module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Order.Quotient | {
"line": 59,
"column": 4
} | {
"line": 59,
"column": 37
} | [
{
"pp": "case h.h.inl\nα : Type u_1\ns : Setoid α\ninst✝¹ : LE α\ninst✝ : Std.Total fun x1 x2 ↦ x1 ≤ x2\nx y : α\nh : x ≤ y\n⊢ ⟦x⟧ ≤ ⟦y⟧ ∨ ⟦y⟧ ≤ ⟦x⟧",
"usedConstants": [
"LE.le",
"Quotient.mk",
"Quotient",
"HasEquiv.Equiv",
"Relation.TransGen.single",
"instHasEquivOfSetoi... | · exact .inl <| .single <| .inl h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.HahnSeries.Multiplication | {
"line": 872,
"column": 6
} | {
"line": 873,
"column": 72
} | [
{
"pp": "case h\nΓ : Type u_1\nR : Type u_3\ninst✝⁶ : AddCommMonoid Γ\ninst✝⁵ : PartialOrder Γ\ninst✝⁴ : IsOrderedCancelAddMonoid Γ\nΓ' : Type u_6\ninst✝³ : AddCommMonoid Γ'\ninst✝² : PartialOrder Γ'\ninst✝¹ : IsOrderedCancelAddMonoid Γ'\ninst✝ : NonUnitalNonAssocSemiring R\nf : Γ ↪o Γ'\nhf : ∀ (x y : Γ), f (x ... | simp only [mem_map, mem_addAntidiagonal,
Function.Embedding.coe_prodMap, mem_support, Prod.exists] at hij | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.HahnSeries.Multiplication | {
"line": 1025,
"column": 4
} | {
"line": 1025,
"column": 43
} | [
{
"pp": "case a.h\nΓ : Type u_1\nR : Type u_3\ninst✝⁴ : AddCommMonoid Γ\ninst✝³ : LinearOrder Γ\ninst✝² : IsOrderedCancelAddMonoid Γ\ninst✝¹ : NonUnitalNonAssocSemiring R\ninst✝ : NoZeroDivisors R\nx y : R⟦Γ⟧\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ (x * y).coeff (x.order + y.order) ≠ 0",
"usedConstants": [
"HahnSe... | simp [coeff_mul_order_add_order x y, *] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Valuation.ValuationSubring | {
"line": 204,
"column": 10
} | {
"line": 204,
"column": 12
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nA : ValuationSubring K\na : ↥A\nh : A.valuation ↑a = 1\nc : ↑a = 0\n⊢ False",
"usedConstants": [
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toIn... | c, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Valuation.ValuationSubring | {
"line": 371,
"column": 6
} | {
"line": 371,
"column": 60
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nA R S : ValuationSubring K\nhR : A ≤ R\nhS : A ≤ S\nh : R ≤ S\nx : ↥A\nhx : x ∈ A.idealOfLE S hS\nc : 1 ≤ S.valuation ↑((A.inclusion R hR) x)\n⊢ False",
"usedConstants": [
"ValuationSubring.valuation_lt_one_iff",
"LinearOrderedCommGroupWithZero.toLinearOrder... | apply not_le_of_gt ((valuation_lt_one_iff S _).1 hx) c | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Order.Ring.StandardPart | {
"line": 202,
"column": 4
} | {
"line": 202,
"column": 47
} | [
{
"pp": "case mk.mk.mk.inl\nK : Type u_1\ninst✝⁶ : LinearOrder K\ninst✝⁵ : Field K\ninst✝⁴ : IsOrderedRing K\nx✝ y✝ : K\nR : Type u_2\ninst✝³ : LinearOrder R\ninst✝² : CommRing R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : Archimedean R\nx y : FiniteElement K\nh✝ : mk x ≤ mk y\nz : FiniteElement K\nh : x ≤ y\n⊢ mk... | · exact mk.monotone' <| add_le_add_left h _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Polynomial.CoeffMem | {
"line": 40,
"column": 8
} | {
"line": 42,
"column": 99
} | [] | span R coeffs(p)
_ = 1 ^ deg(p) * span R coeffs(p) := by simp
_ ≤ spanCoeffs(q) ^ deg(p) * spanCoeffs(p) := by gcongr; exacts [le_sup_left, le_sup_right] | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Algebra.Order.Ring.StandardPart | {
"line": 342,
"column": 4
} | {
"line": 342,
"column": 41
} | [
{
"pp": "case inr\nK : Type u_1\ninst✝² : LinearOrder K\ninst✝¹ : Field K\ninst✝ : IsOrderedRing K\nx y : K\nhx : 0 < mk x\nhy : mk y < 0\n⊢ mk (x + y) ≠ 0",
"usedConstants": [
"Eq.mpr",
"IsDomain.to_noZeroDivisors",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUni... | rw [mk_add_eq_mk_right (hy.trans hx)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Order.Ring.StandardPart | {
"line": 468,
"column": 6
} | {
"line": 468,
"column": 33
} | [
{
"pp": "case h.e'_3.h.e'_3\nK : Type u_1\ninst✝² : LinearOrder K\ninst✝¹ : Field K\ninst✝ : IsOrderedRing K\nf : ℝ →+*o K\nx : K\nhx : mk x < 0\nhr : ∀ {r : ℝ}, mk x < mk (f r)\nh : x < 0\n⊢ {r | x < f r} = Set.univ",
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"congrArg",... | rw [Set.eq_univ_iff_forall] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegree | {
"line": 66,
"column": 2
} | {
"line": 67,
"column": 72
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝ : Semiring R\np : R[X]\nn : ℕ\nhn : n ≠ 0\nhp : p.IsMonicOfDegree n\n⊢ p = X ^ n + p.eraseLead",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Polynomial.IsMonicOfDegree.natDegree_eq",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingH... | · nth_rewrite 1 [← p.eraseLead_add_C_mul_X_pow]
rw [add_comm, hp.natDegree_eq, hp.leadingCoeff_eq, map_one, one_mul] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegree | {
"line": 115,
"column": 6
} | {
"line": 115,
"column": 23
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nm n : ℕ\nhp : p.IsMonicOfDegree m\nhpq : (p * q).IsMonicOfDegree (m + n)\nH : Nontrivial R\nh₂ : q.Monic\nthis : (p * q).natDegree = m + n\nh : p.leadingCoeff * q.leadingCoeff ≠ 0\n⊢ q.natDegree = n",
"usedConstants": [
"HMul.hMul",
... | natDegree_mul' h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.IsMonicOfDegree | {
"line": 130,
"column": 6
} | {
"line": 130,
"column": 23
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\nm n : ℕ\nhq : q.IsMonicOfDegree n\nhpq : (p * q).IsMonicOfDegree (m + n)\nH : Nontrivial R\nh₂ : p.Monic\nthis : (p * q).natDegree = m + n\nh : p.leadingCoeff * q.leadingCoeff ≠ 0\n⊢ p.natDegree = m",
"usedConstants": [
"HMul.hMul",
... | natDegree_mul' h, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.DenomsClearable | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 78
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\ninst✝¹ : LinearOrder K\ninst✝ : IsStrictOrderedRing K\nf : ℤ[X]\na b : ℤ\nb0 : 0 < b\nfab : eval (↑a / ↑b) (Polynomial.map (algebraMap ℤ K) f) ≠ 0\nev : ℤ\nbi : K\nbu : bi * (algebraMap ℤ K) b = 1\nhF :\n (algebraMap ℤ K) ev =\n (algebraMap ℤ K) b ^ f.natDegree * eva... | rw [eq_one_div_of_mul_eq_one_left bu, eq_intCast, eq_intCast, abs_mul] at Fa | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Smeval | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semiring R\nS : Type u_2\ninst✝² : AddCommMonoid S\ninst✝¹ : Pow S ℕ\ninst✝ : MulActionWithZero R S\nx : S\n⊢ smeval 0 x = 0",
"usedConstants": [
"congrArg",
"AddMonoid.toAddZeroClass",
"Polynomial.sum",
"AddZeroClass.toAddZero",
"Polynomial.smul... | simp only [smeval_eq_sum, sum_zero_index] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Polynomial.Smeval | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semiring R\nS : Type u_2\ninst✝² : AddCommMonoid S\ninst✝¹ : Pow S ℕ\ninst✝ : MulActionWithZero R S\nx : S\n⊢ smeval 0 x = 0",
"usedConstants": [
"congrArg",
"AddMonoid.toAddZeroClass",
"Polynomial.sum",
"AddZeroClass.toAddZero",
"Polynomial.smul... | simp only [smeval_eq_sum, sum_zero_index] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Smeval | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semiring R\nS : Type u_2\ninst✝² : AddCommMonoid S\ninst✝¹ : Pow S ℕ\ninst✝ : MulActionWithZero R S\nx : S\n⊢ smeval 0 x = 0",
"usedConstants": [
"congrArg",
"AddMonoid.toAddZeroClass",
"Polynomial.sum",
"AddZeroClass.toAddZero",
"Polynomial.smul... | simp only [smeval_eq_sum, sum_zero_index] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Smeval | {
"line": 130,
"column": 22
} | {
"line": 130,
"column": 63
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semiring R\np q : R[X]\nS : Type u_2\ninst✝² : AddCommMonoid S\ninst✝¹ : Pow S ℕ\ninst✝ : Module R S\nx : S\nc : R\nf : R[X]\n⊢ (c • f).smeval x = (RingHom.id R) c • f.smeval x",
"usedConstants": [
"instHSMul",
"Semiring.toModule",
"congrArg",
"Distrib... | simp only [smeval_smul, RingHom.id_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Polynomial.Smeval | {
"line": 130,
"column": 22
} | {
"line": 130,
"column": 63
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semiring R\np q : R[X]\nS : Type u_2\ninst✝² : AddCommMonoid S\ninst✝¹ : Pow S ℕ\ninst✝ : Module R S\nx : S\nc : R\nf : R[X]\n⊢ (c • f).smeval x = (RingHom.id R) c • f.smeval x",
"usedConstants": [
"instHSMul",
"Semiring.toModule",
"congrArg",
"Distrib... | simp only [smeval_smul, RingHom.id_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Smeval | {
"line": 130,
"column": 22
} | {
"line": 130,
"column": 63
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semiring R\np q : R[X]\nS : Type u_2\ninst✝² : AddCommMonoid S\ninst✝¹ : Pow S ℕ\ninst✝ : Module R S\nx : S\nc : R\nf : R[X]\n⊢ (c • f).smeval x = (RingHom.id R) c • f.smeval x",
"usedConstants": [
"instHSMul",
"Semiring.toModule",
"congrArg",
"Distrib... | simp only [smeval_smul, RingHom.id_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.PartialFractions | {
"line": 236,
"column": 6
} | {
"line": 236,
"column": 71
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nι : Type u_2\ninst✝ : DecidableEq ι\ng : ι → R[X]\nq₁ q₂ : R[X]\nr₁ r₂ : ι → R[X]\ni : ι\ns : Finset ι\nhi : i ∉ s\nih :\n (∀ i ∈ s, (g i).Monic) →\n ((↑s).Pairwise fun i j ↦ IsCoprime (g i) (g j)) →\n (∀ i ∈ s, (r₁ i).degree < (g i).degree) →\n (∀ i ∈... | refine (degree_sum_le _ _).trans_lt ((Finset.sup_lt_iff ?_).2 ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Polynomial.SumIteratedDerivative | {
"line": 219,
"column": 13
} | {
"line": 219,
"column": 15
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_3\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : Nontrivial A\ninst✝ : NoZeroDivisors A\np : R[X]\nq : ℕ\nhq : 0 < q\ninj_amap : Function.Injective ⇑(algebraMap R A)\np0 : p ≠ 0\nc : ℕ → R[X] := fun k ↦ if hk : q ≤ k then ⋯.choose else 0\nc_le : ∀... | c, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Polynomial.SumIteratedDerivative | {
"line": 241,
"column": 47
} | {
"line": 241,
"column": 90
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nA : Type u_3\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : Nontrivial A\ninst✝ : NoZeroDivisors A\np : R[X]\nq : ℕ\nhq : 0 < q\ninj_amap : Function.Injective ⇑(algebraMap R A)\np0 : p ≠ 0\nc : ℕ → R[X] := fun k ↦ if hk : q ≤ k then ⋯.choose else 0\nc_le : ∀... | tsub_add_cancel_of_le (Nat.one_le_of_lt hq) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.PresentedMonoid.Basic | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 24
} | [
{
"pp": "α : Type u_2\nM : Type u_3\ninst✝ : Monoid M\nrels : FreeMonoid α → FreeMonoid α → Prop\nφ ψ : PresentedMonoid rels →* M\nhx : ∀ (x : α), φ (of rels x) = ψ (of rels x)\n⊢ EqOn (⇑φ) (⇑ψ) (range (of rels))",
"usedConstants": [
"_private.Mathlib.Algebra.PresentedMonoid.Basic.0.PresentedMonoid.ex... | grind [Set.eqOn_range] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.Algebra.QuadraticAlgebra.Defs | {
"line": 245,
"column": 25
} | {
"line": 245,
"column": 61
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\na b r : R\nx y : QuadraticAlgebra R a b\ninst✝³ : SMul S R\ninst✝² : SMul T R\ns✝ : S\ninst✝¹ : SMul Sᵐᵒᵖ R\ninst✝ : IsCentralScalar S R\ns : S\nz : QuadraticAlgebra R a b\n⊢ MulOpposite.op s • z = s • z",
"usedConstants": [
"QuadraticAlgebra.re",
... | by ext <;> exact op_smul_eq_smul _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.QuadraticAlgebra.Basic | {
"line": 136,
"column": 4
} | {
"line": 136,
"column": 44
} | [
{
"pp": "K : Type u_1\nR : Type u_2\na b : R\ninst✝² : CommSemiring R\nA : Type u_3\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nf : QuadraticAlgebra R a b →ₐ[R] A\n⊢ f ω * f ω = a • 1 + b • f ω",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"QuadraticAlgebra.instSMul",
"instHSM... | simp [← map_mul, omega_mul_omega_eq_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.QuadraticAlgebra.Basic | {
"line": 136,
"column": 4
} | {
"line": 136,
"column": 44
} | [
{
"pp": "K : Type u_1\nR : Type u_2\na b : R\ninst✝² : CommSemiring R\nA : Type u_3\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nf : QuadraticAlgebra R a b →ₐ[R] A\n⊢ f ω * f ω = a • 1 + b • f ω",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"QuadraticAlgebra.instSMul",
"instHSM... | simp [← map_mul, omega_mul_omega_eq_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.QuadraticAlgebra.Basic | {
"line": 136,
"column": 4
} | {
"line": 136,
"column": 44
} | [
{
"pp": "K : Type u_1\nR : Type u_2\na b : R\ninst✝² : CommSemiring R\nA : Type u_3\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nf : QuadraticAlgebra R a b →ₐ[R] A\n⊢ f ω * f ω = a • 1 + b • f ω",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"QuadraticAlgebra.instSMul",
"instHSM... | simp [← map_mul, omega_mul_omega_eq_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.QuadraticAlgebra.Basic | {
"line": 181,
"column": 43
} | {
"line": 181,
"column": 84
} | [
{
"pp": "K : Type u_1\nR : Type u_2\na b : R\ninst✝ : CommRing R\nx✝¹ x✝ : QuadraticAlgebra R a b\n⊢ (star (x✝¹ + x✝)).re = (star x✝¹ + star x✝).re",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"QuadraticAlgebra.re",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne... | simp only [re_star, re_add, im_add]; ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.QuadraticAlgebra.Basic | {
"line": 181,
"column": 43
} | {
"line": 181,
"column": 84
} | [
{
"pp": "K : Type u_1\nR : Type u_2\na b : R\ninst✝ : CommRing R\nx✝¹ x✝ : QuadraticAlgebra R a b\n⊢ (star (x✝¹ + x✝)).re = (star x✝¹ + star x✝).re",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"QuadraticAlgebra.re",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne... | simp only [re_star, re_add, im_add]; ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.RuleOfSigns | {
"line": 168,
"column": 91
} | {
"line": 206,
"column": 6
} | [
{
"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\n⊢ ((X - C η) * P).eraseLead.signVariations = ((X - C η) * P.eraseLead).signVariations",
"usedConstants": [
"sub_neg",
"IsRig... | by
obtain ⟨d, hd⟩ := Nat.exists_eq_add_one.mpr (natDegree_pos_of_nextCoeff_ne_zero hc.ne)
have hndxP : natDegree ((X - C η) * P) = P.natDegree + 1 := by
have hPn0 : P ≠ 0 :=
leadingCoeff_ne_zero.mp hP₀.ne'
rw [natDegree_mul (X_sub_C_ne_zero η) hPn0, natDegree_X_sub_C, add_comm]
have hndxeP : natDegr... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Star.Free | {
"line": 64,
"column": 21
} | {
"line": 64,
"column": 83
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nX : Type u_2\na b : FreeAlgebra R X\n⊢ (MulOpposite.unop ∘ ⇑((lift R) (MulOpposite.op ∘ ι R))) (a + b) =\n (MulOpposite.unop ∘ ⇑((lift R) (MulOpposite.op ∘ ι R))) a +\n (MulOpposite.unop ∘ ⇑((lift R) (MulOpposite.op ∘ ι R))) b",
"usedConstants": [
... | simp only [Function.comp_apply, map_add, MulOpposite.unop_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Star.Free | {
"line": 64,
"column": 21
} | {
"line": 64,
"column": 83
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nX : Type u_2\na b : FreeAlgebra R X\n⊢ (MulOpposite.unop ∘ ⇑((lift R) (MulOpposite.op ∘ ι R))) (a + b) =\n (MulOpposite.unop ∘ ⇑((lift R) (MulOpposite.op ∘ ι R))) a +\n (MulOpposite.unop ∘ ⇑((lift R) (MulOpposite.op ∘ ι R))) b",
"usedConstants": [
... | simp only [Function.comp_apply, map_add, MulOpposite.unop_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Star.Free | {
"line": 64,
"column": 21
} | {
"line": 64,
"column": 83
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nX : Type u_2\na b : FreeAlgebra R X\n⊢ (MulOpposite.unop ∘ ⇑((lift R) (MulOpposite.op ∘ ι R))) (a + b) =\n (MulOpposite.unop ∘ ⇑((lift R) (MulOpposite.op ∘ ι R))) a +\n (MulOpposite.unop ∘ ⇑((lift R) (MulOpposite.op ∘ ι R))) b",
"usedConstants": [
... | simp only [Function.comp_apply, map_add, MulOpposite.unop_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Star.CentroidHom | {
"line": 88,
"column": 26
} | {
"line": 88,
"column": 56
} | [
{
"pp": "α : Type u_1\ninst✝¹ : NonUnitalNonAssocSemiring α\ninst✝ : StarRing α\nz : ↥(NonUnitalStarSubsemiring.center α)\na : α\n⊢ a * ↑(star z) = ↑(star z) * a",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"NonUnitalStarSubsemiring.instStarMemClass",
"Membership.mem... | by rw [(star z).property.comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 154,
"column": 85
} | {
"line": 155,
"column": 14
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝ : AddMonoid k\np : G →₀ k\n⊢ { toFinsupp := p }.support = p.support",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"AddMonoid.toAddZeroClass",
"Finsupp.support",
"AddZeroClass.toAddZero",
"id",
"SkewMonoidAlgebr... | by
rw [support] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 157,
"column": 91
} | {
"line": 158,
"column": 14
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝ : AddMonoid k\np : SkewMonoidAlgebra k G\n⊢ p.toFinsupp.support = p.support",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"AddMonoid.toAddZeroClass",
"Finsupp.support",
"AddZeroClass.toAddZero",
"id",
"SkewMono... | by
rw [support] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Tropical.BigOperators | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝¹ : LinearOrder R\ninst✝ : OrderTop R\ns : Multiset R\n⊢ trop s.inf = (map trop s).sum",
"usedConstants": [
"Multiset.sum",
"Multiset.map_cons",
"Multiset.map",
"Tropical.instAddCommMonoidTropical",
"congrArg",
"AddMonoid.toAddZeroClass",
... | induction s using Multiset.induction with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.Tropical.BigOperators | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝¹ : LinearOrder R\ninst✝ : OrderTop R\ns : Multiset (Tropical R)\n⊢ untrop s.sum = (map untrop s).inf",
"usedConstants": [
"Multiset.sum",
"Lattice.toSemilatticeSup",
"Multiset.map_cons",
"Multiset.map",
"Tropical.instAddCommMonoidTropical",
"c... | induction s using Multiset.induction with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 1051,
"column": 56
} | {
"line": 1051,
"column": 60
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Monoid G\ninst✝ : MulSemiringAction G k\nf g : SkewMonoidAlgebra k G →+* k\nh₁ : ∀ (b : k), f (single 1 b) = g (single 1 b)\nh_of : ∀ (a : G), f (single a 1) = g (single a 1)\nthis : ∀ {a : G} {b₁ b₂ : k}, single 1 b₁ * single a b₂ = single a (b... | h_of | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Vertex.HVertexOperator | {
"line": 135,
"column": 2
} | {
"line": 137,
"column": 30
} | [
{
"pp": "Γ : Type u_5\nΓ' : Type u_6\ninst✝⁸ : PartialOrder Γ\ninst✝⁷ : PartialOrder Γ'\nR : Type u_7\ninst✝⁶ : CommRing R\nU : Type u_8\nV : Type u_9\nW : Type u_10\ninst✝⁵ : AddCommGroup U\ninst✝⁴ : Module R U\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddCommGroup W\ninst✝ : Module R W\nA : HVer... | ext
simp only [compHahnSeries_coeff, map_smul, coeff_apply_apply, HahnSeries.coeff_smul]
rw [← HahnSeries.coeff_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Vertex.HVertexOperator | {
"line": 135,
"column": 2
} | {
"line": 137,
"column": 30
} | [
{
"pp": "Γ : Type u_5\nΓ' : Type u_6\ninst✝⁸ : PartialOrder Γ\ninst✝⁷ : PartialOrder Γ'\nR : Type u_7\ninst✝⁶ : CommRing R\nU : Type u_8\nV : Type u_9\nW : Type u_10\ninst✝⁵ : AddCommGroup U\ninst✝⁴ : Module R U\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddCommGroup W\ninst✝ : Module R W\nA : HVer... | ext
simp only [compHahnSeries_coeff, map_smul, coeff_apply_apply, HahnSeries.coeff_smul]
rw [← HahnSeries.coeff_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.RingedSpace.Stalks | {
"line": 196,
"column": 2
} | {
"line": 196,
"column": 36
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nX Y : PresheafedSpace C\nf : X ⟶ Y\nx y : ↑↑X\nh : x ⤳ y\n⊢ colimit.desc ((OpenNhds.inclusion ((TopCat.Hom.hom f.base) y)).op ⋙ Y.presheaf)\n { pt := colimit ((OpenNhds.inclusion ((TopCat.Hom.hom f.base) x)).op ⋙ Y.presheaf),\n ... | refine colimit.hom_ext fun j => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Geometry.RingedSpace.SheafedSpace | {
"line": 284,
"column": 2
} | {
"line": 284,
"column": 40
} | [
{
"pp": "case left_cancellation\nC : Type u\ninst✝⁶ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type v\ninst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninstCC : ConcreteCategory C FC\ninst✝⁴ : HasColimits C\ninst✝³ : HasLimits C\ninst✝² : PreservesLimits (CategoryTheory.forget C)\ninst✝¹ : Preserv... | apply_fun InducedCategory.Hom.hom at e | Mathlib.Tactic._aux_Mathlib_Tactic_ApplyFun___elabRules_Mathlib_Tactic_applyFun_1 | Mathlib.Tactic.applyFun |
Mathlib.Geometry.RingedSpace.LocallyRingedSpace | {
"line": 413,
"column": 6
} | {
"line": 413,
"column": 22
} | [
{
"pp": "X Y : LocallyRingedSpace\ne : X ≅ Y\ny : ↑Y.toTopCat\n⊢ Hom.stalkMap e.hom ((ConcreteCategory.hom e.inv.base) y) ≫ Hom.stalkMap e.inv y = Y.presheaf.stalkSpecializes ⋯",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.LocallyRingedSpace.stalkMap_hom_inv._proof_1",
"AlgebraicGeomet... | ← stalkMap_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.RingedSpace.LocallyRingedSpace | {
"line": 426,
"column": 6
} | {
"line": 426,
"column": 22
} | [
{
"pp": "X Y : LocallyRingedSpace\ne : X ≅ Y\nx : ↑X.toTopCat\n⊢ Hom.stalkMap e.inv ((ConcreteCategory.hom e.hom.base) x) ≫ Hom.stalkMap e.hom x = X.presheaf.stalkSpecializes ⋯",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.PresheafedSpace.carrier",
"CategoryTheory.CategoryStruct.toQuiv... | ← stalkMap_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.RingedSpace.Basic | {
"line": 227,
"column": 2
} | {
"line": 227,
"column": 32
} | [
{
"pp": "case h\nX : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\ns : Set ↑(X.presheaf.obj (op U))\ni : ↑(X.presheaf.obj (op U))\na✝ : i ∈ s\n⊢ IsClosed (↑(X.basicOpen i))ᶜ",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeomet... | simp only [isClosed_compl_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.RingedSpace.PresheafedSpace | {
"line": 324,
"column": 14
} | {
"line": 324,
"column": 36
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : PresheafedSpace C\n⊢ X.presheaf = (Presheaf.pushforward C (X.ofRestrict ⋯).base).obj (X.restrict ⋯).presheaf",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"AlgebraicGeometry.PresheafedSpace.carrier",
"CompleteLatt... | restrict_top_presheaf, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 55,
"column": 18
} | {
"line": 62,
"column": 35
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX : LocallyRingedSpace\nh : ∀ (x : ↑X.toTopCat), ∃ R f, x ∈ Set.range ⇑(ConcreteCategory.hom f.base) ∧ IsOpenImmersion f\n⊢ ∀ (x : ↑X.toTopCat), ∃ U R, Nonempty (X.restrict ⋯ ≅ Spec.toLocallyRingedSpace.obj (op R))",
"usedConstants": [
"Opposite",
... | by
intro x
obtain ⟨R, f, h₁, h₂⟩ := h x
refine ⟨⟨⟨_, h₂.base_open.isOpen_range⟩, h₁⟩, R, ⟨?_⟩⟩
apply LocallyRingedSpace.isoOfSheafedSpaceIso
refine SheafedSpace.forgetToPresheafedSpace.preimageIso ?_
apply PresheafedSpace.IsOpenImmersion.isoOfRangeEq (PresheafedSpace.ofRestrict _ _) f.1
exac... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 612,
"column": 2
} | {
"line": 614,
"column": 68
} | [
{
"pp": "X Y Z : Scheme\nf : X ⟶ Z\ng : Y ⟶ Z\nH : IsOpenImmersion f\n⊢ Set.range ⇑(pullback.fst g f ≫ g) = Set.range ⇑g ∩ Set.range ⇑f",
"usedConstants": [
"ContinuousMap.continuous",
"Set.range_comp",
"Eq.mpr",
"CategoryTheory.Limits.pullback",
"AlgebraicGeometry.SheafedSpace... | rw [Scheme.Hom.comp_base, TopCat.coe_comp, Set.range_comp, range_pullbackFst,
Opens.map_obj, Opens.coe_mk, Set.image_preimage_eq_inter_range,
Set.inter_comm, Opens.carrier_eq_coe, Scheme.Hom.coe_opensRange] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 612,
"column": 2
} | {
"line": 614,
"column": 68
} | [
{
"pp": "X Y Z : Scheme\nf : X ⟶ Z\ng : Y ⟶ Z\nH : IsOpenImmersion f\n⊢ Set.range ⇑(pullback.fst g f ≫ g) = Set.range ⇑g ∩ Set.range ⇑f",
"usedConstants": [
"ContinuousMap.continuous",
"Set.range_comp",
"Eq.mpr",
"CategoryTheory.Limits.pullback",
"AlgebraicGeometry.SheafedSpace... | rw [Scheme.Hom.comp_base, TopCat.coe_comp, Set.range_comp, range_pullbackFst,
Opens.map_obj, Opens.coe_mk, Set.image_preimage_eq_inter_range,
Set.inter_comm, Opens.carrier_eq_coe, Scheme.Hom.coe_opensRange] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 612,
"column": 2
} | {
"line": 614,
"column": 68
} | [
{
"pp": "X Y Z : Scheme\nf : X ⟶ Z\ng : Y ⟶ Z\nH : IsOpenImmersion f\n⊢ Set.range ⇑(pullback.fst g f ≫ g) = Set.range ⇑g ∩ Set.range ⇑f",
"usedConstants": [
"ContinuousMap.continuous",
"Set.range_comp",
"Eq.mpr",
"CategoryTheory.Limits.pullback",
"AlgebraicGeometry.SheafedSpace... | rw [Scheme.Hom.comp_base, TopCat.coe_comp, Set.range_comp, range_pullbackFst,
Opens.map_obj, Opens.coe_mk, Set.image_preimage_eq_inter_range,
Set.inter_comm, Opens.carrier_eq_coe, Scheme.Hom.coe_opensRange] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Cover.MorphismProperty | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 29
} | [
{
"pp": "K : Precoverage Scheme\ninst✝ : JointlySurjective K\nX : Scheme\n𝒰 : Cover K X\n⊢ ⋃ i, Set.range ⇑(𝒰.f i) = Set.univ",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Scheme",
"CategoryTheory.PreZeroHypercover.f",
"AlgebraicGeometry.PresheafedSpace.carrier",
"congrAr... | rw [Set.eq_univ_iff_forall] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.RingedSpace.OpenImmersion | {
"line": 728,
"column": 2
} | {
"line": 728,
"column": 82
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX Y Z : SheafedSpace C\nf : X ⟶ Z\ng : Y ⟶ Z\nH : IsOpenImmersion f\nthis : (preservesLimitIso forget (cospan g f)).hom ≫ limit.π (cospan g f ⋙ forget) left = (pullback.fst g f).hom\n⊢ PresheafedSpace.IsOpenImmersion ((preservesLimitIso forget (cospan g f)).hom ≫ ... | have := HasLimit.isoOfNatIso_hom_π (diagramIsoCospan (cospan g f ⋙ forget)) left | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.AlgebraicGeometry.Restrict | {
"line": 112,
"column": 59
} | {
"line": 114,
"column": 29
} | [
{
"pp": "X : Scheme\nU : X.Opens\nW : (↑U).Opens\n⊢ U.ι ''ᵁ W ≤ U",
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.opensFunctor",
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"Lattice.toSemilatticeSup",
"AlgebraicGeometry.PresheafedSpace.carr... | by
simp_rw [← U.ι_image_top]
exact U.ι.image_mono le_top | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Restrict | {
"line": 418,
"column": 40
} | {
"line": 418,
"column": 49
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v, u₁} C\nX✝ : Scheme\nU✝ : X✝.Opens\nX : Scheme\nU V : X.Opens\ne : U = V\n⊢ Set.range ⇑U.ι = Set.range ⇑V.ι",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.PresheafedSpace.carrier",
"congrArg",
"CategoryTheory.ConcreteCategory.hom",
"... | by rw [e] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Restrict | {
"line": 642,
"column": 4
} | {
"line": 642,
"column": 44
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nX Y U : Scheme\nf : X ⟶ Y\ng : U ⟶ Y\ninst✝ : IsOpenImmersion g\nV : Y.Opens := Scheme.Hom.opensRange g\ne : U ≅ ↑V := IsOpenImmersion.isoOfRangeEq g V.ι ⋯\nt : pullback f g ⟶ pullback f V.ι := pullback.map f g f V.ι (𝟙 X) e.hom (𝟙 Y) ⋯ ⋯\n⊢ t ≫ (pullbackRest... | pullbackRestrictIsoRestrict_hom_ι_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Restrict | {
"line": 765,
"column": 46
} | {
"line": 765,
"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 stalkMap U.ι ((resLE f U V e) x) ≫ stalkMap (resLE f U V e) x",
"usedConstants": [
"Eq.mpr",
... | ← stalkMap_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 348,
"column": 6
} | {
"line": 348,
"column": 16
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nf g : R\nU : Opens ↑(PrimeSpectrum.Top R)\nhu₁ : U ≤ basicOpen g\nhu₂ : U ≤ basicOpen f\n⊢ const f g U hu₁ * const g f U hu₂ = 1",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
"Semiring.toModule",
"Opposite",
"HMul.hMul",
... | const_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 352,
"column": 6
} | {
"line": 352,
"column": 16
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nf g₁ g₂ : R\nU : Opens ↑(PrimeSpectrum.Top R)\nhu₁ : U ≤ basicOpen g₁\nhu₂ : U ≤ basicOpen g₂\n⊢ const f g₁ U hu₁ * const g₁ g₂ U hu₂ = const f g₂ U hu₂",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
"Semiring.toModule",
"Opposite",
... | const_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 653,
"column": 2
} | {
"line": 660,
"column": 19
} | [
{
"pp": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx : ↑(PrimeSpectrum.Top R)\nf : R\nhf : x ∈ basicOpen f\n⊢ IsUnit ((algebraMap R (Module.End R ↑((structurePresheafInModuleCat R M).stalk x))) f)",
"usedConstants": [
"Eq.mpr",
"RingHom.instRingHomClass",
... | have := (isUnit_toStalk x f hf).map (algebraMap _
(Module.End ((structurePresheafInCommRingCat R).stalk x)
((structurePresheafInModuleCat R M).stalk x)))
rw [Module.End.isUnit_iff] at this ⊢
convert! this
ext a
simp only [Module.algebraMap_end_apply]
rw [toStalk_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 653,
"column": 2
} | {
"line": 660,
"column": 19
} | [
{
"pp": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx : ↑(PrimeSpectrum.Top R)\nf : R\nhf : x ∈ basicOpen f\n⊢ IsUnit ((algebraMap R (Module.End R ↑((structurePresheafInModuleCat R M).stalk x))) f)",
"usedConstants": [
"Eq.mpr",
"RingHom.instRingHomClass",
... | have := (isUnit_toStalk x f hf).map (algebraMap _
(Module.End ((structurePresheafInCommRingCat R).stalk x)
((structurePresheafInModuleCat R M).stalk x)))
rw [Module.End.isUnit_iff] at this ⊢
convert! this
ext a
simp only [Module.algebraMap_end_apply]
rw [toStalk_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.GammaSpecAdjunction | {
"line": 484,
"column": 63
} | {
"line": 488,
"column": 51
} | [
{
"pp": "X : Scheme\nr : ↑Γ(X, ⊤)\n⊢ X.toSpecΓ ⁻¹ᵁ PrimeSpectrum.basicOpen r = X.basicOpen r",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.Scheme",
"Lattice.toSemilatticeSup",
... | by
rw [← basicOpen_eq_of_affine, Scheme.preimage_basicOpen, ← Scheme.Hom.appTop]
congr
rw [Scheme.toSpecΓ_appTop]
exact Iso.inv_hom_id_apply (C := CommRingCat) _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 735,
"column": 4
} | {
"line": 743,
"column": 64
} | [
{
"pp": "R M A : Type u\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nx : ↑(PrimeSpectrum.Top R)\n⊢ stalkToLocalizationₗ R M x ≫ localizationtoStalkₗ R M x = 𝟙 ((structurePresheafInModuleCat R M).stalk x)",
"usedConstants": [
"Eq.mpr",
... | apply stalk_hom_ext
intro U hxU
ext s
obtain ⟨g, hxg, igU, f, hs⟩ :=
exists_const _ s x hxU
rw [germ_stalkToLocalizationₗ_assoc, Category.comp_id, ← germ_res_apply _ igU.hom _ hxg]
refine congr(localizationtoStalkₗ R M x (openToLocalizationₗ R M _ x hxg $hs)).symm.trans ?_
refine (localiza... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 735,
"column": 4
} | {
"line": 743,
"column": 64
} | [
{
"pp": "R M A : Type u\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nx : ↑(PrimeSpectrum.Top R)\n⊢ stalkToLocalizationₗ R M x ≫ localizationtoStalkₗ R M x = 𝟙 ((structurePresheafInModuleCat R M).stalk x)",
"usedConstants": [
"Eq.mpr",
... | apply stalk_hom_ext
intro U hxU
ext s
obtain ⟨g, hxg, igU, f, hs⟩ :=
exists_const _ s x hxU
rw [germ_stalkToLocalizationₗ_assoc, Category.comp_id, ← germ_res_apply _ igU.hom _ hxg]
refine congr(localizationtoStalkₗ R M x (openToLocalizationₗ R M _ x hxg $hs)).symm.trans ?_
refine (localiza... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 386,
"column": 6
} | {
"line": 386,
"column": 24
} | [
{
"pp": "X : Scheme\n𝒰 : X.OpenCover\ni : (gluedCover 𝒰).J\nx : ↥((gluedCover 𝒰).U i)\nthis : Hom.stalkMap ((gluedCover 𝒰).ι i ≫ fromGlued 𝒰) x = (X.presheaf.stalkCongr ⋯).hom ≫ Hom.stalkMap (𝒰.f i) x\n⊢ IsIso (Hom.stalkMap (fromGlued 𝒰) (((gluedCover 𝒰).ι i) x))",
"usedConstants": [
"Algebrai... | Hom.stalkMap_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Limits | {
"line": 269,
"column": 6
} | {
"line": 269,
"column": 68
} | [
{
"pp": "ι : 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))\nthis : Topology.IsOpenEmbedding (⇑(Sigma.desc α) ∘ ⇑(sigmaMk f))\n⊢ Topology.IsOpenEmbedding ⇑(Sigma.desc α)",
"usedConstants": [
... | convert! this.comp (sigmaMk f).symm.isOpenEmbedding; ext; simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Limits | {
"line": 269,
"column": 6
} | {
"line": 269,
"column": 68
} | [
{
"pp": "ι : 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))\nthis : Topology.IsOpenEmbedding (⇑(Sigma.desc α) ∘ ⇑(sigmaMk f))\n⊢ Topology.IsOpenEmbedding ⇑(Sigma.desc α)",
"usedConstants": [
... | convert! this.comp (sigmaMk f).symm.isOpenEmbedding; ext; simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 292,
"column": 27
} | {
"line": 292,
"column": 57
} | [
{
"pp": "case hf.e_a.h₀.e_a\nX Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g\ns : PullbackCone f g\ni j : (Precoverage.ZeroHypercover.pullback₁ s.fst 𝒰).I₀\n⊢ (pullbackSymmetry s.fst (𝒰.f j)).inv ≫ pullback.snd s.fst (𝒰.f j) =\n pullback.map (𝒰.f ... | pullbackSymmetry_inv_comp_snd, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 293,
"column": 51
} | {
"line": 293,
"column": 69
} | [
{
"pp": "case hf.e_a.h₁\nX Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g\ns : PullbackCone f g\ni j : (Precoverage.ZeroHypercover.pullback₁ s.fst 𝒰).I₀\n⊢ pullback.fst ((Precoverage.ZeroHypercover.pullback₁ s.fst 𝒰).f i)\n ((Precoverage.ZeroHype... | pullback.lift_snd, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Gluing | {
"line": 767,
"column": 8
} | {
"line": 767,
"column": 63
} | [
{
"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 ⋙ forgetToLocallyRingedSpace)\ni j : (glueData F).toLocallyRingedSpaceGlueData.J\... | ← cancel_epi (Hom.isoOpensRange (F.map _)).hom.toLRSHom | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.RingedSpace.PresheafedSpace.Gluing | {
"line": 410,
"column": 2
} | {
"line": 410,
"column": 42
} | [
{
"pp": "case h\nC : Type u\ninst✝¹ : Category.{v, u} C\nD : GlueData C\ninst✝ : HasLimits C\ni : D.J\nU : Opens ↑↑(D.U i)\n⊢ { pt := (D.U i).presheaf.obj (op U), π := { app := fun j ↦ D.ιInvAppπApp U (unop j), naturality := ⋯ } }.π.app\n (op (WalkingMultispan.right i)) =\n (D.U i).presheaf.map (eqToHom... | change D.opensImagePreimageMap i i U = _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 360,
"column": 26
} | {
"line": 360,
"column": 44
} | [
{
"pp": "case e_a.h₁\nX Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g\ni : 𝒰.I₀\nj : (Precoverage.ZeroHypercover.pullback₁ (pullback.fst (p1 𝒰 f g) (𝒰.f i)) (gluing 𝒰 f g).openCover).I₀\n⊢ pullback.fst (pullback.fst (p1 𝒰 f g) (𝒰.f i)) ((gluing 𝒰 ... | pullback.lift_snd, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 378,
"column": 31
} | {
"line": 378,
"column": 49
} | [
{
"pp": "case hom_inv_id.h₁\nX Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g\ns : PullbackCone f g\ni : 𝒰.I₀\n⊢ pullback.lift (pullback.snd (p1 𝒰 f g) (𝒰.f i)) (pullback.fst (p1 𝒰 f g) (𝒰.f i) ≫ p2 𝒰 f g) ⋯ ≫\n pullback.lift ((gluing 𝒰 f g).ι... | pullback.lift_snd, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.Limits | {
"line": 662,
"column": 20
} | {
"line": 662,
"column": 42
} | [
{
"pp": "ι : Type u\nX : Scheme\ninst✝ : Finite ι\nU : ι → X.Opens\nhU : ∀ (i : ι), IsAffineOpen (U i)\nhU' : _root_.Pairwise (Disjoint on U)\ni j : ι\ne : i ≠ j\n⊢ (Disjoint on fun x ↦ Set.range ⇑((fun i ↦ (U i).ι) x)) i j",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopol... | convert! hU' e using 0 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 380,
"column": 50
} | {
"line": 380,
"column": 68
} | [
{
"pp": "case inv_hom_id.h₀\nX Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g\ns : PullbackCone f g\ni : 𝒰.I₀\n⊢ pullback.lift ((gluing 𝒰 f g).ι i) (pullback.fst (𝒰.f i ≫ f) g) ⋯ ≫ pullback.snd (p1 𝒰 f g) (𝒰.f i) =\n 𝟙 (pullback (𝒰.f i ≫ f) g) ≫... | pullback.lift_snd, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.Morphisms.Basic | {
"line": 324,
"column": 2
} | {
"line": 326,
"column": 98
} | [
{
"pp": "P : MorphismProperty Scheme\ninst✝² : IsZariskiLocalAtSource P\nX Y : Scheme\nf : X ⟶ Y\ninst✝¹ : IsZariskiLocalAtTarget P\ninst✝ : P.RespectsRight IsOpenImmersion\nU : ↥X → Y.Opens\nV : ↥X → X.Opens\nhxU : ∀ (x : ↥X), x ∈ (V x).carrier\ne : ∀ (x : ↥X), V x ≤ f ⁻¹ᵁ U x\nhf : ∀ (x : ↥X), P (Scheme.Hom.r... | · intro x
rw [← Scheme.Hom.resLE_comp_ι _ (e x)]
exact MorphismProperty.RespectsRight.postcomp (Q := @IsOpenImmersion) _ inferInstance _ (hf x) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 432,
"column": 37
} | {
"line": 432,
"column": 55
} | [
{
"pp": "case create.h.e_a.h₀\nX Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g\ns✝ s : PullbackCone f g\nm : s.pt ⟶ (PullbackCone.mk (p1 𝒰 f g) (p2 𝒰 f g) ⋯).pt\nh₁ : m ≫ p1 𝒰 f g = s.fst\nh₂ : m ≫ p2 𝒰 f g = s.snd\ni : (Precoverage.ZeroHypercover.pu... | pullback.lift_snd, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 434,
"column": 52
} | {
"line": 434,
"column": 70
} | [
{
"pp": "case create.h.e_a.h₁\nX Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (i : 𝒰.I₀), HasPullback (𝒰.f i ≫ f) g\ns✝ s : PullbackCone f g\nm : s.pt ⟶ (PullbackCone.mk (p1 𝒰 f g) (p2 𝒰 f g) ⋯).pt\nh₁ : m ≫ p1 𝒰 f g = s.fst\nh₂ : m ≫ p2 𝒰 f g = s.snd\ni : (Precoverage.ZeroHypercover.pu... | pullback.lift_snd, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.Morphisms.Constructors | {
"line": 221,
"column": 10
} | {
"line": 221,
"column": 84
} | [
{
"pp": "case of_sSup_eq_top.a.refine_2\nP : MorphismProperty Scheme\nhP₂ : ∀ {X Y : Scheme} (f : X ⟶ Y) {ι : Type u} (U : ι → Y.Opens), IsOpenCover U → (∀ (i : ι), P (f ∣_ U i)) → P f\nX Y : Scheme\nf : X ⟶ Y\nι : Type u\nU : ι → Y.Opens\nhU : iSup U = ⊤\nH : ∀ (i : ι), P.universally (f ∣_ U i)\nX' Y' : Scheme... | simp only [Category.assoc, morphismRestrict_ι, Scheme.isoOfEq_hom_ι_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.RingHom.Surjective | {
"line": 57,
"column": 60
} | {
"line": 57,
"column": 88
} | [
{
"pp": "case add\nR S T : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : CommRing T\ninst✝¹ : Algebra R S\ninst✝ : Algebra R T\nh : Function.Surjective ⇑(algebraMap R T)\nx y : S\n⊢ ∃ a, (algebraMap S (S ⊗[R] T)) a = (algebraMap S (S ⊗[R] T)) x + (algebraMap S (S ⊗[R] T)) y",
"usedConstants":... | exact ⟨x + y, map_add _ x y⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 802,
"column": 2
} | {
"line": 802,
"column": 37
} | [
{
"pp": "M S T : Scheme\ninst✝¹ : M.Over S\nf : T ⟶ S\ninst✝ : MonObj (M.asOver S)\n⊢ MonObj ((pullback (M ↘ S) f).asOver T)",
"usedConstants": [
"CategoryTheory.Limits.pullback",
"CategoryTheory.Over",
"AlgebraicGeometry.Scheme",
"inferInstance",
"AlgebraicGeometry.Scheme.cano... | unfold asOver OverClass.asOver at * | Lean.Elab.Tactic.evalUnfold | Lean.Parser.Tactic.unfold |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 806,
"column": 2
} | {
"line": 806,
"column": 37
} | [
{
"pp": "M S T : Scheme\ninst✝² : M.Over S\nf : T ⟶ S\ninst✝¹ : MonObj (M.asOver S)\ninst✝ : IsCommMonObj (M.asOver S)\n⊢ IsCommMonObj ((pullback (M ↘ S) f).asOver T)",
"usedConstants": [
"CategoryTheory.Limits.pullback",
"CategoryTheory.Over",
"AlgebraicGeometry.Scheme",
"AlgebraicG... | unfold asOver OverClass.asOver at * | Lean.Elab.Tactic.evalUnfold | Lean.Parser.Tactic.unfold |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 809,
"column": 2
} | {
"line": 809,
"column": 37
} | [
{
"pp": "M S T : Scheme\ninst✝¹ : M.Over S\nf : T ⟶ S\ninst✝ : GrpObj (M.asOver S)\n⊢ GrpObj ((pullback (M ↘ S) f).asOver T)",
"usedConstants": [
"CategoryTheory.Limits.pullback",
"CategoryTheory.Over",
"AlgebraicGeometry.Scheme",
"inferInstance",
"AlgebraicGeometry.Scheme.cano... | unfold asOver OverClass.asOver at * | Lean.Elab.Tactic.evalUnfold | Lean.Parser.Tactic.unfold |
Mathlib.AlgebraicGeometry.Pullbacks | {
"line": 815,
"column": 2
} | {
"line": 815,
"column": 37
} | [
{
"pp": "M S T : Scheme\ninst✝¹ : M.Over S\nf : T ⟶ S\ninst✝ : MonObj (M.asOver S)\n⊢ IsMonHom (Hom.asOver (pullback.fst (M ↘ S) (𝟙 S)) S)",
"usedConstants": [
"CategoryTheory.Limits.pullback",
"CategoryTheory.Over",
"AlgebraicGeometry.Scheme.Hom.asOver",
"AlgebraicGeometry.Scheme",... | unfold asOver OverClass.asOver at * | Lean.Elab.Tactic.evalUnfold | Lean.Parser.Tactic.unfold |
Mathlib.AlgebraicGeometry.Morphisms.Preimmersion | {
"line": 105,
"column": 7
} | {
"line": 105,
"column": 77
} | [
{
"pp": "X Y Z : Scheme\nf : X ⟶ Z\ng : Y ⟶ Z\nx✝¹ : HasPullback f g\nx✝ : IsPreimmersion g\nthis : SurjectiveOnStalks (pullback.fst f g)\nx : ↥(pullback f g)\n⊢ ((pullback.fst f g) x, (pullback.snd f g) x) ∈ {x | f x.1 = g x.2}",
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.Hom",
"Cate... | simp only [Set.mem_setOf, ← Scheme.Hom.comp_apply, pullback.condition] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.AlgebraicGeometry.Morphisms.Preimmersion | {
"line": 105,
"column": 7
} | {
"line": 105,
"column": 77
} | [
{
"pp": "X Y Z : Scheme\nf : X ⟶ Z\ng : Y ⟶ Z\nx✝¹ : HasPullback f g\nx✝ : IsPreimmersion g\nthis : SurjectiveOnStalks (pullback.fst f g)\nx : ↥(pullback f g)\n⊢ ((pullback.fst f g) x, (pullback.snd f g) x) ∈ {x | f x.1 = g x.2}",
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.Hom",
"Cate... | simp only [Set.mem_setOf, ← Scheme.Hom.comp_apply, pullback.condition] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.Preimmersion | {
"line": 105,
"column": 7
} | {
"line": 105,
"column": 77
} | [
{
"pp": "X Y Z : Scheme\nf : X ⟶ Z\ng : Y ⟶ Z\nx✝¹ : HasPullback f g\nx✝ : IsPreimmersion g\nthis : SurjectiveOnStalks (pullback.fst f g)\nx : ↥(pullback f g)\n⊢ ((pullback.fst f g) x, (pullback.snd f g) x) ∈ {x | f x.1 = g x.2}",
"usedConstants": [
"AlgebraicGeometry.PresheafedSpace.Hom",
"Cate... | simp only [Set.mem_setOf, ← Scheme.Hom.comp_apply, pullback.condition] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.QuasiCompact | {
"line": 138,
"column": 55
} | {
"line": 139,
"column": 61
} | [
{
"pp": "X : Scheme\n⊢ CompactSpace ↥X ↔ QuasiCompact (terminal.from X)",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.Scheme",
"AlgebraicGeometry.PresheafedSpace.carrier",
"CategoryTheory.CategoryStruct.to... | by
rw [HasAffineProperty.iff_of_isAffine (P := @QuasiCompact)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 294,
"column": 2
} | {
"line": 295,
"column": 69
} | [
{
"pp": "P : MorphismProperty Scheme\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\ninst✝² : HasRingHomProperty P Q\nX Y : Scheme\nf : X ⟶ Y\nH : P f\ninst✝¹ : IsAffine X\ninst✝ : IsAffine Y\n⊢ Q (CommRingCat.Hom.hom (Scheme.Hom.appTop f))",
"usedConstants": [
"A... | rw [Scheme.Hom.appTop, Scheme.Hom.app_eq_appLE]
exact appLE P f H ⟨_, isAffineOpen_top _⟩ ⟨_, isAffineOpen_top _⟩ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 294,
"column": 2
} | {
"line": 295,
"column": 69
} | [
{
"pp": "P : MorphismProperty Scheme\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\ninst✝² : HasRingHomProperty P Q\nX Y : Scheme\nf : X ⟶ Y\nH : P f\ninst✝¹ : IsAffine X\ninst✝ : IsAffine Y\n⊢ Q (CommRingCat.Hom.hom (Scheme.Hom.appTop f))",
"usedConstants": [
"A... | rw [Scheme.Hom.appTop, Scheme.Hom.app_eq_appLE]
exact appLE P f H ⟨_, isAffineOpen_top _⟩ ⟨_, isAffineOpen_top _⟩ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 76,
"column": 8
} | {
"line": 80,
"column": 15
} | [
{
"pp": "case mpr.refine_2.refine_2\nX : Scheme\nH : ∀ (U V : ↑X.affineOpens), IsCompact (↑↑U ∩ ↑↑V)\nU : X.Opens\nhU : IsCompact U.carrier\nV✝ : X.Opens\nhV : IsCompact V✝.carrier\nS : X.Opens\nx✝ : IsCompact S.carrier\nV : ↑X.affineOpens\n⊢ ∀ (S : X.Opens),\n IsCompact S.carrier → ∀ (U : ↑X.affineOpens), I... | intro S _ W hW
change IsCompact ((S.1 ∪ W.1) ∩ V.1)
rw [Set.union_inter_distrib_right]
apply hW.union
apply H | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.QuasiSeparated | {
"line": 76,
"column": 8
} | {
"line": 80,
"column": 15
} | [
{
"pp": "case mpr.refine_2.refine_2\nX : Scheme\nH : ∀ (U V : ↑X.affineOpens), IsCompact (↑↑U ∩ ↑↑V)\nU : X.Opens\nhU : IsCompact U.carrier\nV✝ : X.Opens\nhV : IsCompact V✝.carrier\nS : X.Opens\nx✝ : IsCompact S.carrier\nV : ↑X.affineOpens\n⊢ ∀ (S : X.Opens),\n IsCompact S.carrier → ∀ (U : ↑X.affineOpens), I... | intro S _ W hW
change IsCompact ((S.1 ∪ W.1) ∩ V.1)
rw [Set.union_inter_distrib_right]
apply hW.union
apply H | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Properties | {
"line": 381,
"column": 23
} | {
"line": 383,
"column": 40
} | [
{
"pp": "X : Scheme\ninst✝¹ : IsIntegral X\ninst✝ : Subsingleton ↥X\n⊢ IsField ↑Γ(X, ⊤)",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.instIsDomainCarrierObjOppositeOpensCarrierCarrierCommRingCatPresheafOpOpensTopOfIsIntegral",
"AlgebraicGeometry.SheafedS... | by
rw [← PrimeSpectrum.t1Space_iff_isField]
apply X.isoSpec.hom.homeomorph.t1Space | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.RingHomProperties | {
"line": 715,
"column": 2
} | {
"line": 720,
"column": 61
} | [
{
"pp": "case inr\nP : MorphismProperty Scheme\nQ : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\ninst✝ : HasRingHomProperty P Q\nX Y : Scheme\nf : X ⟶ Y\nQ' : {R S : Type u} → [inst : CommRing R] → [inst_1 : CommRing S] → (R →+* S) → Prop\nhQ' : RespectsIso fun {R S} [CommRin... | · obtain ⟨U, hU, hfx, _⟩ := Opens.isBasis_iff_nbhd.mp Y.isBasis_affineOpens
(Opens.mem_top <| f x)
obtain ⟨V, hV, hx, e⟩ := Opens.isBasis_iff_nbhd.mp X.isBasis_affineOpens
(show x ∈ f ⁻¹ᵁ U from hfx)
rw [← hQ'.arrow_mk_iso_iff (Scheme.Hom.resLEStalkMap f e ⟨x, hx⟩)]
exact this (IsZariskiLocalAtS... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.Morphisms.IsIso | {
"line": 48,
"column": 98
} | {
"line": 52,
"column": 53
} | [
{
"pp": "⊢ HasAffineProperty (isomorphisms Scheme) fun X x f x_1 ↦ IsAffine X ∧ IsIso (Scheme.Hom.appTop f)",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier"... | by
convert! HasAffineProperty.of_isZariskiLocalAtTarget (isomorphisms Scheme) with X Y f hY
exact ⟨fun ⟨_, _⟩ ↦ (arrow_mk_iso_iff (isomorphisms _) (arrowIsoSpecΓOfIsAffine f)).mpr
(inferInstanceAs (IsIso (Spec.map (f.appTop)))),
fun (_ : IsIso f) ↦ ⟨.of_isIso f, inferInstance⟩⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 293,
"column": 2
} | {
"line": 293,
"column": 17
} | [
{
"pp": "X : Scheme\nI : X.IdealSheafData\nU V : ↑X.affineOpens\nx : ↥X\nhxV : x ∈ ↑↑V\nhxU : x ∈ ↑↑U\nhx : ∀ f ∈ I.ideal U, x ∉ X.basicOpen f\n⊢ ∀ f ∈ I.ideal V, x ∉ X.basicOpen f",
"usedConstants": [
"Opposite",
"CommRingCat.carrier",
"AlgebraicGeometry.PresheafedSpace.carrier",
"T... | intro s hfU hxs | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.AlgebraicGeometry.Stalk | {
"line": 250,
"column": 33
} | {
"line": 251,
"column": 95
} | [
{
"pp": "R : CommRingCat\ninst✝ : IsLocalRing ↑R\n⊢ (Spec R).presheaf.germ ⊤ (closedPoint ↑R) trivial ≫ (stalkClosedPointIso R).hom = (Scheme.ΓSpecIso R).hom",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.assoc",
"AlgebraicGeometry.Spec",
"AlgebraicGeometry.SheafedSpace.inst... | by
rw [← ΓSpecIso_hom_stalkClosedPointIso_inv, Category.assoc, Iso.inv_hom_id, Category.comp_id] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.IdealSheaf.Basic | {
"line": 585,
"column": 12
} | {
"line": 585,
"column": 87
} | [
{
"pp": "case a\nX : Scheme\nI : X.IdealSheafData\nZ : Closeds ↥X\nU : ↑X.affineOpens\nf : ↑Γ(X, ↑U)\nF : Γ(X, ↑U) ⟶ Γ(X, X.basicOpen f) := X.presheaf.map (homOfLE ⋯).op\nthis✝¹ : Algebra ↑Γ(X, ↑U) ↑Γ(X, ↑(X.affineBasicOpen f)) := (CommRingCat.Hom.hom F).toAlgebra\nthis✝ : IsLocalization.Away f ↑Γ(X, X.basicOpe... | ← U.2.map_fromSpec (X.affineBasicOpen f).2 (homOfLE (X.basicOpen_le f)).op, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.PullbackCarrier | {
"line": 394,
"column": 2
} | {
"line": 395,
"column": 81
} | [
{
"pp": "case h.a.snd\nX Y S : Scheme\nf : X ⟶ S\ng : Y ⟶ S\nx : (fun X ↦ X) (Types.PullbackObj (forget.map f) (forget.map g))\nz : ↥(pullback f g)\nh1 : (pullback.fst f g) z = (↑x).1\nh2 : (pullback.snd f g) z = (↑x).2\n⊢ (↑((⇑(ConcreteCategory.hom (Types.pullbackIsoPullback (forget.map f) (forget.map g)).hom)... | · simp only [Function.comp_apply, Types.pullbackIsoPullback_hom_snd]
rwa [← types_comp_apply (g := pullback.snd _ _), pullbackComparison_comp_snd] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.Morphisms.Affine | {
"line": 315,
"column": 2
} | {
"line": 316,
"column": 48
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\n⊢ IsAffineHom (pullback.diagonal f) ↔\n ∀ (U : Y.Opens),\n IsAffineOpen U → ∀ V₁ ≤ f ⁻¹ᵁ U, ∀ V₂ ≤ f ⁻¹ᵁ U, IsAffineOpen V₁ → IsAffineOpen V₂ → IsAffineOpen (V₁ ⊓ V₂)",
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
... | refine congr($(HasAffineProperty.eq_targetAffineLocally
(.diagonal @IsAffineHom)) f).to_iff.trans ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
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