module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.Noetherian.Basic | {
"line": 180,
"column": 2
} | {
"line": 183,
"column": 85
} | {
"line": 185,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\ninst✝⁸ : Ring R\ninst✝⁷ : Ring S\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : AddCommGroup P\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : Module S P\nι : Type u_6\ninst✝ : Finite ι\n⊢ ∀ {M : ι → Type u_7} [inst ... | [] | apply Finite.induction_empty_option _ _ _ ι
· exact fun e h ↦ isNoetherian_of_linearEquiv (LinearEquiv.piCongrLeft R _ e)
· infer_instance
· exact fun ih ↦ isNoetherian_of_linearEquiv (LinearEquiv.piOptionEquivProd R).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Noetherian.Basic | {
"line": 180,
"column": 2
} | {
"line": 183,
"column": 85
} | {
"line": 185,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\ninst✝⁸ : Ring R\ninst✝⁷ : Ring S\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : AddCommGroup P\ninst✝³ : Module R M\ninst✝² : Module R N\ninst✝¹ : Module S P\nι : Type u_6\ninst✝ : Finite ι\n⊢ ∀ {M : ι → Type u_7} [inst ... | [] | apply Finite.induction_empty_option _ _ _ ι
· exact fun e h ↦ isNoetherian_of_linearEquiv (LinearEquiv.piCongrLeft R _ e)
· infer_instance
· exact fun ih ↦ isNoetherian_of_linearEquiv (LinearEquiv.piOptionEquivProd R).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.OrzechProperty | {
"line": 89,
"column": 34
} | {
"line": 89,
"column": 50
} | {
"line": 89,
"column": 51
} | [
{
"pp": "R : Type u\ninst✝⁶ : Semiring R\ninst✝⁵ : OrzechProperty R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module.Finite R M\nN : Type w\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\ni f : N →ₗ[R] M\nhi : Injective ⇑i\nhf : Surjective ⇑f\nn : ℕ\ng : (Fin n → R) →ₗ[R] M\nhg : Surje... | [
"R : Type u\ninst✝⁶ : Semiring R\ninst✝⁵ : OrzechProperty R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module.Finite R M\nN : Type w\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\ni f : N →ₗ[R] M\nhi : Injective ⇑i\nhf : Surjective ⇑f\nn : ℕ\ng : (Fin n → R) →ₗ[R] M\nhg : Surjective ⇑g\nth... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.OrzechProperty | {
"line": 91,
"column": 35
} | {
"line": 91,
"column": 51
} | {
"line": 91,
"column": 52
} | [
{
"pp": "R : Type u\ninst✝⁶ : Semiring R\ninst✝⁵ : OrzechProperty R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module.Finite R M\nN : Type w\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\ni f : N →ₗ[R] M\nhf : Surjective ⇑f\nn : ℕ\ng : (Fin n → R) →ₗ[R] M\nhg : Surjective ⇑g\nthis✝² : ... | [
"R : Type u\ninst✝⁶ : Semiring R\ninst✝⁵ : OrzechProperty R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module.Finite R M\nN : Type w\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\ni f : N →ₗ[R] M\nhf : Surjective ⇑f\nn : ℕ\ng : (Fin n → R) →ₗ[R] M\nhg : Surjective ⇑g\nthis✝² : Small.{u, v}... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.OrzechProperty | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 18
} | {
"line": 92,
"column": 19
} | [
{
"pp": "R : Type u\ninst✝⁶ : Semiring R\ninst✝⁵ : OrzechProperty R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module.Finite R M\nN : Type w\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\ni f : N →ₗ[R] M\nn : ℕ\ng : (Fin n → R) →ₗ[R] M\nhg : Surjective ⇑g\nthis✝² : Small.{u, v} M\nthis... | [
"R : Type u\ninst✝⁶ : Semiring R\ninst✝⁵ : OrzechProperty R\nM : Type v\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module.Finite R M\nN : Type w\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\ni f : N →ₗ[R] M\nn : ℕ\ng : (Fin n → R) →ₗ[R] M\nhg : Surjective ⇑g\nthis✝² : Small.{u, v} M\nthis✝¹ : AddComm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Noetherian.Orzech | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 13
} | {
"line": 73,
"column": 14
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type w\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : IsNoetherian R M\nf : M × N →ₗ[R] M\ni : Injective ⇑f\nx y : N\nh : Injective ⇑(LinearMap.fst R M N)\n⊢ x = y",
"ppTerm": "?m.46",
"assign... | [
"R : Type u_1\nM : Type u_2\nN : Type w\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\ninst✝ : IsNoetherian R M\nf : M × N →ₗ[R] M\ni : Injective ⇑f\nx y : N\nh : Injective ⇑(LinearMap.fst R M N)\n⊢ x = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Noetherian.Basic | {
"line": 196,
"column": 22
} | {
"line": 196,
"column": 38
} | {
"line": 196,
"column": 38
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\ninst✝¹⁰ : Ring R\ninst✝⁹ : Ring S\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : AddCommGroup P\ninst✝⁵ : Module R M\ninst✝⁴ : Module R N\ninst✝³ : Module S P\nι : Type u_6\ninst✝² : Finite ι\nα✝ : Type u_6\ninst✝¹ : Fin... | [
"R : Type u_1\nS : Type u_2\nM : Type u_3\nN : Type u_4\nP : Type u_5\ninst✝¹⁰ : Ring R\ninst✝⁹ : Ring S\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup N\ninst✝⁶ : AddCommGroup P\ninst✝⁵ : Module R M\ninst✝⁴ : Module R N\ninst✝³ : Module S P\nι : Type u_6\ninst✝² : Finite ι\nα✝ : Type u_6\ninst✝¹ : Fintype α✝\nih ... | rw [iSup_option] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Noetherian.Basic | {
"line": 215,
"column": 68
} | {
"line": 218,
"column": 49
} | {
"line": 220,
"column": 0
} | [
{
"pp": "R : Type u_1\nN : Type u_4\ninst✝² : Ring R\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nS : Submodule R N\n⊢ IsNoetherian R N ↔ IsNoetherian R ↥S ∧ IsNoetherian R (N ⧸ S)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"RingHomSurjective.... | [] | by
refine ⟨fun _ ↦ ⟨inferInstance, inferInstance⟩, fun ⟨_, _⟩ ↦ ?_⟩
apply isNoetherian_of_range_eq_ker S.subtype S.mkQ
rw [Submodule.ker_mkQ, Submodule.range_subtype] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Noetherian.Basic | {
"line": 303,
"column": 2
} | {
"line": 303,
"column": 40
} | {
"line": 303,
"column": 41
} | [
{
"pp": "R : 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 → (partialSups f) n = (partialSups f) m\nm : ℕ\np : n ≤ m\n⊢ (partialSups... | [
"R : 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 → (partialSups f) n = (partialSups f) m\nm : ℕ\np : n ≤ m\n⊢ (partialSups f) m ⊔ f (m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Maps | {
"line": 387,
"column": 4
} | {
"line": 387,
"column": 25
} | {
"line": 387,
"column": 26
} | [
{
"pp": "case inl\nι : Type u_4\nR : ι → Type u_5\ninst✝ : (i : ι) → Semiring (R i)\nJ : Ideal ((i : ι) → R i)\nx : (i : ι) → R i\nhxJ : x ∈ J\nj : ι\nI : Ideal (R j)\nh : I.IsMaximal\nle : comap (Pi.evalRingHom R j) I ≤ J\nhxI : x ∉ comap (Pi.evalRingHom R j) I\nr y : R j\nhy : y ∈ I\neq : r * (Pi.evalRingHom ... | [
"case inl\nι : Type u_4\nR : ι → Type u_5\ninst✝ : (i : ι) → Semiring (R i)\nJ : Ideal ((i : ι) → R i)\nx : (i : ι) → R i\nhxJ : x ∈ J\nj : ι\nI : Ideal (R j)\nh : I.IsMaximal\nle : comap (Pi.evalRingHom R j) I ≤ J\nhxI : x ∉ comap (Pi.evalRingHom R j) I\nr y : R j\nhy : y ∈ I\neq : r * (Pi.evalRingHom R j) x + y =... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Noetherian.Basic | {
"line": 404,
"column": 2
} | {
"line": 404,
"column": 17
} | {
"line": 404,
"column": 18
} | [
{
"pp": "case h\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : Small.{v, u} R\nM : Type v\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nm : ℕ\nf' : (Fin m → R) →ₗ[R] M\nhf' : Surjective ⇑f'\nf : (Fin m →₀ Shrink.{v, u} R) →ₗ[R] M := ⋯\n⊢ Surjective ⇑f",
"ppTerm": "?h",
"assigned"... | [
"case h\nR : Type u\ninst✝⁴ : CommRing R\ninst✝³ : Small.{v, u} R\nM : Type v\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nm : ℕ\nf' : (Fin m → R) →ₗ[R] M\nhf' : Surjective ⇑f'\nf : (Fin m →₀ Shrink.{v, u} R) →ₗ[R] M :=\n f' ∘ₗ ↑(Finsupp.mapRange.linearEquiv (Shrink.linearEquiv R R) ≪≫... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 81,
"column": 4
} | {
"line": 81,
"column": 31
} | {
"line": 81,
"column": 32
} | [
{
"pp": "R : Type u\ninst✝ : EuclideanDomain R\na : R\na0 : ¬a = 0\n⊢ 0 / a = 0",
"ppTerm": "?m.23",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : EuclideanDomain R\na : R\na0 : ¬a = 0\n⊢ 0 / a = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 28
} | {
"line": 85,
"column": 29
} | [
{
"pp": "R : Type u\ninst✝ : EuclideanDomain R\na : R\na0 : a ≠ 0\n⊢ a / a = 1",
"ppTerm": "?m.11",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : EuclideanDomain R\na : R\na0 : a ≠ 0\n⊢ a / a = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 152,
"column": 15
} | {
"line": 152,
"column": 30
} | {
"line": 152,
"column": 31
} | [
{
"pp": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na b : R\nh : gcd a b = 0\n⊢ a = 0 ∧ b = 0",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na b : R\nh : gcd a b = 0\n⊢ a = 0 ∧ b = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 157,
"column": 37
} | {
"line": 157,
"column": 69
} | {
"line": 157,
"column": 70
} | [
{
"pp": "R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na b c x✝¹ : R\nx✝ : c ∣ 0\nH : c ∣ x✝¹\n⊢ c ∣ gcd 0 x✝¹",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"CommRing.toNonUnitalCommRing",
"congrArg",
"CommSemiring.toSem... | [
"R : Type u\ninst✝¹ : EuclideanDomain R\ninst✝ : DecidableEq R\na b c x✝¹ : R\nx✝ : c ∣ 0\nH : c ∣ x✝¹\n⊢ c ∣ x✝¹"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Maps | {
"line": 505,
"column": 4
} | {
"line": 507,
"column": 85
} | {
"line": 508,
"column": 2
} | [
{
"pp": "case mp\nR : Type u\nS : Type v\ninst✝³ : Semiring R\ninst✝² : Semiring S\nE : Type u_4\ninst✝¹ : EquivLike E R S\ninst✝ : RingEquivClass E R S\ne : E\nI : Ideal R\ny : S\n⊢ y ∈ map e I → ∃ x ∈ I, e x = y",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Equiv.apply_symm_apply"... | [] | intro h
simp_rw [show map e I = _ from map_comap_of_equiv (RingEquivClass.toRingEquiv e : R ≃+* S)] at h
exact ⟨(EquivLike.toEquiv e).symm y, h, (EquivLike.toEquiv e).apply_symm_apply y⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Maps | {
"line": 505,
"column": 4
} | {
"line": 507,
"column": 85
} | {
"line": 508,
"column": 2
} | [
{
"pp": "case mp\nR : Type u\nS : Type v\ninst✝³ : Semiring R\ninst✝² : Semiring S\nE : Type u_4\ninst✝¹ : EquivLike E R S\ninst✝ : RingEquivClass E R S\ne : E\nI : Ideal R\ny : S\n⊢ y ∈ map e I → ∃ x ∈ I, e x = y",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Equiv.apply_symm_apply"... | [] | intro h
simp_rw [show map e I = _ from map_comap_of_equiv (RingEquivClass.toRingEquiv e : R ≃+* S)] at h
exact ⟨(EquivLike.toEquiv e).symm y, h, (EquivLike.toEquiv e).apply_symm_apply y⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Maps | {
"line": 537,
"column": 4
} | {
"line": 537,
"column": 70
} | {
"line": 537,
"column": 71
} | [
{
"pp": "R : Type u\nS : Type v\nF : Type u_1\ninst✝⁴ : Semiring R\ninst✝³ : Semiring S\ninst✝² : FunLike F R S\nf : F\nI✝ J : Ideal R\nK✝ L : Ideal S\nG : Type u_2\ninst✝¹ : FunLike G S R\ninst✝ : RingHomClass F R S\nι : Sort u_3\nhf : Function.Bijective ⇑f\nI : Ideal R\nK x✝¹ x✝ : Ideal S\nh :\n { toFun := c... | [
"R : Type u\nS : Type v\nF : Type u_1\ninst✝⁴ : Semiring R\ninst✝³ : Semiring S\ninst✝² : FunLike F R S\nf : F\nI✝ J : Ideal R\nK✝ L : Ideal S\nG : Type u_2\ninst✝¹ : FunLike G S R\ninst✝ : RingHomClass F R S\nι : Sort u_3\nhf : Function.Bijective ⇑f\nI : Ideal R\nK x✝¹ x✝ : Ideal S\nh :\n { toFun := comap f, invF... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Maps | {
"line": 599,
"column": 4
} | {
"line": 599,
"column": 40
} | {
"line": 599,
"column": 41
} | [
{
"pp": "case refine_1\nR : Type u\nS : Type v\nF : Type u_1\ninst✝³ : Ring R\ninst✝² : Ring S\ninst✝¹ : FunLike F R S\ninst✝ : RingHomClass F R S\nf : F\nhf : Function.Surjective ⇑f\nI : Ideal S\nφ : Ideal S ↪o Ideal R := ⋯\nJ : Ideal S\nK : Ideal R\nh : φ J < K\n⊢ ¬map f K ≤ map f (φ J)",
"ppTerm": "?refi... | [
"case refine_1\nR : Type u\nS : Type v\nF : Type u_1\ninst✝³ : Ring R\ninst✝² : Ring S\ninst✝¹ : FunLike F R S\ninst✝ : RingHomClass F R S\nf : F\nhf : Function.Surjective ⇑f\nI : Ideal S\nφ : Ideal S ↪o Ideal R := orderEmbeddingOfSurjective f hf\nJ : Ideal S\nK : Ideal R\nh : φ J < K\n⊢ ¬K ≤ comap f J"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Operations | {
"line": 54,
"column": 2
} | {
"line": 54,
"column": 69
} | {
"line": 56,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝ : AddCommGroup M\na i : M\n⊢ i ∈ (ℤ ∙ a).toAddSubgroup ↔ i ∈ AddSubgroup.zmultiples a",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Submodule",
"instHSMul",
"AddSubgroup.mem_mk._simp_1",
"Submodule.toAddSubmonoid",
"_private.Mat... | [] | simp [Submodule.mem_span_singleton, AddSubgroup.mem_zmultiples_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Ideal.Maps | {
"line": 911,
"column": 4
} | {
"line": 911,
"column": 15
} | {
"line": 911,
"column": 16
} | [
{
"pp": "case mp\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\nh : ∀ (m : ι →₀ M), r • m = 0\ni : ι\n⊢ ∀ (m : M), r • m = 0",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
... | [
"case mp\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\nh : ∀ (m : ι →₀ M), r • m = 0\ni : ι\n⊢ ∀ (m : M), r • m = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Maps | {
"line": 922,
"column": 14
} | {
"line": 922,
"column": 25
} | {
"line": 922,
"column": 26
} | [
{
"pp": "case mp\nR : Type u_1\ninst✝² : Semiring R\nι : Type u_4\nM : ι → Type u_5\ninst✝¹ : (i : ι) → AddCommMonoid (M i)\ninst✝ : (i : ι) → Module R (M i)\nr : R\nh : ∀ (m : Π₀ (i : ι), M i), r • m = 0\ni : ι\nm : M i\n⊢ r • m = 0",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"us... | [
"case mp\nR : Type u_1\ninst✝² : Semiring R\nι : Type u_4\nM : ι → Type u_5\ninst✝¹ : (i : ι) → AddCommMonoid (M i)\ninst✝ : (i : ι) → Module R (M i)\nr : R\nh : ∀ (m : Π₀ (i : ι), M i), r • m = 0\ni : ι\nm : M i\n⊢ r • m = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Maps | {
"line": 929,
"column": 14
} | {
"line": 929,
"column": 25
} | {
"line": 929,
"column": 26
} | [
{
"pp": "case mp\nR : Type u_1\ninst✝² : Semiring R\nι : Type u_4\nM : ι → Type u_5\ninst✝¹ : (i : ι) → AddCommMonoid (M i)\ninst✝ : (i : ι) → Module R (M i)\nr : R\nh : ∀ (m : (i : ι) → M i), r • m = 0\ni : ι\nm : M i\n⊢ r • m = 0",
"ppTerm": "?mp",
"assigned": false,
"usedConstants": [],
"used... | [
"case mp\nR : Type u_1\ninst✝² : Semiring R\nι : Type u_4\nM : ι → Type u_5\ninst✝¹ : (i : ι) → AddCommMonoid (M i)\ninst✝ : (i : ι) → Module R (M i)\nr : R\nh : ∀ (m : (i : ι) → M i), r • m = 0\ni : ι\nm : M i\n⊢ r • m = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Maps | {
"line": 992,
"column": 2
} | {
"line": 1004,
"column": 35
} | {
"line": 1006,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set M\nr : R\n⊢ r ∈ (span R s).annihilator ↔ ∀ (n : ↑s), r • ↑n = 0",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"instHSMul",
"S... | [] | rw [Submodule.mem_annihilator]
constructor
· intro h n
exact h _ (Submodule.subset_span n.prop)
· intro h n hn
refine Submodule.span_induction ?_ ?_ ?_ ?_ hn
· intro x hx
exact h ⟨x, hx⟩
· exact smul_zero _
· intro x y _ _ hx hy
rw [smul_add, hx, hy, zero_add]
· intro a x _ hx
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Maps | {
"line": 992,
"column": 2
} | {
"line": 1004,
"column": 35
} | {
"line": 1006,
"column": 0
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\ns : Set M\nr : R\n⊢ r ∈ (span R s).annihilator ↔ ∀ (n : ↑s), r • ↑n = 0",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"instHSMul",
"S... | [] | rw [Submodule.mem_annihilator]
constructor
· intro h n
exact h _ (Submodule.subset_span n.prop)
· intro h n hn
refine Submodule.span_induction ?_ ?_ ?_ ?_ hn
· intro x hx
exact h ⟨x, hx⟩
· exact smul_zero _
· intro x y _ _ hx hy
rw [smul_add, hx, hy, zero_add]
· intro a x _ hx
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 357,
"column": 6
} | {
"line": 357,
"column": 18
} | {
"line": 357,
"column": 18
} | [
{
"pp": "R : Type u\ninst✝ : EuclideanDomain R\nx y z : R\nh1 : z ≠ 0\nh2 : z ∣ y\n⊢ (x * z + y) / z = x + y / z",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"instHDiv",
"NonUnitalCommRing.toN... | [
"R : Type u\ninst✝ : EuclideanDomain R\nx y z : R\nh1 : z ≠ 0\nh2 : z ∣ y\n⊢ (z * x + y) / z = x + y / z"
] | mul_comm x z | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 366,
"column": 6
} | {
"line": 366,
"column": 18
} | {
"line": 366,
"column": 18
} | [
{
"pp": "R : Type u\ninst✝ : EuclideanDomain R\nx y z : R\nh1 : z ≠ 0\nh2 : z ∣ y\n⊢ (x * z - y) / z = x - y / z",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"instHDiv",
"NonUnitalCommRing.toN... | [
"R : Type u\ninst✝ : EuclideanDomain R\nx y z : R\nh1 : z ≠ 0\nh2 : z ∣ y\n⊢ (z * x - y) / z = x - y / z"
] | mul_comm x z | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Ideal.Maps | {
"line": 1082,
"column": 4
} | {
"line": 1082,
"column": 49
} | {
"line": 1083,
"column": 4
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : FunLike F R S\nrc : RingHomClass F R S\nA : Set (Ideal R)\nf : F\nhf : Function.Surjective ⇑f\nh : ∀ J ∈ A, RingHom.ker f ≤ J\nj : Ideal S\nhj : j ∈ map f '' A\ny : S\nhy : y ∈ map f (sInf A)\nx : R\nhx :... | [
"case refine_1\nR : Type u_1\nS : Type u_2\nF : Type u_3\ninst✝² : Ring R\ninst✝¹ : Ring S\ninst✝ : FunLike F R S\nrc : RingHomClass F R S\nA : Set (Ideal R)\nf : F\nhf : Function.Surjective ⇑f\nh : ∀ J ∈ A, RingHom.ker f ≤ J\nj : Ideal S\nhj : j ∈ map f '' A\ny : S\nhy : y ∈ map f (sInf A)\nx : R\nhx : x ∈ sInf A ... | obtain ⟨J, hJ⟩ := (Set.mem_image _ _ _).mp hj | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Ideal.Maps | {
"line": 1088,
"column": 38
} | {
"line": 1088,
"column": 49
} | {
"line": 1088,
"column": 50
} | [
{
"pp": "R : 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\ny : S\nhy : y ∈ sInf (map f '' A)\nx : R\nhx : f x = y\n⊢ ∀ I ∈ A, y ∈ map f I",
"ppTerm... | [
"R : 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\ny : S\nhy : y ∈ sInf (map f '' A)\nx : R\nhx : f x = y\n⊢ ∀ I ∈ A, y ∈ map f I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Operations | {
"line": 223,
"column": 2
} | {
"line": 223,
"column": 13
} | {
"line": 223,
"column": 14
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nr : R\nN : Submodule R M\n⊢ span R ({r} • ↑N) = r • N",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule.pointwiseDistribMulAction",
"Submodule",
... | [
"R : Type u\nM : Type v\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nr : R\nN : Submodule R M\n⊢ span R (r • ↑N) = r • N"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 434,
"column": 29
} | {
"line": 434,
"column": 40
} | {
"line": 434,
"column": 41
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹ : EuclideanDomain R\ninst✝ : CommRing S\ne : S ≃+* R\na b : S\nhb : b ≠ 0\n⊢ e b ≠ 0",
"ppTerm": "?m.98",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring",
"RingEquiv.instEquivLike",
"RingEq... | [
"R : Type u_1\nS : Type u_2\ninst✝¹ : EuclideanDomain R\ninst✝ : CommRing S\ne : S ≃+* R\na b : S\nhb : b ≠ 0\n⊢ ¬b = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.EuclideanDomain.Basic | {
"line": 437,
"column": 29
} | {
"line": 437,
"column": 40
} | {
"line": 437,
"column": 41
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝¹ : EuclideanDomain R\ninst✝ : CommRing S\ne : S ≃+* R\na b : S\nhb : b ≠ 0\n⊢ e b ≠ 0",
"ppTerm": "?m.123",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring",
"RingEquiv.instEquivLike",
"RingE... | [
"R : Type u_1\nS : Type u_2\ninst✝¹ : EuclideanDomain R\ninst✝ : CommRing S\ne : S ≃+* R\na b : S\nhb : b ≠ 0\n⊢ ¬b = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Maps | {
"line": 1095,
"column": 4
} | {
"line": 1095,
"column": 37
} | {
"line": 1095,
"column": 38
} | [
{
"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\nx : R\nJ : Submodule R R\nhJ : J ∈ A\nx' : R\nhx' : x' ∈ J\nhy : f x' ∈ sInf ... | [
"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\nx : R\nJ : Submodule R R\nhJ : J ∈ A\nx' : R\nhx' : x' ∈ J\nhy : f x' ∈ sInf (map f '' A)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GCDMonoid.Multiset | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 15
} | {
"line": 214,
"column": 16
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\ns t : Multiset α\nhs : ∃ x, x ∈ s ∧ x ≠ 0\nht : s = map (fun x ↦ s.gcd * x) t\n⊢ Associated (s.gcd * 1) (s.gcd * t.gcd)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case refine_1\nα : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\ns t : Multiset α\nhs : ∃ x, x ∈ s ∧ x ≠ 0\nht : s = map (fun x ↦ s.gcd * x) t\n⊢ Associated s.gcd (s.gcd * t.gcd)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Operations | {
"line": 477,
"column": 4
} | {
"line": 477,
"column": 60
} | {
"line": 477,
"column": 61
} | [
{
"pp": "R : Type u\ninst✝¹ : Semiring R\nI : Ideal R\nι : Type u_1\ns : Finset ι\nJ : ι → Ideal R\ninst✝ : ∀ (i : ι), (J i).IsTwoSided\nh : ∀ i ∈ s, I ⊔ J i = ⊤\ni : ι\nt : Finset ι\nhis : i ∈ s\nhts : t ⊆ s\nhit : i ∉ t\neq_top : I ⊔ ⨅ i ∈ t, J i = ⊤\n⊢ ⊤ ≤ I ⊔ (⨅ x ∈ t, J x) * J i",
"ppTerm": "?m.71",
... | [
"R : Type u\ninst✝¹ : Semiring R\nI : Ideal R\nι : Type u_1\ns : Finset ι\nJ : ι → Ideal R\ninst✝ : ∀ (i : ι), (J i).IsTwoSided\nh : ∀ i ∈ s, I ⊔ J i = ⊤\ni : ι\nt : Finset ι\nhis : i ∈ s\nhts : t ⊆ s\nhit : i ∉ t\neq_top : I ⊔ ⨅ i ∈ t, J i = ⊤\n⊢ ⊤ ≤ I ⊔ ⨅ x ∈ t, J x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Maps | {
"line": 1335,
"column": 2
} | {
"line": 1335,
"column": 47
} | {
"line": 1335,
"column": 48
} | [
{
"pp": "case right\nA : Type u_1\nB : Type u_2\ninst✝¹ : Ring A\ninst✝ : Ring B\nf : A →+* B\nsurj : Function.Surjective ⇑f\nI : Ideal B\nJ : Ideal A\neq : comap f I = RingHom.ker f ⊔ J\nx : B\nx' : A\nhx : x' ∈ RingHom.ker f ⊔ J\nhx' : f x' = x\ny : A\nhy : y ∈ RingHom.ker f\nz : A\nhz : z ∈ J\nhyz : y + z = ... | [
"case right\nA : Type u_1\nB : Type u_2\ninst✝¹ : Ring A\ninst✝ : Ring B\nf : A →+* B\nsurj : Function.Surjective ⇑f\nI : Ideal B\nJ : Ideal A\neq : comap f I = RingHom.ker f ⊔ J\nx : B\nx' : A\nhx : x' ∈ RingHom.ker f ⊔ J\nhx' : f x' = x\ny : A\nhy : y ∈ RingHom.ker f\nz : A\nhz : z ∈ J\nhyz : y + z = x'\n⊢ y ∈ Ri... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Maps | {
"line": 1343,
"column": 2
} | {
"line": 1343,
"column": 20
} | {
"line": 1343,
"column": 21
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹ : CommRing A\ninst✝ : CommRing B\nf : A →+* B\nsurj : Function.Surjective ⇑f\nI : Ideal B\nJ : Ideal A\neq : comap f I = RingHom.ker f ⊔ J\nx : B\nhx : x ∈ I\ny : A\nmem : y ∈ J\nhy : f y = x\n⊢ x ∈ map f J",
"ppTerm": "?m.91",
"assigned": true,
"usedConst... | [
"A : Type u_1\nB : Type u_2\ninst✝¹ : CommRing A\ninst✝ : CommRing B\nf : A →+* B\nsurj : Function.Surjective ⇑f\nI : Ideal B\nJ : Ideal A\neq : comap f I = RingHom.ker f ⊔ J\nx : B\nhx : x ∈ I\ny : A\nmem : y ∈ J\nhy : f y = x\n⊢ f y ∈ map f J"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Operations | {
"line": 492,
"column": 39
} | {
"line": 492,
"column": 62
} | {
"line": 492,
"column": 63
} | [
{
"pp": "case succ.inl\nR : Type u\ninst✝¹ : Semiring R\nI J : Ideal R\ninst✝ : J.IsTwoSided\nh : I ⊔ J = ⊤\nih : I ⊔ J ^ 0 = ⊤\n⊢ I ⊔ J ^ (0 + 1) = ⊤",
"ppTerm": "?succ.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"Semiring.toModule",
"con... | [
"case succ.inl\nR : Type u\ninst✝¹ : Semiring R\nI J : Ideal R\ninst✝ : J.IsTwoSided\nh : I ⊔ J = ⊤\nih : I ⊔ J ^ 0 = ⊤\n⊢ I ⊔ J = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GCDMonoid.Finset | {
"line": 229,
"column": 13
} | {
"line": 229,
"column": 24
} | {
"line": 229,
"column": 25
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\ns✝ : Finset β\nf : β → α\na : α\nb : β\ns : Finset β\nhbs : b ∉ s\nh : Associated (s.gcd fun x ↦ a * f x) (a * s.gcd f)\n⊢ Associated ((insert b s).gcd fun x ↦ a * f x) (a * (insert b s).gcd f)",
"ppTerm": "?m... | [
"α : Type u_2\nβ : Type u_3\ninst✝¹ : CommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\ns✝ : Finset β\nf : β → α\na : α\nb : β\ns : Finset β\nhbs : b ∉ s\nh : Associated (s.gcd fun x ↦ a * f x) (a * s.gcd f)\n⊢ Associated (GCDMonoid.gcd (a * f b) (s.gcd fun x ↦ a * f x)) (a * GCDMonoid.gcd (f b) (s.gcd f))"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Operations | {
"line": 527,
"column": 4
} | {
"line": 527,
"column": 19
} | {
"line": 528,
"column": 2
} | [
{
"pp": "case inl\nR : Type u\ninst✝ : Semiring R\nI : Ideal R\nn : ℕ\nh : I = ⊤\n⊢ I ^ n = ⊤",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"id",
"Submodule.instTop",
"Ideal",
"IsScalarTower.left",
... | [] | rw [h, top_pow] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Ideal.Operations | {
"line": 527,
"column": 4
} | {
"line": 527,
"column": 19
} | {
"line": 528,
"column": 2
} | [
{
"pp": "case inl\nR : Type u\ninst✝ : Semiring R\nI : Ideal R\nn : ℕ\nh : I = ⊤\n⊢ I ^ n = ⊤",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"id",
"Submodule.instTop",
"Ideal",
"IsScalarTower.left",
... | [] | rw [h, top_pow] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Ideal.Operations | {
"line": 527,
"column": 4
} | {
"line": 527,
"column": 19
} | {
"line": 528,
"column": 2
} | [
{
"pp": "case inl\nR : Type u\ninst✝ : Semiring R\nI : Ideal R\nn : ℕ\nh : I = ⊤\n⊢ I ^ n = ⊤",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
"id",
"Submodule.instTop",
"Ideal",
"IsScalarTower.left",
... | [] | rw [h, top_pow] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Ideal.Operations | {
"line": 544,
"column": 2
} | {
"line": 544,
"column": 19
} | {
"line": 544,
"column": 20
} | [
{
"pp": "R : Type u\ninst✝ : Semiring R\nI : Ideal R\nn m : ℕ\nhnI : I ^ n = 0\nhmn : n ≤ m\nx : R\nhx : x ∈ I\n⊢ x ^ m = 0",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : Semiring R\nI : Ideal R\nn m : ℕ\nhnI : I ^ n = 0\nhmn : n ≤ m\nx : R\nhx : x ∈ I\n⊢ x ^ m = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Operations | {
"line": 655,
"column": 24
} | {
"line": 655,
"column": 41
} | {
"line": 655,
"column": 41
} | [
{
"pp": "case hI\nR : Type u\ninst✝¹ : CommSemiring R\nι : Type u_2\ninst✝ : Fintype ι\nI : ι → R\nhI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j)\n⊢ Set.univ.Pairwise (Function.onFun IsCoprime I)",
"ppTerm": "?hI",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Function.onFun",
"... | [
"case hI\nR : Type u\ninst✝¹ : CommSemiring R\nι : Type u_2\ninst✝ : Fintype ι\nI : ι → R\nhI : ∀ (i j : ι), i ≠ j → IsCoprime (I i) (I j)\n⊢ Pairwise (Function.onFun IsCoprime I)"
] | Set.pairwise_univ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 532,
"column": 30
} | {
"line": 532,
"column": 61
} | {
"line": 532,
"column": 62
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\na b : α\nh : normalize b = b\n⊢ gcd a b = b ↔ b ∣ a",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"congrArg",
"semigroupDvd",
"SemigroupWithZero.toSemig... | [
"α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\na b : α\nh : normalize b = b\n⊢ gcd b a = b ↔ b ∣ a"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.UniqueFactorizationDomain.Defs | {
"line": 151,
"column": 2
} | {
"line": 152,
"column": 36
} | {
"line": 154,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\na : α\n⊢ a ≠ 0 → ∃ f, (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Eq.mpr",
"congrArg",
"Multiset.prod"... | [] | simp_rw [← UniqueFactorizationMonoid.irreducible_iff_prime]
apply WfDvdMonoid.exists_factors a | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.UniqueFactorizationDomain.Defs | {
"line": 151,
"column": 2
} | {
"line": 152,
"column": 36
} | {
"line": 154,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\na : α\n⊢ a ≠ 0 → ∃ f, (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Eq.mpr",
"congrArg",
"Multiset.prod"... | [] | simp_rw [← UniqueFactorizationMonoid.irreducible_iff_prime]
apply WfDvdMonoid.exists_factors a | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.UniqueFactorizationDomain.Defs | {
"line": 218,
"column": 4
} | {
"line": 218,
"column": 52
} | {
"line": 218,
"column": 53
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\na : α\nha : Irreducible a\nhf : factors a = 0\n⊢ False",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\na : α\nha : Irreducible a\nhf : factors a = 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 883,
"column": 30
} | {
"line": 883,
"column": 61
} | {
"line": 883,
"column": 62
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\na b : α\nh : normalize b = b\n⊢ lcm a b = b ↔ a ∣ b",
"ppTerm": "?m.12",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\na b : α\nh : normalize b = b\n⊢ lcm a b = b ↔ a ∣ b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.InvariantBasisNumber | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 61
} | {
"line": 136,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u\ninst✝ : Semiring R\nh : StrongRankCondition R\nn : ℕ\nf : (Fin (n + 1) → R) →ₗ[R] Fin n → R\nhf : Injective ⇑f\n⊢ False",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"le_of_fin_injective",
"Nat.succ",
"Nat.not_succ_le_self"
],... | [] | exact Nat.not_succ_le_self n (le_of_fin_injective R f hf) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.InvariantBasisNumber | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 61
} | {
"line": 136,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u\ninst✝ : Semiring R\nh : StrongRankCondition R\nn : ℕ\nf : (Fin (n + 1) → R) →ₗ[R] Fin n → R\nhf : Injective ⇑f\n⊢ False",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"le_of_fin_injective",
"Nat.succ",
"Nat.not_succ_le_self"
],... | [] | exact Nat.not_succ_le_self n (le_of_fin_injective R f hf) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.InvariantBasisNumber | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 61
} | {
"line": 136,
"column": 2
} | [
{
"pp": "case refine_1\nR : Type u\ninst✝ : Semiring R\nh : StrongRankCondition R\nn : ℕ\nf : (Fin (n + 1) → R) →ₗ[R] Fin n → R\nhf : Injective ⇑f\n⊢ False",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"le_of_fin_injective",
"Nat.succ",
"Nat.not_succ_le_self"
],... | [] | exact Nat.not_succ_le_self n (le_of_fin_injective R f hf) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.InvariantBasisNumber | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 13
} | {
"line": 149,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝² : Semiring R\ninst✝¹ : Nontrivial R\ninst✝ : OrzechProperty R\nn : ℕ\ni : (Fin (n + 1) → R) →ₗ[R] Fin n → R\nhi : Injective ⇑i\nh : 0 = update 0 (Fin.last n) 1\n⊢ False",
"ppTerm": "?m.35",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"R : Type u\ninst✝² : Semiring R\ninst✝¹ : Nontrivial R\ninst✝ : OrzechProperty R\nn : ℕ\ni : (Fin (n + 1) → R) →ₗ[R] Fin n → R\nhi : Injective ⇑i\nh : 0 = update 0 (Fin.last n) 1\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1075,
"column": 6
} | {
"line": 1075,
"column": 59
} | {
"line": 1075,
"column": 60
} | [
{
"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\nthis : a * b * ↑(Classical.choose ⋯) = a * ↑(Classical.choose ⋯) * (b * ↑(Classical.choose ⋯))\n⊢ a * ... | [
"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\nthis : a * b * ↑(Classical.choose ⋯) = a * ↑(Classical.choose ⋯) * (b * ↑(Classical.choose ⋯))\n⊢ a * (b * ↑(Class... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.InvariantBasisNumber | {
"line": 190,
"column": 24
} | {
"line": 190,
"column": 35
} | {
"line": 190,
"column": 36
} | [
{
"pp": "R : Type u\ninst✝⁴ : Semiring R\ninst✝³ : RankCondition R\nM : Type u_1\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nn : ℕ\nf : (Fin n → R) →ₗ[R] M\nhf : Surjective ⇑f\ng : M →ₗ[R] Fin (n + 1) → R\nhg : Surjective ⇑g\n⊢ False",
"ppTerm": "?m.43",
"assigned": false,... | [
"R : Type u\ninst✝⁴ : Semiring R\ninst✝³ : RankCondition R\nM : Type u_1\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : Module.Finite R M\nn : ℕ\nf : (Fin n → R) →ₗ[R] M\nhf : Surjective ⇑f\ng : M →ₗ[R] Fin (n + 1) → R\nhg : Surjective ⇑g\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Operations | {
"line": 1003,
"column": 2
} | {
"line": 1003,
"column": 13
} | {
"line": 1003,
"column": 14
} | [
{
"pp": "R : Type u\ninst✝ : CommSemiring R\nI : Ideal R\nhI : I.IsPrime\ns : Multiset R\n⊢ s.prod ∈ I ↔ ∃ p ∈ s, p ∈ I",
"ppTerm": "?m.21",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u\ninst✝ : CommSemiring R\nI : Ideal R\nhI : I.IsPrime\ns : Multiset R\n⊢ s.prod ∈ I ↔ ∃ p ∈ s, p ∈ I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PrincipalIdealDomain | {
"line": 105,
"column": 34
} | {
"line": 105,
"column": 93
} | {
"line": 107,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : AddCommMonoid M\ninst✝² : Semiring R\ninst✝¹ : Module R M\nS : Submodule R M\ninst✝ : S.IsPrincipal\n⊢ S = ⊥ ↔ generator S = 0",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"congrArg",
"Iff.rfl",
... | [] | rw [← @span_singleton_eq_bot R M, span_singleton_generator] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.PrincipalIdealDomain | {
"line": 105,
"column": 34
} | {
"line": 105,
"column": 93
} | {
"line": 107,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : AddCommMonoid M\ninst✝² : Semiring R\ninst✝¹ : Module R M\nS : Submodule R M\ninst✝ : S.IsPrincipal\n⊢ S = ⊥ ↔ generator S = 0",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"congrArg",
"Iff.rfl",
... | [] | rw [← @span_singleton_eq_bot R M, span_singleton_generator] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PrincipalIdealDomain | {
"line": 105,
"column": 34
} | {
"line": 105,
"column": 93
} | {
"line": 107,
"column": 0
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : AddCommMonoid M\ninst✝² : Semiring R\ninst✝¹ : Module R M\nS : Submodule R M\ninst✝ : S.IsPrincipal\n⊢ S = ⊥ ↔ generator S = 0",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"congrArg",
"Iff.rfl",
... | [] | rw [← @span_singleton_eq_bot R M, span_singleton_generator] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PrincipalIdealDomain | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 48
} | {
"line": 142,
"column": 49
} | [
{
"pp": "R : Type u\ninst✝¹ : CommSemiring R\nS : Ideal R\ninst✝ : IsPrincipal S\nis_prime : S.IsPrime\nne_bot : S ≠ ⊥\nx✝¹ x✝ : R\n⊢ generator S ∣ x✝¹ * x✝ → generator S ∣ x✝¹ ∨ generator S ∣ x✝",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"Semiri... | [
"R : Type u\ninst✝¹ : CommSemiring R\nS : Ideal R\ninst✝ : IsPrincipal S\nis_prime : S.IsPrime\nne_bot : S ≠ ⊥\nx✝¹ x✝ : R\n⊢ x✝¹ * x✝ ∈ S → x✝¹ ∈ S ∨ x✝ ∈ S"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1144,
"column": 6
} | {
"line": 1144,
"column": 42
} | {
"line": 1144,
"column": 43
} | [
{
"pp": "case neg\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... | [
"case neg\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_gcd : ∀ (a b ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Operations | {
"line": 1015,
"column": 2
} | {
"line": 1016,
"column": 41
} | {
"line": 1017,
"column": 2
} | [
{
"pp": "case pos\nR : Type u\ninst✝ : CommSemiring R\nI P : Ideal R\nhP : P.IsPrime\nn : ℕ\nh : I ^ n ≤ P\nhn : n = 0\n⊢ I ≤ P",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Submodule",
"MulOne.toOne",
"Ideal.one_eq_top",
"Semiring.toModule",
"IsScalarTower... | [
"case neg\nR : Type u\ninst✝ : CommSemiring R\nI P : Ideal R\nhP : P.IsPrime\nn : ℕ\nh : I ^ n ≤ P\nhn : ¬n = 0\n⊢ I ≤ P"
] | · rw [hn, pow_zero, one_eq_top] at h
exact fun ⦃_⦄ _ ↦ h Submodule.mem_top | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.GCDMonoid.Basic | {
"line": 1195,
"column": 6
} | {
"line": 1195,
"column": 16
} | {
"line": 1196,
"column": 6
} | [
{
"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 :... | [
"case neg\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 := fun a b ↦ ... | · exact ac | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Dimension.StrongRankCondition | {
"line": 125,
"column": 4
} | {
"line": 125,
"column": 15
} | {
"line": 125,
"column": 16
} | [
{
"pp": "case i.hf\nR : Type u\nM : Type v\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : RankCondition R\nι : Type u_2\ninst✝¹ : Fintype ι\nb : Basis ι R M\nw : Set M\ninst✝ : Fintype ↑w\ns : span R w = ⊤\n⊢ span R (range Subtype.val) = ⊤",
"ppTerm": "?i.hf",
"assigned": ... | [
"case i.hf\nR : Type u\nM : Type v\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : RankCondition R\nι : Type u_2\ninst✝¹ : Fintype ι\nb : Basis ι R M\nw : Set M\ninst✝ : Fintype ↑w\ns : span R w = ⊤\n⊢ span R w = ⊤"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.PrincipalIdealDomain | {
"line": 537,
"column": 4
} | {
"line": 537,
"column": 77
} | {
"line": 537,
"column": 78
} | [
{
"pp": "case neg\nR : Type u\ninst✝ : Semiring R\nhR : ¬IsPrincipalIdealRing R\nc : Set (Ideal R)\nhs : c ⊆ nonPrincipals R\nhchain : IsChain (fun x1 x2 ↦ x1 ≤ x2) c\nH : ¬c.Nonempty\n⊢ ∃ I ∈ nonPrincipals R, ∀ J ∈ c, J ≤ I",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case neg\nR : Type u\ninst✝ : Semiring R\nhR : ¬IsPrincipalIdealRing R\nc : Set (Ideal R)\nhs : c ⊆ nonPrincipals R\nhchain : IsChain (fun x1 x2 ↦ x1 ≤ x2) c\nH : ¬c.Nonempty\n⊢ ∃ I, ¬IsPrincipal I"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.StrongRankCondition | {
"line": 210,
"column": 4
} | {
"line": 210,
"column": 15
} | {
"line": 210,
"column": 16
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : StrongRankCondition R\nι : Type u_2\nv : ι → M\ni : LinearIndependent R v\nw : Set M\ninst✝ : Finite ↑w\ns : range v ⊆ ↑(span R w)\nthis : Fintype ↑w := Fintype.ofFinite ↑w\nt : Finset ι\nv' : ↑↑t → M :... | [
"R : Type u\nM : Type v\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : StrongRankCondition R\nι : Type u_2\nv : ι → M\ni : LinearIndependent R v\nw : Set M\ninst✝ : Finite ↑w\ns : range v ⊆ ↑(span R w)\nthis : Fintype ↑w := Fintype.ofFinite ↑w\nt : Finset ι\nv' : ↑↑t → M := fun x ↦ v ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.StrongRankCondition | {
"line": 239,
"column": 2
} | {
"line": 239,
"column": 58
} | {
"line": 239,
"column": 59
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : StrongRankCondition R\nι : Type u_2\nv : ι → M\ni : LinearIndependent R v\nw : Finset M\ns : span R ↑w = ⊤\n⊢ #ι ≤ ↑w.card",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"us... | [
"R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : StrongRankCondition R\nι : Type u_2\nv : ι → M\ni : LinearIndependent R v\nw : Finset M\ns : span R ↑w = ⊤\n⊢ #ι ≤ ↑w.card"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.StrongRankCondition | {
"line": 306,
"column": 2
} | {
"line": 306,
"column": 13
} | {
"line": 306,
"column": 14
} | [
{
"pp": "R : Type u_2\ninst✝¹ : Semiring R\ninst✝ : StrongRankCondition R\nn m : ℕ\nv : Fin m → Fin n → R\nh : LinearIndependent R v\n⊢ m ≤ n",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"R : Type u_2\ninst✝¹ : Semiring R\ninst✝ : StrongRankCondition R\nn m : ℕ\nv : Fin m → Fin n → R\nh : LinearIndependent R v\n⊢ m ≤ n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.StrongRankCondition | {
"line": 356,
"column": 2
} | {
"line": 356,
"column": 57
} | {
"line": 356,
"column": 58
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : StrongRankCondition R\nι : Type u_2\ninst✝¹ : Fintype ι\nb : Basis ι R M\nι' : Type u_3\ninst✝ : Fintype ι'\nv : ι' → M\nhv : LinearIndependent R v\n⊢ Fintype.card ι' ≤ Fintype.card ι",
"ppTerm": "?... | [
"R : Type u\nM : Type v\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : StrongRankCondition R\nι : Type u_2\ninst✝¹ : Fintype ι\nb : Basis ι R M\nι' : Type u_3\ninst✝ : Fintype ι'\nv : ι' → M\nhv : LinearIndependent R v\n⊢ Fintype.card ι' ≤ Fintype.card ι"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.StrongRankCondition | {
"line": 405,
"column": 4
} | {
"line": 405,
"column": 15
} | {
"line": 405,
"column": 16
} | [
{
"pp": "R✝ : Type u\nS : Type u_1\nM✝ : Type v\ninst✝⁹ : Semiring R✝\ninst✝⁸ : AddCommMonoid M✝\ninst✝⁷ : Module R✝ M✝\nι : Type w\nι' : Type w'\ninst✝⁶ : StrongRankCondition R✝\nR : Type ?u.18\nM : Type ?u.22\ninst✝⁵ : Ring R\ninst✝⁴ : StrongRankCondition R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝... | [
"R✝ : Type u\nS : Type u_1\nM✝ : Type v\ninst✝⁹ : Semiring R✝\ninst✝⁸ : AddCommMonoid M✝\ninst✝⁷ : Module R✝ M✝\nι : Type w\nι' : Type w'\ninst✝⁶ : StrongRankCondition R✝\nR : Type ?u.18\nM : Type ?u.22\ninst✝⁵ : Ring R\ninst✝⁴ : StrongRankCondition R\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : IsDomain... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Matrix.Diagonal | {
"line": 71,
"column": 17
} | {
"line": 71,
"column": 28
} | {
"line": 71,
"column": 29
} | [
{
"pp": "n : Type u_3\nα : Type v\ninst✝¹ : DecidableEq n\ninst✝ : Zero α\nd₁ d₂ : n → α\nh : diagonal d₁ = diagonal d₂\ni : n\n⊢ d₁ i = d₂ i",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : Type u_3\nα : Type v\ninst✝¹ : DecidableEq n\ninst✝ : Zero α\nd₁ d₂ : n → α\nh : diagonal d₁ = diagonal d₂\ni : n\n⊢ d₁ i = d₂ i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Matrix.Diagonal | {
"line": 75,
"column": 36
} | {
"line": 75,
"column": 47
} | {
"line": 75,
"column": 48
} | [
{
"pp": "n : Type u_3\nα : Type v\ninst✝¹ : DecidableEq n\ninst✝ : Zero α\nd₁ d₂ : n → α\nh : diagonal d₁ = diagonal d₂\ni : n\n⊢ d₁ i = d₂ i",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n : Type u_3\nα : Type v\ninst✝¹ : DecidableEq n\ninst✝ : Zero α\nd₁ d₂ : n → α\nh : diagonal d₁ = diagonal d₂\ni : n\n⊢ d₁ i = d₂ i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.StrongRankCondition | {
"line": 448,
"column": 34
} | {
"line": 448,
"column": 47
} | {
"line": 448,
"column": 48
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nι : Type w\ninst✝ : StrongRankCondition R\nh : Basis ι R M\n⊢ finrank R M = toNat (Cardinal.lift.{v, w} #ι)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Module.Bas... | [
"R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\nι : Type w\ninst✝ : StrongRankCondition R\nh : Basis ι R M\n⊢ finrank R M = toNat (Cardinal.lift.{w, v} (Module.rank R M))"
] | h.mk_eq_rank, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Dimension.StrongRankCondition | {
"line": 466,
"column": 67
} | {
"line": 466,
"column": 83
} | {
"line": 466,
"column": 83
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : StrongRankCondition R\nι : Type w\nb : Finset ι\nh : Basis (↥b) R M\n⊢ Fintype.card ↥b = b.card",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [
"R : Type u\nM : Type v\ninst✝³ : Semiring R\ninst✝² : AddCommMonoid M\ninst✝¹ : Module R M\ninst✝ : StrongRankCondition R\nι : Type w\nb : Finset ι\nh : Basis (↥b) R M\n⊢ b.card = b.card"
] | Fintype.card_coe | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Matrix.Diagonal | {
"line": 345,
"column": 11
} | {
"line": 345,
"column": 29
} | {
"line": 345,
"column": 29
} | [
{
"pp": "n : Type u_3\nα : Type v\ninst✝¹ : DecidableEq n\ninst✝ : Zero α\nM : Matrix n n α\nv : n → α\n⊢ M = (diagonal v)ᵀ ↔ M = diagonal v",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Matrix",
"id",
"Iff",
"Matrix.diagonal_tra... | [
"n : Type u_3\nα : Type v\ninst✝¹ : DecidableEq n\ninst✝ : Zero α\nM : Matrix n n α\nv : n → α\n⊢ M = diagonal v ↔ M = diagonal v"
] | diagonal_transpose | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.Dimension.StrongRankCondition | {
"line": 615,
"column": 35
} | {
"line": 615,
"column": 65
} | {
"line": 615,
"column": 66
} | [
{
"pp": "case refine_2\nR : Type u\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\nx✝ : ∃ n > 0, Module.rank R (Fin n → R) = 0 ∨ ℵ₀ ≤ Module.rank R (Fin n → R)\nn : ℕ\npos : n > 0\neq : Module.rank R (Fin n → R) ≠ 0 → ℵ₀ ≤ Module.rank R (Fin n → R)\n⊢ ℵ₀ ≤ Module.rank R (Fin n → R)",
"ppTerm": "?refine_2",
... | [
"case refine_2\nR : Type u\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\nx✝ : ∃ n > 0, Module.rank R (Fin n → R) = 0 ∨ ℵ₀ ≤ Module.rank R (Fin n → R)\nn : ℕ\npos : n > 0\neq : 1 ≤ Module.rank R (Fin n → R) → ℵ₀ ≤ Module.rank R (Fin n → R)\n⊢ ℵ₀ ≤ Module.rank R (Fin n → R)"
] | ← Cardinal.one_le_iff_ne_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Matrix.Basis | {
"line": 204,
"column": 6
} | {
"line": 204,
"column": 17
} | {
"line": 204,
"column": 18
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nα : Type u_7\ninst✝⁶ : DecidableEq m\ninst✝⁵ : DecidableEq n\ninst✝⁴ : AddCommMonoid α\ninst✝³ : Finite m\ninst✝² : Finite n\ninst✝¹ : Nonempty m\ninst✝ : Nonempty n\nP : Matrix m n α → Prop\nM : Matrix m n α\nh_add : ∀ (p q : Matrix m n α), P p → P q → P (p + q)\nh_std_basi... | [
"m : Type u_2\nn : Type u_3\nα : Type u_7\ninst✝⁶ : DecidableEq m\ninst✝⁵ : DecidableEq n\ninst✝⁴ : AddCommMonoid α\ninst✝³ : Finite m\ninst✝² : Finite n\ninst✝¹ : Nonempty m\ninst✝ : Nonempty n\nP : Matrix m n α → Prop\nM : Matrix m n α\nh_add : ∀ (p q : Matrix m n α), P p → P q → P (p + q)\nh_std_basis : ∀ (i : m... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Matrix.Mul | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 69
} | {
"line": 82,
"column": 70
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nα : Type v\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : NonUnitalSemiring α\nu : m → α\nw : n → α\nv : Matrix m n α\n⊢ (fun j ↦ u ⬝ᵥ fun i ↦ v i j) ⬝ᵥ w = u ⬝ᵥ fun i ↦ v i ⬝ᵥ w",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"F... | [
"m : Type u_2\nn : Type u_3\nα : Type v\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : NonUnitalSemiring α\nu : m → α\nw : n → α\nv : Matrix m n α\n⊢ ∑ x, ∑ x_1, u x_1 * (v x_1 x * w x) = ∑ x, ∑ i, u x * (v x i * w i)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Matrix.Mul | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 36
} | {
"line": 140,
"column": 37
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nα : Type v\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : NonUnitalNonAssocSemiring α\nu : m → α\nx : n → α\ne : n ≃ m\n⊢ u ⬝ᵥ x ∘ ⇑e.symm = u ∘ ⇑e ⬝ᵥ x",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"m : Type u_2\nn : Type u_3\nα : Type v\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : NonUnitalNonAssocSemiring α\nu : m → α\nx : n → α\ne : n ≃ m\n⊢ u ⬝ᵥ x ∘ ⇑e.symm = u ∘ ⇑e ⬝ᵥ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Matrix.Basis | {
"line": 414,
"column": 15
} | {
"line": 414,
"column": 26
} | {
"line": 414,
"column": 27
} | [
{
"pp": "n : Type u_3\nα : Type u_7\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : Semiring α\nx✝ : Nonempty n\ni : n\nx : α\nhx : ∀ (g : Matrix n n α), g * (scalar n) x = (scalar n) x * g\n⊢ (∀ (g : α), g * x = x * g) ∧ (scalar n) x = (scalar n) x",
"ppTerm": "?m.121",
"assigned": true,
"used... | [
"n : Type u_3\nα : Type u_7\ninst✝² : DecidableEq n\ninst✝¹ : Fintype n\ninst✝ : Semiring α\nx✝ : Nonempty n\ni : n\nx : α\nhx : ∀ (g : Matrix n n α), g * (scalar n) x = (scalar n) x * g\n⊢ ∀ (g : α), g * x = x * g"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Matrix.Mul | {
"line": 272,
"column": 2
} | {
"line": 272,
"column": 44
} | {
"line": 272,
"column": 45
} | [
{
"pp": "m : Type u_2\nα : Type v\ninst✝³ : Fintype m\ninst✝² : CommRing α\ninst✝¹ : Nontrivial m\ninst✝ : Nontrivial α\na : m → α\nh : Function.Injective (dotProduct a)\nb : m → α\nhb : b ≠ 0\nhba : b ⬝ᵥ a = 0\n⊢ False",
"ppTerm": "?m.32",
"assigned": false,
"usedConstants": [],
"usedFVars": []... | [
"m : Type u_2\nα : Type v\ninst✝³ : Fintype m\ninst✝² : CommRing α\ninst✝¹ : Nontrivial m\ninst✝ : Nontrivial α\na : m → α\nh : Function.Injective (dotProduct a)\nb : m → α\nhb : b ≠ 0\nhba : b ⬝ᵥ a = 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Matrix.Mul | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 23
} | {
"line": 278,
"column": 24
} | [
{
"pp": "m : Type u_2\nα : Type v\ninst✝³ : Fintype m\ninst✝² : CommRing α\ninst✝¹ : Nontrivial m\ninst✝ : Nontrivial α\na : m → α\nh : Function.Injective fun x ↦ x ⬝ᵥ a\nb : m → α\nhb : b ≠ 0\nhba : b ⬝ᵥ a = 0\n⊢ False",
"ppTerm": "?m.35",
"assigned": false,
"usedConstants": [],
"usedFVars": []... | [
"m : Type u_2\nα : Type v\ninst✝³ : Fintype m\ninst✝² : CommRing α\ninst✝¹ : Nontrivial m\ninst✝ : Nontrivial α\na : m → α\nh : Function.Injective fun x ↦ x ⬝ᵥ a\nb : m → α\nhb : b ≠ 0\nhba : b ⬝ᵥ a = 0\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Matrix.Mul | {
"line": 587,
"column": 34
} | {
"line": 587,
"column": 49
} | {
"line": 587,
"column": 50
} | [
{
"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\nR : Type u_10\ninst✝¹ : NonAssocSemiring R\ninst✝ : IsStablyFiniteRing R\nf : R →* Matrix (Fin 1) (Fin 1) R := { toFun := fun r ↦ diagonal fun... | [
"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\nR : Type u_10\ninst✝¹ : NonAssocSemiring R\ninst✝ : IsStablyFiniteRing R\nf : R →* Matrix (Fin 1) (Fin 1) R := { toFun := fun r ↦ diagonal fun x ↦ r, map_... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Ideal.Operations | {
"line": 1221,
"column": 4
} | {
"line": 1221,
"column": 15
} | {
"line": 1221,
"column": 16
} | [
{
"pp": "R : Type u_2\nι : Type u_3\ninst✝ : CommRing R\ns : Set ι\nhs : s.Finite\nf : ι → Ideal R\na b : ι\nhp : ∀ i ∈ s, i ≠ a → i ≠ b → (f i).IsPrime\nI : Ideal R\nt : Finset ι\nht : ∀ (a : ι), a ∈ t ↔ a ∈ s\nx✝ : R\n⊢ x✝ ∈ ⋃ i ∈ s, ↑(f i) ↔ x✝ ∈ ⋃ i ∈ t, ↑(f i)",
"ppTerm": "?m.75",
"assigned": true,... | [
"R : Type u_2\nι : Type u_3\ninst✝ : CommRing R\ns : Set ι\nhs : s.Finite\nf : ι → Ideal R\na b : ι\nhp : ∀ i ∈ s, i ≠ a → i ≠ b → (f i).IsPrime\nI : Ideal R\nt : Finset ι\nht : ∀ (a : ι), a ∈ t ↔ a ∈ s\nx✝ : R\n⊢ (∃ i ∈ s, x✝ ∈ f i) ↔ ∃ i ∈ t, x✝ ∈ f i"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Algebra.Order.Group.Finset | {
"line": 48,
"column": 2
} | {
"line": 48,
"column": 63
} | {
"line": 48,
"column": 64
} | [
{
"pp": "case refine_3\nα : Type u_1\ninst✝ : DecidableEq α\ns : Multiset α\na : α\nH : s.card ≠ 0 ∧ s = s.card • {a}\n⊢ s.toFinset = {a}",
"ppTerm": "?refine_3",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"case refine_3\nα : Type u_1\ninst✝ : DecidableEq α\ns : Multiset α\na : α\nH : s.card ≠ 0 ∧ s = s.card • {a}\n⊢ s.toFinset = {a}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Matrix.Mul | {
"line": 974,
"column": 2
} | {
"line": 974,
"column": 13
} | {
"line": 974,
"column": 14
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nα : Type v\ninst✝¹ : NonAssocSemiring α\ninst✝ : Fintype n\nA B : Matrix m n α\nh : A.mulVec = B.mulVec\ni : m\nj : n\n⊢ A i j = B i j",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"m : Type u_2\nn : Type u_3\nα : Type v\ninst✝¹ : NonAssocSemiring α\ninst✝ : Fintype n\nA B : Matrix m n α\nh : A.mulVec = B.mulVec\ni : m\nj : n\n⊢ A i j = B i j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Matrix.Mul | {
"line": 983,
"column": 2
} | {
"line": 983,
"column": 13
} | {
"line": 983,
"column": 14
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nα : Type v\ninst✝¹ : NonAssocSemiring α\ninst✝ : Fintype m\nA B : Matrix m n α\nh : (fun x v ↦ v ᵥ* x) A = (fun x v ↦ v ᵥ* x) B\ni : m\nj : n\n⊢ A i j = B i j",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
... | [
"m : Type u_2\nn : Type u_3\nα : Type v\ninst✝¹ : NonAssocSemiring α\ninst✝ : Fintype m\nA B : Matrix m n α\nh : (fun x v ↦ v ᵥ* x) A = (fun x v ↦ v ᵥ* x) B\ni : m\nj : n\n⊢ A i j = B i j"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Matrix.Mul | {
"line": 1103,
"column": 2
} | {
"line": 1103,
"column": 33
} | {
"line": 1103,
"column": 34
} | [
{
"pp": "m : Type u_2\nn : Type u_3\nα : Type v\ninst✝² : NonUnitalCommSemiring α\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nA : Matrix m n α\nx : n → α\ny : m → α\n⊢ x ᵥ* Aᵀ ⬝ᵥ y = y ᵥ* A ⬝ᵥ x",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"m : Type u_2\nn : Type u_3\nα : Type v\ninst✝² : NonUnitalCommSemiring α\ninst✝¹ : Fintype m\ninst✝ : Fintype n\nA : Matrix m n α\nx : n → α\ny : m → α\n⊢ x ᵥ* Aᵀ ⬝ᵥ y = y ᵥ* A ⬝ᵥ x"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.SetTheory.Cardinal.Finsupp | {
"line": 23,
"column": 2
} | {
"line": 23,
"column": 13
} | {
"line": 23,
"column": 14
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : Fintype α\ninst✝ : Zero β\n⊢ #(α →₀ β) = lift.{u, v} #β ^ Fintype.card α",
"ppTerm": "?m.8",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u\nβ : Type v\ninst✝¹ : Fintype α\ninst✝ : Zero β\n⊢ #(α →₀ β) = lift.{u, v} #β ^ Fintype.card α"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Finsupp.Multiset | {
"line": 55,
"column": 24
} | {
"line": 55,
"column": 40
} | {
"line": 55,
"column": 40
} | [
{
"pp": "α : Type u_1\na : α\nn : ℕ\n⊢ ((single a n).sum fun a n ↦ n • {a}) = n • {a}",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"instHSMul",
"congrArg",
"Finsupp.sum_single_index",
"AddMonoid.toNSMul",
"Finsu... | [
"α : Type u_1\na : α\nn : ℕ\n⊢ n • {a} = n • {a}",
"α : Type u_1\na : α\nn : ℕ\n⊢ 0 • {a} = 0"
] | sum_single_index | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Finsupp.Multiset | {
"line": 84,
"column": 2
} | {
"line": 87,
"column": 20
} | {
"line": 89,
"column": 0
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝ : CommMonoid α\nf : α →₀ ℕ\n⊢ ∀ (a : α) (b : ℕ) (f : α →₀ ℕ),\n a ∉ f.support →\n b ≠ 0 →\n ((toMultiset f).prod = f.prod fun a n ↦ a ^ n) →\n (toMultiset (single a b + f)).prod = (single a b + f).prod fun a n ↦ a ^ n",
"ppTerm": "?refine_2... | [] | · intro a n f _ _ ih
rw [toMultiset_add, Multiset.prod_add, ih, toMultiset_single, Multiset.prod_nsmul,
Finsupp.prod_add_index' pow_zero pow_add, Finsupp.prod_single_index, Multiset.prod_singleton]
exact pow_zero a | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Matrix.Basic | {
"line": 929,
"column": 17
} | {
"line": 929,
"column": 49
} | {
"line": 931,
"column": 0
} | [
{
"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✝⁴ : CommSemiring R\ninst✝³ : CommSemiring α\ninst✝² : Fintype m\ninst✝¹ : DecidableEq m\ninst✝ : Algeb... | [] | by simp [algebraMap_eq_diagonal] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Dimension.Free | {
"line": 98,
"column": 73
} | {
"line": 98,
"column": 84
} | {
"line": 98,
"column": 85
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Free R M\nN : Type v\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nι : Type v\nv : ι → N\nhv : LinearIndependent R v\ncnd : Module.rank R M ≤ #ι\n⊢ lift.{v, v} (Module.rank R M) ≤ lift.{v, v} #ι",
... | [
"R : Type u\nM : Type v\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Free R M\nN : Type v\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\nι : Type v\nv : ι → N\nhv : LinearIndependent R v\ncnd : Module.rank R M ≤ #ι\n⊢ Module.rank R M ≤ #ι"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Dimension.Free | {
"line": 109,
"column": 49
} | {
"line": 109,
"column": 60
} | {
"line": 109,
"column": 61
} | [
{
"pp": "R : Type u\nM : Type v\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Free R M\nN : Type v\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\ncnd : Module.rank R M < Module.rank R N\n⊢ lift.{v, v} (Module.rank R M) < lift.{v, v} (Module.rank R N)",
"ppTerm": "?m.50",
... | [
"R : Type u\nM : Type v\ninst✝⁵ : Semiring R\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Free R M\nN : Type v\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\ncnd : Module.rank R M < Module.rank R N\n⊢ Module.rank R M < Module.rank R N"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TwoSidedIdeal.Basic | {
"line": 93,
"column": 13
} | {
"line": 93,
"column": 56
} | {
"line": 93,
"column": 57
} | [
{
"pp": "case mpr\nR : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nI : TwoSidedIdeal R\nx y : R\nh : I.ringCon (x - y) 0\n⊢ I.ringCon x y",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"RingCon.instFunLikeForallProp",
"HEq.refl",
"NonUnitalNonAssocRing.toAd... | [
"case e'_3\nR : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nI : TwoSidedIdeal R\nx y : R\nh : I.ringCon (x - y) 0\n⊢ x = x - y + y",
"case e'_4\nR : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nI : TwoSidedIdeal R\nx y : R\nh : I.ringCon (x - y) 0\n⊢ y = 0 + y"
] | convert! I.ringCon.add h (I.ringCon.refl y) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.RingTheory.TwoSidedIdeal.Basic | {
"line": 169,
"column": 27
} | {
"line": 169,
"column": 38
} | {
"line": 169,
"column": 39
} | [
{
"pp": "R : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nI : TwoSidedIdeal R\ncarrier : Set R\nzero_mem : 0 ∈ carrier\nadd_mem : ∀ {x y : R}, x ∈ carrier → y ∈ carrier → x + y ∈ carrier\nneg_mem : ∀ {x : R}, x ∈ carrier → -x ∈ carrier\nmul_mem_left : ∀ {x y : R}, y ∈ carrier → x * y ∈ carrier\nmul_mem_right : ∀ ... | [
"R : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nI : TwoSidedIdeal R\ncarrier : Set R\nzero_mem : 0 ∈ carrier\nadd_mem : ∀ {x y : R}, x ∈ carrier → y ∈ carrier → x + y ∈ carrier\nneg_mem : ∀ {x : R}, x ∈ carrier → -x ∈ carrier\nmul_mem_left : ∀ {x y : R}, y ∈ carrier → x * y ∈ carrier\nmul_mem_right : ∀ {x y : R}, x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TwoSidedIdeal.Basic | {
"line": 170,
"column": 27
} | {
"line": 170,
"column": 38
} | {
"line": 170,
"column": 39
} | [
{
"pp": "R : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nI : TwoSidedIdeal R\ncarrier : Set R\nzero_mem : 0 ∈ carrier\nadd_mem : ∀ {x y : R}, x ∈ carrier → y ∈ carrier → x + y ∈ carrier\nneg_mem : ∀ {x : R}, x ∈ carrier → -x ∈ carrier\nmul_mem_left : ∀ {x y : R}, y ∈ carrier → x * y ∈ carrier\nmul_mem_right : ∀ ... | [
"R : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nI : TwoSidedIdeal R\ncarrier : Set R\nzero_mem : 0 ∈ carrier\nadd_mem : ∀ {x y : R}, x ∈ carrier → y ∈ carrier → x + y ∈ carrier\nneg_mem : ∀ {x : R}, x ∈ carrier → -x ∈ carrier\nmul_mem_left : ∀ {x y : R}, y ∈ carrier → x * y ∈ carrier\nmul_mem_right : ∀ {x y : R}, x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TwoSidedIdeal.Basic | {
"line": 172,
"column": 10
} | {
"line": 172,
"column": 67
} | {
"line": 172,
"column": 68
} | [
{
"pp": "R : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nI : TwoSidedIdeal R\ncarrier : Set R\nzero_mem : 0 ∈ carrier\nadd_mem : ∀ {x y : R}, x ∈ carrier → y ∈ carrier → x + y ∈ carrier\nneg_mem : ∀ {x : R}, x ∈ carrier → -x ∈ carrier\nmul_mem_left : ∀ {x y : R}, y ∈ carrier → x * y ∈ carrier\nmul_mem_right : ∀ ... | [
"R : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nI : TwoSidedIdeal R\ncarrier : Set R\nzero_mem : 0 ∈ carrier\nadd_mem : ∀ {x y : R}, x ∈ carrier → y ∈ carrier → x + y ∈ carrier\nneg_mem : ∀ {x : R}, x ∈ carrier → -x ∈ carrier\nmul_mem_left : ∀ {x y : R}, y ∈ carrier → x * y ∈ carrier\nmul_mem_right : ∀ {x y : R}, x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TwoSidedIdeal.Basic | {
"line": 258,
"column": 29
} | {
"line": 258,
"column": 57
} | {
"line": 258,
"column": 58
} | [
{
"pp": "R : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nI I' J' : TwoSidedIdeal R\n⊢ { toFun := op, invFun := unop, left_inv := ⋯, right_inv := ⋯ } I' ≤\n { toFun := op, invFun := unop, left_inv := ⋯, right_inv := ⋯ } J' ↔\n I' ≤ J'",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
... | [
"R : Type u_1\ninst✝ : NonUnitalNonAssocRing R\nI I' J' : TwoSidedIdeal R\n⊢ I'.ringCon.op ≤ J'.ringCon.op ↔ I'.ringCon ≤ J'.ringCon"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.TwoSidedIdeal.Kernel | {
"line": 50,
"column": 4
} | {
"line": 50,
"column": 50
} | {
"line": 50,
"column": 51
} | [
{
"pp": "case mp\nR : 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\nh : ker f = ⊥\nx y : R\nhxy : f x = f y\n⊢ x = y",
"ppTerm": "?mp",
"assigned": false,
"usedConstant... | [
"case mp\nR : 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\nh : ker f = ⊥\nx y : R\nhxy : f x = f y\n⊢ x = y"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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