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.Algebra.Star.CHSH | {
"line": 196,
"column": 37
} | {
"line": 196,
"column": 64
} | [
{
"pp": "R : Type u\ninst✝⁶ : Ring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : StarRing R\ninst✝³ : StarOrderedRing R\ninst✝² : Algebra ℝ R\ninst✝¹ : IsOrderedModule ℝ R\ninst✝ : StarModule ℝ R\nA₀ A₁ B₀ B₁ : R\nT : IsCHSHTuple A₀ A₁ B₀ B₁\nM : ∀ (m : ℤ) (a : ℝ) (x : R), m • a • x = (↑m * a) • x\nP : R := (√2)⁻¹ • (A₁... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.CHSH | {
"line": 199,
"column": 2
} | {
"line": 199,
"column": 68
} | [
{
"pp": "case a\nR : Type u\ninst✝⁶ : Ring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : StarRing R\ninst✝³ : StarOrderedRing R\ninst✝² : Algebra ℝ R\ninst✝¹ : IsOrderedModule ℝ R\ninst✝ : StarModule ℝ R\nA₀ A₁ B₀ B₁ : R\nT : IsCHSHTuple A₀ A₁ B₀ B₁\nM : ∀ (m : ℤ) (a : ℝ) (x : R), m • a • x = (↑m * a) • x\nP : R := ⋯\nQ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.Subsemiring | {
"line": 122,
"column": 50
} | {
"line": 122,
"column": 66
} | [
{
"pp": "R : Type v\ninst✝¹ : NonAssocSemiring R\ninst✝ : StarRing R\nS : StarSubsemiring R\ns : Set R\nhs : s = ↑S\na : R\nha : a ∈ (S.copy s hs).carrier\n⊢ a ∈ S.carrier",
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"Submonoid.toSubsemigroup",
"Membership.mem",
"S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.Subsemiring | {
"line": 122,
"column": 50
} | {
"line": 122,
"column": 66
} | [
{
"pp": "R : Type v\ninst✝¹ : NonAssocSemiring R\ninst✝ : StarRing R\nS : StarSubsemiring R\na : R\nhs : ↑S = ↑S\n__Subsemiring✝ : Subsemiring R := S.copy (↑S) hs\nha : a ∈ (S.copy (↑S) hs).carrier\n⊢ a ∈ S.carrier",
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"Submonoid.toSubs... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 170,
"column": 2
} | {
"line": 170,
"column": 23
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝¹ : AddMonoid k\ninst✝ : DecidableEq G\np q : SkewMonoidAlgebra k G\n⊢ (p + q).support ⊆ p.support ∪ q.support",
"usedConstants": [
"Eq.mpr",
"Finset.instUnion",
"congrArg",
"Finset",
"AddMonoid.toAddZeroClass",
"Finsupp.support",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 209,
"column": 2
} | {
"line": 209,
"column": 21
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝ : AddMonoid k\nf g : G →₀ k\n⊢ { toFinsupp := f } = { toFinsupp := g } ↔ ∀ (n : G), { toFinsupp := f }.coeff n = { toFinsupp := g }.coeff n",
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"congrArg",
"AddMonoid.toAddZeroClass",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 332,
"column": 2
} | {
"line": 332,
"column": 32
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝² : One G\ninst✝¹ : AddMonoidWithOne k\na : G\ninst✝ : Decidable (a = 1)\n⊢ coeff 1 a = if a = 1 then 1 else 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"AddMonoid.toAddZeroClass",
"Classical.propDecidable",
"AddZeroClass.toAddZero",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.UnitaryStarAlgAut | {
"line": 95,
"column": 41
} | {
"line": 95,
"column": 66
} | [
{
"pp": "R : Type u_3\nS : Type u_4\ninst✝⁸ : Ring R\ninst✝⁷ : StarMul R\ninst✝⁶ : CommRing S\ninst✝⁵ : StarMul S\ninst✝⁴ : Algebra S R\ninst✝³ : StarModule S R\ninst✝² : Algebra.IsCentral S R\ninst✝¹ : IsCancelMulZero S\ninst✝ : Module.IsTorsionFree S R\nu v : ↥(unitary R)\nx✝ : ∃ y, y • 1 = star ↑v * ↑u\ny : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.UnitaryStarAlgAut | {
"line": 96,
"column": 46
} | {
"line": 96,
"column": 71
} | [
{
"pp": "R : Type u_3\nS : Type u_4\ninst✝⁸ : Ring R\ninst✝⁷ : StarMul R\ninst✝⁶ : CommRing S\ninst✝⁵ : StarMul S\ninst✝⁴ : Algebra S R\ninst✝³ : StarModule S R\ninst✝² : Algebra.IsCentral S R\ninst✝¹ : IsCancelMulZero S\ninst✝ : Module.IsTorsionFree S R\nu v : ↥(unitary R)\nx✝ : ∃ y, y • 1 = star ↑v * ↑u\ny : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 471,
"column": 21
} | {
"line": 471,
"column": 32
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝ : AddCommMonoid k\ninstNonempty : Nonempty G\np : SkewMonoidAlgebra k G → Prop\nf : SkewMonoidAlgebra k G\nsingle : ∀ (g : G) (a : k), p (SkewMonoidAlgebra.single g a)\nadd : ∀ (f g : SkewMonoidAlgebra k G), p f → p g → p (f + g)\n⊢ p 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Tropical.Basic | {
"line": 286,
"column": 23
} | {
"line": 286,
"column": 34
} | [
{
"pp": "R : Type u\ninst✝ : LinearOrder R\nx y : Tropical R\nh : x ≤ y\n⊢ untrop (x + y) = untrop x",
"usedConstants": [
"Eq.mpr",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
"DistribLattice.toLattice",
"id",
"SemilatticeInf.toMin",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Tropical.Basic | {
"line": 290,
"column": 23
} | {
"line": 290,
"column": 34
} | [
{
"pp": "R : Type u\ninst✝ : LinearOrder R\nx y : Tropical R\nh : y ≤ x\n⊢ untrop (x + y) = untrop y",
"usedConstants": [
"Eq.mpr",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
"DistribLattice.toLattice",
"id",
"inf_eq_right._simp_1",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 704,
"column": 42
} | {
"line": 704,
"column": 58
} | [
{
"pp": "case single\nk : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Monoid G\ninst✝ : MulSemiringAction G k\ng : G\na : k\n⊢ ((single 1 1).sum fun a₁ b₁ ↦ (single g a).sum fun a₂ b₂ ↦ single (a₁ * a₂) (b₁ * a₁ • b₂)) = single g a",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAdd... | sum_single_index | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 708,
"column": 60
} | {
"line": 708,
"column": 76
} | [
{
"pp": "case single\nk : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Monoid G\ninst✝ : MulSemiringAction G k\ng : G\na : k\n⊢ ((single 1 1).sum fun a₂ b₂ ↦ single (g * a₂) (a * g • b₂)) = single g a",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOn... | sum_single_index | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Tropical.BigOperators | {
"line": 49,
"column": 29
} | {
"line": 49,
"column": 40
} | [
{
"pp": "R : Type u_1\ninst✝ : AddCommMonoid R\ns : Multiset R\n⊢ ∀ (a : List R), trop (sum ⟦a⟧) = (map trop ⟦a⟧).prod",
"usedConstants": [
"Multiset.sum",
"Tropical.instCommMonoidTropical",
"Multiset.map",
"Multiset.prod",
"id",
"Quotient.mk",
"Tropical",
"Li... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Tropical.BigOperators | {
"line": 65,
"column": 29
} | {
"line": 65,
"column": 40
} | [
{
"pp": "R : Type u_1\ninst✝ : AddCommMonoid R\ns : Multiset (Tropical R)\n⊢ ∀ (a : List (Tropical R)), untrop (prod ⟦a⟧) = (map untrop ⟦a⟧).sum",
"usedConstants": [
"Multiset.sum",
"Tropical.instCommMonoidTropical",
"Multiset.map",
"Multiset.prod",
"id",
"Quotient.mk",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Tropical.BigOperators | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 35
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝ : ConditionallyCompleteLinearOrder R\ns : Finset S\nf : S → Tropical (WithTop R)\n⊢ untrop (∑ i ∈ s, f i) = ⨅ i, untrop (f ↑i)",
"usedConstants": [
"WithTop.instInfSet",
"Eq.mpr",
"Lattice.toSemilatticeSup",
"iInf",
"Tropical.untrop_in... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 881,
"column": 8
} | {
"line": 881,
"column": 90
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Mul G\ninst✝ : SMulZeroClass G k\nf g : SkewMonoidAlgebra k G\nx : G\ns : Finset (G × G)\nhs : ∀ {p : G × G}, p ∈ s ↔ p.1 * p.2 = x\nF : G × G → k := fun p ↦ if p.1 * p.2 = x then f.coeff p.1 * p.1 • g.coeff p.2 else 0\np : G × G\nhps : p ∈ s\nh... | simp only [Finset.mem_filter, mem_support_iff, not_and, Classical.not_not] at hp ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 906,
"column": 8
} | {
"line": 906,
"column": 90
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Mul G\ninst✝ : SMulZeroClass G k\nf g : SkewMonoidAlgebra k G\nx : G\nthis : ({p | p.1 * p.2 = x} ∩ Function.support fun p ↦ f.coeff p.1 * p.1 • g.coeff p.2).Finite\ns : Finset (G × G) := this.toFinset\nF : G × G → k := fun p ↦ if p.1 * p.2 = x ... | simp only [Finset.mem_filter, mem_support_iff, not_and, Classical.not_not] at hp ⊢ | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 946,
"column": 17
} | {
"line": 946,
"column": 33
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : Mul G\ninst✝ : SMulZeroClass G k\nr : k\ng g' : G\nx : SkewMonoidAlgebra k G\nh : ¬∃ d, g' = g * d\n⊢ ((single g r).sum fun a₁ b₁ ↦ x.sum fun a₂ b₂ ↦ if a₁ * a₂ = g' then b₁ * a₁ • b₂ else 0) = 0",
"usedConstants": [
"Eq.mpr",
"N... | sum_single_index | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 1119,
"column": 2
} | {
"line": 1119,
"column": 13
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝³ : Semiring k\ninst✝² : Monoid G\ninst✝¹ : MulSemiringAction G k\ninst✝ : Nontrivial k\na b : G\nh : (single a 1).toFinsupp = (single b 1).toFinsupp\n⊢ a = b",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.SkewMonoidAlgebra.Basic | {
"line": 1143,
"column": 57
} | {
"line": 1143,
"column": 68
} | [
{
"pp": "k : Type u_1\nG : Type u_2\ninst✝² : Semiring k\ninst✝¹ : MulOneClass G\nR : Type u_3\ninst✝ : Semiring R\nf : k →+* R\ng : G →* R\nc : k\nφ : SkewMonoidAlgebra k G\nthis : (liftNC ↑f ⇑g).comp ((smulAddHom k (SkewMonoidAlgebra k G)) c) = (AddMonoidHom.mulLeft (f c)).comp (liftNC ↑f ⇑g)\n⊢ (liftNC ↑f ⇑g... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.LinearMap | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 13
} | [
{
"pp": "R : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝¹³ : Semiring R\ninst✝¹² : InvolutiveStar R\ninst✝¹¹ : AddCommMonoid E\ninst✝¹⁰ : Module R E\ninst✝⁹ : StarAddMonoid E\ninst✝⁸ : StarModule R E\ninst✝⁷ : AddCommMonoid F\ninst✝⁶ : Module R F\ninst✝⁵ : StarAddMonoid F\ninst✝⁴ : StarModule R F\nG : Type u_4\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.LinearMap | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 13
} | [
{
"pp": "R : Type u_5\nE : Type u_6\nF : Type u_7\nG : Type u_8\nH : Type u_9\ninst✝¹⁷ : CommSemiring R\ninst✝¹⁶ : StarRing R\ninst✝¹⁵ : AddCommMonoid E\ninst✝¹⁴ : StarAddMonoid E\ninst✝¹³ : Module R E\ninst✝¹² : StarModule R E\ninst✝¹¹ : AddCommMonoid F\ninst✝¹⁰ : StarAddMonoid F\ninst✝⁹ : Module R F\ninst✝⁸ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.LinearMap | {
"line": 210,
"column": 13
} | {
"line": 210,
"column": 24
} | [
{
"pp": "R : Type u_5\ninst✝⁸ : CommSemiring R\ninst✝⁷ : StarRing R\nn : Type u_8\ninst✝⁶ : DecidableEq n\nB : n → Type u_9\ninst✝⁵ : (i : n) → AddCommMonoid (B i)\ninst✝⁴ : (i : n) → Module R (B i)\ninst✝³ : (i : n) → StarAddMonoid (B i)\ninst✝² : ∀ (i : n), StarModule R (B i)\ninst✝¹ : Fintype n\ninst✝ : (i :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Star.LinearMap | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 18
} | [
{
"pp": "R : Type u_1\nE : Type u_2\ninst✝⁵ : Semiring R\ninst✝⁴ : InvolutiveStar R\ninst✝³ : AddCommMonoid E\ninst✝² : Module R E\ninst✝¹ : StarAddMonoid E\ninst✝ : StarModule R E\nf : WithConv (End R E)\nhf : IsUnit f.ofConv\nu : (End R E)ˣ\nhu : ↑u = f.ofConv\nthis : IsUnit (star (toConv ↑u)).ofConv\n⊢ IsUni... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.RingedSpace.Stalks | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 87
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nU : TopCat\nX : PresheafedSpace C\nf : U ⟶ ↑X\nh : IsOpenEmbedding ⇑(ConcreteCategory.hom f)\nV : Opens ↑U\nx : ↑U\nhx : x ∈ V\n⊢ X.presheaf.germ (h.functor.obj V) ((ConcreteCategory.hom f) x) ⋯ ≫ (X.restrictStalkIso h x).inv =\n (X.rest... | rw [← restrictStalkIso_hom_eq_germ, Category.assoc, Iso.hom_inv_id, Category.comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Geometry.RingedSpace.Stalks | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 87
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nU : TopCat\nX : PresheafedSpace C\nf : U ⟶ ↑X\nh : IsOpenEmbedding ⇑(ConcreteCategory.hom f)\nV : Opens ↑U\nx : ↑U\nhx : x ∈ V\n⊢ X.presheaf.germ (h.functor.obj V) ((ConcreteCategory.hom f) x) ⋯ ≫ (X.restrictStalkIso h x).inv =\n (X.rest... | rw [← restrictStalkIso_hom_eq_germ, Category.assoc, Iso.hom_inv_id, Category.comp_id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.RingedSpace.Stalks | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 87
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasColimits C\nU : TopCat\nX : PresheafedSpace C\nf : U ⟶ ↑X\nh : IsOpenEmbedding ⇑(ConcreteCategory.hom f)\nV : Opens ↑U\nx : ↑U\nhx : x ∈ V\n⊢ X.presheaf.germ (h.functor.obj V) ((ConcreteCategory.hom f) x) ⋯ ≫ (X.restrictStalkIso h x).inv =\n (X.rest... | rw [← restrictStalkIso_hom_eq_germ, Category.assoc, Iso.hom_inv_id, Category.comp_id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.RingedSpace.Basic | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 13
} | [
{
"pp": "case h\nX : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj (op U))\nx : ↥U\nh : (ConcreteCategory.hom (X.presheaf.germ U ↑x ⋯)) f = 0\nh1 : (ConcreteCategory.hom (X.presheaf.germ U ↑x ⋯)) f = (ConcreteCategory.hom (X.presheaf.germ U ↑x ⋯)) 0\nV : Opens ↑↑X.toPresheafedSpace\nhv : ↑x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.RingedSpace.Basic | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 13
} | [
{
"pp": "case h\nX : RingedSpace\nU : Opens ↑↑X.toPresheafedSpace\nf : ↑(X.presheaf.obj (op U))\nx : ↑↑X.toPresheafedSpace\nhx : x ∈ U\nh : IsUnit ((ConcreteCategory.hom (X.presheaf.germ U x hx)) f)\nV : Opens ↑↑X.toPresheafedSpace\nhxV : x ∈ V\ng : ToType (X.presheaf.obj (op V))\nW : Opens ↑↑X.toPresheafedSpac... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.RingedSpace.SheafedSpace | {
"line": 287,
"column": 4
} | {
"line": 287,
"column": 15
} | [
{
"pp": "C : 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 : PreservesFilteredColimits (Cate... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.RingedSpace.PresheafedSpace | {
"line": 220,
"column": 4
} | {
"line": 220,
"column": 84
} | [
{
"pp": "case h.w\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX Y : PresheafedSpace C\nH : ↑X ≅ ↑Y\nα : (Presheaf.pushforward C H.hom).obj X.presheaf ≅ Y.presheaf\nU✝ : Opens ↑↑Y\n⊢ α.inv.app (op U✝) ≫\n X.presheaf.map (eqToHom ⋯) ≫\n ((Presheaf.pushforward C H.hom).obj X.presheaf).map ((eqToHom ⋯... | simp only [eqToHom_map, eqToHom_app, eqToHom_trans_assoc, eqToHom_refl, id_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.RingedSpace.PresheafedSpace | {
"line": 232,
"column": 4
} | {
"line": 232,
"column": 15
} | [
{
"pp": "case w\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX Y : PresheafedSpace C\nH : X ≅ Y\nU : Opens ↑↑Y\n⊢ H.hom.c.app (op U) ≫ (Presheaf.pushforwardToOfIso ((forget C).mapIso H).symm H.inv.c).app (op U) =\n (𝟙 Y.presheaf).app (op U)",
"usedConstants": [
"CategoryTheory.Functor.op",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.RingedSpace.PresheafedSpace | {
"line": 310,
"column": 4
} | {
"line": 310,
"column": 15
} | [
{
"pp": "case right_cancellation.h.w\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nU : TopCat\nX : PresheafedSpace C\nf : U ⟶ ↑X\nhf : IsOpenEmbedding ⇑(ConcreteCategory.hom f)\nthis✝¹ : Mono f\nZ : PresheafedSpace C\ng₁ g₂ : Z ⟶ X.restrict hf\neq : g₁ ≫ X.ofRestrict hf = g₂ ≫ X.ofRestrict hf\nV : Opens ↑↑(X.re... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Sheaves.LocalPredicate | {
"line": 115,
"column": 6
} | {
"line": 115,
"column": 17
} | [
{
"pp": "X : TopCat\nT✝ : ↑X → Type u_1\nT : Type ?u.4489\ninst✝ : TopologicalSpace T\nU : Opens ↑X\nf : ↥U → T\nx : ↥U\nV : Opens ↑X\nm : ↑x ∈ V\ni : V ⟶ U\nw : ContinuousAt (fun x ↦ f (i x)) ⟨↑x, m⟩\n⊢ ContinuousAt f x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Spec | {
"line": 178,
"column": 4
} | {
"line": 178,
"column": 15
} | [
{
"pp": "case h.a\nX : RingedSpace\nR : CommRingCat\nα β : X ⟶ sheafedSpaceObj R\nw : α.hom.base = β.hom.base\nh :\n ∀ (r : ↑R),\n let U := PrimeSpectrum.basicOpen r;\n (CommRingCat.ofHom (algebraMap (↑R) ((structureSheafInType ↑R ↑R).obj.obj (op U))) ≫ α.hom.c.app (op U)) ≫\n X.presheaf.map (eqTo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Scheme | {
"line": 247,
"column": 42
} | {
"line": 247,
"column": 53
} | [
{
"pp": "X Y : Scheme\ntoLRSHom'✝¹ toLRSHom'✝ : X.Hom Y.toLocallyRingedSpace\nh_base : { toLRSHom' := toLRSHom'✝¹ }.base = { toLRSHom' := toLRSHom'✝ }.base\nh_app :\n ∀ (U : Y.Opens),\n { toLRSHom' := toLRSHom'✝¹ }.app U ≫ X.presheaf.map (eqToHom ⋯).op = { toLRSHom' := toLRSHom'✝ }.app U\nU : TopologicalSpa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Scheme | {
"line": 838,
"column": 2
} | {
"line": 838,
"column": 66
} | [
{
"pp": "X : Scheme\nU : X.Opens\nι : Type u_1\nf : ι → Set ↑Γ(X, U)\n⊢ X.zeroLocus (⋃ i, f i) = ⋂ i, X.zeroLocus (f i)",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"Opposite",
"CommRingCat.carrier",
"AlgebraicGeometry.Pres... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Scheme | {
"line": 846,
"column": 18
} | {
"line": 846,
"column": 29
} | [
{
"pp": "X : Scheme\nU : X.Opens\nI : Ideal ↑Γ(X, U)\nx : ↥X\nH : ∀ f ∈ I, x ∉ X.basicOpen f\nf : ↑Γ(X, U)\nhx : x ∈ X.basicOpen f\nhn : f ^ 0 ∈ I\n⊢ f ∈ I",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 13
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nH : IsOpenImmersion f\nU : X.Opens\n⊢ f ''ᵁ U ≤ opensRange f",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 13
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nH : IsOpenImmersion f\nU V : X.Opens\nhUV : (fun x ↦ f ''ᵁ x) U = (fun x ↦ f ''ᵁ x) V\n⊢ U = V",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 614,
"column": 4
} | {
"line": 614,
"column": 19
} | [
{
"pp": "X Y Z : Scheme\nf : X ⟶ Z\ng : Y ⟶ Z\nH : IsOpenImmersion f\n⊢ (Scheme.Hom.opensRange f).carrier ∩ Set.range ⇑g = Set.range ⇑g ∩ Set.range ⇑f",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.PresheafedSpace.carrier",
"congrArg",
"CategoryTheory.ConcreteCategory.hom",
... | Set.inter_comm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.OpenImmersion | {
"line": 706,
"column": 8
} | {
"line": 706,
"column": 39
} | [
{
"pp": "U V X Y : Scheme\ng : U ⟶ V\niU : U ⟶ X\niV : V ⟶ Y\nf : X ⟶ Y\ninst✝¹ : IsOpenImmersion iU\ninst✝ : IsOpenImmersion iV\nH : iU ≫ f = g ≫ iV\nH' : f ⁻¹ᵁ Scheme.Hom.opensRange iV = Scheme.Hom.opensRange iU\n⊢ Set.range ⇑(pullback.snd iV f) = Set.range ⇑iU",
"usedConstants": [
"Eq.mpr",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Cover.MorphismProperty | {
"line": 217,
"column": 4
} | {
"line": 217,
"column": 18
} | [
{
"pp": "K : Precoverage Scheme\nX✝ Y Z : Scheme\n𝒰✝ : Cover K X✝\nf : X✝ ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰✝.I₀), HasPullback (𝒰✝.f x ≫ f) g\nP Q : MorphismProperty Scheme\nX : Scheme\n𝒰 : AffineCover P X\nx : ↥X\ny : ↥(Spec (𝒰.X (𝒰.idx x)))\nhy : (𝒰.f (𝒰.idx x)) y = x\n⊢ ∃ i, x ∈ Set.range ⇑({ I₀ := 𝒰.... | use 𝒰.idx x, y | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Geometry.RingedSpace.OpenImmersion | {
"line": 332,
"column": 8
} | {
"line": 336,
"column": 45
} | [
{
"pp": "case op.op\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\nunop✝¹ unop✝ : Opens ↑↑X\ni : op unop✝¹ ⟶ op unop✝\n⊢ X.presheaf.map i ≫\n invApp f (unop (op uno... | simp only [(inv_naturality_assoc), restrict_carrier, restrict_presheaf,
TopCat.Presheaf.pushforward_obj_obj, Functor.comp_obj, Functor.op_obj,
TopCat.Presheaf.pushforward_obj_map, Functor.comp_map, Functor.op_map, Quiver.Hom.unop_op,
NatTrans.naturality_assoc, TopCat.Presheaf.pushforward_o... | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.RingedSpace.OpenImmersion | {
"line": 343,
"column": 4
} | {
"line": 343,
"column": 15
} | [
{
"pp": "case refine_1\nC : Type u\ninst✝ : Category.{v, u} C\nX Y Z : PresheafedSpace C\nf : X ⟶ Z\nhf : IsOpenImmersion f\ng : Y ⟶ Z\n⊢ (pullbackConeOfLeftFst f g ≫ f).base = (Y.ofRestrict ⋯ ≫ g).base",
"usedConstants": [
"CategoryTheory.Limits.hasFiniteLimits_of_hasLimits",
"CategoryTheory.Li... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Cover.Open | {
"line": 234,
"column": 4
} | {
"line": 234,
"column": 15
} | [
{
"pp": "case hcover\nX : Scheme\nU : X.Opens\nf g : ↑Γ(X, U)\n𝒰 : X.OpenCover\nh : ∀ (i : 𝒰.I₀), (ConcreteCategory.hom (Hom.app (𝒰.f i) U)) f = (ConcreteCategory.hom (Hom.app (𝒰.f i) U)) g\nx : ↥X\nhx : x ∈ U\n⊢ ∃ x_1 y, (𝒰.f (Cover.idx 𝒰 x_1)) y = x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Cover.Open | {
"line": 269,
"column": 10
} | {
"line": 274,
"column": 20
} | [
{
"pp": "X Y Z : Scheme\n𝒰 : X.OpenCover\nf : X ⟶ Z\ng : Y ⟶ Z\ninst✝ : ∀ (x : 𝒰.I₀), HasPullback (𝒰.f x ≫ f) g\nR : CommRingCat\n⊢ { I₀ := ↑R, X := fun r ↦ Spec (CommRingCat.of (Localization.Away r)),\n f := fun r ↦ Spec.map (CommRingCat.ofHom (algebraMap (↑R) (Localization.Away r))) }.presieve₀ ∈\n ... | by
rw [presieve₀_mem_precoverage_iff]
refine ⟨fun x ↦ ⟨1, ?_⟩, AlgebraicGeometry.Scheme.isOpenImmersion_SpecMap_localizationAway⟩
rw [Set.range_eq_univ.mpr ((TopCat.epi_iff_surjective _).mp _)]
· exact trivial
· infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Restrict | {
"line": 236,
"column": 35
} | {
"line": 236,
"column": 46
} | [
{
"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.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"SetLike.coe_subset_coe._simp_1",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.RingedSpace.OpenImmersion | {
"line": 1278,
"column": 6
} | {
"line": 1278,
"column": 75
} | [
{
"pp": "X Y : LocallyRingedSpace\nf : X ⟶ Y\nH : IsOpenImmersion f\ninst✝ : Epi f.base\n⊢ IsIso f",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.IsIso",
"AlgebraicGeometry.SheafedSpace",
"congrArg",
"CommRingCat",
"CommRingCat.instCategory",
"id",
"AlgebraicG... | ← isIso_iff_of_reflects_iso _ LocallyRingedSpace.forgetToSheafedSpace | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.Restrict | {
"line": 274,
"column": 27
} | {
"line": 274,
"column": 55
} | [
{
"pp": "X : Scheme\nU V : X.Opens\ne : U ≤ V\nW : (↑V).Opens\ny : ↥X\nhyU : y ∈ U\nhyW : ⟨y, hyU⟩ ∈ ↑(X.homOfLE e ⁻¹ᵁ W)\n⊢ ⟨y, ⋯⟩ ∈ ↑W",
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"AlgebraicGeometry.PresheafedSpace.carrier",
"CommRingCat",
"TopologicalSpace.Opens... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Restrict | {
"line": 276,
"column": 26
} | {
"line": 276,
"column": 54
} | [
{
"pp": "X : Scheme\nU V : X.Opens\ne : U ≤ V\nW : (↑V).Opens\ny : ↥↑V\nhyW : y ∈ ↑W\nhyU : (ConcreteCategory.hom (LocallyRingedSpace.Hom.toShHom (Hom.toLRSHom V.ι)).hom.base) y ∈ ↑U\n⊢ ⟨↑y, hyU⟩ ∈ ↑(X.homOfLE e ⁻¹ᵁ W)",
"usedConstants": [
"Eq.mpr",
"SetLike.mem_coe._simp_1",
"AlgebraicGeo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Submonoid.Inverses | {
"line": 87,
"column": 2
} | {
"line": 93,
"column": 62
} | [
{
"pp": "M : Type u_1\ninst✝ : Monoid M\nS : Submonoid M\nhS : S ≤ IsUnit.submonoid M\n⊢ S.leftInv.leftInv = S",
"usedConstants": [
"Units.val",
"Eq.mpr",
"Monoid.toMulOneClass",
"congrArg",
"InvolutiveInv.toInv",
"Group.toDivisionMonoid",
"Membership.mem",
"U... | refine le_antisymm S.leftInv_leftInv_le ?_
intro x hx
have : x = ((hS hx).unit⁻¹⁻¹ : Mˣ) := by
rw [inv_inv (hS hx).unit]
rfl
rw [this]
exact S.leftInv.unit_mem_leftInv _ (S.unit_mem_leftInv _ hx) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Submonoid.Inverses | {
"line": 87,
"column": 2
} | {
"line": 93,
"column": 62
} | [
{
"pp": "M : Type u_1\ninst✝ : Monoid M\nS : Submonoid M\nhS : S ≤ IsUnit.submonoid M\n⊢ S.leftInv.leftInv = S",
"usedConstants": [
"Units.val",
"Eq.mpr",
"Monoid.toMulOneClass",
"congrArg",
"InvolutiveInv.toInv",
"Group.toDivisionMonoid",
"Membership.mem",
"U... | refine le_antisymm S.leftInv_leftInv_le ?_
intro x hx
have : x = ((hS hx).unit⁻¹⁻¹ : Mˣ) := by
rw [inv_inv (hS hx).unit]
rfl
rw [this]
exact S.leftInv.unit_mem_leftInv _ (S.unit_mem_leftInv _ hx) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Submonoid.Inverses | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 56
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\nS : Submonoid M\nhS : S ≤ IsUnit.submonoid M\nx : ↥S.leftInv\n⊢ ↑((S.leftInvEquiv hS) x) * ↑x = 1",
"usedConstants": [
"MulOne.toOne",
"MulEquiv.instEquivLike",
"Submonoid.mul",
"HMul.hMul",
"Monoid.toMulOneClass",
"Membership.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Submonoid.Inverses | {
"line": 181,
"column": 2
} | {
"line": 181,
"column": 61
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoid M\nS : Submonoid M\nhS : S ≤ IsUnit.submonoid M\nx : ↥S\n⊢ ↑x * ↑((S.leftInvEquiv hS).symm x) = 1",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"MulEquiv.instEquivLike",
"Submonoid.mul",
"HMul.hMul",
"Monoid.toMulOneClass",
... | convert! S.leftInvEquiv_mul hS ((S.leftInvEquiv hS).symm x) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.AlgebraicGeometry.Restrict | {
"line": 603,
"column": 2
} | {
"line": 603,
"column": 60
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nU : Y.Opens\nV : (↑U).Opens\n⊢ Hom.app (f ∣_ U) (U.ι ⁻¹ᵁ U.ι ''ᵁ V) = Hom.app f (U.ι ''ᵁ U.ι ⁻¹ᵁ U.ι ''ᵁ V) ≫ X.presheaf.map (eqToHom ⋯).op",
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.opensFunctor",
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Restrict | {
"line": 796,
"column": 6
} | {
"line": 796,
"column": 17
} | [
{
"pp": "case refine_2.refine_1\nX Y S T : Scheme\nf : T ⟶ S\ng : Y ⟶ X\niX : X ⟶ S\niY : Y ⟶ T\nH : IsPullback g iY iX f\nUS : S.Opens\nUT : T.Opens\nUX : X.Opens\nhUST : UT ≤ f ⁻¹ᵁ US\nhUSX : UX ≤ iX ⁻¹ᵁ US\nUY : Y.Opens\nhUY : UY = g ⁻¹ᵁ UX ⊓ iY ⁻¹ᵁ UT\n⊢ IsPullback (g ∣_ UX) (resLE iY (f ⁻¹ᵁ US) (g ⁻¹ᵁ UX) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 41
} | [
{
"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\nP : ↑(PrimeSpectrum.Top R)\nU : Opens ↑(PrimeSpectrum.Top R)\nr : ↥(sectionsSubalgebra R U)\na : (x : ↥U) → Localizations M ↑x\nha✝ : a ∈ (sectionsSubmodule M U).carrier\nx : ↥U\... | obtain ⟨hrsy, hry⟩ := hr ⟨y.1, y.2.1⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 291,
"column": 2
} | {
"line": 292,
"column": 43
} | [
{
"pp": "R 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 : R\nhfg : ∀ (x ... | refine ⟨g' * g, ?_, ?_, g' • f, Subtype.ext <| funext fun ⟨y, hy⟩ ↦ ?_⟩ <;>
simp only [PrimeSpectrum.basicOpen_mul] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 322,
"column": 2
} | {
"line": 322,
"column": 13
} | [
{
"pp": "R A : Type u\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nU : Opens ↑(PrimeSpectrum.Top R)\n⊢ const 1 1 U ⋯ = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 433,
"column": 31
} | {
"line": 433,
"column": 66
} | [
{
"pp": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : R\na : M\nb : ↥(Submonoid.powers f)\nc : M\nd : ↥(Submonoid.powers f)\nh_eq : (toBasicOpenₗ R M f) (LocalizedModule.mk a b) = (toBasicOpenₗ R M f) (LocalizedModule.mk c d)\nn : ℕ\nhn : f ^ n ∈ ⊥.colon {d • a - b • c}\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.LocallyDirected | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 13
} | [
{
"pp": "case cond\nJ : Type u_1\ninst✝ : Category.{v_1, u_1} J\nF : Discrete J ⥤ Type u_2\ni : J\n⊢ ∀ (xi xj : F.obj { as := i }),\n (ConcreteCategory.hom (F.map { down := { down := ⋯ } })) xi =\n (ConcreteCategory.hom (F.map { down := { down := ⋯ } })) xj →\n ∃ l fli flj x, (ConcreteCategory.ho... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.LocallyDirected | {
"line": 62,
"column": 4
} | {
"line": 62,
"column": 15
} | [
{
"pp": "case cond.id.id\nJ : Type u_1\ninst✝¹ : Category.{v_1, u_1} J\nF : WidePushoutShape J ⥤ Type u_2\ninst✝ : ∀ (i : J), Mono (F.map (WidePushoutShape.Hom.init i))\ni : WidePushoutShape J\n⊢ ∀ (xi xj : F.obj i),\n (ConcreteCategory.hom (F.map (WidePushoutShape.Hom.id i))) xi =\n (ConcreteCategory... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equalizer | {
"line": 32,
"column": 38
} | {
"line": 32,
"column": 49
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasEqualizer f g\ninst✝ : HasBinaryProduct Y Y\ns : PullbackCone (prod.lift f g) (prod.lift (𝟙 Y) (𝟙 Y))\n⊢ s.fst ≫ f = s.snd",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equalizer | {
"line": 33,
"column": 38
} | {
"line": 33,
"column": 49
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasEqualizer f g\ninst✝ : HasBinaryProduct Y Y\ns : PullbackCone (prod.lift f g) (prod.lift (𝟙 Y) (𝟙 Y))\nH₁ : s.fst ≫ f = s.snd\n⊢ s.fst ≫ g = s.snd",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equalizer | {
"line": 36,
"column": 21
} | {
"line": 36,
"column": 32
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasEqualizer f g\ninst✝ : HasBinaryProduct Y Y\ns : PullbackCone (prod.lift f g) (prod.lift (𝟙 Y) (𝟙 Y))\n⊢ equalizer.lift s.fst ⋯ ≫ equalizer.ι f g ≫ f = s.snd",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 476,
"column": 15
} | {
"line": 476,
"column": 26
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 476,
"column": 50
} | {
"line": 476,
"column": 61
} | [
{
"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... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equalizer | {
"line": 47,
"column": 38
} | {
"line": 47,
"column": 49
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasCoequalizer f g\ninst✝ : HasBinaryCoproduct X X\ns : PushoutCocone (coprod.desc f g) (coprod.desc (𝟙 X) (𝟙 X))\n⊢ f ≫ s.inl = s.inr",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equalizer | {
"line": 48,
"column": 38
} | {
"line": 48,
"column": 49
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasCoequalizer f g\ninst✝ : HasBinaryCoproduct X X\ns : PushoutCocone (coprod.desc f g) (coprod.desc (𝟙 X) (𝟙 X))\nH₁ : f ≫ s.inl = s.inr\n⊢ g ≫ s.inl = s.inr",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.Equalizer | {
"line": 51,
"column": 21
} | {
"line": 51,
"column": 32
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nX Y : C\nf g : X ⟶ Y\ninst✝¹ : HasCoequalizer f g\ninst✝ : HasBinaryCoproduct X X\ns : PushoutCocone (coprod.desc f g) (coprod.desc (𝟙 X) (𝟙 X))\n⊢ (f ≫ coequalizer.π f g) ≫ coequalizer.desc s.inl ⋯ = s.inr",
"usedConstants": [
"Eq.mpr",
"Catego... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 497,
"column": 6
} | {
"line": 498,
"column": 66
} | [
{
"pp": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : R\ns : (structureSheafInType R M).obj.obj (op (basicOpen f))\nι : Type u\nw✝ : Fintype ι\na : ι → M\nb : ι → R\nibU : ∀ (i : ι), basicOpen (b i) ≤ basicOpen f\niU : basicOpen f ≤ ⨆ i, basicOpen (b i)\nhab : ∀ (i j : ι),... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 501,
"column": 24
} | {
"line": 501,
"column": 35
} | [
{
"pp": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : R\ns : (structureSheafInType R M).obj.obj (op (basicOpen f))\nι : Type u\nw✝ : Fintype ι\na : ι → M\nb : ι → R\nibU : ∀ (i : ι), basicOpen (b i) ≤ basicOpen f\niU : basicOpen f ≤ ⨆ i, basicOpen (b i)\nhab : ∀ (i j : ι),... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 530,
"column": 25
} | {
"line": 530,
"column": 36
} | [
{
"pp": "R M : Type u\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nthis : IsLocalizedModule ⊥ (toOpenₗ R M ⊤)\nx y : M\ne : (toOpenₗ R M ⊤) x = (toOpenₗ R M ⊤) y\n⊢ x = y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.MorphismProperty.Local | {
"line": 204,
"column": 2
} | {
"line": 204,
"column": 20
} | [
{
"pp": "C : Type u\ninst✝⁴ : Category.{v, u} C\ninst✝³ : HasEqualizers C\ninst✝² : HasPullbacks C\nX Y S : C\nf g : X ⟶ Y\ns : X ⟶ S\nt : Y ⟶ S\nhf : f ≫ t = s\nhg : g ≫ t = s\nJ : Precoverage C\n𝒰 : J.ZeroHypercover S\ninst✝¹ : J.IsStableUnderBaseChange\ninst✝ : (MorphismProperty.isomorphisms C).IsLocalAtTar... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 979,
"column": 24
} | {
"line": 979,
"column": 51
} | [
{
"pp": "R M A : Type u\ninst✝⁷ : CommRing R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\nS : Type u\ninst✝² : CommRing S\nN : Type u\ninst✝¹ : AddCommGroup N\ninst✝ : Module S N\nσ : R →+* S\nf : M →ₛₗ[σ] N\ny : ↑(PrimeSpectrum.Top S)\nthis✝¹ : Module R N := Module.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.GlueData | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 13
} | [
{
"pp": "C : Type u₁\ninst✝ : Category.{v, u₁} C\nD : GlueData C\ni j : D.J\neq :\n (pullbackSymmetry (D.f i i) (D.f i j)).hom = pullback.snd (D.f i i) (D.f i j) ≫ inv (pullback.fst (D.f i j) (D.f i i))\nthis :\n D.t i j ≫ D.t j i =\n (inv (pullback.fst (D.f i j) (D.f i i)) ≫ 𝟙 (pullback (D.f i j) (D.f i ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.GlueData | {
"line": 121,
"column": 49
} | {
"line": 121,
"column": 60
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v, u₁} C\nC' : Type u₂\ninst✝ : Category.{v, u₂} C'\nD : GlueData C\ni j k : D.J\n⊢ (D.t' j k i ≫ D.t' k i j) ≫ D.t' i j k = 𝟙 (pullback (D.f j k) (D.f j i))",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Category.assoc",
"CategoryTheory.Limits.pul... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.GlueData | {
"line": 228,
"column": 20
} | {
"line": 228,
"column": 49
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v, u₁} C\nC' : Type u₂\ninst✝¹ : Category.{v, u₂} C'\nD : GlueData C\nF : C ⥤ C'\ninst✝ : ∀ (i j k : D.J), PreservesLimit (cospan (D.f i j) (D.f i k)) F\ni j k : D.J\n⊢ ((PreservesPullback.iso F (D.f i j) (D.f i k)).inv ≫\n F.map (D.t' i j k) ≫ (PreservesPullback.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.StructureSheaf | {
"line": 1184,
"column": 4
} | {
"line": 1184,
"column": 41
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\nS : Type u\ninst✝ : CommRing S\nf : R →+* S\nx : R\n⊢ (comap f (basicOpen x) (basicOpen (f x)) ⋯).comp (algebraMap R ↑((structureSheaf R).obj.obj (op (basicOpen x)))) =\n (IsLocalization.map (↑((structureSheaf S).obj.obj (op (basicOpen (f x))))) f ⋯).comp\n (alg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits | {
"line": 145,
"column": 4
} | {
"line": 145,
"column": 47
} | [
{
"pp": "case inst\nX Y : LocallyRingedSpace\nf g : X ⟶ Y\nU : Opens ↑↑(coequalizer (Hom.toShHom f) (Hom.toShHom g)).toPresheafedSpace\nthis✝ :\n coequalizer.π f.toHom g.toHom ≫\n (PreservesCoequalizer.iso SheafedSpace.forgetToPresheafedSpace (Hom.toShHom f) (Hom.toShHom g)).hom =\n (coequalizer.π (Hom... | · apply CommRingCat.equalizer_ι_isLocalHom' | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits | {
"line": 182,
"column": 2
} | {
"line": 183,
"column": 61
} | [
{
"pp": "X Y : LocallyRingedSpace\nf g : X ⟶ Y\nU : Opens ↑↑(coequalizer (Hom.toShHom f) (Hom.toShHom g)).toPresheafedSpace\ns : ↑((coequalizer (Hom.toShHom f) (Hom.toShHom g)).presheaf.obj (op U))\n⊢ ⇑(ConcreteCategory.hom (coequalizer.π (Hom.toShHom f) (Hom.toShHom g)).hom.base) ⁻¹'\n ⇑(ConcreteCategory.... | fapply Types.coequalizer_preimage_image_eq_of_preimage_eq (↾f.base)
(↾g.base) (↾(coequalizer.π f.toShHom g.toShHom).hom.base) | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_tacticFapply__1 | Batteries.Tactic.tacticFapply_ |
Mathlib.Geometry.RingedSpace.LocallyRingedSpace.HasColimits | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 50
} | [
{
"pp": "case H\nX Y : LocallyRingedSpace\nf g : X ⟶ Y\nU : Opens ↑↑(coequalizer (Hom.toShHom f) (Hom.toShHom g)).toPresheafedSpace\ns : ↑((coequalizer (Hom.toShHom f) (Hom.toShHom g)).presheaf.obj (op U))\n⊢ unop ((Opens.map (Hom.toShHom g ≫ coequalizer.π (Hom.toShHom f) (Hom.toShHom g)).hom.base).op.obj (op U... | rw [coequalizer.condition f.toShHom g.toShHom] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Gluing | {
"line": 407,
"column": 4
} | {
"line": 407,
"column": 42
} | [
{
"pp": "case right.right\nα : Type u\ninst✝ : TopologicalSpace α\nJ : Type u\nU : J → Opens α\ns : Set ↑(ofOpenSubsets U).glued\nhs : ∀ (i : (ofOpenSubsets U).J), IsOpen (⇑(ConcreteCategory.hom ((ofOpenSubsets U).ι i)) ⁻¹' s)\ni : (ofOpenSubsets U).J\nx : ↑(of α)\nhx' : x ∈ U i\nhx : (ConcreteCategory.hom ((of... | refine ⟨Set.mem_image_of_mem _ hx, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 624,
"column": 41
} | {
"line": 624,
"column": 79
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : IsAffineOpen U\nV : X.Opens\nx : ↥V\nh : ↑x ∈ U\nthis : IsAffine ↑U\nr : ↑Γ(↑U, ⊤)\nh₁ : ⟨↑x, h⟩ ∈ ↑((↑U).basicOpen r)\nh₂ : (↑U).basicOpen r ≤ U.ι ⁻¹ᵁ V\n⊢ ↑x ∈ X.basicOpen r",
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 626,
"column": 4
} | {
"line": 626,
"column": 40
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : IsAffineOpen U\nV : X.Opens\nx : ↥V\nh : ↑x ∈ U\nthis : IsAffine ↑U\nr : ↑Γ(↑U, ⊤)\nh₂ : (↑U).basicOpen r ≤ U.ι ⁻¹ᵁ V\nh₁ : ↑x ∈ X.basicOpen r\n⊢ X.basicOpen r ≤ V",
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"La... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 775,
"column": 2
} | {
"line": 781,
"column": 6
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nx : ↥X\nU : Y.Opens\nhU : IsAffineOpen U\nV : X.Opens\nhV : IsAffineOpen V\nhVU : V ≤ f ⁻¹ᵁ U\nhx : x ∈ V\n⊢ PrimeSpectrum.comap (CommRingCat.Hom.hom (Scheme.Hom.appLE f U V hVU)) (hV.primeIdealOf ⟨x, hx⟩) =\n hU.primeIdealOf ⟨f x, ⋯⟩",
"usedConstants": [
"Algebrai... | change Spec.map (f.appLE U V hVU) (hV.primeIdealOf ⟨x, hx⟩) = (hU.primeIdealOf ⟨f x, hVU hx⟩)
simp only [IsAffineOpen.primeIdealOf, ← Scheme.Hom.comp_apply, IsAffineOpen.isoSpec_hom,
Scheme.Opens.toSpecΓ_SpecMap_appLE]
simp only [Scheme.Hom.comp_apply]
congr 1
apply Subtype.ext
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 775,
"column": 2
} | {
"line": 781,
"column": 6
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\nx : ↥X\nU : Y.Opens\nhU : IsAffineOpen U\nV : X.Opens\nhV : IsAffineOpen V\nhVU : V ≤ f ⁻¹ᵁ U\nhx : x ∈ V\n⊢ PrimeSpectrum.comap (CommRingCat.Hom.hom (Scheme.Hom.appLE f U V hVU)) (hV.primeIdealOf ⟨x, hx⟩) =\n hU.primeIdealOf ⟨f x, ⋯⟩",
"usedConstants": [
"Algebrai... | change Spec.map (f.appLE U V hVU) (hV.primeIdealOf ⟨x, hx⟩) = (hU.primeIdealOf ⟨f x, hVU hx⟩)
simp only [IsAffineOpen.primeIdealOf, ← Scheme.Hom.comp_apply, IsAffineOpen.isoSpec_hom,
Scheme.Opens.toSpecΓ_SpecMap_appLE]
simp only [Scheme.Hom.comp_apply]
congr 1
apply Subtype.ext
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 787,
"column": 4
} | {
"line": 788,
"column": 11
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : IsAffineOpen U\nx : ↥U\nhx : IsClosed {↑x}\n⊢ IsClosed {x}",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 841,
"column": 2
} | {
"line": 841,
"column": 12
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : IsAffineOpen U\ns : ↑Γ(X, U)\nI : Ideal ↑Γ(X, U)\nH :\n ∀ (x : ↥X) (h : x ∈ U),\n (ConcreteCategory.hom (X.presheaf.germ U x h)) s ∈ Ideal.map (CommRingCat.Hom.hom (X.presheaf.germ U x h)) I\nthis✝ : (x : ↥(Spec Γ(X, U))) → Algebra ↑Γ(X, U) ↑(X.presheaf.stalk (hU.fromS... | intro P hP | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Monoidal.Cartesian.Over | {
"line": 36,
"column": 42
} | {
"line": 36,
"column": 53
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasPullbacks C\nX : C\nY : Over X\nm : Y ⟶ mk (𝟙 X)\n⊢ Hom.left m = Hom.left ((fun Y ↦ homMk Y.hom ⋯) Y)",
"usedConstants": [
"CategoryTheory.Over",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 1060,
"column": 43
} | {
"line": 1060,
"column": 77
} | [
{
"pp": "R S : CommRingCat\nX : Scheme\nφ : R ⟶ S\nhφ : Function.Injective ⇑(ConcreteCategory.hom φ)\nf g : Spec R ⟶ X\nU : X.Opens\nhU : IsAffineOpen U\nhUf : f ⁻¹ᵁ U = ⊤\nhUg : g ⁻¹ᵁ U = ⊤\nH : Spec.map φ ≫ f = Spec.map φ ≫ g\nthis : Mono φ\n⊢ Set.range ⇑f ⊆ Set.range ⇑U.ι",
"usedConstants": [
"Eq.m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 1061,
"column": 41
} | {
"line": 1061,
"column": 75
} | [
{
"pp": "R S : CommRingCat\nX : Scheme\nφ : R ⟶ S\nhφ : Function.Injective ⇑(ConcreteCategory.hom φ)\nf g : Spec R ⟶ X\nU : X.Opens\nhU : IsAffineOpen U\nhUf : f ⁻¹ᵁ U = ⊤\nhUg : g ⁻¹ᵁ U = ⊤\nH : Spec.map φ ≫ f = Spec.map φ ≫ g\nthis : Mono φ\n⊢ Set.range ⇑g ⊆ Set.range ⇑U.ι",
"usedConstants": [
"Eq.m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 1111,
"column": 4
} | {
"line": 1111,
"column": 84
} | [
{
"pp": "case refine_1\nX : Scheme\ninst✝ : IsAffine X\ns : Set ↥X\nhs : IsClosed s\nZ : Set ↥(Spec Γ(X, ⊤)) := X.toΓSpecFun '' s\nhZ : IsClosed Z\n⊢ ∃ I, s = X.zeroLocus ↑I",
"usedConstants": [
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"Lattice.toSemilatticeSup",
... | obtain ⟨I, (hI : Z = _)⟩ := (PrimeSpectrum.isClosed_iff_zeroLocus_ideal _).mp hZ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 1144,
"column": 4
} | {
"line": 1144,
"column": 57
} | [
{
"pp": "X : Scheme\nU : X.Opens\nhU : IsAffineOpen U\ns : Set ↑Γ(X, U)\nx : ↥(Spec Γ(X, U))\nthis : (∀ f ∈ s, ¬f ∉ x.asIdeal) ↔ s ⊆ ↑x.asIdeal\n⊢ x ∈ ⇑hU.fromSpec ⁻¹' X.zeroLocus s ↔ x ∈ PrimeSpectrum.zeroLocus s",
"usedConstants": [
"AlgebraicGeometry.Scheme.Hom.opensFunctor",
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 1161,
"column": 6
} | {
"line": 1161,
"column": 23
} | [
{
"pp": "case pos\nX : Scheme\nU : X.Opens\nI J : Ideal ↑Γ(X, U)\nthis : U.carrier ↓∩ X.zeroLocus ↑(I ⊓ J) = U.carrier ↓∩ (X.zeroLocus ↑I ∪ X.zeroLocus ↑J)\nx : ↥X\nhxU : x ∈ U\n⊢ x ∈ X.zeroLocus ↑(I ⊓ J) ↔ x ∈ X.zeroLocus ↑I ∪ X.zeroLocus ↑J",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Al... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 1191,
"column": 2
} | {
"line": 1191,
"column": 13
} | [
{
"pp": "X : Scheme\nU : X.Opens\nι : Type u_1\nI : ι → Ideal ↑Γ(X, U)\ninst✝ : Finite ι\n⊢ X.zeroLocus ↑(⨅ i, I i) = (⋃ i, X.zeroLocus ↑(I i)) ∪ (↑U)ᶜ",
"usedConstants": [
"Eq.mpr",
"Submodule",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"iInf",
"Semi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.AffineScheme | {
"line": 1197,
"column": 2
} | {
"line": 1197,
"column": 13
} | [
{
"pp": "X : Scheme\nU : X.Opens\nι : Type u_1\nI : ι → Ideal ↑Γ(X, U)\ninst✝¹ : Finite ι\ninst✝ : Nonempty ι\n⊢ X.zeroLocus ↑(⨅ i, I i) = ⋃ i, X.zeroLocus ↑(I i)",
"usedConstants": [
"Eq.mpr",
"Submodule",
"iInf",
"Semiring.toModule",
"Opposite",
"CommRingCat.carrier",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.