module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable | {
"line": 67,
"column": 4
} | {
"line": 69,
"column": 94
} | [
{
"pp": "case refine_2\nτ : ℂ\nhτ : 0 < τ.im\nn : ℤ\ny : ℝ := rexp (-π * τ.im)\nh : y < 1\n⊢ y ^ n ^ 2 ≤ rexp (-π * τ.im) ^ n.natAbs",
"usedConstants": [
"zpow_natCast",
"instPowNat",
"Real.instIsOrderedRing",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.par... | have : n ^ 2 = (n.natAbs ^ 2 :) := by rw [Nat.cast_pow, Int.natAbs_sq]
rw [this, zpow_natCast]
exact pow_le_pow_of_le_one (exp_pos _).le h.le ((sq n.natAbs).symm ▸ n.natAbs.le_mul_self) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable | {
"line": 74,
"column": 30
} | {
"line": 74,
"column": 39
} | [
{
"pp": "τ : ℂ\nhτ : 0 < τ.im\nthis : HasSum (fun n ↦ cexp (2 * ↑π * I * ↑n * 0 + ↑π * I * ↑n ^ 2 * τ)) (jacobiTheta₂ 0 τ)\n⊢ HasSum (fun n ↦ cexp (↑π * I * (↑n + 1) ^ 2 * τ)) ((jacobiTheta τ - 1) / 2)",
"usedConstants": [
"Int.cast",
"NormedCommRing.toSeminormedCommRing",
"NonUnitalCommRi... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.ModularForms.JacobiTheta.OneVariable | {
"line": 78,
"column": 31
} | {
"line": 78,
"column": 40
} | [
{
"pp": "τ : ℂ\nhτ : 0 < τ.im\nthis✝ : HasSum (fun n ↦ cexp (↑π * I * ↑n ^ 2 * τ)) (jacobiTheta τ)\nthis :\n HasSum (fun n ↦ cexp (↑π * I * ↑(n + 1) ^ 2 * τ) + cexp (↑π * I * (-↑(n + 1)) ^ 2 * τ))\n (jacobiTheta τ + cexp (↑π * I * (0 * 0) * τ) - (cexp (↑π * I * (0 * 0) * τ) + cexp (↑π * I * (0 * 0) * τ)))\n... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.Multiplicity | {
"line": 121,
"column": 27
} | {
"line": 121,
"column": 36
} | [
{
"pp": "case zero\nR : Type u_1\ninst✝ : CommRing R\na b : R\np : ℕ\nhp : Odd p\nh1 :\n ∀ (i : ℕ),\n (Ideal.Quotient.mk (span {↑p ^ 2})) ((a + ↑p * b) ^ i) =\n (Ideal.Quotient.mk (span {↑p ^ 2})) (a ^ (i - 1) * (↑p * b) * ↑i + a ^ i)\ns : R := ↑p ^ 2\nhx : 0 ∈ range p\n⊢ a ^ (0 - 1 + (p - 1 - 0)) * ↑p... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.NormTrace | {
"line": 126,
"column": 59
} | {
"line": 126,
"column": 75
} | [
{
"pp": "𝒢 ℋ : Subgroup (GL (Fin 2) ℝ)\nF : Type u_1\nf : F\ninst✝³ : FunLike F ℍ ℂ\nk : ℤ\ninst✝² : 𝒢.IsFiniteRelIndex ℋ\ninst✝¹ : ℋ.HasDetPlusMinusOne\ninst✝ : ModularFormClass F 𝒢 k\nhf : ∏ q, quotientFunc f q = 0\n⊢ ⇑f = 0",
"usedConstants": [
"Real",
"Subgroup.subgroupOf",
"Fintype... | prod_eq_zero_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace | {
"line": 278,
"column": 6
} | {
"line": 278,
"column": 93
} | [
{
"pp": "case refine_2\nι : Type u_1\nR : ι → Type u_2\nA : (i : ι) → Set (R i)\ninst✝¹ : (i : ι) → TopologicalSpace (R i)\nS : Set ι\ninst✝ : ∀ (i : ι), WeaklyLocallyCompactSpace (R i)\nhS : Sᶜ.Finite\nhAcompact : ∀ i ∈ S, IsCompact (A i)\nx : Πʳ (i : ι), [R i, A i]_[𝓟 S]\nK : (i : ι) → Set (R i)\nK_compact :... | simpa only [isEmbedding_coe_of_principal.nhds_eq_comap] using preimage_mem_comap U_nhds | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace | {
"line": 278,
"column": 6
} | {
"line": 278,
"column": 93
} | [
{
"pp": "case refine_2\nι : Type u_1\nR : ι → Type u_2\nA : (i : ι) → Set (R i)\ninst✝¹ : (i : ι) → TopologicalSpace (R i)\nS : Set ι\ninst✝ : ∀ (i : ι), WeaklyLocallyCompactSpace (R i)\nhS : Sᶜ.Finite\nhAcompact : ∀ i ∈ S, IsCompact (A i)\nx : Πʳ (i : ι), [R i, A i]_[𝓟 S]\nK : (i : ι) → Set (R i)\nK_compact :... | simpa only [isEmbedding_coe_of_principal.nhds_eq_comap] using preimage_mem_comap U_nhds | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.RestrictedProduct.TopologicalSpace | {
"line": 278,
"column": 6
} | {
"line": 278,
"column": 93
} | [
{
"pp": "case refine_2\nι : Type u_1\nR : ι → Type u_2\nA : (i : ι) → Set (R i)\ninst✝¹ : (i : ι) → TopologicalSpace (R i)\nS : Set ι\ninst✝ : ∀ (i : ι), WeaklyLocallyCompactSpace (R i)\nhS : Sᶜ.Finite\nhAcompact : ∀ i ∈ S, IsCompact (A i)\nx : Πʳ (i : ι), [R i, A i]_[𝓟 S]\nK : (i : ι) → Set (R i)\nK_compact :... | simpa only [isEmbedding_coe_of_principal.nhds_eq_comap] using preimage_mem_comap U_nhds | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Units.Regulator | {
"line": 283,
"column": 2
} | {
"line": 284,
"column": 79
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ regulator K = regOfFamily (fundSystem K)",
"usedConstants": [
"NumberField.Units.basisUnitLattice",
"Eq.mpr",
"Pi.Function.module",
"InnerProductSpace.toNormedSpace",
"instDecidableNot",
"NormedCommRing.toS... | rw [regOfFamily_of_isMaxRank (isMaxRank_fundSystem K), regulator,
← (basisUnitLattice K).ofZLatticeBasis_span ℝ, basisOfIsMaxRank_fundSystem] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.Units.Regulator | {
"line": 283,
"column": 2
} | {
"line": 284,
"column": 79
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ regulator K = regOfFamily (fundSystem K)",
"usedConstants": [
"NumberField.Units.basisUnitLattice",
"Eq.mpr",
"Pi.Function.module",
"InnerProductSpace.toNormedSpace",
"instDecidableNot",
"NormedCommRing.toS... | rw [regOfFamily_of_isMaxRank (isMaxRank_fundSystem K), regulator,
← (basisUnitLattice K).ofZLatticeBasis_span ℝ, basisOfIsMaxRank_fundSystem] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Units.Regulator | {
"line": 283,
"column": 2
} | {
"line": 284,
"column": 79
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ regulator K = regOfFamily (fundSystem K)",
"usedConstants": [
"NumberField.Units.basisUnitLattice",
"Eq.mpr",
"Pi.Function.module",
"InnerProductSpace.toNormedSpace",
"instDecidableNot",
"NormedCommRing.toS... | rw [regOfFamily_of_isMaxRank (isMaxRank_fundSystem K), regulator,
← (basisUnitLattice K).ofZLatticeBasis_span ℝ, basisOfIsMaxRank_fundSystem] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | {
"line": 156,
"column": 23
} | {
"line": 156,
"column": 32
} | [
{
"pp": "case h\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nc : ℝ\nx✝ : { w // w ≠ w₀ }\n⊢ ↑(↑x✝).mult * 0 = 0 x✝",
"usedConstants": [
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"MulZeroClass.toMul",
"Real.instZero",
... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.FundamentalCone | {
"line": 403,
"column": 37
} | {
"line": 403,
"column": 71
} | [
{
"pp": "case a\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx y : mixedSpace K\nc : ℝ\na b : ↑(integerSet K)\n⊢ integerSetToAssociates K b = integerSetToAssociates K a ↔ (MulAction.orbitRel ↥(torsion K) ↑(integerSet K)) a b",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"NumberFie... | integerSetToAssociates_eq_iff b a, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 299,
"column": 80
} | {
"line": 299,
"column": 89
} | [
{
"pp": "case h\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\nhx : mixedEmbedding.norm (mixedSpaceOfRealSpace (↑expMap x)) = 1\nx✝ : { w // w ≠ w₀ }\n⊢ x ↑x✝ - ↑(↑x✝).mult * 0 = x ↑x✝",
"usedConstants": [
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCo... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 242,
"column": 33
} | {
"line": 242,
"column": 54
} | [
{
"pp": "k : ℕ\nhk : 3 ≤ k\nhk2 : Even k\nz : ℍ\nH :\n ∀ (b : ℕ+),\n ∑' (n : ℤ), ((↑↑b * ↑z + ↑n) ^ k)⁻¹ =\n (-(2 * ↑π * I)) ^ k / ↑(k - 1)! * ∑' (n : ℕ+), ↑↑n ^ (k - 1) * cexp (2 * ↑π * I * (↑↑b * ↑z)) ^ ↑n\nm n : ℕ+\n⊢ ↑↑n ^ (k - 1) * cexp (↑↑n * (2 * ↑π * I * (↑↑m * ↑z))) = ↑↑n ^ (k - 1) * cexp (↑(↑... | by push_cast; ring_nf | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 394,
"column": 2
} | {
"line": 407,
"column": 16
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ LinearIndependent ℝ (completeFamily K)",
"usedConstants": [
"NumberField.Units.basisUnitLattice",
"NormedCommRing.toNormedRing",
"linearIndependent_option",
"NumberField.InfinitePlace.sum_mult_eq",
"Units.val",
... | classical
have h₁ : LinearIndependent ℝ (fun w : {w // w ≠ w₀} ↦ completeFamily K w.1) := by
refine LinearIndependent.of_comp realSpaceToLogSpace ?_
simp_rw [Function.comp_def, realSpaceToLogSpace_completeFamily_of_ne]
convert (((basisUnitLattice K).ofZLatticeBasis ℝ _).reindex equivFinRank).linearIndepen... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 394,
"column": 2
} | {
"line": 407,
"column": 16
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ LinearIndependent ℝ (completeFamily K)",
"usedConstants": [
"NumberField.Units.basisUnitLattice",
"NormedCommRing.toNormedRing",
"linearIndependent_option",
"NumberField.InfinitePlace.sum_mult_eq",
"Units.val",
... | classical
have h₁ : LinearIndependent ℝ (fun w : {w // w ≠ w₀} ↦ completeFamily K w.1) := by
refine LinearIndependent.of_comp realSpaceToLogSpace ?_
simp_rw [Function.comp_def, realSpaceToLogSpace_completeFamily_of_ne]
convert (((basisUnitLattice K).ofZLatticeBasis ℝ _).reindex equivFinRank).linearIndepen... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 394,
"column": 2
} | {
"line": 407,
"column": 16
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\n⊢ LinearIndependent ℝ (completeFamily K)",
"usedConstants": [
"NumberField.Units.basisUnitLattice",
"NormedCommRing.toNormedRing",
"linearIndependent_option",
"NumberField.InfinitePlace.sum_mult_eq",
"Units.val",
... | classical
have h₁ : LinearIndependent ℝ (fun w : {w // w ≠ w₀} ↦ completeFamily K w.1) := by
refine LinearIndependent.of_comp realSpaceToLogSpace ?_
simp_rw [Function.comp_def, realSpaceToLogSpace_completeFamily_of_ne]
convert (((basisUnitLattice K).ofZLatticeBasis ℝ _).reindex equivFinRank).linearIndepen... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Factorization | {
"line": 55,
"column": 6
} | {
"line": 55,
"column": 43
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : Fintype K\np f n : ℕ\nP : K[X]\nhK : Fintype.card K = p ^ f\nhn : p.Coprime n\nhp : Fact (Nat.Prime p)\nhP : P ∣ cyclotomic n K\nhPirr : Irreducible P\nhPmo : P.Monic\nthis✝¹ : Fact (Irreducible P)\nthis✝ : Module.Finite K (AdjoinRoot P)\nthis : Finite (AdjoinRoo... | have := charP_of_card_eq_prime_pow hK | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 345,
"column": 4
} | {
"line": 345,
"column": 22
} | [
{
"pp": "case h.e'_6\nk : ℕ\nhk : 3 ≤ k\nhk2 : Even k\nm : ℕ\nβ : ℂ := -(2 * ↑k / ↑(bernoulli k))\nc : ℕ → ℂ := fun m ↦ if m = 0 then 1 else β * ↑((σ (k - 1)) m)\nτ : ℍ\nhS : Summable fun n ↦ ↑((σ (k - 1)) (n + 1)) * cexp (2 * ↑π * I * ↑τ) ^ (n + 1)\nhval : (E hk) τ - 1 = β * ∑' (n : ℕ), ↑((σ (k - 1)) (n + 1)) ... | rw [tsum_mul_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 550,
"column": 36
} | {
"line": 550,
"column": 45
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\nw : InfinitePlace K\nhw : w ≠ w₀\n⊢ (∑ x_1, if x_1 = ⟨w, hw⟩ then x ↑x_1 else x ↑x_1 * 0) ∈ Set.Ico 0 1 ↔ x w ∈ Set.Ico 0 1",
"usedConstants": [
"NormedCommRing.toNormedRing",
"AddGroup.toSubtractionMonoid",
"... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 345,
"column": 4
} | {
"line": 345,
"column": 22
} | [
{
"pp": "case h.e'_6\nk : ℕ\nhk : 3 ≤ k\nhk2 : Even k\nm : ℕ\nβ : ℂ := -(2 * ↑k / ↑(bernoulli k))\nc : ℕ → ℂ := fun m ↦ if m = 0 then 1 else β * ↑((σ (k - 1)) m)\nτ : ℍ\nhS : Summable fun n ↦ ↑((σ (k - 1)) (n + 1)) * cexp (2 * ↑π * I * ↑τ) ^ (n + 1)\nhval : (E hk) τ - 1 = β * ∑' (n : ℕ), ↑((σ (k - 1)) (n + 1)) ... | rw [tsum_mul_left] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ModularForms.EisensteinSeries.QExpansion | {
"line": 345,
"column": 4
} | {
"line": 345,
"column": 22
} | [
{
"pp": "case h.e'_6\nk : ℕ\nhk : 3 ≤ k\nhk2 : Even k\nm : ℕ\nβ : ℂ := -(2 * ↑k / ↑(bernoulli k))\nc : ℕ → ℂ := fun m ↦ if m = 0 then 1 else β * ↑((σ (k - 1)) m)\nτ : ℍ\nhS : Summable fun n ↦ ↑((σ (k - 1)) (n + 1)) * cexp (2 * ↑π * I * ↑τ) ^ (n + 1)\nhval : (E hk) τ - 1 = β * ∑' (n : ℕ), ↑((σ (k - 1)) (n + 1)) ... | rw [tsum_mul_left] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Factorization | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 72
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : Fintype K\np f n : ℕ\nP : K[X]\nhK : Fintype.card K = p ^ f\nhn : p.Coprime n\nhp : Fact (Nat.Prime p)\nhP : P ∣ cyclotomic n K\nhPdeg : P.natDegree = orderOf (unitOfCoprime (p ^ f) ⋯)\n⊢ Irreducible P",
"usedConstants": [
"Polynomial.cyclotomic_ne_zero... | have hP0 : P ≠ 0 := ne_zero_of_dvd_ne_zero (cyclotomic_ne_zero n K) hP | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Polynomial.Cyclotomic.Factorization | {
"line": 152,
"column": 2
} | {
"line": 152,
"column": 66
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\ninst✝¹ : Fintype K\np f n : ℕ\nhK : Fintype.card K = p ^ f\nhn : p.Coprime n\nhp : Fact (Nat.Prime p)\ninst✝ : DecidableEq K\nh : Associated (normalizedFactors (cyclotomic n K)).prod (cyclotomic n K)\nthis : ∀ P ∈ normalizedFactors (cyclotomic n K), P.natDegree = orderOf... | rw [← H, mul_div_left _ (orderOf_pos _), toFinset_card_of_nodup] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Factorization | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 41
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\ninst✝¹ : Fintype K\np f n : ℕ\nhK : Fintype.card K = p ^ f\nhn : p.Coprime n\nhp : Fact (Nat.Prime p)\ninst✝ : DecidableEq K\nh : Associated (normalizedFactors (cyclotomic n K)).prod (cyclotomic n K)\nthis : ∀ P ∈ normalizedFactors (cyclotomic n K), P.natDegree = orderOf... | have := charP_of_card_eq_prime_pow hK | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.RootsOfUnity.CyclotomicUnits | {
"line": 56,
"column": 35
} | {
"line": 56,
"column": 53
} | [
{
"pp": "case h\nn✝ j : ℕ\nA : Type u_1\nζ : A\ninst✝¹ : CommRing A\ninst✝ : IsDomain A\nn : ℕ\nhζ : IsPrimitiveRoot ζ (n + 2)\nhj : j.Coprime (n + 2)\nm : ℕ\nhm : j * m % (n + 2) = 1\n⊢ ζ - 1 = ζ ^ (j * m) - 1",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"AddGroupWithOne.toAddGroup",
... | ← pow_mod_orderOf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 706,
"column": 8
} | {
"line": 706,
"column": 17
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nn : ℕ\nhn✝ : 0 < n\nhn : 0 < ↑n\n⊢ ENNReal.ofReal (Real.exp (↑n * 0) / ↑n) = (↑n)⁻¹",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"MulZeroCla... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.CanonicalEmbedding.NormLeOne | {
"line": 784,
"column": 2
} | {
"line": 784,
"column": 43
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nx : realSpace K\nhx₀ : x = 0\n⊢ x ∈ compactSet K ↔ x ∈ ↑expMapBasis '' closure (paramSet K) ∪ {0}",
"usedConstants": [
"Eq.mpr",
"Real",
"NumberField.mixedEmbedding.realSpace",
"Pi.topologicalSpace",
"Rea... | · simpa [hx₀] using zero_mem_compactSet K | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.NumberField.Ideal.Asymptotics | {
"line": 45,
"column": 2
} | {
"line": 53,
"column": 91
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nC : ClassGroup (𝓞 K)\nJ : ↥(Ideal (𝓞 K))⁰\ns : ℝ\nhJ : ClassGroup.mk0 J = C⁻¹\n⊢ Nat.card { I // ↑(absNorm ↑I) ≤ s ∧ ClassGroup.mk0 I = C } =\n Nat.card { I // ↑J ∣ ↑I ∧ IsPrincipal ↑I ∧ ↑(absNorm ↑I) ≤ s * ↑(absNorm ↑J) }",
"usedConstants... | simp_rw [← nonZeroDivisors_dvd_iff_dvd_coe]
refine Nat.card_congr ?_
refine ((Equiv.dvd J).subtypeEquiv fun I ↦ ?_).trans
(Equiv.subtypeSubtypeEquivSubtypeInter (fun I : (Ideal (𝓞 K))⁰ ↦ J ∣ I) _)
rw [← ClassGroup.mk0_eq_one_iff (SetLike.coe_mem _)]
simp_rw [Equiv.dvd_apply, Submonoid.coe_mul, ← Submonoid.... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Ideal.Asymptotics | {
"line": 45,
"column": 2
} | {
"line": 53,
"column": 91
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nC : ClassGroup (𝓞 K)\nJ : ↥(Ideal (𝓞 K))⁰\ns : ℝ\nhJ : ClassGroup.mk0 J = C⁻¹\n⊢ Nat.card { I // ↑(absNorm ↑I) ≤ s ∧ ClassGroup.mk0 I = C } =\n Nat.card { I // ↑J ∣ ↑I ∧ IsPrincipal ↑I ∧ ↑(absNorm ↑I) ≤ s * ↑(absNorm ↑J) }",
"usedConstants... | simp_rw [← nonZeroDivisors_dvd_iff_dvd_coe]
refine Nat.card_congr ?_
refine ((Equiv.dvd J).subtypeEquiv fun I ↦ ?_).trans
(Equiv.subtypeSubtypeEquivSubtypeInter (fun I : (Ideal (𝓞 K))⁰ ↦ J ∣ I) _)
rw [← ClassGroup.mk0_eq_one_iff (SetLike.coe_mem _)]
simp_rw [Equiv.dvd_apply, Submonoid.coe_mul, ← Submonoid.... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 73,
"column": 4
} | {
"line": 73,
"column": 52
} | [
{
"pp": "case neg\np k : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nhK : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nh : ¬p = 2\n⊢ Associated ((Algebra.norm ℤ) (hζ.toInteger - 1)) ↑p",
"usedConstants": [
"Eq.mpr",
... | rw [hζ.norm_toInteger_sub_one_of_prime_ne_two h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 73,
"column": 4
} | {
"line": 73,
"column": 52
} | [
{
"pp": "case neg\np k : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nhK : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nh : ¬p = 2\n⊢ Associated ((Algebra.norm ℤ) (hζ.toInteger - 1)) ↑p",
"usedConstants": [
"Eq.mpr",
... | rw [hζ.norm_toInteger_sub_one_of_prime_ne_two h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 73,
"column": 4
} | {
"line": 73,
"column": 52
} | [
{
"pp": "case neg\np k : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nhK : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nζ : K\nhζ : IsPrimitiveRoot ζ (p ^ (k + 1))\nh : ¬p = 2\n⊢ Associated ((Algebra.norm ℤ) (hζ.toInteger - 1)) ↑p",
"usedConstants": [
"Eq.mpr",
... | rw [hζ.norm_toInteger_sub_one_of_prime_ne_two h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.SiegelsLemma | {
"line": 106,
"column": 2
} | {
"line": 115,
"column": 14
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Fintype α\ninst✝ : Fintype β\nA : Matrix α β ℤ\ni : α\n⊢ N i ≤ P i + 1",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"Int.instAddCommMonoid",
"mul_nonneg",
"NormedCommRing.toSeminormedCommRing",
... | calc N i
_ ≤ 0 := by
apply Finset.sum_nonpos
intro j _
simp only [mul_neg, Left.neg_nonpos_iff]
positivity
_ ≤ P i + 1 := by
apply le_trans (Finset.sum_nonneg _) (Int.le_add_one (le_refl P i))
intro j _
positivity | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.NumberTheory.SiegelsLemma | {
"line": 106,
"column": 2
} | {
"line": 115,
"column": 14
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Fintype α\ninst✝ : Fintype β\nA : Matrix α β ℤ\ni : α\n⊢ N i ≤ P i + 1",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"Int.instAddCommMonoid",
"mul_nonneg",
"NormedCommRing.toSeminormedCommRing",
... | calc N i
_ ≤ 0 := by
apply Finset.sum_nonpos
intro j _
simp only [mul_neg, Left.neg_nonpos_iff]
positivity
_ ≤ P i + 1 := by
apply le_trans (Finset.sum_nonneg _) (Int.le_add_one (le_refl P i))
intro j _
positivity | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.SiegelsLemma | {
"line": 106,
"column": 2
} | {
"line": 115,
"column": 14
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Fintype α\ninst✝ : Fintype β\nA : Matrix α β ℤ\ni : α\n⊢ N i ≤ P i + 1",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"Int.instAddCommMonoid",
"mul_nonneg",
"NormedCommRing.toSeminormedCommRing",
... | calc N i
_ ≤ 0 := by
apply Finset.sum_nonpos
intro j _
simp only [mul_neg, Left.neg_nonpos_iff]
positivity
_ ≤ P i + 1 := by
apply le_trans (Finset.sum_nonneg _) (Int.le_add_one (le_refl P i))
intro j _
positivity | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.SiegelsLemma | {
"line": 163,
"column": 23
} | {
"line": 163,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : Fintype α\ninst✝² : Fintype β\nA : Matrix α β ℤ\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nhn : m < n\nhm : 0 < m\n⊢ (↑n * max 1 ‖A‖ * ↑B + 1) ^ m ≤ (↑n * max 1 ‖A‖ * (↑B + 1)) ^ m",
"usedConstants": [
"Distrib.leftDistribClass",
"Norm.norm",
... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Cyclotomic.Ideal | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 92
} | [
{
"pp": "p k : ℕ\nhp : Fact (Nat.Prime p)\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nhK : IsCyclotomicExtension {p ^ (k + 1)} ℚ K\nthis✝ : IsGalois ℚ K\nthis : 𝒑 ≠ ⊥\n⊢ (𝒑.primesOver (𝓞 K)).ncard = 1",
"usedConstants": [
"NormedCommRing.toNormedRing",
"NumberField.RingOfIntegers.... | have h_main := ncard_primesOver_mul_ramificationIdxIn_mul_inertiaDegIn this (𝓞 K) Gal(K/ℚ) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.NumberTheory.NumberField.House | {
"line": 53,
"column": 2
} | {
"line": 53,
"column": 47
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nα β : K\n⊢ house (α * β) ≤ house α * house β",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"RingHom.instRingHomClass",
"Real.instLE",
... | simp only [house, map_mul]; apply norm_mul_le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.House | {
"line": 53,
"column": 2
} | {
"line": 53,
"column": 47
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nα β : K\n⊢ house (α * β) ≤ house α * house β",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"RingHom.instRingHomClass",
"Real.instLE",
... | simp only [house, map_mul]; apply norm_mul_le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.SiegelsLemma | {
"line": 184,
"column": 2
} | {
"line": 211,
"column": 17
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Fintype α\ninst✝ : Fintype β\nA : Matrix α β ℤ\nhn : m < n\nhm : 0 < m\n⊢ ∃ t, t ≠ 0 ∧ A *ᵥ t = 0 ∧ ‖t‖ ≤ (↑n * max 1 ‖A‖) ^ e",
"usedConstants": [
"Int.instAddCommGroup",
"Iff.mpr",
"NormedCommRing.toNormedRing",
"AddGroup.toSubtractionM... | classical
-- Pigeonhole
rcases Finset.exists_ne_map_eq_of_card_lt_of_maps_to
(card_S_lt_card_T A hn hm) (image_T_subset_S A)
with ⟨x, hxT, y, hyT, hneq, hfeq⟩
-- Proofs that x - y ≠ 0 and x - y is a solution
refine ⟨x - y, sub_ne_zero.mpr hneq, by simp only [mulVec_sub, sub_eq_zero, hfeq], ?_⟩
-- Ineq... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.NumberTheory.SiegelsLemma | {
"line": 184,
"column": 2
} | {
"line": 211,
"column": 17
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Fintype α\ninst✝ : Fintype β\nA : Matrix α β ℤ\nhn : m < n\nhm : 0 < m\n⊢ ∃ t, t ≠ 0 ∧ A *ᵥ t = 0 ∧ ‖t‖ ≤ (↑n * max 1 ‖A‖) ^ e",
"usedConstants": [
"Int.instAddCommGroup",
"Iff.mpr",
"NormedCommRing.toNormedRing",
"AddGroup.toSubtractionM... | classical
-- Pigeonhole
rcases Finset.exists_ne_map_eq_of_card_lt_of_maps_to
(card_S_lt_card_T A hn hm) (image_T_subset_S A)
with ⟨x, hxT, y, hyT, hneq, hfeq⟩
-- Proofs that x - y ≠ 0 and x - y is a solution
refine ⟨x - y, sub_ne_zero.mpr hneq, by simp only [mulVec_sub, sub_eq_zero, hfeq], ?_⟩
-- Ineq... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.SiegelsLemma | {
"line": 184,
"column": 2
} | {
"line": 211,
"column": 17
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Fintype α\ninst✝ : Fintype β\nA : Matrix α β ℤ\nhn : m < n\nhm : 0 < m\n⊢ ∃ t, t ≠ 0 ∧ A *ᵥ t = 0 ∧ ‖t‖ ≤ (↑n * max 1 ‖A‖) ^ e",
"usedConstants": [
"Int.instAddCommGroup",
"Iff.mpr",
"NormedCommRing.toNormedRing",
"AddGroup.toSubtractionM... | classical
-- Pigeonhole
rcases Finset.exists_ne_map_eq_of_card_lt_of_maps_to
(card_S_lt_card_T A hn hm) (image_T_subset_S A)
with ⟨x, hxT, y, hyT, hneq, hfeq⟩
-- Proofs that x - y ≠ 0 and x - y is a solution
refine ⟨x - y, sub_ne_zero.mpr hneq, by simp only [mulVec_sub, sub_eq_zero, hfeq], ?_⟩
-- Ineq... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.SiegelsLemma | {
"line": 219,
"column": 2
} | {
"line": 219,
"column": 28
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Fintype α\ninst✝ : Fintype β\nA : Matrix α β ℤ\nhn : m < n\nhm : 0 < m\nhA : A ≠ 0\nthis : ∃ t, t ≠ 0 ∧ A *ᵥ t = 0 ∧ ‖t‖ ≤ (↑n * max 1 ‖A‖) ^ e\n⊢ ∃ t, t ≠ 0 ∧ A *ᵥ t = 0 ∧ ‖t‖ ≤ (↑n * ‖A‖) ^ e",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Nor... | rwa [max_eq_right] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.NumberTheory.Ostrowski | {
"line": 343,
"column": 24
} | {
"line": 343,
"column": 32
} | [
{
"pp": "f : AbsoluteValue ℚ ℝ\nn₀ : ℕ\nhn₀ : 1 < n₀\nh : f ↑n₀ ≤ 1\nn : ℕ\nh_ineq1 : ∀ {m : ℕ}, 1 ≤ m → f ↑m ≤ ↑n₀ * (logb ↑n₀ ↑m + 1)\nh₀ : n ≠ 0\nk : ℕ\nhk : 0 < k\nthis : 0 ≤ logb ↑n₀ ↑n\n⊢ (↑n₀ * (logb (↑n₀) (↑n ^ k) + 1)) ^ (↑k)⁻¹ = (↑n₀ * (↑k * logb ↑n₀ ↑n + 1)) ^ (↑k)⁻¹",
"usedConstants": [
"E... | logb_pow | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Ostrowski | {
"line": 399,
"column": 77
} | {
"line": 399,
"column": 85
} | [
{
"pp": "f : AbsoluteValue ℚ ℝ\nm n : ℕ\nhm : 1 < m\nhn : 1 < n\nnotbdd : ¬∀ (n : ℕ), f ↑n ≤ 1\nk : ℕ\nhk : k ≠ 0\nh_ineq1 : ∀ {m n : ℕ}, 1 < m → 1 < n → f ↑n ≤ ↑m * f ↑m / (f ↑m - 1) * f ↑m ^ logb ↑m ↑n\n⊢ ↑m * f ↑m / (f ↑m - 1) * f ↑m ^ logb (↑m) (↑n ^ k) = ↑m * f ↑m / (f ↑m - 1) * f ↑m ^ (↑k * logb ↑m ↑n)",
... | logb_pow | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.Complex | {
"line": 164,
"column": 4
} | {
"line": 164,
"column": 26
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\n⊢ Valued.v ↑p = 1 / ↑p",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHDiv",
"NormedRing.toRing",
... | PadicAlgCl.valuation_p | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.MahlerBasis | {
"line": 185,
"column": 2
} | {
"line": 196,
"column": 95
} | [
{
"pp": "case refine_2\np : ℕ\nhp : Fact (Nat.Prime p)\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : Module ℤ_[p] E\ninst✝¹ : IsBoundedSMul ℤ_[p] E\ninst✝ : IsUltrametricDist E\nf : C(ℤ_[p], E)\ns t : ℕ\nhst : ∀ (x y : ℤ_[p]), ‖x - y‖ ≤ ↑p ^ (-↑t) → ‖f x - f y‖ ≤ ‖f‖ / ↑p ^ s\nn : ℕ\n⊢ ‖∑ k ∈ range (n ... | · -- Bounding the sum over `range (n + 1)`: every term is small by the choice of `t`
refine norm_sum_le_of_forall_le_of_nonempty nonempty_range_add_one (fun i _ ↦ ?_)
calc ‖((-1 : ℤ) ^ (n - i) * n.choose i) • (f (i + ↑(p ^ t)) - f i)‖
_ ≤ ‖((-1 : ℤ) ^ (n - i) * n.choose i : ℤ_[p])‖ * ‖(f (i + ↑(p ^ t)) - f ... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.Pell | {
"line": 204,
"column": 32
} | {
"line": 204,
"column": 41
} | [
{
"pp": "d : ℤ\na : Solution₁ d\nha : 1 < a.x\nhy : a.y = 0\nprop : a.x ^ 2 - d * 0 = 1\n⊢ False",
"usedConstants": [
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"MulZeroClass.toMul",
"congrArg",
"Pell.Solution₁.x",
... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Pell | {
"line": 230,
"column": 21
} | {
"line": 230,
"column": 30
} | [
{
"pp": "d : ℤ\na : Solution₁ d\nH : a.y = 0\nprop : a.x ^ 2 - d * (0 * 0) = 1\n⊢ a = 1 ∨ a = -1",
"usedConstants": [
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"MulZeroClass.toMul",
"congrArg",
"Pell.Solution₁.x",
... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Pell | {
"line": 230,
"column": 31
} | {
"line": 230,
"column": 40
} | [
{
"pp": "d : ℤ\na : Solution₁ d\nH : a.y = 0\nprop : a.x ^ 2 - d * 0 = 1\n⊢ a = 1 ∨ a = -1",
"usedConstants": [
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"MulZeroClass.toMul",
"congrArg",
"Pell.Solution₁.x",
"HSub.... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Pell | {
"line": 382,
"column": 2
} | {
"line": 384,
"column": 71
} | [
{
"pp": "case refine_2\nd : ℤ\nh₀ : 0 < d\nhd : ¬IsSquare d\nξ : ℝ := √↑d\nhξ : Irrational ξ\nM : ℤ\nhM₁ : 2 * |ξ| + 1 < ↑M\nhM : {q | |q.num ^ 2 - d * ↑q.den ^ 2| < M}.Infinite\nm : ℤ\nhm : {q | q.num ^ 2 - d * ↑q.den ^ 2 = m}.Infinite\nthis : NeZero m.natAbs\nf : ℚ → ZMod m.natAbs × ZMod m.natAbs := fun q ↦ (... | · qify [hd₂]
refine div_ne_zero_iff.mpr ⟨?_, hm₀⟩
exact mod_cast mt sub_eq_zero.mp (mt Rat.eq_iff_mul_eq_mul.mpr hne) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.Padics.WithVal | {
"line": 57,
"column": 17
} | {
"line": 57,
"column": 40
} | [
{
"pp": "p : ℕ\ninst✝ : Fact (Nat.Prime p)\nhp0' : 0 < ↑p\nhp0 : 0 < (↑p)⁻¹\nhp1' : 1 < ↑p\nhp1 : (↑p)⁻¹ < 1\nn : ℕ\nhn : Valued.v (↑p ^ n) = exp (-↑n)\n⊢ ¬Valued.v.restrict (↑p ^ n) = 0",
"usedConstants": [
"Int.instAddCommGroup",
"NormedCommRing.toNormedRing",
"Eq.mpr",
"Int.instAd... | Valuation.restrict_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Pell | {
"line": 600,
"column": 38
} | {
"line": 600,
"column": 47
} | [
{
"pp": "case h.inl.hx\nd : ℤ\na₁ : Solution₁ d\nh : IsFundamental a₁\nx : ℕ\nih : ∀ m < x, ∀ {a : Solution₁ d}, 0 ≤ a.y → ↑m = a.x → 0 < ↑m → ∃ n, a = a₁ ^ n\na : Solution₁ d\nhay : 0 ≤ a.y\nhax' : ↑x = a.x\nhax : 0 < ↑x\nhy : 0 = a.y\nprop : a.x ^ 2 - d * 0 = 1\n⊢ a.x = 1",
"usedConstants": [
"NonUn... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Padics.HeightOneSpectrum | {
"line": 116,
"column": 28
} | {
"line": 116,
"column": 68
} | [
{
"pp": "R✝ : Type u_1\ninst✝⁷ : CommRing R✝\ninst✝⁶ : Algebra R✝ ℚ\ninst✝⁵ : IsIntegralClosure R✝ ℤ ℚ\nR : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : Algebra R ℚ\ninst✝² : IsIntegralClosure R ℤ ℚ\ninst✝¹ : IsDedekindDomain R\ninst✝ : IsFractionRing R ℚ\np : Nat.Primes\n⊢ Prime (Ideal.span {↑↑p})",
"usedConsta... | by simp [← Nat.prime_iff_prime_int, p.2] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.Rayleigh | {
"line": 71,
"column": 2
} | {
"line": 72,
"column": 87
} | [
{
"pp": "r s : ℝ\nhrs : r.HolderConjugate s\nj k : ℤ\nh₁ : ↑j / r ≤ ↑k ∧ ↑k < (↑j + 1) / r\nm : ℤ\nh₂ : beattySeq' s m = j\n⊢ False",
"usedConstants": [
"Int.cast",
"Real",
"HMul.hMul",
"congrArg",
"HSub.hSub",
"Eq.mp",
"Real.instFloorRing",
"Real.instRing",
... | rw [beattySeq', sub_eq_iff_eq_add, Int.ceil_eq_iff, Int.cast_add, Int.cast_one,
add_sub_cancel_right, ← div_lt_iff₀ hrs.symm.pos, ← le_div_iff₀ hrs.symm.pos] at h₂ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Zsqrtd.GaussianInt | {
"line": 196,
"column": 35
} | {
"line": 196,
"column": 54
} | [
{
"pp": "x y : ℤ[i]\nthis : |2⁻¹| = 2⁻¹\n⊢ |(↑(toComplex x / toComplex y).re - ↑↑(round (toComplex x / toComplex y).re) +\n (↑(toComplex x / toComplex y).im - ↑(toComplex (x / y)).im) * I).re| ≤\n |(1 / 2 + 1 / 2 * I).re|",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"GroupWithZer... | simp [normSq, this] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.Zsqrtd.GaussianInt | {
"line": 197,
"column": 35
} | {
"line": 197,
"column": 54
} | [
{
"pp": "x y : ℤ[i]\nthis : |2⁻¹| = 2⁻¹\n⊢ |(↑(toComplex x / toComplex y).re - ↑(toComplex (x / y)).re +\n (↑(toComplex x / toComplex y).im - ↑↑(round (toComplex x / toComplex y).im)) * I).im| ≤\n |(1 / 2 + 1 / 2 * I).im|",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"GroupWithZer... | simp [normSq, this] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.SumFourSquares | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 19
} | [
{
"pp": "m a b c d : ℤ\nh : a ^ 2 + b ^ 2 + c ^ 2 + d ^ 2 = 2 * m\nthis :\n ∀ (f : Fin 4 → ZMod 2),\n f 0 ^ 2 + f 1 ^ 2 + f 2 ^ 2 + f 3 ^ 2 = 0 →\n ∃ i, f i ^ 2 + f ((swap i 0) 1) ^ 2 = 0 ∧ f ((swap i 0) 2) ^ 2 + f ((swap i 0) 3) ^ 2 = 0\nf : Fin 4 → ℤ := ![a, b, c, d]\ni : Fin 4\nhσ : ↑(f i) ^ 2 + ↑(f... | set σ := swap i 0 | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.NumberTheory.Transcendental.Lindemann.AnalyticalPart | {
"line": 60,
"column": 29
} | {
"line": 60,
"column": 37
} | [
{
"pp": "f : ℂ[X]\ns : ℂ\n⊢ P f s = cexp s * (-(cexp (-s) * eval s (sumIDeriv f)) + eval 0 (sumIDeriv f))",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Polynomial.eval",
"_private.Mathlib.NumberTheory.Transcendental.Lindemann.AnalyticalPart.0.LindemannWeierstrass.P",
... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.SumTwoSquares | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 44
} | [
{
"pp": "n : ℕ\nhn : Squarefree n\nH : ∀ q ∈ n.primeFactors, q % 4 ≠ 3\n⊢ IsSquare (-1)",
"usedConstants": [
"Nat.dvd_of_mem_primeFactors",
"Iff.mpr",
"Nat.Coprime",
"NegZeroClass.toNeg",
"Nat.Prime",
"Dvd.dvd",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | induction n using induction_on_primes with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.NumberTheory.SumTwoSquares | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 44
} | [
{
"pp": "n : ℕ\nh : IsSquare (-1)\n⊢ ∃ x y, n = x ^ 2 + y ^ 2",
"usedConstants": [
"NegZeroClass.toNeg",
"Nat.Prime",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"ZMod.exists_sq_eq_neg_one_iff",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"ZMod.commRing",
... | induction n using induction_on_primes with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.NumberTheory.SumTwoSquares | {
"line": 200,
"column": 42
} | {
"line": 200,
"column": 61
} | [
{
"pp": "n x y : ℕ\nh : n = x ^ 2 + y ^ 2\nhxy : x = 0 ∧ y = 0\n⊢ 1 = 0",
"usedConstants": [
"Eq.mpr",
"ZMod.commRing",
"Fin.one_eq_zero_iff",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"id",
"Fin.instOfNat",
"SubtractionMonoid.toSubNegZeroMonoid",
... | Fin.one_eq_zero_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleNumber | {
"line": 167,
"column": 6
} | {
"line": 169,
"column": 22
} | [
{
"pp": "case h\nm : ℕ\nhm : 0 < m\nk p_k : ℕ\nh_k : partialSum (↑m) k = ↑p_k / ↑(m ^ k !)\n⊢ ↑p_k * ↑m ^ (k + 1)! * ↑(m ^ (k + 1)!) + ↑(m ^ k !) * 1 * ↑(m ^ (k + 1)!) =\n ↑(p_k * m ^ ((k + 1)! - k !) + 1) * (↑(m ^ k !) * ↑m ^ (k + 1)!)",
"usedConstants": [
"Nat.cast_mul._simp_1",
"add_mul",
... | norm_cast
rw [add_mul, one_mul, Nat.factorial_succ, add_mul, one_mul, add_tsub_cancel_right, pow_add]
simp [mul_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleNumber | {
"line": 167,
"column": 6
} | {
"line": 169,
"column": 22
} | [
{
"pp": "case h\nm : ℕ\nhm : 0 < m\nk p_k : ℕ\nh_k : partialSum (↑m) k = ↑p_k / ↑(m ^ k !)\n⊢ ↑p_k * ↑m ^ (k + 1)! * ↑(m ^ (k + 1)!) + ↑(m ^ k !) * 1 * ↑(m ^ (k + 1)!) =\n ↑(p_k * m ^ ((k + 1)! - k !) + 1) * (↑(m ^ k !) * ↑m ^ (k + 1)!)",
"usedConstants": [
"Nat.cast_mul._simp_1",
"add_mul",
... | norm_cast
rw [add_mul, one_mul, Nat.factorial_succ, add_mul, one_mul, add_tsub_cancel_right, pow_add]
simp [mul_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.Irrational | {
"line": 77,
"column": 2
} | {
"line": 86,
"column": 23
} | [
{
"pp": "x : ℝ\nhx : Irrational x\nn : ℕ\n⊢ ∀ᶠ (ε : ℝ) in 𝓝 0, ∀ (m : ℤ), ε ≤ dist x (↑m / ↑n)",
"usedConstants": [
"Filter.instMembership",
"Int.cast",
"GroupWithZero.toMonoidWithZero",
"Nat.not_irrational._simp_1",
"False",
"MonoidWithZero.toMulActionWithZero",
"... | have A : IsClosed (range (fun m => (n : ℝ)⁻¹ * m : ℤ → ℝ)) :=
((isClosedMap_smul₀ (n⁻¹ : ℝ)).comp Int.isClosedEmbedding_coe_real.isClosedMap).isClosed_range
have B : x ∉ range (fun m => (n : ℝ)⁻¹ * m : ℤ → ℝ) := by
rintro ⟨m, rfl⟩
simp at hx
rcases Metric.mem_nhds_iff.1 (A.isOpen_compl.mem_nhds B) with ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.Irrational | {
"line": 77,
"column": 2
} | {
"line": 86,
"column": 23
} | [
{
"pp": "x : ℝ\nhx : Irrational x\nn : ℕ\n⊢ ∀ᶠ (ε : ℝ) in 𝓝 0, ∀ (m : ℤ), ε ≤ dist x (↑m / ↑n)",
"usedConstants": [
"Filter.instMembership",
"Int.cast",
"GroupWithZero.toMonoidWithZero",
"Nat.not_irrational._simp_1",
"False",
"MonoidWithZero.toMulActionWithZero",
"... | have A : IsClosed (range (fun m => (n : ℝ)⁻¹ * m : ℤ → ℝ)) :=
((isClosedMap_smul₀ (n⁻¹ : ℝ)).comp Int.isClosedEmbedding_coe_real.isClosedMap).isClosed_range
have B : x ∉ range (fun m => (n : ℝ)⁻¹ * m : ℤ → ℝ) := by
rintro ⟨m, rfl⟩
simp at hx
rcases Metric.mem_nhds_iff.1 (A.isOpen_compl.mem_nhds B) with ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith | {
"line": 66,
"column": 27
} | {
"line": 66,
"column": 64
} | [
{
"pp": "x : ℝ\nn : ℕ\nhn : 0 < n\nhn' : 0 < ↑n\nthis : x < ↑(⌊x * ↑n⌋ + 1) / ↑n\n⊢ ↑(⌊x * ↑n⌋ + 1) ≤ x * ↑n + 1",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"Int.floor",
"congrArg",
"covarian... | push_cast; gcongr; apply Int.floor_le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith | {
"line": 66,
"column": 27
} | {
"line": 66,
"column": 64
} | [
{
"pp": "x : ℝ\nn : ℕ\nhn : 0 < n\nhn' : 0 < ↑n\nthis : x < ↑(⌊x * ↑n⌋ + 1) / ↑n\n⊢ ↑(⌊x * ↑n⌋ + 1) ≤ x * ↑n + 1",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"Int.floor",
"congrArg",
"covarian... | push_cast; gcongr; apply Int.floor_le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Baire.LocallyCompactRegular | {
"line": 42,
"column": 6
} | {
"line": 42,
"column": 76
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\ns : Set X\ninst✝¹ : R1Space X\ninst✝ : LocallyCompactSpace X\nf : ℕ → Set X\nho : ∀ (n : ℕ), IsOpen (f n)\nhd : ∀ (n : ℕ), Dense (f n)\nU : Set X\nU_open : IsOpen U\nU_nonempty : U.Nonempty\nK₀ : PositiveCompacts X\nhK₀ : closure ↑K₀ ⊆ U\nn : ℕ\nK : PositiveCo... | exact (hd n).inter_open_nonempty _ isOpen_interior K.interior_nonempty | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith | {
"line": 200,
"column": 2
} | {
"line": 200,
"column": 40
} | [
{
"pp": "p x : ℝ\nn : ℕ\n⊢ LiouvilleWith p (x + ↑n) ↔ LiouvilleWith p x",
"usedConstants": [
"Eq.mpr",
"Real",
"DivisionRing.toRatCast",
"congrArg",
"Iff.rfl",
"Real.instRatCast",
"Rat",
"AddGroupWithOne.toAddMonoidWithOne",
"LiouvilleWith.add_rat_iff",
... | rw [← Rat.cast_natCast n, add_rat_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith | {
"line": 200,
"column": 2
} | {
"line": 200,
"column": 40
} | [
{
"pp": "p x : ℝ\nn : ℕ\n⊢ LiouvilleWith p (x + ↑n) ↔ LiouvilleWith p x",
"usedConstants": [
"Eq.mpr",
"Real",
"DivisionRing.toRatCast",
"congrArg",
"Iff.rfl",
"Real.instRatCast",
"Rat",
"AddGroupWithOne.toAddMonoidWithOne",
"LiouvilleWith.add_rat_iff",
... | rw [← Rat.cast_natCast n, add_rat_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Transcendental.Liouville.LiouvilleWith | {
"line": 200,
"column": 2
} | {
"line": 200,
"column": 40
} | [
{
"pp": "p x : ℝ\nn : ℕ\n⊢ LiouvilleWith p (x + ↑n) ↔ LiouvilleWith p x",
"usedConstants": [
"Eq.mpr",
"Real",
"DivisionRing.toRatCast",
"congrArg",
"Iff.rfl",
"Real.instRatCast",
"Rat",
"AddGroupWithOne.toAddMonoidWithOne",
"LiouvilleWith.add_rat_iff",
... | rw [← Rat.cast_natCast n, add_rat_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Comparable | {
"line": 281,
"column": 2
} | {
"line": 281,
"column": 36
} | [
{
"pp": "α : Type u_1\na b : α\nr : α → α → Prop\n⊢ ¬SymmGen r a b ↔ IncompRel r a b",
"usedConstants": [
"congrArg",
"iff_self",
"And",
"Iff",
"not_or._simp_1",
"True",
"of_eq_true",
"congrFun'",
"Relation.SymmGen",
"Not",
"Eq.trans"
]
... | simp [Relation.SymmGen, IncompRel] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.Comparable | {
"line": 281,
"column": 2
} | {
"line": 281,
"column": 36
} | [
{
"pp": "α : Type u_1\na b : α\nr : α → α → Prop\n⊢ ¬SymmGen r a b ↔ IncompRel r a b",
"usedConstants": [
"congrArg",
"iff_self",
"And",
"Iff",
"not_or._simp_1",
"True",
"of_eq_true",
"congrFun'",
"Relation.SymmGen",
"Not",
"Eq.trans"
]
... | simp [Relation.SymmGen, IncompRel] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Comparable | {
"line": 281,
"column": 2
} | {
"line": 281,
"column": 36
} | [
{
"pp": "α : Type u_1\na b : α\nr : α → α → Prop\n⊢ ¬SymmGen r a b ↔ IncompRel r a b",
"usedConstants": [
"congrArg",
"iff_self",
"And",
"Iff",
"not_or._simp_1",
"True",
"of_eq_true",
"congrFun'",
"Relation.SymmGen",
"Not",
"Eq.trans"
]
... | simp [Relation.SymmGen, IncompRel] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.CompleteSublattice | {
"line": 138,
"column": 4
} | {
"line": 138,
"column": 44
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : CompleteLattice α\ninst✝ : CompleteLattice β\nf : CompleteLatticeHom α β\nL✝ L : CompleteSublattice α\nt : Set α\nht : t ⊆ ↑L\nhs : ⇑f '' t ⊆ ⇑f '' ↑L\n⊢ f (sSup t) ∈ ⇑f '' ↑L",
"usedConstants": [
"CompleteLatticeHom.instFunLike",
"CompleteSublattice... | exact mem_image_of_mem f (sSupClosed ht) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Interval.Set.SurjOn | {
"line": 58,
"column": 2
} | {
"line": 59,
"column": 61
} | [
{
"pp": "case inr.inr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : PartialOrder β\nf : α → β\nh_mono : Monotone f\nh_surj : Surjective f\na b : α\nhab : a ≤ b\np : β\nhp : p ∈ Icc (f a) (f b)\nhp' : p ∈ Ioo (f a) (f b)\n⊢ p ∈ f '' Icc a b",
"usedConstants": [
"Set.Ioo_subset_Icc_self",... | · exact image_mono Ioo_subset_Icc_self <|
surjOn_Ioo_of_monotone_surjective h_mono h_surj a b hp' | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.Monotone.MonovaryOrder | {
"line": 50,
"column": 4
} | {
"line": 50,
"column": 43
} | [
{
"pp": "case refine_1\nι : Type u_1\nα : Type u_3\nβ : Type u_4\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : ι → α\ng : ι → β\ns : Set ι\nthis : LinearOrder ι := linearOrderOfSTO (MonovaryOrder f g)\nhfg : MonovaryOn f g s\ni : ι\nhi : i ∈ s\nj : ι\nhj : j ∈ s\nhij : i < j\n⊢ f i ≤ f j",
"usedConsta... | obtain h | ⟨h, -⟩ := Prod.lex_iff.1 hij | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Order.Partition.Basic | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 48
} | [
{
"pp": "α : Type u_1\ninst✝ : CompleteLattice α\nP : Partition ⊥\nx : α\nhxP : x ∈ P\n⊢ False",
"usedConstants": [
"le_bot_iff",
"Lattice.toSemilatticeSup",
"CompleteLattice.toLattice",
"OrderBot.toBot",
"PartialOrder.toPreorder",
"Preorder.toLE",
"CompleteLattice.... | obtain rfl := le_bot_iff.mp <| P.le_of_mem hxP | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Order.PrimeSeparator | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 20
} | [
{
"pp": "case h\nα : Type u_1\ninst✝¹ : DistribLattice α\ninst✝ : BoundedOrder α\nF : PFilter α\nI : Ideal α\nhFI : Disjoint ↑F ↑I\nS : Set (Set α) := {J | IsIdeal J ∧ ↑I ≤ J ∧ Disjoint (↑F) J}\nIinS : ↑I ∈ S\nchainub : ∀ c ⊆ S, IsChain (fun x1 x2 ↦ x1 ⊆ x2) c → c.Nonempty → ∃ ub ∈ S, ∀ s ∈ c, s ⊆ ub\nJset : Se... | clear chainub IinS | Lean.Elab.Tactic.evalClear | Lean.Parser.Tactic.clear |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 623,
"column": 6
} | {
"line": 623,
"column": 31
} | [
{
"pp": "Ω : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\na b : ℝ\nf : ℕ → Ω → ℝ\nN : ℕ\nℱ : Filtration ℕ m0\ninst✝ : IsFiniteMeasure μ\nhf : Submartingale f ℱ μ\nhfN : ∀ (ω : Ω), a ≤ f N ω\nhfzero : 0 ≤ f 0\nhab : a < b\n⊢ (b - a) * ∫ (x : Ω), ↑(upcrossingsBefore a b f N x) ∂μ ≤\n ∫ (x : Ω), (∑ k ∈ Fins... | rw [← integral_const_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Process.HittingTime | {
"line": 301,
"column": 4
} | {
"line": 303,
"column": 61
} | [
{
"pp": "case mpr\nΩ : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝¹ : ConditionallyCompleteLinearOrder ι\nu : ι → Ω → β\ns : Set β\nn : ι\nω : Ω\ninst✝ : WellFoundedLT ι\nm i : ι\nhi : i ≤ m\nh' : ∃ j ∈ Set.Ico n i, u j ω ∈ s\n⊢ hittingBtwn u s n m ω < i",
"usedConstants": [
"lt_of_le_of_lt",
"P... | obtain ⟨k, hk₁, hk₂⟩ := h'
refine lt_of_le_of_lt ?_ hk₁.2
exact hittingBtwn_le_of_mem hk₁.1 (hk₁.2.le.trans hi) hk₂ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Process.HittingTime | {
"line": 301,
"column": 4
} | {
"line": 303,
"column": 61
} | [
{
"pp": "case mpr\nΩ : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝¹ : ConditionallyCompleteLinearOrder ι\nu : ι → Ω → β\ns : Set β\nn : ι\nω : Ω\ninst✝ : WellFoundedLT ι\nm i : ι\nhi : i ≤ m\nh' : ∃ j ∈ Set.Ico n i, u j ω ∈ s\n⊢ hittingBtwn u s n m ω < i",
"usedConstants": [
"lt_of_le_of_lt",
"P... | obtain ⟨k, hk₁, hk₂⟩ := h'
refine lt_of_le_of_lt ?_ hk₁.2
exact hittingBtwn_le_of_mem hk₁.1 (hk₁.2.le.trans hi) hk₂ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 665,
"column": 6
} | {
"line": 665,
"column": 21
} | [
{
"pp": "case pos\nΩ : Type u_1\na b : ℝ\nf : ℕ → Ω → ℝ\nN n : ℕ\nhab : a < b\nhab' : 0 < b - a\nhf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω\nhf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a\nk : ℕ\nih :\n upperCrossingTime 0 (b - a) (fun n ω ↦ (f n ω - a)⁺) N k = upperCrossingTime a b f N k ... | simp_rw [hf' ω] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 665,
"column": 6
} | {
"line": 665,
"column": 21
} | [
{
"pp": "case pos\nΩ : Type u_1\na b : ℝ\nf : ℕ → Ω → ℝ\nN n : ℕ\nhab : a < b\nhab' : 0 < b - a\nhf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω\nhf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a\nk : ℕ\nih :\n upperCrossingTime 0 (b - a) (fun n ω ↦ (f n ω - a)⁺) N k = upperCrossingTime a b f N k ... | simp_rw [hf' ω] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 665,
"column": 6
} | {
"line": 665,
"column": 21
} | [
{
"pp": "case pos\nΩ : Type u_1\na b : ℝ\nf : ℕ → Ω → ℝ\nN n : ℕ\nhab : a < b\nhab' : 0 < b - a\nhf : ∀ (ω : Ω) (i : ℕ), b - a ≤ (f i ω - a)⁺ ↔ b ≤ f i ω\nhf' : ∀ (ω : Ω) (i : ℕ), (f i ω - a)⁺ ≤ 0 ↔ f i ω ≤ a\nk : ℕ\nih :\n upperCrossingTime 0 (b - a) (fun n ω ↦ (f n ω - a)⁺) N k = upperCrossingTime a b f N k ... | simp_rw [hf' ω] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Martingale.OptionalStopping | {
"line": 106,
"column": 6
} | {
"line": 106,
"column": 50
} | [
{
"pp": "Ω : Type u_1\nm0 : MeasurableSpace Ω\nμ : Measure Ω\n𝒢 : Filtration ℕ m0\nf : ℕ → Ω → ℝ\nτ : Ω → ℕ∞\ninst✝ : SigmaFiniteFiltration μ 𝒢\nh : Submartingale f 𝒢 μ\nhτ : IsStoppingTime 𝒢 τ\n⊢ Submartingale (stoppedProcess f τ) 𝒢 μ",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNor... | submartingale_iff_expected_stoppedValue_mono | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 746,
"column": 31
} | {
"line": 746,
"column": 55
} | [
{
"pp": "case neg\nΩ : Type u_1\na b : ℝ\nf : ℕ → Ω → ℝ\nN : ℕ\nω : Ω\nhab : a < b\nhN : ¬N = 0\nh₁ : ∀ k ∈ Finset.Ico 1 (upcrossingsBefore a b f N ω + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 k = 1\nh₂ :\n ∀ k ∈ Finset.Ico (upcrossingsBefore a b f N ω + 1) (N + 1), {n | upperCrossingTime a b f ... | Finset.sum_congr rfl h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 747,
"column": 39
} | {
"line": 747,
"column": 48
} | [
{
"pp": "case neg\nΩ : Type u_1\na b : ℝ\nf : ℕ → Ω → ℝ\nN : ℕ\nω : Ω\nhab : a < b\nhN : ¬N = 0\nh₁ : ∀ k ∈ Finset.Ico 1 (upcrossingsBefore a b f N ω + 1), {n | upperCrossingTime a b f N n ω < N}.indicator 1 k = 1\nh₂ :\n ∀ k ∈ Finset.Ico (upcrossingsBefore a b f N ω + 1) (N + 1), {n | upperCrossingTime a b f ... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 786,
"column": 4
} | {
"line": 788,
"column": 23
} | [
{
"pp": "case mp\nΩ : Type u_1\na b : ℝ\nf : ℕ → Ω → ℝ\nω : Ω\n⊢ upcrossings a b f ω < ∞ → ∃ k, upcrossings a b f ω ≤ ↑k",
"usedConstants": [
"Eq.mpr",
"ENNReal.canLift",
"ENNReal.ofNNReal",
"Preorder.toLT",
"le_rfl",
"congrArg",
"PartialOrder.toPreorder",
"Pr... | · intro h
lift upcrossings a b f ω to ℝ≥0 using h.ne with r hr
exact ⟨r, le_rfl⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Martingale.Upcrossing | {
"line": 795,
"column": 9
} | {
"line": 795,
"column": 31
} | [
{
"pp": "case mp\nΩ : Type u_1\na b : ℝ\nf : ℕ → Ω → ℝ\nω : Ω\nthis : upcrossings a b f ω < ∞ ↔ ∃ k, upcrossings a b f ω ≤ ↑k\nk : ℝ≥0\nhk : ∀ (i : ℕ), ↑(upcrossingsBefore a b f i ω) ≤ ↑k\nm : ℕ\nhm : k ≤ ↑m\nN : ℕ\n⊢ ↑k ≤ ↑m",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOn... | ← ENNReal.coe_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Process.Stopping | {
"line": 206,
"column": 8
} | {
"line": 206,
"column": 30
} | [
{
"pp": "case h.refine_1.coe\nΩ : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝³ : LinearOrder ι\nf : Filtration ι m\nτ : Ω → WithTop ι\ninst✝² : TopologicalSpace ι\ninst✝¹ : OrderTopology ι\ninst✝ : FirstCountableTopology ι\nhτ : IsStoppingTime f τ\ni : ι\nh_lub : IsLUB (Set.Iio i) i\nhi_min : ¬IsMin i\... | norm_cast at hk_lt_i ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticNorm_cast___1 | Lean.Parser.Tactic.tacticNorm_cast__ |
Mathlib.Probability.Process.Stopping | {
"line": 695,
"column": 2
} | {
"line": 695,
"column": 48
} | [
{
"pp": "Ω : Type u_1\nι : Type u_3\nm : MeasurableSpace Ω\ninst✝³ : LinearOrder ι\nf : Filtration ι m\nτ π : Ω → WithTop ι\ninst✝² : TopologicalSpace ι\ninst✝¹ : SecondCountableTopology ι\ninst✝ : OrderTopology ι\nhτ : IsStoppingTime f τ\nhπ : IsStoppingTime f π\ns : Set Ω\nhs : MeasurableSet s\n⊢ MeasurableSe... | simp_rw [IsStoppingTime.measurableSet] at hs ⊢ | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Probability.Kernel.Disintegration.CondCDF | {
"line": 102,
"column": 4
} | {
"line": 102,
"column": 28
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\ninst✝ : IsFiniteMeasure ρ\ns : Set α\nhs : MeasurableSet s\nh_empty : ρ (s ×ˢ ∅) = 0\nh_neg : Tendsto (fun r ↦ ρ (s ×ˢ Iic ↑(-r))) atTop (𝓝 (ρ (⋂ r, s ×ˢ Iic ↑(-r))))\nh_inter_eq : ⋂ r, s ×ˢ Iic ↑(-r) = ⋂ r, s ×ˢ Iic ↑r\n⊢ Tendsto (fun r ↦ ρ (... | rw [h_inter_eq] at h_neg | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.Disintegration.CondCDF | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 26
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\nr : ℚ\ns : Set α\nhs : MeasurableSet s\ninst✝ : IsFiniteMeasure ρ\nthis : ∀ (r : ℚ), ∫⁻ (x : α) in s, preCDF ρ r x ∂ρ.fst = ∫⁻ (x : α) in s, (preCDF ρ r * 1) x ∂ρ.fst\n⊢ Measurable fun x ↦ 1",
"usedConstants": [
"ENNReal.measurableSpa... | exact measurable_const | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 142,
"column": 6
} | {
"line": 144,
"column": 39
} | [
{
"pp": "case e_f.h\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\nhρ_zero : ¬(ν a).restrict s = 0\nb : β\n⊢ (stieltjesOf... | congr with y : 1
simp only [mem_Iic, mem_iInter, Subtype.forall]
exact le_iff_forall_lt_rat_imp_le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Disintegration.CDFToKernel | {
"line": 142,
"column": 6
} | {
"line": 144,
"column": 39
} | [
{
"pp": "case e_f.h\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α (β × ℝ)\nν : Kernel α β\nf : α × β → ℚ → ℝ\ninst✝ : IsFiniteKernel κ\nhf : IsRatCondKernelCDF f κ ν\na : α\nx : ℝ\ns : Set β\nhs : MeasurableSet s\nhρ_zero : ¬(ν a).restrict s = 0\nb : β\n⊢ (stieltjesOf... | congr with y : 1
simp only [mem_Iic, mem_iInter, Subtype.forall]
exact le_iff_forall_lt_rat_imp_le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Disintegration.CondCDF | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 72
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nρ : Measure (α × ℝ)\nr : ℚ\ninst✝ : IsFiniteMeasure ρ\n⊢ ∫ (x : α), (preCDF ρ r x).toReal ∂ρ.fst = (ρ.IicSnd ↑r).real univ",
"usedConstants": [
"MeasureTheory.Measure.IicSnd",
"ProbabilityTheory.preCDF",
"Eq.mpr",
"InnerProductSpace.toNo... | rw [← setIntegral_univ, setIntegral_preCDF_fst ρ _ MeasurableSet.univ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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