module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1158,
"column": 55
} | {
"line": 1158,
"column": 64
} | {
"line": 1158,
"column": 65
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : IsCancelMulZero α\ninst✝¹ : NormalizationMonoid α\ninst✝ : DecidableEq α\nlcm : α → α → α\ndvd_lcm_left : ∀ (a b : α), a ∣ lcm a b\ndvd_lcm_right : ∀ (a b : α), b ∣ lcm a b\nlcm_dvd : ∀ {a b c : α}, c ∣ a → b ∣ a → lcm c b ∣ a\nnormalize_l... | [
"case pos\nα : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : IsCancelMulZero α\ninst✝¹ : NormalizationMonoid α\ninst✝ : DecidableEq α\nlcm : α → α → α\ndvd_lcm_left : ∀ (a b : α), a ∣ lcm a b\ndvd_lcm_right : ∀ (a b : α), b ∣ lcm a b\nlcm_dvd : ∀ {a b c : α}, c ∣ a → b ∣ a → lcm c b ∣ a\nnormalize_lcm : ∀ (a b ... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1159,
"column": 58
} | {
"line": 1159,
"column": 67
} | {
"line": 1159,
"column": 68
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : IsCancelMulZero α\ninst✝¹ : NormalizationMonoid α\ninst✝ : DecidableEq α\nlcm : α → α → α\ndvd_lcm_left : ∀ (a b : α), a ∣ lcm a b\ndvd_lcm_right : ∀ (a b : α), b ∣ lcm a b\nlcm_dvd : ∀ {a b c : α}, c ∣ a → b ∣ a → lcm c b ∣ a\nnormalize_l... | [
"case pos\nα : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : IsCancelMulZero α\ninst✝¹ : NormalizationMonoid α\ninst✝ : DecidableEq α\nlcm : α → α → α\ndvd_lcm_left : ∀ (a b : α), a ∣ lcm a b\ndvd_lcm_right : ∀ (a b : α), b ∣ lcm a b\nlcm_dvd : ∀ {a b c : α}, c ∣ a → b ∣ a → lcm c b ∣ a\nnormalize_lcm : ∀ (a b ... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1296,
"column": 4
} | {
"line": 1296,
"column": 44
} | {
"line": 1297,
"column": 2
} | [
{
"pp": "α : Type u_1\nG₀ : Type u_2\ninst✝¹ : CommGroupWithZero G₀\ninst✝ : DecidableEq G₀\na b : G₀\n⊢ Associated ((if a = 0 ∧ b = 0 then 0 else 1) * if a = 0 ∨ b = 0 then 0 else 1) (a * b)",
"ppTerm": "?m.151",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWith... | [] | split_ifs <;> simp_all [Associated.comm] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1296,
"column": 4
} | {
"line": 1296,
"column": 44
} | {
"line": 1297,
"column": 2
} | [
{
"pp": "α : Type u_1\nG₀ : Type u_2\ninst✝¹ : CommGroupWithZero G₀\ninst✝ : DecidableEq G₀\na b : G₀\n⊢ Associated ((if a = 0 ∧ b = 0 then 0 else 1) * if a = 0 ∨ b = 0 then 0 else 1) (a * b)",
"ppTerm": "?m.151",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWith... | [] | split_ifs <;> simp_all [Associated.comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1296,
"column": 4
} | {
"line": 1296,
"column": 44
} | {
"line": 1297,
"column": 2
} | [
{
"pp": "α : Type u_1\nG₀ : Type u_2\ninst✝¹ : CommGroupWithZero G₀\ninst✝ : DecidableEq G₀\na b : G₀\n⊢ Associated ((if a = 0 ∧ b = 0 then 0 else 1) * if a = 0 ∨ b = 0 then 0 else 1) (a * b)",
"ppTerm": "?m.151",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWith... | [] | split_ifs <;> simp_all [Associated.comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1300,
"column": 49
} | {
"line": 1300,
"column": 67
} | {
"line": 1300,
"column": 68
} | [
{
"pp": "α : Type u_1\nG₀ : Type u_2\ninst✝¹ : CommGroupWithZero G₀\ninst✝ : DecidableEq G₀\na b : G₀\nh : a = 0 ∧ b = 0\n⊢ normalize (if a = 0 ∧ b = 0 then 0 else 1) = if a = 0 ∧ b = 0 then 0 else 1",
"ppTerm": "?m.188",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
... | [] | by simp [if_pos h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1301,
"column": 49
} | {
"line": 1301,
"column": 67
} | {
"line": 1301,
"column": 68
} | [
{
"pp": "α : Type u_1\nG₀ : Type u_2\ninst✝¹ : CommGroupWithZero G₀\ninst✝ : DecidableEq G₀\na b : G₀\nh : a = 0 ∨ b = 0\n⊢ normalize (if a = 0 ∨ b = 0 then 0 else 1) = if a = 0 ∨ b = 0 then 0 else 1",
"ppTerm": "?m.190",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
... | [] | by simp [if_pos h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.DedekindFinite | {
"line": 19,
"column": 4
} | {
"line": 20,
"column": 39
} | {
"line": 21,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝¹ : Monoid M\ninst✝ : Finite M\na b : M\nhab : a * b = 1\n⊢ b * a = 1",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Exists",
"id",
"MulOn... | [] | have ⟨c, hbc⟩ := Finite.surjective_of_injective (isLeftRegular_of_mul_eq_one hab) 1
rwa [left_inv_eq_right_inv hab hbc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.DedekindFinite | {
"line": 19,
"column": 4
} | {
"line": 20,
"column": 39
} | {
"line": 21,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝¹ : Monoid M\ninst✝ : Finite M\na b : M\nhab : a * b = 1\n⊢ b * a = 1",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Exists",
"id",
"MulOn... | [] | have ⟨c, hbc⟩ := Finite.surjective_of_injective (isLeftRegular_of_mul_eq_one hab) 1
rwa [left_inv_eq_right_inv hab hbc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Dimension.StrongRankCondition | {
"line": 611,
"column": 2
} | {
"line": 611,
"column": 47
} | {
"line": 612,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\n⊢ (∃ n, ℵ₀ ≤ Module.rank R (Fin n → R)) ↔ ∃ n > 0, finrank R (Fin n → R) ≤ 0",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"Nat.instMulZeroOneClass",
"Semiring.toModule... | [
"R : Type u\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\n⊢ (∃ n, ℵ₀ ≤ Module.rank R (Fin n → R)) ↔ ∃ n > 0, Module.rank R (Fin n → R) = 0 ∨ ℵ₀ ≤ Module.rank R (Fin n → R)"
] | simp_rw [finrank, Nat.le_zero, toNat_eq_zero] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Data.Set.UnionLift | {
"line": 116,
"column": 24
} | {
"line": 116,
"column": 28
} | {
"line": 116,
"column": 29
} | [
{
"pp": "α : Type u_1\nι : Sort u_2\nβ : Sort u_3\nS : ι → Set α\nf : (i : ι) → ↑(S i) → β\nhf : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩\nui : (i : ι) → ↑(S i) → ↑(S i)\nuβ : β → β\nh : ∀ (i : ι) (x : ↑(S i)), f i (ui i x) = uβ (f i x)\nu : ↑(iUnion S) → ↑(iUnion S)\nhui... | [
"α : Type u_1\nι : Sort u_2\nβ : Sort u_3\nS : ι → Set α\nf : (i : ι) → ↑(S i) → β\nhf : ∀ (i j : ι) (x : α) (hxi : x ∈ S i) (hxj : x ∈ S j), f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩\nui : (i : ι) → ↑(S i) → ↑(S i)\nuβ : β → β\nh : ∀ (i : ι) (x : ↑(S i)), f i (ui i x) = uβ (f i x)\nu : ↑(iUnion S) → ↑(iUnion S)\nhui : ∀ (i : ι)... | hui, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Data.Finsupp.Multiset | {
"line": 73,
"column": 4
} | {
"line": 77,
"column": 7
} | {
"line": 79,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\nβ : Type u_2\nf : α →₀ ℕ\ng : α → β\n⊢ ∀ (a : α) (b : ℕ) (f : α →₀ ℕ),\n a ∉ f.support →\n b ≠ 0 →\n Multiset.map g (toMultiset f) = toMultiset (mapDomain g f) →\n Multiset.map g (toMultiset (single a b + f)) = toMultiset (mapDomain g (single a b + f... | [] | intro a n f _ _ ih
rw [toMultiset_add, Multiset.map_add, ih, mapDomain_add, mapDomain_single,
toMultiset_single, toMultiset_add, toMultiset_single, ← Multiset.coe_mapAddMonoidHom,
(Multiset.mapAddMonoidHom g).map_nsmul]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finsupp.Multiset | {
"line": 73,
"column": 4
} | {
"line": 77,
"column": 7
} | {
"line": 79,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\nβ : Type u_2\nf : α →₀ ℕ\ng : α → β\n⊢ ∀ (a : α) (b : ℕ) (f : α →₀ ℕ),\n a ∉ f.support →\n b ≠ 0 →\n Multiset.map g (toMultiset f) = toMultiset (mapDomain g f) →\n Multiset.map g (toMultiset (single a b + f)) = toMultiset (mapDomain g (single a b + f... | [] | intro a n f _ _ ih
rw [toMultiset_add, Multiset.map_add, ih, mapDomain_add, mapDomain_single,
toMultiset_single, toMultiset_add, toMultiset_single, ← Multiset.coe_mapAddMonoidHom,
(Multiset.mapAddMonoidHom g).map_nsmul]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Matrix.Basic | {
"line": 948,
"column": 2
} | {
"line": 948,
"column": 77
} | {
"line": 949,
"column": 2
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nα : Type u_11\nι : Type u_14\ninst✝¹ : NonUnitalNonAssocSemiring α\ninst✝ : Fintype n\nx : n → α\ns : Finset ι\ny : ι → Matrix n m α\nx✝ : m\n⊢ (x ᵥ* ∑ i ∈ s, y i) x✝ = (∑ i ∈ s, x ᵥ* y i) x✝",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr... | [
"m : Type u_2\nn : Type u_3\nα : Type u_11\nι : Type u_14\ninst✝¹ : NonUnitalNonAssocSemiring α\ninst✝ : Fintype n\nx : n → α\ns : Finset ι\ny : ι → Matrix n m α\nx✝ : m\n⊢ ∑ x_1, ∑ i ∈ s, x x_1 * y i x_1 x✝ = ∑ x_1 ∈ s, ∑ x_2, x x_2 * y x_1 x_2 x✝"
] | simp only [vecMul, dotProduct, sum_apply, Finset.mul_sum, Finset.sum_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.TwoSidedIdeal.Basic | {
"line": 129,
"column": 53
} | {
"line": 130,
"column": 50
} | {
"line": 132,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nI : TwoSidedIdeal R\nx y : R\nhy : y ∈ I\n⊢ x * y ∈ I",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"RingCon.instFunLikeForallProp",
"congrArg",
"NonUnitalNonAssocSemiring.toMulZeroClass",
... | [] | by
simpa using! I.ringCon.mul (I.ringCon.refl x) hy | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Dimension.Free | {
"line": 357,
"column": 27
} | {
"line": 357,
"column": 42
} | {
"line": 358,
"column": 2
} | [
{
"pp": "R✝ : Type u\nS : Type u_1\nM✝ M₁ : Type v\nM' : Type v'\ninst✝¹⁵ : Semiring R✝\ninst✝¹⁴ : AddCommMonoid M✝\ninst✝¹³ : Module R✝ M✝\ninst✝¹² : Free R✝ M✝\ninst✝¹¹ : AddCommMonoid M'\ninst✝¹⁰ : Module R✝ M'\ninst✝⁹ : Free R✝ M'\ninst✝⁸ : AddCommMonoid M₁\ninst✝⁷ : Module R✝ M₁\ninst✝⁶ : Free R✝ M₁\ninst✝... | [] | simp [mul_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Algebra.Subalgebra.Lattice | {
"line": 434,
"column": 8
} | {
"line": 434,
"column": 16
} | {
"line": 434,
"column": 17
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\ns✝ : Subalgebra R S\nM✝ : Submonoid S\nH✝ : M✝ ≤ s✝.toSubmonoid\ns : Subalgebra R S\nM : Submonoid S\nH : M ≤ s.toSubmonoid\na b m : S\nhm : m ∈ M\nha : m * a ∈ s\nn : S\nhn : n ∈ M\nhb : n * b ∈ s\n⊢ n *... | [
"R : Type u_1\nS : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\ns✝ : Subalgebra R S\nM✝ : Submonoid S\nH✝ : M✝ ≤ s✝.toSubmonoid\ns : Subalgebra R S\nM : Submonoid S\nH : M ≤ s.toSubmonoid\na b m : S\nhm : m ∈ M\nha : m * a ∈ s\nn : S\nhn : n ∈ M\nhb : n * b ∈ s\n⊢ n * m * a + n *... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.DirectSum.Finsupp | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 13
} | {
"line": 68,
"column": 14
} | [
{
"pp": "case single\nR : 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✝ :... | [] | | single => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.LinearAlgebra.DirectSum.Finsupp | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 13
} | {
"line": 101,
"column": 14
} | [
{
"pp": "case single\nR : 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✝ :... | [] | | single => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Data.Matrix.Block | {
"line": 243,
"column": 53
} | {
"line": 245,
"column": 39
} | {
"line": 247,
"column": 0
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn : Type u_3\no : Type u_4\nα : Type u_12\ninst✝² : Fintype n\ninst✝¹ : Fintype o\ninst✝ : NonUnitalNonAssocSemiring α\nA : Matrix n l α\nB : Matrix n m α\nC : Matrix o l α\nD : Matrix o m α\nx : n ⊕ o → α\n⊢ x ᵥ* fromBlocks A B C D = Sum.elim (x ∘ Sum.inl ᵥ* A + x ∘ Sum.inr... | [] | by
ext i
cases i <;> simp [vecMul, dotProduct] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.FreeModule.PID | {
"line": 196,
"column": 8
} | {
"line": 196,
"column": 37
} | {
"line": 197,
"column": 8
} | [
{
"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 ... | [
"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 N < ψ.submod... | simp only [map_sum, map_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.OreLocalization.Basic | {
"line": 52,
"column": 47
} | {
"line": 52,
"column": 56
} | {
"line": 52,
"column": 57
} | [
{
"pp": "case c\nR : Type u_1\ninst✝¹ : MonoidWithZero R\nS : Submonoid R\ninst✝ : OreSet S\nr : R\ns : ↥S\n⊢ r * 0 /ₒ s = 0 /ₒ 1",
"ppTerm": "?c",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"MulZeroClass.toMul",
"Monoid.toMulOneClass",
"congrArg",
... | [
"case c\nR : Type u_1\ninst✝¹ : MonoidWithZero R\nS : Submonoid R\ninst✝ : OreSet S\nr : R\ns : ↥S\n⊢ 0 /ₒ s = 0 /ₒ 1"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.OreLocalization.Basic | {
"line": 127,
"column": 4
} | {
"line": 127,
"column": 28
} | {
"line": 129,
"column": 0
} | [
{
"pp": "case h\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\nr₂ : X\ns₂ : ↥S\nr₁' : X\ns₁' : ↥S\nr₁ : X\ns₁ sb : ↥S\nrb : R\nhb : sb • r₁ = rb • r₁'\nhb' : ↑sb * ↑s₁ = rb * ↑s₁'\nrc : R\nsc : ↥S\nhc : ↑sc * ↑s₁' = rc * ↑s... | [] | rw [this, hc, mul_assoc] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.GroupTheory.MonoidLocalization.MonoidWithZero | {
"line": 40,
"column": 11
} | {
"line": 40,
"column": 20
} | {
"line": 40,
"column": 21
} | [
{
"pp": "M : Type u_1\ninst✝¹ : CommMonoidWithZero M\nS : Submonoid M\nN : Type u_2\ninst✝ : CommMonoidWithZero N\nf : S.LocalizationMap N\nx✝ : Subsingleton N\nc : ↥S\neq : ↑c * 0 = ↑c * 1\n⊢ 0 ∈ S",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
... | [
"M : Type u_1\ninst✝¹ : CommMonoidWithZero M\nS : Submonoid M\nN : Type u_2\ninst✝ : CommMonoidWithZero N\nf : S.LocalizationMap N\nx✝ : Subsingleton N\nc : ↥S\neq : 0 = ↑c * 1\n⊢ 0 ∈ S"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.MonoidLocalization.MonoidWithZero | {
"line": 96,
"column": 46
} | {
"line": 96,
"column": 55
} | {
"line": 96,
"column": 56
} | [
{
"pp": "M : Type u_1\ninst✝² : CommMonoidWithZero M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoidWithZero N\nP : Type u_3\ninst✝ : CommMonoidWithZero P\nf : S.LocalizationMap N\ng : M →*₀ P\nhg : ∀ (y : ↥S), IsUnit (g ↑y)\n⊢ ↑g (f.sec 0).1 = ↑g ↑(f.sec 0).2 * 0",
"ppTerm": "?m.48",
"assigned": tr... | [
"M : Type u_1\ninst✝² : CommMonoidWithZero M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoidWithZero N\nP : Type u_3\ninst✝ : CommMonoidWithZero P\nf : S.LocalizationMap N\ng : M →*₀ P\nhg : ∀ (y : ↥S), IsUnit (g ↑y)\n⊢ ↑g (f.sec 0).1 = 0"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.OreLocalization.Ring | {
"line": 149,
"column": 10
} | {
"line": 149,
"column": 18
} | {
"line": 149,
"column": 19
} | [
{
"pp": "case c.c\nR : Type u_1\ninst✝⁴ : Semiring R\nS : Submonoid R\ninst✝³ : OreSet S\nX : Type u_2\ninst✝² : AddCommMonoid X\ninst✝¹ : Module R X\nT : Type u_3\ninst✝ : Semiring T\nf : R →+* T\nfS : ↥S →* Tˣ\nhf : ∀ (s : ↥S), f ↑s = ↑(fS s)\nr₁ : R\ns₁ : ↥S\nr₂ : R\ns₂ : ↥S\nr₃ : R\ns₃ : ↥S\nh₃ : ↑s₃ * ↑s₁ ... | [
"case c.c\nR : Type u_1\ninst✝⁴ : Semiring R\nS : Submonoid R\ninst✝³ : OreSet S\nX : Type u_2\ninst✝² : AddCommMonoid X\ninst✝¹ : Module R X\nT : Type u_3\ninst✝ : Semiring T\nf : R →+* T\nfS : ↥S →* Tˣ\nhf : ∀ (s : ↥S), f ↑s = ↑(fS s)\nr₁ : R\ns₁ : ↥S\nr₂ : R\ns₂ : ↥S\nr₃ : R\ns₃ : ↥S\nh₃ : ↑s₃ * ↑s₁ = r₃ * ↑s₂\n... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.FreeModule.PID | {
"line": 357,
"column": 4
} | {
"line": 357,
"column": 46
} | {
"line": 358,
"column": 4
} | [
{
"pp": "ι : Type u_1\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nb : ι → M\ninst✝³ : IsPrincipalIdealRing R\ninst✝² : IsDomain R\ninst✝¹ : Fintype ι\ns : ι → M\nhs : span R (range s) = ⊤\ninst✝ : IsTorsionFree R M\nthis : ∃ s_1, LinearIndepOn R s s_1 ∧ ∀ i ∉ ... | [
"ι : Type u_1\nR : Type u_2\ninst✝⁶ : CommRing R\nM : Type u_3\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nb : ι → M\ninst✝³ : IsPrincipalIdealRing R\ninst✝² : IsDomain R\ninst✝¹ : Fintype ι\ns : ι → M\nhs : span R (range s) = ⊤\ninst✝ : IsTorsionFree R M\nthis : ∃ s_1, LinearIndepOn R s s_1 ∧ ∀ i ∉ s_1, ∃ a, a ... | let φ : M →ₗ[R] M := LinearMap.lsmul R M A | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 1183,
"column": 4
} | {
"line": 1183,
"column": 21
} | {
"line": 1185,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝⁸ : Fintype ι\ninst✝⁷ : DecidableEq ι\nR : Type u_2\ninst✝⁶ : CommSemiring R\nA : Type u_3\ninst✝⁵ : Semiring A\ninst✝⁴ : Algebra R A\nM : Type u_4\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module A M\ninst✝ : IsScalarTower R A M\nr : R\ni✝ j✝ : ι\nx✝ : M\n⊢ (if i✝ = j... | [] | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.RingTheory.Localization.Defs | {
"line": 466,
"column": 30
} | {
"line": 466,
"column": 38
} | {
"line": 466,
"column": 39
} | [
{
"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\nx₁ x₂ : R\ny₁ y₂ : ↥M\n⊢ (algebraMap R S) ↑(y₁ * y₂) * (mk' S x₁ y₁ + mk' S x₂ y₂) = (algebraMap R S) (x₁ * ↑y₂ + x₂ * ↑y₁)",
"ppTerm": "?m.76",
"assi... | [
"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\nx₁ x₂ : R\ny₁ y₂ : ↥M\n⊢ (algebraMap R S) ↑(y₁ * y₂) * mk' S x₁ y₁ + (algebraMap R S) ↑(y₁ * y₂) * mk' S x₂ y₂ =\n (algebraMap R S) (x₁ * ↑y₂ + x₂ * ↑y₁)"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.Defs | {
"line": 499,
"column": 8
} | {
"line": 499,
"column": 16
} | {
"line": 499,
"column": 17
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\ng : R →+* P\nhg : ∀ (y : ↥M), IsUnit (g ↑y)\nx y : S\n⊢ ↑g.toMonoidWithZeroHom ((toLocalizationMap M S).sec (x + y)).1 ... | [
"R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\ng : R →+* P\nhg : ∀ (y : ↥M), IsUnit (g ↑y)\nx y : S\n⊢ ↑g.toMonoidWithZeroHom ((toLocalizationMap M S).sec (x + y)).1 =\n ↑g.to... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.FreeModule.PID | {
"line": 499,
"column": 2
} | {
"line": 499,
"column": 87
} | {
"line": 500,
"column": 2
} | [
{
"pp": "case neg\nι : Type u_1\nR : Type u_2\ninst✝⁵ : CommRing R\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : IsDomain R\ninst✝¹ : IsPrincipalIdealRing R\ninst✝ : Finite ι\nb : Basis ι R M\nval✝ : Fintype ι\nM0 : Submodule R M\nih :\n ∀ N' ≤ M0,\n ∀ x ∈ M0,\n (∀ (c : R), ∀ y ... | [
"case neg\nι : Type u_1\nR : Type u_2\ninst✝⁵ : CommRing R\nM : Type u_3\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : IsDomain R\ninst✝¹ : IsPrincipalIdealRing R\ninst✝ : Finite ι\nb : Basis ι R M\nval✝ : Fintype ι\nM0 : Submodule R M\nih :\n ∀ N' ≤ M0,\n ∀ x ∈ M0,\n (∀ (c : R), ∀ y ∈ N', c • x ... | obtain ⟨n', m', hn'm', bM', bN', as', has'⟩ := ih M' M'_le_M y hy y_ortho N' N'_le_M' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Localization.Defs | {
"line": 710,
"column": 10
} | {
"line": 710,
"column": 68
} | {
"line": 710,
"column": 68
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁶ : CommSemiring S\ninst✝⁵ : Algebra R S\nP : Type u_3\ninst✝⁴ : CommSemiring P\ninst✝³ : IsLocalization M S\ng : R →+* P\nhg : ∀ (y : ↥M), IsUnit (g ↑y)\nT : Submonoid P\nQ : Type u_4\ninst✝² : CommSemiring Q\ninst✝¹ : Algebra ... | [
"R : Type u_1\ninst✝⁷ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝⁶ : CommSemiring S\ninst✝⁵ : Algebra R S\nP : Type u_3\ninst✝⁴ : CommSemiring P\ninst✝³ : IsLocalization M S\ng : R →+* P\nhg : ∀ (y : ↥M), IsUnit (g ↑y)\nT : Submonoid P\nQ : Type u_4\ninst✝² : CommSemiring Q\ninst✝¹ : Algebra P Q\ninst✝ :... | Submonoid.comap_map_eq_of_injective (j : R ≃* P).injective | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.Defs | {
"line": 751,
"column": 4
} | {
"line": 752,
"column": 39
} | {
"line": 753,
"column": 4
} | [
{
"pp": "case exists_of_eq\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := ((algebraMap R S).comp h.symm.toRingHom).toAlgebra\nx y : P\n⊢ (a... | [
"case exists_of_eq\nR : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_3\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\nh : R ≃+* P\nthis : Algebra P S := ((algebraMap R S).comp h.symm.toRingHom).toAlgebra\nx y : P\n⊢ (∃ c, ↑c * h.s... | rw [RingHom.algebraMap_toAlgebra, RingHom.comp_apply, RingHom.comp_apply,
IsLocalization.eq_iff_exists M S] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Star.Pointwise | {
"line": 119,
"column": 78
} | {
"line": 121,
"column": 41
} | {
"line": 123,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Group α\ninst✝ : StarMul α\ns : Set α\n⊢ s⁻¹⋆ = s⋆⁻¹",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Set.ext",
"_private.Mathlib.Algebra.Star.Pointwise.0.Set.star_inv._simp_1_1",
"Set.star",
"DivInvOneMonoid.toInvOneClass",
"Mo... | [] | by
ext
simp only [mem_star, mem_inv, star_inv] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Algebra.Spectrum.Basic | {
"line": 109,
"column": 35
} | {
"line": 111,
"column": 54
} | {
"line": 113,
"column": 0
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝³ : CommSemiring R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : Nontrivial A\nr : R\na : A\n⊢ r ∈ σ a ↔ resolvent a r = 0",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Algebra.algebraMap",
"spectrum",
"AddGroupWi... | [] | by
refine ⟨resolvent_zero_of_mem_spectrum, fun hr ↦ ?_⟩
simpa [mem_iff, Ring.not_isUnit_iff_inverse_eq_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Algebra.Spectrum.Basic | {
"line": 174,
"column": 8
} | {
"line": 174,
"column": 19
} | {
"line": 174,
"column": 20
} | [
{
"pp": "case pos\nR : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nr : Rˣ\ns : R\na : A\nh : ¬IsUnit (s • 1 - a)\n⊢ r • (s • 1 - a)⁻¹ʳ = ((r⁻¹ • s) • 1 - r⁻¹ • a)⁻¹ʳ",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiri... | [
"case pos\nR : Type u\nA : Type v\ninst✝² : CommSemiring R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\nr : Rˣ\ns : R\na : A\nh : ¬IsUnit (s • 1 - a)\n⊢ r • (s • 1 - a)⁻¹ʳ = (r⁻¹ • s • 1 - r⁻¹ • a)⁻¹ʳ"
] | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.TensorProduct.Basic | {
"line": 512,
"column": 15
} | {
"line": 512,
"column": 77
} | {
"line": 512,
"column": 77
} | [
{
"pp": "R : 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✝⁴ : CommSemiring R\ninst✝³ : Semiring A\ninst✝² : Algebra R A\ninst✝¹ : CommSemiring B\ninst✝ : Algebra R B\na : A\nb : B\nx : A ⊗[R] B\nx✝¹ : A\nx✝ : B\n⊢ a • b • x... | [] | by simp [Algebra.smul_def, right_algebraMap_apply, smul_tmul'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Algebra.Spectrum.Basic | {
"line": 353,
"column": 2
} | {
"line": 353,
"column": 85
} | {
"line": 354,
"column": 2
} | [
{
"pp": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : Nontrivial A\nk : 𝕜\nhk : k ≠ 0\n⊢ k ∈ resolventSet 𝕜 0",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"Units.instMulAction",
"Algebra... | [
"𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : Nontrivial A\nk : 𝕜\nhk : k ≠ 0\nthis : IsUnit (Units.mk0 k hk • 1)\n⊢ k ∈ resolventSet 𝕜 0"
] | have : IsUnit (Units.mk0 k hk • (1 : A)) := IsUnit.smul (Units.mk0 k hk) isUnit_one | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Algebra.Spectrum.Basic | {
"line": 354,
"column": 2
} | {
"line": 354,
"column": 62
} | {
"line": 356,
"column": 0
} | [
{
"pp": "𝕜 : Type u\nA : Type v\ninst✝³ : Field 𝕜\ninst✝² : Ring A\ninst✝¹ : Algebra 𝕜 A\ninst✝ : Nontrivial A\nk : 𝕜\nhk : k ≠ 0\nthis : IsUnit (Units.mk0 k hk • 1)\n⊢ k ∈ resolventSet 𝕜 0",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCo... | [] | simpa [mem_resolventSet_iff, Algebra.algebraMap_eq_smul_one] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Algebra.Unitization | {
"line": 638,
"column": 10
} | {
"line": 638,
"column": 18
} | {
"line": 638,
"column": 19
} | [
{
"pp": "case inl_add_inr\nS : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁸ : CommSemiring S\ninst✝⁷ : CommSemiring R\ninst✝⁶ : NonUnitalSemiring A\ninst✝⁵ : Module R A\ninst✝⁴ : IsScalarTower R A A\ninst✝³ : SMulCommClass R A A\ninst✝² : Algebra S R\ninst✝¹ : DistribMulAction S A\ninst✝ : IsScalarTower S R A\n... | [
"case inl_add_inr\nS : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁸ : CommSemiring S\ninst✝⁷ : CommSemiring R\ninst✝⁶ : NonUnitalSemiring A\ninst✝⁵ : Module R A\ninst✝⁴ : IsScalarTower R A A\ninst✝³ : SMulCommClass R A A\ninst✝² : Algebra S R\ninst✝¹ : DistribMulAction S A\ninst✝ : IsScalarTower S R A\ns : S\nr✝ : ... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.Unitization | {
"line": 643,
"column": 10
} | {
"line": 643,
"column": 18
} | {
"line": 643,
"column": 19
} | [
{
"pp": "case inl_add_inr\nS : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁸ : CommSemiring S\ninst✝⁷ : CommSemiring R\ninst✝⁶ : NonUnitalSemiring A\ninst✝⁵ : Module R A\ninst✝⁴ : IsScalarTower R A A\ninst✝³ : SMulCommClass R A A\ninst✝² : Algebra S R\ninst✝¹ : DistribMulAction S A\ninst✝ : IsScalarTower S R A\n... | [
"case inl_add_inr\nS : Type u_1\nR : Type u_2\nA : Type u_3\ninst✝⁸ : CommSemiring S\ninst✝⁷ : CommSemiring R\ninst✝⁶ : NonUnitalSemiring A\ninst✝⁵ : Module R A\ninst✝⁴ : IsScalarTower R A A\ninst✝³ : SMulCommClass R A A\ninst✝² : Algebra S R\ninst✝¹ : DistribMulAction S A\ninst✝ : IsScalarTower S R A\ns : S\nr✝ : ... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.Unitization | {
"line": 747,
"column": 21
} | {
"line": 747,
"column": 29
} | {
"line": 747,
"column": 30
} | [
{
"pp": "case inl_add_inr.inl_add_inr\nS : 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 : A... | [
"case inl_add_inr.inl_add_inr\nS : 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\n... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 217,
"column": 12
} | {
"line": 217,
"column": 21
} | {
"line": 217,
"column": 22
} | [
{
"pp": "R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\na : R\nha : a ∈ AddSubmonoid.closure (range fun s ↦ star s * s)\nc : R\n⊢ 0 ≤ star c * 0 * c",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\na : R\nha : a ∈ AddSubmonoid.closure (range fun s ↦ star s * s)\nc : R\n⊢ 0 ≤ 0 * c"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Star.Basic | {
"line": 224,
"column": 22
} | {
"line": 224,
"column": 30
} | {
"line": 224,
"column": 31
} | [
{
"pp": "R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\na : R\nha : a ∈ AddSubmonoid.closure (range fun s ↦ star s * s)\nc x y : R\nx✝¹ : x ∈ AddSubmonoid.closure (range fun s ↦ star s * s)\nx✝ : y ∈ AddSubmonoid.closure (range fun s ↦ star s... | [
"R : Type u_1\ninst✝³ : NonUnitalSemiring R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\na : R\nha : a ∈ AddSubmonoid.closure (range fun s ↦ star s * s)\nc x y : R\nx✝¹ : x ∈ AddSubmonoid.closure (range fun s ↦ star s * s)\nx✝ : y ∈ AddSubmonoid.closure (range fun s ↦ star s * s)\nhx : ... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.StrictPositivity | {
"line": 136,
"column": 2
} | {
"line": 136,
"column": 58
} | {
"line": 137,
"column": 2
} | [
{
"pp": "A : Type u_1\n𝕜 : Type u_2\ninst✝⁵ : Ring A\ninst✝⁴ : PartialOrder A\ninst✝³ : CommSemiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : Algebra 𝕜 A\ninst✝ : NonnegSpectrumClass 𝕜 A\na : A\nha : IsStrictlyPositive a\nx : 𝕜\nhx : x ∈ spectrum 𝕜 a\nh₁ : 0 ≤ x\n⊢ 0 < x",
"ppTerm": "?m.29",
"assigne... | [
"A : Type u_1\n𝕜 : Type u_2\ninst✝⁵ : Ring A\ninst✝⁴ : PartialOrder A\ninst✝³ : CommSemiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : Algebra 𝕜 A\ninst✝ : NonnegSpectrumClass 𝕜 A\na : A\nha : IsStrictlyPositive a\nx : 𝕜\nhx : x ∈ spectrum 𝕜 a\nh₁ : 0 ≤ x\nh₂ : x ≠ 0\n⊢ 0 < x"
] | have h₂ : x ≠ 0 := by grind [= spectrum.zero_notMem_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Order.Star.Basic | {
"line": 404,
"column": 4
} | {
"line": 404,
"column": 63
} | {
"line": 406,
"column": 0
} | [
{
"pp": "R : Type u_1\nA : Type u_2\ninst✝¹¹ : Semiring R\ninst✝¹⁰ : PartialOrder R\ninst✝⁹ : StarRing R\ninst✝⁸ : StarOrderedRing R\ninst✝⁷ : NonUnitalSemiring A\ninst✝⁶ : StarRing A\ninst✝⁵ : PartialOrder A\ninst✝⁴ : StarOrderedRing A\ninst✝³ : Module R A\ninst✝² : StarModule R A\ninst✝¹ : IsScalarTower R A A... | [] | exact ⟨r • a, smul_mem_closure_star_mul hr ha, add_smul ..⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Algebra.Subalgebra.Directed | {
"line": 58,
"column": 59
} | {
"line": 88,
"column": 35
} | {
"line": 91,
"column": 0
} | [
{
"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
let compat :
∀ (i j) (x : A) (hxi : x ∈ (K i : Set A)) (hxj : x ∈ (K j : Set A)),
f i ⟨x, hxi⟩ = f j ⟨x, hxj⟩ := by
intro i j x hxi hxj
rcases dir i j with ⟨k, hik, hjk⟩
simp [hf i k hik, hf j k hjk]
let liftSup : ((iSup K : Subalgebra R A)) →ₐ[R] B :=
{ toFun :=
Set.iUnionL... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Algebra.Subalgebra.Centralizer | {
"line": 99,
"column": 36
} | {
"line": 99,
"column": 44
} | {
"line": 99,
"column": 45
} | [
{
"pp": "case mpr.add\nR : Type u_1\ninst✝⁵ : CommSemiring R\nA : Type u_2\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_3\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nx : A\nhx : x ∈ S\ny z : ↥(centralizer R S) ⊗[R] B\nhy :\n includeLeft x * (Algebra.TensorProduct.m... | [
"case mpr.add\nR : Type u_1\ninst✝⁵ : CommSemiring R\nA : Type u_2\ninst✝⁴ : Semiring A\ninst✝³ : Algebra R A\nB : Type u_3\ninst✝² : Semiring B\ninst✝¹ : Algebra R B\nS : Set A\ninst✝ : Module.Free R B\nx : A\nhx : x ∈ S\ny z : ↥(centralizer R S) ⊗[R] B\nhy :\n includeLeft x * (Algebra.TensorProduct.map (centrali... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Dimension.Finite | {
"line": 120,
"column": 25
} | {
"line": 125,
"column": 32
} | {
"line": 127,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Free R M\nn : ℕ\nh : Module.rank R M = ↑n\n⊢ Module.Finite R M",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Nontrivial",
"Preorder.toLT",
"LE.le.trans_eq",
... | [] | by
nontriviality R
obtain ⟨⟨ι, b⟩⟩ := Module.Free.exists_basis (R := R) (M := M)
have := mk_lt_aleph0_iff.mp <|
b.linearIndependent.cardinal_le_rank |>.trans_eq h |>.trans_lt natCast_lt_aleph0
exact Module.Finite.of_basis b | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Dimension.Constructions | {
"line": 178,
"column": 25
} | {
"line": 178,
"column": 72
} | {
"line": 178,
"column": 73
} | [
{
"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✝ : Free R M\nι : Type w\nfst✝ : Type v\nbs : Basis fst✝ R M\n⊢ Module.rank R (ι →₀ M) = lift.{v, w} #ι * lift.{w, v} #fst✝",
"ppTerm": "?m.53",
"assigned": true,
... | [
"R : Type u\nM : Type v\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : StrongRankCondition R\ninst✝ : Free R M\nι : Type w\nfst✝ : Type v\nbs : Basis fst✝ R M\n⊢ #((_ : ι) × fst✝) = lift.{v, w} #ι * lift.{w, v} #fst✝"
] | ← (Finsupp.basis fun _ : ι => bs).mk_eq_rank'', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Dimension.Constructions | {
"line": 597,
"column": 64
} | {
"line": 599,
"column": 58
} | {
"line": 601,
"column": 0
} | [
{
"pp": "R : Type u_2\nV : Type u_3\ninst✝² : CommRing R\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\nW : Submodule R V\nm : Type u_4\nn : Type u_5\nbW : Basis m R ↥W\nbQ : Basis n R (V ⧸ W)\nj : n\n⊢ Submodule.Quotient.mk ((bW.sumQuot bQ) (Sum.inr j)) = bQ j",
"ppTerm": "?m.35",
"assigned": true,
... | [] | by
simpa only [sumQuot, Basis.coe_mk, Sum.elim_inr, Function.comp_apply, ← W.mkQ_apply]
using Function.rightInverse_surjInv W.mkQ_surjective _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 149,
"column": 8
} | {
"line": 149,
"column": 34
} | {
"line": 150,
"column": 8
} | [
{
"pp": "case a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : Ring R\ninst✝⁷ : Ring S\ninst✝⁶ : Ring T\nσ : R →+* S\nτ : S →+* T\nE : Type u_4\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_5\ninst✝³ : AddCommGroup F\ninst✝² : Module S F\nG : Type u_6\ninst✝¹ : AddCommGroup G\ninst✝ : Module T ... | [
"case a\nR : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : Ring R\ninst✝⁷ : Ring S\ninst✝⁶ : Ring T\nσ : R →+* S\nτ : S →+* T\nE : Type u_4\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\nF : Type u_5\ninst✝³ : AddCommGroup F\ninst✝² : Module S F\nG : Type u_6\ninst✝¹ : AddCommGroup G\ninst✝ : Module T G\nx : E\ny ... | simp only [mul_smul, this] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Dimension.FreeAndStrongRankCondition | {
"line": 303,
"column": 7
} | {
"line": 303,
"column": 41
} | {
"line": 303,
"column": 42
} | [
{
"pp": "F : Type u_1\nE : Type u_2\ninst✝⁴ : CommRing F\ninst✝³ : StrongRankCondition F\ninst✝² : Ring E\ninst✝¹ : Algebra F E\ninst✝ : Free F E\na✝ : Nontrivial E\nh : finrank F E = 1\n⊢ Function.Surjective ⇑(algebraMap F E)",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Eq.mpr",... | [
"F : Type u_1\nE : Type u_2\ninst✝⁴ : CommRing F\ninst✝³ : StrongRankCondition F\ninst✝² : Ring E\ninst✝¹ : Algebra F E\ninst✝ : Free F E\na✝ : Nontrivial E\nh : finrank F E = 1\n⊢ ⊤ = ⊥"
] | Algebra.surjective_algebraMap_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MonoidAlgebra.Basic | {
"line": 68,
"column": 19
} | {
"line": 68,
"column": 30
} | {
"line": 68,
"column": 31
} | [
{
"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\nt' : R\na... | [
"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\nt' : R\na : R[M]\n⊢ (... | smul_assoc, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.FreeAlgebra | {
"line": 290,
"column": 8
} | {
"line": 290,
"column": 19
} | {
"line": 290,
"column": 20
} | [
{
"pp": "case e_a\nR✝ : Type u_1\nX : Type u_2\ninst✝⁷ : CommSemiring R✝\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : CommSemiring A\ninst✝³ : SMul R S\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\nr : R\ns : S\nx : FreeAlgebra... | [
"case e_a\nR✝ : Type u_1\nX : Type u_2\ninst✝⁷ : CommSemiring R✝\nR : Type u_3\nS : Type u_4\nA : Type u_5\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring S\ninst✝⁴ : CommSemiring A\ninst✝³ : SMul R S\ninst✝² : Algebra R A\ninst✝¹ : Algebra S A\ninst✝ : IsScalarTower R S A\nr : R\ns : S\nx : FreeAlgebra A X\n⊢ r • ... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.LinearPMap | {
"line": 330,
"column": 2
} | {
"line": 330,
"column": 42
} | {
"line": 331,
"column": 2
} | [
{
"pp": "R : 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\nf g : E →ₛₗ.[σ] F\nh : ∀ (x : ↥f.domain) (y : ↥g.domain), ↑x = ↑y → ↑f x = ↑g y\n⊢ f ≤ f.sup g h",
"ppTer... | [
"R : 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\nf g : E →ₛₗ.[σ] F\nh : ∀ (x : ↥f.domain) (y : ↥g.domain), ↑x = ↑y → ↑f x = ↑g y\nz₁ : ↥f.domain\nz₂ : ↥(f.sup g h).domain... | refine ⟨le_sup_left, fun z₁ z₂ hz => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Data.Nat.Prime.Defs | {
"line": 352,
"column": 8
} | {
"line": 352,
"column": 59
} | {
"line": 352,
"column": 60
} | [
{
"pp": "n : ℕ\npos : 0 < n\nnp : n < 2\nh1 : n = n.minFac\n⊢ n.minFac ≤ n / n.minFac",
"ppTerm": "?m.146",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHDiv",
"congrArg",
"Nat.succ_le_of_lt",
"id",
"Nat.minFac",
"HDiv.hDiv",
"instOfNatNat",
... | [
"n : ℕ\npos : 0 < n\nnp : n < 2\nh1 : n = n.minFac\n⊢ minFac 1 ≤ 1 / minFac 1"
] | le_antisymm (le_of_lt_succ np) (succ_le_of_lt pos), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Eval.Defs | {
"line": 454,
"column": 6
} | {
"line": 454,
"column": 14
} | {
"line": 454,
"column": 15
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\n⊢ (p * (X + ↑n)).comp q = p.comp q * (q + ↑n)",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"id",
"Polynomial.instAdd",
"Na... | [
"R : Type u\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\n⊢ (p * X + p * ↑n).comp q = p.comp q * (q + ↑n)"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Coeff | {
"line": 113,
"column": 2
} | {
"line": 115,
"column": 79
} | {
"line": 117,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\n⊢ (p * q).coeff n = ∑ x ∈ antidiagonal n, p.coeff x.1 * q.coeff x.2",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Finset",
"Nat.instA... | [] | rcases p with ⟨p⟩; rcases q with ⟨q⟩
simp_rw [← ofFinsupp_mul, coeff]
exact AddMonoidAlgebra.mul_apply_antidiagonal p q n _ Finset.mem_antidiagonal | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Coeff | {
"line": 113,
"column": 2
} | {
"line": 115,
"column": 79
} | {
"line": 117,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\nn : ℕ\n⊢ (p * q).coeff n = ∑ x ∈ antidiagonal n, p.coeff x.1 * q.coeff x.2",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Finset",
"Nat.instA... | [] | rcases p with ⟨p⟩; rcases q with ⟨q⟩
simp_rw [← ofFinsupp_mul, coeff]
exact AddMonoidAlgebra.mul_apply_antidiagonal p q n _ Finset.mem_antidiagonal | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Coeff | {
"line": 201,
"column": 73
} | {
"line": 201,
"column": 82
} | {
"line": 201,
"column": 83
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nk m : ℕ\nhkm : k ≠ m\nx y : R\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ x + y * 0 ≠ 0 ∧ x * 0 + y ≠ 0",
"ppTerm": "?m.74",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"id",
"Distrib.toAdd",
"Ne",
"MulZero... | [
"R : Type u\ninst✝ : Semiring R\nk m : ℕ\nhkm : k ≠ m\nx y : R\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ x + 0 ≠ 0 ∧ 0 + y ≠ 0"
] | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 131,
"column": 12
} | {
"line": 131,
"column": 23
} | {
"line": 131,
"column": 23
} | [
{
"pp": "case pos\nR : Type u\ninst✝ : Semiring R\np q : R[X]\nh : q.coeff p.natDegree ≠ 0\nhp : p = 0\n⊢ degree 0 ≤ q.degree",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"WithBot",
"congrArg",
"Preorder.toLE",
"id",
... | [
"case pos\nR : Type u\ninst✝ : Semiring R\np q : R[X]\nh : q.coeff p.natDegree ≠ 0\nhp : p = 0\n⊢ ⊥ ≤ q.degree"
] | degree_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Coeff | {
"line": 210,
"column": 75
} | {
"line": 210,
"column": 84
} | {
"line": 210,
"column": 85
} | [
{
"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⊢ x + y * 0 + z * 0 ≠ 0 ∧ x * 0 + y + z * 0 ≠ 0 ∧ x * 0 + y * 0 + z ≠ 0",
"ppTerm": "?m.106",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"co... | [
"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⊢ x + 0 + 0 ≠ 0 ∧ 0 + y + 0 ≠ 0 ∧ 0 + 0 + z ≠ 0"
] | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 199,
"column": 53
} | {
"line": 199,
"column": 64
} | {
"line": 199,
"column": 64
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nn : ℕ\na : R\nthis : DecidableEq R := Classical.decEq R\nh : a = 0\n⊢ degree 0 ≤ ↑n",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"WithBot",
"congrArg",
"AddMonoid.toAddZeroClass",
... | [
"R : Type u\ninst✝ : Semiring R\nn : ℕ\na : R\nthis : DecidableEq R := Classical.decEq R\nh : a = 0\n⊢ ⊥ ≤ ↑n"
] | degree_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 248,
"column": 2
} | {
"line": 248,
"column": 36
} | {
"line": 250,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nh : p ≠ 0\n⊢ (↑p.natDegree).succ = p.natDegree + 1",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"WithBot.succ_coe",
"Nat.instSuccOrder",
"Nat.instPreorder",
"Nat",
"Polynomial.natDegree",
"Nat.instOrd... | [] | exact WithBot.succ_coe p.natDegree | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Polynomial.Coeff | {
"line": 258,
"column": 6
} | {
"line": 258,
"column": 23
} | {
"line": 258,
"column": 24
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nn : ℕ\n⊢ (X * p).coeff (n + 1) = p.coeff n",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"id",
"instOfNatNat",
"Polynomial",
"Polynomial.coeff",
"instHAdd... | [
"R : Type u\ninst✝ : Semiring R\np : R[X]\nn : ℕ\n⊢ (p * X).coeff (n + 1) = p.coeff n"
] | (commute_X p).eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.Defs | {
"line": 411,
"column": 10
} | {
"line": 411,
"column": 24
} | {
"line": 411,
"column": 25
} | [
{
"pp": "case succ\nR : Type u\ninst✝ : Semiring R\np : R[X]\na : WithBot ℕ\nhp : p.degree ≤ a\nn : ℕ\nhn : (p ^ n).degree ≤ ↑n * a\n⊢ (p ^ (n + 1)).degree ≤ ↑(n + 1) * a",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"WithBot.addMonoidWithOne",
"WithBot.instPreorder",
"... | [
"case succ\nR : Type u\ninst✝ : Semiring R\np : R[X]\na : WithBot ℕ\nhp : p.degree ≤ a\nn : ℕ\nhn : (p ^ n).degree ≤ ↑n * a\n⊢ (p ^ (n + 1)).degree ≤ (↑n + 1) * a"
] | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Coeff | {
"line": 315,
"column": 2
} | {
"line": 317,
"column": 23
} | {
"line": 318,
"column": 2
} | [
{
"pp": "case mp\nR : Type u\ninst✝ : Semiring R\nr : R\nφ : R[X]\n⊢ C r ∣ φ → ∀ (i : ℕ), r ∣ φ.coeff i",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Semigroup.toMul",
"Dvd.dvd",
"HMul.hMul",
"congrArg",
"semigroupDvd",
... | [
"case mpr\nR : Type u\ninst✝ : Semiring R\nr : R\nφ : R[X]\n⊢ (∀ (i : ℕ), r ∣ φ.coeff i) → C r ∣ φ"
] | · rintro ⟨φ, rfl⟩ c
rw [coeff_C_mul]
apply dvd_mul_right | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 54,
"column": 29
} | {
"line": 54,
"column": 40
} | {
"line": 54,
"column": 40
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\np : R[X]\ninst✝ : Subsingleton R\n⊢ degree 0 = ⊥",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"WithBot",
"congrArg",
"id",
"Bot.bot",
"Polynomial.degree",
"Polynomial.degree_zero",
"Polyn... | [
"R : Type u\ninst✝¹ : Semiring R\np : R[X]\ninst✝ : Subsingleton R\n⊢ ⊥ = ⊥"
] | degree_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 64,
"column": 4
} | {
"line": 64,
"column": 15
} | {
"line": 66,
"column": 0
} | [
{
"pp": "R : Type u\nn : ℕ\ninst✝ : Semiring R\nh : coeff 0 n ≠ 0\n⊢ False",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
"Polynomial",
"Polynomial.coeff",
"Zero.toOfNat0",
"Polynomial.instZero",
"OfNat.ofNat",
"rfl"
],
"usedFVars": [
"h",... | [] | exact h rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Nat.WithBot | {
"line": 29,
"column": 2
} | {
"line": 33,
"column": 26
} | {
"line": 35,
"column": 0
} | [
{
"pp": "n m : WithBot ℕ\n⊢ n + m = 0 ↔ n = 0 ∧ m = 0",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"WithBot.some",
"WithBot",
"congrArg",
"Nat.add_eq_zero_iff._simp_1",
"WithBot.zero",
"false_and",
"W... | [] | cases n
· simp [WithBot.bot_add]
cases m
· simp [WithBot.add_bot]
simp [← WithBot.coe_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.WithBot | {
"line": 29,
"column": 2
} | {
"line": 33,
"column": 26
} | {
"line": 35,
"column": 0
} | [
{
"pp": "n m : WithBot ℕ\n⊢ n + m = 0 ↔ n = 0 ∧ m = 0",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"False",
"Nat.instMulZeroClass",
"WithBot.some",
"WithBot",
"congrArg",
"Nat.add_eq_zero_iff._simp_1",
"WithBot.zero",
"false_and",
"W... | [] | cases n
· simp [WithBot.bot_add]
cases m
· simp [WithBot.add_bot]
simp [← WithBot.coe_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MonoidAlgebra.Degree | {
"line": 409,
"column": 45
} | {
"line": 409,
"column": 63
} | {
"line": 409,
"column": 64
} | [
{
"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\np q : R[A]\nD : A → B\ninst✝ : AddZeroClass A\nh : supDegree D q < supDegree D p\na : A\nhe : D a = supDegree D p\n⊢ (p + q) (Function.invFun D (supDegree D p)) = leadingCoeff D p",
"ppTerm":... | [
"R : Type u_1\nA : Type u_3\nB : Type u_5\ninst✝³ : Semiring R\ninst✝² : LinearOrder B\ninst✝¹ : OrderBot B\np q : R[A]\nD : A → B\ninst✝ : AddZeroClass A\nh : supDegree D q < supDegree D p\na : A\nhe : D a = supDegree D p\n⊢ p (Function.invFun D (supDegree D p)) + q (Function.invFun D (supDegree D p)) = leadingCoe... | Finsupp.add_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 232,
"column": 2
} | {
"line": 237,
"column": 23
} | {
"line": 239,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\nH : (p + q).natDegree < p.natDegree\n⊢ p.natDegree = q.natDegree",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"False",
"congrArg",
"Nat.lt_or_lt_of_ne",
"lt_asymm",
"Eq.mp",
"Polynomial.natDegree_ad... | [] | by_contra h
cases Nat.lt_or_lt_of_ne h with
| inl h => exact lt_asymm h (by rwa [natDegree_add_eq_right_of_natDegree_lt h] at H)
| inr h =>
rw [natDegree_add_eq_left_of_natDegree_lt h] at H
exact LT.lt.false H | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 232,
"column": 2
} | {
"line": 237,
"column": 23
} | {
"line": 239,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\nH : (p + q).natDegree < p.natDegree\n⊢ p.natDegree = q.natDegree",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"False",
"congrArg",
"Nat.lt_or_lt_of_ne",
"lt_asymm",
"Eq.mp",
"Polynomial.natDegree_ad... | [] | by_contra h
cases Nat.lt_or_lt_of_ne h with
| inl h => exact lt_asymm h (by rwa [natDegree_add_eq_right_of_natDegree_lt h] at H)
| inr h =>
rw [natDegree_add_eq_left_of_natDegree_lt h] at H
exact LT.lt.false H | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 289,
"column": 12
} | {
"line": 289,
"column": 28
} | {
"line": 289,
"column": 28
} | [
{
"pp": "case refine_1\nR : Type u\ninst✝ : Semiring R\np q : R[X]\ni j : ℕ\nh₁ : (i, j) ∈ antidiagonal (p.natDegree + q.natDegree)\nh₂ : (i, j) ≠ (p.natDegree, q.natDegree)\n⊢ p.coeff (i, j).1 * q.coeff (i, j).2 = 0",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"AddMonoid.toAd... | [
"case refine_1\nR : Type u\ninst✝ : Semiring R\np q : R[X]\ni j : ℕ\nh₁ : (i, j).1 + (i, j).2 = p.natDegree + q.natDegree\nh₂ : (i, j) ≠ (p.natDegree, q.natDegree)\n⊢ p.coeff (i, j).1 * q.coeff (i, j).2 = 0"
] | mem_antidiagonal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 308,
"column": 12
} | {
"line": 308,
"column": 28
} | {
"line": 308,
"column": 28
} | [
{
"pp": "case refine_2\nR : Type u\ninst✝ : Semiring R\np q : R[X]\nH : (p.natDegree, q.natDegree) ∉ antidiagonal (p.natDegree + q.natDegree)\n⊢ (p.natDegree, q.natDegree) ∈ antidiagonal (p.natDegree + q.natDegree)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"case refine_2\nR : Type u\ninst✝ : Semiring R\np q : R[X]\nH : (p.natDegree, q.natDegree) ∉ antidiagonal (p.natDegree + q.natDegree)\n⊢ (p.natDegree, q.natDegree).1 + (p.natDegree, q.natDegree).2 = p.natDegree + q.natDegree"
] | mem_antidiagonal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 360,
"column": 65
} | {
"line": 360,
"column": 76
} | {
"line": 361,
"column": 4
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nn : ℕ\nh : 0 ≠ 0\nthis : DecidableEq R := Classical.decEq R\nhp0 : ¬p = 0\nhpn0 : p ^ n = 0\nh1 : p.leadingCoeff ^ n ≠ 0\n⊢ False",
"ppTerm": "?m.119",
"assigned": true,
"usedConstants": [
"Zero.toOfNat0",
"OfNat.ofNat",
"rfl",
... | [] | exact h rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.MonoidAlgebra.Degree | {
"line": 528,
"column": 61
} | {
"line": 528,
"column": 70
} | {
"line": 528,
"column": 71
} | [
{
"pp": "case inl\nR : Type u_1\nA : Type u_3\nB : Type u_5\ninst✝⁶ : Semiring R\ninst✝⁵ : LinearOrder B\ninst✝⁴ : OrderBot B\np q : R[A]\nD : A → B\ninst✝³ : AddZeroClass A\ninst✝² : Add B\ninst✝¹ : AddLeftStrictMono B\ninst✝ : AddRightStrictMono B\nhD : Function.Injective D\nhadd : ∀ (a1 a2 : A), D (a1 + a2) ... | [
"case inl\nR : Type u_1\nA : Type u_3\nB : Type u_5\ninst✝⁶ : Semiring R\ninst✝⁵ : LinearOrder B\ninst✝⁴ : OrderBot B\np q : R[A]\nD : A → B\ninst✝³ : AddZeroClass A\ninst✝² : Add B\ninst✝¹ : AddLeftStrictMono B\ninst✝ : AddRightStrictMono B\nhD : Function.Injective D\nhadd : ∀ (a1 a2 : A), D (a1 + a2) = D a1 + D a... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.MonoidAlgebra.Degree | {
"line": 537,
"column": 32
} | {
"line": 537,
"column": 41
} | {
"line": 537,
"column": 42
} | [
{
"pp": "case inr.inl\nR : Type u_1\nA : Type u_3\nB : Type u_5\ninst✝⁷ : Semiring R\ninst✝⁶ : LinearOrder B\ninst✝⁵ : OrderBot B\np : R[A]\nD : A → B\ninst✝⁴ : AddZeroClass A\ninst✝³ : Add B\ninst✝² : AddLeftStrictMono B\ninst✝¹ : AddRightStrictMono B\ninst✝ : NoZeroDivisors R\nhD : Function.Injective D\nhadd ... | [
"case inr.inl\nR : Type u_1\nA : Type u_3\nB : Type u_5\ninst✝⁷ : Semiring R\ninst✝⁶ : LinearOrder B\ninst✝⁵ : OrderBot B\np : R[A]\nD : A → B\ninst✝⁴ : AddZeroClass A\ninst✝³ : Add B\ninst✝² : AddLeftStrictMono B\ninst✝¹ : AddRightStrictMono B\ninst✝ : NoZeroDivisors R\nhD : Function.Injective D\nhadd : ∀ (a1 a2 :... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 446,
"column": 8
} | {
"line": 446,
"column": 24
} | {
"line": 446,
"column": 24
} | [
{
"pp": "case h\nR : Type u\ninst✝ : Semiring R\ndf dg : ℕ\nf g : R[X]\nhdf : f.natDegree ≤ df\nhdg : g.natDegree ≤ dg\n⊢ (df, dg) ∈ antidiagonal (df + dg)",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"AddMonoid.toAddSemigroup",
"congrArg",
"Finset",
... | [
"case h\nR : Type u\ninst✝ : Semiring R\ndf dg : ℕ\nf g : R[X]\nhdf : f.natDegree ≤ df\nhdg : g.natDegree ≤ dg\n⊢ (df, dg).1 + (df, dg).2 = df + dg"
] | mem_antidiagonal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 458,
"column": 2
} | {
"line": 458,
"column": 56
} | {
"line": 459,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\nS : Type u_2\ninst✝ : SMulZeroClass S R\na : S\np : R[X]\n⊢ (a • p).degree ≤ p.degree",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"WithBot.instPreorder",
"WithBot",
"instHSMul",
"Preorder.toLT",
"W... | [
"R : Type u\ninst✝¹ : Semiring R\nS : Type u_2\ninst✝ : SMulZeroClass S R\na : S\np : R[X]\nm : ℕ\nhm : p.degree < ↑m\n⊢ (a • p).coeff m = 0"
] | refine (degree_le_iff_coeff_zero _ _).2 fun m hm => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Polynomial.Eval.Degree | {
"line": 80,
"column": 22
} | {
"line": 80,
"column": 30
} | {
"line": 80,
"column": 31
} | [
{
"pp": "S : Type v\ninst✝ : CommRing S\nd : ℕ\ny : S\ncast_succ : ↑d + 1 = ↑d.succ\n⊢ ↑d.succ * (∑ x ∈ range d, ↑(d.choose x) * y ^ x + ↑(d.choose d) * y ^ d) - ↑d.succ * y ^ d =\n ∑ x_1 ∈ range (d + 1), ↑((d + 1).choose x_1) * (↑x_1 * y ^ (x_1 - 1))",
"ppTerm": "?m.120",
"assigned": true,
"used... | [
"S : Type v\ninst✝ : CommRing S\nd : ℕ\ny : S\ncast_succ : ↑d + 1 = ↑d.succ\n⊢ ↑d.succ * ∑ x ∈ range d, ↑(d.choose x) * y ^ x + ↑d.succ * (↑(d.choose d) * y ^ d) - ↑d.succ * y ^ d =\n ∑ x_1 ∈ range (d + 1), ↑((d + 1).choose x_1) * (↑x_1 * y ^ (x_1 - 1))"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Eval.Degree | {
"line": 81,
"column": 55
} | {
"line": 81,
"column": 64
} | {
"line": 81,
"column": 65
} | [
{
"pp": "S : Type v\ninst✝ : CommRing S\nd : ℕ\ny : S\ncast_succ : ↑d + 1 = ↑d.succ\n⊢ ∑ i ∈ range d, ↑d.succ * (↑(d.choose i) * y ^ i) =\n ∑ k ∈ range d, ↑((d + 1).choose (k + 1)) * (↑(k + 1) * y ^ (k + 1 - 1)) + ↑((d + 1).choose 0) * 0",
"ppTerm": "?m.161",
"assigned": true,
"usedConstants": [
... | [
"S : Type v\ninst✝ : CommRing S\nd : ℕ\ny : S\ncast_succ : ↑d + 1 = ↑d.succ\n⊢ ∑ i ∈ range d, ↑d.succ * (↑(d.choose i) * y ^ i) =\n ∑ k ∈ range d, ↑((d + 1).choose (k + 1)) * (↑(k + 1) * y ^ (k + 1 - 1)) + 0"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Degree.Operations | {
"line": 619,
"column": 44
} | {
"line": 619,
"column": 56
} | {
"line": 619,
"column": 56
} | [
{
"pp": "R : Type u\ninst✝¹ : Nontrivial R\ninst✝ : Semiring R\nn : ℕ\nhn : 0 < n\na : R\nthis : (C a).degree < (X ^ n).degree\n⊢ (X ^ n).degree = ↑n",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"WithBot",
"congrArg",
"WithBot.instNatCast",
"id",
... | [
"R : Type u\ninst✝¹ : Nontrivial R\ninst✝ : Semiring R\nn : ℕ\nhn : 0 < n\na : R\nthis : (C a).degree < (X ^ n).degree\n⊢ ↑n = ↑n"
] | degree_X_pow | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Eval.Degree | {
"line": 141,
"column": 29
} | {
"line": 141,
"column": 65
} | {
"line": 141,
"column": 65
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\np : R[X]\nhp : p.Monic\nh : ∀ (x : R), f x = 0\nn : ℕ\n⊢ (map f p).coeff n = coeff 0 n",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"Polynomial.coeff_map",
"congrArg",
"RingHom",
... | [] | simp only [h, coeff_map, coeff_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Polynomial.Eval.Degree | {
"line": 141,
"column": 29
} | {
"line": 141,
"column": 65
} | {
"line": 141,
"column": 65
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\np : R[X]\nhp : p.Monic\nh : ∀ (x : R), f x = 0\nn : ℕ\n⊢ (map f p).coeff n = coeff 0 n",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"Polynomial.coeff_map",
"congrArg",
"RingHom",
... | [] | simp only [h, coeff_map, coeff_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Eval.Degree | {
"line": 141,
"column": 29
} | {
"line": 141,
"column": 65
} | {
"line": 141,
"column": 65
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\np : R[X]\nhp : p.Monic\nh : ∀ (x : R), f x = 0\nn : ℕ\n⊢ (map f p).coeff n = coeff 0 n",
"ppTerm": "?m.83",
"assigned": true,
"usedConstants": [
"Polynomial.coeff_map",
"congrArg",
"RingHom",
... | [] | simp only [h, coeff_map, coeff_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Eval.Degree | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 56
} | {
"line": 148,
"column": 2
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\np : R[X]\n⊢ (map f p).degree ≤ p.degree",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"WithBot.instPreorder",
"WithBot",
"Preorder.toLT",
"WithBot.instNatCast",... | [
"R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\np : R[X]\nm : ℕ\nhm : p.degree < ↑m\n⊢ (map f p).coeff m = 0"
] | refine (degree_le_iff_coeff_zero _ _).2 fun m hm => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Polynomial.Eval.Degree | {
"line": 157,
"column": 34
} | {
"line": 157,
"column": 45
} | {
"line": 157,
"column": 45
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\np : R[X]\nhp : map f p = 0\nhp₀ : p ≠ 0\nhpq : (map f p).degree = p.degree\n⊢ degree 0 = ⊥",
"ppTerm": "?m.102",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"WithBot",
"congrArg",
"id",
... | [
"R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nf : R →+* S\np : R[X]\nhp : map f p = 0\nhp₀ : p ≠ 0\nhpq : (map f p).degree = p.degree\n⊢ ⊥ = ⊥"
] | degree_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Adjoin.Polynomial.Basic | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 26
} | {
"line": 48,
"column": 26
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\np : R[X]\n_hp : p ∈ ⊤\nS : Subalgebra R R[X] := R[ X]\n⊢ p ∈ S",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Subalgebra.instSetLike",
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring",
"P... | [
"R : Type u\ninst✝ : CommSemiring R\np : R[X]\n_hp : p ∈ ⊤\nS : Subalgebra R R[X] := R[ X]\n⊢ (p.sum fun n a ↦ (monomial n) a) ∈ S"
] | rw [← sum_monomial_eq p] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Polynomial.AlgebraMap | {
"line": 613,
"column": 4
} | {
"line": 613,
"column": 90
} | {
"line": 614,
"column": 4
} | [
{
"pp": "case pos\nS : Type v\ninst✝ : CommRing S\nz p : S\nf : S[X]\ni : ℕ\ndvd_eval : p ∣ ∑ n ∈ f.support, (RingHom.id S) (f.coeff n) * z ^ n\ndvd_terms : ∀ (j : ℕ), j ≠ i → p ∣ f.coeff j * z ^ j\nhi : i ∈ f.support\n⊢ p ∣ f.coeff i * z ^ i",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [... | [
"case pos\nS : Type v\ninst✝ : CommRing S\nz p : S\nf : S[X]\ni : ℕ\ndvd_eval : p ∣ (RingHom.id S) (f.coeff i) * z ^ i + ∑ x ∈ f.support.erase i, (RingHom.id S) (f.coeff x) * z ^ x\ndvd_terms : ∀ (j : ℕ), j ≠ i → p ∣ f.coeff j * z ^ j\nhi : i ∈ f.support\n⊢ p ∣ f.coeff i * z ^ i"
] | rw [← Finset.insert_erase hi, Finset.sum_insert (Finset.notMem_erase _ _)] at dvd_eval | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 470,
"column": 57
} | {
"line": 472,
"column": 5
} | {
"line": 474,
"column": 0
} | [
{
"pp": "R : Type u\nσ : Type u_1\na : R\ns : σ →₀ ℕ\ninst✝ : CommSemiring R\nh : Decidable (a = 0)\n⊢ ((monomial s) a).support = if a = 0 then ∅ else {s}",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"Semiring.toModule",
"instSub... | [] | by
rw [← Subsingleton.elim (Classical.decEq R a 0) h]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 311,
"column": 2
} | {
"line": 316,
"column": 13
} | {
"line": 318,
"column": 0
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\nS : Type u_2\nτ : Type u_3\nx : τ → S\ninst✝ : CommSemiring S\nf : R →+* MvPolynomial τ S\ng : σ → MvPolynomial τ S\np : MvPolynomial σ R\n⊢ (eval x) (eval₂ f g p) = eval₂ ((eval x).comp f) (fun s ↦ (eval x) (g s)) p",
"ppTerm": "?m.38",
"assig... | [] | apply induction_on p
· simp
· intro p q hp hq
simp [hp, hq]
· intro p n hp
simp [hp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Eval | {
"line": 311,
"column": 2
} | {
"line": 316,
"column": 13
} | {
"line": 318,
"column": 0
} | [
{
"pp": "R : Type u\nσ : Type u_1\ninst✝¹ : CommSemiring R\nS : Type u_2\nτ : Type u_3\nx : τ → S\ninst✝ : CommSemiring S\nf : R →+* MvPolynomial τ S\ng : σ → MvPolynomial τ S\np : MvPolynomial σ R\n⊢ (eval x) (eval₂ f g p) = eval₂ ((eval x).comp f) (fun s ↦ (eval x) (g s)) p",
"ppTerm": "?m.38",
"assig... | [] | apply induction_on p
· simp
· intro p q hp hq
simp [hp, hq]
· intro p n hp
simp [hp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.DFinsupp.Lex | {
"line": 198,
"column": 2
} | {
"line": 200,
"column": 58
} | {
"line": 202,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ninst✝² : (i : ι) → Zero (α i)\ninst✝¹ : LinearOrder ι\ninst✝ : (i : ι) → PartialOrder (α i)\na b : Π₀ (i : ι), α i\nh : a ≤ b\nhne : ¬toLex a = toLex b\n⊢ toLex a < toLex b",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Finset.min'",
"... | [] | exact ⟨Finset.min' _ (nonempty_neLocus_iff.2 hne),
fun j hj ↦ notMem_neLocus.1 fun h ↦ (Finset.min'_le _ _ h).not_gt hj,
(h _).lt_of_ne (mem_neLocus.1 <| Finset.min'_mem _ _)⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.DFinsupp.Lex | {
"line": 198,
"column": 2
} | {
"line": 200,
"column": 58
} | {
"line": 202,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ninst✝² : (i : ι) → Zero (α i)\ninst✝¹ : LinearOrder ι\ninst✝ : (i : ι) → PartialOrder (α i)\na b : Π₀ (i : ι), α i\nh : a ≤ b\nhne : ¬toLex a = toLex b\n⊢ toLex a < toLex b",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Finset.min'",
"... | [] | exact ⟨Finset.min' _ (nonempty_neLocus_iff.2 hne),
fun j hj ↦ notMem_neLocus.1 fun h ↦ (Finset.min'_le _ _ h).not_gt hj,
(h _).lt_of_ne (mem_neLocus.1 <| Finset.min'_mem _ _)⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.DFinsupp.Lex | {
"line": 198,
"column": 2
} | {
"line": 200,
"column": 58
} | {
"line": 202,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ninst✝² : (i : ι) → Zero (α i)\ninst✝¹ : LinearOrder ι\ninst✝ : (i : ι) → PartialOrder (α i)\na b : Π₀ (i : ι), α i\nh : a ≤ b\nhne : ¬toLex a = toLex b\n⊢ toLex a < toLex b",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Finset.min'",
"... | [] | exact ⟨Finset.min' _ (nonempty_neLocus_iff.2 hne),
fun j hj ↦ notMem_neLocus.1 fun h ↦ (Finset.min'_le _ _ h).not_gt hj,
(h _).lt_of_ne (mem_neLocus.1 <| Finset.min'_mem _ _)⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.Basic | {
"line": 798,
"column": 2
} | {
"line": 800,
"column": 23
} | {
"line": 801,
"column": 2
} | [
{
"pp": "case mp\nR : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nr : R\nφ : MvPolynomial σ R\n⊢ C r ∣ φ → ∀ (i : σ →₀ ℕ), r ∣ coeff i φ",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Nat.instMulZeroClass",
"Semigroup.toMul",
... | [
"case mpr\nR : Type u\nσ : Type u_1\ninst✝ : CommSemiring R\nr : R\nφ : MvPolynomial σ R\n⊢ (∀ (i : σ →₀ ℕ), r ∣ coeff i φ) → C r ∣ φ"
] | · rintro ⟨φ, rfl⟩ c
rw [coeff_C_mul]
apply dvd_mul_right | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.MvPolynomial.Variables | {
"line": 87,
"column": 12
} | {
"line": 87,
"column": 60
} | {
"line": 89,
"column": 0
} | [
{
"pp": "R : Type u\nσ : Type u_1\nr : R\ninst✝ : CommSemiring R\n⊢ (C r).vars = ∅",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Multiset.toFinset",
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Nat.instMulZeroClass",
"congrArg",
"CommSemiring.toSemiring",
... | [] | rw [vars_def, degrees_C, Multiset.toFinset_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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