module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Data.ENNReal.Inv | {
"line": 301,
"column": 2
} | {
"line": 301,
"column": 28
} | [
{
"pp": "a b : ℝ≥0∞\n⊢ a⁻¹ ≤ b ↔ b⁻¹ ≤ a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.IsLUB | {
"line": 253,
"column": 51
} | {
"line": 253,
"column": 92
} | [
{
"pp": "α : Type u_3\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\na b : α\nx✝ : a ≤ b\nf : Ultrafilter α\nhfab : Icc a b ∈ ↑f\nhf : ∀ x ∈ Icc a b, ¬↑f ≤ 𝓝 x\nhpt : ∀ x ∈ Icc a b, {x} ∉ f\ns : Set α := {x | x ∈ Icc a b ∧ Icc a x ∉ f}\nhsb : b ∈ upperBounds... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 304,
"column": 2
} | {
"line": 304,
"column": 28
} | [
{
"pp": "a b : ℝ≥0∞\n⊢ a ≤ b⁻¹ ↔ b ≤ a⁻¹",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.IsLUB | {
"line": 253,
"column": 51
} | {
"line": 253,
"column": 100
} | [
{
"pp": "α : Type u_3\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\na b : α\nx✝ : a ≤ b\nf : Ultrafilter α\nhfab : Icc a b ∈ ↑f\nhf : ∀ x ∈ Icc a b, ¬↑f ≤ 𝓝 x\nhpt : ∀ x ∈ Icc a b, {x} ∉ f\ns : Set α := {x | x ∈ Icc a b ∧ Icc a x ∉ f}\nhsb : b ∈ upperBounds... | simpa [nhds_eq_order, eq_true this] using hf c hc | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.Order.IsLUB | {
"line": 253,
"column": 51
} | {
"line": 253,
"column": 100
} | [
{
"pp": "α : Type u_3\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\na b : α\nx✝ : a ≤ b\nf : Ultrafilter α\nhfab : Icc a b ∈ ↑f\nhf : ∀ x ∈ Icc a b, ¬↑f ≤ 𝓝 x\nhpt : ∀ x ∈ Icc a b, {x} ∉ f\ns : Set α := {x | x ∈ Icc a b ∧ Icc a x ∉ f}\nhsb : b ∈ upperBounds... | simpa [nhds_eq_order, eq_true this] using hf c hc | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.IsLUB | {
"line": 253,
"column": 51
} | {
"line": 253,
"column": 100
} | [
{
"pp": "α : Type u_3\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\na b : α\nx✝ : a ≤ b\nf : Ultrafilter α\nhfab : Icc a b ∈ ↑f\nhf : ∀ x ∈ Icc a b, ¬↑f ≤ 𝓝 x\nhpt : ∀ x ∈ Icc a b, {x} ∉ f\ns : Set α := {x | x ∈ Icc a b ∧ Icc a x ∉ f}\nhsb : b ∈ upperBounds... | simpa [nhds_eq_order, eq_true this] using hf c hc | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ENNReal.Inv | {
"line": 388,
"column": 23
} | {
"line": 388,
"column": 39
} | [
{
"pp": "a b c : ℝ≥0∞\nh : a ≤ b * c\nh0 : c = 0\n⊢ a = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Monotone | {
"line": 413,
"column": 6
} | {
"line": 413,
"column": 73
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ninst✝⁵ : LinearOrder α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : OrderTopology α\ninst✝² : ConditionallyCompleteLinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nf : α → β\nx y : α\nh_nonempty : (Ioo y x).Nonempty\nMf : MonotoneOn f (Ioo y x)\nh_bdd : BddAbov... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 445,
"column": 89
} | {
"line": 450,
"column": 43
} | [
{
"pp": "a b : ℝ≥0∞\nha : a ≠ ∞\nhb₀ : b ≠ 0\n⊢ ∃ c, 0 < c ∧ c * a < b",
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"zero_le",
"Eq.mpr",
"ENNReal.instIsOrderedRing",
"ENNReal.instAdd",
"Preorder.toLT",
"instHDiv",
"HMul.hMul",
"IsOrderedRing... | by
obtain rfl | hb := eq_or_ne b ∞
· exact ⟨1, by simpa [lt_top_iff_ne_top]⟩
refine ⟨b / (a + 1), ENNReal.div_pos hb₀ (by finiteness), ENNReal.mul_lt_of_lt_div ?_⟩
gcongr
exact ENNReal.lt_add_right ha one_ne_zero | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.ENNReal.Inv | {
"line": 453,
"column": 6
} | {
"line": 453,
"column": 16
} | [
{
"pp": "a b : ℝ≥0∞\nh₁ : b = ∞ → a ≠ 0\nh₂ : a = ∞ → b ≠ 0\n⊢ a⁻¹ ≤ b ↔ 1 ≤ a * b",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiring",
"id",
"H... | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.ENNReal.Inv | {
"line": 458,
"column": 6
} | {
"line": 458,
"column": 16
} | [
{
"pp": "a b : ℝ≥0∞\n⊢ a ≤ b⁻¹ ↔ a * b ≤ 1",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiring",
"id",
"HDiv.hDiv",
"DivInvMonoid.toMonoid"... | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.ENNReal.Inv | {
"line": 472,
"column": 6
} | {
"line": 472,
"column": 18
} | [
{
"pp": "a b : ℝ≥0∞\nh : a * b = 1\n⊢ a = b⁻¹",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"id",
"MulOne.toMul",
"ENNReal.instCommSemiring",
"MulZeroOneClass.toMulOneClass",
"instMulZeroOneClass... | ← mul_one a, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.IsLUB | {
"line": 256,
"column": 27
} | {
"line": 256,
"column": 58
} | [
{
"pp": "α : Type u_3\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\na b : α\nx✝ : a ≤ b\nf : Ultrafilter α\nhfab : Icc a b ∈ ↑f\nhf : ∀ x ∈ Icc a b, ¬↑f ≤ 𝓝 x\nhpt : ∀ x ∈ Icc a b, {x} ∉ f\ns : Set α := {x | x ∈ Icc a b ∧ Icc a x ∉ f}\nhsb : b ∈ upperBounds... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Monotone | {
"line": 431,
"column": 6
} | {
"line": 431,
"column": 68
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ninst✝⁵ : LinearOrder α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : OrderTopology α\ninst✝² : ConditionallyCompleteLinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nf : α → β\nx y : α\nh_nonempty : (Ioo x y).Nonempty\nMf : MonotoneOn f (Ioo x y)\nh_bdd : BddBelo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Monotone | {
"line": 443,
"column": 6
} | {
"line": 443,
"column": 73
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ninst✝⁵ : LinearOrder α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : OrderTopology α\ninst✝² : ConditionallyCompleteLinearOrder β\ninst✝¹ : TopologicalSpace β\ninst✝ : OrderTopology β\nf : α → β\nx : α\nMf : MonotoneOn f (Iio x)\nh_bdd : BddAbove (f '' Iio x)\nh : (Iio x).Nonempty\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 579,
"column": 6
} | {
"line": 579,
"column": 16
} | [
{
"pp": "⊢ 1 - 2⁻¹ = 2⁻¹",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"DivInvMonoid.toInv",
"instHDiv",
"Monoid.toMulOneClass",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"HSub.hSub",
"id",
"HDiv.hDiv",
"DivInvMonoid.toMonoid",
"AddMo... | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.ENNReal.Inv | {
"line": 608,
"column": 2
} | {
"line": 608,
"column": 94
} | [
{
"pp": "case refine_1\na b c d : ℝ≥0∞\nr p q : ℝ≥0\nx y : ℝ≥0∞\nhxy : x < y\n⊢ (fun x ↦ ⟨(x⁻¹ + 1)⁻¹, ⋯⟩) x < (fun x ↦ ⟨(x⁻¹ + 1)⁻¹, ⋯⟩) y",
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"Iff.mpr",
"Eq.mpr",
"ENNReal.instAdd",
"Preorder.toLT",
"PartialOrder.toP... | · simpa only [Subtype.mk_lt_mk, ENNReal.inv_lt_inv, ENNReal.add_lt_add_iff_right one_ne_top] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Order.IsLUB | {
"line": 386,
"column": 2
} | {
"line": 386,
"column": 13
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : DenselyOrdered α\ninst✝ : FirstCountableTopology α\nx y : α\nhy : x < y\n⊢ ∃ u, StrictAnti u ∧ (∀ (n : ℕ), u n ∈ Ioo x y) ∧ Tendsto u atTop (𝓝 x)",
"usedConstants": [
"Eq.mpr",
"Preord... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.IsLUB | {
"line": 415,
"column": 2
} | {
"line": 415,
"column": 13
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : DenselyOrdered α\ninst✝ : FirstCountableTopology α\ns : Set α\nhs : Dense s\nx y : α\nhy : x < y\n⊢ ∃ u, StrictAnti u ∧ (∀ (n : ℕ), u n ∈ Ioo x y ∩ s) ∧ Tendsto u atTop (𝓝 x)",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.IsLUB | {
"line": 426,
"column": 2
} | {
"line": 426,
"column": 13
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : LinearOrder α\ninst✝³ : OrderTopology α\nβ : Type u_3\ninst✝² : LinearOrder β\ninst✝¹ : DenselyOrdered α\ninst✝ : FirstCountableTopology α\nf : β → α\nx y : α\nhf : DenseRange f\nhmono : Monotone f\nhlt : x < y\n⊢ ∃ u, StrictAnti u ∧ (∀ (n : ℕ), f (u ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 687,
"column": 4
} | {
"line": 687,
"column": 15
} | [
{
"pp": "case ofNat\na : ℝ≥0∞\nha : a ≠ 0\nh'a : a ≠ ∞\na✝ : ℕ\n⊢ 0 < a ^ Int.ofNat a✝",
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"DivInvMonoid.toZPow",
"id",
"DivInvMonoid.toMonoid",
"Int.ofNat... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 693,
"column": 4
} | {
"line": 693,
"column": 15
} | [
{
"pp": "case ofNat\na : ℝ≥0∞\nha : a ≠ 0\nh'a : a ≠ ∞\na✝ : ℕ\n⊢ a ^ Int.ofNat a✝ < ∞",
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"DivInvMonoid.toZPow",
"id",
"DivInvMonoid.toMonoid",
"Int.ofNat... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 704,
"column": 23
} | {
"line": 704,
"column": 57
} | [
{
"pp": "x : ℝ≥0\nhx : ↑x ≠ 0\ny : ℝ≥0\nhy : 1 < ↑y\n⊢ y ≠ 0",
"usedConstants": [
"id",
"NNReal",
"Ne",
"NNReal.instZero",
"Zero.toOfNat0",
"OfNat.ofNat"
]
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 707,
"column": 4
} | {
"line": 707,
"column": 38
} | [
{
"pp": "x : ℝ≥0\nhx : ↑x ≠ 0\ny : ℝ≥0\nhy : 1 < ↑y\nA : y ≠ 0\n⊢ x ≠ 0",
"usedConstants": [
"id",
"NNReal",
"Ne",
"NNReal.instZero",
"Zero.toOfNat0",
"OfNat.ofNat"
]
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.LiminfLimsup | {
"line": 409,
"column": 4
} | {
"line": 409,
"column": 15
} | [
{
"pp": "case inl\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 i r) cofi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.IsLUB | {
"line": 452,
"column": 6
} | {
"line": 452,
"column": 52
} | [
{
"pp": "case h\nα : Type u_1\nγ : Type u_2\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : FirstCountableTopology α\nl : Filter γ\ninst✝ : CountableInterFilter l\nf : γ → α\na d : α\nhd : IsGLB (Ioi a) d\nH0 : Iic a = Iio d\nh : ∀ᶠ (x : γ) in l, f x < d\nx : γ\nhx : f x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 716,
"column": 23
} | {
"line": 716,
"column": 57
} | [
{
"pp": "x : ℝ≥0\nhx : ↑x ≠ 0\ny : ℝ≥0\nhy : 1 < ↑y\n⊢ y ≠ 0",
"usedConstants": [
"id",
"NNReal",
"Ne",
"NNReal.instZero",
"Zero.toOfNat0",
"OfNat.ofNat"
]
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 719,
"column": 4
} | {
"line": 719,
"column": 38
} | [
{
"pp": "x : ℝ≥0\nhx : ↑x ≠ 0\ny : ℝ≥0\nhy : 1 < ↑y\nA : y ≠ 0\n⊢ x ≠ 0",
"usedConstants": [
"id",
"NNReal",
"Ne",
"NNReal.instZero",
"Zero.toOfNat0",
"OfNat.ofNat"
]
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.LiminfLimsup | {
"line": 427,
"column": 68
} | {
"line": 427,
"column": 79
} | [
{
"pp": "ι : 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 i r) cofinite) atTo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 749,
"column": 4
} | {
"line": 749,
"column": 85
} | [
{
"pp": "case negSucc.negSucc.a\nx : ℝ≥0∞\nhx : 1 ≤ x\na b : ℕ\nh : Int.negSucc a ≤ Int.negSucc b\n⊢ b + 1 ≤ a + 1",
"usedConstants": [
"Eq.mpr",
"Preorder.toLE",
"id",
"instOfNatNat",
"Int",
"LE.le",
"Nat.cast",
"instHAdd",
"_private.Mathlib.Data.ENNRea... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 757,
"column": 27
} | {
"line": 757,
"column": 61
} | [
{
"pp": "m n : ℤ\nx : ℝ≥0\nhx : ↑x ≠ 0\n⊢ x ≠ 0",
"usedConstants": [
"id",
"NNReal",
"Ne",
"NNReal.instZero",
"Zero.toOfNat0",
"OfNat.ofNat"
]
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.LiminfLimsup | {
"line": 463,
"column": 44
} | {
"line": 463,
"column": 55
} | [
{
"pp": "ι : 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 i r) cofinite) atTo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 781,
"column": 23
} | {
"line": 781,
"column": 34
} | [
{
"pp": "x : ℝ≥0∞\nhx : 1 ≤ x\nm : ℤ\nhn : -m ≤ 0\n⊢ 0 ≤ m",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 814,
"column": 4
} | {
"line": 815,
"column": 11
} | [
{
"pp": "case neg.inr.inl\nι : Sort u_1\nf : ι → ℝ≥0∞\nhf : ¬∀ (i : ι), f i = 0\nha₀ : ∞ ≠ 0\ni : ι\nhi : ¬f i = 0\n⊢ ∞ * ⨆ i, f i = ⨆ i, ∞ * f i",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"False",
"HMul.hMul",
"eq_false",
"congrArg",
"CommSemiring.toSemiring",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.LiminfLimsup | {
"line": 512,
"column": 6
} | {
"line": 513,
"column": 13
} | [
{
"pp": "R : Type u_4\nS : Type u_5\ninst✝⁶ : ConditionallyCompleteLinearOrder R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : OrderTopology R\ninst✝³ : ConditionallyCompleteLinearOrder S\ninst✝² : TopologicalSpace S\ninst✝¹ : OrderTopology S\nF : Filter R\ninst✝ : F.NeBot\nf : R → S\nf_decr : Antitone f\nf_cont : Con... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 851,
"column": 40
} | {
"line": 851,
"column": 69
} | [
{
"pp": "ι : Sort u_1\nf : ι → ℝ≥0∞\na : ℝ≥0∞\nhinfty : a = ∞ → ⨅ i, f i = 0 → ∃ i, f i = 0\nh₀ : a = 0 → Nonempty ι\n⊢ (⨅ i, f i) * a = ⨅ i, f i * a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.LiminfLimsup | {
"line": 534,
"column": 6
} | {
"line": 534,
"column": 25
} | [
{
"pp": "R : Type u_4\nS : Type u_5\ninst✝⁶ : ConditionallyCompleteLinearOrder R\ninst✝⁵ : TopologicalSpace R\ninst✝⁴ : OrderTopology R\ninst✝³ : ConditionallyCompleteLinearOrder S\ninst✝² : TopologicalSpace S\ninst✝¹ : OrderTopology S\nF : Filter R\ninst✝ : F.NeBot\nf : R → S\nf_decr : Antitone f\nf_cont : Con... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 914,
"column": 2
} | {
"line": 914,
"column": 31
} | [
{
"pp": "ι : Sort u_1\nκ : Sort u_2\nf : ι → ℝ≥0∞\na : ℝ≥0∞\ng : κ → ℝ≥0∞\nhf : ∃ i, f i ≠ ∞\nhg : ∃ j, g j ≠ ∞\nha : ∀ (i : ι) (j : κ), a ≤ f i * g j\n⊢ a ≤ (⨅ i, f ↑i) * ⨅ j, g j",
"usedConstants": [
"Iff.mpr",
"Exists",
"Subtype",
"Ne",
"nonempty_subtype",
"ENNReal",
... | have := nonempty_subtype.2 hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.ENNReal.Inv | {
"line": 916,
"column": 34
} | {
"line": 916,
"column": 45
} | [
{
"pp": "ι : Sort u_1\nκ : Sort u_2\nf : ι → ℝ≥0∞\na : ℝ≥0∞\ng : κ → ℝ≥0∞\nhf : ∃ i, f i ≠ ∞\nhg : ∃ j, g j ≠ ∞\nha : ∀ (i : ι) (j : κ), a ≤ f i * g j\nthis✝ : Nonempty { a // f a ≠ ∞ }\nthis : Nonempty κ\n⊢ ⨅ j, g j ≠ ∞",
"usedConstants": [
"iInf_eq_top._simp_1",
"Eq.mpr",
"iInf",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 919,
"column": 2
} | {
"line": 919,
"column": 44
} | [
{
"pp": "ι : Sort u_1\nκ : Sort u_2\nf : ι → ℝ≥0∞\na : ℝ≥0∞\ng : κ → ℝ≥0∞\nhf : ∃ i, f i ≠ ∞\nha : ∀ (i : ι) (j : κ), a ≤ f i * g j\nthis✝ : Nonempty { a // f a ≠ ∞ }\nthis : Nonempty κ\nhg : ⨅ j, g j ≠ ∞\ni : ι\nhi : f i ≠ ∞\n⊢ a ≤ f ↑⟨i, hi⟩ * ⨅ j, g j",
"usedConstants": [
"Eq.mpr",
"iInf",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.Defs | {
"line": 458,
"column": 33
} | {
"line": 458,
"column": 44
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nx y : α\nε : ℝ≥0∞\nh : y ∈ eball x ε\n⊢ 0 < ε - edist y x",
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"Eq.mpr",
"tsub_pos_iff_lt._simp_1",
"Preorder.toLT",
"ENNReal.instOrderedSub",
"ENNReal.instAddCommMonoi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.LiminfLimsup | {
"line": 648,
"column": 2
} | {
"line": 648,
"column": 57
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace α\ninst✝² : OrderTopology α\ninst✝¹ : DenselyOrdered α\ninst✝ : CompleteLattice β\nf : α → β\nhf : Monotone f\na : α\nhb : ∃ b, a < b\n⊢ limsup f (𝓝[>] a) = ⨅ r, ⨅ (_ : r > a), f r",
"usedConstants": [
"Eq.mpr",
... | rw [(nhdsGT_basis_of_exists_gt hb).limsup_eq_iInf_iSup] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.EMetricSpace.Basic | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 47
} | [
{
"pp": "α : Type u_2\ninst✝ : EMetricSpace α\n⊢ Subsingleton α ↔ IndiscreteTopology α",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.Basic | {
"line": 251,
"column": 4
} | {
"line": 251,
"column": 38
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nhs : ∀ ε > 0, ∃ t, t.Countable ∧ ⋃ x ∈ t, closedEBall x ε = univ\n⊢ ∀ ε > 0, ∃ t, t.Countable ∧ univ ⊆ ⋃ x ∈ t, closedEBall x ε",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"_private.Mathlib.Topology.EMetricSpace.Basic.0.EMetric.secondC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.Basic | {
"line": 325,
"column": 2
} | {
"line": 326,
"column": 9
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\ns : Set α\nι : Sort u_2\nc : ι → Set α\nhs : IsCompact s\nhc₁ : ∀ (i : ι), IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] (c i)\nhc₂ : s ⊆ ⋃ i, c i\n⊢ ∃ δ > 0, ∀ x ∈ s, ∃ i, eball x δ ⊆ c i",
"usedConstants": [
"PseudoEMetricSpace.edist_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.Basic | {
"line": 330,
"column": 2
} | {
"line": 331,
"column": 9
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\ns : Set α\nc : (x : α) → x ∈ s → Set α\nhs : IsCompact s\nhc : ∀ (x : α) (hx : x ∈ s), c x hx ∈ 𝓝 x\n⊢ ∃ δ > 0, ∀ x ∈ s, ∃ y, eball x δ ⊆ c ↑y ⋯",
"usedConstants": [
"PseudoEMetricSpace.edist_comm",
"Eq.mpr",
"PseudoEMetricSpace.toWeakPse... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.Basic | {
"line": 335,
"column": 2
} | {
"line": 336,
"column": 9
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\ns : Set α\nc : α → Set α\nhs : IsCompact s\nhc : ∀ x ∈ s, c x ∈ 𝓝 x\n⊢ ∃ δ > 0, ∀ x ∈ s, ∃ y, eball x δ ⊆ c y",
"usedConstants": [
"PseudoEMetricSpace.edist_comm",
"Eq.mpr",
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
"Preorder.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.Basic | {
"line": 341,
"column": 2
} | {
"line": 342,
"column": 9
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\ns : Set α\nc : (x : α) → x ∈ s → Set α\nhs : IsCompact s\nhc : ∀ (x : α) (hx : x ∈ s), c x hx ∈ 𝓝[s] x\n⊢ ∃ δ > 0, ∀ x ∈ s, ∃ y, eball x δ ∩ s ⊆ c ↑y ⋯",
"usedConstants": [
"PseudoEMetricSpace.edist_comm",
"Eq.mpr",
"PseudoEMetricSpace.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.Diam | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 13
} | [
{
"pp": "X : Type u_2\ns t : Set X\ninst✝ : PseudoEMetricSpace X\nh : (s ∩ t).Nonempty\nx : X\nxs : x ∈ s\nxt : x ∈ t\n⊢ ediam (s ∪ t) ≤ ediam s + ediam t",
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"Set.instUnion",
"id",
"LE.le",
"Metric.ediam",
"instHAdd",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.Basic | {
"line": 346,
"column": 2
} | {
"line": 347,
"column": 9
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\ns : Set α\nc : α → Set α\nhs : IsCompact s\nhc : ∀ x ∈ s, c x ∈ 𝓝[s] x\n⊢ ∃ δ > 0, ∀ x ∈ s, ∃ y, eball x δ ∩ s ⊆ c y",
"usedConstants": [
"PseudoEMetricSpace.edist_comm",
"Eq.mpr",
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
"Pre... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.Basic | {
"line": 351,
"column": 32
} | {
"line": 351,
"column": 43
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\ns : Set α\nc : Set (Set α)\nhs : IsCompact s\nhc₁ : ∀ t ∈ c, IsOpen[PseudoEMetricSpace.toUniformSpace.toTopologicalSpace] t\nhc₂ : s ⊆ ⋃ i, ↑i\n⊢ ∃ δ > 0, ∀ x ∈ s, ∃ t ∈ c, eball x δ ⊆ t",
"usedConstants": [
"Eq.mpr",
"PseudoEMetricSpace.toWeakP... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.Lipschitz | {
"line": 249,
"column": 2
} | {
"line": 249,
"column": 55
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\nf : α → β\nKf : ℝ≥0\nhf : LipschitzWith Kf f\ng : α → γ\nKg : ℝ≥0\nhg : LipschitzWith Kg g\nx y : α\n⊢ edist ((fun x ↦ (f x, g x)) x) ((fun x ↦ (f x, g x)) y) ≤ ↑(max Kf Kg) *... | rw [ENNReal.coe_mono.map_max, Prod.edist_eq, max_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.EMetricSpace.Lipschitz | {
"line": 253,
"column": 2
} | {
"line": 253,
"column": 45
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\na : α\n⊢ LipschitzWith 1 (Prod.mk a)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.Lipschitz | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 44
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nb : β\n⊢ LipschitzWith 1 fun a ↦ (a, b)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.Lipschitz | {
"line": 269,
"column": 12
} | {
"line": 269,
"column": 39
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nK : ℝ≥0\nf : α → α\nhf : LipschitzWith K f\n⊢ LipschitzWith (K ^ 0) f^[0]",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"LipschitzWith",
"Monoid.toMulOneClass",
"congrArg",
"id",
"NNReal",
"instOfNatNat",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.Lipschitz | {
"line": 275,
"column": 2
} | {
"line": 275,
"column": 36
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nK : ℝ≥0\nf : α → α\nhf : LipschitzWith K f\nx : α\nn : ℕ\n⊢ edist (f^[n] x) ((f^[n] ∘ f) x) ≤ ↑K ^ n * edist x (f x)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.Lipschitz | {
"line": 285,
"column": 13
} | {
"line": 285,
"column": 24
} | [
{
"pp": "α : Type u\nι : Type x\ninst✝ : PseudoEMetricSpace α\nf : ι → Function.End α\nK : ι → ℝ≥0\nh : ∀ (i : ι), LipschitzWith (K i) (f i)\n⊢ LipschitzWith (List.map K []).prod (List.map f []).prod",
"usedConstants": [
"MulOne.toOne",
"LipschitzWith",
"Function.End",
"Monoid.toMulO... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.Lipschitz | {
"line": 292,
"column": 12
} | {
"line": 292,
"column": 39
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nf : Function.End α\nK : ℝ≥0\nh : LipschitzWith K f\n⊢ LipschitzWith (K ^ 0) (f ^ 0)",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"LipschitzWith",
"Function.End",
"Monoid.toMulOneClass",
"congrArg",
"id",
"NN... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.Lipschitz | {
"line": 336,
"column": 2
} | {
"line": 336,
"column": 55
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\ninst✝² : PseudoEMetricSpace α\ninst✝¹ : PseudoEMetricSpace β\ninst✝ : PseudoEMetricSpace γ\ns : Set α\nf : α → β\ng : α → γ\nKf Kg : ℝ≥0\nhf : LipschitzOnWith Kf f s\nhg : LipschitzOnWith Kg g s\nx✝ : α\nhx : x✝ ∈ s\ny✝ : α\nhy : y✝ ∈ s\n⊢ edist ((fun x ↦ (f x, g x))... | rw [ENNReal.coe_mono.map_max, Prod.edist_eq, max_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.EMetricSpace.Lipschitz | {
"line": 402,
"column": 12
} | {
"line": 402,
"column": 39
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nf : α → α\nhf : LocallyLipschitz f\n⊢ LocallyLipschitz f^[0]",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.EMetricSpace.Lipschitz | {
"line": 410,
"column": 12
} | {
"line": 410,
"column": 39
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nf : Function.End α\nh : LocallyLipschitz f\n⊢ LocallyLipschitz (f ^ 0)",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Function.End",
"Monoid.toMulOneClass",
"congrArg",
"id",
"instOfNatNat",
"LocallyLipschitz... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.Finset | {
"line": 64,
"column": 41
} | {
"line": 64,
"column": 90
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ns : Set (Finset α)\nt : Finset α\nH : ∀ b ≥ t, b ∈ s\nb : Finset (α ⊕ β)\nhb : b ≥ t.disjSum ∅\n⊢ b.toLeft ≥ t",
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"Finset.toLeft",
"Finset",
"PartialOrder.toPreorder",
"Preorder.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.Finset | {
"line": 71,
"column": 42
} | {
"line": 71,
"column": 91
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\ns : Set (Finset β)\nt : Finset β\nH : ∀ b ≥ t, b ∈ s\nb : Finset (α ⊕ β)\nhb : b ≥ ∅.disjSum t\n⊢ b.toRight ≥ t",
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"_private.Mathlib.Order.Filter.AtTopBot.Finset.0.Filter.tendsto_toRight_atTop._simp... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.Finset | {
"line": 85,
"column": 33
} | {
"line": 85,
"column": 44
} | [
{
"pp": "α : Type u_3\ninst✝¹ : Preorder α\ninst✝ : LocallyFiniteOrderBot α\nh✝ : Nonempty α\nh : IsDirectedOrder α\ns : Finset α\na : α\nha : ∀ i ∈ s, i ≤ a\nb : α\nhb : a ≤ b\nc : α\nhc : c ∈ s\n⊢ c ∈ Finset.Iic b",
"usedConstants": [
"Eq.mpr",
"Finset.mem_Iic._simp_1",
"Finset",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Filter.AtTopBot.Finset | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 13
} | [
{
"pp": "α : Type u_3\ni : α\n⊢ ∀ᶠ (s : Finset α) in atTop, i ∈ s",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset.inhabitedFinset",
"Finset",
"Filter.Eventually",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Membership.mem",
"Exists",
"GE.ge",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 38,
"column": 2
} | {
"line": 38,
"column": 18
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nL : SummationFilter β\ninst✝² : CommGroup α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalGroup α\nf : β → α\na : α\nh : HasProd f a L\n⊢ HasProd (fun b ↦ (f b)⁻¹) a⁻¹ L",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 46,
"column": 2
} | {
"line": 46,
"column": 28
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nL : SummationFilter β\ninst✝² : CommGroup α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalGroup α\nf : β → α\nhf : Multipliable (fun b ↦ (f b)⁻¹) L\n⊢ Multipliable f L",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 35
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nL : SummationFilter β\ninst✝² : CommGroup α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalGroup α\nf g : β → α\nhg : Multipliable g L\nhfg : Multipliable (fun b ↦ f b / g b) L\n⊢ Multipliable f L",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 71,
"column": 31
} | {
"line": 71,
"column": 57
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nL : SummationFilter β\ninst✝² : CommGroup α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalGroup α\nf g : β → α\nhfg : Multipliable (fun b ↦ f b / g b) L\nhf : Multipliable f L\n⊢ Multipliable (fun b ↦ g b / f b) L",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 56
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : CommGroup α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalGroup α\nf : β → α\na₁ a₂ : α\ns : Set β\nhf : HasProd (s.mulIndicator f) a₁\nh : HasProd f (a₁ * a₂)\n⊢ HasProd (f * (s.mulIndicator f)⁻¹) a₂",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 36
} | [
{
"pp": "case h.e'_6\nα : Type u_1\nβ : Type u_2\nL : SummationFilter β\ninst✝⁴ : CommGroup α\ninst✝³ : TopologicalSpace α\ninst✝² : IsTopologicalGroup α\nf : β → α\na : α\ninst✝¹ : L.LeAtTop\ninst✝ : DecidableEq β\nhf : HasProd f a L\nb : β\n⊢ a / f b = 1 / f b * a",
"usedConstants": [
"Eq.mpr",
... | rw [div_mul_eq_mul_div, one_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 36
} | [
{
"pp": "case h.e'_6\nα : Type u_1\nβ : Type u_2\nL : SummationFilter β\ninst✝⁴ : CommGroup α\ninst✝³ : TopologicalSpace α\ninst✝² : IsTopologicalGroup α\nf : β → α\na : α\ninst✝¹ : L.LeAtTop\ninst✝ : DecidableEq β\nhf : HasProd f a L\nb : β\n⊢ a / f b = 1 / f b * a",
"usedConstants": [
"Eq.mpr",
... | rw [div_mul_eq_mul_div, one_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 36
} | [
{
"pp": "case h.e'_6\nα : Type u_1\nβ : Type u_2\nL : SummationFilter β\ninst✝⁴ : CommGroup α\ninst✝³ : TopologicalSpace α\ninst✝² : IsTopologicalGroup α\nf : β → α\na : α\ninst✝¹ : L.LeAtTop\ninst✝ : DecidableEq β\nhf : HasProd f a L\nb : β\n⊢ a / f b = 1 / f b * a",
"usedConstants": [
"Eq.mpr",
... | rw [div_mul_eq_mul_div, one_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 214,
"column": 2
} | {
"line": 215,
"column": 96
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : UniformSpace α\ninst✝¹ : CommGroup α\ninst✝ : IsUniformGroup α\nf : β → α\n⊢ (CauchySeq fun s ↦ ∏ b ∈ s, f b) ↔ ∀ e ∈ 𝓝 1, ∃ s, ∀ (t : Finset β), Disjoint t s → ∏ b ∈ t, f b ∈ e",
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"instHDiv... | simp only [CauchySeq, cauchy_map_iff, prod_atTop_atTop_eq,
uniformity_eq_comap_nhds_one α, tendsto_comap_iff, Function.comp_def, atTop_neBot, true_and] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 69
} | [
{
"pp": "case h\nα : Type u_1\nβ : Type u_2\ninst✝² : UniformSpace α\ninst✝¹ : CommGroup α\ninst✝ : IsUniformGroup α\nf : β → α\nh✝ : ∀ s ∈ 𝓝 1, ∃ a, ∀ b ≥ a, (∏ b ∈ b.2, f b) / ∏ b ∈ b.1, f b ∈ s\ne : Set α\nhe : e ∈ 𝓝 1\ns₁ s₂ t : Finset β\nht : Disjoint t (s₁ ∪ s₂)\nh : (∏ b ∈ (s₁ ∪ s₂, s₁ ∪ s₂ ∪ t).2, f b... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.NatInt | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 35
} | [
{
"pp": "M : Type u_1\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\nm : M\ninst✝ : ContinuousMul M\nf : ℕ → M\nh : HasProd (fun n ↦ f (n + 1)) m\n⊢ HasProd f (f 0 * m)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.NatInt | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 33
} | [
{
"pp": "M : Type u_1\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\nm m' : M\ninst✝ : ContinuousMul M\nf : ℕ → M\nhe : HasProd (fun k ↦ f (2 * k)) m\nthis : Injective fun x ↦ 2 * x\nho : HasProd (fun x ↦ f ↑x) m'\n⊢ IsCompl (Set.range fun x ↦ 2 * x) (Set.range ((fun x ↦ x + 1) ∘ fun x ↦ 2 * x))",
"us... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 295,
"column": 2
} | {
"line": 295,
"column": 48
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝³ : UniformSpace α\ninst✝² : CommGroup α\ninst✝¹ : IsUniformGroup α\nf : β → α\ninst✝ : CompleteSpace α\ni : γ → β\nhf : Multipliable f\nhi : Injective i\n⊢ Multipliable (f ∘ i)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 370,
"column": 4
} | {
"line": 370,
"column": 15
} | [
{
"pp": "α : Type u_1\nG : Type u_4\ninst✝² : TopologicalSpace G\ninst✝¹ : CommGroup G\ninst✝ : IsTopologicalGroup G\nf : α → G\nhf : Multipliable f\ne : Set G\nhe : e ∈ 𝓝 1\ns : Finset α\nhs : ∀ (t : Finset α), Disjoint t s → ∏ k ∈ t, f k ∈ e\nx : α\nhx : x ∉ s\n⊢ f x ∈ e",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 390,
"column": 2
} | {
"line": 390,
"column": 29
} | [
{
"pp": "α : Type u_1\nG : Type u_4\ninst✝⁴ : TopologicalSpace G\ninst✝³ : CommGroup G\ninst✝² : IsTopologicalGroup G\nf : α → G\ninst✝¹ : FirstCountableTopology G\ninst✝ : T1Space G\nhf : Multipliable f\n⊢ (mulSupport f).Countable",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 399,
"column": 6
} | {
"line": 399,
"column": 41
} | [
{
"pp": "β : Type u_2\nG : Type u_4\ninst✝⁴ : TopologicalSpace G\ninst✝³ : CommGroup G\ninst✝² : IsTopologicalGroup G\ninst✝¹ : Infinite β\ninst✝ : T2Space G\na : G\nh : Multipliable fun x ↦ a\nha : ¬a = 1\nthis : {a}ᶜ ∈ 𝓝 1\n⊢ Finite β",
"usedConstants": [
"Eq.mpr",
"Set.univ",
"Finite",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.NatInt | {
"line": 196,
"column": 2
} | {
"line": 196,
"column": 35
} | [
{
"pp": "M : Type u_1\ninst✝³ : CommMonoid M\ninst✝² : TopologicalSpace M\ninst✝¹ : T2Space M\ninst✝ : ContinuousMul M\nf : ℕ → M\nhf : Multipliable fun n ↦ f (n + 1)\n⊢ ∏' (b : ℕ), f b = f 0 * ∏' (b : ℕ), f (b + 1)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 414,
"column": 6
} | {
"line": 414,
"column": 42
} | [
{
"pp": "case inr.inr.h\nβ : Type u_2\nG : Type u_4\ninst✝³ : TopologicalSpace G\ninst✝² : CommGroup G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : T2Space G\na : G\nhβ : Infinite β\nha : a ≠ 1\n⊢ ¬Multipliable fun b ↦ a",
"usedConstants": [
"Eq.mpr",
"InvOneClass.toOne",
"DivisionCommMonoid.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.NatInt | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 49
} | [
{
"pp": "G : Type u_2\ninst✝² : CommGroup G\ng : G\ninst✝¹ : TopologicalSpace G\ninst✝ : IsTopologicalGroup G\nf : ℕ → G\nk : ℕ\n⊢ HasProd (fun n ↦ f (n + k)) g ↔ HasProd f (g * ∏ i ∈ range k, f i)",
"usedConstants": [
"HMul.hMul",
"Monoid.toMulOneClass",
"Finset",
"HasProd",
"... | refine Iff.trans ?_ (range k).hasProd_compl_iff | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Algebra.InfiniteSum.NatInt | {
"line": 253,
"column": 4
} | {
"line": 253,
"column": 36
} | [
{
"pp": "case pos\nG : Type u_2\ninst✝³ : CommGroup G\ninst✝² : TopologicalSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : T2Space G\nf : ℕ → G\nhf : Multipliable f\nh₀ : (fun i ↦ (∏' (i : ℕ), f i) / ∏ j ∈ range i, f j) = fun i ↦ ∏' (k : ℕ), f (k + i)\nh₁ : Tendsto (fun x ↦ ∏' (i : ℕ), f i) atTop (𝓝 (∏' (i : ℕ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Order | {
"line": 157,
"column": 6
} | {
"line": 157,
"column": 54
} | [
{
"pp": "case pos\nι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝³ : CommMonoid α\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf : ι → α\na₂ : α\nha₂ : 1 ≤ a₂\nh : ∀ (s : Finset ι), ∏ i ∈ s, f i ≤ a₂\nhL : ¬L.NeBot\nhf : (mulSupport f).Finite\n⊢ ∏'[L] (i : ι), f i ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Order | {
"line": 173,
"column": 6
} | {
"line": 173,
"column": 32
} | [
{
"pp": "case neg\nι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝⁴ : CommMonoid α\ninst✝³ : Preorder α\ninst✝² : IsOrderedMonoid α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\ng : ι → α\nh : ∀ (i : ι), 1 ≤ g i\nhg : Multipliable g L\nhL : ¬L.NeBot\n⊢ 1 ≤ ∏'[L] (i : ι), g i",
"usedC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.NatInt | {
"line": 328,
"column": 31
} | {
"line": 328,
"column": 42
} | [
{
"pp": "M : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : TopologicalSpace M\nm : M\nf : ℤ → M\nhf : HasProd f m\nthis : Injective Int.negSucc\nu : Finset ℤ\nv' : Finset ℕ\nhv' : u.preimage Nat.cast ⋯ ∪ u.preimage Int.negSucc ⋯ ⊆ v'\na✝ : ℕ\nhx : Int.ofNat a✝ ∈ u\n⊢ a✝ ∈ u.preimage Nat.cast ⋯ ∪ u.preimage Int.negSu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.NatInt | {
"line": 329,
"column": 31
} | {
"line": 329,
"column": 42
} | [
{
"pp": "M : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : TopologicalSpace M\nm : M\nf : ℤ → M\nhf : HasProd f m\nthis : Injective Int.negSucc\nu : Finset ℤ\nv' : Finset ℕ\nhv' : u.preimage Nat.cast ⋯ ∪ u.preimage Int.negSucc ⋯ ⊆ v'\na✝ : ℕ\nhx : Int.negSucc a✝ ∈ u\n⊢ a✝ ∈ u.preimage Nat.cast ⋯ ∪ u.preimage Int.neg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Order | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 49
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝⁷ : CommGroup α\ninst✝⁶ : PartialOrder α\ninst✝⁵ : IsOrderedMonoid α\ninst✝⁴ : TopologicalSpace α\ninst✝³ : IsTopologicalGroup α\ninst✝² : OrderClosedTopology α\nf g : ι → α\na₁ a₂ : α\ni : ι\ninst✝¹ : L.NeBot\ninst✝ : L.LeAtTop\nh : f ≤ g\nhi : f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Ring | {
"line": 40,
"column": 2
} | {
"line": 40,
"column": 18
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝² : NonUnitalNonAssocSemiring α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalSemiring α\nf : ι → α\na₁ a₂ : α\nh : HasSum f a₁ L\n⊢ HasSum (fun i ↦ a₂ * f i) (a₂ * a₁) L",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Ring | {
"line": 43,
"column": 2
} | {
"line": 43,
"column": 18
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝² : NonUnitalNonAssocSemiring α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalSemiring α\nf : ι → α\na₁ a₂ : α\nhf : HasSum f a₁ L\n⊢ HasSum (fun i ↦ f i * a₂) (a₁ * a₂) L",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Ring | {
"line": 97,
"column": 14
} | {
"line": 97,
"column": 55
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝² : DivisionSemiring α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalSemiring α\nf : ι → α\na₁ a₂ : α\nh : a₂ ≠ 0\nH : HasSum (fun i ↦ a₂ * f i) (a₂ * a₁) L\n⊢ HasSum f a₁ L",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Ring | {
"line": 100,
"column": 14
} | {
"line": 100,
"column": 56
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝² : DivisionSemiring α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalSemiring α\nf : ι → α\na₁ a₂ : α\nh : a₂ ≠ 0\nH : HasSum (fun i ↦ f i * a₂) (a₁ * a₂) L\n⊢ HasSum f a₁ L",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Ring | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 35
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝² : DivisionSemiring α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalSemiring α\nf : ι → α\na₁ a₂ : α\nh : a₂ ≠ 0\n⊢ HasSum (fun i ↦ f i / a₂) (a₁ / a₂) L ↔ HasSum f a₁ L",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Ring | {
"line": 107,
"column": 14
} | {
"line": 107,
"column": 55
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝² : DivisionSemiring α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalSemiring α\nf : ι → α\na : α\nh : a ≠ 0\nH : Summable (fun i ↦ a * f i) L\n⊢ Summable f L",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Ring | {
"line": 110,
"column": 14
} | {
"line": 110,
"column": 56
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝² : DivisionSemiring α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalSemiring α\nf : ι → α\na : α\nh : a ≠ 0\nH : Summable (fun i ↦ f i * a) L\n⊢ Summable f L",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Ring | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 35
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝² : DivisionSemiring α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalSemiring α\nf : ι → α\na : α\nh : a ≠ 0\n⊢ Summable (fun i ↦ f i / a) L ↔ Summable f L",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Order | {
"line": 326,
"column": 4
} | {
"line": 326,
"column": 15
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\ninst✝⁵ : AddCommGroup α\ninst✝⁴ : LinearOrder α\ninst✝³ : IsOrderedAddMonoid α\ninst✝² : TopologicalSpace α\ninst✝¹ : Archimedean α\ninst✝ : OrderClosedTopology α\nb : α\nhb : 0 < b\nhf : Summable fun x ↦ b\ns : Finset ι\n⊢ #s • b ≤ ∑' (x : ι), b",
"usedConstants": []
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.InfiniteSum.Ring | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 35
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝³ : DivisionSemiring α\ninst✝² : TopologicalSpace α\ninst✝¹ : IsTopologicalSemiring α\nf : ι → α\na : α\ninst✝ : T2Space α\n⊢ ∑'[L] (x : ι), f x / a = (∑'[L] (x : ι), f x) / a",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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