module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.GroupTheory.Nilpotent | {
"line": 757,
"column": 61
} | {
"line": 765,
"column": 49
} | {
"line": 767,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nH : Type u_2\ninst✝¹ : Group H\nf : G →* H\nhf1 : f.ker ≤ center G\ninst✝ : IsNilpotent H\n⊢ nilpotencyClass G ≤ nilpotencyClass H + 1",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"le_bot_iff",
"Eq.mpr",
"Subgroup.map",
"M... | [] | by
have : IsNilpotent G := isNilpotent_of_ker_le_center f hf1
rw [← lowerCentralSeries_length_eq_nilpotencyClass]
classical apply Nat.find_min'
refine lowerCentralSeries_succ_eq_bot ⊤
(le_trans ((Subgroup.map_eq_bot_iff _).mp ?_) hf1)
rw [map_lowerCentralSeries, ← le_bot_iff,
← lowerCentralSeries_nilp... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Nilpotent | {
"line": 1032,
"column": 4
} | {
"line": 1032,
"column": 20
} | {
"line": 1033,
"column": 4
} | [
{
"pp": "case refine_3\nG : Type u_1\ninst✝ : Group G\na : ℕ\nh2 : a ≤ nilpotencyClass G\n⊢ ∀ (i j : ℕ), i < a + 1 → j < a + 1 → upperCentralSeries G i = upperCentralSeries G j → i = j",
"ppTerm": "?refine_3",
"assigned": true,
"usedConstants": [
"instOfNatNat",
"instHAdd",
"HAdd.h... | [
"case refine_3\nG : Type u_1\ninst✝ : Group G\na : ℕ\nh2 : a ≤ nilpotencyClass G\ni j : ℕ\nhi : i < a + 1\nhj : j < a + 1\n⊢ upperCentralSeries G i = upperCentralSeries G j → i = j"
] | intros i j hi hj | Lean.Elab.Tactic.evalIntros | Lean.Parser.Tactic.intros |
Mathlib.GroupTheory.Nilpotent | {
"line": 1075,
"column": 65
} | {
"line": 1078,
"column": 49
} | {
"line": 1080,
"column": 0
} | [
{
"pp": "G₁ : Type u_2\nG₂ : Type u_3\ninst✝³ : Group G₁\ninst✝² : Group G₂\ninst✝¹ : IsNilpotent G₁\ninst✝ : IsNilpotent G₂\n⊢ nilpotencyClass (G₁ × G₂) = max (nilpotencyClass G₁) (nilpotencyClass G₂)",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
... | [] | by
refine eq_of_forall_ge_iff fun k => ?_
simp only [max_le_iff, ← lowerCentralSeries_eq_bot_iff_nilpotencyClass_le,
top_lowerCentralSeries_prod, prod_eq_bot_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.GroupAction.Iwasawa | {
"line": 72,
"column": 6
} | {
"line": 72,
"column": 17
} | {
"line": 72,
"column": 18
} | [
{
"pp": "M : Type u_1\ninst✝² : Group M\nα : Type u_2\ninst✝¹ : MulAction M α\nIwaS : IwasawaStructure M α\ninst✝ : IsQuasiPreprimitive M α\nN : Subgroup M\nnN : N.Normal\nhNX : fixedPoints (↥N) α ≠ Set.univ\nis_transN : IsPretransitive (↥N) α\nntα : Nontrivial α\na : α\n⊢ N ⊔ IwaS.T a = ⊤",
"ppTerm": "?m.4... | [
"M : Type u_1\ninst✝² : Group M\nα : Type u_2\ninst✝¹ : MulAction M α\nIwaS : IwasawaStructure M α\ninst✝ : IsQuasiPreprimitive M α\nN : Subgroup M\nnN : N.Normal\nhNX : fixedPoints (↥N) α ≠ Set.univ\nis_transN : IsPretransitive (↥N) α\nntα : Nontrivial α\na : α\n⊢ ⊤ ≤ N ⊔ IwaS.T a"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.Primitive | {
"line": 181,
"column": 8
} | {
"line": 183,
"column": 44
} | {
"line": 183,
"column": 45
} | [
{
"pp": "case inr\nG : Type u_1\nX : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G X\na : X\nha : a ∉ fixedPoints G X\nH : ∀ ⦃B : Set X⦄, a ∈ B → IsBlock G B → IsTrivialBlock B\nthis : IsPretransitive G X\nB : Set X\nhB : IsBlock G B\nb : X\nhb : b ∈ B\n⊢ IsTrivialBlock B",
"ppTerm": "?inr",
"assigned... | [] | obtain ⟨g, hg⟩ := exists_smul_eq G b a
rw [← IsTrivialBlock.smul_iff g]
exact H ⟨b, hb, hg⟩ (hB.translate g) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupAction.Primitive | {
"line": 181,
"column": 8
} | {
"line": 183,
"column": 44
} | {
"line": 183,
"column": 45
} | [
{
"pp": "case inr\nG : Type u_1\nX : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G X\na : X\nha : a ∉ fixedPoints G X\nH : ∀ ⦃B : Set X⦄, a ∈ B → IsBlock G B → IsTrivialBlock B\nthis : IsPretransitive G X\nB : Set X\nhB : IsBlock G B\nb : X\nhb : b ∈ B\n⊢ IsTrivialBlock B",
"ppTerm": "?inr",
"assigned... | [] | obtain ⟨g, hg⟩ := exists_smul_eq G b a
rw [← IsTrivialBlock.smul_iff g]
exact H ⟨b, hb, hg⟩ (hB.translate g) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 65,
"column": 40
} | {
"line": 65,
"column": 46
} | {
"line": 65,
"column": 46
} | [
{
"pp": "⊢ -1 ≠ 1",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"MulOne.toOne",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRing",
"of_decide_eq_true",
"Monoid.toMulOneClass",
"Units.instNeg",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 65,
"column": 40
} | {
"line": 65,
"column": 46
} | {
"line": 65,
"column": 46
} | [
{
"pp": "⊢ -1 ≠ 1",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"MulOne.toOne",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRing",
"of_decide_eq_true",
"Monoid.toMulOneClass",
"Units.instNeg",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 65,
"column": 40
} | {
"line": 65,
"column": 46
} | {
"line": 65,
"column": 46
} | [
{
"pp": "⊢ -1 ≠ 1",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"MulOne.toOne",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRing",
"of_decide_eq_true",
"Monoid.toMulOneClass",
"Units.instNeg",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 8
} | {
"line": 103,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nl : List (Perm α)\nhl : ∀ g ∈ l, g.IsSwap\n⊢ -1 ≠ 1",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"MulOne.toOne",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNon... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 149,
"column": 37
} | {
"line": 155,
"column": 59
} | {
"line": 157,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nhα : Nat.card α ≤ 3\n⊢ IsCyclic ↥(alternatingGroup α)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"MulOne.toOne",
"Nat.instMulZeroClass",
"Preorder.toLT",
"Dvd.d... | [] | by
cases subsingleton_or_nontrivial α
· infer_instance
have : 1 < Nat.card α := Finite.one_lt_card
apply isCyclic_of_card_dvd_prime (p := 3)
rw [nat_card_alternatingGroup]
interval_cases (Nat.card α) <;> simp [Nat.factorial_succ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.GroupAction.SubMulAction.OfStabilizer | {
"line": 181,
"column": 31
} | {
"line": 181,
"column": 44
} | {
"line": 181,
"column": 44
} | [
{
"pp": "case left\nG : 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 : α\nhx : x ∈ ofStabilizer G a\ny : α\nhy : y ∈ ofStabilizer G a\nhxy : ↑((conjMap hg) ⟨x, hx⟩) = ↑((conjMap hg) ⟨y, hy⟩)\n⊢ (fun x ↦ g • x) x = (fun x ↦ g • x) y",
"ppTerm": "?left",
... | [
"case left\nG : 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 : α\nhx : x ∈ ofStabilizer G a\ny : α\nhy : y ∈ ofStabilizer G a\nhxy : g • ↑⟨x, hx⟩ = ↑((conjMap hg) ⟨y, hy⟩)\n⊢ (fun x ↦ g • x) x = (fun x ↦ g • x) y"
] | conjMap_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Embedding | {
"line": 71,
"column": 2
} | {
"line": 73,
"column": 81
} | {
"line": 74,
"column": 2
} | [
{
"pp": "α : Type u_1\nm n : ℕ\nhn : ↑m + ↑n ≤ ENat.card α\nx : Fin m ↪ α\n⊢ ∃ a, (fun x ↦ (castAddEmb n).trans x) a = x",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Fin.castAddEmb",
"Eq.mpr",
"Fintype.card_fin",
"ChainCompletePartialOrder.instOfCompleteLattice"... | [
"α : Type u_1\nm n : ℕ\nhn : ↑m + ↑n ≤ ENat.card α\nx : Fin m ↪ α\ny : Fin n ↪ α\nhxy : Disjoint (range ⇑x) (range ⇑y)\n⊢ ∃ a, (fun x ↦ (castAddEmb n).trans x) a = x"
] | obtain ⟨y, hxy⟩ :=
exists_embedding_disjoint_range_of_add_le_ENat_card (s := range x)
(by simpa [← Nat.card_coe_set_eq, Nat.card_range_of_injective x.injective]) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 312,
"column": 30
} | {
"line": 312,
"column": 36
} | {
"line": 312,
"column": 36
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh3 : 3 ≤ Nat.card α\n⊢ 1 < 3",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat.instPreorder",
"Nat",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 312,
"column": 30
} | {
"line": 312,
"column": 36
} | {
"line": 312,
"column": 36
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh3 : 3 ≤ Nat.card α\n⊢ 1 < 3",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat.instPreorder",
"Nat",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 312,
"column": 30
} | {
"line": 312,
"column": 36
} | {
"line": 312,
"column": 36
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh3 : 3 ≤ Nat.card α\n⊢ 1 < 3",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Preorder.toLT",
"of_decide_eq_true",
"id",
"instOfNatNat",
"Bool.true",
"Nat.instPreorder",
"Nat",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 338,
"column": 10
} | {
"line": 338,
"column": 16
} | {
"line": 339,
"column": 2
} | [
{
"pp": "g : Perm (Fin 5)\nh1 : g ≠ 1\nh2 : g.cycleType = Multiset.replicate g.cycleType.card 2\nh✝ : g.cycleType.card * 2 ≤ card (Fin 5)\nh : g.cycleType.card ≤ 3\nha : (-1) ^ g.cycleType.card = 1\n⊢ -1 ≠ 1",
"ppTerm": "?m.184",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 338,
"column": 10
} | {
"line": 338,
"column": 16
} | {
"line": 339,
"column": 2
} | [
{
"pp": "g : Perm (Fin 5)\nh1 : g ≠ 1\nh2 : g.cycleType = Multiset.replicate g.cycleType.card 2\nh✝ : g.cycleType.card * 2 ≤ card (Fin 5)\nh : g.cycleType.card ≤ 3\nha : (-1) ^ g.cycleType.card = 1\n⊢ -1 ≠ 1",
"ppTerm": "?m.184",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 338,
"column": 10
} | {
"line": 338,
"column": 16
} | {
"line": 339,
"column": 2
} | [
{
"pp": "g : Perm (Fin 5)\nh1 : g ≠ 1\nh2 : g.cycleType = Multiset.replicate g.cycleType.card 2\nh✝ : g.cycleType.card * 2 ≤ card (Fin 5)\nh : g.cycleType.card ≤ 3\nha : (-1) ^ g.cycleType.card = 1\n⊢ -1 ≠ 1",
"ppTerm": "?m.184",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 343,
"column": 37
} | {
"line": 343,
"column": 43
} | {
"line": 344,
"column": 4
} | [
{
"pp": "g : Perm (Fin 5)\nh1 : g ≠ 1\nh_1 : g.cycleType.card = 2\nh2 : g.cycleType = Multiset.replicate 2 2\nh✝ : 2 * 2 ≤ card (Fin 5)\nh : 2 ≤ 3\nha : Even 2\n⊢ 0 ≠ 4",
"ppTerm": "?m.282",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"instDecidabl... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 344,
"column": 37
} | {
"line": 344,
"column": 43
} | {
"line": 345,
"column": 4
} | [
{
"pp": "g : Perm (Fin 5)\nh1 : g ≠ 1\nh_1 : g.cycleType.card = 2\nh2 : g.cycleType = Multiset.replicate 2 2\nh✝ : 2 * 2 ≤ card (Fin 5)\nh : 2 ≤ 3\nha : Even 2\nh04 : 0 ≠ 4\n⊢ 1 ≠ 3",
"ppTerm": "?m.292",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 349,
"column": 6
} | {
"line": 349,
"column": 12
} | {
"line": 350,
"column": 2
} | [
{
"pp": "case «2»\ng : Perm (Fin 5)\nh1 : g ≠ 1\nh_1 : g.cycleType.card = 2\nh2 : g.cycleType = Multiset.replicate 2 2\nh✝ : 2 * 2 ≤ card (Fin 5)\nh : 2 ≤ 3\nha : Even 2\nh04 : 0 ≠ 4\nh13 : 1 ≠ 3\n⊢ _root_.Disjoint {0, 4} {1, 3}",
"ppTerm": "?«2»",
"assigned": true,
"usedConstants": [
"Finset.... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 465,
"column": 9
} | {
"line": 465,
"column": 20
} | {
"line": 465,
"column": 21
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nG : Subgroup (Perm α)\nhG : G.index = 2\na✝ : Nontrivial α\nh : ∀ (g : Perm α), g.IsSwap → g ∈ G\n⊢ G = ⊤",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"eq_top_iff",
"congrArg",
"PartialOr... | [
"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nG : Subgroup (Perm α)\nhG : G.index = 2\na✝ : Nontrivial α\nh : ∀ (g : Perm α), g.IsSwap → g ∈ G\n⊢ ⊤ ≤ G"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 462,
"column": 73
} | {
"line": 465,
"column": 52
} | {
"line": 466,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nG : Subgroup (Perm α)\nhG : G.index = 2\na✝ : Nontrivial α\n⊢ ∃ g, g.IsSwap ∧ g ∉ G",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Push.not_exists._simp_1",
"Eq.mpr",
"Mathlib.Tactic.Push.n... | [] | by
by_contra! h
suffices G = ⊤ by rw [this, Subgroup.index_top] at hG; cases hG
rwa [eq_top_iff, ← closure_isSwap, G.closure_le] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.GroupAction.MultipleTransitivity | {
"line": 250,
"column": 2
} | {
"line": 251,
"column": 11
} | {
"line": 252,
"column": 2
} | [
{
"pp": "case inl\nG : Type u_1\nα : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G α\nh2 : IsMultiplyPretransitive G α 2\nthis : IsPretransitive G α\nB : Set α\nhB : IsBlock G B\nh : B.Subsingleton\n⊢ IsTrivialBlock B",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Set.univ",
... | [
"case inr\nG : Type u_1\nα : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G α\nh2 : IsMultiplyPretransitive G α 2\nthis : IsPretransitive G α\nB : Set α\nhB : IsBlock G B\nh : B.Nontrivial\n⊢ IsTrivialBlock B"
] | · left
exact h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup | {
"line": 528,
"column": 18
} | {
"line": 528,
"column": 34
} | {
"line": 528,
"column": 34
} | [
{
"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| s",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"congrArg",
"id",
"Set.image",
"Eq.symm",
"Set.image_id",
"Eq.trans",
"... | [
"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| id '' s"
] | ← Set.image_id s | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup | {
"line": 525,
"column": 43
} | {
"line": 530,
"column": 50
} | {
"line": 532,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nα : Type u_2\ninst✝ : MulAction G α\ns : Set α\n⊢ fixingSubgroup G s ≤ stabilizer G s",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"congrArg",
"fixingSubgroup",
"Membership.mem",
"Eq.rec"... | [] | by
intro k hk
rw [mem_stabilizer_iff]
conv_rhs => rw [← Set.image_id s]
apply Set.image_congr
simpa only [mem_fixingSubgroup_iff, id] using hk | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.GroupAction.MultiplePrimitivity | {
"line": 355,
"column": 10
} | {
"line": 355,
"column": 32
} | {
"line": 356,
"column": 8
} | [
{
"pp": "M : Type u_1\nα : Type u_2\ninst✝³ : Group M\ninst✝² : MulAction M α\nN : Type u_3\nβ : Type u_4\ninst✝¹ : Group N\ninst✝ : MulAction N β\nφ : M → N\nhφ : Function.Surjective φ\nf : α →ₑ[φ] β\nhf : Function.Bijective ⇑f\nn : ℕ\nH : IsMultiplyPreprimitive N β n\ns : Set α\nhs : s.encard + 1 = ↑n\nt : Se... | [] | exact ⟨⟨y, this⟩, rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.GroupAction.MultiplePrimitivity | {
"line": 355,
"column": 10
} | {
"line": 355,
"column": 32
} | {
"line": 356,
"column": 8
} | [
{
"pp": "M : Type u_1\nα : Type u_2\ninst✝³ : Group M\ninst✝² : MulAction M α\nN : Type u_3\nβ : Type u_4\ninst✝¹ : Group N\ninst✝ : MulAction N β\nφ : M → N\nhφ : Function.Surjective φ\nf : α →ₑ[φ] β\nhf : Function.Bijective ⇑f\nn : ℕ\nH : IsMultiplyPreprimitive N β n\ns : Set α\nhs : s.encard + 1 = ↑n\nt : Se... | [] | exact ⟨⟨y, this⟩, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupAction.MultiplePrimitivity | {
"line": 355,
"column": 10
} | {
"line": 355,
"column": 32
} | {
"line": 356,
"column": 8
} | [
{
"pp": "M : Type u_1\nα : Type u_2\ninst✝³ : Group M\ninst✝² : MulAction M α\nN : Type u_3\nβ : Type u_4\ninst✝¹ : Group N\ninst✝ : MulAction N β\nφ : M → N\nhφ : Function.Surjective φ\nf : α →ₑ[φ] β\nhf : Function.Bijective ⇑f\nn : ℕ\nH : IsMultiplyPreprimitive N β n\ns : Set α\nhs : s.encard + 1 = ↑n\nt : Se... | [] | exact ⟨⟨y, this⟩, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Perm.MaximalSubgroups | {
"line": 92,
"column": 6
} | {
"line": 92,
"column": 17
} | {
"line": 92,
"column": 18
} | [
{
"pp": "M : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns : Set α\nhs : ∀ a ∈ s, ∀ b ∈ s, ∃ g, g • a = b\nhs' : ∀ a ∈ sᶜ, ∀ b ∈ sᶜ, ∃ g, g • a = b\nhM : ∀ (a b : α) (g : M), a ∈ s → b ∈ sᶜ → g • a ≠ b\n⊢ stabilizer M s = ⊤",
"ppTerm": "?m.156",
"assigned": true,
"usedConstants"... | [
"M : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\ns : Set α\nhs : ∀ a ∈ s, ∀ b ∈ s, ∃ g, g • a = b\nhs' : ∀ a ∈ sᶜ, ∀ b ∈ sᶜ, ∃ g, g • a = b\nhM : ∀ (a b : α) (g : M), a ∈ s → b ∈ sᶜ → g • a ≠ b\n⊢ ⊤ ≤ stabilizer M s"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups | {
"line": 178,
"column": 6
} | {
"line": 178,
"column": 17
} | {
"line": 178,
"column": 18
} | [
{
"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\n⊢ G = ⊤",
"ppTerm": "?m.38",
... | [
"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\n⊢ ⊤ ≤ G"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Alternating.MaximalSubgroups | {
"line": 181,
"column": 2
} | {
"line": 181,
"column": 71
} | {
"line": 182,
"column": 2
} | [
{
"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\n⊢ alternatingGroup α ≤ Subgroup.map ... | [
"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\n⊢ g ∈ Subgroup.map (alternatingGroup α).subtype ... | apply alternatingGroup_le_of_isPreprimitive_of_isThreeCycle_mem _ hg3 | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.GroupTheory.GroupAction.SubMulAction.Combination | {
"line": 286,
"column": 6
} | {
"line": 288,
"column": 11
} | {
"line": 289,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_2\nn : ℕ\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα : 3 ≤ Nat.card α\nthis✝ :\n ∀ {α : Type u_2} {n : ℕ} [inst : DecidableEq α] [inst_1 : Fintype α],\n 3 ≤ Nat.card α → 2 * n ≤ Nat.card α → IsPretransitive ↥(alternatingGroup α) ↑(powersetCard α n)\nhn : Nat.card α < 2 * n\n... | [] | apply IsPretransitive.of_surjective_map
(mulActionHom_compl_bijective (alternatingGroup α) α _).surjective this
aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupAction.SubMulAction.Combination | {
"line": 286,
"column": 6
} | {
"line": 288,
"column": 11
} | {
"line": 289,
"column": 4
} | [
{
"pp": "case pos\nα : Type u_2\nn : ℕ\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα : 3 ≤ Nat.card α\nthis✝ :\n ∀ {α : Type u_2} {n : ℕ} [inst : DecidableEq α] [inst_1 : Fintype α],\n 3 ≤ Nat.card α → 2 * n ≤ Nat.card α → IsPretransitive ↥(alternatingGroup α) ↑(powersetCard α n)\nhn : Nat.card α < 2 * n\n... | [] | apply IsPretransitive.of_surjective_map
(mulActionHom_compl_bijective (alternatingGroup α) α _).surjective this
aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Perm.Cycle.PossibleTypes | {
"line": 67,
"column": 4
} | {
"line": 67,
"column": 23
} | {
"line": 68,
"column": 4
} | [
{
"pp": "case h.right.right\nα : Type u_2\ninst✝ : Fintype α\nc : List ℕ\nhc : c.sum ≤ Fintype.card α\nklift : (n : ℕ) → n < Fintype.card α → Fin (Fintype.card α) := ⋯\nklift' : (l : List ℕ) → (∀ a ∈ l, a < Fintype.card α) → List (Fin (Fintype.card α)) := ⋯\nhc'_lt : ∀ l ∈ c.ranges, ∀ n ∈ l, n < Fintype.card α\... | [
"case h.right.right\nα : Type u_2\ninst✝ : Fintype α\nc : List ℕ\nhc : c.sum ≤ Fintype.card α\nklift : (n : ℕ) → n < Fintype.card α → Fin (Fintype.card α) := fun n hn ↦ ⟨n, hn⟩\nklift' : (l : List ℕ) → (∀ a ∈ l, a < Fintype.card α) → List (Fin (Fintype.card α)) := fun l hl ↦ pmap klift l hl\nhc'_lt : ∀ l ∈ c.ranges... | intro u hu v hv huv | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 534,
"column": 4
} | {
"line": 538,
"column": 91
} | {
"line": 540,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\n⊢ _root_.Disjoint ofSubtype.range (noncommPiCoprod ⋯).range",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Subgroup.closure_eq",
"Eq.mpr",
"Equiv.Perm.support",
"Monoi... | [] | rw [noncommPiCoprod_range, ← ofSubtype.range.closure_eq]
simp only [zpowers_eq_closure, ← closure_iUnion]
apply disjoint_closure_of_disjoint_support
rintro - ⟨a, rfl⟩ - ⟨-, ⟨b, rfl⟩, ⟨-⟩⟩
exact (ofSubtype_support_disjoint a).mono_right (mem_cycleFactorsFinset_support_le b.2) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 534,
"column": 4
} | {
"line": 538,
"column": 91
} | {
"line": 540,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\n⊢ _root_.Disjoint ofSubtype.range (noncommPiCoprod ⋯).range",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Subgroup.closure_eq",
"Eq.mpr",
"Equiv.Perm.support",
"Monoi... | [] | rw [noncommPiCoprod_range, ← ofSubtype.range.closure_eq]
simp only [zpowers_eq_closure, ← closure_iUnion]
apply disjoint_closure_of_disjoint_support
rintro - ⟨a, rfl⟩ - ⟨-, ⟨b, rfl⟩, ⟨-⟩⟩
exact (ofSubtype_support_disjoint a).mono_right (mem_cycleFactorsFinset_support_le b.2) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 617,
"column": 2
} | {
"line": 620,
"column": 93
} | {
"line": 621,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\nk : Perm ↑(Function.fixedPoints ⇑g)\nv : (c : ↥g.cycleFactorsFinset) → ↥(zpowers ↑c)\nU : Set ↥g.cycleFactorsFinset := ↑Finset.univ\nhU : U.Pairwise fun i j ↦ (↑(v i)).Disjoint ↑(v j)\n⊢ ((kerParam g) (k, v)).cycleType = k.cycleType +... | [
"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\nk : Perm ↑(Function.fixedPoints ⇑g)\nv : (c : ↥g.cycleFactorsFinset) → ↥(zpowers ↑c)\nU : Set ↥g.cycleFactorsFinset := ↑Finset.univ\nhU : U.Pairwise fun i j ↦ (↑(v i)).Disjoint ↑(v j)\n⊢ (ofSubtype k).cycleType + ∑ i ∈ g.cycleFactorsFinset.attach... | rw [kerParam, MonoidHom.noncommCoprod_apply, ← Prod.fst_mul_snd ⟨k, v⟩, Prod.mk_mul_mk, mul_one,
one_mul, Finset.univ_eq_attach,
Disjoint.cycleType_mul (disjoint_ofSubtype_noncommPiCoprod g k v),
Subgroup.noncommPiCoprod_apply, Disjoint.cycleType_noncommProd hU, Finset.univ_eq_attach] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.SchurZassenhaus | {
"line": 281,
"column": 2
} | {
"line": 281,
"column": 28
} | {
"line": 282,
"column": 2
} | [
{
"pp": "case neg\nG : Type u\ninst✝¹ : Group G\nN : Subgroup G\ninst✝ : N.Normal\nhN : (Nat.card ↥N).Coprime N.index\nhN1 : ¬Nat.card ↥N = 0\n⊢ ∃ H, N.IsComplement' H",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Exists",
"Subgroup",
"instOfNatNat",
"dite",
... | [
"case pos\nG : Type u\ninst✝¹ : Group G\nN : Subgroup G\ninst✝ : N.Normal\nhN : (Nat.card ↥N).Coprime N.index\nhN1 : ¬Nat.card ↥N = 0\nhN2 : N.index = 0\n⊢ ∃ H, N.IsComplement' H",
"case neg\nG : Type u\ninst✝¹ : Group G\nN : Subgroup G\ninst✝ : N.Normal\nhN : (Nat.card ↥N).Coprime N.index\nhN1 : ¬Nat.card ↥N = 0... | by_cases hN2 : N.index = 0 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 8
} | {
"line": 103,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\nthis : Nontrivial α\n⊢ Nat.factorial 4 / 2 = 12",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"instHDiv",
"of_decide_eq_true",
"id",
"HDiv.hDiv",
"instOfNatNat",
"B... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.GroupTheory.SpecificGroups.Alternating.Simple | {
"line": 80,
"column": 8
} | {
"line": 80,
"column": 19
} | {
"line": 80,
"column": 20
} | [
{
"pp": "α : Type u_1\ninst✝² : Finite α\ninst✝¹ : DecidableEq α\ninst✝ : (s : Set α) → DecidablePred fun x ↦ x ∈ s\n⊢ ⨆ s, ofSubtype.range = ⊤",
"ppTerm": "?m.2243",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.range",
"eq_top_iff",
"congrArg",
"iSup",
... | [
"α : Type u_1\ninst✝² : Finite α\ninst✝¹ : DecidableEq α\ninst✝ : (s : Set α) → DecidablePred fun x ↦ x ∈ s\n⊢ ⊤ ≤ ⨆ s, ofSubtype.range"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour | {
"line": 107,
"column": 55
} | {
"line": 107,
"column": 61
} | {
"line": 107,
"column": 61
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\nS : Sylow 2 ↥(alternatingGroup α)\nthis : 12 = 2 ^ 2 * 3\n⊢ (2 ^ 2).Coprime 3",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Nat.Coprime",
"of_decide_eq_true",
"Nat.instMonoid",
... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour | {
"line": 107,
"column": 55
} | {
"line": 107,
"column": 61
} | {
"line": 107,
"column": 61
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\nS : Sylow 2 ↥(alternatingGroup α)\nthis : 12 = 2 ^ 2 * 3\n⊢ (2 ^ 2).Coprime 3",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Nat.Coprime",
"of_decide_eq_true",
"Nat.instMonoid",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour | {
"line": 107,
"column": 55
} | {
"line": 107,
"column": 61
} | {
"line": 107,
"column": 61
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\nS : Sylow 2 ↥(alternatingGroup α)\nthis : 12 = 2 ^ 2 * 3\n⊢ (2 ^ 2).Coprime 3",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Nat.Coprime",
"of_decide_eq_true",
"Nat.instMonoid",
... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour | {
"line": 109,
"column": 45
} | {
"line": 109,
"column": 51
} | {
"line": 109,
"column": 51
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\nS : Sylow 2 ↥(alternatingGroup α)\nthis : 12 = 2 ^ 2 * 3\n⊢ ¬2 ∣ 3",
"ppTerm": "?m.86",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"Dvd.dvd",
"of_decide_eq_true",
"Nat.decida... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour | {
"line": 109,
"column": 45
} | {
"line": 109,
"column": 51
} | {
"line": 109,
"column": 51
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\nS : Sylow 2 ↥(alternatingGroup α)\nthis : 12 = 2 ^ 2 * 3\n⊢ ¬2 ∣ 3",
"ppTerm": "?m.86",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"Dvd.dvd",
"of_decide_eq_true",
"Nat.decida... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour | {
"line": 109,
"column": 45
} | {
"line": 109,
"column": 51
} | {
"line": 109,
"column": 51
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\nS : Sylow 2 ↥(alternatingGroup α)\nthis : 12 = 2 ^ 2 * 3\n⊢ ¬2 ∣ 3",
"ppTerm": "?m.86",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"Dvd.dvd",
"of_decide_eq_true",
"Nat.decida... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.Alternating.Simple | {
"line": 112,
"column": 8
} | {
"line": 112,
"column": 19
} | {
"line": 112,
"column": 20
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n⊢ ⨆ s, (ofSubtype ↑s).range = ⊤",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.range",
"eq_top_iff",
"congrArg",
"iSup",
"Finset",
"PartialOrder.toPreorder",
... | [
"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n⊢ ⊤ ≤ ⨆ s, (ofSubtype ↑s).range"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour | {
"line": 129,
"column": 4
} | {
"line": 129,
"column": 10
} | {
"line": 131,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\nS : Sylow 2 ↥(alternatingGroup α)\n⊢ (1 +\n if ({2, 2}.sum ≤ 4 ∧ ∀ a ∈ {2, 2}, 2 ≤ a) ∧ Even ({2, 2}.sum + {2, 2}.card) then\n Nat.factorial 4 /\n ((4 - {2, 2}.sum).factorial * ({2, 2}.pro... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.GroupTheory.SpecificGroups.Alternating.Simple | {
"line": 130,
"column": 6
} | {
"line": 130,
"column": 17
} | {
"line": 130,
"column": 18
} | [
{
"pp": "α : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nhα : 5 ≤ Nat.card α\nhα' : Nat.card α ≠ 6\nN : Subgroup ↥(alternatingGroup α)\ninst✝ : N.Normal\nhN : Nontrivial ↥N\nthis : IsPreprimitive ↥(alternatingGroup α) ↑(Set.powersetCard α 3)\n⊢ N = ⊤",
"ppTerm": "?m.39",
"assigned": true,
... | [
"α : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nhα : 5 ≤ Nat.card α\nhα' : Nat.card α ≠ 6\nN : Subgroup ↥(alternatingGroup α)\ninst✝ : N.Normal\nhN : Nontrivial ↥N\nthis : IsPreprimitive ↥(alternatingGroup α) ↑(Set.powersetCard α 3)\n⊢ ⊤ ≤ N"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour | {
"line": 185,
"column": 2
} | {
"line": 185,
"column": 8
} | {
"line": 187,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\nthis : Monoid.exponent ↥(kleinFour α) = 1 ∨ Monoid.exponent ↥(kleinFour α) = 2\n⊢ ¬4 ≤ 1",
"ppTerm": "?m.406",
"assigned": true,
"usedConstants": [
"instDecidableNot",
"of_decide_eq_true",
"id",... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.GroupTheory.SpecificGroups.Alternating.Simple | {
"line": 168,
"column": 8
} | {
"line": 168,
"column": 19
} | {
"line": 168,
"column": 20
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nh5 : 5 ≤ Nat.card α\n⊢ ⨆ s, Subgroup.map (ofSubtype ↑s) (kleinFour ↥↑s) = ⊤",
"ppTerm": "?m.59",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"eq_top_iff",
"Subgroup.map",
"congrArg",
"iSup",
"Fin... | [
"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nh5 : 5 ≤ Nat.card α\n⊢ ⊤ ≤ ⨆ s, Subgroup.map (ofSubtype ↑s) (kleinFour ↥↑s)"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Alternating.Simple | {
"line": 185,
"column": 6
} | {
"line": 185,
"column": 17
} | {
"line": 185,
"column": 18
} | [
{
"pp": "α : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nhα : 5 ≤ Nat.card α\nhα' : Nat.card α ≠ 8\nN : Subgroup ↥(alternatingGroup α)\ninst✝ : N.Normal\nhN : Nontrivial ↥N\nthis : IsPreprimitive ↥(alternatingGroup α) ↑(Set.powersetCard α 4)\n⊢ N = ⊤",
"ppTerm": "?m.39",
"assigned": true,
... | [
"α : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nhα : 5 ≤ Nat.card α\nhα' : Nat.card α ≠ 8\nN : Subgroup ↥(alternatingGroup α)\ninst✝ : N.Normal\nhN : Nontrivial ↥N\nthis : IsPreprimitive ↥(alternatingGroup α) ↑(Set.powersetCard α 4)\n⊢ ⊤ ≤ N"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 112,
"column": 6
} | {
"line": 112,
"column": 42
} | {
"line": 114,
"column": 0
} | [
{
"pp": "case xa\nn : ℕ\ni : ZMod (2 * n)\n⊢ (xa i).inv * xa i = 1",
"ppTerm": "?xa",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"ZMod.commRing",
"sub_self",
"AddGroupWithOne.toAddGroup",
"CommSemiring.toSemiring",
"AddGroupWithOne.toAddMonoidWithOne",
... | [] | exact congr_arg a (sub_self (n + i)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 112,
"column": 6
} | {
"line": 112,
"column": 42
} | {
"line": 114,
"column": 0
} | [
{
"pp": "case xa\nn : ℕ\ni : ZMod (2 * n)\n⊢ (xa i).inv * xa i = 1",
"ppTerm": "?xa",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"ZMod.commRing",
"sub_self",
"AddGroupWithOne.toAddGroup",
"CommSemiring.toSemiring",
"AddGroupWithOne.toAddMonoidWithOne",
... | [] | exact congr_arg a (sub_self (n + i)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 112,
"column": 6
} | {
"line": 112,
"column": 42
} | {
"line": 114,
"column": 0
} | [
{
"pp": "case xa\nn : ℕ\ni : ZMod (2 * n)\n⊢ (xa i).inv * xa i = 1",
"ppTerm": "?xa",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"ZMod.commRing",
"sub_self",
"AddGroupWithOne.toAddGroup",
"CommSemiring.toSemiring",
"AddGroupWithOne.toAddMonoidWithOne",
... | [] | exact congr_arg a (sub_self (n + i)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.Alternating.Simple | {
"line": 226,
"column": 31
} | {
"line": 226,
"column": 37
} | {
"line": 226,
"column": 37
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n⊢ sign (swap 0 4 * swap 1 3) = 1",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"MonoidHom.instFunLike",
"HMul.hMul",
"of_decide_eq_true",
"MonoidHom",
"Monoid.toMulOneClass",
"instDecida... | [] | decide | Lean.Elab.Tactic.evalDecide | Lean.Parser.Tactic.decide |
Mathlib.GroupTheory.SpecificGroups.Alternating.Simple | {
"line": 226,
"column": 31
} | {
"line": 226,
"column": 37
} | {
"line": 226,
"column": 37
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n⊢ sign (swap 0 4 * swap 1 3) = 1",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"MonoidHom.instFunLike",
"HMul.hMul",
"of_decide_eq_true",
"MonoidHom",
"Monoid.toMulOneClass",
"instDecida... | [] | decide | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.Alternating.Simple | {
"line": 226,
"column": 31
} | {
"line": 226,
"column": 37
} | {
"line": 226,
"column": 37
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n⊢ sign (swap 0 4 * swap 1 3) = 1",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"MonoidHom.instFunLike",
"HMul.hMul",
"of_decide_eq_true",
"MonoidHom",
"Monoid.toMulOneClass",
"instDecida... | [] | decide | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer | {
"line": 282,
"column": 8
} | {
"line": 282,
"column": 19
} | {
"line": 282,
"column": 20
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh5 : 5 ≤ Nat.card α\nthis : closure {b | (↑b).IsThreeCycle} = ⊤\n⊢ _root_.commutator ↥(alternatingGroup α) = ⊤",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"eq_top_iff",
"congrArg",
"Part... | [
"α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nh5 : 5 ≤ Nat.card α\nthis : closure {b | (↑b).IsThreeCycle} = ⊤\n⊢ ⊤ ≤ _root_.commutator ↥(alternatingGroup α)"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.ZGroup | {
"line": 152,
"column": 4
} | {
"line": 173,
"column": 53
} | {
"line": 175,
"column": 0
} | [
{
"pp": "case ind.inr\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Finite G\ninst✝ : IsZGroup G\nH : Subgroup G\nhH : ∀ y < H, IsCyclic ↥⁅y, y⁆\nh : H ≠ ⊥\n⊢ IsCyclic ↥⁅H, H⁆",
"ppTerm": "?ind.inr",
"assigned": true,
"usedConstants": [
"Subgroup.instFiniteSubtypeMem",
"Iff.mpr",
"isCy... | [] | specialize hH ⁅H, H⁆ (IsSolvable.commutator_lt_of_ne_bot h)
replace hH : IsCyclic (⁅commutator H, commutator H⁆ : Subgroup H) := by
let f := Subgroup.equivMapOfInjective ⁅commutator H, commutator H⁆ _ H.subtype_injective
rw [Subgroup.map_commutator, Subgroup.map_subtype_commutator] at f
exact isCy... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.ZGroup | {
"line": 152,
"column": 4
} | {
"line": 173,
"column": 53
} | {
"line": 175,
"column": 0
} | [
{
"pp": "case ind.inr\nG : Type u_1\ninst✝² : Group G\ninst✝¹ : Finite G\ninst✝ : IsZGroup G\nH : Subgroup G\nhH : ∀ y < H, IsCyclic ↥⁅y, y⁆\nh : H ≠ ⊥\n⊢ IsCyclic ↥⁅H, H⁆",
"ppTerm": "?ind.inr",
"assigned": true,
"usedConstants": [
"Subgroup.instFiniteSubtypeMem",
"Iff.mpr",
"isCy... | [] | specialize hH ⁅H, H⁆ (IsSolvable.commutator_lt_of_ne_bot h)
replace hH : IsCyclic (⁅commutator H, commutator H⁆ : Subgroup H) := by
let f := Subgroup.equivMapOfInjective ⁅commutator H, commutator H⁆ _ H.subtype_injective
rw [Subgroup.map_commutator, Subgroup.map_subtype_commutator] at f
exact isCy... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.ZGroup | {
"line": 192,
"column": 4
} | {
"line": 192,
"column": 25
} | {
"line": 193,
"column": 2
} | [
{
"pp": "case pos\nG : Type u_1\ninst✝⁴ : Group G\np : ℕ\ninst✝³ : Fact (Nat.Prime p)\ninst✝² : IsCyclic G\nhG : IsPGroup p G\nK : Type u_4\ninst✝¹ : Group K\ninst✝ : MulDistribMulAction K G\nhGK : Subsingleton K ∧ Nonempty K\nhc : Nat.card G = 0\n⊢ (∀ (g : G) (k : K), k • g * g⁻¹ = 1) ∨ ∀ (g : G), ∃ k q, k • q... | [] | simp [← hGK.1.elim 1] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.SpecificGroups.ZGroup | {
"line": 258,
"column": 76
} | {
"line": 258,
"column": 87
} | {
"line": 259,
"column": 6
} | [
{
"pp": "case refine_1\nG : Type u_1\ninst✝³ : Group G\np : ℕ\ninst✝² : Fact (Nat.Prime p)\ninst✝¹ : Finite G\nP : Sylow p G\ninst✝ : IsCyclic ↥↑P\nQ : Sylow p ↥(Subgroup.normalizer ↑P) := P.subtype ⋯\nthis✝ : (↑Q).Normal\nthis : IsCyclic ↥↑Q\nh : Subgroup.centralizer ↑↑Q = ⊤\n⊢ Subgroup.normalizer ↑P ≤ Subgrou... | [
"case refine_1\nG : Type u_1\ninst✝³ : Group G\np : ℕ\ninst✝² : Fact (Nat.Prime p)\ninst✝¹ : Finite G\nP : Sylow p G\ninst✝ : IsCyclic ↥↑P\nQ : Sylow p ↥(Subgroup.normalizer ↑P) := P.subtype ⋯\nthis✝ : (↑Q).Normal\nthis : IsCyclic ↥↑Q\nh : ⊤ ≤ Subgroup.centralizer ↑↑Q\n⊢ Subgroup.normalizer ↑P ≤ Subgroup.centralize... | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Defs | {
"line": 189,
"column": 19
} | {
"line": 189,
"column": 71
} | {
"line": 191,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : IsEmpty β\nκ : Kernel α β\n⊢ κ.bound = 0",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure",
"congrArg",
"iSup",
"Set.univ",
"ProbabilityTheory.Ke... | [] | by simp [bound, Subsingleton.elim _ (0 : Measure β)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.SpecificGroups.ZGroup | {
"line": 327,
"column": 4
} | {
"line": 327,
"column": 65
} | {
"line": 328,
"column": 2
} | [
{
"pp": "case refine_1\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : Finite G\nhG : IsZGroup G\nH : Subgroup G\nhH : (commutator G).IsComplement' H\nh1 : Abelianization G ≃* ↥H\n⊢ (Nat.card ↥(commutator G)).Coprime (Nat.card ↥H)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Nat.Cop... | [] | exact Nat.card_congr h1.toEquiv ▸ hG.coprime_commutator_index | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Probability.Kernel.Composition.Comp | {
"line": 232,
"column": 11
} | {
"line": 232,
"column": 30
} | {
"line": 232,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nη : Kernel β γ\ninst✝¹ : IsSFiniteKernel η\nκ : Kernel α β\ninst✝ : IsSFiniteKernel κ\n⊢ IsSFiniteKernel (η ∘ₖ κ)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nη : Kernel β γ\ninst✝¹ : IsSFiniteKernel η\nκ : Kernel α β\ninst✝ : IsSFiniteKernel κ\n⊢ IsSFiniteKernel (η ∘ₖ Kernel.sum κ.seq)"
] | ← kernel_sum_seq κ, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Kernel.Composition.ParallelComp | {
"line": 154,
"column": 2
} | {
"line": 160,
"column": 43
} | {
"line": 162,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nmδ : MeasurableSpace δ\nκ : Kernel α β\nη : Kernel γ δ\nx : α × γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\n⊢ IsFiniteKernel (κ ∥ₖ η)",
"ppTerm": "?m.25",
... | [] | refine ⟨⟨κ.bound * η.bound, ENNReal.mul_lt_top κ.bound_lt_top η.bound_lt_top, fun a ↦ ?_⟩⟩
calc (κ ∥ₖ η) a Set.univ
_ = κ a.1 Set.univ * η a.2 Set.univ := parallelComp_apply_univ
_ ≤ κ.bound * η.bound := by
gcongr
· exact measure_le_bound κ a.1 Set.univ
· exact measure_le_bound η a.2 Set.univ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Composition.ParallelComp | {
"line": 154,
"column": 2
} | {
"line": 160,
"column": 43
} | {
"line": 162,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nmδ : MeasurableSpace δ\nκ : Kernel α β\nη : Kernel γ δ\nx : α × γ\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\n⊢ IsFiniteKernel (κ ∥ₖ η)",
"ppTerm": "?m.25",
... | [] | refine ⟨⟨κ.bound * η.bound, ENNReal.mul_lt_top κ.bound_lt_top η.bound_lt_top, fun a ↦ ?_⟩⟩
calc (κ ∥ₖ η) a Set.univ
_ = κ a.1 Set.univ * η a.2 Set.univ := parallelComp_apply_univ
_ ≤ κ.bound * η.bound := by
gcongr
· exact measure_le_bound κ a.1 Set.univ
· exact measure_le_bound η a.2 Set.univ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Composition.ParallelComp | {
"line": 171,
"column": 11
} | {
"line": 171,
"column": 30
} | {
"line": 171,
"column": 31
} | [
{
"pp": "case pos\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nmδ : MeasurableSpace δ\nκ : Kernel α β\nη : Kernel γ δ\nx : α × γ\nh✝ : IsSFiniteKernel κ\nh : IsSFiniteKernel η\n⊢ IsSFiniteKernel (κ ∥ₖ η)",
"ppTerm": "?pos✝",... | [
"case pos\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nmδ : MeasurableSpace δ\nκ : Kernel α β\nη : Kernel γ δ\nx : α × γ\nh✝ : IsSFiniteKernel κ\nh : IsSFiniteKernel η\n⊢ IsSFiniteKernel (Kernel.sum κ.seq ∥ₖ η)"
] | ← kernel_sum_seq κ, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 148,
"column": 2
} | {
"line": 153,
"column": 49
} | {
"line": 155,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nκ : Kernel α β\nη η' : Kernel (α × β) γ\ninst✝¹ : IsSFiniteKernel η\ninst✝ : IsSFiniteKernel η'\nh : ∀ (a : α), ∀ᵐ (b : β) ∂κ a, η (a, b) = η' (a, b)\n⊢ κ ⊗ₖ η = κ ⊗ₖ η'",
"ppTerm": "?m... | [] | by_cases hκ : IsSFiniteKernel κ
swap; · simp_rw [compProd_of_not_isSFiniteKernel_left _ _ hκ]
ext a s hs
rw [compProd_apply hs, compProd_apply hs]
refine lintegral_congr_ae ?_
filter_upwards [h a] with b hb using by rw [hb] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 148,
"column": 2
} | {
"line": 153,
"column": 49
} | {
"line": 155,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nκ : Kernel α β\nη η' : Kernel (α × β) γ\ninst✝¹ : IsSFiniteKernel η\ninst✝ : IsSFiniteKernel η'\nh : ∀ (a : α), ∀ᵐ (b : β) ∂κ a, η (a, b) = η' (a, b)\n⊢ κ ⊗ₖ η = κ ⊗ₖ η'",
"ppTerm": "?m... | [] | by_cases hκ : IsSFiniteKernel κ
swap; · simp_rw [compProd_of_not_isSFiniteKernel_left _ _ hκ]
ext a s hs
rw [compProd_apply hs, compProd_apply hs]
refine lintegral_congr_ae ?_
filter_upwards [h a] with b hb using by rw [hb] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Composition.KernelLemmas | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 51
} | {
"line": 90,
"column": 51
} | [
{
"pp": "case pos.e_f\nX : Type u_1\nY : Type u_2\nZ : Type u_3\nT : Type u_4\nmX : MeasurableSpace X\nmY : MeasurableSpace Y\nmZ : MeasurableSpace Z\nmT : MeasurableSpace T\nκ : Kernel X Y\nX' : Type u_5\nmX' : MeasurableSpace X'\nη : Kernel X' Z\ninst✝¹ : IsSFiniteKernel η\nξ : Kernel Z T\ninst✝ : IsSFiniteKe... | [
"case pos.e_f\nX : Type u_1\nY : Type u_2\nZ : Type u_3\nT : Type u_4\nmX : MeasurableSpace X\nmY : MeasurableSpace Y\nmZ : MeasurableSpace Z\nmT : MeasurableSpace T\nκ : Kernel X Y\nX' : Type u_5\nmX' : MeasurableSpace X'\nη : Kernel X' Z\ninst✝¹ : IsSFiniteKernel η\nξ : Kernel Z T\ninst✝ : IsSFiniteKernel ξ\nhκ :... | comp_apply' _ _ _ (measurable_prodMk_left hs) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Probability.Kernel.Composition.Prod | {
"line": 251,
"column": 2
} | {
"line": 252,
"column": 60
} | {
"line": 254,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nγ : Type u_4\nmγ : MeasurableSpace γ\nδ : Type u_6\nmδ : MeasurableSpace δ\nκ : Kernel α β\ninst✝² : IsSFiniteKernel κ\nμ : Measure γ\ninst✝¹ : SFinite μ\nη : Kernel β δ\ninst✝ : IsSFiniteKernel η\nx : α\ns : Set (γ × δ)\nms : ... | [] | simp_rw [comp_apply' _ _ _ ms, prod_apply, Measure.prod_apply_symm ms, const_apply,
lintegral_comp _ _ _ (measurable_measure_prodMk_right ms)] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 298,
"column": 2
} | {
"line": 298,
"column": 48
} | {
"line": 299,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nκ : Kernel α β\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝ : IsSFiniteKernel η\ns : Set β\nhs : MeasurableSet s\n⊢ κ.restrict hs ⊗ₖ η = (κ ⊗ₖ η).restrict ⋯",
"ppTerm": "?m.... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nκ : Kernel α β\ninst✝¹ : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝ : IsSFiniteKernel η\ns : Set β\nhs : MeasurableSet s\n⊢ κ.restrict hs ⊗ₖ η = κ.restrict hs ⊗ₖ η.restrict ⋯"
] | rw [← compProd_restrict hs MeasurableSet.univ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Probability.Kernel.Composition.IntegralCompProd | {
"line": 228,
"column": 83
} | {
"line": 228,
"column": 96
} | {
"line": 228,
"column": 96
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nE : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝³ : NormedAddCommGroup E\na : α\nκ : Kernel α β\ninst✝² : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝¹ : IsSFiniteKernel η\ninst✝ : NormedSpace ℝ E\ng : ↥(Lp E 1 ((κ ⊗... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nE : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝³ : NormedAddCommGroup E\na : α\nκ : Kernel α β\ninst✝² : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝¹ : IsSFiniteKernel η\ninst✝ : NormedSpace ℝ E\ng : ↥(Lp E 1 ((κ ⊗ₖ η) a))\nth... | ← ofReal_zero | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.Decomposition.IntegralRNDeriv | {
"line": 86,
"column": 4
} | {
"line": 88,
"column": 10
} | {
"line": 89,
"column": 4
} | [
{
"pp": "𝓧 : Type u_1\nm𝓧 : MeasurableSpace 𝓧\nμ ν : Measure 𝓧\nf : ℝ → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nhf_cvx : ConvexOn ℝ (Ici 0) f\nhf_cont : ContinuousWithinAt f (Ici 0) 0\nhf_int : Integrable (fun x ↦ f (μ.rnDeriv ν x).toReal) ν\nhμν : μ ≪ ν\nhν : ¬ν = 0\nthis✝ : NeZero ν\nμ' ... | [
"𝓧 : Type u_1\nm𝓧 : MeasurableSpace 𝓧\nμ ν : Measure 𝓧\nf : ℝ → ℝ\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nhf_cvx : ConvexOn ℝ (Ici 0) f\nhf_cont : ContinuousWithinAt f (Ici 0) 0\nhf_int : Integrable (fun x ↦ f (μ.rnDeriv ν x).toReal) ν\nhμν : μ ≪ ν\nhν : ¬ν = 0\nthis✝ : NeZero ν\nμ' : Measure 𝓧... | have h1 : μ'.rnDeriv ν' =ᵐ[ν] (ν univ)⁻¹ • μ.rnDeriv ν' := by
rwa [Measure.ae_ennreal_smul_measure_eq] at h1'
simp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Kernel.Composition.IntegralCompProd | {
"line": 286,
"column": 74
} | {
"line": 287,
"column": 86
} | {
"line": 287,
"column": 86
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nE : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : NormedAddCommGroup E\na : α\nκ : Kernel α β\nη : Kernel β γ\ninst✝ : NormedSpace ℝ E\nf : γ → E\nhf : AEStronglyMeasurable f ((η ∘ₖ κ) a)\n⊢ (fun x ↦ ∫ (y : γ), f y ∂... | [] | by
filter_upwards [ae_ae_of_ae_comp hf.ae_eq_mk] with _ hx using integral_congr_ae hx | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Composition.IntegralCompProd | {
"line": 409,
"column": 79
} | {
"line": 409,
"column": 92
} | {
"line": 409,
"column": 92
} | [
{
"pp": "case refine_1\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nE : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : NormedAddCommGroup E\na : α\nκ : Kernel α β\nη : Kernel β γ\ninst✝ : NormedSpace ℝ E\ng : ↥(Lp E 1 ((η ∘ₖ κ) a))\nthis : ∀ (i : ↥(Lp E 1 ((η ∘ₖ κ) a)))... | [
"case refine_1\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nE : Type u_4\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\ninst✝¹ : NormedAddCommGroup E\na : α\nκ : Kernel α β\nη : Kernel β γ\ninst✝ : NormedSpace ℝ E\ng : ↥(Lp E 1 ((η ∘ₖ κ) a))\nthis : ∀ (i : ↥(Lp E 1 ((η ∘ₖ κ) a))), Measurable... | ← ofReal_zero | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 565,
"column": 25
} | {
"line": 565,
"column": 42
} | {
"line": 565,
"column": 43
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nκ : Kernel α β\nη : Kernel (α × β) γ\ninst✝¹ : IsSFiniteKernel κ\ninst✝ : IsSFiniteKernel η\nx : α\ns : Set β\nhs : MeasurableSet s\nh_eq : ∀ (b : β), (η (x, b)) {c | b ∈ s} = s.indicator (... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nκ : Kernel α β\nη : Kernel (α × β) γ\ninst✝¹ : IsSFiniteKernel κ\ninst✝ : IsSFiniteKernel η\nx : α\ns : Set β\nhs : MeasurableSet s\nh_eq : ∀ (b : β), (η (x, b)) {c | b ∈ s} = s.indicator (fun b ↦ (η (... | Set.mem_setOf_eq, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.Tilted | {
"line": 369,
"column": 2
} | {
"line": 375,
"column": 35
} | {
"line": 377,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\ninst✝ : SigmaFinite μ\nhf : Integrable (fun x ↦ rexp (f x)) μ\n⊢ (fun x ↦ log ((μ.tilted f).rnDeriv μ x).toReal) =ᵐ[μ] fun x ↦ f x - log (∫ (x : α), rexp (f x) ∂μ)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
... | [] | cases eq_zero_or_neZero μ with
| inl h => simp_rw [h, ae_zero, Filter.EventuallyEq]; exact Filter.eventually_bot
| inr h0 =>
have hf' : AEMeasurable f μ := aemeasurable_of_aemeasurable_exp hf.1.aemeasurable
filter_upwards [rnDeriv_tilted_left_self hf'] with x hx
rw [hx, ENNReal.toReal_ofReal (by positiv... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.MeasureTheory.Measure.Tilted | {
"line": 369,
"column": 2
} | {
"line": 375,
"column": 35
} | {
"line": 377,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\ninst✝ : SigmaFinite μ\nhf : Integrable (fun x ↦ rexp (f x)) μ\n⊢ (fun x ↦ log ((μ.tilted f).rnDeriv μ x).toReal) =ᵐ[μ] fun x ↦ f x - log (∫ (x : α), rexp (f x) ∂μ)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
... | [] | cases eq_zero_or_neZero μ with
| inl h => simp_rw [h, ae_zero, Filter.EventuallyEq]; exact Filter.eventually_bot
| inr h0 =>
have hf' : AEMeasurable f μ := aemeasurable_of_aemeasurable_exp hf.1.aemeasurable
filter_upwards [rnDeriv_tilted_left_self hf'] with x hx
rw [hx, ENNReal.toReal_ofReal (by positiv... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Tilted | {
"line": 369,
"column": 2
} | {
"line": 375,
"column": 35
} | {
"line": 377,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\ninst✝ : SigmaFinite μ\nhf : Integrable (fun x ↦ rexp (f x)) μ\n⊢ (fun x ↦ log ((μ.tilted f).rnDeriv μ x).toReal) =ᵐ[μ] fun x ↦ f x - log (∫ (x : α), rexp (f x) ∂μ)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
... | [] | cases eq_zero_or_neZero μ with
| inl h => simp_rw [h, ae_zero, Filter.EventuallyEq]; exact Filter.eventually_bot
| inr h0 =>
have hf' : AEMeasurable f μ := aemeasurable_of_aemeasurable_exp hf.1.aemeasurable
filter_upwards [rnDeriv_tilted_left_self hf'] with x hx
rw [hx, ENNReal.toReal_ofReal (by positiv... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.LogLikelihoodRatio | {
"line": 136,
"column": 2
} | {
"line": 139,
"column": 6
} | {
"line": 141,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\nc : ℝ≥0\nhc : c ≠ 0\n⊢ llr μ (c • ν) =ᵐ[μ] fun x ↦ llr μ ν x - log ↑c",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"ENNReal.coe_ne_top._s... | [] | rw [← Measure.coe_nnreal_smul]
filter_upwards [llr_smul_right hμν (c : ℝ≥0∞) (by simpa) (by simp)] with x hx
rw [hx]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.LogLikelihoodRatio | {
"line": 136,
"column": 2
} | {
"line": 139,
"column": 6
} | {
"line": 141,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\nc : ℝ≥0\nhc : c ≠ 0\n⊢ llr μ (c • ν) =ᵐ[μ] fun x ↦ llr μ ν x - log ↑c",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"ENNReal.coe_ne_top._s... | [] | rw [← Measure.coe_nnreal_smul]
filter_upwards [llr_smul_right hμν (c : ℝ≥0∞) (by simpa) (by simp)] with x hx
rw [hx]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.Trim | {
"line": 40,
"column": 13
} | {
"line": 40,
"column": 30
} | {
"line": 40,
"column": 31
} | [
{
"pp": "α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nf : α → ℝ≥0∞\nhf : Measurable f\na : ℝ≥0∞\nh_meas_eq : μ {x | ¬f x ≤ a} = (μ.trim hm) {x | ¬f x ≤ a}\n⊢ a ∈ {a | ∀ᵐ (n : α) ∂μ.trim hm, f n ≤ a} ↔ a ∈ {a | ∀ᵐ (n : α) ∂μ, f n ≤ a}",
"ppTerm": "?m.75",
"assigned": true,
"use... | [
"α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nf : α → ℝ≥0∞\nhf : Measurable f\na : ℝ≥0∞\nh_meas_eq : μ {x | ¬f x ≤ a} = (μ.trim hm) {x | ¬f x ≤ a}\n⊢ (∀ᵐ (n : α) ∂μ.trim hm, f n ≤ a) ↔ ∀ᵐ (n : α) ∂μ, f n ≤ a"
] | Set.mem_setOf_eq, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.InformationTheory.KullbackLeibler.Basic | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 8
} | {
"line": 136,
"column": 4
} | [
{
"pp": "case neg\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nhμν : μ ≪ ν\nh_int_iff :\n ∫⁻ (a : α), ENNReal.ofReal (klFun (μ.rnDeriv ν a).toReal) ∂ν = ∞ ↔\n ¬Integrable (fun x ↦ klFun (μ.rnDeriv ν x).toReal) ν\nh_int : ¬Integrable (llr μ ν)... | [
"case neg\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nhμν : μ ≪ ν\nh_int_iff :\n ∫⁻ (a : α), ENNReal.ofReal (klFun (μ.rnDeriv ν a).toReal) ∂ν = ∞ ↔\n ¬Integrable (fun x ↦ klFun (μ.rnDeriv ν x).toReal) ν\nh_int : ¬Integrable (llr μ ν) μ\n⊢ (if μ ... | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.InformationTheory.KullbackLeibler.Basic | {
"line": 172,
"column": 2
} | {
"line": 177,
"column": 42
} | {
"line": 179,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nh : μ ≪ ν\n⊢ (klDiv μ ν).toReal = ∫ (x : α), klFun (μ.rnDeriv ν x).toReal ∂ν",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"InformationTheory.integrable_klFun_rnDe... | [] | by_cases h_int : Integrable (llr μ ν) μ
· rw [klDiv_eq_integral_klFun, if_pos ⟨h, h_int⟩, ENNReal.toReal_ofReal]
exact integral_nonneg fun _ ↦ klFun_nonneg ENNReal.toReal_nonneg
· rw [integral_undef]
· rw [klDiv_of_not_integrable h_int, ENNReal.toReal_top]
· rwa [integrable_klFun_rnDeriv_iff h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.InformationTheory.KullbackLeibler.Basic | {
"line": 172,
"column": 2
} | {
"line": 177,
"column": 42
} | {
"line": 179,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nh : μ ≪ ν\n⊢ (klDiv μ ν).toReal = ∫ (x : α), klFun (μ.rnDeriv ν x).toReal ∂ν",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"InformationTheory.integrable_klFun_rnDe... | [] | by_cases h_int : Integrable (llr μ ν) μ
· rw [klDiv_eq_integral_klFun, if_pos ⟨h, h_int⟩, ENNReal.toReal_ofReal]
exact integral_nonneg fun _ ↦ klFun_nonneg ENNReal.toReal_nonneg
· rw [integral_undef]
· rw [klDiv_of_not_integrable h_int, ENNReal.toReal_top]
· rwa [integrable_klFun_rnDeriv_iff h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.InformationTheory.KullbackLeibler.Basic | {
"line": 232,
"column": 6
} | {
"line": 232,
"column": 43
} | {
"line": 232,
"column": 44
} | [
{
"pp": "case neg\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nhμν : μ ≪ ν\nh_int : Integrable (llr μ ν) μ\nc : ℝ≥0\nhc : ¬c = 0\n⊢ (klDiv (c • μ) (c • ν)).toReal = ↑c * (klDiv μ ν).toReal",
"ppTerm": "?neg✝",
"assigned": true,
"usedC... | [
"case neg\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nhμν : μ ≪ ν\nh_int : Integrable (llr μ ν) μ\nc : ℝ≥0\nhc : ¬c = 0\n⊢ ↑c * (klDiv (c⁻¹ • c • μ) ν).toReal = ↑c * (klDiv μ ν).toReal",
"case neg.hμν\nα : Type u_1\nmα : MeasurableSpace α\nμ ν : M... | toReal_klDiv_smul_right_eq_smul_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.AEMeasurable | {
"line": 345,
"column": 2
} | {
"line": 345,
"column": 83
} | {
"line": 346,
"column": 2
} | [
{
"pp": "α : Type u_1\nF : Type u_2\np : ℝ≥0∞\ninst✝¹ : NormedAddCommGroup F\nm m0 : MeasurableSpace α\nμ : Measure α\none_le_p : Fact (1 ≤ p)\ninst✝ : NormedSpace ℝ F\nhm : m ≤ m0\nf : α → F\nhf : MemLp f p (μ.trim hm)\n⊢ ↑↑↑((lpMeasToLpTrimLie F ℝ p μ hm).symm (MemLp.toLp f hf)) =ᵐ[μ] ↑↑(MemLp.toLp f ⋯)",
... | [
"α : Type u_1\nF : Type u_2\np : ℝ≥0∞\ninst✝¹ : NormedAddCommGroup F\nm m0 : MeasurableSpace α\nμ : Measure α\none_le_p : Fact (1 ≤ p)\ninst✝ : NormedSpace ℝ F\nhm : m ≤ m0\nf : α → F\nhf : MemLp f p (μ.trim hm)\n⊢ ↑↑↑(lpTrimToLpMeas F ℝ p μ hm (MemLp.toLp f hf)) =ᵐ[μ] ↑↑(MemLp.toLp f ⋯)"
] | change lpTrimToLpMeas F ℝ p μ hm (MemLp.toLp f hf) =ᵐ[μ] (MemLp.toLp f _ : α → F) | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2 | {
"line": 127,
"column": 2
} | {
"line": 127,
"column": 6
} | {
"line": 128,
"column": 2
} | [
{
"pp": "α : Type u_1\nE : Type u_2\n𝕜 : Type u_7\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nf g : ↥(Lp E 2 μ)\nhg : AEStronglyMeasurable (↑↑g) μ\n⊢ ⟪↑((condExpL2 E 𝕜 hm) f), g⟫ = ⟪f, g⟫",
... | [
"α : Type u_1\nE : Type u_2\n𝕜 : Type u_7\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nm m0 : MeasurableSpace α\nμ : Measure α\nhm : m ≤ m0\nf g : ↥(Lp E 2 μ)\nhg : AEStronglyMeasurable (↑↑g) μ\n⊢ ⟪f, g⟫ = ⟪↑((condExpL2 E 𝕜 hm) f), g⟫"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.InformationTheory.KullbackLeibler.Basic | {
"line": 349,
"column": 2
} | {
"line": 358,
"column": 58
} | {
"line": 360,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nhμν : μ ≪ ν\nh_int : Integrable (llr μ ν) μ\n⊢ μ.real univ * log (μ.real univ / ν.real univ) + ν.real univ - μ.real univ ≤ (klDiv μ ν).toReal",
"ppTerm": "?m.51",
"assigned": true,
... | [] | by_cases hμ : μ = 0
· simp [hμ, measureReal_def]
by_cases hν : ν = 0
· refine absurd ?_ hμ
rw [hν] at hμν
exact Measure.absolutelyContinuous_zero_iff.mp hμν
refine (le_of_eq ?_).trans (mul_klFun_le_toReal_klDiv hμν h_int)
have : ν.real univ * (μ.real univ / ν.real univ) = μ.real univ := by
rw [mul... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.InformationTheory.KullbackLeibler.Basic | {
"line": 349,
"column": 2
} | {
"line": 358,
"column": 58
} | {
"line": 360,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nhμν : μ ≪ ν\nh_int : Integrable (llr μ ν) μ\n⊢ μ.real univ * log (μ.real univ / ν.real univ) + ν.real univ - μ.real univ ≤ (klDiv μ ν).toReal",
"ppTerm": "?m.51",
"assigned": true,
... | [] | by_cases hμ : μ = 0
· simp [hμ, measureReal_def]
by_cases hν : ν = 0
· refine absurd ?_ hμ
rw [hν] at hμν
exact Measure.absolutelyContinuous_zero_iff.mp hμν
refine (le_of_eq ?_).trans (mul_klFun_le_toReal_klDiv hμν h_int)
have : ν.real univ * (μ.real univ / ν.real univ) = μ.real univ := by
rw [mul... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic | {
"line": 495,
"column": 60
} | {
"line": 499,
"column": 40
} | {
"line": 501,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_3\nm m₀ : MeasurableSpace α\nμ : Measure α\nf : α → E\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : CompleteSpace E\ninst✝³ : PartialOrder E\ninst✝² : ClosedIciTopology E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : IsOrderedModule ℝ E\nhf : 0 ≤ᵐ[μ] f\n⊢ 0 ≤ᵐ[μ]... | [] | by
by_cases hfint : Integrable f μ
· rw [(condExp_zero.symm : (0 : α → E) = μ[0 | m])]
exact condExp_mono (integrable_zero _ _ _) hfint hf
· rw [condExp_of_not_integrable hfint] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.AffineSpace.Matrix | {
"line": 81,
"column": 8
} | {
"line": 81,
"column": 19
} | {
"line": 81,
"column": 20
} | [
{
"pp": "ι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : AffineSpace V P\ninst✝⁵ : Ring k\ninst✝⁴ : Module k V\nb : AffineBasis ι k P\nι' : Type u_1\ninst✝³ : Finite ι\ninst✝² : Fintype ι'\ninst✝¹ : DecidableEq ι\ninst✝ : Nontrivial k\np : ι' → P\nA : Matrix ι ι' k\nhA : A ... | [
"ι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : AffineSpace V P\ninst✝⁵ : Ring k\ninst✝⁴ : Module k V\nb : AffineBasis ι k P\nι' : Type u_1\ninst✝³ : Finite ι\ninst✝² : Fintype ι'\ninst✝¹ : DecidableEq ι\ninst✝ : Nontrivial k\np : ι' → P\nA : Matrix ι ι' k\nhA : A * b.toMatrix... | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Shift | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 40
} | {
"line": 140,
"column": 41
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring k\ninst✝⁴ : AddCommGroup V\ninst✝³ : AddTorsor V P\ninst✝² : Module k V\nι : Type u_4\ninst✝¹ : Fintype ι\ninst✝ : Nontrivial ι\np : ι → P\nh : AffineIndependent k p\ni : ι\nw : ι → k\nhw : ∑ i, w i = 1\nc : k\nj : ι\nhj : j ≠ i\nt : Finset ι\nw' ... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring k\ninst✝⁴ : AddCommGroup V\ninst✝³ : AddTorsor V P\ninst✝² : Module k V\nι : Type u_4\ninst✝¹ : Fintype ι\ninst✝ : Nontrivial ι\np : ι → P\nh : AffineIndependent k p\ni : ι\nw : ι → k\nhw : ∑ i, w i = 1\nc : k\nj : ι\nhj : j ≠ i\nt : Finset ι\nw' : ι → k\nht✝... | weightedVSub_vadd_affineCombination, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Shift | {
"line": 225,
"column": 2
} | {
"line": 227,
"column": 71
} | {
"line": 228,
"column": 2
} | [
{
"pp": "case refine_2\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : LinearOrder k\ninst✝ : IsOrderedRing k\nn : ℕ\ni : Fin n\nx : k\nhxpos : 0 < x\nhx1 : x ≤ 1\nw : Fin n → k\nhw : ∑ i, w i = 1\nhi : w i = 1 - x\nhj : ∀ (j : Fin n), w j ∈ Set.Icc 0 1\nj : Fin n\nhji : j ≠ i\n⊢ x⁻¹ * w j ≤ 1",
"ppTerm": "?refin... | [
"case refine_3\nk : Type u_1\ninst✝² : Field k\ninst✝¹ : LinearOrder k\ninst✝ : IsOrderedRing k\nn : ℕ\ni : Fin n\nx : k\nhxpos : 0 < x\nhx1 : x ≤ 1\nw : Fin n → k\nhw : ∑ i, w i = 1\nhi : w i = 1 - x\nhj : ∀ (j : Fin n), j ≠ i → x⁻¹ * w j ∈ Set.Icc 0 1\n⊢ ∀ (j : Fin n), w j ∈ Set.Icc 0 1"
] | · rw [eq_sub_iff_add_eq, add_comm, ← eq_sub_iff_add_eq] at hi
rw [inv_mul_le_one₀ hxpos, hi, le_sub_iff_add_le, ← hw]
exact add_le_sum (fun i _ ↦ (hj i).1) (mem_univ j) (mem_univ i) hji | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Shift | {
"line": 222,
"column": 2
} | {
"line": 231,
"column": 34
} | {
"line": 233,
"column": 0
} | [
{
"pp": "k : Type u_1\ninst✝² : Field k\ninst✝¹ : LinearOrder k\ninst✝ : IsOrderedRing k\nn : ℕ\ni : Fin n\nx : k\nhxpos : 0 < x\nhx1 : x ≤ 1\nw : Fin n → k\nhw : ∑ i, w i = 1\n⊢ (∀ (j : Fin n), w j ∈ Set.Icc 0 1) ∧ w i = 1 - x ↔\n (∀ (j : Fin n), j ≠ i → x⁻¹ * w j ∈ Set.Icc 0 1) ∧ x⁻¹ * (w i - 1) + 1 = 0",
... | [] | rw [show x⁻¹ * (w i - 1) + 1 = 0 ↔ w i = 1 - x by grind]
refine and_congr_left fun hi ↦ ⟨fun hj j hji ↦ ⟨?_, ?_⟩, fun hj ↦ ?_⟩
· exact mul_nonneg (by simpa using hxpos.le) (hj j).1
· rw [eq_sub_iff_add_eq, add_comm, ← eq_sub_iff_add_eq] at hi
rw [inv_mul_le_one₀ hxpos, hi, le_sub_iff_add_le, ← hw]
exact a... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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