module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Data.NNReal.Basic | {
"line": 204,
"column": 2
} | {
"line": 204,
"column": 15
} | {
"line": 205,
"column": 2
} | [
{
"pp": "ι : Sort u_3\ninst✝ : Nonempty ι\na : ℝ≥0\ng : ι → ℝ≥0\nh : ℝ≥0\nH : ∀ (i : ι), a ≤ g i * h\n⊢ a ≤ iInf g * h",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"iInf",
"HMul.hMul",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
... | [
"ι : Sort u_3\ninst✝ : Nonempty ι\na : ℝ≥0\ng : ι → ℝ≥0\nh : ℝ≥0\nH : ∀ (i : ι), a ≤ g i * h\n⊢ a ≤ ⨅ i, g i * h"
] | rw [iInf_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Order.DenselyOrdered | {
"line": 255,
"column": 2
} | {
"line": 255,
"column": 60
} | {
"line": 256,
"column": 2
} | [
{
"pp": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : DenselyOrdered α\na✝ b✝ : α\ns✝ : Set α\nx : α\ninst✝ : Nontrivial α\ns : Set α\nhs : s ∈ 𝓝[≠] x\nu : Set α\nu_open : IsOpen[inst✝⁴] u\nxu : x ∈ u\nus : u ∩ {x}ᶜ ⊆ s\na b : α\n... | [
"case inr\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : DenselyOrdered α\na✝ b✝ : α\ns✝ : Set α\nx : α\ninst✝ : Nontrivial α\ns : Set α\nhs : s ∈ 𝓝[≠] x\nu : Set α\nu_open : IsOpen[inst✝⁴] u\nxu : x ∈ u\nus : u ∩ {x}ᶜ ⊆ s\na b : α\na_lt_b : a <... | obtain ⟨z, hz⟩ : ∃ z, a < z ∧ z < x := exists_between hy.1 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.Order.Compact | {
"line": 241,
"column": 4
} | {
"line": 241,
"column": 37
} | {
"line": 242,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_2\nβ : Type u_3\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : ClosedIicTopology α\ninst✝ : NoMaxOrder α\nf : β → α\na : α\nhs : IsCompact ∅\nhf : ContinuousOn f ∅\nhf' : ∀ b ∈ ∅, a < f b\na' : α\nha' : a < a'\n⊢ ∃ a', a < a' ∧ ∀ b ∈ ∅, ... | [] | exact ⟨a', ha', fun _ a ↦ a.elim⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Order.Compact | {
"line": 240,
"column": 4
} | {
"line": 241,
"column": 37
} | {
"line": 242,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_2\nβ : Type u_3\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : ClosedIicTopology α\ninst✝ : NoMaxOrder α\nf : β → α\na : α\nhs : IsCompact ∅\nhf : ContinuousOn f ∅\nhf' : ∀ b ∈ ∅, a < f b\n⊢ ∃ a', a < a' ∧ ∀ b ∈ ∅, a' ≤ f b",
"ppTerm... | [] | obtain ⟨a', ha'⟩ := exists_gt a
exact ⟨a', ha', fun _ a ↦ a.elim⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.Compact | {
"line": 240,
"column": 4
} | {
"line": 241,
"column": 37
} | {
"line": 242,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_2\nβ : Type u_3\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : ClosedIicTopology α\ninst✝ : NoMaxOrder α\nf : β → α\na : α\nhs : IsCompact ∅\nhf : ContinuousOn f ∅\nhf' : ∀ b ∈ ∅, a < f b\n⊢ ∃ a', a < a' ∧ ∀ b ∈ ∅, a' ≤ f b",
"ppTerm... | [] | obtain ⟨a', ha'⟩ := exists_gt a
exact ⟨a', ha', fun _ a ↦ a.elim⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.IntermediateValue | {
"line": 274,
"column": 2
} | {
"line": 278,
"column": 30
} | {
"line": 279,
"column": 2
} | [
{
"pp": "case neg\nα : Type u\ninst✝² : TopologicalSpace α\ninst✝¹ : ConditionallyCompleteLinearOrder α\ninst✝ : OrderTopology α\ns : Set α\nhs : IsPreconnected s\nhne : s.Nonempty\nhs' : IsConnected s\nhb : BddBelow s\nha : ¬BddAbove s\n⊢ s ∈\n {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sS... | [
"case pos\nα : Type u\ninst✝² : TopologicalSpace α\ninst✝¹ : ConditionallyCompleteLinearOrder α\ninst✝ : OrderTopology α\ns : Set α\nhs : IsPreconnected s\nhne : s.Nonempty\nhs' : IsConnected s\nhb : ¬BddBelow s\nha : BddAbove s\n⊢ s ∈\n {Icc (sInf s) (sSup s), Ico (sInf s) (sSup s), Ioc (sInf s) (sSup s), Ioo (... | · refine Or.inr <| Or.inr <| Or.inr <| Or.inr ?_
rcases mem_Ici_Ioi_of_subset_of_subset (hs.Ioi_csInf_subset hb ha) fun x hx ↦
csInf_le hb hx with hs | hs
· exact Or.inl hs
· exact Or.inr (Or.inl hs) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 265,
"column": 6
} | {
"line": 265,
"column": 22
} | {
"line": 265,
"column": 23
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na b : E\nr : ℝ\n⊢ b ∈ closedBall a r ↔ ‖a⁻¹ * b‖ ≤ r",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Mono... | [
"E : Type u_5\ninst✝ : SeminormedGroup E\na b : E\nr : ℝ\n⊢ dist a b ≤ r ↔ ‖a⁻¹ * b‖ ≤ r"
] | mem_closedBall', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 425,
"column": 2
} | {
"line": 425,
"column": 33
} | {
"line": 427,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na : E\n⊢ nndist a 1 = ‖a‖₊",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"NNDist.nndist",
"InvOneClass.toOne",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"congrArg",
"NNN... | [] | simp [nndist_eq_nnnorm_inv_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 425,
"column": 2
} | {
"line": 425,
"column": 33
} | {
"line": 427,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na : E\n⊢ nndist a 1 = ‖a‖₊",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"NNDist.nndist",
"InvOneClass.toOne",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"congrArg",
"NNN... | [] | simp [nndist_eq_nnnorm_inv_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 425,
"column": 2
} | {
"line": 425,
"column": 33
} | {
"line": 427,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na : E\n⊢ nndist a 1 = ‖a‖₊",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"NNDist.nndist",
"InvOneClass.toOne",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Monoid.toMulOneClass",
"congrArg",
"NNN... | [] | simp [nndist_eq_nnnorm_inv_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 469,
"column": 58
} | {
"line": 469,
"column": 89
} | {
"line": 471,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na : E\n⊢ nndist 1 a = ‖a‖₊",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"NNDist.nndist",
"InvOneClass.toOne",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"inv_one",
"Monoid.toMulOneClass",
"cong... | [] | simp [nndist_eq_nnnorm_inv_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 469,
"column": 58
} | {
"line": 469,
"column": 89
} | {
"line": 471,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na : E\n⊢ nndist 1 a = ‖a‖₊",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"NNDist.nndist",
"InvOneClass.toOne",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"inv_one",
"Monoid.toMulOneClass",
"cong... | [] | simp [nndist_eq_nnnorm_inv_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 469,
"column": 58
} | {
"line": 469,
"column": 89
} | {
"line": 471,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\na : E\n⊢ nndist 1 a = ‖a‖₊",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"NNDist.nndist",
"InvOneClass.toOne",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"inv_one",
"Monoid.toMulOneClass",
"cong... | [] | simp [nndist_eq_nnnorm_inv_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 882,
"column": 6
} | {
"line": 882,
"column": 22
} | {
"line": 882,
"column": 23
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedCommGroup E\na b : E\nr : ℝ\n⊢ b ∈ closedBall a r ↔ ‖a / b‖ ≤ r",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"congrArg",
"SeminormedCommGroup.toPs... | [
"E : Type u_5\ninst✝ : SeminormedCommGroup E\na b : E\nr : ℝ\n⊢ dist a b ≤ r ↔ ‖a / b‖ ≤ r"
] | mem_closedBall', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 235,
"column": 90
} | {
"line": 240,
"column": 38
} | {
"line": 242,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝ : PseudoMetricSpace α\nf : β → α\nhf : Tendsto (Prod.map f f) (Filter.cofinite ×ˢ Filter.cofinite) (𝓤 α)\n⊢ Bornology.IsBounded (range f)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Set.instSProd",
"Eq.mpr",
"R... | [] | by
rcases (hasBasis_cofinite.prod_self.tendsto_iff uniformity_basis_dist).1 hf 1 zero_lt_one with
⟨s, hsf, hs1⟩
rw [← image_union_image_compl_eq_range]
refine (hsf.image f).isBounded.union (isBounded_image_iff.2 ⟨1, fun x hx y hy ↦ ?_⟩)
exact le_of_lt (hs1 (x, y) ⟨hx, hy⟩) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 439,
"column": 2
} | {
"line": 440,
"column": 94
} | {
"line": 442,
"column": 0
} | [
{
"pp": "α : Type u\nx y z : α\ninst✝ : PseudoMetricSpace α\n⊢ diam {x, y, z} = max (max (dist x y) (dist x z)) (dist y z)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
"Real",
"Metric.ediam_triple",
"... | [] | simp only [diam, ediam_triple, dist_edist]
rw [ENNReal.toReal_max, ENNReal.toReal_max] <;> apply_rules [ne_of_lt, edist_lt_top, max_lt] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 439,
"column": 2
} | {
"line": 440,
"column": 94
} | {
"line": 442,
"column": 0
} | [
{
"pp": "α : Type u\nx y z : α\ninst✝ : PseudoMetricSpace α\n⊢ diam {x, y, z} = max (max (dist x y) (dist x z)) (dist y z)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
"Real",
"Metric.ediam_triple",
"... | [] | simp only [diam, ediam_triple, dist_edist]
rw [ENNReal.toReal_max, ENNReal.toReal_max] <;> apply_rules [ne_of_lt, edist_lt_top, max_lt] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.AtTopBot.Archimedean | {
"line": 264,
"column": 45
} | {
"line": 264,
"column": 92
} | {
"line": 266,
"column": 0
} | [
{
"pp": "α : Type u_1\nR : Type u_2\nl : Filter α\nr : R\ninst✝³ : AddCommGroup R\ninst✝² : LinearOrder R\ninst✝¹ : IsOrderedAddMonoid R\ninst✝ : Archimedean R\nf : α → ℤ\nhr : r < 0\nhf : Tendsto f l atTop\n⊢ Tendsto (fun x ↦ f x • r) l atBot",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": ... | [] | simpa using hf.atTop_zsmul_const (neg_pos.2 hr) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Order.Filter.AtTopBot.Archimedean | {
"line": 264,
"column": 45
} | {
"line": 264,
"column": 92
} | {
"line": 266,
"column": 0
} | [
{
"pp": "α : Type u_1\nR : Type u_2\nl : Filter α\nr : R\ninst✝³ : AddCommGroup R\ninst✝² : LinearOrder R\ninst✝¹ : IsOrderedAddMonoid R\ninst✝ : Archimedean R\nf : α → ℤ\nhr : r < 0\nhf : Tendsto f l atTop\n⊢ Tendsto (fun x ↦ f x • r) l atBot",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": ... | [] | simpa using hf.atTop_zsmul_const (neg_pos.2 hr) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.AtTopBot.Archimedean | {
"line": 264,
"column": 45
} | {
"line": 264,
"column": 92
} | {
"line": 266,
"column": 0
} | [
{
"pp": "α : Type u_1\nR : Type u_2\nl : Filter α\nr : R\ninst✝³ : AddCommGroup R\ninst✝² : LinearOrder R\ninst✝¹ : IsOrderedAddMonoid R\ninst✝ : Archimedean R\nf : α → ℤ\nhr : r < 0\nhf : Tendsto f l atTop\n⊢ Tendsto (fun x ↦ f x • r) l atBot",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": ... | [] | simpa using hf.atTop_zsmul_const (neg_pos.2 hr) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Order.Archimedean | {
"line": 80,
"column": 91
} | {
"line": 84,
"column": 54
} | {
"line": 86,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : CommGroup G\ninst✝⁴ : LinearOrder G\ninst✝³ : IsOrderedMonoid G\ninst✝² : TopologicalSpace G\ninst✝¹ : OrderTopology G\ninst✝ : MulArchimedean G\nS : Subgroup G\n⊢ Dense ↑S ∨ ∃ a, S = closure {a}",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Mathlib... | [] | by
refine (em _).imp (dense_of_not_isolated_one S) fun h => ?_
push Not at h
rcases h with ⟨ε, ε1, hε⟩
exact cyclic_of_isolated_one ε1 (disjoint_left.2 hε) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Ring.Real | {
"line": 124,
"column": 2
} | {
"line": 125,
"column": 48
} | {
"line": 127,
"column": 0
} | [
{
"pp": "case some.some\nα : Type u\nβ : Type v\nγ : Type w\na b : ℝ≥0\n⊢ ContinuousAt (fun p ↦ p.1 + p.2) (some a, some b)",
"ppTerm": "?some.some",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"ENNReal.instAdd",
"ENNReal.ofNNReal",
"ENNReal.tendsto_co... | [] | simp only [ContinuousAt, some_eq_coe, nhds_coe_coe, ← coe_add, tendsto_map'_iff,
Function.comp_def, tendsto_coe, tendsto_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.Filter.IsBounded | {
"line": 115,
"column": 4
} | {
"line": 116,
"column": 57
} | {
"line": 118,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u_1\nβ : Type u_2\ninst✝ : Preorder α\nf : Filter β\nu : β → α\n⊢ (∃ s, BddAbove (u '' s) ∧ ∀ᶠ (x : β) in f, x ∈ s) → IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"congrArg",
"Filter.map",
"Filter.Even... | [] | rintro ⟨s, ⟨b, hb⟩, hs⟩
exact ⟨b, hs.mono <| by simpa [upperBounds] using hb⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.IsBounded | {
"line": 115,
"column": 4
} | {
"line": 116,
"column": 57
} | {
"line": 118,
"column": 0
} | [
{
"pp": "case mpr\nα : Type u_1\nβ : Type u_2\ninst✝ : Preorder α\nf : Filter β\nu : β → α\n⊢ (∃ s, BddAbove (u '' s) ∧ ∀ᶠ (x : β) in f, x ∈ s) → IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"congrArg",
"Filter.map",
"Filter.Even... | [] | rintro ⟨s, ⟨b, hb⟩, hs⟩
exact ⟨b, hs.mono <| by simpa [upperBounds] using hb⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.IsBounded | {
"line": 573,
"column": 4
} | {
"line": 573,
"column": 26
} | {
"line": 574,
"column": 4
} | [
{
"pp": "case inl\nα : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder β\nf : Filter α\nu v : α → β\nh' : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f u\n⊢ IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun a ↦ max (u a) (v a)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSu... | [
"case inl\nα : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder β\nf : Filter α\nu v : α → β\nb : β\nhb : ∀ (a : β), (∀ᶠ (x : β) in map u f, (fun x1 x2 ↦ x1 ≤ x2) x a) → (fun x1 x2 ↦ x1 ≤ x2) b a\n⊢ IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun a ↦ max (u a) (v a)"
] | rcases h' with ⟨b, hb⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Order.Filter.IsBounded | {
"line": 573,
"column": 4
} | {
"line": 573,
"column": 26
} | {
"line": 574,
"column": 4
} | [
{
"pp": "case inr\nα : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder β\nf : Filter α\nu v : α → β\nh' : IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f v\n⊢ IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun a ↦ max (u a) (v a)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSu... | [
"case inr\nα : Type u_1\nβ : Type u_2\ninst✝ : LinearOrder β\nf : Filter α\nu v : α → β\nb : β\nhb : ∀ (a : β), (∀ᶠ (x : β) in map v f, (fun x1 x2 ↦ x1 ≤ x2) x a) → (fun x1 x2 ↦ x1 ≤ x2) b a\n⊢ IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun a ↦ max (u a) (v a)"
] | rcases h' with ⟨b, hb⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.Order.MonotoneConvergence | {
"line": 134,
"column": 41
} | {
"line": 134,
"column": 99
} | {
"line": 136,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝³ : Preorder ι\ninst✝² : TopologicalSpace α\ninst✝¹ : ConditionallyCompletePartialOrderInf α\ninst✝ : InfConvergenceClass α\nf : ι → α\nh_mono : Monotone f\nhbdd : BddBelow (range f)\n⊢ Tendsto f atBot (𝓝 (⨅ i, f i))",
"ppTerm": "?m.21",
"assigned": true,
"... | [] | convert! tendsto_atTop_ciSup h_mono.dual hbdd.dual using 1 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Topology.Order.MonotoneConvergence | {
"line": 134,
"column": 41
} | {
"line": 134,
"column": 99
} | {
"line": 136,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝³ : Preorder ι\ninst✝² : TopologicalSpace α\ninst✝¹ : ConditionallyCompletePartialOrderInf α\ninst✝ : InfConvergenceClass α\nf : ι → α\nh_mono : Monotone f\nhbdd : BddBelow (range f)\n⊢ Tendsto f atBot (𝓝 (⨅ i, f i))",
"ppTerm": "?m.21",
"assigned": true,
"... | [] | convert! tendsto_atTop_ciSup h_mono.dual hbdd.dual using 1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.MonotoneConvergence | {
"line": 134,
"column": 41
} | {
"line": 134,
"column": 99
} | {
"line": 136,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_3\ninst✝³ : Preorder ι\ninst✝² : TopologicalSpace α\ninst✝¹ : ConditionallyCompletePartialOrderInf α\ninst✝ : InfConvergenceClass α\nf : ι → α\nh_mono : Monotone f\nhbdd : BddBelow (range f)\n⊢ Tendsto f atBot (𝓝 (⨅ i, f i))",
"ppTerm": "?m.21",
"assigned": true,
"... | [] | convert! tendsto_atTop_ciSup h_mono.dual hbdd.dual using 1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.EReal.Operations | {
"line": 533,
"column": 2
} | {
"line": 533,
"column": 38
} | {
"line": 535,
"column": 0
} | [
{
"pp": "a b c a' : EReal\nhc' : c - b < a'\nha' : a' < a\nhc : b = ⊥\n⊢ a + b ≤ c",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",
"EReal",
"EReal.add_bot",
"Preorder.toLE",
"Eq.rec",
"Bot.bot",
"LE.le",
"instAddCom... | [] | exact hc ▸ (add_bot a).symm ▸ bot_le | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.EReal.Operations | {
"line": 654,
"column": 6
} | {
"line": 655,
"column": 64
} | {
"line": 656,
"column": 4
} | [
{
"pp": "case coe.top\na✝ : ℝ\nh : ↑a✝ ≠ 0 ∧ ⊤ ≠ 0\n⊢ ↑a✝ * ⊤ ≠ 0",
"ppTerm": "?coe.top",
"assigned": true,
"usedConstants": [
"lt_or_gt_of_ne",
"False",
"Preorder.toLT",
"HMul.hMul",
"congrArg",
"PartialOrder.toPreorder",
"EReal",
"EReal.mul_top_of_ne... | [] | rcases lt_or_gt_of_ne h.1 with (h | h)
<;> simp [EReal.mul_top_of_neg, EReal.mul_top_of_pos, h] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Data.EReal.Operations | {
"line": 654,
"column": 6
} | {
"line": 655,
"column": 64
} | {
"line": 656,
"column": 4
} | [
{
"pp": "case coe.top\na✝ : ℝ\nh : ↑a✝ ≠ 0 ∧ ⊤ ≠ 0\n⊢ ↑a✝ * ⊤ ≠ 0",
"ppTerm": "?coe.top",
"assigned": true,
"usedConstants": [
"lt_or_gt_of_ne",
"False",
"Preorder.toLT",
"HMul.hMul",
"congrArg",
"PartialOrder.toPreorder",
"EReal",
"EReal.mul_top_of_ne... | [] | rcases lt_or_gt_of_ne h.1 with (h | h)
<;> simp [EReal.mul_top_of_neg, EReal.mul_top_of_pos, h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.EReal.Operations | {
"line": 654,
"column": 6
} | {
"line": 655,
"column": 64
} | {
"line": 656,
"column": 4
} | [
{
"pp": "case coe.top\na✝ : ℝ\nh : ↑a✝ ≠ 0 ∧ ⊤ ≠ 0\n⊢ ↑a✝ * ⊤ ≠ 0",
"ppTerm": "?coe.top",
"assigned": true,
"usedConstants": [
"lt_or_gt_of_ne",
"False",
"Preorder.toLT",
"HMul.hMul",
"congrArg",
"PartialOrder.toPreorder",
"EReal",
"EReal.mul_top_of_ne... | [] | rcases lt_or_gt_of_ne h.1 with (h | h)
<;> simp [EReal.mul_top_of_neg, EReal.mul_top_of_pos, h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.LiminfLimsup | {
"line": 1009,
"column": 4
} | {
"line": 1015,
"column": 36
} | {
"line": 1016,
"column": 4
} | [
{
"pp": "case h₁\nα : Type u_1\nι : Type u_4\nι' : Type u_5\ninst✝² : ConditionallyCompleteLinearOrder α\nv : Filter ι\np : ι' → Prop\ns : ι' → Set ι\ninst✝¹ : Countable (Subtype p)\ninst✝ : Nonempty (Subtype p)\nhv : v.HasBasis p s\nf : ι → α\nhs : ∀ (j : Subtype p), (s ↑j).Nonempty\nj0 : Subtype p\nhj0 : BddB... | [
"case h₂\nα : Type u_1\nι : Type u_4\nι' : Type u_5\ninst✝² : ConditionallyCompleteLinearOrder α\nv : Filter ι\np : ι' → Prop\ns : ι' → Set ι\ninst✝¹ : Countable (Subtype p)\ninst✝ : Nonempty (Subtype p)\nhv : v.HasBasis p s\nf : ι → α\nhs : ∀ (j : Subtype p), (s ↑j).Nonempty\nj0 : Subtype p\nhj0 : BddBelow (range ... | · apply iUnion_subset (fun j ↦ ?_)
by_cases hj : j ∈ m
· have : j = liminf_reparam f s p j := by simp only [m, liminf_reparam, hj, ite_true]
conv_lhs => rw [this]
apply subset_iUnion _ j
· simp only [m, mem_setOf_eq, ← nonempty_iInter_Iic_iff, not_nonempty_iff_eq_empty] at hj
s... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.Filter.AtTopBot.BigOperators | {
"line": 60,
"column": 2
} | {
"line": 72,
"column": 61
} | {
"line": 74,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Type u_3\ninst✝ : CommMonoid M\ng : α → β\nhg : Injective g\nf : β → M\nhf : ∀ x ∉ Set.range g, f x = 1\n⊢ Filter.map (fun s ↦ ∏ i ∈ s, f (g i)) atTop = Filter.map (fun s ↦ ∏ i ∈ s, f i) atTop",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"I... | [] | haveI := Classical.decEq β
apply le_antisymm <;> refine map_atTop_finsetProd_le_of_prod_eq fun s => ?_
· refine ⟨s.preimage g hg.injOn, fun t ht => ?_⟩
refine ⟨t.image g ∪ s, Finset.subset_union_right, ?_⟩
rw [← Finset.prod_image hg.injOn]
refine (prod_subset subset_union_left ?_).symm
simp only [Fi... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.AtTopBot.BigOperators | {
"line": 60,
"column": 2
} | {
"line": 72,
"column": 61
} | {
"line": 74,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Type u_3\ninst✝ : CommMonoid M\ng : α → β\nhg : Injective g\nf : β → M\nhf : ∀ x ∉ Set.range g, f x = 1\n⊢ Filter.map (fun s ↦ ∏ i ∈ s, f (g i)) atTop = Filter.map (fun s ↦ ∏ i ∈ s, f i) atTop",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"I... | [] | haveI := Classical.decEq β
apply le_antisymm <;> refine map_atTop_finsetProd_le_of_prod_eq fun s => ?_
· refine ⟨s.preimage g hg.injOn, fun t ht => ?_⟩
refine ⟨t.image g ∪ s, Finset.subset_union_right, ?_⟩
rw [← Finset.prod_image hg.injOn]
refine (prod_subset subset_union_left ?_).symm
simp only [Fi... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 355,
"column": 4
} | {
"line": 355,
"column": 42
} | {
"line": 356,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_1\nG : Type u_4\ninst✝² : TopologicalSpace G\ninst✝¹ : CommGroup G\ninst✝ : IsTopologicalGroup G\nf : α → G\nH : Multipliable f\ne : Set G\nhe : e ∈ 𝓝 1\n⊢ e ∈ Filter.map (fun s ↦ ∏' (a : { x // x ∉ s }), f ↑a) atTop",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstant... | [
"case pos\nα : Type u_1\nG : Type u_4\ninst✝² : TopologicalSpace G\ninst✝¹ : CommGroup G\ninst✝ : IsTopologicalGroup G\nf : α → G\nH : Multipliable f\ne : Set G\nhe : e ∈ 𝓝 1\ns : Finset α\nhs : ∀ (t : Set α), Disjoint t ↑s → ∏' (b : ↑t), f ↑b ∈ e\n⊢ e ∈ Filter.map (fun s ↦ ∏' (a : { x // x ∉ s }), f ↑a) atTop"
] | obtain ⟨s, hs⟩ := H.tprod_vanishing he | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 575,
"column": 2
} | {
"line": 575,
"column": 55
} | {
"line": 577,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝³ : CommMonoid α\ninst✝² : TopologicalSpace α\nι : Type u_4\nS : Type u_5\ns : S\ninst✝¹ : SetLike S α\ninst✝ : SubmonoidClass S α\nh_closed : IsClosed[inst✝²] ↑s\nf : ι → α\nh : ∀ (i : ι), f i ∈ s\nhf : ¬Multipliable f\n⊢ ∏' (i : ι), f i ∈ s",
"ppTerm": "?neg✝",
"a... | [] | · simp [tprod_eq_one_of_not_multipliable hf, one_mem] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.MetricSpace.Isometry | {
"line": 150,
"column": 2
} | {
"line": 151,
"column": 19
} | {
"line": 153,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nf : α → β\nh : Isometry f\nx : α\nr : ℝ≥0∞\n⊢ f ⁻¹' Metric.closedEBall (f x) r = Metric.closedEBall x r",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Set.ext",
"PseudoEMetricSpace.t... | [] | ext y
simp [h.edist_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Isometry | {
"line": 150,
"column": 2
} | {
"line": 151,
"column": 19
} | {
"line": 153,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nf : α → β\nh : Isometry f\nx : α\nr : ℝ≥0∞\n⊢ f ⁻¹' Metric.closedEBall (f x) r = Metric.closedEBall x r",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Set.ext",
"PseudoEMetricSpace.t... | [] | ext y
simp [h.edist_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Isometry | {
"line": 158,
"column": 2
} | {
"line": 159,
"column": 19
} | {
"line": 161,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nf : α → β\nh : Isometry f\nx : α\nr : ℝ≥0∞\n⊢ f ⁻¹' Metric.eball (f x) r = Metric.eball x r",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Set.ext",
"PseudoEMetricSpace.toWeakPseudoE... | [] | ext y
simp [h.edist_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Isometry | {
"line": 158,
"column": 2
} | {
"line": 159,
"column": 19
} | {
"line": 161,
"column": 0
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nf : α → β\nh : Isometry f\nx : α\nr : ℝ≥0∞\n⊢ f ⁻¹' Metric.eball (f x) r = Metric.eball x r",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Set.ext",
"PseudoEMetricSpace.toWeakPseudoE... | [] | ext y
simp [h.edist_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Metrizable.Uniformity | {
"line": 136,
"column": 6
} | {
"line": 136,
"column": 16
} | {
"line": 137,
"column": 6
} | [
{
"pp": "X : Type u_1\nd : X → X → ℝ≥0\ndist_self : ∀ (x : X), d x x = 0\ndist_comm : ∀ (x y : X), d x y = d y x\nhd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))\nthis : IsTrans X fun x y ↦ d x y = 0\nx y : X\nl : List X\nihn : ∀ m < l.length, ∀ (x y : X) (l : List X), l.length =... | [
"X : Type u_1\nd : X → X → ℝ≥0\ndist_self : ∀ (x : X), d x x = 0\ndist_comm : ∀ (x y : X), d x y = d y x\nhd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))\nthis : IsTrans X fun x y ↦ d x y = 0\nx y : X\nl : List X\nihn : ∀ m < l.length, ∀ (x y : X) (l : List X), l.length = m → d x y ≤... | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 277,
"column": 2
} | {
"line": 277,
"column": 55
} | {
"line": 278,
"column": 2
} | [
{
"pp": "β : Type u_4\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\nf : β → ℝ≥0∞\nhf : Antitone f\n⊢ Tendsto f atTop (𝓝 0) ↔ ∀ (ε : ℝ≥0∞), 0 < ε → ∃ n, f n < ε",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"congrArg",
"PartialOrder.to... | [
"β : Type u_4\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\nf : β → ℝ≥0∞\nhf : Antitone f\n⊢ (∀ (ε : ℝ≥0∞), 0 < ε → ∃ n, f n ≤ ε) ↔ ∀ (ε : ℝ≥0∞), 0 < ε → ∃ n, f n < ε"
] | rw [ENNReal.tendsto_atTop_zero_iff_le_of_antitone hf] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.InfiniteSum.Constructions | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 63
} | {
"line": 117,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : CommMonoid α\ninst✝² : TopologicalSpace α\ninst✝¹ : ContinuousMul α\ninst✝ : RegularSpace α\nγ : β → Type u_4\nf : (b : β) × γ b → α\ng : β → α\na : α\nha : HasProd f a\nhf : ∀ (b : β), HasProd (fun c ↦ f ⟨b, c⟩) (g b)\n⊢ HasProd g a",
"ppTerm": "?m.21",
"as... | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : CommMonoid α\ninst✝² : TopologicalSpace α\ninst✝¹ : ContinuousMul α\ninst✝ : RegularSpace α\nγ : β → Type u_4\nf : (b : β) × γ b → α\ng : β → α\na : α\nha : HasProd f a\nhf : ∀ (b : β), HasProd (fun c ↦ f ⟨b, c⟩) (g b)\n⊢ ∀ (ib : Set α), ib ∈ 𝓝 a ∧ IsClosed[inst✝²] ib → ∃ ia, ... | refine (atTop_basis.tendsto_iff (closed_nhds_basis a)).mpr ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Algebra.InfiniteSum.Constructions | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 47
} | {
"line": 119,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : CommMonoid α\ninst✝² : TopologicalSpace α\ninst✝¹ : ContinuousMul α\ninst✝ : RegularSpace α\nγ : β → Type u_4\nf : (b : β) × γ b → α\ng : β → α\na : α\nha : HasProd f a\nhf : ∀ (b : β), HasProd (fun c ↦ f ⟨b, c⟩) (g b)\ns : Set α\nhs : s ∈ 𝓝 a\nhsc : IsClosed[inst✝... | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : CommMonoid α\ninst✝² : TopologicalSpace α\ninst✝¹ : ContinuousMul α\ninst✝ : RegularSpace α\nγ : β → Type u_4\nf : (b : β) × γ b → α\ng : β → α\na : α\nha : HasProd f a\nhf : ∀ (b : β), HasProd (fun c ↦ f ⟨b, c⟩) (g b)\ns : Set α\nhs : s ∈ 𝓝 a\nhsc : IsClosed[inst✝²] s\nu : Fi... | rcases mem_atTop_sets.mp (ha hs) with ⟨u, hu⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.Metrizable.Uniformity | {
"line": 168,
"column": 8
} | {
"line": 168,
"column": 30
} | {
"line": 168,
"column": 30
} | [
{
"pp": "case refine_3.inr\nX : Type u_1\nd : X → X → ℝ≥0\ndist_self : ∀ (x : X), d x x = 0\ndist_comm : ∀ (x y : X), d x y = d y x\nhd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))\nthis : IsTrans X fun x y ↦ d x y = 0\nx y : X\nl : List X\nihn : ∀ m < l.length, ∀ (x y : X) (l : ... | [
"case refine_3.inr\nX : Type u_1\nd : X → X → ℝ≥0\ndist_self : ∀ (x : X), d x x = 0\ndist_comm : ∀ (x y : X), d x y = d y x\nhd : ∀ (x₁ x₂ x₃ x₄ : X), d x₁ x₄ ≤ 2 * max (d x₁ x₂) (max (d x₂ x₃) (d x₃ x₄))\nthis : IsTrans X fun x y ↦ d x y = 0\nx y : X\nl : List X\nihn : ∀ m < l.length, ∀ (x y : X) (l : List X), l.l... | cons_getElem_drop_succ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.InfiniteSum.Constructions | {
"line": 275,
"column": 60
} | {
"line": 276,
"column": 57
} | {
"line": 278,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_4\nX : α → Type u_5\ninst✝¹ : (x : α) → CommMonoid (X x)\ninst✝ : (x : α) → TopologicalSpace (X x)\nL : SummationFilter ι\nf : ι → (x : α) → X x\ng : (x : α) → X x\n⊢ HasProd f g L ↔ ∀ (x : α), HasProd (fun i ↦ f i x) (g x) L",
"ppTerm": "?m.15",
"assigned": true,
"... | [] | by
simp only [HasProd, tendsto_pi_nhds, Finset.prod_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.UniformSpace.Completion | {
"line": 114,
"column": 6
} | {
"line": 114,
"column": 45
} | {
"line": 115,
"column": 4
} | [
{
"pp": "case hh\nα : Type u\ninst✝ : UniformSpace α\n⊢ Monotone fun s ↦ s ○ s",
"ppTerm": "?hh",
"assigned": true,
"usedConstants": [
"SetRel",
"PartialOrder.toPreorder",
"monotone_id",
"CompleteLattice.toConditionallyCompleteLattice",
"Monotone.relComp",
"id",
... | [] | · exact monotone_id.relComp monotone_id | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.UniformSpace.Completion | {
"line": 118,
"column": 6
} | {
"line": 118,
"column": 45
} | {
"line": 119,
"column": 6
} | [
{
"pp": "case hg\nα : Type u\ninst✝ : UniformSpace α\n⊢ Monotone fun s ↦ s ○ s",
"ppTerm": "?hg",
"assigned": true,
"usedConstants": [
"SetRel",
"PartialOrder.toPreorder",
"monotone_id",
"CompleteLattice.toConditionallyCompleteLattice",
"Monotone.relComp",
"id",
... | [
"case hh\nα : Type u\ninst✝ : UniformSpace α\n⊢ Monotone gen"
] | · exact monotone_id.relComp monotone_id | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 552,
"column": 29
} | {
"line": 552,
"column": 70
} | {
"line": 552,
"column": 70
} | [
{
"pp": "ι : Type u_4\nfi : Filter ι\nf : ι → ℝ≥0∞\nhf : ∀ (i : ι), f i ≠ ∞\n⊢ Tendsto (fun n ↦ (f n).toReal) fi (𝓝 (ENNReal.toReal 0)) ↔ Tendsto f fi (𝓝 0)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"PseudoMetricSpace.toUniformS... | [
"ι : Type u_4\nfi : Filter ι\nf : ι → ℝ≥0∞\nhf : ∀ (i : ι), f i ≠ ∞\n⊢ Tendsto f fi (𝓝 0) ↔ Tendsto f fi (𝓝 0)"
] | tendsto_toReal_iff hf ENNReal.zero_ne_top | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 623,
"column": 2
} | {
"line": 623,
"column": 68
} | {
"line": 623,
"column": 69
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\nf : α → ℝ≥0∞\nC : ℝ≥0∞\nhC : C ≠ ∞\nh : ∀ (x y : α), f x ≤ f y + C * edist x y\nx : α\nε : ℝ≥0∞\nε0 : ε > 0\nδ : ℝ≥0\nδ0 : δ > 0\nhδ : C * ↑δ < ε\ny : α\nhy : y ∈ closedEBall x ↑δ\n⊢ f y ∈ Icc (f x - ε) (f x + ε)",
"ppTerm": "?m.91",
"assigned": true,... | [
"case refine_1\nα : Type u_1\ninst✝ : PseudoEMetricSpace α\nf : α → ℝ≥0∞\nC : ℝ≥0∞\nhC : C ≠ ∞\nh : ∀ (x y : α), f x ≤ f y + C * edist x y\nx : α\nε : ℝ≥0∞\nε0 : ε > 0\nδ : ℝ≥0\nδ0 : δ > 0\nhδ : C * ↑δ < ε\ny : α\nhy : y ∈ closedEBall x ↑δ\n⊢ f y + C * edist x y ≤ f y + ε",
"case refine_2\nα : Type u_1\ninst✝ : P... | refine ⟨tsub_le_iff_right.2 <| (h x y).trans ?_, (h y x).trans ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 627,
"column": 41
} | {
"line": 627,
"column": 47
} | {
"line": 627,
"column": 47
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\n⊢ 2 ≠ ∞",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"LinearOrder.toDecidableEq",
"Nat.instAtLeastTwoHAddOfNat",
"id",
"AddMonoidWithOne.toNatCast",
"N... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 627,
"column": 41
} | {
"line": 627,
"column": 47
} | {
"line": 627,
"column": 47
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\n⊢ 2 ≠ ∞",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"LinearOrder.toDecidableEq",
"Nat.instAtLeastTwoHAddOfNat",
"id",
"AddMonoidWithOne.toNatCast",
"N... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 627,
"column": 41
} | {
"line": 627,
"column": 47
} | {
"line": 627,
"column": 47
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoEMetricSpace α\n⊢ 2 ≠ ∞",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"LinearOrder.toDecidableEq",
"Nat.instAtLeastTwoHAddOfNat",
"id",
"AddMonoidWithOne.toNatCast",
"N... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Order.Lattice | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 24
} | {
"line": 77,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝³ : NormedAddCommGroup α\ninst✝² : Lattice α\ninst✝¹ : HasSolidNorm α\ninst✝ : IsOrderedAddMonoid α\na b : α\nh : b ⊓ -b ≤ a ⊓ -a\n⊢ a ⊔ -a ≤ |b|",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZero... | [
"α : Type u_1\ninst✝³ : NormedAddCommGroup α\ninst✝² : Lattice α\ninst✝¹ : HasSolidNorm α\ninst✝ : IsOrderedAddMonoid α\na b : α\nh : b ⊓ -b ≤ a ⊓ -a\n⊢ - -a ⊔ -a ≤ |b|"
] | nth_rw 1 [← neg_neg a] | Mathlib.Tactic._aux_Mathlib_Tactic_NthRewrite___macroRules_Mathlib_Tactic_tacticNth_rw______1 | Mathlib.Tactic.tacticNth_rw_____ |
Mathlib.Topology.MetricSpace.IsometricSMul | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 45
} | {
"line": 107,
"column": 0
} | [
{
"pp": "M : Type u\ninst✝² : DivInvMonoid M\ninst✝¹ : PseudoEMetricSpace M\ninst✝ : IsIsometricSMul Mᵐᵒᵖ M\na b c : M\n⊢ edist (a / c) (b / c) = edist a b",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
"DivInvMonoid.toInv",
... | [] | simp only [div_eq_mul_inv, edist_mul_right] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.MetricSpace.IsometricSMul | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 45
} | {
"line": 107,
"column": 0
} | [
{
"pp": "M : Type u\ninst✝² : DivInvMonoid M\ninst✝¹ : PseudoEMetricSpace M\ninst✝ : IsIsometricSMul Mᵐᵒᵖ M\na b c : M\n⊢ edist (a / c) (b / c) = edist a b",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
"DivInvMonoid.toInv",
... | [] | simp only [div_eq_mul_inv, edist_mul_right] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.IsometricSMul | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 45
} | {
"line": 107,
"column": 0
} | [
{
"pp": "M : Type u\ninst✝² : DivInvMonoid M\ninst✝¹ : PseudoEMetricSpace M\ninst✝ : IsIsometricSMul Mᵐᵒᵖ M\na b c : M\n⊢ edist (a / c) (b / c) = edist a b",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
"DivInvMonoid.toInv",
... | [] | simp only [div_eq_mul_inv, edist_mul_right] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.IsometricSMul | {
"line": 228,
"column": 2
} | {
"line": 228,
"column": 21
} | {
"line": 229,
"column": 2
} | [
{
"pp": "G : Type v\ninst✝² : Group G\ninst✝¹ : PseudoEMetricSpace G\ninst✝ : IsIsometricSMul Gᵐᵒᵖ G\na b : G\nr : ℝ≥0∞\n⊢ (fun x ↦ x * a) ⁻¹' eball b r = eball (b / a) r",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
... | [
"G : Type v\ninst✝² : Group G\ninst✝¹ : PseudoEMetricSpace G\ninst✝ : IsIsometricSMul Gᵐᵒᵖ G\na b : G\nr : ℝ≥0∞\n⊢ (fun x ↦ x * a) ⁻¹' eball b r = eball (b * a⁻¹) r"
] | rw [div_eq_mul_inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.MetricSpace.IsometricSMul | {
"line": 239,
"column": 2
} | {
"line": 239,
"column": 21
} | {
"line": 240,
"column": 2
} | [
{
"pp": "G : Type v\ninst✝² : Group G\ninst✝¹ : PseudoEMetricSpace G\ninst✝ : IsIsometricSMul Gᵐᵒᵖ G\na b : G\nr : ℝ≥0∞\n⊢ (fun x ↦ x * a) ⁻¹' closedEBall b r = closedEBall (b / a) r",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv... | [
"G : Type v\ninst✝² : Group G\ninst✝¹ : PseudoEMetricSpace G\ninst✝ : IsIsometricSMul Gᵐᵒᵖ G\na b : G\nr : ℝ≥0∞\n⊢ (fun x ↦ x * a) ⁻¹' closedEBall b r = closedEBall (b * a⁻¹) r"
] | rw [div_eq_mul_inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.MetricSpace.IsometricSMul | {
"line": 409,
"column": 2
} | {
"line": 409,
"column": 21
} | {
"line": 410,
"column": 2
} | [
{
"pp": "G : Type v\ninst✝² : Group G\ninst✝¹ : PseudoMetricSpace G\ninst✝ : IsIsometricSMul Gᵐᵒᵖ G\na b : G\nr : ℝ\n⊢ (fun x ↦ x * a) ⁻¹' ball b r = ball (b / a) r",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMu... | [
"G : Type v\ninst✝² : Group G\ninst✝¹ : PseudoMetricSpace G\ninst✝ : IsIsometricSMul Gᵐᵒᵖ G\na b : G\nr : ℝ\n⊢ (fun x ↦ x * a) ⁻¹' ball b r = ball (b * a⁻¹) r"
] | rw [div_eq_mul_inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.MetricSpace.IsometricSMul | {
"line": 420,
"column": 2
} | {
"line": 420,
"column": 21
} | {
"line": 421,
"column": 2
} | [
{
"pp": "G : Type v\ninst✝² : Group G\ninst✝¹ : PseudoMetricSpace G\ninst✝ : IsIsometricSMul Gᵐᵒᵖ G\na b : G\nr : ℝ\n⊢ (fun x ↦ x * a) ⁻¹' closedBall b r = closedBall (b / a) r",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
... | [
"G : Type v\ninst✝² : Group G\ninst✝¹ : PseudoMetricSpace G\ninst✝ : IsIsometricSMul Gᵐᵒᵖ G\na b : G\nr : ℝ\n⊢ (fun x ↦ x * a) ⁻¹' closedBall b r = closedBall (b * a⁻¹) r"
] | rw [div_eq_mul_inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 429,
"column": 2
} | {
"line": 429,
"column": 12
} | {
"line": 430,
"column": 2
} | [
{
"pp": "E : Type u_2\ninst✝ : SeminormedCommGroup E\nu v : ℕ → E\nN : ℕ\nhuv : ∀ n ≥ N, u n = v n\nhv : CauchySeq fun n ↦ ∏ k ∈ range (n + 1), v k\nd : ℕ → E := fun n ↦ ∏ k ∈ range (n + 1), u k / v k\nn : ℕ\nhn : n ≥ N\n⊢ ∀ n ≥ N + 1, u n / v n = 1",
"ppTerm": "?m.142",
"assigned": true,
"usedConst... | [
"E : Type u_2\ninst✝ : SeminormedCommGroup E\nu v : ℕ → E\nN : ℕ\nhuv : ∀ n ≥ N, u n = v n\nhv : CauchySeq fun n ↦ ∏ k ∈ range (n + 1), v k\nd : ℕ → E := fun n ↦ ∏ k ∈ range (n + 1), u k / v k\nn : ℕ\nhn : n ≥ N\nm : ℕ\nhm : m ≥ N + 1\n⊢ u m / v m = 1"
] | intro m hm | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Normed.Field.Basic | {
"line": 120,
"column": 4
} | {
"line": 121,
"column": 13
} | {
"line": 122,
"column": 2
} | [
{
"pp": "case mp\n𝕜 : Type u_5\ninst✝¹ : NormedDivisionRing 𝕜\ninst✝ : DiscreteTopology 𝕜\nx : 𝕜\nhx : ‖x‖ = 1\n⊢ x ≠ 0",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Norm.norm",
"False",
"Real.partialOrder",
"Real",
"FloorRing.toFloorSemiring",
"Rea... | [] | contrapose! hx
simp [hx] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Field.Basic | {
"line": 120,
"column": 4
} | {
"line": 121,
"column": 13
} | {
"line": 122,
"column": 2
} | [
{
"pp": "case mp\n𝕜 : Type u_5\ninst✝¹ : NormedDivisionRing 𝕜\ninst✝ : DiscreteTopology 𝕜\nx : 𝕜\nhx : ‖x‖ = 1\n⊢ x ≠ 0",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Norm.norm",
"False",
"Real.partialOrder",
"Real",
"FloorRing.toFloorSemiring",
"Rea... | [] | contrapose! hx
simp [hx] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.DilationEquiv | {
"line": 129,
"column": 4
} | {
"line": 131,
"column": 39
} | {
"line": 132,
"column": 2
} | [
{
"pp": "case pos\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : PseudoEMetricSpace X\ninst✝¹ : PseudoEMetricSpace Y\ninst✝ : PseudoEMetricSpace Z\ne : X ≃ᵈ Y\ne' : Y ≃ᵈ Z\nhX : ∀ (x y : X), edist x y = 0 ∨ edist x y = ∞\n⊢ ratio (e.trans e') = ratio e * ratio e'",
"ppTerm": "?pos✝",
"assigned": tr... | [] | have hY : ∀ x y : Y, edist x y = 0 ∨ edist x y = ∞ := e.surjective.forall₂.2 fun x y ↦ by
refine (hX x y).imp (fun h ↦ ?_) fun h ↦ ?_ <;> simp [*, Dilation.ratio_ne_zero]
simp [Dilation.ratio_of_trivial, *] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.DilationEquiv | {
"line": 129,
"column": 4
} | {
"line": 131,
"column": 39
} | {
"line": 132,
"column": 2
} | [
{
"pp": "case pos\nX : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : PseudoEMetricSpace X\ninst✝¹ : PseudoEMetricSpace Y\ninst✝ : PseudoEMetricSpace Z\ne : X ≃ᵈ Y\ne' : Y ≃ᵈ Z\nhX : ∀ (x y : X), edist x y = 0 ∨ edist x y = ∞\n⊢ ratio (e.trans e') = ratio e * ratio e'",
"ppTerm": "?pos✝",
"assigned": tr... | [] | have hY : ∀ x y : Y, edist x y = 0 ∨ edist x y = ∞ := e.surjective.forall₂.2 fun x y ↦ by
refine (hX x y).imp (fun h ↦ ?_) fun h ↦ ?_ <;> simp [*, Dilation.ratio_ne_zero]
simp [Dilation.ratio_of_trivial, *] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Ring.Basic | {
"line": 248,
"column": 2
} | {
"line": 248,
"column": 23
} | {
"line": 250,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝ : NonUnitalSeminormedRing α\nx y : α\n⊢ ‖(AddMonoidHom.mulRight x) y‖ ≤ ‖y‖ * ‖x‖",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"norm_mul_le"
],
"usedFVars": [
"α",
"inst✝",
"y",
"x"
],
"usedGoals": []
}
] | [] | exact norm_mul_le y x | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.MulAction | {
"line": 182,
"column": 37
} | {
"line": 182,
"column": 66
} | {
"line": 183,
"column": 6
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : NormedDivisionRing α\ninst✝² : SeminormedAddGroup β\ninst✝¹ : MulActionWithZero α β\ninst✝ : IsBoundedSMul α β\nr : α\nx : β\nh : ¬r = 0\n⊢ ‖r‖ * ‖r⁻¹ • r • x‖ ≤ ‖r‖ * (‖r⁻¹‖ * ‖r • x‖)",
"ppTerm": "?m.95",
"assigned": true,
"usedConstants": [
"nor... | [] | by gcongr; apply norm_smul_le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Field.Lemmas | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 40
} | {
"line": 151,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : NormedDivisionRing α\nX : Type u_4\nι : Type u_5\ninst✝ : TopologicalSpace X\ns : Set X\nF : ι → X → α\nf : X → α\nl : Filter ι\nhF : ∀ x ∈ s, Tendsto (fun y ↦ (f y.2, F y.1 y.2)) (l ×ˢ 𝓝[s] x) (𝓤 α)\nhf : ∀ x ∈ s, Disjoint (map f (𝓝[s] x)) (𝓝 0)\nx : X\nhx : x ∈ s\nU : Set X... | [] | simp_all [(half_lt_self hr₀).trans_le] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Normed.Field.Lemmas | {
"line": 318,
"column": 2
} | {
"line": 318,
"column": 30
} | {
"line": 320,
"column": 0
} | [
{
"pp": "case mpr.inr\nK : Type u_4\nval✝ : NontriviallyNormedField K\nh : IsComplete (closedBall 0 1)\nu : ℕ → K\nhu : CauchySeq u\nk : ℝ\nhk : k ∈ upperBounds (Set.range fun n ↦ ‖u n‖)\nkpos : 0 ≤ k\nx : K\nhx : k < ‖x‖\nhu' : CauchySeq ((fun x_1 ↦ x_1 / x) ∘ u)\nhb : ∀ (n : ℕ), ((fun x_1 ↦ x_1 / x) ∘ u) n ∈ ... | [] | simp [div_mul_cancel₀ _ hx'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.MeasurableSpace.Constructions | {
"line": 360,
"column": 12
} | {
"line": 360,
"column": 15
} | {
"line": 361,
"column": 2
} | [
{
"pp": "β : Type u_2\ninst✝ : MeasurableSpace β\nx y : β\nhx : x ∉ measurableAtom y\nz : β\nhzx : z ∈ measurableAtom x\n⊢ z ∉ measurableAtom y",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Membership.mem",
"measurableAtom",
"Set.instMembership",
"Set"
],
... | [
"β : Type u_2\ninst✝ : MeasurableSpace β\nx y : β\nhx : x ∉ measurableAtom y\nz : β\nhzx : z ∈ measurableAtom x\nhzy : z ∈ measurableAtom y\n⊢ False"
] | hzy | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.OuterMeasure.Operations | {
"line": 348,
"column": 30
} | {
"line": 348,
"column": 84
} | {
"line": 350,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\ns : Set α\nhs : s.Nonempty\n⊢ ((comap f) ⊤) s = ⊤ s",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Semiring.toModule",
"IsScalarTower.right",
"CompleteLattice.toLattice",
... | [] | rw [comap_apply, top_apply hs, top_apply (hs.image _)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.OuterMeasure.Operations | {
"line": 348,
"column": 30
} | {
"line": 348,
"column": 84
} | {
"line": 350,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\ns : Set α\nhs : s.Nonempty\n⊢ ((comap f) ⊤) s = ⊤ s",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Semiring.toModule",
"IsScalarTower.right",
"CompleteLattice.toLattice",
... | [] | rw [comap_apply, top_apply hs, top_apply (hs.image _)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.OuterMeasure.Operations | {
"line": 348,
"column": 30
} | {
"line": 348,
"column": 84
} | {
"line": 350,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\ns : Set α\nhs : s.Nonempty\n⊢ ((comap f) ⊤) s = ⊤ s",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Semiring.toModule",
"IsScalarTower.right",
"CompleteLattice.toLattice",
... | [] | rw [comap_apply, top_apply hs, top_apply (hs.image _)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Interval.Set.Instances | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 6
} | {
"line": 86,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\nx : ↑(Icc 0 1)\n⊢ ↑x = 0 ↔ x = 0",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Set.Icc.instZero",
"PartialOrder.toPreorder",
"... | [
"R : Type u_1\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\nx : ↑(Icc 0 1)\n⊢ x = 0 ↔ ↑x = 0"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.Order.Interval.Set.Instances | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 6
} | {
"line": 94,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\nx : ↑(Icc 0 1)\n⊢ ↑x = 1 ↔ x = 1",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"PartialOrder.toPreorder",
"Membership.mem",
"Se... | [
"R : Type u_1\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\nx : ↑(Icc 0 1)\n⊢ x = 1 ↔ ↑x = 1"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.Order.Interval.Set.Instances | {
"line": 188,
"column": 2
} | {
"line": 188,
"column": 6
} | {
"line": 189,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : IsOrderedRing R\ninst✝ : Nontrivial R\nx : ↑(Ico 0 1)\n⊢ ↑x = 0 ↔ x = 0",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Set.Ico.instZero",
"PartialOrd... | [
"R : Type u_1\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : IsOrderedRing R\ninst✝ : Nontrivial R\nx : ↑(Ico 0 1)\n⊢ x = 0 ↔ ↑x = 0"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Algebra.Order.Interval.Set.Instances | {
"line": 250,
"column": 2
} | {
"line": 250,
"column": 6
} | {
"line": 251,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsStrictOrderedRing R\nx : ↑(Ioc 0 1)\n⊢ ↑x = 1 ↔ x = 1",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Set.Ioc",
"NonAssocSemiring.toAddCommMonoidWithOne",
"PartialOrder.toPreorder",
"Mem... | [
"R : Type u_1\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsStrictOrderedRing R\nx : ↑(Ioc 0 1)\n⊢ x = 1 ↔ ↑x = 1"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Data.Sign.Defs | {
"line": 93,
"column": 18
} | {
"line": 93,
"column": 24
} | {
"line": 94,
"column": 2
} | [
{
"pp": "⊢ ∀ (a b c : SignType), a * b * c = a * (b * c)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"of_decide_eq_true",
"id",
"instDecidableEqSignType",
"Bool.true",
"instFintypeSignType",
"Bool",
"SignType",
"Fintype... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.Sign.Defs | {
"line": 93,
"column": 18
} | {
"line": 93,
"column": 24
} | {
"line": 94,
"column": 2
} | [
{
"pp": "⊢ ∀ (a b c : SignType), a * b * c = a * (b * c)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"of_decide_eq_true",
"id",
"instDecidableEqSignType",
"Bool.true",
"instFintypeSignType",
"Bool",
"SignType",
"Fintype... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Sign.Defs | {
"line": 93,
"column": 18
} | {
"line": 93,
"column": 24
} | {
"line": 94,
"column": 2
} | [
{
"pp": "⊢ ∀ (a b c : SignType), a * b * c = a * (b * c)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"of_decide_eq_true",
"id",
"instDecidableEqSignType",
"Bool.true",
"instFintypeSignType",
"Bool",
"SignType",
"Fintype... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Sign.Defs | {
"line": 92,
"column": 17
} | {
"line": 92,
"column": 23
} | {
"line": 93,
"column": 2
} | [
{
"pp": "⊢ ∀ (a b : SignType), a * b = b * a",
"ppTerm": "?m.120",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"of_decide_eq_true",
"id",
"instDecidableEqSignType",
"Bool.true",
"instFintypeSignType",
"Bool",
"SignType",
"Fintype.decidableF... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.Sign.Defs | {
"line": 92,
"column": 17
} | {
"line": 92,
"column": 23
} | {
"line": 93,
"column": 2
} | [
{
"pp": "⊢ ∀ (a b : SignType), a * b = b * a",
"ppTerm": "?m.120",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"of_decide_eq_true",
"id",
"instDecidableEqSignType",
"Bool.true",
"instFintypeSignType",
"Bool",
"SignType",
"Fintype.decidableF... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Sign.Defs | {
"line": 92,
"column": 17
} | {
"line": 92,
"column": 23
} | {
"line": 93,
"column": 2
} | [
{
"pp": "⊢ ∀ (a b : SignType), a * b = b * a",
"ppTerm": "?m.120",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"of_decide_eq_true",
"id",
"instDecidableEqSignType",
"Bool.true",
"instFintypeSignType",
"Bool",
"SignType",
"Fintype.decidableF... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Sign.Defs | {
"line": 101,
"column": 17
} | {
"line": 101,
"column": 23
} | {
"line": 102,
"column": 2
} | [
{
"pp": "⊢ ∀ (a b c : SignType), a ≤ b → b ≤ c → a ≤ c",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"forall_prop_decidable",
"LE.le",
"SignType.instDecidableLE",
"Bool.true",
"instFintypeSignType",
"Bool",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.Sign.Defs | {
"line": 101,
"column": 17
} | {
"line": 101,
"column": 23
} | {
"line": 102,
"column": 2
} | [
{
"pp": "⊢ ∀ (a b c : SignType), a ≤ b → b ≤ c → a ≤ c",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"forall_prop_decidable",
"LE.le",
"SignType.instDecidableLE",
"Bool.true",
"instFintypeSignType",
"Bool",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Sign.Defs | {
"line": 101,
"column": 17
} | {
"line": 101,
"column": 23
} | {
"line": 102,
"column": 2
} | [
{
"pp": "⊢ ∀ (a b c : SignType), a ≤ b → b ≤ c → a ≤ c",
"ppTerm": "?m.58",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"forall_prop_decidable",
"LE.le",
"SignType.instDecidableLE",
"Bool.true",
"instFintypeSignType",
"Bool",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Sign.Defs | {
"line": 100,
"column": 20
} | {
"line": 100,
"column": 26
} | {
"line": 101,
"column": 2
} | [
{
"pp": "⊢ ∀ (a b : SignType), a ≤ b → b ≤ a → a = b",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"forall_prop_decidable",
"LE.le",
"SignType.instDecidableLE",
"instDecidableEqSignType",
"Bool.true",
"instFintype... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.Sign.Defs | {
"line": 100,
"column": 20
} | {
"line": 100,
"column": 26
} | {
"line": 101,
"column": 2
} | [
{
"pp": "⊢ ∀ (a b : SignType), a ≤ b → b ≤ a → a = b",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"forall_prop_decidable",
"LE.le",
"SignType.instDecidableLE",
"instDecidableEqSignType",
"Bool.true",
"instFintype... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Sign.Defs | {
"line": 100,
"column": 20
} | {
"line": 100,
"column": 26
} | {
"line": 101,
"column": 2
} | [
{
"pp": "⊢ ∀ (a b : SignType), a ≤ b → b ≤ a → a = b",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"forall_prop_decidable",
"LE.le",
"SignType.instDecidableLE",
"instDecidableEqSignType",
"Bool.true",
"instFintype... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Sign.Defs | {
"line": 99,
"column": 17
} | {
"line": 99,
"column": 23
} | {
"line": 100,
"column": 2
} | [
{
"pp": "⊢ ∀ (a b : SignType), a ≤ b ∨ b ≤ a",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"LE.le",
"SignType.instDecidableLE",
"Bool.true",
"instFintypeSignType",
"Bool",
"SignType",
"Fintype.decidableForal... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.Sign.Defs | {
"line": 99,
"column": 17
} | {
"line": 99,
"column": 23
} | {
"line": 100,
"column": 2
} | [
{
"pp": "⊢ ∀ (a b : SignType), a ≤ b ∨ b ≤ a",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"LE.le",
"SignType.instDecidableLE",
"Bool.true",
"instFintypeSignType",
"Bool",
"SignType",
"Fintype.decidableForal... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Sign.Defs | {
"line": 99,
"column": 17
} | {
"line": 99,
"column": 23
} | {
"line": 100,
"column": 2
} | [
{
"pp": "⊢ ∀ (a b : SignType), a ≤ b ∨ b ≤ a",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"LE.le",
"SignType.instDecidableLE",
"Bool.true",
"instFintypeSignType",
"Bool",
"SignType",
"Fintype.decidableForal... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Sign.Basic | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 8
} | {
"line": 66,
"column": 0
} | [
{
"pp": "⊢ Finset.univ = {0, -1, 1}",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"Finset.univ",
"SignType.instOne",
"Finset",
"id",
"Insert.insert",
"SignType.instNeg",
"Finset.decidableEq",
"SignType.instZero",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.Data.Sign.Basic | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 8
} | {
"line": 66,
"column": 0
} | [
{
"pp": "⊢ Finset.univ = {0, -1, 1}",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"Finset.univ",
"SignType.instOne",
"Finset",
"id",
"Insert.insert",
"SignType.instNeg",
"Finset.decidableEq",
"SignType.instZero",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Sign.Basic | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 8
} | {
"line": 66,
"column": 0
} | [
{
"pp": "⊢ Finset.univ = {0, -1, 1}",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"Finset.univ",
"SignType.instOne",
"Finset",
"id",
"Insert.insert",
"SignType.instNeg",
"Finset.decidableEq",
"SignType.instZero",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.EReal.Inv | {
"line": 126,
"column": 4
} | {
"line": 127,
"column": 20
} | {
"line": 129,
"column": 0
} | [
{
"pp": "case mpr\nx y : EReal\n⊢ x = y → x.abs = y.abs ∧ sign x = sign y",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"EReal.abs",
"PartialOrder.toPreorder",
"SignType.instLinearOrder",
"EReal",
"SemilatticeInf.toPartialOrder",
"DistribLattice.toLatti... | [] | rintro rfl
exact ⟨rfl, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.EReal.Inv | {
"line": 126,
"column": 4
} | {
"line": 127,
"column": 20
} | {
"line": 129,
"column": 0
} | [
{
"pp": "case mpr\nx y : EReal\n⊢ x = y → x.abs = y.abs ∧ sign x = sign y",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"EReal.abs",
"PartialOrder.toPreorder",
"SignType.instLinearOrder",
"EReal",
"SemilatticeInf.toPartialOrder",
"DistribLattice.toLatti... | [] | rintro rfl
exact ⟨rfl, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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