module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.GroupTheory.FiniteAbelian.Duality | {
"line": 37,
"column": 4
} | {
"line": 38,
"column": 11
} | {
"line": 38,
"column": 12
} | [
{
"pp": "ι : Type u_1\nG : Type u_2\ninst✝ : Monoid G\nn : ι → ℕ\ne : G ≃* ((i : ι) → Multiplicative (ZMod (n i)))\ni : ι\n⊢ n i = orderOf (e.symm (Pi.mulSingle i (Multiplicative.ofAdd 1)))",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulEquiv.instEquivLike",
... | [
"ι : Type u_1\nG : Type u_2\ninst✝ : Monoid G\nn : ι → ℕ\ne : G ≃* ((i : ι) → Multiplicative (ZMod (n i)))\ni : ι\n⊢ n i = addOrderOf 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FiniteAbelian.Duality | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 60
} | {
"line": 76,
"column": 4
} | [
{
"pp": "G : Type u_1\nM : Type u_2\ninst✝² : CommGroup G\ninst✝¹ : Finite G\ninst✝ : CommMonoid M\nhM : HasEnoughRootsOfUnity M (Monoid.exponent G)\ng g' : G\nh : ∀ (φ : G →* Mˣ), φ g = φ g'\n⊢ g = g'",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals":... | [
"G : Type u_1\nM : Type u_2\ninst✝² : CommGroup G\ninst✝¹ : Finite G\ninst✝ : CommMonoid M\nhM : HasEnoughRootsOfUnity M (Monoid.exponent G)\ng g' : G\nh : ∀ (φ : G →* Mˣ), φ g = φ g'\n⊢ g = g'"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FiniteAbelian.Duality | {
"line": 89,
"column": 6
} | {
"line": 90,
"column": 13
} | {
"line": 90,
"column": 14
} | [
{
"pp": "G : Type u_1\nM : Type u_2\ninst✝² : CommGroup G\ninst✝¹ : Finite G\ninst✝ : CommMonoid M\nhM : HasEnoughRootsOfUnity M (Monoid.exponent G)\nι : Type\nw✝ : Fintype ι\nn : ι → ℕ\nh₁ : ∀ (i : ι), 1 < n i\nh₂ : Nonempty (G ≃* ((i : ι) → Multiplicative (ZMod (n i))))\ne : G ≃* ((i : ι) → Multiplicative (ZM... | [
"G : Type u_1\nM : Type u_2\ninst✝² : CommGroup G\ninst✝¹ : Finite G\ninst✝ : CommMonoid M\nhM : HasEnoughRootsOfUnity M (Monoid.exponent G)\nι : Type\nw✝ : Fintype ι\nn : ι → ℕ\nh₁ : ∀ (i : ι), 1 < n i\nh₂ : Nonempty (G ≃* ((i : ι) → Multiplicative (ZMod (n i))))\ne : G ≃* ((i : ι) → Multiplicative (ZMod (n i))) :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FiniteIndexNormalSubgroup | {
"line": 133,
"column": 6
} | {
"line": 133,
"column": 17
} | {
"line": 133,
"column": 18
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nH : Type u_2\nN : Type u_3\ninst✝¹ : Group H\ninst✝ : Group N\nf : G →* H\nK : FiniteIndexNormalSubgroup H\ng : G →* H ⧸ K.toSubgroup := (QuotientGroup.mk' K.toSubgroup).comp f\n⊢ Subgroup.comap f K.toSubgroup = g.ker",
"ppTerm": "?m.51",
"assigned": false,
"... | [
"G : Type u_1\ninst✝² : Group G\nH : Type u_2\nN : Type u_3\ninst✝¹ : Group H\ninst✝ : Group N\nf : G →* H\nK : FiniteIndexNormalSubgroup H\ng : G →* H ⧸ K.toSubgroup := (QuotientGroup.mk' K.toSubgroup).comp f\n⊢ Subgroup.comap f K.toSubgroup = g.ker"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FiniteIndexNormalSubgroup | {
"line": 134,
"column": 4
} | {
"line": 134,
"column": 22
} | {
"line": 134,
"column": 23
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nH : Type u_2\nN : Type u_3\ninst✝¹ : Group H\ninst✝ : Group N\nf : G →* H\nK : FiniteIndexNormalSubgroup H\ng : G →* H ⧸ K.toSubgroup := (QuotientGroup.mk' K.toSubgroup).comp f\nhker : Subgroup.comap f K.toSubgroup = g.ker\n⊢ (Subgroup.comap f K.toSubgroup).FiniteIndex",... | [
"G : Type u_1\ninst✝² : Group G\nH : Type u_2\nN : Type u_3\ninst✝¹ : Group H\ninst✝ : Group N\nf : G →* H\nK : FiniteIndexNormalSubgroup H\ng : G →* H ⧸ K.toSubgroup := (QuotientGroup.mk' K.toSubgroup).comp f\nhker : Subgroup.comap f K.toSubgroup = g.ker\n⊢ g.ker.FiniteIndex"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.CoprodI | {
"line": 1035,
"column": 8
} | {
"line": 1035,
"column": 19
} | {
"line": 1035,
"column": 20
} | [
{
"pp": "ι : Type u_1\ninst✝² : Nontrivial ι\nG : Type u_1\ninst✝¹ : Group G\na : ι → G\nα : Type u_4\ninst✝ : MulAction G α\nX Y : ι → Set α\nhXnonempty : ∀ (i : ι), (X i).Nonempty\nhXdisj : Pairwise (Disjoint on X)\nhYdisj : Pairwise (Disjoint on Y)\nhXYdisj : ∀ (i j : ι), Disjoint (X i) (Y j)\nhX : ∀ (i : ι)... | [
"ι : Type u_1\ninst✝² : Nontrivial ι\nG : Type u_1\ninst✝¹ : Group G\na : ι → G\nα : Type u_4\ninst✝ : MulAction G α\nX Y : ι → Set α\nhXnonempty : ∀ (i : ι), (X i).Nonempty\nhXdisj : Pairwise (Disjoint on X)\nhYdisj : Pairwise (Disjoint on Y)\nhXYdisj : ∀ (i j : ι), Disjoint (X i) (Y j)\nhX : ∀ (i : ι), a i • (Y i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Transfer | {
"line": 192,
"column": 2
} | {
"line": 203,
"column": 89
} | {
"line": 205,
"column": 0
} | [
{
"pp": "case neg\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\ng : G\nkey : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g₀⁻¹ * g ^ k * g₀ = g ^ k\nhH : ¬H.index = 0\nthis : Fintype (G ⧸ H) := fintypeOfIndexNeZero hH\n⊢ g ^ H.index ∈ H",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
... | [] | classical
replace key : ∀ (k : ℕ) (g₀ : G), g₀⁻¹ * g ^ k * g₀ ∈ H → g ^ k ∈ H := fun k g₀ hk =>
(congr_arg (· ∈ H) (key k g₀ hk)).mp hk
replace key : ∀ q : G ⧸ H, g ^ Function.minimalPeriod (g • ·) q ∈ H := fun q =>
key (Function.minimalPeriod (g • ·) q) q.out
(QuotientGroup.out_conj_pow_min... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.GroupTheory.FixedPointFree | {
"line": 59,
"column": 31
} | {
"line": 59,
"column": 60
} | {
"line": 59,
"column": 61
} | [
{
"pp": "F : Type u_1\nG : Type u_2\ninst✝³ : Group G\ninst✝² : FunLike F G G\ninst✝¹ : MonoidHomClass F G G\nφ : F\ninst✝ : Finite G\nhφ : FixedPointFree ⇑φ\nh2 : (⇑φ)^[2] = _root_.id\ng : G\n⊢ g * φ g = 1",
"ppTerm": "?m.35",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGo... | [
"F : Type u_1\nG : Type u_2\ninst✝³ : Group G\ninst✝² : FunLike F G G\ninst✝¹ : MonoidHomClass F G G\nφ : F\ninst✝ : Finite G\nhφ : FixedPointFree ⇑φ\nh2 : (⇑φ)^[2] = _root_.id\ng : G\n⊢ g * φ g = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Focal | {
"line": 100,
"column": 27
} | {
"line": 100,
"column": 38
} | {
"line": 100,
"column": 39
} | [
{
"pp": "case mul\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nn g : G\nhg : g ∈ H\nx✝ y✝ : G\nhx✝ : x✝ ∈ closure {g | g ∈ H ∧ ∃ x ∈ H, ∃ u, g = ⁅x, u⁆}\nhy✝ : y✝ ∈ closure {g | g ∈ H ∧ ∃ x ∈ H, ∃ u, g = ⁅x, u⁆}\nIHa : g * x✝ * g⁻¹ ∈ H.focalSubgroup\nIHb : g * y✝ * g⁻¹ ∈ H.focalSubgroup\n⊢ g * (x✝ * y✝) * g⁻... | [
"case mul\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nn g : G\nhg : g ∈ H\nx✝ y✝ : G\nhx✝ : x✝ ∈ closure {g | g ∈ H ∧ ∃ x ∈ H, ∃ u, g = ⁅x, u⁆}\nhy✝ : y✝ ∈ closure {g | g ∈ H ∧ ∃ x ∈ H, ∃ u, g = ⁅x, u⁆}\nIHa : g * x✝ * g⁻¹ ∈ H.focalSubgroup\nIHb : g * y✝ * g⁻¹ ∈ H.focalSubgroup\n⊢ g * (x✝ * y✝) * g⁻¹ ∈ H.focalS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Focal | {
"line": 101,
"column": 18
} | {
"line": 101,
"column": 41
} | {
"line": 101,
"column": 42
} | [
{
"pp": "case inv\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nn g : G\nhg : g ∈ H\nx✝ : G\nhx✝ : x✝ ∈ closure {g | g ∈ H ∧ ∃ x ∈ H, ∃ u, g = ⁅x, u⁆}\nIH : g * x✝ * g⁻¹ ∈ H.focalSubgroup\n⊢ g * x✝⁻¹ * g⁻¹ ∈ H.focalSubgroup",
"ppTerm": "?inv",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case inv\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nn g : G\nhg : g ∈ H\nx✝ : G\nhx✝ : x✝ ∈ closure {g | g ∈ H ∧ ∃ x ∈ H, ∃ u, g = ⁅x, u⁆}\nIH : g * x✝ * g⁻¹ ∈ H.focalSubgroup\n⊢ g * (x✝⁻¹ * g⁻¹) ∈ H.focalSubgroup"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Focal | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 13
} | {
"line": 198,
"column": 14
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\nP : Sylow p G\ninst✝ : (↑P).FiniteIndex\ng : ↥P\nhQ : IsPGroup p (↥↑P ⧸ (↑P).focalSubgroupOf)\n⊢ ↑g ^ (↑P).index = 1 ↔ g ∈ (↑P).focalSubgroupOf",
"ppTerm": "?m.68",
"assigned": false,
"usedConstants": [],
"usedFVars": [... | [
"G : Type u_1\ninst✝² : Group G\np : ℕ\ninst✝¹ : Fact (Nat.Prime p)\nP : Sylow p G\ninst✝ : (↑P).FiniteIndex\ng : ↥P\nhQ : IsPGroup p (↥↑P ⧸ (↑P).focalSubgroupOf)\n⊢ ↑g ^ (↑P).index = 1 ↔ g ∈ (↑P).focalSubgroupOf"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Order.Radical | {
"line": 50,
"column": 4
} | {
"line": 50,
"column": 15
} | {
"line": 50,
"column": 16
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝¹ : CompleteLattice α\ninst✝ : IsCoatomic α\na : α\nh : a ⊔ radical α = ⊤\nm : α\nc : IsCoatom m\nle : a ≤ m\nq : m = ⊤\n⊢ a = ⊤",
"ppTerm": "?inr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case inr\nα : Type u_1\ninst✝¹ : CompleteLattice α\ninst✝ : IsCoatomic α\na : α\nh : a ⊔ radical α = ⊤\nm : α\nc : IsCoatom m\nle : a ≤ m\nq : m = ⊤\n⊢ a = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeGroup.CyclicallyReduced | {
"line": 76,
"column": 16
} | {
"line": 76,
"column": 39
} | {
"line": 76,
"column": 40
} | [
{
"pp": "α : Type u\nL : List (α × Bool)\nn✝ n : ℕ\nhead : α × Bool\ntail : List (α × Bool)\nh : IsCyclicallyReduced (head :: tail)\n⊢ ∀ l ∈ replicate (n + 1) (head :: tail), IsChain (fun a b ↦ a.1 = b.1 → a.2 = b.2) l",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"α : Type u\nL : List (α × Bool)\nn✝ n : ℕ\nhead : α × Bool\ntail : List (α × Bool)\nh : IsCyclicallyReduced (head :: tail)\n⊢ IsChain (fun a b ↦ a.1 = b.1 → a.2 = b.2) (head :: tail)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeGroup.CyclicallyReduced | {
"line": 142,
"column": 19
} | {
"line": 142,
"column": 30
} | {
"line": 142,
"column": 31
} | [
{
"pp": "α : Type u\nL : List (α × Bool)\ninst✝ : DecidableEq α\na : α × Bool\nl : List (α × Bool)\nb : α × Bool\nih : IsReduced l → IsCyclicallyReduced (reduceCyclically l)\nh : IsReduced (a :: (l ++ [b]))\nh' : ¬(b.1 = a.1 ∧ (!b.2) = a.2)\n⊢ b.1 = a.1 → b.2 = a.2",
"ppTerm": "?m.71",
"assigned": false... | [
"α : Type u\nL : List (α × Bool)\ninst✝ : DecidableEq α\na : α × Bool\nl : List (α × Bool)\nb : α × Bool\nih : IsReduced l → IsCyclicallyReduced (reduceCyclically l)\nh : IsReduced (a :: (l ++ [b]))\nh' : ¬(b.1 = a.1 ∧ (!b.2) = a.2)\n⊢ b.1 = a.1 → b.2 = a.2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Nilpotent | {
"line": 232,
"column": 12
} | {
"line": 232,
"column": 50
} | {
"line": 232,
"column": 51
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nH : Type u_2\ninst✝ : Group H\ne : H ≃* G\n⊢ comap (↑e) (upperCentralSeries G 0) = upperCentralSeries H 0",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulEquiv.instEquivLike",
"_private.Mathlib.GroupTheory.Nilpotent.0... | [
"G : Type u_1\ninst✝¹ : Group G\nH : Type u_2\ninst✝ : Group H\ne : H ≃* G\n⊢ Function.Injective ⇑e"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeGroup.CyclicallyReduced | {
"line": 232,
"column": 6
} | {
"line": 232,
"column": 78
} | {
"line": 232,
"column": 79
} | [
{
"pp": "α : Type u\nL L₁ L₂ L₃ : List (α × Bool)\nn : ℕ\nhn : n ≠ 0\nx y : FreeGroup α\nheq : (fun a ↦ a ^ n) x = (fun a ↦ a ^ n) y\nf : FreeGroup α → ℕ → ℕ :=\n fun a n ↦ (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length + (conjugator a.toWord).length)\ng : FreeGroup α → ℕ → List (α × Bo... | [
"α : Type u\nL L₁ L₂ L₃ : List (α × Bool)\nn : ℕ\nhn : n ≠ 0\nx y : FreeGroup α\nheq : (fun a ↦ a ^ n) x = (fun a ↦ a ^ n) y\nf : FreeGroup α → ℕ → ℕ :=\n fun a n ↦ (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length + (conjugator a.toWord).length)\ng : FreeGroup α → ℕ → List (α × Bool) :=\n fu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeGroup.CyclicallyReduced | {
"line": 234,
"column": 6
} | {
"line": 234,
"column": 78
} | {
"line": 234,
"column": 79
} | [
{
"pp": "α : Type u\nL L₁ L₂ L₃ : List (α × Bool)\nn : ℕ\nhn : n ≠ 0\nx y : FreeGroup α\nf : FreeGroup α → ℕ → ℕ :=\n fun a n ↦ (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length + (conjugator a.toWord).length)\ng : FreeGroup α → ℕ → List (α × Bool) :=\n fun a k ↦ conjugator a.toWord ++ ((... | [
"α : Type u\nL L₁ L₂ L₃ : List (α × Bool)\nn : ℕ\nhn : n ≠ 0\nx y : FreeGroup α\nf : FreeGroup α → ℕ → ℕ :=\n fun a n ↦ (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length + (conjugator a.toWord).length)\ng : FreeGroup α → ℕ → List (α × Bool) :=\n fun a k ↦ conjugator a.toWord ++ ((replicate k ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeGroup.CyclicallyReduced | {
"line": 235,
"column": 35
} | {
"line": 235,
"column": 50
} | {
"line": 235,
"column": 51
} | [
{
"pp": "α : Type u\nL L₁ L₂ L₃ : List (α × Bool)\nn : ℕ\nhn : n ≠ 0\nx y : FreeGroup α\nf : FreeGroup α → ℕ → ℕ :=\n fun a n ↦ (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length + (conjugator a.toWord).length)\ng : FreeGroup α → ℕ → List (α × Bool) :=\n fun a k ↦ conjugator a.toWord ++ ((... | [
"α : Type u\nL L₁ L₂ L₃ : List (α × Bool)\nn : ℕ\nhn : n ≠ 0\nx y : FreeGroup α\nf : FreeGroup α → ℕ → ℕ :=\n fun a n ↦ (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length + (conjugator a.toWord).length)\ng : FreeGroup α → ℕ → List (α × Bool) :=\n fun a k ↦ conjugator a.toWord ++ ((replicate k ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeGroup.CyclicallyReduced | {
"line": 236,
"column": 48
} | {
"line": 236,
"column": 63
} | {
"line": 236,
"column": 64
} | [
{
"pp": "α : Type u\nL L₁ L₂ L₃ : List (α × Bool)\nn : ℕ\nhn : n ≠ 0\nx y : FreeGroup α\nf : FreeGroup α → ℕ → ℕ :=\n fun a n ↦ (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length + (conjugator a.toWord).length)\ng : FreeGroup α → ℕ → List (α × Bool) :=\n fun a k ↦ conjugator a.toWord ++ ((... | [
"α : Type u\nL L₁ L₂ L₃ : List (α × Bool)\nn : ℕ\nhn : n ≠ 0\nx y : FreeGroup α\nf : FreeGroup α → ℕ → ℕ :=\n fun a n ↦ (conjugator a.toWord).length + (n * (reduceCyclically a.toWord).length + (conjugator a.toWord).length)\ng : FreeGroup α → ℕ → List (α × Bool) :=\n fun a k ↦ conjugator a.toWord ++ ((replicate k ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeGroup.Orbit | {
"line": 46,
"column": 2
} | {
"line": 46,
"column": 13
} | {
"line": 46,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\na b : α × Bool\nh : ∀ (x : FreeGroup α), x.toWord[0]? = some a ↔ x.toWord[0]? = some b\n⊢ a = b",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝ : DecidableEq α\na b : α × Bool\nh : ∀ (x : FreeGroup α), x.toWord[0]? = some a ↔ x.toWord[0]? = some b\n⊢ a = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeGroup.NielsenSchreier | {
"line": 177,
"column": 2
} | {
"line": 177,
"column": 33
} | {
"line": 179,
"column": 0
} | [
{
"pp": "G : Type u\ninst✝² : Groupoid G\ninst✝¹ : IsFreeGroupoid G\nT : WideSubquiver (Symmetrify (Generators G))\ninst✝ : Arborescence (WideSubquiver.toType (Symmetrify (Generators G)) T)\na : G\np : Path (root (WideSubquiver.toType (Symmetrify (Generators G)) T)) a\n⊢ treeHom T a = homOfPath T p",
"ppTer... | [] | rw [treeHom, Unique.default_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.FreeGroup.NielsenSchreier | {
"line": 177,
"column": 2
} | {
"line": 177,
"column": 33
} | {
"line": 179,
"column": 0
} | [
{
"pp": "G : Type u\ninst✝² : Groupoid G\ninst✝¹ : IsFreeGroupoid G\nT : WideSubquiver (Symmetrify (Generators G))\ninst✝ : Arborescence (WideSubquiver.toType (Symmetrify (Generators G)) T)\na : G\np : Path (root (WideSubquiver.toType (Symmetrify (Generators G)) T)) a\n⊢ treeHom T a = homOfPath T p",
"ppTer... | [] | rw [treeHom, Unique.default_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.FreeGroup.NielsenSchreier | {
"line": 177,
"column": 2
} | {
"line": 177,
"column": 33
} | {
"line": 179,
"column": 0
} | [
{
"pp": "G : Type u\ninst✝² : Groupoid G\ninst✝¹ : IsFreeGroupoid G\nT : WideSubquiver (Symmetrify (Generators G))\ninst✝ : Arborescence (WideSubquiver.toType (Symmetrify (Generators G)) T)\na : G\np : Path (root (WideSubquiver.toType (Symmetrify (Generators G)) T)) a\n⊢ treeHom T a = homOfPath T p",
"ppTer... | [] | rw [treeHom, Unique.default_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.FreeGroup.Orbit | {
"line": 84,
"column": 38
} | {
"line": 84,
"column": 68
} | {
"line": 84,
"column": 69
} | [
{
"pp": "α : Type u_1\nX : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : MulAction (FreeGroup α) X\nx : X\nw : α × Bool\ng : FreeGroup α\nhg : g ∈ startsWith w\nl : List (α × Bool) := g.toWord\nh : ⟨g, hg⟩ = ⟨mk g.toWord, ⋯⟩\na : α × Bool\nhl : [a] = g.toWord\n⊢ a = w",
"ppTerm": "?m.189",
"assigned": false... | [
"α : Type u_1\nX : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : MulAction (FreeGroup α) X\nx : X\nw : α × Bool\ng : FreeGroup α\nhg : g ∈ startsWith w\nl : List (α × Bool) := g.toWord\nh : ⟨g, hg⟩ = ⟨mk g.toWord, ⋯⟩\na : α × Bool\nhl : [a] = g.toWord\n⊢ a = w"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeGroup.Orbit | {
"line": 88,
"column": 28
} | {
"line": 88,
"column": 58
} | {
"line": 88,
"column": 59
} | [
{
"pp": "α : Type u_1\nX : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : MulAction (FreeGroup α) X\nx : X\nw : α × Bool\ng : FreeGroup α\nhg : g ∈ startsWith w\nl✝ : List (α × Bool) := g.toWord\nh : ⟨g, hg⟩ = ⟨mk g.toWord, ⋯⟩\na b : α × Bool\nl : List (α × Bool)\nhl : a :: b :: l = g.toWord\n⊢ a = w",
"ppTerm":... | [
"α : Type u_1\nX : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : MulAction (FreeGroup α) X\nx : X\nw : α × Bool\ng : FreeGroup α\nhg : g ∈ startsWith w\nl✝ : List (α × Bool) := g.toWord\nh : ⟨g, hg⟩ = ⟨mk g.toWord, ⋯⟩\na b : α × Bool\nl : List (α × Bool)\nhl : a :: b :: l = g.toWord\n⊢ a = w"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Nilpotent | {
"line": 651,
"column": 2
} | {
"line": 651,
"column": 13
} | {
"line": 651,
"column": 14
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nS : Subgroup G\nn : ℕ\n⊢ S.lowerCentralSeries n ≤ S",
"ppTerm": "?m.9",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝ : Group G\nS : Subgroup G\nn : ℕ\n⊢ S.lowerCentralSeries n ≤ S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.FreeGroup.NielsenSchreier | {
"line": 249,
"column": 8
} | {
"line": 249,
"column": 17
} | {
"line": 250,
"column": 8
} | [
{
"pp": "case refine_1\nG : Type u\ninst✝³ : Groupoid G\ninst✝² : IsFreeGroupoid G\nT : WideSubquiver (Symmetrify (Generators G))\ninst✝¹ : Arborescence (WideSubquiver.toType (Symmetrify (Generators G)) T)\nX : Type u\ninst✝ : Group X\nf : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ → X\nf' : Lab... | [
"case refine_1\nG : Type u\ninst✝³ : Groupoid G\ninst✝² : IsFreeGroupoid G\nT : WideSubquiver (Symmetrify (Generators G))\ninst✝¹ : Arborescence (WideSubquiver.toType (Symmetrify (Generators G)) T)\nX : Type u\ninst✝ : Group X\nf : ↑(wideSubquiverEquivSetTotal (wideSubquiverSymmetrify T))ᶜ → X\nf' : Labelling (Gene... | intro a p | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.GroupTheory.Goursat | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 44
} | {
"line": 80,
"column": 45
} | [
{
"pp": "case mp\nG : Type u_1\nH : Type u_2\ninst✝¹ : Group G\ninst✝ : Group H\nI : Subgroup (G × H)\nhI₁ : Surjective (Prod.fst ∘ ⇑I.subtype)\nhI₂ : Surjective (Prod.snd ∘ ⇑I.subtype)\nx y : G × H\nhx : x ∈ I\nhy : y ∈ I\nthis✝ : I.goursatFst.Normal\nthis : I.goursatSnd.Normal\nh : (y.1 / x.1, 1) ∈ I\n⊢ (1, x... | [
"case mp\nG : Type u_1\nH : Type u_2\ninst✝¹ : Group G\ninst✝ : Group H\nI : Subgroup (G × H)\nhI₁ : Surjective (Prod.fst ∘ ⇑I.subtype)\nhI₂ : Surjective (Prod.snd ∘ ⇑I.subtype)\nx y : G × H\nhx : x ∈ I\nhy : y ∈ I\nthis✝ : I.goursatFst.Normal\nthis : I.goursatSnd.Normal\nh : (y.1 / x.1, 1) ∈ I\n⊢ (1, x.2 / y.2) ∈ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Goursat | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 44
} | {
"line": 81,
"column": 45
} | [
{
"pp": "case mpr\nG : Type u_1\nH : Type u_2\ninst✝¹ : Group G\ninst✝ : Group H\nI : Subgroup (G × H)\nhI₁ : Surjective (Prod.fst ∘ ⇑I.subtype)\nhI₂ : Surjective (Prod.snd ∘ ⇑I.subtype)\nx y : G × H\nhx : x ∈ I\nhy : y ∈ I\nthis✝ : I.goursatFst.Normal\nthis : I.goursatSnd.Normal\nh : (1, x.2 / y.2) ∈ I\n⊢ (y.1... | [
"case mpr\nG : Type u_1\nH : Type u_2\ninst✝¹ : Group G\ninst✝ : Group H\nI : Subgroup (G × H)\nhI₁ : Surjective (Prod.fst ∘ ⇑I.subtype)\nhI₂ : Surjective (Prod.snd ∘ ⇑I.subtype)\nx y : G × H\nhx : x ∈ I\nhy : y ∈ I\nthis✝ : I.goursatFst.Normal\nthis : I.goursatSnd.Normal\nh : (1, x.2 / y.2) ∈ I\n⊢ (y.1 / x.1, 1) ∈... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Goursat | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 13
} | {
"line": 87,
"column": 14
} | [
{
"pp": "G : Type u_1\nH : Type u_2\ninst✝¹ : Group G\ninst✝ : Group H\nI : Subgroup (G × H)\ng : G\nh : H\nhg : (g, h).1 ∈ ↑I.goursatFst.toSubmonoid\nhh : (g, h).2 ∈ ↑I.goursatSnd.toSubmonoid\n⊢ (g, h) ∈ I",
"ppTerm": "?m.53",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGo... | [
"G : Type u_1\nH : Type u_2\ninst✝¹ : Group G\ninst✝ : Group H\nI : Subgroup (G × H)\ng : G\nh : H\nhg : (g, h).1 ∈ ↑I.goursatFst.toSubmonoid\nhh : (g, h).2 ∈ ↑I.goursatSnd.toSubmonoid\n⊢ (g, h) ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Nilpotent | {
"line": 776,
"column": 4
} | {
"line": 779,
"column": 62
} | {
"line": 781,
"column": 0
} | [] | [] | ⊤ = f.range := symm (f.range_eq_top_of_surjective hf)
_ = Subgroup.map f ⊤ := MonoidHom.range_eq_map _
_ = Subgroup.map f (upperCentralSeries G n) := by rw [hn]
_ ≤ upperCentralSeries G' n := upperCentralSeries.map hf n | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.GroupTheory.Nilpotent | {
"line": 794,
"column": 4
} | {
"line": 797,
"column": 62
} | {
"line": 799,
"column": 0
} | [] | [] | ⊤ = f.range := symm (f.range_eq_top_of_surjective hf)
_ = Subgroup.map f ⊤ := MonoidHom.range_eq_map _
_ = Subgroup.map f (upperCentralSeries G n) := by rw [hn]
_ ≤ upperCentralSeries G' n := upperCentralSeries.map hf n | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.GroupTheory.Goursat | {
"line": 159,
"column": 6
} | {
"line": 163,
"column": 13
} | {
"line": 163,
"column": 14
} | [
{
"pp": "case mp\nG : Type u_1\nH : Type u_2\ninst✝¹ : Group G\ninst✝ : Group H\nI : Subgroup (G × H)\nG' : Subgroup G := map (MonoidHom.fst G H) I\nH' : Subgroup H := map (MonoidHom.snd G H) I\nP : ↥I →* ↥G' := (MonoidHom.fst G H).subgroupMap I\nQ : ↥I →* ↥H' := (MonoidHom.snd G H).subgroupMap I\nI' : Subgroup... | [
"case mp\nG : Type u_1\nH : Type u_2\ninst✝¹ : Group G\ninst✝ : Group H\nI : Subgroup (G × H)\nG' : Subgroup G := map (MonoidHom.fst G H) I\nH' : Subgroup H := map (MonoidHom.snd G H) I\nP : ↥I →* ↥G' := (MonoidHom.fst G H).subgroupMap I\nQ : ↥I →* ↥H' := (MonoidHom.snd G H).subgroupMap I\nI' : Subgroup (↥G' × ↥H')... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Nilpotent | {
"line": 940,
"column": 2
} | {
"line": 940,
"column": 62
} | {
"line": 941,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\n⊢ ⊤.lowerCentralSeries 1 = ⊥ ↔ IsMulCommutative G",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Monoid.toMulOneClass",
"congrArg",
"Subgroup.upperCentralSeries",
"id",
"MulOne.toMul",
"DivInvMono... | [
"G : Type u_1\ninst✝ : Group G\n⊢ upperCentralSeries G 1 = ⊤ ↔ IsMulCommutative G"
] | rw [lowerCentralSeries_eq_bot_iff_upperCentralSeries_eq_top] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Nilpotent | {
"line": 1040,
"column": 14
} | {
"line": 1040,
"column": 25
} | {
"line": 1040,
"column": 26
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : Nontrivial G\ninst✝ : IsNilpotent G\n⊢ ⊥ ≠ center G",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"id",
"Subgroup",
"Ne",
"Subgroup.center",
"Bot.bot",
"Subgroup.instBot"
],
"usedFVars": [
... | [
"G : Type u_1\ninst✝² : Group G\ninst✝¹ : Nontrivial G\ninst✝ : IsNilpotent G\n⊢ ¬⊥ = center G"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Nilpotent | {
"line": 1187,
"column": 46
} | {
"line": 1187,
"column": 57
} | {
"line": 1187,
"column": 58
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\ninst✝ : IsNilpotent G\nih : ∀ (H : Subgroup (G ⧸ center G)), normalizer ↑H = H → H = ⊤\nH : Subgroup G\nhH : normalizer ↑H = H\nhch : center G ≤ H\n⊢ (mk' (center G)).ker ≤ H",
"ppTerm": "?m.78",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Mon... | [
"G : Type u_1\ninst✝¹ : Group G\ninst✝ : IsNilpotent G\nih : ∀ (H : Subgroup (G ⧸ center G)), normalizer ↑H = H → H = ⊤\nH : Subgroup G\nhH : normalizer ↑H = H\nhch : center G ≤ H\n⊢ center G ≤ H"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 193,
"column": 27
} | {
"line": 193,
"column": 38
} | {
"line": 193,
"column": 39
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝ : SMul G X\nB : Set X\ng₁ g₂ : G\nhB : IsBlock G B\nhg : g₁ • B ⊆ g₂ • B\nhg' : g₁ • B ≠ g₂ • B\n⊢ B = ∅",
"ppTerm": "?m.36",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\nX : Type u_2\ninst✝ : SMul G X\nB : Set X\ng₁ g₂ : G\nhB : IsBlock G B\nhg : g₁ • B ⊆ g₂ • B\nhg' : g₁ • B ≠ g₂ • B\n⊢ B = ∅"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 246,
"column": 2
} | {
"line": 246,
"column": 13
} | {
"line": 246,
"column": 14
} | [
{
"pp": "M : Type u_1\nX : Type u_2\ninst✝¹ : Monoid M\ninst✝ : MulAction M X\nB : Set X\ns : Set M\nhB : IsBlock M B\nhs : ¬B ⊆ s • B\n⊢ Disjoint B (s • B)",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_1\nX : Type u_2\ninst✝¹ : Monoid M\ninst✝ : MulAction M X\nB : Set X\ns : Set M\nhB : IsBlock M B\nhs : ¬B ⊆ s • B\n⊢ Disjoint B (s • B)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 246,
"column": 53
} | {
"line": 246,
"column": 64
} | {
"line": 246,
"column": 65
} | [
{
"pp": "M : Type u_1\nX : Type u_2\ninst✝¹ : Monoid M\ninst✝ : MulAction M X\nB : Set X\ns : Set M\nhB : IsBlock M B\nhs : ¬B ⊆ s • B\n⊢ ¬1 • B ⊆ s • B",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"instHSMul",
"Monoid.toMulOneClass",
... | [
"M : Type u_1\nX : Type u_2\ninst✝¹ : Monoid M\ninst✝ : MulAction M X\nB : Set X\ns : Set M\nhB : IsBlock M B\nhs : ¬B ⊆ s • B\n⊢ ¬B ⊆ s • B"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 261,
"column": 24
} | {
"line": 261,
"column": 35
} | {
"line": 261,
"column": 36
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nX : Type u_2\ninst✝ : MulAction G X\nB : Set X\nhB : IsBlock G B\ng : G\n⊢ g • B ≠ B → Disjoint (g • B) B",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"instHSMul",
"ChainCompletePartialOrder.instOfCompleteLattice",
"CompleteBool... | [
"G : Type u_1\ninst✝¹ : Group G\nX : Type u_2\ninst✝ : MulAction G X\nB : Set X\nhB : IsBlock G B\ng : G\n⊢ ¬g • B = B → Disjoint (g • B) B"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 417,
"column": 4
} | {
"line": 417,
"column": 44
} | {
"line": 417,
"column": 45
} | [
{
"pp": "case nonempty\nG : Type u_1\ninst✝¹ : Group G\nX : Type u_2\ninst✝ : MulAction G X\nB : Set X\nhGX : IsPretransitive G X\nhB : IsBlock G B\nhBe : B.Nonempty\ng : G\nhg : g • B = ∅\n⊢ B = ∅",
"ppTerm": "?nonempty",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals":... | [
"case nonempty\nG : Type u_1\ninst✝¹ : Group G\nX : Type u_2\ninst✝ : MulAction G X\nB : Set X\nhGX : IsPretransitive G X\nhB : IsBlock G B\nhBe : B.Nonempty\ng : G\nhg : g • B = ∅\n⊢ B = ∅"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.IndexNormal | {
"line": 51,
"column": 2
} | {
"line": 52,
"column": 36
} | {
"line": 53,
"column": 2
} | [
{
"pp": "case pos\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nhHp : H.index = (Nat.card G).minFac\nhG0 : ¬Nat.card G = 0\nhG1 : Nat.card G = 1\n⊢ H.Normal",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"congrArg",
"Subgroup.normal_of_index_eq_one",
"Nat.minFac_one",
... | [
"case neg\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nhHp : H.index = (Nat.card G).minFac\nhG0 : ¬Nat.card G = 0\nhG1 : ¬Nat.card G = 1\n⊢ H.Normal"
] | · rw [hG1, minFac_one] at hHp
exact normal_of_index_eq_one hHp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.GroupAction.Primitive | {
"line": 193,
"column": 2
} | {
"line": 193,
"column": 67
} | {
"line": 194,
"column": 2
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G X\nHnt : fixedPoints G X ≠ ⊤\nH : ∀ {B : Set X}, IsBlock G B → IsTrivialBlock B\n⊢ IsPreprimitive G X",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Exists",
"Eq.mp",
"Div... | [
"G : Type u_1\nX : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G X\nH : ∀ {B : Set X}, IsBlock G B → IsTrivialBlock B\nHnt : ∃ a, a ∉ fixedPoints G X\n⊢ IsPreprimitive G X"
] | simp only [Set.top_eq_univ, Set.ne_univ_iff_exists_notMem] at Hnt | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.GroupAction.SubMulAction.OfStabilizer | {
"line": 112,
"column": 46
} | {
"line": 112,
"column": 62
} | {
"line": 112,
"column": 63
} | [
{
"pp": "G✝ : Type u_1\ninst✝³ : Group G✝\nα✝ : Type u_2\ninst✝² : MulAction G✝ α✝\nG : Type u_3\ninst✝¹ : AddGroup G\nα : Type u_4\ninst✝ : AddAction G α\ng : G\na b : α\nhg : b = g +ᵥ a\nx : ↥(SubAddAction.ofStabilizer G a)\nhy : g +ᵥ ↑x ∈ {b}\n⊢ ↑x ∈ {a}",
"ppTerm": "?m.66",
"assigned": true,
"us... | [
"G✝ : Type u_1\ninst✝³ : Group G✝\nα✝ : Type u_2\ninst✝² : MulAction G✝ α✝\nG : Type u_3\ninst✝¹ : AddGroup G\nα : Type u_4\ninst✝ : AddAction G α\ng : G\na b : α\nhg : b = g +ᵥ a\nx : ↥(SubAddAction.ofStabilizer G a)\nhy : g +ᵥ ↑x ∈ {b}\n⊢ ↑x = a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.SubMulAction.OfStabilizer | {
"line": 122,
"column": 45
} | {
"line": 122,
"column": 61
} | {
"line": 122,
"column": 62
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nα : Type u_2\ninst✝ : MulAction G α\ng : G\na b : α\nhg : b = g • a\nx : ↥(ofStabilizer G a)\nhy : g • ↑x ∈ {b}\n⊢ ↑x ∈ {a}",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"SubMulAction.instSetLike",
"Eq.mpr",
"Membership.mem",
... | [
"G : Type u_1\ninst✝¹ : Group G\nα : Type u_2\ninst✝ : MulAction G α\ng : G\na b : α\nhg : b = g • a\nx : ↥(ofStabilizer G a)\nhy : g • ↑x ∈ {b}\n⊢ ↑x = a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.Primitive | {
"line": 333,
"column": 30
} | {
"line": 333,
"column": 59
} | {
"line": 333,
"column": 60
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝⁶ : Group G\ninst✝⁵ : MulAction G X\nH : Type u_3\nY : Type u_4\ninst✝⁴ : Group H\ninst✝³ : MulAction H Y\nφ : G → H\nf : X →ₑ[φ] Y\ninst✝² : Finite Y\ninst✝¹ : IsPretransitive H Y\ninst✝ : IsPreprimitive G X\nhf' : Nat.card Y < 2 * (Set.range ⇑f).ncard\nB : Set Y\nhB :... | [
"G : Type u_1\nX : Type u_2\ninst✝⁶ : Group G\ninst✝⁵ : MulAction G X\nH : Type u_3\nY : Type u_4\ninst✝⁴ : Group H\ninst✝³ : MulAction H Y\nφ : G → H\nf : X →ₑ[φ] Y\ninst✝² : Finite Y\ninst✝¹ : IsPretransitive H Y\ninst✝ : IsPreprimitive G X\nhf' : Nat.card Y < 2 * (Set.range ⇑f).ncard\nB : Set Y\nhB : IsBlock H B... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.SubMulAction.OfStabilizer | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 13
} | {
"line": 173,
"column": 14
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nα : Type u_2\ninst✝ : MulAction G α\ng h k : G\na b c : α\nhg : b = g • a\nhh : c = h • b\nhk : c = k • a\nH : k = h * g\nx : ↥(ofStabilizer G a)\n⊢ ↑(((conjMap hh).comp (conjMap hg)) x) = ↑((conjMap hk) x)",
"ppTerm": "?m.86",
"assigned": true,
"usedConstant... | [
"G : Type u_1\ninst✝¹ : Group G\nα : Type u_2\ninst✝ : MulAction G α\ng h k : G\na b c : α\nhg : b = g • a\nhh : c = h • b\nhk : c = k • a\nH : k = h * g\nx : ↥(ofStabilizer G a)\n⊢ (conjMap hh) ((conjMap hg) x) = (conjMap hk) x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.Primitive | {
"line": 379,
"column": 4
} | {
"line": 379,
"column": 62
} | {
"line": 379,
"column": 63
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝² : Group G\ninst✝¹ : MulAction G X\ninst✝ : IsPreprimitive G X\nA : Set X\nhfA : A.Finite\nhA : A.Nonempty\nhA' : A ≠ Set.univ\na b : X\nh : a ≠ b\nB : Set X := ⋂ g, ⋂ (_ : a ∈ g • A), g • A\nthis : ¬∀ (i : G), b ∈ ⋂ (_ : a ∈ i • A), i • A\n⊢ ∃ g, a ∈ g • A ∧ b ∉ g • A... | [
"G : Type u_1\nX : Type u_2\ninst✝² : Group G\ninst✝¹ : MulAction G X\ninst✝ : IsPreprimitive G X\nA : Set X\nhfA : A.Finite\nhA : A.Nonempty\nhA' : A ≠ Set.univ\na b : X\nh : a ≠ b\nB : Set X := ⋂ g, ⋂ (_ : a ∈ g • A), g • A\nthis : ¬∀ (i : G), b ∈ ⋂ (_ : a ∈ i • A), i • A\n⊢ ∃ g, a ∈ g • A ∧ b ∉ g • A"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Embedding | {
"line": 46,
"column": 45
} | {
"line": 60,
"column": 80
} | {
"line": 62,
"column": 0
} | [
{
"pp": "α : Type u_1\nn : ℕ\ns : Set α\ninst✝ : Finite ↑s\nhs : ↑s.ncard + ↑n ≤ ENat.card α\n⊢ ∃ y, Disjoint s (range ⇑y)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Fintype.card_fin",
"Fintype.ofFinite",
"Subtype.coe_prop",
"Nat.instIsOrderedA... | [] | by
rsuffices ⟨y⟩ : Nonempty (Fin n ↪ (sᶜ : Set α))
· use y.trans (subtype _)
rw [Set.disjoint_right]
rintro _ ⟨i, rfl⟩
simpa only [← mem_compl_iff] using! Subtype.coe_prop (y i)
rcases finite_or_infinite α with hα | hα
· let _ : Fintype α := Fintype.ofFinite α
classical
apply nonempty_of_car... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 394,
"column": 2
} | {
"line": 394,
"column": 77
} | {
"line": 394,
"column": 78
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nH : IsMulCommutative ↥(alternatingGroup α)\nh : 3 < Nat.card α\n⊢ Subsingleton ↥(alternatingGroup α)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.instFunLike",
"MonoidHom",
"Mo... | [
"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nH : IsMulCommutative ↥(alternatingGroup α)\nh : 3 < Nat.card α\n⊢ Subsingleton { x // 1 = sign x }"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.VectorBundle.Riemannian | {
"line": 394,
"column": 6
} | {
"line": 394,
"column": 17
} | {
"line": 394,
"column": 18
} | [
{
"pp": "case inr\nB✝ : Type u_1\ninst✝¹⁸ : TopologicalSpace B✝\nF✝ : Type u_2\ninst✝¹⁷ : NormedAddCommGroup F✝\ninst✝¹⁶ : NormedSpace ℝ F✝\nE✝ : B✝ → Type u_3\ninst✝¹⁵ : TopologicalSpace (TotalSpace F✝ E✝)\ninst✝¹⁴ : (x : B✝) → NormedAddCommGroup (E✝ x)\ninst✝¹³ : (x : B✝) → InnerProductSpace ℝ (E✝ x)\ninst✝¹²... | [
"case inr\nB✝ : Type u_1\ninst✝¹⁸ : TopologicalSpace B✝\nF✝ : Type u_2\ninst✝¹⁷ : NormedAddCommGroup F✝\ninst✝¹⁶ : NormedSpace ℝ F✝\nE✝ : B✝ → Type u_3\ninst✝¹⁵ : TopologicalSpace (TotalSpace F✝ E✝)\ninst✝¹⁴ : (x : B✝) → NormedAddCommGroup (E✝ x)\ninst✝¹³ : (x : B✝) → InnerProductSpace ℝ (E✝ x)\ninst✝¹² : FiberBund... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.MultipleTransitivity | {
"line": 357,
"column": 2
} | {
"line": 381,
"column": 18
} | {
"line": 383,
"column": 0
} | [
{
"pp": "G : Type u_1\nα : Type u_2\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : IsPretransitive G α\nn : ℕ\na : α\n⊢ IsMultiplyPretransitive G α n.succ ↔ IsMultiplyPretransitive (↥(stabilizer G a)) (↥(ofStabilizer G a)) n",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"SubMul... | [] | refine ⟨fun hn ↦ ⟨fun x y ↦ ?_⟩, fun hn ↦ ⟨fun x y ↦ ?_⟩⟩
· obtain ⟨g, hgxy⟩ := exists_smul_eq G (ofStabilizer.snoc x) (ofStabilizer.snoc y)
have hg : g ∈ stabilizer G a := by
rw [DFunLike.ext_iff] at hgxy
convert! hgxy (last n)
simp [ofStabilizer.snoc_last]
use ⟨g, hg⟩
ext i
simp on... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupAction.MultipleTransitivity | {
"line": 357,
"column": 2
} | {
"line": 381,
"column": 18
} | {
"line": 383,
"column": 0
} | [
{
"pp": "G : Type u_1\nα : Type u_2\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : IsPretransitive G α\nn : ℕ\na : α\n⊢ IsMultiplyPretransitive G α n.succ ↔ IsMultiplyPretransitive (↥(stabilizer G a)) (↥(ofStabilizer G a)) n",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"SubMul... | [] | refine ⟨fun hn ↦ ⟨fun x y ↦ ?_⟩, fun hn ↦ ⟨fun x y ↦ ?_⟩⟩
· obtain ⟨g, hgxy⟩ := exists_smul_eq G (ofStabilizer.snoc x) (ofStabilizer.snoc y)
have hg : g ∈ stabilizer G a := by
rw [DFunLike.ext_iff] at hgxy
convert! hgxy (last n)
simp [ofStabilizer.snoc_last]
use ⟨g, hg⟩
ext i
simp on... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.MultiplePrimitivity | {
"line": 132,
"column": 6
} | {
"line": 132,
"column": 17
} | {
"line": 132,
"column": 18
} | [
{
"pp": "case mpr.right\nM : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\nh : IsPreprimitive M α\n⊢ ∀ {s : Set α}, s.encard + 1 = ↑1 → IsPreprimitive ↥(fixingSubgroup M s) ↥(ofFixingSubgroup M s)",
"ppTerm": "?mpr.right",
"assigned": true,
"usedConstants": [
"SubMulAction.i... | [
"case mpr.right\nM : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\nh : IsPreprimitive M α\n⊢ IsPreprimitive ↥(fixingSubgroup M ∅) ↥(ofFixingSubgroup M ∅)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup | {
"line": 204,
"column": 4
} | {
"line": 204,
"column": 75
} | {
"line": 204,
"column": 76
} | [
{
"pp": "case left\nM : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\na : α\ns : Set ↥(ofStabilizer M a)\nx : α\nhx : x ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))\ny : α\nhy : y ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))\nh : (ofFixingSubgroup_insert_map a s) ⟨x, hx⟩ = (ofFixi... | [
"case left\nM : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\na : α\ns : Set ↥(ofStabilizer M a)\nx : α\nhx : x ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))\ny : α\nhy : y ∈ ofFixingSubgroup M (insert a (Subtype.val '' s))\nh : (ofFixingSubgroup_insert_map a s) ⟨x, hx⟩ = (ofFixingSubgroup_i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup | {
"line": 217,
"column": 2
} | {
"line": 222,
"column": 10
} | {
"line": 224,
"column": 0
} | [
{
"pp": "M : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns t : Set α\ng : M\nhg : g • t = s\nk : M\nhk : k ∈ fixingSubgroup M t\n⊢ (MulAut.conj g) k ∈ fixingSubgroup M s",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"Div... | [] | simp only [mem_fixingSubgroup_iff] at hk ⊢
intro y hy
rw [MulAut.conj_apply, eq_comm, mul_smul, mul_smul, ← inv_smul_eq_iff, eq_comm]
apply hk
rw [← Set.mem_smul_set_iff_inv_smul_mem, hg]
exact hy | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup | {
"line": 217,
"column": 2
} | {
"line": 222,
"column": 10
} | {
"line": 224,
"column": 0
} | [
{
"pp": "M : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns t : Set α\ng : M\nhg : g • t = s\nk : M\nhk : k ∈ fixingSubgroup M t\n⊢ (MulAut.conj g) k ∈ fixingSubgroup M s",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"Div... | [] | simp only [mem_fixingSubgroup_iff] at hk ⊢
intro y hy
rw [MulAut.conj_apply, eq_comm, mul_smul, mul_smul, ← inv_smul_eq_iff, eq_comm]
apply hk
rw [← Set.mem_smul_set_iff_inv_smul_mem, hg]
exact hy | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup | {
"line": 289,
"column": 4
} | {
"line": 289,
"column": 38
} | {
"line": 289,
"column": 39
} | [
{
"pp": "case left\nM : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns t : Set α\ng : M\nhst : g • s = t\nx y : ↥(ofFixingSubgroup M s)\nhxy : (conjMap_ofFixingSubgroup hst) x = (conjMap_ofFixingSubgroup hst) y\n⊢ x = y",
"ppTerm": "?left",
"assigned": true,
"usedConstants": [
... | [
"case left\nM : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns t : Set α\ng : M\nhst : g • s = t\nx y : ↥(ofFixingSubgroup M s)\nhxy : (conjMap_ofFixingSubgroup hst) x = (conjMap_ofFixingSubgroup hst) y\n⊢ ↑x = ↑y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup | {
"line": 345,
"column": 10
} | {
"line": 345,
"column": 61
} | {
"line": 345,
"column": 62
} | [
{
"pp": "M : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns✝ s t : Set α\nx : ↥(ofFixingSubgroup M (s ∪ t))\nhx : ⟨↑x, ⋯⟩ ∈ Subtype.val ⁻¹' t\n⊢ ↑x ∈ t",
"ppTerm": "?m.162",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns✝ s t : Set α\nx : ↥(ofFixingSubgroup M (s ∪ t))\nhx : ⟨↑x, ⋯⟩ ∈ Subtype.val ⁻¹' t\n⊢ ↑x ∈ t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.MultipleTransitivity | {
"line": 502,
"column": 2
} | {
"line": 502,
"column": 56
} | {
"line": 503,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nα : Type u_2\ninst✝¹ : MulAction G α\ninst✝ : Finite α\ns : Set α\nhMk : IsMultiplyPretransitive G α s.ncard\n⊢ (fixingSubgroup G s).index = (Nat.card α).choose s.ncard * s.ncard !",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Nat.choose",
... | [
"G : Type u_1\ninst✝² : Group G\nα : Type u_2\ninst✝¹ : MulAction G α\ninst✝ : Finite α\ns : Set α\nhMk : IsMultiplyPretransitive G α s.ncard\n⊢ (fixingSubgroup G s).index * ?m.21! = (Nat.card α).choose s.ncard * s.ncard ! * ?m.21!",
"G : Type u_1\ninst✝² : Group G\nα : Type u_2\ninst✝¹ : MulAction G α\ninst✝ : F... | apply Nat.eq_of_mul_eq_mul_right (Nat.factorial_pos _) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup | {
"line": 426,
"column": 20
} | {
"line": 426,
"column": 54
} | {
"line": 426,
"column": 55
} | [
{
"pp": "M : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns t : Set α\nhst : s = t\nx✝¹ x✝ : ↥(ofFixingSubgroup M s)\nhxy : (ofFixingSubgroup_of_eq M hst) x✝¹ = (ofFixingSubgroup_of_eq M hst) x✝\n⊢ x✝¹ = x✝",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"SubMulAc... | [
"M : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns t : Set α\nhst : s = t\nx✝¹ x✝ : ↥(ofFixingSubgroup M s)\nhxy : (ofFixingSubgroup_of_eq M hst) x✝¹ = (ofFixingSubgroup_of_eq M hst) x✝\n⊢ ↑x✝¹ = ↑x✝"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup | {
"line": 449,
"column": 2
} | {
"line": 449,
"column": 17
} | {
"line": 449,
"column": 18
} | [
{
"pp": "M : Type u_1\nα : Type u_2\ninst✝² : Group M\ninst✝¹ : MulAction M α\ns : Set α\nn : ℕ\ninst✝ : Finite ↑s\nx : Fin n ↪ ↥(ofFixingSubgroup M s)\nthis : Nonempty (Fin s.ncard ≃ ↑s)\ny : Fin s.ncard ↪ ↑s := (Classical.choice this).toEmbedding\nj : Fin s.ncard\ni : Fin n\nH : ↑(y j) = ↑(x i)\n⊢ ↑(x i) ∈ s"... | [
"M : Type u_1\nα : Type u_2\ninst✝² : Group M\ninst✝¹ : MulAction M α\ns : Set α\nn : ℕ\ninst✝ : Finite ↑s\nx : Fin n ↪ ↥(ofFixingSubgroup M s)\nthis : Nonempty (Fin s.ncard ≃ ↑s)\ny : Fin s.ncard ↪ ↑s := (Classical.choice this).toEmbedding\nj : Fin s.ncard\ni : Fin n\nH : ↑(y j) = ↑(x i)\n⊢ ↑(x i) ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.MultiplePrimitivity | {
"line": 224,
"column": 28
} | {
"line": 224,
"column": 52
} | {
"line": 224,
"column": 52
} | [
{
"pp": "M : Type u_1\nα : Type u_2\ninst✝³ : Group M\ninst✝² : MulAction M α\nm n : ℕ\ninst✝¹ : IsMultiplyPreprimitive M α n\ns : Set α\ninst✝ : Finite ↑s\nhs : s.ncard + m = n\nt : Set ↥(ofFixingSubgroup M s)\nht : t.encard + 1 = ↑m\nt' : Set α := Subtype.val '' t\nhtt' : t = Subtype.val ⁻¹' t'\n⊢ s.encard + ... | [
"M : Type u_1\nα : Type u_2\ninst✝³ : Group M\ninst✝² : MulAction M α\nm n : ℕ\ninst✝¹ : IsMultiplyPreprimitive M α n\ns : Set α\ninst✝ : Finite ↑s\nhs : s.ncard + m = n\nt : Set ↥(ofFixingSubgroup M s)\nht : t.encard + 1 = ↑m\nt' : Set α := Subtype.val '' t\nhtt' : t = Subtype.val ⁻¹' t'\n⊢ s.encard + ↑m = s.encar... | Set.Finite.cast_ncard_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.Jordan | {
"line": 189,
"column": 10
} | {
"line": 189,
"column": 49
} | {
"line": 189,
"column": 50
} | [
{
"pp": "n : ℕ\nhrec :\n ∀ m < n,\n ∀ {G : Type u_1} {α : Type u_2} [inst : Group G] [inst_1 : MulAction G α],\n IsPreprimitive G α →\n ∀ {s : Set α},\n s.ncard = m + 1 →\n m + 2 < Nat.card α →\n (IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → Is... | [
"n : ℕ\nhrec :\n ∀ m < n,\n ∀ {G : Type u_1} {α : Type u_2} [inst : Group G] [inst_1 : MulAction G α],\n IsPreprimitive G α →\n ∀ {s : Set α},\n s.ncard = m + 1 →\n m + 2 < Nat.card α →\n (IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPret... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup | {
"line": 532,
"column": 2
} | {
"line": 532,
"column": 47
} | {
"line": 532,
"column": 48
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nα : Type u_2\ninst✝ : MulAction G α\ns : Set α\nk : G\nhk : k ∈ fixingSubgroup G s\n⊢ ∀ a ∈ s, k • a = id a",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"instHSMul",
"Membership.mem",
"id",
"DivInvMonoid.toMonoid",
"... | [
"G : Type u_1\ninst✝¹ : Group G\nα : Type u_2\ninst✝ : MulAction G α\ns : Set α\nk : G\nhk : k ∈ fixingSubgroup G s\n⊢ ∀ a ∈ s, k • a = a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.Period | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 27
} | {
"line": 124,
"column": 28
} | [
{
"pp": "case h\nα : Type v\nM : Type u\ninst✝¹ : Monoid M\ninst✝ : MulAction M α\nexp_pos : 0 < Monoid.exponent M\nm : M\n⊢ Monoid.exponent M ∈ upperBounds (Set.range fun a ↦ period m a)",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.ofPred",
... | [
"case h\nα : Type v\nM : Type u\ninst✝¹ : Monoid M\ninst✝ : MulAction M α\nexp_pos : 0 < Monoid.exponent M\nm : M\n⊢ ∀ (a : α), period m a ≤ Monoid.exponent M"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.Jordan | {
"line": 207,
"column": 8
} | {
"line": 207,
"column": 80
} | {
"line": 207,
"column": 81
} | [
{
"pp": "n : ℕ\nhrec :\n ∀ m < n,\n ∀ {G : Type u_1} {α : Type u_2} [inst : Group G] [inst_1 : MulAction G α],\n IsPreprimitive G α →\n ∀ {s : Set α},\n s.ncard = m + 1 →\n m + 2 < Nat.card α →\n (IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → Is... | [
"n : ℕ\nhrec :\n ∀ m < n,\n ∀ {G : Type u_1} {α : Type u_2} [inst : Group G] [inst_1 : MulAction G α],\n IsPreprimitive G α →\n ∀ {s : Set α},\n s.ncard = m + 1 →\n m + 2 < Nat.card α →\n (IsPretransitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s) → IsMultiplyPret... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.Jordan | {
"line": 256,
"column": 12
} | {
"line": 256,
"column": 23
} | {
"line": 256,
"column": 24
} | [
{
"pp": "case zero\nG : Type u_1\nα : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G α\nhG : IsPreprimitive G α\ns : Set α\nhsn : s.ncard = 0 + 1\nhsn' : 0 + 2 < Nat.card α\nhprim : IsPreprimitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s)\nhα : Finite α\n⊢ IsMultiplyPreprimitive G α (0 + 2)",
"ppTerm":... | [
"case zero\nG : Type u_1\nα : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G α\nhG : IsPreprimitive G α\ns : Set α\nhsn : s.ncard = 0 + 1\nhsn' : 0 + 2 < Nat.card α\nhprim : IsPreprimitive ↥(fixingSubgroup G s) ↥(ofFixingSubgroup G s)\nhα : Finite α\n⊢ IsMultiplyPreprimitive G α 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.Perm.MaximalSubgroups | {
"line": 243,
"column": 4
} | {
"line": 245,
"column": 28
} | {
"line": 246,
"column": 4
} | [
{
"pp": "M : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns B : Set α\nhB_ss_sc : B ⊂ s\nhB : IsBlock M B\nhG : Function.Surjective toPerm\nthis : IsPreprimitive ↥(stabilizer M s) ↑s\nφ' : ↥(stabilizer M s) → M := Subtype.val\n⊢ IsBlock (↥(stabilizer M s)) (Subtype.val ⁻¹' B)",
"ppTerm":... | [
"M : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns B : Set α\nhB_ss_sc : B ⊂ s\nhB : IsBlock M B\nhG : Function.Surjective toPerm\nthis : IsPreprimitive ↥(stabilizer M s) ↑s\nφ' : ↥(stabilizer M s) → M := Subtype.val\nf' : ↑s →ₑ[φ'] α := { toFun := Subtype.val, map_smul' := ⋯ }\n⊢ IsBlock (↥(st... | let f' : (s : Set α) →ₑ[φ'] α := {
toFun := Subtype.val
map_smul' _ _ := rfl } | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups | {
"line": 183,
"column": 4
} | {
"line": 183,
"column": 72
} | {
"line": 183,
"column": 73
} | [
{
"pp": "case h\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh4 : 4 < Nat.card α\nG : Subgroup ↥(alternatingGroup α)\nhG' : IsPreprimitive (↥G) α\ns : Set α\nhG : stabilizer (↥(alternatingGroup α)) s ≤ G\ng : Perm α\nhg : g ∈ stabilizer (Perm α) s\nhg3 : g.IsThreeCycle\n⊢ ⟨g, ⋯⟩ ∈ ↑G ∧ (alternating... | [
"case h\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh4 : 4 < Nat.card α\nG : Subgroup ↥(alternatingGroup α)\nhG' : IsPreprimitive (↥G) α\ns : Set α\nhG : stabilizer (↥(alternatingGroup α)) s ≤ G\ng : Perm α\nhg : g ∈ stabilizer (Perm α) s\nhg3 : g.IsThreeCycle\n⊢ ⟨g, ⋯⟩ ∈ G"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups | {
"line": 188,
"column": 4
} | {
"line": 189,
"column": 28
} | {
"line": 191,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh4 : 4 < Nat.card α\nG : Subgroup ↥(alternatingGroup α)\nhG' : IsPreprimitive (↥G) α\ns : Set α\nhG : stabilizer (↥(alternatingGroup α)) s ≤ G\ng : Perm α\nhg : g ∈ stabilizer (Perm α) s\nhg3 : g.IsThreeCycle\nφ : ↥G →* ↥(Subgroup.map (alternatin... | [] | rwa [← isPreprimitive_congr (f := f) ((alternatingGroup α).subtype.subgroupMap_surjective G)
Function.bijective_id] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.GroupTheory.GroupAction.SubMulAction.Combination | {
"line": 84,
"column": 39
} | {
"line": 84,
"column": 66
} | {
"line": 84,
"column": 67
} | [
{
"pp": "α : Type u_2\ninst✝² : DecidableEq α\nG : Type u_3\ninst✝¹ : AddGroup G\ninst✝ : AddAction G α\nn : ℕ\nhn : 1 ≤ n\nhα : ↑n < ENat.card α\ng : G\nh : ¬AddAction.toPerm g = 1\n⊢ ∃ a, g +ᵥ a ≠ a",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"AddMonoid.toAddSemigroup",
"... | [
"α : Type u_2\ninst✝² : DecidableEq α\nG : Type u_3\ninst✝¹ : AddGroup G\ninst✝ : AddAction G α\nn : ℕ\nhn : 1 ≤ n\nhα : ↑n < ENat.card α\ng : G\nh : ¬AddAction.toPerm g = 1\n⊢ ∃ a, ¬g +ᵥ a = a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupExtension.Defs | {
"line": 282,
"column": 4
} | {
"line": 282,
"column": 16
} | {
"line": 282,
"column": 17
} | [
{
"pp": "N : Type u_1\nE : Type u_2\nG : Type u_3\ninst✝² : Group N\ninst✝¹ : Group E\ninst✝ : Group G\nS : GroupExtension N E G\n⊢ Function.Injective fun s ↦ (↑s.toMonoidHom).toFun",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"MonoidHom.instFunLike",
"... | [
"N : Type u_1\nE : Type u_2\nG : Type u_3\ninst✝² : Group N\ninst✝¹ : Group E\ninst✝ : Group G\nS : GroupExtension N E G\ntoMonoidHom✝ : G →* E\nrightInverse_rightHom✝ : Function.RightInverse (↑toMonoidHom✝).toFun ⇑S.rightHom\n⊢ ∀ ⦃a₂ : S.Splitting⦄,\n (fun s ↦ (↑s.toMonoidHom).toFun) { toMonoidHom := toMonoidHo... | intro ⟨_, _⟩ | Lean.Elab.Tactic.evalIntro | null |
Mathlib.GroupTheory.GroupExtension.Basic | {
"line": 64,
"column": 12
} | {
"line": 64,
"column": 44
} | {
"line": 64,
"column": 44
} | [
{
"pp": "N : Type u_1\nG : Type u_2\ninst✝² : Group N\ninst✝¹ : Group G\nE : Type u_3\ninst✝ : Group E\nS : GroupExtension N E G\nσ σ' : S.Section\ng : G\nn : N\nhn : S.inl n = σ g * (σ' g)⁻¹\n⊢ σ g = S.inl n * σ' g",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Div... | [] | by rw [hn, inv_mul_cancel_right] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.GroupAction.SubMulAction.Combination | {
"line": 112,
"column": 38
} | {
"line": 112,
"column": 65
} | {
"line": 112,
"column": 66
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nα : Type u_2\ninst✝¹ : MulAction G α\nn : ℕ\ninst✝ : DecidableEq α\nhn : 1 ≤ n\nhα : ↑n < ENat.card α\ng : G\nh : ¬toPerm g = 1\n⊢ ∃ a, g • a ≠ a",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"instHSMul",
"Exists",
"id",
"D... | [
"G : Type u_1\ninst✝² : Group G\nα : Type u_2\ninst✝¹ : MulAction G α\nn : ℕ\ninst✝ : DecidableEq α\nhn : 1 ≤ n\nhα : ↑n < ENat.card α\ng : G\nh : ¬toPerm g = 1\n⊢ ∃ a, ¬g • a = a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.Jordan | {
"line": 349,
"column": 2
} | {
"line": 349,
"column": 33
} | {
"line": 349,
"column": 34
} | [
{
"pp": "α : Type u_1\nK : Type u_2\ninst✝¹ : Group K\ninst✝ : MulAction K α\nhα : Nat.card α = 2\nhK : fixedPoints K α ≠ _root_.Set.univ\nn : ℕ\nthis✝ : Finite α\nthis : Fintype α\nφ : K →* Perm α := toPermHom K α\nf : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }\nhf : Function.Bijective ⇑f\nH : Subsingleton ... | [
"α : Type u_1\nK : Type u_2\ninst✝¹ : Group K\ninst✝ : MulAction K α\nhα : Nat.card α = 2\nhK : fixedPoints K α ≠ _root_.Set.univ\nn : ℕ\nthis✝ : Finite α\nthis : Fintype α\nφ : K →* Perm α := toPermHom K α\nf : α →ₑ[⇑φ] α := { toFun := id, map_smul' := ⋯ }\nhf : Function.Bijective ⇑f\nH : Subsingleton ↥φ.range\na ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupExtension.Basic | {
"line": 128,
"column": 6
} | {
"line": 128,
"column": 44
} | {
"line": 128,
"column": 45
} | [
{
"pp": "N : Type u_1\nG : Type u_2\ninst✝³ : Group N\ninst✝² : Group G\nE : Type u_3\ninst✝¹ : Group E\nS : GroupExtension N E G\nE' : Type u_4\ninst✝ : Group E'\nS' : GroupExtension N E' G\nf : E →* E'\ncomp_inl : f.comp S.inl = S'.inl\nrightHom_comp : S'.rightHom.comp f = S.rightHom\ne : E\n⊢ (S.rightHom (Fu... | [
"N : Type u_1\nG : Type u_2\ninst✝³ : Group N\ninst✝² : Group G\nE : Type u_3\ninst✝¹ : Group E\nS : GroupExtension N E G\nE' : Type u_4\ninst✝ : Group E'\nS' : GroupExtension N E' G\nf : E →* E'\ncomp_inl : f.comp S.inl = S'.inl\nrightHom_comp : S'.rightHom.comp f = S.rightHom\ne : E\n⊢ (S.rightHom e)⁻¹ * S.rightH... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupExtension.Basic | {
"line": 128,
"column": 6
} | {
"line": 128,
"column": 74
} | {
"line": 129,
"column": 4
} | [
{
"pp": "N : Type u_1\nG : Type u_2\ninst✝³ : Group N\ninst✝² : Group G\nE : Type u_3\ninst✝¹ : Group E\nS : GroupExtension N E G\nE' : Type u_4\ninst✝ : Group E'\nS' : GroupExtension N E' G\nf : E →* E'\ncomp_inl : f.comp S.inl = S'.inl\nrightHom_comp : S'.rightHom.comp f = S.rightHom\ne : E\n⊢ (S.rightHom (Fu... | [] | simpa only [Function.surjInv_eq] using inv_mul_cancel (S.rightHom e) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.GroupTheory.GroupExtension.Basic | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 42
} | {
"line": 137,
"column": 43
} | [
{
"pp": "N : Type u_1\nG : Type u_2\ninst✝³ : Group N\ninst✝² : Group G\nE : Type u_3\ninst✝¹ : Group E\nS : GroupExtension N E G\nE' : Type u_4\ninst✝ : Group E'\nS' : GroupExtension N E' G\nf : E →* E'\ncomp_inl : f.comp S.inl = S'.inl\nrightHom_comp : S'.rightHom.comp f = S.rightHom\ne' : E'\n⊢ (S.rightHom (... | [
"N : Type u_1\nG : Type u_2\ninst✝³ : Group N\ninst✝² : Group G\nE : Type u_3\ninst✝¹ : Group E\nS : GroupExtension N E G\nE' : Type u_4\ninst✝ : Group E'\nS' : GroupExtension N E' G\nf : E →* E'\ncomp_inl : f.comp S.inl = S'.inl\nrightHom_comp : S'.rightHom.comp f = S.rightHom\ne' : E'\n⊢ (S'.rightHom e')⁻¹ * S'.r... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupExtension.Basic | {
"line": 169,
"column": 4
} | {
"line": 171,
"column": 97
} | {
"line": 172,
"column": 2
} | [
{
"pp": "N : Type u_1\nG : Type u_2\ninst✝² : Group N\ninst✝¹ : Group G\nE : Type u_3\ninst✝ : Group E\nS : GroupExtension N E G\ns : S.Splitting\n⊢ ⇑{\n toFun := fun x ↦\n match x with\n | ⟨n, g⟩ => S.inl n * s g,\n invFun := fun e ↦ ⟨Function.invFun (⇑S.inl) (e * (s (S.... | [] | ext n
simp only [SemidirectProduct.toGroupExtension, Function.comp_apply, MulEquiv.coe_mk,
Equiv.coe_fn_mk, SemidirectProduct.left_inl, SemidirectProduct.right_inl, map_one, mul_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupExtension.Basic | {
"line": 169,
"column": 4
} | {
"line": 171,
"column": 97
} | {
"line": 172,
"column": 2
} | [
{
"pp": "N : Type u_1\nG : Type u_2\ninst✝² : Group N\ninst✝¹ : Group G\nE : Type u_3\ninst✝ : Group E\nS : GroupExtension N E G\ns : S.Splitting\n⊢ ⇑{\n toFun := fun x ↦\n match x with\n | ⟨n, g⟩ => S.inl n * s g,\n invFun := fun e ↦ ⟨Function.invFun (⇑S.inl) (e * (s (S.... | [] | ext n
simp only [SemidirectProduct.toGroupExtension, Function.comp_apply, MulEquiv.coe_mk,
Equiv.coe_fn_mk, SemidirectProduct.left_inl, SemidirectProduct.right_inl, map_one, mul_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.Jordan | {
"line": 395,
"column": 4
} | {
"line": 395,
"column": 49
} | {
"line": 396,
"column": 4
} | [
{
"pp": "case inl\nα : Type u_1\nG : Subgroup (Perm α)\ninst✝¹ : DecidableEq α\ninst✝ : Finite α\nhG : IsPreprimitive (↥G) α\ng : Perm α\nh2g : g.IsSwap\nhg : g ∈ G\nthis : Fintype α\nhα3 : Nat.card α ≤ 2\n⊢ Fintype.card ↥G = Fintype.card (Perm α)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants... | [
"case inl\nα : Type u_1\nG : Subgroup (Perm α)\ninst✝¹ : DecidableEq α\ninst✝ : Finite α\nhG : IsPreprimitive (↥G) α\ng : Perm α\nh2g : g.IsSwap\nhg : g ∈ G\nthis : Fintype α\nhα3 : Nat.card α ≤ 2\n⊢ Fintype.card (Perm α) ≤ Fintype.card ↥G"
] | apply le_antisymm (Fintype.card_subtype_le _) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.GroupTheory.IsPerfect | {
"line": 51,
"column": 6
} | {
"line": 51,
"column": 26
} | {
"line": 51,
"column": 27
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH : Subgroup G\n⊢ IsPerfect ↥H ↔ ⁅H, H⁆ = H",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Group.isPerfect_def",
"Bracket.bracket",
"Membership.mem",
"id",
"Subtype",
"Subgroup",... | [
"G : Type u_1\ninst✝ : Group G\nH : Subgroup G\n⊢ _root_.commutator ↥H = ⊤ ↔ ⁅H, H⁆ = H"
] | Group.isPerfect_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.SubMulAction.Combination | {
"line": 312,
"column": 73
} | {
"line": 312,
"column": 84
} | {
"line": 312,
"column": 85
} | [
{
"pp": "α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nn : ℕ\nh_three_le : 3 ≤ n\nhn : n < Nat.card α\nhα : Nat.card α ≠ 2 * n\nthis : IsPretransitive ↥(alternatingGroup α) ↑(powersetCard α n)\n⊢ ↑n < ENat.card α",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nn : ℕ\nh_three_le : 3 ≤ n\nhn : n < Nat.card α\nhα : Nat.card α ≠ 2 * n\nthis : IsPretransitive ↥(alternatingGroup α) ↑(powersetCard α n)\n⊢ n < Fintype.card α"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.GroupAction.SubMulAction.Combination | {
"line": 318,
"column": 4
} | {
"line": 318,
"column": 68
} | {
"line": 318,
"column": 69
} | [
{
"pp": "case h1\nα : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nn : ℕ\nh_three_le : 3 ≤ n\nhn : n < Nat.card α\nhα : Nat.card α ≠ 2 * n\nthis✝ : IsPretransitive ↥(alternatingGroup α) ↑(powersetCard α n)\nthis : Nontrivial ↑(powersetCard α n)\ns : ↑(powersetCard α n)\n⊢ (↑s)ᶜ.Nonempty",
"ppTerm": ... | [
"case h1\nα : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nn : ℕ\nh_three_le : 3 ≤ n\nhn : n < Nat.card α\nhα : Nat.card α ≠ 2 * n\nthis✝ : IsPretransitive ↥(alternatingGroup α) ↑(powersetCard α n)\nthis : Nontrivial ↑(powersetCard α n)\ns : ↑(powersetCard α n)\n⊢ ¬n = Fintype.card α"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.IndexNSmul | {
"line": 42,
"column": 6
} | {
"line": 43,
"column": 13
} | {
"line": 43,
"column": 14
} | [
{
"pp": "M : Type u_1\ninst✝² : AddCommGroup M\ninst✝¹ : Free ℤ M\ninst✝ : Module.Finite ℤ M\nn : ℕ\n⊢ (nsmulAddMonoidHom n).range.index = (nsmulAddMonoidHom n).range.index",
"ppTerm": "?m.35",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"M : Type u_1\ninst✝² : AddCommGroup M\ninst✝¹ : Free ℤ M\ninst✝ : Module.Finite ℤ M\nn : ℕ\n⊢ (nsmulAddMonoidHom n).range.index = (nsmulAddMonoidHom n).range.index"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.IndexNSmul | {
"line": 82,
"column": 47
} | {
"line": 82,
"column": 58
} | {
"line": 82,
"column": 59
} | [
{
"pp": "M : Type u_1\ninst✝³ : AddCommGroup M\ninst✝² : Module.Finite ℤ M\ninst✝¹ : IsTorsionFree ℤ M\nA : AddSubgroup M\ninst✝ : A.FiniteIndex\nthis : finrank ℤ ↥(DistribSMul.toLinearMap ℤ M A.index).range = finrank ℤ M\nm : M\nhm : m ∈ toIntSubmodule.symm (DistribSMul.toLinearMap ℤ M A.index).range\n⊢ ∃ x, A... | [
"M : Type u_1\ninst✝³ : AddCommGroup M\ninst✝² : Module.Finite ℤ M\ninst✝¹ : IsTorsionFree ℤ M\nA : AddSubgroup M\ninst✝ : A.FiniteIndex\nthis : finrank ℤ ↥(DistribSMul.toLinearMap ℤ M A.index).range = finrank ℤ M\nm : M\nhm : m ∈ toIntSubmodule.symm (DistribSMul.toLinearMap ℤ M A.index).range\n⊢ ∃ x, A.index • x =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.HNNExtension | {
"line": 129,
"column": 21
} | {
"line": 129,
"column": 32
} | {
"line": 129,
"column": 33
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nmotive : HNNExtension G A B φ → Prop\nx : HNNExtension G A B φ\nof : ∀ (g : G), motive (HNNExtension.of g)\nt : motive HNNExtension.t\nmul : ∀ (x y : HNNExtension G A B φ), motive x → motive y → motive (x * y)\ninv : ∀ (x : HNNExtension G A... | [
"G : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nmotive : HNNExtension G A B φ → Prop\nx : HNNExtension G A B φ\nof : ∀ (g : G), motive (HNNExtension.of g)\nt : motive HNNExtension.t\nmul : ∀ (x y : HNNExtension G A B φ), motive x → motive y → motive (x * y)\ninv : ∀ (x : HNNExtension G A B φ), motiv... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.IsSubnormal | {
"line": 138,
"column": 21
} | {
"line": 138,
"column": 36
} | {
"line": 138,
"column": 37
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH✝ H K : Subgroup G\nHK : H ≤ K\nhS : K.IsSubnormal\nhN : (H.subgroupOf K).Normal\nK' : Subgroup G\nHK' : K < K'\nhS' : K'.IsSubnormal\nhN' : (K.subgroupOf K').Normal\nhH : H ≠ K\n⊢ H < K ∧ K.IsSubnormal ∧ (H.subgroupOf K).Normal",
"ppTerm": "?m.156",
"assigned": ... | [
"G : Type u_1\ninst✝ : Group G\nH✝ H K : Subgroup G\nHK : H ≤ K\nhS : K.IsSubnormal\nhN : (H.subgroupOf K).Normal\nK' : Subgroup G\nHK' : K < K'\nhS' : K'.IsSubnormal\nhN' : (K.subgroupOf K').Normal\nhH : H ≠ K\n⊢ H < K"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.IsSubnormal | {
"line": 130,
"column": 4
} | {
"line": 130,
"column": 29
} | {
"line": 131,
"column": 6
} | [
{
"pp": "case step\nG : Type u_1\ninst✝ : Group G\nH✝ H K : Subgroup G\nHK : H ≤ K\nhS : K.IsSubnormal\nhN : (H.subgroupOf K).Normal\nih : K = ⊤ ∨ ∃ K_1, K < K_1 ∧ K_1.IsSubnormal ∧ (K.subgroupOf K_1).Normal\n⊢ H = ⊤ ∨ ∃ K, H < K ∧ K.IsSubnormal ∧ (H.subgroupOf K).Normal",
"ppTerm": "?step",
"assigned":... | [] | | step H K HK hS hN ih => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.GroupTheory.HNNExtension | {
"line": 179,
"column": 4
} | {
"line": 179,
"column": 79
} | {
"line": 180,
"column": 4
} | [
{
"pp": "case inr\nG : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\na : ↥(toSubgroup A B (-1))\n⊢ ↑((toSubgroupEquiv φ (- -1)) ((toSubgroupEquiv φ (-1)) a)) = ↑a",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulEquiv.instEquivLike",
"NonUnitalCom... | [
"case inr\nG : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\na : ↥(toSubgroup A B (-1))\n⊢ (toSubgroupEquiv φ (- -1)) (φ.symm a) = a"
] | simp only [toSubgroup_neg_one, toSubgroupEquiv_neg_one, SetLike.coe_eq_coe] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.IsSubnormal | {
"line": 184,
"column": 6
} | {
"line": 185,
"column": 50
} | {
"line": 186,
"column": 4
} | [
{
"pp": "case top\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\n⊢ ∃ n f, Monotone f ∧ (∀ (i : ℕ), ((f i).subgroupOf (f (i + 1))).Normal) ∧ f 0 = ⊤ ∧ f n = ⊤",
"ppTerm": "?top",
"assigned": true,
"usedConstants": [
"Subgroup.subgroupOf",
"Subgroup.subgroupOf_self",
"congrArg",
... | [] | use 0, fun _ ↦ ⊤, ?_, (by simp)
exact monotone_nat_of_le_succ fun _ ↦ le_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.IsSubnormal | {
"line": 184,
"column": 6
} | {
"line": 185,
"column": 50
} | {
"line": 186,
"column": 4
} | [
{
"pp": "case top\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\n⊢ ∃ n f, Monotone f ∧ (∀ (i : ℕ), ((f i).subgroupOf (f (i + 1))).Normal) ∧ f 0 = ⊤ ∧ f n = ⊤",
"ppTerm": "?top",
"assigned": true,
"usedConstants": [
"Subgroup.subgroupOf",
"Subgroup.subgroupOf_self",
"congrArg",
... | [] | use 0, fun _ ↦ ⊤, ?_, (by simp)
exact monotone_nat_of_le_succ fun _ ↦ le_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.HNNExtension | {
"line": 312,
"column": 10
} | {
"line": 312,
"column": 21
} | {
"line": 312,
"column": 22
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nH : Type u_2\ninst✝¹ : Group H\nM : Type u_3\ninst✝ : Monoid M\nd : TransversalPair G A B\nmotive : NormalWord d → Sort u_4\nofGroup : (g : G) → motive (NormalWord.ofGroup g)\ncons :\n (g : G) →\n (u : ℤˣ) →\n (w : NormalWord d) →... | [
"G : Type u_1\ninst✝² : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nH : Type u_2\ninst✝¹ : Group H\nM : Type u_3\ninst✝ : Monoid M\nd : TransversalPair G A B\nmotive : NormalWord d → Sort u_4\nofGroup : (g : G) → motive (NormalWord.ofGroup g)\ncons :\n (g : G) →\n (u : ℤˣ) →\n (w : NormalWord d) →\n (h... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.IsSubnormal | {
"line": 280,
"column": 2
} | {
"line": 280,
"column": 13
} | {
"line": 280,
"column": 14
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nhH : H.IsSubnormal\nhK : K.IsSubnormal\n⊢ (H ⊓ K).IsSubnormal",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nhH : H.IsSubnormal\nhK : K.IsSubnormal\n⊢ (H ⊓ K).IsSubnormal"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.NoncommCoprod | {
"line": 54,
"column": 24
} | {
"line": 54,
"column": 35
} | {
"line": 54,
"column": 36
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : Mul M\ninst✝¹ : Mul N\ninst✝ : Semigroup P\nf : M →ₙ* P\ng : N →ₙ* P\ncomm : ∀ (m : M) (n : N), Commute (f m) (g n)\nmn mn' : M × N\n⊢ f (mn * mn').1 * g (mn * mn').2 = f mn.1 * g mn.2 * (f mn'.1 * g mn'.2)",
"ppTerm": "?m.32",
"assigned": true... | [
"M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : Mul M\ninst✝¹ : Mul N\ninst✝ : Semigroup P\nf : M →ₙ* P\ng : N →ₙ* P\ncomm : ∀ (m : M) (n : N), Commute (f m) (g n)\nmn mn' : M × N\n⊢ f mn.1 * f mn'.1 * (g mn.2 * g mn'.2) = f mn.1 * g mn.2 * (f mn'.1 * g mn'.2)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.NoncommCoprod | {
"line": 136,
"column": 4
} | {
"line": 136,
"column": 15
} | {
"line": 136,
"column": 16
} | [
{
"pp": "case refine_1\nM : Type u_4\nN : Type u_5\nP : Type u_6\ninst✝² : Group M\ninst✝¹ : Group N\ninst✝ : Group P\nf : M →* P\ng : N →* P\ncomm : ∀ (m : M) (n : N), Commute (f m) (g n)\nh : ∀ (a : M) (b : N), f a * g b = 1 → a = 1 ∧ b = 1\nx : M\n⊢ f x = 1 → x = 1",
"ppTerm": "?refine_1",
"assigned"... | [
"case refine_1\nM : Type u_4\nN : Type u_5\nP : Type u_6\ninst✝² : Group M\ninst✝¹ : Group N\ninst✝ : Group P\nf : M →* P\ng : N →* P\ncomm : ∀ (m : M) (n : N), Commute (f m) (g n)\nh : ∀ (a : M) (b : N), f a * g b = 1 → a = 1 ∧ b = 1\nx : M\n⊢ f x = 1 → x = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.NoncommCoprod | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 15
} | {
"line": 137,
"column": 16
} | [
{
"pp": "case refine_2\nM : Type u_4\nN : Type u_5\nP : Type u_6\ninst✝² : Group M\ninst✝¹ : Group N\ninst✝ : Group P\nf : M →* P\ng : N →* P\ncomm : ∀ (m : M) (n : N), Commute (f m) (g n)\nh : ∀ (a : M) (b : N), f a * g b = 1 → a = 1 ∧ b = 1\nx : N\n⊢ g x = 1 → x = 1",
"ppTerm": "?refine_2",
"assigned"... | [
"case refine_2\nM : Type u_4\nN : Type u_5\nP : Type u_6\ninst✝² : Group M\ninst✝¹ : Group N\ninst✝ : Group P\nf : M →* P\ng : N →* P\ncomm : ∀ (m : M) (n : N), Commute (f m) (g n)\nh : ∀ (a : M) (b : N), f a * g b = 1 → a = 1 ∧ b = 1\nx : N\n⊢ g x = 1 → x = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.HNNExtension | {
"line": 403,
"column": 2
} | {
"line": 403,
"column": 13
} | {
"line": 403,
"column": 14
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nd : TransversalPair G A B\nu : ℤˣ\nw : NormalWord d\nhw : w.head ∈ toSubgroup A B u\nx : G\nh2 : ∀ u' ∈ Option.map Prod.fst (some (-u, x)), w.head ∈ toSubgroup A B u → u = u'\nhx : w.toList.head? = some (-u, x)\n⊢ False",
"ppTerm": "?m.53",
"assi... | [
"G : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nd : TransversalPair G A B\nu : ℤˣ\nw : NormalWord d\nhw : w.head ∈ toSubgroup A B u\nx : G\nh2 : ∀ u' ∈ Option.map Prod.fst (some (-u, x)), w.head ∈ toSubgroup A B u → u = u'\nhx : w.toList.head? = some (-u, x)\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.GroupTheory.MonoidLocalization.UniqueFactorization | {
"line": 29,
"column": 58
} | {
"line": 41,
"column": 71
} | {
"line": 43,
"column": 0
} | [
{
"pp": "M : Type u_1\nN : Type u_2\ninst✝¹ : CommMonoidWithZero M\ninst✝ : CommMonoidWithZero N\nS : Submonoid M\nf : S.LocalizationMap N\nm : M\nprime : Prime m\nn0 : f m ≠ 0\nnu : ¬IsUnit (f m)\n⊢ Prime (f m)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCo... | [] | by
refine ⟨n0, nu, fun n₁ n₂ dvd ↦ ?_⟩
have ⟨⟨m₁, s₁⟩, eq₁⟩ := f.surj n₁
have ⟨⟨m₂, s₂⟩, eq₂⟩ := f.surj n₂
have := (f.map_units (s₁ * s₂)).dvd_mul_right.mpr dvd
rw [Submonoid.mul_def, map_mul, mul_mul_mul_comm, eq₁, eq₂, ← map_mul, f.map_dvd_map] at this
have ⟨s, hs, dvd⟩ := this
rw [← mul_assoc] at dvd
... | [anonymous] | Lean.Parser.Term.byTactic |
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