module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 100,
"column": 2
} | {
"line": 101,
"column": 93
} | {
"line": 103,
"column": 0
} | [
{
"pp": "c : ℂ\nR θ : ℝ\n⊢ HasDerivAt (circleMap c R) (circleMap 0 R θ * I) θ",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"NormedCommRing.toNormedRing",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSemi... | [] | simpa only [mul_assoc, one_mul, ofRealCLM_apply, circleMap, ofReal_one, zero_add]
using! (((ofRealCLM.hasDerivAt (x := θ)).mul_const I).cexp.const_mul (R : ℂ)).const_add c | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 100,
"column": 2
} | {
"line": 101,
"column": 93
} | {
"line": 103,
"column": 0
} | [
{
"pp": "c : ℂ\nR θ : ℝ\n⊢ HasDerivAt (circleMap c R) (circleMap 0 R θ * I) θ",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"NormedCommRing.toNormedRing",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSemi... | [] | simpa only [mul_assoc, one_mul, ofRealCLM_apply, circleMap, ofReal_one, zero_add]
using! (((ofRealCLM.hasDerivAt (x := θ)).mul_const I).cexp.const_mul (R : ℂ)).const_add c | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 100,
"column": 2
} | {
"line": 101,
"column": 93
} | {
"line": 103,
"column": 0
} | [
{
"pp": "c : ℂ\nR θ : ℝ\n⊢ HasDerivAt (circleMap c R) (circleMap 0 R θ * I) θ",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"instInnerProductSpaceRealComplex",
"NormedCommRing.toNormedRing",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSemi... | [] | simpa only [mul_assoc, one_mul, ofRealCLM_apply, circleMap, ofReal_one, zero_add]
using! (((ofRealCLM.hasDerivAt (x := θ)).mul_const I).cexp.const_mul (R : ℂ)).const_add c | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 625,
"column": 48
} | {
"line": 625,
"column": 50
} | {
"line": 626,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nμ : Measure ℝ\nf : ℝ → E\na₀ : ℝ\nhf : IntegrableOn f (Ioi a₀) μ\na : ℝ\n⊢ a ∈ Ici a₀ → AEStronglyMeasurable ((Ioi a).indicator f) μ",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"Real",
"Set.Ici",... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nμ : Measure ℝ\nf : ℝ → E\na₀ : ℝ\nhf : IntegrableOn f (Ioi a₀) μ\na : ℝ\nha : a ∈ Ici a₀\n⊢ AEStronglyMeasurable ((Ioi a).indicator f) μ"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 628,
"column": 48
} | {
"line": 628,
"column": 50
} | {
"line": 629,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nμ : Measure ℝ\nf : ℝ → E\na₀ : ℝ\nhf : IntegrableOn f (Ioi a₀) μ\na : ℝ\n⊢ a ∈ Ici a₀ → ∀ᵐ (a_2 : ℝ) ∂μ, ‖(Ioi a).indicator f a_2‖ ≤ (Ioi a₀).indicator (norm ∘ f) a_2",
"ppTerm": "?m.128",
"assigned": true,
"usedConstants... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nμ : Measure ℝ\nf : ℝ → E\na₀ : ℝ\nhf : IntegrableOn f (Ioi a₀) μ\na : ℝ\nha : a ∈ Ici a₀\n⊢ ∀ᵐ (a_1 : ℝ) ∂μ, ‖(Ioi a).indicator f a_1‖ ≤ (Ioi a₀).indicator (norm ∘ f) a_1"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 636,
"column": 75
} | {
"line": 636,
"column": 77
} | {
"line": 636,
"column": 78
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nμ : Measure ℝ\nf : ℝ → E\na₀ : ℝ\nhf : IntegrableOn f (Ioi a₀) μ\nx : ℝ\nhx : a₀ < x\na : ℝ\n⊢ a ∈ Iio x → (if a₀ < x then f x else 0) = if a < x then f x else 0",
"ppTerm": "?m.247",
"assigned": true,
"usedConstants": [
... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nμ : Measure ℝ\nf : ℝ → E\na₀ : ℝ\nhf : IntegrableOn f (Ioi a₀) μ\nx : ℝ\nhx : a₀ < x\na : ℝ\nha : a ∈ Iio x\n⊢ (if a₀ < x then f x else 0) = if a < x then f x else 0"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 637,
"column": 50
} | {
"line": 637,
"column": 52
} | {
"line": 637,
"column": 53
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nμ : Measure ℝ\nf : ℝ → E\na₀ : ℝ\nhf : IntegrableOn f (Ioi a₀) μ\nx : ℝ\nhx : ¬a₀ < x\na : ℝ\n⊢ a ∈ Ici a₀ → (if a₀ < x then f x else 0) = if a < x then f x else 0",
"ppTerm": "?m.269",
"assigned": true,
"usedConstants": ... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nμ : Measure ℝ\nf : ℝ → E\na₀ : ℝ\nhf : IntegrableOn f (Ioi a₀) μ\nx : ℝ\nhx : ¬a₀ < x\na : ℝ\nha : a ∈ Ici a₀\n⊢ (if a₀ < x then f x else 0) = if a < x then f x else 0"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 660,
"column": 48
} | {
"line": 660,
"column": 50
} | {
"line": 661,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nμ : Measure ℝ\nf : ℝ → E\na₀ : ℝ\nhf : IntegrableOn f (Iio a₀) μ\na : ℝ\n⊢ a ∈ Iic a₀ → AEStronglyMeasurable ((Iio a).indicator f) μ",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"Real",
"Membershi... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nμ : Measure ℝ\nf : ℝ → E\na₀ : ℝ\nhf : IntegrableOn f (Iio a₀) μ\na : ℝ\nha : a ∈ Iic a₀\n⊢ AEStronglyMeasurable ((Iio a).indicator f) μ"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 663,
"column": 48
} | {
"line": 663,
"column": 50
} | {
"line": 664,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nμ : Measure ℝ\nf : ℝ → E\na₀ : ℝ\nhf : IntegrableOn f (Iio a₀) μ\na : ℝ\n⊢ a ∈ Iic a₀ → ∀ᵐ (a_2 : ℝ) ∂μ, ‖(Iio a).indicator f a_2‖ ≤ (Iio a₀).indicator (norm ∘ f) a_2",
"ppTerm": "?m.128",
"assigned": true,
"usedConstants... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nμ : Measure ℝ\nf : ℝ → E\na₀ : ℝ\nhf : IntegrableOn f (Iio a₀) μ\na : ℝ\nha : a ∈ Iic a₀\n⊢ ∀ᵐ (a_1 : ℝ) ∂μ, ‖(Iio a).indicator f a_1‖ ≤ (Iio a₀).indicator (norm ∘ f) a_1"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 671,
"column": 75
} | {
"line": 671,
"column": 77
} | {
"line": 671,
"column": 78
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nμ : Measure ℝ\nf : ℝ → E\na₀ : ℝ\nhf : IntegrableOn f (Iio a₀) μ\nx : ℝ\nhx : x < a₀\na : ℝ\n⊢ a ∈ Ioi x → (if x < a₀ then f x else 0) = if x < a then f x else 0",
"ppTerm": "?m.247",
"assigned": true,
"usedConstants": [
... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nμ : Measure ℝ\nf : ℝ → E\na₀ : ℝ\nhf : IntegrableOn f (Iio a₀) μ\nx : ℝ\nhx : x < a₀\na : ℝ\nha : a ∈ Ioi x\n⊢ (if x < a₀ then f x else 0) = if x < a then f x else 0"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 672,
"column": 50
} | {
"line": 672,
"column": 52
} | {
"line": 672,
"column": 53
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nμ : Measure ℝ\nf : ℝ → E\na₀ : ℝ\nhf : IntegrableOn f (Iio a₀) μ\nx : ℝ\nhx : ¬x < a₀\na : ℝ\n⊢ a ∈ Iic a₀ → (if x < a₀ then f x else 0) = if x < a then f x else 0",
"ppTerm": "?m.269",
"assigned": true,
"usedConstants": ... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nμ : Measure ℝ\nf : ℝ → E\na₀ : ℝ\nhf : IntegrableOn f (Iio a₀) μ\nx : ℝ\nhx : ¬x < a₀\na : ℝ\nha : a ∈ Iic a₀\n⊢ (if x < a₀ then f x else 0) = if x < a then f x else 0"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 608,
"column": 8
} | {
"line": 608,
"column": 88
} | {
"line": 609,
"column": 6
} | [
{
"pp": "n : ℤ\nhn : n ≠ -1\nc w : ℂ\nR : ℝ\nH : w ∈ sphere c |R| → -1 ≤ n\nz : ℂ\nhne : z ≠ w ∨ -1 ≤ n\n⊢ ↑n + 1 ≠ 0",
"ppTerm": "?m.486",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast_neg",
"Int.cast",
"Eq.mpr",
"NegZeroClass.toNeg",
... | [] | rwa [Ne, ← eq_neg_iff_add_eq_zero, ← Int.cast_one, ← Int.cast_neg, Int.cast_inj] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 665,
"column": 8
} | {
"line": 665,
"column": 15
} | {
"line": 665,
"column": 15
} | [
{
"pp": "case inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ≥0\nn : ℕ\nhR : ↑R ^ n ≠ 0\n⊢ ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c (↑R) θ)‖) * (↑R)⁻¹ ^ n * ↑R ^ n ≤\n (2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c (↑R) θ)‖",
"ppTerm": "?i... | [
"case inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ≥0\nn : ℕ\nhR : ↑R ^ n ≠ 0\n⊢ ((2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c (↑R) θ)‖) * (↑R ^ n)⁻¹ * ↑R ^ n ≤\n (2 * π)⁻¹ * ∫ (θ : ℝ) in 0..2 * π, ‖f (circleMap c (↑R) θ)‖"
] | inv_pow | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 745,
"column": 18
} | {
"line": 745,
"column": 33
} | {
"line": 745,
"column": 34
} | [
{
"pp": "c w : ℂ\nR : ℝ\nhw : w ∈ ball c R\nhR : 0 < R\nH : ∀ (n : ℕ), n ≠ 0 → ∮ (z : ℂ) in C(c, R), (z - c) ^ (-↑n - 1) = 0\nthis : ∮ (z : ℂ) in C(c, R), ((w - c) / (z - c)) ^ 0 * (z - c)⁻¹ = 2 * ↑π * I\nn : ℕ\nhn : n ≠ 0\nz : ℂ\nhz✝ : z ∈ sphere c R\n⊢ (z - c)⁻¹ ^ (n + 1) = (z - c) ^ (-↑n - 1)",
"ppTerm":... | [
"c w : ℂ\nR : ℝ\nhw : w ∈ ball c R\nhR : 0 < R\nH : ∀ (n : ℕ), n ≠ 0 → ∮ (z : ℂ) in C(c, R), (z - c) ^ (-↑n - 1) = 0\nthis : ∮ (z : ℂ) in C(c, R), ((w - c) / (z - c)) ^ 0 * (z - c)⁻¹ = 2 * ↑π * I\nn : ℕ\nhn : n ≠ 0\nz : ℂ\nhz✝ : z ∈ sphere c R\n⊢ (z - c)⁻¹ ^ ↑(n + 1) = (z - c) ^ (-↑n - 1)"
] | ← zpow_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Prod | {
"line": 458,
"column": 4
} | {
"line": 458,
"column": 70
} | {
"line": 459,
"column": 2
} | [
{
"pp": "case h_closed\nα : Type u_1\nβ : Type u_2\nE : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝³ : NormedAddCommGroup E\ninst✝² : SFinite ν\ninst✝¹ : NormedSpace ℝ E\ninst✝ : SFinite μ\nhE : CompleteSpace E\n⊢ IsClosed {f | ∫ (z : α × β), ↑↑f z ∂μ.pr... | [] | exact isClosed_eq continuous_integral continuous_integral_integral | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.Prod | {
"line": 458,
"column": 4
} | {
"line": 458,
"column": 70
} | {
"line": 459,
"column": 2
} | [
{
"pp": "case h_closed\nα : Type u_1\nβ : Type u_2\nE : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝³ : NormedAddCommGroup E\ninst✝² : SFinite ν\ninst✝¹ : NormedSpace ℝ E\ninst✝ : SFinite μ\nhE : CompleteSpace E\n⊢ IsClosed {f | ∫ (z : α × β), ↑↑f z ∂μ.pr... | [] | exact isClosed_eq continuous_integral continuous_integral_integral | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Prod | {
"line": 458,
"column": 4
} | {
"line": 458,
"column": 70
} | {
"line": 459,
"column": 2
} | [
{
"pp": "case h_closed\nα : Type u_1\nβ : Type u_2\nE : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝³ : NormedAddCommGroup E\ninst✝² : SFinite ν\ninst✝¹ : NormedSpace ℝ E\ninst✝ : SFinite μ\nhE : CompleteSpace E\n⊢ IsClosed {f | ∫ (z : α × β), ↑↑f z ∂μ.pr... | [] | exact isClosed_eq continuous_integral continuous_integral_integral | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Prod | {
"line": 504,
"column": 4
} | {
"line": 504,
"column": 64
} | {
"line": 505,
"column": 2
} | [
{
"pp": "E : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nF : ℝ → ℝ → E\na b c d : ℝ\nh : IntegrableOn (uncurry F) (uIoc a b ×ˢ uIoc c d) volume\n⊢ ∫ (x : ℝ) in c..d, (if a ≤ b then 1 else -1) • ∫ (y : ℝ) in uIoc a b, F y x ∂volume =\n ∫ (y : ℝ) in c..d, ∫ (x : ℝ) in a..b, F x y",
"p... | [] | simp_rw [intervalIntegral.intervalIntegral_eq_integral_uIoc] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Integral.Prod | {
"line": 504,
"column": 4
} | {
"line": 504,
"column": 64
} | {
"line": 505,
"column": 2
} | [
{
"pp": "E : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nF : ℝ → ℝ → E\na b c d : ℝ\nh : IntegrableOn (uncurry F) (uIoc a b ×ˢ uIoc c d) volume\n⊢ ∫ (x : ℝ) in c..d, (if a ≤ b then 1 else -1) • ∫ (y : ℝ) in uIoc a b, F y x ∂volume =\n ∫ (y : ℝ) in c..d, ∫ (x : ℝ) in a..b, F x y",
"p... | [] | simp_rw [intervalIntegral.intervalIntegral_eq_integral_uIoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Prod | {
"line": 504,
"column": 4
} | {
"line": 504,
"column": 64
} | {
"line": 505,
"column": 2
} | [
{
"pp": "E : Type u_3\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nF : ℝ → ℝ → E\na b c d : ℝ\nh : IntegrableOn (uncurry F) (uIoc a b ×ˢ uIoc c d) volume\n⊢ ∫ (x : ℝ) in c..d, (if a ≤ b then 1 else -1) • ∫ (y : ℝ) in uIoc a b, F y x ∂volume =\n ∫ (y : ℝ) in c..d, ∫ (x : ℝ) in a..b, F x y",
"p... | [] | simp_rw [intervalIntegral.intervalIntegral_eq_integral_uIoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 258,
"column": 35
} | {
"line": 259,
"column": 96
} | {
"line": 259,
"column": 96
} | [
{
"pp": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nz w : ℂ\nHd : DifferentiableOn ℝ f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHi : IntegrableOn (fun z ↦ I • (fderiv ℝ f z) 1 - (fderiv ℝ f z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]]) volume\nx : ℂ\nhx : x ∈ Ioo (min z.re w.re) (max z... | [] | by
simpa only [← mem_interior_iff_mem_nhds, interior_reProdIm, uIcc, interior_Icc] using hx.1 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.DivergenceTheorem | {
"line": 274,
"column": 2
} | {
"line": 274,
"column": 49
} | {
"line": 275,
"column": 2
} | [
{
"pp": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : s.Countable\nHc : ContinuousOn f (Set.Icc a b)\nHd : ... | [
"case inl\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : s.Countable\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ ... | rcases em (∃ i, a i = b i) with (⟨i, hi⟩ | hne) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 80,
"column": 51
} | {
"line": 85,
"column": 76
} | {
"line": 88,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nS : Type u_2\ninst✝³ : Monoid S\ninst✝² : DistribMulAction S R\ninst✝¹ : IsScalarTower S R R\ninst✝ : SMulCommClass S R R\ns : S\nn : ℕ\nm : Multiset R\n⊢ s ^ n • m.esymm n = (map (fun x ↦ s • x) m).esymm n",
"ppTerm": "?m.30",
"assigned": true,
"usedC... | [] | by
rw [esymm, smul_sum, map_map]
trans ((powersetCard n m).map (fun x : Multiset R ↦ s ^ card x • x.prod)).sum
· refine congr_arg _ (map_congr rfl (fun x hx ↦ ?_))
rw [Function.comp_apply, (mem_powersetCard.1 hx).2]
· simp_rw [smul_prod, esymm, powersetCard_map, map_map, Function.comp_def] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.PolynomialGaloisGroup | {
"line": 361,
"column": 36
} | {
"line": 379,
"column": 47
} | {
"line": 381,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝¹ : Field F\np : F[X]\ninst✝ : CharZero F\np_irr : Irreducible p\np_deg : Nat.Prime p.natDegree\n⊢ p.natDegree ∣ Nat.card p.Gal",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"WithBot.instPreorder",
"Eq.mpr",
"IntermediateFiel... | [] | by
rw [Gal.card_of_separable p_irr.separable]
have hp : p.degree ≠ 0 := fun h =>
Nat.Prime.ne_zero p_deg (natDegree_eq_zero_iff_degree_le_zero.mpr (le_of_eq h))
let α : p.SplittingField :=
rootOfSplits (SplittingField.splits p) (by rwa [degree_map])
have hα : IsIntegral F α := .of_finite F α
use Modul... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Polynomial | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 70
} | {
"line": 215,
"column": 4
} | [
{
"pp": "case inr\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\nB : ℝ\nd : ℕ\nf : F →+* K\np : F[X]\nh1 : p.Monic\nh2 : (map f p).Splits\nh3 : p.natDegree ≤ d\nh4 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B\ni : ℕ\nhB : B < 0\n⊢ ‖(map f p).coeff i‖ ≤ max B 1 ^ d * ↑(d.choose (d / 2))",
"ppTe... | [
"case inr\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\nB : ℝ\nd : ℕ\nf : F →+* K\np : F[X]\nh1 : p.Monic\nh2 : (map f p).Splits\nh3 : p.natDegree ≤ d\nh4 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B\ni : ℕ\nhB : B < 0\n⊢ ‖if i = 0 then 1 else 0‖ ≤ max B 1 ^ d * ↑(d.choose (d / 2))"
] | rw [eq_one_of_roots_le hB h1 h2 h4, Polynomial.map_one, coeff_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 62
} | {
"line": 66,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝⁴ : Field K\nA : Type u_2\ninst✝³ : Field A\ninst✝² : CharZero A\ninst✝¹ : NumberField K\ninst✝ : IsAlgClosed A\n⊢ Nonempty (K →+* A)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Rat",
"RingHom",
"Rat.commSe... | [
"K : Type u_1\ninst✝⁴ : Field K\nA : Type u_2\ninst✝³ : Field A\ninst✝² : CharZero A\ninst✝¹ : NumberField K\ninst✝ : IsAlgClosed A\n⊢ 0 < finrank ℚ K"
] | rw [← Fintype.card_pos_iff, NumberField.Embeddings.card K A] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Valuation.Archimedean | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 22
} | {
"line": 62,
"column": 2
} | [
{
"pp": "case refine_1\nF : Type u_1\nΓ₀ : Type u_2\nO : Type u_3\ninst✝⁴ : Field F\ninst✝³ : LinearOrderedCommGroupWithZero Γ₀\ninst✝² : CommRing O\ninst✝¹ : Algebra O F\nv : Valuation F Γ₀\ninst✝ : Nontrivial (↥(MonoidHom.mrange v))ˣ\nhv : v.Integers O\nh : (Set.range (⇑v ∘ ⇑(algebraMap O F))).WellFoundedOn f... | [] | exact ⟨y, by simp⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.DivergenceTheorem | {
"line": 514,
"column": 36
} | {
"line": 536,
"column": 51
} | {
"line": 538,
"column": 0
} | [
{
"pp": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf g : ℝ × ℝ → E\nf' g' : ℝ × ℝ → ℝ × ℝ →L[ℝ] E\na₁ a₂ b₁ b₂ : ℝ\ns : Set (ℝ × ℝ)\nhs : s.Countable\nHcf : ContinuousOn f ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHcg : ContinuousOn g ([[a₁, b₁]] ×ˢ [[a₂, b₂]])\nHdf : ∀ x ∈ Set.Ioo (min a₁ b₁) (max ... | [] | by
wlog h₁ : a₁ ≤ b₁ generalizing a₁ b₁
· specialize this b₁ a₁
rw [uIcc_comm b₁ a₁, min_comm b₁ a₁, max_comm b₁ a₁] at this
simp only [intervalIntegral.integral_symm b₁ a₁]
refine (congr_arg Neg.neg (this Hcf Hcg Hdf Hdg Hi (le_of_not_ge h₁))).trans ?_; abel
wlog h₂ : a₂ ≤ b₂ generalizing a₂ b₂
· s... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 54
} | {
"line": 101,
"column": 0
} | [
{
"pp": "S : Type u_1\ninst✝¹ : SeminormedGroup S\ninst✝ : IsUltrametricDist S\nx y : S\nh : ‖x‖ ≠ ‖y‖\n⊢ max (dist x⁻¹ 1) (dist 1 y) = max ‖x⁻¹⁻¹‖ ‖y‖",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOn... | [] | simp only [dist_one_right, dist_one_left, norm_inv'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Algebra.Valued.NormedValued | {
"line": 83,
"column": 16
} | {
"line": 83,
"column": 57
} | {
"line": 83,
"column": 58
} | [
{
"pp": "case neg\nK : Type u_1\nhK : NormedField K\ninst✝ : IsUltrametricDist K\nU : Set K\nε : ℝ\nhε : ε > 0\nh : Metric.ball 0 ε ⊆ U\nH : Subsingleton (ofClass valuation).ValueGroup₀ˣ\nx : K\nhx : valuation.restrict x < 1\nhx0 : ¬x = 0\n⊢ False",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstant... | [
"case neg\nK : Type u_1\nhK : NormedField K\ninst✝ : IsUltrametricDist K\nU : Set K\nε : ℝ\nhε : ε > 0\nh : Metric.ball 0 ε ⊆ U\nH : Subsingleton (ofClass valuation).ValueGroup₀ˣ\nx : K\nhx : valuation.restrict x < 1\nhx0 : ¬valuation.restrict x = 0\n⊢ False"
] | ← (valuation (K := K)).restrict.zero_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.LexOrder | {
"line": 147,
"column": 4
} | {
"line": 149,
"column": 52
} | {
"line": 150,
"column": 2
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝² : Semiring R\ninst✝¹ : LinearOrder σ\ninst✝ : WellFoundedGT σ\nφ ψ : MvPowerSeries σ R\nd : σ →₀ ℕ\nhd : ↑(toLex d) < φ.lexOrder + ψ.lexOrder\nu v : σ →₀ ℕ\nh : u + v = d\nthis : ↑(toLex u) < φ.lexOrder ∨ ↑(toLex v) < ψ.lexOrder\n⊢ (coeff u) φ * (coeff v) ψ = 0",
... | [] | rcases this with (hu | hv)
· rw [coeff_eq_zero_of_lt_lexOrder hu, zero_mul]
· rw [coeff_eq_zero_of_lt_lexOrder hv, mul_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.LexOrder | {
"line": 147,
"column": 4
} | {
"line": 149,
"column": 52
} | {
"line": 150,
"column": 2
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝² : Semiring R\ninst✝¹ : LinearOrder σ\ninst✝ : WellFoundedGT σ\nφ ψ : MvPowerSeries σ R\nd : σ →₀ ℕ\nhd : ↑(toLex d) < φ.lexOrder + ψ.lexOrder\nu v : σ →₀ ℕ\nh : u + v = d\nthis : ↑(toLex u) < φ.lexOrder ∨ ↑(toLex v) < ψ.lexOrder\n⊢ (coeff u) φ * (coeff v) ψ = 0",
... | [] | rcases this with (hu | hv)
· rw [coeff_eq_zero_of_lt_lexOrder hu, zero_mul]
· rw [coeff_eq_zero_of_lt_lexOrder hv, mul_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Inverse | {
"line": 216,
"column": 6
} | {
"line": 217,
"column": 61
} | {
"line": 217,
"column": 61
} | [
{
"pp": "σ : Type u_1\nk : Type u_3\ninst✝ : Field k\nφ : MvPowerSeries σ k\nh : constantCoeff φ = 0\nn : σ →₀ ℕ\n⊢ (if n = 0 then (constantCoeff φ)⁻¹\n else -(constantCoeff φ)⁻¹ * ∑ x ∈ antidiagonal n, if x.2 < n then (coeff x.1) φ * (coeff x.2) φ⁻¹ else 0) =\n (coeff n) 0",
"ppTerm": "?m.52",
"a... | [] | split_ifs <;>
simp only [h, map_zero, zero_mul, inv_zero, neg_zero] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.NumberTheory.NumberField.Completion.FinitePlace | {
"line": 292,
"column": 2
} | {
"line": 292,
"column": 56
} | {
"line": 294,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝⁶ : Field K\nR : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : Algebra R K\ninst✝³ : IsDedekindDomain R\ninst✝² : IsFractionRing R K\nv : HeightOneSpectrum R\ninst✝¹ : Module.Finite ℤ R\ninst✝ : Module.Free ℤ R\nx : R\n⊢ (HeightOneSpectrum.adicAbv K v) ((algebraMap R K) x) ≤ 1",
"ppTer... | [] | exact v.adicAbv_coe_le_one (one_lt_absNorm_nnreal v) x | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.Completion.FinitePlace | {
"line": 313,
"column": 6
} | {
"line": 313,
"column": 21
} | {
"line": 313,
"column": 22
} | [
{
"pp": "K : Type u_1\ninst✝⁶ : Field K\nR : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : Algebra R K\ninst✝³ : IsDedekindDomain R\ninst✝² : IsFractionRing R K\nv : HeightOneSpectrum R\ninst✝¹ : Module.Finite ℤ R\ninst✝ : Module.Free ℤ R\nx : R\nh_x_nezero : x ≠ 0\n⊢ ↑(absNorm v.asIdeal) ^ Multiplicative.toAdd (unze... | [
"K : Type u_1\ninst✝⁶ : Field K\nR : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : Algebra R K\ninst✝³ : IsDedekindDomain R\ninst✝² : IsFractionRing R K\nv : HeightOneSpectrum R\ninst✝¹ : Module.Finite ℤ R\ninst✝ : Module.Free ℤ R\nx : R\nh_x_nezero : x ≠ 0\n⊢ ↑(absNorm v.asIdeal) ^ Multiplicative.toAdd (unzero ⋯) *\n ... | ← zpow_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic | {
"line": 413,
"column": 10
} | {
"line": 413,
"column": 12
} | {
"line": 413,
"column": 13
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Projective R\na b : R\nh :\n b ^ 2 + W'.toAffine.a₁ * a * b + W'.toAffine.a₃ * b -\n (a ^ 3 + W'.toAffine.a₂ * a ^ 2 + W'.toAffine.a₄ * a + W'.toAffine.a₆) =\n 0\n⊢ W'.a₁ * b - (3 * a ^ 2 + 2 * W'.a₂ * a + W'.a₄) = 0 →\n 2 * b + W'.toAffine.a₁ * a + W'.t... | [
"R : Type r\ninst✝ : CommRing R\nW' : Projective R\na b : R\nh :\n b ^ 2 + W'.toAffine.a₁ * a * b + W'.toAffine.a₃ * b -\n (a ^ 3 + W'.toAffine.a₂ * a ^ 2 + W'.toAffine.a₄ * a + W'.toAffine.a₆) =\n 0\nha : W'.a₁ * b - (3 * a ^ 2 + 2 * W'.a₂ * a + W'.a₄) = 0\n⊢ 2 * b + W'.toAffine.a₁ * a + W'.toAffine.a₃ = ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point | {
"line": 457,
"column": 4
} | {
"line": 459,
"column": 74
} | {
"line": 460,
"column": 2
} | [
{
"pp": "case neg\nF : Type u\ninst✝ : Field F\nW : Projective F\nP : Fin 3 → F\nu : F\nhu : IsUnit u\nhP : W.Nonsingular P\nhPz : ¬P z = 0\n⊢ toAffine W (u • P) = toAffine W P",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"WeierstrassCurve.Projecti... | [] | · rw [toAffine_of_Z_ne_zero ((nonsingular_smul P hu).mpr hP) <| mul_ne_zero hu.ne_zero hPz,
toAffine_of_Z_ne_zero hP hPz, Affine.Point.some.injEq]
simp only [smul_fin3_ext, mul_div_mul_left _ _ hu.ne_zero, and_self] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.GluingOneHypercover | {
"line": 57,
"column": 28
} | {
"line": 61,
"column": 35
} | {
"line": 63,
"column": 0
} | [
{
"pp": "D : GlueData\ni₁ i₂ : D.J\nW : Scheme\np₁ : W ⟶ D.U i₁\np₂ : W ⟶ D.U i₂\nfac : p₁ ≫ D.ι i₁ = p₂ ≫ D.ι i₂\n⊢ { I₀ := D.J, X := D.U, f := D.ι, I₁ := fun x x_1 ↦ PUnit.{u + 1}, Y := fun i₁ i₂ x ↦ D.V (i₁, i₂),\n p₁ := fun i₁ i₂ x ↦ D.f i₁ i₂, p₂ := fun i₁ i₂ x ↦ D.t i₁ i₂ ≫ D.f i₂ i₁, w := ⋯ }.si... | [] | by
refine zariskiTopology.superset_covering (fun T g _ ↦ ?_) (zariskiTopology.top_mem _)
have ⟨φ, h₁, h₂⟩ := PullbackCone.IsLimit.lift' (D.vPullbackConeIsLimit i₁ i₂)
(g ≫ p₁) (g ≫ p₂) (by simpa using g ≫= fac)
exact ⟨⟨⟩, φ, h₁.symm, h₂.symm⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Smooth.Locus | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 50
} | {
"line": 131,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : FinitePresentation R A\nx : PrimeSpectrum A\nhx : x ∈ smoothLocus R A\nf : A\nhxf : x ∈ ↑(basicOpen f)\nhf : ↑(basicOpen f) ⊆ Module.freeLocus A Ω[A⁄R]\n⊢ ∃ t ⊆ smoothLocus R A, IsOpen t ∧ x ∈ t",
"p... | [
"R : Type u_1\nA : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : FinitePresentation R A\nx : PrimeSpectrum A\nhx : x ∈ smoothLocus R A\nf : A\nhxf : x ∈ ↑(basicOpen f)\nhf : Module.Projective (Localization.Away f) (LocalizedModule.Away f Ω[A⁄R])\n⊢ ∃ t ⊆ smoothLocus R A, IsOpen t... | rw [Module.basicOpen_subset_freeLocus_iff] at hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Unramified.Pi | {
"line": 47,
"column": 39
} | {
"line": 47,
"column": 82
} | {
"line": 48,
"column": 4
} | [
{
"pp": "R : Type u_1\nI : Type u_2\ninst✝⁵ : Finite I\nf : I → Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : (i : I) → CommRing (f i)\ninst✝² : (i : I) → Algebra R (f i)\nval✝ : Fintype I\nH : ∀ (i : I), FormallyUnramified R (f i)\nB : Type (max u_2 u_3)\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nJ : Ideal B\nhJ : J... | [] | simp [IsIdempotentElem, e, ← Pi.single_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Etale.Field | {
"line": 87,
"column": 4
} | {
"line": 88,
"column": 18
} | {
"line": 89,
"column": 2
} | [
{
"pp": "case h\nK : Type u_1\nL : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : Algebra.IsSeparable K L\ninst✝ : EssFiniteType K L\nthis✝ : FormallyUnramified K L\nthis : Module.Finite K L\nB : Type (max u_1 u_2)\nx✝¹ : CommRing B\nx✝ : Algebra K B\nI : Ideal B\nh : I ^ 2 = ⊥\nf ... | [] | apply pb.algHom_ext
simp [hx, hx'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Etale.Field | {
"line": 87,
"column": 4
} | {
"line": 88,
"column": 18
} | {
"line": 89,
"column": 2
} | [
{
"pp": "case h\nK : Type u_1\nL : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\ninst✝¹ : Algebra.IsSeparable K L\ninst✝ : EssFiniteType K L\nthis✝ : FormallyUnramified K L\nthis : Module.Finite K L\nB : Type (max u_1 u_2)\nx✝¹ : CommRing B\nx✝ : Algebra K B\nI : Ideal B\nh : I ^ 2 = ⊥\nf ... | [] | apply pb.algHom_ext
simp [hx, hx'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Smooth.Fiber | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 64
} | {
"line": 87,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nP : Type u_3\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing S\ninst✝¹² : Algebra R S\ninst✝¹¹ : Module.Flat R S\ninst✝¹⁰ : CommRing P\ninst✝⁹ : Algebra R P\ninst✝⁸ : Algebra P S\ninst✝⁷ : IsScalarTower R P S\ninst✝⁶ : IsLocalRing R\ninst✝⁵ : IsLocalRing S\ninst✝⁴ : IsLocalHom (al... | [
"R : Type u_1\nS : Type u_2\nP : Type u_3\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing S\ninst✝¹² : Algebra R S\ninst✝¹¹ : Module.Flat R S\ninst✝¹⁰ : CommRing P\ninst✝⁹ : Algebra R P\ninst✝⁸ : Algebra P S\ninst✝⁷ : IsScalarTower R P S\ninst✝⁶ : IsLocalRing R\ninst✝⁵ : IsLocalRing S\ninst✝⁴ : IsLocalHom (algebraMap R S... | have := IsScalarTower.of_algebraMap_eq' φ.comp_algebraMap.symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Smooth.Fiber | {
"line": 125,
"column": 2
} | {
"line": 127,
"column": 77
} | {
"line": 129,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nP : Type u_3\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing S\ninst✝¹² : Algebra R S\ninst✝¹¹ : Module.Flat R S\ninst✝¹⁰ : CommRing P\ninst✝⁹ : Algebra R P\ninst✝⁸ : Algebra P S\ninst✝⁷ : IsScalarTower R P S\ninst✝⁶ : IsLocalRing R\ninst✝⁵ : IsLocalRing S\ninst✝⁴ : IsLocalHom (al... | [
"R : Type u_1\nS : Type u_2\nP : Type u_3\ninst✝¹⁴ : CommRing R\ninst✝¹³ : CommRing S\ninst✝¹² : Algebra R S\ninst✝¹¹ : Module.Flat R S\ninst✝¹⁰ : CommRing P\ninst✝⁹ : Algebra R P\ninst✝⁸ : Algebra P S\ninst✝⁷ : IsScalarTower R P S\ninst✝⁶ : IsLocalRing R\ninst✝⁵ : IsLocalRing S\ninst✝⁴ : IsLocalHom (algebraMap R S... | let eₗ : 𝓀[S] ⊗[Pp] RingHom.ker φ ≃ₗ[S] 𝓀[S] ⊗[P] RingHom.ker (algebraMap P S) :=
(AlgebraTensorModule.congr (.refl 𝓀[S] 𝓀[S]) e₁).restrictScalars S ≪≫ₗ
(AlgebraTensorModule.cancelBaseChange P Pp Sp 𝓀[S] _).restrictScalars S | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 305,
"column": 76
} | {
"line": 307,
"column": 89
} | {
"line": 309,
"column": 0
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Projective F\ninst✝ : DecidableEq F\nP Q : Fin 3 → F\nhP : W.Equation P\nhQ : W.Equation Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z = Q x * P z\nhy : P y * Q z ≠ W.negY Q * P z\n⊢ W.dblX P / W.dblZ P =\n W.toAffine.addX (P x / P z) (Q x / Q z) (W.toAffine.slope... | [] | by
rw [dblX_eq hP hPz, dblZ, toAffine_slope_of_eq hPz hQz hx hy, ← (X_eq_iff hPz hQz).mp hx,
toAffine_addX_of_eq hPz <| sub_ne_zero.mpr <| Y_ne_negY_of_Y_ne' hP hQ hPz hQz hx hy] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Extension.Presentation.Submersive | {
"line": 244,
"column": 88
} | {
"line": 246,
"column": 74
} | {
"line": 248,
"column": 0
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nr : R\ninst✝ : IsLocalization.Away r S\n⊢ (localizationAway S r).jacobian = (algebraMap R S) r",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
... | [] | by
rw [jacobian_eq_jacobiMatrix_det, localizationAway_jacobiMatrix]
simp [show Fintype.card (localizationAway r (S := S)).rels = 1 from rfl] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Extension.Presentation.Submersive | {
"line": 274,
"column": 2
} | {
"line": 275,
"column": 45
} | {
"line": 276,
"column": 2
} | [
{
"pp": "R : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\nι' : Type u_1\nσ' : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing T\ninst✝⁶ : Algebra R T\ninst✝⁵ : Algebra S T\ninst✝⁴ : IsScalarTower R S T\nQ : PreSubmersivePresentation S T ι' σ'\nP : PreSub... | [
"R : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\nι' : Type u_1\nσ' : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing T\ninst✝⁶ : Algebra R T\ninst✝⁵ : Algebra S T\ninst✝⁴ : IsScalarTower R S T\nQ : PreSubmersivePresentation S T ι' σ'\nP : PreSubmersivePrese... | have : Nat.card σ ≤ Nat.card ι :=
card_relations_le_card_vars_of_isFinite P | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Smooth.StandardSmoothCotangent | {
"line": 182,
"column": 6
} | {
"line": 182,
"column": 29
} | {
"line": 182,
"column": 29
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nι : Type u_3\nσ : Type u_4\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Finite σ\nP : SubmersivePresentation R S ι σ\nx✝ : P.toExtension.Cotangent\n⊢ P.sectionCotangent (P.toExtension.cotangentComplex x✝) = x✝",
"ppTerm": "?m.71",
"assigne... | [
"R : Type u_1\nS : Type u_2\nι : Type u_3\nσ : Type u_4\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Finite σ\nP : SubmersivePresentation R S ι σ\nx✝ : P.toExtension.Cotangent\n⊢ ∀ (i : σ), (P.cotangentSpaceBasis.repr (P.toExtension.cotangentComplex x✝)) (P.map i) = P.cotangentComplexAux... | sectionCotangent_eq_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Smooth.StandardSmoothCotangent | {
"line": 190,
"column": 6
} | {
"line": 190,
"column": 29
} | {
"line": 190,
"column": 29
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nι : Type u_3\nσ : Type u_4\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Finite σ\nP : SubmersivePresentation R S ι σ\ni : ι\nhi : ¬P.sectionCotangent (P.cotangentSpaceBasis i) = 0\n⊢ i ∈ Set.range P.map",
"ppTerm": "?m.42",
"assigned": tru... | [
"R : Type u_1\nS : Type u_2\nι : Type u_3\nσ : Type u_4\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Finite σ\nP : SubmersivePresentation R S ι σ\ni : ι\nhi : ¬∀ (i_1 : σ), (P.cotangentSpaceBasis.repr (P.cotangentSpaceBasis i)) (P.map i_1) = P.cotangentComplexAux 0 i_1\n⊢ i ∈ Set.range P... | sectionCotangent_eq_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Extension.Cotangent.LocalizationAway | {
"line": 53,
"column": 6
} | {
"line": 53,
"column": 21
} | {
"line": 53,
"column": 22
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nι : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\ng : S\ninst✝ : IsLocalization.Away g T\nP : Generators R S ι\nf : P.Ring\nh : (algebraMa... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\nι : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\ng : S\ninst✝ : IsLocalization.Away g T\nP : Generators R S ι\nf : P.Ring\nh : (algebraMap P.Ring S) ... | Hom.toAlgHom_X, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 97,
"column": 8
} | {
"line": 97,
"column": 40
} | {
"line": 97,
"column": 41
} | [
{
"pp": "case refine_3\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nσ✝ : Type u_3\nι : Type u_4\nP : Generators R S ι\nσ : Type u_5\nb : Module.Basis σ S P.toExtension.Cotangent\nD : Aux P b\n⊢ (algebraMap P.Ring S) D.g = 1",
"ppTerm": "?refine_3",
"assigne... | [
"case refine_3\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nσ✝ : Type u_3\nι : Type u_4\nP : Generators R S ι\nσ : Type u_5\nb : Module.Basis σ S P.toExtension.Cotangent\nD : Aux P b\n⊢ (algebraMap P.Ring S) D.g = (algebraMap P.Ring S) 1"
] | ← map_one (algebraMap P.Ring S), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.RingHom.StandardSmooth | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 64
} | {
"line": 216,
"column": 2
} | [
{
"pp": "n : ℕ\nR : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Algebra.IsStandardSmoothOfRelativeDimension n R S\nthis : (α : Type) → [_root_.Finite α] → Fintype α := Fintype.ofFinite\nι σ : Type\nw✝¹ : _root_.Finite σ\nw✝ : _root_.Finite ι\nP : Algebra.Submersiv... | [
"n : ℕ\nR : Type u\nS : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : Algebra.IsStandardSmoothOfRelativeDimension n R S\nthis✝ : (α : Type) → [_root_.Finite α] → Fintype α := Fintype.ofFinite\nι σ : Type\nw✝¹ : _root_.Finite σ\nw✝ : _root_.Finite ι\nP : Algebra.SubmersivePresentati... | have := IsScalarTower.of_algebraMap_eq' φ.comp_algebraMap.symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 68,
"column": 56
} | {
"line": 80,
"column": 18
} | {
"line": 83,
"column": 0
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝² : Field k\ninst✝¹ : Field K\ninst✝ : Algebra k K\na : ι → K\nF : MvPolynomial ι k\nhF : Irreducible F\ni : ι\nH : AlgebraicIndependent k fun x ↦ a ↑x\n⊢ Irreducible (F.toPolynomialAdjoinImageCompl a i)",
"ppTerm": "?m.45",
"assigned": true,
"... | [] | by
classical
unfold toPolynomialAdjoinImageCompl
have hc : a '' {i}ᶜ = Set.range (fun x : {j | j ≠ i} ↦ a x) := by ext; simp
let d : {j // j ≠ i} ≃ {j | j ≠ i} := .subtypeEquivRight (by simp)
refine (congrArg Irreducible ?_).mp <|
hF.map (renameEquiv k ((Equiv.optionSubtypeNe i).symm)) |>.map
(optio... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 657,
"column": 2
} | {
"line": 658,
"column": 7
} | {
"line": 660,
"column": 0
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Projective R\nP Q : Fin 3 → R\nu v : R\n⊢ W'.negAddY (u • P) (v • Q) = (u * v) ^ 2 * W'.negAddY P Q",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
... | [] | simp only [negAddY, smul_fin3_ext]
ring1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 657,
"column": 2
} | {
"line": 658,
"column": 7
} | {
"line": 660,
"column": 0
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Projective R\nP Q : Fin 3 → R\nu v : R\n⊢ W'.negAddY (u • P) (v • Q) = (u * v) ^ 2 * W'.negAddY P Q",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
... | [] | simp only [negAddY, smul_fin3_ext]
ring1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 123,
"column": 4
} | {
"line": 127,
"column": 20
} | {
"line": 129,
"column": 0
} | [
{
"pp": "case refine_2\nk : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝² : Field k\ninst✝¹ : Field K\ninst✝ : Algebra k K\na : ι → K\nF : MvPolynomial ι k\nHF : ∀ (F' : MvPolynomial ι k), F' ≠ 0 → (aeval a) F' = 0 → F.totalDegree ≤ F'.totalDegree\nσ : ι →₀ ℕ\nhσ : σ ∈ F.support\ni : ι\nhσi : σ i ≠ 0\nH : (F.toP... | [] | apply_fun Subalgebra.val _ at H
simp_rw [toPolynomialAdjoinImageCompl, Polynomial.coe_mapAlgHom, Polynomial.coeff_map,
AlgHom.coe_toRingHom, map_zero] at H
simp_rw [← H, ← AlgHom.comp_apply]
congr; ext; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 123,
"column": 4
} | {
"line": 127,
"column": 20
} | {
"line": 129,
"column": 0
} | [
{
"pp": "case refine_2\nk : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝² : Field k\ninst✝¹ : Field K\ninst✝ : Algebra k K\na : ι → K\nF : MvPolynomial ι k\nHF : ∀ (F' : MvPolynomial ι k), F' ≠ 0 → (aeval a) F' = 0 → F.totalDegree ≤ F'.totalDegree\nσ : ι →₀ ℕ\nhσ : σ ∈ F.support\ni : ι\nhσi : σ i ≠ 0\nH : (F.toP... | [] | apply_fun Subalgebra.val _ at H
simp_rw [toPolynomialAdjoinImageCompl, Polynomial.coe_mapAlgHom, Polynomial.coeff_map,
AlgHom.coe_toRingHom, map_zero] at H
simp_rw [← H, ← AlgHom.comp_apply]
congr; ext; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.MorphismProperty.Descent | {
"line": 85,
"column": 51
} | {
"line": 88,
"column": 44
} | {
"line": 90,
"column": 0
} | [
{
"pp": "C : Type u_1\ninst✝³ : Category.{v_1, u_1} C\nP Q W : MorphismProperty C\nA✝ X✝ Y✝ Z✝ : C\nfst✝ : A✝ ⟶ X✝\nsnd✝ : A✝ ⟶ Y✝\nf✝ : X✝ ⟶ Z✝\ng✝ : Y✝ ⟶ Z✝\ninst✝² : Q.IsStableUnderBaseChange\ninst✝¹ : P.HasOfPrecompProperty Q\ninst✝ : P.RespectsRight Q\nA X Y Z : C\nfst : A ⟶ X\nsnd : A ⟶ Y\nf : X ⟶ Z\ng : ... | [] | by
apply P.of_precomp (W' := Q) _ _ (Q.of_isPullback h hf)
rw [← h.1.1]
exact RespectsRight.postcomp _ hf _ hfst | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.FlatDescent | {
"line": 59,
"column": 8
} | {
"line": 59,
"column": 76
} | {
"line": 59,
"column": 76
} | [
{
"pp": "R S : CommRingCat\nY : Scheme\nφ : R ⟶ S\ng : Y ⟶ Spec R\nx✝¹ : (@Surjective ⊓ @Flat) (Spec.map φ)\nhfst : UniversallyClosed (pullback.fst (Spec.map φ) g)\nleft✝ : Surjective (Spec.map φ)\nright✝ : Flat (Spec.map φ)\nT : Scheme\nf : T ⟶ Spec R\nx✝ : HasPullback f g\ns : Set ↥(pullback f g)\nhs : IsClos... | [
"R S : CommRingCat\nY : Scheme\nφ : R ⟶ S\ng : Y ⟶ Spec R\nx✝¹ : (@Surjective ⊓ @Flat) (Spec.map φ)\nhfst : UniversallyClosed (pullback.fst (Spec.map φ) g)\nleft✝ : Surjective (Spec.map φ)\nright✝ : Flat (Spec.map φ)\nT : Scheme\nf : T ⟶ Spec R\nx✝ : HasPullback f g\ns : Set ↥(pullback f g)\nhs : IsClosed s\np : pu... | ← Scheme.image_preimage_eq_of_isPullback (isPullback_map_snd_snd ..) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.FlatDescent | {
"line": 78,
"column": 8
} | {
"line": 78,
"column": 76
} | {
"line": 78,
"column": 76
} | [
{
"pp": "R S : CommRingCat\nY : Scheme\nφ : R ⟶ S\ng : Y ⟶ Spec R\nx✝¹ : (@Surjective ⊓ @Flat) (Spec.map φ)\nhfst : UniversallyOpen (pullback.fst (Spec.map φ) g)\nleft✝ : Surjective (Spec.map φ)\nright✝ : Flat (Spec.map φ)\nT : Scheme\nf : T ⟶ Spec R\nx✝ : HasPullback f g\ns : Set ↥(pullback f g)\nhs : IsOpen s... | [
"R S : CommRingCat\nY : Scheme\nφ : R ⟶ S\ng : Y ⟶ Spec R\nx✝¹ : (@Surjective ⊓ @Flat) (Spec.map φ)\nhfst : UniversallyOpen (pullback.fst (Spec.map φ) g)\nleft✝ : Surjective (Spec.map φ)\nright✝ : Flat (Spec.map φ)\nT : Scheme\nf : T ⟶ Spec R\nx✝ : HasPullback f g\ns : Set ↥(pullback f g)\nhs : IsOpen s\np : pullba... | ← Scheme.image_preimage_eq_of_isPullback (isPullback_map_snd_snd ..) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.FlatDescent | {
"line": 99,
"column": 2
} | {
"line": 112,
"column": 29
} | {
"line": 113,
"column": 2
} | [
{
"pp": "case H\nR S : CommRingCat\nY : Scheme\nφ : R ⟶ S\ng : Y ⟶ Spec R\nh : (@Surjective ⊓ @Flat) (Spec.map φ)\nhfst : IsIso (pullback.fst (Spec.map φ) g)\n⊢ isomorphisms Scheme g",
"ppTerm": "?H",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Flat",
"CategoryTheory.Morphism... | [
"case H\nR S : CommRingCat\nY : Scheme\nφ : R ⟶ S\ng : Y ⟶ Spec R\nh : (@Surjective ⊓ @Flat) (Spec.map φ)\nhfst : IsIso (pullback.fst (Spec.map φ) g)\nthis : IsAffine Y\n⊢ isomorphisms Scheme g"
] | have : IsAffine Y :=
have : UniversallyInjective g :=
of_pullback_fst_of_descendsAlong (P := @UniversallyInjective) (f := Spec.map φ)
(Q := @Surjective ⊓ @Flat ⊓ @QuasiCompact) ⟨h, inferInstance⟩ inferInstance
have : Surjective g :=
of_pullback_fst_of_descendsAlong (P := @Surjective) (f := S... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.RingHom.QuasiFinite | {
"line": 89,
"column": 10
} | {
"line": 89,
"column": 12
} | {
"line": 90,
"column": 2
} | [
{
"pp": "R S : Type u_6\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf✝ : R →+* S\ns : Finset S\nhs : Ideal.span ↑s = ⊤\nH : ∀ (r : ↥s), (fun {R S} [CommRing R] [CommRing S] ↦ QuasiFinite) ((algebraMap S (Localization.Away ↑r)).comp f✝)\nalgInst✝ : Algebra R S := f✝.toAlgebra\nP : Ideal R\nx✝ : P.IsPrime\nthis✝ : ... | [
"R S : Type u_6\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf✝ : R →+* S\ns : Finset S\nhs : Ideal.span ↑s = ⊤\nH : ∀ (r : ↥s), (fun {R S} [CommRing R] [CommRing S] ↦ QuasiFinite) ((algebraMap S (Localization.Away ↑r)).comp f✝)\nalgInst✝ : Algebra R S := f✝.toAlgebra\nP : Ideal R\nx✝ : P.IsPrime\nthis✝ : ∀ (r : ↥s), ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Conductor | {
"line": 97,
"column": 12
} | {
"line": 97,
"column": 14
} | {
"line": 98,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p ∈ comap (algebraMap R S) (conductor R x)\nl : R →₀ S\nH : l ∈ Finsupp.supported S S ↑I\nH' : (l.sum fun i a ↦ a • (algebraMap R S) i) = z\na : R\n⊢ a ∈ I → l a • (algebraM... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nx : S\nI : Ideal R\np : R\nz : S\nhp : p ∈ comap (algebraMap R S) (conductor R x)\nl : R →₀ S\nH : l ∈ Finsupp.supported S S ↑I\nH' : (l.sum fun i a ↦ a • (algebraMap R S) i) = z\na : R\nha : a ∈ I\n⊢ l a • (algebraMap R S) ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Valuation.IsTrivialOn | {
"line": 85,
"column": 6
} | {
"line": 85,
"column": 80
} | {
"line": 85,
"column": 80
} | [
{
"pp": "Γ : Type u_1\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nA : Type u_2\ninst✝² : CommRing A\nK : Type u_3\ninst✝¹ : Field K\ninst✝ : Algebra A K\nv : Valuation K Γ\nhv : IsTrivialOn A v\ny : K\nh0 : ¬y = 0\nhy : ¬v y = 1\nhlt : 1 < v y\nthis : IsAlgebraic A y\np : A[X]\nhpnt : p ≠ 0\nhp : (aeval ((algeb... | [
"Γ : Type u_1\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nA : Type u_2\ninst✝² : CommRing A\nK : Type u_3\ninst✝¹ : Field K\ninst✝ : Algebra A K\nv : Valuation K Γ\nhv : IsTrivialOn A v\ny : K\nh0 : ¬y = 0\nhy : ¬v y = 1\nhlt : 1 < v y\nthis : IsAlgebraic A y\np : A[X]\nhpnt : p ≠ 0\nhp : (aeval ((algebraMap K K) y... | ← valuation_aeval_eq_valuation_X_pow_natDegree_of_one_lt_valuation_X _ hlt | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.RatFunc.AsPolynomial | {
"line": 187,
"column": 4
} | {
"line": 187,
"column": 29
} | {
"line": 187,
"column": 30
} | [
{
"pp": "case pos\nK : Type u\ninst✝¹ : Field K\nL : Type u\ninst✝ : Field L\nf : K →+* L\na : L\nx y : K⟮X⟯\nhx : Polynomial.eval₂ f a x.denom ≠ 0\nhy : Polynomial.eval₂ f a y.denom ≠ 0\nhxy : Polynomial.eval₂ f a (x + y).denom = 0\nthis : Polynomial.eval₂ f a x.denom * Polynomial.eval₂ f a y.denom = 0\n⊢ Poly... | [
"case pos.inl\nK : Type u\ninst✝¹ : Field K\nL : Type u\ninst✝ : Field L\nf : K →+* L\na : L\nx y : K⟮X⟯\nhx : Polynomial.eval₂ f a x.denom ≠ 0\nhy : Polynomial.eval₂ f a y.denom ≠ 0\nhxy : Polynomial.eval₂ f a (x + y).denom = 0\nthis : Polynomial.eval₂ f a x.denom * Polynomial.eval₂ f a y.denom = 0\nh✝ : Polynomia... | cases mul_eq_zero.mp this | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.FieldTheory.RatFunc.AsPolynomial | {
"line": 206,
"column": 4
} | {
"line": 206,
"column": 29
} | {
"line": 206,
"column": 30
} | [
{
"pp": "case pos\nK : Type u\ninst✝¹ : Field K\nL : Type u\ninst✝ : Field L\nf : K →+* L\na : L\nx y : K⟮X⟯\nhx : Polynomial.eval₂ f a x.denom ≠ 0\nhy : Polynomial.eval₂ f a y.denom ≠ 0\nhxy : Polynomial.eval₂ f a (x * y).denom = 0\nthis : Polynomial.eval₂ f a x.denom * Polynomial.eval₂ f a y.denom = 0\n⊢ Poly... | [
"case pos.inl\nK : Type u\ninst✝¹ : Field K\nL : Type u\ninst✝ : Field L\nf : K →+* L\na : L\nx y : K⟮X⟯\nhx : Polynomial.eval₂ f a x.denom ≠ 0\nhy : Polynomial.eval₂ f a y.denom ≠ 0\nhxy : Polynomial.eval₂ f a (x * y).denom = 0\nthis : Polynomial.eval₂ f a x.denom * Polynomial.eval₂ f a y.denom = 0\nh✝ : Polynomia... | cases mul_eq_zero.mp this | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 92,
"column": 4
} | {
"line": 92,
"column": 72
} | {
"line": 93,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsDomain R\ninst✝ : FaithfulSMul R S\np✝ : S[X]\nhp✝ : IsAlmostIntegral R[X] p✝\ni : ℕ\nq : S[X]\np : R[X]\nhp : p ∈ R[X]⁰\nhp' : ∀ (n : ℕ), p • q ^ n ∈ (algebraMap R[X] S[X]).range\nn : ℕ\nr : R[X]\nhr... | [
"R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : IsDomain R\ninst✝ : FaithfulSMul R S\np✝ : S[X]\nhp✝ : IsAlmostIntegral R[X] p✝\ni : ℕ\nq : S[X]\np : R[X]\nhp : p ∈ R[X]⁰\nhp' : ∀ (n : ℕ), p • q ^ n ∈ (algebraMap R[X] S[X]).range\nn : ℕ\nr : R[X]\nhr : map (alge... | simp only [Algebra.smul_def, algebraMap_def, coe_mapRingHom] at hr ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.QuasiFinite.Polynomial | {
"line": 78,
"column": 2
} | {
"line": 80,
"column": 49
} | {
"line": 81,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nf : R[X] →ₐ[R] S\nhf : Function.Surjective ⇑f\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : Algebra.WeaklyQuasiFiniteAt R P\nH : RingHom.ker f ≤ Ideal.map C (Ideal.under R P)\nalgInst✝ : Algebra R[X] S := f.toAlgebr... | [
"R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nf : R[X] →ₐ[R] S\nhf : Function.Surjective ⇑f\nP : Ideal S\ninst✝¹ : P.IsPrime\ninst✝ : Algebra.WeaklyQuasiFiniteAt R P\nH : RingHom.ker f ≤ Ideal.map C (Ideal.under R P)\nalgInst✝ : Algebra R[X] S := f.toAlgebra\np : Ideal... | let g' : p.ResidueField[X] ≃ₐ[p.ResidueField] p.Fiber S :=
.trans ((AlgEquiv.quotientBot _ _).symm.trans (Ideal.quotientEquivAlgOfEq _ H'.symm))
(Polynomial.fiberEquivQuotient f hf _).symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.QuasiFinite.Weakly | {
"line": 235,
"column": 6
} | {
"line": 235,
"column": 37
} | {
"line": 235,
"column": 37
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\nq : Ideal S\ninst✝⁴ : q.IsPrime\ninst✝³ : p.IsPrime\ninst✝² : q.LiesOver p\nQ : Ideal (p.Fiber S)\ninst✝¹ : Q.IsPrime\nhQ : Ideal.comap TensorProduct.includeRight.toRingHom Q = q\ninst✝ : QuasiFinit... | [
"R : Type u_1\nS : Type u_2\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\np : Ideal R\nq : Ideal S\ninst✝⁴ : q.IsPrime\ninst✝³ : p.IsPrime\ninst✝² : q.LiesOver p\nQ : Ideal (p.Fiber S)\ninst✝¹ : Q.IsPrime\nhQ : Ideal.comap TensorProduct.includeRight.toRingHom Q = q\ninst✝ : QuasiFiniteAt p.Residu... | Algebra.weaklyQuasiFiniteAt_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 441,
"column": 8
} | {
"line": 441,
"column": 83
} | {
"line": 442,
"column": 6
} | [
{
"pp": "case inl\nK : Type u\ninst✝⁵ : CommRing K\nG₀ : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst✝⁴ : CommGroupWithZero G₀\ninst✝³ : Field L\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : FunLike F R[X] S[X]\nφ : R[X] →+* L\nhφ : R[X]⁰ ≤ Submonoid.comap φ L⁰\nx y : R⟮X⟯\nh✝ : S... | [] | rw [Subsingleton.elim (x + y) y, Subsingleton.elim x 0, map_zero, zero_add] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 441,
"column": 8
} | {
"line": 441,
"column": 83
} | {
"line": 442,
"column": 6
} | [
{
"pp": "case inl\nK : Type u\ninst✝⁵ : CommRing K\nG₀ : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst✝⁴ : CommGroupWithZero G₀\ninst✝³ : Field L\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : FunLike F R[X] S[X]\nφ : R[X] →+* L\nhφ : R[X]⁰ ≤ Submonoid.comap φ L⁰\nx y : R⟮X⟯\nh✝ : S... | [] | rw [Subsingleton.elim (x + y) y, Subsingleton.elim x 0, map_zero, zero_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 441,
"column": 8
} | {
"line": 441,
"column": 83
} | {
"line": 442,
"column": 6
} | [
{
"pp": "case inl\nK : Type u\ninst✝⁵ : CommRing K\nG₀ : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst✝⁴ : CommGroupWithZero G₀\ninst✝³ : Field L\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : FunLike F R[X] S[X]\nφ : R[X] →+* L\nhφ : R[X]⁰ ≤ Submonoid.comap φ L⁰\nx y : R⟮X⟯\nh✝ : S... | [] | rw [Subsingleton.elim (x + y) y, Subsingleton.elim x 0, map_zero, zero_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 452,
"column": 12
} | {
"line": 452,
"column": 25
} | {
"line": 452,
"column": 26
} | [
{
"pp": "case inr.H.H\nK : Type u\ninst✝⁵ : CommRing K\nG₀ : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst✝⁴ : CommGroupWithZero G₀\ninst✝³ : Field L\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : FunLike F R[X] S[X]\nφ : R[X] →+* L\nhφ : R[X]⁰ ≤ Submonoid.comap φ L⁰\nh✝ : Nontrivia... | [
"case inr.H.H\nK : Type u\ninst✝⁵ : CommRing K\nG₀ : Type u_1\nL : Type u_2\nR : Type u_3\nS : Type u_4\nF : Type u_5\ninst✝⁴ : CommGroupWithZero G₀\ninst✝³ : Field L\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : FunLike F R[X] S[X]\nφ : R[X] →+* L\nhφ : R[X]⁰ ≤ Submonoid.comap φ L⁰\nh✝ : Nontrivial R\np : R[X... | mul_comm _ p, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 870,
"column": 21
} | {
"line": 870,
"column": 23
} | {
"line": 871,
"column": 6
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nx : K⟮X⟯\np q a : K[X]\nhq : q ≠ 0\n⊢ a ≠ 0 →\n (if a * q = 0 then (0, 1)\n else\n let r := gcd (a * p) (a * q);\n (C (a * q / r).leadingCoeff⁻¹ * (a * p / r), C (a * q / r).leadingCoeff⁻¹ * (a * q / r))) =\n if q = 0 then (0, 1)\n else\n ... | [
"K : Type u\ninst✝ : Field K\nx : K⟮X⟯\np q a : K[X]\nhq : q ≠ 0\nha : a ≠ 0\n⊢ (if a * q = 0 then (0, 1)\n else\n let r := gcd (a * p) (a * q);\n (C (a * q / r).leadingCoeff⁻¹ * (a * p / r), C (a * q / r).leadingCoeff⁻¹ * (a * q / r))) =\n if q = 0 then (0, 1)\n else\n let r := gcd p q;\n ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 152,
"column": 2
} | {
"line": 177,
"column": 9
} | {
"line": 180,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : StandardEtalePair R\nI : Ideal S\nhI : I ^ 2 = ⊥\nx : S\nhx : P.HasMap ((Ideal.Quotient.mk I) x)\n⊢ ∃! ε, ε ∈ I ∧ P.HasMap (x + ε)",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"... | [] | have hf := Ideal.Quotient.eq_zero_iff_mem.mp
((aeval_algHom_apply (Ideal.Quotient.mkₐ R I) _ _).symm.trans hx.1)
obtain ⟨⟨_, a, ha, -⟩, rfl⟩ := hx.2
obtain ⟨a, rfl⟩ := Ideal.Quotient.mk_surjective a
simp_rw [← Ideal.Quotient.mkₐ_eq_mk R, aeval_algHom_apply, ← map_mul, ← map_one
(Ideal.Quotient.mkₐ R I), I... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Etale.StandardEtale | {
"line": 152,
"column": 2
} | {
"line": 177,
"column": 9
} | {
"line": 180,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nP : StandardEtalePair R\nI : Ideal S\nhI : I ^ 2 = ⊥\nx : S\nhx : P.HasMap ((Ideal.Quotient.mk I) x)\n⊢ ∃! ε, ε ∈ I ∧ P.HasMap (x + ε)",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"... | [] | have hf := Ideal.Quotient.eq_zero_iff_mem.mp
((aeval_algHom_apply (Ideal.Quotient.mkₐ R I) _ _).symm.trans hx.1)
obtain ⟨⟨_, a, ha, -⟩, rfl⟩ := hx.2
obtain ⟨a, rfl⟩ := Ideal.Quotient.mk_surjective a
simp_rw [← Ideal.Quotient.mkₐ_eq_mk R, aeval_algHom_apply, ← map_mul, ← map_one
(Ideal.Quotient.mkₐ R I), I... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.RatFunc.Basic | {
"line": 1118,
"column": 48
} | {
"line": 1118,
"column": 63
} | {
"line": 1118,
"column": 63
} | [
{
"pp": "K : Type u\ninst✝⁴ : Field K\nR : Type u_1\nF : Type u_2\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : FunLike F K[X] R[X]\ninst✝ : MonoidHomClass F K[X] R[X]\nφ : F\nhφ : K[X]⁰ ≤ Submonoid.comap φ R[X]⁰\nf : K⟮X⟯\n⊢ (algebraMap R[X] R⟮X⟯) (φ f.num) / (algebraMap R[X] R⟮X⟯) (φ f.denom) =\n (al... | [
"K : Type u\ninst✝⁴ : Field K\nR : Type u_1\nF : Type u_2\ninst✝³ : CommRing R\ninst✝² : IsDomain R\ninst✝¹ : FunLike F K[X] R[X]\ninst✝ : MonoidHomClass F K[X] R[X]\nφ : F\nhφ : K[X]⁰ ≤ Submonoid.comap φ R[X]⁰\nf : K⟮X⟯\n⊢ (algebraMap R[X] R⟮X⟯) (φ f.num) / (algebraMap R[X] R⟮X⟯) (φ f.denom) =\n (algebraMap R[X... | num_div_denom f | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Smooth.IntegralClosure | {
"line": 166,
"column": 71
} | {
"line": 171,
"column": 40
} | {
"line": 172,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nM : Submonoid R\ninst✝ : IsLocalization M S\nx✝ : ↥(integralClosure R B)\nx : B\nhx : x ∈ integralClosure R B\n⊢ (Algebra.TensorProduct.includeRight.comp ... | [] | by
refine .of_comp (f := algebraMap R S) ?_
convert!
RingHom.IsIntegralElem.map hx
(Algebra.TensorProduct.includeRight : B →ₐ[R] S ⊗[R] B).toRingHom
simp [← IsScalarTower.algebraMap_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Smooth.IntegralClosure | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 64
} | {
"line": 174,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nM : Submonoid R\ninst✝ : IsLocalization M S\nφ : ↥(integralClosure R B) →ₐ[R] ↥(integralClosure S (S ⊗[R] B)) :=\n (Algebra.TensorProduct.includeRight.co... | [
"R : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nM : Submonoid R\ninst✝ : IsLocalization M S\nφ : ↥(integralClosure R B) →ₐ[R] ↥(integralClosure S (S ⊗[R] B)) :=\n (Algebra.TensorProduct.includeRight.comp (integral... | have := IsScalarTower.of_algebraMap_eq' φ.comp_algebraMap.symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Smooth.IntegralClosure | {
"line": 293,
"column": 56
} | {
"line": 293,
"column": 94
} | {
"line": 293,
"column": 95
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : Algebra.IsStandardEtale R S\na : S ⊗[R] B\nhx : IsIntegral S a\n𝓟 : StandardEtalePresentation R S\ne : S ≃ₐ[R] Localization.Away ((AdjoinRoot.mk ... | [] | by simp [RingHom.algebraMap_toAlgebra] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Smooth.IntegralClosure | {
"line": 303,
"column": 4
} | {
"line": 304,
"column": 40
} | {
"line": 306,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nB : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : Algebra.IsStandardEtale R S\na : S ⊗[R] B\nhx : IsIntegral S a\n𝓟 : StandardEtalePresentation R S\ne : S ≃ₐ[R] Localization.Away ((AdjoinRoot.mk ... | [] | simp [RingHom.algebraMap_toAlgebra, e, StandardEtalePair.equivAwayAdjoinRoot,
← aeval_def, ← aeval_algHom_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Unramified.LocalStructure | {
"line": 268,
"column": 6
} | {
"line": 268,
"column": 85
} | {
"line": 269,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nQ : Ideal S\ninst✝⁴ : Q.IsPrime\ninst✝³ : Module.Finite R S\ninst✝² : IsUnramifiedAt R Q\ninst✝¹ : Algebra (Localization.AtPrime (Ideal.under R Q)) (Localization.AtPrime Q)\ninst✝ : Localization.AtPrime.IsLiesOv... | [] | simpa [smul_def, ← IsScalarTower.algebraMap_apply, eq_inv_mul_iff_mul_eq₀ hsQ'] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 246,
"column": 2
} | {
"line": 246,
"column": 24
} | {
"line": 247,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\np : R[X]\nhRS : integralClosure R S = ⊥\nhφ : φ.Finite\nhp : φ p * t ∈ conductor R (φ X)\nalgInst✝ : Algebra R[X] S := φ.toAlgebra\nalgebraizeInst✝ : Module.Finite R[X] S\nthis✝ : IsScala... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nφ : R[X] →ₐ[R] S\nt : S\np : R[X]\nhRS : integralClosure R S = ⊥\nhφ : φ.Finite\nhp : φ p * t ∈ conductor R (φ X)\nalgInst✝ : Algebra R[X] S := φ.toAlgebra\nalgebraizeInst✝ : Module.Finite R[X] S\nthis : IsScalarTower R R[X]... | choose n hn using this | Mathlib.Tactic.Choose._aux_Mathlib_Tactic_Choose___elabRules_Mathlib_Tactic_Choose_choose_1 | Mathlib.Tactic.Choose.choose |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 65,
"column": 8
} | {
"line": 65,
"column": 79
} | {
"line": 65,
"column": 80
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nX✝ Y✝ Z✝ : Y.Opensᵒᵖ\ni : X✝ ⟶ Y✝\nj : Y✝ ⟶ Z✝\n⊢ CommRingCat.ofHom\n ((CommRingCat.Hom.hom (X.presheaf.map (homOfLE ⋯).op)).restrict\n (integralClosure ↑Γ(Y, Opposite.unop X✝) ↑Γ(X, f ⁻¹ᵁ Opposite.unop X✝))\n (integralClosure ↑Γ(Y, Opposite.unop Z✝) ↑Γ(X, f ... | [
"X Y : Scheme\nf : X ⟶ Y\nX✝ Y✝ Z✝ : Y.Opensᵒᵖ\ni : X✝ ⟶ Y✝\nj : Y✝ ⟶ Z✝\n⊢ CommRingCat.ofHom\n ((CommRingCat.Hom.hom (X.presheaf.map (homOfLE ⋯ ≫ homOfLE ⋯).op)).restrict\n (integralClosure ↑Γ(Y, Opposite.unop X✝) ↑Γ(X, f ⁻¹ᵁ Opposite.unop X✝))\n (integralClosure ↑Γ(Y, Opposite.unop Z✝) ↑Γ(X, f ... | ← homOfLE_comp (f.preimage_mono j.unop.le) (f.preimage_mono i.unop.le), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 148,
"column": 8
} | {
"line": 148,
"column": 40
} | {
"line": 148,
"column": 40
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : QuasiCompact f\ninst✝ : QuasiSeparated f\nU V : Y.AffineZariskiSite\ni : U ⟶ V\n⊢ (pullbackRestrictIsoRestrict f ↑U).inv ≫\n Cover.trans (Precoverage.ZeroHypercover.pullback₁ f (directedCover Y)) i ≫\n (pullbackRestrictIsoRestrict f ↑V).hom =\n X.homOfLE... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : QuasiCompact f\ninst✝ : QuasiSeparated f\nU V : Y.AffineZariskiSite\ni : U ⟶ V\n⊢ ((pullbackRestrictIsoRestrict f ↑U).inv ≫\n Cover.trans (Precoverage.ZeroHypercover.pullback₁ f (directedCover Y)) i ≫\n (pullbackRestrictIsoRestrict f ↑V).hom) ≫\n (f ⁻¹ᵁ ↑V).ι ... | ← cancel_mono (Scheme.Opens.ι _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.ZariskisMainTheorem | {
"line": 585,
"column": 4
} | {
"line": 594,
"column": 73
} | {
"line": 595,
"column": 4
} | [
{
"pp": "case succ\nn : ℕ\nIH :\n ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : Ideal S) [inst_3 : p.IsPrime]\n [WeaklyQuasiFiniteAt R p] (f : MvPolynomial (Fin n) R →ₐ[R] S), f.Finite → ZariskisMainProperty R p\nR S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S... | [
"case succ\nn : ℕ\nIH :\n ∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : Ideal S) [inst_3 : p.IsPrime]\n [WeaklyQuasiFiniteAt R p] (f : MvPolynomial (Fin n) R →ₐ[R] S), f.Finite → ZariskisMainProperty R p\nR S : Type u\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : A... | have : Algebra.WeaklyQuasiFiniteAt R (p.under R') := by
let := Localization.AtPrime.algebraOfLiesOver (p.under R') p
let e : Localization.AtPrime (p.under R') ≃ₐ[R] Localization.AtPrime p :=
.ofBijective (IsScalarTower.toAlgHom _ _ _) <| by
refine Localization.localRingHom_bijective_of_sat... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 361,
"column": 2
} | {
"line": 362,
"column": 6
} | {
"line": 364,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\ng : R[X]\nm n : ℕ\nr : R\n⊢ (C r).resultant g m n = (-1) ^ (m * n) * g.coeff n ^ m * r ^ n",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"NegZeroClass.toNeg",
"Polynomial.resultant_add_left_deg",... | [] | rw [← zero_add m, resultant_add_left_deg _ _ _ _ _ (by simp), mul_comm n m]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 361,
"column": 2
} | {
"line": 362,
"column": 6
} | {
"line": 364,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\ng : R[X]\nm n : ℕ\nr : R\n⊢ (C r).resultant g m n = (-1) ^ (m * n) * g.coeff n ^ m * r ^ n",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"NegZeroClass.toNeg",
"Polynomial.resultant_add_left_deg",... | [] | rw [← zero_add m, resultant_add_left_deg _ _ _ _ _ (by simp), mul_comm n m]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Normalization | {
"line": 343,
"column": 6
} | {
"line": 343,
"column": 33
} | {
"line": 343,
"column": 34
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : QuasiCompact f\ninst✝ : QuasiSeparated f\nthis : ↑(ker (toNormalization f)).support = ↑⊥.support\n⊢ IsDominant (toNormalization f)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.IdealSheafData.support",
"Alge... | [
"X Y : Scheme\nf : X ⟶ Y\ninst✝¹ : QuasiCompact f\ninst✝ : QuasiSeparated f\nthis : ↑(ker (toNormalization f)).support = ↑⊤\n⊢ IsDominant (toNormalization f)"
] | IdealSheafData.support_bot, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Quotient.Over | {
"line": 31,
"column": 4
} | {
"line": 31,
"column": 67
} | {
"line": 32,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R S\ninst✝² : Algebra R T\nf : S →ₐ[R] T\nHf : Function.Surjective ⇑f\nP : Ideal R\nP' : Ideal S\ninst✝¹ : P'.IsPrime\ninst✝ : P'.LiesOver P\nhkP' : ¬RingHom.ker f.to... | [] | exact ⟨inferInstance, inferInstanceAs ((q.comap f).LiesOver _)⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 464,
"column": 8
} | {
"line": 464,
"column": 42
} | {
"line": 465,
"column": 4
} | [
{
"pp": "n : ℕ\nIH :\n ∀ m < n,\n ∀ {K : Type u_3} [inst : Field K] (f g : K[X]),\n f.Monic →\n g.Monic →\n f.Splits →\n g.Splits →\n g.natDegree ≤ f.natDegree →\n f.natDegree + g.natDegree = m →\n f.resultant g = (Multiset.map (fu... | [] | simp [hg'.natDegree_eq_card_roots] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 464,
"column": 8
} | {
"line": 464,
"column": 42
} | {
"line": 465,
"column": 4
} | [
{
"pp": "n : ℕ\nIH :\n ∀ m < n,\n ∀ {K : Type u_3} [inst : Field K] (f g : K[X]),\n f.Monic →\n g.Monic →\n f.Splits →\n g.Splits →\n g.natDegree ≤ f.natDegree →\n f.natDegree + g.natDegree = m →\n f.resultant g = (Multiset.map (fu... | [] | simp [hg'.natDegree_eq_card_roots] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 464,
"column": 8
} | {
"line": 464,
"column": 42
} | {
"line": 465,
"column": 4
} | [
{
"pp": "n : ℕ\nIH :\n ∀ m < n,\n ∀ {K : Type u_3} [inst : Field K] (f g : K[X]),\n f.Monic →\n g.Monic →\n f.Splits →\n g.Splits →\n g.natDegree ≤ f.natDegree →\n f.natDegree + g.natDegree = m →\n f.resultant g = (Multiset.map (fu... | [] | simp [hg'.natDegree_eq_card_roots] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 382,
"column": 15
} | {
"line": 383,
"column": 79
} | {
"line": 384,
"column": 6
} | [
{
"pp": "case tmul.refine_2.C\nR : Type u_1\ninst✝ : CommRing R\nn m k : ℕ\nhn : n = m + k\nthis✝¹ : Algebra (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) :=\n (universalFactorizationMap R n m k hn).toAlgebra\nthis✝ : IsDomain (MvPolynomial (Fin m) ℤ ⊗[ℤ] MvPolynomial (Fin k) ℤ)... | [] | simpa [← algebraMap_eq, ← algebraMap_apply, Algebra.algebraMap_eq_smul_one] using!
(universalFactorizationMap R n m k hn).isIntegralElem_map (x := .C a) | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 382,
"column": 15
} | {
"line": 383,
"column": 79
} | {
"line": 384,
"column": 6
} | [
{
"pp": "case tmul.refine_2.C\nR : Type u_1\ninst✝ : CommRing R\nn m k : ℕ\nhn : n = m + k\nthis✝¹ : Algebra (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) :=\n (universalFactorizationMap R n m k hn).toAlgebra\nthis✝ : IsDomain (MvPolynomial (Fin m) ℤ ⊗[ℤ] MvPolynomial (Fin k) ℤ)... | [] | simpa [← algebraMap_eq, ← algebraMap_apply, Algebra.algebraMap_eq_smul_one] using!
(universalFactorizationMap R n m k hn).isIntegralElem_map (x := .C a) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 382,
"column": 15
} | {
"line": 383,
"column": 79
} | {
"line": 384,
"column": 6
} | [
{
"pp": "case tmul.refine_2.C\nR : Type u_1\ninst✝ : CommRing R\nn m k : ℕ\nhn : n = m + k\nthis✝¹ : Algebra (MvPolynomial (Fin n) R) (MvPolynomial (Fin m) R ⊗[R] MvPolynomial (Fin k) R) :=\n (universalFactorizationMap R n m k hn).toAlgebra\nthis✝ : IsDomain (MvPolynomial (Fin m) ℤ ⊗[ℤ] MvPolynomial (Fin k) ℤ)... | [] | simpa [← algebraMap_eq, ← algebraMap_apply, Algebra.algebraMap_eq_smul_one] using!
(universalFactorizationMap R n m k hn).isIntegralElem_map (x := .C a) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Resultant.Basic | {
"line": 753,
"column": 6
} | {
"line": 753,
"column": 69
} | {
"line": 753,
"column": 69
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhg : g.natDegree ≠ 0\nthis :\n ∀ {R : Type u_1} [inst : CommRing R] (f g : R[X]),\n g.natDegree ≠ 0 →\n IsDomain R →\n (f.scaleRoots g.leadingCoeff).resultant g.integralNormalization f.natDegree g.natDegree =\n g.leadingCo... | [
"case neg\nR : Type u_1\ninst✝ : CommRing R\nf g : R[X]\nhg : g.natDegree ≠ 0\nthis :\n ∀ {R : Type u_1} [inst : CommRing R] (f g : R[X]),\n g.natDegree ≠ 0 →\n IsDomain R →\n (f.scaleRoots g.leadingCoeff).resultant g.integralNormalization f.natDegree g.natDegree =\n g.leadingCoeff ^ (f.nat... | this f' g' (by simpa [natDegree_eq_natDegree eg]) inferInstance | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.UniversalFactorizationRing | {
"line": 673,
"column": 43
} | {
"line": 673,
"column": 58
} | {
"line": 673,
"column": 58
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nn m k : ℕ\nhn : n = m + k\np : MonicDegreeEq R n\nP : Ideal R\ninst✝ : P.IsPrime\nf : MonicDegreeEq P.ResidueField m\ng : MonicDegreeEq P.ResidueField k\nH : map (algebraMap R P.ResidueField) ↑p = ↑f * ↑g\nHpq : IsCoprime ↑f ↑g\nφ : 𝓡' →ₐ[R] P.ResidueField := (homEqu... | [] | by simp [e, φ'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.ZariskisMainTheorem | {
"line": 59,
"column": 2
} | {
"line": 59,
"column": 36
} | {
"line": 60,
"column": 2
} | [
{
"pp": "X S : Scheme\nf : X ⟶ S\ninst✝¹ : LocallyOfFiniteType f\ninst✝ : IsSeparated f\nx : ↥X\ns : ↥S\nh : f x = s\nhx : Scheme.Hom.QuasiFiniteAt f x\nU : TopologicalSpace.Opens ↥S\nhU : U ∈ S.affineOpens\nhxU : f x ∈ ↑U\nV : TopologicalSpace.Opens ↥X\nhV : V ∈ X.affineOpens\nhxV : x ∈ ↑V\nhUV : V ≤ f ⁻¹ᵁ U\n... | [
"X S : Scheme\nf : X ⟶ S\ninst✝¹ : LocallyOfFiniteType f\ninst✝ : IsSeparated f\nx : ↥X\ns : ↥S\nh : f x = s\nhx : Scheme.Hom.QuasiFiniteAt f x\nU : TopologicalSpace.Opens ↥S\nhU : U ∈ S.affineOpens\nhxU : f x ∈ ↑U\nV : TopologicalSpace.Opens ↥X\nhV : V ∈ X.affineOpens\nhxV : x ∈ ↑V\nhUV : V ≤ f ⁻¹ᵁ U\nthis : (Comm... | algebraize [(f.appLE U V hUV).hom] | Mathlib.Tactic._aux_Mathlib_Tactic_Algebraize___elabRules_Mathlib_Tactic_tacticAlgebraize___1 | Mathlib.Tactic.tacticAlgebraize__ |
Mathlib.CategoryTheory.Monoidal.Cartesian.CommMon_ | {
"line": 34,
"column": 4
} | {
"line": 34,
"column": 21
} | {
"line": 34,
"column": 22
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : CartesianMonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\nF : Cᵒᵖ ⥤ CommMonCat\nα : (F ⋙ forget CommMonCat).RepresentableBy X\nthis✝ : MonObj X := MonObj.ofRepresentableBy X (F ⋙ forget₂ CommMonCat MonCat) α\nthis : μ = α.homEquiv'.symm (α.homEquiv'... | [
"C : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : CartesianMonoidalCategory C\ninst✝ : BraidedCategory C\nX : C\nF : Cᵒᵖ ⥤ CommMonCat\nα : (F ⋙ forget CommMonCat).RepresentableBy X\nthis✝ : MonObj X := MonObj.ofRepresentableBy X (F ⋙ forget₂ CommMonCat MonCat) α\nthis : μ = α.homEquiv'.symm (α.homEquiv' (fst X X) *... | braiding_hom_fst, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
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