module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.Matrix.RowCol | {
"line": 144,
"column": 53
} | {
"line": 146,
"column": 5
} | [
{
"pp": "m : Type u_2\nα : Type v\nι : Type u_6\ninst✝ : Star α\nv : m → α\n⊢ (replicateRow ι v)ᴴ = replicateCol ι (star v)",
"usedConstants": [
"Pi.instStarForall",
"Matrix.replicateCol",
"Eq.refl",
"Matrix.replicateRow",
"Matrix.conjTranspose",
"Matrix.ext",
"Star... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.RowCol | {
"line": 150,
"column": 68
} | {
"line": 152,
"column": 5
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nα : Type v\nι : Type u_6\ninst✝¹ : Fintype m\ninst✝ : NonUnitalNonAssocSemiring α\nM : Matrix m n α\nv : m → α\n⊢ replicateRow ι (v ᵥ* M) = replicateRow ι v * M",
"usedConstants": [
"HMul.hMul",
"Matrix",
"Matrix.instHMulOfFintypeOfMulOfAddCommMonoid",
... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.RowCol | {
"line": 155,
"column": 71
} | {
"line": 157,
"column": 5
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nα : Type v\nι : Type u_6\ninst✝¹ : Fintype m\ninst✝ : NonUnitalNonAssocSemiring α\nM : Matrix m n α\nv : m → α\n⊢ replicateCol ι (v ᵥ* M) = (replicateRow ι v * M)ᵀ",
"usedConstants": [
"HMul.hMul",
"Matrix",
"Matrix.instHMulOfFintypeOfMulOfAddCommMonoid... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.RowCol | {
"line": 161,
"column": 68
} | {
"line": 163,
"column": 5
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nα : Type v\nι : Type u_6\ninst✝¹ : Fintype n\ninst✝ : NonUnitalNonAssocSemiring α\nM : Matrix m n α\nv : n → α\n⊢ replicateCol ι (M *ᵥ v) = M * replicateCol ι v",
"usedConstants": [
"HMul.hMul",
"Matrix",
"Matrix.instHMulOfFintypeOfMulOfAddCommMonoid",
... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.RowCol | {
"line": 166,
"column": 71
} | {
"line": 168,
"column": 5
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nα : Type v\nι : Type u_6\ninst✝¹ : Fintype n\ninst✝ : NonUnitalNonAssocSemiring α\nM : Matrix m n α\nv : n → α\n⊢ replicateRow ι (M *ᵥ v) = (M * replicateCol ι v)ᵀ",
"usedConstants": [
"HMul.hMul",
"Matrix",
"Matrix.instHMulOfFintypeOfMulOfAddCommMonoid... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.FreeModule.PID | {
"line": 222,
"column": 4
} | {
"line": 222,
"column": 35
} | [
{
"pp": "ι : 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.submoduleImage ... | exact ⟨⟨x, N_le_M xN⟩, hx, rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Field.Subfield.Defs | {
"line": 150,
"column": 29
} | {
"line": 150,
"column": 74
} | [
{
"pp": "K : Type u\nL : Type v\nM : Type w\ninst✝² : DivisionRing K\ninst✝¹ : DivisionRing L\ninst✝ : DivisionRing M\np q : Subfield K\nh : (fun s ↦ s.carrier) p = (fun s ↦ s.carrier) q\n⊢ p = q",
"usedConstants": [
"Subring.toSubsemiring",
"GroupWithZero.toDivisionMonoid",
"Subring.instS... | cases p; cases q; congr; exact SetLike.ext' h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Field.Subfield.Defs | {
"line": 150,
"column": 29
} | {
"line": 150,
"column": 74
} | [
{
"pp": "K : Type u\nL : Type v\nM : Type w\ninst✝² : DivisionRing K\ninst✝¹ : DivisionRing L\ninst✝ : DivisionRing M\np q : Subfield K\nh : (fun s ↦ s.carrier) p = (fun s ↦ s.carrier) q\n⊢ p = q",
"usedConstants": [
"Subring.toSubsemiring",
"GroupWithZero.toDivisionMonoid",
"Subring.instS... | cases p; cases q; congr; exact SetLike.ext' h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Field.Subfield.Basic | {
"line": 339,
"column": 18
} | {
"line": 339,
"column": 77
} | [
{
"pp": "K : Type u\ninst✝ : DivisionRing K\ns : Set K\np : (x : K) → x ∈ closure s → Prop\nmem : ∀ (x : K) (hx : x ∈ s), p x ⋯\none : p 1 ⋯\nadd : ∀ (x y : K) (hx : x ∈ closure s) (hy : y ∈ closure s), p x hx → p y hy → p (x + y) ⋯\nneg : ∀ (x : K) (hx : x ∈ closure s), p x hx → p (-x) ⋯\ninv : ∀ (x : K) (hx :... | by rintro _ _ ⟨_, hx⟩ ⟨_, hy⟩; exact ⟨_, mul _ _ _ _ hx hy⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.OreLocalization.Basic | {
"line": 191,
"column": 4
} | {
"line": 191,
"column": 93
} | [
{
"pp": "case c.c.c.e_a.e_a\nR : Type u_1\ninst✝³ : Monoid R\nS : Submonoid R\ninst✝² : OreSet S\nX : Type u_2\ninst✝¹ : AddMonoid X\ninst✝ : DistribMulAction R X\nr₁ : X\ns₁ : ↥S\nr₂ : X\ns₂ : ↥S\nr₃ : X\ns₃ : ↥S\nra : R\nsa : ↥S\nha : ↑sa * ↑s₁ = ra * ↑s₂\nrc : R\nsc : ↥S\nhc : ↑sc * ↑(sa * s₁) = rc * ↑s₃\n⊢ ... | rw [OreLocalization.expand r₃ s₃ rc (hc.symm ▸ (sc * (sa * s₁)).2)]; congr; ext; exact hc | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.OreLocalization.Basic | {
"line": 191,
"column": 4
} | {
"line": 191,
"column": 93
} | [
{
"pp": "case c.c.c.e_a.e_a\nR : Type u_1\ninst✝³ : Monoid R\nS : Submonoid R\ninst✝² : OreSet S\nX : Type u_2\ninst✝¹ : AddMonoid X\ninst✝ : DistribMulAction R X\nr₁ : X\ns₁ : ↥S\nr₂ : X\ns₂ : ↥S\nr₃ : X\ns₃ : ↥S\nra : R\nsa : ↥S\nha : ↑sa * ↑s₁ = ra * ↑s₂\nrc : R\nsc : ↥S\nhc : ↑sc * ↑(sa * s₁) = rc * ↑s₃\n⊢ ... | rw [OreLocalization.expand r₃ s₃ rc (hc.symm ▸ (sc * (sa * s₁)).2)]; congr; ext; exact hc | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.OreLocalization.Basic | {
"line": 200,
"column": 6
} | {
"line": 200,
"column": 20
} | [
{
"pp": "case c\nR : Type u_1\ninst✝³ : Monoid R\nS : Submonoid R\ninst✝² : OreSet S\nX : Type u_2\ninst✝¹ : AddMonoid X\ninst✝ : DistribMulAction R X\nr✝ : X\ns✝ : ↥S\n⊢ 0 + r✝ /ₒ s✝ = r✝ /ₒ s✝",
"usedConstants": [
"Eq.mpr",
"congrArg",
"OreLocalization",
"AddMonoid.toAddZeroClass",... | ← zero_oreDiv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Field.Subfield.Basic | {
"line": 567,
"column": 26
} | {
"line": 575,
"column": 77
} | [
{
"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\na✝ b✝ : K\nx_mem : a✝ ∈ {z | ∃ x ∈ Subring.closure s, ∃ y ∈ Subring.closure s, x / y = z}\ny_mem : b✝ ∈ {z | ∃ x ∈ Subr... | by
-- Use `id` in the next 2 `obtain`s so that assumptions stay there for the `rwa`s below
obtain ⟨nx, hnx, dx, hdx, rfl⟩ := id x_mem
obtain ⟨ny, hny, dy, hdy, rfl⟩ := id y_mem
by_cases hx0 : dx = 0; · rwa [hx0, div_zero, zero_add]
by_cases hy0 : dy = 0; · rwa [hy0, div_zero, add_zero]
exact
... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.OreLocalization.Basic | {
"line": 204,
"column": 6
} | {
"line": 204,
"column": 20
} | [
{
"pp": "case c\nR : Type u_1\ninst✝³ : Monoid R\nS : Submonoid R\ninst✝² : OreSet S\nX : Type u_2\ninst✝¹ : AddMonoid X\ninst✝ : DistribMulAction R X\nr✝ : X\ns✝ : ↥S\n⊢ r✝ /ₒ s✝ + 0 = r✝ /ₒ s✝",
"usedConstants": [
"Eq.mpr",
"congrArg",
"OreLocalization",
"AddMonoid.toAddZeroClass",... | ← zero_oreDiv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.MonoidLocalization.MonoidWithZero | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 29
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoidWithZero M\nS : Submonoid M\np : Type u_4\nf : M → ↥S → p\nH : ∀ {a c : M} {b d : ↥S}, (r S) (a, b) (c, d) → f a b = f c d\n⊢ liftOn 0 f H = f 0 1",
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Eq.mpr",
"Localization.mk",
"congrArg",... | rw [← mk_zero 1, liftOn_mk] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.MonoidLocalization.MonoidWithZero | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 29
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoidWithZero M\nS : Submonoid M\np : Type u_4\nf : M → ↥S → p\nH : ∀ {a c : M} {b d : ↥S}, (r S) (a, b) (c, d) → f a b = f c d\n⊢ liftOn 0 f H = f 0 1",
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Eq.mpr",
"Localization.mk",
"congrArg",... | rw [← mk_zero 1, liftOn_mk] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.MonoidLocalization.MonoidWithZero | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 29
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoidWithZero M\nS : Submonoid M\np : Type u_4\nf : M → ↥S → p\nH : ∀ {a c : M} {b d : ↥S}, (r S) (a, b) (c, d) → f a b = f c d\n⊢ liftOn 0 f H = f 0 1",
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Eq.mpr",
"Localization.mk",
"congrArg",... | rw [← mk_zero 1, liftOn_mk] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 718,
"column": 85
} | {
"line": 719,
"column": 55
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommSemiring R\nm : Type u_3\nn : Type u_4\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\nM₂ : Type u_6\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M₂\ninst✝ : Finite m\nv₁ : Basis n R R\nv₂ : Basis m R M₂\nx : M₂\n⊢ (toMatrix v₁ v₂) (toSpanSingleton R M₂ x) = vecMulVec ⇑(v₂.repr... | by
ext; simp [toMatrix_apply, vecMulVec_apply, mul_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 738,
"column": 2
} | {
"line": 739,
"column": 48
} | [
{
"pp": "case h₀\nR : Type u_1\ninst✝⁷ : CommSemiring R\nm : Type u_3\nn : Type u_4\ninst✝⁶ : Fintype n\ninst✝⁵ : DecidableEq n\nM₁ : Type u_5\nM₂ : Type u_6\ninst✝⁴ : AddCommMonoid M₁\ninst✝³ : AddCommMonoid M₂\ninst✝² : Module R M₁\ninst✝¹ : Module R M₂\nv₁ : Basis n R M₁\nv₂ : Basis m R M₂\ninst✝ : Fintype m... | · intro i' _ i'_ne
rw [Finsupp.single_eq_of_ne i'_ne, mul_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 1195,
"column": 12
} | {
"line": 1196,
"column": 92
} | [
{
"pp": "ι✝ : Type u_1\ninst✝¹⁰ : Fintype ι✝\ninst✝⁹ : DecidableEq ι✝\nR : Type u_2\ninst✝⁸ : CommSemiring R\nA : Type u_3\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\nM : Type u_4\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\nι : Type u_5\ninst✝¹ : Finite ι\n... | rw [← MulOpposite.isStablyFiniteRing_iff,
← RingEquiv.isStablyFiniteRing_iff (matrixRingEquivEndVecMulOpposite (ι := ι) (A := A))] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 1195,
"column": 12
} | {
"line": 1196,
"column": 92
} | [
{
"pp": "ι✝ : Type u_1\ninst✝¹⁰ : Fintype ι✝\ninst✝⁹ : DecidableEq ι✝\nR : Type u_2\ninst✝⁸ : CommSemiring R\nA : Type u_3\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\nM : Type u_4\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\nι : Type u_5\ninst✝¹ : Finite ι\n... | rw [← MulOpposite.isStablyFiniteRing_iff,
← RingEquiv.isStablyFiniteRing_iff (matrixRingEquivEndVecMulOpposite (ι := ι) (A := A))] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 1195,
"column": 12
} | {
"line": 1196,
"column": 92
} | [
{
"pp": "ι✝ : Type u_1\ninst✝¹⁰ : Fintype ι✝\ninst✝⁹ : DecidableEq ι✝\nR : Type u_2\ninst✝⁸ : CommSemiring R\nA : Type u_3\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\nM : Type u_4\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : Module R M\ninst✝³ : Module A M\ninst✝² : IsScalarTower R A M\nι : Type u_5\ninst✝¹ : Finite ι\n... | rw [← MulOpposite.isStablyFiniteRing_iff,
← RingEquiv.isStablyFiniteRing_iff (matrixRingEquivEndVecMulOpposite (ι := ι) (A := A))] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Localization.Defs | {
"line": 423,
"column": 28
} | {
"line": 423,
"column": 46
} | [
{
"pp": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalization M S\nx y : ↥M\n⊢ mk' S (↑y * ↑x) (y * x) = 1",
"usedConstants": [
"HMul.hMul",
"CommSemiring.toNonUnitalCommSemiring",
"CommSemiring.toSemiri... | exact mk'_self _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Localization.FractionRing | {
"line": 69,
"column": 24
} | {
"line": 72,
"column": 9
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommRing S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommRing P\nA : Type u_4\ninst✝ : CommRing A\nK : Type u_5\nx y : ℤ\n⊢ (algebraMap ℤ ℚ) x = (algebraMap ℤ ℚ) y → ∃ c, ↑c * x = ↑c * y",
"usedConstants": [
"Int.c... | by
rw [eq_intCast, eq_intCast, Int.cast_inj]
rintro rfl
use 1 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Localization.FractionRing | {
"line": 291,
"column": 2
} | {
"line": 292,
"column": 93
} | [
{
"pp": "A : Type u_4\ninst✝¹⁹ : CommRing A\nB : Type u_6\ninst✝¹⁸ : CommRing B\ninst✝¹⁷ : Algebra A B\nK₁ : Type u_8\nK₂ : Type u_9\ninst✝¹⁶ : Field K₁\ninst✝¹⁵ : Field K₂\ninst✝¹⁴ : Algebra A K₁\ninst✝¹³ : Algebra A K₂\ninst✝¹² : IsFractionRing A K₁\nL₁ : Type u_10\nL₂ : Type u_11\ninst✝¹¹ : Field L₁\ninst✝¹⁰... | simp_rw [map_div₀, AlgHom.commutes, ← IsScalarTower.algebraMap_apply,
IsScalarTower.algebraMap_apply A B L₁, AlgHom.commutes, ← IsScalarTower.algebraMap_apply] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.TensorProduct.Finite | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 80
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R M\ninst✝ : Module R N\nI : Submodule R N\ns : Set (↥I ⊗[R] M)\nhs : s.Finite\n⊢ ∃ J, ∃ (_ : J.FG) (hle : J ≤ I), s ⊆ ↑(rTensor M (inclusion hle)).range",
"usedCon... | choose J fg hle y eq using exists_fg_le_eq_rTensor_inclusion (M := M) (I := I) | Mathlib.Tactic.Choose._aux_Mathlib_Tactic_Choose___elabRules_Mathlib_Tactic_Choose_choose_1 | Mathlib.Tactic.Choose.choose |
Mathlib.Algebra.Star.Pointwise | {
"line": 96,
"column": 6
} | {
"line": 96,
"column": 25
} | [
{
"pp": "α : Type u_1\ninst✝ : InvolutiveStar α\ns t : Set α\n⊢ s⋆ ⊆ t ↔ s ⊆ t⋆",
"usedConstants": [
"Eq.mpr",
"Set.star",
"congrArg",
"id",
"HasSubset.Subset",
"Iff",
"propext",
"Set.star_subset_star",
"InvolutiveStar.toStar",
"Eq.symm",
"Eq... | ← star_subset_star, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Star.StarRingHom | {
"line": 144,
"column": 49
} | {
"line": 146,
"column": 5
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Star A\ninst✝¹ : NonUnitalNonAssocSemiring B\ninst✝ : Star B\nf : A →⋆ₙ+* B\nh₁ : ∀ (x y : A), f (x * y) = f x * f y\nh₂ : { toFun := ⇑f, map_mul' := h₁ }.toFun 0 = 0\nh₃ :\n ∀ (x y : A),\n { toFun := ⇑f, map_mul' := h₁ }.to... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Algebra.Spectrum.Basic | {
"line": 295,
"column": 4
} | {
"line": 295,
"column": 98
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nx : R\n⊢ x ∈ resolventSet R (-a) ↔ x ∈ -resolventSet R a",
"usedConstants": [
"NegZeroClass.toNeg",
"_private.Mathlib.Algebra.Algebra.Spectrum.Basic.0.resolventSet_neg._simp_1_4",
"RingHom.in... | simp only [mem_neg, mem_resolventSet_iff, map_neg, ← neg_add', IsUnit.neg_iff, sub_neg_eq_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Algebra.Spectrum.Basic | {
"line": 295,
"column": 4
} | {
"line": 295,
"column": 98
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nx : R\n⊢ x ∈ resolventSet R (-a) ↔ x ∈ -resolventSet R a",
"usedConstants": [
"NegZeroClass.toNeg",
"_private.Mathlib.Algebra.Algebra.Spectrum.Basic.0.resolventSet_neg._simp_1_4",
"RingHom.in... | simp only [mem_neg, mem_resolventSet_iff, map_neg, ← neg_add', IsUnit.neg_iff, sub_neg_eq_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Algebra.Spectrum.Basic | {
"line": 295,
"column": 4
} | {
"line": 295,
"column": 98
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nx : R\n⊢ x ∈ resolventSet R (-a) ↔ x ∈ -resolventSet R a",
"usedConstants": [
"NegZeroClass.toNeg",
"_private.Mathlib.Algebra.Algebra.Spectrum.Basic.0.resolventSet_neg._simp_1_4",
"RingHom.in... | simp only [mem_neg, mem_resolventSet_iff, map_neg, ← neg_add', IsUnit.neg_iff, sub_neg_eq_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Star.StarAlgHom | {
"line": 162,
"column": 57
} | {
"line": 164,
"column": 5
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁶ : Monoid R\ninst✝⁵ : NonUnitalNonAssocSemiring A\ninst✝⁴ : DistribMulAction R A\ninst✝³ : Star A\ninst✝² : NonUnitalNonAssocSemiring B\ninst✝¹ : DistribMulAction R B\ninst✝ : Star B\nf : A →⋆ₙₐ[R] B\nh₁ : ∀ (m : R) (x : A), f (m • x) = (MonoidHom.id R) m... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Star.StarAlgHom | {
"line": 387,
"column": 62
} | {
"line": 389,
"column": 5
} | [
{
"pp": "R : Type u_2\nA : Type u_3\nB : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\ninst✝³ : Star A\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\ninst✝ : Star B\nf : A →⋆ₐ[R] B\nh₁ : f 1 = 1\nh₂ :\n ∀ (x y : A),\n { toFun := ⇑f, map_one' := h₁ }.toFun (x * y) =\n { toFu... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Algebra.Unitization | {
"line": 733,
"column": 17
} | {
"line": 733,
"column": 76
} | [
{
"pp": "S : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝¹² : CommSemiring S\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : NonUnitalSemiring A\ninst✝⁹ : Module R A\ninst✝⁸ : SMulCommClass R A A\ninst✝⁷ : IsScalarTower R A A\nB : Type u_4\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra S B\ninst✝⁴ : Algebra S R\ninst✝³ : DistribMu... | simp only [fst_one, map_one, snd_one, φ.map_zero, add_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Algebra.Unitization | {
"line": 733,
"column": 17
} | {
"line": 733,
"column": 76
} | [
{
"pp": "S : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝¹² : CommSemiring S\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : NonUnitalSemiring A\ninst✝⁹ : Module R A\ninst✝⁸ : SMulCommClass R A A\ninst✝⁷ : IsScalarTower R A A\nB : Type u_4\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra S B\ninst✝⁴ : Algebra S R\ninst✝³ : DistribMu... | simp only [fst_one, map_one, snd_one, φ.map_zero, add_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Algebra.Unitization | {
"line": 733,
"column": 17
} | {
"line": 733,
"column": 76
} | [
{
"pp": "S : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝¹² : CommSemiring S\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : NonUnitalSemiring A\ninst✝⁹ : Module R A\ninst✝⁸ : SMulCommClass R A A\ninst✝⁷ : IsScalarTower R A A\nB : Type u_4\ninst✝⁶ : Semiring B\ninst✝⁵ : Algebra S B\ninst✝⁴ : Algebra S R\ninst✝³ : DistribMu... | simp only [fst_one, map_one, snd_one, φ.map_zero, add_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 383,
"column": 4
} | {
"line": 384,
"column": 73
} | [
{
"pp": "case h.e'_3\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⊢ quasispectrum R (a * b) ∩ {r | IsUnit r} = quasispectrum R (b * a) ∩ {r | IsUnit r}",
"usedConstants": [
"Monoid",... | simpa [Set.inter_comm _ {r | IsUnit r}, Unitization.quasispectrum_eq_spectrum_inr,
Unitization.inr_mul] using spectrum.setOf_isUnit_inter_mul_comm _ _ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 383,
"column": 4
} | {
"line": 384,
"column": 73
} | [
{
"pp": "case h.e'_3\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⊢ quasispectrum R (a * b) ∩ {r | IsUnit r} = quasispectrum R (b * a) ∩ {r | IsUnit r}",
"usedConstants": [
"Monoid",... | simpa [Set.inter_comm _ {r | IsUnit r}, Unitization.quasispectrum_eq_spectrum_inr,
Unitization.inr_mul] using spectrum.setOf_isUnit_inter_mul_comm _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Algebra.Spectrum.Quasispectrum | {
"line": 383,
"column": 4
} | {
"line": 384,
"column": 73
} | [
{
"pp": "case h.e'_3\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⊢ quasispectrum R (a * b) ∩ {r | IsUnit r} = quasispectrum R (b * a) ∩ {r | IsUnit r}",
"usedConstants": [
"Monoid",... | simpa [Set.inter_comm _ {r | IsUnit r}, Unitization.quasispectrum_eq_spectrum_inr,
Unitization.inr_mul] using spectrum.setOf_isUnit_inter_mul_comm _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Star.Basic | {
"line": 235,
"column": 46
} | {
"line": 236,
"column": 62
} | [
{
"pp": "R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\na : R\nha : 0 ≤ a\nc : R\nhc : IsSelfAdjoint c\n⊢ 0 ≤ c * a * c",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"PartialOrder.toPreorder",
"NonU... | by
nth_rewrite 2 [← hc]; exact star_right_conjugate_nonneg ha c | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Algebra.Subalgebra.Directed | {
"line": 62,
"column": 19
} | {
"line": 64,
"column": 88
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁵ : CommSemiring R\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Subalgebra R A\nι : Type u_4\ninst✝ : Nonempty ι\nK : ι → Subalgebra R A\ndir : Directed (fun x1 x2 ↦ x1 ≤ x2) K\nf : (i : ι) → ↥(K i) →ₐ[R] B\nhf... | by
dsimp
exact Set.iUnionLift_const _ (fun i : ι => (0 : K i)) (fun _ => rfl) _ (by simp) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Dimension.Finite | {
"line": 266,
"column": 65
} | {
"line": 266,
"column": 80
} | [
{
"pp": "ι : Type w\nR : Type u\nM : Type v\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\ninst✝³ : IsDomain R\ninst✝² : IsTorsionFree R M\ninst✝¹ : Module.Finite R M\ninst✝ : StrongRankCondition R\np : ι → Submodule R M\nhp : iSupIndep p\n⊢ lift.{w, v} (Module.rank R M) = lift.{w, v} ↑(finrank... | finrank_eq_rank | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.TensorProduct.Maps | {
"line": 524,
"column": 2
} | {
"line": 524,
"column": 13
} | [
{
"pp": "R : Type uR\nA : Type uA\nB : Type uB\nC : Type uC\nD : Type uD\ninst✝⁸ : CommSemiring R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : Semiring B\ninst✝⁴ : Algebra R B\ninst✝³ : Semiring C\ninst✝² : Algebra R C\ninst✝¹ : Semiring D\ninst✝ : Algebra R D\nf : A →ₐ[R] C\ng : B →ₐ[R] D\n⊢ (↑(TensorP... | ext <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.RingTheory.TensorProduct.Maps | {
"line": 524,
"column": 2
} | {
"line": 524,
"column": 13
} | [
{
"pp": "R : Type uR\nA : Type uA\nB : Type uB\nC : Type uC\nD : Type uD\ninst✝⁸ : CommSemiring R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : Semiring B\ninst✝⁴ : Algebra R B\ninst✝³ : Semiring C\ninst✝² : Algebra R C\ninst✝¹ : Semiring D\ninst✝ : Algebra R D\nf : A →ₐ[R] C\ng : B →ₐ[R] D\n⊢ (↑(TensorP... | ext <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.TensorProduct.Maps | {
"line": 524,
"column": 2
} | {
"line": 524,
"column": 13
} | [
{
"pp": "R : Type uR\nA : Type uA\nB : Type uB\nC : Type uC\nD : Type uD\ninst✝⁸ : CommSemiring R\ninst✝⁷ : Semiring A\ninst✝⁶ : Algebra R A\ninst✝⁵ : Semiring B\ninst✝⁴ : Algebra R B\ninst✝³ : Semiring C\ninst✝² : Algebra R C\ninst✝¹ : Semiring D\ninst✝ : Algebra R D\nf : A →ₐ[R] C\ng : B →ₐ[R] D\n⊢ (↑(TensorP... | ext <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.TensorProduct.Maps | {
"line": 712,
"column": 67
} | {
"line": 712,
"column": 78
} | [
{
"pp": "R : Type uR\nS : Type uS\nA : Type uA\nB : Type uB\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Semiring B\ninst✝³ : CommSemiring S\ninst✝² : Algebra R A\ninst✝¹ : Algebra R B\ninst✝ : Algebra R S\nf : A →ₐ[R] S\ng : B →ₐ[R] S\n⊢ (lmul' R).comp (map f g) = lift f g ⋯",
"usedConstants": [... | ext <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.RingTheory.TensorProduct.Maps | {
"line": 712,
"column": 67
} | {
"line": 712,
"column": 78
} | [
{
"pp": "R : Type uR\nS : Type uS\nA : Type uA\nB : Type uB\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Semiring B\ninst✝³ : CommSemiring S\ninst✝² : Algebra R A\ninst✝¹ : Algebra R B\ninst✝ : Algebra R S\nf : A →ₐ[R] S\ng : B →ₐ[R] S\n⊢ (lmul' R).comp (map f g) = lift f g ⋯",
"usedConstants": [... | ext <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.TensorProduct.Maps | {
"line": 712,
"column": 67
} | {
"line": 712,
"column": 78
} | [
{
"pp": "R : Type uR\nS : Type uS\nA : Type uA\nB : Type uB\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Semiring B\ninst✝³ : CommSemiring S\ninst✝² : Algebra R A\ninst✝¹ : Algebra R B\ninst✝ : Algebra R S\nf : A →ₐ[R] S\ng : B →ₐ[R] S\n⊢ (lmul' R).comp (map f g) = lift f g ⋯",
"usedConstants": [... | ext <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Dimension.Constructions | {
"line": 478,
"column": 2
} | {
"line": 478,
"column": 56
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : StrongRankCondition R\ns : Finset M\nhs : LinearIndepOn R id ↑s\n⊢ finrank R ↥(span R ↑s) = s.card",
"usedConstants": [
"finrank_span_set_eq_card",
"Eq.mpr",
"Submodule",
"HEq... | convert finrank_span_set_eq_card (s := (s : Set M)) hs | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___elabRules_Mathlib_Tactic_convert_1 | Mathlib.Tactic.convert |
Mathlib.RingTheory.TensorProduct.Maps | {
"line": 736,
"column": 2
} | {
"line": 736,
"column": 13
} | [
{
"pp": "R : Type uR\nS : Type uS\nA : Type uA\nB : Type uB\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Semiring B\ninst✝³ : CommSemiring S\ninst✝² : Algebra R A\ninst✝¹ : Algebra R B\ninst✝ : Algebra R S\nf : A →ₐ[R] S\ng : B →ₐ[R] S\n⊢ productMap f g = (lmul' R).comp (map f g)",
"usedConstants... | ext <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.RingTheory.TensorProduct.Maps | {
"line": 736,
"column": 2
} | {
"line": 736,
"column": 13
} | [
{
"pp": "R : Type uR\nS : Type uS\nA : Type uA\nB : Type uB\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Semiring B\ninst✝³ : CommSemiring S\ninst✝² : Algebra R A\ninst✝¹ : Algebra R B\ninst✝ : Algebra R S\nf : A →ₐ[R] S\ng : B →ₐ[R] S\n⊢ productMap f g = (lmul' R).comp (map f g)",
"usedConstants... | ext <;> rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.TensorProduct.Maps | {
"line": 736,
"column": 2
} | {
"line": 736,
"column": 13
} | [
{
"pp": "R : Type uR\nS : Type uS\nA : Type uA\nB : Type uB\ninst✝⁶ : CommSemiring R\ninst✝⁵ : Semiring A\ninst✝⁴ : Semiring B\ninst✝³ : CommSemiring S\ninst✝² : Algebra R A\ninst✝¹ : Algebra R B\ninst✝ : Algebra R S\nf : A →ₐ[R] S\ng : B →ₐ[R] S\n⊢ productMap f g = (lmul' R).comp (map f g)",
"usedConstants... | ext <;> rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Star.NonUnitalSubalgebra | {
"line": 666,
"column": 44
} | {
"line": 666,
"column": 59
} | [
{
"pp": "F : Type v'\nR' : Type u'\nR : Type u\nA : Type v\nB : Type w\nC : Type w'\ninst✝¹³ : CommSemiring R\ninst✝¹² : StarRing R\ninst✝¹¹ : NonUnitalSemiring A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : Module R A\ninst✝⁸ : NonUnitalSemiring B\ninst✝⁷ : StarRing B\ninst✝⁶ : Module R B\ninst✝⁵ : FunLike F A B\ninst✝⁴ : ... | Set.union_star, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition | {
"line": 136,
"column": 32
} | {
"line": 136,
"column": 43
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝⁴ : Ring K\ninst✝³ : StrongRankCondition K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ns : Submodule K V\ninst✝ : Free K ↥s\nx✝ : ∃ v₀, v₀ ≠ 0 ∧ ∀ (v : ↥s), ∃ r, r • v₀ = v\nv₀ : V\nhv₀ : v₀ ∈ s\nh : ∀ (v : ↥s), ∃ r, r • ⟨v₀, hv₀⟩ = v\nh' : v₀ = 0\nH : ¬⟨0, ⋯⟩ = 0\n⊢ Fal... | exact H rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.FiniteDimensional.Basic | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 64
} | [
{
"pp": "K : Type u\nV : Type v\ninst✝³ : DivisionRing K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : FiniteDimensional K V\nS : Submodule K V\nh : finrank K ↥S = finrank K V\nbS : Basis (↑(Basis.ofVectorSpaceIndex K ↥S)) K ↥S := Basis.ofVectorSpace K ↥S\nbS_eq : bS = Basis.ofVectorSpace K ↥S\nthis✝¹ ... | rw [bS_eq, Basis.coe_ofVectorSpace, Subtype.range_coe] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition | {
"line": 246,
"column": 4
} | {
"line": 246,
"column": 53
} | [
{
"pp": "case pos\nF : Type u_1\nE : Type u_2\ninst✝⁴ : CommRing F\ninst✝³ : StrongRankCondition F\ninst✝² : Ring E\ninst✝¹ : Algebra F E\nS : Subalgebra F E\nh : Module.rank F ↥S ≤ 1\ninst✝ : Free F ↥S\na✝ : Nontrivial E\nκ : Type u_2\nb : Basis κ F ↥S\nh1 : Module.rank F ↥S = 1\n⊢ S = ⊥",
"usedConstants":... | refine bot_unique fun x hx ↦ Algebra.mem_bot.2 ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 586,
"column": 8
} | {
"line": 586,
"column": 31
} | [
{
"pp": "case w.refine_1\nR : Type u_1\ninst✝⁴ : Ring R\nE : Type u_2\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\nF : Type u_3\ninst✝¹ : AddCommGroup F\ninst✝ : Module R F\nc : Set (E →ₗ.[R] F)\nhc : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) c\ncne : c.Nonempty\nhdir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) (domain '' c)\n... | f_eq ⟨p, hpc⟩ x x' rfl, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Eval.Defs | {
"line": 250,
"column": 2
} | {
"line": 251,
"column": 5
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nx : R\n⊢ eval x p = p.sum fun e a ↦ a * x ^ e",
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"HMul.hMul",
"congrArg",
"Polynomial.sum",
"RingHom",
"id",
"Polynomial.eval₂",
"Monoid.toPow",
"Poly... | rw [eval, eval₂_eq_sum]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Eval.Defs | {
"line": 250,
"column": 2
} | {
"line": 251,
"column": 5
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nx : R\n⊢ eval x p = p.sum fun e a ↦ a * x ^ e",
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
"HMul.hMul",
"congrArg",
"Polynomial.sum",
"RingHom",
"id",
"Polynomial.eval₂",
"Monoid.toPow",
"Poly... | rw [eval, eval₂_eq_sum]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Eval.Defs | {
"line": 777,
"column": 2
} | {
"line": 777,
"column": 95
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\np q : R[X]\nn : ℕ\n⊢ (p * (X - ↑n)).comp q = p.comp q * (q - ↑n)",
"usedConstants": [
"Nat.cast_comm",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"Ring.toNonAssocRing",
"mul_sub",
"congrArg",
"NonUnit... | rw [mul_sub, sub_comp, mul_X_comp, ← Nat.cast_comm, natCast_mul_comp, Nat.cast_comm, mul_sub] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Eval.Defs | {
"line": 777,
"column": 2
} | {
"line": 777,
"column": 95
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\np q : R[X]\nn : ℕ\n⊢ (p * (X - ↑n)).comp q = p.comp q * (q - ↑n)",
"usedConstants": [
"Nat.cast_comm",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"Ring.toNonAssocRing",
"mul_sub",
"congrArg",
"NonUnit... | rw [mul_sub, sub_comp, mul_X_comp, ← Nat.cast_comm, natCast_mul_comp, Nat.cast_comm, mul_sub] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Eval.Defs | {
"line": 777,
"column": 2
} | {
"line": 777,
"column": 95
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\np q : R[X]\nn : ℕ\n⊢ (p * (X - ↑n)).comp q = p.comp q * (q - ↑n)",
"usedConstants": [
"Nat.cast_comm",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"Ring.toNonAssocRing",
"mul_sub",
"congrArg",
"NonUnit... | rw [mul_sub, sub_comp, mul_X_comp, ← Nat.cast_comm, natCast_mul_comp, Nat.cast_comm, mul_sub] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MonoidAlgebra.Basic | {
"line": 78,
"column": 14
} | {
"line": 78,
"column": 24
} | [
{
"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✝⁵ : Semiring R\ninst✝⁴ : Mul M\ninst✝³ : NonUnitalNonAssocSemiring A\ninst✝² : Module R A\ninst✝¹ : IsScalarTower R A A\ninst✝ : SMulCommClass R A A\nf : M →ₙ* A\na₁ a₂ : R... | ← add_smul | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Prime.Defs | {
"line": 90,
"column": 2
} | {
"line": 95,
"column": 18
} | [
{
"pp": "p : ℕ\npp : Prime p\nm : ℕ\nhm : m ∣ p\n⊢ m = 1 ∨ m = p",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Dvd.dvd",
"HMul.hMul",
"congrArg",
"Nat.instMonoid",
"IsUnit",
"Nat.instMulOneClass",
"Eq.mp",
"id",
"MulOne.toMul",
"instM... | obtain ⟨n, hn⟩ := hm
have := pp.isUnit_or_isUnit hn
rw [Nat.isUnit_iff, Nat.isUnit_iff] at this
apply Or.imp_right _ this
rintro rfl
rw [hn, mul_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Prime.Defs | {
"line": 90,
"column": 2
} | {
"line": 95,
"column": 18
} | [
{
"pp": "p : ℕ\npp : Prime p\nm : ℕ\nhm : m ∣ p\n⊢ m = 1 ∨ m = p",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Dvd.dvd",
"HMul.hMul",
"congrArg",
"Nat.instMonoid",
"IsUnit",
"Nat.instMulOneClass",
"Eq.mp",
"id",
"MulOne.toMul",
"instM... | obtain ⟨n, hn⟩ := hm
have := pp.isUnit_or_isUnit hn
rw [Nat.isUnit_iff, Nat.isUnit_iff] at this
apply Or.imp_right _ this
rintro rfl
rw [hn, mul_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.CharP.Defs | {
"line": 282,
"column": 28
} | {
"line": 282,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NonAssocSemiring R\ninst✝ : CharP R 1\nr : R\n⊢ 1 * r = ↑1 * r",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"congrArg",
"id",
"AddMonoidWithOne.toNatCast",
"instOfNatNat",
"AddCommMonoid... | rw [Nat.cast_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.CharP.Defs | {
"line": 282,
"column": 28
} | {
"line": 282,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NonAssocSemiring R\ninst✝ : CharP R 1\nr : R\n⊢ 1 * r = ↑1 * r",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"congrArg",
"id",
"AddMonoidWithOne.toNatCast",
"instOfNatNat",
"AddCommMonoid... | rw [Nat.cast_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.CharP.Defs | {
"line": 282,
"column": 28
} | {
"line": 282,
"column": 45
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NonAssocSemiring R\ninst✝ : CharP R 1\nr : R\n⊢ 1 * r = ↑1 * r",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"congrArg",
"id",
"AddMonoidWithOne.toNatCast",
"instOfNatNat",
"AddCommMonoid... | rw [Nat.cast_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 70
} | [
{
"pp": "case pos\nR : Type u\ninst✝ : Semiring R\np : R[X]\na : R\nhp : p ≠ 0\nhpd : p.degree ≤ 0\n⊢ (p + C a).natDegree = p.natDegree",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"congrArg",
"RingHom",
"Polynomial.natDegree_C",
"id",
"Distrib.toAdd",
"inst... | rw [eq_C_of_degree_le_zero hpd, ← C_add, natDegree_C, natDegree_C] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 70
} | [
{
"pp": "case pos\nR : Type u\ninst✝ : Semiring R\np : R[X]\na : R\nhp : p ≠ 0\nhpd : p.degree ≤ 0\n⊢ (p + C a).natDegree = p.natDegree",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"congrArg",
"RingHom",
"Polynomial.natDegree_C",
"id",
"Distrib.toAdd",
"inst... | rw [eq_C_of_degree_le_zero hpd, ← C_add, natDegree_C, natDegree_C] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 207,
"column": 4
} | {
"line": 207,
"column": 70
} | [
{
"pp": "case pos\nR : Type u\ninst✝ : Semiring R\np : R[X]\na : R\nhp : p ≠ 0\nhpd : p.degree ≤ 0\n⊢ (p + C a).natDegree = p.natDegree",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"congrArg",
"RingHom",
"Polynomial.natDegree_C",
"id",
"Distrib.toAdd",
"inst... | rw [eq_C_of_degree_le_zero hpd, ← C_add, natDegree_C, natDegree_C] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 337,
"column": 2
} | {
"line": 337,
"column": 73
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\nh : p.leadingCoeff * q.leadingCoeff ≠ 0\n⊢ (p * q).leadingCoeff = p.leadingCoeff * q.leadingCoeff",
"usedConstants": [
"HMul.hMul",
"congrArg",
"Polynomial.natDegree_mul'",
"Polynomial.leadingCoeff",
"Polynomial",
"Poly... | simp [← coeff_natDegree, natDegree_mul' h, coeff_mul_degree_add_degree] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 337,
"column": 2
} | {
"line": 337,
"column": 73
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\nh : p.leadingCoeff * q.leadingCoeff ≠ 0\n⊢ (p * q).leadingCoeff = p.leadingCoeff * q.leadingCoeff",
"usedConstants": [
"HMul.hMul",
"congrArg",
"Polynomial.natDegree_mul'",
"Polynomial.leadingCoeff",
"Polynomial",
"Poly... | simp [← coeff_natDegree, natDegree_mul' h, coeff_mul_degree_add_degree] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 337,
"column": 2
} | {
"line": 337,
"column": 73
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\nh : p.leadingCoeff * q.leadingCoeff ≠ 0\n⊢ (p * q).leadingCoeff = p.leadingCoeff * q.leadingCoeff",
"usedConstants": [
"HMul.hMul",
"congrArg",
"Polynomial.natDegree_mul'",
"Polynomial.leadingCoeff",
"Polynomial",
"Poly... | simp [← coeff_natDegree, natDegree_mul' h, coeff_mul_degree_add_degree] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Eval.Coeff | {
"line": 63,
"column": 6
} | {
"line": 63,
"column": 82
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\n⊢ (p.sum fun e a ↦ a * 0 ^ e) = p.coeff 0 * 0 ^ 0",
"usedConstants": [
"MulOne.toOne",
"HMul.hMul",
"Classical.not_not._simp_1",
"Monoid.toMulOneClass",
"congrArg",
"Finset",
"AddMonoid.toAddZeroClass",
"NonUn... | exact Finset.sum_eq_single _ (fun b _ hb => by simp [zero_pow hb]) (by simp) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Polynomial.AlgebraMap | {
"line": 445,
"column": 49
} | {
"line": 445,
"column": 66
} | [
{
"pp": "R : Type u\nA : Type z\nB : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nx : A × B\np : R[X]\n⊢ (aeval x) p = ((aeval x.1) p, (aeval x.2) p)",
"usedConstants": [
"congrArg",
"CommSemiring.toSemiring",
"AlgH... | simp [aeval_prod] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Polynomial.AlgebraMap | {
"line": 445,
"column": 49
} | {
"line": 445,
"column": 66
} | [
{
"pp": "R : Type u\nA : Type z\nB : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nx : A × B\np : R[X]\n⊢ (aeval x) p = ((aeval x.1) p, (aeval x.2) p)",
"usedConstants": [
"congrArg",
"CommSemiring.toSemiring",
"AlgH... | simp [aeval_prod] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.AlgebraMap | {
"line": 445,
"column": 49
} | {
"line": 445,
"column": 66
} | [
{
"pp": "R : Type u\nA : Type z\nB : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Semiring B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nx : A × B\np : R[X]\n⊢ (aeval x) p = ((aeval x.1) p, (aeval x.2) p)",
"usedConstants": [
"congrArg",
"CommSemiring.toSemiring",
"AlgH... | simp [aeval_prod] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.AlgebraMap | {
"line": 673,
"column": 73
} | {
"line": 675,
"column": 40
} | [
{
"pp": "R : Type u\ninst✝² : CommRing R\nM : Type u_3\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : M →ₗ[R] M\nv : M\np : R[X]\n⊢ ((aeval f) p) v = p.sum fun n b ↦ b • (f ^ n) v",
"usedConstants": [
"LinearMap.applyₗ",
"Eq.mpr",
"instHSMul",
"Module.End.instMonoid",
"HMul.... | by
rw [aeval_def, eval₂_eq_sum]
exact map_sum (LinearMap.applyₗ v) _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 321,
"column": 4
} | {
"line": 321,
"column": 32
} | [
{
"pp": "case zero\nR : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nmotive : MvPolynomial σ R → Prop\nC : ∀ (a : R), motive (MvPolynomial.C a)\nmul_X : ∀ (p : MvPolynomial σ R) (n : σ), motive p → motive (p * X n)\ns : σ →₀ ℕ\na : R\n⊢ motive ((monomial 0) a)",
"usedConstants": [
"Nat.instMulZeroCla... | change motive (monomial 0 a) | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 16
} | [
{
"pp": "case C\nR : Type u\nS₁ : Type v\nσ : Type u_1\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S₁\np : MvPolynomial σ R\nf : R →+* S₁\ng : σ → S₁\n⊢ ∀ (a : R) {s : σ →₀ ℕ} {a_1 : R},\n eval₂ f g (C a * (monomial s) a_1) = eval₂ f g (C a) * f a_1 * s.prod fun n e ↦ g n ^ e",
"usedConstants": []
}... | intro a' s a | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 550,
"column": 28
} | {
"line": 550,
"column": 63
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\ni : σ\nh : Finsupp.single i 1 = 0\n⊢ False",
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"HEq.refl",
"False.elim",
"Finsupp.single_eq_zero",
"noConfusion_of_Nat",
"Eq.casesOn",
"instOfNatNat",
... | by cases Finsupp.single_eq_zero.1 h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 1036,
"column": 4
} | {
"line": 1036,
"column": 55
} | [
{
"pp": "case refine_1\nR : Type u_2\nS : Type u_3\nσ : Type u_4\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\nM N : Submodule R S\nr : S\nhr : r ∈ M * N\ns : σ →₀ ℕ\n⊢ (monomial s) r ∈ coeffsIn σ M * coeffsIn σ N",
"usedConstants": [
"MvPolynomial.monomial_mem_coeffsIn._simp... | induction hr using Submodule.mul_induction_on' with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Algebra.MvPolynomial.Degrees | {
"line": 252,
"column": 26
} | {
"line": 254,
"column": 25
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\ni : σ\nf : MvPolynomial σ R\nm : σ →₀ ℕ\nh_m : m ∈ f.support\n⊢ m i ≤ degreeOf i f",
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.instMulZeroClass",
"Nat.instLattice",
"Lattice.toSemilatticeSup",
"cong... | by
rw [degreeOf_eq_sup i]
apply Finset.le_sup h_m | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Polynomial.Degree.TrailingDegree | {
"line": 99,
"column": 6
} | {
"line": 99,
"column": 45
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nn : ℕ\nhp : p ≠ 0\n⊢ p.trailingDegree = ↑n ↔ p.natTrailingDegree = n",
"usedConstants": [
"Eq.mpr",
"ENat.instNatCast",
"congrArg",
"id",
"Nat.cast",
"Iff",
"Nat",
"ENat",
"Polynomial.trailingDegree",
... | trailingDegree_eq_natTrailingDegree hp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.TrailingDegree | {
"line": 163,
"column": 7
} | {
"line": 163,
"column": 46
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nn : ℕ\nhp : p ≠ 0\nH : ↑n ≤ p.trailingDegree\n⊢ n ≤ p.natTrailingDegree",
"usedConstants": [
"ENat.instNatCast",
"congrArg",
"Eq.mp",
"LE.le",
"Nat.cast",
"instLEENat",
"ENat",
"Polynomial.trailingDegree",
... | trailingDegree_eq_natTrailingDegree hp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.Lemmas | {
"line": 188,
"column": 10
} | {
"line": 188,
"column": 15
} | [
{
"pp": "case insert.inl\nR : Type u\nS : Type v\ninst✝ : Semiring R\nf : S → R[X]\nx : S\ns : Finset S\nhx : x ∉ s\nh : {i | i ∈ insert x s ∧ f i ≠ 0}.Pairwise (Ne on degree ∘ f)\nIH : (s.sum f).degree = s.sup fun i ↦ (f i).degree\nH : (f x).degree < (s.sum f).degree\n⊢ (f x + ∑ x ∈ s, f x).degree = max (f x).... | ← IH, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.TrailingDegree | {
"line": 330,
"column": 6
} | {
"line": 330,
"column": 45
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\nh : p.trailingCoeff * q.trailingCoeff ≠ 0\nhp : p ≠ 0\nhq : q ≠ 0\n⊢ (p * q).trailingDegree ≤ p.trailingDegree + q.trailingDegree",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"ENat.instNatCast",
"instLinearOrderENat",
"congrAr... | trailingDegree_eq_natTrailingDegree hp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.Lemmas | {
"line": 202,
"column": 10
} | {
"line": 202,
"column": 15
} | [
{
"pp": "case insert.inr.inr\nR : Type u\nS : Type v\ninst✝ : Semiring R\nf : S → R[X]\nx : S\ns : Finset S\nhx : x ∉ s\nh : {i | i ∈ insert x s ∧ f i ≠ 0}.Pairwise (Ne on degree ∘ f)\nIH : (s.sum f).degree = s.sup fun i ↦ (f i).degree\nH : (s.sum f).degree < (f x).degree\n⊢ (f x + ∑ x ∈ s, f x).degree = max (f... | ← IH, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Reverse | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 35
} | [
{
"pp": "n o n' : ℕ\nhn : n ≤ n + n'\no' : ℕ\nho : o ≤ o + o'\n⊢ n + o + (n' + o') - (n + o) = n + n' - n + (o + o' - o)",
"usedConstants": [
"Eq.mpr",
"Nat.instOrderedSub",
"Nat.instIsOrderedAddMonoid",
"congrArg",
"instIsLeftCancelAddOfAddLeftReflectLE",
"HSub.hSub",
... | repeat' rw [add_tsub_cancel_left] | Lean.Elab.Tactic.evalRepeat' | Lean.Parser.Tactic.repeat' |
Mathlib.Algebra.Polynomial.Reverse | {
"line": 156,
"column": 36
} | {
"line": 156,
"column": 60
} | [
{
"pp": "case zero.zero\nR : Type u_1\ninst✝ : Semiring R\nN O : ℕ\nf : R[X]\nCf : #f.support ≤ Nat.succ 0\nNf : f.natDegree ≤ N\ng : R[X]\nCg : #g.support ≤ Nat.succ 0\nOg : g.natDegree ≤ O\n⊢ reflect (N + O) (C f.leadingCoeff * X ^ f.natDegree * g) =\n reflect N (C f.leadingCoeff * X ^ f.natDegree) * refle... | ← C_mul_X_pow_eq_self Cg | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 162,
"column": 4
} | {
"line": 162,
"column": 89
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\npq : q.degree < p.degree\nn : ℕ\nnd : n = p.natDegree\n⊢ 0 = (p.eraseLead + q).coeff p.natDegree",
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"WithBot",
"congrA... | simpa using (coeff_eq_zero_of_degree_lt (lt_of_lt_of_le pq degree_le_natDegree)).symm | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 176,
"column": 4
} | {
"line": 176,
"column": 89
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : Semiring R\np q : R[X]\npq : p.degree < q.degree\nn : ℕ\nnd : n = q.natDegree\n⊢ 0 = (p + q.eraseLead).coeff q.natDegree",
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"WithBot",
"congrA... | simpa using (coeff_eq_zero_of_degree_lt (lt_of_lt_of_le pq degree_le_natDegree)).symm | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Polynomial.Degree.Lemmas | {
"line": 362,
"column": 77
} | {
"line": 362,
"column": 93
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\np : R[X]\ninst✝ : IsLeftCancelMulZero R\nh : p.natDegree = 1\nr₁ : R\n⊢ ∃ a, a ≠ 0 ∧ ∃ b, C a * X + C b = p",
"usedConstants": [
"Polynomial.C",
"HMul.hMul",
"congrArg",
"Polynomial.natDegree_eq_one",
"NonUnitalNonAssocSemiring.toMulZer... | natDegree_eq_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 241,
"column": 76
} | {
"line": 241,
"column": 92
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nf : R[X]\nhnext : f.nextCoeff = 0\nhlead : (∃ x, C x = f) ∨ f.natDegree = 1\n⊢ f.eraseLead = 0",
"usedConstants": [
"Polynomial.C",
"HMul.hMul",
"congrArg",
"Polynomial.natDegree_eq_one",
"NonUnitalNonAssocSemiring.toMulZeroClass",
... | natDegree_eq_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Reverse | {
"line": 255,
"column": 4
} | {
"line": 255,
"column": 69
} | [
{
"pp": "case neg.a.h\nR : Type u_1\ninst✝ : Semiring R\nf : R[X]\nhf : ¬f = 0\n⊢ f.coeff (f.natDegree - f.reverse.natDegree) ≠ 0",
"usedConstants": [
"Polynomial.reverse_eq_zero",
"NonUnitalNonAssocSemiring.toMulZeroClass",
"mt",
"Polynomial.leadingCoeff",
"Polynomial",
... | have key := mt leadingCoeff_eq_zero.mp (mt reverse_eq_zero.mp hf) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Polynomial.Monic | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 21
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\nhp : p.Monic\nhq : q ≠ 0\n⊢ (p * q).natDegree = p.natDegree + q.natDegree",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Polynomial.natDegree_mul'",
"id",
"Polynomial",
"instHAdd",
"HAdd.hAdd",
... | rw [natDegree_mul'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.EraseLead | {
"line": 326,
"column": 4
} | {
"line": 326,
"column": 51
} | [
{
"pp": "case h.e'_1\nR : Type u_1\ninst✝ : Semiring R\nmotive : R[X] → Prop\nN : ℕ\nzero : motive 0\nC_mul_pow : ∀ (n : ℕ) (r : R), r ≠ 0 → n ≤ N → motive (C r * X ^ n)\nadd : ∀ (f g : R[X]), f.natDegree < g.natDegree → g.natDegree ≤ N → motive f → motive g → motive (f + g)\nf : R[X]\ndf : f.natDegree ≤ N\nhf ... | simpa [support_eq_empty, card_eq_zero] using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.MvPolynomial.Equiv | {
"line": 555,
"column": 93
} | {
"line": 556,
"column": 41
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nn : ℕ\np : MvPolynomial (Fin (n + 1)) R\n⊢ (finSuccEquiv R n) p =\n (eval₂Hom (Polynomial.C.comp C) fun i ↦ Fin.cases Polynomial.X (fun k ↦ Polynomial.C (X k)) i) p",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Polynomial.C",
... | by
rw [← finSuccEquiv_eq, RingHom.coe_coe] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MvPolynomial.Equiv | {
"line": 580,
"column": 2
} | {
"line": 599,
"column": 39
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nn : ℕ\nm : Fin n →₀ ℕ\nf : MvPolynomial (Fin (n + 1)) R\ni : ℕ\n⊢ coeff m (((finSuccEquiv R n) f).coeff i) = coeff (cons i m) f",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Finsupp.instFunLike",
"Eq.mpr",
"Polynomial.C",
"AlgEqu... | induction f using MvPolynomial.induction_on' generalizing i m with
| add p q hp hq => simp only [map_add, Polynomial.coeff_add, coeff_add, hp, hq]
| monomial j r =>
simp only [finSuccEquiv_apply, coe_eval₂Hom, eval₂_monomial, RingHom.coe_comp, Finsupp.prod_pow,
Polynomial.coeff_C_mul, coeff_C_mul, coeff_m... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
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