module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 413,
"column": 21
} | {
"line": 413,
"column": 32
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nx✝ y✝ : CauSeq β abv\nf : (x✝ - y✝).LimZero\nε : α\nhε : ε > 0\n⊢ ∃ i, ∀ j ≥ i, abv (↑(y✝ - x✝) j) < ε",
"usedConstants": [
"Eq.mpr"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 414,
"column": 20
} | {
"line": 414,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nx✝ y✝ z✝ : CauSeq β abv\nfg : (x✝ - y✝).LimZero\ngh : (y✝ - z✝).LimZero\n⊢ (x✝ - z✝).LimZero",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 417,
"column": 28
} | {
"line": 417,
"column": 65
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf1 f2 g1 g2 : CauSeq β abv\nhf : f1 ≈ f2\nhg : g1 ≈ g2\n⊢ f1 + g1 ≈ f2 + g2",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 420,
"column": 2
} | {
"line": 420,
"column": 29
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nhf : f ≈ g\n⊢ -f ≈ -g",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 423,
"column": 28
} | {
"line": 423,
"column": 61
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf1 f2 g1 g2 : CauSeq β abv\nhf : f1 ≈ f2\nhg : g1 ≈ g2\n⊢ f1 - g1 ≈ f2 - g2",
"usedConstants": [
"CauSeq.addGroup",
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 433,
"column": 15
} | {
"line": 433,
"column": 26
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nh : f ≈ g\nl : f.LimZero\n⊢ g.LimZero",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 433,
"column": 70
} | {
"line": 433,
"column": 81
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nh : f ≈ g\nl : g.LimZero\n⊢ f.LimZero",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 474,
"column": 30
} | {
"line": 474,
"column": 41
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nthis : (f * g - 0).LimZero\nhlz : (f * g).LimZero\n⊢ ¬f.LimZero",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 475,
"column": 30
} | {
"line": 475,
"column": 41
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : CauSeq β abv\nhf : ¬f ≈ 0\nhg : ¬g ≈ 0\nthis : (f * g - 0).LimZero\nhlz : (f * g).LimZero\nhf' : ¬f.LimZero\n⊢ ¬g.LimZero",
"usedCon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 671,
"column": 62
} | {
"line": 671,
"column": 78
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nJ I : Ideal R\nhIJ : J * I ≤ J\nhJ : ¬J = 0\nhI : ¬I = 0\ns : Finset (HeightOneSpectrum R) := ⋯.toFinset\nthis✝ : ∀ p ∈ s, J * ∏ q ∈ s, q.asIdeal < J * ∏ q ∈ s \\ {p}, q.asIdeal\na : HeightOneSpectrum R → R\nha : ∀ p ∈ s, a p ∈ J * ∏ q ∈ s ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 494,
"column": 2
} | {
"line": 495,
"column": 9
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf1 f2 g1 g2 : CauSeq β abv\nhf : f1 ≈ f2\nhg : g1 ≈ g2\n⊢ f1 * g1 ≈ f2 * g2",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 499,
"column": 2
} | {
"line": 499,
"column": 44
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁶ : Field α\ninst✝⁵ : LinearOrder α\ninst✝⁴ : IsStrictOrderedRing α\ninst✝³ : Ring β\nabv : β → α\ninst✝² : IsAbsoluteValue abv\nG : Type u_3\ninst✝¹ : SMul G β\ninst✝ : IsScalarTower G β β\nf1 f2 : CauSeq β abv\nc : G\nhf : f1 ≈ f2\n⊢ c • f1 ≈ c • f2",
"usedConstan... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 505,
"column": 17
} | {
"line": 505,
"column": 45
} | [
{
"pp": "case succ\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf1 f2 : CauSeq β abv\nhf : f1 ≈ f2\nn : ℕ\nih : f1 ^ n ≈ f2 ^ n\n⊢ f1 ^ (n + 1) ≈ f2 ^ (n + 1)",
"usedConstants": [
"Eq.m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 517,
"column": 68
} | {
"line": 517,
"column": 79
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁵ : Field α\ninst✝⁴ : LinearOrder α\ninst✝³ : IsStrictOrderedRing α\ninst✝² : Ring β\ninst✝¹ : IsDomain β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nh : const abv 1 ≈ const abv 0\nthis : ∀ ε > 0, ∃ i, ∀ (k : ℕ), i ≤ k → abv (1 - 0) < ε\nh2 : 0 < abv 1\ni : ℕ\nhi : ∀ (k ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 554,
"column": 21
} | {
"line": 554,
"column": 93
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : CauSeq β abv\nhf : ¬f.LimZero\nε : α\nε0 : ε > 0\nK : α\nK0 : K > 0\ni : ℕ\nH : ∀ j ≥ i, K ≤ abv (↑f j)\nj : ℕ\nij : j ≥ i\n⊢ abv ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 558,
"column": 21
} | {
"line": 558,
"column": 93
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf : CauSeq β abv\nhf : ¬f.LimZero\nε : α\nε0 : ε > 0\nK : α\nK0 : K > 0\ni : ℕ\nH : ∀ j ≥ i, K ≤ abv (↑f j)\nj : ℕ\nij : j ≥ i\n⊢ abv ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 561,
"column": 47
} | {
"line": 561,
"column": 60
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : DivisionRing β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nx : β\nhx : x ≠ 0\n⊢ ¬(const abv x).LimZero",
"usedConstants": [
"Eq.mpr",
"congrArg",
"CauSeq.const_limZero",
... | const_limZero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 688,
"column": 4
} | {
"line": 688,
"column": 75
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\na b c : FractionalIdeal R⁰ K\nhac : a ≤ c\nha : a ≠ 0\nhb : b ≠ 0\nthis :\n ∀ {R : Type u_1} [inst : CommRing R] {K : Type u_2} [inst_1 : Field K] [... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 815,
"column": 2
} | {
"line": 815,
"column": 36
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b : CauSeq α abs\nh : b ≤ a\n⊢ a ⊔ b ≈ a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.CauSeq.Basic | {
"line": 818,
"column": 2
} | {
"line": 818,
"column": 36
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b : CauSeq α abs\nh : a ≤ b\n⊢ a ⊓ b ≈ a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 697,
"column": 8
} | {
"line": 697,
"column": 19
} | [
{
"pp": "R✝ : Type u_1\ninst✝⁸ : CommRing R✝\nK✝ : Type u_2\ninst✝⁷ : Field K✝\ninst✝⁶ : Algebra R✝ K✝\ninst✝⁵ : IsFractionRing R✝ K✝\nR : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\na c : FractionalIdeal R⁰ K\nhac... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Basic | {
"line": 339,
"column": 4
} | {
"line": 339,
"column": 15
} | [
{
"pp": "case h.h.h\nx : ℝ\ny✝² y✝¹ y✝ : CauSeq ℚ abs\n⊢ mk y✝² ≤ mk y✝¹ → mk y✝¹ ≤ mk y✝ → mk y✝² ≤ mk y✝",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"abs",
"Rat",
"Rat.linearOrder",
"id",
"CauSeq.instLEAbs",
"Rat.instLattice",
"Rat.inst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Basic | {
"line": 331,
"column": 4
} | {
"line": 331,
"column": 15
} | [
{
"pp": "case h.h\nx : ℝ\ny✝¹ y✝ : CauSeq ℚ abs\n⊢ mk y✝¹ < mk y✝ ↔ mk y✝¹ ≤ mk y✝ ∧ ¬mk y✝ ≤ mk y✝¹",
"usedConstants": [
"Eq.mpr",
"CauSeq.instLTAbs",
"Real.instLE",
"Real",
"abs",
"congrArg",
"Rat",
"Rat.linearOrder",
"Real.instLT",
"id",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Basic | {
"line": 343,
"column": 4
} | {
"line": 343,
"column": 23
} | [
{
"pp": "case h.h\nx : ℝ\ny✝¹ y✝ : CauSeq ℚ abs\n⊢ mk y✝¹ ≤ mk y✝ → mk y✝ ≤ mk y✝¹ → mk y✝¹ = mk y✝",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"abs",
"IsAbsoluteValue.abs_isAbsoluteValue",
"Rat",
"Rat.linearOrder",
"id",
"CauSeq.instLEAbs",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Basic | {
"line": 377,
"column": 4
} | {
"line": 377,
"column": 46
} | [
{
"pp": "case h.h\nx : ℝ\ny✝¹ y✝ : CauSeq ℚ abs\n⊢ 0 < mk y✝¹ → 0 < mk y✝ → 0 < mk y✝¹ * mk y✝",
"usedConstants": [
"CauSeq.Pos",
"Eq.mpr",
"Real.partialOrder",
"Real",
"Preorder.toLT",
"HMul.hMul",
"Real.instZero",
"abs",
"congrArg",
"IsAbsoluteVa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Basic | {
"line": 486,
"column": 4
} | {
"line": 486,
"column": 15
} | [
{
"pp": "case h.h\nx : ℝ\ny✝¹ y✝ : CauSeq ℚ abs\n⊢ mk y✝¹ ≤ mk y✝ ∨ mk y✝ ≤ mk y✝¹",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"abs",
"congrArg",
"Rat",
"Rat.linearOrder",
"id",
"CauSeq.instLEAbs",
"Rat.instLattice",
"Rat.instDivisi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Basic | {
"line": 514,
"column": 74
} | {
"line": 514,
"column": 90
} | [
{
"pp": "x : ℝ\nq : ℚ\n⊢ { cauchy := ↑q.num } / ↑q.den = ↑q.num / ↑q.den",
"usedConstants": [
"Semiring.toNatCast",
"Int.cast",
"Eq.mpr",
"Real",
"Rat.num",
"instHDiv",
"abs",
"congrArg",
"Real.instDivInvMonoid",
"IsAbsoluteValue.abs_isAbsoluteValu... | ofCauchy_intCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Real.Basic | {
"line": 569,
"column": 58
} | {
"line": 570,
"column": 48
} | [
{
"pp": "b : ℕ\nhb : ∀ {a : ℝ}, 0 < a → a * ↑b + 1 ≤ (a + 1) ^ b\na : ℝ\nha' : 0 < a\n⊢ a * ↑(b + 1) + 1 = (0 + 1) ^ b * a + (a * ↑b + 1)",
"usedConstants": [
"one_pow",
"Distrib.leftDistribClass",
"MulOne.toOne",
"Real",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"ad... | by
simp [mul_add, add_assoc, add_left_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Real.Archimedean | {
"line": 73,
"column": 15
} | {
"line": 73,
"column": 26
} | [
{
"pp": "s : Set ℝ\nL : ℝ\nhL : L ∈ s\nU : ℝ\nhU : U ∈ upperBounds s\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y ∈ s, ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y ∈ s, ↑m ≤ y * ↑d} ∧ ∀ z ∈ {m | ∃ y ∈ s, ↑m ≤ y * ↑d}, z ≤ f d\nn : ℕ\nn0 : n > 0\ny : ℝ\nyS : y ∈ s\nhy : ↑(f n) ≤ y * ↑n\n⊢ ↑(↑(f n) / ↑n) ≤ y",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Archimedean | {
"line": 93,
"column": 4
} | {
"line": 93,
"column": 15
} | [
{
"pp": "s : Set ℝ\nL : ℝ\nhL : L ∈ s\nU : ℝ\nhU : U ∈ upperBounds s\nthis : ∀ (d : ℕ), BddAbove {m | ∃ y ∈ s, ↑m ≤ y * ↑d}\nf : ℕ → ℤ\nhf : ∀ (d : ℕ), f d ∈ {m | ∃ y ∈ s, ↑m ≤ y * ↑d} ∧ ∀ z ∈ {m | ∃ y ∈ s, ↑m ≤ y * ↑d}, z ≤ f d\nhf₁ : ∀ n > 0, ∃ y ∈ s, ↑(↑(f n) / ↑n) ≤ y\nhf₂ : ∀ n > 0, ∀ y ∈ s, y - (↑n)⁻¹ < ↑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 760,
"column": 18
} | {
"line": 760,
"column": 29
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI J J' : FractionalIdeal R⁰ K\nh : J ≤ I\nhJ' : J' ≠ 0\nhI : I ≠ 0\nH : I * J' = 0 * J\nh' : J' ≤ 0\nthis : (J' ⊓ spanSingleton R⁰ (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Archimedean | {
"line": 374,
"column": 2
} | {
"line": 374,
"column": 13
} | [
{
"pp": "case h\nx✝ : ℝ\n⊢ x✝ ∈ {x | ∀ ⦃a : ℝ⦄, (a ∈ range fun x ↦ ↑x) → a ≤ x} ↔ x✝ ∈ ∅",
"usedConstants": [
"Eq.mpr",
"False",
"Real.instLE",
"Real",
"Preorder.toLT",
"iff_false",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"Real.instRatCast",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Archimedean | {
"line": 380,
"column": 2
} | {
"line": 380,
"column": 13
} | [
{
"pp": "case h\nx✝ : ℝ\n⊢ x✝ ∈ {x | ∀ ⦃a : ℝ⦄, (a ∈ range fun x ↦ ↑x) → x ≤ a} ↔ x✝ ∈ ∅",
"usedConstants": [
"Eq.mpr",
"False",
"Real.instLE",
"Real",
"Preorder.toLT",
"iff_false",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"Real.instRatCast",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 779,
"column": 4
} | {
"line": 779,
"column": 36
} | [
{
"pp": "case refine_3\nR : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI J I' J' : FractionalIdeal R⁰ K\nH : I * J' = I' * J\nh : J ≤ I\nh' : J' ≤ I'\nhJ' : J' ≠ 0\nhI : I ≠ 0\nthis : J' ⊓... | by_cases H' : I'.divMod I J' = 0 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 780,
"column": 33
} | {
"line": 780,
"column": 49
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI J I' J' : FractionalIdeal R⁰ K\nH : I * J' = I' * J\nh : J ≤ I\nh' : J' ≤ I'\nhJ' : J' ≠ 0\nhI : I ≠ 0\nthis : J' ⊓ spanSingleton ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 783,
"column": 37
} | {
"line": 783,
"column": 76
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI J I' J' : FractionalIdeal R⁰ K\nH : I * J' = I' * J\nh : J ≤ I\nh' : J' ≤ I'\nhJ' : J' ≠ 0\nhI : I ≠ 0\nthis : J' ⊓ spanSingleton ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 784,
"column": 44
} | {
"line": 784,
"column": 83
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI J I' J' : FractionalIdeal R⁰ K\nH : I * J' = I' * J\nh : J ≤ I\nh' : J' ≤ I'\nhJ' : J' ≠ 0\nhI : I ≠ 0\nthis :\n (spanSingleton R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Real | {
"line": 265,
"column": 2
} | {
"line": 265,
"column": 46
} | [
{
"pp": "a : ℝ\nb : ℝ≥0\n⊢ ENNReal.ofReal a ≤ ↑b ↔ a ≤ (↑b).toReal",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"ENNReal.ofNNReal",
"ENNReal.ofReal",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"NNReal",
"LE.le",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Real | {
"line": 270,
"column": 2
} | {
"line": 270,
"column": 46
} | [
{
"pp": "a : ℝ\nha : 0 ≤ a\nb : ℝ≥0\n⊢ ENNReal.ofReal a < ↑b ↔ a < (↑b).toReal",
"usedConstants": [
"Eq.mpr",
"Real",
"ENNReal.ofNNReal",
"Preorder.toLT",
"ENNReal.ofReal",
"congrArg",
"PartialOrder.toPreorder",
"Real.instLT",
"id",
"NNReal",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Real | {
"line": 281,
"column": 2
} | {
"line": 281,
"column": 46
} | [
{
"pp": "b : ℝ\nhb : 0 ≤ b\na : ℝ≥0\n⊢ ↑a ≤ ENNReal.ofReal b ↔ (↑a).toReal ≤ b",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"ENNReal.ofNNReal",
"ENNReal.ofReal",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"NNReal",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Real | {
"line": 291,
"column": 2
} | {
"line": 291,
"column": 46
} | [
{
"pp": "b : ℝ\na : ℝ≥0\n⊢ ↑a < ENNReal.ofReal b ↔ (↑a).toReal < b",
"usedConstants": [
"Eq.mpr",
"Real",
"ENNReal.ofNNReal",
"Preorder.toLT",
"ENNReal.ofReal",
"congrArg",
"PartialOrder.toPreorder",
"Real.instLT",
"id",
"NNReal",
"ENNReal.co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 838,
"column": 4
} | {
"line": 839,
"column": 63
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁷ : CommRing R\ninst✝⁶ : IsDedekindDomain R\nS : Type u_3\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra S R\ninst✝³ : Algebra.IsIntegral S R\ninst✝² : IsDomain S\ninst✝¹ : Module.IsTorsionFree S R\np : Ideal S\ninst✝ : p.IsMaximal\nhp : p ≠ 0\nh : map (algebraMap S R) p ≠ 0\nhF : Fi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Real | {
"line": 358,
"column": 2
} | {
"line": 358,
"column": 40
} | [
{
"pp": "p : ℝ≥0∞\n⊢ p = 0 ∨ p = ∞ ∨ 0 < p.toReal",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"Real.instLT",
"id",
"_private.Mathlib.Data.ENNReal.Real.0.ENNReal.trichotomy._simp_1_1",
"ENNReal.toReal",
"LT.lt",
"ENNReal",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Real | {
"line": 367,
"column": 4
} | {
"line": 367,
"column": 15
} | [
{
"pp": "case inl\nq : ℝ≥0∞\nhpq : 0 ≤ q\n⊢ 0 = 0 ∧ q = 0 ∨\n 0 = 0 ∧ q = ∞ ∨\n 0 = 0 ∧ 0 < q.toReal ∨\n 0 = ∞ ∧ q = ∞ ∨ 0 < ENNReal.toReal 0 ∧ q = ∞ ∨ 0 < ENNReal.toReal 0 ∧ 0 < q.toReal ∧ ENNReal.toReal 0 ≤ q.toReal",
"usedConstants": [
"Eq.mpr",
"False",
"Real.instLE",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 910,
"column": 2
} | {
"line": 911,
"column": 9
} | [
{
"pp": "R : Type u_3\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nI : Ideal R\nhI : I ≠ ⊥\n⊢ ∏ᶠ (p : HeightOneSpectrum R), p.asIdeal ^ multiplicity p.asIdeal I = I",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Real | {
"line": 369,
"column": 4
} | {
"line": 369,
"column": 15
} | [
{
"pp": "case inr.inl\np : ℝ≥0∞\nhp : 0 < p\nhpq : p ≤ ∞\n⊢ p = 0 ∧ ∞ = 0 ∨\n p = 0 ∧ ∞ = ∞ ∨\n p = 0 ∧ 0 < ∞.toReal ∨ p = ∞ ∧ ∞ = ∞ ∨ 0 < p.toReal ∧ ∞ = ∞ ∨ 0 < p.toReal ∧ 0 < ∞.toReal ∧ p.toReal ≤ ∞.toReal",
"usedConstants": [
"Eq.mpr",
"False",
"Real.instLE",
"Real",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Real | {
"line": 376,
"column": 4
} | {
"line": 376,
"column": 15
} | [
{
"pp": "p : ℝ≥0∞\ninst✝ : Fact (1 ≤ p)\n⊢ p = ∞ ∨ 0 < p.toReal ∧ 1 ≤ p.toReal",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Basic | {
"line": 561,
"column": 2
} | {
"line": 561,
"column": 13
} | [
{
"pp": "b a : ℝ≥0\nh : ↑a ≤ ↑b\n⊢ (↑a).toReal ≤ ↑b",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"ENNReal.ofNNReal",
"PartialOrder.toPreorder",
"Preorder.toLE",
"NNReal.coe_le_coe._simp_1",
"id",
"NNReal",
"LE.le",
"NNReal.instPartia... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.NNReal.Defs | {
"line": 547,
"column": 2
} | {
"line": 547,
"column": 29
} | [
{
"pp": "r : ℝ\n⊢ r.toNNReal = 0 ↔ r ≤ 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Basic | {
"line": 666,
"column": 2
} | {
"line": 666,
"column": 13
} | [
{
"pp": "b a : ℝ≥0\nh : ↑a = ∞ → ↑b = ∞\nh_nnreal : ↑a ≠ ∞ → ↑b ≠ ∞ → (↑a).toNNReal ≤ (↑b).toNNReal\nhlt : ↑b < ↑a\n⊢ ↑a ≤ ↑b",
"usedConstants": [
"Eq.mpr",
"ENNReal.ofNNReal",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"NNReal",
"LE.le",
"NNReal.instPa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.NNReal.Defs | {
"line": 574,
"column": 2
} | {
"line": 574,
"column": 13
} | [
{
"pp": "r : ℝ\n⊢ r.toNNReal ≤ 1 ↔ r ≤ 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.NNReal.Defs | {
"line": 578,
"column": 2
} | {
"line": 578,
"column": 27
} | [
{
"pp": "r : ℝ\n⊢ 1 < r.toNNReal ↔ 1 < r",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.NNReal.Defs | {
"line": 582,
"column": 2
} | {
"line": 582,
"column": 13
} | [
{
"pp": "r : ℝ\nn : ℕ\n⊢ r.toNNReal ≤ ↑n ↔ r ≤ ↑n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Basic | {
"line": 726,
"column": 2
} | {
"line": 726,
"column": 32
} | [
{
"pp": "s : Set ℝ\nh : s.OrdConnected\n⊢ (ENNReal.ofReal '' s).OrdConnected",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.NNReal.Defs | {
"line": 586,
"column": 2
} | {
"line": 586,
"column": 27
} | [
{
"pp": "r : ℝ\nn : ℕ\n⊢ ↑n < r.toNNReal ↔ ↑n < r",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.NNReal.Defs | {
"line": 625,
"column": 2
} | {
"line": 625,
"column": 13
} | [
{
"pp": "r : ℝ\n⊢ 1 ≤ r.toNNReal ↔ 1 ≤ r",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.NNReal.Defs | {
"line": 632,
"column": 2
} | {
"line": 632,
"column": 40
} | [
{
"pp": "n : ℕ\nr : ℝ\n⊢ ↑n ≤ r.toNNReal ↔ ↑n ≤ r ∨ n = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.NNReal.Defs | {
"line": 636,
"column": 2
} | {
"line": 636,
"column": 31
} | [
{
"pp": "n : ℕ\nr : ℝ\n⊢ r.toNNReal < ↑n ↔ r < ↑n ∧ n ≠ 0",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"Preorder.toLT",
"PartialOrder.toPreorder",
"Real.instLT",
"id",
"AddMonoidWithOne.toNatCast",
"NNReal",
"Ne",
"instOf... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.NNReal.Defs | {
"line": 716,
"column": 2
} | {
"line": 716,
"column": 49
} | [
{
"pp": "a b : ℝ≥0\nha : 0 < a\nhb : b < 1\n⊢ ∃ n, b ^ n < a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.NNReal.Defs | {
"line": 810,
"column": 6
} | {
"line": 810,
"column": 16
} | [
{
"pp": "x : ℝ≥0\nhx : x ≠ 0\n⊢ x⁻¹ < 1 ↔ 1 < x",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"DivInvMonoid.toInv",
"Preorder.toLT",
"instHDiv",
"GroupWithZero.toDivInvMonoid",
"Monoid.toMulOneClass",
"congrArg",
"PartialOrder.toPreorder",
"Divisi... | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.NNReal.Defs | {
"line": 927,
"column": 42
} | {
"line": 927,
"column": 53
} | [
{
"pp": "r : ℝ\n⊢ 0 ≤ r ∨ 0 ≤ -r",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Real.instLE",
"Real",
"Real.instZero",
"congrArg",
"id",
"Real.instAddGroup",
"SubtractionMonoid.toSubNegZeroMonoid",
"LE.le",
"SubNegZeroMonoid.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Action | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 60
} | [
{
"pp": "r : ℝ≥0\ns : ℝ≥0∞\n⊢ (r • s).toReal = r • s.toReal",
"usedConstants": [
"Eq.mpr",
"Real",
"ENNReal.ofNNReal",
"instHSMul",
"instSMulOfMul",
"HMul.hMul",
"ENNReal.smul_def",
"NNReal.instSMulOfReal",
"congrArg",
"CommSemiring.toSemiring",
... | rw [ENNReal.smul_def, smul_eq_mul, toReal_mul, coe_toReal] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.ENNReal.Operations | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 13
} | [
{
"pp": "a b : ℝ≥0∞\nha₀ : a ≠ 0\nha : a ≠ ∞\n⊢ a * b = a ↔ b = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Operations | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 13
} | [
{
"pp": "a b : ℝ≥0∞\nhb₀ : b ≠ 0\nhb : b ≠ ∞\n⊢ a * b = b ↔ a = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Operations | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 36
} | [
{
"pp": "a : ℝ≥0∞\n⊢ a ≠ 0 → ∀ (n : ℕ), a ^ n ≠ 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Operations | {
"line": 165,
"column": 2
} | {
"line": 165,
"column": 39
} | [
{
"pp": "a b c : ℝ≥0∞\nhle : a ≤ b + c\nhb : b = ∞ → a = ∞\nhc : c = ∞ → a = ∞\n⊢ b + c = ∞ → a = ∞",
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"id",
"instHAdd",
"And",
"HAdd.hAdd",
"ENNReal.add_eq_top._simp_1",
"implies_congr",
"ENNReal",
"_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Operations | {
"line": 176,
"column": 53
} | {
"line": 176,
"column": 89
} | [
{
"pp": "a b : ℝ≥0∞\n⊢ a + b ≠ ∞ ↔ a ≠ ∞ ∧ b ≠ ∞",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Operations | {
"line": 346,
"column": 2
} | {
"line": 346,
"column": 32
} | [
{
"pp": "case neg\na b c : ℝ≥0∞\nhab : b ≤ a\nb_ne_top : b ≠ ∞\nc_top : ¬c = ∞\n⊢ a - b + (b + c) = a + c",
"usedConstants": [
"Eq.mpr",
"ENNReal.instAdd",
"AddMonoid.toAddSemigroup",
"ENNReal.instAddCommMonoid",
"congrArg",
"HSub.hSub",
"id",
"AddCommMonoidWi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Operations | {
"line": 495,
"column": 76
} | {
"line": 495,
"column": 90
} | [
{
"pp": "x y : ℝ≥0\n⊢ ofNNReal '' uIoc x y = uIoc ↑x ↑y",
"usedConstants": [
"Set.Ioc",
"ENNReal.ofNNReal",
"Lattice.toSemilatticeSup",
"congrArg",
"PartialOrder.toPreorder",
"Set.uIoc",
"SemilatticeInf.toPartialOrder",
"SemilatticeSup.toMax",
"DistribLa... | by simp [uIoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.ENNReal.Operations | {
"line": 511,
"column": 4
} | {
"line": 511,
"column": 38
} | [
{
"pp": "case inr\nι : Sort u_1\nh✝ : Nonempty ι\nf : ι → ℝ≥0\n⊢ (⨅ i, ↑(f i)).toNNReal = ⨅ i, ((fun i ↦ ↑(f i)) i).toNNReal",
"usedConstants": [
"Eq.mpr",
"ENNReal.ofNNReal",
"iInf",
"congrArg",
"id",
"_private.Mathlib.Data.ENNReal.Operations.0.ENNReal.toNNReal_iInf._sim... | simp_rw [← coe_iInf, toNNReal_coe] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Data.ENNReal.Operations | {
"line": 516,
"column": 2
} | {
"line": 516,
"column": 78
} | [
{
"pp": "s : Set ℝ≥0∞\nhs : ∀ r ∈ s, r ≠ ∞\nhf : ∀ (i : { x // x ∈ s }), ↑i ≠ ∞\n⊢ (sInf s).toNNReal = sInf (ENNReal.toNNReal '' s)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Operations | {
"line": 597,
"column": 2
} | {
"line": 597,
"column": 13
} | [
{
"pp": "x y z : ℝ≥0∞\nh : ∀ y' > y, ∀ z' > z, x ≤ y' + z'\n⊢ x ≤ y + z",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Operations | {
"line": 615,
"column": 2
} | {
"line": 615,
"column": 78
} | [
{
"pp": "s : Set ℝ≥0∞\nhs : ∀ r ∈ s, r ≠ ∞\nhf : ∀ (i : { x // x ∈ s }), ↑i ≠ ∞\n⊢ (sSup s).toNNReal = sSup (ENNReal.toNNReal '' s)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.T5 | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 68
} | [
{
"pp": "X : Type u_1\ninst✝² : LinearOrder X\ninst✝¹ : TopologicalSpace X\ninst✝ : OrderTopology X\na : X\ns t : Set X\nhd : Disjoint s (closure[inst✝¹] t)\nha : a ∈ s\nhd' : Disjoint (⇑ofDual ⁻¹' s) (closure[instTopologicalSpace] (⇑ofDual ⁻¹' t))\nha' : toDual a ∈ ⇑ofDual ⁻¹' s\n⊢ (s.ordSeparatingSet t).ordCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Operations | {
"line": 712,
"column": 2
} | {
"line": 712,
"column": 13
} | [
{
"pp": "x y z : ℝ≥0∞\nhy : y ≠ 0\nhz : z ≠ 0\nh : ∀ y' < y, ∀ z' < z, y' + z' ≤ x\n⊢ y + z ≤ x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Basic | {
"line": 367,
"column": 2
} | {
"line": 367,
"column": 13
} | [
{
"pp": "x : ℝ\nhx : 0 < ↑x\nh'x : ↑x ≠ ⊤\n⊢ 0 < (↑x).toReal",
"usedConstants": [
"Real",
"Real.instZero",
"Real.instLT",
"id",
"LT.lt",
"EReal.toReal",
"Zero.toOfNat0",
"OfNat.ofNat",
"Real.toEReal"
]
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Basic | {
"line": 371,
"column": 2
} | {
"line": 371,
"column": 13
} | [
{
"pp": "x : ℝ\nhx : ↑x < 0\nh'x : ↑x ≠ ⊥\n⊢ (↑x).toReal < 0",
"usedConstants": [
"Real",
"Real.instZero",
"Real.instLT",
"id",
"LT.lt",
"EReal.toReal",
"Zero.toOfNat0",
"OfNat.ofNat",
"Real.toEReal"
]
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Basic | {
"line": 378,
"column": 4
} | {
"line": 378,
"column": 15
} | [
{
"pp": "case h.mp\ny : ℝ\nhy0 : 0 < ↑y\nright✝ : ↑y < ⊤\n⊢ (↑y).toReal ∈ Ioi 0",
"usedConstants": [
"Eq.mpr",
"Real",
"Set.Ioi",
"Preorder.toLT",
"Real.instZero",
"Membership.mem",
"Set.mem_Ioi._simp_1",
"id",
"LT.lt",
"EReal.toReal",
"Zero.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Basic | {
"line": 381,
"column": 4
} | {
"line": 381,
"column": 15
} | [
{
"pp": "case h\nx : ℝ\nhx : x ∈ Ioi 0\n⊢ ↑x ∈ Ioo 0 ⊤ ∧ (↑x).toReal = x",
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"and_true",
"Real.instZero",
"congrArg",
"PartialOrder.toPreorder",
"EReal",
"Real.instLT",
"Membership.mem",
"in... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Basic | {
"line": 388,
"column": 4
} | {
"line": 388,
"column": 15
} | [
{
"pp": "case h.mp\ny : ℝ\nleft✝ : ⊥ < ↑y\nhy0 : ↑y < 0\n⊢ (↑y).toReal ∈ Iio 0",
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"Real.instZero",
"Membership.mem",
"id",
"LT.lt",
"EReal.toReal",
"Zero.toOfNat0",
"Set.mem_Iio._simp_2",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Basic | {
"line": 391,
"column": 4
} | {
"line": 391,
"column": 15
} | [
{
"pp": "case h\nx : ℝ\nhx : x ∈ Iio 0\n⊢ ↑x ∈ Ioo ⊥ 0 ∧ (↑x).toReal = x",
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"and_true",
"Real.instZero",
"congrArg",
"PartialOrder.toPreorder",
"EReal",
"Real.instLT",
"Membership.mem",
"id... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Basic | {
"line": 397,
"column": 2
} | {
"line": 397,
"column": 13
} | [
{
"pp": "x : ℝ\nhx : ↑x ≠ ⊥\ny : ℝ\nhy : ↑y ≠ ⊤\nh : ↑x ≤ ↑y\n⊢ (↑x).toReal ≤ (↑y).toReal",
"usedConstants": [
"Real.instLE",
"Real",
"id",
"LE.le",
"EReal.toReal",
"Real.toEReal"
]
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Basic | {
"line": 817,
"column": 11
} | {
"line": 817,
"column": 22
} | [
{
"pp": "a : ℝ\nx✝ : ↑a < ⊤\nb : ℚ\nhab : a < ↑b\n⊢ ↑a < ↑↑b",
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"Real.instRatCast",
"PartialOrder.toPreorder",
"EReal",
"Real.instLT",
"EReal.coe_lt_coe_iff._simp_1",
"id",
"Rat.cast",
"LT.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Basic | {
"line": 821,
"column": 25
} | {
"line": 821,
"column": 36
} | [
{
"pp": "a : ℝ\nx✝ : ⊥ < ↑a\nb : ℚ\nhab : ↑b < a\n⊢ ↑↑b < ↑a",
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"Real.instRatCast",
"PartialOrder.toPreorder",
"EReal",
"Real.instLT",
"EReal.coe_lt_coe_iff._simp_1",
"id",
"Rat.cast",
"LT.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.EReal.Basic | {
"line": 840,
"column": 6
} | {
"line": 840,
"column": 36
} | [
{
"pp": "x✝ : ↑{⊥, ⊤}ᶜ\nx : EReal\nhx : x ∈ {⊥, ⊤}ᶜ\n⊢ x ≠ ⊤ ∧ x ≠ ⊥",
"usedConstants": [
"Eq.mpr",
"EReal",
"_private.Mathlib.Data.EReal.Basic.0.EReal.neTopBotEquivReal._simp_5",
"instTopEReal",
"id",
"Ne",
"Bot.bot",
"And",
"Top.top",
"Eq",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.NNReal.Basic | {
"line": 53,
"column": 2
} | {
"line": 53,
"column": 13
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\na : α\nb : ℝ≥0\nc : α\n⊢ ↑(Pi.mulSingle a b c) = Pi.mulSingle a (↑b) c",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.NNReal.Basic | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 13
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\na : α\nb : ℝ≥0\nc : α\n⊢ ↑(Pi.single a b c) = Pi.single a (↑b) c",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Module.Field | {
"line": 30,
"column": 4
} | {
"line": 30,
"column": 24
} | [
{
"pp": "𝕜 : Type u_1\nG : Type u_2\ninst✝⁶ : Semifield 𝕜\ninst✝⁵ : LinearOrder 𝕜\ninst✝⁴ : IsStrictOrderedRing 𝕜\ninst✝³ : AddCommGroup G\ninst✝² : PartialOrder G\ninst✝¹ : MulAction 𝕜 G\ninst✝ : PosSMulMono 𝕜 G\n_a : 𝕜\nha : 0 < _a\nb₁ b₂ : G\nh : _a • b₁ ≤ _a • b₂\n⊢ b₁ ≤ b₂",
"usedConstants": []
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Module.Field | {
"line": 35,
"column": 32
} | {
"line": 35,
"column": 52
} | [
{
"pp": "𝕜 : Type u_1\nG : Type u_2\ninst✝⁶ : Semifield 𝕜\ninst✝⁵ : LinearOrder 𝕜\ninst✝⁴ : IsStrictOrderedRing 𝕜\ninst✝³ : AddCommGroup G\ninst✝² : PartialOrder G\ninst✝¹ : MulActionWithZero 𝕜 G\ninst✝ : PosSMulStrictMono 𝕜 G\na : 𝕜\nha : 0 < a\nb₁ b₂ : G\nh : a • b₁ < a • b₂\n⊢ b₁ < b₂",
"usedConst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.NNReal.Basic | {
"line": 162,
"column": 2
} | {
"line": 162,
"column": 29
} | [
{
"pp": "ι : Sort u_2\nf : ι → ℝ≥0\na : ℝ≥0\n⊢ a * iInf f = ⨅ i, a * f i",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.MonotoneConvergence | {
"line": 283,
"column": 4
} | {
"line": 283,
"column": 31
} | [
{
"pp": "case left\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : Preorder α\ninst✝³ : OrderClosedTopology α\ninst✝² : Preorder β\ninst✝¹ : IsDirectedOrder β\ninst✝ : Nonempty β\nf : β → α\na : α\nhf : Monotone f\nha : Tendsto f atTop (𝓝 a)\nb : β\n⊢ f b ≤ a",
"usedConstants": [
"... | exact hf.ge_of_tendsto ha b | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.ENNReal.Inv | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 44
} | [
{
"pp": "r p : ℝ≥0\n⊢ ↑(p / r) ≤ ↑p / ↑r",
"usedConstants": [
"Eq.mpr",
"ENNReal.ofNNReal",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"GroupWithZero.toDivInvMonoid",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiring",
"DivisionSemir... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 90,
"column": 18
} | {
"line": 90,
"column": 65
} | [
{
"pp": "a b c d : ℝ≥0∞\nr p q : ℝ≥0\n⊢ 1⁻¹ = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Monotone | {
"line": 93,
"column": 2
} | {
"line": 93,
"column": 13
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace α\ninst✝² : OrderTopology α\ninst✝¹ : LinearOrder β\nf : α → β\ninst✝ : SecondCountableTopology α\nhf : MonotoneOn f univ\n⊢ {c | ∃ x y, x < y ∧ f x = c ∧ f y = c}.Countable",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.Monotone | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 39
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁶ : LinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : LinearOrder β\nf : α → β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : SecondCountableTopology β\nhf : Monotone f\n⊢ {x | ¬ContinuousAt f x}.Countable",
"usedConstan... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Order.IsLUB | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 49
} | [
{
"pp": "γ : Type u_2\nα : Type u_3\ninst✝³ : TopologicalSpace α\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : ClosedIicTopology α\nf : γ → α\ninst✝ : TopologicalSpace γ\nS : Set γ\nhS : Dense S\nhf : Continuous[inst✝, inst✝³] f\n⊢ ⨆ s, f ↑s = ⨆ i, f i",
"usedConstants": [
"iSup",
"Part... | by_cases h : BddAbove (range (fun x : S ↦ f x)) | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Topology.Order.IsLUB | {
"line": 219,
"column": 4
} | {
"line": 219,
"column": 49
} | [
{
"pp": "case pos\nγ : Type u_2\nα : Type u_3\ninst✝³ : TopologicalSpace α\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : ClosedIicTopology α\nf : γ → α\ninst✝ : TopologicalSpace γ\nS : Set γ\nhS : Dense S\nhf : Continuous[inst✝, inst✝³] f\nh : BddAbove (range fun x ↦ f ↑x)\n⊢ range f ⊆ closure[inst✝³] ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 271,
"column": 28
} | {
"line": 271,
"column": 39
} | [
{
"pp": "a b c : ℝ≥0∞\nh : 0 < b → b < a → c ≠ 0\n⊢ 0 < b → b < a → c⁻¹ ≠ ∞",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder",
"ENNReal.inv_eq_top._simp_1",
"id",
"Ne",
"Inv.inv",
"implies_c... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 291,
"column": 2
} | {
"line": 291,
"column": 28
} | [
{
"pp": "a b : ℝ≥0∞\n⊢ a⁻¹ < b ↔ b⁻¹ < a",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.ENNReal.Inv | {
"line": 294,
"column": 2
} | {
"line": 294,
"column": 28
} | [
{
"pp": "a b : ℝ≥0∞\n⊢ a < b⁻¹ ↔ b < a⁻¹",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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