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.Data.Finset.NAry | {
"line": 237,
"column": 4
} | {
"line": 238,
"column": 34
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\nγ : Type u_5\ninst✝ : DecidableEq γ\nf : α → β → γ\ns : Finset α\nt : Finset β\n⊢ ↑(t.biUnion fun b ↦ image (fun a ↦ f a b) s) = ↑(image₂ f s t)",
"usedConstants": [
"Eq.mpr",
"Finset.coe_biUnion",
"Iff.of_eq",
"congrArg",
"Finset",
"M... | push_cast
exact Set.iUnion_image_right _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finset.NAry | {
"line": 237,
"column": 4
} | {
"line": 238,
"column": 34
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\nγ : Type u_5\ninst✝ : DecidableEq γ\nf : α → β → γ\ns : Finset α\nt : Finset β\n⊢ ↑(t.biUnion fun b ↦ image (fun a ↦ f a b) s) = ↑(image₂ f s t)",
"usedConstants": [
"Eq.mpr",
"Finset.coe_biUnion",
"Iff.of_eq",
"congrArg",
"Finset",
"M... | push_cast
exact Set.iUnion_image_right _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finset.NAry | {
"line": 500,
"column": 4
} | {
"line": 501,
"column": 38
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → α → β\ns t : Finset α\nhf : ∀ (a b : α), f a b = f b a\n⊢ ↑(image₂ f (s ∪ t) (s ∩ t)) ⊆ ↑(image₂ f s t)",
"usedConstants": [
"Eq.mpr",
"Finset.instUnion",
"congrArg",
"Finset",
"Set.inst... | push_cast
exact image2_union_inter_subset hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finset.NAry | {
"line": 500,
"column": 4
} | {
"line": 501,
"column": 38
} | [
{
"pp": "α : Type u_1\nβ : Type u_3\ninst✝¹ : DecidableEq α\ninst✝ : DecidableEq β\nf : α → α → β\ns t : Finset α\nhf : ∀ (a b : α), f a b = f b a\n⊢ ↑(image₂ f (s ∪ t) (s ∩ t)) ⊆ ↑(image₂ f s t)",
"usedConstants": [
"Eq.mpr",
"Finset.instUnion",
"congrArg",
"Finset",
"Set.inst... | push_cast
exact image2_union_inter_subset hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.LinearIndependent.Lemmas | {
"line": 777,
"column": 2
} | {
"line": 822,
"column": 43
} | [
{
"pp": "K : Type u_3\nV : Type u\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\ns : Set V\nt : Finset V\nhs : LinearIndepOn K id s\nhst : s ⊆ ↑(span K ↑t)\n⊢ ∃ t', ↑t' ⊆ s ∪ ↑t ∧ s ⊆ ↑t' ∧ t'.card = t.card",
"usedConstants": [
"subset_refl._simp_1",
"Eq.mpr",
"Subm... | have :
∀ t : Finset V,
∀ s' : Finset V,
↑s' ⊆ s →
s ∩ ↑t = ∅ →
s ⊆ (span K ↑(s' ∪ t) : Submodule K V) →
∃ t' : Finset V, ↑t' ⊆ s ∪ ↑t ∧ s ⊆ ↑t' ∧ t'.card = (s' ∪ t).card :=
fun t =>
Finset.induction_on t
(fun s' hs' _ hss' =>
have : s = ↑s' := ... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.MonoidAlgebra.NoZeroDivisors | {
"line": 113,
"column": 13
} | {
"line": 113,
"column": 21
} | [
{
"pp": "case H.inr\nR : Type u_1\nA : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : IsCancelAdd R\ninst✝² : IsLeftCancelMulZero R\ninst✝¹ : Mul A\ninst✝ : UniqueProds A\nf : R[A]\nhf : f ≠ 0\ng₁ g₂ : R[A]\nih :\n ∀ s ∈ g₁.support ∪ g₂.support,\n ∀ {g₁_1 g₂_1 : R[A]},\n (fun x ↦ f * x) g₁_1 = (fun x ↦ f * x)... | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 471,
"column": 48
} | {
"line": 471,
"column": 88
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝¹ : Semiring R\nr₁ r₂ : R\nm₁ m₂ : M\ninst✝ : Mul M\nhm : Commute m₁ m₂\nhr : Commute r₁ r₂\n⊢ Commute (single m₁ r₁) (single m₂ r₂)",
"usedConstants": [
"HMul.hMul",
"congrArg",
"MonoidAlgebra.instMul",
"MonoidAlgebra.single_mul_single",
... | simp [Commute, SemiconjBy, hm.eq, hr.eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 471,
"column": 48
} | {
"line": 471,
"column": 88
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝¹ : Semiring R\nr₁ r₂ : R\nm₁ m₂ : M\ninst✝ : Mul M\nhm : Commute m₁ m₂\nhr : Commute r₁ r₂\n⊢ Commute (single m₁ r₁) (single m₂ r₂)",
"usedConstants": [
"HMul.hMul",
"congrArg",
"MonoidAlgebra.instMul",
"MonoidAlgebra.single_mul_single",
... | simp [Commute, SemiconjBy, hm.eq, hr.eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MonoidAlgebra.Defs | {
"line": 471,
"column": 48
} | {
"line": 471,
"column": 88
} | [
{
"pp": "R : Type u_1\nM : Type u_4\ninst✝¹ : Semiring R\nr₁ r₂ : R\nm₁ m₂ : M\ninst✝ : Mul M\nhm : Commute m₁ m₂\nhr : Commute r₁ r₂\n⊢ Commute (single m₁ r₁) (single m₂ r₂)",
"usedConstants": [
"HMul.hMul",
"congrArg",
"MonoidAlgebra.instMul",
"MonoidAlgebra.single_mul_single",
... | simp [Commute, SemiconjBy, hm.eq, hr.eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.UniqueProds.Basic | {
"line": 369,
"column": 4
} | {
"line": 369,
"column": 51
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Semigroup G\ninst✝ : IsCancelMul G\nh : ∀ {A : Finset G}, A.Nonempty → ∃ a1 ∈ A, ∃ a2 ∈ A, UniqueMul A A a1 a2\nA B : Finset G\nhA : A.Nonempty\nhB : B.Nonempty\ng1 : G\nh1 : g1 ∈ B * A\ng2 : G\nh2 : g2 ∈ B * A\nhu : UniqueMul (B * A) (B * A) g1 g2\n⊢ ∃ a0 ∈ A, ∃ b0 ∈ B, UniqueMu... | obtain ⟨b1, hb1, a1, ha1, rfl⟩ := mem_mul.mp h1 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Group.Pointwise.Finset.Basic | {
"line": 982,
"column": 23
} | {
"line": 982,
"column": 38
} | [
{
"pp": "α : Type u_2\ninst✝¹ : DecidableEq α\ninst✝ : DivisionMonoid α\ns : Finset α\n⊢ IsUnit ↑s ↔ ∃ a, s = {a} ∧ IsUnit a",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"IsUnit",
"Exists",
"_private.Mathlib.Algebra.Group.Pointwise.Finset.Basic.0.Finset.isUnit_coe... | Set.isUnit_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Finset.Sort | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 25
} | [
{
"pp": "α : Type u_1\nr : α → α → Prop\ninst✝⁴ : DecidableRel r\ninst✝³ : IsTrans α r\ninst✝² : Std.Antisymm r\ninst✝¹ : Std.Total r\ninst✝ : DecidableEq α\nl : List α\nhl : l.Nodup\nh : l.toFinset.sort r = l\n⊢ List.Pairwise r (l.toFinset.sort r)",
"usedConstants": [
"Finset.pairwise_sort",
"L... | exact pairwise_sort _ r | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.Finset.Sort | {
"line": 94,
"column": 44
} | {
"line": 99,
"column": 25
} | [
{
"pp": "α : Type u_1\nr : α → α → Prop\ninst✝⁴ : DecidableRel r\ninst✝³ : IsTrans α r\ninst✝² : Std.Antisymm r\ninst✝¹ : Std.Total r\ninst✝ : DecidableEq α\nl : List α\nhl : l.Nodup\n⊢ l.toFinset.sort r = l ↔ List.Pairwise r l",
"usedConstants": [
"Eq.mpr",
"List.Pairwise",
"congrArg",
... | by
refine ⟨?_, ((sort_perm_toList _ r).trans (List.toFinset_toList hl)).eq_of_pairwise'
(pairwise_sort _ _)⟩
intro h
rw [← h]
exact pairwise_sort _ r | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.UniqueProds.Basic | {
"line": 593,
"column": 34
} | {
"line": 593,
"column": 39
} | [
{
"pp": "case refine_1\nG : Type u\nH : Type v\ninst✝⁴ : Mul G\ninst✝³ : Mul H\ninst✝² : IsRightCancelMul G\ninst✝¹ : LinearOrder G\ninst✝ : MulLeftStrictMono G\nA B : Finset G\nhc : 1 < #(A ×ˢ B)\nhA : A.Nonempty\nhB : B.Nonempty\na0 : G\nha0 : a0 ∈ A\nb0 : G\nhb0 : b0 ∈ B\nhe0 : a0 * b0 = (A * B).max' ⋯\na1 :... | he.1, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.UniqueProds.Basic | {
"line": 593,
"column": 46
} | {
"line": 593,
"column": 49
} | [
{
"pp": "case refine_1\nG : Type u\nH : Type v\ninst✝⁴ : Mul G\ninst✝³ : Mul H\ninst✝² : IsRightCancelMul G\ninst✝¹ : LinearOrder G\ninst✝ : MulLeftStrictMono G\nA B : Finset G\nhc : 1 < #(A ×ˢ B)\nhA : A.Nonempty\nhB : B.Nonempty\na0 : G\nha0 : a0 ∈ A\nb0 : G\nhb0 : b0 ∈ B\na1 : G\nha1 : a1 ∈ A\nb1 : G\nhe0 : ... | he1 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Basic | {
"line": 849,
"column": 2
} | {
"line": 849,
"column": 27
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np q : R[X]\n⊢ p * q = ∑ i ∈ p.support, q.sum fun j a ↦ (monomial (i + j)) (p.coeff i * a)",
"usedConstants": [
"Semiring.toModule",
"HMul.hMul",
"Polynomial.sum",
"Polynomial.toFinsupp_injective",
"LinearMap.instFunLike",
"Polynomi... | apply toFinsupp_injective | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Algebra.Polynomial.Basic | {
"line": 876,
"column": 2
} | {
"line": 877,
"column": 78
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\nS : Type u_1\ninst✝ : AddCommMonoid S\np q : R[X]\nf : ℕ → R → S\nhf : ∀ (i : ℕ), f i 0 = 0\nh_add : ∀ (a : ℕ) (b₁ b₂ : R), f a (b₁ + b₂) = f a b₁ + f a b₂\n⊢ (p + q).sum f = p.sum f + q.sum f",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMon... | rw [show p + q = ⟨p.toFinsupp + q.toFinsupp⟩ from rfl]
exact Finsupp.sum_add_index (fun i _ ↦ hf i) (fun a _ b₁ b₂ ↦ h_add a b₁ b₂) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Basic | {
"line": 876,
"column": 2
} | {
"line": 877,
"column": 78
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\nS : Type u_1\ninst✝ : AddCommMonoid S\np q : R[X]\nf : ℕ → R → S\nhf : ∀ (i : ℕ), f i 0 = 0\nh_add : ∀ (a : ℕ) (b₁ b₂ : R), f a (b₁ + b₂) = f a b₁ + f a b₂\n⊢ (p + q).sum f = p.sum f + q.sum f",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMon... | rw [show p + q = ⟨p.toFinsupp + q.toFinsupp⟩ from rfl]
exact Finsupp.sum_add_index (fun i _ ↦ hf i) (fun a _ b₁ b₂ ↦ h_add a b₁ b₂) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Basic | {
"line": 1007,
"column": 2
} | {
"line": 1008,
"column": 38
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nn : ℕ\n⊢ p.update n 0 = erase n p",
"usedConstants": [
"Eq.mpr",
"Polynomial.ext",
"congrArg",
"Polynomial.coeff_erase",
"Polynomial.update",
"Polynomial.coeff_update_apply",
"NonUnitalNonAssocSemiring.toMulZeroClas... | ext
rw [coeff_update_apply, coeff_erase] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Basic | {
"line": 1007,
"column": 2
} | {
"line": 1008,
"column": 38
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\np : R[X]\nn : ℕ\n⊢ p.update n 0 = erase n p",
"usedConstants": [
"Eq.mpr",
"Polynomial.ext",
"congrArg",
"Polynomial.coeff_erase",
"Polynomial.update",
"Polynomial.coeff_update_apply",
"NonUnitalNonAssocSemiring.toMulZeroClas... | ext
rw [coeff_update_apply, coeff_erase] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Finsupp.Span | {
"line": 131,
"column": 9
} | {
"line": 131,
"column": 22
} | [
{
"pp": "case refine_2\nR : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nS : Set (Submodule R M)\nm : M\nx✝ : ∃ s, m ∈ ⨆ i ∈ s, ↑i\ns : Finset (Subtype (Membership.mem S))\nhs : m ∈ ⨆ i ∈ s, ↑i\n⊢ m ∈ ⨆ i, ⨆ (hi : i ∈ S), ⨆ (_ : ⟨i, hi⟩ ∈ s), i",
"usedConstants"... | iSup_subtype' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.Submodule.Pointwise | {
"line": 493,
"column": 6
} | {
"line": 494,
"column": 46
} | [
{
"pp": "case mp.smul₁\nR : Type u_2\nM : Type u_3\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nS : Type u_4\ninst✝² : Monoid S\ninst✝¹ : DistribMulAction S M\nN : Submodule R M\ninst✝ : SMulCommClass R S M\nr : S\nx : M\nt : R\nn : M\nmem : n ∈ {r} • N\nh : ∃ m ∈ N, n = r • m\n⊢ ∃ m ∈ N... | rcases h with ⟨n, hn, rfl⟩
exact ⟨t • n, by aesop, smul_comm _ _ _⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Submodule.Pointwise | {
"line": 493,
"column": 6
} | {
"line": 494,
"column": 46
} | [
{
"pp": "case mp.smul₁\nR : Type u_2\nM : Type u_3\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\nS : Type u_4\ninst✝² : Monoid S\ninst✝¹ : DistribMulAction S M\nN : Submodule R M\ninst✝ : SMulCommClass R S M\nr : S\nx : M\nt : R\nn : M\nmem : n ∈ {r} • N\nh : ∃ m ∈ N, n = r • m\n⊢ ∃ m ∈ N... | rcases h with ⟨n, hn, rfl⟩
exact ⟨t • n, by aesop, smul_comm _ _ _⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Finiteness.Cardinality | {
"line": 34,
"column": 2
} | {
"line": 35,
"column": 27
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\n⊢ N.FG ↔ ∃ n f, f.range = N",
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"Submodule",
"RingHomSurjective.ids",
"Semiring.toModule",
"Pi.addCo... | simp_rw [fg_iff_exists_fin_generating_family, ← ((Pi.basisFun R _).constr ℕ).exists_congr_right]
simp [Basis.constr_range] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Finiteness.Cardinality | {
"line": 34,
"column": 2
} | {
"line": 35,
"column": 27
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\n⊢ N.FG ↔ ∃ n f, f.range = N",
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"Submodule",
"RingHomSurjective.ids",
"Semiring.toModule",
"Pi.addCo... | simp_rw [fg_iff_exists_fin_generating_family, ← ((Pi.basisFun R _).constr ℕ).exists_congr_right]
simp [Basis.constr_range] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Submodule.Bilinear | {
"line": 140,
"column": 2
} | {
"line": 141,
"column": 35
} | [
{
"pp": "ι : Sort uι\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : AddCommMonoid P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N →ₗ[R] P\ns : ι → Submodule R M\nt : Submodule R N\n⊢ ma... | suffices map₂ f (⨆ i, span R (s i : Set M)) (span R t) = ⨆ i, map₂ f (span R (s i)) (span R t) by
simpa only [span_eq] using this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Algebra.Module.Submodule.Bilinear | {
"line": 153,
"column": 35
} | {
"line": 153,
"column": 57
} | [
{
"pp": "case h\nR : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : AddCommMonoid P\ninst✝² : Module R M\ninst✝¹ : Module R N\ninst✝ : Module R P\nf : M →ₗ[R] N →ₗ[R] P\nm : M\ns : Submodule R N\n⊢ span R (image2 (fun m n... | image2_singleton_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Ring.Submonoid.Pointwise | {
"line": 165,
"column": 41
} | {
"line": 165,
"column": 51
} | [
{
"pp": "R : Type u_2\ninst✝ : NonUnitalNonAssocSemiring R\nM N : AddSubmonoid R\n⊢ M * N = M * closure ↑N",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"AddSubmonoid.mul",
"AddMonoid.toAddZeroClass",
"id",
"AddSubmonoid",
"AddSubmonoid.closure_eq",
... | closure_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Ring.Submonoid.Pointwise | {
"line": 282,
"column": 21
} | {
"line": 282,
"column": 31
} | [
{
"pp": "R : Type u_2\ninst✝ : Semiring R\ns : AddSubmonoid R\nn : ℕ\n⊢ s ^ n = closure ↑s ^ n",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"congrArg",
"AddMonoid.toAddZeroClass",
"id",
"AddSubmonoid",
"AddCommMonoidWithOne.toAddMonoidWi... | closure_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.GroupAction.SubMulAction.Pointwise | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 24
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : Monoid R\ninst✝³ : MulAction R M\ninst✝² : Monoid M\ninst✝¹ : IsScalarTower R M M\ninst✝ : SMulCommClass R M M\np : SubMulAction R M\n⊢ 1 ⊆ ↑1",
"usedConstants": [
"MulOne.toOne",
"Monoid.toMulOneClass",
"MulOneClass.toMulOne",
"SubMulAct... | exact subset_coe_one | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Coprime.Basic | {
"line": 62,
"column": 48
} | {
"line": 62,
"column": 57
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nx : R\nx✝ : IsCoprime 0 x\na b : R\nH : a * 0 + b * x = 1\n⊢ x * b = 1",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"CommSemiring.toSemiring",
"NonUnitalNonAssoc... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Coprime.Basic | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 33
} | [
{
"pp": "R : Type u\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\nh : IsCoprime (0 0) (0 1)\n⊢ False",
"usedConstants": [
"not_isCoprime_zero_zero"
]
}
] | exact not_isCoprime_zero_zero h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Coprime.Basic | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 33
} | [
{
"pp": "case a\nR : Type u\ninst✝¹ : CommSemiring R\ninst✝ : Nontrivial R\nh : IsCoprime 0 0\n⊢ False",
"usedConstants": [
"not_isCoprime_zero_zero"
]
}
] | exact not_isCoprime_zero_zero h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Coprime.Basic | {
"line": 97,
"column": 24
} | {
"line": 97,
"column": 32
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nx y z : R\nH1 : IsCoprime x z\nH2 : x ∣ y * z\na b : R\nH : a * x + b * z = 1\n⊢ x ∣ y * (a * x + b * z)",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Dvd.dvd",
"HMul.hMul",
"CommSemiring.toNonUnitalCommSemiring",
... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Coprime.Basic | {
"line": 415,
"column": 2
} | {
"line": 415,
"column": 33
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nha hb : 0 ^ 2 = 0\nh : IsCoprime 0 0\nh' : 0 ^ 2 + 0 ^ 2 = 0\n⊢ False",
"usedConstants": [
"AddGroupWithOne.toAddMonoidWithOne",
"SemilatticeInf.toPartialOrder",
"not_isCoprime_zero_zero",
... | exact not_isCoprime_zero_zero h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Coprime.Lemmas | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 37
} | [
{
"pp": "case insert\nR : Type u\nI : Type v\ninst✝ : CommSemiring R\nz : R\ns : I → R\nt : Finset I\na : I\nr : Finset I\nhar : a ∉ r\nih : (↑r).Pairwise (IsCoprime on s) → (∀ i ∈ r, s i ∣ z) → ∏ x ∈ r, s x ∣ z\nHs : (↑(insert a r)).Pairwise (IsCoprime on s)\nHs1 : ∀ i ∈ insert a r, s i ∣ z\n⊢ s a * ∏ x ∈ r, s... | refine IsCoprime.mul_dvd ?_ ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.RingTheory.Ideal.Prod | {
"line": 85,
"column": 6
} | {
"line": 86,
"column": 15
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nI : Ideal R\nJ : Ideal S\nx : S\n⊢ (∃ x_1 ∈ I.prod J, (RingHom.snd R S) x_1 = x) → x ∈ J",
"usedConstants": [
"Submodule",
"Semiring.toModule",
"RingHom",
"Membership.mem",
"Exists",
"Prod.instSemir... | rintro ⟨x, ⟨h, rfl⟩⟩
exact h.2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Prod | {
"line": 85,
"column": 6
} | {
"line": 86,
"column": 15
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nI : Ideal R\nJ : Ideal S\nx : S\n⊢ (∃ x_1 ∈ I.prod J, (RingHom.snd R S) x_1 = x) → x ∈ J",
"usedConstants": [
"Submodule",
"Semiring.toModule",
"RingHom",
"Membership.mem",
"Exists",
"Prod.instSemir... | rintro ⟨x, ⟨h, rfl⟩⟩
exact h.2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Prod | {
"line": 104,
"column": 4
} | {
"line": 104,
"column": 41
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝¹ : Semiring R\ninst✝ : Semiring S\nI✝ : Ideal R\nJ✝ : Ideal S\nI J : Ideal (R × S)\nh : map (RingHom.fst R S) I ≤ map (RingHom.fst R S) J ∧ map (RingHom.snd R S) I ≤ map (RingHom.snd R S) J\n⊢ I ≤ J",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
... | rw [ideal_prod_eq I, ideal_prod_eq J] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Algebra.Operations | {
"line": 587,
"column": 2
} | {
"line": 587,
"column": 26
} | [
{
"pp": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM : Submodule R A\nn : ℕ\n⊢ M ^ n = span R (↑M ^ n)",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule",
"IsScalarTower.right",
"Submodule.span_... | rw [← span_pow, span_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Algebra.Operations | {
"line": 587,
"column": 2
} | {
"line": 587,
"column": 26
} | [
{
"pp": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM : Submodule R A\nn : ℕ\n⊢ M ^ n = span R (↑M ^ n)",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule",
"IsScalarTower.right",
"Submodule.span_... | rw [← span_pow, span_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Algebra.Operations | {
"line": 587,
"column": 2
} | {
"line": 587,
"column": 26
} | [
{
"pp": "R : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nM : Submodule R A\nn : ℕ\n⊢ M ^ n = span R (↑M ^ n)",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule",
"IsScalarTower.right",
"Submodule.span_... | rw [← span_pow, span_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 40
} | [
{
"pp": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na✝ b✝ a b : R\nx✝¹ : a ≠ 0\nx✝ : gcd (b % a) a ∣ b % a ∧ gcd (b % a) a ∣ a\nIH₁ : gcd (b % a) a ∣ b % a\nIH₂ : gcd (b % a) a ∣ a\n⊢ gcd (b % a) a ∣ a ∧ gcd (b % a) a ∣ b",
"usedConstants": [
"Dvd.dvd",
"CommRing.toNonUnitalC... | exact ⟨IH₂, (dvd_mod_iff IH₂).1 IH₁⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 208,
"column": 65
} | {
"line": 208,
"column": 74
} | [
{
"pp": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na b : R\n⊢ a = a + b * 0",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"MulZeroClass.toMul",
"congrArg",
"NonUnital... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 209,
"column": 34
} | {
"line": 209,
"column": 43
} | [
{
"pp": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na b : R\n⊢ b = a * 0 + b",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"MulZeroClass.toMul",
"congrArg",
"NonUnital... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Algebra.Operations | {
"line": 911,
"column": 4
} | {
"line": 913,
"column": 45
} | [
{
"pp": "case refine_1.refine_1\nR : Type u\ninst✝⁶ : CommSemiring R\nS : Type u_1\nM : Type u_2\ninst✝⁵ : CommSemiring S\ninst✝⁴ : Algebra S R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : Module S M\ninst✝ : IsScalarTower S R M\ns : Set S\nN : Submodule R M\nx : M\nx_in : x ∈ ⇑(algebraMap S R) '' s... | · rintro _ x ⟨r, r_in, rfl⟩ x_in
rw [algebraMap_smul]
exact mem_set_smul_of_mem_mem r_in x_in | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 284,
"column": 60
} | {
"line": 284,
"column": 69
} | [
{
"pp": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\nx : R\n⊢ x * 0 / gcd x 0 = 0",
"usedConstants": [
"Eq.mpr",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"MulZeroClass.toMul",
"congr... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 336,
"column": 6
} | {
"line": 336,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝ : EuclideanDomain R\nx y z : R\nh1 : y ≠ 0\nh2 : y ∣ x\n⊢ y * (x / y + z) = x + y * z",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommR... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 354,
"column": 6
} | {
"line": 354,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝ : EuclideanDomain R\nx y z : R\nh1 : z ≠ 0\nh2 : z ∣ y\n⊢ z * (x + y / z) = z * x + y",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommR... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 386,
"column": 6
} | {
"line": 386,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝ : EuclideanDomain R\nx y z t : R\nh1 : y ≠ 0\nh2 : t ≠ 0\nh3 : y ∣ x\nh4 : t ∣ z\n⊢ t * y * (x / y + z / t) = t * x + y * z",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.h... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Maps | {
"line": 93,
"column": 34
} | {
"line": 93,
"column": 77
} | [
{
"pp": "R : Type u\nF : Type u_1\ninst✝⁴ : Semiring R\nS : Type u_2\ninst✝³ : CommSemiring S\ninst✝² : FunLike F R S\ninst✝¹ : RingHomClass F R S\nf : F\nP : Ideal R\ninst✝ : P.IsPrime\nI : Ideal S\nle : comap f I ≤ P\nx✝ : S\n⊢ x✝ ∈ ↑I → x✝ ∉ ↑(Submonoid.map f P.primeCompl)",
"usedConstants": [
"Fal... | by rintro hI ⟨r, hp, rfl⟩; exact hp (le hI) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.Maps | {
"line": 367,
"column": 4
} | {
"line": 367,
"column": 37
} | [
{
"pp": "R : Type u\nS : Type v\nF : Type u_1\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S\ninst✝⁴ : FunLike F R S\nf✝ : F\nI✝ J : Ideal R\nK L : Ideal S\nG : Type u_2\ninst✝³ : FunLike G S R\ninst✝² : RingHomClass F R S\nι : Sort u_3\nf : R →+* S\ninst✝¹ : RingHomSurjective f\nI : Ideal R\ninst✝ : I.IsTwoSided\na... | rw [map_eq_submodule_map] at ha ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Ideal.Maps | {
"line": 881,
"column": 9
} | {
"line": 881,
"column": 34
} | [
{
"pp": "case h\nR : Type u_1\nM : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nι : Type u_4\ninst✝ : Nonempty ι\nr : R\n⊢ r ∈ annihilator R (ι →₀ M) ↔ r ∈ annihilator R M",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Semiring.toModule",
"Module.annihi... | simp_rw [mem_annihilator] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.PrincipalIdealDomain | {
"line": 289,
"column": 12
} | {
"line": 290,
"column": 45
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝ : EuclideanDomain R\nS : Ideal R\nh : ¬{x | x ∈ S ∧ x ≠ 0}.Nonempty\na : R\n⊢ a ∈ S ↔ a = 0",
"usedConstants": [
"Submodule",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
"CommRing.toNonUnitalCommRing",
"CommSemiri... | exact ⟨fun haS => by_contra fun ha0 => h ⟨a, ⟨haS, ha0⟩⟩,
fun h₁ => h₁.symm ▸ S.zero_mem⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.PrincipalIdealDomain | {
"line": 552,
"column": 62
} | {
"line": 560,
"column": 21
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nc : Set (Ideal R)\nhs : c ⊆ {I | ¬IsPrincipal I}\nhchain : IsChain (fun x1 x2 ↦ x1 ≤ x2) c\nK : Ideal R\nhKmem : K ∈ c\n⊢ ∃ I ∈ {I | ¬IsPrincipal I}, ∀ J ∈ c, J ≤ I",
"usedConstants": [
"Submodule",
"SetLike.mem_coe._simp_1",
"False",
"NonUnit... | by
refine ⟨sSup c, ?_, fun J hJ => le_sSup hJ⟩
rintro ⟨x, hx⟩
have hxmem : x ∈ sSup c := hx.symm ▸ Submodule.mem_span_singleton_self x
obtain ⟨J, hJc, hxJ⟩ := (Submodule.mem_sSup_of_directed ⟨K, hKmem⟩ hchain.directedOn).1 hxmem
have hsSupJ : sSup c = J := le_antisymm (by simp [hx, Ideal.span_le, hxJ]) (le_sS... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 102,
"column": 45
} | {
"line": 102,
"column": 54
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : NormalizationMonoid α\nx y : α\nhx : ¬x = 0\nhy : y = 0\n⊢ x * 0 * ↑(normUnit (x * 0)) = x * ↑(normUnit x) * (0 * ↑(normUnit 0))",
"usedConstants": [
"Units.val",
"Eq.mpr",
"HMul.hMul",
"MulZeroClass.toMul",
"congrAr... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 949,
"column": 8
} | {
"line": 949,
"column": 16
} | [
{
"pp": "case h\nα : Type u_1\ninst✝¹ : CommRing α\ninst✝ : NormalizedGCDMonoid α\na b c d : α\nhd : b - c = a * d\ne : α\nhe : c = gcd a c * e\nf : α\nhf : a = gcd a c * f\n⊢ b = gcd a c * (e + f * d)",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Semigroup.toMul",
"Non... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1008,
"column": 12
} | {
"line": 1008,
"column": 21
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : IsCancelMulZero α\ninst✝ : DecidableEq α\ngcd : α → α → α\ngcd_dvd_left : ∀ (a b : α), gcd a b ∣ a\ngcd_dvd_right : ∀ (a b : α), gcd a b ∣ b\ndvd_gcd : ∀ {a b c : α}, a ∣ c → a ∣ b → a ∣ gcd c b\na b : α\na0 : a = 0\n⊢ Associated (gcd a b ... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1061,
"column": 12
} | {
"line": 1061,
"column": 21
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : IsCancelMulZero α\ninst✝¹ : NormalizationMonoid α\ninst✝ : DecidableEq α\ngcd : α → α → α\ngcd_dvd_left : ∀ (a b : α), gcd a b ∣ a\ngcd_dvd_right : ∀ (a b : α), gcd a b ∣ b\ndvd_gcd : ∀ {a b c : α}, a ∣ c → a ∣ b → a ∣ gcd c b\nnormalize_g... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1076,
"column": 13
} | {
"line": 1076,
"column": 22
} | [
{
"pp": "α : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : IsCancelMulZero α\ninst✝¹ : NormalizationMonoid α\ninst✝ : DecidableEq α\ngcd : α → α → α\ngcd_dvd_left : ∀ (a b : α), gcd a b ∣ a\ngcd_dvd_right : ∀ (a b : α), gcd a b ∣ b\ndvd_gcd : ∀ {a b c : α}, a ∣ c → a ∣ b → a ∣ gcd c b\nnormalize_gcd : ∀ (a ... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Operations | {
"line": 536,
"column": 8
} | {
"line": 536,
"column": 22
} | [
{
"pp": "case succ.succ\nR : Type u\ninst✝ : Semiring R\nn : ℕ\nih : n + 1 ≠ 0 → ↑(n + 1) = ⊤\nhn : n + 1 + 1 ≠ 0\n⊢ ↑(n + 1 + 1) = ⊤",
"usedConstants": [
"Eq.mpr",
"Nat.cast_succ",
"Submodule.instAddCommMonoidWithOne",
"Semiring.toModule",
"AddMonoid.toAddSemigroup",
"co... | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Operations | {
"line": 571,
"column": 4
} | {
"line": 572,
"column": 56
} | [
{
"pp": "case neg\nR : Type u\ninst✝ : CommSemiring R\nI J : Ideal R\nn m i : ℕ\nhi : i ∈ Finset.range (n + m + 1)\nhn : ¬n ≤ i\n⊢ I ^ i * J ^ (n + m - i) * ↑((n + m).choose i) ≤ I ^ n + J ^ m",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule",
"Ideal.mul_le_right"... | refine (Ideal.mul_le_right.trans (Ideal.mul_le_left.trans
((Ideal.pow_le_pow_right ?_).trans le_sup_right))) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.Dimension.StrongRankCondition | {
"line": 589,
"column": 2
} | {
"line": 589,
"column": 47
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"Nat.instMulZeroOneClass",
"Semiring.toModule",
"Pi.addCommMonoid",
"Cardinal"... | 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.Matrix.Diagonal | {
"line": 209,
"column": 2
} | {
"line": 210,
"column": 46
} | [
{
"pp": "n : Type u_3\nα : Type v\ninst✝¹ : DecidableEq n\ninst✝ : Zero α\nd : n → α\ni : n\n⊢ (diagonal d).col i = Pi.single i (d i)",
"usedConstants": [
"Equiv.instEquivLike",
"congrArg",
"Matrix",
"Pi.single_apply",
"Matrix.of",
"Equiv",
"Pi.single",
"funex... | ext
simp +contextual [diagonal, Pi.single_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Matrix.Diagonal | {
"line": 209,
"column": 2
} | {
"line": 210,
"column": 46
} | [
{
"pp": "n : Type u_3\nα : Type v\ninst✝¹ : DecidableEq n\ninst✝ : Zero α\nd : n → α\ni : n\n⊢ (diagonal d).col i = Pi.single i (d i)",
"usedConstants": [
"Equiv.instEquivLike",
"congrArg",
"Matrix",
"Pi.single_apply",
"Matrix.of",
"Equiv",
"Pi.single",
"funex... | ext
simp +contextual [diagonal, Pi.single_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Matrix.Diagonal | {
"line": 266,
"column": 8
} | {
"line": 266,
"column": 22
} | [
{
"pp": "l : Type u_1\nm : Type u_2\nn✝ : Type u_3\no : Type u_4\nm' : o → Type u_5\nn' : o → Type u_6\nR : Type u_7\nS : Type u_8\nα : Type v\nβ : Type w\nγ : Type u_9\ninst✝¹ : DecidableEq n✝\ninst✝ : AddMonoidWithOne α\nn : ℕ\n⊢ (diagonal fun x ↦ ↑(n + 1)) = (diagonal fun x ↦ ↑n) + 1",
"usedConstants": [... | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.DedekindFinite | {
"line": 19,
"column": 4
} | {
"line": 20,
"column": 39
} | [
{
"pp": "M : Type u_1\ninst✝¹ : Monoid M\ninst✝ : Finite M\na b : M\nhab : a * b = 1\n⊢ b * a = 1",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Exists",
"id",
"MulOne.toMul",
"left_inv_eq_right_inv",
... | 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
} | [
{
"pp": "M : Type u_1\ninst✝¹ : Monoid M\ninst✝ : Finite M\na b : M\nhab : a * b = 1\n⊢ b * a = 1",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Exists",
"id",
"MulOne.toMul",
"left_inv_eq_right_inv",
... | 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.Algebra.GCDMonoid.Basic | {
"line": 1088,
"column": 55
} | {
"line": 1088,
"column": 64
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : IsCancelMulZero 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\nexists_gcd : ∀ (a b : α), lcm a b ∣ a * b :... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Operations | {
"line": 1025,
"column": 11
} | {
"line": 1025,
"column": 39
} | [
{
"pp": "R : Type u\nι : Type u_1\ninst✝ : CommSemiring R\ns : Finset ι\nx : ι → R\np : Ideal R\nhp : p.IsPrime\n⊢ ∏ i ∈ s, x i ∈ p ↔ ∃ i ∈ s, x i ∈ p",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring",
"Finset",
"PartialOrder.toPreo... | ← span_singleton_le_iff_mem, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1158,
"column": 55
} | {
"line": 1158,
"column": 64
} | [
{
"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... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1159,
"column": 58
} | {
"line": 1159,
"column": 67
} | [
{
"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... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1296,
"column": 4
} | {
"line": 1296,
"column": 44
} | [
{
"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)",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"MulOne.toOne",
"False",
... | 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
} | [
{
"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)",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"MulOne.toOne",
"False",
... | 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
} | [
{
"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)",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"MulOne.toOne",
"False",
... | 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
} | [
{
"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",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"InvOneClass.toOne",
"HMul.hMul",
... | by simp [if_pos h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1301,
"column": 49
} | {
"line": 1301,
"column": 67
} | [
{
"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",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"InvOneClass.toOne",
"HMul.hMul",
... | by simp [if_pos h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Set.UnionLift | {
"line": 116,
"column": 24
} | {
"line": 116,
"column": 28
} | [
{
"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... | hui, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Data.Finsupp.Multiset | {
"line": 73,
"column": 4
} | {
"line": 77,
"column": 7
} | [
{
"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
} | [
{
"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
} | [
{
"pp": "case h\nm : 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✝",
"usedConstants": [
"Eq.mpr",
"Finset.mul_sum",
"Pi.... | simp only [vecMul, dotProduct, sum_apply, Finset.mul_sum, Finset.sum_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Dimension.Free | {
"line": 301,
"column": 27
} | {
"line": 301,
"column": 42
} | [
{
"pp": "case h\nR✝ : Type u\nS : Type u_1\nM✝ M₁ : Type v\nM' : Type v'\ninst✝¹⁵ : Semiring R✝\ninst✝¹⁴ : StrongRankCondition R✝\ninst✝¹³ : AddCommMonoid M✝\ninst✝¹² : Module R✝ M✝\ninst✝¹¹ : Free R✝ M✝\ninst✝¹⁰ : AddCommMonoid M'\ninst✝⁹ : Module R✝ M'\ninst✝⁸ : Free R✝ M'\ninst✝⁷ : AddCommMonoid M₁\ninst✝⁶ :... | simp [mul_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Algebra.Subalgebra.Lattice | {
"line": 435,
"column": 8
} | {
"line": 435,
"column": 16
} | [
{
"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 *... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.DirectSum.TensorProduct | {
"line": 139,
"column": 2
} | {
"line": 143,
"column": 42
} | [
{
"pp": "R : Type u\ninst✝⁹ : CommSemiring R\nS : Type u_1\ninst✝⁸ : Semiring S\ninst✝⁷ : Algebra R S\nι₂ : Type v₂\ninst✝⁶ : DecidableEq ι₂\nM₁' : Type w₁'\nM₂ : ι₂ → Type w₂\ninst✝⁵ : AddCommMonoid M₁'\ninst✝⁴ : (i₂ : ι₂) → AddCommMonoid (M₂ i₂)\ninst✝³ : Module R M₁'\ninst✝² : (i₂ : ι₂) → Module R (M₂ i₂)\ni... | suffices (DirectSum.component S ι₂ _ i).restrictScalars R ∘ₗ
(directSumRight R S M₁' M₂).toLinearMap.restrictScalars R ∘ₗ
(TensorProduct.mk R M₁' (⨁ i, M₂ i) m) =
(TensorProduct.mk R M₁' (M₂ i) m) ∘ₗ (DirectSum.component R ι₂ M₂ i) by
simpa using LinearMap.congr_fun this n | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.LinearAlgebra.DirectSum.Finsupp | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 13
} | [
{
"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
} | [
{
"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
} | [
{
"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
} | [
{
"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 ... | simp only [map_sum, map_smul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.FreeModule.PID | {
"line": 357,
"column": 4
} | {
"line": 357,
"column": 46
} | [
{
"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 ∉ ... | 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.FreeModule.PID | {
"line": 499,
"column": 2
} | {
"line": 499,
"column": 87
} | [
{
"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 ... | 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.OreLocalization.Basic | {
"line": 52,
"column": 47
} | {
"line": 52,
"column": 56
} | [
{
"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",
"usedConstants": [
"Eq.mpr",
"MonoidWithZero.toMulActionWithZero",
"HMul.hMul",
"MulZeroClass.toMul",
"Monoid.toMulOneClass",
"congrArg",
... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.OreLocalization.Basic | {
"line": 127,
"column": 4
} | {
"line": 127,
"column": 28
} | [
{
"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
} | [
{
"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",
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"MulOne.toOne",
"HMul.hMul",
... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.GroupTheory.MonoidLocalization.MonoidWithZero | {
"line": 96,
"column": 46
} | {
"line": 96,
"column": 55
} | [
{
"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",
"usedConstants": [
"CommMonoidW... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 1174,
"column": 4
} | {
"line": 1174,
"column": 21
} | [
{
"pp": "case a.h\nι : 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⊢ ... | split_ifs <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.RingTheory.OreLocalization.Ring | {
"line": 149,
"column": 10
} | {
"line": 149,
"column": 18
} | [
{
"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 ... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.Defs | {
"line": 466,
"column": 30
} | {
"line": 466,
"column": 38
} | [
{
"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)",
"usedConstants": [
"Di... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.Defs | {
"line": 499,
"column": 8
} | {
"line": 499,
"column": 16
} | [
{
"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 ... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.Defs | {
"line": 710,
"column": 10
} | {
"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 ... | Submonoid.comap_map_eq_of_injective (j : R ≃* P).injective | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Localization.Defs | {
"line": 760,
"column": 4
} | {
"line": 761,
"column": 39
} | [
{
"pp": "case toIsLocalizationMap.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).toAl... | 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 |
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