module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.Algebra.Module.LinearMap.Polynomial | {
"line": 550,
"column": 4
} | {
"line": 550,
"column": 17
} | [
{
"pp": "R : Type u_1\nL : Type u_2\nM : Type u_3\ninst✝⁹ : CommRing R\ninst✝⁸ : AddCommGroup L\ninst✝⁷ : Module R L\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\nφ : L →ₗ[R] End R M\ninst✝⁴ : Free R M\ninst✝³ : Module.Finite R M\ninst✝² : Module.Finite R L\ninst✝¹ : Free R L\ninst✝ : IsDomain R\nh : ↑(finrank... | by_contra! h₀ | Mathlib.Tactic.ByContra._aux_Mathlib_Tactic_ByContra___macroRules_Mathlib_Tactic_ByContra_byContra!_1 | Mathlib.Tactic.ByContra.byContra! |
Mathlib.LinearAlgebra.SymplecticGroup | {
"line": 121,
"column": 2
} | {
"line": 122,
"column": 10
} | [
{
"pp": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nh : A ∈ symplecticGroup l R\n⊢ -A ∈ symplecticGroup l R",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"instFintypeSum",
"NonUnitalCommRing.toNonUn... | rw [mem_iff] at h ⊢
simp [h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.SymplecticGroup | {
"line": 121,
"column": 2
} | {
"line": 122,
"column": 10
} | [
{
"pp": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nh : A ∈ symplecticGroup l R\n⊢ -A ∈ symplecticGroup l R",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"instFintypeSum",
"NonUnitalCommRing.toNonUn... | rw [mem_iff] at h ⊢
simp [h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Basis | {
"line": 323,
"column": 49
} | {
"line": 323,
"column": 60
} | [
{
"pp": "case smul\nι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝⁴ : Finite ι\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\nb : Basis ι R L\ninst✝ : Fintype ι\ni : ι\nx : L\nt : R\nu : L\nhu : u ∈ b.cartan\nhv' : ∀ (hx : u ∈ b.cartan), (b.baseSupp i) ⟨u, hx⟩ • b.e i = ⁅u, b.e i⁆\nhx : t •... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Basis | {
"line": 322,
"column": 4
} | {
"line": 323,
"column": 75
} | [
{
"pp": "case smul\nι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝⁴ : Finite ι\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\nb : Basis ι R L\ninst✝ : Fintype ι\ni : ι\nx : L\nt : R\nu : L\nhu : u ∈ Submodule.span R (range b.h)\nhv' : ∀ (hx : u ∈ b.cartan), (b.baseSupp i) ⟨u, hx⟩ • b.e i = ... | rw [← coe_cartan_eq_span, LieSubalgebra.mem_toSubmodule] at hu
rw [← SetLike.mk_smul_mk _ t u hu, map_smul, smul_assoc, hv', smul_lie] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Basis | {
"line": 322,
"column": 4
} | {
"line": 323,
"column": 75
} | [
{
"pp": "case smul\nι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝⁴ : Finite ι\ninst✝³ : CommRing R\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra R L\nb : Basis ι R L\ninst✝ : Fintype ι\ni : ι\nx : L\nt : R\nu : L\nhu : u ∈ Submodule.span R (range b.h)\nhv' : ∀ (hx : u ∈ b.cartan), (b.baseSupp i) ⟨u, hx⟩ • b.e i = ... | rw [← coe_cartan_eq_span, LieSubalgebra.mem_toSubmodule] at hu
rw [← SetLike.mk_smul_mk _ t u hu, map_smul, smul_assoc, hv', smul_lie] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.CartanExists | {
"line": 192,
"column": 34
} | {
"line": 195,
"column": 72
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nhLK : ↑(finrank K L) ≤ #K\nU : LieSubalgebra K L\nx : L\nhxU : x ∈ U\ny : L\nhyU : y ∈ U\nEx : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩\nEy : ↑{x | ∃ y ∈ U, engel K y = x} :=... | by
simp_rw [r, ← lieCharpoly_natDegree K E x' u] at this ⊢
rw [(lieCharpoly_monic K E x' u).eq_X_pow_iff_natDegree_le_natTrailingDegree]
exact le_natTrailingDegree (lieCharpoly_monic K E x' u).ne_zero this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Derivation.Lie | {
"line": 60,
"column": 13
} | {
"line": 60,
"column": 73
} | [
{
"pp": "case H\nR : Type u_1\ninst✝² : CommRing R\nA : Type u_2\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nD1 D2 : Derivation R A A\na✝ : A\nr : R\nd e : Derivation R A A\na : A\n⊢ ⁅d, r • e⁆ a = (r • ⁅d, e⁆) a",
"usedConstants": [
"Derivation",
"LieAlgebra.toModule",
"Algebra.to_smulComm... | simp only [commutator_apply, map_smul, smul_sub, smul_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Lie.CartanExists | {
"line": 352,
"column": 4
} | {
"line": 352,
"column": 26
} | [
{
"pp": "case h.refine_1\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nhLK : ↑(finrank K L) ≤ #K\nU : LieSubalgebra K L\nx : L\nhxU : x ∈ U\ny : L\nhyU : y ∈ U\nEx : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩\nEy : ↑{x | ∃ y ∈ U, ... | apply Nat.find_min hz' | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.GroupTheory.Perm.Cycle.Concrete | {
"line": 420,
"column": 8
} | {
"line": 420,
"column": 31
} | [
{
"pp": "case mk.a\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx✝ : α\nval✝ : Cycle α\ns : List α\nhn✝ : Cycle.Nodup (Quot.mk (⇑(IsRotated.setoid α)) s)\nht : Cycle.Nontrivial (Quot.mk (⇑(IsRotated.setoid α)) s)\nx : α\nhl : 2 ≤ s.length\nhn : s.Nodup\nhx : x ∈ s\n⊢ s.formPerm.toCycle ... | toCycle_eq_toList _ _ x | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Loop | {
"line": 152,
"column": 4
} | {
"line": 167,
"column": 67
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nL : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\ninst✝² : CommRing A\ninst✝¹ : IsAddTorsionFree R\ninst✝ : Algebra A R\nΦ : LinearMap.BilinForm R L\nhΦ : LinearMap.BilinForm.lieInvariant L Φ\nhΦs : Φ.IsSymm\n⊢ twoCochainOfBilinear R A L Φ hΦs ∈... | apply (LieModule.Cohomology.mem_twoCocycle_iff ..).mpr
ext a x b y c z
suffices
b • Φ (Finsupp.single (a + c) ⁅x, z⁆ (-b)) y =
c • Φ (Finsupp.single (a + b) ⁅x, y⁆ (-c)) z +
a • Φ (Finsupp.single (b + c) ⁅y, z⁆ (-a)) x by
simpa [sub_eq_zero, neg_add_eq_iff_eq_add, ← LinearEquiv.map... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Loop | {
"line": 152,
"column": 4
} | {
"line": 167,
"column": 67
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nL : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\ninst✝² : CommRing A\ninst✝¹ : IsAddTorsionFree R\ninst✝ : Algebra A R\nΦ : LinearMap.BilinForm R L\nhΦ : LinearMap.BilinForm.lieInvariant L Φ\nhΦs : Φ.IsSymm\n⊢ twoCochainOfBilinear R A L Φ hΦs ∈... | apply (LieModule.Cohomology.mem_twoCocycle_iff ..).mpr
ext a x b y c z
suffices
b • Φ (Finsupp.single (a + c) ⁅x, z⁆ (-b)) y =
c • Φ (Finsupp.single (a + b) ⁅x, y⁆ (-c)) z +
a • Φ (Finsupp.single (b + c) ⁅y, z⁆ (-a)) x by
simpa [sub_eq_zero, neg_add_eq_iff_eq_add, ← LinearEquiv.map... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.LieTheorem | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 38
} | [
{
"pp": "case h\nk : Type u_1\ninst✝¹⁰ : Field k\nL : Type u_2\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra k L\nV : Type u_3\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : LieRingModule L V\ninst✝⁴ : LieModule k L V\ninst✝³ : CharZero k\ninst✝² : Module.Finite k V\ninst✝¹ : IsTriangularizable k L V\nA : L... | refine nontrivial_of_ne ⟨v, ?_⟩ 0 ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Lie.LieTheorem | {
"line": 201,
"column": 50
} | {
"line": 201,
"column": 61
} | [
{
"pp": "k : Type u_1\ninst✝¹⁰ : Field k\nL : Type u_2\ninst✝⁹ : LieRing L\ninst✝⁸ : LieAlgebra k L\nV : Type u_3\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : LieRingModule L V\ninst✝⁴ : LieModule k L V\ninst✝³ : CharZero k\ninst✝² : Module.Finite k V\ninst✝¹ : IsTriangularizable k L V\nA : LieIdeal ... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.LieTheorem | {
"line": 249,
"column": 2
} | {
"line": 249,
"column": 38
} | [
{
"pp": "case h\nk : Type u_1\ninst✝¹¹ : Field k\nL : Type u_2\ninst✝¹⁰ : LieRing L\ninst✝⁹ : LieAlgebra k L\nV : Type u_3\ninst✝⁸ : AddCommGroup V\ninst✝⁷ : Module k V\ninst✝⁶ : LieRingModule L V\ninst✝⁵ : LieModule k L V\ninst✝⁴ : CharZero k\ninst✝³ : Module.Finite k V\ninst✝² : Nontrivial V\ninst✝¹ : IsSolva... | refine nontrivial_of_ne ⟨v, ?_⟩ 0 ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.LinearRecurrence | {
"line": 205,
"column": 36
} | {
"line": 205,
"column": 53
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nE : LinearRecurrence R\na✝ : Nontrivial R\n⊢ 1 - (∑ i, (monomial ↑i) (E.coeffs i)).coeff E.order = 1",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModul... | finset_sum_coeff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.ChainOfDivisors | {
"line": 176,
"column": 4
} | {
"line": 176,
"column": 32
} | [
{
"pp": "M : Type u_1\ninst✝¹ : CommMonoidWithZero M\ninst✝ : UniqueFactorizationMonoid M\nq r : Associates M\nn : ℕ\nhn : n ≠ 0\nc : Fin (n + 1) → Associates M\nh₁ : StrictMono c\nh₂ : ∀ {r : Associates M}, r ≤ q ↔ ∃ i, r = c i\nhr : r ∣ q\nhq : q ≠ 0\ni : ℕ := (normalizedFactors r).card\nhi : normalizedFactor... | simp only [Finset.mem_image] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.ChainOfDivisors | {
"line": 329,
"column": 2
} | {
"line": 329,
"column": 92
} | [
{
"pp": "M : Type u_1\ninst✝⁴ : CommMonoidWithZero M\ninst✝³ : IsCancelMulZero M\nN : Type u_2\ninst✝² : CommMonoidWithZero N\ninst✝¹ : UniqueFactorizationMonoid N\ninst✝ : UniqueFactorizationMonoid M\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.Iic ... | refine le_antisymm (emultiplicity_prime_le_emultiplicity_image_by_factor_orderIso hp d) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.FractionalIdeal.Basic | {
"line": 141,
"column": 2
} | {
"line": 142,
"column": 52
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝³ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝² : CommRing P\ninst✝¹ : Algebra R P\ninst✝ : FaithfulSMul R P\nI : FractionalIdeal S P\nreg : IsSMulRegular P I.den\nx✝ : P\nhx : x✝ ∈ ↑I\n⊢ (DistribSMul.toLinearMap R P I.den) x✝ ∈ Submodule.map (Algebra.linearMap R... | · rw [← den_mul_self_eq_num]
exact Submodule.smul_mem_pointwise_smul _ _ _ hx | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 589,
"column": 62
} | {
"line": 589,
"column": 90
} | [
{
"pp": "case a\nR : Type u_1\ninst✝³ : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝² : CommRing P\ninst✝¹ : Algebra R P\ninst✝ : IsLocalization S P\nI : FractionalIdeal S P\n⊢ (R ∙ (algebraMap R P) ↑I.den) * ↑I = ↑↑I.num",
"usedConstants": [
"Eq.mpr",
"Submodule.pointwiseDistribMulAction",
... | Submodule.span_singleton_mul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Basic | {
"line": 327,
"column": 6
} | {
"line": 327,
"column": 84
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Field K\ninst✝² : Algebra A K\ninst✝¹ : IsFractionRing A K\ninst✝ : IsDedekindDomain A\nI✝ : Ideal A\nI : FractionalIdeal A⁰ (FractionRing A)\nhI : I ≠ ⊥\na : A\nJ : Ideal A\nha : a ≠ 0\nhJ : I = spanSingleton ... | exact ⟨spanSingleton A⁰ (algebraMap _ _ a) * (J : FractionalIdeal A⁰ _)⁻¹, h₂⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 490,
"column": 8
} | {
"line": 491,
"column": 53
} | [
{
"pp": "case refine_2\nK : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsKilling K L\ninst✝ : IsTriangularizable K (↥H) L\nq : Submodule K (Dual K ↥H)\n... | have : ↑(-β) ∈ q := by rw [Weight.toLinear_neg]; exact q.neg_mem hβ_mem
exact le_iSup₂_of_le _ (hne (-β) this) le_rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 490,
"column": 8
} | {
"line": 491,
"column": 53
} | [
{
"pp": "case refine_2\nK : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsKilling K L\ninst✝ : IsTriangularizable K (↥H) L\nq : Submodule K (Dual K ↥H)\n... | have : ↑(-β) ∈ q := by rw [Weight.toLinear_neg]; exact q.neg_mem hβ_mem
exact le_iSup₂_of_le _ (hne (-β) this) le_rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 485,
"column": 4
} | {
"line": 493,
"column": 60
} | [
{
"pp": "case mp\nK : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsKilling K L\ninst✝ : IsTriangularizable K (↥H) L\nq : Submodule K (Dual K ↥H)\nhq : ∀... | have h_le : J.restr H ≤ ⨆ (χ : H → K) (_ : χ ≠ (α : Weight K H L)), genWeightSpace L χ := by
refine iSup_le fun ⟨β, hβ_mem, hβ_nz⟩ ↦ ?_
rw [sl2SubmoduleOfRoot_eq_sup]
refine sup_le (sup_le ?_ ?_) ?_
· exact le_iSup₂_of_le _ (hne β hβ_mem) le_rfl
· have : ↑(-β) ∈ q := by rw [Weight.toLinear... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 385,
"column": 6
} | {
"line": 385,
"column": 31
} | [
{
"pp": "case a\nT : Type u_4\ninst✝¹ : CommRing T\ninst✝ : IsDedekindDomain T\nI J : Ideal T\nhI : I ≠ ⊥\nhJ : J ≠ ⊥\nH : normalizedFactors (normalizedFactors I ∩ normalizedFactors J).prod = normalizedFactors I ∩ normalizedFactors J\nthis : (normalizedFactors I ∩ normalizedFactors J).prod ≠ 0\na : Ideal T\n⊢ c... | rw [Multiset.count_inter] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Lie.Weights.IsSimple | {
"line": 519,
"column": 2
} | {
"line": 521,
"column": 84
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁷ : Field K\ninst✝⁶ : CharZero K\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra K L\ninst✝³ : FiniteDimensional K L\nH : LieSubalgebra K L\ninst✝² : H.IsCartanSubalgebra\ninst✝¹ : IsKilling K L\ninst✝ : IsTriangularizable K (↥H) L\nI : LieIdeal K L\nhI : ∀ (α : ↥LieSubalgebra... | have h_eq : ∀ α : H.root, α ∈ J.rootSet ↔ α ∈ I.rootSet := fun α ↦ by
rw [mem_rootSet_invtSubmoduleToLieIdeal, rootSystem_root_apply]
exact ⟨I.mem_rootSet_of_mem_rootSpan, fun h ↦ Submodule.subset_span ⟨α, h, rfl⟩⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 583,
"column": 4
} | {
"line": 583,
"column": 28
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nK : Type u_3\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Field K\ninst✝ : IsDedekindDomain A\nI : Ideal R\nJ : Ideal A\nf : R ⧸ I →+* A ⧸ J\nhf : Function.Surjective ⇑f\n⊢ Monotone fun X ↦ ⟨comap (Ideal.Quotient.mk J) (map f (map (Ideal.Quotient.mk I) ↑X)), ⋯⟩",
... | rintro ⟨X, hX⟩ ⟨Y, hY⟩ h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 199,
"column": 4
} | {
"line": 199,
"column": 29
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : Module.Finite R M\nl : M →ₗ[R] N\nhl : Function.Surjective ⇑l\ns : Finset N\nhs : Submodule.span R ↑s = ⊤\nhs' : (linearCombination R Subtype.va... | obtain ⟨y, hy⟩ := H (l x) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.MvPolynomial.Derivation | {
"line": 95,
"column": 83
} | {
"line": 95,
"column": 94
} | [
{
"pp": "case monomial\nσ : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid A\ninst✝² : Module R A\ninst✝¹ : Module (MvPolynomial σ R) A\ninst✝ : IsScalarTower R (MvPolynomial σ R) A\nD : MvPolynomial σ R →ₗ[R] A\nh₁ : D 1 = 0\nH : ∀ (s : σ →₀ ℕ) (i : σ), D ((monomial s) 1 ... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 218,
"column": 6
} | {
"line": 218,
"column": 87
} | [
{
"pp": "case hs1\nR : Type u_1\nM : Type u_3\nN : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : FinitePresentation R N\nl : M →ₗ[R] N\nhl : Function.Surjective ⇑l\ninst✝ : FinitePresentation R ↥l.ker\n⊢ (Submodule.map l ⊤).FG",
... | rw [Submodule.map_top, LinearMap.range_eq_top.mpr hl]; exact Module.Finite.fg_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 218,
"column": 6
} | {
"line": 218,
"column": 87
} | [
{
"pp": "case hs1\nR : Type u_1\nM : Type u_3\nN : Type u_2\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : FinitePresentation R N\nl : M →ₗ[R] N\nhl : Function.Surjective ⇑l\ninst✝ : FinitePresentation R ↥l.ker\n⊢ (Submodule.map l ⊤).FG",
... | rw [Submodule.map_top, LinearMap.range_eq_top.mpr hl]; exact Module.Finite.fg_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 922,
"column": 6
} | {
"line": 922,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : IsPrincipalIdealRing R\ninst✝ : NormalizationMonoid R\na b : R\nha : a ∈ normalizedFactors b\nhb : ¬b = 0\nthis : Prime (span {a})\nc : Ideal R\nhc : c ∈ normalizedFactors (span {b})\nhc' : Associated (span {a}) c\n⊢ span {a} ∈ normalized... | rwa [associated_iff_eq.mp hc'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 374,
"column": 2
} | {
"line": 380,
"column": 16
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_4\nN' : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\ninst✝² : AddCommGroup N'\ninst✝¹ : Module R N'\nS : Submonoid R\nf : N →ₗ[R] N'\ninst✝ : IsLocalizedModule S f\ng : M →ₗ[R] N'\nσ : F... | have hi : f ∘ₗ Finsupp.linearCombination R i = (s₀ • g) ∘ₗ π := by
ext j
simp only [LinearMap.coe_comp, Function.comp_apply, Finsupp.lsingle_apply,
linearCombination_single, one_smul, LinearMap.map_smul_of_tower, ← hs, LinearMap.smul_apply,
i, s₀, π]
rw [← mul_smul, Finset.prod_erase_mul]
ex... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.MvPolynomial.PDeriv | {
"line": 83,
"column": 31
} | {
"line": 83,
"column": 40
} | [
{
"pp": "case pos\nR : Type u\nσ : Type v\ninst✝ : CommSemiring R\ni : σ\nm : σ →₀ ℕ\nr : R\nh : m i = 0\n⊢ (monomial (single i 1 + (m - single i 1))) (1 * (r * 0)) = (monomial m) (0 • r)",
"usedConstants": [
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"NonAssocSemiring.toAddCommMonoidWithO... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Extension.Presentation.Basic | {
"line": 218,
"column": 27
} | {
"line": 218,
"column": 46
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nr : R\ninst✝ : IsLocalization.Away r S\nthis :\n aeval (Generators.localizationAway S r).val =\n (↑(mvPolynomialQuotientEquiv S r)).comp (Ideal.Quotient.mkₐ R (Ideal.span {C r * X () - 1}))\n⊢ RingHom.ker ((↑↑(m... | ← RingHom.comap_ker | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Extension.Presentation.Basic | {
"line": 385,
"column": 4
} | {
"line": 385,
"column": 25
} | [
{
"pp": "case add\nR : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nι' : Type u_1\nσ' : Type u_2\nT : Type u_3\ninst✝¹ : CommRing T\ninst✝ : Algebra S T\nQ : Presentation S T ι' σ'\nP : Presentation R S ι σ\nq : MvPolynomial ι' S\nhq : ∃ a, (Q.aux P... | obtain ⟨b, rfl⟩ := hq | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Extension.Generators | {
"line": 111,
"column": 6
} | {
"line": 111,
"column": 24
} | [
{
"pp": "R : Type u\nS : Type v\nι : Type w\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : Generators R S ι\nx y : S\ne : P.σ x = P.σ y\n⊢ x = y",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"MvPolynomial.aeval",
"congr... | ← P.aeval_val_σ x, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Extension.Generators | {
"line": 622,
"column": 4
} | {
"line": 623,
"column": 95
} | [
{
"pp": "case a\nR : Type u\nS : Type v\nι : Type w\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nι' : Type u_3\nT : Type u_7\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nQ : Generators S T ι'\nP : Generators R S ι\nthis : DecidableEq (ι' →₀ ℕ... | convert_to monomial (e.symm (i, 0)) 1 * (Q.toComp P).toAlgHom.toRingHom
(∑ j ∈ (support x).map e.toEmbedding with j.1 = i, monomial j.2 (coeff (e.symm j) x)) ∈ _ | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___elabRules_Mathlib_Tactic_convertTo_1 | Mathlib.Tactic.convertTo |
Mathlib.RingTheory.Extension.Basic | {
"line": 343,
"column": 4
} | {
"line": 344,
"column": 48
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\nP : Extension R S\nR' : Type ?u.172771\nS' : Type ?u.172774\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\ninst✝³ : Algebra R' S'\nP' : Extension R' S'\nR'' : Type ?u.172898\nS'' : Type ?u.172901\ninst✝² : CommRing R'... | have := smul_eq_zero_of_mem (P.σ (r + s) - (P.σ r + P.σ s) : P.Ring) (by simp) x
simpa only [sub_smul, add_smul, sub_eq_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Extension.Basic | {
"line": 343,
"column": 4
} | {
"line": 344,
"column": 48
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\nP : Extension R S\nR' : Type ?u.172771\nS' : Type ?u.172774\ninst✝⁵ : CommRing R'\ninst✝⁴ : CommRing S'\ninst✝³ : Algebra R' S'\nP' : Extension R' S'\nR'' : Type ?u.172898\nS'' : Type ?u.172901\ninst✝² : CommRing R'... | have := smul_eq_zero_of_mem (P.σ (r + s) - (P.σ r + P.σ s) : P.Ring) (by simp) x
simpa only [sub_smul, add_smul, sub_eq_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.Finiteness | {
"line": 63,
"column": 45
} | {
"line": 63,
"column": 56
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\nM : Submonoid R\nS' : Type u_4\ninst✝⁵ : CommRing S'\ninst✝⁴ : Algebra S S'\ninst✝³ : Algebra R S\ninst✝² : Algebra R S'\ninst✝¹ : IsScalarTower R S S'\ninst✝ : IsLocalization (Submonoid.map (algebraMap R S) M) S'\nx : S\ns : Finset ... | smul_assoc, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Localization.Finiteness | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 56
} | [
{
"pp": "case h\nR : Type u\ninst✝¹¹ : CommSemiring R\nS : Submonoid R\nRₚ : Type v\ninst✝¹⁰ : CommSemiring Rₚ\ninst✝⁹ : Algebra R Rₚ\ninst✝⁸ : IsLocalization S Rₚ\nM : Type w\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\nMₚ : Type t\ninst✝⁵ : AddCommMonoid Mₚ\ninst✝⁴ : Module R Mₚ\ninst✝³ : Module Rₚ Mₚ\nins... | simpa using span_eq_top_of_isLocalizedModule Rₚ S f hT | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Extension.Cotangent.Basic | {
"line": 166,
"column": 6
} | {
"line": 166,
"column": 25
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing S\ninst✝⁷ : Algebra R S\nP : Extension R S\nR' : Type u'\nS' : Type v'\ninst✝⁶ : CommRing R'\ninst✝⁵ : CommRing S'\ninst✝⁴ : Algebra R' S'\nP' : Extension R' S'\ninst✝³ : Algebra R R'\ninst✝² : Algebra S S'\ninst✝¹ : Algebra R S'\ninst✝ : I... | CotangentSpace.map, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Prod.TProd | {
"line": 88,
"column": 75
} | {
"line": 88,
"column": 100
} | [
{
"pp": "ι : Type u\nα : ι → Type v\ni j : ι\nl : List ι\ninst✝ : DecidableEq ι\nhj : j ∈ i :: l\nhji : j ≠ i\nv : TProd α (i :: l)\n⊢ v.elim hj = TProd.elim v.2 ⋯",
"usedConstants": [
"of_eq_false",
"eq_false",
"congrArg",
"Membership.mem",
"List.TProd.elim",
"Or.resolve... | by simp [TProd.elim, hji] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.MeasurableSpace.Defs | {
"line": 215,
"column": 2
} | {
"line": 217,
"column": 25
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\np q : α → Prop\nhs : MeasurableSet {x | p x}\nht : MeasurableSet {x | q x}\n⊢ MeasurableSet {x | p x → q x}",
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
"congrArg",
"Compl.compl",
"setOf",
"Set.instUnion",
"id",
... | have h_eq : {x | p x → q x} = {x | p x}ᶜ ∪ {x | q x} := by grind
rw [h_eq]
exact hs.compl.union ht | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.MeasurableSpace.Defs | {
"line": 215,
"column": 2
} | {
"line": 217,
"column": 25
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\np q : α → Prop\nhs : MeasurableSet {x | p x}\nht : MeasurableSet {x | q x}\n⊢ MeasurableSet {x | p x → q x}",
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
"congrArg",
"Compl.compl",
"setOf",
"Set.instUnion",
"id",
... | have h_eq : {x | p x → q x} = {x | p x}ᶜ ∪ {x | q x} := by grind
rw [h_eq]
exact hs.compl.union ht | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.MeasurableSpace.Basic | {
"line": 319,
"column": 2
} | {
"line": 321,
"column": 22
} | [
{
"pp": "case mp\nα : Type u_1\nβ : Type u_2\ns : Set α\nm : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝² : Zero β\ninst✝¹ : MeasurableSingletonClass β\nb : β\ninst✝ : NeZero b\nh : Measurable (s.indicator fun x ↦ b)\n⊢ MeasurableSet s",
"usedConstants": [
"Set.ext",
"Eq.mpr",
"False"... | · convert h (MeasurableSet.singleton (0 : β)).compl
ext a
simp [NeZero.ne b] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.MeasurableSpace.Basic | {
"line": 362,
"column": 2
} | {
"line": 362,
"column": 51
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nC : Set (Set α)\nD : Set (Set β)\ns : ℕ → Set α\nh1s : ∀ (n : ℕ), s n ∈ C\nh2s : ⋃ n, s n = univ\nt : ℕ → Set β\nh1t : ∀ (n : ℕ), t n ∈ D\nh2t : ⋃ n, t n = univ\n⊢ ⋃ n, (fun n ↦ s (Nat.unpair n).1 ×ˢ t (Nat.unpair n).2) n = univ",
"usedConstants": [
"Set.instSProd"... | rw [iUnion_unpair_prod, h2s, h2t, univ_prod_univ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 52,
"column": 8
} | {
"line": 52,
"column": 30
} | [
{
"pp": "case top\nα : Type u_6\ninst✝ : MeasurableSpace α\nf : α → ℕ∞\nh : ∀ (n : ℕ), MeasurableSet (f ⁻¹' {↑n})\n⊢ MeasurableSet (f ⁻¹' {⊤})",
"usedConstants": [
"Eq.mpr",
"MeasurableSet",
"instTopENat",
"congrArg",
"Set.instSingletonSet",
"id",
"Option.none",
... | ← WithTop.none_eq_top, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 551,
"column": 2
} | {
"line": 551,
"column": 60
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nm : MeasurableSpace α\nmβ : MeasurableSpace β\nι : Type u_6\ninst✝¹ : Countable ι\ninst✝ : Nonempty ι\nt : ι → Set α\nt_meas : ∀ (n : ι), MeasurableSet (t n)\ng : ι → α → β\nhg : ∀ (n : ι), Measurable (g n)\nht : Pairwise fun i j ↦ EqOn (g i) (g j) (t i ∩ t j)\ninhabited_h :... | set f : (⋃ i, t i) → β := iUnionLift t g' ht' _ Subset.rfl | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated | {
"line": 170,
"column": 2
} | {
"line": 172,
"column": 73
} | [
{
"pp": "α : Type u_1\ninst✝³ : MeasurableSpace α\nι : Type u_5\ninst✝² : Preorder ι\ninst✝¹ : IsDirectedOrder ι\ninst✝ : atTop.IsCountablyGenerated\ns : ι → Set α\nhsm : Monotone s\nhs : ∀ (i : ι), MeasurableSet (s i)\n⊢ MeasurableSet (⋃ i, s i)",
"usedConstants": [
"MeasurableSet",
"Filter.Fre... | cases isEmpty_or_nonempty ι with
| inl _ => simp
| inr _ => exact .iUnion_of_monotone_of_frequently hsm <| .of_forall hs | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated | {
"line": 170,
"column": 2
} | {
"line": 172,
"column": 73
} | [
{
"pp": "α : Type u_1\ninst✝³ : MeasurableSpace α\nι : Type u_5\ninst✝² : Preorder ι\ninst✝¹ : IsDirectedOrder ι\ninst✝ : atTop.IsCountablyGenerated\ns : ι → Set α\nhsm : Monotone s\nhs : ∀ (i : ι), MeasurableSet (s i)\n⊢ MeasurableSet (⋃ i, s i)",
"usedConstants": [
"MeasurableSet",
"Filter.Fre... | cases isEmpty_or_nonempty ι with
| inl _ => simp
| inr _ => exact .iUnion_of_monotone_of_frequently hsm <| .of_forall hs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.MeasurableSpace.MeasurablyGenerated | {
"line": 170,
"column": 2
} | {
"line": 172,
"column": 73
} | [
{
"pp": "α : Type u_1\ninst✝³ : MeasurableSpace α\nι : Type u_5\ninst✝² : Preorder ι\ninst✝¹ : IsDirectedOrder ι\ninst✝ : atTop.IsCountablyGenerated\ns : ι → Set α\nhsm : Monotone s\nhs : ∀ (i : ι), MeasurableSet (s i)\n⊢ MeasurableSet (⋃ i, s i)",
"usedConstants": [
"MeasurableSet",
"Filter.Fre... | cases isEmpty_or_nonempty ι with
| inl _ => simp
| inr _ => exact .iUnion_of_monotone_of_frequently hsm <| .of_forall hs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 764,
"column": 11
} | {
"line": 764,
"column": 33
} | [
{
"pp": "case nil\nδ : Type u_4\nX : δ → Type u_6\ninst✝ : (i : δ) → MeasurableSpace (X i)\n⊢ Measurable (TProd.mk [])",
"usedConstants": [
"TProd.instMeasurableSpace",
"MeasurableSpace.pi",
"List.TProd",
"measurable_const",
"PUnit.unit",
"List.nil"
]
}
] | exact measurable_const | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 764,
"column": 11
} | {
"line": 764,
"column": 33
} | [
{
"pp": "case nil\nδ : Type u_4\nX : δ → Type u_6\ninst✝ : (i : δ) → MeasurableSpace (X i)\n⊢ Measurable (TProd.mk [])",
"usedConstants": [
"TProd.instMeasurableSpace",
"MeasurableSpace.pi",
"List.TProd",
"measurable_const",
"PUnit.unit",
"List.nil"
]
}
] | exact measurable_const | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 764,
"column": 11
} | {
"line": 764,
"column": 33
} | [
{
"pp": "case nil\nδ : Type u_4\nX : δ → Type u_6\ninst✝ : (i : δ) → MeasurableSpace (X i)\n⊢ Measurable (TProd.mk [])",
"usedConstants": [
"TProd.instMeasurableSpace",
"MeasurableSpace.pi",
"List.TProd",
"measurable_const",
"PUnit.unit",
"List.nil"
]
}
] | exact measurable_const | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 884,
"column": 30
} | {
"line": 884,
"column": 67
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\np q : α → Prop\nhp : Measurable p\nhq : Measurable q\n⊢ MeasurableSet {a | p a ↔ q a}",
"usedConstants": [
"_private.Mathlib.MeasureTheory.MeasurableSpace.Constructions.0.Measurable.iff._simp_1_1",
"Eq.mpr",
"MeasurableSet",
"congrArg... | simp_rw [iff_iff_implies_and_implies] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Order.Interval.Set.OrdConnectedComponent | {
"line": 180,
"column": 4
} | {
"line": 180,
"column": 48
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : LinearOrder α\ns t : Set α\nx a : α\nhas : a ∈ s\nha : [[a, x]] ⊆ tᶜ ∧ [[a, x]] ⊆ (s.ordSeparatingSet t).ordConnectedSectionᶜ\nb : α\nhbt : b ∈ t\nhb : [[b, x]] ⊆ sᶜ ∧ [[b, x]] ⊆ (t.ordSeparatingSet s).ordConnectedSectionᶜ\nH :\n ∀ {α : Type u_1} [inst : LinearOrder α] ... | exact H b hbt hb a has ha (le_of_not_ge hab) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Interval.Set.OrdConnectedComponent | {
"line": 180,
"column": 4
} | {
"line": 180,
"column": 48
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : LinearOrder α\ns t : Set α\nx a : α\nhas : a ∈ s\nha : [[a, x]] ⊆ tᶜ ∧ [[a, x]] ⊆ (s.ordSeparatingSet t).ordConnectedSectionᶜ\nb : α\nhbt : b ∈ t\nhb : [[b, x]] ⊆ sᶜ ∧ [[b, x]] ⊆ (t.ordSeparatingSet s).ordConnectedSectionᶜ\nH :\n ∀ {α : Type u_1} [inst : LinearOrder α] ... | exact H b hbt hb a has ha (le_of_not_ge hab) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Interval.Set.OrdConnectedComponent | {
"line": 180,
"column": 4
} | {
"line": 180,
"column": 48
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : LinearOrder α\ns t : Set α\nx a : α\nhas : a ∈ s\nha : [[a, x]] ⊆ tᶜ ∧ [[a, x]] ⊆ (s.ordSeparatingSet t).ordConnectedSectionᶜ\nb : α\nhbt : b ∈ t\nhb : [[b, x]] ⊆ sᶜ ∧ [[b, x]] ⊆ (t.ordSeparatingSet s).ordConnectedSectionᶜ\nH :\n ∀ {α : Type u_1} [inst : LinearOrder α] ... | exact H b hbt hb a has ha (le_of_not_ge hab) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Interval.Set.OrdConnectedComponent | {
"line": 199,
"column": 2
} | {
"line": 201,
"column": 46
} | [
{
"pp": "case inr.inr\nα✝ : Type u_1\ninst✝¹ : LinearOrder α✝\ns✝ t✝ : Set α✝\nα : Type u_1\ninst✝ : LinearOrder α\ns t : Set α\na : α\nhas : a ∈ s\nb : α\nhbt : b ∈ t\nhab : a ≤ b\nhsub : [[a, b]] ⊆ (s.ordSeparatingSet t).ordConnectedSectionᶜ\nx : { x // x ∈ s.ordSeparatingSet t }\nha : [[a, ↑x]] ⊆ tᶜ\nhb : [[... | have sol1 := fun (hya : y < a) =>
(disjoint_left (t := ordSeparatingSet s t)).1 disjoint_left_ordSeparatingSet has
(hy <| Icc_subset_uIcc' ⟨hya.le, hax⟩) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.LocalRing.Module | {
"line": 184,
"column": 2
} | {
"line": 214,
"column": 72
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsLocalRing R\ninst✝ : FinitePresentation R M\nι : Type u_5\nv : ι → M\nhli : LinearIndependent k (⇑((TensorProduct.mk R k M) 1) ∘ v)\nhsp : Submodule.span k (Set.range (⇑((TensorProduct.mk R k M) 1)... | let iequiv : (ι →₀ R) ≃ₗ[R] M := by
refine LinearEquiv.ofBijective i ⟨?_, hi⟩
-- By Nakayama's lemma, it suffices to show that `k ⊗ ker(i) = 0`.
rw [← LinearMap.ker_eq_bot, ← Submodule.subsingleton_iff_eq_bot,
← IsLocalRing.subsingleton_tensorProduct (R := R), subsingleton_iff_forall_eq 0]
have : ... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Topology.Order.LiminfLimsup | {
"line": 299,
"column": 4
} | {
"line": 299,
"column": 25
} | [
{
"pp": "case inr.refine_1\nι : Type u_1\nα : Type u_7\nβ : Type u_8\ninst✝³ : ConditionallyCompleteLattice α\ninst✝² : CompleteLinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nu : ι → α → β\nc : β\nh_all : ∀ (i : ι), Tendsto (u i) atTop (𝓝 c)\nh_limsup : Tendsto (fun r ↦ limsup (fun i ↦ u ... | filter_upwards with r | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Topology.MetricSpace.Basic | {
"line": 63,
"column": 65
} | {
"line": 63,
"column": 79
} | [
{
"pp": "γ : Type w\ninst✝² : MetricSpace γ\nα : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : DiscreteTopology α\nε : ℝ\nhε : 0 < ε\nf : α → γ\nhf : Pairwise fun x y ↦ ε ≤ dist (f x) (f y)\n⊢ Pairwise fun x y ↦ (f x, f y) ∉ {p | dist p.1 p.2 < ε}",
"usedConstants": [
"Eq.mpr",
"Real",
"P... | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.MetricSpace.Basic | {
"line": 63,
"column": 65
} | {
"line": 63,
"column": 79
} | [
{
"pp": "γ : Type w\ninst✝² : MetricSpace γ\nα : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : DiscreteTopology α\nε : ℝ\nhε : 0 < ε\nf : α → γ\nhf : Pairwise fun x y ↦ ε ≤ dist (f x) (f y)\n⊢ Pairwise fun x y ↦ (f x, f y) ∉ {p | dist p.1 p.2 < ε}",
"usedConstants": [
"Eq.mpr",
"Real",
"P... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Basic | {
"line": 63,
"column": 65
} | {
"line": 63,
"column": 79
} | [
{
"pp": "γ : Type w\ninst✝² : MetricSpace γ\nα : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : DiscreteTopology α\nε : ℝ\nhε : 0 < ε\nf : α → γ\nhf : Pairwise fun x y ↦ ε ≤ dist (f x) (f y)\n⊢ Pairwise fun x y ↦ (f x, f y) ∉ {p | dist p.1 p.2 < ε}",
"usedConstants": [
"Eq.mpr",
"Real",
"P... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Basic | {
"line": 70,
"column": 66
} | {
"line": 70,
"column": 80
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nβ : Type u_2\nε : ℝ\nhε : 0 < ε\nf : β → α\nhf : Pairwise fun x y ↦ ε ≤ dist (f x) (f y)\n⊢ Pairwise fun x y ↦ (f x, f y) ∉ {p | dist p.1 p.2 < ε}",
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"congrArg",
"PartialOrder... | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.MetricSpace.Basic | {
"line": 70,
"column": 66
} | {
"line": 70,
"column": 80
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nβ : Type u_2\nε : ℝ\nhε : 0 < ε\nf : β → α\nhf : Pairwise fun x y ↦ ε ≤ dist (f x) (f y)\n⊢ Pairwise fun x y ↦ (f x, f y) ∉ {p | dist p.1 p.2 < ε}",
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"congrArg",
"PartialOrder... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Basic | {
"line": 70,
"column": 66
} | {
"line": 70,
"column": 80
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nβ : Type u_2\nε : ℝ\nhε : 0 < ε\nf : β → α\nhf : Pairwise fun x y ↦ ε ≤ dist (f x) (f y)\n⊢ Pairwise fun x y ↦ (f x, f y) ∉ {p | dist p.1 p.2 < ε}",
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"congrArg",
"PartialOrder... | simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Basic | {
"line": 226,
"column": 54
} | {
"line": 226,
"column": 71
} | [
{
"pp": "X : Type u_2\nm : PseudoEMetricSpace X\nd : X → X → ℝ≥0∞\nhd : d = edist\n⊢ m.replaceEDist d hd = m",
"usedConstants": [
"PseudoEMetricSpace.ext",
"EDist.ext",
"PseudoEMetricSpace.toEDist",
"PseudoEMetricSpace.replaceEDist"
]
}
] | ext : 2; exact hd | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Basic | {
"line": 226,
"column": 54
} | {
"line": 226,
"column": 71
} | [
{
"pp": "X : Type u_2\nm : PseudoEMetricSpace X\nd : X → X → ℝ≥0∞\nhd : d = edist\n⊢ m.replaceEDist d hd = m",
"usedConstants": [
"PseudoEMetricSpace.ext",
"EDist.ext",
"PseudoEMetricSpace.toEDist",
"PseudoEMetricSpace.replaceEDist"
]
}
] | ext : 2; exact hd | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Basic | {
"line": 252,
"column": 52
} | {
"line": 252,
"column": 69
} | [
{
"pp": "X : Type u_2\nm : PseudoMetricSpace X\nd : X → X → ℝ\nhd : d = dist\n⊢ m.replaceDist d hd = m",
"usedConstants": [
"PseudoMetricSpace.ext",
"Dist.ext",
"PseudoMetricSpace.replaceDist",
"PseudoMetricSpace.toDist"
]
}
] | ext : 2; exact hd | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Basic | {
"line": 252,
"column": 52
} | {
"line": 252,
"column": 69
} | [
{
"pp": "X : Type u_2\nm : PseudoMetricSpace X\nd : X → X → ℝ\nhd : d = dist\n⊢ m.replaceDist d hd = m",
"usedConstants": [
"PseudoMetricSpace.ext",
"Dist.ext",
"PseudoMetricSpace.replaceDist",
"PseudoMetricSpace.toDist"
]
}
] | ext : 2; exact hd | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Basic | {
"line": 282,
"column": 54
} | {
"line": 282,
"column": 71
} | [
{
"pp": "X : Type u_2\nm : EMetricSpace X\nd : X → X → ℝ≥0∞\nhd : d = edist\n⊢ m.replaceEDist d hd = m",
"usedConstants": [
"EMetricSpace.toPseudoEMetricSpace",
"EMetricSpace.ext",
"EDist.ext",
"PseudoEMetricSpace.toEDist",
"EMetricSpace.replaceEDist"
]
}
] | ext : 2; exact hd | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Basic | {
"line": 282,
"column": 54
} | {
"line": 282,
"column": 71
} | [
{
"pp": "X : Type u_2\nm : EMetricSpace X\nd : X → X → ℝ≥0∞\nhd : d = edist\n⊢ m.replaceEDist d hd = m",
"usedConstants": [
"EMetricSpace.toPseudoEMetricSpace",
"EMetricSpace.ext",
"EDist.ext",
"PseudoEMetricSpace.toEDist",
"EMetricSpace.replaceEDist"
]
}
] | ext : 2; exact hd | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Basic | {
"line": 305,
"column": 52
} | {
"line": 305,
"column": 69
} | [
{
"pp": "X : Type u_2\nm : MetricSpace X\nd : X → X → ℝ\nhd : d = dist\n⊢ m.replaceDist d hd = m",
"usedConstants": [
"MetricSpace.ext",
"MetricSpace.replaceDist",
"Dist.ext",
"MetricSpace.toPseudoMetricSpace",
"PseudoMetricSpace.toDist"
]
}
] | ext : 2; exact hd | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Basic | {
"line": 305,
"column": 52
} | {
"line": 305,
"column": 69
} | [
{
"pp": "X : Type u_2\nm : MetricSpace X\nd : X → X → ℝ\nhd : d = dist\n⊢ m.replaceDist d hd = m",
"usedConstants": [
"MetricSpace.ext",
"MetricSpace.replaceDist",
"Dist.ext",
"MetricSpace.toPseudoMetricSpace",
"PseudoMetricSpace.toDist"
]
}
] | ext : 2; exact hd | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Antilipschitz | {
"line": 200,
"column": 2
} | {
"line": 200,
"column": 15
} | [
{
"pp": "β : Type u_2\ninst✝² : PseudoEMetricSpace β\nK : ℝ≥0\nα : Type u_4\ninst✝¹ : EMetricSpace α\ninst✝ : Nontrivial α\nf : α → β\nhf : AntilipschitzWith K f\n⊢ 0 < K",
"usedConstants": [
"Preorder.toLT",
"PartialOrder.toPreorder",
"NNReal",
"NNReal.instZero",
"NNReal.instP... | by_contra! h₀ | Mathlib.Tactic.ByContra._aux_Mathlib_Tactic_ByContra___macroRules_Mathlib_Tactic_ByContra_byContra!_1 | Mathlib.Tactic.ByContra.byContra! |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 84,
"column": 77
} | {
"line": 87,
"column": 40
} | [
{
"pp": "α : Type u_1\na : ℝ≥0∞\nf : α → ℝ≥0∞\nu : Filter α\nha : a ≠ ∞\nhf : ∀ (x : α), f x ≠ ∞\n⊢ Tendsto (ENNReal.toNNReal ∘ f) u (𝓝 a.toNNReal) ↔ Tendsto f u (𝓝 a)",
"usedConstants": [
"NNReal.instTopologicalSpace",
"Eq.mpr",
"ENNReal.ofNNReal",
"congrArg",
"Function.comp... | by
refine ⟨fun h => ?_, fun h => (ENNReal.tendsto_toNNReal ha).comp h⟩
rw [← coe_comp_toNNReal_comp hf]
exact (tendsto_coe_toNNReal ha).comp h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 141,
"column": 46
} | {
"line": 141,
"column": 68
} | [
{
"pp": "α : Type u_1\nm : α → ℝ≥0∞\nf : Filter α\nh : ∀ (n : ℕ), ∀ᶠ (a : α) in f, ↑n < m a\nx : ℝ≥0\nn : ℕ\nhn : x < ↑n\nx✝ : α\n⊢ ↑x < ↑n",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"ENNReal.ofNNReal",
"Preorder.toLT",
"congrArg",
"PartialO... | ← ENNReal.coe_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.OuterMeasure.Basic | {
"line": 161,
"column": 2
} | {
"line": 163,
"column": 41
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝³ : FunLike F (Set α) ℝ≥0∞\ninst✝² : OuterMeasureClass F α\nμ : F\ninst✝¹ : TopologicalSpace α\ninst✝ : SecondCountableTopology α\ns : Set α\nhs : μ s ≠ 0\n⊢ ∃ x ∈ s, ∀ t ∈ 𝓝[s] x, 0 < μ t",
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"Mathl... | contrapose! hs
simp only [nonpos_iff_eq_zero] at hs
exact measure_null_of_locally_null s hs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.OuterMeasure.Basic | {
"line": 161,
"column": 2
} | {
"line": 163,
"column": 41
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝³ : FunLike F (Set α) ℝ≥0∞\ninst✝² : OuterMeasureClass F α\nμ : F\ninst✝¹ : TopologicalSpace α\ninst✝ : SecondCountableTopology α\ns : Set α\nhs : μ s ≠ 0\n⊢ ∃ x ∈ s, ∀ t ∈ 𝓝[s] x, 0 < μ t",
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"Mathl... | contrapose! hs
simp only [nonpos_iff_eq_zero] at hs
exact measure_null_of_locally_null s hs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 439,
"column": 4
} | {
"line": 442,
"column": 80
} | [
{
"pp": "case neg\na : ℝ≥0∞\na_infty : ¬a = ∞\n⊢ Continuous fun x ↦ x - a",
"usedConstants": [
"Eq.mpr",
"False",
"Continuous",
"eq_false",
"congrArg",
"ContinuousOn.comp_continuous",
"ENNReal.continuousOn_sub",
"continuous_const",
"setOf",
"_priva... | rw [show (fun x => x - a) = (fun p : ℝ≥0∞ × ℝ≥0∞ => p.fst - p.snd) ∘ fun x => ⟨x, a⟩ by rfl]
apply continuousOn_sub.comp_continuous (by fun_prop)
intro x
simp only [a_infty, Ne, mem_setOf_eq, Prod.mk_inj, and_false, not_false_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 439,
"column": 4
} | {
"line": 442,
"column": 80
} | [
{
"pp": "case neg\na : ℝ≥0∞\na_infty : ¬a = ∞\n⊢ Continuous fun x ↦ x - a",
"usedConstants": [
"Eq.mpr",
"False",
"Continuous",
"eq_false",
"congrArg",
"ContinuousOn.comp_continuous",
"ENNReal.continuousOn_sub",
"continuous_const",
"setOf",
"_priva... | rw [show (fun x => x - a) = (fun p : ℝ≥0∞ × ℝ≥0∞ => p.fst - p.snd) ∘ fun x => ⟨x, a⟩ by rfl]
apply continuousOn_sub.comp_continuous (by fun_prop)
intro x
simp only [a_infty, Ne, mem_setOf_eq, Prod.mk_inj, and_false, not_false_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 727,
"column": 74
} | {
"line": 728,
"column": 60
} | [
{
"pp": "a b : ℝ\nh : a ≤ b\n⊢ Metric.diam (Icc a b) = b - a",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Real.instLE",
"Real",
"ENNReal.ofReal",
"congrArg",
"Real.instSub",
"covariant_swap_add_of_covariant_add",
"HSub.hSub",
"R... | by
simp [Metric.diam, ENNReal.toReal_ofReal (sub_nonneg.2 h)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 730,
"column": 74
} | {
"line": 731,
"column": 60
} | [
{
"pp": "a b : ℝ\nh : a ≤ b\n⊢ Metric.diam (Ico a b) = b - a",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Real.instLE",
"Real",
"ENNReal.ofReal",
"congrArg",
"Real.instSub",
"covariant_swap_add_of_covariant_add",
"HSub.hSub",
"R... | by
simp [Metric.diam, ENNReal.toReal_ofReal (sub_nonneg.2 h)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 733,
"column": 74
} | {
"line": 734,
"column": 60
} | [
{
"pp": "a b : ℝ\nh : a ≤ b\n⊢ Metric.diam (Ioc a b) = b - a",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Set.Ioc",
"Real.instLE",
"Real",
"ENNReal.ofReal",
"congrArg",
"Real.instSub",
"covariant_swap_add_of_covariant_add",
"Rea... | by
simp [Metric.diam, ENNReal.toReal_ofReal (sub_nonneg.2 h)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 736,
"column": 74
} | {
"line": 737,
"column": 60
} | [
{
"pp": "a b : ℝ\nh : a ≤ b\n⊢ Metric.diam (Ioo a b) = b - a",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Real.instLE",
"Real",
"ENNReal.ofReal",
"congrArg",
"Real.instSub",
"covariant_swap_add_of_covariant_add",
"HSub.hSub",
"R... | by
simp [Metric.diam, ENNReal.toReal_ofReal (sub_nonneg.2 h)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 206,
"column": 4
} | {
"line": 209,
"column": 41
} | [
{
"pp": "case pos\nx✝ : EReal\nh_top : ¬x✝ = ⊤\nx : EReal\nhx : x ∈ {⊤}ᶜ\nh_bot : x = ⊥\n⊢ ContinuousAt (fun x ↦ ENNReal.ofReal x.toReal) x",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Preorder.toLT",
"Real.instZero",
"ENNReal.ofReal",
"congrArg",
"E... | refine tendsto_nhds_of_eventually_eq ?_
rw [h_bot, nhds_bot_basis.eventually_iff]
simpa [toReal_bot, ENNReal.ofReal_zero, ENNReal.ofReal_eq_zero, true_and] using
⟨0, fun _ hx ↦ toReal_nonpos hx.le⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 206,
"column": 4
} | {
"line": 209,
"column": 41
} | [
{
"pp": "case pos\nx✝ : EReal\nh_top : ¬x✝ = ⊤\nx : EReal\nhx : x ∈ {⊤}ᶜ\nh_bot : x = ⊥\n⊢ ContinuousAt (fun x ↦ ENNReal.ofReal x.toReal) x",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Preorder.toLT",
"Real.instZero",
"ENNReal.ofReal",
"congrArg",
"E... | refine tendsto_nhds_of_eventually_eq ?_
rw [h_bot, nhds_bot_basis.eventually_iff]
simpa [toReal_bot, ENNReal.ofReal_zero, ENNReal.ofReal_eq_zero, true_and] using
⟨0, fun _ hx ↦ toReal_nonpos hx.le⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Semicontinuity.Defs | {
"line": 711,
"column": 11
} | {
"line": 711,
"column": 37
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\ns : Set α\n⊢ LowerHemicontinuousOn f s ↔ ∀ x ∈ s, ∀ (t : Set β), IsClosed t → (∃ᶠ (x' : α) in 𝓝[s] x, f x' ⊆ t) → f x ⊆ t",
"usedConstants": [
"Eq.mpr",
"congrArg",
"nhdsWithin",
... | lowerHemicontinuousOn_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 305,
"column": 2
} | {
"line": 314,
"column": 46
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\n⊢ limsup (fun x ↦ c * u x) f = c * limsup u f",
"usedConstants": [
"Eq.mpr",
"LE.le.eq_or_lt",
"EReal.instDivInvMonoid",
"False",
"Filter.limsup_const",
"Preorder.toLT"... | obtain rfl | h₃ := h₁.eq_or_lt
· simp
simp_rw [EReal.mul_comm (x := c)]
apply eq_of_le_of_ge
· rw [limsup_le_iff]
simpa [← EReal.lt_div_iff (by aesop) (by aesop)]
using fun _ ↦ eventually_lt_of_limsup_lt
· rw [le_limsup_iff]
simpa [← EReal.div_lt_iff (by aesop) (by aesop)]
using fun _ ↦ fr... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 305,
"column": 2
} | {
"line": 314,
"column": 46
} | [
{
"pp": "α : Type u_3\nf : Filter α\nu : α → EReal\ninst✝ : f.NeBot\nc : EReal\nh₁ : 0 ≤ c\nh₂ : c ≠ ⊤\n⊢ limsup (fun x ↦ c * u x) f = c * limsup u f",
"usedConstants": [
"Eq.mpr",
"LE.le.eq_or_lt",
"EReal.instDivInvMonoid",
"False",
"Filter.limsup_const",
"Preorder.toLT"... | obtain rfl | h₃ := h₁.eq_or_lt
· simp
simp_rw [EReal.mul_comm (x := c)]
apply eq_of_le_of_ge
· rw [limsup_le_iff]
simpa [← EReal.lt_div_iff (by aesop) (by aesop)]
using fun _ ↦ eventually_lt_of_limsup_lt
· rw [le_limsup_iff]
simpa [← EReal.div_lt_iff (by aesop) (by aesop)]
using fun _ ↦ fr... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.InfiniteSum.ENNReal | {
"line": 451,
"column": 72
} | {
"line": 454,
"column": 32
} | [
{
"pp": "α : Type u_1\nf g : α → ℝ≥0\nsf sg : ℝ≥0\ni : α\nh : ∀ (a : α), f a ≤ g a\nhi : f i < g i\nhf : HasSum f sf\nhg : HasSum g sg\n⊢ sf < sg",
"usedConstants": [
"NNReal.instTopologicalSpace",
"Iff.mpr",
"Real.partialOrder",
"Real.instLE",
"Real",
"Preorder.toLT",
... | by
have A : ∀ a : α, (f a : ℝ) ≤ g a := fun a => NNReal.coe_le_coe.2 (h a)
have : (sf : ℝ) < sg := hasSum_lt A (NNReal.coe_lt_coe.2 hi) (hasSum_coe.2 hf) (hasSum_coe.2 hg)
exact NNReal.coe_lt_coe.1 this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 509,
"column": 2
} | {
"line": 509,
"column": 63
} | [
{
"pp": "case h.refine_2.refine_2.refine_2\nx : ℝ\n⊢ (⊤, ⊤).2 ∈ Ioi 1",
"usedConstants": [
"Eq.mpr",
"Real",
"Set.Ioi",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"EReal",
"Membership.mem",
"Set.mem_Ioi",
"instTopEReal",
"id",
... | · rw [Set.mem_Ioi, ← EReal.coe_one]; exact EReal.coe_lt_top 1 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 564,
"column": 4
} | {
"line": 564,
"column": 34
} | [
{
"pp": "case top.top\nh₁ : (⊤, ⊤).1 ≠ 0 ∨ (⊤, ⊤).2 ≠ ⊥\nh₂ : (⊤, ⊤).1 ≠ 0 ∨ (⊤, ⊤).2 ≠ ⊤\nh₃ : (⊤, ⊤).1 ≠ ⊥ ∨ (⊤, ⊤).2 ≠ 0\nh₄ : (⊤, ⊤).1 ≠ ⊤ ∨ (⊤, ⊤).2 ≠ 0\n⊢ ContinuousAt (fun p ↦ p.1 * p.2) (⊤, ⊤)",
"usedConstants": [
"_private.Mathlib.Topology.Instances.EReal.Lemmas.0.EReal.continuousAt_mul_top_t... | exact continuousAt_mul_top_top | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 564,
"column": 4
} | {
"line": 564,
"column": 34
} | [
{
"pp": "case top.top\nh₁ : (⊤, ⊤).1 ≠ 0 ∨ (⊤, ⊤).2 ≠ ⊥\nh₂ : (⊤, ⊤).1 ≠ 0 ∨ (⊤, ⊤).2 ≠ ⊤\nh₃ : (⊤, ⊤).1 ≠ ⊥ ∨ (⊤, ⊤).2 ≠ 0\nh₄ : (⊤, ⊤).1 ≠ ⊤ ∨ (⊤, ⊤).2 ≠ 0\n⊢ ContinuousAt (fun p ↦ p.1 * p.2) (⊤, ⊤)",
"usedConstants": [
"_private.Mathlib.Topology.Instances.EReal.Lemmas.0.EReal.continuousAt_mul_top_t... | exact continuousAt_mul_top_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.EReal.Lemmas | {
"line": 564,
"column": 4
} | {
"line": 564,
"column": 34
} | [
{
"pp": "case top.top\nh₁ : (⊤, ⊤).1 ≠ 0 ∨ (⊤, ⊤).2 ≠ ⊥\nh₂ : (⊤, ⊤).1 ≠ 0 ∨ (⊤, ⊤).2 ≠ ⊤\nh₃ : (⊤, ⊤).1 ≠ ⊥ ∨ (⊤, ⊤).2 ≠ 0\nh₄ : (⊤, ⊤).1 ≠ ⊤ ∨ (⊤, ⊤).2 ≠ 0\n⊢ ContinuousAt (fun p ↦ p.1 * p.2) (⊤, ⊤)",
"usedConstants": [
"_private.Mathlib.Topology.Instances.EReal.Lemmas.0.EReal.continuousAt_mul_top_t... | exact continuousAt_mul_top_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Semicontinuity.Basic | {
"line": 85,
"column": 2
} | {
"line": 93,
"column": 28
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\nf : α → β\ninst✝ : LinearOrder β\ns : Set α\nne_s : s.Nonempty\nhs : IsCompact s\nhf : LowerSemicontinuousOn f s\nx✝¹ : Nonempty α\nx✝ : Nonempty ↑s\nφ : β → Filter α := fun b ↦ 𝓟 (s ∩ f ⁻¹' Iic b)\nℱ : Filter α := ⨅ a, φ (f ↑a)\n⊢ ∃ a ∈ s, ∀ x ... | have : ℱ.NeBot := by
apply iInf_neBot_of_directed _ _
· change Directed GE.ge (fun x ↦ (φ ∘ (fun (a : s) ↦ f ↑a)) x)
exact Directed.mono_comp GE.ge (fun x y hxy ↦
principal_mono.mpr (inter_subset_inter_right _ (preimage_mono <| Iic_subset_Iic.mpr hxy)))
(Std.Total.directed _)
· intro x... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.OuterMeasure.OfFunction | {
"line": 217,
"column": 4
} | {
"line": 218,
"column": 23
} | [
{
"pp": "case refine_1\nα : Type u_1\nm : Set α → ℝ≥0∞\nm_empty : m ∅ = 0\nβ : Type u_2\nf : α → β\nhf : Injective f\ns : Set β\nt : ℕ → Set α\nht : f ⁻¹' s ⊆ iUnion t\n⊢ s ⊆ ⋃ n, (range f)ᶜ ∪ f '' t n",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"Set.image_mono",
... | rw [← union_iUnion, ← inter_subset, ← image_preimage_eq_inter_range, ← image_iUnion]
exact image_mono ht | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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