nf : α → β\nx✝ : β\n⊢ x✝ ∈ f '' univ ↔ x✝ ∈ range f",
"tactic": "simp [image, range]"
}
] |
[
696,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
694,
1
] |
Mathlib/RingTheory/Ideal/Operations.lean
|
Ideal.map_inf_le
|
[] |
[
1503,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1502,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
Subsemiring.prod_mono
|
[] |
[
1058,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1056,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
MonoidHom.eq_of_eqOn_top
|
[] |
[
2912,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2911,
1
] |
Mathlib/Data/Real/NNReal.lean
|
Real.coe_toNNReal_le
|
[] |
[
1075,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1074,
1
] |
Mathlib/Topology/MetricSpace/Basic.lean
|
Metric.complete_of_cauchySeq_tendsto
|
[] |
[
1339,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1337,
1
] |
Mathlib/GroupTheory/MonoidLocalization.lean
|
Submonoid.LocalizationMap.isUnit_comp
|
[] |
[
882,
96
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
880,
1
] |
Mathlib/Algebra/Module/LinearMap.lean
|
LinearMap.one_eq_id
|
[] |
[
1022,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1021,
1
] |
Mathlib/Topology/MetricSpace/EMetricSpace.lean
|
EMetric.diam_empty
|
[] |
[
910,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
909,
1
] |
Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean
|
regionBetween_subset
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nf g : α → ℝ\ns : Set α\n⊢ regionBetween f g s ⊆ s ×ˢ univ",
"tactic": "simpa only [prod_univ, regionBetween, Set.preimage, setOf_subset_setOf] using fun a => And.left"
}
] |
[
474,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
473,
1
] |
Mathlib/GroupTheory/Abelianization.lean
|
abelianizationCongr_trans
|
[] |
[
244,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
242,
1
] |
Mathlib/Algebra/CharP/Basic.lean
|
sub_pow_char_of_commute
|
[
{
"state_after": "R : Type u_1\ninst✝² : Ring R\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP R p\nx y : R\nh : Commute x y\n⊢ (x - y + y) ^ p = x ^ p",
"state_before": "R : Type u_1\ninst✝² : Ring R\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP R p\nx y : R\nh : Commute x y\n⊢ (x - y) ^ p = x ^ p - y ^ p",
"tactic": "rw [eq_sub_iff_add_eq, ← add_pow_char_of_commute _ _ _ (Commute.sub_left h rfl)]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝² : Ring R\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP R p\nx y : R\nh : Commute x y\n⊢ (x - y + y) ^ p = x ^ p",
"tactic": "simp"
}
] |
[
263,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
260,
1
] |
Mathlib/Topology/Sets/Opens.lean
|
TopologicalSpace.Opens.iSup_def
|
[] |
[
236,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
235,
1
] |
Mathlib/LinearAlgebra/UnitaryGroup.lean
|
Matrix.mem_unitaryGroup_iff'
|
[
{
"state_after": "n : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα : Type v\ninst✝¹ : CommRing α\ninst✝ : StarRing α\nA : Matrix n n α\nhA : star A * A = 1\n⊢ A * star A = 1",
"state_before": "n : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα : Type v\ninst✝¹ : CommRing α\ninst✝ : StarRing α\nA : Matrix n n α\n⊢ A ∈ unitaryGroup n α ↔ star A * A = 1",
"tactic": "refine' ⟨And.left, fun hA => ⟨hA, _⟩⟩"
},
{
"state_after": "no goals",
"state_before": "n : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα : Type v\ninst✝¹ : CommRing α\ninst✝ : StarRing α\nA : Matrix n n α\nhA : star A * A = 1\n⊢ A * star A = 1",
"tactic": "rwa [mul_eq_mul, mul_eq_one_comm] at hA"
}
] |
[
76,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/FieldTheory/Normal.lean
|
AlgHom.restrictNormal_comp
|
[
{
"state_after": "no goals",
"state_before": "F : Type u_1\nK : Type ?u.298025\ninst✝¹⁸ : Field F\ninst✝¹⁷ : Field K\ninst✝¹⁶ : Algebra F K\nK₁ : Type u_5\nK₂ : Type u_3\nK₃ : Type u_4\ninst✝¹⁵ : Field F\ninst✝¹⁴ : Field K₁\ninst✝¹³ : Field K₂\ninst✝¹² : Field K₃\ninst✝¹¹ : Algebra F K₁\ninst✝¹⁰ : Algebra F K₂\ninst✝⁹ : Algebra F K₃\nϕ : K₁ →ₐ[F] K₂\nχ : K₁ ≃ₐ[F] K₂\nψ : K₂ →ₐ[F] K₃\nω : K₂ ≃ₐ[F] K₃\nE : Type u_2\ninst✝⁸ : Field E\ninst✝⁷ : Algebra F E\ninst✝⁶ : Algebra E K₁\ninst✝⁵ : Algebra E K₂\ninst✝⁴ : Algebra E K₃\ninst✝³ : IsScalarTower F E K₁\ninst✝² : IsScalarTower F E K₂\ninst✝¹ : IsScalarTower F E K₃\ninst✝ : Normal F E\nx✝ : E\n⊢ ↑(algebraMap E K₃) (↑(comp (restrictNormal ψ E) (restrictNormal ϕ E)) x✝) =\n ↑(algebraMap E K₃) (↑(restrictNormal (comp ψ ϕ) E) x✝)",
"tactic": "simp only [AlgHom.comp_apply, AlgHom.restrictNormal_commutes]"
}
] |
[
340,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
337,
1
] |
Mathlib/Topology/UniformSpace/AbstractCompletion.lean
|
AbstractCompletion.uniformContinuous_compareEquiv_symm
|
[] |
[
286,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
285,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.zero_eq_one_iff
|
[
{
"state_after": "case mp\nn m : ℕ\ninst✝ : NeZero n\n⊢ 0 = 1 → n = 1\n\ncase mpr\nn m : ℕ\ninst✝ : NeZero n\n⊢ n = 1 → 0 = 1",
"state_before": "n m : ℕ\ninst✝ : NeZero n\n⊢ 0 = 1 ↔ n = 1",
"tactic": "constructor"
},
{
"state_after": "case mp\nn m : ℕ\ninst✝ : NeZero n\nh : 0 = 1\n⊢ n = 1",
"state_before": "case mp\nn m : ℕ\ninst✝ : NeZero n\n⊢ 0 = 1 → n = 1",
"tactic": "intro h"
},
{
"state_after": "case mp\nn m : ℕ\ninst✝ : NeZero n\nh : 0 = 1\nthis : ↑0 = ↑1\n⊢ n = 1",
"state_before": "case mp\nn m : ℕ\ninst✝ : NeZero n\nh : 0 = 1\n⊢ n = 1",
"tactic": "have := congr_arg ((↑) : Fin n → ℕ) h"
},
{
"state_after": "case mp\nn m : ℕ\ninst✝ : NeZero n\nh : 0 = 1\nthis : n ∣ 1\n⊢ n = 1",
"state_before": "case mp\nn m : ℕ\ninst✝ : NeZero n\nh : 0 = 1\nthis : ↑0 = ↑1\n⊢ n = 1",
"tactic": "simp only [val_zero, val_one', @eq_comm _ 0, ← Nat.dvd_iff_mod_eq_zero] at this"
},
{
"state_after": "no goals",
"state_before": "case mp\nn m : ℕ\ninst✝ : NeZero n\nh : 0 = 1\nthis : n ∣ 1\n⊢ n = 1",
"tactic": "exact eq_one_of_dvd_one this"
},
{
"state_after": "case mpr\nm : ℕ\ninst✝ : NeZero 1\n⊢ 0 = 1",
"state_before": "case mpr\nn m : ℕ\ninst✝ : NeZero n\n⊢ n = 1 → 0 = 1",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "case mpr\nm : ℕ\ninst✝ : NeZero 1\n⊢ 0 = 1",
"tactic": "rfl"
}
] |
[
854,
8
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
847,
1
] |
Mathlib/Data/Finsupp/ToDfinsupp.lean
|
Finsupp.toDfinsupp_zero
|
[] |
[
155,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/Control/Basic.lean
|
guard_false
|
[
{
"state_after": "no goals",
"state_before": "α β γ : Type u\nF : Type → Type v\ninst✝ : Alternative F\nh : Decidable False\n⊢ guard False = failure",
"tactic": "simp [guard, if_neg not_false]"
}
] |
[
206,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Mathlib/Analysis/BoxIntegral/Basic.lean
|
BoxIntegral.Integrable.sum_integral_congr
|
[
{
"state_after": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\nI J : Box ι\nπ : TaggedPrepartition I\ninst✝¹ : Fintype ι\nl : IntegrationParams\nf g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\nc c₁ c₂ : ℝ≥0\nε ε₁ ε₂ : ℝ\nπ₁✝ π₂✝ : TaggedPrepartition I\ninst✝ : CompleteSpace F\nh : Integrable I l f vol\nπ₁ π₂ : Prepartition I\nhU : Prepartition.iUnion π₁ = Prepartition.iUnion π₂\n⊢ Tendsto (integralSum f vol) (toFilteriUnion l I π₁) (𝓝 (∑ J in π₂.boxes, integral J l f vol))",
"state_before": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\nI J : Box ι\nπ : TaggedPrepartition I\ninst✝¹ : Fintype ι\nl : IntegrationParams\nf g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\nc c₁ c₂ : ℝ≥0\nε ε₁ ε₂ : ℝ\nπ₁✝ π₂✝ : TaggedPrepartition I\ninst✝ : CompleteSpace F\nh : Integrable I l f vol\nπ₁ π₂ : Prepartition I\nhU : Prepartition.iUnion π₁ = Prepartition.iUnion π₂\n⊢ ∑ J in π₁.boxes, integral J l f vol = ∑ J in π₂.boxes, integral J l f vol",
"tactic": "refine' tendsto_nhds_unique (h.tendsto_integralSum_sum_integral π₁) _"
},
{
"state_after": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\nI J : Box ι\nπ : TaggedPrepartition I\ninst✝¹ : Fintype ι\nl : IntegrationParams\nf g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\nc c₁ c₂ : ℝ≥0\nε ε₁ ε₂ : ℝ\nπ₁✝ π₂✝ : TaggedPrepartition I\ninst✝ : CompleteSpace F\nh : Integrable I l f vol\nπ₁ π₂ : Prepartition I\nhU : Prepartition.iUnion π₁ = Prepartition.iUnion π₂\n⊢ Tendsto (integralSum f vol) (toFilteriUnion l I π₂) (𝓝 (∑ J in π₂.boxes, integral J l f vol))",
"state_before": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\nI J : Box ι\nπ : TaggedPrepartition I\ninst✝¹ : Fintype ι\nl : IntegrationParams\nf g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\nc c₁ c₂ : ℝ≥0\nε ε₁ ε₂ : ℝ\nπ₁✝ π₂✝ : TaggedPrepartition I\ninst✝ : CompleteSpace F\nh : Integrable I l f vol\nπ₁ π₂ : Prepartition I\nhU : Prepartition.iUnion π₁ = Prepartition.iUnion π₂\n⊢ Tendsto (integralSum f vol) (toFilteriUnion l I π₁) (𝓝 (∑ J in π₂.boxes, integral J l f vol))",
"tactic": "rw [l.toFilteriUnion_congr _ hU]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\nI J : Box ι\nπ : TaggedPrepartition I\ninst✝¹ : Fintype ι\nl : IntegrationParams\nf g : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\ny y' : F\nc c₁ c₂ : ℝ≥0\nε ε₁ ε₂ : ℝ\nπ₁✝ π₂✝ : TaggedPrepartition I\ninst✝ : CompleteSpace F\nh : Integrable I l f vol\nπ₁ π₂ : Prepartition I\nhU : Prepartition.iUnion π₁ = Prepartition.iUnion π₂\n⊢ Tendsto (integralSum f vol) (toFilteriUnion l I π₂) (𝓝 (∑ J in π₂.boxes, integral J l f vol))",
"tactic": "exact h.tendsto_integralSum_sum_integral π₂"
}
] |
[
653,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
648,
1
] |
Mathlib/LinearAlgebra/Matrix/DotProduct.lean
|
Matrix.dotProduct_stdBasis_eq_mul
|
[
{
"state_after": "case h₀\nR : Type v\nn : Type w\ninst✝² : Semiring R\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nv : n → R\nc : R\ni : n\n⊢ ∀ (b : n), b ∈ Finset.univ → b ≠ i → v b * ↑(LinearMap.stdBasis R (fun x => R) i) c b = 0\n\ncase h₁\nR : Type v\nn : Type w\ninst✝² : Semiring R\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nv : n → R\nc : R\ni : n\n⊢ ¬i ∈ Finset.univ → v i * ↑(LinearMap.stdBasis R (fun x => R) i) c i = 0",
"state_before": "R : Type v\nn : Type w\ninst✝² : Semiring R\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nv : n → R\nc : R\ni : n\n⊢ v ⬝ᵥ ↑(LinearMap.stdBasis R (fun x => R) i) c = v i * c",
"tactic": "rw [dotProduct, Finset.sum_eq_single i, LinearMap.stdBasis_same]"
},
{
"state_after": "case h₁\nR : Type v\nn : Type w\ninst✝² : Semiring R\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nv : n → R\nc : R\ni : n\n⊢ ¬i ∈ Finset.univ → v i * ↑(LinearMap.stdBasis R (fun x => R) i) c i = 0",
"state_before": "case h₀\nR : Type v\nn : Type w\ninst✝² : Semiring R\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nv : n → R\nc : R\ni : n\n⊢ ∀ (b : n), b ∈ Finset.univ → b ≠ i → v b * ↑(LinearMap.stdBasis R (fun x => R) i) c b = 0\n\ncase h₁\nR : Type v\nn : Type w\ninst✝² : Semiring R\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nv : n → R\nc : R\ni : n\n⊢ ¬i ∈ Finset.univ → v i * ↑(LinearMap.stdBasis R (fun x => R) i) c i = 0",
"tactic": "exact fun _ _ hb => by rw [LinearMap.stdBasis_ne _ _ _ _ hb, MulZeroClass.mul_zero]"
},
{
"state_after": "no goals",
"state_before": "case h₁\nR : Type v\nn : Type w\ninst✝² : Semiring R\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nv : n → R\nc : R\ni : n\n⊢ ¬i ∈ Finset.univ → v i * ↑(LinearMap.stdBasis R (fun x => R) i) c i = 0",
"tactic": "exact fun hi => False.elim (hi <| Finset.mem_univ _)"
},
{
"state_after": "no goals",
"state_before": "R : Type v\nn : Type w\ninst✝² : Semiring R\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nv : n → R\nc : R\ni x✝¹ : n\nx✝ : x✝¹ ∈ Finset.univ\nhb : x✝¹ ≠ i\n⊢ v x✝¹ * ↑(LinearMap.stdBasis R (fun x => R) i) c x✝¹ = 0",
"tactic": "rw [LinearMap.stdBasis_ne _ _ _ _ hb, MulZeroClass.mul_zero]"
}
] |
[
50,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
46,
1
] |
Mathlib/NumberTheory/Padics/PadicNumbers.lean
|
PadicSeq.stationary
|
[
{
"state_after": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ (padicNorm p)\nhf : ¬f ≈ 0\nthis✝ : ∃ ε, ε > 0 ∧ ∃ N1, ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nε : ℚ\nhε : ε > 0\nN1 : ℕ\nhN1 : ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nN2 : ℕ\nhN2 : ∀ (j : ℕ), j ≥ N2 → ∀ (k : ℕ), k ≥ N2 → padicNorm p (↑f j - ↑f k) < ε\nn m : ℕ\nhn : max N1 N2 ≤ n\nhm : max N1 N2 ≤ m\nthis : padicNorm p (↑f n - ↑f m) < ε\n⊢ padicNorm p (↑f m) = padicNorm p (↑f n)",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ (padicNorm p)\nhf : ¬f ≈ 0\nthis : ∃ ε, ε > 0 ∧ ∃ N1, ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nε : ℚ\nhε : ε > 0\nN1 : ℕ\nhN1 : ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nN2 : ℕ\nhN2 : ∀ (j : ℕ), j ≥ N2 → ∀ (k : ℕ), k ≥ N2 → padicNorm p (↑f j - ↑f k) < ε\nn m : ℕ\nhn : max N1 N2 ≤ n\nhm : max N1 N2 ≤ m\n⊢ padicNorm p (↑f m) = padicNorm p (↑f n)",
"tactic": "have : padicNorm p (f n - f m) < ε := hN2 _ (max_le_iff.1 hn).2 _ (max_le_iff.1 hm).2"
},
{
"state_after": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ (padicNorm p)\nhf : ¬f ≈ 0\nthis✝¹ : ∃ ε, ε > 0 ∧ ∃ N1, ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nε : ℚ\nhε : ε > 0\nN1 : ℕ\nhN1 : ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nN2 : ℕ\nhN2 : ∀ (j : ℕ), j ≥ N2 → ∀ (k : ℕ), k ≥ N2 → padicNorm p (↑f j - ↑f k) < ε\nn m : ℕ\nhn : max N1 N2 ≤ n\nhm : max N1 N2 ≤ m\nthis✝ : padicNorm p (↑f n - ↑f m) < ε\nthis : padicNorm p (↑f n - ↑f m) < padicNorm p (↑f n)\n⊢ padicNorm p (↑f m) = padicNorm p (↑f n)",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ (padicNorm p)\nhf : ¬f ≈ 0\nthis✝ : ∃ ε, ε > 0 ∧ ∃ N1, ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nε : ℚ\nhε : ε > 0\nN1 : ℕ\nhN1 : ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nN2 : ℕ\nhN2 : ∀ (j : ℕ), j ≥ N2 → ∀ (k : ℕ), k ≥ N2 → padicNorm p (↑f j - ↑f k) < ε\nn m : ℕ\nhn : max N1 N2 ≤ n\nhm : max N1 N2 ≤ m\nthis : padicNorm p (↑f n - ↑f m) < ε\n⊢ padicNorm p (↑f m) = padicNorm p (↑f n)",
"tactic": "have : padicNorm p (f n - f m) < padicNorm p (f n) :=\n lt_of_lt_of_le this <| hN1 _ (max_le_iff.1 hn).1"
},
{
"state_after": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ (padicNorm p)\nhf : ¬f ≈ 0\nthis✝² : ∃ ε, ε > 0 ∧ ∃ N1, ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nε : ℚ\nhε : ε > 0\nN1 : ℕ\nhN1 : ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nN2 : ℕ\nhN2 : ∀ (j : ℕ), j ≥ N2 → ∀ (k : ℕ), k ≥ N2 → padicNorm p (↑f j - ↑f k) < ε\nn m : ℕ\nhn : max N1 N2 ≤ n\nhm : max N1 N2 ≤ m\nthis✝¹ : padicNorm p (↑f n - ↑f m) < ε\nthis✝ : padicNorm p (↑f n - ↑f m) < padicNorm p (↑f n)\nthis : padicNorm p (↑f n - ↑f m) < max (padicNorm p (↑f n)) (padicNorm p (↑f m))\n⊢ padicNorm p (↑f m) = padicNorm p (↑f n)",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ (padicNorm p)\nhf : ¬f ≈ 0\nthis✝¹ : ∃ ε, ε > 0 ∧ ∃ N1, ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nε : ℚ\nhε : ε > 0\nN1 : ℕ\nhN1 : ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nN2 : ℕ\nhN2 : ∀ (j : ℕ), j ≥ N2 → ∀ (k : ℕ), k ≥ N2 → padicNorm p (↑f j - ↑f k) < ε\nn m : ℕ\nhn : max N1 N2 ≤ n\nhm : max N1 N2 ≤ m\nthis✝ : padicNorm p (↑f n - ↑f m) < ε\nthis : padicNorm p (↑f n - ↑f m) < padicNorm p (↑f n)\n⊢ padicNorm p (↑f m) = padicNorm p (↑f n)",
"tactic": "have : padicNorm p (f n - f m) < max (padicNorm p (f n)) (padicNorm p (f m)) :=\n lt_max_iff.2 (Or.inl this)"
},
{
"state_after": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ (padicNorm p)\nhf : ¬f ≈ 0\nthis✝² : ∃ ε, ε > 0 ∧ ∃ N1, ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nε : ℚ\nhε : ε > 0\nN1 : ℕ\nhN1 : ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nN2 : ℕ\nhN2 : ∀ (j : ℕ), j ≥ N2 → ∀ (k : ℕ), k ≥ N2 → padicNorm p (↑f j - ↑f k) < ε\nn m : ℕ\nhn : max N1 N2 ≤ n\nhm : max N1 N2 ≤ m\nthis✝¹ : padicNorm p (↑f n - ↑f m) < ε\nthis✝ : padicNorm p (↑f n - ↑f m) < padicNorm p (↑f n)\nthis : padicNorm p (↑f n - ↑f m) < max (padicNorm p (↑f n)) (padicNorm p (↑f m))\nhne : ¬padicNorm p (↑f m) = padicNorm p (↑f n)\n⊢ False",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ (padicNorm p)\nhf : ¬f ≈ 0\nthis✝² : ∃ ε, ε > 0 ∧ ∃ N1, ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nε : ℚ\nhε : ε > 0\nN1 : ℕ\nhN1 : ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nN2 : ℕ\nhN2 : ∀ (j : ℕ), j ≥ N2 → ∀ (k : ℕ), k ≥ N2 → padicNorm p (↑f j - ↑f k) < ε\nn m : ℕ\nhn : max N1 N2 ≤ n\nhm : max N1 N2 ≤ m\nthis✝¹ : padicNorm p (↑f n - ↑f m) < ε\nthis✝ : padicNorm p (↑f n - ↑f m) < padicNorm p (↑f n)\nthis : padicNorm p (↑f n - ↑f m) < max (padicNorm p (↑f n)) (padicNorm p (↑f m))\n⊢ padicNorm p (↑f m) = padicNorm p (↑f n)",
"tactic": "by_contra hne"
},
{
"state_after": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ (padicNorm p)\nhf : ¬f ≈ 0\nthis✝² : ∃ ε, ε > 0 ∧ ∃ N1, ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nε : ℚ\nhε : ε > 0\nN1 : ℕ\nhN1 : ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nN2 : ℕ\nhN2 : ∀ (j : ℕ), j ≥ N2 → ∀ (k : ℕ), k ≥ N2 → padicNorm p (↑f j - ↑f k) < ε\nn m : ℕ\nhn : max N1 N2 ≤ n\nhm : max N1 N2 ≤ m\nthis✝¹ : padicNorm p (↑f n - ↑f m) < ε\nthis✝ : padicNorm p (↑f n - ↑f m) < padicNorm p (↑f n)\nthis : padicNorm p (↑f n - ↑f m) < max (padicNorm p (↑f n)) (padicNorm p (↑f m))\nhne : ¬padicNorm p (-↑f m) = padicNorm p (↑f n)\n⊢ False",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ (padicNorm p)\nhf : ¬f ≈ 0\nthis✝² : ∃ ε, ε > 0 ∧ ∃ N1, ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nε : ℚ\nhε : ε > 0\nN1 : ℕ\nhN1 : ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nN2 : ℕ\nhN2 : ∀ (j : ℕ), j ≥ N2 → ∀ (k : ℕ), k ≥ N2 → padicNorm p (↑f j - ↑f k) < ε\nn m : ℕ\nhn : max N1 N2 ≤ n\nhm : max N1 N2 ≤ m\nthis✝¹ : padicNorm p (↑f n - ↑f m) < ε\nthis✝ : padicNorm p (↑f n - ↑f m) < padicNorm p (↑f n)\nthis : padicNorm p (↑f n - ↑f m) < max (padicNorm p (↑f n)) (padicNorm p (↑f m))\nhne : ¬padicNorm p (↑f m) = padicNorm p (↑f n)\n⊢ False",
"tactic": "rw [← padicNorm.neg (f m)] at hne"
},
{
"state_after": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ (padicNorm p)\nhf : ¬f ≈ 0\nthis✝² : ∃ ε, ε > 0 ∧ ∃ N1, ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nε : ℚ\nhε : ε > 0\nN1 : ℕ\nhN1 : ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nN2 : ℕ\nhN2 : ∀ (j : ℕ), j ≥ N2 → ∀ (k : ℕ), k ≥ N2 → padicNorm p (↑f j - ↑f k) < ε\nn m : ℕ\nhn : max N1 N2 ≤ n\nhm : max N1 N2 ≤ m\nthis✝¹ : padicNorm p (↑f n - ↑f m) < ε\nthis✝ : padicNorm p (↑f n - ↑f m) < padicNorm p (↑f n)\nthis : padicNorm p (↑f n - ↑f m) < max (padicNorm p (↑f n)) (padicNorm p (↑f m))\nhne : ¬padicNorm p (-↑f m) = padicNorm p (↑f n)\nhnam : padicNorm p (-↑f m + ↑f n) = max (padicNorm p (-↑f m)) (padicNorm p (↑f n))\n⊢ False",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ (padicNorm p)\nhf : ¬f ≈ 0\nthis✝² : ∃ ε, ε > 0 ∧ ∃ N1, ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nε : ℚ\nhε : ε > 0\nN1 : ℕ\nhN1 : ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nN2 : ℕ\nhN2 : ∀ (j : ℕ), j ≥ N2 → ∀ (k : ℕ), k ≥ N2 → padicNorm p (↑f j - ↑f k) < ε\nn m : ℕ\nhn : max N1 N2 ≤ n\nhm : max N1 N2 ≤ m\nthis✝¹ : padicNorm p (↑f n - ↑f m) < ε\nthis✝ : padicNorm p (↑f n - ↑f m) < padicNorm p (↑f n)\nthis : padicNorm p (↑f n - ↑f m) < max (padicNorm p (↑f n)) (padicNorm p (↑f m))\nhne : ¬padicNorm p (-↑f m) = padicNorm p (↑f n)\n⊢ False",
"tactic": "have hnam := add_eq_max_of_ne hne"
},
{
"state_after": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ (padicNorm p)\nhf : ¬f ≈ 0\nthis✝² : ∃ ε, ε > 0 ∧ ∃ N1, ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nε : ℚ\nhε : ε > 0\nN1 : ℕ\nhN1 : ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nN2 : ℕ\nhN2 : ∀ (j : ℕ), j ≥ N2 → ∀ (k : ℕ), k ≥ N2 → padicNorm p (↑f j - ↑f k) < ε\nn m : ℕ\nhn : max N1 N2 ≤ n\nhm : max N1 N2 ≤ m\nthis✝¹ : padicNorm p (↑f n - ↑f m) < ε\nthis✝ : padicNorm p (↑f n - ↑f m) < padicNorm p (↑f n)\nthis : padicNorm p (↑f n - ↑f m) < max (padicNorm p (↑f n)) (padicNorm p (↑f m))\nhne : ¬padicNorm p (-↑f m) = padicNorm p (↑f n)\nhnam : padicNorm p (-↑f m + ↑f n) = max (padicNorm p (↑f n)) (padicNorm p (↑f m))\n⊢ False",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ (padicNorm p)\nhf : ¬f ≈ 0\nthis✝² : ∃ ε, ε > 0 ∧ ∃ N1, ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nε : ℚ\nhε : ε > 0\nN1 : ℕ\nhN1 : ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nN2 : ℕ\nhN2 : ∀ (j : ℕ), j ≥ N2 → ∀ (k : ℕ), k ≥ N2 → padicNorm p (↑f j - ↑f k) < ε\nn m : ℕ\nhn : max N1 N2 ≤ n\nhm : max N1 N2 ≤ m\nthis✝¹ : padicNorm p (↑f n - ↑f m) < ε\nthis✝ : padicNorm p (↑f n - ↑f m) < padicNorm p (↑f n)\nthis : padicNorm p (↑f n - ↑f m) < max (padicNorm p (↑f n)) (padicNorm p (↑f m))\nhne : ¬padicNorm p (-↑f m) = padicNorm p (↑f n)\nhnam : padicNorm p (-↑f m + ↑f n) = max (padicNorm p (-↑f m)) (padicNorm p (↑f n))\n⊢ False",
"tactic": "rw [padicNorm.neg, max_comm] at hnam"
},
{
"state_after": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ (padicNorm p)\nhf : ¬f ≈ 0\nthis✝² : ∃ ε, ε > 0 ∧ ∃ N1, ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nε : ℚ\nhε : ε > 0\nN1 : ℕ\nhN1 : ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nN2 : ℕ\nhN2 : ∀ (j : ℕ), j ≥ N2 → ∀ (k : ℕ), k ≥ N2 → padicNorm p (↑f j - ↑f k) < ε\nn m : ℕ\nhn : max N1 N2 ≤ n\nhm : max N1 N2 ≤ m\nthis✝¹ : padicNorm p (↑f n - ↑f m) < ε\nthis✝ : padicNorm p (↑f n - ↑f m) < padicNorm p (↑f n)\nthis : padicNorm p (-↑f m + ↑f n) < padicNorm p (-↑f m + ↑f n)\nhne : ¬padicNorm p (-↑f m) = padicNorm p (↑f n)\nhnam : padicNorm p (-↑f m + ↑f n) = max (padicNorm p (↑f n)) (padicNorm p (↑f m))\n⊢ False",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ (padicNorm p)\nhf : ¬f ≈ 0\nthis✝² : ∃ ε, ε > 0 ∧ ∃ N1, ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nε : ℚ\nhε : ε > 0\nN1 : ℕ\nhN1 : ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nN2 : ℕ\nhN2 : ∀ (j : ℕ), j ≥ N2 → ∀ (k : ℕ), k ≥ N2 → padicNorm p (↑f j - ↑f k) < ε\nn m : ℕ\nhn : max N1 N2 ≤ n\nhm : max N1 N2 ≤ m\nthis✝¹ : padicNorm p (↑f n - ↑f m) < ε\nthis✝ : padicNorm p (↑f n - ↑f m) < padicNorm p (↑f n)\nthis : padicNorm p (↑f n - ↑f m) < max (padicNorm p (↑f n)) (padicNorm p (↑f m))\nhne : ¬padicNorm p (-↑f m) = padicNorm p (↑f n)\nhnam : padicNorm p (-↑f m + ↑f n) = max (padicNorm p (↑f n)) (padicNorm p (↑f m))\n⊢ False",
"tactic": "rw [← hnam, sub_eq_add_neg, add_comm] at this"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nf : CauSeq ℚ (padicNorm p)\nhf : ¬f ≈ 0\nthis✝² : ∃ ε, ε > 0 ∧ ∃ N1, ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nε : ℚ\nhε : ε > 0\nN1 : ℕ\nhN1 : ∀ (j : ℕ), j ≥ N1 → ε ≤ padicNorm p (↑f j)\nN2 : ℕ\nhN2 : ∀ (j : ℕ), j ≥ N2 → ∀ (k : ℕ), k ≥ N2 → padicNorm p (↑f j - ↑f k) < ε\nn m : ℕ\nhn : max N1 N2 ≤ n\nhm : max N1 N2 ≤ m\nthis✝¹ : padicNorm p (↑f n - ↑f m) < ε\nthis✝ : padicNorm p (↑f n - ↑f m) < padicNorm p (↑f n)\nthis : padicNorm p (-↑f m + ↑f n) < padicNorm p (-↑f m + ↑f n)\nhne : ¬padicNorm p (-↑f m) = padicNorm p (↑f n)\nhnam : padicNorm p (-↑f m + ↑f n) = max (padicNorm p (↑f n)) (padicNorm p (↑f m))\n⊢ False",
"tactic": "apply _root_.lt_irrefl _ this"
}
] |
[
104,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Std/Data/String/Lemmas.lean
|
String.Iterator.ValidFor.prevn
|
[
{
"state_after": "no goals",
"state_before": "l r : List Char\nit : Iterator\nh : ValidFor l r it\nx✝ : 0 ≤ List.length l\n⊢ ValidFor (List.drop 0 l) (List.reverse (List.take 0 l) ++ r) (Iterator.prevn it 0)",
"tactic": "simp [h, Iterator.prevn]"
},
{
"state_after": "l r : List Char\nit : Iterator\nh : ValidFor l r it\nn : Nat\nhn : n + 1 ≤ List.length l\n⊢ ValidFor (List.drop (n + 1) l) (List.reverse (List.take (n + 1) l) ++ r) (Iterator.prevn (Iterator.prev it) n)",
"state_before": "l r : List Char\nit : Iterator\nh : ValidFor l r it\nn : Nat\nhn : n + 1 ≤ List.length l\n⊢ ValidFor (List.drop (n + 1) l) (List.reverse (List.take (n + 1) l) ++ r) (Iterator.prevn it (n + 1))",
"tactic": "simp [h, Iterator.prevn]"
},
{
"state_after": "l✝ r : List Char\nit : Iterator\nn : Nat\na : Char\nl : List Char\nh : ValidFor (a :: l) r it\nhn : n + 1 ≤ List.length (a :: l)\n⊢ ValidFor (List.drop (n + 1) (a :: l)) (List.reverse (List.take (n + 1) (a :: l)) ++ r)\n (Iterator.prevn (Iterator.prev it) n)",
"state_before": "l r : List Char\nit : Iterator\nh : ValidFor l r it\nn : Nat\nhn : n + 1 ≤ List.length l\n⊢ ValidFor (List.drop (n + 1) l) (List.reverse (List.take (n + 1) l) ++ r) (Iterator.prevn (Iterator.prev it) n)",
"tactic": "have a::l := l"
},
{
"state_after": "no goals",
"state_before": "l✝ r : List Char\nit : Iterator\nn : Nat\na : Char\nl : List Char\nh : ValidFor (a :: l) r it\nhn : n + 1 ≤ List.length (a :: l)\n⊢ ValidFor (List.drop (n + 1) (a :: l)) (List.reverse (List.take (n + 1) (a :: l)) ++ r)\n (Iterator.prevn (Iterator.prev it) n)",
"tactic": "simpa using h.prev.prevn _ (Nat.le_of_succ_le_succ hn)"
}
] |
[
619,
59
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
613,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.lt_succAbove_iff
|
[
{
"state_after": "case refine'_1\nn m : ℕ\np : Fin (n + 1)\ni : Fin n\n⊢ p < ↑(succAbove p) i → p ≤ ↑castSucc i\n\ncase refine'_2\nn m : ℕ\np : Fin (n + 1)\ni : Fin n\n⊢ p ≤ ↑castSucc i → p < ↑(succAbove p) i",
"state_before": "n m : ℕ\np : Fin (n + 1)\ni : Fin n\n⊢ p < ↑(succAbove p) i ↔ p ≤ ↑castSucc i",
"tactic": "refine' Iff.intro _ _"
},
{
"state_after": "case refine'_1\nn m : ℕ\np : Fin (n + 1)\ni : Fin n\nh : p < ↑(succAbove p) i\n⊢ p ≤ ↑castSucc i",
"state_before": "case refine'_1\nn m : ℕ\np : Fin (n + 1)\ni : Fin n\n⊢ p < ↑(succAbove p) i → p ≤ ↑castSucc i",
"tactic": "intro h"
},
{
"state_after": "case refine'_1.inl\nn m : ℕ\np : Fin (n + 1)\ni : Fin n\nh : p < ↑(succAbove p) i\nH : ↑castSucc i < p\n⊢ p ≤ ↑castSucc i\n\ncase refine'_1.inr\nn m : ℕ\np : Fin (n + 1)\ni : Fin n\nh : p < ↑(succAbove p) i\nH : p ≤ ↑castSucc i\n⊢ p ≤ ↑castSucc i",
"state_before": "case refine'_1\nn m : ℕ\np : Fin (n + 1)\ni : Fin n\nh : p < ↑(succAbove p) i\n⊢ p ≤ ↑castSucc i",
"tactic": "cases' succAbove_lt_ge p i with H H"
},
{
"state_after": "case refine'_1.inl\nn m : ℕ\np : Fin (n + 1)\ni : Fin n\nh : p < ↑castSucc i\nH : ↑castSucc i < p\n⊢ p ≤ ↑castSucc i",
"state_before": "case refine'_1.inl\nn m : ℕ\np : Fin (n + 1)\ni : Fin n\nh : p < ↑(succAbove p) i\nH : ↑castSucc i < p\n⊢ p ≤ ↑castSucc i",
"tactic": "rw [succAbove_below _ _ H] at h"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.inl\nn m : ℕ\np : Fin (n + 1)\ni : Fin n\nh : p < ↑castSucc i\nH : ↑castSucc i < p\n⊢ p ≤ ↑castSucc i",
"tactic": "exact le_of_lt h"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.inr\nn m : ℕ\np : Fin (n + 1)\ni : Fin n\nh : p < ↑(succAbove p) i\nH : p ≤ ↑castSucc i\n⊢ p ≤ ↑castSucc i",
"tactic": "exact H"
},
{
"state_after": "case refine'_2\nn m : ℕ\np : Fin (n + 1)\ni : Fin n\nh : p ≤ ↑castSucc i\n⊢ p < ↑(succAbove p) i",
"state_before": "case refine'_2\nn m : ℕ\np : Fin (n + 1)\ni : Fin n\n⊢ p ≤ ↑castSucc i → p < ↑(succAbove p) i",
"tactic": "intro h"
},
{
"state_after": "case refine'_2\nn m : ℕ\np : Fin (n + 1)\ni : Fin n\nh : p ≤ ↑castSucc i\n⊢ p < succ i",
"state_before": "case refine'_2\nn m : ℕ\np : Fin (n + 1)\ni : Fin n\nh : p ≤ ↑castSucc i\n⊢ p < ↑(succAbove p) i",
"tactic": "rw [succAbove_above _ _ h]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nn m : ℕ\np : Fin (n + 1)\ni : Fin n\nh : p ≤ ↑castSucc i\n⊢ p < succ i",
"tactic": "exact lt_of_le_of_lt h (castSucc_lt_succ i)"
}
] |
[
2112,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2103,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Inverse.lean
|
Real.arcsin_lt_pi_div_two
|
[
{
"state_after": "no goals",
"state_before": "x : ℝ\n⊢ x < sin (π / 2) ↔ x < 1",
"tactic": "rw [sin_pi_div_two]"
}
] |
[
234,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
232,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
Finset.measure_zero
|
[] |
[
3320,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3318,
1
] |
Mathlib/Algebra/Module/LinearMap.lean
|
LinearMap.coe_comp
|
[] |
[
549,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
548,
1
] |
Mathlib/Algebra/Order/Group/Defs.lean
|
div_le_div_left'
|
[] |
[
787,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
786,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
|
Real.one_lt_rpow_iff_of_pos
|
[
{
"state_after": "no goals",
"state_before": "x y z : ℝ\nhx : 0 < x\n⊢ 1 < x ^ y ↔ 1 < x ∧ 0 < y ∨ x < 1 ∧ y < 0",
"tactic": "rw [rpow_def_of_pos hx, one_lt_exp_iff, mul_pos_iff, log_pos_iff hx, log_neg_iff hx]"
}
] |
[
568,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
567,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
inv_lt_inv
|
[] |
[
278,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
277,
1
] |
Mathlib/Algebra/Periodic.lean
|
Function.Periodic.sub_int_mul_eq
|
[] |
[
249,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
248,
1
] |
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