module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.SimpleModule.Isotypic | {
"line": 417,
"column": 38
} | {
"line": 417,
"column": 40
} | {
"line": 418,
"column": 4
} | [
{
"pp": "R : Type u_2\nM : Type u\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsSemisimpleModule R M\nm : Submodule R M\nh : m.IsFullyInvariant\nS : Submodule R M\nle : S ≤ m\nx✝¹ : IsSimpleModule R ↥S\nS' : Submodule R M\nx✝ : S' ∈ {m | Nonempty (↥m ≃ₗ[R] ↥S)}\ne : ↥S' ≃ₗ[R] ↥S\n⊢ S... | [] | by | [anonymous] | by |
Mathlib.RingTheory.SimpleModule.WedderburnArtin | {
"line": 54,
"column": 35
} | {
"line": 54,
"column": 37
} | {
"line": 55,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\ninst✝ : IsSimpleRing R\ntfae_1_to_2 : IsSemisimpleRing R → IsArtinianRing R\ntfae_2_to_3 : IsArtinianRing R → ∃ I, IsAtom I\nx✝ : ∃ I, IsAtom I\nI : Ideal R\nhI : IsAtom I\n⊢ IsSemisimpleRing R",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Non... | [] | by | [anonymous] | by |
Mathlib.RingTheory.SimpleModule.WedderburnArtin | {
"line": 51,
"column": 71
} | {
"line": 51,
"column": 73
} | {
"line": 52,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\ninst✝ : IsSimpleRing R\n⊢ [IsSemisimpleRing R, IsArtinianRing R, ∃ I, IsAtom I].TFAE",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Iff.mpr",
"List.IsChain.cons_cons",
"Submodule",
"LinearEquiv.symm",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.SimpleModule.Isotypic | {
"line": 426,
"column": 8
} | {
"line": 426,
"column": 10
} | {
"line": 426,
"column": 11
} | [
{
"pp": "R : Type u_2\nM : Type u\nS : Type u_4\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup S\ninst✝³ : Module R M\ninst✝² : Module R S\ninst✝¹ : IsSimpleModule R S\ninst✝ : IsSemisimpleModule R M\nm : Submodule R M\nne : m ≠ ⊥\n⊢ m = isotypicComponent R M S → IsIsotypicOfType R (↥m) S ∧ m.... | [] | by | [anonymous] | by |
Mathlib.RingTheory.SimpleModule.Isotypic | {
"line": 448,
"column": 8
} | {
"line": 448,
"column": 10
} | {
"line": 448,
"column": 11
} | [
{
"pp": "R : Type u_2\nM : Type u\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsSemisimpleModule R M\nm : Submodule R M\n⊢ m ∈ isotypicComponents R M → IsIsotypic R ↥m ∧ m.IsFullyInvariant ∧ m ≠ ⊥",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Submodule"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.SimpleModule.Isotypic | {
"line": 462,
"column": 35
} | {
"line": 462,
"column": 37
} | {
"line": 462,
"column": 38
} | [
{
"pp": "R₀ : Type u_1\nR : Type u_2\nM : Type u\nN : Type u_3\nS : Type u_4\ninst✝⁹ : CommSemiring R₀\ninst✝⁸ : Ring R\ninst✝⁷ : Algebra R₀ R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : AddCommGroup S\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : Module R S\ninst✝ : IsSemisimpleModule R M... | [] | by | [anonymous] | by |
Mathlib.RingTheory.SimpleModule.Isotypic | {
"line": 463,
"column": 67
} | {
"line": 463,
"column": 69
} | {
"line": 464,
"column": 4
} | [
{
"pp": "R₀ : Type u_1\nR : Type u_2\nM : Type u\nN : Type u_3\nS✝ : Type u_4\ninst✝⁹ : CommSemiring R₀\ninst✝⁸ : Ring R\ninst✝⁷ : Algebra R₀ R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : AddCommGroup S✝\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : Module R S✝\ninst✝ : IsSemisimpleModule ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.SimpleModule.WedderburnArtin | {
"line": 72,
"column": 17
} | {
"line": 72,
"column": 19
} | {
"line": 72,
"column": 20
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\nright✝ : IsArtinianRing R\nthis : IsSemisimpleRing R\nx✝ : (Nontrivial R ∧ ∀ (I : Ideal R), I.IsTwoSided → I = ⊥ ∨ I = ⊤) ∧ IsArtinianRing R\nleft✝ : Nontrivial R ∧ ∀ (I : Ideal R), I.IsTwoSided → I = ⊥ ∨ I = ⊤\n⊢ (∀ (N : Submodule R R), Ideal.IsTwoSided N → N = ⊥ ∨ N = ⊤) ∧... | [] | by | [anonymous] | by |
Mathlib.RingTheory.SimpleModule.WedderburnArtin | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 35
} | {
"line": 73,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u\ninst✝ : Ring R\nright✝ : IsArtinianRing R\nthis : IsSemisimpleRing R\nx✝ : (Nontrivial R ∧ ∀ (I : Ideal R), I.IsTwoSided → I = ⊥ ∨ I = ⊤) ∧ IsArtinianRing R\nleft✝ : Nontrivial R ∧ ∀ (I : Ideal R), I.IsTwoSided → I = ⊥ ∨ I = ⊤\n⊢ IsSemisimpleRing R ∧ (∀ (N : Submodule R R), I... | [] | exact ⟨this, by rwa [and_comm]⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.SimpleModule.WedderburnArtin | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 35
} | {
"line": 73,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u\ninst✝ : Ring R\nright✝ : IsArtinianRing R\nthis : IsSemisimpleRing R\nx✝ : (Nontrivial R ∧ ∀ (I : Ideal R), I.IsTwoSided → I = ⊥ ∨ I = ⊤) ∧ IsArtinianRing R\nleft✝ : Nontrivial R ∧ ∀ (I : Ideal R), I.IsTwoSided → I = ⊥ ∨ I = ⊤\n⊢ IsSemisimpleRing R ∧ (∀ (N : Submodule R R), I... | [] | exact ⟨this, by rwa [and_comm]⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.SimpleModule.WedderburnArtin | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 35
} | {
"line": 73,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u\ninst✝ : Ring R\nright✝ : IsArtinianRing R\nthis : IsSemisimpleRing R\nx✝ : (Nontrivial R ∧ ∀ (I : Ideal R), I.IsTwoSided → I = ⊥ ∨ I = ⊤) ∧ IsArtinianRing R\nleft✝ : Nontrivial R ∧ ∀ (I : Ideal R), I.IsTwoSided → I = ⊥ ∨ I = ⊤\n⊢ IsSemisimpleRing R ∧ (∀ (N : Submodule R R), I... | [] | exact ⟨this, by rwa [and_comm]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.SimpleModule.WedderburnArtin | {
"line": 67,
"column": 94
} | {
"line": 67,
"column": 96
} | {
"line": 68,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\n⊢ IsSimpleRing R ∧ IsArtinianRing R ↔ IsSemisimpleRing R ∧ IsIsotypic R R ∧ Nontrivial R",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Iff.mpr",
"Eq.mpr",
"Submodule",
"Semiring.toModule",
"_private.Ma... | [] | by | [anonymous] | by |
Mathlib.RingTheory.SimpleModule.Isotypic | {
"line": 471,
"column": 69
} | {
"line": 471,
"column": 71
} | {
"line": 472,
"column": 2
} | [
{
"pp": "R : Type u_2\nM : Type u\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsSemisimpleModule R M\nm : Submodule R M\n⊢ m.IsFullyInvariant ↔ ∃ s ⊆ isotypicComponents R M, m = sSup s",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Comple... | [] | by | [anonymous] | by |
Mathlib.RingTheory.SimpleModule.WedderburnArtin | {
"line": 89,
"column": 69
} | {
"line": 89,
"column": 71
} | {
"line": 90,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝² : Ring R\ninst✝¹ : IsSimpleRing R\ninst✝ : IsArtinianRing R\n⊢ ∃ n, ∃ (_ : NeZero n), ∃ I, ∃ (_ : IsSimpleModule R ↥I), Nonempty (R ≃+* Matrix (Fin n) (Fin n) (Module.End R ↥I)ᵐᵒᵖ)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"IsIsotypic.linearEquiv_fun... | [] | by | [anonymous] | by |
Mathlib.RingTheory.SimpleModule.WedderburnArtin | {
"line": 98,
"column": 51
} | {
"line": 98,
"column": 53
} | {
"line": 99,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝² : Ring R\ninst✝¹ : IsSimpleRing R\ninst✝ : IsArtinianRing R\n⊢ ∃ n, ∃ (_ : NeZero n), ∃ D x, Nonempty (R ≃+* Matrix (Fin n) (Fin n) D)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"Matrix.add",
"Semiring.toModule",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.SimpleModule.WedderburnArtin | {
"line": 107,
"column": 2
} | {
"line": 109,
"column": 93
} | {
"line": 111,
"column": 0
} | [
{
"pp": "R₀ : Type u_1\nR : Type u\ninst✝⁴ : CommSemiring R₀\ninst✝³ : Ring R\ninst✝² : Algebra R₀ R\ninst✝¹ : IsSimpleRing R\ninst✝ : IsArtinianRing R\n⊢ ∃ n,\n ∃ (_ : NeZero n), ∃ I, ∃ (_ : IsSimpleModule R ↥I), Nonempty (R ≃ₐ[R₀] Matrix (Fin n) (Fin n) (Module.End R ↥I)ᵐᵒᵖ)",
"ppTerm": "?m.23",
"a... | [] | have ⟨n, hn, S, hS, ⟨e⟩⟩ := (isIsotypic R R).linearEquiv_fun
refine ⟨n, hn, S, hS, ⟨.trans (.opOp R₀ R) <| .trans (.op ?_) (.symm .mopMatrix)⟩⟩
exact .trans (.moduleEndSelf R₀) <| .trans (e.conjAlgEquiv R₀) (endVecAlgEquivMatrixEnd ..) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.SimpleModule.WedderburnArtin | {
"line": 107,
"column": 2
} | {
"line": 109,
"column": 93
} | {
"line": 111,
"column": 0
} | [
{
"pp": "R₀ : Type u_1\nR : Type u\ninst✝⁴ : CommSemiring R₀\ninst✝³ : Ring R\ninst✝² : Algebra R₀ R\ninst✝¹ : IsSimpleRing R\ninst✝ : IsArtinianRing R\n⊢ ∃ n,\n ∃ (_ : NeZero n), ∃ I, ∃ (_ : IsSimpleModule R ↥I), Nonempty (R ≃ₐ[R₀] Matrix (Fin n) (Fin n) (Module.End R ↥I)ᵐᵒᵖ)",
"ppTerm": "?m.23",
"a... | [] | have ⟨n, hn, S, hS, ⟨e⟩⟩ := (isIsotypic R R).linearEquiv_fun
refine ⟨n, hn, S, hS, ⟨.trans (.opOp R₀ R) <| .trans (.op ?_) (.symm .mopMatrix)⟩⟩
exact .trans (.moduleEndSelf R₀) <| .trans (e.conjAlgEquiv R₀) (endVecAlgEquivMatrixEnd ..) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.SimpleModule.WedderburnArtin | {
"line": 106,
"column": 72
} | {
"line": 106,
"column": 74
} | {
"line": 107,
"column": 2
} | [
{
"pp": "R₀ : Type u_1\nR : Type u\ninst✝⁴ : CommSemiring R₀\ninst✝³ : Ring R\ninst✝² : Algebra R₀ R\ninst✝¹ : IsSimpleRing R\ninst✝ : IsArtinianRing R\n⊢ ∃ n,\n ∃ (_ : NeZero n), ∃ I, ∃ (_ : IsSimpleModule R ↥I), Nonempty (R ≃ₐ[R₀] Matrix (Fin n) (Fin n) (Module.End R ↥I)ᵐᵒᵖ)",
"ppTerm": "?m.23",
"a... | [] | by | [anonymous] | by |
Mathlib.RingTheory.SimpleModule.WedderburnArtin | {
"line": 115,
"column": 54
} | {
"line": 115,
"column": 56
} | {
"line": 116,
"column": 2
} | [
{
"pp": "R₀ : Type u_1\nR : Type u\ninst✝⁴ : CommSemiring R₀\ninst✝³ : Ring R\ninst✝² : Algebra R₀ R\ninst✝¹ : IsSimpleRing R\ninst✝ : IsArtinianRing R\n⊢ ∃ n, ∃ (_ : NeZero n), ∃ D x x_1, Nonempty (R ≃ₐ[R₀] Matrix (Fin n) (Fin n) D)",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"I... | [] | by | [anonymous] | by |
Mathlib.RingTheory.SimpleModule.WedderburnArtin | {
"line": 123,
"column": 80
} | {
"line": 123,
"column": 82
} | {
"line": 124,
"column": 2
} | [
{
"pp": "R₀ : Type u_1\nR : Type u\ninst✝⁵ : CommSemiring R₀\ninst✝⁴ : Ring R\ninst✝³ : Algebra R₀ R\ninst✝² : IsSimpleRing R\ninst✝¹ : IsArtinianRing R\ninst✝ : Module.Finite R₀ R\n⊢ ∃ n, ∃ (_ : NeZero n), ∃ D x x_1, ∃ (_ : Module.Finite R₀ D), Nonempty (R ≃ₐ[R₀] Matrix (Fin n) (Fin n) D)",
"ppTerm": "?m.2... | [] | by | [anonymous] | by |
Mathlib.RingTheory.SimpleModule.WedderburnArtin | {
"line": 142,
"column": 85
} | {
"line": 142,
"column": 87
} | {
"line": 143,
"column": 2
} | [
{
"pp": "R₀ : Type u_1\nR : Type u\ninst✝⁸ : CommSemiring R₀\ninst✝⁷ : Ring R\ninst✝⁶ : Algebra R₀ R\nM : Type v\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R₀ M\ninst✝³ : Module R M\ninst✝² : IsScalarTower R₀ R M\ninst✝¹ : IsSemisimpleModule R M\ninst✝ : Module.Finite R M\n⊢ ∃ n S d,\n (∀ (i : Fin n), IsSimpl... | [] | by | [anonymous] | by |
Mathlib.RingTheory.SimpleModule.WedderburnArtin | {
"line": 159,
"column": 77
} | {
"line": 159,
"column": 79
} | {
"line": 160,
"column": 2
} | [
{
"pp": "R₀ : Type u_1\nR : Type u\ninst✝⁸ : CommSemiring R₀\ninst✝⁷ : Ring R\ninst✝⁶ : Algebra R₀ R\nM : Type v\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R₀ M\ninst✝³ : Module R M\ninst✝² : IsScalarTower R₀ R M\ninst✝¹ : IsSemisimpleModule R M\ninst✝ : Module.Finite R M\n⊢ ∃ n D d x x_1,\n (∀ (i : Fin n), N... | [] | by | [anonymous] | by |
Mathlib.RingTheory.SimpleModule.WedderburnArtin | {
"line": 194,
"column": 71
} | {
"line": 194,
"column": 73
} | {
"line": 195,
"column": 2
} | [
{
"pp": "R₀ : Type u_1\nR : Type u\ninst✝³ : CommSemiring R₀\ninst✝² : Ring R\ninst✝¹ : Algebra R₀ R\ninst✝ : IsSemisimpleRing R\n⊢ ∃ n D d x x_1, (∀ (i : Fin n), NeZero (d i)) ∧ Nonempty (R ≃ₐ[R₀] (i : Fin n) → Matrix (Fin (d i)) (Fin (d i)) (D i))",
"ppTerm": "?m.22",
"assigned": true,
"usedConsta... | [] | by | [anonymous] | by |
Mathlib.RingTheory.SimpleModule.WedderburnArtin | {
"line": 210,
"column": 81
} | {
"line": 210,
"column": 83
} | {
"line": 210,
"column": 84
} | [
{
"pp": "R₀ : Type u_1\nR : Type u\ninst✝⁴ : CommSemiring R₀\ninst✝³ : Ring R\ninst✝² : Algebra R₀ R\ninst✝¹ : IsSemisimpleRing R\ninst✝ : Module.Finite R₀ R\nn : ℕ\nD : Fin n → Type u\nd : Fin n → ℕ\nw✝¹ : (i : Fin n) → DivisionRing (D i)\nw✝ : (i : Fin n) → Algebra R₀ (D i)\nhd : ∀ (i : Fin n), NeZero (d i)\n... | [] | by | [anonymous] | by |
Mathlib.RingTheory.SimpleModule.WedderburnArtin | {
"line": 204,
"column": 71
} | {
"line": 204,
"column": 73
} | {
"line": 205,
"column": 2
} | [
{
"pp": "R₀ : Type u_1\nR : Type u\ninst✝⁴ : CommSemiring R₀\ninst✝³ : Ring R\ninst✝² : Algebra R₀ R\ninst✝¹ : IsSemisimpleRing R\ninst✝ : Module.Finite R₀ R\n⊢ ∃ n D d x x_1,\n ∃ (_ : ∀ (i : Fin n), Module.Finite R₀ (D i)),\n (∀ (i : Fin n), NeZero (d i)) ∧ Nonempty (R ≃ₐ[R₀] (i : Fin n) → Matrix (Fin ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Spectrum.Maximal.Topology | {
"line": 36,
"column": 90
} | {
"line": 36,
"column": 92
} | {
"line": 37,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\n⊢ range toPrimeSpectrum = {x | IsClosed {x}}",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Set.ext",
"Eq.mpr",
"PrimeSpectrum.mk",
"congrArg",
"CommSemiring.toSemiring",
"Set.ofPred",
"PrimeSpectrum.cases... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Spectrum.Maximal.Topology | {
"line": 48,
"column": 80
} | {
"line": 48,
"column": 82
} | {
"line": 49,
"column": 6
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nx : MaximalSpectrum R\n⊢ toPrimeSpectrum ⁻¹' {x.toPrimeSpectrum} = {x}",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.instSingletonSet",
"id",
"MaximalSpectrum",
"MaximalSpectrum.toPrim... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Spectrum.Prime.IsOpenComapC | {
"line": 38,
"column": 51
} | {
"line": 38,
"column": 53
} | {
"line": 39,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : R[X]\n⊢ IsOpen (imageOfDf f)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring",
"Set.ofPred",
"AlgebraicGeometry.Polynomial.imageOfDf",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Spectrum.Prime.IsOpenComapC | {
"line": 53,
"column": 78
} | {
"line": 53,
"column": 80
} | {
"line": 54,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nf : R[X]\n⊢ imageOfDf f = PrimeSpectrum.comap C '' (zeroLocus {f})ᶜ",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Set.ext",
"Set.singleton_subset_iff",
"Eq.mpr",
"Polynomial.C",
"PrimeSpectrum.mk",
"Ideal.sub... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Spectrum.Prime.IsOpenComapC | {
"line": 68,
"column": 80
} | {
"line": 68,
"column": 82
} | {
"line": 69,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\n⊢ IsOpenMap (PrimeSpectrum.comap C)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Set.compl_iInter",
"PrimeSpectrum.zeroLocus",
"compl_compl",
"congrArg",
"CommSemiring.toSemir... | [] | by | [anonymous] | by |
Mathlib.RingTheory.TwoSidedIdeal.BigOperators | {
"line": 25,
"column": 25
} | {
"line": 25,
"column": 27
} | {
"line": 26,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nI : TwoSidedIdeal R\nι : Type u_2\nl : List ι\nf : ι → R\nhl : ∀ x ∈ l, f x ∈ I\n⊢ (List.map f l).sum ∈ I",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"RingCon.instFunLikeForallProp",
"congrArg",
"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.TwoSidedIdeal.BigOperators | {
"line": 30,
"column": 25
} | {
"line": 30,
"column": 27
} | {
"line": 31,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nI : TwoSidedIdeal R\nι : Type u_2\ns : Multiset ι\nf : ι → R\nhs : ∀ x ∈ s, f x ∈ I\n⊢ (Multiset.map f s).sum ∈ I",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Multiset.sum",
"Eq.mpr",
"Multiset.map",
"RingC... | [] | by | [anonymous] | by |
Mathlib.RingTheory.TwoSidedIdeal.BigOperators | {
"line": 35,
"column": 19
} | {
"line": 35,
"column": 21
} | {
"line": 36,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nI : TwoSidedIdeal R\nι : Type u_2\ns : Finset ι\nf : ι → R\nhs : ∀ x ∈ s, f x ∈ I\n⊢ s.sum f ∈ I",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"RingCon.instFunLikeForallProp",
"congrArg",
"TwoSidedI... | [] | by | [anonymous] | by |
Mathlib.RingTheory.TwoSidedIdeal.BigOperators | {
"line": 59,
"column": 26
} | {
"line": 59,
"column": 28
} | {
"line": 60,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nI : TwoSidedIdeal R\nι : Type u_2\nl : List ι\nf : ι → R\nhl : ∃ x ∈ l, f x ∈ I\n⊢ (List.map f l).prod ∈ I",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"TwoSidedIdeal.mul_mem_right",
"TwoSidedIdeal.mul_mem... | [] | by | [anonymous] | by |
Mathlib.RingTheory.TwoSidedIdeal.BigOperators | {
"line": 76,
"column": 2
} | {
"line": 77,
"column": 37
} | {
"line": 79,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : TwoSidedIdeal R\nι : Type u_2\ns : Multiset ι\nf : ι → R\nhs : ∃ x ∈ s, f x ∈ I\n⊢ (Multiset.map f s).prod ∈ I",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCo... | [] | rcases s
simpa using listProd_mem (hl := hs) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.TwoSidedIdeal.BigOperators | {
"line": 76,
"column": 2
} | {
"line": 77,
"column": 37
} | {
"line": 79,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : TwoSidedIdeal R\nι : Type u_2\ns : Multiset ι\nf : ι → R\nhs : ∃ x ∈ s, f x ∈ I\n⊢ (Multiset.map f s).prod ∈ I",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCo... | [] | rcases s
simpa using listProd_mem (hl := hs) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.TwoSidedIdeal.BigOperators | {
"line": 75,
"column": 26
} | {
"line": 75,
"column": 28
} | {
"line": 76,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : TwoSidedIdeal R\nι : Type u_2\ns : Multiset ι\nf : ι → R\nhs : ∃ x ∈ s, f x ∈ I\n⊢ (Multiset.map f s).prod ∈ I",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCo... | [] | by | [anonymous] | by |
Mathlib.RingTheory.TwoSidedIdeal.BigOperators | {
"line": 80,
"column": 20
} | {
"line": 80,
"column": 22
} | {
"line": 81,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nI : TwoSidedIdeal R\nι : Type u_2\ns : Finset ι\nf : ι → R\nhs : ∃ x ∈ s, f x ∈ I\n⊢ s.prod f ∈ I",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRing",
"M... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Spectrum.Prime.Homeomorph | {
"line": 41,
"column": 44
} | {
"line": 41,
"column": 46
} | {
"line": 42,
"column": 4
} | [
{
"pp": "R : Type u_3\nS : Type u_4\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nH : ∀ (x : S), ∃ n > 0, x ^ n ∈ f.range\nhker : RingHom.ker f ≤ nilradical R\n⊢ Function.Injective (comap f)",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PrimeSpectrum.ext",... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Spectrum.Prime.Homeomorph | {
"line": 60,
"column": 46
} | {
"line": 60,
"column": 48
} | {
"line": 60,
"column": 49
} | [
{
"pp": "R : Type u_3\nS : Type u_4\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nH : ∀ (x : S), ∃ n > 0, x ^ n ∈ f.range\nhker : RingHom.ker f ≤ nilradical R\nh1 : Function.Injective (comap f)\nhint : f.kerLift.IsIntegral\nhbij : Function.Bijective (comap f)\ns : S\nn : ℕ\nhn : n > 0\nr : R\nhr : f r ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Spectrum.Prime.Homeomorph | {
"line": 40,
"column": 70
} | {
"line": 40,
"column": 72
} | {
"line": 41,
"column": 2
} | [
{
"pp": "R : Type u_3\nS : Type u_4\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nH : ∀ (x : S), ∃ n > 0, x ^ n ∈ f.range\nhker : RingHom.ker f ≤ nilradical R\n⊢ IsHomeomorph (comap f)",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"PrimeSpectrum.basicOpen_pow",
"Topo... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Spectrum.Prime.Homeomorph | {
"line": 66,
"column": 55
} | {
"line": 66,
"column": 57
} | {
"line": 67,
"column": 2
} | [
{
"pp": "k : Type u_1\nK : Type u_2\nR : Type u_3\ninst✝⁵ : Field k\ninst✝⁴ : Field K\ninst✝³ : Algebra k K\ninst✝² : CommRing R\ninst✝¹ : Algebra k R\ninst✝ : IsPurelyInseparable k K\n⊢ IsHomeomorph (comap (algebraMap R (R ⊗[k] K)))",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"E... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Spectrum.Prime.Homeomorph | {
"line": 84,
"column": 69
} | {
"line": 84,
"column": 71
} | {
"line": 85,
"column": 4
} | [
{
"pp": "K : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝⁹ : Field K\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R K\ninst✝⁵ : Algebra R S\nL : Type u_5\ninst✝⁴ : Field L\ninst✝³ : Algebra R L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsPurelyInseparable K L\ne : L ⊗[R] S ≃ₐ[K]... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Spectrum.Prime.Homeomorph | {
"line": 78,
"column": 95
} | {
"line": 78,
"column": 97
} | {
"line": 79,
"column": 2
} | [
{
"pp": "K : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝⁹ : Field K\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : Algebra R K\ninst✝⁵ : Algebra R S\nL : Type u_5\ninst✝⁴ : Field L\ninst✝³ : Algebra R L\ninst✝² : Algebra K L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsPurelyInseparable K L\n⊢ IsHomeomorph (co... | [] | by | [anonymous] | by |
Mathlib.RingTheory.TensorProduct.IsBaseChangeRightExact | {
"line": 54,
"column": 92
} | {
"line": 54,
"column": 94
} | {
"line": 55,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝²⁰ : CommRing R\nS : Type u_2\ninst✝¹⁹ : CommRing S\ninst✝¹⁸ : Algebra R S\nM₁ : Type u_3\nM₂ : Type u_4\nM₃ : Type u_5\nN₁ : Type u_6\nN₂ : Type u_7\nN₃ : Type u_8\ninst✝¹⁷ : AddCommGroup M₁\ninst✝¹⁶ : AddCommGroup M₂\ninst✝¹⁵ : AddCommGroup M₃\ninst✝¹⁴ : AddCommGroup N₁\ninst✝¹³ : ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.UniqueFactorizationDomain.Moebius | {
"line": 39,
"column": 84
} | {
"line": 39,
"column": 86
} | {
"line": 40,
"column": 2
} | [
{
"pp": "n : ℕ\n⊢ moebius n = ArithmeticFunction.moebius n",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"UniqueFactorizationMonoid.normalizedFactors",
"Eq.mpr",
"UniqueFactorizationMonoid.factors_eq_normalizedFactors",
"Nat.instMulZeroClass",
"instSubsinglet... | [] | by | [anonymous] | by |
Mathlib.RingTheory.UniqueFactorizationDomain.Moebius | {
"line": 52,
"column": 61
} | {
"line": 52,
"column": 63
} | {
"line": 53,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : UniqueFactorizationMonoid α\ninst✝ : Nontrivial α\n⊢ moebius 0 = 0",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"False",
"congrArg",
"not_squarefree_zero._simp_1",
"Int",
"CommMonoidWithZero.toM... | [] | by | [anonymous] | by |
Mathlib.RingTheory.UniqueFactorizationDomain.Moebius | {
"line": 55,
"column": 45
} | {
"line": 55,
"column": 47
} | {
"line": 56,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\n⊢ moebius 1 = 1",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"UniqueFactorizationMonoid.factors_one",
"CommMonoidWithZero.toCommMonoid",
"MulOne.toOne",
"Monoid.toMulOneCla... | [] | by | [anonymous] | by |
Mathlib.RingTheory.UniqueFactorizationDomain.Moebius | {
"line": 58,
"column": 85
} | {
"line": 58,
"column": 87
} | {
"line": 59,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\na b : α\nh : Associated a b\n⊢ moebius a = moebius b",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Classical.propDecidable",
"UniqueFactorizationMonoi... | [] | by | [anonymous] | by |
Mathlib.RingTheory.UniqueFactorizationDomain.Moebius | {
"line": 61,
"column": 68
} | {
"line": 61,
"column": 70
} | {
"line": 62,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\na : α\nha : IsUnit a\n⊢ moebius a = 1",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"UniqueFactorizationMonoid.moebius_one",
"MulOne.toOne",
"Mono... | [] | by | [anonymous] | by |
Mathlib.RingTheory.UniqueFactorizationDomain.Moebius | {
"line": 64,
"column": 79
} | {
"line": 64,
"column": 81
} | {
"line": 65,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\na : α\nha : Irreducible a\n⊢ moebius a = -1",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Eq.mpr",
"congrArg",
"Irreducible.squarefree",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.UniqueFactorizationDomain.Moebius | {
"line": 68,
"column": 47
} | {
"line": 68,
"column": 49
} | {
"line": 69,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\na b : α\nh : IsRelPrime a b\n⊢ moebius (a * b) = moebius a * moebius b",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Nontrivial",
"CommMonoidWithZero.toCommMonoid",
"Iff.mpr",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.AlgebraInstances | {
"line": 47,
"column": 80
} | {
"line": 47,
"column": 82
} | {
"line": 48,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nv : Valuation K (WithZero (Multiplicative ℤ))\nL : Type u_2\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : ↥(integralClosure (↥v.valuationSubring) L)\n⊢ IsIntegral (↥v.valuationSubring) x",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Subalgebr... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.AlgebraInstances | {
"line": 64,
"column": 33
} | {
"line": 64,
"column": 35
} | {
"line": 64,
"column": 36
} | [
{
"pp": "K : Type u_1\ninst✝⁶ : Field K\nv : Valuation K (WithZero (Multiplicative ℤ))\nL : Type u_2\ninst✝⁵ : Field L\ninst✝⁴ : Algebra K L\nE : Type ?u.16\ninst✝³ : Field E\ninst✝² : Algebra K E\ninst✝¹ : Algebra L E\ninst✝ : IsScalarTower K L E\n⊢ ↑⟨(algebraMap L E) ↑1, ⋯⟩ = ↑1",
"ppTerm": "?m.82",
"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.AlgebraInstances | {
"line": 68,
"column": 23
} | {
"line": 68,
"column": 25
} | {
"line": 68,
"column": 26
} | [
{
"pp": "K : Type u_1\ninst✝⁶ : Field K\nv : Valuation K (WithZero (Multiplicative ℤ))\nL : Type u_2\ninst✝⁵ : Field L\ninst✝⁴ : Algebra K L\nE : Type ?u.16\ninst✝³ : Field E\ninst✝² : Algebra K E\ninst✝¹ : Algebra L E\ninst✝ : IsScalarTower K L E\nx y : ↥(integralClosure (↥v.valuationSubring) L)\n⊢ ↑⟨(algebraM... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.AlgebraInstances | {
"line": 63,
"column": 23
} | {
"line": 63,
"column": 25
} | {
"line": 63,
"column": 26
} | [
{
"pp": "K : Type u_1\ninst✝⁶ : Field K\nv : Valuation K (WithZero (Multiplicative ℤ))\nL : Type u_2\ninst✝⁵ : Field L\ninst✝⁴ : Algebra K L\nE : Type ?u.16\ninst✝³ : Field E\ninst✝² : Algebra K E\ninst✝¹ : Algebra L E\ninst✝ : IsScalarTower K L E\n⊢ ↑⟨(algebraMap L E) ↑0, ⋯⟩ = ↑0",
"ppTerm": "?m.84",
"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.AlgebraInstances | {
"line": 66,
"column": 23
} | {
"line": 66,
"column": 25
} | {
"line": 66,
"column": 26
} | [
{
"pp": "K : Type u_1\ninst✝⁶ : Field K\nv : Valuation K (WithZero (Multiplicative ℤ))\nL : Type u_2\ninst✝⁵ : Field L\ninst✝⁴ : Algebra K L\nE : Type ?u.16\ninst✝³ : Field E\ninst✝² : Algebra K E\ninst✝¹ : Algebra L E\ninst✝ : IsScalarTower K L E\nx y : ↥(integralClosure (↥v.valuationSubring) L)\n⊢ ↑⟨(algebraM... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 35,
"column": 90
} | {
"line": 35,
"column": 92
} | {
"line": 36,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_2\nN : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nI : Ideal R\nf : M →ₗ[R] N\nsurj : Function.Surjective ⇑f\ninst✝ : Module.Flat R N\nx : M\nhx : x ∈ f.ker ⊓ I • ⊤\ny : TensorProduct R (↥I) M\nhy : ... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Valuation.Minpoly | {
"line": 40,
"column": 89
} | {
"line": 40,
"column": 91
} | {
"line": 41,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝³ : Field K\nΓ₀ : Type u_2\ninst✝² : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation K Γ₀\nL : Type u_3\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : K\n⊢ v ((minpoly K ((algebraMap K L) x)).coeff 0) = v x",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 38,
"column": 69
} | {
"line": 38,
"column": 71
} | {
"line": 39,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_2\nN : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nI : Ideal R\nf : M →ₗ[R] N\nsurj : Function.Surjective ⇑f\ninst✝ : Module.Flat R N\nx : M\nhx : x ∈ f.ker ⊓ I • ⊤\ny : TensorProduct R (↥I) M\nhy : ... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Valuation.Minpoly | {
"line": 49,
"column": 86
} | {
"line": 49,
"column": 88
} | {
"line": 50,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝⁴ : Field K\nΓ₀ : Type u_2\ninst✝³ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation K Γ₀\nL : Type u_3\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : FiniteDimensional K L\nx : L\nhx : IsUnit x\n⊢ v ((minpoly K x).coeff 0) ^ (finrank K L / (minpoly K x).natDegree) ≠ 0",
"ppTe... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 41,
"column": 36
} | {
"line": 41,
"column": 38
} | {
"line": 42,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_2\nN : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nI : Ideal R\nf : M →ₗ[R] N\nsurj : Function.Surjective ⇑f\ninst✝ : Module.Flat R N\nx : M\nhx : x ∈ f.ker ⊓ I • ⊤\ny : TensorProduct R (↥I) M\nhy : ... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 47,
"column": 70
} | {
"line": 47,
"column": 72
} | {
"line": 48,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_2\nN : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nI : Ideal R\nf : M →ₗ[R] N\nsurj : Function.Surjective ⇑f\ninst✝ : Module.Flat R N\nx : M\nhx : x ∈ f.ker ⊓ I • ⊤\ny : TensorProduct R (↥I) M\nhy : ... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Valuation.Quotient | {
"line": 33,
"column": 33
} | {
"line": 33,
"column": 35
} | {
"line": 33,
"column": 36
} | [
{
"pp": "R : Type u_1\nΓ₀ : Type u_2\ninst✝¹ : CommRing R\ninst✝ : LinearOrderedCommMonoidWithZero Γ₀\nv : Valuation R Γ₀\nJ : Ideal R\nhJ : J ≤ v.supp\nq : R ⧸ J\na b : R\nh : (Submodule.quotientRel J) a b\n⊢ v a = v (b + -(-a + b))",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"n... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.Quotient | {
"line": 51,
"column": 48
} | {
"line": 51,
"column": 50
} | {
"line": 52,
"column": 2
} | [
{
"pp": "R : Type u_1\nΓ₀ : Type u_2\ninst✝¹ : CommRing R\ninst✝ : LinearOrderedCommMonoidWithZero Γ₀\nJ : Ideal R\nv : Valuation (R ⧸ J) Γ₀\n⊢ J ≤ (comap (Ideal.Quotient.mk J) v).supp",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"RingHom.instRingHomClass",
"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.Quotient | {
"line": 58,
"column": 9
} | {
"line": 58,
"column": 11
} | {
"line": 59,
"column": 4
} | [
{
"pp": "R : Type u_1\nΓ₀ : Type u_2\ninst✝¹ : CommRing R\ninst✝ : LinearOrderedCommMonoidWithZero Γ₀\nJ : Ideal R\nv : Valuation (R ⧸ J) Γ₀\n⊢ ∀ (r : R ⧸ J), ((comap (Ideal.Quotient.mk J) v).onQuot ⋯) r = v r",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.Quotient | {
"line": 64,
"column": 63
} | {
"line": 64,
"column": 65
} | {
"line": 65,
"column": 2
} | [
{
"pp": "R : Type u_1\nΓ₀ : Type u_2\ninst✝¹ : CommRing R\ninst✝ : LinearOrderedCommMonoidWithZero Γ₀\nv : Valuation R Γ₀\nJ : Ideal R\nhJ : J ≤ v.supp\n⊢ (v.onQuot hJ).supp = Ideal.map (Ideal.Quotient.mk J) v.supp",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Idea... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.Quotient | {
"line": 73,
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} | {
"line": 73,
"column": 57
} | {
"line": 74,
"column": 2
} | [
{
"pp": "R : Type u_1\nΓ₀ : Type u_2\ninst✝¹ : CommRing R\ninst✝ : LinearOrderedCommMonoidWithZero Γ₀\nv : Valuation R Γ₀\n⊢ (v.onQuot ⋯).supp = 0",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"le_rfl",
"congrArg",
"CommSemirin... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.ValuativeRel.Trivial | {
"line": 39,
"column": 19
} | {
"line": 39,
"column": 21
} | {
"line": 39,
"column": 22
} | [
{
"pp": "R✝ Γ : Type\ninst✝⁶ : Ring R✝\ninst✝⁵ : DecidableEq R✝\ninst✝⁴ : IsDomain R✝\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nR : Type\ninst✝² : Semiring R\ninst✝¹ : DecidableEq R\ninst✝ : IsDomain R\nx✝¹ x✝ : R\n⊢ (if x✝ = 0 then x✝¹ = 0 else True) ∨ if x✝¹ = 0 then x✝ = 0 else True",
"ppTerm": "?m.45"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.ValuativeRel.Trivial | {
"line": 40,
"column": 19
} | {
"line": 40,
"column": 21
} | {
"line": 40,
"column": 22
} | [
{
"pp": "R✝ Γ : Type\ninst✝⁶ : Ring R✝\ninst✝⁵ : DecidableEq R✝\ninst✝⁴ : IsDomain R✝\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nR : Type\ninst✝² : Semiring R\ninst✝¹ : DecidableEq R\ninst✝ : IsDomain R\nz✝ y✝ x✝² : R\nx✝¹ : if y✝ = 0 then x✝² = 0 else True\nx✝ : if z✝ = 0 then y✝ = 0 else True\n⊢ if z✝ = 0 th... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.ValuativeRel.Trivial | {
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"column": 17
} | {
"line": 41,
"column": 19
} | {
"line": 41,
"column": 20
} | [
{
"pp": "R✝ Γ : Type\ninst✝⁶ : Ring R✝\ninst✝⁵ : DecidableEq R✝\ninst✝⁴ : IsDomain R✝\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nR : Type\ninst✝² : Semiring R\ninst✝¹ : DecidableEq R\ninst✝ : IsDomain R\nx✝² y✝ z✝ : R\nx✝¹ : if z✝ = 0 then x✝² = 0 else True\nx✝ : if z✝ = 0 then y✝ = 0 else True\n⊢ if z✝ = 0 th... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.ValuativeRel.Trivial | {
"line": 42,
"column": 26
} | {
"line": 42,
"column": 28
} | {
"line": 42,
"column": 29
} | [
{
"pp": "R✝ Γ : Type\ninst✝⁶ : Ring R✝\ninst✝⁵ : DecidableEq R✝\ninst✝⁴ : IsDomain R✝\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nR : Type\ninst✝² : Semiring R\ninst✝¹ : DecidableEq R\ninst✝ : IsDomain R\nx✝² y✝ : R\nx✝¹ : if y✝ = 0 then x✝² = 0 else True\nx✝ : R\n⊢ if y✝ * x✝ = 0 then x✝² * x✝ = 0 else True",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.ValuativeRel.Trivial | {
"line": 43,
"column": 22
} | {
"line": 43,
"column": 24
} | {
"line": 43,
"column": 25
} | [
{
"pp": "R✝ Γ : Type\ninst✝⁶ : Ring R✝\ninst✝⁵ : DecidableEq R✝\ninst✝⁴ : IsDomain R✝\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nR : Type\ninst✝² : Semiring R\ninst✝¹ : DecidableEq R\ninst✝ : IsDomain R\nx✝¹ y✝ z✝ : R\nx✝ : ¬if 0 = 0 then z✝ = 0 else True\n⊢ (if y✝ * z✝ = 0 then x✝¹ * z✝ = 0 else True) → if y✝... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.ValuativeRel.Trivial | {
"line": 44,
"column": 22
} | {
"line": 44,
"column": 24
} | {
"line": 44,
"column": 25
} | [
{
"pp": "R✝ Γ : Type\ninst✝⁶ : Ring R✝\ninst✝⁵ : DecidableEq R✝\ninst✝⁴ : IsDomain R✝\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nR : Type\ninst✝² : Semiring R\ninst✝¹ : DecidableEq R\ninst✝ : IsDomain R\n⊢ ¬if 0 = 0 then 1 = 0 else True",
"ppTerm": "?m.173",
"assigned": true,
"usedConstants": [
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.ValuativeRel.Trivial | {
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"column": 24
} | {
"line": 45,
"column": 26
} | {
"line": 45,
"column": 27
} | [
{
"pp": "R✝ Γ : Type\ninst✝⁶ : Ring R✝\ninst✝⁵ : DecidableEq R✝\ninst✝⁴ : IsDomain R✝\ninst✝³ : LinearOrderedCommGroupWithZero Γ\nR : Type\ninst✝² : Semiring R\ninst✝¹ : DecidableEq R\ninst✝ : IsDomain R\nx✝¹ x✝ : R\n⊢ if x✝ * x✝¹ = 0 then x✝¹ * x✝ = 0 else True",
"ppTerm": "?m.186",
"assigned": true,
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.ValuativeRel.Trivial | {
"line": 48,
"column": 72
} | {
"line": 48,
"column": 74
} | {
"line": 49,
"column": 2
} | [
{
"pp": "R Γ : Type\ninst✝³ : Ring R\ninst✝² : DecidableEq R\ninst✝¹ : IsDomain R\ninst✝ : LinearOrderedCommGroupWithZero Γ\nh : ValuativeRel R\nhv : Valuation.Compatible 1\n⊢ h = trivialRel",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZe... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.ValuativeRel.Trivial | {
"line": 56,
"column": 65
} | {
"line": 56,
"column": 67
} | {
"line": 57,
"column": 2
} | [
{
"pp": "R Γ : Type\ninst✝³ : Ring R\ninst✝² : DecidableEq R\ninst✝¹ : IsDomain R\ninst✝ : LinearOrderedCommGroupWithZero Γ\n⊢ trivialRel = ofValuation 1",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero"... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 32,
"column": 73
} | {
"line": 32,
"column": 75
} | {
"line": 33,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_2\nN : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : AddCommGroup N\ninst✝² : Module R M\ninst✝¹ : Module R N\nI : Ideal R\nf : M →ₗ[R] N\nsurj : Function.Surjective ⇑f\ninst✝ : Module.Flat R N\n⊢ f.ker ⊓ I • ⊤ = I • f.ker",
"ppTerm": "?m.59",
"assign... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 70,
"column": 70
} | {
"line": 70,
"column": 72
} | {
"line": 71,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹⁰ : CommRing R\nS : Type u_2\ninst✝⁹ : CommRing S\nR' : Type u_3\nS' : Type u_4\ninst✝⁸ : CommRing R'\ninst✝⁷ : CommRing S'\ninst✝⁶ : Algebra R S\ninst✝⁵ : Algebra R R'\ninst✝⁴ : Algebra R' S'\ninst✝³ : Algebra S S'\ninst✝² : Algebra R S'\ninst✝¹ : IsScalarTower R S S'\ninst✝ : IsSc... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.ValuativeRel.Trivial | {
"line": 63,
"column": 44
} | {
"line": 63,
"column": 46
} | {
"line": 64,
"column": 2
} | [
{
"pp": "R Γ : Type\ninst✝⁵ : Ring R\ninst✝⁴ : DecidableEq R\ninst✝³ : IsDomain R\ninst✝² : LinearOrderedCommGroupWithZero Γ\ninst✝¹ : ValuativeRel R\ninst✝ : Valuation.Compatible 1\n⊢ Subsingleton (ValueGroupWithZero R)ˣ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Units.val",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 80,
"column": 84
} | {
"line": 80,
"column": 86
} | {
"line": 81,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nJ I : Ideal R\nsq : I * I = ⊥\nf : J.Cotangent →ₗ[R] J.Cotangent\nle : f.range ≤ Submodule.map J.toCotangent (Submodule.comap (Submodule.subtype J) (I * J))\nx : R\nh : x ∈ I * J\n⊢ f (J.toCotangent ⟨x, ⋯⟩) = 0",
"ppTerm": "?m.173",
"assigned": true,
"usedC... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Valuation.Extension | {
"line": 79,
"column": 98
} | {
"line": 79,
"column": 100
} | {
"line": 80,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nΓR : Type u_3\nΓA : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : Ring A\ninst✝³ : LinearOrderedCommMonoidWithZero ΓR\ninst✝² : LinearOrderedCommMonoidWithZero ΓA\ninst✝¹ : Algebra R A\nvR : Valuation R ΓR\nvA : Valuation A ΓA\ninst✝ : vR.HasExtension vA\nx y : R\n⊢ vA ((algebraMa... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.Extension | {
"line": 85,
"column": 77
} | {
"line": 85,
"column": 79
} | {
"line": 86,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nΓR : Type u_3\nΓA : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : Ring A\ninst✝³ : LinearOrderedCommMonoidWithZero ΓR\ninst✝² : LinearOrderedCommMonoidWithZero ΓA\ninst✝¹ : Algebra R A\nvR : Valuation R ΓR\nvA : Valuation A ΓA\ninst✝ : vR.HasExtension vA\nx : R\n⊢ vA ((algebraMap ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.Extension | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 67
} | {
"line": 91,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nΓR : Type u_3\nΓA : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : Ring A\ninst✝³ : LinearOrderedCommMonoidWithZero ΓR\ninst✝² : LinearOrderedCommMonoidWithZero ΓA\ninst✝¹ : Algebra R A\nvR : Valuation R ΓR\nvA : Valuation A ΓA\ninst✝ : vR.HasExtension vA\nx : R\n⊢ vA ((algebraMap ... | [] | simpa only [map_one, not_le] using (val_map_le_iff vR vA 1 x).not | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Valuation.Extension | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 67
} | {
"line": 91,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nΓR : Type u_3\nΓA : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : Ring A\ninst✝³ : LinearOrderedCommMonoidWithZero ΓR\ninst✝² : LinearOrderedCommMonoidWithZero ΓA\ninst✝¹ : Algebra R A\nvR : Valuation R ΓR\nvA : Valuation A ΓA\ninst✝ : vR.HasExtension vA\nx : R\n⊢ vA ((algebraMap ... | [] | simpa only [map_one, not_le] using (val_map_le_iff vR vA 1 x).not | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.Extension | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 67
} | {
"line": 91,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nΓR : Type u_3\nΓA : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : Ring A\ninst✝³ : LinearOrderedCommMonoidWithZero ΓR\ninst✝² : LinearOrderedCommMonoidWithZero ΓA\ninst✝¹ : Algebra R A\nvR : Valuation R ΓR\nvA : Valuation A ΓA\ninst✝ : vR.HasExtension vA\nx : R\n⊢ vA ((algebraMap ... | [] | simpa only [map_one, not_le] using (val_map_le_iff vR vA 1 x).not | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.Extension | {
"line": 88,
"column": 77
} | {
"line": 88,
"column": 79
} | {
"line": 89,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nΓR : Type u_3\nΓA : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : Ring A\ninst✝³ : LinearOrderedCommMonoidWithZero ΓR\ninst✝² : LinearOrderedCommMonoidWithZero ΓA\ninst✝¹ : Algebra R A\nvR : Valuation R ΓR\nvA : Valuation A ΓA\ninst✝ : vR.HasExtension vA\nx : R\n⊢ vA ((algebraMap ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.Extension | {
"line": 91,
"column": 77
} | {
"line": 91,
"column": 79
} | {
"line": 92,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nΓR : Type u_3\nΓA : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : Ring A\ninst✝³ : LinearOrderedCommMonoidWithZero ΓR\ninst✝² : LinearOrderedCommMonoidWithZero ΓA\ninst✝¹ : Algebra R A\nvR : Valuation R ΓR\nvA : Valuation A ΓA\ninst✝ : vR.HasExtension vA\nx : R\n⊢ vA ((algebraMap ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 78,
"column": 69
} | {
"line": 78,
"column": 71
} | {
"line": 79,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nJ I : Ideal R\nsq : I ^ 2 = ⊥\nf : J.Cotangent →ₗ[R] J.Cotangent\nle : f.range ≤ Submodule.map J.toCotangent (Submodule.comap (Submodule.subtype J) (I * J))\n⊢ Submodule.map J.toCotangent (Submodule.comap (Submodule.subtype J) (I * J)) ≤ f.ker",
"ppTerm": "?m.141",... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.ValuativeRel.Trivial | {
"line": 73,
"column": 24
} | {
"line": 73,
"column": 26
} | {
"line": 74,
"column": 2
} | [
{
"pp": "R Γ : Type\ninst✝⁵ : Ring R\ninst✝⁴ : DecidableEq R\ninst✝³ : IsDomain R\ninst✝² : LinearOrderedCommGroupWithZero Γ\ninst✝¹ : ValuativeRel R\ninst✝ : Valuation.Compatible 1\n⊢ ¬IsNontrivial R",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Units.val",
"GroupWithZero.t... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.Extension | {
"line": 98,
"column": 23
} | {
"line": 98,
"column": 25
} | {
"line": 99,
"column": 4
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nΓR : Type u_3\nΓA : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : Ring A\ninst✝² : LinearOrderedCommMonoidWithZero ΓR\ninst✝¹ : LinearOrderedCommMonoidWithZero ΓA\ninst✝ : Algebra R A\nvR : Valuation R ΓR\nvA : Valuation A ΓA\n⊢ vR.IsEquiv (comap (algebraMap R R) vR)",
"ppTerm... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.ValuativeRel.Trivial | {
"line": 81,
"column": 20
} | {
"line": 81,
"column": 22
} | {
"line": 82,
"column": 2
} | [
{
"pp": "R Γ : Type\ninst✝⁵ : Ring R\ninst✝⁴ : DecidableEq R\ninst✝³ : IsDomain R\ninst✝² : LinearOrderedCommGroupWithZero Γ\ninst✝¹ : ValuativeRel R\ninst✝ : Valuation.Compatible 1\n⊢ IsDiscrete R",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"Gr... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Smooth.Quotient | {
"line": 113,
"column": 23
} | {
"line": 113,
"column": 25
} | {
"line": 113,
"column": 26
} | [
{
"pp": "R : Type u_1\ninst✝¹¹ : CommRing R\nS : Type u_2\ninst✝¹⁰ : CommRing S\nR' : Type u_3\nS' : Type u_4\ninst✝⁹ : CommRing R'\ninst✝⁸ : CommRing S'\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra R R'\ninst✝⁵ : Algebra R' S'\ninst✝⁴ : Algebra S S'\ninst✝³ : Algebra R S'\ninst✝² : IsScalarTower R S S'\ninst✝¹ : Is... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.Extension | {
"line": 114,
"column": 23
} | {
"line": 114,
"column": 25
} | {
"line": 115,
"column": 4
} | [
{
"pp": "A : Type u_2\ninst✝⁴ : Ring A\nK : Type u_5\ninst✝³ : Field K\ninst✝² : Algebra K A\nΓA : Type u_7\nΓK : Type u_8\ninst✝¹ : LinearOrderedCommGroupWithZero ΓK\ninst✝ : LinearOrderedCommGroupWithZero ΓA\nvK : Valuation K ΓK\nvA : Valuation A ΓA\nh : Subring.comap (algebraMap K A) vA.integer = vK.integer\... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.Extension | {
"line": 123,
"column": 5
} | {
"line": 123,
"column": 7
} | {
"line": 123,
"column": 8
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nΓR✝ : Type u_3\nΓA✝ : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Ring A\ninst✝⁸ : LinearOrderedCommMonoidWithZero ΓR✝\ninst✝⁷ : LinearOrderedCommMonoidWithZero ΓA✝\ninst✝⁶ : Algebra R A\nvR✝ : Valuation R ΓR✝\nvA✝ : Valuation A ΓA✝\nK : Type u_5\ninst✝⁵ : Field K\ninst✝⁴ : Alg... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.Extension | {
"line": 128,
"column": 98
} | {
"line": 128,
"column": 100
} | {
"line": 129,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : Ring A\ninst✝³ : Algebra R A\nΓR : Type u_6\nΓA : Type u_7\ninst✝² : LinearOrderedCommGroupWithZero ΓR\ninst✝¹ : LinearOrderedCommGroupWithZero ΓA\nvR : Valuation R ΓR\nvA : Valuation A ΓA\ninst✝ : vR.HasExtension vA\nr : ↥vR.integer\na : ↥vA.in... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.Extension | {
"line": 138,
"column": 77
} | {
"line": 138,
"column": 79
} | {
"line": 139,
"column": 2
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : Ring A\ninst✝³ : Algebra R A\nΓR : Type u_6\nΓA : Type u_7\ninst✝² : LinearOrderedCommGroupWithZero ΓR\ninst✝¹ : LinearOrderedCommGroupWithZero ΓA\nvR : Valuation R ΓR\nvA : Valuation A ΓA\ninst✝ : vR.HasExtension vA\nr : ↥vR.integer\n⊢ ↑((algeb... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.Extension | {
"line": 142,
"column": 22
} | {
"line": 142,
"column": 24
} | {
"line": 143,
"column": 4
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nΓR✝ : Type u_3\nΓA✝ : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Ring A\ninst✝⁸ : LinearOrderedCommMonoidWithZero ΓR✝\ninst✝⁷ : LinearOrderedCommMonoidWithZero ΓA✝\ninst✝⁶ : Algebra R A\nvR✝ : Valuation R ΓR✝\nvA✝ : Valuation A ΓA✝\nK : Type u_5\ninst✝⁵ : Field K\ninst✝⁴ : Alg... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.Extension | {
"line": 147,
"column": 63
} | {
"line": 147,
"column": 65
} | {
"line": 147,
"column": 66
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nΓR✝ : Type u_3\nΓA✝ : Type u_4\ninst✝¹² : CommRing R\ninst✝¹¹ : Ring A\ninst✝¹⁰ : LinearOrderedCommMonoidWithZero ΓR✝\ninst✝⁹ : LinearOrderedCommMonoidWithZero ΓA✝\ninst✝⁸ : Algebra R A\nvR✝ : Valuation R ΓR✝\nvA✝ : Valuation A ΓA✝\nK : Type u_5\ninst✝⁷ : Field K\ninst✝⁶ : A... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Valuation.Extension | {
"line": 158,
"column": 22
} | {
"line": 158,
"column": 24
} | {
"line": 159,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nΓR : Type u_6\ninst✝⁵ : LinearOrderedCommGroupWithZero ΓR\nvR : Valuation R ΓR\nS : Type u_9\nΓS : Type u_10\ninst✝⁴ : CommRing S\ninst✝³ : LinearOrderedCommGroupWithZero ΓS\ninst✝² : Algebra R S\ninst✝¹ : IsLocalHom (algebraMap R S)\nvS : Valuation S ΓS\ninst✝ : vR.H... | [] | by | [anonymous] | by |
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