module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 236,
"column": 2
} | {
"line": 236,
"column": 43
} | {
"line": 238,
"column": 0
} | [
{
"pp": "S : Type u_4\nσ : Type u_5\nR : Type u_6\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Fintype σ\nn : ℕ\nf : R →+* S\n⊢ (map f) (esymm σ R n) = esymm σ S n",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"RingHom... | [] | simp_rw [esymm, map_sum, map_prod, map_X] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 236,
"column": 2
} | {
"line": 236,
"column": 43
} | {
"line": 238,
"column": 0
} | [
{
"pp": "S : Type u_4\nσ : Type u_5\nR : Type u_6\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Fintype σ\nn : ℕ\nf : R →+* S\n⊢ (map f) (esymm σ R n) = esymm σ S n",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"RingHom... | [] | simp_rw [esymm, map_sum, map_prod, map_X] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 236,
"column": 2
} | {
"line": 236,
"column": 43
} | {
"line": 238,
"column": 0
} | [
{
"pp": "S : Type u_4\nσ : Type u_5\nR : Type u_6\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Fintype σ\nn : ℕ\nf : R →+* S\n⊢ (map f) (esymm σ R n) = esymm σ S n",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"RingHom... | [] | simp_rw [esymm, map_sum, map_prod, map_X] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.PolynomialGaloisGroup | {
"line": 318,
"column": 8
} | {
"line": 318,
"column": 16
} | {
"line": 319,
"column": 6
} | [
{
"pp": "case pos.inl\nF : Type u_1\ninst✝ : Field F\np q : F[X]\nhq : q.natDegree ≠ 0\nP : F[X] → Prop := fun r ↦ (map (algebraMap F (r.comp q).SplittingField) r).Splits\nkey1 : ∀ {r : F[X]}, Irreducible r → P r\np₁ p₂ : F[X]\nhp₁ : P p₁\nhp₂ : P p₂\nh₁ : p₁.comp q = 0\nh : p₁ = 0\n⊢ P 0",
"ppTerm": "?pos.... | [] | simp [P] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 329,
"column": 11
} | {
"line": 329,
"column": 17
} | {
"line": 329,
"column": 18
} | [
{
"pp": "τ : Type u_2\nσ : Type u_5\nR : Type u_6\ninst✝⁴ : CommSemiring R\ninst✝³ : Fintype σ\ninst✝² : Fintype τ\ninst✝¹ : DecidableEq σ\ninst✝ : DecidableEq τ\nn : ℕ\ne : σ ≃ τ\n⊢ (rename ⇑e) (hsymm σ R n) = hsymm τ R n",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Nat.instMulZ... | [
"τ : Type u_2\nσ : Type u_5\nR : Type u_6\ninst✝⁴ : CommSemiring R\ninst✝³ : Fintype σ\ninst✝² : Fintype τ\ninst✝¹ : DecidableEq σ\ninst✝ : DecidableEq τ\nn : ℕ\ne : σ ≃ τ\n⊢ (rename ⇑e) (∑ s, (Multiset.map X ↑s).prod) = ∑ s, (Multiset.map X ↑s).prod"
] | hsymm, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Maps.Proper.CompactlyGenerated | {
"line": 38,
"column": 8
} | {
"line": 38,
"column": 13
} | {
"line": 38,
"column": 13
} | [
{
"pp": "case e'_3\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\ninst✝ : CompactlyCoherentSpace Y\nf : X → Y\nx✝ : Continuous f ∧ ∀ ⦃K : Set Y⦄, IsCompact K → IsCompact (f ⁻¹' K)\nhf : Continuous f\nh : ∀ ⦃K : Set Y⦄, IsCompact K → IsCompact (f ⁻¹' K)... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Valuation.RankOne | {
"line": 62,
"column": 70
} | {
"line": 91,
"column": 6
} | {
"line": 93,
"column": 0
} | [
{
"pp": "R : Type u_1\nΓ₀ : Type u_2\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\ninst✝ : v.IsNontrivial\n⊢ Nonempty v.RankOne ↔ MulArchimedean (ofClass v).ValueGroup₀",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoi... | [] | by
constructor
· intro h
obtain hv := Nonempty.some h
exact MulArchimedean.comap hv.hom'.toMonoidHom hv.strictMono'
· intro _
obtain ⟨f, hf⟩ :=
Archimedean.exists_orderAddMonoidHom_real_injective (Additive (ValueGroup₀ (.ofClass v))ˣ)
let e := AddMonoidHom.toMultiplicativeRight (α := (ValueG... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Valued.NormedValued | {
"line": 192,
"column": 8
} | {
"line": 192,
"column": 28
} | {
"line": 193,
"column": 8
} | [
{
"pp": "case h\nL : Type u_1\ninst✝¹ : Field L\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nval : Valued L Γ₀\nhv : v.RankOne\nthis : Nonempty { ε // ε > 0 }\nU : Set (L × L)\n⊢ Directed (fun x1 x2 ↦ x1 ≥ x2) fun x ↦ 𝓟 {p | v.norm (p.1 - p.2) < ↑x}",
"ppTerm": "?h✝",
"assigned": true,
... | [
"case h\nL : Type u_1\ninst✝¹ : Field L\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nval : Valued L Γ₀\nhv : v.RankOne\nthis : Nonempty { ε // ε > 0 }\nU : Set (L × L)\n⊢ ∀ (x y : { ε // ε > 0 }),\n ∃ z,\n 𝓟 {p | v.norm (p.1 - p.2) < ↑x} ≥ 𝓟 {p | v.norm (p.1 - p.2) < ↑z} ∧\n 𝓟 {p | v... | simp only [Directed] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors | {
"line": 167,
"column": 6
} | {
"line": 167,
"column": 47
} | {
"line": 168,
"column": 6
} | [
{
"pp": "case pos\nσ : Type u_1\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\nw : σ → ℕ\nf g : MvPowerSeries σ R\nhf : weightedOrder w f < ⊤\nhg : weightedOrder w g < ⊤\np : ℕ := (weightedOrder w f).toNat\nhp : ↑p = weightedOrder w f\nq : ℕ := (weightedOrder w g).toNat\nhq : ↑q = weightedOrder w... | [
"case pos\nσ : Type u_1\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\nw : σ → ℕ\nf g : MvPowerSeries σ R\nhf : weightedOrder w f < ⊤\nhg : weightedOrder w g < ⊤\np : ℕ := (weightedOrder w f).toNat\nhp : ↑p = weightedOrder w f\nq : ℕ := (weightedOrder w g).toNat\nhq : ↑q = weightedOrder w g\nthis : (... | rw [← hp, ← hq, ← Nat.cast_add, ← not_lt] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.PowerSeries.Inverse | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 9
} | {
"line": 70,
"column": 2
} | [
{
"pp": "case neg.e_a.hi\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : R⟦X⟧\nh✝ : ¬n = 0\n⊢ ∀ a ∈ antidiagonal n,\n (match a with\n | (a, b) => (single () a, single () b)) ∈\n antidiagonal (single () n)",
"ppTerm": "?neg.e_a.hi✝",
"assigned": true,
"usedConstants": [
"Finsupp.i... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.PowerSeries.Inverse | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 9
} | {
"line": 70,
"column": 2
} | [
{
"pp": "case neg.e_a.hi\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : R⟦X⟧\nh✝ : ¬n = 0\n⊢ ∀ a ∈ antidiagonal n,\n (match a with\n | (a, b) => (single () a, single () b)) ∈\n antidiagonal (single () n)",
"ppTerm": "?neg.e_a.hi✝",
"assigned": true,
"usedConstants": [
"Finsupp.i... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Inverse | {
"line": 69,
"column": 4
} | {
"line": 69,
"column": 9
} | {
"line": 70,
"column": 2
} | [
{
"pp": "case neg.e_a.hi\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : R⟦X⟧\nh✝ : ¬n = 0\n⊢ ∀ a ∈ antidiagonal n,\n (match a with\n | (a, b) => (single () a, single () b)) ∈\n antidiagonal (single () n)",
"ppTerm": "?neg.e_a.hi✝",
"assigned": true,
"usedConstants": [
"Finsupp.i... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.Inverse | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 9
} | {
"line": 71,
"column": 2
} | [
{
"pp": "case neg.e_a.hj\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : R⟦X⟧\nh✝ : ¬n = 0\n⊢ ∀ a ∈ antidiagonal (single () n),\n (match a with\n | (f, g) => (f (), g ())) ∈\n antidiagonal n",
"ppTerm": "?neg.e_a.hj✝",
"assigned": true,
"usedConstants": [
"Finsupp.instHasAntidiag... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.PowerSeries.Inverse | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 9
} | {
"line": 71,
"column": 2
} | [
{
"pp": "case neg.e_a.hj\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : R⟦X⟧\nh✝ : ¬n = 0\n⊢ ∀ a ∈ antidiagonal (single () n),\n (match a with\n | (f, g) => (f (), g ())) ∈\n antidiagonal n",
"ppTerm": "?neg.e_a.hj✝",
"assigned": true,
"usedConstants": [
"Finsupp.instHasAntidiag... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Inverse | {
"line": 70,
"column": 4
} | {
"line": 70,
"column": 9
} | {
"line": 71,
"column": 2
} | [
{
"pp": "case neg.e_a.hj\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : R⟦X⟧\nh✝ : ¬n = 0\n⊢ ∀ a ∈ antidiagonal (single () n),\n (match a with\n | (f, g) => (f (), g ())) ∈\n antidiagonal n",
"ppTerm": "?neg.e_a.hj✝",
"assigned": true,
"usedConstants": [
"Finsupp.instHasAntidiag... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.Inverse | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 9
} | {
"line": 72,
"column": 2
} | [
{
"pp": "case neg.e_a.left_neg\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : R⟦X⟧\nh✝ : ¬n = 0\n⊢ ∀ a ∈ antidiagonal n,\n (match\n match a with\n | (a, b) => (single () a, single () b) with\n | (f, g) => (f (), g ())) =\n a",
"ppTerm": "?neg.e_a.left_neg✝",
"assigned": true... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.PowerSeries.Inverse | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 9
} | {
"line": 72,
"column": 2
} | [
{
"pp": "case neg.e_a.left_neg\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : R⟦X⟧\nh✝ : ¬n = 0\n⊢ ∀ a ∈ antidiagonal n,\n (match\n match a with\n | (a, b) => (single () a, single () b) with\n | (f, g) => (f (), g ())) =\n a",
"ppTerm": "?neg.e_a.left_neg✝",
"assigned": true... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Inverse | {
"line": 71,
"column": 4
} | {
"line": 71,
"column": 9
} | {
"line": 72,
"column": 2
} | [
{
"pp": "case neg.e_a.left_neg\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : R⟦X⟧\nh✝ : ¬n = 0\n⊢ ∀ a ∈ antidiagonal n,\n (match\n match a with\n | (a, b) => (single () a, single () b) with\n | (f, g) => (f (), g ())) =\n a",
"ppTerm": "?neg.e_a.left_neg✝",
"assigned": true... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.Inverse | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 9
} | {
"line": 73,
"column": 2
} | [
{
"pp": "case neg.e_a.right_neg\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : R⟦X⟧\nh✝ : ¬n = 0\n⊢ ∀ a ∈ antidiagonal (single () n),\n (match\n match a with\n | (f, g) => (f (), g ()) with\n | (a, b) => (single () a, single () b)) =\n a",
"ppTerm": "?neg.e_a.right_neg✝",
"a... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.PowerSeries.Inverse | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 9
} | {
"line": 73,
"column": 2
} | [
{
"pp": "case neg.e_a.right_neg\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : R⟦X⟧\nh✝ : ¬n = 0\n⊢ ∀ a ∈ antidiagonal (single () n),\n (match\n match a with\n | (f, g) => (f (), g ()) with\n | (a, b) => (single () a, single () b)) =\n a",
"ppTerm": "?neg.e_a.right_neg✝",
"a... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Inverse | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 9
} | {
"line": 73,
"column": 2
} | [
{
"pp": "case neg.e_a.right_neg\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : R⟦X⟧\nh✝ : ¬n = 0\n⊢ ∀ a ∈ antidiagonal (single () n),\n (match\n match a with\n | (f, g) => (f (), g ()) with\n | (a, b) => (single () a, single () b)) =\n a",
"ppTerm": "?neg.e_a.right_neg✝",
"a... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.Inverse | {
"line": 74,
"column": 4
} | {
"line": 74,
"column": 32
} | {
"line": 75,
"column": 4
} | [
{
"pp": "case neg.e_a.h\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : R⟦X⟧\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ antidiagonal n\n⊢ (if (i, j).2 < n then (coeff (i, j).1) φ * (coeff (i, j).2) (MvPowerSeries.inv.aux a φ) else 0) =\n if\n (match (i, j) with\n | (a, b) => (single () a, sing... | [
"case neg.e_a.h.inl\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : R⟦X⟧\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ antidiagonal n\nH : n ≤ j\n⊢ (if (i, j).2 < n then (coeff (i, j).1) φ * (coeff (i, j).2) (MvPowerSeries.inv.aux a φ) else 0) =\n if\n (match (i, j) with\n | (a, b) => (single () a, s... | obtain H | H := le_or_gt n j | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.PowerSeries.Inverse | {
"line": 75,
"column": 6
} | {
"line": 75,
"column": 11
} | {
"line": 76,
"column": 4
} | [
{
"pp": "case neg.e_a.h.inl\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : R⟦X⟧\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ antidiagonal n\nH : n ≤ j\n⊢ (if (i, j).2 < n then (coeff (i, j).1) φ * (coeff (i, j).2) (MvPowerSeries.inv.aux a φ) else 0) =\n if\n (match (i, j) with\n | (a, b) => (si... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.PowerSeries.Inverse | {
"line": 75,
"column": 6
} | {
"line": 75,
"column": 11
} | {
"line": 76,
"column": 4
} | [
{
"pp": "case neg.e_a.h.inl\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : R⟦X⟧\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ antidiagonal n\nH : n ≤ j\n⊢ (if (i, j).2 < n then (coeff (i, j).1) φ * (coeff (i, j).2) (MvPowerSeries.inv.aux a φ) else 0) =\n if\n (match (i, j) with\n | (a, b) => (si... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Inverse | {
"line": 75,
"column": 6
} | {
"line": 75,
"column": 11
} | {
"line": 76,
"column": 4
} | [
{
"pp": "case neg.e_a.h.inl\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : R⟦X⟧\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ antidiagonal n\nH : n ≤ j\n⊢ (if (i, j).2 < n then (coeff (i, j).1) φ * (coeff (i, j).2) (MvPowerSeries.inv.aux a φ) else 0) =\n if\n (match (i, j) with\n | (a, b) => (si... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Completion.FinitePlace | {
"line": 279,
"column": 90
} | {
"line": 280,
"column": 61
} | {
"line": 282,
"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⊢ ‖(embedding v) ((algebraMap R K) x)‖ = ↑((toNNReal ⋯) (v.intValuation... | [] | by
simp [norm_embedding, adicAbv_def, valuation_of_algebraMap] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.ModelsWithJ | {
"line": 148,
"column": 19
} | {
"line": 148,
"column": 49
} | {
"line": 148,
"column": 49
} | [
{
"pp": "F : Type u_2\ninst✝¹ : Field F\ninst✝ : DecidableEq F\nh3 : 3 = 0\n⊢ 1728 = 0",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Mathlib.Tactic.Ring.Common.neg_zero",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithO... | [] | by linear_combination 576 * h3 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point | {
"line": 146,
"column": 91
} | {
"line": 147,
"column": 94
} | {
"line": 149,
"column": 0
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Projective R\nP : Fin 3 → R\nhP : W'.Equation P\n⊢ W'.addY P (W'.neg P) = -W'.dblZ P",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"HMul.hMul",
"AddGroupWithOne.toAddGroup",
"congrArg",
"Com... | [] | by
simp only [addY, addX_neg, negAddY_neg hP, addZ_neg, negY, fin3_def_ext, mul_zero, sub_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes | {
"line": 87,
"column": 9
} | {
"line": 87,
"column": 43
} | {
"line": 87,
"column": 43
} | [
{
"pp": "case hS\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Limits.HasPullbacks C\nK : Precoverage C\ninst✝ : K.IsStableUnderBaseChange\nS : C\nF : Cᵒᵖ ⥤ Type u_2\n𝒰 : K.ZeroHypercover S\n𝒱 : K.ZeroHypercover S\nf : Hom K 𝒰 𝒱\nH₁ : Presieve.IsSheafFor F (Presieve.ofArrows 𝒰.X 𝒰.f)\nH₂ : ∀ {X ... | [
"case hS\nC : Type u_1\ninst✝² : Category.{v_1, u_1} C\ninst✝¹ : Limits.HasPullbacks C\nK : Precoverage C\ninst✝ : K.IsStableUnderBaseChange\nS : C\nF : Cᵒᵖ ⥤ Type u_2\n𝒰 : K.ZeroHypercover S\n𝒱 : K.ZeroHypercover S\nf : Hom K 𝒰 𝒱\nH₁ : Presieve.IsSheafFor F (Presieve.ofArrows 𝒰.X 𝒰.f)\nH₂ : ∀ {X : C} (f : X ... | ← Presieve.isSheafFor_iff_generate | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 85
} | {
"line": 158,
"column": 2
} | [
{
"pp": "case refine_1\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : C\nE : PreOneHypercover X\nF : Cᵒᵖ ⥤ Type u_2\nh : E.IsStronglySheafFor F\ns t : (E.multifork F).pt\nhst : (E.multifork F).toSections s = (E.multifork F).toSections t\n⊢ s = t",
"ppTerm": "?refine_1",
"assigned": true,
"usedCon... | [
"case refine_2\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : C\nE : PreOneHypercover X\nF : Cᵒᵖ ⥤ Type u_2\nh : E.IsStronglySheafFor F\ns : (E.multicospanIndex F).sections\n⊢ ∃ a, (E.multifork F).toSections a = s"
] | · exact h.isSheafFor_presieve₀.isSeparatedFor.ext fun _ _ ⟨i⟩ ↦ congr($(hst).val i) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point | {
"line": 277,
"column": 4
} | {
"line": 290,
"column": 73
} | {
"line": 292,
"column": 0
} | [
{
"pp": "case neg\nF : Type u\ninst✝ : Field F\nW : Projective F\nP Q : Fin 3 → F\nhP : W.Nonsingular P\nhQ : W.Nonsingular Q\nhPz : ¬P z = 0\nhQz : ¬Q z = 0\n⊢ W.Nonsingular (W.add P Q)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"WeierstrassCur... | [] | · by_cases hxy : P x * Q z = Q x * P z ∧ P y * Q z = W.negY Q * P z
· by_cases hy : P y * Q z = Q y * P z
· simp only [add_of_Y_eq hP.left hPz hQz hxy.left hy hxy.right, nonsingular_smul _ <|
isUnit_dblU_of_Y_eq hP hPz hQz hxy.left hy hxy.right, nonsingular_zero]
· simp only [add_of_... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic | {
"line": 429,
"column": 2
} | {
"line": 430,
"column": 90
} | {
"line": 432,
"column": 0
} | [
{
"pp": "R : Type r\ninst✝¹ : CommRing R\nW' : Projective R\ninst✝ : NoZeroDivisors R\nP : Fin 3 → R\nhP : W'.Nonsingular P\nhPz : P z = 0\nhPy : P y = 0\n⊢ False",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"False",
"Nat.instMulZeroClass",
... | [] | simp only [nonsingular_of_Z_eq_zero hPz, X_eq_zero_of_Z_eq_zero hP.left hPz, hPy, add_zero,
sub_zero, mul_zero, zero_pow two_ne_zero, or_self, ne_self_iff_false, and_false] at hP | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic | {
"line": 512,
"column": 74
} | {
"line": 513,
"column": 53
} | {
"line": 515,
"column": 0
} | [
{
"pp": "R : Type r\ninst✝¹ : CommRing R\nW' : Projective R\nS : Type s\ninst✝ : CommRing S\nf : R →+* S\n⊢ (W'.map f).polynomialX = (MvPolynomial.map f) W'.polynomialX",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Derivation",
"Finsupp.instAddZeroClass",
"Nat.instMulZ... | [] | by
simp only [polynomialX, map_polynomial, pderiv_map] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic | {
"line": 560,
"column": 57
} | {
"line": 561,
"column": 89
} | {
"line": 563,
"column": 0
} | [
{
"pp": "R : Type r\ninst✝¹⁰ : CommRing R\nW' : Projective R\nS : Type s\ninst✝⁹ : CommRing S\nA : Type u\ninst✝⁸ : CommRing A\nB : Type v\ninst✝⁷ : CommRing B\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra S A\ninst✝³ : IsScalarTower R S A\ninst✝² : Algebra R B\ninst✝¹ : Algebra S B\ninst✝ : IsS... | [] | by
rw [← RingHom.coe_coe, ← map_nonsingular _ hf, AlgHom.toRingHom_eq_coe, map_baseChange] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Etale.Field | {
"line": 155,
"column": 4
} | {
"line": 156,
"column": 81
} | {
"line": 158,
"column": 0
} | [
{
"pp": "case refine_6\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Algebra.IsSeparable K L\nB : Type (max u_1 u_2)\nx✝¹ : CommRing B\nx✝ : Algebra K B\nI : Ideal B\nh : I ^ 2 = ⊥\nf : L →ₐ[K] B ⧸ I\ng : (k : L) → ↥K⟮k⟯ →ₐ[K] B\nhg₁ : ∀ (k : L), (fun g ↦ (Ideal.... | [] | ext x
simpa using AlgHom.congr_fun (hg₁ x) (IntermediateField.AdjoinSimple.gen K x) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Etale.Field | {
"line": 155,
"column": 4
} | {
"line": 156,
"column": 81
} | {
"line": 158,
"column": 0
} | [
{
"pp": "case refine_6\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Algebra.IsSeparable K L\nB : Type (max u_1 u_2)\nx✝¹ : CommRing B\nx✝ : Algebra K B\nI : Ideal B\nh : I ^ 2 = ⊥\nf : L →ₐ[K] B ⧸ I\ng : (k : L) → ↥K⟮k⟯ →ₐ[K] B\nhg₁ : ∀ (k : L), (fun g ↦ (Ideal.... | [] | ext x
simpa using AlgHom.congr_fun (hg₁ x) (IntermediateField.AdjoinSimple.gen K x) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Kaehler.TensorProduct | {
"line": 299,
"column": 2
} | {
"line": 303,
"column": 77
} | {
"line": 305,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\ns : S\na : A\n⊢ (tensorKaehlerEquiv R S A (S ⊗[R] A)).symm ((D S (S ⊗[R] A)) (s ⊗ₜ[R] a)) =\n (algebraMap S (S ⊗[R] A)) s ⊗ₜ[A] (D R A) a",
"ppTerm"... | [] | apply (tensorKaehlerEquiv R S A _).symm_apply_eq.mpr ?_
simp only [Algebra.TensorProduct.algebraMap_apply, Algebra.algebraMap_self, RingHom.id_apply,
tensorKaehlerEquiv_tmul_D]
rw [show s ⊗ₜ 1 = algebraMap S (S ⊗ A) s by simp, Algebra.TensorProduct.right_algebraMap_apply,
algebraMap_smul, ← Derivation.map_s... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Kaehler.TensorProduct | {
"line": 299,
"column": 2
} | {
"line": 303,
"column": 77
} | {
"line": 305,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\ns : S\na : A\n⊢ (tensorKaehlerEquiv R S A (S ⊗[R] A)).symm ((D S (S ⊗[R] A)) (s ⊗ₜ[R] a)) =\n (algebraMap S (S ⊗[R] A)) s ⊗ₜ[A] (D R A) a",
"ppTerm"... | [] | apply (tensorKaehlerEquiv R S A _).symm_apply_eq.mpr ?_
simp only [Algebra.TensorProduct.algebraMap_apply, Algebra.algebraMap_self, RingHom.id_apply,
tensorKaehlerEquiv_tmul_D]
rw [show s ⊗ₜ 1 = algebraMap S (S ⊗ A) s by simp, Algebra.TensorProduct.right_algebraMap_apply,
algebraMap_smul, ← Derivation.map_s... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.AdicCompletion.Algebra | {
"line": 458,
"column": 66
} | {
"line": 459,
"column": 39
} | {
"line": 461,
"column": 0
} | [
{
"pp": "S : Type u_5\ninst✝¹ : CommRing S\nI : Ideal S\ninst✝ : IsAdicComplete I S\nx : AdicCompletion I S\n⊢ (Ideal.Quotient.mk I) ((ofAlgEquiv I).symm x) = (evalOneₐ I) x",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"IsScalarTower.right",
"AlgEq... | [] | by
simp [evalOneₐ, ← mk_ofAlgEquiv_symm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Extension.Presentation.Core | {
"line": 172,
"column": 76
} | {
"line": 172,
"column": 94
} | {
"line": 172,
"column": 94
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nι : Type u_3\nσ : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nP : Presentation R S ι σ\nR₀ : Type u_5\ninst✝⁴ : CommRing R₀\ninst✝³ : Algebra R₀ R\ninst✝² : Algebra R₀ S\ninst✝¹ : IsScalarTower R₀ R S\ninst✝ : P.HasCoeffs R₀\nx : MvPolynomial ι ... | [
"R : Type u_1\nS : Type u_2\nι : Type u_3\nσ : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nP : Presentation R S ι σ\nR₀ : Type u_5\ninst✝⁴ : CommRing R₀\ninst✝³ : Algebra R₀ R\ninst✝² : Algebra R₀ S\ninst✝¹ : IsScalarTower R₀ R S\ninst✝ : P.HasCoeffs R₀\nx : MvPolynomial ι R₀\n⊢ (P.ten... | ← quotientEquiv_mk | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Smooth.StandardSmoothCotangent | {
"line": 174,
"column": 57
} | {
"line": 174,
"column": 85
} | {
"line": 174,
"column": 85
} | [
{
"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.CotangentSpace\ny : P.toExtension.Cotangent\n⊢ (∀ (x_1 : σ),\n P.cotangentEquiv\n (P.cotangentEqu... | [
"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.CotangentSpace\ny : P.toExtension.Cotangent\n⊢ (∀ (x_1 : σ),\n (Finsupp.linearEquivFunOnFinite S S σ) ((Finsupp.lcomap... | LinearEquiv.apply_symm_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 486,
"column": 17
} | {
"line": 486,
"column": 33
} | {
"line": 486,
"column": 34
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nP Q : Fin 3 → F\nhQz : Q z = 0\n⊢ -(P y * 0 - Q y * P z) ^ 3 / (P z * 0) = 0",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"instHDiv",
"HMul.hMul",
"MulZeroClass.toMul",
"AddGroupWit... | [
"F : Type u\ninst✝ : Field F\nP Q : Fin 3 → F\nhQz : Q z = 0\n⊢ -(P y * 0 - Q y * P z) ^ 3 / 0 = 0"
] | mul_zero <| P z, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.RingHom.Etale | {
"line": 87,
"column": 43
} | {
"line": 87,
"column": 80
} | {
"line": 87,
"column": 80
} | [
{
"pp": "R : Type u_3\nS : Type u_4\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nalgInst✝ : Algebra R S := f.toAlgebra\n⊢ Algebra.Etale R S ↔ Module.Flat R S ∧ Algebra.FormallyUnramified R S ∧ (algebraMap R S).FinitePresentation",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
... | [
"R : Type u_3\nS : Type u_4\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nalgInst✝ : Algebra R S := f.toAlgebra\n⊢ Algebra.Etale R S ↔ Module.Flat R S ∧ Algebra.FormallyUnramified R S ∧ Algebra.FinitePresentation R S"
] | RingHom.finitePresentation_algebraMap | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 175,
"column": 4
} | {
"line": 176,
"column": 19
} | {
"line": 178,
"column": 0
} | [
{
"pp": "case refine_2\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\ni : σ\n⊢ ((AlgebraTensorModule.curry (D.tensorCotangentInv ∘ₗ D.tensorCot... | [] | simp [-toExtension_commRing, -toExtension_Ring, -toExtension_algebra₂, tensorCotangentHom_tmul,
kerGen, D.hf] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 291,
"column": 4
} | {
"line": 294,
"column": 98
} | {
"line": 295,
"column": 4
} | [
{
"pp": "case inl\nR : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : FinitePresentation R S\nα : Type u_4\nP : Generators R S α\ninst✝ : Finite α\nσ : Type u_5\nb₀ : Module.Basis σ S P.toExtension.Cotangent\nh✝ : Subsingleton S\n⊢ ∃ P' b, P'.val ∘ Sum.inr = P.v... | [
"case inl\nR : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : FinitePresentation R S\nα : Type u_4\nP : Generators R S α\ninst✝ : Finite α\nσ : Type u_5\nb₀ : Module.Basis σ S P.toExtension.Cotangent\nh✝ : Subsingleton S\nP' : Presentation R S (Unit ⊕ α) (Unit ⊕ σ) ... | let P' : Presentation R S (Unit ⊕ α) (Unit ⊕ σ) :=
{ toGenerators := .ofSurjective (fun i : Unit ⊕ α ↦ 0) (Function.surjective_to_subsingleton _)
relation _ := 1
span_range_relation_eq_ker := by simpa using (RingHom.ker_eq_top_of_subsingleton _).symm } | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 296,
"column": 24
} | {
"line": 296,
"column": 39
} | {
"line": 296,
"column": 39
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : FinitePresentation R S\nα : Type u_4\nP : Generators R S α\ninst✝ : Finite α\nσ : Type u_5\nb₀ : Module.Basis σ S P.toExtension.Cotangent\nh✝ : Subsingleton S\nP' : Presentation R S (Unit ⊕ α) (Unit ⊕ σ... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 296,
"column": 41
} | {
"line": 296,
"column": 56
} | {
"line": 296,
"column": 56
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : FinitePresentation R S\nα : Type u_4\nP : Generators R S α\ninst✝ : Finite α\nσ : Type u_5\nb₀ : Module.Basis σ S P.toExtension.Cotangent\nh✝ : Subsingleton S\nP' : Presentation R S (Unit ⊕ α) (Unit ⊕ σ... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 308,
"column": 4
} | {
"line": 310,
"column": 63
} | {
"line": 311,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : FinitePresentation R S\nα : Type u_4\nP : Generators R S α\ninst✝ : Finite α\nσ : Type u_5\nb₀ : Module.Basis σ S P.toExtension.Cotangent\nh✝ : Nontrivial S\nf : P.toExtension.Cotangent → ↥P.toExtension... | [] | obtain ⟨g, hgmem, hg⟩ := Submodule.exists_sub_one_mem_and_smul_le_of_fg_of_le_sup hJfg le_rfl hJ
let D : Aux P b₀ := { f := f, hf := hf, g := g, hgmem := hgmem, hg := hg }
exact ⟨D.pres, D.basis, D.pres_val_comp_inr, D.basis_apply⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 308,
"column": 4
} | {
"line": 310,
"column": 63
} | {
"line": 311,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : FinitePresentation R S\nα : Type u_4\nP : Generators R S α\ninst✝ : Finite α\nσ : Type u_5\nb₀ : Module.Basis σ S P.toExtension.Cotangent\nh✝ : Nontrivial S\nf : P.toExtension.Cotangent → ↥P.toExtension... | [] | obtain ⟨g, hgmem, hg⟩ := Submodule.exists_sub_one_mem_and_smul_le_of_fg_of_le_sup hJfg le_rfl hJ
let D : Aux P b₀ := { f := f, hf := hf, g := g, hgmem := hgmem, hg := hg }
exact ⟨D.pres, D.basis, D.pres_val_comp_inr, D.basis_apply⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 349,
"column": 24
} | {
"line": 349,
"column": 39
} | {
"line": 349,
"column": 39
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : FinitePresentation R S\nα : Type u_4\nP : Generators R S α\ninst✝¹ : Finite α\ninst✝ : Module.Free S P.toExtension.Cotangent\nh✝ : Subsingleton S\nP' : Presentation R S (Unit ⊕ α) (Unit ⊕ Fin (Module.fi... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Extension.Cotangent.Basis | {
"line": 349,
"column": 41
} | {
"line": 349,
"column": 56
} | {
"line": 349,
"column": 56
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : FinitePresentation R S\nα : Type u_4\nP : Generators R S α\ninst✝¹ : Finite α\ninst✝ : Module.Free S P.toExtension.Cotangent\nh✝ : Subsingleton S\nP' : Presentation R S (Unit ⊕ α) (Unit ⊕ Fin (Module.fi... | [] | by subsingleton | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 741,
"column": 4
} | {
"line": 741,
"column": 67
} | {
"line": 741,
"column": 67
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nW : Projective 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\n⊢ W.addY P Q = W.addY P Q * (P z * Q z) ^ 3 / (P z * Q z) ^ (2 + 1)",
"ppTerm": "?m.88",
"assigned": true,
"usedConstants": [
... | [
"F : Type u\ninst✝ : Field F\nW : Projective 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\n⊢ W.addY P Q = W.addY P Q"
] | mul_div_cancel_right₀ _ <| pow_ne_zero 3 <| mul_ne_zero hPz hQz | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Smooth.NoetherianDescent | {
"line": 184,
"column": 6
} | {
"line": 184,
"column": 21
} | {
"line": 185,
"column": 4
} | [
{
"pp": "case refine_2.refine_1\nR : Type u_1\ninst✝⁴ : CommRing R\nA : Type u\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : CommRing B\ninst✝ : Algebra A B\nD : DescentAux A B\np : D.vars → MvPolynomial D.vars ↥(subalgebra R D)\nhp : ∀ (i : D.vars), (MvPolynomial.map (algebraMap (↥(subalge... | [] | exact D.hqhom i | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 83
} | {
"line": 158,
"column": 4
} | [
{
"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\np : ℕ\nhp : Nat.Prime p\na : ι → K\nF : MvPolynomial ι k\nHF : ∀ (F' : MvPolynomial ι k), F' ≠ 0 → (aeval a) F' = 0 → F.totalDegree ≤ F'.totalDegree\nhF0 : F ≠ 0\nhFa : (aeval a) F = 0\nthis : ∀ (i : ι), ... | [
"k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝² : Field k\ninst✝¹ : Field K\ninst✝ : Algebra k K\np : ℕ\nhp : Nat.Prime p\na : ι → K\nF : MvPolynomial ι k\nHF : ∀ (F' : MvPolynomial ι k), F' ≠ 0 → (aeval a) F' = 0 → F.totalDegree ≤ F'.totalDegree\nhF0 : F ≠ 0\nhFa : (aeval a) F = 0\nthis : ∀ (i : ι), ∀ σ ∈ F.supp... | refine ⟨.univ, (F.coeff ·), ?_, by simpa [MvPolynomial.eq_zero_iff] using! hF0⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 829,
"column": 60
} | {
"line": 831,
"column": 10
} | {
"line": 833,
"column": 0
} | [
{
"pp": "R : Type r\nS : Type s\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nW' : Projective R\nf : R →+* S\nP : Fin 3 → R\n⊢ (W'.map f).dblY (⇑f ∘ P) = f (W'.dblY P)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"WeierstrassCurve.Projective.map_dblZ",
"WeierstrassCurve.Projecti... | [] | by
simp only [dblY, negY_eq, map_negDblY, map_dblX, map_dblZ]
map_simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 216,
"column": 65
} | {
"line": 216,
"column": 70
} | {
"line": 217,
"column": 4
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\np : ℕ\nhp : Nat.Prime p\nH : ∀ (s : Finset K), LinearIndepOn k id ↑s → LinearIndepOn k (fun x ↦ x ^ p) ↑s\na : ι → K\nn : ι\ninst✝ : ExpChar k p\nha' : IsTranscendenceBasis k fun i ↦ a ↑i\nS : Set (MvPol... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 220,
"column": 27
} | {
"line": 220,
"column": 32
} | {
"line": 221,
"column": 2
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\np : ℕ\nhp : Nat.Prime p\nH : ∀ (s : Finset K), LinearIndepOn k id ↑s → LinearIndepOn k (fun x ↦ x ^ p) ↑s\na : ι → K\nn : ι\ninst✝ : ExpChar k p\nha' : IsTranscendenceBasis k fun i ↦ a ↑i\nS : Set (MvPol... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.Morphisms.FormallyUnramified | {
"line": 195,
"column": 24
} | {
"line": 195,
"column": 53
} | {
"line": 195,
"column": 54
} | [
{
"pp": "X Y Z' Z : Scheme\ni : Z' ⟶ Z\ninst✝¹ : IsClosedImmersion i\nf : X ⟶ Y\ninst✝ : FormallyUnramified f\ng₁ g₂ : Z ⟶ X\nhig : i ≫ g₁ = i ≫ g₂\nhgf : g₁ ≫ f = g₂ ≫ f\nn : ℕ\nhn : Scheme.Hom.ker i ^ n = 0\n⊢ DenseRange ⇑i",
"ppTerm": "?m.60",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"X Y Z' Z : Scheme\ni : Z' ⟶ Z\ninst✝¹ : IsClosedImmersion i\nf : X ⟶ Y\ninst✝ : FormallyUnramified f\ng₁ g₂ : Z ⟶ X\nhig : i ≫ g₁ = i ≫ g₂\nhgf : g₁ ≫ f = g₂ ≫ f\nn : ℕ\nhn : Scheme.Hom.ker i ^ n = 0\n⊢ closure (Set.range ⇑i) = Set.univ"
] | denseRange_iff_closure_range, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 890,
"column": 25
} | {
"line": 890,
"column": 38
} | {
"line": 890,
"column": 39
} | [
{
"pp": "R : Type r\nS : Type s\nA : Type u\nB : Type v\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\nW' : Projective R\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra S A\ninst✝³ : IsScalarTower R S A\ninst✝² : Algebra R B\ninst✝¹ : Algebra S B\ninst✝ : IsS... | [
"R : Type r\nS : Type s\nA : Type u\nB : Type v\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommRing S\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\nW' : Projective R\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra R A\ninst✝⁴ : Algebra S A\ninst✝³ : IsScalarTower R S A\ninst✝² : Algebra R B\ninst✝¹ : Algebra S B\ninst✝ : IsScalarTower R... | ← map_dblXYZ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 250,
"column": 20
} | {
"line": 250,
"column": 25
} | {
"line": 250,
"column": 25
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\np : ℕ\nhp : Nat.Prime p\nH : ∀ (s : Finset K), LinearIndepOn k id ↑s → LinearIndepOn k (fun x ↦ x ^ p) ↑s\na : ι → K\ninst✝ : ExpChar k p\ns : Set ι\nn : ι\nha : IsTranscendenceBasis k fun i ↦ a ↑i\nhn :... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 250,
"column": 20
} | {
"line": 250,
"column": 25
} | {
"line": 250,
"column": 25
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\np : ℕ\nhp : Nat.Prime p\nH : ∀ (s : Finset K), LinearIndepOn k id ↑s → LinearIndepOn k (fun x ↦ x ^ p) ↑s\na : ι → K\ninst✝ : ExpChar k p\ns : Set ι\nn : ι\nha : IsTranscendenceBasis k fun i ↦ a ↑i\nhn :... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 250,
"column": 20
} | {
"line": 250,
"column": 25
} | {
"line": 250,
"column": 25
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\np : ℕ\nhp : Nat.Prime p\nH : ∀ (s : Finset K), LinearIndepOn k id ↑s → LinearIndepOn k (fun x ↦ x ^ p) ↑s\na : ι → K\ninst✝ : ExpChar k p\ns : Set ι\nn : ι\nha : IsTranscendenceBasis k fun i ↦ a ↑i\nhn :... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 250,
"column": 44
} | {
"line": 250,
"column": 49
} | {
"line": 250,
"column": 49
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\np : ℕ\nhp : Nat.Prime p\nH : ∀ (s : Finset K), LinearIndepOn k id ↑s → LinearIndepOn k (fun x ↦ x ^ p) ↑s\na : ι → K\ninst✝ : ExpChar k p\ns : Set ι\nn : ι\nha : IsTranscendenceBasis k fun i ↦ a ↑i\nhn :... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 250,
"column": 44
} | {
"line": 250,
"column": 49
} | {
"line": 250,
"column": 49
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\np : ℕ\nhp : Nat.Prime p\nH : ∀ (s : Finset K), LinearIndepOn k id ↑s → LinearIndepOn k (fun x ↦ x ^ p) ↑s\na : ι → K\ninst✝ : ExpChar k p\ns : Set ι\nn : ι\nha : IsTranscendenceBasis k fun i ↦ a ↑i\nhn :... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 250,
"column": 44
} | {
"line": 250,
"column": 49
} | {
"line": 250,
"column": 49
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\np : ℕ\nhp : Nat.Prime p\nH : ∀ (s : Finset K), LinearIndepOn k id ↑s → LinearIndepOn k (fun x ↦ x ^ p) ↑s\na : ι → K\ninst✝ : ExpChar k p\ns : Set ι\nn : ι\nha : IsTranscendenceBasis k fun i ↦ a ↑i\nhn :... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 250,
"column": 55
} | {
"line": 250,
"column": 60
} | {
"line": 250,
"column": 60
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\np : ℕ\nhp : Nat.Prime p\nH : ∀ (s : Finset K), LinearIndepOn k id ↑s → LinearIndepOn k (fun x ↦ x ^ p) ↑s\na : ι → K\ninst✝ : ExpChar k p\ns : Set ι\nn : ι\nha : IsTranscendenceBasis k fun i ↦ a ↑i\nhn :... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 250,
"column": 55
} | {
"line": 250,
"column": 60
} | {
"line": 250,
"column": 60
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\np : ℕ\nhp : Nat.Prime p\nH : ∀ (s : Finset K), LinearIndepOn k id ↑s → LinearIndepOn k (fun x ↦ x ^ p) ↑s\na : ι → K\ninst✝ : ExpChar k p\ns : Set ι\nn : ι\nha : IsTranscendenceBasis k fun i ↦ a ↑i\nhn :... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 250,
"column": 55
} | {
"line": 250,
"column": 60
} | {
"line": 250,
"column": 60
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\np : ℕ\nhp : Nat.Prime p\nH : ∀ (s : Finset K), LinearIndepOn k id ↑s → LinearIndepOn k (fun x ↦ x ^ p) ↑s\na : ι → K\ninst✝ : ExpChar k p\ns : Set ι\nn : ι\nha : IsTranscendenceBasis k fun i ↦ a ↑i\nhn :... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.SeparablyGenerated | {
"line": 255,
"column": 64
} | {
"line": 255,
"column": 69
} | {
"line": 256,
"column": 2
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nι : Type u_3\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\np : ℕ\nhp : Nat.Prime p\nH : ∀ (s : Finset K), LinearIndepOn k id ↑s → LinearIndepOn k (fun x ↦ x ^ p) ↑s\na : ι → K\ninst✝ : ExpChar k p\ns : Set ι\nn : ι\nha : IsTranscendenceBasis k fun i ↦ a ↑i\nhn :... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Algebraic.StronglyTranscendental | {
"line": 56,
"column": 67
} | {
"line": 56,
"column": 88
} | {
"line": 57,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : CommRing T\ninst✝ : Algebra R T\nx : S\nf : S →ₐ[R] T\nhf : Function.Injective ⇑f\nh : IsStronglyTranscendental R (f x)\nu : S\np : R[X]\nhp : (aeval x) p * u = 0\nthis : map (↑f) (map (al... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : CommRing T\ninst✝ : Algebra R T\nx : S\nf : S →ₐ[R] T\nhf : Function.Injective ⇑f\nh : IsStronglyTranscendental R (f x)\nu : S\np : R[X]\nhp : (aeval x) p * u = 0\nthis : map (↑f) (map (algebraMap R S... | ← Polynomial.map_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.IntegralClosure.Algebra.Ideal | {
"line": 40,
"column": 25
} | {
"line": 40,
"column": 30
} | {
"line": 41,
"column": 2
} | [
{
"pp": "case add\nR : Type u_3\ninst✝ : CommRing R\nI : Ideal R\nP x y : R[X]\nhx : x ∈ Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}\nhy : y ∈ Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}\na✝¹ : ∀ (i : ℕ), x.coeff i ∈ I ^ i\na✝ : ∀ (i : ℕ), y.coeff i ∈ I ^ i\ni : ℕ\n⊢ (x + y).coeff i ∈ I ^ i",
"ppTerm": "?ad... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.IntegralClosure.Algebra.Ideal | {
"line": 40,
"column": 25
} | {
"line": 40,
"column": 30
} | {
"line": 41,
"column": 2
} | [
{
"pp": "case add\nR : Type u_3\ninst✝ : CommRing R\nI : Ideal R\nP x y : R[X]\nhx : x ∈ Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}\nhy : y ∈ Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}\na✝¹ : ∀ (i : ℕ), x.coeff i ∈ I ^ i\na✝ : ∀ (i : ℕ), y.coeff i ∈ I ^ i\ni : ℕ\n⊢ (x + y).coeff i ∈ I ^ i",
"ppTerm": "?ad... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.IntegralClosure.Algebra.Ideal | {
"line": 40,
"column": 25
} | {
"line": 40,
"column": 30
} | {
"line": 41,
"column": 2
} | [
{
"pp": "case add\nR : Type u_3\ninst✝ : CommRing R\nI : Ideal R\nP x y : R[X]\nhx : x ∈ Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}\nhy : y ∈ Algebra.adjoin R {x | ∃ r ∈ I, C r * X = x}\na✝¹ : ∀ (i : ℕ), x.coeff i ∈ I ^ i\na✝ : ∀ (i : ℕ), y.coeff i ∈ I ^ i\ni : ℕ\n⊢ (x + y).coeff i ∈ I ^ i",
"ppTerm": "?ad... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Conductor | {
"line": 104,
"column": 4
} | {
"line": 106,
"column": 58
} | {
"line": 107,
"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\nha : a ∈ I\n⊢ (algebraMap... | [
"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⊢ (algebraMap R ↥R[x]) a ... | case h =>
rw [mul_comm]
exact mem_conductor_iff.mp (Ideal.mem_comap.mp hp) _ | Lean.Elab.Tactic.evalCase | Lean.Parser.Tactic.case |
Mathlib.RingTheory.Conductor | {
"line": 112,
"column": 2
} | {
"line": 113,
"column": 27
} | {
"line": 114,
"column": 2
} | [
{
"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\nlem :\n ∀ {a : R},\n a ∈ I →... | [
"case refine_1\nR : 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\nlem :\n ∀ {a : R},\n a ∈ ... | refine Finset.sum_induction _ (fun u => u ∈ algebraMap R<x> S '' I.map (algebraMap R R<x>))
(fun a b => ?_) ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.RingHom.QuasiFinite | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 57
} | {
"line": 127,
"column": 0
} | [
{
"pp": "R : Type u_6\nS : Type u_7\nT : Type u_8\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : CommRing T\nf : R →+* S\nhf : f.IsIntegral\ng : S →+* T\nhg✝ : g.IsStandardOpenImmersion\nhg : (g.comp f).FiniteType\nalgInst✝² : Algebra R S := f.toAlgebra\nalgInst✝¹ : Algebra S T := g.toAlgebra\nalgInst✝ : Al... | [] | exact Algebra.QuasiFinite.of_isIntegral_of_finiteType s | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Valuation.IsTrivialOn | {
"line": 47,
"column": 2
} | {
"line": 58,
"column": 48
} | {
"line": 60,
"column": 0
} | [
{
"pp": "Γ : Type u_1\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nA : Type u_2\ninst✝² : CommSemiring A\nB : Type u_3\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nv : Valuation B Γ\nhv : Valuation.IsTrivialOn A v\nw : B\nhpos : 1 < v w\np : A[X]\nhp : p ≠ 0\n⊢ v ((aeval w) p) = v w ^ p.natDegree",
"ppTerm": "?m.3... | [] | rw [← valuation_aeval_monomial_eq_valuation_pow _ _ (leadingCoeff_ne_zero.mpr hp)]
nth_rw 1 [as_sum_range p, map_sum]
apply Valuation.map_sum_eq_of_lt _ (by simp)
intro i hi
simp only [Finset.mem_sdiff, Finset.mem_range, Nat.lt_add_one_iff, Finset.mem_singleton,
← lt_iff_le_and_ne] at hi
simp only [← C_mu... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.IsTrivialOn | {
"line": 47,
"column": 2
} | {
"line": 58,
"column": 48
} | {
"line": 60,
"column": 0
} | [
{
"pp": "Γ : Type u_1\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nA : Type u_2\ninst✝² : CommSemiring A\nB : Type u_3\ninst✝¹ : Ring B\ninst✝ : Algebra A B\nv : Valuation B Γ\nhv : Valuation.IsTrivialOn A v\nw : B\nhpos : 1 < v w\np : A[X]\nhp : p ≠ 0\n⊢ v ((aeval w) p) = v w ^ p.natDegree",
"ppTerm": "?m.3... | [] | rw [← valuation_aeval_monomial_eq_valuation_pow _ _ (leadingCoeff_ne_zero.mpr hp)]
nth_rw 1 [as_sum_range p, map_sum]
apply Valuation.map_sum_eq_of_lt _ (by simp)
intro i hi
simp only [Finset.mem_sdiff, Finset.mem_range, Nat.lt_add_one_iff, Finset.mem_singleton,
← lt_iff_le_and_ne] at hi
simp only [← C_mu... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 42,
"column": 84
} | {
"line": 42,
"column": 89
} | {
"line": 42,
"column": 89
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nι : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : ι → S[X]\na : ι\ns : Finset ι\nhas : a ∉ s\nIH : (∀ i ∈ s, ∀ (j : ℕ), IsIntegral R ((p i).coeff j)) → ∀ (j : ℕ), IsIntegral R ((s.prod p).coeff j)\nH : ∀ i ∈ insert a s, ∀ (j : ℕ), IsIntegral R ... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 42,
"column": 84
} | {
"line": 42,
"column": 89
} | {
"line": 42,
"column": 89
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nι : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : ι → S[X]\na : ι\ns : Finset ι\nhas : a ∉ s\nIH : (∀ i ∈ s, ∀ (j : ℕ), IsIntegral R ((p i).coeff j)) → ∀ (j : ℕ), IsIntegral R ((s.prod p).coeff j)\nH : ∀ i ∈ insert a s, ∀ (j : ℕ), IsIntegral R ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 42,
"column": 84
} | {
"line": 42,
"column": 89
} | {
"line": 42,
"column": 89
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nι : Type u_3\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : ι → S[X]\na : ι\ns : Finset ι\nhas : a ∉ s\nIH : (∀ i ∈ s, ∀ (j : ℕ), IsIntegral R ((p i).coeff j)) → ∀ (j : ℕ), IsIntegral R ((s.prod p).coeff j)\nH : ∀ i ∈ insert a s, ∀ (j : ℕ), IsIntegral R ... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 72,
"column": 16
} | {
"line": 72,
"column": 91
} | {
"line": 72,
"column": 91
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\np : R[X]\nq : S[X]\nhp : p.Monic\nhq : q.Monic\nH : q ∣ map (algebraMap R S) p\ni : ℕ\na✝ : Nontrivial S\nT : Type u_2\nw✝⁴ : CommRing T\nw✝³ : Algebra S T\nw✝² : Module.Finite S T\nw✝¹ : Module.Free S T\nw✝ : No... | [] | by simpa using! aeval_eq_zero_of_dvd_aeval_eq_zero (x := x) H (by simp_all) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 95,
"column": 43
} | {
"line": 95,
"column": 48
} | {
"line": 96,
"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... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 228,
"column": 26
} | {
"line": 228,
"column": 31
} | {
"line": 228,
"column": 31
} | [
{
"pp": "case e'_5\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nσ : Type w\nhσ : Finite σ\nα β : Type w\ne : α ≃ β\nIH :\n ∀ {f : MvPolynomial α S},\n (algebraMap (MvPolynomial α R) (MvPolynomial α S)).IsIntegralElem f → ∀ (n : α →₀ ℕ), IsIntegral R (coeff n f)... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 228,
"column": 26
} | {
"line": 228,
"column": 31
} | {
"line": 228,
"column": 31
} | [
{
"pp": "case e'_6\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nσ : Type w\nhσ : Finite σ\nα β : Type w\ne : α ≃ β\nIH :\n ∀ {f : MvPolynomial α S},\n (algebraMap (MvPolynomial α R) (MvPolynomial α S)).IsIntegralElem f → ∀ (n : α →₀ ℕ), IsIntegral R (coeff n f)... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 237,
"column": 72
} | {
"line": 237,
"column": 77
} | {
"line": 238,
"column": 6
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nσ : Type w\nhσ : Finite σ\nf : MvPolynomial PEmpty.{w + 1} S\nH : (algebraMap (MvPolynomial PEmpty.{w + 1} R) (MvPolynomial PEmpty.{w + 1} S)).IsIntegralElem f\n⊢ constantCoeff = (isEmptyAlgEquiv S PEmpty.{?u.342... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 238,
"column": 6
} | {
"line": 239,
"column": 64
} | {
"line": 240,
"column": 2
} | [
{
"pp": "case e'_6\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nσ : Type w\nhσ : Finite σ\nf : MvPolynomial PEmpty.{w + 1} S\nH : (algebraMap (MvPolynomial PEmpty.{w + 1} R) (MvPolynomial PEmpty.{w + 1} S)).IsIntegralElem f\nthis : constantCoeff = (isEmptyAlgEquiv ... | [] | simpa [-EmbeddingLike.apply_eq_iff_eq, -isEmptyAlgEquiv_apply] using!
congr((isEmptyAlgEquiv S PEmpty.{w + 1}).symm ($this f)) | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.RingTheory.QuasiFinite.Weakly | {
"line": 78,
"column": 2
} | {
"line": 79,
"column": 83
} | {
"line": 81,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : CommRing T\ninst✝² : Algebra R T\np : Ideal R\nq : Ideal S\ninst✝¹ : q.IsPrime\ninst✝ : QuasiFiniteAt R q\n⊢ WeaklyQuasiFiniteAt R q",
"ppTerm": "?m.19",
"assigned": true,
"use... | [] | rw [weaklyQuasiFiniteAt_iff]
exact .of_surjective_algHom (Ideal.Quotient.mkₐ _ _) Ideal.Quotient.mk_surjective | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.QuasiFinite.Weakly | {
"line": 78,
"column": 2
} | {
"line": 79,
"column": 83
} | {
"line": 81,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : CommRing T\ninst✝² : Algebra R T\np : Ideal R\nq : Ideal S\ninst✝¹ : q.IsPrime\ninst✝ : QuasiFiniteAt R q\n⊢ WeaklyQuasiFiniteAt R q",
"ppTerm": "?m.19",
"assigned": true,
"use... | [] | rw [weaklyQuasiFiniteAt_iff]
exact .of_surjective_algHom (Ideal.Quotient.mkₐ _ _) Ideal.Quotient.mk_surjective | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 247,
"column": 12
} | {
"line": 247,
"column": 17
} | {
"line": 248,
"column": 10
} | [
{
"pp": "case e'_5.hC\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nσ : Type w\nhσ : Finite σ\nα : Type w\ninst✝ : Fintype α\nIH :\n ∀ {f : MvPolynomial α S},\n (algebraMap (MvPolynomial α R) (MvPolynomial α S)).IsIntegralElem f → ∀ (n : α →₀ ℕ), IsIntegral R (... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 247,
"column": 12
} | {
"line": 247,
"column": 17
} | {
"line": 248,
"column": 10
} | [
{
"pp": "case e'_5.hC\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nσ : Type w\nhσ : Finite σ\nα : Type w\ninst✝ : Fintype α\nIH :\n ∀ {f : MvPolynomial α S},\n (algebraMap (MvPolynomial α R) (MvPolynomial α S)).IsIntegralElem f → ∀ (n : α →₀ ℕ), IsIntegral R (... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 247,
"column": 12
} | {
"line": 247,
"column": 17
} | {
"line": 248,
"column": 10
} | [
{
"pp": "case e'_5.hC\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nσ : Type w\nhσ : Finite σ\nα : Type w\ninst✝ : Fintype α\nIH :\n ∀ {f : MvPolynomial α S},\n (algebraMap (MvPolynomial α R) (MvPolynomial α S)).IsIntegralElem f → ∀ (n : α →₀ ℕ), IsIntegral R (... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 248,
"column": 24
} | {
"line": 248,
"column": 29
} | {
"line": 249,
"column": 8
} | [
{
"pp": "case e'_5.hX.none\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nσ : Type w\nhσ : Finite σ\nα : Type w\ninst✝ : Fintype α\nIH :\n ∀ {f : MvPolynomial α S},\n (algebraMap (MvPolynomial α R) (MvPolynomial α S)).IsIntegralElem f → ∀ (n : α →₀ ℕ), IsIntegra... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 248,
"column": 24
} | {
"line": 248,
"column": 29
} | {
"line": 249,
"column": 8
} | [
{
"pp": "case e'_5.hX.some\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nσ : Type w\nhσ : Finite σ\nα : Type w\ninst✝ : Fintype α\nIH :\n ∀ {f : MvPolynomial α S},\n (algebraMap (MvPolynomial α R) (MvPolynomial α S)).IsIntegralElem f → ∀ (n : α →₀ ℕ), IsIntegra... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 249,
"column": 10
} | {
"line": 249,
"column": 15
} | {
"line": 249,
"column": 15
} | [
{
"pp": "case e'_6\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nσ : Type w\nhσ : Finite σ\nα : Type w\ninst✝ : Fintype α\nIH :\n ∀ {f : MvPolynomial α S},\n (algebraMap (MvPolynomial α R) (MvPolynomial α S)).IsIntegralElem f → ∀ (n : α →₀ ℕ), IsIntegral R (coe... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 249,
"column": 10
} | {
"line": 249,
"column": 15
} | {
"line": 249,
"column": 15
} | [
{
"pp": "case e'_6\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nσ : Type w\nhσ : Finite σ\nα : Type w\ninst✝ : Fintype α\nIH :\n ∀ {f : MvPolynomial α S},\n (algebraMap (MvPolynomial α R) (MvPolynomial α S)).IsIntegralElem f → ∀ (n : α →₀ ℕ), IsIntegral R (coe... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.IsIntegral | {
"line": 249,
"column": 10
} | {
"line": 249,
"column": 15
} | {
"line": 249,
"column": 15
} | [
{
"pp": "case e'_6\nR : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nσ : Type w\nhσ : Finite σ\nα : Type w\ninst✝ : Fintype α\nIH :\n ∀ {f : MvPolynomial α S},\n (algebraMap (MvPolynomial α R) (MvPolynomial α S)).IsIntegralElem f → ∀ (n : α →₀ ℕ), IsIntegral R (coe... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.LocalRing.ResidueField.Polynomial | {
"line": 112,
"column": 31
} | {
"line": 112,
"column": 36
} | {
"line": 112,
"column": 36
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R S\nI : Ideal R\ninst✝⁵ : I.IsPrime\nJ : Ideal R[X]\ninst✝⁴ : J.IsPrime\ninst✝³ : J.LiesOver I\ninst✝² : Algebra (Localization.AtPrime I) (Localization.AtPrime J)\ninst✝¹ : Localization.AtPrime.IsLiesOverAlgebra I J... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski | {
"line": 124,
"column": 4
} | {
"line": 124,
"column": 42
} | {
"line": 125,
"column": 4
} | [
{
"pp": "X : Scheme\nU : X.AffineZariskiSite\nS : Sieve U\nx : ↥X\nhxU : x ∈ (toOpensFunctor X).obj U\n⊢ (∃ U_1 f, (Sieve.functorPushforward (toOpensFunctor X) S).arrows f ∧ x ∈ U_1) → ∃ V f, S.arrows f ∧ x ∈ V.toOpens",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"AlgebraicGeometr... | [
"X : Scheme\nU : X.AffineZariskiSite\nS : Sieve U\nx : ↥X\nhxU : x ∈ (toOpensFunctor X).obj U\nV : TopologicalSpace.Opens ↥X\nhxV : x ∈ V\nW : X.AffineZariskiSite\ng : W ⟶ U\nh : V ⟶ (toOpensFunctor X).obj W\nhg : S.arrows g\n⊢ ∃ V f, S.arrows f ∧ x ∈ V.toOpens"
] | rintro ⟨V, f, ⟨W, g, h, hg, rfl⟩, hxV⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.AlgebraicGeometry.Sites.SmallAffineZariski | {
"line": 229,
"column": 4
} | {
"line": 229,
"column": 56
} | {
"line": 230,
"column": 4
} | [
{
"pp": "X : Scheme\nU V : (directedCover X).I₀\nx : ↥(pullback ((directedCover X).f U) ((directedCover X).f V))\na : ↥X := (pullback.fst ((directedCover X).f U) ((directedCover X).f V) ≫ (↑U).ι) x\n⊢ ∃ k hki hkj y, (pullback.lift (X.homOfLE ⋯) (X.homOfLE ⋯) ⋯) y = x",
"ppTerm": "?m.53",
"assigned": tru... | [
"X : Scheme\nU V : (directedCover X).I₀\nx : ↥(pullback ((directedCover X).f U) ((directedCover X).f V))\na : ↥X := (pullback.fst ((directedCover X).f U) ((directedCover X).f V) ≫ (↑U).ι) x\nhaU : a ∈ ↑U\n⊢ ∃ k hki hkj y, (pullback.lift (X.homOfLE ⋯) (X.homOfLE ⋯) ⋯) y = x"
] | have haU : a ∈ U.1 := (pullback.fst U.1.ι V.1.ι x).2 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
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