module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Order.Ring.StandardPart | {
"line": 285,
"column": 4
} | {
"line": 288,
"column": 17
} | {
"line": 289,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝² : LinearOrder K\ninst✝¹ : Field K\ninst✝ : IsOrderedRing K\nx : K\nh : mk x ≠ 0\n⊢ stdPart x = 0",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"AddValuation.toValuation",
"_private.Mathlib.Algebra.Order.Ring.Stand... | [] | obtain h | h := h.lt_or_gt
· exact dif_neg h.not_ge
· rw [stdPart, dif_pos h.le, OrderRingHom.comp_apply, FiniteResidueField.mk_eq_zero.2 h,
map_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Ring.StandardPart | {
"line": 285,
"column": 4
} | {
"line": 288,
"column": 17
} | {
"line": 289,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝² : LinearOrder K\ninst✝¹ : Field K\ninst✝ : IsOrderedRing K\nx : K\nh : mk x ≠ 0\n⊢ stdPart x = 0",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"AddValuation.toValuation",
"_private.Mathlib.Algebra.Order.Ring.Stand... | [] | obtain h | h := h.lt_or_gt
· exact dif_neg h.not_ge
· rw [stdPart, dif_pos h.le, OrderRingHom.comp_apply, FiniteResidueField.mk_eq_zero.2 h,
map_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Derivation.MapCoeffs | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 23
} | {
"line": 169,
"column": 2
} | [
{
"pp": "case h\nA : Type u_1\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : Differential A\nR : Type u_2\ninst✝⁹ : CommRing R\ninst✝⁸ : Differential R\ninst✝⁷ : Algebra A R\ninst✝⁶ : DifferentialAlgebra A R\nR' : Type u_3\ninst✝⁵ : CommRing R'\ninst✝⁴ : Differential R'\ninst✝³ : Algebra A R'\ninst✝² : DifferentialAlgebra A ... | [
"case h\nA : Type u_1\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : Differential A\nR : Type u_2\ninst✝⁹ : CommRing R\ninst✝⁸ : Differential R\ninst✝⁷ : Algebra A R\ninst✝⁶ : DifferentialAlgebra A R\nR' : Type u_3\ninst✝⁵ : CommRing R'\ninst✝⁴ : Differential R'\ninst✝³ : Algebra A R'\ninst✝² : DifferentialAlgebra A R'\ninst✝¹ :... | rw [← deriv_aeval_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Homogenize | {
"line": 183,
"column": 2
} | {
"line": 183,
"column": 38
} | {
"line": 184,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\np q : R[X]\nm n : ℕ\nhm : p.natDegree ≤ m\nhn : q.natDegree ≤ n\n⊢ (p * q).homogenize (m + n) = p.homogenize m * q.homogenize n",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"Polynomial.homogenize_eq_of_is... | [
"case hq\nR : Type u_1\ninst✝ : CommSemiring R\np q : R[X]\nm n : ℕ\nhm : p.natDegree ≤ m\nhn : q.natDegree ≤ n\n⊢ (p.homogenize m * q.homogenize n).IsHomogeneous (m + n)",
"case hpq\nR : Type u_1\ninst✝ : CommSemiring R\np q : R[X]\nm n : ℕ\nhm : p.natDegree ≤ m\nhn : q.natDegree ≤ n\n⊢ (MvPolynomial.aeval ![X, ... | apply homogenize_eq_of_isHomogeneous | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Polynomial.UnitTrinomial | {
"line": 225,
"column": 2
} | {
"line": 226,
"column": 29
} | {
"line": 227,
"column": 2
} | [
{
"pp": "case h\np : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ m + (n - m + k) ∉ Set.Ioo (k + n) (n + n)",
"ppTerm": "?h✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"Nat.in... | [
"case h\np : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ m + k ∉ Set.Ioo (k + n) (n + n)",
"case h\np : ℤ[X]\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nu v w : ℤˣ\nhp : p = trinomial k m n ↑u ↑v ↑w\nkey : n - m + k < n\n⊢ k + n ∉ Set.Ioo (k + n) (n +... | · rw [← add_assoc, add_tsub_cancel_of_le hmn.le, add_comm]
exact fun h => h.1.ne rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 284,
"column": 2
} | {
"line": 284,
"column": 50
} | {
"line": 285,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : Nontrivial R\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\nf : R[X]\ng : ι → R[X]\nhg : ∀ i ∈ s, (g i).Monic\nhgg : (↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)\nn : ι → ℕ\nq : R[X]\nr : ι → R[X]\nhf : f = q * ∏ i ∈ s, g i ^ n i + ∑ i ∈ s, r i * ∏ k ∈ ... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : Nontrivial R\nι : Type u_2\ninst✝ : DecidableEq ι\ns : Finset ι\nf : R[X]\ng : ι → R[X]\nhg : ∀ i ∈ s, (g i).Monic\nhgg : (↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)\nn : ι → ℕ\nq : R[X]\nr : ι → R[X]\nhf : f = q * ∏ i ∈ s, g i ^ n i + ∑ i ∈ s, r i * ∏ k ∈ s.erase i, g... | refine ⟨q + ∑ i ∈ s, q' i, r', hr', hf.trans ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Polynomial.RuleOfSigns | {
"line": 177,
"column": 4
} | {
"line": 177,
"column": 77
} | {
"line": 178,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nP : R[X]\nη : R\nhη : 0 < η\nhP₀ : 0 < P.leadingCoeff\nhc : P.nextCoeff < 0\nd : ℕ\nhd : P.natDegree = d + 1\nhndxP : ((X - C η) * P).natDegree = P.natDegree + 1\nhePn0 : P.eraseLead ≠ 0\n⊢ ((X - C η) * P.eraseLead).n... | [
"R : Type u_1\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nP : R[X]\nη : R\nhη : 0 < η\nhP₀ : 0 < P.leadingCoeff\nhc : P.nextCoeff < 0\nd : ℕ\nhd : P.natDegree = d + 1\nhndxP : ((X - C η) * P).natDegree = P.natDegree + 1\nhePn0 : P.eraseLead ≠ 0\n⊢ P.eraseLead.natDegree + 1 = P.natDegree... | rw [natDegree_mul (X_sub_C_ne_zero η) hePn0, natDegree_X_sub_C, add_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.RuleOfSigns | {
"line": 227,
"column": 2
} | {
"line": 270,
"column": 68
} | {
"line": 272,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nP : R[X]\nη : R\nh₁ : 0 < P.leadingCoeff\nh₂ : P.nextCoeff ≠ 0\n⊢ ∃ c₀ cs,\n ((X - C η) * P).coeffList = P.leadingCoeff :: c₀ :: cs ∧ ((X - C η) * P.eraseLead).coeffList = P.nextCoeff :: cs",
"ppTerm": "?m.78",... | [] | have h₃ := leadingCoeff_ne_zero.mp h₁.ne'
have h₄ := natDegree_eraseLead_add_one h₂
have h₅ : (X - C η) ≠ 0 := X_sub_C_ne_zero η
have h₆ : P.eraseLead ≠ 0 := mt nextCoeff_eq_zero_of_eraseLead_eq_zero h₂
obtain ⟨d, hd⟩ := Nat.exists_eq_add_of_lt (natDegree_pos_of_nextCoeff_ne_zero h₂)
apply leadingCoeff_eraseL... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.RuleOfSigns | {
"line": 227,
"column": 2
} | {
"line": 270,
"column": 68
} | {
"line": 272,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : Ring R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nP : R[X]\nη : R\nh₁ : 0 < P.leadingCoeff\nh₂ : P.nextCoeff ≠ 0\n⊢ ∃ c₀ cs,\n ((X - C η) * P).coeffList = P.leadingCoeff :: c₀ :: cs ∧ ((X - C η) * P.eraseLead).coeffList = P.nextCoeff :: cs",
"ppTerm": "?m.78",... | [] | have h₃ := leadingCoeff_ne_zero.mp h₁.ne'
have h₄ := natDegree_eraseLead_add_one h₂
have h₅ : (X - C η) ≠ 0 := X_sub_C_ne_zero η
have h₆ : P.eraseLead ≠ 0 := mt nextCoeff_eq_zero_of_eraseLead_eq_zero h₂
obtain ⟨d, hd⟩ := Nat.exists_eq_add_of_lt (natDegree_pos_of_nextCoeff_ne_zero h₂)
apply leadingCoeff_eraseL... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Quandle | {
"line": 161,
"column": 40
} | {
"line": 161,
"column": 90
} | {
"line": 163,
"column": 0
} | [
{
"pp": "S : Type u_1\ninst✝ : UnitalShelf S\nx : S\n⊢ x ◃ x = x",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"UnitalShelf.toOne",
"Shelf.self_distrib",
"congrArg",
"UnitalShelf.toShelf",
"id",
"UnitalShelf.act_one",
"Shelf.act",
... | [] | by rw [← act_one x, ← Shelf.self_distrib, act_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Quandle | {
"line": 283,
"column": 8
} | {
"line": 283,
"column": 30
} | {
"line": 283,
"column": 31
} | [
{
"pp": "R : Type u_1\ninst✝ : Rack R\nx y : R\nh : x ◃ x = y ◃ y\n⊢ x = (x ◃ x) ◃⁻¹ x ◃ x",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Rack.toShelf",
"id",
"Rack.left_cancel",
"Shelf.act",
"propext",
"Eq.symm",
... | [
"R : Type u_1\ninst✝ : Rack R\nx y : R\nh : x ◃ x = y ◃ y\n⊢ (x ◃ x) ◃ x = (x ◃ x) ◃ (x ◃ x) ◃⁻¹ x ◃ x"
] | ← left_cancel (x ◃ x), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Quandle | {
"line": 430,
"column": 4
} | {
"line": 430,
"column": 11
} | {
"line": 431,
"column": 2
} | [
{
"pp": "Q : Type u_1\ninst✝ : Quandle Q\nn : ℕ\nx y z : Dihedral n\n⊢ 2 * x - (2 * y - z) = 2 * (2 * x - y) - (2 * x - z)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddC... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Algebra.Quandle | {
"line": 437,
"column": 4
} | {
"line": 437,
"column": 11
} | {
"line": 439,
"column": 0
} | [
{
"pp": "Q : Type u_1\ninst✝ : Quandle Q\nn : ℕ\nx : Dihedral n\n⊢ 2 * x - x = x",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.R... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Algebra.Quandle | {
"line": 681,
"column": 10
} | {
"line": 681,
"column": 66
} | {
"line": 682,
"column": 10
} | [
{
"pp": "case inv\nR : Type u_1\ninst✝¹ : Rack R\nG : Type u_2\ninst✝ : Group G\nF : EnvelGroup R →* G\nx✝ : EnvelGroup R\nx : PreEnvelGroup R\nih_x :\n ((fun f ↦ { toFun := fun x ↦ Quotient.liftOn x (mapAux f) ⋯, map_one' := ⋯, map_mul' := ⋯ })\n ((fun F ↦ (Quandle.Conj.map F).comp (toEnvelGroup R)) F)... | [
"case inv\nR : Type u_1\ninst✝¹ : Rack R\nG : Type u_2\ninst✝ : Group G\nF : EnvelGroup R →* G\nx✝ : EnvelGroup R\nx : PreEnvelGroup R\nih_x :\n ((fun f ↦ { toFun := fun x ↦ Quotient.liftOn x (mapAux f) ⋯, map_one' := ⋯, map_mul' := ⋯ })\n ((fun F ↦ (Quandle.Conj.map F).comp (toEnvelGroup R)) F))\n ⟦x⟧... | have hm : ⟦x.inv⟧ = @Inv.inv (EnvelGroup R) _ ⟦x⟧ := rfl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.QuaternionBasis | {
"line": 125,
"column": 81
} | {
"line": 135,
"column": 37
} | {
"line": 137,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nc₁ c₂ c₃ : R\nq : Basis A c₁ c₂ c₃\nx y : ℍ[R,c₁,c₂,c₃]\n⊢ q.lift (x * y) = q.lift x * q.lift y",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
... | [] | by
simp only [lift, Algebra.algebraMap_eq_smul_one]
simp_rw [add_mul, mul_add, smul_mul_assoc, mul_smul_comm, one_mul, mul_one, smul_smul]
simp only [i_mul_i, j_mul_j, i_mul_j, j_mul_i, i_mul_k, k_mul_i, k_mul_j, j_mul_k, k_mul_k]
simp only [smul_smul, smul_neg, sub_eq_add_neg, ← add_assoc, neg_smul]
simp onl... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.SkewMonoidAlgebra.Lift | {
"line": 88,
"column": 4
} | {
"line": 89,
"column": 21
} | {
"line": 91,
"column": 0
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝⁵ : CommSemiring k\ninst✝⁴ : Monoid G\nA : Type u_4\ninst✝³ : Semiring A\ninst✝² : Algebra k A\ninst✝¹ : MulSemiringAction G k\ninst✝ : SMulCommClass G k k\nF : SkewMonoidAlgebra k G →ₐ[k] A\nf : SkewMonoidAlgebra k G\n| F f",
"ppTerm": "?m.39",
"assigned": true... | [
"k : Type u_1\nG : Type u_2\ninst✝⁵ : CommSemiring k\ninst✝⁴ : Monoid G\nA : Type u_4\ninst✝³ : Semiring A\ninst✝² : Algebra k A\ninst✝¹ : MulSemiringAction G k\ninst✝ : SMulCommClass G k k\nF : SkewMonoidAlgebra k G →ₐ[k] A\nf : SkewMonoidAlgebra k G\n| f.sum fun a b ↦ b • F (single a 1)"
] | rw [lift_unique' F]
simp [lift_apply] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.Algebra.SkewMonoidAlgebra.Lift | {
"line": 88,
"column": 4
} | {
"line": 89,
"column": 21
} | {
"line": 91,
"column": 0
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝⁵ : CommSemiring k\ninst✝⁴ : Monoid G\nA : Type u_4\ninst✝³ : Semiring A\ninst✝² : Algebra k A\ninst✝¹ : MulSemiringAction G k\ninst✝ : SMulCommClass G k k\nF : SkewMonoidAlgebra k G →ₐ[k] A\nf : SkewMonoidAlgebra k G\n| F f",
"ppTerm": "?m.39",
"assigned": true... | [
"k : Type u_1\nG : Type u_2\ninst✝⁵ : CommSemiring k\ninst✝⁴ : Monoid G\nA : Type u_4\ninst✝³ : Semiring A\ninst✝² : Algebra k A\ninst✝¹ : MulSemiringAction G k\ninst✝ : SMulCommClass G k k\nF : SkewMonoidAlgebra k G →ₐ[k] A\nf : SkewMonoidAlgebra k G\n| f.sum fun a b ↦ b • F (single a 1)"
] | rw [lift_unique' F]
simp [lift_apply] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.Algebra.RingQuot | {
"line": 387,
"column": 41
} | {
"line": 389,
"column": 5
} | {
"line": 392,
"column": 0
} | [
{
"pp": "R : Type uR\ninst✝¹ : Semiring R\nT : Type uT\ninst✝ : Semiring T\nf : R →+* T\nr : R → R → Prop\nw : ∀ ⦃x y : R⦄, r x y → f x = f y\nx : R\n⊢ (lift ⟨f, w⟩) ((mkRingHom r) x) = f x",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"RingQuot.definition._proof_1.... | [] | by
simp_rw [lift_def, preLift_def, mkRingHom_def]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.SkewPolynomial.Basic | {
"line": 374,
"column": 60
} | {
"line": 374,
"column": 82
} | {
"line": 376,
"column": 0
} | [
{
"pp": "R : Type u_1\nn : ℕ\ninst✝ : Semiring R\na : R\nh : n ≠ 0\n⊢ (C a).coeff n = 0",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"congrArg",
"AddMonoid.toAddZeroClass",
"SkewMonoidAlgebra.instAddMonoi... | [] | rw [coeff_C, if_neg h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.SkewPolynomial.Basic | {
"line": 374,
"column": 60
} | {
"line": 374,
"column": 82
} | {
"line": 376,
"column": 0
} | [
{
"pp": "R : Type u_1\nn : ℕ\ninst✝ : Semiring R\na : R\nh : n ≠ 0\n⊢ (C a).coeff n = 0",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"congrArg",
"AddMonoid.toAddZeroClass",
"SkewMonoidAlgebra.instAddMonoi... | [] | rw [coeff_C, if_neg h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.SkewPolynomial.Basic | {
"line": 374,
"column": 60
} | {
"line": 374,
"column": 82
} | {
"line": 376,
"column": 0
} | [
{
"pp": "R : Type u_1\nn : ℕ\ninst✝ : Semiring R\na : R\nh : n ≠ 0\n⊢ (C a).coeff n = 0",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"congrArg",
"AddMonoid.toAddZeroClass",
"SkewMonoidAlgebra.instAddMonoi... | [] | rw [coeff_C, if_neg h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Tropical.Basic | {
"line": 293,
"column": 6
} | {
"line": 293,
"column": 19
} | {
"line": 293,
"column": 20
} | [
{
"pp": "R : Type u\ninst✝ : LinearOrder R\nx y : Tropical R\n⊢ x + y = x ↔ x ≤ y",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
"DistribLattice.toLattice"... | [
"R : Type u\ninst✝ : LinearOrder R\nx y : Tropical R\n⊢ trop (min (untrop x) (untrop y)) = x ↔ x ≤ y"
] | trop_add_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Tropical.Basic | {
"line": 296,
"column": 6
} | {
"line": 296,
"column": 19
} | {
"line": 296,
"column": 20
} | [
{
"pp": "R : Type u\ninst✝ : LinearOrder R\nx y : Tropical R\n⊢ x + y = y ↔ y ≤ x",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
"DistribLattice.toLattice"... | [
"R : Type u\ninst✝ : LinearOrder R\nx y : Tropical R\n⊢ trop (min (untrop x) (untrop y)) = y ↔ y ≤ x"
] | trop_add_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Tropical.Basic | {
"line": 302,
"column": 6
} | {
"line": 302,
"column": 19
} | {
"line": 302,
"column": 20
} | [
{
"pp": "R : Type u\ninst✝ : LinearOrder R\nx y z : Tropical R\n⊢ x + y = z ↔ x = z ∧ x ≤ y ∨ y = z ∧ y ≤ x",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
... | [
"R : Type u\ninst✝ : LinearOrder R\nx y z : Tropical R\n⊢ trop (min (untrop x) (untrop y)) = z ↔ x = z ∧ x ≤ y ∨ y = z ∧ y ≤ x"
] | trop_add_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Tropical.BigOperators | {
"line": 105,
"column": 19
} | {
"line": 105,
"column": 77
} | {
"line": 107,
"column": 0
} | [
{
"pp": "case cons\nR : Type u_1\ninst✝¹ : LinearOrder R\ninst✝ : OrderTop R\ns : Tropical R\nx : Multiset (Tropical R)\nIH : untrop x.sum = (map untrop x).inf\n⊢ untrop (s ::ₘ x).sum = (map untrop (s ::ₘ x)).inf",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"Multiset.sum",
"... | [] | simp only [sum_cons, untrop_add, map_cons, inf_cons, ← IH] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Tropical.BigOperators | {
"line": 105,
"column": 19
} | {
"line": 105,
"column": 77
} | {
"line": 107,
"column": 0
} | [
{
"pp": "case cons\nR : Type u_1\ninst✝¹ : LinearOrder R\ninst✝ : OrderTop R\ns : Tropical R\nx : Multiset (Tropical R)\nIH : untrop x.sum = (map untrop x).inf\n⊢ untrop (s ::ₘ x).sum = (map untrop (s ::ₘ x)).inf",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"Multiset.sum",
"... | [] | simp only [sum_cons, untrop_add, map_cons, inf_cons, ← IH] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Tropical.BigOperators | {
"line": 105,
"column": 19
} | {
"line": 105,
"column": 77
} | {
"line": 107,
"column": 0
} | [
{
"pp": "case cons\nR : Type u_1\ninst✝¹ : LinearOrder R\ninst✝ : OrderTop R\ns : Tropical R\nx : Multiset (Tropical R)\nIH : untrop x.sum = (map untrop x).inf\n⊢ untrop (s ::ₘ x).sum = (map untrop (s ::ₘ x)).inf",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"Multiset.sum",
"... | [] | simp only [sum_cons, untrop_add, map_cons, inf_cons, ← IH] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Sites.Pullback | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 34
} | {
"line": 98,
"column": 0
} | [
{
"pp": "C : Type u₂\ninst✝⁸ : Category.{v₂, u₂} C\nD : Type u₃\ninst✝⁷ : Category.{v₃, u₃} D\nG : C ⥤ D\nA : Type u₁\ninst✝⁶ : Category.{v₁, u₁} A\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\ninst✝⁵ : G.IsContinuous J K\ninst✝⁴ : ∀ (F : Cᵒᵖ ⥤ A), G.op.HasLeftKanExtension F\ninst✝³ : RepresentablyFl... | [] | apply comp_preservesFiniteLimits | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Geometry.RingedSpace.PresheafedSpace.HasColimits | {
"line": 317,
"column": 4
} | {
"line": 317,
"column": 57
} | {
"line": 318,
"column": 4
} | [
{
"pp": "case app\nJ : Type u'\ninst✝⁵ : Category.{v', u'} J\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasColimitsOfShape J TopCat\ninst✝² : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)\ninst✝¹ : HasLimitsOfShape Jᵒᵖ C\nF : J ⥤ PresheafedSpace C\ninst✝ : HasColimit F\nU : Opens ↑↑(Limits.colimit ... | [
"case app\nJ : Type u'\ninst✝⁵ : Category.{v', u'} J\nC : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasColimitsOfShape J TopCat\ninst✝² : ∀ (X : TopCat), HasLimitsOfShape Jᵒᵖ (Presheaf C X)\ninst✝¹ : HasLimitsOfShape Jᵒᵖ C\nF : J ⥤ PresheafedSpace C\ninst✝ : HasColimit F\nU : Opens ↑↑(Limits.colimit F)\nX : Jᵒᵖ\... | simp only [colimitCocone, colimit, ← TopCat.comp_app] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Sheaves.LocalPredicate | {
"line": 261,
"column": 6
} | {
"line": 269,
"column": 21
} | {
"line": 273,
"column": 4
} | [
{
"pp": "case refine_1\nX : TopCat\nT : ↑X → Type u_1\nP : LocalPredicate T\nι : Type u_2\nU : ι → Opens ↑X\nsf : (i : ι) → ToType ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i)))\nsf_comp : (subpresheafToTypes P.toPrelocalPredicate).IsCompatible U sf\nsf' : (i : ι) → (X.presheafToTypes T).obj (op (U... | [] | apply P.locality
rintro ⟨x, mem⟩
-- Once we're at a particular point `x`, we can select some open set `x ∈ U i`.
choose i hi using Opens.mem_iSup.mp mem
-- We claim that the predicate holds in `U i`
use U i, hi, Opens.leSupr U i
-- This follows, since our original family `sf` satisfi... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sheaves.LocalPredicate | {
"line": 261,
"column": 6
} | {
"line": 269,
"column": 21
} | {
"line": 273,
"column": 4
} | [
{
"pp": "case refine_1\nX : TopCat\nT : ↑X → Type u_1\nP : LocalPredicate T\nι : Type u_2\nU : ι → Opens ↑X\nsf : (i : ι) → ToType ((subpresheafToTypes P.toPrelocalPredicate).obj (op (U i)))\nsf_comp : (subpresheafToTypes P.toPrelocalPredicate).IsCompatible U sf\nsf' : (i : ι) → (X.presheafToTypes T).obj (op (U... | [] | apply P.locality
rintro ⟨x, mem⟩
-- Once we're at a particular point `x`, we can select some open set `x ∈ U i`.
choose i hi using Opens.mem_iSup.mp mem
-- We claim that the predicate holds in `U i`
use U i, hi, Opens.leSupr U i
-- This follows, since our original family `sf` satisfi... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.RingedSpace.OpenImmersion | {
"line": 332,
"column": 18
} | {
"line": 333,
"column": 80
} | {
"line": 333,
"column": 80
} | [
{
"pp": "case a\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : PresheafedSpace C\nf : X ⟶ Z\nhf : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\ng : Y ⟶ Z\nU : (Opens ↑↑X)ᵒᵖ\n⊢ {\n carrier :=\n ⇑(ConcreteCategory.hom (pullback.snd f.base g.base)) ''\... | [] | · rintro _ ⟨x, h₁, rfl⟩
exact ⟨_, h₁, CategoryTheory.congr_fun pullback.condition x⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.RingedSpace.OpenImmersion | {
"line": 580,
"column": 37
} | {
"line": 580,
"column": 44
} | {
"line": 580,
"column": 44
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nX Y : SheafedSpace C\nf : X ⟶ Y\ninst✝ : SheafedSpace.IsOpenImmersion f\n⊢ toSheafedSpace Y f.hom = X",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.IsOpenImmersion",
"AlgebraicGeometry.SheafedSpa... | [
"case mk\nC : Type u\ninst✝¹ : Category.{v, u} C\nY : SheafedSpace C\ntoPresheafedSpace✝ : PresheafedSpace C\nIsSheaf✝ : toPresheafedSpace✝.presheaf.IsSheaf\nf : { toPresheafedSpace := toPresheafedSpace✝, IsSheaf := IsSheaf✝ } ⟶ Y\ninst✝ : SheafedSpace.IsOpenImmersion f\n⊢ toSheafedSpace Y f.hom = { toPresheafedSpa... | cases X | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 409,
"column": 2
} | {
"line": 409,
"column": 9
} | {
"line": 409,
"column": 9
} | [
{
"pp": "X Y : Scheme\nH : X.toLocallyRingedSpace = Y.toLocallyRingedSpace\n⊢ X = Y",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.mk",
"AlgebraicGeometry.Scheme",
"Opposite",
"TopologicalSpace.Opens.instPartialOrder",
"CommRingCat",... | [
"case mk\nY : Scheme\ntoLocallyRingedSpace✝ : LocallyRingedSpace\nlocal_affine✝ :\n ∀ (x : ↑toLocallyRingedSpace✝.toTopCat),\n ∃ U R, Nonempty (toLocallyRingedSpace✝.restrict ⋯ ≅ Spec.toLocallyRingedSpace.obj (op R))\nH :\n { toLocallyRingedSpace := toLocallyRingedSpace✝, local_affine := local_affine✝ }.toLoca... | cases X | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 298,
"column": 45
} | {
"line": 298,
"column": 52
} | {
"line": 300,
"column": 0
} | [
{
"pp": "case refine_3\nR M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nU : Opens ↑(PrimeSpectrum.Top R)\ns : (structureSheafInType R M).obj.obj (op U)\nx : ↑(PrimeSpectrum.Top R)\nhx : x ∈ U\nV : Opens ↑(PrimeSpectrum.Top R)\nhxV : ↑⟨x, hx⟩ ∈ V\niVU : V ⟶ unop (op U)\nf : M\ng :... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 464,
"column": 86
} | {
"line": 464,
"column": 93
} | {
"line": 464,
"column": 93
} | [
{
"pp": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nU : Opens ↑(PrimeSpectrum.Top R)\nhU : IsCompact ↑U\ns : (structureSheafInType R M).obj.obj (op U)\ng : ↥U → R\nhxg : ∀ (x : ↥U), ↑x ∈ basicOpen (g x)\nigU : ∀ (x : ↥U), basicOpen (g x) ≤ U\nf : ↥U → M\nH :\n ∀ (x : ↥U),\n... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.AlgebraicGeometry.Restrict | {
"line": 586,
"column": 97
} | {
"line": 592,
"column": 55
} | {
"line": 594,
"column": 0
} | [
{
"pp": "X : Scheme\nU V W : X.Opens\nhU : U ≤ W\nhV : V ≤ W\n⊢ IsPullback (X.homOfLE ⋯) (X.homOfLE ⋯) (X.homOfLE hU) (X.homOfLE hV)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.opensFunctor",
"Eq.mpr",
"CategoryTheory.Category.assoc",
... | [] | by
refine (isPullback_morphismRestrict (X.homOfLE hV) (W.ι ⁻¹ᵁ U)).of_iso (V.ι.isoImage _ ≪≫
X.isoOfEq ?_) (W.ι.isoImage _ ≪≫ X.isoOfEq ?_) (Iso.refl _) (Iso.refl _) ?_ ?_ ?_ ?_
· rw [← TopologicalSpace.Opens.map_comp_obj, ← Scheme.Hom.comp_base, Scheme.homOfLE_ι]
exact V.functor_map_eq_inf U
· exact (W.f... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Restrict | {
"line": 843,
"column": 49
} | {
"line": 843,
"column": 62
} | {
"line": 843,
"column": 62
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v, u₁} C\nX Y : Scheme\nf : X ⟶ Y\nU U' : Y.Opens\nV V' : X.Opens\ne : V ≤ f ⁻¹ᵁ U\nx : ↥V\n⊢ (Y.presheaf.stalkSpecializes ⋯ ≫ stalkMap f ↑x) ≫ stalkMap V.ι x =\n (Y.presheaf.stalkCongr ⋯).hom ≫ stalkMap (V.ι ≫ f) x",
"ppTerm": "?m.125",
"assigned": true,
"... | [
"C : Type u₁\ninst✝ : Category.{v, u₁} C\nX Y : Scheme\nf : X ⟶ Y\nU U' : Y.Opens\nV V' : X.Opens\ne : V ≤ f ⁻¹ᵁ U\nx : ↥V\n⊢ (Y.presheaf.stalkSpecializes ⋯ ≫ stalkMap f ↑x) ≫ stalkMap V.ι x =\n (Y.presheaf.stalkCongr ⋯).hom ≫ stalkMap f (V.ι x) ≫ stalkMap V.ι x"
] | stalkMap_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.GammaSpecAdjunction | {
"line": 155,
"column": 4
} | {
"line": 157,
"column": 59
} | {
"line": 158,
"column": 2
} | [
{
"pp": "case mp\nX : LocallyRingedSpace\nr : ↑(Γ.obj (op X))\nf : (structureSheaf ↑(Γ.obj (op X))).obj.obj (op (basicOpen r)) ⟶ X.presheaf.obj (op (X.toΓSpecMapBasicOpen r))\nloc_inst : IsLocalization.Away r ↑((structureSheaf ↑(Γ.obj (op X))).obj.obj (op (basicOpen r)))\n⊢ ConcreteCategory.hom\n (CommRi... | [] | intro h
ext : 1
exact IsLocalization.ringHom_ext (Submonoid.powers r) h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.GammaSpecAdjunction | {
"line": 155,
"column": 4
} | {
"line": 157,
"column": 59
} | {
"line": 158,
"column": 2
} | [
{
"pp": "case mp\nX : LocallyRingedSpace\nr : ↑(Γ.obj (op X))\nf : (structureSheaf ↑(Γ.obj (op X))).obj.obj (op (basicOpen r)) ⟶ X.presheaf.obj (op (X.toΓSpecMapBasicOpen r))\nloc_inst : IsLocalization.Away r ↑((structureSheaf ↑(Γ.obj (op X))).obj.obj (op (basicOpen r)))\n⊢ ConcreteCategory.hom\n (CommRi... | [] | intro h
ext : 1
exact IsLocalization.ringHom_ext (Submonoid.powers r) h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.GammaSpecAdjunction | {
"line": 209,
"column": 4
} | {
"line": 209,
"column": 51
} | {
"line": 209,
"column": 51
} | [
{
"pp": "X : LocallyRingedSpace\nx : ↑X.toTopCat\n⊢ (X.toToΓSpecMapBasicOpen 1 ≫\n ((pushforward CommRingCat X.toΓSpecSheafedSpace.hom.base).obj X.presheaf).germ (basicOpen 1)\n ((ConcreteCategory.hom X.toΓSpecSheafedSpace.hom.base) x) ⋯) ≫\n stalkPushforward CommRingCat X.toΓSpecSheafedSpa... | [
"X : LocallyRingedSpace\nx : ↑X.toTopCat\n⊢ (X.toToΓSpecMapBasicOpen 1 ≫\n ((pushforward CommRingCat X.toΓSpecSheafedSpace.hom.base).obj X.presheaf).germ (basicOpen 1)\n ((ConcreteCategory.hom X.toΓSpecSheafedSpace.hom.base) x) ⋯) ≫\n stalkPushforward CommRingCat X.toΓSpecSheafedSpace.hom.base ... | ← dsimp% stalkPushforward_germ _ _ X.presheaf ⊤ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.GlueData | {
"line": 324,
"column": 2
} | {
"line": 324,
"column": 41
} | {
"line": 325,
"column": 2
} | [
{
"pp": "C : Type u₁\ninst✝⁵ : Category.{v, u₁} C\nC' : Type u₂\ninst✝⁴ : Category.{v, u₂} C'\nD : GlueData C\nF : C ⥤ C'\ninst✝³ : HasMulticoequalizer D.diagram\ninst✝² : PreservesColimit D.diagram.multispan F\ninst✝¹ : ∀ (i j k : D.J), PreservesLimit (cospan (D.f i j) (D.f i k)) F\ni j : D.J\ninst✝ : Reflects... | [
"C : Type u₁\ninst✝⁵ : Category.{v, u₁} C\nC' : Type u₂\ninst✝⁴ : Category.{v, u₂} C'\nD : GlueData C\nF : C ⥤ C'\ninst✝³ : HasMulticoequalizer D.diagram\ninst✝² : PreservesColimit D.diagram.multispan F\ninst✝¹ : ∀ (i j k : D.J), PreservesLimit (cospan (D.f i j) (D.f i k)) F\ni j : D.J\ninst✝ : ReflectsLimit (cospa... | apply IsLimit.postcomposeHomEquiv e _ _ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.AlgebraicGeometry.GammaSpecAdjunction | {
"line": 538,
"column": 2
} | {
"line": 538,
"column": 79
} | {
"line": 539,
"column": 2
} | [
{
"pp": "R S : CommRingCat\nφ ψ : R ⟶ S\n⊢ map φ = map ψ ↔ φ = ψ",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.Scheme",
"Opposite",
"Quiver.opposite",
"CategoryTheory.CategoryStruct.toQuiver",
... | [
"R S : CommRingCat\nφ ψ : R ⟶ S\n⊢ Scheme.Spec.map φ.op = Scheme.Spec.map ψ.op ↔ map φ = map ψ"
] | rw [iff_comm, ← Quiver.Hom.op_inj.eq_iff, ← Scheme.Spec.map_injective.eq_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits | {
"line": 73,
"column": 4
} | {
"line": 75,
"column": 99
} | {
"line": 77,
"column": 0
} | [
{
"pp": "ι : Type v\ninst✝ : Small.{u, v} ι\nF : Discrete ι ⥤ LocallyRingedSpace\ni : Discrete ι\ny : ↑↑((F ⋙ forgetToSheafedSpace).obj i).toPresheafedSpace\nthis : IsLocalRing ↑(((F ⋙ forgetToSheafedSpace).obj i).presheaf.stalk y)\n⊢ IsLocalRing\n ↑((colimit (F ⋙ forgetToSheafedSpace)).presheaf.stalk\n ... | [] | exact
(asIso ((colimit.ι (C := SheafedSpace.{u + 1, u, u} CommRingCat.{u})
(F ⋙ forgetToSheafedSpace) i :).hom.stalkMap y)).symm.commRingCatIsoToRingEquiv.isLocalRing | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 326,
"column": 80
} | {
"line": 329,
"column": 33
} | {
"line": 331,
"column": 0
} | [
{
"pp": "X : Scheme\n𝒰 : X.OpenCover\nx y z : 𝒰.I₀\n⊢ gluedCoverT' 𝒰 x y z ≫ gluedCoverT' 𝒰 y z x ≫ gluedCoverT' 𝒰 z x y =\n 𝟙 (pullback (pullback.fst (𝒰.f x) (𝒰.f y)) (pullback.fst (𝒰.f x) (𝒰.f z)))",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Catego... | [] | by
apply pullback.hom_ext <;> simp_rw [Category.id_comp, Category.assoc]
· apply glued_cover_cocycle_fst
· apply glued_cover_cocycle_snd | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 544,
"column": 2
} | {
"line": 544,
"column": 76
} | {
"line": 545,
"column": 2
} | [
{
"pp": "J : Type w\ninst✝² : Category.{v, w} J\nF : J ⥤ Scheme\ninst✝¹ : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)\ninst✝ : (F ⋙ forget).IsLocallyDirected\ni j k : J\nx : ↥(pullback (V F i j).ι (V F i k).ι)\n⊢ ∃ l fi fj fk α z,\n IsOpenImmersion α ∧\n α ≫ pullback.fst (V F i j).ι (V F i k).ι =... | [
"J : Type w\ninst✝² : Category.{v, w} J\nF : J ⥤ Scheme\ninst✝¹ : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)\ninst✝ : (F ⋙ forget).IsLocallyDirected\ni j k : J\nx : ↥(pullback (V F i j).ι (V F i k).ι)\nk₁ : (k : J) × (k ⟶ i) × (k ⟶ j)\ny₁ : ↥(F.obj k₁.fst)\nhy₁ : (F.map k₁.snd.1) y₁ = ↑((pullback.fst (V F i... | obtain ⟨k₁, y₁, hy₁⟩ := mem_iSup.mp ((pullback.fst (C := Scheme) _ _) x).2 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 558,
"column": 4
} | {
"line": 558,
"column": 84
} | {
"line": 559,
"column": 4
} | [
{
"pp": "case inst\nJ : Type w\ninst✝² : Category.{v, w} J\nF : J ⥤ Scheme\ninst✝¹ : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)\ninst✝ : (F ⋙ forget).IsLocallyDirected\ni j k : J\nx : ↥(pullback (V F i j).ι (V F i k).ι)\nk₁ : (k : J) × (k ⟶ i) × (k ⟶ j)\nk₂ : (k_1 : J) × (k_1 ⟶ i) × (k_1 ⟶ k)\nl : J\nhl... | [
"case inst\nJ : Type w\ninst✝² : Category.{v, w} J\nF : J ⥤ Scheme\ninst✝¹ : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)\ninst✝ : (F ⋙ forget).IsLocallyDirected\ni j k : J\nx : ↥(pullback (V F i j).ι (V F i k).ι)\nk₁ : (k : J) × (k ⟶ i) × (k ⟶ j)\nk₂ : (k_1 : J) × (k_1 ⟶ i) × (k_1 ⟶ k)\nl : J\nhli : l ⟶ k₁.f... | · exact (inferInstance : IsOpenImmersion (pullback.fst (V F i j).ι (V F i k).ι)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 581,
"column": 2
} | {
"line": 590,
"column": 57
} | {
"line": 591,
"column": 2
} | [
{
"pp": "J : Type w\ninst✝³ : Category.{v, w} J\nF : J ⥤ Scheme\ninst✝² : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)\ninst✝¹ : (F ⋙ forget).IsLocallyDirected\ninst✝ : Quiver.IsThin J\ni j : J\nk₁ k₂ : (k : J) × (k ⟶ i) × (k ⟶ j)\nU : (F.obj i).Opens\nh₁ : Hom.opensRange (F.map k₁.snd.1) ≤ U\nh₂ : Hom.op... | [
"J : Type w\ninst✝³ : Category.{v, w} J\nF : J ⥤ Scheme\ninst✝² : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)\ninst✝¹ : (F ⋙ forget).IsLocallyDirected\ninst✝ : Quiver.IsThin J\ni j : J\nk₁ k₂ : (k : J) × (k ⟶ i) × (k ⟶ j)\nU : (F.obj i).Opens\nh₁ : Hom.opensRange (F.map k₁.snd.1) ≤ U\nh₂ : Hom.opensRange (F.... | obtain ⟨l, hli, hlj, y, hy₁, hy₂⟩ := (F ⋙ forget).exists_map_eq_of_isLocallyDirected k₁.2.1 k₂.2.1
((pullback.fst _ _ ≫ (F.map k₁.2.1).isoOpensRange.inv) x)
((pullback.snd _ _ ≫ (F.map k₂.2.1).isoOpensRange.inv) x) (by
simp only [Functor.comp_obj, forget_obj, Functor.comp_map, forget_map,
Concrete... | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.Limits | {
"line": 269,
"column": 4
} | {
"line": 270,
"column": 68
} | {
"line": 271,
"column": 4
} | [
{
"pp": "case left\nι : Type u\nf : ι → Scheme\nX : Scheme\nα : (i : ι) → f i ⟶ X\ninst✝ : ∀ (i : ι), IsOpenImmersion (α i)\nhα : _root_.Pairwise (Disjoint on fun x ↦ Set.range ⇑(α x))\n⊢ Topology.IsOpenEmbedding ⇑(Sigma.desc α)",
"ppTerm": "?left",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"case left\nι : Type u\nf : ι → Scheme\nX : Scheme\nα : (i : ι) → f i ⟶ X\ninst✝ : ∀ (i : ι), IsOpenImmersion (α i)\nhα : _root_.Pairwise (Disjoint on fun x ↦ Set.range ⇑(α x))\n⊢ Topology.IsOpenEmbedding (⇑(Sigma.desc α) ∘ ⇑(sigmaMk f))"
] | suffices Topology.IsOpenEmbedding (Sigma.desc α ∘ sigmaMk f) by
convert! this.comp (sigmaMk f).symm.isOpenEmbedding; ext; simp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.AlgebraicGeometry.Limits | {
"line": 290,
"column": 4
} | {
"line": 290,
"column": 95
} | {
"line": 292,
"column": 0
} | [
{
"pp": "case right.refine_2\nι : Type u\nf : ι → Scheme\nX : Scheme\nα : (i : ι) → f i ⟶ X\ninst✝ : ∀ (i : ι), IsOpenImmersion (α i)\nhα : _root_.Pairwise (Disjoint on fun x ↦ Set.range ⇑(α x))\nx : ↥(∐ f)\ny :\n ↥((Scheme.IsLocallyDirected.openCover (Discrete.functor f)).X\n (Scheme.Cover.idx (Scheme.Is... | [] | · simp [← Scheme.Hom.stalkMap_comp, Scheme.Hom.stalkMap_congr_hom _ _ (colimit.ι_desc _ _)] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.Limits | {
"line": 387,
"column": 58
} | {
"line": 393,
"column": 47
} | {
"line": 395,
"column": 0
} | [
{
"pp": "X Y : Scheme\nx : ↥Y\n⊢ (coprodMk X Y) (Sum.inr x) = coprod.inr x",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"CategoryTheory.Limits.ColimitCocone.cocone",
"CategoryTheory.Limits.BinaryCofan.inr",
"CategoryTheory.Functor",
"AlgebraicGeometry.SheafedSpac... | [] | by
change ((TopCat.binaryCofan X Y).inr ≫
(colimit.isoColimitCocone ⟨_, TopCat.binaryCofanIsColimit _ _⟩).inv ≫ _) x =
Scheme.forgetToTop.map coprod.inr x
congr 2
refine (colimit.isoColimitCocone_ι_inv_assoc ⟨_, TopCat.binaryCofanIsColimit _ _⟩ _ _).trans ?_
exact coprodComparison_inr Scheme.forgetToT... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Limits | {
"line": 410,
"column": 6
} | {
"line": 410,
"column": 67
} | {
"line": 411,
"column": 4
} | [
{
"pp": "case h.inl\nι : Type u\nf : ι → Scheme\nσ : Type v\ng : σ → Scheme\nX Y : Scheme\nx : ↥X\n⊢ ∃ y,\n (Sum.rec (fun x ↦ coprod.inl) (fun x ↦ coprod.inr)\n (Sum.elim (fun x ↦ Sum.inl PUnit.unit) (fun x ↦ Sum.inr PUnit.unit) (Sum.inl x)))\n y =\n (coprodMk X Y) (Sum.inl x)",
"ppT... | [] | simp only [Sum.elim_inl, coprodMk_inl, exists_apply_eq_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.Limits | {
"line": 410,
"column": 6
} | {
"line": 410,
"column": 67
} | {
"line": 411,
"column": 4
} | [
{
"pp": "case h.inl\nι : Type u\nf : ι → Scheme\nσ : Type v\ng : σ → Scheme\nX Y : Scheme\nx : ↥X\n⊢ ∃ y,\n (Sum.rec (fun x ↦ coprod.inl) (fun x ↦ coprod.inr)\n (Sum.elim (fun x ↦ Sum.inl PUnit.unit) (fun x ↦ Sum.inr PUnit.unit) (Sum.inl x)))\n y =\n (coprodMk X Y) (Sum.inl x)",
"ppT... | [] | simp only [Sum.elim_inl, coprodMk_inl, exists_apply_eq_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Limits | {
"line": 410,
"column": 6
} | {
"line": 410,
"column": 67
} | {
"line": 411,
"column": 4
} | [
{
"pp": "case h.inl\nι : Type u\nf : ι → Scheme\nσ : Type v\ng : σ → Scheme\nX Y : Scheme\nx : ↥X\n⊢ ∃ y,\n (Sum.rec (fun x ↦ coprod.inl) (fun x ↦ coprod.inr)\n (Sum.elim (fun x ↦ Sum.inl PUnit.unit) (fun x ↦ Sum.inr PUnit.unit) (Sum.inl x)))\n y =\n (coprodMk X Y) (Sum.inl x)",
"ppT... | [] | simp only [Sum.elim_inl, coprodMk_inl, exists_apply_eq_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 669,
"column": 6
} | {
"line": 669,
"column": 71
} | {
"line": 669,
"column": 71
} | [
{
"pp": "J : Type w\ninst✝⁴ : Category.{v, w} J\nF : J ⥤ Scheme\ninst✝³ : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)\ninst✝² : (F ⋙ forget).IsLocallyDirected\ninst✝¹ : Quiver.IsThin J\ninst✝ : Small.{u, w} J\ni j k : Shrink.{u, w} J\nl : J\nfi : failed to pretty print expression (use 'set_option pp.rawO... | [] | exact TopologicalSpace.Opens.mem_iSup.mpr ⟨⟨l, fj, fk⟩, ⟨z, rfl⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 752,
"column": 4
} | {
"line": 753,
"column": 7
} | {
"line": 755,
"column": 0
} | [
{
"pp": "J : Type w\ninst✝⁴ : Category.{v, w} J\nF : J ⥤ Scheme\ninst✝³ : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)\ninst✝² : (F ⋙ forget).IsLocallyDirected\ninst✝¹ : Quiver.IsThin J\ninst✝ : Small.{u, w} J\ns : Cocone F\nm : (cocone F).pt ⟶ s.pt\nhm : ∀ (j : J), (cocone F).ι.app j ≫ m = s.ι.app j\ni :... | [] | simp [← hm ↓i, cocone, reassoc_of% glueDataι_naturality]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 752,
"column": 4
} | {
"line": 753,
"column": 7
} | {
"line": 755,
"column": 0
} | [
{
"pp": "J : Type w\ninst✝⁴ : Category.{v, w} J\nF : J ⥤ Scheme\ninst✝³ : ∀ {i j : J} (f : i ⟶ j), IsOpenImmersion (F.map f)\ninst✝² : (F ⋙ forget).IsLocallyDirected\ninst✝¹ : Quiver.IsThin J\ninst✝ : Small.{u, w} J\ns : Cocone F\nm : (cocone F).pt ⟶ s.pt\nhm : ∀ (j : J), (cocone F).ι.app j ≫ m = s.ι.app j\ni :... | [] | simp [← hm ↓i, cocone, reassoc_of% glueDataι_naturality]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 620,
"column": 10
} | {
"line": 620,
"column": 69
} | {
"line": 620,
"column": 69
} | [
{
"pp": "X Y Z W : Scheme\nfWX : W ⟶ X\nfWY : W ⟶ Y\nfXZ : X ⟶ Z\nfYZ : Y ⟶ Z\n𝒰 : X.OpenCover\nH :\n ∀ (i : 𝒰.toPreZeroHypercover.1),\n IsPullback (Cover.pullbackHom 𝒰 fWX i) ((Precoverage.ZeroHypercover.pullback₁ fWX 𝒰).f i ≫ fWY) (𝒰.f i ≫ fXZ) fYZ\nh : fWX ≫ fXZ = fWY ≫ fYZ\ni : (openCoverOfLeft 𝒰 ... | [
"X Y Z W : Scheme\nfWX : W ⟶ X\nfWY : W ⟶ Y\nfXZ : X ⟶ Z\nfYZ : Y ⟶ Z\n𝒰 : X.OpenCover\nH :\n ∀ (i : 𝒰.toPreZeroHypercover.1),\n IsPullback (Cover.pullbackHom 𝒰 fWX i) ((Precoverage.ZeroHypercover.pullback₁ fWX 𝒰).f i ≫ fWY) (𝒰.f i ≫ fXZ) fYZ\nh : fWX ≫ fXZ = fWY ≫ fYZ\ni : (openCoverOfLeft 𝒰 fXZ fYZ).toP... | ← IsPullback.paste_vert_iff this.flip (by ext <;> simp [f]) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 70
} | {
"line": 49,
"column": 2
} | [
{
"pp": "X Y Z : Scheme\nf✝ : X ⟶ Y\ng : Y ⟶ Z\nα✝ β✝ : Type u_1\ninst✝¹ : TopologicalSpace α✝\ninst✝ : TopologicalSpace β✝\nf : α✝ → β✝\nι : Type u_1\nU : ι → Opens β✝\nH : IsOpenCover U\nx✝ : Continuous[inst✝¹, inst✝] f\nhf : ∀ (i : ι), Function.Injective ((U i).carrier.restrictPreimage f)\nx₁ x₂ : α✝\ne : f ... | [
"X Y Z : Scheme\nf✝ : X ⟶ Y\ng : Y ⟶ Z\nα✝ β✝ : Type u_1\ninst✝¹ : TopologicalSpace α✝\ninst✝ : TopologicalSpace β✝\nf : α✝ → β✝\nι : Type u_1\nU : ι → Opens β✝\nH : IsOpenCover U\nx✝ : Continuous[inst✝¹, inst✝] f\nhf : ∀ (i : ι), Function.Injective ((U i).carrier.restrictPreimage f)\nx₁ x₂ : α✝\ne : f x₁ = f x₂\ni... | obtain ⟨i, hxi⟩ : ∃ i, f x₁ ∈ U i := by simpa using congr(f x₁ ∈ $H) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.Morphisms.UnderlyingMap | {
"line": 254,
"column": 3
} | {
"line": 254,
"column": 75
} | {
"line": 254,
"column": 75
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝ : IsDominant f\nhf : IsClosed (Set.range ⇑f)\n⊢ Function.Surjective ⇑f",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Set.range_eq_univ",
"AlgebraicGeometry.Scheme.Hom.denseRange",
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.i... | [] | by rw [← Set.range_eq_univ, ← hf.closure_eq, f.denseRange.closure_range] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.Constructors | {
"line": 201,
"column": 2
} | {
"line": 204,
"column": 96
} | {
"line": 206,
"column": 0
} | [
{
"pp": "case refine_2\nP : MorphismProperty Scheme\ninst✝² : P.HasOfPostcompProperty IsOpenImmersion\ninst✝¹ : P.RespectsRight IsOpenImmersion\ninst✝ : IsZariskiLocalAtSource P\ng : {X Y : Scheme} → (f : X ⟶ Y) → (U : X.Opens) → pullback (U.ι ≫ f) (U.ι ≫ f) ⟶ pullback f f :=\n fun {X Y} f U ↦ pullback.map (U.... | [] | · change P _
refine IsZariskiLocalAtSource.of_iSup_eq_top U hU fun i ↦ ?_
rw [pullback.comp_diagonal]
exact RespectsRight.postcomp (P := P) (Q := @IsOpenImmersion) (g _ _) inferInstance _ (hf i) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.RingHom.Locally | {
"line": 251,
"column": 68
} | {
"line": 272,
"column": 17
} | {
"line": 274,
"column": 0
} | [
{
"pp": "P : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nhPa : StableUnderCompositionWithLocalizationAwayTarget fun {R S} [CommRing R] [CommRing S] ↦ P\n⊢ StableUnderCompositionWithLocalizationAwayTarget fun {R S} [CommRing R] [CommRing S] ↦\n Locally fun {R S} [CommRing ... | [] | by
intro R S T _ _ _ _ t _ f hf
obtain ⟨s, hsone, hs⟩ := hf
refine ⟨algebraMap S T '' s, ?_, ?_⟩
· rw [← Ideal.map_span, hsone, Ideal.map_top]
· rintro - ⟨a, ha, rfl⟩
letI : Algebra (Localization.Away a) (Localization.Away (algebraMap S T a)) :=
(IsLocalization.Away.map _ _ (algebraMap S T) a).toAlg... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.SurjectiveOnStalks | {
"line": 86,
"column": 2
} | {
"line": 89,
"column": 40
} | {
"line": 91,
"column": 0
} | [
{
"pp": "X Y Z : Scheme\nf : X ⟶ Y\ng : Y ⟶ Z\ninst✝ : SurjectiveOnStalks (f ≫ g)\n⊢ SurjectiveOnStalks f",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme",
"CommRingCat.carrier",
"AlgebraicGeometry.PresheafedSpace.carrier",
"CategoryTheory.... | [] | refine ⟨fun x ↦ ?_⟩
have := (f ≫ g).stalkMap_surjective x
rw [Scheme.Hom.stalkMap_comp] at this
exact Function.Surjective.of_comp this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.SurjectiveOnStalks | {
"line": 86,
"column": 2
} | {
"line": 89,
"column": 40
} | {
"line": 91,
"column": 0
} | [
{
"pp": "X Y Z : Scheme\nf : X ⟶ Y\ng : Y ⟶ Z\ninst✝ : SurjectiveOnStalks (f ≫ g)\n⊢ SurjectiveOnStalks f",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme",
"CommRingCat.carrier",
"AlgebraicGeometry.PresheafedSpace.carrier",
"CategoryTheory.... | [] | refine ⟨fun x ↦ ?_⟩
have := (f ≫ g).stalkMap_surjective x
rw [Scheme.Hom.stalkMap_comp] at this
exact Function.Surjective.of_comp this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.SurjectiveOnStalks | {
"line": 133,
"column": 4
} | {
"line": 136,
"column": 9
} | {
"line": 137,
"column": 4
} | [
{
"pp": "case e'_5.fst\nX Y S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\ninst✝ : SurjectiveOnStalks g\nL : ↥(pullback f g) → ↥X × ↥Y := ⋯\nR A B : CommRingCat\niX : Spec A ⟶ X\niY : Spec B ⟶ Y\niS : Spec R ⟶ S\na✝² : IsOpenImmersion iX\na✝¹ : IsOpenImmersion iY\na✝ : IsOpenImmersion iS\nφ : R ⟶ A\ne₁ : Spec.map φ ≫ iS = i... | [
"case e'_5.snd\nX Y S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\ninst✝ : SurjectiveOnStalks g\nL : ↥(pullback f g) → ↥X × ↥Y := ⋯\nR A B : CommRingCat\niX : Spec A ⟶ X\niY : Spec B ⟶ Y\niS : Spec R ⟶ S\na✝² : IsOpenImmersion iX\na✝¹ : IsOpenImmersion iY\na✝ : IsOpenImmersion iS\nφ : R ⟶ A\ne₁ : Spec.map φ ≫ iS = iX ≫ f\nψ : R... | · simp only [L, ← Scheme.Hom.comp_apply, pullback.lift_fst, Iso.symm_hom,
Iso.inv_hom_id]
erw [← Scheme.Hom.comp_apply, pullbackSpecIso_inv_fst_assoc]
rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 306,
"column": 6
} | {
"line": 307,
"column": 60
} | {
"line": 308,
"column": 4
} | [
{
"pp": "P : MorphismProperty Scheme\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\ninst✝¹ : HasRingHomProperty P Q\nX Y Z : Scheme\nf : X ⟶ Y\ng : Y ⟶ Z\ninst✝ : IsOpenImmersion f\nH :\n ∀ (U : ↑Z.affineOpens) (V : ↑Y.affineOpens) (e : ↑V ≤ g ⁻¹ᵁ ↑U),\n Q (CommRingCat... | [
"P : MorphismProperty Scheme\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\ninst✝¹ : HasRingHomProperty P Q\nX Y Z : Scheme\nf : X ⟶ Y\ng : Y ⟶ Z\ninst✝ : IsOpenImmersion f\nH :\n ∀ (U : ↑Z.affineOpens) (V : ↑Y.affineOpens) (e : ↑V ≤ g ⁻¹ᵁ ↑U),\n Q (CommRingCat.Hom.hom (Sc... | ← Scheme.Hom.appLE_comp_appLE _ _ _ (f ''ᵁ V) V.1
(Set.image_subset_iff.mpr e) (f.preimage_image_eq _).ge, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Height | {
"line": 92,
"column": 25
} | {
"line": 94,
"column": 46
} | {
"line": 96,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nI : Ideal R\ninst✝¹ : I.FiniteHeight\ninst✝ : I.IsPrime\n⊢ I.primeHeight < ⊤",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ideal.height_lt_top",
"instTopENat",
"congrArg",
"CommSemiring.toSemiring",
... | [] | by
rw [← I.height_eq_primeHeight]
exact Ideal.height_lt_top ‹I.IsPrime›.ne_top | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.Height | {
"line": 224,
"column": 2
} | {
"line": 225,
"column": 30
} | {
"line": 226,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : I.IsPrime\n⊢ I.height = 0 ↔ I ∈ minimalPrimes R",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PrimeSpectrum.mk",
"minimalPrimes_eq_minimals",
"Semiring.toModule",
"congrArg",
"... | [
"R : Type u_1\ninst✝¹ : CommRing R\nI : Ideal R\ninst✝ : I.IsPrime\n⊢ IsMin { asIdeal := I, isPrime := inst✝ } ↔ I ∈ {x | Minimal IsPrime x}"
] | rw [Ideal.height_eq_primeHeight, Ideal.primeHeight, Order.height_eq_zero,
minimalPrimes_eq_minimals] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 292,
"column": 4
} | {
"line": 292,
"column": 96
} | {
"line": 293,
"column": 4
} | [
{
"pp": "X : Scheme\nS : ↑X.affineOpens\nU₁ U₂ : X.Opens\nn₁ n₂ : ℕ\ny₁ : ↑Γ(X, U₁)\ny₂ : ↑Γ(X, U₂)\nf : ↑Γ(X, U₁ ⊔ U₂)\nx : ↑Γ(X, X.basicOpen f)\nh₁ : ↑S ≤ U₁\nh₂ : ↑S ≤ U₂\ne₁ :\n (y₁ |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯ =\n ((f |_ U₁) ⋯ |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯ ^ n₁ * (x |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯... | [
"case h₇₅\nX : Scheme\nS : ↑X.affineOpens\nU₁ U₂ : X.Opens\nn₁ n₂ : ℕ\ny₁ : ↑Γ(X, U₁)\ny₂ : ↑Γ(X, U₂)\nf : ↑Γ(X, U₁ ⊔ U₂)\nx : ↑Γ(X, X.basicOpen f)\nh₁ : ↑S ≤ U₁\nh₂ : ↑S ≤ U₂\ne₁ :\n (y₁ |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯ =\n ((f |_ U₁) ⋯ |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯ ^ n₁ * (x |_ X.basicOpen ((f |_ U₁) ⋯)) ⋯\n... | apply exists_eq_pow_mul_of_is_compact_of_quasi_separated_space_aux_aux (e₁ := e₁) (e₂ := e₂) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Ideal.Height | {
"line": 344,
"column": 37
} | {
"line": 352,
"column": 71
} | {
"line": 354,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nS : Type u_2\ninst✝ : CommRing S\ne : R ≃+* S\nI : Ideal S\n⊢ (comap e I).height = I.height",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"RingHom.instRingHomClass",
"RingEquiv.surjective",
"RingEquiv.idealComa... | [] | by
refine (Equiv.iInf_congr e.idealComapOrderIso fun J ↦ (Equiv.iInf_congr ?_ fun h ↦ ?_).symm).symm
· refine .ofIff ?_
rw [← Ideal.comap_coe,
Ideal.comap_minimalPrimes_eq_of_surjective (f := (↑e : R →+* S)) e.surjective]
exact e.idealComapOrderIso.injective.mem_set_image.symm
· have : J.IsPrime := ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Properties | {
"line": 227,
"column": 4
} | {
"line": 228,
"column": 55
} | {
"line": 229,
"column": 2
} | [
{
"pp": "case inr\nX : Scheme\ninst✝ : IsReduced X\nthis :\n ∀ (X : Scheme) (x : ↥X),\n closure {x} ∈ irreducibleComponents ↥X → ∀ [IsReduced X], (∃ R, X = Spec R) → IsField ↑(X.presheaf.stalk x)\nhX : ¬∃ R, X = Spec R\ni : X.affineCover.I₀\nx : ↥(X.affineCover.X i)\nhx : closure {(X.affineCover.f i) x} ∈ i... | [] | exact preimage_mem_irreducibleComponents hx (X.affineCover.f i).isOpenEmbedding
⟨X.affineCover.f i x, subset_closure rfl, _, rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Properties | {
"line": 268,
"column": 2
} | {
"line": 268,
"column": 42
} | {
"line": 269,
"column": 2
} | [
{
"pp": "X : Scheme\ninst✝ : IsIntegral X\nH : ∃ x x_1, ∃ (_ : IsClosed x) (_ : IsClosed x_1) (_ : Set.univ ⊆ x ∪ x_1), ¬Set.univ ⊆ x ∧ ¬Set.univ ⊆ x_1\n⊢ False",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"False",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierC... | [
"X : Scheme\ninst✝ : IsIntegral X\nS T : Set ↥X\nhS : IsClosed S\nhT : IsClosed T\nh₁ : Set.univ ⊆ S ∪ T\nh₂ : ¬Set.univ ⊆ S\nh₃ : ¬Set.univ ⊆ T\n⊢ False"
] | rcases H with ⟨S, T, hS, hT, h₁, h₂, h₃⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 641,
"column": 41
} | {
"line": 641,
"column": 69
} | {
"line": 642,
"column": 4
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\nU : ↑X.affineOpens\n⊢ ↑I.support ∩ Set.range ⇑(IsAffineOpen.fromSpec ⋯) = ⇑(IsAffineOpen.fromSpec ⋯) '' PrimeSpectrum.zeroLocus ↑(I.ideal U)",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Scheme.IdealSheaf... | [
"X : Scheme\nI : X.IdealSheafData\nU : ↑X.affineOpens\n⊢ ↑I.support ∩ ↑↑U = ⇑(IsAffineOpen.fromSpec ⋯) '' PrimeSpectrum.zeroLocus ↑(I.ideal U)"
] | IsAffineOpen.range_fromSpec, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Stalk | {
"line": 349,
"column": 50
} | {
"line": 349,
"column": 66
} | {
"line": 349,
"column": 66
} | [
{
"pp": "X : Scheme\nR : CommRingCat\ninst✝ : IsLocalRing ↑R\nf : Spec R ⟶ X\nU : TopologicalSpace.Opens ↥X\nhU : U ∈ X.affineOpens\nhxU : f (closedPoint ↑R) ∈ ↑U\nthis : (Spec R).isoSpec.hom ≫ Spec.map (Hom.appLE f U ⊤ ⋯) ≫ IsAffineOpen.fromSpec hU = f\n⊢ Spec.map (stalkClosedPointTo f) ≫ X.fromSpecStalk (f (c... | [
"X : Scheme\nR : CommRingCat\ninst✝ : IsLocalRing ↑R\nf : Spec R ⟶ X\nU : TopologicalSpace.Opens ↥X\nhU : U ∈ X.affineOpens\nhxU : f (closedPoint ↑R) ∈ ↑U\nthis : Spec.map (ΓSpecIso R).hom ≫ Spec.map (Hom.appLE f U ⊤ ⋯) ≫ IsAffineOpen.fromSpec hU = f\n⊢ Spec.map (stalkClosedPointTo f) ≫ X.fromSpecStalk (f (closedPo... | isoSpec_Spec_hom | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 803,
"column": 2
} | {
"line": 804,
"column": 32
} | {
"line": 805,
"column": 2
} | [
{
"pp": "case inl\nX Y : Scheme\nf : X.Hom Y\ninst✝¹ : QuasiCompact f\n𝒰 : X.OpenCover\ninst✝ : Finite 𝒰.I₀\nh✝ : IsEmpty 𝒰.I₀\n⊢ ⋃ i, ↑(ker (𝒰.f i ≫ f)).support = ↑f.ker.support",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.IdealSheafData.support",
... | [
"case inr\nX Y : Scheme\nf : X.Hom Y\ninst✝¹ : QuasiCompact f\n𝒰 : X.OpenCover\ninst✝ : Finite 𝒰.I₀\nh✝ : Nonempty 𝒰.I₀\n⊢ ⋃ i, ↑(ker (𝒰.f i ≫ f)).support = ↑f.ker.support"
] | · have : IsEmpty X := Function.isEmpty 𝒰.idx
simp [ker_eq_top_of_isEmpty] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 255,
"column": 58
} | {
"line": 255,
"column": 74
} | {
"line": 255,
"column": 74
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\nU V W U₀ : ↑X.affineOpens\nhU₀ : ↑U ⊓ ↑W ≤ ↑U₀\n⊢ Set.range ⇑(↑U ⊓ ↑W).ι ⊆ ↑↑U₀",
"ppTerm": "?m.240",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"ChainCompletePartial... | [] | simpa using! hU₀ | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 255,
"column": 58
} | {
"line": 255,
"column": 74
} | {
"line": 255,
"column": 74
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\nU V W U₀ : ↑X.affineOpens\nhU₀ : ↑U ⊓ ↑W ≤ ↑U₀\n⊢ Set.range ⇑(↑U ⊓ ↑W).ι ⊆ ↑↑U₀",
"ppTerm": "?m.240",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"ChainCompletePartial... | [] | simpa using! hU₀ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 255,
"column": 58
} | {
"line": 255,
"column": 74
} | {
"line": 255,
"column": 74
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\nU V W U₀ : ↑X.affineOpens\nhU₀ : ↑U ⊓ ↑W ≤ ↑U₀\n⊢ Set.range ⇑(↑U ⊓ ↑W).ι ⊆ ↑↑U₀",
"ppTerm": "?m.240",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"ChainCompletePartial... | [] | simpa using! hU₀ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 881,
"column": 4
} | {
"line": 881,
"column": 48
} | {
"line": 882,
"column": 4
} | [
{
"pp": "case a\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : QuasiCompact f\n⊢ closure (Set.range ⇑f) ⊆ ↑(ker f).support",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Scheme.IdealSheafData.support",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCar... | [
"case a\nX Y : Scheme\nf : X ⟶ Y\ninst✝ : QuasiCompact f\n⊢ Set.range ⇑f ⊆ ↑(ker f).support"
] | rw [(support _).isClosed.closure_subset_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.IdealSheaf.Subscheme | {
"line": 434,
"column": 2
} | {
"line": 435,
"column": 32
} | {
"line": 436,
"column": 2
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\nU : ↑X.affineOpens\n⊢ IsPreimmersion (I.gluedTo ∣_ ↑U)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty.IsLocalAtTarget.toRespects",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCa... | [
"X : Scheme\nI : X.IdealSheafData\nU : ↑X.affineOpens\n⊢ IsPreimmersion (I.glueDataObjι U)"
] | rw [← MorphismProperty.cancel_left_of_respectsIso @IsPreimmersion (I.glueDataObjIso U).hom,
glueDataObjIso_hom_restrict] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 323,
"column": 2
} | {
"line": 323,
"column": 91
} | {
"line": 325,
"column": 0
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\nhU : IsAffineOpen U\nV₁ : X.Opens\nhV₁ : V₁ ≤ f ⁻¹ᵁ U\nV₂ : X.Opens\nhV₂ : V₂ ≤ f ⁻¹ᵁ U\n⊢ IsAffineOpen (f ⁻¹ᵁ U ⊓ V₁) → IsAffineOpen (f ⁻¹ᵁ U ⊓ V₂) → IsAffineOpen (f ⁻¹ᵁ U ⊓ (V₁ ⊓ V₂)) ↔\n IsAffineOpen V₁ → IsAffineOpen V₂ → IsAffineOpen (V₁ ⊓ V₂)",
"ppTerm... | [] | rw [inf_eq_right.mpr hV₁, inf_eq_right.mpr hV₂, inf_eq_right.mpr (inf_le_left.trans hV₁)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion | {
"line": 159,
"column": 6
} | {
"line": 160,
"column": 43
} | {
"line": 160,
"column": 43
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝ : IsClosedImmersion f\nthis : IsClosedImmersion (Scheme.Hom.toImage f)\n⊢ Function.Surjective ⇑(Scheme.Hom.toImage f)",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Set.range_eq_univ",
"AlgebraicGeometry.Scheme.Hom.denseRange",
"E... | [] | rw [← Set.range_eq_univ, ← f.toImage.isClosedEmbedding.isClosed_range.closure_eq]
exact f.toImage.denseRange.closure_eq | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion | {
"line": 159,
"column": 6
} | {
"line": 160,
"column": 43
} | {
"line": 160,
"column": 43
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝ : IsClosedImmersion f\nthis : IsClosedImmersion (Scheme.Hom.toImage f)\n⊢ Function.Surjective ⇑(Scheme.Hom.toImage f)",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Set.range_eq_univ",
"AlgebraicGeometry.Scheme.Hom.denseRange",
"E... | [] | rw [← Set.range_eq_univ, ← f.toImage.isClosedEmbedding.isClosed_range.closure_eq]
exact f.toImage.denseRange.closure_eq | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.IdealSheaf.Functorial | {
"line": 173,
"column": 24
} | {
"line": 173,
"column": 50
} | {
"line": 173,
"column": 50
} | [
{
"pp": "case a\nX Y : Scheme\nf : X ⟶ Y\nZ : TopologicalSpace.Closeds ↥X\n⊢ ⇑f '' ↑Z ⊆ ⇑f '' ↑(vanishingIdeal Z).support",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Scheme.IdealSheafData.support",
"AlgebraicGeometry.SheafedSpace.instTopolog... | [
"case a\nX Y : Scheme\nf : X ⟶ Y\nZ : TopologicalSpace.Closeds ↥X\n⊢ ⇑f '' ↑Z ⊆ ⇑f '' ↑Z"
] | coe_support_vanishingIdeal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Charpoly.BaseChange | {
"line": 36,
"column": 72
} | {
"line": 39,
"column": 60
} | {
"line": 41,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Module.Free R M\ninst✝ : Module.Finite R M\nf : M →ₗ[R] M\n⊢ LinearMap.det f = (-1) ^ Module.finrank R M * f.charpoly.coeff 0",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
... | [] | by
nontriviality R
rw [← LinearMap.det_toMatrix (Module.Free.chooseBasis R M), Matrix.det_eq_sign_charpoly_coeff,
← Module.finrank_eq_card_chooseBasisIndex, charpoly_def] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Finiteness.FinitePresentationLocal | {
"line": 100,
"column": 4
} | {
"line": 104,
"column": 10
} | {
"line": 105,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\ns✝ : Set S\nhs✝ : Ideal.span s✝ = ⊤\ns : Finset S\nh₁ : ↑s ⊆ s✝\nhs : Ideal.span ↑s = ⊤\nh : ∀ (i : ↥s), FinitePresentation R (Localization.Away ↑i)\nhfintype : FiniteType R S\nn : ℕ\nf : MvPolynomial (Fin n) R →... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\ns✝ : Set S\nhs✝ : Ideal.span s✝ = ⊤\ns : Finset S\nh₁ : ↑s ⊆ s✝\nhs : Ideal.span ↑s = ⊤\nh : ∀ (i : ↥s), FinitePresentation R (Localization.Away ↑i)\nhfintype : FiniteType R S\nn : ℕ\nf : MvPolynomial (Fin n) R →ₐ[R] S\nhf :... | have : ∑ g : { x // x ∈ s }, g' g * h' g = (1 : A) := by
apply eq_of_sub_eq_zero
rw [← map_one (Ideal.Quotient.mk I), ← map_sub, Ideal.Quotient.eq_zero_iff_mem]
apply Ideal.subset_span
simp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.Morphisms.ClosedImmersion | {
"line": 448,
"column": 2
} | {
"line": 448,
"column": 48
} | {
"line": 449,
"column": 2
} | [
{
"pp": "R S : CommRingCat\ninst✝ : Subsingleton ↥(Spec S)\nφ : S ⟶ R\nψ : R ⟶ S\nhg : φ ≫ ψ = 𝟙 S\n⊢ Function.Surjective ⇑(ConcreteCategory.hom ψ)",
"ppTerm": "?m.409",
"assigned": true,
"usedConstants": [
"CommRingCat.carrier",
"CommSemiring.toSemiring",
"CategoryTheory.Concrete... | [
"R S : CommRingCat\ninst✝ : Subsingleton ↥(Spec S)\nφ : S ⟶ R\nψ : R ⟶ S\nhg : φ ≫ ψ = 𝟙 S\n⊢ Function.LeftInverse ⇑(ConcreteCategory.hom ψ) ⇑(ConcreteCategory.hom φ)"
] | apply Function.LeftInverse.surjective (g := φ) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.AlgebraicGeometry.Noetherian | {
"line": 225,
"column": 36
} | {
"line": 225,
"column": 64
} | {
"line": 225,
"column": 65
} | [
{
"pp": "X : Scheme\ninst✝ : IsLocallyNoetherian X\nU V : ↑X.affineOpens\nhInd : Topology.IsInducing ⇑(IsAffineOpen.fromSpec ⋯)\n⊢ IsCompact (⇑(IsAffineOpen.fromSpec ⋯) ⁻¹' (↑↑U ∩ ↑↑V ∩ Set.range ⇑(IsAffineOpen.fromSpec ⋯)))",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"X : Scheme\ninst✝ : IsLocallyNoetherian X\nU V : ↑X.affineOpens\nhInd : Topology.IsInducing ⇑(IsAffineOpen.fromSpec ⋯)\n⊢ IsCompact (⇑(IsAffineOpen.fromSpec ⋯) ⁻¹' (↑↑U ∩ ↑↑V ∩ ↑↑U))"
] | IsAffineOpen.range_fromSpec, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Morphisms.Integral | {
"line": 112,
"column": 2
} | {
"line": 114,
"column": 41
} | {
"line": 115,
"column": 2
} | [
{
"pp": "case inr\nZ S X Y : Scheme\nf : X ⟶ Y\nthis : ∀ ⦃X Y : Scheme⦄ (f : X ⟶ Y), (∃ R, Y = Spec R) → IsIntegralHom f → topologically (@IsClosedMap) f\nhY : ¬∃ R, Y = Spec R\n⊢ IsIntegralHom f → topologically (@IsClosedMap) f",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Algebra... | [
"Z S X Y : Scheme\nf : X ⟶ Y\nhY : ∃ R, Y = Spec R\n⊢ IsIntegralHom f → topologically (@IsClosedMap) f"
] | · rw [IsZariskiLocalAtTarget.iff_of_openCover (P := @IsIntegralHom) Y.affineCover,
IsZariskiLocalAtTarget.iff_of_openCover (P := topologically _) Y.affineCover]
exact fun a i ↦ this _ ⟨_, rfl⟩ (a i) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.Morphisms.Finite | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 87
} | {
"line": 127,
"column": 2
} | [
{
"pp": "X✝ Y✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y : Scheme\nf : X ⟶ Y\nhY : IsAffine Y\nx✝ : IsAffine X ∧ (CommRingCat.Hom.hom (Scheme.Hom.appTop f)).Finite\nleft✝ : IsAffine X\nright✝ : (CommRingCat.Hom.hom (Scheme.Hom.appTop f)).Finite\n⊢ Epi (Scheme.Hom.appTop f) ↔ Mono (Scheme.Hom.app f ⊤).op",
"ppTerm": "?m.1... | [] | exact ⟨fun h ↦ inferInstance, fun h ↦ show Epi (f.app ⊤).op.unop by infer_instance⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Morphisms.Finite | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 87
} | {
"line": 127,
"column": 2
} | [
{
"pp": "X✝ Y✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y : Scheme\nf : X ⟶ Y\nhY : IsAffine Y\nx✝ : IsAffine X ∧ (CommRingCat.Hom.hom (Scheme.Hom.appTop f)).Finite\nleft✝ : IsAffine X\nright✝ : (CommRingCat.Hom.hom (Scheme.Hom.appTop f)).Finite\n⊢ Epi (Scheme.Hom.appTop f) ↔ Mono (Scheme.Hom.app f ⊤).op",
"ppTerm": "?m.1... | [] | exact ⟨fun h ↦ inferInstance, fun h ↦ show Epi (f.app ⊤).op.unop by infer_instance⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.Finite | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 87
} | {
"line": 127,
"column": 2
} | [
{
"pp": "X✝ Y✝ : Scheme\nf✝ : X✝ ⟶ Y✝\nX Y : Scheme\nf : X ⟶ Y\nhY : IsAffine Y\nx✝ : IsAffine X ∧ (CommRingCat.Hom.hom (Scheme.Hom.appTop f)).Finite\nleft✝ : IsAffine X\nright✝ : (CommRingCat.Hom.hom (Scheme.Hom.appTop f)).Finite\n⊢ Epi (Scheme.Hom.appTop f) ↔ Mono (Scheme.Hom.app f ⊤).op",
"ppTerm": "?m.1... | [] | exact ⟨fun h ↦ inferInstance, fun h ↦ show Epi (f.app ⊤).op.unop by infer_instance⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.Finite | {
"line": 184,
"column": 2
} | {
"line": 184,
"column": 99
} | {
"line": 185,
"column": 2
} | [
{
"pp": "case inr\nS : CommRingCat\ninst✝² : JacobsonSpace ↥(Spec S)\nR : CommRingCat\ninst✝¹ : Subsingleton ↥(Spec R)\ninst✝ : IsReduced (Spec R)\nf : Spec R ⟶ Spec S\nh✝ : Nonempty ↥(Spec R)\nthis : IrreducibleSpace ↥(Spec R)\n⊢ IsFinite f ↔ LocallyOfFiniteType f",
"ppTerm": "?inr",
"assigned": true,
... | [
"case inr\nS : CommRingCat\ninst✝² : JacobsonSpace ↥(Spec S)\nR : CommRingCat\ninst✝¹ : Subsingleton ↥(Spec R)\ninst✝ : IsReduced (Spec R)\nf : Spec R ⟶ Spec S\nh✝ : Nonempty ↥(Spec R)\nthis✝ : IrreducibleSpace ↥(Spec R)\nthis : IsDomain ↑R\n⊢ IsFinite f ↔ LocallyOfFiniteType f"
] | have : IsDomain R := (affine_isIntegral_iff R).mp (isIntegral_of_irreducibleSpace_of_isReduced _) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.QuasiAffine | {
"line": 47,
"column": 2
} | {
"line": 49,
"column": 16
} | {
"line": 51,
"column": 0
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝ : X.IsQuasiAffine\n⊢ IsOpenImmersion X.toSpecΓ",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.IsQuasiAffine.toIsImmersion",
"Eq.mpr",
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.SheafedSpace.instTopo... | [] | have : IsIso X.toSpecΓ.imageι := by delta Hom.imageι Hom.image; rw [X.ker_toSpecΓ]; infer_instance
rw [← X.toSpecΓ.toImage_imageι]
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.QuasiAffine | {
"line": 47,
"column": 2
} | {
"line": 49,
"column": 16
} | {
"line": 51,
"column": 0
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝ : X.IsQuasiAffine\n⊢ IsOpenImmersion X.toSpecΓ",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.IsQuasiAffine.toIsImmersion",
"Eq.mpr",
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.SheafedSpace.instTopo... | [] | have : IsIso X.toSpecΓ.imageι := by delta Hom.imageι Hom.image; rw [X.ker_toSpecΓ]; infer_instance
rw [← X.toSpecΓ.toImage_imageι]
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 263,
"column": 63
} | {
"line": 263,
"column": 77
} | {
"line": 263,
"column": 77
} | [
{
"pp": "n : ℕ\nP : (R : Type u) → [inst : CommRing R] → InductionObj R n → Prop\nhP₁ : ∀ (R : Type u) [inst : CommRing R], P R { val := 0 }\nhP₂ :\n ∀ (R : Type u) [inst : CommRing R] (e : InductionObj R n) (i : Fin n),\n (e.val i).Monic → (∀ (j : Fin n), j ≠ i → e.val j = 0) → P R e\nhP₃ :\n ∀ (R : Type ... | [] | simp [H] at hj | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 263,
"column": 63
} | {
"line": 263,
"column": 77
} | {
"line": 263,
"column": 77
} | [
{
"pp": "n : ℕ\nP : (R : Type u) → [inst : CommRing R] → InductionObj R n → Prop\nhP₁ : ∀ (R : Type u) [inst : CommRing R], P R { val := 0 }\nhP₂ :\n ∀ (R : Type u) [inst : CommRing R] (e : InductionObj R n) (i : Fin n),\n (e.val i).Monic → (∀ (j : Fin n), j ≠ i → e.val j = 0) → P R e\nhP₃ :\n ∀ (R : Type ... | [] | simp [H] at hj | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Spectrum.Prime.ChevalleyComplexity | {
"line": 263,
"column": 63
} | {
"line": 263,
"column": 77
} | {
"line": 263,
"column": 77
} | [
{
"pp": "n : ℕ\nP : (R : Type u) → [inst : CommRing R] → InductionObj R n → Prop\nhP₁ : ∀ (R : Type u) [inst : CommRing R], P R { val := 0 }\nhP₂ :\n ∀ (R : Type u) [inst : CommRing R] (e : InductionObj R n) (i : Fin n),\n (e.val i).Monic → (∀ (j : Fin n), j ≠ i → e.val j = 0) → P R e\nhP₃ :\n ∀ (R : Type ... | [] | simp [H] at hj | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.AffineSpace | {
"line": 433,
"column": 4
} | {
"line": 433,
"column": 87
} | {
"line": 435,
"column": 0
} | [
{
"pp": "case a\nn : Type u\nS : Scheme\ninst✝ : IsEmpty n\n⊢ isomorphisms Scheme (terminal.from (Spec (CommRingCat.of (ULift.{u, 0} ℤ))))",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.Scheme",
"AlgebraicGeometry.specULiftZIsTe... | [] | exact isIso_of_isTerminal specULiftZIsTerminal terminalIsTerminal (terminal.from _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.AffineSpace | {
"line": 433,
"column": 4
} | {
"line": 433,
"column": 87
} | {
"line": 435,
"column": 0
} | [
{
"pp": "case a\nn : Type u\nS : Scheme\ninst✝ : IsEmpty n\n⊢ isomorphisms Scheme (terminal.from (Spec (CommRingCat.of (ULift.{u, 0} ℤ))))",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.Scheme",
"AlgebraicGeometry.specULiftZIsTe... | [] | exact isIso_of_isTerminal specULiftZIsTerminal terminalIsTerminal (terminal.from _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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