module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Combinatorics.Matroid.Map | {
"line": 213,
"column": 6
} | {
"line": 213,
"column": 27
} | {
"line": 213,
"column": 28
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nN : Matroid β\nB : Set α\n⊢ (N.comap f).IsBase B ↔ N.IsBasis (f '' B) (f '' f ⁻¹' N.E) ∧ InjOn f B ∧ B ⊆ f ⁻¹' N.E",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"Matroid.isBasis_gro... | [
"α : Type u_1\nβ : Type u_2\nf : α → β\nN : Matroid β\nB : Set α\n⊢ (N.comap f).IsBasis B (N.comap f).E ↔ N.IsBasis (f '' B) (f '' f ⁻¹' N.E) ∧ InjOn f B ∧ B ⊆ f ⁻¹' N.E"
] | ← isBasis_ground_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 158,
"column": 2
} | {
"line": 158,
"column": 64
} | {
"line": 160,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI : Set α\ne : α\nhI : M.Indep I\nhInt : I.Nontrivial\nhe : e ∈ M.closure I\nh : ∀ f ∈ I, e ∉ M.closure (I \\ {f})\nheI : e ∈ I\nf : α\nhf : f ∈ I\nhne : f ≠ e\n⊢ False",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"False",
"eq_false",
... | [] | exact h f hf (mem_closure_of_mem' _ (by simp [heI, hne.symm])) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Matroid.Map | {
"line": 485,
"column": 2
} | {
"line": 485,
"column": 79
} | {
"line": 486,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf✝ : α → β\nE I : Set α\nM : Matroid α\nN : Matroid β\ninst✝ : M.Finitary\nf : α → β\nhf : InjOn f M.E\nI₀ : Set α\nhI₀E : I₀ ⊆ M.E\nhI : ∀ J ⊆ f '' I₀, J.Finite → (M.map f hf).Indep J\nh' : f '' I₀ ⊆ f '' M.E\n⊢ ∃ I₀_1, M.Indep I₀_1 ∧ f '' I₀ = f '' I₀_1",
"ppTerm": "?m... | [
"α : Type u_1\nβ : Type u_2\nf✝ : α → β\nE I : Set α\nM : Matroid α\nN : Matroid β\ninst✝ : M.Finitary\nf : α → β\nhf : InjOn f M.E\nI₀ : Set α\nhI₀E : I₀ ⊆ M.E\nhI : ∀ J ⊆ f '' I₀, J.Finite → (M.map f hf).Indep J\nh' : f '' I₀ ⊆ f '' M.E\nJ₀ : Set α\nhJ₀I₀ : J₀ ⊆ I₀\nhJ₀ : J₀.Finite\n⊢ M.Indep J₀"
] | refine ⟨I₀, indep_of_forall_finite_subset_indep _ fun J₀ hJ₀I₀ hJ₀ ↦ ?_, rfl⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 47
} | {
"line": 144,
"column": 0
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\n⊢ M.closure X = ⋂₀ {F | M.IsFlat F ∧ X ⊆ F}",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"setOf",
"Matroid.closure.eq_1",
"id",
"LE.le",
"Set... | [] | rw [closure, inter_eq_self_of_subset_left hX] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 47
} | {
"line": 144,
"column": 0
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\n⊢ M.closure X = ⋂₀ {F | M.IsFlat F ∧ X ⊆ F}",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"setOf",
"Matroid.closure.eq_1",
"id",
"LE.le",
"Set... | [] | rw [closure, inter_eq_self_of_subset_left hX] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 47
} | {
"line": 144,
"column": 0
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\n⊢ M.closure X = ⋂₀ {F | M.IsFlat F ∧ X ⊆ F}",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"setOf",
"Matroid.closure.eq_1",
"id",
"LE.le",
"Set... | [] | rw [closure, inter_eq_self_of_subset_left hX] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 235,
"column": 8
} | {
"line": 235,
"column": 26
} | {
"line": 235,
"column": 26
} | [
{
"pp": "case refine_1\nα : Type u_1\nM : Matroid α\nX : Set α\nn : ℕ∞\nh : n ≤ M.eRk X\nJ : Set α\nhJ : M.IsBasis' J X\n⊢ ∃ I ⊆ X, M.Indep I ∧ I.encard = n",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Set.encard",
"congrArg",
"Matroid.IsBasis'",
"Matroid.Is... | [
"case refine_1\nα : Type u_1\nM : Matroid α\nX : Set α\nn : ℕ∞\nJ : Set α\nh : n ≤ J.encard\nhJ : M.IsBasis' J X\n⊢ ∃ I ⊆ X, M.Indep I ∧ I.encard = n"
] | ← hJ.encard_eq_eRk | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 441,
"column": 6
} | {
"line": 441,
"column": 27
} | {
"line": 441,
"column": 28
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nB : Set α\n⊢ M.IsBase B ↔ M.Indep B ∧ M.closure B = M.E",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"Matroid.isBasis_ground_iff",
"Matroid.IsBase",
"Matroid.Indep",
"id",
... | [
"α : Type u_2\nM : Matroid α\nB : Set α\n⊢ M.IsBasis B M.E ↔ M.Indep B ∧ M.closure B = M.E"
] | ← isBasis_ground_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 651,
"column": 2
} | {
"line": 651,
"column": 29
} | {
"line": 653,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\ne : α\nhe : M.IsColoop e\n⊢ e ∈ M.closure X ↔ e ∈ X",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Iff.rfl",
"Membership.mem",
"id",
"Iff",
"propext",
"Matroid.closure",... | [] | rw [he.mem_closure_iff_mem] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 486,
"column": 2
} | {
"line": 487,
"column": 89
} | {
"line": 488,
"column": 2
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nI : Set α\nJs : Set (Set α)\nhI : M.Indep I\nhne : Js.Nonempty\nhIs : ∀ J ∈ Js, J ⊆ I\nhiX : ⋂₀ Js ⊆ I\nhiI : M.Indep (⋂₀ Js)\ne : α\nhe : ∀ i ∈ Js, e ∈ M.closure i\nhe' : M.Indep (insert e (⋂₀ Js))\nheEI : e ∈ M.E \\ I\nJ : Set α\nhJI : M.IsBasis J (insert e I)\nheJ : inse... | [
"α : Type u_2\nM : Matroid α\nJs : Set (Set α)\nhne : Js.Nonempty\nhiI : M.Indep (⋂₀ Js)\ne : α\nhe : ∀ i ∈ Js, e ∈ M.closure i\nhe' : M.Indep (insert e (⋂₀ Js))\nJ : Set α\nheJ : insert e (⋂₀ Js) ⊆ J\nf : α\nhI : M.Indep (insert f (J \\ {e}))\nhIs : ∀ J_1 ∈ Js, J_1 ⊆ insert f (J \\ {e})\nhiX : ⋂₀ Js ⊆ insert f (J ... | obtain rfl := hI.eq_of_isBasis (hfb.isBasis_subset (insert_subset hfIJ.1
(by (rw [sdiff_subset_iff, singleton_union]; exact hJI.subset))) (subset_insert _ _)) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 551,
"column": 2
} | {
"line": 554,
"column": 99
} | {
"line": 555,
"column": 2
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX Y : Set α\ninst✝ : M.Finitary\nhX : X.Finite\nhXY : X ⊆ M.closure Y\n⊢ ∃ I ⊆ Y, I.Finite ∧ M.Indep I ∧ X ⊆ M.closure I",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Matroid.IsBasis'.closure_eq_closure",
"Eq.mpr",
"ChainCompletePa... | [
"α : Type u_1\nM : Matroid α\nX Y : Set α\ninst✝ : M.Finitary\nhX : X.Finite\nhXY : X ⊆ M.closure Y\n⊢ ∃ T ⊆ Y, T.Finite ∧ X ⊆ M.closure T"
] | suffices aux : ∃ T ⊆ Y, T.Finite ∧ X ⊆ M.closure T by
obtain ⟨T, hT, hTfin, hXT⟩ := aux
obtain ⟨I, hI⟩ := M.exists_isBasis' T
exact ⟨_, hI.subset.trans hT, hTfin.subset hI.subset, hI.indep, by rwa [hI.closure_eq_closure]⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 648,
"column": 2
} | {
"line": 648,
"column": 25
} | {
"line": 649,
"column": 2
} | [
{
"pp": "α : Type u_2\nM : Matroid α\ne f : α\nI : Set α\nhI : M.Indep I\nhfI : f ∈ M.closure I\nhe : e ∈ M.closure (insert f I \\ {e})\nheI : e = f ∨ e ∈ I\n⊢ M.Indep (insert f I \\ {e})",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Matroid.Indep",
"... | [
"case inl\nα : Type u_2\nM : Matroid α\ne : α\nI : Set α\nhI : M.Indep I\nhfI : e ∈ M.closure I\nhe : e ∈ M.closure (insert e I \\ {e})\n⊢ M.Indep (insert e I \\ {e})",
"case inr\nα : Type u_2\nM : Matroid α\ne f : α\nI : Set α\nhI : M.Indep I\nhfI : f ∈ M.closure I\nhe : e ∈ M.closure (insert f I \\ {e})\nheI : ... | obtain rfl | heI := heI | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 672,
"column": 77
} | {
"line": 675,
"column": 63
} | {
"line": 677,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\n⊢ M✶.RankPos ↔ ∃ C, M.IsCircuit C",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Matroid.rankPos_iff",
"Eq.mpr",
"ChainCompletePartialOrder.instOfCompleteLattice",
"instReflLe",
"congrArg",
"Matroid.E",
"Parti... | [] | by
rw [rankPos_iff, dual_isBase_iff, sdiff_empty, not_iff_comm, not_exists,
← ground_indep_iff_isBase, indep_iff_forall_subset_not_isCircuit]
exact ⟨fun h C _ ↦ h C, fun h C hC ↦ h C hC.subset_ground hC⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 683,
"column": 75
} | {
"line": 684,
"column": 68
} | {
"line": 686,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\n⊢ M.RankPos ↔ ∃ K, M.IsCocircuit K",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Iff.rfl",
"Matroid.dual_rankPos_iff_exists_isCircuit",
"Matroid.dual",
"Exists",
"id",
"Matroid.dua... | [] | by
rw [← dual_dual M, dual_rankPos_iff_exists_isCircuit, dual_dual M] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 851,
"column": 12
} | {
"line": 851,
"column": 27
} | {
"line": 851,
"column": 27
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nS T : Set α\nhS : M.Spanning S\nhST : S ⊆ T\nhT : T ⊆ M.E\n⊢ M.E ⊆ M.closure T",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ChainCompletePartialOrder.instOfCompleteLattice",
"congrArg",
"Matroid.E",
"PartialO... | [
"α : Type u_2\nM : Matroid α\nS T : Set α\nhS : M.Spanning S\nhST : S ⊆ T\nhT : T ⊆ M.E\n⊢ M.closure S ⊆ M.closure T"
] | ← hS.closure_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Algebraic.MvPolynomial | {
"line": 70,
"column": 7
} | {
"line": 70,
"column": 22
} | {
"line": 70,
"column": 23
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : CommRing R\ni : σ\nf : R[X]\nhf : Transcendental R f\nthis : Transcendental (↥(supported R ∅)) ((Polynomial.aeval (X i)) f)\ng : R ≃ₐ[R] ↥(supported R ∅) := (Algebra.botEquivOfInjective ⋯).symm.trans ((supported R ∅).equivOfEq ⊥ ⋯).symm\n⊢ Transcendental R ((Polynomi... | [
"σ : Type u_1\nR : Type u_2\ninst✝ : CommRing R\ni : σ\nf : R[X]\nhf : Transcendental R f\nthis : Transcendental (↥(supported R ∅)) ((Polynomial.aeval (X i)) f)\ng : R ≃ₐ[R] ↥(supported R ∅) := (Algebra.botEquivOfInjective ⋯).symm.trans ((supported R ∅).equivOfEq ⊥ ⋯).symm\n⊢ ¬IsAlgebraic R ((Polynomial.aeval (X i)... | Transcendental, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 909,
"column": 2
} | {
"line": 909,
"column": 87
} | {
"line": 911,
"column": 0
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nI : Set α\nhI : M.Indep I\nhIs : M.Spanning I\n⊢ M.IsBase I",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.IsBase.eq_of_subset_indep",
"Exists",
"Matroid.IsBase",
"id",
"LE.le",
... | [] | obtain ⟨B, hB, hBI⟩ := hIs.exists_isBase_subset; rwa [← hB.eq_of_subset_indep hI hBI] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 909,
"column": 2
} | {
"line": 909,
"column": 87
} | {
"line": 911,
"column": 0
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nI : Set α\nhI : M.Indep I\nhIs : M.Spanning I\n⊢ M.IsBase I",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.IsBase.eq_of_subset_indep",
"Exists",
"Matroid.IsBase",
"id",
"LE.le",
... | [] | obtain ⟨B, hB, hBI⟩ := hIs.exists_isBase_subset; rwa [← hB.eq_of_subset_indep hI hBI] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Algebraic.MvPolynomial | {
"line": 76,
"column": 11
} | {
"line": 76,
"column": 26
} | {
"line": 76,
"column": 27
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : CommRing R\ni : σ\nf : R[X]\n⊢ Transcendental R ((Polynomial.aeval (X i)) f) → Transcendental R f",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"CommSemiring.toSemiring",
... | [
"σ : Type u_1\nR : Type u_2\ninst✝ : CommRing R\ni : σ\nf : R[X]\n⊢ ¬IsAlgebraic R ((Polynomial.aeval (X i)) f) → ¬IsAlgebraic R f"
] | Transcendental, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 944,
"column": 70
} | {
"line": 952,
"column": 100
} | {
"line": 954,
"column": 0
} | [
{
"pp": "α : Type u_2\nM M' : Matroid α\nh : M.E = M'.E\nhsp : ∀ S ⊆ M.E, M.Spanning S ↔ M'.Spanning S\n⊢ M = M'",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ChainCompletePartialOrder.instOfCompleteLattice",
"congrArg",
"Matroid.E",
"Iff.rfl",
... | [] | by
have hsp' : M.Spanning = M'.Spanning := by
ext S
refine (em (S ⊆ M.E)).elim (fun hSE ↦ by rw [hsp _ hSE])
(fun hSE ↦ iff_of_false (fun h ↦ hSE h.subset_ground)
(fun h' ↦ hSE (h'.subset_ground.trans h.symm.subset)))
rw [← dual_inj, ext_iff_indep, dual_ground, dual_ground, and_iff_right h]
in... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 1025,
"column": 4
} | {
"line": 1025,
"column": 58
} | {
"line": 1026,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_2\nβ : Type u_3\nM : Matroid β\nf : α → β\nX I : Set α\nhI : (M.comap f).IsBasis' I X\nhI' : M.IsBasis' (f '' I) (f '' X)\nhIinj : InjOn f I\nx : α\nhxE : f x ∈ M.E\nhxI : x ∈ I\n⊢ M.Indep (f '' insert x I) → InjOn f (insert x I) → x ∈ I ↔ M.Indep (f '' insert x I) → ∃ x_1 ∈ I, f x... | [] | simp [hxI, show ∃ y ∈ I, f y = f x from ⟨x, hxI, rfl⟩] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 1025,
"column": 4
} | {
"line": 1025,
"column": 58
} | {
"line": 1026,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_2\nβ : Type u_3\nM : Matroid β\nf : α → β\nX I : Set α\nhI : (M.comap f).IsBasis' I X\nhI' : M.IsBasis' (f '' I) (f '' X)\nhIinj : InjOn f I\nx : α\nhxE : f x ∈ M.E\nhxI : x ∈ I\n⊢ M.Indep (f '' insert x I) → InjOn f (insert x I) → x ∈ I ↔ M.Indep (f '' insert x I) → ∃ x_1 ∈ I, f x... | [] | simp [hxI, show ∃ y ∈ I, f y = f x from ⟨x, hxI, rfl⟩] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 1025,
"column": 4
} | {
"line": 1025,
"column": 58
} | {
"line": 1026,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_2\nβ : Type u_3\nM : Matroid β\nf : α → β\nX I : Set α\nhI : (M.comap f).IsBasis' I X\nhI' : M.IsBasis' (f '' I) (f '' X)\nhIinj : InjOn f I\nx : α\nhxE : f x ∈ M.E\nhxI : x ∈ I\n⊢ M.Indep (f '' insert x I) → InjOn f (insert x I) → x ∈ I ↔ M.Indep (f '' insert x I) → ∃ x_1 ∈ I, f x... | [] | simp [hxI, show ∃ y ∈ I, f y = f x from ⟨x, hxI, rfl⟩] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.FixedPoints | {
"line": 221,
"column": 66
} | {
"line": 221,
"column": 81
} | {
"line": 221,
"column": 82
} | [
{
"pp": "α : Type u_1\nG : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G α\ng h : G\ncomm : Commute g h\nx : α\n⊢ (g * h⁻¹) • x = h⁻¹ • x ↔ x ∈ fixedBy α g",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"instHSMul",
"HMul.hMul",
... | [
"α : Type u_1\nG : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G α\ng h : G\ncomm : Commute g h\nx : α\n⊢ (h⁻¹ * g) • x = h⁻¹ • x ↔ x ∈ fixedBy α g"
] | comm.inv_right, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 446,
"column": 7
} | {
"line": 446,
"column": 16
} | {
"line": 446,
"column": 17
} | [
{
"pp": "ι : Type u\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : IsDomain S\nthis : lift.{u, max u v} (trdeg S (MvPolynomial ι S)) = lift.{max u v, u} #ι\n⊢ trdeg S (MvPolynomial ι S) = lift.{v, u} #ι",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"Cardina... | [
"ι : Type u\nS : Type v\ninst✝¹ : CommRing S\ninst✝ : IsDomain S\nthis : trdeg S (MvPolynomial ι S) = lift.{max u v, u} #ι\n⊢ trdeg S (MvPolynomial ι S) = lift.{v, u} #ι"
] | lift_id', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Galois.Basic | {
"line": 479,
"column": 6
} | {
"line": 479,
"column": 17
} | {
"line": 479,
"column": 18
} | [
{
"pp": "case adjoin_rootSet'\nF : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : FiniteDimensional F E\ninst✝ : IsGalois F E\nα : E\nh1 : F⟮α⟯ = ⊤\n⊢ Algebra.adjoin F ((minpoly F α).rootSet E) = ⊤",
"ppTerm": "?adjoin_rootSet'",
"assigned": true,
"usedCon... | [
"case adjoin_rootSet'\nF : Type u_1\ninst✝⁴ : Field F\nE : Type u_2\ninst✝³ : Field E\ninst✝² : Algebra F E\ninst✝¹ : FiniteDimensional F E\ninst✝ : IsGalois F E\nα : E\nh1 : F⟮α⟯ = ⊤\n⊢ ⊤ ≤ Algebra.adjoin F ((minpoly F α).rootSet E)"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.FilterBasis | {
"line": 331,
"column": 8
} | {
"line": 331,
"column": 20
} | {
"line": 332,
"column": 8
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : TopologicalSpace R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nB : ModuleFilterBasis R M\ninst✝ : DiscreteTopology R\nx₀ : R\nU : Set M\nh : U = {0}\n⊢ ∃\n V ∈\n (have this := default;\n this).sets,\n V ⊆ (fun x ↦ x₀ • x) ⁻¹... | [
"case right\nR : Type u_1\nM : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : TopologicalSpace R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nB : ModuleFilterBasis R M\ninst✝ : DiscreteTopology R\nx₀ : R\nU : Set M\nh : U = {0}\n⊢ {0} ⊆ (fun x ↦ x₀ • x) ⁻¹' U"
] | use {0}, rfl | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.FieldTheory.SeparableDegree | {
"line": 468,
"column": 2
} | {
"line": 469,
"column": 42
} | {
"line": 471,
"column": 0
} | [
{
"pp": "case neg.refine_2.inr\nF : Type u\ninst✝ : Field F\nf g : F[X]\nh : ¬f = 0 ∧ ¬g = 0\nx : AlgebraicClosure F\nhf : f ≠ 0 ∧ (aeval x) f = 0\nhg : g ≠ 0 ∧ (aeval x) g = 0\nu v : F[X]\nhfg : u * f + v * g = 1\n⊢ False",
"ppTerm": "?neg.refine_2.inr✝",
"assigned": true,
"usedConstants": [
... | [] | simpa only [map_add, map_mul, map_one, hf.2, hg.2, mul_zero, add_zero,
zero_ne_one] using congr(aeval x $hfg) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.Algebra.OpenSubgroup | {
"line": 247,
"column": 2
} | {
"line": 247,
"column": 44
} | {
"line": 248,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : SeparatelyContinuousMul G\nH : Subgroup G\ng : G\nhg : ↑H ∈ 𝓝 g\n⊢ IsOpen[inst✝¹] ↑H",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Iff.mpr",
"Membership.mem",
... | [
"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : SeparatelyContinuousMul G\nH : Subgroup G\ng : G\nhg : ↑H ∈ 𝓝 g\nx : G\nhx : x ∈ ↑H\n⊢ ↑H ∈ 𝓝 x"
] | refine isOpen_iff_mem_nhds.2 fun x hx ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.FieldTheory.SeparableClosure | {
"line": 342,
"column": 27
} | {
"line": 342,
"column": 44
} | {
"line": 342,
"column": 45
} | [
{
"pp": "F : Type u\ninst✝ : Field F\n⊢ Cardinal.toNat (insepDegree F F) = 1",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroOneClass",
"Cardinal.instOne",
"Cardinal",
"congrArg",
"CommSemiring.toSemiring",
"Cardinal.commS... | [
"F : Type u\ninst✝ : Field F\n⊢ Cardinal.toNat 1 = 1"
] | insepDegree_self, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.SeparableClosure | {
"line": 389,
"column": 7
} | {
"line": 389,
"column": 24
} | {
"line": 389,
"column": 25
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nthis : Cardinal.lift.{u, v} (insepDegree F ↥⊥) = Cardinal.lift.{v, u} (insepDegree F F)\n⊢ insepDegree F ↥⊥ = 1",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
... | [
"F : Type u\nE : Type v\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nthis : Cardinal.lift.{u, v} (insepDegree F ↥⊥) = Cardinal.lift.{v, u} 1\n⊢ insepDegree F ↥⊥ = 1"
] | insepDegree_self, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed | {
"line": 134,
"column": 2
} | {
"line": 134,
"column": 6
} | {
"line": 134,
"column": 6
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : IsDomain R\ninst✝³ : Algebra R S\ninst✝² : IsIntegrallyClosed R\ninst✝¹ : IsDomain S\ninst✝ : IsTorsionFree R S\ns : S\nP : R[X]\nhmo : P.Monic\nhP : (Polynomial.aeval s) P = 0\nPmin : ∀ (Q : R[X]), Q.Monic → (Polynomial.aev... | [
"R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : IsDomain R\ninst✝³ : Algebra R S\ninst✝² : IsIntegrallyClosed R\ninst✝¹ : IsDomain S\ninst✝ : IsTorsionFree R S\ns : S\nP : R[X]\nhmo : P.Monic\nhP : (Polynomial.aeval s) P = 0\nPmin : ∀ (Q : R[X]), Q.Monic → (Polynomial.aeval s) Q = 0 ... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.RingTheory.Polynomial.GaussLemma | {
"line": 244,
"column": 2
} | {
"line": 246,
"column": 94
} | {
"line": 247,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nK : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Algebra R K\ninst✝² : IsFractionRing R K\ninst✝¹ : IsDomain R\ninst✝ : IsGCDMonoid R\nf : R[X]\nhf : f.IsPrimitive\ng : K[X]\nthis : NormalizedGCDMonoid R := ⋯.some\nx✝ : g * map (algebraMap R K) f ∈ lifts (algebraMap R K)\nk :... | [
"R : Type u_1\ninst✝⁵ : CommRing R\nK : Type u_2\ninst✝⁴ : Field K\ninst✝³ : Algebra R K\ninst✝² : IsFractionRing R K\ninst✝¹ : IsDomain R\ninst✝ : IsGCDMonoid R\nf : R[X]\nhf : f.IsPrimitive\ng : K[X]\nthis : NormalizedGCDMonoid R := ⋯.some\nx✝ : g * map (algebraMap R K) f ∈ lifts (algebraMap R K)\nk : R[X]\nhk : ... | have g'_mul_f : g' * f = b • k := by
apply map_injective (algebraMap R K) (FaithfulSMul.algebraMap_injective R K)
rw [Polynomial.map_smul, algebraMap_smul, hk, ← smul_mul_assoc, ← hb₂, Polynomial.map_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.Galois.Infinite | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 72
} | {
"line": 243,
"column": 2
} | [
{
"pp": "k : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\nL : IntermediateField k K\ninst✝ : IsGalois k K\n⊢ IsOpen L.fixingSubgroup.carrier ↔ FiniteDimensional k ↥L",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"instSMulOfMul",
"Monoid.to... | [
"k : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\nL : IntermediateField k K\ninst✝ : IsGalois k K\nh : IsOpen L.fixingSubgroup.carrier\n⊢ FiniteDimensional k ↥L"
] | refine ⟨fun h ↦ ?_, fun h ↦ IntermediateField.fixingSubgroup_isOpen L⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.FieldTheory.PurelyInseparable.Basic | {
"line": 652,
"column": 4
} | {
"line": 652,
"column": 47
} | {
"line": 653,
"column": 2
} | [
{
"pp": "case h\nF : Type u\nE : Type v\ninst✝⁸ : Field F\ninst✝⁷ : Field E\ninst✝⁶ : Algebra F E\nK : Type w\ninst✝⁵ : Field K\ninst✝⁴ : Algebra F K\nR : Type u_1\nA : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring A\ninst✝¹ : Algebra R A\np : ℕ\ninst✝ : ExpChar A p\nx y : A\nn : ℕ\nhx : x ^ p ^ n ∈ ... | [] | exact add_mem (pow_mem hx _) (pow_mem hy _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.NumberTheory.NumberField.Basic | {
"line": 346,
"column": 21
} | {
"line": 346,
"column": 50
} | {
"line": 346,
"column": 50
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : NumberField K\nM : Type u_3\ninst✝ : MulOneClass M\nf : M →* K\nh : ∀ (x : M), IsIntegral ℤ (f x)\nx y : M\n⊢ restrict (⇑f) h (x * y) = restrict (⇑f) h x * restrict (⇑f) h y",
"ppTerm": "?m.28",
"assigned": true,
"used... | [
"K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : NumberField K\nM : Type u_3\ninst✝ : MulOneClass M\nf : M →* K\nh : ∀ (x : M), IsIntegral ℤ (f x)\nx y : M\n⊢ ⟨f x * f y, ⋯⟩ = ⟨f x, ⋯⟩ * ⟨f y, ⋯⟩"
] | simp only [restrict, map_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.NumberField.Basic | {
"line": 445,
"column": 26
} | {
"line": 445,
"column": 83
} | {
"line": 447,
"column": 0
} | [
{
"pp": "f : Polynomial ℚ\nhf : Fact (Irreducible f)\n⊢ FiniteDimensional ℚ (AdjoinRoot f)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AdjoinRoot",
"Algebra.algebraMap",
"CommSemiring.toSemiring",
"Rat",
"Rat.commSemiring",
"Field.to... | [] | by convert! (AdjoinRoot.powerBasis hf.out.ne_zero).finite | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Trace.Basic | {
"line": 317,
"column": 36
} | {
"line": 317,
"column": 47
} | {
"line": 317,
"column": 47
} | [
{
"pp": "case neg.succ.hnc\nK : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nH : ¬Algebra.IsSeparable K L\np : ℕ\nhp : ExpChar K p\nthis✝ : p ≠ 0\nx : L\nh₀ : FiniteDimensional K L\nhx : ¬IsSeparable K x\ng : K[X]\nhg₁ : g.Separable\nn : ℕ\nhg₂ : (expand K (p ^ (n + 1))) g = ... | [
"case neg.succ.hnc\nK : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nH : ¬Algebra.IsSeparable K L\np : ℕ\nhp : ExpChar K p\nthis✝ : p ≠ 0\nx : L\nh₀ : FiniteDimensional K L\nhx : ¬IsSeparable K x\ng : K[X]\nhg₁ : g.Separable\nn : ℕ\nhg₂ : (expand K (p ^ (n + 1))) g = minpoly K x\... | Nat.dvd_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Finite.Basic | {
"line": 101,
"column": 68
} | {
"line": 101,
"column": 74
} | {
"line": 101,
"column": 74
} | [
{
"pp": "R : Type u_2\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Fintype R\nf g : R[X]\nhf2 : f.degree = 2\nhg2 : g.degree = 2\nhR : Fintype.card R % 2 = 1\nthis : DecidableEq R := Classical.decEq R\nhd : Disjoint (image (fun x ↦ eval x f) univ) (image (fun x ↦ eval x (-g)) univ)\n⊢ 0 < 2",
"ppTerm"... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.FieldTheory.Finite.Basic | {
"line": 103,
"column": 64
} | {
"line": 103,
"column": 70
} | {
"line": 103,
"column": 70
} | [
{
"pp": "R : Type u_2\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : Fintype R\nf g : R[X]\nhf2 : f.degree = 2\nhg2 : g.degree = 2\nhR : Fintype.card R % 2 = 1\nthis : DecidableEq R := Classical.decEq R\nhd : Disjoint (image (fun x ↦ eval x f) univ) (image (fun x ↦ eval x (-g)) univ)\n⊢ 0 < 2",
"ppTerm"... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.FieldTheory.Finite.Basic | {
"line": 210,
"column": 4
} | {
"line": 210,
"column": 67
} | {
"line": 211,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝² : CommRing K\ninst✝¹ : NoZeroDivisors K\nG : Subgroup Kˣ\ninst✝ : Fintype ↥G\nk : ℕ\nk_pos : k ≠ 0\nk_lt_card_G : k < Nat.card ↥G\na✝ : Nontrivial K\nthis : IsDomain K\na : ↥G\nha : a ^ k ≠ 1\nh_multiset_map : Multiset.map (fun x ↦ ↑↑x ^ k) univ.val = Multiset.map (fun x ↦ ↑↑x ^ k ... | [] | rw [sub_mul, mul_comm, ← h_multiset_map_sum, one_mul, sub_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.Finite.Basic | {
"line": 308,
"column": 47
} | {
"line": 308,
"column": 55
} | {
"line": 308,
"column": 55
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : Fintype K\ninst✝ : DecidableEq K\ni : ℕ\nφ : Kˣ →* K := { toFun := fun x ↦ ↑x ^ i, map_one' := ⋯, map_mul' := ⋯ }\nthis✝ : Decidable (φ = 1)\nthis : q - 1 ∣ i ↔ φ = 1\nh✝ : φ = 1\n⊢ 0 - 1 = -1",
"ppTerm": "?pos✝",
"assigned": true,
"usedCon... | [
"case pos\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : Fintype K\ninst✝ : DecidableEq K\ni : ℕ\nφ : Kˣ →* K := { toFun := fun x ↦ ↑x ^ i, map_one' := ⋯, map_mul' := ⋯ }\nthis✝ : Decidable (φ = 1)\nthis : q - 1 ∣ i ↔ φ = 1\nh✝ : φ = 1\n⊢ -1 = -1",
"case pos\nK : Type u_1\ninst✝² : Field K\ninst✝¹ : Fintype K\ninst✝ :... | zero_sub | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Finite.Basic | {
"line": 472,
"column": 16
} | {
"line": 472,
"column": 24
} | {
"line": 472,
"column": 24
} | [
{
"pp": "case e'_4\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : Fintype K\naux : X ^ q - X ≠ 0\nthis : (X ^ q - X).roots.toFinset = univ\n⊢ 0 - 1 = -1",
"ppTerm": "?e'_4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | [
"case e'_4\nK : Type u_1\ninst✝¹ : Field K\ninst✝ : Fintype K\naux : X ^ q - X ≠ 0\nthis : (X ^ q - X).roots.toFinset = univ\n⊢ -1 = -1"
] | zero_sub | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.Finite.Basic | {
"line": 511,
"column": 84
} | {
"line": 511,
"column": 90
} | {
"line": 511,
"column": 90
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ZMod p\nhp_odd : p % 2 = 1\nf : (ZMod p)[X] := X ^ 2\ng : (ZMod p)[X] := X ^ 2 - C x\n⊢ 0 < 2",
"ppTerm": "?m.430",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.FieldTheory.Finite.Basic | {
"line": 511,
"column": 84
} | {
"line": 511,
"column": 90
} | {
"line": 511,
"column": 90
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ZMod p\nhp_odd : p % 2 = 1\nf : (ZMod p)[X] := X ^ 2\ng : (ZMod p)[X] := X ^ 2 - C x\n⊢ 0 < 2",
"ppTerm": "?m.430",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Finite.Basic | {
"line": 511,
"column": 84
} | {
"line": 511,
"column": 90
} | {
"line": 511,
"column": 90
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nx : ZMod p\nhp_odd : p % 2 = 1\nf : (ZMod p)[X] := X ^ 2\ng : (ZMod p)[X] := X ^ 2 - C x\n⊢ 0 < 2",
"ppTerm": "?m.430",
"assigned": true,
"usedConstants": [
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat",
"LT... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.Finite.Basic | {
"line": 662,
"column": 13
} | {
"line": 662,
"column": 44
} | {
"line": 662,
"column": 45
} | [
{
"pp": "case pos\np x y : ℕ\nhp : Nat.Prime p\nh : (p - 1) * ((x - y) / (p - 1)) = x - y\nhxy : y ≤ x\nhy : 0 < y\nn : ℤ\nhn : n ≡ 0 [ZMOD ↑p]\n⊢ 0 ^ x ≡ 0 ^ y [ZMOD ↑p]",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"instOfNatNat",
"Int",
... | [
"case pos\np x y : ℕ\nhp : Nat.Prime p\nh : (p - 1) * ((x - y) / (p - 1)) = x - y\nhxy : y ≤ x\nhy : 0 < y\nn : ℤ\nhn : n ≡ 0 [ZMOD ↑p]\n⊢ 0 ≡ 0 ^ y [ZMOD ↑p]"
] | zero_pow (hy.trans_le hxy).ne', | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.LinearAlgebra.FreeModule.Finite.CardQuotient | {
"line": 116,
"column": 36
} | {
"line": 122,
"column": 29
} | {
"line": 124,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝³ : AddCommGroup E\ninst✝² : Module ℚ E\nL₁ L₂ : AddSubgroup E\nH : L₁ ≤ L₂\nι : Type u_2\ninst✝¹ : DecidableEq ι\ninst✝ : Fintype ι\nb₁ b₂ : Basis ι ℚ E\nh₁ : L₁ = closure (Set.range ⇑b₁)\nh₂ : L₂ = closure (Set.range ⇑b₂)\n⊢ ↑(L₁.relIndex L₂) = |b₂.det ⇑b₁|",
"ppTerm": "?m.45",... | [] | by
rw [AddSubgroup.relIndex_eq_natAbs_det L₁ L₂ H (b₁.addSubgroupOfClosure L₁ h₁)
(b₂.addSubgroupOfClosure L₂ h₂), Nat.cast_natAbs, Int.cast_abs]
change |algebraMap ℤ ℚ _| = _
rw [Basis.det_apply, Basis.det_apply, RingHom.map_det]
congr; ext
simp [Basis.toMatrix_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.Finite.Basic | {
"line": 834,
"column": 2
} | {
"line": 834,
"column": 67
} | {
"line": 835,
"column": 2
} | [
{
"pp": "F : Type u_3\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na : F\nha : a ≠ 0\n⊢ IsSquare (Units.mk0 a ha) ↔ IsSquare a",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Units.val",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"HMul.hMul",
"... | [
"F : Type u_3\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na : F\nha : a ≠ 0\n⊢ (∃ r, a = ↑r * ↑r) ↔ ∃ r, a = r * r"
] | simp only [IsSquare, Units.ext_iff, Units.val_mk0, Units.val_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Invariant.Basic | {
"line": 502,
"column": 2
} | {
"line": 502,
"column": 41
} | {
"line": 503,
"column": 2
} | [
{
"pp": "G : Type u_1\nA : Type u_2\nB : Type u_3\nK : Type u_4\nL : Type u_5\ninst✝¹⁸ : Group G\ninst✝¹⁷ : CommRing A\ninst✝¹⁶ : CommRing B\ninst✝¹⁵ : Algebra A B\ninst✝¹⁴ : Field K\ninst✝¹³ : Field L\ninst✝¹² : Algebra K L\ninst✝¹¹ : Algebra A K\ninst✝¹⁰ : Algebra B L\ninst✝⁹ : Algebra A L\ninst✝⁸ : IsFractio... | [
"case h\nG : Type u_1\nA : Type u_2\nB : Type u_3\nK : Type u_4\nL : Type u_5\ninst✝¹⁸ : Group G\ninst✝¹⁷ : CommRing A\ninst✝¹⁶ : CommRing B\ninst✝¹⁵ : Algebra A B\ninst✝¹⁴ : Field K\ninst✝¹³ : Field L\ninst✝¹² : Algebra K L\ninst✝¹¹ : Algebra A K\ninst✝¹⁰ : Algebra B L\ninst✝⁹ : Algebra A L\ninst✝⁸ : IsFractionRin... | use algebraMap A K b / algebraMap A K a | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 385,
"column": 2
} | {
"line": 411,
"column": 92
} | {
"line": 413,
"column": 0
} | [
{
"pp": "S : Type u_1\ninst✝³ : CommRing S\ninst✝² : IsDedekindDomain S\ninst✝¹ : Free ℤ S\ninst✝ : Module.Finite ℤ S\np : ℤ\nhp : Prime p\nI : Ideal S\nhI : p ∣ ↑(absNorm I)\n⊢ ∃ P, P.IsMaximal ∧ under ℤ P = span {p} ∧ P ∣ I",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Submodule... | [] | have : IsAddTorsionFree S := .of_isTorsionFree ℤ _
have := CharZero.of_isAddTorsionFree S S
have hpMax : (Ideal.span {p}).IsMaximal :=
((Ideal.span_singleton_prime hp.ne_zero).mpr hp).isMaximal (by simpa using hp.ne_zero)
induction I using UniqueFactorizationMonoid.induction_on_prime with
| h₁ =>
obtain... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Norm.AbsNorm | {
"line": 385,
"column": 2
} | {
"line": 411,
"column": 92
} | {
"line": 413,
"column": 0
} | [
{
"pp": "S : Type u_1\ninst✝³ : CommRing S\ninst✝² : IsDedekindDomain S\ninst✝¹ : Free ℤ S\ninst✝ : Module.Finite ℤ S\np : ℤ\nhp : Prime p\nI : Ideal S\nhI : p ∣ ↑(absNorm I)\n⊢ ∃ P, P.IsMaximal ∧ under ℤ P = span {p} ∧ P ∣ I",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Submodule... | [] | have : IsAddTorsionFree S := .of_isTorsionFree ℤ _
have := CharZero.of_isAddTorsionFree S S
have hpMax : (Ideal.span {p}).IsMaximal :=
((Ideal.span_singleton_prime hp.ne_zero).mpr hp).isMaximal (by simpa using hp.ne_zero)
induction I using UniqueFactorizationMonoid.induction_on_prime with
| h₁ =>
obtain... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HopkinsLevitzki | {
"line": 72,
"column": 2
} | {
"line": 74,
"column": 92
} | {
"line": 76,
"column": 0
} | [
{
"pp": "case succ.succ.refine_2\nR₀ : Type u_1\nR : Type u_2\ninst✝⁷ : Ring R₀\ninst✝⁶ : Ring R\ninst✝⁵ : Module R₀ R\ninst✝⁴ : IsSemiprimaryRing R\nP : (M : Type u) → [inst : AddCommGroup M] → [Module R₀ M] → [Module R M] → Prop\nss : IsSemisimpleRing (R ⧸ Ring.jacobson R)\nJac : Ideal R := Ring.jacobson R\nh... | [] | · rw [← SetLike.coe_subset_coe, ← Module.isTorsionBySet_iff_subset_annihilator,
Module.isTorsionBySet_quotient_iff]
exact fun m i hi ↦ Submodule.smul_mem_smul (Ideal.pow_le_self n.succ_ne_zero hi) trivial | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.JacobsonSpace | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 59
} | {
"line": 114,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : JacobsonSpace X\nS : Set X\nhS : IsLocallyClosed S\n⊢ closure[inst✝¹] (S ∩ closedPoints X) = closure[inst✝¹] S",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"LE.le.antisymm",
"CompleteLattice.toConditionallyCompleteL... | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : JacobsonSpace X\nS : Set X\nhS : IsLocallyClosed S\n⊢ closure[inst✝¹] S ⊆ closure[inst✝¹] (S ∩ closedPoints X)"
] | refine (closure_mono (Set.inter_subset_left)).antisymm ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Spectrum.Prime.Jacobson | {
"line": 92,
"column": 4
} | {
"line": 92,
"column": 60
} | {
"line": 93,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsJacobsonRing R\nx : PrimeSpectrum R\ntfae_1_to_2 : IsOpen {x} → IsClopen {x}\ntfae_2_to_3 : IsClopen {x} → IsClosed {x} ∧ StableUnderGeneralization {x}\nh₁ : IsClosed {x}\nh₂ : StableUnderGeneralization {x}\n⊢ IsOpen {x}",
"p... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsJacobsonRing R\nx : PrimeSpectrum R\ntfae_1_to_2 : IsOpen {x} → IsClopen {x}\ntfae_2_to_3 : IsClopen {x} → IsClosed {x} ∧ StableUnderGeneralization {x}\nh₁ : IsMax x\nh₂ : StableUnderGeneralization {x}\n⊢ IsOpen {x}"
] | rw [isClosed_singleton_iff_isMaximal, ← isMax_iff] at h₁ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.LocalRing.Length | {
"line": 84,
"column": 90
} | {
"line": 100,
"column": 99
} | {
"line": 102,
"column": 0
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nM : Type u_3\ninst✝⁸ : CommRing A\ninst✝⁷ : CommRing B\ninst✝⁶ : IsLocalRing A\ninst✝⁵ : IsLocalRing B\ninst✝⁴ : Algebra A B\ninst✝³ : IsLocalHom (algebraMap A B)\ninst✝² : AddCommGroup M\ninst✝¹ : Module A M\ninst✝ : Flat A B\np q : Submodule A M\nh : p ⋖ q\n⊢ length B ↥(Su... | [] | by
-- Reduce the statement to ℓ_B(B ⊗[A] p) = ℓ_B(B ⊗[A] q) + ℓ_B(B ⧸ m_A B)
rw [← (toBaseChange.toLinearEquiv B p).length_eq, ← (toBaseChange.toLinearEquiv B q).length_eq]
-- Identify q / p with A / m_A, so (B ⊗[A] p ⧸ B ⊗[A] q) ≃ₗ B ⧸ m_A B
let f : p →ₗ[A] q := inclusion h.le
have key : IsSimpleModule A (q ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.LocalRing.ResidueField.Instances | {
"line": 39,
"column": 43
} | {
"line": 50,
"column": 71
} | {
"line": 52,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝¹² : CommRing R\ninst✝¹¹ : CommRing A\ninst✝¹⁰ : CommRing B\ninst✝⁹ : Algebra R A\ninst✝⁸ : Algebra A B\ninst✝⁷ : Algebra R B\ninst✝⁶ : IsScalarTower R A B\np : Ideal A\nq : Ideal B\ninst✝⁵ : q.LiesOver p\ninst✝⁴ : p.IsMaximal\ninst✝³ : q.IsMaximal\ninst✝²... | [] | by
refine Algebra.IsSeparable.of_equiv_equiv
(.symm <| .ofBijective _ p.bijective_algebraMap_quotient_residueField)
(.symm <| .ofBijective _ q.bijective_algebraMap_quotient_residueField) ?_
apply RingHom.ext fun x ↦ ?_
obtain ⟨x, rfl⟩ :=
(RingEquiv.ofBijective _ p.bijective_algebraMap_quotient_residue... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.QuasiFinite.Basic | {
"line": 123,
"column": 80
} | {
"line": 139,
"column": 52
} | {
"line": 141,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : QuasiFinite R S\ninst✝¹ : IsArtinianRing R\ninst✝ : IsLocalRing R\n⊢ Module.Finite R S",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Ideal.Quotient.isScalarTower",
"... | [] | by
let e : (maximalIdeal R).Fiber S ≃ₐ[R] S ⧸ (maximalIdeal R).map (algebraMap R S) :=
(Algebra.TensorProduct.congr (.symm <| .ofBijective _
(Ideal.bijective_algebraMap_quotient_residueField (maximalIdeal R))) .refl).trans <|
(Algebra.TensorProduct.comm _ _ _).trans
((Algebra.TensorProduct.quotIdeal... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Unramified.Finite | {
"line": 168,
"column": 16
} | {
"line": 168,
"column": 92
} | {
"line": 169,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : FormallyUnramified R S\ninst✝¹ : EssFiniteType R S\ninst✝ : Free R S\nI : Type u_2 := Free.ChooseBasisIndex R S\nb : Basis I R S := Free.chooseBasis R S\nb' : Basis I S (S ⊗[R] S) := ((Basis.singleton P... | [
"R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : FormallyUnramified R S\ninst✝¹ : EssFiniteType R S\ninst✝ : Free R S\nI : Type u_2 := Free.ChooseBasisIndex R S\nb : Basis I R S := Free.chooseBasis R S\nb' : Basis I S (S ⊗[R] S) := ((Basis.singleton PUnit.{1} S).... | rw [← b'.linearCombination_repr (elem R S), Finsupp.linearCombination_apply] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.RingTheory.Unramified.Finite | {
"line": 168,
"column": 16
} | {
"line": 168,
"column": 92
} | {
"line": 169,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : FormallyUnramified R S\ninst✝¹ : EssFiniteType R S\ninst✝ : Free R S\nI : Type u_2 := Free.ChooseBasisIndex R S\nb : Basis I R S := Free.chooseBasis R S\nb' : Basis I S (S ⊗[R] S) := ((Basis.singleton P... | [
"R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : FormallyUnramified R S\ninst✝¹ : EssFiniteType R S\ninst✝ : Free R S\nI : Type u_2 := Free.ChooseBasisIndex R S\nb : Basis I R S := Free.chooseBasis R S\nb' : Basis I S (S ⊗[R] S) := ((Basis.singleton PUnit.{1} S).... | rw [← b'.linearCombination_repr (elem R S), Finsupp.linearCombination_apply] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.RingTheory.Unramified.Finite | {
"line": 168,
"column": 16
} | {
"line": 168,
"column": 92
} | {
"line": 169,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : FormallyUnramified R S\ninst✝¹ : EssFiniteType R S\ninst✝ : Free R S\nI : Type u_2 := Free.ChooseBasisIndex R S\nb : Basis I R S := Free.chooseBasis R S\nb' : Basis I S (S ⊗[R] S) := ((Basis.singleton P... | [
"R : Type u_1\nS : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : CommRing S\ninst✝³ : Algebra R S\ninst✝² : FormallyUnramified R S\ninst✝¹ : EssFiniteType R S\ninst✝ : Free R S\nI : Type u_2 := Free.ChooseBasisIndex R S\nb : Basis I R S := Free.chooseBasis R S\nb' : Basis I S (S ⊗[R] S) := ((Basis.singleton PUnit.{1} S).... | rw [← b'.linearCombination_repr (elem R S), Finsupp.linearCombination_apply] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.RingTheory.Smooth.Basic | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 31
} | {
"line": 138,
"column": 4
} | [
{
"pp": "case h₂\nR : Type u\nA : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\nB : Type u_1\ninst✝¹ : CommRing B\ninst✝ : FormallySmooth R A\nI : Ideal B\nhI : IsNilpotent I\n⊢ ∀ ⦃S : Type u_1⦄ [inst : CommRing S] (I J : Ideal S),\n I ≤ J →\n (∀ [inst_1 : Algebra R S], Functio... | [
"case h₂\nR : Type u\nA : Type v\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\nB✝ : Type u_1\ninst✝³ : CommRing B✝\ninst✝² : FormallySmooth R A\nI✝ : Ideal B✝\nhI : IsNilpotent I✝\nB : Type u_1\ninst✝¹ : CommRing B\nI J : Ideal B\nhIJ : I ≤ J\nh₁ : ∀ [inst : Algebra R B], Function.Surjective (Ide... | intro B _ I J hIJ h₁ h₂ _ g | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.Smooth.Basic | {
"line": 317,
"column": 2
} | {
"line": 326,
"column": 48
} | {
"line": 328,
"column": 0
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nP : Extension R A\ninst✝ : FormallySmooth R P.Ring\n⊢ FormallySmooth R A ↔ ∃ l, l ∘ₗ P.cotangentComplex = LinearMap.id",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"LinearMap.id",
... | [] | refine (Algebra.FormallySmooth.iff_split_injection P.algebraMap_surjective).trans ?_
let e : P.ker.Cotangent ≃ₗ[P.Ring] P.Cotangent :=
{ __ := AddEquiv.refl _, map_smul' r m := by ext1; simp; rfl }
constructor
· intro ⟨l, hl⟩
exact ⟨(e.comp l).extendScalarsOfSurjective P.algebraMap_surjective,
Linea... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Smooth.Basic | {
"line": 317,
"column": 2
} | {
"line": 326,
"column": 48
} | {
"line": 328,
"column": 0
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\nP : Extension R A\ninst✝ : FormallySmooth R P.Ring\n⊢ FormallySmooth R A ↔ ∃ l, l ∘ₗ P.cotangentComplex = LinearMap.id",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
"LinearMap.id",
... | [] | refine (Algebra.FormallySmooth.iff_split_injection P.algebraMap_surjective).trans ?_
let e : P.ker.Cotangent ≃ₗ[P.Ring] P.Cotangent :=
{ __ := AddEquiv.refl _, map_smul' r m := by ext1; simp; rfl }
constructor
· intro ⟨l, hl⟩
exact ⟨(e.comp l).extendScalarsOfSurjective P.algebraMap_surjective,
Linea... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Kaehler.JacobiZariski | {
"line": 259,
"column": 25
} | {
"line": 259,
"column": 89
} | {
"line": 261,
"column": 0
} | [
{
"pp": "case add\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\nQ : Generators S T ι\ny : Q.Ring\nx₁ x₂ : MvPolynomial ι S\nhx₁ : (δAux R Q) (x₁ * ... | [] | simp only [add_mul, map_add, hx₁, hx₂, add_smul, smul_add]; abel | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Kaehler.JacobiZariski | {
"line": 259,
"column": 25
} | {
"line": 259,
"column": 89
} | {
"line": 261,
"column": 0
} | [
{
"pp": "case add\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\nQ : Generators S T ι\ny : Q.Ring\nx₁ x₂ : MvPolynomial ι S\nhx₁ : (δAux R Q) (x₁ * ... | [] | simp only [add_mul, map_add, hx₁, hx₂, add_smul, smul_add]; abel | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Etale.Kaehler | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 14
} | {
"line": 161,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R S\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nP : Extension R S\nQ : Extension R T\nf : P.Hom Q\nalg : Algebra P.Ring Q.Ring\nhalg : algebraMap P.Rin... | [
"R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R S\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nP : Extension R S\nQ : Extension R T\nf : P.Hom Q\nalg : Algebra P.Ring Q.Ring\nhalg : algebraMap P.Ring Q.Ring = f... | clear x hx | Lean.Elab.Tactic.evalClear | Lean.Parser.Tactic.clear |
Mathlib.RingTheory.Etale.Kaehler | {
"line": 183,
"column": 58
} | {
"line": 183,
"column": 68
} | {
"line": 183,
"column": 69
} | [
{
"pp": "case tmul.tmul\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R S\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nP : Extension R S\nQ : Extension R T\nf : P.Hom Q\nalg : Algebra P.Ring Q.Ring\nhalg : ... | [
"case tmul.tmul\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing T\ninst✝³ : Algebra R S\ninst✝² : Algebra R T\ninst✝¹ : Algebra S T\ninst✝ : IsScalarTower R S T\nP : Extension R S\nQ : Extension R T\nf : P.Hom Q\nalg : Algebra P.Ring Q.Ring\nhalg : algebraMap P... | tmul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Etale.Kaehler | {
"line": 248,
"column": 8
} | {
"line": 248,
"column": 32
} | {
"line": 248,
"column": 33
} | [
{
"pp": "case left\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\nP : Extension R S\nQ : Extension R T\nf : P.Hom Q\nalg : Algebra P.Ring Q.Ring\nhalg : alge... | [
"case left\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\nP : Extension R S\nQ : Extension R T\nf : P.Hom Q\nalg : Algebra P.Ring Q.Ring\nhalg : algebraMap P.Rin... | ← h1Cotangentι.map_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Etale.Kaehler | {
"line": 286,
"column": 4
} | {
"line": 286,
"column": 73
} | {
"line": 286,
"column": 73
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\nM : Submonoid S\ninst✝ : IsLocalization M T\nP : Extension R S := (Generators.self R S).toExtension\nM... | [] | by simpa using IsLocalization.map_units T ⟨algebraMap P.Ring S y, hy⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Etale.Kaehler | {
"line": 359,
"column": 4
} | {
"line": 359,
"column": 73
} | {
"line": 359,
"column": 73
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing T\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R T\ninst✝² : Algebra S T\ninst✝¹ : IsScalarTower R S T\nM : Submonoid S\ninst✝ : IsLocalization M T\nx : H1Cotangent R S\nP : Extension R S := (Generators.sel... | [] | by simpa using IsLocalization.map_units T ⟨algebraMap P.Ring S y, hy⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Enumerative.InclusionExclusion | {
"line": 59,
"column": 27
} | {
"line": 59,
"column": 44
} | {
"line": 59,
"column": 44
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\ns : Finset ι\nhs : s.Nonempty\nS : ι → Set α\na : α\nha' : a ∈ ⋃ i ∈ s, S i\ni : ι\nhi : i ∈ s\nha : a ∈ S i\n⊢ (⋃ i ∈ s, S i).indicator 1 a - (S i).indicator 1 a = 0",
"ppTerm": "?m.94",
"assigned": true,
"usedConstants": [
"sub_self",
"congrArg",
... | [] | by simp [ha, ha'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Enumerative.InclusionExclusion | {
"line": 72,
"column": 4
} | {
"line": 72,
"column": 8
} | {
"line": 73,
"column": 4
} | [
{
"pp": "case neg\nι : Type u_1\nα : Type u_2\nG : Type u_3\ninst✝ : AddCommGroup G\ns : Finset ι\nS : ι → Set α\nf : α → G\na : α\nha : a ∉ ⋃ i ∈ s, S i\n⊢ 0 = ∑ x ∈ s.powerset with x.Nonempty, -((-1) ^ #x • (⋂ i ∈ x, S i).indicator f a)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
... | [
"case neg\nι : Type u_1\nα : Type u_2\nG : Type u_3\ninst✝ : AddCommGroup G\ns : Finset ι\nS : ι → Set α\nf : α → G\na : α\nha : a ∉ ⋃ i ∈ s, S i\n⊢ ∑ x ∈ s.powerset with x.Nonempty, -((-1) ^ #x • (⋂ i ∈ x, S i).indicator f a) = 0"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Data.NNReal.Defs | {
"line": 471,
"column": 18
} | {
"line": 471,
"column": 27
} | {
"line": 471,
"column": 28
} | [
{
"pp": "ι : Sort u_2\ns : ι → ℝ≥0\n⊢ ↑(sSup (Set.range fun i ↦ s i)) = sSup (Set.range fun i ↦ ↑(s i))",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"NNReal.coe_sSup",
"congrArg",
"Real.instSupSet",
"id",
"NNReal",
"Condi... | [
"ι : Sort u_2\ns : ι → ℝ≥0\n⊢ sSup (toReal '' Set.range fun i ↦ s i) = sSup (Set.range fun i ↦ ↑(s i))"
] | coe_sSup, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.NNReal.Defs | {
"line": 663,
"column": 15
} | {
"line": 663,
"column": 43
} | {
"line": 665,
"column": 0
} | [
{
"pp": "r p : ℝ\nhr : 0 ≤ r\nhp : 0 ≤ p\n⊢ ↑(r + p).toNNReal = ↑(r.toNNReal + p.toNNReal)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Real",
"Real.instZero",
"Real.instAddMonoid",
"congrArg",
"Real.instSemilatticeSup",
"AddMonoid.toAddZeroClass",
... | [] | by simp [hr, hp, add_nonneg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Archimedean.Real.Basic | {
"line": 137,
"column": 6
} | {
"line": 137,
"column": 15
} | {
"line": 137,
"column": 16
} | [
{
"pp": "s : Set ℝ\nh₁ : s.Nonempty\nh₂ : BddBelow s\n⊢ IsGLB s (sInf s)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"congrArg",
"Real.instSupSet",
"id",
"Real.instInfSet",
"IsGLB",
"Real.instNeg",
... | [
"s : Set ℝ\nh₁ : s.Nonempty\nh₂ : BddBelow s\n⊢ IsGLB s (-sSup (-s))"
] | sInf_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.NNReal.Defs | {
"line": 928,
"column": 6
} | {
"line": 928,
"column": 25
} | {
"line": 928,
"column": 26
} | [
{
"pp": "n : ℤ\n⊢ ↑↑n.natAbs = ↑(nnabs ↑n)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"Real.nnabs",
"congrArg",
"MonoidWithZeroHom.funLike",
"Real.semiring",
"... | [
"n : ℤ\n⊢ ↑n.natAbs = ↑(nnabs ↑n)"
] | NNReal.coe_natCast, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Group.Seminorm | {
"line": 329,
"column": 26
} | {
"line": 329,
"column": 36
} | {
"line": 329,
"column": 37
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst✝² : Group E\ninst✝¹ : Group F\ninst✝ : Group G\np✝ q : GroupSeminorm E\nf✝ : F →* E\np : GroupSeminorm E\nf : F →* E\n⊢ p (f 1) = 0",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Mu... | [
"R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst✝² : Group E\ninst✝¹ : Group F\ninst✝ : Group G\np✝ q : GroupSeminorm E\nf✝ : F →* E\np : GroupSeminorm E\nf : F →* E\n⊢ p 1 = 0"
] | f.map_one, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Order.Archimedean.Real.Basic | {
"line": 411,
"column": 2
} | {
"line": 414,
"column": 28
} | {
"line": 416,
"column": 0
} | [
{
"pp": "b : ℝ\nhb : 0 < b\n⊢ ∃ n, 0 < n ∧ (↑n)⁻¹ < b",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"Pre... | [] | refine (exists_nat_gt b⁻¹).imp fun k hk ↦ ?_
have := (inv_pos_of_pos hb).trans hk
refine ⟨Nat.cast_pos.mp this, ?_⟩
rwa [inv_lt_comm₀ this hb] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Archimedean.Real.Basic | {
"line": 411,
"column": 2
} | {
"line": 414,
"column": 28
} | {
"line": 416,
"column": 0
} | [
{
"pp": "b : ℝ\nhb : 0 < b\n⊢ ∃ n, 0 < n ∧ (↑n)⁻¹ < b",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"Pre... | [] | refine (exists_nat_gt b⁻¹).imp fun k hk ↦ ?_
have := (inv_pos_of_pos hb).trans hk
refine ⟨Nat.cast_pos.mp this, ?_⟩
rwa [inv_lt_comm₀ this hb] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 379,
"column": 6
} | {
"line": 379,
"column": 38
} | {
"line": 379,
"column": 39
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R⁰ K\nhI : I ≠ 0\nhI' : I' ≠ 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a ≠ 0\nh... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI I' : FractionalIdeal R⁰ K\nhI : I ≠ 0\nhI' : I' ≠ 0\nhv : Irreducible (Associates.mk v.asIdeal)\na : R\nJ : Ideal R\nha : a ≠ 0\nhaJ : I = spa... | spanSingleton_mul_spanSingleton, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 411,
"column": 4
} | {
"line": 411,
"column": 81
} | {
"line": 412,
"column": 4
} | [
{
"pp": "case insert\nR : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nι : Type u_3\nI : ι → FractionalIdeal R⁰ K\ni : ι\ns : Finset ι\nhi : i ∉ s\nhrec : (∀ i ∈ s, I i ≠ 0) → count K v (∏ i... | [
"case insert\nR : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nι : Type u_3\nI : ι → FractionalIdeal R⁰ K\ni : ι\ns : Finset ι\nhi : i ∉ s\nhrec : (∀ i ∈ s, I i ≠ 0) → count K v (∏ i ∈ s, I i) =... | have hS' : ∀ i ∈ s, I i ≠ 0 := fun j hj => hS j (Finset.mem_insert_of_mem hj) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.ENNReal.Operations | {
"line": 213,
"column": 4
} | {
"line": 213,
"column": 33
} | {
"line": 213,
"column": 34
} | [
{
"pp": "case mpr\na b : ℝ≥0∞\n⊢ a < ∞ ∧ b < ∞ ∨ a = 0 ∨ b = 0 → a * b < ∞",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"HMul.hMul",
"CommSemiring.toSemiring",
"PartialOrder.toPreorder",
"ENNReal.instCommSemiring",
"Or.casesOn",
... | [
"case mpr.inl\na b : ℝ≥0∞\nha : a < ∞\nhb : b < ∞\n⊢ a * b < ∞",
"case mpr.inr.inl\nb : ℝ≥0∞\n⊢ 0 * b < ∞",
"case mpr.inr.inr\na : ℝ≥0∞\n⊢ a * 0 < ∞"
] | rintro (⟨ha, hb⟩ | rfl | rfl) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Data.ENNReal.Operations | {
"line": 438,
"column": 2
} | {
"line": 438,
"column": 30
} | {
"line": 439,
"column": 2
} | [
{
"pp": "p q : ℝ\nhq : 0 ≤ q\n⊢ ENNReal.ofReal (p - q) = ENNReal.ofReal p - ENNReal.ofReal q",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Real",
"ENNReal.ofReal",
"Real.instSub",
"PartialOrder.toPreorder",
"HSub.hSub",
"Preorder.toLE",
"le_tota... | [
"case inl\np q : ℝ\nhq : 0 ≤ q\nh : p ≤ q\n⊢ ENNReal.ofReal (p - q) = ENNReal.ofReal p - ENNReal.ofReal q",
"case inr\np q : ℝ\nhq : 0 ≤ q\nh : q ≤ p\n⊢ ENNReal.ofReal (p - q) = ENNReal.ofReal p - ENNReal.ofReal q"
] | obtain h | h := le_total p q | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.UniformSpace.OfFun | {
"line": 42,
"column": 39
} | {
"line": 42,
"column": 43
} | {
"line": 42,
"column": 43
} | [
{
"pp": "X : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : PartialOrder M\nd : X → X → M\nrefl : ∀ (x : X), d x x = 0\nsymm : ∀ (x y : X), d x y = d y x\ntriangle : ∀ (x y z : X), d x z ≤ d x y + d y z\nhalf : ∀ ε > 0, ∃ δ > 0, ∀ x < δ, ∀ y < δ, x + y < ε\nr : M\nx✝ : r > 0\nx : X × X\nhx : x ∈ {x |... | [
"X : Type u_1\nM : Type u_2\ninst✝¹ : AddCommMonoid M\ninst✝ : PartialOrder M\nd : X → X → M\nrefl : ∀ (x : X), d x x = 0\nsymm : ∀ (x y : X), d x y = d y x\ntriangle : ∀ (x y z : X), d x z ≤ d x y + d y z\nhalf : ∀ ε > 0, ∃ δ > 0, ∀ x < δ, ∀ y < δ, x + y < ε\nr : M\nx✝ : r > 0\nx : X × X\nhx : x ∈ {x | d x.1 x.2 <... | symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.EReal.Basic | {
"line": 424,
"column": 2
} | {
"line": 425,
"column": 20
} | {
"line": 426,
"column": 2
} | [
{
"pp": "case mp\nx : EReal\n⊢ x = ⊥ → ∀ (y : ℝ), x < ↑y",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLT",
"EReal.bot_lt_coe",
"PartialOrder.toPreorder",
"EReal",
"Bot.bot",
"LT.lt",
"Eq.ndrec",
"instPartialOrderERe... | [
"case mpr\nx : EReal\n⊢ (∀ (y : ℝ), x < ↑y) → x = ⊥"
] | · rintro rfl
exact bot_lt_coe | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.EReal.Basic | {
"line": 490,
"column": 2
} | {
"line": 490,
"column": 31
} | {
"line": 491,
"column": 2
} | [
{
"pp": "x : ℝ\n⊢ WithBot.some ∘ WithTop.some ⁻¹' Ici ↑↑x = Ici x",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Real",
"WithBot.some",
"WithBot",
"Set.Ici",
"WithTop.instPreorder",
"Function.comp",
"Set.preimage_com... | [
"x : ℝ\n⊢ WithTop.some ⁻¹' WithBot.some ⁻¹' Ici ↑↑x = Ici x"
] | refine preimage_comp.trans ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Data.EReal.Basic | {
"line": 496,
"column": 2
} | {
"line": 496,
"column": 31
} | {
"line": 497,
"column": 2
} | [
{
"pp": "x : ℝ\n⊢ WithBot.some ∘ WithTop.some ⁻¹' Ioi ↑↑x = Ioi x",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Real",
"Set.Ioi",
"WithBot.some",
"WithBot",
"WithTop.instPreorder",
"Function.comp",
"Set.preimage_com... | [
"x : ℝ\n⊢ WithTop.some ⁻¹' WithBot.some ⁻¹' Ioi ↑↑x = Ioi x"
] | refine preimage_comp.trans ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Data.EReal.Basic | {
"line": 502,
"column": 2
} | {
"line": 502,
"column": 31
} | {
"line": 503,
"column": 2
} | [
{
"pp": "⊢ WithBot.some ∘ WithTop.some ⁻¹' Ioi ⊥ = univ",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Real",
"Set.Ioi",
"WithBot.some",
"WithBot",
"WithTop.instPreorder",
"Set.univ",
"Function.comp",
"Bot.bot"... | [
"⊢ WithTop.some ⁻¹' WithBot.some ⁻¹' Ioi ⊥ = univ"
] | refine preimage_comp.trans ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Data.EReal.Basic | {
"line": 508,
"column": 2
} | {
"line": 508,
"column": 31
} | {
"line": 509,
"column": 2
} | [
{
"pp": "y : ℝ\n⊢ WithBot.some ∘ WithTop.some ⁻¹' Iic ↑↑y = Iic y",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Real",
"WithBot.some",
"WithBot",
"WithTop.instPreorder",
"Function.comp",
"Set.preimage_comp",
"WithTo... | [
"y : ℝ\n⊢ WithTop.some ⁻¹' WithBot.some ⁻¹' Iic ↑↑y = Iic y"
] | refine preimage_comp.trans ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Data.EReal.Basic | {
"line": 514,
"column": 2
} | {
"line": 514,
"column": 31
} | {
"line": 515,
"column": 2
} | [
{
"pp": "y : ℝ\n⊢ WithBot.some ∘ WithTop.some ⁻¹' Iio ↑↑y = Iio y",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Real",
"WithBot.some",
"WithBot",
"WithTop.instPreorder",
"Function.comp",
"Set.preimage_comp",
"WithTo... | [
"y : ℝ\n⊢ WithTop.some ⁻¹' WithBot.some ⁻¹' Iio ↑↑y = Iio y"
] | refine preimage_comp.trans ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Data.EReal.Basic | {
"line": 520,
"column": 2
} | {
"line": 520,
"column": 31
} | {
"line": 521,
"column": 2
} | [
{
"pp": "⊢ WithBot.some ∘ WithTop.some ⁻¹' Iio ↑⊤ = univ",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Real",
"WithBot.some",
"WithBot",
"WithTop.instPreorder",
"Set.univ",
"Function.comp",
"Set.preimage_comp",
... | [
"⊢ WithTop.some ⁻¹' WithBot.some ⁻¹' Iio ↑⊤ = univ"
] | refine preimage_comp.trans ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Data.EReal.Basic | {
"line": 675,
"column": 2
} | {
"line": 678,
"column": 57
} | {
"line": 680,
"column": 0
} | [
{
"pp": "x : ℝ≥0\n⊢ ↑(∞ * ↑x) = ⊤ * ↑↑x",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Real.instLE",
"Real",
"ENNReal.ofNNReal",
"Preorder.toLT",
"HMul.hMul",
"EReal.instMulZeroOneClass",
"Real.instZero",
"c... | [] | rcases eq_or_ne x 0 with (rfl | h0)
· simp
· rw [ENNReal.top_mul (ENNReal.coe_ne_zero.2 h0)]
exact Eq.symm <| if_pos <| NNReal.coe_pos.2 h0.bot_lt | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.EReal.Basic | {
"line": 675,
"column": 2
} | {
"line": 678,
"column": 57
} | {
"line": 680,
"column": 0
} | [
{
"pp": "x : ℝ≥0\n⊢ ↑(∞ * ↑x) = ⊤ * ↑↑x",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Real.instLE",
"Real",
"ENNReal.ofNNReal",
"Preorder.toLT",
"HMul.hMul",
"EReal.instMulZeroOneClass",
"Real.instZero",
"c... | [] | rcases eq_or_ne x 0 with (rfl | h0)
· simp
· rw [ENNReal.top_mul (ENNReal.coe_ne_zero.2 h0)]
exact Eq.symm <| if_pos <| NNReal.coe_pos.2 h0.bot_lt | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 600,
"column": 2
} | {
"line": 601,
"column": 19
} | {
"line": 602,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nI : FractionalIdeal R⁰ K\nhI : I = 0\n⊢ ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, count K v I = 0",
"ppTerm": "?pos✝",
"assigned": t... | [
"case neg\nR : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nI : FractionalIdeal R⁰ K\nhI : ¬I = 0\n⊢ ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, count K v I = 0"
] | · simp only [hI, count_zero, Filter.eventually_cofinite, not_true_eq_false, setOf_false,
finite_empty] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 603,
"column": 4
} | {
"line": 603,
"column": 29
} | {
"line": 605,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nI : FractionalIdeal R⁰ K\nhI : ¬I = 0\nx✝ : HeightOneSpectrum R\n⊢ count K x✝ I =\n ↑((Associates.mk x✝.asIdeal).count (Associates.mk (choose ⋯)).factors)... | [] | rw [count_ne_zero K _ hI] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.ENNReal.Inv | {
"line": 320,
"column": 53
} | {
"line": 320,
"column": 84
} | {
"line": 322,
"column": 0
} | [
{
"pp": "a : ℝ≥0∞\n⊢ 1 < a⁻¹ ↔ a < 1",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"inv_one",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
"id",
... | [] | rw [lt_inv_iff_lt_inv, inv_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.ENNReal.Inv | {
"line": 320,
"column": 53
} | {
"line": 320,
"column": 84
} | {
"line": 322,
"column": 0
} | [
{
"pp": "a : ℝ≥0∞\n⊢ 1 < a⁻¹ ↔ a < 1",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"inv_one",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
"id",
... | [] | rw [lt_inv_iff_lt_inv, inv_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ENNReal.Inv | {
"line": 320,
"column": 53
} | {
"line": 320,
"column": 84
} | {
"line": 322,
"column": 0
} | [
{
"pp": "a : ℝ≥0∞\n⊢ 1 < a⁻¹ ↔ a < 1",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"inv_one",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
"id",
... | [] | rw [lt_inv_iff_lt_inv, inv_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 529,
"column": 6
} | {
"line": 529,
"column": 22
} | {
"line": 529,
"column": 23
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nx y : α\nε : ℝ\n⊢ x ∈ closedBall y ε ↔ y ∈ closedBall x ε",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"congrArg",
"Membership.mem",
"id",
"LE.le",
"Metric.... | [
"α : Type u\ninst✝ : PseudoMetricSpace α\nx y : α\nε : ℝ\n⊢ dist y x ≤ ε ↔ y ∈ closedBall x ε"
] | mem_closedBall', | Lean.Elab.Tactic.evalRewriteSeq | null |
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