module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.Lie.Basis | {
"line": 163,
"column": 2
} | {
"line": 189,
"column": 74
} | {
"line": 190,
"column": 2
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝³ : Finite ι\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nb : Basis ι R L\ny z : L\nhy : y ∈ lieSpan R L (range b.e)\nhz : z ∈ lieSpan R L (range b.f)\n⊢ ⁅y, z⁆ ∈ b.cartan.toLieSubmodule ⊔ b.borelLower ⊔ b.borelUpper",
"ppTerm": "?... | [
"ι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝³ : Finite ι\ninst✝² : CommRing R\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nb : Basis ι R L\ny z : L\nhy : y ∈ lieSpan R L (range b.e)\nhz : z ∈ lieSpan R L (range b.f)\nthis : ∀ (i : ι), ∀ x ∈ lieSpan R L (range b.f), ⁅b.e i, x⁆ ∈ b.cartan.toLieSubmodule ⊔ b.borel... | have (i : ι) (x : L) (hx : x ∈ lieSpan R L (range b.f)) :
⁅b.e i, x⁆ ∈ b.cartan.toLieSubmodule ⊔ b.borelLower := by
induction hx using LieSubalgebra.lieSpan_induction with
| mem u hu =>
obtain ⟨j, rfl⟩ := hu
rcases eq_or_ne i j with rfl | hij
· rw [(b.sl2 i).lie_e_f]
apply LieSub... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Lie.CartanCriterion | {
"line": 153,
"column": 4
} | {
"line": 153,
"column": 52
} | {
"line": 154,
"column": 4
} | [
{
"pp": "L : Type u_2\nM : Type u_3\ninst✝⁹ : LieRing L\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : LieRingModule L M\nK : Type u_4\ninst✝⁶ : Field K\ninst✝⁵ : CharZero K\ninst✝⁴ : IsAlgClosed K\ninst✝³ : LieAlgebra K L\ninst✝² : Module K M\ninst✝¹ : LieModule K L M\ninst✝ : FiniteDimensional K M\nh : traceForm K L M = ... | [
"L : Type u_2\nM : Type u_3\ninst✝⁹ : LieRing L\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : LieRingModule L M\nK : Type u_4\ninst✝⁶ : Field K\ninst✝⁵ : CharZero K\ninst✝⁴ : IsAlgClosed K\ninst✝³ : LieAlgebra K L\ninst✝² : Module K M\ninst✝¹ : LieModule K L M\ninst✝ : FiniteDimensional K M\nh : traceForm K L M = 0\nφ : L →ₗ⁅... | apply q.aeval_apply_smul_mem_of_le_comap hz _ ?_ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 245,
"column": 45
} | {
"line": 247,
"column": 32
} | {
"line": 249,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝¹ : CommSemiring R\nσ : Type u_3\ninst✝ : AddCommMonoid M\nφ : MvPolynomial σ R\nn : M\nw : σ → M\nhφ : IsWeightedHomogeneous w φ n\nd : σ →₀ ℕ\nhd : (weight w) d ≠ n\n⊢ coeff d φ = 0",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Finsupp.i... | [] | by
have aux := mt (@hφ d) hd
rwa [Classical.not_not] at aux | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 534,
"column": 4
} | {
"line": 535,
"column": 51
} | {
"line": 537,
"column": 0
} | [
{
"pp": "case convert_3\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\nσ : Type u_3\ninst✝¹ : AddCommMonoid M\nw : σ → M\ninst✝ : DecidableEq M\nx : ⨁ (i : M), ↥(weightedHomogeneousSubmodule R w i)\nm : M\n⊢ m ∉ DFinsupp.support x → (weightedHomogeneousComponent w m) ↑(x m) = 0",
"ppTerm": "?convert_... | [] | rw [DFinsupp.notMem_support_iff]
intro hm; rw [hm, Submodule.coe_zero, map_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.WeightedHomogeneous | {
"line": 534,
"column": 4
} | {
"line": 535,
"column": 51
} | {
"line": 537,
"column": 0
} | [
{
"pp": "case convert_3\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\nσ : Type u_3\ninst✝¹ : AddCommMonoid M\nw : σ → M\ninst✝ : DecidableEq M\nx : ⨁ (i : M), ↥(weightedHomogeneousSubmodule R w i)\nm : M\n⊢ m ∉ DFinsupp.support x → (weightedHomogeneousComponent w m) ↑(x m) = 0",
"ppTerm": "?convert_... | [] | rw [DFinsupp.notMem_support_iff]
intro hm; rw [hm, Submodule.coe_zero, map_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Eigenspace.Zero | {
"line": 44,
"column": 36
} | {
"line": 49,
"column": 60
} | {
"line": 51,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_3\ninst✝⁵ : CommRing R\ninst✝⁴ : IsDomain R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Module.Finite R M\ninst✝ : Free R M\nφ : End R M\nh : IsNilpotent φ\n⊢ LinearMap.charpoly φ = X ^ finrank R M",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": ... | [] | by
rw [← sub_eq_zero]
apply IsNilpotent.eq_zero
rw [finrank_eq_card_chooseBasisIndex]
apply Matrix.isNilpotent_charpoly_sub_pow_of_isNilpotent
exact h.map (LinearMap.toMatrixAlgEquiv (chooseBasis R M)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Lie.CartanExists | {
"line": 216,
"column": 6
} | {
"line": 216,
"column": 28
} | {
"line": 216,
"column": 29
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nhLK : ↑(finrank K L) ≤ #K\nU : LieSubalgebra K L\nx : L\nhxU : x ∈ U\ny : L\nhyU : y ∈ U\nEx : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩\nEy : ↑{x | ∃ y ∈ U, engel K y = x} :=... | [
"K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nhLK : ↑(finrank K L) ≤ #K\nU : LieSubalgebra K L\nx : L\nhxU : x ∈ U\ny : L\nhyU : y ∈ U\nEx : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩\nEy : ↑{x | ∃ y ∈ U, engel K y = x} := ⟨engel K y,... | ← constantCoeff_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.CartanExists | {
"line": 246,
"column": 8
} | {
"line": 246,
"column": 30
} | {
"line": 246,
"column": 31
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nhLK : ↑(finrank K L) ≤ #K\nU : LieSubalgebra K L\nx : L\nhxU : x ∈ U\ny : L\nhyU : y ∈ U\nEx : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩\nEy : ↑{x | ∃ y ∈ U, engel K y = x} :=... | [
"K : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nhLK : ↑(finrank K L) ≤ #K\nU : LieSubalgebra K L\nx : L\nhxU : x ∈ U\ny : L\nhyU : y ∈ U\nEx : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩\nEy : ↑{x | ∃ y ∈ U, engel K y = x} := ⟨engel K y,... | ← constantCoeff_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.CartanExists | {
"line": 346,
"column": 4
} | {
"line": 346,
"column": 26
} | {
"line": 346,
"column": 27
} | [
{
"pp": "case inr\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nhLK : ↑(finrank K L) ≤ #K\nU : LieSubalgebra K L\nx : L\nhxU : x ∈ U\ny : L\nhyU : y ∈ U\nEx : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩\nEy : ↑{x | ∃ y ∈ U, engel K... | [
"case inr\nK : Type u_1\nL : Type u_2\ninst✝³ : Field K\ninst✝² : LieRing L\ninst✝¹ : LieAlgebra K L\ninst✝ : Module.Finite K L\nhLK : ↑(finrank K L) ≤ #K\nU : LieSubalgebra K L\nx : L\nhxU : x ∈ U\ny : L\nhyU : y ∈ U\nEx : ↑{x | ∃ x_1 ∈ U, engel K x_1 = x} := ⟨engel K x, ⋯⟩\nEy : ↑{x | ∃ y ∈ U, engel K y = x} := ⟨... | ← constantCoeff_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 503,
"column": 6
} | {
"line": 503,
"column": 33
} | {
"line": 503,
"column": 34
} | [
{
"pp": "m : Type um\nn : Type un\nR : Type uR\ninst✝² : Fintype n\ninst✝¹ : Field R\ninst✝ : Fintype m\nM : Matrix m n R\nh : LinearIndependent R M.row\n⊢ M.rank = Fintype.card m",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"Submodule",... | [
"m : Type um\nn : Type un\nR : Type uR\ninst✝² : Fintype n\ninst✝¹ : Field R\ninst✝ : Fintype m\nM : Matrix m n R\nh : LinearIndependent R M.row\n⊢ finrank R ↥(span R (range M.row)) = Fintype.card m"
] | M.rank_eq_finrank_span_row, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Basis | {
"line": 389,
"column": 2
} | {
"line": 389,
"column": 31
} | {
"line": 391,
"column": 0
} | [
{
"pp": "case «2»\nι : Type u_1\nR : Type u_2\nL : Type u_3\ninst✝⁶ : Finite ι\ninst✝⁵ : CommRing R\ninst✝⁴ : LieRing L\ninst✝³ : LieAlgebra R L\nb : Basis ι R L\ninst✝² : Fintype ι\ninst✝¹ : IsDomain R\ninst✝ : CharZero R\n⊢ ![b.cartan.toLieSubmodule, b.borelLower, b.borelUpper] ((fun i ↦ i) ⟨2, ⋯⟩) ≤\n ![r... | [] | · exact b.borelUpper_le_biSup | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Lie.Free | {
"line": 172,
"column": 4
} | {
"line": 172,
"column": 20
} | {
"line": 173,
"column": 4
} | [
{
"pp": "R : Type u\nX : Type v\ninst✝ : CommRing R\n⊢ ∀ (t : R) (x y : FreeLieAlgebra R X), ⁅x, t • y⁆ = t • ⁅x, y⁆",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"instHSMul",
"Semiring.toModule",
"IsScalarTower.right",
"FreeLieAlgebra.instLieRing",
"FreeNonU... | [
"case mk.mk\nR : Type u\nX : Type v\ninst✝ : CommRing R\nt : R\nx✝ : FreeLieAlgebra R X\na : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\ny✝ : FreeLieAlgebra R X\nc : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\n⊢ ... | rintro t ⟨a⟩ ⟨c⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Lie.SemiDirect | {
"line": 120,
"column": 55
} | {
"line": 120,
"column": 72
} | {
"line": 122,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : LieRing K\ninst✝² : LieAlgebra R K\nL : Type u_3\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nψ : L →ₗ⁅R⁆ LieDerivation R K K\n⊢ Function.Injective ⇑(inl ψ)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"and_true",
... | [] | intro; simp [inl] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Lie.SemiDirect | {
"line": 120,
"column": 55
} | {
"line": 120,
"column": 72
} | {
"line": 122,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : LieRing K\ninst✝² : LieAlgebra R K\nL : Type u_3\ninst✝¹ : LieRing L\ninst✝ : LieAlgebra R L\nψ : L →ₗ⁅R⁆ LieDerivation R K K\n⊢ Function.Injective ⇑(inl ψ)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"and_true",
... | [] | intro; simp [inl] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.GroupTheory.Perm.Cycle.Concrete | {
"line": 84,
"column": 6
} | {
"line": 84,
"column": 11
} | {
"line": 85,
"column": 6
} | [
{
"pp": "case h.right\nα : Type u_1\ninst✝ : DecidableEq α\ny : α\nl : List α\ntail_ih✝ : ∀ (x : α), (x :: l).Nodup → 2 ≤ (x :: l).length → (x :: l).formPerm.IsCycle\nx : α\nhl : (x :: y :: l).Nodup\nhn : 2 ≤ (x :: y :: l).length\nk : ℕ\nhk : k < (x :: y :: l).length\nhw : (x :: y :: l).formPerm (x :: y :: l)[k... | [
"case h\nα : Type u_1\ninst✝ : DecidableEq α\ny : α\nl : List α\ntail_ih✝ : ∀ (x : α), (x :: l).Nodup → 2 ≤ (x :: l).length → (x :: l).formPerm.IsCycle\nx : α\nhl : (x :: y :: l).Nodup\nhn : 2 ≤ (x :: y :: l).length\nk : ℕ\nhk : k < (x :: y :: l).length\nhw : (x :: y :: l).formPerm (x :: y :: l)[k] ≠ (x :: y :: l)[... | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.GroupTheory.Perm.Cycle.Concrete | {
"line": 251,
"column": 6
} | {
"line": 251,
"column": 28
} | {
"line": 251,
"column": 28
} | [
{
"pp": "case neg.zero.succ\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx : α\nhc : (p.cycleOf x).IsCycle\nhn✝ : 0 < (p.toList x).length\nhn : 0 < orderOf (p.cycleOf x)\nm : ℕ\nhx : (p.cycleOf x ^ m.succ) x ≠ x\nhm✝ : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\n⊢ (... | [
"case neg.zero.succ\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx : α\nhc : (p.cycleOf x).IsCycle\nhn✝ : 0 < (p.toList x).length\nhn : 0 < orderOf (p.cycleOf x)\nm : ℕ\nhx : (p ^ m.succ) x ≠ x\nhm✝ : m + 1 < (p.toList x).length\nhm : m + 1 < orderOf (p.cycleOf x)\n⊢ (p.cycleOf x ^ 0) x = (... | cycleOf_pow_apply_self | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.Perm.Cycle.Concrete | {
"line": 254,
"column": 6
} | {
"line": 254,
"column": 28
} | {
"line": 254,
"column": 28
} | [
{
"pp": "case neg.succ.zero\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx : α\nhc : (p.cycleOf x).IsCycle\nn : ℕ\nhx : (p.cycleOf x ^ n.succ) x ≠ x\nhn✝ : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nhm✝ : 0 < (p.toList x).length\nhm : 0 < orderOf (p.cycleOf x)\n⊢ (... | [
"case neg.succ.zero\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\np : Perm α\nx : α\nhc : (p.cycleOf x).IsCycle\nn : ℕ\nhx : (p ^ n.succ) x ≠ x\nhn✝ : n + 1 < (p.toList x).length\nhn : n + 1 < orderOf (p.cycleOf x)\nhm✝ : 0 < (p.toList x).length\nhm : 0 < orderOf (p.cycleOf x)\n⊢ (p.cycleOf x ^ (n + 1))... | cycleOf_pow_apply_self | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Lie.Extension | {
"line": 211,
"column": 27
} | {
"line": 211,
"column": 57
} | {
"line": 211,
"column": 57
} | [
{
"pp": "R : Type u_1\nN : Type u_2\nL : Type u_3\nM : Type u_4\ninst✝⁶ : CommRing R\ninst✝⁵ : LieRing L\ninst✝⁴ : LieAlgebra R L\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : LieRingModule L M\ninst✝ : LieModule R L M\nc : ↥(twoCocycle R L M)\nr : R\nx y : ofTwoCocycle c\n⊢ (⁅((ofProd c).symm x).1, (... | [] | by simp [← smul_add, smul_sub] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.Perm.Cycle.Concrete | {
"line": 468,
"column": 2
} | {
"line": 469,
"column": 32
} | {
"line": 471,
"column": 0
} | [
{
"pp": "case refine_3\nα : Type u_1\ninst✝¹ : Finite α\ninst✝ : DecidableEq α\nf : Perm α\nhf : f.IsCycle\ns : Cycle α\nhn : s.Nodup\nhs : (↑⟨s, hn⟩).formPerm ⋯ = f\nhs' : ∀ (y : { s // s.Nodup }), (fun s ↦ (↑s).formPerm ⋯ = f) y → y = ⟨s, hn⟩\n⊢ ∀ (y : { s // s.Nodup ∧ s.Nontrivial }), (fun s ↦ (↑s).formPerm ... | [] | · rintro ⟨t, ht, ht'⟩ ht''
simpa using hs' ⟨t, ht⟩ ht'' | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.FractionalIdeal.Basic | {
"line": 310,
"column": 4
} | {
"line": 311,
"column": 35
} | {
"line": 311,
"column": 35
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nx : P\nx✝ : x ∈ 0\nx' : R\nx'_mem_zero : x' ∈ ↑0\nx'_eq_x : (Algebra.linearMap R P) x' = x\n⊢ x = 0",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"RingHom.instRingHo... | [] | have x'_eq_zero : x' = 0 := x'_mem_zero
simp [x'_eq_x.symm, x'_eq_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.FractionalIdeal.Basic | {
"line": 310,
"column": 4
} | {
"line": 311,
"column": 35
} | {
"line": 311,
"column": 35
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\nS : Submonoid R\nP : Type u_2\ninst✝¹ : CommRing P\ninst✝ : Algebra R P\nx : P\nx✝ : x ∈ 0\nx' : R\nx'_mem_zero : x' ∈ ↑0\nx'_eq_x : (Algebra.linearMap R P) x' = x\n⊢ x = 0",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"RingHom.instRingHo... | [] | have x'_eq_zero : x' = 0 := x'_mem_zero
simp [x'_eq_x.symm, x'_eq_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.FractionalIdeal.Inverse | {
"line": 107,
"column": 53
} | {
"line": 108,
"column": 70
} | {
"line": 110,
"column": 0
} | [
{
"pp": "K : Type u_3\ninst✝⁴ : Field K\nR₁ : Type u_4\ninst✝³ : CommRing R₁\ninst✝² : IsDomain R₁\ninst✝¹ : Algebra R₁ K\ninst✝ : IsFractionRing R₁ K\nx : K\nhx : x ≠ 0\n⊢ spanSingleton R₁⁰ x / spanSingleton R₁⁰ x = 1",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [] | by
rw [spanSingleton_div_spanSingleton, div_self hx, spanSingleton_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.ChainOfDivisors | {
"line": 293,
"column": 4
} | {
"line": 293,
"column": 59
} | {
"line": 294,
"column": 4
} | [
{
"pp": "case refine_2\nM : Type u_1\ninst✝⁴ : CommMonoidWithZero M\ninst✝³ : IsCancelMulZero M\nN : Type u_2\ninst✝² : CommMonoidWithZero N\ninst✝¹ : UniqueFactorizationMonoid N\ninst✝ : UniqueFactorizationMonoid M\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m... | [
"case refine_2\nM : Type u_1\ninst✝⁴ : CommMonoidWithZero M\ninst✝³ : IsCancelMulZero M\nN : Type u_2\ninst✝² : CommMonoidWithZero N\ninst✝¹ : UniqueFactorizationMonoid N\ninst✝ : UniqueFactorizationMonoid M\nm p : Associates M\nn : Associates N\nhn : n ≠ 0\nhp : p ∈ normalizedFactors m\nd : ↑(Set.Iic m) ≃o ↑(Set.I... | rw [← Subtype.coe_mk (p := (· ≤ n)) b this, ← hx] at hb | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 191,
"column": 2
} | {
"line": 192,
"column": 81
} | {
"line": 193,
"column": 2
} | [
{
"pp": "A : Type u_2\ninst✝¹ : CommRing A\ninst✝ : IsDedekindDomain A\nP I : Ideal A\nP_prime : P.IsPrime\nhP : P ≠ ⊥\ni : ℕ\nhlt : normalizedFactors I ≤ Multiset.replicate i (normalize P)\nhle : I ≤ P ^ i\nP_prime' : Prime P\nh1 : I ≠ ⊥\nthis : P ^ i ≠ 0\nh3 : P ^ (i + 1) ≠ 0\n⊢ P ^ i ≤ I",
"ppTerm": "?m.... | [
"A : Type u_2\ninst✝¹ : CommRing A\ninst✝ : IsDedekindDomain A\nP I : Ideal A\nP_prime : P.IsPrime\nhP : P ≠ ⊥\ni : ℕ\nhlt : normalizedFactors I ≤ Multiset.replicate i (normalize P)\nhle : I ≤ P ^ i\nP_prime' : Prime P\nh1 : I ≠ ⊥\nthis : P ^ i ≠ 0\nh3 : P ^ (i + 1) ≠ 0\n⊢ normalizedFactors I ≤ Multiset.replicate i... | rw [← dvd_iff_le, dvd_iff_normalizedFactors_le_normalizedFactors, normalizedFactors_pow,
normalizedFactors_irreducible P_prime'.irreducible, Multiset.nsmul_singleton] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 459,
"column": 2
} | {
"line": 465,
"column": 28
} | {
"line": 467,
"column": 0
} | [
{
"pp": "T : Type u_4\ninst✝¹ : CommRing T\ninst✝ : IsDedekindDomain T\nI : Ideal T\nhI : I ≠ ⊥\nP : Ideal T\nhpm : P.IsMaximal\n⊢ ∃ Q, P ⊔ Q = ⊤ ∧ I = P ^ count P (normalizedFactors I) * Q",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"UniqueFactorizationMonoid.normalizedFactors",... | [] | use (filter (¬ P = ·) (normalizedFactors I)).prod
constructor
· refine P.sup_multiset_prod_eq_top (fun p hpi ↦ ?_)
have hp : Prime p := prime_of_normalized_factor p (filter_subset _ (normalizedFactors I) hpi)
exact hpm.coprime_of_ne ((isPrime_of_prime hp).isMaximal hp.ne_zero) (of_mem_filter hpi)
· nth_rw... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 459,
"column": 2
} | {
"line": 465,
"column": 28
} | {
"line": 467,
"column": 0
} | [
{
"pp": "T : Type u_4\ninst✝¹ : CommRing T\ninst✝ : IsDedekindDomain T\nI : Ideal T\nhI : I ≠ ⊥\nP : Ideal T\nhpm : P.IsMaximal\n⊢ ∃ Q, P ⊔ Q = ⊤ ∧ I = P ^ count P (normalizedFactors I) * Q",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"UniqueFactorizationMonoid.normalizedFactors",... | [] | use (filter (¬ P = ·) (normalizedFactors I)).prod
constructor
· refine P.sup_multiset_prod_eq_top (fun p hpi ↦ ?_)
have hp : Prime p := prime_of_normalized_factor p (filter_subset _ (normalizedFactors I) hpi)
exact hpm.coprime_of_ne ((isPrime_of_prime hp).isMaximal hp.ne_zero) (of_mem_filter hpi)
· nth_rw... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.FractionalIdeal.Operations | {
"line": 1011,
"column": 2
} | {
"line": 1012,
"column": 83
} | {
"line": 1013,
"column": 2
} | [
{
"pp": "R : Type u_5\nS : Type u_6\nK : Type u_7\nL : Type u_8\ninst✝⁹ : CommRing R\ninst✝⁸ : IsDomain R\ninst✝⁷ : CommRing S\ninst✝⁶ : IsDomain S\ninst✝⁵ : CommRing K\ninst✝⁴ : CommRing L\ninst✝³ : Algebra R K\ninst✝² : Algebra S L\ninst✝¹ : IsFractionRing R K\ninst✝ : IsFractionRing S L\nf : R ≃+* S\nx : K\n... | [
"R : Type u_5\nS : Type u_6\nK : Type u_7\nL : Type u_8\ninst✝⁹ : CommRing R\ninst✝⁸ : IsDomain R\ninst✝⁷ : CommRing S\ninst✝⁶ : IsDomain S\ninst✝⁵ : CommRing K\ninst✝⁴ : CommRing L\ninst✝³ : Algebra R K\ninst✝² : Algebra S L\ninst✝¹ : IsFractionRing R K\ninst✝ : IsFractionRing S L\nf : R ≃+* S\nx : K\ny : L\n⊢ (∃ ... | simp only [← FractionalIdeal.mem_coe, coe_mk, mem_map_equiv, coe_spanSingleton,
Submodule.mem_span_singleton, (semilinearEquivOfRingEquiv K L f).eq_symm_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Module.Lattice | {
"line": 95,
"column": 61
} | {
"line": 95,
"column": 82
} | {
"line": 95,
"column": 82
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : CommRing R\nA : Type u_2\ninst✝⁶ : CommRing A\ninst✝⁵ : Algebra R A\nV : Type u_3\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module R V\ninst✝² : Module A V\ninst✝¹ : IsScalarTower R A V\nM : Submodule R V\ninst✝ : IsLattice A M\na : Aˣ\n⊢ a • span A ↑M = ⊤",
"ppTerm": "?m.144",
... | [
"R : Type u_1\ninst✝⁷ : CommRing R\nA : Type u_2\ninst✝⁶ : CommRing A\ninst✝⁵ : Algebra R A\nV : Type u_3\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module R V\ninst✝² : Module A V\ninst✝¹ : IsScalarTower R A V\nM : Submodule R V\ninst✝ : IsLattice A M\na : Aˣ\n⊢ a • ⊤ = ⊤"
] | IsLattice.span_eq_top | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.Lattice | {
"line": 104,
"column": 21
} | {
"line": 104,
"column": 42
} | {
"line": 104,
"column": 42
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : CommRing R\nA : Type u_2\ninst✝⁶ : CommRing A\ninst✝⁵ : Algebra R A\nV : Type u_3\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module R V\ninst✝² : Module A V\ninst✝¹ : IsScalarTower R A V\nM N : Submodule R V\nhle : M ≤ N\ninst✝ : IsLattice A M\nhfg : N.FG\n⊢ ⊤ ≤ span A ↑M",
"ppTerm":... | [
"R : Type u_1\ninst✝⁷ : CommRing R\nA : Type u_2\ninst✝⁶ : CommRing A\ninst✝⁵ : Algebra R A\nV : Type u_3\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module R V\ninst✝² : Module A V\ninst✝¹ : IsScalarTower R A V\nM N : Submodule R V\nhle : M ≤ N\ninst✝ : IsLattice A M\nhfg : N.FG\n⊢ ⊤ ≤ ⊤"
] | IsLattice.span_eq_top | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.Lattice | {
"line": 181,
"column": 2
} | {
"line": 181,
"column": 73
} | {
"line": 183,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹⁰ : CommRing R\nK : Type u_2\ninst✝⁹ : Field K\ninst✝⁸ : Algebra R K\nV : Type u_3\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\ninst✝⁵ : Module R V\ninst✝⁴ : IsScalarTower R K V\ninst✝³ : IsDomain R\ninst✝² : IsPrincipalIdealRing R\ninst✝¹ : IsFractionRing R K\nM : Submodule R V\n... | [] | rw [rank_eq_card_basis b, ← rank_eq_card_basis (b.extendOfIsLattice K)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Module.FinitePresentation | {
"line": 258,
"column": 51
} | {
"line": 258,
"column": 73
} | {
"line": 258,
"column": 73
} | [
{
"pp": "R : Type u_2\nM : Type u_4\nN : Type u_3\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\nP : Type u_1\ninst✝² : AddCommGroup P\ninst✝¹ : Module R P\ninst✝ : FinitePresentation R N\nl : P →ₗ[R] N\nthis : Module.Finite R P\ne : N ≃ₗ[R] M × P\n... | [
"R : Type u_2\nM : Type u_4\nN : Type u_3\ninst✝⁷ : Ring R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommGroup N\ninst✝³ : Module R N\nP : Type u_1\ninst✝² : AddCommGroup P\ninst✝¹ : Module R P\ninst✝ : FinitePresentation R N\nl : P →ₗ[R] N\nthis : Module.Finite R P\ne : N ≃ₗ[R] M × P\nhf : Functio... | ← LinearMap.range_comp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.LinearMap.FiniteRange | {
"line": 209,
"column": 6
} | {
"line": 209,
"column": 45
} | {
"line": 209,
"column": 46
} | [
{
"pp": "K : Type u_1\nV : Type u_2\nV₂ : Type u_4\ninst✝⁵ : CommRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : IsNoetherianRing K\nf : V →ₗ[K] V₂\n⊢ f ∈ finiteRange K V V₂ ↔ f.HasFiniteRange",
"ppTerm": "?m.33",
"assigned": true,
"usedC... | [
"K : Type u_1\nV : Type u_2\nV₂ : Type u_4\ninst✝⁵ : CommRing K\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module K V\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module K V₂\ninst✝ : IsNoetherianRing K\nf : V →ₗ[K] V₂\n⊢ f.HasNoetherianRange ↔ f.HasFiniteRange"
] | mem_finiteRange_iff_hasNoetherianRange, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Ideal.Lemmas | {
"line": 1048,
"column": 36
} | {
"line": 1048,
"column": 63
} | {
"line": 1048,
"column": 63
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsPrincipalIdealRing R\na b : R\nh : ¬FiniteMultiplicity a b\nthis : ¬FiniteMultiplicity (span {a}) (span {b})\n⊢ emultiplicity (span {a}) (span {b}) = ⊤",
"ppTerm": "?m.105",
"assigned": true,
"usedConstants": [
"Iff.mpr... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\ninst✝ : IsPrincipalIdealRing R\na b : R\nh : ¬FiniteMultiplicity a b\nthis : ¬FiniteMultiplicity (span {a}) (span {b})\n⊢ ⊤ = ⊤"
] | emultiplicity_eq_top.2 this | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.LocalizedModule.Int | {
"line": 127,
"column": 2
} | {
"line": 127,
"column": 58
} | {
"line": 128,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommSemiring R\nS : Submonoid R\nM : Type u_2\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nM' : Type u_3\ninst✝³ : AddCommMonoid M'\ninst✝² : Module R M'\nf : M →ₗ[R] M'\ninst✝¹ : IsLocalizedModule S f\ninst✝ : DecidableEq M\nx : M\ns : Finset M'\nhx : f x ∈ Submodule.span R ↑... | [
"R : Type u_1\ninst✝⁶ : CommSemiring R\nS : Submonoid R\nM : Type u_2\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nM' : Type u_3\ninst✝³ : AddCommMonoid M'\ninst✝² : Module R M'\nf : M →ₗ[R] M'\ninst✝¹ : IsLocalizedModule S f\ninst✝ : DecidableEq M\nx : M\ns : Finset M'\nhx : f x ∈ Submodule.span R ↑s\ny : ↥S :=... | let y : S := IsLocalizedModule.commonDenomOfFinset S f s | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Algebra.Module.Presentation.Cokernel | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 43
} | {
"line": 68,
"column": 0
} | [
{
"pp": "A : Type u\ninst✝⁶ : Ring A\nM₁ : Type v₁\nM₂ : Type v₂\nM₃ : Type v₃\ninst✝⁵ : AddCommGroup M₁\ninst✝⁴ : Module A M₁\ninst✝³ : AddCommGroup M₂\ninst✝² : Module A M₂\ninst✝¹ : AddCommGroup M₃\ninst✝ : Module A M₃\npres₂ : Presentation A M₂\nf : M₁ →ₗ[A] M₂\nι : Type w₁\ng₁ : ι → M₁\ns : M₂ → pres₂.G →₀... | [] | exact ⟨CokernelData.ofSection _ _ _ s hs⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.MvPolynomial.Localization | {
"line": 71,
"column": 6
} | {
"line": 72,
"column": 10
} | {
"line": 73,
"column": 4
} | [
{
"pp": "case refine_1\nσ : Type u_1\nR : Type u_2\ninst✝³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nr : R\ninst✝ : Away r S\np : MvPolynomial Unit R\nhp : p ∈ Ideal.span {C r * X () - 1}\n⊢ ∀ x ∈ {C r * X () - 1}, (aeval fun x ↦ invSelf r) x = 0",
"ppTerm": "?r... | [] | rintro p ⟨q, rfl⟩
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.Localization | {
"line": 71,
"column": 6
} | {
"line": 72,
"column": 10
} | {
"line": 73,
"column": 4
} | [
{
"pp": "case refine_1\nσ : Type u_1\nR : Type u_2\ninst✝³ : CommRing R\nM : Submonoid R\nS : Type u_3\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\nr : R\ninst✝ : Away r S\np : MvPolynomial Unit R\nhp : p ∈ Ideal.span {C r * X () - 1}\n⊢ ∀ x ∈ {C r * X () - 1}, (aeval fun x ↦ invSelf r) x = 0",
"ppTerm": "?r... | [] | rintro p ⟨q, rfl⟩
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Extension.Presentation.Basic | {
"line": 271,
"column": 6
} | {
"line": 271,
"column": 54
} | {
"line": 272,
"column": 6
} | [
{
"pp": "R : Type u\nS : Type v\nι : Type w\nσ : Type t\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nT : Type u_1\ninst✝¹ : CommRing T\ninst✝ : Algebra R T\nP : Presentation R S ι σ\nx : MvPolynomial ι T\nH :\n RingHom.ker (TensorProduct.map (AlgHom.id R T) (IsScalarTower.toAlgHom R P.Ring ... | [] | induction x using MvPolynomial.induction_on with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Extension.Generators | {
"line": 511,
"column": 2
} | {
"line": 511,
"column": 50
} | {
"line": 512,
"column": 2
} | [
{
"pp": "R : Type u\nS : Type v\nι : Type w\ninst✝¹⁹ : CommRing R\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : Algebra R S\nP : Generators R S ι\nR' : Type u_1\nS' : Type u_2\nι' : Type u_3\ninst✝¹⁶ : CommRing R'\ninst✝¹⁵ : CommRing S'\ninst✝¹⁴ : Algebra R' S'\nP' : Generators R' S' ι'\nR'' : Type u_4\nS'' : Type u_5\nι'' ... | [] | induction x using MvPolynomial.induction_on with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.Extension.Basic | {
"line": 544,
"column": 2
} | {
"line": 544,
"column": 62
} | {
"line": 545,
"column": 2
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝²² : CommRing R\ninst✝²¹ : CommRing S\ninst✝²⁰ : Algebra R S\nP : Extension R S\nR' : Type ?u.16\nS' : Type ?u.18\ninst✝¹⁹ : CommRing R'\ninst✝¹⁸ : CommRing S'\ninst✝¹⁷ : Algebra R' S'\nP' : Extension R' S'\nR'' : Type ?u.29\nS'' : Type ?u.31\ninst✝¹⁶ : CommRing R''\ninst✝¹... | [
"case refine_1\nR : Type u\nS : Type v\ninst✝²² : CommRing R\ninst✝²¹ : CommRing S\ninst✝²⁰ : Algebra R S\nP : Extension R S\nR' : Type ?u.16\nS' : Type ?u.18\ninst✝¹⁹ : CommRing R'\ninst✝¹⁸ : CommRing S'\ninst✝¹⁷ : Algebra R' S'\nP' : Extension R' S'\nR'' : Type ?u.29\nS'' : Type ?u.31\ninst✝¹⁶ : CommRing R''\nins... | refine .ofBijective (Cotangent.mk.liftBaseChange _) ⟨?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.TensorProduct.Vanishing | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 72
} | {
"line": 125,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type u_3\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u_4\ninst✝ : Fintype ι\nm : ι → M\nn : ι → N\nhm : span R (Set.range m) = ⊤\nhmn : ∑ i, m i ⊗ₜ[R] n i = 0\nG : (ι →₀ R) →ₗ[R] M := linearCo... | [
"R : Type u_1\ninst✝⁵ : CommRing R\nM : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\nN : Type u_3\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nι : Type u_4\ninst✝ : Fintype ι\nm : ι → M\nn : ι → N\nhm : span R (Set.range m) = ⊤\nhmn : ∑ i, m i ⊗ₜ[R] n i = 0\nG : (ι →₀ R) →ₗ[R] M := linearCombination R ... | have G_basis_eq (i : ι) : G (Finsupp.single i 1) = m i := by simp [hG] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Flat.EquationalCriterion | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 9
} | {
"line": 143,
"column": 4
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ntfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))\ntfae_3_iff_2 :\n (∀ {l : ℕ} {f : Fin l → R} {x : Fin l → M}, ∑ i, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x)... | [
"case h\nR : Type u_1\nM : Type u_2\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\ntfae_1_iff_2 : Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(rTensor M (Submodule.subtype I))\ntfae_3_iff_2 :\n (∀ {l : ℕ} {f : Fin l → R} {x : Fin l → M}, ∑ i, f i ⊗ₜ[R] x i = 0 → VanishesTrivially R f x) ↔\n... | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.RingTheory.Flat.EquationalCriterion | {
"line": 282,
"column": 2
} | {
"line": 282,
"column": 20
} | {
"line": 284,
"column": 0
} | [
{
"pp": "case h\nR : Type u_1\nM : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : Flat R M\nP : Type u_3\ninst✝² : AddCommGroup P\ninst✝¹ : Module R P\ninst✝ : FinitePresentation R P\nh₁ : P →ₗ[R] M\nw✝ : ℕ\nK : Submodule R (Fin w✝ → R)\nϕ : P ≃ₗ[R] (Fin w✝ → R) ⧸ K\nhK : ... | [] | simpa [comp_assoc] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.LinearDisjoint | {
"line": 290,
"column": 36
} | {
"line": 290,
"column": 84
} | {
"line": 290,
"column": 84
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring S\ninst✝ : Algebra R S\nM N : Submodule R S\nH : ∀ N' ≤ N, N'.FG → M.LinearDisjoint N'\nx y : ↥M ⊗[R] ↥N\nhxy : (M.mulMap N) x = (M.mulMap N) y\nN' : Submodule R S\nhN : N' ≤ N\nhFG : N'.FG\nh : {x, y} ⊆ ↑(LinearMap.lTensor (↥M) (inclus... | [] | simp [← mulMap_comp_lTensor _ hN, hx', hy', hxy] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.LinearDisjoint | {
"line": 290,
"column": 36
} | {
"line": 290,
"column": 84
} | {
"line": 290,
"column": 84
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring S\ninst✝ : Algebra R S\nM N : Submodule R S\nH : ∀ N' ≤ N, N'.FG → M.LinearDisjoint N'\nx y : ↥M ⊗[R] ↥N\nhxy : (M.mulMap N) x = (M.mulMap N) y\nN' : Submodule R S\nhN : N' ≤ N\nhFG : N'.FG\nh : {x, y} ⊆ ↑(LinearMap.lTensor (↥M) (inclus... | [] | simp [← mulMap_comp_lTensor _ hN, hx', hy', hxy] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.LinearDisjoint | {
"line": 290,
"column": 36
} | {
"line": 290,
"column": 84
} | {
"line": 290,
"column": 84
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Semiring S\ninst✝ : Algebra R S\nM N : Submodule R S\nH : ∀ N' ≤ N, N'.FG → M.LinearDisjoint N'\nx y : ↥M ⊗[R] ↥N\nhxy : (M.mulMap N) x = (M.mulMap N) y\nN' : Submodule R S\nhN : N' ≤ N\nhFG : N'.FG\nh : {x, y} ⊆ ↑(LinearMap.lTensor (↥M) (inclus... | [] | simp [← mulMap_comp_lTensor _ hN, hx', hy', hxy] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.AssociatedPrime.Basic | {
"line": 80,
"column": 39
} | {
"line": 80,
"column": 80
} | {
"line": 81,
"column": 6
} | [
{
"pp": "case mp\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nN : Submodule R M\ninst✝ : IsNoetherianRing R\nx : M\nhx : (N.colon {x}).radical.IsPrime\n⊢ (N.colon {x}).radical ∈ (N.colon {x}).radical.minimalPrimes",
"ppTerm": "?mp",
"assigned": tru... | [
"case mp\nR : Type u_1\nM : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nN : Submodule R M\ninst✝ : IsNoetherianRing R\nx : M\nhx : (N.colon {x}).radical.IsPrime\n⊢ (N.colon {x}).radical ∈ {(N.colon {x}).radical}"
] | Ideal.minimalPrimes_eq_subsingleton_self, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.AssociatedPrime.Finiteness | {
"line": 139,
"column": 4
} | {
"line": 139,
"column": 63
} | {
"line": 140,
"column": 4
} | [
{
"pp": "A : Type u\ninst✝³ : CommRing A\ninst✝² : IsNoetherianRing A\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nx✝ : Module.Finite A M\nmotive : (N : Type v) → [inst : AddCommGroup N] → [inst_1 : Module A N] → [Module.Finite A N] → Prop\nsubsingleton :\n ∀ (N : Type v) [inst : AddCommGroup N] [... | [
"A : Type u\ninst✝³ : CommRing A\ninst✝² : IsNoetherianRing A\nM : Type v\ninst✝¹ : AddCommGroup M\ninst✝ : Module A M\nx✝ : Module.Finite A M\nmotive : (N : Type v) → [inst : AddCommGroup N] → [inst_1 : Module A N] → [Module.Finite A N] → Prop\nsubsingleton :\n ∀ (N : Type v) [inst : AddCommGroup N] [inst_1 : Mod... | replace H : motive s.last := H s.length s.length.lt_add_one | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.RingTheory.LocalRing.Module | {
"line": 320,
"column": 2
} | {
"line": 323,
"column": 79
} | {
"line": 325,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst✝¹⁰ : CommRing R\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : AddCommGroup N\ninst✝⁷ : Module R M\ninst✝⁶ : Module R N\ninst✝⁵ : AddCommGroup P\ninst✝⁴ : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\ninst✝³ : IsLocalRing R\nhg : Surjective ⇑g\nh : Exact ⇑f ... | [] | apply lTensor_injective_of_exact_of_exact_of_rTensor_injective
h hg (LinearMap.exact_subtype_mkQ 𝔪) (Submodule.mkQ_surjective _)
((LinearMap.lTensor_inj_iff_rTensor_inj _ _).mp hf)
(Module.Flat.lTensor_preserves_injective_linearMap _ Subtype.val_injective) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.LocallyConstant.Basic | {
"line": 531,
"column": 4
} | {
"line": 531,
"column": 52
} | {
"line": 532,
"column": 4
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\nα : Type u_4\ninst✝¹ : TopologicalSpace X\nC₁ C₂ : Set X\nh₁ : IsClosed[inst✝¹] C₁\nh₂ : IsClosed[inst✝¹] C₂\nh : C₁ ∪ C₂ = univ\ninst✝ : DecidablePred fun x ↦ x ∈ C₁\ndZ : TopologicalSpace Z := ⊥\nthis : DiscreteTopology Z\nf : ↑C₁ → Z\nhf : IsLocallyConstant ... | [
"X : Type u_1\nY : Type u_2\nZ : Type u_3\nα : Type u_4\ninst✝¹ : TopologicalSpace X\nC₁ C₂ : Set X\nh₁ : IsClosed[inst✝¹] C₁\nh₂ : IsClosed[inst✝¹] C₂\nh : C₁ ∪ C₂ = univ\ninst✝ : DecidablePred fun x ↦ x ∈ C₁\ndZ : TopologicalSpace Z := ⊥\nthis : DiscreteTopology Z\nf : ↑C₁ → Z\nhf✝ : IsLocallyConstant f\nhf : Con... | rw [IsLocallyConstant.iff_continuous] at hf hg ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.Matroid.Basic | {
"line": 1014,
"column": 4
} | {
"line": 1014,
"column": 13
} | {
"line": 1014,
"column": 14
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI : Set α\ne : α\nhI : M.Indep I\n⊢ (∀ (e_1 : α), e_1 = e ∧ e_1 ∉ I ∨ False → ¬M.Indep (insert e_1 I) ∧ e_1 ∈ M.E) ↔\n ¬M.Indep (insert e I) ∧ e ∈ M.E ∨ e ∈ I",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"cong... | [
"α : Type u_1\nM : Matroid α\nI : Set α\ne : α\nhI : M.Indep I\n⊢ (∀ (e_1 : α), e_1 = e ∧ e_1 ∉ I → ¬M.Indep (insert e_1 I) ∧ e_1 ∈ M.E) ↔ ¬M.Indep (insert e I) ∧ e ∈ M.E ∨ e ∈ I"
] | or_false, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.FieldTheory.PrimitiveElement | {
"line": 402,
"column": 2
} | {
"line": 403,
"column": 46
} | {
"line": 404,
"column": 2
} | [
{
"pp": "F : Type u_3\nE : Type u_4\ninst✝⁶ : Field F\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\ninst✝³ : FiniteDimensional F E\ninst✝² : Algebra.IsSeparable F E\nA : Type u_5\ninst✝¹ : Field A\ninst✝ : Algebra F A\nhA : ∀ (x : E), (Polynomial.map (algebraMap F A) (minpoly F x)).Splits\nα : E\nφ : E →ₐ[F] A\n⊢ F⟮... | [
"F : Type u_3\nE : Type u_4\ninst✝⁶ : Field F\ninst✝⁵ : Field E\ninst✝⁴ : Algebra F E\ninst✝³ : FiniteDimensional F E\ninst✝² : Algebra.IsSeparable F E\nA : Type u_5\ninst✝¹ : Field A\ninst✝ : Algebra F A\nhA : ∀ (x : E), (Polynomial.map (algebraMap F A) (minpoly F x)).Splits\nα : E\nφ : E →ₐ[F] A\nh : ∀ (ψ : E →ₐ[... | refine ⟨fun h ψ hψ ↦ (Field.primitive_element_iff_algHom_eq_of_eval' F A hA α).mp h hψ,
fun h ↦ eq_of_le_of_finrank_eq' le_top ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.Matroid.IndepAxioms | {
"line": 145,
"column": 16
} | {
"line": 149,
"column": 26
} | {
"line": 150,
"column": 2
} | [
{
"pp": "α : Type u_1\nM : IndepMatroid α\n⊢ ∀ ⦃I : Set α⦄, M.Indep I ↔ ∃ B, Maximal M.Indep B ∧ I ⊆ B",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"ChainCompletePartialOrder.instOfCompleteLattice",
"congrArg",
"IndepMatroid.subset_ground",
"PartialOrder.toPreord... | [] | by
refine fun I ↦ ⟨fun h ↦ ?_, fun ⟨B, ⟨h, _⟩, hIB'⟩ ↦ M.indep_subset h hIB'⟩
obtain ⟨J, hIJ, hmax⟩ := M.indep_maximal M.E rfl.subset I h (M.subset_ground I h)
rw [maximal_and_iff_right_of_imp M.subset_ground] at hmax
exact ⟨J, hmax.1, hIJ⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.IndepAxioms | {
"line": 152,
"column": 4
} | {
"line": 152,
"column": 59
} | {
"line": 153,
"column": 4
} | [
{
"pp": "α : Type u_1\nM : IndepMatroid α\nB : Set α\nhB : Maximal (fun K ↦ M.Indep K ∧ K ⊆ M.E) B\n⊢ ∃ B, Maximal M.Indep B",
"ppTerm": "?m.143",
"assigned": true,
"usedConstants": [
"congrArg",
"IndepMatroid.subset_ground",
"Eq.mp",
"LE.le",
"maximal_and_iff_right_of_... | [
"α : Type u_1\nM : IndepMatroid α\nB : Set α\nhB : Maximal M.Indep B ∧ B ⊆ M.E\n⊢ ∃ B, Maximal M.Indep B"
] | rw [maximal_and_iff_right_of_imp M.subset_ground] at hB | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.PicardGroup | {
"line": 748,
"column": 6
} | {
"line": 748,
"column": 55
} | {
"line": 749,
"column": 6
} | [
{
"pp": "case mp.refine_1\nR : Type u\nA : Type u_4\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : FaithfulSMul R A\nI : (Submodule R A)ˣ\nh✝ : I ∈ (unitsToPic R A).ker\ne : R ≃ₗ[R] ↥↑I\ne' : R ≃ₗ[R] ↥↑I⁻¹\nh : ↑I = R ∙ ↑(↑e 1)\nh' : ↑I⁻¹ = R ∙ ↑(↑e' 1)\nthis : R ∙ ↑(e 1) * ↑(e' 1)... | [
"case mp.refine_1\nR : Type u\nA : Type u_4\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : FaithfulSMul R A\nI : (Submodule R A)ˣ\nh✝ : I ∈ (unitsToPic R A).ker\ne : R ≃ₗ[R] ↥↑I\ne' : R ≃ₗ[R] ↥↑I⁻¹\nh : ↑I = R ∙ ↑(↑e 1)\nh' : ↑I⁻¹ = R ∙ ↑(↑e' 1)\nthis : R ∙ ↑(e 1) * ↑(e' 1) = 1\nr : Rˣ... | have ⟨r, hr⟩ := span_singleton_eq_one_iff.mp this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.PicardGroup | {
"line": 751,
"column": 6
} | {
"line": 751,
"column": 55
} | {
"line": 752,
"column": 6
} | [
{
"pp": "case mp.refine_2\nR : Type u\nA : Type u_4\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : FaithfulSMul R A\nI : (Submodule R A)ˣ\nh✝ : I ∈ (unitsToPic R A).ker\ne : R ≃ₗ[R] ↥↑I\ne' : R ≃ₗ[R] ↥↑I⁻¹\nh : ↑I = R ∙ ↑(↑e 1)\nh' : ↑I⁻¹ = R ∙ ↑(↑e' 1)\nthis : R ∙ ↑(e' 1) * ↑(e 1)... | [
"case mp.refine_2\nR : Type u\nA : Type u_4\ninst✝³ : CommSemiring R\ninst✝² : Semiring A\ninst✝¹ : Algebra R A\ninst✝ : FaithfulSMul R A\nI : (Submodule R A)ˣ\nh✝ : I ∈ (unitsToPic R A).ker\ne : R ≃ₗ[R] ↥↑I\ne' : R ≃ₗ[R] ↥↑I⁻¹\nh : ↑I = R ∙ ↑(↑e 1)\nh' : ↑I⁻¹ = R ∙ ↑(↑e' 1)\nthis : R ∙ ↑(e' 1) * ↑(e 1) = 1\nr : Rˣ... | have ⟨r, hr⟩ := span_singleton_eq_one_iff.mp this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.Matroid.Constructions | {
"line": 54,
"column": 19
} | {
"line": 54,
"column": 67
} | {
"line": 55,
"column": 2
} | [
{
"pp": "α✝ : Type u_1\nM : Matroid α✝\nE B I X R J : Set α✝\nα : Type u_2\n⊢ ∀ X ⊆ ∅, ExistsMaximalSubsetProperty (fun x ↦ x = ∅) X",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"ChainCompletePartialOrder.instOfCompleteLattice",
"instReflLe",
"congrArg",
"and_sel... | [] | rintro _ _ _ rfl -; exact ⟨∅, by simp [Maximal]⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Matroid.Constructions | {
"line": 54,
"column": 19
} | {
"line": 54,
"column": 67
} | {
"line": 55,
"column": 2
} | [
{
"pp": "α✝ : Type u_1\nM : Matroid α✝\nE B I X R J : Set α✝\nα : Type u_2\n⊢ ∀ X ⊆ ∅, ExistsMaximalSubsetProperty (fun x ↦ x = ∅) X",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"ChainCompletePartialOrder.instOfCompleteLattice",
"instReflLe",
"congrArg",
"and_sel... | [] | rintro _ _ _ rfl -; exact ⟨∅, by simp [Maximal]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Matroid.Constructions | {
"line": 249,
"column": 2
} | {
"line": 250,
"column": 66
} | {
"line": 251,
"column": 2
} | [
{
"pp": "α : Type u_1\nI E R : Set α\n⊢ uniqueBaseOn I E ↾ R = uniqueBaseOn (I ∩ R ∩ E) R",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matroid.uniqueBaseOn_indep_iff'._simp_1",
"congrArg",
"Matroid.E",
"Matroid.Indep",
"id",
"LE.le",
... | [
"α : Type u_1\nI E R : Set α\n⊢ ∀ ⦃I_1 : Set α⦄, I_1 ⊆ R → ((I_1 ⊆ I ∧ I_1 ⊆ E) ∧ I_1 ⊆ R ↔ ((I_1 ⊆ I ∧ I_1 ⊆ R) ∧ I_1 ⊆ E) ∧ I_1 ⊆ R)"
] | simp_rw [ext_iff_indep, restrict_ground_eq, uniqueBaseOn_ground, true_and,
restrict_indep_iff, uniqueBaseOn_indep_iff', subset_inter_iff] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Combinatorics.Matroid.IndepAxioms | {
"line": 249,
"column": 8
} | {
"line": 252,
"column": 36
} | {
"line": 253,
"column": 8
} | [
{
"pp": "α : Type u_1\nE : Set α\nIndep : Set α → Prop\nindep_empty : Indep ∅\nindep_subset : ∀ ⦃I J : Set α⦄, Indep J → I ⊆ J → Indep I\nindep_aug :\n ∀ ⦃I J : Set α⦄, Indep I → I.Finite → Indep J → J.Finite → I.ncard < J.ncard → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)\nindep_compact : ∀ (I : Set α), (∀ J ⊆ I, J.... | [
"α : Type u_1\nE : Set α\nIndep : Set α → Prop\nindep_empty : Indep ∅\nindep_subset : ∀ ⦃I J : Set α⦄, Indep J → I ⊆ J → Indep I\nindep_aug :\n ∀ ⦃I J : Set α⦄, Indep I → I.Finite → Indep J → J.Finite → I.ncard < J.ncard → ∃ e ∈ J, e ∉ I ∧ Indep (insert e I)\nindep_compact : ∀ (I : Set α), (∀ J ⊆ I, J.Finite → Ind... | refine ⟨iUnion f ∪ (B₀ ∩ I),
union_subset (iUnion_subset (fun i ↦ (hf i).1)) inter_subset_right,
(finite_iUnion fun i ↦ (hf i).2.1).union (hB₀fin.subset inter_subset_left),
fun x ⟨hxB₀, hxn⟩ hi ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.Matroid.Map | {
"line": 284,
"column": 69
} | {
"line": 285,
"column": 30
} | {
"line": 287,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nE✝ I✝ : Set α\nM : Matroid α\nN : Matroid β\nE B I : Set α\ninst✝ : N.Finitary\n⊢ (N.comapOn E f).Finitary",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Matroid.comap_finitary",
"Eq.mpr",
"Matroid.comapOn.eq_1",
"Mat... | [] | by
rw [comapOn]; infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Map | {
"line": 287,
"column": 75
} | {
"line": 288,
"column": 30
} | {
"line": 290,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\nE✝ I✝ : Set α\nM : Matroid α\nN : Matroid β\nE B I : Set α\ninst✝ : N.RankFinite\n⊢ (N.comapOn E f).RankFinite",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matroid.comapOn.eq_1",
"Matroid.comapOn",
"congrA... | [] | by
rw [comapOn]; infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 190,
"column": 45
} | {
"line": 191,
"column": 19
} | {
"line": 193,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\n⊢ M.closure (M.loops ∪ X) = M.closure X",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"congrArg",
"Set.instUnion",
"Set.union_comm",
"Matroid.closure_union_loops_eq",
"True",
"eq_self",
"Matroid.clo... | [] | by
simp [union_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 255,
"column": 2
} | {
"line": 255,
"column": 60
} | {
"line": 256,
"column": 2
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nI : Set α\ne : α\nhI : M.Indep I\nhecl : e ∈ M.closure I\nheI : e ∉ I\naux : ⋂₀ {J | J ⊆ I ∧ e ∈ M.closure J} ⊆ I\n⊢ M.IsCircuit (insert e (⋂₀ {J | J ⊆ I ∧ e ∈ M.closure J}))",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"setOf",
"Matroid... | [
"case refine_1\nα : Type u_1\nM : Matroid α\nI : Set α\ne : α\nhI : M.Indep I\nhecl : e ∈ M.closure I\nheI : e ∉ I\naux : ⋂₀ {J | J ⊆ I ∧ e ∈ M.closure J} ⊆ I\n⊢ e ∉ ⋂₀ {J | J ⊆ I ∧ e ∈ M.closure J}",
"case refine_2\nα : Type u_1\nM : Matroid α\nI : Set α\ne : α\nhI : M.Indep I\nhecl : e ∈ M.closure I\nheI : e ∉ ... | refine (hI.subset aux).insert_isCircuit_of_forall ?_ ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 289,
"column": 2
} | {
"line": 289,
"column": 89
} | {
"line": 290,
"column": 2
} | [
{
"pp": "case inr\nα : Type u_1\nM : Matroid α\nC I : Set α\ne : α\nhC : M.IsCircuit C\nhI : M.Indep I\nhCs : C ⊆ insert e I\nheC : e ∈ C\nhCeI : C \\ {e} ⊆ I\nheCcl : e ∈ M.closure (C \\ {e})\nheI : e ∈ M.closure I\n⊢ insert e (⋂₀ {J | J ⊆ I ∧ e ∈ M.closure J}) ⊆ C",
"ppTerm": "?inr",
"assigned": true,... | [
"case inr\nα : Type u_1\nM : Matroid α\nC I : Set α\ne : α\nhC : M.IsCircuit C\nhI : M.Indep I\nhCs : C ⊆ insert e I\nheC : e ∈ C\nhCeI : C \\ {e} ⊆ I\nheCcl : e ∈ M.closure (C \\ {e})\nheI : e ∈ M.closure I\n⊢ C \\ {e} ∈ {J | J ⊆ I ∧ e ∈ M.closure J}"
] | refine insert_subset heC <| (sInter_subset_of_mem (t := C \ {e}) ?_).trans sdiff_subset | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 326,
"column": 4
} | {
"line": 328,
"column": 72
} | {
"line": 329,
"column": 2
} | [
{
"pp": "α : Type u_2\nM : Matroid α\nI : Set α\nhI : M.Indep I\nF : Set α := {x | M.IsBasis I (insert x I)}\nhIF : M.IsBasis I F\nJ X : Set α\nhJF : M.IsBasis J F\nhJX : M.IsBasis J X\ne : α\nheX : e ∈ X\n⊢ M.IsBasis I (insert e I)",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Ch... | [] | exact (hIF.isBasis_of_isBasis_of_subset_of_subset (hJX.isBasis_union hJF) hJF.subset
(hIF.subset.trans subset_union_right)).isBasis_subset (subset_insert _ _)
(insert_subset (Or.inl heX) (hIF.subset.trans subset_union_right)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 433,
"column": 2
} | {
"line": 434,
"column": 66
} | {
"line": 435,
"column": 2
} | [
{
"pp": "case inr\nα : Type u_1\nM : Matroid α\nι : Type u_2\nJ : Set α\nx : ι → α\nI : ι → Set α\nz : α\nhxI : ∀ (i : ι), x i ∉ I i\nhC : ∀ (i : ι), M.IsCircuit (insert (x i) (I i))\nhJx : M.IsCircuit (J ∪ range x)\nhzJ : z ∈ J\nhzI : ∀ (i : ι), z ∉ I i\nhι : Nonempty ι\nhC' : ∀ (i : ι), M.closure (I i) = M.cl... | [
"case inr\nα : Type u_1\nM : Matroid α\nι : Type u_2\nJ : Set α\nx : ι → α\nI : ι → Set α\nz : α\nhxI : ∀ (i : ι), x i ∉ I i\nhC : ∀ (i : ι), M.IsCircuit (insert (x i) (I i))\nhJx : M.IsCircuit (J ∪ range x)\nhzJ : z ∈ J\nhzI : ∀ (i : ι), z ∉ I i\nhι : Nonempty ι\nhC' : ∀ (i : ι), M.closure (I i) = M.closure (inser... | refine mem_of_mem_of_subset (hJx.mem_closure_sdiff_singleton_of_mem (.inl hzJ))
(M.closure_subset_closure (subset_trans ?_ subset_union_left)) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 376,
"column": 69
} | {
"line": 376,
"column": 78
} | {
"line": 377,
"column": 4
} | [
{
"pp": "α : Type u_2\nM : Matroid α\ne : α\nI : Set α\nhI : M.Indep I\n⊢ M.Dep (insert e I) ↔ M.Dep (insert e I) ∧ e ∉ I ∨ False",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Matroid.Dep",
"False",
"congrArg",
"Membership.mem",
"id",
... | [
"α : Type u_2\nM : Matroid α\ne : α\nI : Set α\nhI : M.Indep I\n⊢ M.Dep (insert e I) ↔ M.Dep (insert e I) ∧ e ∉ I"
] | or_false, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Rank.ENat | {
"line": 375,
"column": 55
} | {
"line": 376,
"column": 40
} | {
"line": 378,
"column": 0
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nX : Set α\n⊢ M.eRk X < ⊤ ↔ M.IsRkFinite X",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Matroid.eRk_ne_top_iff",
"Eq.mpr",
"Preorder.toLT",
"instCompleteLinearOrderENat",
"ChainCompletePartialOrder.instOfCompleteLattice"... | [] | by
rw [lt_top_iff_ne_top, eRk_ne_top_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Matroid.Loop | {
"line": 807,
"column": 2
} | {
"line": 807,
"column": 54
} | {
"line": 808,
"column": 2
} | [
{
"pp": "α : Type u_1\nM : Matroid α\nx✝ : ∃ x, M.IsCircuit x ∧ x.Subsingleton\nC : Set α\nhC : M.IsCircuit C\nhCs : C.Subsingleton\n⊢ ∃ x ∈ M.E, M.IsLoop x",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"Matroid.E",
"Membership.mem",
"Exists",
"Set.instSingletonSe... | [
"case inl\nα : Type u_1\nM : Matroid α\nx✝ : ∃ x, M.IsCircuit x ∧ x.Subsingleton\nhC : M.IsCircuit ∅\nhCs : ∅.Subsingleton\n⊢ ∃ x ∈ M.E, M.IsLoop x",
"case inr\nα : Type u_1\nM : Matroid α\nx✝ : ∃ x, M.IsCircuit x ∧ x.Subsingleton\ne : α\nhC : M.IsCircuit {e}\nhCs : {e}.Subsingleton\n⊢ ∃ x ∈ M.E, M.IsLoop x"
] | obtain (rfl | ⟨e, rfl⟩) := hCs.eq_empty_or_singleton | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.Matroid.Circuit | {
"line": 719,
"column": 2
} | {
"line": 720,
"column": 19
} | {
"line": 721,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\nX : Set α\ne : α\nM : Matroid α\nheX : e ∉ X\nhe : e ∈ M.E\n⊢ M.fundCocircuit e X = {e}",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matroid.E",
"Matroid.dual",
"Membership.mem",
"Set.instSinglet... | [
"case neg\nα : Type u_1\nX : Set α\ne : α\nM : Matroid α\nheX : e ∉ X\nhe : e ∉ M.E\n⊢ M.fundCocircuit e X = {e}"
] | · rw [fundCocircuit, fundCircuit_eq_of_mem]
exact ⟨he, heX⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.Matroid.Rank.Cardinal | {
"line": 336,
"column": 2
} | {
"line": 337,
"column": 72
} | {
"line": 338,
"column": 2
} | [
{
"pp": "case neg\nα : Type u\nβ : Type v\nf : α → β\nM : Matroid α\nI J B✝ B'✝ X Y : Set α\ninst✝ : M.Finitary\nB B' : Set α\nN : Matroid α\nhfin : N.Finitary\nhB : N.IsBase B\nhB' : N.IsBase B'\nh : Infinite ↑(B' \\ B)\nS : (a : α) → a ∈ B' \\ B → Set α\nS_fin : ∀ (a : α) (ha : a ∈ B' \\ B), Finite ↑(S a ha)\... | [
"case neg\nα : Type u\nβ : Type v\nf : α → β\nM : Matroid α\nI J B✝ B'✝ X Y : Set α\ninst✝ : M.Finitary\nB B' : Set α\nN : Matroid α\nhfin : N.Finitary\nhB : N.IsBase B\nhB' : N.IsBase B'\nh : Infinite ↑(B' \\ B)\nS : (a : α) → a ∈ B' \\ B → Set α\nS_fin : ∀ (a : α) (ha : a ∈ B' \\ B), Finite ↑(S a ha)\nhSB : ∀ (a ... | have hUB : (B ∩ B') ∪ U ⊆ B :=
union_subset inter_subset_left (iUnion_subset fun e ↦ (hSB e.1 e.2)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.AlgebraicIndependent.Basic | {
"line": 203,
"column": 35
} | {
"line": 203,
"column": 54
} | {
"line": 203,
"column": 55
} | [
{
"pp": "ι : Type u\nR : Type u_2\nA : Type v\nx : ι → A\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A\ninst✝⁷ : Algebra R A\nS : Type u_3\nB : Type u_4\nFRS : Type u_5\nFAB : Type u_6\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra S B\ninst✝³ : FunLike FRS R S\ninst✝² : RingHomClass FRS R S\ninst✝¹ : ... | [
"ι : Type u\nR : Type u_2\nA : Type v\nx : ι → A\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A\ninst✝⁷ : Algebra R A\nS : Type u_3\nB : Type u_4\nFRS : Type u_5\nFAB : Type u_6\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra S B\ninst✝³ : FunLike FRS R S\ninst✝² : RingHomClass FRS R S\ninst✝¹ : FunLike FAB ... | ← eval₂Hom_map_hom, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Matroid.Closure | {
"line": 1041,
"column": 4
} | {
"line": 1041,
"column": 30
} | {
"line": 1042,
"column": 4
} | [
{
"pp": "case mp\nα : Type u_2\nβ : Type u_3\nM : Matroid α\nf : α → β\nhf : InjOn f M.E\nX : Set β\nI : Set α\nhI : M.Indep I\ne : β\n⊢ ((∃ x ∈ M.E, f x = e) ∧ ∀ (x : Set α), M.Indep x → insert e (f '' I) = f '' x → ∃ x ∈ I, f x = e) →\n ∃ x, (x ∈ M.E ∧ (M.Indep (insert x I) → x ∈ I)) ∧ f x = e",
"ppTer... | [
"case mp\nα : Type u_2\nβ : Type u_3\nM : Matroid α\nf : α → β\nhf : InjOn f M.E\nX : Set β\nI : Set α\nhI : M.Indep I\nx : α\nhxE : x ∈ M.E\nh2 : ∀ (x_1 : Set α), M.Indep x_1 → insert (f x) (f '' I) = f '' x_1 → ∃ x_2 ∈ I, f x_2 = f x\n⊢ ∃ x_1, (x_1 ∈ M.E ∧ (M.Indep (insert x_1 I) → x_1 ∈ I)) ∧ f x_1 = f x"
] | rintro ⟨⟨x, hxE, rfl⟩, h2⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.RingTheory.AlgebraicIndependent.Basic | {
"line": 213,
"column": 70
} | {
"line": 213,
"column": 89
} | {
"line": 214,
"column": 4
} | [
{
"pp": "ι : Type u\nR : Type u_2\nA : Type v\nx : ι → A\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A\ninst✝⁷ : Algebra R A\nS : Type u_3\nB : Type u_4\nFRS : Type u_5\nFAB : Type u_6\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra S B\ninst✝³ : FunLike FRS R S\ninst✝² : RingHomClass FRS R S\ninst✝¹ : ... | [
"ι : Type u\nR : Type u_2\nA : Type v\nx : ι → A\ninst✝⁹ : CommRing R\ninst✝⁸ : CommRing A\ninst✝⁷ : Algebra R A\nS : Type u_3\nB : Type u_4\nFRS : Type u_5\nFAB : Type u_6\ninst✝⁶ : CommRing S\ninst✝⁵ : CommRing B\ninst✝⁴ : Algebra S B\ninst✝³ : FunLike FRS R S\ninst✝² : RingHomClass FRS R S\ninst✝¹ : FunLike FAB ... | ← eval₂Hom_map_hom, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.AlgebraicClosure | {
"line": 71,
"column": 2
} | {
"line": 71,
"column": 38
} | {
"line": 73,
"column": 0
} | [
{
"pp": "F : Type u_1\nE : Type u_2\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\nK : Type u_3\ninst✝¹ : Field K\ninst✝ : Algebra F K\ni : E →ₐ[F] K\nx : E\n⊢ x ∈ comap i (algebraicClosure F K) ↔ x ∈ algebraicClosure F E",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"m... | [] | exact map_mem_algebraicClosure_iff i | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.GroupTheory.GroupAction.FixedPoints | {
"line": 274,
"column": 2
} | {
"line": 275,
"column": 58
} | {
"line": 277,
"column": 0
} | [
{
"pp": "α : Type u_1\nG : Type u_2\ninst✝² : Group G\ninst✝¹ : MulAction G α\ninst✝ : FaithfulSMul G α\ng h : G\ndisjoint : Disjoint (fixedBy α g)ᶜ (h • (fixedBy α g)ᶜ)\ncomm : Commute g h\n⊢ g = 1",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"instHSMul",
... | [] | rwa [movedBy_mem_fixedBy_of_commute comm, disjoint_self, Set.bot_eq_empty, ← Set.compl_univ,
compl_inj_iff, fixedBy_eq_univ_iff_eq_one] at disjoint | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 406,
"column": 94
} | {
"line": 408,
"column": 65
} | {
"line": 410,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type w\ninst✝⁴ : CommRing R\ninst✝³ : CommRing A\ninst✝² : Algebra R A\ninst✝¹ : IsDomain A\ninst✝ : FaithfulSMul R A\ns : Set A\n⊢ Algebra.IsAlgebraic (↥(adjoin R s)) A ↔ ∃ t ⊆ s, IsTranscendenceBasis R Subtype.val",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [... | [] | by
simp_rw [← matroid_spanning_iff, ← matroid_isBase_iff, and_comm (a := _ ⊆ s)]
exact Matroid.spanning_iff_exists_isBase_subset (subset_univ _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 586,
"column": 4
} | {
"line": 589,
"column": 46
} | {
"line": 590,
"column": 2
} | [
{
"pp": "ι : Type u\nR : Type u_1\nS : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : FaithfulSMul R S\ninst✝ : NoZeroDivisors S\ns : Set ι\ni j : ι\nv : ι → S\nhj✝ : j ∈ insert i s\nH₁ : IsTranscendenceBasis R fun x ↦ v ↑x\nthis✝² : Nontrivial ↥(adjoin R (v '' (insert i s \\ {... | [] | refine Matroid.closure_subset_closure _ ?_ H₂
rintro x ⟨k, ⟨rfl | hks, hkj⟩, rfl⟩
· exact ⟨his, ne⟩
· exact ⟨⟨k, hks, rfl⟩, inj.ne hks hj hkj⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.AlgebraicIndependent.TranscendenceBasis | {
"line": 586,
"column": 4
} | {
"line": 589,
"column": 46
} | {
"line": 590,
"column": 2
} | [
{
"pp": "ι : Type u\nR : Type u_1\nS : Type v\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : FaithfulSMul R S\ninst✝ : NoZeroDivisors S\ns : Set ι\ni j : ι\nv : ι → S\nhj✝ : j ∈ insert i s\nH₁ : IsTranscendenceBasis R fun x ↦ v ↑x\nthis✝² : Nontrivial ↥(adjoin R (v '' (insert i s \\ {... | [] | refine Matroid.closure_subset_closure _ ?_ H₂
rintro x ⟨k, ⟨rfl | hks, hkj⟩, rfl⟩
· exact ⟨his, ne⟩
· exact ⟨⟨k, hks, rfl⟩, inj.ne hks hj hkj⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.SeparableDegree | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 74
} | {
"line": 161,
"column": 4
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\nK : Type w\ninst✝¹ : Field K\ninst✝ : Algebra F K\ni : E ≃ₐ[F] K\nx✝ : Algebra E K := (↑i).toAlgebra\nx : K\n⊢ IsAlgebraic E x",
"ppTerm": "?m.75",
"assigned": true,
"usedConstants": [
"AlgEquiv.toAlgHom... | [
"F : Type u\nE : Type v\ninst✝⁴ : Field F\ninst✝³ : Field E\ninst✝² : Algebra F E\nK : Type w\ninst✝¹ : Field K\ninst✝ : Algebra F K\ni : E ≃ₐ[F] K\nx✝ : Algebra E K := (↑i).toAlgebra\nx : K\nh : IsAlgebraic E ((algebraMap E K) (↑i.symm x))\n⊢ IsAlgebraic E x"
] | have h := isAlgebraic_algebraMap (R := E) (A := K) (i.symm.toAlgHom x) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.Galois.Basic | {
"line": 225,
"column": 2
} | {
"line": 227,
"column": 39
} | {
"line": 229,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝³ : Field F\nE : Type u_2\ninst✝² : Field E\ninst✝¹ : Algebra F E\nH : Subgroup Gal(E/F)\ninst✝ : FiniteDimensional F E\n⊢ finrank (↥(fixedField H)) E = Nat.card ↥H",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Subgroup.instFiniteSubtypeMem",
"Fix... | [] | have := Fintype.ofFinite H
rw [Nat.card_eq_fintype_card]
exact FixedPoints.finrank_eq_card H E | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.Galois.Basic | {
"line": 225,
"column": 2
} | {
"line": 227,
"column": 39
} | {
"line": 229,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝³ : Field F\nE : Type u_2\ninst✝² : Field E\ninst✝¹ : Algebra F E\nH : Subgroup Gal(E/F)\ninst✝ : FiniteDimensional F E\n⊢ finrank (↥(fixedField H)) E = Nat.card ↥H",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Subgroup.instFiniteSubtypeMem",
"Fix... | [] | have := Fintype.ofFinite H
rw [Nat.card_eq_fintype_card]
exact FixedPoints.finrank_eq_card H E | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.SeparableDegree | {
"line": 379,
"column": 6
} | {
"line": 379,
"column": 13
} | {
"line": 379,
"column": 14
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\nf : F[X]\ninst✝ : DecidableEq E\nh : (map (algebraMap F E) f).Splits\n⊢ f.natSepDegree = (f.aroots E).toFinset.card",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Multiset.toFinset",
"E... | [
"F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\nf : F[X]\ninst✝ : DecidableEq E\nh : (map (algebraMap F E) f).Splits\n⊢ f.natSepDegree = (map (algebraMap F E) f).roots.toFinset.card"
] | aroots, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.OpenSubgroup | {
"line": 248,
"column": 2
} | {
"line": 248,
"column": 60
} | {
"line": 249,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : SeparatelyContinuousMul G\nH : Subgroup G\ng : G\nhg : ↑H ∈ 𝓝 g\nx : G\nhx : x ∈ ↑H\n⊢ ↑H ∈ 𝓝 x",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Subgroup",
"mem_of_mem_nhds",... | [
"G : Type u_1\ninst✝² : Group G\ninst✝¹ : TopologicalSpace G\ninst✝ : SeparatelyContinuousMul G\nH : Subgroup G\ng : G\nhg : ↑H ∈ 𝓝 g\nx : G\nhx : x ∈ ↑H\nhg' : g ∈ H\n⊢ ↑H ∈ 𝓝 x"
] | have hg' : g ∈ H := SetLike.mem_coe.1 (mem_of_mem_nhds hg) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Algebra.OpenSubgroup | {
"line": 526,
"column": 6
} | {
"line": 526,
"column": 11
} | {
"line": 527,
"column": 6
} | [
{
"pp": "G : Type u_2\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : CompactSpace G\nW : Set G\nWClopen : IsClopen W\neinW : 1 ∈ W\nV : Set G\nhV : mulInvClosureNhd V W\nx✝ : G\nha : x✝ ∈ ⋃ n, V ^ (n + 1)\nk : ℕ\nhk : x✝ ∈ V ^ (k + 1)\n⊢ ∃ i, x✝⁻¹ ∈ V ^ (i + 1)",
"ppT... | [
"case h\nG : Type u_2\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : CompactSpace G\nW : Set G\nWClopen : IsClopen W\neinW : 1 ∈ W\nV : Set G\nhV : mulInvClosureNhd V W\nx✝ : G\nha : x✝ ∈ ⋃ n, V ^ (n + 1)\nk : ℕ\nhk : x✝ ∈ V ^ (k + 1)\n⊢ x✝⁻¹ ∈ V ^ (k + 1)"
] | use k | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.FieldTheory.SeparableClosure | {
"line": 355,
"column": 2
} | {
"line": 362,
"column": 17
} | {
"line": 363,
"column": 2
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : Algebra.EssFiniteType F E\nd : Finset E → ℕ := fun s ↦ finInsepDegree (↥(adjoin F ↑s)) E\n⊢ ∃ s,\n MaximalFor (fun t ↦ IsTranscendenceBasis F Subtype.val)\n (fun t ↦ restrictScalars F (separableClosure (↥(... | [
"F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : Algebra.EssFiniteType F E\nd : Finset E → ℕ := fun s ↦ finInsepDegree (↥(adjoin F ↑s)) E\nHexists : {s | IsTranscendenceBasis F Subtype.val}.Nonempty\n⊢ ∃ s,\n MaximalFor (fun t ↦ IsTranscendenceBasis F Subtype.val)\n ... | have Hexists : {s : Finset E | IsTranscendenceBasis F ((↑) : s → E)}.Nonempty := by
have ⟨s, hs⟩ := IntermediateField.fg_top F E
have : Algebra.IsAlgebraic (Algebra.adjoin F (s : Set E)) E := by
rw [← isAlgebraic_adjoin_iff_top, hs, Algebra.isAlgebraic_iff_isIntegral]
refine Algebra.isIntegral_of_su... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.Galois.Infinite | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 18
} | {
"line": 155,
"column": 2
} | [
{
"pp": "k : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\nH : ClosedSubgroup Gal(K/k)\ninst✝ : IsGalois k K\nσ : Gal(K/k)\nhσ : σ ∈ (fixedField ↑H).fixingSubgroup\nh : σ ∉ ↑H\nb : Set Gal(K/k)\nsub : b ⊆ (fun y ↦ σ * y) ⁻¹' (↑H).carrierᶜ\ngp : Subgroup Gal(K/k)\neq : (fun g ... | [
"k : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\nH : ClosedSubgroup Gal(K/k)\ninst✝ : IsGalois k K\nσ : Gal(K/k)\nhσ : σ ∈ (fixedField ↑H).fixingSubgroup\nh : σ ∉ ↑H\nb : Set Gal(K/k)\nsub : b ⊆ (fun y ↦ σ * y) ⁻¹' (↑H).carrierᶜ\ngp : Subgroup Gal(K/k)\nL : IntermediateField k ... | rw [← eq'] at eq | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.Galois.Infinite | {
"line": 268,
"column": 4
} | {
"line": 269,
"column": 98
} | {
"line": 270,
"column": 4
} | [
{
"pp": "case refine_1\nk : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\nL : IntermediateField k K\ninst✝ : IsGalois k K\nh : L.fixingSubgroup.Normal\ng : (x : K) → Subgroup Gal(↥(adjoin k {x}).toIntermediateField/k) :=\n fun x ↦ Subgroup.map (restrictNormalHom ↥(adjoin k {... | [
"case refine_1\nk : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\nL : IntermediateField k K\ninst✝ : IsGalois k K\nh : L.fixingSubgroup.Normal\ng : (x : K) → Subgroup Gal(↥(adjoin k {x}).toIntermediateField/k) :=\n fun x ↦ Subgroup.map (restrictNormalHom ↥(adjoin k {x}).toInterm... | have (l : L) : Normal k (f l) :=
Normal.of_algEquiv <| IntermediateField.liftAlgEquiv <| IntermediateField.fixedField (g l.1) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.PurelyInseparable.Basic | {
"line": 174,
"column": 2
} | {
"line": 174,
"column": 44
} | {
"line": 176,
"column": 0
} | [
{
"pp": "F : Type u_1\nE : Type u_2\ninst✝⁴ : CommRing F\ninst✝³ : Ring E\ninst✝² : Algebra F E\nL : Subalgebra F E\ninst✝¹ : IsPurelyInseparable F ↥L\ninst✝ : Algebra.IsSeparable F ↥L\nx : E\nhx : x ∈ L\ny : F\nhy : (algebraMap F ↥L) y = ⟨x, hx⟩\n⊢ x ∈ ⊥",
"ppTerm": "?m.40",
"assigned": true,
"used... | [] | exact ⟨y, congr_arg (Subalgebra.val _) hy⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.PurelyInseparable.Basic | {
"line": 360,
"column": 4
} | {
"line": 360,
"column": 58
} | {
"line": 361,
"column": 4
} | [
{
"pp": "case neg\nF : Type u\nE : Type v\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nhdeg : finSepDegree F E = 1\nH : ¬Algebra.IsAlgebraic F E\n⊢ IsPurelyInseparable F E",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"congrArg",
"Algebra.Transcendental",
"... | [
"case neg\nF : Type u\nE : Type v\ninst✝² : Field F\ninst✝¹ : Field E\ninst✝ : Algebra F E\nhdeg : finSepDegree F E = 1\nH : Algebra.Transcendental F E\n⊢ IsPurelyInseparable F E"
] | rw [← Algebra.transcendental_iff_not_isAlgebraic] at H | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.FieldTheory.PurelyInseparable.Basic | {
"line": 562,
"column": 9
} | {
"line": 562,
"column": 39
} | {
"line": 562,
"column": 40
} | [
{
"pp": "F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : IsAlgClosure F E\nH : IsPurelyInseparable F E\nthis : IsAlgClosed E\n⊢ IsSepClosed F",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"CompleteLattice.t... | [
"F : Type u\nE : Type v\ninst✝³ : Field F\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : IsAlgClosure F E\nH : separableClosure F E = ⊥\nthis : IsAlgClosed E\n⊢ IsSepClosed F"
] | ← separableClosure.eq_bot_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Trace.Basic | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 60
} | {
"line": 83,
"column": 2
} | [
{
"pp": "S : Type u_2\ninst✝³ : CommRing S\nK : Type u_4\ninst✝² : Field K\ninst✝¹ : Algebra K S\ninst✝ : Nontrivial S\npb : PowerBasis K S\nd_pos : 0 < pb.dim\n⊢ (Algebra.trace K S) pb.gen = -(minpoly K pb.gen).nextCoeff",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"S : Type u_2\ninst✝³ : CommRing S\nK : Type u_4\ninst✝² : Field K\ninst✝¹ : Algebra K S\ninst✝ : Nontrivial S\npb : PowerBasis K S\nd_pos : 0 < pb.dim\nd_pos' : 0 < (minpoly K pb.gen).natDegree\n⊢ (Algebra.trace K S) pb.gen = -(minpoly K pb.gen).nextCoeff"
] | have d_pos' : 0 < (minpoly K pb.gen).natDegree := by simpa | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Trace.Basic | {
"line": 166,
"column": 4
} | {
"line": 171,
"column": 49
} | {
"line": 172,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : CommRing R\nL : Type u_5\ninst✝⁶ : Field L\nF : Type u_6\ninst✝⁵ : Field F\ninst✝⁴ : Algebra R L\ninst✝³ : Algebra L F\ninst✝² : Algebra R F\ninst✝¹ : IsScalarTower R L F\ninst✝ : FiniteDimensional L F\nx : F\nhx : IsIntegral R x\nhx' : IsIntegral L x\n⊢ IsIntegral R (finrank (↥L... | [] | refine (IsIntegral.multiset_sum ?_).nsmul _
intro y hy
rw [mem_roots_map (minpoly.ne_zero hx')] at hy
use minpoly R x, minpoly.monic hx
rw [← aeval_def] at hy ⊢
exact minpoly.aeval_of_isScalarTower R x y hy | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Trace.Basic | {
"line": 166,
"column": 4
} | {
"line": 171,
"column": 49
} | {
"line": 172,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : CommRing R\nL : Type u_5\ninst✝⁶ : Field L\nF : Type u_6\ninst✝⁵ : Field F\ninst✝⁴ : Algebra R L\ninst✝³ : Algebra L F\ninst✝² : Algebra R F\ninst✝¹ : IsScalarTower R L F\ninst✝ : FiniteDimensional L F\nx : F\nhx : IsIntegral R x\nhx' : IsIntegral L x\n⊢ IsIntegral R (finrank (↥L... | [] | refine (IsIntegral.multiset_sum ?_).nsmul _
intro y hy
rw [mem_roots_map (minpoly.ne_zero hx')] at hy
use minpoly R x, minpoly.monic hx
rw [← aeval_def] at hy ⊢
exact minpoly.aeval_of_isScalarTower R x y hy | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Basic | {
"line": 269,
"column": 49
} | {
"line": 272,
"column": 93
} | {
"line": 274,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝ : Field K\nx : 𝓞 K\n⊢ IsIntegral ℤ x",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"AddCommGroup.intIsScalarTower",
"Algebra.algebraMap",
"NumberField.instCommRingRingOfIntegers",
"congrArg",
"CommSemiring.toSemiring",
"Pol... | [] | by
obtain ⟨P, hPm, hP⟩ := x.isIntegral_coe
refine ⟨P, hPm, ?_⟩
rwa [IsScalarTower.algebraMap_eq (S := 𝓞 K), ← Polynomial.hom_eval₂, coe_eq_zero_iff] at hP | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Trace.Basic | {
"line": 231,
"column": 4
} | {
"line": 231,
"column": 48
} | {
"line": 233,
"column": 0
} | [
{
"pp": "case h\nK : Type u_4\nL : Type u_5\ninst✝⁴ : Field K\ninst✝³ : Field L\ninst✝² : Algebra K L\nE : Type u_7\ninst✝¹ : Field E\ninst✝ : Algebra K E\npb : PowerBasis K L\nhE : (Polynomial.map (algebraMap K E) (minpoly K pb.gen)).Splits\nhfx : IsSeparable K pb.gen\nthis✝ : DecidableEq E := Classical.decEq ... | [] | rw [PowerBasis.liftEquiv'_apply_coe, id_def] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Trace.Basic | {
"line": 259,
"column": 4
} | {
"line": 259,
"column": 53
} | {
"line": 260,
"column": 2
} | [
{
"pp": "K : Type u_4\nL : Type u_5\ninst✝⁷ : Field K\ninst✝⁶ : Field L\ninst✝⁵ : Algebra K L\nE : Type u_7\ninst✝⁴ : Field E\ninst✝³ : Algebra K E\ninst✝² : IsAlgClosed E\ninst✝¹ : FiniteDimensional K L\ninst✝ : Algebra.IsSeparable K L\nx : L\nhx : IsIntegral K x\npb : PowerBasis K ↥K⟮x⟯ := adjoin.powerBasis h... | [] | exact (sum_embeddings_eq_finrank_mul L E pb).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Trace.Basic | {
"line": 259,
"column": 4
} | {
"line": 259,
"column": 53
} | {
"line": 260,
"column": 2
} | [
{
"pp": "K : Type u_4\nL : Type u_5\ninst✝⁷ : Field K\ninst✝⁶ : Field L\ninst✝⁵ : Algebra K L\nE : Type u_7\ninst✝⁴ : Field E\ninst✝³ : Algebra K E\ninst✝² : IsAlgClosed E\ninst✝¹ : FiniteDimensional K L\ninst✝ : Algebra.IsSeparable K L\nx : L\nhx : IsIntegral K x\npb : PowerBasis K ↥K⟮x⟯ := adjoin.powerBasis h... | [] | exact (sum_embeddings_eq_finrank_mul L E pb).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Trace.Basic | {
"line": 259,
"column": 4
} | {
"line": 259,
"column": 53
} | {
"line": 260,
"column": 2
} | [
{
"pp": "K : Type u_4\nL : Type u_5\ninst✝⁷ : Field K\ninst✝⁶ : Field L\ninst✝⁵ : Algebra K L\nE : Type u_7\ninst✝⁴ : Field E\ninst✝³ : Algebra K E\ninst✝² : IsAlgClosed E\ninst✝¹ : FiniteDimensional K L\ninst✝ : Algebra.IsSeparable K L\nx : L\nhx : IsIntegral K x\npb : PowerBasis K ↥K⟮x⟯ := adjoin.powerBasis h... | [] | exact (sum_embeddings_eq_finrank_mul L E pb).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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