module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.GroupTheory.Coprod.Basic | {
"line": 673,
"column": 8
} | {
"line": 673,
"column": 19
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\n⊢ (Coprod.lift (inl.comp inl) (Coprod.map inr (MonoidHom.id P))).comp\n (Coprod.lift (Coprod.map (MonoidHom.id M) inl) (inr.comp inr)) =\n MonoidHom.id ((M ∗ N) ∗ P)",
"usedConstants": [
... | ext <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.GroupTheory.Coprod.Basic | {
"line": 673,
"column": 8
} | {
"line": 673,
"column": 19
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\n⊢ (Coprod.lift (inl.comp inl) (Coprod.map inr (MonoidHom.id P))).comp\n (Coprod.lift (Coprod.map (MonoidHom.id M) inl) (inr.comp inr)) =\n MonoidHom.id ((M ∗ N) ∗ P)",
"usedConstants": [
... | ext <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Coprod.Basic | {
"line": 673,
"column": 8
} | {
"line": 673,
"column": 19
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\n⊢ (Coprod.lift (inl.comp inl) (Coprod.map inr (MonoidHom.id P))).comp\n (Coprod.lift (Coprod.map (MonoidHom.id M) inl) (inr.comp inr)) =\n MonoidHom.id ((M ∗ N) ∗ P)",
"usedConstants": [
... | ext <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Coprod.Basic | {
"line": 673,
"column": 25
} | {
"line": 673,
"column": 36
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\n⊢ (Coprod.lift (Coprod.map (MonoidHom.id M) inl) (inr.comp inr)).comp\n (Coprod.lift (inl.comp inl) (Coprod.map inr (MonoidHom.id P))) =\n MonoidHom.id (M ∗ (N ∗ P))",
"usedConstants": [
... | ext <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.GroupTheory.Coprod.Basic | {
"line": 673,
"column": 25
} | {
"line": 673,
"column": 36
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\n⊢ (Coprod.lift (Coprod.map (MonoidHom.id M) inl) (inr.comp inr)).comp\n (Coprod.lift (inl.comp inl) (Coprod.map inr (MonoidHom.id P))) =\n MonoidHom.id (M ∗ (N ∗ P))",
"usedConstants": [
... | ext <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Coprod.Basic | {
"line": 673,
"column": 25
} | {
"line": 673,
"column": 36
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nP : Type u_3\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid P\n⊢ (Coprod.lift (Coprod.map (MonoidHom.id M) inl) (inr.comp inr)).comp\n (Coprod.lift (inl.comp inl) (Coprod.map inr (MonoidHom.id P))) =\n MonoidHom.id (M ∗ (N ∗ P))",
"usedConstants": [
... | ext <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Coxeter.Basic | {
"line": 463,
"column": 18
} | {
"line": 463,
"column": 39
} | [
{
"pp": "B : Type u_1\ni j : B\np k : ℕ\nh' : k < 2 * p → take k (alternatingWord i j (2 * p)) = if Even k then alternatingWord i j k else alternatingWord j i k\nh : k + 1 < 2 * p\nh_even : ¬Even k\nhk : take k (alternatingWord i j (2 * p)) = alternatingWord j i k\n⊢ Odd (2 * p + k)",
"usedConstants": [
... | apply Nat.odd_add.mpr | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.GroupTheory.CoprodI | {
"line": 1004,
"column": 4
} | {
"line": 1004,
"column": 49
} | [
{
"pp": "case hpp\nι : Type u_1\ninst✝² : Nontrivial ι\nG : Type u_1\ninst✝¹ : Group G\na : ι → G\nα : Type u_4\ninst✝ : MulAction G α\nX Y : ι → Set α\nhXnonempty : ∀ (i : ι), (X i).Nonempty\nhXdisj : Pairwise (Disjoint on X)\nhYdisj : Pairwise (Disjoint on Y)\nhXYdisj : ∀ (i j : ι), Disjoint (X i) (Y j)\nhX :... | simp only [map_zpow, FreeGroup.lift_apply_of] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.Coxeter.Inversion | {
"line": 346,
"column": 2
} | {
"line": 346,
"column": 74
} | [
{
"pp": "B : Type u_1\nW : Type u_2\ninst✝ : Group W\nM : CoxeterMatrix B\ncs : CoxeterSystem M W\nω : List B\nt : W\nht : t ∈ cs.leftInvSeq ω\n⊢ cs.IsReflection t",
"usedConstants": [
"congrArg",
"Membership.mem",
"Eq.mp",
"_private.Mathlib.GroupTheory.Coxeter.Inversion.0.CoxeterSys... | simp only [leftInvSeq_eq_reverse_rightInvSeq_reverse, mem_reverse] at ht | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.DoubleCoset | {
"line": 169,
"column": 2
} | {
"line": 178,
"column": 58
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\na : G\n⊢ ⋃ k, op (a * ↑k) • ↑H = doubleCoset a ↑H ↑K",
"usedConstants": [
"mul_inv_cancel_right",
"Set.ext",
"Eq.mpr",
"Semigroup.toMul",
"DivInvMonoid.toInv",
"instHSMul",
"HMul.hMul",
"DivInvOneMonoid... | ext x
simp only [mem_rightCoset_iff, mul_inv_rev, Set.mem_iUnion, mem_doubleCoset,
SetLike.mem_coe]
constructor
· rintro ⟨y, h_h⟩
refine ⟨x * (y⁻¹ * a⁻¹), h_h, y, y.2, ?_⟩
simp only [← mul_assoc, inv_mul_cancel_right, InvMemClass.coe_inv]
· rintro ⟨x, hx, y, hy, hxy⟩
refine ⟨⟨y, hy⟩, ?_⟩
sim... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.DoubleCoset | {
"line": 169,
"column": 2
} | {
"line": 178,
"column": 58
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\na : G\n⊢ ⋃ k, op (a * ↑k) • ↑H = doubleCoset a ↑H ↑K",
"usedConstants": [
"mul_inv_cancel_right",
"Set.ext",
"Eq.mpr",
"Semigroup.toMul",
"DivInvMonoid.toInv",
"instHSMul",
"HMul.hMul",
"DivInvOneMonoid... | ext x
simp only [mem_rightCoset_iff, mul_inv_rev, Set.mem_iUnion, mem_doubleCoset,
SetLike.mem_coe]
constructor
· rintro ⟨y, h_h⟩
refine ⟨x * (y⁻¹ * a⁻¹), h_h, y, y.2, ?_⟩
simp only [← mul_assoc, inv_mul_cancel_right, InvMemClass.coe_inv]
· rintro ⟨x, hx, y, hy, hxy⟩
refine ⟨⟨y, hy⟩, ?_⟩
sim... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.DoubleCoset | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 20
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nt : Finset (Quotient ↑H ↑K)\nht : ⋃ q ∈ t, doubleCoset (out q) ↑H ↑K ≠ Set.univ\nx : G\ny : Quotient ↑H ↑K\nhy : y ∈ t\nq : G\nhq : q ∈ doubleCoset (out y) ↑H ↑K\nhx : Quot.mk (⇑(leftRel K)) q = Quot.mk (⇑(leftRel K)) x\n⊢ ∃ i ∈ t, x ∈ doubleCoset (out i... | refine ⟨y, hy, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.GroupTheory.DoubleCoset | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 20
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nH K : Subgroup G\nt : Finset (Quotient ↑H ↑K)\nht : ⋃ q ∈ t, doubleCoset (out q) ↑H ↑K ≠ Set.univ\nx : G\ny : Quotient ↑H ↑K\nhy : y ∈ t\nq : G\nhq : q ∈ doubleCoset (out y) ↑H ↑K\nhx : Quot.mk (⇑(rightRel H)) q = Quot.mk (⇑(rightRel H)) x\n⊢ ∃ i ∈ t, x ∈ doubleCoset (out... | refine ⟨y, hy, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.GroupTheory.DivisibleHull | {
"line": 146,
"column": 2
} | {
"line": 146,
"column": 74
} | [
{
"pp": "M : Type u_1\ninst✝ : AddCommMonoid M\na : ℚ≥0\nm : M\ns : ℕ+\n⊢ a • mk m s = mk (a.num • m) (⟨a.den, ⋯⟩ * s)",
"usedConstants": [
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"False",
"instHSMul",
"Submonoid.mul",
"HMul.hMul",
"instLinearOrderPNat",
"L... | convert! LocalizedModule.mk'_smul_mk ℚ≥0 a.num m ⟨a.den, by simp⟩ (↑ⁿ s) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.GroupTheory.DivisibleHull | {
"line": 158,
"column": 2
} | {
"line": 158,
"column": 42
} | [
{
"pp": "M : Type u_1\ninst✝¹ : AddCommMonoid M\ninst✝ : IsAddTorsionFree M\ns : ℕ+\nm n : M\nh : (fun m ↦ mk m s) m = (fun m ↦ mk m s) n\n⊢ m = n",
"usedConstants": [
"_private.Mathlib.GroupTheory.DivisibleHull.0.DivisibleHull.mk_left_injective._simp_1_1",
"PNat.val",
"instHSMul",
"... | simp_rw [mk_eq_mk_iff_smul_eq_smul] at h | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.GroupTheory.FreeGroup.CyclicallyReduced | {
"line": 184,
"column": 2
} | {
"line": 185,
"column": 87
} | [
{
"pp": "case zero\nα : Type u\nL : List (α × Bool)\ninst✝ : DecidableEq α\nh : IsReduced L\n⊢ reduce (replicate (0 + 1) L).flatten =\n conjugator L ++ (replicate (0 + 1) (reduceCyclically L)).flatten ++ invRev (conjugator L)",
"usedConstants": [
"Eq.mpr",
"List.replicate",
"congrArg",
... | case zero =>
simpa [← append_assoc, conj_conjugator_reduceCyclically, ← isReduced_iff_reduce_eq] | Lean.Elab.Tactic.evalCase | Lean.Parser.Tactic.case |
Mathlib.GroupTheory.Nilpotent | {
"line": 124,
"column": 33
} | {
"line": 124,
"column": 54
} | [
{
"pp": "case h\nG : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : H.Normal\nx✝ y : G\n⊢ ⁅x✝, y⁆ ∈ H ↔ x✝ * y * (x✝⁻¹ * y⁻¹) ∈ H",
"usedConstants": [
"Eq.mpr",
"commutatorElement_def",
"DivInvMonoid.toInv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Bra... | commutatorElement_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Nilpotent | {
"line": 330,
"column": 4
} | {
"line": 330,
"column": 9
} | [
{
"pp": "case mpr\nG : Type u_1\ninst✝ : Group G\nn : ℕ\nH : ℕ → Subgroup G\nhH : IsAscendingCentralSeries H\nhn : H n = ⊤\n⊢ Group.IsNilpotent G",
"usedConstants": [
"Subgroup.upperCentralSeries",
"Subgroup",
"Nat",
"Exists.intro",
"Subgroup.instTop",
"Top.top",
"E... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.GroupTheory.Nilpotent | {
"line": 464,
"column": 4
} | {
"line": 464,
"column": 9
} | [
{
"pp": "case mp\nG : Type u_1\ninst✝ : Group G\nn : ℕ\nH : ℕ → Subgroup G\nhn : H n = ⊥\nh0 : H 0 = ⊤\nhs : ∀ (x : G) (n : ℕ), x ∈ H n → ∀ (g : G), ⁅x, g⁆ ∈ H (n + 1)\n⊢ ∃ n, lowerCentralSeries G n = ⊥",
"usedConstants": [
"Subgroup",
"Bot.bot",
"Subgroup.lowerCentralSeries",
"Nat",... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.GroupTheory.Nilpotent | {
"line": 610,
"column": 2
} | {
"line": 610,
"column": 7
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nH✝ : Subgroup G\ninst✝ : H✝.Normal\nH : Subgroup G\nn : ℕ\nhG : lowerCentralSeries G n = ⊥\n⊢ ∃ n, lowerCentralSeries (↥H) n = ⊥",
"usedConstants": [
"Membership.mem",
"Subtype",
"Subgroup",
"Bot.bot",
"Subgroup.lowerCentralSeries",
... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.GroupTheory.Nilpotent | {
"line": 698,
"column": 2
} | {
"line": 698,
"column": 7
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nG' : Type u_2\ninst✝ : Group G'\nf : G →* G'\nhf : Function.Surjective ⇑f\nn : ℕ\nhn : upperCentralSeries G n = ⊤\n⊢ IsNilpotent G'",
"usedConstants": [
"Subgroup.upperCentralSeries",
"Subgroup",
"Nat",
"Exists.intro",
"Subgroup.instTop"... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.GroupTheory.Nilpotent | {
"line": 803,
"column": 4
} | {
"line": 813,
"column": 70
} | [
{
"pp": "case pos.succ\nG : Type u_1\ninst✝ : Group G\nhH : IsNilpotent G\nn : ℕ\nhn : nilpotencyClass G = n + 1\n⊢ nilpotencyClass (G ⧸ center G) = n + 1 - 1",
"usedConstants": [
"Eq.mpr",
"Nat.instOrderedSub",
"le_of_add_le_add_right",
"Trans.trans",
"Preorder.toLT",
"N... | suffices Group.nilpotencyClass (G ⧸ center G) = n by simpa
apply le_antisymm
· apply upperCentralSeries_eq_top_iff_nilpotencyClass_le.mp
apply comap_injective (f := (mk' (center G))) Quot.mk_surjective
rw [comap_upperCentralSeries_quotient_center, comap_top, Nat.succ_eq_add_one, ← hn]
exact up... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Nilpotent | {
"line": 803,
"column": 4
} | {
"line": 813,
"column": 70
} | [
{
"pp": "case pos.succ\nG : Type u_1\ninst✝ : Group G\nhH : IsNilpotent G\nn : ℕ\nhn : nilpotencyClass G = n + 1\n⊢ nilpotencyClass (G ⧸ center G) = n + 1 - 1",
"usedConstants": [
"Eq.mpr",
"Nat.instOrderedSub",
"le_of_add_le_add_right",
"Trans.trans",
"Preorder.toLT",
"N... | suffices Group.nilpotencyClass (G ⧸ center G) = n by simpa
apply le_antisymm
· apply upperCentralSeries_eq_top_iff_nilpotencyClass_le.mp
apply comap_injective (f := (mk' (center G))) Quot.mk_surjective
rw [comap_upperCentralSeries_quotient_center, comap_top, Nat.succ_eq_add_one, ← hn]
exact up... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Nilpotent | {
"line": 1062,
"column": 2
} | {
"line": 1062,
"column": 7
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Group G\nH : Subgroup G\ninst✝ : H.Normal\nh : Group.IsNilpotent G\nn : ℕ\nhn : lowerCentralSeries G n = ⊥\n⊢ IsSolvable G",
"usedConstants": [
"Subgroup",
"Bot.bot",
"derivedSeries",
"Nat",
"Exists.intro",
"Subgroup.instBot",
"Eq",
... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.GroupTheory.GroupAction.Transitive | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 36
} | [
{
"pp": "M : Type u_3\nN : Type u_4\nα : Type u_5\nβ : Type u_6\ninst✝³ : Monoid M\ninst✝² : Monoid N\ninst✝¹ : MulAction M α\ninst✝ : MulAction N β\nφ : M → N\nf : α →ₑ[φ] β\nhf : Function.Surjective ⇑f\nh : IsPretransitive M α\n⊢ IsPretransitive N β",
"usedConstants": [
"MulAction.IsPretransitive.mk... | apply MulAction.IsPretransitive.mk | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.GroupTheory.GroupAction.Primitive | {
"line": 248,
"column": 4
} | {
"line": 248,
"column": 26
} | [
{
"pp": "case mp.inr\nG : Type u_3\ninst✝³ : Group G\nX : Type u_4\ninst✝² : MulAction G X\ninst✝¹ : IsPretransitive G X\ninst✝ : Nontrivial X\na : X\nh : IsSimpleOrder (BlockMem G a)\nh_bot_or_top : ∀ (a_1 : BlockMem G a), a_1 = ⊥ ∨ a_1 = ⊤ := eq_bot_or_eq_top\nB : Set X\nhaB : a ∈ B\nhB : IsBlock G B\nhB' : B... | · right; rw [hB']; rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.GroupAction.Primitive | {
"line": 299,
"column": 45
} | {
"line": 299,
"column": 50
} | [
{
"pp": "G : Type u_1\nX : Type u_2\ninst✝⁴ : Group G\ninst✝³ : MulAction G X\nM : Type u_3\ninst✝² : Group M\nα : Type u_4\ninst✝¹ : MulAction M α\ninst✝ : IsPreprimitive M α\nN : Subgroup M\nx✝ : N.Normal\na : α\nha : a ∉ fixedPoints (↥N) α\nh : (orbit (↥N) a).Subsingleton\nn : ↥N\nthis : orbit (↥N) a = {a}\n... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 426,
"column": 2
} | {
"line": 426,
"column": 26
} | [
{
"pp": "case hB\nG : Type u_1\ninst✝¹ : Group G\nX : Type u_2\ninst✝ : MulAction G X\nB : Set X\ng : G\nhB : IsBlock (↥⊤) B\n⊢ IsBlock (↥(Subgroup.comap ↑(MulAut.conj g) ⊤)) B",
"usedConstants": [
"Eq.mpr",
"MulEquiv.instEquivLike",
"MonoidHom.instFunLike",
"MulAction.IsBlock",
... | rwa [Subgroup.comap_top] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.GroupTheory.GroupAction.Blocks | {
"line": 655,
"column": 6
} | {
"line": 655,
"column": 20
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nX : Type u_2\ninst✝¹ : MulAction G X\nB : Set X\ninst✝ : Nontrivial X\na : X\n⊢ Nontrivial (BlockMem G a)",
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"congrArg",
"Exists",
"id",
"Ne",
"nontrivial_iff",
"MulAction.Block... | nontrivial_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Cardinal.Embedding | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 62
} | [
{
"pp": "α : Type u_1\nm k : ℕ\nhmn : m ≤ m + k\nhn : ↑(m + k) ≤ ENat.card α\n⊢ Surjective fun x ↦ (castLEEmb hmn).trans x",
"usedConstants": [
"Fin.Embedding.restrictSurjective_of_add_le_ENatCard"
]
}
] | exact Fin.Embedding.restrictSurjective_of_add_le_ENatCard hn | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 380,
"column": 20
} | {
"line": 380,
"column": 39
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nhα4 : 4 ≤ Nat.card α\ng : Perm α\nhg : g ∈ alternatingGroup α\nhg' : ⟨g, hg⟩ ∈ center ↥(alternatingGroup α)\na : α\nha : a ∈ g.support\nhab : g a ≠ a\nc d : α\nhcd : c ≠ d\nhc : c ≠ a ∧ c ≠ g a\nhd : d ≠ a ∧ d ≠ g a\nk : Perm α := swap (g a) d * ... | exact hc.right.symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.SpecificGroups.Alternating | {
"line": 411,
"column": 6
} | {
"line": 411,
"column": 77
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\ns : Finset α\n⊢ Function.Injective ⇑(ofSubtype s)",
"usedConstants": [
"Eq.mpr",
"MonoidHom.instFunLike",
"MonoidHom",
"Monoid.toMulOneClass",
"congrArg",
"Finset",
"Subgroup.subtype",
"Subgroup... | ← Function.Injective.of_comp_iff (alternatingGroup α).subtype_injective | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.MultiplePrimitivity | {
"line": 143,
"column": 2
} | {
"line": 143,
"column": 38
} | [
{
"pp": "M : Type u_1\nα : Type u_2\ninst✝³ : Group M\ninst✝² : MulAction M α\ninst✝¹ : IsPretransitive M α\nn : ℕ\na : α\ninst✝ : IsMultiplyPreprimitive M α n.succ\n⊢ IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(ofStabilizer M a)) n",
"usedConstants": [
"instOfNatNat",
"Nat",
"Nat.lt_or_... | rcases Nat.lt_or_ge n 1 with h0 | h1 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.GroupTheory.GroupAction.MultiplePrimitivity | {
"line": 198,
"column": 6
} | {
"line": 198,
"column": 70
} | [
{
"pp": "case mpr.right\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\nH : IsMultiplyPreprimitive (↥(stabilizer M a)) (↥(ofStabilizer M a)) n\ns : Set α\nhs : s.encard + 1 = ↑n.succ\nb : α\nhb : b ∈ s\ng : M\nhg : g • b = a\ns' : Set... | apply IsMultiplyPreprimitive.isPreprimitive_ofFixingSubgroup _ n | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.GroupTheory.GroupAction.MultiplePrimitivity | {
"line": 225,
"column": 4
} | {
"line": 225,
"column": 68
} | [
{
"pp": "M : Type u_1\nα : Type u_2\ninst✝³ : Group M\ninst✝² : MulAction M α\nm n : ℕ\ninst✝¹ : IsMultiplyPreprimitive M α n\ns : Set α\ninst✝ : Finite ↑s\nhs : s.ncard + m = n\nt : Set ↥(ofFixingSubgroup M s)\nht : t.encard + 1 = ↑m\nt' : Set α := Subtype.val '' t\nhtt' : t = Subtype.val ⁻¹' t'\n⊢ IsPreprimit... | apply IsMultiplyPreprimitive.isPreprimitive_ofFixingSubgroup _ n | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.GroupTheory.GroupAction.MultiplePrimitivity | {
"line": 238,
"column": 2
} | {
"line": 239,
"column": 34
} | [
{
"pp": "case inl\nM : Type u_1\nα : Type u_2\ninst✝² : Group M\ninst✝¹ : MulAction M α\nn : ℕ\nhα : ↑n.succ ≤ ENat.card α\ninst✝ : IsMultiplyPretransitive M α n.succ\nhn : n = 0\n⊢ IsMultiplyPreprimitive M α n",
"usedConstants": [
"Eq.mpr",
"MulAction.is_zero_preprimitive",
"congrArg",
... | · rw [hn]
exact is_zero_preprimitive M α | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.GroupAction.MultipleTransitivity | {
"line": 503,
"column": 43
} | {
"line": 503,
"column": 81
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\nα : Type u_2\ninst✝¹ : MulAction G α\ninst✝ : Finite α\ns : Set α\nhMk : IsMultiplyPretransitive G α s.ncard\n⊢ (Nat.card α)! = (Nat.card α).choose s.ncard * s.ncard ! * (Nat.card α - s.ncard)!",
"usedConstants": [
"Eq.mpr",
"Nat.choose",
"HMul.hMul... | Nat.choose_mul_factorial_mul_factorial | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.MultipleTransitivity | {
"line": 520,
"column": 44
} | {
"line": 520,
"column": 87
} | [
{
"pp": "ι : Type u_2\ninst✝ : Finite ι\nβ : Type u_3\nx y : ι ↪ β\nσ : Perm β\nhσ : ∀ (a : ι), σ (x a) = y a\ni : ι\n⊢ (σ • x) i = y i",
"usedConstants": [
"Equiv.Perm.applyMulAction",
"instHSMul",
"congrArg",
"Function.Embedding.smul",
"Function.Embedding",
"Equiv.Perm.... | by simp [Function.Embedding.smul_apply, hσ] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.GroupAction.Jordan | {
"line": 79,
"column": 8
} | {
"line": 79,
"column": 22
} | [
{
"pp": "case a.right\nG : Type u_1\nα : Type u_2\ninst✝¹ : Group G\ninst✝ : MulAction G α\nhsn' : 2 < ENat.card α\nhG' : IsMultiplyPretransitive G α 2\na : α\nthis✝¹ : IsPretransitive G α\nthis✝ : Nontrivial α\nhGa : IsCoatom (stabilizer G a)\nhyp : ↑(normalClosure ↑(stabilizer G a)) ⊆ ↑(stabilizer G a)\nthis ... | nontrivial_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.MultiplePrimitivity | {
"line": 320,
"column": 4
} | {
"line": 357,
"column": 34
} | [
{
"pp": "case right\nM : 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\n⊢ ∀ {s : Set α}, s.encard + 1... | intro s hs
let t := f '' s
let ψ : fixingSubgroup M s → fixingSubgroup N t := fun ⟨g, hg⟩ ↦ ⟨φ g, by
simp only [mem_fixingSubgroup_iff] at hg ⊢
intro y hy
suffices ∃ x ∈ s, y = f x by
obtain ⟨x, hx, rfl⟩ := this
rwa [← map_smulₛₗ, hg]
obtain ⟨x, rfl⟩ := hf.surjective y
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.GroupAction.MultiplePrimitivity | {
"line": 320,
"column": 4
} | {
"line": 357,
"column": 34
} | [
{
"pp": "case right\nM : 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\n⊢ ∀ {s : Set α}, s.encard + 1... | intro s hs
let t := f '' s
let ψ : fixingSubgroup M s → fixingSubgroup N t := fun ⟨g, hg⟩ ↦ ⟨φ g, by
simp only [mem_fixingSubgroup_iff] at hg ⊢
intro y hy
suffices ∃ x ∈ s, y = f x by
obtain ⟨x, hx, rfl⟩ := this
rwa [← map_smulₛₗ, hg]
obtain ⟨x, rfl⟩ := hf.surjective y
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.GroupAction.Jordan | {
"line": 326,
"column": 36
} | {
"line": 326,
"column": 38
} | [
{
"pp": "case neg\nα : Type u_1\nK : Type u_2\ninst✝¹ : Group K\ninst✝ : MulAction K α\nhα : Nat.card α = 2\nhK : fixedPoints K α ≠ _root_.Set.univ\nn : ℕ\nthis✝ : Finite α\nthis : Fintype α\nh2 : IsMultiplyPretransitive K α 2\nhn : ¬n ≤ 2\n⊢ ¬n ≤ Nat.card α",
"usedConstants": [
"Eq.mpr",
"congr... | hα | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.SubMulAction.Combination | {
"line": 252,
"column": 2
} | {
"line": 252,
"column": 70
} | [
{
"pp": "α : Type u_2\ninst✝ : DecidableEq α\nn : ℕ\nh_one_le : 1 ≤ n\nhn : n < Nat.card α\nhα : Nat.card α ≠ 2 * n\n⊢ IsPreprimitive (Perm α) ↑(powersetCard α n)",
"usedConstants": [
"Finite",
"Nat.card",
"Nat.ne_zero_of_lt",
"Nat.finite_of_card_ne_zero"
]
}
] | have : Finite α := Nat.finite_of_card_ne_zero (Nat.ne_zero_of_lt hn) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.GroupTheory.GroupAction.SubMulAction.Combination | {
"line": 283,
"column": 6
} | {
"line": 283,
"column": 19
} | [
{
"pp": "α : 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⊢ IsPretran... | apply this hα | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.GroupTheory.IsSubnormal | {
"line": 270,
"column": 4
} | {
"line": 270,
"column": 49
} | [
{
"pp": "case step\nG : Type u_1\ninst✝¹ : Group G\nG' : Type u_2\ninst✝ : Group G'\nH' : Subgroup G'\nf : G →* G'\nH K : Subgroup G'\nh_le : H ≤ K\nhSubn : K.IsSubnormal\nhN : (H.subgroupOf K).Normal\nih : (comap f K).IsSubnormal\n⊢ (comap f H).IsSubnormal",
"usedConstants": [
"Subgroup.IsSubnormal.s... | apply step _ (comap f K) (comap_mono h_le) ih | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.GroupTheory.Perm.ClosureSwap | {
"line": 137,
"column": 83
} | {
"line": 138,
"column": 72
} | [
{
"pp": "α : Type u_2\ninst✝² : DecidableEq α\ninst✝¹ : Finite α\nS : Set (Equiv.Perm α)\nhS : ∀ σ ∈ S, σ.IsSwap\ninst✝ : IsPretransitive (↥(closure S)) α\n⊢ closure S = ⊤",
"usedConstants": [
"Equiv.Perm.applyMulAction",
"Equiv.instEquivLike",
"Subgroup.closure",
"congrArg",
"... | by
simp [eq_top_iff', mem_closure_isSwap hS, orbit_eq_univ, Set.toFinite] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 355,
"column": 67
} | {
"line": 355,
"column": 83
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\na : g.Basis\nc : ↥g.cycleFactorsFinset\nτ✝ τ : ↥(range_toPermHom' g)\nx : α\n⊢ a.ofPermHomFun 1 x = x",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"Finset",
... | ofPermHomFun_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 356,
"column": 68
} | {
"line": 356,
"column": 84
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\na : g.Basis\nc : ↥g.cycleFactorsFinset\nτ✝ τ : ↥(range_toPermHom' g)\nx : α\n⊢ a.ofPermHomFun 1 x = x",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"Finset",
... | ofPermHomFun_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 523,
"column": 4
} | {
"line": 523,
"column": 58
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\ng : Perm α\n⊢ Function.Injective ⇑(noncommPiCoprod ⋯)",
"usedConstants": [
"Finset",
"Subgroup.subtype",
"Equiv.Perm.cycleFactorsFinset",
"Membership.mem",
"Subtype",
"Subgroup",
"Equiv... | apply MonoidHom.injective_noncommPiCoprod_of_iSupIndep | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.GroupTheory.Schreier | {
"line": 81,
"column": 8
} | {
"line": 81,
"column": 79
} | [
{
"pp": "case refine_3\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nR S : Set G\nhR : IsComplement (↑H) R\nhR1 : 1 ∈ R\nhS : closure S = ⊤\nf : G → ↑R := hR.toRightFun\nU : Set G := (fun g ↦ g * (↑(f g))⁻¹) '' (R * S)\ng : G\nx✝ : g ∈ ⊤\ns : G\nhs : s ∈ S\nu : G\nhu : u ∈ ↑(closure U)\nr : G\nhr : r ∈ R\n⊢ (... | show u * r * s⁻¹ = u * (f (r * s⁻¹) * s * r⁻¹)⁻¹ * f (r * s⁻¹) by group | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Schreier | {
"line": 80,
"column": 4
} | {
"line": 88,
"column": 45
} | [
{
"pp": "case refine_3\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nR S : Set G\nhR : IsComplement (↑H) R\nhR1 : 1 ∈ R\nhS : closure S = ⊤\nf : G → ↑R := hR.toRightFun\nU : Set G := (fun g ↦ g * (↑(f g))⁻¹) '' (R * S)\ng : G\nx✝ : g ∈ ⊤\n⊢ ∀ x ∈ closure S, ∀ y ∈ S, x ∈ ↑(closure U) * R → x * y⁻¹ ∈ ↑(closure ... | rintro - - s hs ⟨u, hu, r, hr, rfl⟩
rw [show u * r * s⁻¹ = u * (f (r * s⁻¹) * s * r⁻¹)⁻¹ * f (r * s⁻¹) by group]
refine Set.mul_mem_mul ((closure U).mul_mem hu ((closure U).inv_mem ?_)) (f (r * s⁻¹)).2
refine subset_closure ⟨f (r * s⁻¹) * s, Set.mul_mem_mul (f (r * s⁻¹)).2 hs, ?_⟩
rw [mul_right_inj, inv... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.Schreier | {
"line": 80,
"column": 4
} | {
"line": 88,
"column": 45
} | [
{
"pp": "case refine_3\nG : Type u_1\ninst✝ : Group G\nH : Subgroup G\nR S : Set G\nhR : IsComplement (↑H) R\nhR1 : 1 ∈ R\nhS : closure S = ⊤\nf : G → ↑R := hR.toRightFun\nU : Set G := (fun g ↦ g * (↑(f g))⁻¹) '' (R * S)\ng : G\nx✝ : g ∈ ⊤\n⊢ ∀ x ∈ closure S, ∀ y ∈ S, x ∈ ↑(closure U) * R → x * y⁻¹ ∈ ↑(closure ... | rintro - - s hs ⟨u, hu, r, hr, rfl⟩
rw [show u * r * s⁻¹ = u * (f (r * s⁻¹) * s * r⁻¹)⁻¹ * f (r * s⁻¹) by group]
refine Set.mul_mem_mul ((closure U).mul_mem hu ((closure U).inv_mem ?_)) (f (r * s⁻¹)).2
refine subset_closure ⟨f (r * s⁻¹) * s, Set.mul_mem_mul (f (r * s⁻¹)).2 hs, ?_⟩
rw [mul_right_inj, inv... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Perm.Centralizer | {
"line": 687,
"column": 2
} | {
"line": 688,
"column": 59
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nm : Multiset ℕ\nhm : ¬(m.sum ≤ Fintype.card α ∧ ∀ a ∈ m, 2 ≤ a)\n⊢ #{g | g.cycleType = m} * ((Fintype.card α - m.sum)! * m.prod * ∏ n ∈ m.toFinset, (Multiset.count n m)!) = 0",
"usedConstants": [
"Multiset.sum",
"Multise... | · -- empty case
rw [(card_of_cycleType_eq_zero_iff α).mpr hm, zero_mul] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.HNNExtension | {
"line": 675,
"column": 2
} | {
"line": 675,
"column": 56
} | [
{
"pp": "G : Type u_1\ninst✝ : Group G\nA B : Subgroup G\nφ : ↥A ≃* ↥B\nw₁ w₂ : ReducedWord G A B\nhprod : ReducedWord.prod φ w₁ = ReducedWord.prod φ w₂\nd : TransversalPair G A B\nw₁' : NormalWord d\nhw₁'1 : ReducedWord.prod φ w₁'.toReducedWord = ReducedWord.prod φ w₁\nhw₁'2 : List.map Prod.fst w₁'.toList = Li... | rwa [← List.head?_map, ← hw₂'2, hw₁'2, List.head?_map] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour | {
"line": 134,
"column": 2
} | {
"line": 136,
"column": 25
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\nS : Sylow 2 ↥(alternatingGroup α)\n⊢ ↑S = kleinFour α",
"usedConstants": [
"Sylow.toSubgroup",
"Subgroup.closure_eq",
"Eq.mpr",
"Equiv.Perm.cycleType",
"Sylow.instSetLike",
"InvOneClas... | rw [kleinFour, ← Subgroup.closure_insert_one,
← Set.singleton_union, ← coe_two_sylow_of_card_eq_four hα4]
exact S.closure_eq.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.SpecificGroups.Alternating.KleinFour | {
"line": 134,
"column": 2
} | {
"line": 136,
"column": 25
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\nhα4 : Nat.card α = 4\nS : Sylow 2 ↥(alternatingGroup α)\n⊢ ↑S = kleinFour α",
"usedConstants": [
"Sylow.toSubgroup",
"Subgroup.closure_eq",
"Eq.mpr",
"Equiv.Perm.cycleType",
"Sylow.instSetLike",
"InvOneClas... | rw [kleinFour, ← Subgroup.closure_insert_one,
← Set.singleton_union, ← coe_two_sylow_of_card_eq_four hα4]
exact S.closure_eq.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.PushoutI | {
"line": 347,
"column": 10
} | {
"line": 347,
"column": 57
} | [
{
"pp": "case neg\nι : Type u_1\nG : ι → Type u_2\nH : Type u_3\ninst✝³ : (i : ι) → Group (G i)\ninst✝² : Group H\nφ : (i : ι) → H →* G i\nd : Transversal φ\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → DecidableEq (G i)\nw : Word G\ni : ι\nh✝ : H\nhw : ∀ (i : ι) (g : G i), ⟨i, g⟩ ∈ w.toList → ↑(⋯.equiv g).2 = g\n... | equiv_one (d.compl i) (one_mem _) (d.one_mem _) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.PushoutI | {
"line": 644,
"column": 2
} | {
"line": 644,
"column": 27
} | [
{
"pp": "ι : Type u_1\nG : ι → Type u_2\nH : Type u_3\ninst✝¹ : (i : ι) → Group (G i)\ninst✝ : Group H\nφ : (i : ι) → H →* G i\nhφ : ∀ (i : ι), Injective ⇑(φ i)\nw : Word G\nhw : Reduced φ w\nd : Transversal φ\nw' : NormalWord d\nhw'prod : w'.prod = ofCoprodI w.prod\nhw'map : List.map Sigma.fst w'.toList = List... | rw [← prod_injective heq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.InformationTheory.Coding.KraftMcMillan | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 37
} | [
{
"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\n⊢ ∃ r ≥ 1, ↑r * ↑(S.sup List.length) < (∑ w ∈ S, (1 / ↑(Fintype.card α)) ^ w.length) ^ r"... | let maxLen : ℕ := S.sup List.length | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Probability.Kernel.Basic | {
"line": 383,
"column": 2
} | {
"line": 385,
"column": 95
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ η : Kernel α β\ns : Set α\nhs : MeasurableSet s\ninst✝² : DecidablePred fun x ↦ x ∈ s\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\n⊢ IsFiniteKernel (Kernel.piecewise hs κ η)",
"usedConstants": [
... | refine ⟨⟨max κ.bound η.bound, max_lt κ.bound_lt_top η.bound_lt_top, fun a => ?_⟩⟩
rw [piecewise_apply']
exact (ite_le_sup _ _ _).trans (sup_le_sup (measure_le_bound _ _ _) (measure_le_bound _ _ _)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Probability.Kernel.Basic | {
"line": 383,
"column": 2
} | {
"line": 385,
"column": 95
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ η : Kernel α β\ns : Set α\nhs : MeasurableSet s\ninst✝² : DecidablePred fun x ↦ x ∈ s\ninst✝¹ : IsFiniteKernel κ\ninst✝ : IsFiniteKernel η\n⊢ IsFiniteKernel (Kernel.piecewise hs κ η)",
"usedConstants": [
... | refine ⟨⟨max κ.bound η.bound, max_lt κ.bound_lt_top η.bound_lt_top, fun a => ?_⟩⟩
rw [piecewise_apply']
exact (ite_le_sup _ _ _).trans (sup_le_sup (measure_le_bound _ _ _) (measure_le_bound _ _ _)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 58
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nκ : Kernel α β\nγ : Type u_4\nmγ : MeasurableSpace γ\nh : ¬IsSFiniteKernel κ\n⊢ κ ⊗ₖ 0 = 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"ProbabilityTheory.Kernel.compProd_of_not_isSFiniteKernel_lef... | · rw [Kernel.compProd_of_not_isSFiniteKernel_left _ _ h] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.Composition.CompProd | {
"line": 480,
"column": 6
} | {
"line": 480,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\nmγ : MeasurableSpace γ\nδ : Type u_4\nmδ : MeasurableSpace δ\nκ : Kernel α β\nη : Kernel (α × β) γ\nξ : Kernel (α × β × γ) δ\nhκ : IsSFiniteKernel κ\nhη : IsSFiniteKernel η\nhξ : ¬IsSFiniteKernel ξ\n⊢ ¬IsSFiniteKe... | refine fun h_sfin ↦ hξ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Measure.Tilted | {
"line": 119,
"column": 82
} | {
"line": 124,
"column": 57
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ : Measure α\ninst✝ : SFinite μ\nf : α → ℝ\ns : Set α\n⊢ (μ.tilted f) s = ENNReal.ofReal (∫ (a : α) in s, rexp (f a) / ∫ (x : α), rexp (f x) ∂μ ∂μ)",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NormedCommRing.toSeminormedC... | by
by_cases hf : Integrable (fun x ↦ exp (f x)) μ
· rw [tilted_apply _ _, ← ofReal_integral_eq_lintegral_ofReal]
· exact hf.integrableOn.div_const _
· exact ae_of_all _ (fun _ ↦ by positivity)
· simp [tilted_of_not_integrable hf, integral_undef hf] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Probability.Kernel.Composition.IntegralCompProd | {
"line": 250,
"column": 4
} | {
"line": 250,
"column": 69
} | [
{
"pp": "case pos.h_ind.e_a.hfm\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✝³ : NormedAddCommGroup E\na : α\nκ : Kernel α β\ninst✝² : IsSFiniteKernel κ\nη : Kernel (α × β) γ\ninst✝¹ : IsSFiniteKernel η\ninst✝ : NormedSpace... | · exact (Kernel.measurable_kernel_prodMk_left' hs _).aemeasurable | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Probability.Kernel.Composition.IntegralCompProd | {
"line": 279,
"column": 11
} | {
"line": 279,
"column": 57
} | [
{
"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\nf : β × γ → E\nt :... | setIntegral_compProd MeasurableSet.univ ht hf, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.LogLikelihoodRatio | {
"line": 228,
"column": 4
} | {
"line": 228,
"column": 60
} | [
{
"pp": "case h.hx\nα : 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).toRe... | · refine (mul_pos (exp_pos _) (integral_exp_pos hf)).ne' | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Unique | {
"line": 133,
"column": 6
} | {
"line": 133,
"column": 62
} | [
{
"pp": "α : Type u_1\nF' : Type u_3\nm m0 : MeasurableSpace α\nμ : Measure α\ninst✝³ : NormedAddCommGroup F'\ninst✝² : NormedSpace ℝ F'\ninst✝¹ : CompleteSpace F'\nhm : m ≤ m0\ninst✝ : SigmaFinite (μ.trim hm)\nf g : α → F'\nhf_int_finite : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → IntegrableOn f s μ\nhg_int_f... | integral_congr_ae (ae_restrict_of_ae hfm.ae_eq_mk.symm), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.InformationTheory.KullbackLeibler.Basic | {
"line": 370,
"column": 8
} | {
"line": 370,
"column": 24
} | [
{
"pp": "case pos\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\n⊢ klDiv μ ν ≠ ∞",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"congrArg",
... | klDiv_ne_top_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.InformationTheory.KullbackLeibler.Basic | {
"line": 381,
"column": 6
} | {
"line": 381,
"column": 22
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nh : klDiv μ ν = 0\nh_ne : klDiv μ ν ≠ ∞\n⊢ μ = ν",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"congrArg",
"PseudoMetricSpace.toUniformSpace",
... | klDiv_ne_top_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic | {
"line": 139,
"column": 76
} | {
"line": 140,
"column": 55
} | [
{
"pp": "α : Type u_1\nE : Type u_3\nm m₀ : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nhm : m ≤ m₀\nhμm : SigmaFinite (μ.trim hm)\nf : α → E\nhf : StronglyMeasurable f\nhfi : Integrable f μ\n⊢ μ[f | m] = f",
"usedConstants": [
"Eq.mpr",
"congrArg",
... | by
rw [condExp_of_sigmaFinite hm, if_pos hfi, if_pos hf] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.ConditionalExpectation.Basic | {
"line": 273,
"column": 6
} | {
"line": 273,
"column": 19
} | [
{
"pp": "case pos\nα : Type u_1\nE : Type u_3\nm₀ : MeasurableSpace α\nμ : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhμ : NeZero μ\nf : α → E\nhμ_finite : IsFiniteMeasure μ\nh_meas : StronglyMeasurable μ[f | ⊥]\nc : E\nh_eq : μ[f | ⊥] = fun x ↦ c\nh_integral : ... | ← h_integral, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.InformationTheory.KullbackLeibler.ChainRule | {
"line": 206,
"column": 2
} | {
"line": 206,
"column": 33
} | [
{
"pp": "𝓧 : Type u_1\n𝓨 : Type u_2\nm𝓧 : MeasurableSpace 𝓧\nm𝓨 : MeasurableSpace 𝓨\nμ ν : Measure 𝓧\nκ η : Kernel 𝓧 𝓨\ninst✝³ : IsFiniteMeasure μ\ninst✝² : IsFiniteMeasure ν\ninst✝¹ : IsMarkovKernel κ\ninst✝ : IsMarkovKernel η\n⊢ klDiv (μ ⊗ₘ κ) (ν ⊗ₘ η) = klDiv μ ν + klDiv (μ ⊗ₘ κ) (μ ⊗ₘ η)",
"use... | by_cases h_ac : μ ⊗ₘ κ ≪ ν ⊗ₘ η | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.LinearAlgebra.AffineSpace.Matrix | {
"line": 61,
"column": 2
} | {
"line": 61,
"column": 27
} | [
{
"pp": "ι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁶ : AddCommGroup V\ninst✝⁵ : AffineSpace V P\ninst✝⁴ : Ring k\ninst✝³ : Module k V\nb : AffineBasis ι k P\nι' : Type u_1\ninst✝² : Fintype ι\ninst✝¹ : Finite ι'\ninst✝ : DecidableEq ι'\np : ι' → P\nA : Matrix ι ι' k\nhA : b.toMatrix p * A = 1\n⊢ ... | cases nonempty_fintype ι' | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.LinearAlgebra.AffineSpace.Matrix | {
"line": 81,
"column": 20
} | {
"line": 81,
"column": 28
} | [
{
"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 ... | ← b.tot, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Alternating.DomCoprod | {
"line": 221,
"column": 65
} | {
"line": 221,
"column": 83
} | [
{
"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' ... | Finset.filter_map, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.LinearAlgebra.Alternating.DomCoprod | {
"line": 221,
"column": 6
} | {
"line": 222,
"column": 33
} | [
{
"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' ... | conv_lhs => rw [← Finset.map_univ_equiv (Equiv.mulLeft σ), Finset.filter_map, Finset.sum_map]
simp [-MonoidHom.mem_range] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Alternating.DomCoprod | {
"line": 221,
"column": 6
} | {
"line": 222,
"column": 33
} | [
{
"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' ... | conv_lhs => rw [← Finset.map_univ_equiv (Equiv.mulLeft σ), Finset.filter_map, Finset.sum_map]
simp [-MonoidHom.mem_range] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.TensorProduct.IsBaseChangeHom | {
"line": 233,
"column": 4
} | {
"line": 236,
"column": 27
} | [
{
"pp": "case inl\nR : Type u_6\ninst✝¹⁰ : CommRing R\nS : Type u_7\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\nM : Type u_8\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nP : Type u_9\ninst✝⁵ : AddCommGroup P\ninst✝⁴ : Module R P\ninst✝³ : Module S P\ninst✝² : IsScalarTower R S P\ninst✝¹ : Free R M\ninst✝ : Mo... | have : f = 1 := by
have : Subsingleton M := Module.subsingleton R M
exact Subsingleton.eq_one f
simp [this, endHom_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.TensorProduct.IsBaseChangeHom | {
"line": 233,
"column": 4
} | {
"line": 236,
"column": 27
} | [
{
"pp": "case inl\nR : Type u_6\ninst✝¹⁰ : CommRing R\nS : Type u_7\ninst✝⁹ : CommRing S\ninst✝⁸ : Algebra R S\nM : Type u_8\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nP : Type u_9\ninst✝⁵ : AddCommGroup P\ninst✝⁴ : Module R P\ninst✝³ : Module S P\ninst✝² : IsScalarTower R S P\ninst✝¹ : Free R M\ninst✝ : Mo... | have : f = 1 := by
have : Subsingleton M := Module.subsingleton R M
exact Subsingleton.eq_one f
simp [this, endHom_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.TensorProduct.Opposite | {
"line": 44,
"column": 36
} | {
"line": 44,
"column": 47
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁸ : CommSemiring R\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Semiring A\ninst✝⁵ : Semiring B\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\ne₁ : Aᵐᵒᵖ ⊗[R] Bᵐᵒᵖ ≃ₗ[S] (A ⊗[R] B)ᵐᵒ... | ext <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.LinearAlgebra.TensorProduct.Opposite | {
"line": 44,
"column": 36
} | {
"line": 44,
"column": 47
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁸ : CommSemiring R\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Semiring A\ninst✝⁵ : Semiring B\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\ne₁ : Aᵐᵒᵖ ⊗[R] Bᵐᵒᵖ ≃ₗ[S] (A ⊗[R] B)ᵐᵒ... | ext <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.TensorProduct.Opposite | {
"line": 44,
"column": 36
} | {
"line": 44,
"column": 47
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁸ : CommSemiring R\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Semiring A\ninst✝⁵ : Semiring B\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\ne₁ : Aᵐᵒᵖ ⊗[R] Bᵐᵒᵖ ≃ₗ[S] (A ⊗[R] B)ᵐᵒ... | ext <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.TensorProduct.Opposite | {
"line": 44,
"column": 53
} | {
"line": 44,
"column": 64
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁸ : CommSemiring R\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Semiring A\ninst✝⁵ : Semiring B\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\ne₁ : Aᵐᵒᵖ ⊗[R] Bᵐᵒᵖ ≃ₗ[S] (A ⊗[R] B)ᵐᵒ... | ext <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.LinearAlgebra.TensorProduct.Opposite | {
"line": 44,
"column": 53
} | {
"line": 44,
"column": 64
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁸ : CommSemiring R\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Semiring A\ninst✝⁵ : Semiring B\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\ne₁ : Aᵐᵒᵖ ⊗[R] Bᵐᵒᵖ ≃ₗ[S] (A ⊗[R] B)ᵐᵒ... | ext <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.TensorProduct.Opposite | {
"line": 44,
"column": 53
} | {
"line": 44,
"column": 64
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁸ : CommSemiring R\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Semiring A\ninst✝⁵ : Semiring B\ninst✝⁴ : Algebra R S\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\ne₁ : Aᵐᵒᵖ ⊗[R] Bᵐᵒᵖ ≃ₗ[S] (A ⊗[R] B)ᵐᵒ... | ext <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 331,
"column": 19
} | {
"line": 331,
"column": 29
} | [
{
"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⊢ (f x * ↑(-h.unit⁻¹)) • v + f (x + (f x * ↑(-h.unit⁻¹)) • v) • v = 0",
"usedConstants": [
"Units.val",
"Eq.mpr",
"instHSMul",
"Semirin... | ← add_smul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 598,
"column": 6
} | {
"line": 599,
"column": 39
} | [
{
"pp": "case h₀\nK : Type u_3\nV : Type u_4\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nf : Dual K V\nv : V\nhfv : ¬f v = 0\nι : Set V\nb : Basis (↑ι) K V\ni : ↑ι\nhv : v = b i\nhf : f = f v • b.coord i\nthis✝ : Fintype ↑ι\nthis : (toMatrix b b) (transvection... | · intro j _ hj
simp [Function.update_of_ne hj] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 613,
"column": 6
} | {
"line": 613,
"column": 26
} | [
{
"pp": "case h.e'_2.h.e'_6\nK : Type u_3\nV : Type u_4\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nf : Dual K V\nv : V\nhfv : ¬f v = 0\nι : Set V\nb : Basis (↑ι) K V\ni : ↑ι\nhv : v = b i\nhf : f = f v • b.coord i\nthis : Fintype ↑ι\nx y : ↑ι\nhxy : ¬x = y\n⊢... | by_cases hxi : x = i | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 661,
"column": 8
} | {
"line": 661,
"column": 13
} | [
{
"pp": "R : Type u_3\nV : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : Free R V\ninst✝ : Module.Finite R V\nf : Dual R V\nv : V\nhR : Nontrivial R\nn : ℕ := finrank R V\nb : Basis (Fin n) R V\nS : Type := MvPolynomial (Fin n ⊕ Fin n) ℤ\nγ : S →+* R := ↑(MvPolynomial.aev... | ← hε, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Transvection.Basic | {
"line": 670,
"column": 12
} | {
"line": 670,
"column": 17
} | [
{
"pp": "case inr\nR : Type u_3\nV : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : Free R V\ninst✝ : Module.Finite R V\nf : Dual R V\nv : V\nhR : Nontrivial R\nn : ℕ := finrank R V\nb : Basis (Fin n) R V\nS : Type := MvPolynomial (Fin n ⊕ Fin n) ℤ\nγ : S →+* R := ↑(MvPoly... | ← hv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Equivs | {
"line": 95,
"column": 29
} | {
"line": 95,
"column": 45
} | [
{
"pp": "case add\nR : Type u_1\ninst✝ : CommRing R\nx₁ x₂ : CliffordAlgebra 0\nhx₁ : reverse x₁ = x₁\nhx₂ : reverse x₂ = x₂\n⊢ reverse (x₁ + x₂) = x₁ + x₂",
"usedConstants": [
"Eq.mpr",
"LinearMap.map_add",
"CliffordAlgebra.reverse",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | reverse.map_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Equivs | {
"line": 206,
"column": 29
} | {
"line": 206,
"column": 45
} | [
{
"pp": "case add\nx₁ x₂ : CliffordAlgebra Q\nhx₁ : reverse x₁ = x₁\nhx₂ : reverse x₂ = x₂\n⊢ reverse (x₁ + x₂) = x₁ + x₂",
"usedConstants": [
"Eq.mpr",
"LinearMap.map_add",
"Real",
"CliffordAlgebra.reverse",
"Semiring.toModule",
"Ring.toNonAssocRing",
"congrArg",
... | reverse.map_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.CliffordAlgebra.Contraction | {
"line": 314,
"column": 81
} | {
"line": 320,
"column": 41
} | [
{
"pp": "R : Type u1\ninst✝² : CommRing R\nM : Type u2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ Q' Q'' : QuadraticForm R M\nB B' : BilinForm R M\nh : BilinMap.toQuadraticMap B = Q' - Q\nh' : BilinMap.toQuadraticMap B' = Q'' - Q'\nx : CliffordAlgebra Q\n⊢ (changeForm h') ((changeForm h) x) = (changeForm ⋯... | by
induction x using CliffordAlgebra.left_induction with
| algebraMap => simp_rw [changeForm_algebraMap]
| add _ _ hx hy => rw [map_add, map_add, map_add, hx, hy]
| ι_mul _ _ hx => rw [changeForm_ι_mul, map_sub, changeForm_ι_mul, changeForm_ι_mul, hx, sub_sub,
LinearMap.add_apply, map_add, LinearMap.add_a... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.CliffordAlgebra.EvenEquiv | {
"line": 228,
"column": 38
} | {
"line": 228,
"column": 55
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nQ Q' : QuadraticForm R M\nh : Q' = -Q\nm : M\n⊢ -((even.ι Q').bilin m) m = (algebraMap R ↥(even Q')) (Q m)",
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
"NegZeroClass.toNeg",
... | EvenHom.contract, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup | {
"line": 102,
"column": 4
} | {
"line": 107,
"column": 31
} | [
{
"pp": "case mem\nR : Type u_1\ninst✝³ : CommRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nQ : QuadraticForm R M\ninst✝ : Invertible 2\nx✝ x : (CliffordAlgebra Q)ˣ\nhx : x ∈ Units.val ⁻¹' Set.range ⇑(ι Q)\nb : M\n⊢ involute ↑x * (ι Q) b * ↑x⁻¹ ∈ (ι Q).range",
"usedConstants": [
... | obtain ⟨a, ha⟩ := hx
letI := x.invertible
letI : Invertible (ι Q a) := by rwa [ha]
letI : Invertible (Q a) := invertibleOfInvertibleι Q a
simp_rw [← invOf_units x, ← ha, involute_ι, neg_mul, ι_mul_ι_mul_invOf_ι Q a b, ← map_neg,
LinearMap.mem_range_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.CliffordAlgebra.SpinGroup | {
"line": 102,
"column": 4
} | {
"line": 107,
"column": 31
} | [
{
"pp": "case mem\nR : Type u_1\ninst✝³ : CommRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nQ : QuadraticForm R M\ninst✝ : Invertible 2\nx✝ x : (CliffordAlgebra Q)ˣ\nhx : x ∈ Units.val ⁻¹' Set.range ⇑(ι Q)\nb : M\n⊢ involute ↑x * (ι Q) b * ↑x⁻¹ ∈ (ι Q).range",
"usedConstants": [
... | obtain ⟨a, ha⟩ := hx
letI := x.invertible
letI : Invertible (ι Q a) := by rwa [ha]
letI : Invertible (Q a) := invertibleOfInvertibleι Q a
simp_rw [← invOf_units x, ← ha, involute_ι, neg_mul, ι_mul_ι_mul_invOf_ι Q a b, ← map_neg,
LinearMap.mem_range_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.TensorProduct.Graded.External | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 67
} | [
{
"pp": "case a.H.h.H.h.a.H.h.H.h.e_a\nR : Type u_1\nι : Type u_2\ninst✝¹¹ : CommSemiring ι\ninst✝¹⁰ : Module ι (Additive ℤˣ)\ninst✝⁹ : DecidableEq ι\n𝒜 : ι → Type u_3\nℬ : ι → Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : (i : ι) → AddCommGroup (𝒜 i)\ninst✝⁶ : (i : ι) → AddCommGroup (ℬ i)\ninst✝⁵ : (i : ι) → Modul... | rw [two_nsmul, uzpow_add, uzpow_add, Int.units_mul_self, one_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.FreeModule.ModN | {
"line": 93,
"column": 4
} | {
"line": 93,
"column": 11
} | [
{
"pp": "case refine_2\nG : Type u_1\nH✝ : Type u_2\nM : Type u_3\ninst✝¹ : AddCommGroup G\nn : ℕ\ninst✝ : NeZero n\nι : Type u_4\nb : Basis ι ℤ G\nψ : G →+ G := zsmulAddGroupHom ↑n\nnG : Submodule ℤ G := ((LinearMap.lsmul ℤ G) ↑n).range\nH : Type u_1 := G ⧸ nG\nφ : G →ₗ[ℤ] H := nG.mkQ\nmod : (ι →₀ ℤ) →ₗ[ℤ] ι →... | ext x b | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.LinearAlgebra.Matrix.Determinant.TotallyUnimodular | {
"line": 53,
"column": 8
} | {
"line": 53,
"column": 49
} | [
{
"pp": "case h.inl\nm : Type u_1\nn : Type u_3\nR : Type u_5\ninst✝ : CommRing R\nA : Matrix m n R\nhA : A.IsTotallyUnimodular\nk : ℕ\nf : Fin k → m\ng : Fin k → n\ni j : Fin k\nhfij : f i = f j\nhij : i ≠ j\n⊢ (A.submatrix f g).det = 0",
"usedConstants": [
"Eq.mpr",
"Matrix.submatrix",
"... | rw [← det_transpose, transpose_submatrix] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.