module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.LinearAlgebra.StdBasis | {
"line": 170,
"column": 23
} | {
"line": 170,
"column": 31
} | [
{
"pp": "case pos\nk : Type u_1\nG : Type u_2\ninst✝³ : CommSemiring k\ninst✝² : NoZeroDivisors k\ninst✝¹ : Nontrivial k\ninst✝ : Finite G\nφ : (G → k) →ₐ[k] k\nthis : Fintype G\nh1 : ∑ x, φ (Pi.single x 1) = 1\ns : G\nhs : φ (Pi.single s 1) ≠ 0\nx✝¹ : G\nx✝ : x✝¹ ≠ s\nu : G\nh✝ : u = s\n⊢ (Pi.single s 1 * Pi.s... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.StdBasis | {
"line": 170,
"column": 23
} | {
"line": 170,
"column": 31
} | [
{
"pp": "case neg\nk : Type u_1\nG : Type u_2\ninst✝³ : CommSemiring k\ninst✝² : NoZeroDivisors k\ninst✝¹ : Nontrivial k\ninst✝ : Finite G\nφ : (G → k) →ₐ[k] k\nthis : Fintype G\nh1 : ∑ x, φ (Pi.single x 1) = 1\ns : G\nhs : φ (Pi.single s 1) ≠ 0\nx✝¹ : G\nx✝ : x✝¹ ≠ s\nu : G\nh✝ : ¬u = s\n⊢ (Pi.single s 1 * Pi.... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.Finsupp.VectorSpace | {
"line": 114,
"column": 8
} | {
"line": 114,
"column": 16
} | [
{
"pp": "case neg\nR : Type u_1\nM : Type u_2\nι : Type u_3\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nφ : ι → Type u_4\nb : (i : ι) → Basis (φ i) R M\nx✝ : (i : ι) × φ i\ni : ι\nx : φ i\nj : ι\ny : φ j\nh : ¬i = j\n⊢ ((Finsupp.basis b).repr (single ⟨i, x⟩.fst ((b ⟨i, x⟩.fst) ⟨i, x⟩.snd... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.Finsupp.VectorSpace | {
"line": 114,
"column": 8
} | {
"line": 114,
"column": 16
} | [
{
"pp": "case neg\nR : Type u_1\nM : Type u_2\nι : Type u_3\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nφ : ι → Type u_4\nb : (i : ι) → Basis (φ i) R M\nx✝ : (i : ι) × φ i\ni : ι\nx : φ i\nj : ι\ny : φ j\nh : ¬i = j\n⊢ ((Finsupp.basis b).repr (single ⟨i, x⟩.fst ((b ⟨i, x⟩.fst) ⟨i, x⟩.snd... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Finsupp.VectorSpace | {
"line": 114,
"column": 8
} | {
"line": 114,
"column": 16
} | [
{
"pp": "case neg\nR : Type u_1\nM : Type u_2\nι : Type u_3\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nφ : ι → Type u_4\nb : (i : ι) → Basis (φ i) R M\nx✝ : (i : ι) × φ i\ni : ι\nx : φ i\nj : ι\ny : φ j\nh : ¬i = j\n⊢ ((Finsupp.basis b).repr (single ⟨i, x⟩.fst ((b ⟨i, x⟩.fst) ⟨i, x⟩.snd... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.StdBasis | {
"line": 175,
"column": 21
} | {
"line": 175,
"column": 29
} | [
{
"pp": "case pos\nk : Type u_1\nG : Type u_2\ninst✝³ : CommSemiring k\ninst✝² : NoZeroDivisors k\ninst✝¹ : Nontrivial k\ninst✝ : Finite G\nφ : (G → k) →ₐ[k] k\nthis : Fintype G\nh1 : ∑ x, φ (Pi.single x 1) = 1\ns : G\nhs : φ (Pi.single s 1) ≠ 0\nh2 : ∀ (t : G), t ≠ s → φ (Pi.single t 1) = 0\nh3 : φ (Pi.single ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.StdBasis | {
"line": 175,
"column": 21
} | {
"line": 175,
"column": 29
} | [
{
"pp": "case neg\nk : Type u_1\nG : Type u_2\ninst✝³ : CommSemiring k\ninst✝² : NoZeroDivisors k\ninst✝¹ : Nontrivial k\ninst✝ : Finite G\nφ : (G → k) →ₐ[k] k\nthis : Fintype G\nh1 : ∑ x, φ (Pi.single x 1) = 1\ns : G\nhs : φ (Pi.single s 1) ≠ 0\nh2 : ∀ (t : G), t ≠ s → φ (Pi.single t 1) = 0\nh3 : φ (Pi.single ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.Finsupp.VectorSpace | {
"line": 158,
"column": 64
} | {
"line": 158,
"column": 99
} | [
{
"pp": "R : Type u_1\nι : Type u_3\ninst✝ : Ring R\nα : Type u_4\nβ : Type u_5\nu : α → ι\nv : β → ι\nhu : Function.Injective u\nh : IsCompl (Set.range u) (Set.range v)\ni : ι\n⊢ i ∈ Set.range u ⊔ Set.range v",
"usedConstants": [
"Eq.mpr",
"Codisjoint",
"Lattice.toSemilatticeSup",
"... | rw [codisjoint_iff.mp h.2]; trivial | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Finsupp.VectorSpace | {
"line": 158,
"column": 64
} | {
"line": 158,
"column": 99
} | [
{
"pp": "R : Type u_1\nι : Type u_3\ninst✝ : Ring R\nα : Type u_4\nβ : Type u_5\nu : α → ι\nv : β → ι\nhu : Function.Injective u\nh : IsCompl (Set.range u) (Set.range v)\ni : ι\n⊢ i ∈ Set.range u ⊔ Set.range v",
"usedConstants": [
"Eq.mpr",
"Codisjoint",
"Lattice.toSemilatticeSup",
"... | rw [codisjoint_iff.mp h.2]; trivial | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Projection | {
"line": 52,
"column": 58
} | {
"line": 52,
"column": 92
} | [
{
"pp": "R : Type u_1\ninst✝² : Ring R\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\np : Submodule R E\nf : E →ₗ[R] ↥p\nhf : ∀ (x : ↥p), f ↑x = x\nx : E\nhx : x ∈ p\n⊢ x = ↑(f x)",
"usedConstants": [
"Subtype.coe_mk",
"Eq.mpr",
"Submodule",
"congrArg",
"AddCommGro... | by rw [hf ⟨x, hx⟩, Subtype.coe_mk] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Projection | {
"line": 677,
"column": 2
} | {
"line": 677,
"column": 95
} | [
{
"pp": "case h\nR : Type u_1\ninst✝² : Ring R\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\np q : E →ₗ[R] E\nhp : IsIdempotentElem p\nhq : IsIdempotentElem q\nx✝ : p.range = q.range ∧ p.ker = q.ker\nhr : p.range = q.range\nhk : p.ker = q.ker\nv : E\nhv : v ∈ p.ker\nw : E\nhw : w ∈ p.range\nright✝... | simp [mem_ker.mp, hv, (hk ▸ hv), (mem_range_iff hp).mp, hw, (mem_range_iff hq).mp, (hr ▸ hw)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Projection | {
"line": 764,
"column": 65
} | {
"line": 764,
"column": 81
} | [
{
"pp": "E : Type u_1\nR : Type u_2\ninst✝² : Ring R\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\nf : E →ₗ[R] E\nhf : IsIdempotentElem f\nT : (E →ₗ[R] E)ˣ\n⊢ Commute f ↑((GeneralLinearGroup.generalLinearEquiv R E) T) ↔\n map (↑((GeneralLinearGroup.generalLinearEquiv R E) T)) f.range = f.range ∧\n map (... | le_antisymm_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Algebra.NonUnitalHom | {
"line": 331,
"column": 41
} | {
"line": 331,
"column": 44
} | [
{
"pp": "R : Type u\nS : Type u₁\nT : Type u_1\ninst✝¹⁰ : Monoid R\ninst✝⁹ : Monoid S\ninst✝⁸ : Monoid T\nφ : R →* S\nA : Type v\nB : Type w\nC : Type w₁\ninst✝⁷ : NonUnitalNonAssocSemiring A\ninst✝⁶ : DistribMulAction R A\ninst✝⁵ : NonUnitalNonAssocSemiring B\ninst✝⁴ : DistribMulAction S B\ninst✝³ : NonUnitalN... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.TensorProduct.Basic | {
"line": 325,
"column": 21
} | {
"line": 325,
"column": 29
} | [
{
"pp": "case add\nR : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\nR' : Type u_4\nR'' : Type u_5\ninst✝⁴⁹ : CommSemiring R\ninst✝⁴⁸ : CommSemiring R₂\ninst✝⁴⁷ : CommSemiring R₃\ninst✝⁴⁶ : Monoid R'\ninst✝⁴⁵ : Semiring R''\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\nA✝ : Type u_6\nM✝ : Type u_7\nN✝ : Type u_8\... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.TensorProduct.Basic | {
"line": 325,
"column": 21
} | {
"line": 325,
"column": 29
} | [
{
"pp": "case add\nR : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\nR' : Type u_4\nR'' : Type u_5\ninst✝⁴⁹ : CommSemiring R\ninst✝⁴⁸ : CommSemiring R₂\ninst✝⁴⁷ : CommSemiring R₃\ninst✝⁴⁶ : Monoid R'\ninst✝⁴⁵ : Semiring R''\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\nA✝ : Type u_6\nM✝ : Type u_7\nN✝ : Type u_8\... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.TensorProduct.Basic | {
"line": 325,
"column": 21
} | {
"line": 325,
"column": 29
} | [
{
"pp": "case add\nR : Type u_1\nR₂ : Type u_2\nR₃ : Type u_3\nR' : Type u_4\nR'' : Type u_5\ninst✝⁴⁹ : CommSemiring R\ninst✝⁴⁸ : CommSemiring R₂\ninst✝⁴⁷ : CommSemiring R₃\ninst✝⁴⁶ : Monoid R'\ninst✝⁴⁵ : Semiring R''\nσ₁₂ : R →+* R₂\nσ₂₃ : R₂ →+* R₃\nσ₁₃ : R →+* R₃\nA✝ : Type u_6\nM✝ : Type u_7\nN✝ : Type u_8\... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.Submodule.Finsupp | {
"line": 66,
"column": 8
} | {
"line": 66,
"column": 16
} | [
{
"pp": "case le.refine_2.h.mpr\nR : Type u_2\nM : Type u_3\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nsR : Set R\nN : Submodule R M\ninst✝ : SMulCommClass R R ↥N\nc : R →₀ ↥N\nhc : c ∈ ↑(Finsupp.supported (↥N) R sR)\nx : M\n⊢ x ∈ sR • N → sR • N ∈ {p | ∀ ⦃r : R⦄ ⦃n : M⦄, r ∈ sR → n ∈ ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.Module.Submodule.Finsupp | {
"line": 66,
"column": 8
} | {
"line": 66,
"column": 16
} | [
{
"pp": "case le.refine_2.h.mpr\nR : Type u_2\nM : Type u_3\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nsR : Set R\nN : Submodule R M\ninst✝ : SMulCommClass R R ↥N\nc : R →₀ ↥N\nhc : c ∈ ↑(Finsupp.supported (↥N) R sR)\nx : M\n⊢ x ∈ sR • N → sR • N ∈ {p | ∀ ⦃r : R⦄ ⦃n : M⦄, r ∈ sR → n ∈ ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Module.Submodule.Finsupp | {
"line": 66,
"column": 8
} | {
"line": 66,
"column": 16
} | [
{
"pp": "case le.refine_2.h.mpr\nR : Type u_2\nM : Type u_3\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nsR : Set R\nN : Submodule R M\ninst✝ : SMulCommClass R R ↥N\nc : R →₀ ↥N\nhc : c ∈ ↑(Finsupp.supported (↥N) R sR)\nx : M\n⊢ x ∈ sR • N → sR • N ∈ {p | ∀ ⦃r : R⦄ ⦃n : M⦄, r ∈ sR → n ∈ ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Ring.NonZeroDivisors | {
"line": 35,
"column": 8
} | {
"line": 35,
"column": 11
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Monoid R\nr : R\ninst✝ : IsMulTorsionFree R\nhx : IsLeftRegular r\nhx' : r ≠ 1\nn✝ m : ℕ\nhnm : (fun n ↦ r ^ n) n✝ = (fun n ↦ r ^ n) m\nn l : ℕ\nh₁ : n ≤ n + l\nh₂ : l = 0\n⊢ n = n + l",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"instOfNatNat",
... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.TensorProduct.Map | {
"line": 79,
"column": 56
} | {
"line": 79,
"column": 81
} | [
{
"pp": "R : Type u_1\ninst✝⁸ : CommSemiring R\nM : Type u_7\nN : Type u_8\nP : Type u_9\nQ : Type u_10\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : AddCommMonoid N\ninst✝⁵ : AddCommMonoid P\ninst✝⁴ : AddCommMonoid Q\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : Module R P\ninst✝ : Module R Q\nf : M →ₗ[R] P\ng : N... | ← map₂_mk_top_top_eq_top, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Coprime.Basic | {
"line": 247,
"column": 31
} | {
"line": 247,
"column": 71
} | [
{
"pp": "R : Type u_1\nG : Type u_2\ninst✝⁴ : CommSemiring R\ninst✝³ : Group G\ninst✝² : MulAction G R\ninst✝¹ : SMulCommClass G R R\ninst✝ : IsScalarTower G R R\nx : G\ny z : R\nx✝ : IsCoprime (x • y) z\na b : R\nh : a * x • y + b * z = 1\n⊢ x • a * y + b * z = 1",
"usedConstants": [
"Eq.mpr",
... | by rwa [smul_mul_assoc, ← mul_smul_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Coprime.Lemmas | {
"line": 181,
"column": 2
} | {
"line": 181,
"column": 80
} | [
{
"pp": "case h.e'_2.a\nR : Type u\nI : Type v\ninst✝³ : CommSemiring R\ns : I → R\ninst✝² : Fintype I\ninst✝¹ : Nonempty I\ninst✝ : DecidableEq I\n⊢ Pairwise (IsCoprime on s) ↔ Pairwise (IsCoprime on fun i ↦ s ↑i)",
"usedConstants": [
"Finset.univ",
"Finset.coe_univ",
"Function.onFun",
... | simp only [pairwise_subtype_iff_pairwise_finset', coe_univ, Set.pairwise_univ] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Algebra.Operations | {
"line": 332,
"column": 77
} | {
"line": 333,
"column": 47
} | [
{
"pp": "R : Type u\ninst✝³ : Semiring R\nA : Type v\ninst✝² : Semiring A\ninst✝¹ : Module R A\ninst✝ : IsScalarTower R A A\nM : Submodule R A\nn : ℕ\nh : n ≠ 0\n⊢ M ^ (n + 1) = M * M ^ n",
"usedConstants": [
"Eq.mpr",
"Submodule",
"HMul.hMul",
"congrArg",
"Submodule.pow_add",
... | by
rw [add_comm, M.pow_add h, Submodule.pow_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Algebra.Operations | {
"line": 818,
"column": 13
} | {
"line": 818,
"column": 25
} | [
{
"pp": "ι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nm n : A\ns t : SetSemiring A\nP : Submodule R A\n⊢ (s * t) • P = s • t • P",
"usedConstants": [
"Submodule",
"SetSemiring.down",
"Semigroup.toMul",
... | HSMul.hSMul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Algebra.Operations | {
"line": 820,
"column": 13
} | {
"line": 820,
"column": 25
} | [
{
"pp": "ι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nm n : A\nP : Submodule R A\n⊢ 1 • P = P",
"usedConstants": [
"Submodule",
"MulOne.toOne",
"SetSemiring.down",
"instHSMul",
"Equiv.instEquivL... | HSMul.hSMul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Algebra.Operations | {
"line": 816,
"column": 13
} | {
"line": 816,
"column": 25
} | [
{
"pp": "ι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nm n : A\ns t : SetSemiring A\nP : Submodule R A\n⊢ (s + t) • P = s • P + t • P",
"usedConstants": [
"Submodule",
"SetSemiring.down",
"instHSMul",
... | HSMul.hSMul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Algebra.Operations | {
"line": 822,
"column": 13
} | {
"line": 822,
"column": 25
} | [
{
"pp": "ι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : CommSemiring A\ninst✝ : Algebra R A\nM N : Submodule R A\nm n : A\nP : Submodule R A\n⊢ 0 • P = 0",
"usedConstants": [
"Submodule",
"SetSemiring.down",
"instHSMul",
"Equiv.instEquivLike",
"HMul.hMul... | HSMul.hSMul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Algebra.Operations | {
"line": 901,
"column": 4
} | {
"line": 901,
"column": 41
} | [
{
"pp": "case h.mp\nR : Type u\ninst✝⁴ : CommSemiring R\nA : Type v\ninst✝³ : CommSemiring A\ninst✝² : Algebra R A\nB : Type u_1\ninst✝¹ : CommSemiring B\ninst✝ : Algebra R B\nI J : Submodule R A\nh : A ≃ₐ[R] B\nx : A\nhx : ∀ y ∈ J, x * y ∈ I\ny : A\nhy : y ∈ J\n⊢ ∃ y_1 ∈ I, h y_1 = h x * h y",
"usedConstan... | exact ⟨x * y, hx _ hy, map_mul h x y⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Noetherian.Basic | {
"line": 269,
"column": 91
} | {
"line": 269,
"column": 99
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : IsNoetherian R M\ninst✝ : Nontrivial R\nι : Type u_5\nv : ι → M\nhv : LinearIndependent R v\ni : ι\nx✝ : R ∙ v i = ⊥\n⊢ v i = 0",
"usedConstants": [
"Submodule.span_eq_bot._simp_1",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Noetherian.Basic | {
"line": 269,
"column": 91
} | {
"line": 269,
"column": 99
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : IsNoetherian R M\ninst✝ : Nontrivial R\nι : Type u_5\nv : ι → M\nhv : LinearIndependent R v\ni : ι\nx✝ : R ∙ v i = ⊥\n⊢ v i = 0",
"usedConstants": [
"Submodule.span_eq_bot._simp_1",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Noetherian.Basic | {
"line": 269,
"column": 91
} | {
"line": 269,
"column": 99
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : IsNoetherian R M\ninst✝ : Nontrivial R\nι : Type u_5\nv : ι → M\nhv : LinearIndependent R v\ni : ι\nx✝ : R ∙ v i = ⊥\n⊢ v i = 0",
"usedConstants": [
"Submodule.span_eq_bot._simp_1",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Noetherian.Basic | {
"line": 299,
"column": 4
} | {
"line": 300,
"column": 37
} | [
{
"pp": "case h.succ\nR : Type u_1\nM : Type u_2\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : IsNoetherian R M\nf : ℕ → Submodule R M\nh : ∀ (n : ℕ), Disjoint ((partialSups f) n) (f (n + 1))\nn : ℕ\nw : ∀ (m : ℕ), n ≤ m → f (m + 1) = ⊥\nm : ℕ\np : n + 1 ≤ m + 1\n⊢ f (m + 1) = ⊥",... | · apply w
exact Nat.succ_le_succ_iff.mp p | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 57,
"column": 4
} | {
"line": 57,
"column": 56
} | [
{
"pp": "R : Type u\ninst✝ : EuclideanDomain R\na b : R\nx✝ : b ∣ a\nc : R\ne : a = b * c\n⊢ a % b = 0",
"usedConstants": [
"Eq.mpr",
"Semigroup.toMul",
"instHDiv",
"HMul.hMul",
"AddLeftCancelSemigroup.toIsLeftCancelAdd",
"CommRing.toNonUnitalCommRing",
"congrArg",
... | rw [e, ← add_left_cancel_iff, div_add_mod, add_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 129,
"column": 2
} | {
"line": 130,
"column": 61
} | [
{
"pp": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na : R\n⊢ gcd a 0 = a",
"usedConstants": [
"Eq.mpr",
"EuclideanDomain.mod_lt",
"congrArg",
"CommSemiring.toSemiring",
"EuclideanDomain.r",
"dif_pos",
"id",
"instHMod",
"EuclideanDomai... | rw [gcd]
split_ifs with h <;> simp only [h, zero_mod, gcd_zero_left] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 129,
"column": 2
} | {
"line": 130,
"column": 61
} | [
{
"pp": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na : R\n⊢ gcd a 0 = a",
"usedConstants": [
"Eq.mpr",
"EuclideanDomain.mod_lt",
"congrArg",
"CommSemiring.toSemiring",
"EuclideanDomain.r",
"dif_pos",
"id",
"instHMod",
"EuclideanDomai... | rw [gcd]
split_ifs with h <;> simp only [h, zero_mod, gcd_zero_left] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Maps | {
"line": 697,
"column": 4
} | {
"line": 698,
"column": 35
} | [
{
"pp": "case refine_1\nR : Type u\ninst✝² : CommSemiring R\nS : Type u_2\ninst✝¹ : CommSemiring S\nf : R →+* S\np : Ideal R\ninst✝ : p.IsPrime\nhp : comap f (map f p) = p\nq : Ideal S\nhq₁ : q.IsPrime\nhq₂ : map f p ≤ q\nhq₃ : Disjoint ↑q ↑(Submonoid.map f p.primeCompl)\n⊢ ∃ q, q.IsPrime ∧ comap f q = p",
... | exact ⟨q, hq₁, le_antisymm (disjoint_map_primeCompl_iff_comap_le.mp hq₃)
(map_le_iff_le_comap.mp hq₂)⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Ideal.Operations | {
"line": 93,
"column": 2
} | {
"line": 97,
"column": 28
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nM' : Submodule R M\ns : Set R\nhs : Ideal.span s = ⊤\nx : M\nH : ∀ (r : ↑s), ↑r • x ∈ M'\n⊢ x ∈ M'",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule",
... | suffices LinearMap.range (LinearMap.toSpanSingleton R M x) ≤ M' by
rw [← LinearMap.toSpanSingleton_apply_one R M x]
exact this (LinearMap.mem_range_self _ 1)
rw [LinearMap.range_eq_map, ← hs, map_le_iff_le_comap, Ideal.span, span_le]
exact fun r hr ↦ H ⟨r, hr⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Operations | {
"line": 93,
"column": 2
} | {
"line": 97,
"column": 28
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nM' : Submodule R M\ns : Set R\nhs : Ideal.span s = ⊤\nx : M\nH : ∀ (r : ↑s), ↑r • x ∈ M'\n⊢ x ∈ M'",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Submodule",
... | suffices LinearMap.range (LinearMap.toSpanSingleton R M x) ≤ M' by
rw [← LinearMap.toSpanSingleton_apply_one R M x]
exact this (LinearMap.mem_range_self _ 1)
rw [LinearMap.range_eq_map, ← hs, map_le_iff_le_comap, Ideal.span, span_le]
exact fun r hr ↦ H ⟨r, hr⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Maps | {
"line": 949,
"column": 50
} | {
"line": 949,
"column": 53
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nι : Sort w\nf : ι → Submodule R M\nr : R\nH : r ∈ ⨅ i, (f i).annihilator\nn : M\nhn : n ∈ ⨆ i, f i\nm₁ m₂ : M\nh₁ : (fun x ↦ r • x = 0) m₁\nh₂ : (fun x ↦ r • x = 0) m₂\n⊢ 0 + r • m₂ = 0",
"usedConstants":... | h₂, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.RingTheory.Ideal.Maps | {
"line": 1072,
"column": 4
} | {
"line": 1082,
"column": 56
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : FunLike F R S\nrc : RingHomClass F R S\nA : Set (Ideal R)\nf : F\nhf : Function.Surjective ⇑f\nh : ∀ J ∈ A, RingHom.ker f ≤ J\n⊢ sInf (map f '' A) ≤ map f (sInf A)",
"usedConstants": [
"Eq.mpr",... | intro y hy
obtain ⟨x, hx⟩ := hf y
refine hx ▸ mem_map_of_mem f ?_
have : ∀ I ∈ A, y ∈ map f I := by simpa using hy
rw [Submodule.mem_sInf]
intro J hJ
rcases (mem_map_iff_of_surjective f hf).1 (this J hJ) with ⟨x', hx', rfl⟩
have : x - x' ∈ J := by
apply h J hJ
rw [RingHom.mem_ker... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Maps | {
"line": 1072,
"column": 4
} | {
"line": 1082,
"column": 56
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : FunLike F R S\nrc : RingHomClass F R S\nA : Set (Ideal R)\nf : F\nhf : Function.Surjective ⇑f\nh : ∀ J ∈ A, RingHom.ker f ≤ J\n⊢ sInf (map f '' A) ≤ map f (sInf A)",
"usedConstants": [
"Eq.mpr",... | intro y hy
obtain ⟨x, hx⟩ := hf y
refine hx ▸ mem_map_of_mem f ?_
have : ∀ I ∈ A, y ∈ map f I := by simpa using hy
rw [Submodule.mem_sInf]
intro J hJ
rcases (mem_map_iff_of_surjective f hf).1 (this J hJ) with ⟨x', hx', rfl⟩
have : x - x' ∈ J := by
apply h J hJ
rw [RingHom.mem_ker... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.GCDMonoid.Nat | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 32
} | [
{
"pp": "a b : ℤ\n⊢ Associated a b ↔ a = b ∨ a = -b",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Int.instNegInt",
"Int",
"Int.instMonoid",
"Iff",
"Associated",
"Nat",
"Int.associated_iff_natAbs",
"Int.natAbs",
"propext",
"Or"... | rw [Int.associated_iff_natAbs] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Ideal.Operations | {
"line": 531,
"column": 4
} | {
"line": 532,
"column": 65
} | [
{
"pp": "case succ\nR : Type u\ninst✝ : Semiring R\nn : ℕ\nih : n ≠ 0 → ↑n = ⊤\nhn : n + 1 ≠ 0\n⊢ ↑(n + 1) = ⊤",
"usedConstants": [
"Eq.mpr",
"Nat.cast_succ",
"Submodule.instAddCommMonoidWithOne",
"Ideal.one_eq_top",
"Lattice.toSemilatticeSup",
"Semiring.toModule",
... | obtain rfl | n := n; · rw [Nat.cast_one, one_eq_top]
rw [Nat.cast_succ, ih n.succ_ne_zero, add_eq_sup, top_sup_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Operations | {
"line": 531,
"column": 4
} | {
"line": 532,
"column": 65
} | [
{
"pp": "case succ\nR : Type u\ninst✝ : Semiring R\nn : ℕ\nih : n ≠ 0 → ↑n = ⊤\nhn : n + 1 ≠ 0\n⊢ ↑(n + 1) = ⊤",
"usedConstants": [
"Eq.mpr",
"Nat.cast_succ",
"Submodule.instAddCommMonoidWithOne",
"Ideal.one_eq_top",
"Lattice.toSemilatticeSup",
"Semiring.toModule",
... | obtain rfl | n := n; · rw [Nat.cast_one, one_eq_top]
rw [Nat.cast_succ, ih n.succ_ne_zero, add_eq_sup, top_sup_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Operations | {
"line": 813,
"column": 6
} | {
"line": 813,
"column": 22
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nI : Ideal R\n⊢ I.radical = I ↔ I.IsRadical",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemiring",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"Submodule.instPartialOrder",
... | le_antisymm_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 374,
"column": 2
} | {
"line": 374,
"column": 10
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : GCDMonoid α\na b : α\nha : a ≠ 0\n⊢ gcd a b ≠ 0",
"usedConstants": [
"False",
"eq_false",
"congrArg",
"false_and",
"id",
"CommMonoidWithZero.toMonoidWithZero",
"GCDMonoid.gcd",
"And",
"MonoidW... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 374,
"column": 2
} | {
"line": 374,
"column": 10
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : GCDMonoid α\na b : α\nha : a ≠ 0\n⊢ gcd a b ≠ 0",
"usedConstants": [
"False",
"eq_false",
"congrArg",
"false_and",
"id",
"CommMonoidWithZero.toMonoidWithZero",
"GCDMonoid.gcd",
"And",
"MonoidW... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 374,
"column": 2
} | {
"line": 374,
"column": 10
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : GCDMonoid α\na b : α\nha : a ≠ 0\n⊢ gcd a b ≠ 0",
"usedConstants": [
"False",
"eq_false",
"congrArg",
"false_and",
"id",
"CommMonoidWithZero.toMonoidWithZero",
"GCDMonoid.gcd",
"And",
"MonoidW... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 373,
"column": 82
} | {
"line": 374,
"column": 10
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : GCDMonoid α\na b : α\nha : a ≠ 0\n⊢ gcd a b ≠ 0",
"usedConstants": [
"False",
"eq_false",
"congrArg",
"false_and",
"id",
"CommMonoidWithZero.toMonoidWithZero",
"GCDMonoid.gcd",
"And",
"MonoidW... | by
simp_all | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 377,
"column": 2
} | {
"line": 377,
"column": 10
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : GCDMonoid α\na b : α\nhb : b ≠ 0\n⊢ gcd a b ≠ 0",
"usedConstants": [
"False",
"eq_false",
"congrArg",
"id",
"CommMonoidWithZero.toMonoidWithZero",
"GCDMonoid.gcd",
"And",
"MonoidWithZero.toMulZeroOn... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 377,
"column": 2
} | {
"line": 377,
"column": 10
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : GCDMonoid α\na b : α\nhb : b ≠ 0\n⊢ gcd a b ≠ 0",
"usedConstants": [
"False",
"eq_false",
"congrArg",
"id",
"CommMonoidWithZero.toMonoidWithZero",
"GCDMonoid.gcd",
"And",
"MonoidWithZero.toMulZeroOn... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 377,
"column": 2
} | {
"line": 377,
"column": 10
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : GCDMonoid α\na b : α\nhb : b ≠ 0\n⊢ gcd a b ≠ 0",
"usedConstants": [
"False",
"eq_false",
"congrArg",
"id",
"CommMonoidWithZero.toMonoidWithZero",
"GCDMonoid.gcd",
"And",
"MonoidWithZero.toMulZeroOn... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 376,
"column": 83
} | {
"line": 377,
"column": 10
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : GCDMonoid α\na b : α\nhb : b ≠ 0\n⊢ gcd a b ≠ 0",
"usedConstants": [
"False",
"eq_false",
"congrArg",
"id",
"CommMonoidWithZero.toMonoidWithZero",
"GCDMonoid.gcd",
"And",
"MonoidWithZero.toMulZeroOn... | by
simp_all | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Ideal.Operations | {
"line": 1074,
"column": 4
} | {
"line": 1074,
"column": 80
} | [
{
"pp": "case zero\nι : Type u_1\nR : Type u\ninst✝ : CommRing R\nf : ι → Ideal R\nI : Ideal R\na b : ι\nh : ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑∅, ↑(f i)\n⊢ I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ ∅, I ≤ f i",
"usedConstants": [
"Finset.coe_empty",
"Semiring.toModule",
"congrArg",
"CommSemiring.toSemir... | rw [Finset.coe_empty, Set.biUnion_empty, Set.union_empty, subset_union] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Ideal.Operations | {
"line": 1087,
"column": 8
} | {
"line": 1087,
"column": 45
} | [
{
"pp": "ι : Type u_1\nR : Type u\ninst✝ : CommRing R\nf : ι → Ideal R\nI : Ideal R\nn : ℕ\nih :\n ∀ {s : Finset ι} {a b : ι},\n (∀ i ∈ s, (f i).IsPrime) →\n s.card = n → ↑I ⊆ ↑(f a) ∪ ↑(f b) ∪ ⋃ i ∈ ↑s, ↑(f i) → I ≤ f a ∨ I ≤ f b ∨ ∃ i ∈ s, I ≤ f i\na b i j : ι\nhfji : f j ≤ f i\nu : Finset ι\nhju : j... | rw [Finset.forall_mem_insert] at hp ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 585,
"column": 4
} | {
"line": 585,
"column": 40
} | [
{
"pp": "case h\nα : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : GCDMonoid α\na b c : α\nhab : IsUnit (gcd a b)\nh✝ : Nontrivial α\nha : ¬a = 0\nhb : ¬b = 0\nh : a * b = 1\n⊢ 1 * ↑(Units.mkOfMulEqOne a b h) = a",
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Units.val",
"Eq.m... | rw [Units.val_mkOfMulEqOne, one_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 650,
"column": 2
} | {
"line": 650,
"column": 99
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : GCDMonoid α\nx y : α\nhx : Irreducible x\n⊢ IsUnit (gcd x y) ↔ ¬x ∣ y",
"usedConstants": [
"not_iff_not",
"Eq.mpr",
"Dvd.dvd",
"congrArg",
"Iff.rfl",
"semigroupDvd",
"IsUnit",
"SemigroupWithZero.toS... | rw [hx.isUnit_iff_not_associated_of_dvd (gcd_dvd_left x y), not_iff_not, associated_gcd_left_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 650,
"column": 2
} | {
"line": 650,
"column": 99
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : GCDMonoid α\nx y : α\nhx : Irreducible x\n⊢ IsUnit (gcd x y) ↔ ¬x ∣ y",
"usedConstants": [
"not_iff_not",
"Eq.mpr",
"Dvd.dvd",
"congrArg",
"Iff.rfl",
"semigroupDvd",
"IsUnit",
"SemigroupWithZero.toS... | rw [hx.isUnit_iff_not_associated_of_dvd (gcd_dvd_left x y), not_iff_not, associated_gcd_left_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 650,
"column": 2
} | {
"line": 650,
"column": 99
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : GCDMonoid α\nx y : α\nhx : Irreducible x\n⊢ IsUnit (gcd x y) ↔ ¬x ∣ y",
"usedConstants": [
"not_iff_not",
"Eq.mpr",
"Dvd.dvd",
"congrArg",
"Iff.rfl",
"semigroupDvd",
"IsUnit",
"SemigroupWithZero.toS... | rw [hx.isUnit_iff_not_associated_of_dvd (gcd_dvd_left x y), not_iff_not, associated_gcd_left_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 979,
"column": 41
} | {
"line": 979,
"column": 51
} | [
{
"pp": "α : Type u_1\ninst✝² : CommMonoidWithZero α\ninst✝¹ : IsCancelMulZero α\ninst✝ : DecidableEq α\nf : Associates α →* α\nhinv : Function.RightInverse (⇑f) Associates.mk\na b : α\nha : a ≠ 0\nhb : b ≠ 0\n⊢ Classical.choose ⋯ = (if a = 0 then 1 else Classical.choose ⋯) * if b = 0 then 1 else Classical.choo... | if_neg ha, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1294,
"column": 32
} | {
"line": 1294,
"column": 40
} | [
{
"pp": "α : Type u_1\nG₀ : Type u_2\ninst✝¹ : CommGroupWithZero G₀\ninst✝ : DecidableEq G₀\na b c : G₀\nhac : a ∣ c\nhab : a ∣ b\n⊢ a ∣ if c = 0 ∧ b = 0 then 0 else 1",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"Dvd.dvd",
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisio... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1294,
"column": 32
} | {
"line": 1294,
"column": 40
} | [
{
"pp": "α : Type u_1\nG₀ : Type u_2\ninst✝¹ : CommGroupWithZero G₀\ninst✝ : DecidableEq G₀\na b c : G₀\nhac : a ∣ c\nhab : a ∣ b\n⊢ a ∣ if c = 0 ∧ b = 0 then 0 else 1",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"Dvd.dvd",
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisio... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1294,
"column": 32
} | {
"line": 1294,
"column": 40
} | [
{
"pp": "α : Type u_1\nG₀ : Type u_2\ninst✝¹ : CommGroupWithZero G₀\ninst✝ : DecidableEq G₀\na b c : G₀\nhac : a ∣ c\nhab : a ∣ b\n⊢ a ∣ if c = 0 ∧ b = 0 then 0 else 1",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"Dvd.dvd",
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisio... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Matrix.Basis | {
"line": 352,
"column": 2
} | {
"line": 352,
"column": 10
} | [
{
"pp": "n : Type u_3\nα : Type u_7\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : Semiring α\ni j k : n\nM : Matrix n n α\nhM : Commute (single i j 1) M\nhkj : k ≠ j\nthis : (single i j 1 * M) i k = (M * single i j 1) i k\n⊢ M j k = 0",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOn... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.Matrix.Basis | {
"line": 357,
"column": 2
} | {
"line": 357,
"column": 10
} | [
{
"pp": "n : Type u_3\nα : Type u_7\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : Semiring α\ni j k : n\nM : Matrix n n α\nhM : Commute (single i j 1) M\nhki : k ≠ i\nthis : (single i j 1 * M) k j = (M * single i j 1) k j\n⊢ M k i = 0",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOn... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.Matrix.Basis | {
"line": 362,
"column": 2
} | {
"line": 362,
"column": 10
} | [
{
"pp": "n : Type u_3\nα : Type u_7\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : Semiring α\ni j : n\nM : Matrix n n α\nhM : Commute (single i j 1) M\nthis : (single i j 1 * M) i j = (M * single i j 1) i j\n⊢ M i i = M j j",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.Finsupp.Multiset | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 90
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\na : α\n⊢ toFinsupp {a} = Finsupp.single a 1",
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.instMulZeroClass",
"Finsupp.ext",
"congrArg",
"Pi.single_apply",
"AddMonoid.toAddZeroClass",
"Nat.instAddMonoid... | ext; rw [toFinsupp_apply, count_singleton, Finsupp.single_eq_pi_single, Pi.single_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finsupp.Multiset | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 90
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\na : α\n⊢ toFinsupp {a} = Finsupp.single a 1",
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.instMulZeroClass",
"Finsupp.ext",
"congrArg",
"Pi.single_apply",
"AddMonoid.toAddZeroClass",
"Nat.instAddMonoid... | ext; rw [toFinsupp_apply, count_singleton, Finsupp.single_eq_pi_single, Pi.single_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Cardinal.Finsupp | {
"line": 66,
"column": 2
} | {
"line": 67,
"column": 71
} | [
{
"pp": "α : Type u\ninst✝ : Nonempty α\n⊢ #(Multiset α) = max #α ℵ₀",
"usedConstants": [
"Nat.instMulZeroClass",
"Lattice.toSemilatticeSup",
"Cardinal",
"AddMonoid.toAddZeroClass",
"Classical.propDecidable",
"Nat.instAddMonoid",
"Cardinal.mk",
"SemilatticeSup... | classical
exact Multiset.toFinsupp.toEquiv.cardinal_eq.trans (mk_finsupp_nat α) | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.SetTheory.Cardinal.Finsupp | {
"line": 66,
"column": 2
} | {
"line": 67,
"column": 71
} | [
{
"pp": "α : Type u\ninst✝ : Nonempty α\n⊢ #(Multiset α) = max #α ℵ₀",
"usedConstants": [
"Nat.instMulZeroClass",
"Lattice.toSemilatticeSup",
"Cardinal",
"AddMonoid.toAddZeroClass",
"Classical.propDecidable",
"Nat.instAddMonoid",
"Cardinal.mk",
"SemilatticeSup... | classical
exact Multiset.toFinsupp.toEquiv.cardinal_eq.trans (mk_finsupp_nat α) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Cardinal.Finsupp | {
"line": 66,
"column": 2
} | {
"line": 67,
"column": 71
} | [
{
"pp": "α : Type u\ninst✝ : Nonempty α\n⊢ #(Multiset α) = max #α ℵ₀",
"usedConstants": [
"Nat.instMulZeroClass",
"Lattice.toSemilatticeSup",
"Cardinal",
"AddMonoid.toAddZeroClass",
"Classical.propDecidable",
"Nat.instAddMonoid",
"Cardinal.mk",
"SemilatticeSup... | classical
exact Multiset.toFinsupp.toEquiv.cardinal_eq.trans (mk_finsupp_nat α) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.TwoSidedIdeal.Kernel | {
"line": 39,
"column": 22
} | {
"line": 39,
"column": 30
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : NonUnitalNonAssocSemiring S\nF : Type u_3\ninst✝¹ : FunLike F R S\ninst✝ : NonUnitalRingHomClass F R S\nf : F\nw✝ : R\n⊢ ∀ {x y z : R}, f w✝ = f x → f y = f z → f (w✝ * y) = f (x * z)",
"usedConstants": [
"HMul.hMul",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.TwoSidedIdeal.Kernel | {
"line": 40,
"column": 22
} | {
"line": 40,
"column": 30
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : NonUnitalNonAssocRing R\ninst✝² : NonUnitalNonAssocSemiring S\nF : Type u_3\ninst✝¹ : FunLike F R S\ninst✝ : NonUnitalRingHomClass F R S\nf : F\nw✝ : R\n⊢ ∀ {x y z : R}, f w✝ = f x → f y = f z → f (w✝ + y) = f (x + z)",
"usedConstants": [
"congrArg",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.Matrix.Basic | {
"line": 651,
"column": 39
} | {
"line": 651,
"column": 71
} | [
{
"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\nT : Type u_9\nA : Type u_10\nα✝ : Type u_11\nβ : Type u_12\nγ : Type u_13\ninst✝⁶ : Fintype m\ninst✝⁵ : DecidableEq m\ninst✝⁴ : NonAssocSemiring α✝\ninst✝³ : NonAssocSemiring β\nins... | ext; simp [transpose, mul_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Matrix.Basic | {
"line": 651,
"column": 39
} | {
"line": 651,
"column": 71
} | [
{
"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\nT : Type u_9\nA : Type u_10\nα✝ : Type u_11\nβ : Type u_12\nγ : Type u_13\ninst✝⁶ : Fintype m\ninst✝⁵ : DecidableEq m\ninst✝⁴ : NonAssocSemiring α✝\ninst✝³ : NonAssocSemiring β\nins... | ext; simp [transpose, mul_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.DirectSum.Basic | {
"line": 288,
"column": 48
} | {
"line": 288,
"column": 63
} | [
{
"pp": "ι✝ : Type v\nβ : ι✝ → Type w\ninst✝² : (i : ι✝) → AddCommMonoid (β i)\nM : Type v\nι : Type u_1\ninst✝¹ : AddCommMonoid M\ninst✝ : Unique ι\nx✝ : ⨁ (x : ι), M\np : ι\nx : M\n⊢ (of (fun x ↦ M) p) ((toAddMonoid fun x ↦ AddMonoidHom.id M) ((of (fun i ↦ M) p) x)) = (of (fun i ↦ M) p) x",
"usedConstants... | toAddMonoid_of, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.TensorProduct.Associator | {
"line": 391,
"column": 2
} | {
"line": 391,
"column": 10
} | [
{
"pp": "case h.add\nR : Type u_11\nA : Type u_12\nA' : Type u_13\nB : Type u_14\nB' : Type u_15\nC : Type u_16\nC' : Type u_17\ninst✝⁶ : CommSemiring R\ninst✝⁵ : AddCommMonoid A'\ninst✝⁴ : AddCommMonoid B'\ninst✝³ : AddCommMonoid C'\ninst✝² : Module R A'\ninst✝¹ : Module R B'\ninst✝ : Module R C'\neA : A ≃ A'\... | | add => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.LinearAlgebra.TensorProduct.Basis | {
"line": 149,
"column": 18
} | {
"line": 149,
"column": 26
} | [
{
"pp": "case zero\nR : Type u_1\nM : Type u_3\nN : Type u_4\nκ : Type u_6\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\ninst✝ : DecidableEq κ\n𝒞 : Basis κ R N\ni : κ\n⊢ ((equivFinsuppOfBasisRight 𝒞) 0) i = (TensorProduct.rid R M) ((lTe... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.TensorProduct.Basis | {
"line": 149,
"column": 18
} | {
"line": 149,
"column": 26
} | [
{
"pp": "case tmul\nR : Type u_1\nM : Type u_3\nN : Type u_4\nκ : Type u_6\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\ninst✝ : DecidableEq κ\n𝒞 : Basis κ R N\ni : κ\nx✝ : M\ny✝ : N\n⊢ ((equivFinsuppOfBasisRight 𝒞) (x✝ ⊗ₜ[R] y✝)) i = (... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.TensorProduct.Basis | {
"line": 149,
"column": 18
} | {
"line": 149,
"column": 26
} | [
{
"pp": "case add\nR : Type u_1\nM : Type u_3\nN : Type u_4\nκ : Type u_6\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\ninst✝ : DecidableEq κ\n𝒞 : Basis κ R N\ni : κ\nx✝ y✝ : M ⊗[R] N\na✝¹ : ((equivFinsuppOfBasisRight 𝒞) x✝) i = (Tensor... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.TensorProduct.Basis | {
"line": 155,
"column": 18
} | {
"line": 155,
"column": 26
} | [
{
"pp": "case zero\nR : Type u_1\nM : Type u_3\nN : Type u_4\nι : Type u_5\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\ninst✝ : DecidableEq ι\nℬ : Basis ι R M\ni : ι\n⊢ ((equivFinsuppOfBasisLeft ℬ) 0) i = (TensorProduct.lid R N) ((rTenso... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.TensorProduct.Basis | {
"line": 155,
"column": 18
} | {
"line": 155,
"column": 26
} | [
{
"pp": "case tmul\nR : Type u_1\nM : Type u_3\nN : Type u_4\nι : Type u_5\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\ninst✝ : DecidableEq ι\nℬ : Basis ι R M\ni : ι\nx✝ : M\ny✝ : N\n⊢ ((equivFinsuppOfBasisLeft ℬ) (x✝ ⊗ₜ[R] y✝)) i = (Ten... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.TensorProduct.Basis | {
"line": 155,
"column": 18
} | {
"line": 155,
"column": 26
} | [
{
"pp": "case add\nR : Type u_1\nM : Type u_3\nN : Type u_4\nι : Type u_5\ninst✝⁵ : CommSemiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : AddCommMonoid N\ninst✝¹ : Module R N\ninst✝ : DecidableEq ι\nℬ : Basis ι R M\ni : ι\nx✝ y✝ : M ⊗[R] N\na✝¹ : ((equivFinsuppOfBasisLeft ℬ) x✝) i = (TensorPro... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.TensorProduct.Basis | {
"line": 179,
"column": 2
} | {
"line": 180,
"column": 74
} | [
{
"pp": "R : Type u_1\nM : Type u_3\nN : Type u_4\nι : Type u_5\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nℬ : Basis ι R M\nx : M ⊗[R] N\n⊢ ∃ c, (c.sum fun i n ↦ ℬ i ⊗ₜ[R] n) = x",
"usedConstants": [
"Finsupp.instFunLike",
... | classical obtain ⟨c, rfl⟩ := (TensorProduct.equivFinsuppOfBasisLeft ℬ).symm.surjective x
exact ⟨c, (TensorProduct.comm R M N).injective <| by simp [Finsupp.sum]⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.TensorProduct.Basis | {
"line": 179,
"column": 2
} | {
"line": 180,
"column": 74
} | [
{
"pp": "R : Type u_1\nM : Type u_3\nN : Type u_4\nι : Type u_5\ninst✝⁴ : CommSemiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nℬ : Basis ι R M\nx : M ⊗[R] N\n⊢ ∃ c, (c.sum fun i n ↦ ℬ i ⊗ₜ[R] n) = x",
"usedConstants": [
"Finsupp.instFunLike",
... | classical obtain ⟨c, rfl⟩ := (TensorProduct.equivFinsuppOfBasisLeft ℬ).symm.surjective x
exact ⟨c, (TensorProduct.comm R M N).injective <| by simp [Finsupp.sum]⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.DirectSum.Finsupp | {
"line": 244,
"column": 79
} | {
"line": 244,
"column": 86
} | [
{
"pp": "case single.single\nR : Type u_1\nS : Type u_2\nM : Type u_3\nN : Type u_4\nι : Type u_5\nκ : Type u_6\ninst✝⁸ : CommSemiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : Module R N\ninst✝³ : Semiring S\ninst✝² : Algebra R S\ninst✝¹ : Module S M\ninst✝ : IsScalar... | ite_and | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Matrix.Block | {
"line": 863,
"column": 65
} | {
"line": 865,
"column": 50
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nR : Type u_14\ninst✝¹ : Zero R\ninst✝ : DecidableEq m\nd : m → Matrix n n R\n⊢ (comp m m n n R) (diagonal d) = (reindex (Equiv.prodComm n m) (Equiv.prodComm n m)) (blockDiagonal d)",
"usedConstants": [
"Matrix.comp",
"Equiv.instEquivLike",
"Matrix.block... | by
ext
simp [diagonal, blockDiagonal, Matrix.ite_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.FreeModule.PID | {
"line": 91,
"column": 11
} | {
"line": 91,
"column": 27
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommSemiring R\nI : Ideal R\ninst✝ : Submodule.IsPrincipal I\nx : R\nhx : x ∈ I\n⊢ x ∣ generator I ↔ I = Ideal.span {x}",
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"Semiring.toModule",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
"Comm... | le_antisymm_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.AffineMonoid.Irreducible | {
"line": 34,
"column": 45
} | {
"line": 34,
"column": 53
} | [
{
"pp": "case inl\nM : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : Subsingleton Mˣ\nS : Set M\nx : M\nhx : x ∈ {p | p ∈ Submonoid.closure S ∧ Irreducible p}\nb : M\nx✝¹ : b ∈ Submonoid.closure S\nhb : (fun x x_1 ↦ Irreducible x → x ∈ S) b x✝¹\nx✝ : 1 ∈ Submonoid.closure S\nha : Irreducible 1 → 1 ∈ S\nh : Irreducib... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Algebra.AffineMonoid.Irreducible | {
"line": 34,
"column": 45
} | {
"line": 34,
"column": 53
} | [
{
"pp": "case inr\nM : Type u_1\ninst✝¹ : CommMonoid M\ninst✝ : Subsingleton Mˣ\nS : Set M\nx : M\nhx : x ∈ {p | p ∈ Submonoid.closure S ∧ Irreducible p}\na : M\nx✝¹ : a ∈ Submonoid.closure S\nha : (fun x x_1 ↦ Irreducible x → x ∈ S) a x✝¹\nx✝ : 1 ∈ Submonoid.closure S\nhb : Irreducible 1 → 1 ∈ S\nh : Irreducib... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.LinearAlgebra.TensorProduct.Tower | {
"line": 893,
"column": 95
} | {
"line": 896,
"column": 6
} | [
{
"pp": "R : Type u_1\nA : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring A\ninst✝⁴ : Algebra R A\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : AddCommGroup N\ninst✝ : Module A N\nf : A ⊗[R] M →ₗ[A] N\n⊢ LinearMap.baseChange A f ∘ₗ ↑(cancelBaseChange R A A A M).sy... | by
rw [cancelBaseChange_self_eq_lid]
ext x
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.OreLocalization.Basic | {
"line": 48,
"column": 47
} | {
"line": 48,
"column": 54
} | [
{
"pp": "case c\nR : Type u_1\ninst✝¹ : MonoidWithZero R\nS : Submonoid R\ninst✝ : OreSet S\nr : R\ns : ↥S\n⊢ 0 * (r /ₒ s) = 0",
"usedConstants": []
}
] | | _ r s
=> | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.RingTheory.OreLocalization.Basic | {
"line": 51,
"column": 47
} | {
"line": 51,
"column": 54
} | [
{
"pp": "case c\nR : Type u_1\ninst✝¹ : MonoidWithZero R\nS : Submonoid R\ninst✝ : OreSet S\nr : R\ns : ↥S\n⊢ r /ₒ s * 0 = 0",
"usedConstants": []
}
] | | _ r s
=> | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.LinearAlgebra.FreeModule.PID | {
"line": 491,
"column": 2
} | {
"line": 503,
"column": 36
} | [
{
"pp": "ι : 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\nN O : Submodule R M\nN_le_O : N ≤ O\n⊢ ∃ n o, ∃ (hno : n ≤ o), ∃ bO bN a, ∀ (i : Fin n), ↑(bN i) = a i • ... | cases nonempty_fintype ι
induction O using inductionOnRank b generalizing N with | ih M0 ih =>
obtain ⟨m, b'M⟩ := M0.basisOfPid b
by_cases N_bot : N = ⊥
· subst N_bot
exact ⟨0, m, Nat.zero_le _, b'M, Basis.empty _, finZeroElim, finZeroElim⟩
obtain ⟨y, hy, a, _, M', M'_le_M, N', _, N'_le_M', y_ortho, _, h⟩... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.FreeModule.PID | {
"line": 491,
"column": 2
} | {
"line": 503,
"column": 36
} | [
{
"pp": "ι : 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\nN O : Submodule R M\nN_le_O : N ≤ O\n⊢ ∃ n o, ∃ (hno : n ≤ o), ∃ bO bN a, ∀ (i : Fin n), ↑(bN i) = a i • ... | cases nonempty_fintype ι
induction O using inductionOnRank b generalizing N with | ih M0 ih =>
obtain ⟨m, b'M⟩ := M0.basisOfPid b
by_cases N_bot : N = ⊥
· subst N_bot
exact ⟨0, m, Nat.zero_le _, b'M, Basis.empty _, finZeroElim, finZeroElim⟩
obtain ⟨y, hy, a, _, M', M'_le_M, N', _, N'_le_M', y_ortho, _, h⟩... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Field.Subfield.Basic | {
"line": 247,
"column": 18
} | {
"line": 251,
"column": 76
} | [
{
"pp": "K : Type u\nL : Type v\nM : Type w\ninst✝² : DivisionRing K\ninst✝¹ : DivisionRing L\ninst✝ : DivisionRing M\nS : Set (Subfield K)\n⊢ ∀ x ∈ (sInf (toSubring '' S)).carrier, x⁻¹ ∈ (sInf (toSubring '' S)).carrier",
"usedConstants": [
"Iff.mpr",
"Subring.toSubsemiring",
"GroupWithZer... | by
rintro x hx
apply Subring.mem_sInf.mpr
rintro _ ⟨p, p_mem, rfl⟩
exact p.inv_mem (Subring.mem_sInf.mp hx p.toSubring ⟨p, p_mem, rfl⟩) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 501,
"column": 25
} | {
"line": 502,
"column": 71
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommSemiring R\nk : Type u_2\nl : Type u_3\nm : Type u_4\nn : Type u_5\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\ninst✝¹ : Fintype m\ninst✝ : DecidableEq m\nM : Matrix m n R\nM' : Matrix n m R\nhMM' : M * M' = 1\nhM'M : M' * M = 1\nx : n → R\n⊢ (toLin' M') ((toLin' M) x) = x",
... | by
rw [← Matrix.toLin'_mul_apply, hM'M, Matrix.toLin'_one, id_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Matrix.ToLin | {
"line": 610,
"column": 59
} | {
"line": 611,
"column": 56
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : CommSemiring R\nm : Type u_3\nn : Type u_4\ninst✝⁶ : Fintype n\ninst✝⁵ : Finite m\ninst✝⁴ : DecidableEq n\nM₁ : Type u_5\nM₂ : Type u_6\ninst✝³ : AddCommMonoid M₁\ninst✝² : AddCommMonoid M₂\ninst✝¹ : Module R M₁\ninst✝ : Module R M₂\nv₁ : Basis n R M₁\nv₂ : Basis m R M₂\nf : M₁ →... | by
rw [← Matrix.toLin_symm, LinearEquiv.apply_symm_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.OreLocalization.Ring | {
"line": 157,
"column": 74
} | {
"line": 157,
"column": 78
} | [
{
"pp": "case c.c.e_a.e_a\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 : ↑(... | ← hf | Lean.Elab.Tactic.evalRewriteSeq | null |
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