module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 81,
"column": 2
} | {
"line": 85,
"column": 50
} | {
"line": 87,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nX : Type u_2\ninst✝ : MulAction G X\n⊢ Setoid.IsPartition (range fun a ↦ orbit G a)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"MulAction.nonempty_orbit",
"False",
"Set.Nonempty.ne_empty",
"Set.PairwiseDisjoint.isPartitio... | [] | apply orbit.pairwiseDisjoint.isPartition_of_exists_of_ne_empty
· intro x
exact ⟨_, ⟨x, rfl⟩, mem_orbit_self x⟩
· rintro ⟨a, ha : orbit G a = ∅⟩
exact (MulAction.nonempty_orbit a).ne_empty ha | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 384,
"column": 6
} | {
"line": 384,
"column": 18
} | {
"line": 385,
"column": 4
} | [
{
"pp": "case f\nG : Type u_1\ninst✝¹ : Group G\nX : Type u_2\ninst✝ : MulAction G X\nB : Set X\nH : Subgroup G\nhB : IsBlock (↥H) B\ng : G\nh' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)\nh : G\nhH : h ∈ H\nhh : g * h * g⁻¹ = ↑h'\nthis : h' • g • B = g • h • B\n⊢ ⟨h, hH⟩ • B = B → g • h • B = g ... | [] | intro; congr | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 384,
"column": 6
} | {
"line": 384,
"column": 18
} | {
"line": 385,
"column": 4
} | [
{
"pp": "case f\nG : Type u_1\ninst✝¹ : Group G\nX : Type u_2\ninst✝ : MulAction G X\nB : Set X\nH : Subgroup G\nhB : IsBlock (↥H) B\ng : G\nh' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)\nh : G\nhH : h ∈ H\nhh : g * h * g⁻¹ = ↑h'\nthis : h' • g • B = g • h • B\n⊢ ⟨h, hH⟩ • B = B → g • h • B = g ... | [] | intro; congr | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 388,
"column": 58
} | {
"line": 388,
"column": 78
} | {
"line": 388,
"column": 78
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nX : Type u_2\ninst✝ : MulAction G X\nB : Set X\nH : Subgroup G\nhB : IsBlock (↥H) B\ng : G\nh' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)\nh : G\nhH : h ∈ H\nhh : g * h * g⁻¹ = ↑h'\n⊢ (g * h * g⁻¹ * g) • B = (g * h) • B",
"ppTerm": "?m.163",
"assi... | [
"G : Type u_1\ninst✝¹ : Group G\nX : Type u_2\ninst✝ : MulAction G X\nB : Set X\nH : Subgroup G\nhB : IsBlock (↥H) B\ng : G\nh' : ↥(Subgroup.map (MulEquiv.toMonoidHom (MulAut.conj g)) H)\nh : G\nhH : h ∈ H\nhh : g * h * g⁻¹ = ↑h'\n⊢ (g * h) • B = (g * h) • B"
] | inv_mul_cancel_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.Primitive | {
"line": 217,
"column": 4
} | {
"line": 217,
"column": 52
} | {
"line": 218,
"column": 2
} | [
{
"pp": "case mp\nM : Type u_3\ninst✝³ : Group M\nα : Type u_4\ninst✝² : MulAction M α\nN : Type u_5\nβ : Type u_6\ninst✝¹ : Group N\ninst✝ : MulAction N β\nφ : M → N\nf : α →ₑ[φ] β\nhφ : Function.Surjective φ\nhf : Function.Bijective ⇑f\na✝ : IsPreprimitive M α\n⊢ IsPreprimitive N β",
"ppTerm": "?mp",
... | [] | apply IsPreprimitive.of_surjective hf.surjective | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 590,
"column": 4
} | {
"line": 592,
"column": 57
} | {
"line": 594,
"column": 0
} | [
{
"pp": "case mpr\nG : Type u_1\ninst✝¹ : Group G\nX : Type u_2\ninst✝ : MulAction G X\nB✝ : Set X\nhtGX : IsPretransitive G X\na : X\nB : Set X\nha : a ∈ B\nhB : IsBlock G B\nB' : Set X\nha' : a ∈ B'\nhB' : IsBlock G B'\n⊢ B ⊆ B' → stabilizer G B ≤ stabilizer G B'",
"ppTerm": "?mpr",
"assigned": true,
... | [] | · intro hBB' g hgB
apply hB'.smul_eq_of_mem ha'
exact hBB' <| hgB.symm ▸ (Set.smul_mem_smul_set ha) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
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": 264,
"column": 64
} | {
"line": 267,
"column": 47
} | {
"line": 269,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\n⊢ {g | g.cycleType = {2, 2}} ⊆ ↑(alternatingGroup α)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Multiset.sum",
"Units.val",
"SetLike.mem_coe._simp_1",
"Equiv.Perm.cycleType",
"Equiv.Perm.s... | [] | by
intro g hg
rw [Set.mem_setOf_eq] at hg
simp [sign_of_cycleType, hg, ← Units.val_inj] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 284,
"column": 21
} | {
"line": 284,
"column": 35
} | {
"line": 284,
"column": 35
} | [
{
"pp": "c g' : Perm (Fin 5)\nh : 3 ∈ (c * g').cycleType\nhd : c.Disjoint g'\nleft✝ : c.IsCycle\nh3 : c.support.card = 3\n⊢ (c * (g' * (g' * c))).IsThreeCycle",
"ppTerm": "?m.80",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"HMul.hMul",
"congrArg",
... | [
"c g' : Perm (Fin 5)\nh : 3 ∈ (c * g').cycleType\nhd : c.Disjoint g'\nleft✝ : c.IsCycle\nh3 : c.support.card = 3\n⊢ (c * (g' * g' * c)).IsThreeCycle"
] | ← mul_assoc g' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.SubMulAction.OfFixingSubgroup | {
"line": 157,
"column": 2
} | {
"line": 157,
"column": 82
} | {
"line": 159,
"column": 0
} | [
{
"pp": "M : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\na : α\ns : Set ↥(ofStabilizer M a)\nm : M\n⊢ m ∈ fixingSubgroup M (insert a ((fun x ↦ ↑x) '' s)) ↔\n m ∈ Subgroup.map (stabilizer M a).subtype (fixingSubgroup (↥(stabilizer M a)) s)",
"ppTerm": "?m.42",
"assigned": true,
... | [] | simp [mem_fixingSubgroup_iff, mem_ofStabilizer_iff, subgroup_smul_def, and_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.GroupAction.MultiplePrimitivity | {
"line": 135,
"column": 6
} | {
"line": 135,
"column": 60
} | {
"line": 136,
"column": 4
} | [
{
"pp": "case mpr.left\nM : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\nh : IsPreprimitive M α\n⊢ IsMultiplyPretransitive M α 1",
"ppTerm": "?mpr.left",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"MulAction.IsPretransitive",
"DivInvMonoid.toMonoid",
... | [] | exact is_one_pretransitive_iff.mpr h.toIsPretransitive | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.GroupAction.MultiplePrimitivity | {
"line": 135,
"column": 6
} | {
"line": 135,
"column": 60
} | {
"line": 136,
"column": 4
} | [
{
"pp": "case mpr.left\nM : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\nh : IsPreprimitive M α\n⊢ IsMultiplyPretransitive M α 1",
"ppTerm": "?mpr.left",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"MulAction.IsPretransitive",
"DivInvMonoid.toMonoid",
... | [] | exact is_one_pretransitive_iff.mpr h.toIsPretransitive | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupAction.MultiplePrimitivity | {
"line": 135,
"column": 6
} | {
"line": 135,
"column": 60
} | {
"line": 136,
"column": 4
} | [
{
"pp": "case mpr.left\nM : Type u_1\nα : Type u_2\ninst✝¹ : Group M\ninst✝ : MulAction M α\nh : IsPreprimitive M α\n⊢ IsMultiplyPretransitive M α 1",
"ppTerm": "?mpr.left",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"MulAction.IsPretransitive",
"DivInvMonoid.toMonoid",
... | [] | exact is_one_pretransitive_iff.mpr h.toIsPretransitive | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.MultiplePrimitivity | {
"line": 169,
"column": 4
} | {
"line": 206,
"column": 32
} | {
"line": 208,
"column": 0
} | [
{
"pp": "case mpr\nM : Type u_1\nα : Type u_2\ninst✝² : Group M\ninst✝¹ : MulAction M α\ninst✝ : IsPretransitive M α\nn : ℕ\nhn : 1 ≤ n\na : α\n⊢ IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(ofStabilizer M a)) n → IsMultiplyPreprimitive M α n.succ",
"ppTerm": "?mpr",
"assigned": true,
"usedConstant... | [] | intro H
rw [isMultiplyPreprimitive_iff]
constructor
· exact ofStabilizer.isMultiplyPretransitive.mpr H.isMultiplyPretransitive
· intro s hs
have : ∃ b : α, b ∈ s := by
rw [← Set.nonempty_def, Set.nonempty_iff_ne_empty]
intro h
apply not_lt.mpr hn
rw [h, Set.encard_e... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupAction.MultiplePrimitivity | {
"line": 169,
"column": 4
} | {
"line": 206,
"column": 32
} | {
"line": 208,
"column": 0
} | [
{
"pp": "case mpr\nM : Type u_1\nα : Type u_2\ninst✝² : Group M\ninst✝¹ : MulAction M α\ninst✝ : IsPretransitive M α\nn : ℕ\nhn : 1 ≤ n\na : α\n⊢ IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(ofStabilizer M a)) n → IsMultiplyPreprimitive M α n.succ",
"ppTerm": "?mpr",
"assigned": true,
"usedConstant... | [] | intro H
rw [isMultiplyPreprimitive_iff]
constructor
· exact ofStabilizer.isMultiplyPretransitive.mpr H.isMultiplyPretransitive
· intro s hs
have : ∃ b : α, b ∈ s := by
rw [← Set.nonempty_def, Set.nonempty_iff_ne_empty]
intro h
apply not_lt.mpr hn
rw [h, Set.encard_e... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.Period | {
"line": 106,
"column": 2
} | {
"line": 106,
"column": 73
} | {
"line": 108,
"column": 0
} | [
{
"pp": "α : Type v\nM : Type u\ninst✝¹ : Monoid M\ninst✝ : MulAction M α\nm : M\na : α\n⊢ period m a ∣ Monoid.exponent M",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"instHSMul",
"Dvd.dvd",
"Monoid.toMulOneClass",
"congrArg"... | [] | rw [← pow_smul_eq_iff_period_dvd, Monoid.pow_exponent_eq_one, one_smul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.GroupAction.Period | {
"line": 106,
"column": 2
} | {
"line": 106,
"column": 73
} | {
"line": 108,
"column": 0
} | [
{
"pp": "α : Type v\nM : Type u\ninst✝¹ : Monoid M\ninst✝ : MulAction M α\nm : M\na : α\n⊢ period m a ∣ Monoid.exponent M",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"instHSMul",
"Dvd.dvd",
"Monoid.toMulOneClass",
"congrArg"... | [] | rw [← pow_smul_eq_iff_period_dvd, Monoid.pow_exponent_eq_one, one_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupAction.Period | {
"line": 106,
"column": 2
} | {
"line": 106,
"column": 73
} | {
"line": 108,
"column": 0
} | [
{
"pp": "α : Type v\nM : Type u\ninst✝¹ : Monoid M\ninst✝ : MulAction M α\nm : M\na : α\n⊢ period m a ∣ Monoid.exponent M",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"instHSMul",
"Dvd.dvd",
"Monoid.toMulOneClass",
"congrArg"... | [] | rw [← pow_smul_eq_iff_period_dvd, Monoid.pow_exponent_eq_one, one_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.MultipleTransitivity | {
"line": 625,
"column": 29
} | {
"line": 625,
"column": 67
} | {
"line": 626,
"column": 4
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nG : Subgroup (Equiv.Perm α)\nhα : Nat.card α ≥ 2\ns : Set α\nhmt : IsMultiplyPretransitive (↥G) α s.ncard\nleft✝ : s ⊆ univ\nhs : s.ncard = Nat.card α - 2\nthis : (fixingSubgroup (↥G) s).index * 2 = (Nat.card α)!\n⊢ Nat.card (Equiv.Perm... | [
"case inr\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nG : Subgroup (Equiv.Perm α)\nhα : Nat.card α ≥ 2\ns : Set α\nhmt : IsMultiplyPretransitive (↥G) α s.ncard\nleft✝ : s ⊆ univ\nhs : s.ncard = Nat.card α - 2\nthis : (fixingSubgroup (↥G) s).index * 2 = (Nat.card α)!\n⊢ Nat.card (Equiv.Perm α) ≤ (fixin... | ← (fixingSubgroup G s).index_mul_card, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.VectorBundle.Riemannian | {
"line": 340,
"column": 4
} | {
"line": 340,
"column": 62
} | {
"line": 341,
"column": 2
} | [
{
"pp": "case e\nB : Type u_1\ninst✝⁷ : TopologicalSpace B\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace ℝ F\nE : B → Type u_3\ninst✝⁴ : TopologicalSpace (TotalSpace F E)\ninst✝³ : (x : B) → NormedAddCommGroup (E x)\ninst✝² : (x : B) → InnerProductSpace ℝ (E x)\ninst✝¹ : FiberBundle F E\nin... | [] | simp [Trivialization.coe_continuousLinearEquivAt_eq _ h'x] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.GroupExtension.Basic | {
"line": 64,
"column": 23
} | {
"line": 64,
"column": 43
} | {
"line": 64,
"column": 43
} | [
{
"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 = σ g * (σ' g)⁻¹ * σ' g",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"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 = σ g"
] | inv_mul_cancel_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.Jordan | {
"line": 371,
"column": 28
} | {
"line": 371,
"column": 48
} | {
"line": 371,
"column": 48
} | [
{
"pp": "case h\nα : Type u_1\nG : Subgroup (Perm α)\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ng : Perm α\nhg : g ∈ G\na : α\nleft✝ : g a ≠ a\nhgc : ∀ ⦃y : α⦄, g y ≠ y → g.SameCycle a y\nhs : ∀ (x : α), g • x ≠ x ↔ x ∈ ofFixingSubgroup (↥G) (↑g.support)ᶜ\nthis : ∀ x ∈ ofFixingSubgroup (↥G) (↑g.support)ᶜ, ∃ k,... | [
"case h\nα : Type u_1\nG : Subgroup (Perm α)\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ng : Perm α\nhg : g ∈ G\na : α\nleft✝ : g a ≠ a\nhgc : ∀ ⦃y : α⦄, g y ≠ y → g.SameCycle a y\nhs : ∀ (x : α), g • x ≠ x ↔ x ∈ ofFixingSubgroup (↥G) (↑g.support)ᶜ\nthis : ∀ x ∈ ofFixingSubgroup (↥G) (↑g.support)ᶜ, ∃ k, x = k • a\n... | inv_mul_cancel_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.SubMulAction.Combination | {
"line": 286,
"column": 4
} | {
"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... | [
"case neg\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\nthis : IsPre... | · apply IsPretransitive.of_surjective_map
(mulActionHom_compl_bijective (alternatingGroup α) α _).surjective this
aesop | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.MonoidLocalization.Lemmas | {
"line": 32,
"column": 77
} | {
"line": 32,
"column": 80
} | {
"line": 32,
"column": 81
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nF : Type u_3\nι : Type u_4\ninst✝⁴ : Finite ι\ninst✝³ : CommMonoid M\ninst✝² : CommMonoid N\ninst✝¹ : FunLike F M N\ninst✝ : MulHomClass F M N\nf : F\nS : Submonoid M\nhf : S.IsLocalizationMap ⇑f\nn : ι → N\nx : N → M × ↥S\nhx : ∀ (z : N), z * f ↑(x z).2 = f (x z).1\nval✝ : ... | [
"M : Type u_1\nN : Type u_2\nF : Type u_3\nι : Type u_4\ninst✝⁴ : Finite ι\ninst✝³ : CommMonoid M\ninst✝² : CommMonoid N\ninst✝¹ : FunLike F M N\ninst✝ : MulHomClass F M N\nf : F\nS : Submonoid M\nhf : S.IsLocalizationMap ⇑f\nn : ι → N\nx : N → M × ↥S\nhx : ∀ (z : N), z * f ↑(x z).2 = f (x z).1\nval✝ : Fintype ι\ni... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.HNNExtension | {
"line": 89,
"column": 75
} | {
"line": 90,
"column": 31
} | {
"line": 92,
"column": 0
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nb : ↥B\n⊢ t⁻¹ * of ↑b = of ↑(φ.symm b) * t⁻¹",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"mul_inv_cancel_right",
"Eq.mpr",
"MulEquiv.instEquivLike",
"MonoidHom.instFunLike",
"HMul.hMul... | [] | by
rw [equiv_symm_eq_conj]; simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.MonoidLocalization.Order | {
"line": 38,
"column": 32
} | {
"line": 38,
"column": 54
} | {
"line": 38,
"column": 55
} | [
{
"pp": "α : Type u_1\ninst✝² : CommMonoid α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedCancelMonoid α\ns : Submonoid α\na₁✝ b₁✝ : α\na₂✝ b₂✝ : ↥s\na b : Localization s\na₁ b₁ : α\na₂ b₂ : ↥s\nc₁ d₁ : α\nc₂ d₂ : ↥s\nhab : (r s) (a₁, a₂) (b₁, b₂)\nhcd : (r s) (c₁, c₂) (d₁, d₂)\ne f : ↥s\nhe : ↑b₂ * a₁ = ↑a₂ * b₁... | [
"α : Type u_1\ninst✝² : CommMonoid α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedCancelMonoid α\ns : Submonoid α\na₁✝ b₁✝ : α\na₂✝ b₂✝ : ↥s\na b : Localization s\na₁ b₁ : α\na₂ b₂ : ↥s\nc₁ d₁ : α\nc₂ d₂ : ↥s\nhab : (r s) (a₁, a₂) (b₁, b₂)\nhcd : (r s) (c₁, c₂) (d₁, d₂)\ne f : ↥s\nhe : ↑b₂ * a₁ = ↑a₂ * b₁\nhf : ↑d₂ *... | ← mul_le_mul_iff_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.HNNExtension | {
"line": 648,
"column": 8
} | {
"line": 648,
"column": 44
} | {
"line": 649,
"column": 8
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nd : TransversalPair G A B\nw : ReducedWord G A B\na : ℤˣ × G\nl : List (ℤˣ × G)\nchain : List.IsChain (fun a b ↦ a.2 ∈ toSubgroup A B a.1 → a.1 = b.1) (a :: l)\nw' : NormalWord d\nhw'1 : ReducedWord.prod φ w'.toReducedWord = ReducedWord.pro... | [
"G : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nd : TransversalPair G A B\nw : ReducedWord G A B\na : ℤˣ × G\nl : List (ℤˣ × G)\nchain : List.IsChain (fun a b ↦ a.2 ∈ toSubgroup A B a.1 → a.1 = b.1) (a :: l)\nw' : NormalWord d\nhw'1 : ReducedWord.prod φ w'.toReducedWord = ReducedWord.prod φ { head :... | rw [mul_mem_cancel_right this] at hS | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 550,
"column": 8
} | {
"line": 552,
"column": 76
} | {
"line": 553,
"column": 4
} | [
{
"pp": "case a.left.refine_2\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\na : Perm ↑(Function.fixedPoints ⇑g)\n⊢ ⟨ofSubtype a, ⋯⟩ ∈ ↑(toPermHom g).ker",
"ppTerm": "?a.left.refine_2✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Equiv.Perm.su... | [] | exact Perm.ext fun x ↦ Subtype.ext (disjoint_iff_disjoint_support.mpr
((ofSubtype_support_disjoint a).mono_right
(mem_cycleFactorsFinset_support_le x.2))).commute.mul_inv_cancel | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 550,
"column": 8
} | {
"line": 552,
"column": 76
} | {
"line": 553,
"column": 4
} | [
{
"pp": "case a.left.refine_2\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\na : Perm ↑(Function.fixedPoints ⇑g)\n⊢ ⟨ofSubtype a, ⋯⟩ ∈ ↑(toPermHom g).ker",
"ppTerm": "?a.left.refine_2✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Equiv.Perm.su... | [] | exact Perm.ext fun x ↦ Subtype.ext (disjoint_iff_disjoint_support.mpr
((ofSubtype_support_disjoint a).mono_right
(mem_cycleFactorsFinset_support_le x.2))).commute.mul_inv_cancel | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 550,
"column": 8
} | {
"line": 552,
"column": 76
} | {
"line": 553,
"column": 4
} | [
{
"pp": "case a.left.refine_2\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\na : Perm ↑(Function.fixedPoints ⇑g)\n⊢ ⟨ofSubtype a, ⋯⟩ ∈ ↑(toPermHom g).ker",
"ppTerm": "?a.left.refine_2✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Equiv.Perm.su... | [] | exact Perm.ext fun x ↦ Subtype.ext (disjoint_iff_disjoint_support.mpr
((ofSubtype_support_disjoint a).mono_right
(mem_cycleFactorsFinset_support_le x.2))).commute.mul_inv_cancel | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 588,
"column": 6
} | {
"line": 588,
"column": 27
} | {
"line": 588,
"column": 28
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng x : Perm α\nhx : x ∈ g.cycleFactorsFinset\n⊢ Fintype.card ↥(zpowers x) = (Finset.card ∘ support) x",
"ppTerm": "?m.103",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Equiv.Perm.support",
"congrArg",
"Finse... | [
"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng x : Perm α\nhx : x ∈ g.cycleFactorsFinset\n⊢ orderOf x = (Finset.card ∘ support) x"
] | Fintype.card_zpowers, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer | {
"line": 127,
"column": 2
} | {
"line": 130,
"column": 43
} | {
"line": 132,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nn : ℕ\nhn : 2 ≤ n\nhα : n ≤ Fintype.card α\n⊢ #{g | (↑g).cycleType = {n}} = if Odd n then (n - 1)! * (Fintype.card α).choose n else 0",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Multiset.sum",
"Eq.mpr",
... | [] | rw [← card_map, map_subtype_of_cycleType, apply_ite Finset.card]
simp only [Multiset.sum_singleton, Multiset.card_singleton, Finset.card_empty]
simp_rw [← Nat.not_odd_iff_even, Nat.odd_add_one, not_not,
Perm.card_of_cycleType_singleton hn hα] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer | {
"line": 127,
"column": 2
} | {
"line": 130,
"column": 43
} | {
"line": 132,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nn : ℕ\nhn : 2 ≤ n\nhα : n ≤ Fintype.card α\n⊢ #{g | (↑g).cycleType = {n}} = if Odd n then (n - 1)! * (Fintype.card α).choose n else 0",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Multiset.sum",
"Eq.mpr",
... | [] | rw [← card_map, map_subtype_of_cycleType, apply_ite Finset.card]
simp only [Multiset.sum_singleton, Multiset.card_singleton, Finset.card_empty]
simp_rw [← Nat.not_odd_iff_even, Nat.odd_add_one, not_not,
Perm.card_of_cycleType_singleton hn hα] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer | {
"line": 162,
"column": 10
} | {
"line": 162,
"column": 13
} | {
"line": 162,
"column": 14
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ng : Perm α\nh : Subgroup.centralizer {g} ≤ alternatingGroup α\nc : Perm α\nhc : c ∈ g.cycleFactorsFinset\nd : Perm α\nhd : d ∈ g.cycleFactorsFinset\nhm : #c.support = #d.support\nhm' : c ≠ d\nτ : Perm ↥g.cycleFactorsFinset := swap ⟨c, h... | [
"case pos\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ng : Perm α\nh : Subgroup.centralizer {g} ≤ alternatingGroup α\nc : Perm α\nhc : c ∈ g.cycleFactorsFinset\nd : Perm α\nhd : d ∈ g.cycleFactorsFinset\nhm : #c.support = #d.support\nhm' : c ≠ d\nτ : Perm ↥g.cycleFactorsFinset := swap ⟨c, hc⟩ ⟨d, hd⟩\n... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour | {
"line": 148,
"column": 6
} | {
"line": 148,
"column": 52
} | {
"line": 148,
"column": 52
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\nS : Sylow 2 ↥(alternatingGroup α)\nx✝ : Subsingleton (Sylow 2 ↥(alternatingGroup α))\n⊢ (kleinFour α).Characteristic",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Sylow.toSubgroup",
"Eq.m... | [
"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\nS : Sylow 2 ↥(alternatingGroup α)\nx✝ : Subsingleton (Sylow 2 ↥(alternatingGroup α))\n⊢ (↑S).Characteristic"
] | ← two_sylow_eq_kleinFour_of_card_eq_four hα4 S | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour | {
"line": 165,
"column": 6
} | {
"line": 165,
"column": 52
} | {
"line": 165,
"column": 52
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\nS : Sylow 2 ↥(alternatingGroup α)\n⊢ ↑(kleinFour α) = {1} ∪ {g | (↑g).cycleType = {2, 2}}",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Sylow.toSubgroup",
"Eq.mpr",
"Equiv.Perm.cycl... | [
"α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\nS : Sylow 2 ↥(alternatingGroup α)\n⊢ ↑↑S = {1} ∪ {g | (↑g).cycleType = {2, 2}}"
] | ← two_sylow_eq_kleinFour_of_card_eq_four hα4 S | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 96,
"column": 39
} | {
"line": 96,
"column": 54
} | {
"line": 96,
"column": 55
} | [
{
"pp": "case a.a.a\nn : ℕ\ni j k : ZMod (2 * n)\n⊢ a i * a j * a k = a i * (a j * a k)",
"ppTerm": "?a.a.a",
"assigned": true,
"usedConstants": [
"Mul.mk",
"HMul.hMul",
"HMul.mk",
"id",
"QuaternionGroup",
"QuaternionGroup.a",
"Mul.mul",
"_private.Math... | [
"case a.a.a\nn : ℕ\ni j k : ZMod (2 * n)\n⊢ a (i + j + k) = a (i + (j + k))"
] | simp only [mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 96,
"column": 39
} | {
"line": 96,
"column": 54
} | {
"line": 96,
"column": 55
} | [
{
"pp": "case a.a.xa\nn : ℕ\ni j k : ZMod (2 * n)\n⊢ a i * a j * xa k = a i * (a j * xa k)",
"ppTerm": "?a.a.xa",
"assigned": true,
"usedConstants": [
"Mul.mk",
"HMul.hMul",
"HMul.mk",
"id",
"QuaternionGroup",
"QuaternionGroup.a",
"Mul.mul",
"_private.... | [
"case a.a.xa\nn : ℕ\ni j k : ZMod (2 * n)\n⊢ xa (k - (i + j)) = xa (k - j - i)"
] | simp only [mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 96,
"column": 39
} | {
"line": 96,
"column": 54
} | {
"line": 96,
"column": 55
} | [
{
"pp": "case a.xa.a\nn : ℕ\ni j k : ZMod (2 * n)\n⊢ a i * xa j * a k = a i * (xa j * a k)",
"ppTerm": "?a.xa.a",
"assigned": true,
"usedConstants": [
"Mul.mk",
"HMul.hMul",
"HMul.mk",
"id",
"QuaternionGroup",
"QuaternionGroup.a",
"Mul.mul",
"_private.... | [
"case a.xa.a\nn : ℕ\ni j k : ZMod (2 * n)\n⊢ xa (j - i + k) = xa (j + k - i)"
] | simp only [mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 96,
"column": 39
} | {
"line": 96,
"column": 54
} | {
"line": 96,
"column": 55
} | [
{
"pp": "case a.xa.xa\nn : ℕ\ni j k : ZMod (2 * n)\n⊢ a i * xa j * xa k = a i * (xa j * xa k)",
"ppTerm": "?a.xa.xa",
"assigned": true,
"usedConstants": [
"Mul.mk",
"HMul.hMul",
"HMul.mk",
"id",
"QuaternionGroup",
"QuaternionGroup.a",
"Mul.mul",
"_priv... | [
"case a.xa.xa\nn : ℕ\ni j k : ZMod (2 * n)\n⊢ a (↑n + k - (j - i)) = a (i + (↑n + k - j))"
] | simp only [mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 96,
"column": 39
} | {
"line": 96,
"column": 54
} | {
"line": 96,
"column": 55
} | [
{
"pp": "case xa.a.a\nn : ℕ\ni j k : ZMod (2 * n)\n⊢ xa i * a j * a k = xa i * (a j * a k)",
"ppTerm": "?xa.a.a",
"assigned": true,
"usedConstants": [
"Mul.mk",
"HMul.hMul",
"HMul.mk",
"id",
"QuaternionGroup",
"QuaternionGroup.a",
"Mul.mul",
"_private.... | [
"case xa.a.a\nn : ℕ\ni j k : ZMod (2 * n)\n⊢ xa (i + j + k) = xa (i + (j + k))"
] | simp only [mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 96,
"column": 39
} | {
"line": 96,
"column": 54
} | {
"line": 96,
"column": 55
} | [
{
"pp": "case xa.a.xa\nn : ℕ\ni j k : ZMod (2 * n)\n⊢ xa i * a j * xa k = xa i * (a j * xa k)",
"ppTerm": "?xa.a.xa",
"assigned": true,
"usedConstants": [
"Mul.mk",
"HMul.hMul",
"HMul.mk",
"id",
"QuaternionGroup",
"QuaternionGroup.a",
"Mul.mul",
"_priv... | [
"case xa.a.xa\nn : ℕ\ni j k : ZMod (2 * n)\n⊢ a (↑n + k - (i + j)) = a (↑n + (k - j) - i)"
] | simp only [mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 96,
"column": 39
} | {
"line": 96,
"column": 54
} | {
"line": 96,
"column": 55
} | [
{
"pp": "case xa.xa.a\nn : ℕ\ni j k : ZMod (2 * n)\n⊢ xa i * xa j * a k = xa i * (xa j * a k)",
"ppTerm": "?xa.xa.a",
"assigned": true,
"usedConstants": [
"Mul.mk",
"HMul.hMul",
"HMul.mk",
"id",
"QuaternionGroup",
"QuaternionGroup.a",
"Mul.mul",
"_priv... | [
"case xa.xa.a\nn : ℕ\ni j k : ZMod (2 * n)\n⊢ a (↑n + j - i + k) = a (↑n + (j + k) - i)"
] | simp only [mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.SpecificGroups.Quaternion | {
"line": 96,
"column": 39
} | {
"line": 96,
"column": 54
} | {
"line": 96,
"column": 55
} | [
{
"pp": "case xa.xa.xa\nn : ℕ\ni j k : ZMod (2 * n)\n⊢ xa i * xa j * xa k = xa i * (xa j * xa k)",
"ppTerm": "?xa.xa.xa",
"assigned": true,
"usedConstants": [
"Mul.mk",
"HMul.hMul",
"HMul.mk",
"id",
"QuaternionGroup",
"Mul.mul",
"_private.Mathlib.GroupTheory... | [
"case xa.xa.xa\nn : ℕ\ni j k : ZMod (2 * n)\n⊢ xa (k - (↑n + j - i)) = xa (i + (↑n + k - j))"
] | simp only [mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.SpecificGroups.ZGroup | {
"line": 156,
"column": 6
} | {
"line": 156,
"column": 59
} | {
"line": 157,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : Finite G\ninst✝ : IsZGroup G\nH : Subgroup G\nhH : IsCyclic ↥⁅⁅H, H⁆, ⁅H, H⁆⁆\nh : H ≠ ⊥\nf : ↥⁅commutator ↥H, commutator ↥H⁆ ≃* ↥⁅⁅H, H⁆, ⁅H, H⁆⁆\n⊢ IsCyclic ↥⁅commutator ↥H, commutator ↥H⁆",
"ppTerm": "?m.119",
"assigned": true,
"usedConstants": [
... | [] | exact isCyclic_of_surjective f.symm f.symm.surjective | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer | {
"line": 221,
"column": 2
} | {
"line": 245,
"column": 79
} | {
"line": 247,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ng : Perm α\n⊢ Subgroup.centralizer {g} ≤ alternatingGroup α ↔\n (∀ c ∈ g.cycleType, Odd c) ∧ Fintype.card α ≤ g.cycleType.sum + 1 ∧ ∀ (i : ℕ), Multiset.count i g.cycleType ≤ 1",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants":... | [] | rw [SetLike.le_def]
constructor
· intro h
exact ⟨odd_of_centralizer_le_alternatingGroup h,
card_le_of_centralizer_le_alternating h,
count_le_one_of_centralizer_le_alternating h⟩
· rintro ⟨h_odd, h_fixed, h_count⟩ x hx
rw [← kerParam_range_eq_centralizer_of_count_le_one h_count] at hx
obtai... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.Alternating.Centralizer | {
"line": 221,
"column": 2
} | {
"line": 245,
"column": 79
} | {
"line": 247,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ng : Perm α\n⊢ Subgroup.centralizer {g} ≤ alternatingGroup α ↔\n (∀ c ∈ g.cycleType, Odd c) ∧ Fintype.card α ≤ g.cycleType.sum + 1 ∧ ∀ (i : ℕ), Multiset.count i g.cycleType ≤ 1",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants":... | [] | rw [SetLike.le_def]
constructor
· intro h
exact ⟨odd_of_centralizer_le_alternatingGroup h,
card_le_of_centralizer_le_alternating h,
count_le_one_of_centralizer_le_alternating h⟩
· rintro ⟨h_odd, h_fixed, h_count⟩ x hx
rw [← kerParam_range_eq_centralizer_of_count_le_one h_count] at hx
obtai... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.ZGroup | {
"line": 204,
"column": 30
} | {
"line": 204,
"column": 91
} | {
"line": 205,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝⁴ : Group G\np✝ : ℕ\ninst✝³ : Fact (Nat.Prime p✝)\ninst✝² : IsCyclic G\nK : Type u_4\ninst✝¹ : Group K\ninst✝ : MulDistribMulAction K G\nhGK : (Nat.card G).Coprime (Nat.card K)\nhc : ¬Nat.card G = 0\nthis : Finite G\nϕ : K →* ZMod (Nat.card G) := MulDistribMulAction.toMonoidHomZModOf... | [] | rw [h p k 0 (by rw [hϕ, sub_self, Int.cast_zero]), zpow_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.SpecificGroups.ZGroup | {
"line": 204,
"column": 30
} | {
"line": 204,
"column": 91
} | {
"line": 205,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝⁴ : Group G\np✝ : ℕ\ninst✝³ : Fact (Nat.Prime p✝)\ninst✝² : IsCyclic G\nK : Type u_4\ninst✝¹ : Group K\ninst✝ : MulDistribMulAction K G\nhGK : (Nat.card G).Coprime (Nat.card K)\nhc : ¬Nat.card G = 0\nthis : Finite G\nϕ : K →* ZMod (Nat.card G) := MulDistribMulAction.toMonoidHomZModOf... | [] | rw [h p k 0 (by rw [hϕ, sub_self, Int.cast_zero]), zpow_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.ZGroup | {
"line": 204,
"column": 30
} | {
"line": 204,
"column": 91
} | {
"line": 205,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝⁴ : Group G\np✝ : ℕ\ninst✝³ : Fact (Nat.Prime p✝)\ninst✝² : IsCyclic G\nK : Type u_4\ninst✝¹ : Group K\ninst✝ : MulDistribMulAction K G\nhGK : (Nat.card G).Coprime (Nat.card K)\nhc : ¬Nat.card G = 0\nthis : Finite G\nϕ : K →* ZMod (Nat.card G) := MulDistribMulAction.toMonoidHomZModOf... | [] | rw [h p k 0 (by rw [hϕ, sub_self, Int.cast_zero]), zpow_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.SpecificGroups.ZGroup | {
"line": 313,
"column": 4
} | {
"line": 315,
"column": 86
} | {
"line": 317,
"column": 0
} | [
{
"pp": "case inr\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝⁵ : Group G\ninst✝⁴ : Group G'\ninst✝³ : Group G''\nf : G →* G'\nf' : G' →* G''\ninst✝² : Finite G\ninst✝¹ : IsZGroup G\ninst✝ : IsZGroup G''\nh_le : f'.ker ≤ f.range\np : ℕ\nhp : Nat.Prime p\nP : Sylow p G'\nthis : Fact (Nat.Prime p)\nh_cop :... | [] | have := (P.2.map f').isCyclic_of_isZGroup
apply isCyclic_of_injective (f'.subgroupMap P)
rwa [← MonoidHom.ker_eq_bot_iff, P.ker_subgroupMap f', Subgroup.subgroupOf_eq_bot] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.ZGroup | {
"line": 313,
"column": 4
} | {
"line": 315,
"column": 86
} | {
"line": 317,
"column": 0
} | [
{
"pp": "case inr\nG : Type u_1\nG' : Type u_2\nG'' : Type u_3\ninst✝⁵ : Group G\ninst✝⁴ : Group G'\ninst✝³ : Group G''\nf : G →* G'\nf' : G' →* G''\ninst✝² : Finite G\ninst✝¹ : IsZGroup G\ninst✝ : IsZGroup G''\nh_le : f'.ker ≤ f.range\np : ℕ\nhp : Nat.Prime p\nP : Sylow p G'\nthis : Fact (Nat.Prime p)\nh_cop :... | [] | have := (P.2.map f').isCyclic_of_isZGroup
apply isCyclic_of_injective (f'.subgroupMap P)
rwa [← MonoidHom.ker_eq_bot_iff, P.ker_subgroupMap f', Subgroup.subgroupOf_eq_bot] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Defs | {
"line": 403,
"column": 2
} | {
"line": 403,
"column": 53
} | {
"line": 404,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : Denumerable ι\nκs : ι → Kernel α β\nhκs : ∀ (n : ι), IsSFiniteKernel (κs n)\n⊢ IsSFiniteKernel (Kernel.sum κs)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Denumerable.prod"... | [
"α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝ : Denumerable ι\nκs : ι → Kernel α β\nhκs : ∀ (n : ι), IsSFiniteKernel (κs n)\ne : ℕ ≃ ι × ℕ := (Denumerable.eqv (ι × ℕ)).symm\n⊢ IsSFiniteKernel (Kernel.sum κs)"
] | let e : ℕ ≃ ι × ℕ := (Denumerable.eqv (ι × ℕ)).symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.InformationTheory.Coding.KraftMcMillan | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 19
} | {
"line": 64,
"column": 2
} | [
{
"pp": "α : Type u_1\nS : Finset (List α)\nh : UniquelyDecodable ↑S\nr : ℕ\n⊢ Function.Injective concatFn",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Finset",
"Membership.mem",
"Subtype",
"List",
"Finset.instSetLike",
"_private.Mathlib.InformationTh... | [
"α : Type u_1\nS : Finset (List α)\nh : UniquelyDecodable ↑S\nr : ℕ\nw₁ w₂ : Fin r → ↥S\nhflat : concatFn w₁ = concatFn w₂\n⊢ w₁ = w₂"
] | intro w₁ w₂ hflat | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Probability.Kernel.MeasurableLIntegral | {
"line": 110,
"column": 2
} | {
"line": 112,
"column": 65
} | {
"line": 114,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\ninst✝ : IsSFiniteKernel κ\nt : Set (α × β)\nht : MeasurableSet t\nc : ℝ≥0∞\n⊢ Measurable fun a ↦ ∫⁻ (b : β), t.indicator (Function.const (α × β) c) (a, b) ∂κ a",
"ppTerm": "?m.30",
"assigned": true,
... | [] | unfold Function.const
simp_rw [lintegral_indicator_const_comp measurable_prodMk_left ht _]
exact Measurable.const_mul (measurable_kernel_prodMk_left ht) c | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.MeasurableLIntegral | {
"line": 110,
"column": 2
} | {
"line": 112,
"column": 65
} | {
"line": 114,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\ninst✝ : IsSFiniteKernel κ\nt : Set (α × β)\nht : MeasurableSet t\nc : ℝ≥0∞\n⊢ Measurable fun a ↦ ∫⁻ (b : β), t.indicator (Function.const (α × β) c) (a, b) ∂κ a",
"ppTerm": "?m.30",
"assigned": true,
... | [] | unfold Function.const
simp_rw [lintegral_indicator_const_comp measurable_prodMk_left ht _]
exact Measurable.const_mul (measurable_kernel_prodMk_left ht) c | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.InformationTheory.Coding.KraftMcMillan | {
"line": 162,
"column": 2
} | {
"line": 162,
"column": 34
} | {
"line": 163,
"column": 2
} | [
{
"pp": "α : Type u_1\nS : Finset (List α)\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nh : UniquelyDecodable ↑S\nh_kraft : 1 < ∑ w ∈ S, (1 / ↑(Fintype.card α)) ^ w.length\nK : ℝ := ∑ w ∈ S, (1 / ↑(Fintype.card α)) ^ w.length\nmaxLen : ℕ := S.sup List.length\nhAbs : |1 / K| < 1\nthis : Tendsto (fun r ↦ ↑r * ↑maxLen... | [
"α : Type u_1\nS : Finset (List α)\ninst✝¹ : Fintype α\ninst✝ : Nonempty α\nh : UniquelyDecodable ↑S\nh_kraft : 1 < ∑ w ∈ S, (1 / ↑(Fintype.card α)) ^ w.length\nK : ℝ := ∑ w ∈ S, (1 / ↑(Fintype.card α)) ^ w.length\nmaxLen : ℕ := S.sup List.length\nhAbs : |1 / K| < 1\nthis✝ : Tendsto (fun r ↦ ↑r * ↑maxLen / K ^ r) a... | have := hr (r + 1) (by linarith) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Probability.Kernel.Composition.MeasureComp | {
"line": 208,
"column": 2
} | {
"line": 208,
"column": 36
} | {
"line": 210,
"column": 0
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nπ : Measure Bool\ns : Set α\nhs : MeasurableSet s\n⊢ ∫⁻ (a : Bool), ((Kernel.boolKernel μ ν) a) s ∂π = (π {true} • ν + π {false} • μ) s",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"ENNReal.instAdd",
"False",
... | [] | simp [lintegral_fintype, mul_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Tilted | {
"line": 319,
"column": 13
} | {
"line": 319,
"column": 83
} | {
"line": 320,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nmα : MeasurableSpace α\nf : α → ℝ\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nhf : Integrable (fun x ↦ rexp (f x)) ν\nh : ν = 0\n⊢ μ.rnDeriv (ν.tilted f) =ᵐ[ν] fun x ↦ ENNReal.ofReal (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * μ.rnDeriv ν x",
"ppTerm": "?... | [] | simp_rw [h, ae_zero, Filter.EventuallyEq]; exact Filter.eventually_bot | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Tilted | {
"line": 319,
"column": 13
} | {
"line": 319,
"column": 83
} | {
"line": 320,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nmα : MeasurableSpace α\nf : α → ℝ\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nhf : Integrable (fun x ↦ rexp (f x)) ν\nh : ν = 0\n⊢ μ.rnDeriv (ν.tilted f) =ᵐ[ν] fun x ↦ ENNReal.ofReal (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * μ.rnDeriv ν x",
"ppTerm": "?... | [] | simp_rw [h, ae_zero, Filter.EventuallyEq]; exact Filter.eventually_bot | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.LogLikelihoodRatio | {
"line": 74,
"column": 6
} | {
"line": 74,
"column": 9
} | {
"line": 74,
"column": 10
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nhμν : μ ≪ ν\nx : α\nhx : -llr μ ν x = llr ν μ x\nhx_exp_log : rexp (llr ν μ x) = (ν.rnDeriv μ x).toReal\n⊢ rexp (-llr μ ν x) = (ν.rnDeriv μ x).toReal",
"ppTerm": "?m.70",
"assigned": true,
... | [
"α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nhμν : μ ≪ ν\nx : α\nhx : -llr μ ν x = llr ν μ x\nhx_exp_log : rexp (llr ν μ x) = (ν.rnDeriv μ x).toReal\n⊢ rexp (llr ν μ x) = (ν.rnDeriv μ x).toReal"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Tilted | {
"line": 364,
"column": 6
} | {
"line": 364,
"column": 9
} | {
"line": 364,
"column": 10
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\ninst✝ : SigmaFinite μ\nhf : AEMeasurable f μ\nx : α\nhx : μ.rnDeriv μ x = 1\n⊢ ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) * μ.rnDeriv μ x =\n ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ)",
"ppTerm": "?m.60",
... | [
"α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\ninst✝ : SigmaFinite μ\nhf : AEMeasurable f μ\nx : α\nhx : μ.rnDeriv μ x = 1\n⊢ ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) * 1 = ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ)"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Tilted | {
"line": 370,
"column": 13
} | {
"line": 370,
"column": 83
} | {
"line": 371,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\ninst✝ : SigmaFinite μ\nhf : Integrable (fun x ↦ rexp (f x)) μ\nh : μ = 0\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": "?inl",
"assigned": true,
"use... | [] | simp_rw [h, ae_zero, Filter.EventuallyEq]; exact Filter.eventually_bot | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Tilted | {
"line": 370,
"column": 13
} | {
"line": 370,
"column": 83
} | {
"line": 371,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\ninst✝ : SigmaFinite μ\nhf : Integrable (fun x ↦ rexp (f x)) μ\nh : μ = 0\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": "?inl",
"assigned": true,
"use... | [] | simp_rw [h, ae_zero, Filter.EventuallyEq]; exact Filter.eventually_bot | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Tilted | {
"line": 374,
"column": 8
} | {
"line": 374,
"column": 11
} | {
"line": 374,
"column": 12
} | [
{
"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\nh0 : NeZero μ\nhf' : AEMeasurable f μ\nx : α\nhx : (μ.tilted f).rnDeriv μ x = ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ)\n⊢ log ((μ.tilted f).rnDeriv μ x).toReal = ... | [
"α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\ninst✝ : SigmaFinite μ\nhf : Integrable (fun x ↦ rexp (f x)) μ\nh0 : NeZero μ\nhf' : AEMeasurable f μ\nx : α\nhx : (μ.tilted f).rnDeriv μ x = ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ)\n⊢ log (ENNReal.ofReal (rexp (f x) / ∫ (x : α), rexp (f... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.LogLikelihoodRatio | {
"line": 173,
"column": 11
} | {
"line": 173,
"column": 25
} | {
"line": 173,
"column": 26
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\n⊢ ∫ (a : α), (μ.rnDeriv ν a).toReal * log (μ.rnDeriv ν a).toReal ∂ν = ∫ (a : α), llr μ ν a ∂μ",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"R... | [
"α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : SigmaFinite μ\ninst✝ : μ.HaveLebesgueDecomposition ν\nhμν : μ ≪ ν\n⊢ ∫ (a : α), (μ.rnDeriv ν a).toReal • log (μ.rnDeriv ν a).toReal ∂ν = ∫ (a : α), llr μ ν a ∂μ"
] | ← smul_eq_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.LogLikelihoodRatio | {
"line": 186,
"column": 13
} | {
"line": 186,
"column": 16
} | {
"line": 186,
"column": 17
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nhμν : μ ≪ ν\nhf : Integrable (fun x ↦ rexp (f x)) μ\nhfν : AEMeasurable f ν\nh0 : NeZero μ\nx : α\nhx : ((μ.tilted f).rnDeriv ν x).toReal = (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) * (μ.rnDer... | [
"α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nhμν : μ ≪ ν\nhf : Integrable (fun x ↦ rexp (f x)) μ\nhfν : AEMeasurable f ν\nh0 : NeZero μ\nx : α\nhx : ((μ.tilted f).rnDeriv ν x).toReal = (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) * (μ.rnDeriv ν x).toRe... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.LogLikelihoodRatio | {
"line": 226,
"column": 13
} | {
"line": 226,
"column": 16
} | {
"line": 226,
"column": 17
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nhμν : μ ≪ ν\nhf : Integrable (fun x ↦ rexp (f x)) ν\nh0 : NeZero ν\nx : α\nhx : (μ.rnDeriv (ν.tilted f) x).toReal = (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * (μ.rnDeriv ν x).toReal\nhx_pos ... | [
"α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\nf : α → ℝ\ninst✝¹ : SigmaFinite μ\ninst✝ : SigmaFinite ν\nhμν : μ ≪ ν\nhf : Integrable (fun x ↦ rexp (f x)) ν\nh0 : NeZero ν\nx : α\nhx : (μ.rnDeriv (ν.tilted f) x).toReal = (rexp (-f x) * ∫ (x : α), rexp (f x) ∂ν) * (μ.rnDeriv ν x).toReal\nhx_pos : 0 < μ.rnDe... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique | {
"line": 94,
"column": 8
} | {
"line": 94,
"column": 71
} | {
"line": 94,
"column": 71
} | [
{
"pp": "α : Type u_1\nE' : Type u_2\n𝕜 : Type u_4\np : ℝ≥0∞\nm m0 : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E'\ninst✝² : InnerProductSpace 𝕜 E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nhm : m ≤ m0\nf g : ↥(Lp E' p μ)\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ... | [
"α : Type u_1\nE' : Type u_2\n𝕜 : Type u_4\np : ℝ≥0∞\nm m0 : MeasurableSpace α\nμ : Measure α\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E'\ninst✝² : InnerProductSpace 𝕜 E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nhm : m ≤ m0\nf g : ↥(Lp E' p μ)\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ∞\nhf_int_fi... | integral_sub' (hf_int_finite s hs hμs) (hg_int_finite s hs hμs) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.InformationTheory.KullbackLeibler.Basic | {
"line": 340,
"column": 7
} | {
"line": 344,
"column": 70
} | {
"line": 346,
"column": 0
} | [] | [] | ν.real univ * klFun (μ.real univ / ν.real univ)
_ ≤ ∫ x, klFun (μ.rnDeriv ν x).toReal ∂ν := by
refine mul_le_integral_rnDeriv_of_ac convexOn_klFun continuous_klFun.continuousWithinAt ?_ hμν
rwa [integrable_klFun_rnDeriv_iff hμν]
_ = (klDiv μ ν).toReal := by rw [toReal_klDiv_eq_integral_klFun hμν] | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2 | {
"line": 138,
"column": 88
} | {
"line": 143,
"column": 95
} | {
"line": 145,
"column": 0
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_7\ninst✝ : RCLike 𝕜\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhm : m ≤ m0\nf : ↥(Lp 𝕜 2 μ)\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\n⊢ ∫ (x : α) in s, ↑↑↑((condExpL2 𝕜 𝕜 hm) f) x ∂μ = ∫ (x : α) in s, ↑↑f x ∂μ",
"ppTerm": "?m.56",
"assigned": true,
"usedC... | [] | by
rw [← L2.inner_indicatorConstLp_one (𝕜 := 𝕜) (hm s hs) hμs f]
have h_eq_inner : ∫ x in s, (condExpL2 𝕜 𝕜 hm f : α → 𝕜) x ∂μ =
⟪indicatorConstLp 2 (hm s hs) hμs (1 : 𝕜), condExpL2 𝕜 𝕜 hm f⟫ := by
rw [L2.inner_indicatorConstLp_one (hm s hs) hμs]
rw [h_eq_inner, ← inner_condExpL2_left_eq_right, ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2 | {
"line": 152,
"column": 2
} | {
"line": 156,
"column": 16
} | {
"line": 157,
"column": 2
} | [
{
"pp": "α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhm : m ≤ m0\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nf : ↥(Lp ℝ 2 μ)\nh_meas : AEStronglyMeasurable (↑↑↑((condExpL2 ℝ ℝ hm) f)) μ := lpMeas.aestronglyMeasurable ((condExpL2 ℝ ℝ hm) f)\ng : α → ℝ := Exists.choose h_meas\nhg_meas : StronglyMe... | [
"α : Type u_1\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhm : m ≤ m0\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nf : ↥(Lp ℝ 2 μ)\nh_meas : AEStronglyMeasurable (↑↑↑((condExpL2 ℝ ℝ hm) f)) μ := lpMeas.aestronglyMeasurable ((condExpL2 ℝ ℝ hm) f)\ng : α → ℝ := Exists.choose h_meas\nhg_meas : StronglyMeasurable g\n... | have hg_nnnorm_eq : (fun x => (‖g x‖₊ : ℝ≥0∞)) =ᵐ[μ.restrict s] fun x =>
(‖(condExpL2 ℝ ℝ hm f : α → ℝ) x‖₊ : ℝ≥0∞) := by
refine hg_eq_restrict.mono fun x hx => ?_
dsimp only
simp_rw [hx] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1 | {
"line": 293,
"column": 2
} | {
"line": 299,
"column": 43
} | {
"line": 301,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : Type u_4\ninst✝² : NormedAddCommGroup G\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\ninst✝¹ : NormedSpace ℝ G\nhm : m ≤ m0\ninst✝ : SigmaFinite (μ.trim hm)\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nc : G\n⊢ (condExpInd G hm μ s) c = indicatorConstLp 1 ⋯ hμs c",
"ppTerm": "?m.4... | [] | ext1
grw [indicatorConstLp_coeFn, condExpInd_ae_eq_condExpIndSMul hm (hm s hs) hμs,
condExpIndSMul_ae_eq_smul]
rw [condExpL2_indicator_of_measurable hm hs hμs (1 : ℝ)]
filter_upwards [@indicatorConstLp_coeFn α _ _ 2 μ _ s (hm s hs) hμs (1 : ℝ)] with x hx
rw [hx]
by_cases hx_mem : x ∈ s <;> simp [hx_mem] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL1 | {
"line": 293,
"column": 2
} | {
"line": 299,
"column": 43
} | {
"line": 301,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : Type u_4\ninst✝² : NormedAddCommGroup G\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\ninst✝¹ : NormedSpace ℝ G\nhm : m ≤ m0\ninst✝ : SigmaFinite (μ.trim hm)\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nc : G\n⊢ (condExpInd G hm μ s) c = indicatorConstLp 1 ⋯ hμs c",
"ppTerm": "?m.4... | [] | ext1
grw [indicatorConstLp_coeFn, condExpInd_ae_eq_condExpIndSMul hm (hm s hs) hμs,
condExpIndSMul_ae_eq_smul]
rw [condExpL2_indicator_of_measurable hm hs hμs (1 : ℝ)]
filter_upwards [@indicatorConstLp_coeFn α _ _ 2 μ _ s (hm s hs) hμs (1 : ℝ)] with x hx
rw [hx]
by_cases hx_mem : x ∈ s <;> simp [hx_mem] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2 | {
"line": 217,
"column": 2
} | {
"line": 218,
"column": 85
} | {
"line": 219,
"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 : ↥(Lp E 2 μ)\nc : E\nh_mem_Lp : MemLp (fun a ↦ ⟪c, ↑↑↑((condExpL2 E 𝕜 hm) f) a⟫) 2 μ\nh_eq : ... | [
"case refine_1\nα : 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 : ↥(Lp E 2 μ)\nc : E\nh_mem_Lp : MemLp (fun a ↦ ⟪c, ↑↑↑((condExpL2 E 𝕜 hm) f) a⟫) 2 μ\nh_eq... | refine Lp.ae_eq_of_forall_setIntegral_eq' 𝕜 hm _ _ two_ne_zero ENNReal.coe_ne_top
(fun s _ hμs => integrableOn_condExpL2_of_measure_ne_top hm hμs.ne _) ?_ ?_ ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2 | {
"line": 320,
"column": 6
} | {
"line": 320,
"column": 30
} | {
"line": 321,
"column": 6
} | [
{
"pp": "α : Type u_1\nE' : Type u_3\n𝕜 : Type u_7\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E'\ninst✝² : InnerProductSpace 𝕜 E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhm : m ≤ m0\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nx : E'\nt : Set α\... | [
"case hf\nα : Type u_1\nE' : Type u_3\n𝕜 : Type u_7\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E'\ninst✝² : InnerProductSpace 𝕜 E'\ninst✝¹ : CompleteSpace E'\ninst✝ : NormedSpace ℝ E'\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\nhm : m ≤ m0\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nx : E'\nt : Set α\nht... | rw [lintegral_mul_const] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic | {
"line": 257,
"column": 62
} | {
"line": 261,
"column": 51
} | {
"line": 263,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_3\nm m₀ : MeasurableSpace α\nμ : Measure α\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : CompleteSpace E\nhm : m ≤ m₀\ninst✝ : SigmaFinite (μ.trim hm)\nf g : α → E\nhf : Integrable f μ\nhg_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn... | [] | by
refine ae_eq_of_forall_setIntegral_eq_of_sigmaFinite' hm hg_int_finite
(fun s _ _ => integrable_condExp.integrableOn) (fun s hs hμs => ?_) hgm
(StronglyMeasurable.aestronglyMeasurable stronglyMeasurable_condExp)
rw [hg_eq s hs hμs, setIntegral_condExp hm hf hs] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.ConditionalExpectation.CondexpL2 | {
"line": 388,
"column": 6
} | {
"line": 388,
"column": 30
} | {
"line": 389,
"column": 6
} | [
{
"pp": "α : Type u_1\nG : Type u_5\ninst✝¹ : NormedAddCommGroup G\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\ninst✝ : NormedSpace ℝ G\nhm : m ≤ m0\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nx : G\nt : Set α\nht : MeasurableSet t\nhμt : μ t ≠ ∞\n⊢ ∫⁻ (a : α) in t, ↑‖↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp ... | [
"case hf\nα : Type u_1\nG : Type u_5\ninst✝¹ : NormedAddCommGroup G\nm m0 : MeasurableSpace α\nμ : Measure α\ns : Set α\ninst✝ : NormedSpace ℝ G\nhm : m ≤ m0\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nx : G\nt : Set α\nht : MeasurableSet t\nhμt : μ t ≠ ∞\n⊢ Measurable fun a ↦ ↑‖↑↑↑((condExpL2 ℝ ℝ hm) (indicatorConstLp 2... | rw [lintegral_mul_const] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.AffineSpace.Matrix | {
"line": 120,
"column": 30
} | {
"line": 120,
"column": 43
} | {
"line": 120,
"column": 44
} | [
{
"pp": "ι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AffineSpace V P\ninst✝³ : Ring k\ninst✝² : Module k V\nb : AffineBasis ι k P\ninst✝¹ : Fintype ι\nb₂ : AffineBasis ι k P\ninst✝ : DecidableEq ι\nl m : ι\n⊢ b₂.coords (b₂ l) m = 1 l m",
"ppTerm": "?m.93",
"assi... | [
"ι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AffineSpace V P\ninst✝³ : Ring k\ninst✝² : Module k V\nb : AffineBasis ι k P\ninst✝¹ : Fintype ι\nb₂ : AffineBasis ι k P\ninst✝ : DecidableEq ι\nl m : ι\n⊢ (b₂.coord m) (b₂ l) = 1 l m"
] | coords_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.TensorProduct.IsBaseChangeHom | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 84
} | {
"line": 144,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹⁶ : CommSemiring R\nS : Type u_2\ninst✝¹⁵ : CommSemiring S\ninst✝¹⁴ : Algebra R S\nM : Type u_3\ninst✝¹³ : AddCommMonoid M\ninst✝¹² : Module R M\nN : Type u_4\ninst✝¹¹ : AddCommMonoid N\ninst✝¹⁰ : Module R N\nP : Type u_5\ninst✝⁹ : AddCommMonoid P\ninst✝⁸ : Module R P\nQ : Type u_6\... | [] | simp [IsBaseChange.equiv_tmul, LinearEquiv.congrLeft, linearMapLeftRightHom_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.FixedSubmodule | {
"line": 40,
"column": 38
} | {
"line": 41,
"column": 23
} | {
"line": 43,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝² : Semiring R\nV : Type u_3\ninst✝¹ : AddCommMonoid V\ninst✝ : Module R V\nf : V →ₗ[R] V\nv : V\n⊢ v ∈ f.fixedSubmodule ↔ f v = v",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Submodule",
"AddSubsemigroup.instSetLike",
"congrArg",
"Ad... | [] | by
simp [fixedSubmodule] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Alternating.DomCoprod | {
"line": 161,
"column": 4
} | {
"line": 175,
"column": 9
} | {
"line": 177,
"column": 0
} | [
{
"pp": "ιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' ... | [] | refine
LinearMap.mk₂ R' domCoprod (fun m₁ m₂ n => ?_) (fun c m n => ?_) (fun m n₁ n₂ => ?_)
fun c m n => ?_ <;>
· ext
simp only [domCoprod_apply, add_apply, smul_apply, ← Finset.sum_add_distrib,
Finset.smul_sum, _root_.sum_apply, domCoprod.summand]
congr
ext σ
induction... | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.LinearAlgebra.Alternating.DomCoprod | {
"line": 161,
"column": 4
} | {
"line": 175,
"column": 9
} | {
"line": 177,
"column": 0
} | [
{
"pp": "ιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' ... | [] | refine
LinearMap.mk₂ R' domCoprod (fun m₁ m₂ n => ?_) (fun c m n => ?_) (fun m n₁ n₂ => ?_)
fun c m n => ?_ <;>
· ext
simp only [domCoprod_apply, add_apply, smul_apply, ← Finset.sum_add_distrib,
Finset.smul_sum, _root_.sum_apply, domCoprod.summand]
congr
ext σ
induction... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Alternating.DomCoprod | {
"line": 161,
"column": 4
} | {
"line": 175,
"column": 9
} | {
"line": 177,
"column": 0
} | [
{
"pp": "ιa : Type u_1\nιb : Type u_2\ninst✝¹⁰ : Fintype ιa\ninst✝⁹ : Fintype ιb\nR' : Type u_3\nMᵢ : Type u_4\nN₁ : Type u_5\nN₂ : Type u_6\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommGroup N₁\ninst✝⁶ : Module R' N₁\ninst✝⁵ : AddCommGroup N₂\ninst✝⁴ : Module R' N₂\ninst✝³ : AddCommMonoid Mᵢ\ninst✝² : Module R' ... | [] | refine
LinearMap.mk₂ R' domCoprod (fun m₁ m₂ n => ?_) (fun c m n => ?_) (fun m n₁ n₂ => ?_)
fun c m n => ?_ <;>
· ext
simp only [domCoprod_apply, add_apply, smul_apply, ← Finset.sum_add_distrib,
Finset.smul_sum, _root_.sum_apply, domCoprod.summand]
congr
ext σ
induction... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 189,
"column": 49
} | {
"line": 191,
"column": 49
} | {
"line": 193,
"column": 0
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\nf : Dual R V\nv : V\nhv : f v = 0\nhv' : f (-v) = 0\n⊢ (transvection hv).symm = transvection hv'",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"LinearMap.transvection",
"LinearM... | [] | by
ext;
simp [symm_apply_eq, comp_of_left_eq_apply hv'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 347,
"column": 2
} | {
"line": 347,
"column": 55
} | {
"line": 349,
"column": 0
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\nf : Dual R V\nv : V\nh : IsUnit (1 + f v)\nx : V\n⊢ (dilatransvection h) x = x + f x • v",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"LinearMap.transvection",
"NegZeroClass.to... | [] | simp [dilatransvection, LinearMap.transvection.apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 347,
"column": 2
} | {
"line": 347,
"column": 55
} | {
"line": 349,
"column": 0
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\nf : Dual R V\nv : V\nh : IsUnit (1 + f v)\nx : V\n⊢ (dilatransvection h) x = x + f x • v",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"LinearMap.transvection",
"NegZeroClass.to... | [] | simp [dilatransvection, LinearMap.transvection.apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 347,
"column": 2
} | {
"line": 347,
"column": 55
} | {
"line": 349,
"column": 0
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\nf : Dual R V\nv : V\nh : IsUnit (1 + f v)\nx : V\n⊢ (dilatransvection h) x = x + f x • v",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"LinearMap.transvection",
"NegZeroClass.to... | [] | simp [dilatransvection, LinearMap.transvection.apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Center | {
"line": 201,
"column": 2
} | {
"line": 201,
"column": 36
} | {
"line": 202,
"column": 2
} | [
{
"pp": "case inr\nR : Type u_1\nV : Type u_2\ninst✝⁵ : Ring R\ninst✝⁴ : IsDomain R\ninst✝³ : StrongRankCondition R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : Free R V\nf : V →ₗ[R] V\nhV1 : finrank R V ≠ 1\nh : ∀ (v : V), ¬LinearIndependent R ![v, f v]\nhV : Nontrivial V\n⊢ ∃ a, f = a • 1",
"ppT... | [
"case inr\nR : Type u_1\nV : Type u_2\ninst✝⁵ : Ring R\ninst✝⁴ : IsDomain R\ninst✝³ : StrongRankCondition R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : Free R V\nf : V →ₗ[R] V\nhV1 : finrank R V ≠ 1\nh : ∀ (v : V), ¬LinearIndependent R ![v, f v]\nhV : Nontrivial V\nι : Type u_2 := Free.ChooseBasisIndex R... | let ι := Free.ChooseBasisIndex R V | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.LinearAlgebra.QuadraticForm.TensorProduct | {
"line": 112,
"column": 22
} | {
"line": 112,
"column": 46
} | {
"line": 112,
"column": 47
} | [
{
"pp": "R : Type uR\nA : Type uA\nM₁ : Type uM₁\nM₂ : Type uM₂\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : AddCommGroup M₁\ninst✝⁸ : AddCommGroup M₂\ninst✝⁷ : Algebra R A\ninst✝⁶ : Module R M₁\ninst✝⁵ : Module A M₁\ninst✝⁴ : SMulCommClass R A M₁\ninst✝³ : IsScalarTower R A M₁\ninst✝² : Module R M₂\ni... | [
"R : Type uR\nA : Type uA\nM₁ : Type uM₁\nM₂ : Type uM₂\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : CommRing A\ninst✝⁹ : AddCommGroup M₁\ninst✝⁸ : AddCommGroup M₂\ninst✝⁷ : Algebra R A\ninst✝⁶ : Module R M₁\ninst✝⁵ : Module A M₁\ninst✝⁴ : SMulCommClass R A M₁\ninst✝³ : IsScalarTower R A M₁\ninst✝² : Module R M₂\ninst✝¹ : Inve... | BilinForm.tensorDistrib, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.QuadraticForm.TensorProduct | {
"line": 163,
"column": 21
} | {
"line": 163,
"column": 35
} | {
"line": 163,
"column": 36
} | [
{
"pp": "R : Type uR\nA : Type uA\nM₂ : Type uM₂\nN₁ : Type uN₁\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : AddCommGroup M₂\ninst✝⁵ : AddCommGroup N₁\ninst✝⁴ : Algebra R A\ninst✝³ : Module R N₁\ninst✝² : Module A N₁\ninst✝¹ : IsScalarTower R A N₁\ninst✝ : Module R M₂\nQ₁ Q₂ : QuadraticMap A (A ⊗[R] M₂) ... | [
"R : Type uR\nA : Type uA\nM₂ : Type uM₂\nN₁ : Type uN₁\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing A\ninst✝⁶ : AddCommGroup M₂\ninst✝⁵ : AddCommGroup N₁\ninst✝⁴ : Algebra R A\ninst✝³ : Module R N₁\ninst✝² : Module A N₁\ninst✝¹ : IsScalarTower R A N₁\ninst✝ : Module R M₂\nQ₁ Q₂ : QuadraticMap A (A ⊗[R] M₂) N₁\nh : ∀ (m... | ← smul_eq_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 91,
"column": 74
} | {
"line": 91,
"column": 77
} | {
"line": 91,
"column": 78
} | [
{
"pp": "case add\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd d' d₁ d₂ : Module.Dual R M\nx✝ y✝ : CliffordAlgebra Q\nhx :\n (foldr' Q (contractLeftAux Q (d₁ + d₂)) ⋯ 0) x✝ =\n (foldr' Q (contractLeftAux Q d₁) ⋯ 0) x✝ + (foldr' Q (cont... | [
"case add\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd d' d₁ d₂ : Module.Dual R M\nx✝ y✝ : CliffordAlgebra Q\nhx :\n (foldr' Q (contractLeftAux Q (d₁ + d₂)) ⋯ 0) x✝ =\n (foldr' Q (contractLeftAux Q d₁) ⋯ 0) x✝ + (foldr' Q (contractLeftAux ... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 101,
"column": 57
} | {
"line": 101,
"column": 60
} | {
"line": 101,
"column": 61
} | [
{
"pp": "case add\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd✝ d' : Module.Dual R M\nc : R\nd : Module.Dual R M\nx✝ y✝ : CliffordAlgebra Q\nhx : (foldr' Q (contractLeftAux Q (c • d)) ⋯ 0) x✝ = c • (foldr' Q (contractLeftAux Q d) ⋯ 0) x✝\n... | [
"case add\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd✝ d' : Module.Dual R M\nc : R\nd : Module.Dual R M\nx✝ y✝ : CliffordAlgebra Q\nhx : (foldr' Q (contractLeftAux Q (c • d)) ⋯ 0) x✝ = c • (foldr' Q (contractLeftAux Q d) ⋯ 0) x✝\nhy : (foldr'... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 191,
"column": 43
} | {
"line": 191,
"column": 46
} | {
"line": 191,
"column": 47
} | [
{
"pp": "case add\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd : Module.Dual R M\nx✝ y✝ : CliffordAlgebra Q\nhx : (contractLeft d) ((contractLeft d) x✝) = 0\nhy : (contractLeft d) ((contractLeft d) y✝) = 0\n⊢ (contractLeft d) ((contractLef... | [
"case add\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd : Module.Dual R M\nx✝ y✝ : CliffordAlgebra Q\nhx : (contractLeft d) ((contractLeft d) x✝) = 0\nhy : (contractLeft d) ((contractLeft d) y✝) = 0\n⊢ 0 + (contractLeft d) ((contractLeft d) y✝)... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 193,
"column": 57
} | {
"line": 193,
"column": 60
} | {
"line": 193,
"column": 61
} | [
{
"pp": "case ι_mul\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd : Module.Dual R M\nx✝ : CliffordAlgebra Q\nm✝ : M\nhx : (contractLeft d) ((contractLeft d) x✝) = 0\n⊢ (contractLeft d) (d m✝ • x✝) - (d m✝ • (contractLeft d) x✝ - (ι Q) m✝ * ... | [
"case ι_mul\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd : Module.Dual R M\nx✝ : CliffordAlgebra Q\nm✝ : M\nhx : (contractLeft d) ((contractLeft d) x✝) = 0\n⊢ (contractLeft d) (d m✝ • x✝) - (d m✝ • (contractLeft d) x✝ - (ι Q) m✝ * 0) = 0"
] | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 630,
"column": 4
} | {
"line": 630,
"column": 42
} | {
"line": 630,
"column": 42
} | [
{
"pp": "R : Type u_3\nV : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module R V\ninst✝² : Free R V\ninst✝¹ : Module.Finite R V\ninst✝ : IsDomain R\nf : Dual R V\nv : V\nK : Type u_3 := FractionRing R\nthis✝ : Field K := inferInstance\nthis : (algebraMap R K) (LinearMap.det (transvection f... | [
"R : Type u_3\nV : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module R V\ninst✝² : Free R V\ninst✝¹ : Module.Finite R V\ninst✝ : IsDomain R\nf : Dual R V\nv : V\nK : Type u_3 := FractionRing R\nthis✝ : Field K := inferInstance\nthis : (algebraMap R K) (LinearMap.det (transvection f v)) = ↑1 + ... | ← algebraMap.coe_one (R := R) (A := K) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 203,
"column": 61
} | {
"line": 203,
"column": 64
} | {
"line": 203,
"column": 65
} | [
{
"pp": "case add\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nx✝ y✝ : CliffordAlgebra Q\nhx : (contractLeft d) ((contractLeft d') x✝) = -(contractLeft d') ((contractLeft d) x✝)\nhy : (contractLeft d) ((contractLeft d... | [
"case add\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nx✝ y✝ : CliffordAlgebra Q\nhx : (contractLeft d) ((contractLeft d') x✝) = -(contractLeft d') ((contractLeft d) x✝)\nhy : (contractLeft d) ((contractLeft d') y✝) = -(c... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 206,
"column": 37
} | {
"line": 206,
"column": 40
} | {
"line": 206,
"column": 41
} | [
{
"pp": "case ι_mul\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nx✝ : CliffordAlgebra Q\nm✝ : M\nhx : (contractLeft d) ((contractLeft d') x✝) = -(contractLeft d') ((contractLeft d) x✝)\n⊢ d' m✝ • (contractLeft d) x✝ +... | [
"case ι_mul\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nd d' : Module.Dual R M\nx✝ : CliffordAlgebra Q\nm✝ : M\nhx : (contractLeft d) ((contractLeft d') x✝) = -(contractLeft d') ((contractLeft d) x✝)\n⊢ d' m✝ • (contractLeft d) x✝ + (ι Q) m✝ * ... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 84
} | {
"line": 147,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ : QuadraticForm R M\nf : M × R →ₗ[R] M × R →ₗ[R] CliffordAlgebra Q :=\n ((Algebra.lmul R (CliffordAlgebra Q)).toLinearMap ∘ₗ\n (ι Q ∘ₗ LinearMap.fst R M R + Algebra.linearMap R (CliffordAlgebra Q) ∘ₗ... | [] | rw [ι_sq_scalar, ← map_mul, ← map_sub, sub_eq_add_neg, Q'_apply, sub_eq_add_neg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 295,
"column": 61
} | {
"line": 295,
"column": 64
} | {
"line": 295,
"column": 65
} | [
{
"pp": "case add\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ Q' : QuadraticForm R M\nB : BilinForm R M\nh : BilinMap.toQuadraticMap B = Q' - Q\nd : Module.Dual R M\nx✝ y✝ : CliffordAlgebra Q\nhx : (changeForm h) ((contractLeft d) x✝) = (contractLeft d) ((chang... | [
"case add\nR : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ Q' : QuadraticForm R M\nB : BilinForm R M\nh : BilinMap.toQuadraticMap B = Q' - Q\nd : Module.Dual R M\nx✝ y✝ : CliffordAlgebra Q\nhx : (changeForm h) ((contractLeft d) x✝) = (contractLeft d) ((changeForm h) x✝)... | hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
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