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Mathlib/Analysis/Seminorm.lean | Seminorm.smul_ball_preimage | [
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Mathlib/Analysis/NormedSpace/Multilinear.lean | ContinuousMultilinearMap.op_norm_le_bound | [] | [
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Mathlib/Data/MvPolynomial/Comap.lean | MvPolynomial.comap_id | [
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Mathlib/Algebra/Order/Interval.lean | Interval.length_add_le | [
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Mathlib/Algebra/Hom/Equiv/Basic.lean | MulEquiv.symm_comp_self | [] | [
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Mathlib/Data/Matrix/Block.lean | Matrix.blockDiagonal'_map | [
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"tactic": "rw [apply_dite f, hf]"
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Mathlib/Data/Polynomial/Monic.lean | Polynomial.Monic.isRegular | [
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Mathlib/RingTheory/Subring/Basic.lean | SubringClass.coe_intCast | [] | [
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Mathlib/Analysis/Convex/Side.lean | AffineSubspace.setOf_wSameSide_eq_image2 | [
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"tactic": "rw [wSameSide_iff_exists_left hp, or_iff_right hx]"
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Mathlib/Analysis/SpecificLimits/Basic.lean | geom_le | [
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138,
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Mathlib/AlgebraicTopology/SimplexCategory.lean | SimplexCategory.δ_comp_σ_succ | [
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{
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"state_before": "case a.h.h.h.mk.mk\nn i : ℕ\nisLt✝¹ : i < n + 1\nj : ℕ\nisLt✝ : j < len [n] + 1\n⊢ ↑(if h :\n { val := i, isLt := (_ : i < Nat.succ (n + 1)) } <\n if\n { val := j, isLt := (_ : j < Nat.succ (n + 1)) } <\n { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ (n + 1)) } then\n { val := j, isLt := (_ : j < Nat.succ (n + 1)) }\n else { val := j + 1, isLt := (_ : Nat.succ j < Nat.succ (n + 1)) } then\n Fin.pred\n (if\n { val := j, isLt := (_ : j < Nat.succ (n + 1)) } <\n { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ (n + 1)) } then\n { val := j, isLt := (_ : j < Nat.succ (n + 1)) }\n else { val := j + 1, isLt := (_ : Nat.succ j < Nat.succ (n + 1)) })\n (_ :\n (if\n { val := j, isLt := (_ : j < Nat.succ (n + 1)) } <\n { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ (n + 1)) } then\n { val := j, isLt := (_ : j < Nat.succ (n + 1)) }\n else { val := j + 1, isLt := (_ : Nat.succ j < Nat.succ (n + 1)) }) ≠\n 0)\n else\n Fin.castLT\n (if\n { val := j, isLt := (_ : j < Nat.succ (n + 1)) } <\n { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ (n + 1)) } then\n { val := j, isLt := (_ : j < Nat.succ (n + 1)) }\n else { val := j + 1, isLt := (_ : Nat.succ j < Nat.succ (n + 1)) })\n (_ :\n ↑(if\n { val := j, isLt := (_ : j < Nat.succ (n + 1)) } <\n { val := i + 1, isLt := (_ : Nat.succ i < Nat.succ (n + 1)) } then\n { val := j, isLt := (_ : j < Nat.succ (n + 1)) }\n else { val := j + 1, isLt := (_ : Nat.succ j < Nat.succ (n + 1)) }) <\n n + 1)) =\n j",
"tactic": "simp only [Fin.mk_lt_mk]"
},
{
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"state_before": "case a.h.h.h.mk.mk\nn i : ℕ\nisLt✝¹ : i < n + 1\nj : ℕ\nisLt✝ : j < len [n] + 1\n⊢ ↑(if h :\n { val := i, isLt := (_ : i < Nat.succ (n + 1)) } <\n if j < i + 1 then { val := j, isLt := (_ : j < Nat.succ (n + 1)) }\n else { val := j + 1, isLt := (_ : Nat.succ j < Nat.succ (n + 1)) } then\n Fin.pred\n (if j < i + 1 then { val := j, isLt := (_ : j < Nat.succ (n + 1)) }\n else { val := j + 1, isLt := (_ : Nat.succ j < Nat.succ (n + 1)) })\n (_ :\n (if j < i + 1 then { val := j, isLt := (_ : j < Nat.succ (n + 1)) }\n else { val := j + 1, isLt := (_ : Nat.succ j < Nat.succ (n + 1)) }) ≠\n 0)\n else\n Fin.castLT\n (if j < i + 1 then { val := j, isLt := (_ : j < Nat.succ (n + 1)) }\n else { val := j + 1, isLt := (_ : Nat.succ j < Nat.succ (n + 1)) })\n (_ :\n ↑(if j < i + 1 then { val := j, isLt := (_ : j < Nat.succ (n + 1)) }\n else { val := j + 1, isLt := (_ : Nat.succ j < Nat.succ (n + 1)) }) <\n n + 1)) =\n j",
"tactic": "split_ifs <;> simp <;> simp at * <;> linarith"
}
] | [
301,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
295,
1
] |
Mathlib/RingTheory/Localization/Ideal.lean | IsLocalization.comap_map_of_isPrime_disjoint | [
{
"state_after": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nhI : Ideal.IsPrime I\nhM : Disjoint ↑M ↑I\n⊢ Ideal.comap (algebraMap R S) (Ideal.map (algebraMap R S) I) ≤ I",
"state_before": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nhI : Ideal.IsPrime I\nhM : Disjoint ↑M ↑I\n⊢ Ideal.comap (algebraMap R S) (Ideal.map (algebraMap R S) I) = I",
"tactic": "refine' le_antisymm _ Ideal.le_comap_map"
},
{
"state_after": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nhI : Ideal.IsPrime I\nhM : Disjoint ↑M ↑I\na : R\nha : a ∈ Ideal.comap (algebraMap R S) (Ideal.map (algebraMap R S) I)\n⊢ a ∈ I",
"state_before": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nhI : Ideal.IsPrime I\nhM : Disjoint ↑M ↑I\n⊢ Ideal.comap (algebraMap R S) (Ideal.map (algebraMap R S) I) ≤ I",
"tactic": "refine' (fun a ha => _)"
},
{
"state_after": "case intro.mk\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nhI : Ideal.IsPrime I\nhM : Disjoint ↑M ↑I\na : R\nha : a ∈ Ideal.comap (algebraMap R S) (Ideal.map (algebraMap R S) I)\nb : { x // x ∈ I }\ns : { x // x ∈ M }\nh : ↑(algebraMap R S) a * ↑(algebraMap R S) ↑(b, s).snd = ↑(algebraMap R S) ↑(b, s).fst\n⊢ a ∈ I",
"state_before": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nhI : Ideal.IsPrime I\nhM : Disjoint ↑M ↑I\na : R\nha : a ∈ Ideal.comap (algebraMap R S) (Ideal.map (algebraMap R S) I)\n⊢ a ∈ I",
"tactic": "obtain ⟨⟨b, s⟩, h⟩ := (mem_map_algebraMap_iff M S).1 (Ideal.mem_comap.1 ha)"
},
{
"state_after": "case intro.mk\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nhI : Ideal.IsPrime I\nhM : Disjoint ↑M ↑I\na : R\nha : a ∈ Ideal.comap (algebraMap R S) (Ideal.map (algebraMap R S) I)\nb : { x // x ∈ I }\ns : { x // x ∈ M }\nh : ↑(algebraMap R S) (↑s * a) = ↑(algebraMap R S) ↑b\n⊢ a ∈ I",
"state_before": "case intro.mk\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nhI : Ideal.IsPrime I\nhM : Disjoint ↑M ↑I\na : R\nha : a ∈ Ideal.comap (algebraMap R S) (Ideal.map (algebraMap R S) I)\nb : { x // x ∈ I }\ns : { x // x ∈ M }\nh : ↑(algebraMap R S) a * ↑(algebraMap R S) ↑(b, s).snd = ↑(algebraMap R S) ↑(b, s).fst\n⊢ a ∈ I",
"tactic": "replace h : algebraMap R S (s * a) = algebraMap R S b := by\n simpa only [← map_mul, mul_comm] using h"
},
{
"state_after": "case intro.mk.intro\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nhI : Ideal.IsPrime I\nhM : Disjoint ↑M ↑I\na : R\nha : a ∈ Ideal.comap (algebraMap R S) (Ideal.map (algebraMap R S) I)\nb : { x // x ∈ I }\ns : { x // x ∈ M }\nh : ↑(algebraMap R S) (↑s * a) = ↑(algebraMap R S) ↑b\nc : { x // x ∈ M }\nhc : ↑c * (↑s * a) = ↑c * ↑b\n⊢ a ∈ I",
"state_before": "case intro.mk\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nhI : Ideal.IsPrime I\nhM : Disjoint ↑M ↑I\na : R\nha : a ∈ Ideal.comap (algebraMap R S) (Ideal.map (algebraMap R S) I)\nb : { x // x ∈ I }\ns : { x // x ∈ M }\nh : ↑(algebraMap R S) (↑s * a) = ↑(algebraMap R S) ↑b\n⊢ a ∈ I",
"tactic": "obtain ⟨c, hc⟩ := (eq_iff_exists M S).1 h"
},
{
"state_after": "case intro.mk.intro\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nhI : Ideal.IsPrime I\nhM : Disjoint ↑M ↑I\na : R\nha : a ∈ Ideal.comap (algebraMap R S) (Ideal.map (algebraMap R S) I)\nb : { x // x ∈ I }\ns : { x // x ∈ M }\nh : ↑(algebraMap R S) (↑s * a) = ↑(algebraMap R S) ↑b\nc : { x // x ∈ M }\nhc : ↑c * (↑s * a) = ↑c * ↑b\nthis : ↑c * ↑s * a ∈ I\n⊢ a ∈ I",
"state_before": "case intro.mk.intro\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nhI : Ideal.IsPrime I\nhM : Disjoint ↑M ↑I\na : R\nha : a ∈ Ideal.comap (algebraMap R S) (Ideal.map (algebraMap R S) I)\nb : { x // x ∈ I }\ns : { x // x ∈ M }\nh : ↑(algebraMap R S) (↑s * a) = ↑(algebraMap R S) ↑b\nc : { x // x ∈ M }\nhc : ↑c * (↑s * a) = ↑c * ↑b\n⊢ a ∈ I",
"tactic": "have : ↑c * ↑s * a ∈ I := by\n rw [mul_assoc, hc]\n exact I.mul_mem_left c b.2"
},
{
"state_after": "no goals",
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"tactic": "exact (hI.mem_or_mem this).resolve_left fun hsc => hM.le_bot ⟨(c * s).2, hsc⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nhI : Ideal.IsPrime I\nhM : Disjoint ↑M ↑I\na : R\nha : a ∈ Ideal.comap (algebraMap R S) (Ideal.map (algebraMap R S) I)\nb : { x // x ∈ I }\ns : { x // x ∈ M }\nh : ↑(algebraMap R S) a * ↑(algebraMap R S) ↑(b, s).snd = ↑(algebraMap R S) ↑(b, s).fst\n⊢ ↑(algebraMap R S) (↑s * a) = ↑(algebraMap R S) ↑b",
"tactic": "simpa only [← map_mul, mul_comm] using h"
},
{
"state_after": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nhI : Ideal.IsPrime I\nhM : Disjoint ↑M ↑I\na : R\nha : a ∈ Ideal.comap (algebraMap R S) (Ideal.map (algebraMap R S) I)\nb : { x // x ∈ I }\ns : { x // x ∈ M }\nh : ↑(algebraMap R S) (↑s * a) = ↑(algebraMap R S) ↑b\nc : { x // x ∈ M }\nhc : ↑c * (↑s * a) = ↑c * ↑b\n⊢ ↑c * ↑b ∈ I",
"state_before": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nhI : Ideal.IsPrime I\nhM : Disjoint ↑M ↑I\na : R\nha : a ∈ Ideal.comap (algebraMap R S) (Ideal.map (algebraMap R S) I)\nb : { x // x ∈ I }\ns : { x // x ∈ M }\nh : ↑(algebraMap R S) (↑s * a) = ↑(algebraMap R S) ↑b\nc : { x // x ∈ M }\nhc : ↑c * (↑s * a) = ↑c * ↑b\n⊢ ↑c * ↑s * a ∈ I",
"tactic": "rw [mul_assoc, hc]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nhI : Ideal.IsPrime I\nhM : Disjoint ↑M ↑I\na : R\nha : a ∈ Ideal.comap (algebraMap R S) (Ideal.map (algebraMap R S) I)\nb : { x // x ∈ I }\ns : { x // x ∈ M }\nh : ↑(algebraMap R S) (↑s * a) = ↑(algebraMap R S) ↑b\nc : { x // x ∈ M }\nhc : ↑c * (↑s * a) = ↑c * ↑b\n⊢ ↑c * ↑b ∈ I",
"tactic": "exact I.mul_mem_left c b.2"
}
] | [
92,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
81,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean | Orthonormal.orthogonalFamily | [
{
"state_after": "no goals",
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"tactic": "simp [inner_smul_left, inner_smul_right, hv.2 hij]"
}
] | [
2002,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2000,
1
] |
Mathlib/MeasureTheory/Integral/IntegrableOn.lean | MeasureTheory.IntegrableAtFilter.inf_ae_iff | [
{
"state_after": "α : Type u_1\nβ : Type ?u.1660656\nE : Type u_2\nF : Type ?u.1660662\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl✝ l' l : Filter α\n⊢ IntegrableAtFilter f (l ⊓ Measure.ae μ) → IntegrableAtFilter f l",
"state_before": "α : Type u_1\nβ : Type ?u.1660656\nE : Type u_2\nF : Type ?u.1660662\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nl✝ l' l : Filter α\n⊢ IntegrableAtFilter f (l ⊓ Measure.ae μ) ↔ IntegrableAtFilter f l",
"tactic": "refine' ⟨_, fun h => h.filter_mono inf_le_left⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.1660656\nE : Type u_2\nF : Type ?u.1660662\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf g : α → E\ns t✝ : Set α\nμ ν : Measure α\nl✝ l' l : Filter α\nt : Set α\nht : t ∈ l\nu : Set α\nhu : u ∈ Measure.ae μ\nhf : IntegrableOn f (t ∩ u)\n⊢ IntegrableAtFilter f l",
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"tactic": "rintro ⟨s, ⟨t, ht, u, hu, rfl⟩, hf⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.1660656\nE : Type u_2\nF : Type ?u.1660662\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf g : α → E\ns t✝ : Set α\nμ ν : Measure α\nl✝ l' l : Filter α\nt : Set α\nht : t ∈ l\nu : Set α\nhu : u ∈ Measure.ae μ\nhf : IntegrableOn f (t ∩ u)\n⊢ IntegrableOn f t",
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"tactic": "refine' ⟨t, ht, _⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.1660656\nE : Type u_2\nF : Type ?u.1660662\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf g : α → E\ns t✝ : Set α\nμ ν : Measure α\nl✝ l' l : Filter α\nt : Set α\nht : t ∈ l\nu : Set α\nhu : u ∈ Measure.ae μ\nhf : IntegrableOn f (t ∩ u)\nv : Set α\nhv : MeasurableSet v\n⊢ ↑↑(Measure.restrict μ t) v ≤ ↑↑(Measure.restrict μ (t ∩ u)) v",
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"tactic": "refine' hf.integrable.mono_measure fun v hv => _"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.1660656\nE : Type u_2\nF : Type ?u.1660662\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf g : α → E\ns t✝ : Set α\nμ ν : Measure α\nl✝ l' l : Filter α\nt : Set α\nht : t ∈ l\nu : Set α\nhu : u ∈ Measure.ae μ\nhf : IntegrableOn f (t ∩ u)\nv : Set α\nhv : MeasurableSet v\n⊢ ↑↑μ (v ∩ t) ≤ ↑↑μ (v ∩ (t ∩ u))",
"state_before": "case intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.1660656\nE : Type u_2\nF : Type ?u.1660662\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf g : α → E\ns t✝ : Set α\nμ ν : Measure α\nl✝ l' l : Filter α\nt : Set α\nht : t ∈ l\nu : Set α\nhu : u ∈ Measure.ae μ\nhf : IntegrableOn f (t ∩ u)\nv : Set α\nhv : MeasurableSet v\n⊢ ↑↑(Measure.restrict μ t) v ≤ ↑↑(Measure.restrict μ (t ∩ u)) v",
"tactic": "simp only [Measure.restrict_apply hv]"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.1660656\nE : Type u_2\nF : Type ?u.1660662\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf g : α → E\ns t✝ : Set α\nμ ν : Measure α\nl✝ l' l : Filter α\nt : Set α\nht : t ∈ l\nu : Set α\nhu : u ∈ Measure.ae μ\nhf : IntegrableOn f (t ∩ u)\nv : Set α\nhv : MeasurableSet v\nx : α\nhx : x ∈ u\n⊢ x ∈ {x | (fun x => (v ∩ t) x ≤ (v ∩ (t ∩ u)) x) x}",
"state_before": "case intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.1660656\nE : Type u_2\nF : Type ?u.1660662\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf g : α → E\ns t✝ : Set α\nμ ν : Measure α\nl✝ l' l : Filter α\nt : Set α\nht : t ∈ l\nu : Set α\nhu : u ∈ Measure.ae μ\nhf : IntegrableOn f (t ∩ u)\nv : Set α\nhv : MeasurableSet v\n⊢ ↑↑μ (v ∩ t) ≤ ↑↑μ (v ∩ (t ∩ u))",
"tactic": "refine' measure_mono_ae (mem_of_superset hu fun x hx => _)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.1660656\nE : Type u_2\nF : Type ?u.1660662\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf g : α → E\ns t✝ : Set α\nμ ν : Measure α\nl✝ l' l : Filter α\nt : Set α\nht : t ∈ l\nu : Set α\nhu : u ∈ Measure.ae μ\nhf : IntegrableOn f (t ∩ u)\nv : Set α\nhv : MeasurableSet v\nx : α\nhx : x ∈ u\n⊢ x ∈ {x | (fun x => (v ∩ t) x ≤ (v ∩ (t ∩ u)) x) x}",
"tactic": "exact fun ⟨hv, ht⟩ => ⟨hv, ⟨ht, hx⟩⟩"
}
] | [
431,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
423,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean | Equiv.Perm.cycleOf_self_apply_pow | [] | [
1064,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1063,
1
] |
Mathlib/Algebra/Group/Units.lean | Units.eq_inv_of_mul_eq_one_right | [] | [
373,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
372,
11
] |
Mathlib/Topology/MetricSpace/Contracting.lean | ContractingWith.efixedPoint_isFixedPt | [] | [
130,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
128,
1
] |
Mathlib/Data/Set/Lattice.lean | Set.sInter_eq_compl_sUnion_compl | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.147900\nγ : Type ?u.147903\nι : Sort ?u.147906\nι' : Sort ?u.147909\nι₂ : Sort ?u.147912\nκ : ι → Sort ?u.147917\nκ₁ : ι → Sort ?u.147922\nκ₂ : ι → Sort ?u.147927\nκ' : ι' → Sort ?u.147932\nS : Set (Set α)\n⊢ ⋂₀ S = (⋃₀ (compl '' S))ᶜ",
"tactic": "rw [← compl_compl (⋂₀ S), compl_sInter]"
}
] | [
1254,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1253,
1
] |
Mathlib/RingTheory/Valuation/ValuationRing.lean | ValuationRing.iff_dvd_total | [
{
"state_after": "case refine_1\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type ?u.522037\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nH : ValuationRing R\na b : R\n⊢ a ∣ b ∨ b ∣ a\n\ncase refine_2\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type ?u.522037\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nH : IsTotal R fun x x_1 => x ∣ x_1\na b : R\n⊢ ∃ c, a * c = b ∨ b * c = a",
"state_before": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type ?u.522037\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\n⊢ ValuationRing R ↔ IsTotal R fun x x_1 => x ∣ x_1",
"tactic": "refine ⟨fun H => ⟨fun a b => ?_⟩, fun H => ⟨fun a b => ?_⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case refine_1\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type ?u.522037\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nH : ValuationRing R\na b : R\n⊢ a ∣ b ∨ b ∣ a",
"tactic": "obtain ⟨c, rfl | rfl⟩ := ValuationRing.cond a b <;> simp"
},
{
"state_after": "no goals",
"state_before": "case refine_2\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDomain R\nK : Type ?u.522037\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nH : IsTotal R fun x x_1 => x ∣ x_1\na b : R\n⊢ ∃ c, a * c = b ∨ b * c = a",
"tactic": "obtain ⟨c, rfl⟩ | ⟨c, rfl⟩ := @IsTotal.total _ _ H a b <;> use c <;> simp"
}
] | [
291,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
287,
1
] |
Mathlib/Data/List/Basic.lean | List.nthLe_succ_scanl | [
{
"state_after": "case zero.nil\nι : Type ?u.243601\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : β → α → β\na : α\nb : β\nh : zero + 1 < length (scanl f b [])\n⊢ nthLe (scanl f b []) (zero + 1) h =\n f (nthLe (scanl f b []) zero (_ : zero < length (scanl f b []))) (nthLe [] zero (_ : zero < length []))\n\ncase zero.cons\nι : Type ?u.243601\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : β → α → β\na : α\nb : β\nhead✝ : α\ntail✝ : List α\nh : zero + 1 < length (scanl f b (head✝ :: tail✝))\n⊢ nthLe (scanl f b (head✝ :: tail✝)) (zero + 1) h =\n f (nthLe (scanl f b (head✝ :: tail✝)) zero (_ : zero < length (scanl f b (head✝ :: tail✝))))\n (nthLe (head✝ :: tail✝) zero (_ : zero < length (head✝ :: tail✝)))",
"state_before": "case zero\nι : Type ?u.243601\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : β → α → β\na : α\nb : β\nl : List α\nh : zero + 1 < length (scanl f b l)\n⊢ nthLe (scanl f b l) (zero + 1) h =\n f (nthLe (scanl f b l) zero (_ : zero < length (scanl f b l))) (nthLe l zero (_ : zero < length l))",
"tactic": "cases l"
},
{
"state_after": "no goals",
"state_before": "case zero.nil\nι : Type ?u.243601\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : β → α → β\na : α\nb : β\nh : zero + 1 < length (scanl f b [])\n⊢ nthLe (scanl f b []) (zero + 1) h =\n f (nthLe (scanl f b []) zero (_ : zero < length (scanl f b []))) (nthLe [] zero (_ : zero < length []))",
"tactic": "simp only [length, zero_add, scanl_nil] at h"
},
{
"state_after": "no goals",
"state_before": "case zero.cons\nι : Type ?u.243601\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : β → α → β\na : α\nb : β\nhead✝ : α\ntail✝ : List α\nh : zero + 1 < length (scanl f b (head✝ :: tail✝))\n⊢ nthLe (scanl f b (head✝ :: tail✝)) (zero + 1) h =\n f (nthLe (scanl f b (head✝ :: tail✝)) zero (_ : zero < length (scanl f b (head✝ :: tail✝))))\n (nthLe (head✝ :: tail✝) zero (_ : zero < length (head✝ :: tail✝)))",
"tactic": "simp [scanl_cons, singleton_append, nthLe_zero_scanl, nthLe_cons]"
},
{
"state_after": "case succ.nil\nι : Type ?u.243601\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : β → α → β\na : α\ni : ℕ\nhi :\n ∀ {b : β} {l : List α} {h : i + 1 < length (scanl f b l)},\n nthLe (scanl f b l) (i + 1) h =\n f (nthLe (scanl f b l) i (_ : i < length (scanl f b l))) (nthLe l i (_ : i < length l))\nb : β\nh : succ i + 1 < length (scanl f b [])\n⊢ nthLe (scanl f b []) (succ i + 1) h =\n f (nthLe (scanl f b []) (succ i) (_ : succ i < length (scanl f b []))) (nthLe [] (succ i) (_ : succ i < length []))\n\ncase succ.cons\nι : Type ?u.243601\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : β → α → β\na : α\ni : ℕ\nhi :\n ∀ {b : β} {l : List α} {h : i + 1 < length (scanl f b l)},\n nthLe (scanl f b l) (i + 1) h =\n f (nthLe (scanl f b l) i (_ : i < length (scanl f b l))) (nthLe l i (_ : i < length l))\nb : β\nhead✝ : α\ntail✝ : List α\nh : succ i + 1 < length (scanl f b (head✝ :: tail✝))\n⊢ nthLe (scanl f b (head✝ :: tail✝)) (succ i + 1) h =\n f (nthLe (scanl f b (head✝ :: tail✝)) (succ i) (_ : succ i < length (scanl f b (head✝ :: tail✝))))\n (nthLe (head✝ :: tail✝) (succ i) (_ : succ i < length (head✝ :: tail✝)))",
"state_before": "case succ\nι : Type ?u.243601\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : β → α → β\na : α\ni : ℕ\nhi :\n ∀ {b : β} {l : List α} {h : i + 1 < length (scanl f b l)},\n nthLe (scanl f b l) (i + 1) h =\n f (nthLe (scanl f b l) i (_ : i < length (scanl f b l))) (nthLe l i (_ : i < length l))\nb : β\nl : List α\nh : succ i + 1 < length (scanl f b l)\n⊢ nthLe (scanl f b l) (succ i + 1) h =\n f (nthLe (scanl f b l) (succ i) (_ : succ i < length (scanl f b l))) (nthLe l (succ i) (_ : succ i < length l))",
"tactic": "cases l"
},
{
"state_after": "case succ.nil\nι : Type ?u.243601\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : β → α → β\na : α\ni : ℕ\nhi :\n ∀ {b : β} {l : List α} {h : i + 1 < length (scanl f b l)},\n nthLe (scanl f b l) (i + 1) h =\n f (nthLe (scanl f b l) i (_ : i < length (scanl f b l))) (nthLe l i (_ : i < length l))\nb : β\nh✝ : succ i + 1 < length (scanl f b [])\nh : succ i < 0\n⊢ nthLe (scanl f b []) (succ i + 1) h✝ =\n f (nthLe (scanl f b []) (succ i) (_ : succ i < length (scanl f b []))) (nthLe [] (succ i) (_ : succ i < length []))",
"state_before": "case succ.nil\nι : Type ?u.243601\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : β → α → β\na : α\ni : ℕ\nhi :\n ∀ {b : β} {l : List α} {h : i + 1 < length (scanl f b l)},\n nthLe (scanl f b l) (i + 1) h =\n f (nthLe (scanl f b l) i (_ : i < length (scanl f b l))) (nthLe l i (_ : i < length l))\nb : β\nh : succ i + 1 < length (scanl f b [])\n⊢ nthLe (scanl f b []) (succ i + 1) h =\n f (nthLe (scanl f b []) (succ i) (_ : succ i < length (scanl f b []))) (nthLe [] (succ i) (_ : succ i < length []))",
"tactic": "simp only [length, add_lt_iff_neg_right, scanl_nil] at h"
},
{
"state_after": "no goals",
"state_before": "case succ.nil\nι : Type ?u.243601\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : β → α → β\na : α\ni : ℕ\nhi :\n ∀ {b : β} {l : List α} {h : i + 1 < length (scanl f b l)},\n nthLe (scanl f b l) (i + 1) h =\n f (nthLe (scanl f b l) i (_ : i < length (scanl f b l))) (nthLe l i (_ : i < length l))\nb : β\nh✝ : succ i + 1 < length (scanl f b [])\nh : succ i < 0\n⊢ nthLe (scanl f b []) (succ i + 1) h✝ =\n f (nthLe (scanl f b []) (succ i) (_ : succ i < length (scanl f b []))) (nthLe [] (succ i) (_ : succ i < length []))",
"tactic": "exact absurd h (not_lt_of_lt Nat.succ_pos')"
},
{
"state_after": "case succ.cons\nι : Type ?u.243601\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : β → α → β\na : α\ni : ℕ\nhi :\n ∀ {b : β} {l : List α} {h : i + 1 < length (scanl f b l)},\n nthLe (scanl f b l) (i + 1) h =\n f (nthLe (scanl f b l) i (_ : i < length (scanl f b l))) (nthLe l i (_ : i < length l))\nb : β\nhead✝ : α\ntail✝ : List α\nh : succ i + 1 < length (scanl f b (head✝ :: tail✝))\n⊢ nthLe ([b] ++ scanl f (f b head✝) tail✝) (succ i + 1) (_ : succ i + 1 < length ([b] ++ scanl f (f b head✝) tail✝)) =\n f (nthLe ([b] ++ scanl f (f b head✝) tail✝) (succ i) (_ : succ i < length ([b] ++ scanl f (f b head✝) tail✝)))\n (nthLe (head✝ :: tail✝) (succ i) (_ : succ i < length (head✝ :: tail✝)))",
"state_before": "case succ.cons\nι : Type ?u.243601\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : β → α → β\na : α\ni : ℕ\nhi :\n ∀ {b : β} {l : List α} {h : i + 1 < length (scanl f b l)},\n nthLe (scanl f b l) (i + 1) h =\n f (nthLe (scanl f b l) i (_ : i < length (scanl f b l))) (nthLe l i (_ : i < length l))\nb : β\nhead✝ : α\ntail✝ : List α\nh : succ i + 1 < length (scanl f b (head✝ :: tail✝))\n⊢ nthLe (scanl f b (head✝ :: tail✝)) (succ i + 1) h =\n f (nthLe (scanl f b (head✝ :: tail✝)) (succ i) (_ : succ i < length (scanl f b (head✝ :: tail✝))))\n (nthLe (head✝ :: tail✝) (succ i) (_ : succ i < length (head✝ :: tail✝)))",
"tactic": "simp_rw [scanl_cons]"
},
{
"state_after": "case succ.cons\nι : Type ?u.243601\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : β → α → β\na : α\ni : ℕ\nhi :\n ∀ {b : β} {l : List α} {h : i + 1 < length (scanl f b l)},\n nthLe (scanl f b l) (i + 1) h =\n f (nthLe (scanl f b l) i (_ : i < length (scanl f b l))) (nthLe l i (_ : i < length l))\nb : β\nhead✝ : α\ntail✝ : List α\nh : succ i + 1 < length (scanl f b (head✝ :: tail✝))\n⊢ nthLe (scanl f (f b head✝) tail✝) (succ i + 1 - length [b])\n (_ : succ i + 1 - length [b] < length (scanl f (f b head✝) tail✝)) =\n f (nthLe ([b] ++ scanl f (f b head✝) tail✝) (succ i) (_ : succ i < length ([b] ++ scanl f (f b head✝) tail✝)))\n (nthLe (head✝ :: tail✝) (succ i) (_ : succ i < length (head✝ :: tail✝)))\n\ncase succ.cons.h₁\nι : Type ?u.243601\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : β → α → β\na : α\ni : ℕ\nhi :\n ∀ {b : β} {l : List α} {h : i + 1 < length (scanl f b l)},\n nthLe (scanl f b l) (i + 1) h =\n f (nthLe (scanl f b l) i (_ : i < length (scanl f b l))) (nthLe l i (_ : i < length l))\nb : β\nhead✝ : α\ntail✝ : List α\nh : succ i + 1 < length (scanl f b (head✝ :: tail✝))\n⊢ length [b] ≤ succ i + 1",
"state_before": "case succ.cons\nι : Type ?u.243601\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : β → α → β\na : α\ni : ℕ\nhi :\n ∀ {b : β} {l : List α} {h : i + 1 < length (scanl f b l)},\n nthLe (scanl f b l) (i + 1) h =\n f (nthLe (scanl f b l) i (_ : i < length (scanl f b l))) (nthLe l i (_ : i < length l))\nb : β\nhead✝ : α\ntail✝ : List α\nh : succ i + 1 < length (scanl f b (head✝ :: tail✝))\n⊢ nthLe ([b] ++ scanl f (f b head✝) tail✝) (succ i + 1) (_ : succ i + 1 < length ([b] ++ scanl f (f b head✝) tail✝)) =\n f (nthLe ([b] ++ scanl f (f b head✝) tail✝) (succ i) (_ : succ i < length ([b] ++ scanl f (f b head✝) tail✝)))\n (nthLe (head✝ :: tail✝) (succ i) (_ : succ i < length (head✝ :: tail✝)))",
"tactic": "rw [nthLe_append_right]"
},
{
"state_after": "case succ.cons\nι : Type ?u.243601\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : β → α → β\na : α\ni : ℕ\nhi :\n ∀ {b : β} {l : List α} {h : i + 1 < length (scanl f b l)},\n nthLe (scanl f b l) (i + 1) h =\n f (nthLe (scanl f b l) i (_ : i < length (scanl f b l))) (nthLe l i (_ : i < length l))\nb : β\nhead✝ : α\ntail✝ : List α\nh : succ i + 1 < length (scanl f b (head✝ :: tail✝))\n⊢ f (nthLe (scanl f (f b head✝) tail✝) i (_ : i < length (scanl f (f b head✝) tail✝)))\n (nthLe tail✝ i (_ : i < length tail✝)) =\n f (nthLe ([b] ++ scanl f (f b head✝) tail✝) (succ i) (_ : succ i < length ([b] ++ scanl f (f b head✝) tail✝)))\n (nthLe (head✝ :: tail✝) (succ i) (_ : succ i < length (head✝ :: tail✝)))",
"state_before": "case succ.cons\nι : Type ?u.243601\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : β → α → β\na : α\ni : ℕ\nhi :\n ∀ {b : β} {l : List α} {h : i + 1 < length (scanl f b l)},\n nthLe (scanl f b l) (i + 1) h =\n f (nthLe (scanl f b l) i (_ : i < length (scanl f b l))) (nthLe l i (_ : i < length l))\nb : β\nhead✝ : α\ntail✝ : List α\nh : succ i + 1 < length (scanl f b (head✝ :: tail✝))\n⊢ nthLe (scanl f (f b head✝) tail✝) (succ i + 1 - length [b])\n (_ : succ i + 1 - length [b] < length (scanl f (f b head✝) tail✝)) =\n f (nthLe ([b] ++ scanl f (f b head✝) tail✝) (succ i) (_ : succ i < length ([b] ++ scanl f (f b head✝) tail✝)))\n (nthLe (head✝ :: tail✝) (succ i) (_ : succ i < length (head✝ :: tail✝)))",
"tactic": "simp only [length, zero_add 1, succ_add_sub_one, hi]"
},
{
"state_after": "no goals",
"state_before": "case succ.cons\nι : Type ?u.243601\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : β → α → β\na : α\ni : ℕ\nhi :\n ∀ {b : β} {l : List α} {h : i + 1 < length (scanl f b l)},\n nthLe (scanl f b l) (i + 1) h =\n f (nthLe (scanl f b l) i (_ : i < length (scanl f b l))) (nthLe l i (_ : i < length l))\nb : β\nhead✝ : α\ntail✝ : List α\nh : succ i + 1 < length (scanl f b (head✝ :: tail✝))\n⊢ f (nthLe (scanl f (f b head✝) tail✝) i (_ : i < length (scanl f (f b head✝) tail✝)))\n (nthLe tail✝ i (_ : i < length tail✝)) =\n f (nthLe ([b] ++ scanl f (f b head✝) tail✝) (succ i) (_ : succ i < length ([b] ++ scanl f (f b head✝) tail✝)))\n (nthLe (head✝ :: tail✝) (succ i) (_ : succ i < length (head✝ :: tail✝)))",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case succ.cons.h₁\nι : Type ?u.243601\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\nf : β → α → β\na : α\ni : ℕ\nhi :\n ∀ {b : β} {l : List α} {h : i + 1 < length (scanl f b l)},\n nthLe (scanl f b l) (i + 1) h =\n f (nthLe (scanl f b l) i (_ : i < length (scanl f b l))) (nthLe l i (_ : i < length l))\nb : β\nhead✝ : α\ntail✝ : List α\nh : succ i + 1 < length (scanl f b (head✝ :: tail✝))\n⊢ length [b] ≤ succ i + 1",
"tactic": "simp only [length, Nat.zero_le, le_add_iff_nonneg_left]"
}
] | [
2663,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2647,
1
] |
Mathlib/Topology/PathConnected.lean | Path.extend_range | [] | [
286,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
284,
1
] |
Mathlib/Algebra/Order/CompleteField.lean | LinearOrderedField.inducedMap_inducedMap | [
{
"state_after": "no goals",
"state_before": "F : Type ?u.23494\nα : Type u_3\nβ : Type u_2\nγ : Type u_1\ninst✝³ : LinearOrderedField α\ninst✝² : ConditionallyCompleteLinearOrderedField β\ninst✝¹ : ConditionallyCompleteLinearOrderedField γ\ninst✝ : Archimedean α\na✝ : α\nb : β\nq✝ : ℚ\na : α\nq : ℚ\n⊢ ↑q < inducedMap β γ (inducedMap α β a) ↔ ↑q < inducedMap α γ a",
"tactic": "rw [coe_lt_inducedMap_iff, coe_lt_inducedMap_iff, Iff.comm, coe_lt_inducedMap_iff]"
}
] | [
242,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
240,
1
] |
Mathlib/Order/Atoms.lean | GaloisInsertion.isAtom_of_u_bot | [] | [
720,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
717,
1
] |
Mathlib/Algebra/Star/Basic.lean | star_inv | [] | [
217,
98
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
216,
1
] |
Mathlib/Algebra/Order/Hom/Basic.lean | map_ne_zero_iff_ne_one | [] | [
302,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
301,
1
] |
Mathlib/Data/Set/Pointwise/Basic.lean | Set.singleton_mul_singleton | [] | [
412,
19
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
411,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean | arity_zero | [] | [
79,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
78,
1
] |
Mathlib/ModelTheory/Syntax.lean | FirstOrder.Language.BoundedFormula.mapTermRel_id_id_id | [
{
"state_after": "case falsum\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.67505\nP : Type ?u.67508\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.67536\nn✝¹ n n✝ : ℕ\n⊢ mapTermRel (fun x => id) (fun x => id) (fun x => id) falsum = falsum\n\ncase equal\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.67505\nP : Type ?u.67508\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.67536\nn✝¹ n n✝ : ℕ\nt₁✝ t₂✝ : Term L (α ⊕ Fin n✝)\n⊢ mapTermRel (fun x => id) (fun x => id) (fun x => id) (equal t₁✝ t₂✝) = equal t₁✝ t₂✝\n\ncase rel\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.67505\nP : Type ?u.67508\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.67536\nn✝¹ n n✝ l✝ : ℕ\nR✝ : Relations L l✝\nts✝ : Fin l✝ → Term L (α ⊕ Fin n✝)\n⊢ mapTermRel (fun x => id) (fun x => id) (fun x => id) (rel R✝ ts✝) = rel R✝ ts✝\n\ncase imp\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.67505\nP : Type ?u.67508\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.67536\nn✝¹ n n✝ : ℕ\nf₁✝ f₂✝ : BoundedFormula L α n✝\nih1 : mapTermRel (fun x => id) (fun x => id) (fun x => id) f₁✝ = f₁✝\nih2 : mapTermRel (fun x => id) (fun x => id) (fun x => id) f₂✝ = f₂✝\n⊢ mapTermRel (fun x => id) (fun x => id) (fun x => id) (imp f₁✝ f₂✝) = imp f₁✝ f₂✝\n\ncase all\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.67505\nP : Type ?u.67508\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.67536\nn✝¹ n n✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : mapTermRel (fun x => id) (fun x => id) (fun x => id) f✝ = f✝\n⊢ mapTermRel (fun x => id) (fun x => id) (fun x => id) (all f✝) = all f✝",
"state_before": "L : Language\nL' : Language\nM : Type w\nN : Type ?u.67505\nP : Type ?u.67508\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.67536\nn✝ n : ℕ\nφ : BoundedFormula L α n\n⊢ mapTermRel (fun x => id) (fun x => id) (fun x => id) φ = φ",
"tactic": "induction' φ with _ _ _ _ _ _ _ _ _ _ _ ih1 ih2 _ _ ih3"
},
{
"state_after": "no goals",
"state_before": "case falsum\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.67505\nP : Type ?u.67508\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.67536\nn✝¹ n n✝ : ℕ\n⊢ mapTermRel (fun x => id) (fun x => id) (fun x => id) falsum = falsum",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case equal\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.67505\nP : Type ?u.67508\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.67536\nn✝¹ n n✝ : ℕ\nt₁✝ t₂✝ : Term L (α ⊕ Fin n✝)\n⊢ mapTermRel (fun x => id) (fun x => id) (fun x => id) (equal t₁✝ t₂✝) = equal t₁✝ t₂✝",
"tactic": "simp [mapTermRel]"
},
{
"state_after": "no goals",
"state_before": "case rel\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.67505\nP : Type ?u.67508\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.67536\nn✝¹ n n✝ l✝ : ℕ\nR✝ : Relations L l✝\nts✝ : Fin l✝ → Term L (α ⊕ Fin n✝)\n⊢ mapTermRel (fun x => id) (fun x => id) (fun x => id) (rel R✝ ts✝) = rel R✝ ts✝",
"tactic": "simp [mapTermRel]"
},
{
"state_after": "no goals",
"state_before": "case imp\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.67505\nP : Type ?u.67508\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.67536\nn✝¹ n n✝ : ℕ\nf₁✝ f₂✝ : BoundedFormula L α n✝\nih1 : mapTermRel (fun x => id) (fun x => id) (fun x => id) f₁✝ = f₁✝\nih2 : mapTermRel (fun x => id) (fun x => id) (fun x => id) f₂✝ = f₂✝\n⊢ mapTermRel (fun x => id) (fun x => id) (fun x => id) (imp f₁✝ f₂✝) = imp f₁✝ f₂✝",
"tactic": "simp [mapTermRel, ih1, ih2]"
},
{
"state_after": "no goals",
"state_before": "case all\nL : Language\nL' : Language\nM : Type w\nN : Type ?u.67505\nP : Type ?u.67508\ninst✝² : Structure L M\ninst✝¹ : Structure L N\ninst✝ : Structure L P\nα : Type u'\nβ : Type v'\nγ : Type ?u.67536\nn✝¹ n n✝ : ℕ\nf✝ : BoundedFormula L α (n✝ + 1)\nih3 : mapTermRel (fun x => id) (fun x => id) (fun x => id) f✝ = f✝\n⊢ mapTermRel (fun x => id) (fun x => id) (fun x => id) (all f✝) = all f✝",
"tactic": "simp [mapTermRel, ih3]"
}
] | [
556,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
549,
1
] |
Mathlib/Computability/Language.lean | Language.one_add_kstar_mul_self_eq_kstar | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.123028\nγ : Type ?u.123031\nl✝ m : Language α\na b x : List α\nl : Language α\n⊢ 1 + l∗ * l = l∗",
"tactic": "rw [mul_self_kstar_comm, one_add_self_mul_kstar_eq_kstar]"
}
] | [
292,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
291,
1
] |
Mathlib/Dynamics/Ergodic/MeasurePreserving.lean | MeasureTheory.MeasurePreserving.quasiMeasurePreserving | [] | [
97,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
95,
11
] |
Mathlib/GroupTheory/Subsemigroup/Operations.lean | MulHom.srangeRestrict_surjective | [] | [
857,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
856,
1
] |
Mathlib/MeasureTheory/Integral/CircleIntegral.lean | circleIntegrable_zero_radius | [
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℂ → E\nc : ℂ\n⊢ CircleIntegrable f c 0",
"tactic": "simp [CircleIntegrable]"
}
] | [
273,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
272,
1
] |
Std/Data/RBMap/Lemmas.lean | Std.RBNode.lowerBound?_mem_lb | [] | [
248,
87
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
246,
1
] |
Mathlib/Analysis/SpecificLimits/Basic.lean | tendsto_inverse_atTop_nhds_0_nat | [] | [
35,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
34,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean | pos_and_pos_or_neg_and_neg_of_mul_pos | [
{
"state_after": "case inl\nα : Type u_1\na b c d : α\ninst✝³ : MulZeroClass α\ninst✝² : LinearOrder α\ninst✝¹ : PosMulMono α\ninst✝ : MulPosMono α\nhab : 0 < a * b\nha : a < 0\n⊢ 0 < a ∧ 0 < b ∨ a < 0 ∧ b < 0\n\ncase inr.inl\nα : Type u_1\nb c d : α\ninst✝³ : MulZeroClass α\ninst✝² : LinearOrder α\ninst✝¹ : PosMulMono α\ninst✝ : MulPosMono α\nhab : 0 < 0 * b\n⊢ 0 < 0 ∧ 0 < b ∨ 0 < 0 ∧ b < 0\n\ncase inr.inr\nα : Type u_1\na b c d : α\ninst✝³ : MulZeroClass α\ninst✝² : LinearOrder α\ninst✝¹ : PosMulMono α\ninst✝ : MulPosMono α\nhab : 0 < a * b\nha : 0 < a\n⊢ 0 < a ∧ 0 < b ∨ a < 0 ∧ b < 0",
"state_before": "α : Type u_1\na b c d : α\ninst✝³ : MulZeroClass α\ninst✝² : LinearOrder α\ninst✝¹ : PosMulMono α\ninst✝ : MulPosMono α\nhab : 0 < a * b\n⊢ 0 < a ∧ 0 < b ∨ a < 0 ∧ b < 0",
"tactic": "rcases lt_trichotomy a 0 with (ha | rfl | ha)"
},
{
"state_after": "case inl\nα : Type u_1\na b c d : α\ninst✝³ : MulZeroClass α\ninst✝² : LinearOrder α\ninst✝¹ : PosMulMono α\ninst✝ : MulPosMono α\nhab : 0 < a * b\nha : a < 0\nhb : 0 ≤ b\n⊢ a * b ≤ 0",
"state_before": "case inl\nα : Type u_1\na b c d : α\ninst✝³ : MulZeroClass α\ninst✝² : LinearOrder α\ninst✝¹ : PosMulMono α\ninst✝ : MulPosMono α\nhab : 0 < a * b\nha : a < 0\n⊢ 0 < a ∧ 0 < b ∨ a < 0 ∧ b < 0",
"tactic": "refine' Or.inr ⟨ha, lt_imp_lt_of_le_imp_le (fun hb => _) hab⟩"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\na b c d : α\ninst✝³ : MulZeroClass α\ninst✝² : LinearOrder α\ninst✝¹ : PosMulMono α\ninst✝ : MulPosMono α\nhab : 0 < a * b\nha : a < 0\nhb : 0 ≤ b\n⊢ a * b ≤ 0",
"tactic": "exact mul_nonpos_of_nonpos_of_nonneg ha.le hb"
},
{
"state_after": "case inr.inl\nα : Type u_1\nb c d : α\ninst✝³ : MulZeroClass α\ninst✝² : LinearOrder α\ninst✝¹ : PosMulMono α\ninst✝ : MulPosMono α\nhab : 0 < 0\n⊢ 0 < 0 ∧ 0 < b ∨ 0 < 0 ∧ b < 0",
"state_before": "case inr.inl\nα : Type u_1\nb c d : α\ninst✝³ : MulZeroClass α\ninst✝² : LinearOrder α\ninst✝¹ : PosMulMono α\ninst✝ : MulPosMono α\nhab : 0 < 0 * b\n⊢ 0 < 0 ∧ 0 < b ∨ 0 < 0 ∧ b < 0",
"tactic": "rw [zero_mul] at hab"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nα : Type u_1\nb c d : α\ninst✝³ : MulZeroClass α\ninst✝² : LinearOrder α\ninst✝¹ : PosMulMono α\ninst✝ : MulPosMono α\nhab : 0 < 0\n⊢ 0 < 0 ∧ 0 < b ∨ 0 < 0 ∧ b < 0",
"tactic": "exact hab.false.elim"
},
{
"state_after": "case inr.inr\nα : Type u_1\na b c d : α\ninst✝³ : MulZeroClass α\ninst✝² : LinearOrder α\ninst✝¹ : PosMulMono α\ninst✝ : MulPosMono α\nhab : 0 < a * b\nha : 0 < a\nhb : b ≤ 0\n⊢ a * b ≤ 0",
"state_before": "case inr.inr\nα : Type u_1\na b c d : α\ninst✝³ : MulZeroClass α\ninst✝² : LinearOrder α\ninst✝¹ : PosMulMono α\ninst✝ : MulPosMono α\nhab : 0 < a * b\nha : 0 < a\n⊢ 0 < a ∧ 0 < b ∨ a < 0 ∧ b < 0",
"tactic": "refine' Or.inl ⟨ha, lt_imp_lt_of_le_imp_le (fun hb => _) hab⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nα : Type u_1\na b c d : α\ninst✝³ : MulZeroClass α\ninst✝² : LinearOrder α\ninst✝¹ : PosMulMono α\ninst✝ : MulPosMono α\nhab : 0 < a * b\nha : 0 < a\nhb : b ≤ 0\n⊢ a * b ≤ 0",
"tactic": "exact mul_nonpos_of_nonneg_of_nonpos ha.le hb"
}
] | [
567,
50
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
559,
1
] |
Mathlib/Order/Cover.lean | Wcovby.sup_eq | [] | [
196,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
194,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean | Finset.map_add_left_Ioo | [
{
"state_after": "ι : Type ?u.226888\nα : Type u_1\ninst✝² : OrderedCancelAddCommMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ ↑(addLeftEmbedding c) '' Set.Ioo a b = Set.Ioo (c + a) (c + b)",
"state_before": "ι : Type ?u.226888\nα : Type u_1\ninst✝² : OrderedCancelAddCommMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ map (addLeftEmbedding c) (Ioo a b) = Ioo (c + a) (c + b)",
"tactic": "rw [← coe_inj, coe_map, coe_Ioo, coe_Ioo]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.226888\nα : Type u_1\ninst✝² : OrderedCancelAddCommMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ ↑(addLeftEmbedding c) '' Set.Ioo a b = Set.Ioo (c + a) (c + b)",
"tactic": "exact Set.image_const_add_Ioo _ _ _"
}
] | [
1086,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1083,
1
] |
Mathlib/Combinatorics/Composition.lean | CompositionAsSet.boundary_length | [
{
"state_after": "case h.e'_3\nn : ℕ\nc : CompositionAsSet n\n⊢ Fin.last n = Finset.max' c.boundaries (_ : Finset.Nonempty c.boundaries)",
"state_before": "n : ℕ\nc : CompositionAsSet n\n⊢ ↑(boundary c) { val := length c, isLt := (_ : length c < Finset.card c.boundaries) } = Fin.last n",
"tactic": "convert Finset.orderEmbOfFin_last rfl c.card_boundaries_pos"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nn : ℕ\nc : CompositionAsSet n\n⊢ Fin.last n = Finset.max' c.boundaries (_ : Finset.Nonempty c.boundaries)",
"tactic": "exact le_antisymm (Finset.le_max' _ _ c.getLast_mem) (Fin.le_last _)"
}
] | [
921,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
919,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean | MeasureTheory.OuterMeasure.exists_measurable_superset_of_trim_eq_zero | [
{
"state_after": "case intro.intro.intro\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nh : ↑(trim m) s = 0\nt : Set α\nhst : s ⊆ t\nht : MeasurableSet t\nhm : ↑m t = ↑(trim m) s\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = 0",
"state_before": "α : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nh : ↑(trim m) s = 0\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = 0",
"tactic": "rcases exists_measurable_superset_eq_trim m s with ⟨t, hst, ht, hm⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nα : Type u_1\ninst✝ : MeasurableSpace α\nm✝ m : OuterMeasure α\ns : Set α\nh : ↑(trim m) s = 0\nt : Set α\nhst : s ⊆ t\nht : MeasurableSet t\nhm : ↑m t = ↑(trim m) s\n⊢ ∃ t, s ⊆ t ∧ MeasurableSet t ∧ ↑m t = 0",
"tactic": "exact ⟨t, hst, ht, h ▸ hm⟩"
}
] | [
1721,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1718,
1
] |
Mathlib/FieldTheory/Adjoin.lean | IntermediateField.adjoin.finiteDimensional | [] | [
869,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
868,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean | Ordinal.le_of_mul_le_mul_left | [] | [
841,
68
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
840,
1
] |
Mathlib/Algebra/FreeMonoid/Basic.lean | FreeMonoid.toList_prod | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.8378\nγ : Type ?u.8381\nM : Type ?u.8384\ninst✝¹ : Monoid M\nN : Type ?u.8390\ninst✝ : Monoid N\nxs : List (FreeMonoid α)\n⊢ ↑toList (List.prod xs) = List.join (List.map (↑toList) xs)",
"tactic": "induction xs <;> simp [*, List.join]"
}
] | [
118,
39
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
117,
1
] |
Mathlib/Data/ZMod/Basic.lean | ZMod.nat_cast_zmod_surjective | [] | [
205,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
204,
1
] |
Mathlib/Order/WithBot.lean | WithTop.strictAnti_iff | [] | [
1146,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1144,
1
] |
Mathlib/Algebra/RingQuot.lean | RingCon.coe_algebraMap | [] | [
49,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
47,
1
] |
Mathlib/Algebra/Module/Submodule/Lattice.lean | Submodule.top_coe | [] | [
145,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
144,
1
] |
Mathlib/Data/Fintype/Fin.lean | Fin.card_filter_univ_succ' | [
{
"state_after": "α : Type ?u.9629\nβ : Type ?u.9632\nn : ℕ\np : Fin (n + 1) → Prop\ninst✝ : DecidablePred p\n⊢ Finset.card (if p 0 then {0} else ∅) +\n Finset.card (filter (p ∘ ↑{ toFun := succ, inj' := (_ : Function.Injective succ) }) univ) =\n (if p 0 then 1 else 0) + Finset.card (filter (p ∘ succ) univ)",
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"tactic": "rw [Fin.univ_succ, filter_cons, card_disjUnion, filter_map, card_map]"
},
{
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"tactic": "split_ifs <;> simp"
}
] | [
78,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
75,
1
] |
Mathlib/Data/Finset/Basic.lean | Finset.erase_insert_of_ne | [
{
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"tactic": "have : x ≠ b ∧ x = a ↔ x = a := and_iff_right_of_imp fun hx => hx.symm ▸ h"
},
{
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"state_before": "α : Type u_1\nβ : Type ?u.210334\nγ : Type ?u.210337\ninst✝ : DecidableEq α\ns✝ t u v : Finset α\na✝ b✝ a b : α\ns : Finset α\nh : a ≠ b\nx : α\nthis : x ≠ b ∧ x = a ↔ x = a\n⊢ x ∈ erase (insert a s) b ↔ x ∈ insert a (erase s b)",
"tactic": "simp only [mem_erase, mem_insert, and_or_left, this]"
}
] | [
1921,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1917,
1
] |
Mathlib/Data/Setoid/Partition.lean | Setoid.eqv_class_mem' | [
{
"state_after": "case h.e'_4.h.e'_2.h.a\nα : Type u_1\nc : Set (Set α)\nH : ∀ (a : α), ∃! b x, a ∈ b\nx x✝ : α\n⊢ Rel (mkClasses c H) x x✝ ↔ Rel (mkClasses c H) x✝ x",
"state_before": "α : Type u_1\nc : Set (Set α)\nH : ∀ (a : α), ∃! b x, a ∈ b\nx : α\n⊢ {y | Rel (mkClasses c H) x y} ∈ c",
"tactic": "convert @Setoid.eqv_class_mem _ _ H x using 3"
},
{
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"tactic": "rw [Setoid.comm']"
}
] | [
163,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
160,
1
] |
Mathlib/Data/Stream/Init.lean | Stream'.any_def | [] | [
117,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
116,
1
] |
Std/Classes/LawfulMonad.lean | SatisfiesM.imp | [
{
"state_after": "m : Type u_1 → Type u_2\nα : Type u_1\np q : α → Prop\ninst✝¹ : Functor m\ninst✝ : LawfulFunctor m\nx✝ : m α\nh✝ : SatisfiesM p x✝\nH : ∀ {a : α}, p a → q a\nx : m { a // p a }\nh : Subtype.val <$> x = x✝\n⊢ (Subtype.val ∘ fun x =>\n match x with\n | { val := a, property := h } => { val := a, property := (_ : q a) }) <$>\n x =\n Subtype.val <$> x",
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"tactic": "rw [← h, ← comp_map]"
},
{
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"state_before": "m : Type u_1 → Type u_2\nα : Type u_1\np q : α → Prop\ninst✝¹ : Functor m\ninst✝ : LawfulFunctor m\nx✝ : m α\nh✝ : SatisfiesM p x✝\nH : ∀ {a : α}, p a → q a\nx : m { a // p a }\nh : Subtype.val <$> x = x✝\n⊢ (Subtype.val ∘ fun x =>\n match x with\n | { val := a, property := h } => { val := a, property := (_ : q a) }) <$>\n x =\n Subtype.val <$> x",
"tactic": "rfl"
}
] | [
89,
82
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
87,
1
] |
Mathlib/Algebra/Star/StarAlgHom.lean | NonUnitalStarAlgHom.coe_mk | [] | [
174,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
172,
1
] |
Mathlib/Order/SymmDiff.lean | inf_sup_symmDiff | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.33973\nα : Type u_1\nβ : Type ?u.33979\nπ : ι → Type ?u.33984\ninst✝ : GeneralizedCoheytingAlgebra α\na b c d : α\n⊢ a ⊓ b ⊔ a ∆ b = a ⊔ b",
"tactic": "rw [sup_comm, symmDiff_sup_inf]"
}
] | [
204,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
204,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean | AEMeasurable.real_toNNReal | [] | [
1759,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1757,
1
] |
Mathlib/Topology/Sets/Compacts.lean | TopologicalSpace.NonemptyCompacts.coe_prod | [] | [
305,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
303,
1
] |
Mathlib/Data/Num/Lemmas.lean | ZNum.mul_comm | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.1029930\n⊢ ∀ (a b : ZNum), a * b = b * a",
"tactic": "transfer"
}
] | [
1491,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1491,
9
] |
Mathlib/MeasureTheory/Integral/SetToL1.lean | MeasureTheory.setToFun_congr_smul_measure | [
{
"state_after": "case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1677900\nG : Type ?u.1677903\n𝕜 : Type ?u.1677906\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nc : ℝ≥0∞\nhc_ne_top : c ≠ ⊤\nhT : DominatedFinMeasAdditive μ T C\nhT_smul : DominatedFinMeasAdditive (c • μ) T C'\nf : α → E\nhc0 : c = 0\n⊢ setToFun μ T hT f = setToFun (c • μ) T hT_smul f\n\ncase neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1677900\nG : Type ?u.1677903\n𝕜 : Type ?u.1677906\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nc : ℝ≥0∞\nhc_ne_top : c ≠ ⊤\nhT : DominatedFinMeasAdditive μ T C\nhT_smul : DominatedFinMeasAdditive (c • μ) T C'\nf : α → E\nhc0 : ¬c = 0\n⊢ setToFun μ T hT f = setToFun (c • μ) T hT_smul f",
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"tactic": "by_cases hc0 : c = 0"
},
{
"state_after": "case neg.refine'_1\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1677900\nG : Type ?u.1677903\n𝕜 : Type ?u.1677906\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nc : ℝ≥0∞\nhc_ne_top : c ≠ ⊤\nhT : DominatedFinMeasAdditive μ T C\nhT_smul : DominatedFinMeasAdditive (c • μ) T C'\nf : α → E\nhc0 : ¬c = 0\n⊢ c⁻¹ ≠ ⊤\n\ncase neg.refine'_2\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1677900\nG : Type ?u.1677903\n𝕜 : Type ?u.1677906\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nc : ℝ≥0∞\nhc_ne_top : c ≠ ⊤\nhT : DominatedFinMeasAdditive μ T C\nhT_smul : DominatedFinMeasAdditive (c • μ) T C'\nf : α → E\nhc0 : ¬c = 0\n⊢ μ = c⁻¹ • c • μ",
"state_before": "case neg\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1677900\nG : Type ?u.1677903\n𝕜 : Type ?u.1677906\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nc : ℝ≥0∞\nhc_ne_top : c ≠ ⊤\nhT : DominatedFinMeasAdditive μ T C\nhT_smul : DominatedFinMeasAdditive (c • μ) T C'\nf : α → E\nhc0 : ¬c = 0\n⊢ setToFun μ T hT f = setToFun (c • μ) T hT_smul f",
"tactic": "refine' setToFun_congr_measure c⁻¹ c _ hc_ne_top (le_of_eq _) le_rfl hT hT_smul f"
},
{
"state_after": "case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1677900\nG : Type ?u.1677903\n𝕜 : Type ?u.1677906\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nc : ℝ≥0∞\nhc_ne_top : c ≠ ⊤\nhT : DominatedFinMeasAdditive μ T C\nhT_smul✝ : DominatedFinMeasAdditive (c • μ) T C'\nf : α → E\nhc0 : c = 0\nhT_smul : DominatedFinMeasAdditive 0 T C'\n⊢ setToFun μ T hT f = setToFun (c • μ) T hT_smul✝ f",
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"tactic": "simp [hc0] at hT_smul"
},
{
"state_after": "case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1677900\nG : Type ?u.1677903\n𝕜 : Type ?u.1677906\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nc : ℝ≥0∞\nhc_ne_top : c ≠ ⊤\nhT : DominatedFinMeasAdditive μ T C\nhT_smul✝ : DominatedFinMeasAdditive (c • μ) T C'\nf : α → E\nhc0 : c = 0\nhT_smul : DominatedFinMeasAdditive 0 T C'\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → T s = 0\n⊢ setToFun μ T hT f = setToFun (c • μ) T hT_smul✝ f",
"state_before": "case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1677900\nG : Type ?u.1677903\n𝕜 : Type ?u.1677906\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nc : ℝ≥0∞\nhc_ne_top : c ≠ ⊤\nhT : DominatedFinMeasAdditive μ T C\nhT_smul✝ : DominatedFinMeasAdditive (c • μ) T C'\nf : α → E\nhc0 : c = 0\nhT_smul : DominatedFinMeasAdditive 0 T C'\n⊢ setToFun μ T hT f = setToFun (c • μ) T hT_smul✝ f",
"tactic": "have h : ∀ s, MeasurableSet s → μ s < ∞ → T s = 0 := fun s hs _ => hT_smul.eq_zero hs"
},
{
"state_after": "case pos.h\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1677900\nG : Type ?u.1677903\n𝕜 : Type ?u.1677906\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nc : ℝ≥0∞\nhc_ne_top : c ≠ ⊤\nhT : DominatedFinMeasAdditive μ T C\nhT_smul✝ : DominatedFinMeasAdditive (c • μ) T C'\nf : α → E\nhc0 : c = 0\nhT_smul : DominatedFinMeasAdditive 0 T C'\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → T s = 0\n⊢ c • μ = 0",
"state_before": "case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\nF' : Type ?u.1677900\nG : Type ?u.1677903\n𝕜 : Type ?u.1677906\np : ℝ≥0∞\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : NormedAddCommGroup G\nm : MeasurableSpace α\nμ : Measure α\ninst✝ : CompleteSpace F\nT T' T'' : Set α → E →L[ℝ] F\nC C' C'' : ℝ\nf✝ g : α → E\nc : ℝ≥0∞\nhc_ne_top : c ≠ ⊤\nhT : DominatedFinMeasAdditive μ T C\nhT_smul✝ : DominatedFinMeasAdditive (c • μ) T C'\nf : α → E\nhc0 : c = 0\nhT_smul : DominatedFinMeasAdditive 0 T C'\nh : ∀ (s : Set α), MeasurableSet s → ↑↑μ s < ⊤ → T s = 0\n⊢ setToFun μ T hT f = setToFun (c • μ) T hT_smul✝ f",
"tactic": "rw [setToFun_zero_left' _ h, setToFun_measure_zero]"
},
{
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Mathlib/Data/Finset/Powerset.lean | Finset.mem_powerset | [
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Std/Data/Nat/Gcd.lean | Nat.gcd_rec | [
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Mathlib/Topology/Instances/ENNReal.lean | continuous_edist | [
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Mathlib/Data/Part.lean | Part.get_eq_iff_mem | [] | [
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Mathlib/Data/Set/NAry.lean | Set.image2_image2_left | [
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Mathlib/Order/Minimal.lean | IsLeast.mem_minimals | [] | [
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Mathlib/Geometry/Manifold/LocalInvariantProperties.lean | StructureGroupoid.LocalInvariantProp.liftPropAt_congr_iff_of_eventuallyEq | [
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"tactic": "simp_rw [nhdsWithin_univ, h₁]"
}
] | [
431,
97
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
429,
1
] |
Mathlib/RingTheory/GradedAlgebra/HomogeneousIdeal.lean | Ideal.homogeneousHull_eq_sInf | [] | [
629,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
627,
1
] |
Mathlib/Topology/LocalHomeomorph.lean | LocalHomeomorph.trans_assoc | [] | [
855,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
853,
1
] |
Mathlib/Topology/Spectral/Hom.lean | SpectralMap.comp_id | [] | [
208,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
207,
1
] |
Mathlib/Data/Set/Image.lean | Set.image_eq_image | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.54851\nι : Sort ?u.54854\nι' : Sort ?u.54857\nf✝ : α → β\ns t : Set α\nf : α → β\nhf : Injective f\neq : f '' s = f '' t\n⊢ s = t",
"tactic": "rw [← preimage_image_eq s hf, ← preimage_image_eq t hf, eq]"
}
] | [
558,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
555,
1
] |
Mathlib/Algebra/CharP/Basic.lean | add_pow_char_of_commute | [
{
"state_after": "R : Type u_1\ninst✝¹ : Semiring R\np : ℕ\nhp : Fact (Nat.Prime p)\ninst✝ : CharP R p\nx y : R\nh : Commute x y\nr : R\nhr : (x + y) ^ p = x ^ p + y ^ p + ↑p * r\n⊢ (x + y) ^ p = x ^ p + y ^ p",
"state_before": "R : Type u_1\ninst✝¹ : Semiring R\np : ℕ\nhp : Fact (Nat.Prime p)\ninst✝ : CharP R p\nx y : R\nh : Commute x y\n⊢ (x + y) ^ p = x ^ p + y ^ p",
"tactic": "let ⟨r, hr⟩ := h.exists_add_pow_prime_eq hp.out"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : Semiring R\np : ℕ\nhp : Fact (Nat.Prime p)\ninst✝ : CharP R p\nx y : R\nh : Commute x y\nr : R\nhr : (x + y) ^ p = x ^ p + y ^ p + ↑p * r\n⊢ (x + y) ^ p = x ^ p + y ^ p",
"tactic": "simp [hr]"
}
] | [
251,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
248,
1
] |
Mathlib/Logic/Equiv/Fin.lean | finRotate_apply_zero | [
{
"state_after": "no goals",
"state_before": "m n : ℕ\n⊢ ↑(finRotate (Nat.succ n)) 0 = 1",
"tactic": "rw [finRotate_succ_apply, zero_add]"
}
] | [
447,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
446,
1
] |
Mathlib/MeasureTheory/Function/AEEqFun.lean | MeasureTheory.AEEqFun.inf_le_right | [
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.465891\nδ : Type ?u.465894\ninst✝⁵ : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace γ\ninst✝² : TopologicalSpace δ\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf g : α →ₘ[μ] β\n⊢ ↑(f ⊓ g) ≤ᵐ[μ] ↑g",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.465891\nδ : Type ?u.465894\ninst✝⁵ : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace γ\ninst✝² : TopologicalSpace δ\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf g : α →ₘ[μ] β\n⊢ f ⊓ g ≤ g",
"tactic": "rw [← coeFn_le]"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.465891\nδ : Type ?u.465894\ninst✝⁵ : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace γ\ninst✝² : TopologicalSpace δ\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf g : α →ₘ[μ] β\na✝ : α\nha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝\n⊢ ↑(f ⊓ g) a✝ ≤ ↑g a✝",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.465891\nδ : Type ?u.465894\ninst✝⁵ : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace γ\ninst✝² : TopologicalSpace δ\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf g : α →ₘ[μ] β\n⊢ ↑(f ⊓ g) ≤ᵐ[μ] ↑g",
"tactic": "filter_upwards [coeFn_inf f g] with _ ha"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.465891\nδ : Type ?u.465894\ninst✝⁵ : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace γ\ninst✝² : TopologicalSpace δ\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf g : α →ₘ[μ] β\na✝ : α\nha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝\n⊢ ↑f a✝ ⊓ ↑g a✝ ≤ ↑g a✝",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.465891\nδ : Type ?u.465894\ninst✝⁵ : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace γ\ninst✝² : TopologicalSpace δ\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf g : α →ₘ[μ] β\na✝ : α\nha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝\n⊢ ↑(f ⊓ g) a✝ ≤ ↑g a✝",
"tactic": "rw [ha]"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.465891\nδ : Type ?u.465894\ninst✝⁵ : MeasurableSpace α\nμ ν : Measure α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace γ\ninst✝² : TopologicalSpace δ\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf g : α →ₘ[μ] β\na✝ : α\nha : ↑(f ⊓ g) a✝ = ↑f a✝ ⊓ ↑g a✝\n⊢ ↑f a✝ ⊓ ↑g a✝ ≤ ↑g a✝",
"tactic": "exact inf_le_right"
}
] | [
515,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
511,
11
] |
Mathlib/GroupTheory/Perm/Fin.lean | support_finRotate_of_le | [
{
"state_after": "case intro\nm : ℕ\nh : 2 ≤ 2 + m\n⊢ support (finRotate (2 + m)) = Finset.univ",
"state_before": "n : ℕ\nh : 2 ≤ n\n⊢ support (finRotate n) = Finset.univ",
"tactic": "obtain ⟨m, rfl⟩ := exists_add_of_le h"
},
{
"state_after": "no goals",
"state_before": "case intro\nm : ℕ\nh : 2 ≤ 2 + m\n⊢ support (finRotate (2 + m)) = Finset.univ",
"tactic": "rw [add_comm, support_finRotate]"
}
] | [
122,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
120,
1
] |
Mathlib/Topology/Filter.lean | Continuous.nhds | [] | [
248,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
247,
18
] |
Mathlib/Topology/Inseparable.lean | specializes_iff_mem_closure | [] | [
140,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
139,
1
] |
Std/Data/Int/Lemmas.lean | Int.natAbs_mul_natAbs_eq | [
{
"state_after": "no goals",
"state_before": "a b : Int\nc : Nat\nh : a * b = ↑c\n⊢ natAbs a * natAbs b = c",
"tactic": "rw [← natAbs_mul, h, natAbs]"
}
] | [
176,
89
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
175,
1
] |
Mathlib/Logic/Equiv/List.lean | Denumerable.denumerable_list_aux | [
{
"state_after": "α : Type u_1\nβ : Type ?u.30505\ninst✝¹ : Denumerable α\ninst✝ : Denumerable β\n⊢ ∃ a, a ∈ some [] ∧ encodeList a = 0",
"state_before": "α : Type u_1\nβ : Type ?u.30505\ninst✝¹ : Denumerable α\ninst✝ : Denumerable β\n⊢ ∃ a, a ∈ decodeList 0 ∧ encodeList a = 0",
"tactic": "rw [decodeList]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.30505\ninst✝¹ : Denumerable α\ninst✝ : Denumerable β\n⊢ ∃ a, a ∈ some [] ∧ encodeList a = 0",
"tactic": "exact ⟨_, rfl, rfl⟩"
},
{
"state_after": "case mk\nα : Type u_1\nβ : Type ?u.30505\ninst✝¹ : Denumerable α\ninst✝ : Denumerable β\nv v₁ v₂ : ℕ\ne : unpair v = (v₁, v₂)\n⊢ ∃ a, a ∈ decodeList (succ v) ∧ encodeList a = succ v",
"state_before": "α : Type u_1\nβ : Type ?u.30505\ninst✝¹ : Denumerable α\ninst✝ : Denumerable β\nv : ℕ\n⊢ ∃ a, a ∈ decodeList (succ v) ∧ encodeList a = succ v",
"tactic": "cases' e : unpair v with v₁ v₂"
},
{
"state_after": "case mk\nα : Type u_1\nβ : Type ?u.30505\ninst✝¹ : Denumerable α\ninst✝ : Denumerable β\nv v₁ v₂ : ℕ\ne : unpair v = (v₁, v₂)\nh : (unpair v).snd ≤ v\n⊢ ∃ a, a ∈ decodeList (succ v) ∧ encodeList a = succ v",
"state_before": "case mk\nα : Type u_1\nβ : Type ?u.30505\ninst✝¹ : Denumerable α\ninst✝ : Denumerable β\nv v₁ v₂ : ℕ\ne : unpair v = (v₁, v₂)\n⊢ ∃ a, a ∈ decodeList (succ v) ∧ encodeList a = succ v",
"tactic": "have h := unpair_right_le v"
},
{
"state_after": "case mk\nα : Type u_1\nβ : Type ?u.30505\ninst✝¹ : Denumerable α\ninst✝ : Denumerable β\nv v₁ v₂ : ℕ\ne : unpair v = (v₁, v₂)\nh : (v₁, v₂).snd ≤ v\n⊢ ∃ a, a ∈ decodeList (succ v) ∧ encodeList a = succ v",
"state_before": "case mk\nα : Type u_1\nβ : Type ?u.30505\ninst✝¹ : Denumerable α\ninst✝ : Denumerable β\nv v₁ v₂ : ℕ\ne : unpair v = (v₁, v₂)\nh : (unpair v).snd ≤ v\n⊢ ∃ a, a ∈ decodeList (succ v) ∧ encodeList a = succ v",
"tactic": "rw [e] at h"
},
{
"state_after": "case mk.intro.intro\nα : Type u_1\nβ : Type ?u.30505\ninst✝¹ : Denumerable α\ninst✝ : Denumerable β\nv v₁ v₂ : ℕ\ne : unpair v = (v₁, v₂)\nh : (v₁, v₂).snd ≤ v\na : List α\nh₁ : a ∈ decodeList v₂\nh₂ : encodeList a = v₂\n⊢ ∃ a, a ∈ decodeList (succ v) ∧ encodeList a = succ v",
"state_before": "case mk\nα : Type u_1\nβ : Type ?u.30505\ninst✝¹ : Denumerable α\ninst✝ : Denumerable β\nv v₁ v₂ : ℕ\ne : unpair v = (v₁, v₂)\nh : (v₁, v₂).snd ≤ v\n⊢ ∃ a, a ∈ decodeList (succ v) ∧ encodeList a = succ v",
"tactic": "rcases have : v₂ < succ v := lt_succ_of_le h\n denumerable_list_aux v₂ with\n ⟨a, h₁, h₂⟩"
},
{
"state_after": "case mk.intro.intro\nα : Type u_1\nβ : Type ?u.30505\ninst✝¹ : Denumerable α\ninst✝ : Denumerable β\nv v₁ v₂ : ℕ\ne : unpair v = (v₁, v₂)\nh : (v₁, v₂).snd ≤ v\na : List α\nh₁ : decodeList v₂ = some a\nh₂ : encodeList a = v₂\n⊢ ∃ a, a ∈ decodeList (succ v) ∧ encodeList a = succ v",
"state_before": "case mk.intro.intro\nα : Type u_1\nβ : Type ?u.30505\ninst✝¹ : Denumerable α\ninst✝ : Denumerable β\nv v₁ v₂ : ℕ\ne : unpair v = (v₁, v₂)\nh : (v₁, v₂).snd ≤ v\na : List α\nh₁ : a ∈ decodeList v₂\nh₂ : encodeList a = v₂\n⊢ ∃ a, a ∈ decodeList (succ v) ∧ encodeList a = succ v",
"tactic": "rw [Option.mem_def] at h₁"
},
{
"state_after": "case mk.intro.intro\nα : Type u_1\nβ : Type ?u.30505\ninst✝¹ : Denumerable α\ninst✝ : Denumerable β\nv v₁ v₂ : ℕ\ne : unpair v = (v₁, v₂)\nh : (v₁, v₂).snd ≤ v\na : List α\nh₁ : decodeList v₂ = some a\nh₂ : encodeList a = v₂\n⊢ ofNat α v₁ :: a ∈ decodeList (succ v) ∧ encodeList (ofNat α v₁ :: a) = succ v",
"state_before": "case mk.intro.intro\nα : Type u_1\nβ : Type ?u.30505\ninst✝¹ : Denumerable α\ninst✝ : Denumerable β\nv v₁ v₂ : ℕ\ne : unpair v = (v₁, v₂)\nh : (v₁, v₂).snd ≤ v\na : List α\nh₁ : decodeList v₂ = some a\nh₂ : encodeList a = v₂\n⊢ ∃ a, a ∈ decodeList (succ v) ∧ encodeList a = succ v",
"tactic": "use ofNat α v₁ :: a"
},
{
"state_after": "no goals",
"state_before": "case mk.intro.intro\nα : Type u_1\nβ : Type ?u.30505\ninst✝¹ : Denumerable α\ninst✝ : Denumerable β\nv v₁ v₂ : ℕ\ne : unpair v = (v₁, v₂)\nh : (v₁, v₂).snd ≤ v\na : List α\nh₁ : decodeList v₂ = some a\nh₂ : encodeList a = v₂\n⊢ ofNat α v₁ :: a ∈ decodeList (succ v) ∧ encodeList (ofNat α v₁ :: a) = succ v",
"tactic": "simp [decodeList, e, h₂, h₁, encodeList, pair_unpair' e]"
}
] | [
263,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
252,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean | CategoryTheory.IsPushout.inl_snd' | [
{
"state_after": "case h\nC : Type u₁\ninst✝² : Category C\nZ X Y P : C\nf : Z ⟶ X\ng : Z ⟶ Y\ninl : X ⟶ P\ninr : Y ⟶ P\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n⊢ IsPushout 0 b.inl 0 b.snd",
"state_before": "C : Type u₁\ninst✝² : Category C\nZ X Y P : C\nf : Z ⟶ X\ng : Z ⟶ Y\ninl : X ⟶ P\ninr : Y ⟶ P\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n⊢ IsPushout b.inl 0 b.snd 0",
"tactic": "apply flip"
},
{
"state_after": "case h\nC : Type u₁\ninst✝² : Category C\nZ X Y P : C\nf : Z ⟶ X\ng : Z ⟶ Y\ninl : X ⟶ P\ninr : Y ⟶ P\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n⊢ IsPushout (0 ≫ 0) 0 0 (b.inr ≫ b.snd)",
"state_before": "case h\nC : Type u₁\ninst✝² : Category C\nZ X Y P : C\nf : Z ⟶ X\ng : Z ⟶ Y\ninl : X ⟶ P\ninr : Y ⟶ P\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n⊢ IsPushout 0 b.inl 0 b.snd",
"tactic": "refine' of_right _ (by simp) (of_isBilimit h)"
},
{
"state_after": "no goals",
"state_before": "case h\nC : Type u₁\ninst✝² : Category C\nZ X Y P : C\nf : Z ⟶ X\ng : Z ⟶ Y\ninl : X ⟶ P\ninr : Y ⟶ P\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n⊢ IsPushout (0 ≫ 0) 0 0 (b.inr ≫ b.snd)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\nZ X Y P : C\nf : Z ⟶ X\ng : Z ⟶ Y\ninl : X ⟶ P\ninr : Y ⟶ P\ninst✝¹ : HasZeroObject C\ninst✝ : HasZeroMorphisms C\nb : BinaryBicone X Y\nh : BinaryBicone.IsBilimit b\n⊢ 0 ≫ 0 = b.inl ≫ b.snd",
"tactic": "simp"
}
] | [
787,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
783,
1
] |
Mathlib/LinearAlgebra/Finsupp.lean | Finsupp.sumFinsuppLEquivProdFinsupp_symm_inr | [] | [
961,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
959,
1
] |
Mathlib/Algebra/GroupWithZero/Basic.lean | eq_zero_of_one_div_eq_zero | [] | [
421,
75
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
420,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean | nnnorm_ofDual | [] | [
2334,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2333,
1
] |
Mathlib/Data/Polynomial/Monic.lean | Polynomial.monic_X_pow_sub | [
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nm n✝ : ℕ\nι : Type y\ninst✝ : Ring R\np : R[X]\nn : ℕ\nH : degree p ≤ ↑n\n⊢ Monic (X ^ (n + 1) - p)",
"tactic": "simpa [sub_eq_add_neg] using monic_X_pow_add (show degree (-p) ≤ n by rwa [← degree_neg p] at H)"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nm n✝ : ℕ\nι : Type y\ninst✝ : Ring R\np : R[X]\nn : ℕ\nH : degree p ≤ ↑n\n⊢ degree (-p) ≤ ↑n",
"tactic": "rwa [← degree_neg p] at H"
}
] | [
380,
99
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
379,
1
] |
Mathlib/Data/Num/Lemmas.lean | Num.zneg_toZNum | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.446504\nn : Num\n⊢ -toZNum n = toZNumNeg n",
"tactic": "cases n <;> rfl"
}
] | [
775,
78
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
775,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean | Class.univ_not_mem_univ | [] | [
1564,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1563,
1
] |
Mathlib/MeasureTheory/Measure/Lebesgue/Basic.lean | Real.volume_pi_Ico_toReal | [
{
"state_after": "no goals",
"state_before": "ι : Type u_1\ninst✝ : Fintype ι\na b : ι → ℝ\nh : a ≤ b\n⊢ ENNReal.toReal (↑↑volume (Set.pi univ fun i => Ico (a i) (b i))) = ∏ i : ι, (b i - a i)",
"tactic": "simp only [volume_pi_Ico, ENNReal.toReal_prod, ENNReal.toReal_ofReal (sub_nonneg.2 (h _))]"
}
] | [
256,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
254,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean | IsCompact.nhdsSet_basis_uniformity | [
{
"state_after": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\n⊢ U ∈ 𝓝ˢ K ↔ ∃ i, p i ∧ (⋃ (x : α) (_ : x ∈ K), ball x (s i)) ⊆ U",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\n⊢ HasBasis (𝓝ˢ K) p fun i => ⋃ (x : α) (_ : x ∈ K), ball x (s i)",
"tactic": "refine' ⟨fun U => _⟩"
},
{
"state_after": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\n⊢ (∀ (x : α), x ∈ K → ∃ i, p i ∧ ball x (s i) ⊆ U) ↔ ∃ i, p i ∧ ∀ (i_1 : α), i_1 ∈ K → ball i_1 (s i) ⊆ U",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\n⊢ U ∈ 𝓝ˢ K ↔ ∃ i, p i ∧ (⋃ (x : α) (_ : x ∈ K), ball x (s i)) ⊆ U",
"tactic": "simp only [mem_nhdsSet_iff_forall, (nhds_basis_uniformity' hU).mem_iff, iUnion₂_subset_iff]"
},
{
"state_after": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nH : ∀ (x : α), x ∈ K → ∃ i, p i ∧ ball x (s i) ⊆ U\n⊢ ∃ i, p i ∧ ∀ (i_1 : α), i_1 ∈ K → ball i_1 (s i) ⊆ U",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\n⊢ (∀ (x : α), x ∈ K → ∃ i, p i ∧ ball x (s i) ⊆ U) ↔ ∃ i, p i ∧ ∀ (i_1 : α), i_1 ∈ K → ball i_1 (s i) ⊆ U",
"tactic": "refine' ⟨fun H => _, fun ⟨i, hpi, hi⟩ x hx => ⟨i, hpi, hi x hx⟩⟩"
},
{
"state_after": "case H\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nH : ∀ (x : α), x ∈ K → ∃ i, p i ∧ ball x (s i) ⊆ U\n⊢ ∀ (x : α), x ∈ K → ∃ i, ball x (s ↑i ○ s ↑i) ⊆ U\n\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nH : ∀ (x : α), x ∈ K → ∃ i, ball x (s ↑i ○ s ↑i) ⊆ U\n⊢ ∃ i, p i ∧ ∀ (i_1 : α), i_1 ∈ K → ball i_1 (s i) ⊆ U",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nH : ∀ (x : α), x ∈ K → ∃ i, p i ∧ ball x (s i) ⊆ U\n⊢ ∃ i, p i ∧ ∀ (i_1 : α), i_1 ∈ K → ball i_1 (s i) ⊆ U",
"tactic": "replace H : ∀ x ∈ K, ∃ i : { i // p i }, ball x (s i ○ s i) ⊆ U"
},
{
"state_after": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nH : ∀ (x : α), x ∈ K → ∃ i, ball x (s ↑i ○ s ↑i) ⊆ U\nthis : Nonempty { a // p a }\n⊢ ∃ i, p i ∧ ∀ (i_1 : α), i_1 ∈ K → ball i_1 (s i) ⊆ U",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nH : ∀ (x : α), x ∈ K → ∃ i, ball x (s ↑i ○ s ↑i) ⊆ U\n⊢ ∃ i, p i ∧ ∀ (i_1 : α), i_1 ∈ K → ball i_1 (s i) ⊆ U",
"tactic": "have : Nonempty { a // p a } := nonempty_subtype.2 hU.ex_mem"
},
{
"state_after": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nthis : Nonempty { a // p a }\nI : α → { i // p i }\nhI : ∀ (x : α), x ∈ K → ball x (s ↑(I x) ○ s ↑(I x)) ⊆ U\n⊢ ∃ i, p i ∧ ∀ (i_1 : α), i_1 ∈ K → ball i_1 (s i) ⊆ U",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nH : ∀ (x : α), x ∈ K → ∃ i, ball x (s ↑i ○ s ↑i) ⊆ U\nthis : Nonempty { a // p a }\n⊢ ∃ i, p i ∧ ∀ (i_1 : α), i_1 ∈ K → ball i_1 (s i) ⊆ U",
"tactic": "choose! I hI using H"
},
{
"state_after": "case intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nthis : Nonempty { a // p a }\nI : α → { i // p i }\nhI : ∀ (x : α), x ∈ K → ball x (s ↑(I x) ○ s ↑(I x)) ⊆ U\nt : Finset α\nhtK : ∀ (x : α), x ∈ t → x ∈ K\nht : K ⊆ ⋃ (x : α) (_ : x ∈ t), ball x (s ↑(I x))\n⊢ ∃ i, p i ∧ ∀ (i_1 : α), i_1 ∈ K → ball i_1 (s i) ⊆ U",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nthis : Nonempty { a // p a }\nI : α → { i // p i }\nhI : ∀ (x : α), x ∈ K → ball x (s ↑(I x) ○ s ↑(I x)) ⊆ U\n⊢ ∃ i, p i ∧ ∀ (i_1 : α), i_1 ∈ K → ball i_1 (s i) ⊆ U",
"tactic": "rcases hK.elim_nhds_subcover (fun x => ball x <| s (I x)) fun x _ =>\n ball_mem_nhds _ <| hU.mem_of_mem (I x).2 with\n ⟨t, htK, ht⟩"
},
{
"state_after": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nthis : Nonempty { a // p a }\nI : α → { i // p i }\nhI : ∀ (x : α), x ∈ K → ball x (s ↑(I x) ○ s ↑(I x)) ⊆ U\nt : Finset α\nhtK : ∀ (x : α), x ∈ t → x ∈ K\nht : K ⊆ ⋃ (x : α) (_ : x ∈ t), ball x (s ↑(I x))\n⊢ ∃ i, p i ∧ s i ⊆ ⋂ (x : α) (_ : x ∈ t), s ↑(I x)\n\ncase intro.intro.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nthis : Nonempty { a // p a }\nI : α → { i // p i }\nhI : ∀ (x : α), x ∈ K → ball x (s ↑(I x) ○ s ↑(I x)) ⊆ U\nt : Finset α\nhtK : ∀ (x : α), x ∈ t → x ∈ K\nht : K ⊆ ⋃ (x : α) (_ : x ∈ t), ball x (s ↑(I x))\ni : ι\nhpi : p i\nhi : s i ⊆ ⋂ (x : α) (_ : x ∈ t), s ↑(I x)\n⊢ ∃ i, p i ∧ ∀ (i_1 : α), i_1 ∈ K → ball i_1 (s i) ⊆ U",
"state_before": "case intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nthis : Nonempty { a // p a }\nI : α → { i // p i }\nhI : ∀ (x : α), x ∈ K → ball x (s ↑(I x) ○ s ↑(I x)) ⊆ U\nt : Finset α\nhtK : ∀ (x : α), x ∈ t → x ∈ K\nht : K ⊆ ⋃ (x : α) (_ : x ∈ t), ball x (s ↑(I x))\n⊢ ∃ i, p i ∧ ∀ (i_1 : α), i_1 ∈ K → ball i_1 (s i) ⊆ U",
"tactic": "obtain ⟨i, hpi, hi⟩ : ∃ i, p i ∧ s i ⊆ ⋂ x ∈ t, s (I x)"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nthis : Nonempty { a // p a }\nI : α → { i // p i }\nhI : ∀ (x : α), x ∈ K → ball x (s ↑(I x) ○ s ↑(I x)) ⊆ U\nt : Finset α\nhtK : ∀ (x : α), x ∈ t → x ∈ K\nht : K ⊆ ⋃ (x : α) (_ : x ∈ t), ball x (s ↑(I x))\ni : ι\nhpi : p i\nhi : s i ⊆ ⋂ (x : α) (_ : x ∈ t), s ↑(I x)\n⊢ ∃ i, p i ∧ ∀ (i_1 : α), i_1 ∈ K → ball i_1 (s i) ⊆ U",
"state_before": "α : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nthis : Nonempty { a // p a }\nI : α → { i // p i }\nhI : ∀ (x : α), x ∈ K → ball x (s ↑(I x) ○ s ↑(I x)) ⊆ U\nt : Finset α\nhtK : ∀ (x : α), x ∈ t → x ∈ K\nht : K ⊆ ⋃ (x : α) (_ : x ∈ t), ball x (s ↑(I x))\n⊢ ∃ i, p i ∧ s i ⊆ ⋂ (x : α) (_ : x ∈ t), s ↑(I x)\n\ncase intro.intro.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nthis : Nonempty { a // p a }\nI : α → { i // p i }\nhI : ∀ (x : α), x ∈ K → ball x (s ↑(I x) ○ s ↑(I x)) ⊆ U\nt : Finset α\nhtK : ∀ (x : α), x ∈ t → x ∈ K\nht : K ⊆ ⋃ (x : α) (_ : x ∈ t), ball x (s ↑(I x))\ni : ι\nhpi : p i\nhi : s i ⊆ ⋂ (x : α) (_ : x ∈ t), s ↑(I x)\n⊢ ∃ i, p i ∧ ∀ (i_1 : α), i_1 ∈ K → ball i_1 (s i) ⊆ U",
"tactic": "exact hU.mem_iff.1 ((biInter_finset_mem t).2 fun x _ => hU.mem_of_mem (I x).2)"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nthis : Nonempty { a // p a }\nI : α → { i // p i }\nhI : ∀ (x : α), x ∈ K → ball x (s ↑(I x) ○ s ↑(I x)) ⊆ U\nt : Finset α\nhtK : ∀ (x : α), x ∈ t → x ∈ K\nht : K ⊆ ⋃ (x : α) (_ : x ∈ t), ball x (s ↑(I x))\ni : ι\nhpi : p i\nhi : ∀ (i_1 : α), i_1 ∈ t → s i ⊆ s ↑(I i_1)\n⊢ ∃ i, p i ∧ ∀ (i_1 : α), i_1 ∈ K → ball i_1 (s i) ⊆ U",
"state_before": "case intro.intro.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nthis : Nonempty { a // p a }\nI : α → { i // p i }\nhI : ∀ (x : α), x ∈ K → ball x (s ↑(I x) ○ s ↑(I x)) ⊆ U\nt : Finset α\nhtK : ∀ (x : α), x ∈ t → x ∈ K\nht : K ⊆ ⋃ (x : α) (_ : x ∈ t), ball x (s ↑(I x))\ni : ι\nhpi : p i\nhi : s i ⊆ ⋂ (x : α) (_ : x ∈ t), s ↑(I x)\n⊢ ∃ i, p i ∧ ∀ (i_1 : α), i_1 ∈ K → ball i_1 (s i) ⊆ U",
"tactic": "rw [subset_iInter₂_iff] at hi"
},
{
"state_after": "case intro.intro.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nthis : Nonempty { a // p a }\nI : α → { i // p i }\nhI : ∀ (x : α), x ∈ K → ball x (s ↑(I x) ○ s ↑(I x)) ⊆ U\nt : Finset α\nhtK : ∀ (x : α), x ∈ t → x ∈ K\nht : K ⊆ ⋃ (x : α) (_ : x ∈ t), ball x (s ↑(I x))\ni : ι\nhpi : p i\nhi : ∀ (i_1 : α), i_1 ∈ t → s i ⊆ s ↑(I i_1)\nx : α\nhx : x ∈ K\n⊢ ball x (s i) ⊆ U",
"state_before": "case intro.intro.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nthis : Nonempty { a // p a }\nI : α → { i // p i }\nhI : ∀ (x : α), x ∈ K → ball x (s ↑(I x) ○ s ↑(I x)) ⊆ U\nt : Finset α\nhtK : ∀ (x : α), x ∈ t → x ∈ K\nht : K ⊆ ⋃ (x : α) (_ : x ∈ t), ball x (s ↑(I x))\ni : ι\nhpi : p i\nhi : ∀ (i_1 : α), i_1 ∈ t → s i ⊆ s ↑(I i_1)\n⊢ ∃ i, p i ∧ ∀ (i_1 : α), i_1 ∈ K → ball i_1 (s i) ⊆ U",
"tactic": "refine' ⟨i, hpi, fun x hx => _⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nthis : Nonempty { a // p a }\nI : α → { i // p i }\nhI : ∀ (x : α), x ∈ K → ball x (s ↑(I x) ○ s ↑(I x)) ⊆ U\nt : Finset α\nhtK : ∀ (x : α), x ∈ t → x ∈ K\nht : K ⊆ ⋃ (x : α) (_ : x ∈ t), ball x (s ↑(I x))\ni : ι\nhpi : p i\nhi : ∀ (i_1 : α), i_1 ∈ t → s i ⊆ s ↑(I i_1)\nx : α\nhx : x ∈ K\nz : α\nhzt : z ∈ t\nhzx : x ∈ ball z (s ↑(I z))\n⊢ ball x (s i) ⊆ U",
"state_before": "case intro.intro.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nthis : Nonempty { a // p a }\nI : α → { i // p i }\nhI : ∀ (x : α), x ∈ K → ball x (s ↑(I x) ○ s ↑(I x)) ⊆ U\nt : Finset α\nhtK : ∀ (x : α), x ∈ t → x ∈ K\nht : K ⊆ ⋃ (x : α) (_ : x ∈ t), ball x (s ↑(I x))\ni : ι\nhpi : p i\nhi : ∀ (i_1 : α), i_1 ∈ t → s i ⊆ s ↑(I i_1)\nx : α\nhx : x ∈ K\n⊢ ball x (s i) ⊆ U",
"tactic": "rcases mem_iUnion₂.1 (ht hx) with ⟨z, hzt : z ∈ t, hzx : x ∈ ball z (s (I z))⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nthis : Nonempty { a // p a }\nI : α → { i // p i }\nhI : ∀ (x : α), x ∈ K → ball x (s ↑(I x) ○ s ↑(I x)) ⊆ U\nt : Finset α\nhtK : ∀ (x : α), x ∈ t → x ∈ K\nht : K ⊆ ⋃ (x : α) (_ : x ∈ t), ball x (s ↑(I x))\ni : ι\nhpi : p i\nhi : ∀ (i_1 : α), i_1 ∈ t → s i ⊆ s ↑(I i_1)\nx : α\nhx : x ∈ K\nz : α\nhzt : z ∈ t\nhzx : x ∈ ball z (s ↑(I z))\n⊢ ball x (s i) ⊆ U",
"tactic": "calc\n ball x (s i) ⊆ ball z (s (I z) ○ s (I z)) := fun y hy => ⟨x, hzx, hi z hzt hy⟩\n _ ⊆ U := hI z (htK z hzt)"
},
{
"state_after": "case H\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nH : ∀ (x : α), x ∈ K → ∃ i, p i ∧ ball x (s i) ⊆ U\nx : α\nhx : x ∈ K\n⊢ ∃ i, ball x (s ↑i ○ s ↑i) ⊆ U",
"state_before": "case H\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nH : ∀ (x : α), x ∈ K → ∃ i, p i ∧ ball x (s i) ⊆ U\n⊢ ∀ (x : α), x ∈ K → ∃ i, ball x (s ↑i ○ s ↑i) ⊆ U",
"tactic": "intro x hx"
},
{
"state_after": "case H.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nH : ∀ (x : α), x ∈ K → ∃ i, p i ∧ ball x (s i) ⊆ U\nx : α\nhx : x ∈ K\ni : ι\nhpi : p i\nhi : ball x (s i) ⊆ U\n⊢ ∃ i, ball x (s ↑i ○ s ↑i) ⊆ U",
"state_before": "case H\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nH : ∀ (x : α), x ∈ K → ∃ i, p i ∧ ball x (s i) ⊆ U\nx : α\nhx : x ∈ K\n⊢ ∃ i, ball x (s ↑i ○ s ↑i) ⊆ U",
"tactic": "rcases H x hx with ⟨i, hpi, hi⟩"
},
{
"state_after": "case H.intro.intro.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nH : ∀ (x : α), x ∈ K → ∃ i, p i ∧ ball x (s i) ⊆ U\nx : α\nhx : x ∈ K\ni : ι\nhpi : p i\nhi : ball x (s i) ⊆ U\nt : Set (α × α)\nht_mem : t ∈ 𝓤 α\nht : t ○ t ⊆ s i\n⊢ ∃ i, ball x (s ↑i ○ s ↑i) ⊆ U",
"state_before": "case H.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nH : ∀ (x : α), x ∈ K → ∃ i, p i ∧ ball x (s i) ⊆ U\nx : α\nhx : x ∈ K\ni : ι\nhpi : p i\nhi : ball x (s i) ⊆ U\n⊢ ∃ i, ball x (s ↑i ○ s ↑i) ⊆ U",
"tactic": "rcases comp_mem_uniformity_sets (hU.mem_of_mem hpi) with ⟨t, ht_mem, ht⟩"
},
{
"state_after": "case H.intro.intro.intro.intro.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nH : ∀ (x : α), x ∈ K → ∃ i, p i ∧ ball x (s i) ⊆ U\nx : α\nhx : x ∈ K\ni : ι\nhpi : p i\nhi : ball x (s i) ⊆ U\nt : Set (α × α)\nht_mem : t ∈ 𝓤 α\nht : t ○ t ⊆ s i\nj : ι\nhpj : p j\nhj : s j ⊆ t\n⊢ ∃ i, ball x (s ↑i ○ s ↑i) ⊆ U",
"state_before": "case H.intro.intro.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nH : ∀ (x : α), x ∈ K → ∃ i, p i ∧ ball x (s i) ⊆ U\nx : α\nhx : x ∈ K\ni : ι\nhpi : p i\nhi : ball x (s i) ⊆ U\nt : Set (α × α)\nht_mem : t ∈ 𝓤 α\nht : t ○ t ⊆ s i\n⊢ ∃ i, ball x (s ↑i ○ s ↑i) ⊆ U",
"tactic": "rcases hU.mem_iff.1 ht_mem with ⟨j, hpj, hj⟩"
},
{
"state_after": "no goals",
"state_before": "case H.intro.intro.intro.intro.intro.intro\nα : Type ua\nβ : Type ub\nγ : Type uc\nδ : Type ud\nι : Sort u_1\ninst✝ : UniformSpace α\np : ι → Prop\ns : ι → Set (α × α)\nhU : HasBasis (𝓤 α) p s\nK : Set α\nhK : IsCompact K\nU : Set α\nH : ∀ (x : α), x ∈ K → ∃ i, p i ∧ ball x (s i) ⊆ U\nx : α\nhx : x ∈ K\ni : ι\nhpi : p i\nhi : ball x (s i) ⊆ U\nt : Set (α × α)\nht_mem : t ∈ 𝓤 α\nht : t ○ t ⊆ s i\nj : ι\nhpj : p j\nhj : s j ⊆ t\n⊢ ∃ i, ball x (s ↑i ○ s ↑i) ⊆ U",
"tactic": "exact ⟨⟨j, hpj⟩, Subset.trans (ball_mono ((compRel_mono hj hj).trans ht) _) hi⟩"
}
] | [
834,
30
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
810,
1
] |
Mathlib/Analysis/Complex/Isometry.lean | LinearIsometry.im_apply_eq_im_or_neg_of_re_apply_eq_re | [
{
"state_after": "f : ℂ →ₗᵢ[ℝ] ℂ\nh₂ : ∀ (z : ℂ), (↑f z).re = z.re\nz : ℂ\nh₁ : ‖↑f z‖ = ‖z‖\n⊢ (↑f z).im = z.im ∨ (↑f z).im = -z.im",
"state_before": "f : ℂ →ₗᵢ[ℝ] ℂ\nh₂ : ∀ (z : ℂ), (↑f z).re = z.re\nz : ℂ\n⊢ (↑f z).im = z.im ∨ (↑f z).im = -z.im",
"tactic": "have h₁ := f.norm_map z"
},
{
"state_after": "f : ℂ →ₗᵢ[ℝ] ℂ\nh₂ : ∀ (z : ℂ), (↑f z).re = z.re\nz : ℂ\nh₁ : Real.sqrt (↑normSq (↑f z)) = Real.sqrt (↑normSq z)\n⊢ (↑f z).im = z.im ∨ (↑f z).im = -z.im",
"state_before": "f : ℂ →ₗᵢ[ℝ] ℂ\nh₂ : ∀ (z : ℂ), (↑f z).re = z.re\nz : ℂ\nh₁ : ‖↑f z‖ = ‖z‖\n⊢ (↑f z).im = z.im ∨ (↑f z).im = -z.im",
"tactic": "simp only [Complex.abs_def, norm_eq_abs] at h₁"
},
{
"state_after": "no goals",
"state_before": "f : ℂ →ₗᵢ[ℝ] ℂ\nh₂ : ∀ (z : ℂ), (↑f z).re = z.re\nz : ℂ\nh₁ : Real.sqrt (↑normSq (↑f z)) = Real.sqrt (↑normSq z)\n⊢ (↑f z).im = z.im ∨ (↑f z).im = -z.im",
"tactic": "rwa [Real.sqrt_inj (normSq_nonneg _) (normSq_nonneg _), normSq_apply (f z), normSq_apply z,\n h₂, add_left_cancel_iff, mul_self_eq_mul_self_iff] at h₁"
}
] | [
104,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
99,
1
] |
Mathlib/Data/Polynomial/Derivative.lean | Polynomial.iterate_derivative_comp_one_sub_X | [
{
"state_after": "case zero\nR : Type u\nS : Type v\nT : Type w\nι : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ p : R[X]\n⊢ (↑derivative^[Nat.zero]) (comp p (1 - X)) = (-1) ^ Nat.zero * comp ((↑derivative^[Nat.zero]) p) (1 - X)\n\ncase succ\nR : Type u\nS : Type v\nT : Type w\nι : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ : R[X]\nk : ℕ\nih : ∀ (p : R[X]), (↑derivative^[k]) (comp p (1 - X)) = (-1) ^ k * comp ((↑derivative^[k]) p) (1 - X)\np : R[X]\n⊢ (↑derivative^[Nat.succ k]) (comp p (1 - X)) = (-1) ^ Nat.succ k * comp ((↑derivative^[Nat.succ k]) p) (1 - X)",
"state_before": "R : Type u\nS : Type v\nT : Type w\nι : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np : R[X]\nk : ℕ\n⊢ (↑derivative^[k]) (comp p (1 - X)) = (-1) ^ k * comp ((↑derivative^[k]) p) (1 - X)",
"tactic": "induction' k with k ih generalizing p"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u\nS : Type v\nT : Type w\nι : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ p : R[X]\n⊢ (↑derivative^[Nat.zero]) (comp p (1 - X)) = (-1) ^ Nat.zero * comp ((↑derivative^[Nat.zero]) p) (1 - X)",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case succ\nR : Type u\nS : Type v\nT : Type w\nι : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : CommRing R\np✝ : R[X]\nk : ℕ\nih : ∀ (p : R[X]), (↑derivative^[k]) (comp p (1 - X)) = (-1) ^ k * comp ((↑derivative^[k]) p) (1 - X)\np : R[X]\n⊢ (↑derivative^[Nat.succ k]) (comp p (1 - X)) = (-1) ^ Nat.succ k * comp ((↑derivative^[Nat.succ k]) p) (1 - X)",
"tactic": "simp [ih (derivative p), iterate_derivative_neg, derivative_comp, pow_succ]"
}
] | [
650,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
646,
1
] |
Mathlib/Data/Set/Lattice.lean | Set.iInter_coe_set | [] | [
994,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
992,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean | intervalIntegral.norm_integral_le_integral_norm | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.12055077\n𝕜 : Type ?u.12055080\nE : Type u_1\nF : Type ?u.12055086\nA : Type ?u.12055089\ninst✝² : NormedAddCommGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : NormedSpace ℝ E\na b : ℝ\nf g : ℝ → E\nμ : MeasureTheory.Measure ℝ\nh : a ≤ b\n⊢ (∫ (x : ℝ) in Ι a b, ‖f x‖ ∂μ) = ∫ (x : ℝ) in a..b, ‖f x‖ ∂μ",
"tactic": "rw [uIoc_of_le h, integral_of_le h]"
}
] | [
556,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
554,
1
] |
Mathlib/Analysis/Calculus/LocalExtr.lean | Polynomial.card_roots_toFinset_le_derivative | [] | [
375,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
372,
1
] |
Mathlib/MeasureTheory/Measure/MutuallySingular.lean | MeasureTheory.Measure.MutuallySingular.sum_right | [] | [
100,
62
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
99,
1
] |
Mathlib/Algebra/Order/Monoid/Units.lean | Units.val_lt_val | [] | [
34,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
33,
1
] |
Mathlib/Order/Filter/Basic.lean | Filter.tendsto_def | [] | [
2829,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2827,
1
] |
Mathlib/Data/List/Duplicate.lean | List.not_duplicate_nil | [] | [
71,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
71,
1
] |
Mathlib/Analysis/Complex/LocallyUniformLimit.lean | Complex.exists_cthickening_tendstoUniformlyOn | [
{
"state_after": "case intro.intro\nE : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU K : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhK : IsCompact K\nhU : IsOpen U\nhKU : K ⊆ U\nδ : ℝ\nhδ : 0 < δ\nhKδ : cthickening δ K ⊆ U\n⊢ ∃ δ, δ > 0 ∧ cthickening δ K ⊆ U ∧ TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K",
"state_before": "E : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU K : Set ℂ\nz : ℂ\nM r δ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhK : IsCompact K\nhU : IsOpen U\nhKU : K ⊆ U\n⊢ ∃ δ, δ > 0 ∧ cthickening δ K ⊆ U ∧ TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K",
"tactic": "obtain ⟨δ, hδ, hKδ⟩ := hK.exists_cthickening_subset_open hU hKU"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nE : Type u_1\nι : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nU K : Set ℂ\nz : ℂ\nM r δ✝ : ℝ\nφ : Filter ι\nF : ι → ℂ → E\nf g : ℂ → E\nhf : TendstoLocallyUniformlyOn F f φ U\nhF : ∀ᶠ (n : ι) in φ, DifferentiableOn ℂ (F n) U\nhK : IsCompact K\nhU : IsOpen U\nhKU : K ⊆ U\nδ : ℝ\nhδ : 0 < δ\nhKδ : cthickening δ K ⊆ U\n⊢ ∃ δ, δ > 0 ∧ cthickening δ K ⊆ U ∧ TendstoUniformlyOn (deriv ∘ F) (cderiv δ f) φ K",
"tactic": "exact ⟨δ, hδ, hKδ, tendstoUniformlyOn_deriv_of_cthickening_subset hf hF hδ hK hU hKδ⟩"
}
] | [
141,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
137,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean | ContinuousLinearEquiv.coe_apply | [] | [
1794,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1793,
1
] |
Mathlib/Data/Real/Irrational.lean | irrational_nrt_of_n_not_dvd_multiplicity | [
{
"state_after": "case inl\nx : ℝ\nm : ℤ\nhm : m ≠ 0\np : ℕ\nhp : Fact (Nat.Prime p)\nhxr : x ^ 0 = ↑m\nhv : Part.get (multiplicity (↑p) m) (_ : multiplicity.Finite (↑p) m) % 0 ≠ 0\n⊢ Irrational x\n\ncase inr\nx : ℝ\nn : ℕ\nm : ℤ\nhm : m ≠ 0\np : ℕ\nhp : Fact (Nat.Prime p)\nhxr : x ^ n = ↑m\nhv : Part.get (multiplicity (↑p) m) (_ : multiplicity.Finite (↑p) m) % n ≠ 0\nhnpos : n > 0\n⊢ Irrational x",
"state_before": "x : ℝ\nn : ℕ\nm : ℤ\nhm : m ≠ 0\np : ℕ\nhp : Fact (Nat.Prime p)\nhxr : x ^ n = ↑m\nhv : Part.get (multiplicity (↑p) m) (_ : multiplicity.Finite (↑p) m) % n ≠ 0\n⊢ Irrational x",
"tactic": "rcases Nat.eq_zero_or_pos n with (rfl | hnpos)"
},
{
"state_after": "case inr\nx : ℝ\nn : ℕ\nm : ℤ\nhm : m ≠ 0\np : ℕ\nhp : Fact (Nat.Prime p)\nhxr : x ^ n = ↑m\nhv : Part.get (multiplicity (↑p) m) (_ : multiplicity.Finite (↑p) m) % n ≠ 0\nhnpos : n > 0\n⊢ ¬∃ y, x = ↑y",
"state_before": "case inr\nx : ℝ\nn : ℕ\nm : ℤ\nhm : m ≠ 0\np : ℕ\nhp : Fact (Nat.Prime p)\nhxr : x ^ n = ↑m\nhv : Part.get (multiplicity (↑p) m) (_ : multiplicity.Finite (↑p) m) % n ≠ 0\nhnpos : n > 0\n⊢ Irrational x",
"tactic": "refine' irrational_nrt_of_notint_nrt _ _ hxr _ hnpos"
},
{
"state_after": "case inr.intro\nn : ℕ\nm : ℤ\nhm : m ≠ 0\np : ℕ\nhp : Fact (Nat.Prime p)\nhv : Part.get (multiplicity (↑p) m) (_ : multiplicity.Finite (↑p) m) % n ≠ 0\nhnpos : n > 0\ny : ℤ\nhxr : ↑y ^ n = ↑m\n⊢ False",
"state_before": "case inr\nx : ℝ\nn : ℕ\nm : ℤ\nhm : m ≠ 0\np : ℕ\nhp : Fact (Nat.Prime p)\nhxr : x ^ n = ↑m\nhv : Part.get (multiplicity (↑p) m) (_ : multiplicity.Finite (↑p) m) % n ≠ 0\nhnpos : n > 0\n⊢ ¬∃ y, x = ↑y",
"tactic": "rintro ⟨y, rfl⟩"
},
{
"state_after": "case inr.intro\nn : ℕ\nm : ℤ\nhm : m ≠ 0\np : ℕ\nhp : Fact (Nat.Prime p)\nhv : Part.get (multiplicity (↑p) m) (_ : multiplicity.Finite (↑p) m) % n ≠ 0\nhnpos : n > 0\ny : ℤ\nhxr : y ^ n = m\n⊢ False",
"state_before": "case inr.intro\nn : ℕ\nm : ℤ\nhm : m ≠ 0\np : ℕ\nhp : Fact (Nat.Prime p)\nhv : Part.get (multiplicity (↑p) m) (_ : multiplicity.Finite (↑p) m) % n ≠ 0\nhnpos : n > 0\ny : ℤ\nhxr : ↑y ^ n = ↑m\n⊢ False",
"tactic": "rw [← Int.cast_pow, Int.cast_inj] at hxr"
},
{
"state_after": "case inr.intro\nn p : ℕ\nhp : Fact (Nat.Prime p)\nhnpos : n > 0\ny : ℤ\nhm : y ^ n ≠ 0\nhv : Part.get (multiplicity (↑p) (y ^ n)) (_ : multiplicity.Finite (↑p) (y ^ n)) % n ≠ 0\n⊢ False",
"state_before": "case inr.intro\nn : ℕ\nm : ℤ\nhm : m ≠ 0\np : ℕ\nhp : Fact (Nat.Prime p)\nhv : Part.get (multiplicity (↑p) m) (_ : multiplicity.Finite (↑p) m) % n ≠ 0\nhnpos : n > 0\ny : ℤ\nhxr : y ^ n = m\n⊢ False",
"tactic": "subst m"
},
{
"state_after": "case inr.intro\nn p : ℕ\nhp : Fact (Nat.Prime p)\nhnpos : n > 0\ny : ℤ\nhm : y ^ n ≠ 0\nhv : Part.get (multiplicity (↑p) (y ^ n)) (_ : multiplicity.Finite (↑p) (y ^ n)) % n ≠ 0\nthis : y ≠ 0\n⊢ False",
"state_before": "case inr.intro\nn p : ℕ\nhp : Fact (Nat.Prime p)\nhnpos : n > 0\ny : ℤ\nhm : y ^ n ≠ 0\nhv : Part.get (multiplicity (↑p) (y ^ n)) (_ : multiplicity.Finite (↑p) (y ^ n)) % n ≠ 0\n⊢ False",
"tactic": "have : y ≠ 0 := by\n rintro rfl\n rw [zero_pow hnpos] at hm\n exact hm rfl"
},
{
"state_after": "case inr.intro\nn p : ℕ\nhp : Fact (Nat.Prime p)\nhnpos : n > 0\ny : ℤ\nhm : y ^ n ≠ 0\nthis : y ≠ 0\nhv : 0 ≠ 0\n⊢ False",
"state_before": "case inr.intro\nn p : ℕ\nhp : Fact (Nat.Prime p)\nhnpos : n > 0\ny : ℤ\nhm : y ^ n ≠ 0\nhv : Part.get (multiplicity (↑p) (y ^ n)) (_ : multiplicity.Finite (↑p) (y ^ n)) % n ≠ 0\nthis : y ≠ 0\n⊢ False",
"tactic": "erw [multiplicity.pow' (Nat.prime_iff_prime_int.1 hp.1) (finite_int_iff.2 ⟨hp.1.ne_one, this⟩),\n Nat.mul_mod_right] at hv"
},
{
"state_after": "no goals",
"state_before": "case inr.intro\nn p : ℕ\nhp : Fact (Nat.Prime p)\nhnpos : n > 0\ny : ℤ\nhm : y ^ n ≠ 0\nthis : y ≠ 0\nhv : 0 ≠ 0\n⊢ False",
"tactic": "exact hv rfl"
},
{
"state_after": "case inl\nx : ℝ\nm : ℤ\nhm : m ≠ 0\np : ℕ\nhp : Fact (Nat.Prime p)\nhxr : m = 1\nhv : Part.get (multiplicity (↑p) m) (_ : multiplicity.Finite (↑p) m) % 0 ≠ 0\n⊢ Irrational x",
"state_before": "case inl\nx : ℝ\nm : ℤ\nhm : m ≠ 0\np : ℕ\nhp : Fact (Nat.Prime p)\nhxr : x ^ 0 = ↑m\nhv : Part.get (multiplicity (↑p) m) (_ : multiplicity.Finite (↑p) m) % 0 ≠ 0\n⊢ Irrational x",
"tactic": "rw [eq_comm, pow_zero, ← Int.cast_one, Int.cast_inj] at hxr"
},
{
"state_after": "no goals",
"state_before": "case inl\nx : ℝ\nm : ℤ\nhm : m ≠ 0\np : ℕ\nhp : Fact (Nat.Prime p)\nhxr : m = 1\nhv : Part.get (multiplicity (↑p) m) (_ : multiplicity.Finite (↑p) m) % 0 ≠ 0\n⊢ Irrational x",
"tactic": "simp [hxr,\n multiplicity.one_right (mt isUnit_iff_dvd_one.1 (mt Int.coe_nat_dvd.1 hp.1.not_dvd_one)),\n Nat.zero_mod] at hv"
},
{
"state_after": "n p : ℕ\nhp : Fact (Nat.Prime p)\nhnpos : n > 0\nhm : 0 ^ n ≠ 0\nhv : Part.get (multiplicity (↑p) (0 ^ n)) (_ : multiplicity.Finite (↑p) (0 ^ n)) % n ≠ 0\n⊢ False",
"state_before": "n p : ℕ\nhp : Fact (Nat.Prime p)\nhnpos : n > 0\ny : ℤ\nhm : y ^ n ≠ 0\nhv : Part.get (multiplicity (↑p) (y ^ n)) (_ : multiplicity.Finite (↑p) (y ^ n)) % n ≠ 0\n⊢ y ≠ 0",
"tactic": "rintro rfl"
},
{
"state_after": "n p : ℕ\nhp : Fact (Nat.Prime p)\nhnpos : n > 0\nhm✝ : 0 ^ n ≠ 0\nhm : 0 ≠ 0\nhv : Part.get (multiplicity (↑p) (0 ^ n)) (_ : multiplicity.Finite (↑p) (0 ^ n)) % n ≠ 0\n⊢ False",
"state_before": "n p : ℕ\nhp : Fact (Nat.Prime p)\nhnpos : n > 0\nhm : 0 ^ n ≠ 0\nhv : Part.get (multiplicity (↑p) (0 ^ n)) (_ : multiplicity.Finite (↑p) (0 ^ n)) % n ≠ 0\n⊢ False",
"tactic": "rw [zero_pow hnpos] at hm"
},
{
"state_after": "no goals",
"state_before": "n p : ℕ\nhp : Fact (Nat.Prime p)\nhnpos : n > 0\nhm✝ : 0 ^ n ≠ 0\nhm : 0 ≠ 0\nhv : Part.get (multiplicity (↑p) (0 ^ n)) (_ : multiplicity.Finite (↑p) (0 ^ n)) % n ≠ 0\n⊢ False",
"tactic": "exact hm rfl"
}
] | [
92,
15
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
73,
1
] |
Mathlib/Algebra/Ring/Equiv.lean | RingEquiv.image_eq_preimage | [] | [
349,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
348,
1
] |
Mathlib/AlgebraicTopology/SimplicialObject.lean | CategoryTheory.SimplicialObject.δ_comp_σ_succ' | [
{
"state_after": "C : Type u\ninst✝ : Category C\nX : SimplicialObject C\nn : ℕ\ni : Fin (n + 1)\n⊢ σ X i ≫ δ X (Fin.succ i) = 𝟙 (X.obj [n].op)",
"state_before": "C : Type u\ninst✝ : Category C\nX : SimplicialObject C\nn : ℕ\nj : Fin (n + 2)\ni : Fin (n + 1)\nH : j = Fin.succ i\n⊢ σ X i ≫ δ X j = 𝟙 (X.obj [n].op)",
"tactic": "subst H"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX : SimplicialObject C\nn : ℕ\ni : Fin (n + 1)\n⊢ σ X i ≫ δ X (Fin.succ i) = 𝟙 (X.obj [n].op)",
"tactic": "rw [δ_comp_σ_succ]"
}
] | [
180,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
177,
1
] |
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