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.RingTheory.Valuation.RankOne | {
"line": 269,
"column": 2
} | {
"line": 271,
"column": 71
} | [
{
"pp": "Γ₀ : Type u_2\ninst✝⁴ : LinearOrderedCommGroupWithZero Γ₀\nR : Type u_3\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\ninst✝¹ : MulArchimedean Γ₀\nv : Valuation R Γ₀\ninst✝ : v.Compatible\n⊢ IsRankLeOne R",
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Eq.mpr",
"GroupWith... | rw [isRankLeOne_iff_mulArchimedean]
exact MulArchimedean.comap (embedding.toMonoidHom.comp (ValueGroupWithZero.embed v).toMonoidHom)
(embedding_strictMono.comp (ValueGroupWithZero.embed_strictMono v)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.RankOne | {
"line": 269,
"column": 2
} | {
"line": 271,
"column": 71
} | [
{
"pp": "Γ₀ : Type u_2\ninst✝⁴ : LinearOrderedCommGroupWithZero Γ₀\nR : Type u_3\ninst✝³ : CommRing R\ninst✝² : ValuativeRel R\ninst✝¹ : MulArchimedean Γ₀\nv : Valuation R Γ₀\ninst✝ : v.Compatible\n⊢ IsRankLeOne R",
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Eq.mpr",
"GroupWith... | rw [isRankLeOne_iff_mulArchimedean]
exact MulArchimedean.comap (embedding.toMonoidHom.comp (ValueGroupWithZero.embed v).toMonoidHom)
(embedding_strictMono.comp (ValueGroupWithZero.embed_strictMono v)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Real.Embedding | {
"line": 164,
"column": 63
} | {
"line": 164,
"column": 74
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : ℝ\nthis : ∀ (y : ℚ), y.num • 1 < 0 → ↑y = x → x ≤ 0\n⊢ x ∈ ⇑(Rat.castHom ℝ) '' {r | r.num • 1 < r.den • 0} → x ≤ 0",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 171,
"column": 65
} | {
"line": 171,
"column": 76
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : ℝ\nthis : (∀ (y : ℚ), y.num • 1 < 0 → ↑y ≤ x) → 0 ≤ x\n⊢ (∀ x_1 ∈ ratLt' 0, x_1 ≤ x) → 0 ≤ x",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 174,
"column": 49
} | {
"line": 174,
"column": 60
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : ℝ\nh : ∀ (y : ℚ), y.num • 1 < 0 → ↑y ≤ x\ny : ℚ\nhy : y < 0\n⊢ y.num < 0",
"usedConstants": [
"Rat.instOfNat",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 177,
"column": 17
} | {
"line": 177,
"column": 28
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : ℝ\nh : ∀ (y : ℚ), y.num • 1 < 0 → ↑y ≤ x\nh' : x < 0\ny : ℚ\nhxy : x < ↑y\nhy : ↑y < 0\n⊢ y < 0",
"usedConstants": [... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 192,
"column": 4
} | {
"line": 192,
"column": 15
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx y : M\nh : x < y\nhyz : 0 < y - x\nhy : y = y - x + x\nn : ℕ\nhn : 1 ≤ n • (y - x)\n⊢ ↑{ num := 1, den := n + 1, den_nz :=... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 215,
"column": 81
} | {
"line": 215,
"column": 92
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nthis : ∀ (x : ℚ), x.num • 1 < ↑x.den • 1 → ↑x ≤ 1\n⊢ ∀ b ∈ ratLt' 1, b ≤ 1",
"usedConstants": [
"Eq.mpr",
"R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 218,
"column": 4
} | {
"line": 218,
"column": 28
} | [
{
"pp": "case a\nM : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : ℚ\nhx : x.num • 1 < ↑x.den • 1\n⊢ x ≤ 1",
"usedConstants": [
"Rat.instOfNat",
"Eq.mpr",
"R... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 222,
"column": 6
} | {
"line": 222,
"column": 17
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nthis : ∀ (x : ℝ), (∀ (y : ℚ), y.num • 1 < ↑y.den • 1 → ↑y ≤ x) → 1 ≤ x\n⊢ ∀ (b : ℝ), (∀ x ∈ ratLt' 1, x ≤ b) → 1 ≤ b",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 225,
"column": 63
} | {
"line": 225,
"column": 74
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : ℝ\nh : ∀ (y : ℚ), y.num • 1 < ↑y.den • 1 → ↑y ≤ x\ny : ℚ\nhy : y < 1\n⊢ y.num < ↑y.den",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 228,
"column": 4
} | {
"line": 228,
"column": 40
} | [
{
"pp": "case a\nM : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : ℝ\nh : ∀ (y : ℚ), y.num • 1 < ↑y.den • 1 → ↑y ≤ x\nh' : x < 1\ny : ℚ\nhxy : x < ↑y\nhy : ↑y < 1\n⊢ ∃ y < 1, x < ... | exact ⟨y, (by norm_cast at hy), hxy⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 49,
"column": 2
} | {
"line": 49,
"column": 48
} | [
{
"pp": "S : Type u_1\ninst✝¹ : SeminormedGroup S\ninst✝ : IsUltrametricDist S\nx y : S\n⊢ ‖x * y‖ ≤ max ‖x‖ ‖y‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"Monoid.toMulOneClass",
"SeminormedGroup.toGroup",
"PartialOrder.to... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 55,
"column": 4
} | {
"line": 55,
"column": 38
} | [
{
"pp": "S' : Type u_2\ninst✝ : SeminormedGroup S'\nh : ∀ (x y : S'), ‖x * y‖ ≤ max ‖x‖ ‖y‖\nx y z : S'\n⊢ dist x z ≤ max (dist x y) (dist y z)",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"HMul.hMul",
"DivInvOneMonoid.toInvOneClass",
"Monoid.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 13
} | [
{
"pp": "R : Type u_4\ninst✝¹ : SeminormedAddCommGroup R\ninst✝ : IsUltrametricDist R\n⊢ IsNonarchimedean fun x ↦ ↑‖x‖₊",
"usedConstants": [
"Real",
"AddCommGroup.toAddCommMonoid",
"SeminormedAddGroup.toNNNorm",
"NNNorm.nnnorm",
"id",
"SeminormedAddCommGroup.toSeminormedA... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 53
} | [
{
"pp": "S : Type u_1\ninst✝¹ : SeminormedGroup S\ninst✝ : IsUltrametricDist S\nx y : S\nh : ‖x‖₊ ≠ ‖y‖₊\n⊢ ‖x * y‖₊ = max ‖x‖₊ ‖y‖₊",
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"NNNorm.nnnorm",
"SeminormedGroup.toGroup",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 122,
"column": 2
} | {
"line": 122,
"column": 39
} | [
{
"pp": "S : Type u_1\ninst✝¹ : SeminormedGroup S\ninst✝ : IsUltrametricDist S\nx y z : S\nh : ‖x / y‖ ≠ ‖y / z‖\n⊢ ‖x / z‖ = max ‖x / y‖ ‖y / z‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 53
} | [
{
"pp": "S : Type u_1\ninst✝¹ : SeminormedGroup S\ninst✝ : IsUltrametricDist S\nx y z : S\nh : ‖x / y‖₊ ≠ ‖y / z‖₊\n⊢ ‖x / z‖₊ = max ‖x / y‖₊ ‖y / z‖₊",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"congrArg",
"NNNorm.nnnorm",
"SeminormedGroup.toGroup",
"Semilat... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 136,
"column": 17
} | {
"line": 136,
"column": 42
} | [
{
"pp": "case succ\nS : Type u_1\ninst✝¹ : SeminormedGroup S\ninst✝ : IsUltrametricDist S\nx : S\nn : ℕ\nhn : ‖x ^ n‖₊ ≤ ‖x‖₊\n⊢ ‖x ^ (n + 1)‖₊ ≤ ‖x‖₊",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"NNNorm.nnnorm",
"SeminormedGroup.toGroup... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 13
} | [
{
"pp": "case ofNat\nS : Type u_1\ninst✝¹ : SeminormedGroup S\ninst✝ : IsUltrametricDist S\nx : S\na✝ : ℕ\n⊢ ‖x ^ Int.ofNat a✝‖₊ ≤ ‖x‖₊",
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"congrArg",
"NNNorm.nnnorm",
"SeminormedGroup.toGroup",
"PartialOrder.toPreorder",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 13
} | [
{
"pp": "case negSucc\nS : Type u_1\ninst✝¹ : SeminormedGroup S\ninst✝ : IsUltrametricDist S\nx : S\na✝ : ℕ\n⊢ ‖x ^ Int.negSucc a✝‖₊ ≤ ‖x‖₊",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"congrArg",
"zpow_negSucc",
"NNNorm.nnnorm",
"SeminormedGroup.toGroup",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 198,
"column": 26
} | {
"line": 199,
"column": 9
} | [
{
"pp": "M : Type u_1\nι : Type u_2\ninst✝¹ : SeminormedCommGroup M\ninst✝ : IsUltrametricDist M\n⊢ ∀ U ∈ nhds 1, ∃ V, ↑V ⊆ U",
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"OpenSubgroup",
"Real",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"Real.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 215,
"column": 6
} | {
"line": 215,
"column": 60
} | [
{
"pp": "case cons.refine_1\nM : Type u_1\nι : Type u_2\ninst✝¹ : SeminormedCommGroup M\ninst✝ : IsUltrametricDist M\ns : Finset ι\nf : ι → M\nj : ι\nt : Finset ι\nhj : j ∉ t\nhs✝ : t.Nonempty\nIH : ∃ b ∈ t, ‖∏ i ∈ t, f i‖ ≤ ‖f b‖\nh : ‖∏ i ∈ t, f i‖ ≤ ‖f j‖\n⊢ ‖f j * ∏ i ∈ t, f i‖ ≤ ‖f j‖",
"usedConstants"... | · exact (norm_mul_le_max _ _).trans (max_eq_left h).le | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 227,
"column": 4
} | {
"line": 228,
"column": 11
} | [
{
"pp": "case inr\nM : Type u_1\nι : Type u_2\ninst✝¹ : SeminormedCommGroup M\ninst✝ : IsUltrametricDist M\ns : Finset ι\nf : ι → M\nhs : s.Nonempty\n⊢ ‖∏ i ∈ s, f i‖₊ ≤ s.sup fun x ↦ ‖f x‖₊",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"NNNorm.nnnorm",
"PartialOrder.toP... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.LexOrder | {
"line": 80,
"column": 40
} | {
"line": 82,
"column": 14
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝² : Semiring R\ninst✝¹ : LinearOrder σ\ninst✝ : WellFoundedGT σ\nφ : MvPowerSeries σ R\nd : σ →₀ ℕ\nh : ↑(toLex d) = φ.lexOrder\nhφ : φ ≠ 0\nne : (⇑toLex '' Function.support φ).Nonempty\nhφ' : toLex d = ⋯.min (⇑toLex '' Function.support φ) ne\nthis : toLex d ∈ ⇑toLex ''... | by
simp only [Set.mem_image_equiv, toLex_symm_eq, ofLex_toLex, Function.mem_support, ne_eq] at this
apply this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 339,
"column": 2
} | {
"line": 339,
"column": 63
} | [
{
"pp": "M : Type u_1\nι : Type u_2\ninst✝¹ : SeminormedCommGroup M\ninst✝ : IsUltrametricDist M\nf : ι → M\n⊢ ‖∏' (i : ι), f i‖₊ ≤ ⨆ i, ‖f i‖₊",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"NNReal.coe_iSup",
"congrArg",
"iSup",
"NNNorm.nn... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 350,
"column": 4
} | {
"line": 350,
"column": 45
} | [
{
"pp": "case inl\nM : Type u_1\nι : Type u_2\ninst✝¹ : SeminormedCommGroup M\ninst✝ : IsUltrametricDist M\nf : ι → M\nC : ℝ\nhC : 0 ≤ C\nh : ∀ (i : ι), ‖f i‖ ≤ C\nh✝ : IsEmpty ι\n⊢ ‖∏' (i : ι), f i‖ ≤ C",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"MulOne.toOne",
"Real.instLE",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 368,
"column": 6
} | {
"line": 368,
"column": 22
} | [
{
"pp": "M : Type u_1\nι : Type u_2\ninst✝¹ : SeminormedCommGroup M\ninst✝ : IsUltrametricDist M\nf : ι → M\na : ι\ns : Finset ι\nha : a ∉ s\nhs : (↑(Finset.cons a s ha)).Pairwise fun i j ↦ ‖f i‖₊ ≠ ‖f j‖₊\nhs' : s.Nonempty\nIH : ‖∏ i ∈ s, f i‖₊ = s.sup fun i ↦ ‖f i‖₊\n⊢ ∃ j ∈ s, ‖∏ i ∈ s, f i‖₊ = ‖f j‖₊",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 379,
"column": 4
} | {
"line": 379,
"column": 34
} | [
{
"pp": "M : Type u_1\nι : Type u_2\ninst✝¹ : SeminormedCommGroup M\ninst✝ : IsUltrametricDist M\ns : Finset ι\nf : ι → M\nhs' : s.Nonempty\nhs : (↑s).Pairwise fun i j ↦ ‖f i‖ ≠ ‖f j‖\n⊢ (↑s).Pairwise fun i j ↦ ‖f i‖₊ ≠ ‖f j‖₊",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"cong... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.LexOrder | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 10
} | [
{
"pp": "case h\nσ : Type u_1\nR : Type u_2\ninst✝² : Semiring R\ninst✝¹ : LinearOrder σ\ninst✝ : WellFoundedGT σ\nφ ψ : MvPowerSeries σ R\nd : σ →₀ ℕ\nhd : ↑(toLex d) < φ.lexOrder + ψ.lexOrder\nu v : σ →₀ ℕ\nh : u + v = d\n⊢ φ.lexOrder ≤ ↑(toLex u) → ¬ψ.lexOrder ≤ ↑(toLex v)",
"usedConstants": [
"Nat... | intro hu | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Topology.Algebra.Valued.NormedValued | {
"line": 87,
"column": 12
} | {
"line": 88,
"column": 40
} | [
{
"pp": "case neg.hab\nK : Type u_1\nhK : NormedField K\ninst✝ : IsUltrametricDist K\nU : Set K\nε : ℝ\nhε : ε > 0\nh : Metric.ball 0 ε ⊆ U\nH : Subsingleton (ValueGroup₀ valuation)ˣ\nx : K\nhx : valuation.restrict x < 1\nhx0 : IsUnit (valuation.restrict x)\n⊢ valuation.restrict x = 1",
"usedConstants": []
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 15
} | [
{
"pp": "case mp\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nn : σ →₀ ℕ\nr : R\nH : (monomial n) r ∈ nonZeroDivisorsLeft (MvPowerSeries σ R)\ns : R\nhrs : r * s = 0\nthis : C s = 0\n⊢ s = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.NormedValued | {
"line": 96,
"column": 10
} | {
"line": 96,
"column": 62
} | [
{
"pp": "case h.a\nK : Type u_1\nhK : NormedField K\ninst✝ : IsUltrametricDist K\nU : Set K\nε : ℝ\nhε : ε > 0\nh : Metric.ball 0 ε ⊆ U\nH : ∀ {γ : ℝ≥0}, γ ≠ 0 → ∃ x, x ≠ 0 ∧ (RankLeOne.hom' valuation) (valuation.restrict x) < γ\nx : K\nhx : x ≠ 0\nhxy : (RankLeOne.hom' valuation) (valuation.restrict x) < ⟨ε, ⋯... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 15
} | [
{
"pp": "case mpr.h\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nn : σ →₀ ℕ\nr : R\nH : r ∈ nonZeroDivisorsLeft R\np : MvPowerSeries σ R\nhrp : (monomial n) r * p = 0\ni : σ →₀ ℕ\nthis : r * (coeff i) p = (coeff (i + n)) 0\n⊢ (coeff i) p = (coeff i) 0",
"usedConstants": [
"Semiring.toModule",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 15
} | [
{
"pp": "case mp\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nn : σ →₀ ℕ\nr : R\nH : (monomial n) r ∈ nonZeroDivisorsRight (MvPowerSeries σ R)\ns : R\nhrs : s * r = 0\nthis : C s = 0\n⊢ s = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.NormedValued | {
"line": 100,
"column": 10
} | {
"line": 100,
"column": 52
} | [
{
"pp": "case refine_2.refine_2\nK : Type u_1\nhK : NormedField K\ninst✝ : IsUltrametricDist K\nU : Set K\nε : (ValueGroup₀ valuation)ˣ\nhε : {x | valuation.restrict x < ↑ε} ⊆ U\nx : K\nhx : x ∈ Metric.ball 0 ↑(embedding ↑ε)\n⊢ x ∈ {x | valuation.restrict x < ↑ε}",
"usedConstants": [
"NormedCommRing.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 15
} | [
{
"pp": "case mpr.h\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nn : σ →₀ ℕ\nr : R\nH : r ∈ nonZeroDivisorsRight R\np : MvPowerSeries σ R\nhrp : p * (monomial n) r = 0\ni : σ →₀ ℕ\nthis : (coeff i) p * r = (coeff (i + n)) 0\n⊢ (coeff i) p = (coeff i) 0",
"usedConstants": [
"Semiring.toModule",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors | {
"line": 144,
"column": 4
} | {
"line": 144,
"column": 86
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\nφ ψ : MvPowerSeries σ R\nh : φ * ψ = 0\nw✝ : LinearOrder σ\nh✝ : WellFoundedGT σ\n⊢ φ = 0 ∨ ψ = 0",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"MvPowerSeries.instZero",
"_private.Mathlib.Rin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.NormedValued | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 91
} | [
{
"pp": "L : Type u_1\ninst✝¹ : Field L\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation L Γ₀\nhv : v.RankOne\nx : L\nhx : v.norm x = 0\n⊢ x = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Inverse | {
"line": 158,
"column": 8
} | {
"line": 158,
"column": 19
} | [
{
"pp": "case inl.h\nσ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsLocalRing R\nφ : MvPowerSeries σ R\nu : Rˣ\nh : ↑u = constantCoeff φ\n⊢ constantCoeff φ = ↑u",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Inverse | {
"line": 158,
"column": 8
} | {
"line": 158,
"column": 19
} | [
{
"pp": "case inr.h\nσ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : IsLocalRing R\nφ : MvPowerSeries σ R\nu : Rˣ\nh : ↑u = 1 - constantCoeff φ\n⊢ constantCoeff (1 - φ) = ↑u",
"usedConstants": [
"Units.val",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"RingHom.instR... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPowerSeries.Inverse | {
"line": 212,
"column": 15
} | {
"line": 212,
"column": 26
} | [
{
"pp": "σ : Type u_1\nk : Type u_3\ninst✝ : Field k\nφ : MvPowerSeries σ k\nh : φ⁻¹ = 0\n⊢ constantCoeff φ = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.NormedValued | {
"line": 175,
"column": 10
} | {
"line": 184,
"column": 61
} | [
{
"pp": "case h.refine_1\nL : Type u_1\ninst✝¹ : Field L\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nval : Valued L Γ₀\nhv : v.RankOne\nthis : Nonempty { ε // ε > 0 }\nU : Set (L × L)\nx✝ : ∃ i, {p | v.restrict (p.2 - p.1) < ↑i} ⊆ U\nε : (ValueGroup₀ v)ˣ\nhε : {p | v.restrict (p.2 - p.1) < ↑ε} ⊆ ... | set δ : ℝ≥0 := hv.hom _ ε with hδ
have hδ_pos : 0 < δ := by
rw [hδ, ← map_zero hv.hom]
exact hv.strictMono _ (Units.zero_lt ε)
use δ, hδ_pos
apply subset_trans _ hε
intro x hx
simp only [mem_setOf_eq, Valuation.norm, hδ, NNReal.coe_lt_coe] at hx
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors | {
"line": 173,
"column": 4
} | {
"line": 173,
"column": 29
} | [
{
"pp": "case neg\nσ : Type u_1\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\nw : σ → ℕ\nf g : MvPowerSeries σ R\nhf : ¬weightedOrder w f < ⊤\n⊢ weightedOrder w (f * g) ≤ weightedOrder w f + weightedOrder w g",
"usedConstants": [
"Preorder.toLT",
"instLinearOrderENat",
"not... | rw [not_lt_top_iff] at hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.Valued.NormedValued | {
"line": 175,
"column": 10
} | {
"line": 184,
"column": 61
} | [
{
"pp": "case h.refine_1\nL : Type u_1\ninst✝¹ : Field L\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nval : Valued L Γ₀\nhv : v.RankOne\nthis : Nonempty { ε // ε > 0 }\nU : Set (L × L)\nx✝ : ∃ i, {p | v.restrict (p.2 - p.1) < ↑i} ⊆ U\nε : (ValueGroup₀ v)ˣ\nhε : {p | v.restrict (p.2 - p.1) < ↑ε} ⊆ ... | set δ : ℝ≥0 := hv.hom _ ε with hδ
have hδ_pos : 0 < δ := by
rw [hδ, ← map_zero hv.hom]
exact hv.strictMono _ (Units.zero_lt ε)
use δ, hδ_pos
apply subset_trans _ hε
intro x hx
simp only [mem_setOf_eq, Valuation.norm, hδ, NNReal.coe_lt_coe] at hx
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Inverse | {
"line": 280,
"column": 2
} | {
"line": 280,
"column": 13
} | [
{
"pp": "case inr\nσ : Type u_1\nk : Type u_3\ninst✝ : Field k\nr : k\nhr : r ≠ 0\n⊢ constantCoeff (C r) ≠ 0",
"usedConstants": [
"MvPowerSeries",
"RingHom",
"MvPowerSeries.constantCoeff",
"id",
"Ne",
"Field.toSemifield",
"MvPowerSeries.instSemiring",
"RingHom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.NoZeroDivisors | {
"line": 65,
"column": 4
} | {
"line": 65,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nh : X = 0\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.NormedValued | {
"line": 247,
"column": 2
} | {
"line": 247,
"column": 28
} | [
{
"pp": "L : Type u_1\ninst✝¹ : Field L\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nval : Valued L Γ₀\nhv : v.RankOne\nx : L\n⊢ ‖x‖ ≤ 1 ↔ v.restrict x ≤ v.restrict 1",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"LinearOrderedCommGroupWi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.NormedValued | {
"line": 252,
"column": 2
} | {
"line": 252,
"column": 28
} | [
{
"pp": "L : Type u_1\ninst✝¹ : Field L\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nval : Valued L Γ₀\nhv : v.RankOne\nx : L\n⊢ ‖x‖ < 1 ↔ v.restrict x < v.restrict 1",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"LinearOrderedCommGroupWi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.NormedValued | {
"line": 257,
"column": 2
} | {
"line": 257,
"column": 28
} | [
{
"pp": "L : Type u_1\ninst✝¹ : Field L\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nval : Valued L Γ₀\nhv : v.RankOne\nx : L\n⊢ 1 ≤ ‖x‖ ↔ v.restrict 1 ≤ v.restrict x",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"LinearOrderedCommGroupWi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Valued.NormedValued | {
"line": 262,
"column": 2
} | {
"line": 262,
"column": 28
} | [
{
"pp": "L : Type u_1\ninst✝¹ : Field L\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nval : Valued L Γ₀\nhv : v.RankOne\nx : L\n⊢ 1 < ‖x‖ ↔ v.restrict 1 < v.restrict x",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"LinearOrderedCommGroupWi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.Inverse | {
"line": 80,
"column": 6
} | {
"line": 80,
"column": 42
} | [
{
"pp": "case neg.e_a.h.inr.hc.refine_1\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : R⟦X⟧\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ antidiagonal n\nH : j < n\n⊢ (match (i, j) with\n | (a, b) => (single () a, single () b)).2\n PUnit.unit ≤\n (single () n) PUnit.unit",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.Inverse | {
"line": 81,
"column": 6
} | {
"line": 81,
"column": 42
} | [
{
"pp": "case neg.e_a.h.inr.hc.refine_2\nR : Type u_1\ninst✝ : Ring R\nn : ℕ\na : R\nφ : R⟦X⟧\nh✝ : ¬n = 0\ni j : ℕ\n_hij : (i, j) ∈ antidiagonal n\nH : j < n\nhh :\n single () n ≤\n (match (i, j) with\n | (a, b) => (single () a, single () b)).2\n⊢ n ≤ j",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PowerSeries.Inverse | {
"line": 278,
"column": 7
} | {
"line": 283,
"column": 42
} | [
{
"pp": "k : Type u_2\ninst✝ : Field k\n⊢ ∀ {x : k⟦X⟧}, x ≠ 0 → ∃ n, Associated (X ^ n) x",
"usedConstants": [
"Units.val",
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiring",
"PowerSeries.divXPowOrder",
"Units",
"id",
... | by
intro f hf
use f.order.toNat
use Unit_of_divided_by_X_pow_order f
simp only [Unit_of_divided_by_X_pow_order_nonzero hf]
exact X_pow_order_mul_divXPowOrder | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 35
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Projective R\nP : Fin 3 → R\nu : R\n⊢ W'.neg (u • P) = u • W'.neg P",
"usedConstants": [
"WeierstrassCurve.Projective.neg._proof_1",
"WeierstrassCurve.Projective.negY",
"Eq.mpr",
"instHSMul",
"instSMulOfMul",
"HMul.hMul",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point | {
"line": 246,
"column": 4
} | {
"line": 246,
"column": 43
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Projective F\ninst✝ : DecidableEq F\nP Q : Fin 3 → F\nhP : W.Equation P\nhQ : W.Equation Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhx : P x * Q z = Q x * P z\nhy : P y * Q z ≠ W.negY Q * P z\n⊢ W.dblXYZ P =\n W.dblZ P •\n ![W.toAffine.addX (P x / P z) (Q x / Q z) (W.toA... | dblXYZ_of_Z_ne_zero hP hQ hPz hQz hx hy | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Completion.FinitePlace | {
"line": 61,
"column": 2
} | {
"line": 61,
"column": 13
} | [
{
"pp": "A : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : IsDedekindDomain A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nv : HeightOneSpectrum A\nhv : Finite (A ⧸ v.asIdeal)\n⊢ ¬(Set.range ⇑(valuation K v)).Subsingleton",
"usedConstants": [
"Int.instAddCommGroup",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Completion.FinitePlace | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 13
} | [
{
"pp": "A : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : IsDedekindDomain A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nv : HeightOneSpectrum A\nhv : Finite (A ⧸ v.asIdeal)\n⊢ ¬(Set.range ⇑Valued.v).Subsingleton",
"usedConstants": [
"Int.instAddCommGroup",
"Eq.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Completion.FinitePlace | {
"line": 349,
"column": 29
} | {
"line": 349,
"column": 52
} | [
{
"pp": "K : Type u_1\ninst✝⁵ : Field K\nR : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : Algebra R K\ninst✝² : IsDedekindDomain R\ninst✝¹ : IsFractionRing R K\nv✝ : HeightOneSpectrum R\ninst✝ : NumberField K\nv : FinitePlace K\nx y : K\n⊢ v (x + y) ≤ v x + v y",
"usedConstants": [
"Real.partialOrder",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.Completion.FinitePlace | {
"line": 377,
"column": 6
} | {
"line": 377,
"column": 29
} | [
{
"pp": "case h\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nv₁ v₂ : HeightOneSpectrum (𝓞 K)\nh : v₁ ≠ v₂\nx : 𝓞 K\nhx1 : x ∈ v₁.asIdeal\nhx2 : x ∉ v₂.asIdeal\n⊢ ‖(embedding v₁) ↑x‖ ≠ ‖(embedding v₂) ↑x‖",
"usedConstants": [
"Int.instAddCommGroup",
"Norm.norm",
"Multiplicative... | ← norm_lt_one_iff_mem K | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Completion.FinitePlace | {
"line": 441,
"column": 2
} | {
"line": 441,
"column": 22
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nv : FinitePlace K\nx y : K\nw : HeightOneSpectrum (𝓞 K)\nhw : place (embedding w) = ↑v\nH : ∀ (x : K), v x = (HeightOneSpectrum.adicAbv K w) x\n⊢ v (x + y) ≤ max (v x) (v y)",
"usedConstants": [
"Eq.mpr",
"NumberField.HeightOneSpec... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Fiber | {
"line": 68,
"column": 4
} | {
"line": 68,
"column": 15
} | [
{
"pp": "case refine_1\nP X Y Z : Scheme\nfst : P ⟶ X\nsnd : P ⟶ Y\nf : X ⟶ Z\ng : Y ⟶ Z\nh : IsPullback fst snd f g\ny : ↥Y\n⊢ IsPullback\n (pullback.map snd (Y.fromSpecResidueField y) f (Z.fromSpecResidueField (g y)) fst\n (Spec.map (Scheme.Hom.residueFieldMap g y)) g ⋯ ⋯ ≫\n pullback.fst f (Z.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyOpen | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 42
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\n⊢ IsZariskiLocalAtTarget (topologically @IsOpenMap).universally",
"usedConstants": [
"IsOpenMap",
"AlgebraicGeometry.topologically",
"AlgebraicGeometry.universally_isZariskiLocalAtTarget"
]
}
] | apply universally_isZariskiLocalAtTarget | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.AlgebraicGeometry.Geometrically.Connected | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 39
} | [
{
"pp": "X S : Scheme\nf : X ⟶ S\ninst✝¹ : GeometricallyConnected f\ninst✝ : ConnectedSpace ↥S\nhf : IsOpenMap ⇑f\n⊢ ConnectedSpace ↥X",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.PresheafedSpace.carrier",
"IsC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyOpen | {
"line": 165,
"column": 4
} | {
"line": 165,
"column": 42
} | [
{
"pp": "case inr\nX Y : Scheme\nf : X ⟶ Y\nS K : CommRingCat\nhK : IsField ↑K\ninst✝¹ : IsIntegral (Spec K)\ninst✝ : Subsingleton ↥(Spec K)\nφ : K ⟶ S\nT : Scheme\ng : T ⟶ Spec K\nx✝ : HasPullback g (Spec.map φ)\nthis :\n ∀ {T : Scheme} (g : T ⟶ Spec K) (x : HasPullback g (Spec.map φ)),\n (∃ R, T = Spec R)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.SchemeTheoreticallyDominant | {
"line": 110,
"column": 8
} | {
"line": 110,
"column": 38
} | [
{
"pp": "X Y : Scheme\nZ : Scheme\nS : Scheme\nf✝ : X ⟶ S\ng✝ : Y ⟶ S\nf : X ⟶ S\ng : Y ⟶ S\ninst✝² : IsSchemeTheoreticallyDominant f\ninst✝¹ : QuasiCompact f\ninst✝ : Flat g\nh𝒰 : TopologicalSpace.IsOpenCover fun V ↦ (↑V).1 :=\n TopologicalSpace.Opens.IsBasis.isOpenCover_mem_and_le (Scheme.isBasis_affineOpen... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Morphisms.SchemeTheoreticallyDominant | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 67
} | [
{
"pp": "case a\nX Y : Scheme\nZ : Scheme\nS : Scheme\nf✝ : X ⟶ S\ng✝ : Y ⟶ S\nf : X ⟶ S\ng : Y ⟶ S\ninst✝² : IsSchemeTheoreticallyDominant f\ninst✝¹ : QuasiCompact f\ninst✝ : Flat g\nh𝒰 : TopologicalSpace.IsOpenCover fun V ↦ (↑V).1 := ⋯\nV : TopologicalSpace.Opens ↥Y\nU : TopologicalSpace.Opens ↥S\nhU : U ∈ S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Geometrically.Irreducible | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 36
} | [
{
"pp": "X S : Scheme\nf : X ⟶ S\ninst✝¹ : GeometricallyIrreducible f\ninst✝ : IrreducibleSpace ↥S\nhf : IsOpenMap ⇑f\n⊢ IrreducibleSpace ↥X",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.SheafedSpace.instTopologicalSpaceCarrierCarrier",
"AlgebraicGeometry.PresheafedSpace.carrier",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Geometrically.Reduced | {
"line": 84,
"column": 2
} | {
"line": 95,
"column": 66
} | [
{
"pp": "X Y : Scheme\nf : X ⟶ Y\ninst✝³ : GeometricallyReduced f\ninst✝² : Flat f\ninst✝¹ : IsReduced Y\ninst✝ : Finite ↑(irreducibleComponents ↥Y)\npt : ↑(irreducibleComponents ↥Y) → CommRingCat := fun Z ↦ Y.presheaf.stalk ⋯.genericPoint\nhpt : ∀ (Z : ↑(irreducibleComponents ↥Y)), IsField ↑(pt Z)\nthis✝¹ : (Z... | have H : IsSchemeTheoreticallyDominant g := by
rw [isSchemeTheoreticallyDominant_iff_isDominant, isDominant_iff, denseRange_iff_closure_range,
Set.eq_univ_iff_forall]
intro y
let z : Z := Sigma.ι (fun Z ↦ Spec (pt Z)) ⟨_, irreducibleComponent_mem_irreducibleComponents y⟩
(IsLocalRing.closedPoint... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes | {
"line": 189,
"column": 4
} | {
"line": 189,
"column": 65
} | [
{
"pp": "case refine_1\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : C\nE : PreOneHypercover X\nF : Cᵒᵖ ⥤ Type u_2\nh₁ : E.IsStronglySheafFor F\nh₂ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSeparatedFor F (Sieve.pullback f E.sieve₀).arrows\nS : Sieve X\nH : ∀ (i : E.I₀), Presieve.IsSheafFor F (Sieve.pullback (E.f... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes | {
"line": 236,
"column": 2
} | {
"line": 236,
"column": 11
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nJ : GrothendieckTopology C\nX : C\nE : J.OneHypercover X\nF : Cᵒᵖ ⥤ Type u_2\nhF : Presieve.IsSheaf J F\nS : Sieve X\nh₁ : ∀ (i : E.I₀), Presieve.IsSheafFor F (Sieve.pullback (E.f i) S).arrows\nh₂ : ∀ ⦃i j : E.I₀⦄ (k : E.I₁ i j), Presieve.IsSeparatedFor F (S... | intro Y f | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.CategoryTheory.Sites.Canonical | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 13
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nP : Cᵒᵖ ⥤ Type w\nPs : Set (Cᵒᵖ ⥤ Type w)\nh : P ∈ Ps\nX : C\nS : Sieve X\nhS : S ∈ (finestTopology Ps) X\n⊢ Presieve.IsSheafFor P S.arrows",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Category.TopCat.GrothendieckTopology | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 13
} | [
{
"pp": "X : TopCat\nE : precoverage.ZeroHypercover X\n⊢ ∀ (x : ↑X), ∃ i, x ∈ Set.range ⇑(ConcreteCategory.hom (E.f i))",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.PreZeroHypercover.f",
"TopCat.precoverage",
"congrArg",
"CategoryTheory.ConcreteCategory.hom",
"TopCat.in... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Category.TopCat.GrothendieckTopology | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 13
} | [
{
"pp": "X : TopCat\nE : precoverage.ZeroHypercover X\n⊢ ∀ (i : E.I₀), Topology.IsOpenEmbedding ⇑(ConcreteCategory.hom (E.f i))",
"usedConstants": [
"CategoryTheory.PreZeroHypercover.f",
"TopCat.precoverage",
"CategoryTheory.ConcreteCategory.hom",
"TopCat.instCategory",
"Contin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Category.TopCat.GrothendieckTopology | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 26
} | [
{
"pp": "X : TopCat\nE : precoverage.ZeroHypercover X\n⊢ ∀ (i : E.I₀), Topology.IsOpenEmbedding ⇑(ConcreteCategory.hom (E.f i))",
"usedConstants": [
"CategoryTheory.PreZeroHypercover.f",
"TopCat.precoverage",
"TopCat.isOpenEmbedding_iff._simp_1",
"CategoryTheory.Presieve",
"Cat... | simpa using E.mem₀.right | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.Category.TopCat.GrothendieckTopology | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 26
} | [
{
"pp": "X : TopCat\nE : precoverage.ZeroHypercover X\n⊢ ∀ (i : E.I₀), Topology.IsOpenEmbedding ⇑(ConcreteCategory.hom (E.f i))",
"usedConstants": [
"CategoryTheory.PreZeroHypercover.f",
"TopCat.precoverage",
"TopCat.isOpenEmbedding_iff._simp_1",
"CategoryTheory.Presieve",
"Cat... | simpa using E.mem₀.right | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Category.TopCat.GrothendieckTopology | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 26
} | [
{
"pp": "X : TopCat\nE : precoverage.ZeroHypercover X\n⊢ ∀ (i : E.I₀), Topology.IsOpenEmbedding ⇑(ConcreteCategory.hom (E.f i))",
"usedConstants": [
"CategoryTheory.PreZeroHypercover.f",
"TopCat.precoverage",
"TopCat.isOpenEmbedding_iff._simp_1",
"CategoryTheory.Presieve",
"Cat... | simpa using E.mem₀.right | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Category.TopCat.GrothendieckTopology | {
"line": 105,
"column": 6
} | {
"line": 105,
"column": 17
} | [
{
"pp": "case refine_1.refine_2\nX Y : TopCat\n𝒰 : precoverage.ZeroHypercover Y\nheq : ∀ (i : 𝒰.I₀) (y : ↑(𝒰.X i)), (ConcreteCategory.hom (𝒰.f i)) y = ↑⟨(ConcreteCategory.hom (𝒰.f i)) y, ⋯⟩\nx : (i : 𝒰.I₀) → (yoneda.obj X).obj (Opposite.op (𝒰.X i))\nhx : Presieve.Arrows.Compatible (yoneda.obj X) 𝒰.f x\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.Sites.BigZariski | {
"line": 97,
"column": 17
} | {
"line": 97,
"column": 28
} | [
{
"pp": "X : Scheme\ni j :\n (X.affineCover.refineOneHypercover fun i j ↦\n (pullback (X.affineCover.f i) (X.affineCover.f j)).affineCover.toPreZeroHypercover).I₀\n⊢ (X.affineCover.refineOneHypercover fun i j ↦\n (pullback (X.affineCover.f i) (X.affineCover.f j)).affineCover.toPreZeroHypercover).... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.AlgebraicGeometry.GluingOneHypercover | {
"line": 60,
"column": 28
} | {
"line": 60,
"column": 39
} | [
{
"pp": "D : GlueData\ni₁ i₂ : D.J\nW : Scheme\np₁ : W ⟶ D.U i₁\np₂ : W ⟶ D.U i₂\nfac : p₁ ≫ D.ι i₁ = p₂ ≫ D.ι i₂\nT : Scheme\ng : T ⟶ W\nx✝ : ⊤.arrows g\n⊢ (g ≫ p₁) ≫ D.ι i₁ = (g ≫ p₂) ≫ D.ι i₂",
"usedConstants": [
"AlgebraicGeometry.Scheme.GlueData.ι",
"Eq.mpr",
"CategoryTheory.Category.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Category.TopCat.GrothendieckTopology | {
"line": 125,
"column": 2
} | {
"line": 137,
"column": 57
} | [
{
"pp": "⊢ precoverage ≤ Precoverage.comap uliftFunctor precoverage",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.PreZeroHypercover.f",
"_private.Mathlib.Topology.Category.TopCat.GrothendieckTopology.0.TopCat.precoverage_le_comap_uliftFunctor._simp_1_5",
"TopCat.precoverage",
... | refine Precoverage.le_of_zeroHypercover fun X E ↦ ?_
refine ⟨?_, ?_⟩
· simp only [Presieve.map_ofArrows, Precoverage.mem_comap_iff,
Types.ofArrows_mem_jointlySurjectivePrecoverage_iff, ConcreteCategory.hom_ofHom,
Set.mem_range, TypeCat.Fun.coe_mk]
intro ⟨x⟩
obtain ⟨i, y, rfl⟩ := exists_mem_zeroH... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Category.TopCat.GrothendieckTopology | {
"line": 125,
"column": 2
} | {
"line": 137,
"column": 57
} | [
{
"pp": "⊢ precoverage ≤ Precoverage.comap uliftFunctor precoverage",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.PreZeroHypercover.f",
"_private.Mathlib.Topology.Category.TopCat.GrothendieckTopology.0.TopCat.precoverage_le_comap_uliftFunctor._simp_1_5",
"TopCat.precoverage",
... | refine Precoverage.le_of_zeroHypercover fun X E ↦ ?_
refine ⟨?_, ?_⟩
· simp only [Presieve.map_ofArrows, Precoverage.mem_comap_iff,
Types.ofArrows_mem_jointlySurjectivePrecoverage_iff, ConcreteCategory.hom_ofHom,
Set.mem_range, TypeCat.Fun.coe_mk]
intro ⟨x⟩
obtain ⟨i, y, rfl⟩ := exists_mem_zeroH... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Unramified.Basic | {
"line": 96,
"column": 6
} | {
"line": 96,
"column": 36
} | [
{
"pp": "R : Type v\ninst✝⁶ : CommRing R\nA : Type u\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\ninst✝³ : Small.{w, u} A\nH :\n ∀ ⦃B : Type w⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B),\n I ^ 2 = ⊥ → Function.Injective (Ideal.Quotient.mkₐ R I).comp\nB : Type u\ninst✝² : CommRing B\ninst✝¹ : S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.Basic | {
"line": 100,
"column": 12
} | {
"line": 100,
"column": 97
} | [
{
"pp": "R : Type v\ninst✝⁶ : CommRing R\nA : Type u\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\ninst✝³ : Small.{w, u} A\nH :\n ∀ ⦃B : Type w⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B),\n I ^ 2 = ⊥ → Function.Injective (Ideal.Quotient.mkₐ R I).comp\nB : Type u\ninst✝² : CommRing B\ninst✝¹ : S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.Basic | {
"line": 113,
"column": 6
} | {
"line": 113,
"column": 17
} | [
{
"pp": "case mpr.subsingleton_kaehlerDifferential.a.refine_2.H.a\nR : Type v\ninst✝³ : CommRing R\nA : Type u\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : Small.{w, u} A\nH :\n ∀ ⦃B : Type u⦄ [inst : CommRing B] [Small.{w, u} B] [inst_2 : Algebra R B] (I : Ideal B),\n I ^ 2 = ⊥ → Function.Injective ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.Basic | {
"line": 179,
"column": 6
} | {
"line": 179,
"column": 17
} | [
{
"pp": "case neg\nR : Type v\ninst✝⁵ : CommRing R\nA : Type u\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\nB : Type w\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\nI : Ideal B\ninst✝ : FormallyUnramified R A\nhI : ⨅ i, I ^ i = ⊥\ng₁ g₂ : A →ₐ[R] B\nH : ∀ (x : A), (Ideal.Quotient.mk I) (g₁ x) = (Ideal.Quotient.mk ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Unramified.Basic | {
"line": 237,
"column": 2
} | {
"line": 237,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝⁹ : CommRing R\nA : Type u_2\ninst✝⁸ : CommRing A\ninst✝⁷ : Algebra R A\nB : Type u_3\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R B\ninst✝⁴ : Algebra A B\ninst✝³ : IsScalarTower R A B\ninst✝² : FormallyUnramified R B\nQ : Type u_3\ninst✝¹ : CommRing Q\ninst✝ : Algebra A Q\nI : Ideal Q\n... | refine FormallyUnramified.ext I ⟨2, e⟩ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Smooth.Basic | {
"line": 90,
"column": 35
} | {
"line": 90,
"column": 46
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\nB : Type u_1\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : FormallySmooth R A\nI : Ideal B\nhI : I ^ 2 = ⊥\nf : A →ₐ[R] B ⧸ I\nP : Generators R A A := Generators.self R A\nhP : Function.Injective ⇑P.toExtensio... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Basic | {
"line": 107,
"column": 4
} | {
"line": 107,
"column": 15
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\nB : Type u_1\ninst✝² : CommRing B\ninst✝¹ : Algebra R B\ninst✝ : FormallySmooth R A\nI : Ideal B\nhI : I ^ 2 = ⊥\nf : A →ₐ[R] B ⧸ I\nP : Generators R A A := Generators.self R A\nhP : Function.Injective ⇑P.toExtensio... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Basic | {
"line": 214,
"column": 8
} | {
"line": 214,
"column": 23
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A\ninst✝⁷ : Algebra R A\nB : Type u_1\nP : Type u_2\nC : Type u_3\ninst✝⁶ : CommRing B\ninst✝⁵ : Algebra R B\ninst✝⁴ : CommRing C\ninst✝³ : Algebra R C\ninst✝² : CommRing P\ninst✝¹ : Algebra R P\nP₁ : Extension R A\nP₂ : Extension R A\ninst... | intro r hr s hs | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.Smooth.Basic | {
"line": 434,
"column": 6
} | {
"line": 434,
"column": 70
} | [
{
"pp": "R : Type u_4\ninst✝² : CommRing R\nA : Type u_6\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nh : Function.Surjective ⇑(algebraMap R A)\n⊢ FormallySmooth R A ↔ IsIdempotentElem (RingHom.ker (algebraMap R A))",
"usedConstants": [
"Eq.mpr",
"RingHom.instRingHomClass",
"Semiring.toModul... | Algebra.FormallySmooth.iff_split_surjection (Algebra.ofId R A) h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Smooth.Kaehler | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 46
} | [
{
"pp": "R : Type u_1\nP : Type u_2\nS : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing P\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R P\ninst✝² : Algebra P S\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R P S\ng✝ : S →ₐ[R] P\nf : P →ₐ[R] S\nhf' : RingHom.ker f ^ 2 = ⊥\ng : S →ₐ[R] P\nhg : f.comp g = AlgHom.id R ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Kaehler.JacobiZariski | {
"line": 95,
"column": 6
} | {
"line": 95,
"column": 76
} | [
{
"pp": "R : Type u₁\nS : Type u₂\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u₃\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nι : Type w₁\nσ : Type w₂\nQ : Generators S T ι\nP : Generators R S σ\nx : (Q.comp P).Ring\nhx' : x ∈ (Q.co... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Kaehler | {
"line": 67,
"column": 4
} | {
"line": 68,
"column": 52
} | [
{
"pp": "case a\nR : Type u_1\nP : Type u_2\nS : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing P\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R P\ninst✝² : Algebra P S\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R P S\ng✝ : S →ₐ[R] P\nf : P →ₐ[R] S\nhf' : RingHom.ker f ^ 2 = ⊥\ng : S →ₐ[R] P\nhg : f.comp g = AlgH... | simp only [Algebra.smul_def, map_mul, AlgHom.commutes,
RingHom.id_apply, Submodule.coe_smul_of_tower] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Smooth.Kaehler | {
"line": 76,
"column": 8
} | {
"line": 76,
"column": 50
} | [
{
"pp": "case hr\nR : Type u_1\nP : Type u_2\nS : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing P\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R P\ninst✝² : Algebra P S\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R P S\ng✝ : S →ₐ[R] P\nf : P →ₐ[R] S\nhf' : RingHom.ker f ^ 2 = ⊥\ng : S →ₐ[R] P\nhg : f.comp g = Alg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Kaehler | {
"line": 77,
"column": 8
} | {
"line": 77,
"column": 50
} | [
{
"pp": "case hs\nR : Type u_1\nP : Type u_2\nS : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing P\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R P\ninst✝² : Algebra P S\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R P S\ng✝ : S →ₐ[R] P\nf : P →ₐ[R] S\nhf' : RingHom.ker f ^ 2 = ⊥\ng : S →ₐ[R] P\nhg : f.comp g = Alg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Smooth.Kaehler | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 44
} | [
{
"pp": "case smul_assoc.a.e_a\nR : Type u_1\nP : Type u_2\nS : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing P\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R P\ninst✝² : Algebra P S\ninst✝¹ : Algebra R S\ninst✝ : IsScalarTower R P S\ng : S →ₐ[R] P\nhf' : RingHom.ker (algebraMap P S) ^ 2 = ⊥\nhg : (IsScalarTower.t... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Kaehler.JacobiZariski | {
"line": 120,
"column": 6
} | {
"line": 120,
"column": 17
} | [
{
"pp": "case h2.refine_2\nR : Type u₁\nS : Type u₂\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u₃\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nι : Type w₁\nσ : Type w₂\nQ : Generators S T ι\nP : Generators R S σ\nx : (Q.comp P).Rin... | rw [map_mk] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Smooth.Kaehler | {
"line": 170,
"column": 18
} | {
"line": 172,
"column": 61
} | [
{
"pp": "R : Type u_1\nP : Type u_2\nS : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing P\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R P\ninst✝² : Algebra P S\nl : S ⊗[P] Ω[P⁄R] →ₗ[P] ↥(RingHom.ker (algebraMap P S))\nhl : l ∘ₗ kerToTensor R P S = LinearMap.id\nσ : S → P\nhσ : ∀ (x : S), (algebraMap P S) (σ x) = x... | by
simp only [sectionOfRetractionKerToTensorAux_prop l hl (σ (a + b)) (σ a + σ b) (by simp [hσ]),
map_add, tmul_add, Submodule.coe_add, add_sub_add_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Kaehler.JacobiZariski | {
"line": 166,
"column": 2
} | {
"line": 178,
"column": 98
} | [
{
"pp": "R : Type u₁\nS : Type u₂\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\nT : Type u₃\ninst✝³ : CommRing T\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nι : Type w₁\nσ : Type w₂\nQ : Generators S T ι\nP : Generators R S σ\n⊢ LinearMap.fst T Q.toExtension.Cota... | classical
apply (Q.comp P).cotangentSpaceBasis.ext
intro i
apply Q.cotangentSpaceBasis.repr.injective
ext j
simp only [compEquiv, LinearMap.coe_comp, LinearEquiv.coe_coe, Function.comp_apply, ofComp_val,
LinearEquiv.trans_apply, Basis.repr_self, LinearMap.fst_apply, repr_CotangentSpaceMap]
obtain (i | i... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
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