module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Algebra.Order.ToIntervalMod | {
"line": 745,
"column": 79
} | {
"line": 746,
"column": 45
} | [
{
"pp": "α : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedAddMonoid α\nhα : Archimedean α\np : α\nhp : 0 < p\na b : α\n⊢ toIcoDiv hp a b = toIcoDiv hp 0 (b - a)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"AddCommGroup.toAddCommMonoid",
"AddMonoid.toAddZ... | by
rw [toIcoDiv_sub_eq_toIcoDiv_add, zero_add] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.ToIntervalMod | {
"line": 883,
"column": 10
} | {
"line": 883,
"column": 53
} | [
{
"pp": "α : Type u_1\ninst✝² : AddCommGroup α\ninst✝¹ : LinearOrder α\ninst✝ : IsOrderedAddMonoid α\nhα : Archimedean α\np : α\nhp : 0 < p\nx₁ x₂ x₃ x₄ : α\nh₁₂₃ : toIcoMod hp x₁ x₂ ≤ toIocMod hp x₁ x₃ ∧ ¬toIcoMod hp x₃ x₂ ≤ toIocMod hp x₃ x₁\nh₂₃₄ : toIcoMod hp x₂ x₄ ≤ toIocMod hp x₂ x₃ ∧ ¬toIcoMod hp x₃ x₄ ≤... | (not_modEq_iff_toIcoMod_eq_toIocMod hp).1 h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.Basic | {
"line": 140,
"column": 2
} | {
"line": 145,
"column": 45
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\nts : TopologicalSpace α\ninst✝¹ : Preorder α\ninst✝ : OrderTopology α\na : α\n⊢ TendstoIxxClass Icc (𝓝 a) (𝓝 a)",
"usedConstants": [
"Filter.HasBasis.inf",
"Eq.mpr",
"Set.Ioi",
"iInf",
"Set.ordConnected_Ioi",
"outParam",
... | simp only [nhds_eq_order, iInf_subtype']
refine
((hasBasis_iInf_principal_finite _).inf (hasBasis_iInf_principal_finite _)).tendstoIxxClass
fun s _ => ?_
refine ((ordConnected_biInter ?_).inter (ordConnected_biInter ?_)).out <;> intro _ _
exacts [ordConnected_Ioi, ordConnected_Iio] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.Basic | {
"line": 140,
"column": 2
} | {
"line": 145,
"column": 45
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\nts : TopologicalSpace α\ninst✝¹ : Preorder α\ninst✝ : OrderTopology α\na : α\n⊢ TendstoIxxClass Icc (𝓝 a) (𝓝 a)",
"usedConstants": [
"Filter.HasBasis.inf",
"Eq.mpr",
"Set.Ioi",
"iInf",
"Set.ordConnected_Ioi",
"outParam",
... | simp only [nhds_eq_order, iInf_subtype']
refine
((hasBasis_iInf_principal_finite _).inf (hasBasis_iInf_principal_finite _)).tendstoIxxClass
fun s _ => ?_
refine ((ordConnected_biInter ?_).inter (ordConnected_biInter ?_)).out <;> intro _ _
exacts [ordConnected_Ioi, ordConnected_Iio] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 474,
"column": 2
} | {
"line": 474,
"column": 10
} | [
{
"pp": "case mk.mk\nα : Type u_1\nβ : Type u_2\ntoFun✝¹ : α → β\ninvFun✝¹ : β → α\nsource✝¹ : Set α\ntarget✝¹ : Set β\nmap_source'✝¹ : ∀ ⦃x : α⦄, x ∈ source✝¹ → toFun✝¹ x ∈ target✝¹\nmap_target'✝¹ : ∀ ⦃x : β⦄, x ∈ target✝¹ → invFun✝¹ x ∈ source✝¹\nleft_inv'✝¹ : ∀ ⦃x : α⦄, x ∈ source✝¹ → invFun✝¹ (toFun✝¹ x) = ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 659,
"column": 4
} | {
"line": 659,
"column": 48
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ne : PartialEquiv α β\ne' : PartialEquiv β γ\ns : Set α\n⊢ ((e.restr s).trans e').source = ((e.trans e').restr s).source",
"usedConstants": [
"congrArg",
"PartialEquiv.restr",
"PartialEquiv.trans",
"Set.instInter",
"Inter.inter"... | simp [trans_source, inter_comm, inter_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 659,
"column": 4
} | {
"line": 659,
"column": 48
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ne : PartialEquiv α β\ne' : PartialEquiv β γ\ns : Set α\n⊢ ((e.restr s).trans e').source = ((e.trans e').restr s).source",
"usedConstants": [
"congrArg",
"PartialEquiv.restr",
"PartialEquiv.trans",
"Set.instInter",
"Inter.inter"... | simp [trans_source, inter_comm, inter_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 659,
"column": 4
} | {
"line": 659,
"column": 48
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ne : PartialEquiv α β\ne' : PartialEquiv β γ\ns : Set α\n⊢ ((e.restr s).trans e').source = ((e.trans e').restr s).source",
"usedConstants": [
"congrArg",
"PartialEquiv.restr",
"PartialEquiv.trans",
"Set.instInter",
"Inter.inter"... | simp [trans_source, inter_comm, inter_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 759,
"column": 25
} | {
"line": 759,
"column": 33
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ne✝ : PartialEquiv α β\ne'✝ : PartialEquiv β γ\ne : PartialEquiv α β\ne' : PartialEquiv γ δ\np : α × γ\nhp : p ∈ e.source ×ˢ e'.source\n⊢ (↑e p.1, ↑e' p.2) ∈ e.target ×ˢ e'.target",
"usedConstants": [
"Set.instSProd",
"SProd.sprod",... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 759,
"column": 25
} | {
"line": 759,
"column": 33
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ne✝ : PartialEquiv α β\ne'✝ : PartialEquiv β γ\ne : PartialEquiv α β\ne' : PartialEquiv γ δ\np : α × γ\nhp : p ∈ e.source ×ˢ e'.source\n⊢ (↑e p.1, ↑e' p.2) ∈ e.target ×ˢ e'.target",
"usedConstants": [
"Set.instSProd",
"SProd.sprod",... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 759,
"column": 25
} | {
"line": 759,
"column": 33
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ne✝ : PartialEquiv α β\ne'✝ : PartialEquiv β γ\ne : PartialEquiv α β\ne' : PartialEquiv γ δ\np : α × γ\nhp : p ∈ e.source ×ˢ e'.source\n⊢ (↑e p.1, ↑e' p.2) ∈ e.target ×ˢ e'.target",
"usedConstants": [
"Set.instSProd",
"SProd.sprod",... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 760,
"column": 25
} | {
"line": 760,
"column": 33
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ne✝ : PartialEquiv α β\ne'✝ : PartialEquiv β γ\ne : PartialEquiv α β\ne' : PartialEquiv γ δ\np : β × δ\nhp : p ∈ e.target ×ˢ e'.target\n⊢ (↑e.symm p.1, ↑e'.symm p.2) ∈ e.source ×ˢ e'.source",
"usedConstants": [
"Set.instSProd",
"SPr... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 760,
"column": 25
} | {
"line": 760,
"column": 33
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ne✝ : PartialEquiv α β\ne'✝ : PartialEquiv β γ\ne : PartialEquiv α β\ne' : PartialEquiv γ δ\np : β × δ\nhp : p ∈ e.target ×ˢ e'.target\n⊢ (↑e.symm p.1, ↑e'.symm p.2) ∈ e.source ×ˢ e'.source",
"usedConstants": [
"Set.instSProd",
"SPr... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 760,
"column": 25
} | {
"line": 760,
"column": 33
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ne✝ : PartialEquiv α β\ne'✝ : PartialEquiv β γ\ne : PartialEquiv α β\ne' : PartialEquiv γ δ\np : β × δ\nhp : p ∈ e.target ×ˢ e'.target\n⊢ (↑e.symm p.1, ↑e'.symm p.2) ∈ e.source ×ˢ e'.source",
"usedConstants": [
"Set.instSProd",
"SPr... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 761,
"column": 25
} | {
"line": 761,
"column": 33
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ne✝ : PartialEquiv α β\ne'✝ : PartialEquiv β γ\ne : PartialEquiv α β\ne' : PartialEquiv γ δ\np : α × γ\nhp : p ∈ e.source ×ˢ e'.source\n⊢ (↑e.symm (↑e p.1, ↑e' p.2).1, ↑e'.symm (↑e p.1, ↑e' p.2).2) = p",
"usedConstants": [
"Set.instSProd"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 761,
"column": 25
} | {
"line": 761,
"column": 33
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ne✝ : PartialEquiv α β\ne'✝ : PartialEquiv β γ\ne : PartialEquiv α β\ne' : PartialEquiv γ δ\np : α × γ\nhp : p ∈ e.source ×ˢ e'.source\n⊢ (↑e.symm (↑e p.1, ↑e' p.2).1, ↑e'.symm (↑e p.1, ↑e' p.2).2) = p",
"usedConstants": [
"Set.instSProd"... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 761,
"column": 25
} | {
"line": 761,
"column": 33
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ne✝ : PartialEquiv α β\ne'✝ : PartialEquiv β γ\ne : PartialEquiv α β\ne' : PartialEquiv γ δ\np : α × γ\nhp : p ∈ e.source ×ˢ e'.source\n⊢ (↑e.symm (↑e p.1, ↑e' p.2).1, ↑e'.symm (↑e p.1, ↑e' p.2).2) = p",
"usedConstants": [
"Set.instSProd"... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 762,
"column": 25
} | {
"line": 762,
"column": 33
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ne✝ : PartialEquiv α β\ne'✝ : PartialEquiv β γ\ne : PartialEquiv α β\ne' : PartialEquiv γ δ\np : β × δ\nhp : p ∈ e.target ×ˢ e'.target\n⊢ (↑e (↑e.symm p.1, ↑e'.symm p.2).1, ↑e' (↑e.symm p.1, ↑e'.symm p.2).2) = p",
"usedConstants": [
"Set.... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 762,
"column": 25
} | {
"line": 762,
"column": 33
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ne✝ : PartialEquiv α β\ne'✝ : PartialEquiv β γ\ne : PartialEquiv α β\ne' : PartialEquiv γ δ\np : β × δ\nhp : p ∈ e.target ×ˢ e'.target\n⊢ (↑e (↑e.symm p.1, ↑e'.symm p.2).1, ↑e' (↑e.symm p.1, ↑e'.symm p.2).2) = p",
"usedConstants": [
"Set.... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 762,
"column": 25
} | {
"line": 762,
"column": 33
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ne✝ : PartialEquiv α β\ne'✝ : PartialEquiv β γ\ne : PartialEquiv α β\ne' : PartialEquiv γ δ\np : β × δ\nhp : p ∈ e.target ×ˢ e'.target\n⊢ (↑e (↑e.symm p.1, ↑e'.symm p.2).1, ↑e' (↑e.symm p.1, ↑e'.symm p.2).2) = p",
"usedConstants": [
"Set.... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 938,
"column": 8
} | {
"line": 938,
"column": 62
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ne : α ≃ β\ne' : β ≃ γ\n⊢ (e.trans e').toPartialEquiv.source = (e.toPartialEquiv.trans e'.toPartialEquiv).source",
"usedConstants": [
"congrArg",
"Set.inter_self",
"Set.univ",
"PartialEquiv.trans",
"Equiv.toPartialEquiv",
... | simp [PartialEquiv.trans_source, Equiv.toPartialEquiv] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 938,
"column": 8
} | {
"line": 938,
"column": 62
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ne : α ≃ β\ne' : β ≃ γ\n⊢ (e.trans e').toPartialEquiv.source = (e.toPartialEquiv.trans e'.toPartialEquiv).source",
"usedConstants": [
"congrArg",
"Set.inter_self",
"Set.univ",
"PartialEquiv.trans",
"Equiv.toPartialEquiv",
... | simp [PartialEquiv.trans_source, Equiv.toPartialEquiv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Logic.Equiv.PartialEquiv | {
"line": 938,
"column": 8
} | {
"line": 938,
"column": 62
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ne : α ≃ β\ne' : β ≃ γ\n⊢ (e.trans e').toPartialEquiv.source = (e.toPartialEquiv.trans e'.toPartialEquiv).source",
"usedConstants": [
"congrArg",
"Set.inter_self",
"Set.univ",
"PartialEquiv.trans",
"Equiv.toPartialEquiv",
... | simp [PartialEquiv.trans_source, Equiv.toPartialEquiv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Order.Field | {
"line": 170,
"column": 71
} | {
"line": 172,
"column": 67
} | [
{
"pp": "𝕜 : Type u_1\nα : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nl : Filter α\nf g : α → 𝕜\nC : 𝕜\nhC : C < 0\nhf : Tendsto f l atBot\nhg : Tendsto g l (𝓝 C)\n⊢ Tendsto (fun x ↦ f x * g x) l atTop",
... | by
have := (tendsto_neg_atBot_atTop.comp hf).atTop_mul_neg hC hg
simpa [Function.comp_def] using tendsto_neg_atBot_atTop.comp this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Order.Basic | {
"line": 441,
"column": 6
} | {
"line": 441,
"column": 84
} | [
{
"pp": "case inr.inr\nα : Type u\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\ns : Set α\nhs : s ∈ 𝓝[≥] a\nha : ∃ b, a < b\nb : α\nhab : a < b\nhbs : Ico a b ⊆ s\nc : α\nhc : c ∈ Ioo a b\n⊢ ∃ b, a ≤ b ∧ Icc a b ∈ 𝓝[≥] a ∧ Icc a b ⊆ s",
"usedConstants": [
"Fil... | exact ⟨c, hc.1.le, Icc_mem_nhdsGE hc.1, (Icc_subset_Ico_right hc.2).trans hbs⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Order.Basic | {
"line": 441,
"column": 6
} | {
"line": 441,
"column": 84
} | [
{
"pp": "case inr.inr\nα : Type u\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\ns : Set α\nhs : s ∈ 𝓝[≥] a\nha : ∃ b, a < b\nb : α\nhab : a < b\nhbs : Ico a b ⊆ s\nc : α\nhc : c ∈ Ioo a b\n⊢ ∃ b, a ≤ b ∧ Icc a b ∈ 𝓝[≥] a ∧ Icc a b ⊆ s",
"usedConstants": [
"Fil... | exact ⟨c, hc.1.le, Icc_mem_nhdsGE hc.1, (Icc_subset_Ico_right hc.2).trans hbs⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.Basic | {
"line": 441,
"column": 6
} | {
"line": 441,
"column": 84
} | [
{
"pp": "case inr.inr\nα : Type u\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\ns : Set α\nhs : s ∈ 𝓝[≥] a\nha : ∃ b, a < b\nb : α\nhab : a < b\nhbs : Ico a b ⊆ s\nc : α\nhc : c ∈ Ioo a b\n⊢ ∃ b, a ≤ b ∧ Icc a b ∈ 𝓝[≥] a ∧ Icc a b ⊆ s",
"usedConstants": [
"Fil... | exact ⟨c, hc.1.le, Icc_mem_nhdsGE hc.1, (Icc_subset_Ico_right hc.2).trans hbs⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.Basic | {
"line": 450,
"column": 38
} | {
"line": 450,
"column": 57
} | [
{
"pp": "α : Type u\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\ns : Set α\nhs : s ∈ 𝓝 a\nb : α\nhba : b ≤ a\nhb_nhds : Icc b a ∈ 𝓝[≤] a\nhbs : Icc b a ⊆ s\nc : α\nhac : a ≤ c\nhc_nhds : Icc a c ∈ 𝓝[≥] a\nhcs : Icc a c ⊆ s\n⊢ Icc b a ∪ Icc a c ∈ 𝓝 a ∧ Icc b a ∪ Icc a... | ← nhdsLE_sup_nhdsGE | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.CharacterModule | {
"line": 52,
"column": 29
} | {
"line": 52,
"column": 37
} | [
{
"pp": "R : Type uR\ninst✝³ : CommRing R\nA : Type uA\ninst✝² : AddCommGroup A\nA' : Type u_1\ninst✝¹ : AddCommGroup A'\nB : Type uB\ninst✝ : AddCommGroup B\nx✝² x✝¹ : CharacterModule A\nx✝ : (fun c ↦ (↑c).toFun) x✝² = (fun c ↦ (↑c).toFun) x✝¹\n⊢ x✝² = x✝¹",
"usedConstants": [
"Rat.instOfNat",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Module.CharacterModule | {
"line": 52,
"column": 29
} | {
"line": 52,
"column": 37
} | [
{
"pp": "R : Type uR\ninst✝³ : CommRing R\nA : Type uA\ninst✝² : AddCommGroup A\nA' : Type u_1\ninst✝¹ : AddCommGroup A'\nB : Type uB\ninst✝ : AddCommGroup B\nx✝² x✝¹ : CharacterModule A\nx✝ : (fun c ↦ (↑c).toFun) x✝² = (fun c ↦ (↑c).toFun) x✝¹\n⊢ x✝² = x✝¹",
"usedConstants": [
"Rat.instOfNat",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.CharacterModule | {
"line": 52,
"column": 29
} | {
"line": 52,
"column": 37
} | [
{
"pp": "R : Type uR\ninst✝³ : CommRing R\nA : Type uA\ninst✝² : AddCommGroup A\nA' : Type u_1\ninst✝¹ : AddCommGroup A'\nB : Type uB\ninst✝ : AddCommGroup B\nx✝² x✝¹ : CharacterModule A\nx✝ : (fun c ↦ (↑c).toFun) x✝² = (fun c ↦ (↑c).toFun) x✝¹\n⊢ x✝² = x✝¹",
"usedConstants": [
"Rat.instOfNat",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.CharacterModule | {
"line": 82,
"column": 22
} | {
"line": 82,
"column": 66
} | [
{
"pp": "R : Type uR\ninst✝⁶ : CommRing R\nA : Type uA\ninst✝⁵ : AddCommGroup A\nA' : Type u_1\ninst✝⁴ : AddCommGroup A'\nB : Type uB\ninst✝³ : AddCommGroup B\ninst✝² : Module R A\ninst✝¹ : Module R A'\ninst✝ : Module R B\nf : A →ₗ[R] B\nr : R\nc : CharacterModule B\n⊢ AddMonoidHom.comp (r • c) f.toAddMonoidHom... | ext x; exact congr(c $(f.map_smul r x)).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.CharacterModule | {
"line": 82,
"column": 22
} | {
"line": 82,
"column": 66
} | [
{
"pp": "R : Type uR\ninst✝⁶ : CommRing R\nA : Type uA\ninst✝⁵ : AddCommGroup A\nA' : Type u_1\ninst✝⁴ : AddCommGroup A'\nB : Type uB\ninst✝³ : AddCommGroup B\ninst✝² : Module R A\ninst✝¹ : Module R A'\ninst✝ : Module R B\nf : A →ₗ[R] B\nr : R\nc : CharacterModule B\n⊢ AddMonoidHom.comp (r • c) f.toAddMonoidHom... | ext x; exact congr(c $(f.map_smul r x)).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.ObjectProperty.EpiMono | {
"line": 78,
"column": 2
} | {
"line": 78,
"column": 28
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nP : ObjectProperty C\ninst✝¹ : P.IsClosedUnderQuotients\ninst✝ : HasZeroMorphisms C\nS : ShortComplex C\nhS : S.ShortExact\nh₂ : P S.X₂\nthis : Epi S.g\n⊢ P S.X₃",
"usedConstants": [
"CategoryTheory.ObjectProperty.prop_of_epi",
"CategoryTheory.Sho... | exact P.prop_of_epi S.g h₂ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Category.Grp.IsFinite | {
"line": 55,
"column": 4
} | {
"line": 58,
"column": 41
} | [
{
"pp": "S : ShortComplex AddCommGrpCat\nhS : S.ShortExact\nh₁ : Finite ↑S.X₁\nh₃ : Finite ↑S.X₃\nhg : Function.Surjective ⇑(ConcreteCategory.hom S.g)\ns : ↑S.X₃ → ↑S.X₂\nhs : Function.RightInverse s ⇑(ConcreteCategory.hom S.g)\n⊢ Finite ↑S.X₂",
"usedConstants": [
"CategoryTheory.Abelian.toPreadditive... | have hφ : Function.Surjective (fun (x₁, x₃) ↦ S.f x₁ + s x₃) := fun x₂ ↦ by
obtain ⟨x₁, hx₁⟩ := (ShortComplex.ab_exact_iff S).1 hS.exact (x₂ - s (S.g x₂))
(by simp [hs (S.g x₂)])
exact ⟨⟨x₁, S.g x₂⟩, by simp [hx₁]⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Instances.AddCircle.Defs | {
"line": 537,
"column": 36
} | {
"line": 537,
"column": 45
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : Field 𝕜\np q r : 𝕜\nm : ℕ\n⊢ ↑(m • (r / q)) = m • ↑(r / q)",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRing",... | coe_nsmul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Instances.AddCircle.Defs | {
"line": 765,
"column": 4
} | {
"line": 774,
"column": 43
} | [
{
"pp": "𝕜 : Type u_1\nB : Type u_2\ninst✝⁵ : AddCommGroup 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsOrderedAddMonoid 𝕜\np a : 𝕜\nhp : Fact (0 < p)\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\n⊢ Continuous[QuotientAddGroup.instTopologicalSpace (zmultiples p), instTopologica... | simp_rw [isQuotientMap_quotient_mk'.continuous_iff, continuous_iff_continuousAt,
continuousAt_iff_continuous_left_right]
intro x; constructor
on_goal 1 => erw [equivIccQuot_comp_mk_eq_toIocMod]
on_goal 2 => erw [equivIccQuot_comp_mk_eq_toIcoMod]
all_goals
apply continuous_quot_mk.continuousA... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.AddCircle.Defs | {
"line": 765,
"column": 4
} | {
"line": 774,
"column": 43
} | [
{
"pp": "𝕜 : Type u_1\nB : Type u_2\ninst✝⁵ : AddCommGroup 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsOrderedAddMonoid 𝕜\np a : 𝕜\nhp : Fact (0 < p)\ninst✝² : Archimedean 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\n⊢ Continuous[QuotientAddGroup.instTopologicalSpace (zmultiples p), instTopologica... | simp_rw [isQuotientMap_quotient_mk'.continuous_iff, continuous_iff_continuousAt,
continuousAt_iff_continuous_left_right]
intro x; constructor
on_goal 1 => erw [equivIccQuot_comp_mk_eq_toIocMod]
on_goal 2 => erw [equivIccQuot_comp_mk_eq_toIcoMod]
all_goals
apply continuous_quot_mk.continuousA... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Coalgebra.Convolution | {
"line": 131,
"column": 93
} | {
"line": 134,
"column": 22
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\nC : Type u_4\ninst✝⁷ : CommSemiring R\ninst✝⁶ : NonUnitalSemiring A\ninst✝⁵ : Module R A\ninst✝⁴ : SMulCommClass R A A\ninst✝³ : IsScalarTower R A A\ninst✝² : AddCommMonoid C\ninst✝¹ : Module R C\ninst✝ : Coalgebra R C\nf g h : WithConv (C →ₗ[R] A)\n⊢ (mul' R A... | by
congr 1
ext
simp [mul_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Grp | {
"line": 178,
"column": 41
} | {
"line": 180,
"column": 89
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\ninst✝³ : CartesianMonoidalCategory C\nA B : C\ninst✝² : GrpObj A\ninst✝¹ : GrpObj B\nf : A ⟶ B\ninst✝ : IsMonHom f\n⊢ lift f (ι ≫ f) ≫ μ = toUnit A ≫ η",
"usedConstants": [
"CategoryTheory.Category.assoc",
"CategoryTheory.GrpObj.inv",
"Ca... | by
have := right_inv A =≫ f
rwa [assoc, IsMonHom.mul_hom, assoc, IsMonHom.one_hom, lift_map_assoc, id_comp] at this | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Monoidal.Grp | {
"line": 305,
"column": 6
} | {
"line": 311,
"column": 82
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : CartesianMonoidalCategory C\nA : C\ninst✝ : GrpObj A\n⊢ ∀ (s : PullbackCone μ μ),\n (fun s ↦ lift (lift (s.snd ≫ fst A A) (lift (s.snd ≫ fst A A ≫ ι) (s.fst ≫ fst A A) ≫ μ)) (s.fst ≫ snd A A)) s ≫\n (α_ A A A).hom ≫ A ◁ μ =\n s.snd",
... | refine fun s => CartesianMonoidalCategory.hom_ext _ _ (by simp) ?_
simp only [lift_lift_associator_hom_assoc, lift_whiskerLeft, lift_snd]
have : lift (s.snd ≫ fst _ _ ≫ ι) (s.fst ≫ fst _ _) ≫ μ =
lift (s.snd ≫ snd _ _) (s.fst ≫ snd _ _ ≫ ι) ≫ μ := by
rw [← assoc s.fst, eq_lift_inv_right, l... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Monoidal.Grp | {
"line": 305,
"column": 6
} | {
"line": 311,
"column": 82
} | [
{
"pp": "C : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : CartesianMonoidalCategory C\nA : C\ninst✝ : GrpObj A\n⊢ ∀ (s : PullbackCone μ μ),\n (fun s ↦ lift (lift (s.snd ≫ fst A A) (lift (s.snd ≫ fst A A ≫ ι) (s.fst ≫ fst A A) ≫ μ)) (s.fst ≫ snd A A)) s ≫\n (α_ A A A).hom ≫ A ◁ μ =\n s.snd",
... | refine fun s => CartesianMonoidalCategory.hom_ext _ _ (by simp) ?_
simp only [lift_lift_associator_hom_assoc, lift_whiskerLeft, lift_snd]
have : lift (s.snd ≫ fst _ _ ≫ ι) (s.fst ≫ fst _ _) ≫ μ =
lift (s.snd ≫ snd _ _) (s.fst ≫ snd _ _ ≫ ι) ≫ μ := by
rw [← assoc s.fst, eq_lift_inv_right, l... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monoidal.Grp | {
"line": 320,
"column": 8
} | {
"line": 320,
"column": 45
} | [
{
"pp": "case refine_2\nC : Type u₁\ninst✝² : Category.{v₁, u₁} C\ninst✝¹ : CartesianMonoidalCategory C\nA : C\ninst✝ : GrpObj A\ns : PullbackCone μ μ\nm : s.pt ⟶ (A ⊗ A) ⊗ A\nhm₁ : m ≫ μ ▷ A = s.fst\nhm₂ : m ≫ (α_ A A A).hom ≫ A ◁ μ = s.snd\nh : m ≫ fst (A ⊗ A) A ≫ fst A A = s.snd ≫ fst A A\nthis : lift (s.snd... | rwa [← assoc s.snd, eq_lift_inv_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Algebra.Category.ModuleCat.Adjunctions | {
"line": 117,
"column": 4
} | {
"line": 118,
"column": 15
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\n⊢ ↟(Finsupp.lsingle PUnit.unit) ≫ ↟(Finsupp.lapply PUnit.unit) = 𝟙 (𝟙_ (ModuleCat R))",
"usedConstants": [
"LinearMap.id",
"NonAssocSemiring.toAddCommMonoidWithOne",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
... | ext
simp [free] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Category.ModuleCat.Adjunctions | {
"line": 117,
"column": 4
} | {
"line": 118,
"column": 15
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\n⊢ ↟(Finsupp.lsingle PUnit.unit) ≫ ↟(Finsupp.lapply PUnit.unit) = 𝟙 (𝟙_ (ModuleCat R))",
"usedConstants": [
"LinearMap.id",
"NonAssocSemiring.toAddCommMonoidWithOne",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
... | ext
simp [free] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ExactSequence | {
"line": 137,
"column": 2
} | {
"line": 137,
"column": 21
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\ninst✝ : HasZeroMorphisms C\nn : ℕ\nS : ComposableArrows C n\nhS : S.Exact\ni : ℕ\nhk : i + 1 + 1 ≤ n\n⊢ (S.sc' ⋯ i (i + 1) (i + 1 + 1) ⋯ ⋯ hk).Exact",
"usedConstants": [
"CategoryTheory.ComposableArrows.Exact.exact"
]
}
] | exact hS.exact i hk | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Homology.ShortComplex.ConcreteCategory | {
"line": 179,
"column": 79
} | {
"line": 179,
"column": 87
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type v\ninst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝⁴ : ConcreteCategory C FC\ninst✝³ : HasForget₂ C Ab\ninst✝² : Abelian C\ninst✝¹ : (forget₂ C Ab).Additive\ninst✝ : (forget₂ C Ab).PreservesHomology\nD : SnakeInput C... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Homology.ShortComplex.ConcreteCategory | {
"line": 179,
"column": 79
} | {
"line": 179,
"column": 87
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type v\ninst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝⁴ : ConcreteCategory C FC\ninst✝³ : HasForget₂ C Ab\ninst✝² : Abelian C\ninst✝¹ : (forget₂ C Ab).Additive\ninst✝ : (forget₂ C Ab).PreservesHomology\nD : SnakeInput C... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.ConcreteCategory | {
"line": 179,
"column": 79
} | {
"line": 179,
"column": 87
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type v\ninst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝⁴ : ConcreteCategory C FC\ninst✝³ : HasForget₂ C Ab\ninst✝² : Abelian C\ninst✝¹ : (forget₂ C Ab).Additive\ninst✝ : (forget₂ C Ab).PreservesHomology\nD : SnakeInput C... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Homology.ShortComplex.ConcreteCategory | {
"line": 181,
"column": 79
} | {
"line": 181,
"column": 87
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type v\ninst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝⁴ : ConcreteCategory C FC\ninst✝³ : HasForget₂ C Ab\ninst✝² : Abelian C\ninst✝¹ : (forget₂ C Ab).Additive\ninst✝ : (forget₂ C Ab).PreservesHomology\nD : SnakeInput C... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Homology.ShortComplex.ConcreteCategory | {
"line": 181,
"column": 79
} | {
"line": 181,
"column": 87
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type v\ninst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝⁴ : ConcreteCategory C FC\ninst✝³ : HasForget₂ C Ab\ninst✝² : Abelian C\ninst✝¹ : (forget₂ C Ab).Additive\ninst✝ : (forget₂ C Ab).PreservesHomology\nD : SnakeInput C... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Homology.ShortComplex.ConcreteCategory | {
"line": 181,
"column": 79
} | {
"line": 181,
"column": 87
} | [
{
"pp": "C : Type u\ninst✝⁶ : Category.{v, u} C\nFC : C → C → Type u_1\nCC : C → Type v\ninst✝⁵ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninst✝⁴ : ConcreteCategory C FC\ninst✝³ : HasForget₂ C Ab\ninst✝² : Abelian C\ninst✝¹ : (forget₂ C Ab).Additive\ninst✝ : (forget₂ C Ab).PreservesHomology\nD : SnakeInput C... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Monad.Limits | {
"line": 205,
"column": 8
} | {
"line": 205,
"column": 20
} | [
{
"pp": "case h\nC : Type u₁\ninst✝³ : Category.{v₁, u₁} C\nT : Monad C\nJ : Type u\ninst✝² : Category.{v, u} J\nD : J ⥤ T.Algebra\nc : Cocone (D ⋙ T.forget)\nt : IsColimit c\ninst✝¹ : PreservesColimit (D ⋙ T.forget) T.toFunctor\ninst✝ : PreservesColimit ((D ⋙ T.forget) ⋙ T.toFunctor) T.toFunctor\nA B : J\nf : ... | rw [comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.CategoryTheory.Monad.Limits | {
"line": 517,
"column": 15
} | {
"line": 517,
"column": 62
} | [
{
"pp": "C : Type u₁\ninst✝⁵ : Category.{v₁, u₁} C\nD✝ : Type u₂\ninst✝⁴ : Category.{v₂, u₂} D✝\nJ : Type u\ninst✝³ : Category.{v, u} J\nT : Comonad C\nD : J ⥤ T.Coalgebra\nc : Cone (D ⋙ T.forget)\nt : IsLimit c\ninst✝² : PreservesLimit (D ⋙ T.forget) T.toFunctor\ninst✝¹ : PreservesLimit ((D ⋙ T.forget) ⋙ T.toF... | ← show _ = _ ≫ c.π.app j from T.ε.naturality _, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.ComposableArrows.Basic | {
"line": 574,
"column": 4
} | {
"line": 574,
"column": 16
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nn m : ℕ\nF✝ G✝ : ComposableArrows C n\nF G : ComposableArrows C (n + 1)\nα : F.obj' 0 ⋯ ≅ G.obj' 0 ⋯\nβ : F.δ₀ ≅ G.δ₀\nw : F.map' 0 1 homMk₁._proof_4 ⋯ ≫ app' β.hom 0 ⋯ = α.hom ≫ G.map' 0 1 homMk₁._proof_4 ⋯\n⊢ F.map (homOfLE ⋯) ≫ 𝟙 (F.obj 1) = F.map (homOf... | rw [comp_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Ideal.Over | {
"line": 380,
"column": 45
} | {
"line": 380,
"column": 84
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\nS : Type u_4\ninst✝³ : Ring S\ninst✝² : Algebra R S\ninst✝¹ : Nontrivial S\ninst✝ : Module.IsTorsionFree R S\np : Ideal R\nhp : p ≠ ⊥\nP : Ideal S\nhP : P ∈ p.primesOver S\nthis : P.LiesOver p\n⊢ P ≠ ⊥",
"usedConstants": [
"Ideal.ne_bot_... | exact ne_bot_of_liesOver_of_ne_bot hp P | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.IsPrimary | {
"line": 85,
"column": 4
} | {
"line": 85,
"column": 26
} | [
{
"pp": "case insert.inr\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nι : Type u_3\nf : ι → Submodule R M\na : ι\ns : Finset ι\nha : a ∉ s\nIH :\n ∀ {i : ι},\n i ∈ s →\n (∀ ⦃y : ι⦄, y ∈ s → (f y).IsPrimary) →\n (∀ ⦃y : ι⦄, y ∈ s → ((f y).col... | simp only [inf_insert] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Ideal.Colon | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 10
} | [
{
"pp": "case add\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\nS : Set M\nr : R\nh : r ∈ N.colon S\ns x✝ y✝ : M\nhx✝ : x✝ ∈ span R S\nhy✝ : y✝ ∈ span R S\na✝¹ : r • x✝ ∈ N\na✝ : r • y✝ ∈ N\n⊢ r • (x✝ + y✝) ∈ N",
"usedConstants": [
... | | add => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Localization.Ideal | {
"line": 146,
"column": 2
} | {
"line": 150,
"column": 35
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nI : Ideal R\nhI : I.IsPrimary\nhM : Disjoint ↑M ↑I\n⊢ Ideal.under R (Ideal.map (algebraMap R S) I) = I",
"usedConstants": [
"Eq.mpr",
"Semirin... | have key : Disjoint (M : Set R) I.radical := by
contrapose hM
rw [Set.not_disjoint_iff] at hM ⊢
obtain ⟨a, ha, k, hk⟩ := hM
exact ⟨a ^ k, pow_mem ha k, hk⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Ideal.Quotient.Nilpotent | {
"line": 50,
"column": 4
} | {
"line": 51,
"column": 49
} | [
{
"pp": "case neg.succ.succ.a\nP : ⦃S : Type u_1⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nS : Type u_1\ninst✝ : CommRing S\nI : Idea... | apply h₁
rw [← Ideal.map_pow, Ideal.map_quotient_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Quotient.Nilpotent | {
"line": 50,
"column": 4
} | {
"line": 51,
"column": 49
} | [
{
"pp": "case neg.succ.succ.a\nP : ⦃S : Type u_1⦄ → [inst : CommRing S] → Ideal S → Prop\nh₁ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I : Ideal S), I ^ 2 = ⊥ → P I\nh₂ : ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S), I ≤ J → P I → P (map (Quotient.mk I) J) → P J\nS : Type u_1\ninst✝ : CommRing S\nI : Idea... | apply h₁
rw [← Ideal.map_pow, Ideal.map_quotient_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.Ideal | {
"line": 401,
"column": 31
} | {
"line": 401,
"column": 69
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsLocalization M S\nR' : Type u_3\nS' : Type u_4\ninst✝² : CommRing R'\ninst✝¹ : CommRing S'\ninst✝ : Algebra R' S'\nf : R →+* R'\nhf : Function.Surjective ⇑f\ng : S →+* S'\nhg : Functi... | simpa [sub_eq_zero, mul_sub] using hr' | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.Localization.Ideal | {
"line": 401,
"column": 31
} | {
"line": 401,
"column": 69
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsLocalization M S\nR' : Type u_3\nS' : Type u_4\ninst✝² : CommRing R'\ninst✝¹ : CommRing S'\ninst✝ : Algebra R' S'\nf : R →+* R'\nhf : Function.Surjective ⇑f\ng : S →+* S'\nhg : Functi... | simpa [sub_eq_zero, mul_sub] using hr' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Localization.Ideal | {
"line": 401,
"column": 31
} | {
"line": 401,
"column": 69
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝⁵ : CommRing S\ninst✝⁴ : Algebra R S\ninst✝³ : IsLocalization M S\nR' : Type u_3\nS' : Type u_4\ninst✝² : CommRing R'\ninst✝¹ : CommRing S'\ninst✝ : Algebra R' S'\nf : R →+* R'\nhf : Function.Surjective ⇑f\ng : S →+* S'\nhg : Functi... | simpa [sub_eq_zero, mul_sub] using hr' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.LocalProperties.Submodule | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 90
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nMₚ : (P : Ideal R) → [P.IsMaximal] → Type u_5\ninst✝² : (P : Ideal R) → [inst : P.IsMaximal] → AddCommMonoid (Mₚ P)\ninst✝¹ : (P : Ideal R) → [inst : P.IsMaximal] → Module R (Mₚ P)\nf : (P : Ideal R) → [... | have ⟨P, mP, le⟩ := (eqIdeal R m m').exists_le_maximal ((Ideal.ne_top_iff_one _).mpr ne) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Category.ModuleCat.ChangeOfRings | {
"line": 719,
"column": 6
} | {
"line": 719,
"column": 98
} | [
{
"pp": "R✝ : Type u₁\nS✝ : Type u₂\ninst✝³ : CommRing R✝\ninst✝² : CommRing S✝\nf✝ : R✝ →+* S✝\nR : Type u₁\nS : Type u₂\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nX : ModuleCat R\nY : ModuleCat S\ns : S\ng : X ⟶ (restrictScalars f).obj Y\nr : R\nx : ↑X\n⊢ { toFun := fun x ↦ s • (ConcreteCategory.h... | rw [AddHom.toFun_eq_coe, AddHom.coe_mk, RingHom.id_apply, map_smul, smul_comm r s (g x : Y)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.PolynomialAlgebra | {
"line": 86,
"column": 40
} | {
"line": 86,
"column": 48
} | [
{
"pp": "case a\nR : Type u_1\nA : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\na₁ a₂ : A\np₁ p₂ : R[X]\nk : ℕ\n⊢ ((p₁ * p₂).sum fun a b ↦ if a = k then a₁ * a₂ * (algebraMap R A) b else 0) =\n ((p₁.sum fun n r ↦ (monomial n) (a₁ * (algebraMap R A) r)) *\n p₂.sum fun ... | sum_def, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.PolynomialAlgebra | {
"line": 136,
"column": 66
} | {
"line": 136,
"column": 74
} | [
{
"pp": "case refine_2\nR : Type u_1\nA : Type u_3\ninst✝² : CommSemiring R\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nx : A ⊗[R] R[X]\na : A\np : R[X]\n⊢ (p.sum fun n a_1 ↦ a ⊗ₜ[R] (a_1 • X ^ n)) = a ⊗ₜ[R] p",
"usedConstants": [
"Polynomial.distribMulAction",
"instHSMul",
"Semiring.toModu... | sum_def, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Category.ModuleCat.ChangeOfRings | {
"line": 892,
"column": 8
} | {
"line": 892,
"column": 16
} | [
{
"pp": "case add\nR✝ : Type u₁\nS✝ : Type u₂\ninst✝³ : CommRing R✝\ninst✝² : CommRing S✝\nf✝ : R✝ →+* S✝\nR : Type u₁\nS : Type u₂\ninst✝¹ : CommRing R\ninst✝ : CommRing S\nf : R →+* S\nX : ModuleCat R\nY : ModuleCat S\ng : X ⟶ (restrictScalars f).obj Y\nx✝ y✝ : ↑((restrictScalars f).obj (of S S)) ⊗[R] ↑X\na✝¹... | | add => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Localization.LocalizationLocalization | {
"line": 101,
"column": 6
} | {
"line": 101,
"column": 27
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝⁸ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nN : Submonoid S\nT : Type u_3\ninst✝⁵ : CommSemiring T\ninst✝⁴ : Algebra R T\ninst✝³ : Algebra S T\ninst✝² : IsScalarTower R S T\ninst✝¹ : IsLocalization M S\ninst✝ : IsLo... | refine ⟨z, c * s, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Localization.LocalizationLocalization | {
"line": 183,
"column": 5
} | {
"line": 186,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁴ : CommSemiring S\ninst✝³ : Algebra R S\nN : Submonoid S\nT : Type u_3\ninst✝² : CommSemiring T\ninst✝¹ : Algebra R T\nx : Ideal R\nH : x.IsPrime\ninst✝ : IsDomain R\n⊢ x.primeCompl ≤ nonZeroDivisors R",
"usedConstants": [
... | by
intro a ha
rw [mem_nonZeroDivisors_iff_ne_zero]
exact fun h => ha (h.symm ▸ x.zero_mem) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Localization.LocalizationLocalization | {
"line": 222,
"column": 10
} | {
"line": 222,
"column": 13
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommSemiring R\nS : Type u_2\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nT : Type u_3\ninst✝⁵ : CommSemiring T\ninst✝⁴ : Algebra R T\nM N : Submonoid R\nh : M ≤ N\ninst✝³ : IsLocalization M S\ninst✝² : IsLocalization N T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nx₁ ... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.EssentialFiniteness | {
"line": 80,
"column": 2
} | {
"line": 86,
"column": 45
} | [
{
"pp": "case mpr\nR : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nσ : Finset S\nhσ : ∀ (s : S), ∃ t ∈ adjoin R ↑σ, IsUnit t ∧ s * t ∈ adjoin R ↑σ\n⊢ IsLocalization (Submonoid.comap (algebraMap (↥(adjoin R ↑σ)) S) (IsUnit.submonoid S)) S",
"usedConstants": [
... | · constructor; constructor
· exact fun y ↦ y.prop
· intro s
obtain ⟨t, ht, ht', h⟩ := hσ s
exact ⟨⟨⟨_, h⟩, ⟨t, ht⟩, ht'⟩, rfl⟩
· intro x y e
exact ⟨1, by simpa using Subtype.ext e⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Localization.AtPrime.Basic | {
"line": 459,
"column": 43
} | {
"line": 459,
"column": 86
} | [
{
"pp": "R : Type u_1\ninst✝¹⁴ : CommSemiring R\nS : Type u_4\ninst✝¹³ : CommSemiring S\ninst✝¹² : Algebra R S\nR' : Type u_5\nS' : Type u_6\nM : Submonoid R\nT : Submonoid S\ninst✝¹¹ : CommSemiring R'\ninst✝¹⁰ : CommSemiring S'\ninst✝⁹ : Algebra R R'\ninst✝⁸ : Algebra S S'\ninst✝⁷ : Algebra R' S'\ninst✝⁶ : Alg... | under_map_of_isPrime_disjoint _ _ ‹_› disj, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Laurent | {
"line": 365,
"column": 17
} | {
"line": 365,
"column": 21
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nmotive : R[T;T⁻¹] → Prop\nf : R[T;T⁻¹]\nmul_T : ∀ (f : R[X]) (n : ℕ), motive (toLaurent f * T (-↑n))\nn : ℕ\nf' : R[X]\nhf : toLaurent f' = f * T ↑n\n⊢ motive (f * T ↑n * T (-↑n))",
"usedConstants": [
"LaurentPolynomial.T",
"Int.instAddSemigroup",
... | ← hf | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Laurent | {
"line": 561,
"column": 2
} | {
"line": 561,
"column": 65
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\nS : Type u_3\ninst✝ : CommSemiring S\nf : R →+* S\nx : Sˣ\nn : ℕ\n⊢ (eval₂ f x) (T ↑n) = ↑x ^ n",
"usedConstants": [
"LaurentPolynomial.T",
"Units.val",
"Eq.mpr",
"Polynomial.eval₂_X_pow",
"congrArg",
"CommSemiring.toSemirin... | rw [← Polynomial.toLaurent_X_pow, eval₂_toLaurent, eval₂_X_pow] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Laurent | {
"line": 561,
"column": 2
} | {
"line": 561,
"column": 65
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\nS : Type u_3\ninst✝ : CommSemiring S\nf : R →+* S\nx : Sˣ\nn : ℕ\n⊢ (eval₂ f x) (T ↑n) = ↑x ^ n",
"usedConstants": [
"LaurentPolynomial.T",
"Units.val",
"Eq.mpr",
"Polynomial.eval₂_X_pow",
"congrArg",
"CommSemiring.toSemirin... | rw [← Polynomial.toLaurent_X_pow, eval₂_toLaurent, eval₂_X_pow] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Laurent | {
"line": 561,
"column": 2
} | {
"line": 561,
"column": 65
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\nS : Type u_3\ninst✝ : CommSemiring S\nf : R →+* S\nx : Sˣ\nn : ℕ\n⊢ (eval₂ f x) (T ↑n) = ↑x ^ n",
"usedConstants": [
"LaurentPolynomial.T",
"Units.val",
"Eq.mpr",
"Polynomial.eval₂_X_pow",
"congrArg",
"CommSemiring.toSemirin... | rw [← Polynomial.toLaurent_X_pow, eval₂_toLaurent, eval₂_X_pow] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Matrix.Invertible | {
"line": 49,
"column": 63
} | {
"line": 49,
"column": 77
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nα : Type u_3\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : Semiring α\nA : Matrix m n α\nB : Matrix n n α\ninst✝ : Invertible B\n⊢ A * 1 = A",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"congrArg",... | Matrix.mul_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Matrix.Invertible | {
"line": 53,
"column": 63
} | {
"line": 53,
"column": 77
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nα : Type u_3\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\ninst✝¹ : Semiring α\nA : Matrix m n α\nB : Matrix n n α\ninst✝ : Invertible B\n⊢ A * 1 = A",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"congrArg",... | Matrix.mul_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Trace | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 66
} | [
{
"pp": "n : Type u_3\nR : Type u_6\ninst✝² : Fintype n\ninst✝¹ : AddCommMonoid R\ninst✝ : Nonempty n\nr : R\ninhabited_h : Inhabited n\n⊢ ∃ a, a.trace = r",
"usedConstants": [
"Inhabited.default",
"Matrix.trace_single_eq_same",
"Matrix",
"AddMonoid.toAddZeroClass",
"Classical.... | exact ⟨single default default r, trace_single_eq_same default r⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 401,
"column": 4
} | {
"line": 407,
"column": 18
} | [
{
"pp": "case pos\n𝕜 : Type u_3\ninst✝ : Field 𝕜\nr : ℕ\nM : Matrix (Fin r ⊕ Unit) (Fin r ⊕ Unit) 𝕜\nhM : M (inr ()) (inr ()) ≠ 0\ni : Fin r\nk n : ℕ\nhn : n < r\nIH : ((List.drop (n + 1) (listTransvecCol M)).prod * M) (inl i) (inr ()) = if n + 1 ≤ ↑i then 0 else M (inl i) (inr ())\nhn' : n < (listTransvecCo... | · have hni : n = i := by
cases i
simp only [n', Fin.mk_eq_mk] at h
simp [h]
simp only [h, transvection_mul_apply_same, IH, ← hni, add_le_iff_nonpos_right,
listTransvecCol_mul_last_row_drop _ _ hn]
simp [field] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Matrix.NonsingularInverse | {
"line": 430,
"column": 78
} | {
"line": 430,
"column": 92
} | [
{
"pp": "n : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA : Matrix n n α\nh : IsUnit A.det\n⊢ A * 1 = A",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"congrArg",
... | Matrix.mul_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.NonsingularInverse | {
"line": 567,
"column": 2
} | {
"line": 574,
"column": 92
} | [
{
"pp": "n : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nv : n → α\n⊢ (diagonal v)⁻¹ = diagonal v⁻¹ʳ",
"usedConstants": [
"Iff.mpr",
"Units.val",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"MulZeroClass.to... | rw [nonsing_inv_eq_ringInverse]
by_cases h : IsUnit v
· have := isUnit_diagonal.mpr h
cases this.nonempty_invertible
cases h.nonempty_invertible
rw [Ring.inverse_invertible, Ring.inverse_invertible, invOf_diagonal_eq]
· have := isUnit_diagonal.not.mpr h
rw [Ring.inverse_non_unit _ h, Pi.zero_def, ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.NonsingularInverse | {
"line": 567,
"column": 2
} | {
"line": 574,
"column": 92
} | [
{
"pp": "n : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nv : n → α\n⊢ (diagonal v)⁻¹ = diagonal v⁻¹ʳ",
"usedConstants": [
"Iff.mpr",
"Units.val",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"MulZeroClass.to... | rw [nonsing_inv_eq_ringInverse]
by_cases h : IsUnit v
· have := isUnit_diagonal.mpr h
cases this.nonempty_invertible
cases h.nonempty_invertible
rw [Ring.inverse_invertible, Ring.inverse_invertible, invOf_diagonal_eq]
· have := isUnit_diagonal.not.mpr h
rw [Ring.inverse_non_unit _ h, Pi.zero_def, ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.NonsingularInverse | {
"line": 730,
"column": 22
} | {
"line": 730,
"column": 35
} | [
{
"pp": "case neg\nm : Type u\nn : Type u'\nα : Type v\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : CommRing α\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nA : Matrix m m α\ne₁ e₂ : n ≃ m\nh : ¬IsUnit A\nthis : ¬IsUnit (A.submatrix ⇑e₁ ⇑e₂)\n⊢ 0 = 0 ⇑e₂ ⇑e₁",
"usedConstants": [
"CommSemiring.t... | Pi.zero_apply | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Matrix.Transvection | {
"line": 686,
"column": 76
} | {
"line": 686,
"column": 90
} | [
{
"pp": "n : Type u_1\n𝕜 : Type u_3\ninst✝² : Field 𝕜\ninst✝¹ : DecidableEq n\ninst✝ : Fintype n\nM : Matrix n n 𝕜\nL L' : List (TransvectionStruct n 𝕜)\nD : n → 𝕜\nh : (List.map toMatrix L).prod * M * (List.map toMatrix L').prod = diagonal D\n⊢ M = M * 1",
"usedConstants": [
"Eq.mpr",
"Non... | Matrix.mul_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.SchurComplement | {
"line": 237,
"column": 2
} | {
"line": 241,
"column": 47
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\nα : Type u_4\ninst✝⁸ : Fintype l\ninst✝⁷ : Fintype m\ninst✝⁶ : Fintype n\ninst✝⁵ : DecidableEq l\ninst✝⁴ : DecidableEq m\ninst✝³ : DecidableEq n\ninst✝² : CommRing α\nA : Matrix m m α\nB : Matrix m n α\nC : Matrix n m α\nD : Matrix n n α\ninst✝¹ : Invertible D\... | convert!
Invertible.copy' _ _
(fromBlocks (⅟(A - B * ⅟D * C)) (-(⅟(A - B * ⅟D * C) * B * ⅟D))
(-(⅟D * C * ⅟(A - B * ⅟D * C))) (⅟D + ⅟D * C * ⅟(A - B * ⅟D * C) * B * ⅟D))
(fromBlocks_eq_of_invertible₂₂ _ _ _ _) _ | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Algebra.Polynomial.Identities | {
"line": 88,
"column": 8
} | {
"line": 88,
"column": 23
} | [
{
"pp": "case e_a.e_a\nR : Type u\nS : Type v\nT : Type w\nι : Type x\nk : Type y\nA : Type z\na b : R\nm n : ℕ\ninst✝ : CommRing R\nf : R[X]\nx y : R\n⊢ (f.sum fun e a ↦ a * ↑e * x ^ (e - 1) * y) = eval x (derivative f) * y",
"usedConstants": [
"Polynomial.derivative",
"Eq.mpr",
"Polynomi... | derivative_eval | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Nilpotent.Basic | {
"line": 149,
"column": 71
} | {
"line": 149,
"column": 94
} | [
{
"pp": "R : Type u_1\nx y : R\ninst✝ : Semiring R\nh_comm : Commute x y\nm n : ℕ\nhx : x ^ m = 0\nhy : y ^ n = 0\n⊢ m + n ≤ m + n - 1 + 1",
"usedConstants": [
"Eq.mpr",
"congrArg",
"HSub.hSub",
"Nat.sub_le_iff_le_add",
"id",
"instSubNat",
"instOfNatNat",
"LE.... | ← Nat.sub_le_iff_le_add | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MatrixAlgebra | {
"line": 269,
"column": 8
} | {
"line": 269,
"column": 16
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nR : Type u_5\nS : Type u_6\nA : Type u_7\nB : Type u_8\nM : Type u_9\nN : Type u_10\ninst✝¹⁷ : CommSemiring R\ninst✝¹⁶ : Semiring A\ninst✝¹⁵ : Semiring B\ninst✝¹⁴ : Algebra R A\ninst✝¹³ : Algebra R B\ninst✝¹² : Fintype n\ninst✝¹¹ : DecidableEq n\n... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.MatrixAlgebra | {
"line": 269,
"column": 8
} | {
"line": 269,
"column": 16
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nR : Type u_5\nS : Type u_6\nA : Type u_7\nB : Type u_8\nM : Type u_9\nN : Type u_10\ninst✝¹⁷ : CommSemiring R\ninst✝¹⁶ : Semiring A\ninst✝¹⁵ : Semiring B\ninst✝¹⁴ : Algebra R A\ninst✝¹³ : Algebra R B\ninst✝¹² : Fintype n\ninst✝¹¹ : DecidableEq n\n... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MatrixAlgebra | {
"line": 269,
"column": 8
} | {
"line": 269,
"column": 16
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nR : Type u_5\nS : Type u_6\nA : Type u_7\nB : Type u_8\nM : Type u_9\nN : Type u_10\ninst✝¹⁷ : CommSemiring R\ninst✝¹⁶ : Semiring A\ninst✝¹⁵ : Semiring B\ninst✝¹⁴ : Algebra R A\ninst✝¹³ : Algebra R B\ninst✝¹² : Fintype n\ninst✝¹¹ : DecidableEq n\n... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MatrixAlgebra | {
"line": 308,
"column": 8
} | {
"line": 308,
"column": 16
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nR : Type u_5\nS : Type u_6\nA : Type u_7\nB : Type u_8\nM : Type u_9\nN : Type u_10\ninst✝¹³ : CommSemiring R\ninst✝¹² : Semiring A\ninst✝¹¹ : Semiring B\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : Algebra R B\ninst✝⁸ : Fintype n\ninst✝⁷ : DecidableEq n\nins... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.MatrixAlgebra | {
"line": 308,
"column": 8
} | {
"line": 308,
"column": 16
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nR : Type u_5\nS : Type u_6\nA : Type u_7\nB : Type u_8\nM : Type u_9\nN : Type u_10\ninst✝¹³ : CommSemiring R\ninst✝¹² : Semiring A\ninst✝¹¹ : Semiring B\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : Algebra R B\ninst✝⁸ : Fintype n\ninst✝⁷ : DecidableEq n\nins... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MatrixAlgebra | {
"line": 308,
"column": 8
} | {
"line": 308,
"column": 16
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\np : Type u_4\nR : Type u_5\nS : Type u_6\nA : Type u_7\nB : Type u_8\nM : Type u_9\nN : Type u_10\ninst✝¹³ : CommSemiring R\ninst✝¹² : Semiring A\ninst✝¹¹ : Semiring B\ninst✝¹⁰ : Algebra R A\ninst✝⁹ : Algebra R B\ninst✝⁸ : Fintype n\ninst✝⁷ : DecidableEq n\nins... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.SpanRank | {
"line": 112,
"column": 2
} | {
"line": 116,
"column": 24
} | [
{
"pp": "case mp\nR : Type u_1\nM : Type u\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\np : Submodule R M\n⊢ ⨅ s, #↑↑s < ℵ₀ → ∃ S, S.Finite ∧ span R S = p",
"usedConstants": [
"Submodule",
"csInf_mem",
"Preorder.toLT",
"iInf",
"instSMulOfMul",
"Card... | · rintro h
obtain ⟨s, hs⟩ : ⨅ (s : {s : Set M // span R s = p}), #s ∈
Set.range (fun (s : {s : Set M // span R s = p}) ↦ #s) := csInf_mem ⟨#p, ⟨⟨p, by simp⟩, rfl⟩⟩
refine ⟨s.1, ?_, s.2⟩
simpa [← hs] using h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Module.SpanRank | {
"line": 222,
"column": 4
} | {
"line": 222,
"column": 12
} | [
{
"pp": "case mp\nR : Type u_1\nM : Type u\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nI : Submodule R M\nh : I.spanRank = 0\ns : Set M\nhs₂ : span R s = I\nhs₁ : s = ∅\n⊢ I = ⊥",
"usedConstants": [
"Submodule",
"congrArg",
"Eq.mp",
"id",
"Bot.bot",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Module.SpanRank | {
"line": 325,
"column": 2
} | {
"line": 325,
"column": 88
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM : Type u\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\nσ : R →+* S\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nL : Type v\ninst✝² : AddCommMonoid L\ninst✝¹ : Module S L\ninst✝ : RingHomSurjective σ\nf : M →ₛₗ[σ] L\nhf : Function.Injective ⇑f\np : Submodule R M\ns : Set M\... | obtain rfl : span R s = p := by simpa [(map_injective_of_injective hf).eq_iff] using e | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Module.SpanRank | {
"line": 320,
"column": 75
} | {
"line": 326,
"column": 87
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM : Type u\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\nσ : R →+* S\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nL : Type v\ninst✝² : AddCommMonoid L\ninst✝¹ : Module S L\ninst✝ : RingHomSurjective σ\nf : M →ₛₗ[σ] L\nhf : Function.Injective ⇑f\np : Submodule R M\n⊢ Cardinal... | by
refine (lift_spanRank_map_le f p).antisymm ?_
obtain ⟨s, hs, e⟩ := (p.map f).exists_span_set_card_eq_spanRank
obtain ⟨s, rfl⟩ : ∃ y, f '' y = s := Set.subset_range_iff_exists_image_eq.mp
((subset_span.trans e.le).trans LinearMap.map_le_range)
obtain rfl : span R s = p := by simpa [(map_injective_of_injec... | [anonymous] | Lean.Parser.Term.byTactic |
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