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.NumberTheory.Real.GoldenRatio | {
"line": 225,
"column": 2
} | {
"line": 233,
"column": 51
} | [
{
"pp": "n : ℕ\n⊢ φ * ↑(Nat.fib (n + 1)) + ↑(Nat.fib n) = φ ^ (n + 1)",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"CharP.cast_eq_zero",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.Ring.Common.div_congr",
"Mathlib.Meta.NormNum.is... | induction n with
| zero => simp
| succ n ih =>
calc
_ = φ * (Nat.fib n) + φ ^ 2 * (Nat.fib (n + 1)) := by
simp only [Nat.fib_add_one (Nat.succ_ne_zero n), Nat.succ_sub_succ_eq_sub,
Nat.cast_add, goldenRatio_sq, Nat.sub_zero]; ring
_ = φ * ((Nat.fib n) + φ * (Nat.fib (n + 1))) := by... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Real.GoldenRatio | {
"line": 225,
"column": 2
} | {
"line": 233,
"column": 51
} | [
{
"pp": "n : ℕ\n⊢ φ * ↑(Nat.fib (n + 1)) + ↑(Nat.fib n) = φ ^ (n + 1)",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"CharP.cast_eq_zero",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.Ring.Common.div_congr",
"Mathlib.Meta.NormNum.is... | induction n with
| zero => simp
| succ n ih =>
calc
_ = φ * (Nat.fib n) + φ ^ 2 * (Nat.fib (n + 1)) := by
simp only [Nat.fib_add_one (Nat.succ_ne_zero n), Nat.succ_sub_succ_eq_sub,
Nat.cast_add, goldenRatio_sq, Nat.sub_zero]; ring
_ = φ * ((Nat.fib n) + φ * (Nat.fib (n + 1))) := by... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 367,
"column": 41
} | {
"line": 367,
"column": 52
} | [
{
"pp": "z : ℂ\nk : ℕ\nhk : 1 ≤ k\nhz : z ∈ ℍₒ\n⊢ Summable fun n ↦ (z + ↑(↑n + 1)) ^ (-1 - ↑k)",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Int.cast_natCast",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"AddMonoid.toAddSemigroup"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 368,
"column": 8
} | {
"line": 368,
"column": 36
} | [
{
"pp": "z : ℂ\nk : ℕ\nhk : 1 ≤ k\nhz : z ∈ ℍₒ\n⊢ Summable fun n ↦ (z + ↑(-(↑n + 1))) ^ (-1 - ↑k)",
"usedConstants": [
"neg_add_rev",
"Int.instAddCommGroup",
"AddGroup.toSubtractionMonoid",
"Int.cast_neg",
"Int.cast",
"Eq.mpr",
"NegZeroClass.toNeg",
"NormedCom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 378,
"column": 2
} | {
"line": 378,
"column": 23
} | [
{
"pp": "z✝ : ℂ\nk : ℕ\nhk : 1 ≤ k\nhz✝ : z✝ ∈ ℍₒ\nz : ℂ\nhz : z ∈ ℍₒ\n⊢ ↑π * (↑π * z).cot - z⁻¹ = ∑' (n : ℕ), cotTerm z n",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"DivInvMonoid.toInv",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Cotangent | {
"line": 386,
"column": 4
} | {
"line": 386,
"column": 45
} | [
{
"pp": "k : ℕ\nz : ℂ\nhz : z ∈ ℍₒ\n⊢ iteratedDerivWithin k (fun x ↦ ↑π * (↑π * x).cot) ℍₒ z + iteratedDerivWithin k (fun x ↦ -(1 / x)) ℍₒ z =\n iteratedDerivWithin k (fun x ↦ ↑π * (↑π * x).cot) ℍₒ z + -((-1) ^ k * ↑k ! * z ^ (-1 + -↑k))",
"usedConstants": [
"Eq.mpr",
"iteratedDerivWithin_con... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SumIntegralComparisons | {
"line": 68,
"column": 18
} | {
"line": 68,
"column": 29
} | [
{
"pp": "a b : ℕ\nf g : ℝ → ℝ\nhab : a ≤ b\nh : ∀ i ∈ Ico a b, ∀ x ∈ Ico ↑i ↑(i + 1), f ↑i ≤ g x\nhg : IntegrableOn g (Ico ↑a ↑b) volume\nA : ∀ i ∈ Finset.Ico a b, IntervalIntegrable g volume ↑i ↑(i + 1)\ni : ℕ\nhi : i ∈ Finset.Ico a b\nx : ℝ\nhx : x ∈ Ioo ↑i ↑(i + 1)\n⊢ i ∈ Ico a b",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SumIntegralComparisons | {
"line": 72,
"column": 18
} | {
"line": 72,
"column": 29
} | [
{
"pp": "a b : ℕ\nf g : ℝ → ℝ\nhab : a ≤ b\nh : ∀ i ∈ Ico a b, ∀ x ∈ Ico ↑i ↑(i + 1), f ↑i ≤ g x\nhg : IntegrableOn g (Ico ↑a ↑b) volume\nA : ∀ i ∈ Finset.Ico a b, IntervalIntegrable g volume ↑i ↑(i + 1)\ni : ℕ\nhi : i ∈ Ico a b\n⊢ i ∈ Finset.Ico a b",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SumIntegralExpDecay | {
"line": 50,
"column": 33
} | {
"line": 50,
"column": 44
} | [
{
"pp": "k M : ℕ\nc : ℝ\nhc : 0 < c\nx : ℝ\nhx : x ∈ Icc ↑0 ↑M\ny : ℝ\nhy : y ∈ Icc ↑0 ↑M\nhxy : x ≤ y\n⊢ 0 ≤ x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SumIntegralComparisons | {
"line": 122,
"column": 28
} | {
"line": 122,
"column": 49
} | [
{
"pp": "a b : ℕ\nf : ℝ → ℝ\nhab : a ≤ b\nhf : AntitoneOn f (Icc ↑a ↑b)\n⊢ ∫ (x : ℝ) in ↑a..↑a + ↑(b - a), f x ≤ ∑ x ∈ Finset.Ico 0 (b - a), f ↑(a + x)",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real.instLE",
"Real",
"Real.instRCLike",
"congrArg",
... | Nat.Ico_zero_eq_range | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 63,
"column": 10
} | {
"line": 63,
"column": 21
} | [
{
"pp": "case hbc\nu : ℕ → ℝ\nl : ℝ\nhmono : Monotone u\nhlim :\n ∀ (a : ℝ),\n 1 < a →\n ∃ c,\n (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧\n Tendsto c atTop atTop ∧ Tendsto (fun n ↦ u (c n) / ↑(c n)) atTop (𝓝 l)\nlnonneg : 0 ≤ l\nε : ℝ\nεpos : 0 < ε\nc : ℕ → ℕ\ncgrowth : ∀ᶠ (n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 64,
"column": 4
} | {
"line": 66,
"column": 40
} | [
{
"pp": "u : ℕ → ℝ\nl : ℝ\nhmono : Monotone u\nhlim :\n ∀ (a : ℝ),\n 1 < a →\n ∃ c,\n (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧\n Tendsto c atTop atTop ∧ Tendsto (fun n ↦ u (c n) / ↑(c n)) atTop (𝓝 l)\nlnonneg : 0 ≤ l\nε : ℝ\nεpos : 0 < ε\nc : ℕ → ℕ\ncgrowth : ∀ᶠ (n : ℕ) in at... | obtain ⟨a, ha⟩ :
∃ a : ℕ, ∀ b : ℕ, a ≤ b → (c (b + 1) : ℝ) ≤ (1 + ε) * c b ∧ u (c b) - c b * l ≤ ε * c b :=
eventually_atTop.1 (cgrowth.and L) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.SumIntegralComparisons | {
"line": 171,
"column": 28
} | {
"line": 171,
"column": 49
} | [
{
"pp": "a b : ℕ\nf : ℝ → ℝ\nhab : a ≤ b\nhf : AntitoneOn f (Icc ↑a ↑b)\n⊢ ∑ x ∈ Finset.Ico 0 (b - a), f ↑(a + x + 1) ≤ ∫ (x : ℝ) in ↑a..↑a + ↑(b - a), f x",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real.instLE",
"Real",
"Real.instRCLike",
"congrAr... | Nat.Ico_zero_eq_range | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 84,
"column": 6
} | {
"line": 84,
"column": 31
} | [
{
"pp": "u : ℕ → ℝ\nl : ℝ\nhmono : Monotone u\nhlim :\n ∀ (a : ℝ),\n 1 < a →\n ∃ c,\n (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧\n Tendsto c atTop atTop ∧ Tendsto (fun n ↦ u (c n) / ↑(c n)) atTop (𝓝 l)\nlnonneg : 0 ≤ l\nε : ℝ\nεpos : 0 < ε\nc : ℕ → ℕ\ncgrowth : ∀ᶠ (n : ℕ) in at... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.VonNeumannAlgebra.Basic | {
"line": 160,
"column": 2
} | {
"line": 160,
"column": 40
} | [
{
"pp": "H : Type u\ninst✝² : NormedAddCommGroup H\ninst✝¹ : InnerProductSpace ℂ H\ninst✝ : CompleteSpace H\ne : H →L[ℂ] H\nhe : IsStarProjection e\nS : VonNeumannAlgebra H\nh : ∀ x ∈ S.commutant, e * (x * e) = x * e\nx : H →L[ℂ] H\nhx : x ∈ S.commutant\n⊢ e * (x * e) = e * x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SumIntegralComparisons | {
"line": 232,
"column": 8
} | {
"line": 232,
"column": 39
} | [
{
"pp": "case h.h₂.a\na b : ℕ\nf g : ℝ → ℝ\nhab : a ≤ b\nhf : MonotoneOn f (Icc ↑a ↑b)\nhg : AntitoneOn g (Icc (↑a - 1) (↑b - 1))\nfpos : 0 ≤ f ↑a\ngpos : 0 ≤ g (↑b - 1)\ni : ℕ\nx : ℝ\nhx : ↑i ≤ x ∧ x < ↑i + 1\nhi : a ≤ i ∧ i < b\nI0 : ↑i ≤ ↑b - 1\nI1 : ↑i ∈ Icc (↑a - 1) (↑b - 1)\nI2 : x ∈ Icc ↑a ↑b\n⊢ x - 1 ≤ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SumIntegralComparisons | {
"line": 244,
"column": 8
} | {
"line": 244,
"column": 19
} | [
{
"pp": "case hf.x\na b : ℕ\nf g : ℝ → ℝ\nhab : a ≤ b\nhf : MonotoneOn f (Icc ↑a ↑b)\nhg : AntitoneOn g (Icc (↑a - 1) (↑b - 1))\nfpos : 0 ≤ f ↑a\ngpos : 0 ≤ g (↑b - 1)\nx : ℝ\nhx : x ∈ Icc ↑a ↑b\ny : ℝ\nhy : y ∈ Icc ↑a ↑b\nhxy : x ≤ y\n⊢ x - 1 ∈ Icc (↑a - 1) (↑b - 1)",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SumIntegralComparisons | {
"line": 245,
"column": 8
} | {
"line": 245,
"column": 19
} | [
{
"pp": "case hf.x\na b : ℕ\nf g : ℝ → ℝ\nhab : a ≤ b\nhf : MonotoneOn f (Icc ↑a ↑b)\nhg : AntitoneOn g (Icc (↑a - 1) (↑b - 1))\nfpos : 0 ≤ f ↑a\ngpos : 0 ≤ g (↑b - 1)\nx : ℝ\nhx : x ∈ Icc ↑a ↑b\ny : ℝ\nhy : y ∈ Icc ↑a ↑b\nhxy : x ≤ y\n⊢ y - 1 ∈ Icc (↑a - 1) (↑b - 1)",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SumIntegralComparisons | {
"line": 246,
"column": 8
} | {
"line": 246,
"column": 19
} | [
{
"pp": "case hf.a\na b : ℕ\nf g : ℝ → ℝ\nhab : a ≤ b\nhf : MonotoneOn f (Icc ↑a ↑b)\nhg : AntitoneOn g (Icc (↑a - 1) (↑b - 1))\nfpos : 0 ≤ f ↑a\ngpos : 0 ≤ g (↑b - 1)\nx : ℝ\nhx : x ∈ Icc ↑a ↑b\ny : ℝ\nhy : y ∈ Icc ↑a ↑b\nhxy : x ≤ y\n⊢ x - 1 ≤ y - 1",
"usedConstants": [
"Eq.mpr",
"Real",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 113,
"column": 10
} | {
"line": 113,
"column": 21
} | [
{
"pp": "case hbc\nu : ℕ → ℝ\nl : ℝ\nhmono : Monotone u\nhlim :\n ∀ (a : ℝ),\n 1 < a →\n ∃ c,\n (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧\n Tendsto c atTop atTop ∧ Tendsto (fun n ↦ u (c n) / ↑(c n)) atTop (𝓝 l)\nlnonneg : 0 ≤ l\nA : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SumIntegralComparisons | {
"line": 278,
"column": 8
} | {
"line": 278,
"column": 39
} | [
{
"pp": "case h.h₂.a\na b : ℕ\nf g : ℝ → ℝ\nhab : a ≤ b\nhf : AntitoneOn f (Icc ↑a ↑b)\nhg : MonotoneOn g (Icc (↑a - 1) (↑b - 1))\nfpos : 0 ≤ f ↑b\ngpos : 0 ≤ g (↑a - 1)\ni : ℕ\nx : ℝ\nhx : ↑i ≤ x ∧ x < ↑i + 1\nhi : a ≤ i ∧ i < b\nI0 : ↑i ≤ ↑b - 1\nI1 : ↑i ∈ Icc (↑a - 1) (↑b - 1)\nI2 : x ∈ Icc ↑a ↑b\n⊢ x - 1 ≤ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SumIntegralComparisons | {
"line": 280,
"column": 35
} | {
"line": 280,
"column": 46
} | [
{
"pp": "a b : ℕ\nf g : ℝ → ℝ\nhab : a ≤ b\nhf : AntitoneOn f (Icc ↑a ↑b)\nhg : MonotoneOn g (Icc (↑a - 1) (↑b - 1))\nfpos : 0 ≤ f ↑b\ngpos : 0 ≤ g (↑a - 1)\ni : ℕ\nx : ℝ\nhx : ↑i ≤ x ∧ x < ↑i + 1\nhi : a ≤ i ∧ i < b\nI0 : ↑i ≤ ↑b - 1\nI1 : ↑i ∈ Icc (↑a - 1) (↑b - 1)\nI2 : x ∈ Icc ↑a ↑b\n⊢ x - 1 ∈ Icc (↑a - 1) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SumIntegralComparisons | {
"line": 280,
"column": 55
} | {
"line": 280,
"column": 66
} | [
{
"pp": "a b : ℕ\nf g : ℝ → ℝ\nhab : a ≤ b\nhf : AntitoneOn f (Icc ↑a ↑b)\nhg : MonotoneOn g (Icc (↑a - 1) (↑b - 1))\nfpos : 0 ≤ f ↑b\ngpos : 0 ≤ g (↑a - 1)\ni : ℕ\nx : ℝ\nhx : ↑i ≤ x ∧ x < ↑i + 1\nhi : a ≤ i ∧ i < b\nI0 : ↑i ≤ ↑b - 1\nI1 : ↑i ∈ Icc (↑a - 1) (↑b - 1)\nI2 : x ∈ Icc ↑a ↑b\n⊢ ↑a - 1 ≤ x - 1",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SumIntegralComparisons | {
"line": 282,
"column": 53
} | {
"line": 282,
"column": 64
} | [
{
"pp": "a b : ℕ\nf g : ℝ → ℝ\nhab : a ≤ b\nhf : AntitoneOn f (Icc ↑a ↑b)\nhg : MonotoneOn g (Icc (↑a - 1) (↑b - 1))\nfpos : 0 ≤ f ↑b\ngpos : 0 ≤ g (↑a - 1)\ni : ℕ\nx : ℝ\nhx : ↑i ≤ x ∧ x < ↑i + 1\nhi : a ≤ i ∧ i < b\nI0 : ↑i ≤ ↑b - 1\nI1 : ↑i ∈ Icc (↑a - 1) (↑b - 1)\nI2 : x ∈ Icc ↑a ↑b\n⊢ ↑b ∈ Icc ↑a ↑b",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SumIntegralComparisons | {
"line": 289,
"column": 8
} | {
"line": 289,
"column": 19
} | [
{
"pp": "case hf.x\na b : ℕ\nf g : ℝ → ℝ\nhab : a ≤ b\nhf : AntitoneOn f (Icc ↑a ↑b)\nhg : MonotoneOn g (Icc (↑a - 1) (↑b - 1))\nfpos : 0 ≤ f ↑b\ngpos : 0 ≤ g (↑a - 1)\nx : ℝ\nhx : x ∈ Icc ↑a ↑b\ny : ℝ\nhy : y ∈ Icc ↑a ↑b\nhxy : x ≤ y\n⊢ x - 1 ∈ Icc (↑a - 1) (↑b - 1)",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SumIntegralComparisons | {
"line": 290,
"column": 8
} | {
"line": 290,
"column": 19
} | [
{
"pp": "case hf.x\na b : ℕ\nf g : ℝ → ℝ\nhab : a ≤ b\nhf : AntitoneOn f (Icc ↑a ↑b)\nhg : MonotoneOn g (Icc (↑a - 1) (↑b - 1))\nfpos : 0 ≤ f ↑b\ngpos : 0 ≤ g (↑a - 1)\nx : ℝ\nhx : x ∈ Icc ↑a ↑b\ny : ℝ\nhy : y ∈ Icc ↑a ↑b\nhxy : x ≤ y\n⊢ y - 1 ∈ Icc (↑a - 1) (↑b - 1)",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SumIntegralComparisons | {
"line": 291,
"column": 8
} | {
"line": 291,
"column": 19
} | [
{
"pp": "case hf.a\na b : ℕ\nf g : ℝ → ℝ\nhab : a ≤ b\nhf : AntitoneOn f (Icc ↑a ↑b)\nhg : MonotoneOn g (Icc (↑a - 1) (↑b - 1))\nfpos : 0 ≤ f ↑b\ngpos : 0 ≤ g (↑a - 1)\nx : ℝ\nhx : x ∈ Icc ↑a ↑b\ny : ℝ\nhy : y ∈ Icc ↑a ↑b\nhxy : x ≤ y\n⊢ x - 1 ≤ y - 1",
"usedConstants": [
"Eq.mpr",
"Real",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 136,
"column": 6
} | {
"line": 136,
"column": 31
} | [
{
"pp": "u : ℕ → ℝ\nl : ℝ\nhmono : Monotone u\nhlim :\n ∀ (a : ℝ),\n 1 < a →\n ∃ c,\n (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧\n Tendsto c atTop atTop ∧ Tendsto (fun n ↦ u (c n) / ↑(c n)) atTop (𝓝 l)\nlnonneg : 0 ≤ l\nA : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u n - ↑n * l ≤... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.DiagramLemmas.KernelCokernelComp | {
"line": 189,
"column": 2
} | {
"line": 189,
"column": 13
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX Y Z : C\nf : X ⟶ Y\ng : Y ⟶ Z\n⊢ δ f g = -kernel.ι g ≫ cokernel.π f",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 144,
"column": 8
} | {
"line": 144,
"column": 23
} | [
{
"pp": "case h.hbc\nu : ℕ → ℝ\nl : ℝ\nhmono : Monotone u\nhlim :\n ∀ (a : ℝ),\n 1 < a →\n ∃ c,\n (∀ᶠ (n : ℕ) in atTop, ↑(c (n + 1)) ≤ a * ↑(c n)) ∧\n Tendsto c atTop atTop ∧ Tendsto (fun n ↦ u (c n) / ↑(c n)) atTop (𝓝 l)\nlnonneg : 0 ≤ l\nA : ∀ (ε : ℝ), 0 < ε → ∀ᶠ (n : ℕ) in atTop, u ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Preadditive | {
"line": 33,
"column": 33
} | {
"line": 33,
"column": 68
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nP : ObjectProperty C\nh𝒢 : P.IsSeparating\nX Y : C\nf : X ⟶ Y\nhf : ∀ (G : C), P G → ∀ (h : G ⟶ X), h ≫ f = 0\n⊢ ∀ (G : C), P G → ∀ (h : G ⟶ X), h ≫ f = h ≫ 0",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Preadditive | {
"line": 34,
"column": 30
} | {
"line": 34,
"column": 82
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nP : ObjectProperty C\nh𝒢 : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), (∀ (G : C), P G → ∀ (h : G ⟶ X), h ≫ f = 0) → f = 0\nX Y : C\nf g : X ⟶ Y\nhfg : ∀ (G : C), P G → ∀ (h : G ⟶ X), h ≫ f = h ≫ g\n⊢ ∀ (G : C), P G → ∀ (h : G ⟶ X), h ≫ (f - g) = 0",
"us... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Preadditive | {
"line": 39,
"column": 33
} | {
"line": 39,
"column": 68
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nP : ObjectProperty C\nh𝒢 : P.IsCoseparating\nX Y : C\nf : X ⟶ Y\nhf : ∀ (G : C), P G → ∀ (h : Y ⟶ G), f ≫ h = 0\n⊢ ∀ (G : C), P G → ∀ (h : Y ⟶ G), f ≫ h = 0 ≫ h",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Preadditive | {
"line": 40,
"column": 30
} | {
"line": 40,
"column": 82
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nP : ObjectProperty C\nh𝒢 : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), (∀ (G : C), P G → ∀ (h : Y ⟶ G), f ≫ h = 0) → f = 0\nX Y : C\nf g : X ⟶ Y\nhfg : ∀ (G : C), P G → ∀ (h : Y ⟶ G), f ≫ h = g ≫ h\n⊢ ∀ (G : C), P G → ∀ (h : Y ⟶ G), (f - g) ≫ h = 0",
"us... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Preadditive | {
"line": 44,
"column": 37
} | {
"line": 44,
"column": 72
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nG : C\nhG : IsSeparator G\nX Y : C\nf : X ⟶ Y\nhf : ∀ (h : G ⟶ X), h ≫ f = 0\n⊢ ∀ (h : G ⟶ X), h ≫ f = h ≫ 0",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Preadditive | {
"line": 46,
"column": 32
} | {
"line": 46,
"column": 84
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nG : C\nhG : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), (∀ (h : G ⟶ X), h ≫ f = 0) → f = 0\nX Y : C\nf g : X ⟶ Y\nhfg : ∀ (h : G ⟶ X), h ≫ f = h ≫ g\n⊢ ∀ (h : G ⟶ X), h ≫ (f - g) = 0",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Preadditive | {
"line": 50,
"column": 37
} | {
"line": 50,
"column": 72
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nG : C\nhG : IsCoseparator G\nX Y : C\nf : X ⟶ Y\nhf : ∀ (h : Y ⟶ G), f ≫ h = 0\n⊢ ∀ (h : Y ⟶ G), f ≫ h = 0 ≫ h",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Generator.Preadditive | {
"line": 52,
"column": 32
} | {
"line": 52,
"column": 84
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Preadditive C\nG : C\nhG : ∀ ⦃X Y : C⦄ (f : X ⟶ Y), (∀ (h : Y ⟶ G), f ≫ h = 0) → f = 0\nX Y : C\nf g : X ⟶ Y\nhfg : ∀ (h : Y ⟶ G), f ≫ h = g ≫ h\n⊢ ∀ (h : Y ⟶ G), (f - g) ≫ h = 0",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 225,
"column": 4
} | {
"line": 225,
"column": 15
} | [
{
"pp": "N : ℕ\nj : ℝ\nhj : 0 < j\nc : ℝ\nhc : 1 < c\ncpos : 0 < c\nA : 0 < c⁻¹ ^ 2\nthis : c ^ 3 = c ^ 2 * c\n⊢ c ≤ c ^ 2",
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"instOfNatNat",
"LE.le",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 266,
"column": 6
} | {
"line": 266,
"column": 67
} | [
{
"pp": "case h\nc : ℝ\nhc : 1 < c\ni : ℕ\ncpos : 0 < c\nhi : i ≠ 0\n⊢ 1 ≤ c ^ i * c⁻¹",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real.instLE",
"Real",
"instHDiv",
"HMul.hMul",
"GroupWithZero.toDivInvMonoid"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 281,
"column": 6
} | {
"line": 290,
"column": 41
} | [
{
"pp": "N : ℕ\nj : ℝ\nhj : 0 < j\nc : ℝ\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c⁻¹\n⊢ ∑ i ∈ range N with j < c ^ i, 1 / ↑⌊c ^ i⌋₊ ^ 2 ≤ ∑ i ∈ range N with j < c ^ i, (1 - c⁻¹)⁻¹ ^ 2 * (1 / (c ^ i) ^ 2)",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Real.instI... | gcongr with i
rw [mul_div_assoc', mul_one, div_le_div_iff₀]; rotate_left
· apply sq_pos_of_pos
refine zero_lt_one.trans_le ?_
simp only [Nat.le_floor, one_le_pow₀, hc.le, Nat.one_le_cast, Nat.cast_one]
· exact sq_pos_of_pos (pow_pos cpos _)
rw [one_mul, ← mul_pow]
gcongr
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 281,
"column": 6
} | {
"line": 290,
"column": 41
} | [
{
"pp": "N : ℕ\nj : ℝ\nhj : 0 < j\nc : ℝ\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c⁻¹\n⊢ ∑ i ∈ range N with j < c ^ i, 1 / ↑⌊c ^ i⌋₊ ^ 2 ≤ ∑ i ∈ range N with j < c ^ i, (1 - c⁻¹)⁻¹ ^ 2 * (1 / (c ^ i) ^ 2)",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Real.instI... | gcongr with i
rw [mul_div_assoc', mul_one, div_le_div_iff₀]; rotate_left
· apply sq_pos_of_pos
refine zero_lt_one.trans_le ?_
simp only [Nat.le_floor, one_le_pow₀, hc.le, Nat.one_le_cast, Nat.cast_one]
· exact sq_pos_of_pos (pow_pos cpos _)
rw [one_mul, ← mul_pow]
gcongr
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecificLimits.FloorPow | {
"line": 276,
"column": 2
} | {
"line": 297,
"column": 11
} | [
{
"pp": "N : ℕ\nj : ℝ\nhj : 0 < j\nc : ℝ\nhc : 1 < c\ncpos : 0 < c\nA : 0 < 1 - c⁻¹\n⊢ ∑ i ∈ range N with j < ↑⌊c ^ i⌋₊, 1 / ↑⌊c ^ i⌋₊ ^ 2 ≤ c ^ 5 * (c - 1)⁻¹ ^ 3 / j ^ 2",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
... | calc
(∑ i ∈ range N with j < ⌊c ^ i⌋₊, (1 : ℝ) / (⌊c ^ i⌋₊ : ℝ) ^ 2) ≤
∑ i ∈ range N with j < c ^ i, (1 : ℝ) / (⌊c ^ i⌋₊ : ℝ) ^ 2 := by
gcongr with k hk; exact Nat.floor_le (by positivity)
_ ≤ ∑ i ∈ range N with j < c ^ i, (1 - c⁻¹)⁻¹ ^ 2 * ((1 : ℝ) / (c ^ i) ^ 2) := by
gcongr with i
r... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.CategoryTheory.Limits.Indization.FilteredColimits | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 17
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\nI : Type v\ninst✝⁴ : SmallCategory I\nF : I ⥤ Cᵒᵖ ⥤ Type v\nJ : Type v\ninst✝³ : SmallCategory J\ninst✝² : FinCategory J\nG : J ⥤ CostructuredArrow yoneda (colimit F)\nK : Type v\ninst✝¹ : SmallCategory K\nH : K ⥤ Over (colimit F)\ninst✝ : IsFiltered K\nh : Nonem... | obtain ⟨t⟩ := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Limits.Indization.ParallelPair | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 13
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nA B : Cᵒᵖ ⥤ Type v₁\nf g : A ⟶ B\nP₁ : IndObjectPresentation A\nP₂ : IndObjectPresentation B\ni : K f g P₁ P₂\n⊢ { pt := A, ι := ι₁ f g P₁ P₂ }.ι.app i ≫ f =\n (whiskerRight (ϕ f g P₁ P₂) yoneda).app i ≫ { pt := B, ι := ι₂ f g P₁ P₂ }.ι.app i",
"usedCons... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Indization.ParallelPair | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 13
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v₁, u₁} C\nA B : Cᵒᵖ ⥤ Type v₁\nf g : A ⟶ B\nP₁ : IndObjectPresentation A\nP₂ : IndObjectPresentation B\ni : K f g P₁ P₂\n⊢ { pt := A, ι := ι₁ f g P₁ P₂ }.ι.app i ≫ g =\n (whiskerRight (ψ f g P₁ P₂) yoneda).app i ≫ { pt := B, ι := ι₂ f g P₁ P₂ }.ι.app i",
"usedCons... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Comma.Final | {
"line": 130,
"column": 2
} | {
"line": 145,
"column": 16
} | [
{
"pp": "A : Type u₁\ninst✝¹⁰ : Category.{v₁, u₁} A\nB : Type u₂\ninst✝⁹ : Category.{v₂, u₂} B\nT : Type u₃\ninst✝⁸ : Category.{v₃, u₃} T\nL : A ⥤ T\nR : B ⥤ T\nA' : Type u₄\ninst✝⁷ : Category.{v₄, u₄} A'\nB' : Type u₅\ninst✝⁶ : Category.{v₅, u₅} B'\nT' : Type u₆\ninst✝⁵ : Category.{v₆, u₆} T'\nL' : A' ⥤ T'\nR'... | haveI := final_of_natIso iR
rw [isConnected_iff_of_equivalence (StructuredArrow.commaMapEquivalence iL.hom iR.inv _)]
have : StructuredArrow.map₂ u₂ iR.hom ≅ StructuredArrow.post j₂ G R' ⋙
StructuredArrow.map₂ (G := 𝟭 _) (F := 𝟭 _) (R' := R ⋙ H) u₂ iR.hom ⋙
StructuredArrow.pre _ R H :=
eqToIso (by... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Comma.Final | {
"line": 130,
"column": 2
} | {
"line": 145,
"column": 16
} | [
{
"pp": "A : Type u₁\ninst✝¹⁰ : Category.{v₁, u₁} A\nB : Type u₂\ninst✝⁹ : Category.{v₂, u₂} B\nT : Type u₃\ninst✝⁸ : Category.{v₃, u₃} T\nL : A ⥤ T\nR : B ⥤ T\nA' : Type u₄\ninst✝⁷ : Category.{v₄, u₄} A'\nB' : Type u₅\ninst✝⁶ : Category.{v₅, u₅} B'\nT' : Type u₆\ninst✝⁵ : Category.{v₆, u₆} T'\nL' : A' ⥤ T'\nR'... | haveI := final_of_natIso iR
rw [isConnected_iff_of_equivalence (StructuredArrow.commaMapEquivalence iL.hom iR.inv _)]
have : StructuredArrow.map₂ u₂ iR.hom ≅ StructuredArrow.post j₂ G R' ⋙
StructuredArrow.map₂ (G := 𝟭 _) (F := 𝟭 _) (R' := R ⋙ H) u₂ iR.hom ⋙
StructuredArrow.pre _ R H :=
eqToIso (by... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Comma.StructuredArrow.CommaMap | {
"line": 39,
"column": 10
} | {
"line": 42,
"column": 49
} | [
{
"pp": "C : Type u₁\ninst✝⁶ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\nT : Type u₃\ninst✝⁴ : Category.{v₃, u₃} T\nL : C ⥤ T\nR : D ⥤ T\nC' : Type u₄\ninst✝³ : Category.{v₄, u₄} C'\nD' : Type u₅\ninst✝² : Category.{v₅, u₅} D'\nT' : Type u₆\ninst✝¹ : Category.{v₆, u₆} T'\nL' : C' ⥤ T'\nR' ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Comma.StructuredArrow.CommaMap | {
"line": 55,
"column": 6
} | {
"line": 55,
"column": 17
} | [
{
"pp": "C : Type u₁\ninst✝⁶ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝⁵ : Category.{v₂, u₂} D\nT : Type u₃\ninst✝⁴ : Category.{v₃, u₃} T\nL : C ⥤ T\nR : D ⥤ T\nC' : Type u₄\ninst✝³ : Category.{v₄, u₄} C'\nD' : Type u₅\ninst✝² : Category.{v₅, u₅} D'\nT' : Type u₆\ninst✝¹ : Category.{v₆, u₆} T'\nL' : C' ⥤ T'\nR' ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Indization.Category | {
"line": 297,
"column": 4
} | {
"line": 297,
"column": 15
} | [
{
"pp": "case refine_3\nC : Type u\ninst✝ : Category.{v, u} C\nA B : Ind C\nf : A ⟶ B\nP : IndParallelPairPresentation ((Ind.inclusion C).map f) ((Ind.inclusion C).map f)\n⊢ (P.parallelPairIsoParallelPairCompIndYoneda.app WalkingParallelPair.zero).hom ≫\n (Arrow.mk ((Ind.lim P.I).map P.φ)).hom =\n (Arro... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.FilteredColimitCommutesProduct | {
"line": 239,
"column": 4
} | {
"line": 240,
"column": 66
} | [
{
"pp": "case refine_1\nα : Type u\nI : α → Type u\ninst✝¹ : (i : α) → SmallCategory (I i)\ninst✝ : ∀ (i : α), IsFiltered (I i)\nF : (i : α) → I i ⥤ Type u\ny y' : (fun X ↦ X) (colimit (pointwiseProduct F))\nhy : (hom (colimitPointwiseProductToProductColimit F)) y = (hom (colimitPointwiseProductToProductColimit... | let yk' : (pointwiseProduct F).obj k :=
(pointwiseProduct F).map (IsFiltered.rightToMax ky ky') yk₀' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Connected | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 13
} | [
{
"pp": "J : Type w\ninst✝⁵ : Category.{w', w} J\ninst✝⁴ : IsConnected J\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasPullbacks C\ninst✝¹ : HasColimitsOfShape J C\ninst✝ : HasExactColimitsOfShape J C\nF : J ⥤ C\nc : Cocone F\nhc : IsColimit c\nX Y : C\nf : X ⟶ c.pt\ng h : c.pt ⟶ Y\nhf : ∀ (j : J), pullb... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Connected | {
"line": 72,
"column": 32
} | {
"line": 72,
"column": 43
} | [
{
"pp": "J : Type w\ninst✝⁶ : Category.{w', w} J\ninst✝⁵ : IsConnected J\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasPullbacks C\ninst✝¹ : HasColimitsOfShape J C\ninst✝ : HasExactColimitsOfShape J C\nF : J ⥤ C\nc : Cocone F\nhc : IsColimit c\nX Y : C\nf : X ⟶ c.pt\ng : c.pt... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.FilteredColimitCommutesProduct | {
"line": 261,
"column": 48
} | {
"line": 261,
"column": 88
} | [
{
"pp": "α : Type u\nI : α → Type u\ninst✝¹ : (i : α) → SmallCategory (I i)\ninst✝ : ∀ (i : α), IsFiltered (I i)\nF : (i : α) → I i ⥤ Type u\nky : (i : α) → I i\nyk₀ : (pointwiseProduct F).obj ky\nky' : (i : α) → I i\nyk₀' : (pointwiseProduct F).obj ky'\nk : (i : α) → I i := IsFiltered.max ky ky'\nyk : ∏ᶜ (Func... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Connected | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 13
} | [
{
"pp": "J : Type w\ninst✝⁵ : Category.{w', w} J\ninst✝⁴ : IsConnected J\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasPushouts C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : HasExactLimitsOfShape J C\nF : J ⥤ C\nc : Cone F\nhc : IsLimit c\nX Y : C\ng h : Y ⟶ c.pt\nf : c.pt ⟶ X\nhf : ∀ (j : J), g ≫ f ≫ pushou... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.GrothendieckAxioms.Connected | {
"line": 109,
"column": 31
} | {
"line": 109,
"column": 42
} | [
{
"pp": "J : Type w\ninst✝⁶ : Category.{w', w} J\ninst✝⁵ : IsConnected J\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasPushouts C\ninst✝¹ : HasLimitsOfShape J C\ninst✝ : HasExactLimitsOfShape J C\nF : J ⥤ C\nc : Cone F\nhc : IsLimit c\nX Y : C\ng : Y ⟶ c.pt\nf : c.pt ⟶ X\nhf ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.FilteredColimitCommutesProduct | {
"line": 269,
"column": 25
} | {
"line": 269,
"column": 49
} | [
{
"pp": "α : Type u\nI : α → Type u\ninst✝¹ : (i : α) → SmallCategory (I i)\ninst✝ : ∀ (i : α), IsFiltered (I i)\nF : (i : α) → I i ⥤ Type u\nx : (fun X ↦ X) (∏ᶜ fun s ↦ colimit (F s))\nk : (s : α) → I s\np : (s : α) → (F s).obj (k s)\nhk : ∀ (s : α), (hom (colimit.ι (F s) (k s))) (p s) = (hom (Pi.π (fun s ↦ co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.Injective.Dimension | {
"line": 105,
"column": 2
} | {
"line": 106,
"column": 9
} | [
{
"pp": "case a\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\nX : C\ninst✝ : HasInjectiveDimensionLT X 0\nthis : HasExt C := ⋯\n⊢ Ext.homEquiv₀.symm (𝟙 X) = Ext.homEquiv₀.symm 0",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
"Equiv.instEquivLike",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.Injective.Dimension | {
"line": 147,
"column": 45
} | {
"line": 147,
"column": 56
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\nX : C\ninst✝ : HasExt C\nh : ∀ ⦃Y : C⦄, Subsingleton (Ext Y X 1)\nX✝ Y✝ : C\nf : X✝ ⟶ X\ng : X✝ ⟶ Y✝\nx✝ : Mono g\nφ : { X₁ := X✝, X₂ := Y✝, X₃ := cokernel g, f := g, g := cokernel.π g, zero := ⋯ }.X₂ ⟶ X\nhφ :\n (Ext.mk₀ { X₁ := X✝, X₂ := Y✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.Injective.Dimension | {
"line": 240,
"column": 29
} | {
"line": 240,
"column": 40
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : Abelian C\ninst✝¹ : HasExt C\ninst✝ : EnoughProjectives C\nX : C\nn : ℕ\nhX : ∀ (Y : C), Subsingleton (Ext Y X n)\nd : ℕ\nY : C\ne : Ext Y X d\nhd : d = n + 0\n⊢ d = n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.ModuleEmbedding.GabrielPopescu | {
"line": 91,
"column": 53
} | {
"line": 91,
"column": 83
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : IsGrothendieckAbelian.{v, v, u} C\nG : C\nhG : IsSeparator G\nA B : C\nM : ModuleCat (End G)ᵐᵒᵖ\ng : M ⟶ ModuleCat.of (End G)ᵐᵒᵖ (G ⟶ A)\nhg : Mono g\nf : M ⟶ ModuleCat.of (End G)ᵐᵒᵖ (G ⟶ B)\nF : Finset (Discrete ↑M)\nh : G ⟶ pullback ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.Injective.Dimension | {
"line": 287,
"column": 73
} | {
"line": 287,
"column": 84
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nn : ℕ\nh : HasInjectiveDimensionLT X (n + 1)\ni : ℕ\nhi : ↑n < ↑i\n⊢ n + 1 ≤ i",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Nat.instOne",
"Order.add_one_le_iff._simp_1",
"id",
"instOfNatNat",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.Injective.Dimension | {
"line": 320,
"column": 19
} | {
"line": 320,
"column": 65
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nd : ℕ\nhd : injectiveDimension X = ↑↑d\n⊢ HasInjectiveDimensionLE X d",
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"WithBot",
"ENat.instNatCast",
"Preorder.toLE",
"instPreorderENat",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.ModuleEmbedding.GabrielPopescu | {
"line": 123,
"column": 4
} | {
"line": 123,
"column": 31
} | [
{
"pp": "case hf.h\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : IsGrothendieckAbelian.{v, v, u} C\nG : C\nhG : IsSeparator G\nA B : C\nf : (preadditiveCoyonedaObj G).obj A ⟶ (preadditiveCoyonedaObj G).obj B\nthis : Epi (Sigma.desc fun f ↦ f)\nh : (kernel.ι (Sigma.desc fun m ↦ m) ≫ Sigma.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.GrothendieckCategory.ModuleEmbedding.GabrielPopescu | {
"line": 138,
"column": 6
} | {
"line": 138,
"column": 21
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : IsGrothendieckAbelian.{v, v, u} C\nG : C\nhG : IsSeparator G\nB : C\nhB : Injective B\nM : Ideal (End G)ᵐᵒᵖ\ng : ↥M →ₗ[(End G)ᵐᵒᵖ] G ⟶ B\nl : G ⟶ B\nhl :\n d (ModuleCat.ofHom { toFun := fun i ↦ MulOpposite.unop ↑i, map_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Square | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 46
} | [
{
"pp": "case refine_1.h.toFun.h.a\nsq₁ : Square (Type v)\nsq₂ : Square (Type u)\ne₁ : sq₁.X₁ ≃ sq₂.X₁\ne₂ : sq₁.X₂ ≃ sq₂.X₂\ne₃ : sq₁.X₃ ≃ sq₂.X₃\ne₄ : sq₁.X₄ ≃ sq₂.X₄\ncomm₁₂ : ⇑e₂ ∘ ⇑(ConcreteCategory.hom sq₁.f₁₂) = ⇑(ConcreteCategory.hom sq₂.f₁₂) ∘ ⇑e₁\ncomm₁₃ : ⇑e₃ ∘ ⇑(ConcreteCategory.hom sq₁.f₁₃) = ⇑(Con... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Square | {
"line": 106,
"column": 4
} | {
"line": 106,
"column": 46
} | [
{
"pp": "case refine_2.h.toFun.h.a\nsq₁ : Square (Type v)\nsq₂ : Square (Type u)\ne₁ : sq₁.X₁ ≃ sq₂.X₁\ne₂ : sq₁.X₂ ≃ sq₂.X₂\ne₃ : sq₁.X₃ ≃ sq₂.X₃\ne₄ : sq₁.X₄ ≃ sq₂.X₄\ncomm₁₂ : ⇑e₂ ∘ ⇑(ConcreteCategory.hom sq₁.f₁₂) = ⇑(ConcreteCategory.hom sq₂.f₁₂) ∘ ⇑e₁\ncomm₁₃ : ⇑e₃ ∘ ⇑(ConcreteCategory.hom sq₁.f₁₃) = ⇑(Con... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Square | {
"line": 107,
"column": 4
} | {
"line": 107,
"column": 46
} | [
{
"pp": "case refine_3.h.toFun.h.a\nsq₁ : Square (Type v)\nsq₂ : Square (Type u)\ne₁ : sq₁.X₁ ≃ sq₂.X₁\ne₂ : sq₁.X₂ ≃ sq₂.X₂\ne₃ : sq₁.X₃ ≃ sq₂.X₃\ne₄ : sq₁.X₄ ≃ sq₂.X₄\ncomm₁₂ : ⇑e₂ ∘ ⇑(ConcreteCategory.hom sq₁.f₁₂) = ⇑(ConcreteCategory.hom sq₂.f₁₂) ∘ ⇑e₁\ncomm₁₃ : ⇑e₃ ∘ ⇑(ConcreteCategory.hom sq₁.f₁₃) = ⇑(Con... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Square | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 46
} | [
{
"pp": "case refine_4.h.toFun.h.a\nsq₁ : Square (Type v)\nsq₂ : Square (Type u)\ne₁ : sq₁.X₁ ≃ sq₂.X₁\ne₂ : sq₁.X₂ ≃ sq₂.X₂\ne₃ : sq₁.X₃ ≃ sq₂.X₃\ne₄ : sq₁.X₄ ≃ sq₂.X₄\ncomm₁₂ : ⇑e₂ ∘ ⇑(ConcreteCategory.hom sq₁.f₁₂) = ⇑(ConcreteCategory.hom sq₂.f₁₂) ∘ ⇑e₁\ncomm₁₃ : ⇑e₃ ∘ ⇑(ConcreteCategory.hom sq₁.f₁₃) = ⇑(Con... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.Injective.Resolution | {
"line": 191,
"column": 4
} | {
"line": 191,
"column": 25
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nI J : InjectiveResolution X\n⊢ Homotopy (desc (𝟙 X ≫ 𝟙 X) I I) (𝟙 I.cocomplex)",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
"HomologicalComplex.instCategory",
"Nat.instOne",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.Injective.Resolution | {
"line": 193,
"column": 4
} | {
"line": 193,
"column": 25
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nX : C\nI J : InjectiveResolution X\n⊢ Homotopy (desc (𝟙 X ≫ 𝟙 X) J J) (𝟙 J.cocomplex)",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive",
"Eq.mpr",
"HomologicalComplex.instCategory",
"Nat.instOne",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.Injective.Resolution | {
"line": 251,
"column": 2
} | {
"line": 251,
"column": 17
} | [
{
"pp": "case h.g_comm\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasInjectiveResolutions C\nX Y : C\nf : X ⟶ Y\nI : InjectiveResolution X\nJ : InjectiveResolution Y\nφ : I.cocomplex ⟶ J.cocomplex\ncomm : I.ι.f 0 ≫ φ.f 0 = f ≫ J.ι.f 0\n⊢ (injectiveResolution X).ι ≫\n desc f (injec... | all_goals aesop | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.CategoryTheory.Abelian.Injective.Ext | {
"line": 203,
"column": 9
} | {
"line": 203,
"column": 89
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX Y : C\nR : InjectiveResolution Y\nn : ℕ\nf : X ⟶ R.cocomplex.X n\nm : ℕ\nhm : n + 1 = m\nhf : f ≫ R.cocomplex.d n m = 0\np : ℕ\nhp : p + 1 = n\nx✝ :\n ∃ g,\n g ≫ (R.cochainComplexXIso (↑p) p ⋯).hom ≫ R.cocomplex.d p n ≫... | simp only [← cancel_mono (R.cochainComplexXIso n n rfl).inv, Category.assoc, hg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.CategoryTheory.Abelian.Injective.Ext | {
"line": 203,
"column": 9
} | {
"line": 203,
"column": 89
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX Y : C\nR : InjectiveResolution Y\nn : ℕ\nf : X ⟶ R.cocomplex.X n\nm : ℕ\nhm : n + 1 = m\nhf : f ≫ R.cocomplex.d n m = 0\np : ℕ\nhp : p + 1 = n\nx✝ :\n ∃ g,\n g ≫ (R.cochainComplexXIso (↑p) p ⋯).hom ≫ R.cocomplex.d p n ≫... | simp only [← cancel_mono (R.cochainComplexXIso n n rfl).inv, Category.assoc, hg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Abelian.Injective.Ext | {
"line": 203,
"column": 9
} | {
"line": 203,
"column": 89
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX Y : C\nR : InjectiveResolution Y\nn : ℕ\nf : X ⟶ R.cocomplex.X n\nm : ℕ\nhm : n + 1 = m\nhf : f ≫ R.cocomplex.d n m = 0\np : ℕ\nhp : p + 1 = n\nx✝ :\n ∃ g,\n g ≫ (R.cochainComplexXIso (↑p) p ⋯).hom ≫ R.cocomplex.d p n ≫... | simp only [← cancel_mono (R.cochainComplexXIso n n rfl).inv, Category.assoc, hg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Abelian.Injective.Ext | {
"line": 213,
"column": 7
} | {
"line": 214,
"column": 61
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX Y : C\nR : InjectiveResolution Y\nn m : ℕ\nhm : n + 1 = m\nf : X ⟶ R.cochainComplex.X ↑n\nhf : f ≫ R.cochainComplex.d ↑n ↑m = 0\n⊢ (f ≫ (R.cochainComplexXIso (↑n) n ⋯).hom) ≫ R.cocomplex.d n m = 0",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.Preradical.Colon | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 13
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : Abelian C\nΦ Ψ : Preradical C\nX : C\n⊢ IsPullback ((Φ.colon Ψ).ι.app X) ((Φ.colonπ Ψ).app X) (Φ.π.app X) (Ψ.ι.app (Φ.quotient.obj X))",
"usedConstants": [
"CategoryTheory.Abelian.Preradical.colonπ",
"CategoryTheory.IsPullback",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.Preradical.Colon | {
"line": 194,
"column": 2
} | {
"line": 195,
"column": 39
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{u_2, u_1} C\ninst✝ : Abelian C\nΦ Ψ : Preradical C\n⊢ IsIso (Φ.toColon Ψ) ↔ IsZero (Φ.quotient ⋙ Ψ.r)",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Over",
"CategoryTheory.Functor",
"_private.Mathlib.CategoryTheory.Abelian.Preradical.Colon.0.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.Injective.Ext | {
"line": 257,
"column": 2
} | {
"line": 258,
"column": 27
} | [
{
"pp": "case h.e_g.e_g.e_f.e_a\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX Y : C\nR : InjectiveResolution Y\nn : ℕ\nf : X ⟶ R.cocomplex.X n\nm : ℕ\nhm : n + 1 = m\nhf : f ≫ R.cocomplex.d n m = 0\nY' : C\nR' : InjectiveResolution Y'\ng : Y ⟶ Y'\nφ : R.Hom R' g\nthis✝ : HasDe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.Pseudoelements | {
"line": 413,
"column": 12
} | {
"line": 413,
"column": 23
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : Abelian C\nP Q : C\nf : P ⟶ Q\nx y : Pseudoelement P\na a' : Over P\nh : pseudoApply f ⟦a⟧ = pseudoApply f ⟦a'⟧\nR : C\np : R ⟶ ((fun g ↦ app f g) a).left\nq : R ⟶ ((fun g ↦ app f g) a').left\nep : Epi p\nw✝¹ : Epi q\ncomm : p ≫ ((fun g ↦ app f g) a).hom ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.Pseudoelements | {
"line": 446,
"column": 2
} | {
"line": 451,
"column": 58
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝ : Ring R\nG : ModuleCat R\nx y : Over G\nP : ModuleCat R\np : P ⟶ x.left\nq : P ⟶ y.left\nhp : Epi p\nhq : Epi q\nH : p ≫ x.hom = q ≫ y.hom\na : ↑G\nha : a ∈ (ModuleCat.Hom.hom x.hom).range\n⊢ a ∈ (ModuleCat.Hom.hom y.hom).range",
"usedConstants": [
"Eq.mpr"... | · obtain ⟨a', ha'⟩ := ha
obtain ⟨a'', ha''⟩ := (ModuleCat.epi_iff_surjective p).1 hp a'
refine ⟨q a'', ?_⟩
dsimp at ha' ⊢
rw [← LinearMap.comp_apply, ← ModuleCat.hom_comp, ← H,
ModuleCat.hom_comp, LinearMap.comp_apply, ha'', ha'] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Abelian.Projective.Ext | {
"line": 213,
"column": 2
} | {
"line": 213,
"column": 30
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX Y : C\nR : ProjectiveResolution X\nn : ℕ\nf : R.complex.X n ⟶ Y\nm : ℕ\nhm : n + 1 = m\nhf : R.complex.d m n ≫ f = 0\np : ℕ\nhp : p + 1 = n\nx✝ :\n ∃ g,\n ((R.cochainComplexXIso (-↑n) n ⋯).hom ≫ R.complex.d n p ≫ (R.coc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.Projective.Ext | {
"line": 223,
"column": 2
} | {
"line": 223,
"column": 50
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\ninst✝ : HasExt C\nX Y : C\nR : ProjectiveResolution X\nn m : ℕ\nhm : n + 1 = m\nf : R.cochainComplex.X (-↑n) ⟶ Y\nhf : R.cochainComplex.d (-↑m) (-↑n) ≫ f = 0\n⊢ (R.cochainComplexXIso (-↑m) m ⋯).hom ≫ R.complex.d m n ≫ (R.cochainComplexXIso (-↑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.RightDerived | {
"line": 311,
"column": 22
} | {
"line": 313,
"column": 49
} | [
{
"pp": "C : Type u\ninst✝⁵ : Category.{v, u} C\nD : Type u_1\ninst✝⁴ : Category.{v_1, u_1} D\ninst✝³ : Abelian C\ninst✝² : HasInjectiveResolutions C\ninst✝¹ : Abelian D\nF : C ⥤ D\ninst✝ : F.Additive\nX Y : C\nf : X ⟶ Y\n⊢ F.map f ≫\n (injectiveResolution Y).toRightDerivedZero' F ≫\n ((F.mapHomolog... | InjectiveResolution.toRightDerivedZero'_naturality_assoc f
(injectiveResolution X) (injectiveResolution Y)
(InjectiveResolution.desc f _ _) (by simp), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Abelian.SerreClass.Localization | {
"line": 60,
"column": 4
} | {
"line": 60,
"column": 46
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : Abelian C\nP : ObjectProperty C\ninst✝ : P.IsSerreClass\nX Y : C\nf : X ⟶ Y\nhf : P (Abelian.image f)\n⊢ P.isoModSerre (kernel.ι f)",
"usedConstants": [
"_private.Mathlib.CategoryTheory.Abelian.SerreClass.Localization.0.CategoryT... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.SerreClass.Localization | {
"line": 112,
"column": 6
} | {
"line": 112,
"column": 58
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Abelian C\nD : Type u'\ninst✝³ : Category.{v', u'} D\nL : C ⥤ D\nP : ObjectProperty C\ninst✝² : P.IsSerreClass\nE : Type u''\ninst✝¹ : Category.{v'', u''} E\ninst✝ : Abelian E\nX' X Y : C\nf₁ f₂ : X ⟶ Y\ns : X' ⟶ X\nhs : P.isoModSerre s\ne... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.SerreClass.Localization | {
"line": 124,
"column": 6
} | {
"line": 124,
"column": 58
} | [
{
"pp": "case refine_2\nC : Type u\ninst✝⁵ : Category.{v, u} C\ninst✝⁴ : Abelian C\nD : Type u'\ninst✝³ : Category.{v', u'} D\nL : C ⥤ D\nP : ObjectProperty C\ninst✝² : P.IsSerreClass\nE : Type u''\ninst✝¹ : Category.{v'', u''} E\ninst✝ : Abelian E\nX Y Y' : C\nf₁ f₂ : X ⟶ Y\ns : Y ⟶ Y'\nhs : P.isoModSerre s\ne... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.SerreClass.Localization | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 13
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : Abelian C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nL : C ⥤ D\nP : ObjectProperty C\ninst✝³ : P.IsSerreClass\ninst✝² : L.IsLocalization P.isoModSerre\ninst✝¹ : Preadditive D\ninst✝ : L.Additive\nX Y : C\nh : P.isoModSerre 0\n⊢ P X",
"usedConstants"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.SerreClass.Localization | {
"line": 143,
"column": 15
} | {
"line": 143,
"column": 47
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : Abelian C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nL : C ⥤ D\nP : ObjectProperty C\ninst✝³ : P.IsSerreClass\ninst✝² : L.IsLocalization P.isoModSerre\ninst✝¹ : Preadditive D\ninst✝ : L.Additive\n⊢ IsZero (L.obj 0)",
"usedConstants": [
"Eq.mpr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.SerreClass.Localization | {
"line": 166,
"column": 4
} | {
"line": 166,
"column": 20
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : Abelian C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nL : C ⥤ D\nP : ObjectProperty C\ninst✝³ : P.IsSerreClass\ninst✝² : L.IsLocalization P.isoModSerre\ninst✝¹ : Preadditive D\ninst✝ : L.Additive\nX Y : C\nf : X ⟶ Y\nthis : L.EssSurj\nx✝ : Mono (L.map f)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.SerreClass.Localization | {
"line": 172,
"column": 63
} | {
"line": 172,
"column": 74
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : Abelian C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nL : C ⥤ D\nP : ObjectProperty C\ninst✝³ : P.IsSerreClass\ninst✝² : L.IsLocalization P.isoModSerre\ninst✝¹ : Preadditive D\ninst✝ : L.Additive\nX Y : C\nf : X ⟶ Y\nthis : L.EssSurj\ntfae_1_to_2 : Mono ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.SerreClass.Localization | {
"line": 199,
"column": 4
} | {
"line": 199,
"column": 20
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : Abelian C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nL : C ⥤ D\nP : ObjectProperty C\ninst✝³ : P.IsSerreClass\ninst✝² : L.IsLocalization P.isoModSerre\ninst✝¹ : Preadditive D\ninst✝ : L.Additive\nX Y : C\nf : X ⟶ Y\nthis : L.EssSurj\nx✝ : Epi (L.map f)\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.SerreClass.Localization | {
"line": 235,
"column": 22
} | {
"line": 235,
"column": 57
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : Abelian C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nL : C ⥤ D\nP : ObjectProperty C\ninst✝³ : P.IsSerreClass\ninst✝² : L.IsLocalization P.isoModSerre\ninst✝¹ : Preadditive D\ninst✝ : L.Additive\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nx✝ : Mono f\n⊢ Mono (L.map f)",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.SerreClass.Localization | {
"line": 238,
"column": 22
} | {
"line": 238,
"column": 56
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : Abelian C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nL : C ⥤ D\nP : ObjectProperty C\ninst✝³ : P.IsSerreClass\ninst✝² : L.IsLocalization P.isoModSerre\ninst✝¹ : Preadditive D\ninst✝ : L.Additive\nX✝ Y✝ : C\nf : X✝ ⟶ Y✝\nx✝ : Epi f\n⊢ Epi (L.map f)",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.SerreClass.Localization | {
"line": 262,
"column": 10
} | {
"line": 262,
"column": 21
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : Abelian C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nL : C ⥤ D\nP : ObjectProperty C\ninst✝³ : P.IsSerreClass\ninst✝² : L.IsLocalization P.isoModSerre\ninst✝¹ : Preadditive D\ninst✝ : L.Additive\nX Y : D\nf : X ⟶ Y\nthis✝ : L.PreservesMonomorphisms\nthi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Action.Concrete | {
"line": 144,
"column": 6
} | {
"line": 144,
"column": 29
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nH N : Subgroup G\ninst✝¹ : Fintype (G ⧸ N)\ninst✝ : N.Normal\nv a b : G\nh : a ≈ b\n⊢ (a * v⁻¹)⁻¹ * (b * v⁻¹) ∈ N",
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"DivInvMonoid.toInv",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.SerreClass.Localization | {
"line": 280,
"column": 4
} | {
"line": 280,
"column": 65
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : Abelian C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nL : C ⥤ D\nP : ObjectProperty C\ninst✝³ : P.IsSerreClass\ninst✝² : L.IsLocalization P.isoModSerre\ninst✝¹ : Preadditive D\ninst✝ : L.Additive\nX✝ Y✝ : D\nf✝ : X✝ ⟶ Y✝\nthis✝ : L.Preserv... | refine ⟨_, _, Abelian.factorThruImage f, inferInstance, ⟨?_⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Abelian.SerreClass.Localization | {
"line": 285,
"column": 10
} | {
"line": 285,
"column": 21
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : Abelian C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nL : C ⥤ D\nP : ObjectProperty C\ninst✝³ : P.IsSerreClass\ninst✝² : L.IsLocalization P.isoModSerre\ninst✝¹ : Preadditive D\ninst✝ : L.Additive\nX Y : D\nf : X ⟶ Y\nthis✝ : L.PreservesEpimorphisms\nthis... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Abelian.SerreClass.Localization | {
"line": 299,
"column": 32
} | {
"line": 299,
"column": 57
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\ninst✝⁵ : Abelian C\nD : Type u'\ninst✝⁴ : Category.{v', u'} D\nL : C ⥤ D\nP : ObjectProperty C\ninst✝³ : P.IsSerreClass\ninst✝² : L.IsLocalization P.isoModSerre\ninst✝¹ : Preadditive D\ninst✝ : L.Additive\nX Y : C\nf : X ⟶ Y\nthis✝¹ : L.PreservesMonomorphisms\nth... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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