module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Combinatorics.Matroid.Loop | {
"line": 243,
"column": 94
} | {
"line": 244,
"column": 19
} | {
"line": 246,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ne : α\nM : Matroid β\nf : α → β\n⊢ (M.comap f).IsLoop e ↔ M.IsLoop (f e)",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"congrArg",
"Membership.mem",
"Set.mem_preimage._simp_1",
"iff_self",
"_private.Mathlib.Combinatorics.M... | [] | by
simp [isLoop_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 218,
"column": 23
} | {
"line": 218,
"column": 34
} | {
"line": 218,
"column": 35
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI : Set α\ne : α\nhe : e ∈ M.closure I\n⊢ {e} ⊆ M.E",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matroid.E",
"Membership.mem",
"Set.instSingletonSet",
"id",
"LE.le",
"Set.instLE",
"Singleton... | [
"α : Type u_1\nM : Matroid α\nI : Set α\ne : α\nhe : e ∈ M.closure I\n⊢ e ∈ M.E"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 328,
"column": 2
} | {
"line": 328,
"column": 13
} | {
"line": 328,
"column": 14
} | [
{
"pp": "α : Type u_1\nM : Matroid α\ne : α\nhe : M.IsNonloop e\n⊢ ∃ B, M.IsBase B ∧ e ∈ B",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nM : Matroid α\ne : α\nhe : M.IsNonloop e\n⊢ ∃ B, M.IsBase B ∧ e ∈ B"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 249,
"column": 25
} | {
"line": 249,
"column": 36
} | {
"line": 249,
"column": 37
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\ne : α\nheX : e ∉ M.E\na : α\nhaX : a ∈ X\nh : ∀ t ⊆ X, M.closure ∅ ⊆ M.closure t → a ∈ t\n⊢ a = e",
"ppTerm": "?m.40",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nM : Matroid α\nX : Set α\ne : α\nheX : e ∉ M.E\na : α\nhaX : a ∈ X\nh : ∀ t ⊆ X, M.closure ∅ ⊆ M.closure t → a ∈ t\n⊢ a = e"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Map | {
"line": 560,
"column": 18
} | {
"line": 560,
"column": 63
} | {
"line": 560,
"column": 64
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nM : Matroid α\nf : α ↪ β\nX : Set α\nhX : ⇑f '' X ⊆ range ⇑f\nI : Set α\nx✝ : M.IsBasis (⇑f ⁻¹' ⇑f '' I) (⇑f ⁻¹' ⇑f '' X) ∧ ⇑f '' I ⊆ ⇑f '' X ∧ ⇑f '' X ⊆ range ⇑f\nhb : M.IsBasis (⇑f ⁻¹' ⇑f '' I) (⇑f ⁻¹' ⇑f '' X)\nhIX : ⇑f '' I ⊆ ⇑f '' X\n⊢ M.IsBasis I X ∧ ⇑f '' I = ⇑f '' I ... | [
"α : Type u_1\nβ : Type u_2\nM : Matroid α\nf : α ↪ β\nX : Set α\nhX : ⇑f '' X ⊆ range ⇑f\nI : Set α\nx✝ : M.IsBasis (⇑f ⁻¹' ⇑f '' I) (⇑f ⁻¹' ⇑f '' X) ∧ ⇑f '' I ⊆ ⇑f '' X ∧ ⇑f '' X ⊆ range ⇑f\nhb : M.IsBasis (⇑f ⁻¹' ⇑f '' I) (⇑f ⁻¹' ⇑f '' X)\nhIX : ⇑f '' I ⊆ ⇑f '' X\n⊢ M.IsBasis I X"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 30
} | {
"line": 126,
"column": 31
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nF : Set α\n⊢ M.IsFlat F ↔ ∃ (h : F ⊆ M.E), M.subtypeClosure.IsClosed ⟨F, h⟩",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Matroid.subtypeClosure._proof_2",
"Eq.mpr",
"ChainCompletePartialOrder.instOfCompleteLattice",
"Iff.of_... | [
"α : Type u_2\nM : Matroid α\nF : Set α\n⊢ M.IsFlat F → F ⊆ M.E"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 422,
"column": 26
} | {
"line": 422,
"column": 54
} | {
"line": 422,
"column": 55
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nι : Type u_2\nJ : Set α\nx : ι → α\nI : ι → Set α\nz : α\nhxI : ∀ (i : ι), x i ∉ I i\nhC : ∀ (i : ι), M.IsCircuit (insert (x i) (I i))\nhJx : M.IsCircuit (J ∪ range x)\nhzJ : z ∈ J\nhzI : ∀ (i : ι), z ∉ I i\nhι : IsEmpty ι\n⊢ M.IsCircuit J",
"ppTerm": "?m.58",
"assi... | [
"α : Type u_1\nM : Matroid α\nι : Type u_2\nJ : Set α\nx : ι → α\nI : ι → Set α\nz : α\nhxI : ∀ (i : ι), x i ∉ I i\nhC : ∀ (i : ι), M.IsCircuit (insert (x i) (I i))\nhJx : M.IsCircuit (J ∪ range x)\nhzJ : z ∈ J\nhzI : ∀ (i : ι), z ∉ I i\nhι : IsEmpty ι\n⊢ M.IsCircuit J"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 515,
"column": 34
} | {
"line": 515,
"column": 45
} | {
"line": 515,
"column": 46
} | [
{
"pp": "α : Type u_1\nM : Matroid α\ne : α\ntfae_1_iff_2 : M.IsColoop e ↔ e ∈ M.coloops\ntfae_1_iff_3 : M.IsColoop e ↔ M.IsCocircuit {e}\nh : ∀ (B : Set α), M✶.IsBase B → e ∈ M✶.E \\ B\nB : Set α\nhB : M.IsBase B\n⊢ ?m.80",
"ppTerm": "?m.85",
"assigned": false,
"usedConstants": [],
"usedFVars":... | [
"α : Type u_1\nM : Matroid α\ne : α\ntfae_1_iff_2 : M.IsColoop e ↔ e ∈ M.coloops\ntfae_1_iff_3 : M.IsColoop e ↔ M.IsCocircuit {e}\nh : ∀ (B : Set α), M✶.IsBase B → e ∈ M✶.E \\ B\nB : Set α\nhB : M.IsBase B\n⊢ ?m.80"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Map | {
"line": 712,
"column": 31
} | {
"line": 712,
"column": 42
} | {
"line": 712,
"column": 43
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nE✝ I✝ : Set α\nM✝ : Matroid α\nN : Matroid β\nE X I : Set α\nM : Matroid α\ninst✝ : M.RankPos\nB : Set ↑M.E\nhB : (M.restrictSubtype M.E).IsBase B\n⊢ M.IsBase (Subtype.val '' B)",
"ppTerm": "?m.20",
"assigned": false,
"usedConstants": [],
"usedFVar... | [
"α : Type u_1\nβ : Type u_2\nf : α → β\nE✝ I✝ : Set α\nM✝ : Matroid α\nN : Matroid β\nE X I : Set α\nM : Matroid α\ninst✝ : M.RankPos\nB : Set ↑M.E\nhB : (M.restrictSubtype M.E).IsBase B\n⊢ M.IsBase (Subtype.val '' B)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Map | {
"line": 713,
"column": 37
} | {
"line": 713,
"column": 48
} | {
"line": 713,
"column": 49
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nE✝ I✝ : Set α\nM✝ : Matroid α\nN : Matroid β\nE X I : Set α\nM : Matroid α\ninst✝ : M.RankPos\nB : Set ↑M.E\nhB : (M.restrictSubtype M.E).IsBase B\nhB' : M.IsBase (Subtype.val '' B)\n⊢ B.Nonempty",
"ppTerm": "?m.46",
"assigned": false,
"usedConstants":... | [
"α : Type u_1\nβ : Type u_2\nf : α → β\nE✝ I✝ : Set α\nM✝ : Matroid α\nN : Matroid β\nE X I : Set α\nM : Matroid α\ninst✝ : M.RankPos\nB : Set ↑M.E\nhB : (M.restrictSubtype M.E).IsBase B\nhB' : M.IsBase (Subtype.val '' B)\n⊢ B.Nonempty"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 522,
"column": 4
} | {
"line": 522,
"column": 74
} | {
"line": 523,
"column": 4
} | [
{
"pp": "α : Type u_1\nM : Matroid α\ne : α\ntfae_1_iff_2 : M.IsColoop e ↔ e ∈ M.coloops\ntfae_1_iff_3 : M.IsColoop e ↔ M.IsCocircuit {e}\ntfae_1_iff_4 : M.IsColoop e ↔ ∀ ⦃B : Set α⦄, M.IsBase B → e ∈ B\ntfae_3_to_5 : M.IsCocircuit {e} → (∀ ⦃C : Set α⦄, M.IsCircuit C → e ∉ C) ∧ e ∈ M.E\nx✝ : (∀ ⦃C : Set α⦄, M.I... | [
"α : Type u_1\nM : Matroid α\ne : α\ntfae_1_iff_2 : M.IsColoop e ↔ e ∈ M.coloops\ntfae_1_iff_3 : M.IsColoop e ↔ M.IsCocircuit {e}\ntfae_1_iff_4 : M.IsColoop e ↔ ∀ ⦃B : Set α⦄, M.IsBase B → e ∈ B\ntfae_3_to_5 : M.IsCocircuit {e} → (∀ ⦃C : Set α⦄, M.IsCircuit C → e ∉ C) ∧ e ∈ M.E\nx✝ : (∀ ⦃C : Set α⦄, M.IsCircuit C →... | obtain ⟨C, -, hC, heC⟩ := (mem_closure_iff_exists_isCircuit heB).1 heE | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 527,
"column": 4
} | {
"line": 528,
"column": 22
} | {
"line": 529,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\nM : Matroid α\ne : α\ntfae_1_iff_2 : M.IsColoop e ↔ e ∈ M.coloops\ntfae_1_iff_3 : M.IsColoop e ↔ M.IsCocircuit {e}\ntfae_1_iff_4 : M.IsColoop e ↔ ∀ ⦃B : Set α⦄, M.IsBase B → e ∈ B\ntfae_3_to_5 : M.IsCocircuit {e} → (∀ ⦃C : Set α⦄, M.IsCircuit C → e ∉ C) ∧ e ∈ M.E\ntfae_5_to... | [
"case refine_2\nα : Type u_1\nM : Matroid α\ne : α\ntfae_1_iff_2 : M.IsColoop e ↔ e ∈ M.coloops\ntfae_1_iff_3 : M.IsColoop e ↔ M.IsCocircuit {e}\ntfae_1_iff_4 : M.IsColoop e ↔ ∀ ⦃B : Set α⦄, M.IsBase B → e ∈ B\ntfae_3_to_5 : M.IsCocircuit {e} → (∀ ⦃C : Set α⦄, M.IsCircuit C → e ∉ C) ∧ e ∈ M.E\ntfae_5_to_4 : (∀ ⦃C :... | · obtain ⟨C, -, hC, heC⟩ := (mem_closure_iff_exists_isCircuit heX').1 heX
exact h.1 hC heC | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 529,
"column": 6
} | {
"line": 529,
"column": 61
} | {
"line": 529,
"column": 62
} | [
{
"pp": "case refine_2\nα : Type u_1\nM : Matroid α\ne : α\ntfae_1_iff_2 : M.IsColoop e ↔ e ∈ M.coloops\ntfae_1_iff_3 : M.IsColoop e ↔ M.IsCocircuit {e}\ntfae_1_iff_4 : M.IsColoop e ↔ ∀ ⦃B : Set α⦄, M.IsBase B → e ∈ B\ntfae_3_to_5 : M.IsCocircuit {e} → (∀ ⦃C : Set α⦄, M.IsCircuit C → e ∉ C) ∧ e ∈ M.E\ntfae_5_to... | [
"case refine_2\nα : Type u_1\nM : Matroid α\ne : α\ntfae_1_iff_2 : M.IsColoop e ↔ e ∈ M.coloops\ntfae_1_iff_3 : M.IsColoop e ↔ M.IsCocircuit {e}\ntfae_1_iff_4 : M.IsColoop e ↔ ∀ ⦃B : Set α⦄, M.IsBase B → e ∈ B\ntfae_3_to_5 : M.IsCocircuit {e} → (∀ ⦃C : Set α⦄, M.IsCircuit C → e ∉ C) ∧ e ∈ M.E\ntfae_5_to_4 : (∀ ⦃C :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 429,
"column": 4
} | {
"line": 429,
"column": 49
} | {
"line": 429,
"column": 50
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nι : Type u_2\nJ : Set α\nx : ι → α\nI : ι → Set α\nz : α\nhxI : ∀ (i : ι), x i ∉ I i\nhC : ∀ (i : ι), M.IsCircuit (insert (x i) (I i))\nhJx : M.IsCircuit (J ∪ range x)\nhzJ : z ∈ J\nhzI : ∀ (i : ι), z ∉ I i\nhι : Nonempty ι\ni : ι\n⊢ M.closure (I i) = M.closure (insert (x i... | [
"α : Type u_1\nM : Matroid α\nι : Type u_2\nJ : Set α\nx : ι → α\nI : ι → Set α\nz : α\nhxI : ∀ (i : ι), x i ∉ I i\nhC : ∀ (i : ι), M.IsCircuit (insert (x i) (I i))\nhJx : M.IsCircuit (J ∪ range x)\nhzJ : z ∈ J\nhzI : ∀ (i : ι), z ∉ I i\nhι : Nonempty ι\ni : ι\n⊢ M.closure (I i) = M.closure (insert (x i) (I i))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 233,
"column": 2
} | {
"line": 233,
"column": 49
} | {
"line": 234,
"column": 2
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nn : ℕ∞\n⊢ n ≤ M.eRk X ↔ ∃ I ⊆ X, M.Indep I ∧ I.encard = n",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Set.encard",
"Exists",
"Matroid.Indep",
"LE.le",
"_private.Mathlib.Combinatorics.Matroid.Rank.ENat.0.Mat... | [
"case refine_1\nα : Type u_1\nM : Matroid α\nX : Set α\nn : ℕ∞\nh : n ≤ M.eRk X\n⊢ ∃ I ⊆ X, M.Indep I ∧ I.encard = n",
"case refine_2\nα : Type u_1\nM : Matroid α\nX : Set α\nn : ℕ∞\nx✝ : ∃ I ⊆ X, M.Indep I ∧ I.encard = n\nI : Set α\nhIX : I ⊆ X\nhI : M.Indep I\nhIc : I.encard = n\n⊢ n ≤ M.eRk X"
] | refine ⟨fun h ↦ ?_, fun ⟨I, hIX, hI, hIc⟩ ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 463,
"column": 4
} | {
"line": 463,
"column": 44
} | {
"line": 463,
"column": 45
} | [
{
"pp": "case refine_1\nα : Type u_1\nM : Matroid α\nι : Type u_2\nC₀ : Set α\nhC₀ : M.IsCircuit C₀\nx : ι → α\nC : ι → Set α\nz : α\nhC : ∀ (i : ι), M.IsCircuit (C i)\nh_mem_C₀ : ∀ (i : ι), x i ∈ C₀\nh_mem : ∀ (i : ι), x i ∈ C i\nh_unique : ∀ ⦃i i' : ι⦄, x i ∈ C i' → i = i'\nhzC₀ : z ∈ C₀\nhzC : ∀ (i : ι), z ∉... | [
"case refine_1\nα : Type u_1\nM : Matroid α\nι : Type u_2\nC₀ : Set α\nhC₀ : M.IsCircuit C₀\nx : ι → α\nC : ι → Set α\nz : α\nhC : ∀ (i : ι), M.IsCircuit (C i)\nh_mem_C₀ : ∀ (i : ι), x i ∈ C₀\nh_mem : ∀ (i : ι), x i ∈ C i\nh_unique : ∀ ⦃i i' : ι⦄, x i ∈ C i' → i = i'\nhzC₀ : z ∈ C₀\nhzC : ∀ (i : ι), z ∉ C i\ni : ι\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 406,
"column": 2
} | {
"line": 406,
"column": 25
} | {
"line": 407,
"column": 2
} | [
{
"pp": "α : Type u_2\nM : Matroid α\ne : α\nI : Set α\n⊢ M.Indep (insert e I) ↔ M.Indep I ∧ (e ∉ I → e ∈ M.E \\ M.closure I)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Matroid.E",
"Classical.propDecidable",
"Membership.mem",
"Matroid.Indep",
"Insert.ins... | [
"case pos\nα : Type u_2\nM : Matroid α\ne : α\nI : Set α\nhI : M.Indep I\n⊢ M.Indep (insert e I) ↔ M.Indep I ∧ (e ∉ I → e ∈ M.E \\ M.closure I)",
"case neg\nα : Type u_2\nM : Matroid α\ne : α\nI : Set α\nhI : ¬M.Indep I\n⊢ M.Indep (insert e I) ↔ M.Indep I ∧ (e ∉ I → e ∈ M.E \\ M.closure I)"
] | by_cases hI : M.Indep I | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 626,
"column": 6
} | {
"line": 626,
"column": 17
} | {
"line": 626,
"column": 18
} | [
{
"pp": "case inr.refine_1\nα : Type u_1\nM : Matroid α\ne : α\nB : Set α\nhB : M.IsBase B\nhe : e ∈ B\nh : ∀ x ∈ M.E \\ B, e ∉ M.fundCircuit x B\nx : α\nhxE : x ∈ M.E\nhne : x ≠ e\nhx : x ∉ B\nh_cct : x ∈ M.closure (M.fundCircuit x B \\ {x})\n⊢ M.fundCircuit x B \\ {x} ⊆ B",
"ppTerm": "?inr.refine_1",
... | [
"case inr.refine_1\nα : Type u_1\nM : Matroid α\ne : α\nB : Set α\nhB : M.IsBase B\nhe : e ∈ B\nh : ∀ x ∈ M.E \\ B, e ∉ M.fundCircuit x B\nx : α\nhxE : x ∈ M.E\nhne : x ≠ e\nhx : x ∉ B\nh_cct : x ∈ M.closure (M.fundCircuit x B \\ {x})\n⊢ M.fundCircuit x B ⊆ insert x B"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 482,
"column": 4
} | {
"line": 482,
"column": 15
} | {
"line": 482,
"column": 16
} | [
{
"pp": "case refine_6\nα : Type u_1\nM : Matroid α\nC₀ : Set α\nhC₀ : M.IsCircuit C₀\nX : Set α\nS : Set (Set α)\nz : α\nhCS : ∀ C ∈ S, M.IsCircuit C\nhXC₀ : X ⊆ C₀\nhzC₀ : z ∈ C₀\nhz✝ : ∀ C ∈ S, z ∉ C\nC : α → Set α\nhC : (∀ x ∈ X, C x ∈ S) ∧ ∀ x ∈ X, C x ∩ X = {x}\nC' : Set α\nhC'ss : C' ⊆ (C₀ ∪ ⋃ i, C ↑i) \... | [
"case refine_6\nα : Type u_1\nM : Matroid α\nC₀ : Set α\nhC₀ : M.IsCircuit C₀\nX : Set α\nS : Set (Set α)\nz : α\nhCS : ∀ C ∈ S, M.IsCircuit C\nhXC₀ : X ⊆ C₀\nhzC₀ : z ∈ C₀\nhz✝ : ∀ C ∈ S, z ∉ C\nC : α → Set α\nhC : (∀ x ∈ X, C x ∈ S) ∧ ∀ x ∈ X, C x ∩ X = {x}\nC' : Set α\nhC'ss : C' ⊆ (C₀ ∪ ⋃ i, C ↑i) \\ range fun ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 410,
"column": 12
} | {
"line": 410,
"column": 23
} | {
"line": 410,
"column": 24
} | [
{
"pp": "case h₂\nα : Type u_1\nM : Matroid α\ne : α\nX : Set α\n⊢ M.eRk {e} ≤ 1",
"ppTerm": "?h₂",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case h₂\nα : Type u_1\nM : Matroid α\ne : α\nX : Set α\n⊢ M.eRk {e} ≤ 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 451,
"column": 4
} | {
"line": 451,
"column": 60
} | {
"line": 451,
"column": 61
} | [
{
"pp": "α : Type u_2\nM : Matroid α\ne f : α\nB : Set α\nhB : M.IsBase B\nhe : e ∈ B\nhf : f ∉ M.closure (B \\ {e})\nhfE : f ∈ M.E\nhne : f ≠ e\n⊢ M.Indep (insert f (B \\ {e})) ∧ f ∉ B",
"ppTerm": "?m.65",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\nM : Matroid α\ne f : α\nB : Set α\nhB : M.IsBase B\nhe : e ∈ B\nhf : f ∉ M.closure (B \\ {e})\nhfE : f ∈ M.E\nhne : f ≠ e\n⊢ M.Indep (insert f (B \\ {e})) ∧ f ∉ B"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 422,
"column": 4
} | {
"line": 422,
"column": 15
} | {
"line": 422,
"column": 16
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX Y : Set α\nh : Y ∩ M.E ⊆ M.closure X\n⊢ M.eRk Y ≤ M.eRk X",
"ppTerm": "?m.37",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nM : Matroid α\nX Y : Set α\nh : Y ∩ M.E ⊆ M.closure X\n⊢ M.eRk Y ≤ M.eRk X"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 483,
"column": 4
} | {
"line": 483,
"column": 15
} | {
"line": 483,
"column": 16
} | [
{
"pp": "case refine_1\nα : Type u_1\nM : Matroid α\nC₀ : Set α\nhC₀ : M.IsCircuit C₀\nX : Set α\nS : Set (Set α)\nz : α\nhCS : ∀ C ∈ S, M.IsCircuit C\nhXC₀ : X ⊆ C₀\nhzC₀ : z ∈ C₀\nhz : ∀ C ∈ S, z ∉ C\nC : α → Set α\nhC : (∀ x ∈ X, C x ∈ S) ∧ ∀ x ∈ X, C x ∩ X = {x}\n⊢ ∀ (i : ↑X), M.IsCircuit (C ↑i)",
"ppTe... | [
"case refine_1\nα : Type u_1\nM : Matroid α\nC₀ : Set α\nhC₀ : M.IsCircuit C₀\nX : Set α\nS : Set (Set α)\nz : α\nhCS : ∀ C ∈ S, M.IsCircuit C\nhXC₀ : X ⊆ C₀\nhzC₀ : z ∈ C₀\nhz : ∀ C ∈ S, z ∉ C\nC : α → Set α\nhC : (∀ x ∈ X, C x ∈ S) ∧ ∀ x ∈ X, C x ∩ X = {x}\n⊢ ∀ a ∈ X, M.IsCircuit (C a)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 430,
"column": 2
} | {
"line": 430,
"column": 56
} | {
"line": 430,
"column": 57
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX Y : Set α\nhX : M.IsRkFinite X\nhXY : X ⊆ Y\nhY : ∀ e ∈ Y \\ X, M.eRk (Insert.insert e X) ≤ M.eRk X\nhlt : M.eRk X < M.eRk Y\nz : α\nhz : z ∈ Y \\ X\nhr : M.eRk (Insert.insert z X) = M.eRk X + 1\n⊢ False",
"ppTerm": "?m.53",
"assigned": false,
"usedConstants":... | [
"α : Type u_1\nM : Matroid α\nX Y : Set α\nhX : M.IsRkFinite X\nhXY : X ⊆ Y\nhY : ∀ e ∈ Y \\ X, M.eRk (Insert.insert e X) ≤ M.eRk X\nhlt : M.eRk X < M.eRk Y\nz : α\nhz : z ∈ Y \\ X\nhr : M.eRk (Insert.insert z X) = M.eRk X + 1\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 484,
"column": 4
} | {
"line": 484,
"column": 15
} | {
"line": 484,
"column": 16
} | [
{
"pp": "case refine_2\nα : Type u_1\nM : Matroid α\nC₀ : Set α\nhC₀ : M.IsCircuit C₀\nX : Set α\nS : Set (Set α)\nz : α\nhCS : ∀ C ∈ S, M.IsCircuit C\nhXC₀ : X ⊆ C₀\nhzC₀ : z ∈ C₀\nhz : ∀ C ∈ S, z ∉ C\nC : α → Set α\nhC : (∀ x ∈ X, C x ∈ S) ∧ ∀ x ∈ X, C x ∩ X = {x}\n⊢ ∀ (i : ↑X), ↑i ∈ C₀",
"ppTerm": "?refi... | [
"case refine_2\nα : Type u_1\nM : Matroid α\nC₀ : Set α\nhC₀ : M.IsCircuit C₀\nX : Set α\nS : Set (Set α)\nz : α\nhCS : ∀ C ∈ S, M.IsCircuit C\nhXC₀ : X ⊆ C₀\nhzC₀ : z ∈ C₀\nhz : ∀ C ∈ S, z ∉ C\nC : α → Set α\nhC : (∀ x ∈ X, C x ∈ S) ∧ ∀ x ∈ X, C x ∩ X = {x}\n⊢ ∀ a ∈ X, a ∈ C₀"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 441,
"column": 2
} | {
"line": 441,
"column": 54
} | {
"line": 442,
"column": 2
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI : Set α\nhI : I.Finite\n⊢ M.Indep I ↔ M.eRk I = I.encard",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard",
"Matroid.Indep.eRk_eq_encard",
"congrArg",
"Matroid.Indep",
"id",
"ENat",
... | [
"α : Type u_1\nM : Matroid α\nI : Set α\nhI : I.Finite\nh : M.eRk I = I.encard\n⊢ M.Indep I"
] | refine ⟨fun h ↦ by rw [h.eRk_eq_encard], fun h ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 442,
"column": 2
} | {
"line": 442,
"column": 39
} | {
"line": 443,
"column": 2
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI : Set α\nhI : I.Finite\nh : M.eRk I = I.encard\n⊢ M.Indep I",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Matroid.IsBasis'",
"Exists",
"Matroid.Indep",
"Exists.casesOn",
"Matroid.exists_isBasis'",
"Set"
],
... | [
"α : Type u_1\nM : Matroid α\nI : Set α\nhI : I.Finite\nh : M.eRk I = I.encard\nJ : Set α\nhJ : M.IsBasis' J I\n⊢ M.Indep I"
] | obtain ⟨J, hJ⟩ := M.exists_isBasis' I | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 489,
"column": 4
} | {
"line": 489,
"column": 28
} | {
"line": 489,
"column": 29
} | [
{
"pp": "case refine_4\nα : Type u_1\nM : Matroid α\nC₀ : Set α\nhC₀ : M.IsCircuit C₀\nX : Set α\nS : Set (Set α)\nz : α\nhCS : ∀ C ∈ S, M.IsCircuit C\nhXC₀ : X ⊆ C₀\nhzC₀ : z ∈ C₀\nhz : ∀ C ∈ S, z ∉ C\nC : α → Set α\nhC : (∀ x ∈ X, C x ∈ S) ∧ ∀ x ∈ X, C x ∩ X = {x}\ne : α\nheX : e ∈ X\nf : α\nhfX : f ∈ X\nhef ... | [
"case refine_4\nα : Type u_1\nM : Matroid α\nC₀ : Set α\nhC₀ : M.IsCircuit C₀\nX : Set α\nS : Set (Set α)\nz : α\nhCS : ∀ C ∈ S, M.IsCircuit C\nhXC₀ : X ⊆ C₀\nhzC₀ : z ∈ C₀\nhz : ∀ C ∈ S, z ∉ C\nC : α → Set α\nhC : (∀ x ∈ X, C x ∈ S) ∧ ∀ x ∈ X, C x ∩ X = {x}\ne : α\nheX : e ∈ X\nf : α\nhfX : f ∈ X\nhef : e ∈ C f\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 490,
"column": 2
} | {
"line": 490,
"column": 13
} | {
"line": 490,
"column": 14
} | [
{
"pp": "case refine_5\nα : Type u_1\nM : Matroid α\nC₀ : Set α\nhC₀ : M.IsCircuit C₀\nX : Set α\nS : Set (Set α)\nz : α\nhCS : ∀ C ∈ S, M.IsCircuit C\nhXC₀ : X ⊆ C₀\nhzC₀ : z ∈ C₀\nhz : ∀ C ∈ S, z ∉ C\nC : α → Set α\nhC : (∀ x ∈ X, C x ∈ S) ∧ ∀ x ∈ X, C x ∩ X = {x}\n⊢ ∀ (i : ↑X), z ∉ C ↑i",
"ppTerm": "?ref... | [
"case refine_5\nα : Type u_1\nM : Matroid α\nC₀ : Set α\nhC₀ : M.IsCircuit C₀\nX : Set α\nS : Set (Set α)\nz : α\nhCS : ∀ C ∈ S, M.IsCircuit C\nhXC₀ : X ⊆ C₀\nhzC₀ : z ∈ C₀\nhz : ∀ C ∈ S, z ∉ C\nC : α → Set α\nhC : (∀ x ∈ X, C x ∈ S) ∧ ∀ x ∈ X, C x ∩ X = {x}\n⊢ ∀ a ∈ X, z ∉ C a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 460,
"column": 2
} | {
"line": 460,
"column": 13
} | {
"line": 460,
"column": 14
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI : Set α\nhX : M.IsRkFinite I\nh : I.encard ≤ M.eRk I\n⊢ I.Finite",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nM : Matroid α\nI : Set α\nhX : M.IsRkFinite I\nh : I.encard ≤ M.eRk I\n⊢ I.Finite"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 533,
"column": 2
} | {
"line": 533,
"column": 13
} | {
"line": 533,
"column": 14
} | [
{
"pp": "case refine_2\nα : Type u_1\nM : Matroid α\nh : ∀ (C : Set α), M.IsCircuit C → C.Finite\nI : Set α\nhI : ∀ J ⊆ I, J.Finite → M.Indep J\nx : α\nhx : x ∈ I\n⊢ x ∈ M.E",
"ppTerm": "?refine_2",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case refine_2\nα : Type u_1\nM : Matroid α\nh : ∀ (C : Set α), M.IsCircuit C → C.Finite\nI : Set α\nhI : ∀ J ⊆ I, J.Finite → M.Indep J\nx : α\nhx : x ∈ I\n⊢ x ∈ M.E"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 542,
"column": 9
} | {
"line": 542,
"column": 20
} | {
"line": 542,
"column": 21
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\ne : α\ninst✝ : M.Finitary\nhe : e ∈ M.closure X\nheY : e ∈ X\nJ : Set α\nhJ : M.IsBasis J {e}\n⊢ e ∈ M.closure J",
"ppTerm": "?m.74",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nM : Matroid α\nX : Set α\ne : α\ninst✝ : M.Finitary\nhe : e ∈ M.closure X\nheY : e ∈ X\nJ : Set α\nhJ : M.IsBasis J {e}\n⊢ e ∈ M.closure J"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 508,
"column": 2
} | {
"line": 508,
"column": 13
} | {
"line": 508,
"column": 14
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI : Set α\nhI : M.Indep I\nhIfin : I.Finite\nh : M.eRank ≤ M.eRk I\n⊢ M.IsBase I",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nM : Matroid α\nI : Set α\nhI : M.Indep I\nhIfin : I.Finite\nh : M.eRank ≤ M.eRk I\n⊢ M.IsBase I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 788,
"column": 2
} | {
"line": 788,
"column": 13
} | {
"line": 788,
"column": 14
} | [
{
"pp": "case inr\nα : Type u_1\nM : Matroid α\ninst✝ : M.Loopless\nx : α\nhI : {x}.Subsingleton\nhIE : {x} ⊆ M.E\n⊢ M.Indep {x}",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Matroid.indep_singleton._simp_1",
"Eq.mpr",
"Matroid.Indep",
"Set.instSingletonSet",
... | [
"case inr\nα : Type u_1\nM : Matroid α\ninst✝ : M.Loopless\nx : α\nhI : {x}.Subsingleton\nhIE : {x} ⊆ M.E\n⊢ M.IsNonloop x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 808,
"column": 4
} | {
"line": 808,
"column": 15
} | {
"line": 808,
"column": 16
} | [
{
"pp": "case inl\nα : Type u_1\nM : Matroid α\nx✝ : ∃ x, M.IsCircuit x ∧ x.Subsingleton\nhC : M.IsCircuit ∅\nhCs : ∅.Subsingleton\n⊢ ∃ x ∈ M.E, M.IsLoop x",
"ppTerm": "?inl",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case inl\nα : Type u_1\nM : Matroid α\nx✝ : ∃ x, M.IsCircuit x ∧ x.Subsingleton\nhC : M.IsCircuit ∅\nhCs : ∅.Subsingleton\n⊢ ∃ x ∈ M.E, M.IsLoop x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 531,
"column": 2
} | {
"line": 531,
"column": 78
} | {
"line": 533,
"column": 0
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nI : Set α\nι : Type u_4\nhI : M.Indep I\nX : ι → Set α\nA : Set ι\nhA : A.Nonempty\nh : ∀ i ∈ A, M.IsBasis (X i ∩ I) (X i)\n⊢ ∀ i ∈ A, ⋂ i ∈ A, X i ⊆ M.closure (X i ∩ I)",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"ChainCompletePartialOrder.i... | [] | exact fun i hiA ↦ (biInter_subset_of_mem hiA).trans (h i hiA).subset_closure | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 537,
"column": 10
} | {
"line": 537,
"column": 21
} | {
"line": 537,
"column": 22
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nι : Sort u_3\nI : Set α\ninst✝ : Nonempty ι\nX : ι → Set α\nhI : M.Indep I\nh : ∀ (i : ι), M.IsBasis (X i ∩ I) (X i)\n⊢ ∀ i ∈ univ, M.IsBasis (X i.down ∩ I) (X i.down)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.mem_univ.... | [
"α : Type u_2\nM : Matroid α\nι : Sort u_3\nI : Set α\ninst✝ : Nonempty ι\nX : ι → Set α\nhI : M.Indep I\nh : ∀ (i : ι), M.IsBasis (X i ∩ I) (X i)\n⊢ ∀ (i : PLift ι), M.IsBasis (X i.down ∩ I) (X i.down)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 617,
"column": 11
} | {
"line": 617,
"column": 42
} | {
"line": 617,
"column": 43
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nh : M.eRank = 0\n⊢ M = loopyOn M.E",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nM : Matroid α\nh : M.eRank = 0\n⊢ M = loopyOn M.E"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 675,
"column": 4
} | {
"line": 675,
"column": 57
} | {
"line": 675,
"column": 58
} | [
{
"pp": "case neg.inl\nα : Type u_2\nM : Matroid α\ne : α\nB : Set α\nhB : M.IsBasis B M.E\nhe : e ∈ M.closure (insert e B \\ {e})\nhf : e ∉ M.E\n⊢ M.IsBasis (insert e B \\ {e}) M.E",
"ppTerm": "?neg.inl✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"eq_false",
"... | [
"case neg.inl\nα : Type u_2\nM : Matroid α\ne : α\nB : Set α\nhB : M.IsBasis B M.E\nhe : e ∈ M.closure (insert e B \\ {e})\nhf : e ∉ M.E\n⊢ M.IsBase B"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 628,
"column": 32
} | {
"line": 628,
"column": 77
} | {
"line": 628,
"column": 77
} | [
{
"pp": "α : Type u_1\nX : Set α\n⊢ (freeOn X).eRk X = X.encard",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Set.encard",
"Matroid.Indep.eRk_eq_encard",
"ChainCompletePartialOrder.instOfCompleteLattice",
"congrArg",
"Partia... | [
"α : Type u_1\nX : Set α\n⊢ X.encard = X.encard"
] | (freeOn_indep_iff.2 rfl.subset).eRk_eq_encard | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 646,
"column": 4
} | {
"line": 646,
"column": 15
} | {
"line": 646,
"column": 16
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\nB : Set α\nhB : M✶.IsBasis B M.E\nhI : M✶.IsBasis (B ∩ X) X\nhB' : M✶.IsBase B\n⊢ M.IsBasis (M.E \\ B ∩ (M.E \\ X)) (M.E \\ X)",
"ppTerm": "?m.87",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nM : Matroid α\nX : Set α\nhX : X ⊆ M.E\nB : Set α\nhB : M✶.IsBasis B M.E\nhI : M✶.IsBasis (B ∩ X) X\nhB' : M✶.IsBase B\n⊢ M.IsBasis (M.E \\ B ∩ (M.E \\ X)) (M.E \\ X)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 654,
"column": 35
} | {
"line": 654,
"column": 59
} | {
"line": 654,
"column": 60
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\n⊢ M✶.eRk X + M.eRank = M.eRk (M.E \\ (X ∩ M.E)) + (X ∩ M.E).encard",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard",
"Set.inter_subset_right._simp_1",
"congrArg",
"Matroid.E",
"Matroi... | [
"α : Type u_1\nM : Matroid α\nX : Set α\n⊢ M✶.eRk X + M.eRank = M✶.eRk (X ∩ M.E) + M.eRank"
] | ← eRk_dual_add_eRank .., | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 759,
"column": 4
} | {
"line": 759,
"column": 21
} | {
"line": 759,
"column": 22
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nB : Set α\nhB : M.IsBase B\ne f : α\nhe : e ∈ M.fundCocircuit f B\nhne : e ≠ f\nhB' : M✶.IsBase (M✶.E \\ B)\nhfE : f ∈ M.E\nhfB : f ∈ B\n⊢ e ∈ M.E \\ B",
"ppTerm": "?m.183",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matroid.E",
"Membershi... | [
"α : Type u_1\nM : Matroid α\nB : Set α\nhB : M.IsBase B\ne f : α\nhe : e ∈ M.fundCocircuit f B\nhne : e ≠ f\nhB' : M✶.IsBase (M✶.E \\ B)\nhfE : f ∈ M.E\nhfB : f ∈ B\n⊢ e ∈ M.E ∧ e ∉ B"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 805,
"column": 2
} | {
"line": 805,
"column": 13
} | {
"line": 805,
"column": 14
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nX : Set α\ne : α\nhe : e ∈ X\nheE : e ∈ M.E\n⊢ e ∈ M.closure (X \\ {e}) ↔ M.closure (X \\ {e}) = M.closure X",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\nM : Matroid α\nX : Set α\ne : α\nhe : e ∈ X\nheE : e ∈ M.E\n⊢ e ∈ M.closure (X \\ {e}) ↔ M.closure (X \\ {e}) = M.closure X"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 813,
"column": 16
} | {
"line": 813,
"column": 27
} | {
"line": 813,
"column": 28
} | [
{
"pp": "α : Type u_2\nM₁ M₂ : Matroid α\nh : ∀ (X : Set α), M₁.closure X = M₂.closure X\n⊢ M₁.E = M₂.E",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_2\nM₁ M₂ : Matroid α\nh : ∀ (X : Set α), M₁.closure X = M₂.closure X\n⊢ M₁.E = M₂.E"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.MvPolynomial | {
"line": 62,
"column": 2
} | {
"line": 63,
"column": 64
} | {
"line": 63,
"column": 65
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : CommRing R\ni : σ\ns : Set σ\nh : i ∉ s\nf : R[X]\ng : MvPolynomial (↑s) R[X] →ₐ[R] MvPolynomial (↑s) R[X] := mapAlgHom (Polynomial.aeval f)\nu : (MvPolynomial (↑s) R)[X] →ₐ[R] MvPolynomial σ R :=\n (((supported R (insert i s)).val.comp\n ↑(((optionEquivR... | [
"σ : Type u_1\nR : Type u_2\ninst✝ : CommRing R\ni : σ\ns : Set σ\nh : i ∉ s\nf : R[X]\ng : MvPolynomial (↑s) R[X] →ₐ[R] MvPolynomial (↑s) R[X] := mapAlgHom (Polynomial.aeval f)\nu : (MvPolynomial (↑s) R)[X] →ₐ[R] MvPolynomial σ R :=\n (((supported R (insert i s)).val.comp\n ↑(((optionEquivRight R ↑s).s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AlgebraicIndependent.Basic | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 33
} | {
"line": 68,
"column": 4
} | [
{
"pp": "ι : Type u\nR : Type u_2\nA : Type v\nx : ι → A\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nhx : AlgebraicIndependent R x\n⊢ Injective ⇑(algebraMap R A)",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u\nR : Type u_2\nA : Type v\nx : ι → A\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\nhx : AlgebraicIndependent R x\n⊢ Injective ⇑(algebraMap R A)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.MvPolynomial | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 13
} | {
"line": 96,
"column": 14
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : CommRing R\ni : σ\ns : Set σ\nh : i ∉ s\n⊢ Transcendental (↥(supported R s)) (X i)",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"σ : Type u_1\nR : Type u_2\ninst✝ : CommRing R\ni : σ\ns : Set σ\nh : i ∉ s\n⊢ Transcendental (↥(supported R s)) (X i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.MvPolynomial | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 13
} | {
"line": 99,
"column": 14
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : CommRing R\ni : σ\n⊢ Transcendental R (X i)",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"σ : Type u_1\nR : Type u_2\ninst✝ : CommRing R\ni : σ\n⊢ Transcendental R (X i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Algebraic.MvPolynomial | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 43
} | {
"line": 104,
"column": 4
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\ni : σ\ns : Set σ\n⊢ Transcendental (↥(supported R s)) (X i) ↔ i ∉ s",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"σ : Type u_1\nR : Type u_2\ninst✝¹ : CommRing R\ninst✝ : Nontrivial R\ni : σ\ns : Set σ\n⊢ Transcendental (↥(supported R s)) (X i) ↔ i ∉ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 950,
"column": 6
} | {
"line": 950,
"column": 17
} | {
"line": 950,
"column": 18
} | [
{
"pp": "α : Type u_2\nM M' : Matroid α\nh : M.E = M'.E\nhsp : ∀ S ⊆ M.E, M.Spanning S ↔ M'.Spanning S\nhsp' : M.Spanning = M'.Spanning\n⊢ M = M'",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.dual",
"id",
"propext",
"Eq.s... | [
"α : Type u_2\nM M' : Matroid α\nh : M.E = M'.E\nhsp : ∀ S ⊆ M.E, M.Spanning S ↔ M'.Spanning S\nhsp' : M.Spanning = M'.Spanning\n⊢ M✶ = M'✶"
] | ← dual_inj, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AlgebraicIndependent.Basic | {
"line": 166,
"column": 24
} | {
"line": 166,
"column": 35
} | {
"line": 166,
"column": 36
} | [
{
"pp": "R : Type u_2\nA : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : Subsingleton R\nthis : Subsingleton A\ns : { s // AlgebraicIndepOn R _root_.id s }\n⊢ Cardinal.mk ↑↑s ≤ 1",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Cardin... | [
"R : Type u_2\nA : Type v\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : Subsingleton R\nthis : Subsingleton A\ns : { s // AlgebraicIndepOn R _root_.id s }\n⊢ (↑s).Subsingleton"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Matroid.Rank.Cardinal | {
"line": 384,
"column": 2
} | {
"line": 384,
"column": 69
} | {
"line": 386,
"column": 0
} | [
{
"pp": "α : Type u\nM : Matroid α\n⊢ M.RankInfinite ↔ ℵ₀ ≤ M.cRank",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Cardinal",
"Matroid.RankInfinite",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
"Preorder.toLE... | [] | rw [← not_lt, ← rankFinite_iff_cRank_lt_aleph0, not_rankFinite_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Rank.Cardinal | {
"line": 384,
"column": 2
} | {
"line": 384,
"column": 69
} | {
"line": 386,
"column": 0
} | [
{
"pp": "α : Type u\nM : Matroid α\n⊢ M.RankInfinite ↔ ℵ₀ ≤ M.cRank",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Cardinal",
"Matroid.RankInfinite",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
"Preorder.toLE... | [] | rw [← not_lt, ← rankFinite_iff_cRank_lt_aleph0, not_rankFinite_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Rank.Cardinal | {
"line": 384,
"column": 2
} | {
"line": 384,
"column": 69
} | {
"line": 386,
"column": 0
} | [
{
"pp": "α : Type u\nM : Matroid α\n⊢ M.RankInfinite ↔ ℵ₀ ≤ M.cRank",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Cardinal",
"Matroid.RankInfinite",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
"Preorder.toLE... | [] | rw [← not_lt, ← rankFinite_iff_cRank_lt_aleph0, not_rankFinite_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.AlgebraicIndependent.Basic | {
"line": 201,
"column": 2
} | {
"line": 203,
"column": 83
} | {
"line": 204,
"column": 2
} | [
{
"pp": "ι : Type u\nR : Type u_2\nA : Type v\nx : ι → A\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A\ninst✝⁷ : Algebra R A\nS : Type u_3\nB : Type u_4\nFRS : Type u_5\nFAB : Type u_6\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra S B\ninst✝³ : FunLike FRS R S\ninst✝² : RingHomClass FRS R S\ninst✝¹ : ... | [
"ι : Type u\nR : Type u_2\nA : Type v\nx : ι → A\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A\ninst✝⁷ : Algebra R A\nS : Type u_3\nB : Type u_4\nFRS : Type u_5\nFAB : Type u_6\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra S B\ninst✝³ : FunLike FRS R S\ninst✝² : RingHomClass FRS R S\ninst✝¹ : FunLike FAB ... | have := H (p.map f) <| by
have : (g : A →+* B) _ = _ := congr(g $hp)
rwa [map_zero, map_aeval, ← h, ← eval₂Hom_map_hom, ← aeval_eq_eval₂Hom] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.AlgebraicIndependent.Transcendental | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 41
} | {
"line": 217,
"column": 42
} | [
{
"pp": "ι : Type u_1\nR : Type u_3\nS : Type u\nA : Type v\nx : ι → A\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R S\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\nhx : AlgebraicIndependent R x\nι' : Type u_4\ny : ι' → A\nhxS : range x ⊆ range ⇑(... | [
"ι : Type u_1\nR : Type u_3\nS : Type u\nA : Type v\nx : ι → A\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R S\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\nhx : AlgebraicIndependent R x\nι' : Type u_4\ny : ι' → A\nhxS : range x ⊆ range ⇑(algebraMap S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AlgebraicIndependent.Basic | {
"line": 417,
"column": 68
} | {
"line": 417,
"column": 79
} | {
"line": 417,
"column": 80
} | [
{
"pp": "R : Type u_2\nA : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\ns : Set (Set A)\nhsn : s.Nonempty\nhs : DirectedOn (fun x1 x2 ↦ x1 ⊆ x2) s\nh : ∀ a ∈ s, AlgebraicIndependent R Subtype.val\nthis : Nonempty ↑s := Nonempty.to_subtype hsn\n⊢ ∀ (i : ↑s), AlgebraicIndependent R Subty... | [
"R : Type u_2\nA : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\ns : Set (Set A)\nhsn : s.Nonempty\nhs : DirectedOn (fun x1 x2 ↦ x1 ⊆ x2) s\nh : ∀ a ∈ s, AlgebraicIndependent R Subtype.val\nthis : Nonempty ↑s := Nonempty.to_subtype hsn\n⊢ ∀ a ∈ s, AlgebraicIndependent R Subtype.val"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.AlgebraicClosure | {
"line": 128,
"column": 2
} | {
"line": 129,
"column": 88
} | {
"line": 129,
"column": 89
} | [
{
"pp": "F : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nL : IntermediateField F E\nhs : ∀ (x : ↥L), IsAlgebraic F x\nx : E\nh : x ∈ L\n⊢ x ∈ algebraicClosure F E",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsDomain",
"P... | [
"F : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nL : IntermediateField F E\nhs : ∀ (x : ↥L), IsAlgebraic F x\nx : E\nh : x ∈ L\n⊢ ∃ p, ¬p = 0 ∧ (aeval ((algebraMap E E) x)) p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.AlgebraicClosure | {
"line": 140,
"column": 23
} | {
"line": 142,
"column": 90
} | {
"line": 142,
"column": 91
} | [
{
"pp": "F : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nL : IntermediateField F E\nh : L ≤ algebraicClosure F E\nx : ↥L\n⊢ IsAlgebraic F x",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsDomain",
"IntermediateField.isScal... | [
"F : Type u_1\nE : Type u_2\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nL : IntermediateField F E\nh : L ≤ algebraicClosure F E\nx : ↥L\n⊢ ∃ p, ¬p = 0 ∧ (aeval ((algebraMap E E) ((RingHom.id E) ↑x))) p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.AlgebraicClosure | {
"line": 171,
"column": 15
} | {
"line": 171,
"column": 47
} | {
"line": 171,
"column": 48
} | [
{
"pp": "F : Type u_1\nE : Type u_2\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : IsAlgClosed E\nh : algebraicClosure F E = ⊥\np : F[X]\nhmon : p.Monic\nhirr : Irreducible p\nx : F\nhx : (aeval ((Algebra.ofId F E).toRingHom x)) p = 0\n⊢ eval x p = 0",
"ppTerm": "?m.110",
"assigned":... | [
"F : Type u_1\nE : Type u_2\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : IsAlgClosed E\nh : algebraicClosure F E = ⊥\np : F[X]\nhmon : p.Monic\nhirr : Irreducible p\nx : F\nhx : (aeval ((Algebra.ofId F E).toRingHom x)) p = 0\n⊢ eval x p = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.FixedPoints | {
"line": 96,
"column": 14
} | {
"line": 96,
"column": 25
} | {
"line": 96,
"column": 26
} | [
{
"pp": "α : Type u_1\nG : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G α\ng : G\na : α\nh : ∀ (j : ℤ), a ∈ fixedBy α (g ^ j)\n⊢ a ∈ fixedBy α g",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"MulAction.fixedBy",
"Membership.mem",
"i... | [
"α : Type u_1\nG : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G α\ng : G\na : α\nh : ∀ (j : ℤ), a ∈ fixedBy α (g ^ j)\n⊢ g • a = a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 54,
"column": 2
} | {
"line": 55,
"column": 9
} | {
"line": 55,
"column": 10
} | [
{
"pp": "R : Type u_1\nA : Type w\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\ns : Set A\nhs : AlgebraicIndepOn R _root_.id s\n⊢ ∃ t, s ⊆ t ∧ IsTranscendenceBasis R Subtype.val",
"ppTerm": "?m.26",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"R : Type u_1\nA : Type w\ninst✝² : CommRing R\ninst✝¹ : CommRing A\ninst✝ : Algebra R A\ns : Set A\nhs : AlgebraicIndepOn R _root_.id s\n⊢ ∃ t, s ⊆ t ∧ IsTranscendenceBasis R Subtype.val"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 60,
"column": 2
} | {
"line": 60,
"column": 13
} | {
"line": 60,
"column": 14
} | [
{
"pp": "R : Type u_1\nA : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : FaithfulSMul R A\n⊢ ∃ s, IsTranscendenceBasis R Subtype.val",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\nA : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : FaithfulSMul R A\n⊢ ∃ s, IsTranscendenceBasis R Subtype.val"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 15
} | {
"line": 97,
"column": 16
} | [
{
"pp": "case right\nι : Type u\nR : Type u_1\nA : Type w\nx : ι → A\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\ni : AlgebraicIndependent R x\nw : Set A\ni' : AlgebraicIndepOn R _root_.id w\nh : range x ⊆ w\np : Surjective fun i ↦ ⟨x i, ⋯⟩\nq : (fun x_1 ↦ ↑⟨x x_1, ⋯⟩) ... | [
"case right\nι : Type u\nR : Type u_1\nA : Type w\nx : ι → A\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\ni : AlgebraicIndependent R x\nw : Set A\ni' : AlgebraicIndepOn R _root_.id w\nh : range x ⊆ w\np : Surjective fun i ↦ ⟨x i, ⋯⟩\nq : (fun x_1 ↦ ↑⟨x x_1, ⋯⟩) '' univ = Su... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.FixedPoints | {
"line": 296,
"column": 2
} | {
"line": 296,
"column": 13
} | {
"line": 296,
"column": 14
} | [
{
"pp": "G : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝² : Monoid G\ninst✝¹ : MulAction G A\ninst✝ : MulAction G B\nf : A →ₑ[id] B\ng : G\na : A\nha : a ∈ MulAction.fixedBy A g\n⊢ f a ∈ MulAction.fixedBy B g",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul"... | [
"G : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝² : Monoid G\ninst✝¹ : MulAction G A\ninst✝ : MulAction G B\nf : A →ₑ[id] B\ng : G\na : A\nha : a ∈ MulAction.fixedBy A g\n⊢ g • f a = f a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.FixingSubgroup | {
"line": 114,
"column": 56
} | {
"line": 114,
"column": 82
} | {
"line": 114,
"column": 83
} | [
{
"pp": "M : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns : Set α\nx✝ : M\nhx : x✝ ∈ (fixingSubmonoid M s).carrier\nz : ↑s\n⊢ x✝⁻¹ • ↑z = ↑z",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHSMul",
"inv_smul_... | [] | rw [inv_smul_eq_iff, hx z] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.GroupAction.FixingSubgroup | {
"line": 114,
"column": 56
} | {
"line": 114,
"column": 82
} | {
"line": 114,
"column": 83
} | [
{
"pp": "M : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns : Set α\nx✝ : M\nhx : x✝ ∈ (fixingSubmonoid M s).carrier\nz : ↑s\n⊢ x✝⁻¹ • ↑z = ↑z",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHSMul",
"inv_smul_... | [] | rw [inv_smul_eq_iff, hx z] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupAction.FixingSubgroup | {
"line": 114,
"column": 56
} | {
"line": 114,
"column": 82
} | {
"line": 114,
"column": 83
} | [
{
"pp": "M : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns : Set α\nx✝ : M\nhx : x✝ ∈ (fixingSubmonoid M s).carrier\nz : ↑s\n⊢ x✝⁻¹ • ↑z = ↑z",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHSMul",
"inv_smul_... | [] | rw [inv_smul_eq_iff, hx z] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 13
} | {
"line": 128,
"column": 14
} | [
{
"pp": "ι : Type u\nR : Type u_1\nA : Type w\nx : ι → A\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\nind : AlgebraicIndependent R x\ns : Set A\nind_s : AlgebraicIndepOn R _root_.id s\nhxs : range x ⊆ s\nalg : Algebra.IsAlgebraic (↥(adjoin R (Subtype.val '' range (Set.i... | [
"ι : Type u\nR : Type u_1\nA : Type w\nx : ι → A\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial R\nind : AlgebraicIndependent R x\ns : Set A\nind_s : AlgebraicIndepOn R _root_.id s\nhxs : range x ⊆ s\nalg : Algebra.IsAlgebraic (↥(adjoin R (Subtype.val '' range (Set.inclusion hxs... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 446,
"column": 2
} | {
"line": 446,
"column": 50
} | {
"line": 449,
"column": 0
} | [
{
"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... | [] | rwa [lift_id', ← lift_lift.{u}, lift_id] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 451,
"column": 2
} | {
"line": 451,
"column": 13
} | {
"line": 451,
"column": 14
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ trdeg R R[X] = 1",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\n⊢ trdeg R R[X] = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.FilterBasis | {
"line": 149,
"column": 14
} | {
"line": 149,
"column": 18
} | {
"line": 150,
"column": 4
} | [
{
"pp": "case hpure\nG : Type u\ninst✝ : Group G\nB : GroupFilterBasis G\nx₀ a : G\nU : Set G\n⊢ U ∈ B.toFilterBasis → a ∈ (fun y ↦ a * y) '' id U",
"ppTerm": "?hpure",
"assigned": true,
"usedConstants": [
"instMembershipSetFilterBasis",
"Membership.mem",
"FilterBasis",
"Grou... | [
"case hpure\nG : Type u\ninst✝ : Group G\nB : GroupFilterBasis G\nx₀ a : G\nU : Set G\nU_in : U ∈ B.toFilterBasis\n⊢ a ∈ (fun y ↦ a * y) '' id U"
] | U_in | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.Algebra.FilterBasis | {
"line": 151,
"column": 14
} | {
"line": 151,
"column": 18
} | {
"line": 152,
"column": 4
} | [
{
"pp": "case hopen\nG : Type u\ninst✝ : Group G\nB : GroupFilterBasis G\nx₀ a : G\nU : Set G\n⊢ U ∈ B.toFilterBasis →\n ∀ᶠ (x : G) in map (fun y ↦ a * y) B.filter, (fun y ↦ a * y) '' id U ∈ map (fun y ↦ x * y) B.filter",
"ppTerm": "?hopen",
"assigned": true,
"usedConstants": [
"instMembers... | [
"case hopen\nG : Type u\ninst✝ : Group G\nB : GroupFilterBasis G\nx₀ a : G\nU : Set G\nU_in : U ∈ B.toFilterBasis\n⊢ ∀ᶠ (x : G) in map (fun y ↦ a * y) B.filter, (fun y ↦ a * y) '' id U ∈ map (fun y ↦ x * y) B.filter"
] | U_in | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.Algebra.FilterBasis | {
"line": 199,
"column": 12
} | {
"line": 199,
"column": 16
} | {
"line": 200,
"column": 4
} | [
{
"pp": "case refine_1\nG : Type u\ninst✝ : Group G\nB✝ B : GroupFilterBasis G\nthis : TopologicalSpace G := B.topology\nbasis : (𝓝 1).HasBasis (fun V ↦ V ∈ B) id\nbasis' : (𝓝 1 ×ˢ 𝓝 1).HasBasis (fun i ↦ i.1 ∈ B ∧ i.2 ∈ B) fun i ↦ id i.1 ×ˢ id i.2\nU : Set G\n⊢ U ∈ B → ∃ V W, (V ∈ B ∧ W ∈ B) ∧ ∀ (a b : G), a... | [
"case refine_1\nG : Type u\ninst✝ : Group G\nB✝ B : GroupFilterBasis G\nthis : TopologicalSpace G := B.topology\nbasis : (𝓝 1).HasBasis (fun V ↦ V ∈ B) id\nbasis' : (𝓝 1 ×ˢ 𝓝 1).HasBasis (fun i ↦ i.1 ∈ B ∧ i.2 ∈ B) fun i ↦ id i.1 ×ˢ id i.2\nU : Set G\nU_in : U ∈ B\n⊢ ∃ V W, (V ∈ B ∧ W ∈ B) ∧ ∀ (a b : G), a ∈ V →... | U_in | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.FieldTheory.SeparableDegree | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 71
} | {
"line": 162,
"column": 72
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\nK : Type w\ninst✝¹ : Field K\ninst✝ : Algebra F K\ni : E ≃ₐ[F] K\nx✝ : Algebra E K := (↑i).toAlgebra\nx : K\nh : IsAlgebraic E (↑i (↑i.symm x))\n⊢ IsAlgebraic E x",
"ppTerm": "?m.123",
"assigned": false,
"used... | [
"F : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\nK : Type w\ninst✝¹ : Field K\ninst✝ : Algebra F K\ni : E ≃ₐ[F] K\nx✝ : Algebra E K := (↑i).toAlgebra\nx : K\nh : IsAlgebraic E (↑i (↑i.symm x))\n⊢ IsAlgebraic E x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.FilterBasis | {
"line": 205,
"column": 12
} | {
"line": 205,
"column": 16
} | {
"line": 206,
"column": 4
} | [
{
"pp": "case refine_2\nG : Type u\ninst✝ : Group G\nB✝ B : GroupFilterBasis G\nthis : TopologicalSpace G := B.topology\nbasis : (𝓝 1).HasBasis (fun V ↦ V ∈ B) id\nbasis' : (𝓝 1 ×ˢ 𝓝 1).HasBasis (fun i ↦ i.1 ∈ B ∧ i.2 ∈ B) fun i ↦ id i.1 ×ˢ id i.2\nU : Set G\n⊢ U ∈ B → ∃ ia ∈ B, ∀ x ∈ id ia, x⁻¹ ∈ id U",
... | [
"case refine_2\nG : Type u\ninst✝ : Group G\nB✝ B : GroupFilterBasis G\nthis : TopologicalSpace G := B.topology\nbasis : (𝓝 1).HasBasis (fun V ↦ V ∈ B) id\nbasis' : (𝓝 1 ×ˢ 𝓝 1).HasBasis (fun i ↦ i.1 ∈ B ∧ i.2 ∈ B) fun i ↦ id i.1 ×ˢ id i.2\nU : Set G\nU_in : U ∈ B\n⊢ ∃ ia ∈ B, ∀ x ∈ id ia, x⁻¹ ∈ id U"
] | U_in | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.FieldTheory.SeparableDegree | {
"line": 273,
"column": 6
} | {
"line": 273,
"column": 60
} | {
"line": 274,
"column": 8
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\nK✝ : Type w\ninst✝¹ : Field K✝\ninst✝ : Algebra F K✝\nH : Algebra.Transcendental F E\nι : Type v\nx : ι → E\nhx : IsTranscendenceBasis F x\nthis : Algebra.IsAlgebraic (↥(adjoin F (Set.range x))) E\ni : ι\nK : Type (max u ... | [
"F : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\nK✝ : Type w\ninst✝¹ : Field K✝\ninst✝ : Algebra F K✝\nH : Algebra.Transcendental F E\nι : Type v\nx : ι → E\nhx : IsTranscendenceBasis F x\nthis : Algebra.IsAlgebraic (↥(adjoin F (Set.range x))) E\ni : ι\nK : Type (max u v) := Fracti... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparableDegree | {
"line": 278,
"column": 4
} | {
"line": 278,
"column": 29
} | {
"line": 278,
"column": 30
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\nK✝ : Type w\ninst✝¹ : Field K✝\ninst✝ : Algebra F K✝\nH : Algebra.Transcendental F E\nι : Type v\nx : ι → E\nhx : IsTranscendenceBasis F x\nthis : Algebra.IsAlgebraic (↥(adjoin F (Set.range x))) E\ni : ι\nK : Type (max u ... | [
"F : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\nK✝ : Type w\ninst✝¹ : Field K✝\ninst✝ : Algebra F K✝\nH : Algebra.Transcendental F E\nι : Type v\nx : ι → E\nhx : IsTranscendenceBasis F x\nthis : Algebra.IsAlgebraic (↥(adjoin F (Set.range x))) E\ni : ι\nK : Type (max u v) := Fracti... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.FilterBasis | {
"line": 212,
"column": 12
} | {
"line": 212,
"column": 16
} | {
"line": 213,
"column": 4
} | [
{
"pp": "case refine_4\nG : Type u\ninst✝ : Group G\nB✝ B : GroupFilterBasis G\nthis : TopologicalSpace G := B.topology\nbasis : (𝓝 1).HasBasis (fun V ↦ V ∈ B) id\nbasis' : (𝓝 1 ×ˢ 𝓝 1).HasBasis (fun i ↦ i.1 ∈ B ∧ i.2 ∈ B) fun i ↦ id i.1 ×ˢ id i.2\nx₀ : G\nU : Set G\n⊢ U ∈ B → ∃ ia ∈ B, ∀ x ∈ id ia, x₀ * x *... | [
"case refine_4\nG : Type u\ninst✝ : Group G\nB✝ B : GroupFilterBasis G\nthis : TopologicalSpace G := B.topology\nbasis : (𝓝 1).HasBasis (fun V ↦ V ∈ B) id\nbasis' : (𝓝 1 ×ˢ 𝓝 1).HasBasis (fun i ↦ i.1 ∈ B ∧ i.2 ∈ B) fun i ↦ id i.1 ×ˢ id i.2\nx₀ : G\nU : Set G\nU_in : U ∈ B\n⊢ ∃ ia ∈ B, ∀ x ∈ id ia, x₀ * x * x₀⁻¹ ... | U_in | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.FieldTheory.SeparableDegree | {
"line": 279,
"column": 2
} | {
"line": 279,
"column": 13
} | {
"line": 279,
"column": 14
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\nK✝ : Type w\ninst✝¹ : Field K✝\ninst✝ : Algebra F K✝\nH : Algebra.Transcendental F E\nι : Type v\nx : ι → E\nhx : IsTranscendenceBasis F x\nthis : Algebra.IsAlgebraic (↥(adjoin F (Set.range x))) E\ni : ι\nK : Type (max u ... | [
"F : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\nK✝ : Type w\ninst✝¹ : Field K✝\ninst✝ : Algebra F K✝\nH : Algebra.Transcendental F E\nι : Type v\nx : ι → E\nhx : IsTranscendenceBasis F x\nthis : Algebra.IsAlgebraic (↥(adjoin F (Set.range x))) E\ni : ι\nK : Type (max u v) := Fracti... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparableDegree | {
"line": 361,
"column": 38
} | {
"line": 361,
"column": 54
} | {
"line": 361,
"column": 54
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nf : F[X]\nhf : f ≠ 0\n⊢ f.natSepDegree = f.natDegree ↔ Fintype.card ↥(f.aroots f.SplittingField).toFinset = f.natDegree",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Multiset.toFinset",
"Eq.mpr",
"congrArg",
"Finset",
"... | [
"F : Type u\ninst✝ : Field F\nf : F[X]\nhf : f ≠ 0\n⊢ f.natSepDegree = f.natDegree ↔ (f.aroots f.SplittingField).toFinset.card = f.natDegree"
] | Fintype.card_coe | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Algebra.FilterBasis | {
"line": 274,
"column": 12
} | {
"line": 274,
"column": 16
} | {
"line": 275,
"column": 4
} | [
{
"pp": "case hmul\nR✝ : Type u\ninst✝¹ : Ring R✝\nB✝ : RingFilterBasis R✝\nR : Type u\ninst✝ : Ring R\nB : RingFilterBasis R\nB' : AddGroupFilterBasis R := B.toAddGroupFilterBasis\nthis✝ : TopologicalSpace R := B'.topology\nbasis : (𝓝 0).HasBasis (fun V ↦ V ∈ B') id\nbasis' : (𝓝 0 ×ˢ 𝓝 0).HasBasis (fun i ↦ ... | [
"case hmul\nR✝ : Type u\ninst✝¹ : Ring R✝\nB✝ : RingFilterBasis R✝\nR : Type u\ninst✝ : Ring R\nB : RingFilterBasis R\nB' : AddGroupFilterBasis R := B.toAddGroupFilterBasis\nthis✝ : TopologicalSpace R := B'.topology\nbasis : (𝓝 0).HasBasis (fun V ↦ V ∈ B') id\nbasis' : (𝓝 0 ×ˢ 𝓝 0).HasBasis (fun i ↦ i.1 ∈ B' ∧ i... | U_in | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.FieldTheory.SeparableDegree | {
"line": 443,
"column": 6
} | {
"line": 444,
"column": 29
} | {
"line": 444,
"column": 30
} | [
{
"pp": "case pos.inr\nF : Type u\ninst✝ : Field F\nf g : F[X]\nh : f = 0 ∨ g = 0\nthis :\n ∀ (f g : F[X]),\n f = 0 ∨ g = 0 → f = 0 → ((f * g).natSepDegree = f.natSepDegree + g.natSepDegree ↔ f = 0 ∧ g = 0 ∨ IsCoprime f g)\nhf : ¬f = 0\n⊢ (f * g).natSepDegree = f.natSepDegree + g.natSepDegree ↔ f = 0 ∧ g = ... | [
"case pos.inr\nF : Type u\ninst✝ : Field F\nf g : F[X]\nh : f = 0 ∨ g = 0\nthis :\n ∀ (f g : F[X]),\n f = 0 ∨ g = 0 → f = 0 → ((f * g).natSepDegree = f.natSepDegree + g.natSepDegree ↔ f = 0 ∧ g = 0 ∨ IsCoprime f g)\nhf : ¬f = 0\n⊢ (f * g).natSepDegree = f.natSepDegree + g.natSepDegree ↔ f = 0 ∧ g = 0 ∨ IsCoprim... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparableDegree | {
"line": 462,
"column": 23
} | {
"line": 462,
"column": 63
} | {
"line": 462,
"column": 64
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nf g : F[X]\nh : ¬f = 0 ∧ ¬g = 0\nH : ∀ (a : AlgebraicClosure F), f ≠ 0 ∧ (aeval a) f = 0 → ∀ (b : AlgebraicClosure F), g ≠ 0 ∧ (aeval b) g = 0 → a ≠ b\nu : F[X]\nhu : Irreducible u\nx✝¹ : u ∣ f\nx✝ : u ∣ g\nv : F[X]\nhf : f = u * v\nw : F[X]\nhg : g = u * w\nx : AlgebraicCl... | [
"F : Type u\ninst✝ : Field F\nf g : F[X]\nh : ¬f = 0 ∧ ¬g = 0\nH : ∀ (a : AlgebraicClosure F), f ≠ 0 ∧ (aeval a) f = 0 → ∀ (b : AlgebraicClosure F), g ≠ 0 ∧ (aeval b) g = 0 → a ≠ b\nu : F[X]\nhu : Irreducible u\nx✝¹ : u ∣ f\nx✝ : u ∣ g\nv : F[X]\nhf : f = u * v\nw : F[X]\nhg : g = u * w\nx : AlgebraicClosure F\nhx ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparableDegree | {
"line": 463,
"column": 17
} | {
"line": 463,
"column": 57
} | {
"line": 463,
"column": 58
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nf g : F[X]\nh : ¬f = 0 ∧ ¬g = 0\nH : ∀ (a : AlgebraicClosure F), f ≠ 0 ∧ (aeval a) f = 0 → ∀ (b : AlgebraicClosure F), g ≠ 0 ∧ (aeval b) g = 0 → a ≠ b\nu : F[X]\nhu : Irreducible u\nx✝¹ : u ∣ f\nx✝ : u ∣ g\nv : F[X]\nhf : f = u * v\nw : F[X]\nhg : g = u * w\nx : AlgebraicCl... | [
"F : Type u\ninst✝ : Field F\nf g : F[X]\nh : ¬f = 0 ∧ ¬g = 0\nH : ∀ (a : AlgebraicClosure F), f ≠ 0 ∧ (aeval a) f = 0 → ∀ (b : AlgebraicClosure F), g ≠ 0 ∧ (aeval b) g = 0 → a ≠ b\nu : F[X]\nhu : Irreducible u\nx✝¹ : u ∣ f\nx✝ : u ∣ g\nv : F[X]\nhf : f = u * v\nw : F[X]\nhg : g = u * w\nx : AlgebraicClosure F\nhx ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparableDegree | {
"line": 468,
"column": 2
} | {
"line": 469,
"column": 22
} | {
"line": 469,
"column": 23
} | [
{
"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": false,
"usedConstants": [],
... | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparableDegree | {
"line": 490,
"column": 2
} | {
"line": 491,
"column": 27
} | {
"line": 491,
"column": 28
} | [
{
"pp": "case prime\nF : Type u\ninst✝ : Field F\nf : F[X]\nq n : ℕ\nhprime : Nat.Prime q\nhchar✝ : CharP F q\nthis : Fact (Nat.Prime q)\n⊢ ((expand F (q ^ n)) f).natSepDegree = f.natSepDegree",
"ppTerm": "?prime",
"assigned": true,
"usedConstants": [
"Multiset.toFinset",
"Eq.mpr",
... | [
"case prime\nF : Type u\ninst✝ : Field F\nf : F[X]\nq n : ℕ\nhprime : Nat.Prime q\nhchar✝ : CharP F q\nthis : Fact (Nat.Prime q)\n⊢ ((expand (AlgebraicClosure F) (q ^ n)) (map (algebraMap F (AlgebraicClosure F)) f)).roots.toFinset.card =\n (map (algebraMap F (AlgebraicClosure F)) f).roots.toFinset.card"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.FilterBasis | {
"line": 403,
"column": 14
} | {
"line": 403,
"column": 18
} | {
"line": 404,
"column": 6
} | [
{
"pp": "R✝ : Type u_1\nM✝ : Type u_2\ninst✝⁶ : CommRing R✝\ninst✝⁵ : TopologicalSpace R✝\ninst✝⁴ : AddCommGroup M✝\ninst✝³ : Module R✝ M✝\nB : ModuleFilterBasis R✝ M✝\nR : Type u_3\nM : Type u_4\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nBR : RingFilterBasis R\nBM : AddGroupFilterBasis ... | [
"R✝ : Type u_1\nM✝ : Type u_2\ninst✝⁶ : CommRing R✝\ninst✝⁵ : TopologicalSpace R✝\ninst✝⁴ : AddCommGroup M✝\ninst✝³ : Module R✝ M✝\nB : ModuleFilterBasis R✝ M✝\nR : Type u_3\nM : Type u_4\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nBR : RingFilterBasis R\nBM : AddGroupFilterBasis M\nsmul : ∀ ... | U_in | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.Algebra.FilterBasis | {
"line": 408,
"column": 17
} | {
"line": 408,
"column": 21
} | {
"line": 409,
"column": 6
} | [
{
"pp": "R✝ : Type u_1\nM✝ : Type u_2\ninst✝⁶ : CommRing R✝\ninst✝⁵ : TopologicalSpace R✝\ninst✝⁴ : AddCommGroup M✝\ninst✝³ : Module R✝ M✝\nB : ModuleFilterBasis R✝ M✝\nR : Type u_3\nM : Type u_4\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nBR : RingFilterBasis R\nBM : AddGroupFilterBasis ... | [
"R✝ : Type u_1\nM✝ : Type u_2\ninst✝⁶ : CommRing R✝\ninst✝⁵ : TopologicalSpace R✝\ninst✝⁴ : AddCommGroup M✝\ninst✝³ : Module R✝ M✝\nB : ModuleFilterBasis R✝ M✝\nR : Type u_3\nM : Type u_4\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nBR : RingFilterBasis R\nBM : AddGroupFilterBasis M\nsmul : ∀ ... | U_in | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.FieldTheory.SeparableDegree | {
"line": 613,
"column": 2
} | {
"line": 614,
"column": 24
} | {
"line": 614,
"column": 25
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nf : F[X]\nq : ℕ\ninst✝ : ExpChar F q\nhm : f.Monic\np : F[X]\nhM : p.Monic\nhI : Irreducible p\nhf✝ : p ∣ f\nhD : p.natSepDegree = 1\nn : ℕ\ny : F\nH✝ : n = 0 ∨ y ∉ (frobenius F q).range\nhp : p = X ^ q ^ n - C y\nhF : FiniteMultiplicity p f\nhne : multiplicity p f ≠ 0\nc ... | [
"F : Type u\ninst✝¹ : Field F\nf : F[X]\nq : ℕ\ninst✝ : ExpChar F q\nhm : f.Monic\np : F[X]\nhM : p.Monic\nhI : Irreducible p\nhf✝ : p ∣ f\nhD : p.natSepDegree = 1\nn : ℕ\ny : F\nH✝ : n = 0 ∨ y ∉ (frobenius F q).range\nhp : p = X ^ q ^ n - C y\nhF : FiniteMultiplicity p f\nhne : multiplicity p f ≠ 0\nc : F[X]\nh : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparableDegree | {
"line": 599,
"column": 81
} | {
"line": 614,
"column": 27
} | {
"line": 616,
"column": 0
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nf : F[X]\nq : ℕ\ninst✝ : ExpChar F q\nhm : f.Monic\nh : f.natSepDegree = 1\n⊢ ∃ m n y, m ≠ 0 ∧ (n = 0 ∨ y ∉ (frobenius F q).range) ∧ f = (X ^ q ^ n - C y) ^ m",
"ppTerm": "?m.73",
"assigned": true,
"usedConstants": [
"IsCoprime.pow_left",
"Iff.mpr",... | [] | by
obtain ⟨p, hM, hI, hf⟩ := exists_monic_irreducible_factor _ <| not_isUnit_of_natDegree_pos _
<| Nat.pos_of_ne_zero <| (natSepDegree_ne_zero_iff _).1 (h.symm ▸ Nat.one_ne_zero)
have hD := (h ▸ natSepDegree_le_of_dvd p f hf hm.ne_zero).antisymm <|
Nat.pos_of_ne_zero <| (natSepDegree_ne_zero_iff _).2 hI.nat... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.OpenSubgroup | {
"line": 482,
"column": 6
} | {
"line": 482,
"column": 20
} | {
"line": 483,
"column": 6
} | [
{
"pp": "G : Type u_1\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : CompactSpace G\nW : Set G\nWClopen : IsClopen W\nU V : Set G\nx✝¹ : ∃ T ∈ 𝓝 1, U * T ⊆ W\nx✝ : ∃ T ∈ 𝓝 1, V * T ⊆ W\nT₁ : Set G\nhT₁ : T₁ ∈ 𝓝 1\nmem1 : U * T₁ ⊆ W\nT₂ : Set G\nhT₂ : T₂ ∈ 𝓝 1\nmem2 : ... | [
"G : Type u_1\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : CompactSpace G\nW : Set G\nWClopen : IsClopen W\nU V : Set G\nx✝¹ : ∃ T ∈ 𝓝 1, U * T ⊆ W\nx✝ : ∃ T ∈ 𝓝 1, V * T ⊆ W\nT₁ : Set G\nhT₁ : T₁ ∈ 𝓝 1\nmem1 : U * T₁ ⊆ W\nT₂ : Set G\nhT₂ : T₂ ∈ 𝓝 1\nmem2 : V * T₂ ⊆ W\n... | rw [union_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.OpenSubgroup | {
"line": 528,
"column": 6
} | {
"line": 528,
"column": 54
} | {
"line": 528,
"column": 55
} | [
{
"pp": "case h\nG : Type u_2\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : CompactSpace G\nW : Set G\nWClopen : IsClopen W\neinW : 1 ∈ W\nV : Set G\nhV : mulInvClosureNhd V W\nx✝ : G\nha : x✝ ∈ ⋃ n, V ^ (n + 1)\nk : ℕ\nhk : x✝ ∈ V ^ (k + 1)\n⊢ x✝⁻¹ ∈ V⁻¹ ^ (k + 1)",
... | [
"case h\nG : Type u_2\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : CompactSpace G\nW : Set G\nWClopen : IsClopen W\neinW : 1 ∈ W\nV : Set G\nhV : mulInvClosureNhd V W\nx✝ : G\nha : x✝ ∈ ⋃ n, V ^ (n + 1)\nk : ℕ\nhk : x✝ ∈ V ^ (k + 1)\n⊢ x✝ ∈ V ^ (k + 1)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparableDegree | {
"line": 794,
"column": 2
} | {
"line": 794,
"column": 44
} | {
"line": 794,
"column": 45
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\nL : IntermediateField F E\ninst✝ : Algebra.IsSeparable F ↥L\nx : E\nh : x ∈ L\n⊢ IsSeparable F x",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Field.toDivisionRing",
"id",
"Divis... | [
"F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\nL : IntermediateField F E\ninst✝ : Algebra.IsSeparable F ↥L\nx : E\nh : x ∈ L\n⊢ (minpoly F x).Separable"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.SeparableDegree | {
"line": 821,
"column": 4
} | {
"line": 821,
"column": 54
} | {
"line": 821,
"column": 55
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁷ : Field F\ninst✝⁶ : Field E\ninst✝⁵ : Algebra F E\nK : Type w\ninst✝⁴ : Field K\ninst✝³ : Algebra F K\ninst✝² : Algebra E K\ninst✝¹ : IsScalarTower F E K\ninst✝ : Algebra.IsSeparable F E\nx : K\nhsep : IsSeparable E x\nf : E[X] := minpoly E x\nhf : f = minpoly E x\nE' : I... | [
"F : Type u\nE : Type v\ninst✝⁷ : Field F\ninst✝⁶ : Field E\ninst✝⁵ : Algebra F E\nK : Type w\ninst✝⁴ : Field K\ninst✝³ : Algebra F K\ninst✝² : Algebra E K\ninst✝¹ : IsScalarTower F E K\ninst✝ : Algebra.IsSeparable F E\nx : K\nhsep : IsSeparable E x\nf : E[X] := minpoly E x\nhf : f = minpoly E x\nE' : IntermediateF... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.KrullTopology | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 13
} | {
"line": 271,
"column": 14
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Algebra.IsIntegral K L\nx : L\nE : IntermediateField K L := K⟮x⟯\nhL : FiniteDimensional K ↥E\ng : Gal(L/K)\n⊢ g ∈ MulAction.stabilizer Gal(L/K) x ↔ g ∈ E.fixingSubgroup",
"ppTerm": "?m.169",
"assigned... | [
"K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : Algebra.IsIntegral K L\nx : L\nE : IntermediateField K L := K⟮x⟯\nhL : FiniteDimensional K ↥E\ng : Gal(L/K)\n⊢ g x = x ↔ ∀ x ∈ E, g x = x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.KrullTopology | {
"line": 298,
"column": 4
} | {
"line": 298,
"column": 47
} | {
"line": 300,
"column": 0
} | [
{
"pp": "case mpr\nk : Type u_1\nE : Type u_2\nK : Type u_3\ninst✝⁷ : Field k\ninst✝⁶ : Field E\ninst✝⁵ : Field K\ninst✝⁴ : Algebra k E\ninst✝³ : Algebra k K\ninst✝² : Algebra E K\ninst✝¹ : IsScalarTower k E K\nL : IntermediateField k E\ninst✝ : Normal k E\nf : Gal(K/k)\nx : E\nhx : x ∈ ↑L.toSubsemiring\nh : (a... | [] | rwa [AlgEquiv.restrictNormal_commutes] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.FieldTheory.Galois.GaloisClosure | {
"line": 56,
"column": 2
} | {
"line": 56,
"column": 29
} | {
"line": 56,
"column": 30
} | [
{
"pp": "k : Type u_1\nK : Type u_2\ninst✝² : Field k\ninst✝¹ : Field K\ninst✝ : Algebra k K\ntoIntermediateField✝¹ : IntermediateField k K\nfiniteDimensional✝¹ : FiniteDimensional k ↥toIntermediateField✝¹\nisGalois✝¹ : IsGalois k ↥toIntermediateField✝¹\ntoIntermediateField✝ : IntermediateField k K\nfiniteDimen... | [
"k : Type u_1\nK : Type u_2\ninst✝² : Field k\ninst✝¹ : Field K\ninst✝ : Algebra k K\ntoIntermediateField✝¹ : IntermediateField k K\nfiniteDimensional✝¹ : FiniteDimensional k ↥toIntermediateField✝¹\nisGalois✝¹ : IsGalois k ↥toIntermediateField✝¹\ntoIntermediateField✝ : IntermediateField k K\nfiniteDimensional✝ : Fi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.Galois.GaloisClosure | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 40
} | {
"line": 143,
"column": 41
} | [
{
"pp": "k : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\ninst✝ : IsGalois k K\nf : K →ₐ[k] K\nx : K\n⊢ adjoin k {f x} = adjoin k {x}",
"ppTerm": "?m.31",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"k : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\ninst✝ : IsGalois k K\nf : K →ₐ[k] K\nx : K\n⊢ adjoin k {f x} = adjoin k {x}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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