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.RingTheory.OrzechProperty | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 37
} | [
{
"pp": "R : Type u\ninst✝⁶ : Semiring R\ninst✝⁵ : OrzechProperty R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module.Finite R M\nN : Type w\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\ni f : N →ₗ[R] M\nhi : Injective ⇑i\nhf : Surjective ⇑f\nn : ℕ\ng : (Fin n → R) →ₗ[R] M\nhg : Surje... | haveI := Module.Finite.equiv j.symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 164,
"column": 37
} | {
"line": 164,
"column": 81
} | [
{
"pp": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na b : R\nh : a ∣ b\n⊢ gcd a b = a",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Dvd.dvd",
"CommRing.toNonUnitalCommRing",
"congrArg",
"CommSemiring.toSemiring",
"semigroupDvd",
"SemigroupWith... | rw [gcd_val, mod_eq_zero.2 h, gcd_zero_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 164,
"column": 37
} | {
"line": 164,
"column": 81
} | [
{
"pp": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na b : R\nh : a ∣ b\n⊢ gcd a b = a",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Dvd.dvd",
"CommRing.toNonUnitalCommRing",
"congrArg",
"CommSemiring.toSemiring",
"semigroupDvd",
"SemigroupWith... | rw [gcd_val, mod_eq_zero.2 h, gcd_zero_left] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 164,
"column": 37
} | {
"line": 164,
"column": 81
} | [
{
"pp": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na b : R\nh : a ∣ b\n⊢ gcd a b = a",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Dvd.dvd",
"CommRing.toNonUnitalCommRing",
"congrArg",
"CommSemiring.toSemiring",
"semigroupDvd",
"SemigroupWith... | rw [gcd_val, mod_eq_zero.2 h, gcd_zero_left] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Maps | {
"line": 127,
"column": 14
} | {
"line": 127,
"column": 68
} | [
{
"pp": "R : Type u\nS : Type v\nF : Type u_1\ninst✝⁵ : Semiring R\ninst✝⁴ : Semiring S\ninst✝³ : FunLike F R S\nf : F\nI J : Ideal R\nK L : Ideal S\nG : Type u_2\ninst✝² : FunLike G S R\ninst✝¹ : RingHomClass F R S\ninst✝ : K.IsTwoSided\na✝ b : R\nha : a✝ ∈ comap f K\n⊢ a✝ * b ∈ comap f K",
"usedConstants"... | by rw [mem_comap, map_mul]; exact mul_mem_right _ _ ha | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.Maps | {
"line": 209,
"column": 23
} | {
"line": 209,
"column": 32
} | [
{
"pp": "R : Type u\nS : Type v\nF : Type u_1\ninst✝³ : Semiring R\ninst✝² : Semiring S\ninst✝¹ : FunLike F R S\nf : F\ninst✝ : RingHomClass F R S\nI : Ideal S\nh : I = ⊤\n⊢ comap f ⊤ = ⊤",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"id",
"Submodule.instTop",... | comap_top | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Maps | {
"line": 367,
"column": 8
} | {
"line": 367,
"column": 28
} | [
{
"pp": "R : Type u\nS : Type v\nF : Type u_1\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : FunLike F R S\nf✝ : F\nI✝ J : Ideal R\nK L : Ideal S\nG : Type u_2\ninst✝³ : FunLike G S R\ninst✝² : RingHomClass F R S\nι : Sort u_3\nf : R →+* S\ninst✝¹ : RingHomSurjective f\nI : Ideal R\ninst✝ : I.IsTwoSided\na... | map_eq_submodule_map | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Maps | {
"line": 682,
"column": 4
} | {
"line": 682,
"column": 65
} | [
{
"pp": "case succ\nR : Type u\nS : Type v\nF : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : FunLike F R S\nrc : RingHomClass F R S\nf : F\nK : Ideal S\nn : ℕ\nn_ih : comap f K ^ n ≤ comap f (K ^ n)\n⊢ comap f K ^ n * comap f K ≤ comap f (K ^ n * K)",
"usedConstants": [
"Semirin... | exact (Ideal.mul_mono_left n_ih).trans (Ideal.le_comap_mul f) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Ideal.Operations | {
"line": 46,
"column": 10
} | {
"line": 46,
"column": 51
} | [
{
"pp": "case smul\nR' : Type u_1\nM' : Type u_2\ninst✝² : CommSemiring R'\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R' M'\ns : Set R'\nN : Submodule R' M'\nr : R'\nn : M'\nhn : n ∈ N\na✝¹ : R'\nc : R' →₀ R'\nhc : ↑c.support ⊆ s\na✝ : (c.sum fun mi r ↦ r • mi) • n ∈ s • N\n⊢ ∑ i ∈ c.support, (a✝¹ • c i • i) • ... | refine Submodule.sum_mem _ fun i hi => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Ideal.Maps | {
"line": 1088,
"column": 38
} | {
"line": 1088,
"column": 47
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : FunLike F R S\nrc : RingHomClass F R S\nf : F\nhf : Function.Surjective ⇑f\nI : Ideal R\nH : I.IsPrime\nhk : RingHom.ker f ≤ I\nh : I ⊔ comap f ⊥ = comap f ⊤\n⊢ ⊤ ≤ I",
"usedConstants": [
"Latti... | comap_top | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Operations | {
"line": 710,
"column": 69
} | {
"line": 721,
"column": 58
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nI J : Ideal R\n⊢ IsCoprime I J ↔ Codisjoint I J",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule",
"Codisjoint",
"Ideal.one_eq_top",
"Lattice.toSemilatticeSup",
"Semiring.toModule",
... | by
rw [IsCoprime, codisjoint_iff]
constructor
· rintro ⟨x, y, hxy⟩
rw [eq_top_iff_one]
apply (show x * I + y * J ≤ I ⊔ J from
sup_le (mul_le_left.trans le_sup_left) (mul_le_left.trans le_sup_right))
rw [hxy]
simp only [one_eq_top, Submodule.mem_top]
· intro h
refine ⟨1, 1, ?_⟩
simp... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1018,
"column": 8
} | {
"line": 1018,
"column": 90
} | [
{
"pp": "α : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : IsCancelMulZero α\ninst✝ : DecidableEq α\ngcd : α → α → α\ngcd_dvd_left : ∀ (a b : α), gcd a b ∣ a\ngcd_dvd_right : ∀ (a b : α), gcd a b ∣ b\ndvd_gcd : ∀ {a b c : α}, a ∣ c → a ∣ b → a ∣ gcd c b\na : α\nh : gcd a 0 * Classical.choose ⋯ = a * 0\na0 :... | exact associated_of_dvd_dvd (dvd_gcd (dvd_refl a) (dvd_zero a)) (gcd_dvd_left _ _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.Dimension.StrongRankCondition | {
"line": 547,
"column": 2
} | {
"line": 548,
"column": 65
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : StrongRankCondition R\ninst✝ : Module.Finite R M\nN : Submodule R M\n⊢ ↑(finrank R ↥N) = Module.rank R ↥N",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Nat.instMulZeroOneClass",
... | rw [finrank, Cardinal.cast_toNat_of_lt_aleph0]
exact lt_of_le_of_lt (Submodule.rank_le N) (rank_lt_aleph0 R M) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Dimension.StrongRankCondition | {
"line": 547,
"column": 2
} | {
"line": 548,
"column": 65
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : StrongRankCondition R\ninst✝ : Module.Finite R M\nN : Submodule R M\n⊢ ↑(finrank R ↥N) = Module.rank R ↥N",
"usedConstants": [
"Eq.mpr",
"Submodule",
"Nat.instMulZeroOneClass",
... | rw [finrank, Cardinal.cast_toNat_of_lt_aleph0]
exact lt_of_le_of_lt (Submodule.rank_le N) (rank_lt_aleph0 R M) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.AlgebraTower | {
"line": 217,
"column": 4
} | {
"line": 219,
"column": 23
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\nB : Type u_4\nC : Type u_5\nD : Type u_6\ninst✝⁸ : CommSemiring A\ninst✝⁷ : CommSemiring C\ninst✝⁶ : CommSemiring D\ninst✝⁵ : Algebra A C\ninst✝⁴ : Algebra A D\ninst✝³ : CommSemiring B\ninst✝² : Algebra A B\ninst✝¹ : Algebra B C\ninst✝ : IsScalarTower A B C\nf✝... | obtain rfl : f = fun x => g (algebraMap B C x) := by
ext x
exact (hg x).symm | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Order.Group.Finset | {
"line": 98,
"column": 4
} | {
"line": 98,
"column": 70
} | [
{
"pp": "case a\nι : Type u_1\nM : Type u_3\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : LinearOrder M\ninst✝³ : CanonicallyOrderedAdd M\ninst✝² : Sub M\ninst✝¹ : AddLeftReflectLE M\ninst✝ : OrderedSub M\ns : Finset ι\nf : ι → M\na : M\nhs : s.Nonempty\n⊢ (s.sup' hs fun i ↦ f i + a) ≤ s.sup' hs f + a",
"usedConstant... | exact Finset.sup'_le _ _ fun i hi ↦ by grw [← Finset.le_sup' _ hi] | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Order.Group.Finset | {
"line": 98,
"column": 4
} | {
"line": 98,
"column": 70
} | [
{
"pp": "case a\nι : Type u_1\nM : Type u_3\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : LinearOrder M\ninst✝³ : CanonicallyOrderedAdd M\ninst✝² : Sub M\ninst✝¹ : AddLeftReflectLE M\ninst✝ : OrderedSub M\ns : Finset ι\nf : ι → M\na : M\nhs : s.Nonempty\n⊢ (s.sup' hs fun i ↦ f i + a) ≤ s.sup' hs f + a",
"usedConstant... | exact Finset.sup'_le _ _ fun i hi ↦ by grw [← Finset.le_sup' _ hi] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Group.Finset | {
"line": 98,
"column": 4
} | {
"line": 98,
"column": 70
} | [
{
"pp": "case a\nι : Type u_1\nM : Type u_3\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : LinearOrder M\ninst✝³ : CanonicallyOrderedAdd M\ninst✝² : Sub M\ninst✝¹ : AddLeftReflectLE M\ninst✝ : OrderedSub M\ns : Finset ι\nf : ι → M\na : M\nhs : s.Nonempty\n⊢ (s.sup' hs fun i ↦ f i + a) ≤ s.sup' hs f + a",
"usedConstant... | exact Finset.sup'_le _ _ fun i hi ↦ by grw [← Finset.le_sup' _ hi] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Matrix.Mul | {
"line": 1119,
"column": 39
} | {
"line": 1119,
"column": 50
} | [
{
"pp": "n : Type u_3\nR : Type u_7\ninst✝² : Semiring R\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nM : Matrix n n R\nk l : ℕ\ni : n\nh : M ^ k *ᵥ Pi.single i 1 = 0\nh' : l = l - k + k\n⊢ M ^ (l - k) *ᵥ 0 = 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matrix",
"HSub.hSub",
"N... | mulVec_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Finsupp.Multiset | {
"line": 141,
"column": 6
} | {
"line": 144,
"column": 19
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝ : DecidableEq α\nf : α →₀ ℕ\na : α\n⊢ ((fun s ↦ { support := s.toFinset, toFun := fun a ↦ count a s, mem_support_toFun := ⋯ })\n ((fun f ↦ Finsupp.toMultiset f) f))\n a =\n f a",
"usedConstants": [
"Multiset.toFinset",
"Fins... | simp only [Finsupp.toMultiset_apply, Finsupp.sum, Multiset.count_sum',
Multiset.count_singleton, mul_boole, Finsupp.coe_mk, Finsupp.mem_support_iff,
Multiset.count_nsmul, Finset.sum_ite_eq, ite_not, ite_eq_right_iff]
exact Eq.symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finsupp.Multiset | {
"line": 141,
"column": 6
} | {
"line": 144,
"column": 19
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝ : DecidableEq α\nf : α →₀ ℕ\na : α\n⊢ ((fun s ↦ { support := s.toFinset, toFun := fun a ↦ count a s, mem_support_toFun := ⋯ })\n ((fun f ↦ Finsupp.toMultiset f) f))\n a =\n f a",
"usedConstants": [
"Multiset.toFinset",
"Fins... | simp only [Finsupp.toMultiset_apply, Finsupp.sum, Multiset.count_sum',
Multiset.count_singleton, mul_boole, Finsupp.coe_mk, Finsupp.mem_support_iff,
Multiset.count_nsmul, Finset.sum_ite_eq, ite_not, ite_eq_right_iff]
exact Eq.symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Dimension.Free | {
"line": 276,
"column": 2
} | {
"line": 276,
"column": 32
} | [
{
"pp": "R : Type u\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : StrongRankCondition R\nS : Type u_2\nT : Type u_3\ninst✝⁹ : Semiring S\ninst✝⁸ : Semiring T\ninst✝⁷ : Module R T\ninst✝⁶ : Module S T\ninst✝⁵ : Module R S\ninst✝⁴ : IsScalarTower R S T\ninst✝³ : IsScalarTower S T T\ninst✝² : FaithfulSMul S T\ninst✝¹ : Module.... | by_cases H : Module.Finite R S | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.RingTheory.TwoSidedIdeal.Kernel | {
"line": 46,
"column": 15
} | {
"line": 46,
"column": 27
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : NonUnitalNonAssocSemiring S\nF : Type u_3\ninst✝¹ : FunLike F R S\ninst✝ : NonUnitalRingHomClass F R S\nf : F\nx : R\n⊢ (ker f).ringCon x 0 ↔ f x = 0",
"usedConstants": [
"Eq.mpr",
"RingCon.instFunLikeForallProp",
... | ker_ringCon, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.TwoSidedIdeal.Lattice | {
"line": 130,
"column": 17
} | {
"line": 130,
"column": 50
} | [
{
"pp": "R : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nx✝ : TwoSidedIdeal R\n⊢ x✝ ≤ ⊤",
"usedConstants": [
"Eq.mpr",
"TwoSidedIdeal.ringCon_le_iff",
"congrArg",
"TwoSidedIdeal",
"PartialOrder.toPreorder",
"Preorder.toLE",
"RingCon.instCompleteLattice",
"RingC... | rw [ringCon_le_iff]; exact le_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.TwoSidedIdeal.Lattice | {
"line": 130,
"column": 17
} | {
"line": 130,
"column": 50
} | [
{
"pp": "R : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nx✝ : TwoSidedIdeal R\n⊢ x✝ ≤ ⊤",
"usedConstants": [
"Eq.mpr",
"TwoSidedIdeal.ringCon_le_iff",
"congrArg",
"TwoSidedIdeal",
"PartialOrder.toPreorder",
"Preorder.toLE",
"RingCon.instCompleteLattice",
"RingC... | rw [ringCon_le_iff]; exact le_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.NonUnitalSubalgebra | {
"line": 672,
"column": 53
} | {
"line": 672,
"column": 93
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\ns : Set A\nx✝⁴ x✝³ y✝ : A\nhx✝ : x✝³ ∈ adjoin R s\nhy✝ : y✝ ∈ adjoin R s\nhpx : x✝³ ∈ span R ↑(Subsemigroup.closure s)\nhpy : y✝ ∈ span ... | by simpa [mul_add] using add_mem hxz hyz | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Algebra.NonUnitalSubalgebra | {
"line": 1144,
"column": 52
} | {
"line": 1144,
"column": 59
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\ns : Set A\nr : R\na : A\nha : a ∈ s.centralizer\nx : A\nhx : x ∈ s\n⊢ r • (x * a) = r • (a * x)",
"usedConstants": [
"Eq.mpr",
... | ha x hx | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.DirectSum.Finsupp | {
"line": 78,
"column": 2
} | {
"line": 81,
"column": 43
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring S\ninst✝⁷ : Algebra R S\nM : Type u_3\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\ninst✝⁴ : Module S M\ninst✝³ : IsScalarTower R S M\nN : Type u_4\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type u_5\ninst✝ : DecidableEq ... | induction t with
| zero => simp
| tmul f n => simp only [finsuppLeft_apply_tmul_apply, rTensor_tmul, Finsupp.lapply_apply]
| add x y hx hy => simp [map_add, hx, hy] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.LinearAlgebra.DirectSum.Finsupp | {
"line": 78,
"column": 2
} | {
"line": 81,
"column": 43
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring S\ninst✝⁷ : Algebra R S\nM : Type u_3\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\ninst✝⁴ : Module S M\ninst✝³ : IsScalarTower R S M\nN : Type u_4\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type u_5\ninst✝ : DecidableEq ... | induction t with
| zero => simp
| tmul f n => simp only [finsuppLeft_apply_tmul_apply, rTensor_tmul, Finsupp.lapply_apply]
| add x y hx hy => simp [map_add, hx, hy] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.DirectSum.Finsupp | {
"line": 78,
"column": 2
} | {
"line": 81,
"column": 43
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁹ : CommSemiring R\ninst✝⁸ : Semiring S\ninst✝⁷ : Algebra R S\nM : Type u_3\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module R M\ninst✝⁴ : Module S M\ninst✝³ : IsScalarTower R S M\nN : Type u_4\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\nι : Type u_5\ninst✝ : DecidableEq ... | induction t with
| zero => simp
| tmul f n => simp only [finsuppLeft_apply_tmul_apply, rTensor_tmul, Finsupp.lapply_apply]
| add x y hx hy => simp [map_add, hx, hy] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Matrix.Block | {
"line": 685,
"column": 40
} | {
"line": 685,
"column": 74
} | [
{
"pp": "o : Type u_4\nm' : o → Type u_7\nn' : o → Type u_8\np' : o → Type u_9\nα : Type u_12\ninst✝³ : DecidableEq o\ninst✝² : NonUnitalNonAssocSemiring α\ninst✝¹ : (i : o) → Fintype (n' i)\ninst✝ : Fintype o\nM : (i : o) → Matrix (m' i) (n' i) α\nN : (i : o) → Matrix (n' i) (p' i) α\nk : o\ni : m' k\nk' : o\n... | by rw [dif_neg hj'.symm, zero_mul] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Matrix.Block | {
"line": 757,
"column": 6
} | {
"line": 757,
"column": 65
} | [
{
"pp": "case inl\no : Type u_4\nm' : o → Type u_7\nα : Type u_12\ninst✝² : Zero α\ninst✝¹ : DecidableEq o\ninst✝ : (i : o) → DecidableEq (m' i)\nd : (i : o) × m' i → α\nk : o\ni : m' k\n⊢ (diagonal d).blockDiag' k i i = diagonal (fun i ↦ d ⟨k, i⟩) i i",
"usedConstants": [
"Eq.mpr",
"congrArg",
... | rw [blockDiag'_apply, diagonal_apply_eq, diagonal_apply_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Matrix.Block | {
"line": 757,
"column": 6
} | {
"line": 757,
"column": 65
} | [
{
"pp": "case inl\no : Type u_4\nm' : o → Type u_7\nα : Type u_12\ninst✝² : Zero α\ninst✝¹ : DecidableEq o\ninst✝ : (i : o) → DecidableEq (m' i)\nd : (i : o) × m' i → α\nk : o\ni : m' k\n⊢ (diagonal d).blockDiag' k i i = diagonal (fun i ↦ d ⟨k, i⟩) i i",
"usedConstants": [
"Eq.mpr",
"congrArg",
... | rw [blockDiag'_apply, diagonal_apply_eq, diagonal_apply_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Matrix.Block | {
"line": 757,
"column": 6
} | {
"line": 757,
"column": 65
} | [
{
"pp": "case inl\no : Type u_4\nm' : o → Type u_7\nα : Type u_12\ninst✝² : Zero α\ninst✝¹ : DecidableEq o\ninst✝ : (i : o) → DecidableEq (m' i)\nd : (i : o) × m' i → α\nk : o\ni : m' k\n⊢ (diagonal d).blockDiag' k i i = diagonal (fun i ↦ d ⟨k, i⟩) i i",
"usedConstants": [
"Eq.mpr",
"congrArg",
... | rw [blockDiag'_apply, diagonal_apply_eq, diagonal_apply_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.FreeModule.PID | {
"line": 240,
"column": 2
} | {
"line": 240,
"column": 59
} | [
{
"pp": "case neg.intro\nι : Type u_1\nR : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : IsPrincipalIdealRing R\ninst✝³ : IsDomain R\ninst✝² : Finite ι\nO : Type u_4\ninst✝¹ : AddCommGroup O\ninst✝ : Module R O\nM N : Submodule R O\nb'M : Basis ι R ↥M\nN_bot : N ≠ ⊥\nN_le_M : N ≤ M\nthis : ∃ ϕ, ∀ (ψ : ↥M →ₗ[R] R), ¬ϕ... | refine ⟨y'_ortho_M', ay'_ortho_N', fun n' bN' ↦ ⟨?_, ?_⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.TensorProduct.Tower | {
"line": 471,
"column": 44
} | {
"line": 471,
"column": 55
} | [
{
"pp": "R : Type uR\nA : Type uA\nN : Type uN\nQ : Type uQ\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Algebra R A\ninst✝³ : AddCommMonoid N\ninst✝² : Module R N\ninst✝¹ : AddCommMonoid Q\ninst✝ : Module R Q\nn : N\nq : Q\na b : A\n⊢ (distribBaseChange R A N Q).toAddEquiv ((a * b) ⊗ₜ[R] (n ⊗ₜ[R... | smul_tmul', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Field.Subfield.Defs | {
"line": 148,
"column": 54
} | {
"line": 148,
"column": 74
} | [
{
"pp": "case mk.mk.e_toSubring\nK : Type u\nL : Type v\nM : Type w\ninst✝² : DivisionRing K\ninst✝¹ : DivisionRing L\ninst✝ : DivisionRing M\ntoSubring✝¹ : Subring K\ninv_mem'✝¹ : ∀ (x : K), x ∈ toSubring✝¹.carrier → x⁻¹ ∈ toSubring✝¹.carrier\ntoSubring✝ : Subring K\ninv_mem'✝ : ∀ (x : K), x ∈ toSubring✝.carri... | exact SetLike.ext' h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.OreLocalization.Basic | {
"line": 61,
"column": 33
} | {
"line": 62,
"column": 58
} | [
{
"pp": "R : Type u_1\ninst✝¹ : MonoidWithZero R\nS : Submonoid R\ninst✝ : OreSet S\n⊢ Nontrivial (OreLocalization S R) ↔ ¬0 ∈ S",
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"congrArg",
"OreLocalization",
"Iff.rfl",
"Membership.mem",
"OreLocalization.subsingleton_if... | by
rw [← not_subsingleton_iff_nontrivial, subsingleton_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Field.Subfield.Basic | {
"line": 576,
"column": 14
} | {
"line": 579,
"column": 38
} | [
{
"pp": "K✝ : Type u\nL : Type v\nM : Type w\ninst✝³ : DivisionRing K✝\ninst✝² : DivisionRing L\ninst✝¹ : DivisionRing M\ns✝¹ : Set K✝\nK : Type u\ninst✝ : Field K\ns✝ : Subfield K\ns : Set K\n⊢ ∀ {a b : K},\n a ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z} →\n b ∈ {z | ∃ x ∈ Subri... | by
rintro _ _ ⟨nx, hnx, dx, hdx, rfl⟩ ⟨ny, hny, dy, hdy, rfl⟩
exact ⟨nx * ny, Subring.mul_mem _ hnx hny, dx * dy, Subring.mul_mem _ hdx hdy,
(div_mul_div_comm _ _ _ _).symm⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.OreLocalization.Ring | {
"line": 103,
"column": 2
} | {
"line": 103,
"column": 63
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semiring R\nS : Submonoid R\ninst✝² : OreSet S\nX : Type u_2\ninst✝¹ : AddCommMonoid X\ninst✝ : Module R X\nn : ℕ\nx : OreLocalization S X\ninst : Module ℕ (OreLocalization S X) := instModuleOfIsScalarTower\n⊢ n • x = n • x",
"usedConstants": [
"Inhabited.default",
... | exact congr($(AddCommMonoid.uniqueNatModule.2 inst).smul n x) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.OreLocalization.Ring | {
"line": 195,
"column": 17
} | {
"line": 200,
"column": 43
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Ring R\nS : Submonoid R\ninst✝ : OreSet S\nhS : S ≤ nonZeroDivisorsLeft R\nr₁ r₂ : R\nh : numeratorHom r₁ = numeratorHom r₂\n⊢ r₁ = r₂",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"OreLocalization.oreDiv_eq_iff",
"instHSMul",
"MonoidHom.instFun... | by
rw [numeratorHom_apply, numeratorHom_apply, oreDiv_eq_iff] at h
rcases h with ⟨u, v, h₁, h₂⟩
simp only [S.coe_one, mul_one, Submonoid.smul_def, smul_eq_mul] at h₁ h₂
rw [← h₂, ← sub_eq_zero, ← mul_sub] at h₁
exact (sub_eq_zero.mp (hS u.2 _ h₁)).symm | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 1028,
"column": 2
} | {
"line": 1028,
"column": 65
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Semiring T\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra S T\ninst✝⁵ : Algebra R T\ninst✝⁴ : IsScalarTower R S T\nm : Type u_4\nn : Type u_5\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : DecidableEq m\ni... | rw [smulTower_leftMulMatrix_algebraMap, blockDiagonal_apply_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 1028,
"column": 2
} | {
"line": 1028,
"column": 65
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Semiring T\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra S T\ninst✝⁵ : Algebra R T\ninst✝⁴ : IsScalarTower R S T\nm : Type u_4\nn : Type u_5\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : DecidableEq m\ni... | rw [smulTower_leftMulMatrix_algebraMap, blockDiagonal_apply_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 1028,
"column": 2
} | {
"line": 1028,
"column": 65
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝¹⁰ : CommSemiring R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Semiring T\ninst✝⁷ : Algebra R S\ninst✝⁶ : Algebra S T\ninst✝⁵ : Algebra R T\ninst✝⁴ : IsScalarTower R S T\nm : Type u_4\nn : Type u_5\ninst✝³ : Fintype m\ninst✝² : Fintype n\ninst✝¹ : DecidableEq m\ni... | rw [smulTower_leftMulMatrix_algebraMap, blockDiagonal_apply_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.Defs | {
"line": 933,
"column": 2
} | {
"line": 933,
"column": 68
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nx₁ x₂ : R\ny₁ y₂ : ↥M\n⊢ mk' S (x₁ * ↑y₂ - x₂ * ↑y₁) (y₁ * y₂) = mk' S x₁ y₁ - mk' S x₂ y₂",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
... | rw [sub_eq_add_neg, sub_eq_add_neg, ← mk'_neg, ← mk'_add, neg_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Localization.Defs | {
"line": 933,
"column": 2
} | {
"line": 933,
"column": 68
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nx₁ x₂ : R\ny₁ y₂ : ↥M\n⊢ mk' S (x₁ * ↑y₂ - x₂ * ↑y₁) (y₁ * y₂) = mk' S x₁ y₁ - mk' S x₂ y₂",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
... | rw [sub_eq_add_neg, sub_eq_add_neg, ← mk'_neg, ← mk'_add, neg_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Localization.Defs | {
"line": 933,
"column": 2
} | {
"line": 933,
"column": 68
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommRing S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nx₁ x₂ : R\ny₁ y₂ : ↥M\n⊢ mk' S (x₁ * ↑y₂ - x₂ * ↑y₁) (y₁ * y₂) = mk' S x₁ y₁ - mk' S x₂ y₂",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
... | rw [sub_eq_add_neg, sub_eq_add_neg, ← mk'_neg, ← mk'_add, neg_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.FractionRing | {
"line": 174,
"column": 4
} | {
"line": 174,
"column": 46
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\n⊢ R⁰ ≤ IsUnit.submonoid R ∧ IsUnit.submonoid R ≤ R⁰ ↔ R⁰ ≤ IsUnit.submonoid R",
"usedConstants": [
"Eq.mpr",
"Monoid.toMulOneClass",
"congrArg",
"CommSemiring.toSemiring",
"isUnit_le_nonZeroDivisors",
"PartialOrder.toPreorder",
... | and_iff_left (isUnit_le_nonZeroDivisors R) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Star.Center | {
"line": 44,
"column": 50
} | {
"line": 44,
"column": 67
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Mul R\ninst✝ : StarMul R\ns : Set R\nhcomm : ∀ x ∈ s, ∀ y ∈ s, y * x = x * y\nhcomm_star : ∀ x ∈ s, ∀ y ∈ s, y * star x = star x * y\n⊢ s ∪ star s ⊆ s.centralizer ∩ star s.centralizer",
"usedConstants": [
"Eq.mpr",
"Set.instUnion",
"_private.Mathlib.Algebra.... | union_subset_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Algebra.Spectrum.Basic | {
"line": 293,
"column": 23
} | {
"line": 293,
"column": 38
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nr x : R\n⊢ x ∈ (fun x ↦ r + x) '' σ a ↔ x ∈ σ (↑ₐ r + a)",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"Algebra.algebraMap",
"spectrum",
... | image_add_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.TensorProduct.Basic | {
"line": 628,
"column": 4
} | {
"line": 652,
"column": 81
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\nM : Type u_4\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Module R M\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Module A M\ninst✝⁵ : Module B M\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\ninst✝² : IsScalarTower R A M\ninst✝¹ : IsScal... | refine TensorProduct.induction_on x ?_ ?_ ?_ <;> refine TensorProduct.induction_on y ?_ ?_ ?_
· simp only [(· • ·), mul_zero, map_zero, LinearMap.zero_apply]
· intro a b
simp only [(· • ·), zero_mul, map_zero, LinearMap.zero_apply]
· intro z w _ _
simp only [(· • ·), zero_mul, map_zero, LinearMa... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.TensorProduct.Basic | {
"line": 628,
"column": 4
} | {
"line": 652,
"column": 81
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\nM : Type u_4\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : AddCommMonoid M\ninst✝⁹ : Module R M\ninst✝⁸ : Semiring A\ninst✝⁷ : Semiring B\ninst✝⁶ : Module A M\ninst✝⁵ : Module B M\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\ninst✝² : IsScalarTower R A M\ninst✝¹ : IsScal... | refine TensorProduct.induction_on x ?_ ?_ ?_ <;> refine TensorProduct.induction_on y ?_ ?_ ?_
· simp only [(· • ·), mul_zero, map_zero, LinearMap.zero_apply]
· intro a b
simp only [(· • ·), zero_mul, map_zero, LinearMap.zero_apply]
· intro z w _ _
simp only [(· • ·), zero_mul, map_zero, LinearMa... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 160,
"column": 12
} | {
"line": 162,
"column": 16
} | [
{
"pp": "case h.e'_2\nR : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : (PreQuasiregular A)ˣ\n⊢ (1 + ↑(PreQuasiregular.equiv.symm ↑x)) * (1 + ↑(PreQuasiregular.equiv.symm ↑x⁻¹)) =\n 1 + ↑((↑x⁻... | simp only [mul_one, PreQuasiregular.equiv_symm_apply, one_mul, mul_add,
add_mul, inr_add, inr_mul]
abel | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 160,
"column": 12
} | {
"line": 162,
"column": 16
} | [
{
"pp": "case h.e'_2\nR : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : (PreQuasiregular A)ˣ\n⊢ (1 + ↑(PreQuasiregular.equiv.symm ↑x)) * (1 + ↑(PreQuasiregular.equiv.symm ↑x⁻¹)) =\n 1 + ↑((↑x⁻... | simp only [mul_one, PreQuasiregular.equiv_symm_apply, one_mul, mul_add,
add_mul, inr_add, inr_mul]
abel | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 166,
"column": 12
} | {
"line": 168,
"column": 16
} | [
{
"pp": "case h.e'_2\nR : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : (PreQuasiregular A)ˣ\n⊢ (1 + ↑(PreQuasiregular.equiv.symm ↑x⁻¹)) * (1 + ↑(PreQuasiregular.equiv.symm ↑x)) =\n 1 + ↑((↑x)... | simp only [mul_one, PreQuasiregular.equiv_symm_apply, one_mul, mul_add,
add_mul, inr_add, inr_mul]
abel | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 166,
"column": 12
} | {
"line": 168,
"column": 16
} | [
{
"pp": "case h.e'_2\nR : Type u_1\nA : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : NonUnitalSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nx : (PreQuasiregular A)ˣ\n⊢ (1 + ↑(PreQuasiregular.equiv.symm ↑x⁻¹)) * (1 + ↑(PreQuasiregular.equiv.symm ↑x)) =\n 1 + ↑((↑x)... | simp only [mul_one, PreQuasiregular.equiv_symm_apply, one_mul, mul_add,
add_mul, inr_add, inr_mul]
abel | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 371,
"column": 65
} | {
"line": 373,
"column": 41
} | [
{
"pp": "R : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝⁸ : Semifield R\ninst✝⁷ : Field S\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : Algebra R S\ninst✝⁴ : Module S A\ninst✝³ : IsScalarTower S A A\ninst✝² : SMulCommClass S A A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R S A\na : A\n⊢ quasispectrum R ↑a = quasispectru... | by
rw [quasispectrum_eq_spectrum_union_zero, quasispectrum_eq_spectrum_inr' R S]
simpa using zero_mem_spectrum_inr _ _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 385,
"column": 32
} | {
"line": 385,
"column": 54
} | [
{
"pp": "case h.e'_4\nR : Type u_3\nA : Type u_4\ninst✝⁴ : CommRing R\ninst✝³ : NonUnitalRing A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\na b : A\n⊢ {r | IsUnit r}ᶜ = quasispectrum R (b * a) ∩ {r | IsUnit r}ᶜ",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
... | Set.inter_eq_right.mpr | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 339,
"column": 2
} | {
"line": 339,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nx : R\n⊢ star x < 1 ↔ x < 1",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder... | simpa using star_lt_star_iff (x := x) (y := 1) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Order.Star.Basic | {
"line": 339,
"column": 2
} | {
"line": 339,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nx : R\n⊢ star x < 1 ↔ x < 1",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder... | simpa using star_lt_star_iff (x := x) (y := 1) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Star.Basic | {
"line": 339,
"column": 2
} | {
"line": 339,
"column": 48
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nx : R\n⊢ star x < 1 ↔ x < 1",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"Preorder.toLT",
"congrArg",
"PartialOrder.toPreorder... | simpa using star_lt_star_iff (x := x) (y := 1) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Star.Basic | {
"line": 420,
"column": 2
} | {
"line": 420,
"column": 28
} | [
{
"pp": "R : Type u_4\nS : Type u_5\ninst✝⁷ : NonUnitalSemiring R\ninst✝⁶ : PartialOrder R\ninst✝⁵ : StarRing R\ninst✝⁴ : StarOrderedRing R\ninst✝³ : NonUnitalSemiring S\ninst✝² : PartialOrder S\ninst✝¹ : StarRing S\ninst✝ : StarOrderedRing S\nf : R →⋆ₙ+* S\nx y : R\nhxy : ∃ p ∈ AddSubmonoid.closure (range fun ... | obtain ⟨p, hp, rfl⟩ := hxy | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Algebra.Subalgebra.Centralizer | {
"line": 44,
"column": 13
} | {
"line": 44,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\nA : Type u_2\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T K : Subalgebra R A\n⊢ K ≤ centralizer R ↑(S ⊔ T) ↔ K ≤ centralizer R ↑S ⊓ centralizer R ↑T",
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
"Lattice.toSemilatticeSup",
... | le_centralizer_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Algebra.Subalgebra.Centralizer | {
"line": 49,
"column": 13
} | {
"line": 49,
"column": 32
} | [
{
"pp": "R : Type u_1\ninst✝² : CommSemiring R\nA : Type u_2\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nι : Sort u_3\nS : ι → Subalgebra R A\nK : Subalgebra R A\n⊢ K ≤ centralizer R ↑(⨆ i, S i) ↔ K ≤ ⨅ i, centralizer R ↑(S i)",
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
"iInf",... | le_centralizer_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.TensorProduct.Maps | {
"line": 349,
"column": 8
} | {
"line": 349,
"column": 19
} | [
{
"pp": "R✝¹ : Type uR\nR' : Type u_1\nS✝ : Type uS\nT✝ : Type u_2\nA✝ : Type uA\nB✝ : Type uB\nC : Type uC\nD : Type uD\nE : Type uE\nF : Type uF\ninst✝⁴⁴ : CommSemiring R✝¹\ninst✝⁴³ : CommSemiring S✝\ninst✝⁴² : Algebra R✝¹ S✝\ninst✝⁴¹ : Semiring A✝\ninst✝⁴⁰ : Algebra R✝¹ A✝\ninst✝³⁹ : Algebra S✝ A✝\ninst✝³⁸ :... | smul_tmul', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Dimension.Finite | {
"line": 177,
"column": 4
} | {
"line": 177,
"column": 71
} | [
{
"pp": "case hbc\nR : Type u\nM : Type v\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : StrongRankCondition R\nι : Type w\ninst✝ : Module.Finite R M\nv : ι → M\nh : LinearIndependent R v\n⊢ lift.{w, v} (Module.rank R M) < lift.{v, w} ℵ₀",
"usedConstants": [
"Eq.mpr",
... | rw [← finrank_eq_rank, Cardinal.lift_aleph0, Cardinal.lift_natCast] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition | {
"line": 218,
"column": 4
} | {
"line": 218,
"column": 59
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝⁵ : Ring K\ninst✝⁴ : StrongRankCondition K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\ninst✝¹ : Free K V\ninst✝ : Module.Finite K V\nthis : Nontrivial K\ns : Type v\nhs : Basis s K V\nt : Finset V\nht : span K ↑t = ⊤\n⊢ Finite s",
"usedConstants": [
"AddCommGro... | exact basis_finite_of_finite_spans t.finite_toSet ht hs | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Star.NonUnitalSubalgebra | {
"line": 1208,
"column": 49
} | {
"line": 1208,
"column": 64
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝⁵ : CommSemiring R\ninst✝⁴ : NonUnitalSemiring A\ninst✝³ : StarRing A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\ns : Set A\n⊢ (↑(centralizer R s) ∪ ↑(centralizer R s)).centralizer = (s ∪ star s).centralizer.centralizer",
"usedConsta... | Set.union_self, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.FiniteDimensional.Basic | {
"line": 600,
"column": 53
} | {
"line": 602,
"column": 33
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nv : V\nnz : v ≠ 0\n⊢ finrank K V = 1 ↔ ∀ (w : V), ∃ c, c • v = w",
"usedConstants": [
"Eq.mpr",
"Submodule",
"instHSMul",
"congrArg",
"DistribMulAction.toDistribSMul",
"... | by
rw [finrank_eq_one_iff_of_nonzero v nz]
apply span_singleton_eq_top_iff | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.FreeAlgebra | {
"line": 338,
"column": 8
} | {
"line": 338,
"column": 42
} | [
{
"pp": "case right_distrib\nR : Type u_1\nX : Type u_2\ninst✝² : CommSemiring R\nA : Type u_3\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nf : X → A\na✝¹ : FreeAlgebra R X\na b a✝ b✝ c✝ : Pre R X\n⊢ liftFun R X f ((a✝ + b✝) * c✝) = liftFun R X f (a✝ * c✝ + b✝ * c✝)",
"usedConstants": [
"HMul.hMul",
... | change (_ + _) * _ = _ * _ + _ * _ | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Algebra.MonoidAlgebra.MapDomain | {
"line": 118,
"column": 9
} | {
"line": 118,
"column": 32
} | [
{
"pp": "case h.hfg\nR : Type u_3\nS : Type u_4\nM : Type u_6\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+ S\nhe : Injective ⇑f\nx✝ : R[M]\nm✝ : M\n⊢ (map f x✝) m✝ = ((⇑coeffEquiv.symm ∘ mapRange ⇑f ⋯ ∘ ⇑coeffEquiv) x✝) m✝",
"usedConstants": [
"Finsupp.instFunLike",
"NonAssocSemiring.toAdd... | simp [ofCoeff_mapRange] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.MonoidAlgebra.MapDomain | {
"line": 125,
"column": 9
} | {
"line": 125,
"column": 32
} | [
{
"pp": "case h.hfg\nR : Type u_3\nS : Type u_4\nM : Type u_6\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+ S\nhe : Surjective ⇑f\nx✝ : R[M]\nm✝ : M\n⊢ (map f x✝) m✝ = ((⇑coeffEquiv.symm ∘ mapRange ⇑f ⋯ ∘ ⇑coeffEquiv) x✝) m✝",
"usedConstants": [
"Finsupp.instFunLike",
"NonAssocSemiring.toAd... | simp [ofCoeff_mapRange] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.FreeAlgebra | {
"line": 423,
"column": 2
} | {
"line": 423,
"column": 18
} | [
{
"pp": "R : Type u_1\nX : Type u_2\ninst✝² : CommSemiring R\nA : Type u_3\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nf : X → A\nx : X\n⊢ ((lift R) f) (ι R x) = f x",
"usedConstants": [
"Eq.mpr",
"FreeAlgebra.Pre.of",
"FreeAlgebra.ι",
"Equiv.instEquivLike",
"FreeAlgebra.instSem... | rw [ι_def, lift] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.MonoidAlgebra.MapDomain | {
"line": 255,
"column": 56
} | {
"line": 255,
"column": 74
} | [
{
"pp": "R : Type u_3\nS : Type u_4\nM : Type u_6\ninst✝² : Semiring R\ninst✝¹ : Semiring S\ninst✝ : Mul M\ne : R ≃+ S\nr : R\nm : M\n⊢ (mapAddEquiv M e) (single m r) = single m (e r)",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"AddMonoidHomClass.toAddMonoidHom",
"congr... | simp [mapAddEquiv] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.MonoidAlgebra.MapDomain | {
"line": 255,
"column": 56
} | {
"line": 255,
"column": 74
} | [
{
"pp": "R : Type u_3\nS : Type u_4\nM : Type u_6\ninst✝² : Semiring R\ninst✝¹ : Semiring S\ninst✝ : Mul M\ne : R ≃+ S\nr : R\nm : M\n⊢ (mapAddEquiv M e) (single m r) = single m (e r)",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"AddMonoidHomClass.toAddMonoidHom",
"congr... | simp [mapAddEquiv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MonoidAlgebra.MapDomain | {
"line": 255,
"column": 56
} | {
"line": 255,
"column": 74
} | [
{
"pp": "R : Type u_3\nS : Type u_4\nM : Type u_6\ninst✝² : Semiring R\ninst✝¹ : Semiring S\ninst✝ : Mul M\ne : R ≃+ S\nr : R\nm : M\n⊢ (mapAddEquiv M e) (single m r) = single m (e r)",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"AddMonoidHomClass.toAddMonoidHom",
"congr... | simp [mapAddEquiv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MonoidAlgebra.Basic | {
"line": 344,
"column": 25
} | {
"line": 344,
"column": 40
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\nA : Type u_4\nB : Type u_5\nC : Type u_6\nM : Type u_7\nN : Type u_8\nO : Type u_9\ninst✝⁷ : CommSemiring R\ninst✝⁶ : Semiring A\ninst✝⁵ : Semiring B\ninst✝⁴ : Algebra R A\ninst✝³ : Algebra R B\ninst✝² : Monoid M\ninst✝¹ : Monoid N\ninst✝ : Monoid O\nx✝¹ x✝ : M... | MulAut.mul_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Prime.Defs | {
"line": 221,
"column": 2
} | {
"line": 221,
"column": 26
} | [
{
"pp": "⊢ minFac 1 = 1",
"usedConstants": [
"False",
"Dvd.dvd",
"HMul.hMul",
"congrArg",
"Nat.decidable_dvd",
"Nat.minFac",
"instMulNat",
"instOfNatNat",
"ite_cond_eq_true",
"Bool.true",
"instHAdd",
"Nat.minFacAux",
"Nat.instDvd"... | simp [minFac, minFacAux] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Nat.Prime.Defs | {
"line": 221,
"column": 2
} | {
"line": 221,
"column": 26
} | [
{
"pp": "⊢ minFac 1 = 1",
"usedConstants": [
"False",
"Dvd.dvd",
"HMul.hMul",
"congrArg",
"Nat.decidable_dvd",
"Nat.minFac",
"instMulNat",
"instOfNatNat",
"ite_cond_eq_true",
"Bool.true",
"instHAdd",
"Nat.minFacAux",
"Nat.instDvd"... | simp [minFac, minFacAux] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Prime.Defs | {
"line": 221,
"column": 2
} | {
"line": 221,
"column": 26
} | [
{
"pp": "⊢ minFac 1 = 1",
"usedConstants": [
"False",
"Dvd.dvd",
"HMul.hMul",
"congrArg",
"Nat.decidable_dvd",
"Nat.minFac",
"instMulNat",
"instOfNatNat",
"ite_cond_eq_true",
"Bool.true",
"instHAdd",
"Nat.minFacAux",
"Nat.instDvd"... | simp [minFac, minFacAux] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 592,
"column": 2
} | {
"line": 592,
"column": 81
} | [
{
"pp": "case inr\nR : Type u_1\nS : Type u_2\ninst✝⁵ : Ring R\ninst✝⁴ : Ring S\nσ : R →+* S\nE : Type u_4\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\nF : Type u_5\ninst✝¹ : AddCommGroup F\ninst✝ : Module S F\nc : Set (E →ₛₗ.[σ] F)\nhc : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) c\ncne : c.Nonempty\nhdir : DirectedOn... | set f : ↥(sSup (domain '' c)) → F := fun x => (P x).val.val ⟨x, (P x).property⟩ | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.Data.Nat.Prime.Defs | {
"line": 378,
"column": 4
} | {
"line": 378,
"column": 28
} | [
{
"pp": "case mpr\n⊢ minFac 1 = 1",
"usedConstants": [
"False",
"Dvd.dvd",
"HMul.hMul",
"congrArg",
"Nat.decidable_dvd",
"Nat.minFac",
"instMulNat",
"instOfNatNat",
"ite_cond_eq_true",
"Bool.true",
"instHAdd",
"Nat.minFacAux",
"Na... | simp [minFac, minFacAux] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 607,
"column": 12
} | {
"line": 607,
"column": 15
} | [
{
"pp": "case h\nR : Type u_1\nS : Type u_2\ninst✝⁵ : Ring R\ninst✝⁴ : Ring S\nσ : R →+* S\nE : Type u_4\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\nF : Type u_5\ninst✝¹ : AddCommGroup F\ninst✝ : Module S F\nc : Set (E →ₛₗ.[σ] F)\nhc : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) c\ncne : c.Nonempty\nhdir : DirectedOn (... | hpc | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 751,
"column": 2
} | {
"line": 752,
"column": 72
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁴ : Ring R\nE : Type u_4\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\nF : Type u_5\ninst✝¹ : AddCommGroup F\ninst✝ : Module R F\nf : E →ₗ.[R] F\nx : E\n⊢ x ∈ Submodule.map (LinearMap.fst R E F) f.graph ↔ x ∈ f.domain",
"usedConstants": [
"_private.Mathlib.LinearAl... | simp only [Submodule.mem_map, mem_graph_iff, Subtype.exists, exists_and_left, exists_eq_left,
LinearMap.fst_apply, Prod.exists, exists_and_right, exists_eq_right] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.CharP.Defs | {
"line": 278,
"column": 58
} | {
"line": 278,
"column": 77
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NonAssocSemiring R\ninst✝ : CharP R 1\nthis : ∀ (r : R), r = 0\na b : R\n⊢ a = b",
"usedConstants": [
"Eq.mpr",
"congrArg",
"NonUnitalNonAssocSemiring.toMulZeroClass",
"id",
"NonAssocSemiring.toNonUnitalNonAssocSemiring",
"Zero.toOfNat0",
... | rw [this a, this b] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.CharP.Defs | {
"line": 278,
"column": 58
} | {
"line": 278,
"column": 77
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NonAssocSemiring R\ninst✝ : CharP R 1\nthis : ∀ (r : R), r = 0\na b : R\n⊢ a = b",
"usedConstants": [
"Eq.mpr",
"congrArg",
"NonUnitalNonAssocSemiring.toMulZeroClass",
"id",
"NonAssocSemiring.toNonUnitalNonAssocSemiring",
"Zero.toOfNat0",
... | rw [this a, this b] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.CharP.Defs | {
"line": 278,
"column": 58
} | {
"line": 278,
"column": 77
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NonAssocSemiring R\ninst✝ : CharP R 1\nthis : ∀ (r : R), r = 0\na b : R\n⊢ a = b",
"usedConstants": [
"Eq.mpr",
"congrArg",
"NonUnitalNonAssocSemiring.toMulZeroClass",
"id",
"NonAssocSemiring.toNonUnitalNonAssocSemiring",
"Zero.toOfNat0",
... | rw [this a, this b] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Coeff | {
"line": 208,
"column": 2
} | {
"line": 211,
"column": 58
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nk m n : ℕ\nhkm : k < m\nhmn : m < n\nx y z : R\nhx : x ≠ 0\nhy : y ≠ 0\nhz : z ≠ 0\n⊢ {k, m, n} ⊆ (C x * X ^ k + C y * X ^ m + C z * X ^ n).support",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Fals... | simp_rw [insert_subset_iff, singleton_subset_iff, mem_support_iff, coeff_add, coeff_C_mul,
coeff_X_pow_self, mul_one, coeff_X_pow, if_neg hkm.ne, if_neg hkm.ne', if_neg hmn.ne,
if_neg hmn.ne', if_neg (hkm.trans hmn).ne, if_neg (hkm.trans hmn).ne', mul_zero, add_zero,
zero_add, Ne, hx, hy, hz, not_false_eq_t... | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 19
} | [
{
"pp": "case neg\nR : Type u\na : R\ninst✝ : Semiring R\nh : ¬a = 0\n⊢ (C a).degree ≤ 0",
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"Polynomial.C",
"le_refl",
"Nat.instMulZeroClass",
"WithBot",
"congrArg",
"WithBot.zero",
"Polynomial.degree_C... | rw [degree_C h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 19
} | [
{
"pp": "case neg\nR : Type u\na : R\ninst✝ : Semiring R\nh : ¬a = 0\n⊢ (C a).degree ≤ 0",
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"Polynomial.C",
"le_refl",
"Nat.instMulZeroClass",
"WithBot",
"congrArg",
"WithBot.zero",
"Polynomial.degree_C... | rw [degree_C h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 19
} | [
{
"pp": "case neg\nR : Type u\na : R\ninst✝ : Semiring R\nh : ¬a = 0\n⊢ (C a).degree ≤ 0",
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"Polynomial.C",
"le_refl",
"Nat.instMulZeroClass",
"WithBot",
"congrArg",
"WithBot.zero",
"Polynomial.degree_C... | rw [degree_C h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Degree.Support | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 47
} | [
{
"pp": "case pos\nR : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : AddCommMonoid S\nf : ℕ → R → S\nhf : ∀ (i : ℕ), f i 0 = 0\nn : ℕ\np : R[X]\nhn : p.degree < ↑n\nhp : p = 0\n⊢ ∑ i, f (↑i) (p.coeff ↑i) = p.sum f",
"usedConstants": [
"Eq.mpr",
"Finset.univ",
"congrArg",
"AddMonoi... | rw [hp, sum_zero_index, Finset.sum_eq_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.MonoidAlgebra.Degree | {
"line": 153,
"column": 78
} | {
"line": 157,
"column": 19
} | [
{
"pp": "R : Type u_1\nA : Type u_3\nT : Type u_4\ninst✝⁶ : SemilatticeInf T\ninst✝⁵ : OrderTop T\ninst✝⁴ : Semiring R\ninst✝³ : AddMonoid A\ninst✝² : AddMonoid T\ninst✝¹ : AddLeftMono T\ninst✝ : AddRightMono T\ndegt : A → T\ndegt0 : 0 ≤ degt 0\ndegtm : ∀ (a b : A), degt a + degt b ≤ degt (a + b)\nn : ℕ\nf : R[... | by
refine OrderDual.ofDual_le_ofDual.mpr <| sup_support_pow_le (OrderDual.ofDual_le_ofDual.mp ?_)
(fun a b => OrderDual.ofDual_le_ofDual.mp ?_) n f
· exact degt0
· exact degtm _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MonoidAlgebra.Degree | {
"line": 460,
"column": 2
} | {
"line": 460,
"column": 69
} | [
{
"pp": "R : Type u_1\nA : Type u_3\nB : Type u_5\ninst✝³ : Semiring R\ninst✝² : LinearOrder B\ninst✝¹ : OrderBot B\nD : A → B\nι : Type u_7\ns : Finset ι\nf : ι → R[A]\ninst✝ : AddZeroClass A\nhd : Set.InjOn (supDegree D ∘ f) ↑s\nj : ι\nhj : j ∈ s\nhne : f j ≠ 0\n⊢ ∑ i ∈ s, f i ≠ 0",
"usedConstants": [
... | obtain ⟨i, hi, he⟩ := exists_mem_eq_sup _ ⟨j, hj⟩ (supDegree D ∘ f) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Polynomial.AlgebraMap | {
"line": 131,
"column": 71
} | {
"line": 133,
"column": 56
} | [
{
"pp": "R : Type u_3\nA : Type u_4\nB : Type u_5\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\np : R[X]\nf : A →ₐ[R] B\na : A\n⊢ f (eval₂ (algebraMap R A) a p) = eval₂ (algebraMap R B) (f a) p",
"usedConstants": [
"Eq.mpr",
"Monoi... | by
simp only [eval₂_eq_sum, sum_def]
simp only [map_sum, map_mul, map_pow, AlgHom.commutes] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.AlgebraMap | {
"line": 598,
"column": 8
} | {
"line": 598,
"column": 19
} | [
{
"pp": "case tmul\nR : Type u\nS : Type v\ninst✝⁴ : CommSemiring R\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nN : Type u_3\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nk : ℕ\ns : S\nn : N\n⊢ X ^ k • C s ⊗ₜ[R] n = (monomial k) s ⊗ₜ[R] n",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
... | smul_tmul', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.AlgebraMap | {
"line": 704,
"column": 25
} | {
"line": 713,
"column": 17
} | [
{
"pp": "R : Type u\nA : Type z\nM : Type u_3\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\nq : Submodule R M\nm : M\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : Module A M\ninst✝ : IsScalarTower R A M\nhm : m ∈ q\np : R[X]\na : A\nhq : q ≤ Submodule.comap ((Algebra.lsmul R R ... | by
induction p using Polynomial.induction_on with
| C a => simpa using SMulMemClass.smul_mem a hm
| add f₁ f₂ h₁ h₂ =>
simp_rw [map_add, add_smul]
exact Submodule.add_mem q h₁ h₂
| monomial n t hmq =>
dsimp only at hmq ⊢
rw [pow_succ', mul_left_comm, map_mul, aeval_X, mul_smul]
solve_by_elim | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.DFinsupp.NeLocus | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 43
} | [
{
"pp": "α : Type u_1\nN : α → Type u_2\ninst✝² : DecidableEq α\ninst✝¹ : (a : α) → DecidableEq (N a)\ninst✝ : (a : α) → AddGroup (N a)\nf g : Π₀ (a : α), N a\n⊢ (f - g).neLocus f = g.support",
"usedConstants": [
"DFinsupp.neLocus_comm",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"ins... | rw [neLocus_comm, neLocus_self_sub_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.DFinsupp.NeLocus | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 43
} | [
{
"pp": "α : Type u_1\nN : α → Type u_2\ninst✝² : DecidableEq α\ninst✝¹ : (a : α) → DecidableEq (N a)\ninst✝ : (a : α) → AddGroup (N a)\nf g : Π₀ (a : α), N a\n⊢ (f - g).neLocus f = g.support",
"usedConstants": [
"DFinsupp.neLocus_comm",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"ins... | rw [neLocus_comm, neLocus_self_sub_right] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.DFinsupp.NeLocus | {
"line": 155,
"column": 2
} | {
"line": 155,
"column": 43
} | [
{
"pp": "α : Type u_1\nN : α → Type u_2\ninst✝² : DecidableEq α\ninst✝¹ : (a : α) → DecidableEq (N a)\ninst✝ : (a : α) → AddGroup (N a)\nf g : Π₀ (a : α), N a\n⊢ (f - g).neLocus f = g.support",
"usedConstants": [
"DFinsupp.neLocus_comm",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"ins... | rw [neLocus_comm, neLocus_self_sub_right] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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