module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Combinatorics.Enumerative.InclusionExclusion | {
"line": 93,
"column": 6
} | {
"line": 96,
"column": 26
} | {
"line": 97,
"column": 4
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nG : Type u_3\ninst✝ : AddCommGroup G\ns : Finset ι\nS : ι → Set α\nf : α → G\na : α\nha : a ∈ ⋃ i ∈ s, S i\n⊢ ∑ t ∈ s.powerset, (-1) ^ #t • (⋂ i ∈ t, S i).indicator f a = (∏ i ∈ s, (1 - (S i).indicator 1 a)) • f a",
"ppTerm": "?m.375",
"assigned": true,
"usedCons... | [] | simp only [Int.reduceNeg, prod_sub, prod_const_one, mul_one, sum_smul]
congr! 1 with t
simp only [prod_const_one, prod_indicator_apply]
simp [Set.indicator] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Enumerative.InclusionExclusion | {
"line": 170,
"column": 10
} | {
"line": 170,
"column": 29
} | {
"line": 170,
"column": 30
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nG : Type u_3\ninst✝² : AddCommGroup G\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ns : Finset ι\nS : ι → Finset α\nf : α → G\n⊢ ∑ a ∈ s.inf fun i ↦ (S i)ᶜ, f a = ∑ a, f a - ∑ a ∈ s.biUnion S, f a",
"ppTerm": "?m.81",
"assigned": true,
"usedConstants": [
"Eq.... | [
"ι : Type u_1\nα : Type u_2\nG : Type u_3\ninst✝² : AddCommGroup G\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ns : Finset ι\nS : ι → Finset α\nf : α → G\n⊢ ∑ a ∈ (s.sup S)ᶜ, f a = ∑ a, f a - ∑ a ∈ s.biUnion S, f a"
] | ← Finset.compl_sup, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Sequences | {
"line": 362,
"column": 2
} | {
"line": 363,
"column": 36
} | {
"line": 364,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝ : UniformSpace X\ns : Set X\nV : Set (X × X)\nV_in : V ∈ 𝓤 X\nh : ∀ (t : Set X), t.Finite → ¬s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ V}\nu : ℕ → X\nu_in : ∀ (n : ℕ), u n ∈ s\nhu : ∀ (n m : ℕ), m < n → u m ∉ ball (u n) V\nx : X\nx✝ : x ∈ s\nφ : ℕ → ℕ\nhφ : StrictMono φ\nhuφ : Tendsto (u ∘ φ) atT... | [
"X : Type u_1\ninst✝ : UniformSpace X\ns : Set X\nV : Set (X × X)\nV_in : V ∈ 𝓤 X\nh : ∀ (t : Set X), t.Finite → ¬s ⊆ ⋃ y ∈ t, {x | (x, y) ∈ V}\nu : ℕ → X\nu_in : ∀ (n : ℕ), u n ∈ s\nhu : ∀ (n m : ℕ), m < n → u m ∉ ball (u n) V\nx : X\nx✝ : x ∈ s\nφ : ℕ → ℕ\nhφ : StrictMono φ\nhuφ : Tendsto (u ∘ φ) atTop (𝓝 x)\nN... | obtain ⟨N, hN⟩ : ∃ N, ∀ p q, p ≥ N → q ≥ N → (u (φ p), u (φ q)) ∈ V :=
huφ.cauchySeq.mem_entourage V_in | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 331,
"column": 15
} | {
"line": 331,
"column": 37
} | {
"line": 331,
"column": 37
} | [
{
"pp": "E : Type u_4\ninst✝ : SeminormedCommGroup E\na : E\ns : Subgroup E\nhg : a ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑s\nb : ℕ → ℝ\nb_pos : ∀ (n : ℕ), 0 < b n\nu : ℕ → E\nu_in : ∀ (n : ℕ), u n ∈ s\nlim_u : Tendsto u atTop (𝓝 a)\n⊢ {x | ‖x⁻¹ * a‖ < b 0} ∈ 𝓝 a",
"ppTerm": "?m.1... | [
"E : Type u_4\ninst✝ : SeminormedCommGroup E\na : E\ns : Subgroup E\nhg : a ∈ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑s\nb : ℕ → ℝ\nb_pos : ∀ (n : ℕ), 0 < b n\nu : ℕ → E\nu_in : ∀ (n : ℕ), u n ∈ s\nlim_u : Tendsto u atTop (𝓝 a)\n⊢ {x | dist x a < b 0} ∈ 𝓝 a"
] | ← dist_eq_norm_inv_mul | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Algebra.UniformMulAction | {
"line": 102,
"column": 4
} | {
"line": 103,
"column": 46
} | {
"line": 105,
"column": 0
} | [
{
"pp": "M : Type v\nX : Type x\nY : Type y\ninst✝⁴ : UniformSpace X\ninst✝³ : UniformSpace Y\ninst✝² : SMul M X\ninst✝¹ : SMul M Y\ninst✝ : UniformContinuousConstSMul M Y\nf : X → Y\nhf : IsUniformInducing f\nhsmul : ∀ (c : M) (x : X), f (c • x) = c • f x\nc : M\n⊢ UniformContinuous fun x ↦ c • x",
"ppTerm... | [] | simpa only [hf.uniformContinuous_iff, Function.comp_def, hsmul]
using! hf.uniformContinuous.const_smul c | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.Topology.Algebra.UniformMulAction | {
"line": 102,
"column": 4
} | {
"line": 103,
"column": 46
} | {
"line": 105,
"column": 0
} | [
{
"pp": "M : Type v\nX : Type x\nY : Type y\ninst✝⁴ : UniformSpace X\ninst✝³ : UniformSpace Y\ninst✝² : SMul M X\ninst✝¹ : SMul M Y\ninst✝ : UniformContinuousConstSMul M Y\nf : X → Y\nhf : IsUniformInducing f\nhsmul : ∀ (c : M) (x : X), f (c • x) = c • f x\nc : M\n⊢ UniformContinuous fun x ↦ c • x",
"ppTerm... | [] | simpa only [hf.uniformContinuous_iff, Function.comp_def, hsmul]
using! hf.uniformContinuous.const_smul c | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.UniformMulAction | {
"line": 102,
"column": 4
} | {
"line": 103,
"column": 46
} | {
"line": 105,
"column": 0
} | [
{
"pp": "M : Type v\nX : Type x\nY : Type y\ninst✝⁴ : UniformSpace X\ninst✝³ : UniformSpace Y\ninst✝² : SMul M X\ninst✝¹ : SMul M Y\ninst✝ : UniformContinuousConstSMul M Y\nf : X → Y\nhf : IsUniformInducing f\nhsmul : ∀ (c : M) (x : X), f (c • x) = c • f x\nc : M\n⊢ UniformContinuous fun x ↦ c • x",
"ppTerm... | [] | simpa only [hf.uniformContinuous_iff, Function.comp_def, hsmul]
using! hf.uniformContinuous.const_smul c | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.Completion | {
"line": 535,
"column": 8
} | {
"line": 535,
"column": 37
} | {
"line": 535,
"column": 38
} | [
{
"pp": "case refine_2\nα✝ : Type u_1\ninst✝³ : UniformSpace α✝\nβ : Type u_2\ninst✝² : UniformSpace β\nγ : Type u_3\ninst✝¹ : UniformSpace γ\nα : Type u\ninst✝ : UniformSpace α\na✝ : Completion α\na : α\n⊢ Completion.extension (lift' coe') (Completion.map SeparationQuotient.mk ↑a) = ↑a",
"ppTerm": "?refine... | [
"case refine_2\nα✝ : Type u_1\ninst✝³ : UniformSpace α✝\nβ : Type u_2\ninst✝² : UniformSpace β\nγ : Type u_3\ninst✝¹ : UniformSpace γ\nα : Type u\ninst✝ : UniformSpace α\na✝ : Completion α\na : α\n⊢ Completion.extension (lift' coe') ↑(SeparationQuotient.mk a) = ↑a"
] | map_coe uniformContinuous_mk, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UniformSpace.Completion | {
"line": 526,
"column": 56
} | {
"line": 536,
"column": 41
} | {
"line": 538,
"column": 0
} | [
{
"pp": "α✝ : Type u_1\ninst✝³ : UniformSpace α✝\nβ : Type u_2\ninst✝² : UniformSpace β\nγ : Type u_3\ninst✝¹ : UniformSpace γ\nα : Type u\ninst✝ : UniformSpace α\n⊢ Completion (SeparationQuotient α) ≃ Completion α",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Unif... | [] | by
refine ⟨Completion.extension (lift' ((↑) : α → Completion α)),
Completion.map SeparationQuotient.mk, fun a ↦ ?_, fun a ↦ ?_⟩
· refine induction_on a (isClosed_eq (continuous_map.comp continuous_extension) continuous_id) ?_
refine SeparationQuotient.surjective_mk.forall.2 fun a ↦ ?_
rw [extension_coe ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Order.Lattice | {
"line": 124,
"column": 2
} | {
"line": 125,
"column": 41
} | {
"line": 128,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝³ : NormedAddCommGroup α\ninst✝² : Lattice α\ninst✝¹ : HasSolidNorm α\ninst✝ : IsOrderedAddMonoid α\nx y : α\n⊢ ‖x ⊔ y‖ ≤ ‖x‖ + ‖y‖",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real.instLE",
"Real",
"Lattice.toSemilattice... | [] | have h : ‖x ⊔ y - 0 ⊔ 0‖ ≤ ‖x - 0‖ + ‖y - 0‖ := norm_sup_sub_sup_le_add_norm x y 0 0
simpa only [sup_idem, sub_zero] using h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Order.Lattice | {
"line": 124,
"column": 2
} | {
"line": 125,
"column": 41
} | {
"line": 128,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝³ : NormedAddCommGroup α\ninst✝² : Lattice α\ninst✝¹ : HasSolidNorm α\ninst✝ : IsOrderedAddMonoid α\nx y : α\n⊢ ‖x ⊔ y‖ ≤ ‖x‖ + ‖y‖",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real.instLE",
"Real",
"Lattice.toSemilattice... | [] | have h : ‖x ⊔ y - 0 ⊔ 0‖ ≤ ‖x - 0‖ + ‖y - 0‖ := norm_sup_sub_sup_le_add_norm x y 0 0
simpa only [sup_idem, sub_zero] using h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 211,
"column": 29
} | {
"line": 211,
"column": 73
} | {
"line": 212,
"column": 4
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedGroup E\nu v : E\n⊢ ‖v‖ = ‖u⁻¹ * (u * v)‖",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"MulOne.toOne",
"Semigroup.toMul",
"Real",
"DivInvMonoid.toInv",
"inv_mul_cancel",
"HMul... | [] | by rw [← mul_assoc, inv_mul_cancel, one_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 1006,
"column": 50
} | {
"line": 1008,
"column": 32
} | {
"line": 1010,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝ : NormedGroup E\na b : E\n⊢ 0 < ‖a / b‖ ↔ a ≠ b",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"instHDiv",
"norm_nonneg'",
"Real.instZero",
... | [] | by
rw [(norm_nonneg' _).lt_iff_ne, ne_comm]
exact norm_div_eq_zero_iff.not | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Field.Basic | {
"line": 251,
"column": 2
} | {
"line": 251,
"column": 61
} | {
"line": 253,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝ : NontriviallyNormedField α\n⊢ (𝓝[{x | IsUnit x}] 0).NeBot",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NormedCommRing.toSeminormedCommRing",
"congrArg",
"Filter.NeBot",
"nhdsWi... | [] | simpa only [isUnit_iff_ne_zero] using! nhdsNE_neBot (0 : α) | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.Analysis.Normed.Field.Basic | {
"line": 251,
"column": 2
} | {
"line": 251,
"column": 61
} | {
"line": 253,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝ : NontriviallyNormedField α\n⊢ (𝓝[{x | IsUnit x}] 0).NeBot",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NormedCommRing.toSeminormedCommRing",
"congrArg",
"Filter.NeBot",
"nhdsWi... | [] | simpa only [isUnit_iff_ne_zero] using! nhdsNE_neBot (0 : α) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Field.Basic | {
"line": 251,
"column": 2
} | {
"line": 251,
"column": 61
} | {
"line": 253,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝ : NontriviallyNormedField α\n⊢ (𝓝[{x | IsUnit x}] 0).NeBot",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NormedCommRing.toSeminormedCommRing",
"congrArg",
"Filter.NeBot",
"nhdsWi... | [] | simpa only [isUnit_iff_ne_zero] using! nhdsNE_neBot (0 : α) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Field.Lemmas | {
"line": 143,
"column": 8
} | {
"line": 143,
"column": 39
} | {
"line": 143,
"column": 39
} | [
{
"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 [closedBall_mem_nhds, hr₀] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Field.Lemmas | {
"line": 143,
"column": 8
} | {
"line": 143,
"column": 39
} | {
"line": 143,
"column": 39
} | [
{
"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 [closedBall_mem_nhds, hr₀] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Field.Lemmas | {
"line": 143,
"column": 8
} | {
"line": 143,
"column": 39
} | {
"line": 143,
"column": 39
} | [
{
"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 [closedBall_mem_nhds, hr₀] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Ring.Basic | {
"line": 468,
"column": 14
} | {
"line": 468,
"column": 34
} | {
"line": 469,
"column": 4
} | [
{
"pp": "case hbc\nα : Type u_2\ninst✝ : SeminormedRing α\na b : αˣ\n⊢ ‖(↑a - 1) * (↑b - 1) - (↑b - 1) * (↑a - 1)‖ ≤ ‖(↑a - 1) * (↑b - 1)‖ + ‖(↑b - 1) * (↑a - 1)‖",
"ppTerm": "?hbc",
"assigned": true,
"usedConstants": [
"Units.val",
"HMul.hMul",
"AddGroupWithOne.toAddGroup",
... | [] | exact norm_sub_le .. | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Module.Basic | {
"line": 64,
"column": 43
} | {
"line": 64,
"column": 54
} | {
"line": 64,
"column": 55
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn : ℤ\nx : E\n⊢ ‖n • x‖ = ‖(n • 1) • x‖",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
... | [
"𝕜 : Type u_1\nE : Type u_3\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nn : ℤ\nx : E\n⊢ ‖n • x‖ = ‖n • 1 • x‖"
] | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Lebesgue.Sub | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 28
} | {
"line": 112,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : Countable β\nf : β → α → ℝ≥0∞\nμ : Measure α\nhμ : μ ≠ 0\nhf : ∀ (b : β), Measurable (f b)\nhf_int : ∀ (b : β), ∫⁻ (a : α), f b a ∂μ ≠ ∞\nh_directed : Directed (fun x1 x2 ↦ x1 ≥ x2) f\n⊢ ∫⁻ (a : α), ⨅ b, f b a ∂μ = ⨅ b, ∫⁻ (a : α), f b a ∂... | [
"case intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : Countable β\nf : β → α → ℝ≥0∞\nμ : Measure α\nhμ : μ ≠ 0\nhf : ∀ (b : β), Measurable (f b)\nhf_int : ∀ (b : β), ∫⁻ (a : α), f b a ∂μ ≠ ∞\nh_directed : Directed (fun x1 x2 ↦ x1 ≥ x2) f\nval✝ : Encodable β\n⊢ ∫⁻ (a : α), ⨅ b, f b a ∂μ = ⨅ b,... | cases nonempty_encodable β | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.MeasureTheory.Measure.Typeclasses.Probability | {
"line": 224,
"column": 2
} | {
"line": 224,
"column": 45
} | {
"line": 226,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nm0 : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nμ : Measure α\ns : Set α\ninst✝ : IsZeroOrProbabilityMeasure μ\np : α → Prop\nf✝ : β → α\nf : α → β\nhf : ¬AEMeasurable f μ\n⊢ IsZeroOrProbabilityMeasure (Measure.map f μ)",
"ppTerm": "?neg✝",
"assigned": ... | [] | · simp [isZeroOrProbabilityMeasure_iff, hf] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 96,
"column": 45
} | {
"line": 96,
"column": 63
} | {
"line": 96,
"column": 63
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure α\nc : β\ns : Set β\nhs : MeasurableSet s\n⊢ μ ((fun x ↦ c) ⁻¹' s) = μ univ * s.indicator 1 c",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSpace β\nμ : Measure α\nc : β\ns : Set β\nhs : MeasurableSet s\n⊢ μ (if c ∈ s then univ else ∅) = μ univ * s.indicator 1 c"
] | Set.preimage_const | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Count | {
"line": 60,
"column": 30
} | {
"line": 60,
"column": 79
} | {
"line": 62,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : MeasurableSpace α\ninst✝ : MeasurableSingletonClass α\ns : Set α\nhs : s.Finite\n⊢ count s = ↑(#hs.toFinset)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"congrArg",
"Finset",
"id",
"A... | [] | by rw [← count_apply_finset, Finite.coe_toFinset] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.UnitInterval | {
"line": 497,
"column": 2
} | {
"line": 497,
"column": 86
} | {
"line": 498,
"column": 2
} | [
{
"pp": "ι : Sort u_1\nc : ι → Set (↑I × ↑I)\nhc₁ : ∀ (i : ι), IsOpen (c i)\nhc₂ : univ ⊆ ⋃ i, c i\nδ : ℝ\nδ_pos : δ > 0\nball_subset : ∀ x ∈ univ, ∃ i, Metric.ball x δ ⊆ c i\nhδ : 0 < δ / 2\nh : 0 ≤ 1\nn m : ℕ\n⊢ ∃ i,\n Icc (addNSMul h (δ / 2) n) (addNSMul h (δ / 2) (n + 1)) ×ˢ Icc (addNSMul h (δ / 2) m) (a... | [
"ι : Sort u_1\nc : ι → Set (↑I × ↑I)\nhc₁ : ∀ (i : ι), IsOpen (c i)\nhc₂ : univ ⊆ ⋃ i, c i\nδ : ℝ\nδ_pos : δ > 0\nball_subset : ∀ x ∈ univ, ∃ i, Metric.ball x δ ⊆ c i\nhδ : 0 < δ / 2\nh : 0 ≤ 1\nn m : ℕ\ni : ι\nhsub : Metric.ball (addNSMul h (δ / 2) n, addNSMul h (δ / 2) m) δ ⊆ c i\n⊢ ∃ i,\n Icc (addNSMul h (δ /... | obtain ⟨i, hsub⟩ := ball_subset (addNSMul h (δ / 2) n, addNSMul h (δ / 2) m) trivial | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 374,
"column": 2
} | {
"line": 375,
"column": 22
} | {
"line": 377,
"column": 0
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : MeasurableSingletonClass α\nf : β → α\ns : Finset α\nμ : Measure β\nhf : AEMeasurable f μ\n⊢ (∀ᵐ (b : β) ∂μ, f b ∈ s) ↔ map f μ = ∑ a ∈ s, μ (f ⁻¹' {a}) • dirac a",
"ppTerm": "?m.51",
"assigned": true,
"used... | [] | rw [← ae_map_iff hf (by measurability), ae_mem_finset_iff]
simp [map_apply₀ hf] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Dirac | {
"line": 374,
"column": 2
} | {
"line": 375,
"column": 22
} | {
"line": 377,
"column": 0
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : MeasurableSingletonClass α\nf : β → α\ns : Finset α\nμ : Measure β\nhf : AEMeasurable f μ\n⊢ (∀ᵐ (b : β) ∂μ, f b ∈ s) ↔ map f μ = ∑ a ∈ s, μ (f ⁻¹' {a}) • dirac a",
"ppTerm": "?m.51",
"assigned": true,
"used... | [] | rw [← ae_map_iff hf (by measurability), ae_mem_finset_iff]
simp [map_apply₀ hf] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Lebesgue.Countable | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 40
} | {
"line": 131,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\na : α\n⊢ ∫⁻ (x : α) in {a}, f x ∂μ = f a * μ {a}",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.restrict_singleton",
"instHSMul",
"MeasureTheory.Measu... | [] | simp [lintegral_dirac' _ hf, mul_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.Lebesgue.Countable | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 40
} | {
"line": 131,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\na : α\n⊢ ∫⁻ (x : α) in {a}, f x ∂μ = f a * μ {a}",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.restrict_singleton",
"instHSMul",
"MeasureTheory.Measu... | [] | simp [lintegral_dirac' _ hf, mul_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Lebesgue.Countable | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 40
} | {
"line": 131,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf : α → ℝ≥0∞\nhf : Measurable f\na : α\n⊢ ∫⁻ (x : α) in {a}, f x ∂μ = f a * μ {a}",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.restrict_singleton",
"instHSMul",
"MeasureTheory.Measu... | [] | simp [lintegral_dirac' _ hf, mul_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Metrizable.CompletelyMetrizable | {
"line": 269,
"column": 8
} | {
"line": 269,
"column": 26
} | {
"line": 270,
"column": 6
} | [
{
"pp": "case inr\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\nx y : X\nh : x ≠ y\n⊢ (if x = y then 0 else 1) = if y = x then 0 else 1",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Real",
"eq_false",
"Real.instZero",
"congr... | [] | · simp [h, h.symm] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 340,
"column": 2
} | {
"line": 340,
"column": 30
} | {
"line": 342,
"column": 0
} | [
{
"pp": "ι : Type u_1\nE : ι → Type u_2\ninst✝ : (i : ι) → MetricSpace (E i)\ni : ι\nx y : E i\n⊢ dist ⟨i, x⟩ ⟨i, y⟩ = dist x y",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"dite_cond_eq_true",
"Real",
"congrArg",
"Classical.propDecidable",
"cast",
"M... | [] | simp [Dist.dist, Sigma.dist] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 340,
"column": 2
} | {
"line": 340,
"column": 30
} | {
"line": 342,
"column": 0
} | [
{
"pp": "ι : Type u_1\nE : ι → Type u_2\ninst✝ : (i : ι) → MetricSpace (E i)\ni : ι\nx y : E i\n⊢ dist ⟨i, x⟩ ⟨i, y⟩ = dist x y",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"dite_cond_eq_true",
"Real",
"congrArg",
"Classical.propDecidable",
"cast",
"M... | [] | simp [Dist.dist, Sigma.dist] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 340,
"column": 2
} | {
"line": 340,
"column": 30
} | {
"line": 342,
"column": 0
} | [
{
"pp": "ι : Type u_1\nE : ι → Type u_2\ninst✝ : (i : ι) → MetricSpace (E i)\ni : ι\nx y : E i\n⊢ dist ⟨i, x⟩ ⟨i, y⟩ = dist x y",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"dite_cond_eq_true",
"Real",
"congrArg",
"Classical.propDecidable",
"cast",
"M... | [] | simp [Dist.dist, Sigma.dist] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 407,
"column": 4
} | {
"line": 407,
"column": 19
} | {
"line": 408,
"column": 4
} | [
{
"pp": "case mpr\nι : Type u_1\nE : ι → Type u_2\ninst✝ : (i : ι) → MetricSpace (E i)\ns : Set ((i : ι) × E i)\nH : ∀ x ∈ s, ∃ ε > 0, ∀ (y : (i : ι) × E i), dist x y < ε → y ∈ s\ni : ι\nx : E i\nhx : x ∈ Sigma.mk i ⁻¹' s\nε : ℝ\nεpos : ε > 0\nhε : ∀ (y : (j : ι) × E j), dist ⟨i, x⟩ y < ε → y ∈ s\ny : E i\nhy :... | [
"case mpr\nι : Type u_1\nE : ι → Type u_2\ninst✝ : (i : ι) → MetricSpace (E i)\ns : Set ((i : ι) × E i)\nH : ∀ x ∈ s, ∃ ε > 0, ∀ (y : (i : ι) × E i), dist x y < ε → y ∈ s\ni : ι\nx : E i\nhx : x ∈ Sigma.mk i ⁻¹' s\nε : ℝ\nεpos : ε > 0\nhε : ∀ (y : (j : ι) × E j), dist ⟨i, x⟩ y < ε → y ∈ s\ny : E i\nhy : y ∈ ball x ... | apply hε ⟨i, y⟩ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 565,
"column": 6
} | {
"line": 565,
"column": 58
} | {
"line": 566,
"column": 6
} | [
{
"pp": "case neg\nX : ℕ → Type u\ninst✝ : (n : ℕ) → MetricSpace (X n)\nf : (n : ℕ) → X n → X (n + 1)\nI : ∀ (n : ℕ), Isometry (f n)\nx y : (n : ℕ) × X n\nm : ℕ\nhx : x.fst ≤ m + 1\nhy : y.fst ≤ m + 1\nh : ¬max x.fst y.fst = m + 1\nthis✝ : max x.fst y.fst ≤ m.succ\nthis : max x.fst y.fst ≤ m\n⊢ inductiveLimitDi... | [
"case neg\nX : ℕ → Type u\ninst✝ : (n : ℕ) → MetricSpace (X n)\nf : (n : ℕ) → X n → X (n + 1)\nI : ∀ (n : ℕ), Isometry (f n)\nx y : (n : ℕ) × X n\nm : ℕ\nhx : x.fst ≤ m + 1\nhy : y.fst ≤ m + 1\nh : ¬max x.fst y.fst = m + 1\nthis✝ : max x.fst y.fst ≤ m.succ\nthis : max x.fst y.fst ≤ m\nxm : x.fst ≤ m\n⊢ inductiveLim... | have xm : x.1 ≤ m := le_trans (le_max_left _ _) this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 627,
"column": 2
} | {
"line": 628,
"column": 42
} | {
"line": 629,
"column": 2
} | [
{
"pp": "X : ℕ → Type u\ninst✝ : (n : ℕ) → MetricSpace (X n)\nf : (n : ℕ) → X n → X (n + 1)\nI : ∀ (n : ℕ), Isometry (f n)\nn : ℕ\nh : PseudoMetricSpace ((n : ℕ) × X n) := inductivePremetric I\nx✝ : TopologicalSpace ((n : ℕ) × X n) := PseudoMetricSpace.toUniformSpace.toTopologicalSpace\nx : X n\n⊢ inductiveLimi... | [
"case h1\nX : ℕ → Type u\ninst✝ : (n : ℕ) → MetricSpace (X n)\nf : (n : ℕ) → X n → X (n + 1)\nI : ∀ (n : ℕ), Isometry (f n)\nn : ℕ\nh : PseudoMetricSpace ((n : ℕ) × X n) := inductivePremetric I\nx✝ : TopologicalSpace ((n : ℕ) × X n) := PseudoMetricSpace.toUniformSpace.toTopologicalSpace\nx : X n\n⊢ ⟨n, x⟩.fst ≤ n",... | rw [inductiveLimitDist_eq_dist I ⟨n.succ, f n x⟩ ⟨n, x⟩ n.succ, leRecOn_self,
leRecOn_succ, leRecOn_self, dist_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 212,
"column": 2
} | {
"line": 222,
"column": 49
} | {
"line": 224,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\ns : Set α\n⊢ AnalyticSet s ↔ ∃ β h, ∃ (_ : PolishSpace β), ∃ f, Continuous[_, inst✝] f ∧ range f = s",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Pi.uniformSpace",
"Eq.mpr",
"DiscreteUniformity.instCompleteSpace",
... | [] | constructor
· intro h
rw [AnalyticSet] at h
rcases h with h | h
· refine ⟨Empty, inferInstance, inferInstance, Empty.elim, continuous_bot, ?_⟩
rw [h]
exact range_eq_empty _
· exact ⟨ℕ → ℕ, inferInstance, inferInstance, h⟩
· rintro ⟨β, h, h', f, f_cont, f_range⟩
rw [← f_range]
exa... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 212,
"column": 2
} | {
"line": 222,
"column": 49
} | {
"line": 224,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\ns : Set α\n⊢ AnalyticSet s ↔ ∃ β h, ∃ (_ : PolishSpace β), ∃ f, Continuous[_, inst✝] f ∧ range f = s",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Pi.uniformSpace",
"Eq.mpr",
"DiscreteUniformity.instCompleteSpace",
... | [] | constructor
· intro h
rw [AnalyticSet] at h
rcases h with h | h
· refine ⟨Empty, inferInstance, inferInstance, Empty.elim, continuous_bot, ?_⟩
rw [h]
exact range_eq_empty _
· exact ⟨ℕ → ℕ, inferInstance, inferInstance, h⟩
· rintro ⟨β, h, h', f, f_cont, f_range⟩
rw [← f_range]
exa... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.OpenPos | {
"line": 119,
"column": 2
} | {
"line": 124,
"column": 53
} | {
"line": 125,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\nm : MeasurableSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\nμ : Measure X\ninst✝ : μ.IsOpenPosMeasure\nU : Set X\nf g : X → Y\nhU : IsOpen[inst✝³] U\nhf : ContinuousOn f U\nhg : ContinuousOn g U\nh : μ {a | a ∈ U ∧ ¬f a = g a} = 0\n⊢ ... | [
"X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\nm : MeasurableSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\nμ : Measure X\ninst✝ : μ.IsOpenPosMeasure\nU : Set X\nf g : X → Y\nhU : IsOpen[inst✝³] U\nhf : ContinuousOn f U\nhg : ContinuousOn g U\nh : μ {a | a ∈ U ∧ ¬f a = g a} = 0\nthis : IsOpen[... | have : IsOpen (U ∩ { a | f a ≠ g a }) := by
refine isOpen_iff_mem_nhds.mpr fun a ha => inter_mem (hU.mem_nhds ha.1) ?_
rcases ha with ⟨ha : a ∈ U, ha' : (f a, g a) ∈ (diagonal Y)ᶜ⟩
exact
(hf.continuousAt (hU.mem_nhds ha)).prodMk_nhds (hg.continuousAt (hU.mem_nhds ha))
(isClosed_diagonal.isOpen... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.MetricSpace.PiNat | {
"line": 499,
"column": 6
} | {
"line": 499,
"column": 37
} | {
"line": 499,
"column": 37
} | [
{
"pp": "E : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed[Pi.topologicalSpace] s\nx y : (n : ℕ) → E n\nhx : x ∉ s\nhy : y ∈ s\nA : ∃ n, Disjoint s (cylinder x n)\nB : Nat.find A ≤ firstDiff x y\n⊢ ¬Disjoint s (cylinder... | [
"E : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed[Pi.topologicalSpace] s\nx y : (n : ℕ) → E n\nhx : x ∉ s\nhy : y ∈ s\nA : ∃ n, Disjoint s (cylinder x n)\nB : Nat.find A ≤ firstDiff x y\n⊢ (s ∩ cylinder x (Nat.find A)).Non... | not_disjoint_iff_nonempty_inter | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 128,
"column": 4
} | {
"line": 131,
"column": 57
} | {
"line": 132,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nν : Measure β\ninst✝ : SFinite ν\nm : ∀ {α : Type ?u.42} {β : Type ?u.41} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x)\n⊢ ∀ (c : ℝ≥0∞) ⦃s : Set (α × β)⦄, MeasurableSet ... | [] | intro c s hs
simp only [← indicator_comp_right]
suffices Measurable fun x => c * ν (Prod.mk x ⁻¹' s) by simpa [lintegral_indicator (m hs)]
exact (measurable_measure_prodMk_left hs).const_mul _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 128,
"column": 4
} | {
"line": 131,
"column": 57
} | {
"line": 132,
"column": 2
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nν : Measure β\ninst✝ : SFinite ν\nm : ∀ {α : Type ?u.42} {β : Type ?u.41} {m : MeasurableSpace α} {mβ : MeasurableSpace β} {x : α}, Measurable (Prod.mk x)\n⊢ ∀ (c : ℝ≥0∞) ⦃s : Set (α × β)⦄, MeasurableSet ... | [] | intro c s hs
simp only [← indicator_comp_right]
suffices Measurable fun x => c * ν (Prod.mk x ⁻¹' s) by simpa [lintegral_indicator (m hs)]
exact (measurable_measure_prodMk_left hs).const_mul _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 335,
"column": 71
} | {
"line": 337,
"column": 53
} | {
"line": 338,
"column": 2
} | [
{
"pp": "α : Type u_3\nβ : Type u_4\nt : TopologicalSpace α\ninst✝⁵ : PolishSpace α\ninst✝⁴ : MeasurableSpace α\ninst✝³ : BorelSpace α\ntβ : TopologicalSpace β\ninst✝² : MeasurableSpace β\ninst✝¹ : OpensMeasurableSpace β\nf : α → β\ninst✝ : SecondCountableTopology ↑(range f)\nhf : Measurable f\nb : Set (Set ↑(r... | [] | by
apply MeasurableSet.isClopenable
exact hf.subtype_mk (hb.isOpen s.2).measurableSet | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 407,
"column": 2
} | {
"line": 407,
"column": 15
} | {
"line": 408,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : OpensMeasurableSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous[_, inst✝³] f\nhg : Continuous[_, inst✝³] g\nh : Disjoint (range f) (range g)\n⊢ MeasurablySeparable (range f) (range g)",
"ppTerm": "?m.25",
... | [
"α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : OpensMeasurableSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous[_, inst✝³] f\nhg : Continuous[_, inst✝³] g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\n⊢ False"
] | by_contra hfg | Batteries.Tactic._aux_Batteries_Tactic_Init___macroRules_Batteries_Tactic_byContra_1 | Batteries.Tactic.byContra |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 659,
"column": 72
} | {
"line": 659,
"column": 85
} | {
"line": 659,
"column": 85
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\nγ : Type u_6\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nμ ν : Measure ((α × β) × γ)\ninst✝ : IsFiniteMeasure μ\n⊢ map (⇑MeasurableEquiv.prodAssoc) μ = map (⇑MeasurableEquiv.prodAssoc) ν ↔\n ∀ {s : Set α} {t : Set β} {u : Set γ},\n Measur... | [
"α : Type u_4\nβ : Type u_5\nγ : Type u_6\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nμ ν : Measure ((α × β) × γ)\ninst✝ : IsFiniteMeasure μ\n⊢ (∀ {s : Set α} {t : Set β} {u : Set γ},\n MeasurableSet s →\n MeasurableSet t →\n MeasurableSet u →\n (map (⇑Me... | ext_prod₃_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 682,
"column": 46
} | {
"line": 682,
"column": 55
} | {
"line": 682,
"column": 56
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝¹ : SFinite ν\ninst✝ : SFinite μ\nthis :\n (sum fun i ↦ map Prod.swap ((sfiniteSeq μ i.1).prod (sfiniteSeq ν i.2))) =\n sum fun i ↦ map Prod.swap ((sfiniteSeq μ i.2).prod (sfiniteS... | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝¹ : SFinite ν\ninst✝ : SFinite μ\nthis :\n (sum fun i ↦ map Prod.swap ((sfiniteSeq μ i.1).prod (sfiniteSeq ν i.2))) =\n sum fun i ↦ map Prod.swap ((sfiniteSeq μ i.2).prod (sfiniteSeq ν i.1))\n... | prod_sum, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 682,
"column": 56
} | {
"line": 682,
"column": 65
} | {
"line": 683,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝¹ : SFinite ν\ninst✝ : SFinite μ\nthis :\n (sum fun i ↦ map Prod.swap ((sfiniteSeq μ i.1).prod (sfiniteSeq ν i.2))) =\n sum fun i ↦ map Prod.swap ((sfiniteSeq μ i.2).prod (sfiniteS... | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝¹ : SFinite ν\ninst✝ : SFinite μ\nthis :\n (sum fun i ↦ map Prod.swap ((sfiniteSeq μ i.1).prod (sfiniteSeq ν i.2))) =\n sum fun i ↦ map Prod.swap ((sfiniteSeq μ i.2).prod (sfiniteSeq ν i.1))\n... | prod_sum, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 750,
"column": 66
} | {
"line": 750,
"column": 75
} | {
"line": 750,
"column": 76
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\nτ : Measure γ\ninst✝² : SFinite ν\ninst✝¹ : SFinite μ\ninst✝ : SFinite τ\nthis :\n (sum fun p ↦ (sfiniteSeq μ p.1).prod ((sfiniteSeq ν p.2.1).prod... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\nτ : Measure γ\ninst✝² : SFinite ν\ninst✝¹ : SFinite μ\ninst✝ : SFinite τ\nthis :\n (sum fun p ↦ (sfiniteSeq μ p.1).prod ((sfiniteSeq ν p.2.1).prod (sfiniteSeq... | prod_sum, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 750,
"column": 76
} | {
"line": 750,
"column": 85
} | {
"line": 751,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\nτ : Measure γ\ninst✝² : SFinite ν\ninst✝¹ : SFinite μ\ninst✝ : SFinite τ\nthis :\n (sum fun p ↦ (sfiniteSeq μ p.1).prod ((sfiniteSeq ν p.2.1).prod... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\nτ : Measure γ\ninst✝² : SFinite ν\ninst✝¹ : SFinite μ\ninst✝ : SFinite τ\nthis :\n (sum fun p ↦ (sfiniteSeq μ p.1).prod ((sfiniteSeq ν p.2.1).prod (sfiniteSeq... | prod_sum, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 751,
"column": 63
} | {
"line": 751,
"column": 72
} | {
"line": 751,
"column": 73
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\nτ : Measure γ\ninst✝² : SFinite ν\ninst✝¹ : SFinite μ\ninst✝ : SFinite τ\nthis :\n (sum fun p ↦ (sfiniteSeq μ p.1).prod ((sfiniteSeq ν p.2.1).prod... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\nτ : Measure γ\ninst✝² : SFinite ν\ninst✝¹ : SFinite μ\ninst✝ : SFinite τ\nthis :\n (sum fun p ↦ (sfiniteSeq μ p.1).prod ((sfiniteSeq ν p.2.1).prod (sfiniteSeq... | prod_sum, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 751,
"column": 73
} | {
"line": 751,
"column": 82
} | {
"line": 751,
"column": 83
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\nτ : Measure γ\ninst✝² : SFinite ν\ninst✝¹ : SFinite μ\ninst✝ : SFinite τ\nthis :\n (sum fun p ↦ (sfiniteSeq μ p.1).prod ((sfiniteSeq ν p.2.1).prod... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nμ : Measure α\nν : Measure β\nτ : Measure γ\ninst✝² : SFinite ν\ninst✝¹ : SFinite μ\ninst✝ : SFinite τ\nthis :\n (sum fun p ↦ (sfiniteSeq μ p.1).prod ((sfiniteSeq ν p.2.1).prod (sfiniteSeq... | prod_sum, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 768,
"column": 4
} | {
"line": 768,
"column": 13
} | {
"line": 768,
"column": 14
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝¹ : SFinite ν\ninst✝ : SFinite μ\ns : Set α\nt : Set β\n⊢ (sum fun i ↦ (sfiniteSeq μ i).restrict s).prod (sum fun i ↦ (sfiniteSeq ν i).restrict t) =\n ((sum (sfiniteSeq μ)).prod (su... | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝¹ : SFinite ν\ninst✝ : SFinite μ\ns : Set α\nt : Set β\n⊢ (sum fun p ↦ ((sfiniteSeq μ p.1).restrict s).prod ((sfiniteSeq ν p.2).restrict t)) =\n ((sum (sfiniteSeq μ)).prod (sum (sfiniteSeq ν)))... | prod_sum, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 768,
"column": 14
} | {
"line": 768,
"column": 23
} | {
"line": 768,
"column": 24
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝¹ : SFinite ν\ninst✝ : SFinite μ\ns : Set α\nt : Set β\n⊢ (sum fun p ↦ ((sfiniteSeq μ p.1).restrict s).prod ((sfiniteSeq ν p.2).restrict t)) =\n ((sum (sfiniteSeq μ)).prod (sum (sfi... | [
"α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝¹ : SFinite ν\ninst✝ : SFinite μ\ns : Set α\nt : Set β\n⊢ (sum fun p ↦ ((sfiniteSeq μ p.1).restrict s).prod ((sfiniteSeq ν p.2).restrict t)) =\n (sum fun p ↦ (sfiniteSeq μ p.1).prod (sfiniteSeq... | prod_sum, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Group.Measure | {
"line": 390,
"column": 2
} | {
"line": 390,
"column": 26
} | {
"line": 391,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : DivisionMonoid G\ninst✝³ : MeasurableMul G\ninst✝² : MeasurableInv G\nμ : Measure G\ninst✝¹ : μ.IsInvInvariant\ninst✝ : μ.IsMulLeftInvariant\ng : G\n⊢ MeasurePreserving (fun t ↦ g / t) μ μ",
"ppTerm": "?m.22",
"assigned": true,
"usedConstan... | [
"G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : DivisionMonoid G\ninst✝³ : MeasurableMul G\ninst✝² : MeasurableInv G\nμ : Measure G\ninst✝¹ : μ.IsInvInvariant\ninst✝ : μ.IsMulLeftInvariant\ng : G\n⊢ MeasurePreserving (fun t ↦ g * t⁻¹) μ μ"
] | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 802,
"column": 85
} | {
"line": 802,
"column": 94
} | {
"line": 803,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝² : SFinite ν\ninst✝¹ : SFinite μ\nν' : Measure β\ninst✝ : SFinite ν'\n⊢ (sum (sfiniteSeq μ)).prod (sum fun n ↦ sfiniteSeq ν n + sfiniteSeq ν' n) =\n (sum (sfiniteSeq μ)).prod (sum ... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝² : SFinite ν\ninst✝¹ : SFinite μ\nν' : Measure β\ninst✝ : SFinite ν'\n⊢ (sum fun p ↦ (sfiniteSeq μ p.1).prod (sfiniteSeq ν p.2 + sfiniteSeq ν' p.2)) =\n (sum fun p ↦ (sfiniteSeq μ p.1).prod (s... | prod_sum, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 810,
"column": 85
} | {
"line": 810,
"column": 94
} | {
"line": 811,
"column": 4
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝² : SFinite ν\ninst✝¹ : SFinite μ\nμ' : Measure α\ninst✝ : SFinite μ'\n⊢ (sum fun n ↦ sfiniteSeq μ n + sfiniteSeq μ' n).prod (sum (sfiniteSeq ν)) =\n (sum (sfiniteSeq μ)).prod (sum ... | [
"α : Type u_1\nβ : Type u_2\ninst✝⁴ : MeasurableSpace α\ninst✝³ : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝² : SFinite ν\ninst✝¹ : SFinite μ\nμ' : Measure α\ninst✝ : SFinite μ'\n⊢ (sum fun p ↦ (sfiniteSeq μ p.1 + sfiniteSeq μ' p.1).prod (sfiniteSeq ν p.2)) =\n (sum fun p ↦ (sfiniteSeq μ p.1).prod (s... | prod_sum, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 829,
"column": 29
} | {
"line": 829,
"column": 38
} | {
"line": 829,
"column": 39
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nδ : Type u_4\ninst✝² : MeasurableSpace δ\nf : α → β\ng : γ → δ\nμa : Measure α\nμc : Measure γ\ninst✝¹ : SFinite μa\ninst✝ : SFinite μc\nhf : Measurable f\nhg : Measurable g\n⊢ ... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace γ\nδ : Type u_4\ninst✝² : MeasurableSpace δ\nf : α → β\ng : γ → δ\nμa : Measure α\nμc : Measure γ\ninst✝¹ : SFinite μa\ninst✝ : SFinite μc\nhf : Measurable f\nhg : Measurable g\n⊢ (sum fun p ↦... | prod_sum, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Group.Prod | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 26
} | {
"line": 216,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul₂ G\nμ : Measure G\ninst✝² : SFinite μ\ninst✝¹ : MeasurableInv G\ninst✝ : μ.IsMulLeftInvariant\ng : G\n⊢ μ ≪ map (fun h ↦ g / h) μ",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul₂ G\nμ : Measure G\ninst✝² : SFinite μ\ninst✝¹ : MeasurableInv G\ninst✝ : μ.IsMulLeftInvariant\ng : G\n⊢ μ ≪ map (fun h ↦ g * h⁻¹) μ"
] | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Constructions.Polish.Basic | {
"line": 496,
"column": 4
} | {
"line": 502,
"column": 56
} | {
"line": 504,
"column": 2
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : OpensMeasurableSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous[_, inst✝³] f\nhg : Continuous[_, inst✝³] g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\... | [] | · refine Disjoint.mono_left ?_ huv.symm
change g '' cylinder y n ⊆ v
rw [image_subset_iff]
apply Subset.trans _ hεy
intro z hz
rw [mem_cylinder_iff_dist_le] at hz
exact hz.trans_lt (hn.trans_le (min_le_right _ _)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 1002,
"column": 2
} | {
"line": 1002,
"column": 82
} | {
"line": 1003,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝ : SFinite ν\nf : α × β → ℝ≥0∞\nhf : AEMeasurable f (μ.bind fun x ↦ map (Prod.mk x) ν)\n⊢ (fun a ↦ ∫⁻ (x : α × β), f x ∂map (Prod.mk a) ν) =ᵐ[μ] fun a ↦ ∫⁻ (y : β), f (a, y) ∂ν",
"... | [
"α : Type u_1\nβ : Type u_2\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝ : SFinite ν\nf : α × β → ℝ≥0∞\nhf : AEMeasurable f (μ.bind fun x ↦ map (Prod.mk x) ν)\na : α\nha : AEMeasurable f (map (Prod.mk a) ν)\n⊢ ∫⁻ (x : α × β), f x ∂map (Prod.mk a) ν = ∫⁻ (y : β), f (a,... | filter_upwards [Measurable.map_prodMk_left.aemeasurable.ae_of_bind hf] with a ha | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Measure.Prod | {
"line": 1119,
"column": 2
} | {
"line": 1122,
"column": 44
} | {
"line": 1124,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝² : MeasurableSpace α\ninst✝¹ : MeasurableSpace β\ninst✝ : MeasurableSpace γ\nX : α → β\nY : α → γ\nμ : Measure α\nhY : AEMeasurable Y μ\nhX : ¬AEMeasurable X μ\n⊢ (map (fun a ↦ (X a, Y a)) μ).fst = map X μ",
"ppTerm": "?neg✝",
"assigned"... | [] | · have : ¬AEMeasurable (fun x ↦ (X x, Y x)) μ := by
contrapose hX
exact measurable_fst.comp_aemeasurable hX
simp [map_of_not_aemeasurable, hX, this] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Group.Prod | {
"line": 439,
"column": 2
} | {
"line": 439,
"column": 26
} | {
"line": 440,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul₂ G\nμ : Measure G\ninst✝² : SFinite μ\ninst✝¹ : MeasurableInv G\ninst✝ : μ.IsMulLeftInvariant\ng : G\n⊢ QuasiMeasurePreserving (fun h ↦ g / h) μ μ",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
... | [
"G : Type u_1\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : Group G\ninst✝³ : MeasurableMul₂ G\nμ : Measure G\ninst✝² : SFinite μ\ninst✝¹ : MeasurableInv G\ninst✝ : μ.IsMulLeftInvariant\ng : G\n⊢ QuasiMeasurePreserving (fun h ↦ g * h⁻¹) μ μ"
] | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 81,
"column": 39
} | {
"line": 81,
"column": 79
} | {
"line": 83,
"column": 0
} | [
{
"pp": "case e_μ\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : SFinite μ\nf : α → ℝ≥0∞\ns : Set α\nt : Set α := toMeasurable μ s\n⊢ μ.restrict t = μ.restrict s",
"ppTerm": "?e_μ",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.restrict_toMeasurable_of_sFinite"
... | [] | exact restrict_toMeasurable_of_sFinite s | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 30
} | {
"line": 142,
"column": 0
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nr : ℝ≥0∞\nf : α → ℝ≥0∞\ns : Set α\nhs : MeasurableSet s\n⊢ ((r • μ).withDensity f) s = (r • μ.withDensity f) s",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure.restrict_smul",
"MeasureTheory.Mea... | [] | simp [withDensity_apply, hs] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 308,
"column": 36
} | {
"line": 308,
"column": 51
} | {
"line": 308,
"column": 51
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\np : α → Prop\nf g : α → ℝ≥0∞\nhg : Measurable g\nhfg : f =ᵐ[μ] g\na : α\nha : f a = g a\nh : g a ≠ 0\n⊢ {x | f x ≠ 0} a",
"ppTerm": "?m.97",
"assigned": true,
"usedConstants": [
"congrArg",
"Eq.mp",
"Ne",
"ENNReal"... | [] | rwa [← ha] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 308,
"column": 36
} | {
"line": 308,
"column": 51
} | {
"line": 308,
"column": 51
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\np : α → Prop\nf g : α → ℝ≥0∞\nhg : Measurable g\nhfg : f =ᵐ[μ] g\na : α\nha : f a = g a\nh : g a ≠ 0\n⊢ {x | f x ≠ 0} a",
"ppTerm": "?m.97",
"assigned": true,
"usedConstants": [
"congrArg",
"Eq.mp",
"Ne",
"ENNReal"... | [] | rwa [← ha] at h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 308,
"column": 36
} | {
"line": 308,
"column": 51
} | {
"line": 308,
"column": 51
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\np : α → Prop\nf g : α → ℝ≥0∞\nhg : Measurable g\nhfg : f =ᵐ[μ] g\na : α\nha : f a = g a\nh : g a ≠ 0\n⊢ {x | f x ≠ 0} a",
"ppTerm": "?m.97",
"assigned": true,
"usedConstants": [
"congrArg",
"Eq.mp",
"Ne",
"ENNReal"... | [] | rwa [← ha] at h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Complex.Basic | {
"line": 329,
"column": 16
} | {
"line": 329,
"column": 58
} | {
"line": 330,
"column": 2
} | [
{
"pp": "⊢ ∀ (x : ℂ), 0 • x = 0",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Real",
"instHSMul",
"Real.instZero",
"Real.instAddMonoid",
"congrArg",
"Complex.im",
"AddMonoid.toAddZeroClass",
"AddMonoid.toNSMul",
"Complex.smul_re",
... | [] | by intros; ext <;> simp [smul_re, smul_im] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Complex.Basic | {
"line": 328,
"column": 17
} | {
"line": 328,
"column": 59
} | {
"line": 329,
"column": 2
} | [
{
"pp": "⊢ ∀ (a : ℂ), 0 • a = 0",
"ppTerm": "?m.67",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Real",
"instHSMul",
"Real.instZero",
"congrArg",
"Complex.im",
"Complex.smul_re",
"Complex.instZero",
"Real.instAddGroup",
... | [] | by intros; ext <;> simp [smul_re, smul_im] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.WithDensity | {
"line": 657,
"column": 2
} | {
"line": 659,
"column": 35
} | {
"line": 660,
"column": 2
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ✝ : Measure α\nf : α → ℝ≥0∞\nhfm : Measurable f\nμ : Measure α\ninst✝ : SFinite μ\nhμ : IsFiniteMeasure μ\ns : Set α := {x | f x = ∞}\nhs : MeasurableSet s\nkey : μ.withDensity f = μ.withDensity (sᶜ.indicator f) + sum fun x ↦ μ.withDensity (s.indicator 1)\nthis : ... | [
"α : Type u_1\nm0 : MeasurableSpace α\nμ✝ : Measure α\nf : α → ℝ≥0∞\nhfm : Measurable f\nμ : Measure α\ninst✝ : SFinite μ\nhμ : IsFiniteMeasure μ\ns : Set α := {x | f x = ∞}\nhs : MeasurableSet s\nkey : μ.withDensity f = μ.withDensity (sᶜ.indicator f) + sum fun x ↦ μ.withDensity (s.indicator 1)\nthis✝ : SigmaFinite... | have : SigmaFinite (μ.withDensity (s.indicator 1)) := by
rw [withDensity_indicator hs]
exact SigmaFinite.withDensity 1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.Complex.Basic | {
"line": 513,
"column": 18
} | {
"line": 515,
"column": 8
} | {
"line": 517,
"column": 0
} | [
{
"pp": "z w : ℂ\n⊢ (z * w).re * (z * w).re + (z * w).im * (z * w).im = (z.re * z.re + z.im * z.im) * (w.re * w.re + w.im * w.im)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Complex.mul_im",
"Mathlib.Tactic.Ring.Common.neg_ze... | [] | by
simp only [mul_re, mul_im]
ring | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Norm | {
"line": 220,
"column": 6
} | {
"line": 220,
"column": 49
} | {
"line": 220,
"column": 50
} | [
{
"pp": "x y : ℝ\nm : ℝ := max |x| |y|\nhm₀ : 0 ≤ m\n⊢ √(m ^ 2 + m ^ 2) = √2 * m",
"ppTerm": "?m.97",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"HMul.hMul",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"t... | [
"x y : ℝ\nm : ℝ := max |x| |y|\nhm₀ : 0 ≤ m\n⊢ 0 ≤ m",
"case hx\nx y : ℝ\nm : ℝ := max |x| |y|\nhm₀ : 0 ≤ m\n⊢ 0 ≤ 2"
] | rw [← two_mul, Real.sqrt_mul, Real.sqrt_sq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.RCLike.Basic | {
"line": 349,
"column": 87
} | {
"line": 349,
"column": 97
} | {
"line": 350,
"column": 4
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ (I * z + -I * (starRingEnd K) z) / 2 = -(I * ((starRingEnd K) z - z)) / 2",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"NegZeroClass.toNeg",
"instHDiv",
"NonUnitalCom... | [
"K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ (I * z + -I * (starRingEnd K) z) / 2 = I * -((starRingEnd K) z - z) / 2"
] | ← mul_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 358,
"column": 54
} | {
"line": 358,
"column": 63
} | {
"line": 358,
"column": 64
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\nx✝ : (starRingEnd K) z = z\nh : (starRingEnd K) z = z := x✝\n⊢ I * 0 / 2 = ↑0",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"Real",
"instHDiv",
"NormedRing.toRing",
... | [
"K : Type u_1\ninst✝ : RCLike K\nz : K\nx✝ : (starRingEnd K) z = z\nh : (starRingEnd K) z = z := x✝\n⊢ 0 / 2 = ↑0"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Complex.Module | {
"line": 304,
"column": 15
} | {
"line": 304,
"column": 23
} | {
"line": 304,
"column": 24
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Ring A\ninst✝ : Algebra ℝ A\nI' : A\nhf : I' * I' = -1\nx₁ y₁ x₂ y₂ : ℝ\n⊢ (algebraMap ℝ A) (x₁ * x₂ - y₁ * y₂) + (x₁ * y₂ + y₁ * x₂) • I' =\n (algebraMap ℝ A) x₁ * ((algebraMap ℝ A) x₂ + y₂ • I') + y₁ • I' * ((algebraMap ℝ A) x₂ + y₂ • I')",
"ppTerm": "?m.216",
"assig... | [
"A : Type u_1\ninst✝¹ : Ring A\ninst✝ : Algebra ℝ A\nI' : A\nhf : I' * I' = -1\nx₁ y₁ x₂ y₂ : ℝ\n⊢ (algebraMap ℝ A) (x₁ * x₂ - y₁ * y₂) + (x₁ * y₂ + y₁ * x₂) • I' =\n (algebraMap ℝ A) x₁ * (algebraMap ℝ A) x₂ + (algebraMap ℝ A) x₁ * y₂ • I' +\n y₁ • I' * ((algebraMap ℝ A) x₂ + y₂ • I')"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Complex.Module | {
"line": 304,
"column": 24
} | {
"line": 304,
"column": 32
} | {
"line": 304,
"column": 33
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Ring A\ninst✝ : Algebra ℝ A\nI' : A\nhf : I' * I' = -1\nx₁ y₁ x₂ y₂ : ℝ\n⊢ (algebraMap ℝ A) (x₁ * x₂ - y₁ * y₂) + (x₁ * y₂ + y₁ * x₂) • I' =\n (algebraMap ℝ A) x₁ * (algebraMap ℝ A) x₂ + (algebraMap ℝ A) x₁ * y₂ • I' +\n y₁ • I' * ((algebraMap ℝ A) x₂ + y₂ • I')",
"pp... | [
"A : Type u_1\ninst✝¹ : Ring A\ninst✝ : Algebra ℝ A\nI' : A\nhf : I' * I' = -1\nx₁ y₁ x₂ y₂ : ℝ\n⊢ (algebraMap ℝ A) (x₁ * x₂ - y₁ * y₂) + (x₁ * y₂ + y₁ * x₂) • I' =\n (algebraMap ℝ A) x₁ * (algebraMap ℝ A) x₂ + (algebraMap ℝ A) x₁ * y₂ • I' +\n (y₁ • I' * (algebraMap ℝ A) x₂ + y₁ • I' * y₂ • I')"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Complex.Module | {
"line": 433,
"column": 79
} | {
"line": 435,
"column": 59
} | {
"line": 437,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝³ : AddCommGroup A\ninst✝² : Module ℂ A\ninst✝¹ : StarAddMonoid A\ninst✝ : StarModule ℂ A\nz : ℂ\na : A\n⊢ ℜ (z • a) = z.re • ℜ a - z.im • ℑ a",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroCla... | [] | by
have := by congrm (ℜ ($((re_add_im z).symm) • a))
simpa [-re_add_im, add_smul, ← smul_smul, sub_eq_add_neg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Complex.Module | {
"line": 507,
"column": 71
} | {
"line": 508,
"column": 74
} | {
"line": 510,
"column": 0
} | [
{
"pp": "A : Type u_1\ninst✝³ : AddCommGroup A\ninst✝² : Module ℂ A\ninst✝¹ : StarAddMonoid A\ninst✝ : StarModule ℂ A\nx : A\n⊢ ℑ x = 0 ↔ IsSelfAdjoint x",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Submodule",
"instTrivialStarReal",
"Real",
"NonUnitalCommRing.t... | [] | by
simpa [-ker_imaginaryPart] using! SetLike.ext_iff.mp ker_imaginaryPart x | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.RCLike.Basic | {
"line": 561,
"column": 49
} | {
"line": 561,
"column": 95
} | {
"line": 563,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ ‖(starRingEnd K) z‖ = ‖z‖",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real",
"congrArg",
"CommSemiring.toSemiring",
"MonoidWithZeroHom.funLike",
"RingHom",
"NormedField.toField",
... | [] | simp only [← sqrt_normSq_eq_norm, normSq_conj] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.RCLike.Basic | {
"line": 561,
"column": 49
} | {
"line": 561,
"column": 95
} | {
"line": 563,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ ‖(starRingEnd K) z‖ = ‖z‖",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real",
"congrArg",
"CommSemiring.toSemiring",
"MonoidWithZeroHom.funLike",
"RingHom",
"NormedField.toField",
... | [] | simp only [← sqrt_normSq_eq_norm, normSq_conj] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.RCLike.Basic | {
"line": 561,
"column": 49
} | {
"line": 561,
"column": 95
} | {
"line": 563,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ ‖(starRingEnd K) z‖ = ‖z‖",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real",
"congrArg",
"CommSemiring.toSemiring",
"MonoidWithZeroHom.funLike",
"RingHom",
"NormedField.toField",
... | [] | simp only [← sqrt_normSq_eq_norm, normSq_conj] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.RCLike.Basic | {
"line": 868,
"column": 50
} | {
"line": 869,
"column": 38
} | {
"line": 871,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx : ℝ\n⊢ 0 < ↑x ↔ 0 < x",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"Real.instZero",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
"Real.instLT",
"Normed... | [] | by
rw [← ofReal_zero, ofReal_lt_ofReal] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.RCLike.Basic | {
"line": 872,
"column": 54
} | {
"line": 873,
"column": 38
} | {
"line": 875,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx : ℝ\n⊢ ↑x < 0 ↔ x < 0",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"Real.instZero",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
"Real.instLT",
"Normed... | [] | by
rw [← ofReal_zero, ofReal_lt_ofReal] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.OpenPartialHomeomorph.Continuity | {
"line": 110,
"column": 2
} | {
"line": 111,
"column": 96
} | {
"line": 113,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ne : OpenPartialHomeomorph X Y\nx : X\np : Y → Prop\ns : Set X\nhx : x ∈ e.source\n⊢ (∀ᶠ (y : Y) in 𝓝[↑e.symm ⁻¹' s] ↑e x, p y) ↔ ∀ᶠ (x : X) in 𝓝[s] x, p (↑e x)",
"ppTerm": "?m.25",
"assigned": true,
"used... | [] | refine Iff.trans ?_ eventually_map
rw [e.map_nhdsWithin_eq hx, e.image_source_inter_eq', e.nhdsWithin_target_inter (e.mapsTo hx)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.OpenPartialHomeomorph.Continuity | {
"line": 110,
"column": 2
} | {
"line": 111,
"column": 96
} | {
"line": 113,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ne : OpenPartialHomeomorph X Y\nx : X\np : Y → Prop\ns : Set X\nhx : x ∈ e.source\n⊢ (∀ᶠ (y : Y) in 𝓝[↑e.symm ⁻¹' s] ↑e x, p y) ↔ ∀ᶠ (x : X) in 𝓝[s] x, p (↑e x)",
"ppTerm": "?m.25",
"assigned": true,
"used... | [] | refine Iff.trans ?_ eventually_map
rw [e.map_nhdsWithin_eq hx, e.image_source_inter_eq', e.nhdsWithin_target_inter (e.mapsTo hx)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.RCLike.Basic | {
"line": 1183,
"column": 26
} | {
"line": 1183,
"column": 45
} | {
"line": 1184,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx y : K\n⊢ ‖re x - re y‖ₑ = ‖re (x - y)‖ₑ",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"AddMonoidHom.instAddMonoidHomClass",
... | [] | rw [map_sub re x y] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.RCLike.Basic | {
"line": 1183,
"column": 26
} | {
"line": 1183,
"column": 45
} | {
"line": 1184,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx y : K\n⊢ ‖re x - re y‖ₑ = ‖re (x - y)‖ₑ",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"AddMonoidHom.instAddMonoidHomClass",
... | [] | rw [map_sub re x y] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.RCLike.Basic | {
"line": 1183,
"column": 26
} | {
"line": 1183,
"column": 45
} | {
"line": 1184,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx y : K\n⊢ ‖re x - re y‖ₑ = ‖re (x - y)‖ₑ",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"AddMonoidHom.instAddMonoidHomClass",
... | [] | rw [map_sub re x y] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Tactic.NormNum.NatFactorial | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 81
} | {
"line": 96,
"column": 0
} | [
{
"pp": "n x l y z : ℕ\nh₁ : IsNat n x\nh₂ : IsNat l y\nh₃ : x = z + y\na : ℕ\np : (z + 1).ascFactorial y = a\n⊢ n.descFactorial l = ↑a",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddMonoid.toAddSemigroup",
"congrArg",
"Nat.ascFactorial",
"id",
... | [] | simpa [h₁.out, h₂.out, ← p, h₃] using Nat.add_descFactorial_eq_ascFactorial _ _ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Complex.Exponential | {
"line": 102,
"column": 4
} | {
"line": 107,
"column": 10
} | {
"line": 109,
"column": 0
} | [
{
"pp": "case e'_3.succ\nε : ℝ\nε0 : ε > 0\nj : ℕ\nhj : j + 1 ≥ 1\n⊢ ‖∑ m ∈ range (j + 1), 0 ^ m / ↑m.factorial - 1‖ = 0",
"ppTerm": "?e'_3.succ",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"MulOne.toOne",
"False",
... | [] | induction j with
| zero => simp
| succ j ih =>
rw [← ih (by simp)]
simp only [sum_range_succ, pow_succ]
simp | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 132,
"column": 6
} | {
"line": 132,
"column": 40
} | {
"line": 132,
"column": 41
} | [
{
"pp": "x y : ℂ\n⊢ sinh (x + y) = sinh x * cosh y + cosh x * sinh y",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Complex.sinh",
"HMul.hMul",
"Field.isDomain",
"CharZero.NeZero.two",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"A... | [
"x y : ℂ\n⊢ 2 * sinh (x + y) = 2 * (sinh x * cosh y + cosh x * sinh y)"
] | ← mul_right_inj' (two_ne_zero' ℂ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 132,
"column": 87
} | {
"line": 132,
"column": 95
} | {
"line": 132,
"column": 96
} | [
{
"pp": "x y : ℂ\n⊢ 2 * (sinh x * cosh y + cosh x * sinh y) = cexp x * cexp y - cexp (-x) * cexp (-y)",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"NegZeroClass.toNeg",
"Complex.sinh",
"HMul.hMul",
"congrArg",
... | [
"x y : ℂ\n⊢ 2 * (sinh x * cosh y) + 2 * (cosh x * sinh y) = cexp x * cexp y - cexp (-x) * cexp (-y)"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 133,
"column": 50
} | {
"line": 133,
"column": 84
} | {
"line": 133,
"column": 85
} | [
{
"pp": "x y : ℂ\n⊢ (cexp x - cexp (-x)) * cosh y + cosh x * (cexp y - cexp (-y)) = cexp x * cexp y - cexp (-x) * cexp (-y)",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Semigroup.toMul",
"HMul.hMul",
"Field.isDomain",
... | [
"x y : ℂ\n⊢ 2 * ((cexp x - cexp (-x)) * cosh y + cosh x * (cexp y - cexp (-y))) = 2 * (cexp x * cexp y - cexp (-x) * cexp (-y))"
] | ← mul_right_inj' (two_ne_zero' ℂ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 133,
"column": 85
} | {
"line": 133,
"column": 93
} | {
"line": 134,
"column": 4
} | [
{
"pp": "x y : ℂ\n⊢ 2 * ((cexp x - cexp (-x)) * cosh y + cosh x * (cexp y - cexp (-y))) = 2 * (cexp x * cexp y - cexp (-x) * cexp (-y))",
"ppTerm": "?m.81",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"NegZeroClass.toNeg",
"Semigroup.toMul",
... | [
"x y : ℂ\n⊢ 2 * ((cexp x - cexp (-x)) * cosh y) + 2 * (cosh x * (cexp y - cexp (-y))) =\n 2 * (cexp x * cexp y - cexp (-x) * cexp (-y))"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 147,
"column": 6
} | {
"line": 147,
"column": 40
} | {
"line": 147,
"column": 41
} | [
{
"pp": "x y : ℂ\n⊢ cosh (x + y) = cosh x * cosh y + sinh x * sinh y",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Complex.sinh",
"HMul.hMul",
"Field.isDomain",
"CharZero.NeZero.two",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"A... | [
"x y : ℂ\n⊢ 2 * cosh (x + y) = 2 * (cosh x * cosh y + sinh x * sinh y)"
] | ← mul_right_inj' (two_ne_zero' ℂ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 147,
"column": 87
} | {
"line": 147,
"column": 95
} | {
"line": 147,
"column": 96
} | [
{
"pp": "x y : ℂ\n⊢ 2 * (cosh x * cosh y + sinh x * sinh y) = cexp x * cexp y + cexp (-x) * cexp (-y)",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"NegZeroClass.toNeg",
"Complex.sinh",
"HMul.hMul",
"congrArg",
... | [
"x y : ℂ\n⊢ 2 * (cosh x * cosh y) + 2 * (sinh x * sinh y) = cexp x * cexp y + cexp (-x) * cexp (-y)"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 148,
"column": 48
} | {
"line": 148,
"column": 82
} | {
"line": 148,
"column": 83
} | [
{
"pp": "x y : ℂ\n⊢ (cexp x + cexp (-x)) * cosh y + (cexp x - cexp (-x)) * sinh y = cexp x * cexp y + cexp (-x) * cexp (-y)",
"ppTerm": "?m.70",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Semigroup.toMul",
"Complex.sinh",
"HMul.hMul",
"F... | [
"x y : ℂ\n⊢ 2 * ((cexp x + cexp (-x)) * cosh y + (cexp x - cexp (-x)) * sinh y) = 2 * (cexp x * cexp y + cexp (-x) * cexp (-y))"
] | ← mul_right_inj' (two_ne_zero' ℂ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 148,
"column": 83
} | {
"line": 148,
"column": 91
} | {
"line": 149,
"column": 4
} | [
{
"pp": "x y : ℂ\n⊢ 2 * ((cexp x + cexp (-x)) * cosh y + (cexp x - cexp (-x)) * sinh y) = 2 * (cexp x * cexp y + cexp (-x) * cexp (-y))",
"ppTerm": "?m.81",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"NegZeroClass.toNeg",
"Semigroup.toMul",
... | [
"x y : ℂ\n⊢ 2 * ((cexp x + cexp (-x)) * cosh y) + 2 * ((cexp x - cexp (-x)) * sinh y) =\n 2 * (cexp x * cexp y + cexp (-x) * cexp (-y))"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
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